diff --git a/data_all_eng_slimpj/shuffled/split2/finalzplt b/data_all_eng_slimpj/shuffled/split2/finalzplt new file mode 100644 index 0000000000000000000000000000000000000000..c421b272ccc86cd0fd3d66f80fa1f18ea86827b4 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzplt @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nCell-free massive MIMO is a new network architecture which has been gaining more attention recently as it has the potential to provide a high capacity per user and per unit area \\cite{7827017, 8422577}. As illustrated in Fig. \\ref{systemmodel}, the service area is not divided into separate cells, and the users are not associated with single base stations, but may communicate via multiple access points (APs) simultaneously. To enable this, the Access Points (APs) are distributed over the whole service area and are connected via fronthaul links to the Central Processing Unit (CPU) so that they can cooperate to detect user signals. \n\nDue to the large number of APs deployed in Cell-Free massive MIMO, distributed detection and precoding are often performed at the APs thus reducing the complexity and improving the scalability of channel state information (CSI) acquisition at the CPU. It also avoids the additional fronthaul load due to CSI transfer. However, this approach sacrifices significant achievable data rates in contrast to the centralized approach where the CSI is available at the CPU \\cite{8422865, 8417560, TVTcorrMBB}. In this case (on the uplink) maximum ratio combining (MRC) must be performed at the APs, and separately weighted estimates of each user's signal transmitted to the CPU. It is shown in [2] that the overall fronthaul load may then be higher, and then overall performance poorer, than the case where the quantized CSI and quantized user signals are transferred to the CPU. Moreover if the CSI is available at the CPU more effective detection algorithms such as zero forcing (ZF) may be used, further improving performance [3].\n\nA natural question is therefore whether one can provide CSI at CPU with low fronthaul requirement and with relative simple CSI acquisition. While there has been a number of studies about CSI acquisition and fronthaul load reduction in the context of Cloud Radio Access Networks (C-RAN) \\cite{7444125}, there are few on Cell-Free massive MIMO \\cite{TVTcorrMBB, 8445960}. For instance, the CSI acquisition strategies taking into account the limited fronthaul capacity for single-antenna APs has been investigated in \\cite{TVTcorrMBB}. In the case of multiple-antenna APs the work in \\cite{8445960} studied CSI acquisition at the CPU by utilizing sophisticated hybrid beamforming, where analog combining is performed prior to one-bit analog-to-digital conversions (ADCs).\n\n\\begin{figure}[!t]\n\\centering\n\\input{modelIlustration}\n\\caption{Illustration of Cell-Free Massive MIMO with $L$ access points, $N$ antenna per access point, $K$ users and finite fronthaul rate $C_l$.}\n\\label{systemmodel}\n\\end{figure}\n\nNevertheless, the above works do not explicitly address the channel correlation. For one-bit co-located massive MIMO the problem of channel estimation with correlation was studied among others in \\cite{Kim2018ChannelEF}. Since the real channel tends to be spatially correlated, in this paper we study CSI acquisition for cell-free massive MIMO with limited fronthaul capacity and assuming spatial correlation at multi-antenna APs. In this case, we use Vector Quantization (VQ) with precision of only a small number of bits, so as simultaneously to exploit the channel correlation and meet the low bit requirement of the fronthaul. By applying a vector quantization at APs we investigate the CSI acquisition strategy \\emph{Quantize-and-Estimate} (QE) based on the Bussgang theorem and its counterpart \\emph{Estimate-and-Quantize} (EQ). Our simulation results demonstrate that this few-bit vector quantization can exploit the channel correlation effectively giving better performance in terms of Mean Squared Error (MSE) compared to utilizing Scalar Quantization (SQ) at individual antennas. Further, the QE scheme, which is relatively simple for the implementation at the APs, is shown to offer a significant performance improvement over EQ particularly at low to moderate SNR.\n\nThe rest of the paper is organized as follows. In section \\ref{SM} the system model is described including the spatial channel correlation model. We describe then our considered fronthaul compression technique in section \\ref{FC}. In section \\ref{CSI A}, we present our CSI acquisition schemes, where we make use of the vector quantization and Bussgang decomposition described in section \\ref{FC}. We evaluate then our schemes numerically in section \\ref{NR} and close with conclusion in section \\ref{Conclusion}.\n\n\\emph{Notation}: Roman letter, lower-case boldface letter and upper-case boldface letter are used respectively to denote a scalar, a column vector and a matrix. The set of all complex and real $M\\times N$ matrix are represented by $\\mathbb{R}^{M \\times N}$ and $\\mathbb{C}^{M \\times N}$ respectively. By $\\langle \\cdot, \\cdot\\rangle$ we denote the inner product with $\\Vert \\cdot \\Vert$ as its corresponding vector norm or Frobenius norm. The expectation of random variable is represented by $\\mathbb{E}\\{\\cdot\\}$. We denote circularly complex Gaussian distribution with mean $\\mathbf{m}$ and covariance matrix $\\mathbf{\\Sigma}$ by $\\mathcal{CN}(\\mathbf{m}, \\mathbf{\\Sigma})$. We use $\\mathbf{I}_N$ for $N\\times N$ identity matrix and $\\mathbf{1}_N$ for all-one vector of dimension $N$. We denote the transpose conjugate by $(\\cdot)^H$. For a vector $\\mathbf{a}$, $\\text{diag}(\\mathbf{a})$ denotes a diagonal matrix with the diagonal elements created from vector $\\mathbf{a}$.\n\n\\section{System Model} \\label{SM}\nWe consider uplink transmission in a cell-free system \\cite{7827017}, where we have $K$ single-antenna users (UEs) and $L$ Access Points (APs) equipped with $N\\geq 1$ antennas. We fix the total number of AP antennas in the system at $M=LN$. The processing of the signals received at the APs is virtualized at the Central Processing Unit (CPU), which is connected to the L APs by $L$ error-free fronthaul links which carry the signals in digitally encoded form. \n \n\\subsection{Channel Model}\nWe denote the channel between the $k$-th user and the $m$-th antenna of the $l$-th AP by $g_{mk}$ where $m=(l-1)N+1, \\dots, lN$ for $l=1,\\dots, L$, and $k=1, \\dots, K$. For a given $l$ and $k$ the channel is specified by the $N\\times 1$ vector $\\mathbf{g}_{lk}\\sim\\mathcal{CN}(\\mathbf{0}_N, \\mathbf{\\Sigma}_{lk})$ where $\\mathbf{\\Sigma}_{lk}\\in\\mathbb{C}^{N \\times N}$ is the covariance matrix including the large scale fading and the spatial correlation given by\n\\begin{align}\n\\mathbf{\\Sigma}_{lk}=\\beta_{lk}\\mathbf{R}_{lk}.\n\\end{align} \nThe large scale fading $\\beta_{lk}$ is a path-loss dependent coefficient whereas the correlation matrix $\\mathbf{R}_{lk}\\in\\mathbb{C}^{N \\times N}$ is dependent on the particular environment between AP and UE. In this case, we follow the local scattering model given in \\cite{SIG-093}, where any user $k$ at the azimuth angle $\\theta$ to the AP $l$ is surrounded by scatterers causing correlation between the multipath signal components received between at the antennas of the AP. Accordingly, the correlation coefficient can be specified by an angle of arrival $\\bar{\\theta}$ which is treated as a random variable with probability density function $f(\\bar{\\theta})$ and the entries of the correlation matrix $\\mathbf{R}_{lk}$ are then determined by\n\\begin{align}\n\\left[\\mathbf{R}_{lk}\\right]_{a,b}= \\int e^{j2\\pi d_H(a-b) sin(\\bar{\\theta})} f(\\bar{\\theta}) d\\bar{\\theta},\n\\end{align}\nwhere $d_H$ is the spacing between antennas $1\\leq a, b \\leq N$. Further, $\\bar{\\theta}$ can be expressed as $\\bar{\\theta}=\\theta+\\delta$, where $\\delta$ is a random angular spread with standard deviation $\\sigma_\\delta$.\n\nUsing the Karhunen-Loeve representation we can describe the correlated channel vector as\n\\begin{equation}\n\\mathbf{g}_{lk}=\\beta_{lk}^{1\/2} \\mathbf{U}_{l} \\Lambda_l^{1\/2} \\mathbf{h}_{lk} , \\label{gmk}\n\\end{equation}\nwhere the vector $\\mathbf{h}_{lk}\\sim\\mathcal{CN}(\\mathbf{0}_N, \\mathbf{I_{N}})$ models the small scale fading between the $k$-th user and the $l$-th AP. The unitary matrix $\\mathbf{U}\\in\\mathbb{C}^{N \\times r}$ and the diagonal matrix $\\Lambda\\in\\mathbb{R}^{r \\times r}$ comprise respectively the eigenvectors and the associated eigenvalues of the correlation matrix $\\mathbf{R}_{lk}$ with rank $r$. The channel vector of the $k$-th user to all $L$ APs is then given by $\\mathbf{g}_{k}\\sim\\mathcal{CN}(\\mathbf{0}_M, \\mathbf{\\Sigma}_{k})$, where $\\mathbf{\\Sigma}_{k}=\\text{diag}\\left(\\mathbf{\\Sigma}_{1k}, \\dots,\\mathbf{\\Sigma}_{Lk}\\right)$. Further, we stack the channel from $K$ users to all $L$ APs in the columns of the $M\\times K$ matrix $\\mathbf{G}=\\left[\\mathbf{g}_{1}, \\dots, \\mathbf{g}_{K}\\right]$, such that under the assumption of perfect fronthaul the received signal at the CPU can be modeled as\n\\begin{align}\n\\mathbf{y}=\\mathbf{G}\\mathbf{x}+\\mathbf{w},\n\\end{align} \nwhere $\\mathbf{x}\\in\\mathbb{C}^{K}$ is the channel input from all $K$ users and $\\mathbf{w}\\sim\\mathcal{CN}(\\mathbf{0}_M, \\mathbf{I}_{M})$ is the i.i.d. additive Gaussian white noise at APs. Later, we will remove the assumption of perfect fronthaul and assume that the $l$-th fronthaul link connecting the $l$-th AP to the CPU can transmit quantized signals reliably at a maximum rate of $C_l$.\n\n\\section{Fronthaul Compression} \\label{FC}\nDue to the limited capacity of the fronthaul link and the high load of the digitally encoded signal we need to compress this data for efficient transmission to the CPU. To simplify our analysis, we consider fronthaul links with equal capacity of $C_l=C$ bits for all $l \\in \\{1,\\dots,L\\}$. \n\n\\subsection{Vector Quantization}\nOur considered compression consists of vector quantization followed by fixed-rate lossless coding. At each AP a vector quantizer $Q$ is applied as interface to the fronthaul with \n\\begin{align}\nQ(\\mathbf{x})= \\sum_{i=0}^{\\mathcal{S}-1} q_i T_i(\\mathbf{x}), \\text{ where } \nT_i(\\mathbf{x})=\n\\begin{cases}\n1& \\text{ if } \\mathbf{x}\\in\\mathcal{C}_i\\\\\n0& \\text{ otherwise. } \n\\end{cases} \\label{VQ}\n\\end{align}\nWhenever the input vector $\\mathbf{x}\\in\\mathbb{R}^N$ falls into the cell $\\mathcal{C}_i$, the index i will be transmitted on the fronthaul link, and the reconstruction value $q_i$ taken from the codebook $\\mathcal{Q}=\\{q_i\\}_{i=0}^{\\mathcal{S}-1}\\subset\\mathbb{R}^N$ will be used at the CPU. The codebook size corresponds to the fronthaul capacity by $\\mathcal{S}=2^{C}$. For $N$-dimensional vector quantization we allocate $C\/N$ bits per dimension. Here, we keep $C\/N$ small, to one or two bits per dimension. For a complex-valued signal we quantize separately the real and imaginary part. We do this for the reason that the correlation affects the real and imaginary part of the channel independently.\n\nThe optimal codebooks can be found using the Linde Buzo Gray (LBG) algorithm for minimum mean squared error \\cite{Gersho:1991:VQS:128857}. This algorithm is the counter part of Lloyd algorithm for vector quantization, where the optimal codebook is obtained by alternating between finding the optimal partition by the nearest neighbour criterion and finding the optimal reconstruction values by the centroid condition. The contribution of our scheme compared to separate quantization is that we take the received signal from $N$ antennas at the AP as the input $\\mathbf{x}$ of our vector quantizer $Q$. By doing so we expect to adapt the codebooks to the spatial channel correlation.\n\n\\subsection{Bussgang Decomposition}\nThe quantizer $Q$ given in (\\ref{VQ}) is in general non-linear and the error $\\mathbf{e}\\triangleq\\mathbf{x}-Q(\\mathbf{x})$ resulting from the quantization process is correlated with the input vector $\\mathbf{x}$. However, using the Bussgang theorem \\cite{Bussgang52} we can express our quantizer as the following linear model\n\\begin{align}\n\\mathbf{x}_q=Q(\\mathbf{x})=\\mathbf{F}\\mathbf{x}+\\mathbf{d}, \\label{LinearBussgang}\n\\end{align}\nwhere for a Gaussian input the distortion $\\mathbf{d}$ is statistically equivalent to the quantization error $\\mathbf{e}$ but uncorrelated with the signal component $\\mathbf{x}$. The linear operator $\\mathbf{F}$, which depends essentially on the given distortion characteristic of $Q$, tells us also about the proportional factor between the input-output covariance of the quantizer expressed as \\cite{Bussgang52}\n\\begin{align}\n\\mathbf{C}_{xx_q}&=\\mathbf{F}\\mathbf{C}_{xx}, \\text{ where } \\\\\n\\mathbf{C}_{xx_q}&=\\mathbb{E}\\{\\mathbf{x}\\mathbf{x}_q^H\\} \\text{ and } \\mathbf{C}_{xx}=\\mathbb{E}\\{\\mathbf{x}\\mathbf{x}^H\\}.\n\\end{align}\nIn this case, finding $\\mathbf{F}$ can be seen as finding the LMMSE estimator for $\\mathbf{x}_q$ from the observation $\\mathbf{x}$ \\cite{capacityQuantMIMO}\n\\begin{align}\n\\mathbf{F}&=\\mathbf{C}_{xx_q}\\mathbf{C}_{xx}^{-1}, \\label{F}\n\\end{align}\nwhere the estimation error $\\mathbf{d}$ is then orthogonal to $\\mathbf{x}$. Using (\\ref{F}) the covariance of the distortion $\\mathbf{d}$ can also be expressed as \n\\begin{align}\n\\mathbf{C}_{dd}&=\\mathbb{E}\\{(\\mathbf{x}_q-\\mathbf{F}\\mathbf{x})(\\mathbf{x}_q-\\mathbf{F}\\mathbf{x})^H\\}\\nonumber\\\\\n&=\\mathbf{C}_{x_qx_q}-\\mathbf{C}_{x_qx}\\mathbf{C}_{xx}^{-1}\\mathbf{C}_{xx_q}.\n\\end{align}\nThe closed form expression of $\\mathbf{F}$ is not yet known for a general quantizer, particularly for vector quantizers. Therefore, we compute $\\mathbf{F}$ numerically whenever it is needed by assuming that we have access to measurements of the input as well as the output of $Q$. We estimate the covariance matrix $\\mathbf{C}_{xx}$ from the sample covariance matrix\n\\begin{align}\n\\hat{\\mathbf{C}}_{xx}=\\frac{1}{N_{t}}\\sum_{n_{t}=1}^{N_{t}} \\mathbf{x}[n_{t}]\\mathbf{x}[n_{t}]^H \\label{SampleCovarince}\n\\end{align}\nand respectively for $\\mathbf{C}_{x_qx_q}$ and $\\mathbf{C}_{xx_q}$. The number of observations $N_{t}$ can be conveniently taken equally to the number of codebooks' training, where $\\hat{\\mathbf{C}}_{xx}$ will approach $\\mathbf{C}_{xx}$ for large $N_{t}$.\n\n\\section{CSI Acquisition Strategies}\t\\label{CSI A}\n\\begin{figure*}[ht!]\\centering\n\\includegraphics[width=\\textwidth]{VoronoiChannelN2Q4Merge} \n\\caption{The voronoi region of codebook $\\mathcal{Q}$ for $N=2$ and for different degree of correlation (i.e. for different angular spread standard deviation $\\sigma_{\\delta}$ of Gaussian distributed $\\delta$).}\n\\label{voronoiFig}\n\\end{figure*}\nWe assume in this work that neither the APs nor the CPU knows a priori the channel realization $\\mathbf{G}$. To enable interference suppression at the CPU, the CSI is needed at the CPU. We consider two strategies: estimate and quantize, and quantize and estimate, which will be discussed in the next two sub-sections. \n\n\\subsection{Estimate-and-Quantize}\nIn this scheme we estimate first the channel at APs and then quantize the resulting CSI with the quantizer given in (\\ref{VQ}) to meet the fronthaul-capacity limit of $C$ bits. The channel estimation is done based on the transmission of known pilots. Every user uses a specific sequence taken from a set $\\Psi$ of orthonormal random sequences $\\varphi_k \\in\\mathbb{C}^{\\tau\\times 1}$ with $\\langle\\varphi_k, \\varphi_k'\\rangle=\\delta_{kk'}$ and $\\Vert\\varphi_k\\Vert ^2=1$, where the sequence length $\\tau$ is assumed to be less or equal than the coherence interval $\\tau_c$. The $k$-th user sends $\\sqrt{\\tau}\\varphi_k$ as its pilot such that the $l$-th AP observes the received pilot $\\mathbf{Y}_{p,l}\\in\\mathbb{C}^{N \\times \\tau}$ from all $K$ users as\n\\begin{equation} \\label{receivePilot}\n\\mathbf{Y}_{p,l}= \\sqrt{\\tau\\rho_p}\\sum_{k=1}^{K} \\mathbf{g}_{lk}\\varphi_k^H + \\mathbf{W}_l,\n\\end{equation}\nwhere $\\rho_p$ is the transmit SNR of the pilot and $\\mathbf{W}_l$ is an additive noise matrix at the $l$-th AP whose entries are uncorrelated with zero mean and unit variance. \n\nTo allow all pilots to be orthogonal for all $K$ users, only $K\\leq \\tau$ users may transmit their pilots simultaneously. In this case, the transmitted pilots satisfy\n\\begin{equation}\n\\Phi^H\\Phi=\\tau\\rho_p \\mathbb{I}_K,\\quad \\text{ where } \\Phi=\\sqrt{\\tau\\rho_p}[\\varphi_1, \\dots,\\varphi_K]. \\label{pilotMatrix}\n\\end{equation}\n\nThe channel vector $\\mathbf{g}_{lk}$ can be estimated at the APs where the received pilot $\\mathbf{Y}_{p,l}$ is projected onto $\\varphi_k$ expressed as\n\\begin{align}\n\\mathbf{r}_{p,lk}&=\\frac{1}{\\sqrt{\\tau\\rho_p}}\\mathbf{Y}_{p,l}\\varphi_k\\nonumber\\\\\n&=\\mathbf{g}_{lk}+\\sum_{k'\\neq k}^{K} \\mathbf{g}_{lk'}\\varphi_{k'}^H\\varphi_{k} + \\frac{1}{\\sqrt{\\tau\\rho_p}}\\mathbf{W}_l\\varphi_{k} \\label{projectPilot}\n\\end{align}\nTo obtain the estimate of $\\mathbf{g}_{lk}$ we use the LMMSE estimator given by \n\\begin{align}\n\\hat{\\mathbf{g}}_{lk}= \\mathbf{\\Gamma}_{lk}\\mathbf{r}_{p,lk} \\label{EstimatorIdeal} \n\\end{align}\nThe gain matrix $\\mathbf{\\Gamma}_{lk}$ is given by \n\\begin{align}\n\\mathbf{\\Gamma}_{lk}&=\\mathbf{\\Sigma}_{lk}\\left(\\mathbf{\\Omega}_{lk}\\right)^{-1}, \\text{ where }\n\\mathbf{\\Sigma}_{lk}=\\mathbb{E}\\{\\mathbf{g}_{lk}\\mathbf{g}_{lk}^H\\} \\text{ and } \\\\\n\\mathbf{\\Omega}_{lk}&=\\mathbb{E}\\{\\mathbf{r}_{p,lk}\\mathbf{r}_{p,lk}^H\\}=\\mathbf{\\Sigma}_{lk}+\\frac{1}{\\tau\\rho_p}\\mathbf{I}_{N}\n\\end{align}\n\nAfter accomplishing the channel estimation the APs quantize the channel estimate $\\hat{\\mathbf{g}}_{lk}$. We assume that the large scale fading $\\beta_{lk}$ is relatively constant over a long period and known at the APs. Thus, we may scale the input to the vector quantizer accordingly with $\\beta_{lk}$ and approximate the distribution as multivariate Gaussian. Consequently, we can optimize the codebook $\\mathcal{Q}$ for each AP off-line and need only update it as the $\\beta_{lk}$ changes. As demonstrated in Fig. \\ref{voronoiFig} for $N=2$ the codebooks can exploit the spatial channel correlation effectively. Due to the LBG algorithm the reconstruction points $\\{q_i\\}$ are placed more densely in the region where the input signals come with high probability. As the correlation increases, the reconstruction points get closer to the diagonal to optimally represent the dependency between input signals. Thus, the distance from the input signals to the points $\\{q_i\\}$ becomes smaller resulting a smaller average distortion.\n\n\\subsection{Quantize-and-Estimate}\nInstead of transferring the quantized CSI we consider here another CSI acquisition strategy where we quantize first the received pilots at the APs and then estimate the channel from the quantized pilots at the CPU. To be more specific, the $l$-th AP quantizes the receive pilots at the $N$ antennas jointly as\n\\begin{align}\n\\mathbf{y}_{qp,l}^{(t)}=Q(\\mathbf{y}_{p,l}^{(t)}) \n&=Q\\left(\\sqrt{\\tau\\rho_p}\\sum_{k=1}^{K} \\mathbf{g}_{lk}{\\varphi_k^{(t)}}^* + \\mathbf{w}_l^{(t)} \\right)\\nonumber \\\\\n&=Q\\left(\\sqrt{\\tau\\rho_p}\\mathbf{G}_l{\\varphi^{(t)}}^H + \\mathbf{w}_l^{(t)} \\right) \\label{quantPilots}\n\\end{align}\nwhere the superscript $t=\\{1, \\dots, \\tau\\}$ denotes the index of the pilot sequence. Accordingly $\\mathbf{y}_{p,l}^{(t)}$ is the $t$-th column of $\\mathbf{Y}_{p,l}$ in (\\ref{receivePilot}) and $\\varphi^{(t)}$ is the $t$-th row of $\\Phi$ in (\\ref{pilotMatrix}). \n\nApplying the Bussgang decomposition to (\\ref{quantPilots}) we obtain\n\\begin{align}\n\\mathbf{y}_{qp,l}^{(t)}&=\\mathbf{F}_{p,l}\\mathbf{y}_{p,l}^{(t)} + \\mathbf{d}_{p,l}^{(t)}\\nonumber \\\\\n&=\\sqrt{\\tau\\rho_p}\\mathbf{F}_{p,l}\\mathbf{G}_l{\\varphi^{(t)}}^H+\\mathbf{F}_{p,l}\\mathbf{w}_l^{(t)} + \\mathbf{d}_{p,l}^{(t)}. \\label{y_qp.l^(t)}\n\\end{align}\nThe CPU receives from all $L$ APs as a stack of (\\ref{y_qp.l^(t)})\n\\begin{align}\n\\mathbf{y}_{qp}^{(t)}=\n\\begin{bmatrix}\n\\mathbf{y}_{qp,1}^{(t)}\\\\\n\\vdots \\\\\n\\mathbf{y}_{qp,L}^{(t)}\n\\end{bmatrix}\n=\\begin{bmatrix}\n\\sqrt{\\tau\\rho_p}\\mathbf{F}_{p,1}\\mathbf{G}_1{\\varphi^{(t)}}^H+\\mathbf{F}_{p,1}\\mathbf{w}_1^{(t)} + \\mathbf{d}_{p,1}^{(t)}. \\\\\n\\vdots \\\\\n\\sqrt{\\tau\\rho_p}\\mathbf{F}_{p,L}\\mathbf{G}_L{\\varphi^{(t)}}^H+\\mathbf{F}_{p,L}\\mathbf{w}_L^{(t)} + \\mathbf{d}_{p,L}^{(t)}.\n\\end{bmatrix}\n\\end{align}\nwhere we can concisely rewrite as a $M\\times \\tau$ matrix for $\\tau$-length sequences of quantized received pilots given by\n\\begin{align}\n\\mathbf{Y}_{qp}=\n\\begin{bmatrix}\n\\mathbf{y}_{qp,1}^{(1)}& \\dots &\\mathbf{y}_{qp,1}^{(\\tau)}\\\\\n\\vdots & \\ddots & \\vdots\\\\\n\\mathbf{y}_{qp,L}^{(1)} & \\dots & \\mathbf{y}_{qp,L}^{(\\tau)}\n\\end{bmatrix}\n=\\sqrt{\\tau\\rho_p}\\mathbf{\\tilde{F}}\\mathbf{G}{\\Phi}^H+\\mathbf{\\tilde{F}}\\mathbf{W} + \\mathbf{D},\n\\end{align}\nwhere the matrix $\\mathbf{\\tilde{F}}$ is a $M\\times M$ diagonal matrix with $\\mathbf{F}_{p,l}\\in \\mathbb{R}^{N\\times N}$ in its block diagonal entries. With similar structure to $\\mathbf{Y}_{qp} \\in \\mathbb{C}^{M\\times \\tau}$ the matrix $\\mathbf{W}$ and $\\mathbf{D}$ denote respectively the noise and distortion matrix.\n\nWe can then project $\\mathbf{Y}_{qp}$ onto $\\varphi_k$ expressed as\n\\begin{align}\n\\mathbf{r}_{qp,k}&=\\frac{1}{\\sqrt{\\tau \\rho_p}}\\mathbf{Y}_{qp}\\varphi_k \\nonumber\\\\\n&=\\mathbf{\\tilde{F}}\\mathbf{g}_{k}+\\mathbf{\\tilde{F}}\\sum_{k'\\neq k}^{K} \\mathbf{g}_{k'}\\varphi_{k'}^H\\varphi_{k} + \\frac{1}{\\sqrt{\\tau\\rho_p}}\\left(\\mathbf{\\tilde{F}}\\mathbf{W}+\\mathbf{D}\\right)\\varphi_{k}. \n\\end{align}\nWe estimate $\\mathbf{g}_{k}$ using LMMSE estimator\n\\begin{align}\n\\hat{\\mathbf{g}}_{qp, k}= \\mathbf{\\Gamma}_{qp,k}\\mathbf{r}_{qp,k}. \\label{EstimatorBussgang}\n\\end{align}\nThe gain matrix $\\mathbf{\\Gamma}_{qp,k}$ is given by \n\\begin{align}\n\\mathbf{\\Gamma}_{qp,k}&=\\mathbf{\\Sigma}_{k}\\mathbf{\\tilde{F}}^H\\left(\\mathbf{\\Omega}_{qp,k}\\right)^{-1}, \\text{ where }\n\\mathbf{\\Sigma}_{k}=\\mathbb{E}\\{\\mathbf{g}_{k}\\mathbf{g}_{k}^H\\} \\text{ and } \\nonumber \\\\\n\\mathbf{\\Omega}_{qp,k}&=\\mathbb{E}\\{\\mathbf{r}_{qp,k}\\mathbf{r}_{qp,k}^H\\}=\\mathbf{\\tilde{F}}\\mathbf{\\Sigma}_{k}\\mathbf{\\tilde{F}}^H+\\frac{1}{\\tau\\rho_p}\\left(\\mathbf{\\tilde{F}}\\mathbf{\\tilde{F}}^H+\\mathbf{D}\\mathbf{D}^H\\right)\n\\end{align}\n\n\\begin{remark}\nWe assume that the received signals at the APs are uncorrelated over $l$ and $t$ such that the Gram matrix $\\mathbf{\\tilde{F}}\\mathbf{\\tilde{F}}^H$ and $\\mathbf{D}\\mathbf{D}^H$ have a block diagonal structure. Further, their submatrices are positive definite since $\\mathbf{F}$ and $\\mathbf{d}$ in (\\ref{LinearBussgang}) are positive definite for large observation in sample covariance matrix (\\ref{SampleCovarince}). Thus, the matrix $\\mathbf{\\Omega}_{qp,k}$ is invertible.\n\\end{remark}\n\\section{Numerical Results} \\label{NR}\nWe provide in this section some numerical simulations to asses the performance of the considered schemes above. We did our simulation with $M=120$ total number of the APs' antennas, $L=120\/N$ APs and $K=20$ number of users distributed uniformly in the area of $1\\times 1 \\text{ km}^2$. This area is wrapped around by its copies so that it resembles a network with infinite area. The channel $\\mathbf{g}_{lk}$ in (\\ref{gmk}) is modeled with the large scale fading $\\beta_{lk}$ given as\n\\begin{align}\n\\beta_{lk}=\\text{PL}_{lk}\\cdot 10^{\\frac{\\sigma_{sh}z_{lk}}{10}},\n\\end{align}\nwhere the factor $10^{\\frac{\\sigma_{sh}z_{lk}}{10}}$ is the uncorrelated shadowing with the standard deviation $\\sigma_{sh}= 8 \\text{ dB}$ and $z_{lk}\\sim \\mathcal{N}(0, 1)$. The path loss coefficient follows the three-slope model according to \n\\begin{align}\n\\text{PL}_{lk}=\n\\begin{cases}\n-\\mathcal{L}-35\\text{log}_{10}(d_{lk}), d_{lk} > d_1 \\\\\n-\\mathcal{L}-15\\text{log}_{10}(d_1)-20\\text{log}_{10}(d_{lk}), d_0v_{surf}} \\sum_{A,Z} A Y_{AZ}(v_{surf}') \\, ,\n\\label{eq:atot}\n\\end{equation}\nwhere the sum runs over the different velocity bins. $A_T(t=0)$ is the total baryon number\nat the initial time of the expanding phase. This initial baryon number is calculated using a fitting procedure of the particle energy spectra. Neutrons are not detected in the experiment, and the free neutron number is deduced from the free proton number and the $^3$H\/$^3$He ratio, assuming chemical equilibrium, as explained in \\cite{BougaultJPG19}, and later in the text.\n\n\nIn this equation, we have considered that particles at velocity $v>v_{surf}$ are emitted earlier \nwith respect to the time $t$ identified by the chosen velocity bin $v_{surf}$, and, therefore,\ndo not contribute to the corresponding statistical ensemble.\n\n The chemical constants can be expressed in terms of the mass fractions as:\n\\begin{eqnarray}\nK_c (A,Z)=\\frac{\\omega_{AZ}}{A\\omega_{11}^{Z} \\omega_{10}^{A-Z}}\\left( \\frac{V_T}{A_T}\\right )^{A-1} \\, , \\label{eq:chemica_final}\n\\end{eqnarray}\nwhere $\\omega_{AZ}$, $\\omega_{11}$ and $\\omega_{10}$ are the mass fractions of cluster (A,Z), proton and neutron respectively.\n\nThe measurement of the equilibrium constants thus requires an estimation of the source volume, at the different times of the expansion. In previous studies \\cite{QinPRL108,BougaultJPG19}, this was done with the Mekjian strategy often used in heavy ion collision analyses \\cite{DasGuptaPR72}. Supposing an ideal gas of classical clusters in thermodynamic equilibrium at temperature $T$ in the grand-canonical ensemble, the differential spectrum of a cluster $(A,Z)$ is given by:\n\\begin{equation}\n\\frac{d^3N_{AZ}}{dp^3}(\\vec p)=\\frac{V_f}{h^3} g_{AZ} \\exp \\left [ \\frac {1}{T} \\left ( - \\frac{p^2}{2M_{AZ}} + \nZ\\mu_p + (A-Z)\\mu_n \\right )\\right ] \\; , \\label{eq:spectra}\n\\end{equation}\nwhere $M_{AZ}=Am-B_{AZ}$ is the mass of cluster $(A,Z)$, $m$ is the nucleon mass, $\\mu_n$ and $\\mu_p$ are chemical potentials, $V_f$ is the free volume, and the internal partition sum reads:\n\\begin{equation}\ng_{AZ}=\\sum_K (2J_K+1)\\exp\\left [ -\\frac{E_{K}}{T}\\right \n\\; ,\n\\end{equation}\nwhere the sum runs over the different eigenstates of the clusters with energy $E_K$ and angular momentum $J_K$.\nIn the experimental sample, the differential spectra (corrected for the Coulomb boost as \n$\\Delta p_A=\\sqrt{2mA(E-ZE_C)}=Am \\Delta v_{surf}$, with $E_C=10$ MeV \\cite{BougaultJPG19}), are linked to the multiplicities by:\n\\begin{equation}\n\\frac{d^3N_{AZ}}{dp^3}(\\vec p_A)\\equiv\\tilde Y_{AZ}= \\frac{Y_{AZ}(v_{surf})}{4\\pi p_A^2 \\Delta p_A} \\; . \\label{eq:dN_Y}\n\\end{equation}\n\nAssuming that Eq.~(\\ref{eq:spectra}) holds, if we normalize the cluster spectrum by the proton and neutron spectra at the same velocity, the unknown chemical potentials cancel and we get: \n\\begin{equation}\n\\frac{\\tilde Y_{AZ}(\\vec {p_A})}{\\tilde Y_{p}^Z(\\vec p)\\tilde Y_{n}^{A-Z}(\\vec p)}=h^{3(A-1)}\n\\frac{2J_{AZ}+1}{2^A}\\frac{1}{V_f^{A-1}} \\exp \\left [ \\frac{B_{AZ}}{T}\\right ] \\; , \\label{eq:ratio}\n\\end{equation}\nwhere $p_A=A\\,p$, and $J_{AZ}, B_{AZ}$ are the angular momentum and binding energy of the ground state of the cluster, and we have neglected the population of excited states. \n\nEq.~(\\ref{eq:ratio}) allows independent estimations of the free volume from the different cluster spectra as:\n\\begin{equation}\n{ V_f\n=h^3 \\exp \\left [ \\frac{B_{AZ}}{T(A-1)}\\right ]\\left (\\frac {2J_{AZ}+1}{2^A}\\frac{\\tilde Y_{11}^Z(\\vec p)\\tilde Y_{10}^{A-Z}(\\vec p)}\n{\\tilde Y_{AZ}(\\vec p_A)} \\right )^{\\frac{1}{A-1}} \\; . \\label{eq:vf}\n\\end{equation}\n\nIn the absence of a direct neutron measurement, we can estimate the free neutron-proton ratio \n$R_{np}$ from the multiplicities of the $A=3$ isobars $^3$H and $^3$He using Eq. (\\ref{eq:spectra}):\n\\begin{equation}\nR_{np}=\\frac{\\omega_{10}}{\\omega_{11}}=\\left(\\frac{Y_{31}}{Y_{32}}\\right)\\exp{\\left[ \\frac {B_{32} - B_{31}}{T}\\right]} \\, , \\label{eq:rnp}\n\\end{equation}\nand Eq. (\\ref{eq:vf}) becomes \n\\begin{equation}\n{ V_f\n=h^3R_{np}^{\\frac{A-Z}{A-1}} \\exp \\left [ \\frac{B_{AZ}}{T(A-1)}\\right ]\\left (\\frac {2J_{AZ}+1}{2^A}\\frac{\\tilde Y_{11}^A(\\vec p)}\n{\\tilde Y_{AZ}(\\vec p_A)} \\right )^{\\frac{1}{A-1}} \\; . \\label{eq:vf1}\n\\end{equation}\nFinally, the total volume entering Eq.~(\\ref{eq:chemica_final}) is computed from the free volume, given by Eq.~(\\ref{eq:vf1}), as:\n\\begin{eqnarray}\nV_T= V_f + \\sum_{AZ}V_{AZ}\\frac{\\omega_{AZ}A_T}{A} \\, ,\n\\label{eq:vt}\n\\end{eqnarray}\nwhere we have added the proper volume of the clusters which belong to the\nsource at a given time, with $V_{AZ}=4\\pi R_{AZ}^3\/3$. $R_{AZ}$ is\nthe experimental radius of each cluster, taken from\nRef.~\\cite{Angeli2013}. In Sec.~\\ref{sec:radius}, we will\ndiscuss how the choice of $ R_{AZ}$ affects the determination of the\ntotal volume, and, as a consequence, the total baryonic density and\nthe cluster densities.\n\nIt is important to stress that, for the consistency of the analysis, the different estimations of $V_f$ that can be obtained from Eq.(\\ref{eq:vf1}) considering different particle species, that is different $(A,Z)$ values, should all coincide within experimental errors.\nIf this is not true as we will show in the following, the validity of either Eq.(\\ref{eq:vf1}) or of the hypothesis of statistical equilibrium should be questioned.\n \n\nTo complete the analysis, the thermodynamic parameters $(T,y_p)$ must be evaluated together with the baryonic density $\\rho$, as a function of the surface velocity. \nThe total global proton fraction of the system is given by\n\\begin{eqnarray}\ny_{p}=\\frac{\\sum_{A,Z} Z Y_{AZ}}{\\sum_{A,Z} A Y_{AZ}} \\, ,\\label{eq:yp}\n\\end{eqnarray}\nwith the neutron yield estimate from the proton yield via Eq.(\\ref{eq:rnp}).\nThe temperature is obtained from the Albergo double ratio formula \\cite{AlbergoNCA89}, that we briefly recall.\nIntegrating Eq.~(\\ref{eq:spectra}) over momentum, and neglecting the excited states, we get:\n\\begin{equation}\nN_{AZ}=V_f \\left(\\frac{M_{AZ}T}{2\\pi\\hbar^2}\\right)^{3\/2} (2J_{AZ}+1)e^{ \\frac { \nZ\\mu_p + (A-Z)\\mu_n + B_{AZ} }{T}}\\; . \\label{eq:mult}\n\\end{equation}\n\nThe volume and chemical potential dependence can be eliminated by taking isobaric double ratios \\cite{AlbergoNCA89}.\nIn particular, using $^2$H, $^3$H, $^3$He and $^4$He, we have:\n\\begin{eqnarray}\nT&=&\\frac{\\Delta B}{\\ln\\left [\\frac{3}{2}\\sqrt{\\frac{9}{8}}R_v \\right ] } \n\\; ,\n \\label{eq:temperature}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\nR_v&=&\\frac{N_{21}N_{42}}{N_{32}N_{31}}= \\frac{Y_{21}Y_{42}}{Y_{32}Y_{31}} \\, ,\\\\\n\\Delta B&=& B_{21}+B_{42}-B_{32}-B_{31} \\, .\n\\end{eqnarray}\nWe notice that a slightly different formula was used in previous analyses \\cite{QinPRL108,HagelPRC62,KowalskiPRC75} with an extra $\\sqrt{9\/8}$ in the argument of the logarithm appearing in Eq.~(\\ref{eq:temperature}) ($3\/2\\sqrt{9\/8}\\sqrt{9\/8}$=$1.59\\sqrt{9\/8}$ instead of $3\/2\\sqrt{9\/8}$). The justification of this extra factor was the fact that the analysis does not use total multiplicities, but particles at a given velocity described by a Maxwellian spectrum. We believe that this argument is not correct, because the multiplicities in a given $v_{surf}$ bin are proportional to the total particle numbers (see Eq.~(\\ref{eq:mult})) and not to differential spectra (see Eq.~(\\ref{eq:dN_Y})).\nHowever, we have verified that the presence (or absence) of an extra $\\sqrt{9\/8}$ factor does not produce any visible effect in all the results shown in this paper.\n\n\n\nChemical constants measured by the NIMROD collaboration \\cite{QinPRL108}, using the method explained above, were compared to statistical models employing different treatments of the nucleon-nucleon and nucleon-cluster interactions \\cite{HempelPRC91,PaisPRC97}, with the purpose of constraining the expected cluster binding energy shifts in dense matter, and the associated Mott densities \\cite{Roepke2015,RopkeNPA867,TypelPRC81}. \nHowever, it was observed in Ref.~\\cite{BougaultJPG19} that the validity of the ideal gas expression, Eq.~(\\ref{eq:spectra}), has to be assumed to determine the thermodynamic state $(\\rho,T,y_p)$, and also the value itself of the chemical constants via the free volume definition Eq.~(\\ref{eq:vf1}). \nIndeed Eq.~(\\ref{eq:spectra}) explicitly assumes that the cluster abundances are uniquely governed by their vacuum properties, notably their vacuum binding energies $B_{AZ}$, which is in contradiction with the very purpose of the analysis.\nMoreover, if in medium corrections were indeed negligible, the measured chemical constants would agree with the ideal gas prediction.\nThis latter can be easily worked out from Eq.~(\\ref{eq:chemical}) considering that, for an ideal gas of clusters, $\\rho_{AZ}=N_{AZ}\/V_f$, with $N_{AZ}$ given by Eq.~(\\ref{eq:mult}):\n\\begin{eqnarray}\nK_c^{id}(A,Z)=\\left(\\frac{2\\pi \\hbar^2}{T}\\right)^{\\frac{3(A-1)}{2}}\\left(\\frac{M_{AZ}}{m^A}\\right)^{3\/2}\\frac{(2J_{AZ}+1)}{2^A} \n\\exp{\\left[ \\frac {B_{AZ}}{T}\\right]} \\, . \\label{eq:ideal}\n\\end{eqnarray}\n\n \\begin{figure\n \\begin{tabular}{c}\n\\includegraphics[width=1\\textwidth]{fig1.eps}\n \\end{tabular}\n \\vspace{-1.cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: The chemical equilibrium\n constants of each cluster as a function of the surface velocity\n $v_{surf}$. The error bars only include statistical and systematic\n experimental errors, see Ref.~\\cite{BougaultJPG19}. The solid lines\n are the ideal gas limit given by Eq.~(\\ref{eq:ideal}). The grey band\n shows the area where data might be contaminated by emission from the spectator source. }\n\\label{fig:start}\n\\end{figure} \n\nChemical constants obtained from the analysis of the $^{124}$Xe$+^{124}$Sn system are displayed in\nFig.~\\ref{fig:start}. \nIn this figure, the value of the free volume used to estimate the baryonic density Eq.(\\ref{eq:rho}) in each $v_{surf}$ bin is given by the arithmetic average of the volumes $ V_{f}^{(AZ)}$ extracted from Eq.(\\ref{eq:vf1}) using the yields $Y_{AZ}$ of the different particle species :\n\\begin{equation}\n\\bar V_f=\\frac{1}{N} \\sum_{AZ} V_{f}^{(AZ)} \\, , \\label{eq:vaverage}\n\\end{equation}\nwith $N$ the total number of cluster species ($N=5$ in the present analysis). In fact, this approach to define the system volume is largely arbitrary, since the different volume estimations from the different particle species are not compatible. In the following, a different approach will be considered for the determination of this key quantity, resulting in smaller volumes and having a two-fold effect on the equilirium constants, i.e. making them smaller and moving them to a higher value of the density.\n\n\nThe only differences with respect to the results\npublished in Ref.~\\cite{BougaultJPG19} are the inclusion of the volume associated to deuterons in the arithmetic average of the present calculation and the normalization. Indeed it has to be noticed that the definition of chemical constants in Ref.~\\cite{BougaultJPG19} differs by a factor $A$ with respect to the one of Ref.~\\cite{QinPRL108}. To allow an easier comparison with previous works, we have adopted the definition of Ref.~\\cite{QinPRL108} in this paper.\nAs already mentioned above, the slightly different\nexpression for the temperature Eq.~(\\ref{eq:temperature})\ndoes not produce any effect on the scale of the figure. The\nerror bars are due to the experimental errors associated with the\nmeasurements. The solid lines represent the ideal gas limit, given\nby Eq.~(\\ref{eq:ideal}). The grey band shows the range where the\nexperimental data might be contaminated by emission from the spectator source, since the proton spectra are\nnot well reproduced by the fit used to deduce the mass of the evolving source. \nOne should also observe that equilibrium constants of $^3$H and $^3$He are so close, that they are almost indistinguishable on the scale of the figure.\nWe can see that the measured chemical constants are systematically lower than the ideal gas prediction, and the effect increases with increasing density, showing that binding energy shifts are necessary.\nA qualitatively similar deviation from the ideal gas limit was also found in NIMROD data \\cite{QinPRL108,HempelPRC91}. \nIt is, therefore, clear that a correction is needed to Eq.~(\\ref{eq:spectra}) for the analysis to be consistent. If in-medium corrections at a given temperature and density only depend on the baryonic number of the particle, then their effect will cancel out when taking isobaric ratios and double ratios as in Eqs.(\\ref{eq:rnp}) and (\\ref{eq:temperature}). However, this is not the case for the volume $V_f$, Eq.~(\\ref{eq:vf1}), which in turn affects both the evaluation of the densities $\\rho_{AZ}$ and the evaluation of the total baryonic density $\\rho$.\n\n\\begin{figure\n \\begin{tabular}{cc}\n\\includegraphics[width=1\\textwidth]{fig2.eps}\n \\end{tabular}\n \\vspace{-1cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: The free volume, as estimated from the yields of different cluster species, as a function of $v_{surf}$. Results for $^3$H and $^3$He are indistinguishable. The grey band shows the area where data might be contaminated by emission from the spectator source. }\n\\label{fig2}\n\\end{figure} \n\n\nThe need of an in-medium correction to the ideal gas expression Eq.~(\\ref{eq:vf1}) is further shown by Fig.~\\ref{fig2}.\nThis figure displays the value of the free volume obtained from Eq.~(\\ref{eq:vf1}) as a function of the sorting variable $v_{surf}$ for the $^{124}$Xe$+^{124}$Sn system, using different particle species. Since a $v_{surf}$ bin represents a specific thermodynamic condition $(\\rho,T,y_p)$, if everything was consistent, we should find the same volume whatever the cluster species considered, which is clearly not the case except for the $A=3$ isobars, which lead to almost identical volume estimations (indistinguishable on the scale of the figure). \nQualitatively similar results were obtained with the NIMROD data \\cite{QinPRL108}, showing that \nthe incompatibility among the different volume estimations is not an experimental problem, but it rather points towards an inconsistency in the analysis method. \nTo solve this inconsistency, in the next section we introduce a modification in Eq.~(\\ref{eq:spectra}) allowing for possible in-medium effects.\n\n\n\n\\section{Bayesian analysis} \\label{sec:analysis}\n\n\nSince the different in-medium effects that contribute to the determination of cluster multiplicities can be seen as a shift of the cluster binding energy \\cite{Roepke2015,RopkeNPA867,TypelPRC81}, it is reasonable to suppose that the correction can be defined as a Boltzmann factor:\n\n\\begin{eqnarray}\nC_{AZ}&=&\\exp \\left [- \\frac{\\Delta_{AZ}}{T(A-1)}\\right ] \\label{eq:correction} \\, ,\n\\end{eqnarray}\nwith $\\Delta_{AZ}$ given by\n\\begin{eqnarray} \\label{eq:DeltaAZ}\n\\Delta_{AZ}=a_1 A^{a_2}+a_3 |I|^{a_4} \\, ,\n\\end{eqnarray}\nwhere $a_i$, $i=1,\\dots,4$ are free parameters.\nThe dependence on the cluster species is given by the term $\\Delta_{AZ}$, which, \n for each thermodynamic condition $(T,\\rho,y_p)$ identified by a bin in $v_{surf}$, \n can in principle depend on the two good quantum numbers of each cluster, $A$ and $I=(2Z-A)\/2$. \nThe influence of the functional expression for the correction $\\Delta_{AZ}$ is studied in Section \\ref{sec:correction}. \n\nIt is important to stress that $\\Delta_{AZ}$ can be interpreted as a reduction of the binding energy only in the framework of the classical simplified ideal expression \nEq.(\\ref{eq:spectra}). Indeed the presence of a nuclear medium affects nucleons both in bound, unbound and resonant states \\cite{TypelPRC81}, and $\\Delta_{AZ}$ should be \nunderstood as a global effective correction accounting for all the missing quantum and interaction effects, such as Pauli blocking, effective masses, couplings to the mesons, etc.\nThis correction modifies the expression of the free volume $V_f$ as it can be estimated from the abundance of a given $(AZ)$ species:\n\\begin{equation}\nV_f=h^3R_{np}^{\\frac{A-Z}{A-1}} C_{AZ} \\exp \\left [ \\frac{B_{AZ}}{T(A-1)}\\right ]\n\\cdot \\left (\\frac {2J_{AZ}+1}{2^A}\\frac{\\tilde Y_{11}^A(\\vec p)}\n{\\tilde Y_{AZ}(\\vec p_A)} \\right )^{\\frac{1}{A-1}} \\; , \\label{eq:vfnew}\n\\end{equation}\nwhere the temperature $T$ is still estimated by eq.(\\ref{eq:temperature}).\nThe unknown parameters $\\vec a =\\{ a_i(\\rho,y_p,T),i=1 - 4\\}$ can be fixed by imposing that the volumes obtained from the experimental spectra $\\tilde Y_{AZ}$ via Eq.(\\ref{eq:vfnew}) of the different $(A,Z)$ nuclear species in a given $v_{surf}$, correspond to compatible values. Because of the presence of experimental uncertainties, we cannot simply solve Eq.~(\\ref{eq:vfnew}) for the $\\vec a$ parameters to impose a strictly identical volume for the different species. Even if the experimental errors were negligible, \nthe correlation between $v_{surf}$ and the volume is not a one-to-one correlation because of the physical dispersion of the $v_{surf}$ variable. For these reasons, we consider the unknown $\\vec a$ parameters as random variables. We take in each $v_{surf}$ bin flat priors, $P_{\\rm prior}(\\vec a)=\\theta(\\vec a_{\\rm min}-\\vec a_{\\rm max})$, within an interval largely covering the physically possible reduction range of the binding energy, $0\\le a_1\\le 15$ MeV, $0\\le a_3\\le a_1$ MeV, $-1\\le a_2 \\le 1$, $0 \\le a_4 \\le 4$. \n\nThe posterior distribution is obtained by imposing the volume observation with a likelihood probability as follows:\n\\begin{eqnarray}\nP_{post}(\\vec a)={\\cal N}\\exp\\left(-\\frac{\\sum_{AZ}(V_{f}^{(AZ)}(\\vec a)-\\bar V_f(\\vec a))^2}{2\\bar V_f(\\vec a)^2}\\right) \\, . \\label{eq:likely}\n\\end{eqnarray} \nHere, ${\\cal N}$ is a normalization, $V_{f}^{(AZ)}(\\vec a)$ is the free volume obtained from the $(A,Z)$ cluster using Eq.(\\ref{eq:vfnew}) with the specific choice $\\vec a$ for the parameter set of the correction, and $\\bar V_f(\\vec a)$ is the average volume corresponding to a given parameter set $\\vec a$ from Eq.(\\ref{eq:vaverage}).\n\n \\begin{figure*\n \\begin{tabular}{cc}\n \\includegraphics[width=0.5\\textwidth]{fig3a.eps} & \\includegraphics[width=0.5\\textwidth]{fig3b.eps}\n \\end{tabular}\n\n\\caption{$^{124}$Xe$+^{124}$Sn system: Left: The prior (purple) and\n posterior (green) probability distributions of the total volume for the bins with $v_{surf}=4.1$ cm\/ns (left) and $v_{surf}=5.9$ cm\/ns (right). In red, the probabilities without the correction, designated by $P(C=0)$, are also shown. Right: The corrected free volumes associated with the clusters as a function of\n $v_{surf}$. The error bars are due to both the correction and the experimental errors. The grey band shows the area where data might be contaminated by emission from the spectator source. }\n\\label{fig3}\n\\end{figure*} \n\n\n\nThe prior (posterior) probability distribution of any physical quantity $X$ is then readily calculated as:\n\\begin{equation}\nP(X=X_0)=\\int d\\vec a P(\\vec a) \\delta\\left ( X(\\vec a) - X_0 \\right ) \\, ,\n\\end{equation}\nwhere $P(\\vec a)$ is the prior (posterior) distribution of the correction parameters.\nSimilarly, expectation values can be calculated as:\n\n\\begin{eqnarray}\n\\langle X \\rangle =\\int d \\vec a P(\\vec a) X(\\vec a) \\, ,\n\\end{eqnarray}\nand the correspondent standard deviations as,\n\\begin{eqnarray}\n\\sigma_X =\\sqrt{\\langle X^2\\rangle - \\langle X\\rangle^2 } \\, .\n\\end{eqnarray}\n\n\n\nThe left part of Fig.~\\ref{fig3} shows the prior and posterior distribution of the total volume $V_T$ in two chosen velocity bins, $v_{surf}=4.1$ cm$\/$ns (6$^{\\rm th}$ bin) and $v_{surf}=5.9$ cm$\/$ns (15$^{\\rm th}$ bin). In red, we also show the probabilities calculated without the correction factor, using Eq.(\\ref{eq:vf1}). In this case, the width of the distribution is only due to the experimental errors.\nThe general effect of allowing for an in-medium correction is a\nreduction of the estimated volume. This can be immediately understood\nby comparing the ideal gas estimation Eq.(\\ref{eq:vf}), and the\nmodified expression Eq.(\\ref{eq:vfnew}), and considering that we\nimpose that the correction $C_{AZ}\\le 1$.\nIndeed, following the microscopic calculations of in-medium effects \\cite{Roepke2015}, we consider that, because of the Pauli blocking effect, the influence of the external nucleon gas goes in the direction of reducing the effective binding in the medium.\n\nWe can see that in the lower velocity bin, corresponding to a later time and larger volume, the prior volume distribution is completely unconstrained, reflecting the large dispersion of the volume estimation which is obtained in the absence of the correction (see Fig.\\ref{fig2}).\nOn the other hand, the condition of compatibility between the volume measurements allows a fair determination of this variable, crucial for the rest of the analysis. In the higher velocity bin, corresponding to earlier times of the expansion and more compact configurations, the volume scale is reduced, meaning that the volume estimation is less sensitive to the importance of the correction.\nStill, the posterior distribution is considerably narrower than the prior one. \nThe corrected expectation values of the volume as estimated from the multiplicities of each cluster from Eq.(\\ref{eq:vfnew}), with the associated standard deviations, are shown as a function of $v_{surf}$ in the right part of Fig.~\\ref{fig3}. It is clear that when we include the correction, the average volumes are systematically lower than the uncorrected results of Fig.\\ref{fig2},\n and the estimations obtained from the different cluster species are compatible within error bars. \n\n\n\\section{Equilibrium constants} \\label{sec:kci}\n\n\n\n\n\n The difference between the volume estimation from the ideal gas\n assumption Eq.(\\ref{eq:vf}), and the one determined by the more\n general expression Eq.(\\ref{eq:vfnew}) with the Bayesian estimation\n Eq.(\\ref{eq:likely}) of the correction parameters $\\vec a$, \nobviously reflects itself on the estimation of the chemical constants, as shown in Figs.~\\ref{fig4} and \\ref{fig5}. \nSimilar to Fig.\\ref{fig3}, we show in Fig.\\ref{fig4} the prior and posterior distribution of the $\\alpha$ chemical constant $K_c(4,2)$ in the same velocity bins analyzed in Fig.\\ref{fig3}, while Fig.~\\ref{fig5} shows the average and variance of the chemical equilibrium constants for all the clusters as\na function of the density for the three different experimental systems.\nComparing the uncorrected (labelled \"P(C=0)\") and corrected (labelled \"Ppost\") distributions in Fig.\\ref{fig4}, \nwe can see that the correction to the ideal gas hypothesis leads to a systematic decrease of the chemical constants with respect to the ideal gas assumption (\\ref{eq:vf1}), and an increased dispersion. The effect is higher at lower density, but a shift as important as a factor 10 is observed also in the highest velocity bin, corresponding to the highest density.\n\n\n\\begin{figure\n \\begin{tabular}{cc}\n\\includegraphics[width=0.7\\textwidth]{fig4.eps} \n \\end{tabular}\n \\caption{ The prior (purple) and posterior (green) probability distributions of the chemical equilibrium constant of the $\\alpha$ cluster for the bins with $v_{surf}=4.1$ cm\/ns (left) and $v_{surf}=5.9$ cm\/ns (right) for the $^{124}$Xe$+^{124}$Sn system. In red, the probabilities without the correction, $P(C=0)$, are also shown. }\n\\label{fig4}\n\\end{figure} \n\n\n\n\\begin{figure\n \\begin{tabular}{cc}\n \\includegraphics[width=0.8\\textwidth]{fig5.eps}\n \\end{tabular}\n \\vspace*{-0.8cm}\n\\caption{(Color online) The chemical equilibrium constant of all the clusters\n as a function of the density for the three experimental systems. \n The error bars are due to the correction and the experimental\n errors. The dashed lines are the ideal gas limit given by Eq.~(\\ref{eq:ideal}). The grey band shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig5}\n\\end{figure} \n\n\n\\begin{figure\n \\begin{tabular}{cc}\n\\includegraphics[width=0.9\\textwidth]{fig6.eps}\n \\end{tabular}\n \\vspace*{-0.8cm}\n\\caption{Symbols: double isotope ratio temperature $T_{HHe}$ (black), global proton fraction (multiplied by a factor of 10) (purple), baryon density (multiplied by a factor of 100) (green) as a function of the experimental quantity $v_{surf}$ for the three experimental systems. The uncertainties reflect both the correction and the experimental errors. The solid lines report the ideal gas limit, and were obtained from eq. (\\ref {eq:vaverage}) with eq. (\\ref{eq:vf1}). The grey band shows the area where\ndata might be contaminated by emission from the spectator source.} \n\\label{fig:thermo}\n\\end{figure}\n\n\n\nThe chemical equilibrium constants of all the clusters measured\nin the three different experimental data sets are displayed in Fig.\\ref{fig5}. The dashed lines show the ideal gas limit, given by Eq.(\\ref{eq:ideal}), which, just like in Fig.~\\ref{fig:start}, give incompatible values with the experimental data. For the corrected case, we can see that the results of all the three systems almost perfectly overlap, confirming the expectation that chemical constants do not depend on the proton fraction of the system. It would be interesting to check this point with a larger proton-neutron asymmetry.\n\n\\begin{figure*\n \\begin{tabular}{cc}\n\\includegraphics[width=1.\\textwidth]{fig7.eps}\n \\end{tabular}\n \\vspace*{-1.2cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system. Average volume estimated from the different \n clusters and standard deviations as a function of $v_{surf}$ for three different functional forms of the correction $\\Delta_{AZ}$. The uncertainties are due to both the correction and the experimental errors. The grey band shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig:corrections-V}\n\\end{figure*} \n\nThe thermodynamic conditions explored in the experiments are displayed in Fig.\\ref{fig:thermo}. The temperature is here estimated through the double isotope formula Eq.(\\ref{eq:temperature}), and is indicated as $T_{HHe}$.\nWe can see that the different collisions explore very similar trajectories in the $(T_{HHe},\\rho)$ plane, the only difference being in the global proton fraction, as expected. \nComparing with the results of Ref.~\\cite{BougaultJPG19}, where the ideal gas equation Eq.(\\ref{eq:vf1}) was used to estimate the volume following Ref.~\\cite{QinPRL108}, (full lines in Fig.\\ref{fig:thermo}), we can see that the existence of an in-medium correction goes in the direction of increasing the density, and the effect is the same for the three systems.\n\n\\begin{figure*\n \\begin{tabular}{cc}\n\\includegraphics[width=1.\\textwidth]{fig8.eps}\n \\end{tabular}\n \\vspace*{-1.2cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system. Posterior estimation of the average and standard deviation of the parameters $a_i$ as a function of $v_{surf}$ for three different functional forms of the correction $\\Delta_{AZ}$. The uncertainties are due to both the correction and the experimental errors. The grey band shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig:corrections-ai}\n\\end{figure*}\n\nStill, it is important to stress that the temperature is evaluated with the Albergo $T_{HHe}$ thermometer of Eq.(\\ref{eq:temperature}). As we have discussed in Section \\ref{sec:form}, this expression corresponds to the true thermodynamical temperature $T$ only if the in-medium corrections to the ideal gas of clusters expression Eq.~(\\ref{eq:mult}) cancel \nin the double ratio. This is in principle not the case if the correction does not scale linearly with the particle numbers. \nOur Bayesian analysis does not allow us to determine the deviation of Eq.(\\ref{eq:temperature}) from the true thermodynamic temperature, and this can only be done in the framework of a specific model.\nOne such model will be considered in Section \\ref{sec:model}.\n\n\n \n\n\\begin{figure*\n \\begin{tabular}{cc}\n\\includegraphics[width=1.\\textwidth]{fig9.eps}\n \\end{tabular}\n \\vspace*{-1.2cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system. $\\alpha-$particle chemical equilibrium constants and standard deviations as a function of $v_{surf}$ for three different functional forms of the correction $\\Delta_{AZ}$. The full lines in the left and middle panels correspond to the average right panel results. The uncertainties are due to both the correction and the experimental errors. The grey band shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig:corrections-Kc}\n\\end{figure*} \n\n\n\n\n\n\n\n\\subsection{Different parameter sets for the in-medium effect correction} \\label{sec:correction}\n\nThe correction given by Eq.(\\ref{eq:DeltaAZ}) is a four-parameter set, function of the number of nucleons $A$ and isospin $I$. \nThis functional form has a certain degree of arbitrariness, and the expression is not unique. \n In principle, the correction can depend on all the good quantum numbers of the clusters, namely $A$, $I$, and the charge $Q$. However, the volumes extracted from $^3$H and $^3$He are fully compatible already in the uncorrected data, meaning that, whatever the in-medium correction, it must be the same for both clusters. As a consequence, the correction cannot depend on $Q$. The compatibility of the volumes for $^3$H and $^3$He also implies that we have four data points in each $v_{surf}$ bin to fit our four parameters. This means that in practice we are extracting independent corrections for the different particle species. This is done on purpose, to insure that we are not introducing unjustified hypotheses on the functional dependence of the in-medium correction. However, the drawback is that we cannot attribute a clear physical meaning to the functional form we have introduced. To progress on this point, the analysis should be extended to other cluster species, to see if a four-parameter set is still enough to describe the whole set of data. In the absence of this extra experimental information, we can see if a reduced number of parameters is sufficient to describe the present data, and if \n varying the number of parameters we can get different results for the equilibrium constants. To this aim, we have introduced three different parameterisations, employing two, three and four parameters, respectively, \n and the results are reported in Figs.~\\ref{fig:corrections-V}, \\ref{fig:corrections-ai}, and \\ref{fig:corrections-Kc}. Fig. ~\\ref{fig:corrections-V} shows the posterior first (data point) and second (error bar) moment of the volume distribution. Fig.~\\ref{fig:corrections-ai} shows the optimal values obtained for the parameters, and Fig.~\\ref{fig:corrections-Kc} presents the posterior chemical equilibrium constant for the $\\alpha-$particle. \n \nThe analysis of Figs.~\\ref{fig:corrections-V} and \\ref{fig:corrections-Kc} reveals\n that, whatever the hypothesis on the functional form, we get fully compatible results for the average volumes and chemical constants, which implies compatible results for all observables. \n In particular, in the left panels of Figs.~\\ref{fig:corrections-V}, \\ref{fig:corrections-ai}, and \\ref{fig:corrections-Kc}, a simple two-parameters prescription $\\Delta_{AZ}=a_1 A^{a_2}$ is employed. We can see that the compatibility between the different volume estimations is comparable to the one which is obtained with the full four-parameters formula presented in the right panels. This means that we do not have a compelling evidence that the in-medium correction is isospin dependent.\nThe comparison between the extracted values of the parameters with the different parametrizations is also instructive. Within the simplest prescription shown in the left panel of Fig.~\\ref{fig:corrections-ai}, the value of the $a_2$ parameter is compatible with zero, meaning that the optimal solution is roughly compatible with a constant, that is an effective correction $\\Delta_{AZ}$ that depends on the thermodynamic condition but does not depend on the particle species.\nThis is confirmed by the comparison between the middle panels of Figs.~\\ref{fig:corrections-V}, \\ref{fig:corrections-ai}, and \\ref{fig:corrections-Kc}, where an isospin independent three-parameter formula $\\Delta_{AZ}=a_1 A^{a_2}+a_3$ is tested. We can see that $a_2$ is still compatible with zero, and the sum $a_1+a_3$ is roughly equal to the $a_1$ value obtained in the simpler prescription. This means that the functional dependence on $A$ of the correction appears relatively robust, and confirms that an $A$ independent correction seems to be sufficient.\nIt is important to stress that this interesting experimental finding \ndoes not imply that the in-medium binding energy shift of light clusters is really a constant; it only shows that the effect does not have any clear monotonic dependence with the baryonic number in the range $A=2-6$. Indeed the posterior volume dispersion is never negligible (see Fig.\\ref{fig3}), suggesting that a dependence on the nuclear species cannot be excluded. Beyond mean-field theoretical calculations are needed to settle this point \\cite{Roepke2015}.\n\nFinally, the right panels of Figs.\\ref{fig:corrections-V}, \\ref{fig:corrections-ai}, and \\ref{fig:corrections-Kc} show the complete four-parameter formula $\\Delta_{AZ}=a_1 A^{a_2}+a_3|I|^{a4}$ that will be used as fiducial expression in the successive figures. We can see that the independence on the cluster size is again confirmed, while the data point towards an (approximately quadratic) isospin dependence. As already mentioned above, this is not however a conclusive argument because we have as many parameters as particle species, and a confirmation of the possible isospin dependence of the in-medium effects would need extra equilibrium constants with different isospin values.\n \n \n \\subsection{Effect of the radius of the clusters} \\label{sec:radius}\n \nAs already observed from Fig.\\ref{fig:thermo}, allowing for the possibility of in-medium corrections \nin each $v_{surf}$ bin, \nglobally leads to smaller volumes (and therefore higher global baryonic densities), with respect to the previous works Refs.\\cite{QinPRL108,BougaultJPG19}, where the hypothesis of an ideal gas of clusters was explicitly made in the data analysis. This density increase stems from the fact that we expect from basic theoretical arguments \\cite{Roepke2015} that the in-medium corrections should reduce the effective binding. This corresponds to a positive $\\Delta_{AZ}$ (see Eq.(\\ref{eq:correction})), and consequently smaller volumes (see Eq.(\\ref{eq:vfnew})). However, in order to quantitatively extract the density value, \nthe proper volume of the clusters has to be added to the free volume, see Eq.(\\ref{eq:vt}), which leads to an extra uncertainty in the analysis.\n\nIn this work we have chosen for the radius of each clusters the\n experimental values given in Ref. \\cite{Angeli2013} but in previous\n works the authors have chosen a fixed value for the radius parameter of 1.3 fm. In this section we want to discuss the effect that this might\n have in both the thermodynamics and the chemical equilibrium\n constants. The temperature and\n proton fraction are not affected by the value of $r_0$, but, as seen In Fig.~\\ref{fig:r0-kc4he}, there is a\n clear effect on the total baryonic density, with smaller values of\n $r_0$ giving larger densities. Due to the influence of this\n parameter, we have decided to take in our data analysis\n the experimental radius of each cluster $R_{AZ}$ taken from\nRef.~\\cite{Angeli2013}, and the results are also shown in the same figure.\nThe use of the experimental value $R_{AZ}$ gives rise to\nthe smallest densities.\nIn Fig.~\\ref{fig:r0-kc4he}, the chemical equilibrium constant for the\n $\\alpha-$particle is plotted as a function of $v_{surf}$ (left panel) and total\n baryonic density (right panel), considering the total volume $V_T$ determined by\n $R_{AZ}$ and by {fixed} $r_0$. For the lowest\n densities, all calculations coincide. At the other $v_{surf}$\n extreme, corresponding to the largest densities, $R_{AZ}$\n estimates smaller maximum densities, and, for a given density, smaller\n equilibrium constants.\n \n \n \\begin{figure}[!htbp]\n \\begin{tabular}{cc}\n\\includegraphics[width=0.7\\textwidth]{fig10.eps}\n \\end{tabular}\n \\vspace*{-0.8cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: The density as a\n function of the experimental quantity $v_{surf}$ for the four\n different radii considered. The uncertainties reflect both the\n correction and the experimental errors. The grey\nband shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig:r0-rho-Vsurf}\n\\end{figure} \n\n\n \\begin{figure}[!htbp]\n \\begin{tabular}{cc}\n\\includegraphics[width=1\\textwidth]{fig11.eps}\n \\end{tabular}\n \\vspace*{-0.7cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: The chemical equilibrium constant for\n the $\\alpha-$particle as a function of $v_{surf}$ (left) and as a\n function of the density (right) for the four\n different radii considered. The uncertainties reflect both the\n correction and the experimental errors. The grey\nband shows the area where data might be contaminated by emission from the spectator source.}\n\\label{fig:r0-kc4he}\n\\end{figure} \n\n\\section{Comparison with a theoretical model} \\label{sec:model}\n\n \n\\begin{figure\n \\begin{tabular}{c}\n \\includegraphics[width=0.8\\textwidth]{fig12.eps}\n \\end{tabular}\n \\vspace*{-0.8cm}\n\\caption{$^{124}$Xe$+^{124}$Sn system: Correlation between the input temperature, $T$, of the theoretical model for different $g_{si}$ at $(\\rho,y_p)_{\\rm exp}$, and the isotopic temperature, $T_{HHe}$ evaluated from the theoretical cluster densities, using experimental values for the density and proton fractions. The dashed line shows $T=T_{\\rm HHe}$. }\n\\label{fig:temp}\n\\end{figure} \n\n\n\\begin{figure*\n\\includegraphics[width=0.9\\textwidth]{fig13.eps}\n\\vspace*{-0.8cm}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: $\\alpha$ chemical equilibrium constants as a function of the density, calculated with the ideal gas prescription for the volume (lower set of data with purple lines) and uncertainties due to experimental errors, and with the corrected one (upper set with black lines). In the last case, the uncertainties are due to both the correction and the\n experimental errors. The homogeneous grey and hatched green bands correspond to the chemical equilibrium\n constants from a calculation \\cite{PaisPRC2019} where we consider homogeneous matter with five light clusters, calculated at the average value of ($T$, $\\rho_{\\rm exp}$, $Y_{p_{\\rm exp}}$), and\n considering different cluster-meson scalar coupling constants $g_{s_i} = x_{s} A_i g_s$. The solid line corresponds to the result with $x_s=1$. The dashed line is the ideal gas result given by Eq. (\\ref{eq:ideal}). The color code associated with the data points represents the temperature in MeV. }\n\\label{fig:alphachem}\n\\end{figure*} \n\n\\begin{figure*\n\\includegraphics[width=0.8\\textwidth]{fig14.eps}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: $^3$He (top) and $^3$H (bottom) chemical equilibrium constants as a function of the density, calculated with the ideal gas prescription for the volume (lower set of data) and uncertianties due to experimental errors, and with the corrected one (upper set). In the last case, the uncertainties are due to both the correction and the experimental errors. The homogeneous grey and hatched green bands correspond to the chemical equilibrium\n constants from a calculation \\cite{PaisPRC2019} where we consider\n homogeneous matter with five light clusters, calculated at the\n average value of ($T$, $\\rho_{\\rm exp}$, $Y_{p_{\\rm exp}}$), and\n considering different cluster-meson scalar coupling constants $\n g_{s_i} = x_s A_i g_s$. The color code associated with the data\n points represents the temperature in MeV. }\n\\label{fig:3He3Hchem}\n\\end{figure*} \n\n\\begin{figure*\n\\includegraphics[width=0.8\\textwidth]{fig15.eps}\n\\caption{(Color online) $^{124}$Xe$+^{124}$Sn system: $^6$He (top) and $^2$H (bottom) chemical equilibrium constants as a function of the density, calculated with the ideal gas prescription for the volume (lower set of data) and uncertianties due to experimental errors, and with the corrected one (upper set). In the last case, the uncertainties are due to both the correction and the experimental errors. The homogeneous grey and hatched green bands correspond to the chemical equilibrium\n constants from a calculation \\cite{PaisPRC2019} where we consider\n homogeneous matter with five light clusters, calculated at the\n average value of ($T$, $\\rho_{\\rm exp}$, $Y_{p_{\\rm exp}}$), and\n considering different cluster-meson scalar coupling constants $\n g_{s_i} = x_s A_i g_s$. The color code associated with the data\n points represents the temperature in MeV. }\n\\label{fig:6He2Hchem}\n\\end{figure*} \n \n\n\nIn Ref.~\\cite{PaisPRC97}, a novel approach for the inclusion of\nin-medium effects in the equation of state for warm stellar matter\nwith light clusters was introduced. This model includes a\nphenomenological modification in the scalar cluster-meson coupling,\nand includes an extra term in the effective mass of the clusters,\nderived in the Thomas-Fermi approximation, which gives rise to\neffects similar to the excluded volume approach.\nThe scalar-meson coupling acting on nucleons bound in cluster $i$ with\nmass numer $A_i$ is defined as $g_{s_i} = x_s A_i g_s$, with $g_s$ the\nscalar-meson coupling to nucleons in homogeneous matter and $x_s$ a free parameter. This parameter was fitted in the low-density regime to the Virial EoS, and a value of $x_s=0.85\\pm 0.05$ was found. \n In that work, only four light clusters were considered, $^2$H,\n $^3$H, $^3$He, and $^4$He. Later, in Ref.~\\cite{PaisPRC2019},\n heavier light clusters, with $A_{\\rm max}=12$, together with an average heavy cluster (pasta phase), were added to the model, as it is expected that heavier unstable clusters also form in neutron rich warm stellar matter. In both works, we compared the chemical equilibrium constants obtained with our model with the NIMROD data \\cite{QinPRL108} analyzed assuming an ideal gas expression for the determination of the nuclear density. \nA satisfactory agreement was obtained for all clusters but the deuteron using the scalar cluster-meson coupling parameter $x_s = 0.85 \\pm 0.05$.\n\nComparing the model of Refs.~\\cite{PaisPRC97,PaisPRC2019} with this new analysis will allow to determine the value of the in-medium parameter $x_s$ in a more consistent way, and at the same time will provide an estimate of the effect of the correction we have introduced, with respect to the analysis method of Refs.\\cite{QinPRL108,BougaultJPG19}. \n\nIn order to make this comparison, the thermodynamic variables\n$(T,\\rho,y_p)$ corresponding to each $v_{surf}$ bin must be\nspecified. Among those variables, the density $\\rho$ was evaluated\nin a model independent way in Section \\ref{sec:analysis} under the\nconstraint that the multiplicities of the different clusters at a\ngiven time of the emission, as measured by the $v_{surf}$ variable,\nshould correspond to the same density. On the other hand, an ideal gas\nassumption was used to determine both the temperature\nvia Eq.(\\ref{eq:temperature}), and the proton fraction\nvia Eqs.(\\ref{eq:yp}),(\\ref{eq:rnp}). We have already discussed the fact\nthat, in the absence of any in-medium correction, the volume extracted\nfrom the $^3$He and $^3$H multiplicities already coincide for all\n$v_{surf}$ bins. This is a strong indication that the associated\nin-medium effects must be very close, meaning that they necessarily\n cancel in the $y_p$ estimation\nEq.(\\ref{eq:rnp}). Moreover, we have also observed that the chemical\nconstants appear to be largely independent of the isospin content of the source (see Fig.\\ref{fig5}). \n An extra confirmation of this hypothesis is given by the fact that we have checked that eq.(\\ref{eq:rnp})\nis very well verified in the theoretical model, whatever the value of $x_s$.\nFor those reasons, we use the experimentally determined $y_p$ value as input of the theoretical model. \n\nConcerning the temperature, we fix it in each $(\\rho,y_p)$ point by imposing that the isotopic thermometer Eq.(\\ref{eq:temperature}) evaluated in the model, correctly reproduces the measured $T_{HHe}$ value.\n \n\n \nIn Fig.~\\ref{fig:temp}, we plot the relation between $T$, the input temperature of the theoretical model, and the double ratio thermometer response $T_{HHe}$, given by Eq.(\\ref{eq:temperature}) using the theoretical particle yields. The density and proton fractions in each point are the ones estimated from the data. Different values for the $x_s$ parameter are considered. \nA deviation from the $T=T_{\\rm HHe}$ limit, given by the dashed line,\nis observed for the largest temperatures, when the density is larger and medium effects more important, and for the highest values of $x_s$, corresponding to larger fractions of clusters.\n This is consistent with the fact that the double ratio thermometer $T_{HHe}$ corresponds to the thermodynamic temperature only in the limit of an ideal gas. However we can see that the correlation between $T$ and $T_{HHe}$ is very good, the in-medium effects largely cancel in the double ratio, and the bias does not exceed 5.75\\%. \n Anticipating our results, the maximum bias observed for the coupling that best reproduces\n the experimental equilibrium constants, $x_s=0.92$, never exceeds 5.75 \\%. We have checked that such a bias does not change the experimental results within the error bars.\n\n \n\n\nFrom this correlation we can extract, for each $x_s$ value, the thermodynamic input sets $(T,\\rho,y_p)$ that are compatible with the $T_{HHe}$. The resulting chemical constants are compared to the\nexperimental ones in Figs.~\\ref{fig:alphachem}, \\ref{fig:3He3Hchem}, and \\ref{fig:6He2Hchem}.\n\n\\begin{figure\n \\begin{tabular}{cc}\n\\includegraphics[width=1\\textwidth]{fig16.eps} \n \\end{tabular}\n\\caption{The mass fractions of clusters as a function of the density from a calculation \\cite{PaisPRC2019} where we consider homogeneous matter with four light clusters, and considering different cluster-meson scalar coupling constants $ g_{s_i} = x_s A_i g_s$, with $T=4$ (left) and 10 (right) MeV. The dashed lines are the Virial EoS.}\n\\label{fig:virial}\n\\end{figure} \n\n The grey (green hatched) regions are the theoretical\n calculations performed at different $x_{s}$, and the bands represent the\n experimental quantities. The experimental data points are\n represented with different colors that indicate the respective temperature which changes from 5 to 10 MeV.\n Very similar results are obtained for all the different experimental\nentrance channels and therefore were not shown, see Fig. \\ref{fig5}.\n\nTo appreciate the effect of the correction, the results obtained in the previous analysis \\cite{BougaultJPG19}, where the ideal gas hypothesis was used to extract the source volume, are displayed as a lower set of data. We can remark again that the correction induced by this new analysis leads to increased values of both the density and the chemical constants. \nThe lower hatched bands in the figures correspond to the calculation of Ref.\\cite{PaisPRC97}, where the in-medium corrections were optimized to the uncorrected data of Ref.\\cite{QinPRL108}.\n\n We can see that we need higher values of $x_s$ in order to fit the data, with respect to the results of Refs.~\\cite{PaisPRC97}, ~\\cite{PaisPRC2019}, and an optimal value can\nbe extracted as $x_s=0.92\\pm 0.02$, which is seen to produce an\nadequate reproduction of the whole set of experimental data. This means\n that the nucleons in the clusters behave closer to the unbound nucleons, which correspond to $x_s=1$, than previously deduced.\n\nStill, we can observe a slight mass effect which does not seem fully accounted by the present calculation. Indeed, the optimal $x_s$ value tends to overestimate the equilibrium constant of the heaviest species ($^6$He) and underestimate the one of the lightest cluster ($^2$H). This might suggest that the hypothesis of the model, namely the fact that the coupling to the meson fields scales linearly with the number of nucleons bound in each cluster, could be not fully correct, and a more ab-initio treatment would be in order. This point is left for future work.\n\n\nIt is also interesting to observe that the uncorrected data set of Ref.\\cite{BougaultJPG19} (lower data set in Figs.~\\ref{fig:alphachem} - \\ref{fig:6He2Hchem}) appear only marginally compatible with the previous estimation $x_s=0.85\\pm 0.05$.\nIndeed, the estimation $x_s=0.85\\pm 0.05$, which nicely reproduced the uncorrected Qin et al. data, cannot describe neither the deuteron, nor the $A=3$ isobars of this new data set. Conversely, the introduction of in-medium effects allows a simultaneous reproduction of the whole data. To confirm the new value of the universal coupling $x_s=0.92\\pm 0.02$, and at the same time test the compatibility of the different data sets, it would be very interesting to apply this new analysis to the NIMROD data, such as to verify if the same value for the $x_s$ parameter is able to reproduce both sets of data, once the source volume is correctly estimated from the Bayesian analysis.\n\n\nAt very low density, some constraints on the in-medium binding energy shifts can be obtained from the ab-initio virial equation of state. The results of our model, with the value of $x_s$ optimized on the new analysis of the chemical constants, is compared to the virial constraint in Fig.~ \\ref{fig:virial}. We can see that the new estimation of the $x_s$ parameter is still within the virial EoS limits. A larger scalar-meson coupling has\n strong implications on the dissolution density of the clusters: \n clusters will survive at larger densities.\n\n\n\n\n\\section{Conclusions} \\label{sec:conc}\n\n\n\nA new analysis, where in-medium effects were included via a correction\nto the internal partition function, was done to the experimental data\non the formation of nuclear clusters in heavy-ion collisions\npresented in Ref.~\\cite{BougaultJPG19}. This was done by including a\n global effective correction to the binding energy of the cluster.\n\nUsing a Bayesian analysis, the probability distribution for the parameters of the in-medium\ncorrection was determined by imposing that the free volumes obtained for the\ndifferent clusters in a given source velocity bin be\ncompatible. The main consequence of including the in-medium\ncorrection was to reduce the free volume, and, as a\nconsequence, increase the density.\n\nIn the absence of any in-medium correction,\nonly the volume extracted from the $^3$He and $^3$H multiplicities \ncoincide for all source velocity bins. This indicates that the associated\nin-medium effects are similar and cancel in the calculation of the\nproton fraction, and suggests that, for a given thermodynamic condition and a given cluster mass, the correction should not \nstrongly depend on the charge of the clusters. Besides, the equilibrium constants obtained showed to\nbe quite independent of the isospin content of the three systems that\nwere analysed. It would be interesting to check this point with a larger isospin\nrange.\n\nDifferent functional forms for the correction as a function of the mass and isospin of the cluster were considered, and we observed that the results, both\n for the thermodynamic conditions and chemical equilibrium constants,\n were similar. \n\n A comparison to a theoretical model \\cite{PaisPRC97,PaisPRC2019} was\n also done. The modification of the experimental chemical constants \ndue to the better evaluation of the source density leads to a modified estimation for\nthe magnitude of the scalar-meson\n coupling $x_s$ for nucleons bound in clusters: the optimal values extracted are $x_s = 0.92 \\pm\n 0.02$, larger than the ones found in \\cite{PaisPRC97},\n $x_s=0.85\\pm 0.05$. Values of $x_s$ closer to the unity mean that\n the effect of the medium is less important than previously estimated, and that clusters will melt at larger densities. In turn, this means that the contribution of clusters in the neutrino opacity in the deleptonization phase of the proto-neutron star should be more important than previously considered \\cite{Fischer}.\n\n\n\nIn a future work, it would be extremely interesting to calculate explicitly the neutrino opacity in the relevant thermodynamic conditions. Prior to that, it will be important to perform a new analysis of the experimental data of Ref.~\\cite{QinPRL108}, including the possibility of in-medium effects in the determination of the effective volume in the same spirit as the one presented in this paper, in order to check the consistency of the different data sets, and to settle the model dependence of the results.\n\n\\ack{\nThis work was partly supported by the FCT (Portugal) Projects No. UID\/FIS\/04564\/2019, UID\/FIS\/04564\/2020 and POCI-01-0145-FEDER-029912, and by PHAROS COST Action CA16214. H.P. acknowledges the grant CEECIND\/03092\/2017 (FCT, Portugal). She is very thankful to F.G. and her group at LPC (Caen) for the kind hospitality during her stay there within a PHAROS STSM, where this work started. This work is part of the INDRA collaboration program. We thank the GANIL staff for providing us the beams and for the technical support during the experiment. We acknowledge support from R\\'egion Normandie under RIN\/FIDNEOS.}\n\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n\\thebibliography{50}\n\n\\bibitem{Sumiyoshi2008} Sumiyoshi K and R\\\"opke G 2008 Phys. Rev. C {\\bf 77} 055804\n\\bibitem{Fischer2014} Fischer T, Hempel M, Sagert I, Suwa Y and Schaffner-Bielich J 2014 Eur. Phys. J. A {\\bf 50} 46\n\\bibitem{Furusawa2013} Furusawa S, Nagakura H, Sumiyoshi K and Yamada S 2013 Astrophys. J. {\\bf 774} 78\n\\bibitem{Furusawa2017} Furusawa S, Sumiyoshi K, Yamada S and Suzuki H 2017 Nucl. Phys. 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A {\\bf 50} 46 \n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWith the absence of any signal of new physics at the large hadron\ncollider (LHC) at present energies, searches of physics beyond the\nStandard Model (BSM) is based on the ability to make very precise\ntheoretical predictions within the Standard Model (SM) and to look\nfor possible deviations between experimental observations and\ntheoretical predictions, as a hint of new physics, within estimated\nuncertainties. In order to constrain the new physics model\nparameters, one needs to also compute the BSM signals to the same level\nof theoretical precision as the SM and compare with the observations\nmade at the LHC. Quantum Chromodymanics (QCD) corrections are large at the LHC and inclusion of\nhigher order terms reduces the theoretical uncertainties substantially.\nMany SM processes have been measured at the LHC and have cross sections\nthat are in excellent agreement with higher order QCD predictions. \nThis has helped\nin the discovery of the Higgs boson by ATLAS \\cite{Aad:2012tfa} and CMS\n\\cite{Chatrchyan:2012xdj}\ncollaborations at the LHC and hence the measurement of the important fundamental\nparameter of the SM, the Higgs mass $m_H$ (see \n\\cite{Harlander:2002wh,Anastasiou:2002yz,Ravindran:2003um}). \nPrecise measurement of the\nHiggs mass is essential for the understanding of the stability of\nelectroweak vacuum \\cite{Degrassi:2012ry}.\n\nIn spite of the fact that the SM is in excellent agreement with \nexperimental observations, we know that there are compelling\nreasons to go beyond the SM. In the context of the discovery\nof a boson at 125 GeV in the di-photon channel, models with\nspin-2 were also necessary to ascertain the spin and parity of\nthe discovered boson. In the mean time the bounds on conventional models\nsuch as the Randall-Sundrum models with warped extra dimensions~\\cite{Randall:1999ee},\n where\nthe spin-2 couples universally\nto the SM energy momentum tensor was much higher. \nA universally-coupled spin-2 particle is heavily constrained\n\\cite{Khachatryan:2016yec,Aaboud:2017eta}.\nModels\nwith non-universal coupling of a spin-2 to SM was hence\na suitable alternative. In this model, the spin-2 couples to, two\nsets of gauge invariant SM tensorial operators with different\ncoupling strengths, but are\nnot individually conserved. The universal coupling would correspond\nto the coupling strength being equal and the tensorial operators\nadding up to the conserved energy momentum tensor. Models\nwith non-universal coupling were incorporated in tools like\nHiggs charaterisation \\cite{Artoisenet:2013puc} to NLO in QCD. \nNon universal coupling \nlead to additional challenges: (a) additional UV renornalisation were\nneeded, (b) in the IR sector, additional double and single pole\nterms had to be cancelled with the counter parts from real emission \nprocess and mass factorisation counter terms, thus demonstrating\nthe IR factorisation to NLO for non-universal coupling \\cite{Artoisenet:2013puc}.\nNote that we take this for granted in perturbative QCD (pQCD) and for universal coupling it is guaranteed\nby the conserved energy-momentum tensor. \n\nRecently, the UV structure of non-universal coupling up to three\nloop order in QCD was investigated \\cite{Ahmed:2016qjf} where in the spin-2\nfields couple to two sets of gauge invariant tensorial operators\nconstructed out of the SM fields (with different coupling strengths).\nThese rank-2 operators are unfortunately not conserved, unlike\nenergy-momentum tensor of QCD~\\cite{Nielsen:1977sy}. \nConsequently, both these operators as well as the couplings get\nadditional UV renormalisation order by order in perturbation theory.\nExploiting the universal IR structure of QCD amplitudes\neven in the case of a non-universal spin-2 coupling, on-shell\nform factors of these operators between quark and gluon states\nhave been computed. These are important ingredients for\nobservables at the LHC, to study models with such interactions.\n\nFor universal coupling, depending on the geometry of extra dimensions,\n{\\em viz}.\\ large extra dimensions or warped extra dimension models,\nstudies have been extensively carried out upto higher orders in QCD in various\nchannels that are relevant for the LHC. In these models, the DY process\nhas been studied to NLO \\cite{Mathews:2004xp,Mathews:2005zs,Kumar:2006id}\nfor various observables. Di-vector boson final\nstate have been studied to NLO level in\n\\cite{Kumar:2008pk, Kumar:2009nn, Agarwal:2009xr, Agarwal:2009zg,\nAgarwal:2010sp, Agarwal:2010sn}.\nTo NLO+parton shower (PS) accuracy all the non-color, di-final states have been studied\n\\cite{Frederix:2012dp, Frederix:2013lga, Das:2014tva} in the aMC@NLO\nframework. \nProduction of a generic spin-2 particle in association with coloured\nparticles, vector bosons and the Higgs boson have been studied in\n\\cite{Das:2016pbk} to NLO+PS accuracy. To the next higher order in QCD\nthe form factor of a spin-2 universally coupled to quarks and gluons up\nto two loops was computed in \\cite{deFlorian:2013sza}. Subsequently the\nnext-to-next-to-leading order (NNLO) computation in the threshold limit was done in \\cite{deFlorian:2013wpa}\nand finally the full NNLO computation in \\cite{Ahmed:2016qhu}.\nProduction of a spin-2 in association with a jet to full two-loop QCD\ncorrections has also been completed recently with the evaluation of\ngeneric spin-2 decaying to $g~ g~ g$ \\cite{Ahmed:2014gla} and \n$ q ~\\bar q ~g$ \\cite{Ahmed:2016yox}.\n\nThe di-lepton final state is the most studied and a very clean\nprocess at the LHC. In BSM scenarios the dilepton signal could \nbe enhanced due to additional contributions from BSM intermediate\nstates that could couple to a di-lepton. For the universal\nspin-2 coupling the DY\nprocess has been evaluated upto NNLO in QCD. This involved\nvarious steps: to begin with NLO corrections were evaluated\n\\cite{Mathews:2004xp}, followed by the two loop quark and\ngluon form factors \\cite{deFlorian:2013sza}, which lead to the\ncomputation of NNLO QCD corrections to the graviton\nproduction in models of TeV-scale gravity, within the soft-virtual\napproximation \\cite{deFlorian:2013wpa}. Finally the complete NNLO QCD corrections to the\nproduction of di-leptons at hadron colliders in large extra\ndimension models with spin-2 particles are reported in\n\\cite{Ahmed:2016qhu}. \n\nThe non-universal coupling of spin-2 to SM has been actively\nconsidered by the ATLAS Collaboration \\cite{Aad:2015mxa,Pedersen:2015jdh} to provide\nexclusion of several non-SM spin hypotheses. This analysis\nhas been done in the Higgs characterisation frame work \\cite{Artoisenet:2013puc,Das:2016pbk} \nto NLO+PS accuracy. With the recent results \\cite{Ahmed:2016qjf} upto three loop form factors of a massive spin-2\nparticle with non-universal coupling, NNLO computation is now\npossible. In this article we look at the phenomenological\nimplications of these models to NNLO at the LHC.\n\n\nThe paper is organised as follows. We discuss the effective action that\ndescribes how spin-2 particle couples to those of the SM through two gauge\ninvariant operators with renormalisable coupling. Using this action, we\ncompute QCD radiative correction to the production of pair of leptons in\nparticular their invariant mass distribution up to NNLO level. A detailed\nphenomenological study on the impact of our results is presented for the\nLHC. Finally we conclude. The relevant form factors are presented in the appendix \nand mass factorised partonic cross sections are given as electronically readable version.\n\n\\section{Theoretical Framework}\n\\label{sec:Theory}\n\n\\subsection{Effective action}\n\nThe interaction part of the effective action describes the non-universal coupling of\nthe spin-2 fields denoted by $h_{\\mu\\nu}$ with those of QCD, consists of\ntwo gauge invariant operators, namely $\\hat {\\cal O}^G_{\\mu\\nu}$ and \n$\\hat {\\cal O}^Q_{\\mu\\nu}$ and is given by \n\\begin{equation}\n\\label{eq:action}\nS = -\\frac{1}{2} \\int d^4x ~ h^{\\mu\\nu}(x) \\left(\\hat \\kappa_G~ \\hat {\\cal O}^{G}_{\\mu\\nu} (x) \n\t+ \\hat \\kappa_Q~\\hat {\\cal O}^{Q}_{\\mu\\nu}(x) \\right) \\,,\n\\end{equation}\nwhere $\\hat \\kappa_{G,Q}$ are dimension full couplings, the pure gauge\nsector is denoted by $G$, while $Q$ denotes the fermionic sector and\nits gauge interaction. This decomposition is not unique as one can\nadjust gauge invariant terms between them. The gauge invariant\noperators $\\hat {\\cal O}^G_{\\mu\\nu}$ and $\\hat {\\cal O}^Q_{\\mu\\nu}$\nare as follows:\n\\begin{eqnarray} \\label{eq:emtensor}\n\\hat {\\cal O}^G_{\\mu\\nu} &=&{1 \\over 4} g_{\\mu\\nu} \\hat F_{\\alpha \\beta}^a \\hat F^{a\\alpha\\beta} \n- \\hat F_{\\mu\\rho}^a \\hat F^{a\\rho}_\\nu\n - \\frac{1}{\\hat \\xi} g_{\\mu\\nu} \\partial^\\rho(\\hat A_\\rho^a\\partial^\\sigma \\hat A_\\sigma^a)\n-{1 \\over 2\\hat \\xi}g_{\\mu\\nu} \\partial_\\alpha \\hat A^{a\\alpha} \\partial_\\beta \\hat A^{a\\beta} \n %\n \\nonumber\\\\\n&& + \\frac{1}{\\hat \\xi}(\\hat A_\\nu^a \\partial_\\mu(\\partial^\\sigma \\hat A_\\sigma^a) + \\hat A_\\mu^a\\partial_\\nu\n (\\partial^\\sigma \\hat A_\\sigma^a))\n+\\partial_\\mu \\overline {\\hat \\omega^a} (\\partial_\\nu \\hat \\omega^a - \\hat g_s f^{abc} \\hat A_\\nu^c \\hat \\omega^b)\n\\nonumber\\\\\n&& +\\partial_\\nu \\overline {\\hat \\omega^a} (\\partial_\\mu \\hat \\omega^a- \\hat g_s f^{abc} \\hat A_\\mu^c \\hat \\omega^b)\n %\n-g_{\\mu\\nu} \\partial_\\alpha \\overline {\\hat \\omega^a} (\\partial^\\alpha \\hat \\omega^a - \\hat g_s f^{abc} \\hat A^{c \\alpha} \\hat \\omega^b) \\,,\n\\\\\n \\hat {\\cal O}^Q_{\\mu\\nu} &= &\n \\frac{i}{4} \\Big[ \\overline {\\hat \\psi} \\gamma_\\mu (\\overrightarrow{\\partial}_\\nu -i \\hat g_s T^a \\hat A^a_\\nu)\\hat \\psi\n -\\overline {\\hat \\psi} (\\overleftarrow{\\partial}_\\nu + i \\hat g_s T^a \\hat A^a_\\nu) \\gamma_\\mu \\hat \\psi\n +\\overline {\\hat \\psi} \\gamma_\\nu (\\overrightarrow{\\partial}_\\mu -i \\hat g_s T^a \\hat A^a_\\mu)\\hat \\psi\n \\nonumber\\\\\n &&-\\overline {\\hat \\psi} (\\overleftarrow{\\partial}_\\mu + i \\hat g_s T^a \\hat A^a_\\mu) \\gamma_\\nu \\hat \\psi\\Big]\n- ig_{\\mu\\nu} \\overline {\\hat \\psi} \\gamma^\\alpha (\\overrightarrow{\\partial}_\\alpha -i \\hat g_s T^a \\hat A^a_\\alpha)\\hat \\psi \\,,\n \\end{eqnarray}\nin the above equations the unrenormalised quantities are denoted by hat $( ~{\\hat{}} ~)$.\n$\\hat g_s$ is the strong coupling constant, ${\\hat \\xi}$ the gauge fixing parameter,\n$\\hat A_\\nu^c$ the gauge field, ${\\hat \\psi}$ the quark field and ${\\hat \\omega^a}$ the\nghost fields. The structure constants of $SU(N)$ gauge group are denoted by $f^{abc}$ and\nthe Gell-Mann matrices by $T^a$.\nThe sum of $\\hat{{ \\cal O}}_G$ and $\\hat{{\\cal O}}_Q$ is the energy momentum tensor of the QCD\npart and is protected by radiative corrections to all orders, thanks to fact that\nit is conserved. The Feynman rules for the non-universal case in contrast to the\nuniversal case \\cite{Han:1998sg,Mathews:2004pi}, would have a\nprefactor $\\kappa_Q$ for the coupling for a spin-2 to a pair of fermions or any fermionic \nSM vertex, while a spin-2 coupling to gluons, ghosts or any SM gauge or ghost vertex \nwould have a prefactor $\\kappa_G$. The individual gauge ${\\cal O}_G$ and fermionic\n${\\cal O}_Q$ operators are not conserved in QCD and hence require additional ultraviolet\n(UV) counter terms\nin order to renormalise them. In \\cite{Ahmed:2016qjf}, we determined these additional UV\nrenormalisation constants up to three loop level in QCD. We obtained them by exploiting\nthe universal infrared properties of on-shell amplitudes involving these composite\noperators. Since we have two operators at our disposal, they mix under renormalisation\nas follows:\n\\begin{equation}\n \\label{eq:Zmat}\n\\begin{bmatrix}\nO^{G} \\\\ O^{Q}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n Z_{GG} & Z_{GQ} \\\\ Z_{QG} & Z_{QQ}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\hat O^{G} \\\\ \\hat O^{Q}\n\\end{bmatrix}\n\\,.\n\\end{equation}\nwhere the renormalisation constants $Z_{IJ}$ in terms of the anomalous dimensions\n$\\gamma_{IJ} = \\sum_{n=1}^{\\infty} a_{s}^{n} \\gamma_{IJ}^{(n)}$ are given by\n\\begin{align}\n Z_{IJ} &= \\delta_{IJ} \n + {a}_{s} \\Bigg[ \\frac{2}{\\epsilon}\n \\gamma_{IJ}^{(1)} \\Bigg] \n + {a}_{s}^{2} \\Bigg[\n \\frac{1}{\\epsilon^{2}} \\Bigg\\{ 2\n \\beta_{0} \\gamma_{IJ}^{(1)} + 2 \\gamma_{IK}^{(1)} \\gamma_{KJ}^{(1)} \\Bigg\\} + \\frac{1}{\\epsilon} \\Bigg\\{ \\gamma_{IJ}^{(2)}\\Bigg\\}\n \\Bigg] \\,,\n\\end{align}\nwhere $I,J=G,Q$, $a_s \\equiv g_s^2\/16 \\pi^2$ and space-time dimension is taken to be $d=4+\\epsilon$.\nThe renormalisation constants $Z_{IJ}$ computed in \n\\cite{Ahmed:2016qjf} are given below up to $a_s^2$ for completeness:\n\\begin{align}\nZ_{GG} &= 1 + a_s\\left[ -\\frac{4}{3\\epsilon}n_f \\right]+a_s^2\\bigg[ \\frac{1}{\\epsilon^2}\\left\\{ -\\frac{44}{9}C_A n_f + \\frac{32}{9}C_F n_f +\\frac{16}{9}n_f^2\\right\\}\n\\nonumber\\\\&\n+\\frac{1}{\\epsilon}\\left\\{-\\frac{35}{27}C_A n_f - \\frac{74}{27}C_F n_f\\right\\} \\bigg]\\,,\n\\nonumber\\\\\nZ_{GQ}&= a_s \\left[\\frac{16}{3\\epsilon}C_F \\right] + a_s^2\\bigg[\\frac{1}{\\epsilon^2}\\left\\{\\frac{176}{9}C_A C_F -\\frac{64}{9}C_F n_f -\\frac{128}{9}C_F^2 \\right\\} \n\\nonumber\\\\&\n+ \\frac{1}{\\epsilon}\\left\\{\\frac{376}{27}C_A C_F -\\frac{104}{27}C_F n_f -\\frac{112}{27}C_F^2 \\right\\} \\bigg]\\,,\n\\nonumber\\\\\nZ_{QG}&= a_s\\left[\\frac{4}{3\\epsilon} n_f\\right] + a_s^2\\bigg[ \\frac{1}{\\epsilon^2}\\left\\{\\frac{44}{9}C_A n_f -\\frac{32}{9}C_F n_f -\\frac{16}{9} n_f^2 \\right\\} \n\\nonumber\\\\&\n+\\frac{1}{\\epsilon}\\left\\{ \\frac{35}{27}C_A n_f +\\frac{74}{27} C_F n_f\\right\\}\\bigg]\\,,\n\\nonumber\\\\\nZ_{QQ}&= 1+ a_s\\left[-\\frac{16}{3\\epsilon} \\right] + a_s^2\\bigg[\\frac{1}{\\epsilon^2}\\left\\{-\\frac{176}{9}C_A C_F +\\frac{64}{9}C_F n_f +\\frac{128}{9}C_F^2 \\right\\}\n\\nonumber\\\\&\n+ \\frac{1}{\\epsilon}\\left\\{-\\frac{376}{27}C_A C_F + \\frac{104}{27}C_F n_f +\\frac{112}{27}C_F^2 \\right\\}\\bigg]\\,,\n\\end{align}\nwhere $C_A=N$ and $C_F=(N^2-1)\/2N$ are the quadratic Casimirs of the $SU(N)$ group and $n_f$\nis the number of quark flavours. The fact that the energy momentum tensor \n$T_{\\mu \\nu} = {\\cal O}^G_{\\mu\\nu} + {\\cal O}^Q_{\\mu\\nu}$ is conserved leads to \n$\\gamma^{(n)}_{QG}=-\\gamma^{(n)}_{GG}$ and $\\gamma^{(n)}_{QQ}=-\\gamma^{(n)}_{GQ}$ or\nequivalently $Z_{GG} = 1-Z_{QG}$ and $Z_{QQ}=1-Z_{GQ}$, which is expected to be true to all\norders in $a_s$. \nAll $\\gamma^{(n)}_{GG}$ are proportional to $n_f$ which is consistent with the\nexpectation that the conserved property of ${\\cal O}^G_{\\mu\\nu}$ breaks down beyond\ntree level due to the presence of quark loops. For pure gauge theory \n$(n_f=0)$ and the energy momentum tensor of the pure gauge theory \n${\\cal O}^G_{\\mu\\nu}$ is hence conserved by itself. \n\nDefining the renormalised $\\kappa_I$ in terms of bare ones through \n$\\hat \\kappa_I = \\sum_{J=G,Q} Z_{IJ} \\kappa_J$ with\n$I,J = G,Q$,\nwe find that the action takes the following form\n\\begin{eqnarray}\n\tS &=& -{1 \\over 2} \\int d^4x ~ h_{\\mu\\nu} \\left(\\kappa_G~ {\\cal O}^{G,\\mu\\nu} + \\kappa_Q~{\\cal O}^{Q,\\mu\\nu} \\right)\\,,\n\\end{eqnarray}\nthe resulting interaction terms expressed in terms of renormalised operators and\nrenormalised couplings are guaranteed to predict UV finite quantities to all\norders in strong coupling. In the rest of the paper, we will use this version of\nthe Lagrangian to study the phenomenology.\n\n\n\n\n\n\\subsection{Lepton pair invariant mass distribution $d\\sigma\/dQ^2$}\n\\label{ss:inv}\nOur next task to use the effective action expressed in terms of renormalised operators ${\\cal O}_I$ and couplings $\\kappa_I$ to\nobtain production cross section for a \npair of leptons $(l^+,l^-)$, through the\nscattering of two protons $H_1,H_2$ at the LHC: \n\\begin{equation}\n\\label{eq:3}\nH_1 (P_1)+ H_2 (P_2) \\rightarrow l^+(l_1) + l^-(l_2) + X(P_X)\\,\n\\end{equation}\nwhere the 4-momenta of the corresponding particles are denoted in the \nparentheses and the final inclusive state is denoted by $X$.\nThe hadronic cross section is related to the partonic subprocess \ncross sections in the QCD improved parton model as\n\\begin{align}\n\\label{eq:4}\n2S \\frac{d\\sigma^{H_1H_2}}{d Q^2} \\big(\\tau, Q^2 \\big) = \\sum_{ab =\n q,\\bar{q},g} \\int_{0}^{1} dx_1 \\int_{0}^{1} dx_2 ~~ \n\t\\hat f_a^{H_1}(x_1)\n\t\\hat f_b^{H_2} (x_2) \n\\nonumber\\\\\n\\times \\int_0^1dz \\, 2 s\\frac{d\\hat{\\sigma}^{ab}}{dQ^2}\\big(z,\n Q^2\\big)\\delta(\\tau - zx_1x_2)\\,, \n\\end{align}\nwhere $Q^2$ is the invariant mass square of the final state leptonic pair \nand $S$ is the square of the hadronic center of mass energy which is\nrelated to the partonic one, $s$, through $s=x_{1}x_{2}S$, \nsimilarly $\\tau \\equiv Q^2\/S$, $z \\equiv Q^2\/s$\nand $\\tau = x_1 x_2z$. The unrenormalised partonic distribution functions of the\npartons $a$ and $b$ are $\\hat f_a$ and $\\hat f_b$ respectively.\nThe partonic sub process corresponding to the hadronic process is\n\\begin{align*}\na(p_{1})+b(p_{2})\n\\rightarrow j(q) + \\sum\\limits_{i=1}^{m} X_{i}(q_{i}) \\,,\n\\end{align*}\nwhere the summation over $i$ corresponds to all the real QCD final state\npartons that could contribute to a particular order in perturbative QCD. The \ninitial state partons $a b \\to j$, a neutral state $j$ which could be a\nphoton ($\\gamma^{*}$), Z-boson ($Z^*$) or spin-2 particle and further decays to pair of leptons\n$j \\to l^+ l^-$. \n\nAt the partonic level, one encounters amplitudes involving both SM vector\nbosons and spin-2 particles as propagators and hence, at the cross section\nlevel, the squared amplitudes contain in addition to contributions from SM\nand spin-2 separately, those from interference of SM and spin-2 amplitudes.\nInterestingly, for the invariant mass distributions, the later one identically\nvanishes for the universal case, which was earlier noted both at NLO and NNLO\nlevels in ~\\cite{Mathews:2004xp,Ahmed:2016qhu}. Hence, at the cross section\nlevel, the SM and spin-2 contributions simply add up as\n\\begin{align}\n2 S{d \\sigma^{H_1H_2} \\over dQ^2}(\\tau,Q^2)&=\n2 S{d \\sigma^{H_1H_2}_{\\rm SM} \\over dQ^2}(\\tau,Q^2)\n+2 S{d \\sigma^{H_1H_2}_{\\rm spin-2} \\over dQ^2}(\\tau,Q^2) \\,,\n\\end{align}\nwhere the SM results are known exactly upto NNLO level for long time (see \\cite{Altarelli:1978id, Matsuura:1987wt, Matsuura:1988sm,Hamberg:1990np}) and result at N$^3$LO in the \nsoft gluon approximation is also available, see \\cite{Ahmed:2014cla}.\nFor the spin-2 case with universal coupling, namely $\\kappa_G=\\kappa_Q=\\kappa$, \nthe results upto NNLO level can be found in \\cite{Mathews:2004xp,Ahmed:2016qhu}. \nIn this article, we have extended this\ncomputation to NNLO QCD \nfor the case of non-universal couplings\ni.e., when $\\kappa_G$ and $\\kappa_Q$ are different.\nWe briefly describe the methodology that we use to obtain\nthe mass factorised partonic cross sections up to NNLO level.\nUnlike the SM, for the spin-2 exchange, at\nleading order (LO) we can have\ngluon initiated sub process in addition to the quark initiated\none: \n\\begin{align}\n\\label{eq:16}\nq+{\\bar q} \\rightarrow l^+l^-\\,, \\quad g+g \\rightarrow l^+l^-\\,.\n\\end{align}\n\\\\\nAt next-to-leading order (NLO) in QCD, we have \n\n\\begin{minipage}{2.5in}\n\\begin{align}\n&q + \\bar{q} \\rightarrow l^+l^-+g\\,,\n\\nonumber\\\\\n&g + g \\rightarrow l^+l^- + g\\,,\n\\nonumber\\\\\n&g + q \\rightarrow l^+l^- + q\\,,\n\\nonumber\n\\end{align}\n\\end{minipage}\n\\begin{minipage}{2.5in}\n\\begin{align}\n\\label{eq:17}\n&q + \\bar{q} \\rightarrow l^+l^-+ \\text{one loop}\\,,\n\\nonumber\\\\\n&g + g \\rightarrow l^+l^- + \\text{one loop}\\,,\n\\nonumber\\\\\n&g +\\bar{q} \\rightarrow l^+l^- + \\bar{q}\\,.\n\\end{align}\n\\end{minipage}\n\\\\\n\n\\noindent\nAt NNLO level, we have double real emission, \n\n\\begin{minipage}{2.5in}\n\\begin{align*}\n&q+\\bar{q} \\rightarrow l^+l^- + q + \\bar{q}\\,,\n\\nonumber\\\\\n&g+g \\rightarrow l^+l^- + g + g\\,,\n\\nonumber\\\\\n&g+q \\rightarrow l^+l^- + g + q\\,,\n\\nonumber\\\\\n&q+q \\rightarrow l^+l^- + q + q\\,,\n\\nonumber\\\\\n&q_{1}+\\bar{q}_{2} \\rightarrow l^+l^- + q_{1} + \\bar{q}_{2}\\,,\n\\nonumber\n\\end{align*}\n\\end{minipage}\n\\begin{minipage}{3.1in}\n\\begin{align}\n\\label{eq:18}\n&q_1+\\bar{q}_1 \\rightarrow l^+l^- + q_2 + \\bar{q}_2\\,,\n\\nonumber\\\\\n&q+\\bar{q} \\rightarrow l^+l^- + g + g\\,,\n\\nonumber\\\\\n&g+g\\rightarrow l^+l^- + q + \\bar{q}\\,,\n\\nonumber\\\\\n&g+\\bar{q} \\rightarrow l^+l^- + g + \\bar{q}\\,,\n\\nonumber\\\\\n&q_{1}+q_{2} \\rightarrow l^+l^- + q_{1} + q_{2}\\,,\n\\end{align}\n\\end{minipage}\n\\\\ \n\n\\noindent\nsingle real emission at one loop, \n\n\\begin{minipage}{2.5in}\n\\begin{align*}\n&q + \\bar{q} \\rightarrow l^+l^- + g +\\text{one loop}\\,,\n\\nonumber\\\\\n&g + q \\rightarrow l^+l^- + q + \\text{one loop}\\,,\n\\nonumber\n\\end{align*} \n\\end{minipage}\n\\begin{minipage}{3.1in}\n\\begin{align}\n\\label{eq:19}\n&g+g \\rightarrow l^+l^- + g + \\text{one loop}\\,,\n\\nonumber\\\\\n&g + \\bar{q} \\rightarrow l^+l^- + \\bar{q} + \\text{one loop} \\,,\n\\end{align} \n\\end{minipage}\n\\\\\n\n\\noindent \nand the pure double virtual diagrams:\n\n\\begin{align}\n\\label{eq:20}\n&q + \\bar{q} \\rightarrow l^+l^- + \\text{two loop}\\,,\n\\nonumber\\\\\n&g + g \\rightarrow l^+l^- + \\text{two loop}\\,.\n\\end{align} \n\\\\\nThe virtual corrections at one and two loop levels are straightforward for this process, the \nphase space integrals are often hard to evaluate.\nIn the first computation of the NNLO QCD correction to the DY pair production\n~\\cite{Hamberg:1990np}, the phase space integrals were performed\nin three different frames to achieve the final result. \nThis method was successfully applied \nin~\\cite{Ravindran:2003um} to obtain inclusive cross section for the Higgs production at NNLO. \nIn~\\cite{Harlander:2002wh}, using a systematic expansion around threshold, all the phase space integrals were performed\nto obtain the partonic cross sections for both DY and Higgs productions at NNLO level. Later on, \nin~\\cite{Anastasiou:2002yz}, an elegant formalism was developed to compute both \nreal emissions as well as virtual corrections\napplying integration by parts \n(IBP)~\\cite{Tkachov:1981wb, Chetyrkin:1981qh} and\nLorentz invariance (LI)~\\cite{Gehrmann:1999as} identities. This approach is famously called \nthe method of reverse unitarity. \nThe resulting master integrals (MIs) were computed using the technique of differential equations.\nThe state-of-the-art result, namely, N$^3$LO QCD corrections to the inclusive Higgs boson\n production~\\cite{Anastasiou:2014vaa, Anastasiou:2015ema, Anastasiou:2016cez} uses the method of reverse unitarity. \n We have systematically used this approach \n~\\cite{Anastasiou:2002yz} to calculate the partonic cross section of\nthe DY pair production \nthrough intermediate spin-2 particle at NNLO QCD. \n\nUltraviolet (UV), soft and\ncollinear (IR) divergences do show up beyond leading order \nand they are regularised in dimensional regularisation where the space-time\ndimensions $d$ is chosen to be equal to $4+\\epsilon$. \nThe soft divergences cancel among virtual and real subprocesses processes \nthanks to Kinoshita-Lee-Nauenberg (KLN) theorem~\\cite{Kinoshita:1962ur, Lee:1964is} \nand the remaining UV divergences as well as\ninitial state collinear divergences are removed in ${\\overline {\\text{MS}}}$ scheme\nusing UV renormalisation constants and mass factorisation kernels denoted by $\\Gamma_{ab}(\\mu_F)$\nrespectively. Here, $\\mu_{F}$ is the factorisation scale.\nFor the UV renormalisation, we need to perform \nrenormalisation for strong coupling constant $a_s = g_s^2\/16 \\pi^2$ through $Z_{a_{s}}$ as well as \nrenormalisation of $\\kappa_I$ through $Z_{IJ}$ listed in\nthe previous section. For the former, we have\n\\begin{align}\n\\label{eq:21}\n{\\hat a}_{s} S_{\\epsilon} = \\left( \\frac{\\mu^{2}}{\\mu_{R}^{2}} \\right)^{\\epsilon\/2}\n Z_{a_{s}} a_{s} \\,,\n\\end{align}\nwhere,\n\\begin{align}\n\\label{eq:22}\n&Z_{a_{s}} = 1+ a_s\\left[\\frac{2}{\\epsilon} \\beta_0\\right]\n + a_s^2 \\left[\\frac{4}{\\epsilon^2 } \\beta_0^2\n + \\frac{1}{\\epsilon} \\beta_1 \\right] + \\cdot \\cdot \\cdot \\,,\n\\end{align} \n$a_{s} \\equiv a_{s}(\\mu_{R}^{2})$,\n$S_{\\epsilon} = {\\rm exp} \\left[ (\\gamma_{E} - \\ln 4\\pi)\\epsilon\/2\n\\right]\\,, \\gamma_{E} = 0.5772\\ldots\\,,$ \nand the scale $\\mu$ is introduced to keep the unrenormalised strong\ncoupling constant ${\\hat a}_{s}$ dimensionless in $n$-dimensions. The\nrenormalisation scale is denoted by $\\mu_{R}$. $\\beta_{i}$'s are the coefficients of QCD\n$\\beta$-function~\\cite{Gross:1973id, Politzer:1973fx, Caswell:1974gg, Tarasov:2013zv, Larin:1993tp}. \nThe mass factorised finite cross section can be obtained using \n\\begin{align}\n\\label{eq:23}\n\t2 s {d \\hat \\sigma_{ab} \\over dQ^2}(z,Q^2,1\/\\epsilon) =\n \\sum_{c,d=q,{\\bar q}, g}\\Gamma_{ca}(z,\\mu_F^2,1\/\\epsilon) \\otimes\n \\Gamma_{db}(z,\\mu_F^2,1\/\\epsilon)\\otimes \n 2 s {d \\sigma_{ab} \\over dQ^2}(z,Q^2,\\mu_{F}^2)\\,,\n\\end{align}\nwhere $\\otimes$ are nothing but Mellin convolution. The mass factorisation kernels take the following form\n\\begin{align}\n\\label{eq:24}\n\t\\Gamma_{ab}(z, \\mu_{F}^{2},1\/\\epsilon) =& \n\\delta_{ab} \\delta(1-z)\n + a_s(\\mu_F^2) \\frac{1}{\\epsilon} P^{(0)}_{ab}(z)\n\\nonumber \\\\\n\t& +a_{s}^2(\\mu_{F}^{2})\n\t\\left[\\frac{1}{\\epsilon^{2}} \\Bigg( \\frac{1}{2} P^{(0)}_{ac} \\otimes\n P^{(0)}_{cb} + \\beta_{0} P^{(0)}_{ab}\\Bigg) + \\frac{1}{\\epsilon}\n\t\\Bigg( \\frac{1}{2} P^{(1)}_{ab}\\Bigg)\\right] + \\cdot \\cdot \\cdot \\,, \n\\end{align}\nwhere $P^{(i)}_{ab}$ are the Altarelli-Parisi splitting\nfunctions~\\cite{Altarelli:1977zs, Floratos:1980hm, Floratos:1980hk, Curci:1980uw, Moch:2004pa,Vogt:2004mw}.\nAfter the mass factorisation, the finite partonic cross sections denoted by $2 s {d \\sigma_{ab}\/ dQ^2}$ can be expressed in terms\n$\\Delta^h_{ab}(z,a_s(\\mu_R^2),Q^2\/\\mu_R^2,\\mu_F^2\/\\mu_R^2)$ by factoring out some overall constants. In terms of these $\\Delta^h_{ab}$, the \nhadronic cross section can be written as\n\\begin{align}\n\t2 S{d \\sigma^{H_1H_2}_{\\rm spin-2} \\over dQ^2}(\\tau,Q^2)&=\n\t\\sum_{q,\\bar q,g}{\\cal F}_{h} \\int_0^1 {d x_1 } \\int_0^1 \n{dx_2} \\int_0^1 dz \\delta(\\tau-z x_1 x_2)\n\\times \\Bigg[ \nH_{q{\\bar q}} \n \\sum\\limits_{k=0}^{2} a_{s}^{k} \\Delta^{h, (k)}_{q{\\bar q}} \n\\nonumber\\\\&\n+\nH_{g g} \\sum\\limits_{k=0}^{2} a_{s}^{k} \\Delta^{h, (k)}_{gg} \n+ \\Big( H_{gq} + H_{qg} \\Big)\n \\sum\\limits_{k=1}^{2} a_{s}^{k} \\Delta^{h, (k)}_{gq}\n\\nonumber\\\\&+\nH_{q q} \\sum\\limits_{k=2}^{2}\n a_{s}^{k} \\Delta^{h, (k)}_{qq} \n+ H_{q_{1} q_{2}} \\sum\\limits_{k=2}^{2}\n a_{s}^{k} \\Delta^{h, (k)}_{q_{1}q_{2}} \n\\Bigg]\\,,\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq:31}\n{\\cal F}_{h}=\\;{\\kappa_Q^2 Q^6 \\over 320 \\pi^2 }|{\\cal D}(Q^2)|^2\\,,\n\\quad \\quad \\quad\n\t\\Delta^{h,(k)}_{ab} ~= \\Delta^{h,(k)}_{ab} \\left(z,{Q^2\\over \\mu_R^2},{\\mu_F^2\\over \\mu_R^2}\\right) \\,.\n\\end{align}\n$\\kappa_Q$ in ${\\cal F}_{h}$ corresponds to the leptonic coupling to the spin-2, while\nthe coupling to quarks and gluons are taken in $\\Delta^{h,(k)}_{ab}$. We have provided analytical expressions for these $\\Delta^{h,(k)}_{ab}$ in Mathematica format as an ancillary file.\n${\\cal D}(Q^2)$ is the propagator of the massive spin-2 particle, with a decay width that has\nto be estimated considering its decay to SM particles.\n$H_{ab}$ are the combinations of the mass factorised partonic distribution functions: \n\\begin{align}\n\\label{eq:32}\nH_{q \\bar q}(x_1,x_2,\\mu_F^2)&=\nf_q^{H_1}(x_1,\\mu_F^2) \nf_{\\bar q}^{H_2}(x_2,\\mu_F^2)\n+f_{\\bar q}^{H_1}(x_1,\\mu_F^2)~ \nf_q^{H_2}(x_2,\\mu_F^2)\\,,\n\\nonumber\\\\\nH_{q q}(x_1,x_2,\\mu_F^2)&=\nf_q^{H_1}(x_1,\\mu_F^2) \nf_{q}^{H_2}(x_2,\\mu_F^2)\n+f_{\\bar q}^{H_1}(x_1,\\mu_F^2)~ \nf_{\\bar q}^{H_2}(x_2,\\mu_F^2)\\,,\n\\nonumber\\\\\nH_{q_1 q_2}(x_1,x_2,\\mu_F^2)&=\nf_{q_1}^{H_1}(x_1,\\mu_F^2) \n\\Big( f_{q_2}^{H_2}(x_2,\\mu_F^2) + f_{\\bar q_2}^{H_2}(x_2,\\mu_F^2) \\Big)\n\\nonumber\\\\&+f_{\\bar q_1}^{H_1}(x_1,\\mu_F^2)~ \n\\Big( f_{q_2}^{H_2}(x_2,\\mu_F^2) + f_{\\bar q_2}^{H_2}(x_2,\\mu_F^2) \\Big)\\,,\n\\nonumber\\\\\nH_{g q}(x_1,x_2,\\mu_F^2)&=\nf_g^{H_1}(x_1,\\mu_F^2) \n\\Big(f_q^{H_2}(x_2,\\mu_F^2)\n+f_{\\bar q}^{H_2}(x_2,\\mu_F^2)\\Big)\\,,\n\\nonumber\\\\\nH_{q g}(x_1,x_2,\\mu_F^2)&=\nH_{g q}(x_2,x_1,\\mu_F^2)\\,,\n\\nonumber\\\\\nH_{g g}(x_1,x_2,\\mu_F^2)&=\nf_g^{H_1}(x_1,\\mu_F^2)~ \nf_g^{H_2}(x_2,\\mu_F^2)\\,.\n\\end{align}\nIn the next section, we study the numerical implication of NNLO QCD corrections \nto a spin-2 coupling non-universally to the SM in the DY process.\n\n\n\\begin{section}{Numerical results}\nIn this section, we present the numerical impact of our NNLO results on the\nproduction of di-leptons at the LHC. We considered a minimal scenario of\nnon-universal couplings\nof spin-2 particle with SM fields, where the spin-2 particle couples to all SM fermions \nwith coupling $\\kappa_Q = \\sqrt{2} k_q\/\\Lambda$ and to all SM gauge bosons with a\ncoupling strength of $\\kappa_G = \\sqrt{2} k_g\/\\Lambda$. Numerical results presented\nin this section are for the default choice of model parameters, namely spin-2 particle\nof mass $m_G=500$ GeV, the scale $\\Lambda=2$ TeV and the couplings $(k_q,k_g) = (0.5,1.0)$. Both the renormalization and factorization scales are set equal to the invariant mass of the di-lepton, i.e.,\n$\\mu_R = \\mu_F = Q$. Throughout, we use MSTW2008nnlo parton distribution functions (PDFs) with the corresponding $a_s$ \n\tprovided from {\\tt LHAPDF} unless otherwise stated. The choose $\\sqrt{S}=13$ TeV, the center of mass\nenergy of the incoming hadrons at the LHC.\n\nIn our analysis, we restricted ourselves to the situation where spin-2 particle\ndecays only to SM fields. The spin-2 particle decay widths for non-universal\ncouplings are same as those given in \\cite{Han:1998sg}. For the scenario taken\nup here, where in all spin-2 coupling to all bosons are taken to be identical,\nwe note that\nspin-2 particle decaying to $Z \\gamma$ vanishes identically \n$\\Gamma(h \\to Z\\gamma) =0$ \\cite{Falkowski:2016glr}. In fig.\\ref{subnlo}, we\npresent the NLO corrections\n(only at order $a_s$) from various subprocess contributions to the di-lepton\nproduction. For our default choice of model parameters, we find that $gg$ subprocess \ncontribution dominates over the rest. In general, the total NLO correction is\nsmaller than the $gg$ contribution because of negative contribution from $qg$\nsubprocess. We also note that the $gg$ has dominant contribution to the total \ndecay width for couplings (0.5, 1.0).\n\nTo estimate the impact of QCD corrections, we define the K-factors as follows:\n\\begin{eqnarray}\n\\text{K}_1 = \\frac{d\\sigma^{\\text{NLO}}\/dQ}{d\\sigma^{\\text{LO}}\/dQ} \n\\qquad \\text{and} \\qquad \n\\text{K}_2 = \\frac{d\\sigma^{\\text{NNLO}}\/dQ}{d\\sigma^{\\text{LO}}\/dQ}.\n\\label{eqnkf}\n\\end{eqnarray}\nIn the left panel of fig.\\ref{nlokf}, we present di-lepton invariant mass distributions to NLO for different choices of non-universal \ncouplings $(k_q,k_g)= (1.0,0.5), (1.0,0.1)$ and $(0.5,0.1)$. It is expected for universal couplings that at the resonance region,\nthe cross sections i.e. the height of the peak will be the same simply because the couplings at the matrix element level will cancel with those\nfrom the decay width of the spin-2 particle. However, for non-universal couplings this is not the case \nand hence cross sections at the resonance for different non-universal couplings will be different.\nThus, the precision as well as the phenomenological studies of the spin-2 particle production in this model\nwill be different from those of the warped extra dimension models.The NLO K-factor ($K_1$) is present in the right panel for various choices of $(k_q,k_g)$ and we observe that the K-factor crucially depends on the choice\nof non-universal couplings. In particular we notice that the K-factors are larger for the choice of couplings (1.0,0.1). To understand\nthis behaviour better, it is helpful to study the percentage contribution of various subprocesses to the total correction\nat NLO level, particularly from $qg$ subprocess due to its large flux at LHC energies. In particular we define the percentage of contribution \nof a given subprocess $ab$ as $R^{(i)}_{ab} = (d\\sigma^{H_1 H_2,(i)}_{ab}\/dQ^2) \/\n\t(d \\sigma^{H_1 H_2,(i)}\/dQ^2) \\times 100$, where the numerator is obtained by using \ncontribution from $\\Delta^{h,(i)}_{ab}$ and for the denominator, we include all the partonic channels.\n\n\nIn fig.\\ref{qgnlo-fr}, we plot $R^{(1)}_{qg}$ for different choices of non-universal couplings and we observe that the \nsign of the $qg$ subprocess crucially depends on the choice of couplings. Moreover, we find that $R^{(1)}_{qg}$ is positive\nand is as large as $70\\%$ for the couplings $(1.0,0.1)$, which explains the reason for the large K-factor at the resonance\nregion. However, the sign of the contribution from other subprocesses $q\\bar{q}$ and $gg$ is found \nto be positive for various couplings.\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_sub_nlo_cs_d8.pdf,width=8.0cm,height=8.0cm,angle=0}\n}\n\\caption{\\sf {First order QCD corrections from different subprocesses to di-lepton production. The choice of the\nmodel parameters is as mentioned in the text.}}\n\\label{subnlo}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_nlo_cs2.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_c0var_nlo_kf2.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Di-lepton invariant mass distributions are presented to NLO QCD for\ndifferent choice of couplings ($k_q, k_g$)} in the left panel. The corresponding K-factors\nare presented in the right panel.}\n\\label{nlokf}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_qg_nlo_fr.pdf,width=7.50cm,height=7.50cm,angle=0}\n}\n\\caption{\\sf {Percentage of $qg$ subprocess contribution $R^{(1)}_{qg}$ as defined in the text for different\nchoice of non-universal couplings.}}\n\\label{qgnlo-fr}\n\\end{figure}\n\nIn fig.\\ref{subnnlo}, we present the second order QCD corrections (at $(a_s^2)$) \nfrom various subprocesses to the di-lepton production for the default choice of couplings\n$(k_q, k_g) = (0.5,1.0)$. Similar to the first order QCD corrections, $gg$ \nsubprocess has the dominant contribution over the rest while $qg$ has a negative contribution\nbut is comparable in magnitude to that of $gg$. Because of this large $qg$ subprocess \ncontribution which can flip its sign for certain couplings, it is necessary to study \nthe percentage of its relative contribution $R^{(2)}_{qg}$ to the total second order \ncorrection.\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_sub_nnlo_cs_d8.pdf,width=7.50cm,height=7.50cm,angle=0}\n}\n\\caption{\\sf {Second order QCD corrections from various subprocess to the di-lepton invariant\nmass distribution.}}\n\\label{subnnlo}\n\\end{figure}\nIn fig.\\ref{fr-qgnnlo}, we present $R^{(2)}_{qg}$ for different\nchoices of couplings. As can be seen from the figure, the $qg$ contribution varies\nfrom about $-70\\%$ to about $35\\%$ for the choice of couplings considered here.\nIn particular, for the couplings $(1.0,0.1)$ and $(0.5, 0.1)$ the $qg$ contribution\nis positive while it is negative for the rest of the couplings as well as in the SM.\nThis implies large K-factors for the choice of $(1.0, 0.1)$ couplings for a wide\nrange of the invariant mass distribution.\nIt is worth mentioning here that in general $qg$ subprocess has a negative contribution\nboth in the SM as well as in the case of universal couplings, irrespective of the value\nof the latter.\n\nWe then present the di-lepton invariant mass distribution to various orders in QCD\nfor a particular choice of couplings $(1.0, 0.5)$ in fig.\\ref{c1order}. In this\ncase, the NLO QCD corrections for the signal (SM+spin-2) are as large as $60\\%$ while those\nat NNLO, they are about $80\\%$ at the resonance. Similar results are presented but for our \ndefault choice of model parameters in fig.\\ref{c2order}. Here, the corresponding NLO\ncorrections to the signal are about $45\\%$ while those of NNLO are about $55\\%$.\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_qg_nnlo_fr.pdf,width=7.50cm,height=7.50cm,angle=0}\n}\n\\caption{\\sf {Percentage of $qg$ contribution $R_{qg}^{(2)}$} as defined in the text.}\n\\label{fr-qgnnlo}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_order_cs1.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_order_kf1.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {cross sections at different orders (left panel) and the corresponding K-factors $K_1$ \nand $K_2$ (right panel) are presented for different couplings.}}\n\\label{c1order}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_order_cs2.pdf, width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_order_kf2.pdf, width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Same as fig.\\ref{c1order} but for a different set of couplings.}}\n\\label{c2order}\n\\end{figure}\n\nNext, we will study the invariant mass distributions of both the\nSM and the signal, in particular the impact of QCD corrections for\ndifferent couplings. In fig.\\ref{c01var},\\ref{c02var},\\ref{c03var}, we \npresent these distributions in the left panel and the corresponding NNLO K-factors \n$(K_2)$ in the right panel for 9 different set of non-universal couplings.\nThe respective K-factors for the signal at the resonance region are found\nto vary from about $1.5$ to about as large as $3.0$, owing to different\ncontributions from $qg$ subprocess to the signal as explained before.\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_cs1.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_c0var_kf1.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Di-lepton invariant mass distributions to NNLO for different choice\nof couplings (left panel) and the corresponding K-factors (right panel) are presented.}}\n\\label{c01var}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_cs2.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_c0var_kf2.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Same as fig.\\ref{c01var} but for a different set of couplings.}}\n\\label{c02var}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_c0var_cs3.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_c0var_kf3.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Same as fig.\\ref{c01var} but for a different set of couplings.}}\n\\label{c03var}\n\\end{figure}\n\nFurther, we depict the dependence of invariant mass distributions to NNLO\nin QCD on the center of mass energy E$_\\text{cm}$ of the protons at the LHC.\nWe present our results for E$_\\text{cm} = 7, 8, 13$ and $14$ TeV energies\nfor two different sets of couplings. In fig.\\ref{ecm-var1}, we present\nthe invariant mass distributions and the corresponding K-factors for\nthe universal couplings of $(1.0, 1.0)$. For default choice of non-universal\ncouplings $(0.5, 1.0)$, similar results are presented in fig.\\ref{ecm-var2}.\nIn both the cases, the K-factors at the resonance region \nare found to be larger for $7$ TeV case and are about $1.6$.\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc_ecmvar_cs_d1.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc_ecmvar_kf_d1.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Dependence of cross sections on the di-lepton invariant mass\ndistribution for universal couplings $(1.0,1.0)$.}}\n\\label{ecm-var1}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc_ecmvar_cs.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc_ecmvar_kf.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Same as fig.\\ref{ecm-var1} but for the default choice of\nnon-universal couplings $(0.5,1.0)$.}}\n\\label{ecm-var2}\n\\end{figure}\n\nIn what follows, we study the renormalization scale $\\mu_R$ and \nthe factorization scale $\\mu_F$ uncertainties in our predictions.\nFor this, we define the ratios $R(\\mu_R,\\mu_F)$ of the invariant mass\ndistributions computed at arbitrary scale to those computed at the\nfixed scale. These are defined as\n\\begin{eqnarray*}\n\\text{R}(\\mu_R,\\mu_F) = \\frac{d\\sigma(\\mu_R, \\mu_F)\/dQ}{d\\sigma(Q_0,Q_0)\/dQ}.\n\\end{eqnarray*}\nFor a systematic study of these scale uncertainties, we use LO (NLO and NNLO) PDFs\nfor LO (NLO and NNLO) cross sections respectively.\nFor convenience, we will study at the resonance region i.e. $Q=M=500$ GeV.\nThe fixed scale is set equal to $Q_0 = M$. In the left panel of fig.\\ref{mu-var},\nwe present $\\text{R}(\\mu_R,Q_0)$ by varying $\\mu_R$ from $0.1 Q$ to $10 Q$ and keeping\n$\\mu_F=Q_0$ fixed. At LO, there is no scale $\\mu_R$ entering the cross section.\nThe corresponding scale uncertainties at NLO and NNLO are respectively,\nabout $19\\%$ and $5\\%$.\n\nIn the right panel of fig.\\ref{mu-var}, we present $\\text{R}(Q_0,\\mu_F)$ by\nvarying $\\mu_F$ from $0.1 Q$ to $10 Q$ and keeping $\\mu_R=Q_0$ fixed. For this\nrange of factorization scale variation, the uncertainties in the distributions\nat LO, NLO and NNLO are respectively about $49\\%$, $31\\%$ and $26\\%$.\n\nFinally, we present $\\text{R}(\\mu,\\mu)$ (where $\\mu_R=\\mu_F =\\mu$) in fig.~\\ref{murf-var} by\nvarying $\\mu$ from $0.1 Q$ to $10 Q$. The corresponding scale uncertainties\nat LO, NLO and NNLO are respectively about $49\\%$, $52\\%$ and $30\\%$.\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_murvar_cs.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_mufvar_cs.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Renormalization (left) and factorization (right) scale dependence \nof the di-lepton invariant mass distribution at LO, NLO and NNLO.} }\n\\label{mu-var}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_murfvar_cs.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf Same as fig.\\ref{mu-var} but with $\\mu_R=\\mu_F=\\mu$.}\n\\label{murf-var}\n\\end{figure}\n\nBefore we summarize, we also study the uncertainties in our predictions\ndue to different choice of PDFs used in the calculation. For this analysis,\nwe make predictions using {\\tt MSTW2008, CT10, NNPDF3.0} and {\\tt ABM12}\nPDFs. The results for the invariant mass distributions for the signal at \nNNLO are presented in the left panel of fig.\\ref{pdf-var} and the corresponding \nK-factors are presented in the right panel of fig.\\ref{pdf-var}. The K-factors\nhere are found to vary from $1.18$ at $Q=400$ GeV to about $1.28$ at $Q=1000$ GeV,\nwhile at the resonance they are about $1.54$.\n\n\\begin{figure}[htb]\n\\centerline{\n\\epsfig{file=lhc13_pdfvar_cs.pdf,width=7.5cm,height=7.5cm,angle=0}\n\\epsfig{file=lhc13_pdfvar_kf.pdf,width=7.5cm,height=7.5cm,angle=0}\n}\n\\caption{\\sf {Di-lepton invariant mass distributions for different choice\nparton distribution functions (PDFs)}. }\n\\label{pdf-var}\n\\end{figure}\n\n\\end{section}\n\n\n\n\n\\section{Conclusion}\nIn this article, we have studied for the first time \nthe impact of NNLO QCD corrections\nto the production of a pair of leptons in the presence of a massive spin-2 \nparticle at the LHC.\nThis is done in a minimal scenario where spin-2 particles couple differently to SM\nfermions and SM bosons.\nThis task has been achieved by using the universal IR structure of QCD amplitudes\nand the additional UV renormalization that is particularly required for the\ncase of non-universal couplings, thanks to the recent computations\nof the form factors in QCD beyond leading order with non-universal couplings.\n\nUnlike the models with universal couplings, here the \nphenomenology is rich and different. For collider phenomenology at the LHC, we \npresent the results for the di-lepton production via spin-2 particle in particular \nfor the invariant mass distribution of a pair of leptons for LHC energies. Even at LO, one\ncan notice that the signal has different cross sections at the resonance region \nin contrast to the gravity mediated models where the signal has the same cross section for\ndifferent universal couplings.\nAt higher orders in QCD, say NLO onwards, the spin-2 exploits its freedom of \nbeing produced with different coupling strengths even for a given subprocess. This \nparticular aspect here makes the QCD radiative corrections crucially dependent\non the choice of the spin-2 coupling strength. Hence the impact of \nQCD corrections here is very much different from those of di-lepton or Higgs production \nin the SM.\n\nWe find from our numerical results that the QCD corrections for \n$(k_q,k_g) = (1.0,0.1)$\nare dominant over the rest of the choice of couplings, making the K-factors as large\nas 2.5 or more. For this choice of couplings, the LO gluon fusion contribution\nis very small although gluon fluxes are high for the kinematic region of producing\na 500 GeV particle. But at higher orders where the spin-2 can be emitted off from a \nquark line with large coupling strength, the large quark-gluon fluxes at LHC energies\ncan potentially enhance the spin-2 production rate, as is evident from the numerical\nresults. For di-lepton production the `sign' of $qg$ subprocess is usually negative both\nin the SM as well as in the models of universal couplings. But here we note that the \n`sign' of $qg$ subprocess contribution changes with the non-universal couplings and \nfor the above choice it is positive.\n\nWe also gave predictions for different center of mass energies of the incoming protons\nat the LHC and found that the K-factors are larger for 7 TeV case. We further quantified\nthe renormalization and factorization scale uncertainties. For the variation of the scales \n$\\mu_R$ and $\\mu_F$ between $0.1 Q$ and $10 Q$, the uncertainties are found to get reduced \nfrom about 50\\% at LO to about 30\\% at NNLO. For completeness, we also quantified the \nuncertainty in our predictions due to different choice of the PDFs.\n\nThese NNLO QCD predictions for the hadroproduction of a massive spin-2 with non-universal\ncouplings will augment the similar results previously computed at NLO level and compliment\nthe earlier results for NNLO QCD corrections in models with spin-2\/graviton universal couplings.\n\n\\section*{Acknowledgement}\n\nWe thank G. Das for helping us to validate a part of the numerical results. We also thank T.\\ Ahmed and N.\\ Rana for useful discussions.\nV.\\ Ravindran would like to thank M.\\ Neubert for the visit at University\nof Mainz where the last part of the work was carried out.\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\n\n\nWe are interested in designing a receding horizon control (RHC) algorithm that is able to readily accommodate interval-wise total energy constraints (ITEC). An ITEC system has the property that the total energy of the control input during a (pre-established) periodic time interval is limited (See Fig. \\ref{description}). Examples of systems subject to ITEC include solar-powered systems that operate on battery power when sunlight is not available (e.g., at night). One can think of solar houses and solar-panel equipped mobile agents performing surveillance tasks for an extended period of time. Other examples of systems subject to ITEC can also be found in avionics and include temperature control of the aircraft cabin, for which power allocation may be limited at certain times, such as during take-off and landing.\n\nThe idea behind RHC is simple: the controller solves a finite-horizon control problem, and the first element of the computed control sequence is applied to the system. At the next time step (or sampling time for continuous-time systems), the procedure is repeated. The RHC strategy is very appealing because in most cases it allows for online tractable computation of the control input that does not suffer from the deficiencies of a pure open-loop strategy. The main drawback is that the direct implementation of RHC can be disastrous because stability is not (immediately) guaranteed.\n\nReceding horizon control (or equivalently, model predictive control) dates back to the 1960s, but not until the late 1980s was the issue of stability dealt rigorously \\cite{Mayne:2000p442}. Until then, industry proponents of RHC had to manually ``fine tune\" parameters to achieve a seemingly stable system. The first modification of RHC that guaranteed stability was the introduction of a zero final state constraint \\cite{Mayne:1990p197},\\cite{Keerthi:1988p725}, which in many cases renders the finite optimization problem unfeasible (or at the very least, it places a heavy burden on the controller). Later relaxations of the zero final state constraint included terminal cost functions (that penalize large final states) and terminal invariant constraint sets \\cite{Chisci:1996p1035}, \\cite{Scokeart:1999p1036}, \\cite{Nicolao:2000p1307}. For a detailed survey of RHC, see \\cite{Mayne:2000p442}.\n\nAny of the traditional implementations of RHC (with step wise receding horizon) cannot readily accommodate the ITEC, as we explain in the next section. To remedy this problem, we implement a strategy that utilizes an initial horizon length of $2N$ that recedes by $N$ steps every $N$ steps (where $N$ is the length for which the ITEC is active) and does not recede at any other time instances. The basic idea is for the optimization horizon to encompass the ITEC in its entirety at all times. To guarantee stability of the system under this strategy, we impose contractive final state constraints during the optimization intervals for which the ITEC is not imposed, along with the assumption that the system is $\\beta$-stabilizable (to be defined in section \\ref{secbeta}).\n\nThe idea of contractive constraints was first suggested by \\cite{Yang:1993p694} and has been later used in other works, such as \\cite{Oliveira:1994p553} and \\cite{Kothare:2000p452}. These papers also make use of a varying horizon strategy, which is also employed in \\cite{Michalska:1993p201}, where a dual-mode controller is employed that steers the system to an invariant set by a horizon length that is computed online, and once the state lies in the invariant set the controller changes mode to drive it to zero. In \\cite{Kothare:2000p452}, two horizons are employed: a shorter horizon used in the computation of the control sequence, and a longer horizon in which the predicted final state (propagated with constant input over the longer horizon) needs to satisfy the contractive constraint. The paper that is most similar to this one is \\cite{Oliveira:1994p553}, where a smaller, non-receding contraction horizon is employed. However, the prediction horizon moves at every step and is not suitable for problems with systems subject to ITEC. Another paper worth to mention is \\cite{Kothare:2003p686}, where a time-varying terminal constraint set is employed. Though the terminal constraint is not (necessarily) contractive and the authors employ a fixed horizon length, we feel this work is relevant to our research due to the time-varying nature of the terminal constraint set.\n\nThe paper is organized as follows: in section \\ref{secprob}, we give the problem statement and discuss the shortcomings of the traditional RHC strategy applied to systems subject to ITEC. In section \\ref{secbeta}, we introduce the idea of $\\beta$-stabilizability. In section \\ref{secvhc}, we describe the new RHC strategy, and provide proof of the stability of the system under the proposed algorithm. In section \\ref{secsim}, we give a concrete example along with simulation results that showcases the new RHC strategy. In section \\ref{seccon}, we conclude. Finally, in the appendix we provide the proof to a lemma used to prove our main result.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=12cm]{figure1.jpg}\n \\caption{ITEC systems: Total energy constraints are enforced only at certain intervals}\n \\label{description}\n\\end{figure}\n\n\n\\section{Problem Statement}\\label{secprob}\nSuppose we have a system with dynamics given by:\n\\begin{equation}\\label{dynamics}\nx(k+1)= \\phi(x(k), u(k)),\n\\end{equation}\n\\noindent where $x(k) \\in \\mathbb{R}^n$ and $u(k) \\in \\mathbb{R}^m$ are the state and the input of the system at time $k$, respectively; and $\\phi: \\mathbb{R}^n \\times \\mathbb{R}^m\\rightarrow \\mathbb{R}^n$, with $\\phi(0,0)=0$. Moreover, the system is constrained by:\n\\begin{align}\nu(k) &\\in \\mathcal{U},~~\\forall k \\label{c2} \\\\\n\\displaystyle \\sum_{k=(2p-1)N}^{(2p)N-1}||u(k)||^2& \\leq C, ~~p \\in \\mathbb{Z}_+, \\label{c3}\n\\end{align}\n\n\\noindent where $\\mathcal{U}$ is the input constraint set. Constraint (\\ref{c3}) is known as the interval-wise total energy constraint (ITEC). In a real world application, this would be imposed during time intervals in which the system needs to rely solely on battery power. During other time intervals the system is powered by other energy sources (such as solar panels) and the total energy constraint is not imposed (see Fig. \\ref{description}).\n\nThe infinite horizon control problem is to find the control sequence that minimizes the quadratic cost:\n\\begin{equation}\n\\begin{array}{ll}\n\\mathrm{minimize}~~&\\displaystyle\\sum_{k=1}^{\\infty} ||x(k)||^2+||u(k-1)||^2\\\\\n\\mathrm{subject~to} & \\mathrm{(\\ref{dynamics}), (\\ref{c2})~and~(\\ref{c3}).}\n\\end{array}\n\\end{equation}\n \nOne way of generating control inputs in a tractable-online fashion is to employ a RHC strategy. The traditional implementation of RHC, however, causes a dilemma: how much energy should the controller allocate to the interval that partially covers the total energy constraint interval? One approach is to proportionally allocate the energy constraints. For example, with a horizon $N=3$, the traditional receding horizon approach would perform a finite optimization at $k=1$ (see Fig. \\ref{rhc}) given by:\n\\begin{equation*}\n\\begin{array}{ll}\n\\mathrm{minimize}~~&\\displaystyle\\sum_{k=2}^{4} ||x(k)||^2+||u(k-1)||^2\\\\\n\\mathrm{subject~to} & \\mathrm{(\\ref{dynamics}), (\\ref{c2})}\\\\\n& ||u(3)||^2 \\leq C\/3, \\label{c3_rhc1}\n\\end{array}\n\\end{equation*}\n\\noindent and at time $k=2$, the total energy constraint would become:\n \\begin{equation*}\n||u(3)||^2+||u(4)||^2 \\leq 2C\/3.\n\\end{equation*}\nIn this paper we propose a modified receding horizon strategy that not only provides analytical stability guarantees, but also does not require a heuristic solution to the ITEC allocation problem. It is important to note that the overlap of the constraint interval at the beginning of the optimization interval is no cause for concern. For example, at time $k=4$ in Fig. \\ref{rhc} the amount of energy spent by $u(3)$ cannot be changed, say $||u(3)||^2=\\gamma_3$. Then, the optimization will be carried out with the total energy constraint equal to: \n \\begin{equation*}\n||u(4)||^2+||u(5)||^2 \\leq C-\\gamma_3.\n\\end{equation*}\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=12cm]{figure2b.pdf}\n \\caption{Traditional RHC with $N=3$. Note that during times $k=1$ and $k=2$, the total energy constraint interval is partially covered by the optimization horizon}\n \\label{rhc}\n\\end{figure}\n\n \\section{$\\beta$-STABILIZABILITY}\\label{secbeta}\n\nThe system is said to be $\\beta$-stabilizable with $N$ steps if there exists an admissible control sequence that reduces the size of the state by a factor of $\\beta$ in $N$ steps. In this paper, we do not require the contraction to occur over the optimization interval where the ITEC is active. A formal definition of $\\beta$-stabilizability with ITEC is given below:\n\n\\begin{mydef} We say that a system $\\phi$ is {$\\beta$-stabilizable} with ITEC parameter $C$ and horizon $N$ in an open ball $B_N$ centered around the origin if there exists $\\sigma>0$ such that:\n\\begin{align}\n\\mathcal{U}_N(x, \\sigma, \\beta) &\\not= 0\\\\\n\\mathcal{U}^C_N(x, \\sigma) &\\not= 0\n\\end{align}\nwhere $x \\in B_N$, and $\\mathcal{U}_N(x(k), \\sigma, \\beta)$, with $0 \\leq \\beta < 1$, is the set of all finite control sequences of length $N$ such that:\n\\begin{align}\n\\sum_{j=k+1}^{k+N} ||x(j)||^2+||u(j-1)||^2 & \\leq \\sigma ||x(k)||^2,\\label{boundedcost}\\\\\nu(k) &\\in \\mathcal{U} \\label{boundedinput}\\\\\n||x(k+N)||^2 &\\leq \\beta ||x(k)||^2\n\\end{align}\nand $\\mathcal{U}^C_N(x(k), \\sigma)$ is the set of all finite control sequences of length $N$ such that (\\ref{boundedcost}) and (\\ref{boundedinput}) hold and \n\\begin{align}\n||x(k+N)||^2 &\\leq ||x(k)||^2 \\label{itecbound}\\\\\n\\displaystyle \\sum_{j=k}^{k+N-1}||u(k)||^2& \\leq C, ~~p \\in \\mathbb{Z}_+, \\label{itec}\n\\end{align}\n\\end{mydef}\\\n\nNote that (\\ref{itecbound}) does not impose contraction of the state norm if we constrain the control sequence to respect (\\ref{itec}). We only require the state not to grow. Moreover, if the system has zero-input dynamics given by $\\phi(x(k), 0) = x(k)$, then (\\ref{itecbound}) is immediately satisfied.\n\n \\section{INTERVAL-WISE HORIZON CONTROL} \\label{secvhc}\n \nIn this section, we present an interval-wise receding horizon strategy (IRHC) that uses varying optimization horizons and contractive constraints. The varying horizon will not only enable the construction of an appropriate strategy for the problem with ITEC, but also guarantee asymptotic infinite horizon stability of the system with the imposition contractive constraints and the assumption that the system is $\\beta$-stabilizable.\n\nIn order to accommodate for the total energy constraint, we employ a receding horizon strategy that entirely encompasses the constraint interval at all times (see Fig. \\ref{vhc}). To accomplish this, the horizon is initially set for $h=2N$. Traditional RHC is performed, except that the horizon does ``recede\" (i.e., $h = h-1$). When the horizon reaches $h=N+1$, it is expanded again to $h=2N$.\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=12cm]{figure3.jpg}\n \\caption{Interval-wise RHC with parameter $N=3$ based on algorithm \\ref{algo1}}\n \\label{vhc}\n\\end{figure}\n\nWe implement a receding horizon strategy in the following way:\n\n\\begin{alg}\\label{algo1} {\\bf IRHC with contractive constraints}\n\\begin{enumerate}\n\\item Set the optimization horizon $h = 2N$, time index $k=0$, energy-used tracking parameter $\\gamma = 0$ and auxiliary indices $i=0$, $f=0$ and $p=N$.\n\\item Read in $x(k)$ and solve the constrained optimization problem:\n\\begin{equation}\n\\begin{array}{ll}\n\\mathrm{minimize}~~&\\displaystyle\\sum_{j=k+1}^{k+h} ||x(j)||^2+||u(j-1)||^2\\\\\n\\mathrm{subject~to} & \\mathrm{(\\ref{dynamics}), (\\ref{c2})}\\\\\n& \\displaystyle \\sum_{j=\\max\\{p,k\\}}^{p+N-1}||u(j)||^2 \\leq C-\\gamma,\\\\\n& ||x(k+h_c)||^2 \\leq \\beta^{(i+1)} ||x(0)||^2\n\\end{array}\n\\end{equation}\nand apply the first element in the resulting control sequence to the system, say $u_1$.\n\\item If $f = 0$ (this means the ITEC sits at the beginning of the optimization horizon), set $\\gamma=\\gamma+||u_1||^2$\n\\item Reduce horizon $h = h-1$. \n\\item If $h=N$, set $h=2N$ (this will push the horizon $N$ steps ahead) and $\\gamma=0$ . If $k = p-1$, set $i=i+1$ (this will contract the constraint at the end of the new horizon), $f =1$; else if $k=p+N-1$, set $p=p+2N$ and $f =0$.\n\\item Set $k=k+1$ and go back to step 2.\n\\end{enumerate}\n\\end{alg}\\\n\n\n\n\nWe are now ready to state our main result:\n\n\\begin{thm}\\label{thm1} If the ITEC system $\\phi$ is $\\beta$-stabilizable with horizon $N$, then the interval-wise receding horizon strategy described in Algorithm \\ref{algo1} will drive it asymptotically stable for any initial condition in $x(0) \\in B_N$. Furthermore, the resulting dynamics will satisfy:\n\\begin{equation}\n\\sum_{j=1}^{\\infty} ||x(j)||^2+||u(j-1)||^2 \\leq \\frac{1+\\beta}{1-\\beta} \\sigma||x(0)||^2.\n\\end{equation}\n\\end{thm}\\\n\n\nWe will prove this result by constructing a sequence of control sequences $v_q(k)$ indexed by $q$ that converges to the control sequence obtained by Algorithm \\ref{algo1} and a bounded sequence $\\Gamma_q$, associated with $v_q(k)$. To proceed, we need to define $v_q(k)$ and $\\Gamma_q$:\n\\begin{mydef} $v_q(k)$ is a $q$-indexed sequence of control sequences defined by:\n\\begin{equation}\\label{defv}\nv_q(k) = \\left\\{ \\begin{array}{ll} v_{q-1}(k) & 0\\leq k< (q-1)N;\\\\\nu_{qN}^{*(q+2)N-1} & (q-1)N \\leq k< (q+1)N;\\\\\n0 & k\\geq (q+1)N\\end{array} \\right. \n\\end{equation}\n\\noindent where $u_{k_1}^{*k_2}$ is the control segment obtained by Algorithm \\ref{algo1} for $k_1 \\leq k \\leq k_2$, and $v_1(k)$ is given by:\n\\begin{equation}\nv_1(k) = \\left\\{ \\begin{array}{ll} u_{0}^{*2N-1} & 0\\leq k< 2N;\\\\\n0 & k \\geq 2N \\end{array} \\right. \n\\end{equation}\n\\end{mydef}\\\n\nIn other words, $v_q(k)$ is indexed with each time the horizon is pushed $N$ steps ahead, and it consists of the past control inputs and the current $2N$ control segment, followed by zeros.\n\n\n\\begin{mydef} $\\Gamma_q$ is defined as:\n\\begin{equation}\\label{gamma}\n\\Gamma_q = \\sum_{j=1}^{(q+1)N}||x(j)||^2+||v_q(j-1)||^2+\\sum_{j=q+1}^{\\infty}\\beta^{\\lceil \\frac{j}{2} \\rceil}\\sigma||x(0)||^2\n\\end{equation}\n\\noindent where $v_q$ is as in (\\ref{defv}) and $x(j)$ is generated by $v_q(j-1)$ and the dynamics (\\ref{dynamics}), and $\\lceil \\cdot \\rceil$ is the ceiling function.\n\\end{mydef}\n\nFinally, the following lemma will be crucial to prove the asymptotic stability of the system under Algorithm \\ref{algo1}:\n\n\\begin{lem}\\label{lemma}If the system is $\\beta$-stabilizable with horizon $N$, then:\n\\begin{equation}\\label{lem1}\n\\Gamma_q \\leq \\frac{1+\\beta}{1-\\beta} \\sigma||x(0)||^2, ~~\\forall q\\geq 1\n\\end{equation}\n\\noindent where $\\Gamma_q$ is as in (\\ref{gamma}).\n\\end{lem}\n\\begin{proof} [Proof of Lemma \\ref{lemma}] See Appendix.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm1}] Let $G$ be defined as the infinite horizon cost incurred by the system under Algorithm \\ref{algo1}, i.e.:\n\n\\begin{equation}\nG \\equiv \\sum_{k=0}^{\\infty} ||x(k+1)||^2+||u^*(k)||^2.\\\\\n\\end{equation}\n\\noindent and note that we can write:\n\\begin{align}\n||u^*(k)||^2 &= ||u^*(k)-v_q(k)+v_q(k)||^2\\\\\n&\\leq ||v_q(k)||^2+||u^*(k)-v_q(k)||^2;\n\\end{align}\n\\noindent and so, G is bounded by:\n\\begin{equation}\nG \\leq \\lim_{q\\to\\infty}\\sum_{k=0}^{\\infty} ||x(k+1)||^2+ ||v_q(k)||^2+||u^*(k)-v_q(k)||^2\n\\end{equation}\nUsing (\\ref{gamma}), the previous inequality can be written as:\n\\begin{align}\nG &\\leq \\lim_{q\\to\\infty} \\Gamma_q +\\lim_{q\\to\\infty}\\sum_{k=0}^{\\infty}||u^*(k)-v_q(k)||^2\\\\\n&= \\lim_{q\\to\\infty} \\Gamma_q \\label{conv} \\\\\n&\\leq \\frac{1+\\beta}{1-\\beta} \\sigma||x(0)||^2\\label{finalstep}\n\\end{align}\n\\noindent where (\\ref{conv}) comes from the fact that the sequence $v_q(k)$ converges to sequence $u^*(k)$, and (\\ref{finalstep}) comes from (\\ref{lem1}). This concludes the proof.\n\\end{proof}\\\n\nNote that the IRHC algorithm can also be applied to system without ITEC, as described in the following corollary:\n\\begin{cor}If the non-ITEC system $\\phi$ is $\\beta$-stabilizable with horizon $N$, then the IRHC strategy described in Algorithm \\ref{algo1} (modified to add contraction at every $N$ steps) will drive it asymptotically stable for any initial condition in $x(0) \\in B_N$. Furthermore, the resulting dynamics will satisfy:\n\\begin{equation}\n\\sum_{j=1}^{\\infty} ||x(j)||^2+||u(j-1)||^2 \\leq \\frac{\\sigma}{1-\\beta} ||x(0)||^2\n\\end{equation}\n\\end{cor}\\\n\nThe proof is similar to that of Theorem \\ref{thm1}.\n\nIt is important to mention that the bound on the infinite horizon cost is only used to prove stability of the system and no claims on tightness are made. \n\n\\section{SIMULATION}\\label{secsim}\nIn this section, we employ the proposed algorithm to stabilize the two-dimensional nonlinear oscillator previously discussed in \\cite{Primbs:1999p697}. The dynamics of the system are given by:\n\n\\begin{align*}\n\\dot{x}_1 = & x_2\\\\\n\\dot{x}_2 = &-x_1\\Big(\\frac{\\pi}{2} +\\tan^{-1}(5x_1)\\Big)\\\\\n& ~~~~~~~~-\\frac{5x_1^2}{2(1+25x_1^2)}+4x_2+3u\n\\end{align*}\n\nWe used Matlab to implement the proposed IRHC algorithm to the discretized version of the system with sampling period $0.05$ and ITEC parameters $C=4.8$, $N=4$, initial state $x(0)=[2; -1]$ and $\\beta=0.8$. The $\\beta$-stabilizability (with $\\beta=0.8$) was determined empirically. Figs. \\ref{fig_x} and \\ref{fig_u} show the evolution of the norms of the state and of the input, respectively, at different time instances. The solid lines (marked with x's) represent past signals while the dashed lines (marked with circles) represent the future prediction. Note that the ITEC constraints are satisfied and that the prediction horizon encompasses the constraint intervals in their entirety. Fig. \\ref{fig_x1x2} shows the plot of $x_1$ vs. $ x_2$. Though the trajectory of the states have a oscillatory nature, the algorithm stabilizes the system.\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=9.3cm]{ex5_x.jpg}\n \\caption{Evolution of the state norm for the ITEC simulation.}\n \\label{fig_x}\n \\includegraphics[width=9.3cm]{ex5_u.jpg}\n \\caption{Evolution of the input norm for the ITEC simulation.}\n \\label{fig_u}\n\\end{figure}\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=9.3cm]{ex5x1x2.jpg}\n \\caption{ITEC simulation: $x_1$ vs $x_2$}\n \\label{fig_x1x2}\n\\end{figure}\n\nSince this example was used in \\cite{Primbs:1999p697} to showcase the dangers of the direct application of a receding horizon strategy, we made further simulations where the ITEC is not imposed. An interesting characteristic of this system (and the reason it was discussed in \\cite{Primbs:1999p697}) is that the direct application of a RHC strategy works well for certain horizons, but is unstable for others. We ran simulations with $N=4$ and $N=5$ for the same initial condition as before. The traditional RHC used a fixed horizon of $2N$ and no endpoint constraints, while the proposed IRHC algorithm used contractive constraint parameter $\\beta = 0.8$ and $\\beta = 0.2$. We also compare the receding horizon strategies to the one obtained by $u=-3x_2$ (see \\cite{Primbs:1999p697}). As we can see from Figs. \\ref{ex4_1} and \\ref{ex4_2}, the RHC performs very well for $N=4$, but the system becomes unstable when the horizon is increased to $N=5$. The proposed IRHC strategy with $N=4$ displays a similar oscillatory behavior as before, but incurs a lower cost. Table \\ref{table1} gives the simulated infinite horizon costs for all cases. Note that the system with the constraint $\\beta=0.8$ yields a lower cost than the system with the tighter constraint $\\beta=0.2$, but convergence is faster. Moreover, it is interesting to note that when the horizon parameter is increased to $N=5$, the total cost for $\\beta = 0.8$ increases while the cost for $\\beta=0.2$ decreases. \n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=9.3cm]{N4b08a.jpg}\n \\caption{Non-ITEC simulation with parameter $N=4$, IRHC with $\\beta =0.8$. Here the traditional RHC is stables and performs well}\n \\label{ex4_1}\n \\includegraphics[width=9.3cm]{N5b08a.jpg}\n \\caption{Non-ITEC simulation with parameter $N=5$, traditional RHC is unstable but IRHC $\\beta =0.8$ with still stabilizes the system}\n \\label{ex4_2}\n\\end{figure}\n\n\n\\begin{table}[tb]\n\\caption{Performance Comparison }\n\\label{table1}\n\\begin{center}\n\\begin{tabular}{|r|c|c|}\n\\hline\n& $N=4$ & $N=5$\\\\\n\\hline\n$u=-3x_2$ & \t\t \\multicolumn{2}{|c|}{$325.4$} \\\\\n\\hline\nRHC & \t\t\t\t$437.2$ & Unstable\\\\\n\\hline\nIRHC($\\beta = 0.8$) & $412.9$ & 539.4\\\\\n\\hline\nIRHC($\\beta = 0.2$) & \t$3947$ &2053 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\section{CONCLUSION}\\label{seccon}\nWe proposed a modified receding horizon strategy that readily handles total energy constraints that are only imposed at certain pre-specified periodic time intervals. Our approach utilizes an interval-wise receding horizon for the online optimization problem and contractive constraints to guarantee boundedness of the infinite horizon cost. The new algorithm is demonstrated by an example and compared to the direct implementation of the traditional step-wise receding horizon when no constraints are imposed. Future work include the use of imperfect state information (in a stochastic setting) and the application of this strategy to decentralized control problems. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nT-splines \\cite{SZBN:2003} have been introduced as a free-form geometric technology and were the first tool of interest in Adaptive Isogeometric Analysis (IGA). Although they are still among the most common techniques in Computer Aided Design, T-splines provide algorithmic difficulties that have motivated a wide range of alternative approaches to mesh-adaptive splines, such as hierarchical B-splines \\cite{Forsey:Bartels:1988,KVZB:2014}, THB-splines \\cite{GJS:2012}, LR splines \\cite{DLP:2013}, hierarchical T-splines \\cite{ESLT:2015}, amongst many others. \n\nOne major difficulty using T-splines for analysis has been pointed out by Buffa, Cho and Sangalli \\cite{BCS:2010}, who showed that general T-spline meshes can induce linear dependent T-spline blending functions. This prohibits the use of T-splines as a basis for analytical purposes such as solving a discretized partial differential equation. \nThis insight motivated the research on T-meshes that guarantee the linear independence of the corresponding T-spline blending functions, referred to as \\emph{analysis-suitable T-meshes}.\nAnalysis-suitability has been characterized in terms of topological mesh properties \\cite{ZSHS:2012} and, in an alternative approach, through the equivalent concept of Dual-Compatibility \\cite{BBCS:2012}.\nWhile Dual-Compatibility has been characterized in arbitrary dimensions \\cite{BBSV:2014}, Analysis-Suitability as a topological criterion for linear independence of the T-spline functions is only available in the two-dimensional setting.\n\nIn this paper, we introduce analysis-suitable T-splines for those 3D meshes that are refinements of tensor-product meshes, and propose an algorithm for their local refinement, based on our preliminary work in \\cite{Morgenstern:Peterseim:2015}. In addition, we generalize the algorithm from \\cite{Morgenstern:Peterseim:2015} by introducing a grading parameter $m$ that represents the number of children in a single elements' refinement. This allows the user to fully control how local the refinement shall be. Choosing $m$ large yields meshes with very local refinement, while a small $m$ will cause more wide-spreaded refinement. The former yields a smaller number of degrees of freedom, while the latter reduces the overlap of the basis functions and hence provides sparser Galerkin and collocation matrices.\n\nThis paper is organized as follows. Section~\\ref{sec: refinement} defines the initial mesh and basic refinement steps and introduces our new refinement algorithm. Section~\\ref{sec: adm meshes} then characterizes the class of `admissible meshes' generated by this algorithm.\nIn Section~\\ref{sec: Tspline def} we give a brief definition of trivariate odd-degree T-splines.\nIn Section~\\ref{sec: AS} we give an abstract definition of Analysis-Suitability in the 3D setting and prove that all admissible meshes are analysis-suitable. In Section~\\ref{sec: DC} we define dual-compatible meshes, and prove that analysis-suitability and dual-compatibility are equivalent, and that all dual-compatible meshes provide linear independent T-spline functions. (Figure~\\ref{fig: overview} illustrates this ``long way'' to linear independence.) Section~\\ref{sec: complexity} proves linear complexity of the refinement procedure, and conclusions and an outlook to future work are finally given in Section~\\ref{sec: conclusions}.\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}{l@{\\quad}c@{\\quad}c}\n& \\small Symbol & \\small Section \\\\\\hline\nrefinement algorithm & $\\refine$ & \\ref{sec: refinement} \\\\\nadmissible meshes & $\\A$ & \\ref{sec: adm meshes} \\\\\nanalysis-suitable meshes & $\\AS$ & \\ref{sec: AS} \\\\\ndual-compatible meshes & $\\DC$ & \\ref{sec: DC}\\\\\\hline\n\\end{tabular}\\bigskip\n\\[\n\\refine(\\A)\\stackrel{\\text{Theorem~\\ref{thm: ref works}}}\\subseteq\n\\A\\stackrel{\\text{Theorem~\\ref{thm: A in AS}}}\\subseteq\n\\AS\\stackrel{\\text{Theorem~\\ref{thm: AS in DC}}}\\EQ\n\\DC\\stackrel{\\text{Theorem~\\ref{thm: DC has dual basis}}}\\subseteq\n\\left[\\parbox{3.25cm}{\\centering \\small meshes with linearly independent T-splines}\\right]\n\\]\n\\caption{How we prove linear independence of the T-splines induced by the generated meshes.}\n\\label{fig: overview}\n\\end{figure}\n\n\\section{Adaptive mesh refinement}\\label{sec: refinement}\nThis section defines the new refinement algorithm and characterizes the class of meshes which are generated by this algorithm.\nThe algorithm is essentially a 3D version of the one introduced in \\cite{Morgenstern:Peterseim:2015}, with the additional feature of variable grading.\nThe initial mesh is assumed to have a very simple structure. In the context of IGA, the partitioned rectangular domain is referred to as \\emph{index domain}. This is, we assume that the \\emph{physical domain} (on which, e.g., a PDE is to be solved) is obtained by a continuous map from the active region (cf.\\ Section~\\ref{sec: DC}), which is a subset of the index domain. Throughout this paper, we focus on the mesh refinement only, and therefore we will only consider the index domain. For the parametrization and refinement of the T-spline blending functions, we refer to \\cite{SLSH:2012}.\n\n\\begin{df}[Initial mesh, element]\nGiven $\\tilde X,\\tilde Y,\\tilde Z\\in\\mathbb N$, the initial mesh $\\G_0$ is a tensor product mesh consisting of closed cubes (also denoted \\emph{elements}) with side length 1, i.e.,\n\\[\\G_0\\sei\\Bigl\\{[x-1,x]\\times[y-1,y]\\times[z-1,z]\\mid x\\in\\{1,\\dots,\\tilde X\\},y\\in\\{1,\\dots,\\tilde Y\\},z\\in\\{1,\\dots,\\tilde Z\\}\\Bigr\\}.\\]\nThe domain partitioned by $\\G_0$ is denoted by $\\Omega\\sei (0,\\tilde X)\\times (0,\\tilde Y)\\times(0,\\tilde Z)$.\n\\end{df}\n\nThe key property of the refinement algorithm will be that refinement of an element $K$ is allowed only if elements in a certain neighbourhood are sufficiently fine. The size of this neighbourhood, which is denoted $(\\mathbf p,m)$-patch and defined through the definitions below, depends on the size of $K$, the polynomial degree $\\mathbf p=(p_1,p_2,p_3)$ of the T-spline functions, and the grading parameter $m$. For the sake of legibility, we assume that $p_1,p_2,p_3$ are odd and greater or equal to 3. (For comments on even polynomial degrees, see Section~\\ref{sec: conclusions}.)\n\n\\begin{df}[Level]\nThe \\emph{level} of an element $K$ is defined by \\[\\ell(K)\\sei-\\log_m|K|,\\] where $m$ is the manually chosen grading parameter, i.e., the number of children in a single elements' refinement, and $|K|$ denotes the volume of $K$.\nThis implies that all elements of the initial mesh have level zero and that the refinement of an element $K$ yields $m$ elements of level $\\ell(K)+1$.\n\\end{df}\n\n\\begin{df}[Vector-valued distance]\\label{df: distance}\nGiven $x\\in\\barOmega$ and an element $K$, we define their distance \nas the componentwise absolute value of the difference between $x$ and the midpoint of $K$,\n\\begin{align*}\n\\Dist(K,x)&\\sei\\operatorname{abs}\\bigl(\\midp(K)-x\\bigr)\\ \\in\\reell^3,\\\\\n\\text{with}\\quad\\operatorname{abs}(y) &\\sei \\bigl(\\lvert y_1\\rvert, \\lvert y_2\\rvert, \\lvert y_3\\rvert\\bigr).\n\\end{align*}\nFor two elements $K_1,K_2$, we define the shorthand notation \n\\[\\Dist(K_1,K_2)\\sei\\operatorname{abs}\\bigl(\\midp(K_1)-\\midp(K_2)\\bigr).\\]\n\\end{df}\n\n\\begin{df}\\label{df: magic patch}\nGiven an element $K$, a grading parameter $m\\ge2$ and the polynomial degree $\\mathbf p=(p_1,p_2,p_3)$, \nwe define the open environment\n\\begin{align*}\n\\U(K)&\\sei\\{x\\in\\Omega\\mid\\Dist(K,x)<\\D(\\ell(K))\\},\n\\shortintertext{where}\n\\D(k)&\\sei\\begin{cases}m^{-k\/3}\\,\\bigl(p_1+\\tfrac32,p_2+\\tfrac32,p_3+\\tfrac32\\bigr)&\\text{if }k=0\\bmod3,\n\\\\[.7ex]\nm^{-(k-1)\/3}\\,\\bigl(\\tfrac{p_1+3\/2}m,p_2+\\tfrac32,p_3+\\tfrac32\\bigr)&\\text{if }k=1\\bmod3,\n\\\\[.7ex]\nm^{-(k-2)\/3}\\,\\bigl(\\tfrac{p_1+3\/2}m,\\tfrac{p_2+3\/2}m,p_3+\\tfrac32\\bigr)&\\text{if }k=2\\bmod3.\\end{cases}\n\\intertext{The $(\\mathbf p,m)$-patch of $K$ is defined as the set of all elements that intersect with environment of $K$,}\n\\patch\\G K &\\sei \\{K'\\in\\G\\mid K'\\cap\\U(K)\\neq\\emptyset\\}.\n\\end{align*}\nNote as a technical detail that this definition does \\emph{not} require that $K\\in\\G$. See also Figure~\\ref{fig: magic patch examples} for examples.\n\\end{df}\n\n\\begin{rem}\nBy definition, the size of the $(\\mathbf p,m)$-patch of an element $K$ scales linearly with the size of $K$ and with the polynomial degree $\\mathbf p$. Since $\\D(k)$ is decreasing in $m$, choosing $m$ large will cause small $(\\mathbf p,m)$-patches and hence more localized refinement.\n\\end{rem}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{refex_12}%\n\\hspace{.049\\textwidth}%\n\\includegraphics[width=.3\\textwidth]{refex_20}%\n\\hspace{.049\\textwidth}%\n\\includegraphics[width=.3\\textwidth]{refex_30}%\n\\caption{Examples for the $(\\mathbf p,m)$-patch of an element $K$, for $\\mathbf p=(3,3,3)$, $m=3$ and $\\ell(K)=2,3,4$.}\n\\label{fig: magic patch examples}\n\\end{figure}\n\n\nIn the subsequent definitions, we will give a detailed description of the elementary subdivision steps and then present the new refinement algorithm. \n\n\\begin{df}[Subdivision of an element]\nGiven an arbitrary element $K=[x,x+\\tilde x]\\times[y,y+\\tilde y]\\times[z,z+\\tilde z]$, where $x, y,z, \\tilde x,\\tilde y,\\tilde z\\in\\mathbb{R}$ and $\\tilde x,\\tilde y,\\tilde z>0$, we define the operators\n\\begin{align*}\n\\subdiv_\\mathrm x(K) &\\sei \\bigl\\{\\,[x+\\tfrac {j-1}m\\tilde x,x+\\tfrac jm\\tilde x]\\times[y,y+\\tilde y]\\times[z,z+\\tilde z]\\mid j\\in\\{1,\\dots,m\\}\\bigr\\},\\\\\n\\enspace\\subdiv_\\mathrm y(K) &\\sei \\bigl\\{\\,[x,x+\\tilde x]\\times[y+\\tfrac {j-1}m\\tilde y,y+\\tfrac jm\\tilde y]\\times[z,z+\\tilde z]\\mid j\\in\\{1,\\dots,m\\}\\bigr\\},\n\\\\\\text{and}\\enspace\n\\enspace\\subdiv_\\mathrm z(K) &\\sei \\bigl\\{\\,[x,x+\\tilde x]\\times[y,y+\\tilde y]\\times[z+\\tfrac {j-1}m\\tilde z,z+\\tfrac jm\\tilde z]\\mid j\\in\\{1,\\dots,m\\}\\bigr\\}.\n\\end{align*}\nThese operators will be used for $x$-, $y$-, and $z$-orthogonal subdivisions in the refinement procedure. Their output is illustrated in Figure~\\ref{fig: elemref}.\n\\end{df}\n\\begin{df}[Subdivision]\\label{df: subdivision\nGiven a mesh $\\G$ and an element $K\\in\\G$, we denote by $\\subdiv(\\G,K)$ the mesh that results from a level-dependent subdivision of $K$,\n\\begin{align*}\n\\subdiv(\\G,K)&\\sei\\G\\setminus\\{K\\}\\cup\\child(K),\\\\\n\\text{with}\\enspace\\child(K)&\\sei\n\\begin{cases}\\subdiv_\\mathrm x(K)&\\text{if }\\ell(K)=0\\bmod3,\\\\\\subdiv_\\mathrm y(K)&\\text{if }\\ell(K)=1\\bmod3,\\\\\\subdiv_\\mathrm z(K)&\\text{if }\\ell(K)=2\\bmod3.\\end{cases}\n\\end{align*}\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{elemref_1}\\qquad\n\\includegraphics{elemref_2}\n\\includegraphics{elemref_3}\n\\caption{Elementary subdivision routines for $m=3$:\n$x$-orthogonal subdivision of an element with level 0 (left),\n$y$-orthogonal subdivision of an element with level 1 (middle), and\n$z$-orthogonal subdivision of an element with level 2 (right).}\n\\label{fig: elemref}\n\\end{figure}\n\\end{df}\n\\begin{df}[Multiple subdivisions]\nWe introduce the shorthand notation $\\subdiv(\\G,\\M)$ for the subdivision of several elements $\\M=\\{K_1,\\dots,K_J\\}\\subseteq\\G$, defined by successive subdivisions in an arbitrary order,\n\\[\\subdiv(\\G,\\M)\\sei\\subdiv(\\subdiv(\\dots\\subdiv(\\G,K_1),\\dots),K_J).\\]\n\\end{df}\n\nWe will now define the new refinement algorithm through the subdivision of a superset $\\clos(\\G,\\M)$ of the marked elements $\\M$. In the remaining part of this section, we characterize the class of meshes generated by this refinement algorithm.\n\n\\begin{alg}[Closure]\\label{alg: closure}\nGiven a mesh $\\G$ and a set of marked elements $\\M\\subseteq\\G$ to be refined, the \\emph{closure} $\\clos(\\G,\\M)$ of $\\M$ is computed as follows.\n\\begin{algorithmic}\n\\STATE $\\Mtilde\\sei \\M$\n\\REPEAT\n\\FORALL{$K\\in\\Mtilde$}\n\\STATE $\\Mtilde\\sei\\Mtilde\\cup\\bigl\\{K'\\in\\patch\\G K\\mid \\ell(K')<\\ell(K)\\bigr\\}$\n\\ENDFOR\n\\UNTIL{$\\Mtilde$ stops growing}\n\\RETURN $\\clos(\\G,\\M)=\\Mtilde$\n\\end{algorithmic}\n\\end{alg}\n\n\\begin{alg}[Refinement]\\label{alg: refinement}\n\nGiven a mesh $\\G$ and a set of marked elements $\\M\\subseteq\\G$ to be refined, $\\refine(\\G,\\M)$ is defined by \\[\\refine(\\G,\\M)\\sei\\subdiv(\\G,\\clos(\\G,\\M)).\\]\nAn example of this algorithm is given in Figure~\\ref{fig: refinement algorithm}.\n\\end{alg}\n\\begin{figure}[ht]\n\\centering\\Large\n\\includegraphics[width=.25\\textwidth]{refex_11}%\n\\makebox[.125\\textwidth]{\\raisebox{.125\\textwidth}{$\\stackrel{1^\\text{st}\\text{ iter.}}\\rightarrow$}}%\n\\includegraphics[width=.25\\textwidth]{refex_13}%\n\\makebox[.125\\textwidth]{\\raisebox{.125\\textwidth}{$\\stackrel{2^\\text{nd}\\text{ iter.}}\\rightarrow$}}%\n\\includegraphics[width=.25\\textwidth]{refex_15}\\\\\n\\makebox[.125\\textwidth]{\\raisebox{.125\\textwidth}{$\\stackrel{3^\\text{rd}\\text{ iter.}}\\rightarrow$}}%\n\\includegraphics[width=.25\\textwidth]{refex_15}%\n\\makebox[.125\\textwidth]{\\raisebox{.125\\textwidth}{$\\stackrel{\\text{subdiv.}}\\rightarrow$}}%\n\\includegraphics[width=.25\\textwidth]{refex_18}\n\\caption{Example for Algorithm~\\ref{alg: refinement}, with $\\mathbf p=(3,3,3)$, $m=3$ and $\\M=\\{K\\}$ with $\\ell(K)=2$. In the first iteration of the \\textbf{for}-loop, all coarser (level 1) elements in the $(\\mathbf p,m)$-patch of $K$ are marked as well. in the second iteration, all coarser (level 0) ``neighbours'' of those elements are also marked. Since there are no elements that are coarser than level 0, the third iteration does not change anything. Hence the \\textbf{for}-loop ends, and all marked elements are subdivided in the directions that correspond to their levels. }\n\\label{fig: refinement algorithm}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.3\\textwidth}\n\\includegraphics[width=\\textwidth]{wuerfel1}\n\\caption{$m=2$}\\label{sfig: refex 2}\n\\end{subfigure}\\quad\n\\begin{subfigure}[b]{.3\\textwidth}\n\\includegraphics[width=\\textwidth]{wuerfel2}\n\\caption{$m=4$}\\label{sfig: refex 4}\n\\end{subfigure}\\quad\n\\begin{subfigure}[b]{.3\\textwidth}\n\\includegraphics[width=\\textwidth]{wuerfel3}\n\\caption{$m=16$}\\label{sfig: refex 16}\n\\end{subfigure}\n\\caption{Refinement examples for $\\mathbf p=(3,3,3)$ and different choices of $m$. In all cases, the initial mesh consists of $4\\times5\\times8$ cubes of size $1\\times1\\times1$, and is refined by marking the lower left front corner element repeatedly until it is of the size $\\tfrac1{16}\\times\\tfrac1{16}\\times\\tfrac1{16}$. }\n\\label{fig: refex}\n\\end{figure}\n\\begin{ex}\nConsider an initial mesh that consists of $4\\times5\\times8$ cubes of size $1\\times1\\times1$.\nWe refine the mesh by marking the lower left front corner element repeatedly until it is of the size $\\tfrac1{16}\\times\\tfrac1{16}\\times\\tfrac1{16}$.\nThe resulting meshes for different choices of $m$ are illustrated in Figure~\\ref{fig: refex}, and the results are listed below.\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\nFigure & $m$ & \\parbox{7em}{\\centering number of \\\\ refinement steps} & \\parbox{6em}{\\centering number of \\\\ new elements} \\\\\\hline\n\\ref{sfig: refex 2} & 2 & 12 & 10728 \\\\\n\\ref{sfig: refex 4} & 4 & 6 & 3175 \\\\\n\\ref{sfig: refex 16} & 16 & 3 & 1030\n\\end{tabular}\n\\end{center}\n\\end{ex}\n\n\\section{Admissible meshes}\\label{sec: adm meshes}\nIn the subsequent definitions, we introduce a class of admissible meshes. We will then prove that this class coindices with the meshes generated by Algorithm~\\ref{alg: refinement}.\n\n\\begin{df}[$(\\mathbf p,m)$-admissible subdivisions]\\label{df: adm. subdivision}%\nGiven a mesh $\\G$ and an element $K\\in\\G$, the subdivision of $K$ is called \\emph{$(\\mathbf p,m)$-admissible} if all $K'\\in\\patch\\G K$ satisfy $\\ell(K')\\ge\\ell(K)$.\n\nIn the case of several elements $\\M=\\{K_1,\\dots,K_J\\}\\subseteq\\G$, the subdivision $\\subdiv(\\G,\\M)$ is $(\\mathbf p,m)$-admissible if there is an ordering $(\\sigma(1),\\dots,\\sigma(J))$ (this is, if there is a permutation $\\sigma$ of $\\{1,\\dots,J\\}$) such that \\[\\subdiv(\\G,\\M)=\\subdiv(\\subdiv(\\dots\\subdiv(\\G,K_{\\sigma(1)}),\\dots),K_{\\sigma(J)})\\] is a concatenation of $(\\mathbf p,m)$-admissible subdivisions.\n\\end{df}\n\\begin{df}[Admissible mesh]\\label{df: admissible mesh}\nA refinement $\\G$ of $\\G_0$ is \\emph{$(\\mathbf p,m)$-admissible} if there is a sequence of meshes $\\G_1,\\dots,\\G_J=\\G$ and markings $\\M_j\\subseteq\\G_j$ for $j=0,\\dots,J-1$, such that $\\G_{j+1}=\\subdiv(\\G_j,\\M_j)$ is an $(\\mathbf p,m)$-admissible subdivision for all $j=0,\\dots,J-1$. The set of all $(\\mathbf p,m)$-admissible meshes, which is the initial mesh and its $(\\mathbf p,m)$-admissible refinements, is denoted by $\\A$. For the sake of legibility, we write `admissible' instead of `$(\\mathbf p,m)$-admissible' throughout the rest of this paper.\n\\end{df}\n\n\\begin{thm}\\label{thm: ref works}\nAny admissible mesh $\\G$ and any set of marked elements $\\M\\subseteq\\G$ satisfy \\[\\refine(\\G,\\M)\\in\\A.\\]\n\\end{thm}\nThe proof of Theorem~\\ref{thm: ref works} given at the end of this section relies on the subsequent results.\n\n\\begin{lma}\\label{lma: magic patches are nested}\nGiven an admissible mesh $\\G$ and two nested elements $K\\subseteq\\hat K$ with $K,\\hat K\\in\\tcup\\A$, the corresponding $(\\mathbf p,m)$-patches are nested in the sense $\\patch\\G K\\subseteq\\patch\\G{\\hat K}$.\n\\end{lma}\nThe proof is given in Appendix~\\ref{apx: magic patches are nested}.\n\n\\begin{lma}[local quasi-uniformity]\\label{lma: levels change slowly}\nGiven $K\\in\\G\\in\\A$, \nany $K'\\in\\patch\\G K$ satisfies $\\ell(K')\\ge\\ell(K)-1$.\n\\end{lma}\nThe proof is given in Appendix~\\ref{apx: levels change slowly}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm: ref works}]\nGiven the mesh $\\G\\in\\A$ and marked elements $\\M\\subseteq\\G$ to be refined, we have to show that there is a sequence of meshes that are subsequent admissible refinements, with $\\G$ being the first and $\\refine(\\G,\\M)$ the last mesh in that sequence.\n\nSet $\\Mtilde\\sei\\clos(\\G,\\M)$ and\n\\begin{alignat}{2}\n\\notag\\overline L&\\sei\\max\\ell(\\Mtilde),\\quad\\underline L\\sei\\min\\ell(\\Mtilde)&&\\\\\n\\notag\\M_j&\\sei\\bigl\\{K\\in\\Mtilde\\mid\\ell(K)=j\\bigr\\}&&\\text{for}\\enspace j=\\underline L,\\dots,\\overline L\\\\\n\\label{eq: ref works 1}\\G_{\\underline L}&\\sei\\G,\\quad \\G_{j+1}\\sei\\subdiv(\\G_j,\\M_j)&\\enspace&\\text{for}\\enspace j=\\underline L,\\dots,\\overline L.\n\\end{alignat}\nIt follows that $\\refine(\\G,\\M)=\\G_{\\overline L+1}$. We will show by induction over $j$ that all subdivisions in \\eqref{eq: ref works 1} are admissible. \n\n\nFor the first step $j=\\underline L$, we know $\\{K'\\in\\Mtilde\\mid\\ell(K')<\\underline L\\}=\\emptyset$, and by construction of $\\Mtilde$ that for each $K\\in\\Mtilde_{\\underline L}$ holds \n$\\{K'\\in\\patch\\G K\\mid\\ell(K')<\\ell(K)\\}\\subseteq\\Mtilde$.\nTogether with $\\ell(K)=\\underline L$, it follows for any $K\\in\\Mtilde_{\\underline L}$ that there is no $K'\\in\\patch\\G K$ with $\\ell(K')<\\ell(K)$. This is, the subdivisions of all $K\\in\\Mtilde_{\\underline L}$ are admissible independently of their order and hence $\\subdiv(\\G_{\\underline L},\\Mtilde_{\\underline L})$ is admissible.\n\nConsider an arbitrary step $j\\in\\{\\underline L,\\dots,\\overline L\\}$ and assume that $\\G_{\\underline L},\\dots,\\G_j$ are admissible meshes. \nAssume for contradiction that there is $K\\in\\M_j$ of which the subdivision is not admissible, i.e., there exists $K'\\in\\smash{\\patch{\\G_j}K}$ with $\\ell(K')<\\ell(K)$ and consequently $K'\\notin\\Mtilde$, because $K'$ has not been refined yet. It follows from the closure Algorithm~\\ref{alg: closure} that $K'\\notin\\G$. Hence, there is $\\hat K\\in\\G$ such that $K'\\subset\\hat K$. \nWe have $\\ell(\\hat K)<\\ell(K')<\\ell(K)$, which implies $\\ell(\\hat K)<\\ell(K)-1$. Note that $K\\in\\G$ because $\\M_j\\subseteq\\Mtilde\\subseteq\\G$.\nFrom $K'\\in\\patch{\\G_j}K$, it follows by definition that $K'\\cap\\U(K)\\neq\\emptyset$, and $K'\\subset\\hat K$ yields \n$\\hat K\\cap\\U(K)\\neq\\emptyset$ and hence $\\hat K\\in\\patch\\G K$.\nTogether with $\\ell(\\hat K)<\\ell(K)-1$, Lemma~\\ref{lma: levels change slowly} implies that $\\G$ is not admissible, which contradicts the assumption.\n\\end{proof}\n\n\\section{T-spline definition}\\label{sec: Tspline def}\nIn this section, we define trivariate T-spline functions corresponding to a given admissible mesh. We roughly follow the definitions from \\cite{Morgenstern:Peterseim:2015}.\n\\begin{df}[Active nodes]\nFor each element $K=[x,x+\\tilde x]\\times[y,y+\\tilde y]\\times[z,z+\\tilde z]$, the corresponding set of vertices is denoted by \\[\\N(K)\\sei\\{x,x+\\tilde x\\}\\times\\{y,y+\\tilde y\\}\\times\\{z,z+\\tilde z\\}.\\] We refer to the elements of $\\N\\sei\\bigcup_{K\\in\\G}\\N(K)$ as \\emph{nodes}.\nWe define the \\emph{active region} \\[\\AR\\sei\\Bigl[\\ceilfrac{p_1}2,\\tilde X-\\ceilfrac{p_1}2\\Bigr]\\times\\Bigl[\\ceilfrac{p_2}2,\\tilde Y-\\ceilfrac{p_2}2\\Bigr]\\times\\Bigl[\\ceilfrac{p_3}2,\\tilde Z-\\ceilfrac{p_3}2\\Bigr]\\]\nand the set of \\emph{active nodes} $\\N_A\\sei\\N\\cap\\AR$. \n\\end{df}\n\\begin{df}[Skeleton]\nGiven a mesh $\\G$, denote the union of all closed $x$-orthogonal element faces by $\\xix\\sei\\tcup_{K\\in\\G}\\xix(K)$, with \n\\begin{align*}\n\\xix(K) &\\sei \\{x,x+\\tilde x\\}\\times[y,y+\\tilde y]\\times [z,z+\\tilde z]\\\\\n\\text{for any } K &= [x,x+\\tilde x]\\times[y,y+\\tilde y]\\times [z,z+\\tilde z]\\in\\G.\n\\end{align*}\nWe call $\\xix$ the \\emph{$x$-orthogonal skeleton}. Analogously, we denote the $y$-orthogonal skeleton by $\\xiy$, and the $z$-orthogonal skeleton by $\\xiz$.\n\\end{df}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=.275\\textwidth]{Xi_1}\\hspace{.08\\textwidth}\n\\includegraphics[width=.275\\textwidth]{Xi_2}\\hspace{.08\\textwidth}\n\\includegraphics[width=.275\\textwidth]{Xi_3}\n\\caption{$x$-orthogonal, $y$-orthogonal and $z$-orthogonal skeleton of the final mesh from Figure~\\ref{fig: refinement algorithm}.}\n\\label{fig: Xi_xyz}\n\\end{figure}\n\n\n\\begin{df}[Global index sets]\nFor any $x,y,z\\in\\reell$, we define\n\\begin{alignat*}{2}\n\\XX(y,z) &\\sei \\bigl\\{\\tilde x\\in[0,&\\tilde X]&\\mid(\\tilde x,y,z)\\in\\xix\\bigr\\},\\\\\n\\YY(x,z) &\\sei \\bigl\\{\\tilde y\\in[0,&\\tilde Y]&\\mid(x,\\tilde y,z)\\in\\xiy\\bigr\\},\\\\\n\\ZZ(x,y) &\\sei \\bigl\\{\\tilde z\\in[0,&\\tilde Z]&\\mid(x,y,\\tilde z)\\in\\xiz\\bigr\\}.\n\\end{alignat*}\nNote that in an admissible mesh, the entries $\\bigl\\{ 0,\\dots,\\ceilfrac {p_1}2-1,\\enspace \\tilde X-\\ceilfrac {p_1}2+1,\\dots,\\tilde X\\bigr\\}$ are always included in $\\XX(y,z)$ (and analogously for $\\YY(x,z)$ and $\\ZZ(x,y)$).\n\\end{df}\n\n\\begin{df}[Local index vectors]\nTo each active node $v=(v_1,v_2,v_3)\\in\\N_A$, we associate a local index vector $\\xx(v)\\in\\reell^{p_1+2}$, which is obtained by taking the unique $p_1+2$ consecutive elements in $\\XX(v_2,v_3)$ having $v_1$ as their $\\tfrac{p_1+3}2$-th (this is, the middle) entry. We analogously define $\\yy(v)\\in\\reell^{p_2+2}$ and $\\zz(v)\\in\\reell^{p_3+2}$.\n\\end{df}\n\\begin{df}[T-spline blending function]\n We associate to each active node $v\\in\\N_A$ a trivariate B-spline function, referred as \\emph{T-spline blending function}, defined as the product of the B-spline functions on the corresponding local index vectors, \\[B_v(x,y,z)\\coloneqq N_{\\xx(v)}(x) \\cdot N_{\\yy(v)}(y)\\cdot N_{\\zz(v)}(z).\\]\n\\end{df}\n\n\\section{Analysis-Suitability}\\label{sec: AS}\nIn this section, we give an abstract definition of Analysis-Suitability. Instead of using T-junction extensions as in the 2D case, we define \\emph{perturbed regions} through the intersection of particular T-spline supports. Analysis-Suitability is then defined as the absence of intersections between these perturbed regions. This idea is comparable to the 2D case, where Analysis-Suitability is defined as the absence of intersections between T-junction extensions.\nSubsequent to these definitions, we prove that all previously defined admissible meshes are analysis-suitable.\n\n\\begin{df}[Perturbed regions]\\label{df: perturbed regions}%\nFor $q,r,s\\in\\reell$ define the slices\n\\begin{align*}\n\\Sx(q) &\\sei\\left\\{(\\tilde x,\\tilde y,\\tilde z)\\in\\AR\\mid \\tilde x=q\\right\\},\\\\\n\\Sy(r) &\\sei\\left\\{(\\tilde x,\\tilde y,\\tilde z)\\in\\AR\\mid \\tilde y=r\\right\\},\\\\\n\\Sz(s) &\\sei\\left\\{(\\tilde x,\\tilde y,\\tilde z)\\in\\AR\\mid \\tilde z=s\\right\\}.\n\\intertext{Moreover, we denote by}\n\\Nx(q) &\\sei\\left\\{(v_1,v_2,v_3)\\in\\N_A\\mid(q,v_2,v_3)\\in\\xix\\right\\}\n\\intertext{the set of all nodes of which the projection on the slice $\\Sx(q)$ lies in some element's face.\nDefine analogously}\n\\Ny(r) &\\sei\\left\\{(v_1,v_2,v_3)\\in\\N_A\\mid(v_1,r,v_3)\\in\\xiy\\right\\},\\\\\n\\Nz(s) &\\sei\\left\\{(v_1,v_2,v_3)\\in\\N_A\\mid(v_1,v_2,s)\\in\\xiz\\right\\}.\\\\\n\\intertext{For any $q,r,s\\in\\reell$ we define \\emph{slice perturbations}}\n\\Rx(q) &\\sei \\Sx(q)\\cap\\mbigcup{v\\in\\Nx(q)}\\supp B_v\\ \\cap \\mbigcup{v\\in\\N_A\\setminus\\Nx(q)}\\supp B_v,\\\\\n\\Ry(r) &\\sei \\Sy(r)\\cap\\mbigcup{v\\in\\Ny(r)}\\supp B_v\\ \\cap \\mbigcup{v\\in\\N_A\\setminus\\Ny(r)}\\supp B_v,\\\\\n\\Rz(s) &\\sei \\Sz(s)\\cap\\mbigcup{v\\in\\Nz(s)}\\supp B_v\\ \\cap \\mbigcup{v\\in\\N_A\\setminus\\Nz(s)}\\supp B_v.\n\\end{align*}\nThe \\emph{perturbed regions} $\\Rx$, $\\Ry$, $\\Rz$ are defined by \n\\[\\Rx\\sei\\bigcup_{q\\in\\reell}\\Rx(q),\\quad\\Ry\\sei\\bigcup_{r\\in\\reell}\\Ry(r),\\quad\\Rz\\sei\\bigcup_{s\\in\\reell}\\Rz(s).\\]\nIn a uniform mesh, the perturbed regions are empty. In a non-uniform mesh, the perturbed regions are a superset of all hanging nodes and edges (this is, all kinds of 3D T-junctions). See Figure~\\ref{fig: AS example 1} for a 2D visualization of these definitions.\n\\end{df}\n\n\\begin{df}[Analysis-suitability]\\label{df: AS}%\nA given mesh $\\G$ is \\emph{analysis-suitable} if the above-defined perturbed regions do not intersect, i.e. if\n\\[\\Rx\\cap\\Ry =\\Ry \\cap \\Rz=\\Rz\\cap \\Rx= \\emptyset.\\]\nThe set of analysis-suitable meshes is denoted by $\\AS$.\n\\end{df}\n\\begin{rem}\nWhen applied in the two-dimensional case, the above definitions may yield perturbed regions that are larger than the T-junction extensions from \\cite{ZSHS:2012, BBCS:2012} (see Fig.~\\ref{fig: AS example 2}). However, this occurs only in meshes that are not analysis-suitable, and \nthe 2D version of Definition~\\ref{df: AS} is, regarding refinements of tensor-product meshes, equivalent to the classical definition of analysis-suitability%\n.\n\\end{rem}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[baseline=0]\n\\draw[line width=3pt, red!40] (1.5,-.5) coordinate (q) -- (1.5,5.5) coordinate (sx);\n\\node[above] at (sx) {$\\Sx(q)$};\n\\node[below] at (q) {$q$};\n\\draw[<->] (-.5,6) node [above] {$y$} |- (5,-.5) node [right] {$x$};\n\\draw (q)++(0,-.1)--++(0,.2);\n\\draw[thick] (0,0) grid (4,5)\n (0,1.5)--(3,1.5) (1.5,0)--(1.5,2);\n\\foreach \\a in {0,...,4}\n{ \\foreach \\b in {0,1,2}\n \\draw[fill=orange!40] (\\a,\\b) circle (3pt);\n \\foreach \\b in {3,4,5}\n \\draw[fill=blue!40] (\\a,\\b) circle (3pt);\n}\n\\foreach \\b in {0,1,2}\n \\draw[fill=orange!40] (1.5,\\b) circle (3pt);\n\\foreach \\b in {0,1,1.5,2,3}\n \\draw[fill=orange!40] (\\b,1.5) circle (3pt);\n\\node[right] (na) at (4.5,3.5) {$\\N_A\\setminus\\Nx(q)$};\n\\node[right] (nx) at (4.5,1.5) {$\\Nx(q)$};\n\\foreach \\a in {3,4,5}\n \\draw[->,shorten >= 7pt] (na) -- (4,\\a);\n\\foreach \\a in {0,1,2}\n \\draw[->,shorten >= 7pt] (nx) -- (4,\\a);\n\\end{tikzpicture}\n\\begin{tikzpicture}[baseline=0]\n\\draw[line width=3pt, orange!40] (-.6,-.5) |- (4.4,4) |- (-.6,-.5);\n\\draw[line width=3pt, blue!40] (4.6,5.5) |- (2,1) |- (-.4,1.5) |- (4.6,5.5);\n\\draw[line width=3pt,red] (1.5,1.5)--(1.5,4) ++(.5,-1.5) coordinate (rx);\n\\node at (rx) {$\\Rx(q)$};\n\\draw[thick] (0,0) grid (4,5)\n (0,1.5)--(3,1.5) (1.5,0)--(1.5,2);\n\\node[left] at (-.6,0) {$\\mbigcup{v\\in\\Nx(q)}\\supp B_v$};\n\\node[left] at (-.4,5) {$\\mbigcup{v\\in\\N_A\\setminus\\Nx(q)}\\supp B_v$};\n\\end{tikzpicture}\n\n\\caption{2D example for the construction of the slice perturbation $\\Rx(q)$ in an analysis-suitable mesh. The left figure illustrates the construction of $\\Nx(q)$ and its complement $\\N_A\\setminus\\Nx(q)$, and the right figure shows the resulting slice perturbation, which coincides with the corresponding classical T-junction extension. }\n\\label{fig: AS example 1}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[baseline=0]\n\\draw[line width=3pt, red!40] (1.5,-.5) coordinate (q) -- (1.5,5.5) coordinate (sx);\n\\node[above] at (sx) {$\\Sx(q)$};\n\\node[below] at (q) {$q$};\n\\draw[<->] (-.5,6) node [above] {$y$} |- (5,-.5) node [right] {$x$};\n\\draw (q)++(0,-.1)--++(0,.2);\n\\draw[thick] (0,0) grid (4,5)\n (0,1.5)--(2,1.5) (1.5,0)--(1.5,2);\n\\foreach \\a in {0,...,4}\n{ \\foreach \\b in {0,1,2}\n \\draw[fill=orange!40] (\\a,\\b) circle (3pt);\n \\foreach \\b in {3,4,5}\n \\draw[fill=blue!40] (\\a,\\b) circle (3pt);\n}\n\\foreach \\b in {0,1,2}\n \\draw[fill=orange!40] (1.5,\\b) circle (3pt);\n\\foreach \\b in {0,1,1.5,2}\n \\draw[fill=orange!40] (\\b,1.5) circle (3pt);\n\\node[right] (na) at (4.5,3.5) {$\\N_A\\setminus\\Nx(q)$};\n\\node[right] (nx) at (4.5,1.5) {$\\Nx(q)$};\n\\foreach \\a in {3,4,5}\n \\draw[->,shorten >= 7pt] (na) -- (4,\\a);\n\\foreach \\a in {0,1,2}\n \\draw[->,shorten >= 7pt] (nx) -- (4,\\a);\n\\end{tikzpicture}\n\\begin{tikzpicture}[baseline=0]\n\\draw[line width=3pt, orange!40] (-.6,-.5) |- (4.4,4) |- (-.6,-.5);\n\\draw[line width=3pt, blue!40] (4.6,5.5) |- (1,1) |- (-.4,1.5) |- (4.6,5.5);\n\\draw[line width=3pt,red] (1.5,1)--(1.5,4) ++(.5,-1.5) coordinate (rx);\n\\node at (rx) {$\\Rx(q)$};\n\\draw[thick] (0,0) grid (4,5)\n (0,1.5)--(2,1.5) (1.5,0)--(1.5,2);\n\\node[left] at (-.6,0) {$\\mbigcup{v\\in\\Nx(q)}\\supp B_v$};\n\\node[left] at (-.4,5) {$\\mbigcup{v\\in\\N_A\\setminus\\Nx(q)}\\supp B_v$};\n\\end{tikzpicture}\n\\caption{2D example for the construction of the slice perturbation $\\Rx(q)$ in a mesh that is not analysis-suitable. The left figure illustrates the construction of $\\Nx(q)$ and its complement $\\N_A\\setminus\\Nx(q)$, and the right figure shows the resulting slice perturbation, which is strictly larger than the corresponding classical T-junction extension. }\n\\label{fig: AS example 2}\n\\end{figure}\n\n\n\\begin{thm}\\label{thm: A in AS}%\n$\\A\\subseteq\\AS$ for all $m\\ge 2$.\n\\end{thm}\n\n\\begin{proof}\nWe prove the claim by induction over admissible subdivisions.\nAssume $K\\in\\G\\in\\A\\cap\\AS$ and let $\\hat\\G\\sei\\subdiv(\\G,K)\\in\\A$ be an admissible subdivision of $\\G$. \nWe have to show that $\\hat\\G\\in\\AS$. We assume without loss of generality that $\\ell(K)=0\\bmod 3$. Hence subdividing $K$ adds $m-1$ faces to the mesh, which are $x$-orthogonal. Set $K\\eqqcolon[x,x+\\tilde x]\\times[y,y+\\tilde y]\\times[z,z+\\tilde z]$ and \n$\\tXi\\sei \\{x+\\tfrac jm\\tilde x\\mid j\\in\\{1,\\dots, m-1\\}\\}$, then the skeletons of $\\hat\\G$ are given by\n\\begin{equation*}\n\\hatxix=\\xix\\cup\\tXi\\times[y,y+\\tilde y]\\times[z,z+\\tilde z] ,\\quad\n\\hatxiy=\\xiy,\\quad\\hatxiz=\\xiz.\n\\end{equation*}\nLet $\\hat v\\in\\hat\\N_A\\setminus\\N_A$ be a new active node.\nUsing the local quasi-uniformity from Lemma~\\ref{lma: levels change slowly}, it can be verified that for all $r\\in\\reell$ such that $\\hat v\\in\\hat\\Ny(r)$ follows $\\Ry(r)\\cap\\supp B_{\\hat v}=\\emptyset$. Consequently, $\\hat\\Ry=\\Ry$ and analogously $\\hat \\Rz=\\Rz$. Moreover, $\\hat\\Rx(q)=\\Rx(q)$ for all $q\\notin\\tXi$. It remains to characterize \n\\[\\hat\\Rx(\\xi) = \\Sx(\\xi)\\cap\\mbigcup{v\\in\\hat\\Nx(\\xi)}\\supp B_v\\cap\\mbigcup{v\\in\\hat\\N_A\\setminus\\hat\\Nx(\\xi)}\\supp B_v\\]\nfor any $\\xi\\in\\tXi$.\nWith \n\\begin{equation}\\label{eq: A in AS: active nodes}\n\\begin{aligned}\n\\hat\\Nx(\\xi)&=\\Nx(\\xi)\\cup\\hat\\N_A\\setminus\\N_A\\\\\\text{and}\\enspace\\hat\\N_A\\setminus\\hat\\Nx(\\xi)&=\\N_A\\setminus\\hat\\Nx(\\xi)=\\N_A\\setminus\\Nx(\\xi),\n\\end{aligned}\n\\end{equation}\nit follows\n\\begin{align*}\n\\hat\\Rx(\\xi) &=\n\\Sx(\\xi)\\cap\\mbigcup{v\\in\\hat\\Nx(\\xi)}\\supp B_v\\ \\cap \\mbigcup{v\\in\\hat\\N_A\\setminus\\hat\\Nx(\\xi)}\\supp B_v,\\\\\n&\\Stackrel{\\eqref{eq: A in AS: active nodes}}= \\Sx(\\xi)\\cap\\Bigl(\\mbigcup{v\\in\\Nx(\\xi)}\\supp B_v\\cup\\mbigcup{v\\in\\hat\\N_A\\setminus\\N_A}\\supp B_v\\Bigr)\\cap\\mbigcup{v\\in\\N_A\\setminus\\Nx(\\xi)}\\supp B_v\\\\\n&= \\Rx (\\xi) \\cup \\Bigl(\\underbrace{\\Sx(\\xi)\\ \\cap\\mbigcup{\\hat v\\in\\hat\\N_A\\setminus\\N_A}\\supp B_{\\hat v}}_\\Sigma\\ \\cap\\mbigcup{v\\in\\N_A\\setminus\\Nx(\\xi)}\\supp B_v\\Bigr).\n\\end{align*}\nWe will prove below that $\\Sigma\\cap\\hat\\Rz=\\Sigma\\cap\\hat\\Ry=\\emptyset$. See Figures \\ref{fig: Sx-levels} and \\ref{fig: Sx-perturbations} for an example with $\\ell(K)=3$ and $m=2$.\nAssume for contradiction that there is $s\\in\\reell$ with $\\hat\\Rz(s)\\cap\\Sigma\\neq\\emptyset$. Then there exist $v\\in\\hat\\Nz(s)$ and $w\\in\\hat\\N_A\\setminus\\hat\\Nz(s)$ such that \n\\begin{equation}\\label{eq: A in AS: eqn to contradict}\n\\Sz(s)\\cap \\supp B_v\\cap\\supp B_w\\cap\\Sigma\\neq\\emptyset.\n\\end{equation}\nSince the subdivision of $K$ is admissible, we know that all elements in $\\patch\\G K$ are at least of level $\\ell(K)$. This implies that all those elements are of equal or smaller size than $K$. \nDenote $\\midp(K)\\eqqcolon(\\sigma,\\nu,\\tau)$ and $\\varepsilon\\sei\\tfrac{m^{-\\ell(K)\/3}}2$.\nIt follows\n\\begin{equation}\\label{eq: A in AS: Sigma in patch}\n\\Sigma\\subseteq\\tcup\\patch\\G K,\n\\end{equation}\nand with \\[\\hat\\N_A\\setminus\\N_A\\ \\subset\\ [\\sigma-\\varepsilon,\\sigma+\\varepsilon]\\times[\\nu-\\varepsilon,\\nu+\\varepsilon]\\times[\\tau-\\varepsilon,\\tau+\\varepsilon],\\]\nwe get more precisely\n\\begin{equation}\\label{eq: A in AS: def Sigma}\n\\Sigma\\subseteq\\left\\{\\xi\\right\\}\\times\\bigl[\\nu-\\varepsilon(p_2+2),\\nu+\\varepsilon(p_2+2)\\bigr]\\times\\bigl[\\tau-\\varepsilon(p_3+2),\\tau+\\varepsilon(p_3+2)\\bigr].\n\\end{equation}\nThe second-order patch $\\patch\\G{\\patch\\G K}\\sei\\bigcup_{K'\\in\\patch\\G K}\\patch\\G{K'}$ consists of elements that may be larger in $z$-direction, but are of same or smaller size than $K$ in $x$- and $y$-direction. For $w=(w_1,w_2,w_3)$, Equation \\eqref{eq: A in AS: eqn to contradict} implies $\\supp B_w\\cap\\Sigma\\neq\\emptyset$, and we conclude from \\eqref{eq: A in AS: def Sigma} that \n\\begin{equation}\\label{eq: A in AS: w is near Sigma}\n(w_1,w_2)\\ \\in\\ \\bigl[\\xi-\\varepsilon(p_1+1),\\xi+\\varepsilon(p_1+1)\\bigr]\\times\\bigl[\\nu-\\varepsilon(2p_2+3),\\nu+\\varepsilon(2p_2+3)\\bigr]\n\\end{equation}\nWe assume that there is no element in $\\G$ with level higher than $\\ell(K)+1$. This is an eligible assumption, since every admissible mesh can be reproduced by a sequence of level-increasing admissible subdivisions; see \\cite[Proposition 4.3]{Morgenstern:Peterseim:2015} for a detailed construction. This assumption implies that the $z$-orthogonal skeleton $\\xiz$ is a subset of the $z$-orthogonal skeleton of a uniform $(\\ell(K)+1)$-leveled mesh, \n\\begin{equation}\n\\xiz(\\G)\\subseteq\\xiz(\\Guni{\\ell(K)+1}), \\label{eq: A in AS: xiz in xiz-uni} \n\\end{equation}\nand with $\\min\\ell(\\patch\\G K)=\\ell(K)$, we have even equality on the patch $\\patch\\G K$,\n\\begin{equation}\\label{eq: A in AS: xiz-patch is xiz-uni-patch}\n\\xiz\\bigl(\\patch\\G K\\bigr)=\\xiz\\bigl(\\patch{\\Guni{\\ell(K)}}K\\bigr)=\\xiz\\bigl(\\patch{\\Guni{\\ell(K)+1}}K\\bigr), \n\\end{equation}\nusing the notation $\\xiz\\bigl(\\patch\\G K\\bigr)\\sei\\xiz(\\G)\\cap\\tcup\\patch\\G K$. \nSince $v\\in\\hat\\Nz(s)$, we know that \\mbox{$\\hat\\Nz(s)\\neq\\emptyset$}, \nwhich means that there are elements in $\\G$ that have $z$-orthogonal faces at the $z$-coordinate $s$, i.e., \n\\mbox{$\\Sz(s)\\cap \\xiz(\\G) \\neq\\emptyset$}. With \\eqref{eq: A in AS: xiz in xiz-uni} we get $\\Sz(s)\\cap \\xiz(\\Guni{\\ell(K)+1}) \\neq\\emptyset$.\nSince $\\Guni{\\ell(K)+1}$ is a tensor-product mesh, its $z$-orthogonal skeleton consists of global domain slices, which yields\n$\\Sz(s)\\subseteq \\xiz(\\Guni{\\ell(K)+1}).$\nThe restriction to the patch $\\patch\\G K$ yields\n\\begin{equation}\\label{eq: A in AS: Szs-GpK in xiz-GpK}\n\\Sz(s)\\cap\\tcup\\patch\\G K\\subseteq \\xiz\\bigl(\\patch{\\Guni{\\ell(K)+1}}K\\bigr) \\stackrel{\\eqref{eq: A in AS: xiz-patch is xiz-uni-patch}}=\\xiz\\bigl(\\patch\\G K\\bigr)\\subseteq\\xiz(\\G).\n\\end{equation}\nEquation \\eqref{eq: A in AS: eqn to contradict} implies that $\\Sz(s)\\cap\\Sigma\\neq\\emptyset$, and\nwith \\eqref{eq: A in AS: Sigma in patch} we get that $\\Sz(s)\\cap\\tcup\\patch\\G K\\neq\\emptyset$. \nHence\n\\begin{align}\n\\Sz(s)\\cap\\tcup\\patch\\G K\\ &\\supseteq\\ \\Sz(s)\\cap\\U(K) \\notag \\\\\n&=\\ \\bigl[\\xi-\\varepsilon(2p_1+3),\\xi+\\varepsilon(2p_1+3)\\bigr]\\times\\bigl[\\nu-\\varepsilon(2p_2+3),\\nu+\\varepsilon(2p_2+3)\\bigr]\\times\\{s\\}.\n\\end{align}\nSince $w\\notin\\hat\\Nz(s)$, we know by definition that $(w_1,w_2,s)\\notin\\xiz$. Then it follows from \\eqref{eq: A in AS: Szs-GpK in xiz-GpK} that $(w_1,w_2,s)\\notin\\Sz(s)\\cap\\tcup\\patch\\G K$,\nand hence \n\\begin{equation}\n(w_1,w_2)\\ \\notin\\ \\bigl[\\xi-\\varepsilon(2p_1+3),\\xi+\\varepsilon(2p_1+3)\\bigr]\\times\\bigl[\\nu-\\varepsilon(2p_2+3),\\nu+\\varepsilon(2p_2+3)\\bigr] \n\\end{equation}\nin contradiction to \\eqref{eq: A in AS: w is near Sigma}. This proves that $\\hat\\Rz\\cap\\Sigma=\\emptyset$. Similar arguments prove that $\\Sigma\\cap\\Ry=\\emptyset$, which concludes the proof.\n\\end{proof}\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[scale=.65]\n\\sf\\scriptsize\n\\fill[blue!30] (6,9.5) rectangle (10.5,12);\n\\draw[ultra thick, blue!70] (5,7) rectangle (11.5,14);\n\\draw (0,0) grid (17,21);\n\\foreach \\a in {5,...,11}\n \\draw (\\a+.5,7)--(\\a+.5,14);\n\\foreach \\a in {9.5,10.5,11.5}\n \\draw (6,\\a)--(10.5,\\a);\n\\foreach \\a in {0,...,20}\n{ \\foreach \\b in {0,1,2,14,15,16}\n \\node at (\\b+.5,\\a+.5) {0};\n}\n\\foreach \\a in {3,...,13}\n{ \\foreach \\b in {0,1,2,18,19,20}\n \\node at (\\a+.5,\\b+.5) {0};\n}\n\\foreach \\a in {3,...,17}\n{ \\foreach \\b in {3,4,12,13}\n \\node at (\\b+.5,\\a+.5) {1};\n}\n\\foreach \\a in {5,...,11}\n{ \\foreach \\b in {3,4,5,6,14,15,16,17}\n \\node at (\\a+.5,\\b+.5) {1};\n}\n\\foreach \\a in {7,...,13}\n{ \\foreach \\b in {5,5.5,10.5,11,11.5}\n \\node at (\\b+.25,\\a+.5) {2};\n}\n\\foreach \\a in {12,...,20}\n{ \\foreach \\b in {7,8,12,13}\n \\node at (\\a\/2+.25,\\b+.5) {2};\n}\n\\foreach \\a in {12,...,20}\n{ \\foreach \\b in {9,9.5,10,11,11.5}\n \\node at (\\a\/2+.25,\\b+.25) {3};\n}\n\\foreach \\a in {12,13,14,15,17,18,19,20}\n \\node at (\\a\/2+.25,10.75) {3};\n\\node at (8.25,10.75) {4};\n\\end{tikzpicture}\n\\caption{$yz$-view on the slice $\\Sx(\\xi)$. The numbers denote element levels, and the element in the center with level 4 is a child of $K$. The patch $\\patch{\\hat\\G}K$ is highlighted in blue, and the second-order patch $\\patch{\\hat\\G}{\\patch{\\hat \\G}K}$ is indicated by a thick blue line.}\n\\label{fig: Sx-levels}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[scale=.5]\n\\fill[red!50] (7,9.5) rectangle (9.5,12)\n(1,1) -- (16,1) -- (16,20) -- (1,20) -- (1,1) -- (4,4) -- (4,17) -- (13,17) -- (13,4) -- (4,4) -- (1,1);\n\\foreach \\a in {5,...,11}\n \\draw[ultra thick, red!50] (\\a+.5,5)--(\\a+.5,8) (\\a+.5,13)--(\\a+.5,16);\n\\foreach \\a in {9.5,10.5,11.5}\n \\draw[ultra thick, red!50] (5,\\a)--(6.5,\\a) (10,\\a)--(11.5,\\a);\n\\draw (0,0) grid (17,21);\n\\foreach \\a in {5,...,11}\n \\draw (\\a+.5,7)--(\\a+.5,14);\n\\foreach \\a in {9.5,10.5,11.5}\n \\draw (6,\\a)--(10.5,\\a);\n\\end{tikzpicture}\n\\caption{$yz$-view on the slice $\\Sx(\\xi)$. $\\Rx$ is indicated by red areas. $\\Ry$ is depicted by horizontal red lines, $\\Rz$ are vertical red lines. At the same time, the squared red area in the center coincides with $\\Sigma$.}\n\\label{fig: Sx-perturbations}\n\\end{figure}\n\\section{Dual-Compatibility}\\label{sec: DC}%\nThis section recalls the concept of Dual-Compatibility, which is a sufficient criterion for linear independence of the T-spline functions, based on dual functionals. We follow the ideas of \\cite{BBSV:2014} for the definitions and for the proof of linear independence. In addition, we prove that all analysis-suitable (and hence all admissible) meshes are dual-compatible and thereby generalize a 2D result from \\cite{BBCS:2012}.\n\n\\begin{prp}[Dual functional, {\\cite[Theorem 4.41]{Schumaker:2007}}]\\label{prp:overlap}%\nGiven the local index vector $X=(x_1,\\dots,x_{p+2})$, there exists an $L^2$-functional $\\lambda_{X}$ with $\\supp\\lambda_X=\\supp N_X$ such that for any $\\tilde X=(\\tilde x_1,\\dots,\\tilde x_{p+2})$ satisfying\n\\begin{equation}\\label{eq: overlap}\n \\begin{alignedat}{4}\n \\forall\\ x&\\in\\{x_1,\\dots,x_{p+2}\\}&&:&\\quad \\tilde x_1&\\leq x\\leq \\tilde x_{p+2}&&\\Rightarrow x\\in \\{\\tilde x_1,\\dots,\\tilde x_{p+2}\\}\\\\\n \\text{and}\\quad\\forall\\ \\tilde x&\\in\\{\\tilde x_1,\\dots,\\tilde x_{p+2}\\}&&:&\\quad x_1&\\leq \\tilde x\\leq x_{p+2}&&\\Rightarrow \\tilde x\\in \\{x_1,\\dots,x_{p+2}\\},\n \\end{alignedat}\n\\end{equation}\nfollows $\\lambda_{X}(N_{\\tilde X}) = \\delta_{X\\tilde X}$.\n\\end{prp}\n\\begin{proof}\nFollowing \\cite{Schumaker:2007}, we construct a dual functional\non the same local knot vector $X$ which we denote by $\\lambda_{X}:L^2\\bigl([0,1]\\bigr)\\to\\reell$. For details, see \\cite[Theorem 4.34, 4.37, and 4.41]{Schumaker:2007}.\nLet $y_j=\\cos\\bigl(\\tfrac{p-j+1}{p+1}\\pi\\bigr)$ for $j=0,\\dots,p+1$.\nUsing divided differences, the perfect B-spline of order $p+1$ is defined by\n\\[B^*_{p+1}(x)\\sei (p+1)\\,(-1)^{p+1}\\bigl[y_0,\\dots,y_{p+1}\\bigr]\\left((x-\\bullet)_+\\right)^p\\]\nand satisfies (amongst other things) $\\int_{-1}^1B^*_{p+1}(x)\\de x=1$ as depicted in Figure~\\ref{pic: perfect B-spline}. \n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=.5\\textwidth]{perfBS}\n \\caption{Plot of the perfect B-splines $B^*_4$ (solid), $B^*_6$ (dotted), $B^*_{10}$ (dashed) and the corresponding antiderivatives.}\n \\label{pic: perfect B-spline}\n\\end{figure}\nSet \\[G_{X}(x) \\sei \\int_{-1}^{\\tfrac{2x-{x_1}-{x_{p+2}}}{{x_{p+2}}-{x_1}}}B^*_{p+1}(t)\\de t \\quad\\text{for }{x_1}\\le x\\le {x_{p+2}}\\] and\n\\[\\phi_{X}(x)=\\tfrac1{p!}\\left(x-{x_2}\\right)\\cdots\\bigl(x-{x_{p+1}}\\bigr).\\] We define the dual functional by \n\\begin{equation}\\label{eq:def_lambda}\n\\lambda_{X}(f) = \\int_{{x_1}}^{{x_{p+2}}}\\mspace{-9mu}f\\, D^{p+1}(G_{X}\\,\\phi_{X})\\de x\\quad\\text{for all }f\\in L^2\\bigl([0,1]\\bigr).\n\\end{equation}\nNote in particular that for all $f\\in L^2(\\reell)$ with $f|_{[x_1,x_{p+2}]}=0$ follows $\\lambda_{X}(f)=0$.\nIf \\eqref{eq: overlap} holds then the claim follows by construction, see \\cite[Theorem 4.41]{Schumaker:2007}.\n\\end{proof}\nWe say that two index vectors verifying \\eqref{eq: overlap} \\emph{overlap}. \nIn order to define the set of T-spline blending functions of which we desire linear independence, we construct local index vectors for each active node.\n\n\\begin{df}\nWe define the functional $\\lambda_v$ by\n \\[\\lambda_v(B_{w})\\sei \\lambda_{\\xx(v)}(N_{\\xx(w)})\\cdot\\lambda_{\\yy(v)}(N_{\\yy(w)})\\cdot\\lambda_{\\zz(v)}(N_{\\zz(w)})\\]\n using the one-dimensional functional $\\lambda_X$ defined in \\eqref{eq:def_lambda}.\n\\end{df}\n\n\\begin{df}\\label{df: partial overlap}%\nWe say that a couple of nodes $v,w\\in\\N$ \\emph{partially overlap} if their index vectors overlap in at least two out of three dimensions; this is, if (at least) two of the pairs \\[\\bigl(\\xx(v),\\xx(w)\\bigr),\\ \\bigl(\\yy(v),\\yy(w)\\bigr),\\ \\bigl(\\zz(v),\\zz(w)\\bigr)\\] overlap in the sense of Proposition~\\ref{prp:overlap}.\\bigskip\n\\end{df}\n\n\\begin{df}\nA mesh $\\G$ is \\emph{dual-compatible (DC)} if any two active nodes $v,w\\in\\N_A$ with $\\bigl|\\supp B_v \\cap\\supp B_w\\bigr|>0$ partially overlap.\nThe set of dual-compatible meshes is denoted by $\\DC$.\n\\end{df}\n\\begin{rem}\nThe above Definition~\\ref{df: partial overlap} fulfills the definition of \\emph{partial overlap} given in \\cite[Def.\\ 7.1]{BBSV:2014}, which is not equivalent. The definition given in \\cite{BBSV:2014} is more general, and the corresponding mesh classes are nested in the sense \n$\\DC\\subseteq\\DC_{\\text{\\cite{BBSV:2014}}}$. However, we do have equivalence of these definitions in the two-dimensional setting.\n\\end{rem}\n\nThe following lemma states that the perturbed regions from Definition~\\ref{df: perturbed regions} indicate non-overlapping knot vectors, and it is applied in the proof of Theroem~\\ref{thm: AS in DC} below.\n\\begin{lma}\\label{lma: perturbed regions indicate non-overlapping}%\nLet $q\\in[0,\\tilde X]$ and $v_1,v_2\\in\\N_A$.\nIf $v_1\\in\\Nx(q)\\notni v_2$ and $\\Sx(q)\\cap\\supp B_{v_1}\\cap\\supp B_{v_2}\\neq\\emptyset$, then $\\xx(v_1)$ and $\\xx(v_2)$ do not overlap in the sense of \\eqref{eq: overlap}.\n\\end{lma}\nThis holds analogously for $\\Ny(r),r\\in[0,\\tilde Y]$ and $\\Nz(s),s\\in[0,\\tilde Z]$.\n\\begin{proof}\nLet $v_1=(x_1,y_1,z_1)$. From $v_1\\in\\Nx(q)$ and Definition~\\ref{df: perturbed regions}, we conclude that $(q,y_1,z_1)\\in\\xix$, and hence $q\\in\\XX(y_1,z_1)$. Let $\\xx(v_1)=(x_1^1,\\dots,x_1^{p_1+2})$ be the local $x$-direction knot vector associated to $v_1$, then \\mbox{$\\supp B_{v_1}\\cap \\Sx(q)\\neq\\emptyset$} implies that $x_1^1\\le q\\le x_1^{p_1+2}$. This and $q\\in\\XX(y_1,z_1)$ yield $q\\in\\xx(v_1)$.\nLet \\mbox{$v_2=(x_2,y_2,z_2)$}. From $v_2\\notin\\Nx(q)$, we get $(q,y_2,z_2)\\notin\\xix$, hence $q\\notin\\XX(y_2,z_2)$, and in particular \\mbox{$q\\notin\\xx(v_2)$}. Let \\mbox{$\\xx(v_2)=(x_2^1,\\dots,x_2^{p_1+2})$} be the local knot vector associated to $v_2$, then \\mbox{$\\supp B_{v_2}\\cap \\Sx(q)\\neq\\emptyset$} implies that $x_2^1\\le q\\le x_2^{p_1+2}$. Together with $\\xx(v_1)\\ni q\\notin\\xx(v_2)$, we see that $v_1$ and $v_2$ do not overlap.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm: AS in DC}%\n$\\AS=\\DC$.\n\\end{thm}\n\n\\begin{proof\n``$\\subseteq$''.\nAssume for contradiction a mesh $\\G$ which is not DC, hence there exist active nodes \\mbox{$v,w\\in\\N_A$} with \\mbox{$\\bigl|\\supp B_v\\cap\\supp B_w\\bigr|>0$} that do not overlap in two dimensions, without loss of generality $x$ and $y$.\nWe will show that there exist two slice perturbations $\\Rx(q)$ and $\\Ry(r)$ with nonempty intersection.\nWe denote $v=(v_1,v_2,v_3)$, $w=(w_1,w_2,w_3)$ and $\\xx(v)=(x^v_1,\\dots,x^v_{p_1+2})$. The elements of $\\yy(v),\\xx(w),\\yy(w)$ are denoted analogously. Moreover we define\n\\begin{alignat*}{2}\nx\\mn&\\sei\\max(x^v_1,x^w_1),&\\enspace x\\mx&\\sei\\min(x^v_{p_1+2},x^w_{p_1+2})\\\\\ny\\mn&\\sei\\max(y^v_1,y^w_1),& y\\mx&\\sei\\min(y^v_{p_2+2},y^w_{p_2+2})\\\\\nz\\mn&\\sei\\max(z^v_1,z^w_1),& z\\mx&\\sei\\min(z^v_{p_3+2},z^w_{p_3+2})\n\\end{alignat*}\nand note that\n\\begin{align*}\n\\supp B_v \\cap \\supp B_w = [x\\mn,x\\mx]\\times[y\\mn,y\\mx]\\times[z\\mn,z\\mx].\n\\end{align*}\nSince $\\xx(v)$ and $\\xx(w)$ do not overlap, there exists $q\\in[x\\mn,x\\mx]$ with either \\mbox{$\\xx(v)\\ni q\\notin\\xx(w)$}\\enspace or \\mbox{$\\xx(v)\\notni q\\in\\xx(w)$}. Without loss of generality we assume $\\xx(v)\\ni q\\notin\\xx(w)$. Since $\\xx(v)\\subseteq \\XX(v_2,v_3)$, it follows by definition that $(q,v_2,v_3)\\in\\xix$ and hence $v\\in\\Nx(q)$.\nSince $q\\notin\\xx(w)$ and hence $(q,w_2,w_3)\\notin\\xix$, it follows that $w\\notin\\Nx(q)$.\nThen\n\\begin{align*}\n\\Rx(q) &= \\Sx(q)\\ \\cap\\ \\mathbox{ \\bigcup_{v'\\in\\Nx(q)} } \\supp B_{v'}\\ \\cap\\ \\mathbox{ \\bigcup_{v'\\in\\N_A\\smallsetminus\\Nx(q)} } \\supp B_{v'}\\\\\n &\\supseteq \\Sx(q)\\ \\cap \\ \\supp B_v\\ \\cap \\ \\supp B_w\\\\\n &= \\{q\\}\\times[y\\mn,y\\mx]\\times[z\\mn,z\\mx].\n \\intertext{Analogously, we have}\n \\Ry(r) &\\supseteq [x\\mn,x\\mx]\\times\\{r\\}\\times[z\\mn,z\\mx]\n \\intertext{and hence}\n \\Rx(q) \\cap \\Ry(r) &\\supseteq \\{q\\}\\times\\{r\\}\\times[z\\mn,z\\mx]\\neq\\emptyset,\n \\end{align*}\nwhich means that the mesh $\\G$ is not analysis-suitable.\\bigskip\n\n``$\\supseteq$''.\nAssume for contradiction that the mesh is not analysis-suitable, and w.l.o.g.\\ that there is \\mbox{$v=(q,r,s)\\in\\reell^3$} such that $\\Rx\\cap\\Ry\\supseteq\\{v\\}\\neq\\emptyset$. Definition~\\ref{df: perturbed regions} implies that there exist\n$v_1,v_2,v_3,v_4\\in\\N_A$ with\n\\mbox{$v_1\\in\\Nx(q)\\notni v_2$} and \\mbox{$v_3\\in\\Ny(r)\\notni v_4$}\\enspace such that\n\\[v\\ \\in\\ \\Sx(q)\\cap\\Sy(r)\\cap\\supp B_{v_1}\\cap\\supp B_{v_2}\\cap\\supp B_{v_3}\\cap\\supp B_{v_4}.\\]\nLemma~\\ref{lma: perturbed regions indicate non-overlapping} yields that $\\xx(v_1)$ and $\\xx(v_2)$ do not overlap, and that $\\yy(v_3)$ and $\\yy(v_4)$ do not overlap.\n\n\\textit{Case 1.} If $v_1\\in\\Ny(r)\\notni v_2$, or $v_1\\notin\\Ny(r)\\ni v_2$, then $v_1$ and $v_2$ do not partially overlap.\n\n\\textit{Case 2.} If $v_1\\in\\Ny(r)$ and $v_4\\notin\\Nx(q)$, then $v_1$ and $v_4$ do not partially overlap.\n\n\\textit{Case 3.} If $v_1\\notin\\Ny(r)$ and $v_3\\notin\\Nx(q)$, then $v_1$ and $v_3$ do not partially overlap.\n\n\\textit{Case 4.} If $v_2\\in\\Ny(r)$ and $v_4\\in\\Nx(q)$, then $v_2$ and $v_4$ do not partially overlap.\n\n\\textit{Case 5.} If $v_2\\notin\\Ny(r)$ and $v_3\\in\\Nx(q)$, then $v_2$ and $v_3$ do not partially overlap.\n\nIn all cases (see the table in Fig.~\\ref{tb: cases}), the mesh is not dual-compatible. This concludes the proof.\n\\end{proof}\n\\begin{figure}[ht]\n\\renewcommand \\arraystretch 1\n\\newcommand \\Hline {\\hhline{*{3}{-|}-||-}}\n\\centering\n\\begin{tabular}{c|c|c|c||l}\n$v_1\\in\\Ny(r)$ & $v_2\\in\\Ny(r)$ & $v_3\\in\\Ny(r)$ & $v_4\\in\\Ny(r)$ & case(s) \\\\\\hhline{*{3}{=:}=::=}\n\\true & \\true & \\true & \\true & 4 \\\\\\Hline\n\\true & \\true & \\true & \\false & 2 \\\\\\Hline\n\\true & \\true & \\false & \\true & 4 \\\\\\Hline \n\\true & \\true & \\false & \\false & 2 \\\\\\Hline \n\\true & \\false & \\true & \\true & 1, 5 \\\\\\Hline \n\\true & \\false & \\true & \\false & 1, 2, 5 \\\\\\Hline \n\\true & \\false & \\false & \\true & 1 \\\\\\Hline \n\\true & \\false & \\false & \\false & 1, 2 \\\\\\Hline \n\\false & \\true & \\true & \\true & 1, 4 \\\\\\Hline \n\\false & \\true & \\true & \\false & 1 \\\\\\Hline \n\\false & \\true & \\false & \\true & 1, 3, 4 \\\\\\Hline \n\\false & \\true & \\false & \\false & 1, 3 \\\\\\Hline \n\\false & \\false & \\true & \\true & 5 \\\\\\Hline \n\\false & \\false & \\true & \\false & 5 \\\\\\Hline \n\\false & \\false & \\false & \\true & 3 \\\\\\Hline \n\\false & \\false & \\false & \\false & 3\n\\end{tabular}\n\\caption{The five cases considered in the proof of Theorem~\\ref{thm: AS in DC} cover all possible configurations.}\n\\label{tb: cases}\n\\end{figure}\n\n\n\\begin{thm}\\label{thm: DC has dual basis}\n Let $\\G$ be a DC T-mesh. Then the set of functionals $\\{\\lambda_v\\mid v\\in\\N_A\\}$ is a set of dual functionals for the set $\\{B_v\\mid v\\in\\N_A\\}$.\n\\end{thm}\nThe proof below follows the ideas of \\cite[Proposition~5.1]{BBCS:2012} and \\cite[Proposition~7.3]{BBSV:2014}.\n\\begin{proof}\nLet $v,w\\in\\N_A$. We need to show that \n\\begin{equation}\\label{eq:DC_claim}\n\\lambda_v(B_w) = \\delta_{vw},\n\\end{equation}\nwith $\\delta$ representing the Kronecker symbol.\n\nIf $\\supp B_v$ and $\\supp B_w$ are disjoint (or have an intersection of empty interior), then at least one of the pairs \n\\[\\bigl(\\supp(N_{\\xx(v)}),\\supp(N_{\\xx(w)})\\bigr),\\ \\bigl(\\supp(N_{\\yy(v)}),\\supp(N_{\\yy(w)})\\bigr),\\ \\bigl(\\supp(N_{\\zz(v)}),\\supp(N_{\\zz(w)})\\bigr)\\]\nhas an intersection with empty interior. Assume w.l.o.g.\\ that $\\left|\\supp(N_{\\xx(v)})\\cap\\supp(N_{\\xx(w)})\\right|=0$, then \n\\[\\lambda_v(B_w) = \\underbrace{\\lambda_{\\xx(v)}(N_{\\xx(w)})}_0\\cdot \\lambda_{\\yy(v)}(N_{\\yy(w)})\\cdot \\lambda_{\\zz(v)}(N_{\\zz(w)})=0.\\]\nAssume that $\\supp B_v$ and $\\supp B_w$ have an intersection with nonempty interior.\nSince the mesh $\\G$ is DC, the two nodes overlap in at least two dimensions. Without loss of generality we may assume the index vectors $\\bigl(\\xx(v),\\xx(w)\\bigr)$ and $\\bigl(\\yy(v),\\yy(w)\\bigr)$ overlap. Proposition~\\ref{prp:overlap} yields \n\\[\\lambda_{\\xx(v)}(N_{\\xx(w)}) = \\delta_{v_1w_1} \\enspace\\text{and}\\enspace \\lambda_{\\yy(v)}(N_{\\yy(w)}) = \\delta_{v_2w_2} .\\]\nThe above identities immediately prove \\eqref{eq:DC_claim} if $v_1\\neq w_1$ or $v_2\\neq w_2$. If on the contrary, $v_1=w_1$ and $v_2\\neq w_2$, then $v$ and $w$ are aligned in $z$-direction, this is, $\\zz(v)$ and $\\zz(w)$ are both vectors of $p+2$ consecutive indices from the same index set \n$\\ZZ(v_1,v_2)=\\ZZ(w_1,w_2)$. \nHence $v$ and $w$ must overlap also in $z$-direction. Again, Proposition~\\ref{prp:overlap} yields\n\\[\\lambda_{\\zz(v)}(N_{\\zz(w)}) = \\delta_{v_3w_3},\\]\nwhich concludes the proof.\n\\end{proof}\n\n\\begin{crl}[{\\cite[Proposition~7.4]{BBSV:2014}}]\nLet $\\G$ be a DC T-mesh. Then the set $\\{B_v\\mid v\\in\\N_A\\}$ is linear independent.\n\\end{crl}\n\\begin{proof}\nAssume \\[\\sum_{v\\in\\N_A}c_vB_v=0\\] for some coefficients $\\{c_v\\}_{v\\in\\N_A}\\subseteq\\reell$. Then, for any $w\\in\\N_A$, applying $\\lambda_w$ to the sum, using linearity and Theorem~\\ref{thm: DC has dual basis}, we get\n\\[c_w = \\lambda_w\\,\\Bigl(\\sum_{v\\in\\N_A}c_vB_v\\Bigr)=0.\\]\n\\raiseqed\n\\end{proof}\n\n\n\\section{Linear Complexity}\\label{sec: complexity}\nThis section is devoted to a complexity estimate in the style of a famous estimate for the Newest Vertex Bisection on triangular meshes given by Binev, Dahmen and DeVore \\cite{BDV:2004} and, in an alternative version, by Stevenson~\\cite{Stevenson:2007}.\nLinear Complexity of the refinement procedure is an inevitable criterion for optimal convergence rates in the Adaptive Finite Element Method (see e.g.\\ \\cite{BDV:2004,Stevenson:2007,CFPP:2014} and \\cite[Conclusions]{Buffa:Giannelli:2015}). The estimate and its proof follow our own work \\cite{Morgenstern:Peterseim:2015,BGMP:2016}, which we generalize now to three dimensions and $m$-graded refinement.\nThe estimate reads as follows.\n\n\\begin{thm}\\label{thm: complexity}\nAny sequence of admissible meshes $\\G_0,\\G_1,\\dots,\\G_J$ with \\[\\G_j=\\refine(\\G_{j-1},\\M_{j-1}),\\quad\\M_{j-1}\\subseteq\\G_{j-1}\\quad\\text{for}\\enspace j\\in\\{1,\\dots,J\\}\\] satisfies\n\\[\\left|\\G_J\\setminus\\G_0\\right|\\ \\le\\ C_{\\mathbf p,m}\\sum_{j=0}^{J-1}|\\M_j|\\ ,\\]\nwith $C_{\\mathbf p,m}=\\tfrac{m^{1\/3}}{1-m^{-1\/3}}\\,\\bigl(4d_1+1\\bigr)\\,\\bigl(4d_2+m^{1\/3}\\bigr)\\,\\bigl(4d_3+m^{2\/3}\\bigr)$ and $d_1,d_2,d_3$ from Lemma~\\ref{lma: K1 in refMS => K2 in S} below.\n\\end{thm}\n\n\\begin{lma}\\label{lma: K1 in refMS => K2 in S}\nGiven $\\M\\subseteq\\G\\in\\A$ and $K\\in\\refine(\\G,\\M)\\setminus\\G$, there exists $K'\\in \\M$ such that $\\ell(K)\\le\\ell(K')+1$ and \\[\\Dist(K,K')\\le m^{-\\ell(K)\/3}(d_1,d_2,d_3),\\]\nwith ``$\\le$'' understood componentwise and constants \n\\begin{align*}\nd_1&\\sei \\tfrac1{1-m^{-1\/3}} \\,\\bigl(p_1+\\tfrac{3+m^{1\/3}}2+\\tfrac{m^{1\/3}-1}{m^2}\\bigr),\\\\\nd_2&\\sei \\tfrac{m^{1\/3}}{1-m^{-1\/3}}\\,\\bigl(p_2+\\tfrac{3+m^{1\/3}}2+\\tfrac{m^{1\/3}-1}{m^2}\\bigr),\\\\\nd_3&\\sei \\tfrac{m^{2\/3}}{1-m^{-1\/3}}\\,\\bigl(p_3+\\tfrac{3+m^{1\/3}}2+\\tfrac{m^{1\/3}-1}{m^2}\\bigr).\n\\end{align*}\n\\end{lma}\nThe proof is given in Appendix~\\ref{apx: K1 in refMS => K2 in S}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm: complexity}]\\ \n\n\\pnumpx For $K\\in\\tcup\\A$ and $\\KM\\in\\M\\sei\\M_0\\cup\\dots\\cup\\M_{J-1}$, define $\\lambda(K,\\KM)$ by \\[\\lambda(K,\\KM)\\sei\\begin{cases}m^{(\\ell(K)-\\ell(\\KM))\/3}&\\text{if }\\ell(K)\\le\\ell(\\KM)+1\\text{ and }\\Dist(K,\\KM)\\le 2m^{-\\ell(K)\/3}(d_1,d_2,d_3),\\\\[.3em]0&\\text{otherwise.}\\end{cases}\\]\n\n\\pnumpx[Main idea of the proof.]\n\\begin{alignat*}{2}\n\\left|\\G_J\\setminus\\G_0\\right| &= \\mathbox[2.5em]{\\sum_{K\\in\\G_J\\setminus\\G_0}}1 &&\\Stackrel{\\numref{sum_lambda > 1}}\\le\\sum_{K\\in\\G_J\\setminus\\G_0}\\sum_{\\KM\\in\\M}\\lambda(K,\\KM) \\\\\n&\\Stackrel{\\numref{summe aller lambdas beschraenkt}}\\le\\sum_{\\KM\\in\\M} C_{\\mathbf p,m} &&=C_{\\mathbf p,m}\\,\\sum_{j=0}^{J-1} |\\M_j|.\n\\end{alignat*}\n\n\\pnumpx[Each $K\\in\\G_J\\setminus\\G_0$ satisfies \\[\\sum_{\\KM\\in\\M}\\lambda(K,\\KM)\\ \\ge\\ 1.\\]]%\n\\label{sum_lambda > 1}%\nConsider $K\\in\\G_J\\setminus\\G_0$. Set $j_1 K2 in S} states the existence of $K_1\\in\\M_{j_1}$ with $\\Dist(K,K_1)\\le m^{-\\ell(K)\/3}(d_1,d_2,d_3)$ and $\\ell(K)\\le\\ell(K_1)+1$. Hence $\\lambda(K,K_1)=m^{\\ell(K)-\\ell(K_1)}>0$.\nThe repeated use of Lemma~\\ref{lma: K1 in refMS => K2 in S} yields $j_1>j_2>j_3>\\dots$ and $K_2,K_3,\\dots$ with $K_{i-1}\\in\\G_{j_i+1}\\setminus\\G_{j_i}$ and $K_i\\in\\M_{j_i}$ such that \n\\begin{equation}\\label{eq: complexity -last}\n\\Dist(K_{i-1},K_i)\\le m^{-\\ell(K_{i-1})\/3}(d_1,d_2,d_3)\\enspace\\text{and}\\enspace\\ell(K_{i-1})\\le\\ell(K_i)+1.\n\\end{equation}\nWe repeat applying Lemma~\\ref{lma: K1 in refMS => K2 in S} as $\\lambda(K,K_i)>0$ and $\\ell(K_i)>0$, and we stop at the first index $L$ with $\\lambda(K,K_L)=0$ or $\\ell(K_L)=0$. \nIf $\\ell(K_L)=0$ and $\\lambda(K,K_L)>0$, then\n\\[\\sum_{\\KM\\in\\M}\\lambda(K,\\KM)\\ge\\lambda(K,K_L)=m^{(\\ell(K)-\\ell(K_L))\/3}\\ge m^{1\/3}.\\]\nIf $\\lambda(K,K_L)=0$ because $\\ell(K)>\\ell(K_L)+1$, then \\eqref{eq: complexity -last} yields $\\ell(K_{L-1})\\le\\ell(K_L)+1<\\ell(K)$ and hence\n\\[\\sum_{\\KM\\in\\M}\\lambda(K,\\KM)\\ge\\lambda(K,K_{L-1})=m^{(\\ell(K)-\\ell(K_{L-1}))\/3}>m^{1\/3}.\\]\nIf $\\lambda(K,K_L)=0$ because $\\Dist(K,K_L)>2m^{-\\ell(K)\/3}(d_1,d_2,d_3)$, then a triangle inequality shows\n\\begin{align*}\n2m^{-\\ell(K)\/3}(d_1,d_2,d_3) &< \\Dist(K,K_1)+\\sum_{i=1}^{L-1}\\Dist(K_i,K_{i+1}) \n\\\\&\\le\\ m^{-\\ell(K)\/3}(d_1,d_2,d_3)+\\sum_{i=1}^{L-1} m^{-\\ell(K_i)\/3}(d_1,d_2,d_3),\n\\end{align*}\nand hence $\\smash{\\displaystyle m^{-\\ell(K)\/3} \\le\\sum_{i=1}^{L-1} m^{-\\ell(K_i)\/3}}$. The proof is concluded with \n\\[ 1\\ \\le\\ \\sum_{i=1}^{L-1} m^{(\\ell(K)-\\ell(K_i))\/3}\\ =\\ \\sum_{i=1}^{L-1} \\lambda(K,K_i)\\ \\le\\ \\sum_{\\KM\\in\\M}\\lambda(K,\\KM).\\]\n\n\\pnumpx[For all $j\\in\\{0,\\dots,J-1\\}$ and $\\KM\\in\\M_j$ holds \\[\\sum_{K\\in\\G_J\\setminus\\G_0}\\lambda(K,\\KM)\\ \\le\\ \\tfrac{m^{1\/3}}{1-m^{-1\/3}}\\,\\bigl(4d_1+1\\bigr)\\,\\bigl(4d_2+m^{1\/3}\\bigr)\\,\\bigl(4d_3+m^{2\/3}\\bigr)\\ =\\ C_{\\mathbf p,m}\\ .\\]]%\n\\label{summe aller lambdas beschraenkt}%\nThis is shown as follows. By definition of $\\lambda$, we have\n\\begin{align*}\n\\mathbox[1cm]{\\sum_{K\\in\\G_J\\setminus\\G_0}}\\lambda(K,\\KM)\n&\\le \\mathbox[1cm]{\\sum_{K\\in\\bigcup\\A\\setminus\\G_0}}\\lambda(K,\\KM)\\\\\n&= \\sum_{j=1}^{\\ell(\\KM)+1}m^{(j-\\ell(\\KM))\/3}\\,\\#\\underbrace{\\bigl\\{K\\in\\tcup\\A\\mid\\ell(K)=j\\text{ and }\\Dist(K,\\KM)\\le 2m^{-j\/3}(d_1,d_2,d_3)\\bigl\\}}_B.\n\\end{align*}\nSince we know by definition of the level that $\\ell(K)=j$ implies $|K|=m^{-j}$, we know that $m^j\\left|\\tcup B\\right|$ is an upper bound of $\\#B$. The cuboidal set $\\tcup B$ is the union of all admissible elements of level $j$ having their midpoints inside a cuboid of size \n\\[4m^{-j\/3}d_1\\,\\times\\,4m^{-j\/3}d_2\\,\\times\\,4m^{-j\/3}d_3.\\]\nAn admissible element of level $j$ is not bigger than $m^{-j\/3}\\,\\times\\,m^{(1-j)\/3}\\,\\times\\,m^{(2-j)\/3}$. \nTogether, we have \\[\\bigl|\\tcup B\\bigr|\\le m^{-j}\\,\\bigl(4d_1+1\\bigr)\\,\\bigl(4d_2+m^{1\/3}\\bigr)\\,\\bigl(4d_3+m^{2\/3}\\bigr),\\] and hence $\\#B\\le \\bigl(4d_1+1\\bigr)\\,\\bigl(4d_2+m^{1\/3}\\bigr)\\,\\bigl(4d_3+m^{2\/3}\\bigr)$. An index substitution $k\\sei1-j+\\ell(\\KM)$ proves the claim with\n\\[\\sum_{j=1}^{\\ell(\\KM)+1}m^{(j-\\ell(\\KM))\/3}=\\sum_{k=0}^{\\ell(\\KM)}m^{(1-k)\/3} m_1$, which leads to $\\Delta m^2_{21} = m_2^2 - m_1^2 > 0$. However, the sign of $\\Delta m^2_{31} = m_3^2 - m_1^2$ is still unknown. It can be either greater than zero i.e., $m_3 > m_1$, giving rise to normal ordering (NO) of the neutrino masses or it can be less than zero i.e., $m_3 < m_1$, giving rise to inverted ordering (IO) of the neutrino masses~\\cite{Gonzalez-Garcia:2021dve}. Amongst the experiments that are currently running and are somewhat sensitive to the neutrino mass ordering are the atmospheric based Super-Kamiokande~\\cite{Super-Kamiokande:2017yvm}, and the accelerator-based T2K~\\cite{T2K:2021xwb} and NO$\\nu$A~\\cite{NOvA:2021nfi}. The combined analysis of the data obtained from these experiments shows a preference for normal ordering over inverted ordering~\\cite{Esteban:2020cvm}. Apart from neutrino mass ordering, the other two unknowns in the three flavour scenarios are: the octant of the atmospheric mixing angle $\\theta_{23}$ which can be either in the lower octant i.e., $\\theta_{23} < 45^\\circ$ or in the higher octant i.e., $\\theta_{23} > 45^\\circ$, and the phase $\\delta_{\\rm CP}$. Regarding the octant measurement of the current experiments, there is a mild preference towards the higher octant but the maximal value i.e., $\\theta_{23} = 45^\\circ$ is also allowed at $3 \\sigma$. Regarding $\\delta_{\\rm CP}$, there is a mismatch between the values obtained from T2K and NO$\\nu$A. T2K provides a best-fit value of $\\delta_{\\rm CP}$ around $-90^\\circ$~\\cite{con_talk_t2k} whereas the best-fit value from NO$\\nu$A is around $\\pm 180^\\circ$~\\cite{con_talk}.\n\nThe major proposed future experiments which are designed to establish this hint to a significant confidence level are: accelerator based T2HK~\\cite{Abe:2016ero} and DUNE~\\cite{Abi:2020evt}, atmospheric based ICAL detector of INO facility~\\cite{ICAL:2015stm} (from now on we will refer to it as ICAL), ORCA~\\cite{KM3Net:2016zxf} and PINGU~\\cite{IceCube:2016xxt}, and reactor based JUNO~\\cite{JUNO:2015zny}. In this paper we study the potential of the future experiments ICAL, JUNO and T2HK to determine the neutrino mass ordering. Though each of them individually have good chances of pinning down the neutrino mass ordering, they also have their limitations. In this work, we highlight those limitations and show how one can overcome those limitations by invoking synergistic combination of two or all three of them. In particular, we will probe the role of $|\\Delta m^2_{31}|$ when we fit the simulated data to test neutrino mass ordering in these experiments. We will see that since different oscillation channels when fitted return different values of $|\\Delta m^2_{31}|$, the overall $\\chi^2$ obtained in a combined analysis of two or all three of these experiments returns a $\\chi^2$ that is significantly larger than a simple sum of the individual $\\chi^2$. \n\nNote that synergistic study between future atmospheric and future reactor experiments, future accelerator and future atmospheric experiments and future accelerator and future reactor experiments to determine neutrino mass ordering has been performed in the past. For example, Ref.~\\cite{IceCube-Gen2:2019fet} studies the synergy between PINGU and JUNO, synergy between ORCA and JUNO is studied in Ref.~\\cite{juno_orca}, Ref.~\\cite{Blennow:2013vta} studies the synergy between reactor based Daya Bay II~\\cite{wang} and PINGU, Ref \\cite{Fukasawa:2016yue} studies the synergies among T2HK, DUNE and Hyper-Kamiokande, Ref.~\\cite{Chakraborty:2019jlv} studies synergy between accelerator based ESSnuSB \\cite{ESSnuSB:2021lre} and ICAL, and synergy in T2K II, NO$\\nu$A II and JUNO can found in Ref.~\\cite{Cao:2020ans}. The synergy in the combined beam and atmospheric data of DUNE is studied in Refs.~\\cite{Barger:2013rha,Barger:2014dfa} and the same for ESSnuSB is studied in~\\cite{Blennow:2019bvl}. Recently in Ref.~\\cite{Choubey:2022gzv}, we have studied the synergistic effect of T2HK and JUNO to measure neutrino mass ordering.\nHowever, to the best of our knowledge, a synergistic study among the accelerator, atmospheric and reactor experiments considering these particular set of experiments i.e., ICAL, JUNO and T2HK has not been performed in the past and this paper is the first of this kind. The mass ordering sensitivity of T2HK experiment is dependent on the values of $\\delta_{\\rm CP}$ and can have mass ordering sensitivity for only favourable values of $\\delta_{\\rm CP}$ i.e., $-180^\\circ < \\delta_{\\rm CP} < 0^\\circ$. For JUNO, the mass ordering sensitivity is dependent on the energy resolution whereas the mass ordering sensitivity of ICAL is hardly dependent on $\\delta_{\\rm CP}$. Therefore, in this work we will show that when these three experiments are combined, one can achieve mass ordering sensitivity at a significant confidence level irrespective of the energy resolution of JUNO or the value of $\\delta_{\\rm CP}$. The main synergy between the reactor experiment and the accelerator\/atmospheric experiment come from their sensitivity to the parameter $\\Delta m^2_{31}$. In addition we will also explore (i) effect of varying energy resolution of JUNO, (ii) the effect of longer run-time of ICAL, (iii) effect of different true values of $\\theta_{23}$ and (iv) effect of octant degeneracy in the determination of neutrino mass ordering.\n\nThe paper is organized as follows. In the next section we will briefly describe the experimental specification which we consider for our simulation. In Section \\ref{res} we will present our detailed results. Finally in Section \\ref{sum} we will summarize our results and conclude.\n\n\n\n\n\\section{Experimental setup and simulation details}\n\\label{spec}\nIn this section we will briefly describe the specifications of the experiments which we use in our simulation. For ICAL, we use the GEANT4-based geometry of ICAL detector with specification given in Ref.~\\cite{ICAL:2015stm}. ICAL is a 50 kton iron calorimeter with $\\le$ 1.5 T magnetic field which makes it world's unique detector which can distinguish between charged current atmospheric neutrino and antineutrino events. We use the Honda atmospheric neutrino fluxes calculated for Theni site in India \\cite{Honda:2015fha}. We use the GENIE event generator \\cite{Andreopoulos:2009rq} developed for ICAL by the INO collaboration. We generate 1000 years data for ICAL and then normalize it to 10 years to reduce the Monte Carlo fluctuations. Neutrino oscillations are calculated numerically and incorporated via the re-weighting algorithm. The detector efficiency, muon energy and angle resolutions, the charge identification efficiency of the muons, and the hadron energy resolution provided via simulation work done by the INO collaboration are included \\cite{muon-reso,hadron-resol}. We bin the simulated events in bins of muon energy ($E_{\\mu}$), muon angle ($\\cos\\theta_{\\mu}$) and hadron energy ($E_{\\rm had}$).\nThe binning scheme used is shown in Table~\\ref{tab:inobin} \\cite{moon:2014,Mohan:2016gxm}. The analysis is performed by defining a $\\chi^2$ as given in \\cite{moon:2014}. For systematic errors we use (i) flux normalization error of 20$\\%$ (ii) cross-section normalization error of 10$\\%$ (iii) 5$\\%$ uncertainty on the zenith angle dependence of the fluxes (iv) 5$\\%$ energy-independent systematic uncertainty and an (v) energy dependent tilt factor. See \\cite{anu-mh} for further details.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{ |c|c|c|c| } \n \\hline\n Observable & Range & Bin width & No. of bins\\\\ \n\\hline\n $E^{\\rm obs}_{\\mu}$(GeV)(15 bins) & [0.5,4] & 0.5 & 7\\\\\n \t\t\t\t\t\t & [4,7] & 1 & 3\\\\ \n\t\t\t\t\t\t& [7,11] & 4 & 1 \\\\\n\t\t\t\t\t\t& [11,12.5] & 1.5 & 1\\\\\n\t\t\t\t\t\t&[12.5,15] & 2.5&1\\\\\n\t\t\t\t\t\t&[15,25] & 5&2\\\\\n$\\cos(\\theta^{\\rm obs}_{\\mu}$ (21 bins) & \t[-1.0,-0.4] &0.0.05&12\\\\\n\t\t\t&[-0.4,0.2]&0.10&2\\\\\n\t\t\t\t\t\t&[-0.2,1.0]&0.2&6\\\\\n$E^{\\rm obs}_{\\rm had}$ (GeV) (4bins) &[0,2]&1&2\\\\\n\t\t\t\t\t\t&[2,4]&2&1\\\\\n\t\t\t\t\t\t&[4,15]&11&1\\\\\n\n \\hline\n\\end{tabular}\n\\caption{Details of the three observable $E_{\\mu}$, $\\theta_{\\mu}$ and $E_{\\rm had}$ which are used in the analysis.}\n\\label{tab:inobin}\n\\end{center}\n\\end{table}\n\nThe experiments T2HK and JUNO are simulated using GLoBES \\cite{Huber:2004ka,Huber:2007ji}. \nFor T2HK, we follow the configuration as given in Ref.~\\cite{Abe:2016ero}. We consider two water-Cerenkov detector tanks having fiducial volume of 187~kton each located at Kamioka which is 295 km from the neutrino source at J-PARC having a beam power of 1.3 MW with a total exposure of $27 \\times 10^{21}$ protons on target, corresponding to 10~years of running. We have divided the total run-time into 5 years in neutrino mode and 5 years in anti-neutrino mode. For systematic errors, we have considered overall normalization error of 4.71\\% (4.13\\%) for the appearance (disappearance) channel in neutrino mode and 4.47\\% (4.15\\%) for the appearance (disappearance) channel in antineutrino mode \\cite{Abe:2016ero}. The systematic error is the same for both signal and background. \n\nFor JUNO, we consider the same configurations as given in Ref. \\cite{JUNO:2015zny}. We consider a liquid scintillator detector having 20 kton fiducial mass located at a distance around 53 km from Yangjiang and Taishan nuclear power plants. We have considered the energy resolution of $3\\%\/\\sqrt{E(MeV)}$ unless otherwise mentioned. In this analysis we consider all the reactor cores are located at the same distance from the detector. We use 200 same-size bins within the energy window of 1.8 MeV and 8.0 MeV. We have taken the backgrounds and systematic uncertainties as presented in \\cite{JUNO:2015zny}. We consider the run-time to be 6 years.\n\n \nMass ordering sensitivity of an experiment is defined by its capability to exclude the wrong mass ordering. As the global data shows a preference towards normal mass ordering we will present our main results for normal ordering. For the estimation of the sensitivity, we use the Poisson log-likelihood:\n\\begin{equation}\n \\chi^2_{{\\rm stat}} = 2 \\sum_{i=1}^n \\bigg[ N^{{\\rm IO}}_i - N^{{\\rm NO}}_i - N^{{\\rm NO}}_i \\log\\bigg(\\frac{N^{{\\rm IO}}_i}{N^{{\\rm NO}}_i}\\bigg) \\bigg]\\,,\n \\label{eq:chi2}\n\\end{equation}\nwhere $N^{{\\rm IO}}$ is the number of events expected for the values of the oscillation parameters tested for, $N^{{\\rm NO}}$ is the number of events expected for the parameter values assumed to be realized in Nature (Asimov dataset) and $i$ is the number of energy bin.\nThe values of the oscillation parameters used in this analysis is given in Table \\ref{Tab:Param}.\n\\begin{table}[h]\n\t\\centering\n\t\\setlength{\\extrarowheight}{0.1cm}\n\t\\begin{tabular}{|c|c|c|}\n\t\t\\hline\n\t\t Parameter & Best-fit value & $3 \\sigma$ range \\\\\n\t\t\\hline\n\t\t$\\theta_{12}$ & $33.44^\\circ$ & - \\\\\n\t\t$\\theta_{13}$ & $8.57^\\circ$ & - \\\\\n\t\t$\\theta_{23}$ & $45^\\circ$ & $40^\\circ$ to $52^\\circ$ \\\\\n\t\t$\\delta_{\\rm CP}$ & $-90^\\circ\/0^\\circ$ & $-180^\\circ$ to $180^\\circ$ \\\\\n\t\t$\\Delta m^2_{21}$ & $7.42\\times10^{-5}$ eV$^2$ & - \\\\\n\t\t$\\Delta m^2_{31}$ & $2.531\\times10^{-3}$ eV$^2$ & (2.435 to 2.598) $\\times 10^{-3}$ eV$^2$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{The best-fit values and $3 \\sigma$ ranges of the oscillation parameters used in our calculation.}\n\t\\label{Tab:Param}\n\\end{table}\nWe have kept the parameters $\\theta_{12}$, $\\theta_{13}$ and $\\Delta m^2_{21}$ fixed in both the true and test spectrum and varied the other parameters in the test spectrum. For values of $\\delta_{\\rm CP}$ in T2HK, we will consider the values of $0^\\circ$ and $-90^\\circ$. Note that as the mass ordering sensitivity of ICAL is only mildly dependent on $\\delta_{\\rm CP}$, and since simulating the ICAL data is computationally expensive, we will only give results corresponding to $\\delta_{\\rm CP} = 0^\\circ$ for ICAL. For $\\theta_{23}$, we will consider the value as $45^\\circ$ corresponding to maximal value of this mixing angle unless otherwise mentioned. For JUNO, the oscillation probability is independent of $\\theta_{23}$ and $\\delta_{\\rm CP}$. \n\n\\section{Results}\n\\label{res}\n\nIn this section we will present our simulation results. We will first begin by discussing the main physics issue that is responsible for the synergy between the different experiments. In order to highlight it, we consider one experiment at a time and show how the best-fit value of $|\\Delta m_{31}^2|$ comes out to be different for each oscillation channel. Since different oscillation channels are relevant for the different experiments, the $|\\Delta m_{31}^2|$ that would give minimum $\\chi^2$ is also different for them. As a result, when we combine data from two or three experiments and perform a joint analysis, the net mass ordering sensitivity obtained is significantly higher than what one would get by simply adding the individual $\\chi^2$. \n\n\\subsection{Mass Ordering Sensitivity - The Role of $|\\Delta m^2_{31}|$}\n\nAs discussed before, in order to determine the statistical significance of the neutrino mass ordering in a given forthcoming experiment, we do the following. We simulate the prospective data for that experiment assuming, say, true normal ordering, and then fit it with a theory of inverted mass ordering. In the fit we allow the value of $|\\Delta m^2_{31}|$ to vary freely within its current $3\\sigma$ range. Hence, the minimum of the fit comes at a certain value of $|\\Delta m^2_{31}|$ that is different from the one assumed in the data. In this subsection we highlight this aspect at the neutrino oscillation probability level. This is crucial to our understanding of the physics behind the synergy between the different experiments that we will see in the forthcoming sections. In particular, we will probe at what value of $|\\Delta m^2_{31}|$ we get the minimum difference between the normal and inverted mass ordering in the oscillation probabilities for the different experiments. \\\\\n\n\\subsubsection{ICAL}\n\nThe proposed ICAL is an atmospheric neutrino experiment which will be sensitive to large energy range (0.5-25 GeV) and different baselines. In Fig.~\\ref{fig:inomin}, we have plotted the difference between the disappearance channel probabilities for the two different mass orderings i.e. $\\Delta P_{\\mu \\mu} = P_{\\mu \\mu} (\\rm NO) - P_{\\mu \\mu} (\\rm IO)$ in the $y$-axis. Note that in ICAL, the sensitivity to mass ordering mainly arises from the disappearance channel. In this figure, we have used the value of $\\Delta m^{2}_{31} (\\rm NO)=2.531 \\times 10^{-3}$ eV$^2$ for normal ordering and the different values of $\\Delta m^{2}_{31}$ (IO) for inverted ordering are plotted in the $x$-axis. The left panel is for neutrinos and the right panel is for antineutrinos. Different curves in the figure correspond to different combinations of baseline and energy, chosen because they have the maximum contribution towards the mass ordering sensitivity~\\cite{Choubey:2005zy,Gandhi:2007td}. If the minimum value of $\\Delta P_{\\mu \\mu}$ occurs at different values of $\\Delta m^{2}_{31}$ (IO) for different bins, then it would give rise to synergy between these bins. From the figure we see that the minimum of $\\Delta P_{\\mu \\mu}$ occurs at $\\Delta m^{2}_{31} (\\rm IO) \\sim -2.3 \\times 10^{-3}$ eV$^2$ (smaller values of $|\\Delta m^{2}_{31}|$) for neutrinos and at $\\Delta m^{2}_{31} (\\rm IO) \\sim -2.6 \\times 10^{-3}$ eV$^2$ (larger values of $|\\Delta m^{2}_{31}|$) for antineutrinos. This can also be seen in Fig.~\\ref{inochi}, where we have plotted the mass ordering sensitivity $\\chi^2$ for ICAL as a function of $\\Delta m^{2}_{31}$(IO). The black curve corresponds to neutrinos and the red curve corresponds to antineutrinos. From these curves we see that for neutrinos the $\\chi^2$ minimum is at $\\Delta m^{2}_{31} (\\rm IO) \\simeq -2.380 \\times 10^{-3}$ eV$^2$ and for the antineutrinos the minimum is at $\\Delta m^{2}_{31} (\\rm IO) \\simeq -2.480 \\times 10^{-3}$ eV$^2$. This is in agreement with what we concluded from the probability level discussions. The combined sensitivity coming from the neutrino data and antineutrino data is shown by the blue curve in Fig.~\\ref{inochi}. From this curve we note that when the neutrino and antineutrino sensitivities are combined, the $\\chi^2$ minimum appears around $\\Delta m^{2}_{31} (\\rm IO) = -2.419 \\times 10^{-3}$ eV$^2$, which is different from the true value of $\\Delta m^{2}_{31} (\\rm NO) = 2.531 \\times 10^{-3}$ eV$^2$.\n\n\\begin{figure}[h!]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{ino-prob-nu.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{ino-prob-anu.pdf}\n\\end{minipage}\n\\caption{Left panel shows the $P_{\\nu_{\\mu}\\nu_{\\mu}}(\\rm NO)-P_{\\nu_{\\mu}\\nu_{\\mu}}$(IO) for neutrino and right panel shows the $ P_{\\bar{\\nu}_{\\mu}\\bar{\\nu}_{\\mu}}(\\rm NO)-P_{\\bar{\\nu}_{\\mu}\\bar{\\nu}_{\\mu}}(\\rm IO)$ for anti-neutrinos. These combinations of the energy and baseline have the maximum contribution towards the mass ordering sensitivity.}\n\\label{fig:inomin}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{ino-chi.pdf}\n \\caption{Mass ordering sensitivity $\\chi^{2}$ as a function of $\\Delta m_{31}^2$(IO) for ICAL in $\\nu$ , $\\bar{\\nu}$ and combined analysis. Red line is for $\\bar{\\nu}_{\\mu}$ and black line for $\\nu_{\\mu}$ data and combined results are shown in blue lines.}\n \\label{inochi}\n\\end{figure}\n\n\\subsubsection{T2HK and JUNO} \n\nThe dependence of $\\chi^2$ minimum with respect to $\\Delta m^{2}_{31}$ (IO) for T2HK and JUNO was recently studied in detail by us in Ref.~\\cite{Choubey:2022gzv}. We explored the synergy \n\\begin{itemize}\n \\item between the appearance and the disappearance channels of T2HK,\n \\item between the different $E$ bins of JUNO, and\n \\item between T2HK and JUNO.\n\\end{itemize}\nWe summarise here the main points for completeness. T2HK has a baseline of 295 km and a narrow-band beam peaked at neutrino energy of 0.6 GeV. Therefore, (1) low $E$ ensures that matter effects in the disappearance channel of T2HK can be neglected for all practical purposes and (2) the narrow-band beam ensures that we can essentially work at the oscillation maxima. It can be shown that for a true value of $\\Delta m^{2}_{31}$(NO), the $\\chi^2$ minimum for $\\Delta m^{2}_{31}$(IO) should appear at\n\\begin{equation}\n\\Delta m_{31}^{2}({\\rm IO})=-\\Delta m_{31}^{2}({\\rm NO}) + x ,\n\\end{equation}\nwith\n\\begin{equation}\nx = \\frac{2|U_{\\mu 2}|^{2}}{|U_{\\mu 1}|^{2}+|U_{\\mu 2}|^{2}}\\Delta m_{21}^{2}, \n\\end{equation}\nwhere $U$ is the PMNS matrix. For our choice of oscillation parameters, we obtain \n\\begin{eqnarray}\n\\Delta m_{31}^{2}(\\rm IO) = -2.45\\times10^{-3} ~{\\rm {eV}}^{2}. \n\\label{eq:bfapprox}\n\\end{eqnarray}\nIt was shown in~\\cite{Choubey:2022gzv} that both at the probability level as well as the $\\chi^2$ level, the minima for the disappearance channel comes very close to the value obtained in Eq.~(\\ref{eq:bfapprox}). Indeed the mass ordering $\\chi^2$ goes to zero at the above value of $\\Delta m^{2}_{31}$(IO). The appearance channel on the other hand, has matter effects and hence for this channel the mass ordering sensitivity never goes to zero. Nevertheless it too has a distinct minima with respect to $\\Delta m^{2}_{31}$(IO), which is very different from the one for the disappearance channel. Hence, when we combine the appearance and the disappearance channels, then the synergy coming from different values of $\\Delta m^{2}_{31}$(IO) at which we get the individual $\\chi^2$ minima results in a significantly enhanced $\\chi^2$ for T2HK. \n\nIn JUNO the situation is more complex due to its wide energy range of 1.8 MeV to 8.0 MeV. The condition for minima is given by the relation \\cite{Choubey:2022gzv} \n\\begin{eqnarray}\n&& \\cos^2\\theta_{12}\\sin 2\\Delta_{31}^{IO}+\\sin^2\\theta_{12}\\sin2(\\Delta_{31}^{IO}-\\Delta_{21}) \\nonumber \\\\ \\nonumber\n&-&\\cos^2\\theta_{12}\\sin 2\\Delta _{31}^{NO}.\\frac{\\Delta_{31}^{NO}}{\\Delta_{31}^{IO}}-\\sin^2\\theta_{12}\\sin2\\Delta_{32}^{NO}\\frac{\\Delta_{32}^{NO}}{\\Delta_{31}^{IO}}=0 ,\n\\label{eq:minrelation}\n\\end{eqnarray}\nwhere $\\Delta_{ij}=\\Delta m_{ij}^{2}L\/4E$. Hence, every energy bin of JUNO has a mass ordering minimum of $\\simeq 0$ but at a different value of $\\Delta m^{2}_{31}$(IO). This leads to a synergy between the 200 bins and when we do the full spectral analysis, we get a sizable sensitivity to mass ordering. \n\nFrom the above discussion we see that all the three experiments ICAL, T2HK and JUNO, exhibit different best-fit values of $\\Delta m^{2}_{31}$(IO) for the same true value of $\\Delta m^{2}_{31}$(NO). For this reason, when these experiments are combined, we expect a significant improvement in the overall mass ordering sensitivity.\n\n\n\\subsection{Combined sensitivity of ICAL and JUNO}\n\nWe start with the combined sensitivity of ICAL and JUNO shown in Fig.~\\ref{fig:inojuno}. In this figure (and all the subsequent figures) the lines corresponding to the different experiments and different combination of experiments are clearly marked on the figure. The analysis procedure used was outlined in the previous section. We have plotted the y-axis up to $\\chi^2 = 250$ to show the multiple local minima of JUNO. The sensitivity of ICAL alone for 10 years exposure is shown as a function of $\\Delta m_{31}^2$(IO) by the red line. We notice that the dependence of the ICAL mass-ordering $\\chi^2$ on $\\Delta m_{31}^2$(IO) is significantly shallow in contrast to the one of JUNO, shown by the green line. The mass-ordering $\\chi^2$ obtained from the combined analysis of ICAL and JUNO is shown by the blue line. We note that the best-fit $\\Delta m_{31}^2$(IO) of the combined fit is mostly determined by JUNO, owing to its sharp dependence on $\\Delta m_{31}^2$(IO). However, the inclusion of ICAL does play a role and brings a synergy, which raises the overall $\\chi^2$. The exact values of $\\Delta m_{31}^2$(IO) for which the mass-ordering $\\chi^2$ is minimum is given in Table~\\ref{tab:inojunobf} while the $\\chi^2$ minimum values are given in Table~\\ref{tab:inojunochi}. We see that the combined analysis gives us about 14\\% increase in the sensitivity as compared to a simple sum of the $\\chi^2$ of the two experiments. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{juno-ino.pdf}\n \\caption{Mass ordering sensitivity expected from a combined analysis of JUNO and ICAL, shown as a function of $\\Delta m^2_{31}$(IO).}\n \\label{fig:inojuno}\n\\end{figure}\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|| c | c | c |c ||}\n\\hline\n$\\Delta m^{2}_{31}$ (True) & $\\Delta m^{2}_{31}$ (JUNO) & $\\Delta m^{2}_{31}$ (ICAL) & $\\Delta m^{2}_{31}$ (Combined) \\\\\n\\hline \n 2.531&-2.503 & -2.419 & -2.503 \\\\\n\\hline\n\\end{tabular}\n\\caption{Column 1 shows the true value of $\\Delta m_{31}^2$(NO) (in units of $10^{-3}$ eV$^{2}$) while columns 2, 3 and 4 show values of $\\Delta m_{31}^{2}$(IO) (in units of $10^{-3}$ eV$^{2}$), for which we get mass-ordering $\\chi^2$ minimum for ICAL, JUNO and (ICAL and T2HK) Combined.}\n\\label{tab:inojunobf}\n\\end{table}\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|| c | c | c |c | c ||}\n\\hline\n$\\chi^2_{\\rm JUNO}$ & $\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm JUNO}+\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm Combined}$ & $\\%$ increase\\\\\n\\hline \n 10.23 & 9.12 & 19.35 & 22.01 & 13.7\\\\\n\\hline\n\\end{tabular}\n\\caption{Mass-ordering $\\chi^2$ for true NO and test IO. The last column shows the percentage increase in sensitivity we get from a combined analysis (column 4) as compared to a simple sum (column 3). \n}\n\\label{tab:inojunochi}\n\\end{table}\n\n\\subsection{Combined sensitivity of ICAL and T2HK}\n\nNext we perform a combined study of long baseline experiment T2HK and atmospheric experiment ICAL. The results are shown in Fig.~\\ref{t2hkino}. As seen before, ICAL gives us 3$\\sigma$ mass ordering sensitivity with 10 year data for $\\theta_{23}$ maximal. T2HK also gives us mass ordering sensitivity but it is $\\delta_{\\rm CP}$ dependent (Fig.~\\ref{t2hkino}). This leads to the well known hierarchy - $\\delta_{\\rm CP}$ degeneracy \\cite{Barger:2001yr,Prakash:2012az,Ghosh:2015ena}. In T2HK we get mass ordering sensitivity below 2$\\sigma$ for $\\delta_{\\rm CP}=0^{\\circ}$ (left panel) and 5$\\sigma$ for $\\delta_{\\rm CP}=-90^{\\circ}$ (right panel). When we combine ICAL and T2HK data we get 3.68$\\sigma$ sensitivity for $\\delta_{\\rm CP}=0^{\\circ}$ and 5.9$\\sigma$ for $\\delta_{\\rm CP}=-90^{\\circ}$ . Synergy effect between ICAL and T2HK is very small. The reason is that the combined best-fit value of $\\Delta m^{2}_{31}(\\rm IO)$ is governed almost entirely by the T2HK and INO is unable to make much impact on it. Thus the synergistic effect is very weak. \n \\begin{figure}[h]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{t2hk-ino-cp-0.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{t2hk-ino-cp--90.pdf}\n\\end{minipage}\n\\caption{Mass ordering sensitivity as a function of $\\Delta m^2_{31}$(IO). Left panel is for $\\delta_{\\rm CP} = 0^\\circ$ and the right panel is for $\\delta_{\\rm CP} =-90^\\circ$ corresponding to T2HK.}\n\\label{t2hkino}\n\\end{figure}\n\\begin{table}[h!]\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{||c| c | c | c |c ||}\n\\hline\n& $\\Delta m^{2}_{31}$ (True) & $\\Delta m^{2}_{31}$ (ICAL) & $\\Delta m^{2}_{31}$ (T2HK) & $\\Delta m^{2}_{31}$ (Combined) \\\\\n\\hline \n$\\delta_{\\rm CP}=0$ &2.531 & -2.419 & -2.431 & -2.431 \\\\\n\\hline\n$\\delta_{\\rm CP}=-90$ & 2.531 & -2.419 & -2.428 & -2.428 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Column 2 shows the true value of $\\Delta m_{31}^2$(NO) (in units of $10^{-3}$ eV$^{2}$) while columns 3, 4 and 5 show values of $\\Delta m_{31}^{2}$(IO) (in units of $10^{-3}$ eV$^{2}$), for which we get mass-ordering $\\chi^2$ minimum for ICAL, T2HK and (ICAL and T2HK) Combined.}\n\\label{tab:inot2hkbf}\n\\end{table}\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{||c|c | c | c |c | c |c ||}\n\\hline\n& $\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm T2HK}$ &$\\chi^2_{\\rm T2HK}+\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm Combined}$ & $\\%$ increase\\\\\n\\hline \n$\\delta_{\\rm CP}=0$ & 9.12 & 3.77 & 12.89 &13.55 & 5 \\\\\n\\hline\n$\\delta_{\\rm CP}=-90$ & 9.12 & 25.09 & 34.21 & 34.84 & 1.8\\\\\n\\hline\n\\end{tabular}\n\\caption{Mass-ordering $\\chi^2$ for true NO and test IO. The last column shows the percentage increase in sensitivity we get from a combined analysis (column 5) as compared to a simple sum (column 4). }\n\\label{tab:inot2hkchi}\n\\end{table}\nWe can see in Fig.~\\ref{t2hkino} and Table~\\ref{tab:inot2hkbf} that the best-fit values are similar for both experiments. Also, ICAL has a very flat curve corresponding to $\\Delta m^{2}_{31}$(IO). So this leads to very small synergy effect as can be seen in Table \\ref{tab:inot2hkchi}. The percentage increase as compared to a simple sum of the $\\chi^{2}$ is 5$\\%$ for $\\delta_{\\rm CP}=0^{\\circ}$ and 2$\\%$ for $\\delta_{\\rm CP}=-90^{\\circ}$. However, it is important to note that by combining T2HK and ICAL, the sensitivity to mass ordering improves to about $4 \\sigma$ even for $\\delta_{\\rm CP} = 0^\\circ$, while for $\\delta_{\\rm CP}=-90^{\\circ}$ we have close to $6\\sigma$ sensitivity. \n\n\n \\subsection{Combined sensitivity of ICAL, JUNO and T2HK}\n\n\nIn this section we will see that by combining atmospheric data (ICAL) with long baseline (T2HK) and reactor (JUNO) data, we can achieve more than 10 $\\sigma$ sensitivity for $\\delta_{\\rm CP}=0^\\circ$ and 11 $\\sigma$ sensitivity for $\\delta_{\\rm CP}=-90^\\circ$. The results are presented in Fig. \\ref{combined}, as a function of $\\Delta m^2_{31}$ (IO). In the left (right) panel we have considered $\\delta_{\\rm CP} = 0^\\circ ~ (-90^\\circ)$. The values of $\\Delta m_{31}^2$(IO) for which the mass-ordering $\\chi^2$ is minimum is given in Table~\\ref{tab:inojunot2hkbf} while the $\\chi^2$ minimum values are given in Table~\\ref{tab:inojunot2hkchi}.\n\n \\begin{figure}[h]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{juno-t2hk-ino-cp-0.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{juno-t2hk-ino-cp--90.pdf}\n\\end{minipage}\n\\caption{Mass ordering sensitivity as a function of $\\Delta m^2_{31}$(IO). Left panel is for $\\delta_{\\rm CP} = 0^\\circ$ and the right panel is for $\\delta_{\\rm CP} =-90^\\circ$ corresponding to T2HK.}\n\\label{combined}\n\\end{figure}\n\n\\begin{table}[h!]\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{||c| c |c | c |c|c ||}\n\\hline\n& $\\Delta m^{2}_{31}$ (True) & $\\Delta m^{2}_{31}$ (ICAL) & $\\Delta m^{2}_{31}$ (JUNO) & $\\Delta m^{2}_{31}$ (T2HK) & $\\Delta m^{2}_{31}$ (Combined) \\\\\n\\hline \n$\\delta_{\\rm CP}=0^\\circ$ & 2.531& -2.419 & -2.503 & -2.431 & -2.48 \\\\\n\\hline\n$\\delta_{\\rm CP}=-90^\\circ$ & 2.531& -2.419 & -2.503 & -2.428 & -2.479 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Column 2 shows the true value of $\\Delta m_{31}^2$(NO) (in units of $10^{-3}$ eV$^{2}$) while columns 3, 4, 5 and 6 show values of $\\Delta m_{31}^{2}$(IO) (in units of $10^{-3}$ eV$^{2}$), for which we get mass-ordering $\\chi^2$ minimum for ICAL, JUNO, T2HK and (ICAL and JUNO and T2HK) Combined.\n}\n\\label{tab:inojunot2hkbf}\n\\end{table}\n\n\\begin{table}[h!]\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{||c| c |c | c | c |c | c |c ||}\n\\hline\n& $\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm JUNO}$ & $\\chi^2_{\\rm T2HK}$ & $\\chi^2_{\\rm JUNO}+\\chi^2_{\\rm T2HK}+\\chi^2_{\\rm ICAL}$ & $\\chi^2_{\\rm Combined}$ & $\\%$ increase\\\\\n\\hline \n$\\delta_{\\rm CP}=0^\\circ$ & 9.12 & 10.23 & 3.77 & 23.12 &99.76 & 331 \\\\\n\\hline\n$\\delta_{\\rm CP}=-90^\\circ$ & 9.12 & 10.23 & 25.09 & 44.44 & 125.79 & 183\\\\\n\\hline\n\\end{tabular}}\n\\caption{Mass-ordering $\\chi^2$ for true NO and test IO. The last column shows the percentage increase in sensitivity we get from a combined analysis (column 6) as compared to a simple sum (column 5). }\n\\label{tab:inojunot2hkchi}\n\\end{table}\n \nAs we discussed earlier, for JUNO the mass-ordering $\\chi^2$ is about 10 and for ICAL the mass ordering sensitivity is about 9. The sensitivity of T2HK is about 4 for $\\delta_{\\rm CP} = 0^\\circ$ and 25 for $\\delta_{\\rm CP} = -90^\\circ$. We again stress upon the fact that though the true value of $\\Delta m^2_{31}$ is same for all these experiments, the $\\chi^2$ minimum occurs at different values of $\\Delta m^2_{31}$ (IO) for different experiments. \n For the combination of ICAL, JUNO and T2HK, the mass-ordering $\\chi^2$ increases to about 100 (126) for $\\delta_{\\rm CP} = 0^\\circ (-90^\\circ)$. \n We had seen that for the combination of JUNO+ICAL and T2HK+ICAL, the improvement due to synergy was rather mild. We had explained that this was because the ICAL $\\chi^2$ was shallow and was therefore unable to really change the best-fit $\\Delta m_{31}^2$(IO) significantly. On the other hand, both JUNO and T2HK have well-defined and sharp $\\chi^2$ with best-fit $\\Delta m_{31}^2$(IO) distinctly different. This was discussed in Ref.~\\cite{Choubey:2022gzv}. As a result, when we combined data from all experiments, we get a big synergistic boost leading to the staggering improvement as can be seen in the last column of Table~\\ref{tab:inojunot2hkchi}. \n \n\n\\subsection{Effect of energy resolution of JUNO}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{Juno-res.pdf}\n \\caption{Neutrino mass ordering sensitivity as a function of energy resolution of JUNO.}\n \\label{fig:mesh1}\n\\end{figure}\nNext we discuss how the mass ordering sensitivity changes if we vary the energy resolution of JUNO. In Fig. \\ref{fig:mesh1}, we have presented the mass-ordering sensitivity as a function of the energy resolution of JUNO. From this figure we clearly see that the sensitivity of JUNO goes from $\\chi^2 = 10$ to less than 1 when the energy resolution is varied from 3$\\%\/\\sqrt{E(MeV)}$ to 5$\\%\/\\sqrt{E(MeV)}$. But when it is combined with other experiments we can see that the sensitivity is significantly improved. For ICAL+JUNO+T2HK the $\\chi^2$ is expected to come around 52 (76) for $\\delta_{\\rm CP} = 0^\\circ (-90^\\circ)$ even if the JUNO energy resolution is 5$\\%\/\\sqrt{E(MeV)}$. This shows that even for poorer resolutions of JUNO, with the combination of T2HK and ICAL, one can achieve a mass ordering sensitivity of $7.2 \\sigma$ even for the unfavourable value of $\\delta_{\\rm CP}$. For JUNO+ICAL, the sensitivity is expected to be around $4 \\sigma$ for JUNO energy resolution of 5$\\%\/\\sqrt{E(MeV)}$.\n\n\\subsection{Effect of ICAL run-time}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{ino-run-time.pdf}\n \\caption{Neutrino mass ordering sensitivity as a function of run-time of ICAL.}\n \\label{fig:mesh2}\n\\end{figure}\nIn this section we explore the effect of extending the run-time of ICAL on the mass ordering sensitivity. As atmospheric neutrinos are available without any cost \\footnote{For the detector of course there will be running costs such as maintaining the magnetic field, electronics and power supply for cooling.}, one can in principle run an atmospheric neutrino experiment for a longer time. In Fig. \\ref{fig:mesh2}, we present the mass-ordering sensitivity as a function of run-time of ICAL. For a 20 years running of ICAL, the $\\chi^2$ for ICAL goes up to 16. When it is added with JUNO the $\\chi^2$ reaches 36. We can see that $5\\sigma$ sensitivity for mass ordering could be achieved with just JUNO and 12 years of running of ICAL. For the combination of ICAL+JUNO+T2HK one can have a mass-ordering $\\chi^2$ of 100 (144) for $\\delta_{\\rm CP} = 0^\\circ (-90^\\circ)$ for 20 years running of ICAL. \n\n\\subsection{Effect of true values $\\theta_{23}$ on combined analysis}\n \n\\begin{figure}[h!]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{t23-true-cp-0.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{t23-true-cp--90.pdf}\n\\end{minipage}\n\\caption{Neutrino mass ordering sensitivity as a function of $\\theta_{23}$ (true). Left panel is for $\\delta_{\\rm CP} = 0^\\circ$ and the right panel is for $\\delta_{\\rm CP} =-90^\\circ$ corresponding to T2HK.}\n\\label{fig:2figsB_1}\n\\end{figure}\nNow let us see how the mass ordering sensitivity varies when we vary the true value of $\\theta_{23}$. In Fig. \\ref{fig:2figsB_1} we have plotted the mass-ordering $\\chi^2$ as a function of $\\theta_{23}$ (true). The value of $\\delta_{\\rm CP}$ for T2HK is $0^\\circ$ in the left panel and $-90^\\circ$ in the right panel. In general as $\\theta_{23}$ increases, the mass ordering sensitivity increases ~\\cite{anu-mh, moon:2014}. We can see this general trend in most of the curves, except for the T2HK+JUNO curve for both the values of $\\delta_{\\rm CP}$ and ICAL+JUNO+T2HK curve for $\\delta_{\\rm CP} = 0^\\circ$. For these curves, the $\\chi^2$ increases as $\\theta_{23}$ increases from $42^\\circ$ to $45^\\circ$ and when $\\theta_{23}$ increases further from $45^\\circ$, the sensitivity decreases. To understand this, in Fig. \\ref{fig:2figsB_2}, we have plotted $\\chi^2$ vs $\\Delta m^2_{31}$(IO) for three true values of $\\theta_{23}$ i.e., $42^\\circ$, $45^\\circ$ and $49^\\circ$.\n\\begin{figure}[h!]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{th23-true-cp-0.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{th23-true-cp--90.pdf}\n\n\\end{minipage}\n\\caption{Mass ordering sensitivity as a function of $\\Delta m^2_{31}$(IO) for different values of $\\theta_{23}$. Left panel is for $\\delta_{\\rm CP} = 0^\\circ$ and the right panel is for $\\delta_{\\rm CP} =-90^\\circ$ corresponding to T2HK.}\n\\label{fig:2figsB_2}\n\\end{figure}\nThe value of $\\delta_{\\rm CP}$ for T2HK is $0^\\circ$ in the left panel and $-90^\\circ$ in the right panel.\n\\begin{itemize}\n \\item For $\\delta_{\\rm CP} = 0^\\circ$, we note that though $\\chi^2$ for T2HK increases as $\\theta_{23}$ increases, the $\\chi^2$ minimum tends to shift towards the left. As a result when it is combined with JUNO, the $\\chi^2$ corresponding to $\\theta_{23} = 49^\\circ$ becomes lower than the $\\chi^2$ for $\\theta_{23}=42^\\circ$. However for ICAL+JUNO+T2HK, the $\\chi^2$ for $\\theta_{23} = 49^\\circ$ is higher than $\\chi^2$ for $\\theta_{23} = 42^\\circ$ but lower than $\\chi^2$ for $\\theta_{23} = 45^\\circ$. \n \n \\item For $\\delta_{\\rm CP} = -90^\\circ$, the shift of the T2HK curve towards left for the value of $\\theta_{23} = 49^\\circ$ is less prominent as compared to $\\delta_{\\rm CP} = 0^\\circ$. As a result, for T2HK+JUNO, the $\\chi^2$ for $\\theta_{23} = 49^\\circ$ is higher than $\\chi^2$ for $\\theta_{23} = 42^\\circ$ but lower than $\\chi^2$ for $\\theta_{23} = 45^\\circ$. However, for the combination of ICAL+JUNO+T2HK, the $\\chi^2$ for $\\theta_{23} = 49^\\circ$ is higher than $\\chi^2$ for $\\theta_{23} = 42^\\circ$ and $45^\\circ$ establishing the general trend.\n\\end{itemize}\n\n\\subsection{Effect of octant degeneracy in ICAL and T2HK on mass ordering }\n \n\\begin{figure}[h!]\n\\centering\n\\parbox{7cm}{\n\\includegraphics[width=8cm]{t23-test-lo.pdf}\n }\n\\qquad\n\\begin{minipage}{7cm}\n\\includegraphics[width=8cm]{t23-test-ho.pdf}\n\n\\end{minipage}\n\\caption{Neutrino mass ordering sensitivity as a function of $\\theta_{23}$ (test). Left panel is for $\\delta_{\\rm CP} = 0^\\circ$ and the right panel is for $\\delta_{\\rm CP} =-90^\\circ$ corresponding to T2HK.}\n\\label{fig:2figsB_3}\n\\end{figure}\nFinally let us discuss the effect of octant degeneracy in the mass ordering sensitivity of T2HK and ICAL. In Fig. \\ref{fig:2figsB_3}, we have plotted the mass-ordering $\\chi^2$ as a function of $\\theta_{23}$ (test). The left panel is for $\\theta_{23}$ (true) $= 42^\\circ$ and the right panel is for $\\theta_{23}$ (true) $= 49^\\circ$. From the panels we see that for true $\\theta_{23}$ in the lower octant, the $\\chi^2$ minimum always occurs with the correct octant for both T2HK and ICAL. However for $\\theta_{23}$ in the higher octant, the $\\chi^2$ minimum occurs at the wrong octant for ICAL. But when ICAL is added with T2HK, the minimum of the combined $\\chi^2$ occurs at the correct octant. Therefore, one can say that for $\\theta_{23}$ in the higher octant, T2HK resolves the octant degeneracy of ICAL to improve the mass ordering sensitivity. However, this effect is very small as the ICAL $\\chi^2$ with respect to $\\theta_{23}$ (test) is almost flat.\n\n\n\\section{Summary and conclusions}\n\\label{sum}\n\nIn this paper we have studied the capability of ICAL, JUNO and T2HK experiments to measure neutrino mass ordering. Though studies on synergy in neutrino oscillation experiments have been performed in the past, in this work we took the opportunity to study the synergy between future accelerator, future atmospheric and future reactor neutrino experiment T2HK, ICAL and JUNO, to determine the neutrino mass ordering. We expounded the synergy in mass ordering determination at these experiments. Mass ordering sensitivity comes from the difference in the oscillation probabilities between normal and inverted ordering. The sensitivity to mass ordering is calculated by simulating the data for an assumed true mass ordering, say normal, and fitted with the wrong mass ordering (say) inverted. In the fit we allow $\\delta_{\\rm CP}$, $\\theta_{23}$ and $|\\Delta m_{31}^2|$ to vary within their allowed ranges and the $\\chi^2$ is minimized over this entire grid. In particular, we showed that the effect of minimization of the $\\chi^2$ over $|\\Delta m_{31}^2|$ leads to synergy between the experiments. The best-fit $|\\Delta m_{31}^2|$ for which the mass ordering $\\chi^2$ is minimized depends on the oscillation channel, neutrino energy, as well as matter effects. Since T2HK, ICAL and JUNO have different specifications with respect to these, they have synergy with respect to neutrino mass ordering. For this reason, the same true value of $\\Delta m^2_{31}$ leads to different values of $\\Delta m^2_{31}$ (fit) for the $\\chi^2$ minimum. Therefore, when different experiments are added, the $\\chi^2$ minimum of the added $\\chi^2$ occurs at a different value of $\\Delta m^2_{31}$ leading to an enhanced mass ordering sensitivity. \n\nWe presented results for joint analyses of ICAL+JUNO, ICAL+T2HK and finally ICAL+JUNO+T2HK. We showed that for ICAL+JUNO we can expect about 14\\% increase in sensitivity due to synergy as compared to a simple sum of the individual $\\chi^2$. However, the enhancement in the case of ICAL+T2HK was not seen to be significant. Finally, when we combined all three ICAL+JUNO+T2HK, we found a staggering increase in the sensitivity. For $\\delta_{\\rm CP}=0^\\circ$ ( $\\delta_{\\rm CP}=-90^\\circ$) this increase was shown to be 331\\% (183\\%). The ICAL sensitivity has a rather flat dependence on $\\Delta m^2_{31}$, while JUNO and T2HK senitivity have a very sharp $\\Delta m^2_{31}$ dependence. As a result adding JUNO and T2HK brings the most spectacular increase in the sensitivity. Nonetheless ICAL does play a role and brings additional enhancement to the sensitivity. \n\nWe studied the effect of JUNO energy resolution on the mass ordering sensitivity. We showed that JUNO mass ordering $\\chi^2$ would plummet from about 10 to less than 1 if the energy resolution was to worsen from 3$\\%\/\\sqrt{E(MeV)}$ to 5$\\%\/\\sqrt{E(MeV)}$. However, when performing a combined analysis, the synergy between experiments stabilizes the $\\chi^2$ and we would have an assured determination of mass ordering at greater than $7\\sigma$ significance even if JUNO has a resolution of 5$\\%\/\\sqrt{E(MeV)}$ and T2HK $\\delta_{\\rm CP} = 0^\\circ$. We also studied the impact of ICAL run-time on mass ordering. We showed that more than $5\\sigma$ sensitivity would be easily achievable from different combinations of these experiments and different run times for ICAL. We showed that if ICAL run-time is increases to 20 years, then the combined $\\chi^2$ for ICAL+JUNO+T2HK reaches 100 (144) for $\\delta_{\\rm CP} = 0^\\circ (-90^\\circ)$. Finally, we studied the impact of $\\theta_{23}$ on the mass ordering sensitivity and the role of doing the combined analysis. We also show that if $\\theta_{23}$ belongs in the upper octant, then $\\chi^2$ minimum for ICAL appears in the wrong octant and the addition of T2HK data resolves this octant degeneracy. \n\nIn summary, we showed that ICAL+JUNO+T2HK is a powerful combination of experiments which can establish the true nature of the mass ordering with a significant confidence level irrespective of true values of $\\delta_{\\rm CP}$ and energy resolution of JUNO.\n\n\\section*{Acknowledgements}\nThis work is performed by the members of the INO-ICAL collaboration. We thank to the members of the INO-ICAL collaboration for their valuable comments\nand constructive inputs. We thank A. Dighe, S. Goswami for useful discussion and valuable comments during the INO analysis meetings. The authors also sincerely thank the ICAL internal referees, Amol Dighe and D. Indumathi for their careful reading of the manuscript and for providing useful suggestions. Authors extend sincere thanks to Suprabh Prakash for providing the .glb file for JUNO and for useful discussions regarding the JUNO experiment. The HRI cluster computing facil-\nity (http:\/\/cluster.hri.res.in) is gratefully acknowledged. MG acknowledges Ramanujan Fellowship of SERB, Govt. of India, through grant no: RJF\/2020\/000082. This project has received funding\/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska -Curie grant agreement No 860881-HIDDeN.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\n\n The collective dynamics of rarefied gases at the mesoscopic scale is described by the celebrated Boltzmann equation.\nBut the practical application of the Boltzmann equation has been restricted by its highly resource-consuming features such as the complicated structure of the collision operator, high dimensionality and stiffness problem.\nIn this regard, Bhatnagar, Gross, Krook \\cite{BGK} and, independently Welander \\cite{Wel}, suggested a model equation by\nreplacing the collision operator with a relaxation operator\nwhich still keeps the most important features of the Boltzmann equation such as the conservation laws, $H$-theorem and the correct hydrodynamic limit to the Euler equation. Ever since it was introduced, the BGK model has been widely used in place of the Boltzmann equation because it reproduces the qualitative features of the Boltzmann dynamics very well\nat much lower computational costs.\n\nBoth the Boltzmann equation and the BGK model are derived under the assumption that the gas consists of monatomic molecules.\nThe necessity of kinetic equations that account for the collisional dynamics of polyatomic molecules is apparent, considering that\nthere are very few elements in the nature which stay stable as monatomic molecules at room temperature.\nAny attempt for the description of kinematics of polyatomic moleculess, however, must allow some simplifying assumptions or phenomenological description because the diversity of the inner configuration of polyatomic molecules make it almost impossible to express the pre-post collision process in an explicit form, except for some special cases.\nOne such formulation is so-called the internal energy formulation where a new variable $I$ is introduced to incorporate the information on the non-translational internal energy due to the\nmolecular structure \\cite{ABLP,ALPP,BL,BDLP,Brull3,Cai,KAG,PRS,RS,RS2,TFA}.\n\nIn this paper, we study the existence and asymptotic behavior for the polyatomic ellipsoidal BGK model, which is a polyatomic generalization of the original BGK model using such internal energy formulation: \\cite{ABLP,ALPP}:\n\\begin{align}\\label{polyatomic ESBGK}\n\\begin{split}\n\\partial_tF+v\\cdot \\nabla_xF&=A_{\\nu,\\theta}(\\mathcal{M}_{\\nu,\\theta}(F)-F),\\cr\n\\qquad F(0,x,v,I)&=F_0(x,v,I).\n\\end{split}\n\\end{align}\nThe velocity-energy distribution function $F(t,x,v,I)$ represents the number density on phase point $(x,v)\\in \\mathbb{T}^3_x\\times \\mathbb{R}^3_v$ with non-translational internal energy $I^{2\/\\delta}~(I\\geq0)$ at time $t\\geq0$.\nThe parameter $\\delta>0$ measures the degree of excitation of non-translational mode of the molecules such as the rotational or vibrational mode.\nThe collision frequency $A_{\\nu,\\theta}$ is given by\n$A_{\\nu,\\theta}=(\\rho^{\\alpha}T^{\\beta}_{\\delta})\/(1-\\nu+\\theta\\nu)$\nfor some $0\\leq \\alpha,\\beta\\leq 1$. ($\\rho$ and $T_{\\delta}$ are defined below.) Throughout this paper, we fix $\\alpha=\\beta=1$ for simplicity.\nThe relaxation parameters $-1\/2<\\nu<1$ and $0\\leq\\theta\\leq1$ are introduced to\nreproduce the correct Prandtl number and the second viscosity coefficient in the Chapman-Enskog expansion \\cite{ALPP}.\nThe number $1\/\\theta$ is interpreted as the relaxation collision number, which is the average number of collisions needed\nto transfer the rotational and vibrational internal energy into the translational energy \\cite{ABLP,Cai}.\n\nWe define the macroscopic density, momentum, stress tensor and total energy by\n\\begin{align*}\n\\rho(t,x)&=\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} F(t,x,v,I) dvdI,\\cr\nU(t,x)&=\\frac{1}{\\rho}\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} vF(t,x,v,I) dvdI, \\cr\n\\Theta(t,x) &= \\frac{1}{\\rho}\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} (v-U)\\otimes(v-U)F(t,x,v,I) dvdI, \\cr\nE(t,x)&=\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} \\left(\\frac{1}{2}|v|^2 +I^{\\frac{2}{\\delta}}\\right)F(t,x,v,I) dvdI.\n\\end{align*}\nThe total energy is decomposed further into the following three parts:\n\\[\nE=E_{kin}+E_{tr}+E_{I,\\delta}\n\\]\nwhere the kinetic energy $E_{kin}$, the internal energy\ndue to the translational motion $E_{tr}$, and the internal energy attributed to the internal configuration of the molecules\n$E_{I,\\delta}$ are given respectively by\n\\begin{align*}\nE_{kin}&=\\frac{1}{2}\\rho |U|^2,\\cr\nE_{tr}&=\\frac{1}{2}\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} |v-U|^2 F(t,x,v,I) dvdI, \\cr\nE_{I,\\delta}&=\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} I^{\\frac{2}{\\delta}} F(t,x,v,I) dvdI.\n\\end{align*}\nWe also define the total internal energy $E_{\\delta}$:\n\\begin{align*}\nE_{\\delta}(t,x)&=E_{tr}+E_{I,\\delta}\\cr\n&=\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} \\left(\\frac{1}{2}|v-U|^2 +I^{\\frac{2}{\\delta}}\\right)F(t,x,v,I) dvdI,\n\\end{align*}\nfrom which we can define the corresponding temperatures $T_{\\delta}$, $T_{tr}$ and $T_{I,\\delta}$ using the equipartition principle:\n\\begin{align*}\nE_{\\delta}&=\\frac{3+\\delta}{2}\\rho T_{\\delta},\\quad E_{tr}=\\frac{3}{2}\\rho T_{tr}, \\quad E_{I,\\delta}= \\frac{\\delta}{2}\\rho T_{I,\\delta}.\n\\end{align*}\nConsequently, $T_{\\delta}$ is represented by a convex combination of $T_{tr}$ and $T_{I,\\delta}$:\n\\begin{eqnarray*}\nT_{\\delta}=\\frac{3}{3+\\delta}T_{tr}+\\frac{\\delta}{3+\\delta}T_{I,\\delta}.\n\\end{eqnarray*}\nFor $-1\/2<\\nu<1$ and $0\\leq\\theta\\leq1$, we define the relaxation temperature $T_{\\nu,\\theta}$ and the corrected temperature tensor $\\mathcal{T}_{\\theta}$ by\n\\begin{align*}\nT_{\\theta}&=\\theta T_{\\delta}+(1-\\theta)T_{I,\\delta},\\cr\n\\mathcal{T}_{\\nu,\\theta}&=\\theta T_{\\delta}Id+(1-\\theta)\\big\\{(1-\\nu)T_{tr}Id+\\nu \\Theta\\big\\}.\n\\end{align*}\nNow, the polyatomic ellipsoidal Maxwellian $\\mathcal{M}_{\\nu,\\theta}$ reads\n\\begin{eqnarray}\\label{poly maxwellian}\n\\mathcal{M}_{\\nu,\\theta}(F)= \\frac{\\rho \\Lambda_{\\delta}}{\\sqrt{\\det (2\\pi \\mathcal{T}_{\\nu,\\theta}}) \\,T_{\\theta}^{\\frac{\\delta}{2}}}\\exp\\left(-\\frac{1}{2}(v-U)^{\\top}\\mathcal{T}^{-1}_{\\nu,\\theta}(v-U)-\\frac{I^{\\frac{2}{\\delta}}}{T_{\\theta}}\\right),\n\\end{eqnarray}\nwhere $\\Lambda_{\\delta}$ is the normalizing factor: $\\Lambda_{\\delta}= 1\/\\int_{\\mathbb{R}_+} e^{-I^{2\/\\delta}} dI$.\n\nThe relaxation operator satisfies the following cancellation property:\n\\begin{eqnarray*}\n\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+} (\\mathcal{M}_{\\nu,\\theta}(F)-F)\n\\left(\n\\begin{array}{c}1\\cr v\\cr\\frac{1}{2}|v|^2 +I^{\\frac{2}{\\delta}}\\end{array}\\right)\n dvdI = 0,\n\\end{eqnarray*}\nwhich leads to the conservation of mass, momentum and energy:\n\\begin{align}\\label{ConservationLawsF}\n\\begin{split}\n\\int F(t)dxdvdI\n&=\\int F_0dxdvdI,\\cr\n\\int F(t)vdxdvdI\n&=\\int F_0vdxdvdI,\\cr\n\\int F(t)\\left(\\frac{|v|^2}{2}+I^{2\/\\delta}\\right)dxdvdI\n&=\\int F_0\\left(\\frac{|v|^2}{2}+I^{2\/\\delta}\\right)dxdvdI.\n\\end{split}\n\\end{align}\nThe $H$-theorem for this model was established in \\cite{ALPP}\n(See also \\cite{Brull3,PY2}):\n\\[\n\\frac{d}{dt}\\int_{\\mathbb{R}^3 \\times \\mathbb{R}^+}F(t)\\ln F(t)dvdI\\leq 0.\n\\]\n\\newline\n\n\n\n\nIn this paper, we study the dynamics of the polyatomic BGK model (\\ref{polyatomic ESBGK}) near a global polyatomic Maxwellian:\n\\begin{align}\\label{poly maxwellian}\nm(v,I)=\\frac{\\Lambda_{\\delta}}{\\sqrt{(2\\pi)^3}}e^{-\\frac{|v|^2}{2}-I^{2\/\\delta}}.\n\\end{align}\nFor this, we define the perturbation $f$ around the equilibrium by\n\\begin{align}\\label{perturbation}\nF=m+\\sqrt{m}f,\\quad F_0=m+\\sqrt{m}f_0\n\\end{align}\nand rewrite (\\ref{polyatomic ESBGK}) as\n\\begin{align*}\n\\partial_t f+v\\cdot\\nabla_xf&=L_{\\nu,\\theta}f+\\Gamma_{\\nu,\\theta}(f),\\cr\nf(0,x,v,I)&=f_0(x,v,I),\n\\end{align*}\nwhere $L_{\\nu,\\theta}$ denotes the linearized relaxation operator and $\\Gamma_{\\nu,\\theta}(f)$ is the nonlinear perturbation.\n(See Section 2.) We then analyze this linearized polyatomic BGK model in the framework of nonlinear energy methods\ndeveloped in, for example, \\cite{Guo whole,Guo VMB, Guo VPB}.\n\nThe most important step is to verify the dissipative nature of the linearized relaxation operator $L_{\\nu,\\theta}$.\nIn this regard, we make a key observation that there exists a dichotomy in the coercive estimate of $L_{\\nu,\\theta}$ (See Section 3.):\n\\[\n-(1-\\nu+\\theta\\nu)\\langle L_{\\nu,\\theta}f,f\\rangle_{L^2_{v,I}}\\geq\\theta \\|(I-P_p)f\\|^2_{L^2_{v,I}}, \\quad (0<\\theta\\leq 1)\n\\]\nand\n\\[\n-(1-\\nu)\\langle L_{\\nu,0}f,f\\rangle_{L^2_{v,I}}\\geq\\left(1-|\\nu|\\right) \\|(I-P_m)f\\|^2_{L^2_{v,I}}. \\quad (\\theta=0)\n\\]\nNote that the coefficient in the l.h.s of the above dissipative estimates change continuously as $\\theta$ goes to $0$, while the coefficient in the right hand side jump from $\\theta$ to $1-|\\nu|$ at $\\theta=0$. More importantly,\nthe macroscopic projection on the right hand side, which determines the degeneracy of the dissipation, changes abruptly from\nthe projection $P_p$ on\n\\begin{eqnarray*}\nspan\\bigg\\{\\sqrt{m },~v\\sqrt{m },~\\frac{(|v|^2-3)+(2I^{2\/\\delta}-\\delta)}{\\sqrt{2(3+\\delta)}}\\sqrt{m}\\bigg\\}, \\quad (0<\\theta\\leq 1)\n\\end{eqnarray*}\nto the projection $P_m$ on\n\\begin{eqnarray*}\nspan\\bigg\\{\\sqrt{m },~v\\sqrt{m },~\\frac{|v|^2-3}{\\sqrt{6}}\\sqrt{m},~ \\frac{I^{2\/\\delta}-\\delta}{\\sqrt{2\\delta}}\\sqrt{m }\\bigg\\}.\n\\qquad (\\theta=0)\n\\end{eqnarray*}\nTherefore, the degeneracy at $\\theta=0$ is strictly stronger than the non zero $\\theta$ case.\n\nThis agrees well with the similar dichotomy in the nonlinear entropy-entropy production estimate\nobserved in \\cite{PY2}, of which the above estimates can be considered as a linearized version:\n\\begin{align*}\nD_{\\nu,\\theta}(f)&\\geq\\theta A_{\\nu,\\theta} H(f|\\mathcal{M}_{0,1}),\\hspace{1cm}(0<\\theta \\leq1),\n\\end{align*}\nand\n\\begin{align*}\nD_{\\nu,0}(f)&\\geq\\min\\{1-\\nu, 1+2\\nu\\} A_{\\nu,0} H(f|\\mathcal{M}_{0,0}).\\quad (\\theta=0)\n\\end{align*}\nHere, $D_{\\nu,\\theta}$ and $H(f|g)$ denote the entropy production functional and the relative entropy for (\\ref{polyatomic ESBGK}) respectively. See \\cite{PY2} for the exact definition of the target polyatomic Maxwellians $\\mathcal{M}_{0,1}$ and $\\mathcal{M}_{0,0}$.\nIn \\cite{PY2}, however, it is not clear whether such dichotomy is an intrinsic property of the model, or\n can be resolved into a better estimate that interpolates the two entropy production estimates.\n\nIt is explicitly shown in Section 3 that the linearized\nrelaxation operator $L_{\\nu,\\theta}$ is divided into a truly polyatomic part and an essentially monatomic-like part. In the case $0<\\theta\\leq 1$, the dissipation is governed by the former, whereas the dissipation for $\\theta=0$ case is governed by the latter. This shows that such dichotomy is intrinsic and cannot be avoided by developing a refined argument.\n\nRecalling that $\\theta^{-1}$ is interpreted as the average number of collisions needed for the non-translational energy\ndue to the molecular configuration to be transferred, we see that such dichotomy has a nice physical interpretation: when $\\theta=0$, the relaxation collision number is infinite, and therefore, no matter how many collisions occur, the exchange between the translational energy and the non-translational energy does not happen, making the kinematics essentially - though not exactly - that of the monatomic gases. (Note that in the kernel of $P_m$, the translational energy and the non-translational energy are completely split, whereas they are given in an entangled form in the kernel of $P_p$)\nWe, however, menstion that such physical interpretation alone does not give any hint that there has to be a discontinuity at $\\theta=0$.\n\nAs a result of such dichotomy, we need to employ two different types of micro-macro decomposition, namely, the polyatomic decomposition:\n\\[\nf=P_pf+(I-P_p)f\\quad (0<\\theta\\leq 1),\n\\]\nand the monotomic-like decomposition:\n\\[\nf=P_mf+(I-P_m)f,\\quad (\\theta=0).\n\\]\nTherefore, we need to study two different sets of micro-macro equations accordingly, in order to fill up the degeneracy and to derive the full coercivity.\n\n\n\n\n\\subsection{Main result}\nWe define the high-order energy functional $\\mathcal{E}(f(t))$ :\n\\begin{eqnarray*}\n\\mathcal{E}\\big(f(t)\\big\n=\\frac{1}{2}\\sum_{|\\alpha|+|\\beta|\\leq N}\\|\\partial^{\\alpha}_{\\beta}f(t)\\|^2_{L^2_{x,v,I}}+\\sum_{|\\alpha|+|\\beta|\\leq N}\\int^t_0\\|\\partial^{\\alpha}_{\\beta}f(s)\\|^2_{L^2_{x,v,I}}ds.\n\\end{eqnarray*}\n\\begin{theorem}\\label{main theorem}\nLet $-1\/2<\\nu<1$, $0\\leq\\theta\\leq1$ and $N\\geq 4$. Suppose that $F_0=m+\\sqrt{m}f_0\\geq 0$ has the same mass, momentum and energy with $m$:\n\\begin{eqnarray}\\label{ConservationLawsf0}\n\\begin{split}\n\\int_{\\mathbb{T}^3_x\\times\\mathbb{R}^3_v\\times\\mathbb{R}^+_I}f_0\\sqrt{m} ~dxdvdI&=0,\\cr\n\\int_{\\mathbb{T}^3_x\\times\\mathbb{R}^3_v\\times\\mathbb{R}^+_I}f_0v\\sqrt{m} ~dxdvdI&=0,\\cr\n\\int_{\\mathbb{T}^3_x\\times\\mathbb{R}^3_v\\times\\mathbb{R}^+_I}f_0\\left\\{\\frac{1}{2}|v|^2+I^{2\/\\delta}\\right\\}\\sqrt{m} ~dxdvdI&=0.\n\\end{split}\n\\end{eqnarray}\n\n Then there exist $\\varepsilon>0$ and $C=C(f_0,N,\\nu,\\theta,\\delta)>0$,\n such that if $\\mathcal{E}(0)<\\varepsilon$, then there exists a unique global in time solution $f$ for (\\ref{LBGK}) satisfying\n \\begin{enumerate}\n \\item The distribution function is non-negative for all $t\\geq 0$:\n \\[\n F=m+\\sqrt{m}\\,f\\geq 0,\n \\]\n and satisfies the conservation laws (\\ref{ConservationLawsf}).\n \\item The high-order energy functional is uniformly bounded:\n \\[\n \\mathcal{E}(t)\\leq C\\mathcal{E}(0).\n \\]\n\\item The initial perturbation decays exponentially fast:\n\\[\n\\sum_{|\\alpha|+|\\beta|\\leq N}\\|\\partial^{\\alpha}_{\\beta}f(t)\\|^2_{L^2_{x,v,I}}\\leq C e^{-Ct}.\n\\]\n\\end{enumerate}\n\\end{theorem}\n\n\nA brief review on the related literature is in order. We start with the original monatomic BGK model.\nThe first mathematical study of the BGK model was made in \\cite{Perthame} where Perthame established the existence of weak solutions under the assumption of finite mass, energy and entropy.\nPerthame and Pulvirenti then studied the existence of unique mild solutions\nin a weighted $L^{\\infty}$ space in \\cite{PP}. These results were extended, for example, to\nCauchy problem for $L^p$ data \\cite{WZ}, plasma \\cite{Zhang} or gases under the influence of external forces \\cite{ZH}.\nUkai studied the stationary problem in a bounded interval with a fixed boundary condition in \\cite{Ukai-BGK}.\nFor the application of BGK type models to various macroscopic limits, see \\cite{B,DMOS,LT,MMM,Mellet,SR1,SR2}.\nThe existence of classical solutions and their asymptotic behavior were studied in \\cite{Bello,Chan,Yun}.\nSome error analysis for numerical schemes for BGK model can be found in \\cite{Issautier,RSY}.\n\nRecently, the interest on the ES-BGK model, which is a generalized version of the monatomic BGK model designed to reproduce the physical Prandtl number, revived after the $H$-theorem was verified for this model in \\cite{ALPP}. (See also \\cite{Brull2,Yun4}). For the existence results of this model in various situations, see \\cite{DWY, PY1,Yun2,Yun3}.\n\nThe study of the ellipsoidal BGK model for polyatomic particles started in \\cite{ALPP} is in its initial stage. The $H$-theorem was shown to hold in \\cite{ALPP,Brull3}. Entropy-entropy production estimate for this model was established in \\cite{PY2}, where the dichotomy in the entropy dissipation mechanism mentioned above, was first observed. The extension of \\cite{PP} arguments to the polyatomic case was made in \\cite{PY}. In the near-equilibrium regime, no existence result is available so far.\n\nWe mention that there has been an alternative approach besides the internal energy formulation to construct BGK type model for polyatomic molecules< where the polyatomic gas is treated as a mixture of monatomic gases endowed with discrete levels of internal energy \\cite{GS,GS2}.\n\nWe omit the reference review on the numerical results on BGK type models (monatomic or polyatomic), since they are huge. Interested readers may refer to \\cite{ABLP,ALPP,Cai,GT,Issautier,KAG,PC,PPuppo,RS,RSY} and references therein. For general review\non the mathematical and physical theory of kinetic equations, see \\cite{C,CIP,GL,Krem,Sone,Sone2,Stru,UT,V}.\n\\newline\n\nThe followings are the notations and conventions kept throughout this paper:\n\\begin{itemize}\n\\item All the constants, usually denoted by $C$ will be defined generically.\n\\item For $\\kappa\\in\\mathbb{R}^3$, $\\kappa^{\\top}$ denotes its transpose.\n\\item For symmetric $n\\times n$ matrices $A$ and $B$, $A\\leq B$ means that $B-A$ satisfies $k^{\\top}\\big\\{B-A\\big\\}k\\geq0$ for all $k\\in \\mathbb{R}^n$.\n\n\\item When there's no risk of confusion, we use $\\mathcal{E}(t)$ instead of $\\mathcal{E}\\big(f(t)\\big)$ for simplicity. The latter notation will be employed when the dependency needs to be clarified.\n\\item We slightly abuse the notation to define the summation on the index set $i0$ such that\n\\begin{eqnarray*}\n&&(1)~|\\rho_{\\eta}(x,t)-1|\\leq C\\sqrt{\\mathcal{E}(t)},\\cr\n&&(2)~|U^i_{\\eta }(x,t)|\\leq C\\sqrt{\\mathcal{E}(t)}, \\hspace{1.81cm}(1\\leq i\\leq 3)\\cr\n&&(3)~|\\mathcal{T}_{\\nu,\\theta\\eta}^{ii}(x,t)-1|\\leq C\\sqrt{\\mathcal{E}(t)},\\hspace{0.89cm}(1\\leq i\\leq 3)\\cr\n&&(4)~|\\mathcal{T}^{ij}_{\\nu,\\theta \\eta}(x,t)|\\leq C\n\\sqrt{\\mathcal{E}(t)},\\hspace{1.49cm}(1\\leq i0$.\n\\end{lemma}\n\\begin{proof}\n(1) A straightforward computation gives\n\\begin{align}\\label{deter}\n\\begin{split}\n\\det(\\mathcal{T}_{\\nu,\\theta\\eta})&=\\mathcal{T}^{11}_{\\nu,\\theta\\eta}\\mathcal{T}^{22}_{\\nu,\\theta\\eta}\\mathcal{T}^{33}_{\\nu,\\theta\\eta}\n+2\\mathcal{T}^{12}_{\\nu,\\theta\\eta}\\mathcal{T}^{23}_{\\nu,\\theta\\eta}\\mathcal{T}^{31}_{\\nu,\\theta\\eta}\\cr\n&-\\big\\{\\mathcal{T}^{23}_{\\nu,\\theta\\eta}\\big\\}^2\\mathcal{T}^{11}_{\\nu,\\theta\\eta}\n-\\big\\{\\mathcal{T}^{31}_{\\nu,\\theta\\eta}\\big\\}^2\\mathcal{T}^{22}_{\\nu,\\theta\\eta}\n-\\big\\{\\mathcal{T}^{12}_{\\nu,\\theta\\eta}\\big\\}\\mathcal{T}^{33}_{\\nu,\\theta\\eta}.\n\\end{split}\n\\end{align}\nTherefore, $\\partial^{\\alpha}\\det\\mathcal{T}_{\\nu,\\theta\\eta}$ takes the following form:\n\\begin{eqnarray*}\n\\partial^{\\alpha}\\det(\\mathcal{T}_{\\nu,\\theta\\eta})=\\sum_{\\alpha=\\alpha_1+\\alpha_2+\\alpha_3}\nC_{ij\\ell kmn}\n\\partial^{\\alpha_1}\\mathcal{T}_{\\nu,\\theta\\eta}^{ij}\\partial^{\\alpha_2}\\mathcal{T}^{\\ell k}_{\\nu,\\theta \\eta}\\partial^{\\alpha_3 }\\mathcal{T}^{mn}_{\\nu,\\theta\\eta}.\n\\end{eqnarray*}\nNow, we recall from Lemma \\ref{ULestimatesofRhoUT} and Lemma \\ref{ULestimatesofRhoUT1} that\n\\begin{align}\\label{44}\n\\mathcal{T}^{ii}_{\\nu,\\theta\\eta }=1+o\\big(\\sqrt{\\mathcal{E}(t)}\\,\\big)~(i=1,2,3),\n\\quad\\mathcal{T}^{ij}_{\\nu,\\theta\\eta }=o~\\big(\\sqrt{\\mathcal{E}(t)}\\,\\big) ~(i\\neq j),\n\\end{align}\nand\n\\begin{align}\\label{45}\n\\big|\\partial^{\\alpha}\\mathcal{T}^{ij}_{\\nu,\\theta\\eta }\\big|\\leq C_{\\alpha}\\sqrt{\\mathcal{E}(t)},\n\\end{align}\nto deduce\n\\begin{align*}\n|\\partial^{\\alpha}\\det\\mathcal{T}_{\\nu,\\theta\\eta}|\n&\\leq C_{\\alpha}\\sqrt{\\mathcal{E}(t)}.\n\\end{align*}\n(2) Inserting (\\ref{44}) and (\\ref{45}) into (\\ref{deter}), we get\n\\begin{align*}\n\\det\\mathcal{T}_{\\nu,\\theta\\eta}&=\\big\\{1+o(\\mathcal{E}(t))\\big\\}^3\n-2\\left\\{o\\big(\\sqrt{\\mathcal{E}(t)}\\,\\big)\\right\\}^3-3\\left\\{o\\big(\\sqrt{\\mathcal{E}(t)}\\,\\big)\\right\\}^2\n\\left\\{1+o\\big(\\sqrt{\\mathcal{E}(t)}\\,\\big)\\right\\}\\cr\n&\\geq 1-C_{\\alpha}\\sqrt{\\mathcal{E}(t)},\n\\end{align*}\nfor sufficiently small $\\mathcal{E}(t)$.\n\\end{proof}\n\n\\begin{lemma}\\label{UniformEst}\nLet $0\\leq \\theta\\leq 1$ and $-1\/2<\\nu<1$. Suppose $\\mathcal{E}(t)$ is sufficiently small. Then, there exist positive constants $C_1, C_2$ such that\n\\begin{align*}\n&(1)~X^{\\top}\\{\\mathcal{T}^{-1}_{\\nu,\\theta\\eta}\\}Y\\leq \\{1-C_1\\mathcal{E}(t)\\}^{-1}\\|X\\|\\|Y\\|,\\cr\n&(2)~X^{\\top}\\{\\mathcal{T}^{-1}_{\\nu,\\theta\\eta}\\}X\\geq \\{1+C_2\\mathcal{E}(t)\\}^{-1}\\|X\\|^2,\n\\end{align*}\nfor any $X$, $Y$ in $\\mathbb{R}^3$.\n\\end{lemma}\n\\begin{proof}\nWe start with proving the following claim:\n\\newline{\\bf $\\bullet$ Claim:} For sufficiently small $\\mathcal{E}(t)$, we have\n\\begin{align}\\label{following claim}\n\\left\\{1-C_1\\sqrt{\\mathcal{E}(t)}\\right\\}Id\\leq \\mathcal{T}_{\\nu,\\theta\\eta}\\leq \\left\\{1+C_2\\sqrt{\\mathcal{E}(t)}\\right\\}Id.\n\\end{align}\n{\\bf Proof of the claim:} For $\\kappa\\in\\mathbb{R}^3$, we have\n\\begin{align*}\n\\kappa^{\\top} \\mathcal{T}_{\\nu,\\theta\\eta}\\kappa=\\sum_{i=1,2,3}\\mathcal{T}_{\\nu,\\theta\\eta}^{ii}\\kappa^2_i+\\sum_{1\\leq i,j\\leq3}\n\\mathcal{T}_{\\nu,\\theta\\eta}^{ij}\\kappa_i\\kappa_j.\n\\end{align*}\nIn view of Lemma \\ref{ULestimatesofRhoUT} (3), (4), this gives\n\\begin{align*}\n\\kappa^{\\top} \\mathcal{T}_{\\nu,\\theta\\eta}\\kappa&=\\sum_{i=1,2,3}\\big\\{1+C\\sqrt{\\mathcal{E}(t)}\\big\\}\\kappa^2_i+C\\sum_{1\\leq i,j\\leq3}\n\\sqrt{\\mathcal{E}(t)}\\kappa_i\\kappa_j\\cr\n&=\\sum_{i=1,2,3}\\kappa^2_i+C\\sqrt{\\mathcal{E}(t)}\\Big\\{\\sum_{1\\leq i\\leq 3}\\kappa^2_i+\\sum_{1\\leq i,j\\leq3}\\kappa_i\\kappa_j\\Big\\}\\cr\n&\\leq \\left\\{1+C_1\\sqrt{\\mathcal{E}(t)}\\right\\}|\\kappa|^2.\n\\end{align*}\nLikewise,\n\\begin{align*}\n\\kappa^{\\top} \\mathcal{T}_{\\nu,\\theta\\eta}\\kappa\\geq \\left\\{1-C_2\\sqrt{\\mathcal{E}(t)}\\right\\}|\\kappa|^2.\n\\end{align*}\nThis completes the proof of the claim.\\newline\n\n\\noindent(1) Let $\\{\\lambda_i\\}$ denote the eigenvalues of $\\mathcal{T}_{\\nu,\\theta\\eta}$ so that we can write\n\\begin{align*}\n\\mathcal{T}_{\\nu,\\theta\\eta}=P^{\\top}diag\\{\\lambda_1,\\cdots,\\lambda_{11}\\}P,\n\\end{align*}\nfor some orthogonal matrix $P$. Here $diag\\{a,b,\\cdots\\}$ denotes the diagonal matrix whose diagonal entries are $a,b,\\cdots$.\nTherefore, since the above claim implies\n\\[\n1-C_1\\mathcal{E}(t)\\leq\\lambda_i\\leq 1+C_2\\mathcal{E}(t) \\qquad(1\\leq i\\leq 11)\n\\]\nfor sufficiently small $\\mathcal{E}(t)$, we have\n\\begin{align*}\nX^{\\top}\\mathcal{T}^{-1}_{\\nu,\\theta \\eta}Y&=X^{\\top}\\left[P^{\\top}diag\\{\\lambda^{-1}_1,\\cdots,\\lambda^{-1}_{11}\\}P\\right]Y\\cr\n&=\\{PX\\}^{\\top}diag\\{\\lambda^{-1}_1,\\cdots,\\lambda^{-1}_{11}\\}\\{PY\\}\\cr\n&= \\sum_i\\lambda^{-1}_i\\{PX\\}_i\\{PY\\}_i\\cr\n&\\leq \\max\\{\\lambda^{-1}_i\\}\\|PX\\|\\|PY\\|\\cr\n&= \\max\\{\\lambda^{-1}_i\\}\\|X\\|\\|Y\\|.\\cr\n&\\leq \\{1-C_1\\mathcal{E}(t)\\}^{-1}\\|X\\|\\|Y\\|.\n\\end{align*}\n\n\\noindent (2) Lower bound can be computed in a similar way as follows:\n\\begin{align*}\nX^{\\top}\\mathcal{T}^{-1}_{\\nu,\\theta \\eta}X&=X^{\\top}P^{\\top}diag\\{\\lambda^{-1}_1,\\cdots,\\lambda^{-1}_{11}\\}PX\\cr\n&= \\sum_i\\lambda^{-1}_i\\left|\\{PX\\}_i\\right|^2\\cr\n&\\geq \\min\\{\\lambda^{-1}_i\\}\\|PX\\|^2\\cr\n&= \\min\\{\\lambda^{-1}_i\\}\\|X\\|^2\\cr\n&\\geq \\{1+C_2\\mathcal{E}(t)\\}^{-1}\\|X\\|^2.\n\\end{align*}\n\\end{proof}\nIn the next lemma, we prove an estimate for derivatives of $\\mathcal{T}_{\\nu,\\theta\\eta}$.\n\\begin{lemma}\\label{UniformEst2}\nLet $0\\leq \\theta\\leq 1$ and $-1\/2<\\nu<1$. Assume $\\mathcal{E}(t)$ is sufficiently small. Then we have\n\\begin{eqnarray*}\n~\\|\\partial^{\\alpha}\\big(\\mathcal{T}^{-1}_{\\nu,\\theta\\eta}\\big)\\|\\leq C_{\\alpha}\\sqrt{\\mathcal{E}(t)}.\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nApplying $\\partial\\left\\{\\mathcal{T}_{\\nu,\\theta,\\eta}^{-1}\\right\\}\n=-\\mathcal{T}_{\\nu,\\theta,\\eta}^{-1}\\left\\{\\partial\\mathcal{T}_{\\nu,\\theta,\\eta}\\right\\}\\mathcal{T}_{\\nu,\\theta,\\eta}^{-1}$ recursively, we can derive\n$\\partial^{\\alpha}\\left\\{\\mathcal{T}_{\\nu,\\theta,\\eta}^{-1}\\right\\}=\nP\\big(\\mathcal{T}_{\\nu,\\theta,\\eta}^{-1},\\partial \\mathcal{T}_{\\nu,\\theta,\\eta},\\cdots,\\partial^{\\alpha}\\mathcal{T}_{\\nu,\\theta,\\eta}\\big)$ for some polynomial $P$. Therefore, by Lemma \\ref{ULestimatesofRhoUT} (4), (5) and Lemma \\ref{ULestimatesofRhoUT1} (3), we get\nthe desired result.\n\\end{proof}\n\n\n\\section{Estimates on nonlinear perturbation and Local existence}\n\nIn this section, we establish the local in time existence of smooth solutions. For this, we first need to estimate the nonlinear part.\n\\begin{lemma}\\label{GammaEstimate} The nonlinear perturbation $\\Gamma_{\\nu,\\theta}(f)$ satisfies:\n\\begin{align*}\n&(1)~ \\left|~\\int \\partial^{\\alpha}_{\\beta}\\Gamma_{\\nu,\\theta}(f)gdvdI\\right|\\leq C\\!\\!\\!\\!\\!\\sum_{|\\alpha_1|+|\\alpha_2|\\leq|\\alpha|}\n\\|\\partial^{\\alpha_1}f\\|_{L^2_{v,I}}\\|\\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}\\cr\n&\\hspace{3.8cm}+ C\\!\\!\\!\\!\\!\\sum_{\\substack{|\\alpha_1|+|\\alpha_2|\\leq|\\alpha|\\\\|\\beta_1|\\leq|\\beta|}}\n\\|\\partial^{\\alpha_1} f\\|_{L^2_{v,I}}\\|\\partial^{\\alpha_2}_{\\beta_1} f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}\\cr\n&\\hspace{3.8cm}+C\\!\\!\\!\\!\\!\\sum_{|\\alpha_1|+|\\alpha_2|+|\\alpha_3|\\leq|\\alpha|}\n\\|\\partial^{\\alpha_1}f\\|_{L^2_{v,I}}\\|\\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\n\\|\\partial^{\\alpha_3}f\\|_{L^2_{v,I}}\\|h\\|_{L^2_{v,I}},\\cr\n&(2)~ \\left\\|\\int\\Gamma_{1,2}(f,g)hdvdI\\right\\|_{L^2_{x}}\\!\\!\\!+\\left\\|\\int\\Gamma_{1,2}(g,h)hdvdI\\right\\|_{L^2_{x}}\\leq C\\sup_{x,v,I}|\\nu_{v,I}h|\\sup_x\\|f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{x,v,I}},\\cr\n&\\hspace{0.6cm} \\left\\|\\int\\Gamma_3(f,g,h)rdvdI\\right\\|_{L^2_{x}}\\!\\!+\\left\\|\\int\\Gamma_3(g,f,h)rdvdI\\right\\|_{L^2_{x}}\\!\\!\n+\\left\\|\\int\\Gamma_3(g,h,f)rdvdI\\right\\|_{L^2_{x}}\\cr\n&\\hspace{1.43cm}\\leq C\\sup_{x,v,I}|\\nu_{v,I}r|\\sup_x\\|f\\|_{L^2_{v,I}}\\sup_x\\|g\\|_{L^2_{v,I}}\\|h\\|_{L^2_{x,v,I}},\n\\end{align*}\nwhere $\\nu_{v,I}=(1+|v|^2)(1+I)$.\n\\end{lemma}\n\\begin{proof}\nWe prove this lemma only for $\\Gamma_2$, Other terms can be treated similarly.\nWe first need to estimate $\\partial^{\\alpha}_{\\beta}Q^{\\mathcal{M}}_{ij}$. \\newline\n\\noindent{\\bf Claim:} $\\displaystyle\\big|\\partial^{\\alpha}_{\\beta}Q^{\\mathcal{M}}_{ij}\\big|\\leq\n C_{\\alpha,\\beta}\n\\exp\\left(-\\frac{|v|^2}{12}-\\frac{I^{2\/\\delta}}{6}\\right)$.\\newline\n\\noindent {\\bf Proof of the claim:} Note that there exist a homogeneous polynomial $P_{\\alpha,\\beta}$ and a monomial $M_{\\alpha,\\beta}$ such that\n\\begin{align*}\n\\left|\\partial^{\\alpha}_{\\beta}\\mathcal{M}_{\\nu,\\theta}\\left(\\rho_{\\eta},U_{\\eta},\\mathcal{T}_{\\nu,\\theta\\eta},T_{\\theta\\eta},\\right)\\right|\n&=\\frac{\\left|P_{\\alpha,\\beta}\\left(\\partial\\rho_{\\eta},\\partial U_{\\eta},\\partial (v-U_{\\eta}), \\partial\\mathcal{T}_{\\nu,\\theta\\eta}, \\partial T_{\\theta\\eta},\\partial I^{2\/\\delta}\\right)\\right|}{M_{\\alpha,\\beta}\\left(\\det(\\mathcal{T}_{\\nu,\\theta\\eta}), T_{\\theta\\eta}\\right)}\\cr\n&\\times\\exp\\left(-\\frac{1}{2}(v-U_{\\eta})^{\\top}\\mathcal{T}^{-1}_{\\nu,\\theta\\eta}(v-U_{\\eta})-\\frac{I^{2\/\\delta}}{T_{\\theta\\eta}}\\right).\n\\end{align*}\nHere we slightly abused the notation to let $\\partial$ denote any of $\\partial^{\\bar{\\alpha}}_{\\bar{\\beta}}$ such that $\\bar{\\alpha}\\leq|\\alpha|$ and $\\bar{\\beta}\\leq|\\beta|$.\nRecalling the upper and lower bound estimates on the macroscopic fields in Lemma \\ref{ULestimatesofRhoUT},\nLemma \\ref{ULestimatesofRhoUT1}, the determinant estimates in Lemma \\ref{Det_Estimate} and the estimates on the temperature tensor made in Lemma \\ref{UniformEst}, Lemma \\ref{UniformEst2}, we have\n\\begin{align*}\n\\frac{\\left|P_{\\alpha,\\beta}\\left(\\partial\\rho_{\\eta},\\partial U_{\\eta},\\partial (v-U_{\\eta}), \\partial\\mathcal{T}_{\\nu,\\theta\\eta}, \\partial T_{\\theta\\eta}, I^{2\\delta}\\right)\\right|}{M_{\\alpha,\\beta}\\left(\\det(\\mathcal{T}_{\\nu,\\theta\\eta}),T_{\\theta\\eta}\\right)}\n\\leq C_{\\alpha,\\beta}(1+|v|^2+I^{2\/\\delta})^{m(\\alpha)}.\n\\end{align*}\nfor some $C_{\\alpha,\\beta}, m(\\alpha)>0$.\nOn the other hand, Lemma \\ref{ULestimatesofRhoUT} (2), (5) and Lemma \\ref{UniformEst} (2) give the following lower bound:\n\\begin{align*}\n\\frac{1}{2}(v-U_{\\eta})^{\\top}\\mathcal{T}^{-1}_{\\nu,\\theta\\eta}(v-U_{\\eta})+\\frac{I^{2\/\\delta}}{T_{\\theta\\eta}}\n&\\geq \\bigg(\\frac{2}{3}+2\\varepsilon\\bigg)\\frac{|v-U_{\\eta}|^2}{2}+\\bigg(\\frac{2}{3}+\\varepsilon\\bigg)I^{2\/\\delta}\\cr\n&\\geq \\bigg(\\frac{1}{3}+\\varepsilon\\bigg)|v|^2+\\bigg(\\frac{2}{3}+\\varepsilon\\bigg)I^{2\/\\delta}+o\\big(\\mathcal{E}(t)\\big),\n\\end{align*}\nfor some small $\\varepsilon>0$. This gives\n\\begin{align}\\label{Q_Estimate}\n\\begin{split}\n\\left|\\partial^{\\alpha}_{\\beta}Q^{\\mathcal{M}}_{ij}\\right|\n&\\leq\nC_{\\alpha,\\beta}\\exp\\left(\\frac{1}{4}|v|^2+\\frac{I^{2\/\\delta}}{2}\\right)(1+|v|^2+I^{2\/\\delta})^{m(\\alpha)}\\cr\n&\\times\\exp\\left(-\\bigg(\\frac{1}{3}+\\varepsilon\\bigg)|v|^2\n-\\bigg(\\frac{2}{3}+\\varepsilon\\bigg)I^{2\/\\delta}+o\\big(\\mathcal{E}(t)\\big)\\right)\\cr\n&\\leq C_{\\alpha,\\beta}\\exp\\left(-\\frac{|v|^2}{12}-\\frac{I^{2\/\\delta}}{6}\\right).\n\\end{split}\n\\end{align}\nThis completes the proof of the claim. We now return to the proof of the lemma.\nIn view of (\\ref{Q_Estimate}), we denote throughout this proof\n\\[\n\\widetilde{m}(v,I)=\\exp\\left(-\\frac{|v|^2}{12}-\\frac{I^{2\/\\delta}}{6}\\right)\n\\]\nfor simplicity.\\newline\n\\noindent(1) From (\\ref{Q_Estimate}) and H\\\"{o}lder inequality, we have\n\\begin{align*}\n&\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}|\\partial^{\\alpha}_{\\beta}\\Gamma_2(f)g|dvdI\\cr\n&\\qquad\\leq \\sum_{\\substack{|\\alpha_1|+|\\alpha_2|+|\\alpha_3|\\\\=|\\alpha|}}\n\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}\\big|\\partial^{\\alpha_1}_{\\beta_1}Q^{\\mathcal{M}}_{ij}\n\\langle \\partial^{\\alpha_2}f,e_i\\rangle_{L^2_{v,I}}\\langle \\partial^{\\alpha_2}f,e_j\\rangle_{L^2_{v,I}}\\,g\\big|dvdI\\cr\n&\\qquad\\leq C\\sum_{\\substack{|\\alpha_1|+|\\alpha_2|+|\\alpha_3|\\\\=|\\alpha|}} \\bigg(\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}\\big|\\partial^{\\alpha_1}_{\\beta_1}Q^{\\mathcal{M}}_{ij}\\,\\,g\\big|dvdI\\bigg)\\,\n\\| \\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\n\\|\\partial^{\\alpha_3}f\\|_{L^2_{v,I}}\\cr\n&\\qquad\\leq C_{\\alpha,\\beta}\\sum_{\\substack{|\\alpha_1|+|\\alpha_2|+|\\alpha_3|\\\\=|\\alpha|}}\n\\bigg(\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}\\widetilde{m}\\,|g|\\,dvdI\\bigg)\\,\n\\| \\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\n\\|\\partial^{\\alpha_3}f\\|_{L^2_{v,I}}\\cr\n&\\qquad\\leq C_{\\alpha,\\beta}\\sum_{\\substack{|\\alpha_1|+|\\alpha_2|+|\\alpha_3|\\\\=|\\alpha|}}\n\\| \\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\n\\|\\partial^{\\alpha_3}f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}\\cr\n&\\qquad\\leq C_{\\alpha,\\beta}\\sum_{|\\alpha_1|+|\\alpha_2|\\leq|\\alpha|}\n\\| \\partial^{\\alpha_1}f\\|_{L^2_{v,I}}\n\\|\\partial^{\\alpha_2}f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}.\n\\end{align*}\nHere we omitted $\\sum_{ij}$ for simplicity of presentation.\\newline\n\\noindent(2) Note that when $\\alpha=\\beta=0$,\nwe have much simpler estimate: $|Q^{\\mathcal{M}}_{ij}|\\leq C\\widetilde{m}$ directly from Lemma \\ref{ULestimatesofRhoUT}, Lemma\n\\ref{Det_Estimate} (2) and Lemma \\ref{UniformEst} (2). Therefore, applying H\\\"{o}ler inequality, we obtain\n\\begin{align*}\n\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}\\Gamma_2(f,g)hdvdI&\\leq C\\int_{\\mathbb{R}^3}\\|f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}\n\\bigg(\\int_{\\mathbb{R}^3\\times\\mathbb{R}_+}\\widetilde{m} |h|dvdI\\bigg)\\cr\n&\\leq C\\|f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}\\|h\\|_{L^2_{v,I}}\\cr\n&\\leq C\\sup_{v,I}|\\nu_{v,I}h||\\sup_x\\|f\\|_{L^2_{v,I}}\\|g\\|_{L^2_{v,I}}.\n\\end{align*}\nWe then take $L^2_x$ norm to get the result. The proof for other terms are similar.\n\\end{proof}\n\\subsection{Local existence:}\nNow, the local existence theorem can be proved by standard arguments (See, e.g \\cite{Guo VMB}).\n\\begin{theorem}\\label{localExistence}\nLet $0\\leq \\theta\\leq 1$ and $-1\/2<\\nu<1$. Let $F_0=m+\\sqrt{m }f_0\\geq 0$ and $f_0$ satisfies (\\ref{ConservationLawsf}).\nThen there exist $M_0>0$, $T_*>0$, such that if $ \\mathcal{E}(0)\\leq\\frac{M_0}{2}$, then there is a unique solution $f(t,x,v,I)$ to (\\ref{LBGK}) defined on $[0,T_*)$, such that\n\\begin{enumerate}\n\\item The high order energy $\\mathcal{E}\\big(f(t)\\big)$ is continuous in $[0,T*)$ and uniformly bounded:\n\\begin{eqnarray*}\n\\sup_{0\\leq t\\leq T_*}\\mathcal{E}\\big(f(t)\\big)\\leq M_0.\n\\end{eqnarray*}\n\\item The distribution function stays non-negative on $[0,T_*)$:\n\\begin{eqnarray*}\nF(t,x,v,I)=m +\\sqrt{m }f(t,x,v,I)\\geq 0.\n\\end{eqnarray*}\n\\item The conservation laws (\\ref{ConservationLawsf}) hold for all $[0,T_*)$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe consider the following scheme:\n\\begin{eqnarray}\\label{Localscheme}\n\\partial_tF^{n+1}+v\\cdot\\nabla_xF^{n+1}=\\frac{\\rho^n T_{\\delta}^n}{1-\\nu+\\theta\\nu}\\left\\{\\mathcal{M}_{\\nu,\\theta}(F^n)-F^{n+1}\\right\\},\n\\end{eqnarray}\nwith\n\\begin{eqnarray*}\n\\mathcal{M}_{\\nu,\\theta}(F^n)=\\frac{\\rho^n\\Lambda_{\\delta}}{\\sqrt{\\det(2\\pi\\mathcal{T}^n_{\\nu,\\theta})}\\{T^{n}_{\\theta}\\}^{\\delta\/2}}\\exp\\left(\\frac{1}{2}(v-U^n)\\left\\{\\mathcal{T}^{n}_{\\nu,\\theta }\\right\\}^{-1}(v-U^n)-\\frac{I^{2\/\\delta}}{T^n_{\\theta}}\\right),\n\\end{eqnarray*}\nwhere $\\rho^n$, $U^n$, $\\mathcal{T}^n_{\\nu,\\theta}$ and $T^n_{\\theta}$ denote the local density, bulk velocity and the temperature tensor associated with $F^n=m +\\sqrt{m }f^n$.\nMaking use of Lemma \\ref{GammaEstimate}, the local existence follows from a standard argument.\n(See \\cite{Guo VMB}).\nThe only difference from the usual proof is that the strict positiveness of $\\mathcal{T}^n_{\\nu,\\theta}$ and $T^n_{\\delta}$ should be secured in each step, so that $\\mathcal{M}_{\\nu,\\theta}(F^n)$ is well-defined. This is guaranteed by Lemma \\ref{ULestimatesofRhoUT} (5) and Lemma \\ref{UniformEst}.\n\\end{proof}\n\\section{Micro-macro system} In this section, we study the micro-macro system of (\\ref{LBGK}) to fill up the degeneracy in the dissipation estimates in Theorem \\ref{degenerate coercivity}. The dichotomy in the dissipation\nestimate observed in Theorem \\ref{degenerate coercivity} indicates that we should employ two different sets of micro-macro decomposition.\n\\subsection{Micro-macro system I {\\bf($0<\\theta\\leq 1)$}:}\nDefine\n\\begin{align*}\na(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}f\\sqrt{m }dvdI,\\cr\nb_i(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}fv_i\\sqrt{m }dvdI, \\quad(i=1,2,3)\\cr\nc(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}f\\left(\\frac{(|v|^2-3)+(2I^{2\/\\delta}-\\delta)}{\\sqrt{2(3+\\delta)}}\\right)\\sqrt{m }dvdI,\n\\end{align*}\nso that the polyatomic projection operator $P_p$ is written\n\\begin{eqnarray*}\nP_pf= a(x,t)\\sqrt{m }+\\sum_{i}b_i(x,t)v_i\\sqrt{m }+c(x,t)\\left(\\frac{(|v|^2-3)+(2I^{2\/\\delta}-\\delta)}{\\sqrt{2(3+\\delta)}}\\right)\\sqrt{m }.\n\\end{eqnarray*}\nSince $L_{\\nu,\\theta}\\{P_pf\\}=0$ for $0<\\theta\\leq1$ by Proposition \\ref{Kernel L},\nthe linearized polyatomic BGK model (\\ref{LBGK}) is decomposed into the macroscopic part and the microscopic parts as follows:\n\\begin{eqnarray*}\n\\{\\partial_t+v\\cdot\\nabla_x\\}\\{P_pf\\}=-\\{\\partial_t+v\\cdot\\nabla_x\\}\\{(I-P_p)f\\}+L_{\\nu,\\theta}\\{(I-P_p)f\\}+\\Gamma_{\\nu,\\theta}(f).\\\n\\end{eqnarray*}\nWe then expand the l.h.s and r.h.s with respect to the following basis $(1\\leq i,j\\leq 3)$:\n\\begin{eqnarray}\\label{basis}\n\\big\\{\n\\sqrt{m },\\,v_i\\sqrt{m }, \\,v_iv_j\\sqrt{m }, \\,v_i^2\\sqrt{m }, \\,v_i\\big\\{ |v|^2+(2I^{2\/\\delta}-\\delta)\\big\\}\\sqrt{m },\n\\,( 2I^{2\/\\delta}-\\delta )\\sqrt{m }\n\\big\\}.\n\\end{eqnarray}\nComparing the coefficients on both sides, we derive the following micro-macro system:\n\\begin{align}\\label{MicroMacro}\n\\begin{split}\n\\partial_t a-3A_{\\delta}\\partial_tc&=\\ell_{a}+h_{a},\\\\\n\\partial_tb_i+\\partial_{x_i} a-3A_{\\delta}\\partial_{x_i}c&=\\ell_{abi}+h_{abi},\\\\\n\\partial_{x_i}b_j+\\partial_{x_j}b_i&=\\ell_{ij}+h_{ij}\\quad (i\\neq j)\\\\\nA_{\\delta}\\partial_tc+\\partial_{x_i} b_i&=\\ell_{bci}+h_{bci},\\\\\nA_{\\delta}\\partial_{x_i}c&=\\ell_{ci}+h_{ci},\\\\\nA_{\\delta}\\partial_{t}c&=\\ell_{ct}+h_{ct},\n\\end{split}\n\\end{align}\nfor $i,j =1,2,3$. Here, $A_{\\delta}=1\/\\sqrt{2(3+\\delta)}$, and\n$\\ell_{a}$, $\\ell_{abci}$, $\\ell_{ij}$, $\\ell_{bci}$, $\\ell_{ci}$ and $\\ell_{ct}$ denote the coefficients of projection of\n$-\\{\\partial_t+v\\cdot\\nabla_x\\}\\{(I-P_m)f\\}+L_{\\nu,\\theta}\\{(I-P_m)f\\}$ onto the basis (\\ref{basis}),\nand $h_{a}$, $h_{abci}$, $h_{ij}$, $h_{bci}$, $h_{ci}$ and $h_{ct}$ are the projection of $\\Gamma_{\\nu,\\theta}(f)$ onto (\\ref{basis}).\nAdding the last two equations to the first and second line, we get\n\\begin{align}\\label{MicroMacro}\n\\partial_t a&=(\\ell_{a}+h_{ct})+3\\{\\ell_{ct}+h_{ct}\\},\\nonumber\\\\\n\\partial_tb_i+\\partial_{x_i}a&=\\big\\{\\ell_{abi}+h_{abi}\\big\\}+3\\{\\ell_{ci}+h_{ci}\\},\\nonumber\\\\\n\\partial_{x_i}b_j+\\partial_{x_j}b_i&=\\ell_{ij}+h_{ij}\\quad (i\\neq j)\\\\\nA_{\\delta}\\partial_{t}c+\\partial_{x_i} b_i&=\\ell_{bci}+h_{bci},\\nonumber\\\\\nA_{\\delta}\\partial_{x_i}c&=\\ell_{ci}+h_{ci},\\nonumber\\\\\nA_{\\delta}\\partial_t c&=\\ell_{ct}+h_{ct},\\nonumber.\n\\end{align}\nThe first 5 lines are, up to constant multiplication on the l.h.s and\nadditional slight complication in r.h.s, identical to the micro-macro system derived in \\cite{Guo whole,Guo VMB, Yun} and the last line is easy to estimate.\nHence, following the same line of argument, we arrive at\n\\begin{align}\\label{MicroMacroEstimatee}\n\\sum_{|\\alpha|\\leq N}\\left\\{\\|\\partial^{\\alpha}a\\|^2_{L^2_{x}}+\\|\\partial^{\\alpha}b\\|^2_{L^2_{x}}\n+C_{\\delta}\\|\\partial^{\\alpha}c\\|^2_{L^2_{x}}\\right\\}\n\\leq C \\sum_{|\\alpha|\\leq N-1}\\|\\partial^{\\alpha}(\\ell_{\\nu,\\theta}+h_{\\nu,\\theta})\\|^2_{L^2_{x}}\n\\end{align}\nNote that we have slightly abused the notation to denote\n$\\ell_{\\nu,\\theta}=(\\ell_{a}, \\ell_{abi}$, $\\ell_{ij}, \\ell_{bci}, \\ell_{ci},\\ell_{ct})$ and $h_{\\nu,\\theta}=(h_{a}, h_{abci}, h_{ij}$, $h_{bci}, h_{ci}, h_{ct})$.\nNow, $\\ell_{\\nu,\\theta}$ and $h_{\\nu,\\theta}$ can be controlled in a standard way as follows:\n\\begin{eqnarray*\n\\sum_{|\\alpha|\\leq N-1}\\|\\partial^{\\alpha}(\\ell_{\\nu,\\theta}+h_{\\nu,\\theta})\\|^2_{L^2_{x}}\n\\leq C\\sum_{|\\alpha|\\leq N}\\|(I-P_p)\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}+\nC\\sqrt{M_0}\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}.\n\\end{eqnarray*}\nCombining this with (\\ref{MicroMacroEstimatee}), we derive\n\\begin{align*}\n\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}P_pf\\|^2_{L^2_{x,v,I}}\n&\\leq \\sum_{|\\alpha|\\leq N}\\left\\{\\|\\partial^{\\alpha}a\\|^2_{L^2_{x,v,I}}+\\|\\partial^{\\alpha}b\\|^2_{L^2_{x,v,I}}+\\|\\partial^{\\alpha}c\\|^2_{L^2_{x,v,I}}\\right\\}\\cr\n&\\leq C\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}(I-P_p)f\\|^2_{L^2_{x,v,I}}\n+C\\sqrt{M}_0\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}},\n\\end{align*}\nwhich gives\n\\begin{eqnarray}\\label{microMacroEstimatee}\n\\sum_{|\\alpha|\\leq N}\\|P_p\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\n\\leq C\\sum_{|\\alpha|\\leq N}\\|(I-P_p)\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}.\n\\end{eqnarray}\nThen, we conclude from Proposition \\ref{degenerate coercivity} and (\\ref{microMacroEstimatee}) that\nthere exists $C_{\\nu,\\theta}>0$ such that\n\\begin{eqnarray}\\label{Coercivity}\n\\sum_{|\\alpha|\\leq N}\\langle L_{\\nu,\\theta}\\partial^{\\alpha}f,\\partial^{\\alpha}f\\rangle_{L^2_{x,v,I}}\\leq -C_{\\nu,\\theta}\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}f(t)\\|^2_{L^2_{x,v,I}}\n\\end{eqnarray}\nfor sufficiently small $\\mathcal{E}(t)$.\n\n\\subsection{Micro-macro system II~{\\bf $(\\theta=0)$:}}\nRecalling Proposition \\ref{Kernel L}, we see that in this case, (\\ref{LBGK}) should be decomposed with respect to the monatomic-like projection $P_mf$.\nIn view of this observation, we define\n\\begin{align*}\na(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}f\\sqrt{m }dvdI,\\cr\nb_i(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}fv_i\\sqrt{m }dvdI, \\quad(i=1,2,3)\\cr\nc(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}f\\left(\\frac{|v|^2-3}{\\sqrt{6}}\\right)\\sqrt{m }dvdI,\\cr\nd(x,t)&=\\int_{\\mathbb{R}^3_v\\times \\mathbb{R}^+_I}f\\left(\\frac{2I^{2\/\\delta}-\\delta}{\\sqrt{2\\delta}}\\right)\\sqrt{m }dvdI,\n\\end{align*}\nto write\n\\begin{align*}\nP_mf&= a\\sqrt{m }+\\sum_{i}b_iv_i\\sqrt{m }\n+c\\left(\\frac{|v|^2-3}{\\sqrt{6}}\\right)\\sqrt{m }\n+d\\left(\\frac{2I^{2\/\\delta}-\\delta}{\\sqrt{2\\delta}}\\right)\\sqrt{m }.\n\\end{align*}\nWe recall from Proposition \\ref{Kernel L} that $L_{\\nu,0}(P_mf)=0$, and divide (\\ref{LBGK}) into the macroscopic part and the microscopic part as follows:\n\\begin{eqnarray*}\n\\{\\partial_t+v\\cdot\\nabla_x\\}\\{P_mf\\}=-\\{\\partial_t+v\\cdot\\nabla_x\\}\\{(I-P_m)f\\}+L_{\\nu,0}\\{(I-P_m)f\\}+\\Gamma_{\\nu,0}(f).\n\\end{eqnarray*}\nComparing coefficients corresponding to the following basis:\n\\begin{eqnarray}\\label{basis2}\n\\big\\{\n\\sqrt{m},v_i\\sqrt{m}, v_iv_j\\sqrt{m}, v_i^2\\sqrt{m}, v_i|v|^2\\sqrt{m}, \\big(2I^{2\/\\delta}-\\delta\\big)\\sqrt{m},\n\\big(2I^{2\/\\delta}-\\delta\\big)v_i\\sqrt{m}\n\\big\\}.\n\\end{eqnarray}\nWe obtain $(i,j=1,2,3)$:\n\\begin{align}\\label{MicroMacro0}\n\\partial_t a-3\/\\sqrt{6}\\partial_tc&=\\ell_{a}+h_{a},\\nonumber\\\\\n\\partial_tb_i+\\partial_{x_i}a-3\/\\sqrt{6}\\partial_{x_i}c&=\\ell_{abci}+h_{abci},\\nonumber\\\\\n\\partial_{x_i}b_j+\\partial_{x_j}b_i&=\\ell_{ij}+h_{ij},\\\\\n\\partial_{x_i} b_i+1\/\\sqrt{6}\\partial_{t}c&=\\ell_{bci}+h_{bci},\\nonumber\\\\\n1\/\\sqrt{6}\\partial_{x_i}c&=\\ell_{ci}+h_{ci},\\nonumber\\\\\n\\partial_td&=\\ell_{dt}+h_{dt},\\nonumber\\\\\n\\partial_{x_i}d&=\\ell_{dxi}+h_{dxi},\\nonumber\n\\end{align}\nwhere $\\ell_{a}$, $\\ell_{abci}$, $\\ell_{ij}$, $\\ell_{bci}$, $\\ell_{ci}$, $\\ell_{dt}$ and $\\ell_{dxi}$ are obtained by taking the inner product of\n$-\\{\\partial_t+v\\cdot\\nabla_x\\}\\{(I-P_m)f\\}+L_{\\nu,0}\\{(I-P_m)f\\}$ with the basis in (\\ref{basis2}),\nand $h_{a}$, $h_{abci}$, $h_{ij}$, $h_{bci}$, $h_{ci}$, $h_{dt}$ and $h_{dxi}$ are the inner product of $\\Gamma_{\\nu,0}(f)$ with (\\ref{basis2}).\nSetting $\\tilde{a}=a-3\/\\sqrt{6}c$ and $\\tilde{c}=1\/\\sqrt{6}c$, we derive from (\\ref{MicroMacro0})\n\\begin{align}\\label{MicroMacro}\n\\partial_t \\tilde{a}&=\\ell_{a}+h_{a},\\nonumber\\\\\n\\partial_tb_i+\\partial_{x_i}\\tilde{a}&=\\ell_{abi}+h_{abi},\\nonumber\\\\\n\\partial_{x_i}b_j+\\partial_{x_j}b_i&=\\ell_{ij}+h_{ij},\\\\\n\\partial_{x_i} b_i+\\partial_{t}\\tilde{c}&=\\ell_{bc1}+h_{bc1},\\nonumber\\\\\n\\partial_{x_i}\\tilde{c}&=\\ell_{ci}+h_{ci},\\nonumber\\\\\n\\partial_td&=\\ell_{dt}+h_{dt},\\nonumber\\\\\n\\partial_{x_i}d&=\\ell_{dxi}+h_{dxi}.\\nonumber\n\\end{align}\nExcept for the last two line, which are decoupled from the other equations, and therefore, estimated easily, this is identical to the micro-macro system\nfor the usual Boltzmann equation or BGK model. Therefore, we can derive\n\\begin{eqnarray*}\n\\sum_{|\\alpha|\\leq N}\\left\\{\\|\\partial^{\\alpha}\\widetilde{a}\\|^2_{L^2_{x}}+\\|\\partial^{\\alpha}b\\|^2_{L^2_{x}}\n+\\|\\partial^{\\alpha}\\widetilde{c}\\|^2_{L^2_{x}}\n+\\|\\partial^{\\alpha}d\\|^2_{L^2_{x}}\\right\\}\n\\leq C \\sum_{|\\alpha|\\leq N-1}\\|\\partial^{\\alpha}(\\ell_{\\nu,0}+h_{\\nu,0})\\|^2_{L^2_{x}}.\n\\end{eqnarray*}\nHere we used the simplified notation again: $\\ell_{\\nu,0}=(\\ell_{a}, \\ell_{abci}, \\ell_{ij}, \\ell_{bci}, \\ell_{ci},\\ell_{dt},\\ell_{dxi})$ and $h_{\\nu,0}=(h_{a}, h_{abci}, h_{ij}$, $h_{bci}, h_{ci},h_{dt},h_{dxi})$.\nWe now take $\\frac{3\\sqrt{6}}{10}<\\varepsilon^2<\\frac{2}{\\sqrt{6}}$ and set\n\\begin{eqnarray*}\nC_{\\varepsilon}=\\min\\left\\{(1-\\sqrt{6}\\varepsilon^2), \\,3\/5-\\sqrt{6}\/(2\\varepsilon^2)\\right\\}>0\n\\end{eqnarray*}\nto obtain\n\\begin{eqnarray*}\nC_{\\varepsilon}\\sum_{|\\alpha|\\leq N}\\left\\{\\|\\partial^{\\alpha}a\\|^2_{L^2_{x}}+\\|\\partial^{\\alpha}b\\|^2_{L^2_{x}}+\\|\\partial^{\\alpha}c\\|^2_{L^2_{x}}\n+\\|\\partial^{\\alpha}d\\|^2_{L^2_x}\\right\\}\n\\leq C \\sum_{|\\alpha|\\leq N-1}\\|\\partial^{\\alpha}(\\ell_{\\nu,0}+h_{\\nu,0})\\|^2_{L^2_{x}}.\n\\end{eqnarray*}\nThen, by a similar argument as in the previous case, we can control $P_mf$ by $(I-P_m)f$:\n\\begin{eqnarray}\\label{microMacroEstimate}\n\\sum_{|\\alpha|\\leq N}\\|P_m\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\n\\leq C\\sum_{|\\alpha|\\leq N}\\|(I-P_m)\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}.\n\\end{eqnarray}\nwhich, combined with the dissipation estimate in Theorem \\ref{degenerate coercivity} (2), implies\n\\begin{eqnarray}\\label{Coercivity}\n\\sum_{|\\alpha|\\leq N}\\langle L_{\\nu,0}\\partial^{\\alpha}f,\\partial^{\\alpha}f\\rangle_{L^2_{x,v,I}}\\leq -C_{\\nu}\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}f(t)\\|^2_{L^2_{x,v,I}}.\n\\end{eqnarray}\n\n\n\\section{Proof of Theorem \\ref{main theorem}}\nWe have derived all the necessary estimates to close the energy estimate.\nLet $f$ be the smooth local in time solution obtained in Theorem \\ref{localExistence}.\nTake derivatives on $x$, $t$ and $I$ of (\\ref{LBGK}):\n\\begin{eqnarray*}\n\\partial_t \\partial^{\\alpha}f+v\\cdot\\nabla_x \\partial^{\\alpha}f=L_{\\nu,\\theta}\\partial^{\\alpha}f+\\partial^{\\alpha}\\Gamma_{\\nu,\\theta}(f),\n\\end{eqnarray*}\nand take inner product with $\\partial^{\\alpha}f$ to get\n\\begin{eqnarray*}\n\\frac{1}{2}\\frac{d}{dt}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\\leq \\langle L_{\\nu,\\theta}\\partial^{\\alpha}f,\\partial^{\\alpha}f\\rangle_{L^2_{x,v,I}}+\\langle\\partial^{\\alpha}\\Gamma_{\\nu,\\theta}(f),\\partial^{\\alpha}f\\rangle_{L^2_{x,v,I}}.\n\\end{eqnarray*}\nMaking use of the coercivity estimate in the previous section yields\n\\begin{eqnarray*}\nE^{\\alpha}_0:\\quad\\frac{1}{2}\\frac{d}{dt}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\\!\\!+C\\sum_{|\\alpha|\\leq N}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\\leq\nC\\sqrt{\\mathcal{E}(t)}\\mathcal{D}(t).\n\\end{eqnarray*}\nFor the energy estimate involving velocity derivatives, we apply $\\partial^{\\alpha}_{\\beta}$ to (\\ref{LBGK})\n\\begin{eqnarray*}\n\\big\\{\\partial_t+v\\cdot\\nabla_x +a_{\\nu,\\theta}\\big\\}\\partial^{\\alpha}_{\\beta}f\n=\\sum_{|\\beta_1|\\neq0}\\partial_{\\beta_1}v\\cdot\\nabla_x\\partial^{\\alpha}_{\\beta-\\beta_1}f\n+\\partial_{\\beta}P_{\\nu,\\theta}\\partial^{\\alpha}f+\\partial^{\\alpha}_{\\beta}\\Gamma_{\\nu,\\theta}(f,f),\n\\end{eqnarray*}\nwhere $a_{\\nu,\\theta}=1\/(1-\\nu+\\nu\\theta)$. Then,\ntake inner product with $\\partial^{\\alpha}_{\\beta}f$ and use H\\\"{o}lder inequality with Lemma \\ref{GammaEstimate} to derive\n\\begin{eqnarray*}\n\\begin{split}\nE^{\\alpha}_{\\beta}:\\quad\\frac{1}{2}\\frac{d}{dt}\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}+a_{\\nu,\\theta}\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\n&\\leq C\\sum_{i}\\|\\partial^{\\alpha+e_i}_{\\beta-e_i}f\\|_{L^2_{x,v,I}}\\|\\partial^{\\alpha}_{\\beta}f\\|_{L^2_{x,v,I}}\\cr\n&+C\\|\\partial^{\\alpha}f\\|_{L^2_{x,v,I}}\\|\\partial^{\\alpha}_{\\beta}f\\|_{L^2_{x,v,I}}+C\\sqrt{\\mathcal{E}(t)}\\mathcal{D}(t),\n\\end{split}\n\\end{eqnarray*}\nwhere $e_i (i=1,2,3)$ is the standard basis of $\\mathbb{R}^3_x$. By Young's inequality, we can split the first two terms in the r.h.s as\n\\begin{align*}\n\\|\\partial^{\\alpha+e_i}_{\\beta-e_i}f\\|_{L^2_{x,v,I}}\\|\\partial^{\\alpha}_{\\beta}f\\|_{L^2_{x,v,I}}&\\leq\nC_{\\varepsilon}\\|\\partial^{\\alpha+e_i}_{\\beta-e_i}f\\|^2_{L^2_{x,v,I}}+\\varepsilon\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\\cr\n\\|\\partial^{\\alpha}f\\|_{L^2_{x,v,I}}\\|\\partial^{\\alpha}_{\\beta}f\\|_{L^2_{x,v,I}}&\\leq\nC_{\\varepsilon}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}+\\varepsilon\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\n\\end{align*}\nwhose $\\varepsilon$ terms can be absorbed in the production term in the l.h.s to get\n\\begin{align*}\nE^{\\alpha}_{\\beta}:\\quad\\frac{1}{2}\\frac{d}{dt}\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}+\\frac{1}{2}a_{\\nu,\\theta}\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\n&\\leq C_{\\varepsilon}\\sum_{i}\\|\\partial^{\\alpha+e_i}_{\\beta-e_i}f\\|^2_{L^2_{x,v,I}}+C_{\\varepsilon}\\|\\partial^{\\alpha}f\\|^2_{L^2_{x,v,I}}\\cr\n&+ C\\sqrt{\\mathcal{E}(t)}\\mathcal{D}(t).\n\\end{align*}\nWe then observe that the r.h.s of $\\sum_{|\\beta|=m+1}E^{\\alpha}_{\\beta}$ can be absorbed into the production terms in the\nlower order estimate:\n$C_{m}\\sum_{|\\beta|\\leq m}E^{\\alpha}_{\\beta}$ for sufficiently large $C_m$. This observation enables one to find constants $C^1_m$, $C^2_m$ inductively such that\n\\begin{eqnarray*}\n\\sum_{\\substack{|\\alpha|+|\\beta|\\leq N,\\cr|\\beta|\\leq m}}\\left\\{C^1_m\\frac{d}{dt}\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\n+C^2_m\\|\\partial^{\\alpha}_{\\beta}f\\|^2_{L^2_{x,v,I}}\\right\\}\\leq C_N\\sqrt{\\mathcal{E}(t)}\\mathcal{D}(t).\n\\end{eqnarray*}\nNow, the standard continuity argument gives the global existence for (\\ref{LBGK}) \\cite{Guo VMB}.\nThis completes the proof.\n\\begin{align*}\n\\end{align*}\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n The chiral ring of a four dimensional $\\mathcal{N}=1$ theory plays a crucial role in understanding the dynamics of the theory. In particular, the chiral ring can be used to determine the structure of the moduli space of vacua and the phase structure \\cite{Cachazo:2002ry}. Moreover, the chiral ring structure seems to be still quite important even if the theory has \na unique vacua \\cite{Xie:2016hny,Buican:2016hnq}.\n\nHowever, little is known about the general structure of the chiral ring of $\\mathcal{N}=1$ theory. The purpose of this \npaper is to study the chiral ring of a superconformal field theory (SCFT). We ask the following question: when is a chiral ring ${\\cal R}$\nthe chiral ring of a SCFT? An obvious necessary condition is that the chiral ring has to be graded, since for a SCFT there is \nalways a $U(1)_R$ symmetry which acts non-trivially on all of the chiral operators. In particular, we really should start with a polarized chiral ring $({\\cal R}, \\zeta)$ and ask whether it is \na chiral ring of a SCFT with $U(1)_R$ symmetry $\\zeta$. On the other hand, It is also known that the existence of a grading is not sufficient.\n\nA second motivation for asking above question is the following: in many studies of supersymmetric field theory we start with an asymptotically free gauge theory ${\\cal T}$ and assume that \nit flows to a SCFT ${\\cal T}_0$ at a certain point of the moduli space (often the most singular point). We can compute the chiral ring ${\\cal R}$ of the theory ${\\cal T}$, and let us denote by ${\\cal R}_{0}$ the chiral ring of ${\\cal T}_0$. Many interesting quantities of the SCFT ${\\cal T}_0$ can be computed if ${\\cal R}= {\\cal R}_0$. For example, we can use $a$ maximization to determine the $U(1)_R$ symmetry \\cite{Intriligator:2003jj} of ${\\cal T}_0$. In general, however, ${\\cal R}_0$ can differ from ${\\cal R}$, for example:\n\\begin{itemize}\n\\item[(a)] It is believed that the chiral ring ${\\cal R}$ of $\\mathcal{N}=1$ $SU(N_c)$ SQCD with ${3\\over 2}N_c0$; b) $R_0$ is invariant with respect to symmetries of $R$ and $\\eta$;\n c): The Hilbert series of ${\\cal R}$ and ${\\cal R}_0$ are the same for the symmetries of ${\\cal R}$, this is the continuity condition on test configuration. \n Using above procedure, we can get infinite number of test chiral rings. \nThe criteria for determining whether a test chiral ring ${\\cal R}_0$ destabilizes a polarized ring $({\\cal R},\\zeta)$ is \n\\begin{definition}\nA test chiral ring ${\\cal R}_0$ destabilizes $({\\cal R},\\zeta)$ if ${\\cal R}_0$ gives \\textbf{no less} central charge $a$ with respect to the space of possible $U(1)_R$ symmetries $a \\zeta+s\\eta,~s\\geq 0$. \n\\label{def2}\n\\end{definition}\nIt is crucial that $s\\geq 0$ so we have the same test chiral ring using the symmetry generator $s\\eta$ on ${\\cal R}$. \nNow we state the definition of stable chiral ring:\n\\begin{definition}\nA polarized chiral ring $({\\cal R},\\zeta)$ is called stable if there is no destabilizing test chiral ring. \n\\end{definition}\nThis definition can be thought of as the \\textbf{generalized a maximization} procedure. For the original \\textbf{a maximization} procedure \\cite{Intriligator:2003jj}, we do not change the chiral ring, namely we \nonly use the symmetry generator of ${\\cal R}$ to generate the test chiral ring, and the flat limit ${\\cal R}_0$ is the same as ${\\cal R}$. The hidden assumption in this process is that the ring ${\\cal R}$ is already the ring of a SCFT, and we would like to \ndetermine the correct $U(1)_R$ symmetry. \n\nOnce we define the notion of stability of chiral ring, we would like to state the main conjecture of this paper: \n\\begin{conjecture}\nA polarized chiral ring $({\\cal R},\\zeta)$ is the chiral ring of a SCFT if and only if it is stable.\n\\end{conjecture}\nThis conjecture answers the question when a chiral ring can be that of a SCFT. \nWe would like to test the above conjecture for general class of $\\mathcal{N}=1$ theories. However, we face several difficulties. First, it is usually not easy to derive \nthe full chiral ring of a theory, and so examples are in short supply. Secondly, we do not know how to characterize the $U(1)_R$ like symmetry from the chiral ring itself. \nThirdly, it is not known how to determine the trial central charge $a(\\zeta)$ for a symmetry $\\zeta$ from the chiral ring itself. \nHowever, these problems are solved for a class of models arising from string theory. Precisely, it is possible to determine the exact chiral ring for $\\mathcal{N}=1$ theories derived from \n$N$ D3 branes probing a three dimensional singularity \\cite{Klebanov:1998hh}: the 3d singularity can be defined by an affine variety $X$ with coordinate ring ${\\cal H}_{X} := \\mathbb{C}[x_1,\\ldots, x_n]\/I$, which\ndetermines the chiral ring ${\\cal R}$ of field theory. We can characterize $U(1)_R$ like symmetries by requiring the top form $\\Omega$ \\footnote{The existence of such form puts restriction on the singularity type.} on $X$ having charge two. \nIn the large N limit, the central charge can be computed from the Hilbert series of $X$ \\cite{Bergman:2001qi,Martelli:2005tp,Martelli:2006yb}. \n\nWe would like to determine whether the chiral ring $({\\cal R},\\zeta)$ of above field theory model is stable or not. Assuming our conjecture relating stability of chiral ring and SCFT, \nthe stability of the chiral ring of these models has the following geometric consequence: \n If the chiral ring is stable, then according to AdS\/CFT dictionary \\cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}, in the large N limit the IR SCFT is dual to type IIB string \ntheory on $AdS_5\\times L_5$ \\cite{Klebanov:1998hh, Morrison:1998cs}, where $L_5$ is a five manifold, defined as \n the link of a 3d singularity $X$, which carries a Sasaki-Einstein (SE) metric. The $U(1)_R$ symmetry $\\zeta$ is identified with the Reeb vector field on $L_5$. \n In other words, the stability of the chiral ring is equivalent to the existence of Sasaki-Einstein metric on $L_5$, or equivalently \n the existence of a Ricci-flat conic metric on $X$. \n\n\nThe existence of Sasaki-Einstein metrics has been studied extensively recently in mathematics literature. Briefly, in the setting of Fano K\\\"ahler manifolds, the Yau-Tian-Donaldson conjecture predicted that the existence of K\\\"ahler-Einstein metrics with positive scalar curvature is equivalent to the algebro-geometric notion of K-stability \\cite{donaldson2002scalar} which is an improvement of the original conjecture \nof Yau \\cite{yau1993open}. \nThis conjecture was recently proved by Chen-Donaldson-Sun \\cite{chen2015kahler,chen2015kahler1,chen2015kahler2}. In our more general context, a notion of K-stability and it's implications for the existence \nof Sasaki-Einstein metrics was studied by the first author and Sz\\'ekelyhidi in \\cite{collins2012k,collins2015sasaki}. The notion of K-stability involves constructions of so called test configurations ${\\cal X}$ and the criteria for determining whether ${\\cal X}$ destabilizes $X$ is determined by the sign of the so-called Donaldson-Futaki invariant.\nOne of major point of this paper is to provide an interpretation of the Donaldson-Futaki invariant as a version of generalized $a$ maximization: \n\\begin{theorem}\nThe K-stability of the affine variety $X$ is equivalent to the stability of the chiral ring of the corresponding field theory. \n\\end{theorem}\n\nThe paper is organized as follows: section two reviews some basic facts about $\\mathcal{N}=1$ chiral ring; section three studies theory engineered using\nD3 brane probing certain three dimensional singularity $X$, and K-stability of $X$ is interpreted as the generalized $a$ maximization procedure introduced above; section four discusses\nsome physical consequences from K stability; finally, a conclusion is given in section five. \n\n\n\n \n\n\n\n\\section{Generality of chiral ring}\nConsider a four dimensional $\\mathcal{N}=1$ supersymmetric field theories. A chiral operator ${\\cal O}_i$ is defined as an operator annihilated by supercharges $\\bar{Q}_{\\dot{\\alpha}}$, and \nis defined modulo cohomology of $\\bar{Q}_{\\dot{\\alpha}}$: ${\\cal O}_i\\sim {\\cal O}_i+[\\bar{Q}_{\\dot{\\alpha}}, \\chi]$ \\cite{Cachazo:2002ry}. Chiral operators have some interesting properties:\n\\begin{itemize}\n\\item The sum of two chiral operators is still a chiral operator, and the product of two chiral operators is still a chiral operator. \n\\item There is an identity operator.\n\\item The expectation value of a product of chiral operators are independent of their positions, and they have the simple OPE structure ${\\cal O}_i{\\cal O}_j=\\sum C_{ij}^k {\\cal O}_k$ with $C_{ij}^k$ constant. \n\\end{itemize}\nThese properties imply that the chiral operators form a commutative ring with an identity. To solve a $\\mathcal{N}=1$ theory,\none would like to determine the full set of chiral operators. That is, one would like to find the generators and relations determining the chiral ring. For example, for $SU(N)$ gauge theory with an adjoint matter field $\\Phi$, the generators of single trace chiral operators are: $\\text{Tr}(\\Phi^k)$, $\\text{Tr}(W_\\alpha \\Phi^k)$, and $\\text{Tr}(W_\\alpha W^\\alpha\\Phi^k)$, \nthe full chiral ring relations are determined in \\cite{Cachazo:2002ry}. The chiral ring relations usually are much harder to determine. Typically, we have the following classical chiral ring relations;\n\\begin{itemize}\n\\item Chiral ring relations \ncome from the finite size of matrices, and we have Caley-Hamilton equation for a matrix. For example, \nfor a chiral field in the adjoint representation of gauge group $SU(N)$, the chiral operators $\\text{Tr}(\\phi^i), i> N$ can be expressed in terms of $\\text{Tr}(\\phi^j)$ \n with $j\\leq N$. \n \\item Chiral ring relations come from the constraints of the superpotential. For example, consider\n$\\mathcal{N}=4$ $SU(N)$ gauge theory: this theory has three chiral fields $X, Y, Z$, and a superpotential $W=\\text{Tr}XYZ-\\text{Tr} X Z Y$. The $F$-term equations from the superpotential are\n \\begin{equation}\n [X,Y]=[Y,Z]=[Z,X]=0,\n \\end{equation}\nand so the matrices $X,Y,Z$ commute, which lead to chiral ring relations of the type $\\text{Tr}(XYZ)=\\text{Tr}(XZY)$ and so on. \n \\end{itemize}\nThese classical chiral ring relations can be modified by quantum effects such as instantons, Konishi anomalies and strongly coupled dynamics, and we have the quantum chiral ring.\nThe determination of the generators of chiral ring and the quantum chiral ring relation is a central task in the study of supersymmetric gauge theory. \nLet's assume that the generators of the chiral ring are $x_1, x_2, \\ldots x_s$, and the chiral ring relations are generated by polynomial relations, \nthen the quantum chiral ring is isomorphic to\n\\begin{equation}\n\\mathbb{C}[x_1, x_2,\\ldots, x_s]\/I,\n\\end{equation}\nwhere $\\mathbb{C}[x_1, x_2,\\ldots, x_s]$ is ring of polynomials with complex coefficients, and $I$ is the ideal generated by the chiral ring relations. \nIn general, the parameters of our theory such as the dynamically \ngenerated scale $\\Lambda$ and masses $m_i$ should be included into the generators and relations of chiral ring. From now on, by the chiral ring we will always mean the quantum chiral ring.\n\n\n\n\\textbf{Example}: Consider $\\mathcal{N}=1$ $SU(N)$ SQCD with $N_f=N$ quarks $Q_i, \\tilde{Q}_j$, the space of\nchiral operators are\n\\begin{align}\n& M_{ij}=Q_i\\tilde{Q}_j, \\nonumber\\\\\n&B=\\epsilon_{\\alpha_1 \\alpha_2 \\ldots \\alpha_N} Q_1^{\\alpha_1}\\ldots Q_N^{\\alpha_N}, \\nonumber\\\\\n&\\tilde{B}=\\epsilon_{\\alpha_1 \\alpha_2 \\ldots \\alpha_N} \\tilde{Q}_1^{\\alpha_1}\\ldots \\tilde{Q}_N^{\\alpha_N}.\n\\end{align}\nThe classical chiral ring relation is $\\text{Det}(M)-B\\tilde{B}=0$, but quantum mechanically, the ring relation is changed to \n\\begin{equation}\nf=\\text{Det}(M)-B\\tilde{B}-\\Lambda^{2N_c}=0.\n\\end{equation}\nHere $\\Lambda$ is the dynamical scale of the theory. The chiral ring is then $C[M, B, \\tilde{B},\\Lambda]\/f$ \\cite{Intriligator:1995au}.\n\nWe are interested in the chiral ring of a SCFT. The $\\mathcal{N}=1$ SCFT has a distinguished $U(1)_R$ symmetry, and the scaling dimension of a chiral operator are related to\nits $U(1)_R$ charge by\n\\begin{equation}\nD({\\cal O})={3\\over 2} R({\\cal O}).\n\\end{equation} \nIn general, it is not easy to determine the $U(1)_R$ symmetry of a SCFT. Intriligator and Wecht found a remarkable $a$-maximization procedure to determine the $R$-symmetry \\cite{Intriligator:2003jj}. Namely, they predicted that the correct $R$-symmetry maximizes the central charge $a$. \n\nOne usually defines a SCFT as the IR limit of a UV quiver gauge theory, and the $U(1)_R$ symmetry of IR SCFT can be determined as follows: \nFirst, find all the anomaly free $U(1)$ symmetries in the UV and \ndefine a trial R symmetry $U(1)_{trial}=\\sum_I s_I F_I$. Second, compute the trial central charge using the formula\n\\begin{equation}\na_{trial}(s_I)={3\\over 32} (3 \\text{Tr}(R_{trial}^3)-\\text{Tr}(R_{trial})).\n\\end{equation}\nThe true $U(1)_R$ symmetry is found by maximizing the central charge and this will fix the coefficients $s_I$. \nA crucial assumption of above procedure is that all the symmetries for the IR SCFT are manifest in the UV description! \nHowever, this is often not the case. For example, two possible scenarios are\n \\begin{itemize}\n \\item The violation of unitarity bound: if a gauge invariant operator ${\\cal O}$ violates the unitarity bound after doing $a$-maximization, it is \n argued that this field becomes free \\cite{Seiberg:1994bz}, and that there is an accidental $U(1)$ symmetry acting on this operator ${\\cal O}$ only.\n \\item Even if there is no violation of unitarity bound, accidental symmetry is still possible as a result of some unknown dynamical effect. \n \\end{itemize}\n \n \\textbf{Example}: Let's illustrate the above point by an example. Consider $\\mathcal{N}=1$ $SU(N_c)$ SQCD with $N_f$ fundamental flavors. There is an unique $U(1)_R$ type symmetry\n such that the quarks $Q$ and antiquarks $\\tilde{Q}$ have the following charges\n \\begin{equation}\n R_{Q}=R_{\\tilde{Q}}={N_f-N_c\\over N_f}.\n \\end{equation}\n We require $N_f>N_c$ so that the $R$ charge is positive. The mesons has $R$ charge $R(M)=2{N_f-N_c\\over N_f}$, and baryons an anti-baryons have \n $R$ charge $R(B)=R(\\tilde{B})={N_c(N_f-N_c)\\over N_f}$. So using this candidate $U(1)_R$ symmetry for the IR SCFT, we have $\\Delta(M)={3\\over2}R(M)<1$ \n if $N_c0$ gives the same degeneration limit $X_0$. On the other hand $l\\eta$ with $l<0$ gives a different degeneration limit-- we get the ring generated by the ideal $w^k=0$, which is not normal!\n\n\n\\subsubsection{Futaki invariant and generalized $a$-maximization}\nNow let's start with a ring $X$ with symmetry $\\zeta$ and we also choose the generators $t_i, i=1,\\ldots, n$ for the Lie algebra $\\mathfrak{t}$ of the maximal torus in the automorphism group $G$ of X. Let us write \n$\\zeta=\\sum_{i=1}^n \\zeta_it_i$, and we may as well assume that $\\zeta$ minimizes the volume over all the possible $U(1)_R$ symmetries parametrized by $\\mathfrak{t}$. \nConsider a test configuration ${\\cal X}$ generated by a symmetry generator $\\eta$ and let $X_0$ denote the central fibre. We would like to determine whether or not $X_0$ destabilizes $X$. The \ncrucial ingredient is the Donaldson-Futaki invariant defined in \\cite{donaldson2002scalar}. \n\nThe ring $(X_0, \\zeta, \\eta)$ is still Gorenstein and klt, and has a at least two dimensional symmetry group generated by $\\zeta$ and $\\eta$. There is only \none dimensional possible $U(1)_R$ symmetry as we need to impose following two conditions\n\\begin{itemize}\n\\item[(a)] The charge on the coordinates $x_i$ is positive\n\\item[(b)] The $(3,0)$ form has charge 2. \n\\end{itemize}\nThe second condition can be fixed by computing the Hilbert series of $X_0$ with respect to symmetry generator and \nimposing the condition $a_0=a_1$. This one dimensional candidate $U(1)_R$ symmetry can be parameterized as\n\\begin{equation}\n\\zeta(\\epsilon)=\\zeta+\\epsilon(\\eta-a\\zeta).\n\\end{equation}\nNotice that we require $\\epsilon>0$ so that the central fibre is the same as the original one if we use the symmetry $\\epsilon(\\eta-a\\zeta)$ to generate the test configuration. \nSubstitute the above parameterization into the equation $a_0=a_1$ and expand it to first order in $\\epsilon$, we have\n\\begin{equation}\na_0(\\zeta+\\epsilon(\\eta-a\\zeta))=a_1(\\zeta+\\epsilon(\\eta-a\\zeta)) \\rightarrow a_0(\\zeta)+\\epsilon (\\eta-a \\zeta)\\cdot a_0^{'} = a_1(\\zeta)+\\epsilon (\\eta-a \\zeta)\\cdot a_1^{'}.\n\\end{equation}\nHere $a_0^{'}$ and $a_1^{'}$ are the vectors defined by the derivative ${da_i(\\vec{x})\\over d {\\vec{x}}}|_{\\vec{x}=\\zeta}$, and $\\vec{x}=\\sum_{i=1}^ns_i t_i+b \\eta$. Using the result $a_0(\\zeta)=a_1(\\zeta)$, we have \n\\begin{equation}\na={\\eta\\cdot (a_0^{'}-a_1^{'}) \\over \\zeta \\cdot (a_0^{'}-a_1^{'})}={\\eta\\cdot (a_1^{'}-a_0^{'}) \\over a_0}={1\\over a_0(\\zeta)}({da_1(\\zeta+\\epsilon \\eta) \\over d \\epsilon }-{d a_0(\\zeta+\\epsilon \\eta) \\over d\\epsilon })|_{\\epsilon=0}.\n\\label{constant}\n\\end{equation} \nWe also use the fact $\\zeta\\cdot a_0^{'}=3 a_0(\\zeta),~\\zeta\\cdot a_1^{'}=2a_1(\\zeta)=2a_0(\\zeta)$. \nNow the Futaki invariant is defined to be \n\\begin{equation}\nF(X,\\zeta, \\eta)=D_\\epsilon a_0(\\zeta(\\epsilon))|_{\\epsilon=0}.\n\\end{equation}\nThis definition is not of the form of the original Futaki invariant defined in \\cite{donaldson2002scalar}, however, we will now show that our definition is equivalent to the original one (see also \\cite{collins2015sasaki} for more discussion). We have\n\\begin{align}\n& F(X,\\zeta,\\eta)=D_\\epsilon a_0(\\zeta+\\epsilon(\\eta-a\\zeta))=(\\eta-a \\zeta) \\cdot a_0^{'} \\nonumber\\\\\n&=D_\\epsilon a_0(\\zeta+\\epsilon \\eta)-{\\eta\\cdot (a_0^{'}-a_1^{'}) \\over a_0}D_\\epsilon a_0(\\zeta+\\epsilon \\zeta) \\nonumber\\\\\n&=D_\\epsilon a_0(\\zeta+\\epsilon \\eta)+3a_0(\\zeta){\\eta\\cdot (a_0^{'}-a_1^{'}) \\over a_0} \\nonumber \\\\\n&= D_\\epsilon a_0(\\zeta+\\epsilon \\eta)+3a_0(\\zeta)D_{\\epsilon}{a_1(\\zeta+\\epsilon \\eta)\\over a_0(\\zeta+\\epsilon \\eta)}. \n\\label{futaki}\n\\end{align}\nWe use the definition from first line to second line, and from second line to third line we use the fact $D_\\epsilon a_0(\\zeta+\\epsilon \\zeta)=-3 a_0(\\zeta)$ (which can be found using the definition of Hilbert series). \nThe formula in the last line is precisely the Futaki invariant defined in \\cite{collins2015sasaki}. Having defined the Futaki invariant, we can now state the definition of K-stability.\n\n\\begin{theorem}\nA polarized ring $(X,\\zeta)$ is stable if for any non-trivial test configuration generated by the symmetry $\\eta$, the Futaki invariant satisfies\n\\begin{equation}\nF(X, \\zeta, \\eta)>0.\n\\end{equation}\nAnd for the trivial test configuration, namely the central fibre $X_0$ is the same as $X$, the Futaki invariant satisfies\n\\begin{equation}\nF(X, \\zeta, \\eta)\\geq0.\n\\end{equation}\n\\end{theorem}\nWe now provide a physical interpretation of the Futaki invariant $F$. Since $a_0$ is inverse proportional to the central charge of the coordinate ring of the central fiber, the Futaki invariant is directly related to the maximization of the central charge. The shape of the function $a_0$ with respect to $\\epsilon$ is \ndrawn in figure. \\ref{futaki}. $F<0$ implies that $a_0(\\epsilon)$ is minimized at $\\epsilon>0$, and the new ring gives larger central charge $a$!\nWhen $F=0$ and $X_0$ is different from X, the two ring gives the same central charge $a$, but the central fiber $X_0$ has a strictly larger symmetry group which then \ndestabilizes $X$. When $F>0$, the new ring gives less central charge over the allowed space of symmetries ($\\epsilon>0$). In summation, Futaki invariant is actually implying generalized $a$-maximization. Namely, a test configuration $X_0$ destabilizes $X$ if it gives no less central charge!\n\\begin{figure}[h]\n\\centering\n \\includegraphics{futaki.eps}\n \\caption{Three situations for Futaki invariant. $F>0$: $a_0(\\epsilon)>a(0)$ for $\\epsilon>0$; $F=0$: the minima of $a_0$ is achieved at $\\epsilon=0$; $F<0$: the minima of $a_0$ is achieved fro $\\epsilon>0$. \n Notice that we only need to look at $a_0$ for $\\epsilon>0$.}\n \\label{futaki}\n \\end{figure}\n\n\n\n\\textbf{Example}: Consider the ring $X$ which is generated by the ideal $x^2+y^2+z^2+w^k=0$, this ring has a symmetry $\\zeta$ with charge $({2k\\over k+2},{2k\\over k+2},{2k\\over k+2},{4\\over k+2})$ on coordinates $(x,y,z,w)$. This symmetry is chosen such that the \n$(3,0)$ form $\\Omega={dx \\wedge dy \\wedge dz \\wedge dw\\over df}$ has charge two. The Hilbert series for $\\zeta$ is \n\\begin{equation}\nHilb(X,\\zeta, t)=\\frac{1-t^{\\frac{4 k}{k+2}}}{\\left(1-t^{\\frac{4}{k+2}}\\right) \\left(1-t^{\\frac{2 k}{k+2}}\\right)^3}.\n\\end{equation}\nExpand around $t=1$, we find $a_0(\\zeta)=a_1(\\zeta)={(2+k)^3\\over 8k^2}$. \nNow consider the test configuration generated by the symmetry $\\eta$ with charges $(0,0,0,1)$. In this case, the central fibre $X_0$ is generated by the\nideal $x^2+y^2+z^2=0$. Using formula \\eqref{constant}, \nthe one parameter possible $U(1)_R$ symmetry is \n\\begin{equation}\n\\zeta(\\epsilon)=\\zeta+\\epsilon(\\eta-{1\\over 2}\\zeta).\n\\end{equation}\nThe Hilbert series with respect to above symmetry is \n\\begin{equation}\nHilb(X_0,\\zeta(\\epsilon),t)={1-t^{(1-{\\epsilon\\over 2}){4k\\over k+2}}\\over (1-t^{(1-{\\epsilon\\over 2}){2k\\over k+2}})^3(1-t^{(1-{\\epsilon\\over 2}){4\\over k+2}+\\epsilon)})}.\n\\end{equation}\nSubstituting $t=exp(-s)$ and expand the Hilbert series around $s=0$, we get \n\\begin{equation}\na_0(\\zeta(\\epsilon))=a_1(\\zeta(\\epsilon))=\\frac{2 (k+2)^3}{(\\epsilon-2)^2 k^2 (\\epsilon k+4)}.\n\\end{equation}\nThe Futaki invariant is computed as \n\\begin{equation}\nF=D_\\epsilon a_0(\\zeta(\\epsilon))|_{\\epsilon=0}=\\frac{(4-k) (k+2)^3}{32 k^2}.\n\\end{equation}\nSo $F\\leq 0$ for $k\\geq 4$. Since $X_0$ is clearly not isomorphic to $X$, we conclude that $X_0$ destabilizes $X$ for $k\\geq 4$. A physical interpretation of this result will be given in the next section. \n\n\\subsubsection{Some discussions}\nChecking K-stability involves two steps. First, finding a test configuration and then computing the Futaki invariant. While the computation of Futaki invariant is straightforward, the set of possible test configurations is in principle infinite. Thus, in order to check K-stability one needs to reducing the \nsets of possible test configurations. There are several simplifications we can make\n\\begin{itemize}\n\\item The first simplification has already been used, namely we require that the central fibre to be normal, Gorenstein and klt. This is simply due to the reason that the central fibre should describe the chiral ring of \na $\\mathcal{N}=1$ field theory. \n\\item Assume that the symmetry group of the ring $X$ is $G$, then one only need to consider the flat families generated by a symmetry which commutes with $G$ \\cite{datar2015k, collins2015sasaki}. This fact is quite useful \nfor singularities with many symmetries. In particular, if the variety has three dimensional symmetries (or in other words, $X$ is toric), then there are no non-trivial test configurations, and hence checking stability reduces to volume minimization (or $a$-maximization). \n\\end{itemize}\n\n\n\n\\section{Some physical consequences}\n\n\\subsection{$a$-maximization}\nLet's assume that the ring $X$ is stable and has more than one dimension worth of possible $U(1)_R$ symmetry. The determination of $U(1)_R$ symmetry is \nsolved by $a$-maximization \\cite{Intriligator:2003jj} or equivalently volume minimization \\cite{Martelli:2005tp,Martelli:2006yb}. We now show that $a$-maximization can be explained using K-stability. \nConsider a test configuration generated by the symmetry vector $\\eta$, and the central fibre $X_0$ is the same as $X$.\nThe Futaki invariant is \n\\begin{equation}\nF=D_\\zeta a_0(\\zeta+\\epsilon \\eta)|_{\\epsilon=0}=\\eta\\cdot a_0^{'}(\\zeta),\n\\end{equation}\nIf $F(X, \\zeta, \\eta)>0$, the test configuration $(X_0, \\zeta, \\eta)$ does not destabilize $X$. But, since $\\eta$ preserves $X$, we can use the symmetry generator $-\\eta$ to generate a test configuration with the same central fibre $X$, \nand the Futaki invariant now is $F(X, \\zeta, -\\eta)=-F(X, \\zeta, \\eta)<0$ which will make the ring unstable. So K-stability implies that the symmetry generator has to satisfy \n\\begin{equation}\na_0^{'}(\\zeta)=0.\n\\end{equation}\nNotice that since $a_0$ is a convex function, the solution of above equation is the minimum, and therefore the central charge $a$ is maximized. \n \n\n\\subsection{Unitarity bound}\nOne can always generate a test chiral ring by using a symmetry acting on a single coordinate $x$ only. The central \nfibre $X_0$ is a new ring with $x$ free \\footnote{Mathematically such test configuration is generated by the so-called Rees algebra.}. The \nFutaki invariant is computed in \\cite{collins2012k}, and the answer is \n\\begin{equation}\nF\\propto (dim(x)-1),\n\\end{equation}\nhere we ignore a positive constant, and $dim(x)$ is the scaling dimension of the chiral scalar operator $x$. $X$ is not \ndestabilized by this particular test configuration if \n\\begin{equation}\ndim(x)>1.\n\\end{equation}\nThis is nothing but the unitarity bound on scalar operator represented by $x$. \n\n\\subsection{Singularity with more than one dimensional symmetries}\nConsider a toric Gorenstein singularity, and the rank of symmetry group is $3$. As we discussed above, there is no non-trivial \ntest configuration, and so toric singularity is stable provided we choose the $U(1)_{R}$ symmetry which minimizes the volume. \nOn the other hand, the existence of Sasaki-Einstein metrics on the link of a toric singularity was established using analytic methods in \\cite{futaki2009transverse}.\nFor the ring $X$ with two dimensional symmetries, one only needs to check finite number of test configurations, see \\cite{collins2015sasaki}.\n\n\\textbf{Example}: Consider the ring defined by the ideal $x^2+y^2+z^p+w^q=0$. This ring has a two dimensional \nsymmetry group. One can characterize all the test chiral rings, and the ring is proven to be stable if $(p,q)$ satisfies the following condition \\cite{collins2015sasaki}: \n\\begin{equation}\np<2q~\\text{and}~q<2p.\n\\end{equation} \nNotice that this is just the requirement of the unitarity bound on the operators represented by $z$ and $w$. However, as we will see later, the unitarity bound is not the only\nobstruction which can appear, even in the case of hypersurface singularities.\n\n\n\n\\subsection{Hypersurface singularity}\nConsider an isolated three-fold hypersurface singularity $f:(C^4,0)\\rightarrow (C,0)$ with a $C^*$ action $\\vec{\\zeta}$:\n\\begin{equation}\nf(\\lambda^{w_i}q_i)=\\lambda f(z_i). \n\\end{equation}\nHere all the charges $w_i$ are positive. The canonical three-form is \n\\begin{equation}\n\\Omega={dz_0\\wedge dz_1\\wedge dz_2\\wedge dz_3\\over d F}.\n\\end{equation}\nThis form has charge $\\sum w_i-1$, and the candidate $U(1)_R$ symmetry is found by requiring $\\Omega$ to have charge two:\n\\begin{equation}\n(\\sum w_i-1)\\delta=2\\rightarrow \\delta={2\\over \\sum w_i-1},\n\\end{equation}\nand the candidate $U(1)_R$ symmetry is $\\zeta^{'}=\\delta \\zeta$. \nTo make the coordinate $z_i$ have positive r charge, we require $\\sum w_i-1>0$, which implies that the singularity is a rational Gorenstein (and hence klt) singularity. \nSuch rational hypersurface singularities have been classified by Yau and Yu \\cite{yau2003classification}. \n\nThe Hilbert series of a hypersurface singularity is easy to compute. It takes the following form\n\\begin{equation}\nHilb(f,t,\\zeta^{'})={1-t^{\\delta}\\over (1-t^{w_0 \\delta})(1-t^{w_1 \\delta})(1-t^{w_2 \\delta})(1-t^{w_3 \\delta})}.\n\\end{equation}\n\nNow let's consider a test configuration which is derived by using a one parameter transformation $\\eta$. For simplicity, let's assume that \nthe action $\\vec{\\eta}$ is diagonal on the coordinates with charges $(v_1,v_2,v_3,v_4)$. In the flat limit, we get a new polynomial $f_0$ which does not \nnecessarily define an isolated singularity. $f_0$ has two dimensional symmetries generated by $(\\zeta^{'}, \\eta)$, however, there is only a one dimensional \nsymmetries which could be the possible $U(1)_R$ symmetry. The one parameter symmetry group can be parameterized as\n\\begin{equation}\n\\eta(\\epsilon)=\\zeta^{'}+\\epsilon(\\eta-a \\zeta^{'}),\n\\end{equation}\nand $a$ can be computed using the formula \\ref{constant}.\nThe Futaki invariant can be computed using formula \\ref{futaki}, and we have\n\\begin{align}\n&F(f,\\zeta^{'}, \\eta)=D_{\\epsilon} a_0(\\zeta(\\epsilon))|_{\\epsilon=0}= \\nonumber\\\\\n&-[(v_4w_1w_2w_3(w_1+w_2+w_3-2w_4-1)+v_3 w_1 w_2 w_4(w_1+w_2+w_4-2w_3-1)+ \\nonumber\\\\\n&v_2w_1w_3w_4(w_1+w_3+w_4-2w_2-1)+v_1w_2w_3w_4(w_2+w_3+w_4-2w_1-1).\n\\label{hyper}\n\\end{align}\nIn the following, we are going to use this formula to test whether a hypersurface singularity is stable or not. \n\n\\subsubsection{Irrelevance of superpotential term}\nRecall that we have already studied the singularity \n\\begin{equation}\nf=z_0^2+z_1^2+z_2^2+z_3^{2k},\n\\end{equation}\nfrom K-stability perspective, and we showed that this ring is unstable for $k\\geq 2$. The destabilizing configuration has the central fibre $X_0= \\{z_0^2+z_1^2+z_2^2=0\\}$; see section $3.2.2$. \n\nLet's interpret this result from field theory point of view. The quiver gauge theory description is found in \\cite{Cachazo:2001sg}; see \nfigure \\ref{quiver} below. \n\\begin{figure}[h]\n\\centering\n \\includegraphics{quiver.eps}\n \\caption{Quiver gauge theory description for D3 brane probing the singularity defined by $z_0^2+z_1^2+z_2^2+z_3^{2k}=0$. The superpotential is described in \\eqref{super}.}\n \\label{quiver}\n \\end{figure}\nWe have the following superpotential term:\n\\begin{equation}\nW=\\text{Tr} (\\phi_1(A_1B_1+A_2B_2))-\\text{Tr}(\\phi_2 (B_1A_1+B_2 A_2))-2{\\text{Tr}\\phi_1^{k+1}\\over k+1}+2{\\text{Tr}\\phi_2^{k+1}\\over k+1}.\n\\label{super}\n\\end{equation}\nThe $U(1)_R$ charge is fixed such that the NSVZ $\\beta$ function is zero, and each term in superpotential $W$ has charge two:\n\\begin{align}\n&{1\\over2}[(R(A_1)-1)+(R(A_2)-1)+(R(B_1)-1)+(R(B_2)-1)]+(R(\\phi_1)-1)+1=0, \\nonumber\\\\\n&{1\\over2}[(R(A_1)-1)+(R(A_2)-1)+(R(B_1)-1)+(R(B_2)-1)]+(R(\\phi_2)-1)+1=0, \\nonumber\\\\\n&R(A_1)+R(B_1)+R(\\phi_1)=2,~~R(A_2)+R(B_2)+R(\\phi_1)=2, \\nonumber\\\\\n&R(A_1)+R(B_1)+R(\\phi_2)=2,~~R(A_2)+R(B_2)+R(\\phi_2)=2, \\nonumber\\\\\n&R(\\phi_1)=R(\\phi_2)={2\\over k+1}. \n\\end{align}\nWe can use symmetry or $a$ maximization to find the following $R$ charges: $R(A_1)=R(A_2)=R(B_1)=R(B_2)={k\\over k+1},~R(\\phi_1)=R(\\phi_2)={2\\over k+1}$. \nThe $F$-term relations from the superpotential are:\n\\begin{align}\n& {\\partial W\\over \\partial A_1}=0:~~B_1\\phi_1-\\phi_2 B_1=0, \\nonumber\\\\\n&{\\partial W\\over \\partial B_1}=0:~~\\phi_1 A_1-A_1\\phi_2=0, \\nonumber\\\\\n&{\\partial W\\over \\partial A_2}=0:~~B_2\\phi_1-\\phi_2 B_2=0, \\nonumber\\\\\n&{\\partial W\\over \\partial B_2}=0:~~\\phi_1 A_2-A_2\\phi_2=0, \\nonumber\\\\\n&{\\partial W\\over \\partial \\phi_1}=0:~~A_1B_1+A_2B_2-2\\phi_1^k=0, \\nonumber\\\\\n&{\\partial W\\over \\partial \\phi_2}=0:~~B_1A_1+B_2A_2-2\\phi_2^k=0. \\nonumber\\\\\n\\end{align}\nThe scalar chiral ring of this theory (which is related to the holomorphic functions on $X$) is generated by the loops in the quiver subject to the above relations. \nThe single trace scalar chiral operators are generated by the simple loops, such as $\\text{Tr}(A_iB_j)$, and \none can order them by their $U(1)_R$ charge.\n\n\nConsider the singularity $X$ defined by the equation $f=z_0^2+z_1^2+z_2^2+z_3^{2k}$, the unique candidate $U(1)_R$ symmetry $\\zeta^{'}$ has charge \n$({2k\\over k+1},{2k\\over k+1},{2k\\over k+1}, {2\\over k+1})$ which is identified as the field theory $U(1)_R$ charge. We can make a holomorphic change of coordinates \nto write $f$ as $f=U^2+V^2+(-W+Z^k)(W+Z^k)$. \nThe holomorphic functions on $X$ can be \nidentified with the field theory chiral operators as follows:\n\\begin{align}\n \\text{Tr}A_1B_2&=U,\\qquad &\\text{Tr}A_2 B_1&=V, \\nonumber\\\\\n \\text{Tr}A_1B_1&=-W+Z^k,\\qquad &\\text{Tr}A_2B_2&=W+Z^k. \\nonumber\\\\\n \\text{Tr}\\phi_1&=Z. \n\\end{align}\nIt can be checked that the full set of scalar chiral operators of the field theory which can get expectation value is captured by the ring $X$.\n\n\nThe properties of the IR SCFT can be derived as follows: The quiver without the superpotential term $\\text{Tr}(\\phi_1^{k+1})$ and $\\text{Tr}(\\phi_2^{k+1})$ \ndefines a four dimensional $\\mathcal{N}=2$ SCFT ${\\cal T}_0$, and all of the elementary fields $A_i, B_i, \\phi_i$ are free with $U(1)_R$ charge ${2\\over 3}$. \nOur $\\mathcal{N}=1$ theory can be thought of as deforming $\\mathcal{N}=2$ SCFT ${\\cal T}_0$ by the \nsuperpotential terms involving the adjoint chiral superfields $\\phi_1$ and $\\phi_2$. \nThe scaling dimensions for the superpotential terms ${\\cal O}_1=\\text{Tr}(\\phi_1^{k+1})$ and ${\\cal O}_2=\\text{Tr}(\\phi_2^{k+1})$ are $\\Delta[{\\cal O}_i]=k+1$, and so they\nare irrelevant for $k>2$; \nthe IR SCFT is just the original SCFT ${\\cal T}_0$. \nFor $k=2$, the superpotential term is marginally irrelevant \\cite{Green:2010da}, and the IR SCFT is also the original SCFT ${\\cal T}_0$.\n\nWe have used K stability to check that the ring is unstable for $k\\geq 2$, and the destabilizing configuration has a central fibre $X_0: f=z_0^2+z_1^2+z_2^2$. \nIt is interesting to note that the IR SCFT (affine $A_1$, $\\mathcal{N}=2$ SCFT) associated with the ring $X$ is actually described by $D3$ branes probing the singularity $X_0$. This fact supports our claim that the central fibre $X_0$ describes the possible chiral ring of the IR SCFT, and the result from K-stability is in agreement with field theory result!\n\n\n\n\\subsubsection{Further obstructions}\nThe unstable example considered so far have been caused by the irrelevance of superpotential terms or the violation of the unitarity bound. \nWe now give an example where the instability of the chiral ring is more subtle. Consider a singularity\n\\begin{equation}\nf=z_0^2+z_1^2+z_2^p+z_2z_3^q. \n\\end{equation}\nThe only possible $U(1)_R$ symmetry has charge $({pq\\over p+q-1},{pq\\over p+q-1},{2q\\over p+q -1},{2(p-1)\\over p+q-1})$. The scaling dimensions of $z_2$ and $z_3$ are\n\\begin{equation}\n[z_2]={3q\\over 2(p+q-1)},~~[z_3]={3}{(p-1)\\over p+q-1}.\n\\end{equation}\nusing the relation $\\Delta({\\cal O})={3\\over2}R({\\cal O})$.\nThe unitarity bound on the scalar operators implies that $[z_2]>1$ and $[z_3]>1$, we find \n\\begin{equation}\np<2q+1~\\&~q<2p-2.\n\\label{uni}\n\\end{equation}\nThe unitarity bound can also be found using the test configuration generated by the symmetry acting on coordinate $z_2$ and $z_3$ only. \n\nConsider a test configuration generated by the symmetry $\\eta$ with charge $(0,0, 1, -1\/q)$. We have the following family generated by $\\eta$;\n\\begin{equation}\nz_0^2+z_1^2+t^pz_2^p+z_2z_3^q=0.\n\\end{equation}\nThe flat limit over $t=0$ is described by the equation $z_0^2+z_1^2+z_2 z_3^q=0$. The Futaki invariant can be computed using the formula \\ref{hyper}:\n\\begin{equation}\nF(X_0, \\zeta, \\eta)=-\\frac{(p+q-1)^2 \\left(p^2-2 p q+q-1\\right)}{2 (p-1)^2 q^2}.\n\\end{equation}\nSo the original ring is stable if \n\\begin{equation}\nq(2p-1)-(p^2-1)>0 \\rightarrow q>{p^2-1\\over 2p-1}. \n\\label{stronger}\n\\end{equation}\nThis bound is stronger than the unitarity bound~\\eqref{uni} for certain range of the parameters. Let's set $p=6$, the unitarity bound from~\\eqref{uni} implies that \n\\begin{equation}\n {5\\over 2}34\/11,\n\\end{equation}\nwhich gives a stronger lower bound, i.e. $q=3$ satisfies the unitarity bound, but is unstable due to some other dynamical reason. The chiral ring of the IR SCFT is \ndescribed by $z_0^2+z_1^2+z_2 z_3^q=0$ for $q=3$ which is also a three dimensional quotient singularity. \n\n\\section{Conclusion}\nWe introduce a notion of stability for $\\mathcal{N}=1$ chiral rings, and conjecture that a chiral ring is the chiral ring of a SCFT if and only if \nit is stable. We test our stability notion for models engineered using D3 brane probing 3-fold singularity, and show\nthat the notion of K-stability for the existence of Ricci-flat conic metric is equivalent to the field theory stability. This notion can be used to explain $a$-maximization, an operator becoming free if \nit violates unitarily bound, and the irrelevance of superpotential terms, etc. In general, our stability notion explains the consequences of accidental symmetries appearing in the study of quiver gauge theory: the chiral ring of the IR SCFT is different form that \nof UV theory if there are accidental symmetries.\nAccidental symmetries cause many problems in studying supersymmetric field theory with four supercharges \\cite{Kutasov:2003iy,Buican:2011ty}. Our study shows the importance \nof the chiral ring, and shows that the generalized notion of $a$-maximization plays a key role. Similar notion of generalized a maximization idea has already been used by Intriligator \nto settle some interesting IR phase questions \\cite{Intriligator:2005if}. It would be interesting to use our stability notion to reconsider those models. \n\nThe stability notion proposed here can be generalized to three dimensional $\\mathcal{N}=2$ theory. Although one does not have the central charge notion in this context, we may replace it by the so-called $F$-function \\cite{Jafferis:2010un}. For the theory engineered by M2 branes probing a four-fold singularity, one still has the notion of K-stability for the four-fold singularity and much of the theory is similar. We leave the details to the interested reader. \nSimilarly, one can also define the a notion of stability for two dimensional $(0,2)$ theory, \nand we hope that the accidental symmetry for $(0,2)$ theory studied in \\cite{Bertolini:2014ela} can be put into the stability framework.\n\nThere are some further questions about the stability of $\\mathcal{N}=1$ chiral ring.\nOf crucial importance is to understand the constraints on set of possible test chiral rings. At present, unless a large symmetry group intervenes, there are infinite number of possible test rings, and it seems computational impossible to check all of them, even in basic examples. It would be nice to have some physical input which could shed some light on this issue. In this paper, we only studied the models engineered using D3 brane, and it will be of great interest to study other $\\mathcal{N}=1$ theories. \n\nOur primary focus has been on testing whether a chiral ring is the chiral ring of a SCFT. If the chiral ring is unstable, it is important to determine the ring of IR SCFT. Our study shows that the central fibre of the destabilizing test configuration should be the candidate chiral ring of the IR SCFT, and it is interesting to \ndetermine the special destabilizing test configuration which would give the chiral ring of IR SCFT. We hope to come to this question in the future. \n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nThe work of S.T Yau is supported by NSF grant DMS-1159412, NSF grant PHY-\n0937443, and NSF grant DMS-0804454. T.C. Collins is supported by NSF grant DMS-1506652\nThe work of D. Xie is supported by Center for Mathematical Sciences and Applications at Harvard University, and in part by the Fundamental Laws Initiative of\nthe Center for the Fundamental Laws of Nature, Harvard University.\n\n\n\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn recent years optical communication through metallic nanostructures has attracted significant attention.\nSince 2000 when Pendry initiated the idea of superlens imaging \\cite{Pendry}, it has became an important research topic in past years. \n Noble metal nanostuctures based on gold and silver have allowed the light to be guides beyond the diffraction limit, which has opened new possibilities for further miniaturization and applications of optoelectronic devices.\n \nIn 2005 Lee \\textit{et al} \\cite{Lee05} presented experimental and theoretical studies of the optical superlens using a thin silver slab and demonstrated optical imaging with resolution well below the diffraction limit.\nThe investigations of different, cheaper material to construct efficient superlens systems basing on different metals have started; Zhao \\emph{et al} \\cite{Zhao16} have studied several superlens materials, including Ni, Cr and also Cu in the 400 nm wavelength range concluding that the dielectrics covering the superlens surfaces have to be characterized by similar permittivities, which influence the imaging quality. It has also been noted that a large permittivity of the dielectric in the front of the superlens is in favour of the imaging effect. Mkhitaryan \\emph{et al} have recently demonstrated that high Q factor plasmonic structures can be realized with copper \\cite{Mkhitaryan21}, further confirming that Cu is a promising candidate for construction of efficient superlens.\n\nThe general theoretical model valid for both homogeneous and\nevanescent waves, transfer-matrix analyses of the near-field metallic superlens and\nthe numerical finite-difference in time-domain (FDTD) simulations has been proposed by Tremblay \\cite{Tremblay} and the detailed\nrole of near-field optics in a superlens image formation has been reviewed recently by Adams et al \\cite{Adams}.\n\nMoreover, the important aspect of producing superlens is the accomplishment of the optimal resolution. It is affected by the finite thickness of the superlens, which generates both long- and short-range surface plasmon modes, they in turn influence a transfer function distorting the image field. Wang et al \\cite{Wang11} found that the presence of loss is not always detrimental to the superlens resolution. \n \nSince the discovery of negative index metamaterials and superlens, there is a strong need for a lens medium that is characterized not only by the necessary negative permittivity, but also one that offers tunability of its optical properties. By coupling various quantum systems with the metamaterial, one can provide such a capability \\cite{Quach2011}. These so-called quantum metamaterials \\cite{Stav2018} are a new class of media that bridges the gap between quantum systems and classical metamaterials. In this paper, we focus on the recently discovered Rydberg excitons (REs) in Cu$_2$O \\cite{Kazimierczuk} and their potential application in copper based superlens.\nREs are highly excited excitonic states characterized by a principal quantum number $n>>$1 for dipole-allowded P-type enveloped wavefunctions. In principle, they have similar scaling as Rydberg atoms \\cite{Heck17} although the physical origin of these similarities in their case bases on a complex valence band structure and different selection rules for excitons. The dimensions of REs reach micrometers and their life-times can reach hundreds of nanoseconds. Nowadays REs become one of the rapidly developing topic in solid-state optics; the studies extend from linear to nonlinear regimes in bulk media and nanostructures \\cite{Orfanakis,my2022,Thomas2022}. Recently, REs interaction with plasmons has been studied, showing possibility of an evident extension of propagation length and bridging plasmonics and Rydberg excitons physics \\cite{my_arxiv}. Here, we expand upon that work by considering the effect of excitons on the dynamics of copper-based superlens.\n \n \n We perform the detailed analysis of Cu and Cu$_2$O, with its particularly high $\\epsilon=7.5$ based superlens taking into account all above mentioned aspects. \n The required matching of permittivity of Cu and Cu$_2$O can be achieved either by tuning the frequency, adjusting the Cu film thickness \\cite{Stenzel19} or using metal-dielectric composites \\cite{Cai05} such as a fine mix of Cu and Cu$_2$O layers with excitons. Tailoring the optical properties of medium opened a new possibility of controlling the image. This is because the propagation dynamics of the field depends on the diffraction and dispersion properties of the medium. In particular, Archambault \\emph{el al}\\cite{Archambault12} pointed out that an improvement of spatial resolution has to be achieved using time-dependent incident pulse. In the spirit of their conclusions we examine the influence of the group velocities of various modes on the image formation. \n \nThe paper is organized as follows. In the first section, a numerical model of the optical properties of copper and copper oxide is presented, which is later used in theoretical calculations and numerical simulations. Then, the system setup is outlined. The third section is devoted to the study of the steady state transmission coefficient of the copper superlens. Next, the dynamic of the image formation in the superlens is investigated. Finally, the conclusions are presented. The detailed description of the numerical simulation method is included in the appendix.\n \n\n \n\\section{Material model}\nFor the numerical description of the optical properties of Cu, we use the Drude-Lorentz model in which the permittivity, in the case of two-oscillator model, is given by \\cite{Okada}\n\\begin{equation}\\label{eq:DL}\n\\epsilon(\\omega)=\\varepsilon_\\infty+\\frac{f_0}{\\omega_0^2-\\omega^2+i\\omega \\Gamma_0}+\\frac{f_1}{\\omega_1^2-\\omega^2+i\\omega \\Gamma_1}, \n\\end{equation}\nwith the numerical value of parameters summarized in the Table \\ref{parTable} in Appendix A. The values of parameters are adjusted to provide a good fit to the experimental data presented in ref. \\cite{Hollstein}. A comparison of the experimentally measured permittivity spectrum and the model values is shown on the Fig. \\ref{Cu_model}. In particular, the model is adjusted to provide a good fit in the energy region between $E \\sim 2140 - 2170$ meV (dashed, vertical lines), where excitonic resonances are present. In particular, we assume that the optical properties of Cu are determined by an ensemble of free electrons. The oscillator strength $f_0$ is determined by density of electron plasma, with so-called plasma frequency $\\omega_p = \\sqrt{f_0}$. Due to the fact that the electrons are not bound, there is no restoring force that would result in some intrinsic resonant frequency of charges and so $\\omega_0=0$. The damping parameter $\\Gamma_0$ represents various dissipative processes. The parameters of the second oscillator $f_1$, $\\omega_1$, $\\Gamma_1$ are used to include the effect of inter-band transitions that are responsible for an increase of absorption around 2.35 eV (see Fig. \\ref{Cu_model}).\n\nFor the copper oxide, the same Eq. (\\ref{eq:DL}) with a different set of parameters can be used. The two main contributions to the Cu$_2$O susceptibility are the constant value of $\\epsilon_\\infty=7.5$ and the effect of 2P exciton resonance, as shown on the Fig. \\ref{Cu2O_model}. This resonance is included as an oscillator with the frequency corresponding to the 2P exciton energy $2147$ meV and the oscillator strength and damping constant adjusted to match the height and width of absorption peak of the exciton \\cite{Raczynska2016}. In the case of Cu$_2$O, the second oscillator in Eq. (\\ref{eq:DL}) is unused.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{cu}\n\\caption{Real (continuous line) and imaginary (dashed line) part of permittivity given by Eq. \\ref{eq:DL}, compared to experimental data (dots).}\\label{Cu_model}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\includegraphics[width=\\linewidth]{cu2o}\n\\caption{Imaginary (left side) and real (right side) part of permittivity. Solid lines is given by theoretical calculation for Cu$_2$O bulk 2P exciton from \\cite{Raczynska2016} and dashed lines given by Eq. \\ref{eq:DL} with parameters depict in Table \\ref{parTable}.}\\label{Cu2O_model}\n\\end{figure}\n\n\\section{System setup}\nLets consider a multilayer system consisting of a thin copper layer surrounded by copper oxide, as shown on the Fig. \\ref{superlensmodel}. The optical axis of the system (horizontal, dashed line) is aligned to the $y$ axis and the source\/image planes (vertical dashed lines) are parallel to the $x$ axis. Blue lines depict a schematic of optical rays connecting the point source with its image (red dots). For an efficient superlens operation, the thickness $L$ needs to be smaller than approximately $0.1 \\lambda$, where $\\lambda$ is the wavelength of the light illuminating the system. \n\\begin{figure}[ht!]\n\\includegraphics[width=0.6\\linewidth]{schemat}\n\\caption{Schematic representation of a copper superlens surrounded by Cu$_2$O.}\\label{superlensmodel}\n\\end{figure}\nThe important parameter determining the superlens resolution is the wave vector $k_0$ and its $x$ component $k_x$. Specifically, to achieve subwavelength resolution, the superlens needs to transmit high spatial frequency components with $k_x>k_0$. In such a case, the $y$ component of the wave vector $k_y = \\sqrt{k_0^2 - k_x^2}$ is imaginary and thus it describes evanescent waves. The key characteristic of the superlens is the capability to amplify these rapidly decaying evanescent waves \\cite{Pendry}.\n\n\\section{Transmission coefficient}\nOne of the primary tools for studying the performance of the superlens is the transmission spectrum in the wave vector domain, e.g. the relation $T(k_x)$. The system is capable of subwavelength imaging when the values of $T(k_x>k_0)$ are significant. Similar to the approach presented in \\cite{Wang11}, we calculate the transmission coefficient (transfer function) of a flat superlens in electrostatic limit, e.g. in the case where the lens thickness is considerably smaller than illuminated wavelength. Let's consider a source and image plane located at a distance $L\/2$ from the superlens surfaces (see Fig. \\ref{superlensmodel}). Transmission coefficient is calculated by multilayer transmission using Parratt recursion relation \\cite{Parratt1954} \n\\begin{equation}\\label{eq:transmisja}\nT=T_0T_1...T_N,\n\\end{equation}\nwhere $T_j$ is transmission coefficient in each layer is given by \n \\begin{equation}\n T_j=e^{i k_z^{(j+1)}L_{j+1}} \\frac{1+F_j}{1+F_jR_{j+1}},\n\\end{equation}\nwith reflection coefficient \n\\begin{equation}\nR_j=e^{2ik_z^{(j)}L_j}\\frac{F_j+R_{j+1}}{1+F_j R_{j+1}},\n\\end{equation}\nand \n\\begin{equation}\nF_j=\\frac{\\epsilon_{j+1}k_z^{j}-\\epsilon_{j}k_z^{j+1}}{\\epsilon_{j+1}k_z^{j}+\\epsilon_{j}k_z^{j+1}}.\n\\end{equation}\nParameter $j=0,1,2$ numerates Cu$_2$O, Cu, Cu$_2$O layers, correspondingly. $k_z^{j}$ is z-component of wavevector in $j$ layer and $L_j$ is its thickness with $L_0=L_{N+1}=0$ and $R_{N+1}=0$. Similar approach one can found in other papers about superlenses, see \\cite{Splawinski2021,Hakkarainen2009}\n\nAs a first step, we can verify the derived theoretical transmission relation by comparing it with the results presented by Wang \\textit{at al} \\cite{Wang11}. In particular, the Fig. \\ref{RT_verif} depicts the transmission coefficient of idealized superlens consisting of a single layer of $\\epsilon=-1$ material, with $\\lambda\/10$ thickness, surrounded by vacuum. There are several characteristic points in the transmission spectrum; for $k_x\/k_0<1$, the transmitted mode corresponds to regular, propagating wave; the transmission coefficient in this range is smaller than 1. For higher values of $k_x$, the transmission coefficient describes evanescent waves that do not carry energy, so one can have $T>1$. In particular, the transmission spectrum usually contains two poles \\cite{Wang11}, which are clearly visible on the Fig. \\ref{RT_verif}. For $k_x\/k_0>9$, there is a cutoff point where the transmission quickly drops. This means that the spatial resolution of the superlens is limited to approximately $\\lambda\/9$; one can increase this resolution by making the lens thinner \\cite{Pendry}. The presence of the sharp poles in the transfer function (transmission coefficient) is generally undesirable as it introduces distortion \\cite{Wang11}; one can see that the introduction of absorption (Fig. \\ref{RT_verif}, blue line) significantly reduces the gain on the second, high $k_x$ pole, so that the transfer function becomes more flat (e.g. all wave modes are amplified to a similar degree). Thus, despite the fact that absorption is in general detrimental to the superlens resolution, its presence can be beneficial to the image quality. Due to this reason, our proposal of Cu-based superlens has some advantages over traditional designs based on noble metals.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{r1}\n\\caption{Transmission coefficient as a function of the x component of wave vector $k_x$, calculated for a model superlens with $\\epsilon=-1$.}\\label{RT_verif}\n\\end{figure}\n\nAs a next step, we consider a copper superlens with a thickness $L=20$ nm, surrounded by Cu$_2$O. From the perspective of practical applications, such a system is potentially simple to fabricate by controlled oxidation of Cu surface \\cite{Rodriguez11}, vapour deposition \\cite{Steinhauer20} or other techniques \\cite{Lynch21}. The transmission coefficient calculated for a range of values of wave vector component $k_x$ and energy $E$ is shown on the Fig. \\ref{RT_1}. \n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{r2}\n\\caption{Transmission coefficient (color) as a function of the x component of wave vector $k_x$ and energy.}\\label{RT_1}\n\\end{figure}\nDue to the fact that copper has a significantly higher absorption coefficient than the idealized superlens material considered previously, there is only a single pole in the transmission coefficient at $k_x \\approx k_0$. Overall, the usable range of wave vector values where transmission $T > 0.1$ is $k_x < 6$, which corresponds to roughly $\\lambda\/7$ resolution and is comparable to silver superlens performance \\cite{Sheng2009,Hakkainen2010}. One can notice that the strongest 2P exciton, which corresponds to 2147 meV (white, dashed line), produces a noticeable change in transmission coefficient (especially visible on the black contours). This means that the excitons in Cu$_2$O are a viable tool for fine-tuning of the lens properties in a very narrow energy range. \n\nOne of the primary considerations in the superlens design is the choice of thickness. In fact in the case of metallic lens with negative permittivity and positive permeability, the thickness needs to be much smaller than illuminated wavelength \\cite{Fang2003}. Also, in contrast to the ideal n=-1 superlens, there is a finite range of the values of wave vector component $k_x$ that are transmitted, as illustrated earlier on the Fig. \\ref{RT_1}. In general, we can expect that high spatial-frequency components of the image decay much faster in a thicker lens \\cite{Wang11}. This is indeed the case in the calculated results shown on the Fig. \\ref{RT_2}. Overall, there is a good match between theoretical and numerical results. The first local transmission maximum, which is also present in the idealized case on the Fig. \\ref{RT_verif}, forms a very sharp peak in theoretical calculations. To further confirm these results, we use the Finite-Difference Time-Domain (FDTD) simulation that is based directly on Maxwell's equations and thus allows for a direct study of superlens optical properties with no simplifying assumptions that are present in theoretical description. The only important limitation of FDTD is the limited time and spatial resolution, which results in a smaller, broader peak in the numerical transmission spectrum (Fig. \\ref{RT_2} b)). As mentioned before, the secondary peak on the Fig. \\ref{RT_verif} located around $k_x=7k_0$ is very sensitive to absorption and in case of Cu, it is almost completely suppressed. One can see that only in the case of $L=12$ nm, a very wide local maximum of transmission is visible on the Fig. \\ref{RT_2} at $k_x \\approx 5 k_0$. Similar to the earlier results, the lack of sharp peaks in the transmission spectrum is beneficial, as it means that all spatial frequency components are amplified to the same degree \\cite{Wang11}. In general, a lens thickness $L$ of the order of 10 nm is favourable, with significant transmission up to $k_x\/k_0 \\approx 10$. As the lens becomes thicker, the transmission cutoff becomes smaller, up to $k_x\/k_0 \\approx 2$ for the thickest considered lens $L=60$ nm.\n\\begin{figure}[ht!]\n\\centering\na)\\includegraphics[width=.6\\linewidth]{t2}\nb)\\includegraphics[width=.6\\linewidth]{t1}\n\\caption{Transmission coefficient as a function of the x component of wave vector $k_x$ a) calculated from Eq. (\\ref{eq:transmisja}) and b) obtained in FDTD simulation.}\\label{RT_2}\n\\end{figure}\n\nAfter establishing the accuracy of the FDTD simulation, one can use the numerical results to investigate various aspects of superlens operation that are not described by the steady state transmission coefficient. As mentioned in the discussion of the Fig. \\ref{RT_1}, the presence of excitons has a limited impact on transmission coefficient; overall, the imaginary part of Cu$_2$O permittivity in the spectral range of 2P exciton resonance is of the order of $Im$ $\\epsilon \\sim 10^{-3}$ (Fig. \\ref{Cu2O_model}) as compared to $Im$ $\\epsilon \\sim 1$ in the case of copper (\\ref{Cu_model}). Therefore, the majority of absorption comes from the metal. On the other hand, it is well known that the surface plasmons-polaritons (SPPs) excited on the metal-dielectric interface are very sensitive to the susceptibility changes in the dielectric. In the case of superlens, these plasmons are excited on both metal-dielectric interfaces and they play a pivotal role in the evanescent field amplification and thus the image formation \\cite{Pendry,Smith03}. At the same time, they tend to distort the image field \\cite{Wang11}. \n\nTo study these plasmons and their influence on superlens operation, let's consider the numerically obtained electric field amplitude in the optical axis of the system e.g., on the line connecting a single point source and its image, shown on the Fig. \\ref{RT_3} a). Naturally, the highest peak of the field amplitude is located at the source. One can also see significant peaks on the Cu-Cu$_2$O interfaces (continuous, vertical lines). These maxima correspond to surface plasmon resonances. The last peak is located on the image plane. Interestingly, while the image peak is almost the same regardless whether the energy is tuned to the 2P excitonic resonance (2147 meV) or not, the local maxima corresponding to SPPs are significantly affected by the presence of excitons. This means that the superlens containing excitons is capable of reconstructing the image while partially suppressing the surface plasmons. As a result, in the image of the point source (Fig. \\ref{RT_3} b)), the side maxima caused by interference with the plasmons are reduced. These findings are consistent with the results by Wang \\textit{et al} \\cite{Wang11}, where it is shown that adding loss in the image region can provide a significant enhancement of the image quality.\n\\begin{figure}[ht!]\n\\centering\na)\\includegraphics[width=.9\\linewidth]{cross}\nb)\\includegraphics[width=.9\\linewidth]{image}\n\\caption{Normalized electric field amplitude a) in the optical axis b) in the image plane.}\\label{RT_3}\n\\end{figure}\nIt should be noted that the influence of excitons on the system is limited to a very narrow part of the spectrum; the largest 2P exciton resonance is characterized by a linewidth of $\\Gamma \\sim 5$ meV, while the energy range where a good match between Cu$_2$O and copper permittivity that allows for superlens operation is wider than the whole excitonic spectrum (see Fig. \\ref{Cu_model}). This means that the system can be fine tuned with small changes of illuminating light frequency without impacting the image quality due to mismatch of permittivities. \n\n\\section{Superlens in time domain}\n\nAn important aspect of the comprehensive description of the superlens operation is the study of the time evolution of the image. The Fig. \\ref{EVO1} depicts the field amplitude in the image plane of a $L=20$ nm superlens illuminated by two point sources. The distance between sources is 40 nm, which is approximately $\\lambda\/5$ (the energy of 2147 meV corresponds to a wavelength of 571 nm in vacuum and 210 nm in Cu$_2$O). The time unit is scaled by the speed of light in Cu$_2$O which is around $ 1.1 \\cdot 10^8$ m\/s. \n\\begin{figure}[ht!]\n\\includegraphics[width=.999\\linewidth]{evo1}\n\\caption{Normalized field amplitude in the image plane as a function of time.}\\label{EVO1}\n\\end{figure}\nOne can see that initial, wide peak forming in the center of the image quickly transforms into two narrow peaks that are images of the radiation sources. The gradual formation of the image is caused by the varying group velocity of specific wave modes. In particular, the high $k_x$ modes that correspond to the fine details of the image are characterized by a low group velocity and thus need longer time to propagate through the lens. Alternatively, one can describe this process in terms of mode amplification; an evanescent wave with very large $k_x$ (e.g large, imaginary wave vector) decays very rapidly in free space. Thus, to reconstruct the image, these modes need to be more amplified by the lens, which takes longer time. \n\nOn the Fig. \\ref{EVO2}, a similar evolution is presented, calculated for two energies. On the left panel the energy is set to match the 2P exciton energy, while the right panel corresponds to the spectral region away from excitonic resonances.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{evo2}\n\\caption{Normalized field amplitude in the image plane (color) as a function of time, for two different energies.}\\label{EVO2}\n\\end{figure}\nThe diagonal dashed lines on the bottom correspond to the casualty limit; no field can propagate in the system faster than $x=ct$. The horizontal dashed line corresponds to the moment when the image field reaches half of its maximum amplitude. In the case where excitonic resonance is present, the group velocity of wave modes is reduced due to the fact that an excitonic resonance corresponds to a sudden increase of normal dispersion. Specifically, we recall that the group velocity is given by\n\\begin{equation}\nV_g = \\frac{c}{n+\\omega\\frac{\\partial n}{\\partial \\omega}}\n\\end{equation}\nwhere $\\omega$ is the radiation frequency and $n=\\sqrt{\\epsilon}$ is the refraction coefficient. In the spectral range of excitonic resonance, the permittivity changes very rapidly with frequency, so that $\\partial n\/\\partial \\omega$ has a significant value and noticeably reduces the group velocity \\cite{my_arxiv}. Due to this, the image formation on the left panel of the Fig. \\ref{EVO2} is slightly delayed. Notably, the presence of excitons has a stabilizing effect on the image; the spurious peak located between the two source images is noticeably weaker in the left panel, confirming the earlier findings that additional absorption in dielectric is beneficial to the image quality.\n\nIt should be also noted that the superlens resolution can be potentially greatly improved by considering time-dependent illumination \\cite{Archambault12}. The use of excitons opens up a new opportunity to provide time-dependent superlens properties; by using a separate control field, one can take advantage of the nonlinear properties of excitons in Cu$_2$O due to the Rydberg blockade effect and change its absorption on demand. Furthermore, we recall that the optical properties of excitons are highly nonlinear \\cite{my2022}; in the case of superlens, the optical field is highly focused in subwavelength volume, further enhancing the system nonlinearity, especially in high power pulsed operation. Finally, one can use an external field to modulate the spatial distribution of excitons and thus induce a spatially nonuniform optical properties of Cu$_2$O. In such a case, it might be possible to realize enhanced transmission of evanescent waves and improvement of resolution that are usually accomplished with lens surface roughness \\cite{Fang2003, Wang11}. In contrast with fabricated roughness, the exciton distribution can be tuned on demand. \n\n\\section{Conclusions}\nSince their recent discovery in 2014, Rydberg excitons in Cu$_2$O are becoming a very valuable tool in creating nonlinear, controllable optical nanostructures. One of their possible applications are tunable plasmonic devices, where copper is used as the metal that facilitates excitation of SPPs and allows for easy fabrication of the structure. We show that a Cu layer surrounded by Cu$_2$O can act as an efficient superlens. It is demonstrated that such a device is capable of subwavelength imaging with a resolution approaching $\\lambda\/9$, which is competitive with traditional silver and gold based setups. Moreover, it is shown that excitons in Cu$_2$O can be used to modify the optical properties of the lens on demand and that the increased absorption due to the excitonic resonances is beneficial to the image quality. Several possible prospects for taking advantage of nonlinear optical properties of Cu$_2$O due to the excitons are outlined in the context of superlens design.\n\n\n\\section{Introduction}\nIn recent years optical communication through metallic nanostructures has attracted significant attention.\nSince 2000 when Pendry initiated the idea of superlens imaging \\cite{Pendry}, it has became an important research topic in past years. \n Noble metal nanostuctures based on gold and silver have allowed the light to be guides beyond the diffraction limit, which has opened new possibilities for further miniaturization and applications of optoelectronic devices.\n \nIn 2005 Lee \\textit{et al} \\cite{Lee05} presented experimental and theoretical studies of the optical superlens using a thin silver slab and demonstrated optical imaging with resolution well below the diffraction limit.\nThe investigations of different, cheaper material to construct efficient superlens systems basing on different metals have started; Zhao \\emph{et al} \\cite{Zhao16} have studied several superlens materials, including Ni, Cr and also Cu in the 400 nm wavelength range concluding that the dielectrics covering the superlens surfaces have to be characterized by similar permittivities, which influence the imaging quality. It has also been noted that a large permittivity of the dielectric in the front of the superlens is in favour of the imaging effect. Mkhitaryan \\emph{et al} have recently demonstrated that high Q factor plasmonic structures can be realized with copper \\cite{Mkhitaryan21}, further confirming that Cu is a promising candidate for construction of efficient superlens.\n\nThe general theoretical model valid for both homogeneous and\nevanescent waves, transfer-matrix analyses of the near-field metallic superlens and\nthe numerical finite-difference in time-domain (FDTD) simulations has been proposed by Tremblay \\cite{Tremblay} and the detailed\nrole of near-field optics in a superlens image formation has been reviewed recently by Adams et al \\cite{Adams}.\n\nMoreover, the important aspect of producing superlens is the accomplishment of the optimal resolution. It is affected by the finite thickness of the superlens, which generates both long- and short-range surface plasmon modes, they in turn influence a transfer function distorting the image field. Wang et al \\cite{Wang11} found that the presence of loss is not always detrimental to the superlens resolution. \n \nSince the discovery of negative index metamaterials and superlens, there is a strong need for a lens medium that is characterized not only by the necessary negative permittivity, but also one that offers tunability of its optical properties. By coupling various quantum systems with the metamaterial, one can provide such a capability \\cite{Quach2011}. These so-called quantum metamaterials \\cite{Stav2018} are a new class of media that bridges the gap between quantum systems and classical metamaterials. In this paper, we focus on the recently discovered Rydberg excitons (REs) in Cu$_2$O \\cite{Kazimierczuk} and their potential application in copper based superlens.\nREs are highly excited excitonic states characterized by a principal quantum number $n>>$1 for dipole-allowded P-type enveloped wavefunctions. In principle, they have similar scaling as Rydberg atoms \\cite{Heck17} although the physical origin of these similarities in their case bases on a complex valence band structure and different selection rules for excitons. The dimensions of REs reach micrometers and their life-times can reach hundreds of nanoseconds. Nowadays REs become one of the rapidly developing topic in solid-state optics; the studies extend from linear to nonlinear regimes in bulk media and nanostructures \\cite{Orfanakis,my2022,Thomas2022}. Recently, REs interaction with plasmons has been studied, showing possibility of an evident extension of propagation length and bridging plasmonics and Rydberg excitons physics \\cite{my_arxiv}. Here, we expand upon that work by considering the effect of excitons on the dynamics of copper-based superlens.\n \n \n We perform the detailed analysis of Cu and Cu$_2$O, with its particularly high $\\epsilon=7.5$ based superlens taking into account all above mentioned aspects. \n The required matching of permittivity of Cu and Cu$_2$O can be achieved either by tuning the frequency, adjusting the Cu film thickness \\cite{Stenzel19} or using metal-dielectric composites \\cite{Cai05} such as a fine mix of Cu and Cu$_2$O layers with excitons. Tailoring the optical properties of medium opened a new possibility of controlling the image. This is because the propagation dynamics of the field depends on the diffraction and dispersion properties of the medium. In particular, Archambault \\emph{el al}\\cite{Archambault12} pointed out that an improvement of spatial resolution has to be achieved using time-dependent incident pulse. In the spirit of their conclusions we examine the influence of the group velocities of various modes on the image formation. \n \nThe paper is organized as follows. In the first section, a numerical model of the optical properties of copper and copper oxide is presented, which is later used in theoretical calculations and numerical simulations. Then, the system setup is outlined. The third section is devoted to the study of the steady state transmission coefficient of the copper superlens. Next, the dynamic of the image formation in the superlens is investigated. Finally, the conclusions are presented. The detailed description of the numerical simulation method is included in the appendix.\n \n\n \n\\section{Material model}\nFor the numerical description of the optical properties of Cu, we use the Drude-Lorentz model in which the permittivity, in the case of two-oscillator model, is given by \\cite{Okada}\n\\begin{equation}\\label{eq:DL}\n\\epsilon(\\omega)=\\varepsilon_\\infty+\\frac{f_0}{\\omega_0^2-\\omega^2+i\\omega \\Gamma_0}+\\frac{f_1}{\\omega_1^2-\\omega^2+i\\omega \\Gamma_1}, \n\\end{equation}\nwith the numerical value of parameters summarized in the Table \\ref{parTable} in Appendix A. The values of parameters are adjusted to provide a good fit to the experimental data presented in ref. \\cite{Hollstein}. A comparison of the experimentally measured permittivity spectrum and the model values is shown on the Fig. \\ref{Cu_model}. In particular, the model is adjusted to provide a good fit in the energy region between $E \\sim 2140 - 2170$ meV (dashed, vertical lines), where excitonic resonances are present. In particular, we assume that the optical properties of Cu are determined by an ensemble of free electrons. The oscillator strength $f_0$ is determined by density of electron plasma, with so-called plasma frequency $\\omega_p = \\sqrt{f_0}$. Due to the fact that the electrons are not bound, there is no restoring force that would result in some intrinsic resonant frequency of charges and so $\\omega_0=0$. The damping parameter $\\Gamma_0$ represents various dissipative processes. The parameters of the second oscillator $f_1$, $\\omega_1$, $\\Gamma_1$ are used to include the effect of inter-band transitions that are responsible for an increase of absorption around 2.35 eV (see Fig. \\ref{Cu_model}).\n\nFor the copper oxide, the same Eq. (\\ref{eq:DL}) with a different set of parameters can be used. The two main contributions to the Cu$_2$O susceptibility are the constant value of $\\epsilon_\\infty=7.5$ and the effect of 2P exciton resonance, as shown on the Fig. \\ref{Cu2O_model}. This resonance is included as an oscillator with the frequency corresponding to the 2P exciton energy $2147$ meV and the oscillator strength and damping constant adjusted to match the height and width of absorption peak of the exciton \\cite{Raczynska2016}. In the case of Cu$_2$O, the second oscillator in Eq. (\\ref{eq:DL}) is unused.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{cu}\n\\caption{Real (continuous line) and imaginary (dashed line) part of permittivity given by Eq. \\ref{eq:DL}, compared to experimental data (dots).}\\label{Cu_model}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\includegraphics[width=\\linewidth]{cu2o}\n\\caption{Imaginary (left side) and real (right side) part of permittivity. Solid lines is given by theoretical calculation for Cu$_2$O bulk 2P exciton from \\cite{Raczynska2016} and dashed lines given by Eq. \\ref{eq:DL} with parameters depict in Table \\ref{parTable}.}\\label{Cu2O_model}\n\\end{figure}\n\n\\section{System setup}\nLets consider a multilayer system consisting of a thin copper layer surrounded by copper oxide, as shown on the Fig. \\ref{superlensmodel}. The optical axis of the system (horizontal, dashed line) is aligned to the $y$ axis and the source\/image planes (vertical dashed lines) are parallel to the $x$ axis. Blue lines depict a schematic of optical rays connecting the point source with its image (red dots). For an efficient superlens operation, the thickness $L$ needs to be smaller than approximately $0.1 \\lambda$, where $\\lambda$ is the wavelength of the light illuminating the system. \n\\begin{figure}[ht!]\n\\includegraphics[width=0.6\\linewidth]{schemat}\n\\caption{Schematic representation of a copper superlens surrounded by Cu$_2$O.}\\label{superlensmodel}\n\\end{figure}\nThe important parameter determining the superlens resolution is the wave vector $k_0$ and its $x$ component $k_x$. Specifically, to achieve subwavelength resolution, the superlens needs to transmit high spatial frequency components with $k_x>k_0$. In such a case, the $y$ component of the wave vector $k_y = \\sqrt{k_0^2 - k_x^2}$ is imaginary and thus it describes evanescent waves. The key characteristic of the superlens is the capability to amplify these rapidly decaying evanescent waves \\cite{Pendry}.\n\n\\section{Transmission coefficient}\nOne of the primary tools for studying the performance of the superlens is the transmission spectrum in the wave vector domain, e.g. the relation $T(k_x)$. The system is capable of subwavelength imaging when the values of $T(k_x>k_0)$ are significant. Similar to the approach presented in \\cite{Wang11}, we calculate the transmission coefficient (transfer function) of a flat superlens in electrostatic limit, e.g. in the case where the lens thickness is considerably smaller than illuminated wavelength. Let's consider a source and image plane located at a distance $L\/2$ from the superlens surfaces (see Fig. \\ref{superlensmodel}). Transmission coefficient is calculated by multilayer transmission using Parratt recursion relation \\cite{Parratt1954} \n\\begin{equation}\\label{eq:transmisja}\nT=T_0T_1...T_N,\n\\end{equation}\nwhere $T_j$ is transmission coefficient in each layer is given by \n \\begin{equation}\n T_j=e^{i k_z^{(j+1)}L_{j+1}} \\frac{1+F_j}{1+F_jR_{j+1}},\n\\end{equation}\nwith reflection coefficient \n\\begin{equation}\nR_j=e^{2ik_z^{(j)}L_j}\\frac{F_j+R_{j+1}}{1+F_j R_{j+1}},\n\\end{equation}\nand \n\\begin{equation}\nF_j=\\frac{\\epsilon_{j+1}k_z^{j}-\\epsilon_{j}k_z^{j+1}}{\\epsilon_{j+1}k_z^{j}+\\epsilon_{j}k_z^{j+1}}.\n\\end{equation}\nParameter $j=0,1,2$ numerates Cu$_2$O, Cu, Cu$_2$O layers, correspondingly. $k_z^{j}$ is z-component of wavevector in $j$ layer and $L_j$ is its thickness with $L_0=L_{N+1}=0$ and $R_{N+1}=0$. Similar approach one can found in other papers about superlenses, see \\cite{Splawinski2021,Hakkarainen2009}\n\nAs a first step, we can verify the derived theoretical transmission relation by comparing it with the results presented by Wang \\textit{at al} \\cite{Wang11}. In particular, the Fig. \\ref{RT_verif} depicts the transmission coefficient of idealized superlens consisting of a single layer of $\\epsilon=-1$ material, with $\\lambda\/10$ thickness, surrounded by vacuum. There are several characteristic points in the transmission spectrum; for $k_x\/k_0<1$, the transmitted mode corresponds to regular, propagating wave; the transmission coefficient in this range is smaller than 1. For higher values of $k_x$, the transmission coefficient describes evanescent waves that do not carry energy, so one can have $T>1$. In particular, the transmission spectrum usually contains two poles \\cite{Wang11}, which are clearly visible on the Fig. \\ref{RT_verif}. For $k_x\/k_0>9$, there is a cutoff point where the transmission quickly drops. This means that the spatial resolution of the superlens is limited to approximately $\\lambda\/9$; one can increase this resolution by making the lens thinner \\cite{Pendry}. The presence of the sharp poles in the transfer function (transmission coefficient) is generally undesirable as it introduces distortion \\cite{Wang11}; one can see that the introduction of absorption (Fig. \\ref{RT_verif}, blue line) significantly reduces the gain on the second, high $k_x$ pole, so that the transfer function becomes more flat (e.g. all wave modes are amplified to a similar degree). Thus, despite the fact that absorption is in general detrimental to the superlens resolution, its presence can be beneficial to the image quality. Due to this reason, our proposal of Cu-based superlens has some advantages over traditional designs based on noble metals.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{r1}\n\\caption{Transmission coefficient as a function of the x component of wave vector $k_x$, calculated for a model superlens with $\\epsilon=-1$.}\\label{RT_verif}\n\\end{figure}\n\nAs a next step, we consider a copper superlens with a thickness $L=20$ nm, surrounded by Cu$_2$O. From the perspective of practical applications, such a system is potentially simple to fabricate by controlled oxidation of Cu surface \\cite{Rodriguez11}, vapour deposition \\cite{Steinhauer20} or other techniques \\cite{Lynch21}. The transmission coefficient calculated for a range of values of wave vector component $k_x$ and energy $E$ is shown on the Fig. \\ref{RT_1}. \n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{r2}\n\\caption{Transmission coefficient (color) as a function of the x component of wave vector $k_x$ and energy.}\\label{RT_1}\n\\end{figure}\nDue to the fact that copper has a significantly higher absorption coefficient than the idealized superlens material considered previously, there is only a single pole in the transmission coefficient at $k_x \\approx k_0$. Overall, the usable range of wave vector values where transmission $T > 0.1$ is $k_x < 6$, which corresponds to roughly $\\lambda\/7$ resolution and is comparable to silver superlens performance \\cite{Sheng2009,Hakkainen2010}. One can notice that the strongest 2P exciton, which corresponds to 2147 meV (white, dashed line), produces a noticeable change in transmission coefficient (especially visible on the black contours). This means that the excitons in Cu$_2$O are a viable tool for fine-tuning of the lens properties in a very narrow energy range. \n\nOne of the primary considerations in the superlens design is the choice of thickness. In fact in the case of metallic lens with negative permittivity and positive permeability, the thickness needs to be much smaller than illuminated wavelength \\cite{Fang2003}. Also, in contrast to the ideal n=-1 superlens, there is a finite range of the values of wave vector component $k_x$ that are transmitted, as illustrated earlier on the Fig. \\ref{RT_1}. In general, we can expect that high spatial-frequency components of the image decay much faster in a thicker lens \\cite{Wang11}. This is indeed the case in the calculated results shown on the Fig. \\ref{RT_2}. Overall, there is a good match between theoretical and numerical results. The first local transmission maximum, which is also present in the idealized case on the Fig. \\ref{RT_verif}, forms a very sharp peak in theoretical calculations. To further confirm these results, we use the Finite-Difference Time-Domain (FDTD) simulation that is based directly on Maxwell's equations and thus allows for a direct study of superlens optical properties with no simplifying assumptions that are present in theoretical description. The only important limitation of FDTD is the limited time and spatial resolution, which results in a smaller, broader peak in the numerical transmission spectrum (Fig. \\ref{RT_2} b)). As mentioned before, the secondary peak on the Fig. \\ref{RT_verif} located around $k_x=7k_0$ is very sensitive to absorption and in case of Cu, it is almost completely suppressed. One can see that only in the case of $L=12$ nm, a very wide local maximum of transmission is visible on the Fig. \\ref{RT_2} at $k_x \\approx 5 k_0$. Similar to the earlier results, the lack of sharp peaks in the transmission spectrum is beneficial, as it means that all spatial frequency components are amplified to the same degree \\cite{Wang11}. In general, a lens thickness $L$ of the order of 10 nm is favourable, with significant transmission up to $k_x\/k_0 \\approx 10$. As the lens becomes thicker, the transmission cutoff becomes smaller, up to $k_x\/k_0 \\approx 2$ for the thickest considered lens $L=60$ nm.\n\\begin{figure}[ht!]\n\\centering\na)\\includegraphics[width=.6\\linewidth]{t2}\nb)\\includegraphics[width=.6\\linewidth]{t1}\n\\caption{Transmission coefficient as a function of the x component of wave vector $k_x$ a) calculated from Eq. (\\ref{eq:transmisja}) and b) obtained in FDTD simulation.}\\label{RT_2}\n\\end{figure}\n\nAfter establishing the accuracy of the FDTD simulation, one can use the numerical results to investigate various aspects of superlens operation that are not described by the steady state transmission coefficient. As mentioned in the discussion of the Fig. \\ref{RT_1}, the presence of excitons has a limited impact on transmission coefficient; overall, the imaginary part of Cu$_2$O permittivity in the spectral range of 2P exciton resonance is of the order of $Im$ $\\epsilon \\sim 10^{-3}$ (Fig. \\ref{Cu2O_model}) as compared to $Im$ $\\epsilon \\sim 1$ in the case of copper (\\ref{Cu_model}). Therefore, the majority of absorption comes from the metal. On the other hand, it is well known that the surface plasmons-polaritons (SPPs) excited on the metal-dielectric interface are very sensitive to the susceptibility changes in the dielectric. In the case of superlens, these plasmons are excited on both metal-dielectric interfaces and they play a pivotal role in the evanescent field amplification and thus the image formation \\cite{Pendry,Smith03}. At the same time, they tend to distort the image field \\cite{Wang11}. \n\nTo study these plasmons and their influence on superlens operation, let's consider the numerically obtained electric field amplitude in the optical axis of the system e.g., on the line connecting a single point source and its image, shown on the Fig. \\ref{RT_3} a). Naturally, the highest peak of the field amplitude is located at the source. One can also see significant peaks on the Cu-Cu$_2$O interfaces (continuous, vertical lines). These maxima correspond to surface plasmon resonances. The last peak is located on the image plane. Interestingly, while the image peak is almost the same regardless whether the energy is tuned to the 2P excitonic resonance (2147 meV) or not, the local maxima corresponding to SPPs are significantly affected by the presence of excitons. This means that the superlens containing excitons is capable of reconstructing the image while partially suppressing the surface plasmons. As a result, in the image of the point source (Fig. \\ref{RT_3} b)), the side maxima caused by interference with the plasmons are reduced. These findings are consistent with the results by Wang \\textit{et al} \\cite{Wang11}, where it is shown that adding loss in the image region can provide a significant enhancement of the image quality.\n\\begin{figure}[ht!]\n\\centering\na)\\includegraphics[width=.9\\linewidth]{cross}\nb)\\includegraphics[width=.9\\linewidth]{image}\n\\caption{Normalized electric field amplitude a) in the optical axis b) in the image plane.}\\label{RT_3}\n\\end{figure}\nIt should be noted that the influence of excitons on the system is limited to a very narrow part of the spectrum; the largest 2P exciton resonance is characterized by a linewidth of $\\Gamma \\sim 5$ meV, while the energy range where a good match between Cu$_2$O and copper permittivity that allows for superlens operation is wider than the whole excitonic spectrum (see Fig. \\ref{Cu_model}). This means that the system can be fine tuned with small changes of illuminating light frequency without impacting the image quality due to mismatch of permittivities. \n\n\\section{Superlens in time domain}\n\nAn important aspect of the comprehensive description of the superlens operation is the study of the time evolution of the image. The Fig. \\ref{EVO1} depicts the field amplitude in the image plane of a $L=20$ nm superlens illuminated by two point sources. The distance between sources is 40 nm, which is approximately $\\lambda\/5$ (the energy of 2147 meV corresponds to a wavelength of 571 nm in vacuum and 210 nm in Cu$_2$O). The time unit is scaled by the speed of light in Cu$_2$O which is around $ 1.1 \\cdot 10^8$ m\/s. \n\\begin{figure}[ht!]\n\\includegraphics[width=.999\\linewidth]{evo1}\n\\caption{Normalized field amplitude in the image plane as a function of time.}\\label{EVO1}\n\\end{figure}\nOne can see that initial, wide peak forming in the center of the image quickly transforms into two narrow peaks that are images of the radiation sources. The gradual formation of the image is caused by the varying group velocity of specific wave modes. In particular, the high $k_x$ modes that correspond to the fine details of the image are characterized by a low group velocity and thus need longer time to propagate through the lens. Alternatively, one can describe this process in terms of mode amplification; an evanescent wave with very large $k_x$ (e.g large, imaginary wave vector) decays very rapidly in free space. Thus, to reconstruct the image, these modes need to be more amplified by the lens, which takes longer time. \n\nOn the Fig. \\ref{EVO2}, a similar evolution is presented, calculated for two energies. On the left panel the energy is set to match the 2P exciton energy, while the right panel corresponds to the spectral region away from excitonic resonances.\n\\begin{figure}[ht!]\n\\includegraphics[width=.9\\linewidth]{evo2}\n\\caption{Normalized field amplitude in the image plane (color) as a function of time, for two different energies.}\\label{EVO2}\n\\end{figure}\nThe diagonal dashed lines on the bottom correspond to the casualty limit; no field can propagate in the system faster than $x=ct$. The horizontal dashed line corresponds to the moment when the image field reaches half of its maximum amplitude. In the case where excitonic resonance is present, the group velocity of wave modes is reduced due to the fact that an excitonic resonance corresponds to a sudden increase of normal dispersion. Specifically, we recall that the group velocity is given by\n\\begin{equation}\nV_g = \\frac{c}{n+\\omega\\frac{\\partial n}{\\partial \\omega}}\n\\end{equation}\nwhere $\\omega$ is the radiation frequency and $n=\\sqrt{\\epsilon}$ is the refraction coefficient. In the spectral range of excitonic resonance, the permittivity changes very rapidly with frequency, so that $\\partial n\/\\partial \\omega$ has a significant value and noticeably reduces the group velocity \\cite{my_arxiv}. Due to this, the image formation on the left panel of the Fig. \\ref{EVO2} is slightly delayed. Notably, the presence of excitons has a stabilizing effect on the image; the spurious peak located between the two source images is noticeably weaker in the left panel, confirming the earlier findings that additional absorption in dielectric is beneficial to the image quality.\n\nIt should be also noted that the superlens resolution can be potentially greatly improved by considering time-dependent illumination \\cite{Archambault12}. The use of excitons opens up a new opportunity to provide time-dependent superlens properties; by using a separate control field, one can take advantage of the nonlinear properties of excitons in Cu$_2$O due to the Rydberg blockade effect and change its absorption on demand. Furthermore, we recall that the optical properties of excitons are highly nonlinear \\cite{my2022}; in the case of superlens, the optical field is highly focused in subwavelength volume, further enhancing the system nonlinearity, especially in high power pulsed operation. Finally, one can use an external field to modulate the spatial distribution of excitons and thus induce a spatially nonuniform optical properties of Cu$_2$O. In such a case, it might be possible to realize enhanced transmission of evanescent waves and improvement of resolution that are usually accomplished with lens surface roughness \\cite{Fang2003, Wang11}. In contrast with fabricated roughness, the exciton distribution can be tuned on demand. \n\n\\section{Conclusions}\nSince their recent discovery in 2014, Rydberg excitons in Cu$_2$O are becoming a very valuable tool in creating nonlinear, controllable optical nanostructures. One of their possible applications are tunable plasmonic devices, where copper is used as the metal that facilitates excitation of SPPs and allows for easy fabrication of the structure. We show that a Cu layer surrounded by Cu$_2$O can act as an efficient superlens. It is demonstrated that such a device is capable of subwavelength imaging with a resolution approaching $\\lambda\/9$, which is competitive with traditional silver and gold based setups. Moreover, it is shown that excitons in Cu$_2$O can be used to modify the optical properties of the lens on demand and that the increased absorption due to the excitonic resonances is beneficial to the image quality. Several possible prospects for taking advantage of nonlinear optical properties of Cu$_2$O due to the excitons are outlined in the context of superlens design.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Eidesstattliche Erkl\u00e4rung}\n\t\n\tHiermit erkl\u00e4re ich, \\textbf{Alexander Kilian} (Matrikelnummer~77856), gegen\u00fcber der Fakult\u00e4t f\u00fcr Informatik und Mathematik der Universit\u00e4t Passau, dass ich die Masterarbeit mit dem Thema \\textit{Infinite-Dimensional Port-Hamiltonian Systems with a Moving Interface} selbstst\u00e4ndig angefertigt und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe. Alle w\u00f6rtlich und sinngem\u00e4\u00df \u00fcbernommenen Ausf\u00fchrungen wurden als solche gekennzeichnet. Weiterhin erkl\u00e4re ich, dass ich diese Arbeit in gleicher oder \u00e4hnlicher Form nicht bereits einer anderen Pr\u00fcfungsbeh\u00f6rde vorgelegt habe.\n\t\n\\vspace{2.5 cm}\n\t\n\tPassau, den 20.12.2021 \\\\\n\\noindent \\begin{flushleft}\n\t\\begin{tabular}{l}\n\t\t\\includegraphics[height=0.7cm]{Signature.png} \\tabularnewline\n\t\tAlexander Kilian \n\t\\end{tabular}\n\t\\par\\end{flushleft}\n\t\\clearpage\n\n\\chapter*{Summary}\t\nThis thesis deals with the formulation and analysis of two systems of conservation laws defined on two complementary intervals and coupled by some moving interface as a single infinite-dimensional port-Hamiltonian system. This approach may be regarded as an extension of the well-established mathematical framework of boundary port-Hamiltonian systems, i.e., infinite-dimensional Hamiltonian systems interacting with its physical environment. \n\\vspace{0.5 cm}\\\\\nA vast variety of multi-physics systems can be modeled as a Hamiltonian system, a special type of ordinary and partial differential equations. Such systems have a Hamiltonian, which describes the internally stored energy of the system. By allowing such systems to be open, one usually refers to this class of systems as (boundary) port-Hamiltonian systems. For the analysis of infinite-dimensional port-Hamiltonian systems, one combines the energy-based approach by means of Hamiltonians with the poweful framework of strongly continuous semigroups applied in infinite-dimensional systems theory. This approach has led to fundamental well-posedness and stability results in the field of infinite-dimensional port-Hamiltonian systems. Furthermore, the power flow at the spatial boundary has been utilized to control port-Hamiltonian systems.\n\\vspace{0.5 cm}\\\\\nThe great success of this system description motivates to extend the port-Hamiltonian formulation to systems of conservation laws that are coupled by some moving interface. In order to keep track of the power flow at the position of the interface, one has to introduce interface port variables. In this thesis, we provide a well-posed port-Hamiltonian model of two distinct systems of conservation laws defined on two complementary, 1-dimensional spatial domains that are coupled by a fixed interface, and illustrate this result by reference to two transmission lines coupled through a resistor. Furthermore, we discuss under which boundary and interface conditions this system is exponentially stable. Since the formulation in case of a moving interface is quite troubled, we present some results that may be regarded as a first step towards a well-posed model, and debate the main issues that have to be tackled in the future.\n\\clearpage\n\n\\chapter*{Zusammenfassung}\nZiel der Masterarbeit ist es, zwei auf komplement\u00e4ren Intervallen definierte Systeme von Energieerhaltungss\u00e4tzen, die durch eine mobile Schnittstelle gekoppelt werden, mathematisch als ein unendlich-dimensionales port-Hamiltonsches System zu modellieren und zu analysieren. Dieses Modell kann als eine Verallgemeinerung der port-Hamiltonschen Formulierung linearer unendlich-dimensionaler Systeme, die mit ihrer Umgebung interagieren, verstanden werden. \n\\vspace{0.5 cm}\\\\\nZahlreiche physikalische Prozesse k\u00f6nnen mathematisch mit Hilfe sogenannter Hamiltonscher Systeme modelliert werden. Diese besitzen eine Hamilton-Funktion, die in vielen Anwendungen die Systemenergie beschreibt. Erlaubt man einen Energieaustausch mit der Umgebung des Systems, so spricht man von port-Hamiltonschen Systemen, die sowohl durch gew\u00f6hnliche als auch partielle Differentialgleichungen beschrieben werden k\u00f6nnen. Bei der Untersuchung unendlich-dimensionaler port-Hamiltonscher Systeme verbindet man den abstrakten, funktionalanalytischen Ansatz mittels der Halbgruppentheorie mit dem auf Hamilton-Funktionen basierten, physikalisch motivierten Ansatz. Mit dieser Methodik lie\u00dfen sich elementare Wohlgestelltheits- und Stabilit\u00e4tsresultate auf dem Gebiet unendlich-dimensionaler port-Hamiltonscher Systeme beweisen. Dar\u00fcber hinaus eignet sich das port-Hamiltonsche Modell aufgrund des Energieaustauschs mit der Umgebung bestens zur energiebasierten Regelung solcher Systeme.\n\\vspace{0.5 cm}\\\\\nAufgrund des beachtlichen Erfolgs dieser Systembeschreibung soll in dieser Arbeit der Versuch einer port-Hamiltonschen Formulierung zweier Systeme von Energieerhaltungss\u00e4tzen, die durch eine mobile Schnittstelle gekoppelt sind, unternommen werden. Dazu werden Schnittstellen-Port-Variablen eingef\u00fchrt, um den Energiefluss an der sich bewegenden Schnitt- stelle zu modellieren. Wir werden ein wohlgestelltes System des eben beschriebenen Szenarios im Falle einer fixen Schnittstellenposition pr\u00e4sentieren und dieses Resultat anhand zweier \u00dcbertragungsleitungen mit unterschiedlichen physikalischen Eigenschaften, die durch einen elektrischen Widerstand gekoppelt werden, illustrieren. Ferner werden wir charakterisieren, unter welchen Rand- und Schnittstellenbedingungen ein solches System exponentiell stabil ist. Die port-Hamiltonsche Formulierung im Falle einer mobilen Schnittstellen ist problembeladen. Wir werden daher Probleme und offene Fragen, die es in Zukunft noch zu kl\u00e4ren gilt, ausf\u00fchrlich diskutieren und f\u00fcr den Wohlgestelltheits-Nachweis essentielle Eigenschaften des modellierten Systems pr\u00e4sentieren. \n\\clearpage\n\n\t\n\t\\pagenumbering{roman}\t\n\t\\tableofcontents\n\n\\listoffigures\n\\listoftables\n\t\n\t\n\t\n\t\\section*{Notation}\n\n\t\\def1.5{1.5}\n\t\\begin{tabular}{l l}\n\t\t\\textbf{Symbol} & \\textbf{Description} \\\\\n\t\t$\\mathbb{N}$, $\\mathbb{R}$, $\\mathbb{C}$ & the set of natural, real, and complex numbers, respectively \\\\\n\t\t$\\mathbb{R}_{>0}$ & the set of positive real numbers \\\\\n\t\t$\\mathbb{R}_+$ & the set of non-negative real numbers \\\\\n\t\t$\\mathbb{F}^{m \\times n}$ & the vector space of $m \\times n$-matrices with entries in $\\mathbb{F}$ \\\\\n\t\t$\\mathbb{F}^n$ & the vector space of $n$-dimensional column vectors with entries in $\\mathbb{F}$ \\\\\n\t\t$\\realpart(z)$ & real part of $z \\in \\mathbb{C}$ \\\\\n\t\t$\\| \\cdot \\|_2$ & Euclidean norm on $\\mathbb{R}^n$ \\\\\n\t\t$\\| \\cdot \\|_{\\operatornorm \\colon D(A) \\to X}$ & operator norm of an operator $A \\colon D(A) \\subset X \\to X$ \\\\\n\t\t$\\| \\cdot \\|_{\\mathcal{L}(X)}$ & operator norm on $\\mathcal{L}(X)$ \\\\\n\t\t$X'$ & dual space of a vector space $X$ \\\\\n\t\t$\\langle \\cdot, \\cdot\\rangle_{X' \\times X}$ & natural pairing between a vector space $X$ and its dual space $X'$ \\\\\n\t\t$\\langle \\cdot \\mid \\cdot \\rangle$ & power pairing defined on a Hilbert space \\\\\n\t\t$\\ll \\cdot , \\cdot \\gg$ & plus pairing associated with a power pairing \\\\\n\t\t$M^{\\top}$, $v^{\\top}$ & transpose of a matrix $M \\in \\mathbb{F}^{m \\times n}$\/a vector $v \\in \\mathbb{F}^n$ \\\\\n\t\t$M > N$ $(M \\geq N)$ & positive (semi-)definiteness of $M - N \\in \\mathbb{F}^{n \\times n}$ \\\\\n\t\t$A^{\\ast}$ & (formal) Hilbert space adjoint of an operator $A$ \\\\\n\t\t$S^{\\bot}$ & orthogonal complement of the space $S$ \\\\\n\t\t$S^{\\perp \\!\\!\\! \\perp}$ & orthogonal complement of $S$ with respect to the plus pairing \\\\\n\t\t$L^1([a,b], \\mathbb{F}^n)$ & vector space of $\\mathbb{F}^n$-valued integrable functions on $[a,b]$ \\\\\n\t\t$L^2([a,b], \\mathbb{F}^n)$ & vector space of $\\mathbb{F}^n$-valued square integrable functions on $[a,b]$ \\\\\n\t\t$\\| \\cdot \\|_{L^2} = \\| \\cdot \\|_{L^2(a,b)}$ & standard $L^2$-norm on $L^2([a,b], \\mathbb{F}^n)$ \\\\\n\t\t$L_{loc}^1([0, \\infty), X)$ & vector space of $X$-valued locally integrable functions on $[0, \\infty)$ \\\\\n\t\t$H^N([a,b], \\mathbb{F}^n)$ & Sobolev space of order $N \\in \\mathbb{N}$ on $[a,b]$ \\\\\n\t\t$H_{loc}^1([0, \\infty), X)$ & vector space of $X$-valued locally $H^1$-functions on $[0, \\infty)$ \\\\\n\t\t\t$\\mathcal{C}([a,b], X)$ & vector space of continuous functions $f \\colon [a,b] \\to X$ with finite norm \\\\\n\t\t& $\\|f\\|_{\\infty} := \\sup\\limits_{x \\in [a,b]} \\|f(x)\\|_X$ \\\\\n\t\t\\end{tabular}\n\\newpage\n\t\\def1.5{1.5}\n\\begin{tabular}{l l}\n\t$\\mathcal{C}^k([a,b], X)$ & vector space of $k$-times continuously differentiable functions $f \\colon [a,b] \\to X$ \\\\\n\t$\\mathcal{C}^{\\infty}(\\mathbb{R}^n, \\mathbb{R})$ & vector space of infinitely differentiable functions \\\\\n\t$\\mathcal{D}(\\Omega)$ & vector space of infinitely differentiable functions with compact \\\\\n\t& support on $\\Omega$ (the set of test functions on $\\Omega$) \\\\\n\t$\\mathcal{D}'(\\Omega)$ & vector space of distributions on $\\Omega$ \\\\\n\t$\\langle u, \\varphi \\rangle_{\\mathcal{D}'(\\Omega)}$ & action of $u \\in \\mathcal{D}'(\\Omega)$ on a test function $\\varphi \\in \\mathcal{D}(\\Omega)$ \\\\\n\t$\\mathcal{L}(X,Y)$ & space of linear bounded operators from $X$ to $Y$ \\\\\n\t$\\mathcal{L}(X)$ & space of linear bounded operators on $X$ \\\\\n\t$I_X$ ($I$) & identity of the vector space $X$ \\\\\n\t$A_{|\\Omega}$ & restriction of $A$ to the set $\\Omega \\subset D(A)$ \\\\\n\t$\\overline{D(A)}$ & closure of $D(A) \\subset X$ with respect to $\\|\\cdot\\|_X$ \\\\\n\t$\\frac{dx}{dz}$ & derivative of $x$ with respect to $z$ \\\\\n\t$\\dot{x} = \\frac{dx}{dt}$ & derivative of $x$ with respect to time $t$ \\\\\n\t$\\ddot{x} = \\frac{d^2 x}{dt^2}$ & second derivative of $x$ with respect to time $t$ \\\\\n\t$\\frac{\\partial w}{\\partial x_i} = \\partial_{x_i} w $ & partial derivative of $w(x_1,\\ldots,x_n)$ with respect to $x_i$, $i = 1,\\ldots,n$ \\\\\n\t$\\nabla H(x)$ & gradient column vector of $H \\colon \\mathbb{R}^n \\to \\mathbb{R}$ \\\\\n\t$\\frac{\\delta}{\\delta x}H = \\delta_x H$ & variational derivative of a functional $H \\colon L^2([a,b],\\mathbb{R}^n) \\to \\mathbb{R}$ \\\\\n\n\n\n\n\n\t\t$T_xX$ & tangent space of $X$ at $x \\in X$ \\\\\n\t\t$T_x'X$ & cotangent space of $X$ at $x \\in X$ \\\\\n\t\t$A^J$ & set of mappings $J \\to A$, with $J$ an index set \\\\\n\t\t$\\rho(A)$, $\\sigma(A)$ & resolvent set and spectrum of an operator $A \\colon D(A) \\subset X \\to X$ \\\\\n\t\t$f(x_0^+) = \\lim\\limits_{x \\searrow x_0}f(x)$ & right-hand limit of $f$ at $x_0$ \\\\\n\t\t$f(x_0^-) = \\lim\\limits_{x \\nearrow x_0}f(x)$ & left-hand limit of $f$ at $x_0$ \\\\\n\t\t$\\mathcal{O}(\\epsilon)$ & big O notation with respect to $\\epsilon$ \\\\\n\t\t$A \\hookrightarrow B$ & embedding of $A \\subset B$ into $B$\n\t\t\n\t\\end{tabular}\n\t\\clearpage\n\t\t\\pagenumbering{arabic\n\\chapter{Introduction}\n\\section{Background}\n\nThe port-Hamiltonian systems theory \\cite{ModCompPhys, JacobZwart, SchaftJeltsema} is a systematic framework for modeling systems comprising various physical domains, such as mechanical, electric, hydraulic, and thermal domains. This approach is realized by classifying every (ideal) system component subject to its main physical characteristics, and by taking energy to be medium of communication between the physical domains in multi-physics systems. The eponymous concepts used for this modeling framework are the system's Hamiltonian (that is, its total energy) stemming from mechanics \\cite{Arnold}, and the port-based modeling approach \\cite{Paynter}, which is based on the power-conserving interconnection of various system components by means of ports. They consist of physical quantities whose product yields power (e.g., current and voltage in an electrical network, or force and velocity in a mechanical system). That way, the power flow between the system's components can be expressed. The resulting network structure can be geometrically described by a power-conserving interconnection structure, called the Dirac structure, which has proved to be highly valuable in the field of port-based modeling and analysis \\cite{Courant, SchaftMaschke95}. \n\\vspace{0.5 cm}\\\\\nThere are three main features of port-Hamiltonian systems worth highlighting. Firstly, the (power-conserving) interconnection of port-Hamiltonian systems is again a port-Hamiltonian system. This is a crucial property and emphasizes the network structure of this framework. Secondly, this framework allows the inclusion of energy-dissipating elements, which is why port-Hamiltonian models are broadly applicable. Lastly, the port-Hamiltonian framework is greatly suitable for control purposes. It allows to model the interaction of the system with its environment by means of external ports which are, in particular, accessible for controller action.\n\\vspace{0.5 cm}\\\\\nBased on the theory of infinite-dimensional Hamiltonian systems in \\cite{Olver}, the port-Hamiltonian formulation has been extended to infinite-dimensional systems \\cite{SchaftMaschke02}. The interconnection structure is generalized to a Stokes-Dirac structure, where the power flow through the boundary of the spatial domain, that is, the interaction with the system's environment, is encompassed. Just as in the finite-dimensional case, this is essential for the interconnection of several Hamiltonian systems, and it is a good starting point for the purpose of control, hence the name boundary port-Hamiltonian systems. For the analysis of infinite-dimensional port-Hamiltonian systems one usually considers abstract Cauchy problems of the form\n\\begin{align*}\n\t\\dot{x}(t) &= A_{\\mathcal{Q}}x(t), \\hspace{0.5 cm} t > 0, \\\\\n\tx(0) &= x_0 \\in X,\n\\end{align*}\nwith $A_{\\mathcal{Q}} \\colon D(A_{\\mathcal{Q}}) \\subset X \\to X$ a linear (usually unbounded) operator on some Hilbert space $X$ that is associated with the port-Hamiltonian system. This port-Hamiltonian operator stores information concerning the interdomain coupling of the modeled multi-physics system as well as the mechanical and physical properties of the system. More precisely, these operators are usually given by $A_{\\mathcal{Q}} = \\mathcal{J}(\\mathcal{Q}\\cdot)$, with $\\mathcal{J}$ a formally skew-symmetric matrix differential operator modeling the interdomain coupling, and $\\mathcal{Q}$ a coercive and bounded multiplication operator representing the physical properties. To prove well-posedness and stability results, one applies the well-documented theory of strongly continuous semigroups \\cite{CurtainZwart, EngelNagel, Pazy83}. The most important class of semigroups in the framework of infinite-dimensional port-Hamiltonian systems are contraction semigroups, which are closely related to energy dissipation. Typically, one incorporates dissipation phenomena into the domain of the operator $A_{\\mathcal{Q}}$ by means of the port variables, and proves that this operator is the infinitesimal generator of a contraction semigroup. For control purposes, it seems natural to let the controls and observations act at the boundary of the spatial domain, leading to the notion of (port-Hamiltonian) boundary control systems \\cite{Fattorini}, \\cite[Chapter 10]{Tucsnak-Weiss}. These systems have been extensively studied in the PhD theses \\cite{Augner16, Villegas}.\n\\vspace{0.5 cm}\\\\ \nDue to the great success of this system description, quite recently it was investigated whether the port-Hamiltonian formulation of infinite-dimensional systems can be extended to first-order systems that are coupled by some moving interface \\cite{Diagne}. There are several instances where a system's spatial domain might be subdivided into areas where the intrinsic physical properties or even the aggregate phase differ. For example, one may consider two distinct gases separated by a moving piston, or liquid-solid phase transitions. The mathematical model of the latter is known as the Stefan problem \\cite{Gupta}. A port-Hamiltonian formulation of the two-phase Stefan problem has already been tested in \\cite{Vincent}. In contrast to boundary port-Hamiltonian systems, this formulation yields an abstract port-Hamiltonian control system with the input being the velocity of the interface, and offers a new field of research in terms of interface control. \n\\vspace{0.5 cm}\\\\\nIn this thesis, we review the port-Hamiltonian formulation of two systems of conservation laws defined on two complementary intervals that are coupled by a moving interface, as elaborated in \\cite{Diagne}, and further examine this class of systems. Our final objective is to collect some useful results concerning the underlying port-Hamiltonian operators by means of the theory of strongly continuous semigroups that may help to prove well-posedness and stability results with respect to the position of the moving interface. The main contributions made in this thesis are the following:\n\\begin{enumerate}[label = $\\bullet$]\n\t\\item We specify the Stokes-Dirac structure of the model proposed in \\cite{Diagne} with respect to a formally skew-symmetric matrix differential operator in case of a fixed interface, see Proposition \\ref{Interface Paper - Proposition Dirac Structure DI}.\n\t\\item If we allow for dissipation at the boundary and at the interface position, then the port-Hamiltonian operator associated with two systems of conservation laws that are coupled by a fixed interface is the infinitesimal generator of a contraction semigroup on the respective energy space. In particular, the underlying model is well-defined, see Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}.\n\t\\item Under some more restrictive assumption concerning the boundary condition, the generated semigroup is exponentially stable, see Theorem \\ref{Interface - Theorem 2 Exponential Stability}.\n\t\\item We consider a simplified model which is closely related to the one formulated in \\cite{Diagne}, and present criteria for the stability of the associated family of port-Hamiltonian operators that encompasses the position of the moving interface, see Subsection \\ref{Subsection Stability of the Family of Infinitesimal Generators}. \n\\end{enumerate}\n\n\\section{Outline of this Thesis}\n The remaining chapters are organized as follows: \n\\vspace{0.5 cm}\\\\\n\\textbf{Chapter \\ref{Chapter Finite-dimensional Port-Hamiltonian Systems}} aims to present the basic concepts and notions of port-based modeling. We deal with the port-Hamiltonian formulation of finite-dimensional multi-physics systems, where we restrict ourselves to linear systems. To that end, we introduce the fundamental concept of a power-conserving interconnection structure, called a Dirac structure, and briefly discuss its constituting port variables. This allows us to define the port-Hamiltonian dynamics by means of the underlying Dirac structure. \n\\vspace{0.5 cm}\\\\\n\\textbf{Chapter \\ref{Chapter Some Background}} covers some well-known notions and results from the literature that are used in the upcoming chapters. We will mainly recall the basics of infinite-dimensional systems theory and, with that said, we will recall the concept of strongly continuous semigroups and some important results that characterize their infinitesimal generators. Furthermore, we will discuss formally adjoint operators of matrix differential operators.\n\\vspace{0.5 cm}\\\\\nIn \\textbf{Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}} we generalize the port-Hamiltonian approach of Chapter \\ref{Chapter Finite-dimensional Port-Hamiltonian Systems} to infinite-dimen- sional systems, where we restrict ourselves to idealized systems with a 1-dimensional spatial domain. We discuss the main differences to finite-dimensional port-Hamiltonian systems, and show how the power flow at the boundary of the spatial domain is incorporated into the definition of the Dirac structure, leading to the notion of a boundary port-Hamiltonian system. This chapter sets a basis for the discussion in the following chapter.\n\\vspace{0.5 cm}\\\\\n\\textbf{Chapter \\ref{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface}} is the centerpiece of this thesis and studies the port-Hamiltonian formulation of a system of two conservation laws with a moving interface. The main goal is to provide a well-posed model for a system of two conservation laws on two complementary intervals separated by an interface, where we take the interface's velocity to be the input variable of the system. It turns out that this formulation poses delicate problems. Therefore, we vastly simplify the original problem, and work out some important properties of the simplified system that may help to further investigate the initial one.\n\\vspace{0.5 cm}\\\\\nIn \\textbf{Chapter \\ref{Chapter Conclusion}} we summarize the covered topics and reflect the main contributions made in this work. Furthermore, we draw our conclusions from the discussion in Chapter \\ref{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface} and give some comments on open problems and potential future work concerning this rather new class of systems.\n\n\n\n\n\n\n\n\\chapter{Finite-Dimensional Port-Hamiltonian Systems}\n\\label{Chapter Finite-dimensional Port-Hamiltonian Systems}\n\nIn this chapter, we want to study basic concepts of port-based network modeling and provide the definition of finite-dimensional port-Hamiltonian systems and their underlying Dirac structures. Section \\ref{Section A Motivating Example} is devoted to the description of the main characteristics of port-based network modeling, which originates in the work of Paynter \\cite{Paynter}. To this end, we consider the model of a simple mass-spring system. This motivates to introduce the port-Hamiltonian formulation of multi-physics systems. The dynamics of port-Hamiltonian systems are geometrically specified by so-called Dirac structures. In Section \\ref{Section Dirac Structures} we will introduce this fundamental concept and give some examples of Dirac structures. In the Sections \\ref{Section Energy-storing Elements}, \\ref{Section Energy-dissipating Elements}, and \\ref{Section External Ports} we briefly discuss energy-storing, energy-dissipating, and external ports, respectively. In the finite-dimensional case, these are the constitutive ports of a Dirac structure.\nAfter that, we are finally able to geometrically define the dynamics of port-Hamiltonian systems, which we are going to do in Section \\ref{Section Port-Hamiltonian Dynmacis}. As mentioned in the beginning, the port-Hamiltonian framework is particularly convenient for control purposes. Therefore, we will provide a state-space representation of a huge class of port-Hamiltonian systems that is suitable to be taken as the starting point for investigating control theoretical aspects. We will mainly focus on linear port-Hamiltonian systems.\n\\vspace{0.5 cm}\\\\\nThe whole chapter relies heavily on the Chapters 2, 3, and 4 in \\cite{SchaftJeltsema}, and on \\cite[Section 2.2]{ModCompPhys}.\n\n\\section{A Motivating Example}\n\\label{Section A Motivating Example}\nIn order to motivate the framework of port-based modeling, we start with a simple example.\n\\begin{example}[Mass-Spring System; Example 2.1 in \\cite{SchaftJeltsema}]\n\t\\label{Example Mass-Spring System}\n Consider a point mass with mass $m \\in \\mathbb{R}_{> 0}$, moving in one direction and without friction under the influence of a\n\tspring force corresponding to a linear spring with spring constant $k \\in \\mathbb{R}_{>0}$. The usual way of modeling this mass-spring system is by means of the second-order differential equation\n\t\\begin{align}\n\t\tm \\ddot{z}(t) = - k(z(t) - z_0),\n\t\t\\label{Example Mass-Spring System - ODE}\n\t\\end{align}\nwhere $z(t) \\in \\mathbb{R}$ denotes the position of the mass at time instant $t \\geq 0$, and $z_0 \\in \\mathbb{R}_{>0}$ is the length of the spring at rest. In order to present the basic principles of port-based network modeling, we consider this problem from a different perspective. The mass-spring system is now regarded as the interconnection of two subsystems storing energy, namely the spring system storing potential energy and the mass system storing kinetic energy, see Figure~\\ref{Figure Mass-Spring-System}. \\\\\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width = 8cm]{Mass-Spring-System.png}\n\t\\caption{Mass-Spring System Regarded as the Interconnection of two Subsystems \\cite[Figure~2.1]{SchaftJeltsema}}\n\t\\label{Figure Mass-Spring-System}\n\t\\end{figure}\n\nFor the spring system the potential energy is expressed in terms of the elongation $q = z - z_0 $ of the spring. Assuming a linear\nspring, according to Hooke's law the potential energy is given as $E_{\\text{pot}}(q) = \\frac{1}{2} kq^2$. This leads to the spring system equations\n\\begin{align*}\n\t\\text{spring system} \\colon \\begin{cases}\n\t\t\\dot{q} = -f_k, \\\\\n\t\te_k = \\frac{d}{dq} E_{\\text{pot}}(q),\n\t\\end{cases}\n\\end{align*}\nwhere $-f_k$ denotes the velocity of the endpoint of the spring where\nit is attached to the mass, and $e_k = kq$ describes the spring force at this endpoint. \\\\\nFor the mass system we obtain similar equations. Denoting the momentum of the mass by $p \\in \\mathbb{R}$, the kinetic\nenergy is given as $E_{\\text{kin}}(p) = \\frac{1}{2m} p^2$, which leads to the following mass system equations:\n\\begin{align*}\n\t\\text{mass system} \\colon \\begin{cases}\n\t\t\\dot{p} = -f_m, \\\\\n\t\te_m = \\frac{d}{dp} E_{\\text{kin}}(p).\n\t\\end{cases}\n\\end{align*}\nHere, $-f_m$ denotes the force exerted on the mass, and $e_m = \\frac{p}{m}$ is the velocity of the mass. \\\\\nThese separately regarded subsystems are now interconnected through Newton's third law in the following way:\n\\begin{align*}\n\t\t-f_k &= e_m, \\\\\n\t\tf_m &= e_k.\n\\end{align*}\nThe system's stored energy is given by the Hamiltonian $H \\colon \\mathbb{R}^2 \\to \\mathbb{R}$ defined by\n\\begin{align}\n\tH(q,p) = E_{\\text{pot}}(q) + E_{\\text{kin}}(p) = \\frac{1}{2}kq^2 + \\frac{1}{2m} p^2.\n\t\\label{Example Mass-Spring System - Hamiltonian}\n\\end{align}\nThe interconnected system can be written in terms of the Hamiltonian as follows:\n\\begin{align}\n\t\\begin{bmatrix}\n\t\t\t\\dot{q} \\\\\n\t\t\t\\dot{p}\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t0 & 1 \\\\\n\t-1 & 0 \n\\end{bmatrix} \\begin{bmatrix}\n\\frac{\\partial H}{\\partial q} (q,p) \\\\\n\\frac{\\partial H}{\\partial p}(q,p) \n\\end{bmatrix}.\n\\label{Example Mass-Spring System - pH System}\n\\end{align}\nThese are the so-called Hamiltonian equations for the mass-spring system. Note that \\eqref{Example Mass-Spring System - pH System} is equivalent to the second-order model \\eqref{Example Mass-Spring System - ODE} by the correspondence $q = z - z_0$: we have\n\\begin{align*}\n\tm \\ddot{z} = m \\ddot{q} = m \\dot{\\left(\\frac{\\partial H}{\\partial p} (q,p)\\right)} = m \\dot{\\frac{p}{m}} = - \\frac{\\partial H}{\\partial q}(q,p) = - kq = -k(z - z_0). \n\\end{align*}\n\\QEDA\n\\end{example}\n\n\\begin{remark}\n\tThe reason for the minus sign in front of $f_k$ in Example \\ref{Example Mass-Spring System} is that we want the product $f_ke_k$ between the spring's velocity and force to be the incoming power with respect to the interconnection. The same applies for the product $f_m e_m$ between the exerted force on the mass and its velocity. This sign convention will be adopted throughout this chapter.\n\\end{remark}\n\nDespite its simplicity, Example \\ref{Example Mass-Spring System} already displays some of the\nmain characteristics of port-based network modeling: we regard the system as the coupling of the energy-storing spring system with the energy-storing mass system through a power-conserving interconnection where the power is routed from the mass system\nto the spring system and vice versa. This approach can be applied to general multi-physics systems, i.e., systems comprising different physical domains.\n\\vspace{0.5 cm}\\\\\nMore generally, in port-based modeling a physical system is considered as the interconnection of three different types of ideal components:\n\\begin{enumerate}\n\t\\item[(i)] \\textit{Energy-storing elements:} typical examples of energy-storing elements are ideal inductors (by means of the magnetic field), capacitors (by means of the electric field), masses, and springs.\n\t\\item[(ii)] \\textit{Energy-dissipating} (or \\textit{resistive}) elements: examples of elements corresponding to energy-dissipation (due to friction, resistance, etc.) are resistors and dampers.\n\t\\item[(iii)] \\textit{Energy-routing elements:} elements such as gyrators and transformers only redirect the power flow in the overall system \\cite[Part XV]{Paynter}.\n\\end{enumerate}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width = 10cm]{Dirac-Structure.png}\n\t\\caption{Port-Hamiltonian System \\cite[Figure 2.2]{SchaftJeltsema}}\n\t\\label{Figure Port-Hamiltonian System}\n\t\\end{figure}\nFor the port-Hamiltonian formulation the energy-storing elements will be grouped into a single object denoted by $\\mathcal{S}$ ('storage'), and correspondingly the energy-dissipating elements\nare grouped into a single object denoted by $\\mathcal{R}$ ('resistive'). Moreover, the\ninterconnection of all energy-routing elements can be considered as one energy-routing structure denoted by $\\mathcal{D}$, which we will call the system's underlying \\emph{Dirac structure}.\n\\vspace{0.5 cm}\\\\\nThe principle of port-Hamiltonian systems modeling is schematically illustrated in Figure~\\ref{Figure Port-Hamiltonian System}: the energy-storing elements $\\mathcal{S}$ and the energy-dissipating elements $\\mathcal{R}$ are linked to a central interconnection (energy-routing) structure $\\mathcal{D}$. This linking is realized via pairs $(f,e)$ of vectors of flow and effort variables of equal dimension. We call a pair $(f,e)$ of vectors of flow and effort variables a port, and the total set of variables $f,e$ is called the set of port variables. The physical meaning of efforts and flows in various physical domains is listed in Table \\ref{Table Flows And Efforts}.\n\\begin{table}\n\\centering\n\\small\n{\\rowcolors{2}{white!100}{black!10!white!90} \n\t\\begin{tabular}{c c c}\n\t\t\\hline\n\t\tPhysical Domain & Flow $f \\in \\mathcal{F}$ & Effort $e \\in \\mathcal{E}$ \\\\ [0.5ex]\n\t\t\\hline \\hline\n\t\telectric & current $I$ & voltage $V$ \\\\\n\t\tmagnetic & voltage $V$ & current $I$ \\\\\n\t\tpotential translation & velocity $v$ & force $F$ \\\\\n\t\tkinetic translation & force $F$ & velocity $v$ \\\\\n\t\tpotential rotation & angular velocity $\\omega$ & torque $T$ \\\\\n\t\tkinetic rotation & torque $T$ & angular velocity $\\omega$ \\\\\n\t\tpotential hydraulic & volume flow $\\phi$ & pressure $p$ \\\\\n\t\tkinetic hydraulic & pressure $p$ & volume flow $\\phi$ \\\\\n\t\tchemical & chemical potential $\\mu$ & molar flow $f_N$ \\\\\n\t\tthermal & temperature $T$ & entropy flow $f_S$ \\\\ [0.5ex]\n\t\t\\hline\n\t\\end{tabular}\n}\n\\caption{Flows and Efforts Corresponding to their Physical Domains \\cite[Table 1.2]{ModCompPhys}}\n\\label{Table Flows And Efforts}\n\\end{table}\nFigure \\ref{Figure Port-Hamiltonian System} shows three ports: the port $(f_S, e_S)$ linking to energy-storage, the port $(f_R,e_R)$ corresponding to energy-dissipation, and the\nexternal port $(f_P,e_P)$, by which the system interacts with its environment.\n\\vspace{0.5 cm}\\\\\nIn the following sections, we will provide more details about the fundamental notion of a Dirac structure on finite-dimensional spaces. To this end, we discuss the constituting ports of Dirac structures in the finite-dimensional case. We will treat successively energy-storing elements, energy-dissipating (resistive) elements, and\nexternal ports. After a short comment on state choice of different physical domains, we finally get to the basic geometric definition of a port-Hamiltonian system. Subsequently, we will present some simple yet important examples of such systems.\n\n\\section{Dirac Structures}\n\\label{Section Dirac Structures}\nThe key property of a Dirac structure is power conservation, meaning that it links the port variables in such a way that the total power entering or leaving a Dirac structure always amounts to zero. For the definition of a Dirac structure, we need to define the spaces of flows and efforts as well as the power flow defined on these spaces. We denote by $\\mathcal{F}$ some finite-dimensional linear \\emph{space of flows}. Its elements will be denoted by $f \\in \\mathcal{F}$, and are called \\emph{flow} (or \\emph{energy}) \\emph{variables}. The \\emph{space of effort} (or \\emph{co-energy}) \\emph{variables} is given by the dual linear space $\\mathcal{E} = \\mathcal{F}'$, and its elements are denoted by $e \\in \\mathcal{E}$. The total space of flow and effort variables is $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$, and is called the \\emph{space of port variables} or the \\emph{bond space}. The power on the total space of port variables is defined by \n\\begin{align}\n\tP \\colon \\mathcal{B} \\to \\mathbb{R}, \\hspace{0.5 cm} (f,e) \\mapsto \\langle e \\mid f \\rangle,\n\t\\label{Function - Power}\n\\end{align}\nwhere $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ is the so-called \\emph{power pairing}. In the finite-dimensional case the power pairing $\\langle \\cdot \\mid \\cdot \\rangle$ equals the natural pairing $\\langle \\cdot , \\cdot \\rangle_{\\mathcal{F}' \\times \\mathcal{F}}$ on the respective vector space, see Section \\ref{Section Operators}. Usually, one may take $\\mathcal{F} = \\mathbb{R}^k$ and thus $\\mathcal{E} = \\mathbb{R}^k$, and the power pairing equals the standard inner product:\n\\begin{align*}\n\t\\langle e \\mid f \\rangle = \\langle e, f \\rangle_2, \\hspace{0.5 cm} (f,e) \\in \\mathcal{B}.\n\\end{align*}\nWith the data provided in Table~\\ref{Table Flows And Efforts} one may check that the (inner) product of flow and effort variables indeed equals power. However, we note that there are interesting cases where it is beneficial not to take $\\mathcal{F} = \\mathbb{R}^k$. For instance, in three-dimensional mechanical systems one usually works with Lie algebras of different matrix groups (see \\cite[Section 1.4]{Olver}). For further details we refer to \\cite[Chapter 3]{SchaftJeltsema}. We will see in the Chapters \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems} and \\ref{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface} that in the infinite-dimensional case, the power pairing does, in general, not coincide with the natural pairing.\n\\vspace{0.5 cm}\\\\\nNow we are able to define the notion of a Dirac structure in the finite-dimensional case. \n\\begin{definition}[Dirac Structure]\n\t\\label{Definition Dirac Structure - Finite-dimensional Case}\n\tConsider a finite-dimensional linear space of flows $\\mathcal{F}$ and its dual space of efforts $\\mathcal{E} = \\mathcal{F}'$. Let $P$ be the power mapping on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ given by \\eqref{Function - Power}. A linear subspace $\\mathcal{D} \\subset \\mathcal\n\t{B}$ is called a (constant) Dirac structure if \n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item \\label{DiracStructureProp1} $P_{|\\mathcal{D}} \\equiv 0$, i.e., $P(f,e) = 0$ for all $(f,e) \\in \\mathcal{D}$, and\n\t\t\\item \\label{DiracStructureProp2} $\\dim (\\mathcal{D}) = \\dim (\\mathcal{F})$. \n\t\\end{enumerate}\n\\end{definition}\nProperty \\ref{DiracStructureProp1} in Definition \\ref{Definition Dirac Structure - Finite-dimensional Case} corresponds to power conservation, and states the previously mentioned\nfact that the total power flow always equals zero. It can be shown that the maximal dimension of any subspace $\\mathcal{D} \\subset \\mathcal{B}$ satisfying property \\ref{DiracStructureProp1} is equal to $\\dim (\\mathcal{F})$. This claim follows immediately from an equivalent definition of a Dirac structure, which we will state next. Another advantage of this equivalent definition of a Dirac structure is that it can be generalized to the case of an infinite-dimensional vector space $\\mathcal{F}$, leading to the definition of an infinite-dimensional Dirac structure, as we will see in Chapter~\\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}.\n\\vspace{0.5 cm}\\\\\nIn order to give this equivalent characterization of a Dirac structure, we define a canonical bilinear form $\\ll \\cdot, \\cdot \\gg$ on the space $\\mathcal{B}$ which is closely related to the power pairing:\n\\begin{align}\n\t\\begin{split}\n\t\\ll \\cdot, \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} &\\to \\mathbb{R}, \\\\\n\t((f^1,e^1),(f^2,e^2)) &\\mapsto \\hspace{0.1 cm}\\ll (f^1,e^1) , (f^2,e^2) \\gg \\hspace{0.1 cm} := \\langle e^1 \\mid f^2 \\rangle + \\langle e^2 \\mid f^1 \\rangle.\n\t\\end{split}\n\t\\label{Plus Pairing}\n\\end{align}\nThis bilinear form is \\emph{non-degenerate}, that is, if $\\ll (f^1,e^1) , (f^2,e^2) \\gg = 0$ for all $(f^2,e^2) \\in \\mathcal{B}$, then necessarily $(f^1,e^1) = 0$. However, it is not positive definite. We call this bilinear form a \\emph{plus pairing}.\n\\vspace{0.5 cm}\\\\\nWe will now show that the following equivalent definition of a Dirac structure indeed satisfies the properties stated in Definition \\ref{Definition Dirac Structure - Finite-dimensional Case}.\n\\begin{proposition}[Proposition 2.1 in \\cite{SchaftJeltsema}]\n\t\\label{Proposition Characterization Dirac Structure}\n\t\tConsider a finite-dimensional linear space of flows $\\mathcal{F}$ and its dual space of efforts $\\mathcal{E} = \\mathcal{F}'$. Let $P$ be the power mapping on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ given by \\eqref{Function - Power}. A linear subspace $\\mathcal{D} \\subset \\mathcal{B}$ is a Dirac structure if and only if \n\t\t\\begin{align}\n\t\t\t\\mathcal{D} = \\mathcal{D}^{\\perp \\!\\!\\! \\perp},\n\t\t\t\\label{Proposition Characterization Dirac Structure - Equivalent Property}\n\t\t\\end{align} where $\\mathcal{D}^{\\perp \\!\\!\\! \\perp}$ denotes the orthogonal complement of $ \\mathcal{D}$ with respect to the plus pairing $\\ll \\cdot, \\cdot \\gg$.\n\\end{proposition}\n\\begin{proof}\n\t''$\\Rightarrow$'': Let $\\mathcal{D}$ be a Dirac structure in the sense of Definition \\ref{Definition Dirac Structure - Finite-dimensional Case}. Let $(f^1,e^1)$, $(f^2,e^2) \\in \\mathcal{D}$ be arbitrary port variables. By linearity, we have $(f^1 + f^2, e^1 + e^2) \\in \\mathcal{D}$. By virtue of property \\ref{DiracStructureProp1} in Definition \\ref{Definition Dirac Structure - Finite-dimensional Case}, we conclude that\n\t\\begin{align*}\n\t\t0 &= \\langle e^1 + e^2 \\mid f^1 + f^2 \\rangle \\\\\n\t\t&= \\langle e^1 \\mid f^1 \\rangle + \\langle e^1 \\mid f^2 \\rangle + \\langle e^2 \\mid f^1 \\rangle + \\langle e^2 \\mid f^2 \\rangle \\\\\n\t\t&= \\langle e^1 \\mid f^2 \\rangle + \\langle e^2 \\mid f^1 \\rangle \\\\\n\t\t&= \\hspace{0.2 cm} \\ll (e^1,f^1), (e^2,f^2) \\gg,\n\t\\end{align*}\nwhence $\\mathcal{D} \\subset \\mathcal{D}^{\\perp \\!\\!\\! \\perp}$. Moreover, as the plus pairing $\\ll \\cdot, \\cdot \\gg$ is non-degenerate, we have\n\\begin{align*}\n\t\\dim (\\mathcal{D}^{\\perp \\!\\!\\! \\perp}) = \\dim (\\mathcal{F} \\times \\mathcal{E}) - \\dim (\\mathcal{D}) = 2 \\dim (\\mathcal{F}) - \\dim (\\mathcal{D}),\n\\end{align*}\nand due to property \\ref{DiracStructureProp2} in Definition \\ref{Definition Dirac Structure - Finite-dimensional Case}, it follows that $ \\dim(\\mathcal{D}) = \\dim(\\mathcal{D}^{\\perp \\!\\!\\! \\perp})$ and thus $\\mathcal{D} = \\mathcal{D}^{\\perp \\!\\!\\! \\perp}$. \n\\vspace{0.5 cm}\\\\\n''$\\Leftarrow$'': Let $\\mathcal{D}$ satisfy (\\ref{Proposition Characterization Dirac Structure - Equivalent Property}). Then for every $(f,e) \\in \\mathcal{D}$ we have in particular that\n\\begin{align*}\n\t0 =\t\\ll (f,e), (f,e) \\gg = 2\\langle e \\mid f \\rangle = 2 P(f,e).\n\\end{align*}\nThus, property \\ref{DiracStructureProp1} is satisfied. Again, as the plus pairing $\\ll \\cdot, \\cdot \\gg$ is non-degenerate, we obtain\n\\begin{align}\n\t\\dim (\\mathcal{D}^{\\perp \\!\\!\\! \\perp}) = 2 \\dim (\\mathcal{F}) - \\dim( \\mathcal{D}).\n\t\\label{Proof Proposition Characterization Dirac Structure - Dimension Equation}\n\\end{align}\nEquation (\\ref{Proposition Characterization Dirac Structure - Equivalent Property}) now implies that $\\dim(\\mathcal{D}) = \\dim (\\mathcal{F})$, and hence property \\ref{DiracStructureProp2} of a Dirac structure is satisfied as well. \n\\end{proof}\nIn order to prove the alternative characterization, we first note that we have shown that property \\ref{DiracStructureProp1} implies $\\mathcal{D} \\subset \\mathcal{D}^{\\perp \\!\\!\\! \\perp}$. Together with equation (\\ref{Proof Proposition Characterization Dirac Structure - Dimension Equation}) this yields that any subspace $\\mathcal{D}$ satisfying property \\ref{DiracStructureProp1} has the property that $\\dim (\\mathcal{D}) \\leq \\dim (\\mathcal{F})$. In conclusion, a Dirac structure indeed is the linear subspace of maximal dimension satisfying property \\ref{DiracStructureProp1} of Definition \\ref{Definition Dirac Structure - Finite-dimensional Case}.\n\\vspace{0.5 cm}\\\\\n There are several examples of constant Dirac structures. We will present only a few of them and refer to \\cite[Section 2.2]{SchaftJeltsema} for more.\n\\begin{example}[Dirac Structures; cf. Section 2.2 in \\cite{SchaftJeltsema}]\n\t\\label{Example Dirac Structures}\n\tLet $\\mathcal{F} = \\mathbb{R}^n$ and $\\mathcal{E} = \\mathcal{F}' = \\mathbb{R}^n$. Consider the following examples of Dirac structures:\n\t\\begin{enumerate}\n\t\t\\item[1.] Let $J \\colon \\mathcal{E} \\to \\mathcal{F},$ $e \\mapsto f = Je$, with $J \\in \\mathbb{R}^{n \\times n}$ a skew-symmetric matrix, i.e., $J = - J^{\\top}$. Then one can readily show that\n\t\t\\begin{align*}\n\t\t\\mathcal{D}_J :=\t\\gra (-J) := \\left\\{ (f,e) \\in \\mathcal{F} \\times \\mathcal{E} \\mid -f = Je \\right\\}\n\t\t\\end{align*}\n\tis a Dirac structure. According to Proposition \\ref{Proposition Characterization Dirac Structure}, we need to show that $\\mathcal{D}_J = \\mathcal{D}_J^{\\perp \\!\\!\\! \\perp}$ with respect to $\\ll \\cdot , \\cdot \\gg$ defined in \\eqref{Plus Pairing}. Due to skew-symmetry of $J$, for $(f^1,e^1),(f^2,e^2) \\in \\mathcal{D}_J$ we have\n\t\\begin{align*}\n\t\t\\ll (f^1,e^1), (f^2,e^2) \\gg &= \\langle e^1, f^2 \\rangle_2 + \\langle e^2 , f^1 \\rangle_2 \\\\\n\t\t&= \\langle e^2, - Je^1 \\rangle_2 + \\langle e^1, - Je^2 \\rangle_2 \\\\\n\t\t&= \\langle e^1, Je^2 \\rangle_2 - \\langle e^1, J e^2 \\rangle_2 \\\\\n\t\t&=0.\n\t\\end{align*} \nWe conclude that $\\mathcal{D}_J \\subset \\mathcal{D}_J^{\\perp \\!\\!\\! \\perp}$. Conversely, let $(f,e) \\in \\mathcal{D}_J^{\\perp \\!\\!\\! \\perp}$, that is, for all $(f^1, e^1) \\in \\mathcal{D}_J$, \n\\begin{align*}\n\t0 &= \\ll (f^1, e^1), (f,e) \\gg \\\\\n\t&= \\langle e^1, f \\rangle_2 + \\langle e, f^1 \\rangle_2 \\\\\n\t&= \\langle e^1, f \\rangle_2 + \\langle e, -Je^1 \\rangle_2 \\\\\n\t&= \\langle e^1,f \\rangle_2 + \\langle e^1, Je \\rangle_2.\n\\end{align*} \nConsequently, for this equation to hold for all $(f^1, e^1) \\in \\mathcal{D}_J$, we need $f = -Je$, whence $(f,e) \\in \\mathcal{D}_J$ and therefore $\\mathcal{D}_J^{\\perp \\!\\!\\! \\perp} \\subset \\mathcal{D}_J$. Altogether, $\\mathcal{D}_J = \\mathcal{D}_J^{\\perp \\!\\!\\! \\perp}$, and $\\mathcal{D}_J$ is a Dirac structure. \n\\vspace{0.5 cm}\\\\\nAs we will see, graphs of skew-symmetric linear maps in the finite-dimensional case, and (formally) skew-symmetric matrix differential operators in the infinite-dimensional case play a substantial role in the port-Hamiltonian framework.\n\\item[2.] Let $\\mathcal{K} \\subset \\mathcal{F}$ be any subspace. Define\n\\begin{align*} \n\t\\mathcal{K}^{\\bot} := \\left\\{ e \\in \\mathcal{E} \\mid \\langle e \\mid f \\rangle = 0 \\hspace{0.1 cm} \\text{for all } f \\in \\mathcal{K} \\right\\}.\n\\end{align*}\nThen $\\mathcal{D} = \\mathcal{K} \\times \\mathcal{K}^{\\bot}$ is a so-called separable Dirac structure (cf. \\cite[Definition 2.2]{SchaftJeltsema}), i.e., for all $(f^{1},e^{1}),(f^{2},e^{2}) \\in \\mathcal{D}$ we have\n\\begin{align*}\n\t\\langle e^{1} \\mid f^{2} \\rangle = 0.\n\\end{align*}\nOne can show that any separable Dirac structure is of the form $\\mathcal{K} \\times \\mathcal{K}^{\\bot}$ for an appropriately chosen subspace $\\mathcal{K} \\subset \\mathcal{F}$, see \\cite[Proposition 2.2]{SchaftJeltsema}. A typical example for a separable Dirac structure is given by transformers. A transformer is an element linking two scalar bonds with flow and effort variables $(f^{1}, e^{1}), (f^{2}, e^{2}) \\in \\mathbb{R}^2$ by\n\\begin{align*}\n\tf^{2} &= \\alpha f^1, \\\\\n\te^1 &= - \\alpha e^2,\n\\end{align*}\nwith $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$ the transformer ratio. More precisely, let $\\mathcal{F} = \\mathcal{E} = \\mathbb{R}^2$ and set $\\mathcal{K}_{\\alpha} = \\left\\{ (f, \\alpha f) \\in \\mathcal{F} \\mid f \\in \\mathbb{R}\\right\\}$. It is clear that $\\mathcal{K}_{\\alpha}^{\\bot} = \\left\\{ (- \\alpha e,e) \\in \\mathcal{E} \\mid e \\in \\mathbb{R} \\right\\}$, and that $\\dim(\\mathcal{K}_{\\alpha} \\times \\mathcal{K}_{\\alpha}^{\\bot}) = \\dim(\\mathcal{F}) = 2$, which is why $\\mathcal{D}_{\\alpha} = \\mathcal{K}_{\\alpha} \\times \\mathcal{K}_{\\alpha}^{\\bot}$ is indeed a separable Dirac structure. This can be easily generalized to the multi-port case.\n\\end{enumerate}\n\t\\QEDA\n\\end{example}\nAs illustrated in Figure \\ref{Figure Port-Hamiltonian System}, the Dirac structure describes how energy is exchanged among various components in a physical system. In the following, we want to specify the port variables that constitute a (finite-dimensional) Dirac structure. We will give an overview concerning port variables related to energy-storing and energy-dissipating system elements as well as external ports.\n\n\\section{Energy-Storing Elements}\n\\label{Section Energy-storing Elements}\nThe energy-storing multi-port element $\\mathcal{S}$ unifies all energy-storing elements of the system. The port variables of the Dirac structure associated with $\\mathcal{S} = \\mathcal{F}_S \\times \\mathcal{E}_S$ will be denoted by $(f_S,e_S) \\in \\mathcal{S}$. The total energy storage of the system is defined by a Hamiltonian function $H \\colon X \\to \\mathbb{R}$ on the state space $X$. We restrict ourselves to the case where the state space is given by $X = \\mathbb{R}^n$. In general, the state space is a smooth (finite-dimensional) manifold, see \\cite[Chapter 3]{SchaftJeltsema}, \\cite{SchaftMaschke20}. In Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems} we will discuss the extension to the infinite-dimensional case.\n\\vspace{0.5 cm}\\\\\nBefore providing more details about energy-storing multi-port elements, we want to clarify briefly how the state variables have to be chosen in various physical domains, as they are closely related to the flows and efforts. In the common classification of domains, many domains are characterized by two types of states, namely the \\emph{generalized displacement} $q \\in X$ and the \\emph{generalized momentum} $p \\in X'$. Depending on the physical domain, one obtains the generalized displacement $q$ by time-integration of the flow variable $f \\in \\mathcal{F}$, whereas the generalized momentum $p$ refers to time-integration of the effort variable $e \\in \\mathcal{E}$. Table~\\ref{Table Flows Efforts States} is an extension of Table~\\ref{Table Flows And Efforts} and gives a summary of the generalized states in the respective physical subdomain. For more background information concerning this topic, we recommend the introductory chapter in \\cite{ModCompPhys}.\n\\vspace{0.5 cm}\n\\begin{table}\n\t\\renewcommand{1.5}{2}\n\t\\tiny\n\t\\centering\n\t{\\rowcolors{3}{black!10!white!90}{white!100}\n\t\t\\begin{tabular}{ c c c c c} \n\t\t\t\\hline\n\t\t\tPhysical Domain & Flow & Effort & Generalized Displacement & Generalized Momentum \\\\ \n\t\t\t& $f \\in \\mathcal{F}$ & $e \\in \\mathcal{E}$ & $q = \\int f \\, dt$ & $p = \\int e \\, dt$ \\\\ [0.5ex]\n\t\t\t\\hline \\hline\n\t\t\telectric & current $I$ & voltage $V$ & charge $Q$ & magnetic flux linkage $\\lambda$ \\\\\n\t\t\tmagnetic & voltage $V$ & current $I$ & magnetic flux linkage $\\lambda$ & charge $Q$ \\\\ \n\t\t\tpotential translation & velocity $v$ & force $F$ & displacement $x$ & momentum $p$ \\\\\n\t\t\tkinetic translation & force $F$ & velocity $v$ & momentum $p$ & displacement $x$ \\\\\n\t\t\tpotential rotation & angular velocity $\\omega$ & torque $T$ & angular displacement $\\theta$ & angular momentum $b$ \\\\\n\t\t\tkinetic rotation & torque $T$ & angular velocity $\\omega$ & angular momentum $b$ & angular displacement $\\theta$ \\\\\n\t\t\tpotential hydraulic & volume flow $\\phi$ & pressure $p$ & volume $V$ & momentum of a flow tube $\\Gamma$ \\\\\n\t\t\tkinetic hydraulic & pressure $p$ & volume flow $\\phi$ & momentum of a flow tube $\\Gamma$ & volume $V$ \\\\ \n\t\t\tchemical & chemical potential $\\mu$ & molar flow $f_N$ & number of moles $N$ & $-$\\\\\n\t\t\tthermal & temperature $T$ & entropy flow $f_S$ & entropy $S$ & $-$ \\\\ [0.5ex]\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t}\n\t\\caption{Flows, Efforts, and Generalized State Variables in Different Physical Domains \\\\ \\cite[Table 1.1]{ModCompPhys}}\n\t\\label{Table Flows Efforts States}\n\\end{table}\n\n\nThe vector of flow variables is given by the rate $\\dot{x}$ of the state $x \\in X$. Thus, for any (time-dependent) state $x \\in X$ the flow vector $\\dot{x}$ will be an element of the linear space $\\mathcal{F}_S= T_xX$, the \\emph{tangent space} of $X$ at $x \\in X$. In the given case $X = \\mathbb{R}^n$, the tangent space $T_xX$ may be identified with $X$.\n\\vspace{0.5 cm}\\\\\nFurthermore, the vector of effort variables of the energy-storing multi-port element is given by the gradient vector $\\nabla H(x) \\in \\mathcal{E}_S = T_x'X$, the dual space of the tangent space $T_xX$, which again may be identified with $X' = \\mathbb{R}^n$. Note that we define $\\nabla H(x)$, $x \\in X$, to be a column vector instead of a row vector. This convention will be applied throughout this work.\n\\vspace{0.5 cm}\\\\\nNow, in case of a time-dependent state $x(t) \\in X$, $t \\geq 0$, we obtain the following power-balance for the energy-storing multi-port element:\n\\begin{align}\n\t\\frac{d}{dt}H(x(t)) = \\nabla H^{\\top}(x(t)) \\dot{x}(t).\n\t\\label{Power Balance Energy-Storing Element}\n\\end{align}\nIn the present case, the bond space is given by $\\mathcal{B} = \\mathcal{S} = \\mathcal{F}_S \\times \\mathcal{E}_S$, and the power-pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E}_S \\times \\mathcal{F}_S \\to \\mathbb{R}$ is given by $\\langle \\cdot , \\cdot \\rangle_2$. The interconnection of the energy-storing elements to the storage port $(f_S, e_S) \\in X \\times X'$ of some Dirac structure $\\mathcal{D} \\subset X \\times X'$ is done by setting\n\\begin{align}\n\tf_S = - \\dot{x} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} e_S = \\nabla H(x).\n\t\\label{Choice of Port Variables fS and eS}\n\\end{align}\nNote that the minus sign in (\\ref{Choice of Port Variables fS and eS}) has been placed in order to have a consistent power flow convention: $\\nabla H^{\\top}(x) \\dot{x}$ is the power flowing into the energy-storing elements, whereas $e_S^{\\top}f_S$ is the power flowing into the Dirac structure. Hence, with \\eqref{Function - Power} and \\eqref{Choice of Port Variables fS and eS} the power-balance (\\ref{Power Balance Energy-Storing Element}) can also be written as\n\\begin{align}\n\t\\frac{d}{dt}H(x(t)) = - e_S^{\\top}(t)f_S(t) = - \\langle e_S(t) \\mid f_S(t) \\rangle = 0, \\hspace{0.5 cm} t \\geq 0,\n\t\\label{Power-Balance Energy-Storing Element with fS and eS}\n\\end{align}\nprovided that $(f_S(t), e_S(t)) \\in \\mathcal{D}$ for $t \\geq 0$. Equation \\eqref{Power-Balance Energy-Storing Element with fS and eS} then expresses power conservation of the system.\n\\section{Energy-Dissipating Elements}\n\\label{Section Energy-dissipating Elements}\nThe second multi-port element $\\mathcal{R}$ corresponds to internal energy dissipation, and its port variables are\ndenoted by $(f_R,e_R)$. These port variables are terminated on a static energy-dissipating (resistive) relation $\\mathcal{R}$. Let $\\mathcal{F}_R = \\mathbb{R}^m$ and $\\mathcal{E}_R = \\mathcal{F}_R' = \\mathbb{R}^m$ be the dual space of $\\mathcal{F}_R$. In general, a resistive relation $\\mathcal{R} \\subset \\mathcal{F}_R \\times \\mathcal{E}_R$ is a set of the form\n\\begin{align}\n\t\\mathcal{R} = \\left\\{ (f_R,e_R) \\in \\mathcal{F}_R \\times \\mathcal{E}_R \\mid R(f_R,e_R) = 0, \\hspace{0.1 cm} \\langle e_R, f_R \\rangle_2 \\leq 0 \\right\\}.\n\t\\label{Resistive Relation}\n\\end{align}\nWe will call the subset $\\mathcal{R}$ an \\emph{energy-dissipating relation} or a \\emph{resistive structure}. If we restrict ourselves to linear resistive structures, the mapping $R \\colon \\mathcal{F}_R \\times \\mathcal{E}_R \\to \\mathbb{R}$ will be of the form\n\\begin{align*}\n\tR(f_R,e_R) = R_ff_R + R_ee_R,\n\\end{align*}\nwhere $R_f,R_e \\in \\mathbb{R}^{m \\times m}$. In order to guarantee that the second constitutive relation in (\\ref{Resistive Relation}) is satisfied whenever the first one is, we need to impose the following conditions on those matrices:\n\\begin{enumerate}\n\t\\item[(i)] $R_fR_e^{\\top} = R_eR_f^{\\top} \\geq 0$ and\n\t\\item[(ii)] $\\rank \\left( \\begin{bmatrix}\n\t\tR_f & R_e\n \t\\end{bmatrix} \\right)= \\dim (\\mathcal{F}_R) = m$. \n\\end{enumerate}\nIf these conditions are satisfied, then we can equivalently rewrite the first constitutive condition in (\\ref{Resistive Relation}) as the requirement that there exists some vector $\\lambda \\in \\mathbb{R}^m$ such that\n\\begin{align*}\n\tf_R = R_e^{\\top} \\lambda \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} e_R = -R_f^{\\top}\\lambda.\n\\end{align*}\nIt is easy to check now that under these conditions we have $\\langle e_R , f_R \\rangle_2 \\leq 0$, and hence the second constitutive condition in (\\ref{Resistive Relation}) can be omitted. Altogether, a linear resistive structure is of the form\n\\begin{align}\n\t\\mathcal{R} = \\left\\{ (f_R,e_R) \\in \\mathcal{F}_R \\times \\mathcal{E}_R \\mid R_f f_R + R_ee_R = 0 \\right\\}\n\t\\label{Resistive Relation - Linear}\n\\end{align}\nwith suitably defined matrices $R_f, R_e \\in \\mathbb{R}^{m \\times m}$. For a system with energy-storing and resistive elements, the bond space is given by $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E} = (X \\times \\mathcal{F}_R) \\times (X' \\times \\mathcal{E}_R)$, and the power pairing again equals the natural inner product. By definition, the Dirac structure $\\mathcal{D} \\subset \\mathcal{B}$ of such a system satisfies the power-balance\n\\begin{align}\n\t\\left\\langle \\begin{pmatrix}\n\t\te_S \\\\\n\t\te_R \n\t\\end{pmatrix} \\, \\Bigg| \\, \\begin{pmatrix}\n\tf_S \\\\\n\tf_R\n\\end{pmatrix} \\right\\rangle = e_S^{\\top}f_S + e_R^{\\top}f_R = 0, \\hspace{0.5 cm} \\left( \\begin{pmatrix}\nf_S \\\\\nf_R \n\\end{pmatrix} , \\begin{pmatrix}\ne_S \\\\\ne_R\n\\end{pmatrix} \\right) \\in \\mathcal{D}.\n\t\\label{Power-Balance with Dissipation}\n\\end{align}\nThis leads, by substitution of (\\ref{Power-Balance Energy-Storing Element with fS and eS}) and (\\ref{Power-Balance with Dissipation}), to\n\\begin{align}\n\t\\frac{d}{dt} H(x(t)) = -e_S^{\\top}(t) f_S(t) = e_R^{\\top}(t) f_R(t) \\leq 0, \\hspace{0.5 cm} t\\geq0.\n\t\\label{Power-Balance Energy-Storing and Energy-Dissipating Elements}\n\\end{align}\nThis inequality expresses the fact that the internally stored energy (the Hamiltonian) monotonically decreases in the presence of resistive elements.\n\\vspace{0.5 cm}\\\\\nAn important special case of energy-dissipating relations occurs when\nthe resistive relation can be expressed as\n\\begin{align*}\n\tf_R = - F(e_R),\n\\end{align*}\nwhere $F \\colon \\mathbb{R}^m \\to \\mathbb{R}^{m}$ satisfies $e_R^{\\top} F(e_R) \\geq 0$ for all $e_R \\in \\mathbb{R}^m$. For linear resistive elements, we have $F(e_R) = Re_R$ with $R \\in \\mathbb{R}^{m \\times m}$ being symmetric and positive semi-definite. A typical example of a linear resistive structure is provided by a linear damper in a spring-mass-damper system, where $\\mathcal{R}_d$ is given by\n\\begin{align}\n\t\\label{Resistive Relation Finite-Dimensional Case}\n\t\\mathcal{R}_d = \\{ (f_R,e_R) \\in \\mathbb{R}^2 \\mid f_R = -de_R \\},\n\\end{align}\nand where $d \\in \\mathbb{R}_+$ is the damping constant.\n\n\\section{External Ports}\n\\label{Section External Ports}\nThe \\emph{external port} $(f_P,e_P) \\in \\mathcal{F}_P \\times \\mathcal{E}_P$, where $\\mathcal{F}_P = \\mathbb{R}^k$ and $\\mathcal{E}_P= \\mathcal{F}_P'$, models the interaction of the system with its environment. This comprises different situations. For instance, there are port variables which are accessible for controller action (see \\cite[Chapter 5]{ModCompPhys}, \\cite[Chapter~7]{SchaftL2}), or there is an external port describing the interaction with the physical environment. Another important type of external port variables corresponds to source connections, as it is often the case in electrical circuits. For instance, in an electrical circuit with voltage source the system's input is the voltage of the source, and the current through the source is set to be the resulting output variable. If one wishes to distinguish between these types of external ports, which in some cases may even be inevitable, one can divide the external port into a control port and an interaction port (cf. \\cite[Subsection 2.2.1]{ModCompPhys}). However, we will not distinguish between them.\n\\vspace{0.5 cm}\\\\\nNow, taking the external port into account, for a Dirac structure\n\\begin{align*}\n\t\\mathcal{D} \\subset X \\times \\mathcal{F}_R \\times \\mathcal{F}_P \\times X' \\times \\mathcal{E}_R \\times \\mathcal{E}_P\n\\end{align*}\nthe power-balance (\\ref{Power-Balance with Dissipation}) extends to\n\\begin{align*}\n\t \\left\\langle \\begin{pmatrix}\n\t \te_S \\\\\n\t \te_R \\\\\n\t \te_P\n\t \\end{pmatrix} \\, \\Bigg| \\, \\begin{pmatrix}\n\t f_S \\\\\n\t f_R \\\\\n\t f_P\n \\end{pmatrix} \\right\\rangle = e_S^{\\top}f_S + e_R^{\\top}f_R + e_P^{\\top}f_P = 0, \\hspace{0.5 cm} \\left( \\begin{pmatrix}\n f_S \\\\\n f_R \\\\\n f_P\n\\end{pmatrix} , \\begin{pmatrix}\ne_S \\\\\ne_R \\\\\ne_P\n\\end{pmatrix} \\right) \\in \\mathcal{D},\n\n\\end{align*}\nand (\\ref{Power-Balance Energy-Storing and Energy-Dissipating Elements}) accordingly extends to\n\\begin{align*}\n\t\\frac{d}{dt}H(x(t)) = - e_S^{\\top}(t) f_S(t) = e_R^{\\top}(t)f_R(t) + e_P^{\\top}(t)f_P(t) \\leq e_P^{\\top}(t)f_P(t), \\hspace{0.5 cm} t \\geq 0.\n\n\\end{align*}\n This inequality states that the increase of the internally stored energy is always less\nthan or equal to the externally supplied power.\n\n\\section{Port-Hamiltonian Dynamics}\n\\label{Section Port-Hamiltonian Dynmacis}\nNow that we have discussed the constitutive ports of a Dirac structure and the corresponding power-balance, we want to present the underlying geometrical structure of port-Hamiltonian systems. We will begin with the definition of linear port-Hamiltonian systems by means of Dirac structures, and give a more familiar representation of a big class of such systems. At the end of this section, we will give some final remarks regarding non-linear port-Hamiltonian systems and port-Hamiltonian differential algebraic systems.\n\\vspace{0.5 cm}\\\\\nWe begin with the definition of linear port-Hamiltonian systems with respect to an underlying Dirac structure.\n\n\\begin{definition}[Linear Port-Hamiltonian System]\n\t\\label{Definition Linear Port-Hamiltonian System}\nConsider the state space $X = \\mathbb{R}^n$ and a Hamiltoninan $H \\colon X \\to \\mathbb{R}$ defining energy storage. Furthermore, let $\\mathcal{F}_R = \\mathcal{E}_R = \\mathbb{R}^m$ and $\\mathcal{F}_P = \\mathcal{E}_P = \\mathbb{R}^k$. A linear port-Hamiltonian system on $X$ is defined as a triple $(\\mathcal{D}, H, \\mathcal{R})$, consisting of a Dirac structure\n\\begin{align*}\n\t\\mathcal{D} \\subset \\mathcal{F} \\times \\mathcal{E} = (X \\times \\mathcal{F}_R \\times \\mathcal{F}_P) \\times (X' \\times \\mathcal{E}_R \\times \\mathcal{E}_P)\n\\end{align*}\ndefined with respect to the power pairing $\\langle \\cdot \\mid \\cdot \\rangle = \\langle \\cdot , \\cdot \\rangle_2 \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$, with energy-storing ports $(f_S,e_S) \\in X \\times X'$, resistive ports $(f_R, e_R) \\in \\mathcal{F}_R \\times\\mathcal{E}_R$, external ports $(f_P, e_P) \\in \\mathcal{F}_p \\times \\mathcal{E}_P$, the Hamiltonian $H$, and a linear resistive structure\n\\begin{align*}\n\t\\mathcal{R} \\subset \\mathcal{F}_R \\times \\mathcal{E}_R\n\\end{align*}\nof the form \\eqref{Resistive Relation - Linear}. Its dynamics are geometrically specified by the requirement that for all $t \\geq 0$ we have\n\\begin{align*}\n\t\\left( \\begin{pmatrix}\n\t\t-\\dot{x}(t) \\\\\n\t\tf_R (t) \\\\\n\t\tf_P (t)\n\t\\end{pmatrix}, \\begin{pmatrix}\n\t\\nabla H(x(t)) \\\\\n\te_R(t) \\\\\n\te_P (t) \n\\end{pmatrix} \\right) \\in \\mathcal{D} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} (f_R(t), e_R(t)) \\in \\mathcal{R}.\n\\end{align*}\n\\end{definition}\nAt the beginning of Chapter \\ref{Chapter Finite-dimensional Port-Hamiltonian Systems}, we have already seen how a mass-spring system is modeled as a port-Hamiltonian system. We revisit this example to show that this is indeed a linear port-Hamiltonian system in the sense of Definition \\ref{Definition Linear Port-Hamiltonian System}.\n\\begin{example}[Mass-spring System Revisited]\n\t\\label{Example Mass-spring System Revisited}\nConsider the mass-spring system from Example \\ref{Example Mass-Spring System}. The dynamics of this system are defined by equation \\eqref{Example Mass-Spring System - pH System}. Defining the state space $X = \\mathbb{R}^2$ with state vector $x = \\begin{bmatrix}\n\tq \\\\\n\tp\n\\end{bmatrix}$, and taking $\\mathcal{F}_S = \\mathcal{E}_S = X$, one immediately sees that the underlying Dirac structure of this system is given by the graph of the skew-symmetric matrix $-J \\in \\mathbb{R}^{2 \\times 2}$ with\n\\begin{align*}\nJ = \\begin{bmatrix}\n\t0 & 1 \\\\\n\t-1 & 0\n\\end{bmatrix}, \\text{ i.e.,} \\hspace{0.3 cm} \t\\mathcal{D}_{J} = \\{ (f_S,e_S) \\in \\mathcal{F}_S \\times \\mathcal{E}_S \\mid - f_S = Je_S \\}.\n\\end{align*}\nHere we have $- f_S = \\begin{bmatrix}\n\t\\dot{q} \\\\\n\t\\dot{p} \n\\end{bmatrix}$ and $e_S = \\begin{bmatrix}\n\\frac{\\partial H}{\\partial q} \\\\\n\\frac{\\partial H}{\\partial p}\n\\end{bmatrix}$, where $H \\colon X \\to \\mathbb{R}$ is the Hamiltonian given by \\eqref{Example Mass-Spring System - Hamiltonian}. Note that the matrix $J$ is the one given on the right-hand side in \\eqref{Example Mass-Spring System - pH System}. As there are neither external ports nor resistive elements, the dynamics of this linear port-Hamiltonian system is geometrically specified by\n\\begin{align*}\n (f_S(t), e_S(t)) =\t(-\\dot{x}(t), \\nabla H(x(t))) \\in \\mathcal{D}, \\hspace{0.5 cm} t \\geq 0.\n\\end{align*}\nMoreover, by skew-symmetry of $J$, the power-balance as expected yields\n\\begin{align*}\n\t\\frac{d}{dt} H(q(t),p(t)) = \\left\\langle \\begin{pmatrix}\n\t\tkq(t) \\\\\n\t\t\\frac{p(t)}{m}\n\t\\end{pmatrix} \\Bigg| \\begin{pmatrix}\n\t\\dot{q}(t) \\\\\n\t\\dot{p}(t)\n\\end{pmatrix} \\right\\rangle = - \\langle e_S(t) \\mid f_S(t) \\rangle = 0, \\hspace{0.5 cm} t \\geq 0.\n\\end{align*}\n\\QEDA\n\\end{example}\nA state-space representation, as established for the mass-spring system in Example \\ref{Example Mass-Spring System}, is quite common. More general, an important class of linear port-Hamiltonian systems is obtained by considering Dirac structures which are the graph of a skew-symmetric map of the form\n\\begin{align*}\n\t\\begin{bmatrix}\n\t\te_S \\\\\n\t\te_R \\\\\n\t\te_P \\\\\n\t\\end{bmatrix} \\mapsto \\begin{bmatrix}\n\t\tf_S \\\\\n\t\tf_R \\\\\n\t\tf_P \n\t\\end{bmatrix} := \\begin{bmatrix}\n\t\t-J & -G_R & -G \\\\\n\t\tG_R^{\\top} & 0 & 0 \\\\\n\t\tG^{\\top} & 0 & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\te_S \\\\\n\t\te_R \\\\\n\t\te_P \\\\\n\t\\end{bmatrix} ,\n\\end{align*}\nwith $J = - J^{\\top} \\in \\mathbb{R}^{n \\times n}$, a linear dissipation relation $f_R = - \\bar{R}e_R$ for some positive-semidefinite matrix $\\bar{R} \\in \\mathbb{R}^{m \\times m}$, and $G_R \\in \\mathbb{R}^{n \\times m}$, $G \\in \\mathbb{R}^{n \\times k}$. This yields so-called \\emph{input-state-output port-Hamiltonian systems} of the form\n\\begin{align}\n\t\\begin{split}\n\t\\dot{x} &= \\left[ J - R \\right] \\nabla H(x) + G u, \\\\\n\ty&= G^{\\top} \\nabla H(x), \n\t\\end{split}\n\t\\label{Input-State-Output port-Hamiltonian System}\n\\end{align}\nwith $R := G_R \\bar{R} G_R^{\\top} \\in \\mathbb{R}^{n \\times n}$, and with $u = e_P$ the input vector and $y = f_P$ the output vector. This is a convenient representation for the investigation of control related problems for port-Hamiltonian systems, see \\cite[Chapter~5]{ModCompPhys}, \\cite[Chapter~7]{SchaftL2}, \\cite[Chapter~15]{SchaftJeltsema}. \n\\vspace{0.5 cm}\\\\\nThe following example aims to emphasize port-based network modeling of multi-physics systems. We also want to point to Example 2.5 in \\cite{SchaftJeltsema}, where a port-Hamiltonian formulation for the dynamics of a DC motor is derived.\n\\begin{example}[Levitated Ball System; Example 2.4 in \\cite{SchaftJeltsema}]\n\t\\label{Example - Levitated Ball System}\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width = 6 cm]{Levitated-Ball_2.png}\n\t\t\\caption{Magnetically Levitated Ball of Example \\ref{Example - Levitated Ball System} \\cite[Figure 2.4]{SchaftJeltsema}}\n\t\t\\label{Figure Levitated Ball System}\n\t\t\\end{figure}\n\tConsider the dynamics of an iron ball that is levitated by the magnetic field of a controlled inductor, as schematically depicted in Figure \\ref{Figure Levitated Ball System}. The Hamiltonian $H \\colon \\mathbb{R}^3 \\to \\mathbb{R}$ of this system - with $q \\in \\mathbb{R}$ the height of the ball, $m >0$ the ball's mass, $p \\in \\mathbb{R}$ the vertical momentum, $ \\varphi \\in \\mathbb{R}$ the magnetic flux-linkage of the inductor, $R_c >0$ the coil's resistance, $g > 0$ the gravitational constant, $L(\\cdot)$ the inductance depending on the height $q$, $V \\in \\mathbb{R}$ the voltage across the inductor, and $I \\in \\mathbb{R}$ the current through the inductor - is defined as\n\t\\begin{align*}\n\t\tH(q,p,\\varphi) = E_{\\text{pot}}(q) + E_{\\text{kin}}(p) + E_{\\text{mag}}(q,\\varphi) = mgq + \\frac{p^2}{2m} + \\frac{\\varphi^2}{2L(q)}.\n\t\\end{align*}\nThe system dynamics are given by\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\\begin{bmatrix}\n\t\t\t\\dot{q} \\\\\n\t\t\t\\dot{p} \\\\\n\t\t\t\\dot{\\varphi} \n\t\t\\end{bmatrix} &= \\begin{bmatrix}\n\t\t0 & 1 & 0 \\\\\n\t\t-1 & 0 & 0 \\\\\n\t\t0 & 0 & - R_c \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\\frac{\\partial H}{\\partial q} \\\\\n\t\\frac{\\partial H}{\\partial p} \\\\\n\t\\frac{\\partial H }{\\partial \\varphi}\n\\end{bmatrix} + \\begin{bmatrix}\n0 \\\\\n0 \\\\\n1\n\\end{bmatrix} V, \\\\\nI &= \\frac{\\partial H}{\\partial \\varphi}.\n\\end{split}\n\\label{Example Levitated Ball System - pH System}\n\t\\end{align}\nHere, $e_P = V$ denotes the input and $f_P = I$ denotes the system's output. Furthermore, the matrices $J$, $R \\in \\mathbb{R}^{3 \\times 3}$, and $G \\in \\mathbb{R}^{3}$ from \\eqref{Input-State-Output port-Hamiltonian System} are given by\n\\begin{align*}\n\tJ = \\begin{bmatrix}\n\t\t0 & 1 & 0 \\\\\n\t\t-1 & 0 & 0 \\\\\n\t\t0 & 0 & 0 \n\t\\end{bmatrix}, \\hspace{0.3 cm} R = \\begin{bmatrix}\n\t0 & 0 & 0 \\\\\n\t0 & 0 & 0 \\\\\n\t0 & 0 & R_c\n\\end{bmatrix}, \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} G = \\begin{bmatrix}\n0 \\\\ \n0 \\\\\n1\n\\end{bmatrix}.\n\\end{align*}\nWhile the system description \\eqref{Example Levitated Ball System - pH System} may at first glance indicate that the mechanical and the magnetic part of the given system are decoupled, they are actually coupled, namely via the Hamiltonian $H$: the magnetic energy $E_{\\text{mag}}$ depends both on the flux $\\varphi$ and the mechanical variable $q$. Consequently, the evolution of the (mechanical) momentum variable $p$ depends on the magnetic variable $\\varphi$, and the evolution of the magnetic variable $\\varphi$ depends on the mechanical variable $q$.\n\\QEDA\n\\end{example}\nNeedless to say, there exist nonlinear port-Hamiltonian systems as well. The major difference to linear port-Hamiltonian systems is that the state space $X$ becomes a smooth manifold, and the Dirac structure is not constant anymore, but is modulated by the state variables. Moreover, the spaces $\\mathcal{F}_S$ and $\\mathcal{E}_S$ become the tangent space and the cotangent space, respectively, as mentioned in Section \\ref{Section Energy-storing Elements}. Furthermore, the input-state-output description \\eqref{Input-State-Output port-Hamiltonian System} of nonlinear port-Hamiltonian systems becomes\n\\begin{align}\n\t\\begin{split}\n\t\\dot{x} &= [J(x) - R(x)]\\nabla H(x) + G(x) u, \\\\\n\ty &= G^{\\top}(x)\\nabla H(x). \n\\end{split}\n\\label{Nonlinear PH-System - Finite Dimensional}\n\\end{align}\n\nFor details and some examples we refer to \\cite[Chapter 6]{SchaftL2}, \\cite[Chapter 3]{SchaftJeltsema}, and the references specified therein. \n\\vspace{0.5 cm}\\\\\nWhen modeling dynamical systems, the appearance of algebraic equations next to differential equations is ubiquitous. For instance, in network modeling of physical systems, the interconnections between different dynamical components of the overall system almost unavoidably lead to algebraic constraints. Such systems are called \\emph{linear port-Hamiltonian descriptor systems} (\\emph{port-Hamiltonian differential-algebraic systems}), or short \\emph{pHDAEs} \\cite{Beattie}. The geometric definition of port-Hamiltonian systems is quite handy, as it can even characterize the dynamics of pHDAEs by choosing the underlying Dirac structure suitably. This is the reason why in Definition \\ref{Definition Linear Port-Hamiltonian System} one usually refers to the triple $(\\mathcal{D}, H, \\mathcal{R})$ as a (linear) port-Hamiltonian differential-algebraic system.\n\\vspace{0.5 cm}\\\\\nIn this chapter, we have presented the basic concepts and main characteristics of the port-Hamiltonian framework on finite-dimensional spaces. We have learned that the centerpiece of this approach is the power-conserving interconnection structure $\\mathcal{D}$, called the Dirac structure. We have presented the basic definition of a Dirac structure and discussed its constituting ports in the finite-dimensional case, namely ports associated with energy-storing and energy-dissipating elements as well as external ports. Lastly, we have geometrically specified the dynamics of linear port-Hamiltonian systems by means of their underlying Dirac structures and have given a state-space representation for a huge class of port-Hamiltonian systems.\n\\vspace{0.5 cm}\\\\\nNow that we have gained a basic understanding of port-based modeling of physical systems, we want to study infinite-dimensional port-Hamiltonian systems in the upcoming chapters, to finally deal with the main topic of this thesis, namely the port-Hamiltonian formulation of a system of two conservation laws with a moving interface. In order to do so, we need to provide some background information concerning semigroup theory, infinite-dimensional systems theory, the concept of admissibility, and evolution equations. We will treat these subjects in the next chapter. \n\n\n\\chapter{Some Background on Functional Analysis and Systems Theory}\n\\label{Chapter Some Background}\nIn preparation for the main part of this thesis, we recall some well-known concepts and results on selected topics. Most of the time, we will give no proofs, but provide references regarding the respective topics. Furthermore, we will omit various topics that are usually covered in the literature, and only focus on aspects that are useful for the discussion in the following chapters.\n\\vspace{0.5 cm}\\\\\nThis chapter is organized as follows. In Section \\ref{Section Operators} we recall the definition of the Hilbert space adjoint of a densely defined operator $A \\colon D(A) \\subset X \\to X$. Moreover, we introduce the notion of the formally adjoint operator of a matrix differential operator, and discuss the difference between the Hilbert space adjoint and the formal adjoint. As we will deal with infinite-dimensional (control) systems, we need a tool for the analysis of such systems. In Section~\\ref{Section Strongly Continuous Semigroups} we are going to study strongly continuous semigroups and characterize their infinitesimal generators. In Section \\ref{Section The Abstract Cauchy Problem} we cover abstract (i.e. Hilbert space-valued) linear differential equations and give sufficient and necessary conditions for well-posedness of such systems. We will see that the framework of strongly continuous semigroups is a powerful tool for the analysis of so-called abstract Cauchy problems. Afterwards, we are going to shortly discuss infinite-dimensional control systems in Section \\ref{Section Linear Control Systems}. For a proper treatment of such systems, we need to introduce the extrapolation space $X_{-1}$ and the concept of admissible control operators. We will give criteria for when the mild solution of an abstract control system is a continuous $X$-valued function. Section \\ref{Section Evolution Equations} is the last section of this chapter and deals with linear, time-variant evolution problems. We will study (two-parameter) evolution systems and define the notion of stability for a family of infinitesimal generators of $C_0$-semigroups. We assume that the reader is acquainted with Riemann integrals of vector-valued functions \\cite[Section 3.1]{Lorenzi}, the Lebesgue theory of vector-valued measurable functions, and Bochner intergrals \\cite[Section 1.1]{Arendt}.\n\\vspace{0.5 cm}\\\\\nUnless stated otherwise, throughout this chapter we assume that $X$ is a real or complex separable Hilbert space with inner product $\\langle \\cdot, \\cdot \\rangle_{X}$.\n\n\\section{Adjoint and Formally Adjoint Operators}\n\\label{Section Operators}\nIn this section, we want to define and highlight the difference of the Hilbert space adjoint of a matrix differential operator to its formal adjoint, and provide a simple example to illustrate this difference. For a detailed, more general discussion on formally adjoint operators we refer to \\cite[Chapter 13]{Aubin}.\n\\vspace{0.5 cm}\\\\\nFor the first part of this section, we follow \\cite[Section 2.9]{Tucsnak-Weiss}.\n\\begin{definition}\n\tLet $V$ and $Z$ be Hilbert spaces. An operator $J \\in \\mathcal{L}(V,Z)$ is called an isomorphism, or a unitary operator, from $V$ to $Z$, if we have\n\\begin{align*}\n\t\\langle J v, Jw \\rangle_Z = \\langle v , w \\rangle_V, \\hspace{0.5 cm} v,w \\in V.\n\\end{align*}\n\\end{definition}\nOne may verify that $J \\in \\mathcal{L}(V,Z)$ is unitary if and only if $\\|Jv\\|_Z = \\|v\\|_V$ holds for all $v \\in V$, and $\\ran(J) = Z$.\n\\vspace{0.5 cm}\\\\\nFor any Hilbert space $V$, we denote by $ \\langle z, v \\rangle_{V' \\times V}$ the functional $z \\in V'$ applied to $v \\in V$, so that $ \\langle \\cdot , \\cdot \\rangle_{V' \\times V}$ is linear in the first component and antilinear in the second component. This pairing is called the \\emph{natural (duality) pairing} on $V$. The norm on $V'$ is defined as\n\\begin{align*}\n\t\\| z \\|_{V'} = \\sup_{\\stackrel{ v \\in V}{\\|v\\|_V = 1}} \\vert \\langle z, v \\rangle_{V' \\times V} \\vert, \\hspace{0.5 cm} z \\in V'. \n\\end{align*}\nThere is an operator $J_R \\colon V \\to V'$ satisfying\n\\begin{align*}\n\t\\langle J_R v, w \\rangle_{V' \\times V} = \\langle v, w \\rangle_V, \\hspace{0.5 cm} v, w \\in V. \n\\end{align*}\nAccording to the Riesz representation theorem, $J_R$ is an isomorphism. We say that we identify the Hilbert space $V$ with its dual $V'$, if we do not distinguish between $v$ and $J_Rv$ for all $v \\in V$. \n\\vspace{0.5 cm}\\\\\nIn the following, we identify $X$ with its dual $X'$. Consider a linear and densely defined operator $A \\colon D(A) \\subset X \\to X$. We wish to associate a \\emph{Hilbert space adjoint} $A^{\\ast}$ to $A$. Its domain is given by\n\\begin{align*}\n\tD(A^{\\ast}) := \\left\\{ y \\in X \\mid \\langle A \\cdot , y \\rangle_X \\colon D(A) \\to \\mathbb{R}, \\, x \\mapsto \\langle Ax, y \\rangle_X \\text{ is continuous} \\right\\}.\n\\end{align*}\nIf $y \\in D(A^{\\ast})$, then the Hahn-Banach theorem extends the functional\n\\begin{align*}\n\tx \\mapsto \\langle A x, y \\rangle_X\n\\end{align*}\nto a continuous linear functional on $X$. The Riesz representation theorem yields that there exists a unique element $z_y \\in X$ such that this functional has the form\n\\begin{align*}\n\t\\langle \\cdot , z_y \\rangle_X. \n\\end{align*}\nFor $y \\in D(A^{\\ast})$ we define $A^{\\ast}y = z_y$. Hence, it holds that\n\\begin{align*}\n\t\\langle A x, y \\rangle_X = \\langle x , A^{\\ast}y \\rangle_X, \\hspace{0.5 cm} x \\in D(A), \\, y \\in D(A^{\\ast}). \n\\end{align*}\nIn other words, the domain of the Hilbert space adjoint $A^{\\ast}$ of an operator $A$ is given by\n\\begin{align}\n\t\\label{Adjoint Operator Domain wrt Frechet Riesz}\n\tD(A^{\\ast}) = \\left\\{ y \\in X \\mid \\exists! \\, z_y \\in X \\colon \\forall x \\in D(A) \\colon \\langle Ax, y \\rangle_X = \\langle x , z_y \\rangle_X \\right\\}.\n\\end{align}\nThe following definitions can be found in Section 3.2 and Section 3.7 in \\cite{Tucsnak-Weiss}, and in \\cite[Appendix~A.3.2]{CurtainZwart}.\n\\begin{definition}\n\t\\label{Definition Symmetric and Adjoint Operators}\n\tLet $A \\colon D(A) \\subset X \\to X$ and $B \\colon D(B) \\subset X \\to X$ be linear and densely defined.\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item $A$ is called symmetric, if \n\t\t\\begin{align*}\n\t\t\t\\langle Ax, y \\rangle_X = \\langle x, Ay \\rangle_X, \\hspace{0.5 cm} x,y \\in D(A).\n\t\t\\end{align*}\n\t\t\\item $A$ is called skew-symmetric, if \n\t\t\\begin{align*}\n\t\t\t\\langle Ax, y \\rangle_X = - \\langle x, Ay \\rangle_X, \\hspace{0.5 cm} x,y \\in D(A).\n\t\t\\end{align*} \n\t\\item If for all $x \\in D(A)$, $y \\in D(B)$ it holds that\n\t\\begin{align*}\n\t\t\\langle Ax, y \\rangle_X = \\langle x, By \\rangle_X,\n\t\t\\end{align*}\n\tthen $B \\subset A^{\\ast}$, that is, $D(B) \\subset D(A^{\\ast})$ and $By = A^{\\ast}y$ for all $y \\in D(B)$.\n\t\t\\item $A$ is called self-adjoint if $A^{\\ast} = A$, that is, $D(A) = D(A^{\\ast})$ and $Ax = A^{\\ast}x$ for all $x \\in D(A)$.\n\t\t\\item $A$ is called skew-adjoint if $A^{\\ast} = -A$.\n\t\t\\item $A$ is called coercive if it is self-adjoint and there exists some $m > 0$ such that $A \\geq mI_X$, i.e.,\n\t\t\\begin{align*}\n\t\t\t\\langle Ax, x \\rangle_X \\geq m \\|x\\|_X^2, \\hspace{0.5 cm} x \\in D(A).\n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{definition}\nFor the rest of this section, we restrict ourselves to matrix differential operators defined on a Sobolev space. For this type of operators, we want to introduce the notion of \\emph{formally adjoint operators}. Roughly speaking, the only difference to the Hilbert space adjoint is that the constitutive relations of its domain are relaxed in a way such that \n\\begin{align*}\n\t\\langle A x, y \\rangle_X = \\langle x, A^{\\ast} y \\rangle_X \\hspace{0.2 cm} + \\text{boundary terms}, \\hspace{0.5 cm} x \\in D(A), \\, y \\in D(A^{\\ast}).\n\\end{align*}\nThe discussion in Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators} will clarify the necessity of allowing boundary terms concerning the relation between an operator and its formal adjoint.\n\\vspace{0.5 cm}\\\\\nNow, let $X = L^2([a,b], \\mathbb{R}^n)$ be endowed with the inner product\n\\begin{align*}\n\t\\langle x , y \\rangle_X = \\langle x , y \\rangle_{L^2} = \\int_{a}^{b} y^{\\top}(z) x(z) \\, dz, \\hspace{0.5 cm} x,y \\in X.\n\\end{align*}\nLet us define the Sobolev spaces\n \\begin{align*}\n\tH^1([a,b], \\mathbb{R}^n) &= \\left\\{ x \\in X \\, \\big| \\, x \\text{ is absolutely continuous}, \\, \\frac{dx}{dz} \\in X \\right\\}, \\\\\n\tH^N([a,b], \\mathbb{R}^n) &= \\left\\{ x \\in X \\, \\big| \\, x \\text{ is absolutely continuous}, \\, \\frac{dx}{dz} \\in H^{N-1}([a,b], \\mathbb{R}^n) \\right\\}, \\hspace{0.5 cm} N \\geq 2,\n\\end{align*}\nas well as\n\\begin{align*}\n\tH_0^1([a,b], \\mathbb{R}^n) &= \\left\\{ x \\in H^1([a,b], \\mathbb{R}^n) \\mid x(a) = x(b) = 0 \\right\\}, \\\\\n\tH_0^N([a,b], \\mathbb{R}^n) &= \\left\\{ x \\in H^N([a,b],\\mathbb{R}^n) \\cap H_0^{N-1}([a,b], \\mathbb{R}^n) \\, \\big| \\, \\frac{dx}{dz} \\in H_0^{N-1}([a,b], \\mathbb{R}^n) \\right\\}, \\hspace{0.5 cm} N \\geq 2.\n\\end{align*} \n\tFurthermore, define the \\emph{boundary trace operator} $\\trace \\colon H^N \\left( [a,b], \\mathbb{R}^n \\right) \\to \\mathbb{R}^{2nN}$ by\n\\begin{align}\n\t\\trace(x) := \\begin{bmatrix}\n\t\tx(b) \\\\\n\t\t\\vdots \\\\\n\t\t\\frac{d^{N-1}x}{d z^{N-1}}(b) \\\\\n\t\tx(a) \\\\\n\t\t\\vdots \\\\\n\t\t\\frac{d^{N-1}x}{d z^{N-1}}(a)\n\t\\end{bmatrix}, \\hspace{0.5 cm} x \\in H^N \\left( [a,b], \\mathbb{R}^n \\right).\n\t\\label{Trace Operator - Chapter 3}\n\\end{align}\nThe boundary trace operator $\\trace$ is linear, bounded, and surjective, see \\cite[Section 7.8]{Aubin}. \nIt allows us to equivalently write for $N \\in \\mathbb{N}$,\n\\begin{align*}\n\tH_0^N([a,b], \\mathbb{R}^n) = \\left\\{ x \\in H^N([a,b], \\mathbb{R}^n) \\mid \\trace(x) = 0 \\right\\}.\n\\end{align*}\nNow, consider the matrix differential operator $A \\colon D(A) \\subset X \\to X$ given by\n\\begin{align}\n\t\\begin{split}\n\tD(A) &= \\left\\{ x \\in X \\mid x \\in H^N([a,b], \\mathbb{R}^n), \\, W \\trace(x) = 0 \\right\\}, \\\\\n\tAx &= \\sum_{i = 0}^{N} P_i \\frac{d^ix}{dz^i}, \\hspace{0.5 cm} x \\in D(A),\n\t\\end{split}\n\\label{Matrix Differential Operator A}\n\\end{align}\nwith $P_i \\in \\mathbb{R}^{n \\times n}$, $i = 0, \\ldots,N$, and $W \\in \\mathbb{R}^{n \\times 2n}$. In particular, we have $H_0^N([a,b], \\mathbb{R}^n) \\subset D(A)$.\n\n\\begin{definition}\n\tFor the domain of the formal adjoint $A_0^{\\ast} \\colon D(A_0^{\\ast}) \\subset X \\to X$ of the matrix differential operator $A$ defined in \\eqref{Matrix Differential Operator A} it holds that\n\t\\begin{align*}\n\tD(A_0^{\\ast}) = \\left\\{ y \\in X \\mid \\langle A \\cdot, y \\rangle_X \\colon H_0^N([a,b], \\mathbb{R}^n) \\to \\mathbb{R}, \\hspace{0.1 cm} x \\mapsto \\langle A x, y \\rangle_X \\text{ is continuous} \\hspace{0.1 cm} \\right\\}.\n\\end{align*}\n\\end{definition} \nIn other words, the formal adjoint of the differential operator $A$ defined in \\eqref{Matrix Differential Operator A} is simply given by the adjoint of its restriction $A_{|H_0^N([a,b],\\mathbb{R}^n)}$. \n\\vspace{0.5 cm}\\\\\nNext, we present the formal pendant of some of the notions introduced in Definition \\ref{Definition Symmetric and Adjoint Operators}.\n\\begin{definition}\n\t\\label{Definition Formal Operator Properties}\n\tLet $A \\colon D(A) \\subset X \\to X$ be the matrix differential operator defined in \\eqref{Matrix Differential Operator A}. \n\t\\begin{enumerate}\n\t\t\\item[(i)] $A$ is called formally symmetric, if\n\t\t\\begin{align*}\n\t\t\t\\langle A x, y \\rangle_X = \\langle x, Ay \\rangle_X, \\hspace{0.5 cm} x,y \\in H_0^N([a,b], \\mathbb{R}^n).\n\t\t\\end{align*}\n\t\t\\item[(ii)] $A$ is called formally skew-symmetric, if\n\t\t\\begin{align*}\n\t\t\t\\langle Ax, y \\rangle_X = - \\langle x, Ay \\rangle_X, \\hspace{0.5 cm} x,y \\in H_0^N([a,b], \\mathbb{R}^n).\n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{definition}\nLet us clarify the difference by reference to some simple examples.\n\\begin{example}\n\tConsider the following examples.\n\t\\begin{enumerate}\n\t\t\\item[(i)][cf. Example 13.4 in \\cite{Rudin}] Let $X = L^2([a,b], \\mathbb{R})$. Consider the operator $A \\colon D(A) \\subset X \\to X$ given by\n\t\t\\begin{align*}\n\t\t\tD(A) &= H^1([a,b], \\mathbb{R}), \\\\\n\t\t\tAx &= \\frac{d}{dz}x, \\hspace{0.5 cm} x \\in D(A).\n\t\t\\end{align*}\n\tWe claim that the Hilbert space adjoint and the formal adjoint of the operator $A$ are given by $A_1$ and $A_2$, respectively, with\n\t\\begin{align*}\nD(A_1) &= H_0^1([a,b], \\mathbb{R}), \\\\\nD(A_2) &= D(A), \\\\\nA_k x &= - \\frac{d}{dz}x, \\hspace{0.5 cm} x \\in D(A_k), \\, k= 1,2. \n\\end{align*}\nNote that for all $x \\in D(A)$, $y \\in D(A_1)$ we have\n\\begin{align*}\n\t\\langle Ax, y \\rangle_X = \\int_{a}^{b} \\frac{d}{dz}x(z) y(z) \\, dz = \\big[ x(z) y(z) \\big]_{a}^{b} - \\int_{a}^{b} x(z) \\frac{d}{dz}y(z) \\, dz = \\langle x, A_1 y \\rangle_X. \n\\end{align*}\nThus, $A_1 \\subset A^{\\ast}$. In particular, by virtue of the Cauchy-Schwarz inequality, for $y \\in D(A_1)$ we have\n\\begin{align*}\n\t\\sup_{\\substack{ x \\in D(A) \\\\\n\t\t\t \\|x\\|_X =1 \n\t\t\t\t\t\t }} \\vert \\langle Ax, y \\rangle_X \\vert \\leq \\sup_{\\substack{ x \\in D(A) \\\\\n\t\t\t\t\t\t \\|x\\|_X =1 }} \\int_{a}^{b} \\vert x(z) \\vert \\left\\vert \\frac{d}{dz} y(z) \\right\\vert \\, dz \\leq \t\\sup_{\\substack{ x \\in D(A) \\\\\n\t\t\t\t\t\t \\|x\\|_X =1 }} \\|x \\|_X \\left\\|\\frac{d}{dz} y \\right\\|_X \\leq C_y.\n\t\\end{align*}\n\t Conversely, suppose that $y \\in D(A^{\\ast}) \\subset X$. Set $\\psi = A^{\\ast}y$ and define $\\Psi = \\int_{a}^{\\cdot} \\psi(r) \\, dr \\in H^1([a,b], \\mathbb{R})$. Then for all $x \\in D(A)$ it holds that\n\\begin{align*}\n\t\\langle A x, y \\rangle_X = \\langle x, A^{\\ast} y \\rangle_X = \\langle x, \\psi \\rangle_X\t= \\langle x, \\frac{d}{dz} \\Psi \\rangle_X = x(b) \\Psi(b) - \\langle Ax, \\Psi \\rangle_X. \n\\end{align*}\nSince this equation holds in particular for all non-zero constant functions, which are obviously contained in $D(A)$, we conclude that $\\Psi(b) = 0$. Hence,\n\\begin{align*}\n\t\\langle A x, y + \\Psi \\rangle_X = 0\n\\end{align*}\nfor all $x \\in D(A)$. As a consequence,\n\\begin{align*}\n\ty + \\Psi \\in (\\ran (A))^{\\bot} = X^{\\bot} = \\left\\{ 0 \\right\\},\n\\end{align*}\nwhence $y = - \\Psi$. This implies that $y \\in H^1([a,b], \\mathbb{R})$, $\\trace(y) = 0$, and \n\\begin{align*}\n\ty = - \\int_{a}^{\\cdot} \\psi (r) \\, dr = - \\int_{a}^{\\cdot} A^{\\ast}y(r) \\, dr. \n\\end{align*} \nBy taking the derivative on both sides, we conclude that $A^{\\ast}y = - \\frac{d}{dz} y$ for all $y \\in H_0^1([a,b], \\mathbb{R})$. Altogether, the Hilbert space adjoint is given by $A^{\\ast} = A_1$. \n\\vspace{0.5 cm}\\\\\nTo prove that the formal adjoint of $A$ is given by $A_2$, we proceed similarly. Note that for all $x \\in H_0^1([a,b], \\mathbb{R})$, $y \\in D(A_2)$ we have\n\\begin{align*}\n\t\\langle Ax, y \\rangle_X = \\langle x, A_2 \\rangle_X.\n\\end{align*}\nConversely, we again suppose that $y \\in X$ lies in the domain of the formal adjoint, and define $\\psi = A^{\\ast}y$ as well as $\\Psi = \\int_{a}^{\\cdot} \\psi(r) \\, dr$. Since $x(b) = 0$ for all $x \\in H_0^1([a,b], \\mathbb{R})$, we find that $y + \\Psi \\in (\\ran (A_{|H_0^1([a,b],\\mathbb{R})}))^{\\bot}$. Note that $\\ran(A_{|H_0^1([a,b],\\mathbb{R})})$ consists of all functions $y \\in X$ satisfying\n\\begin{align*}\n\t\\int_{a}^{b} y(z) \\, dz = 0. \n\\end{align*}\nDenoting by $Y$ the one-dimensional subspace of $X$ that contains the constant functions on $[a,b]$, we conclude that\n\\begin{align*}\n\t\\ran (A_{|H_0^1([a,b],\\mathbb{R})}) = Y^{\\bot}.\n\\end{align*}\nHence, $y + \\Psi \\in Y$. As a result, $y$ is absolutely continuous with $\\frac{d}{dz} y \\in X$, and therefore lies in $H^1([a,b], \\mathbb{R})$. Summing up, the formal adjoint is given by $A_2$. We see that we obtain the domain of the formal adjoint by simply dropping the boundary conditions from the domain of the Hilbert space adjoint. \n\\item[(ii)] Let $X = L^2([a,b], \\mathbb{R}^2)$. Consider the operator $A \\colon D(A) \\subset X \\to X$ given by\n\\begin{align*}\n\tD(A) &= \\left\\{ x \\in X \\mid x \\in H^1([a,b], \\mathbb{R}^2), \\, x(a) = 0 \\right\\}, \\\\\n\tAx &= \\begin{bmatrix}\n\t\t0 & -1 \\\\\n\t\t-1 & 0\n\t\t\\end{bmatrix} \\frac{d}{dz} x, \\hspace{0.5 cm} x \\in D(A). \n\\end{align*}\nThe boundary condition is therefore given by\n\\begin{align*}\n\t\\begin{bmatrix}\n\t\t0 & 0 & 1 & 0 \\\\\n\t\t0 & 0 & 0 & 1\n\t\\end{bmatrix} \\trace(x) = 0, \\hspace{0.5 cm} x \\in D(A). \n\\end{align*}\nFor all $x,y \\in D(A)$ it holds that\n\\begin{align*}\n\t\\langle A x, y \\rangle_X &= - \\int_{a}^{b} y_1 (z) \\frac{d}{dz} x_2(z) + y_2(z) \\frac{d}{dz} x_1(z) \\, dz \\\\\n\t&= - \\big[ x_1(z) y_2(z) + y_1(z) x_2(z) \\big]_{a}^{b} + \\int_{a}^{b} \\frac{d}{dz}y_1(z) x_2(z) + \\frac{d}{dz}y_2(z) x_1(z) \\, dz \\\\\n\t&= - x_1(b) y_2(b) - y_1(b) x_2(b) - \\langle x, Ay \\rangle_X. \n\\end{align*}\nThus, $A$ is not skew-symmetric, but formally skew-symmetric. \n\\end{enumerate}\n\\QEDA\n\\end{example}\nAfter clarifying the difference between Hilbert space adjoints and formally adjoint operators of matrix differential operators, we are going to introduce the powerful concept of strongly continuous semigroups. \n\n\n\\section{Strongly Continuous Semigroups}\n\\label{Section Strongly Continuous Semigroups}\nThe theory of (one-parameter) semigroups of linear operators has a broad variety of applications, and has proven to be a successful tool for the analysis of linear infinite-dimensional (control) systems. For a matrix operator $A \\in \\mathbb{R}^{n \\times n}$, we know that the family $( e^{At})_{t \\geq 0}$ of matrix exponentials describes the evolution of the state subject to the linear system\n\\begin{align*}\n\t\\dot{x}(t) = Ax(t).\n\\end{align*}\nIn particular, the mapping $t \\mapsto e^{At}$ is differentiable and satisfies\n\\begin{align*}\n\t\\frac{d}{dt} e^{At} = Ae^{At}.\n\\end{align*}\nThe concept of strongly continuous semigroups is commonly used to generalize the above scenario in order to study (linear) systems with values in an arbitrary Hilbert space.\n\\vspace{0.5 cm}\\\\\nIn this section, we give a concise overview concerning strongly continuous semigroups, and collect some facts about the infinitesimal generators of such semigroups as well as some important criteria regarding the generation of strongly continuous semigroups. The material covered in this section can be found in \\cite{Arendt}, \\cite{CurtainZwart}, \\cite{EngelNagel}, \\cite{Pazy83}, or \\cite{Tucsnak-Weiss}, to name a few.\n\\subsection{Strongly Continuous Semigroups and their Generators}\nWe start by introducing the necessary terminology.\n\\begin{definition}[Semigroup]\n\tA family $(T(t))_{t \\geq 0} \\in \\mathcal{L}(X)^{[0, \\infty)}$ is called a semigroup of bounded linear operators on $X$, or short semigroup, if it satisfies\n\\begin{enumerate}\n\t\\item[(i)] $T(0) = I_X$, and\n\t\\item[(ii)] the semigroup property:\n\t\\begin{align*}\n\t\tT(t+s) = T(t)T(s), \\hspace{0.5 cm} t,s \\geq 0. \n\t\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\n\tA semigroup $(T(t))_{t \\geq 0}$ on $X$ is called \n\t\\begin{enumerate}\n\t\t\\item[(i)] strongly continuous (or a $C_0$-semigroup) if for all $x \\in X$, \n\t\t\\begin{align*}\n\t\t\t\\lim_{t \\searrow 0} \\|T(t)x - x \\|_X = 0, \n\t\t\\end{align*}\n\t\\item[(ii)] uniformly continuous if it holds that\n\t\\begin{align*}\n\t\t\\lim_{t \\searrow 0} \\|T(t) - I_X \\|_{\\mathcal{L}(X)} = 0. \n\t\\end{align*}\n\t\\end{enumerate}\n\\end{definition}\n\nThe strong continuity of $(T(t))_{t \\geq 0}$ is equivalent to the mapping $t \\mapsto T(t)x$ being continuous for all $x \\in X$. Analogously, the mapping $t \\mapsto T(t) \\in \\mathcal{L}(X)$ is continuous provided that $(T(t))_{t \\geq 0}$ is uniformly continuous. \n\\vspace{0.5 cm}\\\\\nIn the following, we restrict ourselves to strongly continuous semigroups, since the presented results hold in particular for uniformly continuous semigroups. An overview of the main differences to strongly continuous semigroups is given in \\cite[Section 1.1]{Pazy83}.\n\\vspace{0.5 cm}\\\\\nAn essential property of $C_0$-semigroups is the following. \n\\begin{theorem}[Theorem 1.2.2 in \\cite{Pazy83}]\n\t\\label{Theorem C0-Semigroup Estimate}\n\tLet $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup on $X$. Then there exist constants $M \\geq 1$ and $\\omega \\in \\mathbb{R}$ such that\n\t\\begin{align*}\n\t\t\\|T(t)\\|_{\\mathcal{L}(X)} \\leq Me^{\\omega t}, \\hspace{0.5 cm} t \\geq 0. \n\t\\end{align*}\nWe say that $(T(t))_{t \\geq 0}$ is of type $C_0(M, \\omega)$.\n\\end{theorem}\n\n\\begin{theorem}[cf. Proposition 2.1.2 in \\cite{Tucsnak-Weiss}]\n\t\\label{Theorem T(t)x is continuous}\n\tFor a strongly continuous semigroup $(T(t))_{t \\geq 0}$ on $X$, the mapping \n\t\\begin{align*}\n\t[0, \\infty) \\times X \\to X, \\hspace{0.3 cm} (t,x) \\mapsto T(t)x\n\t\\end{align*}\nis continuous. \n\\end{theorem}\n\n\\begin{proof}\n\tLet $(t_n)_{n \\in \\mathbb{N}} \\in \\mathbb{R}_+^{\\mathbb{N}}$ and $(x_n)_{n \\in \\mathbb{N}} \\in X^{\\mathbb{N}}$ be sequences converging to $t_0 \\in \\mathbb{R}_+$ and $x_0 \\in X$, respectively. Theorem \\ref{Theorem C0-Semigroup Estimate} yields \n\t\\begin{align*}\n\t\t\\|T(t_n)x_n - T(t_0) x_0 \\|_X &\\leq \\|T(t_n)(x_n - x_0) \\|_X + \\|T(t_n) x_0 - T(t_0)x_0 \\|_X \\\\\n\t\t&\\leq Me^{\\omega t_n} \\|x_n - x_0\\|_X + Me^{\\omega \\min \\{ t_n, t_0\\}}\\|T(\\vert t_n - t_0 \\vert) x_0 - x_0 \\|_X. \n\t\\end{align*}\nSince the sequence $\\left( e^{\\omega t_n} \\right)_{n \\in \\mathbb{N}}$ is bounded, and since $x_n \\to x_0$ by assumption, the first term converges to $0$. The second term converges to $0$ as well, because by the strong continuity of $(T(t))_{ t \\geq 0}$ it holds that\n\\begin{align*}\n\t\\lim_{n \\to \\infty}T(\\vert t_n - t_0 \\vert) x_0 = x_0.\n\\end{align*}\nThis proves the claim. \n\\end{proof}\n\nTheorem \\ref{Theorem C0-Semigroup Estimate} gives rise to the notion of the exponential growth bound for $C_0$-semigroups. \n\n\n\\begin{definition}[Growth Bound]\n\t\\label{Definition Growth Bound}\n\tLet $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup on $X$. Its (exponential) growth bound is defined as\n\t\\begin{align*}\n\t\t\\omega (T(\\cdot)) := \\inf \\left\\{ \\omega \\in \\mathbb{R} \\mid \\exists M_{\\omega} \\geq 1\\colon \\forall t \\geq 0\\colon \\, \\|T(t)\\|_{\\mathcal{L}(X)} \\leq M_{\\omega} e^{\\omega t} \\right\\}.\n\t\\end{align*}\n\\end{definition}\nIn this context, we want to introduce the notion of exponential stability for strongly continuous semigroups, which is one of the most important stability concepts in this framework. For an extensive discussion on stability in general and some well-known characterizations of exponential stability we refer to \\cite[Chapter~4]{CurtainZwart}.\n\\begin{definition}[Exponential Stability]\n\t\\label{Definition Exponential Stability}\n\tA $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$ is called exponentially stable if there exist constants $M\\geq 1$ and $\\alpha > 0$ such that $(T(t))_{t \\geq 0}$ is of type $C_0(M, - \\alpha)$. \n\\end{definition}\n\nNext, we want to discuss what it means for a linear operator $A \\colon D(A) \\subset X \\to X$ to generate a strongly continuous semigroup, and collect some properties of such an operator as well as some relations between a $C_0$-semigroup and its infinitesimal generator.\n\n\\begin{definition}[Infinitesimal Generator]\n\t\\label{Definition Infinitesimal Generator}\n\tLet $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup on $X$. The infinitesimal generator $A \\colon D(A) \\subset X \\to X$ of $(T(t))_{t \\geq 0}$ is defined by\n\t\\begin{align*}\n\t\tD(A) &= \\left\\{ x \\in X \\mid \\lim_{t \\searrow 0} \\frac{T(t) - I}{t}x \\, \\text{exists} \\right\\}, \\\\\n\t\tAx &= \\lim_{t \\searrow 0} \\frac{T(t) - I}{t}x, \\hspace{0.5 cm} x \\in D(A). \n\t\\end{align*}\n\\end{definition}\n\n\\begin{theorem}[Theorem 1.2.4 in \\cite{Pazy83}]\n\t\\label{Theorem Properties of C0-Semigroups}\n\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. Then the following assertions are true:\n\t\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item \\label{C0SemigroupProp1} For all $x \\in X$ and $t \\geq 0$ it holds that\n\t\t\\begin{align*}\n\t\t\t\\lim_{h \\searrow 0} \\frac{1}{h} \\int_{t}^{t + h} T(\\tau)x \\, d\\tau = T(t)x. \n\t\t\\end{align*} \n\t\\item \\label{C0SemigroupProp2} We have $\\int_{0}^{t} T(\\tau) x \\, d\\tau \\in D(A)$ for all $x \\in X$, $ t \\geq 0$, with\n\t\\begin{align*}\n\t\tA \\left( \\int_{0}^{t} T(\\tau) x \\, d\\tau \\right) = T(t)x - x.\n\t\\end{align*}\n\\item \\label{C0SemigroupProp3} It holds that $T(t) D(A) \\subset D(A)$ for all $t \\geq 0$. Moreover, for all $x \\in D(A)$ we have\n\\begin{align*}\n\t\\frac{d}{dt}T(t)x = AT(t)x = T(t)Ax. \n\\end{align*}\n\\item \\label{C0SemigroupProp4} For all $x \\in D(A)$ and for all $0 \\leq \\tau \\leq t$ we have\n\\begin{align*}\n\tT(t)x - T(\\tau)x = \\int_{\\tau}^{t} T(r) Ax \\, dr = \\int_{\\tau}^{t} AT(r) x \\, dr. \n\\end{align*}\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\tAd \\ref{C0SemigroupProp1}: By continuity of the mapping $t \\mapsto T(t)x$ we have for all $ x\\in X$, $t \\geq 0$, \n\t\\begin{align*}\n\t\t\\hspace{0.5 cm} \\lim_{h \\searrow 0} \\left\\| \\frac{1}{h} \\int_{t}^{t+h} T(\\tau) x \\, d\\tau - T(t)x \\right\\|_X\t&= \\lim_{h \\searrow 0} \\left\\| \\frac{1}{h} \\int_{t}^{t+h} T(\\tau) x - T(t) x \\, d\\tau \\right\\|_X \\\\\n\t\t&\\leq \\lim_{h \\searrow 0} \\frac{1}{h} h \\sup_{\\tau \\in [t, t+ h]} \\| T(\\tau)x - T(t)x \\|_X \\\\\n\t&= 0. \n\t\\end{align*}\n\tAd \\ref{C0SemigroupProp2}: Let $h > 0$. As $T(t) \\in \\mathcal{L}(X)$, it is closed for all $t \\geq 0$. Thus, we have for all $x \\in X$,\n\t\\begin{align*}\n\t\t\\frac{1}{h}(T(h) - I) \\int_{0}^{t} T(\\tau) x \\, d\\tau &= \\frac{1}{h} \\int_{0}^{t} T(\\tau + h) x \\, d\\tau - \\frac{1}{h} T(\\tau) x \\, d\\tau \\\\\n\t\t&= \\frac{1}{h} \\int_{t}^{t +h} T(\\tau) x \\, d\\tau - \\frac{1}{h} \\int_{0}^{h} T(\\tau)x \\, d\\tau. \n\t\\end{align*}\nBy applying property \\ref{C0SemigroupProp1}, this expression converges to $T(t)x - x$ as $h \\searrow 0$. This shows \\ref{C0SemigroupProp2}. \n\t\\vspace{0.5 cm}\\\\\n\tAd \\ref{C0SemigroupProp3}: Let $x \\in D(A)$ and $h > 0$. Then\n\t\\begin{align*}\n\t\t\\frac{1}{h} (T(h) - I)T(t)x = T(t) \\left( \\frac{T(h) - I}{h} \\right)x \\to T(t) A x\n\t\\end{align*}\nas $h \\searrow 0$. Thus, by definition, $T(t)x \\in D(A)$ with $AT(t)x = T(t)Ax$ for all $t \\geq 0$. Moreover, we have for the right-hand derivative that\n\\begin{align*}\n\t\\frac{d^+}{dt}T(t)x = AT(t)x = T(t)Ax. \n\\end{align*}\nNow, let $t,h> 0$ such that $t - h \\geq 0$. Then\n\\begin{align*}\n\t&\\hspace{0.48 cm} \\left\\| \\frac{1}{h} (T(t)x - T(t-h)x) - T(t)Ax \\right\\|_X \\\\\n\t&= \\left\\| \\frac{1}{h} (T(t)x - T(t-h)x) + T(t-h)Ax - T(t-h)Ax - T(t)Ax \\right\\|_X \\\\\n\t&\\leq \\|T(t-h)\\|_{\\mathcal{L}(X)} \\left\\| \\frac{1}{h} (T(h)x - x) - Ax \\right\\|_X + \\|T(t-h)Ax - T(t)Ax \\|_X\n\\end{align*}\nBy strong continuity and the fact that $A$ generates $(T(t))_{t \\geq 0}$, taking the limit $h\\searrow 0$ yields the same relation for the left-hand derivative.\n\t\\vspace{0.5 cm}\\\\\n\tAd \\ref{C0SemigroupProp4}: This follows from property \\ref{C0SemigroupProp3} by integration over $[\\tau,t]$. \n\\end{proof}\nFrom these properties one infers:\n\\begin{corollary}[Corollary 1.2.5 in \\cite{Pazy83}]\n\t\\label{Corollary - Generators Are Closed and Densely Defined}\n\tIf $A \\colon D(A) \\subset X \\to X$ is the infinitesimal generator of a strongly continuous semigroup $(T(t))_{t \\geq 0}$ on $X$, then it is densely defined and closed. \n\\end{corollary}\n\nLastly, we note that the generator of a strongly continuous semigroup is uniquely determined.\n\n\\begin{theorem}[Theorem 1.2.6 in \\cite{Pazy83}]\nLet $(T(t))_{t \\geq 0}$ and $(S(t))_{t \\geq 0}$ be $C_0$-semigroups on $X$ with infinitesimal generators $A$ and $B$, respectively. If $A = B$, then\n\\begin{align*}\n\tT(t) = S(t), \\hspace{0.5 cm } t \\geq 0. \n\\end{align*}\n\\end{theorem}\n\n\\subsection{Spectrum and Resolvent of Infinitesimal Generators}\nIn the following, we want to characterize infinitesimal generators of $C_0$-semigroups. To this end, we introduce the notion of the resolvent operator, and collect some results relevant for the proof of the aforementioned characterization. \n\\begin{definition}[Resolvent]\n\t\\label{Definition Resolvent}\n\tLet $A \\colon D(A) \\subset X \\to X$ be a linear operator. The resolvent set $\\rho(A)$ is the set of elements $\\lambda \\in \\mathbb{C}$ which satisfy the following:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\overline{(\\lambda I - A)(D(A))} = X$. \n\t\t\\item[(ii)] $(\\lambda I - A)^{-1}$ exists and is bounded in\n\t\t\\begin{align*}\n\t\t\tD((\\lambda I - A)^{-1}) := (\\lambda I - A)(D(A))\n\t\t\\end{align*} \n\twith respect to the norm on $X$ induced by $(\\lambda I - A)(D(A))$. \n\t\\end{enumerate} \nIf $\\rho(A) \\neq \\emptyset$, for $\\lambda \\in \\rho(A)$ we call\n\\begin{align*} \n\tR(\\lambda, A) := (\\lambda I - A)^{-1} \\in \\mathcal{L}(X)\n\\end{align*}\nthe resolvent operator of $A$ in $\\lambda$. The spectrum of $A$ is\n\\begin{align*}\n\t\\sigma(A) := \\mathbb{C} \\setminus \\rho(A).\n\\end{align*}\n\\end{definition}\n\n\\begin{definition}\n\t\tLet $A \\colon D(A) \\subset X \\to X$ be a linear operator. The spectrum of $A$ is the disjoint union of the following sets:\n\t\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\t\\item The set\n\t\t\t\\begin{align*}\n\t\t\t\t\\sigma_p(A) := \\left\\{ \\lambda \\in \\mathbb{C} \\mid (\\lambda I - A) \\text{ is not injective} \\right\\}\n\t\t\t\\end{align*}\n\t\t\tis called the point spectrum of $A$. Its elements are called eigenvalues, and an element $x_{\\lambda} \\in D(A)$ with $Ax_{\\lambda} = \\lambda x_{\\lambda}$ is called an eigenfunction with eigenvalue $\\lambda \\in \\sigma_p(A)$.\n\t\t\t\\item The set\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\sigma_c(A) := & \\left\\{ \\lambda \\in \\mathbb{C} \\mid \\overline{(\\lambda I - A) (D(A))} =X, \\right. \\\\\n\t\t\t\t\t& \\qquad \\hspace{0.7cm} \\left. (\\lambda I - A) \\text{ is injective}, \\, (\\lambda I - A)^{-1} \\text{ is unbounded} \\right\\}\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\t\tis called the continuous spectrum of $A$.\n\t\t\t\\item The set\n\t\t\t\\begin{align*}\n\t\t\t\t\\sigma_r(A) := \\left\\{ \\lambda \\in \\mathbb\n\t\t\t\tC \\mid \\overline{(\\lambda I - A) (D(A))} \\neq X, \\, (\\lambda I - A) \\text{ is injective} \\right\\}\n\t\t\t\\end{align*}\n\t\t\tis called the residual spectrum of $A$.\n\t\t\t\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}[Proposition 2.8.4 in \\cite{Tucsnak-Weiss}]\n\t\\label{Proposition 2.8.4 in Tucsnak}\n\tLet $A \\colon D(A) \\subset X \\to X$ be linear and densely defined, and let $\\lambda \\in \\rho(A)$. Then $\\overline{\\lambda} \\in \\rho(A^{\\ast})$ and\n\t\\begin{align*}\n\t\t\\left[ (\\lambda I - A)^{-1} \\right]^{\\ast} = (\\overline{\\lambda}I - A^{\\ast})^{-1}. \n\t\\end{align*}\n\\end{proposition}\nAn important feature of infinitesimal generators is that the corresponding resolvent operator can be written as the \\emph{Laplace transform} of the respective $C_0$-semigroup.\n\n\\begin{theorem}[Proposition 2.3.1 in \\cite{Tucsnak-Weiss}]\n\t\\label{Theorem Laplace Transform}\n\tLet $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup on $X$. Let $A \\colon D(A) \\subset X \\to X$ be its infinitesimal generator. Then the following assertions are true:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\mathbb{C}_{\\omega(T(\\cdot))} = \\left\\{ z \\in \\mathbb{C} \\mid \\realpart(z) > \\omega(T(\\cdot)) \\right\\} \\subset \\rho(A)$.\n\t\t\\item[(ii)] For all $x\\in X$, $\\lambda \\in \\mathbb{C}_{\\omega(T(\\cdot))}$ it holds that\n\t\t\\begin{align*}\n\t\t\t(\\lambda I - A)^{-1} x = \\int_{0}^{\\infty} e^{-\\lambda t}T(t)x \\, dt. \n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\tIf $\\alpha = \\realpart(\\lambda) > \\omega(T(\\cdot))$, then there are some constants $M \\geq1$ and $\\epsilon > 0$ such that\n\t\\begin{align*}\n\t\t\\|T(t) \\|_{\\mathcal{L}(X)} \\leq M e^{(\\alpha - \\epsilon)t}, \\qquad t \\geq 0.\n\t\\end{align*}\n\tThus, the improper Riemann integral exists, since the integrand converges exponentially to $0$ as $t \\to \\infty$. Now, define\n\t\\begin{align*}\n\t\tZ(\\lambda)x := \\int_{0}^{\\infty} e^{-\\lambda t} T(t)x \\, dt, \\qquad x \\in X, \\, \\lambda \\in \\mathbb{C}_{\\omega(T(\\cdot))}.\n\t\\end{align*}\n\tThen we have for $\\lambda \\in \\mathbb{C}_{\\omega(T(\\cdot))}$ that\n\t\\begin{align*}\n\t\t\\|Z(\\lambda) x \\|_X \\leq \\int_{0}^{\\infty} e^{- (\\realpart(\\lambda))t} M e^{(\\realpart(\\lambda)- \\epsilon)t} \\|x\\|_X \\, dt = \\frac{M}{\\epsilon} \\|x\\|_X.\n\t\\end{align*}\n\tThus, $Z(\\lambda) \\in \\mathcal{L}(X)$. Furthermore, by substituting $\\eta = t+h$ and by Theorem~\\ref{Theorem Properties of C0-Semigroups}~\\ref{C0SemigroupProp1}, we have\n\t\\begin{align*}\n\t\t\\frac{1}{h} \\left(T(h) - I \\right) Z(\\lambda) x & = \\frac{1}{h} \\int_{0}^{\\infty} e^{-\\lambda t} \\left( T(t+h) - T(t) \\right) x \\, dt \\\\\n\t\t& = \\frac{e^{\\lambda h}}{h} \\int_{0}^{\\infty} e^{-\\lambda \\eta} T(\\eta)x \\, d\\eta - \\frac{e^{\\lambda h}}{h} \\int_{0}^{h} e^{-\\lambda \\eta} T(\\eta) x \\, d\\eta \\\\\n\t\t& \\qquad - \\frac{1}{h} \\int_{0}^{\\infty} e^{-\\lambda \\eta} T(\\eta) x \\, d\\eta \\\\\n\t\t& = \\frac{e^{\\lambda h}-1}{h} \\int_{0}^{\\infty} e^{-\\lambda t} T(t) x \\, dt - \\frac{e^{\\lambda h}}{h} \\int_{0}^{h} e^{-\\lambda t} T(t)x \\, dt \\\\\n\t\t& \\to \\lambda Z(\\lambda) x - x\n\t\\end{align*}\n\tas $h \\to 0$. We conclude that $ Z(\\lambda)x \\in D(A)$ with\n\t\\begin{align*}\n\t\tAZ(\\lambda) x = \\lambda Z(\\lambda) x - x.\n\t\\end{align*}\n\tHence, $(\\lambda I - A) Z(\\lambda) x = x$ for all $x \\in X$. As $A$ is closed, for all $x \\in D(A)$ the following holds:\n\t\\begin{align*}\n\t\tZ(\\lambda) A x & = \\int_{0}^{\\infty} e^{-\\lambda t} T(t)Ax \\, dt \\\\\n\t\t& = \\int_{0}^{\\infty} e^{-\\lambda t} A T(t)x \\, dt \\\\\n\t\t& = A \\left( \\int_{0}^{\\infty} e^{-\\lambda t} T(t)x \\, dt \\right) \\\\\n\t\t& = AZ(\\lambda) x.\n\t\\end{align*}\n\tThis yields\n\t\\begin{align*}\n\t\tZ(\\lambda) (\\lambda I -A) x = (\\lambda I - A)Z(\\lambda) x = x.\n\t\\end{align*}\n\tIn conclusion, we have that $Z(\\lambda) = (\\lambda I -A)^{-1}$, and so $\\lambda \\in \\rho(A)$.\n\\end{proof}\n\nIn the following, we want to present some characteristics of infinitesimal generators of strongly continuous semigroups. To this end, we need to introduce the \\emph{Yosida approximation} of a closed and densely defined operator $A \\colon D(A) \\subset X \\to X$. Assume that there exist constants $M > 0$ and $\\omega > 0$ such that $(\\omega, \\infty) \\subset \\rho(A)$ and such that for all $n \\in \\mathbb{N}$, \n\\begin{align*}\n\t\\|R^n(\\lambda,A) \\|_{\\mathcal{L}(X)} \\leq \\frac{M}{(\\lambda - \\omega)^n}, \\hspace{0.5 cm} \\lambda > \\omega.\n\\end{align*}\nIf so, it has been shown that for every $\\lambda \\in \\rho(A)$, the Yosida approximation defined as\n\\begin{align*}\n\tA_{\\lambda} := \\lambda^2 R(\\lambda,A) - \\lambda I = \\lambda AR(\\lambda,A) \\in \\mathcal{L}(X)\n\\end{align*}\nhas some useful properties which are exploited in order to prove the necessity of the following characterization of infinitesimal generators of $C_0$-semigroups. The sufficiency flollows from Corollary \\ref{Corollary - Generators Are Closed and Densely Defined} and Theorem \\ref{Theorem Laplace Transform}.\n\\begin{theorem}[Hille-Yosida, cf. Theorem II.3.8 in \\cite{EngelNagel}]\n\t\\label{Theorem Hille-Yosida - C0 Semigroup}\n\tA linear operator $A \\colon D(A) \\subset X \\to X$ is the infinitesimal generator of a $C_0$-semigroup on $X$ of type $C_0(M, \\omega)$ if and only if the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $A$ is closed and densely defined.\n\t\t\\item[(ii)] $(\\omega, \\infty) \\subset \\rho(A)$.\n\t\t\\item[(iii)] For all $\\lambda > \\omega$ and for all $n \\in \\mathbb{N}$ one has\n\t\t\\begin{align*}\n\t\t\t\\|R^n(\\lambda, A) \\|_{\\mathcal{L}(X)} \\leq \\frac{M}{(\\lambda - \\omega)^n}.\n\t\t\\end{align*}\n\t\\end{enumerate} \n\\end{theorem}\n\nA similar result can be proved for an important class of strongly continuous semigroups, called contraction semigroups. They are especially interesting in the port-Hamiltonian framework of infinite-dimensional systems. \n\n\\begin{definition}[Contraction Semigroup]\n\t\\label{Definition Contraction Semigroup}\n\tLet $(T(t))_{t \\geq0}$ be a $C_0$-semigroup on $X$.\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\\item It is called a contraction semigroup, if it is of type $C_0(1,0)$, that is,\n\t\\begin{align*}\n\t\t\\|T(t)\\|_{\\mathcal{L}(X)} \\leq 1, \\hspace{0.5 cm} t \\geq 0.\n\t\\end{align*}\n\\item It is called a unitary semigroup, if for all $x \\in X$ it holds that\n\\begin{align*}\n\t\\|T(t)x \\|_X = \\|x\\|_X, \\hspace{0.5 cm} t \\geq 0.\n\t\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\\begin{theorem}[Hille-Yosida, Theorem 1.3.1 in \\cite{Pazy83}]\n\t\\label{Theorem Hille-Yosida - Contraction Semigroup}\n\tA linear operator $A \\colon D(A) \\subset X \\to X$ is the infinitesimal generator of a contraction semigroup on $X$ if and only if the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $A$ is closed and $\\overline{D(A)} = X$. \n\t\t\\item[(ii)] $(0, \\infty) \\subset \\rho(A)$, and for every $\\lambda > 0$, \n\t\t\\begin{align*}\n\t\t\t\\|R(\\lambda,A) \\|_{\\mathcal{L}(X)} \\leq \\frac{1}{\\lambda}.\n\t\t\\end{align*}\n\\end{enumerate}\n\\end{theorem}\n\nThere is another important characterization for when a linear operator generates a contraction semigroup. To this end, we need to clarify what it means that a linear operator is dissipative. \n\n\\begin{definition}[Dissipative Operator]\n\t\\label{Definition Dissipative Operator}\n\tA linear operator $A \\colon D(A) \\subset X \\to X$ is called dissipative if for all $x \\in D(A)$, \n\t\\begin{align*}\n\t\t\\realpart \\langle Ax, x \\rangle_X \\leq 0.\n\t\\end{align*}\n\\end{definition}\nThe dissipativity of a linear operator can be characterized in the following way. \n\n\\begin{lemma}[Proposition 6.1.5 in \\cite{JacobZwart}]\n\t\\label{Lemma Characterization Dissipativity}\nA linear operator $A \\colon D(A) \\subset X \\to X$ is dissipative if and only if for all $x \\in D(A)$, $\\lambda >0$ it holds that\n\\begin{align*}\n\t\\| (\\lambda I - A)x \\|_X \\geq \\lambda \\| x \\|_X. \n\\end{align*}\n\t\\end{lemma}\n\n\n\\begin{lemma}[Lemma 6.1.6 in \\cite{JacobZwart}]\n\t\\label{Lemma Auxiliary Lemma Lumer Phillips}\n\tLet $A \\colon D(A) \\subset X \\to X$ be a closed, dissipative operator. Then $\\ran (\\lambda I - A)$ is closed for all $\\lambda > 0$. \n\\end{lemma}\n\nApplying Theorem \\ref{Theorem Hille-Yosida - Contraction Semigroup} as well as Lemma \\ref{Lemma Characterization Dissipativity} and Lemma \\ref{Lemma Auxiliary Lemma Lumer Phillips}, one can prove that generators of contraction semigroups can be characterized as follows. \n\n\\begin{theorem}[Lumer-Phillips, Theorem 6.1.7 in \\cite{JacobZwart}]\n\t\\label{Theorem Lumer-Phillips}\n\tA linear operator $A \\colon D(A) \\subset X \\to X$ is the infinitesimal generator of a contraction semigroup on $X$ if and only if the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $A$ is dissipative.\n\t\t\\item[(ii)] $\\ran(I-A) = X$.\n\t\\end{enumerate}\n\\end{theorem}\n\nNow that we have examined some properties of strongly continuous semigroups and their infinitesimal generators, we can finally deal with solving Hilbert space-valued initial value problems. It turns out that $C_0$-semigroups are a vital tool for the analysis of infinite-dimensional systems. \n\n\\section{The Abstract Cauchy Problem}\n\\label{Section The Abstract Cauchy Problem}\nIn this section, we introduce the abstract Cauchy problem, i.e., a linear inital value problem with values in a Hilbert space. We will clarify the notion of a solution of such an abstract differential equation, and characterize well-posedness of this class of systems. Again, this is standard material and is covered in \\cite[Chapter 5]{CurtainZwart}, \\cite[Section II.6]{EngelNagel}, \\cite[Chatper 4]{Pazy83}.\n\\subsection{Linear Homogeneous Initial Value Problems}\n\\label{Subsection Linear Homogeneous Initial Value Problems}\nWe begin with the definition of an abstract Cauchy problem.\n\\begin{definition}[Abstract Cauchy Problem]\n\tThe linear initial value problem \n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\\dot{x}(t) &= Ax(t), \\hspace{0.5 cm} t > 0, \\\\\n\t\tx(0) &= x_0 \\in X,\n\t\t\\end{split}\n\t\\label{General Abstract Cauchy Problem}\n\t\\end{align}\nwith $A \\colon D(A) \\subset X \\to X$ a linear operator, $x_0$ the initial value and $x \\colon [0, \\infty) \\to X$ a time-dependent, Hilbert space-valued function is called an abstract Cauchy problem. The function $x$ is called a (classical) solution, if the following holds:\n\\begin{enumerate}\n\t\\item[(i)] $x \\in \\mathcal{C}([0, \\infty), X)$.\n\t\\item[(ii)] $x_{| (0,\\infty)} \\in \\mathcal{C}^1((0, \\infty) , X)$. \n\t\\item[(iii)] $x(t) \\in D(A)$ for all $ t > 0$.\n\t\\item[(iv)] $x$ satisfies \\eqref{General Abstract Cauchy Problem}. \n\\end{enumerate}\n\\end{definition}\nWe need to agree on a concept of well-posedness for the abstract Cauchy problem \\eqref{General Abstract Cauchy Problem}. Just as in the finite-dimensional case, we require for its solutions existence, uniqueness, and continuous dependence on the initial data. This guides us to the following definition. \n\\begin{definition}[Well-posedness]\n\t\\label{Definition Well-posedness of an Abstract Cauchy Problem}\n\tThe abstract Cauchy problem \\eqref{General Abstract Cauchy Problem} is called well-posed, if the following assertions are true:\n\t\\begin{enumerate}\n\t\t\\item[(i)] The operator $A$ is densely defined.\n\t\t\\item[(ii)] For all $x_0 \\in D(A)$, there exists a unique classical solution $x(\\cdot;x_0) \\colon [0, \\infty) \\to X$.\n\t\t\\item[(iii)] The classical solution depends continuously on the initial value, that is, for all $\\epsilon > 0$, $\\tau > 0$, and $x_0 \\in D(A)$ there exists some $\\delta > 0$ such that for all $x_1 \\in D(A)$ the following holds:\n\t\t\\begin{align*}\n\t\t\t\\| x_0 - x_1 \\|_X \\leq \\delta \\hspace{0.3 cm} \\Rightarrow \\hspace{0.3 cm} \\sup_{t \\in [0,\\tau]} \\| x(t; x_0) - x(t;x_1) \\|_X \\leq \\epsilon. \n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{definition}\n\nThere are weaker notions of well-posedness as well, but in this thesis, we work with the concept suggested in Definition \\ref{Definition Well-posedness of an Abstract Cauchy Problem}. \n\\vspace{0.5 cm}\\\\\nThe following theorem shows that if the operator $A$ associated with \\eqref{General Abstract Cauchy Problem} is the infinitesimal generator of a $C_0$-semigroup, then one can easily construct a classical solution. \n\\begin{theorem}[Proposition II.6.2 in \\cite{EngelNagel}]\n\t\\label{Theorem A Generator yields Unique Solution}\n\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. Then, for every initial value $x_0 \\in D(A)$, the function \n\t\\begin{align}\n\t\tx = x(\\cdot; x_0) \\colon [0, \\infty) \\to X, \\hspace{0.3 cm} t \\mapsto T(t)x_0,\n\t\t\\label{Theorem Unique Classical Solutiono of ACP - Unique Solution}\n\t\\end{align}\n\tis the unique classical solution of \\eqref{General Abstract Cauchy Problem}, with $x$ depending continuously on the initial data. \n\\end{theorem}\n\\begin{proof}\n\tThe proof that $x$ defined in \\eqref{Theorem Unique Classical Solutiono of ACP - Unique Solution} is a classical solution follows immediately from property (iii) of Theorem \\ref{Theorem Properties of C0-Semigroups}. Next, we want to show that $x$ is the only solution of \\eqref{General Abstract Cauchy Problem}.\n\t\\vspace{0.5 cm}\\\\\n\tAssume that $y \\colon [0, \\infty) \\to X$ solves \\eqref{General Abstract Cauchy Problem} as well. For $t > 0$, define the mapping\n\t\\begin{align*}\n\t\t\\psi \\colon \\tau \\mapsto T(t - \\tau)y(\\tau), \\hspace{0.5 cm} \\tau \\in [0,t]. \n\t\\end{align*}\nBy assumption, $y$ is continuously differentiable and $y(\\tau) \\in D(A)$ for all $\\tau \\in (0,t)$. Thus, for all $\\tau, h > 0$ such that $\\tau + h < t$ it holds that\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\frac{\\psi(\\tau + h) - \\psi(\\tau)}{h} \\\\\n\t &= \\frac{T(t - \\tau - h) y(\\tau + h) - T(t - \\tau) y(\\tau)}{h} \\\\\n\t&= \\frac{T(t - \\tau - h) y(\\tau + h) - T(t - \\tau - h)y(\\tau) + T(t - \\tau - h) y(\\tau) - T(t - \\tau) y(\\tau)}{h} \\\\\n\t&= T(t - \\tau - h) \\frac{y(\\tau +h) - y(\\tau)}{h} + \\frac{T(t - \\tau - h )y(\\tau) - T(t - \\tau)y(\\tau)}{h}.\n\\end{align*} \nAgain, since $y(\\tau) \\in D(A)$ for all $\\tau \\in (0,t)$, the second term converges to $- AT(t-\\tau)y(s)$ as $h \\searrow 0$. For the first term, we may find the limit by writing\n\\begin{align*}\n\t T(t - \\tau - h) \\frac{y(\\tau +h) - y(\\tau)}{h} &= T(t - \\tau) \\frac{y(\\tau + h) - y(\\tau)}{h} \\\\\n\t &\\hspace{0.5 cm} + \\left[ T(t - \\tau - h) - T(t - \\tau) \\right] \\left( \\frac{y(\\tau + h) - y(\\tau)}{h} - \\frac{d}{d\\tau}y(\\tau) \\right) \\\\\n\t &\\hspace{0.5 cm} + \\left[ T(t - \\tau - h) - T(t - \\tau) \\right] \\frac{d}{d\\tau}y(\\tau).\n\\end{align*} \nNow, since $(T(t))_{t \\geq 0}$ is strongly continuous, $T(t - \\tau - h) - T(t - \\tau)$ is uniformly bounded on compact (time) intervals. Consequently, the second term tends to zero as $h \\searrow 0$. By virtue of the strong continuity, the same holds for the last expression. Hence, this sum converges to $T(t - \\tau) \\frac{d}{d \\tau}y(\\tau)$ as $h \\searrow 0$. Altogether, it holds that\n\\begin{align*}\n\t\\frac{d}{d \\tau} \\psi(\\tau) &= T(t - \\tau) \\frac{d}{d\\tau} y(\\tau) - A T(t - \\tau) y(\\tau) \\\\\n\t&= T(t - \\tau) Ay(\\tau) - AT(t - \\tau) y(\\tau) \\\\\n\t&=0. \n\\end{align*}\nThe last equation follows from Theorem \\ref{Theorem Properties of C0-Semigroups} \\ref{C0SemigroupProp3}. Integration yields that\n\\begin{align*}\n\t0 = \\int_{0}^{t} \\frac{d}{d\\tau} (T(t- \\tau)y(\\tau)) \\, d\\tau = y(t) - T(t)x_0,\n\\end{align*}\nwhence $y(t) = T(t)x_0 = x(t)$ for all $t \\geq 0$. This proves the uniqueness of the (classical) solution of \\eqref{General Abstract Cauchy Problem}. \n\\vspace{0.5 cm}\\\\\nLastly, we need to show the continuous dependence of the solution on the initial data. As $(T(t))_{t \\geq 0}$ is strongly continuous, it is of type $C_0(M, \\omega)$ for some $M \\geq 1$, $\\omega \\in \\mathbb{R}$ (see Theorem~\\ref{Theorem C0-Semigroup Estimate}). Thus, we have for $\\tau > 0$ and $x_0,x_1 \\in D(A)$, \n\\begin{align*}\n\t\\sup_{t \\in [0,\\tau]} \\|x(t; x_0) - x(t; x_1) \\|_X = \\sup_{t \\in [0,\\tau]} \\|T(t) (x_0 - x_1) \\|_X \\leq Me^{\\vert \\omega\\vert \\tau} \\| x_0 - x_1 \\|_X. \n\\end{align*}\nThis shows that the solution is continuously depending on the initial value and, hence, completes the proof. \n\\end{proof}\nTheorem \\ref{Theorem A Generator yields Unique Solution} shows in particular that it is sufficient for well-posedness of the abstract Cauchy problem \\eqref{General Abstract Cauchy Problem} that the associated operator $A$ is the infinitesimal generator of a $C_0$-semigroup. In fact, it is necessary as well. However, we will not prove the necessity and refer to \\cite[Theorem II.6.7]{EngelNagel}. \n\n\\begin{theorem}[Corollary II.6.9 in \\cite{EngelNagel}]\n\t\\label{Theorem Characterization of Well-Posedness}\n\tLet $A \\colon D(A) \\subset X \\to X$ be a linear and closed operator. Then the initial value problem \\eqref{General Abstract Cauchy Problem} is well-posed in the sense of Definition \\ref{Definition Well-posedness of an Abstract Cauchy Problem} if and only if $A$ is the infinitesimal generator of a strongly continuous semigroup $(T(t))_{t \\geq 0}$ on $X$. If so, then for any $x_0 \\in D(A)$, the unique classical solution is given by \\eqref{Theorem Unique Classical Solutiono of ACP - Unique Solution}.\n\\end{theorem}\n\nTheorem \\ref{Theorem Characterization of Well-Posedness} emphasizes the importance of the concept of strongly continuous semigroups, since they are well-suited for the study of abstract Cauchy problems. The starting point for the analysis of abstract Cauchy problems is therefore to check whether the associated operator $A$ generates a strongly continuous semigroup. In this context, let us quickly come back to dissipative operators, and consider the following example.\n\\begin{example}\n\t\\label{Example Contraction Semigroup}\n Consider the abstract initial value problem \\eqref{General Abstract Cauchy Problem} associated with an operator $A \\colon D(A) \\subset X \\to X$ generating a contraction semigroup $(T(t))_{t \\geq 0}$, and with initial value $x_0 \\in D(A)$. Then, by Theorem \\ref{Theorem Characterization of Well-Posedness}, the system has the unique solution $x = T(\\cdot)x_0$. Moreover, according to Theorem \\ref{Theorem Lumer-Phillips}, $A$ is dissipative. This yields for all $t > 0$, \n\\begin{align*}\n\t\\frac{d}{dt} \\|x(t)\\|_X^2 &= \\langle \\dot{x}(t), x(t) \\rangle_X + \\langle x(t), \\dot{x}(t) \\rangle_X \\\\\n\t&= \\langle Ax(t), x(t) \\rangle_X + \\langle x(t), Ax(t) \\rangle_X \\\\\n\t&= 2 \\realpart \\langle Ax(t), x(t) \\rangle_X \\\\\n\t&\\leq 0. \n\\end{align*}\nIntegration on both sides over $[0, \\tau]$ yields $\\| x(\\tau) \\|_X^2 \\leq \\| x(0)\\|_X^2$ for all $\\tau \\geq 0$. In many physical problems, the value $\\|x\\|_X^2$, $x \\in X$, represents the energy level of the state $x$. Thus, for solutions of abstract Cauchy problems associated with a dissipative operator, the energy of the corresponding solution is dissipating. As we will see, this property is essential in the port-Hamiltonian framework of infinite-dimensional systems. \n\\QEDA\n\\end{example}\n\n\\subsection{Linear Inhomogeneous Initial Value Problems}\n\nAs a preparation for the analysis of abstract control systems, in this section, we shortly discuss inhomogeneous initial value problems of the following form. \n\\begin{definition}[Inhomogeneous Abstract Cauchy Problem]\n\tThe initial value problem \n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\t\\dot{x}(t) &= Ax(t) + f(t), \\hspace{0.5 cm} t > 0, \\\\\n\t\t\tx(0) &= x_0 \\in X,\n\t\t\\end{split}\n\t\t\\label{General Abstract Cauchy Problem - Inhomogeneous}\n\t\\end{align}\n\twith $A \\colon D(A) \\subset X \\to X$ a linear operator, $x_0$ the initial value, and $f \\colon [0, \\infty) \\to X$, $x \\colon [0, \\infty) \\to X$ time-dependent, $X$-valued functions is called an inhomogeneous abstract Cauchy problem. Let $\\tau > 0$. The function $x$ is called a (classical) solution on $[0, \\tau)$, if the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $x_{|[0,\\tau]} \\in \\mathcal{C}([0, \\tau], X)$.\n\t\t\\item[(ii)] $x_{| (0,\\tau)} \\in \\mathcal{C}^1((0, \\tau) , X)$. \n\t\t\\item[(iii)] $x(t) \\in D(A)$ for all $ t \\in [0,\\tau]$.\n\t\t\\item[(iv)] $x$ satisfies \\eqref{General Abstract Cauchy Problem - Inhomogeneous} on $(0,\\tau)$. \n\t\\end{enumerate}\n\\end{definition} \n\nFrom the discussion in Subsection \\ref{Subsection Linear Homogeneous Initial Value Problems} it makes sense to assume that the operator $A$ associated with \\eqref{General Abstract Cauchy Problem - Inhomogeneous} generates a $C_0$-semigroup, so that the equation with $f \\equiv 0$ has a uniquely determined classical solution. \n\\vspace{0.5 cm}\\\\\nWe want to show first that if \\eqref{General Abstract Cauchy Problem - Inhomogeneous} has a solution, then it has a particular form. More precisely, it is given by the \\emph{variation of constants} formula. \n\\begin{lemma}[cf. Corollary 4.2.2 in \\cite{Pazy83}]\n\t\\label{Lemma Mild Solution}\n\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. Let $f \\in \\mathcal{C}([0, \\tau], X)$, $x_0 \\in D(A)$, and assume that $x \\in \\mathcal{C}^1([0,\\tau], X)$ is a classical solution of \\eqref{General Abstract Cauchy Problem - Inhomogeneous} on $[0, \\tau)$. Then $x$ is of the form\n\t\\begin{align}\n\t\t\\label{Mild Solution}\n\t\tx(t) = T(t)x_0 + \\int_{0}^{t} T(t-s) f(s) \\, ds, \\hspace{0.5 cm} t \\in [0,\\tau].\n\t\\end{align}\n\\end{lemma}\n\\begin{proof}\n\tFor $ 0 < t < \\tau$, define the mapping\n\t\\begin{align*}\n\t\t\\psi \\colon s \\mapsto T(t - s)x(s), \\hspace{0.5 cm} s \\in [0, t]. \n\t\\end{align*} \nWith similar arguments as in the proof of Theorem \\ref{Theorem A Generator yields Unique Solution} one can show that $\\psi \\in \\mathcal{C}^1((0,t), X)$, and its derivative is given by\n\\begin{align*}\n\t\\frac{d}{ds} \\psi(s) = T(T-s)\\dot{x}(s) - AT(t-s)x(s), \\hspace{0.5 cm} s \\in (0,t).\n\\end{align*}\nUsing Theorem \\ref{Theorem Properties of C0-Semigroups} \\ref{C0SemigroupProp3} and the fact that $x$ is a classical solution, we find\n\\begin{align}\n\t\\label{xyz}\n\t\\frac{d}{ds}\\psi(s) = T(t-s)(Ax(s) + f(s)) - T(t-s)Ax(s) = T(t-s)f(s). \n\\end{align}\nDue to the continuity of $f$, the mapping $s \\mapsto T(t-s)f(s)$ is integrable. Thus, by integration of \\eqref{xyz} and by definition of $\\psi$, we obtain\n\\begin{align*}\nx(t) - T(t)x(0) =\\psi(t) - \\psi(0) = \\int_{0}^{t} T(t-s)f(s) ds.\n\\end{align*}\nIn conclusion, $x$ is of the form \\eqref{Mild Solution}. This proves the claim.\n\\end{proof}\nThe map \\eqref{Mild Solution} can be defined for any $x_0 \\in X$ and any $f \\in L^1([0, \\tau], X)$. In this case, it is an element of $\\mathcal{C}([0,\\tau], X)$ and is called the \\emph{mild solution} of \\eqref{General Abstract Cauchy Problem - Inhomogeneous}. It is clear that, in general, the mild solution \\eqref{Mild Solution} is not a classical solution of \\eqref{General Abstract Cauchy Problem - Inhomogeneous}. Furthermore, the continuity of $f$ we required in Lemma \\ref{Lemma Mild Solution} is not sufficient to guarantee that there is a classical solution, even if $x_0 \\in D(A)$. To see this, let $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup with generator $A$ and the property that there exist some $\\hat{t} > 0$ and $\\hat{x} \\in X$ such that $T(\\hat{t}) \\hat{x} \\notin D(A)$. Now, consider the initial value problem\n\\begin{align*}\n\t\\dot{x}(t) &= Ax(t) + T(t)\\hat{x}, \\hspace{0.5 cm} t> 0, \\\\\n\tx(0) &= 0. \n\\end{align*}\nOne may show that the mild solution is given by $t \\mapsto tT(t)\\hat{x}$, which is not differentiable on $(0, \\infty)$, since $\\hat{t}T(\\hat{t})\\hat{x} \\notin D(A)$. Therefore, by Lemma \\ref{Lemma Mild Solution}, for this system there does not exist a classical solution at all. \n\\vspace{0.5 cm}\\\\\nIn the following, we want to present criteria that guarantee the existence of classical solutions of \\eqref{General Abstract Cauchy Problem - Inhomogeneous} for any initial value $x_0 \\in D(A)$. \n\\begin{theorem}[Theorem 4.2.4 in \\cite{Pazy83}]\n\t\\label{Theorem Criteria Existence of IACP}\n\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. Let $f \\in L^1([0,\\tau], X) \\cap \\mathcal{C}((0,\\tau], X)$ and define\n\t\\begin{align*}\n\t\tV(t) = \\int_{0}^{t} T(t-s)f(s) \\, ds, \\hspace{0.5 cm} t \\in [0, \\tau]. \n\t\\end{align*}\nThen \\eqref{General Abstract Cauchy Problem - Inhomogeneous} has a classical solution on $[0,\\tau)$ for all $x_0 \\in D(A)$ if one of the following properties hold:\n\\begin{enumerate}[label = (\\roman*)]\n\t\\item \\label{IHCP1} $V_{|(0,\\tau)} \\in \\mathcal{C}^1((0,\\tau), X)$.\n\t\\item \\label{IHCP2} For all $t \\in (0,\\tau)$ it holds that $V(t) \\in D(A)$ and $AV(\\cdot)$ is continuous on $(0,\\tau)$.\n\\end{enumerate}\nConversely, if \\eqref{General Abstract Cauchy Problem - Inhomogeneous} has a solution on $[0,\\tau)$ for some $x_0 \\in D(A)$, then it satisfies both \\ref{IHCP1} and \\ref{IHCP2}. \n\\end{theorem}\nFrom this theorem we may deduce the following two corollaries, which we will not prove. \n\\begin{corollary}[Corollary 4.2.5 in \\cite{Pazy83}]\n\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. If $f \\in \\mathcal{C}^1([0, \\tau], X)$, then for every $x_0 \\in D(A)$ the initial value problem \\eqref{General Abstract Cauchy Problem - Inhomogeneous} has a unique classical solution on $[0, \\tau)$. \n\\end{corollary}\n\n\\begin{corollary}[Corollary 4.2.6 in \\cite{Pazy83}]\n\t\tLet $A \\colon D(A) \\subset X \\to X$ be the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X$. Let $f \\in L^1([0,\\tau], X) \\cap \\mathcal{C}((0,\\tau), X)$. If $f(s) \\in D(A)$ for all $s \\in (0, \\tau)$ and $Af(\\cdot) \\in L^1([0,\\tau], X)$, then for every $x_0 \\in D(A)$ the initial value problem \\eqref{General Abstract Cauchy Problem - Inhomogeneous} has a unique classical solution on $[0, \\tau)$. \n\\end{corollary}\nNext, we want to have a quick discussion on linear control systems. \n\n\\section{Linear Control Systems}\n\\label{Section Linear Control Systems}\n\nThe inhomogeneous system \\eqref{General Abstract Cauchy Problem - Inhomogeneous} can be written as an abstract control system, if we replace the function $f(\\cdot)$ by $Bu(\\cdot) \\colon [0, \\tau] \\to X$, where $u \\colon [0, \\infty) \\to U$ is some input with the input space $U$ being a separable Hilbert space, and $B \\in \\mathcal{L}(U,X)$ is the input operator. However, the assumption that the input operator is a bounded linear operator from $U$ to $X$ is quite restrictive. In many applications, e.g., boundary and point control problems, the input operators are unbounded. \n\\vspace{0.5 cm}\\\\\nFor a proper treatment of systems with an unbounded input operator, we make use of the concept of admissible control operators. To this end, we lift the state space $X$ to a larger space $X_{-1}$, called the extrapolation space, that has the property that the $C_0$-semigroup $(T(t))_{t \\geq 0}$ generated by $A \\colon D(A) \\subset X \\to X$ extends uniquely to a $C_0$-semigroup $(T_{-1}(t))_{t \\geq 0}$ on $X_{-1}$, and its generator $A_{-1}$ is the extension of $A$. We then want to find those $X_{-1}$-valued input operators for which the lifted control system admits uniquely determined, $X$-valued solutions. Hence, admissibility is a well-suited concept to prove well-posedness of abstract control systems.\n\\vspace{0.5 cm}\\\\\nWe proceed as follows. In Subsection \\ref{Subsection The Spaces X1 and X-1} we introduce the necessary terminology as well as the concept of duality with respect to a pivot space, where we follow the exposition from Section 2.9 and Section 2.10 in \\cite{Tucsnak-Weiss}. Then, in Subsection \\ref{Subsection Abstract Control Systems and Admissibility} we give a concise overview about $X_{-1}$-valued control systems and the significance of admissible control operators, which is based on Section 4.1 and Section 4.2 in \\cite{Tucsnak-Weiss}.\n\n\\subsection{The Spaces $\\mathbf{X}_1$ and $\\mathbf{X}_{-1}$}\n\\label{Subsection The Spaces X1 and X-1}\nAs a preparation for abstract control systems and the definition of a solution of such systems, we are going to introduce the interpolation space and the extrapolation space for a linear operator $A \\colon D(A) \\subset X \\to X$. Besides that, we need to explain what it means that two Hilbert spaces are dual with respect to a pivot space.\n\\vspace{0.5 cm}\\\\\nLet $V$ and $H$ be Hilbert spaces with $V \\subset H$. We say that the \\emph{embedding} $V \\hookrightarrow H$ is \\emph{continuous} provided that there exists some $m \\geq 0$ such that for all $v \\in V$ it holds that\n\\begin{align*}\n\t\\|v\\|_H \\leq m \\| v\\|_V. \n\\end{align*}\nIn other words, the identity operator from $V$ to $H$ is an element of $\\mathcal{L}(V,H)$. \n\n\\begin{theorem}[Proposition 2.9.2 in \\cite{Tucsnak-Weiss}]\n\t\\label{Theorem Pivot Space}\n\tLet $V$ and $H$ be Hilbert spaces with $V \\subset H$. Assume that the embedding $V \\hookrightarrow H$ is continuous and densely defined. Define the function $\\| \\cdot \\|_{\\ast} \\colon H \\to \\mathbb{R}$ by\n\t\\begin{align}\n\t\t\\label{Norm ast}\n\t\t\\|z \\|_{\\ast} = \\sup_{\\stackrel{v \\in V}{\\|v\\|_V = 1}} \\vert \\langle z , v \\rangle_H \\vert, \\hspace{0.5 cm} z \\in H. \n\t\\end{align}\nThen $\\| \\cdot \\|_{\\ast}$ is a norm on $H$. Let $V^{\\ast}$ denote the completion of $H$ with respect to $\\| \\cdot \\|_{\\ast}$. Define the operator $J \\colon V^{\\ast} \\to V'$ as follows: for any $z \\in V^{\\ast}$, \n\\begin{align}\n\t\\label{Theorem Pivot Space - Limit}\n\t\\langle Jz, v \\rangle_{V' \\times V} = \\lim_{n \\to \\infty} \\langle z_n, v \\rangle_H, \\hspace{0.5 cm} v \\in V, \n\\end{align}\nwith $(z_n)_{n \\in \\mathbb{N}} \\in H^{\\mathbb{N}}$ a sequence converging to $z$ in $V^{\\ast}$. Then $J$ is an isomorphism from $V^{\\ast}$ to $V'$. \n\\end{theorem}\n\n\\begin{proof}[Sketch of the proof of Theorem \\ref{Theorem Pivot Space}]\n\tWe will skip the proofs that $\\| \\cdot \\|_{\\ast}$ is a norm, that the limit \\eqref{Theorem Pivot Space - Limit} exists and is independent of the choice of the sequence $(z_n)_{n \\in \\mathbb{N}}$, as long as $z_n \\to z$ in $V^{\\ast}$. We will show that $\\langle Jz, v \\rangle_{V' \\times V}$ depends continuously on $v \\in V$, and thus $Jz \\in V'$. From the definition of $\\| \\cdot \\|_{\\ast}$ we deduce\n\t\\begin{align*}\n\t\\vert \\langle z, v \\rangle_H \\vert \\leq \\|z\\|_{\\ast} \\| v \\|_V , \\hspace{0.5 cm} z \\in H, \\, v \\in V. \n\t\\end{align*}\nHence, for any $z \\in V^{\\ast}$ and any $v \\in V$, \n\\begin{align*}\n\t\\vert \\langle Jz, v \\rangle_{V' \\times V} \\vert = \\lim_{n \\to \\infty} \\vert \\langle z_n, v \\rangle_H \\vert \\leq \\lim_{n \\to \\infty} \\|z_n\\|_{\\ast} \\|v \\|_V = \\|z\\|_{\\ast} \\| v \\|_V. \n\\end{align*}\nAs a consequence, $Jz \\in V'$. In particular, $\\| Jz \\|_{V'} \\leq \\| z \\|_{\\ast}$ for all $z \\in V^{\\ast}$, whence $J \\in \\mathcal{L}(V^{\\ast}, V')$. From the definition of $J$ we infer that\n\\begin{align*}\n\\langle\tJz, v \\rangle_{V' \\times V} = \\langle z , v \\rangle_H, \\hspace{0.5 cm} z \\in H, \\, v \\in V, \n\\end{align*}\nand so $\\| J z\\|_{V'} = \\|z \\|_{\\ast}$ for all $z \\in H$. Since $H$ is dense in $V^{\\ast}$ and $J$ is continuous, we conclude that $\\| J z\\|_{V'} = \\|z \\|_{\\ast}$ even holds for all $z \\in V^{\\ast}$. The verification that $J$ is onto can be looked up in the proof of Proposition 2.9.2 in \\cite{Tucsnak-Weiss}. Altogether, $J$ is an isomorphism from $V^{\\ast}$ to $V'$. \n\\end{proof}\nIf $V$, $H$, and $V^{\\ast}$ are as in Theorem \\ref{Theorem Pivot Space}, then we say that we identify $V^{\\ast}$ with $V'$ if we do not distinguish between $z$ and $Jz$ for all $z \\in V^{\\ast}$. Thus, we have\n\\begin{align*}\n\tV \\subset H \\subset V',\n\\end{align*}\nwith continuous and densely defined embeddings. We call $V'$ the \\emph{dual of} $V$ \\emph{with respect to the pivot space} $H$. Moreover, the norm $\\| \\cdot \\|_{\\ast}$ defined in \\eqref{Norm ast} is called the \\emph{dual norm of} $\\| \\cdot \\|_V$ \\emph{with respect to the pivot space} $H$. \n\\vspace{0.5 cm}\\\\\nNote that the space $V$ is uniquely determined by $V'$, as it consists of all $v \\in H$ for which the mapping $z \\mapsto \\langle z, v \\rangle_H$ has a continuous extension to $V'$. We therefore call $V$ the \\emph{dual of} $V'$ \\emph{with respect to the pivot space} $H$. \n\\vspace{0.5 cm}\\\\\nNow that we have clarified duality with respect to a pivot space, we can finally turn to the definition of the interpolation space and the extrapolation space of a linear operator. \n\n\\begin{definition}[Interpolation Space $\\mathbf{X_1}$]\n\t\\label{Definition Interpolation Space}\n\tLet $A \\colon D(A) \\subset X \\to X$ be linear and densely defined with $\\rho(A) \\neq \\emptyset$. For a fixed $\\lambda \\in \\rho(A)$, define the interpolation space $X_1$ = $(D(A), \\|\\cdot\\|_1)$, where \n\t\\begin{align*}\n\t\t\\|x\\|_1 := \\| (\\lambda I - A)x \\|_X, \\hspace{0.5 cm} x \\in D(A). \n\t\\end{align*}\n\t\\end{definition}\n\tAccording to \\cite[Proposition 2.10.1]{Tucsnak-Weiss} the interpolation space $X_1$ is a Hilbert space which is continuously embedded in $X$. Furthermore, for every $\\lambda \\in \\rho(A)$, $\\| \\cdot \\|_1$ is equivalent to the \\emph{graph norm} $\\| \\cdot \\|_{D(A)}$ of $A$, and if $L \\in \\mathcal{L}(X)$ is such that $L D(A) \\subset D(A)$, then $L \\in \\mathcal{L}(X_1)$. \n\t\\vspace{0.5 cm}\\\\\n\tFor the adjoint operator $A^{\\ast} \\colon D(A^{\\ast}) \\subset X \\to X$ of $A$ one can analogously define the Hilbert space $X_1^d = (D(A^{\\ast}), \\|\\cdot\\|_1^d)$ with\n\t\\begin{align*}\n\t\t\\|x\\|_1^d := \\| (\\overline{\\lambda} I - A^{\\ast})x \\|_X, \\hspace{0.5 cm} x \\in D(A^{\\ast}), \n\t\\end{align*}\n\twhere $\\overline{\\lambda} \\in \\rho(A^{\\ast})$ ($\\Leftrightarrow \\lambda \\in \\rho(A)$, see Proposition \\ref{Proposition 2.8.4 in Tucsnak}).\n\t\\begin{definition}[Extrapolation Space $\\mathbf{X_{-1}}$]\n\t\t\\label{Definition Extrapolation Space}\n\t\tLet $A \\colon D(A) \\subset X \\to X$ be linear and densely defined with $\\rho(A) \\neq \\emptyset$. For a fixed $\\lambda \\in \\rho(A)$, define the extrapolation space $X_{-1}$ as the completion of $X$ with respect to the norm\n\t\\begin{align*}\n\t\t\\|x\\|_{-1} := \\|(\\lambda I - A)^{-1}x\\|_X ,\\hspace{0.5 cm} x \\in X. \n\t\\end{align*}\n\\end{definition}\n\tAccording to \\cite[Proposition 2.10.2]{Tucsnak-Weiss} the extrapolation space $X_{-1}$ is again a Hilbert space, and the norms for different $\\lambda \\in \\rho(A)$ are equivalent. In particular, $X_{-1}$ is independent of the choice of $\\lambda$. Moreover, $X_{-1}$ is the dual of $ X_1^d$ with respect to the pivot space $X$. To see the last assertion, define $\\|\\cdot \\|_{-1}$ and $\\| \\cdot \\|_1^d$ with respect to the same $\\lambda \\in \\rho(A)$, and apply Proposition \\ref{Proposition 2.8.4 in Tucsnak}. Then, for every $z \\in X$, \n\t\\begin{align*}\n\\|z \\|_{-1} &= \\|(\\lambda I - A)^{-1}z \\|_X \\\\\n\t\t\t&= \\sup_{\\stackrel{x \\in X}{\\|x\\|_X = 1}} \\vert \\langle (\\lambda I -A)^{-1}z, x \\rangle_X \\vert \\\\\n\t\t\t&= \\sup_{\\stackrel{x \\in X}{\\|x\\|_X = 1}} \\vert \\langle z, (\\overline{\\lambda}I - A^{\\ast})^{-1} x \\rangle_X \\vert \\\\\n\t\t\t&= \\sup_{\\stackrel{y \\in X_1^d}{\\| y \\|_1^d = 1}} \\vert \\langle z , y \\rangle_X \\vert. \n\t\\end{align*} \nNext, we present an important result for the upcoming section. Assume that $A$ generates a $C_0$-semigroup. Then this semigroup has a unique extension to $\\mathcal{L}(X_{-1})$. \nSubsequently, we show an example where we specify the extrapolation space $X_{-1}$ for an infinitesimal generator. \n\n\\begin{proposition}[Proposition 2.10.3 in \\cite{Tucsnak-Weiss}]\n\tLet $A \\colon D(A) \\subset X \\to X$ be linear and densely defined with $\\rho(A) \\neq \\emptyset$. For $\\lambda \\in \\rho(A)$, define $X_1$ and $X_{-1}$ as in Definition \\ref{Definition Interpolation Space} and \\ref{Definition Extrapolation Space}, respectively. Then $A \\in \\mathcal{L}(X_1, X)$ and $A$ has a unique extension $A_{-1} \\in \\mathcal{L}(X, X_{-1})$. Moreover, \n\t\\begin{align*}\n\t\t(\\lambda I - A)^{-1} \\in \\mathcal{L}(X, X_1) \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} (\\lambda I - A_{-1})^{-1} \\in \\mathcal{L}(X_{-1}, X). \n\t\\end{align*}\nIn particular, $\\lambda \\in \\rho(A_{-1})$, and these operators are unitary. \n\\end{proposition}\n\n\\begin{proposition}[Proposition 2.10.4 in \\cite{Tucsnak-Weiss}]\n\t\tLet $A \\colon D(A) \\subset X \\to X$ be the generator of a strongly continuous semigroup $(T(t))_{t \\geq 0}$ on $X$. For $\\lambda \\in \\rho(A)$, define $X_1$ and $X_{-1}$ as in Definition \\ref{Definition Interpolation Space} and \\ref{Definition Extrapolation Space}, respectively. For $t \\geq 0$, the restriction of $T(t)$ to $X_1$ (considered as an operator in $\\mathcal{L}(X_1)$) is the image of $T(t) \\in \\mathcal{L}(X)$ through the unitary operator $\t(\\lambda I - A)^{-1} \\in \\mathcal{L}(X, X_1)$. Therefore, these operators form a strongly continuous semigroup on $X_1$ whose generator is $A_{|D(A^2)}$. The operator $T_{-1}(t) \\in \\mathcal{L}(X_{-1})$ is the image of $T(t) \\in \\mathcal{L}(X)$ through the unitary operator $(\\lambda I - A_{-1})^{-1} \\in \\mathcal{L}(X_{-1}, X)$. Therefore, these extended operators form a strongly continuous semigroup $(T_{-1}(t))_{t \\geq 0}$ on $X_{-1}$, whose generator is $A_{-1} \\colon D(A_{-1}) \\subset X_{-1} \\to X_{-1}$ with $D(A_{-1}) = X$. \n\\end{proposition}\n\n\n\\begin{example}[Example 2.10.7 in \\cite{Tucsnak-Weiss}]\n\tConsider the left-shift semigroup $(T(t))_{t \\geq 0}$ on $X = L^2([0, \\infty), \\mathbb{C})$ defined by\n\t\\begin{align*}\n\t\tT(t)x(\\cdot) = x(\\cdot + t), \\hspace{0.5 cm} t\\geq 0, \\, x \\in X.\n\t\\end{align*}\n\tIts generator $A \\colon D(A) \\subset X \\to X$ is given by\n\t\\begin{align*}\n\t\tD(A) &= H^1((0,\\infty), \\mathbb{C}), \\\\\n\t\tAx &= \\frac{d}{dz}x, \\hspace{0.5 cm} x \\in D(A).\n\t\\end{align*}\nWe want to determine the extrapolation space $X_{-1}$ of this semigroup. To this end, we define the Hilbert space adjoint $A^{\\ast} \\colon D(A^{\\ast}) \\subset X \\to X$ of $A$, which is given by\n\t\\begin{align*}\n\t\tD(A^{\\ast}) &= H_0^1((0, \\infty), \\mathbb{C}), \\\\\n\t\tA^{\\ast} x &= - \\frac{d}{dz}x , \\hspace{0.5 cm} x \\in D(A^{\\ast}).\n\t\\end{align*}\nFrom the preceding discussion, we know that the dual of $X_1^d = (D(A^{\\ast}), \\|\\cdot\\|_1^d)$ with respect to the pivot space $X$ is given by $X_{-1}$. According to Definition 13.4.7 in \\cite{Tucsnak-Weiss}, the dual space of $H_0^1((0, \\infty), \\mathbb{C})$ is therefore given by the Sobolev space $H^{-1}((0, \\infty), \\mathbb{C})$, i.e., \n\t\\begin{align*}\n\t\tX_{-1} = (H_0^1((0, \\infty), \\mathbb{C}))' = H^{-1}((0, \\infty), \\mathbb{C}).\n\t\\end{align*}\nNext, we want to specify the norm on $X_{-1}$. Choosing $\\lambda = 1 \\in \\rho(A)$, Theorem \\ref{Theorem Laplace Transform} yields\n\t\\begin{align*}\n\t\t(\\lambda I - A)^{-1} x(z) = \\int_{0}^{\\infty} e^{-t} x(z + t) \\, dt, \\hspace{0.5 cm} x \\in X, \\, z \\in (0, \\infty). \n\t\\end{align*}\n\tThus, the norm $\\|\\cdot\\|_{-1}$ can be presented as\n\t\\begin{align*}\n\t\t\\|x\\|_{-1} &= \\left\\| \\int_{0}^{\\infty} e^{-t} x(\\cdot + t) \\, dt \\right\\|_X, \\hspace{0.5 cm} x \\in X.\n\t\\end{align*}\n\\QEDA\n\\end{example}\nHaving introduced the necessary concepts, we can have a quick discussion on $X_{-1}$-valued control systems in the following segment.\n\\subsection{Abstract Control Systems and Admissibility}\n\\label{Subsection Abstract Control Systems and Admissibility}\nIn this section, we collect some facts about existence and uniqueness of solutions of abstract control systems with values in the extrapolation space $X_{-1}$ of the form\n\\begin{align*}\n\t\\dot{x}(t) &= A_{-1}x(t) + Bu(t), \\hspace{0.5 cm} t >0, \\\\\n\tx(0) &= x_0 \\in X_{-1}. \n\\end{align*}\nIn the following, we will denote the extension $A_{-1}$ of $A \\colon D(A) \\subset X \\to X$ and the extension $T_{-1}(t)$ of $T(t)$, $t \\geq 0$, by $A$ and $T(t)$ again, respectively. We will not give any proofs and refer to \\cite[Chapter 4]{Tucsnak-Weiss}. \n\\vspace{0.5 cm}\\\\\nFirst, we want to clarify what we want to understand under a solution in $X_{-1}$. \n\n\\begin{definition}[Solution in $\\mathbf{X_{-1}}$]\n\t\\label{Definition Solution in X-1}\n\tConsider the differential equation\n\t\\begin{align}\n\t\t\\label{Abstract System in X-1}\n\t\t\\dot{x}(t) = Ax(t) + Bu(t), \\hspace{0.5 cm} t > 0, \n\t\\end{align}\nwith $A \\colon D(A) \\subset X_{-1} \\to X_{-1}$, $u \\in L_{\\text{loc}}^2([0, \\infty), U)$, $B \\in \\mathcal{L}(U, X_{-1})$, and with $U$ a separable Hilbert space. A solution of \\eqref{Abstract System in X-1} is a function\n\\begin{align*}\n\tx \\in L_{loc}^1([0,\\infty), X) \\cap \\mathcal{C}([0, \\infty), X_{-1})\n\\end{align*}\nwhich satisfies the following equations in $X_{-1}$: \n\\begin{align*}\n\tx(t) - x(0) = \\int_{0}^{t} Ax(s) + Bu(s) \\, ds, \\hspace{0.5 cm } t \\geq 0. \n\\end{align*}\n\\end{definition}\nJust as it is the case for the inhomogeneous Cauchy problem, if \\eqref{Abstract System in X-1} has a solution, then it has a particular form. \n\n\\begin{proposition}[Proposition 4.1.4 in \\cite{Tucsnak-Weiss}]\n\tWith the notation in Definition \\ref{Definition Solution in X-1}, suppose that $x$ is a solution of \\eqref{Abstract System in X-1} in $X_{-1}$. Then $x$ is given by \n\t\\begin{align*}\n\t\tx(t) = T(t)x_0 + \\int_{0}^{t} T(t-s) Bu(s) \\, ds, \\hspace{0.5 cm} t \\geq 0, \n\t\\end{align*}\nand is called the mild solution of \\eqref{Abstract System in X-1}. In particular, for every $x_0 \\in X$ there exists at most one solution in $X_{-1}$ of \\eqref{Abstract System in X-1} satisfying the initial condition $x(0) = x_0$. \n\\end{proposition}\nNext, we give a criterion for the initial value $x_0$ and the function $f$ that guarantees the existence of a unique solution satisfying the initial condition $x(0) = x_0$. \n\n\\begin{theorem}[Theorem 4.1.6 in \\cite{Tucsnak-Weiss}]\n\tIf $x_0 \\in X$ and $Bu(\\cdot) \\in H_{loc}^1([0,\\infty), X_{-1})$, then \\eqref{Abstract System in X-1} has a unique solution in $X_{-1}$, denoted by $x$, that satisfies $x(0) = x_0$. Moreover, this solution is such that\n\t\\begin{align*}\n\t\tx \\in \\mathcal{C}([0, \\infty), X) \\cap \\mathcal{C}^1([0, \\infty), X_{-1}), \n\t\\end{align*}\nand for all $t \\geq 0$ it satisfies \\eqref{Abstract System in X-1} in a classical sense. \n\\end{theorem}\nNow, let us consider the following abstract control system:\n\\begin{align}\n\\begin{split}\n\t\\dot{x}(t) &= Ax(t) + Bu(t), \\hspace{0.5 cm} t >0, \\\\\n\tx(0) &= x_0 \\in X.\n\\end{split}\n\\label{Abstract Control System}\n\\end{align}\nMotivated by the study of systems of the form \\eqref{Abstract Control System}, the concept of admissibility was introduced. Generally speaking, one wants to specify all those control operators $B$ for which all mild solutions $x$ of \\eqref{Abstract Control System} are continuous $X$-valued functions. Operators $B$ with this property are called admissible. \n\n\\begin{definition}[Admissible Control Operator]\n\t\\label{Definition Admissible Control Operator}\n\tLet $(T(t))_{t \\geq 0}$ be a $C_0$-semigroup on $X_{-1}$. Let $\\tau \\geq 0$. Define the mapping $\\Phi_{\\tau} \\in \\mathcal{L}(L^2([0, \\infty), U), X_{-1})$ by\n\t\\begin{align}\n\t\t\\Phi_{\\tau} u := \\int_{0}^{\\tau} T(\\tau- \\eta) Bu(\\eta) \\, d\\eta, \\hspace{0.5 cm} u \\in L^2([0, \\infty), U).\n\t\t\\label{Phi tau u}\n\t\\end{align}\n\tAn operator $B \\in \\mathcal{L}(U, X_{-1})$ is called an admissible control operator for $(T(t))_{t \\geq 0}$ on $X_{-1}$ if for some $\\tau > 0$ it holds that\n\t\\begin{align*}\n\t\t\\ran (\\Phi_{\\tau}) \\subset X.\n\t\\end{align*}\n\\end{definition}\nIn other words: if $B$ is admissible, then in \\eqref{Phi tau u} the integrand is $X_{-1}$-valued, but the integral itself is an element of $X$. \n\n\\begin{proposition}[Proposition 4.2.2 in \\cite{Tucsnak-Weiss}]\n\t Suppose that $B \\in \\mathcal{L}(U, X_{-1})$ is admissible for the $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X_{-1}$. Then for every $t \\geq 0$ it holds that\n\t \\begin{align*}\n\t \t\\Phi_t \\in \\mathcal{L}(L^2([0, \\infty), U), X). \n\t \\end{align*}\n\\end{proposition}\nThe operators $\\Phi_t$, $t \\geq 0$, are called the \\emph{input maps corresponding to the system} \\eqref{Abstract Control System}, and have the following property.\n\\begin{proposition}[Proposition 4.2.4 in \\cite{Tucsnak-Weiss}]\n\t Suppose that $B \\in \\mathcal{L}(U, X_{-1})$ is admissible for the $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X_{-1}$. Then the mapping\n\t \\begin{align*}\n\t \t[0, \\infty) \\times L^2([0, \\infty) ,U), \\hspace{0.3 cm} (t,u) \\mapsto \\Phi_t u \n\t \\end{align*}\n is continuous. \n\\end{proposition}\nThe main result of this section states under which criteria the mild solutions of the abstract control system \\eqref{Abstract Control System} are continuous $X$-valued functions, and is given below.\n\n\\begin{theorem}[Proposition 4.2.5 in \\cite{Tucsnak-Weiss}]\n\t\\label{Proposition Well-Posedenss of Abstract Control Systems}\n\t Suppose that $B \\in \\mathcal{L}(U, X_{-1})$ is admissible for the $C_0$-semigroup $(T(t))_{t \\geq 0}$ on $X_{-1}$ generated by $A \\colon D(A) \\subset X_{-1} \\to X_{-1}$. Then for every $x_0 \\in X$ and every $u \\in L_{loc}^2([0, \\infty), U)$, the initial value problem \\eqref{Abstract Control System} has a unique solution in $X_{-1}$. This solution is given by\n\t \\begin{align*}\n\t \tx(t) = T(t)x_0 + \\Phi_t u, \\hspace{0.5 cm} t \\geq 0, \n\t \\end{align*}\n and satisfies\n \\begin{align*}\n \tx \\in \\mathcal{C}([0, \\infty), X) \\cap H_{loc}^1((0, \\infty), X_{-1}). \n \\end{align*}\n\\end{theorem}\nThis theorem shows that admissibility is a valuable tool to prove well-posedness of the abstract control system \\eqref{Abstract Control System}. In \\cite[Section 4.2]{Tucsnak-Weiss} one can find an example where an admissible operator has been determined for the unilateral right-shift semigroup.\n\\vspace{0.5 cm}\\\\\nThe following and last section of this chapter deals with the generalization of the abstract Cauchy problem discussed in Section \\ref{Section The Abstract Cauchy Problem}, where the operator associated with the system changes over time as well. \n\n\n\\section{Evolution Equations}\n\\label{Section Evolution Equations}\nIn this section, we want to give a concise overview concerning evolution problems. Roughly speaking, evolution problems are abstract, time-variant Cauchy problems, that is, the operator associated with the Cauchy problem is changing over time. We will introduce the necessary terminology and study the concept of stability for this class of systems, as this is a key property that has to be satisfied for proving well-posedness. This section relies substantially on \\cite[Chapter 5]{Pazy83}.\n\\vspace{0.5 cm}\\\\\nAs in the autonomous case, let us begin with the definition of an evolution problem.\n\\begin{definition}[Evolution Problem]\n\t\\label{Definition Classical Solution Evolution Problem}\nLet $0 \\leq t \\leq \\tau$, and let $A(t) \\colon D(A(t)) \\subset X \\to X$ be a linear operator. The time-variant initial value problem\n\\begin{align}\n\t\\begin{split}\n\t\\dot{x}(t) &= A(t) x(t), \\hspace{0.5 cm} 0 \\leq s < t \\leq \\tau, \\\\\n\tx(s) &= x_0 \\in X\n\\end{split}\n\\label{Evolution Problem}\n\\end{align}\nis called an evolution problem. A function $x \\colon [s,\\tau] \\to X$ is called a classical solution of the evolution problem \\eqref{Evolution Problem} if the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $x \\in \\mathcal{C}([s,\\tau], X)$. \n\t\t\\item[(ii)] $x_{|(s,\\tau)} \\in \\mathcal{C}^1((s,\\tau), X)$.\n\t\t\\item[(iii)] $x(t) \\in D(A(t))$ for all $s < t \\leq \\tau$.\n\t\t\\item[(iv)] $x$ satisfies \\eqref{Evolution Problem}. \n\t\\end{enumerate}\n\\end{definition}\nThe proof of the following theorem is based on using Picard's iteration method and applying the Banach contraction principle. \n\\begin{theorem}[Theorem 5.1.1 in \\cite{Pazy83}]\n\t\\label{Theorem 5.1.1 in Pazy}\n\t\\label{Theorem Classical Solution Homogeneous Evolution Problem}\n\tFor $0 \\leq t \\leq \\tau$, let $A(t) \\in \\mathcal{L}(X)$. If the mapping \n\t\\begin{align*}\n\t\t[0,\\tau] \\to \\mathcal{L}(X), \\hspace{0.3 cm} t \\mapsto A(t)\n\t\\end{align*}\nis continuous, then for every $x_0 \\in X$ the initial value problem \\eqref{Evolution Problem} has a unique classical solution. \n\\end{theorem}\n\n\\begin{proof}\n\tWe make use of Picard's iteration theorem. Since the mapping $ t \\mapsto A(t)$ is continuous, we may define\n\t\\begin{align*}\n\t\t\\alpha = \\max_{t \\in [0, \\tau]} \\|A(t)\\|_{\\mathcal{L}(X)}. \n\t\\end{align*}\nFurthermore, for $x_0 \\in X$ define the mapping\n\\begin{align*}\nS_{x_0} \\colon \\mathcal{C}([s,\\tau], X) &\\to \\mathcal{C}([s,\\tau], X), \\\\\n\tu &\\mapsto x_0 + \\int_{s}^{\\cdot} A(\\eta) u(\\eta) \\, d\\eta, \n\\end{align*}\nwhere $\\mathcal{C}([s,\\tau], X)$ is endowed with the norm \n\\begin{align*}\n\t\\|u\\|_{\\infty} = \\max_{t \\in [s,\\tau]} \\|u(t) \\|_X, \\hspace{0.5 cm} u \\in \\mathcal{C}([s,\\tau], X), \n\\end{align*}\nmaking $(\\mathcal{C}([s,\\tau], X), \\|\\cdot\\|_{\\infty})$ a Banach space. \nFor all $u,v \\in \\mathcal{C}([s,\\tau], X)$ and for all $s \\leq t \\leq \\tau$ the following estimate holds:\n\\begin{align*}\n\t\\| S_{x_0}u(t) - S_{x_0} v(t) \\|_X &= \\left\\| x_0 + \\int_{s}^{t} A(\\eta) u(\\eta) \\, d\\eta - x_0 - \\int_{s}^{t} A(\\eta) v(\\eta) \\, d\\eta \\right\\|_X \\\\\n\t&= \\left\\| \\int_{s}^{t} A(\\eta) \\left[ u(\\eta) - v(\\eta) \\right] \\, d\\eta \\right\\|_X \\\\\n\t&\\leq \\int_{s}^{t} \\|A(\\eta)\\|_{\\mathcal{L}(X)} \\| u(\\eta) - v(\\eta) \\|_X \\, d\\eta \\\\\n\t&\\leq \\alpha (t-s) \\|u - v\\|_{\\infty}. \n\\end{align*}\nFurthermore, we have\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\| S_{x_0}^2u(t) - S_{x_0}^2v(t) \\|_X \\\\\n\t&= \\left\\| S_{x_0} \\left( x_0 + \\int_{s}^{\\cdot} A(\\eta_1) u(\\eta_1) \\, d\\eta_1 \\right)(t) - S_{x_0} \\left( x_0 + \\int_{s}^{\\cdot} A(\\eta_1) v(\\eta_1) \\, d\\eta_1 \\right) (t) \\right\\|_X \\\\\n\t&= \\left\\| \\int_{s}^{t} A(\\eta_2) \\left( x_0 + \\int_{s}^{\\eta_2} A(\\eta_1) u(\\eta_1) \\, d\\eta_1 \\right) \\, d\\eta_2 - \\int_{s}^{t} A(\\eta_2) \\left( x_0 + \\int_{s}^{\\eta_2} A(\\eta_1) v(\\eta_1) \\, d\\eta_1 \\right) \\, d\\eta_2 \\right\\|_X \\\\\n\t&= \\left\\| \\int_{s}^{t} A(\\eta_2) \\left( \\int_{s}^{\\eta_2} A(\\eta_1) \\left[ u(\\eta_1) - v(\\eta_1) \\right] d\\eta_1 \\right) \\, d\\eta_2 \\right\\|_X \\\\\n\t&\\leq \\alpha \\int_{s}^{t} \\int_{s}^{\\eta_2} \\| A(\\eta_1) \\left[ u(\\eta_1) - v(\\eta_1) \\right] \\|_X \\, d\\eta_1 \\, d\\eta_2 \\\\\n\t&\\leq \\alpha^2 \\|u - v\\|_{\\infty} \\int_{s}^{t} \\int_{s}^{\\eta_2} 1 \\, d\\eta_1 \\, d\\eta_2 \\\\\n\t&= \\frac{\\alpha^2 (t-s)^2}{2} \\| u - v\\|_{\\infty}.\n\\end{align*}\nBy induction it follows that \n\\begin{align*}\n\t\\|S_{x_0}^n u(t) - S_{x_0}^n v(t) \\|_X \\leq \\frac{\\alpha^n (\\tau - s)^n}{n!} \\|u - v\\|_{\\infty} \n\\end{align*}\nfor all $n \\in \\mathbb{N}$, $u,v \\in \\mathcal{C}([s,\\tau], X)$ and $s \\leq t \\leq \\tau$. Consequently, we have\n\\begin{align*}\n\t\\|S_{x_0}^n u - S_{x_0}^n v \\|_{\\infty} \\leq \\frac{\\alpha^n (\\tau - s)^n}{n!} \\|u - v\\|_{\\infty}.\n\\end{align*}\nNote that for $N \\in \\mathbb{N}$ large enough we get $\\frac{\\alpha^N (\\tau - s)^N}{N!} < 1$, and so $S_{x_0}^N$ is a contraction mapping. By the Banach contraction principle, $S_{x_0}^{N}$ has a unique fixed point $x = x(\\cdot;x_0) \\in \\mathcal{C}([s,\\tau], X)$. One can readily see that\n\\begin{align*}\n\tS_{x_0}^N(S_{x_0}(x)) = S_{x_0}(S_{x_0}^N(x)) =S_{x_0}(x).\n\\end{align*}\nThus, $S_{x_0}(x)$ is a fixed point of $S_{x_0}^N$. As $x$ is the unique fixed point of $S_{x_0}^N$, we infer that $S_{x_0}(x) = x$ and, hence, $x$ is a fixed point of $S_{x_0}$ as well. Finally, the uniqueness of the fixed point of $S_{x_0}^N$ together with the fact that every fixed point of $S_{x_0}$ is a fixed point of $S_{x_0}^N$ yield that $x$ is the unique fixed point of $S_{x_0}$ as well, and therefore satisfies\n\\begin{align}\n\t\\label{Theorem 5.1.1 in Pazy - Solution}\n\tx(t) = x_0 + \\int_{s}^{t} A(\\eta) x(\\eta) \\, d\\eta, \\hspace{0.5 cm} t \\in [s,\\tau]. \n\\end{align} \nNow, since $x$ is continuous, the right-hand side of \\eqref{Theorem 5.1.1 in Pazy - Solution} is differentiable. Thus, $x$ is differentiable with derivative\n\\begin{align*}\n\t\\frac{d}{dt}x(t) = \\frac{d}{dt} \\left( x_0 + \\int_{s}^{t} A(\\eta) x(\\eta) \\, d\\eta \\right) = A(t)x(t), \\hspace{0.5 cm} t \\in [s,\\tau]. \n\\end{align*}\nFurthermore, we have $x(s) = x_0$, and so $x$ is a solution of the evolution problem \\eqref{Evolution Problem}. Note that every solution of \\eqref{Evolution Problem} is in particular a solution of \\eqref{Theorem 5.1.1 in Pazy - Solution}. Due to the uniqueness of the fixed point $x$ of the mapping $S_{x_0}$, we conclude that the solution of \\eqref{Evolution Problem} is unique. \n\\end{proof}\nRecall that in Theorem \\ref{Theorem 5.1.1 in Pazy} we required the operators $A(t)$, $0 \\leq t \\leq \\tau$, to be bounded linear operators on $X$. However, this result gives us some intuition regarding the behavior of the solutions of the evolution problem \\eqref{Evolution Problem}. Using the representation \\eqref{Theorem 5.1.1 in Pazy - Solution} of such a solution, we may define a solution operator $U(t,s) \\colon X \\to X$, $0\\leq s \\leq t \\leq \\tau$, of the initial value problem \\eqref{Evolution Problem} as follows:\n\\begin{align}\n\t\\label{Pazy - Two-Parameter Family of Operators}\n\tU(t,s) x_0 = x(t;x_0) = x_0 + \\int_{s}^{t} A(\\eta) x(\\eta) \\, d\\eta, \\hspace{0.5 cm} 0 \\leq s \\leq t \\leq \\tau, \\, x_0 \\in X.\n\\end{align}\nThe family $(U(t,s))_{0 \\leq s \\leq t \\leq \\tau}$ is a two-parameter family of operators on $X$.\nThe main properties of the family of solution operators are given in the next theorem. \n\\begin{theorem}[Theorem 5.1.2 in \\cite{Pazy83}]\n\t\\label{Theorem 5.1.2 in Pazy}\n\tFor $0 \\leq t \\leq \\tau$, let $A(t) \\in \\mathcal{L}(X)$, and let the mapping $t \\mapsto A(t)$ be continuous on $[0,\\tau]$. Then for every $0 \\leq s \\leq t \\leq \\tau$, the operator $U(t,s)$ defined in \\eqref{Pazy - Two-Parameter Family of Operators} is an element of $\\mathcal{L}(X)$ and it satisfies the following:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\|U(t,s)\\|_{\\mathcal{L}(X)} \\leq \\exp \\left( \\int_{s}^{t} \\|A(\\eta) \\|_{\\mathcal{L}(X)} \\, d\\eta \\right)$.\n\t\t\\item[(ii)] $U(t,t) = I_X$ and for $0 \\leq s \\leq r \\leq t \\leq \\tau$ we have\n\t\t\\begin{align*}\n\t\t\tU(t,s) = U(t,r) U(r,s).\n\t\t\\end{align*}\n\t\\item[(iii)] The mapping $(t,s) \\mapsto U(t,s)$ is continuous for $0 \\leq s \\leq t \\leq \\tau$. \n\t\\item[(iv)] For $0 \\leq s \\leq t \\leq \\tau$ we have\n\t\\begin{align*}\n\t\t\\frac{\\partial}{\\partial t}U(t,s) = A(t) U(t,s). \n\t\\end{align*}\n\\item[(v)] For $0 \\leq s \\leq t \\leq \\tau$ we have\n\\begin{align*}\n\t\\frac{\\partial}{\\partial s} U(t,s) = - U(t,s) A(s). \n\\end{align*}\n\t\\end{enumerate}\n\\end{theorem}\nThese properties resemble the ones of (uniformly continuous) one-parameter semigroups studied in Section \\ref{Section Strongly Continuous Semigroups}. Indeed, in the time-variant case the two-parameter family of operators $(U(t,s))_{0 \\leq s \\leq t\\leq \\tau}$ replaces the one-paramter semigroup $(U(t))_{t \\geq 0}$ of the autonomous case, which is the $C_0$-semigroup generated by the (autonomous) operator $A$. This motivates to introduce the notion of evolution systems, which play a substantial role in the framework of evolution problems.\n\\begin{definition}[Evolution System]\n\t\\label{Definition Evolution System}\n\tA two-parameter family of operators $U(t,s) \\in \\mathcal{L}(X)$, $0 \\leq s \\leq t \\leq \\tau$, is called an evolution system if the following holds:\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item \\label{EvolutionSystemProp1} $U(s,s) = I_X$ and $U(t,s) = U(t,r) U(r,s)$ for $0 \\leq s \\leq r \\leq t \\leq \\tau$. \n\t\t\\item \\label{EvolutionSystemProp2} The mapping $(t,s) \\mapsto U(t,s)$ is strongly continuous for $0 \\leq s \\leq t \\leq \\tau$. \n\t\t\\end{enumerate}\n\\end{definition}\nJust as for one-paramter semigroups, the strong continuity of $(U(t,s))_{0 \\leq s \\leq t \\leq \\tau}$ is equivalent to the mapping $(t,s) \\mapsto U(t,s)x$ being continuous for all $x \\in X$. \n\\vspace{0.5 cm}\\\\\nIn the following, we want to present one important criterion on a given family of (usually unbouded) operators $(A(t))_{t \\in [0,\\tau]}$ on $X$ that guarantees the existence of a unique classical solution of the evolution problem \\eqref{Evolution Problem} for a dense subset of initial values in $X$. The existence of a unique solution allows us to specify an evolution system associated with the family $(A(t))_{t \\in [0,\\tau]}$. Note that the uniqueness of the solution yields property \\ref{EvolutionSystemProp1}, whereas the continuity of the solution at the initial data yields property \\ref{EvolutionSystemProp2} of the corresponding evolution system. The aforementioned criterion is called stability, and is defined as follows.\n\\begin{definition}[Stable Family of Generators]\n\t\\label{Definition Stable Family of Generators}\n\tA family $(A(t))_{t \\in [0, \\tau]}$ of infinitesimal generators of $C_0$-semigroups on $X$ is called stable if there exist constants $M \\geq 1$ and $\\omega \\in \\mathbb{R}$, called stability constants, such that the following holds:\n\t\\begin{enumerate}\n\t\t\\item[(i)] For $0 \\leq t \\leq \\tau$ we have $(\\omega, \\infty) \\subset \\rho(A(t))$.\n\t\t\\item[(ii)] For every finite sequence $0 \\leq t_1 \\leq t_2 \\leq \\ldots \\leq t_k \\leq \\tau$, $k \\in \\mathbb{N}$, it holds that\n\t\t\\begin{align}\n\t\t\t \\label{Definition Stable Family of Generators - Condition 2}\n\t\t\t\\left\\| \\prod_{j = 1}^{k} R(\\lambda, A(t_j)) \\right\\|_{\\mathcal{L}(X)} \\leq \\frac{M}{(\\lambda - \\omega)^k}, \\hspace{0.5 cm} \\lambda > \\omega.\n\t\t\t\\end{align}\n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n\t\\label{Remark Stable Familiy of Generators}\n\tWe give some short remarks concerning Definition \\ref{Definition Stable Family of Generators}.\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item In general, the operators $R(\\lambda, A(t_j))$ do not commute. Hence, the order of the terms given in \\eqref{Definition Stable Family of Generators - Condition 2} is important.\n\t\t\\item The stability of $(A(t))_{t \\in [0,\\tau]}$ is preserved when the norm $\\|\\cdot\\|_X$ is replaced by an equivalent one. However, the stability constants depend on the particular norm $X$ is endowed with. \n\t\t\\item If for $ 0 \\leq t \\leq \\tau$, $A(t)$ generates a $C_0$-semigroup of type $C_0(1, \\omega)$ (see Theorem \\ref{Theorem C0-Semigroup Estimate}), then one can readily see that the family $(A(t))_{t \\in [0,\\tau]}$ is stable with stability constants $M=1$ and $\\omega \\in \\mathbb{R}$. In particular, every family $(A(t))_{t \\in [0,\\tau]}$ of generators of contraction semigroups is stable.\n\t\\end{enumerate}\n\\end{remark}\nIn general, it is not easy to determine whether a given family $(A(t))_{t \\in [0, \\tau]}$ of infinitesimal generators is stable. In the next theorem we give a characterization regarding the stability of such a family. For the proof, one exploits Theorem \\ref{Theorem Laplace Transform}. \n\\begin{theorem}[Theorem 5.2.2 in \\cite{Pazy83}]\t\n\t\\label{Theorem 5.2.2 in Pazy}\n\tFor $0 \\leq t \\leq \\tau$, let $A(t)$ be the generator of the $C_0$-semigroup $(S_t(s))_{s \\geq 0}$ on $X$. Then the family $(A(t))_{t \\in [0, \\tau]}$ is stable if and only if there exist constants $M\\geq 1$ and $\\omega \\in \\mathbb{R}$ such that $(\\omega, \\infty) \\subset \\rho(A(t))$ for all $t \\in [0, \\tau]$, and either one of the following conditions is satisfied:\n\t\\begin{enumerate}\n\t\t\\item[(i)] For every finite sequence $0 \\leq t_1 \\leq t_2 \\leq \\ldots \\leq t_k \\leq \\tau$, $k \\in \\mathbb{N}$, and for $s_j \\geq 0$, $j = 1,\\ldots,k$, \n\t\t\\begin{align*}\n\t\t\t\\left\\| \\prod_{j = 1}^{k} S_{t_j}(s_j) \\right\\|_{\\mathcal{L}(X)} \\leq M \\exp \\left( \\omega \\sum_{j=1}^{k} s_j \\right). \n\t\t\\end{align*} \n\t\\item[(ii)] For every finite sequence $0 \\leq t_1 \\leq t_2 \\leq \\ldots \\leq t_k \\leq \\tau$, $k \\in \\mathbb{N}$, and for $\\lambda_j > \\omega$, $j = 1,\\ldots,k$,\n\t\\begin{align*}\n\t\t\\left\\| \\prod_{j = 1}^{k} R(\\lambda_j, A(t_j)) \\right\\|_{\\mathcal{L}(X)} \\leq M \\prod_{j=1}^{k} \\frac{1}{\\lambda_j - \\omega}.\n\t\\end{align*}\n\t\\end{enumerate}\n\\end{theorem} \nWe want to emphasize once again that the stability of a family $(A(t))_{t \\in [0,\\tau]}$ is not the only condition that has to be satisfied in order to guarantee the existence of a unique solution of the evolution problem \\eqref{Evolution Problem} associated with $(A(t))_{t \\in [0,\\tau]}$. For instance, there has to exist a densely and continuously embedded subspace $Y$ of $X$ such that for all $t \\in [0,\\tau]$, the space $Y$ is an invariant subspace of the semigroup $(S_t(s))_{s \\geq 0}$ generated by $A(t)$, and the family $(\\tilde{A}(t))_{t \\in [0, \\tau]}$ given by $\\tilde{A}(t) = A(t)_{|D(A(t)) \\cap Y}$, $t \\in [0, \\tau]$, has to be stable on $Y$. Together with another condition related to $Y$, one may show that there exists a unique $Y$-valued solution of \\eqref{Evolution Problem}. This holds equally for inhomogeneous evolution problems of the form\n\\begin{align*}\n\t\t\\dot{x}(t) &= A(t) x(t) + f(t), \\hspace{0.5 cm} 0 \\leq s < t \\leq \\tau, \\\\\n\tx(s) &= x_0 \\in X,\n\\end{align*} \nprovided that $f \\in \\mathcal{C}([s,\\tau], X)$. However, we will not provide any further details concerning this topic. Instead, we refer to \\cite[Chapter 5]{Pazy83} for more information. \n\\vspace{0.5 cm}\\\\\nIn this chapter, we have defined formally adjoint operators of differential operators defined on Sobolev spaces, and collected some facts about operator semigroups and their infinitesimal generators. Furthermore, we have examined the abstract Cauchy problem associated with operators generating a $C_0$-semigroup. After a quick discussion on abstract control systems and the concept of admissible control operators, we have studied the extension of abstract Cauchy problems to evolution problems, where the associated operator is replaced by a time-dependent family of infinitesimal generators, and presented one important criterion for well-posedness of such systems. We have now set the foundation for a comprehensive analysis of the systems considered in the following chapters, namely infinite-dimensional port-Hamiltonian systems in Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}, and said systems of order 1 which are coupled by a moving interface in Chapter \\ref{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface}. \n\n\n\n\n\n\\chapter{Infinite-Dimensional Port-Hamiltonian Systems}\n\\label{Chapter Infinite-dimensional Port-Hamiltonian Systems}\n\nIn this chapter, we are going to investigate infinite-dimensional port-Hamiltonian systems. Throughout the chapter, we will restrict ourselves to systems with a 1-dimensional spatial domain $[a,b] \\subset \\mathbb{R}$. We start again with some motivational examples in Section \\ref{Section Motivating Examples}, specifically the model of a lossless transmission line in Subsection \\ref{Subsection The Lossless Transmission Line} and the model of a vibrating string in Subsection \\ref{Subsection The Vibrating String}. We will see that the evaluation of the co-energy variables at the boundary of the spatial domain play a fundamental role. Hence, in contrast to the finite-dimensional case, we need to introduce further port variables in order to derive a port-Hamiltonian formulation of infinite-dimensional systems, which will be called boundary port-Hamiltonian systems. The examples can be found in \\cite[Chapter~4]{ModCompPhys}, \\cite{LeGorrec}, \\cite[Chapter~14]{SchaftJeltsema}, or in the introductory chapter of \\cite{Villegas}. Subsequently, in Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators} we will generalize the obtained systems considered in Section \\ref{Section Motivating Examples} and study boundary port-Hamiltonian systems which are associated with formally skew-symmetric differential operators. To that end, we will define suitable matrices and operators in order to give a proper representation of the boundary port variables constituting the infinite-dimensional Dirac structure. Following this, we are able to define the bond space and a suitable power pairing to finally define the underlying Dirac structure associated with such systems, and to specify the dynamics with respect to these so-called Stokes-Dirac structures. This section heavily relies on \\cite{LeGorrec} and the Sections 2.1 and 2.2 in \\cite{Villegas}. At the end of this chapter, we will shortly discuss the analysis of boundary port-Hamiltonian systems.\n\\vspace{0.5 cm}\\\\\nWe want to stress that we will not introduce an abstract mathematical object or a particular solution concept for the systems considered in this chapter. As a consequence, the notion of a system should be understood as a formal concept rather than a general definition.\n\\vspace{0.5 cm}\\\\\nUnless stated otherwise, we consider the Hilbert space $X = L^2([a,b], \\mathbb{R}^n)$ equipped with the inner product\n\\begin{align}\n\t\\label{L^2 inner product}\n\t\\langle f, g \\rangle_{L^2} = \\int_{a}^{b} g^{\\top}(z) f(z) \\, dz, \\hspace{0.5 cm} f,g \\in X.\n\\end{align}\n\n\n\n\\section{Motivating Examples}\n\\label{Section Motivating Examples}\nWe want to look for a common structure in the models describing the dynamics of various systems. We seek to rewrite the system models as an abstract Cauchy problem of the form\n\\begin{align*}\n\t\\dot{x}(t) &= Ax(t), \\hspace{0.5 cm} t > 0, \\\\\t\n\tx(0) &= x_0 \\in X, \n\\end{align*}\nwhere $x \\colon [0, \\infty) \\to X$ is the state variable, and $A \\colon D(A) \\subset X \\to X$ is a linear operator, as we have discussed in Section \\ref{Section The Abstract Cauchy Problem} in more detail. For the analysis of such systems we use the framework of operator semigroups. In this section, we want to present two examples from which we will infer a general structure for a class of partial differential equations.\n\\vspace{0.5 cm}\\\\\nBeforehand, we want to clarify the notion of the variational derivative for a special class of functionals based on \\cite[Section 4.1]{Olver}, and give a quick example.\n\\begin{definition}[Variational Derivative]\nLet $H \\colon X \\to \\mathbb{R}$ be a functional defined by\n\\begin{align}\n\t\\label{General Functional a Variational Derivative is Applied to}\n\tH(x) = \\int_{a}^{b} \\mathcal{H}(x(z)) \\, dz, \\hspace{0.5 cm} x \\in X,\n\\end{align} \nwith $\\mathcal{H} \\in \\mathcal{C}^\\infty({\\mathbb{R}^n, \\mathbb{R}})$. The variational derivative $\\frac{\\delta H}{\\delta x} \\colon X \\to X$ of $H$ given by\n\\begin{align*}\n\t\\frac{\\delta H}{\\delta x}(x) = \\left( \\frac{\\delta H}{\\delta x_1}(x), \\ldots, \\frac{\\delta H}{\\delta x_n}(x) \\right), \\hspace{0.5 cm} x \\in X,\n\\end{align*}\nis an operator uniquely determined by the requirement that it satisfies\n\\begin{align*}\n\tH(x + \\epsilon \\eta) = H(x) + \\epsilon \\int_{a}^{b} \\left( \\frac{\\delta H}{\\delta x}(x) \\right)^{\\top}(z) \\eta (z) \\, dz + \\mathcal{O}(\\epsilon^2)\n\\end{align*}\nfor any $x \\in X,$ and for any variation $\\epsilon \\eta$, where $\\epsilon \\in \\mathbb{R}$, $\\eta \\in X$, and $\\mathcal{O}(\\epsilon^2)$ is the big O notation with respect to $\\epsilon^2$. \n\\end{definition}\n\n\\begin{example}[cf. Section 1.6 in \\cite{Villegas}]\n\t\\label{Example Variational Derivative}\nConsider the functional\n\\begin{align}\n\tH(x) = \\frac{1}{2} \\int_{a}^{b} x^{\\top}(z) \\left( \\mathcal{Q} x \\right)(z) \\, dz, \\hspace{0.5 cm} x \\in X,\n\t\\label{Subsection Lossless Transmission Line - Standard Hamiltonian Form}\n\\end{align}\nwhere $\\mathcal{Q} \\in \\mathcal{L}(X)$ is a coercive operator on $X$. Then we have for all $x,\\eta \\in X$, $\\epsilon \\in \\mathbb{R}$, \n\\begin{align*}\n\tH(x + \\epsilon \\eta) &= \\frac{1}{2} \\int_{a}^{b} (x(z) + \\epsilon \\eta(z))^{\\top} \\left( \\mathcal{Q} (x + \\epsilon \\eta \\right)(z) \\, dz \\\\\n\t&= \\frac{1}{2} \\int_{a}^{b} x^{\\top}(z) \\left( \\mathcal{Q} x \\right) (z) + \\epsilon \\left( x^{\\top}(z) \\left( \\mathcal{Q} \\eta \\right)(z) + \\eta^{\\top}(z) \\left( \\mathcal{Q}x \\right)(z) \\right) + \\epsilon^2 \\eta^{\\top}(z) \\left( \\mathcal{Q} \\eta \\right)(z) \\, dz \\\\\n\t&= H(x) + \\epsilon \\int_{a}^{b} \\left(\\mathcal{Q} x \\right)^{\\top}(z) \\eta(z) \\, dz + \\mathcal{O}(\\epsilon^2).\n\\end{align*}\nFrom this we deduce that $\\frac{\\delta H}{\\delta x}(x)= \\mathcal{Q}x$.\n\\QEDA\n\\end{example}\nFor functionals of the form \\eqref{General Functional a Variational Derivative is Applied to} one can show that the variational derivative is induced by the gradient of the integrand $\\mathcal{H}$ in the following way:\n\\begin{align*}\n\t\\frac{\\delta}{\\delta x}H(x)(z) = \\nabla \\mathcal{H}(x(z)), \\hspace{0.5 cm} x \\in X, \\, z \\in [a,b]. \n\\end{align*}\n\n\n\\subsection{The Lossless Transmission Line}\n\\label{Subsection The Lossless Transmission Line}\n\t\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width = 8cm]{Transmission-Line.png}\n\t\\caption{Transmission Line \\cite[Figure 7.2]{JacobZwart}}\n\t\\label{Figure Transmission Line}\n\\end{figure}\nConsider an ideal transmission line on the spatial interval $[a,b] \\subset \\mathbb{R}$, as depicted in Figure~\\ref{Figure Transmission Line}. Telegrapher's equations constitute the following system:\n\\begin{align}\n\t\\begin{split}\n\t\\frac{\\partial Q}{\\partial t}(z,t) &= - \\frac{\\partial I}{\\partial z} (z,t) = - \\frac{\\partial }{\\partial z} \\frac{\\varphi(z,t)}{L(z)}, \\\\\n\t\\frac{\\partial \\varphi}{\\partial t}(z,t) &= - \\frac{\\partial V}{\\partial z}(z,t) = - \\frac{\\partial}{\\partial z} \\frac{Q(z,t)}{C(z)}.\n\t\\end{split}\n\\label{Subsection Lossless Transmission Line - PDE}\n\\end{align}\nHere, $Q(z,t) \\in \\mathbb{R}$ and $\\varphi(z,t) \\in \\mathbb{R}$ denote the charge and the magnetic flux-linkage at position $z \\in [a,b]$ and at time $t \\geq 0$, respectively, and $C(z) >0$ and $L(z) >0$ are the distributed capacity and the distributed inductance of the transmission line at $z \\in [a,b]$, repsectively. Furthermore, the current and the voltage are given by $I = \\frac{\\varphi}{L}$ and $V = \\frac{Q}{C}$, respectively. The electro-magnetic energy density $\\mathcal{H} \\in \\mathcal{C}^{\\infty}(\\mathbb{R}^2, \\mathbb{R})$ is given by\n\\begin{align*}\n\t\\mathcal{H}(Q,\\varphi) = \\frac{1}{2} \\left(\\frac{Q^2}{C} + \\frac{\\varphi^2}{L} \\right), \\hspace{0.5 cm} (Q, \\varphi) \\in \\mathbb{R}^2.\n\n\\end{align*}\n The system's Hamiltonian $H \\colon X \\to \\mathbb{R}$, describing the internally stored energy, is the integral of the electro-magnetic energy density $\\mathcal{H}$: letting $x = (Q, \\varphi) \\in X$ we have\n\\begin{align}\n\tH(Q,\\varphi) = \\frac{1}{2} \\int_{a}^{b} \\frac{Q(z)^2}{C(z)} + \\frac{\\varphi(z)^2}{L(z)} \\, dz = \\frac{1}{2} \\int_{a}^{b} \\begin{bmatrix}\n\t\tQ(z) & \\varphi(z)\n\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\t\\frac{1}{C(z)} & 0 \\\\\n\t\t\t0 & \\frac{1}{L(z)} \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\tQ(z) \\\\\n\t\t\\varphi (z) \n\t\\end{bmatrix} \\, dz.\n\\label{Subsection Lossless Transmission Line - Hamiltonian H}\n\\end{align}\nAssuming that both the capacity $C$ and the inductance $L$ are bounded on $[a,b]$, we may define the coercive multiplication operator $\\mathcal{Q} \\in \\mathcal{L}(X)$ as follows:\n\\begin{align*}\n\t(\\mathcal{Q}x)(z) = \\mathcal{Q}(z)x(z) = \\begin{bmatrix}\n\t\t\\frac{1}{C(z)} & 0 \\\\\n\t\t0 & \\frac{1}{L(z)} \n\t\\end{bmatrix} \\begin{bmatrix}\n\tQ(z) \\\\\n\t\\varphi(z)\n\\end{bmatrix}, \\hspace{0.5 cm} x \\in X, \\, z \\in [a,b]. \n\\end{align*} \nThus, the Hamiltonian $H$ from \\eqref{Subsection Lossless Transmission Line - Hamiltonian H} can be written as $H(x) = \\frac{1}{2} \\langle x , \\mathcal{Q} x \\rangle_{L^2}$ for $x \\in X$.\n\\vspace{0.5 cm}\\\\\nAs we have seeen in the finite-dimensional case (cf. Section \\ref{Section Energy-storing Elements}), the co-energy variables correspond to the gradient of the system's Hamiltonian function. In the infinite-dimensional case, this generalizes to the variational derivative of the Hamiltonian functional.\nAs shown in Example \\ref{Example Variational Derivative}, if a functional is of the form \\eqref{Subsection Lossless Transmission Line - Standard Hamiltonian Form}, then the variational derivative is given by\n\\begin{align*}\n\t\\frac{\\delta H}{\\delta x} (x) = \\mathcal{Q}x = \\begin{bmatrix}\n\t\t\\frac{Q}{C} \\\\\n\t\t\\frac{\\varphi}{L}\n\t\\end{bmatrix}, \\hspace{0.5 cm} x =(Q, \\varphi) \\in X.\n\\end{align*}\nAltogether, we may equivalently rewrite the system (\\ref{Subsection Lossless Transmission Line - PDE}) as\n\\begin{align}\n\t\\frac{\\partial }{\\partial t} \\begin{bmatrix}\n\t\tQ \\\\\n\t\t\\varphi \n\t\\end{bmatrix} = \\begin{bmatrix}\n\t0 & - \\frac{\\partial }{\\partial z} \\\\\n\t- \\frac{\\partial }{\\partial z} & 0 \n\\end{bmatrix} \\begin{bmatrix}\n\\frac{\\delta H}{\\delta Q}(Q, \\varphi) \\\\\n\\frac{\\delta H}{\\delta \\varphi}(Q, \\varphi)\n\\end{bmatrix}.\n\\label{Subsection Lossless Transmission Line - pH System}\n\\end{align}\nThis system is called an \\emph{infinite-dimensional Hamiltonian system} defined with respect to the matrix differential operator $\\mathcal{J} \\colon D(\\mathcal{J}) \\subset X \\to X$ given by\n\\begin{align*}\n\tD(\\mathcal{J}) &= H^1([a,b], \\mathbb{R}^2), \\\\\n\t\\mathcal{J}x &= \\begin{bmatrix}\n\t\t0 & - \\frac{d}{dz} \\\\\n\t\t- \\frac{d}{dz} & 0 \n\t\\end{bmatrix}x, \\hspace{0.5 cm} x \\in D(\\mathcal{J}),\n\\end{align*}\nand with respect to the Hamiltonian $H$ defined in \\eqref{Subsection Lossless Transmission Line - Hamiltonian H}. Necessarily, the matrix differential operator $\\mathcal{J}$ has to satisfy two properties, namely formal skew-symmetry and the Jacobi identity, as explained in \\cite[Chapter 7]{Olver}. We will only verify the formal skew-symmetry in the sense of Definition \\ref{Definition Formal Operator Properties}. To this end, let $e =(e_1,e_2)$, $\\tilde{e} = (\\tilde{e}_1,\\tilde{e}_2) \\in H_0^1([a,b], \\mathbb{R}^2)$ be some vectors vanishing at the boundary. Then integration by parts yields\n\\begin{align}\n\t\\begin{split}\n\t&\\hspace{0.5 cm} \\langle \\mathcal{J} e, \\tilde{e}\\rangle_{L^2} + \\langle e, \\mathcal{J} \\tilde{e} \\rangle_{L^2} \\\\\n\t&= - \\int_{a}^{b} \\tilde{e}_1(z) \\frac{d e_2}{dz}(z) + \\tilde{e}_2(z) \\frac{d e_1}{dz}(z) + e_1(z) \\frac{d \\tilde{e}_2}{dz}(z) + e_2(z) \\frac{d \\tilde{e}_1}{dz} (z) \\, dz \\\\\n\t&= - \\big[ e_1(z)\\tilde{e}_2(z) + e_2(z) \\tilde{e}_1(z) \\big]_{a}^{b} \\\\\n\t&= 0.\n\t\\end{split}\n\t\\label{Subsection Lossless Transmission Line - Skew-Symmetry Test}\n\\end{align}\nSince $\\mathcal{J}$ is not modulated by the state variable $x$, Corollary 7.5 in \\cite{Olver} yields that $\\mathcal{J}$ is a Hamiltonian operator (in the sense of \\cite[Definition 7.1]{Olver}).\n\\vspace{0.5 cm}\\\\\nAnalogously to the finite-dimensional case, the Hamiltonian structure provides energy conservation: assume that $x \\colon [0, \\infty) \\to X$ is a classical solution of \\eqref{Subsection Lossless Transmission Line - pH System} which is vanishing at the boundary. Then by symmetry of the operator $\\mathcal{Q}$ and by formal skew-symmetry of $\\mathcal{J}$ we obtain \n\\begin{align}\n\t\\begin{split}\n\t\\frac{d}{dt} H(x(\\cdot,t)) &= \\frac{1}{2} \\frac{d}{dt} \\langle x(\\cdot,t), \\mathcal{Q} x(\\cdot,t) \\rangle_{L^2} \\\\\n\t &= \\frac{1}{2} \\left\\langle \\frac{\\partial x}{\\partial t}(\\cdot,t) , \\mathcal{Q}x (\\cdot,t) \\right\\rangle_{L^2} + \\frac{1}{2} \\left\\langle \\mathcal{Q} x (\\cdot,t), \\frac{\\partial x}{\\partial t}(\\cdot,t) \\right\\rangle_{L^2} \\\\\n\t&= \\frac{1}{2} \\langle \\mathcal{J} \\mathcal{Q} x (\\cdot,t), \\mathcal{Q} x (\\cdot,t) \\rangle_{L^2} + \\frac{1}{2} \\langle \\mathcal{Q} x (\\cdot,t), \\mathcal{J} \\mathcal{Q} x(\\cdot,t) \\rangle_{L^2} \\\\\n\t&= 0.\n\\end{split}\n\\label{Subsection Lossless Transmission Line - Energy Conservation}\n\\end{align}\nWe see that the Hamiltonian formulation imposes that there must not occur any energy exchange at the boundaries of the transmission line with the rest of the circuit, and therefore the total energy is conserved. However, this assumption is quite restrictive, and for the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take the interaction with\nits environment into account. This motivates to introduce a new type of port variables, called \\emph{boundary port variables}, and thus to extend the (port-)Hamiltonian framework to boundary port-Hamiltonian systems in the infinite-dimensional case.\n\\vspace{0.5 cm}\\\\\nNow, if the state variables do not vanish at the boundary of the spatial domain, then from (\\ref{Subsection Lossless Transmission Line - Skew-Symmetry Test}) we find that some boundary terms appear. Indeed, the energy is not conserved in general, but satisfies the following energy-balance equation, which again can be derived from (\\ref{Subsection Lossless Transmission Line - Skew-Symmetry Test}) and (\\ref{Subsection Lossless Transmission Line - Energy Conservation}):\n\\begin{align}\n\t\\begin{split}\n\t\\frac{d}{dt} H(x(\\cdot,t)) &= \\frac{1}{2} \\langle \\mathcal{J} \\mathcal{Q} x (\\cdot,t), \\mathcal{Q} x (\\cdot,t) \\rangle_{L^2} + \\frac{1}{2} \\langle \\mathcal{Q} x (\\cdot,t), \\mathcal{J} \\mathcal{Q} x(\\cdot,t) \\rangle_{L^2} \\\\\n\t &=- \\frac{1}{2} \\left[ 2 \\frac{Q(z,t)}{C(z)} \\frac{\\varphi (z,t)}{L(z)} \\right]_{a}^{b} \\\\\n\t&= - \\big[ V(z,t) I (z,t) \\big]_{a}^{b}.\n\t\\end{split}\n\\label{Subsection Lossless Transmission Line - Energy Balance Equation}\n\\end{align}\nNote that the product of voltage and current equals power. Hence, equation (\\ref{Subsection Lossless Transmission Line - Energy Balance Equation}) states that the change of internally stored energy solely occurs via the spatial boundary. This suggests to introduce the evaluation not of the state variables, but of the co-energy variables $ V = \\frac{Q}{C}$ and $I = \\frac{\\varphi}{L}$ at the spatial boundary as new port variables in order to involve the power flow through the boundary of the system (\\ref{Subsection Lossless Transmission Line - pH System}). We will postpone the definition of the boundary port variables to Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}.\n\\begin{remark}[Energy Space] \n\t\\label{Remark Energy Space}\n\tAs $H(x) = \\frac{1}{2} \\langle x, \\mathcal{Q}x \\rangle_{L^2}$, $x \\in X$, and as $\\mathcal{Q} \\in \\mathcal{L}(X)$ is a bounded and coercive operator, one usually endows the state space $X$ with the inner product $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{Q}}$ defined by\n\t\\begin{align}\n\t\t\\langle x, y \\rangle_{\\mathcal{Q}} := \\frac{1}{2} \\langle x, \\mathcal{Q} y \\rangle_{L^2}, \\hspace{0.5 cm} x, y \\in X.\n\t\t\\label{Subsection The Lossless Transmission Line - Inner Product WRT Q}\n\t\\end{align}\n\tNote that for all $z \\in [a,b]$ we have $mI \\leq \\mathcal{Q}(z) \\leq MI$ for some $0 0$, $a,b \\in \\mathbb{R}$, held stationary at both ends and free to vibrate transversely. The scenario is illustrated in Figure \\ref{Figure Vibrating String}. The vibration of the string can be modeled by\n\t\\begin{align}\n\t\t\\frac{\\partial w^2}{\\partial t^2}(z,t) = \\frac{1}{(\\rho A)(z)} \\frac{\\partial}{\\partial z} \\left( T(z) \\frac{\\partial w}{\\partial z}(z,t) \\right), \n\t\t\\label{Example Vibrating String - Vibrating String Equation}\n\t\\end{align}\nwhere $w(z,t) \\in \\mathbb{R}$ denotes the vertical displacement of the string at position $z \\in [a,b]$, and at time $t \\geq 0$, and $T(z) >0$ and $(\\rho A) (z) > 0$, $z \\in [a,b]$ denote the Young's modulus of the string and the linear mass density, respectively, which are assumed to be bounded. We take the momentum $p = (\\rho A) \\frac{\\partial w}{\\partial t}$ and the strain $\\epsilon = \\frac{\\partial w}{\\partial z} $ to constitute the state variable $x=(p,\\epsilon) \\in X$. The Hamiltonian $H \\colon X \\to \\mathbb{R}$ of this system is then given by the sum of the kinetic and the elastic energies in the following way: \n\\begin{align}\n\tH(p,\\epsilon) = \\frac{1}{2} \\int_{a}^{b} \\frac{1}{(\\rho A)(z)} p(z)^2 + T(z) \\epsilon(z)^2 \\, dz = \\frac{1}{2} \\int_{a}^{b} \\begin{bmatrix}\n\t\tp(z) & \\epsilon(z)\n\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\t\\frac{1}{(\\rho A)(z)} & 0 \\\\\n\t\t\t0 & T(z)\n\\end{bmatrix} \\begin{bmatrix}\np(z) \\\\\n\\epsilon(z)\n\\end{bmatrix} \\, dz.\n\t\\label{Subsection The Vibrating String - Energy}\n\\end{align}\nAgain, we may define the coercive multiplication operator $\\mathcal{Q} \\in \\mathcal{L}(X)$ given by\n\\begin{align*}\n\t(\\mathcal{Q}x)(z) = \\mathcal{Q}(z) x(z) = \\begin{bmatrix}\n\t\t\\frac{1}{(\\rho A)(z)} & 0 \\\\\n\t\t0 & T(z)\n\t\\end{bmatrix} \\begin{bmatrix}\n\tp(z) \\\\\n\t\\epsilon(z)\n\\end{bmatrix}, \\hspace{0.5 cm} x \\in X, \\, z \\in [a,b].\n\\end{align*}\nAdditionally, as the Hamiltonian is once again of the form (\\ref{Subsection Lossless Transmission Line - Standard Hamiltonian Form}), the variational derivative of the Hamiltonian $H$ equals the coercive matrix operator $\\mathcal{Q}$:\n\\begin{align*}\n\t\\frac{\\delta H}{\\delta x}(x) = \\begin{bmatrix}\n\t\t\\frac{\\delta H}{\\delta p}(p,\\epsilon) \\\\\n\t\t\\frac{\\delta H}{\\delta \\epsilon}(p,\\epsilon)\n\t\\end{bmatrix} = \\begin{bmatrix} \n\t\\frac{p}{\\rho A} \\\\\nT\\epsilon\n\\end{bmatrix} = \\begin{bmatrix}\n\\frac{\\partial w}{\\partial t} \\\\\nT \\frac{\\partial w}{\\partial z}\n\\end{bmatrix}, \\hspace{0.5 cm} x \\in X. \n\\end{align*}\nAltogether, we may equivalently rewrite equation (\\ref{Example Vibrating String - Vibrating String Equation}) as an infinite-dimensional, first-order Hamiltonian system of the form\n\\begin{align}\n\t\\frac{\\partial}{\\partial t} \\begin{bmatrix}\n\t\t\tp \\\\\n\t\t\t\\epsilon\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\t\t0 & \\frac{\\partial}{\\partial z} \\\\\n\t\t\t\\frac{\\partial }{\\partial z} & 0 \n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\t\\frac{\\delta H}{\\delta p}(p,\\epsilon) \\\\\n\t\t\t\\frac{\\delta H}{\\delta \\epsilon}(p,\\epsilon)\n\t\\end{bmatrix} \n\t\\label{Subsection The Vibrating String - Vibrating String Equation in pH form}\n\\end{align}\nwith respect to the (Hamiltonian) matrix differential operator $\\mathcal{J} \\colon D(\\mathcal{J}) \\subset X \\to X$ given by\n\\begin{align*}\n\tD(\\mathcal{J}) &= H^1([a,b], \\mathbb{R}^2), \\\\\n\t\\mathcal{J}x &= \\begin{bmatrix}\n\t\t0 & \\frac{d}{dz} \\\\\n\t\t\\frac{d}{dz} & 0 \n\t\\end{bmatrix}x, \\hspace{0.5 cm} x\\in D(\\mathcal{J}),\n\\end{align*}\nand with respect to the Hamiltonian $H$ given in \\eqref{Subsection The Vibrating String - Energy}. The proof that $\\mathcal{J}$ is indeed a Hamiltonian operator (see again \\cite[Definition 7.1]{Olver}) works exactly as in the previous example. For a classical solution $x \\colon [0, \\infty) \\to X$ of \\eqref{Subsection The Vibrating String - Vibrating String Equation in pH form}, the energy-balance equation of the system \\eqref{Subsection The Vibrating String - Vibrating String Equation in pH form} is given by\n\\begin{align}\n\t\\frac{d}{dt} H(x(\\cdot,t)) = \\left[ \\frac{p(z,t)}{(\\rho A)(z)} T(z) \\epsilon(z,t) \\right]_{a}^{b},\n\t\\label{Subsection The Vibrating String - Balance Equation}\n\\end{align}\nwhere the product of the co-energy variables, namely the velocity $\\frac{\\delta H}{\\delta p}(p,\\epsilon) = \\frac{p}{\\rho A}$ and the stress $\\frac{\\delta H}{\\delta \\epsilon}(p,\\epsilon) = T \\epsilon$, again equals power. In our particular scenario, the ends of the string are both fixed, that is, the velocity at the boundary is zero. In particular, $p(a,t) = p(b,t) = 0$, $t \\geq 0$, which is why in this case, the energy is conserved. However, if at least one end of the string may move freely or some force may be applied to it, the power flow through the boundary has to be encompassed, as depicted in \\eqref{Subsection The Vibrating String - Balance Equation}.\n\\begin{remark}\n\t\\label{Remark Skew-Symmetry J and Operator Q}\nWe note that the model \\eqref{Subsection The Vibrating String - Vibrating String Equation in pH form} of this system has the same structure as the model \\eqref{Subsection Lossless Transmission Line - pH System} of the lossless transmission line: the multiplication operator $ \\mathcal{Q} \\in \\mathcal{L}(X)$ is coercive and represents the mechanical and physical properties of the system. Furthermore, the Hamiltonian operators can be written as $\\mathcal{J} = P_1 \\frac{d}{dz}$, where the matrix $P_1 \\in \\mathbb{R}^{2 \\times 2}$ is symmetric (and invertible). Given the vibrating string example, the formal skew-symmetry of the differential operator $\\mathcal{J}$ implies that the kinetic energy is transformed into elastic energy and vice versa. Hence, the internally stored energy is conserved in the absence of any power flow through the boundary. The formally skew-symmetric differential operators can be regarded as the infinite-dimensional counterpart of the skew-symmetric matrices in the finite-dimensional case.\n\\end{remark}\nJust as in the finite-dimensional case there may also occur (internal) energy dissipation in infinite-dimensional systems. We refer to \\cite[Subsection 4.1.3]{ModCompPhys} and \\cite[Section 1.4]{Villegas} and the references mentioned there. In the sequel, we will generalize the systems considered in Section \\ref{Section Motivating Examples} and only study idealized systems where any dissipation phenomena are neglected.\n\n\n\n\n\n\n\\section{Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}\n\\label{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}\nWe infer from the preceding examples that it is possible to describe a variety of PDEs with the following structure:\n\\begin{align}\n\t\\begin{split}\n\t\t\\frac{\\partial x}{\\partial t} (z,t) &= \\mathcal{J} (\\mathcal{Q}(z)x(z,t)), \\hspace{0.5 cm} t > 0, \\\\\n\t\tx(z,0) &= x_0(z)\n\t\\end{split}\n\t\\label{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case}\n\\end{align}\non the state space $X = L^2 ([a,b], \\mathbb{R}^n)$, where $ \\frac{\\delta}{\\delta x} H =\\mathcal{Q} \\in \\mathcal{L}(X)$ is the variational derivative of the Hamiltonian $H \\colon X \\to \\mathbb{R}$ given by\n\\begin{align}\n\tH (x) = \\int_{a}^{b} \\mathcal{H}(x(z)) \\, dz = \\frac{1}{2} \\int_{a}^{b} x^{\\top}(z) (\\mathcal{Q} x)(z) \\, dz,\n\t\\label{General Lossless Port-Hamiltonian System - Hamiltonian H}\n\\end{align} \nwhere $\\mathcal{H} \\in \\mathcal{C}^{\\infty}(\\mathbb{R}^n, \\mathbb{R})$ is the Hamiltonian density. Furthermore, the (multiplication) operator $\\mathcal{Q}$ is assumed to be coercive.\nLet the differential operator $\\mathcal{J} \\colon D(\\mathcal{J}) \\subset X \\to X$ be given by\n\\begin{align}\n\t\\begin{split}\n\tD(\\mathcal{J}) &= H^N \\left( [a,b], \\mathbb{R}^n \\right), \\\\\n\t\\mathcal{J} x &= \\sum_{i=0}^{N} P_i \\frac{d^i x}{d z^i}, \\hspace{0.5 cm} x \\in D(\\mathcal{J}),\n\t\\end{split}\n\t\\label{Skew-Symmetric Differential Operator J - N-Dimensional}\n\\end{align}\n with $P_i \\in \\mathbb{R}^{n \\times n}$, $i = 0,\\ldots,N$, where $P_N$ is supposed to be invertible. Furthermore, we assume that\n\\begin{align}\n\tP_i = (-1)^{i+1} P_i^{\\top}, \\hspace{0.5 cm} i =0,\\ldots,N,\n\t\\label{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}\n\\end{align} \nthat is, the constituting matrices of $\\mathcal{J}$ are skew-symmetric and symmetric in an alternating fashion. We will see that condition (\\ref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}) implies that the differential operator $\\mathcal{J}$ is formally skew-symmetric. Recall from Remark \\ref{Remark Skew-Symmetry J and Operator Q} that the formal skew-symmetry corresponds to an energy-conserving exchange of energy between different physical domains. Finally, following Remark \\ref{Remark Energy Space}, we endow the state space $X$ with the inner product $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{Q}}$ defined in (\\ref{Subsection The Lossless Transmission Line - Inner Product WRT Q}), so that the Hamiltonian $H$ can be rewritten as in (\\ref{Subsection The Lossless Transmission Line - Representation Hamiltonian WRT Q norm}).\n\\vspace{0.5 cm}\\\\\nIn this section, we want to define the Dirac structure which is associated with systems of the form \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case}. As the examples in Section \\ref{Section Motivating Examples} suggested, we need to introduce boundary port variables to encompass the power flow through the boundary as well. In Subsection~\\ref{Subsection Boundary Port Variables} we will define this new type of port variables by means of suitably chosen matrices and operators, and give a representation of the evaluation of the effort variables at the spatial boundary with respect to the boundary port variables. Subsection \\ref{Subsection Dirac Structure and Boundary Port-Hamiltonian Systems} deals with the right choice of the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ and a suitable power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ constituting the bilinear form (the plus pairing) $\\ll \\cdot, \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ in order to define the underlying Stokes-Dirac structure associated with infinite-dimensional linear systems of the form \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case}, leading to the notion of boundary port-Hamiltonian systems. \n\n\n\\subsection{Boundary Port Variables}\n\\label{Subsection Boundary Port Variables}\nLet us, to begin with, show that the differential operator $\\mathcal{J}$ defined by \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices} is indeed formally skew-symmetric. \n\\begin{theorem}[Theorem 2.1 in \\cite{Villegas}]\n\t\\label{Theorem Stokes-Like Theorem}\n\tLet $\\mathcal{J}$ be the differential operator defined in \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}. Then for any two functions $e_1,e_2 \\in D(\\mathcal{J})$, we have\n\t\\begin{align}\n\t\\langle \\mathcal{J} e_1 , e_2 \\rangle_{L^2} + \\langle e_1 , \\mathcal{J} e_2 \\rangle_{L^2} = \\left[ \\begin{bmatrix}\n\t\t\te_1^{\\top}(z), \\ldots, \\frac{d^{N-1} e_1^{\\top}}{d z^{N-1}} (z)\n\t\t\\end{bmatrix} \\mathcal{P} \\begin{bmatrix}\n\t\te_2(z) \\\\\n\t\t\\vdots \\\\\n\t\t\\frac{d^{N-1} e_2}{d z^{N-1}}(z)\n\t\\end{bmatrix} \\right]_{a}^{b},\n\\label{Theorem Stokes-Like Theorem - Equation}\n\t\\end{align}\nwith the symmetric matrix $\\mathcal{P} \\in \\mathbb{R}^{nN \\times nN}$ given by\n\\begin{align}\n\t\\mathcal{P} = \\begin{bmatrix}\n\t\tP_1 & P_2 & P_3 & \\cdots & P_{N-1} & P_N \\\\\n\t\t-P_2 & -P_3 & - P_4 & \\cdots & - P_{N} & 0 \\\\\n\t\tP_3 & P_4 & \\ddots & \\ddots & 0 & 0 \\\\\n\t\t\\vdots & \\ddots & \\ddots & \\ddots & & \\vdots \\\\\n\t\t(-1)^{N-1}P_N & 0 & \\cdots & \\cdots & & 0 \n\t\\end{bmatrix}.\n\\label{Matrix P - Distributed-Parameter}\n\\end{align} \nIn particular, $\\mathcal{J}$ is formally skew-symmetric.\n\\end{theorem}\n\\begin{proof}\n\tThe proof is rather tedious. The claim basically follows from iteratively applying integration by parts. We refer to \\cite[Theorem 3.1]{Memorandum}.\n\\end{proof}\nIn order to define a power-conserving interconnection structure in the infinite-dimensional case, we need to incorporate the evaluation of the effort variables at the spatial boundary into the definition of the respective power pairing, which we will specify later on. To this end, we will write the right-hand side of \\eqref{Theorem Stokes-Like Theorem - Equation} subject to the boundary port variables. Note that these port variables will depend on the constituting matrices of $\\mathcal{J}$, which are captured in the matrix $\\mathcal{P}$ defined in \\eqref{Matrix P - Distributed-Parameter}. Since we suppose that the matrix $P_N$ is invertible, it is easy to check that $\\mathcal{P}$ is invertible as well. \n\\vspace{0.5 cm}\\\\\nIn order to define the boundary port variables, we need to introduce further matrices. \nFirst, let the matrix $\\mathcal{P}_{\\extern} \\in \\mathbb{R}^{2nN \\times 2nN}$ associated with the differential operator $\\mathcal{J}$ be defined as\n\t\\begin{align*}\n\t\t\\mathcal{P}_{\\extern} = \\begin{bmatrix}\n\t\t\t\\mathcal{P} & 0 \\\\\n\t\t\t0 & -\\mathcal{P}\n\t\t\t\\end{bmatrix},\n\t\\end{align*}\nwhere $\\mathcal{P}$ is the symmetric and invertible matrix from (\\ref{Matrix P - Distributed-Parameter}). Next, we factorize $\\mathcal{P}_{\\extern}$ in such a way that it allows us to define the port variables, and simultaneously guarantees that the bilinear form - which has yet to be defined - becomes independent of the (matrix) coefficients of the operator $\\mathcal{J}$, as we will see shortly.\n\\begin{lemma}[Lemma 2.4 in \\cite{Villegas}]\n\t\\label{Lemma Auxillary Lemma Port-Boundary Variables}\n\tThe matrix $R_{\\extern} \\in \\mathbb{R}^{2nN \\times 2nN}$ defined as\n\t\\begin{align}\n\t\tR_{\\extern} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n\t\t\t\\mathcal{P} & -\\mathcal{P} \\\\\n\t\t\tI & I \n\t\t\\end{bmatrix}\n\t\\label{Matrix Rext}\n\t\\end{align}\nis invertible and satisfies\n\\begin{align*}\n\t\\mathcal{P}_{\\extern} = R_{\\extern}^{\\top} \\Sigma R_{\\extern},\n\n\\end{align*}\nwhere\n\\begin{align}\n\t\\label{Matrix Sigma}\n\t\\Sigma = \\begin{bmatrix}\n\t\t0 & I \\\\\n\t\tI & 0\n\t\\end{bmatrix} \\in \\mathbb{R}^{2nN \\times 2nN}.\n\t\\end{align}\nFurthermore, all possible matrices $R \\in \\mathbb{R}^{2nN \\times 2nN}$ satisfying $\\mathcal{P}_{\\extern} = R^{\\top} \\Sigma R$ are of the form\n\\begin{align*}\n\tR = U R_{\\extern},\n\\end{align*}\nwith $U \\in \\mathbb{R}^{2nN \\times 2nN}$ satisfying $U^{\\top} \\Sigma U = \\Sigma$. \n\\end{lemma}\nThe proof of Lemma \\ref{Lemma Auxillary Lemma Port-Boundary Variables} is fairly easy. For details we refer to the proof of \\cite[Lemma 3.4]{LeGorrec}.\n\\vspace{0.5 cm}\\\\\nThe essential part of defining the Dirac structure associated with the operator $\\mathcal{J}$ is to introduce the boundary port variables, which we are finally able to do.\n\n\n\\begin{definition}\n\\label{Definition Boundary Port Variables in The Linear Case}\nThe boundary port variables associated with the differential operator $\\mathcal{J}$ given by \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices} are the vectors $f_{\\partial}, e_{\\partial} \\in \\mathbb{R}^{nN}$ defined by \n\\begin{align}\n\t\\label{Boundary Flow and Effort Linear Case} \n\t\\begin{bmatrix}\n\t\tf_{\\partial,e} \\\\ \n\t\te_{\\partial,e} \n\t\\end{bmatrix} := R_{\\extern} \\trace (e), \\hspace{0.5 cm} e \\in D(\\mathcal{J}),\n\\end{align}\nwhere $\\trace \\colon H^N([a,b], \\mathbb{R}^n) \\to \\mathbb{R}^{2nN}$ is the boundary trace operator defined in \\eqref{Trace Operator - Chapter 3}.\n\\end{definition}\nWe usually write $\\begin{bmatrix}\n\tf_{\\partial} \\\\\n\te_{\\partial}\n\\end{bmatrix}$ instead of $\\begin{bmatrix}\n\tf_{\\partial, e} \\\\\n\te_{\\partial, e}\n\\end{bmatrix}$ whenever the dependence of the boundary port variables on the variable $e \\in D(\\mathcal{J})$ is obvious. We call $f_{\\partial} \\in \\mathbb{R}^{nN}$ and $e_{\\partial} \\in \\mathbb{R}^{nN}$ the \\emph{boundary flow} and the \\emph{boundary effort}, respectively. \n\\vspace{0.5 cm}\\\\\nUsing Definition \\ref{Definition Boundary Port Variables in The Linear Case} and Lemma \\ref{Lemma Auxillary Lemma Port-Boundary Variables}, we can rewrite the right-hand side of \\eqref{Theorem Stokes-Like Theorem - Equation} in terms of the boundary port variables in the following way: for $e_1,e_2 \\in H^N \\left( [a,b], \\mathbb{R}^n \\right)$ we have\n\\begin{align*}\n\\begin{bmatrix}\n\t\tf_{\\partial,e_1}^{\\top} & e_{\\partial, e_1}^{\\top}\n\t\\end{bmatrix} \\Sigma \\begin{bmatrix}\n\t\tf_{\\partial, e_2} \\\\\n\t\te_{\\partial, e_2}\n\t\\end{bmatrix} &= \\trace(e_1)^{\\top} R_{\\extern}^{\\top} \\Sigma R_{\\extern} \\trace(e_2) \\\\\n&= \\trace(e_1)^{\\top} \\mathcal{P}_{\\extern} \\trace(e_2) \\\\\n&= \\trace(e_1)^{\\top} \\begin{bmatrix}\n\\mathcal{P} & 0 \\\\\n0 & -\\mathcal{P}\n\\end{bmatrix} \\trace(e_2) \\\\\n&= \\left[ \\begin{bmatrix}\n\te_1^{\\top}(z) & \\cdots & \\frac{d^{N-1} e_1^{\\top}}{d z^{N-1}} (z)\n\\end{bmatrix} \\mathcal{P} \\begin{bmatrix}\n\te_2(z) \\\\\n\t\\vdots \\\\\n\t\\frac{d^{N-1} e_2}{d z^{N-1}}(z)\n\\end{bmatrix} \\right]_{a}^{b}.\n\\end{align*} \nThus, for all $e_1, e_2 \\in D(\\mathcal{J})$, equation (\\ref{Theorem Stokes-Like Theorem - Equation}) becomes\n\\begin{align}\n\t\\begin{split}\n\t\t\\langle \\mathcal{J}e_1 , e_2 \\rangle_{L^2} + \\langle e_1, \\mathcal{J} e_2 \\rangle_{L^2} &= \\begin{bmatrix}\n\t\t\tf_{\\partial,e_1}^{\\top} & e_{\\partial, e_1}^{\\top}\n\t\t\\end{bmatrix} \\Sigma \\begin{bmatrix}\n\t\tf_{\\partial, e_2} \\\\\n\t\te_{\\partial, e_2}\n\t\\end{bmatrix} \\\\\n&= f_{\\partial, e_1}^{\\top} e_{\\partial, e_2} + e_{\\partial, e_1}^{\\top} f_{\\partial, e_2}. \n\\end{split}\n\\label{Theorem Stokes-Like Theorem - Equation Representation 2}\n\\end{align}\n\nNow, we are able to define the Dirac structure on a suitable bond space corresponding to the formally skew-symmetric operator $\\mathcal{J}$.\n\n\\subsection{Dirac Structure and Boundary Port-Hamiltonian Systems}\n\\label{Subsection Dirac Structure and Boundary Port-Hamiltonian Systems}\nWe have already discussed that in the finite-dimensional case the space of flows $\\mathcal{F}$ and the space of efforts $\\mathcal{E}$ correspond to a finite-dimensional vector space and its dual space, with the power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ of an element of the flow space and of an element of the effort space being equal to physical power (see Section \\ref{Section Dirac Structures}). In the present case the flow space $\\mathcal{F}$ is now an infinite-dimensional Hilbert space, and the effort space $\\mathcal{E}$ is again to be defined as the dual space of the space of flows, $\\mathcal{E} = \\mathcal{F}'$. Throughout, we will identify $\\mathcal{F}$ with its dual $\\mathcal{E}$. \n\\vspace{0.5 cm}\\\\\nIn order to define a Dirac structure, we need to endow the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ not with the canonical inner product $\\langle \\cdot , \\cdot \\rangle_{\\mathcal{B}}$, but with the plus pairing $\\ll \\cdot, \\cdot \\gg$ defined in \\eqref{Plus Pairing}, i.e., \n\\begin{align}\n\t\\ll (f^1,e^1), (f^2, e^2) \\gg \\hspace{0.1 cm}= \\langle e^1 \\mid f^2 \\rangle + \\langle e^2 \\mid f^1 \\rangle, \\hspace{0.5 cm} ((f^1, e^1), (f^2, e^2)) \\in \\mathcal{B} \\times \\mathcal{B}.\n\t\\label{Plus Pairing - Distributed-Parameter}\n\\end{align}\nThis symmetric bilinear form depends on the choice of the power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$, which we will, together with the flow and effort spaces, specify shortly. \n\\vspace{0.5 cm}\\\\\nWe may now define the notion of a Dirac structure on an infinite-dimensional Hilbert space.\n\\begin{definition}\n\t\\label{Definition Dirac Structure - Infinite-dimensional Case}\n\tLet the flow space $\\mathcal{F}$ be an infinite-dimensional Hilbert space, and let the effort space $\\mathcal{E} = \\mathcal{F}'$ be its dual space. A linear subspace $\\mathcal{D} \\subset \\mathcal{B}$ of the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ is called a Dirac structure if \n\t\\begin{align*}\n\t\t\\mathcal{D} = \\mathcal{D}^{\\perp \\!\\!\\! \\perp}, \n\t\\end{align*}\n\twhere $\\mathcal{D}^{\\perp \\!\\!\\! \\perp}$ denotes the orthogonal complement of $\\mathcal{D}$ with respect to the plus pairing $\\ll \\cdot, \\cdot \\gg$ defind in \\eqref{Plus Pairing - Distributed-Parameter}.\n\\end{definition}\nNote that this is precisely the characterization of a Dirac structure in the finite-dimensional case (cf. Proposition \\ref{Proposition Characterization Dirac Structure}), and that this implies that the Dirac structure $\\mathcal{D}$ captures the power-conserving interconnection structure of a system. For a general definition of an infinite-dimensional port-Hamiltonian system with total energy \\eqref{General Lossless Port-Hamiltonian System - Hamiltonian H}, see \\cite[Definition~1.13]{Villegas}. \n\\vspace{0.5 cm}\\\\\nIn Subsection \\ref{Subsection Boundary Port Variables} we have defined boundary port variables which we need to incorporate into the definition of the power pairing and thus the plus pairing. By Definition \\ref{Definition Dirac Structure - Infinite-dimensional Case}, this is necessary for defining the Dirac structure associated with $\\mathcal{J}$ defined by \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}. First, we need to select the flow space and the effort space appropriately. Because the bilinear form contains both state space elements and their evaluation at the boundary, we set\n\\begin{align}\n\t\\mathcal{F} = \\mathcal{E} = L^2 \\left( [a,b], \\mathbb{R}^n \\right) \\times \\mathbb{R}^{nN},\n\t\\label{Flow And Effort Space - Linear Infinite-dimensional Systems}\n\\end{align}\nendowed with the canonical inner product:\n\\begin{align*}\n\t\\langle (f^1,f_{\\partial}^1), (f^2,f_{\\partial}^2) \\rangle_{\\mathcal{F}} = \\langle f^1, f^2 \\rangle_{L^2} + \\langle f_{\\partial}^1, f_{\\partial}^2 \\rangle_2, \\hspace{0.5 cm} (f^1, f_{\\partial}^1), (f^2,f_{\\partial}^2) \\in \\mathcal{F}.\n\\end{align*}\nIn contrast to the finite-dimensional case, the power pairing does not necessarily coincide with the canonical duality pairing\/inner product, but is slightly modified. Here, we choose the power pairing\n\\begin{align}\n\t\\left\\langle \\begin{pmatrix}\n\t\te \\\\\n\t\te_{\\partial}\n\t\\end{pmatrix} \\Bigg| \\begin{pmatrix}\n\tf \\\\\n\tf_{\\partial}\n\\end{pmatrix} \\right\\rangle = \\langle e , f \\rangle_{L^2} - \\langle e_{\\partial}, f_{\\partial} \\rangle_2, \\hspace{0.5 cm} \\left( \\begin{pmatrix}\ne \\\\\ne_{\\partial}\n\\end{pmatrix}, \\begin{pmatrix}\nf \\\\\nf_{\\partial}\n\\end{pmatrix} \\right) \\in \\mathcal{E} \\times \\mathcal{F}. \n\\label{Power Pairing - Infinite-Dimensional Case}\n\\end{align}\nThe reason behind that modification will be clear soon. Altogether, we endow the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ with the following plus pairing:\n\\begin{align}\n\t\\ll (f^1, f_{\\partial}^1, e^1, e_{\\partial}^1), (f^2, f_{\\partial}^2, e^2, e_{\\partial}^2) \\gg \\hspace{0.1 cm} := \\langle e^2, f^1 \\rangle_{L^2} - \\langle e_{\\partial}^2, f_{\\partial}^1 \\rangle_2 + \\langle e^1, f^2 \\rangle_{L^2} - \\langle e_{\\partial}^1, f_{\\partial}^2 \\rangle_2, \n\t\\label{Plus Pairing - Infinite-Dimensional Case}\n\\end{align}\nwhere $(f^i, f_{\\partial}^i, e^i, e_{\\partial}^i) \\in \\mathcal{B}$, $i=1,2$.\n\\vspace{0.5 cm}\\\\\nNow we have everything required to define the Dirac structure associated with the formally skew-symmetric operator $\\mathcal{J}$ given by (\\ref{Skew-Symmetric Differential Operator J - N-Dimensional})-(\\ref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}).\n\\begin{theorem}[Theorem 2.7 in \\cite{Villegas}]\n\t\\label{Theorem Dirac Structure Associated With Skew-Symmetric Operator J}\n\tConsider the formally skew-symmetric operator\n\t$\\mathcal{J}$ given by \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}. Let $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ be the bond space with respect to the flow space $\\mathcal{F}$ and the effort space $\\mathcal{E}$ defined in \\eqref{Flow And Effort Space - Linear Infinite-dimensional Systems}. Then the subspace $\\mathcal{D}_{\\mathcal{J}} \\subset \\mathcal{B}$ given by\n\t\\begin{align}\n\t\t\\mathcal{D}_{\\mathcal{J}} := \\left\\{ \\left( \\begin{pmatrix}\n\t\t\tf \\\\\n\t\t\tf_{\\partial} \\end{pmatrix},\n\t\t\\begin{pmatrix}\n\t\t\te \\\\\n\t\t\te_{\\partial}\n\t\t\\end{pmatrix} \\right) \\in \\mathcal{B} \\hspace{0.1 cm} \\Bigg| \\hspace{0.1 cm} e \\in H^N \\left( [a,b], \\mathbb{R}^n \\right), \\hspace{0.1 cm} f = \\mathcal{J}e , \\hspace{0.1 cm} \\begin{bmatrix}\n\t\tf_{\\partial, e} \\\\\n\t\te_{\\partial, e}\n\t\\end{bmatrix} = R_{\\extern} \\trace (e) \\right\\}\n\\label{Stokes-Dirac structure - Linear PH Systems}\n\t\\end{align}\nis a Dirac structure with respect to the plus pairing\n$\\ll\\cdot,\\cdot\\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ defined in \\eqref{Plus Pairing - Infinite-Dimensional Case}, and is called the Stokes-Dirac structure.\n\\end{theorem}\nFor the proof one has to verify that $\\mathcal{D}_{\\mathcal{J}} = \\mathcal{D}_{\\mathcal{J}}^{\\perp \\!\\!\\! \\perp}$ holds with respect to the plus pairing $\\ll \\cdot, \\cdot \\gg$, according to Definition \\ref{Definition Dirac Structure - Infinite-dimensional Case}. We refer to the proof of Theorem 3.6 in \\cite{LeGorrec}.\n\\vspace{0.5 cm}\\\\\nThe Stokes-Dirac structure $\\mathcal{D}_{\\mathcal{J}}$ is the underlying Dirac structure of the class of systems of the form \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case}. Consequently, and analogously to the finite-dimensional case, the dynamics of these systems can be geometrically specified with respect to $\\mathcal{D}_{\\mathcal{J}}$, which we are going to do next.\n\\vspace{0.5 cm}\\\\\nBeforehand, we explain the reason behind the specific choice of the power pairing \\eqref{Power Pairing - Infinite-Dimensional Case}: assume that\n$(f^i, f_{\\partial}^i , e^i, e_{\\partial}^i) \\in \\mathcal{D}_{\\mathcal{J}}$ for $i = 1, 2$. As $\\mathcal{D} = \\mathcal{D}^{\\perp \\!\\!\\! \\perp}$, we get\n\\begin{align*}\n\t0 &= \\hspace{0.1 cm} \\ll (f^1, f_{\\partial}^1, e^1, e_{\\partial}^1), (f^2, f_{\\partial}^2, e^2, e_{\\partial}^2) \\gg \\\\\n\t&= \\hspace{0.1 cm}\t\\ll (\\mathcal{J}e^1, f_{\\partial}^1, e^1, e_{\\partial}^1), (\\mathcal{J}e^2, f_{\\partial}^2, e^2, e_{\\partial}^2) \\gg \\\\\n\t&= \\langle \\mathcal{J}e^1, e^2 \\rangle_{L^2} + \\langle e^1, \\mathcal{J}e^2 \\rangle_{L^2} - \\langle e_{\\partial}^2, f_{\\partial}^1 \\rangle_2 - \\langle e_{\\partial}^1, f_{\\partial}^2 \\rangle_2 \\\\\n\t&= \\langle \\mathcal{J}e^1, e^2 \\rangle_{L^2} + \\langle e^1, \\mathcal{J}e^2 \\rangle_{L^2} - \\begin{bmatrix}\n\t\tf_{\\partial,e^1}^{\\top} & e_{\\partial, e^1}^{\\top}\n\t\\end{bmatrix} \\Sigma \\begin{bmatrix}\n\t\tf_{\\partial, e^2} \\\\\n\t\te_{\\partial, e^2}\n\t\\end{bmatrix},\n\\end{align*}\nwith $\\Sigma$ defined in \\eqref{Matrix Sigma}.\nThis is exactly equation \\eqref{Theorem Stokes-Like Theorem - Equation Representation 2}, which, in particular, has been exploited in the proof of Theorem \\ref{Theorem Dirac Structure Associated With Skew-Symmetric Operator J}. Now, comparing the system description \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case} and the constitutive relations \\eqref{Stokes-Dirac structure - Linear PH Systems} of $\\mathcal{D}_{\\mathcal{J}}$, we infer:\n\\begin{enumerate}\n\t\\item[(i)] Define the temporal change of the state variables as the flow variables, i.e., $f = \\frac{\\partial x}{\\partial t}$. \n\t\\item[(ii)] Define the variational derivative of $H$ as the effort variables, $e = \\frac{\\delta H}{\\delta x}(x) = \\mathcal{Q}x$. \n\t\\item[(iii)] Encompass the two external boundary port variables $f_{\\partial,e}, e_{\\partial,e}$ defined with respect to the effort variables $e$, see Definition \\ref{Definition Boundary Port Variables in The Linear Case}.\n\\end{enumerate}\nThus, we are able to specify the dynamics of the system \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case} with respect to the Stokes-Dirac structure $\\mathcal{D}_{\\mathcal{J}}$ in \\eqref{Stokes-Dirac structure - Linear PH Systems}. By doing so, the power flow through the boundary of the spatial domain is encompassed, which is why we call such systems \\emph{boundary port-Hamiltonian systems}. \n\\begin{corollary}\n\tConsider the Hamiltonian system \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case} with Hamiltonian \\eqref{General Lossless Port-Hamiltonian System - Hamiltonian H}, which is augmented with the boundary port variables given in Definition \\ref{Definition Boundary Port Variables in The Linear Case}. The dynamics of the resulting boundary port-Hamiltonian system are geometrically specified by the requirement that for all $t >0$ we have\n\t\\begin{align*}\n\t\t\\left( \\begin{pmatrix}\n\t\t\t\\frac{\\partial}{\\partial t}x(\\cdot,t) \\\\\n\t\t\tf_{\\partial}(t) \n\t\t\\end{pmatrix}, \\begin{pmatrix}\n\t\t\\mathcal{Q}x(\\cdot,t) \\\\\n\t\te_{\\partial}(t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\mathcal{J}},\n\t\\end{align*}\nwith $f_{\\partial}(t) = f_{\\partial, \\mathcal{Q}x(\\cdot,t)}$, $e_{\\partial}(t) = e_{\\partial, \\mathcal{Q}x(\\cdot,t)}$, and $\\mathcal{D}_{\\mathcal{J}}$ the Stokes-Dirac structure \\eqref{Stokes-Dirac structure - Linear PH Systems} defined with respect to the plus pairing \\eqref{Plus Pairing - Infinite-Dimensional Case}. \n\\end{corollary}\nRecall that the energy space $X$ is endowed with the inner product $\\langle \\cdot , \\cdot \\rangle_{\\mathcal{Q}}$ defined in \\eqref{Subsection The Lossless Transmission Line - Inner Product WRT Q}. Thus, for a classical solution $x \\colon [0, \\infty) \\to X$ of the system \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case} with the Hamiltonian $H \\colon X \\to \\mathbb{R}$ given in \\eqref{General Lossless Port-Hamiltonian System - Hamiltonian H} we obtain the following balance equation with respect to the boundary port variables: applying \\eqref{Theorem Stokes-Like Theorem - Equation Representation 2}, we have for all $t \\geq 0$,\n\\begin{align}\n\t\\begin{split}\n\t\\frac{d}{dt} \\|x(\\cdot,t) \\|_{\\mathcal{Q}}^2 = \\frac{d}{dt} H(x(\\cdot,t)) &= \\frac{1}{2} \\frac{d}{dt} \\langle x(\\cdot,t), \\mathcal{Q}x(\\cdot,t) \\rangle_{L^2} \\\\\n\t&= \\frac{1}{2} \\langle \\mathcal{J}\\mathcal{Q}x(\\cdot,t), \\mathcal{Q}x(\\cdot,t) \\rangle_{L^2} + \\frac{1}{2} \\langle \\mathcal{Q}x(\\cdot,t), \\mathcal{J}\\mathcal{Q}x(\\cdot,t) \\rangle_{L^2} \\\\\n\t&= \\frac{1}{2} \\begin{bmatrix} \n\t\tf_{\\partial}^{\\top}(t) & e_{\\partial}^{\\top}(t)\n\t\\end{bmatrix}\n\\Sigma \\begin{bmatrix}\n\tf_{\\partial}(t) \\\\\n\te_{\\partial}(t)\n\\end{bmatrix} \\\\\n&= \\langle e_{\\partial}(t), f_{\\partial}(t) \\rangle_2. \n\\end{split}\n\\label{Power Balance for General Linear PH-Systems - Infinite Dimensions}\n\\end{align}\nThis equation expresses the fact that the change of internal power solely happens at the boundary of the spatial domain, and therefore generalizes the balance equations \\eqref{Subsection Lossless Transmission Line - Energy Balance Equation} and \\eqref{Subsection The Vibrating String - Balance Equation} we have derived in the introductory section of this chapter. \n\\begin{remark}\n\tIn Section \\ref{Section Energy-storing Elements} we pointed out that, in order to distinguish between the power flowing into the energy-storing elements and the power flowing into the Dirac structure, we need to put a minus sign in front of the rate of change of the state $x$, see \\eqref{Choice of Port Variables fS and eS}. In the infinite-dimensional case, the power flowing into the energy-storing elements would correspond to the inner product $\\langle \\frac{\\delta H}{\\delta x}(x), \\frac{\\partial x}{\\partial t} \\rangle_{L^2}$, and the power flow into the Dirac structure would correspond to the inner product $\\langle e_S, f_S \\rangle_{L^2}$, leading to the relations $f = f_S = - \\frac{\\partial x}{\\partial t}$ and $e = e_S =\\frac{\\delta H}{\\delta x}(x)$. Necessarily, some minor changes have to be made concerning the Dirac structure and the plus pairing. To be more accurate, the plus pairing in \\eqref{Plus Pairing - Infinite-Dimensional Case} would need to be changed to \n\t\\begin{align*}\n\t\t\\ll (f^1, f_{\\partial}^1, e^1, e_{\\partial}^1), (f^2, f_{\\partial}^2, e^2, e_{\\partial}^2) \\gg \\hspace{0.1 cm} := \\langle f^1, e^2 \\rangle_{L^2} + \\langle f_{\\partial}^1, e_{\\partial}^2 \\rangle_2 + \\langle e^1, f^2 \\rangle_{L^2} + \\langle e_{\\partial}^1, f_{\\partial}^2 \\rangle_2, \n\t\\end{align*}\nthat is, the plus pairing is induced by the canonical power pairing $\\langle \\cdot \\mid \\cdot \\rangle$ on the Hilbert space $\\mathcal{F} = \\mathcal{E}$, and not by the one defined in \\eqref{Power Pairing - Infinite-Dimensional Case}. Moreover, the Stokes-Dirac structure given in \\eqref{Stokes-Dirac structure - Linear PH Systems} would be of the form\n\t\\begin{align*}\n\t\\mathcal{D}_{-\\mathcal{J}} = \\left\\{ \\left( \\begin{pmatrix}\n\t\tf \\\\\n\t\tf_{\\partial} \\end{pmatrix},\n\t\\begin{pmatrix}\n\t\te \\\\\n\t\te_{\\partial}\n\t\\end{pmatrix} \\right) \\in \\mathcal{B} \\hspace{0.1 cm} \\Bigg| \\hspace{0.1 cm} e \\in H^N \\left( [a,b], \\mathbb{R}^n \\right), \\hspace{0.1 cm} - f = \\mathcal{J}e , \\hspace{0.1 cm} \\begin{bmatrix}\n\t\tf_{\\partial, e} \\\\\n\t\te_{\\partial, e}\n\t\\end{bmatrix} = R_{\\extern} \\trace (e) \\right\\}.\n\\end{align*}\nWith these simple adjustments one could continue the aforementioned power flow convention from Chapter \\ref{Chapter Finite-dimensional Port-Hamiltonian Systems}. The proof that $\\mathcal{D}_{- \\mathcal{J}}$ is indeed a Dirac structure with respect to the modified plus pairing works analogously. However, as both sign conventions are common in the literature (cf. \\cite{LeGorrec, MaschkeLagrange} and \\cite{SchaftMaschke02, SchaftMaschke20}), we will stick with the sign convention applied throughout this chapter for the rest of the thesis. \n\\end{remark}\nThis is a good starting point for the analysis of the system \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case} with Hamiltonian \\eqref{General Lossless Port-Hamiltonian System - Hamiltonian H}. To this end, one defines the (maximal) \\emph{port-Hamiltonian operator} $A_{\\mathcal{Q}} \\colon D(A_{\\mathcal{Q}}) \\subset X \\to X$ given by\n\\begin{align*}\n\tD(A_{\\mathcal{Q}}) &= \\left\\{ x \\in X \\mid \\mathcal{Q}x \\in D(\\mathcal{J}) \\right\\}, \\\\\n\tA_{\\mathcal{Q}}x &= \\mathcal{J}(\\mathcal{Q}x), \\hspace{0.5 cm} x \\in D(A_{\\mathcal{Q}}), \n\\end{align*}\nwhere $\\mathcal{J}$ is the formally skew-symmetric operator defined in \\eqref{Skew-Symmetric Differential Operator J - N-Dimensional}-\\eqref{Skew-Symmetric Differential Operator J - N-Dimensional - Corresponding Matrices}, and studies the abstract Cauchy problem\n\\begin{align}\n\t\\begin{split}\n\t\t\\dot{x}(t) &= A_{\\mathcal{Q}}x(t), \\hspace{0.5 cm} t >0, \\\\\n\t\tx(0) &= x_0 \\in X,\n\t\\end{split}\n\t\\label{Evolution Equation wrt pH-Operator AQ}\n\\end{align}\nas discussed in the beginning of Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}. As for every partial differential equation, it is clear that one has to impose boundary conditions in order to guarantee well-posedness of the system \\eqref{Evolution Equation wrt pH-Operator AQ}. Indeed, in \\cite[Section 4]{LeGorrec} it has been shown that formulating the boundary conditions with respect to the boundary port variables, one can characterize those boundary conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. \n\\vspace{0.5 cm}\\\\\nWhat we may deduce from equation \\eqref{Power Balance for General Linear PH-Systems - Infinite Dimensions} is that the change of energy, and therefore the change of the state, is equal to the power flow at the boundary. Hence, it seems natural to let some input (or at least a part of it) act on the boundary of the spatial domain, and to define boundary control systems associated with boundary port-Hamiltonian systems, see \\cite[Chapter 11]{JacobZwart}. Some control related problems of such systems have been successfully studied in the PhD theses \\cite{Augner16} and \\cite{Villegas}. \n\\vspace{0.5 cm}\\\\\nIn this chapter, we have worked out the main differences of port-based modeling of finite-dimensional systems to infinite-dimensional systems on a 1-dimensional spatial domain. We give a short overview of the major changes:\n\\begin{enumerate}\n\t\\item[(i)] The gradient of the Hamiltonian function is replaced by the variational derivative $\\frac{\\delta H}{\\delta x}$ of the Hamiltonian functional (the gradient $\\nabla \\mathcal{H}$ of the Hamiltonian density). \n\t\\item[(ii)] The skew-symmetric matrix $J$ is replaced by the formally skew-symmetric differential operator $\\mathcal{J}$. \n\t\\item[(iii)] One has to incorporate the power flow at the boundary of the spatial domain (by means of the boundary port variables $f_{\\partial}$ and $e_{\\partial}$) into the definition of the (Stokes-) Dirac structure. \n\t\\item[(iv)] The power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ does, in general, not coincide with the canonical inner product on $\\mathcal{F} = \\mathcal{E}$, but is slightly modified. \n\\end{enumerate}\nMoreover, in Theorem \\ref{Theorem Dirac Structure Associated With Skew-Symmetric Operator J} we have defined the underlying power-conserving interconnection structure, namely the Stokes-Dirac structure $\\mathcal{D}_{\\mathcal{J}}$, of the system \\eqref{General Lossless Port-Hamiltonian System - Infinite-Dimensional Case}. \n\\vspace{0.5 cm}\\\\\nNow that we have gained a basic understanding concerning port-based modeling of infinite-dimensional systems, we can finally move on to the main part of this thesis. We want to define two first order Hamiltonian systems that are coupled through a moving interface as a single boundary port-Hamiltonian system.\n\n\n\nAt this stage it is significant to mention two points:\n\n\n\n\n\n\n\n\n\\chapter{Boundary Port-Hamiltonian Systems with a Moving Interface}\n\\label{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface}\nThe main chapter of this thesis is devoted to the formulation of two systems of two conservation laws defined on two complementary intervals that are coupled by some moving interface as a single boundary port-Hamiltonian system on the composed domain. Given the great success of the study of boundary port-Hamiltonian systems from Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}, in \\cite{Diagne} one wanted to investigate whether the port-Hamiltonian formulation can be extended to two systems that are coupled by a moving interface. As we have already mentioned in the introductory chapter of this thesis, this approach has been motivated by the fact that there are several cases where the system is heterogeneous in the considered spatial domain, and thus may be subdivided in several phases. Due to the distinct physical properties on the respective subdomains, there might arise some discontinuities of some variables at the interface, e.g., some pressure discontinuities of two gases separated by a moving piston, or different heat fluxes at the interface of a liquid-solid transition. These discontinuities are modeled by a set of interface relations, and result in a change of the position of the interface. According to \\cite{Diagne}, the port-Hamiltonian formulation of this scenario yields an abstract control system, where the input is defined as the velocity of the moving interface position. \n\\vspace{0.5 cm}\\\\\n\\emph{Our utmost objective is to rewrite the systems defined in \\cite{Diagne} as evolution problems and to analyze them by means of the framework of strongly continuous semigroups}, see Chapter \\ref{Chapter Some Background}. It should be mentioned beforehand that we will not be able to cover all problems regarding this fairly new system description, as this is beyond the scope of this thesis. Furthermore, the concept of a system used in this chapter remains purely formal. However, we seek to provide some useful results that may help to prove well-posedness of this class of systems, which one may call boundary port-Hamiltonian interface control systems. This chapter is organized as follows. In Section \\ref{Section Port-Hamiltonian System of two Conservation Laws} we recall the port-Hamiltonian formulation of a system of two conservation laws with an arbitrary Hamiltonian functional. Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface} deals with the port-Hamiltonian formulation of two systems that are coupled by some interface at a fixed position. We will introduce interface port variables associated with the predefined interface relations, and so-called color functions in order to express the state variables on the total spatial domain. Furthermore, we will define the interconnected system as a port-Hamiltonian system associated with a formally skew-symmetric operator, and present the underyling Dirac structure of this class of systems. Section \\ref{Section Port-Hamiltonian Systems Coupled through a Moving Interface} aims to repeat the preceding procedure in case of a moving interface. We will see that this approach yields a system that has the form of an abstract control system, with the input being the velocity of the displacement of the interface position. However, there are several difficulties concerning the proper formulation and analysis of this class of systems. This is why we tackle some of these problems in the upfollowing sections. In Section \\ref{Section A Simplified System} we will greatly simplify the systems obtained in Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface}, and show that, under certain boundary and interface conditions, the associated port-Hamiltonian operator generates a contraction semigroup, see Theorem~\\ref{Interface - Theorem A Generates a Contraction Semigroup}. Afterwards, we present criteria for when this operator even generates an exponentially stable (contraction) semigroup. Furthermore, we specify its Hilbert space adjoint and the associated resolvent operator. Lastly, in Subsection \\ref{Subsection Stability of the Family of Infinitesimal Generators} we will come back to the case where the interface is moving over time. Using the simplified system discussed throughout Section~\\ref{Section A Simplified System}, we define a family of port-Hamiltonian operators that encompasses the position of the moving interface, and prove the stability of this family of operators.\n\\vspace{0.5 cm}\\\\\nWe assume that some basic concepts about weak derivatives and distributions are known to the reader. For a brief overview, we refer to the first two sections in \\cite[Chapter 13]{Tucsnak-Weiss}. For a comprehensive exposition we suggest \\cite{Vladimirov}. We will denote the space of test functions by $\\mathcal{D}(a,b) := \\mathcal{C}_{c}^{\\infty}((a,b),\\mathbb{R})$, and the vector space of distributions by $\\mathcal{D}'(a,b)$. Throughout, we will identify a function $f \\in L^2([a,b], \\mathbb{R})$ with its regular distribution $\\Lambda_{f} \\colon \\mathcal{D}(a,b) \\to \\mathbb{R}$. In particular, we will sometimes regard the space $L^2([a,b], \\mathbb{R})$ as a subspace of $\\mathcal{D}'(a,b)$. The action of a regular distribution $\\Lambda_{f}$ on a test function $\\varphi \\in \\mathcal{D}(a,b)$ is denoted by \n\\begin{align*}\n\t\\langle \\Lambda_f , \\varphi \\rangle_{\\mathcal{D}'(a,b)} := \\langle f, \\varphi \\rangle_{L^2} = \\int_{a}^{b} f(z) \\varphi(z) \\, dz. \n\\end{align*}\n\n\n\\section{Port-Hamiltonian System of two Conservation Laws}\n\\label{Section Port-Hamiltonian System of two Conservation Laws}\n\nIn this section, we briefly recall the port-Hamiltonian formulation of a system of two conservation laws describing an energy-conserving exchange of energy between different physical domains. We have already come across this type of systems in Section \\ref{Section Motivating Examples}, namely the first-order systems of the lossless transmission line and the vibrating string. However, in this section, we will slightly generalize these systems by considering arbitrary Hamiltonian functionals.\n\\vspace{0.5 cm}\\\\\nLet $X = L^2([a,b], \\mathbb{R}^2)$ be endowed with the standard $L^2$-inner product $\\langle \\cdot, \\cdot \\rangle_{L^2}$ defined in \\eqref{L^2 inner product}. As mentioned before, let $H \\colon X \\to \\mathbb{R}$ be some arbitrary Hamiltonian functional of the form\n\\begin{align}\n\tH(x) = \\int_{a}^{b} \\mathcal{H}(x(z)) \\, dz, \\hspace{0.5 cm} x \\in X,\n\t\\label{Interface Paper - Hamiltonian H}\n\\end{align}\nwith Hamiltonian density $\\mathcal{H} \\in \\mathcal{C}^{\\infty}(\\mathbb{R}^2, \\mathbb{R})$ and variational derivative $\\frac{\\delta }{\\delta x}H \\colon X \\to X$, as described in the beginning of Section \\ref{Section Motivating Examples}. Now, consider a system of two conservation laws\n\\begin{align}\n\t\\frac{\\partial}{\\partial t} x(z,t) + \\frac{\\partial}{\\partial z} \\mathcal{N}(x(\\cdot,t))(z) = 0, \\hspace{0.5 cm} z \\in [a,b], \\, t >0,\n\t\\label{Interface Paper - Standard PDE - Conservation Law}\n\\end{align}\non the state space $X$, with the effort variables defined as\n\\begin{align*}\n\t\\mathcal{N}(x) = \\begin{bmatrix}\n\t\t\\mathcal{N}_1(x) \\\\\n\t\t\\mathcal{N}_2(x)\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\t0 & 1 \\\\\n\t\t1 & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\\delta_{x_1} H(x) \\\\\n\t\\delta_{x_2} H(x)\n\\end{bmatrix}, \\hspace{0.5 cm} x \\in X.\n\\end{align*}\nThen the system of conservation laws \\eqref{Interface Paper - Standard PDE - Conservation Law} may be equivalently rewritten as the Hamiltonian system\n\\begin{align}\n\\frac{\\partial}{\\partial t} x(z,t) = \\mathcal{J} \\frac{\\delta}{\\delta x} H(x(\\cdot,t))(z)\n\t\\label{Interface Paper - Standard PDE}\n\\end{align}\ngenerated by the Hamiltonian \\eqref{Interface Paper - Hamiltonian H} and defined with respect to the differential operator \n$\\mathcal{J} \\colon D(\\mathcal{J}) \\subset X \\to X$ given by\n\\begin{align*}\n\t\\begin{split}\n\tD(\\mathcal{J}) &= H^1([a,b], \\mathbb{R}^2), \\\\\n\t\\mathcal{J}x &= \\begin{bmatrix}\n\t\t0 & - \\frac{d}{dz} \\\\\n\t\t-\\frac{d}{dz} & 0 \n\t\\end{bmatrix}x, \\hspace{0.5 cm} x \\in D(\\mathcal{J}).\n\\end{split}\n\\end{align*}\nSince the matrix\n\\begin{align}\nP_1 = \\begin{bmatrix}\n\t0 & - 1 \\\\\n\t-1 & 0\n\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 2}\n\\label{Interface Paper - Matrix P1}\n\\end{align} \nis symmetric, recall that, by Theorem \\ref{Theorem Stokes-Like Theorem}, the matrix differential operator $\\mathcal{J} = P_1 \\frac{d}{dz}$ is formally skew-symmetric.\n\\vspace{0.5 cm}\\\\\nNext, we need to augment the Hamiltonian system \\eqref{Interface Paper - Standard PDE} with boundary port variables. In contrast to \\cite{Diagne}, we will stick with the definition from Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}. This will have an impact on the definition of the Dirac structure and the corresponding plus pairing $\\langle \\cdot \\mid \\cdot \\rangle$, and hence slightly deviates from the execution in \\cite{Diagne}. Now, since $N=1$, the matrix $\\mathcal{P}$ defined in \\eqref{Matrix P - Distributed-Parameter} reduces to $\\mathcal{P} = P_1$, and the invertible matrix $R_{\\extern}$ defined in \\eqref{Matrix Rext} is therefore given by\n\\begin{align}\n\t\\label{Interface Paper - Matrix Rext}\n\tR_{\\extern} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n\t\tP_1 & - P_1 \\\\\n\t\tI & I \n\t\\end{bmatrix} \\in \\mathbb{R}^{4 \\times 4}. \n\\end{align}\nAccording to \\eqref{Boundary Flow and Effort Linear Case}, the boundary flow variable $f_{\\partial} = f_{\\partial, e} \\in \\mathbb{R}^2$ and boundary effort variable $e_{\\partial} = e_{\\partial, e} \\in \\mathbb{R}^2$ are for all effort variables $ e \\in D(\\mathcal{J})$ given by\n\\begin{align*}\n\n\t\\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = R_{\\extern} \\trace(e) = \\begin{bmatrix} \\frac{1}{\\sqrt{2}} (P_1e(b) - P_1 e(a)) \\\\\n\t\t\\frac{1}{\\sqrt{2}} ( e(b) + e(a))\n\t\\end{bmatrix}. \n\\end{align*}\nThe only difference is that the geometric specification of the dynamics of the system \\eqref{Interface Paper - Standard PDE} with respect to the Stokes-Dirac structure $\\mathcal{D}_{\\mathcal{J}}$ defined in \\eqref{Stokes-Dirac structure - Linear PH Systems} becomes\n\t\\begin{align*}\n\t\t\\left( \\begin{pmatrix}\n\t\t\t\\partial_t x_1 (\\cdot,t) \\\\\n\t\t\t\\partial_t x_2 (\\cdot,t) \\\\\n\t\t\tf_{\\partial}(t) \n\t\t\\end{pmatrix}, \\begin{pmatrix}\n\t\t\\delta_{x_1} H(x(\\cdot,t)) \\\\\n\t\t\\delta_{x_2} H(x(\\cdot,t)) \\\\\n\t\te_{\\partial}(t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\mathcal{J}}, \\hspace{0.5 cm} t >0,\n\t\\end{align*}\nwhere $f_{\\partial}(t) = f_{\\partial, \\delta_xH(x(\\cdot,t))}$ and $e_{\\partial}(t) = e_{\\partial, \\delta_x H(x(\\cdot,t))}$. \n\\vspace{0.5 cm}\\\\\nDue to the constitutive relation of the Dirac structure $\\mathcal{D}_{\\mathcal{J}}$, the rate of energy change along a solution $x \\colon [0, \\infty) \\to X$ of \\eqref{Interface Paper - Standard PDE} is still given for all $t \\geq 0$ by the following balance equation:\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\frac{d}{dt} H(x(\\cdot,t)) \\\\\n\t&= \\frac{d}{dt} \\int_{a}^{b} \\mathcal{H}(x(z,t)) \\, dz \\\\\n\t&= \\int_{a}^{b} \\frac{\\partial}{\\partial t} \\mathcal{H}(x(z,t)) \\, dz \\\\\n\t&= \\int_{a}^{b} \\delta_{x_1} H(x(z,t)) \\frac{\\partial}{\\partial t} x_1(z,t) + \\delta_{x_2} H(x(z,t)) \\frac{\\partial}{\\partial t} x_2 (z,t) \\, dz \\\\\n\t&= \\int_{a}^{b} - \\delta_{x_1} H(x(z,t)) \\frac{\\partial}{\\partial z} \\big( \\delta{x_2} H(x(z,t)) \\big) - \\delta_{x_2} H(x(z,t)) \\frac{\\partial}{\\partial z} \\big( \\delta_{x_1}H(x(z,t)) \\big) \\, dz \\\\\n\t&= \\int_{a}^{b} - \\frac{\\partial}{\\partial z} \\left[ \\delta_{x_1 }H(x(z,t)) \\delta_{x_2}H(x(z,t)) \\right] \\, dz \\\\\n\t&= - \\big[ \\delta_{x_1}H(x(z,t)) \\delta_{x_2}H(x(z,t)) \\big]_{a}^{b} \\\\\n\t&= \\frac{1}{2} \\big[ \\delta_x H(x(z,t))^{\\top} P_1 \\delta_x H(x(z,t)) \\big]_{a}^{b} \\\\\n\t&= \\langle e_{\\partial}(t) , f_{\\partial}(t) \\rangle_2,\n\\end{align*}\nwhere $\\delta_x H(x(z,t)) = \\delta_x H(x(\\cdot,t))(z)$. So, just as in Subsection \\ref{Subsection Dirac Structure and Boundary Port-Hamiltonian Systems}, this equality states that the system's change of energy equals the power supplied to the system through the boundary. \n\\vspace{0.5 cm}\\\\\nNow that we have clarified the port-Hamiltonian formulation of systems of two conservation laws defined by an arbitrary Hamiltonian functional $H$ of the form \\eqref{Interface Paper - Hamiltonian H}, we may deal with our main concern next. \n\n\n\n\\section{Two Port-Hamiltonian Systems Coupled by an Interface}\n\\label{Section Two Port-Hamiltonian Systems Coupled by an Interface}\n\nIn the next two sections we examine two boundary port-Hamiltonian systems defined on two complementary intervals which are coupled by a moving interface. We seek to derive a formulation of the interconnected system as a boundary port-Hamiltonian system. It is well-known that the (power-conserving) interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. This interconnection is realized by the composition of the underyling Dirac structures through a number of mutually shared port variables. The internally stored energy of the interconnected system, described by the Hamiltonian on the product space, is simply the sum of the Hamiltonians of the constituting subsystems. We refer to \\cite[Chapter 6]{SchaftJeltsema} for the finite-dimensional case as well as to \\cite[Chapter 7]{Villegas} and the references specified therein for the infinite-dimensional case. In \\cite[Section 2.2]{Diagne} it was depicted how port-Hamiltonian systems are coupled through their (fixed) boundaries.\n\\vspace{0.5 cm}\\\\\nHowever, as we consider moving interfaces, we will have to deal with time-varying spatial domains, and thus we have to take a different approach. In order to keep track of the interface position, we are going to take the characteristic functions of the time-varying spatial domains of each subsystem to be part of the state variables. In preparation for the time-varying case studied in Section \\ref{Section Port-Hamiltonian Systems Coupled through a Moving Interface}, this section is devoted to the aforementioned formulation as a boundary port-Hamiltonian system in case of a fixed interface position, and is organized as follows: In Subsection \\ref{Subsection Prolongation with Color Functions} we introduce the characteristic functions of the complementary subdomains of the interconnected subsystems, called color-functions, as part of the state variables. Furthermore, we define the Hamiltonian and its variational derivative on the extended state space, and we specify interface port variables with respect to some predefined interface relations. In Subsection \\ref{Subsection Balance Equations of the State Variables} we state the balance equations of the constituting state variables. The main difficulty relies on the integration of the interface relations into the definition of these balance equations. Finally, in Subsection \\ref{Subsection Port-Hamiltonian Formulation and Dirac Structure} we are going to write the system of balance equations as a (generalized) Hamiltonian system with respect to some formally skew-symmetric operator on the extended state space. Moreover, by taking the interface and boundary port variables into account, we may define the underlying Dirac structure and state a balance equation of the total system with respect to the boundary and the interface port variables.\n\n\n\\subsection{Color Functions and Interface Port Variables}\n\\label{Subsection Prolongation with Color Functions}\nOur starting situation is as follows: let $H^- \\colon L^2((a,0), \\mathbb{R}^2) \\to \\mathbb{R}$ and\n$H^+ \\colon L^2((0,b), \\mathbb{R}^2) \\to \\mathbb{R}$ be two Hamiltonian functionals given by\n\\begin{align}\n\tH^-(x^-) &= \\int_{a}^{0} \\mathcal{H}^-(x^-(z)) \\, dz, \\hspace{0.5 cm} x^- \\in L^2((a,0), \\mathbb{R}^2), \\hspace{0.3 cm} \\text{and} \\label{Interface Paper - Hamiltonian H-} \\\\\n\tH^+(x^+) &= \\int_{0}^{b} \\mathcal{H}^+(x^+(z)) \\, dz, \\hspace{0.5 cm} x^+ \\in L^2((0,b), \\mathbb{R}^2), \\label{Interface Paper - Hamiltonian H+}\n\\end{align}\nwith the distinct energy densities $\\mathcal{H}^{\\pm} \\in \\mathcal{C}^{\\infty}(\\mathbb{R}^2, \\mathbb{R})$, respectively, indicating that the physical properties of the system differ on the respective domain. We assume that we may, in a way, extend these Hamiltonians to Hamiltonian functionals $H^{\\pm} \\colon X \\to \\mathbb{R}$. Now, consider the two systems of conservation laws\n\\begin{align}\n\t\\begin{split}\n\t\t\\frac{\\partial}{\\partial t} x^- + \\frac{\\partial}{\\partial z} \\mathcal{N}^-(x^-) &= 0, \\hspace{0.5 cm} z \\in [a,0), \\hspace{0.1 cm} t >0, \\\\\n\t\t\\frac{\\partial}{\\partial t} x^+ + \\frac{\\partial}{\\partial z} \\mathcal{N}^+(x^+) &= 0, \\hspace{0.5 cm} z \\in (0,b], \\hspace{0.1 cm} t > 0,\n\t\\end{split} \n\t\\label{Interface Paper - Two Systems of Conservation Laws on Respective Domain} \n\\end{align}\ndefined on the respective interval $[a,0)$ or $(0,b]$, where\n\\begin{align}\n\t\\mathcal{N}^-(x^-) = \\begin{bmatrix}\n\t\t0 & 1 \\\\\n\t\t1 & 0\n\t\\end{bmatrix} \\frac{\\delta}{\\delta x^-}H^-(x^-) \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\mathcal{N}^+(x^+) = \\begin{bmatrix}\n\t0 & 1 \\\\\n\t1 & 0 \n\\end{bmatrix} \\frac{\\delta}{\\delta x^+}H^+(x^+)\n\\label{Interface Paper - Relation Flux Variable and Variational Derivative}\n\\end{align} are the \\emph{flux variables} corresponding to the Hamiltonians $H^-$ and $H^+$ of the respective system defined in \\eqref{Interface Paper - Hamiltonian H-} and \\eqref{Interface Paper - Hamiltonian H+}, respectively. Our objective is to interconnect these two systems to a single port-Hamiltonian system on the composed spatial domain $[a,b]$, where the position of the interface is assumed to be at the fixed position $z=0$. In order to do so, we need to introduce suitable state variables defined on the composed domain, and we need to define some \\emph{interface relations} at $z =0$. These interface relations simultaneously constitute the \\emph{interface port variables} $(e_I, f_I) \\in \\mathbb{R}^2$ which we will need do define the underyling Dirac structure of the coupled system. These are given by\n\\begin{align}\n\tf_I &= \\mathcal{N}_1^+(x^+)(0^+) = \\mathcal{N}_1^-(x^-)(0^-), \\label{Interface Paper - Interface Variable fI} \\\\\n\t-e_I &= \\mathcal{N}_2^+(x^+)(0^+) - \\mathcal{N}_2^-(x^-)(0^-). \\label{Interface Paper - Interface Variable eI}\n\\end{align}\nEquation \\eqref{Interface Paper - Interface Variable fI} is a continuity equation of the flux variable $\\mathcal{N}_1$, called the privileged variable, and \\eqref{Interface Paper - Interface Variable eI} is a balance equation of the flux variable $\\mathcal{N}_2$ with external (source) term $e_I$.\n\\vspace{0.5 cm}\\\\\nIn order to keep track of the interface position, we use the characteristic functions of the domains of the two systems given in \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain}, called \\emph{color functions}:\n\\begin{align}\n\tc_0(z,t) = \\begin{cases} \n1, & z \\in [a,0), \\\\\n0, & z \\in [0,b],\n\\end{cases} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\overline{c}_0(z,t) = \\begin{cases} \n0, & z \\in [a,0], \\\\\n1, & z \\in (0,b].\n\\end{cases}\n\\label{Interface Paper - Characteristic Functions c0 and c0overline}\n\\end{align}\nThis allows us to write the state variables of the coupled system as the sum of prolongations of the variables of each subsystem to the composed domain $[a,b]$:\n\\begin{align}\n\t\\begin{split}\n\tx(z,t) &= x^-(z,t) + x^+(z,t) \\\\\n\t&= c_0(z,t)x(z,t) + \\overline{c}_0(z,t)x(z,t).\n\\end{split}\n\\label{Interface Paper - State Variable x Split}\n\\end{align}\nNext, we want to define the Hamiltonian and, hence, the flux variables of the total system. To this end, we introduce the extended state space $\\tilde{X} =L^2([a,b], \\mathbb{R}^4)$. The extended state variable is denoted by\n\\begin{align}\n\t\\tilde{x} = (x, c, \\overline{c}) \\in \\tilde{X},\n\t\\label{Interface Paper - Extended State Variable xtilde}\n\\end{align}\nwhere we want to stress that $\\overline{c}$ is fully independent of $c$. While in the fixed interface scenario, it might be unnecessary to consider the extended state space $\\tilde{X}$ (the color functions $c_0$ and $\\overline{c}_0$ are time-independent), this is essential in case of a moving interface discussed in Section~\\ref{Section Port-Hamiltonian Systems Coupled through a Moving Interface}. So, in preparation for the moving interface scenario, we will (for now) formulate the coupled system on the extended state space $\\tilde{X}$.\n\\vspace{0.5 cm}\\\\ \nWe start with the energy density $\\mathcal{H} \\in \\mathcal{C}^{\\infty}(\\mathbb{R}^4, \\mathbb{R})$, which is given by\n\\begin{align}\n\t\\mathcal{H}(x,c, \\overline{c}) = c \\mathcal{H}^-(x) + \\overline{c} \\mathcal{H}^+(x), \\hspace{0.5 cm} \\tilde{x} \\in \\mathbb{R}^4.\n\t\\label{Interface Paper - Hamiltonian Density Split}\n\\end{align}\nNext, define $H \\colon \\tilde{X} \\to \\mathbb{R}$ as\n\\begin{align}\n\tH(x,c, \\overline{c}) = \\int_{a}^{b} \\mathcal{H}(x(z), c(z), \\overline{c}(z)) \\, dz, \\hspace{0.5 cm} \\tilde{x} \\in \\tilde{X}.\n\t\\label{Interface Paper - Hamiltonian H of Coupled System}\n\\end{align}\nSince $H^{\\pm} \\colon X \\to \\mathbb{R}$ are functionals, one can readily see that $H$ is a functional as well. The variational derivative $\\delta_{\\tilde{x}} H \\colon \\tilde{X} \\to \\tilde{X}$ is again induced by the gradient of the energy density $\\mathcal{H}$ defined in \\eqref{Interface Paper - Hamiltonian Density Split}, i.e., \n\\begin{align}\n\t\\label{Interface Paper - Extended Variational Derivatives}\n\t\\delta_{\\tilde{x}} H(\\tilde{x}) = \\begin{bmatrix}\n\t\t\\delta_{x} H(x, c, \\overline{c}) \\\\\n\t\t\\delta_{c} H(x, c, \\overline{c}) \\\\\n\t\t\\delta_{\\overline{c}} H(x, c, \\overline{c})\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\tc \\delta_{x} H^-(x) + \\overline{c} \\delta_{x} H^+(x) \\\\\n\t\t\\mathcal{H}^-(x) \\\\\n\t\t\\mathcal{H}^+(x)\n\t\\end{bmatrix}, \\hspace{0.5 cm } \\tilde{x} \\in \\tilde{X}.\n\\end{align}\nIf we now choose $c$ and $\\overline{c}$ as the color functions defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline}, then we obtain for all $(x, c_0, \\overline{c}_0) \\in \\tilde{X}$,\n\\begin{align*}\n\tH(x, c_0, \\overline{c}_0) &= \\int_{a}^{b} c_0(z) \\mathcal{H}^-(x(z)) + \\overline{c}_0(z) \\mathcal{H}^+(x(z)) \\, dz \\\\\n\t&= \\int_{a}^{0} \\mathcal{H}^-(x(z)) \\, dz + \\int_{0}^{b} \\mathcal{H}^+(x(z)) \\, dz \\\\\n\t&= H^-(x^-) + H^+(x^+). \n\\end{align*}\nSo, $H(\\cdot ,c_0, \\overline{c}_0)$ describes the internally stored energy, which is given by the sum of the Hamiltonians of the subsystems defined in \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain}, and therefore is the Hamiltonian of the coupled system.\n\\vspace{0.5 cm}\\\\\nThe vector of flux variables of the coupled system becomes\n\\begin{align}\n\t\\mathcal{N}(x,c_0, \\overline{c}_0) = c_0 \\mathcal{N}^-(x) + \\overline{c}_0 \\mathcal{N}^+(x), \\hspace{0.5 cm} x \\in X, \n\t\\label{Interface Paper - Flux Variable N Split}\n\\end{align}\nwhich obviously satisfies \n\\begin{align*}\n\tc_0 \\mathcal{N}(x,c_0, \\overline{c}_0) = c_0 \\mathcal{N}^-(x) \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \t\\overline{c}_0 \\mathcal{N}(x,c_0, \\overline{c}_0) = \\overline{c}_0 \\mathcal{N}^+(x).\n\\end{align*} \nFor the sake of brevity, for $i = 1,2,$ we will sometimes write $\\mathcal{N}_i^{\\pm}(z,t)$ instead of $\\mathcal{N}_i^{\\pm}(x)(z,t)$, and $\\mathcal{N}_i(z,t)$ instead of\n\\begin{align*}\n\t \\mathcal{N}_i(x,c_0, \\overline{c}_0)(z,t) = \\mathcal{N}_i(x(\\cdot,t), c_0(\\cdot,t), \\overline{c}_0(\\cdot,t))(z). \n\\end{align*} \nNote that the first row of \\eqref{Interface Paper - Extended Variational Derivatives} corresponds to the definition of the flux variable (\\ref{Interface Paper - Flux Variable N Split}) in the following way:\n\\begin{align}\n\t\\begin{split}\n\t\\begin{bmatrix}\n\t\t0 & 1 \\\\\n\t\t1 & 0 \n\t\\end{bmatrix} \\delta_x H(x, c_0, \\overline{c}_0) &= \\begin{bmatrix}\n\t0 & 1 \\\\\n\t1 & 0 \n\\end{bmatrix} c_0 \\delta_x H^-(x) + \\overline{c}_0 \\delta_x H^+ (x) \\\\\n &= c_0 \\mathcal{N}^-(x) + \\overline{c}_0 \\mathcal{N}^+(x) \\\\\n &= \\mathcal{N}(x, c_0, \\overline{c}_0). \n\\end{split}\n\\label{Interface Paper - Relation Flux Variable and Variational Derivative - Composed Domain}\n\\end{align}\nIn order to define the coupled system as a Hamiltonian-like system on the extended state space $\\tilde{X}$, we need to define the balance equation of each variable constituting the extended state variable $\\tilde{x} = (x, c_0, \\overline{c}_0)$, and to include the interface relations \\eqref{Interface Paper - Interface Variable fI}-\\eqref{Interface Paper - Interface Variable eI}. This is what we are going to do next.\n\n\n\\subsection{Balance Equations of the State Variables}\n\\label{Subsection Balance Equations of the State Variables}\n\n\nThe coupling of the two systems of conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} at the (interface) position $z=0$ with the interface relations \\eqref{Interface Paper - Interface Variable fI}-\\eqref{Interface Paper - Interface Variable eI} led us to the introduction of the prolongated flux variables \\eqref{Interface Paper - Flux Variable N Split} defined on the composed domain $[a,b]$. Consequently, the flux variable $\\mathcal{N}_1(x, c_0, \\overline{c}_0)$ satisfies a continuity equation at the interface, whereas the flux variable $\\mathcal{N}_2(x, c_0, \\overline{c}_0)$ satisfies a balance equation at the interface. In the following, we want to state a conservation law for each of the variables $x_1$, $x_2$, $c_0$, and $\\overline{c}_0$ on the whole domain $[a,b]$ having regard to the interface relations.\n\\paragraph*{Conservation laws of $\\mathbf{c_0}$ and $\\mathbf{\\overline{c}_0}$.} First of all, let us state the balance equations of the color functions $c_0$ and $\\overline{c}_0$ defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline} which are part of the extended state variable $\\tilde{x} \\in \\tilde{X} = L^2([a,b], \\mathbb{R}^4)$ defined in \\eqref{Interface Paper - Extended State Variable xtilde} as well. It is clear that, for all $t \\geq 0$, they satisfy the trivial conservation laws\n\\begin{align}\n\t\\partial_t c_0 = 0 \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\partial_t \\overline{c}_0 = 0,\n\t\\label{Interface Paper - Conservation Laws of Color Functions with Fixed Interface Position}\n\\end{align}\nrespectively.\n\\paragraph*{Conservation law of $\\mathbf{x_1}$.} Next, consider the conservation law of the state variable $x_1$ on $[a,b]$. Assuming that the individual conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} are satisfied, with the aid of the flux variable \\eqref{Interface Paper - Flux Variable N Split}, it may be formally written as\n\\begin{align}\n\t\\begin{split}\n\t\\partial_t x_1 &= - \\partial_z \\mathcal{N}_1(x, c_0, \\overline{c}_0) \\\\\n\t&= - \\partial_z ( c_0 \\mathcal{N}_1^-(x) + \\overline{c}_0 \\mathcal{N}_1^+(x) ) \\\\\n\t&= - \\partial_z \\left( c_0 \\mathcal{N}_1(x, c_0, \\overline{c}_0) + \\overline{c}_0 \\mathcal{N}_1(x, c_0, \\overline{c}_0) \\right) \\\\\n\t&= \\underbrace{- \\left[ \\partial_z ( c_0 \\cdot) + \\partial_z (\\overline{c}_0 \\cdot) \\right]}_{=: \\mathbf{d}_0} \\mathcal{N}_1(x, c_0, \\overline{c}_0).\n\\end{split}\n\\label{Interface Paper - Conservation Law of x1 - Nonmoving Interface}\n\\end{align}\nTo begin with, for all $t >0$, equation (\\ref{Interface Paper - Conservation Law of x1 - Nonmoving Interface}) holds in $\\mathcal{D}'(a,b)$: let $\\varphi \\in \\mathcal{D}(a,b)$. Then, due to \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} and the continuity condition \\eqref{Interface Paper - Interface Variable fI}, the following holds for all $t > 0$:\n\t\\begin{align*}\n\t\t\\allowdisplaybreaks\n\n\t&\\hspace{0.5 cm} \\int_{a}^{b} \\left[ \\mathbf{d}_0 \\mathcal{N}_1(x,c_0, \\overline{c}_0) (z,t) \\right] \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{b} - \\left[ \\partial_z (c_0(z,t) \\mathcal{N}_1(z,t)) + \\partial_z (\\overline{c}_0 (z,t) \\mathcal{N}_1(z,t)) \\right] \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{b} - \\left[ \\partial_z (c_0(z,t) \\mathcal{N}_1 (z,t)) \\right] \\varphi(z) \\, dz + \\int_{a}^{b} - \\left[ \\partial_z (\\overline{c}_0 (z,t) \\mathcal{N}_1(z,t)) \\right] \\varphi(z) \\, dz \\\\\n\\text{(part. int.)} \\hspace{0.3 cm} &= \\int_{a}^{b} c_0(z,t) \\mathcal{N}_1(z,t) \\frac{d}{dz} \\varphi(z) \\, dz + \\int_{a}^{b} \\overline{c}_0(z,t) \\mathcal{N}_1(z,t) \\frac{d}{dz} \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{0} \\mathcal{N}_1^-(z,t) \\frac{d}{dz} \\varphi(z) \\, dz + \\int_{0}^{b} \\mathcal{N}_1^+(z,t) \\frac{d}{dz} \\varphi(z) \\, dz \\\\\n\\text{(part. int.)} \\hspace{0.3 cm}\t&= \\mathcal{N}_1^-(0^-,t) \\varphi(0) + \\int_{a}^{0} - \\partial_z \\mathcal{N}_1^-(z,t) \\varphi(z) \\, dz \\\\\n\t&\\hspace{0.4 cm} - \\mathcal{N}_1^+(0^+,t) \\varphi(0) + \\int_{0}^{b} - \\partial_z \\mathcal{N}_1^+(z,t) \\varphi(z) \\, dz \\\\\n\\text{\\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain}} \\hspace{0.3 cm}\t&= \\mathcal{N}_1^-(0^-,t) \\varphi(0) + \\int_{a}^{0} \\partial_t x_1^-(z,t) \\varphi(z) \\, dz \\color{white}\\text{abcdefghijklmnopqrstuvw.} \\\\\n\t&\\hspace{0.4 cm} - \\mathcal{N}_1^+(0^+,t) \\varphi(0) + \\int_{0}^{b} \\partial_t x_1^+(z,t) \\varphi(z) \\, dz \\\\\n\\text{\\eqref{Interface Paper - Interface Variable fI}} \\hspace{0.3 cm}\t&= \\int_{a}^{b} \\partial_t x_1(z,t) \\varphi(z) \\, dz.\n\\end{align*}\n\n\nNote that for all $t >0$, $\\varphi \\in \\mathcal{D}(a,b)$, one computes\n\\begin{align*}\n- \\int_{a}^{b} \\frac{\\partial}{\\partial z} c_0(z,t) \\mathcal{N}_1(z,t) \\varphi(z) \\, dz = \\int_{a}^{0} \\frac{\\partial}{\\partial z} (\\mathcal{N}_1^-(z,t) \\varphi(z)) \\, dz = \\mathcal{N}_1^-(0^-,t) \\varphi(0), \n\\end{align*}\nand\n\\begin{align*}\n\t- \\int_{a}^{b} \\frac{\\partial}{\\partial z} \\overline{c}_0(z,t) \\mathcal{N}_1(z,t) \\varphi(z) \\, dz = \\int_{0}^{b} \\frac{\\partial}{\\partial z} (\\mathcal{N}_1^+(z,t) \\varphi(z)) \\, dz = - \\mathcal{N}_1^+(0^+,t) \\varphi(0). \n\\end{align*}\nThis implies that, due to the continuity condition \\eqref{Interface Paper - Interface Variable fI}, the evaluation of the flux variables of the respective subdomain at the interface cancel each other out, and equation \\eqref{Interface Paper - Conservation Law of x1 - Nonmoving Interface} becomes\n\\begin{align}\n\t\\begin{split}\n\t\\partial_t x_1(\\cdot,t) &= \\mathbf{d}_0 \\mathcal{N}_1(x, c_0, \\overline{c}_0)(\\cdot,t) \\\\\n\t&= - c_0(\\cdot,t) \\partial_z \\mathcal{N}_1^-(x)(\\cdot,t) - \\partial_z c_0(\\cdot,t)\\mathcal{N}_1(x, c_0, \\overline{c}_0)(\\cdot,t) \\\\\n\t&\\hspace{0.475 cm} - \\overline{c}_0(\\cdot,t) \\partial_z \\mathcal{N}_1^+(x)(\\cdot,t) - \\partial_z \\overline{c}_0(\\cdot,t) \\mathcal{N}_1(x, c_0, \\overline{c}_0)(\\cdot,t) \\\\\n\t&= - c_0(\\cdot,t) \\partial_z \\mathcal{N}_1^-(x)(\\cdot,t) - \\overline{c}_0(\\cdot,t) \\partial_z \\mathcal{N}_1^+(x)(\\cdot,t).\n\\end{split}\n\t\\label{Interface Paper - Conservation Law of x1 - Nonmoving Interface - Version 2 in L2}\n\\end{align}\nAs a consequence, equation \\eqref{Interface Paper - Conservation Law of x1 - Nonmoving Interface - Version 2 in L2} holds, in fact, in $L^2([a,b], \\mathbb{R})$. This motivates to define the operator $\\mathbf{d}_0 \\colon D(\\mathbf{d}_0) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R})$ in the following way:\n\\begin{align}\n\t D(\\mathbf{d}_0) &= \\left\\{ x \\in L^2([a,b], \\mathbb{R}) \\mid x_{|(a,0)} \\in H^1((a,0), \\mathbb{R}), \\hspace{0.1 cm} x_{|(0,b)} \\in H^1((0,b), \\mathbb{R}), \\hspace{0.1 cm} x \\in \\mathcal{C}([a,b], \\mathbb{R}) \\right\\} \\nonumber \\\\\n\t &\\hspace{0.12 cm} = H^1([a,b], \\mathbb{R}), \\label{Interface Paper - Operator d0} \\\\\n\t\\mathbf{d}_0 x &= - \\left[ \\frac{d}{dz} (c_0 x) + \\frac{d}{dz} (\\overline{c}_0 x) \\right], \\hspace{0.5 cm} x \\in D(\\mathbf{d}_0). \\nonumber\n\\end{align}\nFor a function $x = c_0 x^- + \\overline{c}_0 x^+ \\in D(\\mathbf{d}_0)$ we have in particular that\n\\begin{align*}\n\\mathbf{d}_0 x = - c_0 \\frac{d}{dz} x^- - \\overline{c}_0 \\frac{d}{dz} x^+. \n\\end{align*} \nThis way, the continuity equation \\eqref{Interface Paper - Interface Variable fI} of the flux variable $\\mathcal{N}_1(x,c_0, \\overline{c}_0)$ is incorporated into the domain of the operator $\\mathbf{d}_0$. This allows us to define classical solutions of the conservation law of the state variable $x_1$ in the following way.\n\\begin{definition}\n\tLet $x = (x_1,x_2) \\colon [0, \\infty) \\to X$. Let $Y = L^2([a,b],\\mathbb{R})$. We call $x_1 \\colon [0,\\infty) \\to Y$ a classical solution of the conservation law\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\t\\partial_t x_1(t) &= \\mathbf{d}_0 \\mathcal{N}_1 (x, c_0, \\overline{c}_0)(\\cdot,t), \\hspace{0.5 cm} t > 0, \\\\\n\t\t\tx_1(0) &= x_{0,1} \\in Y, \n\t\t\\end{split}\n\t\\label{Interface Paper - Definition Conservation Law of x1 - Nonmoving Interface}\n\t\\end{align}\n\twhere $\\mathbf{d}_0$ is the operator given by \\eqref{Interface Paper - Operator d0}, and with the flux variable\n\t\\begin{align*}\n\t\t\\mathcal{N}_1(x, c_0, \\overline{c}_0) = c_0 \\mathcal{N}_1^-(x) + \\overline{c}_0 \\mathcal{N}_1^+(x)\n\t\\end{align*}\n\tof the systems of conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} coupled at $z = 0$, if:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $x_1 \\in \\mathcal{C}([0,\\infty), Y)$. \n\t\t\\item[(ii)] $x_{1_{ |(0,\\infty) }} \\in \\mathcal{C}^1((0, \\infty), Y)$.\n\t\t\\item[(iii)] $\\mathcal{N}_1(x, c_0, \\overline{c}_0)(\\cdot,t) \\in D(\\mathbf{d}_0)$ for all $t > 0$.\n\t\t\\item[(iv)] $x_1$ satisfies \\eqref{Interface Paper - Definition Conservation Law of x1 - Nonmoving Interface}.\n\t\\end{enumerate}\n\\end{definition}\n\nNote that the operator $\\mathbf{d}_0$ acts as the differential operator $-\\frac{d}{dz}$ on each subdomain $[a',b'] \\subset [a,b]$ not containing the interface, i.e., with either $a \\leq a' < b' < 0$, or with $0 < a' < b' \\leq b$. Indeed, according to \\eqref{Interface Paper - Conservation Law of x1 - Nonmoving Interface - Version 2 in L2}, for all $t > 0$ it holds that\n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} x_1(z,t) \\, dz = \\int_{a'}^{b'} \\partial_t x_1^- (z,t) \\, dz = \\int_{a'}^{b'} - \\partial_z \\mathcal{N}_1^-(x) (z,t) \\, dz = \\mathcal{N}_1^-(a',t) - \\mathcal{N}_1^- (b',t)\n\\end{align*}\nor\n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} x_1(z,t) \\, dz = \\int_{a'}^{b'} \\partial_t x_1^+ (z,t) \\, dz = \\int_{a'}^{b'} - \\partial_z \\mathcal{N}_1^+(x)(z,t) \\, dz = \\mathcal{N}_1^+(a',t) - \\mathcal{N}_1^+ (b',t),\n\\end{align*}\nrespectively. Now, if we consider an interval $[a',b']$ with $a \\leq a' < 0 < b' \\leq b$, then the continuity of the flux variable $\\mathcal{N}_1(x, c_0 , \\overline{c}_0)$ at the interface implies that\n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} x_1(z,t) \\, dz\t&= \\int_{a'}^{b'} \\mathbf{d}_0 \\mathcal{N}_1(x, c_0, \\overline{c}_0)(z,t) \\, dz \\, \\\\\n\t&= \\int_{a'}^{0} - \\partial_z \\mathcal{N}_1^-(x)(z,t) \\, dz + \\int_{0}^{b'} - \\partial_z \\mathcal{N}_1^+(x)(z,t) \\, dz \\\\\n\t&= \\mathcal{N}_1^-(a',t) \\underbrace{- \\mathcal{N}_1^-(0^-,t) + \\mathcal{N}_1^+ (0^+,t)}_{\\stackrel{(\\ref{Interface Paper - Interface Variable fI})}{=0}} - \\mathcal{N}_1^+(b',t) \\\\\n\t&= \\mathcal{N}_1^-(a',t) - \\mathcal{N}_1^+(b',t) \\\\\n\t&= \\mathcal{N}_1(a',t) - \\mathcal{N}_1(b',t).\n\\end{align*}\n\\paragraph*{Conservation law of $\\mathbf{x_2}$.} Lastly, let us consider the conservation law of the state variable $x_2$. Recall that at the interface, the associated flux variable $\\mathcal{N}_2(x, c_0, \\overline{c}_0) = c_0 \\mathcal{N}_2^-(x) + \\overline{c}_0 \\mathcal{N}_2^+(x)$ is supposed to satisfy the balance equation \\eqref{Interface Paper - Interface Variable eI}, where $e_I \\in \\mathbb{R}$ is defined as the difference of the flux variables $\\mathcal{N}_2^+(x)$ and $\\mathcal{N}_2^-(x)$ localized at the interface. In order to formulate this conservation law, we have to calculate the formal adjoint $\\mathbf{d}_0^{\\ast} \\colon D(\\mathbf{d}_0^{\\ast}) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R})$ of the operator $\\mathbf{d}_0$ defined in \\eqref{Interface Paper - Operator d0}. This will allow us to define a formally skew-symmetric differential operator, as we will see over the course of this chapter.\n\\vspace{0.5 cm}\\\\\nRecall from Section \\ref{Section Operators} that the domain of the adjoint operator is given by\n\\begin{align*}\n\t\\left\\{ y \\in X \\mid \\langle \\mathbf{d}_0 \\cdot, y \\rangle_{L^2} \\colon D(\\mathbf{d}_0) \\to \\mathbb{R}, \\hspace{0.1 cm} x \\mapsto \\langle \\mathbf{d}_0 x, y \\rangle_{L^2} \\text{ is continuous} \\right\\}.\n\\end{align*}\nHowever, we need to find the formally adjoint operator, as the system is considered to be open. Since, in contrast to the differential operators considered in Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}, this encompasses both the power flow at the boundary and the interface, we seek an operator $\\mathbf{d}_0^{\\ast}$ such that\n\\begin{align*}\n\t\\langle \\mathbf{d}_0 x, y \\rangle_{L^2} = \\langle x, \\mathbf{d}_0^{\\ast} y \\rangle_{L^2} \\hspace{0.2 cm} + \\text{boundary and interface terms}, \\hspace{0.3 cm} x \\in D(\\mathbf{d}_0), \\, y \\in D(\\mathbf{d}_0^{\\ast}).\n\\end{align*}\nTherefore, the general domain of the formal adjoint $\\mathbf{d}_0^{\\ast}$ is given by\n\\begin{align}\n\t\\label{Interface Paper - Operator d_0ast General Domain}\n\tD(\\mathbf{d}_0^{\\ast}) = \\left\\{ y \\in X \\mid \\langle \\mathbf{d}_0 \\cdot, y \\rangle_{L^2} \\colon \\mathbf{H}_0^1(a,b) \\to \\mathbb{R}, \\hspace{0.1 cm} x \\mapsto \\langle \\mathbf{d}_0 x, y \\rangle_{L^2} \\text{ is continuous} \\right\\},\n\\end{align}\nwhere \n\\begin{align}\n\t\\label{Interface Paper - Subspace H(a,b)}\n\t\\mathbf{H}_0^1(a,b) := \\left\\{ f \\in H_0^1([a,b],\\mathbb{R}) \\mid f(0) = 0 \\right\\} \\subset D(\\mathbf{d}_0).\n\\end{align}\nNote that the set $\\mathbf{H}_0^1(a,b)$ is dense in $L^2([a,b], \\mathbb{R})$. Hence, the formal adjoint of $\\mathbf{d}_0$, as we wish to define it, actually exists. \n\\vspace{0.5 cm}\\\\\nWe first want to specify the domain $D(\\mathbf{d}_0^{\\ast})$ given in \\eqref{Interface Paper - Operator d_0ast General Domain}. To that end, let $x \\in \\mathbf{H}_0^1(a,b)$ and let $y \\in L^2([a,b], \\mathbb{R})$. Then we get\n\\begin{align*}\n\t\\langle \\mathbf{d}_0 x, y \\rangle_{L^2} &= \\int_{a}^{b} (\\mathbf{d}_0 x(z)) y(z) \\, dz \\\\\n\t&= \\int_{a}^{b} - \\left[ \\frac{d}{dz} (c_0 x)(z) + \\frac{d}{dz} (\\overline{c}_0x )(z) \\right] y(z) \\, dz \\\\\n\t&= - \\int_{a}^{0} \\frac{d}{dz} (c_0 x)(z) y(z) \\, dz - \\int_{0}^{b} \\frac{d}{dz} (\\overline{c}_0 x)(z) y (z) \\, dz.\n\\end{align*}\nIf we additionally assume that $y_{|(a,0)} \\in H^1((a,0),\\mathbb{R})$ and $y_{|(0,b)} \\in H^1((0,b), \\mathbb{R})$, then integration by parts yields\n\\begin{align}\n\t\\begin{split} \n\t\\langle \\mathbf{d}_0 x, y \\rangle_{L^2}\t&= -x(0^-) y(0^-) + x(a) y(a) + x(0^+) y(0^+) - x(b) y(b) \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{0} (x c_0 )(z) \\frac{d}{dz} y(z) \\, dz + \\int_{0}^{b} (x \\overline{c}_0)(z) \\frac{d}{dz} y(z) \\, dz \\\\\n\t&= - \\underbrace{\\big[ (c_0 + \\overline{c}_0)(z) x(z) y(z) \\big]_{a}^{b}}_{=0} + \\underbrace{x(0)}_{=0} \\left[ y(0^+) - y(0^-) \\right] \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{0} x(z) \\frac{d}{dz} y(z) \\, dz + \\int_{0}^{b} x(z) \\frac{d}{dz} y(z) \\, dz. \n\t\\end{split}\n\\label{Interface Paper - Operator d_0ast Preparation}\n\\end{align}\n\tBy virtue of H\u00f6lder's inequality, we have for all $x \\in \\mathbf{H}_0^1(a,b)$ and for all such $y \\in L^2([a,b], \\mathbb{R})$,\n\\begin{align*}\n\t\\vert \\langle \\mathbf{d}_0 x, y \\rangle_{L^2} \\vert & \\leq \\int_{0}^{a} \\left\\vert x(z) \\frac{d}{dz} y(z) \\right\\vert \\, dz + \\int_{0}^{b} \\left\\vert x(z) \\frac{d}{dz} y(z) \\right\\vert \\, dz \\\\\n\t&\\leq \\|x\\|_{L^2(a,0)} \\left\\| \\frac{d}{dz} y\\right \\|_{L^2(a,0)} + \\|x \\|_{L^2(0,b)} \\left\\| \\frac{d}{dz} y \\right\\|_{L^2(0,b)} \\\\\n\t&\\leq \\|x \\|_{L^2(a,b)} \\left( \\left\\| \\frac{d}{dz} y \\right\\|_{L^2(a,0)} + \\left\\| \\frac{d}{dz} y \\right\\|_{L^2(0,b)} \\right) \\\\\n\t&\\leq C_y \\|x\\|_{L^2(a,b)}, \n\\end{align*}\nwhence\n\\begin{align*}\n\t\\| \\langle \\mathbf{d}_0 \\cdot , y \\rangle_{L^2} \\|_{\\text{op} \\colon \\mathbf{H}_0^1(a,b) \\to \\mathbb{R}} = \\sup_{\\substack{ x \\in \\mathbf{H}_0^1(a,b) \\\\\n\t\\|x\\|_{L^2} =1}} \\vert \\langle \\mathbf{d}_0 x, y \\rangle_{L^2} \\vert \\leq C_y.\n\\end{align*}\nConsequently, each element $y \\in L^2([a,b], \\mathbb{R})$ that is an $H^1$-function on the respective subdomains is an element of the domain $D(\\mathbf{d}_0^{\\ast})$. From the preceding calculations one can readily see that we cannot relax this requirement. As a result, the (maximal) domain of the operator $\\mathbf{d}_0^{\\ast}$ can be specified as follows:\n\\begin{align*}\n\t\t\tD(\\mathbf{d}_0^{\\ast}) = \\left\\{ y \\in L^2([a,b], \\mathbb{R}) \\mid y_{|(a,0)} \\in H^1((a,0),\\mathbb{R}), \\hspace{0.1cm} y_{|(0,b)} \\in H^1((0,b), \\mathbb{R}) \\hspace{0.1 cm} \\right\\}.\t\n\\end{align*}\nIt is left to determine the elements $\\mathbf{d}_0^{\\ast}y \\in L^2([a,b], \\mathbb{R})$ for $y \\in D(\\mathbf{d}_0^{\\ast})$. We have already calculated in \\eqref{Interface Paper - Operator d_0ast Preparation} that for all $x \\in D(\\mathbf{d}_0)$ and for all $y \\in D(\\mathbf{d}_0^{\\ast})$ we get\n\\begin{align*}\n\t\\langle \\mathbf{d}_0 x , y \\rangle_{L^2} \t&= - \\big[ (c_0 + \\overline{c}_0)(z) x(z) y(z) \\big]_{a}^{b} + x(0) \\left[ y(0^+) - y(0^-) \\right] \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{0} x (z) \\frac{d}{dz} y(z) \\, dz + \\int_{0}^{b} x(z) \\frac{d}{dz} y(z) \\, dz \\\\\n\t&\\stackrel{!}{=} - \\big[ (c_0 + \\overline{c}_0)(z) x(z) y(z) \\big]_{a}^{b} + x(0) \\left[ y(0^+) - y(0^-) \\right] + \\langle x, \\mathbf{d}_0^{\\ast} y \\rangle_{L^2}.\n\\end{align*}\nThus, our objective is to rewrite the second line as a single $L^2$-inner product on the composed domain $[a,b]$ with respect to $x$ and $\\mathbf{d}_0^{\\ast}y$. We claim that $\\mathbf{d}_0^{\\ast}y \\in L^2([a,b], \\mathbb{R}) $ is given as\n\\begin{align*}\n\t\\mathbf{d}_0^{\\ast}y = \\left[ \\frac{d}{dz} (c_0 y) + \\frac{d}{dz} (\\overline{c}_0 y) \\right] - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz}\\overline{c}_0 \\right] y , \\hspace{0.5 cm} y \\in D(\\mathbf{d}_0^{\\ast}). \n\\end{align*}\nAlthough at first glance, the co-domain of this operator seems to be $\\mathcal{D}'(a, b)$, we will see that the distributional terms of this expression, that is, the Dirac masses arising from taking the derivatives of the characteristic functions $c_0$ and $\\overline{c}_0$, cancel each other out, so that the remaining terms lie again in $L^2([a, b], \\mathbb{R}) \\subset \\mathcal{D}'(a,b)$.\n\\vspace{0.5 cm}\\\\\nInitially, let $\\varphi \\in \\mathcal{D}(a,b)$, which is a dense subset of $D(\\mathbf{d}_0) = H^1([a,b], \\mathbb{R})$. Then for all $y \\in D(\\mathbf{d}_0^{\\ast})$ it holds that (omitting the dependence on the variable $z$)\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{a}^{b} \\left( \\left[ \\frac{d}{dz} (c_0y) + \\frac{d}{dz} (\\overline{c}_0 y) \\right] - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz} \\overline{c}_0 \\right] y \\right) \\varphi \\, dz \\\\\n\t&= \\int_{a}^{b} \\frac{d}{dz} (c_0 y) \\varphi \\, dz + \\int_{a}^{b} \\frac{d}{dz} (\\overline{c}_0 y) \\varphi \\, dz - \\int_{a}^{b} \\frac{d}{dz} c_0 \\varphi y \\, dz - \\int_{a}^{b} \\frac{d}{dz} \\overline{c}_0 \\varphi y \\, dz \\\\\n\t(\\text{part. int.}) \\hspace{0.3 cm}\t&= \\big[ (c_0 + \\overline{c}_0) \\varphi y \\big]_{a}^{b} - \\int_{a}^{0} c_0 y \\frac{d}{dz} \\varphi \\, dz - \\int_{0}^{b} \\overline{c}_0 y \\frac{d}{dz} \\varphi \\, dz \\\\ \n\t&\\hspace{0.45 cm} - \\big[ (c_0 + \\overline{c}_0) \\varphi y \\big]_{a}^{b} + \\int_{a}^{0} \\frac{d}{dz} (\\varphi y) \\, dz + \\int_{0}^{b} \\frac{d}{dz} (\\varphi y) \\, dz \\\\\n\t&= - \\int_{a}^{0} y \\frac{d}{dz} \\varphi \\, dz - \\int_{0}^{b} y \\frac{d}{dz} \\varphi \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{0} \\varphi \\frac{d}{dz} y + y \\frac{d}{dz} \\varphi \\, dz + \\int_{0}^{b} \\varphi \\frac{d}{dz} y + y \\frac{d}{dz} \\varphi \\, dz \\\\ \n\n\t&= \\int_{a}^{0} \\varphi \\frac{d}{dz} y \\, dz + \\int_{0}^{b} \\varphi \\frac{d}{dz} y \\, dz.\n\\end{align*}\n\nNext, we want to show that for all $x \\in D(\\mathbf{d}_0)$, $y \\in D(\\mathbf{d}_0^{\\ast})$ the inner product $\\langle x, \\mathbf{d}_0^{\\ast}y \\rangle_{L^2}$ exists in a classical sense by proving that the operation $x \\mapsto \\langle x , \\mathbf{d}_0^{\\ast} y \\rangle_{L^2}$ is closed. Let $y \\in D(\\mathbf{d}_0^{\\ast})$, pick any $x \\in D(\\mathbf{d}_0)$, and let $(\\varphi_n)_{n \\in \\mathbb{N}} \\in \\mathcal{D}(a,b)^{\\mathbb{N}}$ be a sequence with $\\lim\\limits_{n \\to \\infty} \\varphi_n = x$ in $L^2$. Since the embedding $L^2([a,b], \\mathbb{R}) \\hookrightarrow L^1([a,b], \\mathbb{R})$ is continuous, this convergence holds in $L^1$ as well. Following \\cite[Chapter 6]{Klenke}, we may apply the dominated convergence theorem. This yields\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\lim_{n \\to \\infty} \\int_{a}^{b} \\left( \\left[ \\frac{d}{dz} (c_0y) + \\frac{d}{dz} (\\overline{c}_0 y) \\right] - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz} \\overline{c}_0 \\right] y \\right) \\varphi_n \\, dz \\\\ \n\t&= \\lim_{n \\to \\infty} \\int_{a}^{0} \\varphi_n \\frac{d}{dz} y \\, dz + \\lim_{n \\to \\infty} \\int_{0}^{b} \\varphi_n \\frac{d}{dz} y \\, dz \\\\\n\t&= \\int_{a}^{0} x \\frac{d}{dz} y \\, dz + \\int_{0}^{b} x \\frac{d}{dz} y \\, dz.\n\\end{align*}\nThus, for all $x \\in D(\\mathbf{d}_0)$, $y \\in D(\\mathbf{d}_0^{\\ast})$ the inner product $\\langle x , \\mathbf{d}_0^{\\ast}y \\rangle_{L^2}$ exists with\n\\begin{align*}\n\t\\langle x , \\mathbf{d}_0^{\\ast}y \\rangle_{L^2} = \\int_{a}^{0} x(z) \\frac{d}{dz} y(z) \\, dz + \\int_{0}^{b} x(z) \\frac{d}{dz} y(z) \\, dz.\n\\end{align*} \nAltogether, the formal adjoint $\\mathbf{d}_0^{\\ast} \\colon D(\\mathbf{d}_0^{\\ast}) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R})$ is given by\n\\begin{align}\n\t\\begin{split}\n\t\tD(\\mathbf{d}_0^{\\ast}) &= \\left\\{ y \\in L^2([a,b], \\mathbb{R}) \\mid y_{|(a,0)} \\in H^1((a,0),\\mathbb{R}), \\hspace{0.1cm} y_{|(0,b)} \\in H^1((0,b), \\mathbb{R}) \\hspace{0.1 cm} \\right\\}, \\\\\n\t\\mathbf{d}_0^{\\ast}y &= \\left[ \\frac{d}{dz} (c_0 y) + \\frac{d}{dz} (\\overline{c}_0 y) \\right] - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz}\\overline{c}_0 \\right] y \\\\\n\t&\\hspace{0.12 cm}= \\left( - \\mathbf{d}_0 - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz} \\overline{c}_0 \\right] \\right)y, \\hspace{0.5 cm} y \\in D(\\mathbf{d}_0^{\\ast}).\n\t\\end{split}\n\t\\label{Interface Paper - Operator d_0ast} \n\\end{align}\nIn particular, for a function $y = c_0 y^- + \\overline{c}_0 y^+ \\in D(\\mathbf{d}_0^{\\ast})$ we have\n\\begin{align*}\n\t\\mathbf{d}_0^{\\ast} y = c_0 \\frac{d}{dz} y^- + \\overline{c}_0 \\frac{d}{dz} y^+. \n\\end{align*}\nMoreover, we obtain the following relation between the operators $\\mathbf{d}_0$ and $\\mathbf{d}_0^{\\ast}$:\n\\begin{align}\n\t\\label{Interface Paper - Relation d_0 and d_0ast}\n\t\\langle \\mathbf{d}_0 x , y \\rangle_{L^2} = - \\big[ x(z) y(z) \\big]_{a}^{b} + x(0) \\left[ y(0^+) - y(0^-) \\right] + \\langle x , \\mathbf{d}_0^{\\ast} y \\rangle_{L^2}, \\hspace{0.5 cm} x \\in D(\\mathbf{d}_0), \\, y \\in D(\\mathbf{d}_0^{\\ast}).\n\\end{align} \nWe are finally able to state the conservation law of the state variable $x_2$ with respect to the operator $\\mathbf{d}_0^{\\ast}$. Assume that for all $t > 0$, the flux variable $\\mathcal{N}_2(x,c_0, \\overline{c}_0)(\\cdot,t)$ on the composed domain is an element of $D(\\mathbf{d}_0^{\\ast})$. From the preceding discussion and due to the individual conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain}, we deduce that for all $\\varphi \\in \\mathcal{D}(a,b)$ we have\n\\begin{align*}\n\n\t &\\hspace{0.5 cm} \\int_{a}^{b} \\left[ - \\mathbf{d}_0^{\\ast} \\mathcal{N}_2(x,c_0, \\overline{c}_0)(z,t) \\right] \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{0} - \\partial_z \\mathcal{N}_2^- (z,t) \\varphi(z) \\, dz + \\int_{0}^{b} - \\partial_z \\mathcal{N}_2^+(z,t) \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{0} \\partial_t x_2^-(z,t) \\varphi(z) \\, dz + \\int_{0}^{b} \\partial_t x_2^+(z,t) \\varphi(z) \\, dz \\\\\n\t&= \\int_{a}^{b} \\partial_t x_2(z,t) \\varphi(z) \\, dz.\n\n\\end{align*} \nIn conclusion, the conservation law of the state variable $x_2$ is given by\n\\begin{align}\n\t\\partial_t x_2 = - \\mathbf{d}_0^{\\ast} \\mathcal{N}_2(x,c_0, \\overline{c}_0),\n\t\\label{Interface Paper - Conservation Law of x2 - Nonmoving Interface} \n\\end{align}\nand it holds in a classical sense.\n\\begin{remark}\n\t\\label{Interface Paper - Remark Balance Equation}\nWe need to point out that the balance equation \\eqref{Interface Paper - Interface Variable eI} the flux variable $\\mathcal{N}_2$ is supposed to satisfy is incorporated neither into the domain of the operator $\\mathbf{d}_0^{\\ast}$ nor into the conservation law \\eqref{Interface Paper - Conservation Law of x2 - Nonmoving Interface} itself. So far, we have only allowed a jump discontinuity of $\\mathcal{N}_2$ at the interface. One possible way is to rewrite the conservation law as\n\\begin{align*}\n\t\\partial_t x_2 = \\mathbf{d}_0 \\mathcal{N}_2(x,c_0,\\overline{c}_0) - e_I \\delta_0,\n\\end{align*}\nwith $\\delta_0 \\in \\mathcal{D}'(a,b)$ the Dirac mass at $z = 0$, and $e_I \\colon [0, \\infty) \\to \\mathbb{R}$. By doing so, $e_I$ may be regarded as an external source term, and the equation holds (in a distributional sense) if and only if $\\mathcal{N}_2$ satisfies the balance equation \\eqref{Interface Paper - Interface Variable eI}. However, since this will only complicate the modeling in the further course and since we want to define a formally skew-symmetric matrix differential operator later on, we will stick with the conservation law \\eqref{Interface Paper - Conservation Law of x2 - Nonmoving Interface} with respect to the formally adjoint operator $\\mathbf{d}_0^{\\ast}$ of $\\mathbf{d}_0$, and take the interface port variable $e_I$ to be implicitly defined by means of the jump discontinuity of $\\mathcal{N}_2(x,c_0, \\overline{c}_0)$ at the interface. As we will see, this does not affect the dynamics of the resulting system over the one proposed in \\cite{Diagne}.\n\\end{remark}\nLet us now define classical solutions of the conservation law of $x_2$ with respect to $\\mathcal{N}_2(x,c_0, \\overline{c}_0)$.\n\n\\begin{definition}\n\t Let $x = (x_1,x_2) \\colon [0, \\infty) \\to X$. Let $Y = L^2([a,b], \\mathbb{R})$. We call $x_2 \\colon [0,\\infty) \\to Y$ a classical solution of the conservation law\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\t\\partial_t x_2(t) &= - \\mathbf{d}_0^{\\ast} \\mathcal{N}_2 (x, c_0, \\overline{c}_0)(\\cdot,t), \\hspace{0.5 cm} t > 0, \\\\\n\t\t\tx_2(0) &= x_{0,2} \\in Y, \n\t\t\\end{split}\n\t\t\\label{Interface Paper - Definition Conservation Law of x2 - Nonmoving Interface}\n\t\\end{align}\n\twhere $\\mathbf{d}_0^{\\ast}$ is the operator given in \\eqref{Interface Paper - Operator d_0ast}, and with the flux variable\n\t\\begin{align*}\n\t\t\\mathcal{N}_2(x, c_0, \\overline{c}_0) = c_0 \\mathcal{N}_2^-(x) + \\overline{c}_0 \\mathcal{N}_2^+(x)\n\t\\end{align*}\n\tof the systems of conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} coupled at $z=0$, if:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $x_2 \\in \\mathcal{C}([0,\\infty), Y)$. \n\t\t\\item[(ii)] $x_{2_{ |(0,\\infty) }} \\in \\mathcal{C}^1((0, \\infty), Y)$.\n\t\t\\item[(iii)] $\\mathcal{N}_2(x, c_0, \\overline{c}_0)(\\cdot,t) \\in D(\\mathbf{d}_0^{\\ast})$ for all $t > 0$.\n\t\t\\item[(iv)] $x_2$ satisfies \\eqref{Interface Paper - Definition Conservation Law of x2 - Nonmoving Interface}.\n\t\\end{enumerate}\n\\end{definition} \nAnalogously to the operator $\\mathbf{d}_0$, its formal adjoint $\\mathbf{d}_0^{\\ast}$ acts as the differential operator $ \\frac{d}{dz}$ on the respective subdomains. Assuming that the flux variable $\\mathcal{N}_2$ satisfies the balance equation \\eqref{Interface Paper - Interface Variable eI}, on an arbitrary interval $[a',b'] \\subset [a,b]$ containing the interface we get the following for all $t>0$:\n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} x_2(z,t) \\, dz &= \\int_{a'}^{b'} \\partial_t x_2(z,t) \\, dz \\\\\n\t&= \\int_{a'}^{b'} -\\mathbf{d}_0^{\\ast} \\mathcal{N}_2(x,c_0, \\overline{c}_0) (z,t) \\, dz \\\\\n\t&= \\int_{a'}^{0} - \\partial_z \\mathcal{N}_2^-(x) (z,t) \\, dz + \\int_{0}^{b'} - \\partial_z \\mathcal{N}_2^+(x)(z,t) \\, dz \\\\\n\t&= \\mathcal{N}_2^-(a',t) - \\mathcal{N}_2^+(b',t) - \\mathcal{N}_2^-(0^-,t) + \\mathcal{N}_2^+(0^+,t) \\\\\n\t&= \\mathcal{N}_2(a',t) - \\mathcal{N}_2(b',t) - e_I(t).\n\\end{align*}\nFinally, we can formulate the interconnection of the conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} as a port-Hamiltonian system defined on the extended state space $\\tilde{X}$, which we are going to do next.\n\n\\subsection{Port-Hamiltonian Formulation and Dirac Structure}\n\\label{Subsection Port-Hamiltonian Formulation and Dirac Structure}\nIn the previous section, we have formulated the dynamics of the state variables $x_1$ and $x_2$ on the composed domain $[a,b]$ through the conservation laws \\eqref{Interface Paper - Definition Conservation Law of x1 - Nonmoving Interface} and \\eqref{Interface Paper - Definition Conservation Law of x2 - Nonmoving Interface} using the operator $\\mathbf{d}_0$ and its formal adjoint $\\mathbf{d}_0^{\\ast}$ given by \\eqref{Interface Paper - Operator d0} and \\eqref{Interface Paper - Operator d_0ast}, respectively. Furthermore, in \\eqref{Interface Paper - Conservation Laws of Color Functions with Fixed Interface Position} we stated the conservation laws for the characteristic functions $c_0$ and $\\overline{c}_0$ defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline} of the two subdomains separated by the interface, which are constituting the extended state variable as well. Now, we want to formulate a Hamiltonian system off of these equation on the extended state space $\\tilde{X}$, and we want to specify the underlying Dirac structure of the fixed interface scenario.\n\\begin{remark}\n\tRecall that both the operator $\\mathbf{d}_0$ and its formal adjoint $\\mathbf{d}_0^{\\ast}$ depend on the color functions $c_0$ and $\\overline{c}_0$. Strictly speaking, the operators $\\mathbf{d}_0$ and $\\mathbf{d}_0^{\\ast}$ are therefore modulated by the state $\\tilde{x} = (x, c_0, \\overline{c}_0) \\in \\tilde{X}$. However, as the color functions remain constant over time, we will neglect this dependence of the aforementioned operators. \n\\end{remark}\nWith the aid of \\eqref{Interface Paper - Relation Flux Variable and Variational Derivative - Composed Domain}, we may define the agumented system \n\\begin{align*}\n\t\\frac{\\partial}{\\partial t} \\begin{bmatrix}\n\t\tx_1 \\\\\n\t\tx_2 \\\\\n\t\tc_0 \\\\\n\t\t\\overline{c}_0\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t0 & \\mathbf{d}_0 & 0 & 0 \\\\\n\t- \\mathbf{d}_0^{\\ast} & 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & 0 \n\\end{bmatrix} \\begin{bmatrix}\nc_0 \\delta_{x_1} H^-(x) + \\overline{c}_0 \\delta_{x_1} H^+(x) \\\\\nc_0 \\delta_{x_2} H^-(x) + \\overline{c}_0 \\delta_{x_2} H^+(x) \\\\\n\\mathcal{H}^-(x) \\\\\n\\mathcal{H}^+(x)\n\\end{bmatrix},\n\\end{align*}\nor short, \n\\begin{align}\n\t\\label{Interface Paper - Augmented System wrt Ja}\n\t\\frac{\\partial}{\\partial t} \\tilde{x} = \\tilde{\\mathcal{J}}_0 \\delta_{\\tilde{x}} H(\\tilde{x}),\n\\end{align}\nwith the Hamiltonian's variational derivative $\\delta_{\\tilde{x}} H$ given by \\eqref{Interface Paper - Extended Variational Derivatives}, and with the operator $\\tilde{\\mathcal{J}}_0 \\colon D(\\tilde{\\mathcal{J}}_0) \\subset \\tilde{X} \\to \\tilde{X}$ given by\n\\begin{align}\n\t\\begin{split}\n\tD(\\tilde{\\mathcal{J}}_0)\t&= D(\\mathbf{d}_0^{\\ast}) \\times D(\\mathbf{d}_0) \\times L^2([a,b], \\mathbb{R}^2), \\\\\n\t\\tilde{\\mathcal{J}}_0 \\tilde{x} &= \\begin{bmatrix}\n\t\t0 & \\mathbf{d}_0 & 0 & 0 \\\\\n\t\t- \\mathbf{d}_0^{\\ast} & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 \n\t\\end{bmatrix} \\tilde{x}, \\hspace{0.5 cm} \\tilde{x} \\in \\tilde{X}.\n\\end{split}\n\\label{Interface Paper - Operator Ja}\n\\end{align}\nBy virtue of the relation \\eqref{Interface Paper - Relation d_0 and d_0ast} between the operators $\\mathbf{d}_0$ and $\\mathbf{d}_0^{\\ast}$, we find for all $\\tilde{e}^1, \\tilde{e}^2 \\in D(\\tilde{\\mathcal{J}}_0)$ that\n\\begin{align}\n\t\\begin{split}\n\t &\\hspace{0.5 cm} \\langle \\tilde{\\mathcal{J}}_0 \\tilde{e}^1, \\tilde{e}^2 \\rangle_{L^2} + \\langle \\tilde{e}^1, \\tilde{\\mathcal{J}}_0 \\tilde{e^2} \\rangle_{L^2} \\\\\n\t &= \\int_{a}^{b} (e^2)^{\\top}(z) \\begin{bmatrix}\n\t \t0 & \\mathbf{d}_0 \\\\\n\t \t-\\mathbf{d}_0^{\\ast} & 0 \n\t \\end{bmatrix} e^1(z) + \\left( \\begin{bmatrix}\n\t 0 & \\mathbf{d}_0 \\\\\n\t -\\mathbf{d}_0^{\\ast} & 0 \n \\end{bmatrix} e^2(z) \\right)^{\\top} e^1(z) \\, dz \\\\\n&= \\int_{a}^{b} e_1^2(z) \\mathbf{d}_0 e_2^1(z) - e_2^2(z) \\mathbf{d}_0^{\\ast} e_1^1(z) + e_1^1(z) \\mathbf{d}_0 e_2^2(z) - e_2^1(z) \\mathbf{d}_0^{\\ast} e_1^2(z) \\, dz \\\\ \n\t &= - \\big[ e_1^1(z)e_2^2(z) \\big]_{a}^{b} - \\big[ e_2^1(z) e_1^2(z) \\big]_{a}^{b} \\\\\n\t &\\hspace{0.5 cm}+ e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] \\\\\n\t &= \\big[(e^1)^{\\top}(z) P_1 e^2(z) \\big]_{a}^{b} + e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right],\n\t\\end{split}\n\\label{Interface Paper - Skew-Symmetry of Ja}\n\\end{align}\nwith $P_1$ defined in \\eqref{Interface Paper - Matrix P1}. Hence, the operator $\\tilde{\\mathcal{J}}_0$ is formally skew-symmetric. We will not check whether $\\tilde{\\mathcal{J}}_0$ satifies the Jacobi identity (see again \\cite[Chapter 7]{Olver}), so we refer to the system \\eqref{Interface Paper - Augmented System wrt Ja} as a \\emph{generalized Hamiltonian system}.\n\\vspace{0.5 cm}\\\\\nIn order to take the energy exchange at the boundary and the power inflow (or outflow) at the interface into account, the generalized Hamiltonian system\n\\eqref{Interface Paper - Augmented System wrt Ja} will now be extended to a boundary port-Hamiltonian system. For this purpose, we need to define a Stokes-Dirac structure similar to the one given in \\eqref{Stokes-Dirac structure - Linear PH Systems}.\n\\vspace{0.5 cm}\\\\\nTo find the underlying Dirac structure of the system \\eqref{Interface Paper - Augmented System wrt Ja}, we need to define a suitable power pairing $ \\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ constituting the plus pairing $\\ll \\cdot , \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$. We will basically proceed as in Section \\ref{Section Boundary Port-Hamiltonian Systems Associated with Skew-symmetric Operators}. \n\\vspace{0.5 cm}\\\\\nTo begin with, consider the operator $\\mathcal{J}_0 \\colon D(\\mathcal{J}_0) \\subset X \\to X$ given by\n\\begin{align*}\n\n\t\tD(\\mathcal{J}_0) &= \\left\\{ x \\in X \\mid x_1 \\in D(\\mathbf{d}_0^{\\ast}), \\hspace{0.1 cm} x_2 \\in D(\\mathbf{d}_0)\\right\\}, \\\\\n\t\t\\mathcal{J}_0 x &= \\begin{bmatrix}\n\t\t\t0 & \\mathbf{d}_0 \\\\\n\t\t\t- \\mathbf{d}_0^{\\ast} & 0 \n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\tx_1 \\\\\n\t\t\tx_2\n\t\t\\end{bmatrix}, \\hspace{0.5 cm} x \\in D(\\mathcal{J}_0).\n\n\n\\end{align*}\nNote that $\\mathcal{J}_0$ is the operator we obtain if we reduce the operator $\\tilde{\\mathcal{J}}_0$ from \\eqref{Interface Paper - Operator Ja} to $X$, i.e., if we ignore the color functions we included in the definition of the extended state space $\\tilde{X}$. This operator will be useful for finding the power pairing. Note that from \\eqref{Interface Paper - Skew-Symmetry of Ja} it immediately follows that $\\mathcal{J}_0$ is formally skew-symmetric as well, and that on any interval $(c,d) \\subset[a,b]$ not containing the interface position $z = 0$, the operator $\\mathcal{J}_0$ simply acts as the matrix differential operator $P_1 \\frac{d}{dz}$. We will come across this operator in Section \\ref{Section A Simplified System} again. \n\\vspace{0.5 cm}\\\\\nFor systems associated with a formally skew-symmetric differential operator, we have already defined the boundary port variables in Definition \\ref{Definition Boundary Port Variables in The Linear Case}. Since the interface position resides in the interior of the spatial domain $[a,b]$, the trace operator defined in \\eqref{Trace Operator - Chapter 3} can be extended to\n\\begin{align*}\n\t\\trace_0 \\colon D(\\mathcal{J}_0) \\to \\mathbb{R}^4, \\hspace{0.3 cm} e \\mapsto \\begin{bmatrix}\n\t\te(b) \\\\\n\t\te(a)\n\t\\end{bmatrix}.\n\\end{align*}\nFollowing \\eqref{Boundary Flow and Effort Linear Case}, the boundary flow variable $f_{\\partial} = f_{\\partial, e} \\in \\mathbb{R}^2$ and the boundary effort variable $e_{\\partial} = e_{\\partial, e} \\in \\mathbb{R}^2$ can, for all co-energy variables $ e \\in D(\\mathcal{J}_0)$, be defined as\n\\begin{align*}\n\n\t\\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = R_{\\extern} \\trace_0(e) = \\begin{bmatrix} \\frac{1}{\\sqrt{2}} (P_1e(b) - P_1 e(a)) \\\\\n\t\t\\frac{1}{\\sqrt{2}} ( e(b) + e(a))\n\t\\end{bmatrix}, \n\\end{align*} \nwith $R_{\\extern}$ defined in \\eqref{Interface Paper - Matrix Rext}. Furthermore, we have introduced the interface relations in \\eqref{Interface Paper - Interface Variable fI} and \\eqref{Interface Paper - Interface Variable eI} with respect to the flux variables. In general, the interface flow $f_I = f_{I,e} \\in \\mathbb{R}$ and the interface effort $e_I = e_{I,e} \\in \\mathbb{R}$ are for all effort variables $ e \\in D(\\mathcal{J}_0)$ given as follows:\n\\begin{align*}\n\tf_I &= e_2 (0^+) = e_2(0^-), \\\\\n\t-e_I &= e_1(0^+) - e_1 (0^-).\n\\end{align*}\nNow, let us choose the power pairing in order to define the Dirac structure associated with the system \\eqref{Interface Paper - Augmented System wrt Ja} that is augmented with the boundary and interface port variables. Define the space of flows $\\mathcal{F}$ and the space of efforts $\\mathcal{E}$ as\n\\begin{align}\n\t\\mathcal{F} = \\mathcal{E} = L^2([a,b], \\mathbb{R}^4) \\times \\mathbb{R} \\times \\mathbb{R}^2,\n\t\\label{Interface Paper - Flow and Effort Space}\n\\end{align}\nendowed with the canonical inner product $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{F}}$. For the right choice of the power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ we make use of equation \\eqref{Interface Paper - Skew-Symmetry of Ja}. Note that for all $ e^1, e^2 \\in D(\\mathcal{J}_0)$ we may write the last line of \\eqref{Interface Paper - Skew-Symmetry of Ja} with respect to the boundary and interface port variables as follows: \n\\begin{align*}\n\t&\\hspace{0.5 cm} \\big[ (e^1)^{\\top}(z) P_1 e^2(z) \\big]_{a}^{b} + e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] \\\\\n\t&= \\langle e_{\\partial, e^2} , f_{\\partial, e^1} \\rangle_2 + \\langle e_{\\partial, e^1} , f_{\\partial, e^2} \\rangle_2 - f_{I,e^1} e_{I, e^2} - f_{I, e^2} e_{I, e^1}. \n\\end{align*}\nThe representaion of the evaluation at the boundary with respect to the boundary flows and efforts can be looked up in Subsection \\ref{Subsection Boundary Port Variables} (see page \\pageref{Theorem Stokes-Like Theorem - Equation Representation 2}). Consequently, we have for all $\\tilde{e}^1, \\tilde{e}^2 \\in D(\\tilde{\\mathcal{J}}_0)$, \n\\begin{align}\n\t\\langle \\tilde{\\mathcal{J}}_0 \\tilde{e}^1, \\tilde{e}^2 \\rangle_{L^2} + \\langle \\tilde{e}^1, \\tilde{\\mathcal{J}}_0 \\tilde{e}^2 \\rangle_{L^2} = \\langle e_{\\partial, e^2} , f_{\\partial, e^1} \\rangle_2 + \\langle e_{\\partial, e^1} , f_{\\partial, e^2} \\rangle_2 - f_{I,e^1} e_{I, e^2} - f_{I, e^2} e_{I, e^1}.\n\t\\label{Interface Paper - Skew-Symmetry of Ja - Representation 2}\n\\end{align}\nThus, we shall define\n\\begin{align}\n\t\\left\\langle \\begin{pmatrix}\n\t\t\\tilde{e} \\\\\n\t\te_I \\\\\n\t\te_{\\partial} \n\t\\end{pmatrix} \\Bigg| \\begin{pmatrix}\n\t\t\\tilde{f} \\\\\n\t\tf_I \\\\\n\t\tf_{\\partial}\n\t\\end{pmatrix} \\right\\rangle := \\langle \\tilde{e}, \\tilde{f} \\rangle_{L^2} +e_I f_I - \\langle e_{\\partial}, f_{\\partial} \\rangle_2, \\hspace{0.5 cm} \\left( \\begin{pmatrix}\n\t\t\\tilde{e} \\\\\n\t\te_I \\\\\n\t\te_{\\partial} \n\t\\end{pmatrix} , \\begin{pmatrix}\n\t\t\\tilde{f} \\\\\n\t\tf_I \\\\\n\t\tf_{\\partial}\n\t\\end{pmatrix} \\right) \\in \\mathcal{E} \\times \\mathcal{F}, \n\t\\label{Interface Paper - Power Pairing}\n\\end{align}\nand the plus pairing $\\ll \\cdot , \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ induced by the power pairing \\eqref{Interface Paper - Power Pairing} is accordingly defined as in \\eqref{Plus Pairing - Distributed-Parameter}. Finally, we can define the Dirac structure associated with the formally skew-symmetric operator $\\tilde{\\mathcal{J}}_0$ given by \\eqref{Interface Paper - Operator Ja}. The idea is based on the proofs of Proposition 4 in \\cite{Diagne} and of Theorem 3.6 in \\cite{LeGorrec}.\n\\begin{proposition}\n\t\\label{Interface Paper - Proposition Dirac Structure DI}\n Let $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ be the bond space with respect to the flow space $\\mathcal{F}$ and the effort space $\\mathcal{E}$ defined in \\eqref{Interface Paper - Flow and Effort Space}. The subspace $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0} \\subset \\mathcal{B}$ defined by \n\t\\begin{align}\n\t\t\\mathcal{D}_{\\tilde{\\mathcal{J}}_0} &:= \n\t\t\t\\left\\{ \\left( \\begin{pmatrix}\n\t\t\t\t\\tilde{f} \\\\\n\t\t\t\tf_I \\\\\n\t\t\t\tf_{\\partial}\n\t\t\t\\end{pmatrix} , \\begin{pmatrix}\n\t\t\t\t\\tilde{e} \\\\\n\t\t\t\te_I \\\\\n\t\t\t\te_{\\partial}\n\t\t\t\\end{pmatrix} \\right) \\in \\mathcal{F} \\times \\mathcal{E} \\hspace{0.1 cm} \\Bigg| \\hspace{0.1 cm} \\tilde{e} \\in D(\\tilde{\\mathcal{J}}_0), \\hspace{0.1 cm}\n\t\t\t\\tilde{f} = \\tilde{\\mathcal{J}}_0 \\tilde{e}, \\hspace{0.1 cm} f_I = e_2(0), \\right. \\label{Interface Paper - Dirac Structure DI} \\\\ \n\t\t\t&\\hspace{5.7 cm} \\left. e_I = e_1(0^-) - e_1(0^+), \\hspace{0.1 cm} \\begin{bmatrix}\n\t\t\t\tf_{\\partial } \\\\\n\t\t\t\te_{\\partial}\n\t\t\t\\end{bmatrix} = R_{\\extern} \\trace_0 (e_1,e_2) \\hspace{0.1 cm} \\right\\}, \\nonumber\n\t\\end{align}\nwith the flow and effort variables \n\\begin{align*}\n\t\\tilde{f} = \\begin{bmatrix}\n\t\tf_1 \\\\\n\t\tf_2 \\\\\n\t\tf_{c} \\\\\n\t\tf_{\\overline{c}}\n\t\\end{bmatrix} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\tilde{e} = \\begin{bmatrix}\n\te_1 \\\\\n\te_2 \\\\\n\te_{c} \\\\\n\te_{\\overline{c}}\n\\end{bmatrix},\n\\end{align*}\n respectively, associated with a system of two conservation laws defined on the\n spatial domain $[a,b] \\subset \\mathbb{R}$ with an interface at the point $z = 0$ which imposes the continuity of the effort variable $e_2$ and allows for the discontinuity of the effort variable $e_1$, and associated with the extended state \\eqref{Interface Paper - Extended State Variable xtilde}, the differential operator $\\tilde{\\mathcal{J}}_0$ defined in \\eqref{Interface Paper - Operator Ja}, the operator $\\mathbf{d}_0$ and its formal adjoint $\\mathbf{d}_0^{\\ast}$ defined in \\eqref{Interface Paper - Operator d0} and \\eqref{Interface Paper - Operator d_0ast}, respectively, is a Dirac structure with respect to the plus pairing $\\ll \\cdot, \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ induced by the power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ given by \\eqref{Interface Paper - Power Pairing}.\n\\end{proposition}\n\n\\begin{proof}\n\tIn the following, we denote by\n\t\\begin{align*}\n\t\tF = \\begin{pmatrix}\n\t\t\t\\tilde{f} \\\\\n\t\t\tf_I \\\\\n\t\t\tf_{\\partial}\n\t\t\\end{pmatrix} \\in \\mathcal{F} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} E = \\begin{pmatrix}\n\t\t\t\\tilde{e} \\\\\n\t\t\te_I \\\\\n\t\t\te_{\\partial}\n\t\t\\end{pmatrix} \\in \\mathcal{E}\n\t\\end{align*}\n\tthe vector of flows and the vector of efforts, respectively, for all $(F,E) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$.\n\tTo prove the claim, we have to verify that the space $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ coincides with its orthogonal complement $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp}$ with respect to the plus pairing $\\ll \\cdot, \\cdot \\gg$, see Definition \\ref{Definition Dirac Structure - Infinite-dimensional Case}.\n\t\\vspace{0.5 cm}\\\\\nWe first show that $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0} \\subset \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp}$. In order to do so, we have to show that for every $(F^1,E^1)$, $(F^2,E^2) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ we have\n\t\\begin{align*}\n\t\\ll (F^1,E^1), (F^2,E^2) \\gg = \\langle E^1 \\mid F^2 \\rangle + \\langle E^2 \\mid F^1 \\rangle \\stackrel{!}{=} 0\n\t\\end{align*}\n\twith respect to the pairing $\\langle \\cdot \\mid \\cdot \\rangle $ defined in \\eqref{Interface Paper - Power Pairing}. Using the constitutive relations of $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ and \\eqref{Interface Paper - Skew-Symmetry of Ja - Representation 2}, we compute \n\t\\begin{align*}\n\t\t&\\hspace{0.5 cm} \\ll (F^1,E^1), (F^2,E^2) \\gg \\\\\n\t\t&= \\langle \\tilde{e}^2, \\tilde{f}^1 \\rangle_{L^2} + \\langle \\tilde{e}^1, \\tilde{f}^2 \\rangle_{L^2} + e_I^1 f_I^2 + e_I^2 f_I^1 - \\langle e_{\\partial}^1 , f_{\\partial}^2 \\rangle_2 - \\langle e_{\\partial}^2 , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t&= \\langle \\tilde{e}^2, \\tilde{\\mathcal{J}}_0 \\tilde{e}^1 \\rangle_{L^2} + \\langle \\tilde{e}^1, \\tilde{\\mathcal{J}}_0 \\tilde{e}^2 \\rangle_{L^2} + e_I^1 f_I^2 + e_I^2 f_I^1 - \\langle e_{\\partial}^1 , f_{\\partial}^2 \\rangle_2 - \\langle e_{\\partial}^2 , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t&= \\langle e_{\\partial}^2, f_{\\partial}^1 \\rangle_2 + \\langle e_{\\partial}^1 , f_{\\partial}^2 \\rangle_2 - f_I^1 e_I^2 - f_I^2 e_I^1 + e_I^1 f_I^2 + e_I^2 f_I^1 - \\langle e_{\\partial}^1 , f_{\\partial}^2 \\rangle_2 - \\langle e_{\\partial}^2 , f_{\\partial}^1 \\rangle_2 \\\\ \n\t\t&= 0.\n\t\\end{align*}\n As $(F^1,E^1), (F^2, E^2) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ can be chosen arbitrarily, we conlcude that $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0} \\subset \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp}$.\n\t\\vspace{0.5 cm}\\\\\n\tNext, we want to show the inclusion $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp} \\subset \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$. To this end, let \n\t\\begin{align*}\n\t(F^2, E^2) = \\left( \\begin{pmatrix}\n\t\\tilde{\\phi} \\\\\n\t\\phi_I \\\\\n\t\\phi_{\\partial}\n\t\\end{pmatrix}, \\begin{pmatrix}\n\t\\tilde{\\epsilon} \\\\\n\t\\epsilon_I \\\\\n\t\\epsilon_{\\partial}\n\\end{pmatrix} \\right) \\in \\mathcal{B} \n\\end{align*}\nsuch that for all $(F^1,E^1) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ it holds that\n\t\\begin{align*}\n\t\t\\ll (F^1, E^1), (F^2, E^2) \\gg = 0. \n\t\\end{align*}\n\tWe need to verify that $(F^2, E^2) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$. For simplicity, we will omit the dependence on the variable $z$. For any $(F^1, E^1) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ we have\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\t0 &= \\int_{a}^{b} (\\tilde{e}^1)^{\\top} \\tilde{\\phi} + \\epsilon_1 (\\mathbf{d}_0 e_2^1) + \\epsilon_2 (- \\mathbf{d}_0^{\\ast} e_1^1) \\, dz + e_I^1 \\phi_I + \\epsilon_I f_I^1 \\\\\n\t\t\t&\\hspace{0.5 cm} - \\langle e_{\\partial}^1, \\phi_{\\partial} \\rangle_{2} - \\langle \\epsilon_{\\partial}, f_{\\partial}^1 \\rangle_{2}. \n\t\t\\end{split}\n\t\t\\label{Interface Paper - Proposition 4 - Variable Determination}\n\t\\end{align} \n\tNote that the choice of the variable $\\tilde{e}^1 = (e_1^1,e_2^1, e_{c}^1, e_{\\overline{c}}^1)$ determines all remaining variables in $(F^1, E^1) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$. Since \\eqref{Interface Paper - Proposition 4 - Variable Determination} has to hold for all $(F^1, E^1) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$, we will consider different instances for the previously mentioned variable in order to verify that $(F^2, E^2) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$. \n\t\\begin{enumerate}\n\t\t\\item[(i)] Let $e_1^1 = 0$, $e_2^1 = 0$, and $e_{\\overline{c}}^1 = 0$. \\\\\n\t\tFrom \\eqref{Interface Paper - Dirac Structure DI} and \\eqref{Interface Paper - Proposition 4 - Variable Determination} we derive\n\t\t\\begin{align*}\n\t\t\t0 = \\int_{a}^{b} (\\tilde{e}^1)^{\\top} \\tilde{\\phi} \\, dz = \\int_{a}^{b} e_{c}^1 \\phi_{c} \\, dz. \n\t\t\\end{align*} \n\t\tAs $e_{c}^1$ may be chosen arbitrarily, we conclude that $\\phi_{c} = 0$. If we change the roles of $e_{c}^1$ and $e_{\\overline{c}}^1$, then we analogously obtain $\\phi_{\\overline{c}} = 0$. \n\t\t\\item[(ii)] Let $e_1^1(a) = e_1^1(b) = 0$, $e_1^1(0^+) - e_1^1(0^-) = 0$, and $e_2^1 = 0$. \\\\\n\t\tThen we deduce from \\eqref{Interface Paper - Proposition 4 - Variable Determination} that\n\t\t\\begin{align*}\n\t\t\t0 = \\int_{a}^{b} e_1^1 \\phi_1 + \\epsilon_2(- \\mathbf{d}_0^{\\ast} e_1^1) \\, dz. \n\t\t\\end{align*}\n\t\tAssume that $\\epsilon_2 \\in D(\\mathbf{d}_0)$. Then, by virtue of \\eqref{Interface Paper - Relation d_0 and d_0ast}, it follows that\n\t\t\\begin{align*}\n\t\t\t0 &= \\int_{a}^{b} e_1^1 \\phi_1 + e_1^1 (- \\mathbf{d}_0 \\epsilon_2) \\, dz - \\big[ e_1^1 \\epsilon_2 \\big]_{a}^{b} + \\epsilon_2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] \\\\ \n\t\t\t&= \\int_{a}^{b} e_1^1 (- \\mathbf{d}_0 \\epsilon_2 + \\phi_1) \\, dz,\n\t\t\\end{align*}\n\t\twhence $\\phi_1 = \\mathbf{d}_0 \\epsilon_2$. \n\t\t\\item[(iii)] Let $e_1^1 = 0$ and $e_2^1(a) = e_2^1(0) = e_2^1(b) = 0$. \\\\\n\t\tIf we assume that $\\epsilon_1 \\in D(\\mathbf{d}_0^{\\ast})$ and if we apply \\eqref{Interface Paper - Relation d_0 and d_0ast} once again, we get\n\t\t\\begin{align*}\n\t\t\t0 &= \\int_{a}^{b} e_2^1 \\phi_2 + \\epsilon_1 (\\mathbf{d}_0 e_2^1) \\, dz \\\\\n\t\t\t&= \\int_{a}^{b} e_2^1 \\phi_2 + e_2^1( \\mathbf{d}_0^{\\ast} \\epsilon_1) \\, dz - \\big[e_2^1 \\epsilon_1 \\big]_{a}^{b} + e_2^1(0) \\left[ \\epsilon_1(0^+) - \\epsilon_1(0^-) \\right] \\\\\n\t\t\t&= \\int_{a}^{b} e_2^1 \\phi_2 + e_2^1( \\mathbf{d}_0^{\\ast} \\epsilon_1) \\, dz,\n\t\t\\end{align*} \n\t\twhence $\\phi_2 = - \\mathbf{d}_0^{\\ast} \\epsilon_1$. \n\t\\end{enumerate}\n\tFrom the preceding instances, we conclude that $\\tilde{\\epsilon} \\in D(\\tilde{\\mathcal{J}}_0)$ and\n\t\\begin{align*}\n\t\t\\tilde{\\phi} = \\tilde{\\mathcal{J}}_0 \\tilde{\\epsilon}.\n\t\\end{align*}\n\tIt is left to show the last three constitutive relations in (\\ref{Interface Paper - Dirac Structure DI}) concerning the boundary and the interface port variables. \n\t\\begin{enumerate}\n\t\t\t\\item[(iv)] Let $e_1^1(a) = e_1^1(b) =0$ and $e_2^1 = 0$. \\\\\n\t\tFrom \\eqref{Interface Paper - Relation d_0 and d_0ast} and \\eqref{Interface Paper - Proposition 4 - Variable Determination} we infer\n\t\t\\begin{align*}\n\t\t\t0 &= \\int_{a}^{b} e_1^1(\\mathbf{d}_0 \\epsilon_2) + \\epsilon_2(-\\mathbf{d}_0^{\\ast} e_1^1) \\, dz + e_I^1 \\phi_I \\\\\n\t\t\t&= \\epsilon_2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] + e_I^1 \\phi_I \\\\\n\t\t\t&= - e_I^1 \\epsilon_2(0) + e_I^1 \\phi_I.\n\t\t\\end{align*}\n\t\tHence, $\\phi_I = \\epsilon_2(0)$. \n\t\t\\item[(v)] Let $e_1^1 = 0$ and $e_2^1(a) = e_2^1(b) = 0$. \\\\\n\t\tWith the same argument we find\n\t\t\\begin{align*}\n\t\t\t0 &= \\int_{a}^{b} e_2^1(-\\mathbf{d}_0^{\\ast} \\epsilon_1) + \\epsilon_1(\\mathbf{d}_0 e_2^1) \\, dz +\\epsilon_I f_I^1 \\\\\n\t\t\t&= e_2^1(0) \\left[ \\epsilon_1(0^+) - \\epsilon_1(0^-) \\right] + \\epsilon_I f_I^1 \\\\\n\t\t\t&= f_I^1 \\left[ \\epsilon_1(0^+) - \\epsilon_1(0^-) \\right] + \\epsilon_I f_I^1.\n\t\t\t\\end{align*}\n\t\tConsequently, $\\epsilon_I = \\epsilon_1(0^-) - \\epsilon_1(0^+)$.\n\t\t\\item[(vi)] Let $e_1^1(0^+) - e_1^1(0^-) = 0$. \\\\\n\t\tFollowing the discussion in Subsection \\ref{Subsection Boundary Port Variables} and using the relation \\eqref{Interface Paper - Skew-Symmetry of Ja}, equation \\eqref{Interface Paper - Proposition 4 - Variable Determination} can be written as\n\t\t\\begin{align*}\n\t\t\t0 \t&= \\langle \\tilde{\\epsilon}, \\tilde{\\mathcal{J}}_0 \\tilde{e}^1 \\rangle_{L^2} + \\langle \\tilde{e}^1, \\tilde{\\mathcal{J}}_0 \\tilde{\\epsilon} \\rangle_{L^2} + \\epsilon_I f_I^1 - \\langle e_{\\partial}^1 , \\phi_{\\partial} \\rangle_2 - \\langle \\epsilon_{\\partial} , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t\t&= \\big[ (e^1)^{\\top} P_1 \\epsilon \\big]_{a}^{b} - f_I^1 \\epsilon_I + \\epsilon_I f_I^1 - \\langle e_{\\partial}^1 , \\phi_{\\partial} \\rangle_2 - \\langle \\epsilon_{\\partial} , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t\t&= \\trace_0(e^1)^{\\top} \\mathcal{\\begin{bmatrix}\n\t\t\t\tP_1 & 0 \\\\\n\t\t\t\t0 & - P_1\n\t\t\t\\end{bmatrix} } \\trace_0(\\epsilon) - \\langle e_{\\partial}^1 , \\phi_{\\partial} \\rangle_2 - \\langle \\epsilon_{\\partial} , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t&= \\trace_0(e^1)^{\\top} \\mathcal{P}_{\\extern} \\trace_0(\\epsilon) - \\langle e_{\\partial}^1 , \\phi_{\\partial} \\rangle_2 - \\langle \\epsilon_{\\partial} , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t&= \\trace_0(e^1)^{\\top} R_{\\extern}^{\\top} \\Sigma R_{\\extern} \\trace_0(\\epsilon) - \\langle e_{\\partial}^1 , \\phi_{\\partial} \\rangle_2 - \\langle \\epsilon_{\\partial} , f_{\\partial}^1 \\rangle_2 \\\\\n\t\t&= \\begin{bmatrix}\n\t\t\t(f_{\\partial}^1)^{\\top } & (e_{\\partial}^1)^{\\top}\n\t\t\\end{bmatrix} \\Sigma R_{\\extern} \\trace_0(\\epsilon) - (e_{\\partial}^1)^{\\top} \\phi_{\\partial} - \\epsilon_{\\partial}^{\\top} f_{\\partial}^1 \\\\\n&= \\begin{bmatrix}\n\t(e_{\\partial}^1)^{\\top } & (f_{\\partial}^1)^{\\top} \\end{bmatrix} \\left[ R_{\\extern} \\trace_0(\\epsilon) - \\begin{bmatrix}\n\t\\phi_{\\partial} \\\\\n\t\\epsilon_{\\partial}\n\\end{bmatrix} \\right].\n\t\t\\end{align*}\nSince a proper choice for the variable $\\tilde{e}$ yields arbitary values for the vectors $e_{\\partial}^1$ and $f_{\\partial}^1$, we conclude that\n\\begin{align*}\n\t\\begin{bmatrix}\n\t\t\\phi_{\\partial} \\\\\n\t\t\\epsilon_{\\partial}\n\t\\end{bmatrix} = R_{\\extern} \\trace_0(\\epsilon).\n\\end{align*}\n\t\\end{enumerate}\n\tThus, we have $(F^2, E^2) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$, and so the inclusion $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp} \\subset \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ holds as well. Summing up, we have $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0} = \\mathcal{D}_{\\tilde{\\mathcal{J}}_0}^{\\perp \\!\\!\\! \\perp}$. This completes the proof. \t\n\\end{proof}\n\n\n\n\nComparing the constitutive relations of the Dirac structure $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ defined in \\eqref{Interface Paper - Dirac Structure DI} with the generalized Hamiltonian system \\eqref{Interface Paper - Augmented System wrt Ja} together with the interface relations \\eqref{Interface Paper - Interface Variable fI}-\\eqref{Interface Paper - Interface Variable eI} imposed on the flux variable $\\mathcal{N}(x, c_0, \\overline{c}_0)$, one may easily see that this system possesses a boundary port-Hamiltonian structure. Note that it is crucial to pick states of the form\n\\begin{align*}\n\t\\tilde{x} = (x, c_0, \\overline{c}_0) \\in \\tilde{X}, \n\\end{align*}\nwith $c_0$ and $\\overline{c}_0$ the color functions defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline}, as the conservation laws derived in Subsection \\ref{Subsection Balance Equations of the State Variables}, which are constituting the generalized Hamiltonian system \\eqref{Interface Paper - Augmented System wrt Ja}, have been determined with this particular choice. Denoting $f_I(t) = f_{I, \\delta_{\\tilde{x}}H(\\tilde{x}(\\cdot,t))}$ and $e_I(t) = e_{I, \\delta_{\\tilde{x}}H(\\tilde{x}(\\cdot,t))}$ for $t \\geq 0$, the following holds. \n\\begin{corollary}\n\t\\label{Interface Paper - Corollary PH System with Nonmoving Interface}\n\tThe generalized Hamiltonian system \\eqref{Interface Paper - Augmented System wrt Ja} of four conservation laws may be defined as a generalized boundary port-Hamiltonian system, and is geometrically specified by the requirement that for all $t > 0$ we have\n\t\\begin{align*}\n\t\t\\left( \\begin{pmatrix}\n\t\t\t\\partial_t \\tilde{x}(\\cdot,t) \\\\\n\t\t\tf_I(t) \\\\\n\t\t\tf_{\\partial}(t)\n\t\t\\end{pmatrix}, \\begin{pmatrix}\n\t\t\\delta_{\\tilde{x}} H(\\tilde{x}(\\cdot,t)) \\\\\n\t\te_I(t) \\\\\n\t\te_{\\partial} (t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\tilde{\\mathcal{J}}_0},\n\t\\end{align*}\nwith $\\mathcal{D}_{\\tilde{\\mathcal{J}}_0}$ the Dirac structure defined in \\eqref{Interface Paper - Dirac Structure DI}, the extended state vector $\\tilde{x} = (x, c_0, \\overline{c}_0)$, where $c_0$ and $\\overline{c}_0$ are precisely the color functions defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline}, the variational derivate \\eqref{Interface Paper - Extended Variational Derivatives} of the Hamiltonian $H\\colon \\tilde{X} \\to \\mathbb{R}$ defined in \\eqref{Interface Paper - Hamiltonian H of Coupled System}, the pair of port variables $(f_I,e_I) \\in \\mathbb{R}^2$ associated with the interface, and the pair of port variables $(f_{\\partial}, e_{\\partial}) \\in \\mathbb{R}^4$ associated with the boundary of the spatial domain $[a,b]$. \n\\end{corollary}\nAs a consequence of the boundary port-Hamiltonian structure, we can state a power balance equation along solutions $\\tilde{x} \\colon [0, \\infty) \\to \\tilde{X}$ of \\eqref{Interface Paper - Augmented System wrt Ja} of the form \n\\begin{align*}\n\t\\tilde{x}(t) = \\begin{bmatrix}\n\t\tx_1(\\cdot, t) \\\\\n\t\tx_2(\\cdot, t) \\\\\n\t\tc_0(\\cdot,t) \\\\\n\t\t\\overline{c}_0(\\cdot,t)\n\t\\end{bmatrix}, \\hspace{0.5 cm} t \\geq 0,\n\\end{align*}\nby means of the boundary and interface port variables: using \\eqref{Interface Paper - Skew-Symmetry of Ja - Representation 2}, for all $t > 0$ it holds that\n\\begin{align*}\n\n\t&\\hspace{0.5 cm} \\frac{d}{dt} H(\\tilde{x}(\\cdot,t)) \\\\\n\t&= \\frac{d}{dt} \\int_{a}^{b} c_0(z,t) \\mathcal{H}^-(x(z,t)) + \\overline{c}_0(z,t) \\mathcal{H}^+(x(z,t)) \\, dz \\\\\n\t&= \\int_{a}^{b} (\\partial_t c_0 (z,t)) \\mathcal{H}^-(x(z,t)) + c_0(z,t) \\partial_t \\mathcal{H}^-(x(z,t)) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{b} (\\partial_t \\overline{c}_0(z,t)) \\mathcal{H}^+(x(z,t)) + \\overline{c}_0(z,t) \\partial_t \\mathcal{H}^+(x(z,t)) \\, dz \\\\\n\t&= \\int_{a}^{b} c_0(z,t) \\left[ \\delta_{x_1}H^-(x(z,t)) \\partial_t x_1(z,t) + \\delta_{x_2}H^-(x(z,t)) \\partial_t x_2(z,t) \\right] \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\int_{a}^{b} \\overline{c}_0(z,t) \\left[ \\delta_{x_1}H^+(x(z,t)) \\partial_t x_1(z,t) + \\delta_{x_2}H^+(x(z,t)) \\partial_t x_2(z,t) \\right] \\, dz \\\\\n\t&= \\int_{a}^{b} c_0(z,t) \\left[ \\delta_{x_1}H^-(x(z,t)) (\\mathbf{d}_0 \\delta_{x_2}H^-(x(z,t))) + \\delta_{x_2}H^-(x(z,t)) \\left( - \\mathbf{d}_0^{\\ast} \\delta_{x_1}H^-(x(z,t)) \\right) \\right] \\, dz \\\\\n\t&\\hspace{0.5 cm } + \\int_{a}^{b} \\overline{c}_0(z,t) \\left[ \\delta_{x_1}H^+(x(z,t)) (\\mathbf{d}_0 \\delta_{x_2}H^+(x(z,t))) + \\delta_{x_2}H^+(x(z,t)) \\left( - \\mathbf{d}_0^{\\ast} \\delta_{x_1}H^+(x(z,t)) \\right) \\right] \\, dz \\\\\n\t&= \\frac{1}{2} \\left( \\langle \\tilde{\\mathcal{J}}_0 \\delta_{\\tilde{x}} H(\\tilde{x}(\\cdot,t)), \\delta_{\\tilde{x}} H(\\tilde{x}(\\cdot,t)) \\rangle_{L^2} + \\langle \\delta_{\\tilde{x}} H(\\tilde{x}(\\cdot,t)) , \\tilde{\\mathcal{J}}_0 \\delta_{\\tilde{x}} H(\\tilde{x}(\\cdot,t)) \\rangle_{L^2} \\right) \\\\\n\t&= \\langle e_{\\partial}(t), f_{\\partial}(t) \\rangle_2 - e_I(t) f_I(t).\n\n\n\\end{align*}\nThis equation expresses the fact that the change of energy is given by the power flow through the boundary minus the power flow at the interface position. An interesting question is whether this class of systems even admits a solution, and whether it is well-posed. We will postpone the answer to questions of this kind to Section \\ref{Section A Simplified System}, as in this section, we only wanted to lay the foundation for the upcoming segment.\n\\vspace{0.5 cm}\\\\\nIn this section, we have formulated two systems of conservation laws that are coupled by a fixed interface as a boundary port-Hamiltonian system with some power flow at the interface. In the following, we repeat this procedure, but we assume that the interface position is not fixed anymore, but moving over time. \n\n\n\\section{Port-Hamiltonian Systems Coupled by a Moving Interface} \n\\label{Section Port-Hamiltonian Systems Coupled through a Moving Interface}\nIn this section, we want to consider two systems of conservation laws that are coupled by a moving interface. We denote by $l\\colon [0, \\infty) \\to (a,b)$ the time-varying position of the interface and its velocity by $\\dot{l}\\colon [0, \\infty) \\to \\mathbb{R}$. In particular, we assume that $l$ is continuously differentiable. We proceed as in the last section. We begin by formulating the conservation laws in Subsection \\ref{Subsection Balance Equations of the State Variables - Moving Interface} and, hence, highlight the difference to the fixed interface case. Afterwards, we briefly describe the port-Hamiltonian formulation in case of a moving system in Subsection \\ref{Subsection Port-Hamiltonian Formulation - Moving Interface}. This yields an abstract control system, where the input is the velocity $\\dot{l}$ of the displacement of the interface. We will see that this system description poses delicate problems. Thus, we will discuss the main issues of this class of systems, and seek a simplified model closely related to the original problem that we are going to analyze in the further course. \n\n\\subsection{Balance Equations of the State Variables}\n\\label{Subsection Balance Equations of the State Variables - Moving Interface}\nSince the position of the interface $l$ is potentially moving, the systems of conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain} are now given by\n\\begin{align}\n\t\\begin{split}\n\t\t\\frac{\\partial}{\\partial t} x^- + \\frac{\\partial}{\\partial z} \\mathcal{N}_{l(t)}^-(x^-) &= 0, \\hspace{0.5 cm} z \\in [a,l(t)), \\hspace{0.1 cm} t >0, \\\\\n\t\t\\frac{\\partial}{\\partial t} x^+ + \\frac{\\partial}{\\partial z} \\mathcal{N}_{l(t)}^+(x^+) &= 0, \\hspace{0.5 cm} z \\in (l(t),b], \\hspace{0.1 cm} t > 0.\n\t\\end{split} \n\t\\label{Interface Paper - Two Systems of Conservation Laws on Respective Domain - Moving Interface} \n\\end{align}\nThis implies that the Hamiltonians of the respective system depend on the time as well. For $t \\geq 0$, the Hamiltonian functionals $H_{l(t)}^- \\colon L^2((a,l(t)), \\mathbb{R}^2) \\to \\mathbb{R}$ and $H_{l(t)}^+ \\colon L^2((l(t),b),\\mathbb{R}^2) \\to \\mathbb{R}$ are given as follows:\n\\begin{align*}\n\tH_{l(t)}^-(x^-) &= \\int_{a}^{l(t)} \\mathcal{H}^-(x^-(z)) \\, dz, \\hspace{0.5 cm} x^- \\in L^2((a,l(t)), \\mathbb{R}^2), \\\\\n\tH_{l(t)}^+(x^+) &= \\int_{l(t)}^{b} \\mathcal{H}^+(x^+(z)) \\, dz, \\hspace{0.5 cm} x^+ \\in L^2((l(t),b),\\mathbb{R}^2).\n\\end{align*}\nThe flux variables $\\mathcal{N}_{l(t)}^-(x^-)$ and $\\mathcal{N}_{l(t)}^+(x^+)$ appearing in \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain - Moving Interface} still satisfy the relation \\eqref{Interface Paper - Relation Flux Variable and Variational Derivative}.\nAs the interface is moving, the two color functions now depend on the time-varying position of the interface. They are given by\n\\begin{align}\n\tc_{l}(z,t) = \\begin{cases}\n\t\t1, & z \\in [a,l(t)), \\\\\n\t\t0, & z \\in [l(t),b],\n\t\\end{cases} \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\overline{c}_{l} (z,t) = \\begin{cases} \n\t0, & z \\in [a,l(t)], \\\\\n\t1, & z \\in (l(t),b],\n\t\\end{cases}\n\\label{Interface Paper - Color Functions Depending on Moving Interface}\n\\end{align}\nrespectively. The prolongated state variables, the Hamiltonian functional, and the prolongated flux variables may be defined similarly to the definitions \\eqref{Interface Paper - State Variable x Split}, \\eqref{Interface Paper - Hamiltonian H of Coupled System}, and \\eqref{Interface Paper - Flux Variable N Split}, respectively: for every $t\\geq 0$, we have\n\\begin{align*}\n\tx(z,t) &= x^-(z,t) + x^+(z,t) \\\\\n\t&= c_l(z,t) x(z,t) + \\overline{c}_l(z,t) x(z,t).\n\\end{align*}\nThe energy density is still given by\n\\begin{align*}\n\t\\mathcal{H}(\\tilde{x}) = \\mathcal{H}(x,c, \\overline{c}) = c \\mathcal{H}^-(x) + \\overline{c} \\mathcal{H}^+(x), \\hspace{0.5 cm} \\tilde{x} \\in \\mathbb{R}^4. \n\n\\end{align*}\nConsequently, if we choose $c = c_l$ and $\\overline{c} = \\overline{c}_l$ as in \\eqref{Interface Paper - Color Functions Depending on Moving Interface}, then the Hamiltonian functional is for all $\\tilde{x} = (x, c_l, \\overline{c}_l) \\in \\tilde{X}$ given by\n \\begin{align*}\n H(\\tilde{x}) &= \\int_{a}^{b} \\mathcal{H}(x(z),c_l(z), \\overline{c}_l(z)) \\, dz \\\\\n &= \\int_{a}^{b} c_l(z) \\mathcal{H}^-(x(z)) + \\overline{c}_l(z) \\mathcal{H}^+(x(z)) \\, dz \\\\\n &= \\int_{a}^{l} \\mathcal{H}^-(x(z)) \\, dz + \\int_{l}^{b} \\mathcal{H}^+(x(z)) \\, dz \\\\\n &= H_l^-(x^-) + H_l^+(x^+).\n\\end{align*}\nThus, for all $t \\geq 0$, the mapping $x \\mapsto H(x, c_l(\\cdot,t), \\overline{c}_l(\\cdot,t))$ describes the internally stored energy of the coupled system at time $t$. For any $(x,c_l, \\overline{c}_l) \\in \\tilde{X}$, the variational derivative is given by\n\\begin{align}\n\t\\label{Interface Paper - Variational Derivative - Moving Interface}\n\t\\delta_{\\tilde{x}} H(\\tilde{x}) = \\begin{bmatrix}\n\t\t\\delta_{x} H(x, c_l, \\overline{c}_l) \\\\\n\t\t\\delta_{c_l} H(x, c_l, \\overline{c}_l) \\\\\n\t\t\\delta_{\\overline{c}_l} H(x, c_l, \\overline{c}_l)\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\tc_l \\delta_{x} H_l^-(x) + \\overline{c}_l \\delta_{x} H_l^+(x) \\\\\n\t\t\\mathcal{H}^-(x) \\\\\n\t\t\\mathcal{H}^+(x)\n\t\\end{bmatrix},\n\\end{align}\nand the flux variables are again defined through\n\\begin{align}\n\t\\begin{split}\n\t\\begin{bmatrix}\n\t\t0 & 1 \\\\\n\t\t1 & 0 \n\t\\end{bmatrix} \\delta_{x} H(x, c_l, \\overline{c}_l) &= \\begin{bmatrix}\n\t\t0 & 1 \\\\\n\t\t1 & 0 \n\t\\end{bmatrix} c_l \\delta_x H_l^-(x) + \\overline{c}_l \\delta_x H_l^+ (x) \\\\\n\t&= c_l \\mathcal{N}_l^-(x) + \\overline{c}_l \\mathcal{N}_l^+(x) \\\\\n\t&= \\mathcal{N}(x, c_l, \\overline{c}_l).\n\\end{split}\n\\label{Interface Paper - Relation Flux Variable and Variational Derivative - Composed Domain - Moving Interface}\n\\end{align}\nNote that by prolongation of the flux variables $\\mathcal{N}_1$ and $\\mathcal{N}_2$ defined on the composed domain $[a,b]$, they do not explicitly depend on the position $l$ of the moving interface, but implicitly through the color functions \\eqref{Interface Paper - Color Functions Depending on Moving Interface}.\n\\vspace{0.5 cm}\\\\ \nDue to the change of the position, the interface relations and, with that said, the interface port variables become time-dependent as well. For $t \\geq 0$, these are given by\n\\begin{align}\n\tf_I(t) &= \\mathcal{N}_{l(t), 1}^+(x^+)(l(t)^+) = \\mathcal{N}_{l(t), 1}^-(x^-)(l(t)^-), \\label{Interface Paper - Interface Flow fI - Moving Interface} \\\\\n\t-e_I(t) &= \\mathcal{N}_{l(t), 2}^+(x^+)(l(t)^+) - \\mathcal{N}_{l(t), 2}^-(x^-)(l(t)^-), \\label{Interface Paper - Interface Effort e_I - Moving Interface} \n\\end{align}\nwhere $l(t) \\in (a,b)$.\n\\vspace{0.5 cm}\\\\\nNow, we may begin with stating the balance equations for the variables constituting the extended state variable $\\tilde{x} = (x,c_l, \\overline{c}_l) \\in \\tilde{X}$ in case of a moving interface. To this end, let $\\Omega := (a,b) \\times (0, \\infty)$, and let $\\mathcal{D}(\\Omega)$ denote the set of test functions on $\\Omega$. Throughout this section, we will show that the balance equations hold in $\\mathcal{D}'(\\Omega)$.\n\n\\paragraph*{Conservation laws of $\\mathbf{c_l}$ and $\\mathbf{\\overline{c}_l}$.} In contrast to the fixed interface case, where the respective color functions $c_0$ and $\\overline{c}_0$ satisfied trivial conservations laws, the color functions \\eqref{Interface Paper - Color Functions Depending on Moving Interface} are the solutions of the transport equations depending on the velocity $\\dot{l}$ of the interface, which are given by\n\\begin{align}\n\t\\partial_t c_l(z,t) = - \\dot{l}(t) \\partial_z c_l(z,t) \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\partial_t \\overline{c}_l(z,t) = - \\dot{l}(t) \\partial_z \\overline{c}_l(z,t)\n\t\\label{Interface Paper - Balance Equation of Color Function with Moving Interface}\n\\end{align}\nwith the initial conditions\n\\begin{align*}\n\tc_l(z,0) = c_{l_0} (z) \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\overline{c}_l(z,0) = \\overline{c}_{l_0}(z),\n\n\\end{align*}\nrespectively, where $l_0 = l(0) \\in (a,b)$. \\\\\n\\begin{figure}[h!]\n\t\\centering\n\n\t{{\\includegraphics[width=7.7764cm]{Char.png} }}%\n\t\\qquad\n\n\t{{\\includegraphics[width=7.7764cm]{Bump.png} }}%\n\t\\caption{Schematic Plots of the Color Function $c_l$ and of $c_l$ Tested Against some Test Function $\\varphi \\in \\mathcal{D}(\\Omega)$, respectively}\n\t\\label{Figure Char and Bump}\n\\end{figure}\n\nThe transport equations given in \\eqref{Interface Paper - Balance Equation of Color Function with Moving Interface} hold in a distributional sense. We only show this for the first equation, as the proof for the second equation works analogously. A plot of the color function $c_l$ defined with respect to some $\\mathcal{C}^1$-function $l$ and of $c_l$ tested against some test function $\\varphi \\in \\mathcal{D}(\\Omega)$ is depicted in Figure \\ref{Figure Char and Bump}.\n\\vspace{0.5 cm}\\\\\nLet $\\varphi \\in \\mathcal{D}(\\Omega)$. Then we have\n\\begin{align*}\n\t& \\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} - \\dot{l}(t) \\partial_z c_l(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t\\text{(part. int.)} \\hspace{0.3 cm}\t&= \\int_{0}^{\\infty} - \\dot{l}(t) \\left( \\big[c_l (z,t) \\varphi(z,t) \\big]_{z=a}^{b} - \\int_{a}^{b} c_l(z,t) \\partial_z \\varphi(z,t) \\, dz \\right) \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\dot{l}(t) \\int_{a}^{l(t)} \\partial_z \\varphi(z,t) \\, dz \\, dt\\\\\n\t&= \\int_{0}^{\\infty} \\dot{l}(t) \\varphi(l(t),t) \\, dt. \n\\end{align*}\nFurthermore, by applying Fubini's theorem and Leibniz's rule we obtain\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t c_l(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{a}^{b} \\big[ c_l(z,t) \\varphi(z,t) \\big]_{t = 0}^{\\infty} - \\int_{0}^{\\infty} c_l(z,t) \\partial_t \\varphi(z,t) \\, dt \\, dz \\\\\n\t&= \\int_{0}^{\\infty} - \\int_{a}^{l(t)} \\partial_t \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} - \\frac{d}{dt} \\int_{a}^{l(t)} \\varphi(z,t) \\, dz + \\varphi(l(t),t) \\dot{l}(t) \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\dot{l}(t) \\varphi(l(t),t) \\, dt \\underbrace{- \\int_{0}^{\\infty} \\frac{d}{dt} \\int_{a}^{l(t)} \\varphi(z,t) \\, dz \\, dt}_{=0} \\\\\n\t&= \\int_{0}^{\\infty} \\dot{l}(t) \\varphi(l(t),t) \\, dt.\n\\end{align*}\nThus, the conservation laws given in \\eqref{Interface Paper - Balance Equation of Color Function with Moving Interface} hold in $\\mathcal{D}'(\\Omega)$.\n\\vspace{0.5 cm}\\\\\nFor all $t >0$ and on any interval $[a',b'] \\subset [a,b]$ not containing the interface position $l(t)$, it holds that\n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} c_l(z,t) \\, dz = \t\\frac{d}{dt} \\int_{a'}^{b'} \\overline{c}_l(z,t) \\, dz = 0. \n\\end{align*}\nIf $a \\leq a' < l(t) < b' \\leq b$, then applying Leibniz's rule once again yields\n\\begin{align*}\n\t\t\\frac{d}{dt} \\int_{a'}^{b'} c_l(z,t) \\, dz = \t\\frac{d}{dt} \\left( \\int_{a'}^{l(t)} 1 \\, dz + \\int_{l(t)}^{b'} 0 \\, dz \\right) = \\dot{l}(t)\n\\end{align*}\nand, analogously, \n\\begin{align*}\n\t\\frac{d}{dt} \\int_{a'}^{b'} \\overline{c}_l(z,t) \\, dz = \\frac{d}{dt} \\left( \\int_{a'}^{l(t)} 0 \\, dz + \\int_{l(t)}^{b'} 1 \\, dz \\right) = - \\dot{l}(t).\n\\end{align*}\nNext, we want to find the conservation laws for the state variables $x_1$ and $x_2$.\n\\paragraph*{Conservation law of $\\mathbf{x_1}$.}\nAs in the fixed interface scenario, the flux variable $\\mathcal{N}_1(x, c_l, \\overline{c}_l)$ is supposed to satisfy the continuity condition \\eqref{Interface Paper - Interface Flow fI - Moving Interface}. For $l \\in (a,b)$, define the operator $\\mathbf{d}_l \\colon D(\\mathbf{d}_l) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R})$ by\n\\begin{align}\n\tD(\\mathbf{d}_l) &= \\left\\{ x \\in L^2([a,b], \\mathbb{R}) \\mid x_{|(a,l)} \\in H^1((a,l), \\mathbb{R}), \\hspace{0.1 cm} x_{|(l,b)} \\in H^1((l,b),\\mathbb{R}), \\hspace{0.1 cm} x \\in \\mathcal{C}([a,b], \\mathbb{R}) \\right\\} \\nonumber \\\\\n\t&= H^1([a,b], \\mathbb{R}), \\label{Interface Paper - Operator dl} \\\\\n\t\\mathbf{d}_lx &= - \\left[ \\frac{d}{dz} (c_l x) + \\frac{d}{dz} (\\overline{c}_l x) \\right], \\hspace{0.5 cm} x \\in D(\\mathbf{d}_l). \\nonumber\n\\end{align}\nAs the color functions \\eqref{Interface Paper - Color Functions Depending on Moving Interface} are changing over time, for $t >0$ the operator \n\\begin{align*}\n\t\\mathbf{d}(t) = \\mathbf{d}(\\tilde{x}(t)) \\colon D(\\mathbf{d}(t)) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R})\n\\end{align*} \nis modulated by the state $\\tilde{x}(t) = (x(\\cdot,t), c_l(\\cdot,t), \\overline{c}_l(\\cdot,t)) \\in \\tilde{X}$, and is given as follows:\n\\begin{align}\n\t\\begin{split}\n\tD(\\mathbf{d}(t)) &= D(\\mathbf{d}_l) = H^1([a,b], \\mathbb{R}), \\\\\n\t\\mathbf{d}(t) x &= - \\left[ \\partial_z (c_l(\\cdot,t) x) + \\partial_z (\\overline{c}_l(\\cdot,t) x) \\right], \\hspace{0.5 cm} x \\in D(\\mathbf{d}(t)).\n\\end{split}\n\\label{Interface Paper - Operator d(t)}\n\\end{align}\nLet us now define the conservation law of the state variable $x_1$ on $[a,b]$ in case of a moving interface. For $t >0$, assume that $\\mathcal{N}_1(x,c_l, \\overline{c}_l)(\\cdot,t) \\in D(\\mathbf{d}(t))$, and let $\\varphi \\in \\mathcal{D}(\\Omega)$. Using the individual conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain - Moving Interface} and the continuity condition \\eqref{Interface Paper - Interface Flow fI - Moving Interface}, which is once again included in the domain of $\\mathbf{d}(t)$, one computes\n\\begin{align}\n\t\\begin{split}\n\t&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} [\\mathbf{d}(t) \\mathcal{N}_1(x, c_l, \\overline{c}_l) (z,t)] \\varphi (z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\mathcal{N}_1^-(l(t)^-,t) \\varphi(l(t),t) - \\mathcal{N}_1^+(l(t)^+,t) \\varphi(l(t),t) \\\\\n\t&\\hspace{0.5 cm} - \\int_{a}^{l(t)} \\partial_z \\mathcal{N}_1^-(z,t) \\varphi (z,t) \\, dz - \\int_{l(t)}^{b} \\partial_z \\mathcal{N}_1^+(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\int_{a}^{l(t)} \\partial_t x_1^-(z,t) \\varphi (z,t) \\, dz +\\int_{l(t)}^{b} \\partial_t x_1^+(z,t) \\varphi(z,t) \\, dz \\, dt.\n\\end{split}\n\\label{Interface Paper - Hilfe1}\n\\end{align}\nFurthermore, we have\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t x_1(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t \\left[ c_l(z,t) x_1(z,t) + \\overline{c}_l(z,t) x_1(z,t) \\right] \\varphi(z,t) \\, dz \\, dt \\\\\n\\text{(Fubini)} \\hspace{0.3 cm}\t&= - \\int_{0}^{\\infty} \\int_{a}^{l(t)} x_1(z,t) \\partial_t \\varphi(z,t) \\, dz \\, dt \\\\\n\t&\\hspace{0.5 cm} - \\int_{0}^{\\infty} \\int_{l(t)}^{b} x_1(z,t) \\partial_t \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= - \\int_{0}^{\\infty} \\int_{a}^{l(t)} \\partial_t (x_1\\varphi)(z,t) - \\partial_t x_1(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t&\\hspace{0.5 cm} - \\int_{0}^{\\infty} \\int_{l(t)}^{b} \\partial_t (x_1\\varphi)(z,t) - \\partial_t x_1(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n\t \\text{(Leibniz)} \\hspace{0.3 cm} &= \\int_{0}^{\\infty} \\int_{a}^{l(t)} \\partial_t x_1^-(z,t) \\varphi(z,t) \\, dz + \\dot{l}(t) x_1^-(l(t)^-,t) \\varphi(l(t),t) \\, dt \\\\\n\t &\\hspace{0.5 cm} + \\int_{0}^{\\infty} \\int_{l(t)}^{b} \\partial_t x_1^+(z,t) \\varphi(z,t) \\, dz - \\dot{l}(t) x_1^+(l(t)^+,t) \\varphi(l(t),t) \\, dt,\n\\end{align*}\nwhere in the last step we additionally used the fact that\n\\begin{align*}\n\t\\int_{0}^{\\infty} \\frac{d}{dt} \\int_{a}^{l(t)} x_1^-(z,t) \\varphi(z,t) \\, dz \\, dt = \n\t\\int_{0}^{\\infty} \\frac{d}{dt} \\int_{l(t)}^{b} x_1^+(z,t) \\varphi(z,t) \\, dz \\, dt = 0. \n\\end{align*}\n\nTogether with \\eqref{Interface Paper - Hilfe1}, we conclude that for all $\\varphi \\in \\mathcal{D}(\\Omega)$ it holds that\n\\begin{align*}\n&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t x_1(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n &= \\int_{0}^{\\infty} \\int_{a}^{b} \\mathbf{d}(t) \\mathcal{N}_1(z,t) \\varphi (z,t) \\, dz - \\dot{l}(t) \\left[ x_1^+(l(t)^+,t) - x_1^-(l(t)^-,t) \\right] \\varphi(l(t),t) \\, dt.\n\\end{align*}\nThus, the equation\n\\begin{align}\n\t\\partial_t x_1 = \\mathbf{d}(t) \\mathcal{N}_1(x, c_l, \\overline{c}_l) - \\dot{l} \\left[ c_l x_1 \\partial_z c_l + \\overline{c}_l x_1 \\partial_z \\overline{c}_l \\right]\n\t\\label{Interface Paper - Conservation Law x1 - Moving Interface}\n\\end{align}\nholds in $\\mathcal{D}'(\\Omega)$. We want to stress that, in \\eqref{Interface Paper - Conservation Law x1 - Moving Interface}, the equation\n\\begin{align*}\n\tc_l \\partial_t x_1^- + \\overline{c}_l \\partial_t x_1^+ = \\mathbf{d}(t) \\mathcal{N}_1(x, c_l, \\overline{c}_l) \n\\end{align*}\nholds in $L^2([a,b], \\mathbb{R})$. Since the color functions $c_l$ and $\\overline{c}_l$ do not satisfy the trivial conservation laws anymore, but the transport equation \\eqref{Interface Paper - Balance Equation of Color Function with Moving Interface}, equation \\eqref{Interface Paper - Conservation Law x1 - Moving Interface} comprises an equality solely holding in $\\mathcal{D}'(\\Omega)$ as well, namely\n\\begin{align*}\n\tx_1 \\partial_t c_l + x_1 \\partial_t \\overline{c}_l = - \\dot{l} \\left[ c_l x_1 \\partial_z c_l + \\overline{c}_l x_1 \\partial_z \\overline{c}_l \\right].\n\\end{align*}\nOnce again, on any interval $[a',b'] \\subset [a,b]$ not containing the interface position $l(t)$, $t >0$, the operator $\\mathbf{d}(t)$ simply acts as the differential operator $- \\frac{d}{dz}$. However, due to the moving interface, on intervals $[a',b']$ with $a \\leq a' < l(t) < b' \\leq b$ there appears an additional term depending on the velocity $\\dot{l}$ of the interface:\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\frac{d}{dt} \\int_{a'}^{b'} x_1(z,t) \\, dz \\\\\n\t&= \\frac{d}{dt} \\left( \\int_{a'}^{l(t)} x_1^-(z,t) \\, dz + \\int_{l(t)}^{b'} x_1^+(z,t) \\, dz \\right) \\\\\n\t\\text{(Leibniz)} \\hspace{0.3 cm}\t&= \\int_{a'}^{l(t)} \\partial_t x_1^-(z,t) \\, dz + \\int_{l(t)}^{b'} \\partial_t x_1^+(z,t) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\dot{l}(t) x_1^-(l(t)^-,t) - \\dot{l}(t) x_1^+(l(t)^+,t) \\\\\n\t&= \\int_{a'}^{b'} \\mathbf{d}(t) \\mathcal{N}_1(x,c_l, \\overline{c}_l)(z,t) \\, dz - \\dot{l}(t) \\left[ x_1^+(l(t)^+,t) - x_1^-(l(t)^-,t) \\right] \\\\\n\t&= \\mathcal{N}_1^-(a',t) - \\mathcal{N}_1^+(b',t) - \\dot{l}(t) \\left[ x_1^+(l(t)^+,t) - x_1^-(l(t)^-,t) \\right].\n\\end{align*}\n\\paragraph*{Conservation law of $\\mathbf{x_2}$.} Lastly, we want to formulate the conservation law of the state variable $x_2$ in case of a moving interface, where the corresponding flux variable $\\mathcal{N}_2(x,c_l, \\overline{c}_l)$ is supposed to satisfy the balance equation \\eqref{Interface Paper - Interface Effort e_I - Moving Interface} at all times. Beforehand, we once again need to determine the formally adjoint operator of $\\mathbf{d}(t)$, since the conservation law of $x_2$ has to be formulated with respect to $\\mathbf{d}^{\\ast}(t)$, $t>0$. Analogously to the operator $\\mathbf{d}_0^{\\ast}$ defined in \\eqref{Interface Paper - Operator d_0ast}, for $l \\in (a,b)$ one defines the formal adjoint $\\mathbf{d}_l^{\\ast} \\colon D(\\mathbf{d}_0^{\\ast}) \\subset X \\to X$ of $\\mathbf{d}_l$ through\n\\begin{align}\n\t\\begin{split}\n\tD(\\mathbf{d}_l^{\\ast}) &= \\left\\{ y \\in X \\mid y_{|(a,l)} \\in H^1((a,l), \\mathbb{R}), \\hspace{0.1 cm} y_{|(l,b)} \\in H^1((l,b), \\mathbb{R}) \\right\\}, \\\\\n\t\\mathbf{d}_l^{\\ast} y &= \\left[ \\frac{d}{dz} (c_l y) + \\frac{d}{dz} (\\overline{c}_l y) \\right] - \\left[ \\frac{d}{dz} c_l + \\frac{d}{dz} \\overline{c}_l \\right] y \\\\\n\t&= \\left( - \\mathbf{d}_l - \\left[ \\frac{d}{dz} c_l + \\frac{d}{dz} \\overline{c}_l \\right] \\right) y. \n\t\\end{split}\n\\label{Interface Paper - Operator dlast}\n\\end{align}\nThe relation between $\\mathbf{d}_l$ and its formal adjoint $\\mathbf{d}_l^{\\ast}$ defined in \\eqref{Interface Paper - Operator dl} and \\eqref{Interface Paper - Operator dlast}, respectively, is given as follows: for $x \\in D(\\mathbf{d}_l)$, $y \\in D(\\mathbf{d}_l^{\\ast})$ we have\n\\begin{align*}\n\t\\langle \\mathbf{d}_l x, y \\rangle_{L^2} = - \\big[ x(z) y(z) \\big]_{a}^{b} + x(l) \\left[ y(l^+) - y(l^-) \\right] + \\langle x, \\mathbf{d}_l^{\\ast} y \\rangle_{L^2}.\n\\end{align*}\n\n\nHence, the operator \n\\begin{align*}\n\\mathbf{d}^{\\ast}(t) = \\mathbf{d}^{\\ast}(\\tilde{x}(t)) \\colon D(\\mathbf{d}^{\\ast}(t)) \\subset L^2([a,b], \\mathbb{R}) \\to L^2([a,b], \\mathbb{R}), \\hspace{0.5 cm} t > 0,\n\\end{align*} \nis modulated by the color functions $c_l$, $\\overline{c}_l$, and thus by the extended state $\\tilde{x} = (x,c_l, \\overline{c}_l) \\in \\tilde{X}$. Unlike $\\mathbf{d}(t)$ defined in \\eqref{Interface Paper - Operator d(t)}, the domain of $\\mathbf{d}^{\\ast}(t)$ potentially changes over time. However, as the interface position $l$ is assumed to be continuously differentiable, the time-dependence of the domain is somewhat regular. For $t > 0$, the operator $\\mathbf{d}^{\\ast}(t)$ is defined by\n\\begin{align}\n\t\\begin{split}\n\tD(\\mathbf{d}^{\\ast}(t)) &= \\left\\{ y \\in X \\mid y_{|(a,l(t))} \\in H^1((a,l(t)), \\mathbb{R}), \\hspace{0.1 cm} y_{|(l(t),b)} \\in H^1((l(t),b), \\mathbb{R}) \\right\\}, \\\\\n\t\\mathbf{d}^{\\ast}(t) y &= \\left[ \\partial_z (c_l(\\cdot,t) y) + \\partial_z (\\overline{c}_l(\\cdot,t) y) \\right] - \\left[ \\partial_z c_l(\\cdot,t) + \\partial_z \\overline{c}_l(\\cdot,t) \\right] y \\\\\n\t&= \\left( - \\mathbf{d}(t) - \\left[ \\partial_z c_l(\\cdot,t) + \\partial_z \\overline{c}_l(\\cdot,t) \\right] \\right) y.\n\\end{split}\n\\label{Interface Paper - Operator d(t)ast}\n\\end{align}\nFor $t >0$, $x \\in D(\\mathbf{d}(t))$, and $y \\in D(\\mathbf{d}^{\\ast}(t))$, the relation between the operators $\\mathbf{d}(t)$ and $\\mathbf{d}^{\\ast}(t)$ is given as follows:\n\\begin{align*}\n\t\\langle \\mathbf{d}(t) x, y \\rangle_{L^2}= - \\big[ x(z) y(z) \\big]_{a}^{b} + x(l(t)) \\left[ y(l(t)^+) - y(l(t)^-) \\right] + \\langle x, \\mathbf{d}^{\\ast}(t) y \\rangle_{L^2}.\n\\end{align*}\n\nNow we are able to derive the conservation law of $x_2$ with respect to the operator $\\mathbf{d}^{\\ast}(t)$, $t>0$. Assume that $\\mathcal{N}_2(x, c_l, \\overline{c}_l)(\\cdot,t) \\in D(\\mathbf{d}^{\\ast}(t))$ for all $t > 0$. By virtue of the individual conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain - Moving Interface} and with the same arguments as for the conservation law of the state variable $x_1$, we have for all $\\varphi \\in \\mathcal{D}(\\Omega)$,\n\\begin{align*}\n &\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t x_2(z,t) \\varphi(z,t) \\, dz \\, dt \\\\\n &= \\int_{0}^{\\infty} \\int_{a}^{l(t)} \\partial_t x_2^-(z,t) \\varphi(z,t) \\, dz + \\dot{l}(t) x_2^-(l(t)^-,t) \\varphi(l(t),t) \\, dt \\\\\n\t&\\hspace{0.5 cm} + \\int_{0}^{\\infty} \\int_{l(t)}^{b} \\partial_t x_2^+(z,t) \\varphi(z,t) \\, dz - \\dot{l}(t) x_2^+(l(t)^+,t) \\varphi(l(t),t) \\, dt.\n\\end{align*}\nFurthermore, we observe that\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} [- \\mathbf{d}^{\\ast}(t) \\mathcal{N}_2(x,c_l, \\overline{c}_l) (z,t)] \\varphi(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\int_{a}^{l(t)} - \\partial_z \\mathcal{N}_2^-(z,t) \\varphi (z,t) \\, dz + \\int_{l(t)}^{b} - \\partial_z \\mathcal{N}_2^+(z,t) \\varphi (z,t) \\, dz \\, dt \\\\\n\t&= \\int_{0}^{\\infty} \\int_{a}^{l(t)} \\partial_t x_2^-(z,t) \\varphi(z,t) \\, dz + \\int_{l(t)}^{b} \\partial_t x_2^+(z,t) \\varphi(z,t) \\, dz \\, dt. \n\\end{align*}\nIn conclusion, the equation\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{0}^{\\infty} \\int_{a}^{b} \\partial_t x_2(z,t) \\varphi (z,t) \\, dz \\, dt \\\\\n\t &= \\int_{0}^{\\infty} \\int_{a}^{b} - \\mathbf{d}^{\\ast}(t) \\mathcal{N}_2(z,t) \\varphi(z,t) \\, dz - \\dot{l}(t) \\left[ x_2^+(l(t)^+,t) - x_2^-(l(t)^-,t) \\right] \\varphi(l(t),t) \\, dt\n\\end{align*}\nholds for all $\\varphi \\in \\mathcal{D}(\\Omega)$. Thus, the conservation law is given by\n\\begin{align}\n\t\\partial_t x_2 = - \\mathbf{d}^{\\ast}(t) \\mathcal{N}_2(x, c_l, \\overline{c}_l) - \\dot{l} \\left[ c_l x_2 \\partial_z c_l + \\overline{c}_l x_2 \\partial_z \\overline{c}_l \\right],\n\t\\label{Interface Paper - Conservation Law x2 - Moving Interface}\n\\end{align}\nand it holds in $\\mathcal{D}'(\\Omega)$. As for the conservation law of the state variable $x_1$, we want to emphasize that the equation \n\\begin{align*}\n\tc_l \\partial_t x_2^- + \\overline{c}_l \\partial_t x_2^+ = - \\mathbf{d}^{\\ast}(t) \\mathcal{N}_2(x, c_l, \\overline{c}_l) \n\\end{align*}\nholds in a classical sense, whereas\n\\begin{align*}\n\tx_2 \\partial_t c_l + x_2 \\partial_t \\overline{c_l } = - \\dot{l} \\left[ c_l x_2 \\partial_z c_l + \\overline{c}_l x_2 \\partial_z \\overline{c}_l \\right]\n\\end{align*}\nmerely holds in $\\mathcal{D}'(\\Omega)$. Recall from Remark \\ref{Interface Paper - Remark Balance Equation} that the conservation law \\eqref{Interface Paper - Conservation Law x2 - Moving Interface} does not encompass the balance equation \\eqref{Interface Paper - Interface Effort e_I - Moving Interface}. As proposed, for simplification of the model we do not consider $e_I(t)$ as an external source term, but define it to quantify the jump discontinuity at the moving interface position $l(t)$, $t \\geq 0$. \n\\vspace{0.5 cm}\\\\\nAs usual, on subdomains $[a',b'] \\subset [a,b]$ not containing $l(t)$, $t>0$, the operator $\\mathbf{d}^{\\ast}(t)$ acts as the differential operator $-\\frac{d}{dz}$. \nHowever, due to the moving interface and the jump discontinuity of the flux variable $\\mathcal{N}_2$ at the interface position, on $[a',b']$ with $a \\leq a' < l(t) < b' \\leq b$ it holds that \\\\\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\frac{d}{dt} \\int_{a'}^{b'} x_2 (z,t) \\, dz \\\\\n\t&= \\frac{d}{dt} \\left( \\int_{a'}^{l(t)} x_2^-(z,t) \\, dz + \\int_{l(t)}^{b'} x_2^+(z,t) \\, dz \\right)\\\\\n\t&= \\int_{a'}^{l(t)} \\partial_t x_2^-(z,t) \\, dz + \\int_{l(t)}^{b'} \\partial_t x_2^+(z,t) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\dot{l}(t) x_2^-(l(t)^-,t) - \\dot{l}(t) x_2^+(l(t)^+,t) \\\\\n\t&= \\int_{a'}^{b'} - \\mathbf{d}^{\\ast}(t) \\mathcal{N}_2(x, c_l, \\overline{c}_l) \\, dz - \\dot{l}(t) \\left[ x_2^+(l(t)^+,t) - x_2^-(l(t)^-,t) \\right] \\\\\n\t&= \\mathcal{N}_2^-(a',t) - \\mathcal{N}_2^+(b',t) - \\mathcal{N}_2^-(l(t)^-,t) + \\mathcal{N}_2^+(l(t)^+,t) - \\dot{l}(t) \\left[ x_2^+(l(t)^+,t) - x_2^-(l(t)^-,t) \\right] \\\\ \n\t&= \\mathcal{N}_2^-(a',t) - N_2^+(b',t) - e_I(t) - \\dot{l}(t) \\left[ x_2^+(l(t)^+,t) - x_2^-(l(t)^-,t) \\right]. \n\\end{align*}\nAfter formulating the conservation laws of the variables constituting the extended state $\\tilde{x} = (x,c_l, \\overline{c}_l) \\in \\tilde{X}$, we can eventually discuss the port-Hamiltonian formulation of the systems of conservation laws \\eqref{Interface Paper - Two Systems of Conservation Laws on Respective Domain - Moving Interface} that are coupled through a moving interface. \n\n\\subsection{Port-Hamiltonian Formulation}\n\\label{Subsection Port-Hamiltonian Formulation - Moving Interface}\n\nIn the previous section, we have derived four conservation laws of the constituting state variables with respect to a moving interface, stated in \\eqref{Interface Paper - Balance Equation of Color Function with Moving Interface}, \\eqref{Interface Paper - Conservation Law x1 - Moving Interface}, and \\eqref{Interface Paper - Conservation Law x2 - Moving Interface}. All of these equations hold in $\\mathcal{D}'(\\Omega)$. We now want to formulate a (generalized) Hamiltonian system which comprises these conservation laws. As opposed to the fixed interface scenario, this yields (formally) an abstract control system. For a complete port-Hamiltonian formulation, we would need to consider the conjugated output as well (cf. \\eqref{Nonlinear PH-System - Finite Dimensional} in the finite-dimensional case). In \\cite{Diagne}, the adjoint output map has been outlined, but in this thesis, we will not address this issue.\n\\vspace{0.5 cm}\\\\ \nOmitting the dependence on the spatial variable $z$, and with the aid of \\eqref{Interface Paper - Relation Flux Variable and Variational Derivative - Composed Domain - Moving Interface} we formally obtain the system\n\\begin{align*}\n\t\\frac{\\partial}{\\partial t} \\begin{bmatrix}\n\t\tx_1(t) \\\\\n\t\tx_2(t)\\\\\n\t\tc_l(t) \\\\\n\t\t\\overline{c}_l(t)\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\t\t0 & \\mathbf{d}(t) & 0 & 0 \\\\\n\t\t\t-\\mathbf{d}^{\\ast}(t) & 0 & 0 & 0 \\\\\n\t\t\t0 & 0 & 0 & 0 \\\\\n\t\t\t0 & 0 & 0 & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\\delta_{x_1}H(x, c_l, \\overline{c}_l)(t) \\\\\n\t\t\\delta_{x_2}H(x, c_l, \\overline{c}_l)(t) \\\\\n\t\t\\delta_{c_l}H(x, c_l, \\overline{c}_l)(t) \\\\\n\t\t\\delta_{\\overline{c}_l} H(x, c_l, \\overline{c}_l)(t) \n\t\\end{bmatrix} - \\begin{bmatrix}\n\t\t\t(c_l x_1)(t) & (\\overline{c}_l x_1)(t) \\\\\n\t\t\t(c_l x_2)(t) & (\\overline{c}_l x_2)(t) \\\\\n\t\t\t1 & 0 \\\\\n\t\t\t0 & 1 \n\t\t\\end{bmatrix} \\partial_z \\begin{bmatrix}\n\t\t\tc_l(t) \\\\\n\t\t\t\\overline{c}_l(t)\n\t\\end{bmatrix} \\dot{l}(t),\n\\end{align*}\nor short,\n\\begin{align}\n\t\\frac{\\partial}{\\partial t}\\tilde{x}(t) = \\tilde{\\mathcal\n\t{J}}(\\tilde{x}(t)) \\delta_{\\tilde{x}}H(\\tilde{x}(t)) + \\tilde{B}(\\tilde{x}(t)) \\dot{l}(t). \n\t\\label{Interface Paper - Augmented System wrt Jl}\n\\end{align}\n\n\nFor $\\tilde{x}(t) = (x_1(t),x_2(t), c_l(t), \\overline{c}_l(t)) \\in \\tilde{X}$, the operator $\\tilde{\\mathcal{J}}(\\tilde{x}(t)) \\colon D(\\tilde{\\mathcal{J}}(\\tilde{x}(t))) \\subset \\tilde{X} \\to \\tilde{X}$ given by\n\\begin{align*}\nD(\\tilde{\\mathcal{J}}(\\tilde{x}(t))) &= D(\\mathbf{d}^{\\ast}(t)) \\times D(\\mathbf{d}(t)) \\times L^2([a,b], \\mathbb{R}^2), \\\\\n\\tilde{\\mathcal{J}}(\\tilde{x}(t)) \\tilde{y} &= \\begin{bmatrix}\n\t0 & \\mathbf{d}(t) & 0 & 0 \\\\\n\t-\\mathbf{d}^{\\ast}(t) & 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & 0 \n\\end{bmatrix} \\tilde{y}, \\hspace{0.5 cm} \\tilde{y} \\in D(\\tilde{\\mathcal{J}}(\\tilde{x}(t))), \n\\end{align*}\nis modulated by the extended state $\\tilde{x}(t) \\in \\tilde{X}$, since the operator $\\mathbf{d}(t)$ and its formal adjoint $\\mathbf{d}^{\\ast}(t)$ defined in \\eqref{Interface Paper - Operator d(t)} and \\eqref{Interface Paper - Operator d(t)ast}, respectively, depend on the time-varying color functions $c_l$ and $\\overline{c}_l$ defined in \\eqref{Interface Paper - Color Functions Depending on Moving Interface}. The variational derivative $\\delta_{\\tilde{x}} H \\colon \\tilde{X} \\to \\tilde{X}$ is defined as in \\eqref{Interface Paper - Variational Derivative - Moving Interface}. \n\\vspace{0.5 cm}\\\\\nIf we take the velocity $\\dot{l}$ of the interface position to be the input of the system \\eqref{Interface Paper - Augmented System wrt Jl}, then one may define the mapping $B(\\tilde{x}(t))$ as the input map with respect to the input $\\dot{l}$, i.e., \n\\begin{align}\n\tB(\\tilde{x}(t)) \\colon \\dot{l} \\mapsto - \\begin{bmatrix}\n\t\t(c_l x_1)(t) & (\\overline{c}_l x_1)(t) \\\\\n\t\t(c_l x_2)(t) & (\\overline{c}_l x_2)(t) \\\\\n\t\t1 & 0 \\\\\n\t\t0 & 1 \n\t\\end{bmatrix} \\partial_z \\begin{bmatrix}\n\t\tc_l(t) \\\\\n\t\t\\overline{c}_l(t)\n\\end{bmatrix} \\dot{l}, \\hspace{0.5 cm} \\dot{l} \\in \\mathbb{R}.\n\\label{Interface Paper - Operator Bx}\n\\end{align}\nIn the following, we want to give some comments on the obtained system description \\eqref{Interface Paper - Augmented System wrt Jl} of two systems of conservation laws that are coupled by some moving interface defined on the extended state space $\\tilde{X} = L^2([a,b], \\mathbb{R}^4)$.\n\\vspace{0.5 cm}\\\\\nFirst and foremost, recall that on any interval not containing the interface position, each conservation law stated in \\eqref{Interface Paper - Balance Equation of Color Function with Moving Interface}, \\eqref{Interface Paper - Conservation Law x1 - Moving Interface}, and \\eqref{Interface Paper - Conservation Law x2 - Moving Interface} holds in a classical sense. However, in contrast to the fixed interface scenario discussed in Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface}, these conservation laws defined on the composed domain $[a,b]$ contain an additional term consisting of evaluations of the extended state variables at the interface position multiplied by the velocity of the moving interface. Regarding the system \\eqref{Interface Paper - Augmented System wrt Jl}, these point evaluations are captured in the entries of the input mapping $B(\\tilde{x})$. Thus, one possible approach to deal with this class of systems is to define another state space comprising the following information:\n\\begin{enumerate}[label = (\\roman*)]\n\t\\item the position of the moving interface $l(t)$ and its velocity $\\dot{l}(t)$, $t > 0$,\n\t\\item the solutions on the respective subdomains $[a,l(t))$ and $(l(t),b]$, $t >0$, and\n\t\\item the right-hand limits $x_1^+(l(t)^+,t)$, $x_2^+(l(t)^+,t)$ and the left-hand limits $x_1^-(l(t)^-,t)$, $x_2^-(l(t)^-,t)$ of the state variables $x_1$ and $x_2$, $t > 0$.\n\\end{enumerate}\nNevertheless, for the analysis, we wish to write the system \\eqref{Interface Paper - Augmented System wrt Jl} as an abstract control system of the form\n\\begin{align*}\n\t\\dot{x}(t) = Ax(t) + Bu(t), \\hspace{0.5 cm} t > 0,\n\\end{align*}\nas discussed in Section \\ref{Section Linear Control Systems}, and check the sufficient conditions to prove well-posedness of the system. Unfortunately, this is not offhandedly possible due to a number of reasons. We have identified the following main issues:\n\\begin{enumerate}[label =(P\\arabic*)]\n\\item \\label{Problem1MovingInterface} From \\eqref{Interface Paper - Augmented System wrt Jl} we infer that the operator $A$ is modulated by the state and, in particular, changes over time. So for each time instance $t >0$ and $\\tilde{x}(t)$, we need to define the operator $\\tilde{A}(\\tilde{x}(t)) \\colon D(\\tilde{A}(\\tilde{x}(t))) \\subset \\tilde{X} \\to \\tilde{X}$ with\n\\begin{align}\n\t\\tilde{A}(\\tilde{x}(t))\\tilde{y} = \\tilde{\\mathcal{J}}(\\tilde{x}(t)) \\delta_{\\tilde{y}} H(\\tilde{y}), \\hspace{0.5 cm} \\tilde{y} \\in D(\\tilde{A}(\\tilde{x}(t))).\n\t\\label{Interface Paper - Operator tildeA}\n\\end{align}\nThis yields a family $(\\tilde{A}( \\tilde{x}(t)))_{t \\geq 0}$ of state-dependent operators. As a consequence, this system does not belong to the class of systems considered in Section \\ref{Section Linear Control Systems}. Furthermore, the state $\\tilde{x}(t)$ has to be of the form\n\\begin{align*}\n\t\\tilde{x}(t) = \\begin{bmatrix}\n\t\tx_1(t) \\\\\n\t\tx_2(t) \\\\\n\t\tc_l(t) \\\\\n\t\t\\overline{c}_l(t)\n\t\\end{bmatrix},\n\\end{align*} \ni.e., it has to be defined with respect to the color functions $c_l$, and $\\overline{c}_l$. That is because the operators $\\mathbf{d}(t)$ and $\\mathbf{d}^{\\ast}(t)$ constituting the matrix differential operator $\\tilde{\\mathcal{J}}(\\tilde{x}(t))$ depend (in order do keep track of the interface position) on the color functions, and each conservation law has been defined with respect to $c_l$ and $\\overline{c}_l$. With that said, the domain $D(\\tilde{A}(\\tilde{x}(t)))$ has to be vastly restricted, such that $\\tilde{A}(\\tilde{x}(t))$ cannot be densely defined, let alone the infinitesimal generator of a $C_0$-semigroup. This is a huge problem, as this is a basic requirement to prove well-posedness of abstract control systems, see Proposition~\\ref{Proposition Well-Posedenss of Abstract Control Systems}.\n\n\\item \\label{Problem2MovingInterface} The input map $B(\\tilde{x}(t))$ is modulated by the state $\\tilde{x}(t)$ as well. Aside from that, there would be several problems even if $B$ were independent of the extended state $\\tilde{x}$. Recall that we claimed that the conservation laws of the constituting state variables derived in Subsection \\ref{Subsection Balance Equations of the State Variables - Moving Interface} hold in a distributional sense. Moreover, it is easy to check that for all $t > 0$ and for a reasonable choice of $x_1$ and $x_2$, each entry of $B(\\tilde{x})$ given in \\eqref{Interface Paper - Operator Bx} lies in $\\mathcal{D}'(a,b)$. Following the discussion in Section \\ref{Section Linear Control Systems}, we know that the assumption that $B$ is a bounded linear operator from $\\mathbb{R}$ to $\\tilde{X}$ is too restrictive. In order to apply the concept of admissibility, we need to check whether $B$ is an element of $\\mathcal{L}(\\mathbb{R}, \\tilde{X}_{-1})$, where we may identify $\\mathcal{L}(\\mathbb{R}, \\tilde{X}_{-1})$ with the extrapolation space $\\tilde{X}_{-1} = (D(\\tilde{A}(\\tilde{x})), \\| \\cdot \\|_{-1})$ associated with the operator $\\tilde{A}(\\tilde{x})$ defined in \\eqref{Interface Paper - Operator tildeA} (see Definition \\ref{Definition Extrapolation Space}). Assuming that there is a way to show that $A(\\tilde{x})$ generates a $C_0$-semigroup (see \\ref{Problem1MovingInterface}), for a further treatment we would need to check whether the vector defined in \\eqref{Interface Paper - Operator Bx} lies in $\\tilde{X}_{-1}$, and, as the case may be, whether $B$ is admissible, see Definition \\ref{Definition Admissible Control Operator}. This is the second condition that has to be satisfied in order to show well-posedness, see again Proposition \\ref{Proposition Well-Posedenss of Abstract Control Systems}. Since in the present scenario it is unavoidable that the input map is modulated by the state, we need to find another approach. Although there is some literature dealing with such input maps (see \\cite{Ball}), the treatment of this problem is once again beyond the scope of this thesis.\n\t\n\\item \\label{Problem3MovingInterface} There are also some difficulties concerning the port-Hamiltonian formulation of the obtained system \\eqref{Interface Paper - Augmented System wrt Jl}. In \\cite{Diagne}, the adjoint output map and the underlying Dirac structure of the system have been outlined. However, according to \\ref{Problem1MovingInterface}-\\ref{Problem2MovingInterface}, the co-domain of $B(\\tilde{x})$ is, at best, the extrapolation space $\\tilde{X}_{-1}$, and it is not clear how one shall incorporate the constitutive relation between the flow and effort variables that represents the dynamics of the system \\eqref{Interface Paper - Augmented System wrt Jl} into the definition of the Dirac structure. Possibly, one has to introduce the notion of a generalized Dirac structure that preserves the main characteristic of Dirac structures, namely the power-conserving interconnection between the system components through the respective ports. Furthermore, the Dirac structure is not constant anymore, but is modulated by the extended state $\\tilde{x}$, as the position of the interface has to be captured at all times.\n\n\\end{enumerate}\nIn conclusion, it does not seem promising to further investigate the quite problematic system description \\eqref{Interface Paper - Augmented System wrt Jl}, since the treatment of (some of) these delicate problems is beyond the scope of this thesis.\n\\vspace{0.5 cm}\\\\ \nAlternatively, we are going to consider a simplified model that is still closely related to the original one, and seek some fruitful results that might help for further investigating the formulation derived in \\eqref{Interface Paper - Augmented System wrt Jl}. A plausible approach is to simply remove the color functions from the state variables again, and to consider the adjusted system description on the state space $X = L^2([a,b], \\mathbb{R}^2)$. This is what we are going to do next. \n\n\n\\section{A Simplified Model}\n\\label{Section A Simplified System}\nFor the remainder of this thesis, we want to study a simplified version of the systems \\eqref{Interface Paper - Augmented System wrt Ja} and \\eqref{Interface Paper - Augmented System wrt Jl}, where we assume that the color functions are not part of the state variable anymore. We seek to find useful results that are beneficial for the further analysis of the respective original problem. Our main focus relies on proving that under certain (passivity) conditions, for arbitrary interface positions $l \\in (a,b)$, the port-Hamiltonian operator associated with the simplified system generates a contraction semigroup on its corresponding energy space, as this result sets the foundation for further investigations.\n\\vspace{0.5 cm}\\\\\nWe start with the fixed interface position $l = 0$. Let $X = L^2([a,b], \\mathbb{R}^2)$. Consider the system\n\\begin{align}\n\t\\label{Interface - Simplified PH-System}\n\t\\partial_t x(z,t) = \\mathcal{J}_0 \\left( \\mathcal{Q}_0(z) x(z,t) \\right), \\hspace{0.5 cm} t >0, \n\\end{align}\nwhere $\\mathcal{Q}_0 = c_0 \\mathcal{Q}^- + \\overline{c}_0 \\mathcal{Q}^+ \\in \\mathcal{L}(X)$ is a coercive matrix multiplication operator. In particular, there exist constants $0 < m \\leq M$ such that $mI \\leq \\mathcal{Q}^{\\pm}(z) \\leq MI$ holds for almost all $z \\in [a,b]$. The functions $c_0$ and $\\overline{c}_0$ are the color functions defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline}. \n\\vspace{0.5 cm}\\\\\nWe endow $X$ with $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{Q}_0} = \\frac{1}{2} \\langle \\cdot, \\mathcal{Q}_0 \\cdot \\rangle_{L^2}$, i.e., \n\\begin{align}\n\t\\langle x , y \\rangle_{\\mathcal{Q}_0} = \\frac{1}{2} \\int_{a}^{b} y^{\\top}(z) \\mathcal{Q}_0(z) x(z) \\, dz, \\hspace{0.5 cm} x, y \\in X.\n\t\\label{Interface - Inner Product wrt Q0}\n\\end{align}\nThe Hamiltonian $H \\colon X \\to \\mathbb{R}$ on the energy space $(X, \\| \\cdot \\|_{\\mathcal{Q}_0})$ is defined as\n\\begin{align}\n\t\\label{Interface - Simplified PH-System - Hamiltonian}\n\tH(x) = \\frac{1}{2} \\int_{a}^{b} x^{\\top}(z) \\mathcal{Q}_0(z) x(z) \\, dz = \\|x\\|_{\\mathcal{Q}_0}^2, \\hspace{0.5 cm} x \\in X.\n\\end{align}\nThe operator $\\mathcal{J}_0 \\colon D(\\mathcal{J}_0) \\subset X \\to X$ has already been introduced in Subsection \\ref{Subsection Port-Hamiltonian Formulation and Dirac Structure}, and is given by\n\\begin{align}\n\t\\begin{split}\n\tD(\\mathcal{J}_0) &= \\left\\{ x \\in X \\mid x_1 \\in D(\\mathbf{d}_0^{\\ast}), \\hspace{0.1 cm} x_2 \\in D(\\mathbf{d}_0)\\right\\}, \\\\\n\t\\mathcal{J}_0 x &= \\begin{bmatrix}\n\t\t0 & \\mathbf{d}_0 \\\\\n\t\t- \\mathbf{d}_0^{\\ast} & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\tx_1 \\\\\n\t\tx_2\n\t\\end{bmatrix}, \\hspace{0.5 cm} x \\in D(\\mathcal{J}_0),\n\\end{split}\n\\label{Interface - Operator J0}\n\\end{align}\nwhere the operator $\\mathbf{d}_0$ and its formal adjoint $\\mathbf{d}_0^{\\ast}$ have been defined in \\eqref{Interface Paper - Operator d0} and \\eqref{Interface Paper - Operator d_0ast}, respectively: they are given by\n\\begin{align*}\n\tD(\\mathbf{d}_0) &= H^1([a,b], \\mathbb{R}), \\\\\n\t\\mathbf{d}_0x &= - \\left[ \\frac{d}{dz} (c_0 x) + \\frac{d}{dz} (\\overline{c}_0 x ) \\right], \\hspace{0.5 cm} x \\in D(\\mathbf{d}_0),\n\\end{align*}\nand\n\\begin{align*}\n\tD(\\mathbf{d}_0^{\\ast}) &= \\left\\{ y \\in L^2([a,b], \\mathbb{R}) \\mid y_{|(a,0)} \\in H^1((a,0), \\mathbb{R}), \\hspace{0.1 cm} y_{|(0,b)} \\in H^1((0,b), \\mathbb{R}) \\right\\}, \\\\\n\t\\mathbf{d}_0^{\\ast}y &= \\left( - \\mathbf{d}_0 - \\left[ \\frac{d}{dz} c_0 + \\frac{d}{dz} \\overline{c}_0 \\right] \\right) y, \\hspace{0.5 cm} y \\in D(\\mathbf{d}_0^{\\ast}).\n\\end{align*}\nNote that on any interval $[a',b'] \\subset[a,b]$ not containing the interface position $l = 0$, the operator $\\mathcal{J}_0$ simply acts as the matrix differential operator $P_1 \\frac{d}{dz}$, where\n\\begin{align}\n\tP_1 = \\begin{bmatrix}\n\t\t0 & -1 \\\\\n\t\t-1 & 0 \n\t\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 2}.\n\\label{Interface - Matrix P1}\n\\end{align}\nWe will proceed as follows: In Subsection \\ref{Subsection Dirac Structure - Infinite Dimensional Case} we begin with the definition of the underlying Dirac structure of the system \\eqref{Interface - Simplified PH-System}. This can be easily generalized to arbitrary interface positions $l \\in (a,b)$. Following this, in Subsection \\ref{Subsection Generation of a Contraction Semigroup} we will show that under certain boundary and interface conditions, the port-Hamiltonian operator associated with the system \\eqref{Interface - Simplified PH-System} generates a contraction semigroup on the energy space. If we add some more assumptions concerning the boundary conditions and the matrix operators constituting $\\mathcal{Q}_0$, we can even guarantee exponential stability of the semigroup, as we will work out in Subsection \\ref{Subsection Exponential Stability}. Furthermore, we determine the adjoint operator of the port-Hamiltonian operator in Subsection \\ref{Subsection The Adjoint Operator}, and the corresponding resolvent operator in Subsection \\ref{Subsection Resolvent}. Lastly, in Subsection \\ref{Subsection Stability of the Family of Infinitesimal Generators} we shall define a family of port-Hamiltonian operators that keeps track of the moving interface position, and analyze the resulting time-variant evolution problem associated with this family. We present sufficient conditions for the stability of this family, as discussed in Section \\ref{Section Evolution Equations}. This is a key property and our last contribution made in this thesis for the further analysis of the system formulation suggested in \\cite{Diagne}.\n\\subsection{Dirac Structure}\n\\label{Subsection Dirac Structure - Infinite Dimensional Case}\nAs usual, to define the underlying Dirac structure of the system \\eqref{Interface - Simplified PH-System}, we need to find a suitable power pairing $ \\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ which will constitute the plus pairing $\\ll \\cdot , \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$. We will mainly repeat the steps we have seen in Subsection \\ref{Subsection Port-Hamiltonian Formulation and Dirac Structure}, and therefore will not go into detail.\n\\vspace{0.5 cm}\\\\\nAnalogously to the verification that the operator $\\tilde{\\mathcal{J}}_0$ given in \\eqref{Interface Paper - Operator Ja} is formally skew-symmetric, see \\eqref{Interface Paper - Skew-Symmetry of Ja}, we can show that the operator $\\mathcal{J}_0$ defined in \\eqref{Interface - Operator J0} is formally skew-symmetric as well. Applying the relation \\eqref{Interface Paper - Relation d_0 and d_0ast} between the operators $\\mathbf{d}_0$ and $\\mathbf{d}_0^{\\ast}$, for $e^1, e^2 \\in D(\\mathcal{J}_0)$ we have\n\\begin{align}\n\t\\begin{split}\n \\langle \\mathcal{J}_0 e^1 , e^2 \\rangle_{L^2} + \\langle e^1 , \\mathcal{J}_0 e^2 \\rangle_{L^2} &= - \\big[ e_1^2(z) e_2^1(z) \\big]_{a}^{b} + e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] \\\\\n\t&\\hspace{0.5 cm} - \\big[ e_1^1(z) e_2^2(z) \\big]_{a}^{b} + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] \\\\\n\t&= \\big[ (e^1)^{\\top}(z) P_1 e^2(z) \\big]_{a}^{b} \\\\\n\t&\\hspace{0.5 cm} + e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right],\n\t\\end{split} \n\\label{Interface - Skew-Symmetry of J0}\n\\end{align}\nwith $P_1$ defined in \\eqref{Interface - Matrix P1}. This relation will help us finding the appropriate power pairing later on. Prior to that, we need to augment the system \\eqref{Interface - Simplified PH-System} with boundary and interface port variables, which we will introduce next. \n\\vspace{0.5 cm}\\\\\nAgain, since the interface position resides in the interior of the spatial domain $[a,b]$, the trace operator defined in \\eqref{Trace Operator - Chapter 3} can be extended to $\\trace_0 \\colon D(\\mathcal{J}_0) \\to \\mathbb{R}^4$ with \n\\begin{align*}\n e \\mapsto \\begin{bmatrix}\n\t\te(b) \\\\\n\t\te(a)\n\t\\end{bmatrix}, \\hspace{0.5 cm} e \\in D(\\mathcal{J}_0).\n\\end{align*}\nConsequently, for all co-energy variables $ e \\in D(\\mathcal{J}_0)$, the boundary flow $f_{\\partial} = f_{\\partial, e} \\in \\mathbb{R}^2$ and the boundary effort $e_{\\partial} = e_{\\partial, e} \\in \\mathbb{R}^2$ are defined as\n\\begin{align}\n\t\\label{Interface - Boundary Flow and Effort}\n\t\\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = R_{\\extern} \\trace_0(e) = \\begin{bmatrix} \\frac{1}{\\sqrt{2}} (P_1e(b) - P_1 e(a)) \\\\\n\t\t\\frac{1}{\\sqrt{2}} ( e(b) + e(a))\n\t\\end{bmatrix}, \n\\end{align} \nwhere\n\\begin{align}\n\t\\label{Interface - Matrix Rext}\n\tR_{\\extern} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n\t\tP_1 & - P_1 \\\\\n\t\tI & I \n\t\\end{bmatrix} \\in \\mathbb{R}^{4 \\times 4}. \n\\end{align}\nNext, we define the interface port variables $e_{I} = e_{I, e}$, $f_I = f_{I, e} \\in \\mathbb{R}$, which have already been introduced in Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface}. For all effort variables $e \\in D(\\mathcal{J}_0)$, the interface flow and the interface effort are given as follows:\n\\begin{align}\n\tf_I &= e_2 (0^+) = e_2(0^-), \\label{Interface - Simplified Continuity Equation} \\\\\n\t-e_I &= e_1(0^+) - e_1 (0^-). \\label{Interface - Simplified Balance Equation}\n\\end{align}\nIt is left to find the power pairing in order to define the Dirac structure associated with the system \\eqref{Interface - Simplified PH-System} that is augmented with the previously defined boundary and interface port variables. To this end, we write the right-hand side of \\eqref{Interface - Skew-Symmetry of J0} with respect to the boundary and interface port variables for all $ e^1, e^2 \\in D(\\mathcal{J}_0)$ as follows: \n\\begin{align}\n\t\\begin{split}\n\t&\\hspace{0.5 cm} \\big[ (e^1)^{\\top}(z) P_1 e^2(z) \\big]_{a}^{b} + e_2^1(0) \\left[ e_1^2(0^+) - e_1^2(0^-) \\right] + e_2^2(0) \\left[ e_1^1(0^+) - e_1^1(0^-) \\right] \\\\\n\t&= \\langle e_{\\partial, e^2} , f_{\\partial, e^1} \\rangle_2 + \\langle e_{\\partial, e^1} , f_{\\partial, e^2} \\rangle_2 - f_{I,e^1} e_{I, e^2} - f_{I, e^2} e_{I, e^1}. \n\t\\end{split}\n\\label{Interface - Auxiliary Equation Power Pairing}\n\\end{align}\nNow, define the space of flows $\\mathcal{F}$ and the space of efforts $\\mathcal{E}$ for the reduced system as\n\\begin{align}\n\t\\mathcal{F} = \\mathcal{E} = L^2([a,b], \\mathbb{R}^2) \\times \\mathbb{R} \\times \\mathbb{R}^2,\n\t\\label{Interface - Flow and Effort Space}\n\\end{align}\nendowed with the canonical inner product $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{F}}$. For the right choice of the power pairing $\\langle \\cdot \\mid \\cdot \\rangle \\colon \\mathcal{E} \\times \\mathcal{F} \\to \\mathbb{R}$ we simply substitute the relation \\eqref{Interface - Auxiliary Equation Power Pairing} into equation \\eqref{Interface - Skew-Symmetry of J0}. This yields for all $e^1, e^2 \\in D(\\mathcal{J}_0)$, \n\\begin{align}\n\t\\langle \\mathcal{J}_0 e^1, e^2 \\rangle_{L^2} + \\langle e^1, \\mathcal{J}_0 e^2 \\rangle_{L^2} = \\langle e_{\\partial, e^2} , f_{\\partial, e^1} \\rangle_2 + \\langle e_{\\partial, e^1} , f_{\\partial, e^2} \\rangle_2 - f_{I,e^1} e_{I, e^2} - f_{I, e^2} e_{I, e^1}.\n\t\\label{Interface - Skew-Symmetry of J0 - Representation 2}\n\\end{align}\nThus, we define\n\\begin{align}\n\t\\left\\langle \\begin{pmatrix}\n\t\te \\\\\n\t\te_I \\\\\n\t\te_{\\partial} \n\t\\end{pmatrix} \\Bigg| \\begin{pmatrix}\n\tf \\\\\n\tf_I \\\\\n\tf_{\\partial}\n\t\\end{pmatrix} \\right\\rangle := \\langle e, f \\rangle_{L^2} +e_I f_I - \\langle e_{\\partial}, f_{\\partial} \\rangle_2, \\hspace{0.5 cm} \\left( \\begin{pmatrix}\n\te \\\\\n\te_I \\\\\n\te_{\\partial} \n\\end{pmatrix} , \\begin{pmatrix}\nf \\\\\nf_I \\\\\nf_{\\partial}\n\\end{pmatrix} \\right) \\in \\mathcal{E} \\times \\mathcal{F}, \n\\label{Interface - Power Pairing}\n\\end{align}\nand the plus pairing $\\ll \\cdot , \\cdot \\gg \\colon \\mathcal{B} \\times \\mathcal{B} \\to \\mathbb{R}$ on the bond space $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ induced by the power pairing \\eqref{Interface - Power Pairing} is accordingly defined as in \\eqref{Plus Pairing - Distributed-Parameter}. The Dirac structure associated with the formally skew-symmetric operator $\\mathcal{J}_0$ defined in \\eqref{Interface - Operator J0} is given as follows.\n\n\\begin{corollary}\n\t\\label{Interface - Corollary Dirac Structure}\n\tConsider the formally skew-symmetric operator $\\mathcal{J}_0$ given by \\eqref{Interface - Operator J0}. Let $\\mathcal{B} = \\mathcal{F} \\times \\mathcal{E}$ be the bond space with respect to the flow space $\\mathcal{F}$ and the effort space $\\mathcal{E}$ defined in \\eqref{Interface - Flow and Effort Space}. Then the subspace $\\mathcal{D}_{\\mathcal{J}_0} \\subset \\mathcal{B}$ defined as\n\t\\begin{align}\n\t\t\\mathcal{D}_{\\mathcal{J}_0} :=& \\left\\{ \\left( \\begin{pmatrix}\n\t\t\tf \\\\\n\t\t\tf_I \\\\\n\t\t\tf_{\\partial}\n\t\t\\end{pmatrix} , \\begin{pmatrix}\n\t\te \\\\\n\t\te_I \\\\\n\t\te_{\\partial}\n\t\\end{pmatrix} \\right) \\in \\mathcal{B} \\hspace{0.1 cm} \\Bigg| \\hspace{0.1 cm} e \\in D(\\mathcal{J}_0), \\, f = \\mathcal{J}_0e, \\, f_I = e_2(0), \\right. \\label{Interface - Dirac Structure DJ0} \\\\\n&\\hspace{4.22 cm} \\left. \\, e_I = e_1(0^-) - e_1(0^+), \\, \\begin{bmatrix}\n\tf_{\\partial} \\\\\n\te_{\\partial}\n\\end{bmatrix} = R_{\\extern} \\trace_0(e) \\right\\} \\nonumber\n\t\\end{align}\nis a Dirac structure with respect to the plus pairing $\\ll \\cdot , \\cdot \\gg$ induced by the power pairing \\eqref{Interface - Power Pairing}. \n\\end{corollary}\n\n\\begin{proof}\n\tThis follows immediately from the proof of Proposition \\ref{Interface Paper - Proposition Dirac Structure DI}.\n\t\\end{proof}\n\nAs usual, we can geometrically specify the dynamics of the system \\eqref{Interface - Simplified PH-System} with respect to its underlying Dirac structure. \n\n\\begin{corollary}\n\tConsider the system \\eqref{Interface - Simplified PH-System} with Hamiltonian \\eqref{Interface - Simplified PH-System - Hamiltonian} augmented with the boundary and interface port variables given in \\eqref{Interface - Boundary Flow and Effort} and \\eqref{Interface - Simplified Continuity Equation}-\\eqref{Interface - Simplified Balance Equation}, respectively. The dynamics of the resulting port-Hamiltonian system are geometrically specified by the requirement that for all $t > 0$ it holds that\n\t\\begin{align*}\n\t\t\\left( \\begin{pmatrix}\n\t\t\t\\frac{\\partial}{\\partial t} x(\\cdot,t) \\\\\n\t\t\tf_I(t) \\\\\n\t\t\tf_{\\partial}(t)\n\t\t\\end{pmatrix} , \\begin{pmatrix}\n\t\t\\mathcal{Q}_0x(\\cdot,t) \\\\\n\t\te_I(t) \\\\\n\t\te_{\\partial}(t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\mathcal{J}_0},\n\t\\end{align*}\nwith $f_I(t) = f_{I, \\mathcal{Q}_0x(\\cdot,t)}$, $e_I(t) = e_{I, \\mathcal{Q}_0 x(\\cdot,t)}$, $f_{\\partial}(t) = f_{\\partial, \\mathcal{Q}x(\\cdot,t)}$, $e_{\\partial}(t) = e_{\\partial, \\mathcal{Q}x(\\cdot,t)}$, and $\\mathcal{D}_{\\mathcal{J}_0}$ the Dirac structure defined in \\eqref{Interface - Dirac Structure DJ0}. \n\\end{corollary}\nLastly, we want to state a balance equation for classical solutions of the first-order system \\eqref{Interface - Simplified PH-System} by means of the boundary and interface port variables.\n\\begin{corollary}\n\t\\label{Interface - Theorem Balance Equation}\n\tLet $x \\colon [0, \\infty) \\to X$ be a classical solution of the system \\eqref{Interface - Simplified PH-System} with Hamiltonian \\eqref{Interface - Simplified PH-System - Hamiltonian}. Then we have for all $t \\geq 0$, \n\t\\begin{align}\n\t\t\\label{Interface - Power Balance Equation wrt Boundary and Interface Port}\n\t\\frac{d}{dt} \\|x(t)\\|_{\\mathcal{Q}_0}^2 =\t\\frac{d}{dt}H(x(t)) = \\langle e_{\\partial}(t), f_{\\partial}(t) \\rangle_2 - e_I(t) f_I(t).\n\t\\end{align}\n\\end{corollary}\n\\begin{proof}\n\tBy exploiting the fact that $x$ satisfies \\eqref{Interface - Simplified PH-System} and by using \\eqref{Interface - Skew-Symmetry of J0 - Representation 2}, we get for every $t \\geq 0$, \n\t\\begin{align*}\n\t\t\\frac{d}{dt}H(x(\\cdot,t)) &= \\frac{1}{2} \\frac{d}{dt} \\langle x(\\cdot,t), \\mathcal{Q}_0x(\\cdot,t) \\rangle_{L^2} \\\\\n\t\t&= \\frac{1}{2} \\langle \\mathcal{J}_0( \\mathcal{Q}_0x(\\cdot,t) ), \\mathcal{Q}_0 x(\\cdot,t) \\rangle_{L^2} + \\frac{1}{2} \\langle \\mathcal{Q}_0 x(\\cdot,t), \\mathcal{J}_0(\\mathcal{Q}_0x(\\cdot,t)) \\rangle_{L^2} \\\\\n\t\t&= \\langle e_{\\partial}(t), f_{\\partial}(t) \\rangle_2 - e_I(t) f_I(t).\n\t\\end{align*}\n\\end{proof}\nEquation \\eqref{Interface - Power Balance Equation wrt Boundary and Interface Port} states that the change of energy is equal to the power flow both at the boundary and at the interface position. Note that this is the same balance equation as for the system \\eqref{Interface Paper - Augmented System wrt Ja}.\n\\vspace{0.5 cm}\\\\\nNow that we have established the port-Hamiltonian formulation of the simplified system \\eqref{Interface - Simplified PH-System}, we want to discuss under which conditions this system is well-defined. \n\\subsection{Generation of a Contraction Semigroup}\n\\label{Subsection Generation of a Contraction Semigroup}\n\nIn this section, we want to prove well-posedness of the system \\eqref{Interface - Simplified PH-System}. To this end, we investigate the port-Hamiltonian operator associated with the system \\eqref{Interface - Simplified PH-System} with Hamiltonian \\eqref{Interface - Simplified PH-System - Hamiltonian}. As mentioned at the end of Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems}, in \\cite[Section 4]{LeGorrec} it has been proved that formulating the boundary conditions with respect to the boundary port variables, one can characterize those boundary conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. We will therefore follow this approach. However, we need to include interface conditions as well, which we want to specify by means of the interface port variables. These conditions will be incorporated into the definition of the domain of the port-Hamiltonian operator. We will show that if we impose a passivity condition at the interface, then these operators generate a contraction semigroup and therefore guarantee well-posedness of the system \\eqref{Interface - Simplified PH-System}. Furthermore, we will show that, under some more restrictive conditions, these operators even generate a unitary semigroup. At the end of this section, we apply the former result by showing well-posedness of the model of two transmission lines with different physical properties that are coupled through a resistor. \n\\vspace{0.5 cm}\\\\\nLet us start with the boundary conditions for the system \\eqref{Interface - Simplified PH-System}. We impose boundary conditions of the form \n\\begin{align}\n\t\\tilde{W}_B \\begin{bmatrix}\n\t\t(\\mathcal{Q}_0x)(b) \\\\\n\t\t(\\mathcal{Q}_0x)(a) \n\t\\end{bmatrix} = 0\n\\label{Interface - Boundary Condition}\n\\end{align}\nfor some $\\tilde{W}_B \\in \\mathbb{R}^{2 \\times 4}$. These are boundary conditions with respect to the co-energy variables of the system. In order to formulate them with respect to the boundary port variables, we just use their definition \\eqref{Interface - Boundary Flow and Effort}. As the matrix $R_{\\extern}$ defined in \\eqref{Interface - Matrix Rext} is invertible, we may equivalently write \\eqref{Interface - Boundary Condition} as\n\\begin{align}\n\tW_B \\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = 0, \n\t\\label{Interface - Boundary Condition wrt Boundary Flow and Effort}\n\\end{align}\nwhere $W_B = \\tilde{W}_B R_{\\extern}^{-1} \\in \\mathbb{R}^{2 \\times 4}$. Recall that we want to show that the port-Hamiltonian operator associated with the system \\eqref{Interface - Simplified PH-System} generates a contraction semigroup. As depicted in Example \\ref{Example Contraction Semigroup}, this implies that the energy of the state $x \\in X$ is dissipating. By virtue of the balance equation \\eqref{Interface - Power Balance Equation wrt Boundary and Interface Port}, we see that we need to choose the matrix $W_B$ in \\eqref{Interface - Boundary Condition wrt Boundary Flow and Effort} such that $\\langle e_{\\partial}, f_{\\partial} \\rangle_{2} \\leq 0$ holds. This is possible, as we will see shortly. \n\\vspace{0.5 cm}\\\\\nNext, we need to impose an interface condition constituted by the interface port variables \\eqref{Interface - Simplified Continuity Equation}-\\eqref{Interface - Simplified Balance Equation}. Again, from the balance equation \\eqref{Interface - Power Balance Equation wrt Boundary and Interface Port} we infer that the interface relation\n\\begin{align}\n\t\\label{Interface - Interface Passivity Relation}\n\tf_I = r e_I\n\\end{align}\nfor some $r \\geq 0$ is a good choice. Recall that the interface port variables are precisely given by\n\\begin{align*}\n\tf_I &= \\left( \\mathcal{Q}^+x\\right)_2 (0^+) = \\left( \\mathcal{Q}^- x \\right)_2 (0^-), \\\\\n\t-e_I &= \\left( \\mathcal{Q}^+x \\right)_1 (0^+) - \\left( \\mathcal{Q}^- x \\right)_1 (0^-). \n\\end{align*}\n\\begin{remark}\nNote that, due to the minus sign on the right-hand side of the balance equation \\eqref{Interface - Power Balance Equation wrt Boundary and Interface Port}, the relation \\eqref{Interface - Interface Passivity Relation} is in fact a passivity relation with passivity constant $r \\geq 0$. This does not contradict the discussion from Section \\ref{Section Energy-dissipating Elements}: If we consider the power pairing \\eqref{Power-Balance with Dissipation} and the balance equation \\eqref{Power-Balance Energy-Storing and Energy-Dissipating Elements}, then it is easy to check that passivity is a matter of sign convention. The important thing is that the inner product (including the sign) of the resistive port variables appearing in the balance equation is less than or equal to $0$, indicating energy dissipation. Thus, in contrast to the resistive structure \\eqref{Resistive Relation Finite-Dimensional Case} in the finite-dimensional case, the set of resistive relations $\\mathcal{R}_r$ concerning the interface port variables is now given by\n\\begin{align*}\n\t\\mathcal{R}_r = \\left\\{ (f_I,e_I) \\in \\mathbb{R}^2 \\mid f_I = re_I\\right\\}.\n\\end{align*}\n\\end{remark}\nNow we are able to define the port-Hamiltonian operator $A_{\\mathcal{Q}_0} \\colon D(A_{\\mathcal{Q}_0}) \\subset X \\to X$ associated with the system \\eqref{Interface - Simplified PH-System} with Hamiltonian \\eqref{Interface - Simplified PH-System - Hamiltonian}. It is given by\n\\begin{align}\n\t\\begin{split}\n\t\tD(A_{\\mathcal{Q}_0}) &= \\left\\{ x \\in X \\, \\big| \\, \\mathcal{Q}_0x \\in D(\\mathcal{J}_0), \\hspace{0.1 cm} f_{I, \\mathcal{Q}_0x} = r e_{I, \\mathcal{Q}_0x}, \\hspace{0.1 cm} W_B \\begin{bmatrix}\n\t\t\tf_{\\partial, \\mathcal{Q}_0 x} \\\\\n\t\t\te_{\\partial, \\mathcal{Q}_0 x}\n\t\t\\end{bmatrix} = 0 \\right\\}, \\\\\n\t\tA_{\\mathcal{Q}_0}x &= \\mathcal{J}_0 (\\mathcal{Q}_0x), \\hspace{0.5 cm} x \\in D(A_{\\mathcal{Q}_0}).\n\t\\end{split}\n\t\\label{Interface - Simplified PH System - Operator A}\n\\end{align}\nWe wish to prove that $A_{\\mathcal{Q}_0}$ generates a contraction semigroup $(T(t))_{t \\geq 0}$ on the energy space $(X, \\| \\cdot \\|_{\\mathcal{Q}_0})$. Before we prove the statement, we need some auxiliary lemmas. The proofs of those lemmas can be found in Section 7.2 and Section 7.3 in \\cite{JacobZwart}.\n\\begin{lemma}[Lemma 7.2.3 in \\cite{JacobZwart}]\n\t\\label{Interface - Auxiliary Lemma 1}\n\tLet $Z$ be a Hilbert space with inner product $\\langle \\cdot, \\cdot \\rangle$, and let $\\mathcal{P} \\in \\mathcal{L}(Z)$ be a coercive operator on $Z$. We define $Z_{\\mathcal{P}}$ as the Hilbert space $Z$ endowed with $\\langle \\cdot , \\cdot \\rangle_{\\mathcal{P}} := \\langle \\cdot , \\mathcal{P} \\cdot \\rangle$. Then the operator $A \\colon D(A) \\subset Z \\to Z$ generates a contraction semigroup on $Z$ if and only if \n\t\\begin{align*}\n\t\tA\\mathcal{P} \\colon D(A\\mathcal{P}) = \\left\\{ x \\in Z \\mid \\mathcal{P}z \\in D(A) \\right\\} \\subset Z_{\\mathcal{P}} \\to Z_{\\mathcal{P}}\n\t\\end{align*}\ngenerates a contraction semigroup on $Z_{\\mathcal{P}}$. \n\\end{lemma}\nMoreover, we need two technical lemmas concerning matrix representations. \n\\begin{lemma}[Lemma 7.3.1 in \\cite{JacobZwart}]\n\t\\label{Interface - Auxiliary Lemma 2}\n\tFor a matrix $W \\in \\mathbb{R}^{n \\times 2n}$, the following are equivalent:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\rank(W) = n$ and $W\\Sigma W^{\\top} \\geq 0$, with $\\Sigma \\in \\mathbb{R}^{4 \\times 4}$ defined in \\eqref{Matrix Sigma}. \n\t\t\\item[(ii)] There exist an invertible matrix $S \\in \\mathbb{R}^{n \\times n}$ and a matrix $V \\in \\mathbb{R}^{n \\times n}$ with $V V^{\\top} \\leq I$ such that\n\t\t\\begin{align}\n\t\t\t\\label{Interface - Auxiliary Lemma 2 - Matrix Representation}\n\t\t\tW = S \\begin{bmatrix}\n\t\t\t\tI + V & I - V\n\t\t\t\\end{bmatrix}.\n\t\t\\end{align}\n\t\\end{enumerate} \n\\end{lemma}\n\n\\begin{lemma}[Lemma 7.3.2 in \\cite{JacobZwart}]\n\t\\label{Interface - Auxiliary Lemma 3}\n\tSuppose that a matrix $W \\in \\mathbb{R}^{n \\times 2n}$ can be written as in \\eqref{Interface - Auxiliary Lemma 2 - Matrix Representation} for some matrices $S, V \\in \\mathbb{R}^{n \\times n}$, where $S$ is invertible. Then it holds that\n\t \\begin{align*}\n\t \t\\ker (W) = \\ran \\left( \\begin{bmatrix}\n\t \t\tI - V \\\\\n\t \t\t-I - V\n\t \t\\end{bmatrix}\\right).\n\t \\end{align*}\n\\end{lemma}\nNow we are able to prove that the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A} generates a contraction semigroup. The idea is based on the proof of Theorem 7.2.4 in \\cite{JacobZwart}. \n\n\\begin{theorem}\n\t\\label{Interface - Theorem A Generates a Contraction Semigroup}\n\tConsider the port-Hamiltonian operator $A_{\\mathcal{Q}_0} \\colon D(A_{\\mathcal{Q}_0}) \\subset X \\to X$ defined in \\eqref{Interface - Simplified PH System - Operator A} with passivity constant $r \\geq 0$ and with a full rank matrix $W_B \\in \\mathbb{R}^{2 \\times 4}$ satisfying $W_B \\Sigma W_B^{\\top} \\geq 0$. Then $A_{\\mathcal{Q}_0}$ is the infinitesimal generator of a contraction semigroup on the energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$. In particular, the abstract Cauchy problem\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\\dot{x}(t) &= A_{\\mathcal{Q}_0}x(t), \\hspace{0.5 cm} t > 0, \\\\\n\t\tx(0) &= x_0 \\in X,\n\t\\end{split}\n\\label{Interface - Theorem A Generates a Contraction Semigroup - ACP}\n\t\\end{align}\nis well-defined.\n\\end{theorem}\n\n\\begin{proof}\n\tFor the proof, we need to show two properties:\n\\begin{enumerate}\n\t\\item[(i)] $A_{\\mathcal{Q}_0}$ is dissipative (see Definition \\ref{Definition Dissipative Operator}).\n\t\\item[(ii)] ran$(I - A_{\\mathcal{Q}_0}) = X$. \n\\end{enumerate}\nApplying Theorem \\ref{Theorem Lumer-Phillips} then yields that $A_{\\mathcal{Q}_0}$ generates a contraction semigroup. The proof of these properties is split into three steps.\n\\vspace{0.5 cm}\\\\\n\\textit{Step 1.} Recall that the energy space $X$ is endowed with the inner product $\\langle \\cdot, \\cdot \\rangle_{\\mathcal{Q}_0} = \\frac{1}{2} \\langle \\cdot , \\mathcal{Q}_0 \\cdot \\rangle_{L^2}$. Since $\\mathcal{Q}_0 \\in \\mathcal{L}(X)$ is a coercive operator, by Lemma \\ref{Interface - Auxiliary Lemma 1} we may assume for the rest of the proof that $\\mathcal{Q}_0 \\equiv I$. In particular, $X$ is endowed with the inner product $\\frac{1}{2} \\langle \\cdot , \\cdot \\rangle_{L^2}$. In the following, we will write $A$ instead of $A_I$. \n\\vspace{0.5 cm}\\\\\n\\textit{Step 2.} \\label{Step 2} We begin by proving that $A$ is dissipative. From \\eqref{Interface - Skew-Symmetry of J0 - Representation 2} one can readily see that for all $x \\in D(A)$ it holds that\n\\begin{align*}\n\t\\langle Ax, x \\rangle_{L^2} + \\langle x, Ax \\rangle_{L^2} &= \\langle \\mathcal{J}_0x , x \\rangle_{L^2} + \\langle x, \\mathcal{J}_0 x \\rangle_{L^2} \\\\\n\t&= 2 \\langle e_{\\partial}, f_{\\partial} \\rangle_2 - 2 e_If_I.\n\\end{align*}\nWe want to estimate the right-hand side. Note that by assumption, for all $x \\in D(A)$ we have $\\begin{bmatrix}\n\tf_{\\partial} \\\\\n\te_{\\partial}\n\\end{bmatrix} \\in \\ker(W_B)$. Since $\\rank(W_B) = 2$ and since we assume that $W_B \\Sigma W_B^{\\top} \\geq 0$, Lemma~\\ref{Interface - Auxiliary Lemma 2} implies that $W_B$ may be written as\n\\begin{align*}\n\tW_B = S \\begin{bmatrix}\n\t\tI+ V & I - V\n\t\\end{bmatrix},\n\\end{align*}\nwith $S \\in \\mathbb{R}^{2 \\times 2}$ invertible and $V \\in \\mathbb{R}^{2 \\times 2}$ such that $VV^{\\top} \\leq I$, or equivalently, $V^{\\top}V \\leq I$. Thus, by Lemma \\ref{Interface - Auxiliary Lemma 3}, for all $x \\in D(A)$ there exists some $\\lambda_x \\in \\mathbb{R}^2$ such that\n\\begin{align}\n\t\\label{Interface - Theorem Contraction Semigroup - Boundary Flow and Effort Representation}\n\t\\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\tI-V \\\\\n\t\t-I - V\n\t\\end{bmatrix} \\lambda_x.\n\\end{align}\nUsing the passivity relation \\eqref{Interface - Interface Passivity Relation} and \\eqref{Interface - Theorem Contraction Semigroup - Boundary Flow and Effort Representation}, we compute for all $x \\in D(A)$,\n\\begin{align*}\n\t \\langle Ax, x \\rangle_{L^2} &= \\langle e_{\\partial}, f_{\\partial} \\rangle_2 - e_If_I \\\\\n\t&= \\lambda_x^{\\top} (-I-V)^{\\top}(I -V) \\lambda_x - e_I f_I \\\\\n\t&= \\lambda_x^{\\top} (-I + V^{\\top}V) \\lambda_x - e_I f_I \\\\\n\t&\\leq - e_I f_I \\\\\n\t&= - r e_I^2 \\\\\n\t&\\leq 0.\n\\end{align*}\nThus, the operator $A$ is dissipative.\n\\vspace{0.5 cm}\\\\\n\\textit{Step 3.} It is left to show that ran$(I - A) = X$. Let $y = c_0y^- + \\overline{c}_0 y^+ \\in X$ and consider the equation $(I - A)x = y$, i.e.,\n\\begin{align}\n\t\\label{Interface - ODE of (I-A)x equals y}\n\t\\begin{bmatrix}\n\t\tx_1 \\\\\n\t\tx_2\n\t\\end{bmatrix} - \\underbrace{\\begin{bmatrix}\n\t\t0 & \\mathbf{d}_0 \\\\\n\t\t- \\mathbf{d}_0^{\\ast} & 0 \n\t\\end{bmatrix}}_{=\\mathcal{J}_0} \\begin{bmatrix}\n\t\tx_1 \\\\\n\t\tx_2\n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\ty_1 \\\\\n\t\ty_2\n\t\\end{bmatrix}.\n\\end{align}\nWe need to show that the solution of this differential equation is an element of $D(A)$. As the operator $\\mathcal{J}_0$ acts as the differential operator $P_1 \\frac{d}{dz}$ on any subdomain not containing the interface position, with $P_1$ given in \\eqref{Interface - Matrix P1}, this can be equivalently written as\n\\begin{align*}\n\tx^- - P_1 \\frac{d}{dz} x^- &= y^- \\hspace{0.3 cm} \\text{on } [a,0), \\\\\n\tx^+ - P_1 \\frac{d}{dz} x^+ &= y^+ \\hspace{0.3 cm} \\text{on } (0,b]\n\\end{align*}\nwith suitable initial conditions at $z =a$ and $z = 0^+$, respectively. We will shortly discuss this in more detail. Noticing that $P_1^{-1} = P_1$ and writing\n\\begin{align*}\n\t\\frac{d}{dz} x^{\\pm} = P_1 (x^{\\pm} - y^{\\pm}),\n\\end{align*}\nthe variation of constants formula yields the solutions\n\\begin{align}\n\t\\varphi_y^-(z ; x(a)) &= e^{P_1(z-a)}x(a) - \\int_{a}^{z} e^{P_1(z-s)} P_1 y^-(s) \\, ds, \\hspace{0.5 cm} z \\in [a,0), \\label{Interface - Solution of ODE left} \\\\\n\t\\varphi_y^+(z; x(0^+)) &= e^{P_1z} x(0^+) - \\int_{0}^{z} e^{P_1(z-s)} P_1 y^+(s) \\, ds, \\hspace{0.5 cm } z \\in (0, b]. \\label{Interface - Solution of ODE right}\n\\end{align}\nIn particular, we have\n\\begin{align*}\n\t\\varphi_y^-(0^-; x(a)) = e^{-P_1a}x(a) - \\int_{a}^{0} e^{-P_1 s} P_1 y^-(s) \\, ds.\n\\end{align*}\nIn the following, we will consider two different instances concerning the passivity constant $r \\geq 0$. \n\\vspace{0.5 cm}\\\\\n\\underline{Case 1: $r > 0$}\n\nDue to the continuity condition \\eqref{Interface - Simplified Continuity Equation}, we have to choose \\begin{align}\n\tx_2(0) = f_I = \\left( \\varphi_y^-(0^-; x(a))\\right)_2.\n\t\\label{Interface - ODE Interface Continuity Equation}\n\\end{align}\nFurthermore, the balance equation \\eqref{Interface - Simplified Balance Equation} is given by\n\\begin{align}\n\t\\label{Interface - ODE Interface Balance Equation}\n\te_I = - \\left( x_1(0^+) - \\left( \\varphi_y^-(0^-; x(a)) \\right)_1 \\right).\n\\end{align}\nDue to the passivity relation \\eqref{Interface - Interface Passivity Relation}, we have to choose $x_1(0^+)$ such that\n\\begin{align*}\n\tf_I = r e_I.\n\\end{align*}\nThus, the initial condition $x(0^+) =(x_1(0^+), x_2(0))$ of $\\varphi_y^+(\\cdot; x(0^+))$ is uniquely determined through the interface relations and conditions, and depends on the choice of $x(a)$. The solution of \\eqref{Interface - ODE of (I-A)x equals y} is therefore given by\n\\begin{align}\n\t\\label{Interface - Solution of ODE complete}\n\t\\varphi_y(z; x(a), x(0^+)) = c_0(z) \\varphi_y^-(z; x(a)) + \\overline{c}_0(z) \\varphi_y^+(z; x(0^+)), \\hspace{0.5 cm} z \\in [a,b].\n\\end{align}\nIt is left to show that $\\varphi_y(\\cdot; x(a),x(0^+))$ is indeed an element of $D(A)$ for some initial value $x(a) \\in \\mathbb{R}^2$. To that end, we need to verify the last constitutive relation of $D(A)$ in \\eqref{Interface - Simplified PH System - Operator A}. By using \\eqref{Interface - Boundary Flow and Effort} and \\eqref{Interface - Boundary Condition wrt Boundary Flow and Effort}, the solution $\\varphi_y(\\cdot; x(a), x(0^+))$ is an element of $D(A)$ if and only if\n\\begin{align}\n\t\\label{Interface - D(A) condition - representation 1}\n\t0 = W_B \\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = W_B R_{\\extern} \\begin{bmatrix}\n\t\t\\varphi_y(b; x(a), x(0^+)) \\\\\n\t\t\\varphi_y(a; x(a), x(0^+))\n\t\\end{bmatrix}. \n\\end{align} \nObviously, $\\varphi(a;x(a), x(0^+)) = x(a)$. We wish to write $\\varphi(b; x(a), x(0^+))$ as $E x(a) + q$ for some $E \\in \\mathbb{R}^{2 \\times 2}$, $q \\in \\mathbb{R}^2$. Note that\n\\begin{align*}\n\t\\varphi_y(b; x(a), x(0^+)) = \\varphi_y^+(b; x(0^+)) = e^{P_1 b}x(0^+) \\underbrace{- \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds}_{=: q_1}.\n\\end{align*}\nUsing the interface conditions \\eqref{Interface - ODE Interface Continuity Equation}-\\eqref{Interface - ODE Interface Balance Equation}, we get\n\\begin{align*}\n\tf_I &= r e_I \\\\\n\t\\Leftrightarrow \\hspace{0.3 cm} \\left(\\varphi_y^-(0^-; x(a)) \\right)_2 &= r \\left[ - \\left( x_1(0^+) - \\left( \\varphi_y^-(0^+; x(a)) \\right)_1 \\right) \\right] \\\\\n\t\\Leftrightarrow \\hspace{1.955 cm} x_1(0^+) &= - \\frac{1}{r} \\left( \\varphi_y^-(0^-; x(a)) \\right)_2 + \\left( \\varphi_y^-(0^-; x(a)) \\right)_1.\n\\end{align*}\nHence, we may write\n\\begin{align*}\n\t&\\hspace{0.5 cm} e^{P_1b}x(0^+) \\\\\n\t&= e^{P_1b}\\begin{bmatrix}\n\t\t- \\frac{1}{r} \\left( e^{-P_1a} x(a) - \\int_{a}^{0} e^{-P_1 s} P_1 y^-(s) \\, ds \\right)_2 + \\left( e^{-P_1a}x(a) - \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_1 \\\\\n\t\t\\left( e^{-P_1a} x(a) - \\int_{a}^{0} e^{-P_1 s} P_1 y^-(s) \\, ds \\right)_2 \n\t\\end{bmatrix} \\\\\n\t&= \\underbrace{ e^{P_1b} \\begin{bmatrix}\n\t\t\t- \\frac{1}{r} \\left( e^{-P_1a}x(a) \\right)_2 + \\left( e^{-P_1a} x(a) \\right)_1 \\\\\n\t\t\t\\left( e^{-P_1a}x(a) \\right)_2\n\t\\end{bmatrix}}_{=: Ex(a)} \\\\\n\t&\\hspace{0.5 cm} \\underbrace{+ e^{P_1b} \\begin{bmatrix}\n\t\t\t- \\frac{1}{r} \\left( - \\int_{a}^{0} e^{-P_1s} P_1 y^-(s) \\, ds \\right)_2 + \\left( - \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_1 \\\\\n\t\t\t- \\left( \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_2\n\t\\end{bmatrix}}_{=: q_2}. \n\\end{align*}\nLetting $q := q_1 + q_2$, we finally have\n\\begin{align*}\n\t\\varphi_y(b; x(a),x(0^+)) = Ex(a) + q, \n\\end{align*}\nwith only $q$ depending on $y \\in X$. Now we can come back to the boundary condition \\eqref{Interface - D(A) condition - representation 1}. Equation \\eqref{Interface - D(A) condition - representation 1} is equivalent to \n\\begin{align}\n\t\\label{Interface - D(A) condition - representation 2}\n\t0 = W_B R_{\\extern} \\begin{bmatrix}\n\t\tE x(a) + q \\\\\n\t\tx(a) \n\t\\end{bmatrix} \\hspace{0.3 cm} \\Leftrightarrow \\hspace{0.3 cm} W_B R_{\\extern} \\begin{bmatrix}\n\t\tE \\\\\n\t\tI\n\t\\end{bmatrix} x(a) = - W_B R_{\\extern} \\begin{bmatrix}\n\t\tq \\\\\n\t\t0\n\t\\end{bmatrix}.\n\\end{align}\nTherefore, we want to show that the matrix $W_B R_{\\extern} \\begin{bmatrix}\n\tE \\\\\n\tI\n\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 2}$ is invertible, since this implies that there exists a unique initial value $x(a) \\in \\mathbb{R}^2$ such that $\\varphi_y(\\cdot; x(a), x(0^+)) \\in D(A)$. As $y \\in X$ was chosen arbitrarily, ran$(I - A) = X$, and we may apply Theorem~\\ref{Theorem Lumer-Phillips}. \n\\vspace{0.5cm}\\\\\nAssume now that there exists $0 \\neq r_0 \\in \\ker\\left( W_B R_{\\extern} \\begin{bmatrix}\n\tE \\\\\n\tI\n\\end{bmatrix} \\right)$, and let $y = 0$. Consider the solution of equation \\eqref{Interface - ODE of (I-A)x equals y} given by\n\\begin{align*}\n\t\\varphi_0(z; r_0, x(0^+)) = c_0(z) e^{P_1(z-a)}r_0 + \\overline{c}_0(z) e^{P_1z} \\begin{bmatrix}\n\t\tx_1(0^+) \\\\\n\t\t\\left( \\varphi^-(0^-; r_0) \\right)_2\n\t\\end{bmatrix},\n\\end{align*}\nwith $x_1(0^+) \\in \\mathbb{R}$ accordingly chosen. Note that $\\varphi_0(\\cdot; r_0, x(0^+)) \\in D(\\mathcal{J}_0)$, and, since $y = 0$, that $q = 0$. Consequently, $\\varphi_0(\\cdot; r_0, x(0^+)) \\in D(A)$, since by assumption and \\eqref{Interface - D(A) condition - representation 2}, we have\n\\begin{align*}\n\tW_B R_{\\extern} \\begin{bmatrix}\n\t\tE \\\\\n\t\tI\n\t\\end{bmatrix} r_0 = 0. \n\\end{align*}\nAs $\\varphi_0(\\cdot; r_0, x(0^+))$ is a non-zero solution of $(I - A)x = 0$, it is, in particular, an eigenfunction of $A$ with eigenvalue $\\lambda = 1$. This contradicts the fact that $A$ is dissipative, and therefore $W_B R_{\\extern} \\begin{bmatrix}\n\tE \\\\\n\tI\n\\end{bmatrix}$ is in fact invertible.\n\\vspace{0.5 cm}\\\\\n\\underline{Case 2: $r = 0$}\n\nThe passivity relation \\eqref{Interface - Interface Passivity Relation} together with the continuity condition \\eqref{Interface - Simplified Continuity Equation} yield that we have to choose\n\\begin{align*}\n\t0 = f_I = x_2(0) = \\left( \\varphi_y^-(0^-; x(a)) \\right)_2. \n\\end{align*}\nNote that for the matrix exponential $e^{P_1 \\cdot}$, where $P_1$ is given in \\eqref{Interface - Matrix P1}, we have\n\\begin{align*}\n\te^{P_1z} = \\frac{1}{2 e^z} \\begin{bmatrix}\n\t\te^{2z} +1 & -e^{2z} +1 \\\\\n\t\t-e^{2z} +1 & e^{2z} +1\n\t\\end{bmatrix}, \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align*}\nHence, it holds that\n\\begin{align*}\n\t0 &= \\left( \\varphi_y^-(0^-; x(a)) \\right)_2 \\\\\n\t&= \\frac{-e^{-2a} +1}{2e^{-a}} x_1(a) + \\frac{e^{-2a} +1}{2e^{-a}} x_2(a) - \\left( \\int_{a}^{0} e^{-P_1s} P_1 y^-(s) \\, ds \\right)_2 \\\\\n\t\\Leftrightarrow \\hspace{0.5 cm} x_2(a) &= \\frac{2 e^{-a}}{e^{-2a} +1} \\left( \\int_{a}^{0} e^{-P_1 s} P_1 y^-(s) \\, ds \\right)_2 - \\frac{- e^{-2a}+1}{e^{-2a} +1} x_1(a). \n\\end{align*}\nIn conclusion, the interface conditions specify the values $x_2(0)$ and $x_2(a)$. Furthermore, $\\varphi_y$ is an element of $D(\\mathcal{J}_0)$ and it satisfies the interface relation. Therefore, it is left to find the values of $x_1(a)$ and $x_1(0^+)$ such that the solution $\\varphi_y$ satisfies the last constitutive relation of $D(A)$, namely\n\\begin{align}\n\t\\label{Interface - Theorem A Generates a Unitary Semigroup - Boundary Condition in Proof}\n\tW_B R_{\\extern} \\begin{bmatrix}\n\t\t\\varphi_y^+(b; x(0^+)) \\\\\n\t\t\\varphi_y^-(a; x(a)) \n\t\\end{bmatrix} = 0, \n\\end{align}\nwhere $R_{\\extern}$ is defined in \\eqref{Interface - Matrix Rext}. We have $\\varphi_y^-(a;x(a)) = x(a)$ and\n\\begin{align*}\n\t\\varphi_y^+(b;x(0^+)) = e^{P_1b}x(0^+) - \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds. \n\\end{align*}\nUsing the definition of $R_{\\extern}$, $e^{P_1 \\cdot}$, and the values $x_2(a)$ and $x_2(0)$, we compute\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\sqrt{2} R_{\\extern} \\begin{bmatrix}\n\t\t\\varphi_y^+(b; x(0^+)) \\\\\n\t\t\\varphi_y^-(a; x(a)) \n\t\\end{bmatrix} \\\\\n\t&= \\begin{bmatrix}\n\t\t0 & -1 & 0 & 1 \\\\\n\t\t-1 & 0 & 1 & 0 \\\\\n\t\t1 & 0 & 1 & 0 \\\\\n\t\t0 & 1 & 0 & 1\n\t\\end{bmatrix} \\begin{bmatrix}\n\t\te^{P_1b}x(0^+) - \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\\\\n\t\tx(a)\n\t\\end{bmatrix} \\\\\n\t&= \\begin{bmatrix}\n\t\t- \\frac{- e^{2b} + 1}{2e^{b}} x_1(0^+) - \\frac{-e^{-2a} +1}{e^{-2a} +1} x_1(a) + \\frac{2e^{-a}}{e^{-2a} +1} \\left( \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_2 + \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_2 \\\\\n\t\t- \\frac{ e^{2b} + 1}{2e^{b}} x_1(0^+) + x_1(a) + \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_1 \\\\\n\t\t\\frac{e^{2b} +1}{2e^{b}} x_1(0^+) + x_1(a) - \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_1 \\\\\n\t\t\\frac{- e^{2b} +1}{2e^{b}} x_1(0^+) - \\frac{-e^{-2a} +1}{e^{-2a} +1} x_1(a) + \\frac{2e^{-a}}{e^{-2a} +1} \\left( \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_2 - \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_2\n\t\\end{bmatrix}.\n\\end{align*}\nNow, let the matrix $M \\in \\mathbb{R}^{2 \\times 2}$ be defined such that\n\\begin{align*}\n\t W_B \\begin{bmatrix}\n\t\t- \\frac{- e^{2b} + 1}{2e^{b}} x_1(0^+) - \\frac{-e^{-2a} +1}{e^{-2a} +1} x_1(a) \\\\\n\t\t- \\frac{ e^{2b} + 1}{2e^{b}} x_1(0^+) + x_1(a) \\\\\n\t\t\\frac{e^{2b} +1}{2e^{b}} x_1(0^+) + x_1(a) \\\\\n\t\t\\frac{- e^{2b} +1}{2e^{b}} x_1(0^+) - \\frac{-e^{-2a} +1}{e^{-2a} +1} x_1(a) \n\t\\end{bmatrix} = M \\begin{bmatrix}\n\tx_1(a) \\\\\n\tx_1(0^+)\n\\end{bmatrix}.\n\\end{align*}\nFurthermore, define $N \\in \\mathbb{R}^2$ by\n\\begin{align*}\n\tN = - W_B \\begin{bmatrix}\n\t\t\\frac{2e^{-a}}{e^{-2a} +1} \\left( \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_2 + \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_2 \\\\\n\t\t\\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_1 \\\\\n\t\t- \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_1 \\\\\n\t\t\\frac{2e^{-a}}{e^{-2a} +1} \\left( \\int_{a}^{0} e^{-P_1s}P_1 y^-(s) \\, ds \\right)_2 - \\left( \\int_{0}^{b} e^{P_1(b-s)} P_1 y^+(s) \\, ds \\right)_2\n\t\\end{bmatrix}.\n\\end{align*}\nThen the boundary condition \\eqref{Interface - Theorem A Generates a Unitary Semigroup - Boundary Condition in Proof} is satisfied if and only if\n\\begin{align}\n\tM \\begin{bmatrix}\n\t\tx_1(a) \\\\\n\t\tx_1(0^+)\n\t\\end{bmatrix} = N. \n\t\\label{Interface - Theorem A Generates a Unitary Semigroup - Linear System of Equations in Proof}\n\\end{align}\nWe shall verify that the matrix $M$ is invertible. If so, then for every $y \\in X$ the initial values are uniquely determined and can be calculated through \\eqref{Interface - Theorem A Generates a Unitary Semigroup - Linear System of Equations in Proof}. Moreover, the corresponding solution $\\varphi_y(\\cdot; x(a), x(0^+))$ of $(I-A)x = y$ is an element of $D(A)$, and so $\\ran(I - A) = X$. The Lumer-Phillips Theorem \\ref{Theorem Lumer-Phillips} then yields the claim. \n\\vspace{0.5 cm}\\\\\nAs in the first case, assume that there is some $0 \\neq r_0 \\in \\ker(M)$, and let $y = 0$. Consider the solution of equation \\eqref{Interface - ODE of (I-A)x equals y} given by\n\\begin{align*}\n\t\\varphi_0(z; x(a), x(0^+)) = c_0(z) e^{P_1(z-a)}x(a) + \\overline{c}_0(z) e^{P_1z} \\begin{bmatrix}\n\t\tx_1(0^+) \\\\\n\t\t0\n\t\\end{bmatrix}, \n\\end{align*}\nwith $r_0 = \\begin{bmatrix}\n\tx_1(a) \\\\\n\tx_1(0^+)\n\\end{bmatrix}$, and $x_2(a) = - \\frac{- e^{-2a}+1}{e^{-2a}+1} x_1(a)$. Then $\\varphi_0(\\cdot; x(a), x(0^+)) \\in D(\\mathcal{J}_0)$, and with this choice of the initial values, the interface relation \\eqref{Interface - Interface Passivity Relation} is satisfied as well. Note that $N = 0$, since $y = 0$. Furthermore, \\eqref{Interface - Theorem A Generates a Unitary Semigroup - Linear System of Equations in Proof} is satisfied, since by assumption we have\n\\begin{align*}\n\tM r_0 = 0.\n\\end{align*}\nAs explained in the first case, this contradicts the dissipativity of $A$, which is why the matrix $M$ is indeed invertible.\n\\vspace{0.5 cm}\\\\ \nIn conclusion, the operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A} is the infinitesimal generator of a contraction semigroup on the energy space $(X, \\| \\cdot \\|_{\\mathcal{Q}_0})$. The fact that for all $x_0 \\in D(A_{\\mathcal{Q}_0})$, the abstract Cauchy problem \\eqref{Interface - Theorem A Generates a Contraction Semigroup - ACP} has a unique classical solution $x \\colon [0, \\infty) \\to X$, depending continuously on the initial data, follows immediately from Theorem \\ref{Theorem Characterization of Well-Posedness}.\n\\end{proof}\nTheorem \\ref{Interface - Theorem A Generates a Contraction Semigroup} shows that, under certain boundary and interface conditions, the port-Hamil- tonian model of two systems of conservation laws coupled by a rigid interface is well-defined. If we compare the system \\eqref{Interface Paper - Augmented System wrt Ja} defined on the extended state space $\\tilde{X}$ in Subsection \\ref{Subsection Port-Hamiltonian Formulation and Dirac Structure} with the simplified system \\eqref{Interface - Simplified PH-System} treated in this section, we notice that no valuable information gets lost if we take the color functions $c_0$ and $\\overline{c}_0$ defined in \\eqref{Interface Paper - Characteristic Functions c0 and c0overline} not to be part of the state variables. For the moving interface scenario, this is not true, as the corresponding color functions \\eqref{Interface Paper - Color Functions Depending on Moving Interface} do not satisfy a trivial conservation law, but the transport equation in a distributional sense, see Subsection \\ref{Subsection Balance Equations of the State Variables - Moving Interface}. We will come back to the moving interface scenario in Subsection \\ref{Subsection Stability of the Family of Infinitesimal Generators}.\n\\vspace{0.5 cm}\\\\\nBased on Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}, we may find conditions which ensure that the internally stored energy of the system is conserved. \n\\begin{corollary}\n\t\\label{Interface - Corollary A Generates a Unitary Semigroup}\n\t\tConsider the port-Hamiltonian operator $A_{\\mathcal{Q}_0} \\colon D(A_{\\mathcal{Q}_0}) \\subset X \\to X$ defined in \\eqref{Interface - Simplified PH System - Operator A} with passivity constant $r=0$ and with a full rank matrix $W_B \\in \\mathbb{R}^{2 \\times 4}$ satisfying $W_B \\Sigma W_B^{\\top} = 0$. Then $A_{\\mathcal{Q}_0}$ is the infinitesimal generator of a unitary semigroup on the energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$.\n\\end{corollary}\n\n\\begin{proof}\n\tTheorem \\ref{Interface - Theorem A Generates a Contraction Semigroup} yields that $A_{\\mathcal{Q}_0}$ generates a contraction semigroup $(T(t))_{t \\geq 0}$ on the energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$. So, we only need to show that $(T(t))_{t \\geq 0}$ is a unitary semigroup.\n\t\\vspace{0.5 cm}\\\\\n\tBy Lemma \\ref{Interface - Auxiliary Lemma 2}, the matrix $W_B$ is of the form \\eqref{Interface - Auxiliary Lemma 2 - Matrix Representation}, with an invertible matrix $S \\in \\mathbb{R}^{2 \\times 2}$ and with $V \\in \\mathbb{R}^{2 \\times 2}$ satisfying $VV^{\\top} \\leq I$. Since we have\n\t\t\\begin{align*}\n\t\t0 = \tW_B \\Sigma W_B^{\\top} = S \\begin{bmatrix}\n\t\t\t\tI + V & I - V\n\t\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\t0 & I \\\\\n\t\t\tI & 0 \n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\tI + V^{\\top} \\\\\n\t\tI- V^{\\top}\n\t\\end{bmatrix} S^{\\top} = 2S (I- VV^{\\top}) S^{\\top}, \n\t\t\\end{align*}\n\twe deduce that $VV^{\\top} = V^{\\top}V = I$. Thus, by repeating Step 2 of the proof of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}, we get for all $x \\in D(A_{\\mathcal{Q}_0})$, \n\t\\begin{align}\n\t\t2 \\langle A_{\\mathcal{Q}_0}x, x \\rangle_{\\mathcal{Q}_0} = \\langle e_{\\partial}, f_{\\partial} \\rangle_2 - e_I f_I = \\lambda_x^{\\top} (-I + V^{\\top}V) \\lambda_x = 0. \n\t\t\\label{Interface - Theorem A Generates a Unitary Semigroup - Dissipativity Property in Proof}\n\t\\end{align}\nNow, for $x_0 \\in D(A_{\\mathcal{Q}_0})$, consider the mapping $t \\mapsto \\| T(t)x_0\\|_{\\mathcal{Q}_0}^2$, $t \\geq 0$. This mapping describes the energy of the solution of the abstract Cauchy problem \\eqref{Interface - Theorem A Generates a Contraction Semigroup - ACP}. Recall from Theorem~\\ref{Theorem Properties of C0-Semigroups}~\\ref{C0SemigroupProp3} that $T(t)x_0 \\in D(A_{\\mathcal{Q}_0})$ for all $t\\geq 0$ and for all $x_0 \\in D(A_{\\mathcal{Q}_0})$. By taking the derivative and by exploiting the dissipativity \\eqref{Interface - Theorem A Generates a Unitary Semigroup - Dissipativity Property in Proof} of $A_{\\mathcal{Q}_0}$, we obtain for all $t > 0$,\n\\begin{align*}\n\t\\frac{d}{dt} \\| T(t)x_0\\|_{\\mathcal{Q}_0}^2 &= \\frac{d}{dt} \\langle T(t)x_0 , T(t)x_0 \\rangle_{\\mathcal{Q}_0} \\\\\n\t&= \\langle A_{\\mathcal{Q}_0}T(t)x_0, T(t)x_0 \\rangle_{\\mathcal{Q}_0} + \\langle T(t)x_0, A_{\\mathcal{Q}_0}T(t)x_0 \\rangle_{\\mathcal{Q}_0} \\\\\n\t&= 2 \\langle A_{\\mathcal{Q}_0}T(t)x_0, T(t)x_0 \\rangle_{\\mathcal{Q}_0} \\\\\n\t&= 0. \n\\end{align*}\nThus, the mapping $t \\mapsto \\| T(t)x_0\\|_{\\mathcal{Q}_0}^2$ is constant for all $x_0 \\in D(A_{\\mathcal{Q}_0})$. Since $\\|T(0) x_0 \\|_{\\mathcal{Q}_0} = \\| x_0\\|_{\\mathcal{Q}_0}$ and since $A_{\\mathcal{Q}_0}$ is densely defined, we have \n\\begin{align*}\n\t\\|T(t) x\\|_{\\mathcal{Q}_0} = \\|x\\|_{\\mathcal{Q}_0}, \\hspace{0.5 cm} t \\geq 0, \\, x \\in X. \n\\end{align*}\nTherefore, $(T(t))_{t \\geq 0}$ is a unitary semigroup. This completes the proof. \n\\end{proof}\n\nTo finish this section, we model two transmission lines that are coupled through a resistor. \n\\begin{example}[Coupled Transmission Lines]\n\t\\label{Interface - Example Coupled Transmission Lines}\n\tConsider two transmission lines defined on the spatial intervals $[a,0)$ and $(0,b]$, respectively. We assume that the distributed capacity and the distributed inductance of the transmission lines are different. Recall from Subsection~\\ref{Subsection The Lossless Transmission Line} that the conservation laws are given by\n\t\\begin{align*}\n\t\t\\frac{\\partial}{\\partial t} \\begin{bmatrix}\n\t\t\tQ^- \\\\\n\t\t\t\\varphi^-\n\t\t\\end{bmatrix} &= \\begin{bmatrix}\n\t\t0 & -\\frac{\\partial}{\\partial z} \\\\\n\t\t- \\frac{\\partial}{\\partial z} & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\\frac{Q^-}{C^-} \\\\\n\t\\frac{\\varphi^-}{L^-}\n\\end{bmatrix}, \\hspace{0.5 cm} z \\in [a,0), \\, t > 0, \\\\\n\t\\frac{\\partial}{\\partial t} \\begin{bmatrix}\n\tQ^+ \\\\\n\t\\varphi^+\n\\end{bmatrix} &= \\begin{bmatrix}\n\t0 & -\\frac{\\partial}{\\partial z} \\\\\n\t- \\frac{\\partial}{\\partial z} & 0 \n\\end{bmatrix} \\begin{bmatrix}\n\t\\frac{Q^+}{C^+} \\\\\n\t\\frac{\\varphi^+}{L^+}\n\\end{bmatrix}, \\hspace{0.5 cm} z \\in (0,b], \\, t > 0.\n\t\t\\end{align*}\n\tLetting $x^{\\pm} = (Q^{\\pm}, \\varphi^{\\pm})$, the Hamiltonian functionals are given by \n\t\\begin{align*}\n\t\tH^-(x^-) &= \\int_{a}^{0} \\mathcal{H}^-(x^-(z)) \\, dz = \\frac{1}{2} \\langle x^-, \\mathcal{Q}^- x^- \\rangle_{L^2(a,0)}, \\hspace{0.5 cm} x^- \\in X^- = L^2((a,0), \\mathbb{R}^2), \\\\\n\t\t\tH^+(x^+) &= \\int_{a}^{0} \\mathcal{H}^+(x^+(z)) \\, dz = \\frac{1}{2} \\langle x^+, \\mathcal{Q}^+ x^+ \\rangle_{L^2(0,b)}, \\hspace{0.5 cm} x^+ \\in X^+ = L^2((0,b), \\mathbb{R}^2), \n\t\\end{align*}\nwhere $\\mathcal{Q}^{\\pm} \\in \\mathcal{L}(X^{\\pm})$ are the coercive multiplication operators given by\n\\begin{align*}\n\t(\\mathcal{Q}^{\\pm}x^{\\pm})(\\cdot) = \\begin{bmatrix}\n\t\t\\frac{1}{C^{\\pm}(\\cdot)} & 0 \\\\\n\t\t0 & \\frac{1}{L^{\\pm}(\\cdot)} \\end{bmatrix} \\begin{bmatrix}\n\t\t\tQ^{\\pm}(\\cdot) \\\\\n\t\t\t\\varphi^{\\pm}(\\cdot)\n\t\t\\end{bmatrix}, \\hspace{0.5 cm} x ^{\\pm} \\in X^{\\pm},\n\\end{align*}\nrespectively. Recall that the co-energy variables are given by the\nvoltages\n\\begin{align*}\n\tV^{\\pm} = \\frac{Q^{\\pm}}{C^{\\pm}} = \\frac{\\delta H^{\\pm}}{\\delta Q^{\\pm}} = \\left( \\mathcal{Q}^{\\pm} x^{\\pm} \\right)_1\n\\end{align*} \nand the currents \n\\begin{align*}\nI^{\\pm} = \\frac{\\varphi^{\\pm}}{L^{\\pm}} = \\frac{\\delta H^{\\pm}}{\\delta Q^{\\pm}} = \\left( \\mathcal{Q}^{\\pm} x^{\\pm} \\right)_2.\n\\end{align*}\nWe want to couple these two transmission lines at $z = 0$, and assume that a resistor with resistance $R_I >0$ is placed at the interface. Furthermore, we set the voltage at $z = a$ to zero, and we put another resistor with resistance $R_b > 0$ at $z = b$. We want to show by means of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup} that the coupled system is well-posed. \n\\vspace{0.5 cm}\\\\\nThe model of the coupling is precisely given by the system \\eqref{Interface - Simplified PH-System} defined on the state space $X = L^2([a,b], \\mathbb{R}^2)$, with $[a,b]$ the composed spatial domain. Defining the coercive matrix operator $\\mathcal{Q}_0 = c_0 \\mathcal{Q}^- + \\overline{c}_0 \\mathcal{Q}^+ \\in \\mathcal{L}(X)$, we may endow $X$ with the inner product $\\langle \\cdot , \\cdot \\rangle_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Inner Product wrt Q0}.\n\\vspace{0.5 cm}\\\\\nDue to the resistor placed at the interface, we expect a voltage drop, and therefore have a discontinuity of the effort variable $V = c_0 V^- + \\overline{c}_0 V^+$, whereas the current $I = c_0 I^- + \\overline{c}_0 I^+$ flows continuously through the resistor. Thus, the interface port variables are given by\n\\begin{align*}\n\tf_{I, \\mathcal{Q}_0 x} &= I(0^+) = I(0^-), \\\\\n\t-e_{I, \\mathcal{Q}_0 x} &= V(0^+) - V(0^-), \n\\end{align*}\nand, by virtue of Ohm's law, the interface relation is given by the passivity relation\n\\begin{align*}\n\tf_I(t) = \\frac{1}{R_I} e_I(t), \\hspace{0.5 cm} t \\geq 0.\n\\end{align*}\nFurthermore, we have the boundary conditions\n\\begin{align*}\nV(a,t) = 0 \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} V(b,t) = R_b I(b,t), \\hspace{0.5 cm} t \\geq 0.\n\\end{align*}\nNeglecting the time-dependence and using \\eqref{Interface - Boundary Flow and Effort}, we may equivalently write\n\\begin{align*}\n\t0 = \\begin{bmatrix}\n\t\t0 & 0 & 1 & 0 \\\\\n\t\t-1 & R_b & 0 & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\tV(b) \\\\\n\tI(b) \\\\\n\tV(a) \\\\\n\tI(a)\n\\end{bmatrix} = \\tilde{W}_B \\begin{bmatrix}\n(\\mathcal{Q}_0x)(b) \\\\\n(\\mathcal{Q}_0x)(a)\n\\end{bmatrix} = W_B \\begin{bmatrix}\nf_{\\partial, \\mathcal{Q}_0 x} \\\\\ne_{\\partial, \\mathcal{Q}_0 x}\n\\end{bmatrix},\n\\end{align*}\nwith $W_B = \\tilde{W}_B R_{\\extern}^{-1} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n\t0 & 1 & 1 & 0 \\\\\n\t-R_b & 1 & -1 & R_b\n\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 4}$. Note that $\\rank(W_B) = 2$. Furthermore, one computes that\n\\begin{align*}\n\tW_B \\Sigma W_B^{\\top} = \\begin{bmatrix}\n\t\t0 & 0 \\\\\n\t\t0 & 2R_b\n\t\\end{bmatrix} \\geq 0.\n\\end{align*}\nSumming up, the constituting relations of the domain of the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ associated with the system of transmission lines coupled through a resistor satisfy the assumptions of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}. Hence, $A_{\\mathcal{Q}_0}$ generates a contraction semigroup on the energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$, and therefore, this system is well-defined. Its dynamics are geometrically specified by the requirement that for all $t> 0$ it holds that\n\t\\begin{align*}\n\t\\left( \\begin{pmatrix}\n\t\t\\frac{\\partial}{\\partial t} x(\\cdot,t) \\\\\n\t\tf_I(t) \\\\\n\t\tf_{\\partial}(t)\n\t\\end{pmatrix} , \\begin{pmatrix}\n\t\t\\mathcal{Q}_0x(\\cdot,t) \\\\\n\t\te_I(t) \\\\\n\t\te_{\\partial}(t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\mathcal{J}_0}, \\hspace{0.3 cm} (f_I(t), e_I(t)) \\in \\mathcal{R}_{\\frac{1}{R_I}},\n\\end{align*}\nwhere the resistive structure $\\mathcal{R}_{\\frac{1}{R_I}}$ is given by\n\\begin{align*}\n\t\\mathcal{R}_{\\frac{1}{R_I}} = \\left\\{ (f_I, e_I) \\in \\mathbb{R}^2 \\mid f_I = \\frac{1}{R_I}e_I \\right\\}.\n\\end{align*}\n\\QEDA\n\\end{example}\n\n\\subsection{Exponential Stability}\n\\label{Subsection Exponential Stability}\nIn this section, we want to find conditions that guarantee the exponential stability (cf. Definition \\ref{Definition Exponential Stability}) of the abstract Cauchy problem \\eqref{Interface - Theorem A Generates a Contraction Semigroup - ACP} associated with the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A}. Throughout this section, we impose the following assumptions:\n\\begin{enumerate}[label = (A\\arabic*)]\n\t\\item \\label{MatrixQ0Assumption1} The matrix operators $\\mathcal{Q}^{\\pm}$ defining the coercive operator $\\mathcal{Q}_0 \\in \\mathcal{L}(X)$ satisfy $\\mathcal{Q}^{\\pm} \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2})$. \n\t\\item \\label{ContractionAssumption} $r \\geq 0$, $\\rank(W_B) = 2$ and $W_B \\Sigma W_B^{\\top} \\geq 0$, with $\\Sigma$ defined in \\eqref{Matrix Sigma}. In other words, the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ generates a contraction semigroup (see Theorem~\\ref{Interface - Theorem A Generates a Contraction Semigroup}).\n\\end{enumerate}\n\nWe begin by showing an essential property in order to verify the exponential stability of the port-Hamiltonian system \\eqref{Interface - Theorem A Generates a Contraction Semigroup - ACP}. The following lemma is a simple extension of \\cite[Lemma~9.1.2]{JacobZwart}.\n\\begin{lemma}\n\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stability}\n\tLet the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold. Denote by $(T(t))_{t \\geq 0}$ the contraction semigroup generated by $A_{\\mathcal{Q}_0}$. Then there exist constants $\\tau >0$ and $C >0$ such that for every initial value $x_0 \\in D(A_{\\mathcal{Q}_0})$, the trajectory $x(\\cdot) = T(\\cdot)x_0$ satisfies\n\t\\begin{align}\n\t\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stablity - Inequality}\n\t\t\\| x(\\tau) \\|_{\\mathcal{Q}_0}^2 &\\leq C \\int_{0}^{\\tau} \\| \\mathcal{Q}^-(a)x(a,t) \\|_2^2 + \\| \\mathcal{Q}^+(b) x(b,t) \\|_2^2 \\, dt.\n\t\\end{align} \n\\end{lemma}\n\n\\begin{proof}\n\tSince this is clear for $x_0 = 0$, assume that $0 \\neq x_0 \\in D(A_{\\mathcal{Q}_0})$. Define the function $F_1 \\colon (0,b] \\to (0, \\infty)$ by\n\t\\begin{align}\n\t\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stability - Function F1}\n\t\tF_1(z) := \\int_{\\gamma_1(b-z)}^{\\tau_1 - \\gamma_1(b-z)} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dt, \\hspace{0.5 cm} z \\in (0,b], \n\t\\end{align}\nwith $\\gamma_1 > 0$ and $\\tau_1 > 0$ chosen such that $\\tau_1 > 2 \\gamma_1 b$. Recall that on domains not containing the interface position, the operator $\\mathcal{J}_0$ defined in \\eqref{Interface - Operator J0} acts as the differential operator $P_1 \\frac{d}{dz}$, where $P_1$ is given in \\eqref{Interface - Matrix P1}. Following the proof of Lemma 9.1.2 in \\cite{JacobZwart}, by exploiting the fact that $x$ satisfies \\eqref{Interface - Simplified PH-System} and that the matrices $\\mathcal{Q}^+(z)$ and $P_1$ are symmetric, we find\n\\begin{align*}\n\t\\frac{d}{dz} F_1(z) &= - \\int_{\\gamma_1(b-z)}^{\\tau_1 - \\gamma_1(b-z)} x^{\\top}(z,t) \\frac{d}{dz} \\mathcal{Q}^+(z) x(z,t) \\, dt \\\\\n\t&\\hspace{0.5 cm} + x^{\\top}(z,\\tau_1 - \\gamma_1(b-z)) \\left( P_1^{-1} + \\gamma_1 \\mathcal{Q}^+(z) \\right) x(z, \\tau_1 - \\gamma_1(b-z)) \\\\\n\t&\\hspace{0.5 cm} + x^{\\top}(z, \\gamma_1(b-z)) \\left( -P_1^{-1} + \\gamma_1 \\mathcal{Q}^+(z) \\right) x(z,\\gamma_1(b-z)). \n\\end{align*}\nSince $\\frac{d}{dz} \\mathcal{Q}^+$ is bounded, we may find a constant $\\kappa_1 >0 $ such that\n\\begin{align*}\n\t \\frac{d}{dz} \\mathcal{Q}^+(z) \\leq \\kappa_1 \\mathcal{Q}^+(z), \\hspace{0.5 cm} z \\in (0,b]. \n\\end{align*}\nFurthermore, we may choose $\\gamma_1$ large enough such that $ P_1^{-1} + \\gamma_1 \\mathcal{Q}^+(z)$ and $ -P_1^{-1} + \\gamma_1 \\mathcal{Q}^+(z)$ are positive definite. The constant $\\tau_1$ has to be changed accordingly. This yields\n\\begin{align*}\n\t\\frac{d}{dz} F_1(z) \\geq - \\kappa_1 \\int_{\\gamma_1(b-z)}^{\\tau_1 - \\gamma_1(b-z)} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dt \\stackrel{\\eqref{Interface - Lemma Auxiliary Lemma for Exponential Stability - Function F1} }{=} - \\kappa_1 F_1(z). \n\\end{align*}\nThis implies that for all $z_1 \\in (0,b]$ it holds that\n\\begin{align*}\n\t\\int_{z_1}^{b} \\frac{\\frac{d}{dz}F_1(z)}{F_1(z)} \\, dz \\geq - \\kappa_1 \\int_{z_1}^{b} 1 \\, dz.\n\\end{align*}\nSimple computations yield that\n\\begin{align}\n\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stability - Estimate F1}\n\tF_1(b) \\geq F_1(z) e^{-\\kappa_1(b-z)} \\geq F_1(z) e^{-\\kappa_1 b}, \\hspace{0.5 cm} z\\in (0,b]. \n\\end{align}\nNext, we define $F_2 \\colon [a,0) \\to (0,\\infty)$ by\n\\begin{align*}\n\tF_2 (z) = \\int_{\\gamma_2 \\frac{b}{\\vert a \\vert}(z-a)}^{\\tau_2 - \\gamma_2 \\frac{b}{\\vert a \\vert }(z-a)} x^{\\top}(z,t) \\mathcal{Q}^-(z) x(z,t) \\, dt, \\hspace{0.5 cm} z \\in [a,0), \n\\end{align*}\nwhere $\\gamma_2 > 0$ and $\\tau_2 > -2\\gamma_2a \\frac{b}{\\vert a \\vert} = 2 \\gamma_2 b$. With similar arguments we may choose $\\gamma_2$, $\\tau_2$, and $\\kappa_2 > 0 $ large enough such that \n\\begin{align}\n\t\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stability - Estimate F2}\n\tF_2(z) \\leq F_2(a) e^{\\kappa_2 (z-a)} \\leq F_2(a) e^{-\\kappa_2 a}, \\hspace{0.5 cm} z \\in [a,0).\n\\end{align}\nNow define $\\gamma = \\max\\{ \\gamma_1, \\gamma_2\\}$, $\\tau = \\max\\{ \\tau_1, \\tau_2\\}$, $\\kappa = \\max\\{ \\kappa_1, \\kappa_2\\}$, and let $F_1$ and $F_2$ be defined with respect to $\\gamma$ and $\\tau$. \n\\vspace{0.5 cm} \\\\\nAs $(T(t))_{t \\geq 0}$ is of type $C_0(1,0)$, we have $\\|x(t_2)\\|_{\\mathcal{Q}_0} \\leq \\| x(t_1) \\|_{\\mathcal{Q}_0}$ for all $t_1 \\leq t_2$. Thus, \n\\begin{align}\n\t\t\\label{Interface - Lemma Auxiliary Lemma for Exponential Stability - Estimate T(tau)}\n\t\\int_{\\gamma b}^{\\tau - \\gamma b} \\|x(t)\\|_{\\mathcal{Q}_0}^2 \\, dt \\geq \\|x(\\tau - \\gamma b )\\|_{\\mathcal{Q}_0}^2 \\int_{\\gamma b}^{\\tau - \\gamma b} 1 \\, dt = (\\tau - 2 \\gamma b) \\| x(\\tau - \\gamma b) \\|_{\\mathcal{Q}_0}^2. \n\\end{align}\nApplying Fubini's theorem, using the estimates \\eqref{Interface - Lemma Auxiliary Lemma for Exponential Stability - Estimate F1}-\\eqref{Interface - Lemma Auxiliary Lemma for Exponential Stability - Estimate T(tau)}, and since $\\mathcal{Q}^+(z), \\mathcal{Q}^-(z) \\geq mI$ for all $z \\in [a,b]$, we infer that \n\\begin{align*}\n\t&\\hspace{0.5 cm} 2(\\tau - 2 \\gamma b) \\|x(\\tau)\\|_{\\mathcal{Q}_0}^2 \\\\\n\t&\\leq 2 (\\tau - 2 \\gamma b) \\| x(\\tau - \\gamma b)\\|_{\\mathcal{Q}_0}^2 \\\\\n\t&\\leq 2 \\int_{\\gamma b}^{\\tau - \\gamma b} \\|x(t) \\|_{\\mathcal{Q}_0}^2 \\, dt \\\\\n\t&= \\int_{\\gamma b}^{\\tau - \\gamma b} \\int_{a}^{0} x^{\\top}(z,t) \\mathcal{Q}^-(z) x(z,t) \\, dz + \\int_{0}^{b} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dz \\, dt \\\\\n\t&= \\int_{a}^{0} \\int_{\\gamma (b-0)}^{\\tau - \\gamma (b-0)} x^{\\top}(z,t) \\mathcal{Q}^-(z) x(z,t) \\, dt \\, dz + \\int_{0}^{b} \\int_{\\gamma (b-0)}^{\\tau - \\gamma (b-0)} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dt \\, dz \\\\\n\t&= \\int_{a}^{0} \\int_{\\gamma \\frac{b}{\\vert a \\vert}(0- a)}^{\\tau - \\gamma \\frac{b}{\\vert a \\vert}(0- a)} x^{\\top}(z,t) \\mathcal{Q}^-(z) x(z,t) \\, dt \\, dz + \\int_{0}^{b} \\int_{\\gamma (b-0)}^{\\tau - \\gamma (b-0)} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dt \\, dz \n\\end{align*}\n\\begin{align*}\n\t& \\leq \\int_{a}^{0} \\int_{\\gamma \\frac{b}{\\vert a \\vert}(z- a)}^{\\tau - \\gamma \\frac{b}{\\vert a \\vert}(z-a)} x^{\\top}(z,t) \\mathcal{Q}^-(z) x(z,t) \\, dt \\, dz + \\int_{0}^{b} \\int_{\\gamma (b-z)}^{\\tau - \\gamma (b-z)} x^{\\top}(z,t) \\mathcal{Q}^+(z) x(z,t) \\, dt \\, dz \\\\\n\t&= \\int_{a}^{0} F_2(z) \\, dz + \\int_{0}^{b} F_1(z) \\, dz \\\\\n\t&\\leq -a F_2(a) e^{- \\kappa a} + b F_1(b) e^{\\kappa b} \\\\\n\t&= -a e^{- \\kappa a} \\int_{0}^{\\tau} x^{\\top}(a,t) \\mathcal{Q}^-(a) x(a,t) \\, dt + b e^{\\kappa b} \\int_{0}^{\\tau} x^{\\top}(b,t) \\mathcal{Q}^+(b) x(b,t) \\, dt \\\\\n\t&\\leq \\frac{-a e^{- \\kappa a}}{m} \\int_{0}^{\\tau} \\| \\mathcal{Q}^- (a) x(a,t) \\|_2^2 \\, dt + \\frac{b e^{\\kappa b}}{m} \\int_{0}^{\\tau} \\| \\mathcal{Q}_0(b) x(b,t) \\|_2^2 \\, dt.\n \\end{align*}\nLetting $C = \\frac{\\max \\left\\{-a e^{- \\kappa a}, b e^{\\kappa b} \\right\\} }{2m (\\tau - 2 \\gamma b)}$, we conclude that\n\t\\begin{align*}\n\t\t\\|x(\\tau) \\|_{\\mathcal{Q}_0}^2 \\leq C \\int_{0}^{\\tau} \\| \\mathcal{Q}^-(a) x(a,t) \\|_2^2 + \\| \\mathcal{Q}^+(b) x(b,t) \\|_2^2 \\, dt.\n\t\\end{align*}\nThis shows \\eqref{Interface - Lemma Auxiliary Lemma for Exponential Stablity - Inequality} and, hence, the claim. \n\\end{proof}\n\nLemma \\ref{Interface - Lemma Auxiliary Lemma for Exponential Stability} shows that there exists some time $\\tau >0 $ such that the energy of a state can be bounded by the sum of the accumulated energies at both boundaries during the time interval $[0, \\tau]$. It allows us to prove the following sufficient condition for exponential stability.\n\\begin{theorem}\n\t\\label{Interface - Theorem Exponential Stability}\n\t\tLet the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold. Assume there exists some $ k > 0$ such that \n\t\\begin{align}\n\t\t\\label{Interface - Theorem Exponential Stability - Inequality}\n\t\t\\langle A_{\\mathcal{Q}_0} x_0, x_0 \\rangle_{\\mathcal{Q}_0} &\\leq - k \\left( \\| (\\mathcal{Q}_0 x_0)(a) \\|_2^2 + \\| (\\mathcal{Q}_0 x_0)(b) \\|_2^2 \\right)\n\t\\end{align}\nholds for all $x_0 \\in D(A_{\\mathcal{Q}_0})$. Then $A_{\\mathcal{Q}_0}$ generates an exponentially stable $C_0$-semigroup. \n\\end{theorem}\n\\begin{proof}\n\tThis follows immediately from Lemma \\ref{Interface - Lemma Auxiliary Lemma for Exponential Stability} and the proof of Theorem 9.1.3 in \\cite{JacobZwart}. \n\\end{proof}\nLastly, we want to present another sufficient condition for the exponential stability of the port-Hamiltonian system \\eqref{Interface - Simplified PH-System}. Once again, we follow the exposition in \\cite[Chapter 9]{JacobZwart}.\n\\begin{theorem}\n\t\\label{Interface - Theorem 2 Exponential Stability}\n\t\tLet the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold. If the matrix $W_B$ satisfies $W_B \\Sigma W_B^{\\top} > 0$, then $A_{\\mathcal{Q}_0}$ generates an exponentially stable $C_0$-semigroup. \n\\end{theorem}\n\n\\begin{proof}\n\tBy Lemma \\ref{Interface - Auxiliary Lemma 2}, we may write\n\t\\begin{align*}\n\t\tW_B = S \\begin{bmatrix}\n\t\t\tI + V & I - V\n\t\t\\end{bmatrix}, \n\t\\end{align*}\nwhere $S \\in \\mathbb{R}^{2 \\times 2}$ is invertible and $V \\in \\mathbb{R}^{2 \\times 2}$ satisfies $VV^{\\top} < I$. Moreover, define the matrix\n\\begin{align*}\n\tW_C := \\begin{bmatrix}\n\t\tI + V^{\\top} & - I + V^{\\top}\n\t\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 4}.\n\\end{align*}\nAs shown in the proof of Lemma 9.1.4 in \\cite{JacobZwart}, the matrix $W_1 := \\begin{bmatrix}\n\tW_B \\\\\n\tW_C\n\\end{bmatrix}$ is invertible. \n\\vspace{0.5 cm}\\\\\nNow, let $ x \\in D(A_{\\mathcal{Q}_0})$. By Lemma \\ref{Interface - Auxiliary Lemma 3}, we have\n\\begin{align*}\n\t\\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = \\begin{bmatrix}\n\tI - V \\\\\n\t-I - V\n\\end{bmatrix} \\lambda_x\n\\end{align*}\nfor some $\\lambda_x \\in \\mathbb{R}^2$. By virtue of \\eqref{Interface - Skew-Symmetry of J0 - Representation 2}, we compute that\n\\begin{align*}\n\t\\langle A_{\\mathcal{Q}_0}x, x \\rangle_{\\mathcal{Q}_0} + \\langle x , A_{\\mathcal{Q}_0}x \\rangle_{\\mathcal{Q}_0} &= \\frac{1}{2} \\left( \\langle \\mathcal{J}_0\\mathcal{Q}_0 x, \\mathcal{Q}_0 x \\rangle_{L^2} + \\langle \\mathcal{Q}_0 x , \\mathcal{J}_0 \\mathcal{Q}_0 x \\rangle_{L^2} \\right) \\\\\n\t&= \\langle e_{\\partial}, f_{\\partial} \\rangle_2 - e_If_I \\\\\n\t&= \\lambda_x^{\\top}(-I + V^{\\top}V) \\lambda_x - e_I f_I. \n\\end{align*}\nFurthermore, it holds that\n\\begin{align*}\n\ty_x := W_C \\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = \\begin{bmatrix}\n\tI + V^{\\top} & -I + V^{\\top}\n\\end{bmatrix} \\begin{bmatrix}\nI - V \\\\\n-I - V\n\\end{bmatrix} \\lambda_x = 2(I- V^{\\top}V) \\lambda_x . \n\\end{align*}\nSince $V^{\\top} V - I <0$, the matrix $-I + V^{\\top} V$ is invertible, and $(-I + V^{\\top}V)^{-1} \\leq -\\hat{m}_1 I$ for some $\\hat{m}_1 > 0$. Combining the preceding equations and using the passivity condition \\eqref{Interface - Interface Passivity Relation} of the interface port variables, we obtain the following estimate:\n\\begin{align}\n\t\\langle A_{\\mathcal{Q}_0}x, x \\rangle_{\\mathcal{Q}_0} = \\frac{1}{8} y_x^{\\top} (-I + V^{\\top}V)^{-1}y_x - \\frac{1}{2}e_If_I \\leq -m_1 \\|y_x\\|_2^2,\n\t\\label{Interface - Lemma Exponential Stability Proof - Estimate 1}\n\\end{align}\nwith $m_1 = \\frac{\\hat{m}_1}{8}$. Using the definition of $y_x$ and the boundary port variables \\eqref{Interface - Boundary Flow and Effort}, we infer\n\\begin{align}\n\t\\begin{bmatrix}\n\t\t0 \\\\\n\t\ty_x\n\t\\end{bmatrix} = \\begin{bmatrix}\n\tW_B \\\\\n\tW_C\n\\end{bmatrix} \\begin{bmatrix}\nf_{\\partial} \\\\\ne_{\\partial}\n\\end{bmatrix} = W_1 R_{\\extern} \\begin{bmatrix}\n\t(\\mathcal{Q}_0x)(b) \\\\\n\t(\\mathcal{Q}_0x)(a)\n\\end{bmatrix} := W \\begin{bmatrix}\n(\\mathcal{Q}_0x)(b) \\\\\n(\\mathcal{Q}_0x)(a)\n\\end{bmatrix}. \n\\label{Interface - Lemma Exponential Stability Proof - Definition W}\n\\end{align}\nAs both $R_{\\extern}$ and $W_1$ are invertible, the matrix $W = W_1 R_{\\extern} \\in \\mathbb{R}^{4 \\times 4}$ is invertible as well, and there exists some $m_2 > 0$ such that $\\|W w \\|_2^2 \\geq m_2 \\|w\\|_2^2$ holds for all $w \\in \\mathbb{R}^4$. Consequently, taking the norm on both sides of \\eqref{Interface - Lemma Exponential Stability Proof - Definition W} yields\n\\begin{align}\n\t\\|y_x\\|_2^2 = \\left\\| W \\begin{bmatrix}\n\t\t(\\mathcal{Q}_0x)(b) \\\\\n\t\t(\\mathcal{Q}_0x)(a)\n\t\\end{bmatrix} \\right\\|_2^2 \\geq m_2 \\left\\| \\begin{bmatrix}\n\t(\\mathcal{Q}_0x)(b) \\\\\n\t(\\mathcal{Q}_0x)(a)\n\\end{bmatrix} \\right\\|_2^2 = m_2 \\left( \\| (\\mathcal{Q}_0x)(a) \\|_2^2 + \\| (\\mathcal{Q}_0x)(b) \\|_2^2 \\right). \n\\label{Interface - Lemma Exponential Stability Proof - Estimate y}\n\\end{align}\nFinally, inserting \\eqref{Interface - Lemma Exponential Stability Proof - Estimate y} into \\eqref{Interface - Lemma Exponential Stability Proof - Estimate 1}, we obtain for all $x \\in D(A_{\\mathcal{Q}_0})$, \n\\begin{align*}\n\t\t\\langle A_{\\mathcal{Q}_0}x, x \\rangle_{\\mathcal{Q}_0} \\leq -m_1 \\|y_x\\|_2^2 \\leq - m_1 m_2 \\left( \\| (\\mathcal{Q}_0x)(a) \\|_2^2 + \\| (\\mathcal{Q}_0x)(b) \\|_2^2 \\right). \n\\end{align*}\nThis is exactly inequality \\eqref{Interface - Theorem Exponential Stability - Inequality}, and so Lemma \\ref{Interface - Theorem Exponential Stability} yields the exponential stability of the contraction semigroup generated by $A_{\\mathcal{Q}_0}$.\n\\end{proof}\nIn this section, we have shown that if we impose certain boundary conditions, then we can ensure that the semigroup $(T(t))_{t \\geq 0}$ generated by the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ is of type $C_0(1, -\\alpha)$ for some $\\alpha > 0$, i.e.,\n\\begin{align*}\n\t\\|T(t)\\|_{\\mathcal{L}(X, \\|\\cdot\\|_{\\mathcal{Q}_0})} \\leq e^{-\\alpha}, \\hspace{0.5 cm} t \\geq 0. \n\\end{align*}\nAs opposed to the boundary conditions, the interface condition plays a minor role in the proof of Theorem \\ref{Interface - Theorem 2 Exponential Stability}. One only has to ensure that there is no power inflow at the interface position.\n\n\n\n\n\\subsection{The Adjoint Operator}\n\\label{Subsection The Adjoint Operator}\nThis section aims to specify the Hilbert space adjoint $A_{\\mathcal{Q}_0}^{\\ast}$ of the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A}. To this end, we need an auxiliary lemma. For the proof, we follow the idea of the second assertion in \\cite[Lemma~7.2.1]{JacobZwart}.\n\\begin{lemma}\n\t\\label{Interface - Auxiliary Lemma Adjoint Operator}\n\t\tConsider the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A}. Then for every vector\n\t\t\\begin{align*}\n\t\t\t\\begin{bmatrix}\n\t\t\t\tf \\\\\n\t\t\t\te\n\t\t\t\\end{bmatrix} \\in \\ker(W_B)\n\t\t\\end{align*}\n\tthere exists an element $x \\in D(A_{\\mathcal{Q}_0})$ such that\n\t\\begin{align}\n\t\t\\begin{bmatrix}\n\t\t\tf_{\\partial, \\mathcal{Q}_0 x} \\\\\n\t\t\te_{\\partial, \\mathcal{Q}_0 x}\n\t\t\\end{bmatrix} = \t\\begin{bmatrix}\n\t\tf \\\\\n\t\te\n\t\\end{bmatrix}.\n\\label{Interface - Auxiliary Lemma Adjoint Operator - Target Equation}\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\begin{bmatrix}\n\tf \\\\\n\te\n\\end{bmatrix} \\in \\ker(W_B)$, and define\n\\begin{align*}\n\t\\begin{bmatrix}\n\t\tu \\\\\n\t\ty\n\t\\end{bmatrix} = R_{\\extern}^{-1} \\begin{bmatrix}\n\t\tf \\\\\n\t\te\n\t\\end{bmatrix},\n\\end{align*}\nwith $R_{\\extern}$ given in \\eqref{Interface - Matrix Rext}. Now, consider $x_0 \\in X$ defined as\n\\begin{align*}\n\tx_0(z) = \\mathcal{Q}_0^{-1}(z) \\left( \\frac{(b-z)z}{(b-a)a} y + \\frac{(z-a)z}{(b-a)b} u \\right), \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align*}\nOne can readily see that $\\mathcal{Q}_0x_0 \\in H^1([a,b], \\mathbb{R}^2) \\subset D(\\mathcal{J}_0)$. Furthermore, we compute\n\\begin{align*}\n\t(\\mathcal{Q}_0x_0)(a) &= y, \\\\\n\t(\\mathcal{Q}_0x_0)(0) &= 0, \\\\\n\t(\\mathcal{Q}_0x_0)(b) &= u.\n\\end{align*}\nThus, $x_0$ satisfies the interface condition \\eqref{Interface - Interface Passivity Relation}, and by definition of the boundary port variables \\eqref{Interface - Boundary Flow and Effort} we have\n\\begin{align*}\n\\begin{bmatrix}\n\tf_{\\partial, \\mathcal{Q}_0x_0} \\\\\n\te_{\\partial, \\mathcal{Q}_0x_0}\n\\end{bmatrix} =\tR_{\\extern}\t\\trace_0 (\\mathcal{Q}_0x_0) = R_{\\extern} \\begin{bmatrix}\n\t\t(\\mathcal{Q}_0x_0)(b) \\\\\n\t\t(\\mathcal{Q}_0x_0)(a)\n\t\\end{bmatrix} = \\begin{bmatrix}\n\tf \\\\\n\te\n\\end{bmatrix} \\in \\ker(W_B).\n\\end{align*}\nAltogether, $x_0$ is an element of $D(A_{\\mathcal{Q}_0})$ and it satisfies \\eqref{Interface - Auxiliary Lemma Adjoint Operator - Target Equation}. \n\\end{proof}\n\n The proof of the following theorem is based on the proof of Theorem 2.24 in \\cite{Villegas}. \n\\begin{theorem}\n\t\\label{Interface - Theorem Adjoint Operator}\n\tConsider the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A}. Assume that the matrix $W_B$ may be written as \\eqref{Interface - Auxiliary Lemma 2 - Matrix Representation}, with $S \\in \\mathbb{R}^{2 \\times 2}$ invertible and $V \\in \\mathbb{R}^{2 \\times 2}$ satisfying $V^{\\top}V \\leq I$. On the energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$, the Hilbert space adjoint $A_{\\mathcal{Q}_0}^{\\ast} \\colon D(A_{\\mathcal{Q}_0}^{\\ast}) \\subset X \\to X$ of the operator $A_{\\mathcal{Q}_0}$ is given by\n\t\\begin{align}\n\t\t\\begin{split}\n\t\tD(A_{\\mathcal{Q}_0}^{\\ast}) &= \\left\\{ y \\in X \\hspace{0.1 cm} \\Big| \\hspace{0.1 cm} \\mathcal{Q}_0y \\in D(\\mathcal{J}_0), \\hspace{0.1 cm} f_{I, \\mathcal{Q}_0y} = -r e_{I, \\mathcal{Q}_0y}, \\hspace{0.1 cm} \\begin{bmatrix}\n\t\t\t-(I+V^{\\top}) & I + V^{\\top}\n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\tf_{\\partial, \\mathcal{Q}_0y} \\\\\n\t\te_{\\partial, \\mathcal{Q}_0 y}\n\t\\end{bmatrix} = 0 \\right\\}, \\\\\n\t\tA_{\\mathcal{Q}_0}^{\\ast} y &= - \\mathcal{J}_0 (\\mathcal{Q}_0 y), \\hspace{0.5 cm} y \\in D(A_{\\mathcal{Q}_0}^{\\ast}). \n\t\\end{split}\n\\label{Interface - Adjoint Operator}\n\t\\end{align}\n\\end{theorem}\n\\begin{proof}\n\tRecall from Section \\ref{Section Operators} that the domain of the Hilbert space adjoint can be written as \\eqref{Adjoint Operator Domain wrt Frechet Riesz}. Let $x \\in \\mathbf{H}_0^1(a,b) \\times H_0^1([a,b], \\mathbb{R}) \\subset D(A_{\\mathcal{Q}_0})$, with $\\mathbf{H}_0^1(a,b)$ the space of $H^1$-functions vanishing both at the boundary and the interface defined in \\eqref{Interface Paper - Subspace H(a,b)}. If $y \\in D(A_{\\mathcal{Q}_0}^{\\ast})$, then the following holds:\n\t\\begin{align*}\n\t \\frac{1}{2} \\langle \\mathcal{J}_0 (\\mathcal{Q}_0 x) , \\mathcal{Q}_0 y \\rangle_{L^2} =\t\\langle A_{\\mathcal{Q}_0} x , y \\rangle_{\\mathcal{Q}_0} = \\langle x, A_{\\mathcal{Q}_0}^{\\ast} y \\rangle_{\\mathcal{Q}_0} = \\frac{1}{2} \\langle \\mathcal{Q}_0 x , A_{\\mathcal{Q}_0}^{\\ast}y \\rangle_{L^2}. \n\t\\end{align*}\nFrom this equation we infer that $\\mathcal{Q}_0 y$ has to be an element of $D(\\mathcal{J}_0)$ for all $y \\in D(A_{\\mathcal{Q}_0}^{\\ast})$. This is the first constitutive relation of $D(A_{\\mathcal{Q}_0}^{\\ast})$, and it allows us to define boundary and interface port variables with respect to $\\mathcal{Q}_0y$. By virtue of equation \\eqref{Interface - Skew-Symmetry of J0 - Representation 2}, we get for all $x \\in D(A_{\\mathcal{Q}_0})$, $y \\in D(A_{\\mathcal{Q}_0}^{\\ast})$,\n\\begin{align}\n\t\\begin{split}\n\t2 \\langle A_{\\mathcal{Q}_0} x, y \\rangle_{\\mathcal{Q}_0} &= \\langle \\mathcal{J}_0 (\\mathcal{Q}_0 x), \\mathcal{Q}_0 y \\rangle_{L^2} \\\\\n\t&= - \\langle \\mathcal{Q}_0 x, \\mathcal{J}_0 (\\mathcal{Q}_0 y) \\rangle_{L^2} + \\langle e_{\\partial, \\mathcal{Q}_0 y} , f_{\\partial, \\mathcal{Q}_0 x} \\rangle_2 + \\langle e_{\\partial, \\mathcal{Q}_0 x}, f_{\\partial, \\mathcal{Q}_0 y} \\rangle_2 \\\\\n\t&\\hspace{0.5 cm} - f_{I, \\mathcal{Q}_0 x} e_{I, \\mathcal{Q}_0 y} - f_{I, \\mathcal{Q}_0 y} e_{I, \\mathcal{Q}_0 x}. \n\t\\end{split} \n\\label{Interface - Theorem Adjoint Operator - Relation in Proof}\n\\end{align}\nWe need to get rid of the boundary and interface terms. With $\\Sigma$ defined in \\eqref{Matrix Sigma}, we may write\n\\begin{align*}\n\t\\langle e_{\\partial, \\mathcal{Q}_0 y} , f_{\\partial, \\mathcal{Q}_0 x} \\rangle_2 + \\langle e_{\\partial, \\mathcal{Q}_0 x}, f_{\\partial, \\mathcal{Q}_0 y} \\rangle_2 = \\begin{bmatrix}\n\t\tf_{\\partial, \\mathcal{Q}_0x}^{\\top} &\n\t\te_{\\partial, \\mathcal{Q}_0y}^{\\top} \n\t\\end{bmatrix} \\Sigma \\begin{bmatrix}\n\t\tf_{\\partial, \\mathcal{Q}_0y} \\\\\n\t\te_{\\partial, \\mathcal{Q}_0y}\n\t\\end{bmatrix}.\n\\end{align*}\nBy assumption, $\\begin{bmatrix} \n\tf_{\\partial, \\mathcal{Q}_0x} \\\\\n\te_{\\partial, \\mathcal{Q}_0x} \n\\end{bmatrix} \\in \\ker (W_B)$ for all $x \\in D(A_{\\mathcal{Q}_0})$. By virtue of Lemma \\ref{Interface - Auxiliary Lemma 3}, we have\n\\begin{align*}\n\t\\begin{bmatrix} \n\t\tf_{\\partial, \\mathcal{Q}_0x} \\\\\n\t\te_{\\partial, \\mathcal{Q}_0x} \n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\tI + V \\\\\n\t\t- (I + V) \n\t\\end{bmatrix} \\lambda_x\n\\end{align*}\nfor some $\\lambda_x \\in \\mathbb{R}^2$. Lemma \\ref{Interface - Auxiliary Lemma Adjoint Operator} yields that for every $\\begin{bmatrix}\n\tf \\\\\n\te\n\\end{bmatrix} \\in \\ker (W_B)$ there exists some $x \\in D(A_{\\mathcal{Q}_0})$ such that $\\begin{bmatrix} \n\tf_{\\partial, \\mathcal{Q}_0x} \\\\\n\te_{\\partial, \\mathcal{Q}_0x} \n\\end{bmatrix} = \\begin{bmatrix}\n\tf \\\\\n\te\n\\end{bmatrix}$. Now, since $\\dim(\\ker(W_B)) = 2$, the equality\n\\begin{align*}\n\t\\lambda_x^{\\top} \\begin{bmatrix}\n\t\tI + V^{\\top} & - (I+V^{\\top}) \n\t\\end{bmatrix} \\Sigma \\begin{bmatrix}\n\t\tf_{\\partial, \\mathcal{Q}_0y } \\\\\n\t\te_{\\partial, \\mathcal{Q}_0y}\n\t\\end{bmatrix} = 0\n\\end{align*}\nhas to hold for every $\\lambda_x \\in \\mathbb{R}^2$. Hence, the second constitutive condition of $D(A_{\\mathcal{Q}_0}^{\\ast})$ is given by\n\\begin{align*}\n\t\\begin{bmatrix} \n\t\tf_{\\partial, \\mathcal{Q}_0y} \\\\\n\t\te_{\\partial, \\mathcal{Q}_0y} \n\t\\end{bmatrix} \\in \\ker \\left( \\begin{bmatrix}\n\t\t-(I + V^{\\top}) & I + V^{\\top}\n\t\\end{bmatrix} \\right).\n\\end{align*}\n\nLastly, we need to get rid of the interface terms. Due to the interface relation \\eqref{Interface - Interface Passivity Relation}, we can write\n\n\\begin{align*}\n f_{I, \\mathcal{Q}_0x} e_{I, \\mathcal{Q}_0y} + f_{I, \\mathcal{Q}_0y} e_{I, \\mathcal{Q}_0x} =\n\t r e_{I, \\mathcal{Q}_0x} e_{I,\\mathcal{Q}_0y} + f_{I, \\mathcal{Q}_0y} e_{I, \\mathcal{Q}_0x} \\stackrel{!}{=} 0.\n\\end{align*}\nHence, the last constitutive condition of $D(A_{\\mathcal{Q}_0}^{\\ast})$ is given by\n\\begin{align*}\n\tf_{I, \\mathcal{Q}_0y} = - r e_{I, \\mathcal{Q}_0y}.\n\\end{align*}\nAltogether, the domain of the adjoint operator $A_{\\mathcal{Q}_0}^{\\ast}$ is precisely given as in \\eqref{Interface - Adjoint Operator}. For all $x \\in D(A_{\\mathcal{Q}_0})$, $y \\in D(A_{\\mathcal{Q}_0}^{\\ast})$, equation \\eqref{Interface - Theorem Adjoint Operator - Relation in Proof} becomes\n\\begin{align*}\n\t\\langle A_{\\mathcal{Q}_0} x , y \\rangle_{\\mathcal{Q}_0} = \\frac{1}{2} \\langle \\mathcal{J}_0 (\\mathcal{Q}_0 x), \\mathcal{Q}_0 y \\rangle_{L^2} = - \\frac{1}{2} \\langle \\mathcal{Q}_0 x, \\mathcal{J}_0 (\\mathcal{Q}_0 y) \\rangle_{L^2} = \\langle x , -\\mathcal{J}_0 (\\mathcal{Q}_0 y) \\rangle_{\\mathcal{Q}_0}. \n\\end{align*}\nThus, it holds that\n\\begin{align*}\n\tA_{\\mathcal{Q}_0}^{\\ast} y = - \\mathcal{J}_0 (\\mathcal{Q}_0 y), \\hspace{0.5 cm} y \\in D(A_{\\mathcal{Q}_0}^{\\ast}).\n\\end{align*}\nThis completes the proof. \n\\end{proof}\n\n\\subsection{Resolvent associated with $\\mathbf{A_{\\mathcal{Q}_0}}$}\n\\label{Subsection Resolvent}\nIn this section, we want to determine the resolvent operator $R(\\lambda, A_{\\mathcal{Q}_0})$ of the port-Hamiltonian operator $A_{\\mathcal{Q}_0}$ defined in \\eqref{Interface - Simplified PH System - Operator A} for $\\lambda \\in (0, \\infty) \\subset \\rho(A)$. For that purpose, we will basically replicate Step 3 in the proof of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}. Recall that for $\\mathcal{Q}_0 \\equiv I$, the solution of $(I-A_I)x = y$ for some $y \\in X$ is given by $\\varphi_y(\\cdot; x(a), x(0^+)) \\in D(A_I)$ defined in \\eqref{Interface - Solution of ODE left}, \\eqref{Interface - Solution of ODE right}, and \\eqref{Interface - Solution of ODE complete}. The initial value $x(a)$ can be calculated through \\eqref{Interface - D(A) condition - representation 2}, and $x(0^+)$ is given by\n\\begin{align*}\n\tx(0^+) = \\begin{bmatrix}\n\t\tx_1(0^+) \\\\\n\t\tx_2(0) \n\t\\end{bmatrix} = \\begin{bmatrix}\n\t\t- \\frac{1}{r} \\left( \\varphi^-(0^-; x(a)) \\right)_2 + \\left( \\varphi^-(0^-; x(a)) \\right)_1 \\\\\n\t\t\\left( \\varphi^-(0^-; x(a)) \\right)_2 \n\t\\end{bmatrix}. \n\\end{align*}\nThis yields\n\\begin{align*}\n\t\\| (I-A)^{-1} y \\|_{L^2}^2 &= \\|\\varphi_y(\\cdot; x(a), x(0^+)) \\|_{L^2}^2 \\\\\n\t&= \\int_{a}^{b} \\|\\varphi_y(z; x(a), x(0^+)) \\|_2^2 \\, dz \\\\\n\t&= \\int_{a}^{b} \\left[ \\left( c_0(z) \\varphi^-(z; x(a)) + \\overline{c}_0(z)\\varphi^+(z; x(0^+)) \\right)_1 \\right]^2 \\\\\n\t&\\hspace{0.5 cm} + \\left[ \\left( c_0(z) \\varphi^-(z; x(a)) + \\overline{c}_0(z)\\varphi^+(z; x(0^+)) \\right)_2 \\right]^2 \\, dz \\\\\n\t&= \\int_{a}^{0} \\| \\varphi^-(z;x(a)) \\|_2^2 \\, dz + \\int_{0}^{b} \\| \\varphi^+(z;x(0^+))\\|_2^2 \\, dz.\n\\end{align*}\nThus, the solution of the equation $(\\lambda I - A_{\\mathcal{Q}_0})x = y$ for some $y \\in X$, $\\lambda >0$ coincides with $R(\\lambda, A_{\\mathcal{Q}_0})y$. Subsequently, we will exploit this fact in order to compute the resolvent operator of $A_{\\mathcal{Q}_0}$.\n\\vspace{0.5 cm}\\\\\nTo this end, consider again the energy space $(X, \\| \\cdot\\|_{\\mathcal{Q}_0})$ with $\\mathcal{Q}_0= c_0 \\mathcal{Q}^- + \\overline{c}_0 \\mathcal{Q}^+ \\in \\mathcal{L}(X)$ a coercive matrix operator, and let the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold, where $r > 0$. Let $y \\in X$ and $\\lambda > 0$ be arbitary. Recall that on the complementary domains $[a,0)$ and $(0,b]$, the operator $\\mathcal{J}_0$ given in \\eqref{Interface - Operator J0} acts as the differential operator $P_1 \\frac{d}{dz}$, with $P_1$ defined in \\eqref{Interface - Matrix P1}. Thus, the equation $(\\lambda I-A_{\\mathcal{Q}_0})x = y$ can be equivalently written as\n\\begin{align*}\n\t\\lambda x^- - P_1 \\frac{d}{dz} (\\mathcal{Q}^- x^-) &= y^- \\hspace{0.5 cm} \\text{on } [a,0), \\\\\n\t\\lambda x^+ - P_1 \\frac{d}{dz}(\\mathcal{Q}^+ x^+) &= y^+ \\hspace{0.5 cm} \\text{on } (0,b].\n\\end{align*}\nSince $\\mathcal{Q}^{\\pm} \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2})$, observe that we may write\n\\begin{align*}\n\t\\frac{d}{dz} x^{\\pm}(z) = (\\mathcal{Q}^{\\pm})^{-1} (z) \\left( \\lambda P_1 - \\frac{d}{dz} \\mathcal{Q}^{\\pm}(z) \\right) x^{\\pm}(z) - (\\mathcal{Q}^{\\pm})^{-1}(z) P_1 y^{\\pm}(z)\n\n\\end{align*}\non the respective subdomain. \nThe variation of constants formula yields\n\\begin{align*}\n\t\\varphi_{\\lambda, y}^-(z; x(a)) = \\Lambda_{\\lambda}^-(z,a) x(a) - \\int_{a}^{z} \\Lambda_{\\lambda}^-(z,s)(\\mathcal{Q}^-)^{-1}(s) P_1 y^-(s) \\, ds, \\hspace{0.5 cm} z \\in [a,0),\n\\end{align*}\nand\n\\begin{align*}\n\t\\varphi_{\\lambda, y}^+(z; x(0^+)) = \\Lambda_{\\lambda}^+(z,0) x(0^+) - \\int_{0}^{z} \\Lambda_{\\lambda}^+(z,s) (\\mathcal{Q}^+)^{-1}(s) P_1 y^+(s) \\, ds, \\hspace{0.5 cm} z \\in (0,b],\n\\end{align*}\nwith $\\Lambda_{\\lambda}^-(\\cdot,a)$ and $\\Lambda_{\\lambda}^+(\\cdot,0^+)$ the respective transition matrices defined on the respective subdomain. \nThe composed solution of $(I-A)x = y$ is therefore given by\n\\begin{align*}\n\t\\varphi_{\\lambda, y}(z; x(a), x(0^+)) = c_0(z) \\varphi_{\\lambda, y}^-(z; x(a)) + \\overline{c}_0(z) \\varphi_{\\lambda, y}^+(z; x(0^+)), \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align*}\nFor the remainder of this section, we will omit indicating the dependence on $\\lambda$ and $y$. Analogously to the proof of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}, the initial value $x(a)$ is chosen such that the boundary condition\n\\begin{align*}\n\tW_B R_0 \\begin{bmatrix}\n\t\t\\mathcal{Q}(b) \\varphi(b; x(a)) \\\\\n\t\t\\mathcal{Q}(a) \\varphi(a; x(a))\n\t\\end{bmatrix} = 0\n\\end{align*}\nis satisfied, and the value $x(0^+)$ is determined by means of the interface relations. In the following, we will compute those values.\n\\vspace{0.5 cm}\\\\\nAccording to \\eqref{Interface - Simplified Continuity Equation}-\\eqref{Interface - Simplified Balance Equation}, the interface port variables are given as follows:\n\t\\begin{align*}\n\t\tf_I &= \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_2 = \\left( \\mathcal{Q}^+(0) x(0^+) \\right)_2 \\\\\n\t\t- e_I &= \\left( \\mathcal{Q}^+(0) x(0^+) \\right)_1 - \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1. \n\t\\end{align*}\n\tMore precisely, \n\t\\begin{align}\n\t\t\\begin{split}\n\t\tf_I &= Q_{12}^-(0) \\left( \\varphi^-(0^-; x(a)) \\right)_1 + Q_{22}^- (0) \\left( \\varphi^-(0^-; x(a)) \\right)_2 \\\\\n\t\t&= Q_{12}^+(0) x_1(0^+) + Q_{22}^+(0) x_2(0^+). \n\t\t\\end{split}\n\t\\label{Interface - Calculation x(0+) - fI}\n\t\\end{align}\n\tThe exact value of $f_I$ is defined through the first equation. Furthermore, \n\t\\begin{align*}\n\t\t- e_I &= Q_{11}^+(0) x_1(0^+) + Q_{12}^+(0) x_2(0^+) \\\\\n\t\t&\\hspace{0.5 cm} - \\left( Q_{11}^-(0) \\left( \\varphi^-(0^-; x(a)) \\right)_1 + Q_{12}^-(0) \\left( \\varphi^-(0^-; x(a)) \\right)_2 \\right). \n\t\\end{align*}\n\tThe entries of the initial value $x(0^+)$ can be determined by exploiting the passivity relation \\eqref{Interface - Interface Passivity Relation}: we have \n\t\\begin{align}\n\t\t&f_I = r e_I \\nonumber \\\\\n\t\t\\Leftrightarrow \\hspace{0.5 cm} &Q_{11}^+(0) x_1(0^+) + Q_{12}^+(0) x_2(0^+) = - \\frac{1}{r} f_I + \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1 \\nonumber \\\\\n\t\t\\Leftrightarrow \\hspace{0.5 cm} &x_1(0^+) = - \\frac{1}{r Q_{11}^+(0)} f_I + \\frac{1}{Q_{11}^+(0)} \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1 - \\frac{Q_{12}^+(0)}{Q_{11}^+(0)} x_2(0^+). \\label{Interface - Calculation x(0+) - x1(0+)}\n\t\\end{align}\n\tPlugging this into the second equation of \\eqref{Interface - Calculation x(0+) - fI} yields\n\t\\begin{align*}\n\t\tf_I &= \\frac{Q_{12}^+(0)}{Q_{11}^+(0)} \\left( - \\frac{1}{r} f_I + \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1 \\right) \\\\\n\t\t&\\hspace{0.5 cm} + \\left( - \\frac{(Q_{12}^+(0))^2}{Q_{11}^+(0)} + Q_{22}^+(0) \\right) x_2(0^+), \n\t\\end{align*}\n\twhence\n\t\\begin{align*}\n\t\tx_2(0^+) = \\frac{Q_{11}^+(0)}{(Q_{12}^+(0))^2 + Q_{11}^+(0) Q_{22}^+(0)} \\left( f_I - \\frac{Q_{12}^+(0)}{Q_{11}^+(0)} \\left( - \\frac{1}{r} f_I + \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1 \\right) \\right). \n\t\\end{align*}\n\tThus, the initial value $x(0^+)$ is uniquely defined and depends on the initial value $x(a)$. Letting $\\tilde{c}_1 = \\frac{Q_{11}^+(0)}{(Q_{12}^+(0))^2 + Q_{11}^+(0) Q_{22}^+(0)} \\left( 1 + \\frac{Q_{12}^+(0)}{r Q_{11}^+(0)} \\right)$, $\\tilde{c}_2 = - \\frac{Q_{12}^+(0)}{(Q_{12}^+(0))^2 + Q_{11}^+(0) Q_{22}^+(0)}$, we may write\n\t\\begin{align*}\n\t\tx_2(0^+) = \\tilde{c}_1 f_I + \\tilde{c}_2 \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1.\n\t\\end{align*}\n\tFurthermore, letting $\\tilde{c}_3 = - \\frac{1 + r Q_{12}^+(0) \\tilde{c}_1}{r Q_{11}^+(0)}$, $\\tilde{c}_4 = \\frac{1 - Q_{12}^+(0) \\tilde{c}_2}{Q_{11}^+(0)}$, and inserting $x_2(0^+)$ in \\eqref{Interface - Calculation x(0+) - x1(0+)}, we get\n\t\\begin{align*}\n\t\tx_1(0^+) = \\tilde{c}_3 f_I + \\tilde{c}_4 \\left( \\mathcal{Q}^-(0) \\varphi^-(0^-; x(a)) \\right)_1. \n\t\\end{align*}\n\tFinally, the choices\n\t\\begin{align*}\n\t\tc_1 &= \\tilde{c}_3 Q_{12}^-(0) + \\tilde{c}_4 Q_{11}^-(0), \\hspace{0.3 cm } c_2 = \\tilde{c}_3 Q_{22}^-(0) + \\tilde{c}_4 Q_{12}^-(0), \\\\\n\t\tc_3 &= \\tilde{c}_1 Q_{12}^-(0) + \\tilde{c}_2 Q_{11}^-(0), \\hspace{0.3 cm} c_4 = \\tilde{c}_1 Q_{22}^-(0) + \\tilde{c}_2 Q_{12}^-(0)\n\t\\end{align*}\n\tyield\n\t\\begin{align}\n\t\tx_1(0^+) &= c_1 \\left( \\varphi^-(0^-; x(a)) \\right)_1 + c_2 \\left( \\varphi^-(0^-; x(a)) \\right)_2 \\label{Interface - x1(0+)} \\\\\n\t\tx_2(0^+) &= c_3 \\left( \\varphi^-(0^-; x(a)) \\right)_1 + c_4 \\left( \\varphi^-(0^-; x(a)) \\right)_2.\n\t\t\\label{Interface - x2(0+)}\n\t\\end{align}\nDefining $C_0 = \\begin{bmatrix}\n\tc_1 & c_2 \\\\\n\tc_3 & c_4\n\\end{bmatrix} \\in \\mathbb{R}^{2 \\times 2}$, we may write\n\\begin{align*}\n\tx(0^+) = C_0 \\varphi^-(0^-; x(a)). \n\\end{align*}\nWe continue with the determination of the initial value $x(a) \\in \\mathbb{R}^2$. It has to be chosen such that\n\t\\begin{align}\n\t\t\\label{Interface - Calculation x(a) Reference}\n\t\t0 = W_B \\begin{bmatrix}\n\t\t\tf_{\\partial} \\\\\n\t\t\te_{\\partial}\n\t\t\\end{bmatrix} = W_B R_{\\extern} \\begin{bmatrix}\n\t\t\t\\mathcal{Q}_0(b)\\varphi^+(b; x(0^+)) \\\\\n\t\t\t\\mathcal{Q}_0(a) \\varphi^-(a; x(a))\n\t\t\\end{bmatrix} = W_B R_{\\extern} \\begin{bmatrix}\n\t\t\t\\mathcal{Q}^+(b) & 0 \\\\\n\t\t\t0 & \\mathcal{Q}^-(a) \n\t\t\t\\end{bmatrix}\n\t\t\\begin{bmatrix}\n\t\t\t\\varphi^+(b; x(0^+)) \\\\\n\t\t\t x(a)\n\t\t\\end{bmatrix}.\n\t\\end{align}\n\tWe again wish to write $\\varphi^+(b; x(0^+))$ as $E x(a) + q$ for some $E \\in \\mathbb{R}^{2 \\times 2}$ and $q \\in \\mathbb{R}^2$. Similarly to the proof of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}, we compute that\n\t\\begin{align*}\n\t\tE x(a)&= \\Lambda^+(b,0) \\begin{bmatrix}\n\t\t\tc_1 \\left( \\Lambda^-(0^-,a) x(a) \\right)_1 + c_2 \\left( \\Lambda^-(0^-,a) x(a) \\right)_2 \\\\\n\t\t\tc_3 \\left( \\Lambda^-(0^-,a) x(a) \\right)_1 + c_4 \\left( \\Lambda^-(0^-,a) x(a) \\right)_2\n\t\t\\end{bmatrix} \\\\\n\t&= \\left( \\Lambda^+(b,0)C_0\\Lambda^-(0^-,a) \\right) x(a)\t\\end{align*}\n\tand \n\t\\begin{align*}\n\t\tq &= \\Lambda^+(b,0) \\begin{bmatrix} c_1 \\left( \\varphi_{\\text{inhom}}^-(0^-, x(a)) \\right)_1 + c_2 \\left( \\varphi_{\\text{inhom}}^-(0^-, x(a)) \\right)_2 \\\\\n\t\t\tc_3 \\left( \\varphi_{\\text{inhom}}^-(0^-, x(a)) \\right)_1 + c_4 \\left( \\varphi_{\\text{inhom}}^-(0^-, x(a)) \\right)_2\n\t\t\\end{bmatrix} \\\\\n\t\t&\\hspace{0.5 cm} - \\int_{0}^{b} \\Lambda^+(b,s) (\\mathcal{Q}^+)^{-1}(s) P_1 y^+(s) \\, ds \\\\\n\t\t&= \\Lambda^+(b,0)C_0\\varphi_{\\text{inhom}}^-(0^-;x(a)) - \\int_{0}^{b} \\Lambda^+(b,s) (\\mathcal{Q}^+)^{-1}(s) P_1 y^+(s) \\, ds, \n\t\\end{align*}\nwhere\n\\begin{align*}\n\t\\varphi_{\\text{inhom}}^-(0^-, x(a)) = - \\int_{a}^{0} \\Lambda^-(0^-,s) (\\mathcal{Q}^-)^{-1}(s) P_1 y^-(s) \\, ds.\n\\end{align*}\n\tFollowing the proof, the initial value $x(a)$ can be calculated from \\eqref{Interface - Calculation x(a) Reference} and is given by\n\t\\begin{align}\n\t\t\\label{Interface - x(a)}\n\t\tx(a) = - \\left( W_B R_{\\extern} \\begin{bmatrix}\n\t\t\t\\mathcal{Q}^+(b) & 0 \\\\\n\t\t\t0 & \\mathcal{Q}^-(a)\n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\tE \\\\\n\t\t\tI\n\t\t\\end{bmatrix} \\right)^{-1} W_B R_{\\extern} \\begin{bmatrix}\n\t\t\t\\mathcal{Q}^+(b) & 0 \\\\\n\t\t\t0 & \\mathcal{Q}^-(a)\n\t\t\\end{bmatrix} \\begin{bmatrix}\n\t\t\tq \\\\\n\t\t\t0\n\t\t\\end{bmatrix}.\n\t\\end{align}\nFinally, we can specify the resolvent operator of $A_{\\mathcal{Q}_0}$. For every $\\lambda > 0$ and for every $y \\in X$ we have\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\|R(\\lambda, A_{\\mathcal{Q}_0})y \\|_{\\mathcal{Q}_0}^2 \\\\\n\t&= \\| \\varphi_{\\lambda, y}(\\cdot; x(a), x(0^+)) \\|_{\\mathcal{Q}_0}^2 \\\\\n\t&= \\frac{1}{2} \\int_{a}^{b} \\left( c_0(z) \\varphi^-(z;x(a)) + \\overline{c}_0(z) \\varphi^+(z; x(0^+)) \\right)^{\\top} \\mathcal{Q}_0(z) \\left( c_0(z) \\varphi^-(z;x(a)) + \\overline{c}_0(z) \\varphi^+(z; x(0^+)) \\right) \\, dz \\\\\n\t&= \\frac{1}{2} \\left( \\int_{a}^{0} \\left( \\varphi^-(z; x(a)) \\right)^{\\top} \\mathcal{Q}^-(z) \\varphi^-(z;x(a)) \\, dz + \\int_{0}^{b} \\left( \\varphi^+(z; x(0^+)) \\right)^{\\top} \\mathcal{Q}^+(z) \\varphi^+(z; x(0^+)) \\, dz \\right),\n\\end{align*}\nwith the initial values $x(a)$ and $x(0^+)$ given by \\eqref{Interface - x(a)} and \\eqref{Interface - x1(0+)}-\\eqref{Interface - x2(0+)}, respectively. \n\\vspace{0.5 cm}\\\\\nWe skip specifying the resolvent in the case that the passivity constant is given by $r =0$. To this end, one would basically need to replicate Case 2 in Step 3 of the proof of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}. Instead, we will come back to the scenario where the interface position is moving over time, and prove a crucial stability property of the resulting system. \n\n\\subsection{Stability of the Family of Infinitesimal Generators}\n\\label{Subsection Stability of the Family of Infinitesimal Generators}\nIn the last segment of this chapter we shall analyze the simplified port-Hamiltonian system \\eqref{Interface - Simplified PH-System} in case of a moving interface. We want to define a family of port-Hamiltonian operators that encompasses the position of the interface, and we want to associate this family with an evolution problem of the form\n\\begin{align*}\n\t\\dot{x}(t) &= A(t)x(t), \\hspace{0.5 cm} 0 \\leq s < t \\leq \\tau, \\\\\n\tx(s) &= x_0 \\in X.\n\\end{align*}\nWe shall impose assumptions such that this family is a stable family in the sense of Definition~\\ref{Definition Stable Family of Generators}.\n\\vspace{0.5 cm}\\\\\nEverything we have proved in Section \\ref{Section A Simplified System} can be analogously shown for arbitrary interface positions $l \\in (a,b)$. We have already come across the operators $\\mathbf{d}_l$ and its formal adjoint $\\mathbf{d}_l^{\\ast}$ in Subsection \\ref{Subsection Balance Equations of the State Variables - Moving Interface}. They are defined by \n\\begin{align*}\n\tD(\\mathbf{d}_l) &= H^1([a,b], \\mathbb{R}), \\\\\n\t\\mathbf{d}_lx &= - \\left[ \\frac{d}{dz} (c_l x) + \\frac{d}{dz} (\\overline{c}_l x ) \\right], \\hspace{0.5 cm} x \\in D(\\mathbf{d}_l),\n\\end{align*}\nand\n\\begin{align*}\n\tD(\\mathbf{d}_l^{\\ast}) &= \\left\\{ y \\in L^2([a,b], \\mathbb{R}) \\mid y_{|(a,l)} \\in H^1((a,l), \\mathbb{R}), \\hspace{0.1 cm} y_{|(l,b)} \\in H^1((l,b), \\mathbb{R}) \\right\\}, \\\\\n\t\\mathbf{d}_l^{\\ast}y &= \\left( - \\mathbf{d}_l - \\left[ \\frac{d}{dz} c_l + \\frac{d}{dz} \\overline{c}_l \\right] \\right) y, \\hspace{0.5 cm} y \\in D(\\mathbf{d}_l^{\\ast}),\n\\end{align*}\nrespectively, where $c_l$ and $\\overline{c}_l$ are the color functions with a jump discontinuity at the interface position $l$. This allows us to define the formally skew-symmetric matrix differential operator $\\mathcal{J}_l \\colon D(\\mathcal{J}_l) \\subset X \\to X$ as follows:\n\\begin{align*}\n\tD(\\mathcal{J}_l) &= \\left\\{ x \\in X \\mid x_1 \\in D(\\mathbf{d}_l^{\\ast}), \\hspace{0.1 cm} x_2 \\in D(\\mathbf{d}_l)\\right\\}, \\\\\n\t\\mathcal{J}_l x &= \\begin{bmatrix}\n\t\t0 & \\mathbf{d}_l \\\\\n\t\t- \\mathbf{d}_l^{\\ast} & 0 \n\t\\end{bmatrix} \\begin{bmatrix}\n\t\tx_1 \\\\\n\t\tx_2\n\t\\end{bmatrix}, \\hspace{0.5 cm} x \\in D(\\mathcal{J}_l).\n\\end{align*}\nThen the port-Hamiltonian operator $A_{\\mathcal{Q}_l} \\colon D(A_{\\mathcal{Q}_l}) \\subset X \\to X$ given by\n\\begin{align*}\n\n\t\tD(A_{\\mathcal{Q}_l}) &= \\left\\{ x \\in X \\mid \\mathcal{Q}_lx \\in D(\\mathcal{J}_l), \\hspace{0.1 cm} f_{\\mathcal{Q}_l x} = r e_{\\mathcal{Q}_l x}, \\hspace{0.1 cm} W_B \\begin{bmatrix}\n\t\t\tf_{\\partial, \\mathcal{Q}_lx} \\\\\n\t\t\te_{\\partial, \\mathcal{Q}_l x}\n\t\t\\end{bmatrix} = 0 \\right\\}, \\\\\n\t\tA_{\\mathcal{Q}_l}x &= \\mathcal{J}_l (\\mathcal{Q}_lx), \\hspace{0.5 cm} x \\in D(A_{\\mathcal{Q}_l}),\n\\end{align*}\nassociated with the abstract Cauchy problem\n\\begin{align*}\n\t\\dot{x}(t) &= A_{\\mathcal{Q}_l}x(t), \\hspace{0.5 cm} t > 0, \\\\\n\tx(0) &= x_0 \\in X,\n\\end{align*}\ngenerates a contraction semigroup on the energy space $(X, \\| \\cdot \\|_{\\mathcal{Q}_l})$, where\n\\begin{align*}\n\t\\mathcal{Q}_l = c_l \\mathcal{Q}^- + \\overline{c}_l \\mathcal{Q}^+ \\in \\mathcal{L}(X)\n\\end{align*}\nis coercive, if the assumptions of Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup} are satisfied. More precisely, it generates a contraction semigroup $(T_l(t))_{t \\geq 0}$ provided that $W_B \\Sigma W_B^{\\top} \\geq 0$, $\\rank(W_B) =2$, and $r \\geq 0$. Let now the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold (see page \\pageref{MatrixQ0Assumption1}). If $W_B \\Sigma W_B^{\\top} > 0$, then the semigroup $(T_l(t))_{t \\geq 0}$ is even of type $C_0(1,- \\alpha)$ for some $\\alpha >0$, see Theorem~\\ref{Interface - Theorem 2 Exponential Stability}. Its Hilbert space adjoint $A_{\\mathcal{Q}_l}^{\\ast} \\colon D(A_{\\mathcal{Q}_l}) \\subset X \\to X$ is accordingly defined as in \\eqref{Interface - Adjoint Operator}. Furthermore, the underlying Dirac structure is given by\n\\begin{align*}\n\t\\mathcal{D}_{\\mathcal{J}_l} :=& \\left\\{ \\left( \\begin{pmatrix}\n\t\tf \\\\\n\t\tf_I \\\\\n\t\tf_{\\partial}\n\t\\end{pmatrix} , \\begin{pmatrix}\n\t\te \\\\\n\t\te_I \\\\\n\t\te_{\\partial}\n\t\\end{pmatrix} \\right) \\in \\mathcal{B} \\hspace{0.1 cm} \\Bigg| \\hspace{0.1 cm} e \\in D(\\mathcal{J}_l), \\, f = \\mathcal{J}_le, \\, f_I = e_2(0), \\right. \\\\\n\t&\\hspace{4.22 cm} \\left. \\, e_I = e_1(0^-) - e_1(0^+), \\, \\begin{bmatrix}\n\t\tf_{\\partial} \\\\\n\t\te_{\\partial}\n\t\\end{bmatrix} = R_{\\extern} \\trace_l(e) \\right\\},\n\\end{align*}\nwhere\n\\begin{align*}\n\t\\trace_l \\colon D(\\mathcal{J}_l) \\to \\mathbb{R}^4, \\hspace{0.3 cm} e \\mapsto \\begin{bmatrix}\n\t\te(b) \\\\\n\t\te(a)\n\t\\end{bmatrix}.\n\\end{align*}\nTogether with the resistive structure\n\\begin{align*}\n\t\\mathcal{R}_r = \\{ (f_I,e_I) \\in \\mathbb{R}^2 \\mid f_I = re_I \\},\n\\end{align*}\nthe dynamics are geometrically specified by the requirement that for all $t> 0$ it holds that\n\\begin{align*}\n\t\\left( \\begin{pmatrix}\n\t\t\\frac{\\partial}{\\partial t} x(\\cdot,t) \\\\\n\t\tf_I(t) \\\\\n\t\tf_{\\partial}(t)\n\t\\end{pmatrix} , \\begin{pmatrix}\n\t\t\\mathcal{Q}_lx(\\cdot,t) \\\\\n\t\te_I(t) \\\\\n\t\te_{\\partial}(t)\n\t\\end{pmatrix} \\right) \\in \\mathcal{D}_{\\mathcal{J}_l}, \\hspace{0.3 cm} (f_I(t), e_I(t)) \\in \\mathcal{R}_{r}.\n\\end{align*}\nLastly, if $r >0$, then for the resolvent it holds that\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\|R(\\lambda, A_{\\mathcal{Q}_l})y \\|_{\\mathcal{Q}_l}^2 \\\\\n\t&= \\frac{1}{2} \\left( \\int_{a}^{l} \\left( \\varphi^-(z; x(a)) \\right)^{\\top} \\mathcal{Q}^-(z) \\varphi^-(z;x(a)) \\, dz + \\int_{l}^{b} \\left( \\varphi^+(z; x(l^+)) \\right)^{\\top} \\mathcal{Q}^+(z) \\varphi^+(z; x(l^+)) \\, dz \\right),\n\\end{align*}\nwith $y \\in X$, $\\lambda >0$, and the solutions $\\varphi^-(\\cdot; x(a))$, $\\varphi^+(\\cdot; x(l^+))$ as well as the values $x(a)$ and $x(l^+)$ accordingly defined, see Subsection \\ref{Subsection Resolvent}.\n\\vspace{0.5 cm}\\\\\nUnfortunately, the state space $(X, \\|\\cdot\\|_{\\mathcal{Q}_l})$ has to be changed for different interface positions $l \\in (a,b)$ to guarantee that the aforementioned statements hold true. However, we have a norm equivalence of the family $(\\|\\cdot\\|_{\\mathcal{Q}_l})_{l \\in (a,b)}$ to the norm $\\| \\cdot \\|_{\\mathcal{Q}_0}$: Since $mI \\leq \\mathcal{Q}^{\\pm}(z) \\leq MI$ for all $z \\in [a,b]$, we get for all $l \\in (a,b)$,\n\\begin{align*}\n\t\\frac{m}{2} \\| x\\|_{L^2}^2 \\leq \\| x \\|_{\\mathcal{Q}_l}^2 \\leq \\frac{M}{2} \\| x \\|_{L^2}^2, \\hspace{0.5 cm} x \\in X.\n\\end{align*}\nIn particular, we have\n\\begin{align*}\n\t\\frac{2}{M} \\| x \\|_{\\mathcal{Q}_0}^2 \\leq \\| x \\|_{L^2}^2 \\leq \\frac{2}{m} \\| x \\|_{\\mathcal{Q}_0}^2, \\hspace{0.5 cm} x \\in X. \n\\end{align*}\nThe norm equivalence between $\\| \\cdot \\|_{\\mathcal{Q}_0}$ and $\\| \\cdot \\|_{\\mathcal{Q}_l}$ for some $l \\in (a,b)$ follows immediately from the preceding inequalities:\n\\begin{align}\n\t\\label{Interface - Norm Equivalence}\n\t\\frac{m}{M} \\| x \\|_{\\mathcal{Q}_0}^2 \\leq \\frac{m}{2} \\| x \\|_{L^2}^2 \\leq \\| x\\|_{\\mathcal{Q}_l}^2 \\leq \\frac{M}{2} \\| x \\|_{L^2}^2 \\leq \\frac{M}{m} \\|x \\|_{\\mathcal{Q}_0}^2, \\hspace{0.5 cm} x \\in X.\n\\end{align}\nNote that the constants $\\underline{M} = \\sqrt{\\frac{m}{M}}$ and $\\overline{M}= \\sqrt{\\frac{M}{m}}$ are independent of the interface position $l \\in (a,b)$.\n\\vspace{0.5 cm}\\\\\nWe now want to show the following: Let $0 < \\tau < \\infty$, and let $l \\colon [0, \\tau] \\to (a,b)$ be continuously differentiable with $\\dot{l} \\in \\mathcal{C}([0,\\tau], \\mathbb{R})$. Let $(A(t))_{t \\in [0, \\tau]}$ be a family of port-Hamiltonian operators, where $A(t) = A_{\\mathcal{Q}_{l(t)}}\\colon D(A(t))\\subset X \\to X$ is given by\n\\begin{align}\n\t\\begin{split}\n\tD(A(t)) &= \\left\\{ x \\in X \\mid \\mathcal{Q}_{l(t)} x \\in D(\\mathcal{J}_{l(t)}), \\hspace{0.1 cm} f_I(t) = re_I(t), \\hspace{0.1 cm} W_B \\begin{bmatrix}\n\t\tf_{\\partial}\\\\\n\t\te_{\\partial}\n\t\t\\end{bmatrix} = 0 \\right\\}, \\\\\n\tA(t)x &= \\mathcal{J}_{l(t)} (\\mathcal{Q}_{l(t)} x) \\\\\n\t&= \\begin{bmatrix}\n\t\t0 & \\mathbf{d}_{l(t)} \\\\\n\t\t- \\mathbf{d}_{l(t)}^{\\ast} & 0 \n\t\\end{bmatrix} (\\mathcal{Q}_{l(t)} x), \\hspace{0.5 cm} x \\in D(A(t)),\n\\end{split}\n\\label{Interface - Operator Family A(t)}\n\\end{align} \nwith $e_I(t) = e_{I, \\mathcal{Q}_{l(t)}x}$, $f_I(t) = f_{I, \\mathcal{Q}_{l(t)}x}$. The operators $\\mathbf{d}_{l(t)}$ and $\\mathbf{d}_{l(t)}^{\\ast}$ constituting the matrix differential operator $\\mathcal{J}_{l(t)} \\colon D(\\mathcal{J}_{l(t)}) \\subset X \\to X$ given by\n\\begin{align}\n\t\\begin{split}\n\t\tD(\\mathcal{J}_{l(t)}) &= D(\\mathbf{d}_{l(t)}^{\\ast}) \\times D(\\mathbf{d}_{l(t)}), \\\\\n\t\t\\mathcal{J}_{l(t)}x &= \\begin{bmatrix}\n\t\t\t0 & \\mathbf{d}_{l(t)} \\\\\n\t\t\t- \\mathbf{d}_{l(t)}^{\\ast} & 0 \n\t\t\\end{bmatrix} x, \\hspace{0.5 cm} x \\in D(\\mathcal{J}_{l(t)}), \n\t\\end{split}\n\\label{Interface - Operator Jl(t)}\n\\end{align} \nare precisely defined as the operators $\\mathbf{d}(t)$ and $\\mathbf{d}^{\\ast}(t)$ given in \\eqref{Interface Paper - Operator d(t)} and \\eqref{Interface Paper - Operator d(t)ast}, respectively. Furthermore, the coercive operator $\\mathcal{Q}_{l(t)} \\in \\mathcal{L}(X)$, $t \\in [0, \\tau]$, is given by \n\\begin{align}\n\t\\mathcal{Q}_{l(t)} = c_{l}(\\cdot,t) \\mathcal{Q}^- + \\overline{c}_{l}(\\cdot,t) \\mathcal{Q}^+.\n\t\\label{Interface - Operator Ql(t)}\n\\end{align}\nWe claim that, under certain conditions, this is a stable family on the state space $(X, \\| \\cdot \\|_{\\mathcal{Q}_0})$.\n\\begin{remark}\n\tWe changed the notation of the operators $\\mathbf{d}(t)$ and $\\mathbf{d}^{\\ast}(t)$ to $\\mathbf{d}_{l(t)}$ and $\\mathbf{d}_{l(t)}^{\\ast}$, respectively, to emphasize that the operators explicitly depend on the moving interface position $l(t)$, $t >0$. Recall that the operators $\\mathbf{d}(t)$ and $\\mathbf{d}^{\\ast}(t)$ introduced in Subsection \\ref{Subsection Balance Equations of the State Variables - Moving Interface} depend implicitly on the interface position through their definition with respect to the color functions $c_l$ and $\\overline{c}_l$ that we took to be part of the extended state variable $\\tilde{x} \\in \\tilde{X} = L^2([a,b], \\mathbb{R}^4)$. Since this is not the case for the simplified system discussed throughout Section \\ref{Section A Simplified System}, the new notation helps to distinguish the different scenarios.\n\\end{remark}\nIf the assumptions \\ref{MatrixQ0Assumption1}-\\ref{ContractionAssumption} hold, then, due to the norm equivalence \\eqref{Interface - Norm Equivalence} of the norms of the respective energy spaces $(X, \\|\\cdot\\|_{\\mathcal{Q}_{l(t)}})$, we can at least guarantee that the family $(A(t))_{t \\in [0,\\tau]}$ is a family of infinitesimal generators of $C_0$-semigroups on $(X, \\| \\cdot \\|_{\\mathcal{Q}_0})$ (or any other state space $(X, \\|\\cdot\\|_{\\mathcal{Q}_{\\hat{l}}})$ with $\\hat{l} \\in (a,b)$), which are of type $C_0(\\overline{M},0)$. However, as we will see, this is not sufficient to prove the stability of this family. Thus, we need to impose more restrictive assumptions. These are given as follows:\n\n\\begin{enumerate}[label =($\\tilde{\\mathrm{A}}$\\arabic*)]\n\t\\item \\label{AssumptionP1MatricesQ} The matrix operators $\\mathcal{Q}^{\\pm} \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2})$ defining the coercive operator $\\mathcal{Q}_{l(t)} \\in \\mathcal{L}(X)$ are diagonal matrix operators, i.e., \n\t\\begin{align*}\n\t\t\\mathcal{Q}^{\\pm}(z) = \\begin{bmatrix}\n\t\t\tQ_{11}^{\\pm}(z) & 0 \\\\\n\t\t\t0 & Q_{22}^{\\pm}(z)\n\t\t\\end{bmatrix}, \\hspace{0.5 cm} z \\in [a,b],\n\t\\end{align*} \nand they satisfy\n\\begin{align*}\n\t\\frac{Q_{11}^+}{Q_{11}^-}(0) = 1 \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\frac{Q_{11}^+}{Q_{11}^-}(z) = \\frac{Q_{22}^+}{Q_{22}^-}(z), \\hspace{0.5 cm} z \\in [a,b].\n\\end{align*}\n\t\\item \\label{AssumptionP2ConservationAtInterface} $r =0 $, $\\rank(W_B) = 2$, and $W_B \\Sigma W_B^{\\top} \\geq 0$.\n\\end{enumerate}\nAssumption \\ref{AssumptionP1MatricesQ} implies that the ratios of the respective physical properties of the subsystems are identical on the spatial domain $[a,b]$, and that the physical properties even coincide in $z = 0$. Further, assumption \\ref{AssumptionP2ConservationAtInterface} implies that $A(t)$ generates a contraction semigroup on the respective energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_{l(t)}})$, where there is no power flow at the interface position, see Theorem~\\ref{Interface - Theorem A Generates a Contraction Semigroup}.\n\\vspace{0.5 cm}\\\\\nSince we can prove the stability neither directly nor by means of Theorem \\ref{Theorem 5.2.2 in Pazy}, we make use of an extension of a Lyapunov theorem \\cite{Datko}.\n\n\\begin{theorem}\n\t\\label{Interface - Theorem Stability of A(t)}\n\tLet $0 < \\tau < \\infty$, and let $l \\colon [0, \\tau] \\to (a,b)$ be continuously differentiable with $\\dot{l} \\in \\mathcal{C}([0,\\tau], \\mathbb{R})$. Let the assumptions \\ref{AssumptionP1MatricesQ}-\\ref{AssumptionP2ConservationAtInterface} hold. Consider the family $(A(t))_{t \\in[0,\\tau]}$ of port-Hamiltonian operators given by \\eqref{Interface - Operator Family A(t)}-\\eqref{Interface - Operator Ql(t)} defined on $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$. Then there exists some $\\omega > 0$ such that for every $t \\in [0, \\tau]$ and for every $x \\in D(A(t))$,\n\t\\begin{align}\n\t\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0} \\leq \\omega \\| x\\|_{\\mathcal{Q}_0}^2.\n\t\t\\label{Interface - Necessary Condition Stablity of Family}\n\t\\end{align} \n\\end{theorem}\n\n\\begin{proof}\nLet $t \\in [0, \\tau]$ such that for the interface position we have $a < l(t) < 0$. Recall that in this case, the operator $\\mathcal{J}_{l(t)}$ defined in \\eqref{Interface - Operator Jl(t)} acts as the operator $P_1 \\frac{d}{dz}$ on the subdomains $[a, l(t))$ and $(l(t),b]$, with $P_1$ defined in \\eqref{Interface - Matrix P1}. Furthermore, as $A(t)$ generates a contraction semigroup on $(X, \\|\\cdot\\|_{\\mathcal{Q}_{l(t)}})$, it is dissipative, see Theorem \\ref{Theorem Lumer-Phillips}. More precisely, for all $x \\in D(A(t))$ it holds that\n\\begin{align*}\n&\\hspace{0.5 cm} \\langle A(t)x, x \\rangle_{\\mathcal{Q}_{l(t)}} \\\\\n&= \\frac{1}{2} \\int_{a}^{b} x^{\\top}(z) \\mathcal{Q}_{l(t)}(z) A(t) x(z) \\, dz \\\\ \n\t&= \\frac{1}{2} \\int_{a}^{l(t)} x^{\\top}(z) \\mathcal{Q}^-(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^-(z) x(z)) \\, dz + \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\mathcal{Q}^+(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+(z) x(z)) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} \\int_{0}^{b} x^{\\top}(z) \\mathcal{Q}^+(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+(z) x(z)) \\, dz \\\\\n\t&\\leq 0.\n\\end{align*}\nThus, we have\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\langle A(t)x, x \\rangle_{\\mathcal{Q}_0} \\\\\n\t&= \\frac{1}{2} \\int_{a}^{l(t)} x^{\\top}(z) \\mathcal{Q}^-(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^-(z) x(z)) \\, dz + \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\mathcal{Q}^-(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+(z) x(z)) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} \\int_{0}^{b} x^{\\top}(z) \\mathcal{Q}^+(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+(z) x(z)) \\, dz \\\\\n\t&= \\langle A(t)x,x \\rangle_{\\mathcal{Q}_{l(t)}} \\\\\n\t&\\hspace{0.5 cm } + \\frac{1}{2} \\int_{l(t)}^{0} (\\mathcal{Q}^-x)^{\\top}(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+x)(z) - (\\mathcal{Q}^+x)^{\\top}(z) P_1 \\frac{d}{dz} (\\mathcal{Q}^+x)(z) \\, dz \\\\\n\t&\\leq \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z) (\\mathcal{Q}^-(z) - \\mathcal{Q}^+(z)) P_1 \\frac{d}{dz} (\\mathcal{Q}^+x)(z) \\, dz \\\\\n\t&= \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z)\\left[ (\\mathcal{Q}^-(z) - \\mathcal{Q}^+(z))P_1 \\frac{d}{dz}\\mathcal{Q}^+(z)\\right] x(z) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\left[ (\\mathcal{Q}^-(z) - \\mathcal{Q}^+(z))P_1 \\mathcal{Q}^+(z) \\right] \\frac{d}{dz} x(z) \\, dz.\n\\end{align*}\nWe may choose $\\omega_1^- > 0$ large enough such that (in the sense of the symmetric part)\n\\begin{align*}\n (\\mathcal{Q}^-(z) - \\mathcal{Q}^+(z))P_1 \\frac{d}{dz}\\mathcal{Q}^+(z) \\leq \\omega_1^- \\mathcal{Q}^-(z), \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align*}\nNow, let us define the matrix operator $\\tilde{Q} \\colon [a,b] \\to \\mathbb{R}^{2 \\times 2}$ by\n\\begin{align}\n\t\\label{Interface - Theorem Stability of A(t) - Matrix TildeQ}\n\t\\tilde{Q}(z) = (\\mathcal{Q}^-(z) - \\mathcal{Q}^+(z))P_1 \\mathcal{Q}^+(z) = - \\begin{bmatrix}\n\t\t0 & Q_{22}^+ (Q_{11}^- - Q_{11}^+) \\\\\n\t\tQ_{11}^+(Q_{22}^- - Q_{22}^+) & 0 \n\t\\end{bmatrix}(z), \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align}\nBy assumption \\ref{AssumptionP1MatricesQ}, we have $\\mathcal{Q}^+, \\mathcal{Q}^- \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2}$), and so the matrix operator $\\tilde{Q}$ is continuously differentiable as well. Moreover, by \\ref{AssumptionP1MatricesQ}, it is easy to check that $\\tilde{Q}(z)$ is symmetric for all $z \\in [a,b]$. Thus, integration by parts yields that\n\\begin{align*}\n\t&\\hspace{0.5 cm} \\int_{l(t)}^{0} x^{\\top}(z) \\tilde{Q}(z) \\frac{d}{dz} x(z) \\, dz \\\\\n\t&= \\frac{1}{2} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} - \\frac{1}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\frac{d}{dz} \\tilde{Q}(z) x(z) \\, dz.\n\\end{align*}\nOnce again, we may choose $\\omega_2^- > 0$ large enough such that\n\\begin{align*}\n\t- \\frac{d}{dz} \\tilde{Q}(z) \\leq \\omega_2^- \\mathcal{Q}^-(z), \\hspace{0.5 cm} z \\in [a,b]. \n\\end{align*}\nAltogether, we obtain for all $x \\in D(A(t))$, \n\\begin{align}\n\t\\begin{split}\t\n\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0}\t&\\leq \\frac{\\omega_1^-}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\mathcal{Q}^-(z) x(z) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{4} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} + \\frac{\\omega_2^-}{4} \\int_{l(t)}^{0} x^{\\top}(z)\\mathcal{Q}^-(z) x(z) \\, dz. \n\\end{split}\n\\label{Interface - Theorem Auxiliary - A(t) is Stable - Proof Estimate 1}\n\\end{align}\nThe interface relation \\eqref{Interface - Interface Passivity Relation} with $r = 0$ together with the assumption that $\\mathcal{Q}^{\\pm}$ are diagonal matrix operators imply that $x_2(l(t)^+) = x_2(l(t)^-) = 0$. Furthermore, by assumption \\ref{AssumptionP1MatricesQ}, $\\mathcal{Q}^{\\pm}$ coincide in $z = 0$. \nConsequently, it holds that\n\\begin{align*}\n\t \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} &= - x_1(0)x_2(0) \\left[ Q_{22}^+ (Q_{11}^- - Q_{11}^+) + \tQ_{11}^+(Q_{22}^- - Q_{22}^+) \\right] (0) \\\\\n\t&\\hspace{0.5 cm} + x_1(l(t)^+) x_2(l(t)^+) \\left[ Q_{22}^+ (Q_{11}^- - Q_{11}^+) + \tQ_{11}^+(Q_{22}^- - Q_{22}^+) \\right] (l(t)) \\\\\n\t&= 0.\n\\end{align*}\nWith the choice $\\omega^- = \\max\\{\\omega_1^-, \\frac{\\omega_2^-}{2}\\}$ we deduce from \\eqref{Interface - Theorem Auxiliary - A(t) is Stable - Proof Estimate 1} that for all $x \\in D(A(t))$, where $t \\in [0, \\tau]$ is chosen such that $a < l(t) < 0$, it holds that\n\\begin{align*}\n\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0} \\leq \\omega^- \\|x\\|_{\\mathcal{Q}_0}^2. \n\\end{align*}\nNow, let $t \\in [0, \\tau]$ such that for the interface position we have $0 < l(t) < b$. Analogously to the previous case, we have for all $x \\in D(A(t))$, \n\\begin{align*}\n\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0} &\\leq \\frac{1}{2} \\int_{0}^{l(t)} x^{\\top}(z) \\left[ (\\mathcal{Q}^+(z) - \\mathcal{Q}^-(z))P_1 \\frac{d}{dz} \\mathcal{Q}^-(z) \\right] x(z) \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} \\int_{0}^{l(t)} x^{\\top}(z) \\left[ (\\mathcal{Q}^+(z) - \\mathcal{Q}^-(z))P_1 \\mathcal{Q}^-(z) \\right] \\frac{d}{dz} x(z) \\, dz. \n\\end{align*}\nOne can readily see that the same arguments as in the former case apply, and so there exists some $\\omega^+ > 0$ such that \n\\begin{align*}\n\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0} \\leq \\omega^+ \\|x\\|_{\\mathcal{Q}_0}^2\n\\end{align*}\nfor all $x \\in D(A(t))$ with $0 < l(t) < b$. If $l(t) = 0$ for some $t \\in [0,\\tau]$, then $A(t)$ is dissipative on $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$. Altogether, the choice $\\omega = \\max\\{\\omega^-, \\omega^+\\}$ yields the estimate \\eqref{Interface - Necessary Condition Stablity of Family} for all $t \\in [0, \\tau]$ and for all $x \\in D(A(t))$. This proves the claim. \n\\end{proof}\n\n\\begin{remark}\n\tWe want to emphasize that the choice of the state space $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$ the family $(A(t))_{t \\in [0, \\tau]}$ is defined on was arbitrary. In fact, we may endow $X$ with any norm of the form $\\|\\cdot\\|_{\\mathcal{Q}_{\\hat{l}}}$, where $\\hat{l} \\in (a,b)$, and change the assumption \\ref{AssumptionP1MatricesQ} as follows:\n\t\\begin{enumerate}[label = ($\\tilde{\\mathrm{A}}$1)]\n\t\t\\item The matrix operators $\\mathcal{Q}^{\\pm} \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2})$ defining the coercive operator $\\mathcal{Q}_{l(t)} \\in \\mathcal{L}(X)$ are diagonal matrix operators and they satisfy\n\t\t\\begin{align*}\n\t\t\t\\frac{Q_{11}^+}{Q_{11}^-}(\\hat{l}) = 1 \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\frac{Q_{11}^+}{Q_{11}^-}(z) = \\frac{Q_{22}^+}{Q_{22}^-}(z), \\hspace{0.5 cm} z \\in [a,b].\n\t\t\\end{align*}\n\t\\end{enumerate}\nIn particular, if there is some $\\hat{l} \\in (a,b)$ such that the matrices $\\mathcal{Q}^-(\\hat{l})$ and $\\mathcal{Q}^+(\\hat{l})$ coincide, then it is advisable to consider the family $(A(t))_{t \\in [0, \\tau]}$ defined on the space $(X, \\|\\cdot\\|_{\\mathcal{Q}_{\\hat{l}}})$. \n\\end{remark}\nNote that the inequality \\eqref{Interface - Necessary Condition Stablity of Family} implies that for all $t \\in [0, \\tau]$, the $C_0$-semigroup $(S_t(s))_{s \\geq 0}$ on $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$ generated by $A(t)$ is of type $C_0(1, \\omega)$: By virtue of \\eqref{Interface - Necessary Condition Stablity of Family}, for all $x \\in D(A(t))$ it holds that\n\\begin{align}\n\t\\begin{split}\n\t \\frac{d}{ds} \\| S_t(s)x \\|_{\\mathcal{Q}_0}^2 &= \\frac{d}{ds} \\langle S_t(s)x, S_t(s)x \\rangle_{\\mathcal{Q}_0} \\\\\n\t &= 2 \\langle AS_t(s)x, S_t(s)x \\rangle_{\\mathcal{Q}_0} \\\\\n\t &\\leq 2\\omega \\|S_t(s)x\\|_{\\mathcal{Q}_0}^2.\n\t\\end{split}\n\\label{Interface - Semigroup Estimate}\n\\end{align} \nSince $s \\mapsto e^{2 \\omega s}\\|x\\|_{\\mathcal{Q}_0}^2$ is the unique solution of the initial value problem\n\\begin{align*}\n\t\\frac{d}{ds} y(s) &= 2 \\omega y(s), \\hspace{0.5 cm} s > 0, \\\\\n\ty(0) & = \\|x\\|_{\\mathcal{Q}_0}^2,\n\\end{align*}\nwe infer from \\eqref{Interface - Semigroup Estimate} that\n\\begin{align*}\n\t\\frac{d}{ds} \\| S_t(s)x \\|_{\\mathcal{Q}_0}^2 \\leq \\frac{d}{ds} \\left( e^{\\omega s} \\|x\\|_{\\mathcal{Q}_0} \\right)^2\n\\end{align*}\nholds for all $x \\in D(A(t))$. As $D(A(t))$ is a dense subset of $X$, we conclude that\n\\begin{align*}\n\t\\|S_t(s)\\|_{\\mathcal{L}(X, \\|\\cdot\\|_{\\mathcal{Q}_0})} \\leq e^{\\omega s}, \\hspace{0.5 cm} s \\geq 0. \n\\end{align*}\n Recall from Remark \\ref{Remark Stable Familiy of Generators} that this yields that the family $(A(t))_{t \\in [0, \\tau]}$ is stable in the sense of Definition \\ref{Definition Stable Family of Generators} with stability constants $M=1$ and $\\omega > 0$. Indeed, by the Hille-Yosida Theorem \\ref{Theorem Hille-Yosida - C0 Semigroup} we have $(\\omega, \\infty) \\subset \\rho(A(t))$ for all $t \\in [0, \\tau]$. Furthermore, for every finite sequence $0 \\leq t_1 \\leq t_2 \\leq \\ldots \\leq t_k \\leq \\tau$, $k \\in \\mathbb{N}$, and for all $\\lambda > \\omega$ it holds that\n\\begin{align*}\n\t\\left\\| \\prod_{j = 1}^{k} R(\\lambda, A(t_j)) \\right\\|_{ \\mathcal{L}(X, \\|\\cdot\\|_{\\mathcal{Q}_0})} &\\leq \\prod_{j=1}^{k} \\left\\| R(\\lambda, A(t_j)) \\right\\|_{ \\mathcal{L}(X, \\|\\cdot\\|_{\\mathcal{Q}_0})} \\\\\n\t&\\leq \\prod_{j=1}^{k} \\frac{1}{\\lambda - \\omega} \\\\\n\t&= \\frac{1}{(\\lambda - \\omega)^k}.\n\\end{align*}\nTogether with Theorem \\ref{Interface - Theorem Stability of A(t)} we can summarize these facts as follows:\n\\begin{corollary}\n\t\\label{Interface - Corollary Stability of A(t)}\n\tLet $0 < \\tau < \\infty$, and let $l \\colon [0, \\tau] \\to (a,b)$ be continuously differentiable with $\\dot{l} \\in \\mathcal{C}([0,\\tau], \\mathbb{R})$. Let the assumptions \\ref{AssumptionP1MatricesQ}-\\ref{AssumptionP2ConservationAtInterface} hold. Then the family $(A(t))_{t \\in[0,\\tau]}$ of port-Hamiltonian operators given by \\eqref{Interface - Operator Family A(t)}-\\eqref{Interface - Operator Ql(t)} is stable on $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$.\n\t\\hfill\\ensuremath{\\blacksquare}\n\\end{corollary}\n\nLastly, we wish to prove the same statement as in Corollary \\ref{Interface - Corollary Stability of A(t)} for the system where we allow for a power flow at the moving interface. We impose the following assumptions:\n\\begin{enumerate}[label =($\\tilde{\\mathrm{B}}$\\arabic*)]\n\t\\item \\label{AssumptionB1MatricesQ} The matrix operators $\\mathcal{Q}^{\\pm} \\in \\mathcal{C}^1([a,b], \\mathbb{R}^{2 \\times 2})$ defining the coercive operator $\\mathcal{Q}_{l(t)} \\in \\mathcal{L}(X)$ are diagonal matrix operators and they satisfy\n\t\\begin{align*}\n\t\t\\frac{Q_{11}^+}{Q_{11}^-}(0) = 1 \\hspace{0.3 cm} \\text{and} \\hspace{0.3 cm} \\frac{Q_{11}^+}{Q_{11}^-}(z) = \\frac{Q_{22}^+}{Q_{22}^-}(z), \\hspace{0.5 cm} z \\in [a,b].\n\t\\end{align*}\n\t\\item \\label{AssumptionB2ExpStability} $r >0 $, $\\rank(W_B) = 2$, and $W_B \\Sigma W_B^{\\top} > 0$.\n\\end{enumerate}\nAssumption \\ref{AssumptionB2ExpStability} implies that $A(t)$ generates an exponentially stable contraction semigroup on the respective energy space $(X, \\|\\cdot\\|_{\\mathcal{Q}_{l(t)}})$, see Theorem \\ref{Interface - Theorem 2 Exponential Stability}.\n\\vspace{0.5 cm}\\\\\nLet $t \\in [0, \\tau]$ such that for the interface position we have $a < l(t) < 0$. From assumption \\ref{AssumptionB2ExpStability} and from the proof of Theorem \\ref{Interface - Theorem 2 Exponential Stability} we deduce that for all $x \\in D(A(t))$ it holds that\n\\begin{align*}\n\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_{l(t)}} &=\\frac{1}{2} \\langle e_{\\partial}, f_{\\partial} \\rangle_2 - \\frac{1}{2} e_I(t)f_I(t) \\\\\n\t&\\leq - k \\|\\trace_{l(t)}(\\mathcal{Q}_{l(t)}x)\\|_{2}^2 - \\frac{1}{2} e_I(t)f_I(t)\n\\end{align*}\nfor some $k > 0$ small enough. With the same arguments as in the proof of Theorem \\ref{Interface - Theorem Stability of A(t)}, the following estimate holds for all $x \\in D(A(t))$:\n\\begin{align}\n\t\\begin{split}\t\n\t\t\\langle A(t)x, x \\rangle_{\\mathcal{Q}_0}\t&\\leq - k \\|\\trace_{l(t)}(\\mathcal{Q}_{l(t)}x)\\|_{2}^2 - \\frac{1}{2} e_I(t)f_I(t) + \\frac{\\omega_1^-}{2} \\int_{l(t)}^{0} x^{\\top}(z) \\mathcal{Q}^-(z) x(z) \\, dz \\\\\n\t\t&\\hspace{0.5 cm} + \\frac{1}{4} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} + \\frac{\\omega_2^-}{4} \\int_{l(t)}^{0} x^{\\top}(z) \\mathcal{Q}^-(z) x(z) \\, dz, \n\t\\end{split}\n\t\\label{Interface - A(t) is Stable - Proof Estimate 1}\n\\end{align}\nwith $\\omega_1^- >0$ and $\\omega_2^- >0$ large enough, and with $\\tilde{Q}$ defined in \\eqref{Interface - Theorem Stability of A(t) - Matrix TildeQ}. In the following, we estimate the expression\n\\begin{align*}\n\t- f_I(t) e_I(t) + \\frac{1}{2} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0}.\n\\end{align*}\nBy assumption \\ref{AssumptionB1MatricesQ}, it holds that\n\\begin{align*}\n\t\\frac{1}{2} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} = \\frac{1}{2} x_1(l(t)^+) x_2(l(t)^+) \\left[ Q_{22}^+ (Q_{11}^- - Q_{11}^+) + \tQ_{11}^+(Q_{22}^- - Q_{22}^+) \\right] (l(t)).\n\\end{align*}\nBy definition of the interface port variables \\eqref{Interface - Simplified Continuity Equation}-\\eqref{Interface - Simplified Balance Equation}, we have \n\\begin{align*}\n\t&\\hspace{0.5 cm} - f_I(t)e_I(t) + \\frac{1}{2} x_1(l(t)^+) x_2(l(t)^+) \\left[ Q_{22}^+ (Q_{11}^- - Q_{11}^+) + \tQ_{11}^+(Q_{22}^- - Q_{22}^+) \\right] (l(t)) \\color{white}\\text{abcdefghii} \\\\\n\t&= f_I(t) \\left[ Q_{11}^+(l(t)) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right] \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} x_1(l(t)^+) x_2(l(t)^+) \\left[ (Q_{11}^-Q_{22}^+)(l(t)) -2 (Q_{11}^+ Q_{22}^+)(l(t)) + \t(Q_{11}^+Q_{22}^-)(l(t)) \\right] \n\\end{align*}\n\\begin{align*}\n\t&= f_I(t) \\left[ Q_{11}^+(l(t)) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right] - Q_{11}^+(l(t)) x_1(l(t)^+) \\underbrace{Q_{22}^+(l(t)) x_2(l(t)^+)}_{= f_I(t)} \\\\\n\t&\\hspace{0.5 cm} + \\frac{1}{2} x_1(l(t)^+) x_2(l(t)^+) \\left[ (Q_{11}^+Q_{22}^-)(l(t)) + \t(Q_{11}^-Q_{22}^+)(l(t)) \\right] \\\\\n\t&= - f_I(t) Q_{11}^-(l(t)) x_1(l(t)^-) + \\frac{1}{2} x_1(l(t)^+) x_2(l(t)^+) \\left[ (Q_{11}^-Q_{22}^+)(l(t)) + \t(Q_{11}^+Q_{22}^-)(l(t)) \\right] \\\\\n\t&= - f_I(t) Q_{11}^-(l(t)) x_1(l(t)^-) + \\frac{1}{2} f_I(t) Q_{11}^-(l(t)) x_1(l(t)^+) + \\frac{1}{2} f_I(t) \\frac{Q_{22}^-}{Q_{22}^+}(l(t)) Q_{11}^+(l(t) x_1(l(t)^+) \\\\\n\t&= f_I(t) \\left[ \\frac{1}{2} \\left( Q_{11}^-(l(t)) + \\frac{Q_{22}^-}{Q_{22}^+}(l(t)) Q_{11}^+(l(t)) \\right) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right].\n\\end{align*}\nLet us define \n\\begin{align}\n\t\\label{Interface - Eta-}\n\t\\eta^-(t) := \\frac{1}{2} \\left( \\frac{Q_{11}^-}{Q_{11}^+}(l(t)) + \\frac{Q_{22}^-}{Q_{22}^+}(l(t)) \\right) \\in \\left[ \\frac{m}{M}, \\frac{M}{m} \\right]. \n\\end{align}\nNow, by virtue of the passivity relation \\eqref{Interface - Interface Passivity Relation}, we compute\n\\begin{align*}\n\t&\\hspace{0.5 cm} f_I(t) \\left[ \\frac{1}{2} \\left( Q_{11}^-(l(t)) + \\frac{Q_{22}^-}{Q_{22}^+}(l(t)) Q_{11}^+(l(t)) \\right) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right] \\\\\n\t&= r e_I(t) \\left[ \\eta^-(t) Q_{11}^+(l(t)) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right] \\\\\n\t&= - r \\left( Q_{11}^+(l(t)) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right) \\left[ \\eta^-(t) Q_{11}^+(l(t)) x_1(l(t)^+) - Q_{11}^-(l(t)) x_1(l(t)^-) \\right] \\\\\n\t&= -r \\left[ \\eta^-(t) (Q_{11}^+(l(t)) x_1(l(t)^+))^2 - (\\eta^-(t) + 1) Q_{11}^+(l(t)) x_1(l(t)^+) Q_{11}^-(l(t)) x_1(l(t)^-) \\right. \\\\\n\t&\\hspace{1.2 cm} \\left. + (Q_{11}^-(l(t)) x_1(l(t)^-))^2 \\right]. \n\\end{align*}\nPut $e_1^+(z) = Q_{11}^+(z) x_1(z)$, $z \\in (l(t),b]$, and $e_1^-(z) = Q_{11}^-(z) x_1(z)$, $z \\in [a,l(t))$. By virtue of the inequality $xy \\leq \\frac{1}{2}(x^2 + y^2)$ for all $x,y \\in \\mathbb{R}$, we find\n\\begin{align*}\n\t&\\hspace{0.5 cm} \tr \\left[ - \\eta^-(t) e_1^+(l(t))^2 - e_1^-(l(t))^2 + (\\eta^-(t) +1) e_1^+(l(t)) e_1^-(l(t)) \\right] \\\\\n\t&\\leq r \\left[ - \\eta^-(t) e_1^+(l(t))^2 - e_1^-(l(t))^2 + \\frac{\\eta^-(t) +1}{2}e_1^+(l(t))^2 + \\frac{\\eta^-(t) +1}{2}e_1^-(l(t))^2 \\right] \\\\\n\t&= r \\left[ \\frac{1 - \\eta^-(t)}{2} e_1^+(l(t))^2 + \\frac{\\eta^-(t) -1}{2}e_1^-(l(t))^2 \\right].\n\\end{align*}\nAltogehter, we have\n\\begin{align*}\n-f_I(t) e_I(t) + \\frac{1}{2} \\big[ x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} = r \\left[ \\frac{1 - \\eta^-(t)}{2} e_1^+(l(t))^2 + \\frac{\\eta^-(t) -1}{2}e_1^-(l(t))^2 \\right].\n\\end{align*}\nWe wish to find some $\\omega_3^- >0$ large enough that guarantees that (cf. \\eqref{Interface - A(t) is Stable - Proof Estimate 1})\n\\begin{align*}\n\t- k \\|\\trace_{l(t)}(\\mathcal{Q}_{l(t)}x)\\|_{2}^2 - \\frac{1}{2} f_I(t) e_I(t) + \\frac{1}{4} \\big[x^{\\top}(z) \\tilde{Q}(z) x(z) \\big]_{l(t)}^{0} \\stackrel{!}{\\leq} \\omega_3^- \\| x|_{\\mathcal{Q}_0}^2.\n\\end{align*}\n\nDefine \n\\begin{align*}\n\te_2^-(z) &= Q_{22}^-(z)x_2(z), \\hspace{0.5 cm} z \\in [a,l(t)], \\\\\n\te_2^+(z) &= Q_{22}^+(z) x_2(z), \\hspace{0.5 cm} z \\in [l(t),b]. \n\\end{align*}\nUsing the definition of $e_1^{\\pm}$ and $e_2^{\\pm}$, we have\n\\begin{align*}\n2 \\|x\\|_{\\mathcal{Q}_0}^2 &= \\int_{a}^{l(t)} Q_{11}^-(z) x_1(z)^2 + Q_{22}^-(z) x_2(z)^2 \\, dz + \\int_{l(t)}^{0} Q_{11}^-(z) x_1(z)^2 + Q_{22}^-(z) x_2(z)^2 \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\int_{0}^{b} Q_{11}^+(z) x_1(z)^2 + Q_{22}^+(z) x_2(z)^2 \\, dz \\\\\n\t&= \\int_{a}^{l(t)} \\frac{1}{Q_{11}^-}(z) e_1^-(z)^2 + \\frac{1}{Q_{22}^-}(z) e_2^-(z)^2 \\, dz + \\int_{l(t)}^{0} \\frac{Q_{11}^-}{(Q_{11}^+)^2}(z) e_1^+(z)^2 + \\frac{Q_{22}^-}{(Q_{22}^+)^2}(z) e_2^+(z)^2 \\, dz \\\\\n\t&\\hspace{0.5 cm} + \\int_{0}^{b} \\frac{1}{Q_{11}^+}(z) e_1^+(z)^2 + \\frac{1}{Q_{22}^+}(z) e_2^+(z)(z)^2 \\, dz \\\\\n\t&\\geq \\delta \\left( \\int_{a}^{l(t)} e_1^-(z)^2 + e_2^-(z)^2 \\, dz + \\int_{l(t)}^{b} e_1^+(z)^2 + e_2^+(z)^2 \\, dz \\right)\n\\end{align*}\nfor some $\\delta >0$ small enough. Hence, it amounts to show that there is some $\\tilde{\\omega}_3^- > 0$ such that for all $x \\in D(A(t))$, \n\\begin{align}\n\t\\begin{split}\n\t\t&\\hspace{0.5 cm} - k \\|\\trace_{l(t)}(\\mathcal{Q}_{l(t)}x)\\|_{2}^2 + \\frac{r}{4} \\left[\t(1 - \\eta^-(t)) e_1^+(l(t))^2 + (\\eta^-(t) - 1) e_1^-(l(t))^2 \\right] \\\\ &\\stackrel{!}{\\leq} \\tilde{\\omega}_3^- \\left( \\int_{a}^{l(t)} e_1^-(z)^2 + e_2^-(z)^2 \\, dz + \\int_{l(t)}^{b} e_1^+(z)^2 + e_2^+(z)^2 \\, dz \\right).\n\t\\end{split}\n\t\\label{Interface - A(t) is Stable - Proof Estimate 2}\n\\end{align}\nNote that the continuity condition \\eqref{Interface - Simplified Continuity Equation} implies that\n\\begin{align*}\n\t&\\hspace{0.5 cm} \t(1 - \\eta^-(t)) e_1^+(l(t))^2 + (\\eta^-(t) - 1) e_1^-(l(t))^2 \\\\\n\t&= \t(1 - \\eta^-(t)) [e_1^+(l(t))^2 + e_2^+(l(t))^2] + (\\eta^-(t) - 1) [e_1^-(l(t))^2 + e_2^-(l(t))^2].\n\\end{align*}\nThus, by using the definition of $e_1^{\\pm}$ and $e_2^{\\pm}$ again, \\eqref{Interface - A(t) is Stable - Proof Estimate 2} is equivalent to\n\\begin{align*}\n\t&\\hspace{0.5 cm} - k \\left( \\| (\\mathcal{Q}^- x)(a) \\|_2^2 + \\| (\\mathcal{Q}^+ x)(b) \\|_2^2 \\right) \\\\\n\t&\\hspace{0.5 cm} + \\frac{r}{4} \\left[ (\\eta^-(t) -1) \\| \\mathcal{Q}^-(l(t)) x(l(t)^-) \\|_2^2 + (1 - \\eta^-(t)) \\|\\mathcal{Q}^+(l(t)) x(l(t)^+) \\|_2^2 \\right] \\\\\n\t&\\stackrel{!}{\\leq} \\tilde{\\omega}_3^- \\left( \\int_{a}^{l(t)} \\|(\\mathcal{Q}^-x)(z)\\|_2^2 \\, dz + \\int_{l(t)}^{b} \\|(\\mathcal{Q}^+x)(z)\\|_2^2 \\, dz \\right) \\\\\n\t&= \\tilde{\\omega}_3^- \\|\\mathcal{Q}_{l(t)}x\\|_{L^2}^2.\n\\end{align*}\nSimilarly, for $t \\in [0, \\tau]$ such that $0 < l(t) < b$, the following estimate has to hold for all $ x \\in D(A(t))$:\n\\begin{align*}\n\t&\\hspace{0.5 cm} - k \\left( \\| (\\mathcal{Q}^- x)(a) \\|_2^2 + \\| (\\mathcal{Q}^+ x)(b) \\|_2^2 \\right) \\\\\n\t&\\hspace{0.5 cm} + \\frac{r}{4} \\left[ (1- \\eta^+(t)) \\| \\mathcal{Q}^-(l(t)) x(l(t)^-) \\|_2^2 + (\\eta^+(t) -1) \\|\\mathcal{Q}^+(l(t)) x(l(t)^+) \\|_2^2 \\right] \\\\\n\t&\\stackrel{!}{\\leq} \\tilde{\\omega}_3^+ \\|\\mathcal{Q}_{l(t)}x\\|_{L^2}^2,\n\\end{align*}\nwhere \n\\begin{align}\n\t\\label{Interface - Eta+}\n\t\\eta^+(t) = \\frac{1}{2} \\left( \\frac{Q_{11}^+}{Q_{11}^-}(l(t)) + \\frac{Q_{22}^+}{Q_{22}^-}(l(t)) \\right) \\in \\left[\\frac{m}{M}, \\frac{M}{m} \\right]. \n\\end{align}\nThis can be summarized as follows:\n\\begin{corollary}\n\t\t\\label{Interface - Corollary 2 Stability of A(t)}\n\tLet $0 < \\tau < \\infty$, and let $l \\colon [0, \\tau] \\to (a,b)$ be continuously differentiable with $\\dot{l} \\in \\mathcal{C}([0,\\tau], \\mathbb{R})$. Let the assumptions \\ref{AssumptionB1MatricesQ}-\\ref{AssumptionB2ExpStability} hold. Let $k > 0$ be chosen such that \\eqref{Interface - Theorem Exponential Stability - Inequality} is satisfied. Then the family $(A(t))_{t \\in[0,\\tau]}$ of port-Hamiltonian operators given by \\eqref{Interface - Operator Family A(t)}-\\eqref{Interface - Operator Ql(t)} is stable on $(X, \\|\\cdot\\|_{\\mathcal{Q}_0})$, if the following assertions are true:\n\t\\begin{enumerate}[label = (\\roman*)]\n\t\t\\item There exists some $\\tilde{\\omega}^- >0$ such that for all $t \\in [0, \\tau]$ with $a < l(t) < 0$ and for all $x \\in D(A(t))$ we have\n\t\t\\begin{align*}\n\t\t\t&\\hspace{0.5 cm} - k \\left( \\| (\\mathcal{Q}^- x)(a) \\|_2^2 + \\| (\\mathcal{Q}^+ x)(b) \\|_2^2 \\right) \\\\\n\t\t\t&\\hspace{0.5 cm} + \\frac{r}{4} \\left[ (\\eta^-(t) -1) \\| \\mathcal{Q}^-(l(t)) x(l(t)^-) \\|_2^2 + (1 - \\eta^-(t)) \\|\\mathcal{Q}^+(l(t)) x(l(t)^+) \\|_2^2 \\right] \\\\\n\t\t\t&\\leq \\tilde{\\omega}^- \\|\\mathcal{Q}_{l(t)}x\\|_{L^2}^2,\n\t\t\\end{align*}\n\twhere $\\eta^-(t)$ is defined in \\eqref{Interface - Eta-}.\n\t\\item There exists some $\\tilde{\\omega}^+ >0$ such that for all $t \\in [0, \\tau]$ with $0 < l(t) < b$ and for all $x \\in D(A(t))$ we have\n\t\\begin{align*}\n\t\t&\\hspace{0.5 cm} - k \\left( \\| (\\mathcal{Q}^- x)(a) \\|_2^2 + \\| (\\mathcal{Q}^+ x)(b) \\|_2^2 \\right) \\\\\n\t\t&\\hspace{0.5 cm} + \\frac{r}{4} \\left[ (1 - \\eta^+(t)) \\| \\mathcal{Q}^-(l(t)) x(l(t)^-) \\|_2^2 + (\\eta^+(t) -1) \\|\\mathcal{Q}^+(l(t)) x(l(t)^+) \\|_2^2 \\right] \\\\\n\t\t&\\leq \\tilde{\\omega}^+ \\|\\mathcal{Q}_{l(t)}x\\|_{L^2}^2,\n\t\\end{align*}\nwhere $\\eta^+(t)$ is defined in \\eqref{Interface - Eta+}.\n\t\\end{enumerate}\n\t\\hfill\\ensuremath{\\blacksquare}\n\\end{corollary}\nAs we have discussed in Section \\ref{Section Evolution Equations}, the stability of the family $(A(t))_{t \\in [0, \\tau]}$ is not sufficient for well-posedness of the corresponding evolution problem. However, we will not further analyze this system description and conclude this segment with the preceding stability results.\n\\vspace{0.5 cm}\\\\\nIn this section, we have studied a simplified model closely related to the one formulated in \\cite{Diagne}. We have defined its underlying Dirac structure, and we have worked out passivity conditions that guarantee that the associated port-Hamiltonian operator generates a contraction semigroup on the respective energy space. Those conditions have been defined with respect to the interface and boundary port variables. Moreover, we have presented criteria for exponential stability of the $C_0$-semigroup generated by the port-Hamiltonian operator, and we have specified the associated Hilbert space adjoint as well as the resolvent. All these results hold in case of a fixed interface. At the end of this section, we have defined a family of port-Hamiltonian operators that keeps track of the position of the moving interface. We have shown under which assumptions this family is stable, and we have learned that the power flow at the position of the moving interface is a critical factor for the stability.\n\n\\chapter{Conclusions and Future Work}\n\\label{Chapter Conclusion}\nIn this thesis, we have provided a functional-analytic foundation for the port-Hamiltonian formulation of two systems of two conservation laws that are coupled by a moving interface worked out in \\cite{Diagne}. Furthermore, we have analyzed a simplified version of the proposed model by means of the powerful framework of strongly continuous semigroups. The results presented in this work can be considered as a firm basis for the further analysis of this quite new system class.\n\\vspace{0.5 cm}\\\\ \nIn Chapter \\ref{Chapter Finite-dimensional Port-Hamiltonian Systems} we have introduced the concept of port-based modeling and the necessary terminology related to this framework by reference to finite-dimensional systems. We have studied some basic properties of the underlying Dirac structure, i.e., the power-conserving interconnection structure, as well as the constituting port variables of a multi-physics system, and depicted how the dynamics of a port-Hamiltonian system can be defined with respect to the Dirac structure. Subsequently, in Chapter \\ref{Chapter Some Background} we have briefly recalled the definition of the Hilbert space adjoint of a linear operator $A \\colon D(A) \\subset X \\to X$ defined on some Hilbert space $X$, and we have highlighted the difference between the Hilbert space adjoint and the formal adjoint of a matrix differential operator. This allowed us to define formally skew-symmetric operators, which play a fundamental role in the framework of infinite-dimensional port-Hamiltonian systems. Afterwards, we have reviewed the basics of infinite-dimensional systems theory. For the analysis we have decided for the theory of semigroups of linear operators, so we gave a concise overview of this framework in Section \\ref{Section Strongly Continuous Semigroups}. Our emphasis was to recall sufficient conditions for well-posedness of abstract control systems (see Section \\ref{Section Linear Control Systems}) and of evolution problems (see Section \\ref{Section Evolution Equations}). In Chapter \\ref{Chapter Infinite-dimensional Port-Hamiltonian Systems} we have extended the port-Hamiltonian framework to infinite-dimensional systems, where we have restricted ourselves to systems with a 1-dimensional spatial domain. We have investigated port-Hamiltonian systems that are associated with formally skew-symmetric matrix differential operators, and we have highlighted the main differences compared to its finite-dimensional counterpart. The most important difference is that one has to take the energy flow at the spatial boundary into account. This is accomplished by introducing boundary port variables, which has allowed us to define the infinite-dimensional power-conserving interconnection structure, called the Stokes-Dirac structure, associated with formally skew-symmetric operators. \n\\vspace{0.5 cm}\\\\\nThe preceding chapters have set the foundation for the main part of this thesis, namely the port-Hamiltonian formulation and the analysis of two first-order port-Hamiltonian systems that are coupled by a moving interface, which we have dealt with in Chapter \\ref{Chapter Boundary Port-Hamiltonian Systems with a Moving Interface}. In Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface} we have started by formulating the aforementioned systems that are coupled by an interface fixed at $z = 0$ as a single boundary port-Hamiltonian system, and we have imposed interface conditions on the co-energy variables. Furthermore, we have introduced so-called color functions, which are characteristic functions of the complementary subdomains that keep track of the position of the interface. We have taken them to constitute the extended state space $\\tilde{X} = L^2([a,b], \\mathbb{R}^4)$. Having regard to the interface relations, we have defined balance equations of the state variables on the composed domain, and we have defined a generalized Hamiltonian system of off these equations. This has allowed us to define a Dirac structure for the fixed interface scenario with respect to a formally skew-symmetric differential operator, see Proposition \\ref{Interface Paper - Proposition Dirac Structure DI}. \n\\vspace{0.5 cm}\\\\\nIn Section \\ref{Section Port-Hamiltonian Systems Coupled through a Moving Interface} we have repeated the exposition from Section \\ref{Section Two Port-Hamiltonian Systems Coupled by an Interface} in case of a moving interface position. As opposed to the fixed interface case, the proper formulation as a port-Hamiltonian system is fraught with problems. We have stated the main issues in \\ref{Problem1MovingInterface}-\\ref{Problem3MovingInterface}. Instead of tackling these problems directly, we have studied a simplified model that is closely related to the original system in Section \\ref{Section A Simplified System}, where we have merely assumed that the color functions are not part of the state variables anymore. Furthermore, we have assumed that the variational derivative of the Hamiltonian functional is linear. For the analysis, we have started again by assuming that the interface is fixed. After specifying the Dirac structure of this system (see Corollary \\ref{Interface - Corollary Dirac Structure}), we have analyzed the port-Hamiltonian operator associated with this system. We have shown that if we impose passivity conditions at the boundary and at the interface, then the port-Hamiltonian operator is the infinitesimal generator of a contraction semigroup. In particular, the coupling of two first-order port-Hamiltonian systems coupled by a fixed interface is well-defined, see Theorem \\ref{Interface - Theorem A Generates a Contraction Semigroup}. We have illustrated this result by reference to the coupling of two transmission lines with distinct physical properties, see Example \\ref{Interface - Example Coupled Transmission Lines}. Furthermore, we have been able to show that under more restrictive assumptions regarding the boundary conditions, the port-Hamiltonian operator generates an exponentially stable contraction semigroup, see Theorem \\ref{Interface - Theorem 2 Exponential Stability}. After specifying the Hilbert space adjoint as well as the resolvent of the port-Hamiltonian operator in Subsection \\ref{Subsection The Adjoint Operator} and Subsection \\ref{Subsection Resolvent}, respectively, we have studied a family of port-Hamiltonian operators that encompasses the position of the moving interface. We have provided criteria that ensure that this family is stable (see Subsection \\ref{Subsection Stability of the Family of Infinitesimal Generators}), which is an essential property for the proof of well-posedness of the evolution problem associated with this family.\n\\vspace{0.5 cm}\\\\ \nOne of these criteria is that we have assumed that there is no power flow at the interface (see Corollary~\\ref{Interface - Corollary Stability of A(t)}), which is quite restrictive. If we allow for a power flow at the interface, there is room for improvement concerning the stability result stated in this thesis (see Corollary~\\ref{Interface - Corollary 2 Stability of A(t)}). Furthermore, there are several problems remaining open. For instance, one needs to check for the sufficient conditions for well-posedness apart from stability. Moreover, we have not discussed the input operator associated with the simplified system in case of a moving interface at all. Hence, plenty of control related problems have to be clarified in the future. Additionally, for a complete port-Hamiltonian formulation, we need to find the output port variable conjugated to the velocity of the interface (the input). This is the major topic for further research. After these problems have been successfully tackled for the simplified system, one may eventually deal with the original system \\eqref{Interface Paper - Augmented System wrt Jl} accompanied by the delicate problems \\ref{Problem1MovingInterface}-\\ref{Problem3MovingInterface} again (see pages \\pageref{Interface Paper - Augmented System wrt Jl}-\\pageref{Problem3MovingInterface}). \n\\vspace{0.5 cm}\\\\\nBesides the demanding task of proving well-posedness of the boundary port-Hamiltonian interface control system, as one may call the model proposed in \\cite{Diagne}, there are countless possibilities to generalize the (simplified) system description, e.g., admitting a broader class of color functions, deploying a passive network at the interface position, etc. In conclusion, the port-Hamiltonian model of two systems coupled by a moving interface offers a vast field of potential research. \n\n\n\n\n\\newpage\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzyww b/data_all_eng_slimpj/shuffled/split2/finalzyww new file mode 100644 index 0000000000000000000000000000000000000000..7dea635cb453761999f252374697485a34297e13 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzyww @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn \\cite{Hi}, Hickerson obtained an explicit formula for Dedekind sum $s(p,q)$ in terms of the elements of the continued fraction of $p\/q$, where the Dedekind sum $s(p,q)$ is defined as \n\\begin{equation*}\ns(p,q):=\\sum_{k=0}^{q-1}\\big(\\big(\\frac{k}{q}\\big)\\big)\\big(\\big(\\frac{pk}{q}\\big)\\big).\n\\end{equation*}\nHere $((-))$ denotes the sawtooth function(ie. for $x\\in \\FR$, $((x)) = x- [x] -\\frac12$ if $x\\not\\in \\FZ$ and $((x))=0$ otherwise).\nHickerson's formula is written as follows:\n\\begin{equation}\\label{hik_formul}\n12s(p,q)=\\begin{cases}\n \\frac{p-q_{n-1}}{q}+\\sum_{i=1}^{n}(-1)^{i+1}a_i & \\text{if $n$ is even }, \\\\\n \\frac{p+q_{n-1}}{q}+\\sum_{i=1}^{n}(-1)^{i+1}a_i -3 & \\text{if $n$ is odd}.\n\\end{cases}\n\\end{equation}\nwhere $p_k\/q_k= [a_0, \\ldots, a_{k}]$ be the $k$-th convergent of $p\/q$ and $a_i$ are the terms of the continued fraction of $p\/q$:\n\\begin{equation}\n\\frac{p}{q} = [a_0,\\ldots,a_n] = a_0 + \\cfrac{1}{a_1+\\cfrac{1}{\\ldots + \\cfrac{1}{a_n}}}\n\\end{equation}\nThe idea of the proof is successive application of the celebrated reciprocity formula of Dedekind sums: \n$$\ns(p,q) + s(q,p) = \\frac{1}{12}(p\/q + q\/p + 1\/pq) -1\/4\n$$\nHickerson used the formula to show the density of \nthe graph of Dedekind sums $\\{(p\/q, s(p,q))\\in \\FR^2| (p,q)=1, q>0\\}$ in \\textit{loc.cit.}. It implies a conjecture of Rademacher saying that $s(p,q)$ is dense in $\\FR$(\\cite{RG}).\n\nIn Hickerson's formula \\eqref{hik_formul}, the Dedekind sum is divided into \n integral and fractional parts. \n\nBoth are interesting in opposite directions. \nIntegral part appears in\nthe formula for $\\zeta$ or $L$-values of real quadratic fields.\nNamely, Meyer's formula for partial zeta value of an ideal $\\mathfrak{a}$ of a real quadratic field at $s=0$ is as follows\n\\begin{equation*}\n\\zeta(\\mathfrak{a},0)=\\begin{cases}\n \\frac{1}{12}\\sum_{i=1}^{n}(-1)^{i+1}a_i & \\text{if $n$ is even }, \\\\\n \\frac{1}{12}\\sum_{i=1}^{n}(-1)^{i+1}a_i -3 & \\text{if $n$ is odd}.\n\\end{cases}\n\\end{equation*}\nwhere $a_i$ are terms of the continued fraction of a rational number $p\/q$ in relation with $\\mathfrak{a}$ (cf. \\cite{Me}, \\cite{Van}, \\cite{J-L2}, \\cite{J-L4}).\nWe are not going to give exact description of the numbers appearing here as well as the definition of the partial zeta function but refer to Sec.\\ref{partial_sec} of this article. \nThe special value is \nexactly the integer part of $12s(p,q)$ as appeared in \\eqref{hik_formul} but a universal denominator $12$ independent of $\\mathfrak{a}$.\nThe occurrence of $12$ is explained in \\cite{J-L3}.\nThe fractional part (mod 1) is given as a Laurent polynomial in $p$ divided by $q$ and is necessary in relating the Kloosterman sum to Dedekind sums:\n\\begin{equation*}\n\\sum_{\\substack{00$ \nand \n$$\nr_N:=\\text{L.C.M.}\\left\\{\\text{Denominator of} \\,\\,\\beta_{N}\\begin{pmatrix}N \\\\i+1\\end{pmatrix} B_{i+1}B_{N-i-1}\\Big| \\text{$i$ odd}, 0\\leq i \\leq {N-2}\\right\\}.\n$$\n\\end{thm}\n\nFor higher dimension, we have an analogue of not only the Dedekind sums but also the generalized Dedekind sums (\\cite{zagier1},\n\\cite{FY}, \\cite{CJL}). \nIn \\cite{CJL}, the decomposition of higher dimensional Dedekind sums into integral and rational parts is given but only the rational part is explicit. \n\nThis paper is composed as follows. \nIn Sec. 2, we briefly recall the notion of Todd series and its cocycle property. \nWe relate the generalized Dedekind sums to the coefficients of Todd series.\nIn Sec. 3, we relate a cone decomposition to a continued fraction. \nIn Sec.4, we use the results of earlier sections to obtain an explicit formula lifting the Hickerson's\nformula to the level of Todd series. In Sec. 5, we give the proof of Thm.\\ref{ma} by descending\nto the level of generalized Dedekind sums. \nSec.6 is devoted to the proof of Thm.\\ref{higher_Meyer_formula} and contains \na table for partial zeta values in our formula. \nFinally in Sec.7, we prove Thm.\\ref{ma2}. \n\n\n\\section*{Acknowlegements}\nWe are grateful to Vincent Maillot and Haesang Sun for various discussions and encouragement.\n\n \n\n\\section{Todd series and generalized Dedekind sums}\\label{sec_Todd}\nLet $M$ be the standard lattice $\\FZ^2$ in $\\FR^2$. \nLet $\\sigma = \\sigma(v_1,v_2)$ be a lattice cone with primitive lattices $v_1,v_2$ bounding $\\sigma$. \nWe take the orientation(ie. the order of the rays) into consideration so that $$\\sigma(v_1,v_2) =-\\sigma(v_2,v_1).$$ \nLet $A_\\sigma$ be an integer coefficient matrix whose columns are the lattice vectors $v_1,v_2$ in $\\FZ^2$.\nWe will often identify with a lattice cone $\\sigma$ with its matrix $A_\\sigma$.\n$M_\\sigma$ denotes the sublattice of $M$ generated by $v_1,v_2$. \n$\\Gamma_\\sigma := M\/M_\\sigma$ is isomorphic to a cyclic group of order $\\left|\\det(A_\\sigma)\\right|$.\n \n\nFor $g \\in M$ representing $\\gamma\\in \\Gamma_\\sigma$, we have rational numbers $a_{\\sigma,i}(g)$, $i=1,2$ such that\n$$\ng = a_{\\sigma,1}(\\gamma)v_1 + a_{\\sigma,2}(\\gamma) v_2 \n$$\n\nSince $a_{i,\\sigma}$ is integral on $M_\\sigma$, we have a character $\\chi_{\\sigma,i}$ on $\\Gamma_\\sigma$\nas \n$$\n\\chi_{\\sigma,i} (\\gamma) := \\bbe\\left(a_{\\sigma, i}(g)\\right),\\quad \\text{for $i=1,2$}.\n$$\n\n\n\\begin{defn}\\label{Todd_series}\nThe Todd power series of $\\sigma$ is defined as:\n\\begin{equation}\\label{todd_definition}\n\\Td_\\sigma (x_1,x_2): = \\sum_{\\gamma\\in \\Gamma_\\sigma} \\frac{x_1}{1-\\chi_{\\sigma,1}(\\gamma) e^{-x_1}}\n\\frac{x_2}{1-\\chi_{\\sigma,2}(\\gamma) e^{-x_2}}.\n\\end{equation}\n\\end{defn}\nThe Todd series is analytic in a neighborhood of $0\\in \\FC^2$ and has Taylor expansion.\nNote that the Todd series is invariant of the $\\GL_2(\\FZ)$ equivalent class of cones by the following proposition\n(ie. for $A\\in \\GL_2(\\FZ)$, \n$\\Td_\\sigma (x_1,x_2) = \\Td_{A\\sigma} (x_1,x_2)$).\n\n\n\n\nIf $p, q>0$ are a pair of nonnegative integers prime to each other, $(1,0)$ and $(p,q)$ are\nprimitive lattice vectors and linearly independent. \nLet $\\sigma_{pq}$ denote the cone generated by\n$(1,0)$ and $(p,q)$. \nWe shall write $\\Td_{pq}$ instead of $\\Td_{\\sigma_{pq}}$ for simplicity.\n\nThen the generalized Dedekind sum $s_{ij}(p,q)$ is identified with the $x_1^i x_2^j$-coefficient of\n$\\Td_{pq}$.\nFor $i=j=1$, it is the classical Dedekind sum: $s_{11}(p,q) = s(p,q)$.\nLet us denote the $x_1^i x_2^j$-coefficient of $\\Td_{pq}(x_1,x_2)$ by $t_{ij}(p,q)$.\n\\begin{equation}\n\\Td_{pq}(x_1,x_2) = \\sum_{i,j \\ge 0} \\frac{t_{ij}(p,q)}{i! j!} x_1^i x_2^j.\n\\end{equation}\nThen we have \n\\begin{equation}\nt_{i j }(p,q)=-(-q)^{i+j-1}\\left(s_{ij}(p,q)+\\delta(i,j)B_i B_j\\right),\n\\end{equation}\\label{equa10}\nwhere \n$\\delta(i,j)=\\begin{cases}1,& i=1 \\text{ or }j=1,\n\\\\0, &\\text{ otherwise}\\end{cases}$ and $B_i$ is the $i$-th\nBernoulli number.\n\n\n\n\n\n\nFor a lattice cone $\\sigma$, its normalized Todd series is defined as follows:\n\\begin{defn}\nThe normalized Todd series $S_\\sigma(x_1,x_2)$ of $\\sigma$ is defined as\n$$\nS_{\\sigma} (x_1,x_2) = \\frac{1}{\\det(A_\\sigma) x_1 x_2 } \\Td_\\sigma(x_1,x_2).\n$$\nSimilarly, $S_{\\sigma_{pq}}$ is abbreviated to $S_{pq}$ as in unnormalized case.\n\\end{defn}\n\n$S_\\sigma(x_1,x_2)$ is a Laurent series in variables $x_1, x_2$ and has simple poles along $x_1=0$ and $x_2=0$.\nAlso note that swapping two rays of the cone interchanges not only the variables but also the \nsign in $S_\\sigma$. Thus the orientation is reflected in $S_\\sigma$ and changes the sign(cf. \\cite{J-L3}).\n\n\\begin{align}\nS_{-\\sigma} (x_1,x_2) &= - S_{\\sigma}(x_2,x_1) \\\\\n\\Td_{-\\sigma} (x_1,x_2) & = \\Td_\\sigma(x_2,x_1)\n\\end{align}\n\n\n\n\\begin{prop}[Todd cocycle]\nIf we set \n$\n\\Phi(\\sigma) = S_{\\sigma} (A_\\sigma^{-1}(x_1,x_2))$, then we have\n$$\n\\Phi(\\sigma_1 + \\sigma_2) = \\Phi(\\sigma_1) + \\Phi(\\sigma_2).$$\n\\end{prop}\n\n\\begin{proof}\nFor the proof, we refer the reader to Thm.3 of \\cite{Pom2}.\n\\end{proof}\n\n\n\\section{Continued fraction and Cone decomposition}\nIn this section, we would like to relate nonsingular decomposition of $\\sigma_{p,q}$ with the continued fraction of $p\/q$\nas is done in \\cite{brion}. \nAccording to the cocycle property of the (normalized) Todd series, \nthe (normalized) Todd series is decomposed into sum of those \nof nonsingular cones.\n\nSuppose $q$ and $p$ be relatively prime positive integers and suppose $q>p$.\nLet $p\/q$ have the following continued fraction expansion:\n$$\n\\frac{q}{p}=a_1+ \\cfrac{1}{a_2 + \\cdots\\cfrac{1}{a_n}},$$\nwhere $a_i\\geq1$ are all positive integers. \nPut $(p_{-1},q_{-1})=(1,0)$,\n$(p_0,q_0)=(0,1)$. \nFor $i\\geq1$, $p_i\/q_i$ be the $i$-th convergent of $p\/q$:\n$$\n\\frac{q_i}{p_i}:=a_1+ \\cfrac{1}{a_2 + \\cdots\\cfrac{1}{a_i}}.\n$$\n$p_i$ and $q_i$ are uniquely determined by assuming $q_i>0$ and $\\gcd(p_i,q_i)=1$.\n$(p_i,q_i)$ is computed by iterating elementary M\\\"obius transformations $A_i =\\left( \\begin{smallmatrix} a_i & 1 \\\\ 1 & 0\\end{smallmatrix}\\right)$ in $\\GL_2(\\FZ)$ in the following manner:\n\\begin{equation}\n\\begin{pmatrix} p_i \\\\ q_i \\end{pmatrix} = A_1 A_2 \\cdots A_i \\begin{pmatrix} 1 \\\\ 0\\end{pmatrix} = A_1 A_2 \\cdots A_{i-1} \n\\begin{pmatrix}\na_i \\\\ 1\n\\end{pmatrix}\n\\end{equation}\nComparing $(p_k,q_k)$, $(p_{k-1},q_{k-1})$ and $(p_{k+1},q_{k+1})$ using above formula, we obtain \n\\begin{equation}\\label{relation_p_k}\n(p_{k+1},q_{k+1}) = (p_{k-1},q_{k-1}) + a_{k+1} (p_k,q_k)\n\\end{equation}\nwhich will be used in the next section. \n\n\nNote that $v_k=(p_k,q_k)$ for $-1\\leq k \\leq n$ is primitive lattice vector in the 1st quadrant.\nWe have the following (virtual) cone decomposition of $\\sigma$\ninto nonsingular cones: \n$$\n\\sigma:= \\sigma_{pq}= \\sigma(v_{-1},v_{n})=\\sum_{k=-1}^{n-1} \\sigma_k,\n$$\nwhere $\\sigma_k := \\sigma(v_k, v_{k+1})$.\n\n\n\\section{Nonsingular decomposition of Todd series}\n\nFrom the additivity of normalized Todd series, we have \n$$\\Phi(\\sigma_{p,q})=\\sum_{k=-1}^{n-2}\\Phi(\\sigma_k),$$\nwhere $$\\Phi(\\sigma)=S_{\\sigma}(A_{\\sigma}^{-1}(x,y)).$$\nFor $\\sigma_e=\\sigma((1,0),(0,1))$, from $\\GL_2(\\FZ)$-invariance of Todd power series,\nwe have \n\\begin{equation*}\n\\begin{split}\n&S_{\\sigma_k}=S_{A_{\\sigma_k }\\sigma_e}=\\frac{1}{\\det{A_{\\sigma_k}}xy} \\Td_{A_{\\sigma_k }\\sigma_e}(x,y)\\\\\n&=\\frac{1}{\\det{A_{\\sigma_k}}xy} \\Td_{\\sigma_e}(x,y)=(-1)^{k+1}\\frac{\\Td(x,y)}{xy}.\n\\end{split}\n\\end{equation*}\nThus we obtain the following expression:\n\\begin{equation}\\label{normalized_Todd}\nS_{pq}(x,y)=\\sum_{k=-1}^{n-1}(-1)^{k+1}F\\left(A_{\\sigma_k}^{-1}A_{\\sigma}(x,y)^{t}\\right),\n\\end{equation}\nwhere\n$$F(x,y)=\\frac{1}{1-e^{-x}}\\frac{1}{1-e^{-y}}=\\frac{\\Td (x,y)}{xy}.$$\n\n\n\nFor $-1\\leq k \\leq n-1$ let $M_k(x,y)$ be linear forms in $x$ and $y$ defined as follows:\n\\begin{equation}\nA_{\\sigma_k}^{-1} A_\\sigma \\begin{pmatrix} x \\\\ y \\end{pmatrix}= \\begin{pmatrix} M_{k+1}(x,y) \\\\ M_k(x,y)\\end{pmatrix} \n\\end{equation}\nSince $A_{\\sigma_k}^{-1}\\in \\GL_2(\\FZ)$, $M_k(x,y)$ has coefficients in $\\FZ$. \n\n\nAs $\\det(A_\\sigma)= q$, by multiplying $qxy$, we have \n\\begin{equation}\\label{S_sigma}\n\\begin{split}\n\\Td_{pq}(x,y)&=qxy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\frac{\\Td(M_k,M_{k+1})}{M_k M_{k+1}}\\\\\n&=qxy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}(-1)^{i+j}\\frac{B_i B_j}{i! j!}M_k^{j-1}M_{k+1}^{i-1}.\n\\end{split}\n\\end{equation}\n\n\nDenote by $\\Td_{\\sigma}^N$ the degree $N$ homogeneous\npart of $\\Td_{\\sigma}$. \nThen from \\eqref{S_sigma}\n$\\Td_\\sigma^{N}$ is given as follows:\n\\begin{equation}\\label{ToddN}\n\\begin{split}\n\\Td_{\\sigma}^{N}\n=&qxy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\sum_{i=0}^{N-2}(-1)^N\\frac{B_{i+1}}{(i+1)!}\\frac{B_{N-i-1}}{(N-i-1)!}M_k^{N-2-i}M_{k+1}^{i}\\\\\n&+(-1)^Nqxy\\frac{B_{N}}{N!}\\sum_{k=-1}^{n-1}(-1)^{k+1}\\frac{M_k^{N}+M_{k+1}^{N}}{M_kM_{k+1}}.\n\\end{split}\n\\end{equation}\n\nFrom \\eqref{relation_p_k}, \nfor $k\\ge 0$ \nwe have\n$$\nM_{k-1}-M_{k+1}=a_{k+1}M_k\n$$\nand \n\\begin{equation}\\label{qxy}\n\\begin{split}\n&qxy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\frac{M_k^{N}+M_{k+1}^{N}}{M_kM_{k+1}}\\\\\n&=qxy\\left(\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\sum_{i=0}^{N-2}M_{k-1}^{N-2-i}M_{k+1}^i\\right)+M_0^{N-1}x+M_{n-1}^{N-1}y.\n\\end{split}\n\\end{equation}\n\nTherefore, plugging \\eqref{qxy} into \\eqref{ToddN}, we obtain\n\\begin{equation}\\label{eq1}\n\\begin{split}\n&\\Td_{pq}^{N}\n=qxy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\sum_{i=0}^{N-2}(-1)^{N}\\frac{B_{i+1}}{(i+1)!}\\frac{B_{N-i-1}}{(N-i-1)!}M_k^{N-2-i}M_{k+1}^{i}\\\\\n&+(-1)^Nq\\frac{B_{N}}{N!}xy\\left(\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\sum_{i=0}^{N-2}M_{k-1}^{N-i-2}M_{k+1}^i\n\\right)+\\frac{B_{N}}{N!}M_0^{N-1}x+\\frac{B_{N}}{N!}M_{n-1}^{N-1}y.\n\\end{split}\n\\end{equation}\n\n\n\n\n\n\\section{Computation of Dedekind sums}\nIn this section, using the result of earlier sections, we give a proof of Thm.\\ref{ma}.\n\nLet $m_k$ and $\\ell_k$ be the coefficients of $M_k$:\n$\nM_k(x,y) = m_k x+\\ell_k y.\n$ \nWe already know that $m_k$ and $\\ell_k$ are determined by the continued fraction of $p\/q$ in the following manner:\n\\begin{equation}\nM_k(x,y) = m_k x+\\ell_k y=\n\\begin{cases}\nqy, & k=-1\\\\\n(-1)^{k}\\left(q_kx+(p q_k-q p_k)y\\right), & 0\\le k \\le n-1\\\\\n(-1)^n qx, & k=n .\n\\end{cases}\n\\end{equation}\nThen each part of \\eqref{eq1} is computed as follows:\n\\begin{itemize}\n\\item\n\\begin{equation*}\n\\begin{split}\n&xy\\sum_{k=-1}^{n-1}(-1)^{k+1}\\sum_{i=0}^{N-2}\\frac{B_{i+1}B_{N-i-1}}{(i+1)!(N-i-1)!}(m_k x+\\ell_ky)^{N-2-i}(m_{k+1} x+\\ell_{k+1}y)^{i}\\\\\n& = \\sum_{k=-1}^{n-1}(-1)^{k+1}\\sum_{\\alpha,\\beta}f_k(\\alpha,\\beta)x^{\\alpha+\\beta+1} y^{N-1-\\alpha-\\beta}\n\\end{split}\n\\end{equation*}\nwhere \n$$\nf_k(\\alpha,\\beta):=\\frac{m_k^{\\beta}m_{k+1}^{\\alpha}}{\\alpha! \\beta!}\\sum_{i=\\alpha}^{N-2-\\beta} \\frac{1}{(N-2-i-\\beta)!}\\frac{1}{(i-\\alpha)!}\\frac{B_{i+1}}{(i+1)}\\frac{B_{N-i-1}}{(N-i-1)}\\ell_k^{N-2-i-\\beta}\\ell_{k+1}^{i-\\alpha} \n$$\n\\item\n\\begin{equation*}\n\\begin{split}\n&xy\\left(\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\sum_{i=0}^{N-2}(m_{k-1}x+\\ell_{k-1}y)^{N-i-2}(m_{k+1}x+\\ell_{k+1}y)^i\n\\right)\\\\\n&=\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\sum_{\\alpha, \\beta}g_k(\\alpha,\\beta)x^{\\alpha+\\beta+1} y^{N-1-\\alpha-\\beta},\n\\end{split}\n\\end{equation*}\nwhere\n$$\ng_k(\\alpha,\\beta):= \\frac{ m_{k-1}^{\\beta}m_{k+1}^{\\alpha}}{\\alpha! \\beta!}\\sum_{i=\\alpha}^{ N-2-\\beta}\\frac{(N-i-2)!}{(N-2-i-\\beta)!}\\frac{i!}{(i-\\alpha)!}\\ell_{k-1}^{N-i-2-\\beta}\\ell_{k+1}^{i-\\alpha}.\n$$\n\\item\n$$\nM_0^{N-1}x=\\sum_{i=0}^{N-1}{N-1 \\choose i}\nm_0^{i}\\ell_0^{j}x^{i+1}y^{j}.\n$$\n\\item\n$$\nM_{n-1}^{N-1}y=\\sum_{i+j=N-1}{N-1 \\choose i} \nm_{n-1}^i \\ell_{n-1}^{j}x^{i}y^{j+1}\n$$\n\\end{itemize}\n\n\\begin{thm}\\label{tcoefficient}\nThe coefficient of $x^{e+1}y^{f+1}$ of $\\Td_{pq}^{N}$ with $e+f=N-2$ is\n\\begin{equation*}\n\\begin{split} \n& q \\sum_{\\alpha+\\beta=e} \\Big[\\sum_{k=-1}^{n-1}(-1)^{k+1}f_k(\\alpha,\\beta)+ \\frac{B_N}{N!}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}g_k(\\alpha,\\beta)\\Big{]} \\\\\n& + \\frac{B_N}{N!}\\binom{N-1}{f+1}p^{f+1}+\\frac{B_N}{N!} \\binom{N-1}{e+1}((-1)^{n-1}q_{n-1})^{e+1}.\n\\end{split}\n\\end{equation*}\n\\end{thm}\nCombining the formula (10) and Thm. \\ref{tcoefficient}, we have the proof of Thm.\\ref{ma}.\n\n\n\\section{Application 1: partial zeta values}\\label{partial_sec}\nIn this section, we apply the formula for generalized Dedekind sums to the formula for partial zeta values of ideals of real quadratic fields at nonpositive integers due to Siegel \nand finish the proof of Thm.\\ref{higher_Meyer_formula}.\n\nLet us recall the definition of the partial zeta function of an ideal of a number field. \nFor an ideal $\\mathfrak{a}$ of a number field $K$, the partial zeta function $\\zeta(\\mathfrak{a}, s)$ is defined as\n$$\n\\zeta(\\mathfrak{a},s) := \\sum_{\\substack{\\mathfrak{b} \\sim \\mathfrak{a} \\\\ \\mathfrak{b} : \\text{integral ideal}}} N\\mathfrak{b}^{-s}, \\quad\\text{for $Re(s)>1$}.\n$$\n$\\zeta(\\mathfrak{a},s)$ is meromorphically continued to the whole complex plane with a unique simple pole at $s=1$. \n\nLet $\\mathfrak{a}=[\\alpha,\\beta]$ be an ideal of a real quadratic field $K$. \nLet $\\omega=-\\frac{\\beta}{\\alpha}>\\omega'$ and $\\epsilon>1$ \nbe the totally positive fundamental unit. \nWith respect to the basis $\\alpha, \\beta$, the multiplication by $\\epsilon^{-1}$ is represented by a hyperbolic element \n$\\begin{pmatrix}\np & r \\\\ q & s\n\\end{pmatrix}\\in \\SL_2(\\FZ)$:\n$$ \\epsilon^{-1} \\begin{pmatrix}\n \\alpha \\\\\n \\beta\n\\end{pmatrix}=\\begin{pmatrix}\n p&q \\\\\n r&s\n\\end{pmatrix}\\begin{pmatrix}\n \\alpha \\\\\n \\beta\n\\end{pmatrix}.$$\n\n\nIn this setting, a reformulation of the formula of Siegel by Coates-Sinnott(Lemma 11 in \\cite{C-S}) is given as follows:\n\\begin{thm}[Siegel's formula]\n\\begin{equation}\n\\begin{split}\n(-1)^N\\frac{2N}{B_{2N}}&\\zeta(\\mathfrak{a},1-N)=\\\\\n&\\frac{1}{q^{2N-1}} \\sum_{k=0}^{2N-2}(-1)^{k } \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+ \\frac{B_{2N}}{2N}\\sum_{k=0}^{2N-2}(-1)^{k + 1} e_{N, k} \\frac{s_{k + 1, 2 N - k - 1}(p,q)}{(k + 1) (2 N - 1 - k)}\n\\end{split}\n\\end{equation}\nwhere $e_{N,k}$ is given in the following way:\n$$\n(z^2+(p+s)z+1)^{N-1}=\\sum_{i=0}^{2N-2}e_{N,k}z^k\n$$\n\\end{thm}\n\nNow we apply our formula in Thm.\\ref{ma} to Siegel's formula.\nSoon we will see that the contribution of the rational part $s_{e,f}^R(p,q)$ of Dedekind sums canceled by themselves. \nIt is essentially due to the following observation. \n\\begin{lem}\\label{rational}\n$$ \\sum_{k=0}^{2N-2}(-1)^{k } \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+\\frac{2N}{B_{2N}}\\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{ e_{N, k}}{(k + 1) (2 N - 1 - k)}s^{R}_{k + 1, 2 N - k - 1}(p,q)=0$$\n\\end{lem}\n\\begin{proof}\nWe note that from Theorem 1.1, we have \n$$\ns^{R}_{e, f}(p,q)= \\frac{B_{e + f}}{(e + f)} \\Big{(}e p^\n f + f \\Big{(}(-1)^{n-1}q_{n-1}\\Big{)}^e\\Big{)}.$$\n Also we have $\\quad s=(-1)^{n-1}q_{n-1}$ (See pp. 39--40, \\cite{Van}).\n \nPutting these into the left hand side of the equality, we have\n \\begin{equation*}\n \\begin{split}\n T:=& \\sum_{k=0}^{2N-2}(-1)^{k } \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+\\frac{2N}{B_{2N}}\\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{ e_{N, k}}{(k + 1) (2 N - 1 - k)}s^{R}_{k + 1, 2 N - k - 1}(p,q)\\\\\n &= \\sum_{k=0}^{2N-2}(-1)^{k } \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+ \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{ e_{N, k}}{(k + 1) (2 N - 1 - k)}((k+1)p^{2N-k-1}+(2N-k-1)s^{k+1}).\n \\end{split}\n \\end{equation*}\n \nWe notice that\n$$\n\\int_{0}^{Z}(z^2+(p+s)z+1)^{N-1}dz=\\sum_{i=0}^{2N-2}\\frac{e_{N,k}}{k+1}Z^{k+1}\n$$\nand \n$$e_{N,2N-2-j}=e_{j},\\,\\,\\,\\,\\,\\, j=0,1,\\cdots,2N-2.$$\n\nNow $T$ can be written as sum of definite integrals as follows: \n \\begin{equation*}\n \\begin{split}\n T&=- \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+ \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{ e_{N, k}}{(k + 1) (2 N - 1 - k)}((k+1)p^{2N-k-1}+(2N-k-1)s^{k+1})\\\\\n &=-\\int_{0}^{-p-s}(z^2+(p+s)z+1)^{N-1}dz+\\int_{0}^{-p}(z^2+(p+s)z+1)^{N-1}dz+\\int_{0}^{-s}(z^2+(p+s)z+1)^{N-1}dz\\\\\n &=-\\int_{-p}^{-p-s}(z^2+(p+s)z+1)^{N-1}dz+\\int_{0}^{-s}(z^2+(p+s)z+1)^{N-1}dz\\\\\n &=-\\int_{0}^{-s}(z^2+(-p+s)z+1-ps)^N-(z^2+(p+s)z+1)^N dz.\n \\end{split}\n \\end{equation*}\n\nFinally it remains to show that \n$$\n\\int_{0}^{-s}(z^2+(-p+s)z+1-ps)^N-(z^2+(p+s)z+1)^N dz =0.\n$$\nBy change of the variable $z=t-s\/2$, the above becomes\n$$\n\\int_{s\/2}^{-s\/2} \\left( (t^2 - p t + A)^N - (t^2 + pt + A)^N \\right)dt\n$$\nwhere $A=1-ps\/2 - s^2\/4$. \nFurther the integrand is simplified as \n$$\n-2pt \\sum_{i+j=N-1} (t^2 -pt +A)^i (t^2+pt +A)^j.\n$$\nIt is easily checked to be an odd function in $t$. \nThus the vanishing of $T$ is obtained and this completes the proof. \n\\end{proof}\nNow simply applying the above Lemma to Siegel's formula, we \nfinish the proof of Thm.\\ref{higher_Meyer_formula} in the following:\n\\begin{equation*}\n\\begin{split}\n&\\frac{(-1)^N2N}{B_{2N}}\\zeta(\\mathfrak{a},1-N)\\\\\n&=\\frac{-1}{q^{2N-1}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}\\\\\n&\\quad+\\frac{1}{q^{2N-1}}\\frac{2N}{B_{2N}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{e_{N, k}}{(k + 1) (2 N - 1 - k)}\\left(q s^I_{k + 1, 2 N - k - 1}(p,q)+ s^R_{k + 1, 2 N - k - 1}(p,q)\\right)\\\\\n&=\\frac{-1}{q^{2N-1}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{e_{N, k}}{k + 1} (p + s)^{k + 1}+\\frac{1}{q^{2N-1}}\\frac{2N}{B_{2N}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} \\frac{e_{N, k}}{(k + 1) (2 N - 1 - k)} s^R_{k + 1, 2 N - k - 1}(p,q)\\\\\n&\\quad+\\frac{(-1)^N}{q^{2N-2}} \\frac{2N}{B_{2N}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} e_{N, k} \n\\frac{s^{I}_{k + 1, 2 N - k - 1}(p,q)}{(k + 1) (2 N - 1 - k)}\\\\\n&=\\frac{(-1)^N}{q^{2N-2}} \\frac{2N}{B_{2N}} \\sum_{k=0}^{2N-2}(-1)^{k + 1} e_{N, k} \\frac{s^{I}_{k + 1, 2 N - k - 1}(p,q)}{(k + 1) (2 N - 1 - k)}\n\\end{split}\n\\end{equation*}\n\nNext comes explicit formula of the partial zeta values for $s=-1$ and $s=-2$ using integral part of Dedekind sum and obtain explicit expression for them using continued fraction of $\\frac{p}{q}.$\n\\begin{exam}\n\\begin{equation*}\n\\begin{split}\n&\\zeta(\\mathfrak{a},-1)=-\\frac{1}{3q^3} s_{1,3}^I(p,q)+\\frac{(p+s)}{4q^3}s_{2,2}^I(p,q)-\\frac{1}{3q^3}s_{3,1}^I(p,q)\\\\\n&\\zeta(\\mathfrak{a},-2)\\\\\n&=\\frac{1}{5q^5}s_{1,5}^I(p,q)-\\frac{1}{4q^5}(p+s)s_{2,4}^I(p,q)+\\frac{1}{9q^5}(p+s)^2s_{3,3}^I(p,q)-\\frac{1}{4q^5}(p+s)s_{4,2}^I(p,q)+\\frac{1}{5q^5}s_{5,1}^I(p,q).\n\\end{split}\n\\end{equation*}\n\n\n\n\n\n\nBelow is the table of explicit formula of $s_{i,j}^I(p,q)$ for $i+j=4$ and $ 6$, respectively.\n\\begin{center}\n \\begin{tabular}{| l | l |}\n \\hline\n $s_{2,2}^I(p,q)$&$\\frac{1}{36}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}D_{k+1}q_{k}+D_k q_{k+1}\\Big{)}$\\\\\n $$&$-\\frac{1}{180}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\Big{(}q_{k+1}(D_{k-1}+2D_{k+1})+q_{k-1}(2D_{k-1}+D_{k+1})\\Big{)}$\\\\\n\n\n \\hline\n $ s_{3,1}^I(p,q)$&$\\frac{1}{24}\\sum_{k=-1}^{n-1}(-1)^{k} q_kq_{k+1}$\\\\\n $$&$-\\frac{1}{120}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\Big{(}q_{k+1}^2+q_{k-1}q_{k+1}+q_{k-1}^2\\Big{)}$\\\\\n\n\n \\hline\n $ s_{1,3}^I(p,q)$&$\\frac{1}{24}\\sum_{k=-1}^{n-1}(-1)^{k} D_kD_{k+1}$\\\\\n $$&$-\\frac{1}{120}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\Big{(}D_{k+1}^2+D_{k-1}D_{k+1}+D_{k-1}^2\\Big{)}$\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\n\n\n\n\n\n\\begin{center}\n \\begin{tabular}{| l | l |}\n \\hline\n$ s_{5,1}^I(p,q)$&$\\frac{1}{72}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}q_{k+1}q_{k}^3+q_k q_{k+1}^3\\Big{)}$\\\\\n$$&$+\\frac{1}{252}\\sum_{k=0}^{n-1}(-1)^{k}a_{k+1}\\Big{(}q_{k-1}^4+q_{k-1}^3q_{k+1}+q_{k-1}^2q_{k+1}^2+q_{k-1}q_{k+1}^3+q_{k+1}^4\\Big{)}$\\\\\n\\hline\n$ s_{4,2}^I(p,q)$&$\\frac{1}{180}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}D_{1 + k} q_k^3 + 3 D_k q_k^2 q_{1 + k} + \n 3 D_{1 + k} q_{k} q_{1 + k}^2 + D_{k} q_{1 + k}^3\\Big{)}$\\\\\n $$&$+\\frac{1}{630}\\sum_{k=0}^{n-1}(-1)^{k}a_{k+1}\\Big{(}4 D_{k-1} q_{ k-1}^3 + D_ {k+1} q_{k-1}^3 + \n 3 D_{ k-1} q_{k-1}^2 q_{k+1} + \n 2 D_{k+1} q_{k-1}^2q_{k+1}$\\\\\n $$&$+2 D_{k-1} q_{k-1} q_{k+1}^2 + \n 3 D_{k+1} q_{k-1} q_{k+1}^2 + D_{k-1} q_{k+1}^3 + \n 4 D_{k+1} q_{k+1}^3\\Big{)}$\\\\\n\\hline\n$ s_{3,3}^I(p,q)$&$\\frac{1}{80}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}D_kD_{1+k}q_k^2+D_k^2q_kq_{k+1}+D_{k+1}^2q_kq_{1+k}+D_kD_{1+k}q_{1+k}^2\\Big{)}+$\\\\\n$$&$\\frac{1}{840}\\sum_{k=0}^{n-1}(-1)^{k}a_{k+1}\\Big{(}6D_{k-1}^2q_{k-1}^2+3D_{k-1}D_{k+1}q_{k-1}^2+D_{k+1}^2q_{k-1}^2+3D_{k-1}^2q_{k-1}q_{1+k}$\\\\\n$$&$+4D_{k-1}D_{k+1}q_{k-1}q_{k+1}+3D_{k+1}^2q_{k-1}q_{k+1}+D_{k-1}^2q_{k+1}^2+3D_{k-1}D_{k+1}q_{k+1}^2+6D_{1+k}^2q_{1+k}^2\n\\Big{)}$\\\\\n\\hline\n$s_{2,4}^I(p,q)$&$\\frac{1}{180}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}3D_k^2D_{k+1}q_k+D_{k+1}^3q_k+D_k^3q_{1+k}+3D_kD_{1+k}^2q_{1+k}\\Big{)}$\\\\\n$$&$+\\frac{1}{630}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\Big{(}4D_{k-1}^3q_{k-1}+3D_{k-1}^2D_{k+1}q_{k-1}+2D_{k-1}D_{k+1}^2q_{k-1}+D_{k+1}^3q_{k-1}$\\\\\n$$&$+D_{k-1}^3q_{k+1}+2D_{k-1}^2D_{k+1}q_{k+1}+3D_{k-1}D_{k+1}^2q_{k+1}+4D_{k+1}^3q_{k+1}\\Big{)}$\\\\\n\\hline\n$s_{1,5}^I(p,q)$&$\\frac{q}{72}\\sum_{k=-1}^{n-1}(-1)^{k+1} \\Big{(}D_{k+1}D_{k}^3+D_k D_{k+1}^3\\Big{)}$\\\\\n$$&$+\\frac{q}{252}\\sum_{k=0}^{n-1}(-1)^ka_{k+1}\\Big{(}D_{k-1}^4+D_{k-1}^3D_{k+1}+D_{k-1}^2D_{k+1}^2+D_{k-1}D_{k+1}^3+D_{k+1}^4\\Big{)}$\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\n\n\n\n\n\n\n\n\n\n\n\n\\end{exam}\n\n\n\n\n\\section{Application 2: equidistribution of the graph of Generalized Dedekind sum}\\label{last_sec}\n\nIn this section, we give a proof for Thm.\\ref{ma2} which describes the distribution of $H_{ij}(p,q)$ for $i+j$ even and a relatively prime pair of integers $(p,q)$ in $\\FR^2$ given as follows:\n$$\nH_{ij}(p,q):=\\Big{(}\\frac{p}{q},R_{i+j} q^{N-2}s_{i,j}(p,q)\\Big{)}\n$$ \nWe will show that the fractional part of the graph $H_{ij}(p,q)$ taken in $[0,1)^2$ are \nequidistributed in the sense of Weyl(\\cite{Weyl}).\nAs a function on the unit torus, it is equivalent to saying that the average of the point-mass supported on the sequence of the vectors $H_{ij}(p,q)$(ordered by increment of $q$) is weakly \nconvergent to the standard Lebesgue measure on the torus.\n\nA sequence $\\alpha_k$ in $[0,1)$ is said to be equidistributed if the average of the point-mass supported on the\ntruncated sequence weakly converges to the standard Lebesgue measure of the unit interval. \nIt is checked by the behavior of the Fourier coefficient of the average of the point-mass. \nLet $\\bbe(x)=\\exp(2\\pi i x)$ for $x\\in \\FR$. \nThe Fourier coefficient of the character $\\bbe(mx)$ is given as\n$$\n\\int_0^1 \\bbe(m x) \\frac1N \\sum_{k\\le N} \\delta_{\\alpha_k} dx = \\frac{1}{N}\\sum_{k\\le N} \\bbe(m \\alpha_k).\n$$\nThen the sequence $\\alpha_k$ is equidistributed iff $a_m(N) \\to 0$ as $N\\to \\infty$ for each nonzero $m\\in \\FZ$.\n\nFor a sequence on a $n$-dimensional torus to be equidistributed, \nthe criterion is changed as follows. \nWe note first that a character of $n$-dimensional is represented by a lattice vector $\\bf{m}\\in \\FZ^n$. \nIt needs that for any $\\bf{m}\\in \\FZ^n\\setminus \\{0\\}$,\n$$\n\\frac1N \\sum_{k\\le N} \\bbe(\\bf{m}\\cdot \\alpha_k) \\to 0 \\quad\\text{as $N\\to \\infty$}.\n$$\n\n\n\n\n\n\n\n\n\n\nNow we return to the case of our interest.\nLet $\\bf{m}\\in \\FZ^{2}$ a lattice vector and let, for a positive real number $x$, \n$E({\\bf m},x)$ be the average of the exponential of $(2\\pi i) {\\bf m}\\cdot H_{ij}(p,q)$:\n\\begin{equation}\nE({\\bf m},x) := \\frac{1}{\\# \\left\\{(p,q)|\\gcd(p,q)=1, p1$ and $m_2\\ne 0$} \\\\\n1 & \\text{if $m_2 =0$}\\\\\n0 & \\text{if $j=1$ and $m_1 + m_2 = 0$}\n\\end{cases}\n$$\nFor any nonzero ${\\bf m}$, \nwe have an upper bound for the exponential sum:\n\\begin{equation}\n\\left| K({\\bf m}\\cdot F_{ij}, p)\\right| \\le C \\cdot p^{1\/2}\n\\end{equation}\nfor a given prime $p$ and $C$ is a constant independent of $p$.\n\n\n\n\n\n\n\n\nNow, we consider the case that $q$ is a power of prime $p$ or product of several prime powers. \nUsing the same reduction process in Section 6 of \\cite{J-L3}, we obtain the following bound for \ngiven positive integer $q$. \n\n\n\\begin{thm\nFor given positive integer $i,j$ such that $N=i+j$ is even and positive integer $q$, we have\n$$\n\\left|K({\\bf m}\\cdot F_{ij}, q)\\right|<< q^{\\frac{3}{4}+\\epsilon}\n$$\nfor all $\\epsilon>0$.\n\\end{thm}\n\n\n\n\nFinally the Weyl criterion for $H_{ij}(p,q)$ comes from the following estimate\n\\begin{equation}\n\\begin{split}\n\\sum_{01$ let $\\phi(x):=\\left|\\left(\\FZ\/[x]\\FZ\\right)^*\\right|$ be the Euler's phi function. \nSince \n$\n\\sum_{q$ 0.25 ML. This suggests that arid environments may favor N$_2$ adsorption, though it is noted that water also acts as a proton source so nitrogen reduction cannot occur if the humidity is too low. One interpretation of the surface coverage probability is that the BEEF-vdW functional is not sufficiently accurate to precisely determine the surface coverage, though the bare surface is the most likely. Another perspective is to view this as a sensitivity analysis of coverage with respect to binding energies of competing surface species. From this view, the results indicate that adsorption sites with slightly higher relative N$_2$ adsorption are likely to have a significant N$_2$ coverage under gas-phase conditions. This could include defect sites or other facets, although the site must have both stronger absolute binding energies and stronger relative adsorption of N$_2$ vs. H$_2$O. There are likely sites that satisfy this criterion, but their prevalence must also be considered. Regardless of the interpretation, the conclusion is that nitrogen coverages are expected to be relatively low on rutile TiO$_2$ (110) which is unsurprising given the inert nature of N$_2$.\n\nThe surface phase diagrams under aqueous conditions (\\ref{fig:surface_diagram}b) indicate that water will be the dominant surface species for a wide range of N$_2$ pressures, with adsorbed N$_2$ becoming dominant at approximately 100 atm. In this case the probability analysis indicates that although the surface coverage of water\/hydroxyl is not well-determined by the BEEF-vdW functional, the probability of a significant N$_2$ coverage is negligible at aqueous conditions. Interestingly, several reports have shown appreciable nitrogen fixation rates under aqueous conditions. \\cite{Augugliaro_1982,Hirakawa_2017} The fact that competitive nitrogen adsorption is not favored under aqueous conditions indicates that highly reactive surface groups or defect sites with higher N$_2$ binding energy but low stability may play a role. The finding that nitrogen adsorption is more favorable in the gas phase suggests that the enhanced photocatalytic activity under gas phase conditions is due to the improved ability to adsorb N$_2$ in the absence of water. This intuitive result corroborates previous findings indicating that a key challenge in photocatalytic nitrogen fixation is getting N$_2$ to adsorb\\cite{Zhu_2013,Vettraino_2002,Schrauzer_2011}, and qualitatively explains the correlation between N$_2$ pressure and photocatalytic nitrogen reduction activity \\cite{Schrauzer_2011,Ali_2016}. The issue of nitrogen adsorption must be addressed in any photo(electro)catalytic system regardless of reaction mechanism, and will be a fundamental problem for low pressure nitrogen fixation processes.\n\n\\subsection{Thermochemistry of nitrogen reduction}\n\n\\begin{figure}[h]\n\\centering\n\n\\includegraphics[width=0.25\\textwidth]{figures\/N2_redox_ladder.pdf}\n\\caption{Redox couples for nitrogen reduction (red) and oxidation (blue) with required driving force ($\\Delta$V) \\cite{Medford_2017}. The difference between hydrogen evolution and nitrogen reduction ($\\mathrm{\\Delta \\Delta V_{HER}}$) is shown to highlight the selectivity challenge for nitrogen reduction. Band edges for rutile TiO$_2$ were obtained from literature \\cite{Nozik_1996} and are shown (black bar) for reference (note scale break at 1.75 V).}\n\\label{fig:redox_ladder}\n\\end{figure}\n\nThe electrochemical nitrogen reduction reaction converts nitrogen to ammonia and occurs at an equilibrium potential of 0.05 V vs. RHE (Fig. \\ref{fig:redox_ladder}). The proximity of this redox couple to the hydrogen evolution reaction (H$^+$\/H$_2$ at 0 V) makes selective reduction of nitrogen to ammonia challenging \\cite{Skulason_2012, Montoya_2015,Singh_2017}. Furthermore, the alignment of the rutile TiO$_2$ conduction band edge (Fig. \\ref{fig:redox_ladder}) indicates that the available overpotential for nitrogen reduction under photocatalytic conditions is relatively low ($<$0.15 V). Nonetheless, selective nitrogen reduction has been observed on TiO$_2$ photocatalysts \\cite{Schrauzer_1977, Schrauzer_1983,Schrauzer_2011,Augugliaro_1982}, indicating that TiO$_2$ is capable of dissociating the strong N-N bond more easily than the much weaker H-H bond. This suggests that identification of the active site for photocatalytic nitrogen reduction may enable the development of improved nitrogen reduction electrocatalysts. In this section we hypothesize that the active site is pristine rutile (110), and examine the binding free energies of intermediates for the dissociative and associative nitrogen reduction pathways.\n\n The mechanism for thermocatalytic nitrogen reduction (i.e. Haber-Bosch catalysis) is established as a dissociative mechanism in which the first step is scission of the N-N bond, followed by hydrogenation of adsorbed mono-nitrogen \\cite{Emmett_1933,Ertl_1976,Spencer_1981,Honkala_2005}; the electrochemical equivalent is shown in equations \\ref{eq:diss_mech_start} - \\ref{eq:diss_mech_finish}. Calculations of N-N scission on rutile oxides shows that they follow ``ideal'' scaling, suggesting that the activation barrier for nitrogen dissociation may be low \\cite{Vojvodic_2014}, hence we first investigate the dissociative mechanism. Figure \\ref{fig:reaction_mechanism_diss} shows the free energy diagram for the dissociative mechanism at standard temperature and pressure at the equilibrium potential. The dissociation energy of N$_2$ is remarkably high ($>$8.5 eV), indicating that this route is not remotely thermodynamically feasible on rutile (110). This extreme barrier is not sensitive to the BEEF-vdW exchange-correlation approximation, and is expected to be prohibitively large even in the presence of solvent stabilization since each adsorbed N* would need to be stabilized by $>$3 eV, significantly more than typical solvent stabilization values\\cite{Karlberg_2007,He_2017,Hellman2017}. Furthermore, this step is not electrochemically driven, so even the application of large overpotentials will not enable direct N-N scission. This provides strong evidence against direct dissociation of the N-N bond on the pristine rutile TiO$_2$ surface. The same stabilization would be required for NH* species, effectively eliminating any pathway involving NH* (e.g. dissociation of NNH). Adsorbed NH$_2$ species are somewhat more stable, and may exist under solvated conditions, opening the possibility of mechanisms involving dissociation of N$_2$H$_{\\mathrm{x}>2}$ species, similar to the associative mechanism that will be discussed subsequently.\n \n\\begin{align}\n\t\\label{eq:diss_mech_start}\n\tN_2(g)+^* & \\rightarrow N_2^{*} \\\\\n N_2^* + ^* & \\rightarrow 2N^*\\\\\n 2N^* + 2(H^+ + e^-) & \\rightarrow 2NH^* \\\\\n 2NH^* + 2(H^+ + e^-) & \\rightarrow 2NH_2^* \\\\\n 2NH_2^* + 2(H^+ + e^-) & \\rightarrow 2NH_3^* \\\\\n NH_3^* & \\rightarrow NH_3(g) + ^{*}\n \\label{eq:diss_mech_finish}\n\\end{align}\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{figures\/dissociative_path_edited.pdf}\n\\caption{Free energy diagram for dissociative nitrogen reduction at the equilibrium potential computed from DFT (0.008 V, compared to 0.05 V from experiment). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in equations \\ref{eq:diss_mech_start} - \\ref{eq:diss_mech_finish}.}\n\\label{fig:reaction_mechanism_diss}\n\\end{figure}\n\nThe associative nitrogen reduction mechanism proceeds via diazene (N$_2$H$_2$) and hydrazine (N$_2$H$_4$), as shown in equations \\ref{eq:assoc_mech_start} - \\ref{eq:assoc_mech_finish}. This mechanism has been proposed to be most relevant for electrochemical nitrogen reduction \\cite{H_skuldsson_2017,Skulason_2012,van_der_Ham_2014}, and the fact that hydrazine has been observed as a photoreduction product on TiO$_2$ \\cite{Schrauzer_2011} suggests that it may be the relevant photocatalytic nitrogen reduction mechanism. The free energy diagram for the associative mechanism at the equilibrium potential is shown in Fig. \\ref{fig:FED_assoc}a, while the free energy diagram at the conduction band edge energy for rutile is shown in Fig. \\ref{fig:FED_assoc}b. Under photocatalytic conditions the conduction band edge is the relevant potential, corresponding to an overpotential of 0.15 V. However, examination of the free energy diagram reveals a thermodynamic limiting potential of 2.5 V due to the unstable NNH* adsorbed intermediate, which is consistent with previous work \\cite{H_skuldsson_2017}. This barrier is significantly higher than the conduction band potential, making the route improbable unless the adsorbates are stabilized significantly ($\\approx$ 2 eV) by solvent\/dipole effects. The rate-limiting hydrogenation of N$_2$ is consistent with studies of electrochemical nitrogen reduction on metals \\cite{Skulason_2012}, nitrides \\cite{Abghoui_2016}, and oxides \\cite{H_skuldsson_2017} indicating a general trend for photo- and electrochemical nitrogen reduction.\n\n\\begin{align}\n\t\\label{eq:assoc_mech_start}\n\tN_2(g)+^* & \\rightarrow N_2^{*} \\\\\n N_2^* + H^+ + e^- & \\rightarrow NNH^* \\\\\n NNH^* + H^+ + e^- & \\rightarrow HNNH^* \\\\\n HNNH^* + H^+ + e^- & \\rightarrow HNNH_2^* \\\\\n HNNH_2^* + H^+ + e^- & \\rightarrow H_2NNH_2^* \\\\\n H_2NNH_2^* + ^* + H^+ + e^- & \\rightarrow NH_2^* + NH_3^* \\\\\n NH_2^* + NH_3^* + H^+ + e^- & \\rightarrow 2NH_3^* \\\\\n 2NH_3^* & \\rightarrow 2NH_3(g) + 2^{*} \n \\label{eq:assoc_mech_finish}\n\\end{align}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/associative_FED.pdf}\n\\caption{Free energy diagram for associative nitrogen reduction at an overpotential ($\\eta$) of zero (a) and at the overpotential due to the conduction band edge ($\\eta=0.15$ V) (b). The equilibrium potential ($\\eta=0$) is computed to be 0.008 V (0.05 V experimentally). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in equations \\ref{eq:assoc_mech_start} - \\ref{eq:assoc_mech_finish}.}\n\\label{fig:FED_assoc}\n\\end{figure}\n\nThe results indicate a prohibitively high barrier for both dissociative and associative nitrogen reduction on pristine rutile TiO$_2$ (110), although the associative pathway is significantly more favorable than the dissociative pathway. These findings are robust to the error of the exchange-correlation approximation employed, and the energetic barriers are significantly larger than the typical magnitude of electrochemical interface effects that have been neglected (solvent, electric field). This evidence refutes the hypothesis that rutile (110) is the active site for photocatalytic nitrogen reduction on TiO$_2$, necessitating the development and testing of alternative hypotheses.\n\n\\subsection{Nitrogen reduction at oxygen vacancies and iron substitutions}\n\nSurface defects are known to play a key role in many types of heterogeneous catalysis \\cite{Yates_1991}, and oxygen vacancies in particular have been shown to participate in numerous catalytic reactions on oxides \\cite{McFarland_2013} including TiO$_2$ \\cite{Kim2016,Kim2014,Diebold2003,Pang2008,Pang_2013}. Oxygen vacancies are typically highly reactive, leading to binding energies that are often substantially stronger than binding at the stoichiometric surface. The unstable nature of N$_2$H on pristine rutile (110) (Fig. \\ref{fig:FED_assoc}) indicates that defect sites may enable nitrogen reduction by enhancing the stability of N$_2$H and other high-energy intermediates. Bridging oxygen (O-br) vacancies are known to occur commonly on rutile (110) surfaces \\cite{Diebold2003,Pang2008,Pang_2013}, and a recent analysis of nitrogen reduction over titania has shown that the reaction rate is proportional to the measured number of oxygen defects.\\cite{Hirakawa_2017}. Due to these considerations the O-br vacancy is a natural starting point for evaluating the effect of surface defects, although it is noted that other intrinsic defects, such as Ti vacancies in the surface or sub-surface may also play a role \\cite{Hobiger_1990}.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/defect_associative_FED.pdf}\n\\caption{Free energy diagram for associative nitrogen reduction at an O-br vacancy site at an overpotential ($\\eta$) of zero (a) and at the overpotential due to the conduction band edge ($\\eta=0.15$ V) (b). The equilibrium potential ($\\eta=0$) is computed to be 0.008 V (0.05 V experimentally). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in the Supporting Information.}\n\\label{fig:FED_defect_assoc}\n\\end{figure}\n\nThe energetics of the associative nitrogen reduction pathway at the rutile (110) O-br defect site are shown in Fig. \\ref{fig:FED_defect_assoc}. Comparison to the energetics of the pristine surface (Fig. \\ref{fig:defect_effects}a) reveals a significant stabilization of the N$_2$H intermediate, corresponding to a thermodynamic limiting potential of 1.21 V. However, examination of the energy diagram at the conduction band potential of rutile TiO$_2$ (Fig. \\ref{fig:FED_defect_assoc}b) indicates that the N$_2$H and N$_2$H$_2$ intermediates are still too unstable to explain the photocatalytic reduction observed, although it is plausible that the thermodynamic barrier of 1.39 eV (energy of N$_2$H$_2$ at the conduction band edge) may be overcome by stabilization if solvent or dipole effects for N$_2$H and N$_2$H$_2$ are considerably larger than typical amounts of $\\approx$0.6 eV\\cite{Karlberg_2007,He_2017,Hellman2017}. Furthermore, other defects such as sub-surface O or Ti vacancies may increase the reactivity of the surface, although as defects become less stable they will also become less prevalent on the surface. This tradeoff between stability and reactivity has been noted previously for amorphous oxides \\cite{Goldsmith_2013}, and suggests that more reactive (unstable) defects will have a limited impact due to low prevalence. The O-br defect considered here has a formation free energy of 1.54 eV at 0V SHE referenced against water, indicating that it will occur with an relatively low probability on thermodynamically equilibrated surfaces. Morphological (e.g. particle edges) and kinetic (e.g. trapped bulk defects) effects will increase the prevalence of vacancies or other defects in real catalysts \\cite{Yan_2013}; these effects are difficult to control and characterize, and may be the source of some discrepancies in the nitrogen photofixation literature \\cite{Medford_2017}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/Fe_defect_associative_FED.pdf}\n\\caption{Free energy diagram for associative nitrogen reduction at an iron substitution site at an overpotential ($\\eta$) of zero (a) and at the overpotential due to the conduction band edge ($\\eta=0.15$ V) (b). The equilibrium potential ($\\eta=0$) is computed to be 0.008 V (0.05 V experimentally). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in the Supporting Information.}\n\\label{fig:FED_FE_assoc}\n\\end{figure}\n\nIn addition, many experimental studies have noted that iron dopants significantly increase the nitrogen photofixation rates.\\cite{Schrauzer_1977,Schrauzer_1983,Augugliaro_1982,Soria_1991} Iron dopants may affect catalyst activity through effects in the bulk or on the surface chemistry. In the bulk, iron has been shown to improve charge separation, reducing recombination of electron-hole pairs. This charge separation enhancement has been suggested to be the dominant effect of iron dopants for photocatalytic nitrogen fixation on titania. \\cite{Soria_1991} Alternatively, an iron substitution at the surface may stabilize the states along the reductive pathway, directly improving the energetics of the process. The latter hypothesis was tested by computing the energetics of the associative mechanism on a slab with an iron substitution defect. Two defects were considered, an Fe$^{4+}$ defect arising from direct substitution of the 5-fold Ti atom, and an Fe$^{2+}$ defect formed by substitution of a Ti atom beneath a bridging O and removal of the bridging O, effectively forming a O-br defect and Fe substitution (see the Supplementary Information for visualization of the slab). The Fe$^{2+}$ defect was found to be more stable, and Fig. \\ref{fig:FED_FE_assoc} shows the energetics of the associative pathway with an iron-substituted rutile (110) surface (a comparison of the free energy path for the Fe$^{4+}$ defect is available in the Supplementary Information). This mechanism is very similar to the mechanism on the pristine and O-br vacancies, with the slight difference that N-NH$_2$ is more stable than HNNH on the Fe defect. The limiting potential of 2.2 eV is comparable to that of the defected surface in Fig. \\ref{fig:FED_defect_assoc} (1.7 eV), but is slightly higher due to the stronger adsorption of N$_2$. \nThe energetics of the associative pathway on Fe-substitution defects are compared directly to O-br defects and pristine rutile (110) in Fig. \\ref{fig:defect_effects}a, illustrating that the Fe-substitution defect has a similar effect to the O-br vacancy. The energy required to form this defect was calculated to be 1.1 eV relative to bulk rutile and BCC iron. \nThis moderate formation energy is lower than that of the O-br defect, and N$_2$ adsorbs with a relatively strong binding energy of -0.5 eV. This suggests that Fe surface defects promote the formation of O-br vacancies and adsorption of N$_2$. Nonetheless, the high limiting potential of 2.2 V indicates that the Fe-substitution defect is not active for photocatalytic nitrogen reduction.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/defect_effects.pdf}\n\\caption{Comparison of free energy pathways over pristine (blue), iron-doped (red) and oxygen-vacancy defect (black) for the associative (a) and dissociative (b) nitrogen reduction pathways at the equilibrium potential. All details are consistent with Figs. \\ref{fig:reaction_mechanism_diss}a - \\ref{fig:FED_FE_assoc}a. Error bars are omitted for clarity.}\n\\label{fig:defect_effects}\n\\end{figure}\n\nAnother possibility is that O-br vacancies and\/or Fe-substitution defects significantly impact the energetics of the dissociative pathway. Iron catalysts are known to activate the N-N bond in the Haber-Bosch process\\cite{Emmett_1933, Honkala_2005}, suggesting that the Fe site may play a role, and Hirakawa et. al. \\cite{Hirakawa_2017} have hypothesized that direct N-N bond scission at O-br vacancies is the mechanism for photocatalytic nitrogen fixation. The energetics of the dissociative pathway for both Fe-substitution and O-br vacancy defects are compared with the energetics of the pristine surface in Fig. \\ref{fig:defect_effects}b. The results show that the Fe-substitution has a relatively small effect on the thermodynamics of N-N bond scission, although a more pronounced effect is seen for NH* intermediates that are stabilized by $>$1 eV. The O-br vacancies have a much larger effect on N-N bond scission, stabilizing adsorbed N* by $>$2 eV per adsorbate. However, the thermodynamic barrier of $\\approx$ 4 eV is still prohibitive at ambient conditions, and significantly higher than the 1.21 V thermodynamic limiting potential needed for the associative mechanism at the O-br vacancy. An alternative possibility is a mixed mechanism proceeding through dissociaton of partially hydrogenated species, since the NH$_{\\mathrm{x}}$ species are stable at the O-br vacancy; however, this would still necessitate the formation of the potential-limiting HNNH* species from the associative mechanism and would be thermodynamically (though not kinetically) equivalent. Furthermore, we note that the free energy diagram in Fig. \\ref{fig:defect_effects}b requires two O-br vacancies, since each N* (or NH$_{\\mathrm{x}}$) is adsorbed at a vacancy. The relatively high formation energy of O-br vacancies suggests that this is improbable, and the experimental investigation of Hirakawa et. al. \\cite{Hirakawa_2017} shows a linear dependence on oxygen vacancies, rather than the quadratic dependence that would be indicative of direct N-N (or HN-NH) scission by two vacancy sites. This finding strongly refutes the proposed hypothesis that direct N-N scission by O-br vacancies is the mechanism of photocatalytic nitrogen fixation on TiO$_2$. However, the fact that the O-br vacancy significantly stabilizes NH$_{\\mathrm{x}}$ species, making NH$_{\\mathrm{x}}$ binding close to exothermic suggests that it can promote nitrogen reduction and ammonia formation after the N-N bond has been cleaved.\n\n\\subsection{Nitrogen oxidation and indirect reduction}\n\nThe majority of experimental investigations of photocatalytic nitrogen fixation on TiO$_2$ catalysts have identified reduced products of ammonia or ammonium. However, several reports have observed nitrates as the main product \\cite{Bickley_1979, Yuan_2013}, and examination of the standard redox potentials and TiO$_2$ band edges (\\ref{fig:redox_ladder}) indicates that the band alignment for nitrogen oxidation to NO is significantly more favorable on TiO$_2$ than the band alignment for nitrogen reduction. The band alignment provides $\\approx$ 1.25 V overpotential for both the oxygen reduction half-reaction and the oxidation of nitrogen for NO. Based on this we hypothesize that the scission of the N-N bond on rutile (110) proceeds via the oxidation of nitrogen to NO, which is subsequently oxidized or reduced depending on the details of the catalyst and reaction conditions.\n\nThe thermodynamics of nitrogen oxidation intermediates on rutile (110) are shown in Fig. \\ref{fig:FED_oxid} at the equilibrium potential (a) and the energy of the TiO$_2$ valence band edge (b). Examination of the free energy profile at the equilibrium potential (Fig. \\ref{fig:FED_oxid}a) reveals that the thermodynamic limiting potential is 0.72 V (surface oxygen formation is the potential-limiting step), considerably lower than the case of nitrogen reduction. In addition, the direct adsorption of N$_2$ to this reactive surface oxygen is exergonic by 0.3 eV, indicating that adsorption of N$_2$ is favored by oxygen-rich surfaces. When the significant driving force provided by the photo-excited hole (1.22 V overpotential) is taken into account (Fig. \\ref{fig:FED_oxid}b) the oxidative path becomes extremely favorable, with all steps being exergonic with the exception of N$_2$O$_2$ dissociation which is very slightly ($<0.1$eV) uphill. We note that this analysis considers only the thermodynamics of adsorbed states, and neglects activation barriers that will ultimately govern the kinetics of the process. These barriers could potentially be significant for N-O coupling, although the strong driving force under photocatalytic conditions will improve the kinetics of any electrochemical step.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/oxidative_FED.pdf}\n\\caption{Free energy diagram for nitrogen oxidation to NO at an overpotential ($\\eta$) of zero (a) and at the overpotential due to the valence band edge ($\\eta=1.22$ V) (b). The equilibrium potential ($\\eta=0$) is computed to be 1.465 V (1.68 V experimentally). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in the Supporting Information.}\n\\label{fig:FED_oxid}\n\\end{figure}\n\nThe computational results provide strong evidence that nitrogen oxidation is thermodynamically feasible on the rutile (110) active site, while N$_2$ reduction is thermodynamically challenging. This is at odds with the experimental observation of reduced products on TiO$_2$ \\cite{Schrauzer_1977,Schrauzer_1983,Augugliaro_1982,Soria_1991,Schrauzer_2011,Hirakawa_2017}. One possible explanation is that nitrogen is first oxidized to NO and subsequently reduced to ammonia. The conversion of NO to NH$_3$ is a 5 electron process with a redox potential at 0.71 V vs. RHE, well below the band gap of TiO$_2$\\cite{Medford_2017}, and has been reported experimentally\\cite{Ranjit_1997} and studied theoretically\\cite{Xie_2017}. The thermodynamic feasibility of this pathway on rutile (110) has been computed and the most thermodynamically favorable path is shown in Fig. \\ref{fig:FED_oxid_red}. The mechanism and energetics are consistent with prior work\\cite{Xie_2017}, and indicate that this is indeed thermodynamically feasible. However, reduction of nitrogen oxides is a complex process that can also form partially reduced species such as N$_2$O or N$_2$. In particular, the reaction of NO to N$_2$O and the reaction of N$_2$O to N$_2$ has been observed under UHV conditions by Yates and colleagues\\cite{Rusu2000,Rusu2001}. Fully understanding the selectivity of photocatalytic NO reduction on TiO$_2$ is beyond the scope of this work, but selectivity should be considered in future studies of NO reduction.\n\n\nExperimentally, titania photocatalysts have been reported to reduce nitrates under aqueous conditions, although selectivity to dinitrogen vs. ammonia varies widely based on preparation conditions and metal dopants \\cite{Kobwittaya_2014, Kominami_2010, Ranjit_1997,Shand_2013,Xie_2017,Lozovskii_2009}. Contrarily, oxidation of ammonia has also been reported for TiO$_2$ photocatalysts \\cite{Pollema_1992, Wang_1994, Kominami_2014}, as well as simultaneous reduction of nitrate and oxidation of ammonium to N$_2$ \\cite{Kominami_2014}. Furthermore, the observation of hydrazine as a product from TiO$_2$ nitrogen photofixation is not explained by this mechanism \\cite{Schrauzer_2011}. These conflicting results suggest that other hypotheses should also be considered. In particular, the formation of hydrazine should be examined more closely to determine if hydrazine could be formed through recombination of NH$_x$ species (consistent with the oxidative pathway) or if it is formed associatively (consistent with the reductive pathway). Isotopic scrambling experiments\\cite{Urabe_1978} and use of photoelectrochemical applied bias experiments may provide insight into this outstanding question.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/NO_reduction_FED.pdf}\n\\caption{Free energy diagram for NO reduction to NH$_3$ at an overpotential ($\\eta$) of zero (a) and at the overpotential due to the valence band edge ($\\eta=0.81$ V) (b). The equilibrium potential ($\\eta=0$) is computed to be 0.59 V (0.71 V experimentally). The blue error bars represent one standard deviation of the BEEF-vdW energy ensemble. Adsorbed states are labeled, and the full reaction mechanism is listed in the Supporting Information.}\n\\label{fig:FED_oxid_red}\n\\end{figure}\n\n\n\\section{Conclusions}\n\n\nThe hypothesis that any of the sites on pristine rutile (110) are the active site for nitrogen reduction was shown to be false based on the results of BEEF-vdW DFT calculations for both the associative and dissociative mechanism, although the associative mechanism was found to be considerably more thermodynamically favorable. The revised hypothesis that oxygen vacancies or Fe substitutions are active sites was shown to be more plausible, and a significant stabilization of NH$_{\\mathrm{x}}$ by O-br vacancies was found. Yet, the predicted thermodynamic barriers are considerably higher than the conduction band edge, and the defects are not predicted to be thermodynamically stable under operating conditions, leading to the conclusion that this hypothesis is also improbable. However, the results indicate that N-N bond cleavage is thermodynamically facile on rutile (110) through an oxidative pathway, particularly with the strong oxidative driving force provided by photogenerated holes. Based on this finding, the hypothesis of an oxidized NO* intermediate in nitrogen reduction is introduced as a possible mechanism for photocatalytic nitrogen fixation on rutile (110). The work provides initial molecular-scale insight into the mechanisms that underly photocatalytic nitrogen fixation on TiO$_2$ by conclusively eliminating several possible explanations for this important process and identifying a novel hypothesis of indirect reduction through an oxidized intermediate.\n\n\\section{Methods}\n\\label{sec:methods}\n\\subsection{Computational}\n\\subsubsection{Density Functional Theory}\nIn the current work the Quantum ESPRESSO software package\\cite{QE-2009} is used in conjunction with the Atomic Simulation Package (ASE)\\cite{ISI:000175131400009} to carry out plane wave density functional theory calculations. The BEEF-vdw functional \\cite{Wellendorff_2012} is used for all work with a plane wave cuttoff of 400 eV and a 4x4x1 Monkhorst-Pack k-point grid is used for all slab models \\cite{Monkhorst_1976}. Spin polarization and a dipole correction\\cite{Bengtsson_1999} (applied along the axis of the slab) are used for all slab calculations. Structures are optimized using the BFGS line search method to a total force of 0.05 eV\/\\AA{}. The geometries and adsorption energies for all species are available in Tables S1 and S3. All gas-phase species are evaluated in a periodic unit cell with 6 \\AA{} of vacuum space and a $\\Gamma$-point k-point sampling. A number of calculated properties of TiO$_2$ (e.g. band gap, water adsorption energy) vary with the number of layers in the model slab \\cite{Sun2010,Bredow2004,Patel2014}; slabs with an even number of layers are known to be more accurate due to a symmetric electronic environment between sets of layers \\cite{Sun2010}. The error due to slab size is expected to be ca. 0.2 eV \\cite{Sun2010}, similar to the error of DFT for adsorption energies \\cite{Wellendorff_2015}. Furthermore, there is debate regarding the accuracy of GGA functionals for TiO$_2$. Several studies have indicated that catalytic properties, even of defects, can be reliably treated with GGA functionals \\cite{Schaub_2001,Li_2008,Zheng_2016}, although other work indicates that hybrid, +U, or double-hybrid methods are necessary, particularly to treat electronic transitions and optical properties \\cite{Morgan_2007,Landmann_2012,De_k_2011, Jauho_2015}. In this work we utilize the error estimation capabilities of the BEEF-vdW functional \\cite{Wellendorff_2012} in order to estimate its accuracy and quantify the uncertainty due to the GGA approximation. Although the quantitative accuracy of BEEF-vdW error estimation in oxide materials is not well-known \\cite{Walker_2016}, the BEEF-vdW ensembles provide a systematic way to assess the sensitivity of conclusions to accuracy of approximations \\cite{Medford_2015a}. The error bars shown in this work correspond to $\\pm 1 \\sigma$ of the ensemble of 2000 energies provided by the BEEF-vdW functional. The uncertainty quantified in the BEEF ensembles is also propagated to phase diagrams (Fig. \\ref{fig:surface_diagram}). The probability of a species on the surface is defined as:\n\\begin{align}\n\tP_i = \\frac{1}{N}\\sum_{l} ^{N} \\frac{\\exp(\\frac{-G_i^l}{kT})}{\\sum_{j} ^{M}\\exp(\\frac{-G^l_j}{kT})}\n\t\\label{eq:prob}\n\\end{align}\nwhere $P_i$ is the probability of species $i$ given DFT uncertainty, $G_i^l$ is the free energy of species $i$ computed from energy $l$ of the BEEF-vdW ensemble, $M$ are the number of total species considered, and $N$ is the total number of energies in the BEEF-vdW ensemble (2000). This is equivalent to the average surface coverage from a ensemble of phase diagrams generated from the energies of the BEEF-vdW ensemble. \n\n\\subsubsection{Thermochemistry}\n\nGround state electronic energies ($E_{ele}$) are obtained from DFT and converted to free energies by including zero point energy (ZPE) and thermal contributions:\n\\begin{align}\n\tG_i^o &= E_{ele} + E_{ZPE} + \\Delta H - T \\Delta S \\label{eq:G_ref}\n\\end{align}\nwhere $E_{ZPE}$ is the zero-point energy and $\\Delta H$ (enthalpy) and $\\Delta S$ (entropy) are thermal contributions computed using the vibrational frequencies computed through a finite difference approximation to the Hessian. Adsorbates are treated using the harmonic approximation with a low-frequency cutoff of 30 cm$^{-1}$, while gas phase molecules were treated as ideal gases. The ASE implementation \\cite{ISI:000175131400009} is used for vibrational analysis and statistical mechanics corrections. Vibrational frequencies for all species are reported in Table S2. Vibrational frequencies of adsorbates at defect sites are assumed to be the same as vibrational frequencies on the pristine surface. All thermodynamics were evaluated at 300 K and gas partial pressures were set approximate atmospheric conditions (0.8 atm N$_2$, 0.2 atm O$_2$). Liquid water was approximated as an ideal gas at saturation pressure at 300 K (0.035 atm). Relative (formation) energies were computed with respect to reference states using the equation:\n\\begin{align}\n\tG_i = G_i^o - \\sum_j n_j \\mu_j\\label{eq:G_sum}\n\\end{align}\nwhere $G_i$ is the formation free energy of species $i$, $G_i^o$ is the ``absolute'' free energy computed from DFT and free energy corrections (see Equation \\ref{eq:G_ref}), $n_i$ is the number of atoms $j$ in species $i$, and $\\mu_i$ is the reference chemical potential. The reference for nitrogen is N$_2$ ($\\mu_N = \\frac{1}{2} G_{N_2}^o$), the reference for oxygen is H$_2$O ($\\mu_{H_2O} = \\frac{1}{2} G_{H_2O}^o - \\mu_H$) and the reference for hydrogen is set by the computational hydrogen electrode ($\\mu_H = \\frac{1}{2} G_{H_2}^o$ + eU). In this work no gas-phase corrections (e.g. O$_2$ \\cite{Norskov_2004}, CO$_2$ \\cite{Peterson_2010}) are applied since it is difficult to rigorously correct the BEEF-vdW ensembles; the quantification and propagation of error with the BEEF-vdW ensembles will capture these errors to some extent, and overpotentials are defined relative to the equilibrium potential computed by DFT in order to minimize the influence of inaccurate equilibrium potentials from DFT (see following section).\n\n\n\\subsubsection{Photoelectrochemistry}\n\\label{sec:PEC}\n\nThe photoelectrochemical model used assumes that excitation, charge transport, and charge transfer are decoupled from electrochemistry and is largely consistent with the approach reviewed by Hellman and Wang \\cite{Hellman2017}. All states involving electrons utilized the CHE model reaction as a reference electrode. This model sets the free energy of reaction of hydrogen splitting, reaction \\ref{eq:HER}, to zero at a potential of zero. This ``computational hydrogen electrode'' (CHE) is conceptually equivalent to the reversible hydrogen electrode (RHE)\\cite{Peterson_2010}.\n\nThe energy of each state involving the addition of an electron is varied by an energy of $eU$, where e is the fundamental change and U is the potential relative to this zero point, at potentials other than zero reaction:\n\n\\begin{equation}\n\t\\label{eq:HER}\n\t\\frac{1}{2}H_2\\rightarrow H^{+}+e^{-}\n\\end{equation}\n\nThis model was also used to assess the effect of photo-excited electrons and holes. The locations of the band edges were obtained from the literature \\cite{Nozik_1996} and used to set the potentials of the electrons and holes relative to a the redox couples of the relevant reactions so that the overpotential is equivalent to the experimentally expected overpotential. This approach causes the \\textit{absolute} potential of electrons\/holes to depend on the reaction in question (due to DFT errors in the reaction energies), while the \\textit{relative} energy of electrons\/holes to a given redox couple is equal to the experimental value. The computed (and experimental) equilibrium potential of a given reaction is provided in the caption of each figure.\n\n\\subsubsection{Determination of adsorption sites}\n\nSeveral adsorption sites were tested for stability for all stable molecular species and adatoms. For most stable molecular species the only viable adsorption site was the 5-fold titanium site; attempts to adsorb species to the bridging oxygen or in plane oxygens either led to desorption or reorientation to the 5-fold titanium site. These findings are in agreement with previous first-principles examinations.\\cite{Sorescu2000,Stodt2013} Adatoms (N*, O*, H*) were found to have multiple adsorption sites. Oxygen was found to adsorb to the 5-fold titanium site or form a diatomic species at the bridging oxygen site. The latter can be idenified as the $\\alpha$-oxygen species reported by Lu.\\cite{Lu1994} Hydrogen adatoms were found to be most stable at the in plane oxygen, and the nitrogen adatom was found to have highest stability over the bridging oxygen. A full list of energies and structures is provided in the Supplementary Information.\n\n\n\n\n\n\n\n\\begin{acknowledgement}\n\nThe authors thank the School of Chemical \\& Biomolecular Engineering \nat the Georgia Institute of Technology for providing start-up funding\nfor this work. The authors are also grateful to Marta C. Hatzell for constructive discussions, and Timothy Doane for providing references regarding photocatalytic nitrogen fixation.\n\n\\end{acknowledgement}\n\\section{Publication Disclosure}\nThis document is the unedited Author's version of a Submitted Work that has been submitted for publication in ACS Sustainable Chemistry and Engineering, copyright \u00a9 American Chemical Society.\n\n\n\\begin{suppinfo}\nThe Supporting Information is available and contains:\n\nAll adsorption energies, vibrational frequencies, and structure images; surface free energy, coverage, and probability diagrams including all species under oxidizing and reducing conditions; reaction equations for all mechanisms and defect formations; comparison of free energy pathway for different Fe substitution models.\n\n\\end{suppinfo}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\nLet $\\Omega \\subset \\mathbb R^m$ be an open, bounded domain with Lipschitz boundary and let $F\\colon \\mathbb R^{m \\times n} \\to [0, \\infty[$ be a~convex function. We consider the functional $\\widetilde{\\mathcal F} \\colon C^1(\\Omega, \\mathbb R^{m \\times n})\\to [0, \\infty]$ given by\n\\begin{equation} \\label{tildeF} \n\t\\widetilde{\\mathcal F}(w) = \\int_{\\Omega} F(w_x) \\,\\mathrm d \\mathcal L^m \n\\end{equation} \nand $\\mathcal F \\colon L^2(\\Omega, \\mathbb R^{m \\times n}) \\to [0, \\infty]$ defined as a lower semicontinuous envelope of $\\widetilde{\\mathcal F}$, i.\\,e. \n\\begin{equation}\\label{L2_env} \n\t\\mathcal F(w) = \\inf\\left\\{\\liminf_{\\ell \\to \\infty} \\widetilde{\\mathcal F}(w^\\ell), \\ (w^\\ell) \\subset C^1(\\Omega, \\mathbb R^{m \\times n}),\\ w^\\ell \\to w \\text{ in } L^1_{loc}(\\Omega, \\mathbb R^n) \\right\\}. \n\\end{equation} \nWe define a generalized Rudin-Osher-Fatemi functional $\\mathcal E \\colon L^2(\\Omega, \\mathbb R^n) \\to [0, \\infty]$ by \n \\begin{equation} \\label{ROF_def} \n \t \\mathcal E_h^\\lambda(w) = \\lambda \\mathcal F(w) + \\frac{1}{2} \\int_{\\Omega} |w - h|^2 \\,\\mathrm d \\mathcal L^m, \n \\end{equation}\nwhere $h \\in L^2(\\Omega, \\mathbb R^n)$, $\\lambda > 0$. We will drop indices $h, \\lambda$ as long as we are not interested in dependence on them. By convexity, this functional is weakly lower semicontinuous. Moreover, it is proper and its sublevel sets are weakly compact. Therefore it admits a minimizer $u$, which is unique since $\\mathcal E$ is strictly convex on its domain (i.\\,e.\\ the subset of $L^2(\\Omega, \\mathbb R^n)$ where $\\mathcal E$ is finite).\n\nIn the superlinear case, where $F(\\xi) \\sim |\\xi|^p$, $p > 1$, the domain of $\\mathcal F$ (and $\\mathcal E$) coincides with $L^2(\\Omega, \\mathbb R^n)\\cap W^{1,p}(\\Omega, \\mathbb R^n)$, and on this set $\\mathcal F$ is given by formula \\eqref{tildeF}. Analogous statement is no longer true if $p = 1$, since $W^{1,1}(\\Omega, \\mathbb R^n)$ is not reflexive. Here we consider general $F$ of \\emph{linear growth}, i.\\,e. \n\\begin{equation} \\label{lin_growth} \n\tF(\\xi) \\leq C_F^+ |\\xi| + B_F^+ \\quad \\text{for } \\xi \\in \\mathbb R^{m \\times n}\n\\end{equation}\nwith $C_F^+ > 0$, $B_F^+ \\in \\mathbb R$. In this case, the domain of $\\mathcal F$ includes $L^2(\\Omega, \\mathbb R^n)\\cap BV(\\Omega, \\mathbb R^n)$. Recall that $BV(\\Omega, \\mathbb R^n)$ is the space of those elements of $L^1(\\Omega, \\mathbb R^n)$ whose distributional derivative $w_x$ belongs to $M(\\Omega, \\mathbb R^{m \\times n})$, the space of vector-valued Radon measures. The characterization of $\\mathcal F$ restricted to $BV(\\Omega, \\mathbb R^n)$ was obtained by Goffman and Serrin in \\cite{GS64} and is described in more detail in Section 2. Given a~$BV$ mapping $w$, one can write decomposition of its derivative $w_x$ into absolutely continuous and singular parts with respect to the Lebesgue measure as $\\,\\mathrm d w_x = w_x^{ac} \\,\\mathrm d \\mathcal L^m + \\,\\mathrm d w_x^s$. Consequently, we can define a measure $F(w_x^s) \\in M(\\Omega)$ by\n\\begin{equation} \\label{Fwx}\n\t\\,\\mathrm d F(w_x) = F(w_x^{ac}) \\,\\mathrm d \\mathcal L^m + F^{\\infty}\\left(\\tfrac{w_x^s}{|w_x^s|}\\right) \\,\\mathrm d |w_x^s|, \\quad \\text{where } F^{\\infty}(\\nu) = \\lim_{t \\to \\infty} \\tfrac{1}{t}F(t \\nu) \\text{ for } \\nu \\in \\mathbb S^{n-1}.\n\\end{equation} \nThen, one can prove that for $w \\in BV(\\Omega, \\mathbb R^n)$\n\\begin{equation}\n \\mathcal F(w) = F(w_x)(\\Omega).\n\\end{equation}\nIf moreover $F$ is \\emph{coercive}, i.\\,e. \n\\begin{equation} \\label{coercivity}\n\tF(\\xi) \\geq C_F^-|\\xi| + B_F^-\\quad \\text{for } \\xi \\in \\mathbb R^{m \\times n} \n\\end{equation} \nwith $C_F^- > 0$, $B_F^- \\in \\mathbb R$, then the domain of $\\mathcal F$ coincides with $L^2(\\Omega, \\mathbb R^n)\\cap BV(\\Omega, \\mathbb R^n)$. In the case $F = |\\cdot|$, $\\mathcal F$ coincides with the standard $BV$ seminorm also known as the total variation, while $\\mathcal E$ coincides with the classical Rudin-Osher-Fatemi functional, see \\cite{rof, ChambolleLions}.\n\nA natural question arises, what can be said about the singular part of $u_x$. So far, this question has been mostly investigated in the scalar-valued case $n=1$. Let us report briefly on known results in this direction. Under the assumption that $\\Omega$ is convex, it has been proved in \\cite{LasicaRybka} that \n\\[F^\\infty(u_x^s)(\\Omega) \\leq F^\\infty(h_x^s)(\\Omega).\\]\nIn particular, if $h \\in W^{1,1}(\\Omega)$, then $u \\in W^{1,1}(\\Omega)$. On the other hand, in the case where $F$ is the $\\ell^1$ norm, one can construct an example of non-convex polygon $\\Omega \\subset \\mathbb R^2$ and $h \\in C^\\infty(\\overline{\\Omega})$ such that $u$ has a jump discontinuity in $\\Omega$ \\cite{tetris}. This sort of singularity does not appear if $F$ is the Euclidean norm, in which case $|u_x^j| \\leq |h_x^j|$ as measures, i.\\;e.\n\\[|u_x^j|(V) \\leq |h_x^j|(V) \\quad \\text{for Borel } V \\subset \\Omega\\]\nfor $h \\in BV(\\Omega) \\cap L^\\infty(\\Omega)$, where $w_x^j$ denotes the jump part of $w_x$ for $w \\in BV(\\Omega)$ \\cite{jalalzaijump}. This estimate has been generalized to a wider class of functions $F$ in \\cite{Valkonen2015}. To our knowledge, it remains an open question whether an analogous estimate holds for the whole singular part of $u_x$. Moreover, no similar estimate on the singular or jump part of $u_x$ is known in the vector-valued setting $n > 1$, except in the one-dimensional case $m=1$. \n\nFrom now on we will restrict ourselves to $m=1$. In the one-dimensional, scalar-valued case $m=n=1$, for $F = |\\cdot|$, it has been proved in \\cite{bcno, BonforteFigalli} that\n\\begin{equation} \\label{1d_homo_est} \n\t|u_x| \\leq |h_x| \\quad \\text{as measures.}\n\\end{equation} \nThe proof in \\cite{bcno} follows by analysis of level sets of $u$, thus depending on the linear order of the range $\\mathbb R$. On the other hand, the reasoning in \\cite{BonforteFigalli} is based on explicit description of minimizers for piecewise constant $h$. Inequality \\eqref{1d_homo_est} was generalized to the vector-valued case $n > 1$ in \\cite{giacomellilasica} using integral estimates. \n\nIn this paper, our goal is to obtain estimates on the singular part of the minimizer in the case $m=1$, $n>1$ for possibly general $F$. We will need a structural assumption formally similar to the one in \\cite{mercier} (note however that in \\cite{mercier} $m>1$, $n=1$ and the results are not closely related). We recall the following definition (compare e.\\,g.\\ \\cite{ChambolleNovaga2015}). We say that a function $\\varphi \\colon \\mathbb R^n \\to [0, +\\infty[$ is an \\emph{anisotropy} if it is convex and positively $1$-homogeneous, i.\\,e. \n\t\\[\\varphi(t p) = t \\varphi(p) \\quad \\text{for } p \\in \\mathbb R^n,\\ t \\geq 0.\\]\n\tFor any anisotropy $\\varphi$, we denote \n\t\\[c_{\\varphi}^+ = \\max_{0 \\neq p \\in \\mathbb R^n} \\frac{\\varphi(p)}{|p| }, \\quad c_{\\varphi}^- = \\min_{0 \\neq p \\in \\mathbb R^n} \\frac{\\varphi(p)}{|p|}. \\]\nAn anisotropy $\\varphi$ will be called \\emph{coercive} if $c_\\varphi^- > 0$. Note that any even anisotropy is a seminorm (and vice versa) and any coercive, even anisotropy is a norm (and vice versa). We prove the following three local estimates on the variation measure of minimizers of $\\mathcal E$ in terms of the variation measure of the datum $h$ under varying assumptions on $F$.\n\\begin{thm} \\label{thm:homo}\n\tLet $F\\equiv\\varphi \\colon \\mathbb R^n \\to [0, \\infty[$ be an anisotropy. Suppose that $h \\in BV(U, \\mathbb R^n)$ for an open interval $U \\subset I$. Then the minimizer $u$ of $\\mathcal E$ satisfies $u \\in BV(U, \\mathbb R^n)$ and $|u_x| \\leq |h_x|$ in the sense of Borel measures on $U$, i.\\,e.\n\t\\begin{equation} \\label{homo_est} \n\t\t|u_x|(E) \\leq |h_x|(E) \\text{ for any Borel set } E \\subset U.\n\t\\end{equation}\n\\end{thm}\n\n\\begin{thm} \\label{thm:constants}\n\tLet $F = f \\circ \\varphi$, where $f\\colon[0, \\infty[ \\to [0, \\infty[$ is convex, increasing and of linear growth, and $\\varphi\\colon \\mathbb R^n \\to \\mathbb R$ is a coercive anisotropy. Suppose that $h \\in BV(U, \\mathbb R^n)$ for an open interval $U \\subset I$. Then the minimizer $u$ of $\\mathcal E$ satisfies $|u_x^{s}| \\leq \\left( \\frac{c_\\varphi^+}{c_\\varphi^-} \\right)^2|h_x^{s}|$ in the sense of Borel measures on $U$, i.\\,e.\n\t\\begin{equation} \\label{eq:constants}\n\t\t|u_x^{s}|(E) \\leq \\left( \\tfrac{c_\\varphi^+}{c_\\varphi^-} \\right)^2 |h_x^{s}|(E) \\text{ for any Borel set } E \\subset U.\n\t\\end{equation}\n\\end{thm}\n\n\\begin{thm} \\label{thm:regular norm} \n\tLet $F = f \\circ \\varphi$, where $f\\colon [0, \\infty[ \\to [0, \\infty[$ is strictly convex, increasing and of linear growth, $\\varphi\\colon \\mathbb R^n \\to \\mathbb R$ is a~coercive anisotropy which is strictly convex in the sense of Reshetnyak (see Definition \\ref{def: Reshetnyak}). Moreover, assume that $\\varphi$ is differentiable in $\\mathbb R^n \\setminus \\{0\\}$ and that $f$ is differentiable with $f'(0) = 0$. Suppose that $h \\in BV(U, \\mathbb R^n)$ for an open interval $U \\subset I$. Then the minimizer $u$ of $\\mathcal E$ satisfies $|u_x^{s}| \\leq |h_x^{s}|$ in the sense of Borel measures on $U$, i.e.\n\t\\begin{equation} \\label{eq: regular norm}\n\t\t|u_x^{s}|(E) \\leq |h_x^{s}|(E) \\text{ for any Borel set } E \\subset U.\n\t\\end{equation}\n\\end{thm}\n\n\nWithout the assumption on coercivity of the anisotropy in Theorem~\\ref{thm:homo}, it is not immediately clear that the minimizer $u$ is a~BV function. This difficulty does not appear in Theorems \\ref{thm:constants} and \\ref{thm:regular norm} since any convex, increasing function $f$ on $[0, \\infty[$ satisfies $f(s) \\geq cs + b$ for some $a > 0$, $b \\in \\mathbb R$. Hence, if $\\varphi$ is a~coercive anisotropy, then $F = f \\circ \\varphi$ satisfies \\eqref{coercivity}. \n\nThe proofs of the three theorems are similar in spirit -- a two-level approximation of the functional $\\mathcal E$ is used. On the more regular level, the minimizer $u^{\\varepsilon, \\eta}$ satisfies the Euler-Lagrange equation in strong sense. We test the equation with a suitable function of form $\\zeta^2 G(u_x^{\\varepsilon, \\eta})$, where $\\zeta \\in C^1_c$, while $G$ is chosen so that $G(w_x)_x \\cdot DF(w_x)_x \\geq 0$ for $w \\in W^{2,2}(\\Omega, \\mathbb R^n)$ and $G(\\xi) \\cdot \\xi \\sim |\\xi|$ for large $|\\xi|$. It is essential for our argument to take $G$ of form $G(\\xi) = g(\\xi) \\xi$. Due to such choice, the bulk of terms involving $\\zeta_x$ vanishes, owing to equality $\\xi \\cdot D^2\\varphi(\\xi) = 0$, which follows from homogeneity of $\\varphi$. In the homogeneous case $F = \\varphi$, performing this procedure with a relatively simple $g$ readily leads to the strongly local estimate \\eqref{homo_est}. \n\nIn the inhomogeneous case $F = f \\circ \\varphi$, the argument becomes more involved and, expectedly, yields a bound only on the singular part of $u_x$. One of essential difficulties in the proof of such estimates is posed by very weak continuity properties of the operator $w \\mapsto w_x^s$ on $BV(I, \\mathbb R^n)$. Notably, maps such as $w \\mapsto |w_x^s|(\\Omega)$ fail to be semicontinuous w.\\,r.\\,t.\\;weak* as well as strict or area-strict convergence in $BV(I, \\mathbb R^n)$. Therefore, we need to work instead with quantities of type \n\\[ (|w_x| - k)_+ = \\int (|w_x^{ac}| - k)_+ \\,\\mathrm d \\mathcal L^1 + |w_x^s| \\]\nand obtain estimates for their decay as $k \\to \\infty$. However, our previously described procedure delivers estimates for more complicated, non-convex functions of $u_x$ which again fail to be lower semicontinuous. \nNonetheless, by exploiting equivalence between $\\varphi$ and $|\\cdot|$, we can deduce estimates for a convex function of $u_x$. This is why in Theorem \\ref{thm:constants} we require coercivity of $\\varphi$ and only obtain an estimate with a multiplicative constant. \n\nWe note that such a constant is undesirable in the context of image processing, since in applications the minimization procedure tends to be iterated many times (in the spirit of the \\emph{minimizing movements scheme} -- we recall that notion in the next paragraph). In the case that $F$ is strictly convex and differentiable, we obtain a better estimate \\eqref{eq: regular norm}. In the proof we use spherical compactification of $\\mathbb R^n$ (see Section~\\ref{sec: prelim}) and the fact that the derivative $DF$ is a~homeomorphism to improve the convergence of approximate minimizers. This is one of the reasons why we need additional assumptions on $F$ in this case. However, we do not know whether \\eqref{eq: regular norm} can fail otherwise. \n\n\nLet us recall that the convex, lower semicontinuous function $\\mathcal F$ on the Hilbert space $L^2(\\Omega, \\mathbb R^n)$ defines a unique gradient flow \\cite{maximaux}. In other words, for any $v_0 \\in L^2(\\Omega, \\mathbb R^n)$ there exists a unique function $v \\in C([0, \\infty[, L^2(\\Omega, \\mathbb R^n))\\cap H^1_{loc}(0,T;L^2(\\Omega, \\mathbb R^n))$ such that \n\\begin{equation} \\label{grad_flow} \n\tv_t(t) \\in - \\partial \\mathcal F(v(t)) \\quad \\text{ for a.\\,e.\\ } t > 0, \\qquad v(0) = v_0. \n\\end{equation} \nThe map associating the minimizer of $\\mathcal E_h^\\lambda$ to a given $h$ coincides with the resolvent operator for the subdifferential $- \\partial \\mathcal F$. This is the basis of the \\emph{minimizing movements scheme}: for a given $N \\in \\mathbb N$ we iteratively define \\[v^N_0 = v_0, \\qquad v^N_j = \\mathrm{arg\\,min}\\, E_{v^N_{j-1}}^{1\/N} \\quad \\text{for } j=1,2,\\ldots\\]\nThen, for a.\\,e.\\ $t > 0$ we have \n\\begin{equation} \\label{grad_flow_conv} \n\tv^N_j \\to v(t) \\quad \\text{in } L^2(I, \\mathbb R^n) \\quad \\text{if } N \\to \\infty,\\ j\/N \\to t.\n\\end{equation} \nLet $F=\\varphi$ is an anisotropy. Then, by Theorem \\ref{thm:homo}, for all $N,j \\in \\mathbb N$, $v^N_j \\in BV(U,\\mathbb R^n)$ and \n\\begin{equation*} \n\t|(v^N_j)_x |(V) \\leq |(v^N_{j-1})_x |(V) \\leq \\ldots \\leq |(v_0)_x |(V)\n\\end{equation*} \nfor any open $V \\subset U$. Using \\eqref{grad_flow_conv} and Theorem \\ref{thm-gs-lsc}, we deduce $|v_x(t)|(V) \\leq |(v_0)_x|(V)$ for any open $V \\subset U$ and a.\\,e.\\ $t>0$. Thus, we obtain\n\\begin{cor}\\label{thm: grad flows} \n\tSuppose that $F = \\varphi$ is an anisotropy. If $v_0 \\in BV(U, \\mathbb R^n)$ with open $U \\subset I$, then $v(t) \\in BV(U, \\mathbb R^n)$ and $|v_x(t)| \\leq |v_{0,x}|$ in the sense of Borel measures on $U$, i.\\,e.\n\t\\begin{equation}\\label{gf_thm_1} \n\t|v_x(t)|(E) \\leq |v_{0,x}|(E) \\quad \\text{for any Borel set } E \\subset U.\n\t\\end{equation} \n\\end{cor}\nWe are unable to transfer Theorems \\ref{thm:constants} and \\ref{thm:regular norm} to the gradient flow setting since $|u_x^s|$ does not have good semicontinuity properties. On the other hand, estimates \\eqref{eq:constants final} and \\eqref{eq: before k limit} do not behave well under iteration. \n\nIn the next section we recall some facts that will be used in the sequel. In Section~\\ref{sec: approximate} approximate functionals are defined and in Sections \\ref{sec:homo}, \\ref{sec:coercive}, \\ref{sec:regular} we prove Theorems \\ref{thm:homo}, \\ref{thm:constants} and\\ref{thm:regular norm}, respectively. \n\n\\section{Preliminaries} \\label{sec: prelim}\n\nThroughout the paper, $\\varrho \\in C^\\infty_c(\\mathbb R^d)$ denotes a~positive, radially symmetric function such that $\\int_{\\mathbb R^d} \\varrho \\,\\mathrm d \\mathcal L^d = 1$ and $\\varrho_\\varepsilon(p) = \\varepsilon^{-d} \\varrho(p\/\\varepsilon)$. We will say that measures $\\mu_j \\in M(\\Omega, \\mathbb R^d)$ converge weakly* to $\\mu \\in M(\\Omega, \\mathbb R^d)$ and write $\\mu_j \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} \\mu$, if for any $\\zeta \\in C_c(\\Omega)$ it is true that $\\lim_{j \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d \\mu_j = \\int_\\Omega \\zeta \\,\\mathrm d \\mu$.\n\n\\subsection*{Functions of measures}\n\nDefinition \\eqref{Fwx} from the Introduction can be applied to any Radon measure, not necessarily a~derivative of a~$BV$ function. Throughout this subsection, $G$ denotes~a convex function $G:\\mathbb R^d \\to [0, \\infty[$ of linear growth. Given such a~function and a~measure $\\mu \\in M(\\Omega, \\mathbb R^d)$, set\n\\begin{equation} \\label{Fmu}\n\t\\,\\mathrm d G(\\mu) = G(\\mu^{ac}) \\,\\mathrm d \\mathcal L^m + G^{\\infty}\\left(\\tfrac{\\mu^s}{|\\mu^s|}\\right) \\,\\mathrm d |\\mu^s|, \\quad \\text{where } G^{\\infty}(\\nu) = \\lim_{t \\to \\infty} \\tfrac{1}{t}F(t \\nu) \\text{ for } \\nu \\in \\mathbb S^{n-1}.\n\\end{equation}\nThe function $G^\\infty$ is often called a~recession function. In the following paragraphs, facts about properties of $G(\\mu)$ are collected.\n\n\\begin{thm}\\cite[p.\\ 172]{GS64} \\label{thm-gs-lsc}\n\tLet $\\mu_j, \\mu \\in M(\\Omega, \\mathbb R^d)$ and assume that $\\mu_j \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} \\mu$ on $\\Omega$. Then for any open $U \\subset \\Omega$,\n\t\\begin{equation*}\n\t\tG(\\mu)(U) \\leq \\liminf_{j \\to \\infty} G(\\mu_j)(U).\n\t\\end{equation*}\n\\end{thm}\n\n\\begin{cor} \\label{cor-gs-lsc-zeta}\n\tLet $\\mu_j, \\mu \\in M(\\Omega, \\mathbb R^n)$ and assume that $\\mu_j \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} \\mu$ on $\\Omega$. Then for any non-negative $\\zeta \\in C_c(\\Omega)$,\n\t\\begin{equation*}\n\t\t\\int_\\Omega \\zeta \\,\\mathrm d G(\\mu) \\leq \\liminf_{j \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_j).\n\t\\end{equation*}\n\\end{cor}\n\n \n\\begin{proof}\n\tSuppose that $\\liminf_{j \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_j) < + \\infty$ (if not, the inequality clearly holds). Choose a~subsequence $\\mu_{j_\\ell}$ such that\n\\begin{equation*}\n\t\\lim_{\\ell \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_{j_\\ell}) = \\liminf_{j \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_j).\n\\end{equation*}\nSince $G$ is of at most linear growth, the sequence of measures $\\left \\{ G(\\mu_{j_\\ell}) \\right \\}_\\ell$ is bounded and hence there exists a~subsequence (which we do not relabel) weakly* convergent to a~measure $\\nu$. Both $\\nu$ and $G(\\mu)$ are finite on $\\Omega$ and therefore there exists only countably many balls $V$ for which $\\nu(\\partial V) >0$ and $G(\\mu)(\\partial V) > 0$. Therefore, at each point $x \\in \\Omega$ there exists a~family of open balls $\\left \\{ V_x^i \\right \\}_{i=1}^{N(x)}$ centered at $x$ with $\\mathrm{diam}\\left ({V_x^i} \\right) \\to 0$ as $i \\to \\infty$ for which \n\\[\n\t\\nu(\\partial V) = G(\\mu)(\\partial V) = 0.\n\\]\n\n\nThe family $\\left \\{ \\overline{V_x^i}: \\, x \\in J, \\, i = 1, \\ldots N(x) \\right \\}$ of closed balls satisfies assumptions of a~corollary to Besicovitch's covering theorem (see \\cite[Section 1.5, Corollary I]{evansgariepy}). For any open $U \\subset \\Omega$, it is then possible to choose a~countable subfamily $\\mathcal{G}$ of disjoint balls such that\n\\begin{equation} \\label{eq: family G}\n\t\\mathsmaller{\\bigcup} \\mathcal{G} \\subset U \\text{ and } \\left [\\nu + G(\\mu) \\right] \\left( U \\setminus \\mathsmaller{\\bigcup} \\mathcal{G} = 0 \\right).\n\\end{equation}\nFor any ball $V$ with $\\overline{V} \\in \\mathcal{G}$, it is true that $G(\\mu_{j_\\ell})(V) \\to \\nu(V)$ and using Theorem \\ref{thm-gs-lsc}, we may write\n\\begin{equation*}\n\t\\nu (\\overline{V}) = \\nu(V) = \\lim_{\\ell \\to \\infty} G(\\mu_{j_\\ell})(V) \\geq G(\\mu)(V) = G(\\mu)(\\overline{V}).\n\\end{equation*}\nSumming over balls from the family $\\mathcal{G}$ and using \\eqref{eq: family G} results in the estimate $\\nu(U) \\geq G(\\mu)(U)$ for any open $U \\subset \\Omega$, which implies that\n\\[ \n\\nu(E) \\geq G(\\mu)(E) \\text{ for any Borel } E \\subset \\Omega.\n\\]\nAs a result, in view of the choice of the subsequence $\\mu_{j_\\ell}$, one gets\n\\[\n\\int_\\Omega \\zeta \\,\\mathrm d G(\\mu) \\leq \\int_\\Omega \\zeta \\,\\mathrm d \\nu = \\lim_{\\ell \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_{j_\\ell}) = \\liminf_{j \\to \\infty} \\int_\\Omega \\zeta \\,\\mathrm d G(\\mu_j).\n\\]\n\\end{proof}\n\n\\begin{lemma}\\cite[p.\\ 171]{GS64} \\label{lem: jensen GS}\nLet $U \\subset \\mathbb R^m$ be an open set and $b: U \\to [0, \\infty[$ a~function with $\\int_U b \\,\\mathrm d \\mathcal L^m \\neq 0$. For $\\mu \\in M(U, \\mathbb R^d)$, it is true that\n\\[\n\tG \\left( \\frac{\\int_U b \\,\\mathrm d \\mu}{\\int_U b \\,\\mathrm d \\mathcal L^m} \\right) \\leq \\frac{\\int_U b \\,\\mathrm d G\\mu}{\\int_U b \\,\\mathrm d \\mathcal L^m}. \n\\]\n\\end{lemma}\n\n\\begin{lemma} \\label{lem: G weaklystar}\nSuppose that $w \\in L^1(\\Omega) \\cap BV(U)$ for some open set $U \\subsetneq \\Omega$ and let $w^\\varepsilon = w \\ast \\varrho_\\varepsilon$. Then, $G(w_x^\\varepsilon) \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} G(w_x)$ on $U$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe statement follows from a small modification of \\cite[Theorem 2.2]{ambrosiofuscopallara}, which requires the use of Lemma \\ref{lem: jensen GS}.\n\\end{proof}\n\nThe following theorem establishes coincidence of the measure $G(w_x)(\\Omega)$ with the following relaxation, as mentioned in the Introduction.\n\\begin{thm}\\cite[Theorem 5]{GS64}\nLet $G: \\mathbb R^{m \\times n} \\to [0, \\infty[$ be a~convex function of linear growth and $w \\in BV(\\Omega, \\mathbb R^n)$. Then, \n\t\\begin{align*}\n\tG(w_x)(\\Omega)\n\t&= \\inf \\left \\{ \\liminf_{\\ell \\to \\infty} \\int_\\Omega G(w_x^\\ell) \\,\\mathrm d \\mathcal L^m: \\, w^\\ell \\in C^1(\\Omega), \\, w^\\ell \\to w \\text{ in } L^1_{\\mathrm{loc}}(\\Omega) \\right \\}.\n\t\\end{align*}\n\\end{thm}\n\nLastly, let us quote the following fact from measure theory.\n\\begin{lemma} \\label{lem: mu leq nu}\n\tLet $\\mu, \\nu$ be positive Radon measures on an open set $U \\subset \\mathbb R^m$ which satisfy \n\\[\n\t\\int_U \\zeta^2 \\,\\mathrm d \\mu \\leq \\int_U \\zeta^2 \\,\\mathrm d \\nu\n\\]\t\nfor any $\\zeta \\in C_c^1(U)$. Then, $\\mu(E) \\leq \\nu(E) \\text{ for any Borel set } E \\subset U.$\n\\end{lemma}\n\n\\begin{proof}\n\tGiven any open set $V \\subset U$, it is possible to find an increasing family $\\{K_i\\}_i$ of compact sets contained in $V$ whose union equals $V$. For each $i$ choose a~non-negative function $\\zeta_i \\in C_c^1(V)$ with $||\\zeta_i||_\\infty \\leq 1$ which equals $1$ on $K_i$. Consequently, the assumption on $\\mu$ and $\\nu$ yields\n\\begin{equation*}\n\\mu(K_i) \\leq \\int_U \\zeta_i^2 \\,\\mathrm d \\mu \\leq \\int_V \\zeta_i^2 \\,\\mathrm d \\nu \\leq \\nu(V),\n\\end{equation*}\nwhich by continuity of measure $|u_x^s|$ means that $\\mu(V) \\leq \\nu(V)$ for any open $V \\subset U$. Now, if $E$ is an arbitrary Borel set, then\n\\begin{equation*}\n\t\\mu(E) = \\inf{ \\left \\{\\mu(V):\\, V \\text{ open, } E \\subset V \\right \\}} \\leq \\inf{ \\left\\{ \\nu(V):\\, V \\text{ open, } E \\subset V \\right \\}} = \\nu(E),\n\\end{equation*}\nwhich finishes the proof.\n\\end{proof}\n\n\\subsection*{Convex functions}\n\n\\begin{defn} \\label{def: Reshetnyak}\nLet $\\psi: \\mathbb R^n \\to [0, \\infty)$ be a~positively $1$-homogeneous function. It is convex if and only if for any two points $p, q \\in \\mathbb R^n$ it is true that\n\\begin{equation} \\label{ineq: convexity}\n\t\\psi(p + q) \\leq \\psi(p) + \\psi(q).\n\\end{equation}\nFollowing Reshetnyak's seminal paper \\cite{resh}, we will say that a~positively $1$-homogeneous function $\\psi$ is \\emph{strictly convex in the sense of Reshetnyak} if and only if inequality \\eqref{ineq: convexity} turns into equality only if $p = 0$ or $q = tp$ for some $t \\geq 0$.\n\\end{defn}\n\n\\begin{lemma} \\label{lem: strict convexity}\nLet $\\psi: \\mathbb R^n \\to [0, \\infty)$ be positively $1$-homogeneous function which vanishes only at the origin and $g:\\mathbb R \\to \\mathbb R$ be strictly convex and increasing. If $\\psi$ is strictly convex in the sense of Reshetnyak, then $G := g \\circ \\psi$ is strictly convex, i.\\;e., $G(\\lambda p + (1 - \\lambda)q) < \\lambda G(p) + (1 - \\lambda)G(q)$ for any distinct $p, q \\in \\mathbb R^n$. If $\\psi$ is not strictly convex in the sense of Reshetnyak, then $G$ is not strictly convex.\n\\end{lemma}\n\n\\begin{proof}\nChoose any distinct $p, q \\in \\mathbb R^n$ and $\\lambda > 0$ and assume strict convexity (in the sense of Reshetnyak) of $\\psi$. If $p = 0$, then the statement holds since $g$ is strictly convex and $\\psi$ vanishes at the origin (and only there). If $q = tp$ for $t > 0$ and $t \\neq 1$, then $\\psi(q) = t\\psi(p) \\neq \\psi(p)$ and again it suffices to use strict convexity of $g$. At last, for points for which inequality \\eqref{ineq: convexity} is strict, one can write\n\\begin{align*}\n\tG\\left(\\lambda p + (1-\\lambda)q \\right) &< g \\left( \\lambda \\psi(p) + (1- \\lambda) \\psi(q) \\right) \\\\\n\t& \\leq \\lambda g(\\psi(p)) + (1-\\lambda)g(\\psi(q)).\n\\end{align*}\nThe first inequality is strict as $g$ under our assumptions is strictly increasing whereas the second is not as it might happen that $\\psi(p) = \\psi(q)$.\n\nSuppose now that $\\psi$ is not strictly convex (in the sense of Reshetnyak) so that there exist non-zero points $p, q$ such that $p \\neq tq$ for any $t > 0$ for which inequality \\eqref{ineq: convexity} yields equality. The same is true for $1\/2p$ and $1\/2q$, which implies that $\\psi$ has to be affine on the segment $[p, q]$. As a~result, for $\\lambda = (1 + \\psi(p)\/\\psi(q))^{-1}$ it is true that\n\\begin{equation*}\n\t\\psi \\left( p + \\frac{\\psi(p)}{\\psi(q)}q \\right) = \\lambda^{-1} \\psi( \\lambda p + (1 - \\lambda)q) = \\psi(p) + \\psi \\left( \\frac{\\psi(p)}{\\psi(q)}q \\right).\n\\end{equation*}\nTherefore, we have found points $\\tilde{p} = p$, $\\tilde{q} = (\\psi(p)\/\\psi(q)) q$ such that $\\tilde{p} \\neq t\\tilde{q}$, whose $\\psi$-norms coincide and for which inequality \\eqref{ineq: convexity} turns into equality. Consequently, for any $\\lambda \\in (0,1)$\n\\begin{equation*}\n\tG\\left(\\lambda \\tilde{p} + (1-\\lambda)\\tilde{q} \\right) = g \\left( \\lambda \\psi(\\tilde{p}) + (1- \\lambda) \\psi(\\tilde{q}) \\right) = g(\\psi(\\tilde{p})) = \\lambda G(\\tilde{p}) + (1- \\lambda) G(\\tilde{q}),\n\\end{equation*}\nwhat finishes the proof.\n\\end{proof}\n\n\n\\begin{remark} \\label{remark: f(t)\/t}\n\tGiven a~convex and differentiable function $f: [0, \\infty[ \\to [0, \\infty[$ of at most linear growth (with constant $C_f^+$) it is true that\n\t\\begin{equation*}\n\t\tf^\\infty := \\lim_{t \\to \\infty} \\frac{f(t)}{t} = \\lim_{t \\to \\infty} f'(t) \\leq C_f^+.\n\t\\end{equation*}\nMoreover, convolution of such $f$ with a~standard mollifier yields a~function which is of at most linear growth with the same constant. This inequality also shows that such $f$ is Lipschitz, so mollifications of $f$ converge uniformly to $f$ (and not only locally). For any convex $\\psi: \\mathbb R^n \\to \\mathbb R$, its mollification $\\psi_\\eta$ with a~radially symmetric mollifier gives $\\psi_\\eta \\geq \\psi$.\n\\end{remark}\n\nGiven $f: [0, +\\infty[ \\to [0, +\\infty[$, set\n\\begin{equation} \\label{eq: f eta}\n\tf_\\eta(t) = \\left( f(t) \\indi_{\\{t>\\eta\\}} + f(\\eta)\\indi_{\\{-\\eta \\leq t \\leq \\eta\\}} \\right) \\ast \\varrho_{\\frac{\\eta}{2}}.\n\\end{equation}\nTo construct a~suitable sequence approximating a~given anisotropy $\\varphi$ we use a~slightly modified mollification procedure which retains homogeneity and which has been described by Schneider in his monograph on convex bodies \\cite{schneider}.\n\n\\begin{lemma} \\label{lem: schneider} \\cite[Theorem 3.3.1]{schneider}\n\tTake $\\varrho(p) := \\rho(|p|)$ with $\\rho: [0, +\\infty[ \\to [0, +\\infty[$ being a~smooth function with support contained in $[1\/2,1]$ such that $\\int_{\\mathbb R^n} \\rho(|z|) \\,\\mathrm d z = 1$. For any anisotropy $\\varphi$, the function\n\t\\begin{equation}\\label{goodnorm_eq}\n\t\t\\overline{\\varphi}_\\eta(p) := \\int_{\\mathbb R^n} \\varphi(p + |p|z) \\varrho_\\eta(z) \\,\\mathrm d z\n\t\\end{equation}\n\tis an anisotropy of class $C^{\\infty}(\\mathbb R^n \\setminus \\{0\\})$.\n\\end{lemma}\n\n\\begin{prop}\\label{goodnorm}\n\tLet $\\varphi$ be an anisotropy on $\\mathbb R^n$. There exists a~sequence of anisotropies $\\{ \\widetilde{\\varphi}_\\eta \\}_\\eta$ with the following properties\n\t\\begin{enumerate}[(i)]\n\t\t\\item $\\widetilde{\\varphi}_\\eta \\in C^\\infty(\\mathbb R^n \\setminus \\{0\\})$ and $D^2 \\left( \\frac{1}{2} \\widetilde{\\varphi}_\\eta^{\\,2} \\right) \\geq \\eta \\mathrm{Id}$ in $\\mathbb R^n \\setminus \\{0\\}$;\n\t\t\\item $\\widetilde{\\varphi}_\\eta \\geq \\varphi$ and $\\widetilde{\\varphi}_\\eta$ converge to $\\varphi$ locally uniformly on $\\mathbb R^n$.\n\t\\end{enumerate}\n\tAssume additionally that $\\varphi \\in C^{1}(\\mathbb R^n \\setminus \\{0\\})$. Then,\n\t\\begin{enumerate}[(i)]\n\t\t\\setcounter{enumi}{2}\n\t\t\\item $D \\widetilde{\\varphi}_\\eta$ converges to $D \\varphi$ uniformly on $\\mathbb R^n \\setminus \\{0\\}$.\n \\end{enumerate}\n If $f: [0, +\\infty[ \\to [0, +\\infty[$ is strictly convex, increasing and differentaible with $f'(0) = 0$,\n \\begin{enumerate}[(i)]\n \\setcounter{enumi}{3}\n\t\t\\item $D(f_\\eta \\circ \\widetilde{\\varphi}_\\eta)$ converges to $D(f \\circ \\varphi)$ uniformly on $\\mathbb R^n$, where $f_\\eta$ is defined in \\eqref{eq: f eta}.\n\t\\end{enumerate}\t\n\\end{prop}\nAnisotropies which satisfy condition (i) are often referred to as anisotropies of class $C^\\infty_+$ (see \\cite{schneider}, \\cite{BellettiniCasellesNovaga2005}).\n\n\\begin{proof}\n\tWe define\n\t\\[\\widetilde{\\varphi}_\\eta(p) = \\sqrt{\\overline{\\varphi}_\\eta^2(p) + \\eta |p|^2}, \\] \n\twhere $\\overline{\\varphi}_\\eta$ is given by \\eqref{goodnorm_eq}. Condition (i) is clearly satisfied whereas $\\widetilde{\\varphi}_\\eta \\geq \\varphi$ follows from the fact that $\\overline{\\varphi}_\\eta \\geq \\varphi$. Indeed, due to homogeneity, it suffices to check it only for $p$ from the unit sphere, for which \\eqref{goodnorm_eq} amounts to the ordinary convolution for which this property can be easily shown to be true. Local uniform convergence of $\\widetilde{\\varphi}_\\eta$ to $\\varphi$ holds since $\\overline{\\varphi}_\\eta$ converges to $\\varphi$ locally uniformly as well.\n\t\n\tThe rest of the proof is devoted to the regular case. The key to uniform convergence of derivatives is the fact that derivative of positively $1$-homogeneous function is $0$-homogeneous, whence it is enough to check it over the unit sphere. Passing with the differentiation under the integral sign gives for $|p| = 1$\n\t\\begin{equation*}\n\t\tD \\overline{\\varphi}_\\eta (p) = \\int_{\\mathbb R^n} D\\varphi(p + z) \\left( \\mathrm{Id} + z \\cdot p \\otimes p \\right) \\varrho_\\eta(|z|) \\,\\mathrm d z.\n\t\\end{equation*}\n\tUniform continuity of $D\\varphi$ on compact sets implies that $D \\overline{\\varphi}_\\eta$ converges uniformly to $D\\varphi$. Then, simple calculations lead to the conclusion that property (iii) is satisfied.\n\t\n\tThe proof of (iv) starts with showing that $f' \\circ \\widetilde{\\varphi}_\\eta$ converges to $f'\\ \\circ \\varphi$ uniformly on $\\mathbb R^n$. Recall that for any $\\delta$ there exists an $M$ such that if only $t \\geq M$, then $f^\\infty - f'(t) \\leq \\delta$. Fix any $\\delta > 0$, choose $M$ accordingly and take any $p \\in \\mathbb R^n$. On the compact ball $\\{p: \\, \\varphi(p) \\leq M\\}$ functions $\\widetilde{\\varphi}_\\eta$ converge uniformly to $\\varphi$, i.e, there exists $\\eta_0$ such that for all $\\eta \\leq \\eta_0$ and $p$ with $\\varphi(p) \\leq M$ it is true that\n\t\\[\n\t|\\widetilde{\\varphi}_\\eta(p) - \\varphi(p)| \\leq \\delta.\n\t\\]\n\tWhenever $\\varphi(p) \\geq M$, then by (ii) $\\widetilde{\\varphi}_\\eta(p) \\geq M$ and as a~result, after adding and subtracting $f^\\infty$, we get\n\t\\begin{equation*}\n\t\t| f'(\\widetilde{\\varphi}_\\eta(p)) - f'(\\varphi(p)) | \\leq 2\\delta,\n\t\\end{equation*}\n\twhich shows the desired claim.\n\t\n\tClearly, $D(f_\\eta \\circ \\widetilde{\\varphi}_\\eta)(0) = D(f \\circ \\varphi)(0) = 0$. Take any $p \\neq 0$ and observe that\n\t\\begin{equation*}\n\t\t|D(f_\\eta \\circ \\widetilde{\\varphi}_\\eta)(p) - D(f \\circ \\varphi)(p)| \\leq f^\\infty |D\\widetilde{\\varphi}_\\eta(p) - D\\varphi(p)| + |f'_\\eta(\\widetilde{\\varphi}_\\eta(p)) - f'(\\varphi(p))||D\\varphi(p)|.\n\t\\end{equation*} \n\tBy (iii), it is evident that the first term converges uniformly to zero. Moreover, $|D\\varphi|$ is bounded. Adding and subtracting $f'(\\widetilde{\\varphi}_\\eta(p))$ in the second term together with the claim established in the previous paragraph and the fact that $f'_\\eta$ converge uniformly to $f'$ allows us to conclude that the second term converges uniformly to zero as well.\n\\end{proof}\n\n\\begin{lemma} \\label{lem: smooth phi sqr}\n\tThere exists a convex function $\\widetilde{\\widetilde{\\varphi}}_\\eta \\in C^2(\\mathbb R^n)$, which equals $\\widetilde{\\varphi}_\\eta^2$ outside $B_\\eta(0)$.\n\\end{lemma}\n\n\\begin{proof}\n\tTo construct such a~function, it is sufficient to mollify $\\widetilde{\\varphi}_\\eta^2$ with $\\varrho_{\\varepsilon}$ for a~sufficiently small $\\varepsilon$ and glue it to $\\widetilde{\\varphi}_\\eta^2$ along a~small annulus in $B_\\eta(0)$. Convexity is achieved thanks to the fact that $D^2\\widetilde{\\varphi}_\\eta^2 \\geq \\eta \\mathrm{Id}$ on the boundary of the ball $B_\\eta(0)$ and the appropriate choice of $\\varepsilon$, as proved in \\cite[Section 2]{ghomi}.\n\\end{proof}\n\n\\subsection*{Spherical compactification}\n\\begin{defn}\nLet $\\overline{\\mathbb R^n} := \\mathbb R^n \\cup \\mathbb{S}^{n-1}_\\infty$, where $\\mathbb{S}^{n-1}_\\infty := \\{ \\infty \\, \\omega: \\, \\omega \\in \\mathbb{S}^{n-1}_1 \\}$. We call $\\overline{\\mathbb R^n}$ the spherical compactification of $\\mathbb R^n$. Furthermore, define $\\Phi: \\overline{\\mathbb R^n} \\to \\overline{\\mathbb{B}^n_1}$ with the formula\n\\begin{equation*}\n\t\\Phi(p) = \\begin{cases}\n\t\t\\frac{p}{1 + |p|} & \\text{if } p \\in \\mathbb R^n, \\\\\n\t\t\\omega & \\text{if } p = \\infty \\omega \\in \\mathbb{S}^{n-1}_\\infty.\n\t\\end{cases}\n\\end{equation*}\nThe space $\\overline{\\mathbb R^n}$ can be equipped with the metric\n\\begin{equation*}\n\t\\overline{d}(p, q) := \\lvert \\Phi(p) - \\Phi(q) \\rvert.\n\\end{equation*}\n\\end{defn}\n\nThe space $\\overline{\\mathbb R^n}$ with topology induced by the norm $\\overline{d}$ is indeed compact since $\\Phi$ is a~homeomorphism between this space and the closed unit ball with the Euclidean subspace topology. Clearly, any point $p$ in $\\mathbb R^n$ can be uniquely represented as $q = r \\omega$ with $\\omega \\in \\mathbb{S}^{n-1}_\\infty$. If one looks at a~sequence of points $p_j = r_j \\omega_j$, it is evident that it converges to a~point $p = \\infty \\omega \\in \\mathbb{S}^{n-1}_\\infty$ if and only if $r_j \\to +\\infty$ and $\\omega_j \\to \\omega$. On the other hand, if a~sequence of points $p_j$ converges in $\\overline{d}$ to a~point $p \\in \\mathbb R^n$, then for all but a~finite number of $j$, $p_j$ lies in $\\mathbb R^n$ as well. Moreover, uniform continuity of $\\Phi^{-1}$ implies that in such case $|p_j - p| \\to 0$. Additionally, since $\\Phi$ is $1$-Lipschitz, a~sequence of points $p_j \\in \\mathbb R^n$ which converges in Euclidean distance to $p \\in \\mathbb R^n$, is also convergent to $p$ in $\\overline{\\mathbb R^n}$ with metric $\\overline{d}$.\n\n\\begin{lemma} \\label{lem: injective on spheres}\nLet $\\varphi: \\mathbb R^n \\to \\mathbb R$ be a~$C^1(\\mathbb R^n \\setminus \\{0\\})$ strictly convex (in the sense of Reshetnyak) anisotropy. Then for any $r>0$ the mapping $D\\varphi: \\{ p\\in \\mathbb R^n: \\, \\varphi(p) = r\\} \\to \\mathbb R^n $ is injective.\n\\end{lemma}\n\n\\begin{proof}\nThe function $\\varphi^2$ is $C^1$ on the whole $\\mathbb R^n$ and is strictly convex as shown in Lemma \\ref{lem: strict convexity}. Therefore, $D\\varphi^2$ is strictly monotone and hence injective. For any $p \\neq q$ such that $\\varphi(p) = \\varphi(q) = r$ for some positive $r > 0$ it is then true that\n\\[\n\t2 r D\\varphi(p) \\neq 2 r D\\varphi(q),\n\\]\nwhich means that $D\\varphi(p) \\neq D \\varphi(q)$.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem: DF homeo}\nThe mapping $\\overline{DF}: \\overline{R^n} \\to \\overline{R^n}$ defined as\n\\begin{equation*}\n\t\\overline{DF} = \\begin{cases}\n\t\tDF(p) & \\text{if } p \\in \\mathbb R^n, \\\\\n\t\tf^\\infty D\\varphi(\\omega) & \\text{if } p = \\infty \\omega \\in \\mathbb{S}^{n-1}_\\infty\n\t\\end{cases}\n\\end{equation*}\nis a homeomorphism.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\overline{\\mathbb R^n}$ is compact, it suffices to prove that $\\overline{DF}$ is continuous and injective. Continuity is easily established by considering three cases. Firstly, take a~point $p_0 \\in \\mathbb R^n$ and any sequence $p_n$ converging to $p_0$ in metric $\\overline{d}$. Then, necessarily, almost all $p_n$ lie in $\\mathbb R^n$ as well, so continuity at such a~point follows from continuity of $DF$. Secondly, if one approaches $p_0 = \\infty \\omega_0 \\in \\mathbb{S}_\\infty^{n-1}$ with a~sequence of points in $R^n$, then writing $p_n = r_n \\omega_n$ gives\n\\[\n\tDF(p_n) = f'(r_n \\varphi(\\omega_n))D\\varphi(\\omega_n) \\to f^\\infty D\\varphi(\\omega_0) = \\overline{DF}(p_0),\n\\]\nas $\\{\\varphi(\\omega_n)\\}_n$ is bounded and $r_n \\to +\\infty$, and $D\\varphi$ is continuous on the unit sphere. Lastly, if one approaches such a~point with a~sequence of points lying on $\\mathbb{S}_{n-1}^\\infty$, then continuity of $\\overline{DF}$ follows directly from continuity of $D\\varphi$ on the unit sphere.\n\nTo prove injectivity, observe that $F$ being strictly convex implies $DF$ being strictly monotone, which means that $DF: \\mathbb R^n \\to \\mathbb R^n$ is one-to-one. Therefore, $\\overline{DF}(p) \\neq \\overline{DF}(q)$ for any distinct $p, q \\in \\mathbb R^n$. In view of Lemma \\ref{lem: injective on spheres}, it suffices to check that $\\overline{DF}(p) \\neq \\overline{DF}(q)$ for $p \\in \\mathbb R^n$ and $q \\in \\mathbb{S}_\\infty^{n-1}$. To this end, let us show that for any $p \\in \\mathbb R^{n}\\setminus \\{0\\}$ it is true that\n\\begin{equation} \\label{dual norm derivative}\n\t\\varphi^* \\left(D\\varphi(p) \\right) = 1, \\text{ where } \\varphi^*(p) := \\sup \\left \\{ p \\cdot q: \\, \\varphi(q) \\leq 1 \\right \\}.\n\\end{equation}\nIndeed, taking $q = p\/\\varphi(p)$ gives $\\varphi^*\\left(D\\varphi(p)\\right) \\geq 1$. On the other hand, the product $q \\cdot D\\varphi(p)$ equals the derivative of $\\varphi$ in direction $q$ and hence by triangle inequality one gets\n\\begin{equation*}\n\t\\varphi^*\\left( D\\varphi(p) \\right) = \\lim_{t \\to 0^+} \\frac{1}{t} \\left( \\varphi(p + tq) - \\varphi(p) \\right) \\leq \\varphi(q) \\leq 1.\n\\end{equation*}\n\nAssume now that $DF(p) = f^\\infty D\\varphi(\\omega)$ for some $p \\in \\mathbb R^n \\setminus \\{0\\}$ and $\\omega \\in \\mathbb{S}^{n-1}_1$. Applying $\\varphi^*$ to both sides and using its positive homogeneity results in\n\\begin{equation*}\n\tf'(\\varphi(p)) \\varphi^* \\left( D \\varphi(p) \\right) = f^\\infty \\varphi^*\\left( D\\varphi(\\omega) \\right).\n\\end{equation*}\nGiven \\eqref{dual norm derivative}, one gets $f'(\\varphi(p)) = f^\\infty$, which contradicts strict convexity of $f$. Therefore, $\\overline{DF}$ is injective.\n\\end{proof}\n\n\\section{Approximate functionals} \\label{sec: approximate}\n\nSuppose $h \\in L^2(I) \\cap BV(U, \\mathbb R^n)$ for an open interval $U \\subset I$ and extend $h$ by zero over a~slightly larger than $I$ interval in order to properly define mollifications of $h$. We set\n\\begin{equation}\n\th^\\varepsilon = h \\ast \\varrho_\\varepsilon.\n\\end{equation}\nWe suppose that $F_{\\eta}$ is given either by \n\\begin{equation} \\label{Feta1def}\n\tF_\\eta = f_\\eta \\circ \\overline{\\varphi}_\\eta + \\frac{\\eta}{2}|\\cdot|^2 \n\\end{equation} \nor \n\\begin{equation} \\label{Feta2def}\n\tF_\\eta = f_\\eta \\circ \\widetilde{\\varphi}_\\eta + \\frac{\\eta}{2} \\widetilde{\\widetilde{\\varphi}}_\\eta.\n\\end{equation}\nIn \\eqref{Feta1def}, \\eqref{Feta2def} we have used $f_\\eta$, $\\overline{\\varphi}_\\eta$, $\\widetilde{\\varphi}_\\eta$ and $\\widetilde{\\widetilde{\\varphi}}_\\eta$ defined in \\eqref{eq: f eta}, Lemma \\ref{lem: schneider}, Proposition \\ref{goodnorm} and Lemma \\ref{lem: smooth phi sqr} for given anisotropy $\\varphi \\colon \\mathbb R^n \\to [0, \\infty[$ and convex, increasing $f \\colon [0, \\infty[ \\to [0, \\infty[$ of linear growth. Observe that $F_\\eta$ is smooth and that $D^2 F_\\eta \\geq \\eta \\mathrm{Id}$. Moreover, since $\\widetilde{\\varphi}_\\eta \\geq \\varphi$ and $f_\\eta \\geq f$ due to convexity of $f$, it is true that $F_\\eta \\geq F$.\n\nWe will use a two-layer approximation of $\\mathcal E$: \n\\begin{align*}\n\t\\mathcal E_{\\varepsilon, \\eta} (w) &= \\lambda \\int_I F_\\eta(w_x) \\,\\mathrm d \\mathcal L^1 + \\frac{1}{2} \\int_I |w - h^{\\varepsilon} |^2 \\,\\mathrm d \\mathcal L^1 \\text{ for } w \\in W^{1,2}(I, \\mathbb R^n) \\text{ with mimimizer } u^{\\varepsilon, \\eta}, \\\\\n\t\\mathcal E_\\varepsilon(w) &= \\lambda \\int_I F(w_x) \\,\\mathrm d \\mathcal L^1 + \\frac{1}{2} \\int_I |w-h^\\varepsilon|^2 \\,\\mathrm d \\mathcal L^1 \\text{ for } w \\in L^2(I, \\mathbb R^n) \\text{ with mimimizer } u^{\\varepsilon}. \n\\end{align*}\nObserve that $F_\\eta$ and $F$ are strictly convex, non-negative, weakly lower semicontinuous in $W^{1,2}$ and $L^2$, respectively. Moreover the sublevels of $\\mathcal E_{\\varepsilon, \\eta}$, $\\mathcal E_\\varepsilon$ are weakly compact in $W^{1,2}$ and $L^2$, respectively. Consequently, there indeed exist unique minimizers of these functionals.\n\n\\begin{lemma} \\label{lem: gamma conv eta}\n\tFunctionals $\\mathcal E_{\\varepsilon, \\eta}$ $\\Gamma$-converge to $\\mathcal E_{\\varepsilon}$ w.\\,r.\\,t.\\;weak $W^{1,2}$ convergence.\n\\end{lemma}\n\\begin{proof}\n\tAs $F_\\eta \\geq F$, then $\\mathcal E_{\\varepsilon, \\eta} \\geq \\mathcal E_\\varepsilon$, whence lower semicontinuity follows from a~standard result \\cite[Chapter 8.2, Theorem 1]{evans}.\n\tAs for the recovery sequence, choose $w_{\\eta} = w$, then one only needs to show that\n\t\\begin{equation} \\label{f eta limit}\n\t\t\\lim_{\\eta \\to 0}\\int_I F_\\eta(w_x) \\,\\mathrm d \\mathcal L^1 = \\int_I F(w_x) \\,\\mathrm d \\mathcal L^1.\n\t\\end{equation}\n\tSince $\\overline{\\varphi}$ (resp.\\ $\\widetilde{\\varphi}_\\eta$) is an anisotropy and $f_\\eta$ is of at most linear growth, we have $f_\\eta \\circ \\overline{\\varphi} (w_x) \\leq C( 1 + |w_x|)$ (resp.\\ $f_\\eta \\circ \\widetilde{\\varphi}_\\eta (w_x) \\leq C( 1 + |w_x|)$), which is integrable over $I$, implying that dominated convergence theorem can be applied. Additionally, we have\n\t\\begin{equation*}\n\t\t\\lim_{\\eta \\to 0} \\frac{\\eta}{2} \\int_I \\widetilde{\\widetilde{\\varphi}}_\\eta (w_x) \\,\\mathrm d \\mathcal L^1 \\leq \\lim_{\\eta \\to 0} \\frac{\\eta}{2} \\int_I C (1 + |w_x|^2) \\,\\mathrm d \\mathcal L^1 = 0,\n\t\\end{equation*}\n\twhich proves \\eqref{f eta limit} and concludes the proof of the lemma.\n\\end{proof}\n\n\\begin{lemma} \\label{lem: gamma conv eps 1}\n\tFunctionals $\\mathcal E_\\varepsilon$ $\\Gamma$-converge to $\\mathcal E$ w.\\,r.\\,t\\;weak convergence in $L^2$.\n\\end{lemma}\n\\begin{proof}\n\tSince $h^\\varepsilon$ converge strongly to $h$ in $L^2(I, \\mathbb R^n)$, for each $w \\in BV(I, \\mathbb R^n)$ there exists a~trivial recovery sequence. \n\t\n\tThen, take $w \\in BV(I, \\mathbb R^n)$ and a~sequence $w^\\varepsilon$ converging to $w$ weakly in $L^2$. Since the norm is lower semicontinuous w.\\,r.\\,t.\\;weak convergence,\n\t\\begin{equation*}\n\t\t\\liminf_{\\varepsilon \\to 0} \\int_I |w^\\varepsilon - h^\\varepsilon|^2 \\,\\mathrm d \\mathcal L^1 \\geq \\int_I |w - h|^2 \\,\\mathrm d \\mathcal L^1. \n\t\\end{equation*}\n\t\n\tLower semicontinuity of $\\mathcal F$ with respect to strong $L^2(I, \\mathbb R^n)$ convergence follows from the definition of $\\mathcal F$, as $L^2$ convergence is stronger than $L^1_{loc}$ convergence. Since $\\mathcal F$ is convex, it is also lower semicontinuous with respect to weak $L^2$ convergence.\n\\end{proof}\n\nSince $\\mathcal E_{\\varepsilon, \\eta}$ is smooth, the minimizer admits weak second derivatives and satisfies the following Euler-Lagrange equation\n \\begin{equation} \\label{Euler-Lagrange-eps-eta}\n\tu^{\\varepsilon, \\eta} - h^\\varepsilon = \\lambda \\left(DF_\\eta(u_x^{\\varepsilon, \\eta})\\right)_x \\text{ in } I, \\quad u_x^{\\varepsilon, \\eta} = 0 \\text{ on } \\partial I.\n\\end{equation}\nDue to the fact that $D^2F_\\eta \\geq \\eta \\mathrm{Id}$, we gather that $u^{\\varepsilon, \\eta} \\in W^{2,2}(I, \\mathbb R^n)$, which in particular means that $u^{\\varepsilon, \\eta} \\in C^{1}(\\bar{I}, \\mathbb R^n)$.\n\n\\begin{lemma} \\label{lem: cpt eta}\nAs $\\eta \\to 0$, the sequence of minimizers $u^{\\varepsilon, \\eta}$ converges strongly in $L^2(I, \\mathbb R^n)$ and weakly in $W^{1,2}(I, \\mathbb R^n)$ to the minimizer $u^\\varepsilon$.\n\\end{lemma}\n\\begin{proof}\nFirstly, let us prove that the sequence of minimizers $u^{\\varepsilon, \\eta}$ is bounded in $W^{1,2}$. The fact that $\\mathcal E_{\\varepsilon, \\eta}(u^{\\varepsilon, \\eta}) \\leq \\mathcal E_{\\varepsilon, \\eta}(0)$ together with triangle inequality for $L^2$ norm show that $u^{\\varepsilon, \\eta}$ are bounded in $L^2$. For the boundedness of the derivatives, let us multiply the Euler-Lagrange equation by $u_{xx}^{\\varepsilon, \\eta}$ and integrate the l.\\,h.\\,s.\\;by parts. The r.\\,h.\\,s\\;is non-negative due to convexity of $F_\\eta$ and the boundary term vanishes, hence\n\t\\begin{equation*}\n\t\t\\int_{I} |u_x^{\\varepsilon, \\eta}|^2 \\,\\mathrm d \\mathcal L^1 \\leq \\int_I u_x^{\\varepsilon, \\eta} \\cdot h_x^\\varepsilon \\,\\mathrm d \\mathcal L^1.\n\t\\end{equation*}\nApplying Young's inequality gives the desired bound for $L^2$ norm of the sequence of derivatives of minimizers.\n\nTherefore, from any subsequence $u^{\\varepsilon, \\eta}$ it is possible to choose a~weakly convergent subsequence $u^{\\varepsilon, \\eta_j}$. In view of the $\\Gamma$-convergence proved in Lemma \\ref{lem: gamma conv eta}, the weak limit of $\\{u^{\\varepsilon, \\eta_j}\\}_j$ must equal the minimizer $u^\\varepsilon$. A standard application of Rellich-Kondrashov theorem allows us to choose a~further subsequence (which we do not relabel) which additionally converges strongly in $L^2$ to $u^\\varepsilon$. Therefore, a~properly convergent subsequence can be chosen from arbitrary subsequence of $\\{u_x^\\varepsilon\\}_\\varepsilon$, which shows the desired convergence of the whole sequence.\n\\end{proof}\n\n\\begin{lemma} \\label{lem: gamma conv eps}\nAs $\\varepsilon \\to 0$, the minimizers $u^\\varepsilon$ converge weakly in $L^2(I, \\mathbb R^n)$ to the minimizer~$u$. If $F$ is coercive (i.e., satisfies \\eqref{coercivity}), then the sequence $u^{\\varepsilon, \\eta}$ converges strongly in $L^2(I, \\mathbb R^n)$ and $u_x^\\varepsilon \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} u_x$. \n\\end{lemma}\n\\begin{proof}\n To see that $u^\\varepsilon$ are bounded in $L^2$, it suffices to write $\\mathcal E_\\varepsilon(u^\\varepsilon) \\leq \\mathcal E_\\varepsilon(0)$ and use triangle inequality for $L^2$ norm. Consequently, from every sequence $u^{\\varepsilon_i}$ it is possible to choose a~further subsequence which, by $\\Gamma$-convergence proved in Lemma \\ref{lem: gamma conv eps 1}, converges weakly in $L^2$ to $u$.\n\n If $F$ is coercive, one can use it to bound $\\int_I F(u^{\\varepsilon_i}_x)$ from below to see that $\\{u^{\\varepsilon_i}\\}_i$ is bounded in $BV$, which implies an existence of a~subsequence strongly convergent in $L^2$ (see \\cite[Corollary 3.49]{ambrosiofuscopallara}) such that the derivatives converge weakly* as measures. \n\\end{proof}\n\n\n\\section{The homogeneous case} \\label{sec:homo}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:homo}]\nLet \n\\begin{equation} \n\tF_\\eta:= f_\\eta \\circ \\overline{\\varphi}_\\eta + \\frac{\\eta}{2} |\\cdot|^2, \n\\end{equation} \nwhere $f = |\\cdot|$ is the absolute value function. We calculate \n\\[\nDF_{\\eta}(p) = f_\\eta'(\\overline{\\varphi}_\\eta(p)) D \\overline{\\varphi}_\\eta(p) + \\eta p, \n\\]\n\\[\n\\left(DF_\\eta(u_x^{\\varepsilon, \\eta})\\right)_x = f_\\eta'(\\overline{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta})) D^2 \\overline{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) u_{xx}^{\\varepsilon, \\eta} + f_\\eta''(\\overline{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta})) \\overline{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta})_x D \\overline{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) + \\eta\\, u_{xx}^{\\varepsilon, \\eta}. \n\\]\nTesting the Euler-Lagrange equation \\eqref{Euler-Lagrange-eps-eta} with $u^{\\varepsilon, \\eta}_{xx}$, integrating by parts on the l.\\,h.\\,s.\\;and using Cauchy's inequality we obtain a uniform estimate\n\\begin{equation} \\label{homo_eta_unif_bd} \n\\int_I |u_x^{\\varepsilon, \\eta}|^2 + 2 \\lambda \\eta \\int_I |u_{xx}^{\\varepsilon, \\eta}|^2 \\leq \\int_I |h_x^{\\varepsilon}|^2 .\n\\end{equation} \n\nWe define \n\\begin{equation} \n\t\\mathfrak G_k (p) = (|p|-k)_+, \\quad \\mathfrak G_{k, \\delta} (p) = \\sqrt{(|p|-k)_+^2 + \\delta^2}.\n\\end{equation} \nTherefore, \n\\[\nD \\mathfrak G_{k, \\delta} (p) = \\frac{(|p|-k)_+}{\\sqrt{(|p|-k)_+^2 + \\delta^2}} \\frac{p}{|p|}\n\\] \nand, given $\\zeta \\in C_c^1(U)$ \n\\begin{multline*}\n\t\\left(D \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}) \\zeta^2\\right)_x = \\frac{(|u_x^{\\varepsilon, \\eta}|-k)_+}{\\sqrt{(|u_x^{\\varepsilon, \\eta}|-k)_+^2 + \\delta^2}} \\frac{1}{|u_x^{\\varepsilon, \\eta}|}\\left(I - \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|} \\otimes \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|}\\right) u_{xx}^{\\varepsilon, \\eta} \\zeta^2 \\\\ + \\frac{\\delta^2 \\indi_{|u_x^{\\varepsilon, \\eta}|>k }}{\\sqrt{(|u_x^{\\varepsilon, \\eta}|-k)_+^2 + \\delta^2}^3} \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|} \\otimes \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|}\\, u_{xx}^{\\varepsilon, \\eta} \\zeta^2 + \\frac{(|u_x^{\\varepsilon, \\eta}|-k)_+}{\\sqrt{(|u_x^{\\varepsilon, \\eta}|-k)_+^2 + \\delta^2}} \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|} \\, 2 \\zeta \\zeta_x.\n\\end{multline*} \nWe observe that \n\\begin{equation}\\label{homo_est1} \n\t\\int_I \\left(D \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}) \\zeta^2\\right)_x \\cdot u_{xx}^{\\varepsilon, \\eta} \\geq \\int_I 2 \\zeta \\zeta_x \\frac{(|u_x^{\\varepsilon, \\eta}|-k)_+}{\\sqrt{(|u_x^{\\varepsilon, \\eta}|-k)_+^2 + \\delta^2}} \\frac{u_x^{\\varepsilon, \\eta}}{|u_x^{\\varepsilon, \\eta}|} \\cdot u_{xx}^{\\varepsilon, \\eta} \\geq - 2 \\max |\\zeta_x| \\left(\\int_I |u_{xx}^{\\varepsilon, \\eta}|^2 \\right)^{\\frac{1}{2}}. \n\\end{equation} \n\nAssuming that $k > \\frac{2}{c_\\varphi^-} \\eta$, we have $|p| \\leq k$ or $\\overline{\\varphi}_\\eta(p) \\geq 2 \\eta$ for all $p \\in \\mathbb R^n$. In the latter case $f'(\\overline{\\varphi}_\\eta(p)) = 1$ and $f''(\\overline{\\varphi}_\\eta(p)) = 0$. Taking into account \\eqref{homo_est1}, \\eqref{homo_eta_unif_bd} and the observation that \n\\[D^2 \\overline{\\varphi}_\\eta(p) \\cdot p = 0 \\quad \\text{for } p \\neq 0,\\]\nwe get \n\\begin{equation} \\label{homo_est2} \n\t\\int_I \\left(D \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}) \\zeta^2\\right)_x \\cdot \\left(DF_\\eta(u_x^{\\varepsilon, \\eta})\\right)_x \\geq - 2 \\eta \\max |\\zeta_x| \\left(\\int_I |u_{xx}^{\\varepsilon, \\eta}|^2 \\right)^{\\frac{1}{2}} \\geq - \\frac{\\sqrt{2\\eta}}{\\sqrt{\\lambda}} \\max |\\zeta_x|\\left(\\int_I |h_x^\\varepsilon|^2 \\right)^{\\frac{1}{2}}. \n\\end{equation} \nOn the other hand, by convexity of $\\mathfrak G_{k, \\delta}$, \n\\begin{multline} \\label{homo_rhs_est} \n\\int_I \\left(D \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}) \\zeta^2\\right)_x \\cdot (u^{\\varepsilon, \\eta} - h^\\varepsilon) = \\int_I \\zeta^2 D \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}) \\cdot (h^\\varepsilon_x - u^{\\varepsilon, \\eta}_x) \\\\ \\leq \\int_I \\zeta^2 \\mathfrak G_{k, \\delta} (h_x^\\varepsilon) - \\int_I \\zeta^2 \\mathfrak G_{k, \\delta} (u_x^{\\varepsilon, \\eta}). \n\\end{multline} \nCollecting \\eqref{homo_est2} and \\eqref{homo_rhs_est} and passing to the limit $\\delta \\to 0^+$ using dominated convergence, we infer that \n\\begin{equation} \\label{homo_est3} \n\t \\int_I \\zeta^2 \\mathfrak G_k (u_x^{\\varepsilon, \\eta}) \\leq \\int_I \\zeta^2 \\mathfrak G_k (h_x^\\varepsilon) + \\frac{\\sqrt{2\\eta}}{\\sqrt{\\lambda}} \\max |\\zeta_x|\\left(\\int_I |h_x^\\varepsilon|^2 \\right)^{\\frac{1}{2}}.\n\\end{equation}\nAs shown in Lemma \\ref{lem: cpt eta}, $u_x^{\\varepsilon, \\eta}$ converge weakly in $L^2$ to $u_x^{\\varepsilon}$ as $\\eta \\to 0^+$. Therefore, by lower semicontinuity of $w_x \\mapsto \\int \\mathfrak G_k(w_x) \\,\\mathrm d \\mathcal L^1$, inequality \\eqref{homo_est3} yields\n\\begin{equation} \\label{homo_est4} \n\t\\int_I \\zeta^2 \\mathfrak G_k (u_x^\\varepsilon) \\leq \\int_I \\zeta^2 \\mathfrak G_k (h_x^\\varepsilon).\n\\end{equation}\n\nWhat is more, Lemma \\ref{lem: mu leq nu} implies that $\\mathfrak{G}_k(u_x^\\varepsilon)(U) \\leq \\mathfrak G_k(h_x^\\varepsilon(U))$. For any $w \\in BV(I, \\mathbb R^n)$ \\[ |w_x| \\leq \\mathfrak{G}_k(w_x) + k \\leq |w_x| + k\\]\nas measures, hence, using the fact that the sequence $\\mathfrak G_k(h_x^\\varepsilon)$ is weakly* convergent by Lemma \\ref{lem: G weaklystar} (and therefore bounded), we deduce that $u^\\varepsilon$ is uniformly bounded in $BV(U, \\mathbb R^n)$. Since by Lemma \\ref{lem: gamma conv eps} we already know that $u^\\varepsilon$ converge weakly in $L^2$ to $u$, it can be inferred that $u \\in BV(U, \\mathbb R^n)$ and $u_x^\\varepsilon \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} u_x$. As a~result, Lemmata \\ref{cor-gs-lsc-zeta} and \\ref{lem: G weaklystar} mean that the following inequality holds\n\\[\n \\int_U \\zeta^2 \\,\\mathrm d \\mathfrak{G}_k(u_x) \\leq \\int_U \\zeta^2 \\mathfrak G_k(h_x).\n\\]\nEmploying Lemma \\ref{lem: mu leq nu} again goes to show that\n\\begin{equation*} \n\t\\mathfrak{G}_k (u_x) \\leq \\mathfrak{G}_k (h_x) \n\\end{equation*}\nas measures or, in other words,\n\\begin{equation*} \n\t\\mathfrak |u_x^s|(V) + \\int_V (|u_x^{ac}|-k)_+ \\,\\mathrm d \\mathcal L^1 \\leq |h_x^s|(V) + \\int_V (|h_x^{ac}|-k)_+ \\,\\mathrm d \\mathcal L^1 \n\\end{equation*}\nfor any $V$ Borel. Passing to the limit $k\\to 0^+$ we obtain the desired assertion.\n\\end{proof}\n\n\\section{The coercive case} \\label{sec:coercive} \n\nLet us now multiply both sides of the Euler-Lagrange equation \\eqref{Euler-Lagrange-eps-eta} by $(\\zeta^2 G_k(u_x^{\\varepsilon, \\eta}))_x$, where $\\zeta \\in C_c^1(U)$ and\n\\begin{equation*}\n\tG_k(p) = g_k \\circ \\tilde{\\varphi}_\\eta(p)p \\text{ and } g_k(\\sigma) = \\frac{(\\sigma - k)_+}{\\sigma^2} f'_\\eta(\\sigma)\n\\end{equation*}\nObserve that $G_k(p)$ is bounded since $f_\\eta$ is of linear growth and $\\widetilde{\\varphi}_\\eta$ is a~coercive anisotropy. After integrating and differentiating by parts on the l.\\,h.\\,s. we get\n\\begin{equation} \\label{eq: after mult}\n\t\\frac{1}{\\lambda} \\int_I \\zeta^2 G_k(u_x^{\\varepsilon, \\eta}) \\cdot (u_x^{\\varepsilon, \\eta} - h_x^{\\varepsilon}) \\,\\mathrm d \\mathcal L^1 = - \\int_I (\\zeta^2 G_k(u_x^{\\varepsilon, \\eta}))_x \\cdot D^2 F_\\eta (u_x^{\\varepsilon, \\eta}) \\cdot u_{xx}^{\\varepsilon, \\eta} \\,\\mathrm d \\mathcal L^1.\n\\end{equation}\n\n\n\\begin{prop} \\label{prop: general lhs eps}\n\tThe l.\\,h.\\,s.\\;of \\eqref{eq: after mult} satisfies\n\\begin{align*}\n\t\\liminf_{\\varepsilon \\to 0} \\liminf_{\\eta \\to 0} & \\int_I \\zeta^2 G_k(u_x^{\\varepsilon, \\eta}) \\cdot (u_x^{\\varepsilon, \\eta} - h_x^{\\varepsilon}) \\,\\mathrm d \\mathcal L^1 \\geq \\\\\n\t& \\frac{r(k)}{ (c_\\varphi^+)^{3}} \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^- |u_x| - k)_+(c_\\varphi^+ |u_x| - k)_+ |u_x|^{-1} - \\frac{f^\\infty}{ c_\\varphi^- c_\\varphi^+} \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^+ |h_x| - k)_+,\n\\end{align*}\nwhere $\\lim_{k \\to \\infty} r(k) = f^\\infty$.\n\\end{prop}\n\n\\begin{proof}\n\tWe will firstly show that for a~fixed $\\varepsilon$, it is true that\n\\begin{align} \\label{eq: lhs general eta}\n\\begin{split}\n\t\\liminf_{\\eta \\to 0} & \\int_I \\zeta^2 G_k(u_x^{\\varepsilon, \\eta}) \\cdot (u_x^{\\varepsilon, \\eta} - h_x^{\\varepsilon}) \\,\\mathrm d \\mathcal L^1 \\geq \\\\\n\t& \\frac{r(k)}{ (c_\\varphi^+)^{3}} \\int_I \\zeta^2 (c_\\varphi^- |u_x^{\\varepsilon}| - k)_+(c_\\varphi^+ |u_x^{\\varepsilon}| - k)_+ |u_x^{\\varepsilon}|^{-1} \\,\\mathrm d \\mathcal L^1 - \\frac{f^\\infty}{ c_\\varphi^- c_\\varphi^+} \\int_I \\zeta^2 (c_\\varphi^+ |h_x^\\varepsilon| - k)_+ \\,\\mathrm d \\mathcal L^1.\n\\end{split}\n\\end{align}\nSince $\\widetilde{\\varphi}_\\eta(p) \\geq \\varphi(p)$ and due to local uniform convergence of $\\widetilde{\\varphi}_\\eta$ to $\\varphi$, one can define a~decreasing function $\\eta(\\delta)$ such that for all $\\eta \\leq \\eta (\\delta)$\n\\begin{equation} \\label{eq: norms}\n\tc_\\varphi^- |p| \\leq \\widetilde{\\varphi}_\\eta(p) \\leq c^+_\\varphi + \\delta.\n\\end{equation}\nThen, it is possible to get a~function $\\delta(\\eta)$, which will converge to zero as $\\eta$ goes to zero. Let us also note a~consequence of convexity of the function $p \\mapsto (C|p| - k)_+$, namely, the inequality\n\t\\begin{equation} \\label{ineq: lhs general}\n\t\tC \\indi_{C|p| > k} \\cdot \\frac{p}{|p|} (p - q) \\geq (C|p| - k)_+ - (C|q| - k)_+.\n\t\\end{equation}\n\nUsing the fact that $(\\widetilde{\\varphi}_\\eta(p) - k)_+ \\, \\indi_{(c^+_\\varphi + \\delta)|p| > k} = (\\widetilde{\\varphi}_\\eta(p) - k)_+$ and \\eqref{ineq: lhs general}, we obtain the following estimate\n\\begin{align*}\n\tG_k(u_x^{\\varepsilon, \\eta}) &\\cdot (u_x^{\\varepsilon, \\eta} - h_x^\\varepsilon) = \\frac{(\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) - k)_+}{\\widetilde{\\varphi}_\\eta^2(u_x^{\\varepsilon, \\eta})} f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta})) u_x^{\\varepsilon, \\eta} \\cdot (u_x^{\\varepsilon, \\eta} - h_x^\\varepsilon) \\\\\n\t\t&\\overset{\\eqref{ineq: lhs general}}{\\geq} \\frac{(\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) - k)_+}{\\widetilde{\\varphi}_\\eta^2(u_x^{\\varepsilon, \\eta})} f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta})) \\frac{|u_x^{\\varepsilon, \\eta}|}{c_\\varphi^+ + \\delta}\\left [ \\left ( (c_\\varphi^+ + \\delta)|u_x^{\\varepsilon, \\eta}| - k \\right)_+ - \\left( (c_\\varphi^+ + \\delta) |h_x^\\varepsilon| - k \\right)_+ \\right ].\n\\end{align*}\nHandling these two terms separately and using \\eqref{eq: norms} repeatedly yields\n\\begin{align} \\label{eq: G k ineq}\n\tG_k&(u_x^{\\varepsilon, \\eta}) \\cdot (u_x^{\\varepsilon, \\eta} - h_x^\\varepsilon) \\geq \\nonumber \\\\\n\t\t&\\frac{f'_\\eta(k)}{(c_\\varphi^+ + \\delta)^3} \\underbrace{\\left( c_\\varphi^-|u_x^{\\varepsilon, \\eta}| - k \\right)_+ \\left( c_\\varphi^+|u_x^{\\varepsilon, \\eta}| - k \\right)_+ \\frac{1}{|u_x^{\\varepsilon, \\eta}|}}_{A_k(|u_x^{\\varepsilon, \\eta}|)} - \\frac{f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}))}{(c_\\varphi^- - \\delta)(c_\\varphi^+ + \\delta)}\\left( (c_\\varphi^+ + \\delta)|h_x^\\varepsilon| - k \\right)_+ \\nonumber \\\\\n\t\t&\\geq \\frac{f(k) - f(0) -1}{k} (c_\\varphi^+ + \\delta)^{-3} A_k(|u_x^{\\varepsilon, \\eta}|) - \\frac{f^\\infty}{(c_\\varphi^- - \\delta)c_\\varphi^+} \\left( (c_\\varphi^+ + \\delta)|h_x^\\varepsilon| - k \\right)_+.\n\\end{align}\n\nThe last inequality essentially follows from convexity of $f$. Indeed, due to uniform convergence of $f_\\eta$ to~$f$, $||f - f_\\eta||_\\infty \\leq 1$ for sufficiently small $\\eta$. Together with convexity of $f_\\eta$ and the fact that $f_\\eta \\geq f$, it means that\n\\[\n\t\\frac{f(t) - f(0) - 1}{t} \\leq \\frac{1}{t}(f_\\eta(t) - f_\\eta(0)) \\leq \\, f'_\\eta(t) \\, \\leq \\frac{1}{t}( f_\\eta(2t) - f_\\eta(t)) \\leq \\frac{f(2t) + 1}{t} - \\frac{f(t)}{t}.\n\\] \nTaking limit with $t \\to \\infty$ yields $\\lim_{t \\to \\infty} f'_\\eta(t) \\leq f^\\infty$, which was used in the estimate in the second term, $f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) \\leq \\lim_{t \\to \\infty} f'_\\eta(t)$. On the other hand, plugging $t = k$ on the l.\\,h.\\,s.\\;of this inequality explains the estimate in the first term. Let $r(k) := (f(k) - f(0) - 1)\/k$ and observe that $r(k) > 0 $ for sufficiently large $k$ and that $\\lim_{k \\to \\infty} r(k) = f^\\infty$.\n\nSince $h_x^\\varepsilon$ is integrable, it follows from dominated convergence theorem that\n\\begin{equation} \\label{eq: lim for h}\n\t\\lim_{\\eta \\to 0} \\frac{f^\\infty}{(c_\\varphi^- - \\delta(\\eta))c_\\varphi^+} \\int_I \\left( (c_\\varphi^+ + \\delta(\\eta))|h_x^{\\varepsilon} | - k \\right)_+ \\,\\mathrm d \\mathcal L^1 = \\frac{f^\\infty}{c_\\varphi^- c_\\varphi^+} \\int_I (c_\\varphi^+|h_x^\\varepsilon| - k)_+ \\,\\mathrm d \\mathcal L^1.\n\\end{equation}\nIn the term involving $u_x^{\\varepsilon, \\eta}$ it will not be possible to take the limit but it will suffice to estimate (recall that Lemma \\ref{lem: cpt eta} established that $u^{\\varepsilon,\\eta}$ converges weakly in $W^{1,2}$ to $u^\\varepsilon$)\n\\begin{align} \\label{eq: liminf for u}\n\\begin{split}\n\t\\liminf_{\\eta \\to 0} r(k)(c_\\varphi^+ + \\delta)^{-3} \\int_I A_k( |u_x^{\\varepsilon, \\eta}|) \\,\\mathrm d \\mathcal L^1 &\\geq r(k) (c_\\varphi^+)^{-3} \\, \\liminf_{\\eta \\to 0} \\int_I A_k(|u_x^{\\varepsilon, \\eta}|) \\,\\mathrm d \\mathcal L^1 \\\\\n\t&\\geq r(k) (c_\\varphi^+)^{-3} \\int_I A_k(|u_x^{\\varepsilon}|) \\,\\mathrm d \\mathcal L^1.\n\\end{split}\n\\end{align}\nThe last inequality follows from convexity of $p \\mapsto A_k(|p|)$ (it is easy to check that $A_k(s)$ is convex and increasing). Indeed, since $u_x^{\\varepsilon, \\eta}$ converge weakly in $L^2$ to $u_x^\\varepsilon$, lower semicontinuity follows from the standard result \\cite[Chapter 8.2, Theorem 1]{evans}.\n\nAt this point, one needs to multiply the inequality \\eqref{eq: G k ineq} by $\\zeta^2$ and integrate it over $I$. Then, take $\\liminf$ with $\\eta \\to 0$ and use \\eqref{eq: lim for h} and \\eqref{eq: liminf for u} to arrive at the desired result \\eqref{eq: lhs general eta}.\n\n\nWe may now turn to the passage with $\\varepsilon \\to 0$. Plugging $G(p) = (c_\\varphi^+|p| - k)_+$ in Lemma \\ref{lem: G weaklystar} means that\n\t\\[\n\t\t\\lim_{\\varepsilon \\to 0} \\int_I \\zeta^2(c_\\varphi^+|h_x^\\varepsilon| - k)_+ \\,\\mathrm d \\mathcal L^1 = \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^+|h_x| - k)_+.\n\t\\]\nAs shown in Lemma \\ref{lem: gamma conv eps}, $u_x^\\varepsilon \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} u_x$, which after applying Corollary \\ref{cor-gs-lsc-zeta} results in the slightly modified lower semicontinuity, i.e., \n\\[\n\\liminf_{\\varepsilon \\to 0} \\int_I \\zeta^2 \\,\\mathrm d A_k(|u_x^{\\varepsilon}|) \\geq \\int_I \\zeta^2 \\,\\mathrm d A_k(|u_x|),\n\\]\nwhich finishes the proof.\n\\end{proof}\n\n\n\\begin{prop} \\label{prop-rhs-eps-eta}\nSet $\\mathcal{R}(\\varepsilon, k) := \\limsup_{\\eta \\to 0} - \\int_I (\\zeta^2 G_k(u_x^{\\varepsilon, \\eta}))_x \\cdot D^2 F_\\eta (u_x^{\\varepsilon, \\eta}) \\cdot u_{xx}^{\\varepsilon, \\eta} \\,\\mathrm d \\mathcal L^1$.\nThen, the r.\\,h.\\,s. of equation \\eqref{eq: after mult} satisfies \n\\[\n\t\\lim_{k \\to \\infty} \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k) = 0.\n\\]\n\\end{prop}\n\n\n\\begin{proof}\nThroughout this proof $u_x^{\\varepsilon, \\eta}$ is denoted without superscripts. We compute the necessary derivatives and deal with the terms coming from $f_\\eta \\circ \\widetilde{\\varphi}_\\eta$ and $\\eta\/2 \\widetilde{\\widetilde{\\varphi}}_\\eta$ separately. \n\\begin{comment}\n\\begin{align*}\n\tD^2 F_\\eta &= f^{''}_\\eta \\circ \\widetilde{\\varphi}_\\eta D \\widetilde{\\varphi}_\\eta \\otimes D \\widetilde{\\varphi}_\\eta + f'_\\eta \\circ \\widetilde{\\varphi}_\\eta D^2 \\widetilde{\\varphi}_\\eta + \\theta(\\eta) \\left ( D\\widetilde{\\varphi}_\\eta \\otimes D \\widetilde{\\varphi}_\\eta + \\widetilde{\\varphi}_\\eta D^2\\widetilde{\\varphi}_\\eta \\right ) \\\\\n\t(\\zeta^2 G_k(u_x))_x &= 2 \\zeta \\zeta' g_k(\\widetilde{\\varphi}_\\eta(u_x)) u_x + \\zeta^2 g_k'(\\widetilde{\\varphi}_\\eta(u_x))(\\widetilde{\\varphi}_\\eta(u_x))_x u_x + \\zeta^2 g_k(\\widetilde{\\varphi}_\\eta(u_x)) u_{xx}\n\\end{align*}\n\\begin{align*}\n\t(\\zeta^2 G_k(u_x))_x &\\cdot D^2 F_\\eta(u_x) \\cdot u_{xx} =\\\\\n\t&\\quad 2\\zeta \\zeta'g_k(\\widetilde{\\varphi}_\\eta(u_x)) f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) (\\widetilde{\\varphi}_\\eta(u_x))_x \\\\\n\t&+ \\zeta^2 g_k'(\\widetilde{\\varphi}_\\eta(u_x))f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) (\\widetilde{\\varphi}_\\eta(u_x))_x^2 \\\\\n\t&+ \\zeta^2 g_k(\\widetilde{\\varphi}_\\eta(u_x)) f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) (\\widetilde{\\varphi}_\\eta(u_x))_x^2 \\\\\n\t&+ \\zeta^2 g_k(\\widetilde{\\varphi}_\\eta(u_x)) f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) u_{xx} \\cdot D^2 \\widetilde{\\varphi}_\\eta \\cdot u_{xx} \\\\\n\t&+ \\theta(\\eta) \\Bigl ( \n\t2\\zeta\\zeta_x g_k(\\widetilde{\\varphi}_\\eta(u_x))u_x \\cdot D\\widetilde{\\varphi}_\\eta \\otimes D\\widetilde{\\varphi}_\\eta \\cdot u_{xx} \\\\\n\t&+ \\zeta^2g_k'(\\widetilde{\\varphi}_\\eta(u_x))(\\widetilde{\\varphi}_\\eta(u_x))_x u_x \\cdot D\\widetilde{\\varphi}_\\eta \\otimes D\\widetilde{\\varphi}_\\eta \\cdot u_{xx} \\\\\n\t&+ \\zeta^2 g_k(\\widetilde{\\varphi}_\\eta(u_x))u_{xx} \\cdot D\\widetilde{\\varphi}_\\eta \\otimes D\\widetilde{\\varphi}_\\eta \\cdot u_{xx} \\\\\n\t&+ \\zeta^2g_k(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) u_{xx} \\cdot D^2\\widetilde{\\varphi}_\\eta \\cdot u_{xx}\n\t\\Bigr) \\\\\n\\end{align*}\n\\end{comment}\n\\begin{align*}\n\t\t(\\zeta^2 G_k(u_x))_x \\cdot D^2(f_\\eta \\circ \\widetilde{\\varphi}_\\eta )(u_x)\\cdot u_{xx} &=\n\t2\\zeta \\zeta_x g_k(\\widetilde{\\varphi}_\\eta(u_x)) f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) (\\widetilde{\\varphi}_\\eta(u_x))_x \\\\\n\t&+ \\zeta^2 f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) (\\widetilde{\\varphi}_\\eta(u_x))_x^2 \\left( g_k'(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) + g_k(\\widetilde{\\varphi}_\\eta(u_x)) \\right)\\\\\n\t&+ \\zeta^2 g_k(\\widetilde{\\varphi}_\\eta(u_x)) f'_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) u_{xx} \\cdot D^2 \\widetilde{\\varphi}_\\eta(u_x) \\cdot u_{xx} \\\\\n\t&\\geq -\\zeta_x^2 f''_\\eta(\\widetilde{\\varphi}_\\eta(u_x)) \\underbrace{\\frac{g_k^2(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta^2(u_x)}{g_k(\\widetilde{\\varphi}_\\eta(u_x)) + g'_k(\\widetilde{\\varphi}_\\eta(u_x))\\widetilde{\\varphi}_\\eta(u_x)}}_{\\mathcal{K}(\\widetilde{\\varphi}_\\eta(u_x))}.\n\\end{align*}\nThe rest vanishes, because $\\widetilde{\\varphi}_\\eta$ is homogeneous and hence $D^2 \\widetilde{\\varphi}_\\eta(u_x) \\cdot u_{x} = 0$. The third term is non-negative, whereas for the first and the second and third we use $2ab \\geq -a^2 - b^2$ inequality, which yields the above estimate. To this end, one needs to make sure that $g_k(\\sigma) + g'_k(\\sigma)\\sigma$ is non-negative for $\\sigma > k$. Clearly, after substituting $\\sigma$ for $\\widetilde{\\varphi}_\\eta(u_x)$, we have\n\\begin{equation*}\n\\mathcal{K}(\\sigma) = \\frac{(\\sigma - k)_+^2 | f'_{\\eta} (\\sigma) |^2}{k f'_{\\eta}(\\sigma) + \\sigma(\\sigma - k) f''_{\\eta}(\\sigma)}.\n\\end{equation*}\nForgetting about the non-negative term $k f'_\\eta$ in the denominator and using linear growth of $f_\\eta$ and the fact that $\\sigma > k$ allows us to conclude that\n\\begin{equation} \\label{ke-small}\n\\zeta^2_x f''_\\eta (\\sigma) \\mathcal{K}(\\sigma) \\leq \\frac{||\\zeta_x||_\\infty^2}{\\sigma}(\\sigma - k)_+ |f'_\\eta(\\sigma)|^2 \\leq \\frac{C ||\\zeta_x||_\\infty^2}{k} (\\sigma - k)_+.\n\\end{equation}\n\nLet us now turn to the neglected terms of the derivative which are multiplied by $\\eta\/2$. Using an analogous method, we will show that they converge to zero when $\\eta$ converges to zero. Recall that $\\widetilde{\\widetilde{\\varphi}}_\\eta = \\widetilde{\\varphi}_\\eta^2$ outside a~small ball, where $(\\zeta^2 G_k(u_x))_x$ vanishes anyway.\n\n\\begin{align*}\n\t\\frac{2}{\\eta}(\\zeta^2 G_k(u_x))_x \\cdot D^2 \\left(\\frac{1}{2}\\eta \\widetilde{\\varphi}_\\eta^2 \\right) \\cdot u_{xx} &=\n\t2\\zeta\\zeta_x g_k(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) (\\widetilde{\\varphi}_\\eta(u_x))_x \\\\\n\t&+ \\zeta^2 \\Bigl (\\widetilde{\\varphi}_\\eta(u_x))_x^2 (g_k'(\\widetilde{\\varphi}_\\eta(u_x))\\widetilde{\\varphi}_\\eta(u_x) + g_k(\\widetilde{\\varphi}_\\eta(u_x) \\Bigr) \\\\\n\t&+ \\zeta^2g_k(\\widetilde{\\varphi}_\\eta(u_x)) \\widetilde{\\varphi}_\\eta(u_x) u_{xx} \\cdot D^2\\widetilde{\\varphi}_\\eta \\cdot u_{xx} \\\\\n\t& \\geq -\\zeta_x^2 \\mathcal{K}(\\widetilde{\\varphi}_\\eta(u_x)).\n\\end{align*}\nForgetting about the term with $f''_\\eta$ in the denominator of $\\mathcal{K}$ and keeping in mind, as before, that $f_\\eta$ is of linear growth, one gets\n\\begin{equation} \\label{ke-big}\n\\zeta^2_x \\mathcal{K}(\\sigma) \\leq \\frac{C ||\\zeta_x||_\\infty^2}{k} (\\sigma - k)_+^2.\n\\end{equation}\n\nUsing estimates \\eqref{ke-big} and \\eqref{ke-small} and careful multiplying by $-1$ lead to the conclusion that\n\\begin{equation*}\n- \\int_I (\\zeta^2 G_k(u_x))_x \\cdot D^2 F_\\eta (u_x) \\cdot u_{xx} \\,\\mathrm d \\mathcal L^1 \\leq \\frac{C||\\zeta_x||_\\infty^2}{k} \\left( \\int_I (\\widetilde{\\varphi}_\\eta(u_x) - k)_+ \\,\\mathrm d \\mathcal L^1 + \\eta \\int_I (\\widetilde{\\varphi}_\\eta(u_x) - k)_+^2 \\,\\mathrm d \\mathcal L^1 \\right).\n\\end{equation*}\n\nEventually, recall that if $\\varepsilon$ is fixed, then $\\left \\{ \\int_I (\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) - k)_+^2 \\,\\mathrm d \\mathcal L^1 \\right \\}$ is bounded w.\\,r.\\,t.\\;$\\eta$ (which was essentially proved in Lemma \\ref{lem: cpt eta}), whence the second term vanishes when $\\eta$ converges to zero. Lastly, the fact that \n\\[\n\\mathcal E_{\\varepsilon, \\eta}(u^{\\varepsilon, \\eta}) \\leq \\int_I |h^\\varepsilon|^2 \\,\\mathrm d \\mathcal L^1 \\leq 2\\int_I |h|^2 \\,\\mathrm d \\mathcal L^1\n\\]\nand convexity of $f$ imply that $\\left \\{\\int_I (\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) - k)_+ \\,\\mathrm d \\mathcal L^1 \\right\\}$ is bounded (w.\\,r.\\,t.\\;both $\\eta$ and $\\varepsilon$).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:constants}]\n\tPropositions \\ref{prop: general lhs eps} and \\ref{prop-rhs-eps-eta} imply that the minimizer $u$ of $\\mathcal E$ satisfies\n\t\\begin{equation} \\label{eq:constants final}\n\t\t\\frac{r(k)}{ (c_\\varphi^+)^{3}} \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^- |u_x| - k)_+(c_\\varphi^+ |u_x| - k)_+ |u_x|^{-1} \\leq \\frac{f^\\infty}{ c_\\varphi^- c_\\varphi^+} \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^+ |h_x| - k)_+ + \\lambda \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k),\n\t\\end{equation}\nwhere $\\lim_{k \\to \\infty} r(k) = f^\\infty$. Let us now pass with $k \\to \\infty$. Since\n\\[\n\t\\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^+|h_x| - k)_+ = \\int_I \\zeta^2 \\,\\mathrm d (c_\\varphi^+|h_x^{ac}| - k)_+ \\,\\mathrm d \\mathcal L^1 + c_\\varphi^+ \\int_I \\zeta^2 \\,\\mathrm d |h_x^s|,\n\\]\nwe can use dominated convergence theorem (as $\\zeta^2 |h_x^{ac}|$ is integrable) to pass with the limit in the first integral and see that it vanishes. Therefore, the r.\\,h.\\,s.\\;of \\eqref{eq:constants final} converges to $f^\\infty(c_\\varphi^-)^{-1} \\int_I \\zeta^2 \\,\\mathrm d |h_x^s|$. In an analogous manner, limit passage on the l.\\,h.\\,s\\;can be performed, which results in\n\\begin{equation} \\label{eq:constants final2}\n\t\\int_I \\zeta^2 \\,\\mathrm d |u_x^s| \\leq \\left ( \\frac{c_\\varphi^+}{c_\\varphi^-} \\right)^2 \\int_I \\zeta^2 \\,\\mathrm d |h_x^s|.\n\\end{equation}\nTo finish the proof, it suffices to apply Lemma \\ref{lem: mu leq nu} for $\\mu = |u_x^s|$ and $\\nu = (c_\\varphi^+)^2(c_\\varphi^-)^{-2}|h_x^s|$.\n\\end{proof}\n\n\\section{The regular case} \\label{sec:regular} \n\nFirstly, let us note that under assumptions of Theorem~\\ref{thm:regular norm}, $F$ is differentiable in $\\mathbb R^n$. Additionally, any differentiable convex function on $\\mathbb R^n$ is in fact continuously differentiable (\\cite[Corollary 25.5.1]{rockafellar}). Since $\\mathbb R^n \\setminus \\{0\\}$ can be easily covered with a~finite number of open convex sets, this fact remains true if one replaces $\\mathbb R^n$ with $\\mathbb R^n \\setminus \\{0\\}$. In view of Lemma \\ref{lem: strict convexity}, $F$ is strictly convex and the assumption of strict convexity in the sense of Reshetnyak for $\\varphi$ is necessary for $F$ to be strictly convex.\n\nWe define $\\mathcal F_M \\colon M(I, \\mathbb R^n) \\to [0, \\infty[$ by \n\\[\\mathcal F_M(\\mu) = F(\\mu)(\\mathcal I).\\]\nClearly, we have $\\mathcal F(w) = \\mathcal F_M(w_x)$ for $w \\in BV(I, \\mathbb R^n)$. \n\n\\begin{thm} \\cite[Theorem 2.4]{anzellotti2} \\label{thm:anzellotti}\n\tFunctional $\\mathcal F_M$ is differentiable at $\\alpha \\in M(I, \\mathbb R^n)$ in direction $\\beta \\in M(I, \\mathbb R^n)$ iff $|\\beta^s| \\ll |\\alpha^s|$ and that the derivative is of the form\n\\begin{equation*}\n\t\\int_I DF(\\alpha^{ac}) \\cdot \\beta^{ac} \\,\\mathrm d \\mathcal L^1 + \\int_I DF^{\\infty} \\left( \\frac{\\alpha^s}{|\\alpha^s|} \\right) \\cdot \\,\\mathrm d \\beta^s,\n\\end{equation*}\nwhere $DF^\\infty(p) = \\lim_{t \\to \\infty} \\tfrac{1}{t}DF(tp)$.\n\\end{thm}\n\n\\begin{lemma} \\label{lem: vanish at endpoints}\n\tThe following statements hold:\n\t\\begin{enumerate}[(i)]\n\t\t\\item $DF_\\eta(u_x^{\\varepsilon, \\eta}) \\in W^{1,2}_0(I, \\mathbb R^n)$ and $\\lambda \\left( DF_\\eta(u_x^{\\varepsilon, \\eta}) \\right)_x = u^{\\varepsilon, \\eta} - h^\\varepsilon$ a.e. in $I$; \n\t\t\\item $DF(u_x^{\\varepsilon}) \\in W^{1,2}_0(I, \\mathbb R^n)$ and $\\lambda \\left( DF(u_x^{\\varepsilon}) \\right)_x = u^\\varepsilon - h^\\varepsilon$ a.e. in $I$;\n\t\t\\item $DF(u_x^{ac}) \\in W^{1,2}_0(I, \\mathbb R^n)$ and $\\lambda \\left( DF (u_x^{ac}) \\right)_x = u - h$ a.e. in $I$.\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe will show only proof of (iii) as it is the most interesting one. Statements (i) and (ii) are proved in the same way only without invoking the additional Theorem \\ref{thm:anzellotti}.\n\n\tFirstly, observe that since $DF$ is bounded, the function $DF(u_x^{ac}) \\in L^2(I, \\mathbb R^n)$. Secondly, applying Theorem \\ref{thm:anzellotti} allows us to write the derivative of $\\mathcal F$ at point $u$ and in direction $\\xi \\in C^\\infty_c(I)$ as \n\\begin{equation*}\n\t\\int_I \\xi_x DF(u_x^{ac}) \\,\\mathrm d \\mathcal L^1,\n\\end{equation*}\nsince the integral with respect to the singular part vanishes. As $u$ is the minimizer of $\\mathcal E$, it is true that\n\\begin{equation*}\n\t- \\lambda \\int_I \\xi_x DF(u_x^{ac}) \\,\\mathrm d \\mathcal L^1 = \\int_I \\xi (u - h) \\,\\mathrm d \\mathcal L^1,\n\\end{equation*}\nwhich means that the distributional derivative of $DF(u_x^{ac})$ equals $(u - h)\/\\lambda \\in L^2(I, \\mathbb R^n)$. Consequently, $DF(u_x^{ac}) \\in W^{1,2}(I, \\mathbb R^n)$, which implies also that there exists a~continuous representative of this function.\n\t\n\tTo show that any continuous representative equals zero at the endpoints of the interval, let $\\xi^\\ell(x) = (-\\ell x + 1 + al) \\cdot \\indi_{[a, a + 1\/\\ell]}$ be a~piece-wise linear function equal $1$ for $x = a$ and $0$ on $[a, a + 1\/\\ell]$. Once again the Euler-Lagrange equation given by Theorem \\ref{thm:anzellotti} yields \n\\begin{equation*}\n\t-\\lambda \\int_I \\xi^\\ell_x DF(u_x^{ac}) \\,\\mathrm d \\mathcal L^1 = \\int_I \\xi^\\ell (u - h) \\,\\mathrm d \\mathcal L^1.\n\\end{equation*}\nAs $\\xi^\\ell_x = -\\ell \\indi_{[a, a + 1\/\\ell]}$, continuity of $DF(u_x^{ac})$ implies that the l.\\,h.\\,s. converges to $DF(u_x^{ac}(a))$ as $\\ell \\to \\infty$. Morover, dominated convergence theorem enables us to pass with the limit under the integral sign in order to get zero on the r.\\,h.\\,s, which implies that $DF(u_x^{ac}(a)) = 0$. To get $DF(u_x^{ac}(b)) = 0$, one needs to repeat this argument with $\\xi^\\ell = (\\ell x + 1 -\\ell b) \\cdot \\indi_{[b-1\/\\ell, b]}$.\n\\end{proof}\n\nObserve that for fixed representatives of $u^{\\varepsilon, \\eta}$ and $DF_\\eta(u_x^{\\varepsilon, \\eta})$ it is true that almost everywhere on $I$, $DF_\\eta(u_x^{\\varepsilon, \\eta})$ is an actual composition of $DF$ and $u_x^{\\varepsilon, \\eta}$, i.e., $DF_\\eta(u_x^{\\varepsilon, \\eta})(x_0) = DF_\\eta(u_x^{\\varepsilon, \\eta}(x_0))$. The same holds for analogous pairs of mappings from Lemma \\ref{lem: vanish at endpoints}.\n\n\\subsection*{Limit passage with $\\eta$}\n\nThanks to the additional regularity of $F$, it is possible to perform a~more subtle limit passage with $\\eta \\to 0$ on the l.\\,h.\\,s\\;than in the previous case.\n\n\\begin{lemma} \\label{lem: a.e. conv eta}\nThere exists a sequence $\\eta_j$ with $\\eta_j \\to 0$ as $j \\to \\infty$ such that $u_x^{\\varepsilon, \\eta_j}$ converges to $u_x^\\varepsilon$ a.\\,e.\\;on $I$.\n\\end{lemma}\n\n\\begin{proof}\nTo begin with, we will show that the sequence $DF_{\\eta}( u_x^{\\varepsilon, \\eta})$ converges to $DF(u_x^\\varepsilon)$ in $C( \\overline{I}, \\mathbb R^n)$. Due to the choice of $h^\\varepsilon$ and properties established in Lemma \\ref{lem: cpt eta}, functions $ u^{\\varepsilon, \\eta} - h^\\varepsilon$ converge in $L^2$ to $u^{\\varepsilon}-h^{\\varepsilon}$. In light of Lemma \\ref{lem: vanish at endpoints}, it is then evident that $(DF_{\\eta}(u_x^{\\varepsilon, \\eta_j}))_x$ converge to $(DF(u_x^{\\varepsilon}))_x$ in $L^2$ as well. For every $x_0 \\in \\overline{I}$, due to the boundary condition established in the same lemma, it is true that\n\\begin{equation*}\n\tDF_{\\eta} (u_x^{\\varepsilon, \\eta})(x_0) = \\int_a^{x_0} \\left(DF_{\\eta} (u_x^{\\varepsilon, \\eta}) \\right)_x \\,\\mathrm d \\mathcal L^1 \\quad \\text{and} \\quad DF (u_x^{\\varepsilon})(x_0) = \\int_a^{x_0} \\left(DF(u_x^{\\varepsilon}) \\right)_x \\,\\mathrm d \\mathcal L^1.\n\\end{equation*}\nThe $L^2$ convergence of the integrands allows to conclude that\n\\begin{equation} \\label{eq: a.e. conv eta}\n\t\\sup_{x_0 \\in \\overline{I}} \\left \\lvert DF_{\\eta}( u_x^{\\varepsilon, \\eta}) (x_0) - DF(u_x^{\\varepsilon})(x_0) \\right \\rvert \\to 0.\n\\end{equation}\n\nSecondly, let us observe that\n\\begin{align} \\label{eq: l1 convergence}\n\\begin{split}\n\t\\int_I \\left| DF_{\\eta} (u_x^{\\varepsilon, \\eta}) - DF(u_x^{\\varepsilon, \\eta}) \\right| \\,\\mathrm d \\mathcal L^1 &\\leq \\int_I \\left| D(f_{\\eta} \\circ \\widetilde{\\varphi}_\\eta)(u_x^{\\varepsilon, \\eta}) - D(f \\circ \\varphi)(u_x^{\\varepsilon, \\eta}) \\right | \\,\\mathrm d \\mathcal L^1 +\\\\\n\t&+ \\frac{\\eta}{2} \\int_I |D \\widetilde{\\widetilde{\\varphi}}_{\\eta} (u_x^{\\varepsilon, \\eta})| \\,\\mathrm d \\mathcal L^1.\n\\end{split}\n\\end{align}\nIn view of property (iv) from Proposition \\ref{goodnorm}, it is evident that the first integral converges to zero. The second one (up to a negligible constant coming from the fact that $\\widetilde{\\widetilde{\\varphi}}$ coincides with $\\widetilde{\\varphi}_{\\eta}$ outside a small ball) does not exceed\n\\[\n\t\\eta \\int_I \\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) |D\\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) | \\,\\mathrm d \\mathcal L^1.\n\\]\nAs $|D\\widetilde{\\varphi}_\\eta|$ is a~bounded function and $\\int_I \\widetilde{\\varphi}_\\eta(u_x^{\\varepsilon, \\eta}) \\,\\mathrm d \\mathcal L^1$ is bounded w.r.t $\\eta$ (see proof of Proposition \\ref{prop-rhs-eps-eta}), one can conclude that \\eqref{eq: l1 convergence} converges to zero. This implies existence of a~subsequence $\\eta_j$ with $\\eta_j \\to 0$ as $j \\to \\infty$ for which\n\\begin{equation*}\n\t\\lim_{j \\to \\infty} \\left| DF_{\\eta_{j}} (u_x^{\\varepsilon, \\eta_j}) - DF(u_x^{\\varepsilon, \\eta_j}) \\right| = 0 \\text{ a.e. on } I.\n\\end{equation*}\n\nIn \\eqref{eq: a.e. conv eta} it has been already checked that the sequence $DF_{\\eta_j} (u_x^{\\varepsilon, \\eta_j})$ converges to $DF (u_x^\\varepsilon)$ in $C( \\overline{I}, \\mathbb R^n)$. As a consequence, \n\\begin{equation*}\n\t\\lim_{j \\to \\infty} \\left| DF (u_x^{\\varepsilon, \\eta_j}) - DF(u_x^{\\varepsilon}) \\right| = 0 \\text{ a.e. on } I.\n\\end{equation*}\nAs $F$ is strictly convex and $C^1$, $DF$ is strictly monotone and continuous. Strict monotonicity of $DF$ implies that the map is injective and hence, by Brouwer's invariance of domain theorem, there exists its continuous inverse $DF^{-1}$. Therefore, at any point in which $DF(u_x^{\\varepsilon, \\eta_j})(x_0) \\to DF(u_x^{\\varepsilon})(x_0)$ and the mappings $DF(u_x^{\\varepsilon, \\eta_j})$ and $DF(u_x^{\\varepsilon})$ are actual compositions,\nwe have\n\\begin{equation*}\n\tu_x^{\\varepsilon, \\eta_j}(x_0) = DF^{-1} \\left( DF(u_x^{\\varepsilon, \\eta_j}(x_0)) \\right) \\to DF^{-1} \\left( DF(u_x^{\\varepsilon} (x_0)) \\right) = u_x^{\\varepsilon}(x_0).\n\\end{equation*}\nSince the set of such points is of full measure, we have established the desired a.e. convergence of $u_x^{\\varepsilon, \\eta_j}$ to $u_x^\\varepsilon$.\n\n\\end{proof}\n\n\\begin{prop} \\label{prop-lhs-eps-eta}\n\tThere exists a sequence $\\eta_j$ with $\\eta_j \\to 0$ as $j \\to \\infty$ for which the l.\\,h.\\,s. of equation \\eqref{eq: after mult} satisfies\n\t\\begin{equation*}\n\t\t\\lim_{j \\to \\infty} \\int_I \\zeta^2 G_k(u_x^{\\varepsilon, \\eta_j}) \\cdot (u_x^{\\varepsilon, \\eta_j} - h_x^{\\varepsilon}) \\,\\mathrm d \\mathcal L^1\n\t\t= \\int_I \\zeta^2 \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^{\\varepsilon})} f'(\\varphi(u_x^\\varepsilon)) (u_x^{\\varepsilon} - h_x^{\\varepsilon}) \\cdot u_x^\\varepsilon \\,\\mathrm d \\mathcal L^1.\n\t\\end{equation*}\n\\end{prop}\n\n\n\\begin{proof}\nLet us choose the sequence from Lemma \\ref{lem: a.e. conv eta}, for which we know that the integrand is a.\\,e convergent. Then, it suffices to prove its uniform integrability and apply the Vitali convergence theorem. Bearing in mind the formula for $G_k(\\cdot)$, the absolute value of the integrand can be estimated from above by $C |u_{x}^{\\varepsilon, \\eta} - h_x^{\\varepsilon}|$ for some constant $C$. As a~result, it suffices to check that\n\\begin{equation*}\n\t\\int_{|u_x^{\\varepsilon, \\eta} - h_x^{\\varepsilon}| \\geq t} |u_x^{\\varepsilon, \\eta} - h_x^{\\varepsilon}| \\,\\mathrm d \\mathcal L^1 \\xrightarrow{t \\to \\infty} 0,\n\\end{equation*}\nwhich follows from H\\\"{o}lder's and Chebyshev's inequalities and again the fact that the sequence $\\{u_x^{\\varepsilon, \\eta}\\}_\\eta$ is bounded in $L^2$.\n\\end{proof}\n\n\n\\begin{prop} \\label{prop: inequality with eps}\n\tRecall that $u^\\varepsilon$ is a~minimizer of the functional $\\mathcal E_\\varepsilon$. The following estimate holds\n\t\\begin{equation*}\n\t\tf'(k) \\int_I \\zeta^2 |u_x^\\varepsilon|^2 \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} \\,\\mathrm d \\mathcal L^1 \\leq f^\\infty \\int_I \\zeta^2 |h_x^\\varepsilon| |u_x^\\varepsilon| \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} \\,\\mathrm d \\mathcal L^1 + \\lambda \\mathcal{R}(\\varepsilon, k),\n\t\\end{equation*}\nwhere $\\mathcal{R}(\\varepsilon, k)$ was defined in Proposition \\ref{prop-rhs-eps-eta}.\n\\end{prop}\n\n\\begin{proof}\nThe proof boils down to observing what has already been proved about the Euler--Lagrange equation \\eqref{Euler-Lagrange-eps-eta} of $\\mathcal E_{\\varepsilon, \\eta}$. Combining Propositions \\ref{prop-rhs-eps-eta} and \\ref{prop-lhs-eps-eta} gives\n\\begin{equation*}\n\t\\int_I \\zeta^2 \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} f'(\\varphi(u_x^\\varepsilon)) |u_x^{\\varepsilon}|^2\\,\\mathrm d \\mathcal L^1 \n\t\\leq\n\t \\int_I \\zeta^2 \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} f'(\\varphi(u_x^\\varepsilon)) h_x^{\\varepsilon} \\cdot u_x^\\varepsilon \\,\\mathrm d \\mathcal L^1 + \\mathcal{R}(\\varepsilon, k).\n\\end{equation*}\nThen, on the l.\\,h.\\,s.\\;one needs to use the fact that $f'$ is non-decreasing and the integration takes place on the set where $\\varphi(u_x^\\varepsilon) > k$, whereas on the r.\\,h.\\,s.\\;the fact that $f'(t) \\leq f^\\infty$ for all~$t$.\n\\end{proof}\n\n\\subsection*{Limit passage with $\\varepsilon$}\n\n\\begin{lemma} \\label{lem: u eps a.e.}\nThere exist functions $\\overline{u}_x^{\\varepsilon_i}, \\overline{u}_x^{ac} \\in C(\\overline{I}, \\overline{R^n})$ s.t.\\;$\\overline{u}_x^{\\varepsilon_i} \\to \\overline{u}_x^{ac}$ in $C(\\overline{I}, \\overline{R^n})$ and for a.\\,e.\\;$x_0 \\in \\overline{I}$, $\\overline{u}_x^{\\varepsilon_i}(x_0) = u_x^{\\varepsilon_i}(x_0)$ and for a.\\,e.\\;$x_0 \\in \\overline{I}$, $\\overline{u}_x^{ac}(x_0) = u_x^{ac}(x_0)$.\n\\end{lemma}\n\\begin{proof}\n\tLemma \\ref{lem: vanish at endpoints} implies (by a~reasoning similar to the beginning of the proof of Lemma \\ref{lem: a.e. conv eta}) that $DF(u_x^\\varepsilon)$ converge to $DF(u_x^{ac})$ in $C(\\overline{I}, \\mathbb R^n)$. This implies convergence in $C(\\overline{I}, \\overline{\\mathbb R^n})$. The map $\\overline{DF}$ is a~homeomorphism, as established in Lemma \\ref{lem: DF homeo}, which prompts the following definition\n\\begin{equation*}\n\t\\overline{u}_x^{\\varepsilon_i} := \\overline{DF}^{-1} \\circ DF(u_x^{\\varepsilon_i}) \\text{ and } \\overline{u}_x^{ac} := \\overline{DF}^{-1} \\circ DF (u_x^{ac}).\n\\end{equation*}\nRequested properties of these functions now follow from continuity of $\\overline{DF}^{-1}$ and the fact that $u_x^{\\varepsilon_i}$ and $u_x^{ac}$ are almost everywhere finite.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:regular norm}]\nLet us recall the inequality proved in Proposition \\ref{prop: inequality with eps}\n\\begin{equation*}\n\t\tL_{\\varepsilon, k} := f'(k) \\int_I \\zeta^2 |u_x^\\varepsilon|^2 \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} \\,\\mathrm d \\mathcal L^1 \\leq f^\\infty \\int_I \\zeta^2 |h_x^\\varepsilon| |u_x^\\varepsilon| \\frac{(\\varphi(u_x^\\varepsilon) - k)_+}{\\varphi^2(u_x^\\varepsilon)} \\,\\mathrm d \\mathcal L^1 + \\lambda \\mathcal{R}(\\varepsilon, k) := R_{\\varepsilon, k},\n\\end{equation*}\nwhere $\\zeta \\in C_c^1(I)$ and $\\lim_{k \\to \\infty} \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k) = 0$ as stated in Proposition \\ref{prop-rhs-eps-eta}. We will firstly pass with $\\varepsilon$ to zero. Define a~continuous function $b_k: \\overline{\\mathbb R^n} \\to \\mathbb R$\n\\begin{equation*}\n\tb_k(p) = \\begin{cases}\n\t\t\\frac{|p| \\left( \\varphi(p) - k \\right)_+}{\\varphi^2(p)} & \\text{ if } p \\in \\mathbb R^n \\vspace{1mm} \\\\\n\t\t\\frac{1}{\\varphi(\\omega)} & \\text{ if } p = \\infty \\omega \\in \\mathbb{S}^{n-1}_\\infty.\n\t\\end{cases}\n\\end{equation*}\nClearly, $u_x^\\varepsilon = \\overline{u}_x^{\\varepsilon}$ $|u_x^{\\varepsilon}|$-a.e. due to absolute continuity of this measure w.r.t $\\mathcal L^1$. Therefore,\n\\begin{equation*}\n\tL_{\\varepsilon, k} = f'(k) \\int_I \\zeta^2 |u_x^{\\varepsilon}| \\left( b_k(\\overline{u}_x^\\varepsilon) - b_k(\\overline{u}_x^{ac}) \\right) \\,\\mathrm d \\mathcal L^1 + f'(k) \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |u_x^{\\varepsilon}|.\n\\end{equation*}\nIn view of Lemma \\ref{lem: u eps a.e.} (ii), it is true that $b_k(\\overline{u}_x^{\\varepsilon_i})$ converges to $\\overline{u}_x^{ac}$ in $C(\\overline{I}, \\mathbb R)$, which implies that the first term converges to zero (recall that $\\{u_x^\\varepsilon\\}$ is bounded in $L^1$). The second term is lower semicontinuous w.\\,r.\\,t\\;weak* convergence of $u_x^\\varepsilon \\to u_x$ (see Corollary \\ref{cor-gs-lsc-zeta}) and whence\n\\begin{equation*}\n\t\\lim_{i \\to \\infty} L_{\\varepsilon_i, k} \\geq f'(k) \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |u_x|.\n\\end{equation*}\nAs $h^\\varepsilon$ has been chosen so that $|h_x^\\varepsilon| \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} |h_x|$, one shows in a~similar manner that\n\\begin{equation*}\n\t\\limsup_{\\varepsilon \\to 0} R_{\\varepsilon, k} \\leq f^\\infty \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |h_x| + \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k).\n\\end{equation*}\nAs a result,\n\\begin{align} \\label{eq: before k limit}\n\tf'(k) \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |u_x^{s}| \t\t&\\leq f^\\infty \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |h_x| + \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k) \\leq \\\\\n\t& \\leq f^\\infty \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |h_x^{ac}| + f^\\infty \\int_I \\zeta^2 b_k(\\overline{u}_x^{ac}) \\,\\mathrm d |h_x^{s}| + \\limsup_{\\varepsilon \\to 0} \\mathcal{R}(\\varepsilon, k). \\nonumber\n\\end{align}\n\nWe will now pass with $k$ to infinity. At any point $x \\in I$ for which $\\overline{u}_x^{ac}(x) \\in \\mathbb{S}^{n-1}_\\infty$, set $\\omega(x) \\in \\mathbb{S}_{1}^{n-1}$ to be such that $\\overline{u}_x^{ac}(x) = \\infty \\, \\omega(x)$. Clearly, for any $x \\in I$\n\\[\n\tb_k(\\overline{u}_x^{ac}(x)) \\xrightarrow{k \\to \\infty} b(x) := \\frac{1}{\\varphi(\\omega(x))} \\indi_{ \\left\\{x: \\, \\overline{u}_x^{ac}(x) \\in \\mathbb{S}_\\infty^{n-1} \\right\\}}(x).\n\\]\nTherefore, the fact that $|\\overline{u}_x^{ac}|$ is a.e.\\;finite together with dominated convergence theorem allow us to conclude that \\eqref{eq: before k limit} after passing with $k$ to infinity and dividing by $f^\\infty$ becomes\n\\begin{equation}\\label{eq: after k limit}\n\t\\int_I \\zeta^2 b \\,\\mathrm d |u_x^s| \\leq \\int_I \\zeta^2 b \\,\\mathrm d |h_x^s|.\n\\end{equation}\nFunction $b$ is measurable w.r.t.\\;any Radon measure as a~pointwise limit of continuous functions, whence both $ b \\,\\mathrm d |u_x^{s}|$ and $ b \\,\\mathrm d |h_x^s|$ are~Radon measures. Lemma \\ref{lem: mu leq nu} for $\\mu = b \\,\\mathrm d |u_x^s|$ and $\\nu = b \\,\\mathrm d |h_x^s|$ implies that\n\\begin{equation*}\n\t\\int_E b \\,\\mathrm d |u_x^s| \\leq \\int_E b \\,\\mathrm d |h_x^s| \\text{ for any Borel } E \\subset U.\n\\end{equation*}\n \nIt is shown in Lemma \\ref{lem: ac sphere} below that $b(x) > 0$ for $|u_x^s|$-a.e. $x$, which implies absolute continuity of $|u_x^s|$ w.r.t $|h_x^s|$. Writing $|u_x^s|\/|h_x^s|$ for the Radon--Nikodym derivative of $|u_x^s|$ w.\\,r.\\,t.\\;$|h_x^s|$, we can see that the measure\n\\[\n\tb\\,\\left( 1 - \\frac{|u_x^s|}{|h_x^s|} \\right) \\,\\mathrm d |h_x^s|\n\\]\nis non-negative on Borel sets. Multiplying it by $\\varphi(\\omega(x)) \\, \\indi_{ \\left \\{x: \\, \\overline{u}_x^{ac}(x) \\in \\mathbb{S}_\\infty^{n-1} \\right\\}}(x)$, which is clearly non-negative, yields a measure\n\\[\n\t\\indi_{ \\left \\{x: \\, \\overline{u}_x^{ac}(x) \\in \\mathbb{S}_\\infty^{n-1} \\right\\}} \\, \\left( 1 - \\frac{|u_x^s|}{|h_x^s|} \\right) \\,\\mathrm d |h_x^s|\n\\]\nwhose non-negativity implies that for all Borel sets $E \\subset U$ one has $|u_x^s|(E) \\leq |h_x^s|(E)$, which concludes the proof. \n\n\n\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{lem: ac sphere}\n\tFor $|u_x^s|$-a.e. $x$ it is true that $\\overline{u}_x^{ac}(x) \\in \\mathbb{S}_\\infty^{n-1}$.\n\\end{lemma}\n\n\\begin{proof}\nChoose $x_0$ such that $|\\overline{u}_x^{ac}(x_0)| < \\infty$. Since $\\overline{u}_x^{ac}$ is continuous, there exists an open interval $J \\subset I$ such that for any $y \\in J$ it is true that $\\overline{u}_x^{ac}(y) \\in \\mathbb R^n$. Then for any $\\xi \\in C_c(J)$,\n\\begin{equation*}\n\t\\left | \\int_J u_x^{\\varepsilon} \\xi \\,\\mathrm d \\mathcal L^1 - \\int_J u_x^{ac} \\xi \\,\\mathrm d \\mathcal L^1 \\right | = \\left | \\int_J \\overline{u}_x^{\\varepsilon} \\xi \\,\\mathrm d \\mathcal L^1 - \\int_J \\overline{u}_x^{ac} \\xi \\,\\mathrm d \\mathcal L^1 \\right | \\leq \\sup_{y \\in J} |\\overline{u}_x^\\varepsilon(y) - \\overline{u}_x^{ac}(y) | \\int_J \\xi \\,\\mathrm d \\mathcal L^1 \\xrightarrow{\\varepsilon \\to 0} 0,\n\\end{equation*}\ndue to the facts that $\\overline{u}_x^{\\varepsilon}$ coincides with $u_x^\\varepsilon$ a.e. and $\\overline{u}_x^{ac}$ coincides with $u_x^{ac}$ a.e. and that $\\overline{u}_x^{\\varepsilon}$ converge in $C(\\overline{I}, \\overline{\\mathbb R^n})$ to $\\overline{u}_x^{ac}$. As a result, $u_x^\\varepsilon \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} u_x^{ac}$ on $J$. On the other hand, Lemma \\ref{lem: gamma conv eps} says that $u_x^\\varepsilon \\mathrel{\\ensurestackMath{\\stackon[0.5pt]{\\rightharpoonup}{\\scriptstyle\\ast}}} u_x$ on $J$, which implies that $u_x^s \\mres J = 0$ and, consequently, that\n\\[\n\t|u_x^s| \\mres J = 0,\n\\]\nwhich in turn means that $x_0 \\notin \\mathrm{supp}\\,|u_x^s|$ (the support of the measure $|u_x^s|$). We have therefore established that if $x_0 \\in \\mathrm{supp}\\,|u_x^s|$, then $\\overline{u}_x^{ac} \\in \\mathbb{S}_\\infty^{n-1}$. It remains to recall that the set $\\mathrm{supp}\\,|u_x^s|$ is of full $|u_x^s|$-measure, which concludes the proof. \\\\\n\\end{proof} \n\n\\noindent \n{\\bf \\Large Acknowledgements} \\\\ \n\nThe second author was partially supported by the grant nr.\\ 2020\/36\/C\/ST1\/00492 of the National Science Center, Poland. Part of this work was created during second author's JSPS Postdoctoral Fellowship at the University of Tokyo. \n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFor the last decade, lattice gauge theory techniques have made \ndramatic progress in increasing our understanding of the\nnon-perturbative regime of QCD \\cite{DeGrand:2007}. \nDespite considerable advances, there are still sources of systematic error in \nlattice data, for example, the finite extent of the lattice and the \nunphysically large quark masses. Fortunately low-energy hadron properties are\ndominated by virtual pion interactions and the systematic treatment of such \ninteractions using chiral perturbation theory ($\\chi$PT) allows one to\nparametrize the lattice volume and quark mass dependence of certain observables. \nThere has been considerable activity to\nunderstand theoretically the quark mass and lattice volume dependence of \nhadronic observables. \nFurther extentions of chiral perturbation theory have been developed to\naccount for quenching and partially quenching \\cite{Bernard:1992,Bernard:1994,\nSharpe:2000,Sharpe:2001fh}, and \ndiscretization errors \\cite{Sharpe:1998,Sharpe:1999}.\nAn example is the nucleon axial charge, $g_A$. \nRecent lattice studies have made impressive strides toward determining \n$g_A$ \\cite{Edwards:2005,Khan:2006}.\nIn tandem, recent $\\chi$PT analyses of the chiral \\cite{\ns&m:2002,Hemmert:2003,Hemmert:2006}, continuum \\cite{s&m:2003,fjj:2007}\nand volume extrapolations \\cite{s&m:2004,d&m:2004,s&w:2007} are poised to \nconnect the data to the physical point.\nWe are beginning to enter a stage in which the combination of lattice \nQCD data and $\\chi$PT will enable the study the \nhadronic properties from the first principles.\n\nA serious issue, however, confronts this program when extended to hyperon \nobservables. Various $SU(3)$ predictions for hyperon properties compare \npoorly to experiment in contrast to the many successful $SU(2)$ \npredictions for the nucleon. While $\\chi$PT can be used to systematically \nincorporate effects from the strange quark mass, the systematic expansion \nin the baryon sector has terms that scale (in the worst case) as \n$\\sim m_\\eta \/ M$, where $m_\\eta$ is the mass of the $\\eta$-meson\nand $M$ is the average hyperon mass. A well known conflict between \n$\\chi$PT analyses and experimental data exist for hyperon decays.\nFor example, the non-leptonic weak decays, $\\Lambda \\to p \\pi^-$\nand $\\Sigma^+ \\to n \\pi^+$, have been extensively investigated \nexperimentally. In particular the $s$- and $p$-wave \ncontributions to these weak decays are determined to high precision.\nAlthough efforts in the framework of $\\chi$PT have been devoted to understand \nthese non-leptonic decays theoretically\n\\cite{Jenkins:1991jv,\nJenkins:1991es,Jenkins:1992,\nSpringer:1995,m&w:1997,Borasoy:1999.0,Springer:1999.1,AbdElHady:1998gw,\nAH:1999,bbps:2005},\nlong-standing disagreement between these theoretical analyses and \nexperimental data remain \\cite{Gasp:2003,Hint:2003,Silas:2006.4}.\n\nOne is thus led to question the efficacy of three-flavor $\\chi$PT\nin the baryon sector. Without this systematic model-independent expansion, \nlattice QCD data for hyperon properties cannot be reliably extrapolated \nto the physical values of the quark masses. Additionally volume and continuum \nextrapolations using three-flavor {$\\chi$PT}\\ cannot be trusted.\nIndeed the first lattice \ncalculation of hyperon axial charges, \n$g_{\\Sigma\\Sigma}$ and $g_{\\Xi \\Xi}$ \\cite{Lin:2007gv,Lin:2007ap},\nshows little evidence for the one-loop predictions from (partially quenched) \n$\\chi$PT \\cite{will:2004}. The lattice, however, can provide a diagnostic tool \nto investigate the condition of three-flavor $\\chi$PT.\nA complete study of baryon axial charges is the natural starting point. \nThese couplings enter in the loop graphs that determine the long-range \nchiral corrections to all baryon observables. Input of these measured \nparameters into $\\chi$PT expressions allows one to numerically assess\nthe behavior of the long-range contributions in the chiral expansion.\nThis information can then be used to address the convergence of the \nchiral expansion. Perhaps the expansion is converging to the wrong answer, \nor perhaps the expansion is not converging at all. \nIf it is the latter case, one can use the lattice to investigate the\ncause. Perhaps certain observables are corrupted by \nlarge values of local contributions that can be isolated\nand determined from lattice data, or perhaps nearby resonances \nare leading to large enhancements.\n\nIn this work, we provide a follow-up to \\cite{will:2004}\nby determining the full set of hyperon matrix elements of flavor changing \naxial currents. We work to next-to-leading order in partially \nquenched heavy baryon chiral perturbation theory to address both the chiral\nbehavior and momentum-transfer dependence of the axial form factors.\nDue to meson pole contributions, the pseudoscalar form factor provides \nan observable \nwell-suited for the investigation of chiral physics in three-flavor theories. \nDespite the accumulation of a large number (eight) of undetermined \nlow-energy constants, we utilize the full set of axial charges to make \nnon-trivial next-to-leading order predictions.\n\nOur paper is organized as follows. First in Sect. II, aspects of\nPQ$\\chi$PT relevant to our calculations are reviewed. \nIn Sect. III, we map the PQQCD axial-vector current onto operators in \nPQ$\\chi$PT up to next-to-leading order. \nThe hyperon axial-current matrix elements are determined for \n$|{\\Delta} I| = 1$ transitions (Sect. III B), and $|{\\Delta} S| = 1$ transitions \n(Sect. III C). Various wavefunction renormalization factors\nare collected in the Appendix.\nNon-trivial next-to-leading order predictions for axial\ncharges, and a discussion of $SU(3)$\nbreaking corrections are presented in Sect. IV, which concludes our paper. \n\n\\section{Partially Quenched Chiral Lagrangian}\n\\label{chipt}\nBefore we detail the calculation of the axial current matrix elements, \nwe briefly review partially quenched chiral \nperturbation theory. \nWe recall the partially quenched chiral Lagrangian in \nthe meson sector first and emphasize the relation between lattice measured\nmeson masses and the parameters of the Lagrangian. The baryon \nLagrangian is then described in detail. \n\n\\subsection{Mesons}\nThe lattice action we consider here is comprised of valence and sea \nquarks, each of which comes in three flavors. In the continuum limit,\nthis action can be described by the partially quenched QCD (PQQCD) \nLagrange density, which is given by\n\\begin{equation}\n{\\mathcal L} = \\ol Q\\, iD\\hskip-0.65em \/ \\, Q - \\ol Q m_Q Q\\, ,\n\\label{pqqcd}\n\\end{equation} \nwhere the quark fields appear in the vector $Q$, which has entries\n\\begin{equation}\nQ = ( u, d, s, j, l, r, \\tilde{u}, \\tilde{d}, \\tilde{s} )^T\\,,\n\\label{3flavor}\n\\end{equation}\nand transforms in the fundamental representation of the graded group\n$SU(6|3)$. \nThe quark components of the field $Q$ satisfy the following \ngraded equal-time commutation relation\n\\begin{eqnarray}\nQ_i^\\alpha ({\\bf x}) Q_k^{\\beta \\dagger} ({\\bf y}) - \n(-)^{\\eta_i\\eta_k}Q_k^{\\beta \\dagger} ({\\bf y})Q_i^\\alpha ({\\bf x})\n& = & \n\\delta^{\\alpha\\beta}\\delta_{ik}\\delta^3({\\bf x}-{\\bf y})\\, ,\n\\label{eq:comm}\n\\end{eqnarray}\nwhere ($\\alpha$,$\\beta$) and ($i$,$k$) are spin and flavor indices \nrespectively.\nThe $\\eta_k$'s appearing above are given by\n$\\eta_k=+1$ for $k=1$--$6$ and $\\eta_k=0$ \nfor $k=7$--$9$. The $\\eta_{k}$ maintain the graded structure of the \nLie algebra. Further, the graded equal-time \ncommutation relations for two $Q$'s or two $Q^\\dagger$'s vanish. \nThe partially quenched generalization of the mass \nmatrix $m_Q$ is given by\n\\begin{equation}\nm_Q = \\text{diag} (m_u, m_d, m_s, m_j, m_l ,m_s, m_u, m_d, m_s)\\,.\n\\label{massmatrix1}\n\\end{equation}\nIn this work, we enforce the isospin limit in both the valence and sea \nsectors so that we have\n\\begin{equation}\nm_Q = \\text{diag} (\\bar{m}, \\bar{m}, m_s, m_j, m_j ,m_s, \\bar{m},\\bar{m}, m_s)\\,.\n\\label{massmatrix2}\n\\end{equation}\nNotice that with\nEq.~(\\ref{massmatrix1}) (and similarly Eq.~(\\ref{massmatrix2})), there is an \nexact cancelation \nbetween valence and ghost quark contributions to the determinant in the\npath integral for the QCD partition function. This cancelation leaves\nonly the contribution from the sea sector.\nWhen $m_Q = 0$, the Lagrangian Eq.~(\\ref{pqqcd})\nhas a graded $U(6|3)_{L} \\otimes U(6|3)_{R}$ symmetry which will reduce to\n$SU(6|3)_{L} \\otimes SU(6|3)_{R} \\otimes U(1)_{V}$ by the axial anomaly\n\\cite{Sharpe:2001fh}. We assume that the chiral symmetry is \nspontaneously broken:\n$SU(6|3)_{L} \\otimes SU(6|3)_{R} \\rightarrow SU(6|3)_{V}$,\nhence an identification between PQQCD and QCD \ncan be made. \nThe low-energy effective theory of PQQCD is written in \nterms of the pseudo-Goldstone mesons emerging from spontaneous chiral symmetry \nbreaking. At leading order in an expansion in momentum and quark mass, \n\\footnote{Here we adopt the standard power counting: \n$\\partial^2 \\sim m_q \\sim \\varepsilon^2$, where $\\varepsilon$ is a small\nparameter.} the \nPQ$\\chi$PT Lagrangian for the \nmesons is given by\n\\begin{equation} \\label{eq:Llead}\n{\\mathcal L} = \n\\frac{f^2}{8}\n\\text{str} \\Big(\\partial^{\\mu}\\Sigma^\\dagger \\partial_\\mu\\Sigma\\Big)\n \\,+ \\,\\lambda\\text{str}\\Big(m_q\\Sigma^\\dagger+m_q^\\dagger\\Sigma\\Big)\\,-\\, \nm_0^2\\Phi_0^2\\,,\n\\end{equation}\nwhere $f=132$~MeV, the str() denotes a graded flavor trace and the meson \nfields is incorporated in $\\Sigma$ through\n\\begin{equation}\\label{meson}\n \\Sigma=\\exp\\Bigg(\\frac{2i\\Phi}{f}\\Bigg)\n = \\xi^2 \\, , \\ \n \\Phi=\n \\Bigg(\n \\begin{array}{cc}\n M & \\chi^{\\dagger} \\\\ \n \\chi & \\tilde{M}\n \\end{array}\n \\Bigg)\n\\,.\\end{equation}\nThe matrices $M$, \n$\\tilde{M}$ in Eq.~(\\ref{meson}) contain bosonic mesons, while $\\chi$ and \n$\\chi^{\\dagger}$ are matrices consisting of fermionic mesons. \nHere $\\Phi_{0}\\,=\\,\\text{str}(\\Phi)\\sqrt{6}$ is the flavor singlet field and is \nincluded as a device to obtain the flavor neutral propagators in PQ$\\chi$PT.\nExpanding the Lagrangian in Eq.~\\eqref{eq:Llead}, \none can determine the meson masses which enter into the calculations of baryon \nobservables. In particular, the masses of mesons at leading order with \nquark content $Q_{i}\\overline{Q}'_{j}$ are\n\\begin{equation}\nm_{Q_{i} Q'_{j}}^2 = \\frac{4\\lambda}{f^2}((m_{Q})_{ii}\\,+\\,(m_{Q'})_{jj})\\, . \n\\label{qmass}\n\\end{equation}\nThe flavor singlet field additionally acquires a mass $m^2_{0}$.\nDue to the strong $U(1)_{A}$ anomaly, this mass can be taken on the \norder of the chiral symmetry breaking scale, $m_{0}\\sim \\Lambda_{\\chi}\\approx\n4\\pi f$. The flavor singlet field can thus be integrated out. \nHowever, the propagator of the flavor neutral fields deviate \nfrom a simple pole form \\cite{Sharpe:2001fh}. For \n$a,b = u,d,s$, the $\\eta_{a} \\eta_{b}$ propagator at leading order is given by \n\\cite{Sharpe:2001fh}\n\\begin{eqnarray}\n{\\cal G}_{\\eta_a\\eta_b} & = & \n{ i\\delta^{ab}\\over q^2-m_{aa}^2+i\\epsilon}\n\\ -\\ {i\\over 3}\n{(q^2-m_{jj}^2+i\\epsilon)(q^2-m_{rr}^2+i\\epsilon)\\over (q^2-m_{aa}^2+i\\epsilon)(q^2-m_{bb}^2+i\\epsilon)(q^2-m^2_{X}+i\\epsilon)}\n\\,,\n\\label{doublepole}\n\\end{eqnarray}\nwhere the masses of valence-valence mesons $m_{aa}^2$, $ m_{bb}^2$ and the\nmasses of the sea-sea mesons $m_{jj}^2$, $m_{rr}^2$ are \ngiven by Eq.~(\\ref{qmass}). In Eq.~(\\ref{doublepole}), the mass $m_X$ is \ndefined as $m^2_{X} = \\frac{1}{3}(m^2_{jj}+2m^2_{rr})$. The flavor \nneutral propagator Eq.~(\\ref{doublepole}) can be conveniently written in \nthe form\n\\begin{eqnarray}\n{\\cal G}_{\\eta_a\\eta_b} & = & \\delta^{ab} P_a\\ +\\ \n{\\cal H}_{aa}( P_a , P_b , P_X)\n\\,,\n\\label{doublecomp}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nP_a & = & { i \\over q^2 - m_{aa}^2+i\\epsilon}\n\\, \\, ,\\, \\, \nP_b \\, = \\, { i \\over q^2 - m_{bb}^2+i\\epsilon}\n\\, \\, ,\\,\\, \nP_X \\, = \\, { i \\over q^2 - m_X^2+i\\epsilon}\\,\\,,\n\\nonumber\\\\\n{\\cal H}_{ab}( A, B, C) & = & \n-{1\\over 3}\\left[\\ \n{(m_{jj}^2-m_{aa}^2)(m_{rr}^2-m_{aa}^2)\\over \n(m_{aa}^2-m_{bb}^2)(m_{aa}^2-m_X^2)}\\ A\n-\n{(m_{jj}^2-m_{bb}^2)(m_{rr}^2-m_{bb}^2)\\over \n(m_{aa}^2-m_{bb}^2)(m_{bb}^2-m_X^2)}\\ B\n\\right.\\nonumber\\\\ & & \\left.\\qquad\n\\ +\\ \n{(m_X^2-m_{jj}^2)(m_X^2-m_{rr}^2)\\over \n(m_X^2-m_{aa}^2)(m_X^2-m_{bb}^2)}\\ C\n\\ \\right]\n\\,.\n\\label{doublecomp1}\n\\end{eqnarray} \nThe above form is convenient for contributions from flavor neutral mixing.\nWhen there is no mixing, i.e. $a \\,=\\, b$ and $A \\,=\\, B$, \nthe hairpin propagator\nhas a double pole and the limit of Eq.~(\\ref{doublecomp1}) must be taken, \nand produces \n\\begin{eqnarray}\n{\\cal H}_{aa}(A,A,C) & = & \n-\\frac{1}{3} \\Bigg[\\, \\frac{\\partial}{\\partial {m^{2}_{aa}}} \\frac{(m^{2}_{jj}-m^{2}_{aa})(m^{2}_{rr}-m^{2}_{aa})}{(m^{2}_{aa}-m^{2}_{X})} A \\nonumber \\\\\n&& \\qquad \\qquad \\quad + \\frac{(m^{2}_{jj}-m^{2}_{X})(m^{2}_{rr}-m^{2}_{X})}{(m^{2}_{X}-m^{2}_{aa})^{2}} C\\,\\Bigg] \\ \\ .\n\\label{doubleisospin}\n\\end{eqnarray}\n\nIn partially quenched simulations, one numerically determines\nthe values of the valence pion $m_{\\pi,\\text{val}}$ and valence kaon\n$m_{K,\\text{val}}$ masses, as well as the sea pion $m_{\\pi,\\text{sea}}$\nand sea kaon $m_{K,\\text{sea}}$ masses. When one uses PQ$\\chi$PT to calculate \nthe meson mass dependence of observables, they are expressed in\nterms of meson masses via the tree-level relation in Eq.~(\\ref{qmass}). \nTo use the lattice determined meson masses in the valence and sea sectors,\nit is straightforward algebra to convert the loop meson masses appearing \nin PQ$\\chi$PT to those measured directly on the lattice.\nExplicitly we have\n\\begin{eqnarray}\nm_{uu}^2\n&=&\nm_{\\eta_u}^2\n=\nm_{\\pi,\\text{val}}^2\\,,\n\\nonumber \\\\\nm_{us}^2\n&=&\nm_{K,\\text{val}}^2\\,,\n\\nonumber \\\\\nm_{ss}^2\n&=&\n m_{\\eta_s}^2\n=\n2 m_{K,\\text{val}}^2 - m_{\\pi,\\text{val}}^2\\,,\n\\nonumber \\\\\nm_{uj}^2\n&=&\n\\frac{1}{2}\n\\left(\nm_{\\pi,\\text{val}}^2\n+\nm_{\\pi,\\text{sea}}^2\n\\right)\\,,\n\\nonumber \\\\\nm_{ur}^2\n&=&\n\\frac{1}{2}\n\\left(\nm_{\\pi,\\text{val}}^2\n-\nm_{\\pi,\\text{sea}}^2\n\\right)\n+\nm_{K,\\text{sea}}^2\\,,\n\\nonumber \\\\\nm_{sj}^2\n&=&\\frac{1}{2}\n\\left(\nm_{\\pi,\\text{sea}}^2\n-\nm_{\\pi,\\text{val}}^2\n\\right)\n+\nm_{K,\\text{val}}^2\\,,\n\\nonumber \\\\\nm_{sr}^2\n&=&\n- \\frac{1}{2}\n\\left(\nm_{\\pi,\\text{val}}^2\n+\nm_{\\pi,\\text{sea}}^2\n\\right)\n+\nm_{K,\\text{val}}^2\n+\nm_{K,\\text{sea}}^2\\,,\n\\nonumber \\\\\nm_{jj}^2\n&=&\nm_{\\pi,\\text{sea}}^2\\,,\n\\nonumber \\\\\nm_{rr}^2\n&=&\n2 m_{K,\\text{sea}}^2\n-\nm_{\\pi,\\text{sea}}^2\\,,\n\\nonumber \\\\\nm_X^2\n&=&\n\\frac{4}{3} m_{K,\\text{sea}}^2\n-\n\\frac{1}{3} m_{\\pi,\\text{sea}}^2\n\\,.\n\\label{convertlattice}\n\\end{eqnarray}\nThese relations must be modified if the source of partial quenching is due\nto mixed lattice actions, see \\cite{Bar:2005tu,Tiburzi:2005is,Chen:2007ug}.\n\\subsection{Baryon}\nIn this section, we discuss the baryon sector of PQ$\\chi$PT in the framework\nof \\cite{Labrenz:1996jy,MSavage:2002,MSavage:2002.1}. \nBuilding blocks for the baryon Lagrangian are the \nsuper-multiplets ${\\mathcal B}_{ijk}$ and ${\\mathcal T}^{\\mu}_{ijk}$. The \n$\\bf{240}$-dimensional super-multiplet of spin-$\\frac{1}{2}$ baryons \n${\\mathcal B}_{ijk}$ satisfies the following \nrelations under the interchange of the flavor indices \\cite{Labrenz:1996jy}\n\\begin{eqnarray}\n{\\cal B}_{ijk} & = & (-)^{1+\\eta_j \\eta_k}\\, {\\cal B}_{ikj}\n\\, , \\,\n{\\cal B}_{ijk} \\, +\\, (-)^{1+\\eta_i \\eta_j}\\, {\\cal B}_{jik}\n\\, +\\, (-)^{1 + \\eta_i\\eta_j + \\eta_j\\eta_k + \\eta_k\\eta_i}\\, \n{\\cal B}_{kji}\\, =\\, 0\\,,\n\\label{eq:bianchi1}\n\\end{eqnarray}\nand the familiar octet baryons are embeded in ${\\mathcal B}_{ijk}$ through\n\\cite{MSavage:2002,MSavage:2002.1}\n\\begin{equation}\n{\\cal B}_{ijk} = \\frac{1}{\\sqrt{6}}(\\epsilon_{ijl}B_{k}^{l} +\n\\epsilon_{ikl}B_{j}^{l})\\,,\n\\end{equation} \nwhere $B$ is the octet baryon matrix\n\\begin{equation}\n B = \n \\begin{pmatrix}\n \\frac{1}{\\sqrt{6}}\\Lambda + \\frac{1}{\\sqrt{2}}\\Sigma^0 &\n \\Sigma^+ & p\\\\\n \\Sigma^- & \\frac{1}{\\sqrt{6}}\\Lambda -\n \\frac{1}{\\sqrt{2}}\\Sigma^0 & n\\\\\n \\Xi^- & \\Xi^0 & -\\frac{2}{\\sqrt{6}}\\Lambda\\\\\n \\end{pmatrix} \\,.\n\\label{eq:octetbaryons}\n\\end{equation}\nThe spin-$\\frac{3}{2}$ resonances are contained in the\nthe $\\bf{138}$-dimensional super-multiplet ${\\mathcal T}_{ijk}$, which\nsatisfies \n\\begin{eqnarray}\n{\\cal T}_{ijk} \\, = \\, \n(-)^{1+\\eta_i\\eta_j} {\\cal T}_{jik}\\, =\\, \n(-)^{1+\\eta_j\\eta_k} {\\cal T}_{ikj} \\,,\n\\label{eq:bianchi2}\n\\end{eqnarray}\nunder the interchange of flavor indices \\cite{Labrenz:1996jy}. \nFurthermore, one embeds the decuplet baryons in ${\\mathcal T}_{ijk}$ by \n\\begin{equation}\n\\ \\ \\ \\ {\\cal T}_{ijk} = T_{ijk} \\,, \n\\end{equation}\nwhere $T$ is totally symmetric tensor containing the decuplet resonances\n\\begin{eqnarray}\nT_{111} \\,& = &\\, \\Delta^{++}\n\\,\\, ,\\,\\,\nT_{112} \\, =\\, {1\\over\\sqrt{3}}\\Delta^+\n\\,\\, ,\\,\\,\nT_{122} \\, =\\, {1\\over\\sqrt{3}}\\Delta^0\n\\,\\, , \\,\\, \nT_{222} \\, =\\, \\Delta^{-}\\,,\n\\nonumber\\\\ \nT_{113} \\,& = &\\, {1\\over\\sqrt{3}}\\Sigma^{*,+}\n\\,\\, ,\\,\\, \nT_{123} \\, =\\, {1\\over\\sqrt{6}}\\Sigma^{*,0}\n\\,\\, ,\\,\\, \nT_{223} \\, =\\, {1\\over\\sqrt{3}}\\Sigma^{*,-}\\,,\n\\nonumber\\\\\nT_{133} \\,& = &\\, {1\\over\\sqrt{3}}\\Xi^{*,0}\n\\,\\, ,\\,\\, \nT_{233} \\, =\\, {1\\over\\sqrt{3}}\\Xi^{*,-}\n\\,\\, , \\,\\, \nT_{333} \\, =\\, \\Omega^{-}\n\\, .\n\\label{eq:decupletbaryon}\n\\end{eqnarray}\nThe free Lagrangian for the $\\bf{240}$-dimensional super-multiplet ${\\mathcal B}_{ijk}$ \nand the $\\bf{138}$-dimensional super-multiplet ${\\mathcal T}_{ijk}$ fields in \n$SU(6|3)$ PQ$\\chi$PT is given by \n\\cite{MSavage:2002.1}\n\\begin{eqnarray} \\label{eqn:L}\n {\\mathcal L}\n &=&\n i\\Big(\\ol{\\mathcal B} v\\cdot{\\mathcal D}{\\mathcal B}\\Big)\n +2{\\alpha}_{M}\\Big(\\ol{\\mathcal B} {\\mathcal B}{\\mathcal M}_+\\Big)\n +2{\\beta}_{M}\\Big(\\ol{\\mathcal B} {\\mathcal M}_+{\\mathcal B}\\Big)\n +2\\sigma_{M}\\Big(\\ol{\\mathcal B}\\cB\\Big)\\text{str}\\Big({\\mathcal M}_+\\Big) \\nonumber \\\\\n &-&i\\Big(\\ol{\\mathcal T}^{\\mu} v\\cdot{\\mathcal D}{\\mathcal T}_\\mu\\Big)\n +{\\Delta}\\Big(\\ol{\\mathcal T}^{\\mu}{\\mathcal T}_\\mu\\Big)\n +2{\\gamma}_{M}\\Big(\\ol{\\mathcal T}^{\\mu} {\\mathcal M}_+{\\mathcal T}_\\mu\\Big)\n -2\\ol\\sigma_{M}\\Big(\\ol{\\mathcal T}^{\\mu}{\\mathcal T}_\\mu\\Big)\\text{str}\\Big({\\mathcal M}_+\\Big)\n \\,,\n\\end{eqnarray}\nwhere the mass operator ${\\mathcal M}_{+}$ is defined by:\n\\begin{equation}\n{\\mathcal M}_+ = \\frac{1}{2}\\left(\\xi^\\dagger m_Q \\xi^\\dagger + \\xi m_Q \\xi\\right)\n\\, .\\end{equation}\nThe parameter $\\Delta$ is the mass splitting between the octet \nand decuplet baryons in the chiral limt.\nPhenomenologically we know $\\Delta \\sim m_{\\phi}$, where $\\phi$ is an $SU(3)$\nmeson, hence the decuplet baryons much be included as dynamical fields in\nEq.~(\\ref{eqn:L}).\nThe parenthesis notation for flavor contractions used in Eq.~\\eqref{eqn:L} \nis that of~\\cite{MSavage:2002.1}.\nThe partially quenched Lagrangian describing the interactions of the \n${\\mathcal B}_{ijk}$ and ${\\mathcal T}^{\\mu}_{ijk}$ with the pseudo-Goldstone mesons is given by\n\\cite{MSavage:2002.1}\n\\begin{eqnarray} \\label{eqn:Linteract}\n {\\cal L} &=& \n\t 2 {\\alpha} \\Big(\\ol {\\mathcal B} S^{\\mu} {\\mathcal B} A_\\mu \\Big)\n\t+ 2 {\\beta} \\Big(\\ol {\\mathcal B} S^{\\mu} A_\\mu {\\mathcal B} \\Big)\n\t+ 2{\\mathcal H}\\Big(\\ol{{\\mathcal T}}^{\\nu} S^{\\mu} A_\\mu {\\mathcal T}_\\nu\\Big) \\nonumber \\\\ \n \t&+& \\sqrt{\\frac{3}{2}}{\\mathcal C}\n \t\t\\Big[\n \t\t\t\\Big(\\ol{{\\mathcal T}}^{\\nu} A_\\nu {\\mathcal B}\\Big)+ \\Big(\\ol {\\mathcal B} A_\\nu {\\mathcal T}^{\\nu}\\Big)\n \t\t\\Big] \\,. \n\\label{interL}\n\\end{eqnarray}\nThe axial-vector and vector meson fields $A_\\mu$ and $V_\\mu$\nare defined by: $ A_{\\mu}=\\frac{i}{2}\n\\left(\\xi\\partial_{\\mu}\\xi^\\dagger-\\xi^\\dagger\\partial_{\\mu}\\xi\\right)$ \nand $V_{\\mu}=\\frac{1}{2} \\left(\\xi\\partial_{\\mu}\\xi^\\dagger+\\xi^\\dagger\\partial_{\\mu}\\xi\\right)$.\nThe latter appears in Eq.~\\eqref{eqn:L} for the\ncovariant derivatives of ${\\mathcal B}_{ijk}$ and ${\\mathcal T}_{ijk}$ \nthat both have the form\n\\begin{equation}\n ({\\mathcal D}^{\\mu} {\\mathcal B})_{ijk}\n =\n \\partial^{\\mu} {\\mathcal B}_{ijk}\n +(V^{\\mu})^{l}_{i}{\\mathcal B}_{ljk}\n +(-)^{\\eta_i(\\eta_j+\\eta_m)}(V^\\mu)^{m}_{j}{\\mathcal B}_{imk}\n +(-)^{(\\eta_i+\\eta_j)(\\eta_k+\\eta_n)}(V^{\\mu})^{n}_{k}{\\mathcal B}_{ijn}\\, .\n\\end{equation}\nThe vector $S_{\\mu}$ is the covariant spin operator \\cite{Jenkins:1991jv,\nJenkins:1991es}. The parameters that appear in the {PQ$\\chi$PT}\\ Lagrangian can be \nrelated to those in {$\\chi$PT}\\ by matching. To be more specific, one restricts to \nthe $q_{\\text{sea}}q_{\\text{sea}}q_{\\text{sea}}$ sector and compares the\n{PQ$\\chi$PT}\\ Lagrangian obtained with that \nof $\\chi$PT. With this matching procedure, one finds that\n$\\alpha = \\frac{2}{3}D + 2F$, $\\beta = -\\frac{5}{3}D + F$, and the other \nparameters ${\\cal C}$ and ${\\cal H}$ appearing above have the same \nnumerical values as in $\\chi$PT \\cite{MSavage:2002.1}.\n\n\n\\section{The Axial-Vector Current} \n\\label{accurent}\n\\subsection{The Axial-Vector Current in PQ$\\chi$PT}\n\\label{accurentpqcpt}\nThe baryon matrix elements of the axial-vector current, \n$j^{a}_{\\mu,5} = \\overline{q}\\lambda^{a}\\gamma_\\mu\\gamma_5 q$, have been studied \nextensively both \non the lattice \\cite{Edwards:2005,Khan:2006} and \n$\\chi$PT \\cite{Jenkins:1991jv,Jenkins:1991es,Kim&Kim:1996,\nBorasoy:1999,Zhu:2001,s&m:2002,s&m:2004,dwill:2002,d&m:2004,will:2004,\nKambor:1999,Bernard:1998gv, Hemmert:2003,Meissner:2006,Hemmert:2006}. \nIn PQQCD, the axial\ncurrent is defined by $J^{a}_{\\mu,5} = \n\\overline{Q}\\,\\overline{\\lambda}{}^{a}\\gamma_\\mu\\gamma_5 Q$. \nIn general, one must worry that the choice of supermatrices\n$\\overline{\\lambda}{}^{a}$ is not unique even after the \nrequirement $\\text{str}(\\overline{\\lambda}{}^{a}) = 0$ \nhas been enforced. To be relevant for any practical lattice calculation, the\nchoice of PQQCD matrices should maintain the cancelation of valence and ghost\nquark loops with an operator insertion \\cite{will:2004,bct0412}. This is\nbecause otherwise the PQQCD theory corresponds to a lattice theory where twice\nthe number of disconnected contractions must be calculated. However \nsince we are only interested in flavor-changing operators, the self \ncontractions of these\noperators automatically vanish. Thus we can decouple the ghost and sea quarks\nsectors from the flavor changing axial current by choosing the upper\n$3\\times3$ block of $\\overline{\\lambda}$ to be the Gell-Mann matrices. This\nchoice merely corresponds to an axial transition operator that only acts in\nthe valence sector and is precisely what is implemented on the\nlattice.\\footnote{Isospin symmetry allows one to relate isospin \ntransition matrix elements to differences of flavor conserving matrix elements.\nThese difference have often been calculated on the lattice.\nFor strangeness transitions, $SU(3)$ is badly broken disallowing\nthe analogous procedure.}\n\nHaving fixed the $\\overline{\\lambda}$ supermatrices, we map the PQQCD axial\ncurrent operator into the heavy baryon PQ$\\chi$PT. At leading order, the \nPQ$\\chi$PT axial current is given by \\cite{s&m:2002}\n\\begin{eqnarray}\nJ_{\\mu,5}^{a}\n& = & \n2\\alpha\\Big(\\overline{\\cal B} S_\\mu {\\cal B}{\\overline{\\lambda}{}^{a}_{\\xi +}}\n\\Big)\\,+\\, \n2\\beta\\Big(\\overline{\\cal B} S_\\mu {\\overline{\\lambda}{}^{a}_{\\xi +}}{\\cal B}\n\\Big)\\,+\\, \n2{\\cal H}\\Big(\\overline{\\cal T}^\\nu S_\\mu {\\overline{\\lambda}{}^{a}_{\\xi +}}{\\cal T}_\\nu\\Big)\\nonumber\\\\\n&\\,+\\,&\n\\sqrt{3\\over 2}{\\cal C} \n\\Big[\\Big(\\overline{\\cal T}_\\mu {\\overline{\\lambda}{}^{a}_{\\xi +}} {\\cal B}\\Big)\n\\,+\\,\\Big(\\overline{\\cal B}\\,{\\overline{\\lambda}{}^{a}_{\\xi +}} {\\cal T}_\\mu\\Big)\n\\Big].\n\\label{eq:LOaxialcurrent}\n\\end{eqnarray}\nwith $\\alpha$, $\\beta$, ${\\cal H}$ and ${\\cal C}$ the same low energy constants\n(LEC's) appearing in Eq.~(\\ref{eqn:Linteract}) and \n$\\overline{\\lambda}{}^a_{\\xi +} = {1\\over 2}(\n\\xi\\overline{\\lambda}{}^a\\xi^\\dagger\n+\\xi^\\dagger\\overline{\\lambda}{}^a\\xi)$. \nSince we work to next-to-leading order (NLO) in the chiral expansion\nand NLO in the momentum expansion, \nwe further require the contributions to the matrix elements from NLO\naxial current. \nAt NLO, there are two contributions to the axial matrix \nelements: one is from the NLO axial current in the baryon sector and the other\nis obtained from the local counterterms involving one insertion of the quark \nmass matrix $m_{Q}$.\nThe former is given by\n\\begin{eqnarray}\nJ^{a}_{\\mu,5} \n&=&\n\\frac{1}{{\\Lambda}_\\chi^2} \n\\Bigg\\{ \n2 n_{\\alpha} \n\\Big[\n\\partial_{\\mu} \\partial_{\\nu} \n\\Big( \\overline{{\\mathcal B}} S^\\nu {\\mathcal B} \\, \\overline{\\lambda}^{a}_{\\xi^{+}} \\Big)\n-\n\\partial^2\n\\Big( \\overline{{\\mathcal B}} S_\\mu {\\mathcal B} \\, \\overline{\\lambda}^{a}_{\\xi^{+}} \\Big)\n\\Big] \\nonumber \\\\\n& + &\n2 n_{\\beta} \n\\Big[\n\\partial_{\\mu} \\partial_{\\nu} \n\\Big( \\overline{{\\mathcal B}} S^\\nu \\, \\overline{\\lambda}^{a}_{\\xi^{+}} {\\mathcal B} \\Big)\n-\n\\partial^2\n\\Big( \\overline{{\\mathcal B}} S_\\mu \\, \\overline{\\lambda}^{a}_{\\xi^{+}} {\\mathcal B} \\Big)\n\\Big]\n\\Bigg\\}\\,.\n\\label{nloPQ}\n\\end{eqnarray}\nwhile the latter reads \\cite{s&m:2002}:\n\\begin{eqnarray}\nJ_{\\mu,5}^{a,m_{Q}}&=& \n16\\frac{\\lambda}{f^2}\\Big[ \nb_1 {\\overline{\\cal B}}^{kji}\\{ \\overline{\\lambda}^a_{\\xi +}\\,,\\, \n{\\cal M}_+\\}^n_i\\ S_\\mu {\\cal B}_{njk}\n+b_2 (-)^{(\\eta_i+\\eta_j)(\\eta_k+\\eta_n)} \n{\\overline{\\cal B}}^{kji}\\{\\overline{\\lambda}^a_{\\xi +}\\,,\\, {\\cal M}_+\\}^n_k \n S_\\mu {\\cal B}_{ijn}\\nonumber\\\\\n&+& b_3(-)^{\\eta_l (\\eta_j+\\eta_n)}\n{\\overline{\\cal B}}^{kji} (\\overline{\\lambda}^a_{\\xi +})^l_i \n({\\cal M}_+)^n_j S_\\mu {\\cal B}_{lnk} \\nonumber\\\\\n&+& b_4(-)^{\\eta_l \\eta_j + 1} \n{\\overline{\\cal B}}^{kji} \\Big( \n(\\overline{\\lambda}^a_{\\xi +})^l_i ( {\\cal M}_+)^n_j\n+( {\\cal M}_+)^l_i (\\overline{\\lambda}^a)^n_j \\Big)S_\\mu {\\cal B}_{nlk}\n\\nonumber\\\\\n&+&b_5 (-)^{\\eta_i(\\eta_l+\\eta_j)}\n{\\overline{\\cal B}}^{kji}(\\overline{\\lambda}^a_{\\xi +})^l_j \n( {\\cal M}_+)^n_i S_\\mu {\\cal B}_{nlk}\n+b_6 {\\overline{\\cal B}}^{kji} (\\overline{\\lambda}^a_{\\xi +})^l_i S_\\mu {\\cal B}_{ljk}\n\\ {\\rm str}( {\\cal M}_+) \\nonumber\\\\\n&+& b_7(-)^{(\\eta_i+\\eta_j)(\\eta_k+\\eta_n)}\n{\\overline{\\cal B}}^{kji} (\\overline{\\lambda}^a_{\\xi +})^n_k S_\\mu {\\cal B}_{ijn}\n\\ {\\rm str}( {\\cal M}_+) \n\\nonumber\\\\\n&+& b_8 {\\overline{\\cal B}}^{kji} S_\\mu {\\cal B}_{ijk} \n\\ {\\rm str}(\\overline{\\lambda}^a_{\\xi +} {\\cal M}_+ ) \\Big]\\,,\n\\label{eq:axcts}\n\\end{eqnarray}\nwhere the coefficients $b_1,b_2.....,b_8$ must be determined from lattice\nsimulations. The relation between the partially quenched parameters \n$n_{\\alpha}\\,,n_{\\beta}$ \nand the physical parameters $n_{D}\\,,n_{F}$ in usual $SU(3)$ $\\chi$PT can be \nobtained by matching: \n$n_{\\alpha} = \\frac{2}{3}n_{D} + 2n_{F}$, \n$n_{\\beta} = -\\frac{5}{3}n_{D} + n_{F}$.\nNotice that the operator $b_8 {\\overline{\\cal B}}^{kji} S_\\mu {\\cal B}_{ijk} \n\\ {\\rm str}(\\overline{\\lambda}^a_{\\xi +} {\\cal M}_+ )$ does not contribute \nto the flavor changing transitions at\ntree-level. This leaves seven independent partially quenched NLO operators, \none more than that in ordinary $SU(3)$. However, because these counterterms only\ncontribute to tree-level, no unphysical combinations will be introduced.\n\\begin{figure}\n\\begin{center}\n\\hbox{\n\\includegraphics[width=0.4\\textwidth]{octet_axial.eps}\n~~~~~~~~\n\\includegraphics[width=0.4\\textwidth]{octet_axial_hairpin.eps}\n}\n\\end{center}\n\\caption{One-loop diagrams which contribute to \nthe leading non-analytic terms of the octet baryon axial form factors. \nMesons are represented by a dashed line while the single \nand double lines are the symbols for an octet and a decuplet respectively. \nThe solid circle is an insertion of the axial current operator and the\nsolid squares are the couplings given in Eq.~(\\ref{interL}). \nThe wave function renormalization diagrams are depicted \nin the bottom row. The diagrams with a cross on the loop meson are \nthe hairpin contributions which arise from the flavor neutral meson \npropagators.}\n\\label{fig0}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\hbox{\n\\includegraphics[width=0.4825\\textwidth]{octet_axial_pseudo.eps}\n~~~~\n\\includegraphics[width=0.4825\\textwidth]{octet_axial_pseudo_hairpin.eps}\n}\n\\end{center}\n\\caption{One-loop diagrams which contribute to \nthe leading non-analytic terms of the octet baryon induced pseudoscalar \nform factors. Diagram elements are the same as FIG.~\\ref{fig0}} \n\\label{fig1}\n\\end{figure}\n\n\\subsection{Isospin Changing Transitions}\n\\label{IST}\nIn the partially quenched theory, the supermatrix for the $\\Delta I \\,=\\,1$ \nisospin changing transitions is\n\\begin{equation} \\label{eqn:tau}\n\\overline{\\lambda}^{1+2i}_{i j} = \n\\begin{cases}\n 1 \\quad i = 1\\, , j = 2 \\\\\n 0 \\quad {\\text{otherwise}}\n\\end{cases}\n\\,.\\end{equation}\nWithin the baryon octet there are six isospin $\\Delta I = 1$ changing \ntransitions, namely\n\\begin{equation}\nn \\rightarrow p\\,\\,,\\,\\,\\Sigma^{0} \\rightarrow \\Sigma^{+}\\,\\,,\\,\\,\n\\Sigma^{-} \\rightarrow \\Sigma^{0}\\,\\,,\\,\\,\\Lambda \n\\rightarrow \\Sigma^{+}\\,\\,,\\,\\,\n\\Sigma^{-} \\rightarrow \\Lambda\\,\\,,\\,\\,\\Xi^{-} \\rightarrow \\Xi^{0}\\,\\,.\n\\end{equation}\nThe neutron to proton axial transition defines the nucleon axial form factor\n$G_{A,NN}(q^2)$\nand induced pseudoscalar form factor $G_{P,NN}(q^2)$ \n\\begin{equation}\n\\langle p (P') | J^{1+2i}_{\\mu,5} | n\n(P)\\rangle\\,=\\,\\overline{U}_{N}(P')\n\\Big[2S_{\\mu}G_{A,NN}(q^2)+\\frac{q_{\\mu} q \\cdot S}{(2m_{N})^2}G_{P,NN}(q^2) \\Big]U_{N}(P)\\,,\n\\end{equation}\nwhere $q_{\\mu} = (P' - P)_{\\mu}$ is the four momentum transfer. \nThe $\\Sigma$$\\Sigma$ transition matrix elements define the $\\Sigma$$\\Sigma$ \naxial form factor $G_{A,\\Sigma\\Sigma}(q^2)$ and induced pseudoscalar form \nfactor $G_{P,\\Sigma\\Sigma}(q^2)$\n\\begin{equation}\n\\langle \\Sigma^{0} (P') | J^{1+2i}_{\\mu,5} | \\Sigma^{-}\n(P)\\rangle\\,=\\,\\frac{1}{\\sqrt{2}}\\overline{U}_{\\Sigma}(P')\n\\Big[2S_{\\mu}G_{A,\\Sigma\\Sigma}(q^2)+\\frac{q_{\\mu} q \\cdot S}{(2m_{\\Sigma})^2}G_{P,\\Sigma\\Sigma}(q^2) \\Big]U_{\\Sigma}(P)\\,.\n\\end{equation}\nWhile there are two $\\Sigma$$\\Sigma$ isospin transitions, their matrix \nelements are related by isospin algebra\n\\begin{equation}\n\\langle \\Sigma^{+}(P') | J^{1+2i}_{\\mu,5} | \n\\Sigma^{0}(P)\\rangle\\,=\\,\n-\\langle \\Sigma^{0} (P')| J^{1+2i}_{\\mu,5} | \\Sigma^{-}(P)\\rangle\\,.\n\\end{equation}\nThe $\\Lambda$$\\Sigma$ transition matrix elements define the \n$\\Lambda$$\\Sigma$ axial form factor $G_{A,\\Lambda\\Sigma}$\nand induced pseudoscalar form factor $G_{P,\\Lambda\\Sigma}(q^2)$\n\\begin{equation}\n\\langle \\Lambda (P') | J^{1+2i}_{\\mu,5} | \\Sigma^{-}\n(P)\\rangle\\,=\\,\\frac{1}{\\sqrt{6}}\\overline{U}_{\\Lambda}(P')\n\\Big[2S_{\\mu}G_{A,\\Lambda\\Sigma}(q^2)+\\frac{q_{\\mu} q \\cdot S}{(m_{\\Lambda}+m_{\\Sigma})^2}G_{P,\\Lambda\\Sigma}(q^2) \\Big]U_{\\Sigma}(P)\\,.\n\\end{equation}\nAlthough there are two $\\Lambda$$\\Sigma$ transition matrix elements, they\nare related by isospin\n\\begin{equation}\n\\langle \\Sigma^{+}(P') | J^{1+2i}_{\\mu,5} | \\Lambda(P)\\rangle\\,=\\,\n\\langle \\Lambda (P')| J^{1+2i}_{\\mu,5} | \\Sigma^{-}(P)\\rangle\\,.\n\\end{equation}\nFinally, the $\\Xi$$\\Xi$ axial form factor $G_{A,\\Xi\\Xi}$ and induced\npseudoscalar form factor $G_{P,\\Lambda\\Sigma}(q^2)$\nappear in the $\\Xi$$\\Xi$ transition matrix element\n\\begin{equation}\n\\langle \\Xi^{0} (P') | J^{1+2i}_{\\mu,5} | \\Xi^{-}\n(P)\\rangle\\,=\\,\\overline{U}_{\\Xi}(P')\n\\Big[2S_{\\mu}G_{A,\\Xi\\Xi}(q^2)+\\frac{q_{\\mu} q \\cdot S}{(2m_{\\Xi})^2}G_{P,\\Xi\\Xi}\n(q^2) \\Big]U_{\\Xi}(P)\\,.\n\\end{equation}\nHere we use heavy baryon spinors and notation. One can easily show \nup to recoil corrections,\n$2\\overline{U}(P') S_{\\mu} U (P)= \\overline{U}(P')\\gamma_{\\mu} \\gamma_{5}\nU(P)$, where on the right-hand side appears ordinary Dirac matrices and \nspinors. Thus the axial charges, $G_{A,B'B}(0)$, are the \nstandard ones. In our power counting, while the tree-level contributions \nfrom LO axial current is of order $\\varepsilon^{0}$, the tree-level \ncontributions obtained from NLO current count $\\varepsilon^2$. In addition, \nat order $\\varepsilon^2$\nthere are leading non-analytic contributions to the matrix elements from the\none-loop diagrams shown in Fig.~\\ref{fig0} and Fig.~\\ref{fig1}. \nThe one-loop diagrams in Fig.~\\ref{fig0} contribute to the axial form factors\nwhile the induced pseudoscalar form factors receive contributions\nfrom the one-loop diagrams in Fig.~\\ref{fig1}. \nEvaluation of the diagrams in Fig.~\\ref{fig0} together with the tree-level \ncontributions yields the following expression for the axial form factor \nof $\\Delta I \\,=\\,1$ isospin transitions\n\\begin{eqnarray}\nG_{A,B'B}(q^2) & = &\\, g_{B'B}\\sqrt{Z_{B'}Z_{B}} + \\frac{1}{16\\pi^2f^2}\n\\Bigg[ g_{B'B} \\sum_{\\phi}C_{\\phi}{\\cal L}(m_{\\phi},\\mu)\n\\nonumber \\\\\n& + & {\\cal H}{\\cal C}^2\\Big(\\sum_{\\phi} E_{\\phi}{\\cal J}(m_{\\phi},\\Delta,\\mu)\n + \\sum_{\\phi\\phi^{\\prime}} \\bar{E}_{\\phi\\phi^{\\prime}}{\\cal T}(\\eta_{\\phi}\n \\eta_{\\phi'},\\Delta,\\mu)\\Big)\\nonumber \\\\\n& + & \n{\\mathcal C}^2 \n\\Big(\\sum_{\\phi}A_{\\phi}{\\cal K}(m_{\\phi},\\Delta,\\mu)\n + \\sum_{\\phi\\phi^{\\prime}} \\bar{A}_{\\phi\\phi^{\\prime}}\n{\\cal S}(\\eta_{\\phi} \\eta_{\\phi^{\\prime}},\\Delta,\\mu)\n\\Big) \\nonumber \\\\\n& + &\\sum_{\\phi} Y_{\\phi}{\\cal L}(m_{\\phi},\\mu)\n + \\sum_{\\phi\\phi^{\\prime}} \\bar{Y}_{\\phi,\\phi^{\\prime}}{\\cal\n R}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu) \\Bigg] \\nonumber \\\\ \n& + & n_{B'B}\\frac{q^2}{\\Lambda_{\\chi}^2} + \\sum_{\\phi}u_{\\phi}m_{\\phi}^2\n\\,. \n\\label{matrixiso}\n\\end{eqnarray}\nIn Eq.~(\\ref{matrixiso}), $B$ ($B'$) stands for\nthe initial (final) octet baryon, $Z_{B}$ and $Z_{B'}$ are \nthe wave function renormalization factors and are given in the\nAppendix. The constants $g_{B'B}$'s are the leading order octet \nbaryon axial charges\n\\begin{eqnarray}\ng_{NN} &=& (D\\,+\\,F)\\,, \\,\\, g_{\\Lambda\\Sigma} = 2D\\,, \n\\,\\, g_{\\Xi\\Xi} = (D\\,-\\,F)\\,,\\,\\,\ng_{\\Sigma\\Sigma} = 2F\\,, \n\\end{eqnarray} \nand the coefficients $n_{B'B}$ are given by\n\\begin{eqnarray}\nn_{NN} = n_{D}\\,+\\,n_{F}\\,,\\,\\,\nn_{\\Lambda\\Sigma} = 2n_{D}\\,,\\,\\, \nn_{\\Xi\\Xi} = n_{D}\\,-\\,n_{F}\\,,\\,\\,\nn_{\\Sigma\\Sigma} = 2n_{F}\\,. \n\\end{eqnarray} \nThe coefficeints $C_{\\phi}$, $E_{\\phi}$, $\\bar{E}_{\\phi\\phi'}$, $A_{\\phi}$, \n$\\bar{A}_{\\phi\\phi'}$, $Y_{\\phi}$, $\\bar{Y}_{\\phi\\phi'}$\nare given in tables \\ref{tabi1}, \\ref{tabi2},\n\\ref{tabi3}, \\ref{tabi5} and \\ref{tabi7} while the \nnon-analytic functions\nappearing in Eq.~(\\ref{matrixiso}), namely, \n${\\cal L}$'s, ${\\cal J}$'s, ${\\cal K}$'s , ${\\cal R}$'s, ${\\cal T}$'s and\n${\\cal S}$'s are given by\n\\begin{eqnarray}\n{\\cal L}(m,\\mu) = m^{2}\\log\\Big(\\frac{m^{2}}{\\mu^2}\\Big) \\,, \\nonumber \\\\\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal K}(m,\\Delta,\\mu) & = & \\Big(m^2-{2\\over 3}\\Delta^2\\Big)\\log\\Big({m^2\\over\\mu^2}\\Big) \\nonumber \\\\\n&& \\quad\\,+\\, {2\\over 3}\\Delta \\sqrt{\\Delta^2-m^2}\n\\log\\Big({\\Delta-\\sqrt{\\Delta^2-m^2+ i \\epsilon}\\over\n\\Delta+\\sqrt{\\Delta^2-m^2+ i \\epsilon}}\\Big)\n\\nonumber\\\\\n& & \\quad \\, +\\, {2\\over 3} {m^2\\over\\Delta} \\Big(\\ \\pi m - \n\\sqrt{\\Delta^2-m^2}\n\\log\\Big({\\Delta-\\sqrt{\\Delta^2-m^2+ i \\epsilon}\\over\n\\Delta+\\sqrt{\\Delta^2-m^2+ i \\epsilon}}\\Big)\n\\Big)\\,,\n\\label{eq:Kdecfun}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal J}(m,\\Delta,\\mu) & = & \\Big(m^2-2\\Delta^2\\Big)\\log\\Big({m^2\\over\\mu^2}\\Big) \\quad \\qquad \\nonumber \\\\\n&&\\quad \\,+\\,2\\Delta\\sqrt{\\Delta^2-m^2}\n\\log\\Big({\\Delta-\\sqrt{\\Delta^2-m^2+ i \\epsilon}\\over\n\\Delta+\\sqrt{\\Delta^2-m^2 + i \\epsilon}}\\Big)\\,,\n\\label{eq:decfun}\n\\end{eqnarray}\n\\begin{eqnarray} \n{\\cal R}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu) &=& {\\cal H}({\\cal\n L}(m_{\\eta_{\\phi}},\\mu),{\\cal L}(m_{\\eta_{\\phi'}},\\mu),\n{\\cal L}(m_{X},\\mu))\\,,\\nonumber \\\\ \n{\\cal T}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu) &=& {\\cal H}({\\cal J}(m_{\\eta_{\\phi}},\\Delta,\\mu),\n{\\cal J}(m_{\\eta_{\\phi'}},\\Delta,\\mu),{\\cal J}(m_{X},\\Delta,\\mu)\\,,\\, \\nonumber \\\\\n{\\cal S}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu) &=& {\\cal H}({\\cal K}(m_{\\eta_{\\phi}},\\Delta,\\mu),{\\cal K}(m_{\\eta_{\\phi'}}\\Delta,\\mu),\n{\\cal K}(m_{X},\\Delta,\\mu))\\,.\n\\label{hairpinoneloop} \n\\end{eqnarray}\nAt NLO, the momentum behavior of the axial form factors\nis purely polynomial. This in turn implies that the axial\nradii $\\langle r^{2}_{B'B} \\rangle$ which are defined by $ \\langle r^{2}_{B'B} \n\\rangle \\equiv \\lim_{q \\rightarrow 0}6\\frac{d}{dq^2}G_{A,B'B}(q^2)$, \nare insensitive to NLO chiral corrections. Since\nno $q^2$ dependence appears in the non-analytic functions from one-loop\ndiagrams, the axial form factors are insensitive to the long-range \neffects introduced by boundary conditions. \nTherefore, flavor twisted boundary conditions can be used to produce momentum\ntransfer between initial and final baryon states without sizable finite \nvolume corrections to the extraction of the axial radii \\cite{twisted13}. \n \nThe one-loop diagrams which contribute at NLO to the pseudoscalar\nform factor are depicted in Fig.2. Additionally there are further\ndiagrams generated by the insertion of local interactions from\nthe fourth-order meson Lagrangian. Despite the large number of\ndiagrams, there are a number of simplifications. In particular\nthe second and fourth diagrams of the second line in Fig. 2\n(along with local insertions on the meson line) lead to the one-loop\nrenormalized pion propagator. Additionally the second and fourth\ndiagrams of the first row in Fig 2 (along with NLO pion axial coupling)\ncontribute to the one-loop value of the pion decay constant.\nThe remaining ten diagrams are generated from vertices in the NLO\nbaryon Lagrangian. These diagrams renormalize the tree-level axial\ncoupling of the pion to the baryons. Carefully accounting for all of \nthese factors, we find\n\\begin{eqnarray}\nG_{P,B'B}(q^2)\\,& = &\\,(m_B+m_{B'})^2\n \\Big(\\,\\frac{f_{\\pi}\/f}{q^2-m^2_{\\pi}}G_{A,B'B}(0)\\sqrt{Z_{\\pi}}\\,\\,\n-\\, \\frac{1}{3}\\langle r^{2}_{B'B} \\rangle\\,\\Big)\\,,\n\\label{pseudoiso}\n\\end{eqnarray}\nwhere in this NLO expression, the axial charge $G_{A,B'B}(0)$,\npion mass $m_{\\pi}$ and pion decay constant $f_\\pi$ are taken to be \ntheir physical values and $Z_{\\pi}$ is the pion wavefunction\nrenormalization which is shown in the Appendix. In fitting lattice\ndata to Eq.~(\\ref{pseudoiso}), one would thus use the lattice measured \nvalues for $G_{A,B'B}(0)$, $m_{\\pi}$ and $f_{\\pi}$. The final \nterm in the pseudoscalar form factor is $\\langle r^{2}_{B'B} \\rangle$, \nwhich is the axial radius. Its\nappearance here was discovered long ago under the guise of PCAC by\nAdler and Dothan \\cite{Adler66}\n\nThe simple structure of the psuedoscalar form factor at NLO\nin both {$\\chi$PT}\\ and {PQ$\\chi$PT}\\, allows one perform an approximate check\nof the Goldberger-Treiman relation. The residue of the pseudoscalar\nform factor at the pion pole is proportional to the pion-baryon-baryon\ncoupling $G_{\\pi B'B}$.\nOne can thus perturbatively investigate the Goldberger-Treiman\nrelation using a lattice determination of the pseudoscalar form\nfactor. This indirect method is considerably simpler\nthan a lattice measurement of baryon-to-baryon-plus-pion correlation\nfunctions which contain final state interactions.\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{cc}\n\\multicolumn{2}{c}{$C_{\\phi}$}\\tabularnewline\n\\hline\n$uj$&\n$ur$\\tabularnewline\n$-2$&\n$-1$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $C_{\\phi}$ \nin PQ$\\chi$PT for the isospin changing axial form factors. \nThe $C_{\\phi}$ are categorized by the loop mesons $\\phi$ with \nmass $m_{\\phi}$ and are the same for all isospin transitions.}\n\\label{tabi1}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{cccccccc|ccc}\n\\multicolumn{8}{c|}{$E_{\\phi}$}&\n\\multicolumn{3}{c}{$\\bar{E}_{\\phi\\phi'}$}\\tabularnewline\n\\cline{1-8} \n\\hline\n&\n$\\eta_{u}$&\n$\\eta_{s}$&\n$us$&\n$uj$&\n$ur$&\n$sj$&\n$sr$&\n$\\eta_{u}\\eta_{u}$&\n$\\eta_{u}\\eta_{s}$&\n$\\eta_{s}\\eta_{s}$\\tabularnewline\n$NN$&\n$-\\frac{20}{27}$&\n$0$&\n$0$&\n$-\\frac{40}{81}$&\n$-\\frac{20}{81}$&\n$0$&\n$0$&\n$0$&\n$0$&\n$0$\\tabularnewline\n$\\Lambda\\Sigma$&\n$-\\frac{10}{27}$&\n$0$&\n$-\\frac{10}{54}$&\n$-\\frac{10}{27}$&\n$-\\frac{10}{54}$&\n$0$&\n$0$&\n$0$&\n$0$&\n$0$\\tabularnewline\n$\\Xi\\Xi$&\n$0$&\n$\\frac{5}{81}$&\n$\\frac{10}{81}$&\n$0$&\n$0$&\n$\\frac{10}{81}$&\n$\\frac{5}{81}$&\n$\\frac{10}{81}$&\n$-\\frac{20}{81}$&\n$\\frac{10}{81}$\\tabularnewline\n$\\Sigma\\Sigma$&\n$-\\frac{10}{81}$&\n$0$&\n$-\\frac{65}{81}$&\n$-\\frac{10}{81}$&\n$-\\frac{5}{81}$&\n$-\\frac{40}{81}$&\n$-\\frac{20}{81}$&\n$-\\frac{20}{81}$&\n$\\frac{40}{81}$&\n$-\\frac{20}{81}$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $E_{\\phi}$ and $\\bar{E}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the isospin changing axial form factors. \nThe $E_{\\phi}$ are categorized by the loop mesons $\\phi$ with \nmass $m_{\\phi}$ and $\\bar{E}_{\\phi\\phi'}$ are listed\nby pairs $\\phi \\phi'$ of $\\eta_q$ mesons.}\n\\label{tabi2}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc|cc}\n\\multicolumn{5}{c|}{$A_{\\phi}$}&\n{$\\bar{A}_{\\phi\\phi'}$}\\tabularnewline \n\\hline \n&\n$\\eta_{u}$ & $\\eta_{s}$ & $us$ & $uj$ & $\\eta_{u}\\eta_{u}$ & $\\eta_{u}\\eta_{s}$\\tabularnewline\n$NN$ & $\\frac{8}{9}D+\\frac{8}{3}F$ & $0$ & $0$ & $\\frac{8}{3}D+\\frac{8}{9}F$\n& $0$ & $0$\\tabularnewline\n$\\Lambda\\Sigma$ & $-\\frac{4}{9}D+\\frac{4}{3}F$ & $0$ & $\\frac{16}{9}D+\\frac{8}{3}F$ & \n $\\frac{16}{9}D$ & $\\frac{16}{9}D-\\frac{8}{3}F$ & $-\\frac{20}{9}D+\\frac{4}{3}F$\\tabularnewline\n$\\Xi\\Xi$ & $0$ & $\\frac{8}{9}F$ & $\\frac{4}{9}D+\\frac{4}{9}F$ \n& $0$ & $\\frac{8}{9}D-\\frac{8}{9}F$ & $-\\frac{8}{9}D-\\frac{8}{9}F$\\tabularnewline\n$\\Sigma\\Sigma$ & $\\frac{8}{9}F$ & $0$ & $\\frac{4}{9}D+\\frac{4}{9}F$ &\n$\\frac{8}{9}D+\\frac{8}{9}F$ & $\\frac{16}{9}F$ & $-\\frac{8}{9}D-\\frac{8}{9}F$ \\tabularnewline\n\\hline \n\\hline\n&\n$ur$ & $sj$ & $sr$ & $$ & $\\eta_{s}\\eta_{s}$ & $$\\tabularnewline\n$NN$ & $\\frac{4}{3}D+\\frac{4}{9}F$ & $0$ & $0$ & $$\n& $0$ & $$\\tabularnewline\n$\\Lambda\\Sigma$ & $\\frac{8}{9}D$ & $\\frac{8}{9}D+\\frac{8}{3}F$ & \n$\\frac{4}{9}D+\\frac{4}{3}F$ & $$ & $\\frac{4}{9}D+\\frac{4}{3}F$ & $$\\tabularnewline\n$\\Xi\\Xi$ & $0$ & $\\frac{16}{9}F$ & $\\frac{8}{9}F$\n& $$ & $\\frac{16}{9}F$ & $$\\tabularnewline\n$\\Sigma\\Sigma$ & $\\frac{4}{9}D+\\frac{4}{9}F$ &\n$\\frac{16}{9}D-\\frac{16}{9}F$ & \n$\\frac{8}{9}D-\\frac{8}{9}F$ & $$ & $\\frac{8}{9}D-\\frac{8}{9}F$ & $$ \\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $A_{\\phi}$ and $\\bar{A}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the isospin changing axial form factors. \nThe $A_{\\phi}$ and $\\bar{A}_{\\phi\\phi'}$ coefficients are categorized\nas in Table \\ref{tabi2}.}\n\\label{tabi3}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc|c}\n\\multicolumn{3}{c|}{$Y_{\\phi}$}&\n{$\\bar{Y}_{\\phi\\phi'}$}\\tabularnewline \n\\hline \n&\n$\\eta_{u}$ & $\\eta_{s}$ & $\\eta_{u}\\eta_{u}$\\tabularnewline\n$NN$ & $-\\frac{4}{3}D^{3}+\\frac{16}{3}D^{2}F-4DF^{2}$ & $0$ & $-D^{3}+5D^{2}F-3DF^{2}-9F^{3}$\\tabularnewline\n$\\Lambda\\Sigma$ & $-\\frac{16}{9}D^{3}+\\frac{16}{3}D^{2}F$ & $0$ & $\\frac{16}{3}D^{2}F-8DF^{2}$\\tabularnewline\n$\\Xi\\Xi$ & $0$ & $-\\frac{2}{9}D^{3}-\\frac{2}{3}D^{2}F-2DF^{2}+2F^{3}$ & $-D^{3}+3D^{2}F-3DF^{2}+F^{3}$\\tabularnewline\n$\\Sigma\\Sigma$ & $-\\frac{8}{9}D^{3}+\\frac{4}{3}D^{2}F-4F^{3}$ & $0$ & $-8F^{3}$\\tabularnewline\n\\hline \n\\hline\n&\n$us$ & $uj$ & $\\eta_{u}\\eta_{s}$\\tabularnewline\n$NN$ & $0$ & $\\frac{4}{3}D^3-\\frac{4}{3}D^2F+4DF^2-4F^3$ & $0$\\tabularnewline\n$\\Lambda\\Sigma$ & $\\frac{4}{9}D^3-\\frac{4}{3}D^2F$ & \n$\\frac{4}{9}D^3-\\frac{8}{3}D^2F+4DF^2$ & $-\\frac{8}{3}D^3+\\frac{16}{3}D^2F-8DF^2$\\tabularnewline\n$\\Xi\\Xi$ & $\\frac{2}{9}D^3-\\frac{2}{3}D^2F+6DF^2-2F^3$ & \n$0$ & $4D^2F-8DF^2+4F^3$\\tabularnewline\n$\\Sigma\\Sigma$ & $-\\frac{4}{9}D^3+\\frac{16}{3}D^2F-8DF^2+4F^3$ & \n$\\frac{8}{9}D^3+\\frac{4}{3}D^2F-4F^3$ & $8DF^2-8F^3$\\tabularnewline\n\\hline \n\\hline\n&\n$ur$ & $sj$ & $\\eta_{s}\\eta_{s}$\\tabularnewline\n$NN$ & $\\frac{2}{3}D^3-\\frac{2}{3}D^2F+2DF^2-2F^3$ & \n$0$ & $0$\\tabularnewline\n$\\Lambda\\Sigma$ & $\\frac{2}{9}D^3-\\frac{4}{3}D^2F+2DF^2$ & \n$\\frac{4}{3}D^3+\\frac{8}{3}D^2F-4DF^2$ & $\\frac{2}{3}D^3+\\frac{4}{3}D^2F-2DF^2$\\tabularnewline\n$\\Xi\\Xi$ & $0$ & \n$\\frac{4}{9}D^3+\\frac{4}{3}D^2F-4DF^2+4F^3$ & $-4DF^2+4F^3$\\tabularnewline\n$\\Sigma\\Sigma$ & $\\frac{4}{9}D^3+\\frac{2}{3}D^2F-2F^3$ & \n$-4D^2F+8DF^2-4F^3$ & $-2D^2F+4DF^2-2F^3$\\tabularnewline\n\\hline \n\\hline\n&\n$sr$ & $$ & $$\\tabularnewline\n$NN$ & $0$ & \n$$ & $$\\tabularnewline\n$\\Lambda\\Sigma$ & $\\frac{2}{3}D^3+\\frac{4}{3}D^2F-2DF^2$ & \n$$ & $$\\tabularnewline\n$\\Xi\\Xi$ & $\\frac{2}{9}D^3+\\frac{2}{3}D^2F-2DF^2+2F^3$ & \n$$ & $$\\tabularnewline\n$\\Sigma\\Sigma$ & $-2D^2F+4DF^2-2F^3$ & \n$$ & $$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $Y_{\\phi}$ and $\\bar{Y}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the isospin changing axial form factors. \nThe $Y_{\\phi}$ and $\\bar{Y}_{\\phi\\phi'}$ coefficients are categorized \nas in Table \\ref{tabi2}.}\n\\label{tabi5}\n\\bigskip\n\\bigskip\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\multicolumn{5}{c}{$u_{\\phi}$}\\tabularnewline\n\\hline\n&\n$\\,\\,\\,uu\\,\\,\\,\\,$&\n$\\,\\,\\,ss\\,\\,\\,\\,$&\n$\\,\\,\\,jj\\,\\,\\,\\,$&\n$\\,\\,\\,rr\\,\\,\\,\\,$\\tabularnewline\n$\\,\\,\\,NN\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{3}b_{1}+\\frac{2}{3}b_2-\\frac{1}{6}b_3+\\frac{1}{6}b_4\n+\\frac{1}{3}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,0\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{2}{3}b_7-\\frac{1}{3}b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{3}b_7-\\frac{1}{6}b_6\\,\\,\\,$\\tabularnewline\n$\\,\\,\\,\\Lambda\\Sigma\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-b_{1}+\\frac{1}{2}b_2-\\frac{1}{4}b_3\n+\\frac{1}{4}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{4}b_3+\\frac{1}{2}b_4\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{2}b_7-b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{4}b_7-\\frac{1}{2}b_6\\,\\,\\,$\\tabularnewline\n$\\,\\,\\,\\Xi\\Xi\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{2}{3}b_{1}-\\frac{1}{6}b_2\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{3}b_3+\\frac{1}{3}b_4-\\frac{1}{12}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{6}b_7-\\frac{2}{3}b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{12}b_7-\\frac{1}{3}b_6\\,\\,\\,$\\tabularnewline\n$\\,\\,\\,\\Sigma\\Sigma\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{3}b_{1}+\\frac{5}{6}b_2+\\frac{1}{12}b_3+\\frac{1}{6}b_4\n+\\frac{1}{12}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{12}b_3-\\frac{1}{3}b_4+\\frac{1}{3}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{5}{6}b_7+\\frac{1}{3}b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{5}{12}b_7+\\frac{1}{6}b_6\\,\\,\\,$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $u_{\\phi}$ \nin PQ$\\chi$PT for the isospin changing axial form factors. The \n$u_{\\phi}$ coefficients are categorized by the mesons with \nmass $m_{\\phi}$.}\n\\label{tabi7}\n\\bigskip\n\\bigskip\n\\end{table}\n\n\n\\subsection{Strangeness Changing Transitions}\n\\label{SCT}\nThe $\\Delta S = -1$ strangeness changing transitions corresponds to\nthe flavor matrix $\\overline{\\lambda}^{4+5i}$ which is given by\n\\begin{equation} \\label{eqn:tau1}\n\\overline{\\lambda}^{4+5i}_{i j} = \n\\begin{cases}\n 1 \\quad i = 1\\, , j = 3 \\\\\n 0 \\quad {\\text{otherwise}}\n\\end{cases}\n\\,,\\end{equation}\nin the partially quenched theory. With Eq.~(\\ref{eqn:tau1}) there exists six \nstrangeness changing transitions among the hyperons\n\\begin{equation}\n\\Sigma^{0} \\rightarrow p\\,\\,,\\,\\,\\Sigma^{-} \\rightarrow n\\,\\,,\\,\\,\n\\Lambda \\rightarrow p\\,\\,,\\,\\,\\Xi^{0}\n\\rightarrow \\Sigma^{+}\\,\\,,\\,\\,\n\\Xi^{-} \\rightarrow \\Sigma^{0}\\,\\,,\\,\\,\\Xi^{-} \\rightarrow \\Lambda\\,\\,.\n\\end{equation}\nThe $N$$\\Lambda$ transition matrix elements define the $N$$\\Lambda$ \naxial form factor $G_{A,N\\Lambda}(q^2)$ and induced pseudoscalar form factor\n$G_{P,N\\Lambda}(q^2)$\n\\begin{equation}\n\\langle p (P')| J^{4+5i}_{\\mu,5} | \\Lambda (P) \\rangle\\,=\\,-\\frac{1}{\\sqrt{6}}\n\\overline{U}_{N}(P')\\Big[ 2S_{\\mu}G_{A,N\\Lambda}(q^2)+\\frac{q_{\\mu} q \\cdot\n S}{(m_N+m_{\\Lambda})^2}G_{P,N\\Lambda}(q^2)\\Big]U_{\\Lambda}(P)\\,,\n\\end{equation}\nwhere, as above $q_{\\mu} = (P' - P)_{\\mu}$ is the four momentum transfer. The\n$\\Lambda$$\\Xi$ transition matrix elements define the \n$\\Lambda$$\\Xi$ axial form factor $G_{A,\\Lambda\\Xi}$ and induced pseudoscalar\nform factor $G_{P,\\Lambda\\Xi}$\n\\begin{equation}\n\\langle \\Lambda (P') | J^{4+5i}_{\\mu,5}| \\Xi^{-} (P)\\rangle\\,=\\,\n\\frac{1}{\\sqrt{6}} \n\\overline{U}_{\\Lambda}(P')\\Big[2S_{\\mu}G_{A,\\Lambda\\Xi}(q^2)\n+ \\frac{q_{\\mu} q \\cdot S}{(m_{\\Lambda}+m_{\\Xi})^2}G_{P,\\Lambda\\Xi}(q^2)\\Big]U_{\\Xi}(P)\\,.\n\\end{equation}\nThe $N$$\\Sigma$ axial transitions defines the $N$$\\Sigma$ axial form factor \n$G_{A,N\\Sigma}(q^2)$ and the induced pseudoscalar form factor $G_{P,N\\Sigma}(q^2)$\n\\begin{equation}\n\\langle n (P') | J^{+}_{\\mu,5} | \\Sigma^{-} (P)\\rangle\\,=\\, \n\\overline{U}_{N}(P')\\Big[2S_{\\mu}G_{A,N\\Sigma}(q^2) + \\frac{q_{\\mu} q \\cdot\n S}{(m_N+m_{\\Sigma})^2}G_{P,N\\Sigma}(q^2)\\Big]U_{\\Sigma}(P)\\,.\n\\end{equation}\nFinally, the $\\Sigma$$\\Xi$ axial form factor $G_{A,\\Sigma\\Xi}(q^2)$ and\nthe induced pseudoscalar form factor $G_{P,\\Sigma\\Xi}(q^2)$ is defined \nthrough the $\\Sigma$$\\Xi$ transition matrix element\n\\begin{equation}\n\\langle \\Sigma^{0} (P') | J^{4+5}_{\\mu,5}| \\Xi^{-} (P)\\rangle\\,=\\,\n\\frac{1}{\\sqrt{2}}\\overline{U}_{\\Sigma}(P')\\Big[2S_{\\mu}G_{A,\\Sigma\\Xi}(q^2)+\n\\frac{q_{\\mu} q \\cdot S}{(m_{\\Sigma}+m_{\\Xi})^2}G_{P,\\Sigma\\xi}(q^2)\\Big]U_{\\Xi}(P)\\,.\n\\end{equation} \nNotice while there are two $N$$\\Sigma$ and two $\\Sigma$$\\Xi$ transitions, \nboth of their matrix elements are \nrelated by isospin factors, namely:\n\\begin{eqnarray}\n\\langle p (P')| J^{4+5i}_{\\mu,5} | \n\\Sigma^{0}(P)\\rangle\\,&=&\\,\n\\frac{1}{\\sqrt{2}}\\langle n (P')| J^{4+5i}_{\\mu,5} | \\Sigma^{-}(P)\\rangle\n\\,, \\nonumber \\\\\n\\langle \\Sigma^{0} (P') | J^{4+5i}_{\\mu,5} | \\Xi^{-} (P)\\rangle\\,&=&\\,\\frac{1}{\\sqrt{2}}\n\\langle \\Sigma^{+} (P')| J^{4+5i}_{\\mu,5} | \\Xi^{0} (P)\\rangle\\,.\n\\end{eqnarray}\nFollowing the same considerations in section \\ref{IST} and \nassembling the LO and NLO contributions, the axial form factors of strangeness\nchanging transitions are given by\n\\begin{eqnarray}\nG_{A,B'B}(q^2) & = &\\, g_{B'B}\\sqrt{Z_{B'}Z_{B}} + \\frac{1}{16\\pi^2f^2}\\Bigg[ g_{B'B}\\Big(\n\\sum_{\\phi}C_{\\phi}{\\cal L}(m_{\\phi},\\mu)\n\\nonumber \\\\\n& + &\\sum_{\\phi\\phi^{\\prime}} \\bar{C}_{\\phi\\phi^{\\prime}}{\\cal\n R}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu)\\Big)\n+ {\\cal H}{\\cal C}^2\\Big(\\sum_{\\phi} E_{\\phi}{\\cal J}(m_{\\phi},\\Delta,\\mu) \\nonumber \\\\\n& + &\\sum_{\\phi\\phi^{\\prime}} \\bar{E}_{\\phi\\phi^{\\prime}}{\\cal T}(\\eta_{\\phi}\n \\eta_{\\phi'},\\Delta,\\mu)\\Big) \\nonumber \\\\\n& + & {\\mathcal C}^2 \\Big( \\sum_{\\phi}A_{\\phi}{\\cal K}(m_{\\phi},\\Delta,\\mu)\n + \\sum_{\\phi\\phi^{\\prime}}\n\\bar{A}_{\\phi\\phi^{\\prime}}\n{\\cal S}(\\eta_{\\phi} \\eta_{\\phi^{\\prime}},\\Delta,\\mu) \\Big) \\nonumber \\\\\n& + &\\sum_{\\phi} Y_{\\phi}{\\cal L}(m_{\\phi},\\mu)\n + \\sum_{\\phi\\phi^{\\prime}} \\bar{Y}_{\\phi,\\phi^{\\prime}}{\\cal\n R}(\\eta_{\\phi}\\eta_{\\phi'},\\Delta,\\mu) \\Bigg] \\nonumber \\\\\n& + & n_{B'B}\\frac{q^2}{\\Lambda_{\\chi}^2} + \\sum_{\\phi}u_{\\phi}m_{\\phi}^2\n\\,. \n\\label{matrixstrange}\n\\end{eqnarray}\nIn Eq.~(\\ref{matrixstrange}), we use $B$ and\n$B'$ to denote the initial and final states of octet baryon, $Z_B$ and\n$Z_{B'}$ are again the wave function renormalization for which the explicit \nexpressions are given in the Appendix. The $g_{B'B}$'s appearing\nabove are the leading order octet baryon axial charges\n\\begin{eqnarray}\ng_{N\\Lambda} &=& (3F\\,+\\,D)\\,,\\,\\,\\, \ng_{\\Lambda\\Xi} = (3F\\,-\\,D)\\,, \n\\nonumber \\\\\ng_{N\\Sigma} &=& (D\\,-\\,F)\\,,\\,\\,\\,g_{\\Sigma\\Xi} = (F\\,+\\,D),\n\\end{eqnarray} \nand the coefficients $n_{B'B}$ are given by a formula of exactly the same form\n\\begin{eqnarray}\nn_{N\\Lambda} &=& 3n_{F}\\,+\\,n_{D}\\,,\\,\\,\nn_{\\Lambda\\Xi} = 3n_{F}\\,-\\,n_{D}\\,,\\nonumber \\\\\nn_{N\\Sigma} &=& n_{D}\\,-\\,n_{F}\\,,\\,\\,\nn_{\\Sigma\\Xi} = n_{D}\\,+\\,n_{F}\\,. \n\\end{eqnarray} \nThe coefficeints $C_{\\phi}$, $\\bar{C}_{\\phi\\phi'}$, $E_{\\phi}$, $\\bar{E}_{\\phi\\phi'}$, $A_{\\phi}$, \n$\\bar{A}_{\\phi\\phi'}$, $Y_{\\phi}$, $\\bar{Y}_{\\phi\\phi'}$\nare given in tables \\ref{tabs1}, \\ref{tabs2},\n\\ref{tabs3}, \\ref{tabs5}. \n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{cccc|ccc}\n\\multicolumn{4}{c|}{$C_{\\phi}$}&\n\\multicolumn{3}{c}{$\\bar{C}_{\\phi\\phi'}$}\\tabularnewline\n\\hline\n$uj$&\n$ur$&\n$sj$&\n$sr$&\n$\\eta_{u}\\eta_{u}$&\n$\\eta_{u}\\eta_{s}$&\n$\\eta_{s}\\eta_{s}$\\tabularnewline\n$-1$&\n$-\\frac{1}{2}$&\n$-1$&\n$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&\n$1$&\n$-\\frac{1}{2}$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $C_{\\phi}$ and $\\bar{C}_{\\phi\\phi'}$ \nin PQ$\\chi$PT, which are all the same for the strangeness changing \naxial form factors. The $C_{\\phi}$ and $\\bar{C}_{\\phi\\phi'}$ coefficients\nare categorized as in Table \\ref{tabi2}.}\n\\label{tabs1}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{cccccccc|ccc}\n\\multicolumn{8}{c|}{$E_{\\phi}$}&\n\\multicolumn{3}{c}{$\\bar{E}_{\\phi\\phi'}$}\\tabularnewline\n\\hline\n&\n$\\eta_{u}$&\n$\\eta_{s}$&\n$us$&\n$uj$&\n$ur$&\n$sj$&\n$sr$&\n$\\eta_{u}\\eta_{u}$&\n$\\eta_{u}\\eta_{s}$&\n$\\eta_{s}\\eta_{s}$\\tabularnewline\n$N\\Lambda$&\n$-\\frac{10}{9}$&\n$0$&\n$-\\frac{10}{18}$&\n$-\\frac{10}{9}$&\n$-\\frac{10}{18}$&\n$0$&\n$0$&\n$0$&\n$0$&\n$0$\\tabularnewline\n$\\Lambda\\Xi$&\n$-\\frac{10}{27}$&\n$0$&\n$-\\frac{20}{27}$&\n$-\\frac{20}{27}$&\n$-\\frac{10}{27}$&\n$0$&\n$0$&\n$0$&\n$0$&\n$0$\\tabularnewline\n$N\\Sigma$&\n$\\frac{10}{81}$&\n$0$&\n$\\frac{5}{81}$&\n$\\frac{10}{81}$&\n$\\frac{5}{81}$&\n$0$&\n$0$&\n$0$&\n$0$&\n$0$\\tabularnewline\n$\\Sigma\\Xi$&\n$-\\frac{10}{81}$&\n$-\\frac{10}{81}$&\n$-\\frac{40}{81}$&\n$-\\frac{20}{81}$&\n$-\\frac{10}{81}$&\n$-\\frac{20}{81}$&\n$-\\frac{10}{81}$&\n$-\\frac{20}{81}$&\n$\\frac{40}{81}$&\n$-\\frac{20}{81}$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $E_{\\phi}$ and $\\bar{E}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the strangeness changing axial form factors. \nThe $E_{\\phi}$ and $\\bar{E}_{\\phi\\phi'}$ coefficients are categorized as in\nTable \\ref{tabi2}.}\n\\label{tabs2}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc|cc}\n\\multicolumn{5}{c|}{$A_{\\phi}$}&\n{$\\bar{A}_{\\phi\\phi'}$}\\tabularnewline \n\\hline \n&\n$\\eta_{u}$ & $\\eta_{s}$ & $us$ & $uj$ & $\\eta_{u}\\eta_{u}$ & $\\eta_{u}\\eta_{s}$\\tabularnewline\n$N\\Lambda$ & $2D+2F$ & $0$ & $-\\frac{2}{3}D+2F$ & $\\frac{16}{3}D$\n& $0$ & $0$\\tabularnewline\n$\\Lambda\\Xi$ & $-\\frac{2}{3}D+\\frac{2}{3}F$ & $\\frac{2}{9}D+\\frac{2}{3}F$\n& $\\frac{4}{9}D-\\frac{4}{3}F$ & \n$\\frac{20}{9}D-4F$ & $\\frac{16}{9}D-\\frac{8}{3}F$ & $-\\frac{20}{9}D+\\frac{4}{3}F$\\tabularnewline\n$N\\Sigma$ & $\\frac{2}{9}D+\\frac{10}{9}F$ & $0$ & $\\frac{2}{9}D+\\frac{2}{9}F$\n& $\\frac{16}{9}F$ & $-\\frac{4}{9}D+\\frac{4}{3}F$ & $\\frac{4}{9}D-\\frac{4}{3}F$\\tabularnewline\n$\\Sigma\\Xi$ & $-\\frac{2}{9}D+\\frac{2}{9}F$ &\n$-\\frac{2}{9}D+\\frac{2}{9}F$ & $\\frac{4}{3}D+\\frac{20}{9}F$ &\n$\\frac{4}{3}D+\\frac{4}{9}F$ & $\\frac{4}{9}D+\\frac{4}{9}F$ & $-\\frac{8}{9}D-\\frac{8}{9}F$ \\tabularnewline\n\\hline \n\\hline\n&\n$ur$ & $sj$ & $sr$ & $$ & $\\eta_{s}\\eta_{s}$ & $$\\tabularnewline\n$N\\Lambda$ & $\\frac{8}{3}D$ & $0$ & $0$ & $$\n& $0$ & $$\\tabularnewline\n$\\Lambda\\Xi$ & $\\frac{10}{9}D-2F$ & $\\frac{4}{9}D+\\frac{4}{3}F$ & \n$\\frac{2}{9}D+\\frac{2}{3}F$ & $$ & $\\frac{4}{9}D+\\frac{4}{3}F$ & $$\\tabularnewline\n$N\\Sigma$ & $\\frac{8}{9}F$ & $0$ & $0$\n& $$ & $0$ & $$\\tabularnewline\n$\\Sigma\\Xi$ & $\\frac{2}{3}D+\\frac{2}{9}F$ &\n$\\frac{4}{3}D+\\frac{4}{9}F$ & \n$\\frac{2}{3}D+\\frac{2}{9}F$ & $$ & $\\frac{4}{9}D+\\frac{4}{9}F$ & $$ \\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $A_{\\phi}$ and $\\bar{A}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the strangeness changing axial form factors. \nThe $A_{\\phi}$ and $\\bar{A}_{\\phi\\phi'}$ coefficients are categorized as\nin Table \\ref{tabi2}.}\n\\bigskip\n\\label{tabs3}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc|c}\n\\multicolumn{3}{c|}{$Y_{\\phi}$}&\n{$\\bar{Y}_{\\phi\\phi'}$}\\tabularnewline \n\\hline \n&\n$\\eta_{u}$ & $\\eta_{s}$ & $\\eta_{u}\\eta_{u}$\\tabularnewline\n$N\\Lambda$ & $\\frac{1}{9}D^3+5D^2F-9DF^2+3F^3$ & $0$ & $-\\frac{4}{3}D^3+2D^2F+12DF^2-18F^3$\\tabularnewline\n$\\Lambda\\Xi$ &\n$-\\frac{5}{3}D^3+\\frac{11}{3}D^2F-5DF^2+3F^3$ & $-\\frac{5}{9}D^3-D^2F+DF^2-3F^3$ &\n$\\frac{4}{3}D^3-\\frac{22}{3}D^2F+12DF^2-6F^3$\\tabularnewline\n$N\\Sigma$ & $-\\frac{1}{9}D^3-D^2F+DF^2+F^3$ & \n$0$ & $2D^2F-8DF^2+6F^3$\\tabularnewline\n$\\Sigma\\Xi$ & $-\\frac{1}{3}D^3+\\frac{7}{3}D^2F-DF^2-F^3$ & \n$-\\frac{1}{3}D^3+\\frac{7}{3}D^2F-DF^2-F^3$ & $2D^2F-2F^{3}$\\tabularnewline\n\\hline \n\\hline\n&\n$us$ & $uj$ & $\\eta_{u}\\eta_{s}$\\tabularnewline\n$N\\Lambda$ & $-\\frac{25}{9}D^3+7D^2F-3DF^2-3F^3$ &\n$\\frac{20}{9}D^3-4D^2F+12DF^2-12F^3$ & \n$\\frac{1}{3}D^3+D^2F-3DF^2-9F^3$\\tabularnewline\n$\\Lambda\\Xi$ & $\\frac{8}{9}D^3+\\frac{16}{3}D^2F-8DF^2$ & \n$-\\frac{2}{3}D^3-\\frac{22}{3}D^2F+14DF^2-6F^3$ & $-\\frac{1}{3}D^3-\\frac{7}{3}D^2F+15DF^2-15F^3$\\tabularnewline\n$N\\Sigma$ & $\\frac{1}{9}D^3-\\frac{1}{3}D^2F+3DF^2-F^3$ & \n$\\frac{4}{9}D^3+\\frac{4}{3}D^2F-4DF^2+4F^3$ & $-D^3+5D^2F-7DF^2+3F^3$\\tabularnewline\n$\\Sigma\\Xi$ & $-\\frac{2}{3}D^3+\\frac{2}{3}D^2F-2DF^2+2F^3$ & \n$\\frac{2}{3}D^3-\\frac{2}{3}D^2F+2DF^2-2F^3$ & $-D^3+D^2F-3DF^2-5F^3$\\tabularnewline\n\\hline \n\\hline\n&\n$ur$ & $sj$ & $\\eta_{s}\\eta_{s}$\\tabularnewline\n$N\\Lambda$ & $\\frac{10}{9}D^3-2D^2F+6DF^2-6F^3$ & \n$0$ & $0$\\tabularnewline\n$\\Lambda\\Xi$ & $-\\frac{1}{3}D^3-\\frac{11}{3}D^2F+7DF^2-3F^3$ & \n$\\frac{10}{9}D^3+\\frac{10}{3}D^2F-2DF^2-6F^3$ & $\\frac{2}{3}D^2F-6F^3$\\tabularnewline\n$N\\Sigma$ & $\\frac{2}{9}D^3+\\frac{2}{3}D^2F-2DF^2+2F^3$ & \n$0$ & $0$\\tabularnewline\n$\\Sigma\\Xi$ & $\\frac{1}{3}D^3-\\frac{1}{3}D^2F+DF^2-F^3$ & \n$\\frac{2}{3}D^3-\\frac{2}{3}D^2F+2DF^2-2F^3$ & $2D^2F-2F^3$\\tabularnewline\n\\hline \n\\hline\n&\n$sr$ & $$ & $$\\tabularnewline\n$N\\Lambda$ & $0$ & \n$$ & $$\\tabularnewline\n$\\Lambda\\Xi$ & $\\frac{5}{9}D^3+\\frac{5}{3}D^2F-DF^2-3F^3$ & \n$$ & $$\\tabularnewline\n$N\\Sigma$ & $0$ & \n$$ & $$\\tabularnewline\n$\\Sigma\\Xi$ & $\\frac{1}{3}D^3-\\frac{1}{3}D^2F+DF^2-F^3$ & \n$$ & $$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $Y_{\\phi}$ and $\\bar{Y}_{\\phi\\phi'}$ \nin PQ$\\chi$PT for the strangeness changing axial form factors. \nThe $Y_{\\phi}$ and $\\bar{Y}_{\\phi\\phi'}$ coefficients are categorized as\nin Table \\ref{tabi2}.}\n\\label{tabs5}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\multicolumn{5}{c}{$u_{\\phi}$}\\tabularnewline\n\\hline\n&\n$uu\\,\\,$&\n$ss\\,\\,$&\n$jj\\,\\,$&\n$rr\\,\\,$\\tabularnewline\n$N\\Lambda\\,\\,$&\n$\\,\\,\\,\\,\\frac{3}{4}b_{2}+\\frac{3}{4}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{3}{4}b_2\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{3}{2}b_7\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{3}{4}b_7$\\tabularnewline\n$\\Lambda\\Xi\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{2}b_{1}+\\frac{1}{2}b_2+\\frac{1}{4}b_3-\\frac{3}{4}b_4\n+\\frac{1}{2}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{2}b_1+\\frac{1}{2}b_2+\\frac{1}{4}b_3+\\frac{1}{4}b_4\\,\\,\\,\\,$&\n$\\,\\,\\,\\,b_7+b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{2}b_7+\\frac{1}{2}b_6$\\tabularnewline\n$N\\Sigma\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{3}b_{1}-\\frac{1}{12}b_2-\\frac{1}{3}b_3\n+\\frac{1}{3}b_4-\\frac{1}{12}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{3}b_1-\\frac{1}{12}b_2\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{6}b_7-\\frac{2}{3}b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{12}b_7-\\frac{1}{3}b_6$\\tabularnewline\n$\\Sigma\\Xi\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{6}b_{1}+\\frac{1}{3}b_2-\\frac{1}{12}b_3+\\frac{1}{12}b_4\n+\\frac{1}{6}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,-\\frac{1}{6}b_{1}+\\frac{1}{3}b_2-\\frac{1}{12}b_3+\\frac{1}{12}b_4\n+\\frac{1}{6}b_5\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{2}{3}b_7-\\frac{1}{3}b_6\\,\\,\\,\\,$&\n$\\,\\,\\,\\,\\frac{1}{3}b_7-\\frac{1}{6}b_6$\\tabularnewline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $u_{\\phi}$ \nin PQ$\\chi$PT for the strangeness changing axial \nform factors. The coefficients $u_{\\phi}$ are \ncategorized by the mesons with mass $m_{\\phi}$.}\n\\label{tabs7}\n\\end{table}\nFinally, the\nnon-analytic functions ${\\cal L}$'s, ${\\cal J}$'s, ${\\cal K}$'s , \n${\\cal R}$'s, ${\\cal T}$'s and ${\\cal S}$'s in above equations \nare defined in section \\ref{IST}. \nEmploying the same argument as one did in deriving the isospin changing\npseudoscalar form factors, one arrives at \na similar expression for the strangeness changing \npseudoscalar form factor \n\\begin{eqnarray}\nG_{P,B'B}(q^2)\\,& = &\\,(m_B+m_{B'})^2\n \\Big(\\,\\frac{f_{K}\/f}{q^2-m^2_{K}}\\,G_{A,B'B}(0)\\sqrt{Z_{K}}\\, \n\\,- \\, \\frac{1}{3}\\langle r^{2}_{B'B} \\rangle\\,\\Big)\\,,\n\\label{pseudostrange}\n\\end{eqnarray} \nwhere in this NLO expression, the axial charge $G_{A,B'B}(0)$,\nkaon mass $m_{K}$ and kaon decay constant $f_{K}$ are taken to be \ntheir physical values and $Z_{K}$ is the kaon wavefunction renormalization\nwhich is shown in the Appendix. In fitting lattice\ndata to Eq.~ (\\ref{pseudostrange}), one would thus use the lattice measured \nvalues for $G_{A,B'B}(0)$, $m_{K}$ and $f_{K}$. The final \nterm in the pseudoscalar form factor is $\\langle r^{2}_{B'B} \\rangle$, \nwhich is the strangeness changing axial radius. \nAs has been shown in section \\ref{IST}, \nthe simple structure of the psuedoscalar form factor at NLO\nin both {$\\chi$PT}\\ and {PQ$\\chi$PT}\\, allows one perform an approximate check\nof the Goldberger-Treiman relation. Here the residue of the pseudoscalar\nform factor at the kaon pole is proportional to the kaon-baryon-baryon\ncoupling $G_{K B'B}$ thus the pseudoscalar form factor provides\nan indirect and simple method to investigate the Goldberger-Treiman\nrelation on the lattice.\n\n\n\\section{Discussion}\n\\label{discussion} \nAbove we have calculated the full set of flavor-changing\naxial-current matrix elements of the hyperons.\nThe expressions will be useful for the study of the \nchiral and momentum behavior of hyperon axial form factors \nusing lattice QCD. \n\nConsidering the axial charges of hyperons and of their \ntransitions, there are only two parameters which survive \nthe chiral limit, namely $D$ and $F$. \nAs there are eight such charges, there are six relations \nbetween them. Focusing just on the isospin and strangeness\ntransitions individually, we have four of the six relations\n\\begin{eqnarray}\n&& \ng_{NN} + g_{\\Xi \\Xi} - g_{\\Lambda \\Sigma} = 0\\,, \n\\label{eq:firstsetone}\n\\\\\n&& \ng_{NN} - g_{\\Xi \\Xi} - g_{\\Sigma \\Sigma} = 0\\,,\n\\\\\n&&\ng_{N \\Lambda} + g_{\\Lambda \\Xi} - 3 ( g_{\\Sigma \\Xi} - g_{N \\Sigma} )\n= 0\\,,\n\\\\\n&&\ng_{N \\Lambda} - g_{\\Lambda \\Xi} - g_{N \\Sigma} - g_{\\Sigma \\Xi}\n= 0\\,\n\\label{eq:lastsetone}\n.\\end{eqnarray}\nCombining isospin and strangeness transitions\ntogether, we arrive at the final two relations\n\\begin{eqnarray}\n&& \ng_{N \\Lambda} + g_{\\Lambda \\Xi} - 3 g_{\\Sigma \\Sigma} = 0\\,,\n\\label{eq:firstsettwo}\n\\\\\n&&\ng_{N \\Lambda} - g_{\\Lambda \\Xi} - g_{\\Lambda \\Sigma} = 0\\,\n\\label{eq:lastsettwo}\n.\\end{eqnarray}\nThese relations hold in $SU(3)$ $\\chi$PT, as well as $SU(6|3)$ \nPQ$\\chi$PT. Of course chiral corrections modify these relations\nand the ``$0$\" should be interpreted as \n$\\mathcal{O}( m_\\phi^2 \/ \\Lambda_\\chi^2)$. \nThe expressions derived above in section \\ref{IST} and \\ref{SCT} \nprovide these $\\mathcal{O} (m_\\phi^2 \/\\Lambda_\\chi^2)$\ncorrections which are generally linear combinations of non-analytic loop\ncontributions and unknown local counterterms.\n\nThe relations in Eqs.~\\eqref{eq:firstsetone}--\\eqref{eq:lastsetone}\nactually apply not only to the axial charges, but \nto the respective hyperon transition matrix elements as a whole. \nThe pseudoscalar form factors do \nnot satisfy Eqs.~\\eqref{eq:firstsettwo} and \\eqref{eq:lastsettwo}\ndue to the difference in pion versus kaon poles. The axial \nform factors, by contrast, satisfy these latter two relations. \n\nApart from the axial couplings $D$, $F$, $\\mathcal{C}$, $\\mathcal{H}$,\nand meson masses, the axial charges depend on six (eight) unknown \nparameters in $\\chi$PT (PQ$\\chi$PT). This lack of predictive power\nhistorically has been overlooked by supposing, what has been termed,\nthe formal dominance of chiral logarithms. One can do slightly \nbetter and attempt model estimation of these parameters, but \nthe fact stands that most chiral analyses have seriously lacked \nthe ability to make complete predictions. This is the great\nadvantage of lattice QCD calculations, which have promise to determine \nthe complete information about the low-energy constants.\n\nDespite the absence of knowledge concerning the eight coupling constants \nin the NLO current Eq.~(\\ref{eq:axcts}), we can make two non-trivial predictions\nat next-to-leading order by eliminating the local terms. \nThe leading local terms are cancelled in the relations Eqs.~\\eqref{eq:firstsetone}--\\eqref{eq:lastsettwo}.\nCombinations of these relations can be used to eliminate the next-to-leading order \nlocal terms. While there are seven contributing terms, only five of them, \n$b_1, \\ldots, b_5$ \ncontribute to the relations. \nThus there must exist at least one combination of Eqs.~\\eqref{eq:firstsetone}--\\eqref{eq:lastsettwo}\nthat is independent of the \n$b_i$. \nRather fortuitously there are two such \nnon-trivial combinations\nof axial charges which are independent of these \ncouplings, \\emph{viz}.\n\\begin{eqnarray}\n\\Delta g \n&\\equiv& \n2 g_{NN} - g_{N \\Lambda} - g_{N \\Sigma} - g_{\\Lambda \\Sigma}\n- g_{\\Sigma \\Sigma} + 2 g_{\\Sigma \\Xi}\\,,\n\\\\\n\\Delta G\n&\\equiv&\n2 g_{NN} + 2 g_{\\Xi\\Xi} - 2 g_{{\\Lambda} {\\Sigma}}\n+ \ng_{N{\\Sigma}} + g_{{\\Lambda} \\Xi} + g_{{\\Sigma} \\Xi} - g_{N {\\Lambda}}\n.\\end{eqnarray}\nThese relations are independent of NLO counterterms in both \nPQ$\\chi$PT\nand \n$\\chi$PT. \nThe expressions for ${\\Delta} g$ and ${\\Delta} G$ in \n$\\chi$PT \nare%\n\\footnote{\nPartially quenched expressions can be obtained \nfrom Eqs.~\\eqref{eq:deltag} and \\eqref{eq:deltaG}\nunder the replacements:\n$\\mathcal{G} [{\\mathcal L}] \\to \\mathcal{G}^{PQ} [{\\mathcal L}, {\\mathcal R}]$,\n$\\mathcal{G} [{\\mathcal K}] \\to \\mathcal{G}^{PQ} [{\\mathcal L}, \\mathcal{S}]$,\nand\n$\\mathcal{G} [{\\mathcal{J}}] \\to \\mathcal{G}^{PQ} [{\\mathcal L}, {\\mathcal T}]$.\nThe partially quenched functional is given by\n\\begin{equation}\n\\mathcal{G}^{PQ} [A,B]\n=\n2 A_\\pi - 4 A_K + 2 A_{\\eta_s} + 2 B_{\\eta_u \\eta_u} - 4 B_{\\eta_u \\eta_s} + 2 B_{\\eta_s \\eta_s}\n\\notag\n,\\end{equation}\nfor any function \n$A_\\phi = A(m_\\phi,{\\Delta},\\mu)$\nand hairpin function\n$B_{\\eta_{\\phi}\\eta_{\\phi'}} = B(\\eta_{\\phi}\\eta_{\\phi'},{\\Delta},\\mu)$.\nThere is no scale dependence in \n$\\mathcal{G}^{PQ} [A,B]$.\n}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.4825\\textwidth]{deltagdeltaGplot.eps}\n\\end{center}\n\\caption{Plot of $\\Delta g$ and ${\\Delta} G$ as a function of the pion mass $m_{\\pi}$.} \n\\label{fig3}\n\\end{figure}\n\\begin{eqnarray}\n(4 \\pi f)^2 \\, \\Delta g &=& \\frac{2}{3} \\Big( D^3 + 5 D^2 F\n- 6 D F^2 - 6 F^3 \\Big) \\, \n\\mathcal{G} [{\\mathcal L}]\n\\nonumber \\\\\n&& - \\mathcal{C}^2 \\Big[ \\frac{10}{81} \\mathcal{H} + \\frac{1}{6}(D + F) \\Big]\\, \n\\mathcal{G} [{\\mathcal{J}}]\n- \\frac{2}{9} \\mathcal{C}^2 (D -F)\\, \n\\mathcal{G} [{\\mathcal K}]\\,,\n\\label{eq:deltag}\n\\\\\n(4 \\pi f)^2 \\, \\Delta G &=& \\frac{4}{3} D ( D^2 - 6 F^2)\\,\n\\mathcal{G} [{\\mathcal L}]\n- \\frac{1}{3} D \\mathcal{C}^2 \\,\n\\mathcal{G} [{\\mathcal{J}}]\n+ \\frac{4}{3} F \\mathcal{C}^2 \\, \n\\mathcal{G} [{\\mathcal K}]\n\\label{eq:deltaG}\n\\,,\\end{eqnarray}\nwhere the Gell-Mann Okubo linear combination functional is defined by\n\\begin{equation}\n\\mathcal{G}[A] = A_{\\pi} - 4 A_{K} + 3 A_{\\eta}\n\\,,\\end{equation}\nfor any function $A_\\phi = A(m_\\phi,{\\Delta},\\mu)$. \nThe quantities ${\\Delta} g$ and ${\\Delta} G$ allow one to test chiral logarithms directly. \nAccordingly these relations only superficially have scale \ndependence, and upon the limit of $SU(3)_V$ symmetry, they vanish. \nWith these symmetry breaking relations, we have separated out the \nshort distance physics and hence isolated long-distance \nchiral corrections. To obtain values for ${\\Delta} g$ and ${\\Delta} G$, as well as the curve shown in \nFig.~\\ref{fig3}, we have fixed the strange quark mass at its physical \nvalue, and used the $SU(6)$ values for the axial couplings: $D = 3\/4$,\n$F = 2\/3 D$, $\\mathcal{C} = - 2 D$, and $\\mathcal{H} = - 3 D$. \nIn particular, we find\n$\\Delta g = -0.0035$ \nand\n$\\Delta G = -0.017$ \nat physical pion mass $m_{\\pi}\\,=\\,140\\, {\\text{MeV}}$.\n\nFour further non-trivial relations, analogous to \n$\\Delta g$ and ${\\Delta} G$ \nabove,\nexist when one includes charges\narising from the strangeness axial current\n$\\ol s \\gamma_\\mu \\gamma_5 s$. \nThese matrix elements have been determined at zero momentum \ntransfer in \nPQ$\\chi$PT~%\n\\cite{will:2004}.\nDetermination of these charges requires the calculation of disconnected\noperator contractions on the lattice. As such contributions are notoriously\ndifficult to determine, \nwe leave it to future work to deduce the remaining symmetry\nbreaking relations. \n\n\n\\begin{acknowledgments}\nWe thank G. Colangelo for discussions. This work is supported in part by \nthe U.S.\\ Dept.~of Energy, Grant Nos.\\ DE-FG02-05ER41368-0 and \nDE-FG02-93ER40762 (B.C.T.) and by the Schweizerischer Nationalfonds (F.-J.J.). \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{I1}\nResearchers have shown the unsuitability of simplistic methods such as exhaustive search for solving optimization search problems. Simplistic methods particularly become increasingly inefficient as the solution space increases, and at times are unable to escape local minima.\n\nOver the years, there have been increased interests in the adaptation of natural or biological phenomena to design more efficient optimization search algorithms. A particular class (population based) of these algorithms which have been shown to be efficient are based on cooperation among agents in a solution space. An agent can been seen as a candidate solution to the optimization problem to be solved. To find optimal solutions, candidate solutions share\/exchange information about the search space as the algorithm progresses.\n\nThe main difference between these algorithms based on agent cooperation lies in the way the agents share information, and how candidate solutions are updated based on the exchanged information.\n\nThere is always a need to identify new or adapt existing meta-heuristic algorithms, which may perform better for some kind of computationally intractable problems. This is why this study proposes the Infrasonic Search Algorithm (ISA); a population based algorithm inspired from the Gravitational Search Algorithm (GSA)\\cite{Rashedi2009}. \n\nIn the GSA, it is assumed that masses represent the fitness of a solution, with heavier masses indicating fitter solutions and based on the law of gravitation, heavier masses attract lighter masses towards itself. In the same vein, in the infrasonic search algorithm higher quality solutions produce higher sound intensity compared to poorer solutions and agents move towards regions producing high intensity infrasonic sounds.\n\nThe difference between the GSA and the mating display algorithm is in the formulation of fitness of solutions and the movement of agents in a solution space. The GSA is based on the law of gravity, while the mating display heuristic is based on sound intensity. In subsequent sections, some population based heuristic algorithms are overviewed, the Infrasonic Search Algorithm is defined and result of solving 23 benchmark functions with the Infrasonic Search Algorithm compared with other population based algorithms is presented.\n\n\n\n\\section{Related Work}\\label{I2}\nIt is necessary to find optimal combinations for the efficient management of scarce resources, which is why optimization methods are useful and have been applied in a number of fields among which include Manufacturing\\cite{Alaouchiche2021,Yildiz2017}, Engineering\\cite{Houssein2020,Feng2021} and Education\\cite{kenekayoro2020population, Kenekayoro2016}.\n\nSolutions to optimization problems which could be constrained or unconstrained, single objective or multi objective have been successfully found using \nalgorithms that could broadly be classified into nature inspired vs non-nature inspired. However, Beheshti, Mariyam and Shamsuddin\\cite{Beheshti2013} \nhave listed other possible classification schemes from different view points. Irrespective of the classification scheme, algorithms can still be \ngrouped to be either nature or non-nature inspired.\n\nIn this research, nature inspired algorithms are those algorithms such as the Genetic Algorithm\\cite{Holland1975}, Ants Colony Optimization algorithm\\cite{Dorigo1999}, Gravitational Search Algorithm\\cite{Rashedi2009} and Particle Swarm Optimization Algorithm\\cite{Venter2003} that mimic a physical or biological process that exist in nature, while non-nature inspired algorithms such as Tabu Search\\cite{Glover1989}, Hill Climbing and \nHarmony Search\\cite{Mahdavi2007} are based on heuristic not present in nature. \n\nPopulation based algorithms are a subset of nature inspired optimization methods where information is shared between candidate solutions in order to \nfind better candidates. The difference between population based techniques is largely in the way information is shared between candidate solutions \nand how new candidate solutions are found based on the shared information. \n\nThe Genetic Algorithm\\cite{Holland1975}, inspired from natural evolution shares information by selection, crossover and mutation genetic operators which have been discussed extensively in researches\\cite{Deb2000,MagalhaesMendes2013}. In the genetic algorithm, a candidate solution is represented as a chromosome \nand then two solutions are combined through the crossover operator or a single allele in a chromosome is changed through mutation to generate \na new candidate solution. Selection techniques among which include tournament and roulette wheel are used to determine chromosomes that will make up a subsequent generation. Razali and Geraghty\\cite{Razali2011} have investigated how different GA selection strategies influence the performance of the genetic algorithm. In principle the good properties of older solutions will be transferred to subsequent generations, ensuring that solutions found in subsequent generations are fitter than solutions found in previous generations. \n\nThe Ants Colony Optimization Algorithm\\cite{Dorigo1999} is inspired from the way ants find a path to a food source through cooperation. Initially, \nagents move randomly but update their path trail with pheromones proportional to the quality of a solution found. Subsequent agents use this pheromone \ntrail as information that guide their traversal to a solution and then if the path is promising, additional pheromones are deposited on the path, thus \nincreasing the likelihood that future agents will follow that path.\n\nAgents in the Gravitational Search Algorithm\\cite{Rashedi2009} share information following the Newtonian laws of gravity and motion. The quality of a solution (fitness of an agent) determines its mass, and agents move towards each other by the gravitational force $F = ma$. Fitter agents have larger masses, thus their movement is less influenced by less fit solutions, whilst unfit solutions move towards heavier masses (fitter solutions).\n\nAgents in the Particle Swarm Optimization algorithm are guided to newer solutions based on the history of the agents' best solution found and the global best solution of all agents in a population.\n\nNumerous population based algorithms have been identified, the majority of which have been exhaustively reviewed\\cite{Beheshti2013}; however the genetic algorithm, particle swarm optimization and the gravitational search algorithms are among the most widely used population based algorithms that is used to solve optimization problems. The algorithm investigated in this research is an adaptation of the GSA, and the results presented in subsequent sections compares the quality of the Infrasonic Search Algorithm with the Genetic Algorithm, Particle Swarm Optimization Algorithm and the Gravitational Search Algorithm.\n\n\\section{Methods}\n\n\n\\subsection{Infrasonic Search Algorithm (ISA)}\nLargely inspired from the Gravitational Search Algorithm (GSA), in the Infrasonic Search Algorithm (ISA) agents aim to creation the loudest sound intensity by listening and adjusting its position based on the sound produced from other agents in the solution space.\n\nPeafowls have been shown to produce infrasonic sounds with their tails to attract mates. Other peacocks respond to this mating sound by increased movement or an increase in a corresponding infrasonic sound.\\cite{Freeman2015} Adapting this for a heuristic search algorithm, high intensity sounds are created from high quality solutions which determine the rate of movement\/exploration of search space from neighbouring agents. Given that sound intensity reduces as the distance between the originating source increases, the rate at which an agent responds to sound from another agent reduces as the euclidean distance \nbetween the two agents increases. The formulation for updating and creating sounds in described in the next section.\n\n\n\\subsubsection{Formulation of the ISA}\nThe intensity of sound from a peafowl and the rate at which the sound is produced is used to attract mates among peafowls. This study uses this principle for a meta-heuristic search algorithm. In adapting the peafowls' infrasonic mating sound display for a meta-heuristic search algorithm, higher sound intensity\/rate are created by fitter agents, and thus result in increased movement\/exploration from other agents. \n\nSimilar to the GSA, in the ISA agents are attracted towards other fitter agents who ostensibly are better mates, and the extent to which an agent will move towards another agent is based on the sound intensity from that agent and the distance between the two agents. \n\nThe sound intensity between two agents represented as vectors $V_i$ and $V_j$ is defined as:\n\\begin{equation}\nI_{i,j} = \\frac{P_i}{A^2_{i,j}}\n\\label{eq1}\n\\end{equation}\nwhere $A_{i,j} = 4 \\pi r^2_{i,j}$ and $r$ is he euclidean distance between $V_i$ and $V_j$ and $P_i$ is the normalised power determined by the fitness of $V_i$.\n\nSound power is the rate at which the energy from a source (agent) is transferred. The normalised fitness at a given time $t$ of the agent is determined by Equation \\ref{eq2} and Equation \\ref{eq3}. \n\n\\begin{equation}\np_i(t) = \\frac{fitness_i(t) - worst(t)}{best(t) - worst(t)}\n\\label{eq2}\n\\end{equation}\n\n\\begin{equation}\nP_i(t) = \\frac{p_i(t)}{ \\sum\\limits_{j=1}^{n} p_j(t) }\n\\label{eq3}\n\\end{equation}\n\nAn agent moves towards its neighbour with the highest intensity by a percentage of the difference between the vector that represents the agent $V_i$ and the vector that represents the neighbour $V_j$ with the highest intensity as shown in Equation \\ref{eq5}. The percentage $per_i(t)$ is computed by Equation \\ref{eq4}, where $\\rho = [0, 100]$ is a variable that controls the extent to which an agent is attracted to the neighbour with the highest intensity. \n\n\\begin{equation}\nper_i(t) =\\begin{cases} \\left(\\frac{fitness_j(t)}{max(fitness_i(t), fitness_j(t))} ~ + ~ rand(0, 1) \\right) \\times \\rho & minimisation \\\\\n \\left(\\frac{fitness_i(t)}{max(fitness_i(t), fitness_j(t))} ~ + ~ rand(0, 1) \\right) \\times \\rho & maximisation\n \\end{cases}\n\\label{eq4}\n\\end{equation}\n\n\\begin{equation}\ndisplacement_i(t) = \\frac{\\left( V_i(t) - V_j(t) \\right) \\times per_i(t)}{100}\n\\label{eq5}\n\\end{equation}\n\nThe displacement computed in Equation \\ref{eq5} is added to the vector representing the agent $V_i$ plus a fraction of its past movement to form a new vector for the subsequent iteration as shown in Equation \\ref{eq7}. The idea is that an agent's memory also has an influence on its future movement, a concept also used in the Gravitational Search Algorithm.\n\n\\begin{equation}\n\\begin{array}{c}\nmovement_i(t) = movement_i(t - 1) \\times rand(0, 1) + displacement_i(t) \\\\\nwhere~movement_i(0)~=~0\n\\end{array}\n\\label{eq6}\n\\end{equation} \n\n\n\\begin{equation}\nV_i(t + 1) = movement_i(t) + V_i(t)\n\\label{eq7}\n\\end{equation}\n\nIn cases when updating an agent's vector result in a vector outside the search domain, values outside the domain are set to a random value in the search domain. \n\nAgents in a population are initialized to random values, for example $F_1(x)$ for \\ref{tab2} an agent will be initialized to a random vector $[d_1, d_2 .... d_{n=50}]$; where $d_1$ is a value in range $[-100, 100]$. In each iteration, the position of agents are adjusted based on equation \\ref{eq5}, equation \\ref{eq6} and equation \\ref{eq7} and then the global best agent is returned in the final iteration. In terms of the time complexity of the ISA, as with the GSA, all agents are compared with others, thus the algorithm is run in $O(n^2)$. The pseudocode for the ISA is shown in Algorithm \\ref{algo1}.\n\nTwo parameters control how the agents are moved towards neighbours; $\\rho$ a value in range $[0 - 100]$ is a percentage that controls the extent to which an agent moves towards its highest intensity neighbour and the random value in equation \\ref{eq6} controls the extent to which an agents' historical movement influences its new position. Grid search is used to find appropriate values for $\\rho$ for each benchmark function. In grid parameters are exhaustively searched to find the most optimal value for a particular algorithm. \n\n \n\\begin{algorithm2e}[H]\n\\DontPrintSemicolon\n \n $pop \\gets initialize~population~to~random~values$\\\\\n $movement \\gets initialize~historical~movement~to~zeroes$\\\\\n $best \\gets fittest~solution~in~pop$\n \n \\For{i = 1 to max iterations} \n { \n Compute displacement (Equation \\ref{eq5})\\\\\n \tCompute the extent to which agents are moved (Equation \\ref{eq6})\\\\\n \tUpdate historical movement\\\\\n \tUpdate the position of agents to form a new population (Equation \\ref{eq7})\\\\\n \tUpdate best solution\\\\\n }\n \n \\Return $best$.\n\\caption{Pseudocode for the Infrasonic Search Algorithm (ISA)}\n\\label{algo1}\n\\end{algorithm2e}\n\n \n\n\\section{Comparative Study}\n\nHussain et al.\\cite{hussain2017common} have reviewed benchmark functions that can be used to evaluate the quality of meta-heuristic algorithms. 23 of these benchmark functions shown in Appendix \\ref{appendix_a} were used to evaluate to performance of the ISA in order to compare with the results of the Genetic Algorithm, Particle Swarm Optimization and the Gravitational Search Algorithm published by Rashedi et al.\\cite{Rashedi2009}. \n\nTable \\ref{tab2} shows the average and median results for the minimization of the 23 benchmark functions in Appendix \\ref{appendix_a} after 1000 runs for $F_1$ to $F_{13}$ and 500 iterations for $F_{14}$ to $F_{23}$. The result for the Genetic Algorithm, Particle Swarm Optimization and the Gravitational Search Algorithm is extracted from the results presented by Rashedi et al.\\cite{Rashedi2009}.\n\n\n\n\n\n\n\\section{Discussion and Conclusion}\nSimplistic methods have been shown to be unable to find adequate results for the majority\nof optimisation problems, which is why this researches have focused on finding more suitable\ntechniques to find optimal solutions to these kind of problems.\n\nEven as a number of search algorithms inspired from natural phenomena and\/or other principles\nthat perform well for optimisation problems have been developed, the no free lunch theorem \nsuggests that no method is guaranteed to outperform all others in all problems. Thus, there is the \nconstant need to adapt previously developed algorithms or identify new algorithms that may perform \nwell for some categories of search problems.\n\nThis research presented an algorithm inspired from the mating behaviour of peafowls. Peafowls create \ninfrasonic sounds to attract possible mates, and the probability that another peafowl responds to to a \ngenerated infrasonic sound is increases based on the sound intensity produced and the distance between \npeafowls. This is a simplistic description on how peafowls identify suitable mates. In reality, infrasonic \ninfrasonic sound generation is just one factor in a more complex process.\n\nFollowing the principle of peafowls using infrasonic sounds to attract suitable mates, an agent in a population \nwill produce sounds with higher intensity based on the quality (fitness) of the agents' solution, and thus attract \nother agents to itself. This idea is identical to the Gravitational Search Algorithm, in that in the GA, fitter \nsolutions have higher masses and thus attracts other solutions. The main difference between the GSA \nand the method presented in this research is in the formulation in which agent move towards each other. In the GA, \nthe formulation is based on the laws of gravity while the method presented in this study (Infrasonic Sound Algorithm), \nis based on the formulations of sound travelling in a medium.\n\nThe results showed in Table \\ref{tab2} suggest that the ISA was able to identify competitive solutions to 23 benchmark unimodal\nand multimodal functions compared to the Genetic Algorithm, Particle Swarm Optimization and the Gravitational Search Algorithm. The ISA \nperformed better than the GSA in 4 of the 23 benchmark functions, and better than the Genetic Algorithm in 10 of the 23 \nbenchmark functions, suggesting that the proposed method may be suitable for some kind of optimization problems. Further work \nwill aim to find those optimization problems that the ISA will be suitable for. \n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzameg b/data_all_eng_slimpj/shuffled/split2/finalzzameg new file mode 100644 index 0000000000000000000000000000000000000000..a3fa39df6473c7e52975e99b01bcbf9998ec60ae --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzameg @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec1}\nIn this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy Littlewood maximal operator on variable $\\ell^{{p(\\cdot)}} ({\\mathbb Z})$ sequence spaces. Inequalities for Hardy Littlewood maximal operator for variable sequence spaces $\\ell^{{p(\\cdot)}} ({\\mathbb Z})$, follows from corresponding inequalities for $\\ell^p({\\mathbb Z})$ spaces and Log Holder continuity. To prove, the inequalities for fractional Hardy Littlewood maximal operator, we use Calderon Zygmund decomposition for sequences and Log Holder continuity. \\\\\nSeveral authors have proved strong type and weak type inequalities for Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator using various methods. In this paper, we provide a method that uses Calderon-Zygmund decomposition for sequences\n to prove strong and weak type inequalities for these maximal operators. Similar proof for the real line can be found in $\\text{\\cite{fior_new_proof_maximal}}$ where boundedness of Hardy-Littlewood maximal operator is proved using Calderon-Zygmund decomposition. \nEven though, we follow same method of proof given in $\\text{\\cite{fior_new_proof_maximal}}$, there are some remarkable differences \nbetween the proofs on integers and proofs on real line. \\\\\nFor detailed history of variable Lebesgue spaces, please refer $\\text{\\cite{fior_var_book}}, \\text{\\cite{lars1}}$. Similar conclusions in broader scope utilizing framework of homogeneous spaces may be found in $\\text{\\cite{shukla}}$. For detailed references on work in spaces of homogeneous type under variable exponent setting, please refer $\\text{\\cite{alme1}}, \\text{\\cite{mizuta1}}, \\text{\\cite{goro1}}, \\text{\\cite{hast1}},\\text{\\cite{kok1}}, \\text{\\cite{kok2}}$.\n\nIn this paper, we use the method of Calderon Zygmund decomposition for sequences and Log Holder continuity. We assume only Log Holder continuity at infinity and local Log Holder continuity follows in case of integers. \n\n\n\n\n\n\n\n\n\n\\section{Definitions and Notation}\n\n\n\n\n\nIn this paper, the following notation are used. Given a bounded sequence $ \\left \\{ p(n): n \\in {\\mathbb Z} \\right \\}$ which takes values in $[1, \\infty ) $, define \n$\\ell^{{p(\\cdot)}} ({\\mathbb Z})$ to be set of all sequences $\\left \\{ a(k) \\right \\}$ on ${\\mathbb Z}$ such that for some $\\lambda>0$, $ \\sum_{ k \\in {\\mathbb Z}} ( \\frac{\\abs{a(k)}}{\\lambda} )^{p(k)}< \\infty$. \\\\\nThroughout this paper, $ \\left \\{ p(n): n \\in {\\mathbb Z} \\right \\}$ denotes a bounded sequence which takes values in $[1,\\infty)$. Define $ {p_{-}} = \\inf \\left \\{ p(n): n \\in {\\mathbb Z} \\right \\}, {p_{+}} = \\sup \\left \\{ p(n): n \\in {\\mathbb Z} \\right \\} $. \\\\\nLet $\\mathcal{S}$ denote the set of all bounded sequences which take values in $[1, \\infty)$ such that ${p_{+}} < \\infty $. \\\\\nFor any set $ A \\subset {\\mathbb Z}$ , $\\abs{A}$ denotes cardinality of A. \n\\\\\nLet \n\\[ \\Omega = \\left \\{ n \\in {\\mathbb Z}: 1 \\leq p(n) < \\infty \\right \\} \\] \n\\[ \\Omega_\\infty = \\left \\{ n \\in {\\mathbb Z}: p(n) = \\infty \\right \\} \\] Then define modular functional associated with ${p(\\cdot)}$ as\n\\[ \\rho_{{p(\\cdot)}}(a) = \\sum_{k \\in {\\mathbb Z} \\setminus \\Omega_\\infty} \\abs{ a(k)}^{p(k)} + \\normb{a}_{\\ell^\\infty(\\Omega_\\infty)} \\]\n\nNote, when ${p_{+}} < \\infty$, modular functional becomes\n\\[ \\rho_{{p(\\cdot)}}(a) = \\sum_{k \\in {\\mathbb Z} } \\abs{ a(k)}^{p(k)} \\]\nThroughout this paper, we assume ${p_{+}} < \\infty$. Further for a given sequence $\\left \\{ a(k): k \\in {\\mathbb Z} \\right \\}$, define norm in $\\ell^{{p(\\cdot)}}({\\mathbb Z})$ as\n \\[ \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = \\inf \\left \\{ \\lambda >0: \\rho_{{p(\\cdot)}}(\\frac{a}{\\lambda}) \\leq 1 \\right \\} \\]\n$\\normb{}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$ is a norm $\\text{\\cite{han}}$. A similar norm on $\\ell^{{p(\\cdot)}}({\\mathbb R}^n)$ is defined in $\\text{\\cite{fior_var_book}}$ and there it is proved it is a norm. \n\n\n\n\\begin{definition}\nGiven a non-negative sequence of real numbers, $ \\left \\{ a(n): n \\in {\\mathbb Z} \\right \\}, 0 \\leq \\alpha<1$, define fractional Hardy-Littlewood Maximal operator as follows:\n\\[ M_{\\alpha} a(n) = \\sup_{n \\in I} \\abs{I}^{\\alpha -1} \\sum_{k \\in I} \\abs{a(k)} \\] where supremum is taken over all intervals of integers which contain $n$. When $\\alpha=0$, fractional Hardy-Littlewood maximal operator becomes Hardy-Littlewood maximal operator.\n\\end{definition}\n\n\n\n\n\n\n\n\\begin{definition}\nGiven a sequence ${p(\\cdot)} \\in \\mathcal{S}$, we say that ${p(\\cdot)}$ is log-Holder continuous at infinity, if there exists positive real constants $C_\\infty, p_\\infty$ such that for all $n \\in {\\mathbb Z}$, \n\\[ \\abs{ p(n) - p_\\infty } \\leq \\frac{ C_\\infty} { \\log(e + \\abs{n})} , n \\in {\\mathbb Z} \\] and $e$ is exponential number. In this case, we write ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$\n\\end{definition}\n\n\nWe will also require Young's inequality in this paper. For any real $ a, b \\in {\\mathbb R}$, and any two real numbers $p, q$ such that $\\frac{1}{p} + \\frac{1}{q}= 1$, Young's inequality $\\text{\\cite{Krey1}}$ is as follows: \n\\[ ab \\leq \\frac{a^{p}}{p} + \\frac{b^{q}}{q } \\]\n\n\\begin{lemma}{\\label{sak_lem_LH_infty}}\nLet $ \\left \\{ q(n) \\right \\}$ be the sequence which satisfies $\\frac{1}{p(n)} + \\frac{1}{q(n)} =1$ $\\forall n \\in {\\mathbb Z}$. Let $1 \\leq {p_{-}} \\leq p(n) \\leq {p_{+}} < \\infty$, $\\forall n \\in {\\mathbb Z}$. The following are equivalent:\n\\begin{enumerate}\n\\item ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$\n\\item $\\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$ \n\\item $\\frac{1}{{q(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$ \n\\item ${q(\\cdot)} \\in LH_\\infty({\\mathbb Z})$ \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(a) We shall prove (1) $\\implies (2).\\\\\n $Let ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$. Note, that when ${p_{+}} < \\infty$, $\\forall n \\in {\\mathbb Z}$,\n\\begin{align*}\n\\abs{ \\frac{1}{p(n)} - \\frac{1}{p_\\infty}} = \\abs{ \\frac{p(n) - p_\\infty}{p(n)p_\\infty}} \\leq \\frac{\\abs{p(n) - p_\\infty}}{{p_{-}} p_\\infty} \\leq \\frac{\\frac{C_\\infty}{{p_{-}} p_\\infty}}{\\log(e+\\abs{n})} \n\\end{align*} \nfor some $LH_\\infty$ constant, which is $k_\\infty = \\frac{C_\\infty}{{p_{-}} p_\\infty}$. \\\\\n(b) We shall prove (2) $\\implies$ (1).\\\\\nLet $\\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$. Then, $\\forall n \\in {\\mathbb Z}$\n\\begin{align*}\n\\abs{p(n)-p_\\infty} &=\\abs{p(n)p_\\infty \\biggl( \\frac{1}{p(n)} - \\frac{1}{p_\\infty} \\biggr) } \\leq \\abs{p(n)p_\\infty} \\frac{C_\\infty}{\\log(e+\\abs{n})} \\\\\n& \\leq {p_{+}} p_\\infty \\frac{C_\\infty }{\\log(e+\\abs{n})} \\\\\n& \\leq \\frac{k_\\infty}{\\log(e+\\abs{n})}\n\\end{align*} for some $LH_\\infty$ constant which is $k_\\infty = {p_{+}} p_\\infty C_\\infty $ .\\\\\n(c) We shall prove that (1) $\\implies$ (3). \\\\\nLet ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$. Note, that when ${p_{+}} < \\infty$, for $n \\in {\\mathbb Z}$,\n\\begin{align*}\n\\abs{ \\frac{1}{q(n)} - \\frac{1}{q_\\infty}} &= \\abs{ \\frac{1}{p(n)} - \\frac{1}{p_\\infty}} = \\abs{ \\frac{p(n) - p_\\infty}{p(n)p_\\infty}} \\leq \\frac{\\abs{p(n) - p(\\infty)}}{{p_{-}} p_\\infty} \\\\\n& \\leq \\frac{\\frac{C_\\infty}{{p_{-}} p_\\infty}}{\\log(e+\\abs{n})} \n\\end{align*} \nfor some $LH_\\infty$ constant, which is $k_\\infty = \\frac{C_\\infty}{{p_{-}} p_\\infty}$. \\\\\n(d) We shall prove (3) $\\implies$ (4) and (1).\\\\\n Let $\\frac{1}{{q(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$. Then $\\frac{1}{\\frac{1}{{q(\\cdot)}}} \\in LH_\\infty({\\mathbb Z})$, which implies that ${q(\\cdot)} \\in LH_\\infty({\\mathbb Z})$. Since $\\frac{1}{{q(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$ and\n\\[ \\abs{ \\frac{1}{p(n)} - \\frac{1}{p_\\infty}} = \\abs{ \\frac{1}{q(n)} - \\frac{1}{q_\\infty}} \\] it follows that $\\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$ and hence, using (b), ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$.\\\\\n\\end{proof}\n\n\\begin{lemma}\nLet $ \\left \\{ q(n) \\right \\}$ be the sequence which satisfies $\\frac{1}{p(n)} + \\frac{1}{q(n)} =1$ $\\forall n \\in {\\mathbb Z}$. Let $1 \\leq {p_{-}} \\leq p(n) \\leq {p_{+}} < \\infty$, $\\forall n \\in {\\mathbb Z}$. Then, ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$ implies $p_\\infty \\geq {p_{-}}$.\n\\end{lemma}\n\\begin{proof}\n Given that ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$ implies $\\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$ using Lemma[$\\ref{sak_lem_LH_infty}$] ,it follows that\n\\begin{align*}\n\\abs{ \\frac{1}{p_\\infty}} & = \\abs{ \\frac{1}{p(n)} + \\frac{1}{p_\\infty} - \\frac{1}{p(n)} } \\\\\n& \\leq \\abs{ \\frac{1}{p(n)}} + \\abs{ \\frac{1}{p_\\infty} - \\frac{1}{p(n) } } \\\\\n& \\leq \\abs{\\frac{1}{{p_{-}}}} + \\frac{C_\\infty}{\\log(e+\\abs{n}) } \n\\end{align*}\nSince this is true for every $n$, $p_\\infty \\geq {p_{-}}$.\n\\end{proof}\n\nThe following lemma will be used as a variation of Holder's inequality. Proof of this lemma in continuous version can be found in $\\text{\\cite{fior_var_book}}$. Same line of proof works here.\n\\begin{lemma}{\\label{sak_holder_lem1}}\nFor $0 \\leq \\alpha < 1$, and $ p,q $ such that $ 1< p< \\frac{1}{\\alpha}, \\frac{1}{p} - \\frac{1}{q} = {\\alpha}$.\nFor every interval $I$ in ${\\mathbb Z}$ and non-negative sequence $\\left \\{ a(k) \\right \\}$ \\\\\n\\[ \\abs{I}^{{\\alpha} -1 } \\sum_{k \\in I} a(k) \\leq \\biggl( \\sum_{k \\in I} a(k)^p \\biggr)^{\\frac{1}{p} - \\frac{1}{q}} \\biggl( \\frac{1}{\\abs{I}} \\sum_{k \\in I} a(k) \\biggr)^{\\frac{p}{q}} \\]\n\\end{lemma}\n\n\n\n\n\\section{Some Results on Variable Lebesgue spaces}\nIn this section, we prove certain results that are used in proving boundedness of maximal operators.The proofs of these results in discrete case are same as corresponding results on real line $\\text{\\cite{fior_var_book}}$. For completeness sake, some of the proofs are included. \\\\\n\n\n\n\nThe modular functional has the following properties defined below $\\text{\\cite{fior_var_book}}$.\n\\begin{lemma}{\\label{sak_modular}}\nLet $ \\left \\{ u(n) \\right \\} $ be a non-negative sequence of real numbers. Let ${p(\\cdot)} \\in \\mathcal{S}$, then: \n\\begin{enumerate}\n\\item For all u, $\\rho_{{p(\\cdot)}}(u) \\geq 0$ and $\\rho_{{p(\\cdot)}}(\\abs{u})=\\rho_{{p(\\cdot)}}(u) $\\\\\n\\item $\\rho_{{p(\\cdot)}}(u) =0$ if and only if $u(k)=0$ for all k $\\in {\\mathbb Z}$ \\\\\n\\item If $\\rho_{{p(\\cdot)}}(u) < \\infty$, then $u(k) < \\infty $ for all k $\\in {\\mathbb Z}$ \\\\\n\\item $\\rho_{{p(\\cdot)}}$ is convex: Given $\\alpha, \\beta \\geq 0, \\alpha + \\beta =1, \\rho_{{p(\\cdot)}}(\\alpha u + \\beta v) \\leq \\alpha \\rho_{{p(\\cdot)}}(u) + \\beta \\rho_{{p(\\cdot)}}(v) $\\\\\n\\item If $\\abs{u(k)} \\geq \\abs{v(k)}$ , then $\\rho_{{p(\\cdot)}}(u) \\geq \\rho_{{p(\\cdot)}}(v) $. \\\\\n\\item If for some $\\delta > 0 , \\rho_{{p(\\cdot)}}(\\frac{u}{\\delta})< \\infty $, then the function $\\lambda \\to \\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda})$ is continuous \nand decreasing on $[\\delta, \\infty)$. Further $\\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda}) \\to 0$ as $\\lambda \\to \\infty $.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nContinuous version of Lemma[$\\ref{sak_modular}$] can be found in $\\text{\\cite{fior_var_book}}$. Same line of proof works here.\n\\end{proof}\n\nThe following lemma is the connection between modular functional and $\\ell^{{p(\\cdot)}}({\\mathbb Z})$ norm.\n\\begin{lemma} \nLet $ \\left \\{ u(n) , n \\in {\\mathbb Z} \\right \\} $ be a non-negative sequence of real numbers. Let ${p(\\cdot)} \\in \\mathcal{S}$. if ${p_{+}} < \\infty$, then $u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$ if and only if\n\\[ \\rho_{{p(\\cdot)}}(u) = \\sum_{k \\in {\\mathbb Z}} {\\abs{u(k)}}^{p(k)} < \\infty \\]\n\\begin{proof}\nNote if $\\rho_{{p(\\cdot)}}(u) < \\infty$, then by definition of norm $u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$. Conversely since $u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$ by definiton of norm, $\\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda}) < \\infty$ for some $\\lambda >0$. Further by (6) of modular functional properties, we have $\\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda}) < \\infty$ for some $\\lambda >1$. So, it follows that\n\\[ \\rho_{{p(\\cdot)}}(u)= \\sum_{k \\in {\\mathbb Z}} \\left( \\frac{\\abs{u(k)}\\lambda}{\\lambda} \\right)^{p(k)} \\leq \\lambda^{{p_{+}} } \\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda}) < \\infty \\]\n\\end{proof}\n\\end{lemma}\n\n\n\\begin{proposition}{\\label{sak_Prop_2.10}}\nLet $ \\left \\{ a(n) , n \\in {\\mathbb Z} \\right \\} $ be a non-negative sequence of real numbers such that $a \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$ and let $p(\\cdot) \\in \\mathcal{S}$.\n\\begin{enumerate}\n\\item For all $\\lambda \\geq 1$, \n\\[ \\lambda^{{p_{-}}} \\rho_{{p(\\cdot)}}(a) \\leq \\rho_{{p(\\cdot)}}(\\lambda a) \\leq \\lambda^{{p_{+}}} \\rho_{{p(\\cdot)}}(a) \\] \n\\item when $0 < \\lambda <1$ the reverse inequalities are true, i.e \n\\[ \\lambda^{{p_{+}}} \\rho_{{p(\\cdot)}}(a) \\leq \\rho_{{p(\\cdot)}}(\\lambda a) \\leq \\lambda^{{p_{-}}} \\rho_{{p(\\cdot)}}(a) \\]\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nFor $\\lambda \\geq 1$ , \n\\begin{align*}\n\\rho_{{p(\\cdot)}}(\\lambda a) = \\sum_{n=1}^\\infty \\biggl( \\lambda a(n) \\biggr)^{p(n)} = \\sum_{n=1}^\\infty {\\lambda}^{p(n)} {a(n)}^{p(n)} \\leq \\lambda^{{p_{+}}} \\rho_{{p(\\cdot)}}(a)\n\\end{align*}\nFurther,\n\\begin{align*}\n{\\lambda}^{{p_{-}}} \\rho_{{p(\\cdot)}}(a) = \\sum_{n=1}^{\\infty} {\\lambda}^{{p_{-}}} {a(n)}^{p(n)} \\leq \\sum_{n=1}^\\infty \\biggl( \\lambda a(n) \\biggr)^{p(n)} = \\rho_{{p(\\cdot)}}(\\lambda a)\n\\end{align*}\n\\end{proof}\n\n\n\\begin{lemma}{\\label{sak_lem7}}[Fatou Property of the Norm].\nGiven non-negative sequence $\\left \\{ u(n) \\right \\}$ of real numbers, let ${p(\\cdot)} \\in \\mathcal{S}$. Further, let $ \\left \\{ u_k \\right \\} \\subset \\ell^{{p(\\cdot)}}({\\mathbb Z})$ be non-negative sequences of real numbers such that $u_k$ increases to a sequence $u$ pointwise. Then either $u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z}) $ and $ \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\to \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$ or $ u \\notin \\ell^{{p(\\cdot)}}({\\mathbb Z})$ and $\\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\to \\infty$.\n\\end{lemma}\n\\begin{proof}\nFrom\n$u_k(n) \\leq u_{k+1}(n) \\implies \\sum_{k \\in {\\mathbb Z}} u_k(n) \\leq \\sum_{k \\in {\\mathbb Z}} u_{k+1}(n) $. So $\\rho_{{p(\\cdot)}}(u_k) \\leq \\rho_{{p(\\cdot)}}(u_{k+1})$. This implies that $\\rho_{{p(\\cdot)}}(\\frac{u_k}{\\lambda}) \\leq \\rho_{{p(\\cdot)}}(\\frac{u_{k+1}}{\\lambda}) $ and hence $\\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\normb{u_{k+1}}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$.\nTherefore, $\\left \\{ u_k \\right \\}$ is an increasing sequence, so is $ \\left \\{ \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\right \\}$, thus it either converges or diverges to $\\infty$. \\\\\n\nIf $ u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$, since $ u_k \\leq u, \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$. Otherwise, if $u_k \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$, then $\\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} < \\infty = \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$. In either case,it will suffice to show that for any $\\lambda < \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$, for all k, sufficiently large $\\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} > \\lambda$. \\\\\n\nFix such a $\\lambda$, by the definition of the norm $ \\rho_{{p(\\cdot)}}(u\/\\lambda) > 1$. Therefore by monotone convergence theorem of the classical Lebesgue spaces,\n\\begin{align*} \\rho_{{p(\\cdot)}}(\\frac{u}{\\lambda}) & = \\sum_{m \\in {\\mathbb Z}} \\biggl( \\frac{ \\abs{u(m) }} {\\lambda } \\biggr)^{p(m)} = \\lim_{k \\to \\infty} \\biggl( \\sum_{m \\in {\\mathbb Z}} \\biggl( \\frac{ \\abs{ u_k(m) } } { \\lambda } \\biggr)^{p(m)} \\biggr) = \\\\\n & = \\lim_{k \\to \\infty} \\rho_{{p(\\cdot)}}(\\frac{u_k} {\\lambda}) \n\\end{align*}\nHence, for all $k $ sufficiently large, $\\rho_{{p(\\cdot)}}(u_k\/ \\lambda) > 1$ and so $ \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} > \\lambda $.\n\\end{proof}\n\n\n\\begin{lemma}{\\label{sak_lem8}}[Fatou's lemma for sequences].\nLet $ \\left \\{ u_k \\right \\}$ be a non-negative sequence of real numbers such that $ \\left \\{ u_k \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$. \n. Let ${p(\\cdot)} \\in \\mathcal{S}$, suppose the sequence $ \\left \\{ u_k \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z}) $ is such that $u_k(n) \\to u(n) $ for every $n$. If\n\\[ \\liminf_{k \\to \\infty } \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} < \\infty \\]\nthen $ u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z}) $ and \n\\[ \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\liminf_{k \\to \\infty } \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\]\n\\end{lemma}\n\n\\begin{proof}\nDefine a new sequence \n\\[ v_k(i) = \\inf_{ m \\geq k} \\abs{u_m(i)} \\]\nThen for all $ m \\geq k, v_k(i) \\leq \\abs{u_m(i) }$ and so $ v_k \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$. Further by the definition $ \\left \\{ v_k \\right \\}$ is an increasing sequence and\n\\[ \\lim_{k \\to \\infty } v_k(i) = \\liminf_{ m \\to \\infty} \\abs{u_m(i)} = \\abs{u(i)}, \\quad i \\in {\\mathbb Z} \\]\nAlso,\n\\[ \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = \\normb{ \\lim_{k \\to \\infty} v_k }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = \\lim_{k \\to \\infty} \\normb{v_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = \\liminf_{k \\to \\infty} \\normb{v_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\]\nTherefore, by Fatou's norm property[$\\ref{sak_lem7}$] for sequences\n\\[ \\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = \\lim_{k \\to \\infty} \\normb{v_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\lim_{k \\to \\infty} ( \\inf_{ m \\geq k} \\normb{u_m}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} ) = \\liminf_{k \\to \\infty} \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\]\nSo, if $ \\liminf_{k \\to \\infty} \\normb{u_k}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} < \\infty$, then $\\normb{u}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} < \\infty$, which implies $ u \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$.\n\\end{proof}\n\n\n\n\\begin{lemma}\nLet $ \\left \\{ a(k) \\right \\} $ be a non-negative sequence of real numbers and ${p(\\cdot)} \\in \\mathcal{S}$\n\\begin{enumerate}\n\\item If $ a \\in \\ell^{p(\\cdot)}({{\\mathbb Z}})$ and $ \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} > 0$, then $\\rho_{{p(\\cdot)}}(\\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z}) }}) \\leq 1$ \\\\\n\\item If ${p_{+}} < \\infty$, then $ \\rho_{{p(\\cdot)}}(\\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z}) }}) =1$ for all nontrivial $a \\in \\ell^{p(\\cdot)}({{\\mathbb Z}})$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSince $\\normb{a}_{\\ell^{{p(\\cdot)}}}({\\mathbb Z}) = \\inf \\left \\{ \\lambda > 0: \\rho_{{p(\\cdot)}}(a\/\\lambda) \\leq 1 \\right \\}$, $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}+ \\frac{1}{n}$ is not an infimum. Therefore, there exists a $\\lambda_n$ such that\n$\\lambda_n \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} + \\frac{1}{n}$ and $ \\rho_{{p(\\cdot)}}(\\frac{a}{\\lambda_n}) \\leq 1$.\nFix such a decreasing sequence $\\left \\{ \\lambda_n \\right \\}$ such that $ \\left \\{ \\lambda_n \\right \\} \\to \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$. Then by Fatou's lemma and the definition of modular,\n\\begin{align*}\n\\rho_{{p(\\cdot)}}(\\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}}) & = \\sum_{k \\in {\\mathbb Z}} \\biggl( \\frac{\\abs{a(k)}}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z}) }} \\biggr)^{p(k)}\\\\\n& = \\sum_{k \\in {\\mathbb Z}} \\lim_{n \\to \\infty} \\biggl( \\frac{\\abs{a(k)}}{\\lambda_n} \\biggr)^{p(k)} \\\\\n& \\leq \\liminf_{n \\to \\infty} \\sum_{k \\in {\\mathbb Z}} \\biggl( \\frac{\\abs{a(k)}}{\\lambda_n} \\biggr)^{p(k)} \\\\\n& = \\liminf_{n \\to \\infty} \\rho_{{p(\\cdot)}}(\\frac{a}{\\lambda_n}) \\leq 1\n\\end{align*}\nNow suppose that ${p_{+}} < \\infty$ but $\\rho(\\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}}) < 1$. Then for all $\\lambda$ such that $0< \\lambda < \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$, by Lemma[\\ref{sak_Prop_2.10}]\n\\[ \\rho_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}(a\/\\lambda) = \\rho_{{p(\\cdot)}}( \\frac{ \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}}{\\lambda} \\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}}) \\leq \\biggl( \\frac{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}}{\\lambda} \\biggr)^{{p_{+}}} \\rho_{{p(\\cdot)}}(\\frac{a}{\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}}) \\]\nTherefore, we can find $\\lambda$ sufficiently close to $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$ such that $\\rho_{{p(\\cdot)}}(a\/\\lambda)<1$. \nBut by the definition of the norm , we must have $\\rho_{{p(\\cdot)}}(a\/\\lambda) \\geq 1$. This leads to contradiction, so the equality holds.\n\n\n\\end{proof}\n\nUsing Fatou's lemma for sequences [$\\ref{sak_lem8}$] and properties of modular functional, following lemma can be proved.\n\\begin{lemma}{\\label{sak_corollary_2}}\nLet ${p(\\cdot)} \\in \\mathcal{S}$ and $ \\left \\{ a(k) \\right \\} $ be a non-negative sequence of real numbers such that $ \\left \\{ a(k) \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$\n\\begin{align*}\n\\text{If} \\quad \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq 1, \\text{then} \\qquad \\rho_{{p(\\cdot)}}(a) \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\\\\n\\text{If} \\quad \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\geq 1, \\text{then} \\qquad \\rho_{{p(\\cdot)}}(a) \\geq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nContinuous version of Lemma[$\\ref{sak_corollary_2}$] can be found in $\\text{\\cite{fior_var_book}}$. Same line of proof works here.\n\\end{proof}\n\nUsing Lemma[$\\ref{sak_Prop_2.10}$], we can prove the following lemma below. Continuous version of Lemma[$\\ref{sak_corollary_3}$] can be found in $\\text{\\cite{fior_var_book}}$. Same line of proof works here.\n\\begin{lemma}{\\label{sak_corollary_3}}\nLet $ \\left \\{ a(k) \\right \\} $ be a non-negative sequence of real numbers such that $ \\left \\{ a(k) \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$ and ${p(\\cdot)} \\in \\mathcal{S}$. Then\n\\begin{enumerate}\n\\item If $ \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} >1$, then $\\rho_{{p(\\cdot)}}(a)^{1\/{p_{+}}} \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\rho_{{p(\\cdot)}}(a)^{1\/{p_{-}}}$ \\\\\n\\item If $ 0 < \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq 1 $, then $\\rho_{{p(\\cdot)}}(a)^{1\/{p_{-}}} \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\leq \\rho_{{p(\\cdot)}}(a)^{1\/{p_{+}}}$ \\\\\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Calderon-Zygmund decompostion for Sequences}\n\\begin{theorem}{\\label{CD_General}}\nLet $1\\leq p < \\infty$ and a $\\in l^p({\\mathbb Z})$. For every $t>0$, and $0\\leq \\alpha<1$, there exists a sequence of disjoint intervals $\\left \\{ I_j^t \\right \\}$ such that\n\\begin{align*}\n(i)& \\quad t < \\frac{1}{ \\abs{I_j^t}^{1-\\alpha} } \\sum_{k \\in I_j^t} \\abs{a(k)} \\leq 2t,\\forall j \\in {\\mathbb Z} \\\\\n(ii)& \\quad \\forall n \\not\\in \\cup_j I_j^t, \\quad \\abs{a(n} \\leq t \\\\\n(iii)& \\quad \\text{If} \\quad t_1 > t_2 , \\quad \\textbf{then each} \\quad I_j^{t_1} \\quad \\textbf{is subinterval of some} \\quad I_m^{t_2}, \\quad \\forall j,m \\in {\\mathbb Z}\n\\end{align*}\n\\end{theorem}\n\\begin{proof}\nFor each positive integer N, consider the collection of disjoint intervals of cardinality $2^N$, \\\\\n\\begin{eqnarray*}\n\\left \\{ I_{N,j} \\right \\} = \\left \\{ [ (j-1)2^N +1, \\dots, j2^N] \\right \\} , j \\in {\\mathbb Z}.\n\\end{eqnarray*} \\\\\nFor each $t>0$, let $N=N_t$ be the smallest positive integer such that \n\\[ \\frac{1}{\\abs{I_{N_t,j} }^{1-\\alpha} } \\sum_{k \\in I_{N_t,j}} \\abs{a(k)} \\leq t \\] \t\\\\\nSuch $N_t$ is possible as $ a\\in \\ell^p ({\\mathbb Z})$. Now consider collection $ \\left \\{ I_{N_t,j} \\right \\}$ and subdivide each of these intervals into two intervals of equal cardinality. If $I$ is one of these intervals either \n\\begin{align*}\n(A)& \\quad \\frac{1}{\\abs{I}^{1-\\alpha} } \\sum_{k \\in I} \\abs{a(k)} >t \\\\ \\quad \\text{or} \\quad\n(B)& \\quad \\frac{1}{\\abs{I}^{1-\\alpha} } \\sum_{k \\in I} \\abs{a(k)} \\leq t \\\\\n\\end{align*}\nIn case (A) we select this interval and include it in a collection $\\left \\{ I_{r,j} \\right \\}$. \\\\\nIn case (B) we subdivide I once again unless I is a singleton and select intervals as above. \nNow the elements which are not included in $\\left \\{ I_{r,j} \\right \\}$ form a set S such that for every $n \\in S$, $\\abs{a(n)} \\leq t$. This proves (i). \\\\\nAlso from the choice of $\\left \\{ I_{r,j} \\right \\}$, note that $\\left \\{ I_{r,j} \\right \\}$ are disjoint and satisfy \n\\begin{eqnarray*}\n\\frac{1}{\\abs{I_{r,j} }^{1-\\alpha} } \\sum_{k \\in I_{r,j}} \\abs{a(k)} >t\n\\end{eqnarray*}\nSince each $I_{r,j}$ is contained in an interval $J_0$ with card $J_0 = \\abs{2I_{r,j} }^{1-\\alpha}$, which is not selected in the previous step, we have\n\\[\n\\frac{1}{\\abs{I_{r,j}}^{1-\\alpha} } \\sum_{K \\in I_{r,j}} \\abs{a(k)} \\leq \\frac{2}{\\abs{J_0 }} \\sum_{k \\in J_0} \\abs{a(k)} \\leq 2t.\n\\]\nThis proves (ii). It remains to prove (iii). \\\\\nIf $t_1 > t_2$ then $N_{t_1} \\leq N_{t_2}$. So each $I_{N_{t_1}, j}$ is contained in some \n$I_{N_{t_2}, j}$. In the subdivision and the selecting process for $t_1$ we have \n\\[\n\\frac{1}{\\abs{I_{N_{t_1,j} } }^{1-\\alpha} } \\sum_{k \\in I_{N_{t_1,j} }} \\abs{a(k)} > t_1 > t_2.\n\\]\nSo, if $I_j^{t_1}$ is not one of the intervals $I_m^{t_2}$, then it must be subinterval of some $I_m^{t_2}$ selected in an earlier step. This completes proof.\n\\end{proof}\n\n\n\n\\begin{theorem}{\\label{CD_Weak}}\nLet $1\\leq p < \\infty$ and a $\\in l^p({\\mathbb Z})$. Let $\\left \\{ I_j^t \\right \\}$ be intervals obtained from Calderon Zygmund decomposition at height $t$ and $ 0\\leq \\alpha <1 $. Then\n\\[ \\left \\{ n: M_{\\alpha}a(n) > 9t \\right \\} \\subseteq \\cup_j 2I_j^t \\]\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe apply Calderon-Zygmund decomposition to the sequence $\\left \\{ a(n), n \\in {\\mathbb Z} \\right \\} $, where $a \\in \\ell^p({\\mathbb Z})$. For every $t > 0$, we get a sequence of disjoint intervals $ \\left \\{ I^t_j \\right \\}$ satisfying the criteria of Calderon - Zygmund decompostion. \nFor each $j$, we have\n\\[ \\frac{1}{{\\abs{I^t_j}}^{1-\\alpha}} \\sum_{k \\in I^t_j} \\abs{a(k)} > t \\]\nTherefore, $\\cup_j I^t_j \\subseteq \\left \\{n : M_{\\alpha} a(n) >t \\right \\}$.\n Let $n \\notin \\cup_j 2 I^t_j$ and $I$ be any interval which contains $n$. Then \n\\[ \\sum_{k \\in I} \\abs{a(k)} = \\sum_{ k \\in I \\cap (\\cup_j I^t_j) } \\abs{a(k)} + \\sum_{ I \\cap (\\cup_j I^t_j)^{\\complement} } \\abs{a(k)} = S_1 + S_2 \\]\nTo estimate $S_1$, we observe a simple geometric fact. If $ I \\cap I^t_j$ is non-empty and I is not contained in $ 2 I^t_j$, then $I^t_j \\subset 4I$. Since $n \\in I$ and $n \\notin 2 I^t_j$, for each $j$, I is not contained in $ 2I^t_j$ for each $j$. Also, note that $S_2 \\leq \\abs{I}$.\nTherefore, \n\\begin{align*}\nS_1 &\\leq \\sum_{ \\left \\{ j: I^t_j \\subseteq 4I \\right \\} } \\sum_{ k \\in I^t_j} \\abs{a(k)} \\\\\n& \\leq \\sum_{ \\left \\{ j: I^t_j \\subseteq 4I \\right \\} } 2t \\abs{I^t_j }\\\\\n& \\leq 2t \\abs{4I} \\\\\n& \\leq 8t \\abs{I}. \n\\end{align*} \nHence, $ \\sum_{ k \\in I} \\abs{a(k)} \\leq S_1 + S_2 \\leq 9t \\abs{I} $. \n\nSince $I$ was an arbritrary interval containing $n$, we have $ M_{\\alpha}a(n) \\leq 9t $. Therefore\n\\[ ( \\cup_j 2 I^t_j )^{\\complement} \\subseteq \\left \\{ n: M_{\\alpha}a(n) \\leq 9t \\right \\} \\]\n\\end{proof}\n\n\\begin{lemma}{\\label{CD_Template}}\nLet $1\\leq p < \\infty, a \\in l^p({\\mathbb Z})$ and ${q(\\cdot)} \\in \\mathcal{S}$. Let $\\left \\{ I^k_j \\right \\}$ be intervals obtained from Calderon-Zygmumd decomposition at height $(9t)^{k-1}$ and $t >0$. For any maximal operator $M_{\\alpha}, 0 \\leq \\alpha < 1$, and for every $0 0$. Define $\\Omega_{k} = \\left \\{ i \\in {\\mathbb Z}: M_{\\alpha}a(i) > A^{k} =(9t)^{k} \\right \\}$. \nFor each integer $k$, apply Theorem[$\\ref{CD_General}$] Calderon-Zygmund decomposition for sequence $\\left \\{ a(i) \\right \\}$, at height $t =A^{k-1}$ to get\npairwise disjoint cubes $ \\left \\{ I_j^k \\right \\} $ such that \n\\begin{align}\n\\Omega_k \\subset \\cup_j 2 I_j^k \\label{eq:eq1}\\tag{CD1} \\\\\n\\frac{1}{ {\\abs{I^k_j} }^{1-\\alpha} } \\sum_{{i \\in I^k_j}} a(i) > A^{k-1} \\label{eq:eq2}\\tag{CD2}\n\\end{align}\n\nFrom \\eqref{eq:eq2} we get \\\\\n\\begin{align*}\n\\frac{1}{\\abs{2I_j^k}^{1-\\alpha}} \\sum_{i \\in 2I_j^k} a(i) > \\frac{1}{2^{1-\\alpha}} A^{k-1} \n\\end{align*}\n\nDefine sets inductively as follows: \n\\begin{align*}\nE_1^k = \\biggl( \\Omega_{k+1} \\setminus \\Omega_{k} \\biggr) \\cap 2I_1^k \\nonumber \\\\\nE_2^k = \\biggl( \\biggl( \\Omega_{k+1} \\setminus \\Omega_{k} \\biggr) \\cap 2I_2^k \\biggr) \\setminus E_1^k \\nonumber\\\\\nE_3^k = \\biggl( \\biggl( \\Omega_{k+1} \\setminus \\Omega_{k}\\biggr) \\cap 2I_3^k \\setminus (E_1^k \\cup E_2^k ) \\biggr) \\nonumber\\\\\n\\ldots\nE_m^k = \\biggl( \\biggl( \\Omega_{k+1} \\setminus \\Omega_{k}\\biggr) \\cap 2I_{m}^k \\setminus (E_1^k \\cup E_2^k \\ldots E_{m-1}^k) \\biggr) \\nonumber\\\\\n\\end{align*}\nThen, sets $E_j^k$ are pairwise disjoint for all $j$ and $k$ and satisfy for every $k$\n\\[\\Omega_{k+1} \\setminus \\Omega_{k} = \\bigcup_j E_j^k \\label{eq:eq4} \\tag{CD3} \\]\nSo, ${\\mathbb Z} = \\bigcup_k \\biggl( \\Omega_{k+1}\\setminus \\Omega_{k} \\biggr) = \\bigcup_k \\bigcup_j E_j^k $.\nFurther, note $\\Omega_{k+1} \\setminus \\Omega_{k} = \\left \\{ A^{k+1} < M_{\\alpha} a(i) < A^k \\right \\}$. \\\\\nWe now estimate $ M_{\\alpha}a$ as follows, noting $A < 1$,\n\\begin{align*}\n \\sum_{i \\in {\\mathbb Z}} {M_{\\alpha} a(i)}^{q(i)}\n & = \\sum_k \\sum_{\\Omega_{k+1} \\setminus \\Omega_{k}} {M a(i)}^{q(i)} \\\\\n & \\leq \\sum_k \\sum_{\\Omega_{k+1} \\setminus \\Omega_{k}} [A^{k}]^{q(i)} \\\\\n & \\leq A^{{q_{-}}}2^{{q_{+}}(\\alpha-1)} \\sum_{k,j} \\sum_{E^k_j} \\biggl( \\frac{1}{\\abs{2I^k_j}^{1-\\alpha}} \\sum_{i \\in 2I^k_j} a(i) \\biggr)^{q(i)}\n\\end{align*}\n\\end{proof}\n\n\\section{Fractional Hardy-Littlewood Maximal operator}\n\n\n\\subsection{Strong $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Fractional Hardy-Littlewood Maximal Operator}\nIn this section, we derive strong $({p(\\cdot)}, {p(\\cdot)})$ inequality for fractional Hardy-Littlewood maximal Operator. In order to prove this theorem, we use Lemma[$\\ref{CD_Template}$].\n \n \\begin{theorem}[Strong $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Fractional Hardy-Littlewood Maximal Operator]\nGiven a non-negative sequence $ \\left \\{ a(i) \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z}), 0\\leq \\alpha < 1$, let $ {p(\\cdot)} \\in \\mathcal{S},{p_{+}}< \\infty, {p_{-}}> 1 , {p(\\cdot)} \\in LH_\\infty({\\mathbb Z}), {q(\\cdot)} \\in LH_\\infty({\\mathbb Z}), \\text{and} \\quad 1< {p_{-}} \\leq {p_{+}} < \\frac{1}{\\alpha} $. Define the exponent function ${q(\\cdot)}$ by \n\\[ \\frac{1}{p(k)} - \\frac{1}{q(k)} = \\alpha , k \\in {\\mathbb Z}\\] Then\n\\[ \\normb{M_{\\alpha} a}_{\\ell^{q(.)}({\\mathbb Z})} \\leq C \\normb{a}_{\\ell^{p(.)}({\\mathbb Z})} \\]\n\\end{theorem}\n \n\n\\begin{proof}\n\n\n\n\n \nNow, we are going to prove $ \\normb{M_{\\alpha} a}_{\\ell^{{q(\\cdot)}}({\\mathbb Z})} \\leq c \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$. We may assume without loss of generality that $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} =1$. We will show that there exist a constant $\\lambda_2=\\lambda_2({p(\\cdot)}) > 0$ such that $\\rho_{{q(\\cdot)}}(M_{\\alpha}\\frac{a}{\\lambda_2}) \\leq 1$. \nFor this it suffices to prove $ \\rho_{{q(\\cdot)}}( \\alpha_2 \\beta_2 \\gamma_2 \\delta_2 M_{\\alpha} a) \\leq \\frac{1}{2}$ form some non-negative real numbers $\\alpha_2,\\beta_2, \\gamma_2,\\delta_2 $.\nLet $ {\\lambda_2}^{-1} = \\alpha_2 \\beta_2 \\gamma_2 \\delta_2 $. Then \n\\[ \\rho_{{q(\\cdot)}}( \\alpha_2 \\beta_2 \\gamma_2 \\delta_2 M_{\\alpha} a) = \\sum_{m \\in {\\mathbb Z}} [\\alpha_2 \\beta_2 \\gamma_2 \\delta_2 M_{\\alpha} a(m)]^{q(m)} \\]\nTo estimate this term we perform \nCalderon Zygmund decomposition for sequences $\\left \\{ a(k) \\right \\}$ and use Lemma[$\\ref{CD_Template}$] for the fractional Hardy-Littlewood maximal operator $M_{\\alpha}$. We will show that $\\rho_{{q(\\cdot)}}( \\alpha_2 \\beta_2 \\gamma_2 \\delta_2 M_{\\alpha} a) \\leq \\frac{1}{2} $ for suitable choices of $ \\alpha_2, \\beta_2, \\gamma_2, \\delta_2$.\nIf we set $\\alpha_2 = A^{q_{-}} 2^{{q_{+}}{(\\alpha-1)}}$ and using Lemma[$\\ref{CD_Template}$] for $M_{\\alpha}$, we get\n\\[\\sum_{m \\in {\\mathbb Z}} [ \\alpha_2 \\beta_2 \\gamma_2 \\delta_2 M_{\\alpha} a(m) ]^{q(m)} \\leq \\sum_{k,j} \\sum_{E^k_j} \\biggl( \\beta_2 \\gamma_2 \\delta_2 \\abs{2I^k_j}^{\\alpha-1} \\sum_{r \\in 2I^k_j} a(r) \\biggr)^{q(m)} \\label{eq:eq3}\\tag{CD3} \\]\n We have to estimate right hand side of $\\eqref{eq:eq3}$. At this point, we note that $q_\\infty < \\infty$. Let $ g_2(r) = {a(r)}^{p(r) } $, then $\\eqref{eq:eq3}$ becomes,\n\\[ \\sum_{k,j} \\sum_{E^k_j} \\biggl( \\beta_2 \\gamma_2 \\delta_2 \\abs{ 2I^k_j}^{\\alpha -1} \\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{q(m)} \\]\n\nNote, since $\\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$, it follows that the exponents $p_\\infty, q_\\infty$ satisfy $ \\frac{1}{p_\\infty} - \\frac{1}{q_\\infty} = \\alpha $. Hence, using Lemma $\\ref{sak_lem1}$ with exponents $p_\\infty, q_\\infty$. \n\\begin{align*}\n\\abs{2I^k_j}^{\\alpha -1} \\sum_{ r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} & \n& \\leq \\biggl( \\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{ p_\\infty } { p(r)}} \\biggr)^{ \\frac{1}{p_\\infty} - \\frac{1}{q_\\infty} } \\biggl( \\frac{1}{\\\n\\abs{2I^k_j} }\\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{\\frac{p_\\infty}{q_\\infty} } \\label{eq:eq4}\\tag{CD4} \n\\end{align*}\n\nFurther, note the following estimates.\n\\begin{enumerate}\n\\item \n${g_2(r)}^{p_\\infty} \\leq 1$. since $g_2(r)^{p_\\infty} = {a(r)}^{p(r) p_\\infty} \\leq 1$ since $ a(r) \\leq 1$ \\\\\n\n\\item \nUsing $R(k) = (e+ \\abs{k})^{-N}$, with $N >1$. By taking $N$ large enough, it follows that\n\\[ \\sum_{m \\in {\\mathbb Z}} {R(m)}^{\\frac{1}{p_\\infty}} \\leq \\sum_{m \\in {\\mathbb Z}} {R(m)}^{\\frac{1}{q_\\infty}} \\leq 1 \\]\nBy taking $N$ large, the sum i.e $ \\sum_{m \\in {\\mathbb Z}} R(m) \\leq 1 $. \\\\\n\n\\item Also, note that \n\\[ \\sum_{r \\in 2I^k_j} g_2(r) \\leq \\sum_{r \\in 2I^k_j} a_2(r)^{p(r)} \\leq \\sum_{r \\in {\\mathbb Z}} a(r)^{p(r)} \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} =1 \\] \\\\\n\n\\item Since ${g_2(k)}^{p_\\infty} \\leq 1$. Put $ F = {g_2(k)}^{p_\\infty} $, so that $F \\leq 1$. Hence, we use following form of Lemma[$\\ref{sak_lem1}$] , where $\\frac{1}{r(\\cdot)} $ is taken as $LH_\\infty$ constant.\n\\[ \\sum_{m \\in {\\mathbb Z}} F(m)^{\\frac{1}{p(m)}} \\leq C \\sum_{m \\in {\\mathbb Z}} F(m)^{\\frac{1}{p_\\infty}} + C \\sum_{m \\in {\\mathbb Z}} {R(m)}^{\\frac{1}{p_\\infty}} \\]\nSince $ \\frac{1}{{p(\\cdot)}} \\in LH_\\infty({\\mathbb Z}) $ and using Lemma[$\\ref{sak_lem1}$] with exponents $p_\\infty, q_\\infty$ with $ F= {g_2(r)}^{p_\\infty} \\leq 1$, based on estimates [1-4], it follows that\n\\begin{align*}\n\\sum_{{\\mathbb Z}} \\biggl( g_2(k)^{p_\\infty} \\biggr)^{\\frac{1}{p(m)} } & \\leq C \\sum_{m \\in {\\mathbb Z}} \\biggl( g_2(k)^{p_\\infty} \\biggr)^{\\frac{1}{p_\\infty}} + C \\sum_{m \\in {\\mathbb Z}} R(m)^{\\frac{1}{p_\\infty}} \\\\\n&= C \\sum_{m \\in {\\mathbb Z}} g_2(k) + C \\sum_{m \\in {\\mathbb Z}} R(m)^{\\frac{1}{p_\\infty}} \\leq C\n\\end{align*}\nTherefore $\\eqref{eq:eq4}$ is\n\\begin{align*}\n \\abs{2I^k_j}^{\\alpha -1} \\sum_{ r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \n& \\leq \\biggl( \\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{ p_\\infty } { p(r)}} \\biggr)^{ \\frac{1}{p_\\infty} - \\frac{1}{q_\\infty} } \\biggl( \\frac{1}{\\\n\\abs{2I^k_j} }\\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{\\frac{p_\\infty}{q_\\infty} } \\\\\n& \\leq C^{\\alpha} \\biggl( \\frac{1}{\\abs{2I^k_j} }\\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{\\frac{p_\\infty}{q_\\infty} }\n\\end{align*}\n\n\n\n\\end{enumerate}\n\n\n\nTherefore, using estimates [1-4], we can choose constant $\\beta_2 >0 $ such that\n\\begin{align*}\n\\sum_{k,j} \\sum_{E^k_j} & \\biggl( \\beta_2 \\gamma_2 \\delta_2 \\abs{ 2I_j}^{\\alpha-1} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{q(m)} \\leq \\sum_{k,j} \\sum_{E^k_j} \\gamma_2 \\delta_2 \\biggl( \\biggl( \\frac{1}{\\abs{2I^k_j}} \\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{1}{p(r)}} \\biggr)^{p_\\infty} \\biggr)^{\\frac{q(m)}{q_\\infty} }\n\\end{align*}\n\nNote, $ \\frac{1}{\\abs{2I^k_j}} \\sum_{r \\in 2I^k_j} {g_2(r)}^{\\frac{p_\\infty} { p(r)} } \\leq \\frac{1}{\\abs{2I^k_j}} \\sum_{r \\in {\\mathbb Z}} {g_2(r)}^{\\frac{p_\\infty} { p(r)} } \\leq 1 $. \\\\\nSo, let $ F = \\biggl( g_2(k)^{\\frac{1}{p(k)}} \\biggr)^{q(k) p_\\infty} \\leq \\biggl( \\sum_{r \\in {\\mathbb Z}} g_2(k)^{\\frac{1}{p(k)}} \\biggr)^{q(k) p_\\infty} \\leq 1$. \\\\\n Using Lemma[$\\ref{sak_lem1}$] and with $\\frac{1}{{q(\\cdot)}} \\in LH_\\infty({\\mathbb Z})$, we get\n\\[ \\sum_{k \\in {\\mathbb Z}} F^{\\frac{1}{q_\\infty}} \\leq C_1 \\sum_{k \\in {\\mathbb Z}} F^{\\frac{1}{q(k)}} + C_2 \\sum_{k \\in {\\mathbb Z}} R^{\\frac{1}{q_\\infty}} \\]\nHence,\n\\begin{align*}\nF^{\\frac{1}{q_\\infty}} &\\leq \\sum_{k \\in {\\mathbb Z}} F^{\\frac{1}{q_\\infty}} \\leq C_1 \\sum_{k \\in {\\mathbb Z}} F^{\\frac{1}{q(k)}} + C_2 \\sum_{k \\in {\\mathbb Z}} R^{\\frac{1}{q_\\infty}} \\\\\n& = C_1 \\biggl( \\sum_{k \\in {\\mathbb Z}} g_2(k)^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} + C_2 \\sum_{k \\in {\\mathbb Z}} R^{\\frac{1}{q_\\infty}} \n\\end{align*}\n\nTherefore,\n\\begin{align*}\n\\biggl( \\sum_{k \\in {\\mathbb Z}} {g_2(k)}^{\\frac{1}{p(k)}} \\biggr)^{q(k) \\frac{p_\\infty}{q_\\infty} } & \\leq C_1 \\biggl( \\sum_{k \\in {\\mathbb Z}} {g_2(k)}^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} + C_2 \\sum_{k \\in {\\mathbb Z}} {R(k)}^{\\frac{1}{q_\\infty}} \\\\\n&\\leq C_1 \\biggl( \\sum_{k \\in {\\mathbb Z}} {g_2(k)}^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} + C_2 \\sum_{k \\in {\\mathbb Z}} {R(k)}^{\\frac{1}{q_\\infty}} \n\\end{align*}\nand so,\n\\begin{align*}\n\\sum_{k,j} \\sum_{E^k_j} \\gamma_2 \\delta_2 & \\biggl( \\biggl( \\frac{1}{\\abs{2I^k_j}} \\sum_{k \\in 2I^k_j} g_2(k)^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} \\biggr)^{\\frac{q(m)}{q_\\infty}} \\\\\n&\\leq \\delta_2 \\biggl( C_1 \\biggl( \\frac{1}{\\abs{2I^k_j}} \\sum_{k \\in 2I^k_j} {g_2(k)}^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} + C_2 \\sum_{k \\in {\\mathbb Z}} {R(k)}^{\\frac{1}{q_\\infty}} \n\\end{align*}\n\nTake $\\gamma_2>0$ such that $C_1 =1, C_2 = \\frac{1}{6}$. Then\n\\begin{align*}\n\\sum_{k,j} \\sum_{E^k_j} & \\gamma_2 \\delta_2 \\biggl( \\biggl( \\frac{1}{\\abs{2I^k_j}} \\sum_{k \\in 2I^k_j} g_2(k)^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} \\biggr)^{\\frac{q(m)}{q_\\infty}} \\\\\n&\\leq \\sum_{k,j} \\sum_{E^k_j} \\delta_2 \\biggl( \\biggl (\\frac{1}{\\abs{2I^k_j}} \\sum_{k \\in 2I^k_j} g_2(k)^{\\frac{1}{p(k)}} \\biggl)^{p_\\infty} \\biggr) + \\frac{1}{6} \\sum_{k \\in {\\mathbb Z}} {R(k)}^{\\frac{1}{q_\\infty}} \\\\\n& \\leq \\sum_{{\\mathbb Z}} \\delta_2 {M g_2(\\cdot)}^{\\frac{1}{{p(\\cdot)}}} (k)^{p_\\infty} + \\frac{1}{6} \n\\end{align*}\n\nNote that the maximal operator is bounded on $\\ell^{p_\\infty} ({\\mathbb Z})$, since $p_\\infty \\geq {p_{-}} > 1$.\nAgain apply Lemma[$\\ref{sak_lem1}$] to get \\\\\n\\[ \\sum_{k \\in {\\mathbb Z}} M( g_2(\\cdot)^{\\frac{1}{{p(\\cdot)}}} ) (k)^{p_\\infty} \\leq C \\sum_{k \\in {\\mathbb Z}} {g_2(\\cdot)}^{\\frac{p_\\infty}{p(k)}} \\leq C \\sum_{k \\in {\\mathbb Z}} g_2(k) + C \\sum_{k \\in {\\mathbb Z}} R(k)^{\\frac{1}{p_\\infty}} \\leq C \\]\nFinally, note $a(\\cdot)$ is $\\ell^{p_\\infty} ({\\mathbb Z})$ integrable, ${g_2(\\cdot)}^{\\frac{1}{{p(\\cdot)}}}$ is also $\\ell^{p_\\infty} ({\\mathbb Z})$ integrable. So, we can choose $\\delta_2 >0$ such that\n\\[ \\sum_{k \\in {\\mathbb Z}} \\delta_2 M(g_2(\\cdot)^{\\frac{1}{{p(\\cdot)}}} ) (k)^{p_\\infty} + \\frac{1}{6} \\leq \\frac{1}{3} + \\frac{1}{6} = \\frac{1}{2} \\]\n\\end{proof}\n\n\\subsection{Weak $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Fractional Hardy-Littlewood Maximal Operator}\n\\begin{theorem}[Weak $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Fractional Hardy-Littlewood Maximal Operator]\nGiven a non-negative sequence $ \\left \\{ a(i) \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z})$, let $ {p(\\cdot)} \\in \\mathcal{S},{p_{+}}< \\infty, {p_{-}}= 1 , 1 \\leq {p_{-}} \\leq {p_{+}} < \\frac{1}{\\alpha} \\quad \\text{and} \\quad {p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$. Then\n\\[ \\sup_{t > 0} t \\normb{ \\chi_{ \\left \\{ M_{\\alpha} a(k\t) >9t \\right \\} }}_{\\ell^{{q(\\cdot)}} ({\\mathbb Z})} \\leq C \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\]\n\\end{theorem}\n\\begin{proof}\nDenote by $ \\Omega = \\left \\{ k \\in {\\mathbb Z}: M_\\alpha a(k) > 9t \\right \\}$.\n Using Calderon-Zygmund decomposition for sequence $\\left \\{ a(k) \\right \\}$ by Lemma[$\\ref{CD_Weak}$], we get for fixed $ t > 0$, \n\\[ \\left \\{ k \\in {\\mathbb Z}: M_\\alpha a(k) > 9t \\right \\} = \\cup_j 2I_j\\]\nNow, consider disjoint sets $E_j$ such that $ E_j \\subset 2I_j$ and $\\Omega = \\cup_j E_j$.\nTo prove the weak inequality, it will suffice to show that for each $k \\in \\Omega$, $ t \\normb{\\chi_{\\Omega}(k)}_{{q(\\cdot)}} \\leq C$ and in turn it will suffice to show that for some $\\alpha_2 >0$,\n\\[ \\rho_{{q(\\cdot)}}(\\alpha_2 t \\chi_{\\Omega} ) = \\sum_{k \\in \\Omega} [\\alpha_2 t]^{q(m)} \\leq 1 \\]\nWe will show that each term on the right is bounded by $\\frac{1}{2}$ for suitable choice of $\\alpha_2$.\nTo estimate\n$ \\sum_{k \\in \\Omega} [\\alpha_2 t]^{q(m)}$, we note from results in previous section, \n\\begin{align*}\n\\sum_{k \\in \\Omega} [ \\alpha_2 t]^{q(m)} & \\leq \\sum_{j} \\sum_{E_j} \\delta_2 \\biggl( \\frac{1}{\\abs{2I_j}} \\sum_{k \\in 2I_j} {g_2(k)}^{\\frac{1}{p(k)}} \\biggr)^{p_\\infty} + \\frac{1}{6} \\\\\n& \\leq \\sum_{j} \\sum_{E_j} \\delta_2 \\biggl( \\frac{1}{\\abs{2I_j}} \\sum_{k \\in 2I_j} {g_2(k)}^{\\frac{p_\\infty}{p(k)}} \\biggr) + \\frac{1}{6} \\\\\n&\\leq \\sum_{k \\in \\Omega} \\delta_2 {g_2(k)}^{\\frac{p_\\infty}{p(k)}} + \\frac{1}{6} \n\\end{align*}\nNow , choose $\\delta_2 >0$ such that right hand side is bounded by $\\frac{1}{2}$.\n\\end{proof}\n\n\n\\section{ Hardy-Littlewood Maximal operator}\nIn this section, we prove boundedness of Hardy-Littlewood maximal operator for $\\ell^{{p(\\cdot)}}({\\mathbb Z})$ spaces where ${p_{-}}>1$. The proof is based on boundedness of Hardy-Littlewood maximal operator on $\\ell^p({\\mathbb Z})$, where $p$ is a fixed number, $1< p< \\infty$. \\\\\n\\begin{remark}\nNote that when $\\alpha =0$ the fractional Hardy-Littlewood maximal operator is nothing but Hardy-Littlewood maximal operator.\nHowever we can prove strong type, weak type inequalities for Hardy-Littlewood maximal operator on $\\ell^{{p(\\cdot)}}({\\mathbb Z})$ directly from the corresponding results for fixed $\\ell^p({\\mathbb Z})$ spaces, $ 1 < p < \\infty$. \nThe key point of the proofs is Lemma[$\\ref{sak_lem1}$].\n\\end{remark}\n\n\n\n\\begin{lemma}\\label{sak_lem1}\nLet $p(\\cdot) : {\\mathbb Z} \\to [0,\\infty) $ be such that $ p(\\cdot) \\in LH_\\infty({\\mathbb Z})$ and $ 0< p_\\infty < \\infty$. Let $R(k) = (e + \\abs{k})^{-N}, N > \\frac{1}{p_\\infty} $. Then there exists a real constant C depending on N and $LH_\\infty({\\mathbb Z})$ constant of $p(\\cdot)$ such that given any set E and any function F with $0\\leq F(y) \\leq 1$ for $ y \\in E$,\n\\end{lemma}\n\\begin{align}\n\\sum_E F(m)^{p(m)} \\leq \\sum_E F(m)^{p_\\infty} + C \\sum_E R(m)^{p_\\infty} \\label{eq:eq5}\\tag{CD5} \\\\ \n\\sum_E F(m)^{p_\\infty} \\leq C \\sum_E F(m)^{p(m)} + C \\sum_E R(m)^{p_\\infty} \\label{eq:eq6}\\tag{CD6} \n\\end{align}\n\\begin{proof}\nThis follows from continuous version $\\text{\\cite{fior_var_book}}$. Same line of proof works here\n\\end{proof}\n\n\n\n\\subsection{Strong $({p(\\cdot)}, {p(\\cdot)})$ Inequality}\n\\begin{theorem}[Strong $({p(\\cdot)}, {p(\\cdot)})$ Inequality]\n\n Given a non-negative sequence $ \\left \\{ a(i) \\right \\} \\in \\ell^{{p(\\cdot)}}({\\mathbb Z}), {p(\\cdot)} \\in \\mathcal{S},{p_{+}}< \\infty, {p_{-}}> 1$ , then\n\\[ \\normb{Ma}_{\\ell^{p(.)}({\\mathbb Z})} \\leq C \\normb{a}_{\\ell^{p(.)}({\\mathbb Z})} \\]\n\\end{theorem}\n\n\n\n\n\n\n\\begin{proof}\nBy homogenity, it is enough to prove the above result with the assumption $\\normb{a}_{{\\ell^{p(\\cdot)}}({\\mathbb Z})} =1 $. \nDue to modular property $\\ref{sak_corollary_2}$, this implies that \n$\\sum_{i \\in {\\mathbb Z}} \\abs{a(i)}^{p(i)} \\leq 1 $. So, it will suffice to prove that\n\\begin{align*}\n\\sum_{i \\in {\\mathbb Z}} \\abs{Ma(i)}^{p(i)} \\leq C \n\\end{align*}\n\nGiven that $ 0 \\leq a(k) \\leq 1 $ , it follows that $ 0 \\leq Ma(k) \\leq 1$. To prove boundedness of $ \\left \\{Ma \\right \\}$, \nwe start with Lemma[\\ref{sak_lem1}] as follows:\n\\[ \\sum_{k \\in {\\mathbb Z}} Ma(k)^{p(k)} \\leq C \\sum_{k \\in {\\mathbb Z}} Ma(k)^{p_\\infty} + C \\sum_{k \\in {\\mathbb Z}} R(k)^{p_\\infty} \\]\nSince $N > \\frac{1}{p_\\infty}$, $ \\sum_{k \\in {\\mathbb Z}} R(k)^{Np_\\infty} = \\sum_{k \\in {\\mathbb Z}} (\\frac{1}{e+\\abs{k}})^{Np_\\infty}$ converges and can be bounded as $\\leq 1$. So, the second integral is a constant depending only on $p_\\infty$ by taking sufficiently large $N > \\frac{1}{p_\\infty}$.\n\nTo bound the first integral, note that $ 1 < {p_{-}} \\leq p_\\infty$. Since $p_\\infty > 1$, M is bounded on $\\ell^{p_\\infty }({\\mathbb Z})$ and by using strong $(p,p)$ inequality valid for classical Lebesgue spaces with index $p_\\infty$, we get using Lemma[$\\ref{sak_lem1}$] and [$\\eqref{eq:eq5}$],\n\\begin{align*}\n\\sum_{k \\in {\\mathbb Z}} Ma(k)^{p_\\infty} \\leq C \\sum_{k \\in {\\mathbb Z}} a(k)^{p_\\infty} \\leq C \\sum_{k \\in{\\mathbb Z}} a(k)^{p(k)} + C \\sum_{k \\in {\\mathbb Z}} R(k)^{p_\\infty} \\leq C\n\\end{align*}\nLike previous case, the term involving summation of $R(k)$ is bounded by a constant depending only on $p_\\infty$ by taking sufficiently large $N > \\frac{1}{p_\\infty} $.\\\\\nTherefore, using above results,\n\\[ \\rho_{{p(\\cdot)}}(Ma) = \\sum_{k \\in {\\mathbb Z}} Ma(k)^{p(k)} \\leq C. \\]\n\\end{proof}\n\n\n\\subsection{Weak $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Hardy-Littlewood Maximal Operator}\n\\begin{theorem}[Weak $({p(\\cdot)}, {p(\\cdot)})$ Inequality for Hardy-Littlewood Maximal Operator]\nGiven ${p(\\cdot)} \\in \\mathcal{S}, {p(\\cdot)} \\geq 1$, if ${p(\\cdot)} \\in LH_\\infty({\\mathbb Z})$, then\n\\[ \\sup_{t > 0} \\normb{ t \\chi_{ \\left \\{ Ma(n) >t \\right \\} }}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\leq c \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\]\nwhere constant depends on the Log-Holder constants of ${p(\\cdot)}, {p_{-}}$ and $p_\\infty$(if this value is finite) .\n\\end{theorem}\n\\begin{proof}\n\n\n\\begin{enumerate}\n\\item Case: ${p_{-}} > 1$ and $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} = 1$. \\\\\nLet $ A = \\left \\{ n \\in {\\mathbb Z} : Ma(n) > t \\right \\} $. Then, by the use of strong $ ({p(\\cdot)}, {p(\\cdot)})$ inequality for Hardy-Littlewood maximal operator from previous section, it follows that\n\\begin{align*}\n \\normb{ t \\chi_{ \\left \\{ n: Ma(n) >t \\right \\} }(k) }_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} & \\leq \\sum_{k \\in {\\mathbb Z}} \\abs{ t \\chi_{ \\left \\{ n: Ma(n) > t \\right \\} } (k) }^{p(k)}\\leq \\sum_{ k \\in A} \\abs{t}^{p(k)} \\\\\n& \\leq \\sum_{k \\in A} {Ma}^{p(k)} \\leq \\sum_{k \\in {\\mathbb Z}} {Ma(k)}^{p(k)} = \\rho_{{p(\\cdot)}}( Ma) \\leq C = C \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\\\\n\\end{align*}\n\\item Case: ${p_{-}} >1 $ and $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\neq 1 $ \\\\\nDenote, $b(n) = \\frac{a(n)}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}}$ , so that $\\normb{b}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}=1$. Let $ A = \\left \\{ n \\in {\\mathbb Z} : Ma(n) > t \\right \\} $\nBy homogenity of the norm, it follows that,\n\\begin{align*}\n\\normb{t \\chi_{ \\left \\{ n: Ma(n) > t \\right \\}}(k) }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} &\\\\\n& = \\normb{t \\chi_{ \\left \\{ n: Mb(n) > \\frac{t}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}} \\right \\}}(k) }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\\\\n& = \\normb{ \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\frac{t}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}} \\chi_{\\left \\{ n: Mb(n) > \\frac{t}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}} \\right \\} }(k) }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\\\\n& = \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\normb{ \\frac{t}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}} \\chi_{\\left \\{ n: Mb(n) > \\frac{t}{\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}} \\right \\} }(k) }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\\\\n& \\leq C \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})}\n\\end{align*}\n\\item Case: ${p_{-}}=1$ and $\\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} =1$ \\\\\nLet $ A = \\left \\{ n \\in {\\mathbb Z} : Ma(n) > t \\right \\} $. Then, by the use of strong $ ({p(\\cdot)}, {p(\\cdot)})$ inequality for Hardy-Littlewood maximal operator from previous section, it follows that\n\\begin{align*}\n \\normb{ t \\chi_{ \\left \\{ n: Ma(n) >t \\right \\} }(k) }_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} & \\leq \\sum_{k \\in {\\mathbb Z}} \\abs{ t \\chi_{ \\left \\{ n: Ma(n) < t \\right \\} } (k) }^{p(k)}= \\sum_{ k \\in A} \\abs{t}^{p(k)} \n\\\\\n=\\begin{cases}\nt^{{p_{+}}} \\abs{A} & t \\geq 1 \\\\\nt^{{p_{-}}} \\abs{A} & t \\leq 1\n\\end{cases}\n\\end{align*}\nNow, by use of weak$({p_{+}}, {p_{+}})$ inequality applicable to $\\ell^{{p_{+}}}({\\mathbb Z})$ spaces when $t \\geq 1$ and weak$({p_{+}}, {p_{+}})$ inequality applicable to $\\ell^{{p_{-}}}({\\mathbb Z})$ spaces when $t \\leq 1$ respectively, it follows that\n\\[\\abs{A} \\leq \n\\begin{cases}\n\\frac{C}{t^{{p_{+}}}} \\normb{a}_{\\ell^{p_{+}} ({\\mathbb Z})} & t \\geq 1 \\\\ \\\\\n\\frac{C}{t^{{p_{-}}}} \\normb{a}_{\\ell^{p_{-}} ({\\mathbb Z})} & t \\leq 1\n\\end{cases}\n\\]\nNote, by Lemma$\\ref{sak_corollary_2}$,\n\\[ (\\normb{a}_{{p_{+}}})^{{p_{+}}} \\leq \\sum_{k \\in {\\mathbb Z}} {\\abs{a(k)}}^{{p_{+}}} \\leq \\sum_{k \\in {\\mathbb Z}} {\\abs{a(k)}}^{p(k)} \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} \\]\nand $\\normb{a}_{{p_{-}}} = \\normb{a}_1 \\leq \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})}$ as $\\ell_p({\\mathbb Z}) \\subset \\ell_1({\\mathbb Z})$. Hence\n\\begin{align*}\n\\abs{A} \\leq \n\\begin{cases}\n\\frac{C}{t^{{p_{+}}}} \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} & t \\geq 1 \\\\\n\\frac{C}{t} \\normb{a}_{\\ell^{{p(\\cdot)}}({\\mathbb Z})} & t \\leq 1\n\\end{cases}\n\\end{align*}\nand therefore,\n\\begin{align*}\n \\normb{ t \\chi_{ \\left \\{ n: Ma(n) >t \\right \\} }(k) }_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} & = \n\\begin{cases}\n\\leq C \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} & t \\geq 1 \\\\\n\\leq C \\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} & t \\leq 1\n\\end{cases}\n\\end{align*}\n\\item Case: ${p_{-}}=1$ and $\\normb{a}_{\\ell^{{p(\\cdot)}} ({\\mathbb Z})} \\neq 1$. The conclusion follows similar to case(2) and case(3).\n\\end{enumerate}\n\\end{proof}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nWe let $O(n)$ denote the orthogonal group of the Euclidean $n$-space $\\mathbb{R}^{n}$ and $\\theta_n$ its Haar probability measure. We metrize $O(n)$ with the usual operator norm. Let also $\\mathcal L^n$ stand for the Lebesgue measure on $\\mathbb{R}^{n}$ and let $\\dim$ stand for the Hausdorff dimension and $\\mathcal H^s$ for $s$-dimensional Hausdorff measure. We shall prove the following theorem:\n\n\n\n\\begin{thm}\\label{theo1}\nLet $s$ and $t$ be positive numbers with $s+t > n+1$. Let $A$ and $B$ be Borel subsets of $\\mathbb{R}^{n}$ with $\\mathcal H^s(A)>0$ and $\\mathcal H^t(B)>0$. Then \nthere is $E\\subset O(n)$ such that \n$$\\dim E\\leq 2n-s-t+(n-1)(n-2)\/2=n(n-1)\/2-(s+t-(n+1))$$ \nand for $g\\in O(n)\\setminus E$,\n\\begin{equation}\\label{eq4}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (g(B)+z)\\geq s+t-n\\})>0.\n\\end{equation}\n\\end{thm}\n\nThe version stated in the abstract concerning the case $\\dim A + \\dim B > n+1$ is slightly weaker than Theorem \\ref{theo1}. Notice that the above upper bound for the dimension of $E$ is at least $(n-1)(n-2)\/2$ which is the dimension of $O(n-1)$. In Section \\ref{Examples} we show that it is needed in the estimates. The assumption $s+t>n+1$ only comes from the fact that the statement is trivial if $s+t\\leq n+1$: then the above upper for $\\dim E$ is at least $n(n-1)\/2=\\dim O(n)$ and we could take $E=O(n)$. \n\nThis is an exceptional set estimate related to the following result of \n\\cite{M2}: (\\ref{eq4}) holds for $\\theta_n$ almost all $g\\in O(n)$ if one of the sets has dimension bigger than $(n+1)\/2$, see also Chapter 13 in \\cite{M4} and Chapter 7 in \\cite{M5}. This of course is satisfied when $s+t>n+1$, as in the theorem. It is expected that this generic result with respect to $\\theta_n$ should hold whenever $\\dim A + \\dim B > n$. Under this condition it was proved (without exceptional set estimates) in \\cite{K} and \\cite{M1} provided the orthogonal group is replaced by a larger transformation group, for example by similarity maps as in \\cite{M1}, or, more generally by Kahane in \\cite{K}, by any closed subgroup of the general linear group acting transitively outside the origin. For the orthogonal group no dimensional restrictions are needed provided one of the sets satisfies some extra condition, for example if it is rectifiable, see \\cite{M1}, or a Salem set, see \\cite{M3}.\n\nIt is easy to see, cf. the remark at the end of \\cite{M2}, that in Theorem \\ref{theo1} the positivity of the Hausdorff measures cannot be relaxed to $\\dim A = s$ and $\\dim B = t$.\n\nIf one of the sets supports a measure with sufficiently fast average decay over spheres for the Fourier transform, we can improve the estimate of Theorem \\ref{theo1}. Then the results even hold for the sum sets provided the dimensions are big enough. This is given in Theorem \\ref{theo3}. It yields immediately the following result in case one of the sets is a Salem set. By definition, $A$ is a Salem set if for every $02n-1$, then \n\\begin{equation}\\label{eq5}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (B+z)\\geq u\\})>0.\n\\end{equation}\n(b)If $\\dim A+\\dim B\\leq 2n-1$, then there is $E\\subset O(n)$ with \n$$\\dim E\\leq n(n-1)\/2-u$$ \nsuch that for $g\\in O(n)\\setminus E$,\n\\begin{equation}\\label{eq7}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (g(B)+z)\\geq u\\})>0.\n\\end{equation}\n\\end{thm}\n\nAnother consequence of Theorem \\ref{theo3} is the following improvement of Theorem \\ref{theo1} in the case where one of the sets has small dimension:\n\n\\begin{thm}\\label{theo6}\nLet $A$ and $B$ be Borel subsets of $\\mathbb{R}^{n}$ and suppose that $\\dim A\\leq (n-1)\/2$. If $00.\n\\end{equation}\n\\end{thm}\n\nThe method used to prove Theorem \\ref{theo1} can easily be modified to other subgroups of the general linear group $GL(n)$ in place of the orthogonal group. For example, let $S(n)$ be the group of similarities, the compositions of orthogonal maps and dilations. Then $\\dim S(n)=n(n-1)\/2+1$ and for any $x,z\\in\\mathbb{R}^{n}\\setminus\\{0\\}$, the dimension of $\\{g\\in S(n): g(z)=x\\}$ is the same as the dimension of $O(n-1)$, that is, $(n-1)(n-2)\/2$. With small changes in the proof of Theorem \\ref{theo1} this leads to \n\n\n\\begin{thm}\\label{theo2}\nLet $s$ and $t$ be numbers with $0 n$. Let $A$ and $B$ be Borel subsets of $\\mathbb{R}^{n}$ such that $\\mathcal H^s(A)>0$ and $\\mathcal H^t(B)>0$. Then \nthere is $E\\subset S(n)$ with \n$$\\dim E\\leq 2n-s-t+(n-1)(n-2)\/2$$ \nand for $g\\in S(n)\\setminus E$,\n\\begin{equation}\\label{eq6}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (g(B)+z)\\geq s+t-n\\})>0.\n\\end{equation}\n\\end{thm}\n\n\n\\section{Peliminaries}\n\nThe proof of Theorem \\ref{theo1} is based on the relationship of the Hausdorff dimension to the energies of measures and their relations to the Fourier transform. For $A\\subset\\mathbb{R}^{n}$ (or $A\\subset O(n)$) we denote by $\\mathcal M(A)$ the set of non-zero Radon measures $\\mu$ on $\\mathbb{R}^{n}$ with compact support $\\spt\\mu\\subset A$. The Fourier transform of $\\mu$ is defined by\n$$\\widehat{\\mu}(x)=\\int e^{-2\\pi ix\\cdot y}\\,d\\mu y,~ x\\in\\mathbb{R}^{n}.$$\nFor $0(n-1)(n-2)\/2$. If $\\theta(B(g,r))\\leq r^{\\a}$ for all $g\\in O(n)$ and $r>0$, then for $x,z\\in\\mathbb{R}^{n}\\setminus\\{0\\}, r>0$,\n\\begin{equation}\\label{eq11}\n\\theta(\\{g:|x-g(z)|< r\\})\\lesssim (r\/|z|)^{\\a-(n-1)(n-2)\/2}.\n\\end{equation}\n\\end{lm}\n\\begin{proof}\nFirst we may clearly assume that $|z|=1$, and then also that $|x|=1$, because $|x-g(z)|< r$ implies $|x\/|x|-g(z)|< 2r$. Then $O_{x,z}:=\\{g\\in O(n): g(z)=x\\}$ can be identified with $O(n-1)$. Hence it is a smooth compact $(n-1)(n-2)\/2$-dimensional submanifold of $O(n)$ which implies that it can be covered with roughly $r^{-(n-1)(n-2)\/2}$ balls of radius $r$. If $g\\in G$ satisfies $|x-g(z)|< r$, then $g$ belongs to the $r$-neighbourhood of $O_{x,z}$. The lemma follows from this.\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{theo1}}\n\nThe key to the proof of Theorem \\ref{theo1} is the following energy estimate. \nFor $\\mu, \\nu\\in\\mathcal M(\\mathbb{R}^{n}), g\\in O(n)$ and $z\\in\\mathbb{R}^{n}$, let $\\nu_{\\varepsilon}=\\psi_{\\varepsilon}\\ast\\nu$ as above and set\n\n\\begin{equation}\\label{eq0}\n\\nu_{g,z,\\varepsilon}(x)=\\nu_{\\varepsilon}(g^{-1}(x)-z),\\ x\\in\\mathbb{R}^{n}.\n\\end{equation}\n\n\\begin{lm}\\label{lemma1} Let $\\beta>0$ and $\\theta\\in\\mathcal M(O(n))$ be such that $\\theta(O(n))\\leq 1$ and for $x,z\\in\\mathbb{R}^{n}\\setminus\\{0\\}, r>0,$ \n\\begin{equation}\\label{eq9}\n\\theta(\\{g\\in O(n):|x-g(z)|n-\\beta$. Let $\\mu, \\nu \\in \\mathcal M(\\mathbb{R}^{n})$. Then\n\\begin{equation}\\label{eq15}\n\\iint I_u(\\nu_{g,z,\\varepsilon}\\mu)\\,d\\mathcal L^nz\\,d\\theta g \\leq C(n,s,t)I_s(\\mu)I_t(\\nu).\n\\end{equation}\n\n\\end{lm}\n\n\\begin{proof}\nWe may assume that $I_s(\\mu)$ and $I_t(\\nu)$ are finite. \nDefine \n$$\\tilde{\\nu}_{g,x,\\varepsilon}(z)=\\nu_{\\varepsilon}(g^{-1}(x)-z),\\ z\\in\\mathbb{R}^{n}.$$\nThen \n$$\\widehat{\\tilde{\\nu}_{g,x,\\varepsilon}}(z)=e^{-2\\pi ig^{-1}(x)\\cdot z}\\widehat{\\nu_{\\varepsilon}}(-z).$$\nHence by Parseval's formula for $x,y\\in\\mathbb{R}^{n}, x\\neq y,$\n\\begin{align*}\n&\\int\\nu_{\\varepsilon}(g^{-1}(x)-z)\\nu_{\\varepsilon}(g^{-1}(y)-z)\\,d\\mathcal L^nz\\\\\n&=\\int\\widehat{\\tilde{\\nu}_{g,x,\\varepsilon}}(z)\\overline{\\widehat{\\tilde{\\nu}_{g,y,\\varepsilon}}(z)}\\,d\\mathcal L^nz\\\\\n&=\\int\\widehat{\\nu_{\\varepsilon}}(-z)\\overline{\\widehat{\\nu_{\\varepsilon}}(-z)}e^{-2\\pi ig^{-1}(x-y)\\cdot z}d\\mathcal L^nz\\\\\n\\end{align*}\nIt follows by Fubini's theorem\t that\n\\begin{align*}\n&I:=\\iint I_u(\\nu_{g,z,\\varepsilon}\\mu)\\,d\\mathcal L^nz\\,d\\theta g\\\\\n&=\\iiiint k_u(x-y)\\nu_{\\varepsilon}(g^{-1}(x)-z)\\nu_{\\varepsilon}(g^{-1}(y)-z)\\,d\\mu x\\,d\\mu y\\,d\\mathcal L^nz\\,d\\theta g\\\\\n&=\\iiint k_u(x-y)\\left(\\int\\nu_{\\varepsilon}(g^{-1}(x)-z)\\nu_{\\varepsilon}(g^{-1}(y)-z)\\,d\\mathcal L^nz\\right)\\,d\\mu x\\,d\\mu y\\,d\\theta g\\\\\n&=\\iiint k_u(x-y)\\left(\\int|\\widehat{\\nu_{\\varepsilon}}(z)|^2 e^{2\\pi ig^{-1}(x-y)\\cdot z}\\,d\\mathcal L^nz\\right)\\,d\\mu x\\,d\\mu y\\,d\\theta g\\\\\n&=\\iiint k_{u,g,z}\\ast\\mu(x)\\,d\\mu x|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\,d\\theta g,\n\\end{align*}\nwhere\n\\begin{equation*}\nk_{u,g,z}(x)=|x|^{-u}e^{2\\pi ig^{-1}(x)\\cdot z}=|x|^{-u}e^{2\\pi ix\\cdot g(z)}\n\\end{equation*}\nOne checks by direct computation that the Fourier transform of $k_{u,g,z}$, in the sense of distributions, is given by \n\\begin{equation*}\n\\widehat{k_{u,g,z}}(x)=\nc(n,u)|x-g(z)|^{u-n}.\n\\end{equation*}\nIt follows that\n$$\\iint k_{u,g,z}\\ast\\mu\\,d\\mu=\\int\\widehat{k_{u,g,z}}|\\widehat{\\mu}|^2\\,d\\mathcal L^n=c(n,u)\\int|x-g(z)|^{u-n}|\\widehat{\\mu}(x)|^2\\,d\\mathcal L^nx.$$\nAs $I_u(\\mu)<\\infty$, this is easily checked approximating $\\mu$ with $\\psi_{\\varepsilon}\\ast\\mu$ and using the Lebesgue dominated convergence theorem. \nThus \n\\begin{equation}\\label{eq8}\nI=c(n,u)\\iiint |x-g(z)|^{u-n}\\,d\\theta g|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz.\n\\end{equation}\nWe first observe that if $|x|\\geq 2|z|$, then\n$$\\int |x-g(z)|^{u-n}\\,d\\theta g\\leq \\theta(O(n))2^{n-u}|x|^{u-n}\\leq 2^{2n}|x|^{s-n}|z|^{t-n}.$$\nSimilarly if $|z|\\geq 2|x|$, then\n$$\\int |x-g(z)|^{u-n}\\,d\\theta g\\leq \\theta(O(n))2^{n-u}|z|^{u-n}\\leq 2^{2n}|x|^{s-n}|z|^{t-n}.$$\nSuppose then that $|z|\/2\\leq|x|\\leq2|z|.$ Then by the assumption\n\\begin{align*}\n&\\int|x-g(z)|^{u-n}\\,d\\theta g = \\int_0^{\\infty}\\theta(\\{g:|x-g(z)|^{u-n}>\\lambda\\})\\,d\\lambda\\\\\n&=(n-u)\\int_0^{\\infty}\\theta(\\{g:|x-g(z)|0$. \nIt follows that \n\\begin{equation}\\label{eq13}\nI\\lesssim \\iint |x|^{s-n}|z|^{t-n}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(y)|^2\\,dx\\,dy\\lesssim I_s(\\mu)I_t(\\nu),\n\\end{equation}\n as required.\n\n\n\n\\end{proof}\n\nNext we show that, with $\\theta$ as above, for $\\theta\\times\\mathcal L^n$ almost all $(g,z)$ the measures $\\nu_{g,z,\\varepsilon}\\mu$ converge weakly as $\\varepsilon\\to 0$. It is immediate that for almost all $(g,z)$ this takes place through some sequence $(\\varepsilon_j)$, depending on $(g,z)$, but we would at least need one sequence which is good for almost all $(g,z)$. The proof of the following theorem was inspired by an argument of Kahane in \\cite{K}.\n\n\\begin{thm}\\label{theo5}\nLet $s,t$ and $u$ be positive numbers with $u=s+t-n>0$ and let $\\mu, \\nu\\in\\mathcal M(\\mathbb{R}^{n})$ with $I_s(\\mu)<\\infty$ and $I_t(\\nu)<\\infty$. Let $\\psi_{\\varepsilon}$ be an approximate identity and $\\nu_{\\varepsilon}=\\psi_{\\varepsilon}\\ast\\nu$. For $g\\in O(n)$ and $z\\in\\mathbb{R}^{n}$, let $\\nu_{g,z,\\varepsilon}$ be as in (\\ref{eq0}). Finally, let $\\theta\\in\\mathcal M(O(n))$ be as in Lemma \\ref{lemma1}. Then for $\\theta\\times\\mathcal L^n$ almost all $(g,z)$, as $\\varepsilon\\to 0$, the measures \n$\\nu_{g,g^{-1}(z),\\varepsilon}\\mu$ converge weakly to a measure $\\lambda_{g,z}$ with the properties\n\\begin{itemize}\n\\item[(a)] $$\\spt\\lambda_{g,z}\\subset\\spt\\mu\\cap(g(\\spt\\nu)+z),$$\n\\item[(b)]$$\\int\\lambda_{g,z}(\\mathbb{R}^{n})\\,d\\mathcal L^nz = \\mu(\\mathbb{R}^{n})\\nu(\\mathbb{R}^{n})\\ \\text{for}\\ \\theta\\ \\text{almost all}\\ g\\in O(n),$$\n\\item[(c)]\n$$\\iint I_u(\\lambda_{g,z})\\,d\\mathcal L^nz\\,d\\theta g\\leq C(n,s,t)I_s(\\mu)I_t(\\nu).$$ \n\\end{itemize} \n\\end{thm}\n\\begin{proof}\nIf the convergence takes place, the support property (a) is clear. Using the change of variable from $z$ to $g^{-1}(z)$ in the appropriate places, it is then sufficient to show that for $\\theta\\times\\mathcal L^n$ almost all $(g,z)$, as $\\varepsilon\\to 0$, the measures \n$\\nu_{g,z,\\varepsilon}\\mu$ converge weakly to a measure $\\tilde{\\lambda}_{g,z}$ such that (b) and (c) hold with $\\lambda_{g,z}$ replaced by $\\tilde{\\lambda}_{g,z}$. \n\nLet $\\phi\\in C^+_0(\\mathbb{R}^{n})$. Then by Lemma \\ref{lemma1},\n$$\\iint (\\int\\nu_{g,z,\\varepsilon}\\phi\\,d\\mu)^2\\,d\\mathcal L^nz\\,d\\theta g\\lesssim I_s(\\mu)I_t(\\nu)<\\infty.$$ \nHence by Fatou's lemma\n$$\\int \\left(\\liminf_{\\varepsilon\\to 0}\\int(\\int\\nu_{g,z,\\varepsilon}\\phi\\,d\\mu)^2\\,d\\mathcal L^nz\\right)\\,d\\theta g\\lesssim I_s(\\mu)I_t(\\nu)<\\infty.$$\nThus for $\\theta$ almost all $g$ there is a sequence $(\\varepsilon_j)$ tending to $0$ such that \n$$\\sup_j\\int(\\int\\nu_{g,z,\\varepsilon_j}\\phi\\,d\\mu)^2\\,d\\mathcal L^nz<\\infty.$$\nOn the other hand, defining the measure $\\mu_{\\phi,g}$ by $\\int h\\,d\\mu_{\\phi,g}=\\int h(-g^{-1}(x))\\phi(x)\\,d\\mu x$, we have\n$$\\int\\nu_{g,z,\\varepsilon}\\phi\\,d\\mu=\\int\\nu_{\\varepsilon}(g^{-1}(x)-z)\\phi(x)\\,d\\mu x=\\mu_{\\phi,g}\\ast\\nu\\ast\\psi_{\\varepsilon}(-z),$$\nand the measures $\\mu_{\\phi,g}\\ast\\nu\\ast\\psi_{\\varepsilon}$ converge weakly to $\\mu_{\\phi,g}\\ast\\nu$. Consequently, $\\mu_{\\phi,g}\\ast\\nu$ is an $L^2$ function on $\\mathbb{R}^{n}$ and the convergence takes place almost everywhere. It follows now that for $\\theta$ almost all $g\\in O(n)$ and for every $\\phi\\in C^+_0(\\mathbb{R}^{n})$ the finite limit\n\\begin{equation}\\label{eq2}\nL_{g,z}\\phi:=\\lim_{\\varepsilon\\to 0}\\int\\nu_{g,z,\\varepsilon}\\phi\\,d\\mu=\\lim_{\\varepsilon\\to 0}\\mu_{\\phi,g}\\ast\\nu\\ast\\psi_{\\varepsilon}(-z)=\\mu_{\\phi,g}\\ast\\nu(-z)\n\\end{equation}\nexists for almost all $z\\in\\mathbb{R}^{n}$. Let $\\mathcal D$ be a countable dense subset of $C^+_0(\\mathbb{R}^{n})$ containing a function $\\phi_0$ which is $1$ on the support of $\\mu$. Then there is a set $E$ of measure zero such that (\\ref{eq2}) holds for all $z\\in\\mathbb{R}^{n}\\setminus E$ for all $\\phi\\in \\mathcal D$, the exceptional set is indenpendent of $\\phi$. Applying (\\ref{eq2}) to $\\phi_0$ we see that\n$$\\sup_{\\varepsilon>0}\\int\\nu_{g,z,\\varepsilon}\\,d\\mu<\\infty.$$\nThen by the Cauchy criterion the denseness of $\\mathcal D$ yields that whenever $z\\in\\mathbb{R}^{n}\\setminus E$, there is the finite limit $L_{g,z}\\phi:=\\lim_{\\varepsilon\\to 0}\\int\\nu_{g,z,\\varepsilon}\\phi\\,d\\mu$ for all $\\phi\\in C^+_0(\\mathbb{R}^{n})$. Hence by the Riesz representation theorem the positive linear functional $L_{g,z}$ corresponds to a Radon measure $\\tilde{\\lambda}_{g,z}$ to which the measures $\\nu_{g,z,\\varepsilon}\\mu$ converge weakly.\n\nThe claim (b) follows from (\\ref{eq2}):\n\\begin{align*}\n\\int\\lambda_{g,z}(\\mathbb{R}^{n})\\,d\\mathcal L^nz = &\\int L_{g,z}\\phi_0\\,d\\mathcal L^nz \n= \\int\\mu_{\\phi_0,g}\\ast\\nu(-z)\\,d\\mathcal L^n\\\\\n&=\\mu_{\\phi_0,g}(\\mathbb{R}^{n})\\nu(\\mathbb{R}^{n})=\\mu(\\mathbb{R}^{n})\\nu(\\mathbb{R}^{n}).\t\n\\end{align*}\nThe claim (c) follows from Lemma \\ref{lemma1}, Fatou's lemma and the lower semicontinuity of the energy-integrals under the weak convergence.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{theo1}]\nTheorem \\ref{theo1} follows from Lemma \\ref{lemma1} and Theorem \\ref{theo5}: Let \n$$G=\\{g\\in O(n): \\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (g(B)+z))\\geq s+t-n\\})=0\\}.$$\nThen $G$ is a Borel set. We leave checking this to the reader. It is a bit easier when $A$ and $B$ are compact. We may assume the compactness since $A$ and $B$ as in the theorem contain compact subsets with positive measure, cf \\cite{Fe}, 2.10.48. Suppose, contrary to what is claimed, that $\\dim G > 2n-s-t+(n-1)(n-2)\/2.$ Let \n$\\dim G > \\a > 2n-s-t + (n-1)(n-2)\/2$. Then by Frostman's lemma, cf. \\cite{M4}, Theorem 8.8, and Lemma \\ref{lemma2} there is $\\theta\\in\\mathcal M(G)$ satisfying (\\ref{eq9}) with $\\beta=\\a-(n-1)(n-2)\/2>2n-s-t$. \nBy Frostman's lemma there are $\\mu\\in\\mathcal M(A)$ and $\\nu\\in\\mathcal M(B)$ such that $\\mu(B(x,r))\\leq r^s$ and $\\nu(B(x,r))\\leq r^t$ for all balls $B(x,r)$. Then by easy estimation, for example, as in the beginning of Chapter 8 in \\cite{M4}, $I_{s'}(\\mu)<\\infty$ and $I_{t'}(\\nu)<\\infty$ for $00\\}$ has positive Lebesgue measure for $\\theta$ almost all $g$. It then follows from Theorem \\ref{theo5}(a) and (c) and (\\ref{eq3}) that for $\\theta$ almost all $g$,~ $\\dim A\\cap(g(B)+z)\\geq s+t-n$ for almost all $z\\in E_g$. This contradicts the definition of $G$ and the fact that $\\theta$ has support in $G$. \n\\end{proof}\n\n\n\n\\section{Intersections and the decay of spherical averages}\\label{decay}\n\nFor $\\mu\\in\\mathcal M(\\mathbb{R}^{n})$ define the spherical averages\n$$\\sigma(\\mu)(r)=r^{1-n}\\int_{S(r)}|\\widehat{\\mu}(x)|^2\\,d\\sigma_r^{n-1}x, r>0,$$\nwhere $\\sigma_r^{n-1}$ is the surface measure on the sphere $S(r)=\\{x\\in\\mathbb{R}^{n}:|x|=r\\}$. \nNotice that if $\\sigma(\\mu)(r)\\lesssim r^{-\\gamma}$ for $r>0$ and for some $\\gamma>0$, then $I_s(\\mu)<\\infty$ for $00, I_{\\gamma}(\\mu)<\\infty$ and $I_t(\\nu)<\\infty$. \n\n(a) If $\\gamma+t>2n-1$, then \n\\begin{equation}\\label{eq16}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim \\spt\\mu\\cap (\\spt\\nu+z)\\geq \\gamma+t-n\\})>0.\n\\end{equation}\n(b) If $\\gamma+t\\leq 2n-1$, then there is $E\\subset O(n)$ with \n$$\\dim E\\leq 2n-1-\\gamma-t+(n-1)(n-2)\/2$$ \nsuch that for $g\\in O(n)\\setminus E$,\n\\begin{equation}\\label{eq17}\n\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim \\spt\\mu\\cap (g(\\spt\\nu)+z))\\geq \\gamma+t-n\\})>0.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof} Let $u=\\gamma+t-n$. \nAs above, we only need to show that the the conclusion of Lemma \\ref{lemma1} holds under the present assumptions, but now the upper bound in (\\ref{eq15}) will be a constant involving $\\Gamma, I_t(\\nu),I_{\\gamma}(\\mu), I_u(\\mu),\\mu(\\mathbb{R}^{n})$ and $\\nu(\\mathbb{R}^{n})$. \nFor the statement (a) there is no $\\theta$ integration (or $\\theta$ is the Dirac measure at the identity map) and $n-12|z|$ or $|z|>2|x|$ we can argue as before. \n\nTo prove (a), suppose $n-12} |x-z|^{u-n}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\lesssim \\Gamma I_t(\\nu).$$\nSince $-11,$ can be estimated by\n\\begin{align*}\n&\\iint_{||x|-|z||\\leq 1, |z|>2} |x-z|^{u-n}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^n\\,z\\\\\n&\\leq \\int_{|z|>2}\\int_{|z|-1}^{|z|+1}|r-|z||^{u-n}\\int_{S(r)}|\\widehat{\\mu}(x)|^2\\,d\\sigma_r^{n-1}x\\,dr|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma\\int_{|z|>2} |z|^{n-1-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\leq \\Gamma\\int |z|^{t-n}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\lesssim \\Gamma I_t(\\nu).\n\\end{align*}\nFor the remaining part we have \n\\begin{align*}\n&\\iint_{||x|-|z|| > 1, 1<|z|\/2\\leq|x|\\leq 2|z|} |x-z|^{u-n}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\\\\n&\\leq\\int\\sum_{1\\leq2^j\\leq3|z|} \\int_{2^j\\leq||x|-|z||\\leq 2^{j+1},|z|\/2\\leq|x|\\leq 2|z|}||x|-|z||^{u-n}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\\\\n&\\lesssim \\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(u-n)}\\int_{2^j\\leq||x|-|z||\\leq 2^{j+1},|z|\/2\\leq|x|\\leq 2|z|}|\\widehat{\\mu}(x)|^2\\,d\\mathcal L^nx|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&= \\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(u-n)}\\int_{2^j\\leq|r-|z||\\leq 2^{j+1},|z|\/2\\leq r\\leq 2|z|}\\int_{S(r)}|\\widehat{\\mu}(x)|^2\\,d\\sigma_r^{n-1}x\\,dr|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma \\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(u-n)}2^j|z|^{n-1-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma\\int |z|^{u-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\lesssim \\Gamma I_t(\\nu).\n\\end{align*}\n\n\nWe establish the statement (b) by showing that \n$$\\iiint_{|z|\/2\\leq |x|\\leq 2|z|,|z|>2} |x-g(z)|^{u-n}\\,d\\theta g|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\lesssim \\Gamma I_t(\\nu),$$\nwhere $\\theta\\in\\mathcal M(O(n))$ is as in Lemma \\ref{lemma1} with $n-1-u<\\beta2} |x-g(z)|^{u-n}\\,d\\theta g|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(x)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^n\\,z\\\\\n&\\lesssim\\int_{|z|>2} \\int_{|z|-1}^{|z|+1}|r-|z||^{\\beta+u-n}|z|^{-\\beta}\\int_{S(r)}|\\widehat{\\mu}(x)|^2\\,d\\sigma_r^{n-1}x\\,dr|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma\\int_{|z|>2} |z|^{-\\beta+n-1-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\leq \\Gamma\\int |z|^{t-n}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\n\\lesssim \\Gamma I_t(\\nu).\n\\end{align*}\nFor the remaining part we have \n\n\\begin{align*}\n&\\iiint_{||x|-|z|| > 1, 1<|z|\/2\\leq|x|\\leq 2|z|} |x-g(z)|^{u-n}\\,d\\theta g|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\\\\n&\\leq\\int\\sum_{1\\leq2^j\\leq3|z|} \\int_{2^j\\leq||x|-|z||\\leq 2^{j+1},|z|\/2\\leq|x|\\leq 2|z|}||x|-|z||^{\\beta+u-n}|z|^{-\\beta}|\\widehat{\\mu}(x)|^2|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nx\\,d\\mathcal L^nz\\\\\n&\\lesssim \\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(\\beta+u-n)}|z|^{-\\beta}\\int_{2^j\\leq||x|-|z||\\leq 2^{j+1},|z|\/2\\leq|x|\\leq 2|z|}|\\widehat{\\mu}(x)|^2\\,d\\mathcal L^nx|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&= \\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(\\beta+u-n)}|z|^{-\\beta}\\int_{2^j\\leq|r-|z||\\leq 2^{j+1},|z|\/2\\leq r\\leq 2|z|}\\int_{S(r)}|\\widehat{\\mu}(x)|^2\\,d\\sigma_r^{n-1}x\\,dr|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma\\int\\sum_{1\\leq2^j\\leq3|z|} 2^{j(\\beta+u-n)}|z|^{-\\beta}2^j|z|^{n-1-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\\\\n&\\lesssim \\Gamma\\int |z|^{u-\\gamma}|\\widehat{\\nu_{\\varepsilon}}(z)|^2\\,d\\mathcal L^nz\\lesssim \\Gamma I_t(\\nu).\n\\end{align*}\n\\end{proof}\n\nFor $00\n\\end{equation}\nholds for all $\\mu\\in\\mathcal M(\\mathbb{R}^{n})$ with support in the unit ball. Estimates for $\\gamma(s)$ are discussed in \\cite{M5}, Chapter 15. For $s\\leq (n-1)\/2$ the optimal, rather easy, result $\\gamma(s)=s$ is valid, see \\cite{M5}, Lemma 3.15. This together with Theorem \\ref{theo3} yields immediately Theorem \\ref{theo6}. For $s>(n-1)\/2$ the optimal estimate fails and Theorem \\ref{theo3} only gives a lower bound for the dimension of intersections which stays below and bounded away from $\\dim A + \\dim B - n$. The deepest estimates are due to Wolff \\cite{W} in the plane and to Erdo\\u gan \\cite{E} in higher dimensions. They give $\\gamma_n(s)\\geq (n+2s-2)4$ for $n\/2\\leq s \\leq (n+2)\/2$. Theorem \\ref{theo3} combined with this leads to the result that if $\\dim A + \\dim B\/2 - (3n+2)\/4>u>0$, then \n$$\\mathcal L^n(\\{z\\in\\mathbb{R}^{n}: \\dim A\\cap (g(B)+z)) > u\\})>0$$\nfor $g\\in O(n)$ outside an exceptional set $E$ with $\\dim E \\leq n(n-1)\/2 - u$. Plugging into Theorem \\ref{theo3} other known estimates for $\\gamma_n(s)$ gives similar rather weak intersection results.\n\n\n\\section{Examples}\\label{Examples}\n\nThe first example here shows that in Theorem \\ref{theo1} the bound $(n-1)(n-2)\/2$ is sharp in the case where both sets have the maximal dimension $n$. This of course does not tell us anything in the plane but it explains the appearance of the dimension of $O(n-1)$. \nIn the following we identify $O(n-1)$ with a subset of $O(n)$ letting $g\\in O(n-1)$ mean the map $(x_1,\\dots,x_n)\\mapsto (g(x_1\\dots,x_{n-1}),x_n)$. \n\\begin{ex}\\label{ex1}\nLet $n\\geq 3$. There are compact sets $A, B\\subset\\mathbb{R}^n$ such that $\\dim A=\\dim B=n$ and for every $g\\in O(n-1)$,~ $\\dim A\\cap (g(B)+z)\\leq n-1$ for all $z\\in\\mathbb{R}^n$.\n\\end{ex}\n\\begin{proof}\nLet $C, D\\subset\\mathbb{R}$ be compact sets such that $\\dim C = \\dim D = 1$ and for every $z\\in\\mathbb{R}$ the intersection $C\\cap(D+z)$ contains at most one point. Such sets were constructed in \\cite{M1}, the construction is explained also in \\cite{M4}, Example 13.18. Let $F$ be the closed unit ball in $\\mathbb{R}^{n-1}$ and take $A=F\\times C$ and $B=F\\times D$. These sets have the required properties.\n\\end{proof}\n\nThe following example shows that we need some additional assumptions, for example as in Theorem \\ref{theo3}, to get any result using only translations:\n\n\\begin{ex}\\label{ex2}\nThere are compact sets $A, B\\subset\\mathbb{R}^n$ such that $\\dim A = \\dim B =n$ and for every $z\\in\\mathbb{R}^n$ the intersection $A\\cap(B+z)$ contains at most one point.\n\\end{ex}\n\\begin{proof}\nLet $C, D\\subset\\mathbb{R}$ be the compact sets of the previous example. Take $A=C^n$ and $B=D^n$. These sets have the required properties.\n\\end{proof}\n\nI do not know what are the sharp bounds for the dimension of exceptional sets of Theorem \\ref{theo1}. For simplicity, let us look at this question in the plane. Let $d(s,t)\\in[0,1], 02,$ be the infimum of the numbers $d>0$ with the property that for all Borel sets $A,B\\subset\\mathbb{R}^2$ with $\\dim A = s, \\dim B = t$ and for all $0u\\})=0\\}\\leq d.$$\nThe problem is to determine $d(s,t)$. We know from Theorem \\ref{theo1} that if $s+t>3$, then $d(s,t)\\leq 4-s-t$. In particular $d(2,2)=0$. This suggests that $d(s,t)$ might be $4-s-t$ when $s+t>3$. However we know from Theorem \\ref{theo6} that $d(s,t)\\leq 3-s-t$ whenever $s\\leq 1\/2$. I would be happy to see some examples sheding light into this question.\n\n\\section{Concluding remarks}\nAs mentioned in the Introduction, intersection problems of this type for general sets were first studied by Kahane in \\cite{K} and by the author in \\cite{M1}, involving transformations such as similarities. For the orthogonal group the result in \\cite{M2} concerned the case where one of the sets has dimension bigger than $(n+1)\/2$. In \\cite{M3} a general method was developed to get dimension estimates for the distance sets and intersections once suitable spherical average estimates\t(\\ref{eq19}) for measures with finite energy are available. Such deep estimates were proved by Wolff in \\cite{W} and Erdo\\u gan in \\cite{E}. They gave the best known results for the distance sets, see \\cite{M5}, Chapters 15 and 16, but only minor progress for the intersections, as mentioned in Section \\ref{decay}. The known estimates for $\\sigma(\\mu)$ are discussed in \\cite{M5}, Chapter 15, see also \\cite{LR} for a recent one. \n\nThe reverse inequality in Theorem \\ref{theo1} fails: for any $0\\leq s\\leq n$ there exists a Borel set $A\\subset \\mathbb{R}^{n}$ such that \n$\\dim A\\cap f(A)=s$ for all similarity maps $f$ of $\\mathbb{R}^{n}$. This follows from \\cite{F2}, see also Example 13.19 in \\cite{M4} and the further references given there. The reverse inequality holds if one of the sets is a reasonably nice integral dimensional set, for example rectifiable, or if $\\dim A\\times B = \\dim A + \\dim B$, see \\cite{M1}. This latter condition is valid if, for example, one of the sets is Ahlfors-David regular, see \\cite{M4}, pp. 115-116. For such reverse inequalities no rotations $g$ are needed (or, equivalently, they hold for every $g$).\n\nExceptional set estimates in the spirit of this paper were first proved for projections by Kaufman in \\cite{Ka}, then continued by Kaufman and the author \\cite{KM} and by Falconer \\cite{F1}. Peres and Schlag \\cite{PS} proved such estimates for large classes of generalized projections. Exceptional set estimates for intersections with planes were first proved by Orponen \\cite{O1} and continued by Orponen and the author \\cite{MO}. In \\cite{O2} Orponen derived estimates for radial projections. All these estimates expect those in \\cite{MO} and some in \\cite{PS} are known to be sharp. Some of these and other related results are also discussed in \\cite{M5}.\n\n\n\nRecently Donoven and Falconer \\cite{DF} investigated Hausdorff dimension of intersections for subsets of certain Cantor sets and Shmerkin and Suomala \\cite{SS} for large classes of random sets.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\\begin{abstract}\nHopfield attractor networks are robust distributed models of human memory. We propose construction rules such that an attractor network may implement an arbitrary finite state machine (FSM), where states and stimuli are represented by high-dimensional random bipolar vectors, and all state transitions are enacted by the attractor network's dynamics. Numerical simulations show the capacity of the model, in terms of the maximum size of implementable FSM, to be linear in the size of the attractor network. We show that the model is robust to imprecise and noisy weights, and so a prime candidate for implementation with high-density but unreliable devices. By endowing attractor networks with the ability to emulate arbitrary FSMs, we propose a plausible path by which FSMs may exist as a distributed computational primitive in biological neural networks.\n\\end{abstract}\n\n\\section{Introduction}\n\nHopfield attractor networks are robust models of human memory, as from a simple Hebbian learning rule they display emergent attractor dynamics which allow for reliable pattern recall, completion, and correction even in noisy situations \\cite{hopfield_neural_1982, amit_modeling_1989, eliasmith_unified_2005}. Attractor models have since found widespread use in neuroscience as a functional and tractable model of human memory \\cite{little_existence_1974, schneidman_weak_2006, chaudhuri_computational_2016, khona_attractor_2022}. The assumption of these models is that the network represents different states by different, usually uncorrelated, global patterns of persistent activity. When the network is presented with an input that closely resembles one of the stored states, the network state switches to the corresponding fixed-point attractor.\n\nThis process of switching between discrete attractor states is thought to be fundamental both to describe biological neural activity, as well as to model higher cognitive decision making processes \\cite{daelli_neural_2010, mante_context-dependent_2013, miller_itinerancy_2016, tajima_task-dependent_2017, brinkman_metastable_2022}. What attractor models currently lack, however, is the ability to perform state-dependent computation, a hallmark of human cognition \\cite{dayan_simple_2008, buonomano_state-dependent_2009, granger_toward_2020}. That is, when the network is presented with an input, the attractor state to which the network switches ought to be dependent both upon the input stimulus as well as the state the network currently inhabits, rather than simply the input.\n\nWe thus seek to endow a classical neural attractor model, the Hopfield network, with the ability to perform state-dependent switching between attractor states, without resorting to the use of biologically implausible mechanisms, such as higher-order weight tensors or training via backpropagation algorithms. The resulting attractor networks will then be able to robustly emulate any arbitrary Finite State Machine (FSM), vastly improving their usefulness as a neural computational primitive.\n\nWe achieve this by leaning heavily on the framework of Vector Symbolic Architectures (VSAs). VSAs treat computation in an entirely distributed manner, by letting symbols be represented by high-dimensional random vectors: hypervectors \\cite{kanerva_fully_2002, plate_holographic_2003, gayler_vector_2004}. When equipped with a few basic operators for binding and superimposing vectors together, corresponding often either to element-wise multiplication or addition respectively, these architectures are able to store primitives such a sets, sequences, graphs and arbitrary data bindings, as well as enabling more complex relations, such as analogical and figurative reasoning \\cite{kanerva_hyperdimensional_2009, kleyko_vector_2021}. Although different VSA implementations often have differing representations and binding operations \\cite{kleyko_survey_2022}, they all share the need for an auto-associative cleanup memory, which can recover a clean version of the most similar stored hypervector, given a noisy version of itself. We here use the recurrent dynamics of a Hopfield-like neural attractor network as a state-holding auto-associative memory \\cite{gritsenko_neural_2017}.\n\nSymbolic FSM states will thus be represented each by a hypervector and stored within the attractor network as a fixed-point attractor. Stimuli will also be represented by hypervectors, which, when input to the attractor network, will trigger the network dynamics to transition between the correct attractor states. We make use of common VSA techniques to construct a weights matrix to acheive these dynamics, where we use the Hadamard product between bipolar vectors $\\{-1,1\\}^N$ as our binding operation. We thus claim that attractor-based FSMs may be a plausible biological computational primitive insofar as Hopfield networks are.\n\nThis represents a computational paradigm that is a departure from conventional von Neumann architectures, wherein the separation of memory and computation is a major limiting factor in current advances in conventional computational performance (the von Neumann bottleneck \\cite{backus_can_1978, indiveri_memory_2015}). Similarly, the high redundancy and lack of reliance on individual components makes this architecture is fit for implementation with novel in-memory computing technologies such as resistive RAM (RRAM) or phase change memory (PCM) devices, which could perform the network's matrix-vector-multiplication (MVM) in a single step \\cite{xia_memristive_2019, ielmini_-memory_2018, zidan_chapter_2020}.\n\\section{Theory}\n\nThroughout this paper, symbols will be represented by high-dimensional randomly generated dense bipolar vectors\n\n\\begin{equation}\n \\vec{x} \\in \\{ -1 , 1\\}^{N}\n\\end{equation}\n\nwhere the number of dimensions $N>10,000$. Unless explicitly stated otherwise, any bold lowercase Latin letter may be assumed to be a new, independently generated hypervector, with the value $X_{i}$ at any index $i$ in $\\vec{x}$ generated according to\n\n\\begin{equation}\n \\text{I\\kern-0.15em P}(X_{i} = 1) = \\text{I\\kern-0.15em P}(X_{i}=-1) = \\frac{1}{2}\n\\label{eqn:probX_half}\n\\end{equation}\n\nFor any two arbitrary hypervectors $\\vec{a}$ and $\\vec{b}$, we define the similarity between the two vectors by the normalised inner product\n\n\\begin{equation}\n d(\\vec{a},\\vec{b}) := \\frac{1}{N} \\vec{a} ^{\\intercal} \\vec{b} = \\frac{1}{N} \\sum_{i = 1}^{N} a_{i}b_{i}\n\\label{eqn:d_simple}\n\\end{equation}\n\nwhere the similarity between a vector and itself $d(\\vec{a},\\vec{a}) = 1$, and $d(\\vec{a},-\\vec{a}) = -1$. Due to the high dimensionality of the vectors, the similarity between any two unrelated (and so independently generated) vectors is the mean of an unbiased random sequence of $-1$ and $1$s\n\n\\begin{equation}\n d(\\vec{a}, \\vec{b}) = 0 \\pm \\frac{1}{\\sqrt{N}} \\approx 0\n\\end{equation}\n\nwhich tends to 0 for $N\\rightarrow \\infty$. It is from this result that we get the requirement of high dimensionality, as it ensures that the inner product between two random vectors is approximately 0. We can say that independently generated vectors are \\textit{psuedo-orthogonal} \\cite{kleyko_vector_2021}. For a set of independently generated states $\\{\\vec{x}^\\mu\\}$, these results can be summarised by\n\n\\begin{equation}\n d(\\vec{x}^{\\mu},\\pm \\vec{x}^{\\nu} ) \\, \\stackrel{N \\rightarrow \\infty}{=} \\, \\pm \\delta^{\\mu \\nu}\n\\end{equation}\n\nwhere $\\delta^{\\mu \\nu}$ is the Kronecker delta. Hypervectors may be combined in a so called \\textit{binding} operation to produce a new vector that is dissimilar to both its constituents. We here choose the Hadamard product, or element-wise multiplication, as our binding operation, denoted \"$\\circ$\".\n\n\\begin{equation}\n (\\vec{a}\\circ \\vec{b})_{i} = a_{i} \\cdot b_{i}\n\\end{equation}\n\nThe statement that the product of two vectors is dissimilar to its constituents is written as\n\n\n\n\\begin{equation}\n\\begin{split}\n d(\\vec{a} \\circ \\vec{b}, \\vec{a}) \\approx 0 \\\\\n d(\\vec{a} \\circ \\vec{b}, \\vec{b}) \\approx 0\n\\end{split}\n\\end{equation}\n\nwhere we implicitly assume that $N$ is large enough that we can ignore the $\\mathcal{O}(\\frac{1}{\\sqrt{N}})$ noise terms.\n\nIf we wish to recover similarity between $\\vec{a} \\circ \\vec{b}$ and $\\vec{a}$, we can mask the system using $\\vec{b}$, such that only dimensions where $b_i = 1$ are remaining. Then, we have\n\n\n\n\\begin{equation}\n\\begin{split}\nd \\big( \\vec{a} \\circ \\vec{b} &, \\vec{a} \\circ H(\\vec{b}) \\big) = \\frac{1}{N} \\sum_{1 \\leq i \\leq N} a_i b_i a_i H(b_i) \\\\\n& = \\frac{1}{N} \\Bigg[ \\sum_{\\substack{1 \\leq i \\leq N \\\\ b_i =1}} a_i^2 H(1) - \\! \\! \\sum_{\\substack{1 \\leq i \\leq N \\\\ b_i =-1}} \\! \\! a_i^2 H(-1) \\Bigg] \\\\\n& = \\frac{1}{N} \\sum_{\\substack{1 \\leq i \\leq N \\\\ b_i =1}} 1 \\\\\n& \\approx \\frac{1}{2} \n\\end{split}\n\\end{equation}\n\n\nwhere we have used the Heaviside step function $H(\\cdot)$ defined by\n\n\\begin{equation}\n \\big( H(\\vec{b}) \\big)_{i} = H(b_{i}) = \\begin{cases} 1 \\quad \\text{if} \\quad b_{i} > 0 \\\\ 0 \\quad \\text{otherwise} \\end{cases}\n\\end{equation}\n\nto create a multiplicative mask $H(\\vec{b})$, setting to 0 all dimensions where $b_i = -1$. In the second line, we have split the summation over all dimensions into summations over dimensions where $b_i = 1$ and $-1$ respectively. The final similarity of $\\frac{1}{2}$ is a consequence of approximately half of all values in a any vector being +1 (\\mbox{Equation \\ref{eqn:probX_half}}).\n\n\nWe choose this as a mechanism for recovering similarity, rather than simply applying another Hadamard multiplication $\\vec{b}\\, \\circ $, as it is an operation that can easily and robustly be realised in a neural attractor network with asynchronous updates, as discussed later.\n\n\n\\subsection{Hopfield networks}\n\nA Hopfield network is a dynamical system defined by its internal state vector $\\vec{z}$ and fixed recurrent weights matrix $\\mat{W}$, with a state update rule given by\n\n\\begin{equation}\n \\vec{z}_{t+1} = \\mathrm{sgn} \\big( \\mat{W} \\vec{z}_t \\big)\n \\label{eqn:update_hopfield}\n\\end{equation}\n\nwhere $\\vec{z}_{t}$ is the network state at discrete time step $t$, and $\\mathrm{sgn}(\\cdot)$ is an element-wise sign function, with zeroes resolving\\footnote{Though this arbitrary choice may seem to incur a bias to a particular state, in practise the postsynaptic sum very rarely equals 0.} to +1.\n\nFrom standard Hopfield theory, we know that if we want to store $P$ uncorrelated patterns $\\{ \\vec{x}^{\\nu} \\}_{\\nu =1}^{P}$ within a Hopfield network, we can construct the weights matrix $\\mat{W}$ according to\n\n\\begin{equation}\n \\mat{W} = \\sum_{\\mathrm{patterns} \\, \\, \\nu}^{P} \\vec{x}^{\\nu} \\vec{x}^{\\nu \\intercal}\n\\end{equation}\n\nthen as long as not too many patterns are stored ($P < 0.14 N$ \\cite{hopfield_neural_1982}), the patterns will become fixed-point attractors of the network's dynamics, and the network can perform robust pattern completion and correction.\n\n\n\n\n\\subsection{Finite State Machines}\n\nA Finite State Machine (FSM) $M$ is a discrete system with a finite state set $Z_{\\text{FSM}} = \\{\\zeta_{1},\\zeta_{2}, \\ldots , \\zeta_{N_{Z}}$ \\}, a finite input stimulus set $S_{\\text{FSM}} = \\{\\sigma_{1}, \\sigma_{2}, \\ldots, \\sigma_{N_{S}} \\}$, and finite output response set $R_{\\text{FSM}} = \\{\\rho_{1}, \\rho_{2}, \\ldots, \\rho_{N_{R}} \\}$. $M$ is then fully defined with the addition of its two characterising functions $F(\\cdot)$ and $G(\\cdot)$\n\n\\begin{equation}\n\\begin{split}\n z_{t+1} & = F(z_{t}, s_{t}) \\\\\n r_{t+1} & = G(z_{t}, s_{t})\n\\end{split}\n\\end{equation}\n\nwhere $z_{t} \\in Z_{\\text{FSM}}$, $r_{t} \\in R_{\\text{FSM}}$ and $s_{t} \\in S_{\\text{FSM}}$ are the state, output and stimulus at time step $t$ respectively. $F(\\cdot)$ thus provides the state update rule, while $G(\\cdot)$ provides the output for any state-stimulus pair.\n\n$M$ can thus be represented by a directed graph, where each node represents a different state $\\zeta$, and every edge has a stimulus $\\sigma$ and optional output $\\rho$ associated with it.\n\n\n\n\n\\section{Attractor network construction}\n\\label{section:methods}\n\nWe now show how a Hopfield-like attractor network may be constructed to emulate an arbitrary FSM, where the states within the FSM are now attractors within the attractor network, and the stimuli for transitions between node states in the FSM trigger all corresponding transitions between attractors. More specifically, for every node $\\zeta \\in Z_{\\text{FSM}}$, an associated hypervector $\\vec{x}$ is randomly generated and stored as an attractor within the attractor network. We use $Z_{\\text{AN}}$ to denote the set of nodal hypervectors stored as attractors within the attractor network. Every unique stimulus $\\sigma \\in S_{\\text{FSM}}$ in the FSM is also now associated with a randomly generated hypervector $\\vec{s} \\in S_{\\text{AN}}$, where $S_{\\text{AN}}$ is the set of hypervectors associated with a unique stimulus. For the FSM edge outputs, a corresponding set of output hypervectors is similarly generated. These correspondences are summarised in \\mbox{Table \\ref{table:notation}}.\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{ll|ll}\n\\multicolumn{2}{c|}{FSM (Symbols)} & \\multicolumn{2}{c}{Attractor Net. (Vectors)} \\\\ \\hline\nNodes & $\\zeta \\in Z_{\\text{FSM}}$ & Attractors & $\\vec{x} \\in Z_{\\text{AN}}$ \\\\\nInput stimuli & $\\sigma \\in S_{\\text{FSM}}$ & Input stimuli & $\\vec{s} \\in S_{\\text{AN}}$ \\\\\nOutputs & $\\rho \\in R_{\\text{FSM}}$ & Output vectors & $\\vec{r} \\in R_{\\text{AN}}$ \n\\end{tabular}\n\\caption{A comparison of the notation used to represent states, inputs and outputs in the FSM picture, and the corresponding hypervectors used to represent the FSM within the attractor network.}\n\\label{table:notation}\n\\end{table}\n\n\n\n\\subsection{Constructing transitions}\n\nWe consider the general situation that we want to initiate a transition from \\textit{source} attractor state $\\vec{x} \\in Z_{\\text{AN}}$ to \\textit{target} attractor state $\\vec{y} \\in Z_{\\text{AN}}$, by imposing some stimulus state $\\vec{s} \\in S_{\\text{AN}}$ as input onto the network.\n\n\\begin{equation}\n \\vec{x} \\stackrel{\\vec{s}}{\\longrightarrow} \\vec{y}\n\\end{equation}\n\n\nCrucial to the functioning of the network transitions is how we model input to the network. We choose to model input to the network as a masking of the network state, such that all dimensions where the stimulus $\\vec{s}$ is -1 are forced to be 0. This may be likened to saying we are considering input to the network that selectively silences half of all neurons according to the stimulus vector. While a stimulus vector $\\vec{s}$ is being imposed upon the network, the modified state update rule is thus\n\n\\begin{equation}\n \\vec{z}_{t+1} = \\mathrm{sgn} \\big( \\mat{W} (\\vec{z}_{t} \\circ H (\\vec{s})) \\big)\n\\label{eqn:update_hop_mask}\n\\end{equation}\n\nwhere the Hadamard multiplication of the network state with $H (\\vec{s})$ enacts the masking operation, and the weights matrix $\\mat{W}$ is constructed such that $\\vec{z}_{t+1}$ will resemble the desired target state.\n\n\n\nFor every edge in the FSM, we generate an \"edge state\" $\\vec{e}$, which is also stored as an attractor within the network. Each edge will use this $\\vec{e}$ state as a \"halfway-house\", en route to $\\vec{y}$. Additionally, each unique edge label will now have \\textit{two} stimulus hypervectors associated with it, $\\vec{s}_a$ and $\\vec{s}_b$ which trigger transitions from source state $\\vec{x}$ to edge state $\\vec{e}$ and edge state $\\vec{e}$ to target state $\\vec{y}$ respectively. A general transition now looks like\n\n\\begin{equation}\n \\vec{x} \\stackrel{\\vec{s}_a}{\\longrightarrow} \\vec{e} \\stackrel{\\vec{s}_b}{\\longrightarrow} \\vec{y}\n\\end{equation}\n\nwhere $\\vec{x}, \\vec{y} \\in Z_{\\text{AN}}$ correspond to nodal states in the FSM but $\\vec{e}$ exists purely to facilitate the transition.\nThe weights matrix is constructed as\n\n\\begin{equation}\n \\mat{W} = \\underbrace{\\frac{1}{N}\\sum_{\\vphantom{\\text{dg}}\\text{nodes $\\nu$}}^{N_{Z}} \\vec{x}^\\nu \\vec{x}^{\\nu \\intercal} }_{\\substack{\\text{ Hopfield}\\\\\\text{attractor terms}}} +\n \\underbrace{\\frac{1}{N}\\sum_{\\text{edges $\\eta$}}^{N_{E}} \\mat{E}^{\\eta}}_{\\substack{\\text{Asymmetric}\\\\\\text{transition terms}}}\n\\label{eqn:W_construction}\n\\end{equation}\n\n\n\nwhere $\\vec{x}^\\nu \\in Z_{\\text{AN}}$ is the state corresponding to the $\\nu$'th node in the graph to be implemented, $N_{Z}$ and $N_{E}$ are the number of nodes and edges respectively, and $\\mat{E}^\\eta$ is the addition to the weights matrix required to implement an individual edge, given by\n\n\n\n\\begin{equation}\n\\begin{split}\n \\mat{E}^{(\\eta)} & = \\vphantom{(}\\vec{e} \\vec{e}^{\\intercal} \\\\\n & +H(\\vec{s}_a) \\circ (\\vec{e}-\\vec{x}) (\\vec{x} \\circ \\vec{s}_a)^{\\intercal} \\\\\n & +H(\\vec{s}_b) \\circ (\\vec{y} -\\vec{e}) (\\vec{ e} \\circ \\vec{s}_b)^{\\intercal} \n \\label{eqn:tran_term}\n\\end{split}\n\\end{equation}\n\n\n\nwhere $\\vec{x}$, $\\vec{e}$ and $\\vec{y}$ are the source, edge, and target states of the edge $\\eta$ respectively, and $\\vec{s}_a$ and $\\vec{s}_b$ are the input stimulus vectors associated with this edge's label. The edge index $\\eta$ has been dropped for brevity. The $\\vec{e}\\vec{e}^{\\intercal}$ term is the attractor we have introduced as a halfway-house for the transition. The second set of terms enacts the $ \\vec{x} \\stackrel{\\vec{s}_a}{\\longrightarrow} \\vec{e}$ transition, by giving a nonzero inner product with the network state only when the network is in state $\\vec{x}$, \\textit{and} the network is being masked by the input $\\vec{s}_a$. This allows terms to be stored in $\\mat{W}$ which are effectively obfuscated, not affecting network dynamics considerably, until a specific stimulus is applied as a mask to the network. Likewise, the third set of terms enacts the $\\vec{e} \\stackrel{\\vec{s}_b}{\\longrightarrow} \\vec{y}$ transition.\n\nIn the absence of input, the network functions like a standard Hopfield attractor network,\n\n\n\\begin{equation}\n \\mat{W}\\vec{x} \\approx \\vec{x} \\pm \\sigma \\vec{n} \\quad \\forall \\quad \\vec{x} \\in Z_{\\text{AN}}\n\\label{eqn:normal_attractor}\n\\end{equation}\nwhere $\\vec{n} \\in \\mathbb{R}^{N}$ is a standard normally distributed random vector, and\n\\begin{equation}\n\\sigma = \\sqrt{\\frac{N_{Z} + 3N_{E}}{N}} \n\\label{eqn:sigma_SNR}\n\\end{equation} \nis the magnitude of noise due to the undesired finite inner product with other stored terms (see Appendix). Thus as long as the magnitude of the noise is not too large, $\\vec{x}$ will be a solution of $\\vec{z} = \\mathrm{sgn}(\\mat W\\vec{z})$ and so a fixed-point attractor of the dynamics.\n\nWhen a valid stimulus is presented as input to the network however, masking the network state, the previously obfuscated asymmetric transition terms become significant and dominate the dynamics. Assuming there is a stored transition term $\\mat{E}$ corresponding to a valid edge with vectors $\\vec{x},\\vec{e},\\vec{y},\\vec{s}_a, \\vec{s}_b$ having the same meaning as in \\mbox{Equation \\ref{eqn:tran_term}}, we have \n\n\n\\begin{equation}\n\\begin{split}\n \\mat{W} \\big( \\vec{x} \\circ H(\\vec{s}_a) \\big) \\appropto & \\, \\, H \\big( \\vec{s}_a \\big) \\circ \\vec{e}+ H(-\\vec{s}_a)\\circ \\vec{x} \\pm \\sqrt{2}\\sigma \\vec{n}\n\\end{split}\n\\end{equation}\n\nwhere $\\appropto$ implies approximate proportionality. The second set of terms can be ignored, as they project only to neurons which are currently being masked. Thus the only significant term is that containing the edge state $\\vec{e}$, which consequently drives the network to the $\\vec{e}$ state, enacting the $\\vec{x} \\stackrel{\\vec{s}_a}{\\longrightarrow} \\vec{e}$ transition. Since the state $\\vec{e}$ is also stored as an attractor within the network, we have\n\n\\begin{equation}\n \\mat{W} \\big( \\vec{e} \\circ H(\\vec{s}_a) \\big) \\, \\appropto \\, \\vec{e} \\pm \\sqrt{2} \\sigma \\vec{n} \n\\end{equation}\n\nand\n\n\\begin{equation}\n \\mat{W}\\vec{e} \\approx \\vec{e} \\pm \\sigma \\vec{n}\n\\end{equation}\n\nthus the edge states $\\vec{e}$ are also fixed-point attractors of the network dynamics. To complete the transition from state $\\vec{x}$ to $\\vec{y}$, the second stimulus $\\vec{s}_b$ is applied, giving\n\n\n\\begin{equation}\n\\begin{split}\n \\mat{W} \\big( \\vec{e} \\circ H(\\vec{s}_b) \\big) & \\appropto H \\big( \\vec{s}_b \\big) \\circ \\vec{y} + H(-\\vec{s}_b) \\circ \\vec{e} \\pm \\sqrt{2}\\sigma \\vec{n}\n\\end{split}\n\\end{equation}\n\n\n\nwhich drives the network state towards $\\vec{y} \\in Z_{\\text{AN}}$, the desired target attractor state. By consecutive application of the inputs $\\vec{s}_a$ and $\\vec{s}_b$, the transition terms $\\mat{E}^\\eta$ stored in $\\mat{W}$ have thus caused the network to controllably transition from the source to target attractor states. Transition terms $\\mat{E}^\\eta$ may be iteratively added to $\\mat{W}$ to achieve any arbitrary transition between attractor states, and so any arbitrary FSM may be implemented within a large enough attractor network.\n\nNote that we have here ignored that the diagonal of $\\mat{W}$ is set to 0 (no self connections), but this does not significantly affect these results.\n\n\n\n\\subsection{Edge outputs}\n\nUntil now we have not mentioned the other critical component of FSMs: the output associated with every edge. We have separated the construction of transitions and edge outputs for clarity, since the two may be effectively decoupled.\nMuch like for the nodes and edges in the FSM to be implemented, for every unique FSM output $\\rho \\in R_{\\text{FSM}}$, we generate a corresponding hypervector $\\vec{r} \\in R_{\\text{AN}}$, where $R_{\\text{AN}}$ is the set of all output vectors. Different however, is that we let these be sparse ternary vectors $\\vec{r} \\in \\{ -1, 0, 1 \\}^{N}$ with coding level $f_r := \\frac{1}{N}\\sum_{i}^N \\lvert r_i \\rvert $, the fraction of nonzero elements. These output states are then embedded in the edge state attractors, altering the $\\vec{e}\\vec{e}^{\\intercal}$ terms in each $\\mat{E}$ term according to\n\n\\begin{equation}\n \\vec{e}\\vec{e}^{\\intercal} \\rightarrow \\vec{e}_{\\vec{r}}\\vec{e}^T := \\Big[ \\vec{e} \\circ \\big( \\vec{1}-H(\\vec{r} \\circ \\vec{r}) \\big) + \\vec{r} \\Big] \\vec{e} ^{\\intercal} \n\\label{eqn:output_attr}\n\\end{equation}\n\nwhere $\\vec{e}_{\\vec{r}}$ is here defined and $\\vec{1}$ is a vector of all ones. As a result of this modification, the edge states $\\vec{e}$ themselves will no longer be exact attractors of the space. The composite state $\\vec{e}_{\\vec{r}}$ will however be stable, in which the presence of $\\vec{r}$ can be easily detected ($\\vec{e}_{\\vec{r}} \\cdot \\vec{r} = Nf_{r}$). This has been achieved without incurring any similarity and thus interference between attractors, which would otherwise alter the dynamics of the previously described transitions.\nA full transition term $\\mat{E}$, including its output, is thus given by\n\n\\begin{equation}\n\\begin{split}\n \\mat{E}^{(\\eta)} & = \\vphantom{(}\\Big[ \\vec{e} \\circ \\big( \\vec{1}-H(\\vec{r} \\circ \\vec{r}) \\big) + \\vec{r} \\Big] \\vec{e} ^{\\intercal} \\\\\n & +H(\\vec{s}_a) \\circ (\\vec{e}-\\vec{x}) (\\vec{x} \\circ \\vec{s}_a)^{\\intercal} \\\\\n & +H(\\vec{s}_b) \\circ (\\vec{y} -\\vec{e}) (\\vec{ e} \\circ \\vec{s}_b)^{\\intercal} \n \\label{eqn:tran_term_final}\n\\end{split}\n\\end{equation}\n\nwhich combined with the network state masking operation is solely responsible for storing the FSM connectivity and enabling the desired inter-attractor transition dynamics.\n\n\n\n\n\\section{Results}\n\n\\subsection{FSM Emulation}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=3in]{figs\/greek_graph.pdf}\n\\caption{An example FSM which we implement within the attractor network. Each node within the graph (e.g. \"Zeus\") is represented by a new hypervector $\\vec{x}^\\mu$ and stored as an attractor within the network. Every edge is labelled by its stimulus (e.g. \"father\\_is\"), for which corresponding hypervectors $\\vec{s}_a$ and $\\vec{s}_b$ are also generated. When a stimulus' hypervector is input to the network, it should allow all corresponding attractor transitions to take place. Each edge may also have an associated output symbol, where we here choose the edges labelled \"type\" to output the generation of the god \\{\"Primordial\", \"Titans\", \"Olympians\"\\}. This graph was chosen as it displays the generality of the embedding: it contains cycles, loops, bidirectional edges and state-dependent transitions.\n}\n\\label{fig:starwars_net}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/walk_normal_greek2_print.pdf}\n\\caption{An attractor network transitioning through attractor states in a state-dependent manner, as a sequence of input stimuli is presented to the network. \\textbf{a)} The input stimuli to the network, where for each unique edge label (e.g. \"father\\_is\") in the FSM to be implemented (\\mbox{\\mbox{Figure \\ref{fig:starwars_net}}}) a pair of hypervectors $\\vec{s}_a$ and $\\vec{s}_b$ have been generated. No stimulus, a stimulus $\\vec{s}_a$, then a stimulus $\\vec{s}_b$ are input for 10 time steps each in sequence. \\textbf{b)} \\& \\textbf{c)} The similarity of the network state $\\vec{z}_t$ to stored node states $\\vec{x} \\in Z_{\\text{AN}}$ and stored edge states $\\vec{e}$ respectively, computed via the inner product (\\mbox{Equation \\ref{eqn:d_simple}}). \\textbf{d)} The similarity of the network state $\\vec{z}_t$ to the sparse output states $\\vec{r} \\in R_{\\text{AN}}$. All similarities have been labelled with the state they represent and the colours are purely illustrative. The state transitions shown here are explicitly state dependent, as can be seen from the repeated input of \"father\\_is\", which results in a transition to state \"Kronos\" when in \"Hades\", but to \"Uranus\" when in \"Kronos\". Additionally, the network is unaffected by nonsense input that does not correspond to a stored edge, as the network remains in the attractor \"Uranus\" when presented with the input \"father\\_is\".\n}\n\\label{fig:walk_standard}\n\\end{figure*}\n\n\n\nTo show the generality of FSM construction, we chose to implement a directed graph representing the relationships between gods in ancient Greek mythology, due to the graph's dense connectivity. The graph and thus FSM to be implemented is shown in \\mbox{\\mbox{Figure \\ref{fig:starwars_net}}}. From the graph it is clear that a state machine representing the graph must explicitly be capable of state-dependent transitions, e.g. the input \"overthrown\\_by\" must result in a transition to state \"Kronos\" when in state \"Uranus\", but to state \"Zeus\" when in state \"Kronos\". To construct $\\mat{W}$, the necessary random hypervectors are first generated. For every node state $\\zeta \\in Z_{\\text{FSM}}$ within the FSM (e.g. \"Zeus\", \"Kronos\") a random bipolar vector $\\vec{x}$ is generated. For every unique edge label $\\sigma \\in S_{\\text{FSM}}$ (e.g. \"overthrown\\_by\", \"father\\_is\") a pair of random stimulus hypervectors $\\vec{s}_a$ and $\\vec{s}_b$ is generated. Similarly, sparse ternary output vectors $\\vec{r}$ are also generated. The weights matrix $\\mat{W}$ is then iteratively constructed as per Equations \\ref{eqn:W_construction} and \\ref{eqn:tran_term_final}, with a new hypervector $\\vec{e}$ also being generated for every edge. The matrix generated from this procedure we denote $\\mat{W}^{\\text{ideal}}$. For all of the following results, the attractor network is first initialised to be in a certain attractor state, in this case, \"Hades\". The network is then allowed to freely evolve for 10 time steps (chosen arbitrarily) as per \\mbox{Equation \\ref{eqn:update_hopfield}}, with every neuron being updated simultaneously on every time step. During this period, it is desired that the network state $\\vec{z}_t$ remains in the attractor state in which it was initialised. An input stimulus $\\vec{s}_a$ is then presented to the network for 10 time steps, during which time the network state is masked by the stimulus vector, and the network evolves synchronously according to \\mbox{Equation \\ref{eqn:update_hop_mask}}. If the stimulus corresponds to a valid edge in the FSM, the network state $\\vec{z}_t$ should then be driven towards the correct edge state attractor $\\vec{e}$. After these 10 time steps, the second stimulus vector $\\vec{s}_b$ for a particular input is presented for 10 time steps. Again, the network evolves according to \\mbox{Equation \\ref{eqn:update_hop_mask}}, and the network should be driven towards the target state attractor $\\vec{y}$, completing the transition. This process is repeated every 30 time steps, causing the network state $\\vec{z}_t$ to travel between attractor states $\\vec{x} \\in Z_{\\text{AN}}$, corresponding to a valid walk between states $\\zeta \\in Z_{\\text{FSM}}$ in the represented FSM. To view the resulting network dynamics, the similarity between the network state $\\vec{z}_t$ and the edge and node states is calculated as per \\mbox{Equation \\ref{eqn:d_simple}}, such that a similarity of 1 between $\\vec{z}_t$ and some attractor state $\\vec{x}^\\nu$ implies $\\vec{z}_t = \\vec{x}^\\nu$ and thus that the network is inhabiting that attractor. The similarity between the network state $\\vec{z}_t$ and the outputs states $\\vec{r} \\in R_{\\text{AN}}$ is also calculated, but due to the output vectors being sparse, the maximum value that the similarity can take is $d(\\vec{z}, \\vec{r}) = f_r$, which would be interpreted as that output symbol being present.\n\nAn attractor network performing a walk is shown in \\mbox{\\mbox{Figure \\ref{fig:walk_standard}}}, with parameters $N = 10,000$, $N f_r = 200$, $N_{Z} = 8$, and $N_E = 16$. This corresponds to the network having a per-neuron noise (the finite size effect resulting from random hypervectors having a nonzero similarity to each-other) of $\\sigma \\approx 0.07$, calculated via \\mbox{Equation \\ref{eqn:sigma_SNR}}. The magnitude of the noise is thus small compared with the desired signal of magnitude 1 (\\mbox{Equation \\ref{eqn:normal_attractor}}), and so we are far away from reaching the memory capacity of the network. The network performs the walk as intended, transitioning between the correct nodal attractor states and corresponding edge states with their associated outputs. The specific sequence of inputs was chosen to show the generality of implementable state transitions. First, there is the explicit state dependence in the repeated input of \"father\\_is, father\\_is\". Second, it contains an input stimulus that does not correspond to a valid edge for the currently inhabited state (\"Zeus overthrown\\_by\"), which should not cause a transition. Third, it contains bidirectional edges (\"consort\\_is\"), whose repeated application causes the network to flip between two states (between \"Kronos\" and \"Rhea\"). And fourthly self-connections, whose target states and source states are identical. Since the network traverses all these edges as expected, we do not expect the precise structure of an FSM's graph to limit whether or not it can be emulated by the attractor network.\n\n\\subsection{Network robustness}\nOne of the advantages of attractor neural networks that make them suitable as plausible biological models is their robustness to imperfect weights \\cite{amit_modeling_1989}. That is, individual synapses may have very few bits of precision or become damaged, yet the relevant brain region must still be able to carry out its functional task. To this end, we subject the network presented here to similar non-idealities, to check that the network retains the feature of global stability and robustness despite being implemented with low-precision and noisy weights. In the first of these tests, the ideal weights matrix $\\mat{W}^{\\text{ideal}}$ was binarised and then additive noise was applied, via\n\n\n\\begin{equation}\n W^{\\text{noisy}}_{ij} = \\mathrm{sgn}\\big( W^{\\text{ideal}}_{ij}) + \\sigma_{\\text{noise}} \\cdot \\chi_{ij}\n \\label{eqn:W_noisy}\n\\end{equation}\n\n\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/walk_binnoise_greek2.pdf}\n\\caption{The attractor network performing a walk as in \\mbox{Figure \\ref{fig:walk_standard}}, but using the damaged weights matrix $\\mat{W}^{\\text{noisy}}$, whose entries have been binarised and then independent additive noise has been applied, as per \\mbox{Equation \\ref{eqn:W_construction}}. \\textbf{a)} The distribution of weights after they have been thusly damaged with noise of magnitude $\\sigma_{\\text{noise}} = 2$. Weights whose ideal values were positive or negative have been plotted separately. \\textbf{b)} The similarity of the network state $\\vec{z}_t$ to stored node states, with the network using the weights from a). Shown above is the sequence of inputs given to the network, identical to in \\mbox{Figure \\ref{fig:walk_standard}}. \\textbf{c)} The distribution of weights damaged with $\\sigma_{\\text{noise}} = 5$. \\textbf{d)} The similarity of the network state to stored node states, but with the network using the damaged weights from c). The network transitions are thus highly robust to unreliable weights, and show a gradual degradation in performance, even when the network's weights are majorly imprecise and noisy. For both b) and d) the edge state and output similarity plots have been omitted for visual clarity.}\n\\label{fig:starwars_smear}\n\\end{figure*}\n\n\n\nwhere $\\chi_{ij} \\in \\mathbb{R}$ are independently sampled standard Gaussian variables, sampled once during matrix construction, and $\\sigma_{\\text{noise}} \\in \\mathbb{R}$ is a scaling factor on the strength of noise being imposed. The $\\mathrm{sgn}(\\cdot)$ function forces the weights to be bipolar, emulating that the synapses may have only 1 bit of precision, while the $\\chi_{ij}$ random variables act as a smearing on the weight state, emulating that the two weight states have a finite width. A $\\sigma_{\\text{noise}}$ value of $2$ thus corresponds to the magnitude of the noise being equal to that of the signal (whether $W^{\\text{ideal}}_{ij} \\geq 0$), and so, for example, for a damaged weight value of $W^{\\text{noisy}}_{ij} = +1$ there is a 38\\% chance that the pre-damaged weight $W^{\\text{ideal}}_{ij} = -1$. This level of degradation is far worse than is expected even from novel binary memory devices \\cite{xia_memristive_2019}, and presumably also for biology. We used the same set of hypervectors and sequence of inputs as in \\mbox{Figure \\ref{fig:walk_standard}}, but this time using the degraded weights matrix $\\mat{W}^{\\text{noisy}}$, to test the network's robustness. The results are shown in \\mbox{Figure \\ref{fig:starwars_smear}} for weight degradation values of $\\sigma_{\\text{noise}} = 2$ and $\\sigma_{\\text{noise}} = 5$. We see that for $\\sigma_{\\text{noise}} = 2$ the attractor network performs the walk just as well as in Figure $\\ref{fig:walk_standard}$, which used the ideal weights matrix, despite the fact that here the binary weight distributions overlap each-other considerably. Furthermore, we have that $d(\\vec{z}_t, \\vec{x}^\\nu) \\approx 1$ where $\\vec{x}^\\nu$ is the attractor that the network should be inhabiting at any time, indicating that the attractor stability and recall accuracy is unaffected by the non-idealities. For $\\sigma_{\\text{noise}} = 5$, a scenario where the realised weight carries very little information about the ideal weight's value, we see that the network nonetheless continues to function, performing the correct walk between attractor states. However, there is a degradation in the recall of stored states, with the network state no longer converging to a similarity of 1 with the stored attractor states. For greater values of $\\sigma_{\\text{noise}}$, the network ceases to perform the correct walk, and indeed does not converge on any stored attractor state (not shown). \n\nA further test of robustness was to restrict the weights matrix to be sparse, as a dense all-to-all connectivity may not be feasible in biology, where synaptic connections are spatially constrained and have an associated chemical cost. Similar to the previous test, the sparse weights matrix was generated via\n\n\\begin{equation}\n W^{\\text{sparse}}_{ij} = \\mathrm{sgn}(W_{ij}^{\\text{ideal}} \\, ) \\cdot H (\\, \\lvert W_{ij} \\rvert - \\theta)\n \\label{eqn:W_sparse}\n\\end{equation}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/walk_sparse_greek2.pdf}\n\\caption{The attractor network performing a walk as in \\mbox{Figure \\ref{fig:walk_standard}}, but using a sparse ternary weights matrix $\\mat{W}^{\\text{sparse}} \\in \\{-1,0,1\\}^{N \\times N}$, generated via \\mbox{Equation \\ref{eqn:W_sparse}}. The weights matrices for \\textbf{a)} and \\textbf{b)} are 98\\% and 99\\% sparse respectively. Shown are the similarities of the network state $\\vec{z}_t$ with stored node states $\\vec{x}^\\nu \\in Z_{\\text{AN}}$, with the stimulus input hypervector to the network at any time shown above. We see that even when 98\\% of the entries in $\\mat{W}$ are zeroes, the network continues to function with negligible loss in stability, as the correct walk between attractor states is performed, and the network converges on stored attractors with similarity $d(\\vec{z}_t, \\vec{x}) \\approx 1$. At 99\\% sparsity there is a degradation in the accuracy of stored attractors, with the network converging on states with $d(\\vec{z}_t, \\vec{x}) < 1$, but with the correct walk still being performed. Beyond 99\\% sparsity the attractor dynamics break down (not shown). Thus although requiring a large number of neurons $N$ to enforce state pseudo-orthogonality, the network requires far fewer than $N^2$ nonzero weights to function robustly.}\n\\label{fig:starwars_sparse}\n\\end{figure*}\n\n\nwhere $\\theta$ is a threshold set such that $\\mat{W}^{\\text{sparse}} \\in \\{-1,0,1\\}^{N \\times N}$ has the desired sparsity. Through this procedure, only the most extreme weight values are allowed to be nonzero. Since the terms inside $\\mat{W}^{\\text{ideal}}$ are symmetrically distributed around 0, there are approximately as many +1 entries in $\\mat{W}^{\\text{sparse}}$ as -1s. Using the same hypervectors and sequence of inputs as before, an attractor network performing a walk using the sparse weights matrix $\\mat{W}^{\\text{sparse}}$ is shown in \\mbox{Figure \\ref{fig:starwars_sparse}}, with sparsities of 98\\% and 99\\%. We see that for the 98\\% sparse case, there is again very little difference with the ideal case shown in \\mbox{Figure \\ref{fig:walk_standard}}, with the network still having a similarity of $d(\\vec{z}_t, \\vec{x}) \\approx 1$ with stored attractor states, and performing the correct walk. When the sparsity is pushed further to 99\\% however, we see that despite the network performing the correct walk, the attractor states are again slightly degraded, with the network converging on states with $d(\\vec{z}_t, \\vec{x}^\\nu) < 1$ with stored attractor states $\\vec{x}^\\nu$. For greater sparsities, the network ceases to perform the correct walk, and again does not converge on any stored attractor state (not shown).\n\nThese two tests thus highlight the extreme robustness of the model to imprecise and unreliable weights. The network may be implemented with 1 bit precision weights, whose weight distributions are entirely overlapping, or set 98\\% of the weights to 0, and still continue to function without any discernible loss in performance. The extent to which the weights matrix may be degraded and the network still remain stable is of course a function not only of the level of degradation, but also of the size of the network $N$, as well as the the number of states $N_Z$ and edges $N_E$ stored within the network. For conventional Hopfield models with Hebbian learning, these two factors are normally theoretically treated alike, as contributing an effective noise to the postsynaptic sum as in \\mbox{Equation \\ref{eqn:sigma_SNR}}, and so the magnitude of withstandable synaptic noise increases with increasing $N$ \\cite{amit_modeling_1989, sompolinsky_theory_1987}. Although a thorough mathematical investigation into the scaling of weight degradation limits is justified, as a first result we have here given numerical data showing stability even in the most extreme cases of non-ideal weights, and expect that any implementation of the network with novel devices would be far away from such extremities.\n\n\n\\subsection{Asynchronous updates}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/walk_async_greek.pdf}\n\\caption{An attractor network performing a shorter walk than in \\mbox{Figure \\ref{fig:walk_standard}}, but where neurons are updated asynchronously, with each neuron having a 10\\% chance of updating on any time step. \\textbf{a)} The similarity of the network state $\\vec{z}_t$ to stored attractor states, with the input stimuli to the network shown above. \\textbf{b)} The evolution of a subset of neurons within the attractor network, where for visual clarity, three nodal states shown have hypervectors taken from columns of the $N$-dimensional Hadamard matrix, rather than being randomly generated. The network functions largely the same as in the synchronous case, but with transitions between states now taking a finite number of time steps to complete. The model is thus not dependent on the precise timing of neuron updates, and should function robustly in asynchronous systems where timing is unreliable.}\n\\label{fig:async}\n\\end{figure*}\n\n\n\nAnother useful property of Hopfield networks is the ability to robustly function even with asynchronously updating neurons, wherein not every neuron experiences a simultaneous state update. This property is especially important for any architecture claiming to be biologically plausible, as biological neurons update asynchronously and largely independent of each-other, without the the need for global clock signals. To this end, we ran a similar experiment to that in \\mbox{Figure \\ref{fig:walk_standard}}, using the undamaged weights matrix $\\mat{W}^{\\text{ideal}}$, but with an asynchronous neuron update rule, wherein on each time step every neuron has only a 10\\% chance of updating its state. The remaining 90\\% of the time, the neuron retains its state from the previous time step, regardless of its postsynaptic sum. There is thus no fixed order of neuron updates, and indeed it is not even a certainty that a neuron will update in any finite time. To account for the slower dynamics of the network state, the time for which inputs were presented to the network, as well as the periods without any input, was increased from 10 to 40 time steps. To be able to easily view the gradual state transition, three of the attractor states were chosen to be columns of the $N$-dimensional Hadamard matrix, rather than being randomly generated. The results are shown in \\mbox{Figure \\ref{fig:async}}, for a shorter sequence of stimulus inputs. We see that the network functions as intended, but with the network now converging on the correct attractors in a finite number of updates rather than in just one. The model proposed here is thus not reliant on synchronous dynamics, which is important not only for biological plausibility, but also when considering possible implementations on asynchronous neuromorphic hardware.\n\n\n\n\n\n\\subsection{Storage capacity}\nIt is well known that the storage capacity of a Hopfield network, the number of patterns $P$ that can be stored and reliably retrieved, is proportional to the size of the network, via $P < 0.14N$ \\cite{hopfield_neural_1982, amit_modeling_1989}. When one tries to store more than $P$ attractors within the network, the so-called memory blackout occurs, after which no pattern can be retrieved. We thus perform numerical simulations for a large range of attractor network and FSM sizes, to see if an analogous relationship exists. Said otherwise, for an attractor network of finite size $N$, what sizes of FSM can the network successfully emulate?\n\nFor a given $N$, number of FSM nodes $N_Z$ and edges $N_E$, a random FSM was generated and an attractor network constructed to represent it as described in Section \\ref{section:methods}. To ensure a reasonable FSM was generated, the FSM's graph was first generated to have all nodes connected in a sequential ring structure, i.e. every state $\\zeta^{\\nu}\\in Z_{\\text{FSM}}$ connects to $\\zeta^{\\nu+1 \\mod N_Z}$. The remaining edges between nodes were selected at random, until the desired number of edges $N_E$ was reached. For each edge an associated stimulus is then required. Although one option would be to allocate as few unique stimuli as possible, so that the state transitions are maximally state dependent, this results in some advantageous cancellation effects between the $\\mat{E}^\\eta$ transition terms and the stored attractors $\\vec{x}^\\nu \\vec{x}^{\\nu \\intercal}$. To instead probe a worst-case scenario, each edge was assigned a unique stimulus.\n\nWith the FSM now generated, an attractor network with $N$ neurons was constructed as previously described. An initial attractor state was chosen at random, and then a random valid walk between states was chosen to be performed (chosen arbitrarily to be of length 6, corresponding to each run taking 180 time steps). The corresponding sequence of stimuli was input to the attractor network via the same procedure as in \\mbox{Figure \\ref{fig:walk_standard}}, each masking the network state. Each run was then evaluated to have either passed or failed, with a pass meaning that the network state inhabited the correct attractor state with overlap $d(\\vec{z}_t, \\vec{x}^\\nu) > 0.75$ in the middle of all intervals when it should be in a certain node attractor state. A pass thus corresponds to the network performing the correct walk between attractor states. The results are shown in \\mbox{Figure \\ref{fig:memcap}}. We see that for a given $N$, there is a linear relationship between the the number of nodes $N_Z$ and number of edges $N_E$ in the FSM that can be implemented before failure. That this trade-off exists is not surprising, since both contribute additively to the SNR within the attractor network (\\mbox{Equation \\ref{eqn:sigma_SNR}}). For each $N$, a linear Support Vector Machine (SVM) was fitted to the data, to find the separating boundary at which failure and success of the walk are approximately equiprobable. The boundary is given by $N_Z + \\beta N_E = c(N)$, where $\\beta$ represents the relative cost of adding nodes and edges, and $c(N)$ is an offset. For all of the fitted boundaries, the value of $\\beta$ was found to be approximately constant, with $\\beta = 2.4 \\pm 0.1$, and so is assumed to be independent of $N$. For every value of $N$, we define the capacity $C$ to be the maximum size of FSM which can be implemented before failure, for which $N_E = N_Z$. The capacity $C$ is then given by $C(N) = \\frac{c(N)}{1+\\beta}$, and is also plotted in \\mbox{Figure \\ref{fig:memcap}}.\nA linear fit reveals an approximate proportionality relationship of $C(N) \\approx 0.025 N$. In all, the boundary which limits the size of FSM which can be emulated is then given by \n\n\\begin{equation}\n N_Z + 2.4 N_E < 0.085 N\n\\end{equation}\n\nIt is expected that additional edges consume more of the network's storage capacity than additional nodes, since for every edge, 5 additional terms are added to $\\mat{W}$ (\\mbox{Equation \\ref{eqn:tran_term_final}}), contributing 3$\\times$ as much cross-talk noise as adding a node would (\\mbox{Equation \\ref{eqn:sigma_SNR}}). We can compare this storage capacity relation with that of the standard Hopfield model, by considering the case $N_E = 0$, i.e. there are no transition terms in the network, and so the network is identical to a standard Hopfield network. In this case, our failure boundary would become $N_Z < 0.085N$, in comparison to Hopfield's $P < 0.14 N$. Although this may seem like a drastic reduction in memory capacity, we must remember that as a result of input stimuli applying a masking operation to our network, the actual size of the network during these periods is actually $N^\\prime :=\\frac{1}{2}N$. In this case, the failure boundary can be rewritten as $N_Z < 0.17 N^\\prime$, which is more in keeping with the Hopfield estimate\\footnote{Although it might seem we are claiming that the network implemented here has a greater storage capacity than standard Hopfield networks, our boundary at $0.17N$ is for equiprobable success and failure, while the $0.14N$ figure is given for overwhelmingly likely success.}.\n\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=4.5in]{figs\/memcap_final.pdf}\n\\caption{The capacity of the attractor network for varying size $N$, in terms of the size of FSM that can be emulated before failure. For a given $N$, a random FSM was generated with number of nodes $N_Z$ and number of edges $N_E$. An attractor network was then constructed as described in Section \\ref{section:methods}, and a sequence of stimuli input to the network that should trigger a specific walk between attractor states. \\textbf{a)} Every coloured square is a successful walk, with no unique $(N_Z,N_E, N)$ triplet being sampled more than once, and lower-$N$ squares occlude higher-$N$ squares. Since only graphs with at least as many edges as nodes were sampled, $N_E - N_Z$ is given on the $y$-axis rather than $N_E$. The overlain black lines are the SVM-fitted decision boundaries, distinguishing between values that succeeded and values that failed. \\textbf{b)} The capacity $C$ for varying Hopfield network sizes $N$, where $C$ is defined to be the maximum size of of FSM which can be implemented before failure, for which $N_E=N_Z$. A linear fit is overlain, and shows a linear relationship in the capacity $C$ in terms of $N$ over the range explored. Assuming that the gradients of the linear fit in a) are equal, the boundary at which failure and success are equiprobable is given by $N_Z + 2.4 N_E = 0.085 N$.}\n\\label{fig:memcap}\n\\end{figure*}\n\n\n\n\\section{Discussion}\n\\label{section:discussion}\n\n\n\\subsection{Hardware implementability}\nSince the network described in this paper differs very little from the conventional Hopfield description, having changed only the prescription for the generation of the weights matrix and the specific way that input is modelled, we can lean heavily on previous works concerning VLSI implementations of Hopfield networks \\cite{howard_associative_1987, verleysen_analog_1989, weinfeld_fully_1989}.\n\nThe main difficulty associated with an implementation would be the size requirements, requiring a high dimensionality $N$ to ensure pseudo-orthogonality of randomly generated hypervectors. Although we may need $N>10^{4}$ neurons, we have shown this does not necessarily mean we need $10^{8}$ synapses, as the network still functions when the weights matrix is sparse (\\mbox{Figure \\ref{fig:starwars_sparse}}). Despite having shown the robust functioning of the network only in the two cases of sparse ternary weights and dense noisy weights, we expect there to be a trade-off between the amount of each non-ideality that the network can withstand before failure. That is, an attractor network with dense noisy weights may withstand a greater degree of noise before failure than a network with sparse noisy weights.\n\nAn efficient and high-density implementation of a dense weights matrix may however be enabled by novel memristive crossbar technologies, which execute the dense matrix-vector-multiplication (MVM) step required in the state update rule in one operation \\cite{ielmini_-memory_2018, xia_memristive_2019, zidan_chapter_2020}. Such devices are already of great interest due to their immediate application in image processing and deep learning acceleration, and hybrid CMOS-memristor hardware implementations have thus already been pursued for their usefulness as an associative memory \\cite{guo_modeling_2015, yang_novel_2017, molahasani_majdabadi_hybrid_2021}, as well as for more direct FSM implementations \\cite{graves_memory_2020, de_lima_stap_2022} using memristive ternary content addressable memory (TCAM) cells. Since we have shown that individual weights may be bistable, incredibly unreliable and noisy without incurring a significant loss to performance, then a large enough crossbar would be a highly suitable platform on which to implement the network presented here.\n\nSince the additions to the weights matrix for each state and transition are composed purely of outer products, fully parallel one-shot in-memory online learning of the weights matrix may be achievable. As long as the updates in the memristors' conductances are sufficiently linear and symmetric, then attractors and transitions may be learned in one-shot by specifying the two vectors at the crossbar's inputs and outputs \\cite{alibart_pattern_2013, li_situ_2021}. \n\n\n\n\\subsection{Relation to other architectures}\nWhile there is a large body of work concerning the equivalence between RNNs and FSMs, their implementations broadly fall into a few categories. There are those that require iterative gradient descent methods to mimic an FSM \\cite{zeng_learning_1993, lee_giles_learning_1995, das_unified_1994}, which makes them difficult to train for large FSMs, and improbable for use in biology. There are those that require creating a new FSM with an explicitly expanded state set, $Z' := Z \\times S$, such that there is a new state for every old state-stimulus pair \\cite{minsky_computation_1967, sanfeliu_algebraic_1999}, which is unfavourable due to the the explosion of (usually one-hot) states needing to be represented, as well as the difficulty of adding new states or stimuli iteratively. There are those that require higher-order weight tensors in order to explicitly provide a weight entry for every unique state-stimulus pair \\cite{omlin_fuzzy_1998, forcada_finite-state_2001, mali_neural_2020}. As well as being non-distributed, the weight tensors require synapses to connect between more than two neurons, which is difficult to implement and non-biological \\cite{krotov_large_2021}. In \\cite{recanatesi_memory_2017} transitions are triggered by adiabatically modulating a global inhibition parameter, such that the network may transition between similar stored patterns. Lacking however is a method to construct a network to perform arbitrary, controllable transitions between states. In \\cite{chen_attractor-state_2020} an in-depth analysis of small populations of rate-based neurons is conducted, wherein synapses with short-term synaptic depression enable a rich behaviour of itinerancy between attractor states, but does not scale to large systems and arbitrary stored memories.\n\nMost closely resembling our approach, however, are earlier works concerned with the related task of creating a sequence of transitions between attractor states in Hopfield-like neural networks. The majority of these efforts rely upon the use of synaptic delays, such that the postsynaptic sum on a time step $t$ depends, for example, also on the network state at time $t-10$, rather than just $t-1$. These delay synapses thus allow attractor cross-terms of the form $\\vec{x}^{\\nu+1}\\vec{x}^{\\nu \\intercal}$ to become influential only after the network has inhabited an attractor state for a certain amount of time, triggering a walk between attractor states \\cite{sompolinsky_temporal_1986, kleinfeld_sequential_1986}. This then also allowed for the construction of networks with state-dependent input-triggered transitions \\cite{gutfreund_processing_1988, amit_neural_1988, drossaers_hopfield_1992}. Similar networks were shown to function without the need for synaptic delays, but require fine tuning of network parameters and suffer from extremely low storage capacity \\cite{buhmann_noise-driven_1987, amit_modeling_1989}. In any case, the need for synaptic delay elements represents a large requirement on any substrate which might implement such a network, and indeed are problematic to implement in neuromorphic systems \\cite{nielsen_compact_2017}.\n\nState-dependent computation in spiking neural networks was realised in \\cite{neftci_synthesizing_2013} and \\cite{liang_neuromorphic_2019}, where they used population attractor dynamics to achieve robust state representations via sustained spiking activity. However, these approaches differ from this work in that the state representations are still fundamentally population-based rather than distributed, and so pose difficulties such as the requirement of finding a new population of neurons to represent any new state.\n\nThis work also differs from conventional methods to implement graphs in VSAs \\cite{kleyko_vector_2021, poduval_graphd_2022}, in that the network state does not need to be read by an outsider in order to implement state-dependent switching. That is, where in previous works a graph is encoded by a hypervector such that it may be reliably decoded by external circuitry, we instead encode the graph's connectivity in the attractor network's weights matrix, such that its recurrent dynamics realise the desired state machine. Our implementation could however have been brought closer to previous works, and indeed made simpler, if there were a way to reliably achieve a flipping of neuron states when input is received. Then, the transition dynamics could be achieved just by storing edge terms like $\\mat{E} = \\vec{y} (\\vec{x} \\circ \\vec{s})^{\\intercal}$. Although achieving a state flip may be easy in a digital synchronous system, it would be very difficult to robustly achieve in an analogue asynchronous system. To avoid flickering, the flip would need to be reliably initiated by a single event. These events would also need to arrive at all neurons simultaneously, lest attractor dynamics take over and correct these apparent errors. Both of these factors prohibit such an operation from existing in biological systems.\n\nThe lack of a flipping mechanism is also discussed in \\cite{rigotti_internal_2010}, wherein they show the necessity of a population of neurons with mixed selectivity, connected to both the input and attractor neurons, in order to achieve the flipping-like behaviour necessary for complex state switching. This requirement arose by demanding that the network state switch to resembling the target state immediately upon receiving a stimulus. We instead show that similar results can be achieved without this extra population, if we relax to instead demanding only that the network soon evolve to the target state.\n\n\n\n\n\\subsection{Biological plausibility}\n\nTransitions between discrete neural attractor states are thought to be a crucial mechanism for performing context-dependent decision making in biological neural systems \\cite{daelli_neural_2010, mante_context-dependent_2013, miller_itinerancy_2016, tajima_task-dependent_2017}. Attractor dynamics enable a temporary retention of received information, and ensure that irrelevant inputs do not produce stable deviations in the neural state. As such, in this work we provide a description of an attractor-based network that can perform controllable context-dependent transitions in a simple manner, while abiding by the principles of distributed representation.\n\nThe procedure for generating the weights matrix $\\mat{W}$, as a result of this simplicity, makes the proposed network more plausible than other more complex approaches, e.g. those utilising gradient descent methods. It can be learned in one-shot in a fully online fashion, since adding a new node or edge involves only an additive contribution to the weights matrix, which does not require knowledge of irrelevant edges, nodes, their hypervectors, or the weight values themselves. Furthermore, as a result of the entirely distributed representation of states and transitions, new behaviours may be added to the weights matrix at a later date, both without having to allocate new hardware, and without having to recalculate $\\mat{W}$ with all previous data. Both of these factors are critical for continual online learning.\n\nEvaluating the local learnability of $\\mat{W}$ to implement transitions is also necessary to evaluate the biological plausibility of the model. In the original Hopfield paper the weights could be learned using the simple Hebbian rule\n\n\\begin{equation}\n \\delta w_{ij} = x^{\\nu}_{i} x^{\\nu}_{j}\n\\end{equation}\n\nwhere $x_i^\\nu$ and $x_j^\\nu$ are the activities of the post- and presynaptic neurons respectively, and $\\delta w_{ij}$ the online synaptic efficacy update \\cite{hebb_organization_1949, hopfield_neural_1982}. While the attractor terms within the network can be learned in this manner, the transition cross-terms that we have introduced require an altered version of the learning rule. If we simplify our network construction by removing the edge state attractors, then the local weight update required to learn a transition between states is given by\n\n\n\\begin{equation}\n \\delta w_{ij} = H(s_i) y_{i} x_{j} s_{j}\n\\end{equation}\n\nwhere $\\vec{y}$, $\\vec{x}$ and $\\vec{s}$ are as previously defined. In removing the edge states, we disallow FSMs with consecutive edges with the same stimulus (e.g. \"father\\_is, father\\_is\"), but this is not a problem if completely general FSM construction is not the goal per se. This state-transition learning rule is just as local as the original Hopfield learning rule, as the weight update from presynaptic neuron $j$ to postsynaptic neuron $i$ is dependent only upon information that may be made directly accessible in the pre- and postsynaptic neurons, and does not depend on information in other neurons to which the synapse is not connected \\cite{zenke_brain-inspired_2021, khacef_spike-based_2022}.\n\n\nThe robust functioning of the network despite noisy and unreliable weights is another prerequisite for the model to plausibly be able to exist in biological systems. The network weights may be considerably degraded without affecting the behaviour of the network, and indeed beyond this the network exhibits a so-called graceful degradation in performance. Furthermore, biological synapses are expected to have only a few bits of precision \\cite{oconnor_graded_2005, bartol_hippocampal_2015, baldassi_learning_2016}, and the network has been shown to function even in the worst case of binary weights. These properties stem from the massive redundancy arising from storing the attractor states across the entire synaptic matrix in a distributed manner, a technique that the brain is expected to utilise \\cite{rumelhart_parallel_1987, crawford_biologically_2016}.\nSince the network is still an attractor network, it retains all of the properties that make attractor networks suitable for modelling cognitive function, such as that the network can perform robust pattern completion and correction, i.e. the recovery of a stored prototypical memory given a damaged, incomplete or noisy version, and thereafter function as a stable working memory \\cite{hopfield_neural_1982, amit_modeling_1989}.\n\n\\section{Conclusion}\n\\label{section:conclusion}\n\nAttractor neural networks are robust abstract models of human memory, but previous attempts to endow them with complex and controllable attractor-switching capabilities have suffered mostly from being either non-distributed, not scalable, or not robust. We have here introduced a simple procedure by which any arbitrary FSM may be embedded into a large enough Hopfield-like attractor network, where states and stimuli are replaced by high-dimensional random vectors, and all information pertaining to FSM transitions is stored in the network's weights matrix in a fully distributed manner. Our method of modelling input to the network as a masking of the network state allows cross-terms between attractors to be stored in the weights matrix in a way that they are effectively obfuscated until the correct state-stimulus pair is present, much in a manner similar to the standard binding-unbinding operation in more conventional VSAs.\n\nWe showed that the network retains many of the features of attractor networks which make them suitable for biology, namely that the network is robust to unreliable and imprecise weights, thus also making them highly suitable for implementation with high-density but noisy devices. We presented numerical results showing that the network capacity in terms of implementable FSM size scales linearly with the size of the attractor network, and also that the network continues to function when the synchronous neuron update rule is replaced with a stochastic, asynchronous variant. \n\nIn summary, we introduced an attractor-based neural state machine which overcomes many of the shortcomings that made previous models unsuitable for use in biology, and propose that attractor-based FSMs may thus represent a plausible path by which FSMs may exist as a distributed computational primitive in biological neural networks.\n\\section*{Acknowledgements}\nWe thank Dr. Federico Corradi, Dr. Nicoletta Risi and Dr. Matthew Cook for their invaluable input and suggestions, as well as their help with proofreading this document. \n\nFunded by the Deutsche Forschungsgemeinschaft (DFG German Research Foundation) - Projects NMVAC (432009531) and MemTDE (441959088).\n\nThe authors would like to acknowledge the financial support of the CogniGron research center and the Ubbo Emmius Funds (Univ. of Groningen).\n\n\\FloatBarrier\n\n\\printbibliography\n\n\\clearpage\n \n\\onecolumn\n \n\\section*{Appendix}\n\n\\subsection{Dynamics without masking}\n\nFor the following calculations we assume that the coding level of the output states $f_r$ is low enough that their effect can be ignored. With this in mind, if we ignore the semantic differences between attractors for node states and attractors for edge states, the two summations over states can be absorbed into one summation over both types of attractor, here both denoted $\\vec{x}^\\nu$. Similarly there is then no difference between the two transition cross-terms within each $\\mat{E}$ term, and they too can be absorbed into one summation. Our simplified expression for $\\mat{W}$ is now given by\n\n\n\\begin{equation}\n \\mat{W} = \\frac{1}{N}\\sum_{\\vphantom{\\text{dg}}\\text{attr's $\\nu$}}^{N_Z + N_E} \\vec{x}^\\nu \\vec{x}^{\\nu\\, \\intercal} +\n \\frac{1}{N}\\sum_{\\text{tran's $\\lambda$}}^{2N_{E}} H(\\vec{s}^{\\pi(\\lambda)}) \\circ (\\vec{x}^{\\upsilon(\\lambda)} - \\vec{x}^{\\chi(\\lambda)}) (\\vec{x}^{\\chi(\\lambda)} \\circ \\vec{s}^{\\pi(\\lambda)})^{\\intercal}\n\\label{eqn:apx_1}\n\\end{equation}\n\nwhere $\\chi(\\lambda)$ and $\\upsilon(\\lambda)$ are functions $\\{1,\\ldots, 2N_E\\} \\rightarrow \\{1,\\ldots,N_Z + N_E \\}$ determining the indices of the source and target states for transition $\\lambda$, and $\\pi(\\lambda) : \\{1, \\ldots, 2N_E\\} \\rightarrow \\{1, \\ldots N_{\\text{stimuli}}\\}$ determines the index of the associated stimulus. We then wish to calculate the statistics of the postsynaptic sum $\\mat{W}\\vec{z}$ while the attractor network is currently in an attractor state. When in an attractor state $\\vec{x}^\\mu$, the postsynaptic sum is given by\n\n\n\\begin{equation}\n\\begin{split}\n\\Big[ \\mat{W}\\vec{x}^\\mu \\Big]_i & = \\frac{1}{N}\\sum_{\\vphantom{\\text{dg}}\\text{attr's $\\nu$}}^{N_Z + N_E} x_i^{\\nu} \\underbrace{\\big[ \\vec{x}^{\\nu} \\cdot \\vec{x}^\\mu \\big]}_{ \\substack{N\\text{ if } \\mu = \\nu \\\\ \\text{ else } \\mathcal{N}(0,N)}} \n + \\frac{1}{N}\\sum_{\\text{tran's $\\lambda$}}^{2N_{E}} H(s_i^{\\pi(\\lambda)}) \\circ (x_i^{ \\upsilon (\\lambda)} - x_i^{ \\chi (\\lambda)}) \\underbrace{\\big[ (\\vec{x}^{ \\chi (\\lambda)} \\circ \\vec{s}^{ \\pi(\\lambda)}) \\cdot \\vec{x}^\\mu \\big]}_{ \\mathcal{N}(0,N)} \\\\\n & = x^\\mu_i + \\sum_{\\vphantom{\\text{dg}}\\substack{\\text{attr's}\\\\ \\nu \\neq \\mu}}^{N_Z + N_E} \\underbrace{x_i^{\\nu}}_{ \\mathrm{Var.} = 1 } \\Big[ \\, \\mathcal{N}^{\\nu} \\big( 0,\\frac{1}{N} \\big) \\Big] \n + \\sum_{\\text{tran's $\\lambda$}}^{2N_{E}} \\underbrace{H(s_i^{ \\pi(\\lambda)}) \\circ (x_i^{ \\upsilon (\\lambda)} - x_i^{ \\chi (\\lambda)})}_{ \\mathrm{Var.}=1 }\\Big[ \\mathcal{N}^{\\lambda} \\big(0,\\frac{1}{N} \\big) \\Big] \\\\\n & \\approx x^\\mu_i + \\mathcal{N}\\Bigg( 0, \\frac{N_Z + N_E -1}{N} \\Bigg) + \\mathcal{N}\\Bigg( 0, \\frac{2N_E}{N} \\Bigg) \\\\\n & \\approx x^\\mu_i + \\mathcal{N}\\Bigg( 0, \\frac{N_Z + 3N_E}{N} \\Bigg)\n\\end{split}\n\\label{eqn:apx_2}\n\\end{equation}\n\nwhere we have used the notation $\\mathcal{N}(\\mu, \\sigma^2)$ to denote a normally-distributed random variable (RV) with mean $\\mu$ and variance $\\sigma^2$. In the third line we have made the approximation in the transition summation that the linear sum of attractor hypervectors, each multiplied by a Gaussian RV, is itself a separate Gaussian RV in each dimension. This holds as long as there are \"many\" attractor terms appearing on the LHS of the transition summation. Said otherwise, if the summation over transition terms has only very few unique attractor terms on the LHS ($N_E \\gg N_Z$), then the noise will be a random linear sum of the same few (masked) hypervectors, each with approximate magnitude $\\frac{1}{\\sqrt{N}}$, and so will be highly correlated between dimensions. Nonetheless we assume we are far away from this regime, and let the effect of the sum of these unwanted terms be approximated by a normally-distributed random vector, and so we have\n\n\\begin{equation}\n \\mat{W}\\vec{x}^\\mu \\approx \\vec{x}^\\mu + \\sigma\\vec{n}\n\\end{equation}\n\nwhere $\\sigma = \\sqrt{\\frac{N_Z + 3N_E}{N}}$ is the strength of cross-talk noise, and $\\vec{n}$ a vector composed of IID standard normally-distributed terms. This procedure of quantifying the signal-to-noise ratio (SNR) is adapted from that in the original Hopfield paper \\cite{hopfield_neural_1982, amit_modeling_1989}.\n\n\\subsection{Dynamics with masking}\n\nWe can similarly calculate the postsynaptic sum when in an attractor state $\\vec{x}^\\mu$, while the network is being masked by a stimulus $\\vec{s}^\\kappa$, with this (state, stimulus) tuple corresponding to a certain valid transition $\\lambda^\\prime$, with source, target, and stimulus vectors $\\vec{x}^\\mu$, $\\vec{x}^\\phi$, and $\\vec{s}^\\kappa$ respectively:\n\n\\begin{equation}\n\\begin{split}\n\\Big[ \\mat{W} \\big( \\vec{x}^\\mu \\circ H(\\vec{s}^\\kappa) \\big) \\Big]_i & = \\frac{1}{N}\\sum_{\\vphantom{\\text{dg}}\\text{attr's $\\nu$}}^{N_Z + N_E} x_i^{\\nu} \\underbrace{\\big[ \\vec{x}^{\\nu} \\cdot \\big( \\vec{x}^\\mu \\circ H (\\vec{s}^\\kappa) \\big) \\big]}_{ \\substack{\\frac{1}{2}N\\text{ if } \\mu = \\nu \\\\ \\text{ else } \\mathcal{N}(0,\\frac{1}{2}N)}} \\\\\n & \\quad \\quad + \\frac{1}{N}\\sum_{\\text{tran's $\\lambda$}}^{2N_{E}} H(s_i^{ \\pi(\\lambda)}) (x_i^{ \\upsilon (\\lambda)} - x_i^{ \\chi (\\lambda)}) \\underbrace{\\big[ (\\vec{x}^{ \\chi(\\lambda)} \\circ \\vec{s}^{ \\pi(\\lambda)}) \\cdot \\big( \\vec{x}^\\mu \\circ H (\\vec{s}^\\kappa) \\big)\\big]}_{ \\substack{\\frac{1}{2}N\\text{ if } \\chi(\\lambda) = \\mu \\text{ and } \\pi(\\lambda) = \\kappa \\\\ \\text{ else } \\mathcal{N}(0,\\frac{1}{2}N)}} \\\\\n & = \\frac{1}{2}x^\\mu_i + \\frac{1}{2}H(s_i^\\kappa) (x_i^\\phi - x_i^\\mu) + \\sum_{\\vphantom{\\text{dg}}\\substack{\\text{attr's}\\\\ \\nu \\neq \\mu}}^{N_Z + N_E} \\underbrace{x_i^{\\nu}}_{ \\mathrm{Var.} = 1 } \\Big[ \\, \\mathcal{N}^{\\nu}(0,\\frac{1}{2N}) \\Big] \\\\\n & \\quad \\quad + \\sum_{\\substack{\\text{tran's} \\\\ \\lambda \\neq \\lambda^\\prime}}^{2N_{E}} \\underbrace{H(s_i^{ \\pi(\\lambda)}) (x_i^{ \\upsilon(\\lambda)} - x_i^{ \\chi (\\lambda)})}_{ \\mathrm{Var.} = 1 } \\Big( \\mathcal{N}^{\\lambda}(0,\\frac{1}{2N}) \\Big) \\\\\n & \\approx \\frac{1}{2} \\big[ H(s_i^\\kappa) + H(-s_i^\\kappa) \\big] x^\\mu_i + \\frac{1}{2}H(s_i^\\kappa)( x_i^\\phi - x_i^\\mu) \n + \\mathcal{N}\\Big( 0, \\frac{N_Z + N_E -1}{2N} \\Big) + \\mathcal{N}\\Big( 0, \\frac{2N_E-1}{2N} \\Big) \\\\\n \n & = \\frac{1}{2}H(s_i^\\kappa) x_i^\\phi + \\frac{1}{2}H(-s_i^\\kappa) x_i^\\mu + \\mathcal{N}\\Big( 0, \\frac{N_Z + 3N_E -2}{2N} \\Big) \\\\\n & \\approx \\frac{1}{2} \\Bigg[ H(s_i^\\kappa) x_i^\\phi + H(-s_i^\\kappa) x_i^\\mu + \\sqrt{2} \\cdot \\mathcal{N}\\Big( 0,\\frac{N_Z + 3N_E}{N} \\Big) \\Bigg] \\\\\n\\end{split}\n\\label{eqn:apx_3}\n\\end{equation}\n\nwhere in the third line we have made the same approximations as previously discussed. The postsynaptic sum is thus approximately $\\vec{x}^\\phi$ in all indices that are not currently being masked, which drives the network towards that (target) attractor. In vector form, the above is written as\n\n\\begin{equation}\n \\mat{W} \\big( \\vec{x}^\\mu \\circ H ( \\vec{s}^\\kappa ) \\big) \\,\\, \\appropto \\,\\, H ( \\vec{s}^ \\kappa) \\circ \\vec{x}^\\phi + H (- \\vec{s}^ \\kappa) \\circ \\vec{x}^\\mu + \\sqrt{2}\\sigma\\vec{n}\n\\end{equation}\n\nwhere it is assumed that there exists a stored transition from state $\\vec{x}^\\mu$ to $\\vec{x}^\\phi$ with stimulus $\\vec{s}^\\kappa$, and $\\appropto$ denotes approximate proportionality. A similar calculation can be performed in the case that a stimulus is imposed which does not correspond to a valid transition for the current state. In this case, no terms of significant magnitude emerge from the transition summation, and we are left with \n\n\\begin{equation}\n \\mat{W} \\big( \\vec{x}^\\mu \\circ H ( \\vec{s}^\\text{invalid} ) \\big) \\,\\, \\appropto \\,\\, \\vec{x}^\\mu + \\sqrt{2}\\sigma\\vec{n}\n\\end{equation}\n\ni.e. the attractor dynamics are largely unaffected. Since we have not distinguished between our above attractor terms being node attractors or edge attractors, or our stimuli from being $\\vec{s}_a$ or $\\vec{s}_b$ stimuli, the above results can be applied to all relevant situations \\textit{mutatis mutandis}.\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAbstract polytopes are combinatorial structures with properties that generalise those of\nclassical polytopes. In many ways they are more fascinating than convex polytopes\nand tessellations. Highly symmetric examples of abstract polytopes include not\nonly classical regular polytopes such as the Platonic solids, and more exotic structures such as\nthe $120$-cell and $600$-cell, but also regular maps on surfaces (such as Klein's quartic).\n\nRoughly speaking, an abstract polytope $\\mathcal {P}$ is a partially ordered set endowed with\na rank function, satisfying certain conditions that arise naturally from a geometric\nsetting. Such objects were proposed by Gr\\\"unbaum in the 1970s, and their definition\n(initially as `incidence polytopes') and theory were developed by Danzer and Schulte.\n\nAn \\emph{automorphism} of an abstract polytope $\\mathcal {P}$ is an order-preserving permutation\nof its elements, and every automorphism of is uniquely determined by its effect\non any maximal chain in $\\mathcal {P}$ (which is known as a `flag' in $\\mathcal {P}$). The\nmost symmetric examples are regular, with all flags lying in a single orbit.\nThe comprehensive book written by Peter McMullen and Egon Schulte~\\cite{ARP} is\nnowadays seen as the principal reference on this subject.\n\nAn interesting class of examples which are not quite regular are the chiral polytopes.\nFor these, the automorphism group has two orbits on flags, with any two flags that\ndiffer in a single element lying in different orbits.\nChirality is a fascinating phenomenon that does not have a counterpart in the classical\ntheory of traditional convex polytopes.\nThe study of chiral abstract polytopes\nwas pioneered by Schulte and Weiss (see~\\cite{ChiralPolytopes, ChiralPolytopesWithPSL} for example),\nbut it has been something of a challenge to find and construct finite examples.\n\nFor quite some time, the only known finite examples of chiral polytopes had ranks $3$ and $4$.\nIn rank $3$, these are are given by the irreflexible (chiral) maps on closed\ncompact surfaces (see Coxeter and Moser~\\cite{BooksCoxeter}).\nSome infinite examples of chiral polytopes of rank $5$ were constructed by Schulte\nand Weiss in~\\cite{FEChiralPolytopes}, and then some finite examples\nof rank $5$ were constructed just over ten years ago by Conder, Hubard and Pisanski~\\cite{ConstructChiralPolytopes}.\n\nMany small examples of chiral polytopes are now known.\nThese include all chiral polytopes with at most $4000$ flags,\nand all that are constructible from an almost simple group $\\Ga$\nof order less than $900000$.\nThese have been assembled in collections, as in~\\cite{atles1, atles2}, for example.\nIn early $2009$ Conder and Devillers devised a construction for chiral polytopes whose facets are simplices,\nand used this to construct examples of finite chiral polytopes of ranks $6$, $7$ and $8$ [unpublished].\n\n At about the same time, Pellicer devised a quite different method for constructing finite chiral polytopes\nwith given regular facets, and used this construction to prove the existence of finite chiral polytopes of every rank $d \\geq 3$; see~\\cite{ConstructChiralPolytopesforanyrank}.\nA few years later, Cunningham and Pellicer proved that every finite chiral $d$-polytope with regular facets\nis itself the facet of a chiral $(d +1)$-polytope; see~\\cite{moreonConstructChiralPolytopesforanyrank}.\n\nThe work of Conder and Devillers was later taken up by Conder, Hubard, O'Reilly Regueiro and Pellicer~\\cite{ChiralPolytopesOfANSN},\nto prove that all but finitely many alternating groups $A_n$ and symmetric groups $S_n$ are the automorphism group\nof a chiral 4-polytope of type $\\{3,3,k\\}$ for some $k$ (dependent on $n$).\nThis has recently been extended to every rank greater than $4$ by the same authors as~\\cite{ChiralPolytopesOfANSN}.\n\nAlso Conder and Zhang in~\\cite{AbelianCoverOfChiralPolytopes} introduced a new covering method\nthat allows the construction of some infinite families of chiral polytopes,\nwith each member of a family having the same rank as the original, but with the size of the members of the family\ngrowing linearly with one (or more) of the parameters making up its `type' (Schl\\\"afli symbol).\nThey have used this method to construct several new infinite families of chiral polytopes of ranks $3, 4, 5$ and $6$.\nFurthermore, Zhang constructed in her PhD thesis~\\cite{Zhang'sPhdThesis} a number of chiral polytopes\nof types $\\{4, 4, 4\\}$, $\\{4, 4, 4, 4\\}$ and $\\{4, 4, 4, 4, 4\\}$, with automorphism groups of orders $2^{10}, 2^{11}, 2^{12}$,\nand $2^{15}, 2^{16}, \\cdots, 2^{22}$, and $2^{18}, 2^{19}$, respectively.\n\n\\smallskip\nNow let $\\mathcal {P}$ be a regular or chiral $4$-polytope. We say that $\\mathcal {P}$ is {\\em locally toroidal} if its facets and its vertex-figures are\nmaps on the $2$-sphere or on the torus, and either its facets or its vertex-figures (or both) are toroidal.\nUp to duality, rank 4 polytopes that are locally toroidal are of type $\\{4,4,3\\}$, $\\{4,4,4\\}$, $\\{6,3,3\\}$,\n$\\{6,3,4\\}$, $\\{6,3,5\\}$, $\\{6,3,6\\}$ or $\\{3,6,3\\}$.\n\nSchulte and Weiss~\\cite{ChiralPolytopesWithPSL} developed a construction that starts with a $3$-dimensional\nregular hyperbolic honeycomb and a faithful representation of its symmetry group as a group of complex M\\\"obius transformations\n(generated by the inversions in four circles that cut one another at the same angles as the corresponding\nreflection planes in hyperbolic space), and then derived chiral $4$-polytopes by applying\nmodular reduction techniques to the corresponding matrix group (see Monson and Schulte~\\cite{BE2009}).\nThey then used the simple group $\\hbox{\\rm PSL}(2, p)$ with $p$ an odd prime to construct infinite families of such polytopes.\n\nSome years later, Breda, Jones and Schulte~\\cite{ChiralPolytopesofPSLextension} developed a method of `mixing' a\nchiral $d$-polytope with a regular $d$-polytope to produce a larger example of a chiral polytope of the same rank $d$.\nThey used this to construct such polytopes with automorphism group $\\hbox{\\rm PSL}(2, p) \\times \\Ga^{+}(\\Omega)$, where $\\Ga^{+}(\\Omega)$\nis the rotation group of a finite regular locally-toroidal $4$-polytope.\nFor example, $\\Ga^{+}(\\Omega)$ could be $A_6 \\times C_2$ or $A_5$, when the corresponding\nchiral polytope has type $\\{4, 4, 3\\}$ or $\\{6, 3, 3\\}$, respectively.\n\nOne can see that almost all of the examples mentioned above involve non-abelian simple groups.\nOn the other hand, there appear to be few known examples of chiral polytopes with solvable automorphism groups, \napart from some of small order, and families of rank 3 polytopes arising from chiral maps on the torus (of type $\\{3,6\\}$, $\\{4,4\\}$ or $\\{6,3\\}$). \nThis was the main motivation for the research leading to this paper.\nIt was also motivated in part by a problem posed by Schulte and Weiss~\\cite{Problem}, namely the following:\n\\begin{prob}\nCharacterize the groups of orders $2^n$, with $n$ a positive integer, which are automorphism groups of regular or chiral polytopes.\n\\end{prob}\n\nHere we construct two infinite families of locally toroidal chiral 4-polytopes of type $\\{4,4,4\\}$, with solvable automorphism groups.\nEach family contains one example with $1024m^2$ or $2048m^2$ automorphisms, respectively, for every integer $m \\ge 1$.\nIn particular, if we let $m$ be an arbitrary power of $2$, say $2^k$ (with $k \\ge 0$), then the automorphism group\nhas order $2^{10+2k}$ or $2^{11+2k}$, which can be expressed as $2^n$ for an arbitrary integer $n \\ge 10$.\n\nThis extends some earlier work by the second and third authors (in~\\cite{HFL} and~\\cite{HFL1}), which showed that\nall conceivable ranks and types can be achieved for {\\em regular\\\/} polytopes with automorphism group of $2$-power order.\nIt also extends both the work by Zhang~\\cite{Zhang'sPhdThesis} mentioned above, and a construction by Cunningham~\\cite{TightChiralPolyhedra}\nof infinite families of tight chiral $3$-polytopes of type $\\{k_1, k_2\\}$ with automorphism group of order $2k_1k_2$\n(considered here for the special cases where $k_1$ and $k_2$ are powers of $2$).\n\n\n\\section{Additional background}\n\\label{background}\n\nIn this section we give some further background that may be helpful for the rest of the paper.\n\n\\subsection{Abstract polytopes: definition, structure and properties}\n\nAn abstract polytope of rank $n$ is a partially ordered set $\\mathcal {P}$ endowed with a strictly monotone rank function\nwith range $\\{-1, 0, \\cdots, n\\}$, which satisfies four conditions, to be given shortly.\n\nThe elements of $\\mathcal {P}$ are called \\emph{faces} of $\\mathcal {P}$. More specifically, the elements of $\\mathcal {P}$ of rank $j$ are called $j$-faces,\nand a typical $j$-face is denoted by $F_j$.\nTwo faces $F$ and $G$ of $\\mathcal {P}$ are said to be \\emph{incident} with each other if $F \\leq G$ or $F \\geq G$ in $\\mathcal {P}$.\nA \\emph{chain} of $\\mathcal {P}$ is a totally ordered subset of $\\mathcal {P}$, and is said to have \\emph{length} $i$ if it contains exactly $i+1$ faces.\nThe maximal chains in $\\mathcal {P}$ are called the \\emph{flags} of $\\mathcal {P}$.\nTwo flags are said to be $j$-\\emph{adjacent} if they differ in just one face of rank $j$,\nor simply \\emph{adjacent} (to each other) if they are $j$-adjacent for some~$j$.\nAlso if $F$ and $G$ are faces of $\\mathcal {P}$ with $F \\leq G$, then the set $\\{\\,H \\in \\mathcal {P} \\ |\\ F \\leq H \\leq G\\,\\}$ is\ncalled a \\emph{section} of $\\mathcal {P}$, and is denoted by $G\/F$. Such a section has rank $m-k-1$,\nwhere $m$ and $k$ are the ranks of $G$ and $F$ respectively.\nA section of rank $d$ is called a $d$-section.\n\n\\smallskip\nWe can now give the four conditions that are required of $\\mathcal {P}$ to make it an abstract polytope.\nThese are listed as (P1) to (P4) below:\n\\begin{itemize}\n\\item [(P1)] $\\mathcal {P}$ contains a least face and a greatest face, denoted by $F_{-1}$ and $F_n$, respectively.\n\\item [(P2)] Each flag of $\\mathcal {P}$ has length $n+1$ (so has exactly $n+2$ faces, including $F_{-1}$ and $F_n$).\n\\item [(P3)] $\\mathcal {P}$ is \\emph{strong flag-connected}, which means that any two flags $\\Phi$ and $\\Psi$ of $P$ can\nbe joined by a sequence of successively adjacent flags $\\Phi=\\Phi_{0}, \\Phi_1, \\cdots, \\Phi_k=\\Psi$, each of which contains $\\Phi \\cap \\Psi$.\n\\item [(P4)] The rank $1$ sections of $\\mathcal {P}$ have a certain homogeneity property known as the \\emph{diamond condition},\nnamely as follows: if $F$ and $G$ are incidence faces of $\\mathcal {P}$, of ranks $i-1$ and $i+1$, respectively, where $0 \\le i \\le n-1$,\nthen there exist precisely \\emph{two} $i$-faces $H$ in $\\mathcal {P}$ such that $F< H< G$.\n\\end{itemize}\n\n\\noindent\nAn easy case of the diamond condition occurs for polytopes of rank 3 (or polyhedra): if $v$ is a vertex of same face $f$,\nthen there are two edges that are incident with both $v$ and $f$.\n\n\\smallskip\nNext, every $2$-section $G\/F$ of $\\mathcal {P}$ is isomorphic to the face lattice of a polygon.\nNow if it happens that the number of sides of every such polygon depends only on the rank of $G$,\nand not on $F$ or $G$ itself, then we say that the polytope $\\mathcal {P}$ is {\\em equivelar}.\nIn this case, if $k_i$ is the number of edges of every $2$-section between an $(i-2)$-face and an $(i+1)$-face of $\\mathcal {P}$,\nfor $1 \\leq i \\leq n$, then the expression $\\{k_1, k_2, \\cdots, k_{n-1}\\}$ is called the Schl\\\"afli type of $\\mathcal {P}$.\n(For example, if $\\mathcal {P}$ has rank 3, then $k_1$ and $k_2$ are the valency of each vertex and the size of each 2-face,\nrespectively.)\n\n\\subsection{Automorphisms of polytopes}\n\nAn \\emph{automorphism} of an abstract polytope $\\mathcal {P}$ is an order-preserving permutation of its elements.\nIn particular, every automorphism preserves the set of faces of any given rank.\nUnder permutation composition, the set of all automorphisms of $\\mathcal {P}$ forms a group, called the automorphism\ngroup of $\\mathcal {P}$, and denoted by $\\hbox{\\rm Aut}}\\def\\Out{\\hbox{\\rm Out}(\\mathcal {P})$ or sometimes more simply as $\\Ga(\\mathcal {P})$.\nAlso it is not difficult to use the diamond condition and strong flag-connectedness to prove that if an automorphism\npreserves of flag of $\\mathcal {P}$, then it fixes every flag of $\\mathcal {P}$ and hence every element of $\\mathcal {P}$.\nIt follows that $\\Ga(\\mathcal {P})$ acts semi-regularly (or fixed-point-freely) on $\\mathcal {P}$.\n\n\\smallskip\nA polytope $\\mathcal {P}$ is said to be \\emph{regular} if its automorphism group $\\Ga(\\mathcal {P})$ acts transitively\n(and hence regularly) on the set of flags of $\\mathcal {P}$.\nIn this case, the number of automorphisms of $\\mathcal {P}$ is as large as possible, and equal to the number of flags of $\\mathcal {P}$.\nIn particular, $\\mathcal {P}$ is equivelar, and the stabiliser in $\\Ga(\\mathcal {P})$ of every 2-section of $\\mathcal {P}$ induces the full dihedral group\non the corresponding polygon.\nMoreover, for a given flag $\\Phi$ and for every $i \\in \\{0,1,\\dots,n-1\\}$, the polytope $\\mathcal {P}$ has a unique\nautomorphism $\\rho_i$ that takes $\\Phi$ to the unique flag $(\\Phi)^{i}$ that differs from $\\Phi$ in precisely its $i$-face,\nand then the automorphisms $\\rho_0,\\rho_1,\\dots,\\rho_{n-1}$ generate $\\Ga(\\mathcal {P})$ and satisfy the defining\nrelations for the string Coxeter group $[k_1, k_2, \\cdots, k_{n-1}]$, where the $k_i$ are as given in the\nprevious subsection for the Schl\\\"afli type of $\\mathcal {P}$.\nThey also satisfy a certain `intersection condition', which follows from the diamond and strong flag-connectedness\nconditions.\nThese and many more properties of regular polytopes may be found in~\\cite{ARP}.\n\n\\smallskip\nWe now turn to chiral polytopes, for which a good reference is~\\cite{ChiralPolytopes}.\n\n\\smallskip\nA polytope $\\mathcal {P}$ said to be {\\em chiral\\\/} if its automorphism group $\\Ga(\\mathcal {P})$ has two orbits on flags,\nwith every two adjacent flags lying in different orbits.\n(Another way of viewing this definition is to consider $\\mathcal {P}$ as admitting no `reflectiing' automorphism\nthat interchanges a flag with an adjacent flag.)\nHere the number of flags of $\\mathcal {P}$ is $2|\\Ga(\\mathcal {P})|$, and $\\Ga(\\mathcal {P})$ acts regularly on each of two orbits.\nAgain $\\mathcal {P}$ is equivelar, with the stabiliser in $\\Ga(\\mathcal {P})$ of every 2-section of $\\mathcal {P}$ inducing the full cyclic group on the corresponding polygon.\n\nNext, for a given flag $\\Phi$ and for every $j \\in \\{1,2,\\dots,n-1\\}$, the chiral polytope $\\mathcal {P}$ admits an\nautomorphism $\\s_j$ that takes $\\Phi$ to the flag $(\\Phi)^{j, j-1}$ which differs from $\\Phi$ in precisely its $(j-1)$-and $j$-faces.\nThese automorphisms $\\s_1,\\s_2,\\dots,\\s_{n-1}$ generate $\\Ga(\\mathcal {P})$, and if $\\mathcal {P}$ has Schl\\\"afli type $\\{k_1, k_2, \\dots, k_{n-1}\\}$,\nthen they satisfy the defining relations for the orientation-preserving subgroup of (index $2$ in) the\nstring Coxeter group $[k_1, k_2, \\cdots, k_{n-1}]$.\nAlso they satisfy a `chiral' form of the intersection condition, which is a variant of the one mentioned earlier for regular polytopes.\n\nChiral polytopes occur in pairs (or {\\em enantiomorphic\\\/} forms), such that each member of the pair is the `mirror image' of the other.\nSuppose one of them is $\\mathcal {P}$, and has Schl\\\"afli type $\\{k_1, k_2, \\cdots, k_{n-1}\\}$. Then $\\Ga(\\mathcal {P})$ is isomorphic\nto the quotient of the orientation-preserving subgroup $\\Lambda^{\\rm o}$ of the Coxeter group $\\Lambda = [k_1, k_2, \\cdots, k_{n-1}]$\nvia some normal subgroup $K$. By chirality, $K$ is not normal in the full Coxeter group $\\Lambda$, but is conjugated by\nany orientation-reversing element $c \\in \\Lambda$ to another normal subgroup $K^c$ which is the kernel of an epimorphism from $\\Lambda^{\\rm o}$\nto the automorphism group $\\Ga(\\mathcal {P}^c)$ of the mirror image $\\mathcal {P}^c$ of $\\mathcal {P}$.\n\nThe automorphism groups of $\\mathcal {P}$ and $\\mathcal {P}^c$ are isomorphic to each other, but their canonical generating sets\nsatisfy different defining relations. In fact, replacing the elements $\\s_1$ and $\\s_2$ in the canonical generating\ntuple $(\\s_1,\\s_2,\\s_3,\\dots,\\s_{n-1})$ by $\\s_1^{-1}$ and $\\s_1^{\\,2}\\s_2$ gives a set of generators for $\\Ga(\\mathcal {P})$\nthat satisfy the same defining relations as a canonical generating tuple for $\\Ga(\\mathcal {P}^c)$,\nbut chirality ensures that there is no automorphism of $\\Ga(\\mathcal {P})$ that takes $(\\s_1,\\s_2)$ to $(\\s_1^{-1},\\s_1^{\\,2}\\s_2)$\nand fixes all the other $\\s_j$.\n\nConversely, any finite group $G$ that is generated by $n-1$ elements $\\s_1,\\s_2,\\dots,\\s_{n-1}$ which satisfy both\nthe defining relations for $\\Lambda^{\\rm o}$ and the chiral form of the intersection condition is the `rotation subgroup' of an\nabstract $n$-polytope $\\mathcal {P}$ that is either regular or chiral.\nIndeed, $\\mathcal {P}$ is regular if and only if $G$ admits a group automorphism $\\rho} \\def\\s{\\sigma} \\def\\t{\\tau} \\def\\om{\\omega} \\def\\k{\\kappa$ of order $2$\nthat takes $(\\s_1,\\s_2,\\s_3,\\dots,\\s_{n-1})$ to $(\\s_1^{-1},\\s_1^{\\,2}\\s_2,\\s_3,\\dots,\\s_{n-1})$.\n\n\\smallskip\nWe now focus our attention on the rank 4 case.\nHere the generators $\\s_1, \\s_2, \\s_3$ for $\\Ga(\\mathcal {P})$ satisfy the canonical relations\n$\\,\\s_1^{k_1} = \\s_2^{k_2} = \\s_3^{k_3} = (\\s_1\\s_2)^2 = (\\s_2\\s_3)^2 = (\\s_1\\s_2\\s_3)^2 = 1$,\nand the chiral form of the intersection condition can be abbreviated to\n$$\\lg \\s_1\\rg \\cap \\lg \\s_2, \\s_3\\rg = \\{1\\} = \\lg \\s_1, \\s_2\\rg \\cap \\lg \\s_3\\rg \\ \\hbox{ and } \\ \\lg \\s_1, \\s_2\\rg \\cap \\lg \\s_2, \\s_3\\rg =\\lg \\s_2\\rg.$$\n\n\nThe following proposition is useful for the groups we will deal with in the proof of our main theorem.\nIt is called the {\\em quotient criterion} for chiral $4$-polytopes.\n\n\\begin{prop}{\\rm \\cite[Lemma 3.2]{ChiralPolytopesofPSLextension}}\\label{quotient criterion}\n$\\,$\nLet $G$ be a group generated by elements $\\s_1, \\s_2, \\s_3$ such that $(\\s_1\\s_2)^2=(\\s_2\\s_3)^2=(\\s_1\\s_2\\s_3)^2=1$,\nand let $\\theta\\!: G \\to H$ be a group homomorphism taking $\\,\\s_j \\mapsto \\ld_j\\,$ for $1\\le j \\le 3$,\nsuch that the restriction of $\\theta$ to either $\\lg \\s_1, \\s_2\\rg$ or $\\lg \\s_2, \\s_3\\rg$ is injective.\nIf $(\\ld_{1}, \\ld_{2}, \\ld_{3})$ is a canonical generating triple for $H$ as the automorphism group of some chiral $4$-polytope,\nthen the triple $(\\s_1, \\s_2, \\s_3)$ satisfies the chiral form of the intersection condition for $G$.\n\\end{prop}\n\n\n\n\\subsection{Group theory}\n\n\nWe use standard notation for group theory, as in~\\cite{GroupBook} for example.\nIn this subsection we briefly describe some of the specific aspects of group theory that we need.\n\n\\smallskip\nLet $G$ be any group. We define the {\\em commutator\\\/} $[x, y]$ of elements $x$ and $y$ of $G$\nby $[x, y]=x^{-1}y^{-1}xy$, and then define the {\\em derived subgroup} (or commutator subgroup) of $G$\nas the subgroup $G'$ of $G$ generated by all such commutators.\nThen for any non-negative integer $n$, we define the $n\\,$th derived group of $G$ by setting\n$$G^{(0)}=G, \\ G^{(1)}=G', \\ \\hbox{and } \\ G^{(n)}=(G^{(n-1)})' \\ \\hbox{when} \\ n \\geq 1.$$\n\nA group $G$ is called {\\em solvable\\\/} if $G^{(n)}=1$ for some $n$.\n(This terminology comes from Galois theory, because a polynomial over a field $\\mathbb{F}$ is solvable by radicals\nif and only if its Galois group over $\\mathbb{F}$ is a solvable group.)\nEvery abelian group and every finite $p$-group is solvable, but every non-abelian simple groups is not solvable.\nIn fact, the smallest non-abelian simple group $A_5$ is also the smallest non-solvable group.\n\nWe also need the following, which are elementary and so we give them without proof.\n\n\\begin{prop}\\label{solvable}\nIf $N$ is a normal subgroup of a group $G$, such that both $N$ and $G\/N$ are solvable, then so is $G$. \n\\end{prop}\n\n\\begin{prop}\\label{freeabelian}\nLet $G$ be the free abelian group $\\mathbb{Z}\\oplus \\mathbb{Z}$ of rank $2$, generated by two elements $x$ and $y$\nsubject to the single defining relation $[x,y] = 1$.\nThen for every positive integer $m$, the subgroup $G_m=\\lg x^m, y^m\\rg$ is characteristic in $G$, with index $|G:G_m|=m^2$.\n\\end{prop}\n\nFinally, we will use some Reidemeister-Schreier theory, which produces a defining presentation\nfor a subgroup $H$ of finite index in a finitely-presented group $G$.\nAn easily readable reference for this is~\\cite[Chapter IV]{Johnson}, but in practice we use its implementation\nas the {\\tt Rewrite} command in the {\\sc Magma} computation system~\\cite{BCP97}.\nWe also found the groups that we use in the next section with the help of {\\sc Magma} in constructing\nand analysing some small examples.\n\n\t\n\\section{Main results}\\label{Main results}\n\n\\begin{theorem}\n\\label{maintheorem}\nFor every positive integer $m \\geq 1$, there exist chiral $4$-polytopes $\\mathcal {P}_m$ and $\\mathcal {Q}_m$ of type $\\{4,4,4\\}$\nwith solvable automorphism groups of order $1024m^2$ and $2048m^2$, respectively.\n\\end{theorem}\n\n\\f {\\bf Proof.}\\hskip10pt\nWe begin by defining ${\\cal U}$ as the finitely-presented group \\\\[-10pt]\n$$\\lg\\, a, b, c \\ | \\ a^4 = b^4 = c^4 = (ab)^2 = (bc)^2 = (abc)^2 = (a^2b^2)^4 = a^2c^2b^2(ac)^2 = [a,c^{-1}]b^2 = 1 \\,\\rg.$$\nThis group ${\\cal U}$ has two normal subgroups of index $1024$ and $2048$, namely the subgroups\ngenerated by $\\{(ac^{-1})^4, (c^{-1}a)^4\\}$ and $\\{(bc^{-1})^4, (c^{-1}b)^4\\}$, respectively.\nThe quotients of ${\\cal U}$ by each of these give the initial members of our two infinite families.\n\n\\medskip\n\\noindent Case (1): Take $N$ as the subgroup of ${\\cal U}$ generated by $x = (ac^{-1})^4$ and $y = (c^{-1}a)^4$.\n\n\\smallskip\nA short computation with {\\sc Magma} shows that $N$ is normal in ${\\cal U}$, with index $1024$. \nIn fact, the defining relations for ${\\cal U}$ can be used to show that \\\\[-15pt]\n\\begin{center}\n\\begin{tabular}{lll}\n$a^{-1}xa = y$, \\qquad & $b^{-1}xb = y$ \\quad & and \\quad $c^{-1}xc = y$, \\\\[+2pt]\n$a^{-1}ya = x$, \\qquad & $b^{-1}yb = x^{-1}$ \\quad & and \\quad $c^{-1}yc = x^{-1}$. \\\\[-3pt]\n\\end{tabular}\n\\end{center}\n\nThe first, third and fifth of these are easy to prove by hand, while the second and fourth can \nbe verified in a number of ways, and the sixth follows from the other five. \nOne way to prove the second and fourth is by hand, which we leave as a challenging exercise for the interested reader.\nAnother is by a partial enumeration of cosets of the identity subgroup in ${\\cal U}$. \nFor example, if this is done using the {\\tt ToddCoxeter} command in {\\sc Magma}, allowing the definition of just 8000 cosets, \nthen multiplication by each of the words $b^{-1}xby^{-1}$ and $a^{-1}yax^{-1}$ is found to fix the trivial coset, \nand therefore $b^{-1}xby^{-1} = 1 = a^{-1}yax^{-1}.$ \n\nIt follows that conjugation by $a$, $b$ and $c$ induce the three permutations $\\,(x,y)(x^{-1},y^{-1})$,\n$\\,(x,y,x^{-1},y^{-1})\\,$ and $\\,(x,y,x^{-1},y^{-1})$ on the set $\\{x,y,x^{-1},y^{-1}\\}$,\nand then $(ac^{-1})^2$ and $(c^{-1}a)^2$ centralise both $x$ and $y$, so $x$ and $y$ centralise each other.\n\nAlso the {\\tt Rewrite} command in {\\sc Magma} gives a defining presentation for $N$, with $[x,y] = 1$ as a single \ndefining relation. Hence the normal subgroup $N$ is free abelian of rank 2. \n\n\n\\smallskip\nThe quotient ${\\cal U}\/N$ is isomorphic to the automorphism group of the chiral 4-polytope\nof type $\\{4,4,4\\}$ with 1024 automorphisms listed at~\\cite{atles1}.\n\n\\smallskip\nNow for any positive integer $m$, let $N_m$ be the subgroup generated by $x^m = (ac^{-1})^{4m}$\nand $y^m = (c^{-1}a)^{4m}$. By Proposition~\\ref{freeabelian}, we know that $N_m$ is characteristic in $N$\nand hence normal in ${\\cal U}$, with index $|\\,{\\cal U}:N_m| = |\\,{\\cal U}:N||N:N_m| = 1024m^2$.\nMoreover, in the quotient $G_m = {\\cal U}\/N_m$, the subgroup $N\/N_m$ is abelian and normal,\nwith quotient $({\\cal U}\/N_m)\/(N\/N_m) \\cong {\\cal U}\/N$ being a $2$-group, and so $G_m$ is solvable,\nby Proposition~\\ref{solvable}.\n\nNext, we use Proposition~\\ref{quotient criterion} to prove that the triple $(\\bar a, \\bar b, \\bar c)$ of\nimages of $a,b,c$ in $G_m$ satisfies the chiral form of the intersection condition.\nTo do this, we observe that the group with presentation $\\lg \\, u,v \\ | \\ u^4 = v^4 = (uv)^2 =(u^{2}v^{2})^4=1 \\, \\rg$\nhas order $2^7 = 128$, as does its image in the group $G_1 = {\\cal U}\/N_1$ under the epimorphism\ntaking $(u,v) \\mapsto (aN_1,bN_1)$. These claims are easily verifiable using {\\sc Magma}.\nThen since $G_1 = {\\cal U}\/N_1$ is a quotient of $G_m = {\\cal U}\/N_m$, the subgroup generated\nby $\\bar a$ and $\\bar b$ in $G_m = {\\cal U}\/N_m$ must have order $128$ as well,\nand hence the restriction to $\\lg \\bar a, \\bar b \\rg$ of the natural homomorphism from $G_m$ to $G_1$ is injective,\nas required.\n\nAccordingly, $G_m$ is the rotation group of a chiral or regular 4-polytope $\\mathcal {P}_m$ of type $\\{4,4,4\\}$.\nAssume for the moment that $\\mathcal {P}_m$ is regular. Then there exists an automorphism $\\rho$ of $G_m$\ntaking $(\\bar a, \\bar b, \\bar c)$ to $(\\bar a^{-1}, \\bar a^{\\,2}\\bar b, \\bar c)$, and it follows from the\nrelation $1 = \\overline{a^2c^2b^2(ac)^2} = \\bar a^2 \\bar c^2 \\bar b^2 (\\bar a \\bar c)^2$ that also\n$1 = (\\bar a^2 \\bar c^2 \\bar b^2 (\\bar a \\bar c)^2)^\\rho = \\bar a^{-2} \\bar c^{2} (\\bar a^{2} \\bar b)^2 (\\bar a^{-1} \\bar c)^2$ in $G_m$.\nBut the image of this element in $G_1$ has order 2, and as this is also the image of $a^{-2}c^{2}(a^{2}b)^{2}(a^{-1}c)^2)$ in $G_1$,\nit follows that $(\\bar a^2 \\bar c^2 \\bar b^2 (\\bar a \\bar c)^2)^\\rho = \\bar a^{-2} \\bar c^{2} (\\bar a^{2} \\bar b)^2 (\\bar a^{-1} \\bar c)^2$\nis non-trivial in $G_m$, a contradiction. \n\n\\smallskip\nThus $\\mathcal {P}_m$ is chiral, with automorphism group $G_m$ of order $1024m^2$.\n\n\\medskip\n\\noindent Case (2): Take $K$ as the subgroup of ${\\cal U}$ generated by $z = (bc^{-1})^4$ and $w = (c^{-1}b)^4$.\n\n\\smallskip\nAnother computation with {\\sc Magma} shows that $K$ is normal in ${\\cal U}$, with index $2048$,\nand moreover, the {\\tt Rewrite} command tells us that $K$ is free abelian of rank $2$. \nIn this case, the defining relations for ${\\cal U}$ give \\\\[-18pt]\n\\begin{center}\n\\begin{tabular}{lll}\n$a^{-1}za = z^{-1}$, \\qquad & $b^{-1}zb = w$ \\quad & and \\quad $c^{-1}zc = w$, \\\\[+2pt]\n$a^{-1}wa = w$, \\qquad & $b^{-1}wb = z^{-1}$ \\quad & and \\quad $c^{-1}wc = z^{-1}$. \\\\[-6pt]\n\\end{tabular}\n\\end{center}\nThe quotient ${\\cal U}\/K$ is the automorphism group of the chiral 4-polytope of type $\\{4,4,4\\}$\nwith 2048 automorphisms found by Zhang in~\\cite{Zhang'sPhdThesis}.\n\n\\smallskip\nNow for any positive integer $m$, let $K_m$ be the subgroup generated by $z^m = (bc^{-1})^{4m}$\nand $w^m = (c^{-1}b)^{4m}$. Using Proposition~\\ref{freeabelian}, we find that $K_m$ is characteristic in $K$\nand hence normal in ${\\cal U}$, with index $|\\,{\\cal U}:K_m| = |\\,{\\cal U}:K||K:K_m| = 2048m^2$.\nAlso the quotient $H_m = {\\cal U}\/K_m$ is solvable, again by Proposition~\\ref{solvable}.\n\nNext, the image of the subgroup generated by $a$ and $b$ in $H_1 = {\\cal U}\/K$ has order $128$,\nso just as in Case (1) above, we can apply Proposition~\\ref{quotient criterion} and find that the triple $(\\bar a, \\bar b, \\bar c)$\nof images of $a,b,c$ in $H_m = {\\cal U}\/K_m$ satisfies the chiral form of the intersection condition.\n\nThus $H_m$ is the rotation group of a chiral or regular 4-polytope $\\mathcal {Q}_m$ of type $\\{4,4,4\\}$.\n\n\\smallskip\nMoreover, the same argument as used in Case (1) shows that $\\mathcal {Q}_m$ is chiral, because\nthe image in $H_1$ of the element $a^{-2}c^{2}(a^{2}b)^{2}(a^{-1}c)^2)$ has order $2$,\nand hence the image of that element in $H_m$ is non-trivial.\n\n\\smallskip\nThus $\\mathcal {Q}_m$ is chiral, with automorphism group $H_m$ of order $2048m^2$.\n \\hfill\\mbox{\\raisebox{0.7ex}{\\fbox{}}} \\vspace{4truemm}\n\n\nAs a special case we have the following Corollary, which is an immediate consequence of Theorem~\\ref{maintheorem}\nwhen $m$ is taken as a power of $2$.\n\n\\begin{cor}\nThere exists a chiral $4$-polytope of type $\\{4,4,4\\}$ with automorphism group of order $2^n,$\nfor every integer $n \\ge 10$.\n\\end{cor}\n\nOn the other hand, an inspection of the lists at \\cite{atles1} shows that there exists no \nsuch chiral polytope with automorphism group of order $2^n$, where $n \\le 9$. \n\n\\newpage\\noindent\n{\\bf\\Large Acknowledgements}\n\n\\medskip\\smallskip\n\\noindent\nThe first author acknowledges the hospitality of Beijing Jiaotong University,\nand partial support from the N.Z. Marsden Fund (project UOA1626). \nThe second and the third authors acknowledge the partial support from \nthe National Natural Science Foundation of China (11731002) and the 111 Project of China (B16002). The three authors also acknowledge the extensive use of the {\\sc Magma} system~\\cite{BCP97} in helping conduct experiments that helped show the way towards the main theorem.\n\\\\[-12pt]\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nConsider the problem of distribution-free predictive inference: given dataset $D_n \\equiv \\{(X_i, Y_i)\\}_{i=1}^n$ drawn i.i.d. from $P_{XY} = P_X \\times P_{Y|X}$ on $\\smash{\\mathcal{X} \\times \\mathcal{Y}}$, and $\\smash{X_{n+1} \\sim P_X}$, we must construct a prediction set $C(D_n,\\alpha,X_{n+1}) \\equiv C(X_{n+1})$ for $Y_{n+1}$ such that\n\\begin{equation}\\label{eq:marginal-validity}\n\\text{ for any distribution } P_{XY}, ~ \\mathbb{P}(Y_{n+1} \\in C(X_{n+1})) \\geq 1-\\alpha, \n\\end{equation}\nwhere the probability is taken over all $n+1$ points and $\\alpha \\in (0,1)$ is a predefined confidence level. In other words, how do we construct $C(X)$ with nonasymptotic coverage that holds without assumptions on $P_{XY}$? \nMethods with property~\\eqref{eq:marginal-validity} are called \\emph{marginally} valid to differentiate them from \\emph{conditional} validity:\n\\[\n\\forall P_{XY},~ \\mathbb{P}(Y_{n+1} \\in C(X_{n+1}) \\mid X_{n+1}=x) \\geq 1-\\alpha, \\text{ for $P_X$-almost all $x$.} \n\\]\nConditional validity is known to be impossible without assumptions on $P_{XY}$; see~\\citet[Section 2.6.1]{balasubramanian2014conformal} and~\\citet{barber2019limits}. Hence, we will only be concerned with developing methods that are (provably) marginally valid, but have reasonable conditional coverage in practice (ie, empirically).\n\n\n\nConformal prediction is a general and universal framework for constructing marginally valid prediction sets\\footnote{The spaces $\\mathcal{X}$ and $\\mathcal{Y}$ are without restriction. For example, take $\\mathcal{X} \\equiv \\mathbb{R}^d$ and let $\\mathcal{Y}$ be a subset of $\\mathbb{R}$, or a discrete space such as in multiclass classification. Though it is not formally required, it may be helpful think of $\\mathcal{Y}$ as a totally ordered set or a metric space.\n}. It acts like a wrapper around any prediction algorithm; in other words, any black-box prediction algorithm can be ``conformalized'' to produce valid prediction sets instead of point predictions. \nWe refer the reader to the works of~\\citet{vovk2005algorithmic} and~\\citet{balasubramanian2014conformal} for details.\nIn this paper, we provide an alternate and equivalent viewpoint for accomplishing the same goals, that we call \\emph{nested conformal prediction}.\n\nConformal prediction starts from a nonconformity score. As we will see with explicit examples later, these nonconformity scores are often rooted in some underlying geometric intuition about how a good prediction set may be discovered from the data. Nested conformal acknowledges this geometric intuition and makes it explicit: instead of starting from a score, it instead starts from a sequence of all possible prediction sets $\\{\\mathcal{F}_t(x)\\}_{t\\in\\mathcal{T}}$ for some ordered set $\\mathcal{T}$. Just as we suppressed the dependence of set $C(\\cdot)$ on the labeled data $D_n$ in property \\eqref{eq:marginal-validity}, here too $\\mathcal{F}_t(\\cdot)$ will actually depend on $D_n$ but we suppress this for notational simplicity. \n\nThese prediction sets are `nested', that is, for every $t_1 \\leq t_2 \\in \\mathcal{T}$, we require that $\\mathcal{F}_{t_1}(x) \\subseteq \\mathcal{F}_{t_2}(x)$; also $\\mathcal{F}_{\\inf \\mathcal{T}}=\\emptyset$ and $\\mathcal{F}_{\\sup \\mathcal{T}}=\\mathcal{Y}$. Thus large values of $t$ correspond to larger prediction regions. Given a tolerance $\\alpha \\in [0, 1]$, we wish to identify the smallest $t \\in \\mathcal{T}$ such that \n\\[\\mathbb{P}(Y \\in \\mathcal{F}_{t}(X)) \\geq 1 - \\alpha. \n\\]\nIn a nutshell, nested conformal learns a data-dependent mapping $\\alpha \\to t(\\alpha)$ using the conformal principle. Note that the mapping must be decreasing; lower tolerance values $\\alpha$ naturally lead to larger prediction regions. \n\n\n\nWe illustrate the above principle in a simple setting involving split\/inductive conformal prediction \\cite{papadopoulos2002inductive}, \\cite{lei2018distribution},~\\cite[Chapter 2.3]{balasubramanian2014conformal}. First, split $D_n$ into a training set $D_1 \\equiv \\{(X_i, Y_i)\\}_{1\\le i\\le m}$ and a calibration set $D_2 \\equiv \\{(X_i, Y_i)\\}_{m < i\\le n}$. Using $D_1$, construct an estimate $\\widehat{\\mu}(\\cdot)$ of the conditional mean of $Y$ given $X$. Then construct the nonconformity score as the residuals of $\\widehat{\\mu}$ on $D_2$: \n$R_i := |Y_i - \\widehat{\\mu}(X_i)|, \\text{for } i \\in D_2.$\nFinally, define\n\\[\nC(X_{n+1}) = \\Big\\{y\\in\\mathbb{R}:\\,|y - \\widehat{\\mu}(X_{n+1})| < Q_{1-\\alpha}(\\{R_i\\}_{i\\in D_2})\\Big\\},\n\\] \nwhere $Q_{1-\\alpha}(A)$ for a finite set $A$ represents the $(1-\\alpha)$-th quantile of elements in $A$ (defined later). The above set is easily shown to be marginally valid due to exchangeability of order statistics.\n\nWe now give an alternate derivation of the above set using nested conformal:\n\\begin{enumerate}\n\t\\item After learning $\\widehat \\mu$ using $D_1$ (as done before), construct a sequence of prediction regions corresponding to symmetric intervals around $\\widehat{\\mu}(x)$: \n \\[\n \\{\\mathcal{F}_t(x)\\}_{t\\ge0} := \\{[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]:t\\ge0\\}.\n \\] \n Note that $\\mathcal{F}_t(x)$ is a random set since it is based on $\\widehat{\\mu}(x)$, and in our earlier notation, we take $\\mathcal{T}=[0,\\infty]$.\n It is clear that regardless of $\\widehat{\\mu}$, for any distribution of the random vector $(X, Y)$, and for any $\\alpha \\in [0,1]$, there exists a (minimal) $t = t(\\alpha)$ such that\n\n$\\mathbb{P}(Y\\in\\mathcal{F}_t(X)) \\ge 1-\\alpha.$\n\n\tHence we can rewrite our nested family $\\{\\mathcal{F}_t(x)\\}$ as\n\n\t$$\\Big\\{[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]:t\\ge0\\Big\\} = \\Big\\{[\\widehat{\\mu}(x) - t(\\alpha), \\widehat{\\mu}(x) + t(\\alpha)]:\\alpha\\in[0,1]\\Big\\}.$$\n\n\t\\item The only issue now is that we do not know the map $\\alpha\\mapsto t(\\alpha)$, that is, given $\\alpha$ we do not know which of these prediction intervals to use. Hence we use the calibration data to ``estimate'' the map $\\alpha\\to t(\\alpha)$. This is done by finding the smallest $t$ such that $\\mathcal{F}_t(x)$ contains $Y_i$ for at least $1-\\alpha$ fraction of the calibration points $(X_i, Y_i)$ (we provide formal details later). Because the sequence $\\{\\mathcal{F}_t(x)\\}$ is increasing in $t$, finding the smallest $t$ leads to the smallest prediction set within the nested family. \n\\end{enumerate}\n\nEmbedding these scores into our nested framework not only allows for easy comparison (like in Table~\\ref{tab:summary_literature}), but allows us to extend these methods beyond the split\/inductive conformal setting that they were originally derived in. Specifically, we seamlessly derive cross-conformal or jackknife+ variants of these and other methods, including our new method called QOOB (pronounced cube). \n\nA final reason that the assumption of nestedness is natural is the fact that the optimal prediction sets are nested: Suppose $Z_1, \\ldots, Z_{n}$ are exchangeable random variables with a common distribution that has density $p(\\cdot)$ with respect to some underlying measure. The ``oracle'' prediction set~\\citep{lei2013distribution} for a future observation $Z_{n+1}$ is given by the \\emph{level set} of the density with valid coverage, that is, $\\{z:p(z) \\ge t(\\alpha)\\}$ with $t(\\alpha)$ defined by largest $t$ such that $\\mathbb{P}(p(Z_{n+1}) \\ge t) \\ge 1-\\alpha$. Because $\\{z:p(z) \\ge t\\}$ is decreasing with $t$, $\\{z:p(z) \\ge t(\\alpha)\\}$ is decreasing with $\\alpha\\in[0,1]$, forming a nested sequence of sets.\n\n\n\\subsection{Organization and contributions}\nFor simplicity, our discussion will assume that $\\mathcal{Y} \\subseteq \\mathbb{R}$, but all ideas are easily extended to other prediction settings. The paper is organized as follows:\n\\begin{enumerate}\n\\item In Section~\\ref{sec:split-conformal}, we formalize the earlier discussion and present split\/inductive conformal~\\citep{papadopoulos2002inductive,lei2018distribution} in the language of nested conformal prediction, and translate various conformity scores developed in the literature for split conformal into nested prediction sets. \n\\item In Section~\\ref{sec:Jackknife-plus}, we rephrase the jackknife+~\\citep{barber2019predictive} (and cross-conformal prediction~\\citep{vovk2015cross}) in terms of the nested framework. \nThis allows the jackknife+ to use many recent score functions, such as those based on quantiles, which were originally developed and deployed in the split framework. \nParticularly, in subsection~\\ref{subsec:compuation-cross} we provide an efficient implementation of cross-conformal that matches the jackknife+ prediction time for a large class of nested sets that includes all standard nested sets. \n\n\\item In Section~\\ref{sec:OOB-conformal}, we extend the Out-of-Bag conformal~\\citep{johansson2014regression} and jackknife+ after bootstrap~\\citep{kim2020predictive} methods to our nested framework. These are based on ensemble methods such as random forests, and are relatively computationally efficient because only a single ensemble needs to be built. \n\\item In Section~\\ref{sec:QOOB}, we consolidate the ideas developed in this paper to construct a novel conformal method called QOOB (Quantile Out-of-Bag, pronounced cube), that is both computationally and statistically ``efficient.'' \nQOOB combines four ideas: quantile regression, cross-conformalization, ensemble methods and out-of-bag predictions. While being theoretically valid, Section~\\ref{sec:numerical-experiments} demonstrates QOOB's excellent empirical performance. \n\\end{enumerate}\n\n\nIn Appendix \\ref{appsec:k-fold} we derive K-fold variants of jackknife+\/cross-conformal in the nested framework and in Appendix \\ref{appsec:sampling-nested}, we develop the other aggregated conformal methods of subsampling and bootstrap in the nested framework. In Appendix \\ref{appsec:empty-case}, we discuss cross-conformal computation and the jackknife+ in the case when our nested sequence could contain empty sets. This is a subtle but important issue to address when extending these methods them to quantile-based nested sets of \\citet{romano2019conformalized}, and thus relevant to QOOB as well. In Appendix~\\ref{appsec:optimal-prediction-set} we interpret optimal conditionally-valid prediction sets (which are the level sets of the conditional density function) in the nested framework. Finally, Appendix \\ref{appsec:proofs} contains all proofs.\n\n\\section{Split conformal based on nested prediction sets}\\label{sec:split-conformal}\nIn the introduction, we showed that in a simple regression setup with the nonconformity scores as held-out residuals, split conformal intervals can be naturally expressed in terms of nested sets. Below, we introduce the general nested framework and recover the usual split conformal method with general scores using this framework. We show how existing nonconformity scores in literature exhibit natural re-interpretations in the nested framework. The following description of split conformal follows descriptions by \\citet{papadopoulos2002inductive} and~\\citet{lei2018distribution} but rewrites it in terms of nested sets. \n\nSuppose $(X_i, Y_i)\\in\\mathcal{X}\\times\\mathcal{Y}, i\\in[n]$ denotes the training dataset. Let $[n] = \\mathcal{I}_1\\cup\\mathcal{I}_2$ be a partition of $[n]$. For $\\mathcal{T}\\subseteq\\mathbb{R}$ and each $x\\in\\mathcal{X}$, let $\\{\\mathcal{F}_t(x)\\}_{t\\in\\mathcal{T}}$ (with $\\mathcal{F}_t(x)\\subseteq\\mathcal{Y}$) denote a nested sequence of sets constructed based on the first split of training data $\\{(X_i, Y_i):i\\in\\mathcal{I}_1\\}$, that is, $\\mathcal{F}_t(x) \\subseteq \\mathcal{F}_{t'}(x)$ for $t \\le t'$. The sets $\\mathcal{F}_t(x)$ in almost all examples are random through the training data, although they are not required to be random. Consider the score \n\\begin{equation}\\label{eq:Score-definition}\nr(x, y) ~:=~ \\inf\\{t\\in\\mathcal{T}:\\,y\\in\\mathcal{F}_t(x)\\},\n\\end{equation}\nwhere $r$ is a mnemonic for ``radius'' and $r(x,y)$ can be informally thought of as the smallest ``radius'' of the set that captures $y$ (and perhaps thinking of a multivariate response, that is $\\mathcal{Y}\\subseteq \\mathbb{R}^d$, and $\\{\\mathcal{F}_t(x)\\}$ as representing appropriate balls\/ellipsoids might help with that intuition).\nDefine the scores for the second split of the training data $\\{r_i = r(X_i, Y_i)\\}_{i\\in\\mathcal{I}_2}$ and set\n\\[\nQ_{1-\\alpha}(r, \\mathcal{I}_2) ~:=~ \\lceil (1-\\alpha)(1 + 1\/|\\mathcal{I}_2|)\\rceil\\mbox{-th quantile of }\\{r_i\\}_{i\\in\\mathcal{I}_2}.\n\\]\n(that is, $Q_{1-\\alpha}(r, \\mathcal{I}_2)$ is the $\\lceil(1-\\alpha)(1 + 1\/|\\mathcal{I}_2|)\\rceil$-th largest element of the set $\\{r_i\\}_{i\\in\\mathcal{I}_2}$). \nThe final prediction set is given by \n\\begin{equation}\nC(x) ~:=~ \\mathcal{F}_{Q_{1-\\alpha}(r,\\mathcal{I}_2)}(x) = \\{y \\in \\mathcal{Y}:\\,r(x, y) \\le Q_{1-\\alpha}(r, \\mathcal{I}_2)\\}.\n\\label{eq:Nested-prediction-set}\n\\end{equation}\nThe following well known sample coverage guarantee holds true \\citep{papadopoulos2002inductive,lei2018distribution}.\n\\begin{prop}\\label{thm:Conformal-main-result}\nIf $\\{(X_i, Y_i)\\}_{i\\in[n]\\cup\\{n+1\\}}$ are exchangeable, then the prediction set $C(\\cdot)$ in \\eqref{eq:Nested-prediction-set} satisfies \n\\[\n\\mathbb{P}\\left(Y_{n+1}\\in C(X_{n+1})\\mid \\{(X_i, Y_i):i\\in\\mathcal{I}_1\\}\\right) ~\\ge~ 1-\\alpha.\n\\]\nIf the scores $\\{r_i, i\\in\\mathcal{I}_2\\}$ are almost surely distinct, then $C(\\cdot)$ also satisfies\n\\begin{equation}\\label{eq:Upper-bound}\n\\mathbb{P}\\left(Y_{n+1}\\in C(X_{n+1})\\mid \\{(X_i, Y_i):i\\in\\mathcal{I}_1\\}\\right) ~\\le~ 1-\\alpha + \\frac{1}{|\\mathcal{I}_2|+1}.\n\\end{equation}\n\\end{prop}\nSee Appendix \\ref{appsec:split-conformal} for a proof. \nEquation~\\eqref{eq:Score-definition} is the key step that converts a sequence of nested sets $\\{\\mathcal{F}_t(x)\\}_{t\\in\\mathcal{T}}$ into a nonconformity score $r$. Through two examples, we demonstrate how natural sequences of nested sets in fact give rise to standard nonconformity scores considered in literature, via equation~\\eqref{eq:Score-definition}. \n\\begin{enumerate}\n \\item \\textbf{Split\/Inductive Conformal \\citep{papadopoulos2002inductive,lei2018distribution}.} Let $\\widehat{\\mu}_{\\mathcal{I}_1}(\\cdot)$ be an estimator of the regression function $\\mathbb{E}[Y|X]$ based on $(X_i, Y_i), i\\in\\mathcal{I}_1$, and consider nested sets corresponding to symmetric intervals around the mean estimate: \n \n\n\n\n \\[\n \\mathcal{F}_t(x) ~:=~ [\\widehat{\\mu}_{\\mathcal{I}_1}(x) - t, \\widehat{\\mu}_{\\mathcal{I}_1}(x) + t], \\ t \\in \\mathcal{T} = \\mathbb{R}^+.\n \\]\n Observe now that\n \\begin{align*}\n \\inf\\{t\\ge0:y\\in\\mathcal{F}_t(x)\\} &= \\inf\\{t\\ge0:\\widehat{\\mu}_{\\mathcal{I}_1}(x) - t \\le y\\le \\widehat{\\mu}_{\\mathcal{I}_1}(x) + t\\}\\\\\n &= \\inf\\{t\\ge0:\\, -t \\le y - \\widehat{\\mu}_{\\mathcal{I}_1}(x) \\le t\\} ~=~ |y - \\widehat{\\mu}_{\\mathcal{I}_1}(x)|,\n \\end{align*}\n which is exactly the nonconformity score of split conformal. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \\item \\textbf{Conformalized Quantiles \\citep{romano2019conformalized}.} \nFor any $\\beta \\in (0,1)$, let the function $q_\\beta(\\cdot)$ be the conditional quantile function. Specifically, for each $x$, define $q_{\\beta}(x):= \\sup\\{a \\in \\mathbb{R}: \\mathbb{P}(Y \\le a \\mid X = x) \\le \\beta$\\}.\n Let $\\widehat{q}_{\\alpha\/2}(\\cdot), \\widehat{q}_{1-\\alpha\/2}(\\cdot)$ be any conditional quantile estimators based on $(X_i, Y_i),i\\in\\mathcal{I}_1$.\n \n \n If the quantile estimates are good, we hope that $\\mathbb{P}(Y \\in [\\widehat{q}_{\\alpha\/2}(X), \\widehat{q}_{1-\\alpha\/2}(X)]) \\approx 1-\\alpha$, but this cannot be guaranteed in a distribution-free or assumption lean manner. However, it may be possible to achieve this with a symmetric expansion or shrinkage of the interval $[\\widehat{q}_{\\alpha\/2}(X), \\widehat{q}_{1-\\alpha\/2}(X)]$ (assuming $\\widehat{q}_{\\alpha\/2}(X) \\leq \\widehat{q}_{1 - \\alpha\/2}(X)$). \n \n Following the intuition, consider\n \\begin{equation}\\label{eq:CQR}\n \\mathcal{F}_t(x) ~:=~ [\\widehat{q}_{\\alpha\/2}(x) - t, \\widehat{q}_{1-\\alpha\/2}(x) + t], \\ t\\in\\mathbb{R}.\n \\end{equation}\n\n Note that the sets in~\\eqref{eq:CQR} are increasing in $t$ if $\\widehat{q}_{\\alpha\/2}(x) \\le \\widehat{q}_{1-\\alpha\/2}(x)$, and\n \\begin{align*}\n \\inf\\{t\\in\\mathbb{R}:\\,y\\in\\mathcal{F}_t(x)\\} &= \\inf\\{t\\in\\mathbb{R}:\\,\\widehat{q}_{\\alpha\/2}(x) - t \\le y \\le \\widehat{q}_{1-\\alpha\/2}(x) + t\\}\\\\\n \n &= \\max\\{\\widehat{q}_{\\alpha\/2}(x) - y, y - \\widehat{q}_{1-\\alpha\/2}(x)\\}.\n \\end{align*}\n Hence $r(X_i, Y_i) = \\max\\{\\widehat{q}_{\\alpha\/2}(X_i) - Y_i,\\,Y_i - \\widehat{q}_{1-\\alpha\/2}(X_i)\\}$ for $i\\in\\mathcal{I}_2$. This recovers exactly the nonconformity score proposed by \\citet{romano2019conformalized}. We believe that it is more intuitive to start with the shape of the predictive region, like we did here, than a nonconformity score. In this sense, nested conformal is a formalized technique to go from statistical\/geometric intuition about the shape of the prediction region to a nonconformity score. \n \n \n\\end{enumerate}\nSee Table~\\ref{tab:summary_literature} for more translations between scores and nested sets. \nAll the examples discussed in this section and listed in Table~\\ref{tab:summary_literature} (except the one from~\\cite{izbicki2019distribution}) result in interval prediction sets. However, optimal prediction sets need not be intervals and furthermore, intervals are not suitable when the response space is not convex (e.g., classification). Optimal predictions sets that are conditionally valid are obtained as level sets of the conditional densities~\\citep{izbicki2019distribution} and are discussed through the nested framework in Appendix~\\ref{appsec:optimal-prediction-set}. \n\n\\begin{table}[h!]\n\\caption{Examples from the literature covered by nested conformal framework. The methods listed are split conformal, locally weighted conformal, CQR, CQR-m, CQR-r, distributional conformal and conditional level-set conformal. Functions $\\widehat{q}_{a}$ represents a conditional quantile estimate at level $a$, and $\\widehat{f}$ represents a conditional density estimate.}\n\\label{tab:summary_literature}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{llll}\\hline\nReference & $\\mathcal{F}_t(x)$ & $\\mathcal{T}$ & Estimates \\\\\\hline\\hline\n\\citet{lei2018distribution} & $[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]$ & $[0, \\infty)$ & $\\widehat{\\mu}$ \\\\\n\\citet{lei2018distribution} & $[\\widehat{\\mu}(x) - t\\widehat{\\sigma}(x),\\widehat{\\mu}(x) + t\\widehat{\\sigma}(x)]$ & $[0, \\infty)$ & $\\widehat{\\mu}, \\widehat{\\sigma}$ \\\\\n\\citet{romano2019conformalized} & $[\\widehat{q}_{\\alpha\/2}(x) - t, \\widehat{q}_{1-\\alpha\/2}(x) + t]$ & $(-\\infty, \\infty)$ & $\\widehat{q}_{\\alpha\/2}, \\widehat{q}_{1-\\alpha\/2}$ \\\\\n\\citet{kivaranovic2019adaptive} & $(1+t)[\\widehat{q}_{\\alpha\/2}(x), \\widehat{q}_{1-\\alpha\/2}(x)] - t\\widehat{q}_{1\/2}(x)$ & $(-\\infty, \\infty)$ & $\\widehat{q}_{\\alpha\/2}, \\widehat{q}_{1-\\alpha\/2}, \\widehat{q}_{1\/2}$ \\\\\n\\citet{sesia2019comparison} & $[\\widehat{q}_{\\alpha\/2}(x), \\widehat{q}_{1-\\alpha\/2}(x)] \\pm t(\\widehat{q}_{1-\\alpha\/2}(x) - \\widehat{q}_{\\alpha\/2}(x))$ & $(-1\/2,\\infty)$ & $\\widehat{q}_{\\alpha\/2}, \\widehat{q}_{1-\\alpha\/2}$ \\\\\n\\citet{chernozhukov2019distributional} & $[\\widehat{q}_{t}(x), \\widehat{q}_{1-t}(x)]$ & $(0,1\/2)$ & $\\{\\widehat{q}_{\\alpha}\\}_{\\alpha\\in[0,1]}$\\\\\n\\citet{izbicki2019distribution} & $\\{y:\\widehat{f}(y|x) \\geq \\widecheck{t}_{\\delta}(x)\\}$\\tablefootnote{$\\ \\widecheck{t}_{\\delta}(x)$ is an estimator of $t_{\\delta}(x)$, where $t_{\\delta}(x)$ is defined the largest $t$ such that $\\mathbb{P}(f(Y|X) \\ge t_{\\delta}(X)\\big|X = x) \\ge 1 - \\delta$; see~\\citep[Definition 3.3]{izbicki2019distribution} for details.} & $[0,1]$ & $\\widehat{f}$\\\\\n\\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nSplit conformal prediction methods are often thought of as being statistically inefficient because they only make use of one split of the data for training the base algorithm, while the rest is held-out for calibration. Recently many extensions have been proposed~\\citep{carlsson2014aggregated,vovk2015cross,johansson2014regression,bostrom2017accelerating,barber2019predictive,kim2020predictive} that make use all of the data for training. \nAll of these methods can be rephrased easily in terms of nested sets; we do so in Sections~\\ref{sec:Jackknife-plus} and \\ref{sec:OOB-conformal}. This understanding also allows us to develop our novel method QOOB in Section~\\ref{sec:QOOB}.\n\n\n\\section{Cross-conformal and Jackknife+ using nested sets}\\label{sec:Jackknife-plus}\n\nIn the previous section, we used a part of training data to construct the nested sets and the remaining part to calibrate them for finite sample validity. This procedure, although computationally efficient, can be statistically inefficient due to the reduction of the sample size used for calibrating. Instead of splitting into two parts, it is statistically more efficient to split the data into multiple parts. In this section, we describe such versions of nested conformal prediction sets and prove their validity.\nThese versions in the score-based conformal framework are called cross-conformal prediction and the jackknife+, and were developed by~\\citet{vovk2015cross} and~\\citet{barber2019predictive}, but the latter only for a specific score function.\n\n\\subsection{Rephrasing leave-one-out cross-conformal using nested sets}\nWe now derive leave-one-out cross-conformal in the language of nested prediction sets.\nSuppose $\\{\\mathcal{F}_{t}^{-i}(x)\\}_{t\\in\\mathcal{T}}$ for each $x\\in\\mathcal{X},\\,i\\in[n]$ denotes a collection of nested sets constructed based only on $\\{(X_j, Y_j)\\}_{\\,j\\in[n]\\setminus\\{i\\}}$. We assume that each set $\\mathcal{F}_{t}^{-i}(x)$ is constructed invariantly to permutations of the input points $\\{(X_j, Y_j)\\}_{\\,j\\in[n]\\setminus\\{i\\}}$. (Note that this is an additional assumption compared to the split conformal version). The $i$-th nonconformity score $r_i$ induced by these nested sets is defined as $r_i(x, y) ~=~ \\inf\\{t\\in\\mathcal{T}:\\,y\\in\\mathcal{F}_t^{-i}(x)\\}.$\n\\ifx false \nThe residual for point $(X_i, Y_i)$ is defined as\n\\[\nr_i(X_i, Y_i) ~:=~ \\inf\\{t\\in\\mathcal{T}:\\,Y_i \\in \\mathcal{F}_t^{-i}(X_i)\\}.\n\\]\n\\fi \nThe leave-one-out cross-conformal prediction set is given by\n\\begin{equation}\\label{eq:LOO}\n{C}^{\\texttt{LOO}}(x) ~:=~ \\left\\{y\\in\\mathbb{R}:\\,\\sum_{i=1}^n \\mathbbm{1}\\{r_i(X_i, Y_i) < r_i(x, y)\\} < (1-\\alpha)(n+1)\\right\\}.\n\\end{equation}\nFor instance, given a conditional mean estimator $\\widehat{\\mu}^{-i}(\\cdot)$ trained on $\\{(X_j, Y_j)\\}_{\\,j\\in[n]\\setminus\\{i\\}}$, we can consider the nested sets $\\mathcal{F}_t^{-i}(x) = [\\widehat{\\mu}^{-i}(x) - t, \\widehat{\\mu}^{-i}(x) + t]$ to realize the absolute deviation residual function $r_i(x, y) = |y - \\widehat{\\mu}^{-i}(x)|$.\nWe now state the coverage guarantee that ${C}^{\\texttt{LOO}}(\\cdot)$ satisfies. \n\n\\begin{thm}\\label{thm:validity-nested-conformal-jackknife-plus}\nIf $\\{(X_i, Y_i)\\}_{i\\in[n+1]}$ are exchangeable and the sets $\\mathcal{F}_{t}^{-i}(x)$ constructed based on $\\{(X_j, Y_j)\\}_{ j\\in[n]\\setminus\\{i\\}}$ are invariant to their ordering, then\n\\[\n\\mathbb{P}(Y_{n+1}\\in{C}^{\\texttt{LOO}}(X_{n+1})) ~\\ge~ 1 - 2\\alpha.\n\\]\n\\end{thm}\nSee Appendix \\ref{appsec:Jackknife-plus} for a proof\n of Theorem~\\ref{thm:validity-nested-conformal-jackknife-plus}, which follows the proof of Theorem 1 in~\\citet{barber2019predictive} except with the new residual defined based on nested sets. In particular, Theorem~\\ref{thm:validity-nested-conformal-jackknife-plus} applies when the nested sets are constructed using conditional quantile estimators as in the conformalized quantile example discussed in Section~\\ref{sec:split-conformal}. The discussion in this section can be generalization to cross-conformal and the CV+ methods of~\\citet{vovk2015cross} and~\\citet{barber2019predictive}, which construct K-fold splits of the data and require training an algorithm only $n\/K$ times (instead of $n$ times in the leave-one-out case). These are discussed in the nested framework in Appendix \\ref{appsec:k-fold}.\n \nIn a regression setting, one may be interested in constructing prediction sets that are intervals (since they are easily interpretable), whereas $C^{\\texttt{LOO}}(x)$ need not be an interval in general. Also, it is not immediately evident how one would algorithmically compute the prediction set defined in~\\eqref{eq:LOO} without trying out all possible values $y \\in \\mathcal{Y}$. We discuss these concerns in subsections~\\ref{subsec:jackknife+} and \\ref{subsec:compuation-cross}. \n\\subsection{Prediction intervals containing $C^{\\texttt{LOO}}(x)$ and jackknife+}\\label{sec:intervals-LOO}\n\\label{subsec:jackknife+}\nIn a regression setting, prediction intervals may be more interpretable or `actionable' than prediction sets that are not intervals. To this end, intervals that contain $C^{\\texttt{LOO}}(x)$ are good candidates for prediction intervals since they inherit the coverage validity of Theorem~\\ref{thm:validity-nested-conformal-jackknife-plus}. For the residual function $r_i(x, y) = |y - \\widehat{\\mu}^{-i}(x)|$, \\citet{barber2019predictive} provided an interval that always contains $C^{\\texttt{LOO}}(x)$, called the jackknife+ prediction interval. In this section, we discuss when the jackknife+ interval can be defined for general nested sets. Whenever jackknife+ can be defined, we argue that another interval can be defined that contains $C^\\texttt{LOO}(x)$ and is guaranteed to be no longer in width than the jackknife+ interval\n\nIn the general formulation of nested sets, an analog of the jackknife+ interval may not always exist. However, in the special case when the nested sets $\\mathcal{F}_t^{-i}(x)$ are themselves either intervals or empty sets, an analog of the jackknife+ interval can be derived. Note that all the examples listed in Table~\\ref{tab:summary_literature} (except for the last one) result in $\\mathcal{F}_t(x)$ being either a nonempty interval or the empty set. For clarity of exposition, we discuss the empty case separately in Appendix \\ref{appsec:empty-case}. Below, suppose $\\mathcal{F}_{r_i(X_i, Y_i)}^{-i}(x) $ is a nonempty interval and define the notation\n\\[\n[\\ell_i(x), u_i(x)]~:=~\\mathcal{F}_{r_i(X_i, Y_i)}^{-i}(x) .\n\\] \nWith this notation, the cross-conformal prediction set can be re-written as \n\\begin{align}\nC^{\\texttt{LOO}}(x) &= \\left\\{y:\\,\\sum_{i=1}^n \\mathbbm{1}\\{y\\notin [\\ell_i(x), u_i(x)]\\} < (1-\\alpha)(n+1)\\right\\} \\nonumber \\\\\n&= \\left\\{y:\\, \\alpha(n+1) - 1 < \\sum_{i= 1}^n \\mathbbm{1}\\{y \\in [\\ell_i(x), u_i(x)]\\} \\right\\}. \\label{eq:loo-interval-definition}\n\\end{align}\nSuppose $y < q_{n,\\alpha}^{-}(\\{\\ell_i(x)\\})$, where \n$q_{n,\\alpha}^{-}(\\{\\ell_i(x)\\})$ denotes the \n$\\lfloor\\alpha(n+1)\\rfloor$-th smallest value of $\\{\\ell_i(x)\\}_{i=1}^n$. Clearly, \n\\[\n\\sum_{i= 1}^n \\mathbbm{1}\\{y \\in [\\ell_i(x), u_i(x)]\\} \\leq \\sum_{i= 1}^n \\mathbbm{1}\\{y \\geq l_i(x)\\} \\leq \\lfloor\\alpha(n+1)\\rfloor - 1,\n\\] \nand hence $y \\notin C^\\texttt{LOO}(x)$. Similarly it can be shown that if $y > q_{n,\\alpha}^{+}(\\{u_i(x)\\})$ (where $q_{n,\\alpha}^{+}(\\{u_i(x)\\})$ denotes the $\\lceil(1-\\alpha)(n+1)\\rceil$-th smallest value of $\\{u_i(x)\\}_{i=1}^n$), $y \\notin C^\\texttt{LOO}(x)$. \nHence, defining\nthe jackknife+ prediction interval as\n\\begin{equation}\nC^{\\texttt{JP}}(x) ~:=~ [q_{n,\\alpha}^{-}(\\{\\ell_i(x)\\}),\\, q_{n,\\alpha}^{+}(\\{u_i(x)\\})],\\label{eq:JP-definition}\n\\end{equation}\nwe conclude\n\\begin{equation}\\label{eq:loo-jp}\nC^{\\texttt{LOO}}(x)~\\subseteq~ C^{\\texttt{JP}}(x) \\quad\\mbox{for all}\\quad x\\in\\mathcal{X}.\n\\end{equation}\nHowever, there exists an even shorter interval containing~$C^{\\texttt{LOO}}(x)$: its convex hull; this does not require the nested sets to be intervals. The convex hull of $C^{\\texttt{LOO}}(x)$ is defined as the smallest interval containing itself. Hence,\n\\begin{equation}\\label{eq:intervals-containing-loo}\nC^{\\texttt{LOO}}(x) ~\\subseteq~ \\mathrm{Conv}(C^{\\texttt{LOO}}(x)) ~\\subseteq~ C^{\\texttt{JP}}(x).\n\\end{equation} \nBecause of~\\eqref{eq:intervals-containing-loo}, the coverage guarantee from Theorem~\\ref{thm:validity-nested-conformal-jackknife-plus} continues to hold for $\\mathrm{Conv}(C^{\\texttt{LOO}}(x))$ and $C^{\\texttt{JP}}(x)$. Interestingly, $C^{\\texttt{LOO}}(x)$ can be empty but $C^{\\texttt{JP}}(x)$ is non-empty if each $\\mathcal{F}_{r_i(X_i, Y_i)}^{-i}(x) $ is non-empty (in particular it contains the medians of $\\{\\ell_i(x)\\}$ and $\\{u_i(x)\\}$). Further, $\\mathrm{Conv}(C^{\\texttt{LOO}}(x))$ can be a strictly smaller interval than $C^{\\texttt{JP}}(x)$; see Subsection~\\ref{subsec:cc-jp} for details.\n\n\\subsection{Efficient computation of the cross-conformal prediction set}\n\\label{subsec:compuation-cross}\nEquation~\\eqref{eq:LOO} defines $C^{\\texttt{LOO}}(x)$ implicitly, and does not address the question of how to compute the mathematically defined prediction set efficiently. If the nested sets $\\mathcal{F}_t^{-i}(x)$ are themselves guaranteed to either be intervals or empty sets, jackknife+ seems like a computationally feasible alternative since it just relies on the quantiles $q_{n,\\alpha}^{-}(\\{\\ell_i(x)\\}),\\, q_{n,\\alpha}^{+}(\\{u_i(x)\\})$ which can be computed efficiently. However, it turns out that $C^{\\texttt{LOO}}(x), \\mathrm{Conv}(C^{\\texttt{LOO}}(x)),$ and $C^{\\texttt{JP}}(x)$ can all be computed in near linear in $n$ time. In this section, we provide an algorithm for near linear time computation of the aforementioned prediction sets. We will assume for simplicity that $\\mathcal{F}_t^{-i}(x)$ is always an interval; the empty case is discussed separately in Appendix \\ref{appsec:empty-case}.\n\nFirst, notice that the inclusion in~\\eqref{eq:LOO} need not be ascertained for every $y \\in \\mathcal{Y}$ but only for a finite set of values in $\\mathcal{Y}$. These values are exactly the ones corresponding to the end-points of the intervals produced by each training point $\\mathcal{F}_{r_i(X_i, Y_i)}^{-i}(x) = [\\ell_i(x), u_i(x)]$. This is because none of the indicators $\\mathbbm{1}\\{r_i(X_i, Y_i) < r_i(x, y)\\}$ change value between two consecutive interval end-points. Since $\\ell_i(x)$ and $u_i(x)$ can be repeated, we define the bag of all these values (see footnote\\footnote{A bag denoted by $\\Lbag \\cdot \\Rbag$ is an unordered set with potentially repeated elements. Bag unions respect the number of occurrences of the elements, eg $\\Lbag 1, 1, 2 \\Rbag \\cup \\Lbag 1, 3 \\Rbag = \\Lbag 1, 1, 1, 2, 3 \\Rbag$. } for $\\Lbag \\cdot \\Rbag$ notation):\n\\begin{equation}\n\\label{eq:y-k-main}\n\\mathcal{Y}^x := \\bigcup_{i=1}^n \\Lbag \\ell_i(x), u_i(x)\\Rbag .\n\\end{equation}\nThus we only need to verify the condition\n\\begin{equation}\n\\label{eq:y-condition}\n\\sum_{i= 1}^n \\mathbbm{1}\\{y \\in [\\ell_i(x), u_i(x)]\\} > \\alpha(n+1) - 1\n\\end{equation}\nfor the $2n$ values of $y \\in \\mathcal{Y}^x$ and construct the prediction sets suitably.\nDone naively, verifying \\eqref{eq:y-condition} itself is an $O(n)$ operation for an overall time of $O(n^2)$. However, \\eqref{eq:y-condition} can be verified for all $y \\in \\mathcal{Y}^x$ in one pass on the sorted $\\mathcal{Y}^x$ for a running time of $O(n\\log n)$; we describe how to do so now\n\\begin{algorithm}[!t]\n\t\\SetAlgoLined\n\t\n\t\\KwInput\n\t\tDesired coverage level $\\alpha$; $\\mathcal{Y}^x$ and $\\mathcal{S}^x$ computed as defined in equations~\\eqref{eq:y-k-main},~\\eqref{eq:s-k-main} using the training data $\\{(X_i, Y_i)\\}_{i=1}^n$, test point $x$ and any sequence of nested sets $\\mathcal{F}^{-i}_t(\\cdot)$ \\;}\n\t\\KwOutput{Prediction set $C^x \\subseteq \\mathcal{Y}$}\n\t\n\t$threshold \\gets \\alpha(n+1) - 1$;\n\tif $threshold < 0$, then return $\\mathbb{R}$ and stop;\\\\\n\n\t\n\n\t$C^x \\gets \\emptyset$; $count \\gets 0$; $left\\_endpoint \\gets 0$\\;\n\t\\For{$i \\gets 1$ \\textbf{to} $|\\mathcal{Y}^x|$}{\n\t\t\\eIf{$s^x_i = 1$}{ \\label{line:condition-1}\n\t\t\t$count \\gets count + 1$\\; \\label{line:increase-count}\n\t\t\t\\If{$count > threshold $ \\textbf{and} $count -1 \\leq threshold$}{ \\label{line:condition-2}\n\t\t\t\t$left\\_endpoint \\gets y^x_i$\\; \\label{line:update-left-endpoint}\n\t\t\t}\n\t\t}{\n\t\t\t\\If{$count > threshold$ \\textbf{and} $count - 1\\leq threshold$}{ \\label{line:condition-3}\n\t\t\t\t$C^x \\gets C^x \\cup \\{[left\\_endpoint, y^x_i]\\}$\\; \\label{line:update-prediction-set}\n\t\t\t}\n\t\t\t$count \\gets count - 1$\\; \\label{line:decrease-count}\n\t\t}\n\t}\n\t\\Return $C^x$\\;\n\t\\caption{Efficient cross-conformal style aggregation}\n\t\\label{alg:efficientCC}\n\\end{algorithm}\n\nLet the sorted order of the points $\\mathcal{Y}^x$ be $y^x_1 \\leq y^x_2 \\leq \\ldots \\leq y^x_{|{\\mathcal{Y}}^x|}$. If $\\mathcal{Y}^x$ contains repeated elements, we require that the left end-points $\\ell_i$ come earlier in the sorted order than the right end-points $u_i$ for the repeated elements. Also define the bag of indicators $\\mathcal{S}^x$ with elements $s^x_1 \\leq s^x_2 \\leq \\ldots \\leq s^x_{|{\\mathcal{Y}^x}|}$, where \n\\begin{equation}\n\\label{eq:s-k-main}\ns^x_i :=\n\\begin{cases}\n1 & \\mbox{if } y^x_i \\mbox{ corresponds to a left end-point}\\\\\n0 & \\mbox{if } y^x_i \\mbox{ corresponds to a right end-point}.\n\\end{cases}\n\\end{equation}\nGiven $\\mathcal{Y}^x$ and $\\mathcal{S}^x$, Algorithm~\\ref{alg:efficientCC} describes how to compute the cross-conformal prediction set in one pass (thus time $O(n)$) for every test point. Thus the runtime (including the sorting) is $O(n\\log n)$ time to compute the predictions $\\mathcal{F}_{r_i(X_i, Y_i)}^{-i}(x)$ for every $i$. If each prediction takes time $\\leq T$, the overall time is $O(n\\log n)+Tn$, which is the same as jackknife+.\\footnote{For jackknife+, using quick-select, we could obtain $O(n) + Tn$ randomized, but the testing time $Tn$ usually dominates the additional $n\\log n$ required to sort.} In Appendix \\ref{appsec:alg-proof} we describe the algorithm and argue its correctness. \n\\begin{prop}\n\t\\label{prop:efficientCC-correctness}\n\tAlgorithm~\\ref{alg:efficientCC} correctly computes the cross-conformal prediction set~\\ref{eq:LOO} given $\\mathcal{Y}^x$ and $\\mathcal{S}^x$. \n\\end{prop}\n\\section{Extending ensemble based out-of-bag conformal methods using nested sets}\n\\label{sec:OOB-conformal}\nCross-conformal, jackknife+, and their K-fold versions perform multiple splits of the data and for every training point $(X_i, Y_i)$, a residual function $r_i$ is defined using a set of training points that does not include $(X_i, Y_i)$. \nIn the previous section, our description required training the base algorithm multiple times on different splits of the data. Often each of these individual algorithms is itself an ensemble algorithm (such as random forests). As described in this section, an ensemble algorithm naturally provide multiple (random) splits of the data from a single run and need not be re-trained on different splits to produce conformal prediction regions. This makes the conformal procedure computationally efficient.\nAt the same time, like cross-conformal, the conformal prediction regions produced here are often shorter than split conformal because they use all of the training data for prediction. \nIn a series of interesting papers \\citep{johansson2014regression, bostrom2017accelerating, linusson2019efficient, kim2020predictive}, many authors have exhibited promising empirical evidence that these ensemble algorithms improve the width of prediction sets without paying a computational cost. We call this method the \\emph{OOB-conformal} method (short for out-of-bag). \n\nWe now describe the procedure formally within the nested conformal framework, thus extending it instantly to residual functions that have hitherto not been considered. Our procedure can be seen as a generalization of the OOB-conformal method~\\citep{linusson2019efficient} or the jackknife+ after bootstrap method ~\\citep{kim2020predictive}:\n\\begin{enumerate}\n\t\\item Let $\\{M_j\\}_{j = 1}^K$ denote $K \\geq 1$ independent and identically distributed random sets drawn uniformly from $\\{M: M\\subset[n], |M| = m\\}$. This is the same as subsampling. Alternatively $\\{M_j\\}_{j=1}^K$ could be i.i.d. random bags, where each bag is obtained by drawing $m$ samples with replacement from $[n]$. This procedure corresponds to bootstrap. \n\t\\item For every $i\\in [n]$, define \n\t\\[M_{-i} := \\{M_j : i \\notin M_j\\},\\]\n\t which contains the training sets that are \\emph{out-of-bag} for the $i$-th data point. \n\t\\item The idea now is to use an ensemble learning method that, for every $i$, aggregates $|M_{-i}|$ many predictions to identify a single collection of nested sets $\\{\\mathcal{F}_t^{-i}(x)\\}_{t \\in \\mathcal{T}}$. For instance, one can obtain an estimate $\\widehat{\\mu}_{j}(\\cdot)$ of the conditional mean based on the training data corresponding to $M_j$, for every $j$, and then construct \n\t\\[\n\t\\mathcal{F}_t^{-i}(x) = [\\widehat{\\mu}_{-i}(x) - t, \\widehat{\\mu}_{-i}(x) + t],\n\t\\]\n\twhere $\\widehat{\\mu}_{-i}(\\cdot)$ is some combination (such as the mean) of $\\{\\widehat{\\mu}_j(\\cdot)\\}_{\\{j: i\\notin M_j\\}}$.\n\n\t\\item The remaining conformalization procedure is identical to $C^{\\texttt{LOO}}(x)$ described in Section~\\ref{sec:Jackknife-plus}. Define the residual score \n\n\t$r_i(x, y) := \\inf\\left\\{t\\in\\mathcal{T}:\\,y\\in\\mathcal{F}_t^{-i}(x)\\right\\}.$\n\n\\end{enumerate}\nUsing the cross-conformal scheme, the prediction set for any $x \\in \\mathcal{X}$ is given as\n\\begin{equation}\nC^{\\texttt{OOB}}(x) := \\left\\{y:\\,\\sum_{i=1}^n \\mathbbm{1}\\{r_i(X_i, Y_i) < r_i(x, y)\\} < (1-\\alpha)(n+1)\\right\\}.\n\\label{eq:def-oob-cross}\n\\end{equation}\nIf $\\mathcal{F}_t^{-i}(x)$ is an interval for all $1\\le i\\le n$ and $x\\in\\mathcal{X}$, then following the discussion in Section~\\ref{sec:intervals-LOO}, we could also derive a jackknife+ style prediction interval that is guaranteed to be non-empty:\n\\begin{equation}\nC^{\\texttt{OOB-JP}}(x) := [q^{-}_{n,\\alpha}(\\ell_i(x)), q^{+}_{n,\\alpha}(u_i(x))].\n\\label{eq:def-oob-jp}\n\\end{equation}\n\nIf $\\mathcal{F}_t^{-i}(x)$ could further contain empty sets, a jackknife+ interval can still be derived following the discussion in Appendix \\ref{appsec:empty-case}, but we skip these details here. Once again, we have that for every $x \\in \\mathcal{X}$, $C^{\\texttt{OOB}}(x) \\subseteq C^{\\texttt{OOB-JP}}(x)$; see Equation~\\eqref{eq:loo-jp} for details. The computational discussion of subsection~\\ref{subsec:compuation-cross} extends to $C^{\\texttt{OOB}}$. \n\n\n\nRecently, \\citet{kim2020predictive} provided a $1-2\\alpha$ coverage guarantee of the OOB-conformal method when $\\mathcal{F}_t^{-i}(x) = [\\widehat{\\mu}_{-i}(x) - t, \\widehat{\\mu}_{-i}(x) + t]$ where $\\widehat{\\mu}_{-i}(\\cdot)$ represents the aggregation of conditional mean estimate from $\\{M_j\\}_{i\\notin M_j}$. We generalize their result to any sequence of nested sets and extend it to the cross-conformal style aggregation scheme. In order to obtain a coverage guarantee, the conformal method must ensure a certain exchangeability requirement is satisfied. To do so, the argument of \\citet{kim2020predictive} required the number of random resamples $K$ to itself be drawn randomly from a binomial distribution. We assert the same requirement in the following theorem (proved in Appendix~\\ref{appsec:OOB-conformal}).\n\\begin{thm}\\label{thm:OOB-cross-validity}\n\tFix a permutation invariant ensemble technique that constructs sets $\\{\\mathcal{F}_t^{-i}\\}_{t\\in \\mathcal{T}}$ given a collection of subsets of $[n]$. Fix integers $\\widetilde{K}, m \\geq 1$ and let \n\t\\begin{align*}\n\tK ~&\\sim~ \\mathrm{Binomial}\\left(\\widetilde{K}, \\left(1 - \\frac{1}{n+1}\\right)^m\\right) \\quad (\\mbox{in the case of bagging}),\\;\\mathrm{or},\\\\\n\tK ~&\\sim~ \\mathrm{Binomial}\\left(\\widetilde{K}, 1 - \\frac{m}{n+1}\\right)\\quad (\\mbox{in the case of subsampling}).\n\t\\end{align*} \n\tThen \n\n\t$\\mathbb{P}(Y_{n+1}\\in C^{\\texttt{OOB}}(X_{n+1})) \\ge 1- 2\\alpha.$\n\n\\end{thm}\nBecause $C^{\\texttt{OOB}}(x) \\subseteq C^{\\texttt{OOB-JP}}(x)$ for every $x \\in \\mathcal{X}$, the validity guarantee continues to hold for $C^{\\texttt{OOB-JP}}(\\cdot)$.\nWhile we can only prove a $1-2\\alpha$ coverage guarantee, it has been observed empirically that the OOB-conformal method with regression forests as the ensemble scheme and nested sets $\\{[\\widehat{\\mu}(x) - t\\widehat{\\sigma}(x), \\widehat{\\mu}(x) + t\\widehat{\\sigma}(x)]\\}_{t \\in \\mathbb{R}^+}$ satisfies $1-\\alpha$ coverage while providing the shortest prediction sets on average~\\citep{bostrom2017accelerating}.\nOn the other hand, the best empirically performing nested sets are the ones introduced by \\citet{romano2019conformalized}: $\\{[\\widehat{q}_{\\beta}(x) - t, \\widehat{q}_{\\beta}(x) + t]\\}_{t \\in \\mathbb{R}}$ (for an appropriately chosen $\\beta$). Using nested conformal, we show how these these two ideas can be seamlessly integrated: quantile based nested sets with an OOB-style aggregation scheme. In Section~\\ref{sec:QOOB} we formally develop our novel method QOOB, and in Section~\\ref{sec:numerical-experiments} we empirically verify that it achieves competitive results in terms of the length of prediction sets. \n\n\n\\section{QOOB: A novel conformal method using nested sets}\n\\label{sec:QOOB}\nThe nested conformal interpretation naturally separates the design of conformal methods into two complementary aspects: \n\\begin{enumerate}[label=(\\alph*)]\n\t\\item identifying an information efficient nonconformity score based on a set of nested intervals, and\n\t\\item performing sample efficient aggregation of the nonconformity scores while maintaining validity guarantees.\n\\end{enumerate}\nIn this section, we leverage this dichotomy to merge two threads of ideas in the conformal literature and develop a novel conformal method that empirically achieves state-of-the-art results in terms of the width of prediction sets. \n\nFirst, we review what is known on aspect (b). While split-conformal based methods are computationally efficient, they lose sample efficiency due to sample splitting. Aggregated conformal methods such as cross-conformal, jackknife+, and OOB-conformal do not have this drawback and are the methods of choice for computationally feasible and sample efficient prediction sets. Among all aggregation techniques, the OOB-conformal method has been observed empirically to be the best aggregation scheme which uses all the training data efficiently~\\citep{bostrom2017accelerating}. \n\nNext, we consider aspect (a), the design of the nested sets. The nested sets considered by \\citet{bostrom2017accelerating} are $\\{[\\widehat{\\mu}(x) - t\\widehat{\\sigma}(x), \\widehat{\\mu}(x) + t\\widehat{\\sigma}(x)]\\}_{t \\in \\mathbb{R}^+}$ based on mean and variance estimates obtained using out-of-bag trees. On the other hand, it has been convincingly demonstrated that nested sets based on quantile estimates $\\widehat{q}_s(x)$ given by $\\{[\\widehat{q}_{\\beta}(x) - t, \\widehat{q}_{1-\\beta}(x) + t]\\}_{t \\in \\mathbb{R}}$ perform better than those based on mean-variance estimates in the split conformal setting~\\citep{romano2019conformalized, sesia2019comparison}. \n\nBuilding on these insights, we make a natural suggestion: Quantile Out-of-Bag (QOOB) conformal; pronounced ``cube'' conformal. This method works in the following way. First, a quantile regression based on random forest~\\citep{meinshausen2006quantile} with $T$ trees is learnt by subsampling or bagging the training data $T$ times. Next, using the out-of-bag trees for every training point $(X_i, Y_i)$, a quantile estimate $\\widehat{q}^{-i}_s(\\cdot)$ is learnt for fixed levels $s = \\beta$ and $ s = 1 - \\beta$. Here $\\beta = k\\alpha$ for some constant $k$. Now for every training point $i$, we define the nested sets as\n\\[\n\\mathcal{F}_t^{-i}(x) ~:=~ [\\widehat{q}_\\beta^{-i}(x) - t, \\widehat{q}_{1-\\beta}^{-i}(x) + t].\n\\]\n\nThe nonconformity scores based on these nested sets are aggregated to provide a prediction set as described by~$C^{\\texttt{OOB}}(x)$ in~\\eqref{eq:def-oob-cross} of Section~\\ref{sec:OOB-conformal}. Algorithm~\\ref{alg:QOOB} describes QOOB procedurally. Following subsection~\\ref{subsec:compuation-cross}, the aggregation step of QOOB (line \\ref{line:qoob-aggregation}, Algorithm~\\ref{alg:QOOB}) can be performed in time $O(n\\log n)$. \n\\begin{algorithm}[!t]\n\t\\SetAlgoLined\n\t\\KwInput{Training data $ \\{(X_i, Y_i)\\}_{i=1}^n$, test point $x$, desired coverage level $\\alpha$, number of trees $T$, nominal quantile level $\\beta$ (default $=2\\alpha$)}\n\t\\KwOutput{Prediction set $C^x \\subseteq \\mathcal{Y}$}\n\t$\\{M_j\\}_{j = 1}^T \\gets$ training bags drawn independently from $[n]$ using subsampling or bootstrap\\;\n\t$M_{-i} \\gets \\{M_j : i \\notin M_j\\}$\\;\n\t\\For{$i \\gets 1$ \\textbf{to} $T$}{\n\t\t$\\phi_i \\gets $ Quantile regression trees learnt using the data-points in $M_i$\\\\\\qquad \\qquad (this step could include subsampling of features)\\;\n\t}\n\t\\For{$i \\gets 1$ \\textbf{to} $n$}{\n\t\t$\\Phi_{-i} \\gets \\{\\phi_j: j \\in M_{-i}\\}$\\;\n\t\t$\\widehat{q}^{-i}(\\cdot) \\gets$ quantile regression forest using the trees $\\Phi_{-i}$\\;\n\t\n\t\t$\\mathcal{F}_t^{-i}(\\cdot) \\gets [\\widehat{q}_\\beta^{-i}(\\cdot) - t, \\widehat{q}_{1-\\beta}^{-i}(\\cdot) + t]$\\; \n\t\t$r_i(\\cdot, \\cdot) \\gets ((x, y) \\to \\inf\\left\\{t\\in\\mathcal{T}:\\,y\\in\\mathcal{F}_t^{-i}(x)\\right\\})$\\;\n\t\t\n\t}\n\t\n\t$C^x \\gets $ OOB prediction set defined in Equation~\\eqref{eq:def-oob-cross}; (call Algorithm~\\ref{alg:efficientCC} with $\\mathcal{Y}^x$, $\\mathcal{S}^x$ computed using $\\mathcal{F}_t^{-i}(\\cdot)$, the training data $\\{(X_i, Y_i)\\}_{i=1}^n$ and test point $x$ as described in equations~\\eqref{eq:y-k-main}, ~\\eqref{eq:s-k-main}) \\label{line:qoob-aggregation}\\\\% \\{(X_i, Y_i)\\}_{i=1}^n$, $x$, and $(r_i)_{i=1}^n$)\\; \n\t\\Return $C^x$\\;\n\t\\caption{Quantile Out-of-Bag conformal (QOOB)}\n\t\\label{alg:QOOB}\n\\end{algorithm}\n\nSince QOOB is a special case of OOB-conformal, it inherits an assumption-free $1-2\\alpha$ coverage guarantee from Theorem~\\ref{thm:OOB-cross-validity} if $K$ is drawn from an appropriate binomial distribution as described in the theorem. In practice, we typically obtain $1-\\alpha$ coverage with a fixed $K$. In Section~\\ref{sec:numerical-experiments}, we empirically demonstrate that QOOB achieves state-of-the-art performance on multiple real-world datasets. We also discuss three aspects of our method: \n\\begin{enumerate}[label=(\\alph*)]\n\t\\item how to select the nominal quantile level $\\beta = k\\alpha$, \n\t\\item the effect of the number of trees $T$ on the performance, and \n\t\\item the performance of the jackknife+ version of our method (QOOB-JP), which corresponds to the OOB-JP style aggregation (equation~\\eqref{eq:def-oob-jp}) of quantile-based nonconformity scores.\n\\end{enumerate}\n\\section{Numerical comparisons}\\label{sec:numerical-experiments}\nWe compare several methods discussed in this paper using synthetic and real datasets. MATLAB code to execute QOOB and reproduce the experiments in this section is provided at \\url{https:\/\/github.com\/AIgen\/QOOB}. Some experiments on synthetic data are discussed in subsection~\\ref{subsec:synthetic}; the rest of this section discusses results on real datasets. We use the following six datasets from the UCI repository: blog feedback, concrete strength, superconductivity, news popularity, kernel performance and protein structure. Metadata and links for these datasets are provided in subsection~\\ref{subsec:experiment-info}. In order to assess the coverage and width properties, we construct multiple version of each of these three datasets. For each dataset, we obtain 100 versions by independently drawing 1000 data points randomly (without replacement) from the full dataset. Then we split each such version into two parts: training and testing of sizes $768$\\footnote{the number of training points is divisible by many factors, which is useful for creating a varying number of folds for K-fold methods} and $232$ respectively. Hence corresponding to each of the six datasets, we get 100 different datasets with 768 training and 232 testing points. \n\nFor each conformal method, we report the following two metrics:\n\\begin{itemize}\n\t\n\t\\item \\emph{Mean-width}: For a prediction set $C(x)\\subseteq \\mathcal{Y} = \\mathbb{R}$ its width is defined as its Lebesgue measure. For instance, if $C(x)$ is an interval, then the width is its length and if $C(x)$ is a union of two or more disjoint intervals, then the width is the sum of the lengths of these disjoint intervals. We report the average over the mean-width given by\n\t\\begin{equation}\\label{eq:mean-width}\n\t\\mbox{Ave-Mean-Width} := \\frac{1}{100}\\sum_{b = 1}^{100}\\, \\left(\\frac{1}{232}\\sum_{i=1}^{232} \\mbox{width}({C}_b(X_i^b))\\right).\n\t\\end{equation}\n\tHere ${C}_b(\\cdot)$ is a prediction region learnt from the $b$-th version of a dataset. The outer mean is the average over 100 versions of a dataset. The inner mean is the average of the width over the testing points in a particular version of a dataset.\n\t\\item \\emph{Mean-coverage}: We have proved finite-sample coverage guarantees for all our methods and to verify (empirically) this property, we also report the average over the mean-coverage given by\n\t\\begin{equation}\\label{eq:mean-coverage}\n\t\\mbox{Ave-Mean-Coverage} := \\frac{1}{100}\\sum_{b=1}^{100}\\,\\left(\\frac{1}{232}\\sum_{i=1}^{232} \\mathbbm{1}\\{Y_i^b \\in {C}_b(X_i^b)\\}\\right).\n\t\\end{equation}\n\t\n\\end{itemize} \nIn addition to reporting the average over versions of a dataset, we also report the estimated standard deviation of the average (to guage the fluctuations). In the rest of the discussion, the qualification `average' may be skipped for succinctness, but all reports and conclusions are to be understood as comments on the average value for mean-width and mean-coverage. \n\nRandom forest (RF) based regressors perform well across different conformal methods and will be used as the base regressor in our experiments, with varying $T$, the number of trees. Each tree is trained on an independently drawn bootstrap sample from the training set (containing about $(1-1\/e)100\\% \\approx 63.2\\%$ of all training points).\nThe numerical comparisons will use the following methods:\n\\begin{enumerate}\n\t\\item SC-$T$: Split conformal~\\citep{papadopoulos2002inductive, lei2018distribution} with nested sets $\\{[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]\\}_{t \\in \\mathbb{R}^+}$ and $T$ trees. \n\t\\item Split-CQR-$T$ ($2\\alpha$): Split conformalized quantile regression~\\citep{romano2019conformalized} with $T$ trees and nominal quantile level $2\\alpha$. This corresponds to the nested sets $\n\t\\{ [\\widehat{q}_{2\\alpha}^{-i}(x) - t, \\widehat{q}_{1-{2\\alpha}}^{-i}(x) + t]\\}_{t \\in \\mathbb{R}}$. Quantile conformal methods require the nominal quantile level to be set carefully, as also noted by~\\citet{sesia2019comparison}. In our experiments, we observe that Split-CQR-$T$ performs well at the nominal quantile level $2\\alpha$. This is discussed more in subsection~\\ref{subsec:nominal-quantile}. \n\t\\item 8-fold-CC-$T$: 8-fold cross-conformal~\\citep{vovk2015cross, barber2019predictive} with $T$\n\ttrees learnt for every fold and the nested sets $\\{[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]\\}_{t \\in \\mathbb{R}^+}$. Leave-one-out cross-conformal is computationally expensive if $T$ trees are to be trained for each fold, and does not lead to significantly improved performance compared to OOB-CC in our experiments. Hence we did not report a detailed comparison across all datasets. \n\t\\item OOB-CC-$T$: OOB-cross-conformal~\\citep{johansson2014regression, kim2020predictive} with $T$\n\ttrees. This method considers the nested sets $\\{[\\widehat{\\mu}(x) - t, \\widehat{\\mu}(x) + t]\\}_{t \\in \\mathbb{R}^+}$ where $\\widehat{\\mu}$ is the average of the mean-predictions for $x$ on out-of-bag trees. \n\t\\item OOB-NCC-$T$: OOB-normalized-cross-conformal~\\citep{bostrom2017accelerating} with $T$ trees. This method considers nested sets $\\{[\\widehat{\\mu}(x) - t\\widehat{\\sigma}(x), \\widehat{\\mu}(x) + t\\widehat{\\sigma}(x)]\\}_{t \\in \\mathbb{R}^+}$ where $\\widehat{\\sigma}(x)$ is the standard deviation of mean-predictions for $x$ on out-of-bag trees.\n\n\t\\item QOOB-$T$ ($2\\alpha$): OOB-quantile-cross-conformal with $T$ trees and nominal quantile level $\\beta = 2\\alpha$. This is our proposed method. In our experiments, we observe that QOOB-T performs well at the nominal quantile level $2\\alpha$. We elaborate more on the nominal quantile selection in subsection~\\ref{subsec:nominal-quantile}.\n\\end{enumerate}\n\n\\begin{table}[!h]\n\t\\caption{Mean-width~\\eqref{eq:mean-width} of conformal methods with regression forests ($\\alpha = 0.1$).\\\\ Average values across 100 simulations are reported with the standard deviation in brackets. }\n\t\\centering\n\t\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{c|c|c|c|c|c|c}\n\t\t\t\\hline\n\t\t\n\t\t\t\\textbf{Method} & {Blog} & {Protein} & {Concrete}& {News} & {Kernel} & {Superconductivity} \\\\ \\hline \\hline \n\t\t\tSC-100 & 25.54 & 16.88 & 22.29 & 12491.84 & 452.71 & 54.46 \\\\[-0.15in] \n\t\t\t& (0.71) & (0.08) & (0.14) & (348.07) & (5.10) & (0.37) \\\\\\hline\n\t\t\tSplit-CQR-100 (2$\\alpha$) & \\textbf{12.22} & 14.20 & 21.45 & \\textbf{7468.15} & \\textbf{295.49} & 39.59 \\\\[-0.15in] \n\t\t\t& (0.35) & (0.09) & (0.12) & (136.93) & (3.09) & (0.27)\\\\\\hline\n\t\t\t8-fold-CC-100 & 24.83 & 16.42 & 19.23 & 12461.40 & 411.81 & 50.30 \\\\[-0.15in] \n\t\t\t& (0.44) & (0.05) & (0.04) & (263.54) & (3.4299) & (0.24) \\\\\\hline\n\t\t\tOOB-CC-100 & 24.76 & 16.38 & 18.69 & 12357.58 & 402.97 & 49.31 \\\\[-0.15in] \n\t\t\t& (0.50) & (0.04) & (0.03) & (213.72) & (3.13) & (0.24) \\\\\\hline\n\t\t\tOOB-NCC-100 & 20.31 & 14.87 & 18.66 & 11500.22 & 353.35 & 39.55 \\\\ [-0.15in]\n\t\t\t& (0.42) & (0.05) & (0.06) & (320.91) & (2.95) & (0.22)\\\\\\hline\n\t\t\tQOOB-100 ($2\\alpha$) & 14.43 & \\textbf{13.74} & \\textbf{18.19} & 7941.19 & 300.04 & \\textbf{37.04} \\\\[-0.15in] \n\t\t\t& (0.38) & (0.05) & (0.05) & (89.21) & (2.70) & (0.18)\\\\\\hline\n\t\\end{tabular}}\n\t\\label{table:overall-comparison-width}\n\\end{table}\n\n\\begin{table}[!h]\n\t\\caption{Mean-coverage~\\eqref{eq:mean-coverage} of conformal methods with regression forests ($\\alpha = 0.1$). The standard deviation of these average mean-widths are zero upto two significant digits. }\n\t\\centering\n\t\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{c|c|c|c|c|c|c}\n\t\t\t\\hline\n\t\t\n\t\t\t\\textbf{Method} & {Blog} & {Protein} & {Concrete}& {News} & {Kernel} & {Superconductivity} \\\\ \\hline \\hline \n\t\t\tSC-100 & 0.90 & 0.90 & 0.90 & 0.90 & 0.90 & 0.90 \\\\ \\hline\n\t\t\tSplit-CQR-100 ($2\\alpha$) & 0.91 & 0.90 & 0.90 & 0.90 & 0.90 & 0.90 \\\\ \\hline\n\t\t\t8-fold-CC-100 & 0.91 & 0.91 & 0.91 & 0.91 & 0.91 & 0.90 \\\\ \\hline\n\t\t\tOOB-CC-100 & 0.90 & 0.91 & 0.91 & 0.91 & 0.91 & 0.90 \\\\ \\hline\n\t\t\tOOB-NCC-100 & 0.92 & 0.91 & 0.91 & 0.92 & 0.93 & 0.91 \\\\ \\hline\n\t\t\tQOOB-100 ($2\\alpha$) & 0.92 & 0.91 & 0.92 & 0.91 & 0.93 & 0.91 \\\\ \\hline\n\t\\end{tabular}}\n\t\\label{table:overall-comparison-coverage}\n\\end{table}\n\\label{subsec:QOOB-main}\n\nTables~\\ref{table:overall-comparison-width} and~\\ref{table:overall-comparison-coverage} report the mean-width and mean-coverage that these conformal methods achieve on 6 datasets. Here, the number of trees $T$ is set to $100$ for all the methods.\nWe draw the following conclusions: \n\n\\begin{itemize}\n\t\\item Our novel method QOOB achieves the shortest or close to the shortest mean-width compared to other methods while satisfying the $1-\\alpha$ coverage guarantee. The closest competitor is Split-CQR. As we further investigate in subsection~\\ref{subsec:QOOB-trees}, even on datasets where Split-CQR performs better than QOOB, if the number of trees are increased beyond 100, QOOB shows a decrease in mean-width while Split-CQR does not improve. For example, on the kernel dataset, QOOB outperforms Split-CQR at 400 trees.\n\t\\item In Table~\\ref{table:overall-comparison-width}, QOOB typically has low values for the standard deviation of the average-mean-width across all methods. This entails more reliability to our method, which may be desirable in some applications. In subsections~\\ref{subsec:nominal-quantile} and~\\ref{subsec:QOOB-trees}, we observe that this property is true across different number of trees and nominal quantile levels as well. \n\t\\item On every dataset, QOOB achieves coverage higher than the prescribed value of $1-\\alpha$, with a margin of 1-3\\%. Surprisingly this is true even if it shortest mean-width among all methods. It may be possible to further improve the performance of QOOB in terms of mean-width by investigating the cause for this over-coverage. \n\t\\item OOB-CC does better than 8-fold-CC with faster running times. Thus, to develop QOOB, we chose to work with out-of-bag conformal.\n\\end{itemize}\n\n\n\nIn the rest of this section, we present more experimental results split\nwith the following key insights:\n\\begin{itemize}\n\n\t\\item In subsection~\\ref{subsec:nominal-quantile}, we discuss the significant impact that nominal quantile selection has on the performance of QOOB and Split-CQR. We observe that $2\\alpha$ is an appropriate general purpose nominal quantile recommendation for both methods. \n\t\\item In subsection~\\ref{subsec:QOOB-trees}, we show that increasing the number of trees $T$ leads to decreasing mean-widths for QOOB, while this is not true for its closest competitor Split-CQR. QOOB also outperforms other competing OOB methods across different values for the number of trees $T$. \n\t\\item In subsection~\\ref{subsec:QOOB-sample-size}, we compare QOOB and Split-CQR in the small sample size (small $n$) regime where we expect sample splitting methods to lose statistical efficiency. We confirm that QOOB significnatly outperforms Split-CQR on all six datasets we have considered for $n \\leq 100$. \n\t\\item In subsection~\\ref{subsec:cc-jp}, we compare the related methods of cross-conformal and jackknife+ and demonstrate that there exist settings where cross-conformal leads to shorter intervals compared to jackknife+, while having a similar computational cost (as discussed in subsection~\\ref{subsec:compuation-cross}). \n\t\\item In subsection~\\ref{subsec:synthetic}, we demonstrate on a simple synthetic dataset that QOOB achieves conditional coverage empirically. We experiment with the data distribution designed by \\citet{romano2019conformalized} for demonstrating the conditional coverage of Split-CQR.\n\\end{itemize}\n\n\n\n\n\n\\subsection{Nominal quantile selection has a significant effect on QOOB}\n\\label{subsec:nominal-quantile}\nQOOB and Split-CQR both use nominal quantiles $\\widehat{q}_\\beta$, $\\widehat{q}_{1-\\beta}$ from an underlying quantile prediction algorithm learned on the training data. In the case of Split-CQR, as observed by \\citet{romano2019conformalized} and \\citet{sesia2019comparison}, tuning $\\beta$ leads to improved performance.\nA quantile-wide comparison is reported in Figure~\\ref{fig:width-nominal-level} for QOOB-100 ($k\\alpha$) and Split-CQR-100 ($k\\alpha$), with OOB-NCC-100 as a fixed baseline (that does not vary with $k$). We observe that the nominal quantile level significantly affects the performance of Split-CQR and QOOB. Both methods perform well at the nominal quantile of about $2\\alpha$, although this identification procedure may violate the exchangeability requirement of Theorem~\\ref{thm:OOB-cross-validity}. We encourage a more detailed study on the theoretical and empirical aspects of nominal quantile selection in future work. We also note that for all values of $k$, QOOB typically has smaller standard deviation of the average-mean-width compared to Split-CQR as denoted by the shaded region, implying more reliability in the predictions. \n\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.9\\textwidth]{legendAlpha.png}\\vspace{-0.8cm}\\\\\n\t\\centering\n\t\\subfloat[Concrete structure.]{\\includegraphics[width=0.33\\textwidth]{concrete_OOBvaryAlpha.pdf}}\n\t\\subfloat[Blog feedback.]{ \\includegraphics[width=0.33\\textwidth]{blog_OOBvaryAlpha.pdf}}\n\t\\subfloat[Protein structure.]{\\includegraphics[width=0.33\\textwidth]{protein_OOBvaryAlpha.pdf}}\\\\\n\t\\subfloat[Superconductivity.]{\\includegraphics[width=0.33\\textwidth]{superconductor_OOBvaryAlpha.pdf}}\n\t\\subfloat[News popularity.]{ \\includegraphics[width=0.33\\textwidth]{news_OOBvaryAlpha.pdf}}\n\t\\subfloat[Kernel performance.]{\\includegraphics[width=0.33\\textwidth]{kernel_OOBvaryAlpha.pdf}}\n\t\\caption{QOOB and Split-CQR are sensitive to the nominal quantile level $\\beta = k\\alpha$. At $\\beta \\approx 2\\alpha$, they perform better than OOB-NCC for all datasets (OOB-NCC does not require nominal quantile tuning and is a constant baseline). For the plots above, $\\alpha = 0.1$ and all methods plotted have empirical mean-coverage at least $1-\\alpha$. The mean-width values are averaged over 100 iterations. The shaded area denotes $\\pm 1$ std-dev for the average of mean-width. }\n\t\\label{fig:width-nominal-level}\n\\end{figure}\n\n\n\\subsection{QOOB has shorter prediction intervals as we increase the number of trees}\n\\label{subsec:QOOB-trees}\nIn this experiment, we investigate the performance of the competitive conformal methods from Table~\\ref{table:overall-comparison-width} as the number of trees $T$ are varied. For QOOB and Split-CQR, we fix the quantile level to $\\beta = 2\\alpha$. We also compare with OOB-NCC and another quantile based OOB method described as follows. Like QOOB, suppose we are given a quantile estimator $\\widehat{q}_s(\\cdot)$. Consider the quantile-based construction of nested sets suggested by~\\citet{chernozhukov2019distributional}: \n\\[\\mathcal{F}_t(x) = [\\widehat{q}_t(x), \\widehat{q}_{1-t}(x)]_{t \\in (0, 1\/2)}.\n\\]\nUsing these nested sets and the OOB conformal scheme (Section~\\ref{sec:OOB-conformal}) leads to the QOOB-D method (for `distributional' conformal prediction as the original authors called it). Since QOOB-D does not require nominal quantile selection, we considered this method as a possible solution to the nominal quantile problem of QOOB and Split-CQR (subsection~\\ref{subsec:nominal-quantile}). The results are reported in Figure~\\ref{fig:width-trees} for $T$ ranging from 50 to 400. \n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{legendTrees.png}\\vspace{-0.6cm}\n\t\\subfloat[Concrete structure.]{\\includegraphics[width=0.33\\textwidth]{concrete_OOBvaryTrees.pdf}}\n\t\\subfloat[Blog feedback.]{ \\includegraphics[width=0.33\\textwidth]{blog_OOBvaryTrees.pdf}}\n\t\\subfloat[Protein structure.]{\\includegraphics[width=0.33\\textwidth]{protein_OOBvaryTrees.pdf}}\\\\\n\t\\subfloat[Superconductivity.]{\\includegraphics[width=0.33\\textwidth]{superconductor_OOBvaryTrees.pdf}}\n\t\\subfloat[News popularity.]{ \\includegraphics[width=0.33\\textwidth]{news_OOBvaryTrees.pdf}}\n\t\\subfloat[Kernel performance.]{\\includegraphics[width=0.33\\textwidth]{kernel_OOBvaryTrees.pdf}}\n\t\\caption{The performance of QOOB ($2\\alpha$) improves with increasing number of trees $T$, while the performance of Split-CQR ($2\\alpha$) does not. QOOB ($2\\alpha$) beats every other method except Split-CQR ($2\\alpha$) for all values of $T$.\n\t\tFor the plots above, $\\alpha = 0.1$ and all methods plotted have empirical mean-coverage at least $1-\\alpha$. The mean-width values are averaged over 100 iterations. The shaded area denotes $\\pm 1$ std-dev for the average of mean-width. }\n\t\\label{fig:width-trees}\n\\end{figure}\n\nWe observe that with increasing $T$, QOOB continues to show improving performance in terms of the width of prediction intervals. Notably, this is not true for Split-CQR, which does not show improving performance beyond 100 trees. In the results reported in Table~\\ref{table:overall-comparison-width}, we noted that Split-CQR-100 outperformed QOOB-100 on the blog feedback, news popularity and kernel performance datasets. \nHowever, from Figure~\\ref{fig:width-trees} we observe that for $T = 400$, QOOB performs almost the same as Split-CQR on blog feedback and news popularity, and in fact does significantly better than Split-CQR on kernel performance. Further, QOOB shows lower values for the standard deviation of the average-mean-width. (The QOOB-D method performs worse than QOOB for every dataset, and hence we did not report it in the other comparisons in this paper.)\n\n\n\\subsection{QOOB outperforms Split-CQR at small sample sizes}\\label{subsec:QOOB-sample-size}\nQOOB needs $n$ times more computation than Split-CQR to produce prediction intervals, since one needs to make $n$ individual predictions. If fast prediction time is desired, our experiments in subsections~\\ref{subsec:nominal-quantile} and \\ref{subsec:QOOB-trees} indicate that Split-CQR is a competitive quick alternative. However, here we demonstrate that at the small sample regime, QOOB significantly outperforms Split-CQR on all six datasets that we have considered.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{legendnTrain.png}\\vspace{-0.6cm}\n\t\\centering\n\t\\subfloat[Concrete structure.]{\\includegraphics[width=0.33\\textwidth]{concrete_OOBvaryN.pdf}}\n\t\\subfloat[Blog feedback.]{ \\includegraphics[width=0.33\\textwidth]{blog_OOBvaryN.pdf}}\n\t\\subfloat[Protein structure.]{\\includegraphics[width=0.33\\textwidth]{protein_OOBvaryN.pdf}}\\\\\n\t\\subfloat[Superconductivity.]{\\includegraphics[width=0.33\\textwidth]{superconductor_OOBvaryN.pdf}}\n\t\\subfloat[News popularity.]{ \\includegraphics[width=0.33\\textwidth]{news_OOBvaryN.pdf}}\n\t\\subfloat[Kernel performance.]{\\includegraphics[width=0.33\\textwidth]{kernel_OOBvaryN.pdf}}\n\t\\caption{The performance of QOOB and Split-CQR with varying number of training points $n$. QOOB has shorter mean-width than Split-CQR across datasets for small $n$ and also smaller standard-deviation of the average mean-width. Here, $\\alpha = 0.1$ and all methods plotted have empirical mean-coverage at least $1-\\alpha$. The mean-width values are averaged over 100 iterations. The shaded area denotes $\\pm 1$ std-dev for the average of mean-width. }\n\t\\label{fig:width-n}\n\\end{figure}\nTo make this comparison, we subsample the datasets to a smaller sample size and consider the mean-width and mean-coverage properties of QOOB ($2\\alpha$) and Split-CQR ($2\\alpha$) with $T = 100$. Figure~\\ref{fig:width-n} contains the results with $n$ ranging from $30$ to $240$. We observe that at small $n$, QOOB does significantly better than Split-CQR. This behavior is expected since at smaller values of $n$, the statistical loss due to sample splitting is most pronounced. \n\nSince the overall computation time decreases as $n$ decreases, QOOB is a significantly better alternative in the small sample regime on all fronts. \n\n\n\n\\subsection{Cross-conformal outperforms jackknife+}\n\\label{subsec:cc-jp}\nCross-conformal prediction sets are always smaller than the corresponding jackknife+ prediction sets by construction; see Subsection~\\ref{sec:intervals-LOO} and~\\eqref{eq:loo-jp}. This implies that in terms of both metrics (mean-width and mean-coverage), cross-conformal is no worse than jackknife+. The fact however that cross-conformal may not be an interval might be of practical importance. The aim of this subsection is to show that the jackknife+ prediction interval can sometimes be strictly larger than even the smallest interval containing the cross-conformal prediction set. \nWe demonstrate this through the performance of QOOB against QOOB-JP and QOOB-Conv on the blog feedback dataset (Table~\\ref{table:cc-jp}). Here QOOB-JP refers to the OOB-JP version~\\eqref{eq:def-oob-jp} (whereas QOOB uses the OOB version~\\eqref{eq:def-oob-cross}), and QOOB-Conv corresponds to the shortest interval completion of QOOB, that is, the convex hull of QOOB prediction set. For each of these, we set the nominal quantile level to $0.5$ instead of $2\\alpha$ as suggested earlier because this led to the most pronounced difference in mean-widths. \n\\begin{table}[H]\n\t\\caption{Mean-width of $C^{\\texttt{OOB}}(x), \\mathrm{Conv}(C^{\\texttt{OOB}}(x)),$ and $C^{\\texttt{OOB-JP}}(x)$ for the blog feedback dataset with QOOB method. The base quantile estimator is quantile regression forests, and $\\alpha = 0.1$. Average values across 100 simulations are reported with the standard deviation in brackets . }\n\t\\centering\n\t\\begin{tabular}{c|c|c}\n\t\t\\hline\n\t\t\\textbf{Method} &\\textbf{Mean-width} & \\textbf{Mean-coverage} \\\\ \\hline\\hline\n\t\tQOOB-100 ($\\beta = $0.5) & \\textbf{14.67 (0.246)} & 0.908 (0.002) \\\\%[-0.1in]& (0.246) & (0.002)\\\\ \n\t\t\\hline\n\t\tQOOB-Conv-100 ($\\beta = $0.5) & 14.73 (0.249) & 0.908 (0.002) \\\\%[-0.1in]& (0.249) & (0.002) \\\\ \n\t\t\\hline\n\t\tQOOB-JP-100 ($\\beta = $0.5) & 15.36 (0.248) & 0.911 (0.002) \\\\%[-0.1in] & (0.248) & (0.002) \\\\ \n\t\t\\hline\n\t\\end{tabular}\n\t\\label{table:cc-jp}\n\\end{table}\nWhile this is a specific setting, our goal is to provide a proof of existence. There may be other settings where QOOB and QOOB-JP are identical or nearly equal. Because the cross-conformal prediction set as well as its convex hull can be computed in nearly the same time (as shown in subsection~\\ref{subsec:compuation-cross}), one should always prefer cross-conformal or its convex hull over jackknife+.\n\n\\subsection{QOOB demonstrates conditional coverage empirically }\n\\label{subsec:synthetic}\nTo demonstrate that Split-CQR exhibits conditional coverage, \\citet[Appendix B]{romano2019conformalized} designed the following data-generating distribution for $P_{XY}$:\n\\begin{align*}\n\t\\epsilon_1 &\\sim N(0, 1), \n\t\\epsilon_2 \\sim N(0, 1), \n\tu \\sim \\text{Unif}[0, 1], \\quad\\mbox{and}\\quad X \\sim \\text{Unif}[0, 1], \\\\\n\tY &\\sim \\text{Pois}(\\sin^2(X) + 0.1) + 0.03 X \\epsilon_1 + 25 \\mathbf{1}\\{u < 0.01\\} \\epsilon_2.\n\\end{align*}\nWe use the same distribution to demonstrate the conditional coverage of QOOB. Additionally, we performed the experiments at a lower sample size than $n \\leq 400$ to understand the effect of sample size on both methods (while the original experiments had $n = 2000$). Figure~\\ref{fig:synthetic} summarizes the experiments. \n\\begin{figure}[!h]\n\t\\hspace{1.2cm}\\includegraphics[width=0.2\\textwidth]{legendSynthetic1.png}\\hspace{1.6cm}\\includegraphics[width=0.18\\textwidth]{legendSynthetic4.png}\\hspace{1.5cm}\\includegraphics[width=0.2\\textwidth]{legendSynthetic3.png}\n\t \\vspace{-1cm}\n\t\\centering\n\t\\subfloat[$n = 100$. (QOOB) MW = 2.16, MC = 0.91. (Split-CQR) MW = 2.23, MC = 0.92.]{\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_100_visual.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_100_coverage.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_100_length.pdf}}\\\\\\vspace{-0.5cm}\n\t\\subfloat[$n = 200$. (QOOB) MW = 1.99, MC = 0.92. (Split-CQR) MW = 2.18, MC = 0.91.]{\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_200_visual.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_200_coverage.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_200_length.pdf}}\\\\\\vspace{-0.5cm}\n\t\\subfloat[$n = 300$. (QOOB) MW = 1.86, MC = 0.89. (Split-CQR) MW = 1.94, MC = 0.89.]{\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_300_visual.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_300_coverage.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_300_length.pdf}}\\\\\\vspace{-0.5cm}\n\n\n\n\t\\subfloat[$n = 400$. (QOOB) MW = 2.10, MC = 0.91. (Split-CQR) MW = 2.09, MC = 0.90.]{\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_400_visual.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_400_coverage.pdf}\\includegraphics[width=0.33\\textwidth]{cqr_qrc_synthetic_400_length.pdf}}\n\t\\caption{The performance of QOOB-100 and Split-CQR-100 on synthetic data with varying number of training points $n$ ($\\alpha = 0.1$). MW refers to mean-width and MC refers to mean-coverage. QOOB shows conditional coverage at smaller values of $n$ than Split-CQR. Subsection~\\ref{subsec:synthetic} contains more experimental details. } \n\t\\label{fig:synthetic}\n\\end{figure}\n\nFor this experiment, the number of trees $T$ are set to $100$ for both methods. To choose the nominal quantile level, we first ran the python notebook at \\url{https:\/\/github.com\/yromano\/cqr} to reproduce the original experiments performed by \\citet{romano2019conformalized}. Their code first learns a nominal quantile level for Split-CQR by cross-validating. On executing their code, we typically observed values near $0.1$ for $\\alpha = 0.1$ and hence we picked this nominal quantile level for our experiments as well (for both Split-CQR and QOOB). Another aspect that we needed to be careful about was the depth of the quantile regression forest. For our simple 1-dimensional distribution, deeper trees lead to wide prediction sets for both Split-CQR and QOOB. As was also done in the original experiments, the performance improves if the minimum number of training data-points in the tree leaves is set to $40$. We do so in our experiment as well. \n\n\n\\subsection{Additional information on experiments}\n\n\nOur experimental results can be reproduced using the QOOB implementation at \\url{https:\/\/github.com\/AIgen\/QOOB}. Details for the datasets used in our experiments are provided in Table~\\ref{table:metadata}. \\label{subsec:experiment-info}\nAll our experiments were conducted in MATLAB using the TreeBagger class\nfor training regression forests and quantile regression forests. Default parameters were used for all datasets apart from the synthetic dataset of subsection~\\ref{subsec:synthetic}. \n\n\\begin{table}\n\t\\caption{Meta-data for the datasets used in our experiments. $N$ refers to the total number of data-points from which we create 100 versions by independently drawing 1000 data points randomly (as described in the beginning of Section~\\ref{sec:numerical-experiments}). $d$ refers to the feature dimension. }\n\t\\centering\n\t\\resizebox{\\textwidth}{!}{\\begin{tabular}{|c|l|c|c|}\n\t\t\t\\hline\n\t\t\t\\textbf{Dataset} & \\textbf{$N$} & \\textbf{$d$} &\\textbf{URL (\\url{http:\/\/archive.ics.uci.edu\/ml\/datasets\/*})} \\\\\\hline \n\t\t\tBlog & 52397 & 280 & \\url{BlogFeedback} \\\\ \\hline\n\t\t\tProtein & 45730 & 8 & \\url{Physicochemical+Properties+of+Protein+Tertiary+Structure} \\\\ \\hline\n\t\t\tConcrete & 1030 & 8 & \\url{Concrete+Compressive+Strength} \\\\ \\hline\n\t\t\tNews & 39644 & 59 & \\url{Online+News+Popularity} \\\\ \\hline\n\t\t\tKernel\\tablefootnote{The GPU kernel dataset contains four output variables corresponding to four measurements of the same entity. The output variable is the average of these values.} & 241600 & 14 &\\url{SGEMM+GPU+kernel+performance} \\\\ \\hline\n\t\t\tSuperconductivity & 21263 & 81& \\url{Superconductivty+Data} \\\\ \\hline\n\t\\end{tabular}}\n\t\\label{table:metadata}\n\\end{table}\n\n\n\n\n\n\\section{Summary}\\label{sec:summary}\n\nThis paper introduced a simple alternative framework to the score-based conformal prediction, for developing distribution-free prediction sets, which is instead based on a sequence of nested prediction sets. We argued that our nested conformal prediction framework is arguably more natural and intuitive. \nWe demonstrated how to translate a variety of existing nonconformity scores into nested prediction sets.\nWe showed how cross-conformal prediction, the jackknife+, and out-of-bag conformal can be described in our nested framework. The interpretation provided by nested conformal opens up a variety of new procedures\nto practitioners. We propose one such method: QOOB, which uses quantile regression forests to perform out-of-bag conformal prediction. We demonstrated empirically that QOOB achieves state-of-the-art performance on multiple real-world datasets. \n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Robin Dunn, Jaehyeok Shin and Aleksandr Podkopaev for comments on an initial version of this paper. AR is indebted to Rina Barber, Emmanuel Cand{\\`e}s and Ryan Tibshirani for several insightful conversations on conformal prediction. This work used the Extreme Science and Engineering Discovery Environment (XSEDE)~\\citep{towns2014xsede}, which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges system~\\citep{Nystrom:2015:BUF:2792745.2792775}, which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC). CG thanks the PSC consultant Roberto Gomez for support with MATLAB issues on the cluster.\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaxnu b/data_all_eng_slimpj/shuffled/split2/finalzzaxnu new file mode 100644 index 0000000000000000000000000000000000000000..99810b03c65f865eaaa7872b6d51d281f4491010 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaxnu @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\n\tNematic liquid crystal (NLC) is a liquid crystal phase with has rod-shaped molecules which tend to align along a particular direction denoted by a unit vector $\\mathbf{n}$, called the optical director axis. In addition to $\\mathbf{n}:\\mathbb{R}^d \\to \\mathbb{R}^3$ , the hydrodynamic of an isothermal and incompressible NLC is also described by its pressure $p:\\mathbb{R}^d\\to \\mathbb{R}$ and velocity $\\mathbf{v}:\\mathbb{R}^d\\to \\mathbb{R}^d$.\nWe refer to \\cite{Chandrasekhar}\nand \\cite{Gennes} for a comprehensive treatment of the physics of\nliquid crystals.\n\t\nUsing the Ericksen and Leslie continuum theory for liquid crystals, see \\cite{Ericksen} and\nLeslie \\cite{Leslie}, F. Lin and C. Liu\n\\cite{Lin-Liu} derived the most basic and simplest form of the dynamical system modeling the motion of a nematic liquid crystal (NLC)\nflowing in $\\mathbb{R}^d (d=2,3)$. This system is given by \n\t\t\\begin{align}\n\t&\td\\mathbf{v}+\\biggl[(\\mathbf{v}\\cdot\\nabla)\\mathbf{v}-\\Delta \\mathbf{v}+\\nabla p\\biggr]dt= -\\nabla\\cdot(\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n})dt+ \\mathbf{f} dt,\\label{eqn-SELE-v} \\\\\n\t&\t\\mathrm{div }\\; \\mathbf{v}= 0,\\label{eqn-SELE-div} \\\\\n\t&\td\\mathbf{n}+(\\mathbf{v}\\cdot\\nabla)\\mathbf{n} dt = \\biggl[\\Delta\n\t\\mathbf{n}+ \\lvert \\nabla \\d \\rvert^2 \\d \\biggr]dt+\\mathbf{g} dt , \\label{eqn-SELE-d}\\\\\n\t& \\lvert \\mathbf{n} \\rvert^2= 1,\\label{eqn-SELE-sphere}\n\t\\end{align}\n\twhere $\\mathbf{f}$ and $\\mathbf{g}$ are forcings acting on the system.\n\t The entries of the matrix $\\nabla \\mathbf{n} \\odot \n\\nabla \\mathbf{n}$ are defined by \n\\begin{equation*}\n[\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n}]_{i,j}=\\sum_{k=1}^3 \\frac{\\partial\n\t\\mathbf{n}^{(k)}}{\\partial x_i}\\frac{\\partial \\mathbf{n}^{(k)}}{\\partial x_j},\\;\\;\n\\mbox{ } i,j=1,\\dots, d.\n\\end{equation*}\nBefore proceeding further, we should mention that \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} is obtained by neglecting several terms such as the viscous Leslie stress tensor in the equation for $\\mathbf{v}$, the stretching and rotational effects for $\\mathbf{d}$. Thus, it is not known whether the models \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} and \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} are thermodynamically stable or consistent. However, these models still retain many mathematical and essential features of the dynamics for NLCs. In the recent papers \\cite{MH+JP-2017} \\cite{MH+JP-2018}, \\cite{MH et al-2014}, \\cite{Lin+Wang-2014} and \\cite{Sun+Liu} several thermodynamically consistent and stable models of NLC have been developed and analysed.\n\t\n\tIn this paper, we fix a bounded domain $\\mathcal{O}\\subset \\mathbb{R}^d$, $d=12,3$ with smooth boundary and we consider the following stochastic system\n\t\\begin{align}\n\t&\td\\mathbf{v}+\\Bigl[(\\mathbf{v}\\cdot\\nabla)\\mathbf{v}-\\Delta \\mathbf{v}+\\nabla p\\Bigr]dt= -\\nabla\\cdot(\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n})dt+ S(\\mathbf{v}) dW_1,\\label{eqn-SLQE-v} \\\\\n\t&\t\\mathrm{div }\\; \\mathbf{v}= 0,\\label{eqn-SLQE-div} \\\\\n\t&\td\\mathbf{n}+(\\mathbf{v}\\cdot\\nabla)\\mathbf{n} dt = \\bigl[\\Delta\n\t\\mathbf{n}+f(\\d)\\bigr]dt+(\\mathbf{n}\\times \\mathbf{h})\\circ\n\tdW_2, \\label{eqn-SLQE-d}\\\\\n\t&\t\\mathbf{v}=0 \\text{ and } \\frac{\\partial \\mathbf{n}}{\\partial \\boldsymbol{\\nu}}=0 \\text{ on }\n\t\\partial \\mathcal{O},\\label{BC}\\\\\n& \\mathbf{v}(0)=\\mathbf{v}_0 \\text{ and } \\mathbf{n}(0)=\\mathbf{n}_0, \\label{Init-Cond}\n\t\\end{align}\n\twhere $\\mathbf{v}_0: \\mathcal{O} \\to \\mathbb{R}^d$, $\\mathbf{n}_0:\\mathcal{O}\\to \\mathbb{R}^3$ are given mappings, $\\boldsymbol{\\nu}$ is the unit outward normal to $\\partial \\mathcal{O}$, $f$ is a polynomial function satisfying some conditions to be fixed later. Here, $W_1$ and $W_2$ are respectively independent cylindrical Wiener process and standard Brownian motion, $(\\mathbf{n}\\times \\mathbf{h})\\circ dW_2$ is understood in\n\tthe Stratonovich sense.\n\n\n\tThe Fr\\'eedericksz transition, which is produced by applying a sufficiently strong external perturbation (e.g. magnetic or electric fields) to an undistorted NLC, and its behaviour under random perturbation have been extensively studied in several physics papers, see \\cite{Horsthemke+Lefever-1984, San Miguel-1985,FS+MSanM}, all of which neglected the fluid velocity. However, it is pointed out in \\cite[Chapter 5]{Gennes} that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. It is this gap in knowledge that is the motivation for our mathematical study which was initiated in the old unpublished preprints \\cite{BHP13} and \\cite{BHP-arxiv}, see also the recent papers \\cite{BHP18} and \\cite{BHP19}.\n\n\n\t\n\tIn this paper, we mainly prove the existence and uniqueness of a maximal local strong solution which is\nunderstood in the sense of stochastic calculus and PDEs. This result is a corollary of several abstract results which are proved in Section \\ref{ABST-STRONG} and are of independent interest. \n In the case $d=2$, we show the non-explosion of the maximal solution by an adaptation and combination of Khashminskii test for non-explosions and an idea of Schmalfu{\\ss} elaborated in \\cite{Bjorn}, see Section \\ref{SLC-Sect4} for more details. Our novelty is the extension of the Schmalfu{\\ss} idea, which has been used so far to prove the uniqueness of solutions of Stochastic Navier-Stokes equations and related problems, to the proof of the global existence of a strong solution to the problem \\eqref{eqn-SLQE-v}-\\eqref{Init-Cond}. In particular, we give another proof of the global existence of 2D stochastic Navier-Stokes equations with multiplicative noise and for initial data with finite enstrophy. Thus, our paper can also be seen as a generalization of the results for the existence and uniqueness of maximal local and global solutions of strong solutions of stochastic Navier-Stokes proved in \\cite{Nathan1}, \\cite{Nathan2} and \\cite{Mikulevicius}.\n\nWe should notice that some of the arguments elaborated in Section \\ref{ABST-STRONG} have been already used in \\cite{BHR-2014} and \\cite{BHP19} which respectively studied the strong solution of some stochastic hydrodynamic equations (NSEs, MHD and 3D Leray $\\alpha$-models) driven by L\\'evy noise, and the existence and uniqueness of a maximal local smooth solution to the stochastic Ericksen-Leslie system \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} on the $d$-dimensional torus. We are also strongly convinced that with these general results it is possible, although it has not been done in detail, to prove the existence of strong solution of several stochastic hydrodynamical models such as the NSEs, MHD equations, $\\alpha$-models for Navier-Stokes and related problems.\n\n\t\n\tWhile the deterministic version of \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} has been the subject of intensive mathematical studies, see \\cite{Rojas-Medar, Hong,Lin-Liu,Lin-Wang,Lin-Wang-2016, Shkoller,Dai+Schonbeck_2014} and \\cite{Cavaterra,Hong-2014,Hong-2012,Huang-2014, Wang}, there are fewer results related to the stochastic system \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d}. The unpublished paper \\cite{BHP13} proved the existence and uniqueness of mxaimal local strong solution to the system \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} with a bounded nonlinear term $f(\\d)=\\mathds{1}_{\\lvert \\d \\rvert\\le 1} (1-\\lvert \\d\\rvert^2)\\d $. The paper \\cite{BHP18} deals only with weak (both in PDEs and stochastic calculus sense) solutions and the maximum principle. Some of the results in \\cite{BHP18} and the current paper have already been used in several papers such as \\cite{ZB+UM+AP}, \\cite{ZB+UM+AP-2019}, \\cite{RZ+GZ}, \\cite{BGuo+GZhou-2019}, \\cite{GZhou-2019} and \\cite{Wang+Wu+Zhou-2019}.\n\n\t\n\t Very recently we have become aware of a recent paper by Feireisl and Petcu \\cite{FeirPetcu_2019}, in which they proved the existence of a \\textit{dissipative martingale}, as well as the existence of a local strong solution and weak-strong uniqueness of the solution of the stochastic Navier-Stokes Allen-Cahn Equations. Note that in \\cite{FeirPetcu_2019} the second unknown $\\mathbf{n}$ is a scalar field, the nonlinear term $f(\\cdot)$ is globally Lipschitz and the derivative of a double-well potential $F(\\cdot)$, and the coefficient of the noise entering the equations for $\\mathbf{n}$ is bounded.\n\tThe paper \\cite{BHP19} is the first paper to deal with the the stochastic counterpart of the Ericksen-Leslie equations \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere}. The results in present manuscript is not covered in \\cite{BHP19} because in contrast to our framework which considers initial condition $(\\v_0, \\d_0)\\in \\mathrm{H}^1 \\times \\mathbf{H}^{2}$, the initial data in \\cite{BHP19} satisfies $(\\v_0, \\d_0)\\in \\mathrm{H}^{\\alpha}\\times \\mathrm{H}^{\\alpha+1}$ for $\\alpha>\\frac d2$, where $d=2,3$ is the space dimension. There is also the papers \\cite{Medjo1} which seeks for a special solution $(\\mathbf{v},\\mathbf{n})$ with the unknown $\\mathbf{n}$ is replaced by an angle $\\theta$ such that $\\mathbf{n}=(\\cos\\theta, \\sin\\theta)$. This model reduction considerably simplify the mathematical analysis of \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-d}.\n\t\n\t\n\tTo close this introduction, we emphasize that the analysis in the present paper might also be of great interest in the numerical study of stochastic Ericksen-Leslie system. In fact, on the one hand our assumptions on the polynomial $f(\\d)$ enable us to consider the typical Ginzburg-Landau function $f_\\varepsilon(\\d)=\\frac{1}{\\varepsilon^2}(1 -\\lvert \\d \\rvert^2) \\d$, see Assumption \\ref{eqn-f} and Remark \\ref{Rem:Ginz-Land-General}. In the numerical context, handling the constraint $\\lvert \\d\\rvert=1$ in the Ericksen-Leslie is a rather challenging task and to overcome this difficulty, one usually use the Ginzburg-Landau approximation, see \\cite{Walkington}. On the other hand, to get convergence and a rate of convergence of a space discretization for parabolic SPDEs, one often has to consider a regular solution in favour of weak solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Preliminary results and notations}\\label{sec-spaces-2}\n\n\t\\subsection{Functional spaces and linear operators}\n\tFollowing \\cite{BHP18} we introduce in this section various notations and results that are frequently used in this paper.\n\t\n\t For two topological spaces $X$ and $Y$ the symbol $X\\hookrightarrow Y$ means that the embedding $X$ is continuously embedded in $Y$. \n\t\n\t\\noindent Let $d\\in \\{2,3\\}$ and assume that $\\mathcal{O} \\subset \\mathbb{R}^d$ is a bounded domain with boundary $\\partial \\mathcal{O}$ of class $\\mathcal{C}^\\infty$.\n\tFor $p\\in [1,\\infty)$ and $k\\in \\mathbb{N}$ the symbols $\\mathrm{L}^p(\\mathcal{O})$ or by $\\mathbb{W}^{k,p}(\\mathcal{O})$ (resp. by $\\mathbf{L}^p(\\mathcal{O})$ or by $\\mathbf{W}^{k,p}(\\mathcal{O})$) respectively denote the Lebesgue and Sobolev spaces of functions $\\mathbf{v}:\\mathbb{R}^d\\to \\mathbb{R}^d$ (resp. $\\mathbf{n}:\\mathbb{R}^d \\to \\mathbb{R}^3$) \n\tFor $p=2$ the function spaces $\\mathbb{W}^{k,2}(\\mathcal{O})$ and $\\mathbf{W}^{k,p}$ are respectively denoted by $\\mathrm{H}^k$ ($\\mathbf{H}^k$) and their norms are \n\tdenoted by $\\lVert \\cdot\n\t\\rVert_k$. The scalar products on $\\mathrm{L}^2$ and $\\mathbf{L}^2$ are denoted by the same symbol $\\langle\n\tu,v\\rangle$ for $u,v\\in \\mathrm{L}^2$ (resp. $u,v\\in \\mathbf{L}^2$ ) and their associated norms is denoted by $\\lVert\n\tu\\rVert$, $u\\in \\mathrm{L}^2$ (resp. $u\\in \\mathrm{L}^2$).\n\tBy $\\mathrm{H}^1_0$ and $\\mathrm{W}^{1,r}_0$, $r>2$, we mean the spaces of functions in $\\mathrm{H}^1$ and $\\mathrm{W}^{1,r}$\n\tthat vanish on the boundary on $\\mathcal{O}$. \n\t It is well known that if $a=\\frac{d}4$, then there exists a constants $c>0$ such that \n\t\\begin{equation} \\label{GAG-l4}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}\\le c \\begin{cases}\n \\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-a} \\lvert \\nabla \\mathbf{u}\n\\rvert^a_{\\mathrm{L}^2} \t\\text{ if } \\mathbf{u}\\in \\mathbb{H}^{1}_{0}\\\\\n\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-a} \\lvert \\mathbf{u}\n\\rvert^a_{\\mathbb{H}^1} \t\\text{ if } \\mathbf{u}\\in \\mathbb{H}^{1},\n\t\\end{cases}\n\t\\end{equation}\n\t\t\\begin{equation} \\label{GAG-LInf}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^\\infty}\\le c \\begin{cases}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}^{1-a} \\lvert \\nabla \\mathbf{u}\n\t\\rvert^a_{\\mathrm{L}^4} \t\\text{ if } \\mathbf{u}\\in \\mathbb{W}^{1,4}_{0}\\\\\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}^{1-a} \\lvert \\mathbf{u}\n\t\\rvert^a_{\\mathbb{W}^{1,4}} \t\\text{ if } \\mathbf{u}\\in \\mathbb{W}^{1,4},\n\t\\end{cases}\n\t\\end{equation}\n\tand since $\\mathrm{H}^{2} \\hookrightarrow \\mathrm{W}^{1,4}$, we have \n\t\\begin{equation}\n\t\\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^\\infty}\\le \\lVert \\mathbf{u} \\rVert_1 ^{1-a}\\lVert\n\t\\mathbf{u} \\rVert_{2}^a, \\; \\mathbf{u}\\in \\mathrm{H}^2 .\\label{GAG-LInf-2}\n\t\\end{equation}\n\t\n\tWe now introduce the following spaces\n\t\\begin{align*}\n\t\\mathcal{V}& =\\left\\{ \\mathbf{u}\\in \\mathcal{C}_{c}^{\\infty }(\\mathcal{O},\\mathbb{R}^d)\\,\\,\\text{such that}%\n\t\\,\\,\\Div \\mathbf{u}=0\\right\\} \\\\\n\t\\mathrm{V}& =\\,\\,\\text{closure of $\\mathcal{V}$ in }\\,\\,\\mathrm{H}_0^{1}(\\mathcal{O}) \\\\\n\t\\mathrm{H}& =\\,\\,\\text{closure of $\\mathcal{V}$ in\n\t}\\,\\,\\mathrm{L}^{2}(\\mathcal{O}).\n\t\\end{align*}\n\tAs usual, we endow $\\mathrm{H}$ with the scalar product and norm of $\\mathrm{L}^2$, and we equip the space $\\mathrm{V}$ with he scalar product\n\t$\\langle \\nabla \\mathbf{u}, \\nabla \\mathbf{v}\\rangle$ which is equivalent to the\n\t$\\mathrm{H}^1(\\mathcal{O})$-scalar product on $\\mathrm{V}$.\n\t\n\tLet $\\Pi: \\mathrm{L}^2 \\rightarrow \\mathrm{H}$ be the Helmholtz-Leray projection\n\tfrom $\\mathrm{L}^2$ onto $\\mathrm{H}$. We denote by $\\mathrm{A}=-\\Pi\\Delta$ the\n\tStokes operator with domain $D(\\mathrm{A})=\\mathrm{V}\\cap \\mathrm{H}^2$, see for instance \\cite[Chapter I, Section 2.6]{Temam} . It is well-known that the spaces $\\mathrm{V}_\\beta:=D(\\mathrm{A}^\\beta)$, $\\beta \\in \\mathbb{R}$, are Hilbert spaces when endowed with the graph inner product and $\\mathrm{V}_\\frac12 = \\mathrm{V}$. It is also well-known that the map $\\mathrm{A}^\\delta: \\mathrm{V}_\\beta \\to \\mathrm{V}_{\\beta-\\delta}$, $\\beta,\\delta \\in \\mathbb{R}$, is a linear isomorphism. For all these facts we refer, for instance, to \\cite{PC+CF}.\n\n\t\n The Neumann Laplacian acting on $\\mathbb{R}^d$-valued\n\tfunction will be denoted by $\\mathrm{A}_1$, that is,\n\t\\begin{equation*}\n\t\\begin{split}\n\tD(\\mathrm{A}_1)&:=\\biggl\\{\\mathbf{u}\\in \\mathbf{H}^2: \\frac{\\partial \\mathbf{u}}{\\partial \\boldsymbol{\\nu}}=0 \\text{ on } \\partial \\mathcal{O}\\biggr\\},\\\\\n\t\\mathrm{A}_1\\mathbf{u}&:=-\\sum_{i=1}^d \\frac{\\partial^2 \\mathbf{u}}{\\partial\n\t\tx_i^2},\\;\\;\\; \\mathbf{u} \\in D(\\mathrm{A}_1).\n\t\\end{split}\n\t\\end{equation*}\n\tNotice that the Neumann Laplacian $\\mathrm{A}_1$ can be viewed as a linear map $\\mathrm{A}_1:\\mathbf{H}^1 \\to (\\mathbf{H}^1)^\\ast$ satisfying \t\n\t\\begin{equation}\\label{Eq:WeakNeumannLaplace}\n\t{}_{(\\mathbf{H}^1)^\\ast}\\langle \\mathrm{A}_1 \\mathbf{u}, \\mathbf{n}\\rangle_{\\mathbf{H}^1}=\\langle \\nabla \\mathbf{u}, \\nabla \\mathbf{n} \\rangle,\\text{ for all } \\mathbf{u},\\mathbf{n} \\in \\mathbf{H}^1.\n\t\\end{equation}\n\tThanks to \\cite[Theorem 5.31]{Haroske} one can define and characterize in standard way the spaces $\\mathbf{X}_\\alpha=D(\\hat{\\mathrm{A}}_1^{\\alpha})$, $\\alpha \\in [0,\\infty)$, where $\\hat{\\mathrm{A}}_1=I+\\mathrm{A}_1$. Also, it can be shown that $\\mathbf{X}_\\alpha \\hookrightarrow \\mathbf{H}^{2\\alpha}$, for all $\\alpha \\geq 0$ and\n\t$\\mathbf{X}:=\\mathbf{X}_{\\frac12}=\\mathbf{H}^1$, see, for instance, \\cite[Sections 4.3.3 \\& 4.9.2]{Triebel}. \n\n\t\n\t\n\tNow, let $\\mathbf{h}\\in\\mathbf{L}^\\infty$ be fixed and define a linear bounded operator $G$ from $\\mathbf{L}^2$ into itself by\n\\begin{equation}\\label{Eq:LinG}\n\t G: \\mathbf{L}^2 \\ni \\mathbf{n} \\mapsto \\mathbf{n}\\times \\mathbf{h}\\in \\mathbf{L}^2.\n\\end{equation}It is straightforward to check that there exists a constant $C>0$ such that\n\t$$ \\lVert G(\\mathbf{n})\\rVert \\le C \\lVert \\mathbf{h}\\rVert_{\\mathbf{L}^\\infty} \\lvert \\mathbf{n} \\rvert_{\\mathbf{L}^2}, \\text{ for all } \\mathbf{n} \\in \\mathbf{L}^2.$$\n\t\n\t\\subsection{The nonlinear terms}\n\t\n\tThroughout this paper $\\mathbf{B}^\\ast$ denotes the dual space of\n\ta Banach space $\\mathbf{B}$. We also denote by $\\langle \\Psi,\n\t\\mathbf{b}\\rangle_{\\mathbf{B}^\\ast,\\mathbf{B}}$ the value of $\\Psi\\in \\mathbf{B}^\\ast$ on\n\t$\\mathbf{b}\\in \\mathbf{B}$. Throughout, $\\partial_{x_i}=\\frac{\\partial}{\\partial x_i} $ and {$\\phi^{(i)}$}\n\tis the $i$-th entry of any vector-valued $\\phi$.\n\t\n\tLet $p,q,r \\in [1,\\infty]$ such that $\\frac 1p +\\frac 1q +\\frac 1r\\le 1.$ Then, we define a trilinear form $b(\\cdot, \\cdot,\\cdot)$ by\n\t\\begin{equation*}\n\tb(\\mathbf{u},\\mathbf{v},\\mathbf{w})=\\sum_{i,j=1}^d \\int_\\mathcal{O}\\mathbf{u}^{(i)}\\frac{\\partial\n\t\t\\mathbf{v}^{(j)}}{\\partial x_i}\\mathbf{w}^{(j)} dx,\\,\\, \\mathbf{u}\\in \\mathrm{L}^p ,\\mathbf{v}\\in\n\t\\mathbb{W}^{1,q}, \\text{ and } \\mathbf{w}\\in \\mathrm{L}^r,\n\t\\end{equation*}\n\tNote if $\\mathbf{v}\\in \\mathbf{W}^{1,q}$ and $\\mathbf{w} \\in \\mathbf{L}^r$, then we have to take the sum over $j$ from $j=1$ to $j=3$.\n\t\n\tIt is well known, see \\cite[Section II.1.2]{Temam}, that there is a bilinear map $B: \\mathrm{V} \\times \\mathrm{V} \\to \\mathrm{V}^\\ast$ such that\n\t\\begin{equation}\\label{DEF-B1}\n\t\\langle\n\tB(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w})\\text{ for } \\mathbf{w}\\in \\mathrm{V},\\text{ and } \\mathbf{u}, \\mathbf{v}\\in \\mathrm{V}.\n\t\\end{equation}\n\tIn a similar way, there is also a bilinear map$\\tilde{B}:\\mathrm{V} \\times \\mathbf{H}^1\\to (\\mathbf{H}^1)^\\ast$\n\tsuch that\n\t\\begin{equation}\\label{DEF-B2}\n\t\\langle\n\t\\tilde{B}(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{(\\mathbf{H}^1)^\\ast,\\mathbf{H}^1}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w})\\,\\, \\text{ for all }\n\t\\mathbf{u} \\in \\mathrm{V}, \\,\\, \\mathbf{v}, \\,\\,\\mathbf{w}\\in \\mathbf{H}^1.\n\t\\end{equation}\n\tThe following lemma was proved in \\cite[Section II.1.2]{Temam}. \n\t\\begin{lem}\\label{LEM-B}\n\t\tThe bilinear map $B(\\cdot, \\cdot)$ maps continuously $\\mathrm{V}\\times\n\t\t\\mathrm{H}^1$ into $\\mathrm{V}^\\ast$ and\n\t\t\\begin{align}\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w}), \\text{ for all }\n\t\t\\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in \\mathrm{H}^1,\\mathbf{w} \\in \\mathrm{V},\\label{B1}\\\\\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=-b(\\mathbf{u},\\mathbf{w},\\mathbf{v})\n\t\t\\text{ for all } \\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in \\mathrm{H}^1,\\mathbf{w} \\in \\mathrm{V},\\label{B2}\\\\\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{v}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=0 \\,\\, \\text{ for all } \\mathbf{u}\\in \\mathrm{V},\n\t\t\\mathbf{v}\\in \\mathrm{V},\\label{B3}\\\\\n\t\t&\\lvert B(\\mathbf{u},\\mathbf{v})\\rvert_{\\mathrm{V}^\\ast}\\le C_0 \\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-\\frac d4}\n\t\t\\lvert\\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^2}^{\\frac d4} \\lvert \\mathbf{v}\\rvert_{\\mathrm{L}^2}^{1-\\frac d4}\\rvert_{\\mathrm{L}^2}\n\t\t\\nabla \\mathbf{v}\\rvert_{\\mathrm{L}^2}^\\frac d4, \\text{ for all } \\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in\n\t\t\\mathrm{H}^1.\\label{B4}\n\t\t\\end{align}\n\t\\end{lem}\n\t\\begin{lem}\\label{LEM-G1}\n\tThere exists a constant $C_1>0$\n\t\tsuch that\n\t\t\\begin{align}\n\t&\t\\lVert \\tilde{B} (\\mathbf{v}, \\mathbf{n})\\lVert \\le C_1 \\lVert \\mathbf{v}\\rVert_{\\mathrm{L}^2}^{1-\\frac\n\t\t\td4}\\rVert \\nabla \\mathbf{v}\\rVert_{\\mathrm{L}^2}^\\frac d4 \\lVert \\mathbf{n}\\rVert_{\\mathbf{H}^1}^{1-\\frac d4}\n\t\t\\lVert \\mathbf{n} \\rVert_{\\mathbf{H}^2}^{\\frac d4}, \\text{ for all } \\mathbf{v} \\in \\mathrm{V},\n\t\t\\mathbf{n}\\in \\mathrm{H}^2,\\label{EST-G1}\\\\\n\t&\t\\langle \\tilde{B}(\\mathbf{v},\\mathbf{n}),\\mathbf{n}\\rangle=0,\\text{\n\t\t\tfor\n\t\t\tany } \\mathbf{v}\\in \\mathrm{V}, \\mathbf{n} \\in \\mathrm{H}^2.\\label{tild-b-0}\n\t\t\\end{align}\n\t\\end{lem}\n\t\\begin{proof}\n\tThe estimate \\eqref{EST-G1} follows from \\cite[Lemma 6.2]{BHP18} and the Gagliardo-Nirenberg estimate \\eqref{GAG-l4}. The proof of \\eqref{tild-b-0} and \\eqref{B3} are the same, see \\cite[Section II.1.2]{Temam}. \n\t\\end{proof}\n\t\n\t\n\tLet $r,\\,p,\\, q\\in(1,\\infty)$ such that $ \\frac 1p\n\t+\\frac 1q+\\frac 1r\\le 1$. For \\text{ $\\d_1\\in \\mathbf{W}^{1,p}$, $\\d_2\\in \\mathbf{W}^{1,q}$ and $\\mathbf{u}\\in\n\t\t\\mathbb{W}^{1,r}$ } we set \n\t\\begin{equation}\\label{INT-md}\n\t\\mathfrak{m}(\\d_1, \\d_2,\\mathbf{u})= -\\sum_{i,j=1}^d\\sum_{k=1}^3 \\int_\\mathcal{O}\n\t\\partial_{x_i}\\mathbf{n}_1^{(k)} \\partial_{x_j}\\mathbf{n}_2^{(k)} \\partial_{x_j}\\mathbf{u}^{(i)}\\,dx. \n\t\\end{equation}\n\tSince $d\\le 4$,\n\tthe integral in \\eqref{INT-md} is also well defined for $\\mathbf{n}_1,\n\t\\d_2 \\in \\mathbf{H}^2$ and $\\mathbf{u}\\in \\mathrm{V}$.\n\t\n\t\n\tWe recall the following proposition which can be found in \\cite[Proposition 2.2 \\& Remark 2.3]{BHP18}.\n\t\\begin{prop}\\label{LEM-M}\n\t\tLet $d\\in [1,4]$. There exists a bilinear map $M:\\mathbf{H}^2\\times \\mathbf{H}^2 \\to \\mathrm{V}^\\ast $ such that\n\t\t\\begin{align}\\label{def-Md}\n\t\t&\\langle M(\\mathbf{n}_1,\\d_2), \\mathbf{u}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}= \\mathfrak{m}(\\d_1,\\d_2,\\mathbf{u}), \\;\\text{$\\d_1, \\, \\d_2 \\in \\mathbf{H}^2$, } \\mathbf{u} \\in \\mathrm{V},\\\\\t\n\t\t&\t\\langle M(\\mathbf{f}, \\mathbf{g}), \\mathbf{v} \\rangle_{\\mathrm{V}^\\ast, \\mathrm{V}}= \\langle \\Pi[\\Div (\\nabla \\mathbf{f}\n\t\t\\odot \\nabla \\mathbf{g})], \\mathbf{v} \\rangle \\text{ for all } \\mathbf{f}, \\mathbf{g}\\in \\mathbf{X}_{1} \\text{ and } \\mathbf{v} \\in \\mathrm{H},\\label{Eq:Identity-M-L2}\\\\\n\t\t&\t\\langle \\tilde{B}(\\mathbf{v},\\mathbf{n}), \\mathrm{A}_1 \\mathbf{n}\\rangle+\\langle\n\t\tM(\\mathbf{n},\\d), \\mathbf{v}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=0, \\text{ for all } \\mathbf{v}\\in \\mathrm{V}, \\mathbf{n} \\in \\mathbf{X}_1,\\label{G1-eq-Md}.\n\t\t\\end{align}\n\t\\end{prop}\n\tIn some places in this manuscript we use the following shorthand notations:\n\t$$ B(\\mathbf{u}):= B(\\mathbf{u},\\mathbf{u}) \\text{ and } M(\\mathbf{n}):= M(\\mathbf{n},\\mathbf{n}),$$ for all $\\mathbf{u}$ and $\\mathbf{n}$ such that the above quantities are meaningful.\n\n\tWe now fix the standing assumptions on the function $f(\\cdot)$.\n\t\\begin{assum}\\label{eqn-f}\n\t\tLet $I_d$ be the set defined by\n\t\t\\begin{equation}\\label{DEG-POL}\n\t\tI_d=\\begin{cases}\n\t\t\\mathbb{N}:=\\{1, 2, 3, \\ldots\\} \\text{ if } d=2,\\\\\n\t\t\\{1\\}, \\text{ if } d=3.\n\t\t\\end{cases}\n\t\t\\end{equation}\n\t\tWe fix $N\\in I_d\n\t\t$ and $a_k\\in \\mathbb{R}$, $k=0,\\ldots, N$,\n\t\twith $a_N<0$. We define a function $\\tilde{f}:[0,\\infty) \\rightarrow \\mathbb{R}$ by\n\t\t$$ \\tilde{f}(r)=\\sum_{k=0}^N a_k r^k, \\text{ for all } r\\in \\mathbb{R}_+.$$\n\t\tWe define a map $f:\\mathbb{R}^3\\rightarrow \\mathbb{R}^3$ by $f(\\d)=\\tilde{f}(\\vert \\d\\vert^2)\\d$ where $\\tilde{f}$ is as above.\n\t\t\n\t\tWe now assume that there exists $F: \\mathbb{R}^3 \\rightarrow \\mathbb{R}$ a Fr\\'echet differentiable map such that $$ F^\\prime(\\d)[\\mathbf{g}]= f(\\d)\\cdot \\mathbf{g},\n\t\\text{ $\\d\\in \\mathbb{R}^d$, $\\mathbf{g}\\in \\mathbb{R}^d$}.$$ Note that if $\\tilde{F}$ is a map such that $\\tilde{F}^\\prime= \\tilde{f}$ and $\\tilde{F}(0)=0$, then, there is $U$ is a polynomial function with $\\text{deg}(U)\\le N$ and $a_{N+1}<0$ such that \n\t$ \\tilde{F}(r)=a_{N+1}r^{N+1}+U(r).$\n\t\\end{assum}\n\n\t\n\t\\begin{Rem}\\label{REM-H2}\n\t\t\\begin{enumerate}[(i)]\n\t\t\\item \\label{Itemi:REM-H1 } There exists a constant $\\ell_3>0$ such that\n\t\t\t\\begin{align}\n\t\t\t\\lvert \\tilde{f}^{\\prime\\prime}(r) \\rvert\\le \\ell_3(1+r^{N-2}), \\,\\, r>0.\\label{ST6-B-2}\n\t\t\t\\end{align}\n\t\t\\item \\label{Itemi:REM-H2} From \\eqref{ST6-B-2}, we infer that there exist $c_0,c_1,c_3>0$ such that\n\t\t\t$$ \\lvert f(\\mathbf{n})\\rvert \\le c_0 (1+\\vert \\d\\vert^{2N+1}), \\text{ } \\lvert f^{\\prime}(\\d)\\rvert\\le\n\t\t\tc_1(1+\\vert \\mathbf{n} \\vert^{2N}) \\text{ and } \\lvert f^{\\prime \\prime}(\\d)\\rvert\\le c_2 (1+\\vert \\mathbf{n} \\vert^{2N-1}) \\text{ for all } \\mathbf{n}\\in \\mathbb{R}^n.$$\n\t\t\t\\item Let $\\tilde{q}=4N+2$. It is easy to show that there exists $C>0$ such that \\mbox{ for all } $\\d\\in \\mathrm{H}^2$\n\t\t\t\\begin{align}\n\t\t\t\\lVert \\mathrm{A}_1 \\d\\rVert^2=&\\lVert \\mathrm{A}_1 \\d +f(\\d)-f(\\d)\\rVert^2\n\t\t\t\\le 2 \\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2+2 \\lVert f(\\d)\\rVert^2,\\nonumber \\\\\n\t\t\t\\le & 2 \\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2+C \\lVert \\d\\rVert^{\\tilde{q}}_{\\mathbf{L}^{\\tilde{q}}}+C.\\label{bigdandel}\n\t\t\t\\end{align}\n\t\n\t\t\t\n\t\t\t\\item Since the norm $\\lVert \\cdot \\rVert_2$ is equivalent to $\\lVert \\cdot \\rVert + \\lVert \\mathrm{A}_1 \\cdot \\rVert $ on $D(\\mathrm{A}_1)$, there exists $C>0$\n\t\t\tsuch that\n\t\t\t\\begin{equation}\\label{bigdanh2}\n\t\t\t\\lVert \\d\\rVert^2_2\\le C (\\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2\n\t\t\t+\\lVert \\d\\rVert^{\\tilde{q}}_{\\mathbf{L}^{\\tilde{q}}}+1), \\mbox{ for all }\\d\\in D(\\mathrm{A}_1).\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\t\\item\\label{Rem-Linf-H1Delta}\n\t\t\tSince $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{4N+2}$, $N \\in I_d$, we infer from \\eqref{bigdanh2} that $\\mathbf{n}\\in \\mathbf{H}^2$ if $\\mathbf{n} \\in \\mathbf{H}^1$ and $\\mathrm{A}_1 \\mathbf{n} -f(\\mathbf{n})\\in \\mathbf{L}^2$.\n\t\t\\end{enumerate}\n\t\\end{Rem}\n\t\t\\begin{Rem}\\label{Rem:Ginz-Land-General}\n\t\tLet $\\varepsilon>0$ and $\\tilde{f}_\\varepsilon(r):=\\frac{1}{\\varepsilon^2}(-r+1)$, $r\\in [0,\\infty) $. Examples of maps $f$ and $F$ satisfying Assumption \\ref{eqn-f} are the following\n\t\t$$f(\\d):=\\tilde{f}(\\vert \\d \\vert^2)\\d=\\frac{1}{\\varepsilon^2}(1-\\lvert \\d \\rvert^2)\\d \\text{ and } F(\\d):=\\frac1{4\\varepsilon^2} [\\tilde{f}(\\vert \\d \\vert^2) ]^2, \\; \\d\\in \\mathbb{R}^d.$$\n\t\\end{Rem}\n\t\\subsection{The assumption on the coefficients of the noise}\n\t\n\t\\begin{assum}\\label{Assum:Usual hypotheses}\n\t\tThroughout this paper we are given a complete filtered probability space $(\\Omega,\n\t\t\\mathcal{F}, \\mathbb{P})$\n\t\twith the filtration $\\mathbb{F}=\\{\\mathcal{F}_t: t\\geq 0\\}$\n\t\tsatisfying the usual hypothesis, \\textit{i.e.},\n\t\tthe filtration is right-continuous and all null sets of $\\mathcal{F}$ are elements of $\\mathcal{F}_0$.\n\t\\end{assum}\n\tThroughout, let $\\mathrm{K}_1$ be a separable Hilbert space, and $W_1=(W_1(t))_{t\\geq 0}$ and \n\t$W_2=(W_2(t))_{t\\geq 0}$ be independent $\\mathrm{K}_1$-cylindrical\n\tWiener process and standard Brownian motion on $(\\Omega,\n\t\\mathcal{F},\\mathbb{F}, \\mathbb{P})$. If $\\mathrm{K}=\\mathrm{K}_1\\times \\mathbb{R}$ then we can assume that $W=(W_1(t),W_2(t))$ is $\\mathrm{K}$-cylindrical\n\tWiener process. \n\t\n\n\tLet $\\tilde{\\mathrm{K}}$ and $\\tilde{\\mathrm{H}}$ be a separable Hilbert and Banach spaces. We denote by $\\gamma(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{ H}})$ the space of $\\gamma$-radonifying operators which generalises the space of Hilbert-Schmidt operators $\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{ H}})$ if $\\tilde{\\mathrm{ H}}$ is a separable Hilbert space, see \\cite{ZB-97}. Let $\\mathscr{M}^2(\\Omega\\times [0,T]; \\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} ))$ the space of all equivalence classes of progressively measurable processes $\\Psi: \\Omega\\times [0,T]\\to\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} )$ satisfying\n\t$$ \\mathbb{E}\\int_0^T \\Vert \\Psi(s)\\Vert^2_{\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} )}ds <\\infty.$$\n\t\tFor a $\\tilde{\\mathrm{K}}$-cylindrical Wiener process $\\tilde{W}$ and $\\Psi \\in\\mathscr{M}^2(\\Omega\\times [0,T]; \\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} ))$ the process $M$ defined by\n\t$ M(t) =\\int_0^t \\Psi(s)d\\tilde{W}(s), t\\in [0,T],$ is a $\\tilde{\\mathrm{H}}$-valued martingale. \n\tFor more detail on the theory of stochastic integration we refer to \\cite[Section 26 ]{Metivier_1982} and \\cite[Chapter 4]{DP+JZ-14}. \n\nLet $G$ be the map defined in \\eqref{Eq:LinG} and $G^2=G\\circ G$. Then, we have the following relation identity\n\tStratonovich and It\\^o's integrals, see \\cite{Brz+Elw_2000},\n\t\\begin{equation*}\n\tG(\\mathbf{n})\\circ dW_2= \\frac 12 G^2(\\mathbf{n}) \\,dt +\n\tG(\\mathbf{n})\\,dW_2.\n\t\\end{equation*}\n\n\t\n\t\n\t\n\tWe now introduce the standing set of hypotheses on the function $S$.\n\t\n\t\\begin{assum}\\label{HYPO-ST}\n\t\tWe assume that $S: \\mathrm{H}\\to \\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})$ is a globally Lipschitz map. In particular,\n\t\tthere exists $\\ell_5\\geq 0$\n\t\tsuch that\n\t\t\\begin{equation}\\label{Eq:Hypo-ST}\n\t\t\\lVert S(\\mathbf{u})\\rVert^2_{\\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})}\\leq \\ell_5 (1+\\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^2}^2),\\;\\; \\mbox{ for all } \\mathbf{u} \\in \\mathrm{H}.\n\t\t\\end{equation}\n\t\\end{assum}\n\t\\begin{Rem}\\label{Rem:HYPO-ST}\n\t\tNotice that the assumption \\eqref{Eq:Hypo-ST} implies that there exists a constant $\\ell_6>0$ such that\n\t\t\\begin{equation}\\label{Eq:Hypo-ST-Rem}\n\t\t\\lVert S(\\mathbf{u})\\rVert^2_{\\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})}\\leq \\ell_5 (1+\\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^2}^2),\\;\\; \\mbox{ for all } \\mathbf{u} \\in \\mathrm{H}.\n\t\t\\end{equation}\n\t\\end{Rem}\n\n\n\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Existence and uniqueness of local and global strong Solution}\\label{SLC-Sect4}\nUsing the notations of Section \\ref{sec-spaces-2}, the system \\eqref{eqn-SLQE-v}-\\eqref{Init-Cond} can be written in the abstract\n\tform\n\t\\begin{align}\n\t&d\\mathbf{v}(t)+\\biggl(\\mathrm{A}\\mathbf{v}(t)+B(\\mathbf{v}(t),\n\t\\mathbf{v}(t))+M(\\mathbf{n}(t))\\biggr)dt=S(\\mathbf{v}(t))dW_1,\\label{ABS-v1}\\\\\n\t&d\\mathbf{n}(t)+\\biggl(\\mathrm{A}_1\\mathbf{n}(t)+ \\tilde{B}(\\mathbf{v}(t),\\mathbf{n}(t))-\n\tf(\\mathbf{n}(t))-\\frac 12 G^2(\\mathbf{n}(t))\\biggr)dt=G(\\mathbf{n}(t))dW_2,\\\\\n\t\\label{ABt-d1}\n\t& \\v(0)=v_0\n\t\\text{ and } \\d(0)=d_0.\n\t\\end{align}\nIn this section we prove the existence and uniqueness of the strong solution to problem \\eqref{ABS-v1}-\\eqref{ABt-d1}. \n\n\t\\subsection{Definition of local solutions}\n\tLet $(B_i, \\lVert \\cdot \\rVert_{B_i})$, $i=1, 2$, be two Banach spaces. We endow $B_1\\times B_2$ with the norm $\n\t\\rVert(b_1,b_2)\\lVert=\\sqrt{\\lVert b_1\\rVert_{B_1}^2 +\\lVert b_2\\rVert_{{B}_2}^2}.$\n\tHenceforth, we put\n\t\\begin{equation}\\label{eqn-spaces}\n\t\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{\\frac12}, \\;\\; \\mathscr{V}=\\mathrm{V}\\times \\mathbf{X}_{1} \\mbox{ and } \\mathscr{E}=\\mathrm{V}_1 \\times \\mathbf{X}_{\\frac32}.\n\t\\end{equation}\n\tNext, we denote by $\\{\\mathbb{S}_1(t)\\}_{t\\geq 0}$ and $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$ the analytic semigroups \n\tgenerated by\n\t$-\\mathrm{A}$ on $\\mathrm{H}$ and by $\\mathrm{A}_1$ on $\\mathbf{L}^2$, respectively. \nIt is well-known that \n\tthe space $\\mathbf{X}_{\\frac12}$ is invariant wrt $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$. The restriction of $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$ to $\\mathbf{X}_{\n\t\t\\frac12}$ is also an analytic semigroup which will be denoted by $\\{\\mathbb{S}_2(t)\\}_{t\\geq 0}$.\n\tThe minus infinitesimal\n\tgenerator $\\tilde{\\mathrm{A}}_1$ of $\\{\\mathbb{S}_2(t)\\}_{t\\geq 0}$ is the part\n\tof $\\mathrm{A}_1$ on $\\mathbf{X}_{\\frac12}$, that is,\n\t\\begin{align*}\n\tD(\\tilde{\\mathrm{A}}_1)=\\{\\mathbf{u} \\in D(\\mathrm{A}_1): \\mathrm{A}_1\\mathbf{u}\\in \\mathbf{X}_{\\frac12}\\},\\;\\;\n\t\\tilde{\\mathrm{A}}_1\\mathbf{u}=\\mathrm{A}_1\\mathbf{u} \\text{ for all } \\mathbf{u} \\in D(\\tilde{\\mathrm{A}}_1).\n\t\\end{align*}\n\tNote that $\\mathbf{X}_{\\frac32}\\subset D(\\tilde{\\mathrm{A}}_1).$\n\tWith all the above notation, the problem \\eqref{ABS-v1}-\\eqref{ABt-d1} can be rewritten as the following stochastic evolution equation in the space $\\mathscr{H}$,\n\t\\begin{equation}\\label{ABSTRACT-LC}\n\td\\mathbf{y}(t) +\\mathbf{A}\\mathbf{y}(t) dt+\\mathbf{F}(\\mathbf{y}(t))\n\tdt+\\mathbf{L}(\\mathbf{y}(t))dt=\\mathbf{G}(\\mathbf{y}(t)) d{W}(t),\n\t\\end{equation}\n\twhere, for $\\mathbf{y}=(\\mathbf{v}, \\d)\\in E$ and $k=(k_1,k_2)\\in \\mathrm{K}$,\n\t\n\t\\begin{equation}\\label{eqn-def-A-F}\n\t\\mathbf{A}\\mathbf{y}=\\begin{pmatrix} \\mathrm{A} \\mathbf{v} \\\\\n\t \\mathrm{A}_1 \\mathbf{n}\n\t\\end{pmatrix},\\;\\;\n\t\\mathbf{F}(\\mathbf{y})=\\begin{pmatrix} B(\\mathbf{v},\\mathbf{v})+M(\\mathbf{n})\\\\\n\t\\tilde{B}(\\mathbf{v},\\mathbf{n})-f(\\mathbf{n})\n\t\\end{pmatrix},\n\t\\end{equation}\n\t\\begin{equation}\\label{eqn-def-L-G}\n\t\\mathbf{L}(\\mathbf{y})=\\begin{pmatrix} 0\\\\ -\\frac 1 2 G^2(\\mathbf{n})\n\t\\end{pmatrix}, \\mathbf{G}(\\mathbf{y})k=\\begin{pmatrix}S(\\mathbf{u})k_1\\\\\n\tG(\\mathbf{n})k_2 \\end{pmatrix}.\n\t\\end{equation}\n\tThe operator $-\\mathbf{A}$ generates an analytic semigroup $\\{\\mathbb{S}(t)\\}_{t\\geq 0}$ on $\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{\\frac12}$\n\tdefined by\n\t\\begin{equation*}\n\t\\mathbb{S}(t)\\begin{pmatrix}\\mathbf{v}\\\\\\d\n\t\\end{pmatrix}=\\begin{pmatrix}\\mathbb{S}_1(t)\\mathbf{v}\\\\\n\t\\mathbb{S}_2(t)\\d\n\t\\end{pmatrix}, \\;\\; (\\mathbf{v},\\d)\\in \\mathscr{H}.\n\t\\end{equation*}\n\n\n\tImportant properties of $\\{\\mathbb{S}(t): t\\ge 0\\}$ are given in the next two lemma.\n\t\\begin{lem}\\label{SEM-1} \\label{SEM-2}\n\t\tLet $T\\in (0,\\infty)$, $\\mathbf{g}=\\begin{pmatrix}\n\t\t\\tilde{g} \\\\ g\n\t\t\\end{pmatrix}\\in L^2(0,T; \\mathrm{H} \\times \\mathbf{X}_{\\frac12})$ and\n\t\t$\n\\begin{pmatrix}\n\t\\mathbf{v}(t)\\\\\n\t\\mathbf{n}(t)\n\t\\end{pmatrix}= \\int_0^t\\mathbb{S}(t-s) \\mathbf{g}(s)\\, ds, \\;\\; t\\ge 0.$ Then, there exists $c_1>0$ such that\n\t\t\\begin{equation*}\n\t\t\\left \\lVert \\begin{pmatrix}\n\t\t\\mathbf{v}\\\\\\mathbf{n}\n\t\t\\end{pmatrix} \\right\\rVert_{C([0,T];\\mathrm{V} \\times \\mathbf{X}_{1})}+\\left \\lVert \\begin{pmatrix}\n\t\t\\mathbf{v}\\\\\\mathbf{n}\n\t\t\\end{pmatrix} \\right\\rVert_{L^2(0,T;D(\\mathrm{A}) \\times \\mathbf{X}_{\\frac32}) }\\le c_1 \\left \\lVert \\begin{pmatrix}\n\t\t\\tilde{g} \\\\ g\n\t\t\\end{pmatrix}\\right \\rVert_{L^2(0,T;\n\t\t\t\\mathrm{H}\\times \\mathbf{X}_{\\frac12})}.\n\t\t\\end{equation*}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThis result is well-known and is a special case of \\cite[Lemma 1.2]{Pardoux}.\n\t\\end{proof}\n\t\\begin{lem}\\label{SEM-3}\n\t\t\t\tLet $T\\in (0,\\infty)$, $\\zeta= \\begin{pmatrix}\n\t\t\t\t\\zeta_1\\\\ \\zeta_2\n\t\t\t\t\\end{pmatrix}\\in \\mathscr{M}^2(0,T;\\mathrm{H}\\times \\mathbf{X}_1) $ and \n\t\t\t\t$ \\begin{pmatrix}\n\t\t\t\t\\mathbf{w}_1(t)\\\\ \\mathbf{w}_2(t)\n\t\t\t\t\\end{pmatrix} = \\int_0^t \\mathbb{S}(t-s) \\zeta(s) dW(s), t \\ge 0. $ Then, \n\t\tthere exists $C>0$ such that \n\t\t\\begin{equation*} \\mathbb{E} \\left \\lVert\n\t\\begin{pmatrix}\n\t\\mathbf{w}_1 \\\\ \\mathbf{w}_2\n\t\\end{pmatrix} \\right \\rVert^2_{C([0,T];\\mathrm{V} \\times \\mathbf{X}_1 )}+\\mathbb{E}\\left \\lVert\n\t\t\\begin{pmatrix}\n\t\t\\mathbf{w}_1 \\\\ \\mathbf{w}_2\n\t\t\\end{pmatrix} \\right \\rVert^2_{L^2(0,T;D(\\mathrm{A}) \\times \\mathbf{X}_{\\frac32})}\\le C \\mathbb{E}\\left \\lVert\n\t\\begin{pmatrix}\n\t\\zeta_1\\\\ \\zeta_2\n\t\\end{pmatrix}\\right \\rVert^2_{L^2(0,T;\\mathrm{V} \\times \\mathbf{X}_1)},\\;\\; T\\ge 0.\n\t\t\\end{equation*}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThis result is also well-known, see \\cite[Lemma 1.4]{Pardoux}.\n\t\\end{proof}\n\n\tLet us recall the following notations\/definition which are borrowed\n\tfrom \\cite{Kunita-90}, see also \\cite{Brz+Elw_2000}.\n\n\n\n\t\\begin{Def}\\label{def-accessible stopping time}\n A random function $\\tau:\\Omega \\to [0,\\infty]$ is called a stopping time, see \\cite[Definition I.2.1]{Kar-Shr-96}, \\cite[Definition 4.1]{Metivier_1982} and \\cite[section III.5]{Elw_1982}, iff for each $t\\geq 0$, the set\n$\\{\\omega \\in \\Omega: t< \\tau(\\omega)\\} \\in \\mathcal{F}_t$ (or equivalently, $\\{\\omega \\in \\Omega: \\tau(\\omega) \\leq t\\} \\in \\mathcal{F}_t$). A stopping time\n\t\t$\\tau:\\Omega \\to [0,\\infty]$ is called accessible, see \\cite[section 2.1, p. 45]{Kunita-90}, iff there exists an increasing\n\t\tsequence\\footnote{In the sense that for all $n\\in \\mathbb{N}$, $\\tau_n \\leq\n\t\t\\tau_{n+1}$, $\\mathbb{P}$-a.s. } of stopping times $\\tau_n{:\\Omega \\to [0,\\infty)}$ such that $\\mathbb{P}$-a.s. \n\t\\begin{inparaenum}\n\t\t\\item[(i)] for all $n\\in \\mathbb{N}$, $\\tau_n <\n\t\t\\tau$;\n\t\t\\item[(ii)] and $\\lim_{n\\to \\infty} \\tau_n =\\tau$.\n\t\\end{inparaenum}\n\nThe sequence $(\\tau_n)_{n\\in \\mathbb{N}}$ as above is usually called an announcing sequence for $\\tau$.\n\t\\end{Def}\n\n\\begin{Rem}\\label{rem-predictable stopping time}\nUnder the Assumption \\ref{Assum:Usual hypotheses} we have the following facts.\n\\begin{trivlist}\n\n\t\t\\item[(i)]\nIt follows from \\cite[Proposition 6.6 (3)]{Metivier_1982} that a stopping time is accessible if and only if it is predictable. Let us recall, see \\cite[Definition 4.9]{Metivier_1982}, that\na stopping time $\\tau$ is predictable iff its graph $[\\tau]:=\\{(t,\\omega)\\in [0,\\infty)\\times\\Omega: t=\\tau(\\omega)\\}$ is a predictable set.\nThe $\\sigma$-field $\\mathcal{P}$ of predictable sets is generated by the family $\\mathcal{R}:=\\{ (s,t]\\times F: 0\\leq s\\leq t, F\\in \\mathcal{F}_s\\}\\cup \\{\\{0\\} \\times F: F\\in \\mathcal{F}_0\\}$, see \\cite[Theorem 3.3]{Metivier_1982}.\n\t\t\\item[(ii)] If $(\\tau_n)_{n \\in \\mathbb{N}}$ is a sequence of accessible stopping times, then $\\sup_{n\\in \\mathbb{N}} \\tau_n$ is also an accessible stopping times, see \\cite[Proposition 6.6]{Metivier_1982}. In particular, If $\\tau$ and $\\sigma$ accessible stopping times with announcing sequences $(\\tau_n)_{n \\in \\mathbb{N}}$ and $(\\sigma_n)_{n \\in \\mathbb{N}}$, then $\\tau\\vee \\sigma$ is an accessible stopping time with announcing sequence $(\\tau_n \\vee \\sigma_n)_{n \\in \\mathbb{N}}$. Furthermore, if $\\mathcal{A}$ is an arbitrary family of accessible stopping times then a family\n\t\t\\[\n\t\t\\mathcal{B}:=\\{ \\sup \\mathcal{C}: \\mathcal{C} \\mbox{ is a finite subset of } \\mathcal{A} \\}\n\t\t\\]\n\t\tis also a family of accessible stopping times such that $\\mathcal{A} \\subset \\mathcal{B}$ and the supremum of each finite subset of $\\mathcal{B}$ belongs to $\\mathcal{B}$. In particular, if $\\Delta$\n\t\tis the family of all accessible stopping times, then the supremum of each finite subset of $\\Delta$ belongs to $\\Delta$.\n\\end{trivlist}\n\n\\end{Rem}\n\n\t\\textbf{Notation}. For a stopping time $\\tau$ we set\n\n\t\\[ \\Omega_t(\\tau) \n\t\\{ \\omega \\in \\Omega : t < \\tau(\\omega)\\},\n\t\\\n\t\\[\n\t[0,\\tau)\\times \\Omega \n\t\\{ (t,\\omega) \\in [0,\\infty)\\times \\Omega: 0\\le t < \\tau(\\omega)\n\t\\}.\n\t\\\n\t\n\t\\begin{Def}\\label{def-adapted process}\n Assume that $X$ is a {topological space}.\n\t\tAn $X$-valued process {local} process $\\eta : [0,\\tau) \\times \\Omega \\to X$ (we will also write $ \\eta(t)$, $t< \\tau$) is\n\t\tadmissible iff\n\t\t\\begin{inparaenum}\n\t\t\t\\item[(i) ] it is adapted, i.e. $\\eta|_{\\Omega_t(\\tau)}: \\Omega_t(\\tau) \\to\n\t\t\tX$ is $\\mathcal{F}_t$ measurable, for all $t\\ge 0$; \\item[(ii)]\n\t\t\tfor almost all $\\omega \\in \\Omega$, the function $[0,\n\t\t\t\\tau(\\omega))\\ni t \\mapsto \\eta(t, \\omega) \\in X$ is continuous.\n\t\t\t\\end{inparaenum}\n\t\t%\n\t\t\n\t\tTwo {local} processes $\\eta_i: [0,\\tau_i) \\times \\Omega \\to X$,\n\t\t$i=1,2$ are called equivalent (and we will write $(\\eta_1,\\tau_1) \\sim\n\t\t(\\eta_2,\\tau_2)$) iff $\\tau_1=\\tau_2$ $\\mathbb{P}$-a.s. and, for all $t>0$,\n\t\tthe following condition holds\n\t\t\\[\n\t\t\\eta_1(\\cdot,\\omega)= \\eta_2(\\cdot,\\omega) \\mbox{ on } [0,t], ; \\mbox{ \t\tfor a.e. $\\omega \\in \\Omega_t(\\tau_1)\\cap \\Omega_t(\\tau_2)$}.\n\t\t\\]\n\t\tNote that if { two local} admissible processes $\\eta_i : [0,\\tau_i) \\times \\Omega \\to\n\t\tX$, $i=1,2$ are such that for all $t>0$\n\t\t$\\eta_1(t)|_{\\Omega_t(\\tau_1)}= \\eta_2(t)|_{\\Omega_t(\\tau_2)}$\n\t\t$\\mathbb{P}$-a.s., then they are equivalent.\n\t\\end{Def}\n\n\\begin{Rem}\\label{rem-adapted local process}\nLet $\\tau$ be an accessible stopping time with an announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ and $\\eta : [0,\\tau) \\times \\Omega \\to X$ is a local process.\n Kunita \\cite[section 2.1, p. 46]{Kunita-90} defined $\\eta$ to be a local adapted process iff\n the following condition is satisfied for every $n$, the stopped process $(\\eta_{t\\wedge \\tau_n})_{t\\geq 0}$ is adapted. We do not know how condition (i) from Definition \\ref{def-adapted process} is related to Kunita's definition. \n\\end{Rem}\n\t\\begin{Def}\\label{def-progrssively measurable process}\nLet $\\tau$ be an accessible stopping time with an announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$. Motivated by \\cite[Proposition 2.18]{Kar-Shr-96}, a local process $\\eta : [0,\\tau) \\times \\Omega \\to X$ is called progressively\nmeasurable iff for every $n$, the stopped process $(\\eta_{t\\wedge \\tau_n})_{t\\geq 0}$ is progressively measurable. \n\\end{Def}\n\tWe now define some concepts of solution to \\eqref{ABSTRACT-LC}, see \\cite[Definition 4.2]{Brz+Millet_2012} or \\cite[Definition\n\t2.1]{Mikulevicius}.\n\t\\begin{Def}\\label{def-local solution} Let $\\mathbf{y}_0:\\Omega \\to V$ be $\\mathcal{F}_0$-measurable random variable satisfying $\\mathbb{E} \\Vert \\mathbf{y}_0\\Vert^2_{\\mathscr{V}}<\\infty$. A local\n\t\tsolution to problem \\eqref{ABSTRACT-LC}(with the initial time\n\t\t$0$) is a pair $(\\mathbf{y},\\tau)$ such that\n\t\t\\begin{enumerate}[(1)]\n\t\t\t\\item $\\tau$ is an accessible stopping time with an announcing sequence $(\\tau_n)_{n\\in \\mathbb{N}}$,\n\t\t\t\\item $\\mathbf{y}: [0,\\tau)\\times \\Omega \\to V$ is an admissible process,\n\t\t\t\\item for every $n\\in \\mathbb{N}$ and $t\\in [0,\\infty)$,\n\t\t\twe have\n\t\t\t\\begin{align}\n\t\t\t\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau_n]} \\Vert\n\t\t\t\t\\mathbf{y}(s)\\Vert^2_{\\mathscr{V}} +\\int_0^{t\\wedge \\tau_n} \\Vert \\mathbf{y}(s)\\Vert_\\mathscr{E}^2 \\,\n\t\t\t\tds\\Big)<\\infty,\\label{eq-locsol_01-a}\n\t\t\t\t\\end{align}\n\t\t\t\tand $\\mathbb{P}$-a.s.\n\t\t\t\t\\begin{align}\n\t\t\t\t\\mathbf{y}(t\\wedge \\tau_n)= \\mathbb{S}(t\\wedge\n\t\t\t\t\\tau_n)\\mathbf{y}_0-\\int_0^{t\\wedge \\tau_n}\n\t\t\t\t\\mathbb{S}(t\\wedge\\tau_n-s)[ \\mathbf{F}(\\mathbf{y}(s\\cnew{\\wedge \\tau_n}))+\\mathbf{L}(\\mathbf{y}(s\\cnew{\\wedge \\tau_n})] \\,ds +I_{\\tau_n}(t\\wedge \\tau_n), \\label{eq-locsol_01-b}\n\t\t\t\\end{align}\nwhere $I_{\\tau_n}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eq-locsol_01-c}\n\t\t\t\\begin{split}\nI_{\\tau_n}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n)}(s)\\mathbb{S}(t-s)\\mathbf{G}(\\mathbf{y}(s \\wedge \\tau_n))\\,\n\t\t\td{W}(s),\\;\\; t\\in[0,\\infty).\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\t\tAlong the lines of the paper \\cite{Brz+Elw_2000}, we say that a\n\t\tlocal solution $\\mathbf{y}(t)$, $t < \\tau$ is global iff\n\t\t$\\tau=\\infty$ $\\mathbb{P}$-a.s.\n\t\\end{Def}\nHereafter, we simply write local solution in place of local mild solution. \n\\begin{Rem}\\label{rem-stochastic interval}\n\\begin{trivlist}\n\\item[(i)] Since $\\tau_n$ is a stopping, the process $\\mathds{1}_{[0,\\tau_n)}(s)$, $s\\in [0,\\infty)$ is well-measurable, see \\cite[Proposition 4.2]{Metivier_1982}. Therefore, since by \\cite[Theorem 1.6]{Metivier_1982}, the $\\sigma$-field of well measurable sets is smaller than the $\\sigma$-field of progressively measurable sets, it follows\nthat the process $\\mathds{1}_{[0,\\tau_n)}(s)$, $s\\in [0,\\infty)$ is progressively measurable. In particular, the integrand in \\eqref{eq-locsol_01-c} is progressively measurable.\n\\item[(ii)] On the other hand, one could use in \\eqref{eq-locsol_01-c} a process $\\mathds{1}_{[0,\\tau_n]}(s)$, $s\\in [0,\\infty)$, which according \\cite[Proposition 4.4]{Metivier_1982}, is predictable. However, here we still stick to the processes as in (i).\n\\end{trivlist}\n\\end{Rem}\n\\begin{Rem}\\label{rem-local solution} Suppose that $\\tau:\\Omega \\to [0,\\infty)$ is an accessible stopping time and $\\mathbf{y}: [0,\\tau]\\times \\Omega \\to V$ is an admissible process such that\nfor every $t \\in [0,\\infty)$\n\t\t\t\\begin{align}\n\t\t&\t\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau]} \\Vert\n\t\t\t\t\\mathbf{y}(s)\\Vert^2_{\\mathscr{V}} +\\int_0^{t\\wedge \\tau} \\Vert \\mathbf{y}(s)\\Vert_\\mathscr{E}^2 \\,\n\t\t\t\tds\\Big)<\\infty,\\label{eq-locsol_01-a2}\\\\ \n\t&\t\t\\mathbf{y}(t\\wedge \\tau)= \\mathbb{S}(t\\wedge\n\t\t\t\\tau)\\mathbf{y}_0-\\int_0^{t\\wedge \\tau}\n\t\t\t\\mathbb{S}(t\\wedge\\tau-s)[ \\mathbf{F}(\\mathbf{y}(s\\wedge \\tau))+\\mathbf{L}(\\mathbf{y}(s \\wedge \\tau)]\\, ds\n +I_{\\tau}(t\\wedge \\tau), \\;\\;\\;\\mbox{ $\\mathbb{P}$-a.s.,} \\label{eq-locsol_01-d2}\n\t\t\t\\end{align}\nwhere $I_{\\tau}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eq-locsol_01-d}\n\t\t\t\\begin{split}\nI_{\\tau}(t):= \\int_0^t\\mathds{1}_{[0,\\tau)}(s)\\mathbb{S}(t-s)\\mathbf{G}(\\mathbf{y}(s \\wedge \\tau))\\,\n\t\t\td{W}(s),\\;\\; t\\in [0,\\infty).\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\nLet us choose an announcing sequence $(\\tau_n)_{n\\in \\mathbb{N}}$ for $\\tau$. Then, by using \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we can show that for every $n$, the conditions\n\\eqref{eq-locsol_01-a} and \\eqref{eq-locsol_01-b} are satisfied with $I_{\\tau_n}$ defined by \\eqref{eq-locsol_01-c}. Therefore we infer that the restriction of the process\n$\\mathbf{y}$ to the open stochastic interval $[0,\\tau)\\times \\Omega$ is a local solution to problem \\eqref{ABSTRACT-LC}.\n\\end{Rem}\n\t\n\tWe now introduce the definition of a maximal local solution.\n\t\n\t\\begin{Def}\\label{Def-maxsol-0}\n\t\tConsider a family $\\mathcal{ LS}$ of all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABSTRACT-LC}. For two\n\t\telements $(u,\\tau), (v,\\sigma) \\in \\mathcal{ LS} $ we write that\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ iff $\\tau \\leq \\sigma$ $\\mathbb{P}$-a.s. and\n\t\t$v_{\\vert [0,\\tau)\\times \\Omega} \\sim u$. Note that if\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ and $(v,\\sigma)\\preceq (u,\\tau)$,\n\t\tthen $(u,\\tau)\\sim (v,\\sigma)$. We write $(u,\\tau)\\prec\n\t\t(v,\\sigma)$ iff $(u,\\tau)\\preceq (v,\\sigma)$ and $(u,\\tau)\\not\\sim\n\t\t(v,\\sigma)$. Then, the pair $(\\mathcal{ LS},\\preceq)$ is \n\t\tpartially ordered. \n\t\tEach maximal element $(u,\\tau)$ in the set $(\\mathcal{\n\t\t\tLS},\\preceq)$\n\t\tis called a maximal local solution to the problem \\eqref{ABSTRACT-LC}. The existence of an upper bound of every non-empty chain of $(\\mathcal{\n\t\t\tLS},\\preceq)$ is justified by Amalgamation\n\t\tLemma \\ref{lem-amalgamation}. \\\\\n\t\tIf $(u, \\tau)$ is a maximal local solution to equation\n\t\\eqref{ABSTRACT-LC}, the stopping time $\\tau$ is called its\n\t\tlifetime.\n\t\\end{Def}\n\t\\subsection{Existence and uniqueness of a maximal local solution: 2D and 3D cases}\n\tBy using Theorem \\ref{thm_local} we will prove in this subsection that the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution. In order to do this, we need to establish several auxiliary results. Throughout this subsection $d=2,3$ and $a=\\frac{d}4$.\n\t\\begin{lem}\\label{Local-LIP-Lem}\n\t\tThere exists $c_2>0$ such that for\n\t\tall $\\mathbf{n}_i\\in \\mathbf{H}^3$, i$=1,2$,\n\t\t\\begin{equation}\\label{local-Lip-F-2}\n\t\t\\begin{split}\n\t\t\\lvert M(\\mathbf{n}_1)-M(\\mathbf{n}_2)\\rvert_{\\mathrm{L}^2} \\le c_2\\biggl(\\lVert \\d_1-\\d_2\\rVert_2\n\t\t\\lVert \\d_1\\rVert_2^{1-a} \\lVert \\d_1\\rVert_3^a+ \\lVert\n\t\t\\d_1-\\d_2\\rVert^{1-a}_2 \\lVert \\d_1-\\d_2\\rVert_3^a \\lVert \\d_2\\rVert_2\n\t\t\\biggr).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThe proof of this result can be found in \\cite[Lemma 6.4]{BHP18}\n\t\\end{proof}\n\n\n\n\t\\begin{lem}\\label{Local-LIP-Lem-2}\n\tThere exist $c_3>0$ such that for all $(\\mathbf{v}_i,\n\t\t\\mathbf{n}_i)\\in \\mathscr{E}$, i$=1,2$,\n\t\t\\begin{equation}\\label{local-Lip-F-3}\n\t\t\\begin{split}\n\t\t\\lVert \\tilde{B}(\\mathbf{v}_1,\\mathbf{n}_1)-\\tilde{B}(\\mathbf{v}_2,\\mathbf{n}_2)\\rVert_1 \\le c_3\n\t\t\\biggl(\\lVert \\nabla(\\mathbf{v}_1-\\mathbf{v}_2)\\rVert \\lVert\n\t\t\\mathbf{n}_1\\rVert_2^{1-a}\\lVert \\d_1\\rVert_3^a\n\t\t+\\lVert(\\mathbf{n}_1-\\mathbf{n}_2)\\lVert^{1-a}_ 2 \\lVert(\\mathbf{n}_1-\\mathbf{n}_2)\\lVert^{a}_3\\lVert\n\t\t\\nabla\\mathbf{v}_2\\lVert\\biggr).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThroughout this proof $C>0$ is an universal constant which may change from one term to the other. \t\n\tLet $(\\mathbf{v}_i,\n\t\\mathbf{n}_i)\\in \\mathscr{E}$, i$=1,2$, and\n\t$(\\mathbf{w},\n\t\\bar{\\d})=(\\mathbf{v}_1-\\mathbf{v}_2, \\d_1-\\d_2). $\nSince $\\tilde{B}$ is bilinear, we have\n\t\\begin{equation}\n\t\\tilde{B}(\\mathbf{v}_1,\\mathbf{n}_1)-\\tilde{B}(\\mathbf{v}_2,\\mathbf{n}_2)=\\tilde{B}(\\mathbf{w},\\d_1)+\\tilde{B}(\\v_2, \\bar{\\d})=J_1 +J_2.\n\t\\end{equation}\n\tIn order to estimate $\\lVert J_i\\rVert_1$,$i=1,2,$, we only focus on estimating $\\lVert \\nabla J_i \\rVert$, because by \\eqref{EST-G1} estimating $\\lVert J_i\\rVert$\n\tis easy. By using the Leibniz rule, the H\\\"older inequality, \\eqref{GAG-l4} and\\eqref{GAG-LInf-2} we infer that\n\t\t\\begin{align*}\n\t\t\\lVert \\nabla J_1\\rVert\\le & C \\biggl(\\lVert \\nabla \\mathbf{w}\\rVert \\lVert \\nabla\n\t\t\\d_1\\rVert_{\\mathrm{L}^\\infty}+\\lVert \\mathbf{w}\\rVert_{\\mathrm{L}^4} \\lVert \\nabla^2\n\t\t\\d_1\\rVert_{\\mathrm{L}^4}\\biggr)\\\\\n\t\t\\le& C \\lVert \\mathbf{w} \\rVert_1 \\biggl(\\lVert \\nabla \\d_1\\rVert^{1-a}_1 \\lVert \\nabla \\d_1\\rVert^a_2 +\n\t\t\\lVert \\nabla^2 \\d_1\\rVert^{1-a}_{L^2} \\lVert \\nabla^2 \\d_1\\rVert_{1}^a \\biggr)\n\\le C \\lVert \\mathbf{v}_1-\\mathbf{v}_2 \\rVert_1 \\lVert \\d_1\\rVert^{1-a}_2 \\lVert \\d_1\\rVert^a_3.\n\t\t\\end{align*}\n\t\tIn a similar way, we can prove that\n\t\t\\begin{equation*}\n\t\t\\lVert \\nabla J_2\\rVert\\le C \\lvert \\nabla\n\t\t\\mathbf{v}_2 \\rvert_{\\mathrm{L}^2} \\lVert \\d_1-\\d_2\\rVert^{1-a}_2 \\lVert \\d_1-\\d_2\\rVert^a_3.\n\t\t\\end{equation*}\n\t\tThe inequality \\eqref{local-Lip-F-3} easily follows from \\eqref{EST-G1} and the estimates for \t$\\lVert \\nabla J_i\\rVert$, $i=1,2$, above.\n\t\\end{proof}\n\t\n\t\\begin{lem}\\label{Local-LIP-Lem-3}\n\t\tLet Assumption \\ref{eqn-f} be satisfied. Then, there exists $c_4>0$\n\t\tsuch that \n\t\t\\begin{equation}\\label{local-Lip-F-4}\n\t\t\\begin{split}\n\t\t\\lVert f(\\mathbf{n}_1)-f(\\mathbf{n}_2)\\rVert_1 \\le c_4\\Big[1+\\lVert \\mathbf{n}_1\\rVert_2^{2N}+\\lVert \\mathbf{n}_2\\rVert_2^{2N}\\Big]\\lVert \\mathbf{n}_1-\\mathbf{n}_2\\rVert_2, \\text{ for all $\\mathbf{n}_1, \\mathbf{n}_2\\in \\mathbf{X}_{1}\\cap \\mathbf{X}_{\\frac32}$}.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\nLet\t$N \\in I_d$, $k\\in \\{0,\\ldots, N\\}$ and $f$ be as in Assumption \\ref{eqn-f}. By the Young inequality and the fact that $\\mathbf{H}^2 $ is an algebra, we infer that there exists $C>0$ such that for all $\\mathbf{n}_1,\\mathbf{n}_2\\in \\mathbf{H}^2$\n\t\t\\begin{equation}\\label{POL-EST-0}\n\t\t\\lVert\\lvert \\mathbf{n}_1 \\rvert^{2k} \\mathbf{n}_2\\rVert_2\\le C \\lVert \\mathbf{n}_1 \\rVert_2^{2k} \\lVert \\mathbf{n}_2\\rVert_2\\le C \\lVert \\mathbf{n}_2\\rVert_2(1+\\lVert \\mathbf{n}_1 \\lVert_2^{2N}.\n\t\t\\end{equation}\n\t\tThus, it is enough to establish \\eqref{local-Lip-F-4} for the leading term $a_N \\lvert \\mathbf{n} \\rvert^{2N}\\mathbf{n}$. For doing so, we have $$\\lvert \\mathbf{n}_1\\rvert^{2N}\\mathbf{n}_1-\\lvert \\mathbf{n}_2\\rvert^{2N}\\mathbf{n}_2= \\vert \\mathbf{n}_1\\vert^{2N}(\\mathbf{n}_1-\\mathbf{n}_2)+\n\t\t\\mathbf{n}_2(\\vert\\mathbf{n}_1\\vert-\\vert\\mathbf{n}_2\\vert)(\\sum_{k=0}^{2N-1}\\vert \\mathbf{n}_1\\vert^{2N-1-k}\\vert \\mathbf{n}_2\\rvert^{k} ),$$\n\t\tfrom which along with \\eqref{POL-EST-0} we infer that \\eqref{local-Lip-F-4} is true for the leading term $a_N \\lvert \\mathbf{n} \\rvert^{2N}\\mathbf{n}$.\n\t\\end{proof}\n\t\\begin{prop}\\label{SLC-ST}\n\tLet $\\alpha=\\frac d4$, $d=2,3$. If Assumption \\ref{eqn-f} is satisfied, then there exists $C_0>0$ such that for all $\\mathbf{y}_i=(\\mathbf{v}_i,\\d_i)$, \\,\\,\n\t\t$i=1,2$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lVert \\mathbf{F}(\\mathbf{y}_1) -\\mathbf{F}(\\mathbf{y}_2)\\rVert_{\\mathscr{H}} \\le C_0 \\Vert \\mathbf{y}_1-\\mathbf{y}_2\\Vert_{\\mathscr{V}}^{1-\\alpha}\n\t\t\\Big[\\Vert \\mathbf{y}_1-\\mathbf{y}_2\\Vert^{\\alpha}_{\\mathscr{V} } \\Vert \\mathbf{y}_1\\Vert^{1-\\alpha}_{\\mathscr{V}} \\Vert \\mathbf{y}_1\\Vert_{\\mathscr{E}}^\\alpha + \\Vert\n\t\t\\mathbf{y}_1-\\mathbf{y}_2\\Vert_{\\mathscr{E}}^\\alpha \\Vert\n\t\t\\mathbf{y}_2\\Vert_{\\mathscr{V}}\\Big]\\\\\n\t\t+c_4\\lVert \\mathbf{y}_1-\\mathbf{y}_2\\rVert_{\\mathscr{V} }\\Big[1+\\lVert \\mathbf{y}_1\\rVert_{\\mathscr{V}}^{2N}+\\lVert \\mathbf{y}_2\\rVert_{\\mathscr{V}}^{2N}\\Big].\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proposition is an easy consequence of Lemmata \\ref{Local-LIP-Lem}-\\ref{Local-LIP-Lem-3}. Thus, we omit its proof.\n\t\\end{proof}\n\t\n\t\n\tUsing Theorems \\ref{Thm:LocalUniqueness}, \n\t\\ref{thm_local} and \\ref{thm_maximal-abstract} we obtain our first main result.\n\t\\begin{thm}\\label{LC-Local-Sol}\n\t\tLet $d=2,3$, $(\\v_0,\\d_0)\\in \\mathrm{L}^2(\\Omega;\\mathrm{V} \\times \\mathbf{X}_{1})$ be $\\mathcal{F}_0$-measurable, $\\mathbf{h}\\in \\mathbf{H}^2$. If Assumptions \\ref{eqn-f}, \\ref{HYPO-ST} and \\ref{Assum:Usual hypotheses} are satisfied, then the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution $((\\mathbf{v};\\mathbf{n}), \\tilde{\\tau}_\\infty)$ satisfying the following properties.\n\t\t\\begin{enumerate}[(1)]\n\t\t\t\\item\\label{item-1} Given $R>0$ and $\\varepsilon >0$ there exists\n\t\t\t$\\tau(\\varepsilon,R)>0$ such that if $\\mathbb{E}\\Vert (\\v_0,\\d_0) \\Vert^{2}_{\\mathrm{V} \\times \\mathbf{X}_{1}} \\leq R^{2}$, then \n\t\t\t\\[{\\mathbb P}\\big(\\tilde{\\tau}_\\infty \\geq \\tau(\\varepsilon,R)\\big) \\geq\n\t\t\t1-\\varepsilon.\\]\n\t\t\t\\item \\label{THM-ii} We also have\n\t\t\t\\begin{align}\n\t\t\t&\\mathbb{P}\\left(\\{ \\tilde{\\tau}_\\infty <\\infty \\}\\cap \\{ \\lvert \\nabla \\mathbf{v}(t) \\rvert_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t) \\rVert_2<\\infty \\} \\right)=0,\\label{MAX-P1}\\\\\n\t\t&\t\\limsup_{t\\toup\\tilde{\\tau}_\\infty } \\lvert \\nabla \\v(t)\\rvert_{\\mathrm{L}^2}^2 +\\lVert \\d(t)\\rVert^2_2+\\int_0^t \\left(\\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2}+ \\lVert \\d(s) \\rVert^2_3 \\right)ds=\\infty \\text{ $\\mathbb{P}$-a.s. on } \\{\\tilde{\\tau}_\\infty<\\infty \\}\\label{MAX-P2}.\n\t\t\t\\end{align}\n\t\t\\end{enumerate}\n\t\t\n\t\\end{thm}\n\t\\begin{proof}\n\t\tLemma \\ref{SEM-1}-\\ref{SEM-3} show that $\\{\\mathbb{S}(t)\\}_{t\\geq 0}$ on\n\t\t$\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{0}$ satisfies Assumption \\ref{assum-01}.\n\t\tThanks to Proposition \\ref{SLC-ST} we can infer by applying Theorems \\ref{Thm:LocalUniqueness},\n\t\t\\ref{thm_local} and \\ref{thm_maximal-abstract} that problem\n\t\t\\eqref{ABSTRACT-LC} has a unique maximal local solution satisfying items \\eqref{item-1} and \\eqref{THM-ii} of Theorem \\ref{LC-Local-Sol}.\n\t\\end{proof}\n\t\\subsection{Existence and uniquness of global strong solution: 2D case}\nBy using the Khashminskii test for non-explosions, see \\cite[Theorem\nIII.4.1]{Kh_1980}, and some arguments from \\cite{Brz+Masl+Seidler_2005}, we prove in this section that if $d=2$ then the problem \\eqref{ABSTRACT-LC} has a unique global solution. \n\t\n\tFor all $(\\mathbf{u}, \\mathbf{d}) \\in C([0,T]; \\mathrm{H}\\times \\mathbf{H}^1) \\bigcap L^2(0,T; \\mathrm{V}\\times \\mathbf{H}^2)$ and $t\\in [0,T]$ we put \n\t\\begin{align}\n\t& \\mathcal{E}[\\mathbf{u},\\mathbf{d}](t)= \\frac{1}{2}\\left(\\lvert \\mathbf{u} (t) \\rvert^2_{\\mathrm{L}^2}+ \\lvert \\mathbf{d}(t)\\rvert_{\\mathbf{L}^2} + \\lvert \\nabla \\mathbf{d}(t) \\rvert^2_{\\mathbf{L}^2} + \\int_\\mathcal{O} F( \\mathbf{d}(t,x)) dx \\right)\\\\\n\t& \\mathscr{D}[\\mathbf{u}, \\mathbf{d} ](t) = \\lvert \\mathrm{A}^\\frac12 \\mathbf{u} (t) \\rvert^2_{\\mathrm{L}^2}+ \\lvert \\mathrm{A}_1\\mathbf{d}(t) - f(\\mathbf{d}(t) ) \\rvert^2_{\\mathbf{L}^2}.\n\t\\end{align}\n\n\t\n\t\\begin{thm}\\label{GLOBAL-ST}\n\t\tLet $d=2$, $N\\in \\mathbb{N}$, $\\mathbf{h} \\in \\mathbf{H}^2$ and $(\\v_0, \\d_0) \\in \\mathrm{L}^2(\\Omega; \\mathrm{V} \\times \\mathbf{X}_{1})$ such that \n\t\t\\begin{equation}\\label{E_0}\n\t\t\\mathbb{E} \\lvert \\mathcal{E}[\\v_0,d_0]\\rvert^{2(4N+2)})= \\mathbb{E} \\biggl(\\vert \\mathbf{v}_0\\vert^2_{\\mathrm{L}^2} + \\vert \\mathbf{n}_0\\vert^2_{\\mathbf{L}^2}+\\vert \\nabla \\mathbf{n}_0 \\vert^2_{\\mathbf{L}^2} + \\int_{\\mathcal{O}}F(\\mathbf{n}_0(x)) dx \\biggr)^{2(4N+2)}<\\infty.\n\t\t\\end{equation}\n\tIf Assumptions \\ref{eqn-f}, \\ref{Assum:Usual hypotheses} and \\ref{HYPO-ST}, \n\t\tthen the problem \\eqref{ABSTRACT-LC} has a unique global strong solution.\n\t\\end{thm}\n\tThe proof of this theorem is given at the end of this subsection.\n\n\t\\begin{prop}\\label{EST1}\n\t\tLet all the assumptions of Theorem \\ref{GLOBAL-ST} be satisfied and $p \\in [2, 2(4N+1)]$. Also, let $(\\tau_k)_{k \\in \\mathbb{N}}$ be the sequence of stopping times defined by\n\t\t\t\\begin{equation}\\label{STOP}\n\t\t\t\t\t\\tau_k=\\inf\\{t \\in [0,\\infty) : \\lvert \\nabla \\v(t)\\rvert_{\\mathrm{L}^2}^2 +\\lVert \\d(t)\\rVert^2_2+\\int_0^t \\left(\\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2}+ \\lVert \\d(s) \\rVert^2_3 \\right)ds >\n\t\t\t\t\tk^2\\}, \\; k\\in \\mathbb{N}.\n\t\t\t\t\t\\end{equation}\n\t\tThen, there exist an increasing function $\\varphi: [0,\\infty) \\to (0,\\infty)$ and $\\kappa_0=\\kappa_0(p,\\lvert \\mathbf{h} \\rvert_{\\mathbf{W}^{1,4}})>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:ESTofVDinWeakerNorm}\n\t\t\\mathbb{E} \\sup_{t\\in [0,T]}\\left( \\mathcal{E}[\\v,\\d] (t\\wedge \\tau_k) \\right)^p + \\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k} \\left(\\mathscr{D}[\\v,\\d](s) -\\frac{a_{N+1}}{2} \\lvert \\d(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)\\, ds \\right] \\le \\kappa_0 \\varphi(T) \\Big( 1 + \\mathbb{E}\\lvert \\mathcal{E}[\\v,\\d](0)\\rvert^p \\Big).\n\t\t\\end{equation}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proof of this proposition will be given in Section \\ref{AppB}.\n\t\\end{proof}\n\tHereafter, we set\n\t\\begin{equation}\\label{Eq:ConstantFrakC0}\n\t\\mathfrak{C}_0= \\kappa_0 \\varphi(T)(1 + \\mathbb{E} \\lvert \\mathcal{E}(\\v,\\d)(0)\\rvert^{2(4N+2)} )\n\t\\end{equation}\n\t\\begin{cor}\n\t\tLet all the assumptions of Proposition \\ref{EST1} be satisfied. Then, there exists $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:EstIntegralOfH2Norm}\n\t\t\\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k}\\lVert \\d(s) \\rVert^2_2 \\right]^2 \\le C (\\mathfrak{C}_0 +1).\n\t\t\\end{equation}\n\t\\end{cor}\n\t\\begin{proof}\n\t\tBy \\eqref{bigdanh2} and $\\mathbf{H}^1 \\hookrightarrow \\mathbf{L}^{4N+2}$, which is valid for $d=2$, there exists a constant $C>0$ such that\n\t\t\\begin{equation}\\label{Eq:Bigdanh2withH1norm}\n\t\t\\lVert \\d \\rVert^2_2 \\le C( \\lvert \\mathrm{A}_1 \\d -f(\\d)\\rvert^2_{\\mathbf{L}^2}+ \\lVert \\d \\rVert^{4N+2}_1+1 ),\n\t\t\\end{equation}\n\t\tfrom which along with \\eqref{Eq:ESTofVDinWeakerNorm} we conclude the proof of the corollary.\n\t\\end{proof}\n\tLet $\\Psi_1: \\mathbf{H}^2 \\to [0, \\infty)$, $\\Psi_2: D(\\mathrm{A}) \\to [0,\\infty)$ and $\\Psi: D(\\mathrm{A})\\times \\mathbf{H}^2 \\to 0,\\infty) $ be defined by\n\t\\begin{align}\n\t& \\Psi_1(\\mathbf{d})= \\frac12 \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2}, \\;\\; \\mathbf{d}\\in \\mathbf{H}^2,\\label{Eq:DefPsi1forD} \\\\\n\t& \\Psi_2(\\mathbf{u})= \\frac12 \\lvert \\nabla \\mathbf{u} \\rvert^2_{\\mathrm{L}^2},\\;\\; \\mathbf{u}\\in D(\\mathrm{A}),\\label{Eq:DefPsi2forv}\\\\\n\t& \\Psi(\\mathbf{u},\\mathbf{d})=\\Psi_1(\\mathbf{d}) + \\Psi_2(\\mathbf{u}), \\text{ } (\\mathbf{u},\\mathbf{d})\\in D(\\mathrm{A}) \\times \\mathbf{H}^2.\\label{Eq:DefFunctionForIto}\n\t\\end{align}\n\tHereafter, $\\Psi_i^\\prime$ and $\\Psi_i^{\\prime\\prime}$, $i=1,2$, are the first and second Fr\\'echet derivatives of $\\Psi_i$, $i=1,2$.\n\n\t\\begin{lem}\\label{Lem:FirstDerAppliedtoVdotNabalaD}\n\t\tThere exists $\\kappa_1>0$ such that for all $\\mathbf{d}\\in \\mathbf{H}^3$ and $\\mathbf{u}\\in D(\\mathrm{A})$ we have\n\t\t\\begin{equation}\\label{Eq:FirstDerAppliedtoVdotNabalaD}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[\\mathbf{u}\\cdot \\nabla \\mathbf{d}] \\le \\kappa_1\\Psi(\\mathbf{u},\\mathbf{d}) \\left[\\lVert \\mathbf{d} \\rVert^2_{1} + 1\\right] \\lVert \\mathbf{d} \\rVert^2_{2} + \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2}.\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tIn this proof $C>0$ is an universal constant which may change from one term to the other.\n\t\tLet $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$. \n\t\tObserve that \n\t\t\\begin{align}\n\t\t\\Psi_1^\\prime(\\mathbf{d})[\\mathbf{g}]=\\langle \\Delta \\mathbf{d} -f(\\mathbf{d}), \\mathrm{A}_1 \\mathbf{g} + f^\\prime(\\mathbf{d})[\\mathbf{g}] \\rangle \\text{ for all } \\mathbf{g}\\in \\mathbf{H}^2, \\label{Eq:FirstDeriveOfPsi}\n\t\t\\end{align}\n\t\tand \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t-\\Delta(\\mathbf{u}\\cdot \\nabla \\mathbf{d})=-(\\Delta \\mathbf{u} \\cdot \\nabla) \\mathbf{d} - (\\mathbf{u}\\cdot \\nabla) \\Delta \\mathbf{d} -2 \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2) \\mathbf{d}\\\\\n\t\t=-(\\Delta \\mathbf{u} \\cdot \\nabla) \\mathbf{d} - 2 \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2) \\mathbf{d}+ (\\mathbf{u}\\cdot \\nabla)[\\Delta \\mathbf{d} -f(\\mathbf{d})] - f^\\prime(\\mathbf{d})[\\mathbf{u}\\cdot \\nabla\\mathbf{d}]\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tHence, by using the identities \\eqref{tild-b-0} and \\eqref{Eq:FirstDeriveOfPsi} we obtain\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[ \\mathbf{u}\\cdot \\nabla \\mathbf{d}]=\\langle \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}), (\\Delta \\mathbf{u}\\cdot \\nabla) \\mathbf{d} \\rangle +2\\langle \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}), \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2)\\mathbf{d}\\rangle,\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhich along with \\eqref{GAG-l4} and the H\\\"older and Young inequalities imply that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[ \\mathbf{u}\\cdot \\nabla \\mathbf{d}]& \\le C \\lvert \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathbf{L}^4} [\\lvert \\mathrm{A} \\mathbf{u} \\rvert_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathbf{L}^4} + \\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^4} \\lvert \\nabla^2 \\mathbf{d} \\rvert_{\\mathbf{L}^2} ]\\\\\n\t\t& \\le \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) ) \\rvert^2_{\\mathbf{L}^2}+ \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}+ C \\lvert \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d})\\rvert_{\\mathbf{L}^2}^2 [ \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathbf{L}^4}^4 + \\lvert \\nabla^2 \\mathbf{d} \\rvert_{\\mathbf{L}^2} ].\n\t\t\\end{split}\n\t\t\\end{equation}\t\n\t\tNow, \\eqref{Eq:FirstDerAppliedtoVdotNabalaD} easily follows from the last line of the the above chain of inequalities.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:FirstDerAppliedtoDeltaD+f(D)}\n\t\tThere exists $\\kappa_2>0$ such that for all $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$ we have\n\t\t\\begin{equation}\\label{Eq:FirstDerAppliedtoDeltaD+f(D)}\n\t\\langle\tf ^\\prime(\\mathbf{d})[\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})], \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rangle\\le \\frac16 \\lvert \\nabla(\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + \\kappa_2 \\Psi(\\mathbf{u},\\mathbf{d}) (1+ \\lVert \\mathbf{d} \\rVert^{4N}_1 ).\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tUsing part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, \\eqref{GAG-l4}, the H\\\"older and Young inequalities, and by $\\mathbf{H}^1\\hookrightarrow \\mathrm{L}^{4N}$ (that is valid for $d=2$) we infer that there exist $C>0$ and $\\kappa_2>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\langle\tf ^\\prime(\\mathbf{d})[\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})], \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d}) \\rangle\n\t\t\t&\\le c_1 \\int_\\mathcal{O} (1+ \\lvert \\mathbf{d} \\rvert^{2N}) \\lvert \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d})\\rvert^2 dx\\\\\n\t\t\t&\\le C \\lvert \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathrm{L}^4}^2 \\lvert (1+ \\lvert \\mathbf{d} \\rvert^{2N}) \\rvert^2_{\\mathbf{L}^2}\\\\\n\t\t&\\le \\frac16 \\lvert \\nabla (\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2}+ \\kappa_2 \\lvert \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathrm{L}^2}^2 (1 + \\lVert\\mathbf{d}\\rVert^{4N}_{1}),\n\t\t\\end{split}\n\t\t\\end{equation}\t\n\t\tfor all $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$. This completes the proof of the lemma.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:SecondDerofPsiAppliedtoGG}\n\t\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. Then, there exists $\\kappa_5=\\kappa_5(\\lVert \\mathbf{h} \\rVert_2)>0$ such that \n\t\t\\begin{equation}\n\t\t\\lvert\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}, \\mathbf{d}\\times \\mathbf{h}] \\rvert+\\lvert\t\\Psi_1^\\prime(\\mathbf{d})[(\\mathbf{d}\\times\\mathbf{h})\\times \\mathbf{h}] \\rvert \\le \\Psi_1(\\mathbf{d})+ \\kappa_5 (1+ \\lVert \\mathbf{d} \\rVert^{4N}_{1}) \\lVert \\mathbf{d} \\rVert^2_{2},\\text{ for all $ \\mathbf{d} D(\\mathrm{A}_1)$.}\n\t\t\\label{Eq:SecondDerofPsiAppliedtoGG}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tLet \\text{ $(\\mathbf{u}, \\mathbf{d})\\in D(\\mathrm{A}) \\times D(\\mathrm{A}_1)$.}\n\t\tWe firstly recall that \n\t\t\\begin{equation}\n\t\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[\\mathbf{g}, \\mathbf{p}]=\\langle \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d}), -f^{\\prime \\prime}(\\mathbf{d})[\\mathbf{g}, \\mathbf{p}] \\rangle + \\langle \\mathrm{A}_1 \\mathbf{p} - f^{\\prime}(\\mathbf{d})[\\mathbf{p}], \\mathrm{A}_1 \\mathbf{g}- f^\\prime(\\mathbf{d})[\\mathbf{g}]\\rangle,\\; \\mathbf{p}, \\mathbf{g} \\in D(\\mathrm{A}_1).\n\t\t\\end{equation}\n\t\tHence, by recalling that $G(\\mathbf{d})=\\mathbf{d}\\times \\mathbf{h}$ we have \n\t\t\\begin{align}\n\t\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[G(\\mathbf{d}),G(\\mathbf{d})]&= \\lvert \\mathrm{A}_1(G(\\mathbf{d}))- f^\\prime(\\mathbf{d})[G(\\mathbf{d})]\\rvert^2_{\\mathbf{L}^2}+ \\langle \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d}), \\mathrm{A}_1(G(\\mathbf{d}))- f^{\\prime\\prime}(\\mathbf{d})[G(\\mathbf{d}), G(\\mathbf{d})]\\rangle\n\t\\nonumber \t\\\\\n\t\t&\\le \\frac14 \\left\\lvert\\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d})\\right\\rvert^2_{\\mathbf{L}^2}+\\left\\lvert \\mathrm{A}_1(G(\\mathbf{d}))-f^{\\prime\\prime}(\\mathbf{d})[G(\\mathbf{d}), G(\\mathbf{d})]\\right\\rvert^2_{\\mathbf{L}^2} + \\lvert \\mathrm{A}_1(G(\\mathbf{d}))-f^\\prime(\\mathbf{d})[G(\\mathbf{d})]\\rvert^2_{\\mathbf{L}^2}\\nonumber \\\\\n\t\t& =\\mathrm{I}_1+ \\mathrm{I}_2+\\mathrm{I}_3\\label{Eq:EstDerA}.\n\t\t\\end{align}\n\t\tSecondly, by the part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, $\\mathbf{H}^2\\hookrightarrow\\mathbf{L}^\\infty$, $\\mathbf{H}^1 \\hookrightarrow \\mathrm{L}^4, \\mathbf{L}^{4N}$ and the H\\\"older and Young inequalities we infer that there exists a constant $C=C(\\lVert \\mathbf{h} \\rVert_2 )>0$ such that \n\t\t\\begin{equation}\\label{Eq:EstDerB}\n\t\t\\begin{split}\n\t\t\\mathrm{I}_2 \\le & \\lvert \\Delta(\\mathbf{d}\\times \\mathbf{h}) \\rvert^2_{\\mathbf{L}^2} + \\lvert \\lvert \\mathbf{d} \\times \\mathbf{h} \\rvert^2 \\lvert f^{\\prime\\prime}(\\mathbf{d})\\rvert \\rvert^2_{\\mathbf{L}^2}\\\\\n\t\t&\\le 4 \\left(\\lvert \\Delta \\mathbf{d}\\times \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}+ 2 \\lvert \\nabla \\mathbf{d} \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} + \\lvert \\mathbf{d} \\times \\Delta \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}\\right) + C (1+ \\lvert \\mathbf{d} \\rvert^{4N}_{\\mathbf{L}^{4N}} )\\lvert \\mathbf{d}\\times \\mathbf{h}\\rvert_{\\mathbf{L}^\\infty}^2 \\\\\n\t\t&\\le C \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tIn a similar way, we prove that there exists a constant $C_7=C(\\lVert \\mathbf{h} \\rVert_2 )>0$ such that \n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\mathrm{I}_3 \\le C \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tCombining this last inequality with \\eqref{Eq:EstDerA} and \\eqref{Eq:EstDerB} proves that $$\\lvert \\Psi_1^{\\prime \\prime}(\\mathbf{d})[G(\\mathbf{d}),G(\\mathbf{d})]\\rvert \\le \\frac12 \\Psi_1(\\mathbf{d}) + C_7 \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).$$ In a similar way, we can also show that \n\t\t$\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[(\\mathbf{d}\\times \\mathbf{h})\\times \\mathbf{h}]\\rvert \\le \\frac12 \\Psi_1(\\mathbf{d}) + C_7 \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).$ One easily conclude the proof of the lemma from the last two estimates.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:Psi1BDGNeeded}\n\t\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. Then, there exists $\\kappa_5=\\kappa_5(\\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $\\mathbf{d} \\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation}\\label{Eq:Psi1BDGNeeded}\n\t\t\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}] \\rvert\\le \\kappa_6 \\left[1+ \\Psi_1(\\mathbf{d}) + \\lVert \\mathbf{d} \\rVert^{4N+1}_{1} +\\lVert \\mathbf{d} \\rVert_1 \\lVert \\mathbf{d}\\rVert_2 \\right].\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tBy part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, \\eqref{GAG-l4}, \\eqref{GAG-LInf}, \\eqref{GAG-LInf-2}, the H\\\"older and Young inequalities and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{4N+2}, \\mathbf{L}^4$, we infer that there exists $\\kappa_6=\\kappa_6(\\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $\\mathbf{d} \\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}] \\rvert&\\le\n\t\t\\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})\\rvert_{\\mathbf{L}^2} \\left(\\lvert [\\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d})]\\times \\mathbf{h} + 2 \\nabla \\mathbf{d} \\times \\nabla \\mathbf{h} + \\mathbf{d} \\times \\Delta \\mathbf{h} -f(\\mathbf{d}) \\times \\mathbf{h} \\rvert_{\\mathbf{L}^2}\\right)\\\\\n\t\t&\t\\le \\kappa_6 \\left( \\Psi_1(\\mathbf{d})+ \\lvert \\nabla \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} + \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^\\infty} + \\lvert \\mathbf{d} \\rvert^{4N+2}_{\\mathbf{L}^{4N+2}} +1\\right),\\\\\n\t\t&\t\\le \\kappa_6 \\left( \\Psi_1(\\mathbf{d})+ \\lVert \\mathbf{d} \\rVert_{1} \\lVert \\mathbf{d} \\rVert_{2} + \\lvert \\mathbf{d} \\rvert^{4N+2}_{1} +1 \\right).\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThis completes the proof of the Lemma \\ref{Lem:Psi1BDGNeeded}.\n\t\\end{proof}\n\tWe will also need to the following results.\n\t\\begin{lem}\\label{Lem:NonlienarNSeItoV}\n\t\tThere exists $\\kappa_3>0$ such that for all $\\mathbf{u}\\in D(\\mathrm{A})$ and $\\mathbf{d}\\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation}\n\t\t-\\Psi^\\prime_2(\\mathbf{u})(B(\\mathbf{u},\\mathbf{u}))=\t-\\langle B(\\mathbf{u},\\mathbf{u}), \\mathrm{A} \\mathbf{u}\\rangle \\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + \\kappa_3 ( \\lvert \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}) \\Psi(\\mathbf{u},\\mathbf{d}).\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tUsing \\eqref{GAG-l4}, the H\\\"older and Young inequalities we infer that $C>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle B(\\mathbf{u},\\mathbf{u}) , \\mathrm{A}\\mathbf{u} \\rangle &=\\langle \\mathbf{u}\\cdot \\nabla \\mathbf{u}, \\mathrm{A}\\mathbf{u}\\rangle\n\t\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert \\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^4} \\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^4}\\\\\n\t\t&\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert^\\frac32 \\lvert \\mathbf{u} \\rvert^\\frac12_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u} \\rvert^\\frac12_{\\mathrm{L}^2}\n\t\t\t\\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + C \\lvert \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} \\Psi_1(\\mathbf{u}),\n\t\t\\end{split}\n\t\t\\end{equation}\n\t for all $(\\mathbf{u}, \\mathbf{d})\\in D(\\mathrm{A})\\times D(\\mathrm{A}_1)$. We easily conclude the proof of Lemma \\ref{Lem:NonlienarNSeItoV} from last line.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:CouplingTermIto}\n\t\tThere exists $\\kappa_4>0$ such that \n\t\t\\begin{equation}\n\t\t-\\Psi^\\prime_2(\\mathbf{u})(M(\\mathbf{d},\\mathbf{d}))=\t-\\langle M(\\mathbf{d},\\mathbf{d}), \\mathrm{A}\\mathbf{u}\\rangle \\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} + \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + \\kappa_4 \\Psi(\\mathbf{u},\\mathbf{d}) \\lVert \\mathbf{d} \\rVert^2_1 \\lVert \\mathbf{d} \\rVert^2_2,\n\t\t\\end{equation}\n\t\tfor all $\\mathbf{d}\\in D(\\mathrm{A}_1)$ satisfying $\\nabla (\\mathrm{A}_1 \\mathbf{d} +f(\\mathbf{d})) \\in \\mathrm{L}^2$, and $\\mathbf{u}\\in D(\\mathrm{A})$.\n\t\\end{lem}\n\t\\begin{proof}\n\t\tIn this proof $C>0$ is an universal constant.\n\t\tLet $\\mathbf{d}\\in D(\\mathrm{A}_1)$ be such that $\\nabla (\\mathrm{A}_1 \\mathbf{d} +f(\\mathbf{d})) \\in \\mathrm{L}^2$, and $\\mathbf{u}\\in D(\\mathrm{A})$.\n\t\tFirstly, since $\\Pi: \\mathrm{L}^2 \\to \\mathrm{H}$ is self-adjoint, $\\nabla F(\\mathbf{d})=\\nabla \\mathbf{d} f(\\mathbf{d})$ and $\\Div \\mathrm{A} \\mathbf{u}=0$, we infer that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle M(\\mathbf{d},\\mathbf{d}) , \\mathrm{A}\\mathbf{u} \\rangle=& \\frac12\\langle \\mathrm{A}\\mathbf{u}, \\nabla \\lvert \\nabla \\mathbf{d}\\rvert^2\\rangle -\\langle \\mathrm{A} \\mathbf{u}, \\nabla \\mathbf{d} \\Delta \\mathbf{d}\\rangle \\\\\n\t\t=&- \\langle \\mathrm{A} \\mathbf{u} \\cdot\\nabla \\mathbf{d}, \\mathrm{A}_1 \\mathbf{d}+f(\\mathbf{d})\\rangle + \\langle \\mathrm{A} \\mathbf{u} , \\nabla F(\\mathbf{d})\\rangle.\n\t\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tSecondly, applying the H\\\"older, the Gagliardo-Nirenberg and the Young inequality yields\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle M(\\mathbf{d},\\mathbf{d}) , \\mathrm{A}\\mathbf{u} \\rangle=& -\\langle \\mathrm{A} \\mathbf{u} \\cdot\\nabla \\mathbf{d}, \\mathrm{A}_1 \\mathbf{d}+f(\\mathbf{d})\\rangle\n\t\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert_{\\mathrm{L}^2} \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert_{\\mathbf{L}^4} \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathrm{L}^4}\\\\\n\t\t\\le &\\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}+ \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + C \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} \\lVert \\mathbf{d} \\rVert^2_1 \\lVert \\mathbf{d} \\rVert^2_2.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tThis completes the proof of Lemma \\ref{Lem:CouplingTermIto}\n\t\\end{proof}\nNow, let $\\kappa_i$, $i=1,\\ldots,4$ be the constants from Lemmata \\ref{Lem:FirstDerAppliedtoVdotNabalaD}, \\ref{Lem:FirstDerAppliedtoDeltaD+f(D)}, \\ref{Lem:NonlienarNSeItoV} and \\ref{Lem:CouplingTermIto}. For all $t\\ge 0 $ we set\n\t\\begin{equation}\\label{Eq:WeightToCancelbadterms}\n\t\\begin{split}\n\t\\Phi(t)=\\exp\\left({-\\int_0^{t}\\left[ (\\kappa_1 +\\kappa_4)(1+ \\lVert \\d(s) \\rVert^2_{1}) \\lVert \\d(s) \\rVert^2_2 + \\kappa_2 (1+ \\lVert \\d(s) \\rVert^{4N}_1) + \\kappa_3 \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\v(s) \\rvert^2_{\\mathrm{L}^2} \\right] ds }\\right).\n\t\\end{split}\n\t\\end{equation}\n\n\n\n\n\tLet $\\mathfrak{C}_0>0$ be the constant defined in \\eqref{Eq:ConstantFrakC0} and $\\mathfrak{C}_1>0$ the constant defined by\n\t\\begin{equation}\\label{Eq:ConstantFrakC1}\n\t\\mathfrak{C}_1= \\Psi(\\v_0,\\d_0) + \\mathfrak{C}_0+1\n\t\\end{equation}\n\t\\begin{prop}\\label{STRONGER-NORM}\n\t\tLet $\\Psi_1$, $\\Phi$, $\\mathfrak{C}_0$ and $\\mathfrak{C}_1$ be defined in \\eqref{Eq:DefPsi1forD}, \\eqref{Eq:WeightToCancelbadterms}, \\eqref{Eq:ConstantFrakC0} and \\eqref{Eq:ConstantFrakC1}, respectively. Let \t$(\\tau_k)_{k \\in \\mathbb{N}}$ be the sequence of stopping times defined in \\eqref{STOP}. \tLet $d=2$, $N\\in \\mathbb{N}$ and $\\mathbf{h} \\in \\mathbf{H}^2$. \n\t\t\n\t\tIf all the other assumptions of Theorem \\ref{GLOBAL-ST} are satisfied, then there exists an increasing function $\\psi:[0,\\infty) \\to (0,\\infty)$ and $\\kappa_9=\\kappa_9(N, \\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{align}\n\t\t&\t\\mathbb{E} \\sup_{s\\in[0,T]} \\Phi(s\\wedge \\tau_k)\\left(\\lvert \\nabla \\v(s\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2}+\\Psi_1(\\d(s\\wedge \\tau_k)) \\right)\\le \\kappa_9 \\psi(T) \\mathfrak{C}_1,\\label{Eq:EstSupofDel+F}\\\\\n\t\t& \\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\left( \\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2} +\\lvert \\nabla (\\mathrm{A}_1\\d(s)+f(\\d(s) )\\rvert^2_{\\mathbf{L}^2} \\right)\\, ds \\le \\kappa_9 \\psi(T) \\mathfrak{C}_1. \\label{Eq:EstInteofNablaDelta+F}\n\t\t\\end{align}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proof of this proposition will be given in Section \\ref{AppB}.\n\t\\end{proof}\n\t\\begin{cor}\n\t\tUnder all the assumptions of Proposition \\ref{STRONGER-NORM}, there exists a $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:EstIntegrofH3NormofD}\n\t\t\\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\lVert \\d(s) \\rVert^2_{3} ds\\le C(\\mathfrak{C}_0 + \\mathfrak{C}_1+1).\n\t\t\\end{equation}\n\t\\end{cor}\n\t\\begin{proof}\n\t\tBy part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, the H\\\"older inequality and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{8N}\\hookrightarrow \\mathbf{L}^4$ we infer that\n\t\t\\begin{align*}\n\t\t\\lvert \\nabla f(\\d) \\rvert^2_{\\mathbf{L}^2} = & \\lvert f^\\prime(\\d)[\\nabla \\d] \\rvert^2_{\\mathbf{L}^2}\\le C(( \\lvert 1+ \\lvert \\d \\rvert^{2N}) \\lvert \\nabla \\d \\rvert \\rvert^2_{\\mathbf{L}^2} )\\\\\n\t\t\\le & \\lvert \\nabla \\d\\rvert^2_{\\mathbf{L}^4}+ \\lvert \\d \\rvert^{4N}_{\\mathbf{L}^{8N}} \\lvert \\nabla \\d \\rvert^2_{\\mathbf{L}^4}\n\t\t\\le C (\\lVert \\d \\rVert^2_{2} + \\lVert \\d \\rVert_1^{8N+2}).\n\t\t\\end{align*}\n\t\tWith this at hand we complete the proof by using \\eqref{Eq:EstIntegralOfH2Norm}, \\eqref{Eq:EstInteofNablaDelta+F} and the fact $$\\lVert \\d \\rVert^2_3\\le \\lVert \\d\\rVert^2_2 + 2 \\lvert \\nabla (\\Delta \\d +f(\\d)) \\rvert^2_{\\mathbf{L}^2} +2 \\lvert \\nabla f(\\d) \\rvert^2_{\\mathbf{L}^2}.$$\n\t\\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\tAfter all these preparations we now proceed to the promised proof of Theorem \\ref{GLOBAL-ST}.\n\t\\begin{proof}[Proof of Theorem \\ref{GLOBAL-ST}]\n\t\tBy Theorem \\ref{LC-Local-Sol} the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution $((\\mathbf{v},\\mathbf{n}); \\tilde{\\tau}_\\infty)$. We shall prove that $\\mathbb{P}\\Big(\\tilde{\\tau}_\\infty<\\infty \\Big)=0 $. For this aim, let $\\{ \\tau_k; k \\in \\mathbb{N}\\}$ be the sequence of stopping times defined in \\eqref{STOP}. \n\t\n\tWe first establish the following chain of inequalities\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\mathbb{P}\\left(\\tau_k k^2\\} } \\right),\\\\\n\t\t&\\le \\frac{1}{k^2}\\mathbb{E}\\biggl(1_{\\{\\tau_k 2 \\log{k}\\} } \\biggr)=:\\mathrm{I}+\\mathrm{II}.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tNow, we estimate $\\mathrm{I}$ and $\\mathrm{II}$ separately.\n\t\tFirstly, from the definition of $\\tau_k$ and $\\Phi$ we have\n\t\t\\begin{align*}\n\t\t\\mathrm{I}\\le &\\frac{1}{k^2}\\mathbb{E} \\left[1_{\\{\\tau_k < t\\}} \\Phi(t\\wedge \\tau_k)\\left( \\lvert \\nabla \\mathbf{v}(t\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t\\wedge \\tau_k) \\rVert^2_2 \\right) \\right]\\\\\n\t\t&\\qquad + \\frac1{k^2}\\mathbb{E} \\left[1_{\\{\\tau_k < t\\}} \\Phi(t\\wedge \\tau_k)\\int_0^{t\\wedge \\tau_k}\\left( \\lvert \\mathrm{A} \\mathbf{v}(s) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(s) \\rVert^2_3 \\right)\\, ds \\right]\\\\\n\t\t\\le & \\frac{1}{k^2}\\mathbb{E} \\left[\\Phi(t\\wedge \\tau_k)\\left( \\lvert \\nabla \\mathbf{v}(t\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t\\wedge \\tau_k) \\rVert^2_2\\right) \\right]+\\frac{1}{k^2}\\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k}\\Phi(s)\\left( \\lvert \\mathrm{A} \\mathbf{v}(s) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(s) \\rVert^2_3 \\right)\\, ds \\right]\n\t\t\\end{align*}\n\t\tFrom \\eqref{Eq:Bigdanh2withH1norm}, \\eqref{Eq:ESTofVDinWeakerNorm} , \\eqref{Eq:EstInteofNablaDelta+F} and \\eqref{Eq:EstIntegrofH3NormofD} we infer that there exists a constant $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation*}\n\t\t\\mathrm{I}\\le \\frac{1}{k^2}C (\\mathfrak{C}_0+\\mathfrak{C}_1+1).\n\t\t\\end{equation*}\n\t\tSecondly, we estimate $\\mathrm{II}$ as follows\n\t\t\\begin{align*}\n\t\t\\mathrm{II}=& \\mathbb{E}\\left(1_{\\{\\tau_k < t\\}} 1_{\\{\\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr > 2\\log{k} \\} }\\right)\n\t\t\\le \\int_{\\{\\tau_k2\\log{k} \\}}\\frac{ \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr}{2\\log{k}} d\\mathbb{P}\\\\\n\t\t\\le & \\frac1{2\\log{k}} \\int_{\\{\\tau_k < t\\}} \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr\\;\\;\\; d\\mathbb{P}\n\t\t\\le \\frac{1}{2\\log{k}} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr.\n\t\t\\end{align*}\n\t\tNow, from \\eqref{Eq:ESTofVDinWeakerNorm} and \\eqref{Eq:EstIntegralOfH2Norm} we infer that there exists a constant $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{align*}\n\t\t&\\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert \\v(s)\\rvert^2_{\\mathrm{L}^2}\\lvert \\nabla \\v(s)\\rvert_{\\mathrm{L}^2}^2 ds\n\t\t\\le \\frac12 \\mathbb{E}\\sup_{0\\le s\\le t\\wedge \\tau_k}\\lvert \\v(s)\\rvert^4_{\\mathrm{L}^2} + \\frac12 \\mathbb{E} \\biggl[\\int_0^{t\\wedge \\tau_k}\n\t\t\\lvert \\nabla \\v(s)\\rvert_{\\mathrm{L}^2}^2 ds\n\t\t\\biggr]^2\\le C (\\mathfrak{C}_0+1),\\\\\n\t\t& \\mathbb{E}\\int_0^{t\\wedge \\tau_k}(1+1 \\lVert \\d(s)\\rVert^{2N}_{\\mathrm{H}^1})^2 ds\\le C (\\mathfrak{C}_0+1),\\\\\n\t\t& \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\Vert \\mathbf{n}(s) \\Vert^2_1 \\Vert \\mathbf{n} (s) \\Vert^2_2 ds \\le \\frac12 \\mathbb{E} \\sup_{0\\le s\\le t\\wedge \\tau_k} \\Vert \\mathbf{n}(s)\\Vert^4_1 + \\frac12 \\mathbb{E} \\biggl[\\int_0^{t\\wedge \\tau_k} \\Vert \\mathbf{n}(s) \\Vert^2_2 ds \\biggr]^2\\le C (\\mathfrak{C}_0+1).\n\t\t\\end{align*}\n\t\tThus, there exists a constant $C>0$ such that for all $k\\in \\mathbb{N}$\n\t\t\\begin{equation*}\n\t\t\\mathrm{II} \\le \\frac1{2\\log{k}} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\phi(s)ds\\le \\frac{C (\\mathfrak{C}_0+1)}{\\log{k}}.\n\t\t\\end{equation*}\n\t\tCollecting the information about $\\mathrm{I}$ and $\\mathrm{II}$ together, we infer that\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lim_{k\\rightarrow \\infty} \\mathbb{P}\\left(\\tau_k0$ such that for all $\\mathbf{d}\\in \\mathrm{ H}^1$ \n\t\t\\begin{equation}\\label{Eq:EstimateItoStratoCorrect}\n\t\t\\lvert \\nabla G(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} + \\langle \\nabla \\mathbf{d}, \\nabla G^2(\\mathbf{d}) \\rvert \\le C \\lVert \\mathbf{d} \\rVert^2_{1}.\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\tLet $\\mathbf{h} \\in \\mathbf{W}^{1,4}$. Using the H\\\"older inequality, the embeddings $\\mathbf{H}^1 \\hookrightarrow \\mathbf{L}^4$ and $\\mathbf{W}^{1,4}\\hookrightarrow \\mathrm{L}^\\infty$, and\n\t\t\\begin{equation}\\label{Eq:MixedScalCrossProducts}\n\t\t\\mathbf{a} \\cdot [(\\mathbf{b} \\times \\mathbf{c}) \\times \\mathbf{d} ]= - (\\mathbf{a} \\times \\mathbf{d}) \\cdot (\\mathbf{b} \\times \\mathbf{c}), \\;\\; \\forall\\;\\; \\mathbf{a}, \\mathbf{b}, \\mathbf{c}, \\mathbf{d}\\in \\mathbb{R}^3,\n\t\t\\end{equation}\n\t\twe infer that for all $\\mathbf{d}\\in \\mathrm{ H}^1$ and $i \\in \\{1,2\\}$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lvert \\partial_i (\\mathbf{d}\\times \\mathbf{h}) \\rvert^2_{\\mathbf{L}^2} + \\langle \\partial_i \\mathbf{d}, \\partial_i [(\\mathbf{d} \\times \\mathbf{h}) \\times \\mathbf{h} ] \\rangle =& \\lvert \\mathbf{d} \\times \\partial_i \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}+ \\langle \\partial_i \\mathbf{d} \\times \\mathbf{h}, \\mathbf{d} \\times \\partial_i \\mathbf{h} \\rangle + \\langle \\partial_i \\mathbf{d}, (\\mathbf{d} \\times \\mathbf{h})\\times \\partial_i \\mathbf{h} \\rangle \\\\\n\t\t&\\le \\lvert \\partial_i \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} \\lvert \\partial_i \\mathbf{h} \\rvert^2_{\\mathbf{L}^4} + \\lvert \\partial_i \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} \\lvert \\mathbf{h} \\rvert_{\\mathbf{L}^\\infty} \\lvert \\partial_i \\mathbf{h} \\rvert_{\\mathbf{L}^4} \\le C \\lVert \\mathbf{d} \\rVert^2_{1} \\lvert \\mathbf{h} \\rvert^2_{\\mathbf{W}^{1,4}}.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tHence, summing over $i$ from $1$ to $2$ imply the desired inequality \\eqref{Eq:EstimateItoStratoCorrect}.\n\t\\end{proof}\n\tWe now give the promised of the proposition.\n\t\\begin{proof}[Proof of Proposition \\ref{EST1} ]\n\t\tWithout loss of generality (Wlog) we only give a proof for $p=2(4N+1)$. Let $\\mathbf{h} \\in \\mathbf{W}^{1,4}$. Throughout this proof $C=C(\\lVert \\mathbf{h} \\rVert_{p, \\mathbf{W}^{1,4}})>0$ is a constant which may change from one term to the next one. Let $k \\in \\mathbb{N}$ be fixed and $\\tau_k$ be defined by \\eqref{STOP} . \n\t\t\n\t\tFirstly, let $\\Lambda:\\mathbf{H}^1 \\to [0,\\infty)$ be the map defined by\n\t\t\\begin{equation}\n\t\t\\Lambda(\\mathbf{d})= \\frac12 \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} + \\lvert \\nabla \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} +\\frac12 \\int_\\mathcal{O} F(\\lvert \\mathbf{d}(x) \\rvert^2) dx, \\;\\; \\mathbf{d}\\in \\mathbf{H}^1.\n\t\t\\end{equation}\n\t\tBy Assumption \\ref{eqn-f} and \\cite[Lemma 8.10]{Brz+Millet_2012} the map $\\Lambda(\\cdot)$ is twice Fr\\'echet differentiable. Moreover, elementary calculations and \\eqref{Eq:MixedScalCrossProducts} imply \n\t\t\\begin{align}\n\t\t& \\Lambda^\\prime(\\mathbf{d})[G(\\mathbf{d})]= \\langle \\nabla \\mathbf{d}, \\mathbf{d}\\times \\nabla \\mathbf{h} \\rangle,\\label{Eq:PsiDerGd}\\\\\n\t\t& \\frac12 \\Lambda^\\prime (G^2(\\mathbf{d})) + \\frac12\\Lambda^{\\prime \\prime}[G(\\mathbf{d}), G(\\mathbf{d}) ]=\\frac12 \\lvert \\nabla G(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} + \\frac12 \\langle \\nabla \\mathbf{d}, \\nabla G^2(\\mathbf{d}) \\rangle. \\label{Eq:ItoStrato-Correct}\n\t\t\\end{align}\n\t\tWe also observe that if $\\mathbf{u} \\in \\mathrm{V}$ such that $\\Div \\mathbf{u}=0$, then\n\t\t\\begin{equation}\n\t\t\\langle \\mathbf{u} \\cdot \\nabla \\mathbf{d}, f(\\mathbf{d}) \\rangle = \\frac12 \\int_\\mathcal{O} \\mathbf{u}(x)\\cdot \\nabla F(\\mathbf{d}(x)) dx=0.\\label{Eq:Divvimply}\n\t\t\\end{equation}\n\t\t\n\t\tSecondly, applying the It\\^o formula to $\\frac12 \\lvert \\v(t\\wedge \\tau_k)\\rvert^2_{\\mathrm{L}^2} + \\Lambda(\\d(t\\wedge \\tau_k)) $ and using \\eqref{tild-b-0}, \\eqref{B3}, \\eqref{G1-eq-Md}, \\eqref{Eq:PsiDerGd}, \\eqref{Eq:ItoStrato-Correct} and \\eqref{Eq:Divvimply} yield\n\t\t\\begin{equation}\\label{Eq:ResultIto}\n\t\t\\begin{split}\n\t\t&\\mathcal{E}[\\v,\\d](t\\wedge \\tau_k)-\\mathcal{E}[\\v,\\d](0) + \\int_0^{t\\wedge \\tau_k} \\mathscr{D}[\\v,\\d] (s)\\, ds - \\int_0^{t\\wedge \\tau_k} \\langle \\d(s) , f(\\d(s)) \\rangle ds-\\frac12 \\int_0^{t\\wedge \\tau_k} \\lvert \\nabla G(\\d(s)) \\rvert^2_{\\mathbf{L}^2} ds \\\\\n\t\t& = \\frac12 \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\mathbf{n}(s), \\nabla G^2(\\mathbf{n}(s)) \\rangle ds +\n\t\t\\int_0^{t\\wedge \\tau_k} \\langle \\v(s), S(\\v(s)) dW_1(s) \\rangle + \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\nabla \\mathbf{h}\\rangle dW_2.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tBefore proceeding further, we should observe that thanks to Assumption \\ref{eqn-f} and \\cite[Lemma 8.7]{Brz+Millet_2012} we infer that there exists exists $c>0$ independent of $k$ such that \n\t\t\\begin{equation*}\n\t\t\\frac{-a_{N+1}}{2} \\int_{\\mathcal{O}} \\lvert \\d(x)\\rvert^{2N+2} dx-c \\int_{\\mathcal{O}} \\lvert \\d(x)\\rvert^2 dx \\le \\langle -f(\\d), \\d\\rangle.\n\t\t\\end{equation*}\n\t\tHence, plugging this inequality and \\eqref{Eq:EstimateItoStratoCorrect} into \\eqref{Eq:ResultIto} implies\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t&\\mathcal{E}[\\v,\\d](t\\wedge \\tau_k)-\\mathcal{E}[\\v,\\d](0) + \\int_0^{t\\wedge \\tau_k} \\mathscr{D}[\\v,\\d] (s)\\, ds -\\frac{a_{N+1}}{2} \\int_0^{t\\wedge \\tau_k} \\lvert \\d(s) \\rvert^{2N+2}_{\\mathrm{L}^{2N+2}} ds\\\\\n\t\t& \\le \\int_0^{t\\wedge \\tau_k} \\langle \\v(s), S(\\v(s)) dW_1(s) \\rangle + \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\nabla \\mathbf{h}\\rangle dW_2+ \\frac12 \\int_0^{t\\wedge \\tau_k} \\mathcal{E}[\\v,\\d](s)ds .\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThirdly, by taking the supremum over $s\\in [0,t]$, raising to the power $p$, taking the mathematical expectation to both sides of the above inequality and applying the H\\\"older inequality we obtain\n\t\t\\begin{equation}\\label{Eq:PowerPofNRJ}\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^p + \\mathbb{E} \\left[ \\int_0^{t\\wedge \\tau_k}\\left(\\mathscr{D}[\\v,\\d](s) - \\frac{a_{N+1}}{2} \\lvert \\mathbf{n}(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)ds \\right]^p\\\\\n\t\t& \\le C\\mathbb{E}\\lvert \\mathscr{E}[\\v,\\d] (0)\\rvert^p+Ct^{p-1}\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\mathbf{n}](s)\\lvert^p ds + C \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\v(s), S(\\v(s)) dW_1(s) \\biggr\\rvert^p\\\\\n\t\t& \\qquad +\n\t\tC \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\mathbf{h} \\rangle dW_2(s) \\biggr\\rvert^p.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tFor the time being let us assume that there exists $C=C(p,\\lVert \\mathbf{h} \\rVert_{\\mathbf{W}^{1,4}})>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t& \\mathscr{M}(t\\wedge \\tau_k):=C \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\v(s), S(\\v(s)) dW_1(s) \\biggr\\rvert^p+\n\t\tC \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\mathbf{h} \\rangle dW_2(s) \\biggr\\rvert^p\\\\\n\t\t&\\qquad \\qquad \\le \\frac12 \\mathbb{E} \\sup_{s\\in [0,t]} \\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k) \\rvert^p + C_1t^\\frac{p}2 + C_1 t^{\\frac{p-2}{2}}\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert \\mathscr{E}[\\v,\\d](s)\\rvert^p ds,\\label{Eq:EstPowerPofMart}\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tfrom which along with \\eqref{Eq:PowerPofNRJ} we infer that there exists $C=C(p,\\lVert \\mathbf{h} \\rVert_{\\mathbf{W}^{1,4}})>0$ such that\n\t\t\\begin{equation\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^p + 2 \\mathbb{E} \\left[ \\int_0^{T\\wedge \\tau_k}\\left(\\mathscr{D}[\\v,\\d](s) - \\frac{a_{N+1}}{2} \\lvert \\mathbf{n}(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)ds \\right]^p\\\\\n\t\t& \\le C \\mathbb{E}\\lvert \\mathscr{E}[\\v,\\d] (0)\\rvert^p+ Ct^\\frac{p}2 + C(t^{p-1}+ t^{\\frac{p-2}{2}}) \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\mathbf{n}](s)\\lvert^p ds.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tWe then apply the Gronwall lemma and obtain the desired \\eqref{Eq:ESTofVDinWeakerNorm}.\\\\ \n\t\tThus, it remains to prove \\eqref{Eq:EstPowerPofMart}. For this purpose, by applying the Burkholder-Davis-Gundy (BDG), the H\\\"older and Young inequalities, and Assumption \\ref{HYPO-ST} (mainly \\eqref{Eq:Hypo-ST}) we infer that \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\mathscr{M}(t\\wedge \\tau_k)& \\le C_4 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2} \\right]^\\frac{p}2+C_4 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\lvert \\nabla \\d(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\d(s) \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} \\right]^\\frac{p}2\\\\\n\t\t& \\le C_4 t^\\frac{p-2}{2} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\d](s)\\rvert^{\\frac{p}{2}} [\\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2} + \\lvert \\d(s) \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} ]^\\frac{p}{2} ds\\\\\n\t\t&\\le \\frac12 \\mathbb{E} \\sup_{s\\in [0,t]} \\mathbb{E} \\rvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^{p} +C_5 t^{p-2} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} [1+ \\lvert\\v(s)\\rvert^2_{\\mathrm{L}^2} + \\lvert \\d(s) \\rvert^2_{\\mathbf{L}^4} \\lvert \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^4} ]^p ds.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThe last line and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^4$ imply \\eqref{Eq:EstPowerPofMart}. This completes the proof of Proposition \\ref{EST1}.\n\t\\end{proof}\n\t\\subsection{Proof of Proposition \\ref{STRONGER-NORM}}\n\tBefore giving the promised proof we firstly state and prove two important lemmata. The first one is inspired by \\cite[Lemma 6.3]{ZB+MO}.\n\t\\begin{lem}\\label{Lem:Abstract-Lem-L(x)}\n\t\tLet $V_1, H_1, \\tilde{V}_1$ be there separable Hilbert spaces such that the embeddings $V_1\\hookrightarrow H_1\\hookrightarrow \\tilde{V}_1$ are dense and continuous. Let $\\mathcal{A}: V_1 \\to \\tilde{V}_1$ be a bounded linear map and $\\mathfrak{f}: [0,T] \\to H_1$ and $g: [0,T] \\to V_1$ measurable and progressively measurable respectively such that\n\t\t\\begin{equation}\\label{Eq:Integrabilitoffandg}\n\t\t\\mathbb{E}\\int_0^T[ \\lvert \\mathfrak{f}(t) \\rvert^2_{\\tilde{V}_1} + \\lvert g(t) \\rvert^2_{H_1} ]dt <\\infty.\n\t\t\\end{equation}\nLet $x: [0,T]\\times \\Omega \\to V_1$ be a progressively measurable and $H_1$-continuous process such that \n\t\\begin{align}\n\t&\t\\mathbb{E}\\int_0^T \\lvert x(s) \\rvert^2_{V_1}ds<\\infty,\\label{Eq:Integrabilityofx}\\\\\n&\tx(t) =x(0) + \\int_0^t \\mathcal{A}x(s)\\, ds + \\int_0^t \\mathfrak{f}(s)\\, ds + \\int_0^t g(s) dW_2(s)\\, \t\\text{ for all $t$ $\\mathbb{P}$-a.s..} \t\\label{Eq:Equationforx}\n\t\\end{align}\n\tNow, let $V_2, H_2$ be two separable Hilbert spaces and $V_2^\\ast$ the dual of $V_2$. We identify $H_2$ with its dual and we assume that the embeddings $V_2\\hookrightarrow H_2\\hookrightarrow V_2^\\ast $ are continuous and dense. Let $\\mathcal{B}: V_2 \\to V_2^\\ast$ be a bounded linear map. Let $L: \\tilde{V}_1 \\to V_2^\\ast$ be a twice Fr\\'echet differentiable map such that:\n\t\\begin{enumerate}\n\t\t\\item $L(V_1)\\subset V_2$ and $L(H_1)\\subset H_2$,\n\t\t\\item there exists $\\mathcal{H}: V_1 \\to H_2$ such that for every $z\\in V_1$\n\t\t\\begin{equation}\\label{Eq:LprimeAz=Bz+H(z)}\n\t\tL^\\prime(z)[\\mathcal{A}z]=\\mathcal{B}L(z) +\\mathcal{H}(z).\n\t\t\\end{equation}\n\t\t\\item The map $L^{\\prime\\prime}$ is bounded on balls.\n\t\\end{enumerate}\n\t\n\tThen for every $t\\in [0,T]$, $\\mathbb{P}$-a.s. the following identity holds in $V_2^\\ast$\n\t\\begin{equation}\\label{Eq:IdentityforL(x)inDualV1}\n\\begin{split}\n\tL(x(t)) =L(x(0)) +\\int_0^t\\mathcal{B}L(x(s))\\, ds + \\int_0^t\\Bigl( L^\\prime(x(s))[\\mathfrak{f}(s)] +\\mathcal{H}(x(s)) \\Bigr)\\, ds \\\\+ \\frac12 \\int_0^t L^{\\prime\\prime}(x(s))[g(s), g(s)] ds+ \\int_0^t L^\\prime(x(s))[g(s)]dW_2(s).\n\\end{split}\n\t\\end{equation}\n\t\\end{lem}\n\\begin{proof}\n\tLet $t\\in [0,T]$ and $\\varphi \\in V_2$. Let $z\\ni L_\\varphi:H_1 \\mapsto L_\\varphi(z):= {}_{V_2^\\ast}\\langle L(z), \\varphi \\rangle_{V_2}\\in \\mathbb{R}$.\n\tBy the assumptions on $L$, $L_\\varphi$ is twice Fr\\'echet differentiable, $L_\\varphi$ and $L^\\prime_\\varphi$ are continuous on $H_1$, $L_\\varphi$, $L_\\varphi^\\prime$ and $L_\\varphi^{\\prime\\prime}$ are locally bounded. Hence, by It\\^o formula, see \\cite[Theorem 3.2]{Pardoux}, we infer that $\\mathbb{P}$-a.s.\n\\begin{equation*}\n\\begin{split}\nL_\\varphi(x(t))=L_\\varphi(x(0)) + \\int_0^t {}_{V_2^\\ast}\\langle L^\\prime(x(s))[\\mathcal{A}(x(s)) +\\mathfrak{f}(s) ], \\varphi \\rangle_{V_2} + \\int_0^t {}_{H_2}\\langle L^\\prime(x(s)) [g(s)],\\varphi \\rangle_{H_2} dW_2(s)\\\\+ \\frac12\\int_0^t {}_{H_2}\\langle L^{\\prime\\prime}(x(s))[g(s), g(s)], \\varphi\\rangle_{H_2} ds\n\\end{split}\n\\end{equation*}\nUsing \\eqref{Eq:LprimeAz=Bz+H(z)} yields\n\\begin{equation*}\n\\begin{split}\nL_\\varphi(x(t))=L_\\varphi(x(0)) + \\int_0^t {}_{V_2^\\ast}\\langle \\mathcal{B}(x(s)), \\varphi \\rangle_{V_2} ds +\\int_0^t {}_{V_2^\\ast}\\langle L^\\prime(x(s))[ \\mathfrak{f}(s)]+H(x(s)) , \\varphi \\rangle_{V_2}ds \\\\+ \\frac12\\int_0^t {}_{H_2}\\langle L^{\\prime\\prime}(x(s))[g(s), g(s)], \\varphi\\rangle_{H_2} ds +\n\\int_0^t {}_{H_2}\\langle L^\\prime(x(s)) [g(s)],\\varphi \\rangle_{H_2} dW_2(s) .\n\\end{split}\n\\end{equation*}\nThis completes the proof of \\eqref{Eq:IdentityforL(x)inDualV1}.\n\\end{proof}\n\t\\begin{lem}\\label{Lem:IdentityforDd+f(d)}\n\tLet $\\mathbf{h}\\in \\mathbf{H}^2$ and $\\{y(t): t\\in [0,\\tilde{\\tau}_\\infty)\\}$ be the local process defined by\n\t\\begin{equation}\\label{Eq:Def-Dd+f(d)}\n\ty(t):=\n\t \\mathrm{A}_1 \\mathbf{n}(t) - f(\\mathbf{n}(t)), \\;\\text{ } t\\in [0,\\tilde{\\tau}_\\infty).\n\t\\end{equation}\n\tThen, for all $t\\in [0,T]$, $k \\in \\mathbb{N}$, $\\mathbb{P}$-a.s. the following equation holds in $(\\mathbf{H}^1)^\\ast$\n\t\\begin{equation}\\label{Eq:IdentityforDd+f(d)}\n\\begin{split}\n&\ty(t\\wedge \\tau_k)+ \\int_0^{t\\wedge \\tau_k} \\Bigl(\\mathrm{A}_1 y(r) + (\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[\\mathbf{v}(r) \\cdot \\nabla \\mathbf{n}(r)] \\Bigr)\\, dr -\\int_0^{t\\wedge \\tau_k} (\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[G(\\mathbf{n}(r))] dW_2\\\\\n&= y(0) +\\frac12 \\int_0^{t\\wedge \\tau_k} \\Bigl(2f^\\prime(\\mathbf{n}(r))y(r) +(\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[G^2(\\d(r))] - f^{\\prime \\prime}(\\d(r))[G(\\d(r)), G(\\d(r))] \\Bigr)\\, dr.\n\\end{split}\n\t\\end{equation}\n\t\\end{lem}\n\\begin{proof}[Proof of Lemma \\ref{Lem:IdentityforDd+f(d)}]\n\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. \n\tLet us put $V_1=D((I+\\mathrm{A}_1)^\\frac32)$, $H_1=D(\\mathrm{A}_1)$, $\\tilde{V}_1=\\mathbf{H}^1$ and $\\mathcal{A}=\\mathcal{B}=-\\mathrm{A}_1$. We also set $V_2=\\mathbf{H}^1$, $H_2=\\mathbf{L}^2$, $V_2^\\ast=(\\mathbf{H}^1)^\\ast$. The map\n\t$L: \\mathbf{H}^1 \\ni z \\mapsto L(z):= \\mathrm{A}_1z-f(z) \\in (\\mathbf{H}^1)^\\ast$satisfies the assumptions of Lemma \\ref{Lem:Abstract-Lem-L(x)}. In particular, if we set $\\mathcal{H}(z)= \\mathrm{A}_1f(z) -f^\\prime(z)[\\mathrm{A}_1z], \\,z \\in V_1 $, then \n\t$$ L^\\prime(z)[\\mathrm{A}_1z]= \\mathrm{A}_1[ \\mathrm{A}_1 z - f(z)] +\\mathrm{A}_1f(z) -f^\\prime(z)[\\mathrm{A}_1z] = \\mathrm{A}_1 L(z) +\\mathcal{H}(z)\\in (\\mathbf{H}^1)^\\ast. $$\n Now, let $k\\in \\mathbb{N}$ and\n \\begin{align}\n &\\mathfrak{f}= \\mathds{1}_{[0,\\tau_k)} [ -\\mathbf{v} \\cdot \\nabla \\mathbf{n} + f(\\mathbf{n}) +\\frac12G^2(\\mathbf{n}) ],\\label{Eq:DefFrakf}\\\\\n &g= \\mathds{1}_{[0,\\tau_k)} G(\\mathbf{n}).\\label{Eq:Defsmallg}\n \\end{align}\nBy Lemmata \\ref{Local-LIP-Lem-2} and\\ref{Local-LIP-Lem-3}, and the definition of $\\tau_k$ we infer that there exists $C>0$ such that\n \\begin{align}\n& \\mathbb{E} \\int_0^{t} \\lvert \\mathfrak{f}(r) \\rvert^2_{\\mathbf{H}^1}\\le \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Bigl(\\lvert \\mathbf{v}(r) \\cdot \\nabla \\mathbf{n}(r)\\rvert^2_{\\mathbf{H}^1} +\\lvert f(\\mathbf{n}(r))\\rvert^2_{\\mathbf{H}^1}+\\lvert \\frac12 (\\mathbf{n}(r) \\times \\mathbf{h}) \\times \\mathbf{h} \\rvert^2_{\\mathbf{H}^1}\\Bigr)\\, dr\\le C,\\label{Eq:Proof of IdDd+f(d)-1}\\\\\n& \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert G(\\mathbf{n}(r)) \\rvert^2_{\\mathbf{H}^2} dr = \t\\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert \\mathbf{n}(r) \\times \\mathbf{h} \\rvert^2_{\\mathbf{H}^2} dr \\le C.\\label{Eq:Proof of IdDd+f(d)-2}\n \\end{align}\n These mean that $\\mathfrak{f}$ and $g$ satisfy \\eqref{Eq:Integrabilitoffandg}. Because the local strong solution $\\mathbf{y}=(\\mathbf{v},\\mathbf{n})$ of \\eqref{ABSTRACT-LC} satisfies \\eqref{eq-locsol_01-a}, the process $x(t)=\\mathbf{n}(t\\wedge \\tau_k)$ satisfies \\eqref{Eq:Integrabilityofx} and \\eqref{Eq:Equationforx} with $\\mathfrak{f}$ and $g$ as defined above. By setting\n $y(t\\wedge \\tau_k)=L(\\mathbf{n}(t\\wedge \\tau_k)), t\\ge 0,$ and applying Lemma \\ref{Lem:Abstract-Lem-L(x)} we obtain\n \\begin{equation}\\label{Eq:AbsFormofAd+f(d)}\n \\begin{split}\n y(t\\wedge \\tau_k)=y(0) +\\int_0^{t\\wedge \\tau_k} \\Bigl( -\\mathrm{A}_1y(r) + L^\\prime(\\mathbf{n}(r)) [\\mathfrak{f}(r)] + \\mathcal{H}(\\boldsymbol{\\nu}(r)) \\Bigr)\\, dr \\\\+ \\frac12\\int_0^{t\\wedge \\tau_k} L^{\\prime \\prime}(\\mathbf{n}(r))[g(r), g(r)] dr + \\int_0^{t\\wedge \\tau_k} L^\\prime(\\mathbf{n}(r)) [g(r) ]dW_2(r).\n \\end{split}\n \\end{equation}\nWe complete the proof of the lemma by taking into account the last line and the following identity \n\\begin{equation}\\label{Eq:1stDer+2ndDerofL}\n L^\\prime(z)=\\mathrm{A}_1 -f^\\prime(z) \\text{ and } L^{\\prime\\prime } (z)= - f^{\\prime\\prime}(z) \\text{ for every } z\\in \\mathbf{H}^1.\n\\end{equation}\n \\end{proof}\n\n\n\tWe now give the promised proof of Proposition \\ref{STRONGER-NORM}.\n\t\\begin{proof}[Proof of Proposition \\ref{STRONGER-NORM}]\n\t\tThroughout, $L(z)=\\mathrm{A}_1 z-f(z)$ and $\\mathcal{H}(z)= \\mathrm{A}_1f(z) - f^\\prime(z)[\\mathrm{A}_1 z]$ be defined as in the proof of Lemma \\ref{Lem:IdentityforDd+f(d)}. \n\tKeeping in mind the notations of \\ref{Lem:IdentityforDd+f(d)}, in particular \\eqref{Eq:DefFrakf}, \\eqref{Eq:Defsmallg} and \\eqref{Eq:1stDer+2ndDerofL},\n\t\twe set\n\t\t\\begin{equation}\n\t\t\\mathfrak{v}= -\\mathds{1}_{[0,\\tau_k]} \\mathrm{A}_1 L(\\mathbf{n}) + L^\\prime(\\mathbf{n})[\\mathfrak{f} ] +\\mathds{1}_{[0,\\tau_k]}\\mathcal{H}(\\mathbf{n}) +\\frac12 L^{\\prime\\prime}(\\mathbf{n})[g, g], \\, k \\in \\mathbb{N}.\n\t\t\\end{equation}\nThen, for all $k \\in \\mathbb{N}$ and $F\\in L^2(\\Omega\\times[0,t]; \\mathbf{L}^2)$, $t\\ge 0$, \n\t\t\t\\begin{equation}\\label{Eq:Proof of IdDd+f(d)-3-b}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^\\prime(\\mathbf{n}(r))[F(r)] \\rvert^2_{\\mathbf{L}^2} dr \\le c_1^2(1+k^2) \\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert F(r) \\rvert^2_{\\mathbf{L}^2}dr.\n\t\t\\end{equation}\n\tIn fact, from Remark \\ref{REM-H2}\\eqref{Itemi:REM-H2}, the embedding $\\mathbf{H}^2\\hookrightarrow \\mathbf{L}^\\infty$ and \\eqref{Eq:Proof of IdDd+f(d)-1} we infer that\n\t\t\\begin{align*}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^\\prime(\\mathbf{n}(r))[F(r)] \\rvert^2_{\\mathbf{L}^2}\\le& \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\int_\\mathcal{O} \\lvert f^\\prime(\\mathbf{n}(r,x))[F(r,x)]\\rvert^2 dxdr\\\\\n\t\t& \\le c_1^2 \\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1 + \\lvert \\mathbf{n}(r,x) \\rvert^{4N})\\lvert F(r,x)\\rvert^2 dxdr\\\\\n\t\t&\\le c_1^2 \\mathbb{E}\\biggl([1+\\sup_{t\\in [0,T]} \\lvert \\mathbf{n}(t\\wedge \\tau_k) \\rvert^{4N}_{\\mathrm{L}^\\infty}]\\int_0^{\\tau_k} \\lvert F(r) \\rvert^2_{\\mathbf{L}^2} dr \\biggr) \\\\\n\t\t&\t\\le c_1^2(1+k^{2N}) \\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert F(r) \\rvert^2_{\\mathbf{L}^2}dr.\n\t\t\\end{align*}\n\t\tIn a similar way, we can prove that for all $k \\in \\mathbb{N}$ and $g=1_{[0,\\tau_k]} \\mathbf{n}\\times \\mathbf{h} $\n\t\t\\begin{equation}\\label{Eq:Proof of IdDd+f(d)-4}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^{\\prime\\prime}(\\mathbf{n}(r))[g(r) , g(r) ]\\rvert^2_{\\mathbf{L}^2} \\le c^2_2 \\lVert \\mathbf{h} \\rVert^4_{2}(1+k^{2N} )\\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert \\mathbf{n}(r) \\rvert^2_{\\mathbf{L}^2}dr .\n\t\t\\end{equation}\n\tFrom the continuity of the linear map $\\mathrm{A}_1:\\mathbf{H}^1 \\to (\\mathbf{H}^1)^\\ast $, the embedding $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^2$, \\eqref{Eq:Proof of IdDd+f(d)-3-b}, \\eqref{Eq:Proof of IdDd+f(d)-4} along with \\eqref{Eq:Proof of IdDd+f(d)-1} and \\eqref{Eq:Proof of IdDd+f(d)-2} we infer that there exits $C=C(k)>0$ such that\n\t\t\\begin{equation}\\label{Eq:DriftsatisfiesPardouxConditions}\n\\mathbb{E}\\int_0^{t} \\lvert \\mathfrak{v}(r) \\rvert^2_{(\\mathbf{H}^1)^\\ast} \\le C \\mathbb{E}\\int_0^{t\\wedge \\tau_k}\\Bigl( \\lvert L(\\mathbf{n}(r)) \\rvert^2_{\\mathrm{L}^2} + \\lvert \\mathfrak{f}(r) \\rvert^2_{\\mathbf{H}^1} +\\lvert f(\\mathbf{n}(r) ) \\rvert^2_{\\mathbf{H}^1}\n+ \\lvert f^{\\prime \\prime}(\\mathbf{n}(r))[g(r), g(r) ] \\rvert^2_{\\mathbf{L}^2} \\Bigr)dr <\\infty.\n\t\t\\end{equation}\n\t\t\n\t\tNext, let $\\{ y(t): t\\in [0,\\tilde{\\tau}_\\infty) \\}$ be the process defined in \\eqref{Eq:Def-Dd+f(d)}. The process $\\{ y(t\\wedge \\tau_k): t \\ge 0 \\}$ is an $\\mathbf{L}^2$-valued process and satisfies the equivalent equations \\eqref{Eq:IdentityforDd+f(d)} and \\eqref{Eq:AbsFormofAd+f(d)}. Hence, by \\eqref{Eq:DriftsatisfiesPardouxConditions} we can apply the It\\^o formula to $ \\frac12\\lvert y(t\\wedge \\tau_k)\\rvert^2_{\\mathbf{L}^2}=\\frac12\\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2}=\\Psi_1(\\mathbf{d})$ (see \\cite[Theorem 3.2]{Pardoux}), and use the fact\n\t\t\\begin{equation*}\n\t\t-(\\mathrm{A}_1-f^\\prime(\\mathbf{n}))[\\mathbf{v} \\cdot \\nabla \\mathbf{n}]+ (f^\\prime(\\mathbf{n})y - (\\mathrm{A}_1-f^\\prime(\\mathbf{n}))[\\frac12G^2(\\d)] - \\frac12f^{\\prime \\prime}(\\d)[G(\\d), G(\\d)] \\in \\mathbf{L}^2,\n\t\t\\end{equation*}\n\tto infer that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation} \\label{Eq:Psi(d)Formnula-1}\n\t\t\\begin{split}\n\t&\t\\Psi_1(\\mathbf{n}(t\\wedge \\tau_k)) + \\int_0^{t\\wedge \\tau_k} \\lvert \\nabla (\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r)) ) \\rvert^2_{\\mathbf{L}^2} dr~+\\int_0^{t\\wedge \\tau_k}\\Psi^\\prime_1(\\d(r))[ \\v(r)\\cdot \\nabla \\d(r)]dr\\\\\n\t&\t=\n\t\t\\Psi_1(\\mathbf{n}_0) +\\int_0^{t\\wedge \\tau_k}\\Psi^\\prime_1(\\d(r))[ \\frac12 G^2(\\d(r))]dr + \\frac12 \\int_0^{t\\wedge \\tau_k} \\Psi_1^{\\prime\\prime}(\\mathbf{n}(r))[G(\\mathbf{n}(r)), G(\\mathbf{n}(r)) ] dr\\\\\n\t&\t- \\int_0^{t\\wedge \\tau_k} \\langle\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r)), f^\\prime(\\mathbf{n}(r))[\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r))] \\rangle dr+ \\int_0^{t\\wedge \\tau_k} \\Psi_1(\\mathbf{n}(r)) [G(\\mathbf{n}(r))]dW_2(r).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tIn preparation of our next step, let us set\n\t\t\\begin{equation*}\n\t\t\\mathrm{M}(t\\wedge \\tau_k)= \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(\\v(s)) \\circ S(\\v(s)) dW_1(s) + \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_1(\\d(s))[G(\\d(s))] dW_2(s),\\, k \\in \\mathbb{N}, t\\ge 0. \n\t\t\\end{equation*}\n\tWith \\eqref{Eq:Psi(d)Formnula-1} and the definitions of $\\Psi_2, \\Psi$ and $\\Phi$ (see \\eqref{Eq:DefPsi2forv}, \\eqref{Eq:DefFunctionForIto} and \\eqref{Eq:WeightToCancelbadterms}) in mind, we\tapply the It\\^o formula to $\\Upsilon(t\\wedge \\tau_k) =\\Phi(t\\wedge \\tau_k) \\Psi(\\mathbf{u},\\mathbf{n})(t\\wedge \\tau_k) $ and obtain that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\Upsilon(t\\wedge \\tau_k) -\\Upsilon(0)=& \\mathrm{M}(t\\wedge \\tau_k)+ \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi^\\prime_2(\\v(s)) [-B(\\v(s),\\v(s ) ) -M(\\d(s), \\d(s)) ] ds\\\\\n\t\t&+\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi^\\prime_1(\\d(s))[ -\\v(s)\\cdot \\nabla \\d(s) +\\frac12 G^2(\\d(s)) ]ds\\\\\n\t\t& +\\frac12 \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_1^{\\prime\\prime}(\\d(s))[G(\\d(s)), G(\\d(s))] ds+\\int_0^{t\\wedge \\tau_k} \\frac{d}{ds}\\Phi(s) \\Psi(\\v(s), \\d(s)))\\, ds\\\\\n\t\t&- \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\langle\\mathrm{A}_1\\mathbf{n}(s) -f(\\mathbf{n}(s)), f^\\prime(\\mathbf{n}(s))[\\mathrm{A}_1\\mathbf{n}(s) -f(\\mathbf{n}(s))] \\rangle ds\\\\\n\t\t&\t-\\int_0^{t\\wedge \\tau_k}\\Phi(s)\\left(\\lvert \\nabla (\\mathrm{A}_1\\mathbf{n}(s))-f(\\mathbf{n}(s)) \\rvert^2_{\\mathbf{L}^2}+ \\lvert \\mathrm{A} \\v(s) \\rvert^2 - \\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2(\\mathrm{K}_1; \\mathrm{V})} \\right)ds.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tBy Assumption \\ref{HYPO-ST}, \\eqref{Eq:Hypo-ST-Rem}, Lemma \\ref{Lem:FirstDerAppliedtoVdotNabalaD}-\\ref{Lem:CouplingTermIto} and the facts $\\lvert \\Phi \\rvert\\le 1$ and\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\frac{d}{ds}\\Phi(s) \\Psi(\\v(s), \\d(s))) = &-\\left[ ((\\kappa_1 +\\kappa_4) \\lVert \\d(s) \\rVert^2_2 + \\kappa_2) (1+ \\lVert \\d(s) \\rVert^2_1) + \\kappa_3 \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\v(s) \\rvert^2_{\\mathrm{L}^2} \\right] \\\\\n\t\t& \\qquad \\qquad \\times \\Phi(s)\\Psi(\\v(s), \\d(s)),\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twe infer that there exists $\\kappa_7>0$ such that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation}\\label{Eq:ItoresultStrongnormMart}\n\t\t\\begin{split}\n\t\t&\\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s\\wedge \\tau_k) + \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\left[\\frac14 \\lvert \\mathrm{A} \\v(s) \\rvert^2_{\\mathrm{L}^2} + \\frac12 \\lvert\\nabla (\\mathrm{A}_1 \\d(s) +f(\\d(s)) ) \\rvert^2_{\\mathrm{L}^2} \\right] ds \\\\\n\t\t& \\le \\Upsilon(0) + \\kappa_7 T+ \\kappa_7 \\mathbb{E} \\int_0^{t\\wedge\\tau_k} \\Upsilon(s)\\, ds + \\kappa_5\\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds + \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathrm{M}(s\\wedge \\tau_k) \\rvert.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\t\\noindent Next, by applying the BDG inequality, taking into account Assumption \\ref{HYPO-ST}, \\eqref{Eq:Psi1BDGNeeded} and the fact $\\lvert \\Phi \\rvert\\le 1 $ we infer that there exists $\\kappa_8>0$ such that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t&\t\\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathrm{M}(s\\wedge \\tau_k) \\rvert \\\\\n\t& \\le \\kappa_8\\mathbb{E} \\left(\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(s)[1+ \\Psi_2(s)] \\Phi(s)\\, ds \\right)^\\frac12 + \\kappa_8\n\t\t\\mathbb{E} \\left(\\int_0^{t\\wedge \\tau_k} \\Phi^2(s) \\lvert \\Psi_1^\\prime(\\d(s))[G(\\d(s))]\\rvert^2 ds \\right)^\\frac12 \\\\\n\t\t& \\le \\frac18 \\mathbb{E} \\sup_{s\\in [0,t]} [\\Phi(s)\\Psi_2(\\v(s)) ]+ \\kappa_8 T +\\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} [\\Phi(s)\\Psi_1(\\d(s))]^2 ds \\right]^\\frac12\\\\\n\t\t& \\qquad \\qquad + \\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+4}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12 +\\kappa_8 \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(\\v(s))ds \\\\\n\t\t&\\le \\frac14 \\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s)+ \\kappa_8 T + \\kappa_8 \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Upsilon(s)\\, ds + \\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tUsing the last inequality and absorbing the term $\\frac14 \\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s)$ in the LHS of \\eqref{Eq:ItoresultStrongnormMart}, and applying Gronwall inequality imply that there exist an increasing function $\\psi:[0,\\infty) \\to (0,\\infty)$ and a constant $\\kappa_9>0$ such that for all $k \\in \\mathbb{N}$ and $T\\ge 0$ \n\t\t\\begin{equation}\\label{Eq:FinalStepEstinStrongNorm}\n\t\t\\begin{split}\n\t\t\\mathbb{E} \\sup_{s\\in [0,T]} \\Upsilon(s\\wedge \\tau_k) +\\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\left[\\frac14 \\lvert \\mathrm{A} \\v(s) \\rvert^2_{\\mathrm{L}^2} + \\frac12 \\lvert\\nabla (\\mathrm{A}_1 \\d(s) +f(\\d(s)) ) \\rvert^2_{\\mathrm{L}^2} \\right] ds\\\\\n\t\t\\le\n\t\t\\psi(T) \\biggl(1+ \\Upsilon(0) + \\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12\\\\\n\t\t+\\mathbb{E} \\int_0^{T\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds \\biggr)\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tUsing the estimate \\eqref{Eq:ESTofVDinWeakerNorm} we easily conclude that there exists a constant $\\tilde{\\kappa}_0>0$ such that\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12\n\t\t+\\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds \\\\\n\t\t& \\qquad \\qquad \\le \\tilde{\\kappa}_0 (\\mathfrak{C}_0+1),\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twhich along with \\eqref{Eq:FinalStepEstinStrongNorm} complete the proof of Proposition \\ref{STRONGER-NORM}\n\t\\end{proof}\n\n\n\n\n\n\t\\section{Strong solution for an abstract stochastic equation}\\label{ABST-STRONG}\nBy a fixed point method we prove in this section general results about the existence and uniqueness of maximal local solution to stochastic evolution equations (SEEs) with \n\tLipschitz coefficients. \n\t\\subsection{Notations and Preliminary}\\label{abstract-framework}\n\tLet $V$, $E$ and $H$ be separable Banach spaces such that\n\t$E\\hookrightarrow V$. We denote the norm in $V$ by $\\Vert\n\t\\cdot \\Vert$ and for $a,b\\in [0,\\infty)$ with $a0$\n\t\tsuch that \t\tfor all $x, y\\in E$.\n\t\t\\begin{equation}\\label{eqn-local Lipschitz-F}\n\t\t\\begin{split}\n\t\t\\vert F(y)-F(x) \\vert_H \\leq C \\sum_{i=1}^N \\Big[ \\Vert y-x\\Vert \\Vert\n\t\ty\\Vert^{p_i-\\alpha_i} \\vert y\\vert_E^{\\alpha_i} + \\vert y-x\\vert_E^{\\alpha_i}\n\t\t\\Vert y-x\\Vert^{1-\\alpha_i} \\Vert x\\Vert^{p_i}\\Big].\n\t\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{assum}\n\t\\begin{assum}\\label{assum-G}\n\t\tAssume that $G: E \\to V$ such that $G(0)=0$ and there exists $k\n\t\t\\geq 1$, $\\beta \\in [0,1)$ and $C_G>0$ such that\n\t\t\\begin{equation}\\label{eqn-local Lipschitz-G}\n\t\t\\Vert G(y)-G(x) \\Vert \\leq C_G \\Big[ \\Vert y-x\\Vert \\Vert\n\t\ty\\Vert^{k-\\beta} \\vert y\\vert_E^\\beta + \\vert y-x\\vert_E^\\beta\n\t\t\\Vert y-x\\Vert^{1-\\beta} \\Vert x\\Vert^k\\Big],\n\t\t\\end{equation}\n\t\tfor all $x, y\\in E$.\n\t\\end{assum}\n\tLet $(\\Omega,\n\t\\mathcal{F}, \\mathbb{P})$ be a complete probability space equipped\n\twith a filtration $\\mathbb{F}=\\{\\mathcal{F}_t: t\\geq 0\\}$\n\tsatisfying the usual hypothesis. By $\\mathscr{M}^2(X_T)$ we denote the Banach space of all $E$-valued processes $u$ that are progressively measurable and with trajectories belonging to $X_T$ $\\mathbb{P}$-a.s., with the norm\n\t\\begin{equation}\\label{eqn-M^2X_T}\n\t\\vert u \\vert_{ \\mathscr{M}^2(X_T)} = \\left(\\mathbb{E}\\Big[ \\sup_{s \\in [0,T]}\n\t\\Vert u(s)\\Vert^2+\\int_0^T \\vert u(s) \\vert_E^2\\, ds\\Big].\\right)^\\frac12\n\t\\end{equation}\n\tLet us also formulate the following assumptions.\n\t\\begin{assum}\\label{assum-01}\n\t\tSuppose that $E\\hookrightarrow V \\hookrightarrow H$. Consider (for\n\t\tsimplicity) a one-dimensional Wiener process $W(t)$.\\\\\n\t\tAssume that $S(t)$, $t\\in [0,\\infty)$, is a family of bounded linear operators on the space $H$ such that: the following properties are satisfied. \n\t\t\\begin{trivlist}\n\t\t\t\t\\item[(i)] For every $T>0$, the linear map $$L^2(0,T;H) \\ni f \\mapsto \\{S\\ast f(t) =\\int_0^{t} S(t-r) f(r)\\, dr; t\\in [0,T] \\} \\in X_T, $$ is continuous. \n\t\t\t\t\n\\item[(ii) ]For every $T>0$, the linear map \n\t\t\t\t$$\\mathscr{M}^2(0,T; V) \\ni \\xi \\mapsto \\{ S \\diamond \\xi(t):= \\int_0^t S(t-r) \\xi(r)\\, dW(r); t \\in [0,T] \\}\\in \\mathscr{M}^2(X_T), $$ is continuous. \n\t\t\t\t\n\t\t\\item[(iii)] For every $T>0$, the linear map $$V\\ni u_0 \\mapsto \\{[0,T]\\ni t \\mapsto Su_0(t):=S(t)u_0 \\} \\in X_T,$$ is continuous.\n\t\t\\end{trivlist}\n\t\\end{assum}\n\tNow let us consider a semigroup $S(t)$, $t\\in [0,\\infty)$ as above\n\tand the abstract SEE\n\t\\begin{equation}\\label{ABS-SPDE-1}\n\tu(t)=S(t)u_0+\\int_0^t S(t-s) F(u(s))\\, ds+\\int_0^t S(t-s) G(u(s))\n\tdW(s),\\;\\; \\mbox{ for all }t>0\n\t\\end{equation}\n\t\n\twhich is a mild version of the problem\n\t\n\t\\begin{equation}\\label{ABS-SPDE-strong}\n\t\\left\\{\\begin{array}{rl} du(t)&= Au(t)\\,dt+ F\\big(u(t)\\big)\\, dt+\n\tG\\big(u(t)\\big)\n\tdW(t),\\;\\;t>0,\\\\\n\tu(0)&=u_0.\n\t\\end{array}\n\t\\right.\n\t\\end{equation}\n\t\\begin{Def}\\label{def-local solution-2}\n\t\tAssume that a $V$-valued $\\mathcal{F}_0$ measurable random variable $u_0$ is given. A local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong} (with the initial time\n\t\t$0$) is a pair $(u,\\tau)$ such that\n\t\t\\begin{enumerate}\n\t\t\t\\item $\\tau$ is an accessible stopping time, \\item\n\t\t\t$u: [0,\\tau)\\times \\Omega \\to V$ is an admissible\\footnote{This\n\t\t\t\talso follows from condition (3) below.} process, \\item there\n\t\t\texists a sequence $(\\tau_m)_{m\\in \\mathbb{N}}$ of\n\t\t\t finite stopping times such that $\\tau_m \\toup \\tau$\n\t\t\t$\\mathbb{P}$-a.s. and, for every $m\\in \\mathbb{N}$ and $t\\ge 0$,\n\t\t\twe have\n\t\t\t\\begin{eqnarray}\\label{eq-locsol_00}\n\t\t\t&&\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau_m]} \\Vert\n\t\t\tu(s)\\Vert^2 +\\int_0^{t\\wedge \\tau_m} \\vert u(s)\\vert_E^2 \\,\n\t\t\tds\\Big)<\\infty,\n\t\t\t\\\\\n\t\t\t\\label{eq-locsol_00-b} u(t\\wedge \\tau_m)&=&S(t\\wedge\n\t\t\t\\tau_m)u_0+\\int_0^{t\\wedge \\tau_m}\n\t\t\tS(t\\wedge\\tau_m-s) F(u(s))\\, ds\\\\\n\t\t\t\\nonumber &+&\\int_0^t\n\t\t\t\\mathds{1}_{[0,\\tau_m)}S(t-s)G(u(s\\wedge\\tau_m))\\, dW(s).\n\t\t\t\\end{eqnarray}\n\t\t\\end{enumerate}\n\t\tAlong the lines of \\cite{Brz+Elw_2000}, we said that a\n\t\tlocal solution $u(t)$, $t < \\tau$ is called global iff\n\t\t$\\tau=\\infty$ $\\mathbb{P}$-a.s.\n\t\\end{Def}\n\n\tLet us first formulate the following useful result.\n\t\\begin{prop}\n\t\t\\label{prop-local solution} Assume that a pair $(u,\\tau)$ is a\n\t\tlocal solution to problem \\eqref{ABS-SPDE-strong}.Then for\n\t\tevery finite stopping time $\\sigma$, a pair $(u_{\\vert [0, \\tau\n\t\t\t\\wedge \\sigma)\\times \\Omega}, \\tau \\wedge \\sigma)$ is also a local\n\t\tmild solution to problem \\eqref{ABS-SPDE-strong}.\n\t\\end{prop}\n\t\n\t\n\t\n\tSecondly, we state the following lemma result which is a generalisation of \\cite[Lemmata III 6A and\n\t6B]{Elw_1982}.\n\n\t\\begin{lem} \\label{lem-amalgamation} \\textbf{(The Amalgamation\t\t\tLemma) }\n\\begin{trivlist}\n\\item[(1)]\nLet $\\Delta$ be a family of accessible stopping\n\t\ttimes taking values in $[0, \\infty ]$. Then a supremum of $\\Delta$, i.e., \n\t\t$ \\tau := \\sup \\, {\\Delta} $, \n\t is an accessible stopping time with values in $[0, \\infty ]$ and\n\t\tthere exists an $\\Delta $-valued increasing sequence $ \\{\n\t\t\\alpha_{n} \\}_{ n= 1 }^\\infty $ such that\n\t\t$ \\tau(\\omega) = \\lim_{n \\to \\infty} \\alpha_n(\\omega)$, for all $\\omega \\in \\Omega$.\n\\item[(2)]\n\t\tAssume also that for each $ \\alpha \\in \\Delta $, $ I_{\n\t\t\t\\alpha } : [ 0, \\alpha ) \\times \\Omega \\to V $ is an admissible\n\t\tprocess such that for all $ \\alpha , \\beta \\in \\Delta $ and\n\t\tevery $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_01} I_{ \\alpha }(t)= I_{ \\beta } (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\alpha \\wedge \\beta) .\n\t\t\\end{equation}\n\t\tThen, there exists an admissible process $ \\mathbf{I}: [0, \\tau )\n\t\t\\times \\Omega \\to V $,\n\t\tsuch that every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_02}\n\t\t\\mathbf{I}(t)= I_{ \\alpha } (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\alpha ).\n\t\t\\end{equation}\n\\item[(3)]\t\tMoreover,\\hdoubtz{This statement is not in Elworthy's book. Ay least I cannot find it.}\n\t\tif ${\\tilde I} :[0,\\tau) \\times \\Omega \\to X$ is any process\n\t\tsatisfying \\eqref{eqn-amalgamation_02} then the process $\\tilde I$\n\t\tis a version of the process $I$, i.e. for all $t\\in [0,\\infty) $\n\t\t\\begin{equation} \\label{eqn-amalgamation_03}\n\t\t\\mathbb{ P}\\left(\\left\\{\\omega \\in \\Omega: t < \\tau(\\omega) , \\;\n\t\tI(t,\\omega)\\not= {\\tilde I}(t,\\omega) \\right\\} \\right) =0 .\n\t\t\\end{equation}\n\t\tIn particular, if in addition ${\\tilde I}$ is an admissible\n\t\tprocess, then\n\t\t\\begin{equation} \\label{eqn-amalgamation_03'}\n\t\t\\mathbf{I}= {\\tilde I}.\n\t\t\\end{equation}\n\\end{trivlist}\t\t\n\t\n\t\t\n\t\\end{lem}\n\t\\begin{Rem}\\label{rem-amalgamation-equiv}\n\t\tLet us note that because both processes $ \\mathbf{I}: [0, \\tau )\n\t\t\\times \\Omega \\to V $ and $ {I}_\\alpha: [0, \\alpha ) \\times \\Omega\n\t\t\\to V $ are admissible (and hence with almost sure continuous\n\t\ttrajectories), and since $\\alpha \\leq \\tau$, condition\n\t\t\\eqref{eqn-amalgamation_02} is equivalent to the following one:\n\t\t\\begin{equation} \\label{eqn-amalgamation_02'}\n\t\t\\mathbf{I}_{\\vert [0,\\alpha)\\times \\Omega }= I_{ \\alpha } .\n\t\t\\end{equation}\n\t\tSimilarly, condition \\eqref{eqn-amalgamation_01} is equivalent\n\t\tto the following one\n\t\t\\begin{equation} \\label{eqn-amalgamation_01'} {I_{ \\alpha }}_{\\vert [0,\\alpha \\wedge \\beta )\\times \\Omega } = {I_{ \\beta }}_{\\vert [0,\\alpha \\wedge \\beta )\\times \\Omega }.\n\t\t\\end{equation}\n\t\t\t\\end{Rem}\n\\begin{proof}[Proof of Lemma \\ref{lem-amalgamation}] Let $\\Delta$ be the family of accessible stopping\n\t\ttimes with values in $[0, \\infty ]$. This set satisfies the assumptions of Lemma \\cite[Lemma III.6A]{Elw_1982}, where the set $\\Delta$ is denoted by $A$. Indeed,\nby Remark \\ref{rem-predictable stopping time}, the supremum of every finite subset of $\\Delta$ belongs to $\\Delta$. Therefore, there exists an $\\mathcal{F}$-measurable function $\\tau:\\Omega\\to [0,\\infty]$ such that\n\\begin{trivlist}\n\\item[(i)] if $\\sigma \\in \\Delta$, then $\\tau\\geq \\sigma$, $\\mathbb{P}$-a.s.;\n\\item[(ii)] if a random variable $\\eta:\\Omega\\to [0,\\infty]$ satisfies $\\tau\\geq \\sigma$, $\\mathbb{P}$-a.s., for all $\\sigma \\in \\Delta$, then\n$\\eta\\geq \\tau$, $\\mathbb{P}$-a.s..\n\\item[(iii)] there exists a sequence $(\\alpha_n)_{n \\in \\mathbb{N}}$ of elements of $\\Delta$ such that \t for all $\\omega \\in \\Omega$, $\\alpha_n(\\omega) \\leq \\alpha_{n+1}(\\omega) \\leq \\tau(\\omega)$ for all $n\\in \\mathbb{N}$ and\n \t$ \\tau(\\omega) = \\lim_{n\\to \\infty} \\alpha_n(\\omega)= \\sup_{n\\in \\mathbb{N}} \\alpha_n(\\omega)$.\n\\end{trivlist}\nMoreover, $\\tau$ is unique in the sense that if $\\hat \\tau$ satisfies the above conditions (i) and (ii), then $\\hat \\tau \\geq \\tau$, $\\mathbb{P}$-a.s..\nHence, since for every $n \\in \\mathbb{N}$, $\\alpha_n$, is an accessible stopping time, by \\cite[Proposition III.5B]{Elw_1982}, Remark \\ref{rem-predictable stopping time} and \\cite[Proposition 4.11]{Metivier_1982} we infer that $\\tau$ an accessible stopping time. This proves part (1) of Lemma \\ref{lem-amalgamation}.\n\nThe proof of parts (2) and (3) is the same as the proof of \\cite[Lemma III 6 B]{Elw_1982}, so we omit it.\n\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\t\t\\begin{Def}\\label{Def-maxsol}\n\t\tConsider a family $\\mathcal{ LS}$ of all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABS-SPDE-strong}. For two\n\t\telements $(u,\\tau), (v,\\sigma) \\in \\mathcal{ LS} $ we write that\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ iff $\\tau \\leq \\sigma$ $\\mathbb{P}$-a.s. and\n\t\t$v_{\\vert [0,\\tau)\\times \\Omega} \\sim u$. Note that if\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ and $(v,\\sigma)\\preceq (u,\\tau)$,\n\t\tthen $(u,\\tau)\\sim (v,\\sigma)$. We write $(u,\\tau)\\prec\n\t\t(v,\\sigma)$ iff $(u,\\tau)\\preceq (v,\\sigma)$ and $(u,\\tau)\\not\\sim\n\t\t(v,\\sigma)$. Then, the pair $(\\mathcal{ LS},\\preceq)$ is \n\t\tpartially ordered. \n\t\tEach maximal element $(u,\\tau)$ in the set $(\\mathcal{\n\t\t\tLS},\\preceq)$\n\t\tis called a maximal local solution to the problem \\eqref{ABS-SPDE-strong}. The existence of an upper bound of every non-empty chain of $(\\mathcal{\n\t\t\tLS},\\preceq)$ is justified by Amalgamation\n\t\tLemma \\ref{lem-amalgamation}. \\\\\n\t\tIf $(u, \\tau)$ is a maximal local solution to equation\n\t\t\\eqref{ABS-SPDE-strong}, the stopping time $\\tau$ is called its\n\t\tlifetime.\n\t\\end{Def}\n\t\n\t\tA priori, there may be many maximal elements in $(\\mathcal{\n\t\t\tLS},\\preceq)$ and hence many maximal local solutions to\n\t\tthe problem \\eqref{ABS-SPDE-strong}. However, if the uniqueness of local solutions holds, then the\n\t\tuniqueness of the maximal local solution will follow.\n\n\t\\begin{Def}\\label{Def-uniq} A local solution $(u,\\tau)$\n\t\tto problem \\eqref{ABS-SPDE-strong} is unique iff for all other\n\t\tlocal solution $(v,\\sigma)$ to \\eqref{ABS-SPDE-strong} the\n\t\trestricted processes $u_{[0, \\tau\\wedge \\sigma)\\times \\Omega}$ and\n\t\t$v_{[0, \\tau\\wedge\\sigma)\\times \\Omega}$ are equivalent.\n\t\\end{Def}\n\t\n\t\\begin{prop}\\label{prop-loc-implies-max}\n\t\tSuppose that $u_0$ is a $V$-valued random variable and $\\mathcal{F}_0$-measurable. Assume that the following two conditions are satisfies:\n\\begin{trivlist}\n\\item[(i)] there\n\t\texist at least one local solution $(u^0,\\tau^0)$ to problem\n\t\t\\eqref{ABS-SPDE-strong}\n\\item[(ii)]\nif $(u^1,\\tau^2)$ and $(u^2,\\tau^2)$ are local solutions, then for every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_04} u^1(t)= u^2 (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\tau^1 \\wedge \\tau^2 ).\n\t\n\t\t\\end{equation}\n\\end{trivlist}\nThen, problem\nproblem \\eqref{ABS-SPDE-strong} has a unique maximal local solution { $(\\hat{u},\\hat{\\tau})$} satisfying $(u^0,\\tau^0)\\preceq (\\hat{u},\\hat{\\tau}) $. \n\t\\end{prop}\n\t\n\t\\begin{Rem}\\label{rem--loc-implies-max-equiv}\n\t\tLet us note that similarly to Remark \\ref{rem-amalgamation-equiv},\n\t\tbecause both the local solutions $u^1$ and $u^2$ are admissible\n\t\t(hence with almost sure continuous trajectories), condition\n\t\t\\eqref{eqn-amalgamation_04} is equivalent to \n\t\t\\begin{equation} \\label{eqn-amalgamation_04'}\n\t\tu^1_{\\vert [0,\\tau^1 \\wedge \\tau^2)\\times \\Omega }= u^2_{\\vert\n\t\t\t[0,\\tau^1 \\wedge \\tau^2)\\times \\Omega } .\n\t\t\\end{equation}\n\t\\end{Rem}\n\t\\begin{proof}[Proof of Proposition \\ref{prop-loc-implies-max}]\n\t\tLet us choose and fix a local solution $(u^0,\\tau^0)$ to problem\n\t\t\\eqref{ABS-SPDE-strong} and let us consider the family $\\mathcal{ LS}$\nof all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABS-SPDE-strong} such that $(u^0,\\tau^0)\\preceq (u,\\tau) $.\nBy assumptions this set is non-empty. Due to the assumptions (i) and (ii) of Proposition \\ref{prop-loc-implies-max}, by the Amalgamation Lemma \\ref{lem-amalgamation} we infer that there exists an accessible stopping time\n\t\t\t\\[\n\t\t\\hat{\\tau} := \\sup\\left\\{ \\tau: (u,\\tau)\\in \\mathcal{ LS}\\right\\}\n\t\t\\]\n\t\tand an admissible process $\\hat{u}: [0, \\hat{\\tau} )\n\t\t\\times \\Omega \\to V $,\n\t\tsuch that for all $(u,\\tau)\\in \\mathcal{LS}$ and for $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_02-a}\n\t\t\\hat{u}(t)= u (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\tau).\n\t\t\\end{equation}\nMoreover, there exists an increasing sequence $(\\tau_n)$ of accessible stopping times such that $ \\tau(\\omega) = \\lim_{n \\to \\infty} \\tau_n(\\omega)$, for all $\\omega \\in \\Omega$.\n\nIn order to complete the proof of the existence of a maximal lcoal solution, we shall prove that $(\\hat{u},\\hat{\\tau})\\in \\mathcal{LS}$. For this aim, we closely follow the proof of \\cite[Theorem 2.26]{Brz+Elw_2000}.\n Let us define an auxiliary process \t$\\hat{\\eta}=\t\\bigl(\\hat{\\eta}(t)\\bigr),$ $t\\in [0,\\tau)$, such that for each $n\\in \\mathbb{N}$ and $t\\geq 0$, the following equality holds $\\mathbb{P}$-a.s.\n\t\t\\begin{equation}\\label{eqn-proof-02}\n\t\\begin{split}\n\t\\hat{\\eta}(t \\wedge \\tau_n)=S(t\\wedge \\tau_n)u_0 +\\int_0^{t\\wedge \\tau_n} S(t-s) F(\\hat{u}(s\\wedge \\tau_n) )\\, ds +I_{\\tau_n}(t\\wedge \\tau_n),\n\t\\end{split}\n\t\t\\end{equation}\nwhere $I_{\\tau_n}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eqn-proof-03}\n\t\t\t\\begin{split}\nI_{\\tau_n}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n)}(s){S}(t-s) G(\\hat{u}(s \\wedge \\tau_m))\\, \t\t\td{W}(s),\\;\\; t\\geq 0.\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\nAssume that $(u,\\tau)\\in \\mathcal{LS}$. \t Define a process $\\eta=\\bigl(\\eta(t)\\bigr)$, $t\\in [0,\\tau)$ by the above formulae \\eqref{eqn-proof-02}-\\eqref{eqn-proof-03} with\n$\\hat{u}$ replaced by $u$ and the announcing sequence $(\\tau_n)$ of the accessible stopping time $\\hat{\\tau}$ replaced by announcing sequence of the accessible stopping time $\\tau$.\nBecause $(u,\\tau)$ is a local solution, we infer that the process\n$\\eta(t)$, $t\\in [0,\\tau)$\n is a version of the process $u(t)$, $t \\in [0,\\tau)$. Since $\\hat{u}$ satisfies \\eqref{eqn-amalgamation_02-a} and assumption (ii) of Proposition \\ref{prop-loc-implies-max} is satisfied,\n {we infer that\n\\begin{equation} \\label{eqn-amalgamation_02-b}\n\t\t\\hat{\\eta}(t)= u (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\tau).\n\t\t\\end{equation} }\n Hence, by the part (3) of Lemma \\ref{lem-amalgamation}, we infer that the process $\\hat{\\eta}(t)$, $t\\in [0,\\hat{\\tau})$, is a version of the process $\\hat{u}(t)$, $t\\in [0,\\hat{\\tau})$ and therefore we can replace $\\hat{\\eta}$ by $\\hat{u}$ on the LHS of \\eqref{eqn-proof-02}. Therefore, we deduce that $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$. This completes the existence of a local maximal solution.\n\nAs a byproduct of the above proof of the existence of a local maximal solution $(\\hat{u}, \\hat{\\tau})$ we showed that $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$. This, in conjunction with the definition of \n\t$\\mathcal{LS}$ implies that $(u^0,\\tau^0)\\preceq (\\hat{u},\\hat{\\tau}) $.\n\nIt remains to prove the uniqueness of the local maximal solutions. For this aim let us suppose that $(u^1,\\tau^1)$ and $(u^2,\\tau^2)$ are two local maximal solutions. Let us put $\\tilde{\\tau} = \\tau^1\\vee \\tau^2$. Then, by part (ii) of Remark \\ref{rem-predictable stopping time} $\\tilde{\\tau}$ is an accessible stopping time with announcing sequence $(\\tilde{\\tau}_n:=\\tau^1_n \\vee \\tau^2_n)_{n \\in \\mathbb{N}}$, where\n$(\\tau^i_n)_{n \\in \\mathbb{N}}$, $i=1,2$ is an announcing sequence of $\\tau^i$. By the uniqueness assumption (ii), we infer that\n\\begin{equation}\\label{eqn-equivalence}\n({u_1}_{\\lvert _{[0,\\tau^1\\wedge \\tau^2) }},\\tau^1\\wedge \\tau^2) \\sim ({u_2}_{\\lvert _{[0,\\tau^1\\wedge \\tau^2) } },\\tau^1\\wedge \\tau^2).\n\\end{equation}\n We shall now prove that $\\tau^1= \\tau^2$ $\\mathbb{P}$-a.s.. Suppose by contradiction that\n\t\t$\\mathbb{P} (\\{\\tau^1\\neq \\tau^2 \\} )>0. $\n\t\tLet $\\Omega_1:= \\{\\tau^1 \\ge \\tau^2 \\}$ and $\\Omega_2:=\\{ \\tau^2 > \\tau^1 \\}$.\n\t\tWe define a process $(\\tilde{u}, \\tilde{\\tau})$ by the following formula\n\t\t\\begin{equation}\\label{eqn-Def-tildeu}\n\t\\tilde{u} (t,\\omega)=\n\t\t\\begin{cases}\n\t\tu^1(t,\\omega) \\text{ if } \\omega \\in \\Omega_1 \\text{ and } t\\in [0, \\tau^1(\\omega) )\\\\\n\t\tu^2(t,\\omega) \\text{ if } \\omega \\in \\Omega_2 \\text{ and } t\\in [0,\\tau^2(\\omega)). \\\\\n\t\t\\end{cases}\n\t\t\\end{equation}\nWe now claim that the process $(\\tilde{u}, \\tilde{\\tau})$ is a local solution to Problem \\eqref{ABS-SPDE-strong}. Let us fix $n \\in \\mathbb{N}$ and $t\\ge 0$. By symmetry, we can assume that $\\tau^1_n(\\omega)\\le \\tau^2_n(\\omega)$ for all $\\omega \\in \\Omega$ and $n\\in \\mathbb{N}$. Firstly, the proof of the admissibility of $\\tilde{u}(t),\\; t \\in [0,\\tilde{\\tau})$ is very similar to the proof in \\cite[Corollary 2.28]{Brz+Elw_2000}.\nSecondly, let us also observe that on $ \\Omega_1$ we have $\\tilde{\\tau}_n< \\tau^1\\wedge \\tau^2$. Hence, we deduce from \\eqref{eqn-Def-tildeu} and \\eqref{eqn-equivalence} that\n\\begin{equation*}\n\\begin{split}\n\\tilde{u}(t\\wedge\\tilde{\\tau}_n)=&\\tilde{u}(t\\wedge \\tau^2_n)=u^1(t\\wedge \\tau^2_n)=u^2(t\\wedge\\tau^2_n)\\\\\n=&S(t\\wedge\\tau^2_n)u_0 +\\int_0^{t\\wedge\\tau^2_n} S(t\\wedge\\tau^2_n-s) F(u^2(s))\\, ds\n+ I^2_{\\tau_n^2}(t \\wedge \\tau^2_n)\\;\\; t\\geq 0,\n\\end{split}\n\\end{equation*}\nwhere\n\\[\n\t\t\tI^2_{\\tau_n^2}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n^2)}(s){S}(t-s) G(u^2(s \\wedge \\tau_n^2))\\, \t\t\td{W}(s),\\;\\; t\\geq 0.\n\t\t\t\\]\n\nThe last equality follows from the fact that $(u^2,\\tau^2)$ is a local solution. Since $\\tau_n^2=\\tilde{\\tau}_n$, by using \\eqref{eqn-equivalence} and \\cite[Proposition 2.10]{Brz+Elw_2000} we deduce that\n\\begin{equation*}\n\\begin{split}\n\\tilde{u}(t\\wedge\\tilde{\\tau}_n)\n=&S(t\\wedge\\tilde{\\tau}_n)u_0 +\\int_0^{t\\wedge\\tilde{\\tau}_n} S(t\\wedge\\tilde{\\tau}_n-s) F(\\tilde{u}(s))\\, ds\n+ \\tilde{I}_{\\tilde{\\tau_n}}(t\\wedge \\tilde{\\tau_n}), \\;\\; t\\geq 0,\n\\end{split}\n\\end{equation*}\nwhere\n\\[\n\\tilde{I}_{\\tilde{\\tau_n}}(t):=\n\\int_0^t 1_{[0,\\tilde{\\tau}_n)} S(t-s) G(\\tilde{u}(s\\wedge\\tilde{\\tau}_n)) dW(s), \\;\\; t\\geq 0.\n\\]\nHence, $(\\tilde{u},\\tilde{\\tau})$ satisfies equation \\eqref{eq-locsol_00-b} on $\\Omega_1$. In a similar way, we can also show that `$(\\tilde{u},\\tilde{\\tau})$ satisfies \\eqref{eq-locsol_00-b} on $\\Omega_2$. Hence, $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$.\n\nNow, by construction we have $(u^i, \\tau^i) \\preceq (\\tilde{u}, \\tilde{\\tau})$ for $i\\in \\{1,2\\}$ and there exists $i_0 \\in \\{1,2\\}$ such that\n$ (u^{i_0},\\tau^{i_0}) \\not \\sim (\\tilde{u},\\tilde{\\tau} )$. This contradicts the maximality of $(u^{i_0},\\tau^{i_0})$ and completes the proof of Proposition \\ref{prop-loc-implies-max}.\n\n\t\\end{proof}\nAs a byproduct of the proof of the above Proposition \\ref{prop-loc-implies-max} we deduce the following general result.\n\\begin{cor}\\label{cor-max of two local solutions} Let $(\\mathbf{x},\\sigma)$ and $(\\mathbf{y},\\tau)$\nbe two local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong} such that for every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_05} \\mathbf{y}(t)= \\mathbf{x} (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\tau \\wedge \\sigma ).\n\t\\end{equation}\nThen the process \t\t $(\\mathbf{z}, \\sigma \\vee \\tau )$ defined by the following formula\n\t\t\\begin{equation}\\label{eqn-Def-tildeu-2}\n\t\\mathbf{z} (t,\\omega)=\n\t\t\\begin{cases}\n\t\t\\mathbf{x}(t,\\omega), \\text{ if } \\sigma(\\omega)\\geq \\tau(\\omega) \\text{ and } t\\in [0, \\sigma(\\omega) ),\\\\\n\t\t\\mathbf{y}(t,\\omega), \\text{ if } \\sigma(\\omega)< \\tau(\\omega) \\text{ and } t\\in [0,\\tau(\\omega)),\n\t\t\\end{cases}\n\t\t\\end{equation}\nis local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}. The process $(\\mathbf{z}, \\sigma \\vee \\tau )$ is called supremum of $(\\mathbf{x},\\sigma)$ and $(\\mathbf{y},\\tau)$.\n\\end{cor}\n\n\n\n\n\n\t\\subsection{An abstract result}\n\tIn this subsection we prove by a fixed point method some results about the \n\texistence and uniqueness of maximal local mild solution to \\eqref{ABS-SPDE-1}. \n\t\n\tLet\n\t$\\theta:\\mathbb{R}_+\\to [0,1]$ be a ${\\mathcal C}^\\infty_c$ non\n\tincreasing function\n\tsuch that\n\t\\begin{equation}\\label{eqn-theta} \\inf_{x\\in\\mathbb{R}_+}\\theta^\\prime(x)\\geq -1, \\quad \\theta(x)=1\\;\n\t\\mbox{\\rm iff } x\\in [0,1]\\quad \\mbox{\\rm and } \\theta(x)=0 \\;\n\t\\mbox{\\rm iff } x\\in [2,\\infty).\n\t\\end{equation}\n\tand for $n\\geq 1$ set $\\theta_n(\\cdot)=\\theta(\\frac{\\cdot}{n})$.\n\tNote that if $h:\\mathbb{R}_+\\to\\mathbb{R}_+$ is a non decreasing\n\tfunction, then\n\t\\begin{equation}\\label{ineq-theta}\n\t\\theta_n(x)h(x) \\leq h(2n),\\quad\n\n\t\\vert \\theta_n(x)-\\theta_n(y)\\vert \\leq \\frac1n |x-y|, \\text{ for every $x,y\\in {\\mathbb R}$}.\n\t\\end{equation}\n\n\n\n\t\\begin{prop}\\label{prop-global Lipschitz-F}\n\t\tLet $F$ be a mapping satisfying Assumption\n\t\t\\ref{assum-F}. Assume that $\\delta\\in [0,T]$, $\\mathbf{a} \\in X_\\delta$. Then the map\n\t\t\\[\n\t\t\\Phi_{{\\delta,T,\\mathbf{a}}}^n: \\hat{X}_{\\delta,T,\\mathbf{a}} \\ni u \\mapsto \\theta_n( \\vert u\n\t\t\\vert_{X_\\cdot}) F(u) \\in L^2(0,T;H).\n\t\t\\]\n\tis globally Lipschitz and moreover, for all\n\t\t$u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{equation}\\label{eqn-global Lipschitz-F}\n\t\t\\vert \\Phi_{\\delta,T,\\mathbf{a}}^n(u_1)-\\Phi_{\\delta,T,\\mathbf{a}}^n(u_2)\\vert_{L^2(0,T;H)} \\leq \t\tC( C +1)\n\\sum_{i=1}^N (2n)^{p_i+2} (T-\\delta)^{(1-\\alpha_i)\/2}\n\\vert u_1 -u_2\n\t\t\\vert_{X_T}.\n\t\t\\end{equation}\n\t\nIn particular, the Lipschitz constant of $\\Phi_{{\\delta,T,\\mathbf{a}}}^n$ is independent of $\\mathbf{a}$.\n\t\\end{prop}\n\n\tThe proof is based on a proof from\n\t\\cite{Brz+Millet_2012} which in turn was based on a proof from \\cite{deBouard+Deb_1999,deBouard+Deb_2003}.\n\tFor simplicity of notation, below we will write $\\Phi_{T}$ instead of $\\Phi_{\\delta,T,\\mathbf{a}}^n$.\n\n\n\t\\begin{proof}[Proof of Proposition \\ref{prop-global Lipschitz-F}] Wlog we can assume that $N=1$ and we will use notation $\\alpha=\\alpha_1$ and $p=p_1$. In this case, the inequality\n\\eqref{eqn-global Lipschitz-F} takes the following form. For every $n\\in \\mathbb{N}$ there exists $C(n)>0$ such that for all $T>\\delta\\geq 0$, all $\\mathbf{a} \\in X_\\delta$ and\nall $u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{equation}\\label{eqn-global Lipschitz-F-simple}\n\t\t\\vert \\Phi_{\\delta,T,\\mathbf{a}}^n(u_1)-\\Phi_{\\delta,T,\\mathbf{a}}^n(u_2)\\vert_{L^2(0,T;H)} \\leq \t\tC(n) (T-\\delta)^{(1-\\alpha)\/2}\n\\vert u_1 -u_2\n\t\t\\vert_{X_T}.\n\t\t\\end{equation}\nIn what follows we will prove \\eqref{eqn-global Lipschitz-F-simple}.\nLet us fix $n\\in \\mathbb{N}$, $T>\\delta\\geq 0$, $\\mathbf{a} \\in X_\\delta$ and $u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\nNote that $\\Phi_T(0)=0$. Assume that $u_1,u_2 \\in X_T$. Denote,\n\t\tfor $i=1,2$,\n\t\t\\[\n\t\t\\tau_i= \\inf\\{t \\in [0,T]: \\vert u_i \\vert_{X_t} \\geq 2n\\}.\n\t\t\\]\n\t\tNote that if the set on the RHS above is empty, i.e. $\\vert u_i \\vert_{X_t} < 2n$ for all $t\\in [0,T]$,\n\t\tthen $\\tau_i=T$.\n\t\t\n\t\tWlog we can assume that $\\tau_1 \\leq\n\t\t\\tau_2$. Because for $i=1,2, \\;\\; \\theta_n( \\vert u_i\n\t\t\\vert_{X_t}) =0 \\mbox{ for } t \\geq \\tau_2$, we have \n\t\\begin{eqnarray*}\n\t\n\t\n\t\t\\vert \\Phi_T(u_1)-\\Phi_T(u_2)\\vert_{L^2(0,T;H)} \t&=& \\Big[ \\int_0^{\\tau_2} \\vert \\theta_n( \\vert u_1 \\vert_{X_t})\n\t\t\tF(u_1(t))-\\theta_n( \\vert u_2 \\vert_{X_t})\n\t\t\tF(u_2(t))\\vert_H^2\\,dt\\Big]^{1\/2}\n\t\t\t\\\\\n\t\t\t& &\\hspace{-3truecm}\\lefteqn{ \\leq\n\t\t\t\t\\Big[ \\int_0^{\\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert_H^2 \\,dt \\Big]^{1\/2} }\\\\\n\t\t\t&&\\hspace{-2truecm}\\lefteqn{+ \\Big[ \\int_0^{\\tau_2} \\vert\n\t\t\t\t\\theta_n( \\vert u_1 \\vert_{X_t})\\big[ F(u_1(t))- F(u_2(t)) \\big]\n\t\t\t\t\\vert_H^2\\,dt\\Big]^{1\/2} =:A+B}\n\t\t\\end{eqnarray*}\n\t\tNext, since $\\theta_n$ is Lipschitz with Lipschitz constant $2n$ and ${u_1}_{\\lvert_{[0,\\delta]}}={u_2}_{\\lvert_{[0,\\delta]}}=\\mathbf{a}$\n\t\twe have\n\t\t\\begin{eqnarray*}\n\t\nA^2 &=& \\int_0^{\\delta \\wedge \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n+ \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n\\\\\n&=& \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n\\le 4n^2 C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\big[ \\vert\n\t\t\t\\vert u_1 \\vert_{X_t} - \\vert u_2 \\vert_{X_t} \\vert\\big]^2 \\vert F(u_2(t))\\vert_H^2 \\,dt\\\\\n\t\n\t\t\t&\\leq& 4n^2 C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert\n\t\t\tu_1 -u_2 \\vert_{X_t}^2 \\vert F(u_2(t))\\vert_H^2 \\,dt \\leq 4n^2 C^2\\vert\n\t\t\tu_1 -u_2 \\vert_{X_T}^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert F(u_2(t))\\vert_H^2 \\,dt.\n\t\t\\end{eqnarray*}\n\t\tNext, by assumptions and some elementary calculations \n\t\t\\begin{align*}\n\t\t\\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert F(u_2(t))\\vert_H^2 \\,dt \\leq & C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\Vert u_2(t)\\Vert^{2p+2-2\\alpha} \\vert u_2(t)\\vert_E^{2\\alpha} \\,dt\\\\ \n\t\t\\leq & C^2 \\sup_{t \\in [\\delta\\wedge \\tau_2,\\tau_2]} \\Vert u_2(t)\\Vert^{2p+2-2\\alpha}\n\t\t\\big( \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert u_2(t)\\vert_E^{2} \\,dt\\big)^\\alpha\n\t\t(\\tau_2-\\delta\\wedge \\tau_2)^{1-\\alpha}\\\\\n\t\t\\leq & C^2 (T-\\delta)^{1-\\alpha} \\vert u_2 \\vert_{X_{\\delta\\wedge\\tau_2, \\tau_2}}^{2p+2} \\leq C^2 (T-\\delta)^{1-\\alpha}(2n)^{2p+2}.\n\t\t\\end{align*}\n\t\tTherefore,\n\t\t\\begin{equation*}\n\t\t\tA\\le C (T-\\delta)^{(1-\\alpha)\/2}(2n)^{p+2} \\vert u_1 -u_2\n\t\t\t\\vert_{X_T} .\n\t\t\\end{equation*}\n\tSince $\\tau_1 \\leq \\tau_2$, $ \\theta_n( \\vert u_1 \\vert_{X_t}) =0$, $ t \\geq\n\t\t\\tau_1$, $\\theta_n( \\vert u_1 \\vert_{X_t}) \\leq 1,\\, t\\in [0,\\tau_1)$ and ${u_1}_{\\lvert_{[0,\\delta]}}={u_2}_{\\lvert_{[0,\\delta]}}=\\mathbf{a}$, we have\n\t\t\\begin{align*}\n\t\t\tB= \\Big[ \\int_0^{\\tau_2} \\vert \\theta_n( \\vert u_1 \\vert_{X_t})\\big[ F(u_1(t))- F(u_2(t)) \\big] \\vert_H^2\\,dt \\Big]^{1\/2}\n\t\n\t\t\t\\leq& \\Big[\\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\vert F(u_1(t))- F(u_2(t)) \\vert_H^2\\,dt\\Big]^{1\/2}\n\t\t\t\\leq \\tilde{B}_{p},\n\t\t\\end{align*}\n\t\twhere\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\tilde{B}_{p}:=\n\t\tC \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\vert u_1(t)-u_2(t)\\vert_E^{2\\alpha} \\Vert u_1(t)-u_2(t)\\Vert^{2-2\\alpha} \\Vert u_2(t)\\Vert^{2p} \\,dt\\Big]^{1\/2}\\\\\n\t\t+C \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\Vert u_1(t)-u_2(t)\\Vert^2 \\Vert u_1(t)\\Vert^{2p-2\\alpha} \\vert u_1(t)\\vert_E^{2\\alpha} \\,dt\\Big]^{1\/2}.\n\t\t\\end{split}\n\t\t\\end{equation*}\nThis term can be estimated as follows\n\t\t\\begin{align*}\n\t\t\\tilde{B}_{p} \t\\leq& C \\sup_{t\\in [0,T]} \\Vert u_1(t)-u_2(t)\\Vert \\sup_{t\\in\n\t\t\t\t[\\delta\\wedge \\tau_1,\\tau_1]} \\Vert u_1(t)\\Vert^{p-\\alpha} \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1}\n\t\t\t\\vert u_1(t)\\vert_E^{2} \\,dt\\Big]^{\\alpha\/2}\n\t\t\t(\\tau_1-\\delta\\wedge \\tau_1)^{(1-\\alpha)\/2}\n\t\t\t\\\\\n\t\t\t+& C \\sup_{t\\in [0,T]} \\Vert u_1(t)-u_2(t)\\Vert^{1-\\alpha}\n\t\t\t\\sup_{t\\in [\\delta\\wedge \\tau_1,\\tau_1]} \\Vert u_2(t)\\Vert^p \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1}\n\t\t\t\\vert u_1(t)-u_2(t)\\vert_E^{2} \\,dt\\Big]^{\\alpha\/2}\n\t\t\t(\\tau_1-\\delta\\wedge \\tau_1)^{(1-\\alpha)\/2}\n\t\t\t\\\\\n\t\t\t\\leq& C \\vert u_1-u_2\\vert_{X_T} \\vert u_1 \\vert_{X_{\\tau_1}}^p (T-\\delta)^{(1-\\alpha)\/2} +C \\vert u_1-u_2\\vert_{X_T} \\vert u_2 \\vert_{X_{\\tau_1}}^p (T-\\delta)^{(1-\\alpha)\/2}\\\\\n\t\n\t\t\t\\leq& C (T-\\delta)^{(1-\\alpha)\/2} \\vert u_1-u_2\\vert_{X_T} \\Big[\n\t\t\t\\vert u_1 \\vert_{X_{\\tau_1}}^p + \\vert u_2\n\t\t\t\\vert_{X_{\\tau_1}}^p\\Big] \\leq C (2n)^{p+1} (T-\\delta)^{(1-\\alpha)\/2}\n\t\t\t\\vert u_1-u_2\\vert_{X_T}.\n\t\t\\end{align*}\n\t\tSumming up, we proved the following inequality \n\t\t\\begin{eqnarray*}\n\t\t\t\\vert \\Phi_T(u_1)-\\Phi_T(u_2)\\vert_{L^2(0,T;H)}\\le C\\Big[ 2n C +1\\Big] (2n)^{p+1} \\tau_2^{(1-\\alpha)\/2} \\vert\n\t\t\tu_1 -u_2 \\vert_{X_T},\n\t\t\\end{eqnarray*}\n\t\twhich competes the proof of the proposition.\n\t\\end{proof}\n\tThe following result is a special case of Proposition\n\t\\ref{prop-global Lipschitz-F} with $H=V$.\n\t\\begin{cor}\\label{cor-global-lip-G}\n\t\tLet $G$ be a nonlinear mapping satisfying Assumption\n\t\t\\ref{assum-G}. Assume that $n\\in \\mathbb{N}$, $T>0$, $\\delta \\in [0,T]$ and $\\mathbf{a}\\in X_{0,\\delta}$.\n Define a map $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ by\n\t\t\\begin{equation}\\label{eqn-Phi_G}\n\t\t\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n: \\hat{X}_{\\delta,T,\\mathbf{a}} \\ni u \\mapsto \\theta_n(\n\t\t\\vert u \\vert_{X_\\cdot}) G(u) \\in L^2(0,T;V).\n\t\t\\end{equation}\n\t\tThen $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ is globally Lipschitz and moreover, for all\n\t\t$u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{eqnarray}\\label{eqn-global LipschitzG}\n\t\t\\vert \\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n(u_1)-\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n(u_2)\\vert_{L^2(0,T;V)} &\\leq\n\t\t& (2n)^{k+2}C_G(C_G +1) {(T-\\delta)^{(1-\\beta)\/2} }\\vert u_1\n\t\t-u_2 \\vert_{X_T}.\n\t\t\\end{eqnarray}\n{In particular, the Lipschitz constant of $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ is independent of $\\mathbf{a}$.}\n\t\\end{cor}\n\n\n\n\t\\begin{prop}\\label{prop-Psi_T-Lipschitz}\n\tAssume that Assumptions \\ref{assum-F} and \\ref{assum-01} hold. Assume that $n\\in \\mathbb{N}$, $T>0$, $\\delta \\in [0,T]$ and $\\mathbf{a}\\in \\mathscr{M}^2(X_{0,\\delta})$.\n\tThen the map $\\Psi^n_{\\delta,T,\\mathbf{a}}$ defined by\n\t\t\\begin{equation}\\label{eqn-Psi_T}\n\\Psi_{\\delta,T,\\mathbf{a}}^n: \\mathscr{M}^2(\\hat{X}_{\\delta,T, \\mathbf{a}}) \\ni u \\mapsto [S(\\cdot)](\\mathbf{a}(0)) + S \\ast \\Phi_{\\delta, T,\\mathbf{a}}^n\n\t\t(u)+ S\\diamond \\hat{\\Phi}_{\\delta,T,\\mathbf{a}}^n(u) \\in \\mathscr{M}^2(X_{T}),\n\t\t\\end{equation}\n\t\t is globally Lipschitz and moreover, for all $u_1,u_2\n\t\t\\in \\mathscr{M}^2(\\hat{X}_{\\delta,T, \\mathbf{a}})$,\n\t\t\\begin{eqnarray}\\label{eqn-global Lipschitz}\n\t\t\\vert \\Psi_{\\delta,T,\\mathbf{a}}^{n}(u_1)-\\Psi_{\\delta,T,\\mathbf{a}}^{n}(u_2)\\vert_{\\mathscr{M}^2(X_T)} \\leq \\hat{C}(n) \\Big[\\max_{1\\le i \\le N}{(T-\\delta)^{1-\\alpha_i}} \\vee (T-\\delta)^{1-\\beta} \\Big]^{\\frac12}\\vert u_1 -u_2\n\t\t\\vert_{\\mathscr{M}^2(X_T)},\n\\nonumber\n\t\t\\end{eqnarray}\n\t\twhere $\\hat{C}(n)$ is dependent only on $n$ and is given by, for some $D>1$,\n\t\t\\[ \\hat{C}(n)= C_1 C_F (C_F +1) \\sum_{i=1}^N(2n)^{p_i+2} + C_2C_G (2n)^{k+2}( C_G +1)\n\\leq C_3n^{D}.\n\t\t\\]\nIn particular, the Lipschitz constant of $\\Psi_{\\delta,T,\\mathbf{a}}^{n}$ is independent of $\\mathbf{a}$.\n\t\\end{prop}\n\t\\begin{proof}[Proof of Proposition \\ref{prop-Psi_T-Lipschitz}]\n\nFor simplicity of notation we will write\n\t\t$\\Psi_{T}$ instead of $\\Psi_{\\delta,T,\\mathbf{a}}^n$. We will also write $\\Phi_F$ (resp. $\\hat{\\Phi}_G$) instead of $\\Phi^n_{\\delta,T,\\mathbf{a}}$ (resp. $\\hat{\\Phi}^n_{\\delta,T,\\mathbf{a}}$).\n\t\tObviously in view of Assumption \\ref{assum-01} the map $\\Psi_T$ is\n\t\twell defined. Let us fix $u_1,u_2 \\in \\mathscr{M}^2(X_{\\delta,T,\\mathbf{a}})$. Then by the Fubini Theorem, Assumption \\ref{assum-01}, Proposition \\ref{prop-global Lipschitz-F} and Corollary \\ref{cor-global-lip-G} we infer that\n\t\t\\begin{eqnarray*}\n\t\t\t\\vert \\Psi_T(u_1)-\\Psi_T(u_2)\\vert_{\\mathscr{M}^2(X_T)} &\\leq & \\vert S\\ast \\Phi_F(u_1)-S\\ast \\Phi_F(u_2)\\vert_{\\mathscr{M}^2(X_T)}\n\t\t\t+ \\vert S \\diamond \\Phi_G(u_1)-S\\diamond \\Phi_G(u_2)\\vert_{\\mathscr{M}^2(X_T)}\\\\\n\t\t\t&\\leq& C_1 \\vert \\Phi_F(u_1)- \\Phi_F(u_2)\\vert_{\\mathscr{M}^2(0,T;H)}+C_2\n\t\t\t\\vert \\Phi_G(u_1)-\\Phi_G(u_2)\\vert_{\\mathscr{M}^2(X_T)}\n\t\t\t\\\\\n\t\t\t&\\leq& \\hat{C}(n)\\Big[\\max_{1\\le i \\le N}{(T-\\delta)^{1-\\alpha_i}} \\vee (T-\\delta)^{1-\\beta} \\Big]^{\\frac12}\\vert u_1 -u_2\n\t\t\t\\vert_{\\mathscr{M}^2(X_T)}.\n\t\t\\end{eqnarray*}\n\t\tThe proof is complete.\n\t\\end{proof}\nSince our method is based on finding fixed points of $\\Psi_{\\delta,T,\\mathbf{a}}^n$, the following auxiliary result is useful.\n\\begin{lem}\\label{lem-fixed points}\nAssume that $n\\in \\mathbb{N}$ and $T>S> 0$ and $\\mathbf{x} \\in V$. Assume that $\\mathbf{a}\\in \\mathscr{M}^2(\\hat{X}_{0,S,\\mathbf{x}})$ is a fixed point of\n$\\Psi_{0,S,\\mathbf{x}}^n$. Then $\\Psi_{S,T,\\mathbf{a}}^n$ maps $\\mathscr{M}^2(\\hat{X}_{S,T, \\mathbf{a}})$ into itself.\n\\end{lem}\n\\begin{proof}[Proof of Lemma \\ref{lem-fixed points}] Let us choose and fix $n\\in \\mathbb{N}$, $T>S> 0$, $\\mathbf{x} \\in V$ and $\\mathbf{a}\\in \\mathscr{M}^2(\\hat{X}_{0,S,\\mathbf{x}})$, a fixed point of\n$\\Psi_{0,S,\\mathbf{x}}^n$.\nWe will show that $\\Psi^n_{S,T,\\mathbf{a} }$ maps $\\mathscr{M}^2(\\hat{X}_{S,T,\\mathbf{a}})$ into itself.\nTake an arbitrary $u\\in \\hat{X}_{S,T,\\mathbf{a}}$. Since by Proposition \\ref{prop-Psi_T-Lipschitz}, $v:=\\bigl[\\Psi_{S,T,\\mathbf{a}}^n \\bigr](u) \\in \\mathscr{M}^2(X_T)$ we only need to show that\n$v_{|[0,S]}=\\mathbf{a}$. For this aim let us observe that by Definition \\ref{eqn-Psi_T}, we have for $t\\in [0,S]$,\n\\begin{align*}\n v(t)= S(t)(\\mathbf{a}(0)) &+ \\int_0^t S(t-r) \\theta_n(\\lvert u\\rvert_{X_r}) F(u(r))\\, dr\n + \\int_0^t S(t-r) \\theta_n( \\lvert u \\rvert_{X_r}) G(u(r))\\, dW(r).\n \\end{align*}\nBecause $u_{|[0,S]}=\\mathbf{a}$, $\\mathbf{a}(0)=\\mathbf{x}$ and, by assumptions $\\Psi_{0,S,\\mathbf{x}}^n(\\mathbf{a})=\\mathbf{a}$ in $\\mathscr{M}^2(\\hat{X}_{0,S, \\mathbf{x}})$, we infer that\n\\begin{align*}\n v(t)= S(t)\\mathbf{x} &+ \\int_0^t S(t-r) \\theta_n(\\lvert a\\rvert_{X_r}) F(a(r))\\, dr\n + \\int_0^t S(t-r) \\theta_n( \\lvert a \\rvert_{X_r}) G(a(r))\\, dW(r)\\\\\n &=[\\Psi_{0,S,\\mathbf{x}}^n(\\mathbf{a})](t)=\\mathbf{a}(t), \\;\\;\\; t\\in [0,S].\n \\end{align*}\nThis completes the proof of Lemma \\ref{lem-fixed points}.\n \\end{proof}\n\n\n\n\\begin{comment}\nBefore proceeding further, we state the following remark.\n\\begin{Rem}\\label{Rem-Global-Lipshitz}\n\t\\begin{trivlist}\n\t\t\\item[(i)] Let $\\mathbf{a}_0\\in V$. Then, the results in Proposition \\ref{prop-global Lipschitz-F} and Corollary \\ref{cor-global-lip-G} remain true with the space $\\hat{X}_{\\delta,T,\\mathbf{a}}$ replaced by\n\t\t$\\hat{X}_{0,T,\\mathbf{a}_0}$.\n\t\t\\item[(ii)] Let $\\mathbf{a}_0\\in L^2(\\Omega,\\mathcal{F}_0;V)$ and $n \\in \\mathbb{N}$. Then, it follows from item (i) and the proof of Proposition \\ref{prop-Psi_T-Lipschitz} that the map $\\Psi_{0,T,\\mathbf{a}_0}^n$ defined by\n\t\t\\begin{equation}\\label{eqn-Psi_T-0}\n\t\t\\Psi_{0,T,\\mathbf{a}_0}:=\\Psi_{0,T,\\mathbf{a}_0}^n: \\mathscr{M}^2(\\hat{X}_{0,T, \\mathbf{a}_0}) \\ni u \\mapsto S\\mathbf{a}_0 + S \\ast \\Phi_{0, T,\\mathbf{a}_0}^n\n\t\t(u)+ S\\diamond \\hat{\\Phi}_{0,T,\\mathbf{a}_0}^n(u) \\in \\mathscr{M}^2(X_{T}),\n\t\t\\end{equation}\n\t\tis globally Lipschitz with the Lipschitz constant being independent of $\\mathbf{a}_0$.\n\t\\end{trivlist}\n\tIn both items (i) and (ii), inequalities \\eqref{eqn-global Lipschitz-F}, \\eqref{eqn-global LipschitzG} and \\eqref{eqn-global Lipschitz} hold with $T-\\delta$ replaced by $T$.\n\t\n\t\\end{Rem}\n\\end{comment}\n\n\n\n\n\tThe \tfirst two main results of this subsection are given in the following two\n\ttheorems.\n\t\\begin{thm}\\label{Thm:LocalUniqueness}\n\t\tSuppose that Assumption \\ref{assum-F}-Assumption \\ref{assum-01} hold. Let $u_0$ be a\t$\\mathcal{F}_{0}$-measurable $V$-valued square integrable random\n\t\tvariable $u_0$ and $(u,\\tau) $ and $(v,\\sigma)$ two local solutions of \\eqref{ABS-SPDE-1}. Then,\n\t\\begin{equation}\n\t(u_{\\lvert_{[0,\\sigma \\wedge \\tau)\\times \\Omega } }, \\sigma \\wedge \\tau)\\sim (v_{\\lvert_{[0,\\sigma \\wedge \\tau)\\times \\Omega}},\\sigma \\wedge \\tau).\n\t\\end{equation}\n\t\\end{thm}\n\\begin{proof}[Proof of Theorem \\ref{Thm:LocalUniqueness}]\n\tWlog we can assume in Assumptions \\ref{assum-F} and \\ref{assum-G} that $N=1$ and $p_i=k$ and will use the notations $\\alpha_1=\\alpha$, $p_1=p=k$.\n\tLet $(u,\\tau) $ and $(v,\\sigma)$ be two local solutions of \\eqref{ABS-SPDE-1}. Let $(\\tau_n)_{n \\in \\mathbb{N}}$ and $(\\sigma_{n \\in \\mathbb{N}})$ be the announcing sequences of $\\tau$ and $\\sigma$, respectively. By \\cite[Propositions 4.3 \\& 4.11 and Theorem 6.6]{Metivier_1982} the stopping time $\\varrho:=\\tau \\wedge \\sigma$ is accessible and it is easy to show that $(\\varrho_n)_{n \\in \\mathbb{N}}:= (\\tau_n \\wedge \\sigma_n)_{n \\in \\mathbb{N}}$ is an announcing sequence of $\\varrho$.\n\t\n\tHereafter we fix $n\\in \\mathbb{N}$. Since $(v,\\sigma)$ is a local solution to \\eqref{ABS-SPDE-strong} and $\\varrho_n \\le \\sigma_n$, by Corollary \\ref{cor-A.2} we infer that for all $t\\ge 0$ $\\mathbb{P}$-a.s.\n\t\\begin{align}\n\tv(t\\wedge \\varrho_n)= & S_{t\\wedge \\varrho_n} u_0 + \\int_0^{t\\wedge \\varrho_n} S_{t\\wedge \\varrho_n-r}F(v(r)) dr + I_{\\sigma_n}(t\\wedge \\varrho_n) \\nonumber \\\\\n\t=& S_{t\\wedge \\varrho_n} u_0 + \\int_0^{t\\wedge \\varrho_n} S_{t\\wedge \\varrho_n-r}F(v(r)) dr + I_{\\varrho_n}(t\\wedge \\varrho_n),\\label{Eq:vstopped is a local sol}\n\t\\end{align}\n\twhere $$ I_{\\sigma_n}(t):=\\int_0^t 1_{[0,\\sigma_n] }(r) S_{t-r}G(v(r))dW(r),\\; t\\ge 0 .$$\n\tThe identity \\eqref{Eq:vstopped is a local sol} proves that $(v,\\sigma\\wedge \\tau)$ is a local solution to \\eqref{ABS-SPDE-strong}. In a similar way, we prove that $(u, \\sigma\\wedge \\tau)$ is a local solution to \\eqref{ABS-SPDE-strong} as well.\n\t\n\tThirdly, for $k \\in \\mathbb{N}$ we put $\\varrho_{n,k}=\\tau_n \\wedge \\sigma_n\\wedge \\tilde{\\tau}_k \\wedge \\tilde{\\sigma}_k$, where\n\t\\begin{align}\n\\tilde{\\tau}_k=\\inf\\{t \\in [0,T]: \\lvert v \\rvert_{X_t} \\ge k \\} \\wedge \\tau \\text{ and } \t\\tilde{\\sigma}_k=\\inf\\{t \\in [0,T]: \\lvert v \\rvert_{X_t} \\ge k \\} \\wedge \\sigma, \\; k \\in \\mathbb{N}.\\nonumber\n\t\\end{align}\nWe observe that for all $k\\in \\mathbb{N}$ $\\varrho_{n,k}\\le \\varrho_{n, k+1}\\le \\sigma_n \\wedge \\tau_n$ $\\mathbb{P}$-a.s. and $\\varrho_{n,k} \\toup \\sigma_n \\wedge \\tau_n$ $\\mathbb{P}$-a.s. if $k \\to \\infty$. Let us now fix $k \\in \\mathbb{N}$. Arguing as in the proof of \\eqref{Eq:vstopped is a local sol} we show that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\t\\begin{align}\n\tv(t\\wedge \\varrho_{n,k})\n\t=&S_{t\\wedge \\varrho_{n,k}} u_0 + \\int_0^{t\\wedge \\varrho_{n,k}} S_{t\\wedge \\varrho_{n,k}-r}F(v(r)) dr + I_{\\varrho_{n,k}}(t\\wedge \\varrho_{n,k}).\n\t\\end{align}\n\tIn a similar way, we prove that the same identity holds with $v$ replaced by $u$. Hence, setting $w=u-v$ we infer that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\t\\begin{equation}\n\tw(t\\wedge \\varrho_{n,k})\n\t= \\int_0^{t\\wedge \\varrho_{n,k}} S_{t\\wedge \\varrho_{n,k}-r}[F(u(r))- F(v(r))] dr + \\tilde{I}_{\\varrho_{n,k}}(t\\wedge \\varrho_{n,k}).\n\t\\end{equation}\n\twhere\n\t$$ \\tilde{I}_{\\varrho_{n,k}}(t):=\\int_0^t 1_{[0,\\varrho_{n,k}] }(r) S_{t-r}[G(u(r))-G(v(r))]dW(r),\\; t\\ge 0 .$$\n\t\n\tHereafter, $c>0$ denotes an universal constant (independent of $n$ and $k$) which may change from one term to the other.\nFollowing the lines of \\cite[Proof of Lemma 3.8]{Brz+Gat_99} or \\cite[Page 134]{Brz+Masl+Seidler_2005} and using Assumptions \\ref{assum-01}, \\ref{assum-F} and \\ref{assum-G} we infer that for all $t\\ge 0$\n\\begin{align}\n\\mathbb{E} \\lvert w \\rvert^2_{X_{t\\wedge \\varrho_{n,k}} }\\le & c \\mathbb{E} \\int_0^{t\\wedge \\varrho_{n,k}} \\Big[ \\lvert F(u(r)) -F(v(r))\\rvert^2_H + \\lVert G(u(r)) -G(v(r))\\rVert^2\\Big]dr\\nonumber\\\\\n\t\\le & c \\mathbb{E} \\int_0^{t\\wedge \\varrho_{n,k}} \\left(\\Vert w(s) \\Vert^2 \\Vert u(s) \\Vert^{2(p-\\alpha)} \\lvert u(s) \\rvert^{2\\alpha}_E \\right) ds+\n\tc \\mathbb{E}\\int_0^{t\\wedge \\varrho_{n,k}} \\left(\\vert w(s) \\vert^{2\\alpha}_E \\Vert w(s) \\Vert^{2(1-\\alpha)} \\lVert v(s) \\rVert^{2p} \\right)ds\\nonumber \\\\\n\t&=:c\\mathbb{E}I_1+c\\mathbb{E} I_2.\\label{Eq:Uniq-Step-0}\n\\end{align}\nThe H\\\"older inequality and the definition of the stopping time $\\varrho_{n,k}$ imply that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\\begin{align}\n I_1\\le & c \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^{\\frac{2}{1-\\alpha}} \\Vert u(s)\\Vert^{\\frac{2(p-\\alpha)}{1-\\alpha}} ds \\right)^{1-\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\vert u(s) \\vert^{2}_E ds \\right)^{\\alpha} \\\\\n\\le & c R^{2\\alpha}\\sup_{s\\in [0,t\\wedge \\varrho_{n,k}]} \\Vert u(s) \\Vert^{2(p-\\alpha)}\\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^{\\frac{2}{1-\\alpha}} ds \\right)^{1-\\alpha}\\\\\n\\le &c R^{2p} \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds \\right)^{1-\\alpha}\n\\le \\frac14 \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2} + c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds .\\label{Eq:Uniq-I_1}\n\\end{align}\nIn a similar way one can prove that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\\begin{align}\n I_2\\le & c \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 \\Vert v(s)\\Vert^{\\frac{2p}{1-\\alpha}} ds \\right)^{1-\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\vert w(s) \\vert^{2}_E ds \\right)^{\\alpha} \\\\\n\\le & \\frac14 \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2} + c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds.\\label{Eq:Uniq-I_2}\n\\end{align}\nHence, plugging \\eqref{Eq:Uniq-I_1} and \\eqref{Eq:Uniq-I_2} in \\eqref{Eq:Uniq-Step-0} and using $ \\Vert w(t\\wedge \\varrho_{n,k} )\\Vert^2\\le \\lvert w\\rvert^2_{X_{t\\wedge \\varrho_{n,k}}},\\;$ we infer that\n\\begin{equation}\n\\mathbb{E} \\lVert w(t\\wedge \\varrho_{n,k} )\\rVert^2 \\le 2 c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s\\wedge \\varrho_{n,k} ) \\Vert^2 ds, \\;\\forall t\\ge0.\n\\end{equation}\nThis along the Gronwall lemma implies that for all $t\\ge 0$, $\n\\mathbb{E} \\Vert w(t\\wedge \\varrho_{n,k}) \\Vert^2=0.$\nHence, by letting $k \\to \\infty$ we infer that for all $t\\ge 0$, $\n\\mathbb{E} \\Vert w(t\\wedge \\tau_n\\wedge \\sigma) \\Vert^2=0,$\n which along with the continuity of $w$ completes the proof of the theorem.\n\\end{proof}\n\n\n\t\\begin{thm}\\label{thm_local} Suppose that Assumptions \\ref{assum-F}-\\ref{assum-01} are satisfied. Then\n\\begin{trivlist}\n\\item[(I)]\n for every\n\t\t$\\mathcal{F}_{0}$-measurable $V$-valued square integrable random\n\t\tvariable $u_0$ there exits a local process $u=\\big(u(t),\n\t\tt\\in[0,T_1) \\big) $ which is the\n\t\tunique local solution to problem \\eqref{ABS-SPDE-strong},\n\\begin{comment}\\item[(II)]\n for all $R>0$ and $\\varepsilon >0$ there exists\n\t\t$\\tau(\\varepsilon,R)>0$, such that for every\n\t\t$\\mathcal{F}_0$-measurable $V$-valued random variable $u_0$\n\t\tsatisfying $\\mathbb{E}\\Vert u_0 \\Vert^{2} \\leq R^{2}$, one has\n\t\t\\begin{equation}\\label{ineq-positive probability}\n{\\mathbb P}\\big(T_1\\geq \\tau(\\varepsilon,R)\\big) \\geq\n\t\t1-\\varepsilon.\\end{equation}\n\\end{comment}\n\\item[(II)]\n if $R>0$ and $\\varepsilon >0$ then there exists a number \n\t\t$T^\\ast(\\varepsilon,R)>0$, such that for every set $\\Omega_1 \\in \\mathcal{F}_0$ and every\n\t\t$\\mathcal{F}_0$-measurable $V$-valued random variable $u_0$ such that\n\t\t \\[\\Vert u_0 \\Vert \\leq R \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_1,\\] one has\n\t\t\\begin{equation}\\label{ineq-positive probability-2}\n{\\mathbb P}\\big(\\{T_1\\geq T^\\ast(\\varepsilon,R)\\} \\cap \\Omega_1 \\big) \\geq\n\t\t(1-\\varepsilon)\\mathbb{P}(\\Omega_1).\\end{equation}\n\n\n\\end{trivlist}\n\t\\end{thm}\n\t\\begin{proof}[Proof of Theorem \\ref{thm_local}]\n\n\t\tWlog we can assume that $N=1$ and we will use notation $\\alpha=\\alpha_1$ and $p=p_1$.\nLet $u_0\\in\n\t\tL^2(\\Omega, \\mathbb{P}; V)$. We also fix a natural number $n\\in \\mathbb{N}$ in Steps \\textbf{1}-\\textbf{5}.\nIn the first part consisting of Steps \\textbf{1}-\\textbf{7} we will prove the part (I) of the Theorem, i.e. the existence and uniqueness of a local solution to problem \\eqref{ABS-SPDE-strong}.\nPart (II) of the Theorem will be proven in Steps \\textbf{8}-\\textbf{9}.\n\n\\noindent \\textit{Proof of part (I)}\n\\begin{trivlist}\n\\item[\\textbf{Step 1.}] Let us fix $n\\in \\mathbb{N}$ and $T>0$.\nLet $\\Psi^n_{0,T,u_0}: \\mathscr{M}^2(\\hat{X}_{0,T,u_0}) \\to \\mathscr{M}^2(\\hat{X}_{0,T,u_0}) $.\nBy Proposition \\ref{prop-Psi_T-Lipschitz} the map $\\Psi^n_{0, T,u_0}$ is well defined\n and for sufficiently small $T=\\delta_n$, and all $\\mathbf{a}_0$,\n it is an $\\frac12$-contraction. \n\t\t\tThus, by the Banach Fixed Point Theorem, there exists a unique $u^{[n,1]}\\in \\mathscr{M}^2(\\hat{X}_{0,\\delta_n,u_0})$ such that\n\t\t\t\\[u^{[n,1]}=\\Psi^n_{0,\\delta_n,u_0}(u^{[n,1]}).\\] We fix $u^{[n,1]}$ for the rest of the proof. We also put $M:=\\frac{T}{\\delta_n}\\in \\mathbb{N}$.\t\t\n\n\\item[\\textbf{Step 2.}]\nBy Lemma \\ref{lem-fixed points} $\\Psi^n_{\\delta_n, 2 \\delta_n, u^{[n,1]} }$ maps $\\mathscr{M}^2(\\hat{X}_{\\delta_n, 2 \\delta_n, u^{[n,1]}})$ into itself\nand by Proposition \\ref{prop-Psi_T-Lipschitz} and inequality \\eqref{eqn-global Lipschitz} it is an $\\frac 12$-contraction. Therefore, we can find a unique\n $u^{[n,2]} \\in \\mathscr{M}^2(\\hat{X}_{\\delta_n, 2\\delta_n, u^{[n,1]}})$, which we fix for the rest of the proof, such that \\[u^{[n,2]}=\\Psi^n_{\\delta_n, 2\\delta_n, u^{[n,1]}}(u^{[n,2]})\\in \\mathscr{M}^2(X_{\\delta, 2\\delta_n, u^{[n,1]}}).\\]\n\n\n\\item[\\textbf{Step 3.}]\n\t\t\t By induction we can construct a sequence $(u^{[n,k]})_{k=1}^\\infty$ such that\n\\[u^{[n,k]}=\\Psi^n_{(k-1)\\delta_n,k\\delta_n, u^{[n,k-1]}}(u^{[n,k]})\\in \\mathscr{M}^2(X_{(k-1)\\delta_n,k\\delta_n, u^{[n,k-1]}}),\\;\\; k=2,\\ldots .\\]\n\t\t\t\nNote that by construction, the restriction of $ u^{[n,k]} $ to interval $[0,(k-1)\\delta_n]$ is equal to $u^{[n,k-1]}$.\n\n\n\\item[\\textbf{Step 4.}]\nBy \\textbf{Step 3} we can define a process $u^n\\in \\mathscr{M}^2(X_T)$ by $u^n(t)=u^{[n,k]}(t)$, if $t \\in [0,k\\delta_n]$. Moreover, for every $t\\in [0,T]$, $\\mathbb{P}$-a.s.,\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation}\n\t\t\tu^n(t)=S_t u_0+\\int_0^t S_{t-r}[\\theta_n (|u^n|_{X_r})F(u^n(r))]dr+\\int_0^t\n\t\t\tS_{t-r}[\\theta_n (|u^n|_{X_r})G(u^n(r))]dW(r).\n\t\t\t\\end{equation}\n\n\n\\item[\\textbf{Step 5.}]\nLet $(\\tau_n)_{n\\in \\mathbb{N}}$ be a sequence of stopping\n\t\t\ttimes defined by\n\t\t\t\\begin{equation}\\label{eqn-stopping time tau_n}\n\t\t\t\\tau_n=\\inf\\{t\\in [0,\\infty): |u^n|_{X_t}\\ge n\\}.\n\t\t\t\\end{equation}\t\t\n\t\n\nLet us fix $n\\in \\mathbb{N}$.\n By \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we infer from \\eqref{Eq:truncated-equation} that for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation-2}\n\t\t\tu^n(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[\\theta_n (|u^n|_{X_r})F(u^n(r))]\\,dr\n+\\tilde{I}^n_{\\tau_n}(t \\wedge \\tau_n),\n\t\t\t\\end{equation}\nwhere $\\tilde{I}^n_{\\tau_n}$ is a continuous $V$-valued\n\\[\n\\tilde{I}^n_{\\tau_n}(t):= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[\\theta_n (|u^n|_{X_r})G(u^n(r))]dW(r),\\;\\; t\\in [0,\\infty).\n\\]\n\tBy the definition of the function $\\theta_n$ we infer that $\\theta_n(|u^n|_{X_{r}})=1$ for $r\\in [0,t\\wedge\n\t\t\t\\tau_n)$. Hence\n\t\t\t$$\\theta_n (|u^n|_{X_r})F(u^n(r)) = F(u^n(r)), r\\in [0,t\\wedge\n\t\t\t\\tau_n), \\; t\\in [0,\\infty).$$\nTherefore, we deduce that for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\\[\n\\tilde{I}^n_{\\tau_n}(t)= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[G(u^n(r))]dW(r)=:I^n_{\\tau_n}(t).\n\\]\nThus, we infer that\n\t\t\t$u^n$ satisfies, for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation-2-A}\n\t\t\tu^n(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[F(u^n(r))]\\,dr\n+I^n_{\\tau_n}(t \\wedge \\tau_n).\n\t\t\t\\end{equation}\n\n\n\n\\item[\\textbf{Step 6.}]\\label{lem-increasing stopping times}\n Arguing as in the proof of proof of \\cite[Lemma 5.1]{Brz+Millet_2012} we can show that for every $n \\in \\mathbb{N}$,\n\\begin{equation}\n\\label{eqn-increasing stopping times}\n\\tau_n < \\tau_{n+1} \\;\\;\\;\\; \\mathbb{P}\\text{-a.s.}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqn-u_n=u_n+1}\nu^n(t)=u^{n+1}(t) \\mbox{ if } t\\in [0,\\tau_n) \\mbox{ and } n \\in \\mathbb{N},\\;\\; \\mathbb{P}\\text{-a.s.}\n\\end{equation}\nBy taking appropriate modifications we can assume that \\eqref{eqn-increasing stopping times} is satisfied on the whole space $\\Omega$.\nHence, the following limit exists\n\n\\begin{equation}\\label{eqn-tau_infty=lim}\n\\tau_\\infty(\\omega)=\\lim_{n\\rightarrow \\infty} \\tau_n(\\omega), \\,\\, \\omega \\in \\Omega.\n\\end{equation}\nSince our probability basis satisfies the usual hypothesis, $\\tau_\\infty$ is an {accessible} stopping time, see \\cite[Proposition 2.3 and Lemma 2.11]{Kar-Shr-96} with $(\\tau_n)$ being the announcing sequence for $\\tau_\\infty$. The two claims made at the beginning of \\textbf{Step 6} enable us to define a local process $(u,\\tau_\\infty)$ in the following way\n\\begin{equation}\n\\label{eqn-local process u}\n u(t,\\omega)=u^n(t,\\omega) \\text{ if } t<\\tau_n(\\omega), \\omega \\in \\Omega.\n\\end{equation}\n\\item[\\textbf{Step 7.}]\\label{Step7} We claim that $(u,\\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}. \\\\\nIndeed, arguing as in \\textbf{Step 5}, in particular using \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we can show that\n\t\t\t$u^n$ satisfies, for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{eqn-local-3}\n\t\t\tu(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[F(u(r))]\\,dr\n+I_{\\tau_n}(t \\wedge \\tau_n).\n\t\t\t\\end{equation}\t\nwhere $I_{\\tau_n}$ is an $V$-valued continuous process defined by\n\\[\nI_{\\tau_n}(t)= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[G(u(r))]\\,dW(r), \\;\\; t\\in [0,\\infty).\n\\]\nSince, as observed above, $\\tau_\\infty$ is an accessible stopping time with the announcing sequence $(\\tau_n)$, by definition \\ref{def-local solution} we infer that\nthe local\t\t\tprocess $(u, \\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}.\n\nThis also ends the proof of the first part of Theorem \\ref{thm_local}.\n\t\t\t\\end{trivlist}\n\n\n\\noindent \t\\textit{Proof of part (III)}\n\t\n\tLet us recall that $\\Omega_1 \\in \\mathcal{F}_0$. Let $i:\\Omega_1 \\hookrightarrow \\Omega$ be the natural embedding and $W^1$ a process given by \n\t\\[\n\tW^1(t):=W(t)\\circ i,\\;\\; t\\geq 0.\n\t\\]\n\tWe define $\\mathcal{F}_t^1$ by\n\t\\[\n\t\\mathcal{F}_t^1:=\\bigl\\{ A\\cap \\Omega_1=i^{-1}(A): A \\in \\mathcal{F}_t \\bigr\\}, \\;\\;\\; t\\geq 0.\n\t\\]\n\tSimilarly we define $\\mathcal{F}^1$. We put $\\mathbb{F}^1=\\bigl(\\mathcal{F}_t^1\\bigr)_{t \\geq 0}$.\n\t\n\tWe also define a measure\n\t\\[\n\t\\mathbb{P}^1: \\mathcal{F}^1 \\ni A\\cap \\Omega_1 \\mapsto \\frac{\\mathbb{P}(A\\cap \\Omega_1)}{\\mathbb{P}(\\Omega_1)} \\in [0,1].\n\t\\]\n\tIt is easy to check that\n\t$\n\t\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)\n\t$\n\tis a filtered probability space satisfying the usual conditions. Moreover, since the Wiener process $W$ is independent of $\\mathcal{F}_0$,\n\t$W^1$ is a Wiener process on $\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)$.\n\t\n\\noindent\tHence, it is sufficient to prove our result in the case when $\\Omega_1=\\Omega$. Indeed, the following holds true.\n\t\\begin{lem}\\label{lem-localization of the probability space}\n\t\tIf\n\t\ta local process $u=\\big(u(t),\n\t\tt\\in[0, \\tau) \\big) $ is a\n\t\tlocal solution to problem \\eqref{ABS-SPDE-strong} on the original probability basis\n\t\t$\\bigl(\\Omega,\\mathcal{F},\\mathbb{P}, \\mathbb{F}\\bigr)$, then the\n\t\tprocess $u^1=\\big(u(t),\t\tt\\in[0, \\tau^1) \\big) $ defined by \n\t\t\\[\n\t\tu^1(t,\\omega_1):=u(t,i(\\omega_1)), \\;\\; t \\in [0,\\tau(i(\\omega_1)),\\;\\; \\omega_1 \\in \\Omega_1,\n\t\t\\]\n\t\tand $\\tau^1:=\\tau\\circ i$, \n\t\tis a local solution to problem \\eqref{ABS-SPDE-strong} on the new probability basis\n\t\t$\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)$.\n\t\\end{lem}\n\t\n\tThus we assume that $u_0$ is a $\\mathcal{F}_0$-measurable $V$-valued random variable such that\n\t\\[\\Vert u_0 \\Vert \\leq R \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega.\\]\n\t\n\t\n\t\n\tOur proof follows the lines of \\cite[Theorem 5.3]{Brz+Millet_2012}. In order to simplify the notation we will write $\\Psi^n_{T}$ instead of $\\Psi^n_{0,T,u_0}$, We will also write $\\Phi^n_F$ (resp. $\\hat{\\Phi}^n_G$) instead of $\\Phi^n_{\\delta,T,\\mathbf{a}}$ (resp. $\\hat{\\Phi}^n_{\\delta,T,\\mathbf{a}}$).\n\t\n\n\tWe modify the initial data $u_0$ by replacing it by $\\tilde{u}_0=u_01_{\\Omega_1}$. Then $ \\Vert \\tilde{u}_0 \\Vert\\leq R$ on $\\Omega$.\n\t\n\t\\begin{trivlist}\n\t\t\\item[\\textbf{Step 8.}] Let us fix $\\varepsilon\n\t\t>0$ and choose $M$\n\t\tsuch that $M\\ge (C_0+1) \\varepsilon^{-\\frac 12}$, where $C_0$ is the constant appearing in inequality \\eqref{ineq-C_0} below.\n\t\tThanks to Propositions \\ref{prop-global Lipschitz-F} and \\ref{prop-Psi_T-Lipschitz}, Corollary \\ref{cor-global-lip-G} and Assumption\n\t\t\\ref{assum-01} we can find $\\tilde{D}_i(n)>0, i=1,2$, $n\\in\\mathbb{N}$ such that for all $u\\in \\mathscr{M}^2(\\hat{X}_{0,T,u_0})$ we have\n\t\t\\begin{align*}\n\t\t\\lvert S\\ast \\Phi_F^n(u)\\rvert_{\\mathscr{M}^2(X_T)}\n\t\t\\le \\tilde{D}_1(n) \\tilde{C}_1 \\tilde{C}_2(2n \\tilde{C}_2+1)(2n)^{p+2}T^{(1-\\alpha)\/2},\\\\\n\t\t\\lvert S\\diamond \\hat{\\Phi}^n_G(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le \\tilde{D}_2(n) \\tilde{C}_3\n\t\t\\tilde{C}_4 (2n)^{k+2} [2n \\tilde{C}_4+1] T^{(1-\\beta)\/2}.\n\t\t\\end{align*}\n\t\tHence,\t\tsince $1-\\alpha, 1-\\beta >0$ we infer\n\t\tthat there exists a sequence $(K_n(T))_n$ of numerical functions\n\t\tsuch that for all $n$, $\\lim_{T\\rightarrow 0} K_n(T)=0$ and\n\t\t\\begin{equation*}\n\t\t\\lvert S\\ast \\Phi^n_T(u)+ S\\diamond\n\t\t\\Phi^n_G(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le K_n(T),\\;\\; u\\in \\mathscr{M}^2(X_T).\n\t\t\\end{equation*}\n\t\tLet us put $n=M R^2$ \n\t\tand\n\t\tchoose $\\delta_{1}(\\varepsilon,R) >0$ such that\n\t\t$K_{n}(\\delta_{1}(\\varepsilon,R)) \\leq R $. Let $\\Psi^n_T$ be the\n\t\tmapping defined by \\eqref{eqn-Psi_T}. Since $\\mathbb{E} \\lVert \\tilde{u}_0\\rVert^2\\leq\n\t\tR^2 $, we infer by the Assumption \\ref{assum-01} that\n\t\t\\begin{align} \\label{ineq-C_0}\n\t\t\\lvert \\Psi^n_T(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le & C_0 R + K_n(T)\n\t\t\\le (C_0+ 1)R ,\n\t\t\\end{align}\n\t\tfor all $T\\le \\delta_1(\\varepsilon,R)$. \n\t\tFurthermore, Propositions \\ref{prop-Psi_T-Lipschitz}\n\t\timplies that there exists $C>0$ such that \n\t\t\\begin{equation*}\n\t\t\\lvert \\Psi^n_T(u_1)-\\Psi^n_T(u_2)\\rvert_{\\mathscr{M}^2(X_T)}\\le C_3\n\t\tM^{D}R^{D} \\Big[T^{(1-\\alpha)\/2} \\vee T^{(1-\\beta)\/2}\\Big] \\lvert\n\t\tu_1-u_2\\rvert_{\\mathscr{M}^2(X_T)}, \\text{ for all $u_1,u_2\\in\n\t\t\t\\mathscr{M}^2(X_T)$}.\n\t\t\\end{equation*}\n\t\tSince $M\\ge (C_0+1) \\varepsilon^\\frac 12$ we infer that\n\t\t\\begin{equation*}\n\t\t\\lvert \\Psi^n_T(u_1)-\\Psi^n_T(u_2)\\rvert_{\\mathscr{M}^2(X_T)}\\le C_3\n\t\tM^{D}R^{D} \\Big[T^{(1-\\alpha)\/2} \\vee T^{(1-\\beta)\/2}\\Big] \\lvert\n\t\tu_1-u_2\\rvert_{\\mathscr{M}^2(X_T)}.\n\t\t\\end{equation*}\n\t\t\n\t\t\n\t\tHence we can find $\\delta_2(\\varepsilon,R)>0$ such that $\\Psi^n_T$ is a\n\t\tstrict contraction for all $T\\le \\delta_2(\\varepsilon,R)$. Thus if one\n\t\tputs $T^\\ast(\\varepsilon,R)=\\delta_1(\\varepsilon,R)\\wedge \\delta_2(\\varepsilon,R)$, the\n\t\tmapping $\\Psi^n_T$ has a unique fixed point ${\\hat{u}^n}$ which satisfies\n\t\t\\begin{equation}\\label{ineq-moment-1}\n\t\t\\mathbb{E} \\lvert {\\hat{u}^n}\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}}\\le (C_0+1)^2 R^2.\n\t\t\\end{equation}\n\t\tSimilarly to \\eqref{eqn-stopping time tau_n} we can define a new stopping time ${\\hat{\\tau}_n}$ by\n\t\t\\[\n\t\t{\\hat{\\tau}_n}:= \\inf\\{t\\in [0,\\infty): |{\\hat{u}^n}|_{X_t}\\ge n\\}.\n\t\t\\]\n\t\tArguing as in \\textbf{Step 5} we can show that \t\t\t${\\hat{u}^n}$ satisfies, for every $t\\in [0,T]$, $\\mathbb{P}$-a.s.\n\t\t\\begin{equation}\\label{Eq:truncated-equation-9-1}\n\t\t{\\hat{u}^n}(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge {\\hat{\\tau}_n}} S_{t\\wedge {\\hat{\\tau}_n} -r}[F({\\hat{u}^n}(r))]\\,dr\n\t\t+{\\hat{I}^n_{\\hat{\\tau}_n}}(t \\wedge {\\hat{\\tau}_n}).\n\t\t\\end{equation}\n\t\twhere ${\\hat{I}^n_{\\hat{\\tau}_n}}$ is a continuous $V$-valued process defined by\n\t\t\\[\n\t\t{\\hat{I}^n_{\\hat{\\tau}_n}}(t):=\\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\tS_{t-r}[G({\\hat{u}^n}(r))]dW(r), \\;\\; t\\in [0,T].\n\t\t\\]\n\t\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\tBy the definition of the stopping time ${\\hat{\\tau}_n}$,\n\t\t$\\{{\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R)\\} \\subset \\{\n\t\t\\lvert u^n\\rvert_{X_{T^\\ast(\\varepsilon,R)}}\\ge n \\}$. Therefore, by the Chebyshev\n\t\tinequality and inequality \\eqref{ineq-moment-1} we infer that \t\t\n\t\t\\begin{align*}\n\t\t\\mathbb{P}({\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R))\\le &\\mathbb{P}(\\lvert\n\t\tu^n\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}}\\ge n ) \\leq \\frac{1}{n}\\mathbb{E}\\lvert\n\t\tu^n\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}} \\leq \\frac{1}{n} (C_0+1)^2R^2.\n\t\t\\end{align*}\n\t\tSince $n=NR^2$ and $N\\ge (C_0+1)^2 \\varepsilon^{-\\frac 12}$ we get\n\t\t\\begin{align*}\n\t\t\\mathbb{P}({\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R))\\leq & (C_0+1)^2 N^{-2} \\leq \\varepsilon.\n\t\t\\end{align*}\n\t\t\n\t\tHence, we have prove \\eqref{ineq-positive probability-2}.\n\t\t\n\t\t\\item[\\textbf{Step 9.}]\\label{Step9}\n\t\t\n\t\tTo conclude the proof, we observe that in view of Remark \\ref{rem-local solution}, it follows from equality \\eqref{Eq:truncated-equation-9-1} that the process\n\t\t${\\hat{u}^n}$ restricted to the open random interval $[0, \\hat{\\tau}^n)\\times \\Omega$ is a local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}. On the other hand, in \\textbf{Step 7} we proved that also $(u,\\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}.\n\t\t{Because local uniqueness holds for problem \\eqref{ABS-SPDE-strong}}, see Theorem \\ref{Thm:LocalUniqueness}, we infer by applying Corollary \\ref{cor-max of two local solutions} that the supremum\n\t\tof $(u,\\tau_\\infty)$ and $({\\hat{u}^n},\\hat{\\tau}^n)$ is another local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}.\n\t\tHence, the stopping time $T_1= \\tau_\\infty \\vee \\hat{\\tau}^n$ satisfies the requirements of the theorem.\n\t\\end{trivlist}\n\tThis concludes the proof of part (III) of Theorem \\ref{thm_local} and thus of the whole theorem.\n\t\t\n\\end{proof}\n\n The next result is about the existence and uniqueness of a maximal solution and the characterization of its lifespan.\n\t\\begin{thm}\\label{thm_maximal-abstract}\n\tLet $u_0\\in L^2(\\Omega,\\mathcal{F}_0,V)$. Then, problem \\eqref{ABS-SPDE-strong} has a unique maximal local solution $(\\hat{u},\\hat{\\tau})$.\n\t\tMoreover, $(u,\\tau_\\infty) \\sim (\\hat{u},\\hat{\\tau})$ and\n\t\\begin{align}\n\t&\t\\lim_{t\\toup \\hat{\\tau}}|\\hat{u}|_{X_t}=\\infty \\;\\;\\; \\mathbb{P}\\mbox{-a.s. on\n\t\t} \\{\\hat{\\tau}<\\infty\\},\\label{eqn-t_infty}\\\\\n&\t\t\\mathbb{ P}\\Big(\\{\\omega \\in\\Omega :\\hat{\\tau} (\\omega)< \\infty \\text{ and }\n\t\t\\sup_{t\\in [0, \\hat{\\tau} (\\omega))} \\lvert \\hat{u}(t)(\\omega)\\rvert_V<\\infty \\}\\Big)=0. \\label{eqn-t_infty-limit}\n\t\t\\end{align}\n\t\n\t\\end{thm}\n\n\n\t\\begin{proof}\n\t\tLet us choose and fix $u_0\\in L^2(\\Omega,\\mathcal{F}_0,V)$. Wlog we can assume that $\\mathbb{P}(\\{\\hat{\\tau}<\\infty\\})>0$.\n\t\t\n\t\tFirstly, we observe that it follows from the proof of Theorem \\ref{thm_local} that the local process $(u,\\tau_\\infty)$ defined in \\eqref{eqn-local process u} is a local solution to \\eqref{ABS-SPDE-strong}.\n In particular, the set of local solutions to problem \\eqref{ABS-SPDE-strong} is non-empty. Since by Theorem \\ref{Thm:LocalUniqueness} the local uniqueness holds for problem \\eqref{ABS-SPDE-strong}, we infer by applying Proposition \\ref{prop-loc-implies-max} that there exists a unique maximal local solution $(\\hat{u},\\hat{\\tau})$ to \\eqref{ABS-SPDE-strong}. Moreover,\n\t\t$(\\hat{u},\\hat{\\tau})$ satisfies the following\n\t\t\\begin{equation}\\label{eqn-similarity}\n\t\t\\hat{\\tau} \\ge \\tau_\\infty \\text{ $\\mathbb{P}$-a.s. and } \\hat{u}_{\\lvert_{[0,\\tau_\\infty)\\times \\Omega} }= u.\n\t\t\\end{equation}\t\t\nSecondly, suppose that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\tau_\\infty \\bigr)>0$. Let $\\bigl(\\hat{\\tau}_n\\bigr)$ be the announcing sequence of $\\hat{\\tau}$. Since $\\hat{\\tau}_n \\toup \\hat{\\tau}$ $\\mathbb{P}$-a.s., we infer that\nthere exists $n \\in \\mathbb{N}$ such that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr)>0$. Thus we infer that there exists $t>0$ such that\n$\\mathbb{P} \\bigl( t> \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr)>0$. \\\\\nOn the other hand, by the definition \\eqref{eqn-stopping time tau_n} of the announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ for $\\tau_\\infty$ and the definition, see \\eqref{eqn-local process u}, of the local process $(u,\\tau_\\infty)$ that the sequence $\\vert u \\vert_{X_{\\tau_n}}$ converges to $\\infty$ on $\\{ \\tau_\\infty< \\infty\\}$. Hence we infer that $\\lvert u\\rvert_{X_{\\tau_\\infty}} = \\infty $ $\\mathbb{P}$-a.s. on\n $\\bigl\\{ t> \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr\\}$. Since the probability of the last set is $>0$, this contradicts condition \\eqref{eq-locsol_00} of Definition\n \\ref{def-local solution-2} of a local solution. This contradiction implies that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\tau_\\infty \\bigr)=0$, and in view of \\eqref{eqn-similarity}, we also have $(u,\\tau_\\infty) \\sim (\\hat{u},\\hat{\\tau})$.\n\n\n\n\n\t\tThirdly, we infer from the definition \\eqref{eqn-stopping time tau_n} of the announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ for $\\tau_\\infty$ and the definition, see \\eqref{eqn-local process u}, of the local process $(u,\\tau_\\infty)$ that the sequence $\\vert u \\vert_{X_{\\tau_n}}$ converges to $\\infty$ on $\\{\\tau_\\infty< \\infty\\}$. Since\nthe function $t \\mapsto \\vert u \\vert_{X_{t}}$ is increasing, by \\eqref{eqn-similarity}, we infer that\n\\begin{equation}\\label{eqn-blow-up similarity}\n\\dela{\\lim_{t\\toup \\hat{\\tau}} \\lvert \\hat{u} \\rvert_{X_t} \\ge\n\\lim_{t\\toup \\tau_\\infty} \\lvert \\hat{u} \\rvert_{X_t} =} \\lim_{t\\toup \\tau_\\infty}\\lvert u \\rvert_{X_t} = \\lim_{n \\toup \\infty} \\lvert u\\rvert_{X_{\\tau_n}} = \\infty \\text{ $\\mathbb{P}$-a.s. on } \\{\\hat{\\tau}< \\infty \\}.\n\\end{equation}\nThis proves\t\\eqref{eqn-t_infty}.\n\nFourthly, we shall prove \\eqref{eqn-t_infty-limit}. By contradiction, we assume that\nthere exists $\\varepsilon>0$ such that\n\\[ {\\mathbb P }\\big(\\{\\tau_\\infty <\\infty\\} \\cap \\{ \\sup_{t\\in [0,\\tau_\\infty)} |\\hat{u}(t)|_{V}<\\infty\n\\}\\big)=4\\varepsilon>0.\\]\nHence we can easily deduce that there exists $R>0$ such that\n\\[ {\\mathbb P }\\big( \\{ |u(t)|_{V}0$ depending only on $\\varepsilon$ and $R$ comes from \npart (II) of Theorem \\ref{thm_local}. By the definition of an announcing sequence \n $\\bigl(\\tau_n\\bigr)$ of $\\tau_\\infty$ we infer that\nfor arbitrary $\\delta>0$ there exists $n_0>0$ such that\n$\\mathbb{P}(\\Omega_0)\\geq (1-\\delta)\\mathbb{P}(\\tilde{\\Omega})$,\nwhere $\\Omega_0:=\\{\\omega \\in \\tilde{\\Omega}: |u(t)|_{V}0$, by applying part (II) of Theorem \\ref{thm_local} we find a local solution $y(t)$, $t\\in\n[\\tau_{n_0},\\tau_{n_0} +T_1)$ to problem \\eqref{ABS-SPDE-1} with the initial condition (starting at $\\tau_{n_0}$) $y(\\tau_{n_0})=y_0$ such that\n$\\mathbb{P}(T_1\\geq T^\\ast(\\varepsilon,R))>1-\\varepsilon$. \nAlso, let $\\Omega_1:= \\Omega_0 \\cap \\{ T_1\\geq T^\\ast(\\varepsilon,R)\\}.$ Since\n$\\mathbb{P}(\\hat{\\tau}-T_0< \\frac 12 T^\\ast(\\varepsilon,R))\\geq 2\\varepsilon$, we infer that\n\\[\\mathbb{P}\\big(\\Omega_1 \\big) \\geq \\varepsilon>0.\\]\n\nBy a generalization of \\cite[Corollary 2.28]{Brz+Elw_2000} to the case of SPDEs we infer that a local stochastic process $v(t)$, $t\\in [0,\\tau_{n_0}(\\omega) +T_1)$ defined by\n\\begin{equation*}\nv(t,\\omega)=\\begin{cases}\nu(t,\\omega) \\text{ if } t \\in [0,\\tau_{n_0}(\\omega)] ,\\\\\ny(t,\\omega) \\text{ if } t\\in [ \\tau_{n_0}(\\omega), \\tau_{n_0}(\\omega) +T_1 ).\\\\\n\\end{cases}\n\\end{equation*}\nis a local solution to problem \\eqref{ABS-SPDE-1} with the initial data $v(0)=u_0$. However, on the $\\Omega_1$, we have \n\\[\n\\tau_{n_0} +T_1-\\tau_\\infty = T_1 -( \\tau_\\infty -\\tau_{n_0} )\\geq 2\\alpha -\\alpha=\\alpha>0,\n\\]\nwhat contradicts the maximality of the solution $(u,\\tau_\\infty)$ proved in the earlier part of the theorem.\nThis completes the proof of \\eqref{eqn-t_infty-limit} as well as the theorem.\n\n\t\\end{proof}\n\t\\subsection*{Acknowledgments}\n\tThis article is part of a project that is currently funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\\l{}odowska-Curie grant agreement No. 791735 ``SELEs\".\n\tThe authors are also grateful for the support they received from the FWF-Austrian Science through the Stand-Alone project P28010.\n\tZ. Brze{\\' z}niak presented a\n\tlecture based on a preliminary version of this paper at the RIMS Symposium on Mathematical Analysis of Incompressible Flow held at Kyoto in February 2013.\n\tHe would like to thank Professor Toshiaki Hishida for the kind invitation. Razafimandimby is also very grateful to the organizers of the conference `` Nonlinear PDEs in Micromagnetism: Analysis, Numerics and Applications'', which was held in ICMS Edinburgh, UK, for their invitation to present a talk based on preliminary result of this paper at this meeting. He is very grateful for the financial support he received from the International Centre for Mathematical Sciences (ICMS) Edinburgh.\n\tLast, but not the least, the authors wish to thank Professor Guoli Zhou for pointing out some gaps in the previous version of this paper \\cite{BHP-arxiv}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{The ATLAS detector}\n\\label{sec:atlas}\n\nThe ATLAS experiment~\\cite{ATLASdetector} is a general-purpose detector consisting of an inner tracker, a calorimeter and a muon spectrometer. \nThe inner detector (ID) directly surrounds the interaction point; \nit consists of a silicon pixel detector, a semiconductor tracker and a transition radiation tracker, and is embedded in an axial~$2$ T magnetic field.\nThe ID covers the pseudorapidity\\footnote{ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) \nin the centre of the detector and the $z$-axis along the beam pipe.\nThe $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upward. Cylindrical \ncoordinates $(r,\\phi)$ are used in the transverse plane,\n$\\phi$ being the azimuthal angle around the beam pipe. The pseudorapidity $\\eta$ is defined in terms of the \npolar angle $\\theta$ as $\\eta=-\\ln\\tan(\\theta\/2)$ and the\ntransverse momentum $p_{\\rm T}$ is defined as $p_{\\rm T}=p\\sin\\theta$. The rapidity is defined as \n$y=0.5\\ln\\left[\\left( E + p_z \\right)\/ \\left( E - p_z \\right)\\right]$,\nwhere $E$ and $p_z$ refer to energy and longitudinal momentum, respectively. The $\\eta$--$\\phi$ distance \nbetween two particles is defined as\n $\\Delta R=\\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2}$.}\nrange $|\\eta| = $ 2.5 and is enclosed by a calorimeter system containing\nelectromagnetic and hadronic sections.\nThe calorimeter is surrounded by a large muon spectrometer (MS) in a toroidal magnet system.\nThe MS consists of monitored drift tubes and cathode strip chambers, designed to provide\nprecise position measurements in the bending plane in the range $|\\eta| <$ 2.7.\nMomentum measurements in the muon spectrometer are based on track segments formed in at least two of the three precision chamber planes.\n\nThe ATLAS trigger system \\cite{ATLAS:trig} is separated into three levels: the hardware-based Level-1 trigger\nand the two-stage High Level Trigger (HLT), comprising the Level-2 trigger and Event\nFilter, which reduce the 20~MHz proton--proton collision rate to several-hundred Hz of events of interest for data recording to mass storage. \nAt Level-1, the muon trigger searches for patterns of hits satisfying different transverse momentum thresholds with a coarse \nposition resolution but a fast response time using resistive-plate chambers and thin-gap chambers in the ranges $|\\eta| <$ 1.05 and $1.05 <|\\eta| < 2.4$, respectively.\nAround these Level-1 hit patterns ``Regions-of-Interest'' (RoI) are defined that\nserve as seeds for the HLT muon reconstruction. \nThe HLT uses dedicated algorithms to incorporate information from both the \nMS and the ID, achieving position and momentum resolution close to that provided by the offline muon reconstruction.\n\n\n\n\\section{Summary and conclusions}\n\\label{sec:conclusion}\n\nThe prompt and non-prompt production cross-sections, the non-prompt production fraction of the $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$\ndecaying into two muons, the ratio of prompt $\\ensuremath{\\psi(2\\mathrm{S})}$ to prompt $\\jpsi$ production, and the ratio of non-prompt $\\ensuremath{\\psi(2\\mathrm{S})}$ to non-prompt $\\jpsi$ production\nwere measured in the rapidity range\n$|y|<2.0$ for transverse momenta between $8$ and $110$~\\GeV. This measurement was carried out using \\mbox{$2.1\\ifb$} (\\mbox{$11.4\\ifb$}) of $pp$\ncollision data at a centre-of-mass energy of $7$ \\TeV\\ ($8$ \\TeV) recorded by the ATLAS experiment at the LHC. It is the latest in a series of \nrelated measurements of the production of charmonium states\nmade by ATLAS.\nIn line with previous measurements, the central values were obtained \nassuming isotropic $\\psi \\to \\mu\\mu$ decays.\nCorrection factors for these cross-sections, computed for a number of extreme spin-alignment scenarios, are between $-35\\%$ and $+100\\%$ at the lowest transverse momenta studied, and between $-14\\%$ and $+9\\%$ at the highest transverse momenta, depending on the specific scenario.\n\nThe ATLAS measurements presented here extend the range of existing measurements to higher transverse momenta, and to a higher collision energy of $\\sqrt{s} = 8$ \\TeV, and, in overlapping phase-space regions, are consistent with previous measurements made by ATLAS and other LHC experiments.\nFor the prompt production mechanism, the predictions from the NRQCD model, which includes colour-octet contributions with various matrix elements \ntuned to earlier collider data, are found to be in good agreement with the observed data points. \nFor the non-prompt production, the fixed-order next-to-leading-logarithm calculations reproduce the data reasonably well, with a slight overestimation of the differential cross-sections at the highest \ntransverse momenta reached in this analysis.\n\n\\section{Candidate selection}\n\\label{sec:data}\n\nThe analysis is based on data recorded at the LHC in 2011 and 2012 during proton--proton collisions at centre-of-mass energies of $7$~\\TeV\\ and $8$~\\TeV, respectively.\nThis data sample corresponds to a total integrated luminosity of \\mbox{$2.1\\ifb$}\\ and \\mbox{$11.4\\ifb$}\\ for $7$~\\TeV\\ data and $8$~\\TeV\\ data, respectively.\n\nEvents were selected using a trigger requiring two oppositely charged muon candidates, each passing the requirement $\\mbox{$p_{\\text{T}}$}>4$~\\GeV. \nThe muons are constrained to originate from a common vertex, which is fitted with the track parameter uncertainties taken into account. \nThe fit is required to satisfy $\\chi^2 < 20$ for the one degree of freedom.\n\nFor $7$~\\TeV\\ data, the Level-1 trigger required only spatial coincidences in the MS~\\cite{ATLAS:2010kba}. For $8$~\\TeV\\ data, a $4$~\\GeV\\ muon $\\mbox{$p_{\\text{T}}$}$ threshold was also applied at Level-1, which reduced the trigger efficiency for low-$\\mbox{$p_{\\text{T}}$}$ muons.\n\nThe offline analysis requires events to have at least two muons, identified by the muon spectrometer and with matching tracks reconstructed in the ID~\\cite{muons}.\nDue to the ID acceptance, muon reconstruction is possible only for $|\\eta| <$ 2.5. \nThe selected muons are further restricted to $|\\eta| <$ 2.3 to ensure high-quality tracking and triggering, and to reduce the contribution from misidentified muons.\nFor the momenta of interest in this analysis (corresponding to muons with a transverse momentum of at most $O(100)$ \\GeV), measurements of the\nmuons are degraded by multiple scattering within the MS and so only the ID tracking\ninformation is considered. \nTo ensure accurate ID measurements, each muon track must fulfil muon reconstruction and selection requirements~\\cite{muons}.\nThe pairs of muon candidates satisfying these quality criteria are required to have opposite charges.\n\nIn order to allow an accurate correction for trigger inefficiencies, each reconstructed muon candidate is required to match a trigger-identified muon\ncandidate within a cone of $\\Delta R = \\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2}=0.01$.\nDimuon candidates are obtained from muon pairs, constrained to originate from a common vertex using ID track parameters and uncertainties, with a \nrequirement of $\\chi^2 < 20$ of the vertex fit for the one degree of freedom.\nAll dimuon candidates with an invariant mass\nwithin $2.6 < m(\\mu\\mu) < 4.0$ \\GeV\\ and within the kinematic range $\\mbox{$p_{\\text{T}}$}(\\mu\\mu) > 8$ \\GeV, $|y(\\mu\\mu)| < 2.0$ are retained for the analysis.\nIf multiple candidates are found in an event (occurring in approximately $10^{-6}$ of selected events), all candidates are retained.\nThe properties of the dimuon system, such as invariant mass $m(\\mu\\mu)$, transverse momentum $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, and rapidity $|y(\\mu\\mu)|$ are determined from the result of the vertex fit.\n\n\n\n\\section{Introduction}\n\\label{intro}\n\nMeasurements of heavy quark--antiquark bound states (quarkonia) production processes provide \nan insight into the nature of quantum\nchromodynamics (QCD) close to the boundary between the perturbative and non-perturbative regimes. \nMore than forty years since the discovery of the $\\jpsi$, the investigation of\nhidden heavy-flavour production in hadronic collisions \nstill presents significant challenges to both theory and experiment.\n\nIn high-energy hadronic collisions, charmonium states can be produced either directly by \nshort-lived QCD sources (``prompt'' production), or by long-lived sources in the decay chains of beauty hadrons (``non-prompt'' production). \nThese can be separated experimentally using the\ndistance between the proton--proton primary interaction and the decay vertex of the quarkonium state.\nWhile {\\em Fixed-Order with Next-to-Leading-Log} (FONLL) calculations~\\cite{FONLL_2001,Cacciari:2012ny}, made within the framework of perturbative QCD, have\nbeen quite successful in describing non-prompt production of various quarkonium states, a satisfactory understanding of the prompt production mechanisms is still to be achieved.\n\n\nThe $\\ensuremath{\\psi(2\\mathrm{S})}$ meson is the only vector charmonium state that \nis produced with no significant contributions from decays of higher-mass quarkonia, \nreferred to as feed-down contributions. This\nprovides a unique opportunity to study production mechanisms specific to\n$J^{PC}=1^{--}$ states~\\cite{CDFjpsianomaly1,Abulencia:2007us,Abelev:2014qha,Chatrchyan:2011kc,Chatrchyan:2013cla,Aaij:2012ag,Aad:2014fpa,Aaij:2013jxj,Khachatryan:2015rra,Aaij:2013yaa}.\nMeasurements of the production of $J^{++}$ states with $J=0, 1, 2$,\n\\cite{CDFjpsianomaly2,ATLAS:2014ala,Chatrchyan:2012ub,Aaij:2013yaa,Aaij:2013dja,LHCb:2012ac},\nstrongly coupled to the two-gluon channel, allow similar studies in the $CP$-even sector, complementary to the $CP$-odd vector sector. \nProduction of $\\jpsi$ mesons\n\\cite{Aad:2011sp,Abulencia:2007us,Aaij:2011jh,Chatrchyan:2011kc,Abelev:2012gx,Chatrchyan:2013cla,Aaij:2013jxj,Abelev:2014qha,CDFjpsianomaly3,CDFjpsianomaly1,D0jpsi1,D0jpsi2,Khachatryan:2015rra,Aad:2014fpa,CDFjpsianomaly2,LHCb:2012af}\narises from a mixture of different sources, receiving contributions from the production of $1^{--}$ and $J^{++}$ states in comparable amounts.\n\n\nEarly attempts to describe the formation of charmonium\n\\cite{CSM1,CSM2,CSM3,CSM4,CSM5,CSM7,CEM1,CEM2}\nusing leading-order perturbative QCD \ngave rise to a variety of models, none of which could explain the\nlarge production cross-sections measured at the Tevatron\n\\cite{CDFjpsianomaly1,CDFjpsianomaly2,CDFjpsianomaly3,D0jpsi1,D0jpsi2}.\nWithin the colour-singlet model (CSM) \\cite{Lansberg:2008gk}, next-to-next-to-leading-order (NNLO) contributions to the hadronic production of S-wave\nquarkonia were calculated without introducing any new phenomenological parameters. However,\ntechnical difficulties have so far made it impossible to perform the full NNLO calculation, or to \nextend those calculations to the P-wave states. \nSo it is not entirely surprising that the predictions of the model underestimate the experimental data for inclusive production of\n\\jpsi\\ and $\\Upsilon$ states, where the feed-down is significant, but offer a better description for \\ensuremath{\\psi(2\\mathrm{S})}\\ production \\cite{Aad:2011sp,Aad2012dlq}.\n\nNon-relativistic QCD (NRQCD) calculations that include colour-octet (CO) contributions \n\\cite{Bodwin:1994jh} introduce a number of phenomenological parameters --- long-distance matrix elements\n(LDMEs) --- which are determined from fits to the experimental data, and can hence\ndescribe the cross-sections and differential spectra satisfactorily \n{\\cite{CO_LDME1}}. \nHowever, the attempts to describe the polarization of S-wave quarkonium states using this approach\nhave not been so successful~\\cite{Gong:2012ug},\nprompting a suggestion~\\cite{Faccioli:2014cqa} that a more coherent approach is needed for the treatment of polarization\nwithin the QCD-motivated models of quarkonium production. \n\nNeither the CSM nor the NRQCD model gives a satisfactory explanation for the measurement of prompt\n\\jpsi\\ production in association with the $W$~\\cite{Aad:2014rua} and $Z$~\\cite{Aad:2014kba} bosons: in both cases,\nthe measured differential cross-section is larger than theoretical expectations~\\cite{Li:2010hc,Lansberg:2013wva,Gong:2012ah, Mao:2011kf}.\nIt is therefore important to broaden the scope of comparisons \nbetween theory and experiment by providing a variety of experimental information\nabout quarkonium production across a wider kinematic range.\nIn this context, ATLAS has measured the inclusive differential cross-section of \\jpsi\\ production, with $2.3$~pb$^{-1}$ of integrated luminosity~\\cite{Aad:2011sp}, \nat $\\sqrt{s} = 7$~\\TeV\\ using the data collected in 2010, \nas well as the differential cross-sections of the production of $\\chi_c$ states~(4.5~fb$^{-1}$)~\\cite{ATLAS:2014ala}, and of the \\ensuremath{\\psi(2\\mathrm{S})}\\ in its $J\/\\psi\\pi\\pi$ decay mode (2.1 fb$^{-1})$~\\cite{Aad:2014fpa}, at $\\sqrt{s} = 7$~\\TeV\\ with data collected in 2011.\nThe cross-section and polarization measurements \nfrom CDF~\\cite{Abulencia:2007us},\nCMS~\\cite{Chatrchyan:2013cla,Khachatryan:2010yr,CMS},\nLHCb~\\cite{Aaij:2012ag,Aaij:2013jxj,Aaij:2013yaa,Aaij:2012asz,Aaij:2013nlm,Aaij:2014qea} \nand ALICE~\\cite{Abelev:2014qha,Abelev:2011md,Aamodt:2011gj},\ncover a considerable variety of charmonium production characteristics in a wide \nkinematic range (transverse momentum $\\mbox{$p_{\\text{T}}$}\\leq 100$ \\GeV\\ and rapidities $|y|<5$), thus providing a wealth of\ninformation for a new generation of theoretical models.\n\nThis paper presents a precise measurement of\n\\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ production in the dimuon decay mode, both at $\\sqrt{s} = 7$~TeV\nand at $\\sqrt{s} = 8$~TeV.\nIt is presented as a double-differential measurement in transverse momentum and rapidity of the quarkonium state, \nseparated into prompt and non-prompt contributions,\ncovering a range of transverse momenta $8 < \\mbox{$p_{\\text{T}}$}\\leq 110$ \\GeV\\ and rapidities $|y|<2.0$. \nThe ratios of $\\ensuremath{\\psi(2\\mathrm{S})}$ to $\\jpsi$ cross-sections for prompt and non-prompt processes are also reported, as well as the \nnon-prompt fractions of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$.\n\\section{Methodology}\n\\label{sec:method}\n\nThe measurements are performed in intervals of dimuon $\\mbox{$p_{\\text{T}}$}$ and absolute value of the rapidity ($|y|$). \nThe term ``prompt'' refers to the $\\jpsi$ or $\\ensuremath{\\psi(2\\mathrm{S})}$ states --- hereafter called $\\ensuremath{\\psi}$ to refer to either --- are produced \nfrom short-lived QCD decays, including feed-down from other charmonium states as long as they are also produced\nfrom short-lived sources. If the decay chain producing a $\\ensuremath{\\psi}$ state includes long-lived particles such as\n$b$-hadrons, then such $\\ensuremath{\\psi}$ mesons are labelled as ``non-prompt''. Using a simultaneous fit to the invariant mass of the dimuon and its ``pseudo-proper decay time'' \n(described below), prompt and non-prompt signal and background contributions can be extracted from the data.\n\n\nThe probability for the decay of a particle as a function of proper decay time $t$ follows an exponential distribution, \n$p(t) = 1\/\\tau_{B}\\cdot e^{-t\/\\tau_{B}}$ where $\\tau_{B}$ is the mean lifetime of the particle. For each decay, the proper decay time \ncan be calculated as $t = L m\/p$,\nwhere $L$ is the distance between the particle production and decay vertices, $p$ is the momentum of the particle, and $m$ is its invariant mass.\nAs the reconstruction of non-prompt $\\ensuremath{\\psi}$ mesons, such as $b$-hadrons, does not fully describe the properties of the parent,\nthe transverse momentum of the \ndimuon system and the reconstructed dimuon invariant mass are used to construct the\n``pseudo-proper decay time'', $\\tau = L_{xy} m(\\mu\\mu)\/\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, where $L_{xy} \\equiv \\vec{L} \\cdot \\vec{\\mbox{$p_{\\text{T}}$}}(\\mu\\mu)\/\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ \nis the signed projection of the distance of the dimuon decay vertex \nfrom the primary vertex, $\\vec{L}$, onto its transverse momentum, $\\vec{\\mbox{$p_{\\text{T}}$}}(\\mu\\mu)$.\nThis is a good approximation of using the parent $b$-hadron information when the $\\psi$ and parent momenta are closely aligned, which is the case for the values of $\\psi$\ntransverse momenta considered here, and $\\tau$\ntherefore can be used to distinguish statistically between the non-prompt and prompt processes (in which the latter are assumed to decay with vanishingly small lifetime).\nIf the event contains multiple primary vertices~\\cite{ATLASdetector}, the primary vertex closest in $z$ to the dimuon decay vertex is selected.\nThe effect of selecting an incorrect vertex has been shown~\\cite{BJpsiPhi} to have a negligible impact on the extraction of prompt and non-prompt contributions.\nIf any of the muons in the dimuon candidate contributes to the construction of the primary vertex, the corresponding tracks are removed and the vertex is refitted.\n\n\n\\subsection{Double differential cross-section determination}\n\\label{sec:s:diff_xSecDet}\n\nThe double differential dimuon prompt and non-prompt production cross-sections times branching ratio \nare measured separately for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons according to the equations:\n\n\\begin{equation}\n\\frac{\\mathrm{d}^2\\sigma(pp \\rightarrow \\psi)}{\\mathrm{d}\\mbox{$p_{\\text{T}}$}\\mathrm{d}y} \\times \\mathcal{B} (\\psi \\rightarrow \\mu^+\\mu^- ) = \\frac{N_{\\psi}^{\\mathrm{p}}}{\\Delta \\mbox{$p_{\\text{T}}$} \\Delta y \\times \\int\\mathcal{L} \\mathrm{d}t},\n\\label{equ:xSecP}\n\\end{equation}\n\\begin{equation}\n\\frac{\\mathrm{d}^2\\sigma(pp \\rightarrow b\\bar{b} \\rightarrow \\psi)}{\\mathrm{d}\\mbox{$p_{\\text{T}}$}\\mathrm{d}y} \\times \\mathcal{B} (\\psi \\rightarrow \\mu^+\\mu^- ) = \\frac{N_{\\psi}^{\\mathrm{np}}}{\\Delta \\mbox{$p_{\\text{T}}$} \\Delta y \\times \\int\\mathcal{L} \\mathrm{d}t},\n\\label{equ:xSecNP}\n\\end{equation}\n\n\\noindent where $\\int\\mathcal{L} dt$ is the integrated luminosity, $\\Delta \\mbox{$p_{\\text{T}}$} $ and $ \\Delta y$ are the interval sizes in terms of dimuon transverse momentum and\nrapidity, respectively, and $N_{\\psi}^{\\mathrm{p(np)}}$ is the number of observed prompt (non-prompt) $\\psi$ mesons in\nthe slice under study, corrected for acceptance, trigger and reconstruction efficiencies. \nThe intervals in $\\Delta y$ combine the data from negative and positive rapidities.\n\nThe determination of the cross-sections proceeds in several steps. First, a weight is determined for each\nselected dimuon candidate equal to the inverse of the total efficiency for each candidate. \nThe total weight, $w_\\mathrm{tot}$, for each dimuon candidate includes three factors: the fraction of produced $\\psi \\rightarrow \\mu^+\\mu^-$ decays with both\nmuons in the fiducial region $\\mbox{$p_{\\text{T}}$} (\\mu) > 4$ \\GeV\\ and $|\\eta(\\mu)| <$ 2.3 (defined as acceptance, $\\mathcal{A}$), the probability that a candidate\nwithin the acceptance satisfies the offline reconstruction selection ($\\epsilon_\\mathrm{reco}$), and \nthe probability that a reconstructed event satisfies the trigger selection \n($\\epsilon_\\mathrm{trig}$). The weight assigned to a given candidate when calculating the cross-sections is therefore given by:\n\n\\begin{equation*}\nw_{\\mathrm{tot}}^{-1} = \\mathcal{A} \\cdot \\epsilon_{\\mathrm{reco}} \\cdot \\epsilon_{\\mathrm{trig}}.\n\\end{equation*}\n\nAfter the weight determination, an unbinned maximum-likelihood fit is performed to these weighted events in each ($\\mbox{$p_{\\text{T}}$} (\\mu\\mu), \\ |y(\\mu\\mu)|$) interval using the \ndimuon invariant mass, $m(\\mu\\mu)$, and pseudo-proper decay time, $\\tau(\\mu\\mu)$, observables. \nThe fitted yields of $\\jpsi \\rightarrow \\mu^+\\mu^-$ and $\\ensuremath{\\psi(2\\mathrm{S})} \\rightarrow \\mu^+\\mu^-$ are determined separately for prompt and non-prompt processes. \nFinally, the differential cross-section times the $\\psi \\rightarrow \\mu^+\\mu^-$ branching fraction is calculated for each\nstate by including the integrated luminosity and the $\\mbox{$p_{\\text{T}}$}$ and rapidity interval widths as shown in Eqs. (\\ref{equ:xSecP}) and (\\ref{equ:xSecNP}).\n\n\n\\subsection{Non-prompt fraction}\n\\label{sec:s:NPFDet}\n\nThe non-prompt fraction $f_{b}^{\\psi}$ is defined as the number of non-prompt $\\psi$ (produced\nvia the decay of a $b$-hadron) divided by the number of inclusively produced $\\psi$ decaying to muon pairs after applying weighting corrections:\n\n\\begin{equation*}\nf_{b}^{\\psi} \\equiv \\frac{pp \\rightarrow b + X \\rightarrow \\psi + X'}{pp \\xrightarrow{\\mathrm{Inclusive}} \\psi + X'} = \\frac{N^{\\mathrm{np}}_{\\psi}}{N^{\\mathrm{np}}_{\\psi} + N^{\\mathrm{p}}_{\\psi}}, \n\\end{equation*}\n\n\\noindent where this fraction is determined separately for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}. \nDetermining the fraction from this ratio is advantageous since acceptance and efficiencies largely cancel and the systematic uncertainty is reduced.\n\n\\subsection{Ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production}\n\\label{sec:s:PNPRatioDet}\n\nThe ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production, in their dimuon decay modes, is defined as:\n\n\\begin{equation*}\nR^{\\mathrm{p(np)}} =\\frac{N^{\\mathrm{p(np)}} _{\\psi(2\\mathrm{S})}}{N^{\\mathrm{p(np)}} _{J\/\\psi}},\n\\end{equation*}\n\n\\noindent where $N_{\\psi}^{\\mathrm{p(np)}}$ is the number of prompt (non-prompt) \\jpsi\\ or \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons decaying into a \nmuon pair in an interval of $\\mbox{$p_{\\text{T}}$}$ and $y$, corrected for selection efficiencies and acceptance.\n\nFor the ratio measurements, similarly to the non-prompt fraction, the acceptance and efficiency corrections \nlargely cancel, thus allowing a more precise measurement.\nThe theoretical uncertainties on such ratios are also smaller, as several dependencies, such as\nparton distribution functions and $b$-hadron production spectra, largely cancel in the ratio.\n\n\\subsection{Acceptance}\n \nThe kinematic acceptance $\\mathcal{A}$ for a $\\psi \\rightarrow \\mu^+\\mu^-$ decay with $\\mbox{$p_{\\text{T}}$}$ and $y$ is given by the \nprobability that both muons pass the fiducial selection ($\\mbox{$p_{\\text{T}}$}(\\mu)>4$ \\GeV\\ and $|\\eta(\\mu)|<2.3$).\nThis is calculated using generator-level ``accept-reject'' simulations, based on the analytic formula described below. \nDetector-level corrections, such as bin migration effects due to detector resolution, \nare found to be small. They are applied to the results and are also considered as part of the systematic uncertainties. \n\nThe acceptance $\\mathcal{A}$ depends on five independent variables (the two muon momenta are constrained by the\n$m(\\mu\\mu)$ mass condition), chosen as the $\\mbox{$p_{\\text{T}}$}$, $|y|$ and azimuthal angle $\\phi$ of the $\\psi$ meson in the laboratory frame,\nand two angles\ncharacterizing the $\\psi \\rightarrow \\mu^+\\mu^-$ decay, $\\theta^{\\star}$ and $\\phi^{\\star}$, described in detail in Ref. \\cite{Faccioli:2010kd}.\nThe angle $\\theta^{\\star}$ is the angle between the direction of the positive-muon momentum in the $\\psi$ rest frame\nand the momentum of the $\\psi$ in the laboratory frame, while $\\phi^{\\star}$ is defined as the angle between the dimuon \nproduction and decay planes in the laboratory frame. \nThe $\\psi$ production plane is defined by the momentum of the $\\psi$ in the laboratory frame and the positive $z$-axis direction.\nThe distributions in $\\theta^{\\star}$ and $\\phi^{\\star}$ \ndiffer for various possible spin-alignment scenarios of the dimuon system.\n\nThe spin-alignment of the $\\psi$ may vary depending on the production mechanism, which in turn affects the angular distribution of the dimuon decay.\nPredictions of various theoretical models are quite contradictory, while the recent experimental measurements~\\cite{Chatrchyan:2013cla} indicate that the angular dependence of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$\ndecays is consistent with being isotropic. \n\nThe coefficients $\\lambda_{\\theta}, \\lambda_{\\phi}$ and~$\\lambda_{\\theta\\phi}$ in\n\\begin{equation}\n\\frac{\\mathrm{d}^2N}{\\mathrm{d}\\cos\\theta^{\\star}\\mathrm{d}\\phi^{\\star}} \\propto 1 + \\lambda_{\\theta} \\cos^2\\theta^{\\star} + \\lambda_{\\phi} \\sin^2\\theta^{\\star}\\cos2\\phi^{\\star} + \\lambda_{\\theta\\phi} \\sin2\\theta^{\\star}\\cos\\phi^{\\star}\n\\label{equ:acc}\n\\end{equation}\n\\noindent are related to the spin-density matrix elements of the dimuon spin wave function. \n\nSince the polarization of the $\\psi$ state may affect acceptance, seven extreme cases that lead to the largest possible variations of \nacceptance within the phase space of this measurement are identified.\nThese cases, described in Table~\\ref{tab:spin}, are used to define a range in which the results may vary under any physically allowed spin-alignment assumptions.\nThe same technique has also been used in other measurements~\\cite{Aad:2014fpa,ATLAS:2014ala,Aad2012dlq}.\nThis analysis adopts the isotropic distribution in both $\\cos\\theta^{\\star}$ and $\\phi^{\\star}$ as nominal,\nand the variation of the results for a number of extreme spin-alignment scenarios is studied and presented as sets of correction factors, detailed further in Appendix~\\ref{sec:spincorrection}.\n \n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Values of angular coefficients describing the considered spin-alignment scenarios.}\n\\vspace{2mm}\n\\begin{tabular}[h]{r|ccc}\n\\hline\\hline\n & \\multicolumn{3}{c}{Angular coefficients} \\\\ \n & $\\lambda_{\\theta}$ & $\\lambda_{\\phi}$ & $\\lambda_{\\theta\\phi}$ \\\\ \\hline\nIsotropic {\\em (central value)} & $0$ & $0$ & $0$ \\\\\nLongitudinal & $-1$ & $0$ & $0$ \\\\\nTransverse positive & $+1$ & $+1$ & $0$ \\\\\nTransverse zero & $+1$ & $0$ & $0$ \\\\\nTransverse negative & $+1$ & $-1$ & $0$ \\\\\nOff-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive & $0$ & $0$ & $+0.5$ \\\\\nOff-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative & $0$ & $0$ & $-0.5$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:spin}\n\\end{center}\n\\end{table}\n\n\n \nFor each of the two mass-points (corresponding to the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ masses), two-dimensional maps are produced\nas a function of dimuon $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and $|y(\\mu\\mu)|$ for the set of spin-alignment hypotheses. \nEach point on the map is determined from a uniform sampling over~$\\phi^{\\star}$ and~$\\cos\\theta^{\\star}$, \naccepting those trials that pass the fiducial selections. To account for various spin-alignment scenarios, all trials are weighted according to Eq.~\\ref{equ:acc}.\nAcceptance maps are defined within the range $8 < \\mbox{$p_{\\text{T}}$}(\\mu\\mu) < 110$~\\GeV and $|y(\\mu\\mu)| < 2.0$, corresponding to the data considered in the analysis. \nThe map is defined by 100 slices in $|y(\\mu\\mu)|$ and 4400 in $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, using 200k trials for each point, resulting in sufficiently high precision that the statistical uncertainty can be neglected.\nDue to the contributions of background, and the detector resolution of the signal, the acceptance for each candidate is determined from a linear interpolation of the two maps, which are generated for the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ known masses, as a function of the reconstructed mass $m(\\mu\\mu)$.\n\nFigure~\\ref{fig:accMapUnpolBoth} shows the acceptance, projected in $\\mbox{$p_{\\text{T}}$}$ for all the spin-alignment hypotheses for the \\jpsi\\ meson.\nThe differences between the acceptance of the $\\ensuremath{\\psi(2\\mathrm{S})}$ and $\\jpsi$ meson, are independent of rapidity, except near $|y|\\approx2$\nat low $\\mbox{$p_{\\text{T}}$}$. Similarly, the only dependence on $\\mbox{$p_{\\text{T}}$}$ is found below $\\mbox{$p_{\\text{T}}$}\\approx9$~\\GeV.\nThe correction factors (as given in Appendix.~\\ref{sec:spincorrection}) vary most at low $\\mbox{$p_{\\text{T}}$}$, ranging from $-35\\%$ under longitudinal, to $+100\\%$ for transverse-positive scenarios.\nAt high $\\mbox{$p_{\\text{T}}$}$, the range is between $-14\\%$ for longitudinal, and $+9\\%$ for transverse-positive scenarios.\nFor the fraction and ratio measurements, the correction factor is determined from the appropriate ratio of the individual correction factors. \n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=0.8\\textwidth]{c_JpsiAcc_All1D_logx.eps}\n \\caption{Projections of the acceptance as a function of $\\mbox{$p_{\\text{T}}$}$ for the \\jpsi\\ meson for various spin-alignment hypotheses.}\n \\label{fig:accMapUnpolBoth}\n \\end{center}\n\\end{figure}\n\n\n\n\\subsection{Muon reconstruction and trigger efficiency determination}\n\n\\noindent The technique for correcting the 7~\\TeV\\ data for trigger and reconstruction inefficiencies is described in detail in Ref.~\\cite{Aad:2014fpa,Aad2012dlq}.\nFor the 8~\\TeV\\ data, a similar technique is used, however different efficiency maps are required for each set of data, and the 8~\\TeV\\ corrections are detailed briefly below.\n\nThe single-muon reconstruction efficiency is determined from a tag-and-probe study in dimuon decays~\\cite{Aad:2014kba}.\nThe efficiency map is calculated as a function of $\\mbox{$p_{\\text{T}}$} (\\mu)$ and $q\\times \\eta(\\mu)$, where $q=\\pm1$ is the electrical charge of the muon, expressed in units of $e$.\n\nThe trigger efficiency correction consists of two components. The first part represents the trigger efficiency for a \nsingle muon in intervals of $\\mbox{$p_{\\text{T}}$} (\\mu)$ and $q\\times \\eta(\\mu)$.\nFor the dimuon system there is a second correction to account for reductions in efficiency due to closely spaced\nmuons firing only a single RoI, \nvertex-quality cuts, and opposite-sign requirements.\nThis correction is performed in three rapidity intervals: 0--1.0, 1.0--1.2 and 1.2--2.3. The correction is a function of $\\Delta R(\\mu\\mu)$ \nin the first two rapidity intervals and a function of $\\Delta R(\\mu\\mu)$ and $|y(\\mu\\mu)|$ in the last interval.\n\nThe combination of the two components (single-muon efficiency map and dimuon corrections) is illustrated \nin Figure~\\ref{fig:EFmu4DataJpsiTandPCmumupolyL2StarBMaxpt50} by plotting the average trigger-weight correction for the events in this analysis\nin terms of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and $|y(\\mu\\mu)|$.\nThe increased weight at low $\\mbox{$p_{\\text{T}}$}$ and $|y|\\approx 1.25$ is caused by the geometrical acceptance of the muon trigger system and the turn-on threshold behaviour of the muon trigger. At high $\\mbox{$p_{\\text{T}}$}$ the weight is increased due to the reduced opening angle between the two muons.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[scale=0.6]{averageTrigWeightMean.eps}\n \\caption{Average dimuon trigger-weight in the intervals of $\\mbox{$p_{\\text{T}}$} (\\mu\\mu) $ and $ |y(\\mu\\mu)|$ studied in this set of measurements. \n }\n \\label{fig:EFmu4DataJpsiTandPCmumupolyL2StarBMaxpt50}\n \\end{center}\n\\end{figure}\n\n\n\\subsection{Fitting technique}\n\\label{sec:method:fit}\nTo extract the corrected yields of prompt and non-prompt \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons, two-dimensional\nweighted unbinned maximum-likelihood fits are performed on the dimuon invariant mass, $m(\\mu\\mu)$,\nand pseudo-proper decay time, $\\tau(\\mu\\mu)$, in intervals of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and~$|y(\\mu\\mu)|$.\nEach interval is fitted independently from all the others.\nIn $m(\\mu\\mu)$, signal processes of $\\ensuremath{\\psi}$ meson decays are statistically distinguished as narrow peaks convolved with the detector resolution, at their respective mass positions, on top of background continuum.\nIn $\\tau(\\mu\\mu)$, decays originating with zero pseudo-proper decay time and those following an exponential decay distribution (both convolved with a detector resolution function) statistically distinguish prompt and non-prompt signal processes, respectively.\nVarious sources of background processes include Drell-Yan processes, mis-reconstructed muon pairs from prompt and non-prompt sources,\nand semileptonic decays from separate $b$-hadrons. \n\nThe probability density function (PDF) for each fit is defined as a normalized sum, \nwhere each term represents a specific signal or background contribution, with a physically motivated mass and $\\tau$ dependence.\nThe PDF can be written in a compact form as\n\\begin{equation}\n\\label{eqn:pdf}\n\\mathrm{PDF}(m,\\tau) = \\sum_{i=1}^{7} \\kappa_i f_i(m) \\ \\cdot h_i(\\tau) \\otimes R(\\tau),\n\\end{equation}\nwhere $\\kappa_i$ represents the relative normalization of the $i^\\mathrm{th}$ term of the seven considered signal and background contributions (such that $\\sum_i \\kappa_i = 1$), \n$f_i(m)$ is the mass-dependent term, and $\\otimes$ represents the convolution of \nthe $\\tau$-dependent function $h_i(\\tau)$ with the $\\tau$ resolution term, $R(\\tau)$. \nThe latter is modelled by a double Gaussian distribution with both means fixed to zero and widths determined from the fit.\n\nTable \\ref{table:fitModel} lists the contributions to the overall PDF with the corresponding $f_i$ and\n$h_i$ functions. Here $G_1$ and $G_2$ are Gaussian functions, $B_1$ and $B_2$ are Crystal Ball\\footnote{The Crystal Ball function is given by:\n\\\\$B(x;\\alpha,n,\\bar x,\\sigma) = N \\cdot \\begin{cases} \\exp\\left(- \\frac{(x - \\bar x)^2}{2 \\sigma^2}\\right), & \\mbox{for }\\frac{x - \\bar x}{\\sigma} > -\\alpha \\\\\n A \\cdot \\left(A' - \\frac{x - \\bar x}{\\sigma}\\right)^{-n}, & \\mbox{for }\\frac{x - \\bar x}{\\sigma} \\leqslant -\\alpha \\end{cases}$ \\\\where $A = \\left(\\frac{n}{\\left| \\alpha \\right|}\\right)^n \\cdot \\exp\\left(- \\frac {\\left| \\alpha \\right|^2}{2}\\right),\nA' = \\frac{n}{\\left| \\alpha \\right|} - \\left| \\alpha \\right|$}\ndistributions~\\cite{CB1}, \nwhile F is a uniform distribution and $C_1$ a first-order Chebyshev polynomial. \nThe exponential functions $E_1$, $E_2$, $E_3$, $E_4$ and $E_5$\nhave different decay constants, where $E_5(|\\tau|)$ is a double-sided exponential with the same decay constant on\neither side of $\\tau = 0$. The parameter $\\omega$ represents the fractional contribution \nof the $B$ and $G$ mass signal functions, while the Dirac delta function, $\\delta(\\tau)$,\nis used to represent the pseudo-proper decay time distribution of the prompt candidates.\n\n\\begin{table}[h!]\n \\centering\n \\caption{Description of the fit model PDF in Eq.~\\ref{eqn:pdf}. Components of the probability density function used to extract the prompt (P) and\nnon-prompt (NP) contributions for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ signal and the P, NP, and incoherent or mis-reconstructed background (Bkg) contributions.}\n \\begin{tabular}{ l c c c c }\n \\hline \\hline\n $i$ & Type & Source & $f_i(m)$ & $h_i(\\tau)$ \\\\ \\hline \\hline\n 1 & \\jpsi\\ & P & $\\omega B_1(m) + (1-\\omega) G_1(m)$ & $\\delta (\\tau)$ \\\\\n 2 & \\jpsi\\ & NP & $\\omega B_1(m) + (1-\\omega) G_1(m)$ & $ E_1(\\tau)$ \\\\\n 3 & \\ensuremath{\\psi(2\\mathrm{S})}\\ & P & $\\omega B_2(m) + (1-\\omega) G_2(m)$ & $\\delta (\\tau)$ \\\\\n 4 & \\ensuremath{\\psi(2\\mathrm{S})}\\ & NP & $\\omega B_2(m) + (1-\\omega) G_2(m)$ & $E_2(\\tau)$ \\\\ \\hline\n 5 & Bkg & P & $ F$ & $\\delta (\\tau)$ \\\\\n 6 & Bkg & NP & $ C_1(m)$ & $E_3 (\\tau)$ \\\\ \n 7 & Bkg & NP & $ E_4(m)$ & $E_5 (|\\tau|)$ \\\\\n \\hline \n \\end{tabular}\n \\label{table:fitModel}\n\\end{table}\n\nIn order to make the fitting procedure more robust and to reduce the number of free parameters, a number of component terms share common parameters, \nwhich led to 22 free parameters per interval.\nIn detail, the signal mass models are described by the sum of a Crystal Ball shape ($B$) and a Gaussian shape ($G$). For each\nof \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}, the $B$ and $G$ share a common mean, and freely determined widths, \nwith the ratio of the $B$ and $G$ widths common to \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}.\nThe $B$ parameters $\\alpha$, and $n$, describing the transition point of the low-edge from a Gaussian to a power-law \nshape, and the shape of the tail, respectively, are fixed, and variations are considered as part of the fit model\nsystematic uncertainties.\nThe width of $G$ for \\ensuremath{\\psi(2\\mathrm{S})}\\ is set to the width for \\jpsi\\ multiplied by a free parameter scaling term.\nThe relative fraction of $B$ and $G$ is left floating, but common to \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}.\n\nThe non-prompt signal decay shapes ($E_1$,$E_2$) are described by an exponential function (for positive $\\tau$ only) convolved with a\ndouble Gaussian function, $R(\\tau)$ describing the pseudo-proper decay time resolution for the non-prompt component, \nand the same Gaussian response functions to\ndescribe the prompt contributions. \nEach Gaussian resolution component has its mean fixed at $\\tau$ = 0 and a free width.\nThe decay constants of the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ are separate free parameters in the fit.\n\nThe background contributions are described by a prompt and non-prompt component, as well as a double-sided exponential function\nconvolved with a double Gaussian function describing mis-reconstructed or non-coherent muon pairs. \nThe same resolution function as in signal is used to describe the background. For the non-resonant mass\nparameterizations, the non-prompt contribution is modelled by a first-order Chebyshev polynomial. The prompt mass\ncontribution follows a flat distribution and the double-sided background uses an exponential function.\nVariations of this fit model are considered as systematic uncertainties.\n\nThe following quantities are extracted directly from the fit in each interval:\nthe fraction of events that are signal (prompt or non-prompt \\jpsi\\ or \\ensuremath{\\psi(2\\mathrm{S})}); the fraction of signal events that are prompt;\nthe fraction of prompt signal that is \\ensuremath{\\psi(2\\mathrm{S})}; and the fraction of non-prompt signal that is \\ensuremath{\\psi(2\\mathrm{S})}. From these\nparameters, and the weighted sum of events, all measured values are calculated.\n\n\nFor $7$~\\TeV\\ data, 168 fits are performed across the range of $8<\\mbox{$p_{\\text{T}}$}<100$~\\GeV\\ ($8<\\mbox{$p_{\\text{T}}$}<60$~\\GeV) for $\\jpsi$ ($\\ensuremath{\\psi(2\\mathrm{S})}$) and $0<|y|<2$.\nFor $8$~\\TeV\\ data, 172 fits are performed across the range of $8<\\mbox{$p_{\\text{T}}$}<110$~\\GeV\\ and $0<|y|<2$,\nexcluding the area where $\\mbox{$p_{\\text{T}}$}$ is less than 10 \\GeV\\ and simultaneously $|y|$ is\ngreater than 0.75. This region is excluded due to a steeply changing low trigger efficiency \ncausing large systematic uncertainties in the measured cross-section.\n\nFigure~\\ref{fig:fitprojmain} shows the fit results for one of the intervals considered in the analysis, projected onto the invariant mass and pseudo-proper decay time distributions, for $7$~\\TeV\\ data, weighted according to the acceptance and efficiency corrections.\nThe fit projections are shown for the total prompt and total non-prompt contributions (shown as curves), and also for the individual contributions of the $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ prompt and non-prompt signal yields (shown as hashed areas of various types). \n\nIn Figure~\\ref{fig:fitprojmainVIII} the fit results are shown for one high-$\\mbox{$p_{\\text{T}}$}$ interval of $8$~\\TeV\\ data.\n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.51\\textwidth]{figures\/c_c_Mass_5_5.eps}\n \\hspace{-0.55cm}\n \\includegraphics[width=0.51\\textwidth]{figures\/c_c_Tau_5_5.eps}\n \\caption{Projections of the fit result over the mass (left) and pseudo-proper decay time (right) distributions for data collected at $7$~\\TeV\\ for one typical interval. The data are shown with error bars in black, superimposed with the individual components of the fit result projections, where the total prompt and non-prompt components are represented by the dashed and dotted lines, respectively, and the shaded areas show the signal $\\psi$ prompt and non-prompt contributions.\\label{fig:fitprojmain}}\n \\end{center}\n\\end{figure} \n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.51\\textwidth]{figures\/c_8TeV_c_Mass_23_0.eps}\n \\hspace{-0.55cm}\n \\includegraphics[width=0.51\\textwidth]{figures\/c_8TeV_c_Tau_23_0.eps}\n \\caption{Projections of the fit result over the mass (left) and pseudo-proper decay time (right) distributions for data collected at $8$~\\TeV\\ for one high-$\\mbox{$p_{\\text{T}}$}$ interval. The data are shown with error bars in black, superimposed with the individual components of the fit result projections, where the total prompt and non-prompt components are represented by the dashed and dotted lines, respectively, and the shaded areas show the signal $\\psi$ prompt and non-prompt contributions.\\label{fig:fitprojmainVIII}}\n \\end{center}\n\\end{figure} \n\n\n\\subsection{Bin migration corrections}\nTo account for bin migration effects due to the detector resolution, \nwhich results in decays of~$\\ensuremath{\\psi}$ in one bin, being identified and accounted for in another,\nthe numbers of acceptance- and efficiency-corrected dimuon decays extracted from\nthe fits in each interval of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and rapidity are corrected for the differences\nbetween the true and reconstructed values of the dimuon $\\mbox{$p_{\\text{T}}$}$.\nThese corrections are derived from data by comparing analytic functions that are \nfitted to the $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ spectra of dimuon events with and without\nconvolution by the experimental resolution in $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ (as determined from the fitted mass resolution and measured muon angular resolutions), as described in Ref.~\\cite{Aad2012dlq}.\n\nThe correction factors applied to the fitted yields deviate from unity by no more than $1.5\\%$, and for the majority of slices are smaller than $1\\%$.\nThe ratio measurement and non-prompt fractions are corrected by the corresponding ratios of\nbin migration correction factors.\nUsing a similar technique, bin migration corrections as a function of $|y|$ are found \nto differ from unity by negligible amounts.\n\n\n\n\\section*{Acknowledgements}\n\\input{Acknowledgements}\n\n\\clearpage\n\\printbibliography\n\\clearpage\n\n\\section{Results}\n\\label{sec:results}\n\n\nThe $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ non-prompt and prompt production cross-sections are presented, corrected for acceptance and detector efficiencies \nwhile assuming isotropic decay, as described in Section~\\ref{sec:s:diff_xSecDet}. \nAlso presented are the ratios of non-prompt production relative to the inclusive production for $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ mesons separately, described in Section~\\ref{sec:s:NPFDet},\nand the ratio of $\\ensuremath{\\psi(2\\mathrm{S})}$ to $\\jpsi$ production for prompt and non-prompt components separately, described in Section~\\ref{sec:s:PNPRatioDet}.\nCorrection factors for various spin-alignment hypotheses for both 7 and 8~\\TeV\\ data can be found in \nTables~\\ref{tab:sa_long_jpsi}--\\ref{tab:sa_offN_psi2s} (in Appendix) and Tables~\\ref{tab:sa_long_jpsi8}--\\ref{tab:sa_offN_psi2s8} (in Appendix) respectively, in terms of \n\\mbox{$p_{\\text{T}}$}\\ and rapidity intervals.\n\n\\setdescription{font=\\normalfont\\itshape}\n\n\\begin{description}[style=unboxed,leftmargin=0cm]\n\n\n\\item[Production cross-sections] \\hfill\n\nFigures \\ref{fig:res:xSecP} and \\ref{fig:res:xSecNP} show respectively the prompt and non-prompt \ndifferential cross-sections of \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ as functions of $\\mbox{$p_{\\text{T}}$}$ and $|y|$, together with the relevant theoretical predictions, \nwhich are described below.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiP_xSec.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SP_xSec.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SP_xSec.eps}\\hfil\n \\caption{The differential prompt cross-section times dimuon branching fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional scaling factor of 10 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers \n the statistical and systematic \n uncertainty (with the same multiplicative scaling applied). \n The NLO NRQCD theory predictions are also shown.}\n \\label{fig:res:xSecP}\n \\end{center}\n\\end{figure} \n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiNP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SNP_xSec.eps}\\hfil\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiNP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SNP_xSec.eps}\\hfil\n \\caption{The differential non-prompt cross-section times dimuon branching fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional scaling factor of 10 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical\n and systematic uncertainty (with the same multiplicative scaling applied).\n The FONLL theory predictions are also shown.}\n \\label{fig:res:xSecNP}\n \\end{center}\n\\end{figure} \n\n\n\\item[Non-prompt production fractions] \\hfill\n\nThe results for the fractions of non-prompt production relative to the inclusive production of \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\, are presented as a function of $\\mbox{$p_{\\text{T}}$}$ for slices of rapidity in Figure~\\ref{fig:res:NPF}. \nIn each rapidity slice, the non-prompt fraction is seen to increase as a function of $\\mbox{$p_{\\text{T}}$}$ and has no strong dependence on either rapidity or centre-of-mass energy.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NPF_Jpsi.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NPF_Psi.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NPF_Jpsi.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NPF_Psi.eps}\\hfil\n \\caption{The non-prompt fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right), as a function of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional factor of 0.2 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical\n and systematic uncertainty (with the same multiplicative scaling applied).}\n \\label{fig:res:NPF}\n \\end{center}\n\\end{figure} \n\n\n\\textit{Production ratios of \\textmd{\\ensuremath{\\psi(2\\mathrm{S})}} to \\textmd{\\jpsi} }\n\nFigure~\\ref{fig:res:PNP_Ratio} shows the ratios of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ decaying to a muon pair in prompt and non-prompt processes,\n presented as a function of $\\mbox{$p_{\\text{T}}$}$ for slices of rapidity. The non-prompt ratio is shown to be relatively flat across the considered range of $\\pT$,\nfor each slice of rapidity.\nFor the prompt ratio, a slight increase as a function of $\\mbox{$p_{\\text{T}}$}$ is observed, with no strong dependence on rapidity or centre-of-mass energy.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_P_Ratio.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NP_Ratio.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_P_Ratio.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NP_Ratio.eps}\\hfil\n \\caption{The ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production times dimuon branching fraction for prompt (left) and non-prompt (right) processes\n as a function of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each of the slices of rapidity. \n For each increasing rapidity slice, an additional factor of 0.1 is applied to the plotted points for visual clarity.\n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical and systematic uncertainty.}\n \\label{fig:res:PNP_Ratio}\n \\end{center}\n\\end{figure} \n\n\\clearpage\n\n\n\\item[Comparison with theory] \\hfill\n\nFor prompt production, as shown in Figure~\\ref{fig:xsecPtheoryRatio}, the ratio of the NLO NRQCD theory calculations~\\cite{NRQCD1} \nto data, as a function of~$\\mbox{$p_{\\text{T}}$}$ and in slices of rapidity, is provided for \\Jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ at both the 7 and 8~\\TeV\\ centre-of-mass energies.\nThe theory predictions are \nbased on the long-distance matrix elements (LDMEs) from Refs.~\\cite{NRQCD1,Ma:2010vd}, with uncertainties originating from the\nchoice of scale, charm quark mass and LDMEs (see Refs.~\\cite{NRQCD1,Ma:2010vd} for more details).\nFigure~\\ref{fig:xsecPtheoryRatio} shows fair agreement between the theoretical calculation and the data points for the whole $\\mbox{$p_{\\text{T}}$}$ range. \nThe ratio of theory to data does not depend on rapidity.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiP_theoryRatio_lin.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SP_theoryRatio_lin.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiP_theoryRatio_lin.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SP_theoryRatio_lin.eps}\\hfil\n \\caption{The ratios of the NRQCD theoretical predictions to data are presented for the differential prompt cross-section of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each rapidity slice. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The error on the data is the relative error of each data point, while the error bars on the theory prediction are the relative error of each theory point.}\n \\label{fig:xsecPtheoryRatio}\n \\end{center}\n\\end{figure} \n\nFor non-prompt $\\psi$ production, comparisons are made to FONLL theoretical predictions~\\cite{FONLL_2001,Cacciari:2012ny}, which describe the production of \n$b$-hadrons followed by their decay into $\\psi+X$.\nFigure~\\ref{fig:xsecNPtheoryRatio} shows the ratios of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ FONLL predictions to data, as a function of $\\mbox{$p_{\\text{T}}$}$ and in slices of rapidity, \nfor centre-of-mass energies of~7 and 8~\\TeV.\nFor $\\jpsi$, agreement is generally good, but the theory predicts slightly harder $\\mbox{$p_{\\text{T}}$}$ spectra than observed in the data.\nFor $\\ensuremath{\\psi(2\\mathrm{S})}$, the shapes of data and theory appear to be in satisfactory agreement, but the theory predicts higher yields than in the data.\nThere is no observed dependence on rapidity in the comparisons between theory and data for non-prompt \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ production.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiNP_theoryRatio_lin.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SNP_theoryRatio_lin.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiNP_theoryRatio_lin.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SNP_theoryRatio_lin.eps}\\hfil\n \\caption{The ratio of the FONLL theoretical predictions to data are presented for the differential non-prompt \n cross-section of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each rapidity slice. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The error on the data is the relative error of each data point, while the error bars on the theory prediction are the relative error of each theory point.}\n \\label{fig:xsecNPtheoryRatio}\n \\end{center}\n\\end{figure} \n\n\n\\item[Comparison of cross-sections 8~\\TeV\\ with 7~\\TeV\\ ] \\hfill\n\nIt is interesting to compare the cross-section results between the two centre-of-mass energies, \nboth for data and the theoretical predictions.\n\nFigure~\\ref{fig:xsecEnergyRatio} shows the 8~\\TeV\\ to 7~\\TeV\\ cross-section ratios of prompt and non-prompt $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ for both data sets.\nFor the theoretical ratios the uncertainties are neglected here, since the high correlation between them results in large cancellations.\n\n\nDue to a finer granularity in $\\mbox{$p_{\\text{T}}$}$ for the 8~\\TeV\\ data, a weighted average of the 8~\\TeV\\ results is taken across equivalent intervals of the 7~\\TeV\\ data to enable direct comparisons.\nBoth data and theoretical predictions agree \nthat the ratios become larger with increasing $\\mbox{$p_{\\text{T}}$}$,\nhowever at the lower edge of the $\\pT$ range the data tends to be slightly below theory.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecP_J.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecP_P.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecNP_J.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecNP_P.eps}\\hfil\n \\caption{The ratio of the 8~\\TeV\\ and 7~\\TeV\\ differential cross-sections\n\t are presented for prompt~(top) and non-prompt~(bottom) $\\jpsi$~(left) and \n\t $\\ensuremath{\\psi(2\\mathrm{S})}$~(right) for both data~(red points with error bars) and theoretical predictions~(green points).\n\t The theoretical predictions used are NRQCD for prompt and FONLL for non-prompt production.\n\t The uncertainty on the data ratio does not account for possible correlations between 7 and 8 \\TeV\\ data, and no uncertainty is shown \n\t for the ratio of theory predictions.\n}\n \\label{fig:xsecEnergyRatio}\n \\end{center}\n\\end{figure} \n\n\n\\end{description}\n\n\\section{Spin-alignment correction factors}\n\\label{sec:spincorrection}\n\nThe measurement presented here assumes an unpolarized spin-alignment hypothesis for determining the correction factor. In principle, the polarization may be non-zero and may vary with $\\mbox{$p_{\\text{T}}$}$.\nIn order to correct these measurements when well-measured $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ polarizations are determined, a set of correction factors are provided \nin Tables~\\ref{tab:sa_long_jpsi}--\\ref{tab:sa_offN_psi2s} for the 7~\\TeV\\ data, and in the Tables~\\ref{tab:sa_long_jpsi8}--\\ref{tab:sa_offN_psi2s8} for the 8~\\TeV\\ data.\nThese tables are created by altering the spin-alignment hypothesis for either the $\\jpsi$ or $\\ensuremath{\\psi(2\\mathrm{S})}$ meson and then determining the ratio of the mean sum-of-weights of the new hypotheses to the original flat hypothesis.\nThe mean weight is calculated from all the events in each dimuon $\\mbox{$p_{\\text{T}}$}$ and rapidity analysis bin, selecting those dimuons within $\\pm2\\sigma$ of the $\\ensuremath{\\psi}$ fitted mean mass position.\nThe choice of spin-alignment hypothesis for each $\\ensuremath{\\psi}$ meson has negligible effect on the results of the other $\\ensuremath{\\psi}$ meson, and therefore these possible permutations are not considered.\nThe definitions of each of the spin-alignment scenarios, which are given in the caption to the table, are defined in Table~\\ref{tab:spin}.\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``longitudinal'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.666 & 0.672 & 0.674 & 0.680 & 0.688 & 0.690 & 0.690 & 0.690 \\\\\n$8.50$--$9.00$ & 0.670 & 0.674 & 0.678 & 0.685 & 0.689 & 0.694 & 0.694 & 0.698 \\\\\n$9.00$--$9.50$ & 0.673 & 0.676 & 0.680 & 0.687 & 0.693 & 0.697 & 0.698 & 0.700 \\\\\n$9.50$--$10.00$ & 0.675 & 0.678 & 0.683 & 0.689 & 0.694 & 0.697 & 0.701 & 0.703 \\\\\n$10.00$--$10.50$ & 0.679 & 0.681 & 0.687 & 0.692 & 0.697 & 0.699 & 0.702 & 0.706 \\\\\n$10.50$--$11.00$ & 0.682 & 0.686 & 0.691 & 0.696 & 0.700 & 0.702 & 0.704 & 0.708 \\\\\n$11.00$--$11.50$ & 0.688 & 0.689 & 0.694 & 0.699 & 0.701 & 0.705 & 0.708 & 0.710 \\\\\n$11.50$--$12.00$ & 0.692 & 0.695 & 0.698 & 0.702 & 0.706 & 0.708 & 0.710 & 0.712 \\\\\n$12.00$--$13.00$ & 0.698 & 0.700 & 0.703 & 0.707 & 0.711 & 0.713 & 0.715 & 0.717 \\\\\n$13.00$--$14.00$ & 0.707 & 0.709 & 0.711 & 0.714 & 0.717 & 0.720 & 0.721 & 0.723 \\\\\n$14.00$--$15.00$ & 0.716 & 0.717 & 0.720 & 0.722 & 0.725 & 0.727 & 0.728 & 0.730 \\\\\n$15.00$--$16.00$ & 0.724 & 0.726 & 0.728 & 0.729 & 0.732 & 0.734 & 0.735 & 0.737 \\\\\n$16.00$--$17.00$ & 0.733 & 0.733 & 0.735 & 0.737 & 0.739 & 0.741 & 0.742 & 0.744 \\\\\n$17.00$--$18.00$ & 0.740 & 0.741 & 0.743 & 0.744 & 0.746 & 0.747 & 0.749 & 0.750 \\\\\n$18.00$--$20.00$ & 0.751 & 0.752 & 0.753 & 0.754 & 0.756 & 0.758 & 0.758 & 0.760 \\\\\n$20.00$--$22.00$ & 0.765 & 0.765 & 0.766 & 0.767 & 0.769 & 0.770 & 0.771 & 0.772 \\\\\n$22.00$--$24.00$ & 0.777 & 0.777 & 0.778 & 0.780 & 0.781 & 0.781 & 0.782 & 0.783 \\\\\n$24.00$--$26.00$ & 0.789 & 0.789 & 0.790 & 0.790 & 0.791 & 0.792 & 0.793 & 0.794 \\\\\n$26.00$--$30.00$ & 0.803 & 0.803 & 0.804 & 0.804 & 0.805 & 0.806 & 0.806 & 0.807 \\\\\n$30.00$--$40.00$ & 0.827 & 0.827 & 0.828 & 0.828 & 0.829 & 0.829 & 0.830 & 0.831 \\\\\n$40.00$--$60.00$ & 0.863 & 0.863 & 0.864 & 0.864 & 0.864 & 0.865 & 0.865 & 0.866 \\\\\n$60.00$--$100.00$ & 0.902 & 0.904 & 0.904 & 0.903 & 0.904 & 0.904 & 0.902 & 0.906 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse zero'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.336 & 1.324 & 1.315 & 1.309 & 1.299 & 1.297 & 1.296 & 1.298 \\\\\n$8.50$--$9.00$ & 1.329 & 1.323 & 1.310 & 1.300 & 1.291 & 1.284 & 1.280 & 1.284 \\\\\n$9.00$--$9.50$ & 1.326 & 1.315 & 1.303 & 1.295 & 1.289 & 1.281 & 1.279 & 1.276 \\\\\n$9.50$--$10.00$ & 1.317 & 1.311 & 1.300 & 1.289 & 1.284 & 1.276 & 1.276 & 1.272 \\\\\n$10.00$--$10.50$ & 1.310 & 1.304 & 1.297 & 1.290 & 1.280 & 1.276 & 1.273 & 1.269 \\\\\n$10.50$--$11.00$ & 1.302 & 1.298 & 1.291 & 1.285 & 1.276 & 1.271 & 1.267 & 1.268 \\\\\n$11.00$--$11.50$ & 1.296 & 1.290 & 1.284 & 1.278 & 1.271 & 1.266 & 1.263 & 1.261 \\\\\n$11.50$--$12.00$ & 1.288 & 1.284 & 1.277 & 1.274 & 1.265 & 1.261 & 1.260 & 1.257 \\\\\n$12.00$--$13.00$ & 1.276 & 1.273 & 1.268 & 1.263 & 1.257 & 1.255 & 1.251 & 1.250 \\\\\n$13.00$--$14.00$ & 1.263 & 1.260 & 1.254 & 1.250 & 1.247 & 1.244 & 1.243 & 1.240 \\\\\n$14.00$--$15.00$ & 1.248 & 1.246 & 1.244 & 1.240 & 1.236 & 1.233 & 1.233 & 1.230 \\\\\n$15.00$--$16.00$ & 1.237 & 1.233 & 1.231 & 1.228 & 1.225 & 1.223 & 1.223 & 1.221 \\\\\n$16.00$--$17.00$ & 1.224 & 1.222 & 1.221 & 1.219 & 1.216 & 1.213 & 1.212 & 1.212 \\\\\n$17.00$--$18.00$ & 1.213 & 1.213 & 1.211 & 1.208 & 1.205 & 1.204 & 1.204 & 1.203 \\\\\n$18.00$--$20.00$ & 1.200 & 1.198 & 1.197 & 1.196 & 1.194 & 1.192 & 1.192 & 1.190 \\\\\n$20.00$--$22.00$ & 1.183 & 1.182 & 1.180 & 1.180 & 1.178 & 1.177 & 1.176 & 1.175 \\\\\n$22.00$--$24.00$ & 1.168 & 1.167 & 1.166 & 1.165 & 1.164 & 1.164 & 1.163 & 1.163 \\\\\n$24.00$--$26.00$ & 1.155 & 1.155 & 1.154 & 1.154 & 1.153 & 1.152 & 1.151 & 1.150 \\\\\n$26.00$--$30.00$ & 1.140 & 1.140 & 1.140 & 1.139 & 1.138 & 1.138 & 1.138 & 1.137 \\\\\n$30.00$--$40.00$ & 1.117 & 1.117 & 1.117 & 1.116 & 1.116 & 1.116 & 1.115 & 1.115 \\\\\n$40.00$--$60.00$ & 1.087 & 1.087 & 1.086 & 1.086 & 1.086 & 1.085 & 1.085 & 1.085 \\\\\n$60.00$--$100.00$ & 1.057 & 1.056 & 1.057 & 1.057 & 1.056 & 1.056 & 1.057 & 1.055 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse positive'' spin-alignment transverse positive hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.693 & 1.694 & 1.700 & 1.711 & 1.727 & 1.720 & 1.720 & 1.747 \\\\\n$8.50$--$9.00$ & 1.561 & 1.564 & 1.564 & 1.568 & 1.568 & 1.568 & 1.571 & 1.673 \\\\\n$9.00$--$9.50$ & 1.468 & 1.468 & 1.465 & 1.466 & 1.470 & 1.466 & 1.471 & 1.519 \\\\\n$9.50$--$10.00$ & 1.418 & 1.416 & 1.417 & 1.417 & 1.421 & 1.417 & 1.423 & 1.453 \\\\\n$10.00$--$10.50$ & 1.383 & 1.383 & 1.387 & 1.389 & 1.390 & 1.389 & 1.391 & 1.406 \\\\\n$10.50$--$11.00$ & 1.360 & 1.362 & 1.365 & 1.364 & 1.364 & 1.362 & 1.362 & 1.380 \\\\\n$11.00$--$11.50$ & 1.344 & 1.342 & 1.342 & 1.344 & 1.344 & 1.344 & 1.346 & 1.355 \\\\\n$11.50$--$12.00$ & 1.326 & 1.326 & 1.327 & 1.329 & 1.327 & 1.327 & 1.329 & 1.334 \\\\\n$12.00$--$13.00$ & 1.307 & 1.308 & 1.307 & 1.308 & 1.308 & 1.308 & 1.308 & 1.312 \\\\\n$13.00$--$14.00$ & 1.285 & 1.287 & 1.285 & 1.285 & 1.285 & 1.286 & 1.285 & 1.288 \\\\\n$14.00$--$15.00$ & 1.266 & 1.266 & 1.267 & 1.267 & 1.266 & 1.266 & 1.266 & 1.268 \\\\\n$15.00$--$16.00$ & 1.250 & 1.249 & 1.250 & 1.251 & 1.250 & 1.249 & 1.250 & 1.250 \\\\\n$16.00$--$17.00$ & 1.234 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 \\\\\n$17.00$--$18.00$ & 1.222 & 1.223 & 1.223 & 1.223 & 1.222 & 1.222 & 1.223 & 1.222 \\\\\n$18.00$--$20.00$ & 1.206 & 1.206 & 1.206 & 1.207 & 1.207 & 1.207 & 1.206 & 1.205 \\\\\n$20.00$--$22.00$ & 1.187 & 1.187 & 1.187 & 1.188 & 1.187 & 1.187 & 1.187 & 1.186 \\\\\n$22.00$--$24.00$ & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 \\\\\n$24.00$--$26.00$ & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.156 \\\\\n$26.00$--$30.00$ & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.141 \\\\\n$30.00$--$40.00$ & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.117 \\\\\n$40.00$--$60.00$ & 1.087 & 1.087 & 1.087 & 1.086 & 1.087 & 1.086 & 1.086 & 1.086 \\\\\n$60.00$--$100.00$ & 1.058 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.058 & 1.056 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.030 & 1.020 & 1.004 & 0.995 & 0.992 & 0.981 & 0.973 & 0.949 \\\\\n$8.50$--$9.00$ & 1.157 & 1.148 & 1.134 & 1.113 & 1.101 & 1.092 & 1.079 & 1.055 \\\\\n$9.00$--$9.50$ & 1.207 & 1.196 & 1.176 & 1.161 & 1.147 & 1.138 & 1.130 & 1.107 \\\\\n$9.50$--$10.00$ & 1.231 & 1.219 & 1.202 & 1.186 & 1.174 & 1.162 & 1.158 & 1.138 \\\\\n$10.00$--$10.50$ & 1.243 & 1.231 & 1.217 & 1.202 & 1.189 & 1.181 & 1.175 & 1.158 \\\\\n$10.50$--$11.00$ & 1.246 & 1.239 & 1.228 & 1.213 & 1.200 & 1.191 & 1.186 & 1.174 \\\\\n$11.00$--$11.50$ & 1.252 & 1.242 & 1.230 & 1.218 & 1.205 & 1.198 & 1.192 & 1.181 \\\\\n$11.50$--$12.00$ & 1.251 & 1.243 & 1.229 & 1.222 & 1.208 & 1.202 & 1.197 & 1.187 \\\\\n$12.00$--$13.00$ & 1.247 & 1.240 & 1.230 & 1.221 & 1.211 & 1.205 & 1.200 & 1.193 \\\\\n$13.00$--$14.00$ & 1.240 & 1.235 & 1.227 & 1.218 & 1.211 & 1.206 & 1.202 & 1.197 \\\\\n$14.00$--$15.00$ & 1.232 & 1.227 & 1.221 & 1.215 & 1.207 & 1.203 & 1.200 & 1.195 \\\\\n$15.00$--$16.00$ & 1.223 & 1.219 & 1.213 & 1.207 & 1.201 & 1.198 & 1.196 & 1.193 \\\\\n$16.00$--$17.00$ & 1.213 & 1.210 & 1.206 & 1.201 & 1.196 & 1.193 & 1.191 & 1.189 \\\\\n$17.00$--$18.00$ & 1.204 & 1.203 & 1.199 & 1.194 & 1.189 & 1.187 & 1.186 & 1.183 \\\\\n$18.00$--$20.00$ & 1.193 & 1.191 & 1.188 & 1.185 & 1.181 & 1.179 & 1.177 & 1.176 \\\\\n$20.00$--$22.00$ & 1.178 & 1.177 & 1.174 & 1.172 & 1.169 & 1.167 & 1.166 & 1.164 \\\\\n$22.00$--$24.00$ & 1.164 & 1.163 & 1.162 & 1.159 & 1.157 & 1.156 & 1.156 & 1.154 \\\\\n$24.00$--$26.00$ & 1.153 & 1.152 & 1.150 & 1.149 & 1.148 & 1.147 & 1.145 & 1.144 \\\\\n$26.00$--$30.00$ & 1.139 & 1.138 & 1.137 & 1.136 & 1.135 & 1.134 & 1.133 & 1.132 \\\\\n$30.00$--$40.00$ & 1.116 & 1.116 & 1.115 & 1.114 & 1.114 & 1.113 & 1.113 & 1.112 \\\\\n$40.00$--$60.00$ & 1.086 & 1.086 & 1.086 & 1.085 & 1.085 & 1.084 & 1.084 & 1.084 \\\\\n$60.00$--$100.00$ & 1.057 & 1.056 & 1.056 & 1.056 & 1.056 & 1.056 & 1.057 & 1.055 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.015 & 1.047 & 1.073 & 1.094 & 1.113 & 1.120 & 1.124 & 1.122 \\\\\n$8.50$--$9.00$ & 1.020 & 1.058 & 1.087 & 1.110 & 1.125 & 1.134 & 1.142 & 1.144 \\\\\n$9.00$--$9.50$ & 1.019 & 1.056 & 1.084 & 1.107 & 1.127 & 1.138 & 1.144 & 1.145 \\\\\n$9.50$--$10.00$ & 1.017 & 1.053 & 1.081 & 1.105 & 1.122 & 1.129 & 1.140 & 1.142 \\\\\n$10.00$--$10.50$ & 1.017 & 1.049 & 1.077 & 1.100 & 1.115 & 1.125 & 1.132 & 1.136 \\\\\n$10.50$--$11.00$ & 1.014 & 1.048 & 1.075 & 1.095 & 1.109 & 1.118 & 1.124 & 1.130 \\\\\n$11.00$--$11.50$ & 1.015 & 1.044 & 1.069 & 1.088 & 1.103 & 1.112 & 1.117 & 1.122 \\\\\n$11.50$--$12.00$ & 1.014 & 1.043 & 1.066 & 1.083 & 1.096 & 1.105 & 1.112 & 1.115 \\\\\n$12.00$--$13.00$ & 1.012 & 1.038 & 1.060 & 1.076 & 1.089 & 1.097 & 1.101 & 1.105 \\\\\n$13.00$--$14.00$ & 1.012 & 1.035 & 1.053 & 1.068 & 1.079 & 1.087 & 1.090 & 1.093 \\\\\n$14.00$--$15.00$ & 1.010 & 1.031 & 1.048 & 1.061 & 1.070 & 1.076 & 1.080 & 1.083 \\\\\n$15.00$--$16.00$ & 1.010 & 1.028 & 1.043 & 1.054 & 1.063 & 1.068 & 1.072 & 1.074 \\\\\n$16.00$--$17.00$ & 1.009 & 1.025 & 1.039 & 1.049 & 1.056 & 1.062 & 1.065 & 1.067 \\\\\n$17.00$--$18.00$ & 1.009 & 1.023 & 1.035 & 1.044 & 1.051 & 1.055 & 1.059 & 1.060 \\\\\n$18.00$--$20.00$ & 1.007 & 1.019 & 1.030 & 1.039 & 1.045 & 1.049 & 1.051 & 1.053 \\\\\n$20.00$--$22.00$ & 1.007 & 1.016 & 1.025 & 1.032 & 1.038 & 1.041 & 1.043 & 1.044 \\\\\n$22.00$--$24.00$ & 1.005 & 1.014 & 1.021 & 1.027 & 1.031 & 1.035 & 1.036 & 1.037 \\\\\n$24.00$--$26.00$ & 1.005 & 1.012 & 1.018 & 1.024 & 1.027 & 1.030 & 1.032 & 1.032 \\\\\n$26.00$--$30.00$ & 1.004 & 1.010 & 1.015 & 1.019 & 1.023 & 1.025 & 1.026 & 1.026 \\\\\n$30.00$--$40.00$ & 1.003 & 1.007 & 1.011 & 1.014 & 1.016 & 1.017 & 1.018 & 1.019 \\\\\n$40.00$--$60.00$ & 1.002 & 1.004 & 1.006 & 1.008 & 1.009 & 1.010 & 1.010 & 1.010 \\\\\n$60.00$--$100.00$ & 1.001 & 1.002 & 1.003 & 1.003 & 1.004 & 1.004 & 1.005 & 1.005 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.984 & 0.956 & 0.932 & 0.920 & 0.912 & 0.904 & 0.898 & 0.905 \\\\\n$8.50$--$9.00$ & 0.981 & 0.948 & 0.925 & 0.910 & 0.898 & 0.892 & 0.886 & 0.891 \\\\\n$9.00$--$9.50$ & 0.983 & 0.950 & 0.925 & 0.910 & 0.901 & 0.893 & 0.889 & 0.888 \\\\\n$9.50$--$10.00$ & 0.983 & 0.951 & 0.929 & 0.912 & 0.903 & 0.894 & 0.892 & 0.891 \\\\\n$10.00$--$10.50$ & 0.985 & 0.953 & 0.932 & 0.918 & 0.907 & 0.900 & 0.896 & 0.894 \\\\\n$10.50$--$11.00$ & 0.984 & 0.957 & 0.936 & 0.922 & 0.910 & 0.904 & 0.900 & 0.899 \\\\\n$11.00$--$11.50$ & 0.985 & 0.958 & 0.939 & 0.927 & 0.915 & 0.909 & 0.906 & 0.903 \\\\\n$11.50$--$12.00$ & 0.987 & 0.961 & 0.942 & 0.929 & 0.919 & 0.912 & 0.910 & 0.907 \\\\\n$12.00$--$13.00$ & 0.987 & 0.963 & 0.945 & 0.934 & 0.925 & 0.920 & 0.915 & 0.913 \\\\\n$13.00$--$14.00$ & 0.989 & 0.968 & 0.951 & 0.940 & 0.932 & 0.927 & 0.924 & 0.922 \\\\\n$14.00$--$15.00$ & 0.990 & 0.971 & 0.957 & 0.946 & 0.938 & 0.934 & 0.931 & 0.929 \\\\\n$15.00$--$16.00$ & 0.992 & 0.974 & 0.961 & 0.951 & 0.944 & 0.940 & 0.937 & 0.936 \\\\\n$16.00$--$17.00$ & 0.991 & 0.976 & 0.964 & 0.955 & 0.949 & 0.945 & 0.943 & 0.941 \\\\\n$17.00$--$18.00$ & 0.992 & 0.978 & 0.968 & 0.959 & 0.953 & 0.949 & 0.948 & 0.946 \\\\\n$18.00$--$20.00$ & 0.993 & 0.981 & 0.971 & 0.964 & 0.959 & 0.956 & 0.954 & 0.953 \\\\\n$20.00$--$22.00$ & 0.994 & 0.984 & 0.976 & 0.970 & 0.965 & 0.962 & 0.961 & 0.960 \\\\\n$22.00$--$24.00$ & 0.994 & 0.986 & 0.979 & 0.974 & 0.970 & 0.968 & 0.966 & 0.965 \\\\\n$24.00$--$26.00$ & 0.995 & 0.988 & 0.982 & 0.977 & 0.974 & 0.972 & 0.971 & 0.970 \\\\\n$26.00$--$30.00$ & 0.996 & 0.990 & 0.985 & 0.981 & 0.978 & 0.977 & 0.976 & 0.975 \\\\\n$30.00$--$40.00$ & 0.997 & 0.993 & 0.990 & 0.987 & 0.985 & 0.983 & 0.983 & 0.982 \\\\\n$40.00$--$60.00$ & 0.998 & 0.996 & 0.994 & 0.993 & 0.991 & 0.991 & 0.990 & 0.990 \\\\\n$60.00$--$100.00$ & 0.999 & 0.998 & 0.997 & 0.997 & 0.996 & 0.996 & 0.995 & 0.995 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``longitudinal'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.670 & 0.678 & 0.685 & 0.692 & 0.701 & 0.707 & 0.713 & 0.709 \\\\\n$8.50$--$9.00$ & 0.676 & 0.681 & 0.688 & 0.698 & 0.703 & 0.709 & 0.712 & 0.713 \\\\\n$9.00$--$9.50$ & 0.678 & 0.683 & 0.691 & 0.700 & 0.708 & 0.713 & 0.717 & 0.718 \\\\\n$9.50$--$10.00$ & 0.680 & 0.684 & 0.693 & 0.699 & 0.708 & 0.710 & 0.720 & 0.722 \\\\\n$10.00$--$10.50$ & 0.684 & 0.687 & 0.695 & 0.704 & 0.707 & 0.713 & 0.720 & 0.725 \\\\\n$10.50$--$11.00$ & 0.687 & 0.691 & 0.698 & 0.705 & 0.712 & 0.714 & 0.719 & 0.728 \\\\\n$11.00$--$11.50$ & 0.692 & 0.695 & 0.701 & 0.709 & 0.713 & 0.717 & 0.722 & 0.728 \\\\\n$11.50$--$12.00$ & 0.696 & 0.700 & 0.704 & 0.711 & 0.717 & 0.719 & 0.724 & 0.729 \\\\\n$12.00$--$13.00$ & 0.701 & 0.705 & 0.710 & 0.716 & 0.720 & 0.724 & 0.727 & 0.731 \\\\\n$13.00$--$14.00$ & 0.711 & 0.714 & 0.718 & 0.722 & 0.727 & 0.730 & 0.732 & 0.734 \\\\\n$14.00$--$15.00$ & 0.719 & 0.722 & 0.725 & 0.730 & 0.732 & 0.736 & 0.739 & 0.742 \\\\\n$15.00$--$16.00$ & 0.727 & 0.729 & 0.733 & 0.735 & 0.740 & 0.741 & 0.745 & 0.745 \\\\\n$16.00$--$17.00$ & 0.736 & 0.738 & 0.740 & 0.743 & 0.746 & 0.748 & 0.749 & 0.753 \\\\\n$17.00$--$18.00$ & 0.742 & 0.744 & 0.748 & 0.750 & 0.753 & 0.754 & 0.759 & 0.760 \\\\\n$18.00$--$20.00$ & 0.753 & 0.755 & 0.758 & 0.760 & 0.762 & 0.763 & 0.764 & 0.769 \\\\\n$20.00$--$22.00$ & 0.766 & 0.767 & 0.770 & 0.773 & 0.774 & 0.776 & 0.776 & 0.778 \\\\\n$22.00$--$24.00$ & 0.778 & 0.782 & 0.780 & 0.784 & 0.785 & 0.782 & 0.790 & 0.788 \\\\\n$24.00$--$26.00$ & 0.791 & 0.791 & 0.795 & 0.795 & 0.795 & 0.799 & 0.798 & 0.798 \\\\\n$26.00$--$30.00$ & 0.806 & 0.805 & 0.805 & 0.809 & 0.808 & 0.810 & 0.810 & 0.812 \\\\\n$30.00$--$40.00$ & 0.829 & 0.830 & 0.830 & 0.830 & 0.828 & 0.832 & 0.830 & 0.830 \\\\\n$40.00$--$60.00$ & 0.864 & 0.865 & 0.867 & 0.864 & 0.868 & 0.867 & 0.861 & 0.953 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse zero'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.328 & 1.311 & 1.300 & 1.284 & 1.274 & 1.267 & 1.261 & 1.265 \\\\\n$8.50$--$9.00$ & 1.318 & 1.309 & 1.293 & 1.279 & 1.268 & 1.263 & 1.252 & 1.259 \\\\\n$9.00$--$9.50$ & 1.317 & 1.303 & 1.287 & 1.273 & 1.267 & 1.256 & 1.249 & 1.250 \\\\\n$9.50$--$10.00$ & 1.310 & 1.301 & 1.286 & 1.275 & 1.262 & 1.255 & 1.248 & 1.247 \\\\\n$10.00$--$10.50$ & 1.303 & 1.294 & 1.283 & 1.271 & 1.265 & 1.257 & 1.248 & 1.243 \\\\\n$10.50$--$11.00$ & 1.295 & 1.289 & 1.279 & 1.271 & 1.259 & 1.254 & 1.246 & 1.240 \\\\\n$11.00$--$11.50$ & 1.289 & 1.282 & 1.273 & 1.264 & 1.254 & 1.249 & 1.242 & 1.238 \\\\\n$11.50$--$12.00$ & 1.282 & 1.276 & 1.267 & 1.260 & 1.249 & 1.246 & 1.240 & 1.234 \\\\\n$12.00$--$13.00$ & 1.271 & 1.266 & 1.259 & 1.250 & 1.244 & 1.241 & 1.236 & 1.232 \\\\\n$13.00$--$14.00$ & 1.258 & 1.252 & 1.246 & 1.239 & 1.234 & 1.232 & 1.229 & 1.226 \\\\\n$14.00$--$15.00$ & 1.244 & 1.240 & 1.237 & 1.230 & 1.228 & 1.221 & 1.219 & 1.216 \\\\\n$15.00$--$16.00$ & 1.234 & 1.229 & 1.224 & 1.221 & 1.216 & 1.214 & 1.211 & 1.211 \\\\\n$16.00$--$17.00$ & 1.220 & 1.217 & 1.213 & 1.211 & 1.208 & 1.205 & 1.205 & 1.202 \\\\\n$17.00$--$18.00$ & 1.211 & 1.210 & 1.205 & 1.202 & 1.198 & 1.197 & 1.193 & 1.193 \\\\\n$18.00$--$20.00$ & 1.197 & 1.194 & 1.191 & 1.190 & 1.188 & 1.187 & 1.186 & 1.181 \\\\\n$20.00$--$22.00$ & 1.181 & 1.180 & 1.176 & 1.174 & 1.174 & 1.171 & 1.172 & 1.169 \\\\\n$22.00$--$24.00$ & 1.167 & 1.163 & 1.164 & 1.161 & 1.160 & 1.164 & 1.155 & 1.159 \\\\\n$24.00$--$26.00$ & 1.153 & 1.154 & 1.149 & 1.148 & 1.149 & 1.145 & 1.147 & 1.147 \\\\\n$26.00$--$30.00$ & 1.137 & 1.139 & 1.138 & 1.135 & 1.136 & 1.134 & 1.135 & 1.133 \\\\\n$30.00$--$40.00$ & 1.115 & 1.115 & 1.115 & 1.115 & 1.116 & 1.114 & 1.116 & 1.116 \\\\\n$40.00$--$60.00$ & 1.086 & 1.085 & 1.083 & 1.086 & 1.083 & 1.084 & 1.089 & 1.028 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 2.009 & 2.007 & 1.986 & 1.994 & 1.964 & 1.936 & 1.949 & 1.967 \\\\\n$8.50$--$9.00$ & 1.614 & 1.617 & 1.617 & 1.613 & 1.618 & 1.624 & 1.606 & 1.872 \\\\\n$9.00$--$9.50$ & 1.504 & 1.502 & 1.496 & 1.493 & 1.500 & 1.499 & 1.494 & 1.741 \\\\\n$9.50$--$10.00$ & 1.445 & 1.443 & 1.440 & 1.445 & 1.440 & 1.441 & 1.436 & 1.621 \\\\\n$10.00$--$10.50$ & 1.404 & 1.401 & 1.403 & 1.401 & 1.412 & 1.406 & 1.400 & 1.507 \\\\\n$10.50$--$11.00$ & 1.374 & 1.377 & 1.378 & 1.378 & 1.377 & 1.378 & 1.372 & 1.447 \\\\\n$11.00$--$11.50$ & 1.355 & 1.352 & 1.352 & 1.353 & 1.352 & 1.353 & 1.350 & 1.409 \\\\\n$11.50$--$12.00$ & 1.335 & 1.334 & 1.335 & 1.335 & 1.332 & 1.336 & 1.331 & 1.375 \\\\\n$12.00$--$13.00$ & 1.314 & 1.313 & 1.312 & 1.312 & 1.313 & 1.314 & 1.312 & 1.343 \\\\\n$13.00$--$14.00$ & 1.289 & 1.289 & 1.287 & 1.286 & 1.286 & 1.288 & 1.288 & 1.311 \\\\\n$14.00$--$15.00$ & 1.268 & 1.267 & 1.269 & 1.267 & 1.269 & 1.265 & 1.265 & 1.280 \\\\\n$15.00$--$16.00$ & 1.253 & 1.250 & 1.249 & 1.252 & 1.250 & 1.250 & 1.248 & 1.262 \\\\\n$16.00$--$17.00$ & 1.234 & 1.234 & 1.234 & 1.234 & 1.235 & 1.235 & 1.236 & 1.241 \\\\\n$17.00$--$18.00$ & 1.224 & 1.224 & 1.222 & 1.222 & 1.220 & 1.222 & 1.218 & 1.224 \\\\\n$18.00$--$20.00$ & 1.206 & 1.205 & 1.204 & 1.205 & 1.205 & 1.207 & 1.206 & 1.204 \\\\\n$20.00$--$22.00$ & 1.187 & 1.187 & 1.186 & 1.184 & 1.186 & 1.184 & 1.186 & 1.186 \\\\\n$22.00$--$24.00$ & 1.171 & 1.169 & 1.171 & 1.169 & 1.169 & 1.174 & 1.166 & 1.170 \\\\\n$24.00$--$26.00$ & 1.157 & 1.158 & 1.155 & 1.155 & 1.156 & 1.153 & 1.156 & 1.156 \\\\\n$26.00$--$30.00$ & 1.140 & 1.142 & 1.142 & 1.139 & 1.141 & 1.140 & 1.141 & 1.139 \\\\\n$30.00$--$40.00$ & 1.116 & 1.117 & 1.117 & 1.117 & 1.119 & 1.117 & 1.119 & 1.119 \\\\\n$40.00$--$60.00$ & 1.087 & 1.086 & 1.084 & 1.087 & 1.085 & 1.085 & 1.091 & 1.029 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.998 & 0.986 & 0.970 & 0.957 & 0.949 & 0.941 & 0.935 & 0.883 \\\\\n$8.50$--$9.00$ & 1.115 & 1.102 & 1.084 & 1.062 & 1.047 & 1.039 & 1.025 & 0.959 \\\\\n$9.00$--$9.50$ & 1.169 & 1.154 & 1.131 & 1.110 & 1.096 & 1.084 & 1.075 & 1.007 \\\\\n$9.50$--$10.00$ & 1.200 & 1.185 & 1.163 & 1.144 & 1.126 & 1.114 & 1.105 & 1.047 \\\\\n$10.00$--$10.50$ & 1.216 & 1.200 & 1.181 & 1.161 & 1.148 & 1.137 & 1.127 & 1.075 \\\\\n$10.50$--$11.00$ & 1.222 & 1.212 & 1.196 & 1.178 & 1.161 & 1.152 & 1.143 & 1.097 \\\\\n$11.00$--$11.50$ & 1.230 & 1.218 & 1.202 & 1.185 & 1.169 & 1.161 & 1.152 & 1.112 \\\\\n$11.50$--$12.00$ & 1.233 & 1.221 & 1.205 & 1.192 & 1.175 & 1.169 & 1.160 & 1.124 \\\\\n$12.00$--$13.00$ & 1.232 & 1.222 & 1.208 & 1.195 & 1.184 & 1.176 & 1.169 & 1.141 \\\\\n$13.00$--$14.00$ & 1.228 & 1.220 & 1.208 & 1.196 & 1.187 & 1.181 & 1.176 & 1.155 \\\\\n$14.00$--$15.00$ & 1.221 & 1.214 & 1.207 & 1.196 & 1.188 & 1.181 & 1.176 & 1.159 \\\\\n$15.00$--$16.00$ & 1.215 & 1.208 & 1.200 & 1.193 & 1.184 & 1.181 & 1.175 & 1.165 \\\\\n$16.00$--$17.00$ & 1.205 & 1.200 & 1.194 & 1.187 & 1.182 & 1.177 & 1.175 & 1.165 \\\\\n$17.00$--$18.00$ & 1.199 & 1.196 & 1.188 & 1.183 & 1.177 & 1.174 & 1.169 & 1.162 \\\\\n$18.00$--$20.00$ & 1.188 & 1.184 & 1.179 & 1.175 & 1.170 & 1.168 & 1.166 & 1.158 \\\\\n$20.00$--$22.00$ & 1.174 & 1.172 & 1.167 & 1.162 & 1.161 & 1.157 & 1.157 & 1.153 \\\\\n$22.00$--$24.00$ & 1.162 & 1.157 & 1.158 & 1.153 & 1.151 & 1.153 & 1.145 & 1.146 \\\\\n$24.00$--$26.00$ & 1.149 & 1.149 & 1.144 & 1.142 & 1.142 & 1.138 & 1.139 & 1.138 \\\\\n$26.00$--$30.00$ & 1.135 & 1.136 & 1.134 & 1.130 & 1.131 & 1.129 & 1.129 & 1.127 \\\\\n$30.00$--$40.00$ & 1.114 & 1.113 & 1.113 & 1.112 & 1.113 & 1.110 & 1.112 & 1.112 \\\\\n$40.00$--$60.00$ & 1.086 & 1.085 & 1.083 & 1.085 & 1.082 & 1.082 & 1.088 & 1.028 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.017 & 1.052 & 1.081 & 1.100 & 1.118 & 1.123 & 1.129 & 1.106 \\\\\n$8.50$--$9.00$ & 1.023 & 1.064 & 1.094 & 1.118 & 1.136 & 1.146 & 1.151 & 1.132 \\\\\n$9.00$--$9.50$ & 1.021 & 1.062 & 1.093 & 1.119 & 1.140 & 1.150 & 1.153 & 1.139 \\\\\n$9.50$--$10.00$ & 1.019 & 1.060 & 1.092 & 1.119 & 1.135 & 1.144 & 1.152 & 1.146 \\\\\n$10.00$--$10.50$ & 1.020 & 1.057 & 1.088 & 1.112 & 1.132 & 1.140 & 1.146 & 1.145 \\\\\n$10.50$--$11.00$ & 1.017 & 1.055 & 1.085 & 1.108 & 1.124 & 1.134 & 1.139 & 1.141 \\\\\n$11.00$--$11.50$ & 1.017 & 1.052 & 1.079 & 1.102 & 1.118 & 1.127 & 1.131 & 1.137 \\\\\n$11.50$--$12.00$ & 1.017 & 1.050 & 1.076 & 1.096 & 1.110 & 1.120 & 1.126 & 1.130 \\\\\n$12.00$--$13.00$ & 1.014 & 1.044 & 1.069 & 1.088 & 1.102 & 1.111 & 1.116 & 1.123 \\\\\n$13.00$--$14.00$ & 1.013 & 1.041 & 1.061 & 1.078 & 1.091 & 1.100 & 1.104 & 1.111 \\\\\n$14.00$--$15.00$ & 1.012 & 1.036 & 1.056 & 1.070 & 1.082 & 1.088 & 1.092 & 1.098 \\\\\n$15.00$--$16.00$ & 1.011 & 1.032 & 1.049 & 1.064 & 1.073 & 1.079 & 1.083 & 1.090 \\\\\n$16.00$--$17.00$ & 1.010 & 1.029 & 1.045 & 1.057 & 1.065 & 1.072 & 1.076 & 1.080 \\\\\n$17.00$--$18.00$ & 1.010 & 1.027 & 1.041 & 1.051 & 1.059 & 1.064 & 1.068 & 1.071 \\\\\n$18.00$--$20.00$ & 1.008 & 1.023 & 1.035 & 1.045 & 1.052 & 1.057 & 1.059 & 1.062 \\\\\n$20.00$--$22.00$ & 1.008 & 1.019 & 1.030 & 1.037 & 1.044 & 1.047 & 1.050 & 1.052 \\\\\n$22.00$--$24.00$ & 1.006 & 1.016 & 1.025 & 1.032 & 1.037 & 1.042 & 1.042 & 1.044 \\\\\n$24.00$--$26.00$ & 1.005 & 1.014 & 1.021 & 1.027 & 1.032 & 1.034 & 1.037 & 1.038 \\\\\n$26.00$--$30.00$ & 1.005 & 1.012 & 1.018 & 1.022 & 1.027 & 1.029 & 1.030 & 1.031 \\\\\n$30.00$--$40.00$ & 1.003 & 1.008 & 1.013 & 1.016 & 1.019 & 1.020 & 1.022 & 1.022 \\\\\n$40.00$--$60.00$ & 1.002 & 1.005 & 1.007 & 1.009 & 1.010 & 1.011 & 1.013 & 1.004 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.983 & 0.950 & 0.931 & 0.916 & 0.908 & 0.902 & 0.902 & 0.911 \\\\\n$8.50$--$9.00$ & 0.979 & 0.944 & 0.919 & 0.904 & 0.892 & 0.887 & 0.882 & 0.898 \\\\\n$9.00$--$9.50$ & 0.981 & 0.943 & 0.919 & 0.901 & 0.894 & 0.886 & 0.883 & 0.891 \\\\\n$9.50$--$10.00$ & 0.981 & 0.945 & 0.922 & 0.903 & 0.894 & 0.886 & 0.885 & 0.891 \\\\\n$10.00$--$10.50$ & 0.982 & 0.948 & 0.925 & 0.910 & 0.897 & 0.891 & 0.888 & 0.890 \\\\\n$10.50$--$11.00$ & 0.982 & 0.951 & 0.929 & 0.913 & 0.901 & 0.895 & 0.891 & 0.892 \\\\\n$11.00$--$11.50$ & 0.983 & 0.953 & 0.931 & 0.918 & 0.906 & 0.900 & 0.897 & 0.894 \\\\\n$11.50$--$12.00$ & 0.985 & 0.955 & 0.934 & 0.920 & 0.910 & 0.903 & 0.901 & 0.897 \\\\\n$12.00$--$13.00$ & 0.985 & 0.958 & 0.938 & 0.925 & 0.915 & 0.911 & 0.906 & 0.903 \\\\\n$13.00$--$14.00$ & 0.988 & 0.963 & 0.944 & 0.932 & 0.924 & 0.918 & 0.915 & 0.910 \\\\\n$14.00$--$15.00$ & 0.988 & 0.966 & 0.950 & 0.939 & 0.930 & 0.926 & 0.923 & 0.918 \\\\\n$15.00$--$16.00$ & 0.990 & 0.969 & 0.955 & 0.944 & 0.937 & 0.932 & 0.929 & 0.924 \\\\\n$16.00$--$17.00$ & 0.991 & 0.972 & 0.959 & 0.949 & 0.941 & 0.937 & 0.935 & 0.932 \\\\\n$17.00$--$18.00$ & 0.991 & 0.975 & 0.963 & 0.953 & 0.947 & 0.942 & 0.941 & 0.938 \\\\\n$18.00$--$20.00$ & 0.992 & 0.978 & 0.967 & 0.959 & 0.953 & 0.949 & 0.947 & 0.945 \\\\\n$20.00$--$22.00$ & 0.993 & 0.981 & 0.972 & 0.965 & 0.960 & 0.957 & 0.955 & 0.953 \\\\\n$22.00$--$24.00$ & 0.994 & 0.984 & 0.975 & 0.970 & 0.966 & 0.962 & 0.961 & 0.959 \\\\\n$24.00$--$26.00$ & 0.995 & 0.986 & 0.979 & 0.974 & 0.970 & 0.968 & 0.966 & 0.965 \\\\\n$26.00$--$30.00$ & 0.995 & 0.988 & 0.983 & 0.978 & 0.975 & 0.973 & 0.972 & 0.971 \\\\\n$30.00$--$40.00$ & 0.997 & 0.992 & 0.988 & 0.984 & 0.982 & 0.981 & 0.979 & 0.979 \\\\\n$40.00$--$60.00$ & 0.998 & 0.995 & 0.993 & 0.991 & 0.990 & 0.989 & 0.988 & 0.996 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_psi2s} \n \\end{table}\n\n\n\\clearpage\n\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``longitudinal'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here. } \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.672 & 0.674 & 0.678 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.670 & 0.673 & 0.678 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.671 & 0.674 & 0.679 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.674 & 0.676 & 0.681 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.676 & 0.678 & 0.683 & 0.686 & 0.691 & 0.694 & 0.695 & 0.696 \\\\ \n 10.50 -- 11.00 & 0.680 & 0.681 & 0.686 & 0.689 & 0.693 & 0.696 & 0.697 & 0.698 \\\\ \n 11.00 -- 11.50 & 0.684 & 0.685 & 0.690 & 0.692 & 0.695 & 0.698 & 0.700 & 0.701 \\\\ \n 11.50 -- 12.00 & 0.688 & 0.688 & 0.693 & 0.695 & 0.698 & 0.701 & 0.702 & 0.704 \\\\ \n 12.00 -- 12.50 & 0.692 & 0.692 & 0.696 & 0.698 & 0.702 & 0.704 & 0.705 & 0.706 \\\\ \n 12.50 -- 13.00 & 0.696 & 0.696 & 0.700 & 0.702 & 0.705 & 0.707 & 0.708 & 0.710 \\\\ \n 13.00 -- 14.00 & 0.702 & 0.703 & 0.705 & 0.707 & 0.710 & 0.712 & 0.713 & 0.715 \\\\ \n 14.00 -- 15.00 & 0.710 & 0.711 & 0.713 & 0.714 & 0.717 & 0.719 & 0.720 & 0.722 \\\\ \n 15.00 -- 16.00 & 0.719 & 0.719 & 0.721 & 0.722 & 0.724 & 0.725 & 0.727 & 0.729 \\\\ \n 16.00 -- 17.00 & 0.726 & 0.727 & 0.729 & 0.729 & 0.732 & 0.733 & 0.734 & 0.735 \\\\ \n 17.00 -- 18.00 & 0.734 & 0.735 & 0.736 & 0.737 & 0.738 & 0.740 & 0.740 & 0.743 \\\\ \n 18.00 -- 20.00 & 0.744 & 0.745 & 0.746 & 0.746 & 0.748 & 0.750 & 0.750 & 0.752 \\\\ \n 20.00 -- 22.00 & 0.758 & 0.759 & 0.760 & 0.759 & 0.761 & 0.762 & 0.763 & 0.764 \\\\ \n 22.00 -- 24.00 & 0.771 & 0.771 & 0.772 & 0.771 & 0.773 & 0.774 & 0.774 & 0.776 \\\\ \n 24.00 -- 26.00 & 0.783 & 0.783 & 0.783 & 0.783 & 0.784 & 0.786 & 0.786 & 0.787 \\\\ \n 26.00 -- 30.00 & 0.797 & 0.798 & 0.798 & 0.797 & 0.798 & 0.799 & 0.800 & 0.800 \\\\ \n 30.00 -- 35.00 & 0.817 & 0.817 & 0.817 & 0.816 & 0.817 & 0.818 & 0.818 & 0.820 \\\\ \n 35.00 -- 40.00 & 0.836 & 0.836 & 0.836 & 0.835 & 0.835 & 0.836 & 0.836 & 0.840 \\\\ \n 40.00 -- 60.00 & 0.862 & 0.862 & 0.861 & 0.861 & 0.861 & 0.862 & 0.862 & 0.863 \\\\ \n 60.00 -- 110.00 & 0.904 & 0.902 & 0.903 & 0.902 & 0.903 & 0.904 & 0.905 & 0.906 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse zero'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.326 & 1.321 & 1.311 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.326 & 1.320 & 1.309 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.322 & 1.316 & 1.306 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.317 & 1.312 & 1.302 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.311 & 1.306 & 1.297 & 1.291 & 1.283 & 1.278 & 1.275 & 1.273 \\\\ \n 10.50 -- 11.00 & 1.304 & 1.300 & 1.292 & 1.286 & 1.279 & 1.274 & 1.272 & 1.269 \\\\ \n 11.00 -- 11.50 & 1.297 & 1.293 & 1.286 & 1.280 & 1.275 & 1.270 & 1.268 & 1.265 \\\\ \n 11.50 -- 12.00 & 1.290 & 1.287 & 1.280 & 1.275 & 1.270 & 1.266 & 1.263 & 1.261 \\\\ \n 12.00 -- 12.50 & 1.283 & 1.280 & 1.274 & 1.270 & 1.264 & 1.261 & 1.259 & 1.257 \\\\ \n 12.50 -- 13.00 & 1.276 & 1.273 & 1.268 & 1.264 & 1.260 & 1.256 & 1.254 & 1.252 \\\\ \n 13.00 -- 14.00 & 1.265 & 1.264 & 1.259 & 1.256 & 1.252 & 1.249 & 1.247 & 1.245 \\\\ \n 14.00 -- 15.00 & 1.253 & 1.251 & 1.247 & 1.245 & 1.241 & 1.238 & 1.237 & 1.235 \\\\ \n 15.00 -- 16.00 & 1.240 & 1.239 & 1.236 & 1.234 & 1.231 & 1.229 & 1.227 & 1.225 \\\\ \n 16.00 -- 17.00 & 1.228 & 1.227 & 1.225 & 1.223 & 1.220 & 1.218 & 1.218 & 1.216 \\\\ \n 17.00 -- 18.00 & 1.218 & 1.217 & 1.215 & 1.213 & 1.211 & 1.209 & 1.209 & 1.206 \\\\ \n 18.00 -- 20.00 & 1.204 & 1.203 & 1.201 & 1.201 & 1.199 & 1.197 & 1.196 & 1.195 \\\\ \n 20.00 -- 22.00 & 1.186 & 1.186 & 1.185 & 1.185 & 1.183 & 1.182 & 1.181 & 1.180 \\\\ \n 22.00 -- 24.00 & 1.172 & 1.171 & 1.171 & 1.171 & 1.169 & 1.168 & 1.168 & 1.167 \\\\ \n 24.00 -- 26.00 & 1.159 & 1.159 & 1.158 & 1.158 & 1.157 & 1.156 & 1.156 & 1.154 \\\\ \n 26.00 -- 30.00 & 1.144 & 1.144 & 1.143 & 1.144 & 1.143 & 1.142 & 1.141 & 1.141 \\\\ \n 30.00 -- 35.00 & 1.125 & 1.125 & 1.125 & 1.125 & 1.124 & 1.124 & 1.124 & 1.122 \\\\ \n 35.00 -- 40.00 & 1.108 & 1.108 & 1.108 & 1.108 & 1.108 & 1.108 & 1.107 & 1.105 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.086 & 1.087 & 1.087 & 1.087 & 1.087 & 1.087 & 1.086 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.926 & 1.933 & 1.930 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.555 & 1.558 & 1.559 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.463 & 1.464 & 1.465 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.416 & 1.418 & 1.418 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.386 & 1.388 & 1.387 & 1.390 & 1.390 & 1.390 & 1.391 & 1.411 \\\\ \n 10.50 -- 11.00 & 1.363 & 1.365 & 1.365 & 1.367 & 1.367 & 1.366 & 1.368 & 1.382 \\\\ \n 11.00 -- 11.50 & 1.345 & 1.347 & 1.346 & 1.348 & 1.348 & 1.348 & 1.349 & 1.358 \\\\ \n 11.50 -- 12.00 & 1.330 & 1.331 & 1.331 & 1.333 & 1.333 & 1.332 & 1.333 & 1.340 \\\\ \n 12.00 -- 12.50 & 1.316 & 1.318 & 1.317 & 1.319 & 1.318 & 1.319 & 1.319 & 1.325 \\\\ \n 12.50 -- 13.00 & 1.304 & 1.305 & 1.305 & 1.307 & 1.307 & 1.307 & 1.306 & 1.311 \\\\ \n 13.00 -- 14.00 & 1.288 & 1.290 & 1.290 & 1.291 & 1.291 & 1.291 & 1.291 & 1.293 \\\\ \n 14.00 -- 15.00 & 1.270 & 1.271 & 1.271 & 1.272 & 1.272 & 1.271 & 1.272 & 1.272 \\\\ \n 15.00 -- 16.00 & 1.253 & 1.254 & 1.254 & 1.255 & 1.255 & 1.255 & 1.254 & 1.255 \\\\ \n 16.00 -- 17.00 & 1.239 & 1.240 & 1.240 & 1.241 & 1.240 & 1.240 & 1.241 & 1.240 \\\\ \n 17.00 -- 18.00 & 1.227 & 1.227 & 1.227 & 1.228 & 1.228 & 1.227 & 1.228 & 1.226 \\\\ \n 18.00 -- 20.00 & 1.211 & 1.211 & 1.211 & 1.212 & 1.212 & 1.211 & 1.211 & 1.210 \\\\ \n 20.00 -- 22.00 & 1.191 & 1.192 & 1.192 & 1.193 & 1.193 & 1.192 & 1.192 & 1.192 \\\\ \n 22.00 -- 24.00 & 1.175 & 1.176 & 1.176 & 1.177 & 1.176 & 1.176 & 1.176 & 1.175 \\\\ \n 24.00 -- 26.00 & 1.162 & 1.162 & 1.162 & 1.163 & 1.162 & 1.162 & 1.162 & 1.161 \\\\ \n 26.00 -- 30.00 & 1.146 & 1.146 & 1.146 & 1.147 & 1.146 & 1.146 & 1.146 & 1.146 \\\\ \n 30.00 -- 35.00 & 1.126 & 1.126 & 1.126 & 1.127 & 1.127 & 1.126 & 1.126 & 1.125 \\\\ \n 35.00 -- 40.00 & 1.109 & 1.109 & 1.109 & 1.110 & 1.110 & 1.109 & 1.109 & 1.107 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.087 & 1.088 & 1.088 & 1.088 & 1.087 & 1.087 & 1.087 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.056 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.026 & 1.017 & 1.005 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.157 & 1.145 & 1.129 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.207 & 1.196 & 1.178 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.231 & 1.220 & 1.203 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.244 & 1.234 & 1.218 & 1.204 & 1.192 & 1.182 & 1.177 & 1.161 \\\\ \n 10.50 -- 11.00 & 1.250 & 1.241 & 1.227 & 1.214 & 1.202 & 1.193 & 1.188 & 1.175 \\\\ \n 11.00 -- 11.50 & 1.252 & 1.244 & 1.231 & 1.220 & 1.209 & 1.200 & 1.195 & 1.184 \\\\ \n 11.50 -- 12.00 & 1.253 & 1.246 & 1.234 & 1.223 & 1.213 & 1.206 & 1.201 & 1.191 \\\\ \n 12.00 -- 12.50 & 1.251 & 1.245 & 1.234 & 1.224 & 1.215 & 1.208 & 1.204 & 1.196 \\\\ \n 12.50 -- 13.00 & 1.248 & 1.243 & 1.233 & 1.224 & 1.216 & 1.210 & 1.206 & 1.199 \\\\ \n 13.00 -- 14.00 & 1.243 & 1.239 & 1.230 & 1.222 & 1.215 & 1.210 & 1.206 & 1.200 \\\\ \n 14.00 -- 15.00 & 1.236 & 1.231 & 1.224 & 1.218 & 1.212 & 1.207 & 1.204 & 1.200 \\\\ \n 15.00 -- 16.00 & 1.226 & 1.223 & 1.217 & 1.212 & 1.207 & 1.203 & 1.200 & 1.197 \\\\ \n 16.00 -- 17.00 & 1.218 & 1.215 & 1.210 & 1.206 & 1.201 & 1.197 & 1.195 & 1.193 \\\\ \n 17.00 -- 18.00 & 1.209 & 1.206 & 1.202 & 1.199 & 1.195 & 1.192 & 1.190 & 1.187 \\\\ \n 18.00 -- 20.00 & 1.197 & 1.195 & 1.192 & 1.189 & 1.186 & 1.183 & 1.182 & 1.180 \\\\ \n 20.00 -- 22.00 & 1.182 & 1.181 & 1.178 & 1.177 & 1.174 & 1.172 & 1.170 & 1.170 \\\\ \n 22.00 -- 24.00 & 1.168 & 1.167 & 1.166 & 1.165 & 1.162 & 1.161 & 1.160 & 1.159 \\\\ \n 24.00 -- 26.00 & 1.156 & 1.156 & 1.154 & 1.153 & 1.152 & 1.150 & 1.150 & 1.148 \\\\ \n 26.00 -- 30.00 & 1.142 & 1.141 & 1.140 & 1.140 & 1.139 & 1.137 & 1.137 & 1.136 \\\\ \n 30.00 -- 35.00 & 1.124 & 1.123 & 1.123 & 1.123 & 1.122 & 1.121 & 1.121 & 1.119 \\\\ \n 35.00 -- 40.00 & 1.107 & 1.107 & 1.107 & 1.107 & 1.107 & 1.106 & 1.106 & 1.103 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.086 & 1.087 & 1.086 & 1.087 & 1.086 & 1.086 & 1.085 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.056 & 1.056 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.016 & 1.048 & 1.074 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.019 & 1.056 & 1.087 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.019 & 1.055 & 1.086 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.018 & 1.053 & 1.083 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.017 & 1.051 & 1.079 & 1.101 & 1.117 & 1.127 & 1.134 & 1.138 \\\\ \n 10.50 -- 11.00 & 1.016 & 1.048 & 1.075 & 1.096 & 1.110 & 1.120 & 1.126 & 1.131 \\\\ \n 11.00 -- 11.50 & 1.015 & 1.045 & 1.071 & 1.090 & 1.104 & 1.113 & 1.119 & 1.124 \\\\ \n 11.50 -- 12.00 & 1.014 & 1.043 & 1.067 & 1.085 & 1.098 & 1.107 & 1.113 & 1.117 \\\\ \n 12.00 -- 12.50 & 1.014 & 1.040 & 1.063 & 1.080 & 1.093 & 1.101 & 1.106 & 1.111 \\\\ \n 12.50 -- 13.00 & 1.013 & 1.038 & 1.059 & 1.076 & 1.087 & 1.095 & 1.100 & 1.104 \\\\ \n 13.00 -- 14.00 & 1.012 & 1.035 & 1.055 & 1.070 & 1.080 & 1.088 & 1.092 & 1.096 \\\\ \n 14.00 -- 15.00 & 1.011 & 1.031 & 1.049 & 1.062 & 1.072 & 1.078 & 1.082 & 1.085 \\\\ \n 15.00 -- 16.00 & 1.010 & 1.028 & 1.044 & 1.056 & 1.065 & 1.070 & 1.074 & 1.076 \\\\ \n 16.00 -- 17.00 & 1.009 & 1.025 & 1.040 & 1.050 & 1.058 & 1.063 & 1.067 & 1.069 \\\\ \n 17.00 -- 18.00 & 1.008 & 1.023 & 1.036 & 1.046 & 1.053 & 1.057 & 1.060 & 1.062 \\\\ \n 18.00 -- 20.00 & 1.007 & 1.020 & 1.031 & 1.040 & 1.046 & 1.050 & 1.053 & 1.054 \\\\ \n 20.00 -- 22.00 & 1.006 & 1.017 & 1.026 & 1.033 & 1.039 & 1.042 & 1.044 & 1.045 \\\\ \n 22.00 -- 24.00 & 1.005 & 1.014 & 1.022 & 1.028 & 1.033 & 1.036 & 1.038 & 1.039 \\\\ \n 24.00 -- 26.00 & 1.004 & 1.012 & 1.019 & 1.024 & 1.028 & 1.030 & 1.032 & 1.033 \\\\ \n 26.00 -- 30.00 & 1.004 & 1.010 & 1.016 & 1.020 & 1.023 & 1.025 & 1.026 & 1.027 \\\\ \n 30.00 -- 35.00 & 1.003 & 1.008 & 1.012 & 1.015 & 1.018 & 1.019 & 1.020 & 1.021 \\\\ \n 35.00 -- 40.00 & 1.002 & 1.006 & 1.009 & 1.012 & 1.013 & 1.015 & 1.015 & 1.015 \\\\ \n 40.00 -- 60.00 & 1.001 & 1.004 & 1.006 & 1.008 & 1.009 & 1.010 & 1.010 & 1.010 \\\\ \n 60.00 -- 110.00 & 1.001 & 1.002 & 1.003 & 1.003 & 1.004 & 1.004 & 1.004 & 1.005 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.985 & 0.957 & 0.936 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.982 & 0.950 & 0.926 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.982 & 0.950 & 0.926 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.983 & 0.952 & 0.929 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.984 & 0.954 & 0.932 & 0.916 & 0.905 & 0.898 & 0.894 & 0.891 \\\\ \n 10.50 -- 11.00 & 0.985 & 0.956 & 0.935 & 0.919 & 0.909 & 0.903 & 0.899 & 0.895 \\\\ \n 11.00 -- 11.50 & 0.985 & 0.959 & 0.938 & 0.923 & 0.913 & 0.907 & 0.903 & 0.900 \\\\ \n 11.50 -- 12.00 & 0.986 & 0.961 & 0.941 & 0.927 & 0.918 & 0.911 & 0.908 & 0.905 \\\\ \n 12.00 -- 12.50 & 0.987 & 0.963 & 0.944 & 0.931 & 0.922 & 0.916 & 0.912 & 0.909 \\\\ \n 12.50 -- 13.00 & 0.988 & 0.965 & 0.947 & 0.934 & 0.925 & 0.920 & 0.916 & 0.913 \\\\ \n 13.00 -- 14.00 & 0.988 & 0.967 & 0.951 & 0.939 & 0.930 & 0.925 & 0.922 & 0.919 \\\\ \n 14.00 -- 15.00 & 0.990 & 0.971 & 0.955 & 0.944 & 0.937 & 0.932 & 0.929 & 0.927 \\\\ \n 15.00 -- 16.00 & 0.991 & 0.974 & 0.960 & 0.950 & 0.943 & 0.938 & 0.936 & 0.934 \\\\ \n 16.00 -- 17.00 & 0.991 & 0.976 & 0.963 & 0.954 & 0.948 & 0.944 & 0.941 & 0.939 \\\\ \n 17.00 -- 18.00 & 0.992 & 0.978 & 0.967 & 0.958 & 0.952 & 0.949 & 0.946 & 0.945 \\\\ \n 18.00 -- 20.00 & 0.993 & 0.981 & 0.971 & 0.963 & 0.958 & 0.954 & 0.952 & 0.951 \\\\ \n 20.00 -- 22.00 & 0.994 & 0.984 & 0.975 & 0.969 & 0.964 & 0.961 & 0.959 & 0.958 \\\\ \n 22.00 -- 24.00 & 0.995 & 0.986 & 0.979 & 0.973 & 0.969 & 0.967 & 0.965 & 0.964 \\\\ \n 24.00 -- 26.00 & 0.996 & 0.988 & 0.982 & 0.977 & 0.973 & 0.971 & 0.970 & 0.969 \\\\ \n 26.00 -- 30.00 & 0.996 & 0.990 & 0.985 & 0.981 & 0.978 & 0.976 & 0.975 & 0.974 \\\\ \n 30.00 -- 35.00 & 0.997 & 0.992 & 0.988 & 0.985 & 0.983 & 0.982 & 0.981 & 0.980 \\\\ \n 35.00 -- 40.00 & 0.998 & 0.994 & 0.991 & 0.989 & 0.987 & 0.986 & 0.985 & 0.985 \\\\ \n 40.00 -- 60.00 & 0.999 & 0.996 & 0.994 & 0.992 & 0.991 & 0.991 & 0.990 & 0.990 \\\\ \n 60.00 -- 110.00 & 0.999 & 0.998 & 0.997 & 0.997 & 0.996 & 0.996 & 0.996 & 0.996 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``longitudinal'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.672 & 0.677 & 0.686 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.674 & 0.680 & 0.689 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.677 & 0.682 & 0.691 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.680 & 0.684 & 0.692 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.683 & 0.688 & 0.695 & 0.702 & 0.709 & 0.713 & 0.717 & 0.721 \\\\ \n 10.50 -- 11.00 & 0.687 & 0.692 & 0.698 & 0.705 & 0.710 & 0.715 & 0.718 & 0.722 \\\\ \n 11.00 -- 11.50 & 0.692 & 0.695 & 0.701 & 0.708 & 0.714 & 0.716 & 0.718 & 0.725 \\\\ \n 11.50 -- 12.00 & 0.695 & 0.698 & 0.704 & 0.710 & 0.715 & 0.718 & 0.721 & 0.725 \\\\ \n 12.00 -- 12.50 & 0.700 & 0.703 & 0.708 & 0.713 & 0.718 & 0.721 & 0.723 & 0.728 \\\\ \n 12.50 -- 13.00 & 0.704 & 0.706 & 0.711 & 0.716 & 0.721 & 0.722 & 0.726 & 0.730 \\\\ \n 13.00 -- 14.00 & 0.710 & 0.713 & 0.717 & 0.722 & 0.725 & 0.727 & 0.730 & 0.733 \\\\ \n 14.00 -- 15.00 & 0.719 & 0.721 & 0.724 & 0.728 & 0.731 & 0.733 & 0.736 & 0.738 \\\\ \n 15.00 -- 16.00 & 0.727 & 0.728 & 0.732 & 0.735 & 0.737 & 0.740 & 0.741 & 0.743 \\\\ \n 16.00 -- 17.00 & 0.735 & 0.737 & 0.739 & 0.742 & 0.743 & 0.746 & 0.748 & 0.750 \\\\ \n 17.00 -- 18.00 & 0.742 & 0.744 & 0.746 & 0.750 & 0.750 & 0.753 & 0.755 & 0.755 \\\\ \n 18.00 -- 20.00 & 0.753 & 0.754 & 0.756 & 0.759 & 0.760 & 0.761 & 0.762 & 0.765 \\\\ \n 20.00 -- 22.00 & 0.767 & 0.768 & 0.769 & 0.771 & 0.773 & 0.773 & 0.775 & 0.775 \\\\ \n 22.00 -- 24.00 & 0.779 & 0.779 & 0.782 & 0.783 & 0.784 & 0.785 & 0.785 & 0.788 \\\\ \n 24.00 -- 26.00 & 0.791 & 0.791 & 0.793 & 0.794 & 0.793 & 0.795 & 0.795 & 0.795 \\\\ \n 26.00 -- 30.00 & 0.805 & 0.804 & 0.806 & 0.807 & 0.808 & 0.809 & 0.809 & 0.811 \\\\ \n 30.00 -- 35.00 & 0.823 & 0.823 & 0.824 & 0.824 & 0.826 & 0.826 & 0.828 & 0.828 \\\\ \n 35.00 -- 40.00 & 0.841 & 0.841 & 0.840 & 0.842 & 0.843 & 0.842 & 0.843 & 0.843 \\\\ \n 40.00 -- 60.00 & 0.866 & 0.867 & 0.866 & 0.868 & 0.868 & 0.866 & 0.868 & 0.870 \\\\ \n 60.00 -- 110.00 & 0.905 & 0.906 & 0.906 & 0.909 & 0.907 & 0.903 & 0.906 & 0.905 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse zero'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.325 & 1.316 & 1.301 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.321 & 1.311 & 1.295 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.316 & 1.307 & 1.291 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.310 & 1.301 & 1.288 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.303 & 1.295 & 1.283 & 1.272 & 1.261 & 1.254 & 1.249 & 1.244 \\\\ \n 10.50 -- 11.00 & 1.296 & 1.289 & 1.278 & 1.267 & 1.259 & 1.252 & 1.247 & 1.241 \\\\ \n 11.00 -- 11.50 & 1.289 & 1.283 & 1.273 & 1.262 & 1.254 & 1.250 & 1.246 & 1.238 \\\\ \n 11.50 -- 12.00 & 1.282 & 1.276 & 1.267 & 1.258 & 1.251 & 1.246 & 1.242 & 1.236 \\\\ \n 12.00 -- 12.50 & 1.274 & 1.270 & 1.261 & 1.253 & 1.247 & 1.242 & 1.239 & 1.233 \\\\ \n 12.50 -- 13.00 & 1.267 & 1.263 & 1.256 & 1.248 & 1.242 & 1.239 & 1.235 & 1.230 \\\\ \n 13.00 -- 14.00 & 1.257 & 1.254 & 1.247 & 1.241 & 1.236 & 1.232 & 1.229 & 1.225 \\\\ \n 14.00 -- 15.00 & 1.244 & 1.241 & 1.236 & 1.230 & 1.227 & 1.223 & 1.220 & 1.217 \\\\ \n 15.00 -- 16.00 & 1.232 & 1.230 & 1.225 & 1.221 & 1.217 & 1.215 & 1.213 & 1.211 \\\\ \n 16.00 -- 17.00 & 1.221 & 1.218 & 1.215 & 1.211 & 1.209 & 1.206 & 1.204 & 1.202 \\\\ \n 17.00 -- 18.00 & 1.210 & 1.208 & 1.206 & 1.202 & 1.200 & 1.197 & 1.195 & 1.195 \\\\ \n 18.00 -- 20.00 & 1.197 & 1.195 & 1.193 & 1.190 & 1.188 & 1.187 & 1.186 & 1.184 \\\\ \n 20.00 -- 22.00 & 1.180 & 1.179 & 1.177 & 1.175 & 1.173 & 1.172 & 1.171 & 1.171 \\\\ \n 22.00 -- 24.00 & 1.165 & 1.165 & 1.163 & 1.162 & 1.161 & 1.159 & 1.159 & 1.157 \\\\ \n 24.00 -- 26.00 & 1.153 & 1.153 & 1.151 & 1.150 & 1.150 & 1.149 & 1.149 & 1.149 \\\\ \n 26.00 -- 30.00 & 1.138 & 1.139 & 1.138 & 1.136 & 1.136 & 1.135 & 1.135 & 1.133 \\\\ \n 30.00 -- 35.00 & 1.121 & 1.121 & 1.120 & 1.119 & 1.119 & 1.118 & 1.117 & 1.117 \\\\ \n 35.00 -- 40.00 & 1.105 & 1.104 & 1.105 & 1.104 & 1.103 & 1.104 & 1.103 & 1.103 \\\\ \n 40.00 -- 60.00 & 1.084 & 1.083 & 1.084 & 1.083 & 1.083 & 1.084 & 1.083 & 1.081 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.055 & 1.055 & 1.053 & 1.054 & 1.057 & 1.055 & 1.056 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 2.029 & 2.023 & 2.022 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.620 & 1.620 & 1.618 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.504 & 1.504 & 1.502 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.444 & 1.444 & 1.443 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.405 & 1.405 & 1.404 & 1.404 & 1.402 & 1.401 & 1.400 & 1.500 \\\\ \n 10.50 -- 11.00 & 1.377 & 1.377 & 1.376 & 1.375 & 1.375 & 1.373 & 1.373 & 1.443 \\\\ \n 11.00 -- 11.50 & 1.354 & 1.354 & 1.354 & 1.352 & 1.351 & 1.353 & 1.353 & 1.403 \\\\ \n 11.50 -- 12.00 & 1.336 & 1.336 & 1.335 & 1.334 & 1.335 & 1.334 & 1.333 & 1.375 \\\\ \n 12.00 -- 12.50 & 1.320 & 1.320 & 1.320 & 1.319 & 1.319 & 1.319 & 1.318 & 1.351 \\\\ \n 12.50 -- 13.00 & 1.306 & 1.307 & 1.306 & 1.305 & 1.304 & 1.306 & 1.304 & 1.331 \\\\ \n 13.00 -- 14.00 & 1.289 & 1.289 & 1.289 & 1.288 & 1.288 & 1.288 & 1.287 & 1.308 \\\\ \n 14.00 -- 15.00 & 1.268 & 1.269 & 1.268 & 1.267 & 1.268 & 1.267 & 1.266 & 1.281 \\\\ \n 15.00 -- 16.00 & 1.251 & 1.251 & 1.250 & 1.251 & 1.251 & 1.250 & 1.250 & 1.261 \\\\ \n 16.00 -- 17.00 & 1.236 & 1.236 & 1.236 & 1.236 & 1.236 & 1.235 & 1.235 & 1.242 \\\\ \n 17.00 -- 18.00 & 1.223 & 1.222 & 1.223 & 1.222 & 1.223 & 1.221 & 1.221 & 1.227 \\\\ \n 18.00 -- 20.00 & 1.206 & 1.206 & 1.206 & 1.205 & 1.206 & 1.206 & 1.206 & 1.208 \\\\ \n 20.00 -- 22.00 & 1.186 & 1.186 & 1.186 & 1.186 & 1.186 & 1.186 & 1.185 & 1.187 \\\\ \n 22.00 -- 24.00 & 1.170 & 1.171 & 1.170 & 1.170 & 1.170 & 1.170 & 1.170 & 1.169 \\\\ \n 24.00 -- 26.00 & 1.157 & 1.157 & 1.156 & 1.156 & 1.157 & 1.157 & 1.157 & 1.158 \\\\ \n 26.00 -- 30.00 & 1.141 & 1.142 & 1.141 & 1.141 & 1.141 & 1.140 & 1.141 & 1.140 \\\\ \n 30.00 -- 35.00 & 1.122 & 1.122 & 1.122 & 1.122 & 1.122 & 1.122 & 1.121 & 1.121 \\\\ \n 35.00 -- 40.00 & 1.106 & 1.105 & 1.106 & 1.106 & 1.105 & 1.106 & 1.105 & 1.105 \\\\ \n 40.00 -- 60.00 & 1.085 & 1.084 & 1.085 & 1.084 & 1.084 & 1.085 & 1.084 & 1.083 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.055 & 1.055 & 1.054 & 1.054 & 1.057 & 1.055 & 1.056 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.995 & 0.986 & 0.970 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.116 & 1.102 & 1.081 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.170 & 1.156 & 1.133 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.199 & 1.185 & 1.163 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.215 & 1.202 & 1.182 & 1.163 & 1.146 & 1.135 & 1.127 & 1.075 \\\\ \n 10.50 -- 11.00 & 1.225 & 1.212 & 1.194 & 1.175 & 1.161 & 1.150 & 1.142 & 1.098 \\\\ \n 11.00 -- 11.50 & 1.230 & 1.218 & 1.201 & 1.184 & 1.170 & 1.161 & 1.155 & 1.114 \\\\ \n 11.50 -- 12.00 & 1.232 & 1.222 & 1.206 & 1.190 & 1.178 & 1.169 & 1.162 & 1.127 \\\\ \n 12.00 -- 12.50 & 1.232 & 1.223 & 1.208 & 1.194 & 1.182 & 1.174 & 1.168 & 1.137 \\\\ \n 12.50 -- 13.00 & 1.231 & 1.223 & 1.210 & 1.196 & 1.185 & 1.178 & 1.172 & 1.146 \\\\ \n 13.00 -- 14.00 & 1.228 & 1.220 & 1.209 & 1.197 & 1.188 & 1.181 & 1.176 & 1.154 \\\\ \n 14.00 -- 15.00 & 1.221 & 1.215 & 1.206 & 1.196 & 1.188 & 1.182 & 1.177 & 1.161 \\\\ \n 15.00 -- 16.00 & 1.214 & 1.209 & 1.200 & 1.193 & 1.186 & 1.181 & 1.177 & 1.165 \\\\ \n 16.00 -- 17.00 & 1.206 & 1.202 & 1.195 & 1.188 & 1.183 & 1.178 & 1.175 & 1.166 \\\\ \n 17.00 -- 18.00 & 1.198 & 1.195 & 1.189 & 1.183 & 1.179 & 1.174 & 1.171 & 1.165 \\\\ \n 18.00 -- 20.00 & 1.188 & 1.184 & 1.180 & 1.175 & 1.171 & 1.168 & 1.166 & 1.161 \\\\ \n 20.00 -- 22.00 & 1.173 & 1.171 & 1.168 & 1.164 & 1.161 & 1.159 & 1.157 & 1.154 \\\\ \n 22.00 -- 24.00 & 1.161 & 1.160 & 1.156 & 1.154 & 1.151 & 1.149 & 1.149 & 1.145 \\\\ \n 24.00 -- 26.00 & 1.150 & 1.149 & 1.146 & 1.144 & 1.143 & 1.141 & 1.141 & 1.140 \\\\ \n 26.00 -- 30.00 & 1.136 & 1.136 & 1.134 & 1.132 & 1.131 & 1.129 & 1.129 & 1.127 \\\\ \n 30.00 -- 35.00 & 1.119 & 1.119 & 1.117 & 1.117 & 1.115 & 1.115 & 1.113 & 1.113 \\\\ \n 35.00 -- 40.00 & 1.104 & 1.103 & 1.103 & 1.102 & 1.101 & 1.101 & 1.100 & 1.100 \\\\ \n 40.00 -- 60.00 & 1.084 & 1.083 & 1.083 & 1.082 & 1.082 & 1.083 & 1.082 & 1.080 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.054 & 1.055 & 1.053 & 1.054 & 1.057 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.018 & 1.053 & 1.081 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.021 & 1.062 & 1.095 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.021 & 1.062 & 1.096 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.020 & 1.060 & 1.094 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.020 & 1.057 & 1.089 & 1.114 & 1.130 & 1.140 & 1.146 & 1.145 \\\\ \n 10.50 -- 11.00 & 1.018 & 1.055 & 1.085 & 1.108 & 1.124 & 1.133 & 1.139 & 1.142 \\\\ \n 11.00 -- 11.50 & 1.017 & 1.052 & 1.080 & 1.102 & 1.117 & 1.127 & 1.133 & 1.137 \\\\ \n 11.50 -- 12.00 & 1.017 & 1.049 & 1.076 & 1.096 & 1.111 & 1.120 & 1.126 & 1.132 \\\\ \n 12.00 -- 12.50 & 1.016 & 1.046 & 1.072 & 1.091 & 1.105 & 1.113 & 1.119 & 1.125 \\\\ \n 12.50 -- 13.00 & 1.015 & 1.043 & 1.068 & 1.086 & 1.099 & 1.108 & 1.112 & 1.119 \\\\ \n 13.00 -- 14.00 & 1.013 & 1.040 & 1.062 & 1.079 & 1.091 & 1.099 & 1.104 & 1.111 \\\\ \n 14.00 -- 15.00 & 1.012 & 1.036 & 1.056 & 1.071 & 1.082 & 1.089 & 1.093 & 1.099 \\\\ \n 15.00 -- 16.00 & 1.011 & 1.032 & 1.050 & 1.064 & 1.073 & 1.080 & 1.084 & 1.090 \\\\ \n 16.00 -- 17.00 & 1.010 & 1.029 & 1.045 & 1.057 & 1.067 & 1.072 & 1.076 & 1.081 \\\\ \n 17.00 -- 18.00 & 1.009 & 1.026 & 1.041 & 1.052 & 1.060 & 1.065 & 1.068 & 1.073 \\\\ \n 18.00 -- 20.00 & 1.008 & 1.023 & 1.036 & 1.045 & 1.053 & 1.057 & 1.060 & 1.063 \\\\ \n 20.00 -- 22.00 & 1.007 & 1.019 & 1.030 & 1.038 & 1.044 & 1.048 & 1.050 & 1.053 \\\\ \n 22.00 -- 24.00 & 1.006 & 1.016 & 1.025 & 1.032 & 1.037 & 1.040 & 1.043 & 1.044 \\\\ \n 24.00 -- 26.00 & 1.005 & 1.014 & 1.022 & 1.028 & 1.032 & 1.035 & 1.037 & 1.038 \\\\ \n 26.00 -- 30.00 & 1.004 & 1.012 & 1.018 & 1.023 & 1.026 & 1.029 & 1.030 & 1.031 \\\\ \n 30.00 -- 35.00 & 1.003 & 1.009 & 1.014 & 1.017 & 1.020 & 1.022 & 1.023 & 1.023 \\\\ \n 35.00 -- 40.00 & 1.002 & 1.007 & 1.010 & 1.013 & 1.015 & 1.017 & 1.017 & 1.018 \\\\ \n 40.00 -- 60.00 & 1.002 & 1.004 & 1.007 & 1.009 & 1.010 & 1.011 & 1.012 & 1.012 \\\\ \n 60.00 -- 110.00 & 1.001 & 1.002 & 1.003 & 1.004 & 1.004 & 1.005 & 1.005 & 1.005 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.983 & 0.952 & 0.931 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.980 & 0.945 & 0.920 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.980 & 0.945 & 0.919 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.981 & 0.946 & 0.921 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.981 & 0.949 & 0.924 & 0.908 & 0.897 & 0.891 & 0.887 & 0.888 \\\\ \n 10.50 -- 11.00 & 0.982 & 0.951 & 0.928 & 0.912 & 0.901 & 0.895 & 0.891 & 0.890 \\\\ \n 11.00 -- 11.50 & 0.983 & 0.953 & 0.931 & 0.916 & 0.906 & 0.899 & 0.895 & 0.893 \\\\ \n 11.50 -- 12.00 & 0.984 & 0.956 & 0.934 & 0.919 & 0.910 & 0.903 & 0.900 & 0.896 \\\\ \n 12.00 -- 12.50 & 0.985 & 0.958 & 0.937 & 0.923 & 0.914 & 0.908 & 0.904 & 0.900 \\\\ \n 12.50 -- 13.00 & 0.986 & 0.960 & 0.940 & 0.927 & 0.918 & 0.911 & 0.908 & 0.904 \\\\ \n 13.00 -- 14.00 & 0.987 & 0.963 & 0.945 & 0.932 & 0.923 & 0.917 & 0.914 & 0.910 \\\\ \n 14.00 -- 15.00 & 0.988 & 0.967 & 0.950 & 0.938 & 0.930 & 0.925 & 0.922 & 0.917 \\\\ \n 15.00 -- 16.00 & 0.989 & 0.970 & 0.955 & 0.944 & 0.936 & 0.931 & 0.928 & 0.924 \\\\ \n 16.00 -- 17.00 & 0.990 & 0.973 & 0.959 & 0.949 & 0.941 & 0.937 & 0.934 & 0.931 \\\\ \n 17.00 -- 18.00 & 0.991 & 0.975 & 0.962 & 0.953 & 0.946 & 0.943 & 0.940 & 0.936 \\\\ \n 18.00 -- 20.00 & 0.992 & 0.978 & 0.967 & 0.958 & 0.953 & 0.949 & 0.946 & 0.944 \\\\ \n 20.00 -- 22.00 & 0.993 & 0.981 & 0.972 & 0.965 & 0.960 & 0.956 & 0.955 & 0.952 \\\\ \n 22.00 -- 24.00 & 0.994 & 0.984 & 0.976 & 0.970 & 0.965 & 0.963 & 0.961 & 0.960 \\\\ \n 24.00 -- 26.00 & 0.995 & 0.986 & 0.979 & 0.974 & 0.970 & 0.967 & 0.966 & 0.965 \\\\ \n 26.00 -- 30.00 & 0.996 & 0.989 & 0.983 & 0.978 & 0.975 & 0.973 & 0.972 & 0.971 \\\\ \n 30.00 -- 35.00 & 0.997 & 0.991 & 0.987 & 0.983 & 0.981 & 0.979 & 0.978 & 0.978 \\\\ \n 35.00 -- 40.00 & 0.998 & 0.993 & 0.990 & 0.987 & 0.985 & 0.984 & 0.983 & 0.983 \\\\ \n 40.00 -- 60.00 & 0.998 & 0.996 & 0.993 & 0.992 & 0.990 & 0.989 & 0.989 & 0.989 \\\\ \n 60.00 -- 110.00 & 0.999 & 0.998 & 0.997 & 0.996 & 0.996 & 0.995 & 0.995 & 0.995 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_psi2s8} \n \\end{table}\n\n\n\\clearpage\n\n\\section{Systematic uncertainties}\n\\label{sec:syst}\n\n\\begin{figure} [!h]\n \\begin{center}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_jpsiP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_jpsiNP.eps}\\\\\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_psi2sP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_psi2sNP.eps} \n \\caption{Statistical and systematic contributions to the fractional uncertainty on the prompt (left column) and non-prompt (right column)\n $\\jpsi$ (top row) and $\\ensuremath{\\psi(2\\mathrm{S})}$ (bottom row) cross-sections for 7 \\TeV, \n shown for the region $0.75<|y|<1.00$.}\n \\label{fig:fracsyst_xs}\n \\end{center}\n\\end{figure}\n\n\\begin{figure} [!h]\n \\begin{center}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_fNPjpsi.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_fNPpsi2s.eps}\\\\\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_ratioP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_ratioNP.eps} \n \\caption{Breakdown of the contributions to the fractional uncertainty on the non-prompt fractions for $\\jpsi$ (top left) and $\\ensuremath{\\psi(2\\mathrm{S})}$ (top right), \n and the prompt (bottom left) and non-prompt (bottom right) ratios for 7 \\TeV, \n shown for the region $0.75<|y|<1.00$.}\n \\label{fig:fracsyst_ratio}\n \\end{center}\n\\end{figure}\n\n\nThe sources of systematic uncertainties that are applied to the $\\psi$ double differential cross-section\nmeasurements are from uncertainties in: the luminosity determination; muon and trigger efficiency corrections; inner detector tracking efficiencies; the fit model parametrization; and due to bin migration corrections.\nFor the non-prompt fraction and ratio measurements the systematic uncertainties are assessed in the same \nmanner as for the uncertainties on the cross-section, \nexcept that in these ratios some systematic uncertainties, such as the luminosity uncertainty, cancel out.\nThe sources of systematic uncertainty evaluated for the prompt and non-prompt $\\ensuremath{\\psi}$ cross-section measurements, along with the minimum, maximum and median values, are listed in Table~\\ref{table:systlist}.\nThe largest contributions, which originate from the trigger and fit model uncertainties, are typically for the \nhigh $\\mbox{$p_{\\text{T}}$}$ intervals and are due to the limited statistics of the efficiency maps (for the trigger), and the data sample (for the fit model).\n\nFigures~\\ref{fig:fracsyst_xs} and~\\ref{fig:fracsyst_ratio} show, for a representative interval,\nthe impact of the considered uncertainties\non the production cross-section, as well as the non-prompt fraction and ratios for $7$~\\TeV\\ data.\nThe impact is very similar at 8 \\TeV.\n\n\n\\begin{table}[h!]\n \\caption{Summary of the minimum and maximum contributions along with the median value of the systematic uncertainties as percentages for the prompt and non-prompt $\\psi$ cross-section results.\n Values are quoted for 7 and 8 \\TeV\\ data.}\n \\centering\n \\begin{tabular}{l || c c c | c c c }\n \\hline \n & \\multicolumn{3}{c|}{7 \\TeV\\ [\\%]} & \\multicolumn{3}{c}{8 \\TeV\\ [\\%]} \\\\\n \\hline \n Source of systematic uncertainty & Min & Median & Max & Min & Median & Max \\\\\n Luminosity & 1.8 & 1.8 & 1.8 & 2.8 & 2.8 & 2.8 \\\\\n Muon reconstruction efficiency & 0.7 & 1.2 & 4.7 & 0.3 & 0.7 & 6.0 \\\\\n Muon trigger efficiency & 3.2 & 4.7 & 35.9 & 2.9 & 7.0 & 23.4 \\\\\n Inner detector tracking efficiency & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\\\\n Fit model parameterizations & 0.5 & 2.2 & 22.6 & 0.26 & 1.07 & 24.9 \\\\\n Bin migrations & 0.01 & 0.1 & 1.4 & 0.01 & 0.3 & 1.5 \\\\ \\hline\n Total & 4.2 & 6.5 & 36.3 & 4.4 & 8.1 & 27.9 \\\\\n \\hline \n \\hline\n \\end{tabular}\n\\label{table:systlist}\n\\end{table}\n\n\n\\setdescription{font=\\normalfont\\itshape}\n\n\\begin{description}[style=unboxed,leftmargin=0cm]\n\\item[Luminosity] \\hfill \\\\\nThe uncertainty on the integrated luminosity is $1.8\\%$ ($2.8\\%$) for the 7~\\TeV\\ (8~\\TeV) data-taking period. \nThe methodology used to determine these uncertainties is described in Ref.~\\cite{Aad:2013ucp}.\nThe luminosity uncertainty is only applied to the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ cross-section results.\n\n\n\\item[Muon reconstruction and trigger efficiencies] \\hfill \\\\\nTo determine the systematic uncertainty on the muon reconstruction and trigger efficiency maps, each of the maps is reproduced in 100 pseudo-experiments.\nThe dominant uncertainty in each bin is statistical and hence any bin-to-bin correlations are neglected.\nFor each pseudo-experiment a new map is created by varying independently each bin content according to a Gaussian distribution about its estimated value, \ndetermined from the original map.\nIn each pseudo-experiment, the total weight is recalculated for each dimuon $\\mbox{$p_{\\text{T}}$}$ and $|y|$ interval of the analysis.\nThe RMS of the total weight pseudo-experiment distributions for each efficiency type is used as the systematic uncertainty,\nwhere any correlation effects between the muon and trigger efficiencies can be neglected.\n\n\nThe ID tracking efficiency is in excess of $99.5\\%$ \\cite{Aad2012dlq}, and\nan uncertainty of 1\\% is applied to account for the ID dimuon reconstruction inefficiency (0.5\\% per muon, added coherently). This\nuncertainty is applied to the differential cross-sections and is assumed to cancel in the fraction of non-prompt to inclusive production for \\jpsi\\ and \n\\ensuremath{\\psi(2\\mathrm{S})}\\ and in the ratios of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production.\n\nFor the trigger efficiency $\\epsilon_\\mathrm{trig}$, in addition to the trigger efficiency map, \nthere is an additional correction term that accounts for inefficiencies due to correlations between the two trigger muons, such as the dimuon opening angle.\nThis correction is varied by its uncertainty, and the shift in the resultant total weight relative to its central value is added in quadrature to the uncertainty from the map.\nThe choice of triggers is known \\cite{Aad:2014cqa} to introduce a small lifetime-dependent efficiency loss but it is determined to have a negligible effect on the prompt and non-prompt yields and no correction is applied in this analysis.\nSimilarly, the muon reconstruction efficiency corrections of prompt and non-prompt signals are found to be consistent within the statistical uncertainties of the efficiency measurements, and no additional uncertainty is applied.\n\n\n \\item[Fit model uncertainty] \\hfill \\\\\nThe uncertainty due to the fit procedure is determined by varying one component at a\ntime in the fit model described in Section \\ref{sec:method:fit}, creating a set of new fit models. For each new\nfit model, all measured quantities are recalculated, and in each $\\mbox{$p_{\\text{T}}$}$ and $|y|$ interval the spread of\nvariations around the central fit model is used as its systematic uncertainty.\nThe variations of the fit model also account for possible uncertainties due to final-state radiation.\nThe following variations to the central model fit are evaluated: \n\n\\begin{itemize}\n\n \\item signal mass model --- using double Gaussian models in place of the Crystal Ball plus Gaussian model; variation of the\n $\\alpha$ and $n$ parameters of the $B$ model, which are originally fixed;\n\n \\item signal pseudo-proper decay time model --- a double exponential function is used to describe the pseudo-proper decay time distribution for the \n $\\ensuremath{\\psi}$ non-prompt signal;\n\n \\item background mass models --- variations of the mass model using exponentials functions, or quadratic Chebyshev polynomials to describe the components of prompt,\n non-prompt and double-sided background terms;\n\n \\item background pseudo-proper decay time model --- a single exponential function was considered for the non-prompt component;\n\n \\item pseudo-proper decay time resolution model --- using a single Gaussian function in place of the double Gaussian function to model the lifetime resolution\n (also prompt lifetime model); and variation of the mixing terms for the two Gaussian components of this term.\n\n\\end{itemize}\n\nOf the variations considered, it is typically the parametrizations of the signal mass model and pseudo-proper decay time resolution model that dominate the contribution to the fit model uncertainty.\n\n \\item[Bin migrations] \\hfill \\\\\nAs the corrections to the results due to bin migration effects are factors close to unity in all regions, the difference between the correction factor and unity is applied as the uncertainty. \n\n\\end{description}\n\nThe variation of the acceptance corrections with spin-alignment is treated\nseparately, and scaling factors supplied in Appendix~\\ref{sec:spincorrection}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThermal contact between two macroscopic bodies initially at different temperatures corresponds to a situation where the heat transfer between the two bodies is ensured by a thin diathermal interface. The latter may be an immaterial interface between two solids or a diathermal wall between two fluids. Over a time window during which the two macroscopic bodies have negligible energy variations, they behave as thermostats with constant thermodynamic temperatures, while the interface is a mesoscopic system with traceable configurations. After a long enough time inside the considered time window the interface tends to a stationary non equilibrium state where the instantaneous heat current which it receives from a thermostat has a non-vanishing mean value. Even if the interface is described by a model where its degrees of freedom obey a (deterministic or stochastic) microscopic dynamics, there is no general framework, such as Gibbs equilibrium ensemble theory, which would allow to determine the probability distribution of the interface configurations and the corresponding mean instantaneous heat current. Therefore it is most valuable to exhibit solvable models which would shed some light into the dependence of the heat instantaneous current upon the model parameters and the temperatures of the two thermostats.\n\n\nSuch a solvable model has been introduced by Racz and Zia in 1994 \\cite{RaczZia1994}. They consider a chain of classical spins with periodic boundary conditions and interacting with a nearest-neighbor ferromagnetic (Ising) interaction. They endow it with a Glauber stochastic dynamics with single-spin flips at a time and such that the spins on odd and even lattice sites are flipped by thermostats at two different temperatures. In our view, the model may be seen as a zig-zag shaped chain inside a very thin strip between two half-planes occupied by two different thermostats, and where the odd (even) sites are located on the left (right) side of the strip. The stationary two-spin correlations at any distance as well as higher order spin correlations have been extensively studied in Refs.\\cite{RaczZia1994,SchmittmannSchmuser2002,MobiliaZiaSchmittmann2004,MobiliaSchmittmannZia2005}.\n\n\n\nWe point out that the latter model indeed satisfies two requirements needed for a correct description of a thermal contact between two macroscopic bodies during a transient time window where their temperature can be considered as constant. First the contact must be mediated by changes in the internal energy of the spin interface. Second, if the energies of the macroscopic bodies were kept tracked of, the transition rates for both the interface configurations and these two energies would obey the detailed balance with the microcanonical equilibrium probability distribution for these variables of the whole system. Then, in the infinite time limit the two bodies and the interface would be at the same temperature; in a time window where the energy variations of the macroscopic bodies are negligible, the transition rates for the interface depend only on the temperatures of the macroscopic bodies and they obey the local detailed balance \\cite{Derrida2007,BauerCornu2015}\\footnote{In Ref.\\cite{Derrida2007} the terminology \"generalized\" detailed balance is used.}. \nIn the case of a spin interface the two corresponding requirements become: 1) a transition between two spin configurations involves only one thermostat; 2) the transition rate involving a given thermostat obey the detailed balance at the same temperature. For the present two-temperature Ising chain the first requirement is obviously fulfilled\\footnote{We notice that the first requirement is not satisfied by the Ising chain model of Ref.\\cite{CornuHilhorst2017}, where two thermostats at different temperatures act on every spin : the latter model does not describe a situation of thermal contact.}, and the\nsimplest transition rates which fulfill the second requirement are those chosen by Glauber in the case of an Ising chain in contact with a unique thermostat\\cite{Glauber1963}. (We recall that Glauber looked for single-spin flip dynamics such that in the infinite time limit the spin chain indeed relaxes to the canonical equilibrium state at the thermostat temperature and then he chose the simplest transition rates.)\n\n\\medskip\n\nOn the other hand, the quantities of interest for exchange processes which have been considered over the last three decades, both theoretically and experimentally, are amounts of microscopically conserved entities (matter, energy, ...), which are exchanged over a very long time (in the case of thermal contact the corresponding quantities are the heat amounts received by the interface from each thermostat during a fixed long time interval). They have been focused on because the scaled generating function for their cumulants per unit of time in the long time limit as well as its Laplace transform, the large deviation function of the corresponding time-integrated current, have been shown to obey generic symmetry relations, the so-called fluctuation relations. The latter relations are derived from properties of the system dynamics and they are a milestone of \nthe stochastic thermodynamics theory (For a review see Ref.\\cite{Seifert2012}.) \n\n\nIn this context it is also interesting to have at hand solvable models where the statistics of the time-integrated currents can be calculated. For instance, such a model has been exhibited in the case where the heat transfer between two thermostats is ensured by a wire. The energy quanta are represented as particles whose stochastic dynamics is a Symmetric Simple Exclusion Process (SSEP) with adequate boundary conditions: particles with hard cores hop on the sites of a one-dimensional lattice with the same hopping rate in both directions, but have different in-coming and out-going rates at the two lattice ends in contact with two particle reservoirs with different chemical potentials. Then all cumulants of the heat coming out of one thermostat per unit of time can be calculated in the long time limit \\cite{DerridaDoucotRoche2004,BodineauDerrida2007} .\n\n\nIn the present paper we consider a generalized version of the diathermal interface model of Ref.\\cite{RaczZia1994}: the two Glauber dynamics which flip the spins on sites with odd or even indices respectively have not only different temperatures but also different kinetic parameters (see Sec.\\ref{secModel}). \nOur aim is to calculate the scaled generating function of the joint cumulants per unit of time for the heats flowing out of the two thermostats in the long time limit. \n\nThe difference between the kinetic constants is relevant for two macroscopic bodies made with different materials. Some of the corresponding kinetic effects have been previously investigated in our study of a very simple model for a one-dimensional interface between two half-planes occupied by bodies at two different temperatures : a model of independent two-spin pairs where the left-side (right-side) spin in each pair is flipped only by the thermostat on the same side according to a Glauber dynamics. The whole statistics for the spin configurations and the heat amounts exchanged by every pair with the thermostats have been calculated explicitly \\cite{CornuBauer2013}.\n\n\n\n \n\\medskip\nThe method which we use to obtain the scaled generating function for the heat cumulants in the present model is the following.\nFrom its definition this function can be obtained as the largest eigenvalue of the modified Markov matrix which rules the evolution of the joint probability for the system configurations and \n the heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ received on each (odd or even) sublattice. \n \nIn order to study energy exchanges more conveniently, instead of considering the spin configurations, we rather formulate the problem in terms of the position configurations for the domain walls, which sit on the dual lattice. In other words we consider the well-known lattice gas representation on the {\\it dual\\,} lattice (`antiparallel adjacent spin pair' $\\leftrightarrow$ `particle' and `parallel adjacent spin pair' $\\leftrightarrow$ `hole') where a particle is in fact a domain wall. The mapping of Glauber spin dynamics to the particle dynamics then includes the possibility for the creation and annihilation of adjacent particle pairs, which correspond to the injection or the loss of energy in the chain respectively\\footnote{The latter correspondance has been used for instance in Ref.\\cite{FaragoPitard2007} for the calculation of the scaled generating function of the energy injected in an Ising spin ring through the random flips of one spin while all other spins evolve according to a Glauber dynamics at zero temperature which dissipates energy along the ring.}. Our model with two temperatures and two kinetic parameters is mapped to a reaction-diffusion system with two different creation (annihilation) rates as well as two different hopping rates on spatially alternating sites\\footnote{In Ref.\\cite{MobiliaSchmittmannZia2005} the mapping has been used in the reverse sense in order to study the relaxation towards the stationary state for the reaction-diffusion system from results obtained for the Ising spin chain dynamics with two temperatures but a unique kinetic parameter.}.\n\n\n\nThe crucial point is that, in a suitable basis for the representation of the configurations of domain wall positions, the Markov matrix for the evolution of the configuration probability involves only products of two operators : the Markov matrix is mapped to a free fermion Hamiltonian \\cite{Schutz2001}.\nMoreover the mapping can be readily generalized for the modified Markov matrix which rules the evolution of the joint probability for a configuration of domain wall positions and the amounts of heat $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$.\nThe latter matrix can be diagonalized by using free fermions techniques: \n Jordan-Wigner transformation and antiperiodic Fourier transform. Thus we obtain a block diagonal matrix, made of $4\\times 4$ blocks, which can be straighforwardly diagonalized by introducing four pseudo-fermion operators. \n (In the case of two pseudo-fermion operators and the associated Bogoliubov-like transformation, see for instance Ref.\n\\cite{GrynbergStinchcombe1995PRL,GrynbergStinchcombe1995PRE}.)\n The largest eigenvalue of the modified Markov matrix is obtained by filling up all pseudo-fermion states associated with an eigenvalue with a positive real part. \n\n \n\n\\medskip\n\n The paper is organized as follows. The model is defined in Sec.\\ref{secModel} and the relaxation time for the mean global magnetization on each sublattice is calculated. Moreover by solving a hierarchy of equations for the stationary global two-spin correlations at any distance (see Appendix \\ref{secTwospincorrelations}), we determine the mean instantaneous energy current which flows from each thermostat into the spin lattice in the stationary state. The mapping to the dynamics of domain walls on the dual lattice and the associated modified matrix is derived in Sec.\\ref{SecMapping}. Its eigenvalues are determined in Sec.\\ref{SecEigenvalues} by free fermion techniques. The cumulants are obtained in Sec.\\ref{SecCumulants} and various physical regimes are discussed. In conclusion we summarize some finite-size effects and their possible cancellation in heat cumulants. \n\n \n\n \n\\section{Model}\n\\label{secModel}\n\n\\subsection{Description of a thermalization process}\n\nWe consider a one-dimensional lattice with a finite even number of sites $L=2N$, where each site $j$ is occupied by a classical spin $s_j$ ($s_j=\\pm 1$, $j=1,\\ldots, 2N$) with periodic boundary condition $s_{j+2N}=s_j$. Spins interact via the Ising ferromagnetic nearest-neighbor interaction with coupling $K>0$: the energy of a spin configuration $\\mathbf{s}$ is\n\\begin{equation}\n\\label{defEnergie}\n{\\cal E}(\\mathbf{s})=-K \\sum_{j=1}^{2N}s_{j}s_{j+1}.\n\\end{equation}\nWhen the spin at site $j$ is flipped, the energy variation of the Ising chain is equal to\n\\begin{equation}\n\\label{varEnergie}\n\\Delta {\\cal E}(s_j\\to -s_j)= s_j\\frac{s_{j-1}+s_{j+1}}{2} \\Delta E,\n\\end{equation}\nwith $\\Delta E=4K$. This variation can take the values $+\\Delta E$, 0, or $-\\Delta E$, \n\n\nThe model is endowed with a stochastic dynamics where the spin flips at odd (even) sites are due to energy exchanges with a macroscopic body at temperature $T_\\text{\\mdseries o}$ ($T_\\text{\\mdseries e}$) in the course of a thermalization process of the two macroscopic bodies. The transition rates must obey local detailed balance \\cite{Derrida2007,BauerCornu2015}: \nthe transition rates $w(s_j \\to -s_j)$ and $w(-s_j \\to s_j)$ for two reversed flips of the spin at site $j$, while all other spins are kept fixed, must obey the ratio\n\\begin{equation}\n\\label{LDB}\n\\frac{w(s_j \\to -s_j)}{w(-s_j \\to s_j)}=\\text{\\mdseries e}^{-\\beta_j \\Delta {\\cal E}(s_j\\to -s_j)},\n\\end{equation}\nwhere $\\beta_j$ is the inverse temperature of the thermostat acting on site $j$ : $\\beta_j=1\/(k_{\\scriptscriptstyle B} T_j)$ where $k_{\\scriptscriptstyle B}$ is Boltzmann constant and $T_j$ is equal either to $T_\\text{\\mdseries o}$ or $T_\\text{\\mdseries e}$, depending on the parity of $j$.\nAs shown by Glauber \\cite{Glauber1963} in the case of a unique temperature, the simplest transition rates which obey the constraint \\eqref{LDB} read\n\\begin{equation}\n\\label{deftauxtrans}\nw(s_j \\to -s_j)=\\frac{\\nu_j}{2} \\left[1-\\gamma_j \\frac{s_j \\left(s_{j-1}+s_{j+1}\\right)}{2}\\right],\n\\end{equation}\nwhile $w(-s_j \\to s_j)$ is given by the latter expression where $s_j$ is replaced by $-s_j$.\nIn \\eqref{deftauxtrans} $\\gamma_j $ is the thermodynamic parameter at site $j$,\n\\begin{equation}\n\\label{defgammaa}\n\\gamma_j =\\tanh\\left(\\frac{\\beta_j \\Delta E}{2}\\right)\n\\end{equation}\nwhere $\\gamma_j=\\gamma_\\text{\\mdseries o}$ or $\\gamma_\\text{\\mdseries e}$, depending on the parity of the site index,\nand $\\nu_j$ is the kinetic parameter $\\nu_j$ at site $j$. The latter is not determined by the local detailed balance; it can be interpreted as the mean frequency at which the macroscopic body tries to flip the spin at site $j$. Therefore we set $\\nu_j=\\nu_\\text{\\mdseries o}$ or $\\nu_\\text{\\mdseries e}$, depending on the parity of the site index (note that when the two kinetic parameters are equal the present model coincides with that of Ref.\\cite{RaczZia1994}).\nSince the time scale is arbitrary, it is convenient to introduce the dimensionless kinetic parameters $\\overline{\\nu}_\\text{\\mdseries a}$, with $\\text{a}=\\text{o}$ or $\\text{e}$, defined as\n\\begin{equation}\n\\label{defDimensionlessNu}\n\\overline{\\nu}_{\\text a}= \\frac{\\nu_{\\text a}}{\\nu_\\text{\\mdseries o} +\\nu_\\text{\\mdseries e}}.\n\\end{equation}\nThey satisfy the relation $\\overline{\\nu}_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e}=1$ and, apart from the arbitrary time scale, the model has only three independent parameters, $\\gamma_\\text{\\mdseries o}$, $\\gamma_\\text{\\mdseries e}$ and $\\overline{\\nu}_\\text{\\mdseries o}$.\n\n\n\nThe probability $P(\\mathbf{s};t)$ for the system to be in spin configuration $\\mathbf{s}$ at time $t$ evolves according to the master equation\n\\begin{equation}\n\\label{MarkovSpins}\n\\frac{\\text{\\mdseries d} P(\\mathbf{s};t)}{\\text{\\mdseries d} t}=\\sum_j^{2N}w(-s_j \\to s_j)P(\\mathbf{s}_j;t)- \\left(\\sum_j^{2N} w(s_j \\to -s_j) \\right) P(\\mathbf{s};t)\n\\end{equation}\nwhere $\\mathbf{s}_j$ denotes the spin configuration obtained from $\\mathbf{s}$ by changing $s_j$ into $-s_j$.\n The number of configurations is finite, and the transition rates allow the system to evolve from any configuration to any other one after a suitable succession of transitions. Therefore there is a unique stationary solution of the master equation. In the following we focus on the mean values of global quantities and denote $\\Esp{\\cdots}$ and $\\Espst{\\cdots}$ the expectation values calculated with the time-dependent probability $P(\\mathbf{s};t)$ and the stationary probability $P_\\text{\\mdseries st}(\\mathbf{s})$ respectively.\n\n\\subsection{Relaxation of the mean global sublattice magnetizations}\n\\label{secMagnetization}\n\n\n The transition rates are invariant under a global flip of the spins, so that\n a configuration and the corresponding one where all spins are flipped have the same probability in the stationary state. As a result all stationary correlations for an odd number of spins vanish identically; in particular $\\Espst{s_j}=0$. As a consequence the mean values of the global magnetizations on the two sublattices, $M_\\text{\\mdseries o}=\\sum_{n=1}^{N}s_{2n-1}$ and $M_\\text{\\mdseries e}=\\sum_{n=1}^{N} s_{2n}$ respectively, vanish in the stationary state,\n \\begin{equation}\n \\Espst{M_\\text{\\mdseries o}}= \\Espst{M_\\text{\\mdseries e}}=0.\n \\end{equation}\n \nThe relaxation of the mean values of global sublattice magnetizations is readily studied.\nAs in the case of the homogeneous spin chain considered by Glauber \\cite{Glauber1963}, the evolution equation for the mean value of the spin at site $j$ reads\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\Esp{s_j}}{\\text{\\mdseries d} t} = - 2 \\Esp{s_j w(s_j \\to -s_j)}.\n\\end{equation}\nAccording to the expression of the transition rates \\eqref{deftauxtrans}\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\Esp{s_j}}{\\text{\\mdseries d} t} = -\\nu_j \\left[\\Esp{s_j} - \\gamma_j \\frac{\\Esp{s_{j-1}}+\\Esp{s_{j+1}}}{2} \\right].\n\\end{equation}\nThen the coupled evolutions of the magnetizations on the two sublattices read\n\\begin{eqnarray}\n\\frac{\\text{\\mdseries d} \\Esp{M_\\text{\\mdseries o}}}{\\text{\\mdseries d} t} &=& - \\nu_\\text{\\mdseries o}\\left[ \\Esp{M_\\text{\\mdseries o}} - \\gamma_\\text{\\mdseries o} \\Esp{M_\\text{\\mdseries e}}\\right]\n\\\\ \\nonumber\n\\frac{\\text{\\mdseries d} \\Esp{M_\\text{\\mdseries e}}}{\\text{\\mdseries d} t} &=& - \\nu_\\text{\\mdseries e}\\left[ \\Esp{M_\\text{\\mdseries e}} - \\gamma_\\text{\\mdseries e} \\Esp{M_\\text{\\mdseries o}}\\right].\n\\end{eqnarray}\nFrom these equations we retrieve that both mean magnetizations vanish in the stationary state, as predicted by symmetry arguments. The matrix associated with this system of linear equations has two strictly negative eigenvalues \n$\\tfrac{1}{2}(\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e})\\left[-1\\pm\\sqrt{(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}\\right]$, each of which is associated with a couple of right and left eigenvectors (the eigenvalues are negative because\n$(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}=1-4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})<1$). For generic values of the initial magnetizations $M_\\text{\\mdseries o}$ and $M_\\text{\\mdseries e}$, the inverse relaxation time $1\/t_\\text{\\mdseries rel}$ to their stationary value is given by the opposite of the negative eigenvalue with the smallest modulus, and the relaxation time $t_\\text{\\mdseries rel}$ reads\n\\begin{equation}\n\\label{trel}\nt_\\text{\\mdseries rel}=\\frac{2}{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}} \\left[ 1-\\sqrt{(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}\\right]^{-1}.\n\\end{equation}\n\n\n\n\n\n\\subsection{Mean global heat current in the stationary state}\n\nThe mean instantaneous heat current $\\Esp{j_k}$ received by the spin chain at site $k$ from the thermostat at temperature $T_k$ is equal to the expectation value of the variation of the chain energy when the spin $s_k$ is flipped times the transition rate for the flip. According to the expressions for the energy variation \\eqref{varEnergie}\nand for the transition rates \\eqref{deftauxtrans}, the mean instantaneous current reads\n\\begin{equation}\n\\Esp{j_k}=K\\nu_k\\left[ -\\gamma_k -\\gamma_k \\Esp{s_{k-1}s_{k+1}} + \\Esp{s_{k-1} s_k} +\\Esp{s_k s_{k+1}}\\right].\n\\end{equation}\nTherefore the stationary mean value of the global heat current coming from the thermostat acting on spins at even sites, namely\n$\nJ_\\text{\\mdseries e}= \\sum_{n=1}^N j_{2n},\n$\nis determined as\n\\begin{equation}\n\\label{ExpressionJe}\n\\Espst{J_\\text{\\mdseries e}}= N K \\nu_\\text{\\mdseries e} \\left[ -\\gamma_\\text{\\mdseries e} -\\gamma_\\text{\\mdseries e} D^\\text{oo}_2 + D^\\text{oe}_1 + D^\\text{eo}_1\\right]\n\\end{equation}\nwith the following definitions : $D^\\text{oo}_2$ is the average over the sublattice of odd sites of the stationary correlation between two spins separated by two sites,\n\\begin{equation}\nD^\\text{oo}_2=\\frac{1}{N}\\sum_{n=1}^N \\Espst{s_{2n-1} s_{2n+1}},\n\\end{equation} \n$D^\\text{oe}_1=(1\/N)\\sum_{n=1}^N \\Espst{s_{2n-1} s_{2n}}$ \nand $D^\\text{eo}_1$ has an analogous definition. Similarly the stationary mean value of the global heat current coming from the thermostat acting on spins at odd sites, $J_\\text{\\mdseries o}= \\sum_{n=1}^N j_{2n-1}$, reads\n\\begin{equation}\n\\label{ExpressionJo}\n\\Espst{J_\\text{\\mdseries o}}= N K \\nu_\\text{\\mdseries o} \\left[ -\\gamma_\\text{\\mdseries o} -\\gamma_\\text{\\mdseries o} D^\\text{ee}_2 + D^\\text{oe}_1 + D^\\text{eo}_1\\right]\n\\end{equation}\nwith $D^\\text{ee}_{2}=(1\/N)\\sum_{n=1}^N \\Espst{s_{2n} s_{2n+2}}$.\n\n\n\n\\medskip\nThe values of the stationary global two-spin correlations $D^\\text{oo}_2$, $D^\\text{ee}_2$, $D^\\text{oe}_1$ and $D^\\text{eo}_1$ can be determined from a hierarchy of equations for similar quantities with two spins at any distance on the lattice. Details are given in Appendix \\ref{secTwospincorrelations} with the results\n \\eqref{relDooDee}, \\eqref{relDoeDeo}, \\eqref{valueDee}, and \\eqref{valueDoe} for any distance between spins. From the latter results we get\n\\begin{equation}\n\\label{valueDee2}\nD^\\text{ee}_2=\\frac{\\gamma \\eta_-}{\\gamma_\\text{\\mdseries o}} \\, \\frac{1 +\\eta_-^{N- 2} }{1+\\eta_-^N} \n\\end{equation}\nwith\n\\begin{equation}\n\\label{defgamma}\n\\gamma= \\overline{\\nu}_\\text{\\mdseries o} \\gamma_\\text{\\mdseries o} + \\overline{\\nu}_\\text{\\mdseries e} \\gamma_\\text{\\mdseries e},\n\\end{equation}\nwhere the dimensionless kinetic parameters have been defined in \\eqref{defDimensionlessNu}, and $\\eta_-$, with $0< \\eta_- <1$, is defined by\n\\begin{equation}\n\\sqrt{\\eta_-}=\\frac{1-\\sqrt{1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}}{\\sqrt{\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}}.\n\\end{equation}\nMoreover $D^\\text{oo}_2=(\\gamma_\\text{\\mdseries o}\/\\gamma_\\text{\\mdseries e}) D^\\text{ee}_2$, while\n\\begin{equation}\n\\label{valueDoe1}\nD^\\text{oe}_1=D^\\text{eo}_1=\\frac{\\gamma \\sqrt{\\eta_-}}{\\sqrt{\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}} \\,\\frac{1 +\\eta_-^{N-1}}{1+\\eta_-^N}.\n\\end{equation}\nWe notice that the dependence upon the kinetic parameters $\\nu_\\text{\\mdseries o}$ and $\\nu_\\text{\\mdseries e}$ occurs only through the parameter $\\gamma$ defined in \\eqref{defgamma}.\n\nEventually the stationary mean value of the global heat current received on even sites can be calculated from expression \\eqref{ExpressionJe} and relation \\eqref{RelRoots};\nwe get \n\\begin{equation}\n\\label{valueJe}\n\\Espst{J_\\text{\\mdseries e}}= N K \\frac{\\nu_\\text{\\mdseries o} \\nu_\\text{\\mdseries e}}{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}).\n\\end{equation}\nSimilarly the stationary mean value of the global heat current received on odd sites can be obtained from the expression \\eqref{ExpressionJo} ; it proves to be opposite to that on even sites, \n$\\Espst{J_\\text{\\mdseries o}}=-\\Espst{J_\\text{\\mdseries e}}$, as it should in the stationary state where the mean energy of the chain is constant.\n\nWe point out the following remarkable property : though $D^\\text{ee}_2$, $D^\\text{oo}_2$, $D^\\text{oe}_1$ and $D^\\text{eo}_1$ involve finite size corrections (see \\eqref{valueDee2} and \\eqref{valueDoe1}), these corrections cancel one another in the value of the mean global current $\\Espst{J_\\text{\\mdseries e}}$ : $\\Espst{J_\\text{\\mdseries e}}$ is exactly proportional to the size $L=2N$ of the ring. Moreover it happens to be equal to $L$ times the mean current received by a spin in the independent pair model of Ref.\\cite{CornuBauer2013}.\n\n\n\n\n\\section{Mapping to a reaction-diffusion system with pair creation-annihilation}\n\\label{SecMapping}\n\nTo prepare the study of the heat amounts exchanged with the thermostats we consider a mapping to another model for which the evolution operator is quadratic. \n\n\\subsection{Domain wall system}\n\nWhen two spins on neighboring sites are antiparallel, one may consider that there is a domain wall between them, whereas there is no domain wall when they are parallel. The domain walls sit on the edges of the initial lattice. Labeling each edge by its mid-point, one gets another lattice which we call the dual lattice in what follows. The edge $(j-1,j)$ and the corresponding site on the dual lattice are labeled by $j$.\n If $s_{j-1}$ and $s_j$ are antiparallel, $s_{j-1}s_j=-1$, then the occupation number by a domain wall at site $j$ on the dual lattice is $n_j=1$, whereas if $s_{j-1}$ and $s_j$ are parallel $n_j=0$. Thus the correspondance reads\n\\begin{equation}\n\\label{CorrespondenceWallSpin}\nn_j=\\frac{1-s_{j-1}s_{j}}{2}.\n\\end{equation}\nOn a ring the number of domain walls is even and $\\sum_{j=1}^L n_j$ is even.\n\nAs a result a spin configuration can be characterized either by the set $\\mathbf{s}=\\{s_1,\\cdots s_L\\}$ of spin configurations or by the knowledge of the value of $s_1$ and the set of the positions of the domains walls, \nnamely the set of occupations numbers $\\mathbf{n}=\\{n_1,\\cdots,n_L\\}$. \nThe energy of the system can be expressed solely in terms of domain walls as\n\\begin{equation}\n{\\cal E}(\\mathbf{n})= - 2N K+2 K \\sum_{j=1}^{2N} n_j.\n\\end{equation}\n\n\n\n\\subsection{Quantum mechanics notations}\n\nIn the following we use the quantum mechanics notations, as commonly done in the literature. Then a column vector is denoted as a ``ket'', $\\ket{\\ldots}$ and a row vector is denoted as a ``bra'' $\\bra{\\ldots}$.\nThe configuration of occupation numbers by domain walls ${\\mathbf{n}}=\\{n_1, \\cdots, n_{L}\\}$ is represented as a tensor product\n\\begin{equation}\n\\label{tensorialprod}\n \\ket{\\mathbf{n}}=\\otimes_{j=1}^{L} \\ket{n_j},\n\\end{equation}\nwhere $\\ket{n_j}$ is a two-component column vector.\nThe convention used for kets associated to vacant and occupied states is \n\\begin{equation}\n\\label{convention-vacant}\n\\veccolumn{1}{0}_j=\\ket{n_j=0}\\quad\\textrm{and}\\quad \\veccolumn{0}{1}_j=\\ket{n_j=1}.\n\\end{equation}\nThis convention is the standard choice of basis in the condensed matter literature on quantum spin chains.\nWith the representation \\eqref{tensorialprod}-\\eqref{convention-vacant} the row-column product $\\brat{\\mathbf{n'}}\\ket{\\mathbf{n}}$ takes the form\n$\n\\brat{\\mathbf{n'}}\\ket{\\mathbf{n}}=\\prod_{j=1}^{L} \\delta_{n'_j,n_j}\n$.\nTherefore the probability of the domain wall configuration $\\mathbf{n}$ at time $t$, $P(\\mathbf{n};t)$,\ncan be represented as a row-column (scalar) product \n$\nP(\\mathbf{n};t)= \\brat{\\mathbf{n}} \\ket{P_t}\n$,\nwhere $\\ket{P_t}$ is the column vector defined as\n\\begin{equation}\n\\label{defketProb}\n\\ket{P_t}=\\sum_{\\mathbf{n}} P(\\mathbf{n};t) \\ket{\\mathbf{n}}.\n\\end{equation}\n\n\nWith the latter definitions the master equation for the stochastic evolution of the probability $P(\\mathbf{n};t)$, which takes the generic form written in \\eqref{MarkovSpins} in the case of $P(\\mathbf{s};t)$, can be represented as the evolution of the column vector $\\ket{P_t}$ under the Markov matrix $\\mathbb{M}$\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\ket{P_t}}{\\text{\\mdseries d} t}=\\mathbb{M} \\ket{P_t}\n\\end{equation}\nwith \n\\begin{eqnarray}\n\\bra{\\mathbf{n}'} \\mathbb{M} \\ket{\\mathbf{n}}&=& w(\\mathbf{n} \\to \\mathbf{n}')\n \\quad\\textrm{if}\\quad \\mathbf{n}'\\neq \\mathbf{n}\n\\\\ \\nonumber\n\\bra{\\mathbf{n}} \\mathbb{M} \\ket{\\mathbf{n}}&=&-\\sum_{\\mathbf{n}'\\neq\\mathbf{n}} w(\\mathbf{n} \\to \\mathbf{n}'),\n\\end{eqnarray}\nwhere $w(\\mathbf{n} \\to \\mathbf{n}')$ denotes the transition rate from configuration $\\mathbf{n}$ to configuration $\\mathbf{n}'$.\n\n\n\n\\subsection{Markov matrix for the model}\n\n\nFor the present model of domain walls the matrix elements of $\\mathbb{M}$ can be expressed in terms of Pauli matrices. Indeed the operator for the occupation number at site $j$ reads \n\\begin{equation}\n\\label{widehatn}\n\\widehat{n}_j=\\tfrac{1}{2} \\left( \\mathbb{1}_j-\\sigma^z_j\\right)\n\\end{equation}\nwhere $\\mathbb{1}_j$ denotes the identity $2\\times 2$ matrix at site $j$, and\n{\\small $\\sigma^z_j=\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}_j$};\n the operator which changes the occupation number at site $j$ is \n{\\small $\n\\sigma^x_j=\n\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix}_j\n$}.\nBy inspection of the transition rates for the spin configurations $w(\\mathbf{s} \\to \\mathbf{s}')$ given by \\eqref{deftauxtrans}, every transition rate for the occupation numbers by domain walls, $w(\\mathbf{n} \\to \\mathbf{n}')$, can be written as a matrix element $\\bra{\\mathbf{n}'} \\mathbb{W} \\ket{\\mathbf{n}}$, where \n\n- for a hop of a domain wall from site $j$ to site $j+1$\n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}\\sigma^x_j \\sigma^x_{j+1} \\widehat{n}_j \\left(\\mathbb{1}_j-\\widehat{n}_{j+1}\\right)\n\\end{equation}\n\n- for a hop of a domain wall from site $j+1$ to site $j$\n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}\\sigma^x_j \\sigma^x_{j+1} \\left( \\mathbb{1}_j-\\widehat{n}_j\\right) \\widehat{n}_{j+1}\n\\end{equation}\n\n- for the annihilation of two domain walls at sites $j$ and $j+1$ \n\\begin{equation}\n\\mathbb{W} = \\frac{\\nu_j}{2}(1+\\gamma_j)\\sigma^x_j \\sigma^x_{j+1} \\widehat{n}_j \\widehat{n}_{j+1}\n\\end{equation}\n\n- for the creation of two domain walls at sites $j$ and $j+1$ \n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}(1-\\gamma_j)\\sigma^x_j \\sigma^x_{j+1} \\left( \\mathbb{1}_j-\\widehat{n}_j\\right) \\left( \\mathbb{1}_j-\\widehat{n}_{j+1}\\right).\n\\end{equation}\nThe latter expressions can be written in a more compact form by using the \n spin-$\\tfrac{1}{2}$ ladder operators \n{\\small $\n\\sigma^+=\n\\begin{pmatrix}\n 0 & 1 \\\\\n 0 & 0\n\\end{pmatrix}_j\n$} and \n{\\small $\n\\sigma^-=\n\\begin{pmatrix}\n 0 & 0 \\\\\n 1 & 0\n\\end{pmatrix}_j\n$}. They are such that $\\sigma^x\\widehat{n}=\\sigma^+$ and $\\sigma^x(\\mathbb{1}-\\widehat{n})=\\sigma^-$.\nWith the convention \\eqref{convention-vacant} $\\sigma^+_j$ annihilate a domain wall at site $j$, while $\\sigma^-_j$ creates a domain wall at site $j$.\n\nEventually the Markov matrix $\\mathbb{M}$ derived from the master equation for the evolution of the probability of spin configurations \\eqref{MarkovSpins} reads\n\\begin{eqnarray}\n&&\\mathbb{M} =\\frac{\\nu_1+\\nu_2}{2}\n\\\\ \\nonumber\n&& \\times\n \\left[ -(1-\\gamma) N \\mathbb{1}\n -\\gamma\\sum_{j=1}^{2N} \\widehat{n}_j \n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[ \\sigma^+_j \\sigma^-_{j+1} + \\sigma^-_j \\sigma^+_{j+1}+\n (1+\\gamma_j)\\sigma^+_j \\sigma^+_{j+1} +(1-\\gamma_j)\\sigma^-_j \\sigma^-_{j+1} \\right]\\right],\n\\end{eqnarray}\nwhere $\\mathbb{1}$ denotes the identity $2N \\times 2N$ matrix and $\\gamma$ has been defined in \\eqref{defgamma}. The advantage of the domain wall representation with respect to the spin representation is that the Markov matrix is quadratic in terms of operators acting on different sites instead of involving three operators acting on different sites (for the latter case see for instance Ref.\\cite{Felderhof1971-A,Felderhof1971-B}).\n\n\n\n\\section{Eigenvalues of the modified Markov matrix}\n\\label{SecEigenvalues}\n \n\\subsection{Modified Markov matrix}\n\\label{CumulantsMaximumEigenvalue}\n\nWe are interested in the joint cumulants per unit of time for the heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ which are received by the chain from the thermostat acting on spins at odd and even sites during a time $t$ in the long time limit. The corresponding scaled generating function is\n\\begin{equation}\n\\label{defgL}\ng_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)= \\lim_{t\\to\\infty} \\frac{1}{t} \\ln\\Esp{ \\text{\\mdseries e}^{\\lambda_\\text{\\mdseries o}\\Heat_\\text{\\mdseries o}+\\lambda_\\text{\\mdseries e}\\Heat_\\text{\\mdseries e}}},\n\\end{equation}\nwhere $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ are real parameters.\nIn fact an evolution equation can be written for the probability \n$P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t)$ for the system to be in configuration $\\mathbf{n}$ at time $t$ and to have received heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ between times $0$ and $t$. Therefore \nthe expectation value in the definition \\eqref{defgL} can be expressed in terms of the discrete Laplace transform of \n$P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t)$, and then $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ reads\n\\begin{equation}\n\\label{defgLBis}\ng_{2N}(\\lambda_\\text{\\mdseries e},\\lambda_\\text{\\mdseries o};t)= \\lim_{t\\to\\infty} \\frac{1}{t} \\ln\\sum_{\\mathbf{n}} \\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)\n\\end{equation}\nwith\n\\begin{equation}\n\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)= \\sum_{\\{\\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e}\\}} \\text{\\mdseries e}^{\\lambda_\\text{\\mdseries o}\\Heat_\\text{\\mdseries o}+\\lambda_\\text{\\mdseries e}\\Heat_\\text{\\mdseries e}}P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t).\n\\end{equation}\n\n\n By inspection of the transition rates \\eqref{deftauxtrans} and according to the correspondence \\eqref{CorrespondenceWallSpin},\n when the spin at site $j$ is flipped under the action of the thermostat at temperature $T_j$, the variation of \n$Q_j$ is equal to $+\\Delta E$ if a pair of domain walls is created at sites $j$ and $j+1$ , $-\\Delta E$ if a pair of domain walls is annihilated at these sites, $0$ if a domain wall jumps either from $j$ to $j+1$ or from $j+1$ to $j$.\nAs a consequence, with a definition for $ \\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$ analogous to that for $\\ket{P_t}$ given in \\eqref{defketProb}, namely $\\brat{\\mathbf{n}}\\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}=\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$, we get the evolution equation\n\\begin{equation}\n\\label{EvLPProb}\n\\frac{\\text{\\mdseries d} \\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}}{\\text{\\mdseries d} t}= \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \\ket{\\widehat{P}_t(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})},\n\\end{equation}\nwhere the so-called modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ reads\n\\begin{equation}\n\\label{defMmodified}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) =-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N} \\widehat{n}_j \n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[ \\sigma^+_j \\sigma^-_{j+1} + \\sigma^-_j \\sigma^+_{j+1}+\nb_j\\sigma^+_j \\sigma^+_{j+1} +c_j \\sigma^-_j\\sigma^-_{j+1} \\right].\n\\end{equation}\nThe coefficient $b_j$ ($c_j$) is equal either to $b_\\text{\\mdseries o}$ or $b_\\text{\\mdseries e}$ ($c_\\text{\\mdseries o}$ or $c_\\text{\\mdseries e}$) according to the parity of $j$; with the notation $\\text{a}=\\text{o}$ or $\\text{e}$\n\\begin{equation}\n\\label{defbj}\nb_\\text{\\mdseries a}=(1+\\gamma_\\text{\\mdseries a})\\,\\text{\\mdseries e}^{-\\lambda_\\text{\\mdseries a} \\Delta E}\n\\end{equation}\nand\n\\begin{equation}\n\\label{defcj}\nc_\\text{\\mdseries a}=(1-\\gamma_\\text{\\mdseries a})\\,\\text{\\mdseries e}^{\\lambda_\\text{\\mdseries a}\\Delta E}.\n\\end{equation}\nAccording to the evolution equation \\eqref{EvLPProb} the Laplace transform $\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ is equal to \\\\ $\\bra{\\mathbf{n}} \\exp[\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) t] \\ket{\\widehat{P}_{t=0} (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$. \nThus the scaled generating function $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ given by \\eqref{defgLBis} is equal to the largest eigenvalue of the matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $ which rules the evolution of $\\ket{\\widehat{P}_t(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$.\n\n\n\n\\subsection{Jordan-Wigner transformation}\n\nIn order to find the eigenvalues of the modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ given by \\eqref{defMmodified} we take advantage of its structure analogous to a free fermion Hamiltonian and we introduce the following Jordan-Wigner transformation \\cite{JordanWigner1928}\n\\begin{equation}\nf^\\dag_j=\\left(\\prod_{k=1}^{j-1}\\sigma^z_k \\right)\\sigma^-_j\n\\quad\\textrm{and}\\quad\nf_j=\\left(\\prod_{k=1}^{j-1}\\sigma^z_k \\right)\\sigma^+_j.\n\\end{equation}\nThe operator $f_j^\\dag$ is indeed the adjoint of $f_j$, because $(\\sigma^z)^\\dag=\\sigma^z$ and $(\\sigma^-)^\\dag=\\sigma^+$.\nOperators $\\sigma$ acting on different sites commute, whereas $\\sigma^z_j$ anticommutes with $\\sigma^+_j$ and $\\sigma^-_j$; moreover $(\\sigma^+_j)^2=0$ and $(\\sigma^-_j)^2=0$. Therefore the operators $f_j$ and $f^\\dag_j$ obey the fermionic anticommutation relations\n\\begin{equation}\n\\{f_j,f_{j'}\\}=0 \\quad \\{f^\\dag_j,f^\\dag_{j'}\\}=0 \\quad \\{f_j,f^\\dag_{j'}\\}=\\delta_{j,j'}.\n\\end{equation}\nThe occupation number of site $j$ by a domain wall, given by \\eqref{widehatn}, also reads $\\widehat{n}_j=\\sigma^-_j\\sigma^+_j=f^\\dag_j f_j$. The expression \\eqref{defMmodified} of the modified matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ is rewritten in terms of fermionic operators as\n\\begin{eqnarray}\n\\nonumber\n\\label{widehatM}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n&=&-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N}f^\\dag_jf_j\n+\\sum_{j=1}^{2N-1} \\overline{\\nu}_j\\left[f^\\dag_j f_{j+1} - f_jf^\\dag_{j+1} +c_j f^\\dag_jf^\\dag_{j+1} -b_j f_jf_{j+1} \\right] \n\\\\\n&-& \\overline{\\nu}_\\text{\\mdseries e} (-1)^{{\\cal N}_\\text{\\mdseries f}}\\left[ f^\\dag_{2N} f_1 - f_{2N}f^\\dag_1 +c_\\text{\\mdseries e} f^\\dag_{2N}f^\\dag_1-b_\\text{\\mdseries e} f_{2N}f_1\\right]\n\\end{eqnarray}\nwhere ${\\cal N}_\\text{\\mdseries f}=\\sum_{j=1}^{2N} f^\\dag_j f_j$ is the total number of fermions.\n\nSince the spin system is on a ring, there can be only an even number of domain walls in the system. As noticed above, the operator for the occupation number by a domain wall $\\widehat{n}_j$ coincides with the number of fermions at site $j$, $f^\\dag_j f_j$. Therefore we have to consider the restriction $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $ of $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ to the sector with an even number of fermions.\nAccording to \\eqref{widehatM} the expression of this rectriction is invariant by translation along the ring if the fermionic operators are chosen to satisfy the antiperiodic boundary conditions\n\\begin{equation}\n\\label{AntiperiodicC}\nf_{2N+1}=-f_1 \\quad\\textrm{and}\\quad f^\\dag_{2N+1}=-f^\\dag_1.\n\\end{equation}\nThen the restriction reads\n\\begin{equation}\n\\label{Mplus}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n=-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N}f^\\dag_jf_j\n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[f^\\dag_j f_{j+1} - f_jf^\\dag_{j+1} +c_j f^\\dag_jf^\\dag_{j+1} -b_j f_jf_{j+1} \\right].\n\\end{equation}\n\n\\clearpage\n\\subsection{Antiperiodic Fourier transform}\n\nThe next step to the diagonalization is to rewrite the fermionic operators as antiperiodic Fourier transforms which satisfy the antiperiodic boundary conditions \\eqref{AntiperiodicC}\nThe wave numbers are of the form $q=(2k+1)\\pi\/(2N)$ and we work with a complete family of representatives in the set \n\\begin{equation}\n\\label{defBod2N}\n{\\cal B}(2N)=\\{q=(2k+1) \\frac{\\pi}{2N}, k=-N,-N+1, \\cdots, -1,0,1, \\cdots N-1 \\},\n\\end{equation}\nnamely\n\\begin{equation}\n{\\cal B}(2N)=\\{-\\pi +\\frac{\\pi}{2N}, -\\pi +\\frac{3\\pi}{2N}, \\cdots -\\frac{\\pi}{2N}, \\frac{\\pi}{2N}, \\cdots \\pi-\\frac{\\pi}{2N} \\}.\n\\end{equation}\nThe operator $f_j$ can be written as the antiperiodic Fourier transform\n\\begin{equation}\n\\label{defAntiFT}\nf_j=\\frac{1}{\\sqrt{2N}} \\sum_{q\\in {\\cal B}(2N)} \\text{\\mdseries e}^{\\imath qj} \\eta_q\n\\end{equation}\nin terms of the wave fermions\n\\begin{equation}\n\\label{defAntiFTInv}\n\\eta_q =\\frac{1}{\\sqrt{2N}} \\sum_{j=1}^{2N} \\text{\\mdseries e}^{-\\imath qj} f_j.\n\\end{equation}\nGoing from \\eqref{defAntiFT} to \\eqref{defAntiFTInv} relies on the identity\n\\begin{equation}\n\\label{SumGeom}\n\\sum_{j=1}^{2N} \\text{\\mdseries e}^{\\imath(q-q')j}=2N \\,\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)},\n\\end{equation}\nwhere $\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)}=1$ if $q-q'$ is equal to $0$ modulo $2\\pi$ and $\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)}=0$ otherwise.\nAll $q'$s in ${\\cal B}(2N)$ satisfy $\\text{\\mdseries e}^{\\imath q 2N}=-1$, and subsequently $f_j$ does obey the antiperiodic boundary conditions \\eqref{AntiperiodicC}.\nThe adjoint operator $f^\\dag_j$ reads\n\\begin{equation}\nf^\\dag_j=\\frac{1}{\\sqrt{2N}} \\sum_{q\\in {\\cal B}(2N)} \\text{\\mdseries e}^{-\\imath qj} \\eta^\\dag_q.\n\\end{equation}\n\nThese representations are inserted in the expression \\eqref{Mplus}. In the term $\\sum_{j=1}^{2N}f^\\dag_jf_j$ there occurs a summation over all sites of the ring and one uses the identity \\eqref{SumGeom}.\n In the other summations one has to distinguish the two sublattices ; for instance one has to consider the sum $\\sum_{n=1}^N f^\\dag_{2n} f^\\dag_{2n+1}$ and then one uses the identity\n\\begin{equation}\n\\sum_{n=1}^{N} \\text{\\mdseries e}^{\\imath(q+q')2n}=N \\,\\mathbb{1}_{2(q+q')\\equiv 0 (2\\pi)}=N \\,\\mathbb{1}_{q+q'\\equiv 0(\\pi)}.\n\\end{equation}\nAccording to definition \\eqref{defBod2N}, the set ${\\cal B}(2N)$ does not contain the value $0$ and the solution of $q+q'=0$ corresponds to two distinct values $q$ and $-q$. \n\n\nIn order to simplify the following discussion, we assume from now on that $N$ \\textit{is even}. Then ${\\cal B}(2N)$ does not contain $\\pi\/2$ and all values $q$ and $\\pi- q$, are also distinct. \nAfter a symmetrization over the values $q$ and $\\pi-q$ the matrix $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ appears as a sum of contributions each of which involves only the operators associated with the wave numbers $q$, $\\pi-q$, $-q$ and $-(q-\\pi)$. Let us introduce \nthe first quadrant in the set ${\\cal B}$ defined as\n\\begin{equation}\n\\label{defQBod}\n{\\cal QB}(2N)=\\{q=(2k+1) \\frac{\\pi}{2N}, k=0,1, \\cdots (N\/2) -1 \\} =\\{ \\frac{\\pi}{2N}, \\frac{3\\pi}{2N}\\cdots \\frac{\\pi}{2}-\\frac{\\pi}{2N}\\}.\n\\end{equation}\nThen the expression \\eqref{Mplus} for\n$\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ can be rewritten as\n\\begin{equation}\n\\label{widehatMBis}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n=-N \\mathbb{1} +\\sum_{q\\in {\\cal QB}} \\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})\\, V_q,\n\\end{equation}\nwhere $V_q$ is the column vector\n\\begin{equation}\n\\label{defVq}\nV_q=\n\\begin{pmatrix}\n\\eta_q \\\\\n\\eta_{q-\\pi} \\\\\n\\eta^\\dag_{-q} \\\\\n\\eta^\\dag_{\\pi-q}\n\\end{pmatrix},\n\\end{equation}\n$\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T}$ denotes the transposed row vector corresponding to the column vector $V^\\dag_q$ built with the adjoints of the components of $V_q$, and\n\\begin{equation}\n\\mathbb{A}_q (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) =\n\\begin{pmatrix}\n -\\gamma + \\cos q & \\imath a_q \\sin q & \\imath c'_q \\sin q & c_q \\cos q \\\\\n - \\imath a_q \\sin q & -\\gamma - \\cos q & -c_q \\cos q & -\\imath c'_q \\sin q \\\\\n- \\imath b'_q \\sin q & - b_q \\cos q & \\gamma - \\cos q & -\\imath a_q \\sin q \\\\\nb_q \\cos q & \\imath b'_q \\sin q & \\imath a_q \\sin q & \\gamma + \\cos q\n\\end{pmatrix},\n\\end{equation}\nwith \n\\begin{eqnarray}\na_q &=& \\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e}\n\\\\ \\nonumber\nb_q&=& \\overline{\\nu}_\\text{\\mdseries o} b_\\text{\\mdseries o} -\\overline{\\nu}_\\text{\\mdseries e} b_\\text{\\mdseries e}\n\\\\ \\nonumber\nb'_q&=& \\overline{\\nu}_\\text{\\mdseries o} b_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e} b_\\text{\\mdseries e} \n\\\\ \\nonumber\nc_q&=& \\overline{\\nu}_\\text{\\mdseries o} c_\\text{\\mdseries o} -\\overline{\\nu}_\\text{\\mdseries e} c_\\text{\\mdseries e}\n\\\\ \\nonumber\nc'_q&=& \\overline{\\nu}_\\text{\\mdseries o} c_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e} c_\\text{\\mdseries e},\n\\end{eqnarray}\nwhere the $b_\\text{\\mdseries a}$'s and the $c_\\text{\\mdseries a}$'s are defined in \\eqref{defbj} and \\eqref{defcj}. \n\n\\subsection{Diagonalization of $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $}\n\nThe characteristic polynomial of $\\mathbb{A}_q$, $\\text{det}\\left(\\mathbb{A}_q-\\alpha \\mathbb{1}_q\\right)$, proves to be a second order polynomial in $\\alpha^2$, with a constant which is a squared quantity,\n\\begin{equation}\n\\label{CharacteristicPol}\n\\text{det}\\left(\\mathbb{A}_q-\\alpha \\mathbb{1}_q\\right)= \\alpha^4 -2 D \\alpha^2 +F^2.\n\\end{equation}\nMoreover both coefficients $D$ and $F^2$ depend on the parameters $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ only through the difference\n\\begin{equation}\n\\label{deflambdab}\n\\overline{\\lambda}=\\left(\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}\\right)\\Delta E.\n\\end{equation}\nThey read\n\\begin{equation}\nD=1 + (\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2 + \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}\\cos^2q + (1-2\\cos^2q) \\theta(\\overline{\\lambda}) \\right]\n\\end{equation}\nand \n\\begin{equation}\nF=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[ 4(1- \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} \\cos^2 q) + \\theta(\\overline{\\lambda})\\right],\n\\end{equation}\nwhere the function $\\theta(\\overline{\\lambda})$ vanishes when $\\overline{\\lambda}$ is set to zero :\n\\begin{equation}\n\\label{deftheta}\n\\theta(\\overline{\\lambda})=2\\left[ (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) (\\cosh \\overline{\\lambda} -1) + (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e})\\sinh\\overline{\\lambda} \\right].\n\\end{equation}\nThe coefficient $F$ can be rewritten as \n\\begin{equation}\nF=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[ 2 + 2 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}(1-2 \\cos^2 q) + (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) \\cosh \\overline{\\lambda} + (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e})\\sinh\\overline{\\lambda}\\right],\n\\end{equation}\nand the property $(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) > \\vert \\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e} \\vert$ for $\\gamma_\\text{\\mdseries o}<1$ and $\\gamma_\\text{\\mdseries e}<1$ ensures that $F>0$.\nThe squared roots of the characteristic polynomial are\n\\begin{equation}\n\\label{valuealpha2}\n\\alpha^2=D \\pm \\sqrt{D^2-F^2} = \\left(\\sqrt{\\frac{D+F}{2}} \\pm \\sqrt{\\frac{D-F}{2}} \\right)^2.\n\\end{equation}\nwhere $\\sqrt{\\cdots}$ denotes a possibly complex square root. Let us introduce the notations\n$R_\\pm(q,\\overline{\\lambda})= \\tfrac{1}{2}(D\\pm F)$, namely\n\\begin{equation}\n\\label{defRplus}\nR_+(q,\\overline{\\lambda})= 1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda}) \\sin^2q\n\\end{equation}\n\\begin{equation}\n\\label{defRminus}\nR_-(q,\\lambda)=(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})\\right]\\cos^2 q.\n\\end{equation}\nWe notice that, since $F$ is positive, $R_+> R_-$.\n\nNote that $1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})>0$ because of the definition \\eqref{deftheta} of $\\theta(\\overline{\\lambda})$ and the identities\n $1-2\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) >0$ and $(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})> \\vert \\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e} \\vert$\n for $\\gamma_\\text{\\mdseries o}<1$ and $\\gamma_\\text{\\mdseries e}<1$. According to \\eqref{defRplus}, $R_+$ can be rewritten as \n$\nR_+=1-\\sin^2 q + \n\\left(1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})\\right)\\sin^2q\n$ and we conclude that \n$R_+>0$. \n\nOn the other hand, by virtue of definitions \\eqref{defgammaa} and \\eqref{deftheta},\n \\begin{equation}\n \\label{relthetacosh}\n4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})=2\\frac{\\cosh\\left((\\beta_\\text{\\mdseries e}+\\beta_\\text{\\mdseries o}) \\Delta E\/2\\right)-\\cosh\\left(\\overline{\\lambda}-(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o}) \\Delta E\/2\\right)}{\\cosh\\left(\\beta_\\text{\\mdseries e} \\Delta E\/2\\right)\\cosh\\left(\\beta_\\text{\\mdseries o} \\Delta E\/2\\right)},\n\\end{equation}\nso that $4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})>0$ only if $-\\beta_\\text{\\mdseries o} \\Delta E<\n\\overline{\\lambda}<\\beta_\\text{\\mdseries e}\\Delta E$, and we infer from \\eqref{defRminus} that $R_-$ can take both signs. \n\nEventually the four eigenvalues of $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ are\n\\begin{equation}\n\\label{defalphan}\n\\alpha_1= \\sqrt{R_+} + \\sqrt{R_-} \\qquad \\alpha_2= \\sqrt{R_+} - \\sqrt{R_-}\\qquad \\alpha_3=-\\alpha_2 \\qquad \\alpha_4=-\\alpha_1,\n\\end{equation}\nIn these expressions $\\sqrt{R_+}$ denotes the usual positive square root of the positive number $R_+$, whereas $\\sqrt{R_-}$ is either real positive or purely imaginary depending on the sign of $R_-$, i.e. on the values of $\\overline{\\lambda}$ and $q$.\n As noticed previously $R_+> R_-$, and in the case where $\\sqrt{R_-}$ is real, all $\\alpha_k$'s, with $k=1,\\ldots,4$, are real and $\\alpha_1>\\alpha_2>0> \\alpha_3 > \\alpha_4$.\n\n\n\n\n\n\\subsection{Largest eigenvalue of the modified matrix $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$}\n\nFor the sake of conciseness we omit all dependences upon $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ in the present section.\nWe denote by $\\mathbb{D}_q$ the diagonal matrix built with the eigenvalues $\\alpha_1(q),\\ldots,\\alpha_4(q)$ of the matrix $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$. The matrix $\\mathbb{A}_q$ reads\n\\begin{equation}\n\\label{relAD}\n\\mathbb{A}_q= \\mathbb{P}_q \\mathbb{D}_q \\mathbb{P}_q^{-1},\n\\end{equation}\nwhere the $k^\\text{th}$ column of $\\mathbb{P}_q$ is made with the components of a (column) right eigenvector of \n$\\mathbb{A}_q$ associated with the eigenvalue $\\alpha_k(q)$, and $\\mathbb{P}_q^{-1}$ is the inverse matrix of $\\mathbb{P}_q$. Let $\\xi_k(q)$ denote the $k^\\text{th}$ component of the column vector $\\mathbb{P}_q^{-1} V_q$, while $\\xi^\\star_k(q)$ denotes the \n$k^\\text{th}$ component of the row vector $\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{P}_q$, where $V_q$ and $\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} $ are defined in \\eqref{defVq}. With these definitions, the relation \\eqref{relAD} implies that\n\\begin{equation}\n\\label{Sumxi}\n\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{A}_q V_q=\\sum_{k=1}^4 \\alpha_k(q) \\, \\xi^\\star_k(q) \\,\\xi_k(q).\n\\end{equation}\nThe operators $\\xi_k$ and $\\xi^\\star_k$ obey the anticommutation rules $\\{\\xi_k, \\xi_{k'}\\}=0$, $\\{\\xi^\\star_k, \\xi^\\star_{k'}\\}=0$ and $\\{\\xi_k, \\xi^\\star_{k'}\\}=\\delta_{k,k'}$. However, since $\\mathbb{A}_q$ is not hermitian (for the usual scalar product), \n$\\mathbb{P}_q$ is not unitary and the operator $\\xi^\\star_k$ is not the adjoint of $\\xi_k$.\n\n\nNevertheless the anticommutation rules are enough to ensure that the spectrum of the operator $\\xi^\\star_k \\xi_k$ is the set $\\{0,1\\}$. Then, according to the expressions \\eqref{defalphan} of the $\\alpha_k$'s, the value of the right-hand side of \\eqref{Sumxi} with the largest real part is equal to the sum of two eigenvalues and proves to be real positive,\n\\begin{equation}\n\\alpha_1 + \\alpha_2=2 \\sqrt{R_+(q)}>0.\n\\end{equation}\nEventually, according to \\eqref{widehatMBis}, the largest eigenvalue of $ \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ is\n\\begin{equation}\n\\label{LargestEigenvalue}\n\\frac{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}}{2}\n\\left[-N + 2\\sum_{q\\in {\\cal QB}} \\sqrt{R_+(q)}\\right],\n\\end{equation}\nwhere $R_+(q)$ is given by \\eqref{defRplus}.\n\nWe notice that if $\\lambda_\\text{\\mdseries o}=\\lambda_\\text{\\mdseries e}=0$, the modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ becomes the usual Markov matrix for the evolution of $P(\\mathbf{n};t)$; since $\\theta(0)=0$, we retrieve that the largest eigenvalue of the Markov matrix is $0$. Moreover the eigenvalue closest to 0 is obtained by setting $ \\xi^\\star_3(q)\\xi_3(q)$ equal to 1 for $q=0$, and we retrieve the value \\eqref{trel} for the inverse relaxation time of the magnetizations on the two sublattices.\n\n\\clearpage\n\\section{Cumulants of heat amounts per unit of time in the long time limit}\n\\label{SecCumulants}\n\n\\subsection{Scaled generating function for joint cumulants}\n\nAccording to the remark at the end of section \\ref{CumulantsMaximumEigenvalue}, the scaled generating function for the joint cumulants of $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ coincides with the largest eigenvalue of $ \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$. By virtue of \\eqref{LargestEigenvalue} it reads\n\\begin{equation}\n\\label{ExpgL}\ng_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})= \\frac{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}}{2}\\left[-N + 2 \\sum_{q\\in {\\cal QB}} \\sqrt{1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda}) \\sin^2q}\\right],\n\\end{equation}\nwhere ${\\cal QB}(2N)$ is defined in \\eqref{defQBod} and $\\theta(\\overline{\\lambda})$ is given in \\eqref{deftheta}.\nThe joint cumulants per unit of time in the long time limit are determined from the relation\n\\begin{equation}\n\\label{JointCumulant}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}_c =\n\\left.\\frac{\\partial^{p+p'} g_{2N}(\\lambda_\\text{\\mdseries e},\\lambda_\\text{\\mdseries o};t)}{\\partial \\lambda_\\text{\\mdseries e}^p \\partial \\lambda_\\text{\\mdseries o}^{p'}}\\right\\vert_{\\lambda_\\text{\\mdseries e}=\\lambda_\\text{\\mdseries o}=0},\n\\end{equation}\nwhere the index $c$ refers to the truncation of the mean value $\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}$ involved in the definition of the cumulant.\nThe fact that $g_{2N}$ depends only on the difference $\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}$ entails the properties\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}_c =(-1)^{p'}\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^{p+p'}}_c \n\\end{equation}\nand, in particular,\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries o}^p}_c =(-1)^{p}\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p}_c.\n\\end{equation}\nThese properties are linked to the fact that the interface energy can take only a finite number of values whereas the cumulants have no upper bounds in the infinite time limit.\n\nFor the sake of completeness we point out that, according to \\eqref{relthetacosh}, $\\theta(\\overline{\\lambda})$ depends on $\\overline{\\lambda}$ through the function \n$\\cosh\\left(\\overline{\\lambda}-(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o})\\Delta E\/2\\right)$; therefore the scaled generating function satisfies the symmetry $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})=g_{2N}(\\beta_\\text{\\mdseries o}-\\lambda_\\text{\\mdseries o}, \\beta_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries e})$, which is in fact a consequence of the local detailed balance \\eqref{LDB}. Since $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ depends only on the difference $\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}$, this entails a symmetry for the scaled generating function for the cumulants of $\\Heat_\\text{\\mdseries e}$, $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})=g_{2N}(0,\\lambda_\\text{\\mdseries e})$, namely the symmetry $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})=g^\\text{e}_{2N}(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o}-\\lambda_\\text{\\mdseries e})$. Then the corresponding large deviation function for the time-integrated current ${\\cal J}_\\text{\\mdseries e}=\\Heat_\\text{\\mdseries e}\/t$, which can be obtained as the Legendre-Fenchel transform of $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})$, obeys the fluctuation relation $f({\\cal J}_\\text{\\mdseries e})-f(-{\\cal J}_\\text{\\mdseries e})=(\\beta_\\text{\\mdseries o}-\\beta_\\text{\\mdseries e}){\\cal J}_\\text{\\mdseries e}$.\n\n\n\\subsection{Cumulants for heat amount $\\Heat_\\text{\\mdseries e}$}\n\nWe give the explicit expressions for the first four cumulants of $\\Heat_\\text{\\mdseries e}$ per unit of time. They are determined as $\\partial^n g_{2N}(0,\\lambda_\\text{\\mdseries e}) \/ \\partial\\lambda_\\text{\\mdseries e}^n\\vert_{\\lambda_\\text{\\mdseries e}=0}=\\Delta E^n \\times \\partial^n g_{2N}(0,\\lambda_\\text{\\mdseries e}) \/ \\partial\\overline{\\lambda}^n\\vert_{\\overline{\\lambda}=0}$, where $\\overline{\\lambda}=\\left(\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}\\right)\\Delta E$ and $\\Delta E=4K$ is the energy gap in the chain energy. The function $\\theta(\\overline{\\lambda})$ defined in \\eqref{deftheta} can be rewritten as\n\\begin{equation}\n\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})= 2A [\\cosh \\overline{\\lambda} -1] + 2B \\sinh \\overline{\\lambda}\n\\end{equation}\nwhere the parameters $A$ and $B$ are those introduced in Ref.\\cite{CornuBauer2013} for a model with only two spins, namely\n\\begin{equation}\nA=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})\n\\end{equation}\n\\begin{equation}\nB= \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}).\n\\end{equation}\nThen, according to \\eqref{ExpgL}, the first four cumulants of $\\Heat_\\text{\\mdseries e}$ read\n\\begin{eqnarray}\n\\label{ExpCumulants}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}}}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{N}{2} B S_2 \\times \\Delta E\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^2}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{N}{2} \\left[A S_2-B^2S_4\\right] \\times \\Delta E^2\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^3}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{N}{2} B \\left[ S_2- 3 A S_4+3B^2 S_6\\right] \\times \\Delta E^3\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^4}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{N}{2} \\left[A S_2- (3 A^2 + 4B^2) S_4 +18 A B^2 S_6 -15 B^4 S_8 \\right] \\times \\Delta E^4,\n\\end{eqnarray}\nwhere\n\\begin{equation}\nS_{2n}(N)=\\frac{2}{N}\\sum_{k=0}^{(N\/2)-1} \\sin^{2n} \\left(\\frac{(2k+1) \\pi}{2N} \\right).\n\\end{equation}\nThe structure of the cumulants in terms of the coefficients $A$ and $B$ is similar to the structure found for the two-spin model of Ref.\\cite{CornuBauer2013} as well as for the ring of Ref.\\cite{CornuHilhorst2017}; indeed, in the three cases the scaled cumulant generating function depends on $\\lambda_\\text{\\mdseries e}$ only through the same function $\\theta(\\lambda_\\text{\\mdseries e} \\Delta E)$.\n\nThe coefficients $S_{2n}(N)$ (with $N$ even) can be calculated explicitly. In the special case $N=2$, a direct calculation leads to $S_{2n}(2)=(1\/2)^n$. For any $N\\geq 2$,\n by extending the summation up to $N$, rewriting $\\sin u=[\\text{\\mdseries e}^{\\imath u}-\\text{\\mdseries e}^{-\\imath u}]\/(2\\imath)$, using the binomial formula and an identity similar to \\eqref{SumGeom}, we get that\n\\begin{equation}\n\\label{Wallis1}\n\\textrm{if $n4$ the first four cumulants are given by \\eqref{ExpCumulants} where the $S_{2n}(N)$ are to be replaced by the corresponding $W_{2n}$ given in \\eqref{valueWallis}.\n\n\nEventually the dependences upon the thermodynamic parameters, $\\gamma_\\text{\\mdseries o}$ and $\\gamma_\\text{\\mdseries e}$, and the kinetic parameters, $\\nu_\\text{\\mdseries o}$ and $\\nu_\\text{\\mdseries e}$, arise only through the coefficients $A$ and $B$. This is in contrast with what happens for another Ising chain model, where both thermostats act on every spin \\cite{CornuHilhorst2017}: for this model the dependence upon the combination $\\gamma$ of the thermodynamic and kinetic parameters defined in \\eqref{defgamma} also arises in the coefficients $\\Sigma_n(N,\\gamma)$ which replace the coefficients $S_n(N)$ in the expressions \\eqref{ExpCumulants} for the cumulants.\n\n\\clearpage\n\\subsection{Various physical regimes}\n\nAccording to the last remark of the previous section, the discussion of the various physical regimes is the same as that performed in the case of the two-spin model of Ref.\\cite{CornuBauer2013}.\n\nAt equilibrium $\\gamma_\\text{\\mdseries o}=\\gamma_\\text{\\mdseries e}$ and, according to \\eqref{deftheta}, $\\theta(\\overline{\\lambda})=2(1-\\gamma_\\text{\\mdseries e}^2) (\\cosh \\overline{\\lambda} -1)$. Therefore only cumulants of even order do not vanish. According to \\eqref{ExpCumulants} the first two cumulants of even order read\n\\begin{eqnarray}\n\\label{ExpCumulantsEq}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^2}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{1}{4} \\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\times N \\Delta E^2\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^4}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{1}{4} \\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\left[1-\\frac{9}{4}\\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\right]\\times N\\Delta E^4.\n\\end{eqnarray}\nThe probability distribution of $\\Heat_\\text{\\mdseries e}$ is not a Gaussian, since all cumulants of even order have non-vanishing values.\n\n\\medskip\n\nWhen a thermostat has a kinetic parameter far larger than the other one, the scaled generating function becomes proportional to $\\theta(\\overline{\\lambda})$, \n\\begin{equation}\ng_{2N}(\\overline{\\lambda})=\\frac{1}{8} N \\nu_\\text{\\mdseries s}\\theta(\\overline{\\lambda}),\n\\end{equation}\nwhere $\\nu_\\text{\\mdseries s}$ is the kinetic parameter of the slower thermostat. As a consequence $g_{2N}(\\overline{\\lambda})$\ncoincides with the scaled generating function of a continuous-time random walk, because $\\theta(\\overline{\\lambda})$ can be rewritten as\n\\begin{equation}\n\\theta(\\overline{\\lambda})= 2 \\left[ p_+ \\text{\\mdseries e}^{\\overline{\\lambda}} +p_- \\text{\\mdseries e}^{-\\overline{\\lambda}} -(p_++p_-)\\right]\n\\end{equation}\nwith the probabilities $p_+=(1+\\gamma_\\text{\\mdseries o})(1-\\gamma_\\text{\\mdseries e})\/2$ and $p_-=(1-\\gamma_\\text{\\mdseries o})(1+\\gamma_\\text{\\mdseries e})\/2$. As a consequence all cumulants of even (odd) order are equal to the same value when they are measured in unit of $\\Delta E$: for all $p\\geq 1$\n\\begin{eqnarray}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{2p-1}}_c}{ t \\, \\Delta E^{2p-1}} &=& \\frac{1}{4} \\nu_\\text{\\mdseries s} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}) \\, N\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{2p}}_c}{t \\, \\Delta E^{2p}} &=& \\frac{1}{4} \\nu_\\text{\\mdseries s} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) \\, N.\n\\end{eqnarray}\nIn the same kinetic regime, if one thermostat is at zero temperature, for instance $\\gamma_\\text{\\mdseries o}=1$, then\nthe scaled generating function coincides with that of a continuous-time Poisson process, because\n\\begin{equation}\n\\theta(\\overline{\\lambda})= 2 (1-\\gamma_\\text{\\mdseries e}) \\left[ \\text{\\mdseries e}^{\\overline{\\lambda}}-1\\right].\n\\end{equation}\nAs a consequence all cumulants of $\\Heat_\\text{\\mdseries e}$ in unit of $\\Delta E$ are equal to the same value: for all $p\\geq 1$\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{p}}_c}{t \\, \\Delta E^p} = \\frac{1}{4} \\nu_\\text{\\mdseries s} (1-\\gamma_\\text{\\mdseries e}) \\, N.\n\\end{equation}\n\n\\section{Conclusion}\n\n\nIn the present paper we have investigated the heat currents in an Ising spin ring where alternating spins are coupled to two macroscopic bodies at different temperatures and with different kinetic parameters. The stationary mean values of the global two-spin correlations at any distance have been calculated. The dependence upon the kinetic parameters arises only through the linear combination $\\gamma$ defined in \\eqref{defgamma}\\footnote{When the kinetic parameters are set equal, our result are compatible with those of Ref.\\cite{SchmittmannSchmuser2002}.}. The finite-size corrections in the global two-spin correlations disappear in the mean instantaneous global heat current coming out of one thermostat.\n\nThe scaled generating function of the joint cumulants per unit of time for the heat amounts exchanged with the two thermostats over a long time have been calculated exactly. At leading order in the ring size they prove to be proportional to the ring size, as it is the case for the model where two thermostats act on the same site \\cite{CornuHilhorst2017}.\n Moreover, if the order of the cumulant is lower than the number of spins connected to one thermostat, the finite-size corrections again disappear exactly, and the cumulant is strictly proportional to the ring size.\n\n We notice that the proportionality to the ring size at leading order in the size has already been observed for the cumulants of two other kinds of cumulative quantities when the system is homogeneous (only one temperature and one kinetic parameter) and seen as a Simple Exclusion Process with pair creation and annihilation \\cite{PopkovSchutz2011}: the two cumulative quantities are the difference between the numbers of domain wall jumps in the clockwise and anticlockwise directions respectively, and the number of pair annihilations. This is in contrast with the case of the purely diffusive Simple Exclusion Process on a ring, where the cumulants for the difference between the numbers of jumps in the two directions and the cumulants for the total number of jumps are proportional to powers of the ring size which increase with the order of the cumulants \\cite{AppertDerridaETAL2008}.\n\n\nAn interesting open problem is the calculation of the heat cumulants in another solvable model for thermal contact : two joined semi-infinite Ising chains \ncoupled to thermostats at two different temperatures \\cite{LavrentovichZia2010,Lavrentovich2012}. Then the mean global current which flows from one thermostat to the other through the junction between the two half-chains is obtained by summing the mean currents received by all spins in a semi-infinite Ising chain. The intrinsic inhomogeneity of these currents\nwould have to be dealt with by specific methods.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUnstable particles (UP's) are described usually by dressed propagator or with the help of the $S$-matrix with complex pole. The problems of these descriptions have been under considerable discussion for many decades\n[\\refcite{1}]--[\\refcite{7}] (and references therein). There are\nalso other approaches such as time-asymmetric QFT of UP's\n[\\refcite{6}], effective theory [\\refcite{8,9}], modified\nperturbative theory approach [\\refcite{10}], and phenomenological\nQF model of UP's with smeared mass [\\refcite{11,12}]. In this\nwork we consider some remarkable properties of the model [\\refcite{11,12}] which\nare caused by mass smearing and lead to the factorization effects\nin the description of the processes with UP in an intermediate\nstate.\n\nThe model under consideration is based on the time-energy\nuncertainty relation (UR). Despite their formal uniformity,\nvarious UR's have different physical nature. This point has been\ndiscussed for many years beginning with Heisenberg's formulating\nthe uncertainty principle (for instance, see\n[\\refcite{13}]--[\\refcite{16}] and references therein). The first\nmodel of UP based on the time-energy UR was suggested in\n[\\refcite{17}]. Time-dependent wave function of UP in its rest\nframe was written in terms of the Fourier transform which may be\ninterpreted as a distribution of mass values, with a spread,\n$\\delta m$, related to the mean lifetime $\\delta \\tau =1\/\\Gamma$\nby uncertainty relation [\\refcite{14,17}]:\n\\begin{equation}\\label{1.1}\n\\delta m\\cdot\\delta\\tau\\sim 1,\\,\\,\\,\\mbox{or}\\,\\,\\,\\delta m\\sim\n\\Gamma\\,\\,\\,(c=\\hbar =1).\n\\end{equation}\nThus, from the time-energy UR for the unstable quantum system, we\nare led to the concept of mass smearing for UP, which is described\nby UR (\\ref{1.1}). Implicitly (indirectly) the time-energy UR, or\ninstability, is usually taken into account by using the complex\npole in $S$-matrix or dressed propagator which describes UP in an\nintermediate state. Explicit account of the relation (\\ref{1.1})\nis taken by describing UP in a final or initial states with the\nhelp of the mass-smearing effect. From Eq. (\\ref{1.1}) it follows\nthat this effect is noticeable if UP has a large width. Mass\nsmearing was considered in the various fields of particle\nphysics---in the decay processes of UP with large width\n[\\refcite{12}], in the boson-pair production [\\refcite{18,19}],\nand in the phenomenon of neutrino oscillations [\\refcite{16,20}].\nIn these papers, the efficiency of the mass-smearing conception\nwas demonstrated in a wide class of processes.\n\nThe effects of exact factorization in the two-particle scattering\nand three-particle decay were shown for the cases of scalar,\nvector, and spinor UP in Refs.~[\\refcite{21,22}]. Factorized\nformulae for cross section and decay width were derived exactly\n(without any approximations) for the tree-level processes in the\nphysical gauge. However, the results can be generalized taking\naccount of principal part of radiative corrections\n[\\refcite{12,18,19,21}] (see Section 3). The factorization method\nthat is based on the exact factorization in the simplest processes\nwas suggested in [\\refcite{22a}]. In this work we generalize the\nresults of [\\refcite{22a}] to the case of UP with spin $J=3\/2$ and\napply them to some complicated processes. In the second section we\ndescribe the main elements of the model that lead to the effects\nof factorization. These effects in the case of the simplest\nprocesses are considered in Section 3 for UP with\n$J=0,\\,1\/2,\\,1,\\,3\/2$. The factorization method based on the results\nof the third section is applied to some more\ncomplicated processes in Section 4. In this section, we also\nconsider the accuracy of the calculations. Some conclusions are\nmade concerning the applicability and advantages of the method in\nthe fifth section.\n\n\\section{Model of UP with a smeared mass}\n\n\nIn this section we present the elements of the model\n[\\refcite{11,12}] that are used directly in the factorization\nmethod. The field function of the UP is a continuous superposition\nof the standard ones defined at a fixed mass with a weight\nfunction of the mass parameter $\\omega(\\mu)$ which describes mass\nsmearing. As a result of this, an amplitude of the process with\nUP in a final or initial state has the form:\n \\begin{equation}\\label{2.1}\n A(k,\\mu)=\\omega(\\mu)A^{st}(k,\\mu),\n \\end{equation}\nwhere $A^{st}(k,\\mu)$ is an amplitude defined in a standard way at\nfixed mass parameter $\\mu$, and $\\omega(\\mu)$ is a model weight\nfunction.\n\nThe model Green's function has a convolution form with respect to\nthe mass parameter $\\mu$ (the Lehmann representation). In the case\nof scalar UP it is as follows:\n \\begin{equation}\\label{2.2}\n D(x)=\\int D(x,\\mu)\\,\\rho(\\mu)\\,d\\mu,\\,\\,\\,\\rho(\\mu)=|\\omega(\\mu)|^2,\n \\end{equation}\nwhere $D(x,\\mu)$ is defined in a standard way for a fixed\n$\\mu = m^2$ and $\\rho(\\mu)$ is a probability density of the mass parameter\n$\\mu$.\n\nThe model propagators of scalar, vector, and spinor unstable\nfields in momentum representation are given by the following\nexpressions [\\refcite{12}]:\n\\begin{align}\\label{2.3}\n D(q)=&i\\int\\frac{\\rho(\\mu)\\,d\\mu}{q^2-\\mu+i\\epsilon},\\,\\,\\,D_{mn}(q)=-i\\int\\frac{g_{mn}-\n q_{m}q_{n}\/\\mu}{q^2-\\mu+i\\epsilon}\\rho(\\mu)\\,d\\mu,\\notag\\\\\n &\\hat{G}=i\\int\\frac{\\hat{q}+\\sqrt{\\mu}}{q^2-\\mu+i\\epsilon}\\rho(\\mu)\\,d\\mu,\n \\qquad q=\\sqrt{(q_iq^i)}.\n\\end{align}\nThe model propagators are completely defined if the function $\\rho(\\mu)$ is\ndetermined.\n\nHere, we generalize the results of the works [\\refcite{12,21,22}]\nto include the case of unstable fields with spin $J=3\/2$. The propagator of\nthis field is defined in [\\refcite{23,24}] and smearing its mass, $M^2\\to\\mu$, gives:\n\\begin{align}\\label{2.4}\n &\\hat{G}_{mn}(q)=\\int\\rho(\\mu)\\,d\\mu \\{\n -\\frac{\\hat{q}+\\sqrt{\\mu}}{q^2-\\mu+i\\epsilon} ( g_{mn} -\\frac{1}{3} \\gamma_{m}\n \\gamma_{n} - \\frac{\\gamma_{m} q_{n} -\\gamma_{n} q_{m}}{3\\sqrt{\\mu}} - \\frac{2}{3}\n \\frac{q_{m} q_{n}}{\\mu})\\}.\n\\end{align}\n\nDetermination of the weight function $\\omega(\\mu)$ or\ncorresponding probability density $\\rho(\\mu)=|\\omega(\\mu)|^2$ can\nbe done with the help of the various methods [\\refcite{12}]. Here\nwe consider the definition of $\\rho(\\mu)$ which leads to the\neffect of exact factorization. We match the model propagator of\nscalar UP to the standard dressed one:\n\\begin{equation}\\label{2.5}\n \\int\\frac{\\rho(\\mu)d\\mu}{k^2-\\mu+i\\epsilon}\\longleftrightarrow\n \\frac{1}{k^2-M^2_0-\\Pi(k^2)}\\,,\n\\end{equation}\nwhere $\\Pi(k^2)$ is a conventional polarization function. It was\nshown in [\\refcite{11,12}] that the correspondence (\\ref{2.5})\nleads to the definition:\n\\begin{equation}\\label{2.6}\n \\rho(\\mu)=\\frac{1}{\\pi}\\,\\frac{Im\\Pi(\\mu)}{[\\mu-M^2(\\mu)]^2+[Im\\Pi(\\mu)]^2}\\,,\n\\end{equation}\nwhere $M^2(\\mu)=M^2_0+Re\\Pi(\\mu)$. Substitution of the expression\n(\\ref{2.6}) into (\\ref{2.3}) and integration over $\\mu$ lead to\nthe results:\n\\begin{equation}\\label{2.7}\n D_{mn}(q)=i\\frac{-g_{mn}+q_m\n q_n\/q^2}{q^2-M^2(q^2)-iIm\\Pi(q^2)}\n\\end{equation}\nand\n\\begin{equation}\\label{2.8}\n \\hat{G}(q)= i\\frac{\\hat{q}+q}{q^2-M^2(q^2)-iq\\Sigma(q^2)}\\,.\n\\end{equation}\nIn analogy with these definitions we get the expression for the\npropagator of vector-spinor unstable field:\n\\begin{equation}\\label{2.9}\n \\hat{G}_{mn}(q)=\n -\\frac{\\hat{q}+q}{q^2-M^2(q^2)-iq\\Sigma(q^2)} \\{ g_{mn} -\\frac{1}{3} \\gamma_{m}\n \\gamma_{n} - \\frac{\\gamma_{m} q_{n} -\\gamma_{n} q_{m}}{3q} - \\frac{2}{3}\n \\frac{q_{m} q_{n}}{q^2}\\}.\n\\end{equation}\nNote that in Eqs. (\\ref{2.8}) and (\\ref{2.9}) we have substituted\n$q\\Sigma(q^2)$ for $\\Pi(q^2)$. The expressions\n(\\ref{2.5})--(\\ref{2.9}) define an effective theory of UP's which\nfollows from the model and the definition (\\ref{2.6}), that is\nfrom the correspondence (\\ref{2.5}). In this theory the numerators\nof the expressions (\\ref{2.7})--(\\ref{2.9}) differ from the\nstandard ones. The correspondence between standard and model\nexpressions for the cases of vector (in unitary gauge) and spinor\nUP is given by the interchange $m \\leftrightarrow q$ in the\nnumerators of the standard and model propagators (here\n$q=\\sqrt{q_i q^i})$. As a result the structures of the model\npropagators lead to the effect of exact factorization, while the\nstandard ones lead to approximate factorization (see the next\nsection).\n\n\\section{Effects of factorization in the processes\\\\ with UP in an intermediate state}\n\nExact factorization is stipulated by the following properties of the model:\\\\\na) smearing the mass shell of UP in accordance with the time-energy UR;\\\\\nb) specific structure of the numerators of the propagators, Eqs.~(\\ref{2.7})--(\\ref{2.9}).\\\\\nThe first factor allows us to describe UP in an intermediate state\nwith the momentum $q$ as a particle in a final or initial states\nwith the variable mass $m^2=q^2$. In this case, UP is described by\nthe following polarization matrices which differ from the standard\non-shell ones by the change $m\\to q$ (see also [\\refcite{12}]):\n\\begin{align}\\label{3.1}\n&\\sum_{a=1}^{3} e^a_m(\\vec{q})\\dot{e}^a_n(\\vec{q})=-g_{mn}+\\frac{q_m q_n}{q^2}\\,\\,\\,\\,\\,\\mbox{(vector UP)};\\notag\\\\\n&\\sum_{a=1}^{2}u^{a,\\mp}_i(\\vec{q})\\bar{u}^{a,\\pm}_k(\\vec{q})=\\frac{1}{2q^0}(\\hat{q}+q)_{ik}\n\\,\\,\\,\\,\\,\\mbox{(spinor UP)};\\notag\\\\\n&\\hat{\\Pi}_{mn}(q)=-\\frac{1}{4} ( \\hat{q} + q ) \\left[g_{mn}\n-\\frac{1}{3} \\gamma_{m} \\gamma_{n}\n - \\frac{\\gamma_m q_n -\\gamma_n q_m}{3q} - \\frac{2}{3} \\frac{q_m q_n}{q^2}\n \\right],\\notag\\\\\n &\\mbox{(vector-spinor UP)}.\n\\end{align}\nThe second factor is the coincidence of the expressions for the\npropagator numerators (\\ref{2.7})--(\\ref{2.9}) and for the\npolarization matrices (\\ref{3.1}). It allows us to represent the\namplitude of the process with UP in an intermediate state (see\nFig.1) in a partially factorized form:\n\\begin{equation}\\label{3.2}\n\\mathit{M}(p,p',q)=K\\sum_{a}\\frac{\\mathit{M}^{(a)}_1(p,q)\\cdot\n\\mathit{M}^{(a)}_2(p',q)}{P(q^2,M^2)},\n\\end{equation}\nwhere $\\mathit{M}^{(a)}$ is a spiral amplitude. The representation\n(\\ref{3.2}) is a precondition of factorization, while full exact\nfactorization occurs in the transition probability. The effect of\nfactorization is illustrated in Fig.1 where UP in an intermediate\nstate is signed by crossed line.\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor1.eps,height=2cm,width=5cm}}\n \\caption{Factorization in the reducible diagram.}\n \\label{fig:Born1}\n\\end{figure}\n\n\nNow, we demonstrate the factorization effect in the case of the\nsimplest basic elements of the tree processes, where UP is in the\n$s-$ channel intermediate state. The vertices are described by the\nsimplest standard Lagrangians for scalar, vector, and spinor\nparticles (see Refs. [\\refcite{21,22}]). Here we add the\nLagrangian that describes the interaction of vector-spinor\nparticles with lower-spin ones [\\refcite{25}]--[\\refcite{27}]:\n\\begin{align}\\label{3.3}\n \\mathscr{L}_{int}&=\n \\frac{f}{m_\\pi} \\bar{\\Psi}^\\mu \\bar{\\Theta}_{\\nu\\mu} (z,\\lambda) N \\pi^{,\\nu}_a +\n \\mbox{h.c.}\\notag\\\\\n &+\\frac{g}{2 M_N} \\bar{\\Psi}^\\mu \\bar{\\Theta}_{\\nu\\mu} (x,\\lambda)\n \\gamma^{\\xi} \\gamma_5 N\\rho_{\\xi}{}^{\\nu} +\n \\mbox{h.c.},\n\\end{align}\nwhere\n\\begin{equation}\\label{3.4}\n \\Theta_{\\mu\\nu} (\\lambda,\\lambda') =\n g_{\\mu\\nu} + \\frac{\\lambda'-\\lambda}{2\\left( 2 \\lambda - 1 \\right)} \\gamma_\\mu\n \\gamma_\\nu\\,,\n\\end{equation}\n$x$, $z$ are off-shell parameters, and $\\lambda$ is the parameter of\nthe Lagrangian of the Rarita-Schwinger free field.\n\nLet us consider the simplest processes which are used further as the\nbasic elements of the method. Note that calculations are\nmade at the \"tree level\" in the framework of the effective\ntheory, which follows from the model with mass smearing and contains self-energy\ncorrections in the function $\\rho(\\mu)$. This effective theory is\nnot a gauge one, but the definitions of the model propagators\n(\\ref{2.3}) are given in analogy with the standard physical\ngauge (unitary gauge for massive fields).\n\nThe first element is two-particle scattering with UP of any type\nin the intermediate state.\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor2.eps,height=2cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $2\\to 2$ scattering diagram}\n \\label{fig:Born2}\\end{figure}\n\nBy straightforward calculation at the tree level it was shown that\nthe cross-section for all permissible combinations of particles\n$(a,b,R,c,d)$ can be represented in the universal factorized form\n[\\refcite{21}]:\n\\begin{equation}\\label{3.5}\n \\sigma(ab\\rightarrow R\\rightarrow cd)=\\frac{16\\pi(2J_R+1)}\n {(2J_a+1)(2J_b+1)\\bar{\\lambda}^2(m_a,m_b;\\sqrt{s})}\n \\frac{\\Gamma^{ab}_R(s)\\Gamma^{cd}_R(s)}{|P_R(s)|^2}.\n\\end{equation}\nHere, $s=(p_a+p_b)^2$, $P_R(s)$ is the denominator of the\npropagator of an unstable particle $R(s)$ which is defined by\nEqs.~ (\\ref{2.7}) and (\\ref{2.8}), $\\lambda(m_a,m_b;\\sqrt{s})$ is\nnormalized K\\\"{a}ll\\'{e}n function and\n$\\Gamma^{ab}_R(s)=\\Gamma(R(s)\\to ab)$ is a partial width of the\nparticle $R(s)$ with variable mass $m=\\sqrt{s}$. Note that exact\nfactorization in the process under consideration always takes\nplace for scalar $R$ in both the standard and model treatment. In\nthe case of vector and spinor $R$ the factorization is exact in\nthe framework of the model only. In the standard treatment the\nexpression (\\ref{3.5}) is valid in the narrow-width approximation\n(see, for instance, Eq.~(37.51) and corresponding comment in\n[\\refcite{27a}]). If the diagram depicted in Fig.2 is included into a more\ncomplicated one as a sub-diagram, where the particle $c$ and\/or $d$\nare unstable, then we have to generalize partial width, for\ninstance, $\\Gamma^{cd(q)}_R (s,q)=\\Gamma(R(s)\\to cd(q))$, where\n$d(q)$ is unstable particle $d$ with mass $m^2=q^2$ (see the next\nsection). Note that Eq.(\\ref{3.5}) differs from the corresponding\nequation (6) in [\\refcite{21}] and equation (42) in [\\refcite{22}]\nwhich contain misprints. Correct expressions can be got in\n[\\refcite{21,22}] by modification $k_a k_b\/k_R\\to k_R\/k_a k_b$,\nwhere $k_p=2J_p+1$ and $J_p$ is spin of the particle $p =\na,\\,b,\\,R$.\n\nIt was shown in Ref.[\\refcite{21}] that Eq.(\\ref{3.5}) is valid\nfor the cases of scalar ($J=0$), vector ($J=1$), and spinor\n($J=1\/2$) unstable particles. In this work we check by direct\ncalculation that Eq.(\\ref{3.5}) is also correct in the case of UP\nwith $J=3\/2$, when Eq.~(\\ref{2.9}) is used (for instance,\n$\\Delta$-resonance production). We should note that the\nexpressions which involve vector-spinor UP are valid for the\nparticles on the mass shell in the framework of the standard\ntreatment. However, in the framework of the model, UP is always on\nits smeared mass shell and these expressions are valid in general\ncase. Moreover, the part of expressions which contains off-shell\nparameters as well as $\\lambda$-parameter disappears in widths and\ncross-sections. Hence the condition of factorization---the\ncoincidence of the polarization matrix and the numerators of the\npropagators---is fulfilled in the case under consideration (see\nEqs. (\\ref{2.9}) and (\\ref{3.1})).\n\nThe second basic element is a three-particle decay with UP in the\nintermediate state $\\Phi\\to\\phi_1R\\to\\phi_1\\phi_2\\phi_3$ (Fig.\\ref{fig:Born3}), where\n$R$ is UP of any kind.\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor3.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $1\\to 3$ decay diagram}\n \\label{fig:Born3}\\end{figure}\n\nBy straightforward calculations it was shown that the\nthree-particle partial width at the tree level can be represented\nin the universal factorized form [\\refcite{22}]:\n\\begin{equation}\\label{3.6}\n \\Gamma(\\Phi\\rightarrow\\phi_1 \\phi_2 \\phi_3)=\\int_{q^2_1}^{q^2_2}\\Gamma(\\Phi\n \\rightarrow\\phi_1 R(q))\\,\\frac{q\\,\\Gamma(R(q)\\rightarrow\\phi_2\\phi_3)}\n {\\pi \\vert P_{R}(q)\\vert ^2}\\,dq^2\\,,\n\\end{equation}\nwhere $R$ is a scalar, vector or spinor UP, $q_1=m_2+m_3$ and\n$q_2=m_{\\Phi}-m_1$. By direct calculations we also check the\nvalidity of the expression (\\ref{3.6}) for the case of the UP with\n$J=3\/2$ (see the remark to Eq.(\\ref{3.5})). It is seen clearly that\nthe formula (\\ref{3.6}) can include any factorizable corrections.\n\nBy summing over decay channels of $R$, from\nEq.(\\ref{3.6}) we get the well-known convolution formula for the\ndecays with UP in a final state [\\refcite{22}]:\n\\begin{equation}\\label{3.7}\n \\Gamma(\\Phi\\rightarrow\\phi_1 R) = \\int_{q^2_1}^{q^2_2} \\Gamma(\\Phi\\rightarrow\n \\phi_1 R(q))\\,\\rho_{R}(q)\\,dq^2\\,.\n\\end{equation}\nIn Eq.(\\ref{3.7}) smearing the mass of unstable state $R$ is\ndescribed by the probability density $\\rho_R(q)$:\n\\begin{equation}\\label{3.8}\n \\rho_{R}(q)=\\frac{q\\,\\Gamma^{tot}_R(q)}{\\pi\\,\\vert P_{R}(q)\\vert^2}.\n\\end{equation}\nThe expression (\\ref{3.8}) is connected with Eq.(\\ref{2.6}) by the\nrelation $Im\\Pi(q)=q\\, \\Gamma^{tot}_R(q)$.\n\nThe factorized expressions (\\ref{3.6}) and (\\ref{3.7}) are\napplied successfully for the description of the decay $B\\to \\rho D$\n[\\refcite{12}], decay properties of $\\phi(1020)$-meson\n[\\refcite{12}] and $t$-quark [\\refcite{31,32,33}], lightest\nchargino and next-to-lightest neutralino [\\refcite{33a}].\nMoreover, the formula (\\ref{3.6}) describes\nthe decays $\\mu\\to e\\bar{\\nu}_e \\nu_{\\mu}$, $\\tau\\to\ne\\bar{\\nu}_e\\nu_{\\tau}$, and $\\tau^-\\to \\nu_{\\tau}\\pi^-\\pi^0$ with great accuracy (see\nthe next section). It should be noted that, in analogy with two-particle\nscattering, the factorization in the expression for the width\n(\\ref{3.6}) is also exact within the framework of the model and\napproximate in the standard treatment (convolution method). Besides,\nwe note that the expressions (\\ref{3.5}) and (\\ref{3.6})\nsignificantly simplify calculations in comparison with the standard\nones.\n\n\\section{Factorization method in the model of UP's\\\\ with smeared mass}\n\nThe method is based on exact factorization of the simplest\nprocesses with UP in an intermediate state that were considered in\nSection 3. The factorization method has applicability to such Feynman diagrams that can be disconnected into two components by cutting some line corresponding to timelike momentum transfer. For instance, it is applicable to the\ncomplicated scattering and decay-chain processes which can be\nreduced to a chain of the basic elements (\\ref{3.5}) and\n(\\ref{3.6}). Next, we consider some examples of such processes.\n\n\\begin{equation}\\label{4.1}\n1)\\, a+b\\to R_1\\to c+R_2\\to c+d+f \\text{ (Fig.\\ref{fig:Born4})}.\n\\end{equation}\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor4.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $2\\to 3$ scattering-decay diagram}\n \\label{fig:Born4}\\end{figure}\n\nThe cross-section of this process is a combination of the\nexpressions (\\ref{3.5}) and (\\ref{3.6}):\n\\begin{align}\\label{4.2}\n &\\sigma(ab\\to R_1\\to cdf)=\\notag\\\\\n &\\frac{16k_{R_1}}{k_a k_b\\bar{\\lambda}^2(m_a,m_b;\\sqrt{s})}\\frac{\\Gamma^{ab}_{R_1}(s)}{|P_{R_1}(s)|^2}\n \\int_{q^2_1}^{q^2_2}\\Gamma(R_1(s)\\to\n c R_2(q))\\,\\frac{q\\,\\Gamma^{df}_{R_2}(q)}{|P_{R_2}(q)|^2}\\,dq^2,\n\\end{align}\nwhere $k_p=2J_p+1$. It should be noted that the factorization\neffectively reduces the number of independent kinematic\nvariables which have to be integrated. In the standard approach\nfor the process $2\\to 3$ the number of the variables, which\nuniquely specify a point in the phase space, in the general case\nis $N=3n-4=5$, from which four variables have to be integrated\n[\\refcite{28}]. Some of this variables can be integrated out if a\nspecific symmetry of the process occur. In the framework of the\napproach suggested the number of integrated variables is always\n$N_M=1$. The same effect of variable reduction takes place for the\ncase of the basic processes of scattering and decay which were\nconsidered in the previous section.\n\nThe expression (\\ref{4.2}) can be used for fast evaluation\nof cross sections of some scattering processes both in cosmology\nand collider physics. For instance, it is valid for the\ndescription of the annihilation process with the lightest\nsupersymmetric particle in a final state [\\refcite{33a}].\n\n\\begin{equation}\\label{4.3}\n2)\\, \\Phi\\to a+R_1\\to a+b+R_2\\to a+b+c+d\\text{ (Fig.\\ref{fig:Born5})}.\n\\end{equation}\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor5.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $1\\to 4$ decay diagram}\n \\label{fig:Born5}\\end{figure}\n\nThe width of this decay-chain process is given by doubling the\nformula (\\ref{3.6}):\n\\begin{align}\\label{4.4}\n\\Gamma(\\Phi\\to abcd)&=\\frac{1}{\\pi^2}\\int_\n{q^2_1}^{q^2_2}\\frac{q\\,\\Gamma(\\Phi\n \\rightarrow a R_1(q))}{\\vert P_{R_1}(q)\\vert ^2}\\times\\notag\\\\&\\times\\int_{g^2_1}^{g^2_2}\\Gamma(R_1(q)\\to bR_2(g))\n \\,\\frac{g\\,\\Gamma(R_2(g)\\rightarrow cd)}\n {\\vert P_{R_2}(g)\\vert ^2}\\,dg^2\\,dq^2.\n\\end{align}\nNote that in the general case of $n$-particle decay the number of\nkinematic variables, which uniquely specify a point in the phase\nspace, is $N=3n-7=5$ [\\refcite{28}], while the\nmethod gives $N_M=2$ (see comment to the previous case). Thus, we\nhave a significant simplification of calculations. Some\nexamples of processes which can be described by the compact\nformula (\\ref{4.4}) are considered in [\\refcite{33}], where the\nrelation between convolution method and decay-chain method is\nanalyzed.\n\n\\begin{equation}\\label{4.5}\n3)\\, a+b\\to c+R\\to c+d+e\\text{ (Fig.\\ref{fig:Born6})}.\n\\end{equation}\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor6.eps,height=3cm,width=5cm}}\n \\caption{Factorization in $a+b\\to c+R\\to c+d+e$ process.}\n \\label{fig:Born6}\n\\end{figure}\n\nThe cross-section of this $t$-channel process is described by\nconvolution of the cross-section $\\sigma(ab\\to cR)$ and the width\n$\\Gamma(R\\to de)$:\n\\begin{equation}\\label{4.6}\n\\sigma(ab\\to cde)=\\frac{1}{\\pi}\\int_{q_1^2}^{q_2^2}\\sigma(ab\\to\ncR(q))\\frac{q\\Gamma(R(q)\\to de)}{\\vert P_{R}(g)\\vert ^2}\\,dq^2.\n\\end{equation}\nThis formula can be applied to the description of the processes\n$e^+e^-\\to \\gamma Z\\to \\gamma f\\bar{f}$ and $eN\\to e\\Delta \\to\ne\\pi N$. The diagram in Fig.6 illustrates a class of processes at\nthe tree level with fermion-antifermion pair in the one-pole\napproximation, i.e. generated from the decay of $R$ only. However,\nthe formula (\\ref{4.6}) can be easily generalized taking account\nof factorizable radiative corrections. For instance, such a\ngeneralization of the expression (\\ref{4.6}) was made in\n[\\refcite{28a}] for the description of the process $e^+e^-\\to\n\\gamma Z\\to \\gamma \\sum_f \\nu_f\\bar\\nu_f$, $f=e,\\,\\mu,\\,\\tau$, with $\\nu\\bar\\nu$-pairs being produced\nby $Z$-decay only (``single-pole'' resonant production). Note that additional non-resonant ``ladder'' diagrams also contribute to the process $e^+e^-\\to\n\\gamma \\nu\\bar\\nu$. The resonant events can be still separated in certain kinematic regions [\\refcite{28aa}]. In our calculations [\\refcite{28a}] (with the kinematic cut corresponding to the event selection of Ref.[\\refcite{28aa}]) we have taken into account ISR and principal part\nof radiative corrections. These corrections do not change the\nstructure of the expression (\\ref{4.6}) and satisfy the condition\nof factorization. The results are in good agreement with the\nexperimental data and SM predictions at\n$\\sqrt{s}=185 - 210\\,\\mbox{GeV}$ [\\refcite{28a}]. We should note,\nhowever, that our calculations are valid in the energy domain, where\nthe selection of the resonant events is possible.\n\n\\begin{equation}\\label{4.7a}\n4)\\, e^+e^-\\to ZZ\\to \\sum_{i,k} \\bar{f}_i f_i \\bar{f}_k f_k\\text{\n(Fig.\\ref{fig:Born7})}.\n\\end{equation}\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor7.eps,height=2cm,width=5cm}}\n \\caption{$Z$-pair production process.}\n \\label{fig:Born7}\n\\end{figure}\nNow, we consider the process of $Z$-pair production (or\nfour-fermion production in the double-pole approximation). Direct\napplication of the model to the process $e^+e^-\\to ZZ$ or using\nthe factorization method for the full process $e^+e^-\\to\nZZ\\to \\sum_{i,k} \\bar{f}_i f_i \\bar{f}_k f_k$ (double-pole\napproach) gives the following expression for cross-section at the\ntree level [\\refcite{18}]:\n\\begin{equation}\\label{4.8a}\n\\sigma^{tr}(e^+e^-\\to ZZ)=\\int\\int\\sigma^{tr}(e^+e^-\\to\nZ_1(m_1)Z_2(m_2))\\,\\rho_Z(m_1)\\,\\rho_Z(m_2)\\,dm_1\\,dm_2,\n\\end{equation}\nwhere $\\sigma^{tr}(e^+e^-\\to Z_1(m_1)Z_2(m_2))$ is defined in\na standard way for the case of fixed boson masses $m_1$ and $m_2$,\nand probability density of mass $\\rho(m)$ is defined by the\nexpression:\n\\begin{equation}\\label{4.7}\n\\rho_Z(m)=\\frac{1}{\\pi}\\,\\frac{m\\,\\Gamma^{tot}_Z(m)}{(m^2-M^2_Z)^2+(m\\,\\Gamma^{tot}_Z(m))^2}.\n\\end{equation}\nSimilar expressions can be written for the processes $e^+e^-\\to\nW^+W^-$ [\\refcite{19}] and $e^+e^-\\to ZH$ [\\refcite{28a}]. To\ndescribe exclusive processes, such as $e^+e^-\\to ZZ\\to\nf_i\\bar{f}_i f_k\\bar{f}_k$, one has to substitute partial\n$q$-dependent width $\\Gamma_Z^i(q)=\\Gamma(Z(q)\\to f_i\\bar{f}_i)$\ninto the expression (\\ref{4.7}) instead of the total width\n$\\Gamma^{tot}_Z(m)$. Note that the expression similar to\n(\\ref{4.6}) can be written in the standard approach as a result of\nintegration over the phase space variables which describe\n$4f$-states in the semi-analytical approximation (SAA)\n[\\refcite{28b}]. Within the framework of the model considered,\nformula (\\ref{4.6}) is derived exactly without any approximations.\nNote also that as a rule the deviation of the standard exact\nresults from the model ones is negligible (see Eq. (\\ref{4.12})).\n\nThe processes $e^+e^-\\to ZZ, W^+W^-, \\gamma Z, ZH$ were considered\nin detail [\\refcite{17,18,28a}] taking account\nof the relevant radiative corrections. It was shown that the\nresults of the model calculations are in good agreement with the\nexperimental LEP II data and coincide with standard Monte-Carlo\nresults with great accuracy. At the same time the factorization\nmethod significantly simplifies the calculation procedures in\ncomparison with the standard ones that should consider about ten thousands\ndiagrams (see, for example, [\\refcite{28c}]\nand comments in [\\refcite{18}]).\n\nUsing the factorization method one can describe complicated decay-chain and scattering\nprocesses in a simple way. The same results can occur within the\nframe of standard treatment as the approximations. Such\napproximations are known as narrow-width approximation (NWA)\n[\\refcite{29,30}], convolution method (CM) [\\refcite{31}]--[\\refcite{33}], decay-chain method (DCM) [\\refcite{33}] and\nsemi-analytical approach (SAA) [\\refcite{28b}]. All these\napproximations get a strict analytical formulation within the\nframework of the factorization method. For instance, NWA includes five assumptions which\nwas considered in detail in [\\refcite{29}]. The factorization method\ncontains just one assumption---non-factorizable corrections are small\n(the fifth assumption of NWA). The method suggested can be applied\nto very complicated decay-chain and scattering processes by combining the expressions considered above. In this case\nwe have not a strict and general standard analog of such\napproximation.\n\nNow, we consider some ways of evaluation of the method error which\nwe define as the deviation of the model results from the strict\nstandard ones. For a scalar UP the error always equals zero in\naccordance with the definition (\\ref{2.5}). For a vector UP the\nerror is caused by the following difference:\n\\begin{equation}\\label{4.8}\n\\delta\\eta_{\\mu\\nu}=\\eta_{\\mu\\nu}(q^2)-\\eta_{\\mu\\nu}(m^2)=q_{\\mu}q_{\\nu}\\frac{m^2-q^2}{m^2q^2},\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}(m^2)$ and $\\eta_{\\mu\\nu}(q^2)$ are standard\nand model numerators of vector propagators in the physical gauge. In\nthe case of meson-pair production $e^+e^-\\to\n\\rho^0,\\omega,...\\to\\pi^+\\pi^-, K^+K^-,\\rho^+\\rho^-,...$ the\ndeviation equals zero too, due to vanishing contribution of the\ntransverse parts of the amplitudes in both cases:\n\\begin{equation}\\label{4.9}\n\\mathit{M}^{trans}(q)\\sim\n\\bar{e}^-(p_1)\\hat{q}e^-(p_2)=\\bar{e}^-(p_1)(\\hat{p}_1+\\hat{p}_2)e^-(p_2)=0.\n\\end{equation}\nIn the case of the high-energy collisions $e^+e^-\\to Z\\to\nf\\bar{f}$ (we neglect $\\gamma-Z$ interference) the transverse part\nof the amplitude is:\n\\begin{equation}\\label{4.10}\n\\mathit{M}^{trans}(q)\\sim\n\\bar{e}^-(p_1)\\hat{q}(c_e-\\gamma_5)e^-(p_2)\\cdot\n\\bar{f}^+(k_1)(c_f-\\gamma_5)f^+(k_2)\n\\end{equation}\nand we get at $q^2\\approx M^2_Z$:\n\\begin{equation}\\label{4.11}\n\\delta \\mathit{M}\\sim \\frac{m_em_f}{M^2_Z}\\frac{M_Z-q}{M_Z}.\n\\end{equation}\nThus, an error of the factorization method at the vicinity of resonance is always small,\nmoreover, it is suppressed by small factor $m_e m_f\/M^2_Z$. The\nsimilar estimations can be easily done for the case of a spinor\nUP.\n\nThe relative deviation of the model cross section of the\nboson-pair production with consequent decay of the bosons to\nfermion pairs is [\\refcite{18}]:\n\\begin{equation}\\label{4.12}\n\\epsilon_f \\sim\n4\\frac{m_f}{M}[1-M\\int_{m^2_f}^{s}\\frac{\\rho(q^2)}{q}\\,dq^2],\n\\end{equation}\nwhere $M$ is a boson mass. For the case $f=\\tau$ a deviation is\nmaximal, $\\epsilon_{\\tau}\\sim 10^{-3}$. It should be noted that\nthe deviations that are caused by the approach at the tree level\nare significantly smaller then the errors caused by the\nuncertainty in taking account of radiative corrections\n[\\refcite{19}]. Thus, the the error of the method at the tree\nlevel in the case of vector UP is, as a rule, negligible.\n\nBy straightforward calculations we evaluate the relative\ndeviations of the model partial width from standard one, that is,\n$\\epsilon =(\\Gamma^M-\\Gamma^{st})\/\\Gamma_M$ for the case of $\\mu$\nand $\\tau$ decays. We get:\n\\begin{equation}\\label{4.13}\n\\epsilon(\\mu\\to e\\nu\\bar{\\nu})\\approx 5\\cdot\n10^{-4};\\,\\,\\,\\epsilon(\\tau\\to e\\nu\\bar{\\nu})\\approx 3\\cdot\n10^{-6};\\,\\,\\,\\epsilon(\\tau\\to \\mu\\nu\\bar{\\nu})\\approx 3\\cdot\n10^{-2}.\n\\end{equation}\nThe deviation is suppressed by the factor $k=m^2_{l1}\/m^2_{l2}$,\nwhich is small in the first and second case and large in the last\ncase. In the case of the decay $\\tau^-\\to \\nu_{\\tau}\\pi^-\\pi^0$,\nsuppression factor is very small,\n$k=(m^2_{\\pi_0}-m^2_{\\pi_-})^2\/m^4_{\\tau} \\sim 10^{-7}$.\n\nIn the case of a spinor UP in an intermediate state a deviation is\nof the order of $(M_f-q)\/M_f$. It can be large when $q$ is far\nfrom the resonance region. However, in analogy with vector UP,\nthis deviation can be suppressed by small factor too, and we have\nto control this effect in every case under consideration. The same\neffect can occur in the case of the vector-spinor UP.\n\n\\section{Conclusion}\n\nThe model of UP's leads to effective theory of UP's with a\nspecific structure of vector, spinor and vector-spinor\npropagators. Such a structure gives rise to the effects of exact\nfactorization in a broad class of the processes with UP's in\nintermediate states. These effects allow us to develop the\nfactorization method for the description of the complicated\nprocesses with participation of an arbitrary type of UP's. The\nmethod suggested is simple and convenient tool for deriving the\nformulae for cross sections and decay rates in the case of\ncomplicated scattering and decay-chain processes. The\nfactorization method can be used as some analytical analog of NWA,\nwhich enables us to evaluate the error of the approach at the tree\nlevel in a simple way. We have shown that these errors, as a rule,\nare significantly smaller then the ones caused by the uncertainty\nin taking account of radiative corrections. It should be noted that the applicability of the method is limited to the energy scales where the non-resonant or non-factorizable contributions can be neglected.\n\nThe factorization method based on the model of UP with a smeared\nmass can be treated in two various ways. On the one hand, it\nfollows from the specific structure of propagators and can be\ninterpreted as some heuristic (irrespective of the model) way to\nevaluate decay rates and cross sections easily with the help of\nthe concise and convenient expressions. On the other hand, the\nmodel, from which the factorization method follows, is based on the fundamental\nproperties of UP---time-energy UR (i.e., smearing the mass). Thus,\nthe method can be also used as some physical basis for development\nof precision tools of rapid and easy calculations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\\emph{Neural architecture search} (NAS) is a rapidly growing area of \\emph{machine learning} (ML) \ndedicated to automatically designing high performing \\emph{deep learning} models.\nRecent breakthroughs, such as \\textit{differentiable search}, e.g., DARTS \\cite{DARTS},\nhave enabled search at limited computing cost and time.\nHowever, state-of-the-art methodologies still suffer from limited interpretability,\nand current evaluation protocols do not always shed light on the contribution of individual components \n(i.e., search space, training pipeline) while reporting performances \\cite{Yang2020NasEvalHard,Lindauer2020BestPractices}.\n\n\nRecently, a \\emph{fitness landscape analysis}-based (FLA) methodology was introduced: the \\emph{Fitness Landscape Footprint} (a.k.a., \\emph{footprint}) \\cite{traore2021fitness}.\nThe \\emph{footprint} attempts to describe \nwhy a search strategy may be successful, struggle or fail on a target application.\nIt also enables comparing search problems of variable configuration \n(i.e., different search space, fitness function, data, etc.).\n\nOur study takes advantage of the \\emph{footprint} to identify the most favorable sensor setting \nfor NAS.\nParticularly, we consider optimizing convolutional neural network (CNN) image classifiers on the search space defined by NASBench-101 \\cite{ying2019nasbench101} for the real-world image classification problem So2Sat LCZ42~\\cite{So2SatDataset}.\nOur results show that disregard the sensor, the longer the training time, the better the performance (fitness) and the flatter the landscape (less ruggedness and deviation in fitness). \nMoreover, Sentinel-2 and fusion (Sentinel-1 and 2) tend to have more favorable search trajectories (smoother, higher persistence).\nTo the best of our knowledge, our study provides the first quantification \nand comparison of search behavior across sensors (including sensor fusion).\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/footprint\/footprint_various_36.png}\n \\caption{\\emph{Fitness landscape footprint} of So2Sat LCZ42 for Sentinel-1 (blue), Sentinel-2 (yellow), and both sensors together (green), on CNN image classifiers NAS problem.} \n \\label{fig:footprnts}\n\\end{figure}\n\nThis article is structured as follows: \nNext section summarizes the related work and Section~\\ref{sec:fitness-landscape} introduces the \\emph{footprint}. \nSection~\\ref{sec:footprint-for-fusion} proposes the methodology to study NAS problems. \nSections \\ref{sec:exp-setup} and~\\ref{sec:results} presents the experimental settings and the results.\nSection~\\ref{sec:conclusion} outlines the conclusions and proposes future work.\n\n\n\n\n\n\n\n\\section{Related Work}\\label{sec:related-work}\n\n\\emph{Computer vision} (CV) and \\emph{Earth observation} (EO) are closely tied~\\cite{ball2017comprehensive,zhu2017deep}.\nDeep learning models in CV have helped tackle several specific EO use-cases such as \nscene classification~\\cite{LiuResNet2019},\nobject detection~\\cite{ZhangRCNN2019}, \nchange detection~\\cite{Mou2019RCNN_change_detection}, and \nsemantic segmentation~\\cite{YUAN2021Review_Semantic_Segm}, among others.\nHowever, the specificity of sensors require domain-specific models~\\cite{R3Net}.\nIn particular, the availability of several sensors to monitor areas \nhas motivated the activity of multi-modal sensor fusion which remains challenging for current models~\\cite{Hong2021MoreDiverseFusion}.\nMoreover, as the design of new vision-based methodologies can be time-consuming (trial and error),\nEO could benefit from \\emph{automated machine learning} (AutoML) algorithms.\n\nIn AutoML, NAS specializes in finding model configurations \nachieving optimal performances for a given dataset~\\cite{elsken2019neural,ojha2017review}.\nNAS methodologies have been proven to be powerful and efficient,\nwith strategies deriving from various families of optimization algorithm, e.g., \n\\emph{differentiable search}~\\cite{DARTS,traore2021esann}, \n\\emph{Bayesian optimization}~\\cite{camero2021bayesian}, \\emph{meta-heuristic}-based approaches~\\cite{stanley2002evolving,camero2020random,traore2021DataDrivenInit}.\nHowever, in practice, the difficulty of a NAS problem is hard to estimate, because the complexity of its components, namely the\nsearch space, search strategy, performance estimator and additional \\emph{tricks}, is hard to quantify~\\cite{elsken2019neural,Yang2020NasEvalHard}.\n\nThe fields of \\emph{evolutionary computation}, \\emph{optimization} and \\emph{complex systems} have long studied optimization processes and provide us with tools to analyze their behavior.\nIn particular, FLA~\\cite{pitzer2012comprehensive} aims at understanding \nand predicting performances of optimization algorithms.\nRecently, Traor\u00e9 et al. proposed the \\emph{Fitness Landscape Footprint}~\\cite{traore2021fitness}, \na framework to characterize NAS problems from the perspective of a search algorithm. \nThe following section introduces the \\emph{footprint}.\n\n\n\n\n\n\n\n\n\\section{A Framework for Comparative Fitness Landscape Analysis}\\label{sec:fitness-landscape}\n\nBefore describing the \\emph{footprint}, it is important to define what a fitness landscape is.\nLet~$S$ be the set of all possibles solutions of an optimization problem, i.e., the search space.\nLet~$f$ be the fitness function, which attributes to each candidate solution~$x \\in S$, \na fitness measurement~$f(x) \\in \\mathbb{R}$.\nLet~$N$ be a function providing a structure to the search space~$S$, the neighborhood relationship operator. \nThen, the fitness landscape~$\\mathcal{L} = (S, f, N)$ consists of combining the three above, in order \nto provide respectively with a set of possibles solutions, a function to evaluate them and another to interconnect them.\n\nGiven this definition, we are interested in a better understanding of a NAS optimization process. \nThe \\emph{footprint}~\\cite{traore2021fitness} serves this purpose \nof gaining insights into the process by describing its fitness landscape \nwith a set of eight (8) metrics measuring aspects such as\nthe distribution of fitness, ruggedness of the landscape or persistence of fitness. \nA~\\emph{footprint} includes: the mean and variance of fitness over~$S$, the ruggedness~$\\tau$, \nan enumeration of local optima, the positive and negative persistence and their area under the curve ($AuC$).\nThe following paragraphs describe these metrics.\n\nThe \\emph{fitness distance correlation} (FDC) is \noften interpreted a measure of the existence of \nsearch trajectories from randomly picked solutions to the known global optimum.\nIn practice, the FDC is not collected as a correlation score, \nbut visualized as the distribution of fitness versus \ndistance to the global optimum. \nIt writes as~$FDC(f,x^*, S)=\\{(d(x^*,y), f(y)) \\mid \\forall y \\in S\\}$, \nwhere~$S$ denotes the search space, \n$x^* \\in S$ is the global optimum, \n~$d$ a distance function. \n\nThe \\emph{ruggedness} of the landscape also helps assessing the difficulty of the process tackled.\nLet's consider a random walk~$RW$ in~$S$ of $N$ steps (models) and its corresponding fitness values.\nThe ruggedness~$\\tau$ consists in the auto-correlation length over ~$RW$ :~$\\tau=\\frac{1}{\\rho(1)}$, where ~$\\rho(k=1)$ is the serial-correlation coefficient for consecutive lags. \n\nIn \\cite{traore2021fitness}, the authors propose the metric of \\emph{persistence} characterizing the behavior of image classification models overtime. It measures the chances of solutions in the search space, \nto keep a rank~$N$ (top or bottom rank, based on fitness in test), as the training time grows.\nThis metric is complemented with its area under the curve ($AuC$), \nmeasuring the evolution of the persistence as $N$ grows.\n\nAnother way to characterize an optimization fitness landscape is to assess the existence of \\emph{local optima}.\nAs some search algorithms might get stuck in such sub-optimal areas of~$S$, \nan enumeration~\\cite{Hernando2012CardOptimaBenchmark} could help measure the difficulty of the search problem.\n\nLast but not least, the~\\emph{footprint} not only characterizes individual NAS landscapes, \nbut also enables the comparison of a handful considering potential changes \nin either components~$S$, ~$f$, or~$N$.\n\n\n\n\n\n\n\\section{Assessing and comparing the landscapes of NAS for various sensors}\\label{sec:footprint-for-fusion}\n\n\nThis study aims at investigating how the process of searching for neural architectures \nis affected by the type of sensor available as input.\nMore precisely, we seek to identify to what extent \ndoes searching with a given sensor \ndiffers from searching with another one.\nIn particular, we consider the case of a fixed search space, \ntraining pipeline (hyperparameters, duration, etc.)\nand evaluation protocol (fitness function). \nIn practice, we propose to tackle these questions by conducting \na comparative landscape study using the \\emph{footprint} (Section~\\ref{sec:fitness-landscape}).\n\nLet~$\\Sigma=\\{s_i, s_j, s_i + s_j\\}$ be the set of sensors available in our ML task.\nLet~$S$ be a search space of CNN image classifiers, each represented by a unique binary vector.\nConsidering this representation, we choose a neighborhood operator~$N(x)$ assigning to each solution of the search space, all the configurations that are one (1) \n\\textit{hamming distance} away from it. This operator~$N(x)$ writes as follows:~$N(x) = \\{ y \\in S \\mid d_{hamming(x, y)} = 1\\}$, \nwhere~$d_{hamming(x, y)}$ is the \\textit{hamming distance} between two solutions~$(x,y) \\in S^2$.\nAdditionally, we use as fitness function~$f_{s_{i}}$, the measurement of accuracy in test\nafter a training budget of~$b_t$, on an input sensor~$s_i$.\n\nSince we have access to various sensors~$s \\in \\Sigma$ for our ML task, \nthe fitness landscapes obtained write as follows:\n~$\\mathcal{L}_{s_i} = (S, f_{s_i}, N)$, ~$\\mathcal{L}_{s_j} = (S, f_{s_j}, N)$, ~$\\mathcal{L}_{s_k} = (S, f_{s_k}, N)$ for the sensor settings $(s_i, s_j, s_k) \\in \\Sigma^3$.\nIn particular, we consider the case of input level sensor fusion as:~$ s_k = s_i + s_j$.\nBesides, as noted above, the aim is a comparative study so we fix the search space~$S$ and the neighborhood operator~$N$ across all settings.\n\n\n\\section{Experimental Setup}\\label{sec:exp-setup}\nThis section introduces the NASBench-101 database, as well as a custom representation used to encode its solutions.\nThen, the So2Sat LCZ42 dataset used to evaluated solutions, followed by details on the evaluation protocol.\n\n\n\\subsection{NASBench-101}\n\nNASBench-101~\\cite{ying2019nasbench101} is a database containing a large pool of neural networks \nand their evaluations on the image classification dataset of CIFAR-10.\nIt aims at providing an exhaustive fitness measurement for all configurations (N=453k) \nin a search space of CNN image classifiers. \nThis search space defines a model configuration as an image classification backbone \nwith a head, body and tail.\nIts body consists of repeating three identical 'block' structures alternated with down-sampling modules.\nRegarding each block, it consist in a sequence of identical and elementary feed-forward units called cells.\nEach cell is represented by a directed DAG with a maximum number of nodes ($V \\leq 7$), \nmaximum number of edges ($E \\leq 9$) and a fixed listed of three (3) operators (Max-pool 3x3, Convolutional layer 1x1 and 3x3) \nlabelling each node. Therefore, a solution of the search space is identified by a cell, \nencoded in practice by both an adjacency matrix of variable size (upper triangular), and its list of operators.\nMoreover, The head of the model is a 3 x 3 convolution with 128 output channels, while the tail is a dense softmax layer.\n\n\\subsection{Custom feature representation}\n\nIn our experiments, \nwe construct a custom representation to enable solutions of the search space to be identified by a single vector.\nFirst, for our representation of the DAG, we do not label nodes.\nInstead, for the five (5) intermediate nodes out of seven (7) (one for IN and OUT), \nwe account for the fact that each could be one of three (3) operators.\nThus, the DAG contains exactly $N_{nodes}=(1+5*3+1) = 17$ nodes. \nThe new adjacency matrix is therefore of fixed length, i.e $L=17*17$ and non upper-triangular.\nFinally, we flatten the adjacency matrix to obtain a binary vector as identifier.\n\nRegarding the sampling of solutions $x \\in S$, we use the Latin Hypercube Sampling (LHS) for ensure fair data collection.\nBecause of a higher complexity of LHS on the large binary representation, we perform it on the intermediate \nrepresentation as a joint sampling of the original matrix and list of operations.\n\n\n\n\\subsection{So2Sat LCZ42}\n\n\n\nThe So2Sat LCZ42~\\cite{So2SatDataset} is a dataset of satellite imagery covering over forty-two (42) cities around the five (5) continents.\nIt provides with co-registered image patches from Sentinel-1 and Sentinel-2 sensors, \neach attributed with a single label.\nThis label associates an image to a class out of seventeen (17) possibilities, \nall representing the diverse Local Climate Zones (LCZ) around the globe.\nThe framework of LCZ proposes a generic way to describe the morphology \nof land use around the world in both urban and non-urban natural sites. \nThe classes are the following: Compact high-rise (1), Compact mid-rise (2), Compact low-rise (3), Open high-rise (4), \nOpen mid-rise (5), Open low-rise (6), Lightweight low-rise (7), Large low-rise (8), Sparsely built (9), and Heavy industry (10), \nDense trees (11), Scattered tree (12), Bush, scrub (13), Low plants (14), Bare rock or paved (15), Bare soil or sand (16), \nand Water (17).\nFigure~\\ref{fig:lcz42-dataset-visual} displays four (4) pairs of Sentinel-1 and Sentinel-2 image patches respectively from classes \n2, 4, 8 and 17.\n\nThe dataset comprises train ($N_t=352,366$), validation ($N_v=24,119$), and test ($N_{test}=24,188$) samples.\nThe training and validation samples originate from the same set of forty-two (42) cites, \nwhile those from the test set were collected in ten (10) additional cities.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/dataset\/description_lcz42_set2.png}\n \\caption{So2Sat LCZ42 samples. The top row contains SAR patches (Sentinel-1), followed by the associated Multi-spectral patches (Sentinel-2) in the bottom row.}\n \\label{fig:lcz42-dataset-visual}\n\\end{figure}\n\n\\subsection{Evaluation protocol}\nFor the purpose of training and evaluating on the same data distribution, \nwe use a custom setting consisting of training and testing sets made by randomly sampling respectively, \n$80\\%$ and $20\\%$ of the image patches of the original training-set. \nMoreover, as in \\cite{traore2021fitness} we speed-up the training procedure by only considering $P=35\\%$ of samples in the training set.\n\nAdditionally, we use the same search space~$S$ for all sensor settings.\nIn particular, we do not adapt the sampled models to use multiple sensors as input, \ninstead we do stack the data at an input level.\nWe trained ~$N=100$ randomly sampled models, once. After inspection and quality control, \nthere remain 100, 88 and 75 samples for Sentinel-1, 2 and both sensors.\nThe fitness is assessed in test using the Kappa-Cohen metric.\n\n\\section{Results}\\label{sec:results}\n\nThe following section presents results of comparison of search landscapes for various input sensors.\nFirst, we provide an analysis of distributions of fitness.\nThen, we show results of fitness distance correlation, followed by \nan analysis of random walks, as well as measurements of fitness persistence.\nLast but not least, we compare the \\emph{footprint} of the sensors.\n\n\\subsection{Density of Fitness}\\label{subsec:dos}\n\nFirst, we assess the ability of the search space in fitting the task with each sensor. \nFigure~\\ref{fig:pdf-36} and ~\\ref{fig:pdf-108e} display the probability density function (PDF) of fitness,\nrespectively after 36 and 108 epochs of training. \nThe first, second and third columns are, respectively, for using Sentinel-1, Sentinel-2 or both sensors as input.\n\nWe first take a look at the PDFs after 36 epochs of training.\nWhen using Sentinel-1, the distribution of fitness is wide and centered around low values ($\\mu=0.47, \\sigma=0.13$). \nSentinel-2 enables the distribution to improve by reaching a higher average and being more narrow ($\\mu=0.94, \\sigma=0.03$).\nUsing both sensors slightly worsens the fitness, providing with a lower mean and larger deviation ($\\mu=0.89, \\sigma=0.05$).\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/dos_test_acc_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/dos_test_acc_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/dos_test_acc_36e.jpeg}\n \\caption{PDF of fitness after 36 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors as input.\n } \n \\label{fig:pdf-36}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/dos_test_acc_108.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/dos_test_acc_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/dos_test_acc_108.jpeg}\n \\caption{PDF of fitness after 108 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors as input.\n } \n \\label{fig:pdf-108e}\n\\end{figure*}\n\nNext, we look at the PDFs after 108 epochs of training.\nOverall, the task is better handled in all sensor configurations.\nUsing Sentinel-1, the distribution improves by $17$ percentage points in mean fitness ($\\mu=0.64, \\sigma=0.13$).\nIn the case of Sentinel-2, most models fit well the data as the mean fitness increases and the deviation decreases ($\\mu=0.97, \\sigma=0.01$).\nWe observe similar results when using both sensors ($\\mu=0.94, \\sigma=0.04$).\n\n\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/pdf_fitting_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/pdf_fitting_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/pdf_fitting_36e.jpeg}\n \\caption{Fitting PDF to fitness with various theoretical functions after 36 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors. \n The red, blue and green colors correspond to Beta, Log-normal and Weibull distributions, respectively.\n } \n \\label{fig:pdf-fitting-36e}\n\\end{figure*} \n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/emp_vs_theoretical_CDFS_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/cdf_36.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/emp_vs_theoretical_CDFS_36e.jpeg}\n \\caption{ Cumulative PDFs of fitness versus theoretical density functions, after 36 epochs of training. \n The plot uses the same color and sensor convention as Figure~\\ref{fig:pdf-fitting-36e}.\n \n } \n \\label{fig:cdf-36e}\n\\end{figure*} \n\n\nBesides, we seek to identify for each sensor, \nif the behavior of the search space follows a specific theoretical distribution. \nFor this, we consider the more challenging scenario \nof selecting models after only 36 epochs of training.\n\nFigure~\\ref{fig:pdf-fitting-36e} and ~\\ref{fig:cdf-36e} show results of fitting empirical distributions of fitness,\nwith various theoretical distributions.\nFigure~\\ref{fig:pdf-fitting-36e} and ~\\ref{fig:cdf-36e} show, respectively, PDFs and cumulative density functions (CDF), \nall fitted with the Beta (red), Weibull (green) and Log-normal (blue) distributions.\nSimilarly, the first, second and third columns are respectively, \nfor using Sentinel-1, Sentinel-2 or both sensors as input.\nTo complement the plots, Table~\\ref{tab:pdf-fitting-error-s1},~\\ref{tab:pdf-fitting-error-s2} and~\\ref{tab:pdf-fitting-error-s12} provide with the respective fitting errors.\n\n\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & 46.28 & 44.52 & \\textbf{53.63} \\\\ \n\\hline\nAIC & -88.56 & -85.04 & \\textbf{-103.26} \\\\ \n\\hline\nBIC &-83.92 & -80.41 & \\textbf{-98.62}\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for Sentinel-1 } \n\\label{tab:pdf-fitting-error-s1}\n\\end{table}\n\n\\vspace{0.2cm}\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & \\textbf{165.18} & 164.70 & 155.05 \\\\ \n\\hline\nAIC & \\textbf{-326.36} & -325.40 & -306.10 \\\\ \n\\hline\nBIC & \\textbf{-321.80} & -320.84 & -301.54\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for Sentinel-2 } \n\\label{tab:pdf-fitting-error-s2}\n\\end{table}\n\n\\vspace{0.2cm}\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & \\textbf{109.72} & 109.04 & 107.78\\\\ \n\\hline\nAIC & \\textbf{-215.44} & -214.08 & -211.57 \\\\ \n\\hline\nBIC & \\textbf{-211.09} & -209.74 & -207.22\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for both sensors as input } \n\\label{tab:pdf-fitting-error-s12}\n\\end{table}\n\nOverall, the empirical distributions are closely fitted with the selected theoretical distributions.\nFor Sentinel-1, the best candidate is the Log-normal with the largest fitting likelihood, and lowest AIC and BIC error scores. When using Sentinel-2 or both sensors, Beta matches the best the empirical distributions. \n\nTo summarize, \nthe capacity of the search space in fitting the task with the Sentinel-1 sensor appears limited from the PDF perspective.\nIndeed, despite longer training time (108 epochs) the fitness distribution remains far worse than using Sentinel-2.\nUsing Sentinel-2 only, the task can be fitted well enough in particular given long training time.\nCombining Sentinel-1 to Sentinel-2 worsens the distribution of fitness (lower mean, larger deviation).\nTherefore, there is no tangible benefits in fitness, from sensor fusion using the current search space.\nMoreover, results indicate the feasibility in modeling the empirical distributions of fitness for each sensor.\n\n\\subsection{Fitness Distance Correlation}\\label{subsec:fdc}\n\nNext, we analyse the fitness landscape for the various sensor configurations.\nFigure~\\ref{fig:profiles-36} and \\ref{fig:profiles-108} show results of FDC for the three (3) input sensor settings.\nThe layout of the plots follows the convention of Figure~\\ref{fig:pdf-36}.\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen1\/profile_36e.jpeg}\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen2\/recollected\/profile_36e.jpeg}\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen12\/profile_36e.jpeg}\n \\caption{Fitness distance correlation after 36 epochs of training.\n From left to right are shown results using Sentinel-1, Sentinel-2 and both sensors. \n } \n \\label{fig:profiles-36}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/profile_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/profile_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/profile_108e.jpeg}\n \\caption{Fitness distance correlation after 108 epochs of training.\n From left to right are shown results using Sentinel-1, Sentinel-2 and both sensors. \n } \n \\label{fig:profiles-108}\n\\end{figure*}\n\nFirst, we consider the FDC after 36 epochs of training (see Figure~\\ref{fig:profiles-36}).\nOverall, for all sensors, we observe that the respective landscapes are rather rough.\nIndeed, the distribution of fitness per hamming distance to the optimum are relatively wide.\nFor instance, when using Sentinel-1 we notice that solutions at the \\textit{Hamming-distances} $d_{hamming}=\\{8, 11, 13\\}$ display up to 35\\% percentage point in fitness difference.\nAlso, We notice a landscape around low fitness values (c.a 47\\%).\nUsing Sentinel-2 provides with a landscape centered at much higher values (c.a 94\\%).\nWe also notice a consistent increase in fitness, \nas the hamming distance to the optimum decreases.\nSimilarly, the landscape associated with using both sensors is of high fitness. \nHowever, the slope of gained fitness per travelled distance to the optimum worsens (less consistent), compared to the one obtained with Sentinel-2 as input.\n\nThen, we consider the FDC after 108 epochs of training (see Figure~\\ref{fig:profiles-108}).\nOverall, the landscape tends to be more flat and with an increased fitness.\nIn particular for Sentinel-2 or both sensors as input, \nthe flatness is indicated by more narrow distribution of fitness \nat the various distances to the optimum.\nThis also is complemented by potential search trajectories \nthat have little improvements in fitness per travelled distance to the global optimum. \nUsing both sensors brings us a similar behaviors, \nexcept for the existence of a set of models providing poorer fitness values, \nall located at $d_{hamming}=7$ from the optimum.\nThe case of Sentinel-1 is rather odd as there appears to be a favorable (negative) slope, as if the landscape had not converged.\n\nTo summarize, the fitness landscape of So2Sat LCZ42 is rougher when training is limited (36 epochs), \nand flatter towards higher fitness when training long enough (108 epochs) solutions in the search space.\nAs observed when analyzing distributions of fitness (Figure~\\ref{fig:pdf-36}, \\ref{fig:pdf-108e}), \nthis NAS problem benefits better from using Sentinel-2 as input, \nwith improvements in slope and overall fitness in its landscape.\nTherefore, these results complement the analysis of PDFs \nby showing benefits in performances, this time from the perspective of potential NAS algorithm trajectories. It also shows that with the current search space, the search behaviour is worse when using both sensors as input.\n\n\n\\subsection{Random Walk Analysis}\\label{subsec:random-walks}\n\n\nFurthermore, we investigate the behavior \nof local search-based algorithm depending on the input sensor.\nMore precisely, this is done by analyzing random walks.\n\n\\begin{figure\n\\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/random_walks\/rw22_various_sensors_36e.png}\n \\caption{A random walk evaluated with the four (4) different sensor settings. \n } \n \\label{fig:random-walks}\n\\end{figure}\n\n\nFigure~\\ref{fig:random-walks} displays the route of a random walk \nevaluated on for four (4) different sensor settings.\nThe walk itself consists of one hundred (100) steps in the search space.\nAt each step, the selected model is evaluated after being trained for 36 epochs. \nThe blue, yellow, green and red curves are, respectively, \nfor evaluating the fitness with Sentinel-1, Sentinel-2, both sensors, or CIFAR-10 as input. \nAll curves were smoothed with a moving average of five (5) steps.\nWe also consider CIFAR-10, since its fitness evaluations were freely \navailable (ground-truth in NAS-Bench-101) and could serve as reference for comparison, and trouble-shooting.\n\n\nThe evaluation of the walk on Sentinel-1 provides with the lowest overall fitness (~$\\mu=0.44$) and the most rugged route (~$\\tau=17.84$).\nOn the other hand, using Sentinel-2 or both sensors together, provides with more smooth paths, at much higher values. Indeed, the respective averages of fitness are ~$\\mu=0.93$ and ~$\\mu=0.89$. \nThe ruggedness values are ~$\\tau=1.54$ and ~$\\tau=6.03$. \nAlso , the curvature of both routes \\textit{visually} look alike.\nRegarding CIFAR-10, we observe intermediate fitness and ruggedness (~$\\mu=0.64$ and ~$\\tau=1.56$).\nThe relatively large amplitude in fitness and similar curvature, despite lower ruggedness, makes its route look more similar to the one evaluated with Sentinel-1.\n\nAs observed when analyzing FDCs (Section~\\ref{subsec:fdc}),\nSentinel-1 provides with poorer trajectories (lower fitness, larger ruggedness), \nsuggesting either a sensor being unsuitable for the task \nor a search space~$S$ not suitable for the sensor.\nSimilar results (curvature, lower fitness) obtained for ground-truth evaluations on CIFAR-10 suggest \nthat a higher ruggedness seems to associate with harder tasks and lower convergence of models in a random walk route.\n\nTo summarize, the use of either Sentinel-2 or both sensors after only 36 epochs\nenables a NAS route to be of higher smoothness and fitness.\n\n\\subsection{Persistence}\\label{subsec:persistence}\n\nNext, we study the behaviour of solutions in the search space, \nfrom the perspective of persistence in their ranking.\n\nFigure~\\ref{fig:persistence-positive} and \\ref{fig:persistence-negative} \nshow measurements of positive and negative persistence.\nWe consider samples collected for experiments related to section ~\\ref{subsec:dos} and \\ref{subsec:fdc}.\nFor each setting, the blue curve represents the reference population: the models at a given~$Rank-N$\nbased on their fitness after 4 epochs of training.\nThe yellow curve display the share of these models maintaining the same~$Rank-N$ after 12 epochs.\nThe green and red show the same (intersection of sets) respectively after 36 and 108 epochs of training.\nThe positive and negative Persistence refer to using the top and bottom ~$N$ rank function (Nth percentile).\n\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_sentinel_1_quantile_based.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_sen2_quantile_other_label.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_both_sentinel_quantile_based.png}\n \\caption{Positive persistence for Sentinel-1, Sentinel-2 and when using both sensors.} \n \\label{fig:persistence-positive}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_sentinel_1_quantile_based.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_sen2_quantile_other_label.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_both_sentinel_quantile_based.png}\n \\caption{Negative persistence for Sentinel-1, Sentinel-2 and when using both sensors.} \n \\label{fig:persistence-negative}\n\\end{figure}\n\nFirst we have a look at the positive persistence (see Figure~\\ref{fig:persistence-positive}).\nOverall, we observe that the larger the fitness a sensor can provide (see Figure~\\ref{fig:pdf-36} \\ref{fig:pdf-108e}), \nthe larger its persistence (N$<$25) across all training budgets.\nMore precisely in terms of~\\textit{Area under the Curve (N$<$25)}, Sentinel-2 ($AuC=0.14$) improves over the use of both sensors ($AuC=0.07$), which also improves over a single Sentinel-1 sensor ($AuC=0.04$).\n\n\nNext we take a look at the negative persistence (see Figure~\\ref{fig:persistence-negative}).\nOverall, Sentinel-2 generates the larger persistence ($P=31.82$, $AuC=0.26$), while the use of Sentinel-1 ($P=4.0$, $AuC=0.01$) or both sensors \n($P=5.26$, $AuC=0.01$) result in average measurements.\n \nTo summarize, we observe \nsimilar trend across sensors for both positive and negative persistence.\nA larger fitting capacity results in a larger persistence (Sentinel-2).\nIn the case of Sentinel-1, the limited ability to fit the sensor \nmight hinder the ability of models to keep their ranking (potential instabilities during training). \nIn turn, this might result in a poorer persistence. \nFor the better fitted sensor (Sentinel-2) chances of finding top-25\\% and bottom-25\\% performers are considerable ($P=13.7$, $P=31.82$). \n\n\n\\subsection{Fitness Landscape Footprint}\\label{subsec:footprint}\nThe results obtained in the previous sections are summarized by the \\textit{footprint} for each data source.\nFigure~\\ref{fig:footprnts} displays the \\textit{footprint} for Sentinel-1 (blue), Sentinel-2 (yellow) and both sensors together (green).\nThis is done using considering only 36 epochs of training.\n\nAs observed in section~\\ref{subsec:dos}, the search space appears better suited to fit Sentinel-2, than Sentinel-1. \nIndeed, Sentinel-2 enables reaching a larger mean fitness ($\\mu=0.94$) and lower standard deviation ($\\sigma=0.03$) than the other sensors.\nPerforming an input-level fusion slightly worsens the fitness ($\\mu=0.89, \\sigma=0.05$).\nSimilarly, the search landscape of Sentinel-2 appears more favourable to search trajectories. \nThis transpires through random walk routes that are smoother and of higher fitness ($\\mu=0.93$, $\\tau=1.54$).\nRegarding the evolution in the ranking of samples, \na larger fitness results in a larger positive and negative persistence.\nIndeed, this transpires in the larger measurements obtained for Sentinel-2 (\\textit{Pos. AuC=0.14, Neg. AuC=0.26}). \n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nIn this study, we investigate the impact of the choice of an input sensor \non the performance of a neural architecture search strategy.\nMore precisely, we want to know to what extent does searching with a given sensor differs from searching with other sensors, \nin the context of neural architecture optimization.\nAre there benefits or drawbacks in searching with fused sensors ?\n\nTo answer such questions, we use the framework of ~\\textit{Fitness Landscape Footprint}, \nthat we apply on the Real World image classification task So2Sat LCZ42, \nand analyze the process of searching CNN image classifiers provided in the NASBench-101 database.\nAfter sampling and evaluating solutions with three different sensor settings (including input level fusion), \nwe provide a comparative analysis assessing for instance the distribution of fitness, fitness distance correlation and ruggedness of the landscapes.\n\nOverall, we observe a consistent improvement in the capacity of fitting all sensors, the longer the training time.\nUsing Sentinel-2 enables larger fitness, over Sentinel-1 or an input-level fusion of both sensors.\nSimilar results are observed when analysing search landscapes. Indeed, the longer the training, \nthe landscape evolve from high ruggedness to flatness. \nMoreover, search strategies might benefit from a deployment on Sentinel-2, \nas it provides with routes that are smoother, of higher fitness, higher gain per distance travelled, \nand higher persistence in ranking of models. \nWhen a sensor can be fit well enough (Sentinel-2, fusion), we observe very similar behaviour in terms of trajectories (smoothness, ruggedness, fitness). \nThis strongly indicates that search trajectories associated to different sensors are comparable \nwhen the search space is able to fit them decently enough. \n\nAs future work, we propose to investigate how to use the gained insights to help build speed-up techniques for NAS strategies.\nSuch technique could rely, for instance, on searching with a sensor (or a subset of given sensors),\nhelping approximate the search with a more expensive to evaluate target sensor. \n \n\n\\section*{ACKNOWLEDGEMENTS}\\label{ACKNOWLEDGEMENTS}\nAuthors acknowledge support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. [ERC-2016-StG-714087], Acronym: \\textit{So2Sat}), by the Helmholtz Association\nthrough the Framework of Helmholtz AI [grant number: ZT-I-PF-5-01] - Local Unit ``Munich Unit @Aeronautics, Space and Transport (MASTr)'' and Helmholtz Excellent Professorship ``Data Science in Earth Observation - Big Data Fusion for Urban Research'' (W2-W3-100), by the German Federal Ministry of Education and Research (BMBF) in the framework of the international future AI lab \"AI4EO -- Artificial Intelligence for Earth Observation: Reasoning, Uncertainties, Ethics and Beyond\" (Grant number: 01DD20001) and the grant DeToL.\n\n{\n\t\\begin{spacing}{1.17}\n\t\t\\normalsize\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdcnd b/data_all_eng_slimpj/shuffled/split2/finalzzdcnd new file mode 100644 index 0000000000000000000000000000000000000000..9705d72417c6ead6cd34cd20196d5bf26c1a5ae2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdcnd @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nFrom an evolutionary perspective, music and dance have played a vital role for the social function of the human species \\cite{wang2015hypothesis}. They are ubiquitous in human activities, ranging from personal entertainment to social functions and ceremonial activities. On top of forming an indispensable tapestry in human culture, they are also integral to modern civilization and contribute immensely to our individual and social well-being. Over the past centuries, music and dance are gradually systematized into two separate art-forms but their intimate and deep connection is unmistakable. Both entail expressing our internal emotions as external movements. For dance, the medium of expression is visual in the form of body movements whereas for music, movements manifest auditorily through melodies and rhythms.\n\nIt is widely acknowledged that music evokes motions and feelings of intentionality \\cite{small1998musicking}. We may find ourselves moving along to the beats and dancing to music, perhaps even unaware of our movements. Neuroscience research elucidates how music and dance activities may involve similar stimuli in our brain \\cite{brown2008neuroscience,levitin2009current}. Recently, the artificial intelligence community has also taken an interest. Several works \\cite{fan2011example,lee2013music,tang2018dance,lee2019dancing,ren2020self,huang2021dance} have investigated the task of generating dance choreographs from music.\n\nReciprocally, dance would appear incomplete and unadorned if there were no accompanying music. Since generating music from dance remains largely overlooked in the literature, we propose to concurrently tackle this dual task. As summarized in Fig. \\ref{fig:task}, we develop a bi-directional generative model for synthesizing realistic and matching dance from music \\emph{and} music from dance. This dual learning has the additional advantage of enhancing the modeling of each modality. Given the popularity of music and dance videos in entertainment and multimedia, our ultimate goal is to enable effective engagement of the public, contribute to the user experience, and benefit the vast community of musicians and dancers.\n\nThere are several challenges in this task.\n\n\\textbf{Cross-domain generation} \\quad Translating between music and dance constitutes a cross-domain sequence-to-sequence generative modeling task. Most source-to-target domain learning tasks entail image-to-image \\cite{isola2017image,yi2017dualgan,zhu2017unpaired,lin2018conditional} or language translation \\cite{he2016dual}, for which the data lies in topologically identical spaces. However, our task is further complicated by the fact that the ambient spaces for our data have totally distinct topological properties. Specifically, music is represented as waveforms, whereas a dance sequence is represented as 3D motion trajectories on a pose manifold. This increases the difficulty of learning a network mapping between music and dance with realistic outputs. \n\n\\textbf{Creativity and Diversity} \\quad Earlier works in dance choreography \\cite{fan2011example,lee2013music} adopted a similarity retrieval approach which simply glues together dance moves from a learned template and is lacking in both innovation and diversity. \\cite{tang2018dance} utilized a sequence-to-sequence model with Long Short Term Memory (LSTM) units which suffered from the limitation of a single output. However, multiple interpretations for music composition or dance choreography is commonplace and diversity should also be reflected in the model. One viable approach \\cite{lee2019dancing,ren2020self} is to employ Generative Adversarial Networks (GANs) that enables a distribution of plausible outputs instead of a single deterministic one. \n\n\\textbf{Consistency between music and dance} \\quad On top of securing the realism of the generated music pieces or dance sequences, we need to ensure harmony between the generated output and the conditioned input. In other words, we need to extract the shared themes and intentionality between the two media as domain-independent features and ensure that such domain-independent abstractions are reflected in the target output. Furthermore, the kinematic beats in dance and the acoustic beats in music should be aligned.\n\nTo address these challenges, our proposed approach employs transformer network architectures \\cite{vaswani2017attention} as encoders and decoders in a sequence-to-sequence framework. We incorporate a full attention mechanism \\cite{devlin2018bert} for feature learning in both the music and dance domain. This has the key advantage of a global level understanding of underlying themes. We instill diversity into our model by concatenating a random vector with the encoded feature. Furthermore, to effectively leverage the dual structure of the problem, we also propose an optimal transport inspired alignment that serves to match cross-domain features. Specifically, we define a Gromov-Wasserstein distance \\cite{memoli2011gromov} which measures the \\emph{relational} distance between intra-domain distances while preserving the domain topological structures. Optimizing the Gromov-Wasserstein objective facilitates feature learning in each encoder network. It therefore promotes the proximity and similarity of the learnt embeddings despite inherent differences in domain topology.\n\nOur contributions may be summarized as follows:\n\\begin{enumerate}\n \\item We consider a new problem of both music-to-dance generation and its dual dance-to-music generation.\n \\item A novel dual learning strategy is proposed, which incorporates a Gromov-Wasserstein distance to facilitate feature learning of each task, as well as promote coherence between input and output.\n \\item Empirical experiments demonstrate the applicability of our approach in delivering realistic, diverse generations of dance choreographs and music compositions faithful to the input. Moreover, superior performance is observed when comparing to the state-of-the-arts in music-to-dance generation.\n\\end{enumerate}\n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\textwidth]{Figures\/Pipeline.png}\n\\caption{A high-level overview of our pipeline. (a) For data preprocessing, we extract MFCC, chroma and beats features from the music waveform raw data, and represent the dance sequence as pose and translation parameters in the SMPL model \\cite{loper2015smpl}. (b) The two generative networks $G_{M\\to D}$ (music-to-dance choreography) and $G_{D\\to M}$ (dance-to-music composition) in our framework comprise a sequence-to-sequence architecture where the encoder and decoder are both transformer networks. (c) We train our network through a Gromov-Wassserstein loss that facilitates dual learning in this cross-domain situation, on top of the reconstruction loss and cycle consistency loss.}\n\\label{fig:pipeline}\n\\end{figure*}\n\n\\section{Related Work}\n\\subsection{Dance generation}\nEarlier works generate dance sequences from fixed templates following a similarity matching for the music source \\cite{fan2011example,lee2013music}. Since the synthesized choreographs simply re-arrange the dance moves from the training data in rigid fashions, there is the drawback of unnatural transitions and a lack of creativity.\n\nWith the advent of deep learning methods, sequence-to-sequence models have been proposed for generating dance sequences from encoded music features. A pioneering work \\cite{tang2018dance} uses a L2 loss for comparing the dance sequences, which suffers from a tendency of motion freezing. To alleviate this, \\cite{huang2021dance} proposes curriculum learning coupled with a L1 loss whereas \\cite{ye2020choreonet} employs a geodesic loss. To enable the generation of diverse dance sequences, \\cite{huang2021dance,ye2020choreonet} introduce a random seed vector alongside the encoded music features, while an alternative approach \\cite{lee2019dancing,ren2020self} utilises GANs. One key remark is that \\cite{lee2019dancing,ren2020self,huang2021dance} focus on 2D motion. The 3D representation in our work allows important cues such as relative positions or invariance on bone lengths to be clearly put into perspective, thus appearing more more realistic, appealing and geometrically rich.\n\nAnother line of work \\cite{li2020learning,li2021learn} employs a cross-modal architecture for generating dance sequences conditional on both music \\emph{and} previous dance moves. In our work, the primary task of music-to-dance generation is only conditional on a single music stream and does not require any additional dance stream.\n\n\\subsection{Music generation}\nThere are two general approaches in computational music generation. The first focuses on symbolic representations \\cite{briot2017deep,huang2018music,huang2020pop}. The second handles music as raw waveforms either in the audio \\cite{dieleman2018challenge,child2019generating,dhariwal2020jukebox} or frequency domain \\cite{vasquez2019melnet}. Ideally, the waveform representation enables a much richer generative landscape, such as allowing generation of human voices or nuances in musical performances which attach an additional layer of interpretation on top of the sheet music. However, this comes at an extremely high computational cost. To put this in context, the waveform representation for a 5 seconds music sequence sampled at 48kHz would incur a 240,000 length spectrogram sequence. Despite recent advancements such as sparse attentions \\cite{child2019generating} or discretized representations \\cite{dieleman2018challenge,dhariwal2020jukebox}, learning the multiple levels of musical structure and hierarchy which manifest at different scales remains extremely computationally expensive. In light of this, we resort to the lightweight symbolic representation for our music generation task.\n\n\\subsection{Dual Learning}\nDual learning \\cite{xia2017dual} is a paradigm which jointly trains a primary task and its dual task. The symmetry and duality at the data or model level \\cite{xia2018model} can be leveraged, ideally improving the performance of both tasks compared to training for each independently. Dual learning can be in a supervised setting such as in image-to-image translations for paired data \\cite{isola2017image} or machines language translation \\cite{he2016dual}. It may likewise be extended to the unsupervised setting \\cite{yi2017dualgan,zhu2017unpaired} whereby the notion of a cycle consistency loss is introduced, compelling the primary task and dual task to learn inverse maps of each other. In our dual learning scheme, on top of the cycle consistency loss, we also introduce a novel Gromov-Wasserstein loss to facilitate feature space alignment.\n\n\\section{Our approach}\nAn overview of our pipeline is outlined in Fig. \\ref{fig:pipeline}. In the following, we present the details of each component.\n\n\\subsection{Data Preprocessing}\n\\label{sec:preprocessing}\n\\textbf{Music} \\quad The raw music input are sampled at 48kHz, thereby obtaining a waveform representation in the form of a time-frequency spectrogram. However, such high sampling rates may well lead to high computational costs and lots of redundancy. To this end, we employ the Librosa toolbox \\cite{mcfee2015librosa} to perform feature engineering and extraction. These extracted features will be used in our networks instead of the raw waveform.\n\nFollowing \\cite{lee2019dancing}, we first extract Mel-frequency cepstral coefficients (MFCC) \\cite{xu2004hmm} and MFCC delta from the spectrogram. However, these low level features (typically used for speech recognition) may be inadequate for conveying high level musical information. Thereafter, we perform a harmonic percussive source separation \\cite{muller2015fundamentals} which decomposes the spectrogram into harmonic components and percussive components. The harmonic components correspond to the pitch and melody of the music from which we extract chroma features. The percussive components provide the rhythmic information from which we extract the beats and onsets. Specifically, for each frame, our music features is a 53 dimensional vector, comprising 20-dim MFCC, 20-dim MFCC delta, 12-dim chroma features, and 1-dim one-hot encoding for beats.\n\n\\textbf{Dance} \\quad We work with two 3D dance sequence datasets from \\cite{tang2018dance,li2021learn}. \\cite{li2021learn} performed a 3D annotation of the AIST Dance database \\cite{tsuchida2019aist} and each 3D pose is parameterized in the Skinned Multi-Person Linear Model \\cite{loper2015smpl}. For consistency, we performed a fitting with SMPLify \\cite{bogo2016keep} on the dataset in \\cite{tang2018dance} to obtain SMPL pose parameters.\n\nThe SMPL model has $24$ skeletal joints and a pose is represented as $24\\times3$ axis-angle parameters, which characterizes the 3D orientation or rotation of each joint. However the axis-angle (and likewise the quaternion) parameterization is not globally continuous over the 3D rotational group $SO(3)$ \\cite{astey1987cobordism}. Furthermore, it is cumbersome to define a geometrically meaningful loss for axis-angle parameters and we adopt a Stiefel manifold representation \\cite{zhou2019continuity}:\n\\begin{equation}\\label{eqn:stiefelmanifold}\n\\begin{aligned}\n\\hat{R} &= \n\\left(\\begin{matrix} \\rule[-1ex]{0.5pt}{2.5ex} & \\rule[-1ex]{0.5pt}{2.5ex} \\\\ \\mathbf{R}_1 & \\mathbf{R}_2 \\\\ \\rule[-1ex]{0.5pt}{2.5ex} & \\rule[-1ex]{0.5pt}{2.5ex} \\end{matrix}\\right),\n\\end{aligned}\n\\end{equation}\nwhich essentially amounts to discarding the last column for a rotation matrix. Such a representation is smooth over $SO(3)$, thus offering empirical advantages for backpropagation. Overall, the skeletal pose is parameterized as a $24\\times6+3=147$-dim vector. \n\n\\subsection{Problem Formulation}\nWe denote the music space as $\\mathcal{X}$ and the dance space as $\\mathcal{Y}$. We sample fixed duration ($T$ frames) music and dance sequence pairs $(x_i,y_i)_{i=1}^N$ \\emph{i.i.d} from $\\mathcal{X}\\times\\mathcal{Y}$. Formally, the problem statement can be formulated as follows.\n\\begin{itemize}\n\\item The primary task is to learn a mapping $G_{M\\to D}:\\mathcal{X}\\to\\mathcal{Y}$ such that $\\sum_{i=1}^N\\mathcal{L}_{\\mathcal{Y}}(G_{M\\to D}(x_i),y_i)$ is minimized. Here $\\mathcal{L}_{\\mathcal{Y}}$ denotes a metric in dance space $\\mathcal{Y}$.\n\\item The dual task is to learn a mapping $G_{D\\to M}:\\mathcal{Y}\\to\\mathcal{X}$ such that $\\sum_{i=1}^N\\mathcal{L}_{\\mathcal{X}}(G_{D\\to M}(y_i),x_i)$ is minimized. Here $\\mathcal{L}_{\\mathcal{X}}$ denotes a metric in music space $\\mathcal{X}$.\n\\end{itemize}\nIn the ideal case, $G_{M\\to D}$ and $G_{D\\to M}$ would be inverse of each other, \\emph{i.e.} $G_{D\\to M}(G_{M\\to D}(x_i))=x_i$ and $G_{M\\to D}(G_{D\\to M}(y_i))=y_i$. The discrepancy from this ideal case may be leveraged in the form of a cycle discrepancy loss \\cite{zhu2017unpaired} in our dual learning context.\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Figures\/GW.png}\n\\caption{In optimizing for the Gromov-Wasserstein loss, the $\\text{Enc}_{M\\to D}$ network parameters are updated, re-positioning the music embedding vectors $\\{z^x_i\\}_{i=1}^4$ such that the discrepancy between intra-space distances is minimized.}\n\\label{fig:gw}\n\\end{figure}\n\n\\subsection{Overall Framework}\n\\label{sec:framework}\n\\textbf{Transformer and attention} \\quad Both the $G_{M\\to D}$ and $G_{D\\to M}$ consist of transformer networks that serve as encoders and decoders as illustrated in Fig. \\ref{fig:pipeline}. We adopt a full attention mechanism \\cite{devlin2018bert} throughout for all transformers. This allows full access to the entire context, without masking of the future contexts. Explicitly, for a given sequence input $\\mathbf{S}$, an attention layer learns the context $\\mathbf{Z}$:\n\\begin{equation}\\label{eqn:attention}\n\\begin{aligned}\n\\mathbf{Z} = \\text{Softmax}\\left(\\frac{\\langle\\mathbf{S}\\mathbf{W}_q,\\mathbf{S}\\mathbf{W}_k\\rangle}{\\sqrt{D}}\\right)\\mathbf{S}\\mathbf{W}_v,\n\\end{aligned}\n\\end{equation}\nwhere $D$ is the number of channels in the attention layer and $\\mathbf{W}_q,\\mathbf{W}_k,\\mathbf{W}_v$ denote the query, key and value weights respectively.\n\nIntuitively, the full attention mechanism aligns with how humans go about composing music for dance or choreographing dance from music. We make a global consideration of the structure and themes in our design. In allowing access to all contextual information, we expect that the structural constructs and hierarchical abstractions can be better modeled in the transformer networks.\n\n\\textbf{Gromov-Wasserstein Loss to Facilitate Dual Learning in Encoders} \\quad Each encoder network learns an embedding map from its respective input space into a latent space. Explicitly, we have:\n\\begin{equation}\\label{eqn:embedding}\n\\begin{aligned}\n\\text{Enc}_{M\\to D}: \\mathcal{X} &\\to \\mathcal{Z}_\\mathcal{X}\\\\\nx&\\mapsto z^x,\\\\\n\\text{Enc}_{D\\to M}: \\mathcal{Y} &\\to \\mathcal{Z}_\\mathcal{Y}\\\\\ny&\\mapsto z^y.\n\\end{aligned}\n\\end{equation}\nThe music space $\\mathcal{X}$ and dance space $\\mathcal{Y}$ have different topological structures. As such, it would be mathematically impossible to embed them into the same space since embeddings are topology-preserving maps. This means that we cannot directly gauge the similarity between a piece of music and a dance sequence by defining a metric on their embedding vectors.\n\nOn the other hand, embeddings do introduce metric distortions \\cite{bourgain1985lipschitz}. This motivates us to propose a Gromov-Wasserstein loss \\cite{memoli2011gromov,peyre2016gromov,alvarez2018gromov} which measures the discrepancy between the metrics on the two embedding spaces instead of directly comparing between cross-domain samples. By defining an \\emph{inter-space} distance between the respective \\emph{intra-space} distances for $\\mathcal{Z}_\\mathcal{X}$ and $\\mathcal{Z}_\\mathcal{Y}$, the Gromov-Wasserstein loss gives a well-defined notion of distance between music and dance pairs. Heuristically, we present in Fig. \\ref{fig:gw} how optimizing this loss amounts to aligning the feature embeddings.\n\nGive two point sets of $m$ embedded vectors $\\{z^x_i\\}_{i=1}^m,\\{z^y_i\\}_{i=1}^m$, we may view them as two discrete empirical distributions $\\mu,\\nu$ as:\n\\begin{equation}\n\\begin{aligned}\n\\mu = \\sum_{i=1}^m\\frac{1}{m}\\delta_{z^x_i}, \\;\\; \\nu = \\sum_{i=1}^m\\frac{1}{m}\\delta_{z^y_i},\n\\end{aligned}\n\\end{equation}\nwhere $\\delta$ denotes the Dirac delta distribution. Formally, the Gromov-Wasserstein distance for our task is given by\n\\begin{equation}\\label{eqn:gw}\n\\begin{aligned}\nGW(\\mu,\\nu)=\\min_{\\pi\\in\\Pi}\\sum_{i,j,k,l} \\left\\lvert \\lVert z^x_i-z^x_k \\rVert_1 - \\lVert z^y_j-z^y_l \\rVert_1 \\right\\rvert^2 \\pi_{ij} \\pi_{kl}.\n\\end{aligned}\n\\end{equation}\nHere $\\Pi$ defines the set of all joint distributions with marginals $\\mu$ and $\\nu$. The goal is to find the optimal transport matrix $\\pi$ that minimizes the squared distance between intra-space L1 costs.\n\nFollowing \\cite{peyre2016gromov}, we introduce an entropic regularization term which allows much more efficient solving of Eq.~\\eqref{eqn:gw}. The entropy regularized Gromov-Wasserstein distance can then be solved via the Sinkhorn algorithm and projected gradient descent \\cite{peyre2016gromov}. We outline the steps in Algorithm~\\ref{algo:GW}.\n\\begin{algorithm}\n\\caption{GW Distance for 2 batches of $m$ samples}\n\\label{algo:GW}\n\\begin{algorithmic}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\Require music embeddings $\\mathbf{Z}^x=\\{z^x_i\\}_{i=1}^m,\\mathbf{Z}'^x=\\{z'^x_i\\}_{i=1}^m$\n\\Require dance embeddings $\\mathbf{Z}^y=\\{z^y_i\\}_{i=1}^m,\\mathbf{Z}'^y=\\{z'^y_i\\}_{i=1}^m$\n\\renewcommand{\\algorithmicrequire}{\\textbf{Hyperparameters:}}\n\\Require regularization $\\varepsilon > 0$, projection iterations $M$, Sinkhorn iterations $L$\n\\renewcommand{\\algorithmicrequire}{\\textbf{Initialize:}}\n\\Require $\\pi^{(0)}_{ij} = \\frac{1}{n}, \\; \\forall{i,j}$\n\\State Cost Matrix for music embeddings $C_{ij}=\\left\\lVert z^x_i-z^x_j \\right\\rVert_1$\n\\State Cost Matrix for dance embeddings $D_{ij} = \\left\\lVert z^y_i-z^y_j \\right\\rVert_1$\n\\For{$l = 1:M$}\n\\State $E = \\frac{1}{m}D^2 \\mathds{1}_m \\mathds{1}_m^\\intercal + \\frac{1}{m} \\mathds{1}_m \\mathds{1}_m^\\intercal C^2 - 2D \\pi^{(l-1)} C^\\intercal$\n\\State $K = \\exp(-E\/\\varepsilon)$\n\\State $\\mathbf{b}^{(0)} = \\mathds{1}_m$\n\\For{$\\ell = 1:L$}\n\\State $\\mathbf{a}^{(\\ell)} = \\mathds{1}_m \\oslash K\\mathbf{b}^{(\\ell-1)}$, \\; $\\mathbf{b}^{(\\ell)} = \\mathds{1}_m \\oslash K^\\intercal\\mathbf{a}^{(\\ell)}$\n\\State where $\\oslash$ denotes component-wise division\n\\EndFor\n\\State $\\pi^{(l)}=\\text{diag}(\\mathbf{a}^{(L)})K\\text{diag}(\\mathbf{b}^{(L)})$\n\\EndFor\n\\Ensure \\small$GW_{\\varepsilon}(\\mathbf{Z}^x,\\mathbf{Z}'^x,\\mathbf{Z}^y,\\mathbf{Z}'^y) = \\displaystyle \\sum_{i,j,k,l} \\left\\lVert D_{ik}-C_{jl} \\right\\rVert^2\\pi_{ij}^{(M)}\\pi_{kl}^{(M)}$\n\\end{algorithmic}\n\\end{algorithm}\n\nOptimizing the Gromov-Wasserstein distance through Algorithm \\ref{algo:GW} updates the music encoder network parameters $\\text{Enc}_{M\\to D}$ through backpropagation. This matching of the music embeddings with the dance embeddings facilitates learning the duality of our two tasks.\n\n\\textbf{Decoding and Generation} \\quad During training phase, the start token for the decoder network is taken from the training data. Specifically, for the dance generation task, the start token is the initial dance pose. During inference phase, we sample a random seed vector for the initial pose instead. For the music generation task, during inference, we randomly sample a chord root from \\{C,C\\#,D,D\\#,E,F,F\\#,G,G\\#,A,A\\#,B\\} and a chord quality among \\{major, minor\\}. These random seeds during the inference phase introduce diversity in the generated dance and music.\n\n\\subsection{Loss Functions}\n\\label{sec:loss}\nAs illustrated in Fig. \\ref{fig:pipeline}, three loss functions are defined. The first is our Gromov-Wasserstein loss, which has been discussed in detail in the previous subsection. Essentially it serves as an auxiliary regularization loss that promotes a better learning of the correspondence between the music and dance embeddings. This facilitates extracting the commonalities and shared structural similarities between the two media, which would enhance the consistency between inputs and generated outputs in our tasks.\n\nThe reconstruction loss has the general form:\n\\begin{equation}\\label{eqn:reconstruction}\n\\begin{aligned}\n\\mathcal{L}^\\text{reconstruction}_{\\text{dance}}&=&\\sum_i\\mathcal{L}_{\\mathcal{Y}}(y_i,G_{M\\to D}(x_i)),\\\\\n\\mathcal{L}^\\text{reconstruction}_{\\text{music}}&=&\\sum_i\\mathcal{L}_{\\mathcal{X}}(x_i,G_{D\\to M}(y_i)).\n\\end{aligned}\n\\end{equation}\n$\\mathcal{L}_{\\mathcal{Y}}$ denotes the metric over the dance space $\\mathcal{Y}$. Specifically, we have $\\mathcal{Y}=(\\mathbb{R}^3\\times \\underbrace{SO(3)\\times\\cdots\\times SO(3)}_{24\\text{ times}})^T$ where $T$ is the total number of frames in our model. A point $y\\in\\mathcal{Y}$ may then be written as $y=\\left[y^{\\text{trans}}_t,y^{\\text{rot}_1}_t,\\cdots,y^{\\text{rot}_{24}}_t\\right]_{t=1}^T$. We define $\\mathcal{L}_{\\mathcal{Y}}:\\mathcal{Y}\\times\\mathcal{Y}\\to\\mathbb{R}_+$ as follows:\n\\begin{flalign}\\label{eqn:dancemetric}\n&\\small{\\mathcal{L}_{\\mathcal{Y}}(y,\\tilde{y})=\\sum_{t=1}^T\\underbrace{\\lVert y^{\\text{trans}}_t-\\tilde{y}^{\\text{trans}}_t\\rVert_1}_{\\text{L1 loss for translation}}+\\sum_{t=1}^T\\sum_{j=1}^{24}\\text{geodesic}(y^{\\text{rot}_j}_t,\\tilde{y}^{\\text{rot}_j}_t)^2}.\n\\end{flalign}\nHere, $\\text{geodesic}:SO(3)\\times SO(3)\\to\\mathbb{R}_+$ defines the shortest distance between two 3D rotations. Recall that $y^{\\text{rot}_j}_t$ constitutes the first two columns of its associated rotation matrix $R^j_t$ as defined in Eq.~\\eqref{eqn:stiefelmanifold}, we can recover its third column through the cross product. The geodesic distance between rotation matrices $R,\\tilde{R}$ is given by:\n\\begin{flalign}\\label{eqn:geodesic}\n\\text{geodesic}(R,\\tilde{R}) = \\left\\lvert \\arccos \\left[\\frac{\\text{Tr}(R\\tilde{R}^\\intercal)-1}{2}\\right]\\right\\rvert.\n\\end{flalign}\n\nFor the music space $\\mathcal{X}$, we compare only the chroma and beats. The metric $\\mathcal{L}_{\\mathcal{X}}:\\mathcal{X}\\times\\mathcal{X}\\to\\mathbb{R}_+$ is defined as:\n\\begin{flalign}\\label{eqn:musicmetric}\n&\\mathcal{L}_{\\mathcal{X}}(x,\\tilde{x})=\\sum_{t=1}^T\\underbrace{\\lVert x^{\\text{chroma}}_t-\\tilde{x}^{\\text{chroma}}_t\\rVert_1}_{\\text{L1 loss for chroma}}+\\underbrace{\\lVert x^{\\text{beats}}_t-\\tilde{x}^{\\text{beats}}_t\\rVert_1}_{\\text{L1 loss for beats}}.\n\\end{flalign}\nThe cycle consistency loss is another regularization term that we incorporate to facilitate dual learning. Whereas the Gromov-Wasserstein loss is applied at an initial stage to update the music embedding network parameters, the cycle consistency loss is applied at the posterior stage to gauge the generated rendition against the input. With the same metrics defined in Eqs.~\\eqref{eqn:dancemetric} and \\eqref{eqn:musicmetric}, the cycle consistency loss measures the discrepancy of the two dual networks $G_{M\\to D}$ and $G_{D\\to M}$ from being inverse to each other:\n\\begin{equation}\\label{eqn:cycleconsistency}\n\\begin{aligned}\n\\mathcal{L}^\\text{cycle}_{\\text{dance}}&=&\\sum_i\\mathcal{L}_{\\mathcal{Y}}(y_i,G_{M\\to D}(G_{D\\to M}(y_i))),\\\\\n\\mathcal{L}^\\text{cycle}_{\\text{music}}&=&\\sum_i\\mathcal{L}_{\\mathcal{X}}(x_i,G_{D\\to M}(G_{M\\to D}(x_i))).\n\\end{aligned}\n\\end{equation}\nOur transformer networks $G_{M\\to D}$ and $G_{D\\to M}$ are trained concurrently and updated according to the following prescriptions: $$\\mathcal{L}^\\text{GW}+\\mathcal{L}^\\text{reconstruction}_{\\text{dance}}+\\mathcal{L}^\\text{cycle}_{\\text{dance}}\\xrightarrow[\\text{}]{\\text{backpropagation}}G_{M\\to D},$$\n$$\\mathcal{L}^\\text{reconstruction}_{\\text{music}}+\\mathcal{L}^\\text{cycle}_{\\text{music}}\\xrightarrow[\\text{}]{\\text{backpropagation}}G_{D\\to M}.$$\n\n\\begin{figure*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{|l||c||c|}\n\\hline\nMethod & Click $\\downarrow$ & Cha Cha dance sequences displayed at 0.2 second intervals\n\\\\ \\hline\nGround Truth &\n\\animategraphics[height=1.2cm]{5}{Figures\/GT\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/GT_Combined.png}\n\\\\ \\hline\nTang et al. (2018) \\cite{tang2018dance} &\n\\animategraphics[height=1.2cm]{5}{Figures\/Tang(2018)\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Tang_2018__Combined.png}\n\\\\ \\hline\nRen et al. (2020) \\cite{ren2020self} &\n\\animategraphics[height=1.2cm]{5}{Figures\/Ren(2020)\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Ren_2020__Combined.png}\n\\\\ \\hline\nHuang et al. (2021) \\cite{huang2021dance} &\n\\animategraphics[height=1.2cm]{5}{Figures\/Huang(2021)\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Huang_2021__Combined.png}\n\\\\ \\hline\nOurs &\n\\animategraphics[height=1.2cm]{5}{Figures\/Ours\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Ours_Combined.png}\n\\\\ \\hline\n\\end{tabular}\n}\n\\caption{Visual comparison of sample generated Cha Cha dance sequences. Best viewed via Adobe Acrobat Reader. Click the figures under `Click $\\downarrow$' to show dance animations (4 seconds at 5 FPS).}\n\\label{fig:realism}\n\\end{figure*}\n\\section{Experiments}\n\\subsection{Dataset Description}\nAs mentioned in subsection \\ref{sec:preprocessing}, we employ two public datasets from \\cite{tang2018dance} and \\cite{tsuchida2019aist,li2021learn}, which consists of 3D dance sequences with accompanying music. Since these two datasets do not overlap in their dance genres, we combine them into a single entity. We first re-parameterized the skeletal pose representation in \\cite{tang2018dance} in the SMPL model \\cite{loper2015smpl} pose parameters (3D positional representation of 17 keypoints $\\to$ rotational pose representation of 24 keypoints) through an inverse kinematics fitting. For consistency in frame rate, we also perform a downsampling (via spherical interpolation) of the frames per second (FPS) in \\cite{li2021learn} from 60 to 25. The statistics for the aggregated dataset is summarized in the following table.\n\\begin{table}[h]\n\\resizebox{1\\linewidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nDance Genre & \\# of Sequences & Frames & Remarks \\\\\\hline\nRumba & 10 & 20950 & \\multirow{4}{*}{\\parbox{3.2cm}{From \\cite{tang2018dance}. Dance sequences are typically much longer at around 150 seconds.}} \\\\\\cline{1-3}\nCha Cha & 8 & 20425 & \\\\\\cline{1-3}\nTango & 9 & 49165 & \\\\\\cline{1-3}\nWaltz & 34 & 43298 & \\\\\\hline\nBreak Dance & 141 & 46526 & \\multirow{10}{*}{\\parbox{3.2cm}{From \\cite{li2021learn}. Dance sequences generally range from 8 to 12 seconds.}} \\\\ \\cline{1-3}\nHouse & 141 & 40050 & \\\\\\cline{1-3}\nBallet Jazz & 141 & 47727 & \\\\\\cline{1-3}\nStreet Jazz & 141 & 47920 & \\\\\\cline{1-3}\nKrump & 141 & 47534 & \\\\\\cline{1-3}\nLA Style Hip Hop & 141 & 48323 & \\\\\\cline{1-3}\nLock & 141 & 47388 & \\\\\\cline{1-3}\nMiddle Hip Hop & 141 & 48276 & \\\\\\cline{1-3}\nPop & 140 & 46749 & \\\\\\cline{1-3}\nWaack & 140 & 47355 & \\\\\\hline\n\\end{tabular}}\n\\caption{Summary of dataset, aggregated from \\cite{tang2018dance} and \\cite{tsuchida2019aist,li2021learn}. For our tasks, the frame rate is set to 25 FPS. During training, the input and output sequences are fixed at 75 frames (or 3 seconds).}\n\\vspace{-2em}\n\\label{tab:dataset}\n\\end{table}\n\n\\subsection{Implementation Details}\nOur framework is implemented with PyTorch. We set the input and output sequence lengths to $T=75$ frames (equivalent to 3 seconds). Both $G_{M\\to D}$ and $G_{D\\to M}$ have 6 transformer layers and 8 attention heads. The hidden units dimensions for the transformer layers is set to 512 for $G_{M\\to D}$ and 256 for $G_{D\\to M}$. For our Gromov-Wasserstein loss, the hyperparameters for Algorithm \\ref{algo:GW} are as follows: the entropic regularization parameter is set to $\\varepsilon=0.2$ while the number of Sinkhorn iterations and projection iterations are respectively set to $L=30$ and $M=20$. During training, we use the Adam optimizer with a batch size of 16 and an initial learning rate of 1e-4 (decays to \\{1e-5, 5e-6\\} after \\{20k, 40k\\} iterations).\n\nDuring inference, the decoder start token may either be sampled randomly from a pool (for diverse generation setting) or provided as an auxiliary input. For inference over 75 frames, we simply feed the last generated frame as the decoder start token. \n\n\\subsection{Dance Generation}\nFor generation of music-to-dance choreography, we compare our work against \\cite{tang2018dance,ren2020self,huang2021dance}. For \\cite{tang2018dance}, we re-implement it since the source code is not available. We also adapt \\cite{ren2020self,huang2021dance} (both implemented for 2D dance sequences) for 3D dance generation. We employ three quantitative metrics, namely the Fr\\'{e}chet Distance, Diversity and Beats Alignment. The details of these metrics are discussed below and the quantitative results are reported in Tab. \\ref{tab:dance_quantitative}. \n\nFurthermore, we engage a qualitative user study to rate the realism and the genre consistency of the generated dances. The results are tabulated in Tab. \\ref{tab:dance_user}.\n\nWe showcase sample dance animations in Fig. \\ref{fig:realism} and \\ref{fig:diversity}. These sequences spanning 10 seconds are animated at 5 FPS (requires Adobe Acrobat Reader). More sample sequences are available in the supplementary video.\n\n\\begin{figure*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{|c||c||c|}\n\\hline\nGenre & Click $\\downarrow$ & Dance sequences displayed at 0.2 second intervals\n\\\\ \\hline\n\\multirow{3}{3cm}{Cha Cha \\includegraphics[width=3cm]{Figures\/ChaCha_Music.png}} &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_ChaCha1\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_ChaCha1_Combined.png}\n\\\\ \\cline{2-3}\n &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_ChaCha2\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_ChaCha2_Combined.png}\n\\\\ \\cline{2-3}\n &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_ChaCha3\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_ChaCha3_Combined.png}\n\\\\ \\hline\n\\multirow{3}{3cm}{Break Dance \\includegraphics[width=3cm]{Figures\/Break_Music.png}} &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_Break1\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_Break1_Combined.png}\n\\\\ \\cline{2-3}\n &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_Break2\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_Break2_Combined.png}\n\\\\ \\cline{2-3}\n &\n\\animategraphics[height=1.2cm]{5}{Figures\/Diversity_Break3\/}{0}{49} & \\includegraphics[height=1.2cm]{Figures\/Diversity_Break3_Combined.png}\n\\\\ \\hline\n\\end{tabular}\n}\n\\caption{Our framework can generate diverse dance sequences given the same input music but a different starting pose. Best viewed via Adobe Acrobat Reader where dance animations will be played upon clicking the figures under `Click $\\downarrow$'.}\n\\label{fig:diversity}\n\\end{figure*}\n\n\\begin{table}[h]\n\\resizebox{1\\linewidth}{!}{\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nMethod & Fr\\'{e}chet Distance & Diversity & Beats Alignment (\\%) \\\\ \\hline\nGround Truth & - & - & 68.7 \\\\ \\hline\nTang et al. (2018) \\cite{tang2018dance} & 986.4 & 10.3 & 31.2 \\\\ \\hline\nRen et al. (2020) \\cite{ren2020self} & 1526.3 & 48.2 & 49.5 \\\\ \\hline\nHuang et al. (2021) \\cite{huang2021dance} & 384.2 & 37.2 & 62.3 \\\\ \\hline\nOurs & \\textbf{140.5} & \\textbf{49.8} & \\textbf{64.5} \\\\ \\hline\n\\end{tabular}}\n\\caption{Quantitative results for dance generation. We adopt three metrics: 1) \\emph{Fr\\'{e}chet distance} measures the difference from the ground truth, 2) \\emph{Diversity} measures the variation in dance moves, and 3) \\emph{Beats Alignment} evaluates the percentage matching of kinematic beats and music beats. Best performance highlighted in bold.}\n\\vspace{-2em}\n\\label{tab:dance_quantitative}\n\\end{table}\n\n\\textbf{Fr\\'{e}chet Distance} \\quad\nWe define the Fr\\'{e}chet distance as the average 3D joint distance of the generated dance sequence from the ground truth. In our experiments, we evaluate the Fr\\'{e}chet distance for 280 sequences (20 per genre, each spanning 9 seconds) in our pre-allocated test set. Our framework achieves significantly better performance than compared methods. This superior matching of the generated dances with the ground truth suggests that the music-to-dance correspondence is better modeled in our approach.\n\nSample generated Cha Cha sequences are shown in Fig. \\ref{fig:realism} for the same initial pose and input music. We observe that our network generates sequences more consistent with the ground truth. \\cite{tang2018dance} has a inclination to gravitate towards motionless states, showing very limited range of motions. On the other hand, while not suffering from lack of motions, \\cite{ren2020self} tends to deviate from the ground truth.\n\n\\begin{table}[h]\n\\resizebox{1\\linewidth}{!}{\n\\begin{tabular}{|l|c|c|}\n\\hline\nMethod & Realism (Ranking) & Genre Consistency (Ranking) \\\\ \\hline\nTang et al. (2018) \\cite{tang2018dance} & 3.9 & 3.80 \\\\ \\hline\nRen et al. (2020) \\cite{ren2020self} & 2.98 & 2.58 \\\\ \\hline\nHuang et al. (2021) \\cite{huang2021dance} & 2.05 & 2.35 \\\\ \\hline\nOurs & \\textbf{1.07} & \\textbf{1.27} \\\\ \\hline\n\\end{tabular}}\n\\caption{User study for dance generation. We report the average ranking of each method for motion realism and dance genre consistency.}\n\\vspace{-3em}\n\\label{tab:dance_user}\n\\end{table}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Figures\/Music.png}\n\\caption{Visualization of sample generated music (light blue) and original music (dark blue). Left: Visualization of music notes played out in time. Right: Histogram of music notes.}\n\\label{fig:music}\n\\end{figure*}\n\n\\textbf{Diversity} \\quad\nRecall that we incorporate randomness into our decoder through a randomly seeded start token. This allows our framework to generate diverse dance moves from the same input music but a different initial pose. We define our diversity metric as the variation in 3D joints position, evaluated over 5 generated dance sequences conditioned on the same music and different initial poses. This is averaged over 20 independent trials for each genre. In Fig. \\ref{fig:diversity}, We showcase the dance choreographs obtained by our model given the same input music. The diverse generation is indicative that our model is adept at learning the abstract movements in music and translating them to kinematic movements in dance.\n\n\\textbf{Beats Alignment} \\quad\nBeats alignment measures the consistency of rhythmic articulation. For simplicity, we follow \\cite{li2021learn} in defining dance kinematics beats as local minima in average joints 3D speed. We then define a matching beat if the occurrence of the kinematics beat occurs within 5 frames or 0.2 seconds of a music beat. The beats alignment ratio can then be defined as the ratio of matching beats to the total dance beats. Our framework again achieves compelling performance for beats matching, demonstrating its effectiveness in learning the rhythmic articulation of the input music.\n\n\\textbf{User Study} \\quad\nWe engage 5 users (amateur dancers) to assess the generated dances in a single blind study. For each dance genre, we generate 3 dance sequences of 12 seconds duration via different methods. Each user then rank the generated dances according to two criteria: 1) the naturalness or realism of the dance poses; 2) the consistency with the dance genre. As shown in Tab. \\ref{tab:dance_user}, the dance sequences generated from our framework is overwhelmingly ranked most preferred for both realism and genre consistency.\n\n\\subsection{Music Generation}\nTo our knowledge, we are the first to investigate the task of composing music for given dance sequences. We propose to quantify the performance of our model via a notes accuracy metric. We first transpose both compared music pieces to either the C major or A minor scale (depending on its original chord quality), before computing the average notes accuracy. Our model achieves an accuracy of 72\\%, as tabulated in Tab. \\ref{tab:ablation}.\n\nHowever, this metric is not ideal since music composition is rather subjective task and is best assessed via human listening tests \\cite{huang2020pop}. To this end, we invite the reader to look over and listen to the sample music compositions in our supplementary file. A sample generated music piece, is shown in tandem with the ground truth in Fig. \\ref{fig:music}. The visualization is done with JuxtaMidi \\cite{JuxtaMidi}.\n\n\\subsection{Ablation Studies}\n\\begin{table}[h]\n\\resizebox{1\\linewidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{GW} & \\multirow{2}{*}{CC} & \\multicolumn{3}{c|}{Dance} & Music \\\\ \\cline{3-6}\n& & Fr\\'{e}chet Distance & Diversity & Beats Alignment (\\%) & Notes Accuracy \\\\ \\hline\n & \\checkmark & 335.4 & 46.3 & 58.5 & 0.51 \\\\ \\hline\n\\checkmark & & 196.2 & \\textbf{50.1} & 60.3 & 0.59 \\\\ \\hline\n & & 376.5 & 46.2 & 56.4 & 0.37 \\\\ \\hline\n\\checkmark & \\checkmark & \\textbf{140.5} & 49.8 & \\textbf{64.5} & \\textbf{0.72} \\\\ \\hline\n\\end{tabular}}\n\\caption{Ablation experiments performed on two components of our framework. \\emph{GW} refers to the Gromov-Wasserstein loss and \\emph{CC} refers to the cycle consistency loss.}\n\\vspace{-3em}\n\\label{tab:ablation}\n\\end{table}\n\nWe perform ablation studies focused on investigating the effectiveness of our dual learning scheme. The first key component in our framework is the Gromov-Wasserstein loss, applied at the early stage of a training iteration to align the music embeddings with dance embeddings. The second component is the cycle consistency loss, optimized at the final stage of a training iteration as an additional regularization to enforce overall consistency.\n\nThe ablation experiments are reported in Tab. \\ref{tab:ablation}. We observe significant performance drops for the Fr\\'{e}chet distance, beats alignment ratio and music notes accuracy upon removal of the Gromov-Wasserstein objective. This provides strong justification of its potency for handling the duality of the tasks. The cycle consistency loss, is also effective, albeit to a lesser extent. Overall, compared to independently training separate task-specific networks, the conjoined learning in our framework delivers significant improvements through a balanced set-up of well-tailored components.\n\n\\section{Conclusion}\nIn this work, we propose a novel problem of simultaneously learning music-to-dance choreography and dance-to-music composition. The crucial ingredient is how to effectively leverage the duality of the tasks and integrate the information from both domains. To overcome the challenge of our cross-domain setting, we design a Gromov-Wasserstein objective for aligning the music embeddings vis-\\`{a}-vis the dance embeddings, coupled to a cycle consistency loss. These auxiliary losses and our dual learning scheme prove capable in boosting the performance of individual tasks. Our framework delivers realistic and genre-consistent dance generations, as well as viable music compositions. For future work, we seek to extend our framework to raw waveform based music composition and explore multi-persons dance choreography.\n\n\\newpage\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nQCD parton distribution functions\nare specified by matrix-elements of nonlocal quark and gluon \ntwist-2 operators separated by a light like distance and \ntaken inside a forward nucleon matrix element.\nAlthough these forward matrix elements were generalized long time\nago to nonforward parton distribution functions \\cite{class},\nwhere nonlocal QCD operators are sandwiched between states of unequal \nmomenta, they became only recently the object of intense\nstudies \\cite{news,Ra97,Bel98}.\nThis interest was driven by the hope that the measurement of \nnonforward parton distributions (\\mbox{NFPDs}) and subsequent extrapolation\nto the forward limit might shed more light on the spin and angular\nmomentum composition of the nucleon \\cite{Ji97}.\n\nContrary to ordinary parton distribution functions, \nnonforward parton distribution functions are probed when the nucleon\nrecoils elastically. Thus \\mbox{NFPDs} provide a natural interpolation\nbetween ordinary parton distribution functions and elastic form factors. \nIt was shown that \\mbox{NFPDs} are in principle measurable in deeply virtual\nCompton scattering or through exclusive production of light vector\nmesons. \n\nIn this paper we shall consider the diffractive leptoproduction of \nquark antiquark pairs\nin a kinematics where the produced quark pair does not form a single\nmeson but fragments into a $\\pi^+\\pi^-$ \nor $K^+K^-$ pair with comparatively large $p_T$ (Fig.~\\ref{momenta}).\n\\begin{figure}\n\\centerline{\\epsfig{file=momenta.eps,width=5cm}}\n\\caption{The process of the quark antiquark pair production to \nlowest order QED.}\n\\label{momenta}\n\\end{figure}\nWe will consider the case of\nlongitudinal polarized electron beam and proton target.\nThis process is quite similar to diffractive leptoproduction of dijets,\nanalyzed in \\cite{Di95,GM98}, which was proposed to allow for a measurement\nof the polarized glue in the nucleon.\nWe notice that at medium energies like in the HERMES experiment dijets \ncannot be observed. In this experiment one can only look for single\nhadrons representing the jets.\nThe idea to access the polarized forward gluon distribution function of the \nnucleon via the measurement of a double hadronic high momentum final state \nhas been proposed in \\cite{BHK98}.\n\nWe treat the fragmentation \nprocess of the two produced quarks into the final hadrons\nusing the LUND string fragmentation model \n\\cite{strgfrg,Sj95}. To be precise we use this model to\nextrapolate from dijet production, where it is known\nto be valid, to the case where the two jets are represented by\ne.g. $\\pi^+$ and $\\pi^-$. While we do not expect the LUND model to\ngive precise results in this few particle process it should allow for\nan order of magnitude estimate.\n\nIn our calculation we will assume that two pion (kaon) \nleptoproduction can be factorized into three parts.\nWe have two soft parts: one describing the fragmentation of the two \npartons, the other the production of two \npartons from the elastic scattering off the nucleon.\nThe second soft part will be given by a nucleon matrix element of a\nlight cone operator parameterized by \\mbox{NFPDs}.\nThe hard part connects the elastic scattering process and\nsubsequent fragmentation.\n\nTechnically, the process we consider is closely related to the \nexclusive production of a light vector meson. \nIn such a case the proof of factorization has been given in\n\\cite{CFS97}. Recall that the proof only holds for \nlongitudinal photons. Meson production by a transversely polarized photon \nis plagued by endpoint singularities as noted in \\cite{BFGMS94}. \nThus a general factorization theorem does not exist in this case.\nFurther the factorization theorem was proven only on the level of twist-2.\nHowever, it was noted that the proof in \\cite{CFS97} can be extended to \ntwo-pion production if these are produced close to threshold, i.e. if\nthe mass of the produced final state is much smaller than the virtuality\nof the probing photon \\cite{Diehl98}. \nIn such a kinematics it is useful to describe the pion pair in terms\nof a two-pion twist-2 distribution amplitude \\cite{Pol98}. \nThe main difference to our calculation then is the treatment of the produced\nmeson in terms of a light cone twist-2 meson wave function rather than in \nterms of the fragmentation model. In our calculation we do not restrict\nourselves to the kinematic region $M^2(h^+ + h^-) 1\\;{\\rm GeV^2}$ (with $p_T$ being the\ntransverse momentum in the center of mass frame of the photon and the\nincoming nucleon) in order to keep the relevant scale\n$\\mu^2=p_T^2(1+\\frac{Q^2}{M^2(h^+ + h^-)})$ sufficiently large.\nFor the remaining variables we used the cuts\n$0.02 < x_{\\rm Bj} < 0.05$,\n$0.29 < y < 0.96$,\n$0.1 < \\zeta < 0.3$, and\n$0.04 < z_f < 0.96$\nwhere the lower cut in $\\zeta$ was dictated by numerical reasons.\nThe cuts in $z_f$ are part of the\n$z-s$ jet scheme which we applied in our simulations. It is also\nused in the standard Monte Carlo leptoproduction programs \\cite{ingelman}.\n\n\\section{Results and Discussion}\n\nIn Table~\\ref{tab} we have given the various \ncontributions to the total\ncross section for meson pair production.\n\\begin{table}\n\\caption{The different contributions to the total cross sections.}\n\\begin{tabular}{ccc}\n\\\\\n\\hline\n& cross sect. $\\pi^+\\pi^-$ (pb) & cross sect. $K^+K^-$ (pb) \\\\\n\\hline\n\\hline\n$\\sigma_{gg}$ & $ \\;\\;3.2\\;10^{-1}$ & $ \\;\\;2.8\\;10^{-1}$ \\\\\n$\\sigma_{\\Delta g\\Delta g}$ & $ \\;\\;4.2\\;10^{-2}$ & $ \\;\\;4.1\\;10^{-2}$ \\\\\n$\\sigma_{g\\Delta g}$ & $ \\;\\;6.3\\;10^{-2}$ & $ \\;\\;5.3\\;10^{-2}$ \\\\\n\\hline\n$\\sigma_{qq}$ & $ \\;\\;5.4\\;10^{-2}$ & $ \\;\\;4.0\\;10^{-2}$ \\\\\n$\\sigma_{\\Delta q\\Delta q}$ & $ \\;\\;3.3\\;10^{-3}$ & $ \\;\\;2.9\\;10^{-3}$ \\\\\n$\\sigma_{q\\Delta q}$ & $ \\;\\;7.4\\;10^{-3}$ & $ \\;\\;4.8\\;10^{-3}$ \\\\\n\\hline\n$\\sigma_{qg}$ & $ -1.2\\;10^{-1}$ & $ -8.8\\;10^{-2}$ \\\\\n$\\sigma_{\\Delta q\\Delta g}$ & $ -4.6\\;10^{-3}$ & $ -7.7\\;10^{-4}$ \\\\\n$\\sigma_{q\\Delta g+ g\\Delta q}$ & $ -4.3\\;10^{-2}$ & $ -2.9\\;10^{-2}$ \\\\\n\\hline\ntotal spin average & $ \\;\\;2.9\\;10^{-1}$ & $ \\;\\;2.8\\;10^{-1}$ \\\\\ntotal spin asymmetry & $ \\;\\;2.8\\;10^{-2}$ & $ \\;\\;2.9\\;10^{-2}$ \\\\\n\\hline\\\\\n\\end{tabular}\n\\label{tab}\n\\end{table}\nContributions from the\npolarized and unpolarized gluonic \\mbox{NFPD} are denoted by the indices\n$g$ and $\\Delta g$,\ncontributions from the quark \\mbox{NFPDs} by $q$ and $\\Delta q$ respectively.\nSince the cross section is quadratic in the parton distributions\nwe find pure gluonic contributions of the form \n$\\sigma_{gg}$, $\\sigma_{\\Delta g \\Delta g}$, and $\\sigma_{g \\Delta g}$,\nsimilar expressions for the pure quarkonic part, and interference\ncontributions $\\sigma_{qg}$, $\\sigma_{\\Delta q \\Delta g}$, and\n$\\sigma_{q \\Delta g + g \\Delta q}$.\nOnly the interference terms of polarized and unpolarized parton\ndensities $\\sigma_{g \\Delta g}$, $\\sigma_{g \\Delta g}$, and\n$\\sigma_{g \\Delta q + q \\Delta g}$\ndepend on the alignment of the spins of the incoming electron and proton and\nthus contribute to the spin asymmetries\n$\\Delta\\sigma:=\\frac{1}{2}\n(\\sigma_{\\uparrow\\downarrow}-\\sigma_{\\uparrow\\uparrow})$.\n($\\sigma_{\\uparrow\\downarrow}$ is the cross section for antiparallel\nlepton spin and nucleon spin.)\nComparing the contributions to the spin average we see that\n$\\sigma_{\\Delta g \\Delta g}$, $\\sigma_{\\Delta q \\Delta q}$, and\n$\\sigma_{\\Delta q \\Delta g}$ are roughly one order of magnitude\nsmaller than $\\sigma_{gg}$, $\\sigma_{qq}$ and $\\sigma_{qg}$ respectively.\n\nFor our \\mbox{NFPD} Models, the gluon contribution dominates the cross section,\nboth in the unpolarized and polarized channel.\nNote that the normalization of the cross section\ndepends crucially on the form factors (\\ref{formfaktor}).\nFor the used input distributions we finally get \na spin asymmetry of order 10\\%.\n\nThe unpolarized and polarized cross sections\nare plotted in the upper pictures in Fig.~\\ref{cross}\nas functions of the skewedness $\\zeta$.\n\\begin{figure}\n\\centerline{\\epsfig{figure=totunpolvszeta.eps,width=6.2cm}\n \\epsfig{figure=totpolvszeta.eps,width=6.2cm}}\n\\vspace{-0.4cm}\n\\centerline{\\epsfig{figure=totunpolvsdeltay.eps,width=6.2cm}\n \\epsfig{figure=totpolvsdeltay.eps,width=6.2cm}}\n\\vspace{-0.4cm}\n\\centerline{\\epsfig{figure=totunpolvsminv.eps,width=6.2cm}\n \\epsfig{figure=totpolvsminv.eps,width=6.2cm}}\n\\caption{The unpolarized (on the left) and polarized (on the right)\ndifferential cross section for diffractive $\\pi^+\\pi^-$ (solid\ncircles) and $K^+K^-$ (open circles) production with respect to the\nvariables $\\zeta$, $\\Delta y :=\n\\min\\{y_{\\rm particle \\;1},y_{\\rm particle\\; 2}\\}\n- y_{\\rm \\;outg.\\; proton}$, and $M^2(h^+ +h^-)$.}\n\\label{cross}\n\\end{figure}\nThe cross sections receive their\ndominating contribution from the small values of $\\zeta$. In our\nanalysis we restrict ourselves on skewedness values larger than\n$\\zeta > 0.1$.\nTo go to smaller values of $\\zeta$ ultimately would lead\nto the requirement that $Q^2 \\to 0$ and thus would lead to \nthe region of photoproduction. In that kinematical domain the scattered\nelectron is not detectable and thus an unambiguous reconstruction of the\nfinal hadronic state is not possible unless the outgoing proton could\nbe detected by a recoil spectrometer.\nOne method to distinguish the elastic events from inelastic ones in\nthis case is provided by the existence of a rapidity gap. The plots in\nthe middle of Fig.~\\ref{cross} show that the difference between the\nminimum of the meson rapidities and the rapidity of the outgoing\nproton $\\Delta y$ is for all diffractive events larger than $2$.\nIn the lower diagrams in Fig.~\\ref{cross} we plotted the cross section\nversus the invariant mass squared of the produced particles. These\nplots show the increase of the ratio between the kaon\nand pion rates with the invariant mass $M(h^+ + h^-)$ as suggested by\nFig.~\\ref{plot1}.\nThe shape of the polarized and unpolarized cross sections\nis similar.\n\nWe have analyzed diffractive $\\pi^+\\pi^-$ and $K^+K^-$ production as a\ntool for the determination of polarized and unpolarized\nnonforward parton distributions. We implemented the realistic cuts for\nthe HERMES experiment and required sufficiently large $p_T$ to justify\nour leading order treatment. With these cuts the cross sections are\ntoo small to make this channel a promising one for HERMES. Successfull\nmeasurements would probably require an ENC-type machine \\cite{GSI}.\n\n\\begin{ack}\nThe authors thank M.~Diehl, P.~Hoyer, L.~Mankiewicz, G.~Piller, M.~Polyakov, \nA.~Radyushkin, M.~Strikman, and O.~Teryaev for useful discussions.\nThis work has been supported by DFG (Graduiertenkolleg), Studienstiftung, and\nBMBF. M.M. thanks the Nordic Insitute for Theoretical Physics\nfor financial support and hospitality.\n\\end{ack}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper we use the method introduced in\n\\cite{VD2013}, extended in\n\\cite{Baskin-Vasy-Wunsch:Radiation} and \\cite{HVsemi},\nto analyze the Feynman propagator on spaces $(M,g)$, called spaces\nwith non-trapping Lorentzian\nscattering metrics (a notion recalled in detail in Section~\\ref{sec:geometry}), that at infinity resemble\nMinkowski space in an appropriate manner. As the Feynman propagator is\nof fundamental importance in quantum field theory, we expect that our\nresult and methods will be useful in a\nsystematic treatment of QFT on curved, non-static, Lorentzian backgrounds.\n\nHere the Feynman propagator is defined as the inverse of the wave\noperator acting as a map between appropriate function spaces that\ngeneralizes the behavior of the standard Feynman propagator on exact\nMinkowski space. Concretely, the distinguishing feature of propagators\nfrom the perspective of function spaces is in terms of the differential order of\nthe Sobolev spaces at the two halves of the (b-)conormal bundle of the\n`light cone at infinity' $S_\\pm$; components at which the differential\norder is higher, resp.\\ lower, than a threshold value determine the inverse one obtains.\nThus, {\\em we set up function spaces which are\nweighted microlocal Sobolev spaces of variable order of an appropriate kind such that\nthe wave operator for any non-trapping Lorentzian scattering metric is Fredholm for all but a discrete set of\nweights -- See Theorem~\\ref{thm:fredholm}} for a precise\nstatement. Indeed, the same statement holds for more general\nperturbations of Lorentzian scattering metrics in the sense of smooth\nsections of $\\Sym^2\\Tsc^*M$, defined below in\nSection~\\ref{sec:geometry}. Further, we prove Theorem\n \\ref{thm:invertibility} below, which we state roughly now.\n\n\n\n\\begin{thm*}[See Theorem~\\ref{thm:invertibility}]\nFor perturbations of Minkowski space, in\nthe sense of smooth\n sections of $\\Sym^2\\Tsc^*M$, the\n Feynman wave operator, described above, is invertible for a\n suitable range of weights (rates of decay or growth of functions\n in the domain). That is to say, its inverse, the Feynman\n propagator, exists for these space-times.\n\\end{thm*}\n\n\n\nIn order to give a rough idea for what {\\em the} Feynman propagator is\nwe recall that in their groundbreaking paper \\cite{FIOII} Duistermaat and H\\\"ormander constructed distinguished parametrices\nfor wave equations, i.e.\\ distinguished solution operators for $\\Box\nu=f$ modulo $\\CI(M^\\circ)$. Recall that by H\\\"ormander's theorem \\cite{Hormander:Existence},\nsingularities of solutions of wave equations propagate along\nbicharacteristics inside the characteristic set in phase space, i.e.\\ $T^*M^\\circ$; the projections of\nthese to the base space are null-geodesics. Here a bicharacteristic is\nan integral curve of the Hamilton vector field of the principal symbol\nof the wave operator, which is the dual metric function on\n$T^*M^\\circ$. For the inhomogeneous wave equation, $\\Box u=f$, if,\nsay, $f$ has wave front set (i.e.\\ is singular) at only one point in\n$T^*M^\\circ$, the different distinguished parametrices produce\nsolutions with different wave front sets, namely either the forward or\nthe backward bicharacteristic through the point in question. Here\nforward and backward are measured relative to the vector field whose\nintegral curves they are, i.e.\\ the Hamilton vector field. Note,\nhowever, that there is a different notion of forward and backward,\nwhich one may call future- or past-orientedness, namely whether the\nunderlying time function is increasing or decreasing along the\nflow. The relative sign between these notions is the opposite in the\ntwo halves of the characteristic set of the wave operator over\neach point. We point out that from the perspective of microlocal\nanalysis the natural direction of propagation {\\em is} given by the Hamilton flow.\n\nAs explained by Duistermaat and H\\\"ormander, a distinguished parametrix is obtained by choosing a\ndirection of propagation (of singularities, or estimates) {\\em in each\n connected component of the characteristic set of the wave\n operator}. Here the direction of propagation is relative to the\nHamilton flow, as above. If the underlying manifold is connected, as one may assume, the characteristic\nset has two connected components, and there are $2^2=4$ choices:\npropagation forward relative to the Hamilton flow everywhere,\npropagation backward along the Hamilton flow everywhere (these are the\nFeynman and anti-Feynman propagators), resp.\\ propagation in the\nfuture direction everywhere (the retarded propagator) and in the past\ndirection everywhere (the advanced propagator).\nA parametrix, however, is only an approximate\ninverse, modulo smoothing --- smoothing operators are not even compact\non such a manifold; for actual applications (such as any computations in physics)\none would need an actual inverse, and most importantly a {\\em notion\n of an inverse}. This is exactly what we provide in Theorem\n\\ref{thm:invertibility} below.\n\nThe historically usual setup for\nwave equations, and more generally evolution equations, is that of\nCauchy problems: one specifies initial data at a time slice, and then\none studies local or global solvability. In this sense wave equations\nare always locally well-posed due to the finite speed of propagation,\nwhich in turn is proved by energy estimates. Global well-posedness follows if the local solutions \ncan be pieced together well: global hyperbolicity is a notion that \nallows one to do so. If one turns this into a\nsetup of inhomogeneous wave equations, $\\Box u=f$, by cutting $f$ into\ntwo pieces, located in the future, resp.\\ the past, of a Cauchy\nsurface, the choice one is making is that the {\\em support} of $u$ be\nin the future, resp.\\ the past, of that of $f$. This necessarily\nimplies, indeed is substantially stronger than, the statement that singularities of solutions are accordingly propagated, so\ntwo of the Duistermaat-H\\\"ormander parametrices correspond to\nthese. Thus, due to the energy estimates, even when one considers\nglobal solutions, the Cauchy problem, or equivalently the future (or\npast) oriented problem, for the wave equation is essentially local in\ncharacter, though, as discussed in \\cite{VD2013,HVsemi}, in order to\nunderstand the global behavior of solutions, it is extremely useful to\nwork directly in a global framework in any case.\n\nWhat we achieve here is to give an analogous well-posedness framework\nfor the Feynman problems (as opposed to the Cauchy problems). These\nproblems are {\\em necessarily} global in character, very much unlike\nthe Cauchy problems. Thus, they behave similarly, in a certain sense,\nto elliptic PDE. Indeed, from our perspective, it is an accident\n(happening for\ngood reasons) that the future\/past oriented wave equations are local;\none should not normally expect this for any PDE. To be more precise,\nsingularities of solutions behave just as predicted by the\nDuistermaat-H\\\"ormander construction, but this has no content for\n$\\CI$ solutions --- the $\\CI$ `part' of solutions is globally\ndetermined.\n\nThere has been extensive work in the mathematical physics literature\non such QFT problems, often from the perspective of trying to make\nsense of division by functions with zeros on the characteristic set:\nfor Minkowski space, the Fourier transform gives rise to a multiplier\n$\\xi_n^2-(\\xi_1^2+\\ldots+\\xi_{n-1}^2)$; in a $\\pm i0$ sense division\nby this is well-behaved away from the origin, but at the origin\ndelicate questions arise. This is usually thought of as a degree of\nfreedom in defining propagators: precisely how one extends the\ndistribution to $0$ even in this constant coefficient setting. (See\n\\cite[Section~5]{Brunetti-Fredenhagen:Microlocal} for a discussion of\nthis in the QFT context, and\n\\cite{Viet:Renormalization} for a recent treatment of renormalization\nas such extensions.) From our perspective, this is due to\ntranslational invariance of the problem being emphasized at the\nexpense of its homogeneity; Mellin transforming in the radial variable\ngives rise to a much better behaved problem. Indeed, a generalization\nof this is what Melrose's framework of b-analysis \\cite{Melrose:Atiyah} relies on; we\nfurther explore it here in the non-elliptic setting following\n\\cite{VD2013,HVsemi}. When the microlocal structure of the function\nspaces on which the wave\noperator acts corresponds to the above propagation statements\n(propagation in the direction of the Hamilton flow in the Feynman\ncase, and in the opposite direction in the anti-Feynman case), the remaining choice is\nthat of a weight: in the case of Minkowski space it turns out that\nweights $l$ with $|l|<\\frac{n-2}{2}$ give rise to invertibility, while\noutside this range the index of the operator changes, with jumps at weight values\ncorresponding to resonances of the Mellin transformed wave operator\nfamily, which in turn correspond to eigenvalues of the Laplacian on\nthe sphere $\\Delta_{\\sphere^{n-1}}$ as we show by a complex scaling\n(Wick rotation) argument in Section \\ref{sec:Wick}.\n\nFor QFT on curved space-times, the work of Duistermaat and H\\\"ormander was\nused to introduce a microlocal characterization of Hadamard states,\nwhich are considered as physical states of non-interacting QFT, by\nRadzikowski \\cite{Radzikowski:Microlocal}. (Indeed, part of the paper\nof Duistermaat and H\\\"ormander was motivated by QFT questions.) This in turn was then extended by\nBrunetti, Fredenhagen and K\\\"ohler\n\\cite{Brunetti-Fredenhagen-Kohler:Microlocal,\n Brunetti-Fredenhagen:Microlocal}. G\\'erard and Wrochna gave a new\npseudodifferential construction of Hadamard states\n\\cite{Gerard-Wrochna:Construction, Gerard-Wrochna:Yang-Mills}. In a different direction, Finster and Strohmaier extended the general theory to Maxwell fields \\cite{Finster-Strohmaier:Gupta}. However, in all these cases, there\nis no way of fixing a preferred state: one is always working modulo\nsmoothing operators. Our framework on the other hand gives exactly\nsuch a preferred choice. Note also that the Feynman propagator we\nconstruct relates to an Hadamard-type condition; see Remark\n\\ref{rem:hadamard} below.\n\nIn the settings with extra structure, involving time-like Killing\nvector fields, one can construct Feynman propagators in terms of\nelliptic operators, e.g.\\ via Cauchy data. Other constructions\n(such as extensions across null-infinity) in similar\nsettings are investigated by Dappiaggi, Moretti and Pinamonti\n\\cite{Dappiaggi-Moretti-Pinamonti:Rigorous, Moretti:Quantum,\n Dappiaggi-Moretti-Pinamonti:Cosmological}. In fact, these latter results\nbear the closest connections to ours in that a canonical state is\nconstructed using the structure on null-infinity. Our results deal\ndirectly with the `bulk', thanks to the Fredholm formulation, with the\nlinear results having considerable perturbation stability in\nparticular. (It is due to the module structure required in\nSection~\\ref{sec:semi} that the non-linear problem is more restrictive.)\n\nAlong with setting up such a Fredholm framework, we also study\nsemilinear wave equations, following the general scheme of\n\\cite{HVsemi}; we think of these as a first step towards\ninteracting QFT in this setting. However, being fully microlocal, the\nnecessary framework requires more sophisticated function spaces than\nthose discussed in \\cite{HVsemi}. We prove small data\nwell-posedness results in the Feynman setting for certain semilinear\nwave equations in Theorems \\ref{thm:semilinear} and\n\\ref{thm:semilinearthree} below. In particular, Theorem\n\\ref{thm:semilinearthree} can be summarized as follows\n\\begin{thm*}\nIn $\\RR^{3 + 1}$, if $g$ is a perturbation of the Minkowski metric for\nwhich both the\ninvertibility statements in Theorem \\ref{thm:invertibility} and\nTheorem \\ref{thm:modulereginverse} hold, the problem\n \\begin{equation*}\n \\Box u + \\lambda u^3 = f\n \\end{equation*}\nis well-posed for small $f$, where $f$ lies in the range, and $u$ in\nthe domain, of the Feynman wave operator, in particular $u =\n\\Box^{-1}_{g, fey}(f - \\lambda u^3)$ where $\\Box^{-1}_{g, fey}$ is the\nFeynman propagator mapping as in \\eqref{eq:feynmanprop} with $l \\ge 0$\nsufficiently small.\n\\end{thm*}\n\nWhile as far as we are aware non-linear problems have not been\nconsidered in the Feynman context,\nfor the usual Cauchy problem, i.e.\\ the retarded and advanced\npropagators, non-linear problems on Minkowski space, as well as\nperturbations of Minkowski space (as opposed to the more general\nLorentzian scattering metrics considered in the linear parts of the\npaper here), have been very well studied. In particular, even\nquasilinear equations are well \nunderstood due to the work of Christodoulou \\cite{Christodoulou:Global} and\nKlainerman \\cite{Klainerman:Uniform, Klainerman:Null}, with their\nbook on the global stability of Einstein's equation\n\\cite{Christodoulou-Klainerman:Global} being one of the main\nachievements. Lindblad and Rodnianski\n\\cite{Lindblad-Rodnianski:Global-existence,\n Lindblad-Rodnianski:Global-Stability} simplified some of their\narguments, and Bieri \\cite{Bieri:Extensions, Bieri-Zipser:Extensions}\nrelaxed some of the decay conditions. We also mention the work of Wang\n\\cite{Wang:Thesis} obtaining asymptotic expansions, of Lindblad \\cite{Lindblad:Global} for results on a\nclass of quasilinear equations, and of Chru\\'sciel and {\\L}{\\c{e}}ski \\cite{Chrusciel-Leski:Polyhomogeneous}\non improvements when there are no derivatives in the non-linearity. H\\\"ormander's book\n\\cite{Hormander:Nonlinear} provides further references in the general\narea, while the work of Hintz and Vasy \\cite{HVsemi} develops the\nanalogue of the framework we use here in the general Lorentzian\nscattering metric setting (but still for the Cauchy problem). Works for the linear problem with implications for\nnon-linear ones, e.g.\\ via Strichartz estimate include the recent work of Metcalfe and Tataru\n\\cite{Metcalfe-Tataru:Global} where a parametrix construction is\npresented in a low\nregularity setting.\n\nThe structure of the paper is as follows. In\nSection~\\ref{sec:geometry} we describe the underlying geometry and\nstudy the wave operator microlocally in the sense of smoothness (as\nopposed to decay). Estimates modulo compact errors, and thus Fredholm\nproperties, are established in Section~\\ref{sec:fred}. In\nSection~\\ref{sec:Wick} we show that in Minkowski space, the Feynman\npropagator is the limit of the inverses of elliptic problems,\nachieved by a `Wick rotation'; this means that from the perspective of\nspectral theory the Feynman and anti-Feynman propagators are the\nnatural replacement for resolvents. This in particular establishes the\ninvertibility of the Minkowski wave operator on the appropriately\nweighted function spaces. Finally, in Section~\\ref{sec:semi} we study\nsemilinear wave equations in the Feynman framework.\n\nThe third author is grateful to Jan Derezi\\'nski, Christian G\\'erard\nand Michal Wrochna for very helpful discussions, and the authors are\ngrateful to Peter Hintz for comments on the manuscript. The authors\nare also grateful to the anonymous referees for comments that helped to improve the presentation.\n\n\\section{Geometry and the d'Alembertian}\\label{sec:geometry}\nThe basic object of interest\nis a manifold $M$ with boundary $\\pa M$ equipped with a Lorentzian\nmetric $g$ (which we take to be signature $(1,n-1)$) in its interior\nwhich has a certain form at the boundary (which is geometrically\ninfinity) modelled on the Minkowski metric. In order to define the\nprecise class of metrics, it is useful to introduce a more general structure. Thus, $\\Tsc^{*}M$ is\nthe scattering cotangent bundle, which we describe presently,\noriginally defined in \\cite{RBMSpec}. If $\\rho$\nis a boundary defining function, meaning a function in $C^\\infty(M)$\nwhich is non-negative, has $\\{ \\rho = 0 \\} = \\p M$, and such that\n$d\\rho$ is non-vanishing on $\\p M$,\nsmooth sections of $\\Tsc^*M$ near the boundary are locally given by $C^{\\infty}(M)$ linear\ncombinations of the differential forms\n\\begin{equation*}\n \\frac{d\\rho}{\\rho^{2}} ,\\ \\frac{dw_{i}}{\\rho},\n\\end{equation*}\nwhere $w_{1}, \\dots, w_{n-1}$\nform local coordinates on $\\p M$.\nA non-degenerate smooth section of $\\Sym^2\\Tsc^*M$ of Lorentzian\nsignature (which we take to be $(1,n-1)$) is called a {\\em Lorentzian sc-metric}. The\nsmooth topology on sc-metrics is the $\\CI$ topology on sections of\n$\\Sym^2\\Tsc^*M$, i.e.\\ locally in $M$ (which recall is a manifold {\\em\n with} boundary, i.e.\\ smoothness is {\\em up to} the boundary) is induced by the $\\CI$ topology\nof the coefficients of the basis\n\\begin{equation}\\label{eq:symmetric-tensor-basis}\n \\frac{d\\rho}{\\rho^{2}}\\otimes \\frac{d\\rho}{\\rho^{2}} ,\\\n \\frac{d\\rho}{\\rho^{2}}\\otimes_s\\frac{dw_{i}}{\\rho},\\\n \\frac{dw_{i}}{\\rho}\\otimes_s\\frac{dw_{j}}{\\rho},\\ i,j=1,\\ldots,n-1,\\\n i\\leq j,\n\\end{equation}\nwhere $\\otimes_s$ is the symmetric tensor product.\n\nWhen $M^{int} = \\mathbb{R}^n$, the objects above can be described\nalong more familiar lines. Indeed, in this case, the radial\ncompactification of $\\RR^n$ to a ball $\\BB^n$ gives the manifold with\nboundary $M$, see \\cite{RBMSpec}, e.g.\\ by using `reciprocal\nspherical coordinates' to glue the sphere at infinity $\\sphere^{n-1}$\nto $\\RR^n$. Then if $z_1, \\dots, z_n$ are the standard Euclidean\ncoordinates, setting as usual $r = (z_1^2 + \\dots + z_n^2)^{1\/2}$, we\ncan take $\\rho = 1\/r$ outside the unit ball $B_1(0)$ (recall\nthat $\\rho$ is to be a smooth function on all of $M$), and thus \n\\begin{equation}\\label{eq:Euclidean-radial-scattering-form}\n\\frac{d\\rho}{\\rho^2} = - dr \\mbox{ on } \\mathbb{R}^n\\setminus B_1(0),\n\\end{equation}\nand the $w_i$ can be taken to be some set of $n-1$ angular variables on\n$\\mathbb{S}^{n-1}$ which give local coordinates on the sphere. Thus in this\ncase $dw_i \/ \\rho = r \\,d\\omega_i$ and one sees that the volume form\n$dz = r^{n-1}\\,dr \\,d\\Vol_{\\mathbb{S}^{n-1}}$ is a smooth non-vanishing section\nof the top degree form bundle, $\\wedge^n\\Tsc^*M$, up to and including $\\p M$. (Put differently, the volume\nform is equal to a wedge product of $d\\rho \/\\rho^2$ and the $dw_i\n\/\\rho$ times a smooth non-vanishing function $a$ which is smooth up to\n$r = \\infty$.)\nMoreover, $\\CI(\\BB^n)$ consists exactly of the\nspace of classical (one step polyhomogeneous) symbols of order $0$,\nwhile the standard coordinate differentials $dz_j$ lift to $\\BB^n$ to\ngive a basis, over $\\CI(\\BB^n)$, of all smooth sections of\n$\\Tsc^*\\BB^n$. In particular, any translation invariant Lorentzian\nmetric on $\\RR^n$ is (after this identification) a sc-metric; and\nremains so under perturbations of its coefficients by classical symbols of order\n$0$. Moreover, $dz_i\\otimes_s dz_j$, $i,j=1,\\ldots,n$, $i\\leq j$,\nforming a basis of $\\Sym^2\\Tsc^*\\BB^n$, the $\\CI$ topology on sections\nof $\\Sym^2\\Tsc^*\\BB^n$ is simply the $\\CI(\\BB^n)$ topology on the\n$n(n+1)\/2$-tuple of coefficients with respect to this basis.\n\nWe next recall the\ndefinition of the more refined structure of a {\\em Lorentzian scattering space} from\n\\cite{Baskin-Vasy-Wunsch:Radiation} (see also \\cite[Section~5]{HVsemi}), of which the Minkowski metric is\nan example via the radial compactification of $\\RR^n$, depicted in\nFigure \\ref{spacetime}. For this, we\nassume that there is a $\\CI$ function $v$ defined near $\\pa M$, with\n$v|_{\\pa M}$ having a\nnon-degenerate differential at the zero-set $S=\\{v=0,\\rho=0\\}$ of $v$\nin $\\pa M$ (which we call the {\\em light cone at infinity}); here\n$\\rho$ is a boundary defining function with the property that the\nscattering normal vector field $V=\\rho^2\\pa_\\rho$ modulo $\\rho\\Vsc(M)$\n(it is well-defined in this sense) satisfies that $g(V,V)$ has the\nsame sign as $v$ at each point in $\\pa M$, $g$ has the form\n\\begin{equation}\\label{eq:metric}\ng=v\\frac{d\\rho^2}{\\rho^4}-\\left(\\frac{d\\rho}{\\rho^2}\\otimes\\frac{\\alpha}{\\rho}+\\frac{\\alpha}{\\rho}\\otimes\\frac{d\\rho}{\\rho^2}\\right)-\\frac{\\tilde g}{\\rho^2},\n\\end{equation}\nwhere $\\tilde g\\in\\CI(M;\\Sym^2 T^*M)$, $\\alpha\\in\\CI(M;T^*M)$,\n$\\alpha|_S=\\frac{1}{2}\\,dv$ and\n$\\tilde g|_{\\Ann(d\\rho,dv)}$ at $S$ is positive definite, where, for a\nset of one forms $\\beta_1, \\dots, \\beta_k$,\n$\\Ann(\\beta_1, \\dots, \\beta_k)$ is the set of vectors in the\nintersection of the kernels of the $\\beta_i$. Apart from the fact\nthat the Minkowski metric is of this form, the assumption in \\eqref{eq:metric} is\nnatural because it guarantees the structure of the Hamiltonian\ndynamics in the cotangent bundle which is required for our analysis.\n\nThis is not quite a statement about $g|_{\\pa M}$ as a metric on $\\Tsc\nM$, i.e.\\ as a section of $\\Sym^2\\Tsc^*M$, because of the implied\nabsence of a\n$O(\\rho)\\frac{d\\rho^2}{\\rho^4}$ term. Adding such a term results in a {\\em\n long-range Lorentzian scattering metric}, the whole theory relevant\nto the discussion below goes\nthrough in this setting, as shown in the work of Baskin, Vasy and\nWunsch \\cite{Baskin-Vasy-Wunsch:Long-range}; e.g.\\ Schwarzschild space-time is of this\nform near the boundary of the light cone at infinity. (The difference\nis in the precise form of the asymptotics of the linear waves; they\nare well-behaved on a logarithmically different blow-up of $M$ at\n$S$.) \n\nNote that a perturbation of a Lorentzian scattering metric in the\nsense of sc-metrics (smooth sections of $\\Sym^2\\Tsc^*M$) is a Lorentzian sc-metric, but it need {\\em not}\nbe (even a long-range) Lorentzian scattering metric, since the above form of the metric\n\\eqref{eq:metric} need not be preserved. However, the subspace of\nsc-metrics of the form \\eqref{eq:metric} is a closed subset in the\n$\\CI$ topology of sc-metrics within the open set of Lorentzian\nsc-metrics (in the space of smooth sections of $\\Sym^2\\Tsc^*M$); by a\nperturbation in the sense of Lorentzian scattering metrics we\nmean a perturbation within this closed subset.\n\nWe remark here that, as\nis generally the case, only finite regularity (not being $\\CI$) is\nrelevant in any of the discussion below, though the specific\nregularity needed would be a priori rather high. However,\nusing the low regularity results of Hintz \\cite{Hintz:Quasilinear} on\nb-pseudodifferential operators one could easily obtain rather precise\nlow-regularity versions of\nthe linear results presented here.\n\nFor statements beyond Fredholm properties, based on the work in Section~\\ref{sec:Wick}, $M$ will be the ball $\\mathbb{B}^n$, i.e.\\ the\nradial compactification of $\\mathbb{R}^n$, equipped with a smooth\nperturbation of the Minkowski metric,\n\\begin{equation}\n \\label{eq:minkowski}\n g = dz_n^{2} - dz_{1}^{2} - dz_2^2 - \\dots\n- dz_{n - 1}^{2},\n\\end{equation}\n with perturbation understood in the set of sc-metrics. (Later, in\n Section~\\ref{sec:semi}, it will be important to have perturbations\n within scattering metrics to preserve the module structure discussed there.)\nTo see that this takes the form in\n\\eqref{eq:metric}, following \\cite[Sect.\\\n3.1]{Baskin-Vasy-Wunsch:Radiation}, write $g$ in the coordinates $(\\rho,\nv, \\omega)$ defined by $z_n = \\rho^{-1} \\cos \\theta,\nz_{j} = \\rho^{-1}\\omega_{j} \\sin\\theta$ for $1 \\le j \\le n - 1$, where $\\rho = |z|^{-1}$,\nwhere $|z| = \\sqrt{z_1^2 + \\dots + z_n^2}$ and $\\omega_{j} = z_{j}\/ (|z|^{2} -\nz_n^{2})^{1\/2}$, and take $v = \\cos 2 \\theta$. In this case $\\alpha = dv \/ 2$ identically.\n\n\nThe main object of study here is the wave operator, defined in local\ncoordinates by\n\\begin{equation}\n \\label{eq:box}\n \\Box_{g} := \\frac{1}{\\sqrt{g}} \\p_{i} G^{ij} \\sqrt{g} \\p_{j},\n\\end{equation}\nwhere $G$ denotes the inverse of $g$, i.e.\\ the dual metric on\n$1-$forms defined by $g$.\n\n\n\n\n\nWe further\nassume that $g$ is \\emph{non-trapping}, which is to say we assume that $S=S_+\\cup S_-$\n(each $S_\\pm$ being the disjoint union of possibly several connected\ncomponents),\n$$\n\\{\\rho=0,\\\nv>0\\}=C_+\\cup C_-,\n$$\n$C_\\pm$ open, $\\pa C_\\pm=S_\\pm$, and such that the\nnull-geodesics of $g$ tend to $S_+$ as the parameter goes to\n$+\\infty$, $S_-$ as the parameter goes to $-\\infty$, or vice versa. We\nalso let\n$$\nC_0=\\{\\rho=0,\\ v<0\\}.\n$$\nWe then consider $\\Box_g$, on functions (or in the future differential forms or various other\nsquares of Dirac-type operators), and we wish to analyze the\ninvertibility of the Feynman propagator.\n\n\nFor this purpose it is convenient, as we explain further in the next paragraph, to consider\n\\begin{equation}\\label{eq:Ldef}\nL=\\rho^{-(n-2)\/2}\\rho^{-2}\\Box_g\\rho^{(n-2)\/2};\n\\end{equation}\nthen $L\\in\\Diffb^2(M)$, the space of b-differential operators, meaning\nthat locally near $\\p M$, using\ncoordinates $(\\rho, w_{1}, \\dots, w_{n - 1})$ where $\\rho$ is the boundary defining function\nfrom \\eqref{eq:metric} and $w_i$ are any coordinates on $\\p M$, there\nare smooth functions $a_{i, \\alpha} \\in C^\\infty(M)$, such that\n\\begin{equation}\\label{eq:b-diff-op-form}\nL = \\sum_{j + |\\alpha| \\le 2} a_{j, \\alpha}(\\rho \\p_\\rho)^j \\p^\\alpha_{w}.\n\\end{equation}\nIts principal symbol is the dual metric $\\hat G$ of the Lorentzian\nb-metric \n\\begin{equation}\\label{eq:bmetric}\n\\hat g=\\rho^2 g.\n\\end{equation}\nIn general, $\\Diffb^*(M)$ is the algebra of differential operators generated by\n\\begin{equation}\\label{eq:b-vector-fields}\n\\Vb := C^\\infty(M; \\Tb(M)),\n\\end{equation}\nwhich more\nconcretely is the\n$C^{\\infty}(M)$ span of the vector fields\n\\begin{equation}\n \\label{eq:bfields}\n \\rho \\p_{\\rho}, \\quad \\p_{w_{i}},\n\\end{equation}\nThat $L$ is indeed in $\\Diffb^2(M)$ can be checked directly from \\eqref{eq:metric} and \\eqref{eq:box}.\nIn the definition of $L$ in \\eqref{eq:Ldef}, $\\rho^{(n-2)\/2}$ is introduced to\nmake $L$ formally self-adjoint with respect to the b-metric $\\hat g$. The conformal factor $\\rho$ merely\nreparameterizes null-bicharacteristics, so our assumption is\nequivalent to the statement that null-bicharacteristics of $L$ tend to\n$S_\\pm$.\n\nWhile we could consider $\\Box_g$ or in fact $\\Box_g+\\lambda$,\n$\\lambda\\in\\Cx$, as a scattering differential operator, corresponding\nto the sc-structure, we instead work with $L$ because $\\Box_g$ is\nrather degenerate as a sc-operator due to the quadratic vanishing of\nits principal symbol at the zero section at $\\pa M$ (i.e.\\\ninfinity). Note that the option of the b-framework is not available if a non-zero spectral\nparameter $\\lambda$ is added (it would result in a singular term), but\non the other hand in these cases there is no degeneracy in the\noperator in the sc-framework! The latter phenomenon appears also in\nanalogous work in the elliptic setting, namely in analysis of\n$\\Delta_g + \\lambda$ for $g$ a \\emph{Riemannian} scattering metric, in\nparticular in Melrose's work on scattering manifolds \\cite{RBMSpec},\nwhich incidentally proves and uses radial points estimates. See\nalso \\cite{GH2008}. \n\nOne of the main features of our analysis, parallel to the recent work\n\\cite{HVquasi, HVsemi} as well as much other work on analysis on non-compact\nspaces going back to Melrose \\cite{Melrose:Atiyah}, is that we use an\nextension of the vector bundle $T^*(M^{int})$ up to the boundary which\nis better suited\nto the analysis than $T^*M$, and for which in particular the beginnings\nand ends of null-bicharacteristics become tractable objects.\nConcretely, we use the b-cotangent bundle, $\\Tb^*M$, the dual bundle of\nthe b-tangent bundle $\\Tb M$, whose local sections near the boundary\nare $C^\\infty(M)$ linear combinations of \n\\begin{equation*}\n \\frac{d\\rho}{\\rho} ,\\qquad dw_{i},\n\\end{equation*}\nwith\ncoordinates as above. Notation as in the paragraph containing\n\\eqref{eq:Euclidean-radial-scattering-form}, in $\\mathbb{R}^n$ we can\nassume that near infinity the\ndifferential form $d\\rho \/ \\rho = - dr \/r$. (The $w_i$ remain\ncoordinates on the sphere.)\n\nFor an operator $P\\in\\Diffb^m(M)$, the (b-)principal symbol\n$\\sigma_{\\bl,m}(P)$ is a smooth function on $\\Tb^*M$ which is a\nhomogeneous polynomial of degree $m$ on the fibers of\n$\\Tb^*M$, extending the standard principal symbol from $T^*M^{int}$ to $\\Tb^*M$.\nNotice that a vector in $\\Tb_q M$, $q\\in M$, defines a linear function on $\\Tb^*_qM$;\nthe principal symbol of a vector field is given by $i$ times this fiber-linear\nfunction; it is extended to differential operators by making it\nmultiplicative, in the process keeping only the leading ($m$th order)\nterms of the operator. Concretely, writing b-covectors as\n$\\sigma\\,\\frac{d\\rho}{\\rho}+\\sum_j\\zeta_j\\,dw_j$,\n$(\\rho,w,\\sigma,\\zeta)$ are local coordinates on $\\Tb^*M$ (global in\nthe fibers), the b-principal symbol of an operator of the form\n\\eqref{eq:b-diff-op-form} is\n$$\n\\sum_{j+ |\\alpha| = 2} a_{j, \\alpha}(\\rho,w)(i\\sigma)^j (i\\zeta)^\\alpha,\n$$\nsince the symbol of $\\rho \\p_\\rho$ is $i \\sigma$ in the coordinates\n$(\\rho,w,\\sigma,\\zeta)$, because $\\rho \\p_\\rho$ acting on $\\sigma\\,\\frac{d\\rho}{\\rho}+\\sum_j\\zeta_j\\,dw_j$\ncovector is $\\sigma$.\n\n\n\nWe describe the structure of the\nnull-bicharacteristics at the boundary in detail now. The Hamilton\nflow on null-bicharacteristics\ncorresponding to $L$ descends from a flow on $T^* (M^{int})$ to a flow\non the spherical cotangent bundle $S^*(M^{int})$. On can think of\n$S^*(M^{int})$ as either the quotient $(T^* (M^{int}) - o) \/ \\mathbb{R}_+$,\nwhere $o$ denotes the zero section and the action of $\\mathbb{R}_+$ is\nthe standard dilation action on the fibers, or as the bundle obtained\nby radially compactifying the fibers of $T^*(M^{int})$ to obtain a\nball bundle $\\overline{T}^* (M^{int})$ and taking the corresponding sphere\nbundle whose fibers are the boundary of fibers of\n$\\overline{T}^*(M^{int})$. The latter process can just as well be done on\non $\\Tb^* M$ to obtain $\\Tbc^*M$ and taking the boundaries of the\nfibers gives the spherical b-conormal bundle $\\Sb^*M$. \n\nFor the convenience of the reader, we will give a brief summary of the\nproperties of the Hamilton flow used in the propagation estimates,\nwhich are discussed in detail in \\cite[Section\n3]{Baskin-Vasy-Wunsch:Radiation}. The null-bicharacteristic flow of\n$\\Box_g$ is the flow of the Hamilton\nvector field $\\mathsf{H} = (\\p_\\zeta p) \\p _z - (\\p_z p) \\p_\\zeta$\nrestricted to $p = 0$, where\n$z$ are the coordinates on $\\mathbb{R}^{n}$, $\\zeta$ is dual\nto $z$, and $p$ is the principal symbol of $\\Box_g$, which is in fact\njust the dual metric function $g^{-1} \\colon T^* M \\lra \\mathbb{R}$,\n$g^{-1}(z, \\zeta) = | \\zeta|^2_{g^{-1}(z)}$. (Thus on Minkowski space $p(\\zeta)\n= \\zeta_n^2 - \\zeta_1^2 - \\dots - \\zeta_{n-1}^2$ and the\nnull-bicharacteristic flow is a straight line flow in phase space\non the space of null vectors, $\\zeta$ with $p(\\zeta) = 0$, keeping\n$\\zeta$ fixed and evolving $z$ affinely.)\nThe b-principal symbol of $L$, $\\sigma_b(L) = \\lambda$, can be understood as the\nextension of the principal symbol of $L$ in the standard sense to the\nb-cotangent bundle $\\Tb^*M$, thus as discussed\n$\\lambda$ is equal to the dual metric function of $\\hat g =\n\\rho^2g$, which extends smoothly to all $\\Tb^*M$.\nConcretely, in the variables\n$\\rho, v, y$, writing forms as \n$$\n\\sigma \\frac{d\\rho}{\\rho} + \\gamma\\, dv + \\eta\\, dy,\n$$ \n$\\lambda$ satisfies\n\\begin{equation}\\begin{aligned}\\label{eq:b-princ-symbol}\n \\lambda=\\sigma_{\\bl} (L) &= g^{\\rho\\rho}\\sigma^{2} - (4v - \\beta v^{2} +\n O(\\rho v) + O(\\rho^{2})) \\gamma^{2} - 2(2-\\alpha v+ O(\\rho))\\sigma\n \\gamma \\\\\n &\\quad + 2g^{\\rho y}\\cdot \\eta \\sigma + \\big(2 v \\Upsilon + O(\\rho)\n \\big) \\cdot \\eta \\gamma + g^{y_{i}y_{j}}\\eta_{i}\\eta_{j},\n\\end{aligned}\\end{equation}\nwhere all the $O(.)$ terms are smooth, and $\\beta,\\Upsilon$ are smooth\nas well as are $g^{\\rho\\rho}$, etc., which are the dual metric\ncomponents (in the b-basis, for $\\hat g$, or equivalently in the\nsc-basis for $g$); see\n\\cite[Equation~(3.18)]{Baskin-Vasy-Wunsch:Radiation}. (Thus, this\nformula specifies the structure of certain dual metric components for\n$\\hat g$,\nsuch as $\\pa_v^2$ and $\\pa_v(\\rho\\pa_\\rho)$.)\n\nConsider the\nb-Hamilton vector field, i.e.\\ the Hamilton vector field of $L$\nthought of as a vector field on $\\Tb^* M$. (More precisely, this is\nthe (unique) smooth extension of the Hamilton vector field of $L$ from\na vector field on $T^*M^{int}$ to $\\Tb^*M$.) \nAlmost as in \\cite{Baskin-Vasy-Wunsch:Radiation} (which used $\\xi$ in\nplace of $\\sigma$),\nthis is given in the variables $(\\rho, v, y, \\sigma, \\gamma, \\eta)$ by\n\\begin{equation}\n \\begin{aligned}\\label{eq:hamilton}\n \\mathsf{H}_\\bl := (\\p_\\sigma \\lambda) \\rho \\p_\\rho + (\\p_{\\gamma} \\lambda)\n \\p_v + (\\p_\\eta \\lambda) \\p_y - (\\rho \\p_\\rho\\lambda) \\p_\\sigma -\n (\\p_v \\lambda)\\p_{\\gamma} - (\\p_y \\lambda) \\p_\\eta\n \\end{aligned}\n\\end{equation}\nwhere $\\lambda$ is the dual metric function of $\\hat g$ on\n$\\Tb^* M$; see \\cite[Equation~(3.20)]{Baskin-Vasy-Wunsch:Radiation}. Note that $\\mathsf{H}_\\bl$ is automatically a vector field tangent to $\\Tb^*_{\\pa\n M}M$, i.e.\\ a b-vector field on $\\Tb^*M$ (thus a section of\n$\\Tb\\Tb^*M$).\nSince taking the Hamilton vector field is a derivation on \nfunctions on the cotangent bundle, the flow of $\\mathsf{H}_\\bl$ (this being the\nHamilton vector field of $\\hat g=\\rho^2 g$) restricted to \nthe set $\\Sigma = \\{ \\lambda = 0 \\} $ over $M^{int}$ (where $\\rho>0$) is a rescaling of the Hamilton flow of $\\Box_g$ restricted to \n$\\Sigma$; the rescaling becomes singular at $\\rho=0$. In our case,\n\\begin{equation}\\label{eq:bHamvf1}\\begin{aligned}\n\\mathsf{H}_\\bl=&\\big(2g^{\\rho\\rho}\\sigma+2g^{\\rho y}\\eta-2\\gamma(2-\\alpha\nv+O(\\rho))\\big)(\\rho\\pa_\\rho)\\\\\n&-2\\big((4v-\\beta v^2+O(\\rho\nv)+O(\\rho^2))\\gamma\\\\\n&\\qquad\\qquad\\qquad+(2-\\alpha\nv+O(\\rho))\\sigma+(v\\Upsilon+O(\\rho))\\eta\\big)\\pa_v\\\\\n&+2\\big(g^{\\rho y}\\sigma+(v\\Upsilon+O(\\rho))\\gamma+g^{y_iy_j}\\eta_j\\big)\\pa_y\\\\\n&-(\\rho\\pa_\\rho \\lambda)\\pa_\\sigma-(\\pa_v \\lambda) \\pa_\\gamma-(\\pa_y \\lambda) \\pa_\\eta;\n\\end{aligned}\\end{equation}\nsee \\cite[Equation~(3.21)]{Baskin-Vasy-Wunsch:Radiation}.\n\n\n\n\nTo describe how the null-bicharacteristic flow acts at infinity we\nmust define and analyze the b-conormal bundle of the submanifold $S$.\nThere is a\nnatural map of the b-tangent space $\\Tb M \\lra TM$\ndefined on sections, i.e.\\ elements of $\\Vb$, by considering a b-vector\nfield as a standard vector field. (Thus the map is not\nsurjective over the boundary; $\\rho \\p_\\rho$ vanishes there.) We can use the dual map $T^* M \\lra\n\\Tb^* M$ to define the b-conormal bundle of submanifolds; specifically,\nfor our submanifold $S$, the conormal bundle $\\Nb^*S$ equal to the\nimage in $\\Tb^* M$ of covectors in $T^*M$ annihilating the image of $T S \\subset T_S\nM$ in $\\Tb M$. It turns out that the null-bicharacteristics of $L$\n(see Figure \\ref{choices}) terminate both at $S_+$ and $S_-$ at the\nspherical b-conormal bundle \n$$\n\\Sb N^* S_{\\pm} = (\\Nb^* S_{\\pm} \\setminus o) \/ \\mathbb{R}_+.\n$$\nBefore we describe this in more detail, we point out that $\\Nb^* S$ in\nfact has one dimensional fibers, since in coordinates $\\rho, v, y$\nwith $\\rho, v$ (so $S = \\{ \\rho = 0 = v \\}$) as above and $y$ local\ncoordinates on $S$, so vectors in $TS$ are multiples of $\\p_{y}$, are annihilated by\nforms $a \\,dv + b \\,d\\rho$ in $T^*M$, which map to forms $a \\,dv + b \\rho\n(\\rho^{-1} \\,d\\rho)$ in $\\Tb^* M$ and thus restrict to $a\\,dv $ since $S$ lies in the boundary $\\rho = 0$.\nMore concretely, the b-conormal bundle of $S$ is generated by $dv$,\ni.e.\\ in the coordinates above is given by the vanishing of\n$\\rho,v,\\sigma,\\eta$, and thus $y,\\gamma$ are (local) coordinates along it.\nThis means that at each point $p \\in S$,\n\\begin{equation}\\label{eq:spherical-conormal-bundle}\n\\Sb N^*_p S = (\\Nb^*_p S \\setminus \\{ 0 \\} )\/ \\mathbb{R}_+ = \\{\n\\gamma \\, dv\n: \\gamma\n\\neq 0 \\} \/ \\mathbb{R}_+, \n\\end{equation}\nso in fact $\\Nb^* S$ is a line bundle over $S$ and $\\Sb N^* S$ is an\n$\\mathbb{S}^0$ (two points) bundle generated by the images of $dv$ and\n$-dv$ in $\\Tb^*_S M$.\n\n\n\n\nThe flow on null-bicharacteristics, in view of the structure of the\noperator at $S_\\pm$, as shown in \\cite[Section~3]{Baskin-Vasy-Wunsch:Radiation},\nsee also \\cite[Section 5]{HVsemi}, makes the two halves of\nthe spherical b-conormal bundle of $S$, $\\SNb^*\nS=\\SNb^*_+S\\cup\\SNb^*_-S$, into a family of\nsources ($-$) or sinks ($+$) for the Hamilton flow, meaning that the null-bicharacteristics approach\n$\\SNb_+^*S_+$ as their parameter goes to $+\\infty$ and $\\SNb^*_-S_-$\nas the parameter goes to $-\\infty$, or $\\SNb_+^*S_-$ as their parameter goes to $+\\infty$ and $\\SNb^*_-S_+$\nas the parameter goes to $-\\infty$. Correspondingly, the\ncharacteristic set $\\Sigma\\subset\\Tb^*M\\setminus o$, which we also\nidentify as a subset of $\\Sb^*M$, of $L$ globally splits into the\ndisjoint union $\\Sigma_+\\cup\\Sigma_-$, with the first class of\nbicharacteristics contained in $\\Sigma_+$, the second in $\\Sigma_-$.\nOne computes, cf.\\ the discussion after \\cite[Equation~(3.22)]{Baskin-Vasy-Wunsch:Radiation},\nthat the b-Hamilton vector field in \\eqref{eq:bHamvf1} at\n$\\Nb^*S$, modulo terms vanishing there quadratically, is\nsatisfies\n$$\n- 4 \\gamma (\\rho \\p_\\rho) - (8v \\gamma + 4\\sigma) \\p_v +2\\mu_i\\p_{y_i} + 4 \\gamma^2 \\p_{\\gamma}.\n$$\nwith $\\mu_i$ vanishing on $\\Nb^*S$. Thus, b-Hamilton vector field is\nindeed radial at $\\Nb^* S$ (it is a multiple of $\\pa_\\gamma$ as\n$\\sigma,\\rho,v,\\mu_i$ vanish there by \\eqref{eq:spherical-conormal-bundle});\nfurthermore within $\\Nb^* S$ one sees that for $\\gamma>0$ fiber\ninfinity is a sink (the flow tends towards it), while for $\\gamma<0$ a source.\nOne also sees that in fact $\\Sb^*NS$ is a source\/sink bundle depending\non the sign of $\\gamma$, i.e.\\ in the normal directions to $\\Nb^*S$ the flow\nbehaves the same way as within $\\Nb^*S$, by checking the eigenvalues\nof the linearization, see \\cite[Equation~(3.23)]{Baskin-Vasy-Wunsch:Radiation}.\n\n\n\n \\begin{figure}\\label{spacetime}\n \\centering\n\\input{spacetime.pdf_tex}\n \\caption{$\\rho$ equals zero exactly on the boundary and has\n non-vanishing differential there.}\n \\end{figure}\n\n\n\n\nRecall that the basic result for elliptic problems on compact\nmanifolds without boundary is elliptic\nregularity estimates, which in turn imply Fredholm properties.\nIndeed, if $P$ is an elliptic operator of order $k$ on a compact manifold without\nboundary $X$, then for any $m' \\frac{1}{2}, m \\geq m'$ and $m$ is nonincreasing along the Hamilton flow in the direction that leaves $\\mathcal R$. Then if $\\WFb^{m', l}(u)$ and $\\WFb^{m - 1, l}(Lu)$ are both disjoint from $\\mathcal R$, then $\\WFb^{m, l}(u)$ is disjoint from $\\mathcal R$.\n\\end{prop}\n\nFor elliptic regularity, the\nvariable order $m$ is completely arbitrary, but for real principal type\nestimates it has to be non-increasing in the direction along the\nHamilton flow in which we wish to propagate the estimates.\n\nSo now fixing $l$ and taking $m$ satisfying $m+l>1\/2$ at exactly one\nof $\\SNb^*_+S_+$ or $\\SNb^*_-S_-$ and $m + l < 1\/2$ at the other\n(e.g. if $> 1\/2$ at $\\SNb^*_+S_+$ then $< 1\/2$ at $\\SNb^*_-S_-$), and\nsimilarly $m+l>1\/2$ at either $\\SNb^*_-S_+$ or $\\SNb^*_+S_-$ and $m +\nl < 1\/2$ at the other,\nwe\nobtain estimates for $u$ in $\\Hb^{m,l}$ in terms of the $\\Hb^{m-1,l}$\nnorm of $Lu$ plus the $\\Hb^{m',l}$ norm of $u$ for\n$m'1\/2$ at exactly one end of each\nbicharacteristic and $m+l<1\/2$ at the other, \\eqref{eq:choices} is\n{\\em equivalent} to\n\\begin{equation}\\label{eq:choices-S-}\n \\begin{split}\n \\mp(m+l-1\/2)&<0 \\mbox{ at } \\SNb^*_-S_-\\ \\text{(sources)}\\ \\mbox{ and }\\\\\n \\mp(m+l-1\/2)&<0 \\mbox{ at } \\SNb^*_+S_-\\ \\text{(sinks)}.\n \\end{split}\n\\end{equation}\nThat is to say, \n\\begin{equation*}\nL_{\\pm \\pm} \\mbox{ denotes \\textit{any} map } L \\colon \\cX^{m,l} \\lra \\cY^{m-1,l}\n\\end{equation*}\nfor which the pair $(m, l)$ satisfy \\eqref{eq:choices} with the given \n$\\pm, \\pm$ combination (the first sign in the first inequality and\nthe second in the second). (To be clear, the fact that we write the\nchoices in \\eqref{eq:choices} as taking place at $S_+$ is arbitrary,\nas we could just as easily make the choices at $S_-$. Whichever signs\nare chosen at $S_+$, the opposite sign is chosen on the other end of\nthe flow at $S_-$. For\nexample, for $L_{+-}$, the condition on $m$ at $S_-$ is that $\n-(m+l-1\/2)<0$ at $\\SNb^*_-S_-$ and $m+l-1\/2 <0$ at $\\SNb^*_+S_-$.)\nStrictly\nspeaking, $L_{\\pm \\pm}$ depends on $m$, but in fact we will see that\nthe choice of $m$ satisfying a particular version of\n\\eqref{eq:choices} is irrelevant. Thus we use the notation\n\\begin{equation}\n \\label{eq:spacenotation}\n \\cX^{m, l}_{\\pm\\pm} = \\cX^{m,l},\\ \\cY^{m,l}_{\\pm\\pm}=\\cY^{m,l} \\mbox{ for \\emph{any} $(m, l)$ satisfying \\eqref{eq:choices} }\n\\end{equation}\nwith the given $\\pm$, $\\pm$ combination. See Figure~\\ref{choices}.\n\n\n\n \\begin{figure}\\label{choices}\n \\centering\n\\input{choices.pdf_tex}\n \\caption{For the operator $L_{+-}$, corresponding to the forward Feynman\n problem, high regularity is imposed at the `beginning' (near\n $\\SNb^*_- S$) of each\n null bicharacteristic, whether they begin at $S_{+}$ or $S_{-}$.}\n \\end{figure}\nWe call $L_{++}$ the forward wave operator\n(corresponding to the forward solution), $L_{--}$ the backward wave\noperator, $L_{+-}$ and $L_{-+}$ the Feynman and anti-Feynman wave\noperators (sometimes we simply call them both Feynman), with\n$L_{+-}$ propagating forward along the Hamilton flow (i.e.\\ to the sinks), and $L_{-+}$\nbackward along the Hamilton flow (i.e.\\ to the sources) in {\\em both} $\\Sigma_+$ and $\\Sigma_-$. Here we point out that either of the forward\nand backward wave operators propagate estimates in the {\\em opposite}\ndirections relative to the Hamilton flow in $\\Sigma_+$, resp.\\\n$\\Sigma_-$; the propagation is in the same direction relative to a\ntime function in the underlying space $M$.\n\n\n \\begin{figure}\\label{smooth}\n \\centering\n\\input{Sigmas2.pdf_tex}\n \\caption{The traces (i.e.\\ projections from the cotangent space) of\n the light rays passing through an arbitrary\n point $p$. In the cotangent space these separate into the\n forward and backward pointing null-bicharacteristics, depicted\n heuristically at right. The operator $L_{+-}$ corresponds to propagation of\n singularities along the flow, and corresponds to the choice of\n $+$ in the first and $-$ in the second inequality in \\eqref{eq:choices}}\n \\end{figure}\n\n\n\\section{Mapping properties of the Feynman propagator}\\label{sec:fred}\nThe main result of this section is Theorem \\ref{thm:fredholm} below,\nwhich asserts that $L_{\\pm \\pm}$ are Fredholm maps between appropriate\nHilbert spaces. As mentioned, the estimates in \\eqref{eq:propagationestimates} are {\\em\n not} sufficient to conclude that $L$ is Fredholm, \nsince the weaker norm does not possess additional decay. Thus the\nmain technical result of this section is the following. For\n$(m, l)$ chosen as in \\eqref{eq:choices} for any choice of signs\n$\\pm \\pm$, and \\textit{for certain choices of} $l$ (see the theorem),\nwe have\n\\begin{equation}\n \\label{eq:propagationestimatesimproved}\n \\begin{split}\n \\norm[\\Hb^{m, l}]{u} &\\le C ( \\norm[\\Hb^{m-1, l} ]{L u} +\n \\norm[\\Hb^{m', l'} ]{u} ) \\\\\n \\norm[\\Hb^{1 - m, -l}]{v} &\\le C ( \\norm[\\Hb^{-m, -l} ]{L v} +\n \\norm[\\Hb^{1 -m'', -l''} ]{v} )\n \\end{split}\n\\end{equation}\nwhere $m' < m < m''$ and $l' < l < l''$. As explained in the proof of\nTheorem \\ref{thm:fredholm}, it is\nthen a simple exercise using the fact that $\\Hb^{m, l} \\subset\n\\Hb^{m', l'} $ is compact provided $m' < m$ and $l' < l$ to show that\n$L_{\\pm \\pm}$ is Fredholm on the spaces in the theorem. \n\nTo obtain the improved estimates in\n\\eqref{eq:propagationestimatesimproved}, as in elliptic problems, we\nalso need to consider the Mellin\ntransformed normal operator $\\hat N(L)(\\sigma)$ of $L$, which is a\nfamily of differential operators on $\\pa M$, parameterized by\n$\\sigma\\in\\Cx$. Given an arbitrary $P \\in \\Diffb^*(M)$ of order $k$, \n\\begin{equation}\n \\label{eq:boparbitrary}\n P = \\sum_{i + \\absv{\\alpha} \\le k} a_{i, \\alpha}(\\rho, x) (\\rho\n \\p_{\\rho})^{i} \\p_{x}^{\\alpha},\n\\end{equation}\nthe normal operator is locally given by\n\\begin{equation}\n \\label{eq:normal}\n N(P) := \\sum_{i + \\absv{\\alpha} \\le k} a_{i, \\alpha}(0, x)\n (\\rho \\p_{\\rho})^{i} \\p_{x}^{\\alpha} \\in \\Diffb^{k}([0,\n \\infty)_{\\rho} \\times \\p M).\n\\end{equation}\nThe Mellin transform is defined, initially on compactly supported smooth\nfunctions $u \\in C^{\\infty}(\\mathbb{R}^{+}; \\mathbb{C})$, by\n\\begin{equation*}\n \\cM(u)(\\sigma) = \\hat u(\\sigma) = \\int_{0}^{\\infty} \\rho^{-i\\sigma} u(\\rho) \\frac{d\\rho}{\\rho}.\n\\end{equation*}\nNote that $\\cM u(\\sigma) = \\mathcal{F} v (\\sigma)$ where $\\mathcal{F}$\nis the Fourier transform and $v(x) = u(e^{x})$. Writing complex\nnumbers $\\sigma = \\xi + i \\eta$, it extends to a unitary isomorphism\n\\begin{equation}\\label{eq:mellindef}\n \\cM \\colon \\rho^{l}L^{2}(\\mathbb{R}^{+}, d\\rho\/\\rho) \\lra L^{2}(\\set{\\im\n \\sigma = - l } , d\\xi).\n\\end{equation}\nThe inverse map of \\eqref{eq:mellindef} is given by\n\\begin{equation}\n \\label{eq:1}\n \\cM_{l}^{-1} f(\\rho) = \\frac{1}{2\\pi} \\int_{\\set{\\im \\sigma = -l}} \\rho^{i \\sigma}\n f(\\rho) d\\sigma.\n\\end{equation}\nMoreover, conjugating $N(P)$ by the Mellin transform in $\\rho$ gives\n\\begin{equation}\n \\label{eq:reducednormal}\n \\hat{N}(P)(\\sigma) = \\sum_{i + \\absv{\\alpha} \\le k} a_{i, \\alpha}(0, x)\n \\sigma^{i} \\p_{x}^{\\alpha}\n\\end{equation}\n\nWe digress briefly to describe following typical example of a\nb-pseudodifferential operator which is elliptic at a point $p \\in \\Tb\nM$ lying over the boundary, and how it relates to the b-wavefront\nset discussed above. If $p \\in \\Tb^*M$ lies over the boundary, then in coordinates $(\\rho, y, \\xi, \\eta)$\non $\\Tb M$ where $\\rho$ is a boundary defining function, $\\xi$ is dual to\n$\\rho$ and $\\eta$ to $y$, we have $p = (0, y_{0}, \\xi_{0}, \\eta_{0})$. We\nobtain a b-pseudodifferential operator that is elliptic at $p$ by\nchoosing a cutoff function $\\chi(\\rho, y)$ with\n$\\chi(0,y_0) \\neq 0$ and such that $\\chi$ is supported in $\\set{\\rho\n < \\epsilon}$ for small $\\epsilon$, in particular small enough so\nthat $\\set{\\rho < \\epsilon} \\simeq \\p M \\times [0, \\epsilon)$. Let\n$\\phi(\\xi,\\eta)$ be a symbol, homogeneous near infinity, non-zero in\nthe cone given by positive multiples of\n$(\\xi_0,\\eta_0)$. With $\\mathcal{F}$ the Fourier transform in the $y$\nvariables, we define\n\\begin{equation}\n \\label{eq:boperator}\n \\begin{split}\n A u &:= \\mathcal{M}^{-1}_{0} \\mathcal{F}^{-1} \\phi \\mathcal{F}\\mathcal{M} (\\chi u).\n\\end{split}\n\\end{equation}\nThen $A \\in \\Psib^{0, 0}(M)$, and the b-principal symbol of $A$ at\norder and weight $(m, l) = (0, 0)$ is:\n\\begin{equation}\n \\label{eq:bsymbol}\n \\sigma_{0, 0}(A) \\colon \\Tb^* M \\lra \\mathbb{C}, \\quad \\sigma_{0,\n 0}(A) = \\chi \\phi\n\\end{equation}\nwhere we think of $\\chi \\phi= \\chi(\\rho, y) \\phi(\\xi,\\eta)$ as a function on $\\Tb^* M$, which\nnear the boundary and with our coordinates is diffeomorphic to $\\{\n\\rho < \\epsilon, \\xi \\} \\times T^* \\p M$, supported on the\nneighborhood of $(0, y_0, \\xi_0, \\eta_0)$ under consideration.\nIn fact, such operators can be used to neatly describe the b-wavefront\nsets of distributions. Given a distribution $u \\in \\Hb^{-N, l}(M)$,\nthen for $m, l \\in \\mathbb{R}$,\n\\begin{equation}\\label{eq:goodwavefrontdef}\n(0, y_0, \\xi_0, \\eta_0) \\not \\in \\WFb^{m, l}(u) \\iff \\exists \\chi, \\phi\n\\mbox{ with } A \\rho^{- l} u \\in \\Hb^{m, 0}(M),\n\\end{equation}\nwhere $A$ is formed from $\\chi$ and $\\phi$ as in \\eqref{eq:boperator}.\n(The $\\rho^{-l}$ in the front is there so that the inverse Mellin\ntransform $\\mathcal{M}_0^{-1}$ of the resulting object is well defined.)\n\n\n\n\n\nThe structure and properties of $\\hat{N}(L)(\\sigma)$ are discussed at length in\n\\cite{HVsemi}. To briefly summarize, for each $\\sigma$,\n$\\hat{N}(L)(\\sigma)$ is a second order differential operator which is\nelliptic in the interior of the regions $C_{\\pm}$, and hyperbolic on\ntheir complement $\\p M \\setminus (C_+ \\cup C_-)$ whose\ncharacteristic set splits into two components $\\wt{\\Sigma}_{\\pm}$,\neach of which contains a Lagrangian submanifold of radial points lying\nover $S = \\p C_+ \\cup \\p C_-$, and which split the conormal bundle\n$N^*S$ (in $\\p M$) into four components\n$N^*_{\\pm} S_{\\pm}$ which are sources ($N^*_-S$) and sinks\n($N^*_+S$) for the Hamilton flow.\n\nThe estimates corresponding to those of the previous section allow one\nto conclude that $\\hat N(L)(\\sigma)$ is Fredholm\nfor each $\\sigma$ on the induced Sobolev spaces, where $\\im\\sigma=-l$,\ni.e.\\ provided \n\\begin{equation}\\label{eq:boundarychoices}\n \\begin{split}\n \\pm(m- \\im \\sigma-1\/2)&<0 \\mbox{ at } N^*_+S_{+} , \\mbox{ and }\\\\\n \\pm(m- \\im \\sigma-1\/2)&<0 \\mbox{ at } N^*_-S_{+}.\n \\end{split}\n\\end{equation}\nMore precisely here $m$ is replaced by $m|_{S^* \\pa M}$, which is a\nwell-defined subbundle of $\\Sb^*_{\\pa M} M$. (Indeed, the map $\\Tb M\n\\lra TM$, restricts to a surjection $\\Tb_{\\p M} M\n\\lra T\\p M$, and the dual injection $T^*\\p M \\lra \\Tb_{\\p M}^* M$\ngives the desired inclusion of $S^* \\p M$ after modding out by the\naction of $\\mathbb{R}_+$). Thinking of $\\sigma$ as\nthe b-dual variable of $\\rho$ (which thus depends on the choice of\n$\\frac{d\\rho}{\\rho}$ in a neighborhood of $\\p M$ but is invariant at $\\pa M$), covectors are of the form\n$\\beta+\\sigma\\frac{d\\rho}{\\rho}$, $\\beta\\in T^*M$, thus (identifying\nfunctions on $\\Sb^*M$ with homogeneous degree $0$ functions on\n$\\Tb^*M\\setminus o$, where $o$ denotes the zero section) for each\n$\\sigma\\neq 0$ one actually has a function on $T^* \\pa M$. One thus\nobtains a family of large parameter norms (as described in the theorem\njust below), analogous to the usual\nsemiclassical norms: for $\\sigma$ in a compact set, the norms are\nuniformly equivalent to each other, but as $\\sigma\\to\\infty$ this\nceases to be the case. In fact, we have the following applications of\n\\cite[Proposition\n5.2]{Baskin-Vasy-Wunsch:Radiation}, \\cite[Theorem 2.14]{VD2013}.\n\n\\begin{thm}\n \\label{thm:indexzeroboundary}(Statement of \\cite[Theorem\n 2.14]{VD2013} in our setting.)\nIn strips in which $\\mathrm{Im} \\,\\sigma$ is bounded, $\\hat\nN(L)(\\sigma)^{-1}$ has finitely many poles. \n\\end{thm}\n\n\\begin{proof}\nSince this is the statement of \\cite[Theorem\n 2.14]{VD2013}, with the underlying analysis being carried out in\n \\cite[Section~2]{VD2013}, we only give a brief sketch.\n\nAs we will see momentarily, our family $\\hat N(L)(\\sigma)$ forms an analytic\n Fredholm family\n \\begin{equation}\n \\label{eq:normalfamily}\n \\hat N(L)(\\sigma) \\colon \\cX^{m}(\\p M) \\lra \\cY^{m-1}(\\p M),\n \\end{equation}\n where $\\cX^{m}(\\p M) = \\{ \\phi : \\phi \\in H^{m}(\\p M), \\hat\n N(L)(\\sigma) \\phi \\in H^{m-1}(\\p M)\\}$, and $\\cY^{m}(\\p M) = \\set{\n \\phi : \\phi \\in H^{m}(\\p M)}$ provided that $\\sigma$ and $m$ are\n related as in \\eqref{eq:boundarychoices}, whose inverse is thus\n meromorphic if $\\hat N(L)(\\sigma)$ is invertible for at least one\n $\\sigma=\\sigma_0$. For bounded $\\im \\sigma$, we can see that\n $\\hat N(L)(\\sigma)$ is invertible for sufficiently large\n $\\re \\sigma$. This follows exactly as in\n \\cite[Proposition 5.2]{Baskin-Vasy-Wunsch:Radiation}, which in turn\n follows directly from \\cite[Theorem\n 2.14]{VD2013}. The key to this is to consider the semiclassical problem\n obtained by letting $h = |\\sigma|^{-1}$ and $z =\n \\frac{\\sigma}{|\\sigma|}$, and, letting $P_\\sigma = \\hat N\n (L)(\\sigma)$, studying\n$$P_{h, z} := h^2 P_{h^{-1} z} \\in \\Psi^2_h(\\partial M),$$\nwhere $\\Psi^{2}_{h}(\\partial M)$ denotes the space of semiclassical\npseudodifferential operators of order $2$ on $\\partial M$.\nThis semiclassical family on $\\partial M$ has Lagrangian submanifolds of radial\npoints (coming from the b-radial points of $L$), and, as described in\n\\cite[Section~2.8]{VD2013}, the standard positive commutator\nproof of propagation of singularities around Lagrangian submanifolds\nof radial points carries over to the semiclassical regime without\ndifficulty. This allows us to obtain estimates\n\\begin{equation*}\n \\begin{split}\n \\| u \\|_{H^m_h} &\\leq C (h^{-1} \\|P_{h, z} u \\|_{H^{m - 1}_h} +\n h\\|u\\|_{H^{-N}_h}) \\\\\n \\| v \\|_{H^{1 - m}_h} &\\leq C (h^{-1} \\|P^*_{h, z} v \\|_{H^{-m}_h} +\n h\\|u\\|_{H^{-N}_h})\n \\end{split}\n\\end{equation*}\nfor arbitrarily large $N$, within strips of\nbounded $\\im \\sigma$. \nAs described at the beginning of the proof of Theorem\n\\ref{thm:fredholm} below, these estimates imply that $N(L)(\\sigma)$\nmapping in \\eqref{eq:normalfamily} is Fredholm. \nHence, for sufficiently small $h$, the\n$-N$ norm can be absorbed into the left hand side, giving\nby the first inequality injectivity and by the second surjectivity.\n(This point is also elaborated in Theorem \\ref{thm:fredholm}.)\nNote that the statement of \\cite[Proposition\n5.2]{Baskin-Vasy-Wunsch:Radiation} is for only the\nforward and backward propagators, as the results come from microlocal\npositive commutator estimates which are sufficiently microlocal, the\nconclusion, with the same proof, also holds for the Feynman operators.\\end{proof}\n\n\\begin{rem}\n We point out that analogues of\nthe estimates used so far go through if $L$ has sufficiently weak\ntrapping with slight modifications: so-called b-normally hyperbolic\ntrapping, as introduced in \\cite{Hintz-Vasy:Normally}, gives essentially the same estimates for\n$\\sigma$ real and large. (However, we do not study this here.)\n\\end{rem}\n\nFollowing \\cite{HVsemi} we will prove the following Fredholm mapping\nresult for $L$, mapping between spaces $\\cX^{m,l}$ and $\\cY^{m-1,l}$\nwhich satisfy not only the threshold properties in \\eqref{eq:choices},\nbut furthermore that in the high regularity regime we assume a full\norder more Sobolev regularity.\nSpecifically, we will assume that $(m, l)$ are chosen as in \\eqref{eq:choices} for any\nchoices $\\pm, \\pm$, with the additional property that when the $-$ sign is valid on the\nleft hand side, i.e.\\ $-(m+l-1\/2)<0$, then in fact $-(m+l-3\/2)<0$ as\nwell, and thus the\ncomplete set of options for $m, l$ shall be\n\\begin{equation}\n \\label{eq:full-reg-list}\n\\begin{tabular}{|c||c|c|c|c|} \\hline \nRegion & Feynman & Anti-Feynman & Retarded & Advanced \\\\ \\hline\\hline\n$\\SNb^*_+S_+$ & \n\t$m + l < 1\/2$ & \n\t$m + l > 3\/2$ & $m + l < 1\/2$ & $m + l > 3\/2$ \\\\ \\hline\n$\\SNb^*_-S_+$ & \n\t$m + l > 3\/2$ &\n\t$m + l < 1\/2$ & $m + l < 1\/2$ & $m + l > 3\/2$\\\\ \\hline\n$\\SNb^*_+S_-$ &\n\t$m + l < 1\/2$ &\n\t$m + l > 3\/2$ & $m + l > 3\/2$ & $m + l < 1\/2$\\\\ \\hline\n$\\SNb^*_-S_-$ &\n\t$m + l > 3\/2$ & \n\t$m + l < 1\/2$ & $m + l > 3\/2$ & $m + l < 1\/2$\\\\ \\hline\n\\end{tabular}\n\\end{equation}\n\\begin{thm}\\label{thm:fredholm}\nAssume that $(m, l)$ satisfy \\eqref{eq:choices}, and in addition the\nproperties of the previous paragraph, i.e.\\ one of the four columns in\n\\eqref{eq:full-reg-list}. Moreover, assume that, subject to this choice,\nthere are no poles of $\\hat N(L)(\\sigma)^{-1}$ on the line\n$\\im\\sigma=-l$, (where $\\hat N(L)$ maps as in \\eqref{eq:normalfamily}\nwith $m = m \\rvert_{T^* \\p M}$.) Then $L$ is Fredholm as a map\n\\begin{equation}\\label{eq;fredholmmap}\nL:\\cX^{m,l}\\to\\cY^{m-1,l}.\n\\end{equation}\n\\end{thm}\n\n\\begin{rem}\nNote that the Fredholm property is stable under b-perturbations of \n$L$, in $\\Psib^{2,0}$, meaning for operators of the form $L + P$ where\n$P \\in \\Psib^{2, 0}$ is small. For $P$ in $\\Diffb^2$ this means that\n$P$ also has an expression as in \\eqref{eq:b-diff-op-form} but with\ncoefficient functions $a_{j, \\alpha}$ which are small in $C^\\infty$. In particular, any perturbation of a Lorentzian \nscattering metric in the sense of sc-metrics gives rise to a similarly \nFredholm problem. Indeed, for such a perturbation, the poles of the Feynman resolvent family for the normal operator of the\nperturbation of $L$ are themselves perturbed in a continuous fashion from the\npoles of those of the normal operator of $L$.\nSince obtaining a Fredholm problem at a weight corresponds to having\na line $\\{ \\im \\zeta = c \\}$ with no poles, the result follows.\n\\end{rem}\n\n\n\n\\begin{rem}\\label{rem:hadamard}\nMicrolocal elliptic regularity states that\n$\\WFb^{m_0,l}(u)\\setminus\\Sigma\\subset\\WFb^{m_0-2,l}(f)$ if $Lu=f$ and\n$u\\in\\Hb^{\\tilde m,l}$ for some $\\tilde m$ (i.e.\\ $u\\in\\Hb^{-\\infty,l}$).\n Propagation of singularities, in the sense of $\\WFb$, implies that if\n $L u = f$, where $u\\in\\Hb^{m,l}$, $f\\in\\Hb^{m-1,l}$ for some $m$, $l$ satisfying\n \\eqref{eq:choices} for the $+-$ signs, then a point\n $\\alpha\\in\\Sigma\\setminus(\\Sb N_+^*S_+\\cup\\Sb N_+^*S_-)$ (i.e.\\\n $\\alpha$ is not at the radial sink, at which the function spaces have\n low regularity) is not in $\\WFb^{m_0,l}(u)$ provided that\n the backward bicharacteristic through $\\alpha$ is disjoint from\n $\\WFb^{m_0-1,l}(f)$ and provided $\\WFb^{m_0-1,l}(f)$ is disjoint\n from $\\Sb N^*_- S_+\\cup\\Sb N^*_-S_-$, i.e.\\ the radial sources at\n which high regularity is imposed. In particular, if\n$f$ is compactly supported in $M^\\circ$, then\n$\\WFb^{m_0,l}(u)\\setminus (\\Sb N_+^*S_+\\cup\\Sb N_+^*S_-)$\n contained in the order $m_0-1$ wavefront set of $f$ together with the flowout of\n the intersection of the wavefront set of $f$ with the characteristic\n set $\\Sigma$ of $L$. Restricted to the interior, where $\\WFb$ is\n just the standard wave front set $\\WF(u)\\subset S^* M^\\circ$, this states that\n \\begin{equation}\n \\label{eq:hadamard-for-single-space}\n \\WF^{m_0}(u)\\subset \\WF^{m_0-1}(f) \\bigcup (\\cup_{t \\ge 0} \\Phi_t(\\WF^{m_0-1}(f) \\cap \\Sigma))\n \\end{equation}\nwhere $\\Phi_t$ is the time $t$ Hamilton flow (on the cosphere bundle).\nIn particular this applies to $u=L_{+-}^{-1}f$ when $L_{+-}$ is\nactually invertible, so within the characteristic set the wave front set of $u$ is a subset of the\nforward flowout of that of $f$.\n\nThere are analogous conclusions for the other choices of signs in\n\\eqref{eq:choices} with the wavefront sets of solutions\ncontained in the direction of the Hamilton flowout\nof the wavefront set of $f$ corresponding to the choice of direction on each\ncomponent of the characteristic set. In particular, for the $-+$ sign, $\\cup_{t \\ge 0} \\Phi_t(\\WF^{m_0-1}(f) \\cap\n\\Sigma)$ is replaced by $\\cup_{t \\le 0} \\Phi_t(\\WF^{m_0-1}(f) \\cap\n\\Sigma)$.\n\nFurther, it is not hard to show that, provided $L_{\\pm\\pm}^{-1}$ exists, the\nSchwartz kernel $K_{\\pm\\pm}$ of $L_{\\pm\\pm}^{-1}$ satisfies a corresponding wave\nfront set conclusion in $M^\\circ\\times M^\\circ$. For instance, for $L_{+-}^{-1}$,\n$\\WF(K_{+-})\\setminus N^*\\mathrm{diag}$ is contained in the\nforward flowout of $N^*\\mathrm{diag}$, the conormal\nbundle of the diagonal, with respect to the Hamilton vector field in the left factor.\n\\end{rem}\n\n\n\\begin{proof}\nWe wish to obtain the improvements to\n \\eqref{eq:propagationestimates} in the\n estimates in\n \\eqref{eq:propagationestimatesimproved}. These estimates imply that\n the map in \\eqref{eq;fredholmmap} is Fredholm. Indeed, using the\n fact that the containment $\\Hb^{m, l} \\subset\n\\Hb^{m', l'} $ is compact provided $m' < m$ and $l' < l$, the first\n estimate in \\eqref{eq:propagationestimatesimproved} shows\nthat the map has closed range and finite dimensional kernel. Assuming\nthat $v$ lies in $(\\mbox{image}(L:\\cX^{m,l}\\to\\cY^{m-1,l}))^\\perp,$\nwhere the orthogonal complement is taken with respect to the $L^2_b$\npairing (see \\eqref{eq:l2bdef}) between $\\Hb^{1 - m, -l}$ and $\\cY^{m - 1, l} = \\Hb^{m - 1,\n l}$, it follows that $Lv = 0$ and thus the second estimate in\n\\eqref{eq:propagationestimatesimproved} shows that the space of such\n$v$ is finite dimensional.\n\n\nThus we need only obtain the improved estimate in\n\\eqref{eq:propagationestimatesimproved}. The proof is essentially the proof of \\cite[Proposition\n 2.3]{HVsemi}, and we recall it briefly for the convenience of the\n reader. The condition on\n $\\hat N(L)(\\sigma)^{-1}$ on the line $\\im(\\sigma) = -l$ implies by\n taking the inverse Mellin transform that the map\n \\begin{equation}\n \\label{eq:15}\n N(L) \\colon \\cX^{m',l}(\\p M \\times \\mathbb{R}_+) \\lra \\cY^{m' -1,\n l}(\\p M \\times \\mathbb{R}_+)\n \\end{equation}\nis bounded and invertible, \\textit{where $m' \\colon T^* \\p M \\lra\n \\mathbb{R}$ is any function satisfying the constraints in\n \\eqref{eq:choices} that $m$ satisfies}. Thus there is a $C$ such that \n\\begin{equation*}\n \\| u \\|_{\\Hb^{m', l}(\\p M \\times \\mathbb{R}_+)} \\le C \\| N(L) u\n \\|_{\\Hb^{m' - 1, l}(\\p M \\times \\mathbb{R}_+)},\n\\end{equation*}\nand we may furthermore choose $m'$ so that it satisfies the constraint\n\\textit{and} that $m' < m$. Choosing a cutoff function $\\chi$ that is supported near $\\p M$ and\nequal to $1$ in a neighborhood thereof, we have (with a constant whose\nvalue changes from line to line)\n\\begin{equation*}\n \\begin{split}\n \\| u \\|_{\\Hb^{m, l}(M)} &\\le C (\\| L u\n \\|_{\\Hb^{m - 1, l}(M)} + \\| u \\|_{\\Hb^{m', l}(M)}) \\\\\n &\\le C (\\| L u\n \\|_{\\Hb^{m - 1, l}(M)} + \\| \\chi u \\|_{\\Hb^{m', l}(M)} + \\| (1 -\n \\chi) u \\|_{\\Hb^{m', l}(M)}) \\\\\n &\\le C (\\| L u\n \\|_{\\Hb^{m - 1, l}(M)} + \\| N(L) \\chi u \\|_{\\Hb^{m' - 1, l}(M)} + \\| u\n \\|_{\\Hb^{m', l'}(M)}).\n \\end{split}\n\\end{equation*}\nNow, writing $N(L) \\chi = [N(L), \\chi] + \\chi(N(L) - L) + \\chi L$, and\nusing $N(L) - L = \\rho P$ where $P \\in \\Diffb^2(M)$, and\n$[N(L), \\chi] = \\rho P'$ where $P' \\in \\Diffb^1(M)$, note that\n\\begin{equation*}\n \\| u \\|_{\\Hb^{m, l}(M)} \\le C (\\| L u\n \\|_{\\Hb^{m - 1, l}(M)} + \\| u \\|_{\\Hb^{m' + 1, l'}(M)}),\n\\end{equation*}\nso to obtain the improved estimate in\n\\eqref{eq:propagationestimatesimproved} we need only make sure that\n$m' + 1 < m$ which can be done due to the $-(m+l-3\/2)<0$ assumption at\nappropriate radial sets. \n\\end{proof}\n\n\nIt is important to remark here that $L_{\\pm\\pm}$ are rather different\noperators for different choices of $\\pm\\pm$, on the other hand the\nchoice of $m,l$ {\\em satisfying the constraints} corresponding to a\ngiven $\\pm$ (i.e.\\ a given one of the two constraints) matter much\nless. For example, the invertibility of the normal operator\n$\\hat N(L)(\\sigma)$ is {\\em independent} of these additional choices, \nso long as the $m$ satisfies that $m-\\im\\sigma-1\/2$ has the correct\nsign at the\nrelevant locations and has the correct monotonicity. In the Feynman\ncase see Proposition\n\\ref{thm:boundaryscaling} below; the regularity theory shows that the\npotential kernel of the operator,\nas well as of the adjoint, is indeed independent of these choices. The\nchoice of $l$ does affect the index of $L$, however as a Fredholm\noperator, as we show for the Feynman operator in Theorem\n\\ref{thm:constantindex}. \n\n\n\nWe also note that the adjoint of $L_{++}$ is $L_{--}$, while\nthat of $L_{+-}$ is $L_{-+}$, so one should {\\em not} think of $L$ as\na self-adjoint operator even though it is of course formally self-adjoint.\n\n\nThe standard setting in which $\\Box_g$ is considered is that of\nevolutionary problems, in which the forward or backward propagator\n$L_{++}^{-1}$ and $L_{--}^{-1}$ are considered. On the other hand, the\nFeynman propagator arises for instance by Wick-rotating suitable Riemannian\nproblems. Here we are interested in the Feynman propagator, but we\nfirst explain the more studied forward and backward problems in order\nto be able to contrast these.\n\nFor the forward or backward problems the usual tools of evolutionary\nproblems, namely standard energy estimates, can be used to compute the\nindex in some cases, as discussed in \\cite[Theorem~5.2]{HVsemi}. Since\nin this paper we focus on the Feynman propagator, we shall be brief. Thus, recall first from\n\\cite[Section~3.2.1]{Baskin-Vasy-Wunsch:Radiation} that the Lorentzian\nscattering metric $g$ in fact induces (even) asymptotically hyperbolic\nmetrics $k_+$, resp.\\ $k_-$, on $C_+$, resp.\\ $C_-$, with $S_+$, resp.\\ $S_-$ being conformal\ninfinity for these. Similarly, an asymptotically de Sitter Lorentzian\nmetric is induced on $C_0$, for which $S_+$ is future and $S_-$ is\npast conformal infinity. One thus can consider the spectral family\n$\\Delta_{C_\\pm}-\\frac{(n-2)^2}{4}-\\sigma^2$, as well as its inverse\n$\\cR_{C_\\pm}(\\sigma)$ for $\\im\\sigma\\gg 0$, which continues\nmeromorphically to the complex plane (in $\\sigma$).\nThen, as discussed in \\cite[Section~7]{Baskin-Vasy-Wunsch:Radiation}\nand more systematically in \\cite{Vasy:Resolvents-AH-dS},\nfor\nthe forward problem, the poles of $\\hat N(L)(\\sigma)^{-1}$ consist of\nthe poles of the meromorphically continued resolvents\n$\\cR_{C_+}(\\sigma)$ (i.e.\\ its resonances) and\n$\\cR_{C_-}(-\\sigma)$ on the asymptotically hyperbolic caps $C_\\pm$, as\nwell as possibly a subset of $i\\ZZ\\setminus \\{0\\}$. (The latter correspond to\npossible differentiated delta distributional resonant states, which\nexist e.g.\\ in even dimensional Minkowski space and which are\nresponsible for the strong Huygens principle on the one hand and for\nthe absence of poles of the meromorphically continued resolvent on odd dimensional\nhyperbolic spaces on the other hand.) Further, the resonant states\nand dual states have a certain support structure (this corresponds to\n$C_0$ being a hyperbolic region), namely for $\\phi$ supported in\n$C_0\\cup\\overline{C_+}$, $\\hat N(L)(\\sigma)^{-1}\\phi$ can only have\npoles if $\\sigma$ is either a pole of $\\cR_{C_+}(\\sigma)$ or is in\n$-i\\Nat^+$, see \\cite{Baskin-Vasy-Wunsch:Radiation,\n Vasy:Resolvents-AH-dS}. Thus, see \\cite{HVsemi}, suppose that $|l|<1$ (one could take\n$l$ larger if one also excludes the possible imaginary integer poles of\n$\\hat N(L)(\\sigma)^{-1}$), and $\\cR_{C_\\pm}(\\sigma)$ have no poles in\n$\\im\\sigma\\geq -|l|$, and that\nthere is a boundary-defining function $\\rho$ which is globally\ntime-like (in the sense that $\\frac{d\\rho}{\\rho}$ is such with respect\nto $\\hat g$) near $\\overline{C_+}\\cup\\overline{C_-}$. (These\nassumptions hold e.g.\\ on perturbations of Minkowski space.) Then any element\nof $\\Ker L$ would be vanishing to infinite order at $\\overline{C_-}$\n(and the same for $\\Ker L^*$, where $L^*$ is the adjoint of $L$ with\nrespect to the $\\Lb^2 = L^2(\\RR^n, \\mu)$ pairing in \\eqref{eq:l2bdef}, with $C_-$ replaced by $C_+$)\nby the first hypothesis and vanishing in a neighborhood of\n$\\overline{C_-}$ by the second. Finally, a result of\nGeroch's \\cite{Geroch:Domain} (relying on a construction of Hawking's)\nshows that $M$ is globally hyperbolic (there is a Cauchy surface for\nwhich every timelike curve intersects it exactly one time) under these assumptions, and in particular $L_{++}$\nand $L_{--}$ are invertible since any element of $\\Ker L_{++}$ would\nvanish globally, and similarly for elements of $\\Ker L_{++}^*$. One can then use the relative index\ntheorem of Melrose \\cite[Chapter~6]{Melrose:Atiyah} to compute the index of $L$ on other weighted spaces.\n\nFor the Feynman propagator there is no simple direct identification of\nthe poles of $\\hat N(L)(\\sigma)$. However, in Minkowski space, one can\ncompute these exactly by virtue of a Wick rotation (Proposition\n\\ref{thm:boundaryscaling}), and further even\nshow the invertibility of $L$ on appropriate weighted spaces (Theorem \\ref{thm:invertibility}). Namely,\nthe poles of $\\hat N(L)(\\sigma)$ are exactly those values of $\\sigma$\nfor which the operator $\\Delta_{\\sphere^{n-1}}+(n-2)^2\/4+\\sigma^2$ is {\\em not}\ninvertible, i.e.\\ $\\sigma$ is of the form $\\pm\ni\\sqrt{\\lambda+(n-2)^2\/4}$, $\\lambda$ an eigenvalue of\n$\\Delta_{\\sphere^{n-1}}$, i.e.\\ $\\lambda=k(k+n-2)$, $k\\in\\Nat$, so\n$\\lambda+(n-2)^2\/4=(k+(n-2)\/2)^2$, and thus $\\sigma=\\pm\ni(\\frac{n-2}{2}+k)$. For future reference, we define\n\\begin{equation}\n \\label{eq:indicialroots}\n \\Lambda = \\set{ \\pm\n(\\frac{n-2}{2}+k) : k \\in \\mathbb{N}_0\n}\n\\end{equation}\nThis gives a gap between\nthe two strings of poles with positive and negative imaginary parts,\nand for $|l|<\\frac{n-2}{2}$, $L_{+-}$ and $L_{-+}$ are invertible (and\nare adjoints of each other on dual spaces). Since the framework we set\nup is stable under {\\em general b-ps.d.o.\\ perturbations}, we conclude\nthat for general sc-metric perturbations $g$ of the Minkowski metric\n$g_0$, $L_{g,+-}$ and $L_{g,-+}$ have the same properties, provided\nthe $|l|$ is taken slightly smaller:\n\n\\begin{thm}\\label{thm:invertibility}\nLet $\\delta\\in (0,\\frac{n-2}{2})$. Then there exists a neighborhood $\\cU$ of the Minkowski\nmetric $g_0$ in $\\CI(M;\\Sym^2\\Tsc^*M)$ (i.e.\\ in the sense of\nsc-metrics, see \\eqref{eq:symmetric-tensor-basis} and the paragraph\nfollowing it) such that for $g\\in\\cU$,\n\\begin{equation}\\label{eq:invertiblemap}\nL_{g,+-} \\colon \\cXpm^{m, l} \\lra \\cYpm^{m - 1,l},\n\\end{equation}\nwith $\\cXpm^{m, l},\\cYpm^{m - 1,l}$ as in \\eqref{eq:spacenotation},\nis invertible for\n$|l|<\\frac{n-2}{2}-\\delta$ and $m$ satisfying the forward Feynman condition\nfor $+-$ in \\eqref{eq:choices}, strengthened as in Theorem~\\ref{thm:fredholm}, and where $\\cXpm^{m,l}$ is the domain\nof the Feynman wave operator defined in \\eqref{eq:spacenotation}. The same is true for $L_{g,-+}$ with\n$+-$ replaced by $-+$ in all the spaces.\n\\end{thm}\n\\begin{proof}\n For the actual Minkowski metric $g_0$, the invertibility is a restatement of\n Theorem \\ref{thm:scalinglemma} below. Since the estimates in\n \\eqref{eq:propagationestimatesimproved} hold uniformly on a\n sufficiently small neighborhood $\\cU'$ of $g_0$, $L_{g, +-}$ defines a\n continuous bounded family mapping as in \\eqref{eq:invertiblemap},\n and thus is invertible on a possibly smaller neighborhood $\\cU$.\n\\end{proof}\n\nTaking into account the construction of $L$ (see \\eqref{eq:Ldef}), for\nmetrics $g$ in the neighborhood $\\cU$ in the theorem, we deduce that \n\\begin{equation}\\label{eq:realinvertiblemap}\n\\Box_{g,+-}:\\cXpm^{m,l+\\frac{n-2}{2}}\\to \\cYpm^{m,l+\\frac{n-2}{2}+2}\n\\end{equation}\nis invertible for $|l|<\\frac{n-2}{2}-\\delta$. Its inverse is indeed\nthe forward Feynman propagator (which\nis well defined on space $\\cYpm^{m,l+\\frac{n-2}{2}+2}$ with weight\n$l$ in the stated range,\n\\begin{equation}\\label{eq:feynmanprop}\n\\Box^{-1}_{g,fey}:\\cYpm^{m,l+\\frac{n-2}{2}}\\to \\cXpm^{m,l+\\frac{n-2}{2}+2}.\n\\end{equation} \nThe same for $+-$ replaced by $-+$ and ``forward'' replaced by ``backward''.\n\n\\begin{rem}\nThe class of perturbations we consider {\\em does not} preserve the\nradial point structure at $\\SNb^*S_\\pm$. Nonetheless, the {\\em\n estimates} the radial point structure implies for $L$ and $L^*$ are\npreserved, much as discussed for Kerr-de Sitter spaces in \\cite{VD2013}.\n\\end{rem}\n\n\n\n\n\\section{Wick rotation (complex scaling)}\\label{sec:Wick}\n\nIn this section we work only with the Minkowski metric, which we\ncontinue to denote by $g$. We now explain Wick rotations in Minkowski\nspace, where it amounts to\nreplacing $\\Box_{g}=D_{z_n}^2-D_{z_1}^2-\\ldots-D_{z_{n-1}}^2$ by\n\\begin{equation}\\label{eq:rotatedbox}\n\\Box_{g,\\theta}=e^{-2\\theta}D_{z_n}^2-D_{z_1}^2-\\ldots-D_{z_{n-1}}^2\n\\end{equation}\nwhere $\\theta$ is a complex parameter. While it may seem that we are\nusing rather sophisticated techniques for a simple operator, this is\nin some ways necessary since we need invertibility on our variable\norder function spaces, which would not be so easy to show using very\nsimple techniques!\n\nConcretely, consider complex\nscaling, corresponding to pull-back by the diffeomorphism\n$\\Phi_\\theta(z)=(z_1,\\ldots,z_{n-1},e^{\\theta}z_n)$ for\n$\\theta\\in\\RR$, i.e.\\ considering\n$U_\\theta^*\\Box(U_\\theta^{-1})^*$, where $U_\\theta = (\\det D\n\\Phi_\\theta)^{1\/2} \\Phi_\\theta^* f$, extending the result to an\nanalytic family of operators in\n$\\theta\\in\\Cx$ (near the reals). This gives rise to the family\n$\\Box_{g,\\theta}$. Letting\n\\begin{equation}\\label{eq:Ldeftheta}\nL_\\theta=\\rho^{-(n-2)\/2}\\rho^{-2}\\Box_{g,\\theta}\\rho^{(n-2)\/2},\n\\end{equation}\nas\nsoon as $\\im\\theta\\in(-\\pi,\\pi)\\setminus\\{0\\}$, $L_\\theta$ \nis a an elliptic b-differential operator; when $\\theta=\\pm i \\pi\/2$,\none obtains the Euclidean\nLaplacian $\\Box_{g, \\pm i \\pi\/2} = \\Delta_{\\RR^n} $. In the elliptic region the corresponding operator $L_\\theta$ satisfies the\nFredholm estimates uniformly for $L_{\\theta,+-}$ (and its adjoint, for which\nthe imaginary part switches sign, but one propagates estimates\nbackwards)\nwhen $\\im\\theta\\geq 0$, and for $L_{\\theta,-+}$ when $\\im\\theta\\leq\n0$. \n\nThe main analytic property that we will use below for the operators\n$L_\\theta$ is that for regularity functions $m$ chosen to satisfy say\nthe forward ($+-$) Feynman condition, the corresponding operators $L_{\\theta,\n +-}$ satisfy estimates\n \\begin{equation}\n \\label{eq:uniformfredholm}\n \\norm[\\Hb^{m, l}]{u} \\le C ( \\norm[\\Hb^{m-1, l} ]{L_{\\theta} u} +\n \\norm[\\Hb^{m', l'} ]{u} ).\n \\end{equation}\n uniformly in $\\theta$ for $m, l$ corresponding to $+-$ and $m'\n < m$, $l' < l$, meaning precisely that there is a constant $C$ such that for $|\\theta| < \\delta_0, \\im\n \\theta \\ge 0$\n for $u \\in \\Hb^{m, l}$,\n \\eqref{eq:uniformfredholm} holds provided $m, l$ satisfy the $+-$\n Feynman condition and $- l \\not \\in \\Lambda$. For $|\\theta| <\n \\delta_0, \\im \\theta \\le 0$ they hold provided $m, l$ satisfy the $-+$\n Feynman condition and $l \\not \\in \\Lambda$. (Note that $\\Lambda = -\n \\Lambda$ so actually the conditions on $l$ are the same.) The\n reason for the uniformity is that all of the ingredients are\n uniform; this is standard for elliptic estimates. On the other hand, it holds for real\n principal type estimates where the imaginary part of the principal\n symbol amounts to complex absorption, provided one propagates\n estimates in the {\\em forward} direction of the Hamilton flow if the\n imaginary part of the principal symbol is $\\leq 0$ (which is the\n case for $\\im\\theta\\geq 0$, $\\theta$ small) and backwards\n along the Hamilton flow if the imaginary part of the principal\n symbol is $\\geq 0$, as shown by Nonnenmacher and\n Zworski \\cite{Nonnenmacher-Zworski:Quantum} and Datchev and Vasy\n \\cite{Datchev-Vasy:Gluing-prop} in the semiclassical microlocal setting\n and, as is directly relevant here, extended to the general b-setting by Hintz and Vasy\n \\cite[Section~2.1.2]{HVsemi}. Moreover, at radial points in the\n standard microlocal setting this was shown by Haber and Vasy\n \\cite{Haber-Vasy:Radial}, and the proof of\n Proposition~\\ref{thm:propofsing} can be easily modified in the same\n manner so that non-real principal symbol is also allowed at the\n b-radial points. Finally, the normal operator constructions are also\n uniform since they rely on estimates for the Mellin transformed\n family which are uniform as we stated; the resonances (poles) of the inverse\n of this family thus a priori vary continuously, so in particular\n near an invertible weight for $\\theta=0$ one has uniform estimates.\n (In fact we will show in Proposition \\ref{thm:boundaryscaling}\n below that the poles of the complex scaled normal families are\n constant, i.e.\\ do not vary with $\\theta$.)\n\nNote that the estimates in \\eqref{eq:uniformfredholm} are \\emph{not}\nthe standard elliptic estimates. Indeed, the term on the left hand\nside is in a space of differentiability order one lower than\nellipticity provides. The point is that the estimates in\n\\eqref{eq:uniformfredholm} are exactly those which are uniform down to\n$\\im \\theta = 0$.\n\nThe family of operators $L_\\theta$ defines a family of Mellin\ntransformed normal operators on the boundary, $\\hat\nN(L_\\theta)(\\sigma)$ as above, and we have, still for $g$ equal to the\nMinkowski metric, that\n\\begin{equation}\n \\label{eq:8}\n \\hat{N}(L_{\\pm i \\pi \/ 2})(\\sigma) = \\Delta_{\\sphere^{n-1}}+(n-2)^2\/4+\\sigma^2.\n\\end{equation}\n\nWe recall the theorem of Melrose describing the behavior of the\nelliptic operators $L_\\theta$ for $\\im \\theta \\neq 0$, which is a\nspecial case of our more general framework in that elliptic operators\nare also Fredholm on variable order Sobolev spaces in view of our results.\n\n\\begin{thm}[Melrose \\cite{Melrose:Atiyah}, with Theorem~\\ref{thm:fredholm} here\n giving the variable order version] \\label{thm:bopfred}\n Let $P$ be an elliptic b-differential operator of order $k$ on a manifold with\n boundary $M$, and assume that $\\hat N(P)^{-1}(\\sigma)$ has no poles on\n the line $\\im \\sigma = -l$. Then the operator $P$ satisfies\n \\begin{equation}\n \\label{eq:Fredholmelliptic}\n \\| u \\|_{\\Hb^{s + k, l}} \\le C(\\| P u \\|_{\\Hb^{s , l}} + \\| u\n \\|_{\\Hb^{- N, l'}}),\n \\end{equation}\nfor any $N > 0$ and some $l' < l$. In particular, \n$$\nP \\colon \\Hb^{s + k, l} \\lra \\Hb^{s, l}\n$$\nis Fredholm.\n\\end{thm}\n\nThus the set $\\Lambda$ in \\eqref{eq:indicialroots} gives the set of\nweights $l$ for which \n$$\n\\Delta_{\\RR^n} \\colon \\Hb^{m + 1, l + (n - 2)\/2} \\lra \\Hb^{m - 1, l +\n (n - 2)\/2 + 2}\n$$\nis Fredholm; indeed by the definition of $L_\\theta$ in \\eqref{eq:Ldeftheta},\nwe see that\n$$\nL_{i\\pi\/2}=\\rho^{-(n-2)\/2}\\rho^{-2}\\Delta_{\\RR^n}\\rho^{(n-2)\/2} \\colon \\Hb^{m + 1, l} \\lra \\Hb^{m - 1, l}\n$$\nis Fredholm exactly when $- l \\not \\in \\Lambda$.\nConsider the elliptic operators $L_{\\theta}$ as maps between forward\nFeynman b-Sobolev spaces\n\\begin{equation}\\label{eq:ellipticmaps}\n L_{\\theta, +-} \\colon \\Hb^{m + 1, l} \\lra \\Hb^{m-1,l}.\n\\end{equation}\nIn Section \\ref{sec:normalfamily} below, we will prove in Proposition\n\\ref{thm:boundaryscaling} that for the\n$\\theta-$dependent family of Mellin transformed normal operators of the\ncomplex scaled Feynman operators,\n$\\hat{N}(L_{\\theta, +-})(\\sigma)$, \\textit{the inverse families have equal\n poles.} Thus the set $\\Lambda$ in \\eqref{eq:indicialroots} is in\nfact the set of all poles of the inverse families $\\hat{N}(L_{\\theta, +-})$ in the forward Feynman setting. The same holds for $-+$.\nAs a corollary to Theorem \\ref{thm:bopfred} and the fact that the\nindex of a continuous family of Fredholm operators is constant (see\n\\cite{TaylorI}), we obtain the following:\n\\begin{lemma}\\label{thm:constantindexelliptic}\nFor $\\Lambda$ as in \\eqref{eq:indicialroots} and $-l \\not \\in\n\\Lambda$, the maps in\n \\eqref{eq:ellipticmaps} form a continuous\n Fredholm family and thus have constant index for $\\theta \\in (0, \\pi\/2]$.\n\\end{lemma}\n\n\n\n \\begin{figure}\\label{scalingspace}\n \\centering\n\\input{scalingspace.pdf_tex}\n \\caption{The index of $L_{\\theta}$ is constant in the grey region\n by the standard continuity of the index for Fredholm families.\n In Theorem \\ref{thm:constantindex} we show the continuity of the\n index extends up to the dashed line at bottom, i.e.\\ to $L_{+-}$.}\n \\end{figure}\n\n\n\\subsection{Index of $L_{\\theta, +-}$}\n\n\n\nTo prove the invertibility theorem, we will first establish the\nfollowing \n\\begin{thm}\\label{thm:constantindex}\nFor fixed $m, l$ satisfying the forward Feynman condition in $L_{+-}$, and such that $l\n\\not \\in \\Lambda$, for $\\mathrm{Im} \\ \\theta \\in (0, \\pi\/2)$,\n \\begin{equation}\n \\label{eq:equalindices}\n \\ind(L_{\\theta} \\colon \\Hb^{m + 1, l} \\lra \\Hb^{m-1,l}) = \\ind(L_{+-}\n \\colon \\cX^{m, l} \\lra \\cY^{m-1,l}).\n \\end{equation}\n\\end{thm}\n\\begin{proof}\n This follows from the mere fact that the estimates in\n \\eqref{eq:uniformfredholm} hold uniformly in $\\theta$ for $m, l$\n corresponding to $+-$ and $m' < m$, $l' < l$.\n\n\n Assume first that $L_{+-}$ on the right hand side of\n \\eqref{eq:equalindices} is invertible. Then one can drop the compact\n error terms, and thus then the estimates take the form\n \\begin{equation}\\label{eq:invest}\n\\|u\\|_{\\Hb^{m,l}}\\leq C\\|Lu\\|_{\\Hb^{m-1,l}},\\ \\|v\\|_{\\Hb^{1-m,-l}}\\leq\nC\\|L^* v\\|_{\\Hb^{-m,-l}},\n\\end{equation}\nwhere again $L^*$ is the adjoint of $L$ with respect to the $\\Lb^2$\npairing (see \\eqref{eq:l2bdef}). To see that the estimate on the\nright follows,\nsince $\\Hb^{-m, -l}$ is dual\nto $\\Hb^{m, l}$ with respect to\nthe $\\Lb^2$ pairing, using the surjectivity of $L$ to go to\nthe second line we have\n \\begin{gather*}\n \\| L^*v \\|_{\\Hb^{-m, -l}} = \\sup_{ \\| w \\|_{\\Hb^{m,l}} = 1} \\langle L^* v,\n w \\rangle_{\\Lb^2} \\ge \\sup_{ \\| w \\|_{\\cX^{m,l}} = 1} \\langle v, L w\n \\rangle_{\\Lb^2} \\\\ \\ge \\frac 1C \\sup_{ \\| g \\|_{\\Hb^{m - 1,l}}} \\langle v, g\n \\rangle_{\\Lb^2} = \\frac 1C \\| v \\|_{\\Hb^{1 - m , -l}}.\n \\end{gather*}\n\n\nWe claim that the estimates in \\eqref{eq:invest} imply the\nanalogous estimates also hold for $L_\\theta$, $\\im\\theta$ small\nwith $\\im \\theta > 0$, namely that\n\\begin{equation}\\label{eq:Lthetainj}\n\\|u\\|_{\\Hb^{m,l}}\\leq C\\|L_\\theta u\\|_{\\Hb^{m-1,l}},\\ \\|v\\|_{\\Hb^{1-m,-l}}\\leq\nC\\|L_\\theta^* v\\|_{\\Hb^{-m,-l}}.\n\\end{equation}\nOtherwise, for example for the first estimate, we would have a\nsequence $\\theta_j \\to 0$ with $\\im \\theta_j > 0$ and $u_j$ with \n$$\n\\|u_j \\|_{\\Hb^{m, l}} = 1 \\mbox{ and } L_{\\theta_j} u_j \\to 0 \\mbox{\n in } \\Hb^{m - 1,\n l}.\n$$\n Extracting a strongly convergent subsequence of the $u_j$ in $\\Hb^{m', l'}$\nfor $m' < m$ and $l' < l$, by the uniform estimates in\n\\eqref{eq:uniformfredholm} we would obtain a limit $\\wt{u}$ with\n$\\wt{u} \\neq 0$ and $L \\wt{u} = 0$, a contradiction. A similar\nargument shows that the second estimate also holds for small $\\theta$\nwith $\\im \\theta > 0$. \n\n\nNow as soon as $\\im\\theta\\neq 0$, these give\nimproved estimates by elliptic regularity, namely\n$$\n\\|u\\|_{\\Hb^{m+1,l}}\\leq C\\|L_\\theta u\\|_{\\Hb^{m-1,l}},\\\n\\|v\\|_{\\Hb^{2-m,-l}}\\leq C\\|L_\\theta^* v\\|_{\\Hb^{-m,-l}}.\n$$\nIndeed these follow since for $\\im \\theta > 0$, $L_\\theta$ and\n$L^*_\\theta$ are Fredholm\nmaps from $\\Hb^{m' + 1, l}$ to $\\Hb^{m' - 1, l}$ for any $m'$ and by\n\\eqref{eq:Lthetainj} are injective for the given $m$ and $l$ and thus\nfor any $m'$ by elliptic regularity.\nThus, for example taking $m = s$ to be constant in the first inequality\nand $m = -s + 1$ in the second inequality gives that $L_\\theta$ is\ninjective and surjective with domain $\\Hb^{m, l}$ (which again by\nelliptic regularity means that $L_\\theta$ is an isomorphism for any\n$m$ and the given $l$). This establishes the theorem in the case that\n$L_{+-}$ is invertible on the spaces under consideration.\n\nIf $L_{+-}$ is not invertible but is Fredholm, one can get back to the\nsame setting by adding finite dimensional function spaces to the\ndomain and target as usual, showing that the index is stable under\nthis deformation. Concretely, let\n\\begin{equation*}\n \\begin{split}\n V &:= \\Ker(L_{+-} \\colon \\cX^{m, l} \\lra \\Hb^{m-1, l}) \\\\\n W &:= \\Coker(L_{+-} \\colon \\cX^{m, l} \\lra \\Hb^{m-1, l}),\n \\end{split}\n\\end{equation*}\nwhere by definition the cokernel in the second line is the orthogonal\ncomplement of the range with respect to some (fixed) inner product. The map $\\wt{L}_\\theta$ from $W \\oplus\n\\cX^{m, l} = W \\oplus V \\oplus V^\\perp $ to $V \\oplus \\Hb^{m\n -1, l} = V \\oplus W \\oplus W^\\perp$ which takes $w + v + v'$ to $v +\nw + L_\\theta v'$ is an isomorphism for $\\theta = 0$, and by the above\nanalysis is also an isomorphism for $\\theta$ small with $\\im \\theta > 0$.\nTherefore the Fredholm index of the Feynman\npropagators for Minkowski space is the same as that of\n$\\Delta_{\\RR^n}$ acting on a weighted b-space with the same\nweight. \\end{proof}\n\n\n\nWe can then use the relative index\ntheorem of Melrose \\cite[Chapter~6]{Melrose:Atiyah}, which expresses\nthe difference in the index of a b-differential operator at different\nweights as the sum of the residues of the normal operator at\nappropriate indicial roots. This can be\nextended from the elliptic setting considered there to ours without\nany difficulties, to compute the index of $L$ on other weighted\nspaces; here in fact because of the Wick rotation we can use the\nelliptic result directly.\n\\begin{cor}\\label{thm:explicitindex}\nUnder assumptions as in Lemma \\ref{thm:constantindexelliptic}, \n \\begin{equation}\n \\label{eq:12}\n \\ind(L_{\\theta, +-} \\colon \\cX^{m, l} \\lra \\cY^{m - 1, l}) =\n - \\sgn(l) N(\\Delta_{\\mathbb{S}^{n -1} } + (n-2)^2\/4 ; l),\n \\end{equation}\nwhere $N(\\Delta_{\\mathbb{S}^{n -1} } + (n-2)^2\/4; l)$ is the number of\neigenvalues $\\lambda$ of $\\Delta_{\\mathbb{S}^{n -1} } + (n-2)^2\/4$\nwith $\\lambda < l^2$. In particular, \n\\begin{equation*}\n|l| < (n - 2)\/2 \\implies \\ind(L_{g,+-} \\colon \\cX^{m, l} \\lra \\cY^{m - 1, l}) = 0.\n\\end{equation*}\n\\end{cor}\n\\begin{proof}\n By Theorem \\ref{thm:constantindex}, we have that\n \\begin{equation*}\n \\begin{split}\n & \\ind(L_{\\theta, +-} \\colon \\cX^{m, l} \\lra \\cY^{m - 1, l}) \\\\\n & \\qquad =\n \\ind(L_{i\\pi\/2} \\colon \\Hb^{m + 1, l} \\lra\n \\Hb^{m - 1, l}) \\\\\n & \\qquad =\n \\ind(\\Delta_{\\RR^{n}} \\colon \\Hb^{m + 1, l + (n - 2)\/2} \\lra\n \\Hb^{m - 1, l + (n - 2)\/2 + 2}),\n \\end{split}\n \\end{equation*}\nand the latter was computed by Melrose, see \\cite[Section\n6.2]{Melrose:Atiyah}, or the interpretation in \\cite[Theorem\n2.1]{GH2008} where it is shown to be exactly the right hand side of \\eqref{eq:12}.\n\\end{proof}\n\n\\subsection{Invertibility of the Feynman problem for $\\Box_{g,\n \\theta}$ down to $\\theta = 0$.}\n\nIt follows from Theorem \\ref{thm:bopfred} and \\eqref{eq:8}, together\nwith the spectral theory of the sphere discussed above, that\n$$\\Delta_{\\RR^n} \\colon \\Hb^{m+1, l + (n-2)\/2} \\lra \\Hb^{m-1, l +\n (n-2)\/2 + 2}$$ is Fredholm as long as $-l \\not \\in \\Lambda$ where\n$\\Lambda$ is defined in \\eqref{eq:indicialroots}. In fact, we have\n\\begin{thm}\\label{thm:laplaceinverse}\nThe map $\\Delta_{\\RR^n} \\colon \\Hb^{m+1, l + (n-2)\/2} \\lra \\Hb^{m-1, l +\n (n-2)\/2 + 2}$ is invertible provided $|l| < (n-2)\/2,$ $m\\in\\CI(\\Sb^*\\overline{\\RR^n})$.\n\\end{thm}\n\n\\begin{proof}\nThis is shown in the proof of \\cite[Lemma 3.2]{CCH2006}, for\n$m\\in\\RR$. Indeed, they show\nusing the maximum principle and elliptic regularity that there can be\nno nullspace of $\\Delta$ in $\\Hb^{m, l}$ for any $l > 0$ (and the same must be\ntrue for the formal adjoint), from which the result follows since the\noperator is Fredholm. Our results give the\ngeneral Fredholm statement for arbitrary\n$m\\in\\CI(S^*\\overline{\\RR^n})$, and elliptic regularity then gives\nthat any element of the kernel is in $\\Hb^{\\infty,l+(n-2)\/2}$, with an\nanalogous statement for the cokernel, and these are trivial in turn by the\nconstant $m$ result.\n\\end{proof}\n\n\nConsider the map\n\\begin{equation}\n \\label{eq:scaledboxmap}\n \\Box_{g, \\theta} \\colon \\cX^{m,(n-2)\/2 + l}_{+-}(\\theta) \\lra\n\\Hb^{m - 1,(n-2)\/2 + l + 2 }\n\\end{equation}\nwhere $\\cX^{m, l}_{+-}(\\theta) = \\{ u \\in\n \\Hb^{m, l} : \\Box_{g, \\theta} u \\in \\Hb^{m - 1, l + 2} \\}$ is a\n $\\theta$-dependent space with the graph norm,\n$$\n\\|u\\|^2_{\\cX^{m,l}_{+,-}(\\theta)}=\\|u\\|^2_{\\Hb^{m,l}}+\\|\\Box_{g, \\theta} u\\|^2_{\\Hb^{m - 1, l + 2}},\n$$\nso by the elliptic\n estimates discussed above, \n$$\n\\cX^{m, l}_{+-}(\\theta) = \\left\\{\n \\begin{array}{ccc}\n \\cX^{m, l + (n - 2)\/2}_{+-} & \\mbox{if} & \\theta \\in \\mathbb{R}\n \\\\\n \\Hb^{m + 1, l + (n - 2)\/2} & \\mbox{if} & \\im \\theta \\in (0, \\pi) \n \\end{array}\n\\right.,\n$$\nwith the equivalence of norms uniform for $\\theta$ in compact\nsubsets of $\\RR\\times(0,\\pi)$.\n(Here the $+-$ is just to remind us that $m + l$ satisfies the\nconditions corresponding to $L_{g, +-}$, although this makes no\ndifference in the elliptic region.) We will now study the set \n$$\n\\mathcal{D}_l = \\{\\theta : \\im(\\theta) \\in [0, \\pi\/2] \\mbox{ and }\n\\Box_{g, \\theta} \\mbox{ mapping as in \\eqref{eq:scaledboxmap} is\n invertible}.\n\\}\n$$\nWe see that for $|l| < (n-2)\/2$, $\\mathcal{D}_l$ contains $i \\pi \/2 $\nand is thus non-empty.\n\\begin{thm}\\label{thm:scalinglemma}\n Let $|l| < (n-2)\/2$. The set $\\mathcal{D}_l$ contains the entire\n closed strip $\\{ \\im \\theta \\in\n [0, \\pi\/2] \\} $. In particular $\\Box_{g, +-}$ mapping as in\n \\eqref{eq:realinvertiblemap} is invertible for $g$\n equal to the Minkowski metric and $|l| < (n - 2)\/2$. \n\\end{thm}\n\n\nWe will prove Theorem \\ref{thm:scalinglemma} by arguing along lines\nsimilar to those in \\cite{MV2004, MV2005}, which in turn follow the\ndevelopment in \\cite{HS1996}. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:scalinglemma}]\n\n\nWe will define a subspace $\\mathcal{A} \\subset L^2 = L^2(\\RR^n)$ of so-called\nanalytic vectors and a family of maps\n\\begin{equation}\n \\label{eq:9}\n U_\\theta \\colon \\mathcal{A} \\lra L^2,\n\\end{equation}\nfor $\\theta$ in an open neighborhood $\\mathcal{D} \\subset \\mathbb{C}$\nof $0$ with the following properties:\n\\begin{enumerate}\n\\item For $\\theta \\in \\mathbb{R}$, $U_\\theta$ is unitary on $L^2$.\n\\item For $f \\in \\mathcal{A}$ and $\\theta \\in\\mathcal{D}$, $$U_\\theta\n \\Box_{g, \\theta_0} U_\\theta^{-1} f = \\Box_{g, \\theta + \\theta_0} f.$$\nIn particular, $U_\\theta$ is injective and $\\mathcal{A}$ is in the\nrange of $U_\\theta$ for $\\theta \\in \\mathcal{D}$.\n\\item $U_\\theta \\mathcal{A}$ is dense in $\\Hb^{m, l}$ for all $\\theta\n \\in \\mathcal{D}$ and any $m \\colon \\Sb^* M \\lra \\mathbb{R}$, $l \\in \\mathbb{R}$.\n\\end{enumerate}\n\nWe will then leverage the properties of $\\mathcal{A}$ and $U_\\theta$\nto prove Theorem \\ref{thm:scalinglemma} as follows. Recall that, by\nTheorem \\ref{thm:constantindex},\n$\\Box_{g, \\theta}$ as in \\eqref{eq:scaledboxmap} is a Fredholm map of\nindex zero.\nSince it is invertible for $\\theta' = i \\pi\/2$, it is invertible for\n$\\theta$ near $\\theta'$. It follows by the analytic Fredholm theorem that \n\\begin{equation}\n \\label{eq:11}\n \\Box_{g, \\theta}^{-1} \\colon \\Hbpm^{m - 1,l} \\lra \\Hbpm^{m + 1,l}\n\\end{equation}\nextends to a meromorphic family of operators in the strip\n$\\{0 < \\im \\theta < \\pi \\}$ with finite rank poles. In particular, if\n$\\wt{\\theta}$ is a putative pole, then for $\\theta$ near $0$,\n\\begin{equation}\n \\label{eq:16}\n \\Box^{-1}_{g, \\wt{\\theta} + \\theta} = \\sum_{j =-N}^{-1} A_j \\theta^j + M_\\theta, \\mbox{ where } M_\\theta \\mbox{ is holomorphic.}\n\\end{equation}\nThus if $\\wt{\\theta}$ is indeed a pole, by the density of\n$\\mathcal{A}$ we may choose $f, h$ such that, e.g.\\ $\\langle f, A_1 h\\rangle\n\\neq 0$ and thus $\\langle f, \\Box^{-1}_{g,\n \\wt{\\theta} + \\theta} h\\rangle$ has a pole at $\\theta = \\wt{\\theta}$.\nOn the other hand the matrix elements satisfy\n\\begin{equation}\\label{eq:crux}\n\\langle f, \\Box^{-1}_{g,\\wt{\\theta} + \\theta} \\notg \\rangle_{L^2} = \\langle U_{\\overline{\\theta}} f ,\n\\Box^{-1}_{g, \\wt{\\theta}} U^{-1}_{\\theta} \\notg\\rangle.\n\\end{equation}\nWe will see that for $h \\in \\mathcal{A}$, both $U_\\theta h$ and\n$U_\\theta^{-1} h$ are analytic for $\\theta \\in \\mathcal{D}$, so the\nmatrix elements of $\\Box_{g,\\wt{\\theta} + \\theta}^{-1}$ are analytic\nfunctions for\n$\\theta \\in \\mathcal{D}$, and thus\n$\\Box_{g,\\wt{\\theta} + \\theta}^{-1}$ has no poles in $0 < \\im \\theta \\le\n\\pi\/2.$\n\nWe have proven that $\\Box_{g, \\theta}$ is invertible only for those $\\theta$ with $0 < \\im\n\\theta < \\pi$. Since $\\Box_{g,\n \\theta}$ is not strictly speaking an analytic Fredholm family on an open set\ncontaining $\\theta = 0$ (the domain changes according to whether\n$\\im \\theta = 0$ or not and $\\theta = 0$ lies on the boundary of $0\n\\le \\im \\theta \\le \\pi \/2$) we need a different argument there. Pick\n$\\theta_0$ close enough to $0$ so that the density statements\nfor $U_\\theta \\mathcal{A}$ hold on an open set including $\\theta= - \\theta_0$. \nAssuming for contradiction that $\\Box_{g, +-}$ is not invertible for\n$l$ in the given range, it will suffice to construct\nelements $f, \\notg \\in \\mathcal{A}$ such that \n\\begin{equation}\\label{eq:moredivergence}\n\\langle f, \\Box^{-1}_{g, \\theta_0 +\n \\theta} \\notg\\rangle_{L^2} \\mbox{ diverges as } \\theta_0 + \\theta \\to 0 \\mbox{\n in } \\im(\\theta\n+ \\theta_0) > 0,\n\\end{equation}\nsince then by\n\\eqref{eq:crux} with $\\wt{\\theta} = \\theta_0$ we will have a contradiction.\nNote that by this assumption there exists $u_0$ lying in $\\Hbpm^{m - 1, l +\n (n-2)\/2 + 2}$ such that\n$$\nu_0 \\not \\in \\Ran(\\Box_{g,+-} \\colon \\cX_{+-}^{m,\n l + (n-2)\/2} \\lra \\Hb^{m - 1, l + (n-2)\/2 + 2})\n$$ \nsince by Theorem \\ref{thm:constantindex} the map has index zero. Since\n\\eqref{eq:scaledboxmap} is\nFredholm and $\\mathcal{A}$ is dense, we may instead choose $\\notg \\in\n\\mathcal{A}$ such that also $\\notg \\not \\in \\Ran(\\Box_{g,\n 0})$. Using the invertibility proved above, for $\\im (\\theta_0 +\n\\theta) > 0$, we consider\n$\\Box^{-1}_{g, \\theta_0 + \\theta} \\notg \\in \\Hb^{m + 1, l} \\subset\n\\cX^{m, l+(n-2)\/2}_b$. We claim that \n$$\n\\| \\Box^{-1}_{g, \\theta_0 + \\theta} \\notg \\|_{\\cX^{m, l + (n-2)\/2}_{+-}}\n\\mbox{ diverges as } \\theta + \\theta_0 \\to 0\n$$ \nIndeed, otherwise $\\Box^{-1}_{g, \\theta_0 + \\theta} \\notg$ converges subsequentially\nto some $u \\in \\cX^{m, l}_b$ weakly, and by a standard\nargument we must have $L_0 u = \\notg$, which is impossible by\nassumption.\n\nNote that this does not guarantee that\n\\eqref{eq:moredivergence} holds for any $f \\in \\mathcal{A}$; this\nrequires a further argument. To see this, we use the uniform Fredholm\nestimates in \\eqref{eq:uniformfredholm}, which in terms of $\\Box_{g,\n \\theta_0 + \\theta}$ and applied to $\\Box_{g,\n \\theta_0 + \\theta}^{-1} \\notg$ take the form\n \\begin{equation}\n \\label{eq:uniformfredholmbox}\n \\begin{split}\n \\|\\Box_{g, \\theta_0 +\n \\theta}^{-1} \\notg \\|_{\\Hb^{m, l + (n- 2)\/2}} &\\le C ( \\norm[\\Hb^{m-1, l + (n - 2)\/2 +\n 2} ]{\\notg} \\\\\n &\\qquad + \\|\\Box_{g, \\theta_0 + \\theta}^{-1} \\notg \\|_{\\Hb^{m', l' + (n - 2)\/2 } } ),\n \\end{split}\n \\end{equation}\nwhere $m' < m$ and $l' < l$. Letting $\\theta_j$ be a sequence with\n$\\theta_j \\to -\\theta_0$, let $c_j = \\|\\Box_{g, \\theta_0 +\n \\theta_j}^{-1} \\notg \\|_{\\Hb^{m, l + (n- 2)\/2}} $, and\n let \n$$\nu_j = c_j^{-1}\\Box_{g, \\theta_0 +\n \\theta_j}^{-1} \\notg.\n$$ \n By \\eqref{eq:uniformfredholmbox} and\n the compact containment of $\\Hb^{s, \\ell} \\subset \\Hb^{s',\n \\ell'}$ when $s' < s$ and $\\ell' < \\ell$,\n the $u_j$ converge subsequentially (dropped from the notation) to a non-zero element $u \\in \\Hb^{m, l +\n (n- 2)\/2}$. It follows that \n $$\n\\langle \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg, u\n \\rangle_{\\Hb^{m, l +\n (n- 2)\/2}} = c_j(1 + o(1)),\n$$\nwhere $o(1) \\to 0$ as $j \\to \\infty$.\nWe claim that there is a $\\delta_0 > 0$ such that for any $\\wt{f}$ with\n $\\|\\wt{f} - u \\|_{\\Hb^{m, l + (n- 2)\/2}} < \\delta_0$,\nthat $\\langle \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg, \\tilde f\n \\rangle_{\\Hb^{m, l +\n (n- 2)\/2}}$ is also divergent.\n Indeed, \n \\begin{equation}\\label{eq:closetou}\n \\begin{split}\n \\langle \\wt{f}, \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg\\rangle_{\\Hb^{m, l +\n (n- 2)\/2}} &= \\langle u, \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg\n \\rangle_{\\Hb^{m, l +\n (n- 2)\/2}} \\\\\n&\\qquad + \\langle \\wt{f} - u, \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg\n \\rangle_{\\Hb^{m, l +\n (n- 2)\/2}} \\\\\n &\\ge c_j(1 + o(1)) - C c_j \\delta_0 \\ge \\frac 12 c_j,\n \\end{split}\n \\end{equation}\nfor $C \\delta_0 < 1\/3$ and $j$ large. This is not exactly the desired\ndivergence in \\eqref{eq:moredivergence} since the inner product is not\n$L^2$. Define\n\\begin{equation}\n \\label{eq:japanese}\n \\langle z \\rangle = (z_1^2 + \\dots + z_n^2 + 1)^{1\/2},\n\\end{equation}\nand let $P \\in \\Psib^{m}(\\mathbb{R}^n)$ be elliptic and\nself-adjoint. Then (since by the paragraph below \\eqref{eq:l2bdef} we have\n$\\Hb^{0, n\/2} = L^2(\\RR^n)$)\n$$\n\\langle z \\rangle^{l - 1} (P + i) \\colon \\Hb^{m, l + (n-2)\/2} \\lra L^2(\\RR^n).\n$$\nand we may take the $\\Hb^{m, l + (n - 2)\/2}$ inner product to be \n$$\n\\langle u, v \\rangle_{\\Hb^{m, l + (n - 2)\/2}} = \\langle \\langle z \\rangle^{l - 1} (P + i) u, \\langle z \\rangle^{l - 1} (P + i) v\\rangle_{L^2}.\n$$ \nUsing the density of $\\mathcal{A}$ in all weighted b-Sobolev spaces,\nwe choose $$\\wt{f} = (P - i)^{-1}\\langle z \\rangle^{-2l + 2} (P + i)^{-1} f$$\nfor some $f \\in \\mathcal{A}$ such that $\\wt{f}$ within $\\delta_0$ of\n$u$ in $\\Hb^{m, l + (n - 2)\/2}$. Thus\n$$\n \\langle \\wt{f}, \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg\\rangle_{\\Hb^{m, l +\n (n- 2)\/2}} = \\langle f, \\Box_{g, \\theta_0 + \\theta_j}^{-1} \\notg\\rangle_{L^2},\n$$\nand \\eqref{eq:moredivergence} is established, which means that up to\nthe construction of $\\cA$ and $U_\\theta$ and showing that the\nproperties claimed for them hold, the proof is complete.\n\n\n\n\n\n\nIt remains to define $\\mathcal{A}$ and $U_\\theta$ and prove that they\nhave the properties i)-iii) stated above. Following \\cite{MV2004}, we define $\\mathcal{A}$ to be the space of $f\n\\in C^\\infty(\\RR^{n-1} \\times \\RR)$ such that, writing $z = (z'',\nz_n)$ with $z'' \\in \\RR^{n-1}$, we have that $f(z'', z_n)$ is the\nrestriction to $\\zeta \\in \\RR$ of an entire function $f(z'', \\zeta)$\nwhich satisfies\n\\begin{equation}\\label{eq:A1}\n \\sup_{|\\re{\\zeta}| < C |\\im{\\zeta}|} |f(z'', \\zeta)| \\langle \\zeta\n \\rangle^{N} < + \\infty,\n\\end{equation}\nfor any $C, N > 0$ where $\\langle \\zeta \\rangle = (1 + |\\zeta|^2)^{1\/2}$, and\nalso assume that\n\\begin{equation}\n \\label{eq:A2}\n \\supp f(z'', \\zeta) \\subset K\\times \\mathbb{C},\n\\end{equation}\nwhere $K \\subset \\RR^{n-1}$ is compact.\nFinally, for $f \\in \\mathcal{A}$ let\n\\begin{equation}\n \\label{eq:uthetadef}\n U_\\theta (f)(z'', z_n) : = e^{\\theta}f(z'',e^{\\theta} z_n).\n\\end{equation}\nBy the proof of \\cite[Proposition 3.6]{MV2004}, for $|\\im \\theta| < \\pi\n\/ 4$, $U_\\theta\n\\mathcal{A}$ is dense in $L^2 = L^2(\\RR^{n}, |dz|),$ where $|dz|$ denotes\nLebesgue measure. Indeed, given $f \\in C^\\infty_c(\\RR^{n})$, let \n\\begin{equation*}\n f_t(z'', z_n) := \\frac{1}{(4 \\pi t)^{1\/2}}\\int_\\RR\n e^{-(z_n -e^{\\theta} y)^2\/4t} e^{\\theta} f(z'', y) dy.\n\\end{equation*}\nThen the reference shows that $f_t \\in \\mathcal{A}$ and $U_\\theta f_t\n\\to f$ in $L^2$ as $t \\to \\infty$. Thus $U_\\theta \\mathcal{A}$ is\ndense in $L^2 = \\Hb^{0, n\/2}$. To see that $U_\\theta \\mathcal{A}$\nis dense in $\\Hb^{M, L}$, for any $f \\in C^\\infty_c$ take a sequence\n$U_\\theta \\wt{f}_i$ with \n\\begin{equation*}\n U_\\theta \\wt{f}_i \\to (e^{2\\theta} z_n^2 + |z''|^2 + i)^{-L - 2M + n\/2}\n (\\Box_{i\\pi\/2 + \\theta}+ i)^{M} f\n \\in \\Hb^{0, n\/2}, \n\\end{equation*}\nand set $F_i := (e^{2\\theta} z_n^2 + |z''|^2 + i)^{L + 2M - n\/2}\n (\\Box_{i\\pi\/2 + \\theta}+ i)^{-M}\n\\wt{f}_i$. Then in fact $F_i = U_\\theta f_i$, where $f_i = (z_n^2 +\n|z''|^2 + i)^{L + 2M - n\/2}\n (\\Delta+ i)^{-M} \\wt{f}_i$ where again $\\Delta = \\Box_{i\\pi\/2}$ is\n the Laplacian on $\\RR^n$. Since $F_i = U_\\theta f_i \\to f$ in $H^{M,\n L}_b$, and since $\\Hb^{M, L}$ is dense in $\\Hb^{m, l}$ provided $M\n \\ge L$ and $L \\ge l$, the desired density is established. \n\n\n\\end{proof}\n\n\\subsection{Complex scaling for $N(L_\\theta)$}\\label{sec:normalfamily}\n\nIn this section we will apply another complex scaling to the normal\noperators corresponding to the $L_\\theta$. Namely, let $m, l$ be\nchosen for the forward Feynman problem $L_{+-}$, and consider\n the operators $L_{\\theta, +-}$ defined in \\eqref{eq:Ldeftheta}. Let\n $H^m(\\p M)$ denote the variable order Sobolev spaces obtained by\n restricting $m$ to $T^* \\p M$ as described above. Consider the\n operators\n \\begin{equation}\n \\label{eq:normaltheta}\n \\hat{N}(L_{\\theta, +-})(\\sigma)\\colon \\cX^{m}_{\\theta}(\\p M) \\lra H^{m-1}(\\p M),\n \\end{equation}\nwhere $\\cX^{m}_{\\theta}(\\p M) = \\{ u \\in H^{m}(\\p M) :\n\\hat{N}(L_\\theta)(\\sigma) u \\in H^{m - 1}(\\p M) \\}$.\n\\begin{prop}\\label{thm:boundaryscaling}\nThe poles of the inverse family $\\hat{N}(L_{\\theta, +-})(\\sigma)^{-1}$ are\nindependent of $\\theta$ for\n$\\im \\theta \\in [0, \\pi \/ 2]$.\n\\end{prop}\n\\begin{proof}\n As in the previous section, we wish to define a set of analytic\n vectors $\\wt{\\mathcal{A}} \\subset L^2(\\p M)$, and a family of maps\n $\\wt{U}_\\theta \\colon \\wt{\\mathcal{A}} \\lra L^2(\\p M)$ defined for\n $\\theta$ in an open set which we also call $\\mathcal{D} \\subset\n \\mathbb{C}$, and such that conditions i), ii), and iii) below\n \\eqref{eq:9} above hold. The Proposition then follows exactly as in\n the proof of Theorem \\ref{thm:scalinglemma} above.\n\n\n Consider homogeneous degree zero functions on $\\mathbb{R}^n$ of the\n form\n \\begin{equation}\n \\label{eq:homozero}\n F = \\frac{p_l(z_1, \\dots, z_n)}{(z_1^2 + z_2^2 + \\dots + e^{2\\omega}z_n^2)^{l\/2}} ,\n \\end{equation}\n where $p_l$ is a homogeneous polynomial of degree $l$ and $\\omega\n \\in \\mathbb{C}$ with $|\\im(\\omega)| < \\pi\/4$ .\n \\begin{equation}\n \\label{eq:analboundary}\n \\wt{\\mathcal{A}} = \\set{\n f \\in C^\\infty(\\p M) : f = \\sum_{i = 1}^k F_i \\rvert_{\\absv{z} = 1}\n },\n \\end{equation}\n or in words, $\\wt{\\mathcal{A}}$ consists of all finite sums of\n restrictions of homogeneous degree zero functions as in\n \\eqref{eq:homozero} to the sphere. Note that $\\wt{\\mathcal{A}}$ is\n dense in every Sobolev space; indeed, $\\wt{\\mathcal{A}}$ contains the\n spherical harmonics, which are restrictions to the sphere of\n harmonic polynomials, and which form a basis of every Sobolev space\n by Fourier series \\cite{taylor:vol2}.\n For $\\theta \\in \\mathbb{R}$, we define\n \\begin{equation}\n \\label{eq:13}\n V_\\theta F = (\\det D \\Phi_\\theta)^{1\/2} \\Phi_\\theta^* F,\n \\end{equation}\n with $\\Phi_\\theta$ as above, i.e.\\ $\\Phi_\\theta(z_1, \\dots, z_{n -\n 1}, z_n) = (z_1, \\dots, z_{n - 1}, e^\\theta z_n)$, and thereby\n define, for $f = \\sum_{i = 1}^k F_i \\rvert_{\\absv{z} = 1}$,\n \\begin{equation}\n \\label{eq:14}\n \\wt{U}_\\theta f = \\sum_{i = 1}^k (V_\\theta F_i) \\rvert_{\\absv{z} = 1}.\n \\end{equation}\nTo see that condition iii) holds, note that given $0 < \\delta < \\pi\/4$ if we define \n$\\wt{\\cA}_{\\delta} \\subset \\wt{\\cA}$ to be those elements where the\nfunctions $F_i$ in the definition of $\\wt{\\cA}$ have parameter $\\omega$\n(in \\eqref{eq:homozero}) with $|\\omega| \\le \\delta$, then\n$\\wt{\\cA}_\\delta$ remains dense in $L^2$.\nsatisfying $|\\omega| < \\delta$, then $\\wt{\\cA}_\\delta$ remains dense\nsince it still contains the spherical harmonics. On the other hand,\nfor $|\\im \\theta| < \\pi \/ 4 $ and $\\delta$ sufficiently small\n$\\wt{\\cA}_\\delta \\subset \\wt{U}_\\theta \\wt{\\cA}$, so the density\nresult holds also for $\\wt{U}_\\theta \\wt{\\cA}$. \n\nProperty ii) in the complex scaling process in this context simply\nsays that for $f \\in \\wt{\\mathcal{A}}$ and $\\theta \\in\\mathcal{D}$, $$\\wt{U}_\\theta\n N(L_{g, \\theta_0}) \\wt{U}_\\theta^{-1} f = N(L_{g, \\theta + \\theta_0})\n f,$$ which follows directly from the corresponding statement for\n $\\Box_{g}$. Thus all three properties hold and the argument proceed\n exactly as in the previous section.\n\n\\end{proof}\n\n\\section{Module regularity and semilinear problems}\\label{sec:semi}\n\nDespite being the {\\em simplest} natural spaces between which the operator $\\Box_g$\nextends to a Fredholm map, the $\\cX_{\\pm\\pm}^{m, l}$ and\n$\\cY_{\\pm\\pm}^{m, l}$ have the following critical\nshortcoming when one wants to analyze semilinear equations. If one wishes for example to do a\nPicard iteration using the inverse of $\\Box_g$ on some space (as we do\nbelow e.g.\\ in the proof of Proposition \\ref{thm:semilinear})\ninvertibility restricts the range of weights $l$. Furthermore,\nnegative weights $l$ are dangerous when the non-linear term involves\npositive powers, as these will decrease the weight.\nA lower bound\non $l$ produces an upper bound on $m$ in the low regularity region $m +\nl < 1\/2$: if for example one wants to take $l > 0$ then $m < 1\/2$.\nThis is exactly the rub: distributions with Sobolev regularity less\nthan $1\/2$ \\textit{cannot} be multiplied together in general, and thus\na good weight forces bad regularity from the perspective of semilinear\nequations. In this section we overcome this problem by adding module\nderivatives to the $\\Hb^{m, l}$ spaces, as we describe now, using\nthese in the end to prove Theorems \\ref{thm:semilinear} and\n\\ref{thm:semilinearthree} below. We remark that our use of these\nspaces is related to the use of the infinitesimal generators of the\nLorentz group by Klainerman \\cite{Klainerman:Uniform} though we work\nin a purely $L^2$-based setting while Klainerman proceeds in a $L^\\infty\nL^2$ setting, and more crucially our module regularity is necessarily\npseudodifferential. As mentioned earlier, results analogous to ours\nfor the Cauchy problem were obtained by Hintz and Vasy\n\\cite{HVsemi}; in that case one can use module\ngenerators that are vector fields, and thus one has\nin a certain sense a geometric generalization of Klainerman's\nvector fields. For the Feynman propagators we then further generalize\nthis to a microlocal setting.\n\n\nElaborating on Proposition \\ref{thm:propofsing}, one can also have a version between spaces\nwith additional module regularity, much as in\n\\cite[Section~5]{HVsemi}. \nThe module regularity is with respect to pseudodifferential\noperators characteristic\non the halves of the conormal bundles of $S_{\\pm}$ toward which we\npropagate regularity, e.g.\\ for $L_{+-}$ they are characteristic on\n$\\SNb^*_+S_+$ and $\\SNb^*_+S_-$.\nConcretely, consider the $\\Psib^{0}$-module $\\cM_{\\pm\\pm}$ in\n$\\Psib^{1}$ consisting of b-pseudodifferential operators $A$ whose\nb-principal symbols $\\sigma_{b,1}(A)$ vanish\n on the components of\n$\\SNb^*_\\pm S_\\pm$ at which the domain $L_{\\pm \\pm}$ has \\emph{low}\nregularity.\nThus, elements in\n\\begin{gather*}\n \\mathcal{M}_{+-} \\mbox{ are characteristic at } \\SNb^*_+S_+ \\cup\n \\SNb^*_+S_-, \\mbox{ and in } \\\\\n \\mathcal{M}_{++} \\mbox{ are characteristic at } \\SNb^*S_+,\n\\end{gather*}\nmeaning their principal symbols vanish on the indicated sets.\nFor an integer $k$ we consider spaces\n\\begin{equation}\n \\label{eq:definitionmodulespaces}\n \\Hbpmpm^{m, l,\n k} := \\{ u \\in \\Hb^{m,l} : \\cM_{\\pm\\pm}^ku \\subset \\Hb^{m,l} \\}.\n\\end{equation}\nThe $\\Hbpp^{m,l,k}$ (and the $--$\nwhose analysis is essentially identical to the $\\Hbpp^{m,l,k}$) thus\nhave module regularity defined by $\\cM_{++}$, which consists of first\norder b-pseudodifferential operators that are characteristic on the\nb-conormal bundle of $S_{+}$ and are allowed to be b-elliptic at\n$S_{-}$. Thus $\\cM_{++}$ admits differential local generators in the following\nsense; let $\\mathcal{V}_{++}$ denote the $\\CI(M)$ module of vector fields $V$\nwhich in the coordinates $\\rho, v, y$ satisfy the two conditions: i) in a neighborhood\nof $S_{-}$, $V$ is in the $\\CI(M)$ span of $\\rho \\p_{\\rho}, \\p_{v},\n\\p_{y}$, i.e.\\ $V$ is locally a b-vector field there, and ii) near\n$S_{+}$, $V$ is in the $\\CI(M)$ span of $\\rho \\p_{\\rho}, \\rho \\p_{v}, v\n\\p_{v}, \\p_{y}$, i.e.\\ $V$ is tangent to $S_{+}$. Then $\\cM_{++}^j u\n\\subset \\Hb^{m,l} \\iff (\\forall i\\leq j)\\ \\mathcal{V}_{++}^i u\n\\subset \\Hb^{m,l}$, and thus membership of a distribution in\n$\\cM_{++}$ can be checked by applying differential and not\npseudodifferential operators. The $\\Hbpp^{m,l,k}$ were\nstudied in \\cite[Section~5]{HVsemi}. Note that if we\nlocalize near $S_{-}$, since elements of $\\cM_{++}$ are not required\nto be characteristic on $SN^{*}S_{-}$, we have full b-regularity\nto order $m + k$ there, which is to say that if $\\chi$ is a cutoff\nfunction supported away from a neighborhood of $S_{+}$, then for $u\n\\in \\Hbpp^{m,l,k}$, $\\chi u \\in \\Hb^{m + k, l}$.\n\n\n\nWe have the following regularity result which says that if the right\nhand side of $L_{\\pm \\pm}u = f$ has module regularity then the solution $u$ has\nthe appropriate corresponding module regularity. As explained\nbelow \\cite[Theorem 5.3]{HVsemi}, the following is a consequence of the\nextension of \\cite[Proposition 4.4]{Baskin-Vasy-Wunsch:Radiation} obtained in\n\\cite[Theorem 6.3]{Haber-Vasy:Radial} in the interior case (i.e.\\ with no\n``b''). (Recall that $\\cY^{m-1,l} = \\Hb^{m - 1, l}$ and that $\\cX^{m,\n l} \\subset \\Hb^{m, l}$.)\n\n\\begin{thm}\\label{thm:modulereginverse}(See \\cite[Theorem\n 5.4]{HVsemi} for the retarded\/advanced propagators; the proof is\n identical in the present general case.)\n Let $g$ be a perturbation of the Minkowski metric in the sense of\n Lorentzian scattering metrics (see Section~\\ref{sec:geometry}). Let $m \\colon \\Sb^*M \\lra \\mathbb{R}$,\n $l \\in \\mathbb{R}$ satisfy the assumptions of Theorem~\\ref{thm:fredholm},\n corresponding to a\n particular choice of $\\pm \\pm$, and let $k \\in \\mathbb{N}_{0}$, and\n assume that $L_{\\pm\\pm}^{-1}:\\cY^{m-1,l}\\to\\cX^{m,l}$ exists\n (automatic if $|l|<\\frac{n-2}{2}$ and the perturbation is small by Theorem~\\ref{thm:invertibility}).\n Then $L_{\\pm\\pm}^{-1}$ restricts to a bounded map\n $$L^{-1}_{\\pm \\pm} \\colon \\Hbpmpm^{m - 1,l,k} \\lra \\Hbpmpm^{m, l, k}.$$\n\\end{thm}\n\n\nThus, $\\Hbpm^{m,l,k}$ is the subspace of $\\Hb^{m,l}$ consisting of $u$ such that with $\\cM_{+-}$\ndenoting 1st order b-pseudodifferential operators characteristic on\n$\\SNb^*_+S_+$ and $\\SNb^*_+S_-$, $\\cM_{+-}^k u\\in\\Hb^{m,l}$. Since\nelements in $\\cM_{+-}$ can be elliptic wherever they are not required\nto be characteristic, elements in $\\Hb^{m, l, k}$ must have $m+k$\nb-derivatives at the other halves of the\nconormal bundles, i.e.\\ at $\\SNb^*_-S_+$ and $\\SNb^*_-S_-$. Also notice\nthat $m+l>1\/2$ at the radial sets from which we propagate estimates\nimplies that $m+l+k>1\/2$ for $k\\in\\Nat$, so the requirements for the\npropagation estimates are satisfied there; for the radial sets towards\nwhich we propagate the estimates we still need, and have, $m+l<1\/2$ as\nthe module derivatives are `free' in the sense that one can take $k$\narbitrarily high without sacrificing invertibility. (One could\nalso use a different normalization, so there are no $k$ additional\nderivatives present at the other halves, but one has to be careful\nthen to make the total weight function behave appropriately; for the\npresent normalization the previous assumptions on $m$ are the\nappropriate ones.)\n\nThese observations also yield the following\nimprovement of the theorem:\n\n\\begin{thm}\\label{thm:modulereginverse-better}\n Let $g$ be a perturbation of the Minkowski metric in the sense of\n Lorentzian scattering metrics (see Section~\\ref{sec:geometry}). Let $m \\colon \\Sb^*M \\lra \\mathbb{R}$,\n $l \\in \\mathbb{R}$, $k \\in \\mathbb{N}_{0}$ satisfy the following weakening of the assumptions of Theorem~\\ref{thm:fredholm}\n corresponding to a\n particular choice of $\\pm \\pm$: when $m+l-3\/2>0$ is assumed in\n Theorem~\\ref{thm:fredholm}, assume merely $m+l+k-3\/2>0$.\n Assume that $L_{\\pm\\pm}^{-1}:\\Hbpmpm^{m-1,l}\\to\\Hbpmpm^{m,l}$ exists\n (automatic if $|l|<\\frac{n-2}{2}$ and the perturbation is small by Theorem~\\ref{thm:invertibility}).\n Then $L_{\\pm\\pm}^{-1}$ restricts to a bounded map\n $$L^{-1}_{\\pm \\pm} \\colon \\Hbpmpm^{m - 1,l,k} \\lra \\Hbpmpm^{m, l,\n k}.$$\n\\end{thm}\nThe assumptions on the regularity function in the theorem thus\ntranslate to\n\\begin{equation}\n \\label{eq:full-reg-list-2}\n\\begin{tabular}{|c||c|c|} \\hline \nRegion & Feynman & Retarded \\\\ \\hline\\hline\n$\\SNb^*_+S_+$ & \n\t$m + l < 1\/2$ & \n$m + l < 1\/2$ \\\\ \\hline\n$\\SNb^*_-S_+$ & \n\t$m + l + k > 3\/2$ &\n$m + l < 1\/2$ \\\\ \\hline\n$\\SNb^*_+S_-$ &\n\t$m + l < 1\/2$ &\n$m + l + k> 3\/2$ \\\\ \\hline\n$\\SNb^*_-S_-$ &\n\t$m + l + k > 3\/2$ & $m + l + k > 3\/2$ \\\\ \\hline\n\\end{tabular},\n\\end{equation}\nwith analogous conditions for the anti-Feynman and advanced propagators.\n\n\\begin{rem}\nNote that if $\\tilde m$ is a function on $\\Sb^*M$ such that $m=\\tilde\nm$ near the radial sets where the regularity is low and $\\tilde\nm=m+k$ where the regularity is high, $\\tilde m$ monotone along the\nbicharacteristics, then $\\Hbpmpm^{m, l, k}\\subset \\Hbpmpm^{\\tilde\n m,l}$, explaining the sense of `restricts' for $L_{\\pm\\pm}^{-1}$.\n\\end{rem}\n\n\\begin{rem}\\label{rem:constant-order}\nA very useful consequence of the theorem is that given any $l$ one may take {\\em\n constant} $m$ with $m+l<1\/2$ and $k$ such that $m+k+l>3\/2$, i.e.\\\none does not need variable order spaces after all. However, note that\nthe variable order spaces are extremely useful in obtaining this\nconclusion: otherwise at the minimum one would have to do estimates on\nthe dual spaces of spaces with module regularity for the adjoint\noperator; these dual spaces are more difficult to work with.\n\\end{rem}\n\n\\begin{proof}\nWe simply note that with $\\tilde m$ as remarked after the statement of\nthe proposition, for $f\\in \\Hbpmpm^{m - 1,l,k}$, $u=L_{\\pm\\pm}^{-1}f\\in\n\\Hb^{\\tilde m,l}$ and microlocally away from the low regularity\nregions in fact $u$ is in $\\Hb^{m+k,l}$ (as $f$ is in $\\Hb^{m+k-1,l}$\nand $m+k+l>1\/2$ there)\nby the standard high regularity radial point statement, Proposition~\\ref{thm:propofsing}, and the propagation of\nsingularities. On the other hand, at the low regularity radial sets,\nthe extension of \\cite[Proposition 4.4]{Baskin-Vasy-Wunsch:Radiation} as\nexplained\nbelow \\cite[Theorem 5.3]{HVsemi} applies, giving that\n$u$is in $\\Hb^{m,l,k}$ microlocally since $f$ is in $\\Hb^{m-1,l,k}$\nmicrolocally, $u$ is in $\\Hb^{m+k,l}$ in a punctured neighborhood of\nthese radial sets, and $m+l<1\/2$. All these membership statements come\nwith estimates, proving the theorem.\n\\end{proof}\n\nOne reason one may want to develop this is to solve nonlinear\nequations, as we do in Section \\ref{sec:semilinear}. To this end we will be forced to restrict the class of\nregularity functions $m$ we consider in the spaces $\\Hb^{m, l, k}$ so\nthat we can keep track of the wavefront sets of products of \ndistributions therein. Specifically, we will assume that, writing $m_+(x) = \\max_{\\Sb^*_xM}\nm$, that\n\\begin{equation}\\begin{aligned}\\label{eq:reg-function-condition}\n(\\forall x) (\\forall s 1\/2$ and $k\n\\in \\Nat$ satisfies $k > (n - 1)\/2$. Then\n\\begin{equation}\\label{eq:lossabovehalf1}\n\\Hbpm^{m,l_1,k}\\Hbpm^{m,l_2,k}\\subset \\Hbpm^{m - 0,l_1+l_2,k},\n\\end{equation}\nwhere $\\Hbpm^{m - 0,l_1+l_2,k} = \\cap_{\\epsilon > 0} \\Hbpm^{m -\n \\epsilon,l_1+l_2,k}$. Thus for any $\\epsilon > 0$ and $m, k$\nsatisfying the stated conditions, there is a constant $C = C(\\epsilon,\nm, k)$ such that\n$$\n\\| u v \\|_{\\Hbpm^{m - \\epsilon,l_1+l_2,k}} \\le C \\| u \\|_{\\Hbpm^{m,l_1,k}}\n\\| v \\|_{\\Hbpm^{m,l_2,k}}.\n$$\n\nFurthermore, if $m>1\/2$ is constant (and thus all assumption on it are\nsatisfied), one can take $\\ep=0$, and drop $-0$ in \\eqref{eq:lossabovehalf1}.\n\\end{prop}\n\nThe proof of Proposition \\ref{thm:algebra} comes at the end of Section\n\\ref{sec:balgebraprops} below. The condition that $m > 1\/2 $ can (and\nwill) be relaxed in Section \\ref{sec:semilinear}, but for the moment\nwe use it to simplify arguments below.\n\nTo use this proposition for the semilinear Feynman problems, we will\nneed to apply it to the spaces $\\Hbpm$. For any $l<0$, one can find\na constant $m>1\/2$ such that $m+l<1\/2$ and a positive integer $k>(n-1)\/2$ such\nthat $m+k+l>3\/2$ so by\nRemark~\\ref{rem:constant-order} the Feynman propagator is applicable\nand Proposition~\\ref{thm:algebra} is also applicable so that\n\\begin{equation}\\label{eq:reallossabovehalf1-const}\n\\Hbpm^{m,l_1,k}\\Hbpm^{m,l_2,k}\\subset \\Hbpm^{m,l_1+l_2,k}.\n\\end{equation}\nIt is\ninteresting to note that in fact the stronger requirements of\nTheorem~\\ref{thm:modulereginverse} can also be arranged:\n\n\\begin{cor}\\label{thm:algebracor}\nFor every weight $\\ell < 0$, there exists a function $m\n\\colon \\Sb^* M \\lra \\RR$ such that: 1) $m > 1\/2$, 2) $m, \\ell$\nsatisfy the same assumptions as the forward Feynman condition in the\nTheorem~\\ref{thm:fredholm}, concretely those in \\eqref{eq:full-reg-list}, and 3) $m$ satisfies the\nproperty on the sublevel sets and minima in\n\\eqref{eq:reg-function-condition}. For such $m, \\ell$ and for $k\n\\in \\Nat$ satisfying $k > (n - 1)\/2$,\n\\begin{equation}\\label{eq:reallossabovehalf1}\n\\Hbpm^{m,l_1,k}\\Hbpm^{m,l_2,k}\\subset \\Hbpm^{m - 0,l_1+l_2,k}.\n\\end{equation}\n\nIn particular, under these assumptions the $p$-fold\nproducts satisfy\n\\begin{equation}\\label{eq:reallossabovehalf2}\n(\\Hbpm^{m,l,k})^p\\subset \\Hbpm^{m - 0,pl,k}.\n\\end{equation}\n\\end{cor}\n\n\\begin{proof}[Proof of Corollary \\ref{thm:algebracor} assuming\n Proposition \\ref{thm:algebra}]\n Fix $\\ell < 0$. The corollary follows from the proposition by construction of a\n regularity function $m$ satisfying the conditions listed in the\n statement of the corollary.\nTo do so, fix a constant $m_+>1\/2-\\ell>1\/2$; indeed to satisfy the\nstrengthened form given in Theorem~\\ref{thm:fredholm} take $m_+>3\/2-\\ell>3\/2$. The function $m$ will be\narranged to be equal to $m_+$ except on a small neighborhood $U_+$ of $\\SNb_+^* S_+$ and $U_-$ of $\\SNb_+^* S_-$, which are the low regularity regions,\nwhere it will be arranged to be smaller. We consider $U_+$; $U_-$ is\nanalogous. Using any (local) defining functions $\\rho_i$, $i=1,\\ldots,n+1$, of $\\SNb_+^* S_+$ in $\\Sb^* M$ and letting $f=\\sum\\rho_i^2$, the Hamilton\nderivative of $f$ is monotone in a neighborhood $U_+$ of $\\SNb_+^* S_+$ due to the non-degenerate linearization in the normal\ndirection (with the size of the neighborhood of course depending on\nthe choice of the $\\rho_i$), with the monotonicity being strict in the\npunctured neighborhood. We may assume that $U_+$ is disjoint from any\nother component of the radial set; note that if one chooses to, one\nmay always shrink $U_+$ to lie in any pre-specified neighborhood of\n$\\SNb_+^*S_+$. One then takes a cutoff function $\\phi$, with $\\phi\\equiv\n1$ near $0$, $\\phi'\\leq 0$ on $[0,\\infty)$, $\\phi$ supported sufficiently close to $0$ so that\n$\\phi\\circ f$ is compactly supported in $U_+$; for $c>0$,\n$m=m_+-c(\\phi\\circ f)$ satisfies all monotonicity requirements along\nthe Hamilton flow, and if $m_+-c<1\/2-\\ell$, i.e.\\ $c>m_++\\ell-1\/2$,\nthen the radial point part of the Feynman condition is also\nachieved. Note that as $1\/2-\\ell>1\/2$, we may arrange in addition that\n$m_-=\\inf m=m_+-c>1\/2$ by choosing $c0\\},\n$$\nfor all sufficiently small $\\alpha>0$, which however certainly\nholds. This completes the proof of the corollary.\n\\end{proof}\n\n\\subsection{Microlocal multiplicative properties of Sobolev spaces}\nBefore we turn to multiplicative properties of b-Sobolev spaces as in\nProposition~\\ref{thm:algebra} we first study the analogous properties\nof standard Sobolev spaces. Here we need to work with variable order\nSobolev spaces because of the microlocal nature of the spaces\n$\\Hbpm^{m,l_1,k}$, in particular as $m$ is a function in this case,\nand as $k$ gives additional regularity for one half of a conormal\nbundle only.\n\nThus, for a distribution $u$ and for $s\\in\\CI(S^*\\RR^n)$ one defines\n$\\WF^s(u)$ as in the case of the b-wave front set (so the definitions\nagree in the interior): $(p,\\xi)\\notin\\WF^s(u)$ if there exists\n$A\\in\\Psi^0(\\RR^n)$ elliptic at $(p,\\xi)$ such that $Au\\in H^s(\\RR^n)$, or\nequivalently, if there exists\n$A\\in\\Psi^s(\\RR^n)$ elliptic at $(p,\\xi)$ such that $Au\\in L^2(\\RR^n)$. We are\nthen interested in questions of the kind: for which functions $r\\geq\ns\\geq s_0$ on $S^*\\RR^n$ does the implication\n$$\nu\\in H^r,\\ v\\in H^{s_0}\\Rightarrow \\WF^s(uv)\\subset\\WF^s(v),\n$$\nhold?\n\n\n\nBy a (Fourier) weight function $w \\colon \\RR^n \\lra \\RR$, we mean a smooth,\nmeasurable, positive\nfunction of polynomial growth, meaning $w \\le C \\langle \\xi \\rangle^N$ for\nsome $C, N > 0$. The variable order Sobolev space $H^{(w)}$\ncorresponding to $w$ is then\n\\begin{equation}\n \\label{eq:variablesobolev}\n H^{(w)} = \\{ u \\in S'(\\RR^n) : w \\hat{u} \\in L^2(\\RR^n) \\}.\n\\end{equation}\nThus $w(\\xi) = \\langle \\xi \\rangle^s$ for $s \\in \\RR$ defines the standard\nSobolev space of order $s$. The most common weight function we use\nbelow is of the form $w(\\xi) = \\langle \\xi \\rangle^{s(\\hat{\\xi})}$ where\n$$\n\\hat{\\xi} = \\xi \/ |\\xi|,\n$$\nso $s$\nis a function on the unit sphere, and thus we let\n\\begin{equation}\\label{eq:realsobolevdef}\nH^s := H^{(w)}, \\mbox{ where } w = \\langle \\xi \\rangle^s,\n\\end{equation}\nthat is, for $s$ of the form $s=s(\\hat\\xi)$ (which is a rather special\ncase of $s\\in\\CI(S^*\\RR^n)$!), $H^s$ is the special case of $H^{(w)}$\nwhere $w$ has this form.\nThe reason such special $s$ are sufficient for us is that\nmultiplication is a local operation on the base space $\\RR^n$, so\nusing the continuity of a general weight in the base variable, up to\narbitrarily small losses in the order, one may assume that all weights\nare in fact dependent only on the Fourier dual variable, cf.\\ the\nproof of Proposition~\\ref{thm:algebra} in\nSection~\\ref{sec:balgebraprops}, explicitly the containment\n\\eqref{eq:variable-to-const-order}.\n\n\nGiven an interior point $p \\in M$ and local coordinates $x$ near $p$, we write the induced coordinates on\nthe cotangent space $T^*_pM$ with the variable $\\xi$. The map $\\xi\n\\mapsto \\hat{\\xi}$ then identifies the spherical cotangent bundle at\n$p$, $$S^*_pM := (T_p M \\setminus o) \/ \\RR^+, $$ with the\nunit sphere in $\\mathbb{R}^n$. Here $o$ denotes the zero section, and the $\\RR^+$\naction is the natural dilation on the fibers. Given $\\hat\n\\xi \\in S^*_pM$ and $s \\in \\RR$, the Sobolev\nwavefront set of order $s$ at $p$ of a distribution $u$, $\\WF^s(u)$,\nsatisfies $(p, \\xi_0) \\not \\in \\WF^s(u)$ if and only if there\nis a cutoff function $\\chi$ on $M$ supported near $p$ so that $w(\\xi)\n\\widehat{\\chi u} \\in L^2$ for a weight function $w$ satisfying\n\\begin{equation}\\label{eq:hsmicro}\nw(\\xi) \\ge \\chi_2(\\hat{\\xi}) \\langle \\xi \\rangle^s\n\\end{equation}\nfor $\\chi_2$ a cutoff function on\nthe unit sphere $\\mathbb{S}^{n - 1}$ with $\\chi_2(\\hat{\\xi}_0) \\equiv\n1$ on a neighborhood of $\\hat{\\xi}_0$ in $\\mathbb{S}^{n-1}$. In particular, $(p, \\xi_0) \\not \\in \\WF^s(u)$ if and only if, for\nsome cutoff function $\\chi$ and some $\\mathsf{s}$ with $\\mathsf{s} =\n\\mathsf{s}(\\hat{\\xi})$, $\\mathsf{s}(\\hat{\\xi}) \\equiv s$ on some open set $U\n\\subset S^{*}_{p}M$ with $\\hat{\\xi}_{0} \\in U$ and $\\mathsf{s} \\ll 0$ off\n$U$, then $\\chi u \\in H^{\\mathsf{s}}$. \n\n\nWe are interested in properties of the Sobolev wavefront sets of\nproducts of distributions, and to this end we will exploit \\cite[Lemma 4.2]{HVsemi}:\n\\begin{lemma}\\label{thm:algcondition}\n Let $w_1, w_2, w$ be weight functions such that one of the\n quantities\n \\begin{equation}\n \\label{eq:algcondition}\n \\begin{split}\n M_+ := \\sup_{\\xi \\in \\RR^n} \\int \\left( \\frac{w(\\xi)}{w_1(\\eta)\n w_2(\\xi - \\eta)}\\right)^2 d\\eta \\\\\n M_- := \\sup_{\\xi \\in \\RR^n} \\int \\left( \\frac{w(\\xi)}{w_1(\\eta)\n w_2(\\xi - \\eta)}\\right)^2 d\\xi \n \\end{split}\n \\end{equation}\nis finite. Then $H^{(w_1)} \\cdot H^{(w_2)} \\subset H^{(w)}$.\n\\end{lemma}\nThe most well-known algebra property of Sobolev spaces is that $H^s$\nis an algebra provided $s > n\/2$. We now ask, for example, under what\nassumption on $r,s,s_0$ does one have\n\\begin{equation}\\label{eq:basicwavefrontproduct}\nu\\in H^r, v\\in H^{s_0} \\implies\n\\WF^s(uv)\\subset \\WF^s(v),\n\\end{equation}\ni.e.\\ if $u$ satisfies an a priori high regularity assumption, and $v$\nhas a priori not too low regularity, can we conclude that\n if $v$ is $H^s$ microlocally, then\n$uv$ is also $H^s$ microlocally? We now provide a partial answer using \nLemma \\ref{thm:algcondition}. (Note that taking $r = s = s_0\n> n\/2$ in the following lemma gives the standard statement that $H^s$\nis an algebra for $s > n\/2$.)\n\\begin{lemma}\n For distributions $u, v$ (which one may assume to be compactly\n supported due to the locality of multiplication) and $r, s,\n s_0\\in\\RR$,\n \\begin{equation}\n \\label{eq:basicassumptionsforproduct}\n r \\ge s \\ge s_0 > 0 \\mbox{ and } r - s + s_0 > n\/2 \\implies \\mbox{\n the containment \\eqref{eq:basicwavefrontproduct} holds.}\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Indeed, let $u, v$ be compactly supported distributions and let\n $\\xi_0 \\neq 0$ have $(x_0, \\xi_0) \\not \\in \\WF^{s}(v)$. By\n definition, there is an open cone $\\mathsf{C} \\subset \\RR^n\\setminus o$\n containing $\\xi_0$ and a function $\\mathsf{s}$ with\n$$\n\\mathsf{s} = \\mathsf{s}(\\hat{\\xi}) \\mbox{ such that }\n \\mathsf{s} \\equiv s \\mbox{ on } \\mathsf{C}, \\mathsf{s} \\ge s_0,\n$$\nand a cutoff function\n $\\chi$ supported near $x_0$ such that $\\chi v \\in H^\\mathsf{s}$, i.e.\\\n $\\widehat{\\chi v} \\langle \\xi \\rangle^\\mathsf{s} \\in L^2$. To see that $(x_0,\n \\xi_0) \\not \\in \\WF^s(uv)$ we choose a conic subset $\\mathsf{K} \\subset\\mathsf{C}$\n with compact cross section and $\\xi_{0} \\in \\mathsf{K}$, and let $s' =\n s'(\\hat{\\xi})$ be such that $s'(\\hat{\\xi}) \\equiv s$ for $\\hat{\\xi}$\n near $\\hat{\\xi}_0$, $s' \\le s$ everywhere, and such that $s'=s_0$ on\n a conic neighborhood of $\\mathsf{K}^c$. Thus if $\\chi uv \\in H^{s'}$ for\n some (possibly different) cutoff $\\chi$ with $\\chi(x_{0}) \\neq 0$,\n then $(x_{0}, \\xi_{0}) \\not \\in \\WF^{s}(uv)$. The argument below\n shows that $uv$ is microlocally $H^{s'}$ outside $\\mathsf{K}$, but we\n concentrate on the statement in $\\mathsf{K}$. We will apply Lemma\n \\ref{thm:algcondition} with $w = \\langle \\xi \\rangle^{s'}, w_1 = \\langle \\xi\n \\rangle^r$, and $w_2 = \\langle \\xi \\rangle^{\\mathsf{s}}$. That is, we will show that\n \\begin{equation}\n \\label{eq:basicmodel}\n H^{r} \\cdot H^\\mathsf{s} \\subset H^{s'},\n \\end{equation}\n for $r, \\mathsf{s}, s'$ chosen as above. Writing\n \\begin{equation}\n \\label{eq:6}\n I_\\xi = \\int \\left( \\frac{w(\\xi)}{w_1(\\eta) w_2(\\xi - \\eta)}\\right)^2 d\\eta,\n \\end{equation}\n we want to show that $\\sup_\\xi I_\\xi \\le C < \\infty$.\n\n We first note that for $\\xi\\in\\mathsf{K}$ this is bounded by the analogous\n expression where $w(\\xi)$ is replaced by $\\langle\\xi\\rangle^s$.\n Thus, we first show\n \\begin{equation}\\label{eq:I-xi-1}\n \\sup_{\\xi \\in \\mathsf{K}} I_\\xi = \\sup_{\\xi\\in\\mathsf{K}}\\int\\Big(\\frac{\\langle\\xi\\rangle^{s}}{\\langle\\eta\\rangle^{\\mathsf{s}(\\eta)} \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta\n \\end{equation}\n is finite. Since $s \\ge 0$ (recall that $s$ is a constant), $\\langle\\xi\\rangle^{2s}\\lesssim\n \\langle\\eta\\rangle^{2s}+\\langle\\xi-\\eta\\rangle^{2s}$, it suffices to\n prove that\n \\begin{equation}\\label{eq:twointegrals}\n \\sup_{\\xi\\in\\mathsf{K}}\\int\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}(\\eta)-2s}\n \\langle\\xi-\\eta\\rangle^{2r}}\\,d\\eta \\quad \n \\mbox{ and } \\quad \\sup_{\\xi\\in\\mathsf{K}}\\int\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}(\\eta)} \\langle\\xi-\\eta\\rangle^{2r-2s}}\\,d\\eta\n \\end{equation}\n are finite. We start by looking at the second of these. We break up\n the integral into $\\langle\\eta\\rangle\\geq\\langle\\xi-\\eta\\rangle$ and\n it complement. In the former region the integral is bounded by\n$$\n\\int\\frac{1}{\\langle\\xi-\\eta\\rangle^{2r-2s+2\\mathsf{s}(\\eta)}}\\,d\\eta\n$$\nand since $r \\ge s$ in the latter region by\n$$\n\\int\\frac{1}{\\langle\\eta\\rangle^{2r-2s+2\\mathsf{s}(\\eta)}}\\,d\\eta,\n$$\nboth of which are finite under the assumption in\n\\eqref{eq:basicassumptionsforproduct} as $\\mathsf{s}\\geq s_0$.\n\n\nTurning to the first integral in \\eqref{eq:twointegrals}, since\n$\\mathsf{s}(\\eta)$ is not necessarily greater than or equal to $s$, the\nargument of the previous paragraph does not go through -- instead, we\nbreak up the integral into one over $\\mathsf{C}$ and one over $\\mathsf{C}^c$. Now,\nusing that $\\mathsf{s} \\equiv s$ on $\\mathsf{C}$,\n$$\n\\int_{\\mathsf{C}}\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}(\\eta)-2s}\n \\langle\\xi-\\eta\\rangle^{2r}}\\,d\\eta=\\int_{\\mathsf{C}}\\frac{1}{\n \\langle\\xi-\\eta\\rangle^{2r}}\\,d\\eta\\leq\\int \\frac{1}{\n \\langle\\xi-\\eta\\rangle^{2r}}\\,d\\eta=\\int \\frac{1}{\n \\langle\\eta\\rangle^{2r}}\\,d\\eta\n$$\nis finite, independent of $\\xi$, if $r>n\/2$ (which is implied by\n$r-s+s_0>n\/2$ and $s \\ge s_0$). For the integral over $\\mathsf{C}^c$, we use\nthat there is a constant $C_0>0$ such that\n$C_0\\langle\\xi-\\eta\\rangle\\geq\\langle\\eta\\rangle$ for $\\xi\\in\\mathsf{K}$ and\n$\\eta\\in\\mathsf{C}^c$. Correspondingly, as $r\\geq 0$,\n$$\n\\int_{\\mathsf{C}^c}\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}(\\eta)-2s}\n \\langle\\xi-\\eta\\rangle^{2r}}\\,d\\eta\\leq C_0^{2r}\n\\int_{\\mathsf{C}^c}\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}(\\eta)-2s+2r}}\\,d\\eta\\leq\nC_0^{2r} \\int\\frac{1}{\\langle\\eta\\rangle^{2s_0-2s+2r}}\\,d\\eta\n$$\nwhich is finite if $r-s+s_0>n\/2$. This proves \\eqref{eq:I-xi-1}.\n\nThe bound for $I_{\\xi}$ with $\\xi \\not \\in \\mathsf{K}$ proceeds along the\nsame lines, where now one can replace $w(\\xi)$ by\n$\\langle\\xi\\rangle^{s_0}$, and $\\langle\\eta\\rangle^{\\mathsf{s}(\\eta)}$ by\n$\\langle\\eta\\rangle^{s_0}$, and is left to the reader.\n\nThis completes the proof of \\eqref{eq:basicmodel} and thus that the\nconditions on $r, s, s_{0}$ in \\eqref{eq:basicassumptionsforproduct}\nimply \\eqref{eq:basicwavefrontproduct}. \n\\end{proof}\n\nFor our applications, i.e.\\ to study the module regularity\ndefining the spaces $\\Hbpmpm^{m,l,k}$ we will first study spaces for\nwhich one has extra regularity in certain directions. To this end,\nwe write $\\RR^n$ as $\\RR^{d + (n - d)}$, i.e.\\ we decompose into $x = (x',\nx'')$ where $x' \\in \\RR^d$ and $x'' \\in \\RR^{n - d}$, and for functions $f$, we write the\nFourier side variable $\\xi$ as $(\\xi', \\xi'')$. Let\n\\begin{equation}\\label{eq:basicmodulereg}\n\\cY^{m,a}_d(\\RR^{d + (n - d)}) = \\{ u : \\hat{u} \\langle \\xi \\rangle^m \\langle \\xi^{''}\n \\rangle^a \\in L^2 \\},\n\\end{equation}\nso elements have $m$ total derivatives (derivatives in all variables)\nand in addition $a$ derivatives in $x''$.\n\n \\begin{lemma}(\\cite[Lemma~4.4]{HVsemi}.)\\label{thm:basicalgebra}\n Let $m,a \\in \\mathbb{R}$. If $m > d \/2$ and $a > (n - d)\/2$ then $\\cY_d^{m, a}$ is an\n algebra. If $a,b\\geq 0$, $a + b > n - d$, then $\\cY_d^{m,a} \\cdot \\cY_d^{m, b}\n \\subset H^{m}$. \n \\end{lemma}\n \\begin{proof}\n We begin with the second statement. This is exactly \\cite[Equation 4.6]{HVsemi}, namely using $\\langle \\xi \\rangle^p \\lesssim \\langle \\eta\n \\rangle^p + \\langle\\xi - \\eta\n \\rangle^p$ for $p \\geq 0$, we have\n \\begin{equation}\n \\label{eq:4.6}\n \\begin{split}\n & \\int \\left( \\frac{\\langle \\xi \\rangle^m}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta \\rangle^m \\langle \\eta''\n \\rangle^b} \\right)^2 d\\eta \\\\\n &\\le \\int \\left( \\frac{1}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta''\n \\rangle^b} \\right)^2 d\\eta \n + \\int \\left( \\frac{1}{ \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta \\rangle^m \\langle \\eta''\n \\rangle^b} \\right)^2 d\\eta \\\\\n & \\le \\int \\left( \\frac{1}{\\langle \\xi' - \\eta' \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta''\n \\rangle^b} \\right)^2 d\\eta \n + \\int \\left( \\frac{1}{ \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta' \\rangle^m \\langle \\eta''\n \\rangle^b} \\right)^2 d\\eta ,\n \\end{split}\n \\end{equation}\nand integrating first in the primed and then the double primed\nvariable shows this integral is uniformly bounded.\n\nWhen $a = b > (n - d)\/2$, estimating the numerator in \\eqref{eq:4.6}\nin the same way, we see that $\\cY_d^{m, a}$ is an algebra\nsince\n \\begin{equation}\n \\label{eq:firstalgebra}\n \\begin{split}\n & \\int \\left( \\frac{\\langle \\xi \\rangle^m\\langle \\xi'' \\rangle^a}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta \\rangle^m \\langle \\eta''\n \\rangle^a} \\right)^2 d\\eta \\\\\n &\\le \\sum_{i, j = 1}^2 \\int \\left( \\frac{f_i g_j}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta''\n \\rangle^a} \\right)^2 d\\eta , \n \\end{split}\n \\end{equation}\nwhere $f_1 = \\langle \\eta \\rangle^m$, $f_2 = \\langle \\xi - \\eta \\rangle^m$ and $g_1 =\n\\langle \\eta'' \\rangle^a$, $g_2 = \\langle \\xi'' - \\eta'' \\rangle^a$. We estimate the\n$f_1 g_1$ term by\n \\begin{equation*}\n \\begin{split}\n & \\int \\left( \\frac{\\langle \\eta \\rangle^m\\langle \\eta'' \\rangle^a}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta \\rangle^m \\langle \\eta''\n \\rangle^a} \\right)^2 d\\eta \\\\\n &\\le \\int \\left( \\frac{1}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta''\n \\rangle^a } \\right)^2 d\\eta \\le \\int \\left( \\frac{1}{\\langle \\xi' - \\eta' \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a } \\right)^2 d\\eta,\n \\end{split}\n \\end{equation*}\nso integrating separately in the primed and double primed variable\nshows this is uniformly bounded. The other terms are bounded in\nexactly the same way.\n \\end{proof}\n\n\n\nThe wavefront set containment in\n\\eqref{eq:basicwavefrontproduct}--\\eqref{eq:basicassumptionsforproduct} can be\n``improved'' when one assumes the distributions lie in the $\\cY^{m,\n a}_d$, in the sense that less total regularity (i.e.\\ smaller $m$)\nis required.\n\\begin{lemma}\n Let $r, a,m \\in \\RR$, and let $u \\in H^r, v \\in \\cY_d^{m, a}$.\n Then, provided, $m >\n d\/2$, and $a > (n - d)\/2$ we have that for any $s \\in\\RR$ with $r \\ge s \\ge m + a$, that\n \\begin{equation}\n \\label{eq:2}\n \\WF^{s}(uv) \\subset \\WF^{s}(v).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nNote first that the conditions on $r, s, m, $ and $a$ imply that $r, s\n> n\/2$ and thus give square integrable weight functions. Let $\\xi_0\n\\neq 0$ and $(x_0, \\xi_0) \\not \\in \\WF^s(v)$. As above, let $\\mathsf{C}\n\\subset \\RR^n$ be an open,\nconic set, such that \n$$\n\\xi_0 \\in \\mathsf{C} \\mbox{ and } \\{x_0\\}\\times\\mathsf{C} \\subset (\\WF^s(v))^{c},\n$$\nThere is a function $\\mathsf{s} = \\mathsf{s}(\\hat{\\xi})$ such that\n$\\mathsf{s} \\rvert_{\\mathsf{C}} =\ns$, and $\\chi v \\in H^{(w_1)}$ where\n$$\nw_1 = \\langle \\xi \\rangle^m \\langle \\xi'' \\rangle^a + \\langle \\xi \\rangle^{\\mathsf{s}} .\n$$\nFurthermore, $\\chi$ can be chosen such that $\\chi u \\in H^{(w_2)}$\nwhere $w_2 = \\langle \\xi \\rangle^r$. To show that $(x_0, \\xi_0) \\not \\in\n\\WF^s(uv),$ we choose $\\mathsf{K}$ a conic subset\ncontaining $\\xi_0$, and a function $s'(\\hat \\xi)$ with\n\\begin{equation}\\label{eq:sfsdescrip}\ns'(\\hat{\\xi}) = s \\mbox{ for } \\hat{\\xi}_0 \\mbox{ near }\n\\hat{\\xi}, \\mbox{ while } s'(\\hat{\\xi}) = \\delta \\mbox{ on a\n neighborhood of } \\mathsf{K}^c, \n\\end{equation}\nwhere $\\delta > 0$ is fixed and small and also $s' \\le \\mathsf{s}$\neverywhere. We\napply Lemma \\ref{thm:algcondition} with this $w_1, w_2$ and $w = \\langle\n\\xi \\rangle^{s'}$. \n\n\nDefining $I_\\xi$ as in \\eqref{eq:6} with the current $w_1, w_2$ and\n$w$ we have\n\\begin{equation}\n \\label{eq:basicmoduleregwave}\nI_\\xi := \\int\\Big(\\frac{\\langle\\xi\\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} )\n \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta ,\n\\end{equation}\nand we want to know that $M_+ = \\sup_\\xi I_\\xi$ is finite. Again we use\nthat $\\langle \\xi \\rangle^p \\lesssim \\langle \\eta \\rangle^p + \\langle \\xi - \\eta \\rangle^p$\nfor $p > 0$ to write $I(\\xi) \\le I_1(\\xi) + I_2(\\xi)$ where\n\\begin{equation}\n \\label{eq:4}\n \\begin{split}\n I_1(\\xi) &= \\int\\Big(\\frac{\\langle\\eta\\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} )\n \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta \\qquad \\mbox{ and }\\\\\n I_2(\\xi) &= \\int\\Big(\\frac{\\langle\\xi - \\eta\\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} )\n \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta.\n \\end{split}\n\\end{equation}\nFor any $\\xi$, since $\\langle\\xi -\n\\eta\\rangle^s\/\\langle\\xi-\\eta\\rangle^r \\le 1$, \n\\begin{equation*}\n \\begin{split}\n I_2(\\xi) &\\le \\int\\Big(\\frac{1}{\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} }\\Big)^2\\,d\\eta \\le \\int\\frac{1}{\\langle \\eta' \\rangle^{2m} \\langle\n \\eta'' \\rangle^{2a}}\\,d\\eta'd\\eta'',\n \\end{split}\n\\end{equation*}\nwhich is bounded since $m > d\/2$ and $a > (n - d)\/2$.\n\nFinally consider $I_1(\\xi)$ for $\\xi \\in \\mathsf{K}$. We break the integral\nup into $\\eta \\in \\mathsf{C}$ and $\\eta \\in \\mathsf{C}^c$. Using that, $s'(\\xi)\n\\le s$, and that $\\mathsf{s} = s$ on $\\mathsf{C}$ with $s \\ge m + a$,\n\\begin{equation*}\n \\begin{split}\n &\\int_{\\mathsf{C}}\\Big(\\frac{\\langle\\eta\\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} )\n \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta \\le \\int_{\\mathsf{C}}\\frac{1}{\n \\langle\\xi-\\eta\\rangle^{2r}} d\\eta ,\n \\end{split}\n\\end{equation*}\nwhich is bounded uniformly. On the other hand, since $\\mathsf{K}$ has\ncompact cross section and $\\xi \\in \\mathsf{K}$, we have $\\langle \\eta \\rangle \\lesssim \\langle\n\\xi - \\eta \\rangle$ for $\\eta \\in \\mathsf{C}^c$, so as $s\\leq r$,\n\\begin{equation*}\n \\begin{split}\n \\int_{\\mathsf{C}}\\Big(\\frac{\\langle\\eta\\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}} )\n \\langle\\xi-\\eta\\rangle^r}\\Big)^2\\,d\\eta &\\lesssim \\int_{\\mathsf{C}}\\frac{1}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{\\mathsf{s}})^2} d\\eta \\\\\n & \\le \\int_{\\mathsf{C}}\\frac{1}{\\langle \\eta' \\rangle^{2m} \\langle\n \\eta'' \\rangle^{2a} } d\\eta'd\\eta'',\n \\end{split}\n\\end{equation*}\nand again the last integral is bounded. Again, we leave the estimate\nfor $\\xi \\not \\in \\mathsf{K}$ to the reader.\n\\end{proof}\n\n\n\nFurthermore, we have\n\\begin{lemma}\\label{thm:reallemma}\n Let $a,m,s \\in \\mathbb{R}$. Let $u, v \\in \\cY_d^{m, a}$. Then, provided $m >\n d\/2$, and $a > (n - d)\/2$ with $s \\ge m +\n a$, we have\n \\begin{equation}\n \\label{eq:waveinsum}\n \\WF^{s}(uv) \\subset (\\WF^{s}(u) + \\WF^{s}(v))\\cup\\WF^s(u)\\cup\\WF^s(v).\n \\end{equation}\n \\begin{rem}\\label{thm:interior-wave-sum-inclusion}\n Lemma \\ref{thm:reallemma} gives the following elaboration on\n \\eqref{eq:basicwavefrontproduct}--\\eqref{eq:basicassumptionsforproduct}.\n If $u, v \\in H^{s_0}$ for $s_0 \\in \\mathbb{R}$, $s_0 > n\/2$, then\n for any $s \\in \\mathbb{R}$, \\eqref{eq:waveinsum} holds. Indeed,\n for $s_0 > n\/2$, $s>s_0$ (the interesting case) and any $d$ one can find $m, a \\in \\RR$ such that $m > d\n \/2$, $a > (n - d)\/2$ and $s \\ge m + a$, and for any such $H^{s_0} \\subset \\cY_d^{m,\n a}$, so the lemma applies to $u, v$.\n \\end{rem}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma \\ref{thm:reallemma}]\n The idea behind the proof is the following. Working above a fixed\n point $x_{0} \\in \\RR^{n}$, given \n \\begin{equation}\\label{eq:notinsum}\n\\xi \\not \\in (\\WF^{s}(u) +\n \\WF^{s}(v))\\cup\\WF^s(u)\\cup\\WF^s(v),\n\\end{equation}\nconsider the integral\n \\begin{equation}\\label{eq:etapart}\n\\int\\Big(\\frac{\\langle \\xi \\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{s_{1}} )\n (\\langle \\xi - \\eta \\rangle^m \\langle\n \\xi'' - \\eta'' \\rangle^a + \\langle \\xi - \\eta \\rangle^{s_{2}} )}\\Big)^2\\,d\\eta,\n\\end{equation}\nwhere $s_{1}$ and $s_{2}$ are chosen so that $u \\in H^{s_{1}}$ and $v\n\\in H^{s_{2}}$ (near $x_{0}$), and such that $s_{i} = s > n\/2$ on sets\nthat are as large as possible. If one could take $s_{1} = s$ on\n$(\\WF^{s}(u))^{c}$ and $s_{2} = s$ on\n$(\\WF^{s}(v))^{c}$ (one cannot) then the integral would be bounded\nsince by \\eqref{eq:notinsum}, either $\\eta \\not \\in \\WF^s(u)$ or $\\xi -\n\\eta \\not \\in \\WF^s(v)$. In the former case for example, the integral\nis bounded by \n$$\n\\int \\frac{1}{\\langle \\xi - \\eta\\rangle^{2m}\\langle \\xi'' - \\eta''\\rangle^{2s}} d\\eta,\n$$\nwhich is bounded uniformly in $\\xi$.\n\nFor the formal argument, note that for any open conic sets $\\wt{\\mathsf{C}}_{1} \\supset\n\\WF^{s}(u)$ and $\\wt{\\mathsf{C}}_{2} \\supset\n\\WF^{s}(v)$ there are functions $s_{1}, s_{2}$ so that $s_{i} \\equiv s$ off\n$\\wt{\\mathsf{C}}_{i}$ and such that $\\chi u \\in H^{s_{1}}, \\chi v \\in\nH^{s_{2}}$ for some cutoff $\\chi$ with $\\chi(x_{0}) = 1$. Given\n$\\xi$ as in \\eqref{eq:notinsum}, assume furthermore that $\\xi \\not \\in\n\\wt{\\mathsf{C}}_1 + \\wt{\\mathsf{C}}_2$, and let $\\mathsf{C}$ be a conic open set with $C\n\\subset (\\wt{\\mathsf{C}}_1 + \\wt{\\mathsf{C}}_2)^c$\nFor $\\xi\n\\in \\mathsf{C}$ where $\\mathsf{C}$ is an arbitrary open subset with $\\mathsf{C} \\subset\n(\\wt{\\mathsf{C}}_{1} + \\wt{\\mathsf{C}}_{2})^{c}$. Writing $\\langle \\xi \\rangle^{2s}\n\\lesssim \\langle \\eta \\rangle^{2s} + \\langle \\xi - \\eta \\rangle^{2s}$, we can bound the\nintegral above by two terms (one with the $\\eta$ in the numerator and\nthe other with the $\\xi - \\eta$). The argument to bound each of these\nis symmetric so we consider only the $\\eta$ term. Then for $\\xi \\in\n\\mathsf{C}$, the integral in \\eqref{eq:etapart}\ncan be broken up as an integral over $\\wt{\\mathsf{C}}_{1}$ and over\n$\\wt{\\mathsf{C}}_{1}^{c}$. Over $\\wt{\\mathsf{C}}_{1}^{c}$, we have that $\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{s_{1}} >(1\/2) \\langle \\eta \\rangle^{s}$, so\n that part of the integral is bounded by\n\\begin{equation}\n \\label{eq:etapart1}\n \\begin{split}\n \\int\\Big(\\frac{1}{\\langle \\xi - \\eta \\rangle^m \\langle\n \\xi'' - \\eta'' \\rangle^a + \\langle \\xi - \\eta \\rangle^{s_{2}} }\\Big)^2\\,d\\eta,\n \\end{split}\n\\end{equation}\nwhich is uniformly bounded. On the other hand, for $\\eta \\in\n\\wt{\\mathsf{C}}_{1}$, we have that $s_{2}(\\widehat{\\xi - \\eta}) = s$, since $\\xi \\in\n\\mathsf{C}$, that is $\\xi = \\xi - \\eta + \\eta \\not \\in \\wt{\\mathsf{C}}_{1} +\n\\wt{\\mathsf{C}}_{2}$, so $\\xi - \\eta \\not \\in \\wt{\\mathsf{C}}_{2}$. Therefore on\nthe $\\wt{\\mathsf{C}}_{1}$ region, the integral in \\eqref{eq:etapart} is\nbounded by \n\\begin{equation}\n \\label{eq:etapart2}\n \\int\\Big(\\frac{\\langle \\eta \\rangle^s}{(\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{s_{1}} )\n \\langle \\xi - \\eta \\rangle^{s} }\\Big)^2\\,d\\eta.\n\\end{equation}\nBut since $\\xi \\in \\mathsf{C}$ (so in particular $\\xi\\notin \\wt{\\mathsf{C}}_{1}$), $\\langle \\eta \\rangle \\lesssim \\langle \\xi - \\eta \\rangle$, so the\nintegral is uniformly bounded.\n\nThus we will be able to apply Lemma \\ref{thm:algcondition} with $w_{i} = \\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^a + \\langle \\eta \\rangle^{s_{i}}$ and $w = \\langle \\xi \\rangle^{\\mathsf{s}}$\n with $\\mathsf{s} = s$ on an arbitrary conic subset $\\mathsf{K}' \\subset \\mathsf{C}$ by\n arguing exactly as in the previous lemmata, namely taking $\\mathsf{s}$\n small but uniformly positive off of $\\mathsf{K}'$ so that both $w\/w_{1}$\n and $w\/w_{2}$ bounded off $\\mathsf{C}$. In summary we have shown that\n for such $w, w_{1}, w_{2}$ that\n \\begin{equation}\n \\label{eq:tobeused}\n H^{(w_{1})} \\cdot H^{(w_{2})} \\subset H^{(w)},\n \\end{equation}\nand thus the lemma follows.\n\\end{proof}\n\n\\subsection{Microlocal multiplicative properties of b-Sobolev spaces and module\n regularity spaces}\\label{sec:balgebraprops}\n\n\nRecall that the b-Sobolev\nspace $\\Hb^{m, 0}$ consists of distributions $u$ which are $H^m$\nlocally in the interior and whose behavior near infinity is as\nfollows. We consider a tubular neighborhood of $\\p M$ in $M$ (so a\ntubular neighborhood of infinity) by $\\{ \\rho < \\epsilon\\}$ for some\nsmall $\\epsilon > 0$ and boundary defining function $\\rho$. Given a boundary point $p \\in \\p\nM$, we can take coordinates near $p$ in this collar to be of the form\n$(\\rho, w)$ where $w$ form coordinates on $\\p M$. Defining the\nfollowing operation on functions $v$ of\n$(\\rho, w)$ by\n\\begin{equation}\\label{eq:expchange}\n\\wt{v}(x, w) = v(e^x, w),\n\\end{equation}\nthen $u \\in \\Hb^{m, 0}$ if in addition to interior regularity, for some cutoff function $\\chi$, $\\wt{\\chi u} \\in H^m$. Here if\n$m \\colon \\Tb^*M \\lra \\RR$ is a homogeneous degree zero function (at\nleast outside a compact set) i.e.\\ a function on $\\Sb^{*}M$, then we mean $H^m$ as defined in\n\\eqref{eq:realsobolevdef}, where $m = m(p, \\xi)$ and $\\xi$ is a\ncoordinate on $\\Tb^*_pM$.\nNote that the Fourier transform of $\\wt{u}$ is\nequal to the distribution obtained by taking the Fourier transform in\nthe $w$ variables and the Mellin transform (see \\eqref{eq:mellindef}) in $\\rho$. The weighted\nb-Sobolev spaces are defined by $\\Hb^{m, l} = \\rho^l \\Hb^{m, 0}$. Given $u \\in \\Hb^{-N, 0}$ for some $N$, a covector $(p,\n\\xi) \\in \\Tb^*_p(M)$ satisfies $(p, \\xi) \\not \\in \\WFb^{m,0}(u)$ if\n$w(\\xi) \\widehat{\\wt{\\chi u}} \\in L^2$ for some cutoff function, where\n$w$ satisfies \\eqref{eq:hsmicro}. Just as stated above\n\\eqref{eq:hsmicro}, this is equivalent to having a $\\chi$ for which $\\chi u \\in H^{\\mathsf{s}}$ where $\\mathsf{s} =\n\\mathsf{s}(\\hat{\\xi})$, $\\mathsf{s}(\\hat{\\xi}_0) \\equiv s$ in a neighborhood of\n$\\hat{\\xi}_0$ and $\\mathsf{s} \\ll 0$ away from\n$\\hat{\\xi}_0$. Finally, for $u \\in \\Hb^{-N, l}$ with $l \\in \\RR$,\n\\begin{equation}\n \\label{eq:weightedwavefront}\n \\WFb^{m,l}(u) := \\WFb^{m, 0}(\\rho^{-l}u).\n\\end{equation}\n\nUsing the previous section we can, for example, easily prove\n\\begin{lemma}\n Given\n$r,s_0,s \\in \\RR$, then\n\\begin{equation}\\label{eq:basicwavefrontproduct2}\nu\\in \\Hb^{r, l_1}, v\\in \\Hb^{s_0, l_2} \\implies uv \\in \\Hb^{s_0, l_1 +\n l_2} \\mbox{ and }\n\\WFb^s(uv)\\subset \\WFb^s(v),\n\\end{equation}\nprovided \\eqref{eq:basicassumptionsforproduct} above holds, i.e.\\\n$ r \\ge s \\ge s_0 \\ge 0$ and $r - s + s_0 > n\/2$.\n\\end{lemma}\n\\begin{proof}\n The proof follows from the paragraph following Lemma\n \\ref{thm:algcondition}, as we explain now. \n\nFirst let $l_1 = 0 = l_2$. Given such $u$ and $v$, we want to show\nfirst that $uv \\in \\Hb^{s_0, l_1 +\n l_2}$. In the interior of $M$\nthis follows from \\eqref{eq:basicwavefrontproduct} and\n\\eqref{eq:basicassumptionsforproduct} directly. For $p \\in \\p M$, by definition, there is a cutoff function\n$\\chi$ so that $\\wt{\\chi u } \\in H^{r}$, $\\wt{\\chi v} \\in\nH^{s_0}$, where the tilded functions are the functions on the cylinder\ndefined in \\eqref{eq:expchange}. Then $\\wt{\\chi u } \\wt{\\chi v} \\in\nH^{s_0}$ by applying \\eqref{eq:basicwavefrontproduct} with $s = s_0$. \n\nNow we show the wavefront set containment, which is almost identical to\nthe paragraph following \\eqref{eq:basicassumptionsforproduct}.\nIndeed, for $(p, \\xi_0) \\not \\in \\WFb^{s, 0}(v)$, let $\\mathsf{C} \\subset \\Tb^{*}_{p}M$ be an open\ncone with $\\xi_{0} \\in \\mathsf{C}$ and $\\mathsf{C} \\cap \\WFb^{s, 0}(v) =\n\\varnothing$. By definition there is\na cutoff $\\chi$ supported near $p$ such that $\\wt{\\chi v} \\in H^\\mathsf{s}$\nfor some $\\mathsf{s}$ with $\\mathsf{s} \\equiv s$ on $\\mathsf{C}$, $\\mathsf{s} \\ge s_0$ and such that $\\wt{\\chi u} \\in H^r$.\nThen let $\\mathsf{K} \\subset \\mathsf{C}$ be a conic set with compact cross section and\n$\\xi_{0} \\in \\mathsf{K}$, and let $s' = s'(\\hat{\\xi})$ be such that\n$s'(\\hat{\\xi}) \\equiv s$ for $\\hat{\\xi}$ near $\\hat{\\xi}_0$ and such\nthat $s'= s_0$ outside $\\mathsf{K}$.\nIt suffices to show that $\\wt{\\chi u} \\wt{ \\chi v} \\in H^{s'}$, but\nthis is exactly \\eqref{eq:basicmodel} above.\n\nThe statement for $l_{1}$ and $l_{2}$ follows by applying the above\nparagraph to $\\rho^{-l_1} u$ and $\\rho^{-l_2}v$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe can now show that the module regularity spaces $\\Hbpp$ have the\nfollowing algebra property which is closely related to \\cite[Section~5.2]{HVsemi}:\n\\begin{lemma}\\label{thm:firstmodulealg}\n Let $m\\equiv m_0\\in\\mathbb{R}$ and let $k \\in \\Nat$. Provided $m > 1\/2$ and $k > (n\n - 1)\/2$, if $u_1 \\in \\Hbpp^{m,l_1,k}, u_2 \\in \n \\Hbpp^{m,l_2,k}$, then $u_1u_2 \\in \\Hbpp^{m,l_1 + l_2,k}.$\n\\end{lemma}\n\\begin{proof}\nAway from the boundary this is just the statement that $H^{m + k}$\nis an algebra, and at the boundary but away from $S_+$ that\n$\\Hb^{m+k,l_j}$ has the stated multiplicative property. Thus we assume that the $u_i$ are supported in a small\nneighborhood of a point $x \\in S_+$. We begin by showing that\n$$\n\\wt{u_i} \\in\n\\cY^{m, k}_1\n$$ \nwhere $\\wt{u}_i$ is defined\nas in \\eqref{eq:expchange} and $\\cY^{m, k}_{1}$ is the spaces\ndefined in \\eqref{eq:basicmodulereg} with $d = 1$, $a = k$. \nIndeed, for our coordinates $(\\rho, v, y)$ \nwhere $\\rho$ is a boundary\ndefining function and $\\rho = 0 = v$ defines $S_{+}$ (see Section\n\\ref{sec:geometry}), recall the\nMellin transform \\eqref{eq:mellindef}, and consider the Mellin-Fourier\ntransform of test functions $\\psi \\in H^{\\infty, \\infty}_{b}$,\n$\\mathcal{M}\\mathcal{F}_{v,y}(\\psi)$, where $\\mathcal{F}_{v,y}$\ndenotes the Fourier transform in the $v, y$ variables. Concretely\n\\begin{equation}\n \\label{eq:MellinFourier}\n \\mathcal{M}\\mathcal{F}_{v,y}(\\psi) = \\int \\rho^{-i \\zeta} e^{-i v\n \\xi' - i y \\cdot \\eta} \\psi(\\rho, v, y) \\rho^{-1}d\\rho dv dy,\n\\end{equation}\nand we write\n\\begin{equation}\n \\label{eq:7}\n \\xi := (\\xi', \\zeta, \\eta), \\qquad \\xi'' := (\\zeta, \\eta),\n\\end{equation}\nso $\\xi$ is the total dual variable and $\\xi'$ is dual to $v$.\nConsider order $m$ elliptic b-pseudodifferential operator $A$ defined by \\begin{equation}\n \\label{eq:melliptic}\n A\\psi = \\mathcal{F}^{-1}\\mathcal{M}^{-1}_{0} \\langle \\xi \\rangle^{m} \\mathcal{M}\\mathcal{F}_{v,y} \\psi,\n\\end{equation}\nand the order $\\leq k$ b-pseudodifferential operator $B_\\alpha$,\n$|\\alpha|\\leq k$, defined by\n\\begin{equation}\n \\label{eq:k}\n B_\\alpha\\psi = \\mathcal{F}^{-1}\\mathcal{M}^{-1}_{0} (\\xi'')^{\\alpha} \\mathcal{M}\\mathcal{F}_{v,y} \\psi.\n\\end{equation} \nBy definition of $\\Hbpp^{m,0,k}$ we have $AB_\\alpha u_i\n\\in L^{2}_{b}$ for all $|\\alpha|\\leq k$, so\nsince the mellin transform of $u$ is the Fourier transform in $x =\n\\log \\rho$ of $\\wt{u}$ we have $\\wt{u_i} \\in \\mathcal{Y}^{m,k}_{1}$\nlocally near $x$, as claimed.\n\nTo prove the lemma, we must show that if $a, b, c > 0$ integers,\n$\\alpha$ a multiindex,\nand $a + b + c+|\\alpha| \\le k$, we have that $(\\rho \\p_\\rho)^a(\\rho \\p_v)^b(v\n\\p_v)^c \\pa_y^\\alpha (u_1 u_2) \\in \\Hb^m$, but this distribution is equal to \n\\begin{equation}\\begin{aligned}\n \\label{eq:leibnitz}\n \\sum_{a'\\leq a, b'\\leq b,c'\\leq c,\\alpha'\\leq\\alpha} C_{a', b', c',\\alpha'} &((\\rho \\p_\\rho)^{a'}(\\rho \\p_v)^{b'}(v\n\\p_v)^{c'} \\pa_y^{\\alpha'}u_1)\\\\\n&\\qquad\\qquad((\\rho \\p_\\rho)^{a - a'}(\\rho \\p_v)^{b - b'}(v\n\\p_v)^{c - c'} \\pa_y^{\\alpha-\\alpha'}u_2),\n\\end{aligned}\\end{equation}\nfor some combinatorial constants $C_{a', b', c',\\alpha'}$ (which depend on $a, b, c,\\alpha$). In\neach of these terms we have the product of two elements $u_1, u_2$,\nwhich $u_i \\in \\Hbpp^{m, 0, k - r_i}$ where $k - r_1 + k - r_2 \\ge\nk$. But by the previous paragraph, locally near $S_+$, the $u_i$\nsatisfy that $\\wt{u}_{i}$ lies in $\\cY_d^{m, k -\n r_i}$. Thus by the first\npart of Lemma \\ref{thm:basicalgebra}, $\\wt{u_1} \\wt{u_{2}} \\in H^{m}$,\nwhich is to say that\n$u_{1} u_{2} \\in \\Hb^{m,0}$, locally near $S_+$, which is what we wanted in the case $l_1\n= l_2 = 0$. For general weights, i.e.\\ $u_i, \\in \\Hbpp^{m, l_i, k}$,\n$i = 1, 2$, apply the above arguements to $\\rho^{- l_1 - l_2} u_1 u_2\n= (\\rho^{-l_1} u_1 )(\\rho^{-l_2} u_2)$.\n\\end{proof}\n\n\n\n\n\nFinally, we can prove Proposition \\ref{thm:algebra} above.\n\n\\begin{proof}[Proof of Proposition \\ref{thm:algebra}]\nUsing that multiplication is local, we will reduce in the end to\nconsidering two compactly supported\ndistributions $u_i$, $i = 1, 2$, with $u_i \\in \\Hb^{m, l_i, k}(M)$\nsupported in a neighborhood of a point $x \\in M$.\n\nWhile in fact the boundary case discussed below handles this as well,\nwe first treat interior points. So assume that the $u_i$ are supported in a\ncoordinate chart in $M^\\circ$.\n Thus the $u_i \\in H^{m + k}(\\mathbb{R}^n)$, and since $m > 1\/2, k > (n - 1)\/2$,\n there is a constant $m_0 \\in \\mathbb{R}$ so that the $u_i \\in H^{m_0 +\n k}(\\mathbb{R}^n)$ and $m_0 + k > n\/2$. Since $H^{m_0 + k}$ is an\n algebra, $u_1 u_2 \\in H^{m_0 + k}$. We claim that $u_1 u_2 \\in H^{m -\n \\epsilon + k}$. Indeed, for any $x$ and for any $s \\in \\mathbb{R}$ with $s - k < m_+ =\n \\max_{S_x^*\\RR^n} m$, the sets $\\WF^s(u_i)$\n satisfy\n\\begin{equation}\\label{eq:variable-to-const-order}\n\\WF^s(u_i) \\cap T^*_x(\\RR^n) \\subset \\{ (x, \\xi ) : m(x,\n\\hat{\\xi}) + k \\le s\\}.\n\\end{equation}\nBy Remark \\ref{thm:interior-wave-sum-inclusion},\n$$\n\\WF^s(u_1 u_2) \\subset (\\WF^s(u_1) + \\WF^s(u_2)) \\cup \\WF^s(u_1) \\cup \\WF^s(u_2),\n$$\nand thus, \\emph{by the assumption that the non-trivial sublevel sets\n are convex,} we conclude that $\\WF^s(u_1 u_2) \\cap T^*_x\\RR^n$ is also a subset of $\\{ (x, \\xi ) : m(x,\n\\hat{\\xi}) + k\\le s\\}$. To see that $u_1 u_2 \\in H^{m - \\epsilon +\n k}$ then, for any $(x,\\xi)$\n let $s = m(x, \\xi) - \\epsilon\/2 + k$, and then note that since $(x, \\xi) \\not\n \\in \\WF^s(u_i)$ for $i = 1, 2$, by what we just said also $(x,\n \\xi) \\not \\in \\WF^s(u_1 u_2)$.\n\nFor $x \\in \\p M$ and the $u_i$ supported near $x$, we first assume\nthat $l_1 = 0 = l_2$. Since such $u_i$ are also contained in\n$\\Hbpp^{m_-, 0, k}$, by Lemma \\ref{thm:firstmodulealg} we know that $u_1 u_2\\in \\Hbpp^{m_-,0,k}$.\nDue to the second assumption in \\eqref{eq:reg-function-condition},\ngiven $\\ep>0$, microlocally near $\\Sb N_+^*S_+$ (with the neighborhood\nsize depending on $\\ep$), $\\Hbpp^{m_-,0,k}$\nis contained in $\\Hbpm^{m-\\ep,0,k}$, so microlocally near $\\Sb\nN_+^*S_+$ we have the conclusion of the proposition (if\n$l_1=l_2=0$). Thus, it remains to consider points $q \\in \\SNb^*\n\\mathbb{R}^n$ away from $\\Sb\nN_+^*S_+$, but there the microlocal membership of $\\Hbpm^{m-\\ep,0,k}$ is\nequivalent to not being an element of $\\WFb^{m+k-\\ep}$.\nNow $(x, \\xi) \\in \\WFb^{s,\n 0}(u_i)$ if and only if $\\xi \\in \\WF^s(\\chi \\wt{u_i})$ for each\n$\\chi$ with $\\chi \\equiv 1$ near $x$. But then by Lemma\n\\ref{thm:reallemma} we have \\eqref{eq:waveinsum} with $\\wt{u}_i$\nreplacing $u, v$. Thus the same statements hold for this product as\nfor the interior case, and the same argument shows that their product\nis in $\\Hb^{m+k- \\epsilon, 0}$, completing the proof of the\nproposition in $l_1=l_2=0$.\nThe weights are multiplicative, establishing the proposition apart\nfrom the constant $m$ case. In the case of constant $m$, we only need\nto observe that in all the arguments we can take $\\ep=0$.\n\\end{proof}\n\n\n\n\n\\subsection{A semilinear problem}\\label{sec:semilinear}\n\nUsing the above, one has a complete analogue of the semilinear\nresults of \\cite{HVsemi}. Concretely, one can conclude that the\nFeynman problem for the equation\n\\begin{equation}\\label{eq:semilinearequ}\n\\Box_g u+\\lambda u^p=f\n\\end{equation}\nis well-posed for appropriate $p \\in \\mathbb{N}$ (and small $f$), which includes $p\\geq 3$ if $n\\geq\n4$, so in particular the not-yet-second-quantized $\\varphi^4$ theory is\nwell-behaved on these curved space-times.\n\n\\begin{thm}\\label{thm:semilinear}\nSuppose $g$ is a perturbation of Minkowski space in the sense of\n Lorentzian scattering metrics (see Section~\\ref{sec:geometry}) so\n that in particular Theorem~\\ref{thm:modulereginverse} holds.\nLet $p\\in \\mathbb{N},\\lambda \\in \\RR$ with $p$ and the dimension $n$\nsatisfying\n\\begin{equation}\n \\label{eq:weightcondition3-thm}\n \\frac{2}{p-1} < \\frac{n - 2}{2}.\n\\end{equation}\nLet $\\lprime < 0$ satisfy\n\\begin{equation}\n \\label{eq:19-other}\n\\lprime \\in \\Big(\\frac{2}{p-1} - \\frac{n - 2}{2} , 0\\Big).\n\\end{equation}\nNote the interval is\nnon-empty. Let $k>\\frac{n-1}{2}$ be an integer and $m$ a function on\n$\\Sb^*M$ given by Corollary~\\ref{thm:algebracor}, or instead take\n$m>1\/2$ constant and\n$k > \\frac{n - 1}{2}$ integer with $m+l<1\/2$, $m+l+k>3\/2$ which exist if \\eqref{eq:19} holds. Then there is a\nconstant $C>0$ such that the small-data Feynman problem for\n\\eqref{eq:semilinearequ}, i.e.\\ given \n$$\nf\\in \\Hbpm^{m-1,\\lprime+(n-2)\/2+2,k} \\mbox{ with norm } < C\n$$ \nfinding $u\\in\\Hbpm^{m,\\lprime+(n-2)\/2,k}$ satisfying the equation, is\nwell-posed, and\n$u$ can be calculated as the limit of a Picard iteration corresponding\nto the perturbation series.\n\nIn particular the above holds for $p \\ge 4$ and $n \\ge 4$.\n\\end{thm}\n\n\\begin{rem}\n As mentioned, the condition on $p$ and $n$ in \\eqref{eq:weightcondition3-thm} holds in\n particular if $p \\ge 4$ and $n \\ge 4$. It holds also if $p = 3$ and\n $n \\ge 5$ and when $n = 3, p \\ge 6$, \\emph{but fails} for $p = 3, n=4$. We tackle this case in\n Theorem \\ref{thm:semilinearthree} below.\n\\end{rem}\n\n\\begin{rem}\\label{rem:semilinear-retarded}\nOur argument also works for the retarded and advanced problems\nconsidered in \\cite[Section~5]{HVsemi}, but it gives a somewhat\ndifferent result since the multiplicative properties we use are\nsomewhat different, as necessitated by the microlocal nature of the\nspaces that have to be used for the Feynman problems. For the retarded\/advanced problems\n\\cite[Section~5.4]{HVsemi} considers general first order\nsemilinearities. In the case of no derivatives an analogous result is\nshown there under the\nconstraint $p>1+\\frac{3}{n-2}$ which is weaker than\n\\eqref{eq:weightcondition3-thm}. For $n=4$ the difference is whether\n$p=3$ is admissible; it borderline fails our inequality. However, as noted above, in\nSection~\\ref{subsec:intricate-semilinear} we improve our result by\nshowing better multiplicative properties to handle $n=4$, $p=3$.\n\\end{rem}\n\n\\begin{rem}\nBy using $m$ constant as in the statement of the theorem we can\nconsider first order derivative nonlinearities as well, provided in\naddition $m>3\/2$ (so that for the first derivatives the b-order is\n$m-1>1\/2$, note that constant $m$ is crucial since we cannot afford\nthe $\\ep>0$ losses in multiplication);\nthe natural assumption is that these are of the form of a finite sum\nof products of vector fields $V_j\\in\\Vsc(M)$ applied to $u$, times $u^p$: $u^p(V_1\nu)\\ldots(V_{q}u)$, $p+q\\geq 2$. Writing these as $V_j=\\rho W_j$, $W_j\\in\\Vb(M)$, we\ncan proceed as in \\cite[Section~5.4]{HVsemi} and as we proceed below\nin the proof, provided that we can take some $l<-1$ (necessitated by\n$m>3\/2$, see Remark \\ref{rem:constant-order}) for the Feynman\npropagator invertibility considerations. This requires $n\\geq 5$ so\nthat $\\frac{n-2}{2}>1$. The numerology paralleling \\eqref{eq:weightcondition1}-\\eqref{eq:weightcondition2} is then that nonlinearities\nsatisfying\n$$\n(p-1)\\frac{n-2}{2}+q\\frac{n}{2}+(p+q)l-2\\geq l\n$$\ncan be handled by the same method. One can satisfy this with $l$ sufficiently close to $-1$,\n$l<-1$, if\n$$\n(p-1)(n-4)+q(n-2)> 2,\n$$\nwhich is weaker than the requirement $(p-1)(n-4)+q(n-2)> 4$ of\n\\cite[Equation~(5.15)]{HVsemi}, but recall that here we need $n\\geq\n5$. In particular, as our results also apply for the advanced\/retarded\nproblems, see Remark~\\ref{rem:semilinear-retarded}, if $n\\geq 5$, this slightly\nimproves the result of \\cite{HVsemi}, allowing e.g.\\ $q=1$ and $p=1$ for\n$n=5$; $q=1,p=1$ is not allowed in \\cite{HVsemi} even after\nimprovements discussed there in Remark~5.16 using analogues of our\nimprovements in Section~\\ref{subsec:intricate-semilinear}.\n\\end{rem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:semilinear}]\n As in \\cite[Section 5]{HVsemi}, moving the $\\lambda\n u^p$ to the right hand side, we rewrite\n \\eqref{eq:semilinearequ} as\n \\begin{equation}\n \\label{eq:Lsemilinearequ}\n L \\wt{u} = \\wt{f} - \\lambda \\rho^{- 2 + (p - 1)(n - 2)\/2}\n \\wt{u}^p, \n \\end{equation}\nwhere $\\wt{u} := \\rho^{-(n - 2)\/2} u, \\wt{f} := \\rho^{- 2 -(n - 2)\/2}\nf$. Assuming that $f \\in \\Hbpm^{m-1,\\lprime+(n-2)\/2+2,k}$, we have $\\wt{f}\n\\in \\Hbpm^{m-1,\\lprime,k}.$ To apply a Picard iteration to\n\\eqref{eq:Lsemilinearequ}, we want the right hand side to be in the\ndomain of the forward Feynman inverse of $L$, \n$L_{+-}^{-1} \\colon \\cY^{m - 1, \\lprime, k} \\lra \\cX^{m, \\lprime, k}$\n(where $\\cY^{m, \\lprime, k}, \\cX^{m, \\lprime, k}$ are defined as\nin \\eqref{eq:spaces} with the $\\Hb^{m, l}$ replaced by $\\Hb^{m, l, k}$), so by Theorem\n\\ref{thm:modulereginverse} we want it in $\\Hbpm^{m - 1, \\lprime, k}$ where $m\n+ \\lprime$ now satisfies the defining properties of the Feynman\npropagator and such that $|\\lprime| < (n - 2)\/2$. Furthermore, we want to apply the algebra properties in\nProposition \\ref{thm:algebra}; in particular we assume that $m\n> 1\/2$ everywhere. Note\nthat\nwe have both\n\\begin{equation}\\label{eq:conditions}\nm > 1\/2 \\mbox{ and } m + \\lprime < 1\/2\\ \\text{near}\\ \\Sb N_+^*S_+\\ \\text{and}\\ \\Sb N_+^*S_-,\n\\end{equation}\ni.e.\\ in the low regularity\n regions.\nThus $\\lprime$ (which is just a real number) \\emph{must be negative}, and furthermore for any $\\lprime <\n0$ there is a function $m$ meeting all of the criteria of\nCorollary~\\ref{thm:algebracor}, in particular\nboth the criteria in\n\\eqref{eq:conditions} and the Feynman criteria (since $m$ increases\nas one approaches the high regularity regions $\\Sb N_-^*S_-$ and\n$\\Sb N_-^*S_+$), as well as the convexity\/minima criteria\n\\eqref{eq:reg-function-condition}. (Indeed, note that by the remarks\npreceding Corollary~\\ref{thm:algebracor}, we could even take $m$\nconstant if we use Theorem~\\ref{thm:modulereginverse-better} in place of\nTheorem~\\ref{thm:modulereginverse}. In this case we can take $\\ep=0$ below.) Under these assumptions, by Proposition\n\\ref{thm:algebra}, for $k > (n - 1)\/2$ we have that $\\wt{u}^p$ lies in\n$\\Hbpm^{m - \\epsilon, p\\lprime, k}$ for any $\\epsilon > 0$, and thus $\\rho^{- 2 + (p - 1)(n - 2)\/2}\n \\wt{u}^p$ lies in $\\Hbpm^{m - \\epsilon, \\llprime, k}$ where\n\\begin{equation}\n \\label{eq:weightcondition1}\n \\llprime = - 2 + (p - 1)(n - 2)\/2 + p \\lprime,\n\\end{equation}\nand $\\Hbpm^{m - \\epsilon, \\llprime, k} \\subset \\Hbpm^{m - 1, \\lprime , k}$ if and\nonly if\n\\begin{equation}\n \\label{eq:weightcondition2}\n\\llprime \\ge \\lprime \\iff \\lprime \\ge \\frac{2}{p-1} - \\frac{n - 2}{2} ,\n\\end{equation}\nwhere again $\\lprime$ is an arbitrary negative number. \n\nFor any $p, n$ such that\n\\begin{equation}\n \\label{eq:weightcondition3}\n \\frac{2}{p-1} < \\frac{n - 2}{2},\n\\end{equation}\ntaking \n\\begin{equation}\n \\label{eq:19}\n \\llprime = \\lprime \\in \\Big(\\frac{2}{p-1} - \\frac{n - 2}{2} , 0\\Big)\n\\end{equation}\nsufficiently small, and $m$ picked correspondingly as above,\nwe claim that for every $\\delta > 0$, there is an $R \\ge 0$ such that\nif both $\\| \\wt{u} \\|_{\\Hb^{m, \\lprime, k}}$ and $\\| \\wt{v} \\|_{\\Hb^{m,\n \\lprime, k}}$ are bounded by $R$ then\n\\begin{equation}\n \\label{eq:17}\n \\| \\rho^{-2 + (p-1)(n-2)\/2} \\wt{u}^p - \\rho^{-2 + (p-1)(n-2)\/2}\n \\wt{v}^p \\|_{\\Hb^{m-1, \\lprime, k}} \\le \\delta \\| \\wt{u} - \\wt{v}\n \\|_{\\Hb^{m, \\lprime, k}}.\n\\end{equation}\nAssuming the claim for the moment, we see that the map \n\\begin{equation*}\n \\wt{u} \\mapsto L_{+-}^{-1} (\\wt{f} + \\lambda \\wt{u}^p \\rho^{-2 + (p-1)(n-2)\/2})\n\\end{equation*}\nis a contraction mapping on $\\Hb^{m, \\lprime, k}$ and thus the Picard\niteration $\\wt{u}_{n + 1} = L_{+-}^{-1} (\\wt{f} + \\lambda \\wt{u}_{n}^p \\rho^{-2 + (p-1)(n-2)\/2})$ with $\\wt{u}_1 = 0$ converges if\n$\\wt{f}$ is sufficiently small in $\\Hb^{m - 1, \\lprime, k}$ (as assumed\nin the theorem).\n\nThus it remains only to prove the claim. For any $\\lprime$ and for any\n$\\ep>0$ we have\n\\begin{equation}\\label{eq:almostdone}\n \\begin{split}\n \\| \\wt{u}^p - \\wt{v}^p \\|_{\\Hb^{m - \\epsilon, p\\lprime, k}} &= \\Big\\| (\\wt{u} - \\wt{v})\n \\sum_{j = 0}^{p-1} \\wt{u}^j \\wt{v}^{p-1-j} \\Big\\|_{\\Hb^{m-\\epsilon, p\\lprime, k}} \\\\\n &\\le C \\| \\wt{u} - \\wt{v} \\|_{\\Hb^{m, \\lprime, k}} \\max(\\| \\wt{u}\n \\|_{\\Hb^{m, \\lprime, k}}, \\| \\wt{u} \\|_{\\Hb^{m, \\lprime, k}})^{p-1}\n \\end{split}\n\\end{equation}\nprovided\n$m-(p-2)\\mu>1\/2$.\nSince $\\lprime$ satisfies \\eqref{eq:weightcondition2} with $\\llprime$ as in\n\\eqref{eq:19}, by bounding the $\\Hb^{m, \\lprime, k}$ norm with the\n$\\Hb^{m, \\llprime, k}$ norm,\n\\begin{equation*}\n \\| \\rho^{-2 + (p-1)(n-2)\/2} \\wt{u}^p - \\rho^{-2 + (p-1)(n-2)\/2}\n \\wt{v}^p \\|_{\\Hb^{m-\\epsilon, \\lprime, k}} \\le \\| \\wt{u}^p - \n \\wt{v}^p \\|_{\\Hb^{m-\\epsilon, p\\lprime, k}},\n\\end{equation*}\nand combining with \\eqref{eq:almostdone} gives the claim once $\\mu>0$\nis taken sufficiently small.\n\\end{proof}\n\n\\subsection{More intricate multiplicative properties and cubic\n semilinear problems for $n=4$}\\label{subsec:intricate-semilinear}\n\nTo extend to $p = 3$, $n = 4$, we need improvements of the regularity\nproperties for products which allow us to take the weight $\\llprime$ to be\ngreater than zero. To do so and still have $m + \\llprime < 1\/2$ in the low\nregularity zone, we need $m < 1\/2$, which is below the regularity\nthreshold in the work in Section \\ref{thm:basicalgebra}; thus we need\nimprovements of the results therein. The necessary improvements are\nbased on the ideas in the following. \n\\begin{lemma}\n Let $s, s' ,s_0\\in \\RR$. Let $u, v \\in H^{s_0}$, then $\\WF^{s'}(uv) \\subset \\WF^s(v)$, provided\n $s \\ge s_0 \\ge s'$ and $s - s' + s_0 > n\/2$. \n\\end{lemma}\nThe point here is that one can take $s_0 < n\/2$, and obtain a result\nfor $uv$ which says it is in a worse Sobolev space then $H^{s_0}$\nmicrolocally provided $v$ is in a better one microlocally.\n\nThe proof in fact follows the first of the product regularity\narguments above, namely that \\eqref{eq:basicassumptionsforproduct}\nimplies \\eqref{eq:basicwavefrontproduct}. Consider a point $\\xi_0$\nwith $(x_0,\\xi_0)\n\\not \\in \\WF^s(v)$, and a function $\\mathsf{s}\\geq s_0$ which equals $s$ on an open\ncone $\\mathsf{C} \\subset (\\WF^s(v))^{comp}$ and take $\\mathsf{s}' \\equiv s_0$\noutside some compact set $\\mathsf{K} \\subset \\mathsf{C}$, $\\mathsf{s}'\\equiv s'$ near\n$\\xi_0$ with $\\mathsf{s}'\\leq\\mathsf{s}$ everywhere; as we show this implies that $H^{\\mathsf{s}} \\cdot\nH^{s_0} \\subset H^{\\mathsf{s}'}$. Indeed, this is analogous to\n\\eqref{eq:basicmodel} above, and we break the relevant integral\n$I_\\xi$ up in the same way as in \\eqref{eq:twointegrals}, so we must bound integrals\n$$\n\\sup_{\\xi \\in K}\\int\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}-2\\mathsf{s}'}\n \\langle\\xi-\\eta\\rangle^{2s_0}}\\,d\\eta \\quad \n\\mbox{ and } \\quad \\sup_{\\xi\\in \\mathsf{K}}\\int\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}} \\langle\\xi-\\eta\\rangle^{2s_0-2\\mathsf{s}'}}\\,d\\eta.\n$$\nThe second integral is bounded by the arguments above, and for the\nfirst integral, the only difference is that over the set $\\mathsf{C}$, using\nthat $\\mathsf{s} - \\mathsf{s}'\\geq 0$ there, we have\n$$\n\\int_{\\mathsf{C}}\\frac{1}{\\langle\\eta\\rangle^{2\\mathsf{s}-2\\mathsf{s}'}\n \\langle\\xi-\\eta\\rangle^{2s_0}}\\,d\\eta \\le \\int_{\\mathsf{C}} \\frac{1}{\n \\langle\\xi-\\eta\\rangle^{2(\\mathsf{s} - \\mathsf{s}' + s_0 )}}\\,d\\eta \\le \\int_{\\mathsf{C}} \\frac{1}{\n \\langle\\xi-\\eta\\rangle^{2(s - s' + s_0 )}}\\,d\\eta ,\n$$\nwhich is finite since $s - s' + s_0 > n\/2$. The rest of the estimates\nare exactly as in the previous case.\n\n\nApplying this line of thinking to the model spaces $\\cY^{m, a}_d$\ndefined in \\eqref{eq:basicmodulereg}, we can obtain a regularity\nresult for products which allows us to dip under the threshold $d \/ 2$\nabove.\n\\begin{lemma}\\label{thm:reallemmaimproved}\n For $m, m', m_0, a \\in \\RR$ such that $m - m' + m_0 > d\/2$, $a > (n\n - d) \/ 2$ and $m \\ge m_0 \\ge m'$, we have $\\cY^{m_0, a}_d \\cdot\n \\cY_d^{m, a} \\subset \\cY_d^{m', a}$. Furthermore,\n \\begin{equation}\n \\label{eq:10}\n \\WF^{m' + a}(uv) \\subset (\\WF^{m + a}(u) + \\WF^{m + a}(v))\\cup \\WF^{m + a}(u) \\cup \\WF^{m + a}(v).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n To see that the first conclusion holds, we argue as in Lemma\n \\ref{thm:basicalgebra}, and thus use the inequality\n \\begin{equation}\n \\label{eq:notalgebra}\n \\begin{split}\n & \\int \\left( \\frac{\\langle \\xi \\rangle^{m'}\\langle \\xi'' \\rangle^a}{\\langle \\xi - \\eta \\rangle^{m_0} \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta \\rangle^{m} \\langle \\eta''\n \\rangle^a} \\right)^2 d\\eta \\\\\n &\\le \\sum_{i, j = 1}^2 \\int \\left( \\frac{f_i g_j}{\\langle \\xi - \\eta \\rangle^m \\langle \\xi'' - \\eta'' \\rangle^a \\langle \\eta''\n \\rangle^a} \\right)^2 d\\eta , \n \\end{split}\n \\end{equation}\nwhere $f_1 = \\langle \\eta \\rangle^{m'}$, $f_2 = \\langle \\xi - \\eta \\rangle^{m'}$ and $g_1 =\n\\langle \\eta'' \\rangle^a$, $g_2 = \\langle \\xi'' - \\eta'' \\rangle^a$. Replacing the\nunprimed variable with primed variables and using that both\n\\begin{equation*}\n \\int \\left( \\frac{\\langle \\eta' \\rangle^{m'}}{\\langle \\xi - \\eta' \\rangle^{m_0} \\langle\n \\eta' \\rangle^{m} } \\right)^2 d\\eta', \\qquad \\int \\left( \\frac{\\langle \\xi' - \\eta' \\rangle^{m'}}{\\langle \\xi - \\eta' \\rangle^{m_0} \\langle \\eta' \\rangle^{m} } \\right)^2 d\\eta'\n\\end{equation*}\nare uniformly bounded under the stated assumptions on $m, m', m_0$\ngives the statement.\n\nThe wavefront set containment is obtained by locating similar\nimprovements in the proof of Lemma \\ref{thm:reallemma}. Indeed, as\nthere, we have that $u \\in H^{(w_1)}, v \\in H^{(w_2)}$ where $w_i(\\xi) =\n\\langle \\xi \\rangle^{m_0}\\langle \\xi'' \\rangle^a + \\langle \\xi \\rangle^{s_i}$ where\n$s_i = s_i(\\xi)$ where the $s_i \\equiv m + a$ off open conic sets\n$\\wt{\\mathsf{C}}_i$ are arbitrary open sets containing, respectively, \n$\\WF^{m + a}(u)$ and $\\WF^{m + a}(v)$. We \nwant to show that given\n$$\n\\xi \\not \\in (\\WF^{m + a}(u) + \\WF^{m +\n a}(v))\\cup \\WF^{m + a}(u) \\cup \\WF^{m +\n a}(v)\n$$\nand a proper choice of function $s$ with $s = m' + a$ near $\\xi$ that $uv \\in H^{s}$,\nwhich amounts to applying Lemma \\ref{thm:algcondition} with $w = s$ and\n$w_1, w_2$ exactly as in Lemma \\ref{thm:reallemma}.\n\nWe thus want to bound an integral similar to \\eqref{eq:etapart},\nnamely\n\\begin{equation*}\n \\int\\Big(\\frac{\\langle \\xi \\rangle^s}{(\\langle \\eta \\rangle^{m_0} \\langle\n \\eta'' \\rangle^k + \\langle \\eta \\rangle^{s_{1}} )\n (\\langle \\xi - \\eta \\rangle^{m_0} \\langle\n \\xi'' - \\eta'' \\rangle^k + \\langle \\xi - \\eta \\rangle^{s_{2}} )}\\Big)^2\\,d\\eta,\n\\end{equation*}\n We choose\n$s$ so that $s \\le m + a$ and bound\n$\\langle \\xi \\rangle^{2s} \\lesssim \\langle \\eta \\rangle^{2(m + a)} + \\langle \\xi - \\eta\n\\rangle^{2(m + a)} $ and as usual break the integral into two parts\ninvolving the two terms on the right of this bound. Again we focus on\nthe $\\langle \\eta \\rangle^{2s}$ term. Integrating first over $\\wt{\\mathsf{C}}_1$ and then\n$(\\wt{\\mathsf{C}}_1)^{comp}$; over $\\wt{\\mathsf{C}}_{1}^{c}$, we have that $\\langle \\eta \\rangle^m \\langle\n \\eta'' \\rangle^k + \\langle \\eta \\rangle^{s_{1}} >(1\/2) \\langle \\eta \\rangle^{s}$, so\n that part of the integral with $\\langle \\eta \\rangle^{2s}$ in the\n numerator over\n $\\wt{\\mathsf{C}}_1$ is bounded by\n \\begin{gather*}\n \\int\\Big(\\frac{1}{\\langle \\eta \\rangle^{s_1 - s} \\langle \\xi - \\eta \\rangle^m\n \\langle \\xi'' - \\eta'' \\rangle^k + \\langle \\xi - \\eta \\rangle^{s_{2}}\n }\\Big)^2\\,d\\eta \\\\\n \\le \\int\\Big(\\frac{1}{\\langle \\eta' \\rangle^{m - m'} \\langle \\xi' - \\eta' \\rangle^{m_0}\n \\langle \\xi'' - \\eta'' \\rangle^k },\n \\Big)^2\\,d\\eta .\n \\end{gather*}\nwhich, by separating into primed and double primed coordinates and\nusing the assumptions on $m, m_0, m'$ is uniformly bounded. The rest\nof the bounds proceed analogously and are left to the reader.\n\\end{proof}\n\nBy reducing locally and arguing exactly as in the proof of\nProposition~\\ref{thm:algebra} and Corollary~\\ref{thm:algebracor}, we obtain\n\\begin{prop}\\label{thm:bwaveimproved}\n For $\\lprime$ sufficiently small, there exists $m \\colon \\Sb^* M \\lra \\mathbb{R}$ satisfying: 1) that $m \\ge 1\/2 -\n \\delta$ for some $\\delta \\in (0, 1\/2)$, 2) $m, \\lprime$ satisfy the forward\n Feynman condition in the strengthened form given in Theorem~\\ref{thm:fredholm}, and 3) $m$ satisfies the condition on the\n sublevel sets and minima in \\eqref{eq:reg-function-condition}.\n Moreover, for $k\n \\in \\Nat$ satisfying $k > (n - 1)\/2$, \n$$\n\\Hbpm^{m,l_1,k}\\Hbpm^{m, l_2,k}\\subset\n\\Hbpm^{m - 2\\delta-0, l_1+l_2, k}.\n$$\nIn particular, for $\\delta$ sufficiently small $(\\Hbpm^{m,l_1,k})^3 \\subset \\Hbpm^{m - 4\\delta-0, l_1+l_2, k}.$\n\\end{prop}\n\n\n\nWe can now finally prove\n\\begin{thm}[$p = 3, n = 4$]\\label{thm:semilinearthree}\nSuppose $g$ is a perturbation of Minkowski space in the sense of\n Lorentzian scattering metrics (see Section~\\ref{sec:geometry}), in particular so that Theorem\n\\ref{thm:modulereginverse} holds.\nIn dimension $n = 4$, given $\\lambda \\in \\RR$ and a weight\n$\\lprime \\ge 0$ and $\\lprime$ sufficiently small, and a regularity\nfunction $m$ as in Proposition \\ref{thm:bwaveimproved}, there\nis $C>0$ such that the small-data Feynman problem, i.e.\\ given \n$$\nf\\in \\Hbpm^{m-1,\\lprime+(n-2)\/2+2,k} \\mbox{ with norm } < C\n$$ \nfinding $u\\in\\Hbpm^{m,\\lprime+(n-2)\/2,k}$ satisfying \n\\begin{equation}\n \\label{eq:smalldata3}\n \\Box_{g,+-} u + \\lambda u^3 = f,\n\\end{equation} \nis\nwell-posed with $k > (n -1 )\/2$, and\n$u$ can again be calculated as the limit of a Picard iteration\ncorresponding\nto the perturbation series. \n\\end{thm}\n\n\\begin{rem}\n Again by Remark \\ref{rem:constant-order}, there is a version of this\n theorem with $m$ constant, $m \\ge 1\/2 - \\delta$, $l \\ge 0$, $m + l\n < 1\/2$ and $k > (n - 1)\/2$ satisfying $m + l + k > 3\/2$. We leave\n the details of this straightforward modification of the following\n proof to the reader.\n\\end{rem}\n\n\\begin{rem}\nAlthough we do not state it here explicitly, Proposition\n \\ref{thm:bwaveimproved} also gives improvements to the statement of Theorem \\ref{thm:semilinear} for\n other $n,p$ in terms of the spaces in which solvability holds (what\n $\\lprime$ can be),\n though not for whether there {\\em is} a space of the kind considered\n there in which solvability holds.\n\\end{rem}\n\n\\begin{proof}\n The proof is identical to that of Theorem \\ref{thm:semilinear}\n incorporating the improvements given by Proposition\n \\ref{thm:bwaveimproved}. We\n take $\\lprime \\ge 0$ and find an $m$ such that $m + \\lprime$ satisfies the\n Feynman condition and $m > 1\/2 - \\delta$ for some small $\\delta >\n 0$. Rewriting the equation as in\n \\eqref{eq:Lsemilinearequ} with $\\wt{f}$ and $\\wt{u}$ defined in the\n same way, and assuming that $\\wt{f} \\in \\Hbpm^{m, \\lprime, k}$, the\n condition that \n$$\n\\rho^{2 - (p - 1)(n-2)\/2}\\wt{u}^p = \\rho^{4 - n}\\wt{u}^3 \\in\n\\Hbpm^{m - 5\\delta, \\lprime, k} \\subset \\Hbpm^{m - 1, \\lprime, k}\n$$\nis now that $\\delta$ be less than $1\/5$ and that\n\\begin{equation}\\label{eq:pthreeweightcondition}\n \\lprime \\le 4 - n + 3 \\lprime \\iff \\lprime \\ge n\/2 - 2.\n\\end{equation}\nIf $n = 4$, we can thus find $\\lprime$ and $m$ satisfying the Feynman\nconditions and \\eqref{eq:pthreeweightcondition} simultaneously. \nFrom now on we assume that $n = 4$. The existence of an $m$\nsatisfying the conditions in \\eqref{eq:reg-function-condition} is a\ntrivial modification of the proof of Corollary \\ref{thm:algebracor}.\n\n\n\n\nThe Picard iteration argument is now identical to that in Theorem\n\\ref{thm:semilinear} except incorporating the loss in Proposition \\ref{thm:bwaveimproved}. In this case, the claim in \\ref{eq:17} is\nsubstituted by the following: for every $\\delta > 0$ sufficiently small, there is an $R \\ge 0$ such that\nif both $\\| \\wt{u} \\|_{\\Hb^{m - \\delta, \\lprime, k}}$ and $\\| \\wt{v}\n\\|_{\\Hb^{m - \\delta,\n \\lprime, k}}$ are bounded by $R$ then\n\\begin{equation*}\n \\|\\wt{u}^3 -\\wt{v}^3 \\|_{\\Hb^{m - 5\\delta, \\lprime, k}} \\le \\delta\n \\| \\wt{u} - \\wt{v} \\|_{\\Hb^{m - \\delta, \\lprime, k}}.\n\\end{equation*}\nThis and the rest of the proof follow exactly as in Theorem\n\\ref{thm:semilinear} using the improvement in Proposition\n\\ref{thm:reallemmaimproved}.\n\\end{proof}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nModels have been widely used to study the properties of the\nhadron spectrum due to the impossibility to solve\nQuantum Chromodynamics (QCD) at the\ncurrent moment.\nIn particular, the quark potential models incorporate\nthe perturbative one-gluon exchange quark-quark ($qq$) potential\n($V_{\\rm OGE}$) derived from QCD as well as a parametrization\nof some nonperturbative effects through a $qq$ confining\npotential ($V_{\\rm con}$) \\cite{RUJ}. Such an {\\em effective\ntheory} [the quark-gluon coupling constant is taken as an\neffective one and the constituent quark masses ($m_q$) are parameters\nfitted from the baryon magnetic moments] provides a reasonable\nunderstanding of the baryon spectrum and the hadron static\nproperties \\cite{YAO}.\n\nThe idea of an interquark potential has been also\nused to study baryon-baryon interactions. More\nexplicitly, the repulsive core of the nucleon-nucleon (NN)\nforce has been shown to arise from the color-spin structure\nof the $V_{\\rm OGE}$ \\cite{FAE}. Nevertheless, the same\nscheme has been proven incapable of describing the long\nrange part of the NN interaction unless pion exchange\nbetween quarks is introduced. From the basic theory,\nthe origin of the one-pion exchange potential ($V_{\\rm OPE}$)\nis associated with the spontaneous chiral symmetry breaking\nof QCD.\nMoreover, the inclusion of the\n$qq$ sigma potential ($V_{\\rm OSE}$) consistently with its\nchiral partner (the pion) allows to reproduce the\nintermediate range $NN$ interaction as well as the\ndeuteron properties \\cite{FER}.\n\nIn this paper we examine the consistency of the two scenarios.\nWe determine a $qq$ interaction by studying two nucleon\nproperties and charge exchange reactions and proceed to\nanalyze the baryon spectrum for this interaction. As we\nshall discuss the predictions for the baryon masses and\nthe baryon wave functions, it is not only a stringent\ntest of the potential but also a consistency test of the\nformalism used to generate the NN interaction\n[resonating group method (RGM)].\n\n\\section{The Quark-Quark Potential}\n\nThe starting point of our description is\na quark-quark interaction of the following form:\n\n\\begin{equation}\nV_{\\rm qq} (\\vec{r}_{ij}) = V_{\\rm con} (\\vec{r}_{ij}) +\nV_{\\rm OGE} (\\vec{r}_{ij})\n+ V_{\\rm OPE} (\\vec{r}_{ij}) + V_{\\rm OSE} (\\vec{r}_{ij}) \\, ,\n\\end{equation}\n\n\\noindent\nwhere $\\vec{r}_{ij}$ is the interquark distance.\n\nThe confinement is chosen to be linear as suggested by the\nmeson spectrum and lattice calculations:\n\n\\begin{equation}\nV_{\\rm con} (\\vec{r}_{ij}) = - a_c \\,\n( \\vec{\\lambda}_i \\cdot \\vec{\\lambda}_j ) \\, r_{ij} \\, ,\n\\end{equation}\n\n\\noindent\nwhere $\\lambda 's$ are the SU(3) color matrices.\n\n$V_{\\rm OGE}$, $V_{\\rm OPE}$ and $V_{\\rm OSE}$ have been derived in detail\nelsewhere \\cite{RUJ,FER} and we shall limit here to write\nthe final expressions:\n\n\\begin{equation}\nV_{\\rm OGE} ({\\vec r}_{ij}) =\n{1 \\over 4} \\, \\alpha_s \\, {\\vec\n\\lambda}_i \\cdot {\\vec \\lambda}_j\n\\Biggl \\lbrace {1 \\over r_{ij}} -\n{1 \\over {4 \\, m^2_q}} \\, \\biggl [ 1 + {2 \\over 3}\n{\\vec \\sigma}_i \\cdot {\\vec\n\\sigma}_j \\biggr ] \\,\\,\n{{e^{-r_{ij}\/r_0}} \\over\n{r_0^2 \\,\\,r_{ij}}}\n- {1 \\over {4 m^2_q \\, r^3_{ij}}}\n\\, S_{ij} \\Biggr \\rbrace \\, ,\n\\end{equation}\n\n\\noindent\nwhere $\\alpha_s$ is the effective quark-quark-gluon coupling constant,\n$r_0$ is the range of a smeared $\\delta$ function in order to avoid\nand unbound spectrum \\cite{BHA},\nthe $\\sigma ' s$\nstand for the spin Pauli matrices and $S_{ij}$ is the quark tensor operator\n$S_{ij} = 3 (\\vec{\\sigma}_i \\, . \\, \\hat{r}_{ij}) (\\vec{\\sigma}_j\n\\, . \\, \\hat{r}_{ij}) - \\vec{\\sigma}_i \\, . \\vec{\\sigma}_j $.\n\n\\begin{eqnarray}\nV_{\\rm OPE} ({\\vec r}_{ij}) & = & {1 \\over 3}\n\\, \\alpha_{ch} {\\Lambda^2 \\over \\Lambda^2 -\nm_\\pi^2} \\, m_\\pi \\, \\Biggr\\{ \\left[ \\,\nY (m_\\pi \\, r_{ij}) - { \\Lambda^3\n\\over m_{\\pi}^3} \\, Y (\\Lambda \\,\nr_{ij}) \\right] {\\vec \\sigma}_i \\cdot\n{\\vec \\sigma}_j + \\nonumber \\\\\n & & \\left[ H( m_\\pi \\, r_{ij}) - {\n\\Lambda^3 \\over m_\\pi^3} \\, H( \\Lambda \\,\nr_{ij}) \\right] S_{ij} \\Biggr\\} \\,\n{\\vec \\tau}_i \\cdot {\\vec \\tau}_j \\, ,\n\\end{eqnarray}\n\n\\begin{equation}\nV_{\\rm OSE} ({\\vec r}_{ij}) = - \\alpha_{ch} \\,\n{4 \\, m_q^2 \\over m_{\\pi}^2}\n{\\Lambda^2 \\over \\Lambda^2 - m_{\\sigma}^2}\n\\, m_{\\sigma} \\, \\left[\nY (m_{\\sigma} \\, r_{ij})-\n{\\Lambda \\over {m_{\\sigma}}} \\,\nY (\\Lambda \\, r_{ij}) \\right] \\, ,\n\\end{equation}\n\n\\noindent\nwhere $m_\\pi$ ($m_\\sigma$) is the pion (sigma) mass,\n$\\alpha_{ch}$ is the chiral coupling constant (related to the\n$\\pi NN$ coupling constant), $\\Lambda$ is a cutoff\nparameter and Y(x), H(x) are the Yukawa functions defined as:\n\n\\begin{equation}\nY(x) \\, = \\, {e^{-x} \\over x} \\,\\,\\,\\, , \\,\\,\\,\\,\nH(x) \\, = \\, \\Bigl( 1 + {3 \\over x} + { 3 \\over {x^2}} \\Bigr) Y(x) \\, ,\n\\end{equation}\n\nWith this interaction, the two baryon system has been studied in\nthe RGM framework assuming\nfor the spatial part of the\nwave function of the quarks a harmonic\noscillator ground state,\n\n\\begin{equation}\n\\eta_{\\rm os} (\\vec{r}_i - \\vec{R} \\, ) =\n\\left( {1 \\over {\\pi b^2}} \\right)^{3\/4} e^{-(\\vec{r}_i\n- \\vec{R}\\,)^2\/ 2b^2} \\, ,\n\\end{equation}\n\n\\noindent\nwhere the parameter $\\vec{R}$\ndetermines the position of the baryon and\n$b$ is the harmonic oscillator constant.\n\n\\begin{figure}[t]\n\\vbox{\n\\vspace*{-2.75cm}\n\\centerline{\\epsfig{file=espe.eps,height=5.2in}}\n\\vspace*{-3.5cm}\n\\caption{Relative energy $N$ and $\\Delta$ spectrum\nup to 0.7 GeV excitation energy. The solid line corresponds\nto the results of our model. The boxes represent the\nexperimental data with the corresponding uncertainties.}\n\\label{espe}\n}\n\\end{figure}\n\n\\begin{table}[hbt]\n\\caption[]{Value of the potential parameters}\n\\label{param}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\tabstrut $m_q$ & $b$ & $\\alpha_s$ & $\\alpha_{ch}$ & $a_c$ &\n$m_\\pi$ & $m_\\sigma$ & $\\Lambda$ \\\\\nMeV & fm & & & ${\\rm MeV} \\cdot {\\rm fm}^{-1}$ & ${\\rm fm}^{-1}$\n& ${\\rm fm}^{-1}$ & ${\\rm fm}^{-1}$ \\\\\n\\hline\n313 & 0.518 & 0.485 & 0.027 & 91.488 & 0.7 & 3.42 & 4.2 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nUsing the same set of parameters given in Table \\ref{param}, the $NN$\nscattering phase shifts, the static and electromagnetic\nproperties of the deuteron \\cite{FER} and reactions that\ntake place with the excitation of the $\\Delta$ resonance\n\\cite{FER2} are reasonably reproduced.\n\n\\section{Results}\n\nTo study the baryonic spectrum with the\npotential just described (parameters\nas in Table \\ref{param})\nwe solve the Schr\\\"{o}dinger equation in\nthe hyperspherical harmonic\napproach \\cite{BAL}. The low energy $N$ and $\\Delta$\nspectrum obtained, a part of which is shown in Fig. \\ref{espe},\nis quite reasonable though some small discrepancy remains\nconcerning the relative energy position of the Roper\nresonance [$N^* (1440)$] and the first negative parity\nstate excitation [$N^- (1535)$], as it is common in two body\npotential models.\n\n\\begin{figure}[t]\n\\vbox{\n\\vspace*{-3.0cm}\n\\centerline{\\epsfig{file=pot.eps,height=4.0in}}\n\\vspace*{-0.1cm}\n\\caption{Adiabatic $NN$ potential for different nucleon\nwave functions for the channel (S,T)=(1,0).\nThe solid line corresponds to the solution\nof the Schr\\\"odinger equation. The others to a gaussian\nansatz with the values of $b$ given in the figure.}\n\\label{pot}\n}\n\\end{figure}\n\nRegarding the wave function and in order to check the internal\nRGM consistency, it makes sense to compare the $NN$ potential\none obtains from the wave function solution of the Schr\\\"odinger\nequation, with the potential obtained from the ansatz RGM\nwave function for different values of the parameter $b$. For the\nsake of technical simplicity we do this in the Born-Oppeheimer\napproximation. The results appear in Fig. \\ref{pot}, where we can see\nthat it is precisely the value of $b$ that gives\nthe best overall fit to the $NN$ data with RGM, the one\nwhich provides a better approximation. This confers to\n$b$ a self-consistent character and solves the controversy\nabout its possible values.\n\nCertainly, more work and further refinements are needed along\nthis direction. Meanwhile, the proposed\nmodel points out the plausibility of a\nunified description of the baryon structure and the\nbaryon-baryon interaction.\n\n\\begin{acknowledge}\nThis work has been partially funded by Direcci\\'on\nGeneral de Investigaci\\'on Cient\\'{\\i}fica y T\\'ecnica\n(DGICYT) under the Contract No. PB91-0119\n\\end{acknowledge}\n\n\\catcode`\\@=11 \\if@amssymbols%\n\\clearpage\n\\else\\relax\\fi\\catcode`\\@=12\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this short note, we consider the following K\\\"ahler-Ricci \nflow \n\\begin{equation}\n\\label{krf}\n\\frac{\\partial\\widetilde\\omega_t}{\\partial t}=-{\\rm Ric}(\\widetilde\\omega\n_t)-\\widetilde\\omega_t, ~~~~\\tilde\\omega_0=\\omega_0.\n\\end{equation}\nover a closed K\\\"ahler manifold $X$ where $\\omega_0$ is \nany K\\\"ahler metric on $X$. The short time existence of the \nsolution is known from either R. Hamilton's general existence \nresult on Ricci flow in \\cite{ham} or the fact that K\\\"ahler-Ricci \nflow is indeed parabolic. \\\\ \n\nBy the optimal existence result on K\\\"ahler-Ricci flow as in \n\\cite{cascini-lanave} or \\cite{t-znote}, we know the classic \nsolution of (\\ref{krf}) exists exactly as long as the cohomology \nclass $[\\widetilde\\omega_t]$ from formal computation remains \nto be K\\\"ahler. The actually meaning will be explained later. \n\nIt then comes down to the study of the behavior of the metric \nsolution when approaching the flow singularity. In this work, \nwe focus on the study of the case when the flow singularity \nhappens at some finite time. Let's state the main theorems \nbelow. \n\n\\begin{theorem}\n\\label{th:R-blows-up}\nK\\\"ahler-Ricci flow (\\ref{krf}) either exists for all time, or \nthe scalar curvature blows up (from above) at some finite \ntime, i. e. \n$$sup_{X\\times[0, T)}|R(\\widetilde\\omega_t)|=+\\infty$$ \nwhere $T$ is the finite singular time. \n\\end{theorem}\n\nThe blow-up would be from above in sight of the classic \nresult on the lower bound of scalar curvature. Let's point \nout that the statement of this theorem also holds for other \nusual versions K\\\"ahler-Ricci flows by simple rescaling \nconsideration. \n\n\\begin{remark}\n\nThere have been some fundamental results regarding \nthe finite time blowing-up of Ricci flow. In fact, it's known \nthat curvature operator blows up by R. Hamilton's work \n\\cite{ham4} and Ricci curvature blows up by N. Sesum's \nwork \\cite{sesum}. \n\n\\end{remark} \n \nIn the case of finite time singularity, suppose we have a \nholomorphic map \n$$F: X\\longrightarrow Y$$ \nwhere $Y$ is an analytic variety smooth near the image \n$F(X)$ and there is a K\\\"ahler metric, $\\omega_{_{M}}$ \nin a neighborhood of $F(X)$ such that $[\\widetilde\\omega\n_t]=[F^*\\omega_{_{M}}]$. Then we have the following \ncontrol of the blowing-up of scalar curvature. \n\n\\begin{theorem}\n\\label{th:R-control}\nIn the above setting, for the flow (\\ref{krf}), \n$$R(\\widetilde\\omega_t)\\leqslant\\frac{C}{(T-t)^2}$$ \nwhere $C$ is a positive constant. \n\\end{theorem} \n\nThe motivation of the setting of this theorem is the \nsemi-ampleness of the cohomology limit at singular \ntime. It is of quite some interest in algebraic geometry \nas explained in \\cite{t-znote}, for example. \n\n\\begin{acknowledgment}\n\nThe author would like to thank R. Lazarsfeld for pointing \nout the result by J. Demailly and M. Paun in \\cite\n{demailly-paun} which is absolutely crucial to conclude \nTheorem \\ref{th:R-blows-up} for general closed (K\\\"ahler) \nmanifolds. The discussion with J. Song is also valuable \nfor this result. The comparison with the result by G. \nPerelman as mentioned in the last section is suggested \nby J. Lott. The author can not thank his advisor, G. Tian, \nenough for introducing him to this interesting topic and \nconstant encouragement along the way. \n\n\\end{acknowledgment}\n\n\\section{Proof of Theorem \\ref{th:R-blows-up}}\n\nThe proof is by contradiction. We assume {\\bf the \nscalar curvature is uniformly bounded along the \nflow with finite time singularity}. One then makes \nuse of J. Song and G. Tian's parabolic Schwarz \nLemma (as in \\cite{song-tian}) and some basic \ncomputations on this K\\\"ahler-Ricci flow to get \nsome uniform control of the flow metric. The \ncontradiction then comes from the general result \non the existence of K\\\"ahler-Ricci flow and the \nnumerical characterization of K\\\"ahler cone on \nclosed K\\\"ahler manifolds by J. Demailly and M. \nPaun. The rest of this section contains the detailed \nargument. \\\\\n\nAs usual when dealing with K\\\"ahler-Ricci flow, we \nneed the scalar version of (\\ref{krf}) described below \nas in \\cite{t-znote}. \n\nLet $\\omega_t=\\omega_\\infty+e^{-t}(\\omega_0-\n\\omega_\\infty)$ where $[\\omega_\\infty]=K_X$ and \n$\\omega_\\infty=-{\\rm Ric}(\\Omega)$ for some smooth \nvolume form $\\Omega$ over $X$. Then set $\\tilde\n\\omega_t=\\omega_t+\\sqrt{-1}\\partial\\bar\\partial u$ and one \nhas the following parabolic evolution equation for \n$u$,\n\\begin{equation}\n\\label{skrf}\n\\frac{\\partial u}{\\partial t}={\\rm log}\\frac{(\\omega_t+\\sqrt{-1}\\partial\\bar\n\\partial u)^n}{\\Omega}-u, ~~~~u(\\cdot, 0)=0.\n\\end{equation}\n\nNow we state the following optimal existence result of \nK\\\"ahler-Ricci flow (as in \\cite{cascini-lanave} and \\cite\n{t-znote}) mentioned in Introduction. \n\\begin{prop}\n\\label{existence}\n(\\ref{krf}) (or (\\ref{skrf}) equivalently) exists as long as \n$[\\omega_t]$ remains K\\\"ahler, i. e. the solution is for \nthe time interval $[0, T)$ where \n$$T=sup\\{t|\\,[\\omega_t]~ \\text{is K\\\"ahler}.\\}.$$\n\\end{prop}\n\nThe finite time singularity means $[\\omega_T]$ is on \nthe boundary of the (open) K\\\"ahler cone, and thus no \nlonger K\\\"ahler. Clearly it's \"numerically effective\" using \nthe natural generalization of the notion from algebraic \ngeometry. \n\nFrom now on, we consider the flow existing only for some \nfinite interval $[0, T)$. As usual, the $C$'s below might \nstand for different positive constants. In case that the \nsituation is more subtle, lower indices are used to tell \nthem apart. The argument is organized into three steps. \\\\ \n\n\\noindent $\\bullet$ {\\bf Volume Form (Lower) Bound} \n\n\\vspace{0.1in}\n\nWith the bounded scalar curvature assumption, we \ncan easily derive the uniform control on the volume \nform along the flow, using the following evolution of \nvolume form, \n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial\\tilde\\omega^n_t}{\\partial t}\n&= n\\frac{\\partial\\widetilde\\omega_t}{\\partial t}\\wedge{\\tilde\n\\omega^{n-1}_t} \\\\\n&= n(-{\\rm Ric}(\\widetilde\\omega_t)-\\widetilde\n\\omega_t)\\wedge\\tilde\\omega^{n-1}_t \\\\\n&= (-R-n)\\tilde\\omega^n_t. \n\\end{split}\n\\end{equation*}\n\nThis gives $|\\frac{\\partial u}{\\partial t}+u|\\leqslant C$ as $\\widetilde\\omega^n_t=e^{\\frac{\\partial u}{\\partial t}+u}\\Omega$. \n\n\\begin{remark}\n\nInstead of the assumption on scalar curvature, one can \nalso directly assume positive lower bound for the volume \nform or equivalently, $\\frac{\\partial u}{\\partial t}\\geqslant -C$ since \nwe are considering the finite time singularity case. This \nsimple observation actually brings up a very intuitive \nunderstanding of Theorem \\ref{th:R-blows-up}: the flow \n(\\ref{skrf}) is stopped at some finite time because the \nterm in $\\log$ is tending to $0$, i. e. no uniform lower \nbound. \n\n\\end{remark}\n\n\\vspace{0.1in}\n\n\\noindent $\\bullet$ {\\bf Metric Estimate}\n\n\\vspace{0.1in}\n\nWe begin with the inequality from parabolic Schwarz \nLemma. In this note, the Laplacian $\\Delta$ without \nlower index, is always with respect to the changing \nmetric along the flow, $\\widetilde\\omega_t$. \n\nLet $\\phi=\\<\\widetilde\\omega_t, \\omega_0\\>>0$. Using \ncomputation for (\\ref{krf}) in \\cite{song-tian}, one has \n\\begin{equation}\n\\label{pschwarz}\n(\\frac{\\partial}{\\partial t}-\\Delta){\\rm log}\\phi\\leqslant C_1\\phi+1,\n\\end{equation}\nwhere $C_1$ is a positive constant depending on the \nbisectional curvature of $\\omega_0$. It's quite irrelevant \nthat $\\omega_0$ is the initial metric for the K\\\"ahler-Ricci \nflow. In fact, it doesn't have to be a metric over $X$ which \nis an interesting part of this lemma as indicated in \\cite\n{song-tian}. This is useful for the proof of Theorem \n\\ref{th:R-control}. \\\\\n\nApplying Maximum Principle to (\\ref{skrf}) gives $u\n\\leqslant C$. Take $t$-derivative to get \n$$\\frac{\\partial}{\\partial t}\\left(\\frac{\\partial u}{\\partial t}\\right)=\\Delta\\left(\n\\frac{\\partial u}{\\partial t}\\right)-e^{-t}\\<\\widetilde\\omega_t, \\omega\n_0-\\omega_\\infty\\>-\\frac{\\partial u}{\\partial t}.$$\nIt can be reformulated into the following two equations, \n$$\\frac{\\partial}{\\partial t}\\left(e^t\\frac{\\partial u}{\\partial t}\\right)=\\Delta\\left(\ne^t\\frac{\\partial u}{\\partial t}\\right)-\\<\\widetilde\\omega_t, \\omega_0-\n\\omega_\\infty\\>,$$\n$$\\frac{\\partial}{\\partial t}\\left(\\frac{\\partial u}{\\partial t}+u\\right)=\\Delta\\left(\n\\frac{\\partial u}{\\partial t}+u\\right)-n+\\<\\widetilde\\omega_t, \\omega_\n\\infty\\>.$$\nTheir difference gives\n\\begin{equation}\n\\label{u-decreasing}\n\\frac{\\partial}{\\partial t}\\left((e^t-1)\\frac{\\partial u}{\\partial t}-u\\right)=\\Delta\n\\left((e^t-1)\\frac{\\partial u}{\\partial t}-u\\right)+n-\\<\\widetilde\\omega_t, \n\\omega_0\\>.\n\\end{equation}\nBy Maximum Principle, this gives\n$$(e^t-1)\\frac{\\partial u}{\\partial t}-u-nt\\leqslant C,$$\nwhich together with the upper bound of $u$ and local \nbound for $\\frac{\\partial u}{\\partial t}$ near $t=0$ would provide \n$$\\frac{\\partial u}{\\partial t}\\leqslant C.$$\nThe upper bounds on $\\frac{\\partial u}{\\partial t}$ and $u$ together \nwith $|\\frac{\\partial u}{\\partial t}+u|\\leqslant C$ from volume control \ngive the uniform (lower) bounds on $\\frac{\\partial u}{\\partial t}$ and \n$u$. \\\\\n\nMultiply (\\ref{u-decreasing}) by a large enough constant \n$C_2>C_1+1$ and combining it with (\\ref{pschwarz}), one \narrives at \n\\begin{equation}\n\\label{main}\n\\begin{split}\n(\\frac{\\partial}{\\partial t}-\\Delta)\\left({\\rm log}\\phi+\n(e^t-1)\\frac{\\partial u}{\\partial t}-u\\right)\n&\\leqslant nC_2+1-(C_2-C_1)\\phi \\\\\n&\\leqslant C_3-\\phi.\n\\end{split}\n\\end{equation}\nApply Maximum Principle for ${\\rm log}\\phi+(e^t-1)\\frac\n{\\partial u}{\\partial t}-u$. Considering the place where it achieves \nmaximum value, one has \n$$\\phi\\leqslant C,$$\nand so \n$${\\rm log}\\phi+(e^t-1)\\frac{\\partial u}{\\partial t}-u\\leqslant C.$$ \n\nHence we conclude $\\<\\widetilde\\omega_t, \\omega_0\\>\n\\leqslant C$ using the bound on $\\frac{\\partial u}{\\partial t}$ \nand $u$. This trace bound, together with $\\tilde\n\\omega^n_t\\leqslant C\\omega^n_0$, provide the \nuniform bound of $\\widetilde\\omega_t$ as metric, i. e. \n$C^{-1}\\omega_0\\leqslant \\widetilde\\omega_t \\leqslant \nC\\omega_0$. \\\\\n\n\\noindent $\\bullet$ {\\bf Contradiction}\n\n\\vspace{0.1in}\n\nThe metric (lower) bound makes sure that for any fixed \nanalytic variety in $X$, the integral of the proper power \nof $\\widetilde\\omega_t$ is bounded away from $0$, and so \nthe limiting class $[\\omega_T]$ would have positive \nintersection with any analytic variety by taking the \ncohomology limit. Then by Theorem 4.1 in \\cite\n{demailly-paun}, we conclude that $[\\omega_T]$ is actually \nK\\\"ahler which contradicts with the assumption of finite \ntime singularity at $T$ in sight of Proposition \n\\ref{existence}. \\\\\n\nHence we have finished the proof of Theorem \\ref\n{th:R-blows-up}. \n\n\\begin{remark}\n\nIn sight of this numerical characterization of K\\\"ahler \ncone for any general closed K\\\"ahler manifold by J. \nDemailly and M. Paun, the blowing-up of curvature \noperator or Ricci curvature in closed K\\\"ahler case \nis fairly obvious. The situation of scalar curvature is \nthe first non-trivial statement. \n\n\\end{remark}\n\n\\section{Proof of Theorem \\ref{th:R-control}}\n\nNow we derive certain control of the blowing-up of \nscalar curvature by mainly following the argument \nin \\cite{b-curvature}. The argument is also organized \nin three steps. \\\\ \n\n\\noindent $\\bullet$ {\\bf $0$-th Order Estimates} \n\n\\vspace{0.1in}\n\n$u\\leqslant C$ is directly from (\\ref{skrf}). $t$-derivative \nof (\\ref{skrf}) is \n$$\\frac{\\partial}{\\partial t}(\\frac{\\partial u}{\\partial t})=\\Delta\n(\\frac{\\partial u}{\\partial t})-e^{-t}\\<\\widetilde\\omega_t, \\omega_0\n-\\omega_\\infty\\>-\\frac{\\partial u}{\\partial t},$$\nwhich has the following variations,\n$$\\frac{\\partial}{\\partial t}(e^t\\frac{\\partial u}{\\partial t})=\\Delta\n(e^t\\frac{\\partial u}{\\partial t})-\\<\\widetilde\\omega_t, \\omega_0-\n\\omega_\\infty\\>,$$ \n$$\\frac{\\partial}{\\partial t}(\\frac{\\partial u}{\\partial t}+u)=\\Delta\n(\\frac{\\partial u}{\\partial t}+u)-n+\\<\\widetilde\\omega_t, \\omega_\n\\infty\\>.$$\nA proper linear combination of these equations provides \nthe following \"finite time version\" of the second equation,\n$$\\frac{\\partial}{\\partial t}\\((1-e^{t-T})\\frac{\\partial u}{\\partial t}\n+u\\)=\\Delta\\((1-e^{t-T})\\frac{\\partial u}{\\partial t}+u\\)-n+\\<\\tilde\n\\omega_t, \\omega_T\\>.$$\nThe difference of the original two equations gives\n$$\\frac{\\partial}{\\partial t}\\((1-e^t)\\frac{\\partial u}{\\partial t}\n+u\\)=\\Delta\\((1-e^t)\\frac{\\partial u}{\\partial t}+u\\)-n+\\<\\tilde\n\\omega_t, \\omega_0\\>,$$\nwhich implies the \"essential decreasing\" of metric potential \nalong the flow, i. e. \n$$\\frac{\\partial u}{\\partial t}\\leqslant\\frac{nt+C}{e^t-1}.$$\n\nNotice that this estimate only depends on the initial value \nof $u$ and its upper bound along the flow. It is uniform \naway from the initial time.\n\n\\vspace{0.1in}\n\nAnother $t$-derivative gives \n$$\\frac{\\partial}{\\partial t}(\\frac{\\partial^2 u}{\\partial t^2})=\n\\Delta(\\frac{\\partial^2 u}{\\partial t^2})+e^{-t}\\<\\tilde\\omega\n_t, \\omega_0-\\omega_\\infty\\>-\\frac{\\partial^2 u}{\\partial \nt^2}-|\\frac{\\partial\\widetilde\\omega_t}{\\partial t}|^2_{\\tilde\n\\omega_t}.$$\nTake summation with the first $t$-derivative to arrive at \n$$\\frac{\\partial}{\\partial t}(\\frac{\\partial^2 u}{\\partial t^2}+\n\\frac{\\partial u}{\\partial t})=\\Delta(\\frac{\\partial^2 u}{\\partial \nt^2}+\\frac{\\partial u}{\\partial t})-(\\frac{\\partial^2 u}{\\partial \nt^2}+\\frac{\\partial u}{\\partial t})-|\\frac{\\partial\\widetilde\n\\omega_t}{\\partial t}|^2_{\\widetilde\\omega_t},$$\nwhich gives\n$$\\frac{\\partial^2 u}{\\partial t^2}+\\frac{\\partial u}{\\partial t}\n\\leqslant Ce^{-t}.$$\nThis implies the \"essential decreasing\" of volume form \nalong the flow, i. e. \n$$\\frac{\\partial}{\\partial t}(\\frac{\\partial u}{\\partial t}+u)\n\\leqslant Ce^{-t},$$\nwhich also induces\n$$\\frac{\\partial u}{\\partial t}\\leqslant Ce^{-t}.$$\n\n\\vspace{0.1in}\n\nLet's rewrite the metric flow equation as follows,\n$${\\rm Ric}(\\widetilde\\omega_t)=-\\sqrt{-1}\\partial\\bar\\partial\n(u+\\frac{\\partial u}{\\partial t})-\\omega_\\infty.$$\n\nTaking trace with respect to $\\widetilde\\omega_t$ for the original \nmetric flow equation and the one above, we have\n$$R=e^{-t}\\<\\widetilde\\omega_t, \\omega_0-\\omega_\\infty\\>-\\Delta\n(\\frac{\\partial u}{\\partial t})-n=-\\Delta(u+\\frac{\\partial u}{\\partial \nt})-\\<\\widetilde\\omega_t, \\omega_\\infty\\>,$$\nwhere $R$ denotes the scalar curvature of $\\widetilde\\omega_t$. \nUsing the equations above, we also have\n$$R=-n-\\frac{\\partial}{\\partial t}(\\frac{\\partial u}{\\partial t}+u),$$\nand so the estimate got for $\\frac{\\partial}{\\partial t}(\\frac\n{\\partial u}{\\partial t}+u)$ before is equivalent to the well \nknown fact for scalar curvature. \\\\\n\nWe only consider smooth solution of K\\\"ahler-Ricci flow \nin $[0, T)\\times X$ with finite time singularity at $T$. At \nthis moment, we only need that the smooth limiting \nbackground form $\\omega_T\\geqslant 0$. It is essentially \nequivalent to assume $[\\omega_T]$ has a smooth \nnon-negative representative and presumably weaker \nthan the class being \"semi-ample\", i.e. the existence of \nmap $F$ before Theorem \\ref{th:R-control}. \n\n\\vspace{0.1in}\n\nRecall the following equation used before \n$$\\frac{\\partial}{\\partial t}\\((1-e^{t-T})\\frac{\\partial u}\n{\\partial t}+u\\)=\\Delta\\((1-e^{t-T})\\frac{\\partial u}\n{\\partial t}+u\\)-n+\\<\\widetilde\\omega_t, \\omega_T\\>$$\nwith the \"$T$\" in the equation chosen to the \"$T$\" \nabove. With $\\omega_T\\geqslant 0$, by Maximum \nPrinciple, one has\n$$(1-e^{t-T})\\frac{\\partial u}{\\partial t}+u\\geqslant -C.$$ \nTogether with the upper bounds, we conclude \n$$|(1-e^{t-T})\\frac{\\partial u}{\\partial t}+u|\\leqslant C.$$\n\n\\vspace{0.1in}\n\n\\noindent $\\bullet$ {\\bf Parabolic Schwarz Estimate}\n\n\\vspace{0.1in}\n\nUse the following setup as in \\cite{song-tian} for the map \n$F$ before the statement of Theorem \\ref{th:R-control}. \nLet $\\varphi=\\<\\widetilde\\omega_t, F^*\\omega_{_{M}}\\>$, \nthen one has, over $[0, T)\\times X$, \n$$(\\frac{\\partial}{\\partial t}-\\Delta)\\varphi\\leqslant \\varphi\n+C\\varphi^2-H,$$\nwhere $C$ is related to the bisectional curvature bound \nof $\\omega_{_{M}}$ near $F(X)$ and $H\\geqslant 0$ is \ndescribed as follows. Using normal coordinates locally \nover $X$ and $Y$, with indices $i, j$ and $\\alpha, \\beta$, \n$\\varphi=|F_i^\\alpha|^2$ and $H=|F_{ij}^\\alpha|^2$ with \nall the summations. Notice that the normal coordinate \nover $X$ is changing along the flow with the metric. \nUsing this inequality, one has \n$$(\\frac{\\partial}{\\partial t}-\\Delta){\\rm log}\\varphi\\leqslant \nC\\varphi+1.$$\n\n\\begin{remark}\n\nFor application, the map $F$ is coming from the class \n$[\\omega_T]$ with $Y$ being some projective space \n$\\mathbb{CP}^N$, and so $\\omega_T$ is $F^*\\omega$ \nwhere $\\omega$ is (some mutiple of) Fubini-Study \nmetric over $Y$. \n\n\\end{remark}\n\nDefine \n$$v:=(1-e^{t-T})\\frac{\\partial u}{\\partial t}+u$$ \nand we know $|v|\\leqslant C$ for the previous step. \nWe also have \n$$(\\frac{\\partial}{\\partial t}-\\Delta)v=-n+\\<\\widetilde\\omega_t, \n\\omega_T\\>=-n+\\varphi.$$\nAfter taking a large enough positive constant $A$, \nthe following inequality is true,\n$$(\\frac{\\partial}{\\partial t}-\\Delta)({\\rm log}\\varphi-Av)\n\\leqslant -C\\varphi+C.$$\nSince $v$ is bounded, Maximum Principle can \nbe used to deduce $\\varphi\\leqslant C$, i. e. \n$$\\<\\widetilde\\omega_t, \\omega_T\\>\\leqslant C.$$\n\n\\vspace{0.1in}\n\n\\noindent $\\bullet$ {\\bf Gradient and Laplacian Estimates}\n\n\\vspace{0.1in}\n\nIn this part, we derive gradient and Laplacian estimates \nfor $v$. Recall that \n$$(\\frac{\\partial}{\\partial t}-\\Delta)v=-n+\\varphi, ~~~ \\varphi\n=\\<\\widetilde\\omega_t, \\omega_T\\>.$$\nStandard computation gives:\n$$(\\frac{\\partial}{\\partial t}-\\Delta)(|\\nabla v|^2)=|\\nabla v|^2-\n|\\nabla\\nabla v|^2-|\\nabla\\bar\\nabla v|^2+2{\\rm Re}(\\nabla\n\\varphi, \\nabla v),$$\n$$(\\frac{\\partial}{\\partial t}-\\Delta)(\\Delta v)=\\Delta v+({\\rm \nRic}(\\widetilde\\omega_t), \\sqrt{-1}\\partial\\bar\\partial v)+\\Delta\n\\varphi.$$\nAgain, all the $\\nabla$, $\\Delta$ and $(\\cdot, \\cdot)$ are \nwith respect to $\\widetilde\\omega_t$ and $\\nabla\\bar\\nabla v$ \nis just $\\partial\\bar\\partial v$. \\\\ \n\nConsider $\\Psi:=\\frac{|\\nabla v|^2}{C-v}$. Since $v$ is \nbounded, one can easily make sure the denominator is \npositive, bounded and also away from $0$. We have the \nfollowing computation,\n\\begin{equation}\n\\begin{split}\n& ~~(\\frac{\\partial}{\\partial t}-\\Delta)\\Psi= (\\frac{\\partial}\n{\\partial t}-\\Delta)(\\frac{|\\nabla v|^2}{C-v}) \\\\\n&= \\frac{1}{C-v}\\cdot\\frac{\\partial}{\\partial t}(|\\nabla v|^2)\n+\\frac{|\\nabla v|^2}{(C-v)^2}\\cdot\\frac{\\partial v}{\\partial t}\n-\\( \\frac{(|\\nabla v|^2)_{\\bar i}}{C-v}+\\frac{v_{\\bar i}\n|\\nabla v|^2}{(C-v)^2}\\)_i \\\\\n&= \\frac{|\\nabla v|^2}{(C-v)^2}\\cdot(\\frac{\\partial}{\\partial \nt}-\\Delta)v+\\frac{1}{C-v}\\cdot(\\frac{\\partial}{\\partial t}-\\Delta)\n(|\\nabla v|^2)-\\frac{v_i\\cdot(|\\nabla v|^2)_{\\bar i}}{(C-v)^2}- \nv_{\\bar i}\\cdot\\(\\frac{|\\nabla v|^2}{(C-v)^2}\\)_i \\\\\n&= \\frac{|\\nabla v|^2}{(C-v)^2}\\cdot(\\frac{\\partial}{\\partial \nt}-\\Delta)v+\\frac{1}{C-v}\\cdot(\\frac{\\partial}{\\partial t}-\\Delta)\n(|\\nabla v|^2)-\\frac{2{\\rm Re}(\\nabla v, \\nabla |\\nabla v|^2)}\n{(C-v)^2}-\\frac{2|\\nabla v|^4}{(C-v)^3}. \\nonumber\n\\end{split}\n\\end{equation} \nPlug in the results from before and rewrite the differential \nequality for $\\Psi$ below,\n\\begin{equation}\n\\label{gradient2}\n\\begin{split}\n& ~~(\\frac{\\partial}{\\partial t}-\\Delta)\\Psi \\\\\n&= \\frac{(-n+\\varphi)|\\nabla v|^2}{(C-v)^2}+\\frac{|\\nabla v|^2-\n|\\nabla\\nabla v|^2-|\\nabla \\bar\\nabla v|^2}{C-v}+\\frac{2{\\rm \nRe}(\\nabla\\varphi, \\nabla v)}{C-v} \\\\\n& ~~~~-\\frac{2{\\rm Re}(\\nabla v, \\nabla |\\nabla v|^2)}{(C-v)\n^2}-\\frac{2|\\nabla v|^4}{(C-v)^3}. \n\\end{split}\n\\end{equation}\n\nThe computations below are useful next. \n\\begin{equation}\n\\begin{split}\n|(\\nabla v, \\nabla |\\nabla v|^2)|\n&= |v_i(v_j v_{\\bar j})_{\\bar i}| \\\\\n&= |v_iv_{\\bar j}v_{j\\bar i}+v_i\nv_jv_{\\bar j\\bar i }| \\\\\n&\\leqslant |\\nabla v|^2(|\\nabla\\nabla v|+|\\nabla\\bar\\nabla \nv|) \\\\\n&\\leqslant \\sqrt2 |\\nabla v|^2(|\\nabla\\nabla v|^2+|\\nabla\n\\bar\\nabla v|^2)^{\\frac{1}{2} }. \\nonumber\n\\end{split}\n\\end{equation}\n$$\\nabla\\Psi=\\nabla\\(\\frac{|\\nabla v|^2}{C-v}\\)=\\frac\n{\\nabla(|\\nabla v|^2)}{C-v}+\\frac{|\\nabla v|^2\\nabla \nv}{(C-v)^2}.$$\n\nTogether with the bounds for $\\varphi$ and $C-v$, we can \nhave the following computation with $\\epsilon$ representing \nsmall positive constant (different from place to place), \n\\begin{equation}\n\\begin{split}\n& ~~(\\frac{\\partial}{\\partial t}-\\Delta)\\Psi \\\\\n&\\leqslant C|\\nabla v|^2+\\epsilon\\cdot|\\nabla \\varphi|^2-C\n(|\\nabla\\nabla v|^2+|\\nabla\\bar\\nabla v|^2)+ \\\\\n& ~~~~~~ -(2-\\epsilon){\\rm Re}\\(\\nabla \\Psi, \\frac{\\nabla v}\n{C-v}\\)-\\epsilon\\cdot\\frac{{\\rm Re}(\\nabla v, \\nabla |\\nabla \nv|^2)}{(C-v)^2}-\\epsilon\\cdot\\frac{|\\nabla v|^4}{(C-v)^3} \\\\\n&\\leqslant C|\\nabla v|^2+\\epsilon\\cdot|\\nabla \\varphi|^2-C\n(|\\nabla\\nabla v|^2+|\\nabla\\bar\\nabla v|^2)+ \\\\\n& ~~~~~~ -(2-\\epsilon){\\rm Re}\\(\\nabla \\Psi, \\frac{\\nabla v}\n{C-v}\\)+\\epsilon\\cdot(|\\nabla\\nabla v|^2+|\\nabla\\bar\\nabla \nv|^2)-\\epsilon\\cdot|\\nabla v|^4 \\\\ \n&\\leqslant C|\\nabla v|^2+\\epsilon\\cdot|\\nabla \\varphi|^2-\n(2-\\epsilon){\\rm Re}\\(\\nabla \\Psi, \\frac{\\nabla v}{C-v}\\)-\n\\epsilon\\cdot|\\nabla v|^4. \\nonumber\n\\end{split}\n\\end{equation}\n\nWe need a few more calculations to set up Maximum \nPrinciple argument. Recall that $\\varphi=\\<\\widetilde\\omega_t, \n\\omega_T\\>$ and, \n$$(\\frac{\\partial}{\\partial t}-\\Delta)\\varphi\\leqslant \\varphi+\nC\\varphi^2-H.$$\nWith the description of $H$ before and the estimate for \n$\\varphi$, i. e. $\\varphi\\leqslant C$ from Schwarz estimate, \nwe can conclude that \n$$H\\geqslant C|\\nabla\\varphi|^2.$$ \nNow one arrives at\n\\begin{equation}\n\\label{schwarz2} \n(\\frac{\\partial}{\\partial t}-\\Delta)\\varphi\\leqslant C-C\n|\\nabla\\varphi|^2.\n\\end{equation}\nWe also have\n\\begin{equation}\n\\label{elementary}\n|\\(\\nabla \\varphi, \\frac{\\nabla v}{C-v}\\)|\\leqslant \\epsilon\n\\cdot|\\nabla \\varphi|^2+C\\cdot|\\nabla v|^2.\n\\end{equation}\n\n\\vspace{0.2in}\n\nNow consider the function $\\Psi+\\varphi$. By choosing $\n\\epsilon>0$ small enough above (which also affects the \nchoices of $C$'s), we have \n$$(\\frac{\\partial}{\\partial t}-\\Delta)(\\Psi+\\varphi)\\leqslant C+\nC|\\nabla v|^2-\\epsilon\\cdot|\\nabla v|^4-(2-\\epsilon){\\rm Re}\n\\(\\nabla(\\Psi+\\varphi), \\frac{\\nabla v}{C-v}\\).$$\nAt the maximum value point of $\\Psi+\\varphi$, we know \n$|\\nabla v|^2$ can not be too large. It's then easy to \nconclude the upper bound for this term, and so for $\\Psi$. \nHence we have bounded the gradient, i. e. \n$$|\\nabla v|\\leqslant C.$$ \n\nNow we want to do similar thing for the Laplacian, $\\Delta \nv$. Define the funtion $\\Phi:=\\frac{C-\\Delta v}{C-v}$. Similar computation as before gives the following\n\\begin{equation}\n\\label{laplacian2}\n\\begin{split}\n& ~~ (\\frac{\\partial}{\\partial t}-\\Delta)\\Phi= (\\frac{\\partial}\n{\\partial t}-\\Delta)(\\frac{C-\\Delta v}{C-v}) \\\\\n&= -\\frac{1}{C-v}\\cdot (\\frac{\\partial}{\\partial t}-\\Delta)\n\\Delta v+\\frac{C-\\Delta v}{(C-v)^2}\\cdot(\\frac{\\partial}\n{\\partial t}-\\Delta)v+\\frac{2{\\rm Re}(\\nabla v, \\nabla\\Delta v)}\n{(C-v)^2}+ \\\\\n&~~~~~~ -\\frac{2|\\nabla v|^2(C-\\Delta v)}{(C-v)^3} \\\\\n&= -\\frac{1}{C-v}\\cdot(\\Delta v+({\\rm Ric}(\\widetilde\\omega_t), \n\\sqrt{-1}\\partial\\bar\\partial v)+\\Delta \\varphi)+\\frac{C-\n\\Delta v}{C-v}\\cdot(-n+\\varphi) \\\\\n&~~~~~~ +\\frac{2{\\rm Re}(\\nabla v, \\nabla\\Delta v)}\n{(C-v)^2}-\\frac{2|\\nabla v|^2(C-\\Delta v)}{(C-v)^3}. \n\\end{split}\n\\end{equation}\n\nWe also have $\\nabla(\\frac{C-\\Delta v}{C-v})=\\frac\n{(C-\\Delta v)\\nabla v}{(C-v)^2}-\\frac{\\nabla\\Delta v}\n{C-v}$. Recall that it is already known $(0\\leqslant) \n\\varphi\\leqslant C$. The following inequality follows \nfrom standard computation as in \\cite{song-tian} \nand has actually been used for parabolic Schwarz \nestimate, \n$$\\Delta \\varphi\\geqslant ({\\rm Ric}(\\widetilde\\omega_t), \n\\omega_T)+H-C\\varphi^2,$$\nwhere $H\\geqslant C|\\nabla \\varphi|^2\\geqslant 0$ \nfrom the bound of $\\varphi$ as mentioned before. \nNow we have \n\\begin{equation}\n\\label{cases}\n({\\rm Ric}(\\widetilde\\omega_t), \\sqrt{-1}\\partial\\bar\\partial \nv)+\\Delta \\varphi\\geqslant ({\\rm Ric}(\\widetilde\\omega_t), \n\\sqrt{-1}\\partial\\bar\\partial v+\\omega_T)-C.\n\\end{equation}\n\n\\vspace{0.1in}\n\nWe are considering the case $T<\\infty$. Recall that \n$v=(1-e^{t-T})\\frac{\\partial u}{\\partial t}+u$ and $\n\\omega_T=\\omega_\\infty+e^{-T}(\\omega_0-\\omega\n_\\infty)$. We have \n\\begin{equation*}\n\\begin{split}\n{\\rm Ric}(\\widetilde\\omega_t)\n&= -\\sqrt{-1}\\partial\\bar\\partial \\(\\frac{\\partial u}{\\partial t}+u\\)-\n\\omega_\\infty \\\\\n&= -\\sqrt{-1}\\partial\\bar\\partial v -\\omega_T-e^{t-T}\\sqrt{-1}\\partial\\bar\n\\partial \\frac{\\partial u}{\\partial t}+e^{-T}(\\omega_0-\\omega_\\infty) \\\\\n&= -\\sqrt{-1}\\partial\\bar\\partial v -\\omega_T-e^{t-T}\\(\\sqrt{-1}\n\\partial\\bar\\partial \\frac{\\partial u}{\\partial t}-e^{-t}(\\omega_0-\\omega_\\infty)\n\\) \\\\\n&= -\\sqrt{-1}\\partial\\bar\\partial v -\\omega_T-e^{t-T}\\frac{\\partial \\tilde\n\\omega_t}{\\partial t} \\\\\n&= -\\sqrt{-1}\\partial\\bar\\partial v -\\omega_T-e^{t-T}\\(-{\\rm Ric}(\\tilde\n\\omega_t)-\\widetilde\\omega_t\\), \n\\end{split}\n\\end{equation*}\nwhich gives \n$$(1-e^{t-T}){\\rm Ric}(\\widetilde\\omega_t)=-\\sqrt{-1}\\partial\\bar\\partial v-\n\\omega_T+e^{t-T}\\widetilde\\omega_t,$$\nand so \n$${\\rm Ric}(\\widetilde\\omega_t)=-\\frac{\\sqrt{-1}\\partial\\bar\\partial v+\\omega\n_T}{1-e^{t-T}}+\\frac{e^{t-T}}{1-e^{t-T}}\\widetilde\\omega_t,$$\n$$(1-e^{t-T})R=-\\Delta v-\\<\\widetilde\\omega_t, \\omega_T\\>\n+ne^{t-T}.$$\n\n\\vspace{0.1in}\n\nAs $R\\geqslant -C$ and $0\\leqslant \\<\\widetilde\\omega_t, \n\\omega_T\\>$, we have $\\Delta v\\leqslant C$. \\\\ \n\nNow we can continue the estimation (\\ref{cases}) as \nfollows. \n\\begin{equation*}\n\\begin{split}\n&~~ ({\\rm Ric}(\\widetilde\\omega_t), \\sqrt{-1}\\partial\\bar\\partial \nv)+\\Delta \\varphi \\\\\n&\\geqslant ({\\rm Ric}(\\widetilde\\omega_t), \\sqrt{-1}\\partial\\bar\n\\partial v+\\omega_T)-C \\\\\n&\\geqslant \\(-\\frac{\\sqrt{-1}\\partial\\bar\\partial v+\\omega_T}{1-e^{t-T}}\n+\\frac{e^{t-T}}{1-e^{t-T}}\\widetilde\\omega_t, \\sqrt{-1}\\partial\n\\bar\\partial v+\\omega_T\\)-C \\\\\n&= -\\frac{|\\sqrt{-1}\\partial\\bar\\partial v+\\omega_T|^2}{1-e^{t-T}}+\n\\frac{e^{t-T}(\\Delta v+\\<\\widetilde\\omega_t, \\omega_T\\>)}\n{1-e^{t-T}}-C. \n\\end{split}\n\\end{equation*}\nAs $C^{-1}(T-t)\\leqslant 1-e^{t-T}\\leqslant C(T-t)$ for \n$t\\in [0, T)$, using $\\Delta v=\\Delta v-C+C$ and $0\n\\leqslant\\<\\widetilde\\omega_t, \\omega_T\\>\\leqslant C$, \nwe have \n\\begin{equation*}\n\\begin{split}\n&~~ ({\\rm Ric}(\\widetilde\\omega_t), \\sqrt{-1}\\partial\\bar\n\\partial v)+\\Delta \\varphi \\\\\n&= -\\frac{|\\sqrt{-1}\\partial\\bar\\partial v+\\omega_T|^2}{1-e^{t-T}}+\n\\frac{e^{t-T}(\\Delta v+\\<\\widetilde\\omega_t, \\omega_T\\>)}\n{1-e^{t-T}}-C \\\\ \n&\\geqslant -\\frac{1}{T-t}\\((1+\\epsilon)|\\sqrt{-1}\\partial\\bar\\partial \nv|^2+C|\\omega_T|^2\\)-\\frac{C}{T-t}(C-\\Delta v)-\\frac{C}\n{T-t} \\\\\n&\\geqslant -\\frac{1+\\epsilon}{T-t}|\\sqrt{-1}\\partial\\bar\\partial v|^2-\n\\frac{C}{T-t}(C-\\Delta v)-\\frac{C}{T-t}. \n\\end{split}\n\\end{equation*}\n \nNow we can continue the computation for $\\Phi$, \n(\\ref{laplacian2}) as follows. \n\\begin{equation*}\n(\\frac{\\partial}{\\partial t}-\\Delta)\\Phi\\leqslant \\frac{C}{T-t}+\n\\frac{C}{T-t}\\cdot(C-\\Delta v)+\\frac{(1+\\epsilon)|\\nabla\\bar\n\\nabla v|^2}{(T-t)(C-v)}-2{\\rm Re}\\(\\nabla \\Phi, \\frac{\\nabla \nv}{C-v}\\). \n\\end{equation*}\n\nUsing $\\Phi=\\frac{C-\\Delta v}{C-v}\\geqslant C(C-\\Delta v)$, \none arrives at \n\\begin{equation*}\n(\\frac{\\partial}{\\partial t}-\\Delta)((T-t)\\Phi)\\leqslant C+C\\cdot \n(C-\\Delta v)+\\frac{(1+\\epsilon)|\\nabla\\bar\\nabla v|^2}{C-v}-\n2{\\rm Re}(\\nabla \\((T-t)\\Phi\\), \\frac{\\nabla v}{C-v}) \n\\end{equation*}\nIn sight of (\\ref{schwarz2}) and (\\ref{elementary}), we have, \n\\begin{equation*}\n(\\frac{\\partial}{\\partial t}-\\Delta)\\varphi\\leqslant C-4{\\rm Re}\n\\(\\nabla\\varphi, \\frac{\\nabla v}{C-v}\\)+C|\\nabla v|^2. \n\\end{equation*}\nAlso, (\\ref{gradient2}) can be rewritten as \n\\begin{equation}\n\\begin{split}\n(\\frac{\\partial}{\\partial t}-\\Delta)\\Psi \n&\\leqslant \\frac{(-n+\\varphi)|\\nabla v|^2}{(C-v)^2}+\\frac\n{|\\nabla v|^2-|\\nabla\\nabla v|^2-|\\nabla \\bar\\nabla v|^2}\n{C-v}+ \\\\\n&~~~~~~ 2{\\rm Re}\\(\\nabla\\varphi, \\frac{\\nabla v}{C-v}\\)\n-2{\\rm Re}\\(\\Psi, \\frac{\\nabla v}{C-v}\\). \n\\nonumber\n\\end{split}\n\\end{equation}\n\nUsing the bound for $|\\nabla v|$ and choosing $\\epsilon<1$, \nwe have \n\\begin{equation*}\n\\begin{split}\n&~~ (\\frac{\\partial}{\\partial t}-\\Delta)\\((T-t)\\Phi+2\\Psi+2\n\\varphi\\) \\\\ \n&\\leqslant C+C\\cdot(C-\\Delta v)-2{\\rm Re}\\(\\nabla\\((T-t)\n\\Phi+2\\Psi+2\\varphi\\), \\frac{\\nabla v}{C-v}\\)-C|\\nabla\\bar\n\\nabla v|^2 \\\\\n&\\leqslant C+C\\cdot(C-\\Delta v)-2{\\rm Re}\\(\\nabla\\((T-t)\n\\Phi+2\\Psi+2\\varphi\\), \\frac{\\nabla v}{C-v}\\)-C(C-\\Delta \nv)^2\n\\end{split}\n\\end{equation*}\nwhere $|\\nabla\\bar\\nabla v|^2\\geqslant C(\\Delta v)^2\n\\geqslant C(C-\\Delta v)^2-C$ is used for the second \n$\\leqslant$. \\\\ \n\nNow we apply Maximum Principle. At maximum value \npoint of the function $(T-t)\\Phi+2\\Psi+2\\varphi$, we have \n$C-\\Delta v\\leqslant C$. Using the bounds on $\\Psi$ and \n$\\varphi$, $(T-t)\\Phi+2\\Psi+2\\varphi\\leqslant C$ and so \n$$\\Phi\\leqslant \\frac{C}{T-t}, ~~\\text{i.e.}~~ \\Delta v\n\\geqslant -\\frac{C}{T-t}.$$\nFinally since $(1-e^{t-T})R=-\\Delta v-\\<\\widetilde\\omega_t, \n\\omega_T\\>+ne^{t-T}$, we conclude that \n$$R\\leqslant \\frac{C}{(T-t)^2}.$$\n\n\\vspace{0.1in}\n\nTheorem \\ref{th:R-control} is proved. \n\n\\section{Further Remarks}\n\nThere are several closely related results worth mentioning. \nThe last two remarks below should give people the idea \nabout the essential difference between finite time and infinite \ntime singular case for K\\\"ahler-Ricci flow. \n\n\\begin{itemize}\n\n\\item In \\cite{sesum-tian}, following Perelman's idea, Sesum \nand Tian proved that for $X$ with $c_1(X)>0$, for any \ninitial K\\\"ahler metric $\\omega$ such that $[\\omega]=\nc_1(X)$, the K\\\"ahler-Ricci flow \n$$\\frac{\\partial \\widehat\\omega_t}{\\partial t}=-{\\rm Ric}(\\widehat\\omega\n_t)+\\widehat\\omega_t$$\nhas uniformly bounded scalar curvature and diameter \nfor $\\widehat\\omega_t$ where $t\\in [0, \\infty)$. Using \nsimply rescaling of time and metric, one can see for \nour flow (\\ref{krf}) with $[\\omega_0]=c_1(X)$, \n$$R(\\widetilde\\omega_t)\\leqslant \\frac{C}{T-t}$$ \nfor $t\\in [0, T)$ where the finite singular time $T=\\log 2$, \nwhich is a better control than Theorem \\ref{th:R-control} \nfor this special case. \n\n\\item For the infinite time limiting case, the scalar curvature \nwould be bounded if the infinite time limiting class provides \na smooth holomorphic fibration for $X$, i.e. the map $F$ as \nin our setting is a smooth fibration. This is actually proved in \n\\cite{song-tian} if one only considers smooth collapsing case. \n\n\\item For the infinite time limiting case, the scalar curvature \nwould also be bounded if the limiting class is \"semi-ample \nand big\", i.e. the (possibly singular) image of the map $F$ \nis of the same dimension as $X$. This result is proved in \n\\cite{b-curvature}. The more recent work of Yuguang Zhang, \n\\cite{miyaoka-yau}, has given a nice application of it. \n\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdnld b/data_all_eng_slimpj/shuffled/split2/finalzzdnld new file mode 100644 index 0000000000000000000000000000000000000000..843551a79aabb3b33372d8d332454910dc69f3d4 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdnld @@ -0,0 +1,5 @@ +{"text":"\\section{Adaptive JSMA}\n\\newcommand\\bff{\\boldsymbol{F}}\n\\newcommand\\bigone{\\,\\,1\\,\\,}\n\\newcommand\\onevector{\\begin{pmatrix}\n \\bigone \\\\ \\bigone \\\\ \\vdots \\\\ \\bigone \n\\end{pmatrix}}\n\\makeatletter\n\\newcommand*\\dashline{\\rotatebox[origin=c]{90}{$\\dabar@\\dabar@\\dabar@$}}\n\\makeatother\n\\subsection{Formulation of Adaptive JSMA with Papernot's notation}\n\\begin{algorithm}\n \\DontPrintSemicolon\n \\SetKwInOut{Input}{Input}\n \\Input{$\\nabla F(X)$, $\\Gamma$, $t$}\n \n $\\mu = - [ \\nabla F_{\\neg t}(X) \\, \\mathbf{1} ] \\odot \\nabla F_t (X) $\\\\\n $\\textrm{mask} = (\\mu > \\mathbf{0})$\\\\\n $S = \\mu\\odot \\textrm{mask} \\odot \\Gamma$\\\\\n $i_{\\textrm{max}}=\\textrm{arg max}_i \\, S_i$\\\\\n \n \\textbf{return} $i_{\\textrm{max}}, \\sign{(\\nabla F_{i,t}(X))}$\\\\\n \\caption{\\textsc{Adaptive JSMA Subroutine} \\newline\n \\label{alg:ajsma:alt}\n \\textbf{$\\nabla F$} is the Jacobian for a given sample $X$,\n \\textbf{$\\Gamma$} is a mask of the search domain, \\textbf{$t$} is\n the target class.} \\algorithmfootnote{In this paper, we adapt the\n convention of \\cite{papernot2016limitations} to express the\n Jacobian matrix $\\nabla F(X) = \\frac{\\partial F}{\\partial X}$ as\n an $m$ by $n$ matrix, the $(i,j)^\\textrm{th}$ entry is $\\frac{\\partial\n F_j}{\\partial X_i}$. We express all of our vectors as column\n vectors. The vector of inputs, $X$, is a 1 by $m$ vector, where\n $n$ is the number of features, e.g., pixels in images. We denote\n $\\neg t$ to be all classes other than class $t$ and $\\odot$ to be\n element-wise multiplication. Finally, we define $\\mathbf{1}$ to\n be a vector of 1's of length $\\lvert \\neg t \\rvert$.}\n\\end{algorithm}\n\\subsection{Our original formulation of Adaptive JSMA}\n\\begin{algorithm}\n \\DontPrintSemicolon\n \\SetKwInOut{Input}{Input}\n \\Input{$\\nabla \\bff(X)$, $\\Gamma$, $t$}\n $\\mu = -\\nabla \\bff(X)_t \\odot (\\mathbf{1}^T \\nabla \\bff(X)_{\\neg t}) $\\\\\n $\\textrm{mask} = (\\mu > \\mathbf{0})$\\\\\n $S = \\mu\\odot \\textrm{mask} \\odot \\Gamma$\\\\\n $i_{\\textrm{max}}=\\textrm{arg max}_i \\, S_i$\\\\\n \\textbf{return} $i_{\\textrm{max}}, \\sign{(\\nabla \\bff(X)_{t,i})}$\\\\\n \\caption{\\textsc{Adaptive JSMA Subroutine} \\newline\n \\label{alg:ajsma}\n \\textbf{$\\nabla \\bff$} is the Jacobian for a given sample $X$,\n \\textbf{$\\Gamma$} is a mask of the search domain, \\textbf{$t$} is\n the target class.} \\algorithmfootnote{For the purposes of\n mathematical notation in this paper, and without loss of\n generality, we assume that $X$ is a 1 by\n \\color{red} $m$ \\color{black} vector, where $n$ is the number of\n features e.g., pixels in images. We denote $\\neg t$ to be all\n classes other than class $t$ and $\\odot$ to be element-wise\n multiplication. Finally, we define $\\mathbf{1}$ to be a column\n vector of 1's of length $\\lvert \\neg t \\rvert$.}\n\\end{algorithm}\n\\section{Papernot notation}\n\\noindent $F: \\mathbb{R}^m \\rightarrow \\mathbb{R}^n$ \\\\\n$J_F(X) = \\nabla F(X) =\\left( \\frac{\\partial F_j}\n{\\partial X_i}\\right)_{i=1\\cdots m, j = 1\\cdots n}$\\\\\n$ X_{m \\cdot 1} = \\begin{pmatrix} X_1\\\\ \\vdots \\\\ X_m \\end{pmatrix}$, \\quad\n$\\mathbf{1}_{(n-1)\\cdot 1} =\n\\onevector$\\\\\n$\\nabla F(X)_{m \\cdot n}$\\\\\n$\\nabla F_t(X)_{m \\cdot 1}$\\\\\n$\\nabla F_{\\neg t}(X)_{m \\cdot (n-1)}$\\\\\n\\begin{equation*}\n \\begin{aligned}\n \\nabla F(X) & =\n \\begin{pmatrix}\n %\n \\frac{\\partial F_1}{\\partial X_1} & \n \\frac{\\partial F_2}{\\partial X_1} &\n \\cdots &\n \\frac{\\partial F_n}{\\partial X_1} \\\\\n %\n \\frac{\\partial F_1}{\\partial X_2} & \n \\frac{\\partial F_2}{\\partial X_2} &\n \\cdots &\n \\frac{\\partial F_n}{\\partial X_2} \\\\\n %\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n %\n \\frac{\\partial F_1}{\\partial X_m} & \n \\frac{\\partial F_2}{\\partial X_m} &\n \\cdots &\n \\frac{\\partial F_n}{\\partial X_m} \\\\\n %\n \\end{pmatrix}\\\\\n & = \\begin{pmatrix}\n \\nabla F_1(X) & \\dashline & \\nabla F_2(X) & \\dashline & \\cdots & \\dashline & \\nabla F_n(X)\n \\end{pmatrix}\n \\end{aligned}\n\\end{equation*}\n\\section{Comparison of JSMA and AJSMA}\nFor AJSMA, we want (compare equation (9) in \\cite{papernot2016limitations}):\\\\\n\\begin{equation*}\n \\begin{aligned}\n S & = J_{i,t}(X)|\\sum_{j\\ne t}J_{i,j}(X)|\n =\\nabla F_{i,t}(X)|\\sum_{j\\ne\n t}\\nabla F_{i,j}(X)| \\\\\n & \\textrm{ whenever }\\nabla F_{i,t}(X) \\sum_{j\\ne t}\\nabla F_{i,j}(X) < 0,\n \\\\\n & \\textrm{ and } S= 0 \\textrm{ otherwise.}\n \\end{aligned}\n\\end{equation*}\n\\noindent For this example, without (much) loss of generality, let's take $t$ to\nbe the $n^\\textrm{th}$ class so that $\\neg t = \\{ 1,2,\\hdots, n-1 \\}$.\n\n\\begin{equation*}\n \\begin{aligned}\n \\nabla F_{\\neg t}(X) \\, \\mathbf{1} & = \n \\begin{pmatrix}\n \\nabla F_1 & \\dashline & \\nabla F_2 & \\dashline & \\cdots & \\dashline & \\nabla F_{n-1}\n \\end{pmatrix}\n \\onevector\\\\\n & = \n \\begin{pmatrix}\n \\sum_{j \\ne t} \\nabla F_{1,j} \\\\\n \\sum_{j \\ne t} \\nabla F_{2,j} \\\\\n \\vdots\\\\\n \\sum_{j \\ne t} \\nabla F_{m,j} \\\\\n \\end{pmatrix}\n \\end{aligned}\n\\end{equation*}\n\n\\noindent The $i^\\textrm{th}$ element is $\\sum_{j \\ne t} \\nabla F_{i,j}$.\n\n\\begin{equation*}\n \\mu_i = - \\left\\{ [ \\nabla F_{\\neg t}(X) \\, \\mathbf{1} ] \\odot \\nabla F_t (X) \\right\\}_i\n = - \\nabla F_{i,t} \\sum_{j \\ne t} \\nabla F_{i,j}\n\\end{equation*}\n\\noindent JSMA requires\n\\begin{equation*}\n \\textrm{mask } = (\\mu>0) \\wedge (\\nabla F_t (X) <0),\n\\end{equation*}\nAJSMA requires\n\\begin{equation*}\n \\textrm{mask } = (\\mu>0).\n\\end{equation*}\n\n\n\\bibliographystyle{IEEEtranS}\n\n\\section{Algorithms} \\label{sec:algorithms}\n\n\\begin{algorithm}\n \\DontPrintSemicolon\n \\SetKwInOut{Input}{Input}\n \\Input{$\\nabla F(X)$, $\\Gamma$, $t$}\n \n $\\mu = - [ \\nabla F_{\\neg t}(X) \\, \\mathbf{1} ] \\odot \\nabla F_t (X) $\\\\\n $S = \\mu\\odot \\varphi_{\\mu>0}(\\Gamma)$\\\\\n \n \n $i_{\\textrm{max}}=\\textrm{arg max}_i \\, S_i$\\\\\n \n \\textbf{return} $i_{\\textrm{max}}, \\sign{(\\nabla F_{i,t}(X))}$\\\\\n \\caption{\\textsc{Adaptive JSMA Subroutine} \\newline\n \\label{alg:ajsma:alt}\n \\textbf{$\\nabla F$} is the Jacobian for a given sample $X$,\n \\textbf{$\\Gamma$} is the search domain, \\textbf{$t$} is\n the target class.} \\algorithmfootnote{In this paper, we adapt\n Papernot's \\cite{pp16limit} convention to express\n the Jacobian matrix $\\nabla F(X) = \\frac{\\partial F}{\\partial X}$\n as an $m$ by $n$ matrix, the $(i,j)^\\textrm{th}$ entry is\n $\\frac{\\partial F_j}{\\partial X_i}$. We express all of our\n vectors as column vectors. The vector of inputs, $X$, is a 1 by\n $m$ vector, where $n$ is the number of features, e.g., pixels in\n images. We denote $\\neg t$ to be all classes other than class $t$\n and $\\odot$ to be element-wise multiplication. We define\n $\\mathbf{1}$ to be a vector of 1's of length $\\lvert \\neg t\n \\rvert$. Finally, the mask $\\varphi_A(X)$ is defined to be the\n predicate $A \\subset X$ \\cite{minskypapert}.}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\DontPrintSemicolon\n \\SetKwInOut{Input}{Input}\n \\Input{H, n}\n $v = \\mathbf{\\emptyset}$\\\\\n \\While{$\\textrm{\\upshape dim } v \\neq n$} \n {\n $i=\\textrm{arg max } \\lvert H_i \\rvert$\\\\\n $v=\\textrm{append}(v, (i, \\textrm{sign}(H_i)))$\\\\\n $H_i=0$\\\\\n }\n \\textbf{return} $v$\n \\caption{\\textsc{Top-n Approximation\\newline}\n \\label{alg:top_n}\n $H$ is a perturbation histogram, $n$ is the desired number of features\n used to build the universal perturbation vector, $v$.\n }\n \\algorithmfootnote{The vector $v$ is initialized to the $0$-dimensional ``empty vector'' \nwhich we denote by $\\mathbf{\\emptyset}$. At each step of the \\textbf{while} loop, the index \nand direction of the feature resulting in the $k^{\\textrm{th}}$ largest perturbation is appended to $v$.}\n\\end{algorithm}\n\n\\section{Appendix} \\label{sec:appendix}\n\n\\begin{table*}[!ht]\\footnotesize\n \\resizebox{0.76\\paperwidth}{!}{ \\centering\n \\begin{tabulary}{\\linewidth}{lLLLL}\n \\toprule\n\n {} & \\multicolumn{4}{c}{Feature Type}\\\\\n \\cmidrule(l{1em}r{1em}){2-5}\n {Protocol} & \\multicolumn{1}{c}{Basic} & \\multicolumn{1}{c}{Content} \n & \\multicolumn{1}{c}{Timing-based} & \\multicolumn{1}{c}{Host-basted}\\\\\n \\midrule\n\n {\\textbf{TCP}} & {duration, service=aol, service=auth, service=bgp,\n service=courier, service=csnet\\_ns, service=ctf, service=daytime,\n service=discard, service=domain, service=echo, service=efs, service=exec,\n service=finger, service=ftp, service=ftp\\_data, service=gopher,\n service=harvest, service=hostnames, service=http, service=http\\_2784,\n service=http\\_443, service=http\\_8001, service=IRC, service=iso\\_tsap,\n service=klogin, service=kshell, service=ldap, service=link, service=login,\n service=mtp, service=name, service=netbios\\_dgm, service=netbios\\_ns,\n service=netbios\\_ssn, service=netstat, service=nnsp, service=nntp,\n service=other, service=pm\\_dump, service=pop\\_2, service=pop\\_3,\n service=printer, service=private, service=remote\\_job, service=rje,\n service=shell, service=smtp, service=sql\\_net, service=ssh, service=sunrpc,\n service=supdup, service=systat, service=telnet, service=time, service=uucp,\n service=uucp\\_path, service=vmnet, service=whois, service=X11,\n service=Z39\\_50, service=34, flag=OTH, flag=REJ, flag=RSTO, flag=RSTOS0,\n flag=RSTR, flag=S0, flag=S1, flag=S2, flag=S3, flag=SF, flag=SH, src\\_bytes,\n dst\\_bytes, land=1, urgent} & \n\n {hot, num\\_failed\\_logins, logged\\_in=1, num\\_compromised, root\\_shell,\n su\\_attempted, num\\_root, num\\_file\\_creations, num\\_shells,\n num\\_access\\_files, is\\_host\\_login=1, is\\_guest\\_login=1} & \n \n {count, srv\\_count, serror\\_rate, srv\\_serror\\_rate, rerror\\_rate,\n srv\\_rerror\\_rate, same\\_srv\\_rate, diff\\_srv\\_rate, srv\\_diff\\_host\\_rate} & \n\n {dst\\_host\\_count, dst\\_host\\_srv\\_count, dst\\_host\\_same\\_srv\\_rate,\n dst\\_host\\_diff\\_srv\\_rate, dst\\_host\\_same\\_src\\_port\\_rate,\n dst\\_host\\_srv\\_diff\\_host\\_rate, dst\\_host\\_serror\\_rate,\n dst\\_host\\_srv\\_serror\\_rate, dst\\_host\\_rerror\\_rate,\n dst\\_host\\_srv\\_rerror\\_rate} \\\\\n\n \\midrule\n\n {\\textbf{UDP}} & {duration, service=domain\\_u, service=ntp\\_u, service=other,\n service=private, service=tftp\\_u, flag=SF, wrong\\_fragment} & \n\n {} & \n\n {count, srv\\_count, serror\\_rate, rerror\\_rate, same\\_srv\\_rate,\n diff\\_srv\\_rate, srv\\_diff\\_host\\_rate} & \n\n {dst\\_host\\_count, dst\\_host\\_srv\\_count, dst\\_host\\_same\\_srv\\_rate,\n dst\\_host\\_diff\\_srv\\_rate, dst\\_host\\_same\\_src\\_port\\_rate,\n dst\\_host\\_srv\\_diff\\_host\\_rate, dst\\_host\\_serror\\_rate,\n dst\\_host\\_rerror\\_rate} \\\\\n\n \\midrule\n\n {\\textbf{ICMP}} & {service=eco\\_i, service=ecr\\_i, service=red\\_i, service=tim\\_i,\n service=urh\\_i, service=urp\\_i, flag=SF, src\\_bytes, wrong\\_fragment} & \n\n {} & \n\n {count, srv\\_count, serror\\_rate, rerror\\_rate, same\\_srv\\_rate,\n diff\\_srv\\_rate, srv\\_diff\\_host\\_rate} & \n\n {dst\\_host\\_count, dst\\_host\\_srv\\_count, dst\\_host\\_same\\_srv\\_rate,\n dst\\_host\\_diff\\_srv\\_rate, dst\\_host\\_same\\_src\\_port\\_rate,\n dst\\_host\\_srv\\_diff\\_host\\_rate, dst\\_host\\_serror\\_rate,\n dst\\_host\\_rerror\\_rate} \\\\\n\n \\bottomrule\n \\end{tabulary}}\n\n \\caption{The constraints extracted from the NSL-KDD - Unlike TCP, UDP and ICMP\n have limited degrees of freedom.} \\label{tab:nsl-constraints}\n\n\\end{table*}\n\n\\begin{table*}[!ht]\\footnotesize\n \\resizebox{0.55\\paperwidth}{!}{ \\centering\n \\begin{tabular}{rrrcccc}\n \\toprule\n \n {Learning Technique} & \\multicolumn{2}{c}{Parameters} & \\multicolumn{4}{c}{Test Accuracy} \\\\\n \\cmidrule(l{0.5em}r){2-3} \\cmidrule(l{0.5em}r{0.5em}){4-7}\n {} & \\multicolumn{1}{c}{Name} & \\multicolumn{1}{c}{Value} & {NSL-KDD} & {UNSW-NB15} & {MNIST} & {GTSRB} \\\\\n \\midrule\n {\\textbf{Logistic Regression}} & \n \\begin{tabular}{@{}r@{}}\\textsc{penalty}\\\\\\textsc{C}\\end{tabular} &\n \\begin{tabular}{@{}r@{}}l2\\\\1.0\\end{tabular} & {74.21\\%} & {67.65\\%} & {92.02\\%} & {84.94\\%}\\\\\n \\cmidrule(l{0.5em}r){2-3} \\cmidrule(l{0.1em}){4-7}\n {\\textbf{Support Vector Machine}} & \n \\begin{tabular}{@{}r@{}}\\textsc{C}\\\\\\textsc{kernel}\\\\\\textsc{degree}\\end{tabular} & \n \\begin{tabular}{@{}r@{}}1.0\\\\rbf\\\\3\\end{tabular} & {77.31\\%} & {69.02\\%} & {94.04\\%} & {84.3\\%}\\\\\n \\cmidrule(l{0.5em}r){2-3} \\cmidrule(l{0.1em}){4-7}\n {\\textbf{Decision Tree Classifier}} &\n \\begin{tabular}{@{}r@{}}\\textsc{criterion}\\\\\\textsc{max\\_depth}\\\\\\textsc{min\\_samples\\_split}\\\\\n \\textsc{min\\_samples\\_leaf}\\\\\\textsc{max\\_features}\\end{tabular} & \n \\begin{tabular}{@{}r@{}}gini\\\\$\\infty$\\\\2\\\\1\\\\$\\infty$\\end{tabular} & {74.52\\%} & {73.27\\%} & {87.73\\%} & {57.20\\%}\\\\\n \\cmidrule(l{0.5em}r){2-3} \\cmidrule(l{0.1em}){4-7}\n {\\textbf{k-Nearest Neighbor}} &\n \\begin{tabular}{@{}r@{}}\\textsc{k}\\\\\\textsc{p}\\\\\\end{tabular} &\n \\begin{tabular}{@{}r@{}}5\\\\2\\end{tabular} & {74.90\\%} & {72.20\\%} & {96.88\\%} & {52.83\\%}\\\\\n\n \\bottomrule\n \\end{tabular}}\n \\caption{Scikit-Learn Model Information}\n \\label{tab:scikit-architecture}\n\n\\end{table*}\n\n\\begin{figure*}[!ht]\n \\begin{subtable}[h]{0.5\\paperwidth}\n \\includegraphics[width=\\textwidth]{figs\/mnist-heatmap.png}\n \\caption{Perturbation Histogram produced by the AJSMA for the MNIST model $M_A$\n as a 2D Heatmap - The target class (``0'') can be visibly seen by combining the\n decreasing features plot (left) with the increasing features plot (right).}\n \\end{subtable}\n \n \n \\begin{subtable}[h]{0.52\\paperwidth}\n \\includegraphics[width=\\textwidth]{figs\/sketches.png}\n \\caption{Sketch Visualization for MNIST and GTSRB for varying values of $n$ - \n As $n$ increases, the target class visibly forms.}\n \\end{subtable}\n \\caption{Perturbation Histogram for MNIST (a) and adversarial sketches for image\n datasets (b).}\n \\label{fig:graphics}\n\\end{figure*}\n\n\\section{Background} \\label{sec:background} Adversarial machine learning research\nin unconstrained domains has been broad. Since the initial observations of Biggio\net al. and Szegedy et al. in deep neural\nnetworks~\\cite{bg13evasion,sg13intriguing} to the robust attacks from Kurakin et\nal. and Sharif et al.~\\cite{kk17physical,sr16accessorize}, adversarial examples\nhave matured from, ``an intriguing property'' to a tangible threat.\n\n\\everypar{\\looseness=-1}\nThe first generation of attacks were formed in the context of ``white-box''\nattacks~\\cite{pp16limit,gf15explain,cl17towards}. Under this threat model,\nadversarial examples are crafted using information (e.g., model parameters)\ndirectly from the model under attack. This represents a worst-case scenario,\nanalogous to an insider threat, since such information would not be easily\naccessible in most practical contexts. Naturally, this motivates the question: Can\nan adversary successfully attack a model, \\textit{without having direct access to\nits parameters?} Papernot et al. and Tram{\\`e}r et al. investigated this question\nby leveraging \\textit{transferability}: an adversarial example crafted from one\nmodel will often be an adversarial example for a different model, even if they are\nusing different training data and\/or learning\ntechniques~\\cite{tm17space,pp16transferability}. Through this ``black-box,''\nthreat model, an adversary trains a \\textit{surrogate} model by using inputs to\ngenerate output labels from the victim model, called an \\textit{oracle}.\nAfterwards, the surrogate model is used to craft adversarial examples which are\nthen (with high probability) ``transferred'' to the victim model.\n\nConcurrently, others have investigated what limitations (if any) exist for\nadversarial examples. Kurakin et al. and Brown et al. explored how adversarial\nexamples can be applied directly to the physical domain, introducing techniques\nthat produce adversarial examples robust to physical distortions, such as\nrotation, scale, and other transformations~\\cite{kk17physical,bn17adversarial}.\nMoreover, Goodfellow et al. and Moosavi-Dezfooli et al. analyzed \\textit{universal\nadversarial perturbations}: single perturbation vectors used to quickly craft\nadversarial examples from many inputs not known in\nadvance~\\cite{gf15explain,df16universal}. These universal perturbations are\nparticularly concerning as they enable adversaries to take computation offline and\namortize computational costs over many inputs. At present, we are observing an\nevolution in the way adversarial examples manifest. Each generation contributes to\na growing threat against deployed machine learning systems.\n\nFor this work, we modify an existing attack, the \\textit{Jacobian-based Saliency\nMap Approach}, introduced by Papernot et al.~\\cite{pp16limit}. The\n``\\textsc{JSMA}'' produces an adversarial example by iteratively applying\nperturbations to the most salient features in an input. The algorithm terminates\nwhen either the input is successfully misclassified or the specified $l_0$\ndistance\\footnotemark is reached. The \\textsc{JSMA}{} greedily selects features to perturb\nby constructing \\textit{saliency maps}, which encode the influence features have\nover misclassifying a particular input. This $l_0$ minimization makes the \\textsc{JSMA}{}\nan attractive candidate for our evaluated domain, as discussed in\n\\S\\ref{sec:methodology}. However, the \\textsc{JSMA}{} is not necessary for crafting\nadversarial examples in constrained domains. Different attack algorithms can be\nused, with some adjustments, as reviewed in \\S\\ref{sec:methodology}, which we\ndefer to future work. For our study in constrained domains, the \\textsc{JSMA}{} was simply\nthe most readily usable algorithm to leverage.\n\n\\footnotetext{\\looseness=-1 Most attack algorithms have upper limits on the allowable distortion\nthey can introduce. This distortion is defined to be the distance between an\nadversarial example and its original counterpart. There are many metrics\n(principally $l_p$ norms~\\cite{cl17towards}) used throughout the literature to\nmeasure this distance.}\n\n\\section{Conclusions} \\label{sec:conclusions} This paper investigated the impact\nof adversarial examples in constrained domains through the perspective of\ntraditional adversarial algorithms and universal adversarial perturbations. In\naddition to this investigation in unique domains like network intrusion detection,\nwe introduced two new algorithms: the \\textsc{Adaptive JSMA}, which produces\nadversarial examples in constrained domains, and the \\textsc{Histogram Sketch\nGeneration}, which generates adversarial sketches: universal adversarial\nperturbations that obey domain constraints. Our work demonstrates how adversaries\ncan craft permissible adversarial examples in constrained domains.\n\nThrough our experiments, we observed how biased distributions coupled with low\ndimensionality can have a significant impact on model vulnerability, even in the\npresence of constraints. Furthermore, we demonstrated how when defender constrains\nan adversary to \\textit{five random} features, adversarial examples can still be\ncrafted with a $\\mathtt{\\sim}50\\%$ success rate.\n\n\\looseness=-1 Prior to our work, the impact of adversarial learning has been\nlargely understood in the context of unconstrained domains. We initially\nhypothesized that systems whose domains were constrained would be more resilient\nto attack algorithms. However, our investigation suggests the inverse. Our two\nalgorithms were able to craft adversarial examples with minimal distortion and\nwith success rates of greater than 95\\% that also transferred to other models at\nrates up to 93\\%.\n\nIndeed, it is unclear if any domain is immune to adversarial machine learning.\nThrough a simple number of transformations, an adversary can wholly control a\nmodel, thereby defeating systems deployed in sensitive domains. \n\n\\section{Discussion} \\label{sec:discussion} In this section, we describe our\nthoughts for future work.\n\n\\begin{figure}[t]\n\n \\includegraphics[width=\\columnwidth]{figs\/correlations.png}\n\n \\caption{Normalized mean of summed absolute values of Pearson Correlation\n Coefficients for five categories of features in the NSL-KDD - Protocols are\n correlated with the majority of features, closely behind timing-based features,\n which are highly correlated amongst themselves.} \\label{fig:correlations}\n\n\\end{figure}\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Identifying Primary Features.} In this work, we\nidentified primary features manually through not only our understanding of the\ndomain, but observations of the data. We noticed that many features were\ncorrelated with the transport layer protocol (as we expected, based on the\ndescriptions of the features). However, we may not always be able to use the\ndescriptions of features to guide us towards identifying primary features. Thus,\nwe hypothesize that primary features could be identified by ranking the features\nthat are most correlated with others. The intuition here is that while secondary\nfeatures will be highly correlated with primary features, primary features will be\nhighly correlated with the majority of secondary features. We performed a simple\nexperiment where we computed Pearson product-moment correlation coefficients for\nall features in the NSL-KDD training set. We took the mean of sum of the absolute\nvalue of the coefficients for four categories of features as their scores and\nfound protocols to indeed be correlated with the majority of features, as shown in\nFigure \\ref{fig:correlations}. At first, it appeared that timing-based features\nhad a higher correlation than protocols among all features. However, we noticed\nthat the timing-based features were highly correlated \\textit{amongst themselves},\nwhich was the largest contributor to their scores (and to no surprise, since many\ntiming-based features are derivatives or direct inverses of one another, e.g.,\n\\textsc{same\\_srv\\_rate} and \\textsc{diff\\_srv\\_rate}). Thus, if we eliminate such\nfunctionally inverse and derivative features, this approach would suggest that\nprimary features could be identified systematically.\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Defenses.} We also hypothesize that constraints\n\\textit{can} be useful in defending against adversarial examples, even though the\nconstraints found in the studied datasets were ineffective. We are interested if\nconstraints in other domains, such as malware and spam, can prove to be effective\nagainst adversarial example generation algorithms. It is intuitive that\nmaintaining the behavior of malware while simultaneously perturbing binary code\nwould be a challenging problem. \n\nIn addition, we also observe that our approach for building adversarial sketches\ncan also be used by a defender to assess model vulnerability. In particular, a\ndefender could design a simple mechanism (driven by the perturbation histograms)\nthat reveals universal directions that would make the model vulnerable. The\ndefender can then use this analysis to detect adversarial examples at deployment.\nHowever, an adversary could circumvent detection by selectively perturbing\nfeatures that have less impact. Naturally, this would come at a cost of\nintroducing additional distortion and control over more features, which may be\nimpractical for an adversary.\n\n\\looseness=-1\nFinally, we note that defenses against adversarial examples, including detection\ntechniques, robust optimization, etc., is an open problem. At present, it\nis unclear whether or not a generalizable defense exists against adversarial\nexamples. \n\n\\section{Evaluation} \\label{sec:evaluation} In this section, we evaluate our\napproach on two network intrusion datasets, the NSL-KDD and the UNSW-NB15, as\nwell as two image recognition datasets, the GTSRB and MNIST. \nIn our evaluation, we answer two questions:\n\n\\begin{enumerate}\n\n \\item Are constrained domains more robust against adversarial examples?\n \\item Do universal adversarial perturbations exist in constrained domains?\n\n\\end{enumerate}\n\nOur experiments revealed that: we can craft adversarial examples with success\nrates greater than \\results{}, even in the presence of constraints; we can compute\nhighly successful adversarial sketches, reaching greater than 80\\%\nmisclassification rates for the majority of learning techniques.\n\nOur experiments were performed on a Dell Precision T7600 with an Intel Xeon\nE5-2630 and a NVIDIA GeForce TITAN X. We used Cleverhans\n2.0.0~\\cite{pp16cleverhans} for training our models and crafting adversarial\nexamples. \n\n\\subsection{Datasets}\\looseness=-1 Before we describe our experiments, we provide\nan overview of the four evaluated datasets and any preprocessing that we\nperformed. Table \\ref{tab:architecture} describes the model architectures,\nhyperparameters, and model accuracies across all four datasets\\footnotemark. \n\n\\footnotetext{Note that while\nthe accuracy of our network intrusion detection models sit around $77\\%$, these\nare consistent with the literature~\\cite{tl09detailed,\nig15performance,ih13comparison,ms17hybrid}. Notably, Tavallaee et al., who \ninitially uncovered the inherent problems with the KDD CUP 99 (which \nresulted in the creation of the NSL-KDD), reported 77.41\\% accuracy with \nMulti-layer Perceptrons~\\cite{tl09detailed}. Furthermore, Moustafa et al., who \ndesigned the UNSW-NB15 to address the shortcomings of the \nNSL-KDD~\\cite{ms15unsw}, reported 81.34\\% accuracy (using the full training \nset) with an artificial neural network~\\cite{mt16evaluation}. These accuracies \nare are comparable to the highest to date.}\n\nWe evaluate our approach on two network intrusion detection datasets for the\napplication of constrained domains and two image classification datasets for\nunconstrained domains. We use these two image classification datasets for\ncomparison with other works in adversarial machine learning as well as a\ndemonstration of the cross-domain applicability of our approach. Note that these\nare unconstrained domains (i.e., images), and thus the \\textsc{AJSMA}{} behaves similarly\nto the original \\textsc{JSMA}{} (other than that it can perturb in either direction).\n\n\\begin{table}[!ht]\n \\resizebox{\\columnwidth}{!}{ \\centering \n \\begin{tabular}{lcccccr}\n \\toprule\n\n {Dataset} & {Architecture} & {Units} & {Batch Size} & {Learning Rate} & {Epochs} & {Testing Acc.}\\\\\n \\midrule\n {\\textbf{NSL-KDD}} & {MLP} & {123, 64, 32, 5} & {200} & {0.01} & {5} & {$77\\%\\pm 1.0\\%$}\\\\\n {\\textbf{UNSW-NB15}} & {MLP} & {196, 98, 49, 10} & {128} & {0.01} & {10} & {$75\\%\\pm 1.2\\%$}\\\\\n {\\textbf{MNIST}} & {CNN} & {784, 128, 128, 10} & {128} & {0.001} & {6} & {$98\\%\\pm 0.1\\%$}\\\\\n {\\textbf{GTSRB}} & {CNN} & {2700, 128, 128, 42} & {128} & {0.001} & {6} & {$82\\%\\pm 2.0\\%$}\\\\\n\n \\bottomrule\n \\end{tabular}}\n \\caption{Model Information}\n \\label{tab:architecture}\n\n\\end{table}\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{NSL-KDD.} The NSL-KDD dataset is an improved variant\nof the KDD Cup99 dataset~\\cite{tl09detailed}. The KDD Cup99 (and its NSL-KDD\nsuccessor) have been used widely in the network intrusion detection community. We\nchose to use the NSL-KDD for the novel application of adversarial examples in this\nfield, familiarity of the dataset within the academic community, and the lack of\nwell-formed network intrusion detection data. \n\n\\looseness=-1\nThe NSL-KDD contains 5 classes, with 4 attack classes and 1 benign class. It\ncontains 125,973 samples for training and 22,543 samples for testing. It contains\n41 features\\footnotemark, separated into four high-level categories of features:\nbasic features of TCP connections, content features within a connection suggested\nby domain knowledge, traffic features that are computed using a two-second time\nwindow, and host-based features. The NSL-KDD has been widely studied and so we\ndefer to prior work~\\cite{tl09detailed, db15aso} for the subtle details of the\ndataset.\n\n\\footnotetext{We used the post-processing options in WEKA~\\cite{hl09weka}, an\nopen-source data mining framework, to convert categorical features to one-hot\nvectors.}\n\n\\looseness=-1 \\vspace{3pt}\\noindent\\textbf{UNSW-NB15.} The UNSW-NB15 dataset was\ndesigned to be an updated version of the NSL-KDD, containing modern attacks that\nexpress a ``low footprint''~\\cite{ms15unsw}. The Australian Centre for Cyber\nSecurity (ACCS) used the \\textit{IXIA PerfectStorm} tool to create a combination\nof normal and abnormal network traffic. The abnormal traffic generated from the\nIXIA PerfectStorm tool is broken down into nine attack types, which we used as our\nsource classes for generating our adversarial examples\\footnotemark. After the\ntraffic is generated, the authors leveraged \\textit{Argus} and \\textit{Bro-IDS}\ntools to construct reliable features. \n\n\\footnotetext{While the authors intended this dataset to be used for benchmarking\n\\textit{anomaly} detection algorithms, we used it as a \\textit{signature-based}\ndataset by using the last feature, ``attack class,'' as the label. We achieved\nhigh classification accuracies, and thus argue that our modification has no impact\non the significance of our results found in the evaluation of our methodology.}\n\nThe UNSW-NB15 contains 10 classes, with 9 attack classes and 1 benign class. It\ncontains 175,341 samples for training and 83,332 samples for testing. The dataset\ncontains 48 features, separated into four high-level categories of features:\nflow-based, basic connection, content, and time-based. We defer to the authors for\na comprehensive description of the dataset~\\cite{ms15unsw} and its similarity with\nthe NSL-KDD~\\cite{mt16evaluation,ms17hybrid}.\n\n\\vspace{3pt}\\noindent\\textbf{MNIST.} The Modified National Institute of Standards\nand Technology database contains hanwritten digits. We chose to use MNIST due to\nits simplicity, to demonstrate cross-domain applicability of our approach, and\nto have a direct comparison with other adversarial machine learning approaches.\n\nThe MNIST database contains 10 classes, with numerical digits from 0 through 9.\nIt contains 60,000 samples for training and 10,000 samples for testing. Unlike the\nnetwork intrusion detection datasets, no preprocessing was required to integrate\nthis dataset into our experimental setup. We were able to use the dataset directly\nas-is. We defer to the authors for the intricate details surrounding the MNIST\ndataset~\\cite{lc10mnist}.\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{GTSRB.} The German Traffic Sign Recognition Benchmark\nis a dataset of common traffic signs found throughout Germany. We chose to use the\nGTSRB to for its increased complexity over MNIST and as a second example of the\ncross-domain applicability of our methodology.\n\n\\looseness=-1\nThe GTSRB contains 42 classes. After preprocessing, our experiments contained\n21,792 samples for training and 6,893 samples for testing. Throughout the dataset,\nthere are identical images of varying sizes. We first cropped the region of\ninterest (which contains the traffic sign) and downsampled to a final size of\n$30\\times30$. For more details concerning the GTSRB, we defer to the\nauthors~\\cite{sk12mvc}.\n\n\\subsection{Experiment Overview} In this subsection, we describe our experiments\nin detail, using the NSL-KDD as an application of our approach.\n\n\\looseness=-1 Through our evaluation on network intrusion detection, we note that\nour experiments emulate a realistic adversary by taking different attacks and\nmasking them as benign traffic.\n\n\\vspace{3pt}\\noindent\\textbf{Data Curation.} We perform a stratified shuffle-split\non the training set into five parts. Splitting our training set into five parts\nlays the foundation for measuring transferability: each partition (which we refer\nto as \\textit{A, B, C, D, E}) is representative of a uniquely trained model (which\nwe refer to as \\textit{$M_A$, $M_B$, $M_C$, $M_D$, $M_E$} respectively), mirroring\nthe setup described in ~\\cite{pp16transferability}. This setup allows us to\nmeasure \\textit{intra-technique} transferability rates: the rate at which\nadversarial examples crafted from one model are also misclassified by another\nmodel with the same learning technique. Furthermore, we can also measure the\nconverse, \\textit{inter-technique} transferability: the same misclassification\nrate except with a model of an entirely different learning technique\\footnotemark.\nNote, that \\textit{transferability} is not the same as a black-box attack; we do\nnot use the output of an oracle to label data for a surrogate model, as shown\nin~\\cite{pp16transferability,iy18black}. While transferability \\textit{enables}\nblack-box attacks, they are not the same, as transferability has been observed\nprior to the inception of black-box attacks~\\cite{sg13intriguing,gf15explain}.\n \n\\footnotetext{\\looseness=-1 For our inter-technique evaluation, we consider four popular\nlearning techniques: Logistic Regression (LR), Decision Trees (DT), Support Vector\nMachine (SVM), and K-Nearest Neighbors (KNN). Each one of these learners\nrepresent different learning paradigms (and are popular in commercial and academic\ncontexts), and are thus appropriate candidates for evaluating inter-technique\ntransferability. Hyperparameters and other details can be found in the\n\\hyperref[sec:appendix]{Appendix}.}\n\n\\looseness=-1 Next, we perform a stratified shuffle-split on the test set. This\ncreates two sets of isolated inputs: we use the \\textsc{AJSMA}{} on one set and the \\textsc{HSG}{}\non the other. This is to ensure that the sketches are only applied on inputs that\nhad no influence in the creation of the sketch (recall that we analyze the\nadversarial examples created by the \\textsc{AJSMA}{} to build sketches). Furthermore, the\nstratified shuffle-split of the test set is to ensure that the \\textsc{AJSMA}{} crafts\nadversarial examples from inputs spanning all classes. This enables us to create\neffective adversarial sketches, as all class-specific information is distilled\ninto the perturbation histogram.\n\n\\looseness=-1 Using the NSL-KDD as an example for the first stage, we built a\nMulti-layer Perceptron with 4 layers: an input layer of 123 units, fully-connected\nto 64 units, fully-connected to 32 units, and finally an output layer with 5\nunits\\footnotemark. The output layer conveys our 5 classes: Normal (Benign),\nProbe, Denial of Service (DoS), User to Root (U2R), and Remote to Local (R2L). We\nused rectified linear units (ReLU) as our chosen activation function for our\nhidden layers and softmax at the output layer. Our models are trained via the Adam\noptimizer~\\cite{km14adam} with a batch size of 200 and a learning rate of 0.01 for\n5 epochs. With our five splits, each model, $M_A$ through $M_E$, is trained with\n$\\mathtt{\\sim}$25,194 inputs. With these hyperparameters, network architecture,\nand training set size, we were able to achieve an average $77\\%\\pm 1.0\\%$ accuracy\non the test set, which is consistent with the literature~\\cite{tl09detailed,\nig15performance,ih13comparison,ms17hybrid}.\n\n\\footnotetext{\\looseness=-1 We note that the number of layers and units was influenced by\nresearch that suggests an optimal upper bound for the number hidden neurons for\nfeed-forward networks~\\cite{hg98upper}. The remainder of our hyperparameter\nselection follows no formal process.}\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Constraint Generation.} With the heuristic described\nin \\S\\ref{sec:methodology}, we learn constraints in the NSL-KDD with\n\\textsc{protocol} as the primary feature. The intuition behind this selection is\nstraightforward: a majority of the features in the NSL-KDD describe metadata\nsurrounding these protocols, e.g., flag information, services, and content-related\nfeatures like \\textsc{FTP} commands. The distribution of the extracted constraints\nfor the NSL-KDD can be found in Table \\ref{tab:nslconst}. We observe that the TCP\nprotocol offers the highest degree of maneuverability by a wide margin (and to no\nsurprise as it constitutes the majority of traffic flows in the dataset). This is\nunlike UDP and ICMP, who are significantly more constrained. Table\n\\ref{tab:nsl-constraints} in the \\hyperref[sec:appendix]{Appendix} shows all of\nthe constraints, sorted by feature type.\n\n\\begin{table}[!t]\n \\resizebox{\\columnwidth}{!}{ \\centering \n \\begin{tabular}{lccccr}\n \\toprule\n\n {} & \\multicolumn{4}{c}{Feature Type} & {}\\\\\n \\cmidrule{2-5}\n {Protocol} & {Basic} & {Content} & {Timing-based} & {Host-based} & {Total}\\\\\n \\midrule\n {\\textbf{TCP}} & {81} & {12} & {9} & {10} & {112}\\\\\n {\\textbf{UDP}} & {12} & {0} & {7} & {8} & {27}\\\\\n {\\textbf{ICMP}} & {14} & {0} & {7} & {8} & {29}\\\\\n\n \\bottomrule\n \\end{tabular}}\n\n \\caption{NSL-KDD constraint distribution, categorized by feature type - Unlike\n TCP, UDP and ICMP have limited degrees of freedom.} \\label{tab:nslconst}\n\n\\end{table}\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{figs\/nsl-histogram.png}\n\n \\caption{NSL-KDD model $M_B$ Perturbation Histogram produced by adversarial\n examples from the \\textsc{AJSMA}{} in log scale - Certain features are\n \\textit{consistently} increased (a) \\& (b) and decreased (c), indifferent of\n the source class.} \\label{fig:nslhisto}\n\n\\end{figure*}\n\n\\vspace{3pt}\\noindent\\textbf{Adversarial Example Generation.} To investigate the\nfirst question in our evaluation, we craft adversarial examples with the \\textsc{AJSMA}{}.\nWith the constraints integrated into the crafting process, we iterate over the\nfirst half of the test set for each model, $M_A$ through $M_E$, and craft\nadversarial examples (the second half of the test set is used to craft adversarial\nexamples via the sketches produced by the \\textsc{HSG}{}). Table \\ref{tab:nslcraft}\ndescribes the output of this stage from an example NSL-KDD run with ``Benign'' (0)\nas the target class. Figure \\ref{fig:nslhisto} demonstrates the perturbation\nhistogram computed from the same run.\n\n\\begin{table}[!t]\n \\resizebox{\\columnwidth}{!}{ \\centering \n \\begin{tabular}{lc}\n \\toprule\n\n \\multicolumn{2}{c}{\\textsc{AJSMA}{} Experiment Results - Target 0 ``Benign''}\\\\\n \\midrule\n {Testing Inputs} & {11,272}\\\\\n {Labeled as Target Class} & {4,856}\\\\\n {Misclassified as Target Class} & {2,532}\\\\\n {Number of Inputs Attacked} & {3,884}\\\\\n {Average Distortion} & {3.39\\% $\\mathtt{\\sim}$4.18 features}\\\\\n {Class Success Rates} & 0:NaN, 1:100\\%, 2:99\\%\\\\\n\t\t{} & {3.99\\%, 4:100\\%}\\\\\n\n \\bottomrule\n \\end{tabular}}\n\n \\caption{Output from crafting adversarial examples with the \\textsc{AJSMA}{} for NSL-KDD\n model, $M_B$ - Even domains with constraints are vulnerable to adversarial\n examples.} \\label{tab:nslcraft}\n\n\\end{table}\n\n\\looseness=-1\nWe note that there are particular features that were consistently perturbed in nearly\nall adversarial examples, namely setting the service as \\textsc{IRC} and\nincreasing \\textsc{num\\_access\\_files}\\footnotemark. To understand why, we used\nWEKA~\\cite{hl09weka} to analyze the distribution of these features, which revealed\na trivial explanation: inputs that use \\textsc{IRC} as the service and have high\nvalues for \\textsc{num\\_access\\_files} are heavily skewed towards our target\nclass, ``Benign''. Figure \\ref{fig:irc-naf} shows these distributions (Note that\nthe ``Benign'' target class is bottom class (0) on the Y-axis).\n\n\\footnotetext{While it may seem this feature is not derived from network data,\nboth the NSL-KDD and UNSW-NB15 have a ``high-level content`` category of features,\nwhich are derrived from payload information.}\n\n\\begin{figure}[t]\n\n \\includegraphics[width=0.49\\columnwidth]{figs\/distribution-IRC.png}\n \\includegraphics[width=0.49\\columnwidth]{figs\/distribution-num_access_files.png}\n\n \\caption{NSL-KDD class distribution for \\textsc{service=IRC} (a) and\n \\textsc{num\\_access\\_files} (b) - These two features are often perturbed due to\n their bias towards the ``Benign'' class, shown as class 0 on the Y-axis.} \\label{fig:irc-naf}\n\n\\end{figure}\n\n\\looseness=-1 \\vspace{3pt}\\noindent\\textbf{Adversarial Sketch Generation.} To\ninvestigate the second question in our evaluation, we use the perturbation\nhistogram (computed from adversarial examples crafted by the \\textsc{AJSMA}{}) to create\nadversarial sketches. As described in \\S\\ref{sec:methodology}, the \\textsc{HSG}{} creates\nadversarial sketches by selecting the top $n$ most perturbed features from the\nperturbation histogram. Finally, we craft adversarial examples by applying the\nsketch to the second half of the test set, for each model, $M_A$ through $M_E$.\n\n\\looseness=-1 \\subsection{Measuring Success} With the adversarial examples crafted\nvia the \\textsc{AJSMA}{} and \\textsc{HSG}{}, we measure the effectiveness of the two algorithms\nthrough white-box attacks and through transferability. We define the success rate\nof white-box attacks as the number of adversarial examples misclassified as the\ntarget class over the total number of attempted inputs, formally:\n\\[ SR_{wb} = \\frac{|\\{\\vect{x} \\in \\mathbb{X}: f(\\vect{x}) = t\\}|}{|\\mathbb{X}|}\\]\n\\looseness=-1\nwhere $\\mathbb{X}$ represents the set of attempted inputs, $f$ is a model, and $t$\nis the target class. Furthermore, we define the transferability success rate to be\nthe number of adversarial examples misclassified as the target class by the target\nmodel over the number of successful adversarial examples crafted from the source\nmodel, formally:\n\\[ SR_{transfer} = \\frac{|\\{\\vect{x} \\in \\mathbb{X}: f'(\\vect{x}) = t\\}|}\n{|\\{\\vect{x} \\in \\mathbb{X}: f(\\vect{x}) = t\\}|} \\]\nwhere $\\mathbb{X}$ again represents the set of attempted inputs from the source\nmodel $f$, and $f'$ represents the target model.\n\n\\vspace{3pt}\\noindent\\textbf{AJSMA Results.} In Table \\ref{tab:results} (a) on the\nleft, we show the NSL-KDD \\textsc{AJSMA}{} success rates for white-box attacks and\ntransferability rates. The labels on the left represent the source model used to\ngenerate adversarial examples and the labels on top represent the target models\nthat were attacked. For the intra-technique case, the white-box results can be\nread along the diagonal (as the source and target are the same model). For both\nintra- and inter-technique cases, the transferability rates are represented in\nall other cells. The \\textsc{AJSMA}{} was broadly successful in creating targeted\nadversarial examples while introducing relative amounts of distortion comparable\nto image-based experiments, even in the presence of constraints. These results\nsuggest that the constraints for our evaluated domain do not offer any robustness\nagainst adversarial examples.\n\n\\looseness=-1\nAdditionally, it is interesting that the adversarial examples produced by the\n\\textsc{AJSMA}{} had notable transferability rates in intra-technique case (an average of\n73\\% across our network intrusion detection experiments). This would suggest that\ntransferability is stronger in lower dimensional spaces.\n\n\\vspace{3pt}\\noindent\\textbf{HSG Results.} In Figure \\ref{fig:nslbbplot}, we show\nNSL-KDD model $M_B$ success rates for intra- and inter-technique transferability\nfor varying values of $n$ between 1 and 11. There are broad regions for values of\n$n$ which have high success rates, reaching 100\\% in white-box settings and\ngreater than 80\\% transferability rates for the majority of learning techniques.\nIndeed, it would appear that the most often perturbed features (by the \\textsc{AJSMA}{})\nare appropriate candidates for building an adversarial sketch. Furthermore, these\nresults confirm the existence of universal adversarial perturbations that obey\ndomain constraints.\n\n\\looseness=-1\nFinally, we are surprised at the fragility of the models trained on network\nintrusion detection data; sometimes perturbing only \\textit{three} features was\nneeded to misclassify greater than 80\\% of inputs in the test set. We believe that\nthis fragility is partly a function of the skewed distribution certain features\ncan have for specific classes, such as the ones shown in Figure \\ref{fig:irc-naf}.\nThis insight is unlike the adversarial examples crafted in image domains, where\nhigh dimensionality and a more balanced class distribution for features appear to\nmitigate this fragility. \n\n\\begin{figure}[!t]\n \\includegraphics[width=\\columnwidth]{figs\/nsl-transferability.png}\n\n \\caption{\\textsc{HSG}{} intra- and inter- transferability rates for NSL-KDD model\n $M_B$ for target ``Benign'' as a function of $l_0$ distortion $n$ - Values\n between 6-9 for $n$ have greater than 70\\% misclassification for most learning\n techniques.} \\label{fig:nslbbplot}\n\n\\end{figure}\n\nTable \\ref{tab:results} shows the success rates for both of our algorithms across\nall four datasets. The values of $n$ listed equal $\\mathtt{\\sim}4\\%$ $l_0$\ndistortion for the \\textsc{HSG}{}. Again, the labels on the left represent the source\nmodel used to generate adversarial examples, while the labels on top represent the\ntarget models that were attacked. For the intra-technique case, the white-box\nresults can be read along the diagonal (as the source and target models are the\nsame). The transferability rates, for both intra- and inter-technique cases, are\nrepresented in all other cells. \n\n\\looseness=-1\nWe observe high success rates for both the \\textsc{AJSMA}{} and the \\textsc{HSG}{} in both\nwhite-box attacks and transferability for several of the datasets. While the\n\\textsc{AJSMA}{} struggled significantly to produce adversarial examples that transferred\nas the dimensionality increased, the \\textsc{HSG}{} transferability rates were more\nresilient (but still affected) to the increased dimensionality\\footnotemark. This\nwould suggest that the distance between decision boundaries increases as model\ncomplexity increases, thus mitigating transferability for our algorithms.\n\n\\footnotetext{We hypothesize that this is partly due to how the \\textsc{AJSMA}{} is\ndesigned. Conceptually, the \\textsc{AJSMA}{} creates adversarial examples that\n\\textit{just} cross over the decision boundaries. While the decision boundaries\namong models of lower dimensionality may be similar (and thus, an adversarial\nexample can cross over the decision boundaries of multiple models), this appears\nto be untrue for models of higher dimensionality, where there can be multiple\nunique ways to separate the data. We could test this by regularizing high\ndimensional models. Furthermore, we could also alter the design of the \\textsc{AJSMA}{} to\ncontinue adding additional perturbations so that it returns an adversarial example\nthat is more confidently misclassified.}\n\n\\newcommand{\\ajsmakdd}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n\n \\multicolumn{1}{c}{\\normalsize \\textsc{Adaptive JSMA}} \\\\\n \\cmidrule(l{11em}r{11.05em}){1-1}\n\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mca{} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{100} & \\cc{69} & \\cc{51} & \\cc{51} & \\cc{61} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{73} & \\cc{99} & \\cc{72} & \\cc{78} & \\cc{67} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{63} & \\cc{69} & \\cc{100} & \\cc{66} & \\cc{69} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{54} & \\cc{93} & \\cc{70} & \\cc{99} & \\cc{64} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{83} & \\cc{79} & \\cc{71} & \\cc{66} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n \n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{34} & \\cc{40} & \\cc{34} & \\cc{20} \\\\ \\hhline{*{4}{|-}}\n \\cc{49} & \\cc{50} & \\cc{39} & \\cc{44} \\\\ \\hhline{*{4}{|-}}\n \\cc{30} & \\cc{34} & \\cc{27} & \\cc{24} \\\\ \\hhline{*{4}{|-}}\n \\cc{24} & \\cc{38} & \\cc{25} & \\cc{22} \\\\ \\hhline{*{4}{|-}}\n \\cc{42} & \\cc{55} & \\cc{41} & \\cc{17} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\n\\end{tabular}}}\n\n\\newcommand{\\ajsmaunsw}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mca{} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{100} & \\cc{97} & \\cc{92} & \\cc{96} & \\cc{96} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{50} & \\cc{100} & \\cc{72} & \\cc{94} & \\cc{71} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{73} & \\cc{81} & \\cc{100} & \\cc{93} & \\cc{87} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{69} & \\cc{64} & \\cc{55} & \\cc{100} & \\cc{59} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{66} & \\cc{80} & \\cc{90} & \\cc{96} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{72} & \\cc{81} & \\cc{29} & \\cc{53} \\\\ \\hhline{*{4}{|-}}\n \\cc{62} & \\cc{62} & \\cc{12} & \\cc{26} \\\\ \\hhline{*{4}{|-}}\n \\cc{71} & \\cc{76} & \\cc{19} & \\cc{49} \\\\ \\hhline{*{4}{|-}}\n \\cc{53} & \\cc{48} & \\cc{8} & \\cc{25} \\\\ \\hhline{*{4}{|-}}\n \\cc{66} & \\cc{69} & \\cc{15} & \\cc{38} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\newcommand{\\ajsmamnist}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mca{} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{100} & \\cc{39} & \\cc{30} & \\cc{19} & \\cc{11} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{6} & \\cc{100} & \\cc{14} & \\cc{6} & \\cc{2} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{12} & \\cc{32} & \\cc{99} & \\cc{9} & \\cc{8} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{17} & \\cc{44} & \\cc{26} & \\cc{100} & \\cc{21} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{20} & \\cc{53} & \\cc{33} & \\cc{20} & \\cc{99} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{21} & \\cc{24} & \\cc{18} & \\cc{5} \\\\ \\hhline{*{4}{|-}}\n \\cc{19} & \\cc{18} & \\cc{14} & \\cc{2} \\\\ \\hhline{*{4}{|-}}\n \\cc{23} & \\cc{22} & \\cc{15} & \\cc{1} \\\\ \\hhline{*{4}{|-}}\n \\cc{21} & \\cc{21} & \\cc{19} & \\cc{5} \\\\ \\hhline{*{4}{|-}}\n \\cc{24} & \\cc{27} & \\cc{18} & \\cc{6} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\newcommand{\\ajsmagtsrb}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mca{} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{97} & \\cc{2} & \\cc{1} & \\cc{3} & \\cc{2} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{1} & \\cc{95} & \\cc{0.5} & \\cc{1} & \\cc{1} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{2} & \\cc{2} & \\cc{98} & \\cc{2} & \\cc{0.8} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{0.5} & \\cc{1} & \\cc{3} & \\cc{99} & \\cc{1} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{0.7} & \\cc{2} & \\cc{1} & \\cc{2} & \\cc{94} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.2} & \\cc{1} & \\cc{3} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.3} & \\cc{2} & \\cc{2} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.1} & \\cc{2} & \\cc{1} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.2} & \\cc{0.8} & \\cc{3} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.1} & \\cc{0.5} & \\cc{2} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\newcommand{\\topnkdd}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n\n \\multicolumn{1}{c}{\\large \\textsc{Histogram Sketch Generation}} \\\\\n \\cmidrule(l{8.5em}r{8.5em}){1-1}\n\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mcr{$n=6$} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} & \\cc{100} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{80} & \\cc{85} & \\cc{16} & \\cc{19} \\\\ \\hhline{*{4}{|-}}\n \\cc{79} & \\cc{82} & \\cc{20} & \\cc{59} \\\\ \\hhline{*{4}{|-}}\n \\cc{65} & \\cc{83} & \\cc{50} & \\cc{34} \\\\ \\hhline{*{4}{|-}}\n \\cc{81} & \\cc{87} & \\cc{22} & \\cc{59} \\\\ \\hhline{*{4}{|-}}\n \\cc{98} & \\cc{93} & \\cc{27} & \\cc{19} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\n\\end{tabular}}}\n\n\\newcommand{\\topnunsw}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mcr{$n=9$} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{88} & \\cc{99} & \\cc{95} & \\cc{96} & \\cc{96} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{99} & \\cc{100} & \\cc{99} & \\cc{100} & \\cc{99} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{93} & \\cc{97} & \\cc{100} & \\cc{77} & \\cc{95} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{80} & \\cc{99} & \\cc{100} & \\cc{100} & \\cc{99} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{98} & \\cc{100} & \\cc{100} & \\cc{94} & \\cc{92} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{99} & \\cc{96} & \\cc{29} & \\cc{37} \\\\ \\hhline{*{4}{|-}}\n \\cc{99} & \\cc{100} & \\cc{20} & \\cc{62} \\\\ \\hhline{*{4}{|-}}\n \\cc{73} & \\cc{100} & \\cc{27} & \\cc{41} \\\\ \\hhline{*{4}{|-}}\n \\cc{74} & \\cc{99} & \\cc{13} & \\cc{30} \\\\ \\hhline{*{4}{|-}}\n \\cc{85} & \\cc{100} & \\cc{28} & \\cc{39} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\newcommand{\\topnmnist}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mcr{$n=41$} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{24} & \\cc{35} & \\cc{18} & \\cc{14} & \\cc{9} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{17} & \\cc{42} & \\cc{14} & \\cc{15} & \\cc{54} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{22} & \\cc{31} & \\cc{41} & \\cc{9} & \\cc{14} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{13} & \\cc{30} & \\cc{14} & \\cc{36} & \\cc{13} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{15} & \\cc{29} & \\cc{11} & \\cc{12} & \\cc{31} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{45} &\\cc{48} & \\cc{14} & \\cc{7} \\\\ \\hhline{*{4}{|-}}\n \\cc{62} &\\cc{61} & \\cc{13} & \\cc{5} \\\\ \\hhline{*{4}{|-}}\n \\cc{60} &\\cc{55} & \\cc{32} & \\cc{5} \\\\ \\hhline{*{4}{|-}}\n \\cc{46} &\\cc{46} & \\cc{33} & \\cc{6} \\\\ \\hhline{*{4}{|-}}\n \\cc{44} &\\cc{58} & \\cc{21} & \\cc{4} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\newcommand{\\topngtsrb}{%\n\\resizebox{0.5\\columnwidth}{!}{ \\centering \n\\begin{tabular}{c c}\n \\begin{tabular}{c|c|c|c|c|c|}\n \\mcr{$n=104$} & \\mca{$M_A$} & \\mca{$M_B$} & \\mca{$M_C$} & \\mca{$M_D$} & \\mca{$M_E$} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_A$} & \\cc{34} & \\cc{6} & \\cc{27} & \\cc{6} & \\cc{7} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_B$} & \\cc{9} & \\cc{27} & \\cc{14} & \\cc{2} & \\cc{8} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_C$} & \\cc{17} & \\cc{4} & \\cc{44} & \\cc{2} & \\cc{7} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_D$} & \\cc{29} & \\cc{12} & \\cc{16} & \\cc{29} & \\cc{12} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\mcb{$M_E$} & \\cc{14} & \\cc{8} & \\cc{15} & \\cc{3} & \\cc{35} \\\\ \\hhline{*1{~}*{5}{|-}}\n \\end{tabular}\n\n \\begin{tabular}{|c|c|c|c|}\n \\mca{$LR$} & \\mca{$SVM$} & \\mca{$DT$} & \\mca{$KNN$} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.2} & \\cc{2} & \\cc{12} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.5} & \\cc{1} & \\cc{6} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.6} & \\cc{1} & \\cc{5} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{1} & \\cc{1} & \\cc{5} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\cc{0.1} & \\cc{1} & \\cc{4} & \\cc{0} \\\\ \\hhline{*{4}{|-}}\n \\end{tabular}\n\\end{tabular}}}\n\n\\begin{table*}[ht]\n \\centering \n \\begin{subtable}[h]{\\textwidth}\n \\ajsmakdd\n \\topnkdd\n \\caption{NSL-KDD}\n \\end{subtable}\n \\begin{subtable}[h]{\\textwidth}\n \\ajsmaunsw\n \\topnunsw\n \\caption{UNSW-NB15}\n \\end{subtable}\n \\begin{subtable}[h]{\\textwidth}\n \\ajsmamnist\n \\topnmnist\n \\caption{MNIST}\n \\end{subtable}\n \\begin{subtable}[h]{\\textwidth}\n \\ajsmagtsrb\n \\topngtsrb\n \\caption{GTSRB}\n \\end{subtable}\n\n \\caption{Results for \\textsc{AJSMA}{} (left) and \\textsc{HSG}{} (right) for all of our\n experiments for target class ``Benign'' - Values of $n$ for our sketches\n represent $\\mathtt{\\sim}4\\%$ $l_0$ distortion.}\n \n \\label{tab:results}\n\\end{table*}\n\nFrom our investigation, we highlight some key takeaways: \n\n\\begin{enumerate} \n\n \\item Constraints, as they stand, are not problematic for crafting adversarial\n examples in the domains we studied. The \\textsc{AJSMA}{} reached $100\\%$ success rates\n for most attempted inputs, with distortion rates comparable to image-based\n experiments, while in the presence of constraints.\n \n \\item Universal adversarial perturbations that obey domain constraints exist.\n The \\textsc{HSG}{} produced Adversarial Sketches that reached 100\\% success rates in\n constrained domains for a values of $n$ that represent $\\mathtt{\\sim}4\\%$\n $l_0$ distortion. Furthermore, they produced adversarial examples that had\n higher transferability rates than the \\textsc{AJSMA}{}.\n\n \\item Network intrusion detection data is highly fragile: a small dimensionality\n and biased distributions enable attack algorithms to alter very few features\n to successfully craft targeted adversarial examples.\n \n \\item Worst-case scenarios (i.e., white-box attacks) are highly vulnerable, not\n surprisingly. With direct access to model parameters, an adversary can have a\n sophisticated level of control over the output of a model.\n\n \\item Network intrusion detection data appears to be highly vulnerable to\n transferability attacks. Even in the presence of disjoint training sets and\n different learning techniques, both attacks produced adversarial examples with\n surprising levels (an average of 73\\% for the \\textsc{AJSMA}{} and 97\\% for the\n \\textsc{HSG}{}, for models of the same learning technique) of transferability.\n\n\\end{enumerate}\n\n\\section{Introduction} \\label{sec:introduction} \\looseness=-1 \nMachine learning algorithms are\nrapidly revolutionizing many industries including\ntransportation ~\\cite{pl91efficient}, finance ~\\cite{bs01business,wg97neural},\nhealth care ~\\cite{kr15machine}, education~\\cite{jl08authorship,sz07machine}, and\nsecurity ~\\cite{db15aso,sm10outside,sc99application,ts09intrusion}. For the past\ndecade, we have seen a revolution in automation as research has focused on\nincreasing the accuracy, problem size, and applicable domains for these automated\nlearners. The results are promising: the latest learning algorithms have shown\npreviously impossible accuracies for problems spanning multiple domains. However,\nwhen an adversary is introduced, the machine learning algorithms and the myriad\ndomains they serve often become vulnerable to an adversary.\n\n\\looseness=-1\nOne of the directions the field of adversarial machine learning explores is the\nimpact of \\textit{adversarial examples}: inputs to machine learning models that an\nattacker has intentionally designed to cause the model to make a mistake, e.g.,\nmisclassify~\\cite{gf17attacking}. Within the scope of this research, there have\nbeen studies that target particular inputs through a variety of threat models,\nwith algorithms such as the \\textsc{Jacobian-Saliency Map\nApproach}~\\cite{pp16limit}, \\textsc{Carlini-Wagner}~\\cite{cl17towards}, and\n\\textsc{Projected Gradient Descent}~\\cite{md18towards}, among others. A recent\nclass of attacks, exposing potentially more serious vulnerability, compute\n\\textit{universal adversarial perturbations}~\\cite{gf15explain,df16universal}.\nThese perturbations can be precomputed and quickly applied at runtime to arbitrary\ninputs and still achieve adversarial goals (e.g., misclassification of arbitrary\ninputs to a class chosen by the adversary). The existence of adversarial examples\npresents a compelling barrier for sensitive domains that use machine learning.\nInvestigations have shown that no domain (thus far) is immune to this phenomenon;\nthe scope of adversarial examples has been expansive, reaching into image\nprocessing~\\cite{pp16limit,gf15explain,cl17towards,ms16deep}, malware\ndetection~\\cite{gs17adversarial,kj18adversarial},\ntext~\\cite{ji17adversarial,eh18hotflip}, and even speech\nrecognition~\\cite{cl18audio}.\n\n\\looseness=-1\nA repeated criticism of adversarial machine learning research is that\ninvestigations have almost completely focused on unconstrained domains. \nImplicit to this argument is that such freedom overestimates the capabilities of an\nadversary in constrained domains where they are often bound by the semantics\nof features \\textit{and} capable of only controlling a subset of features. It stands to\nreason that such algorithms used to generate adversarial examples would be less\neffective in constrained domains. In this work, we consider network intrusion\ndetection as our exemplary constrained domain, using existing network intrusion\ndetection datasets: the NSL-KDD~\\cite{tl09detailed} and UNSW-NB15~\\cite{ms15unsw}.\nHere, \\textit{constraints} are defined by the following three characteristics: the\nvalues within a feature may be fixed (binary vs continuous), the values of\ndifferent features may be correlated (TCP flags in packets and TCP as the\ntransport protocol), and some features may not be controllable by an adversary\n(round-trip times).\n\n\\looseness=-1\nFurthermore, while there has been research that has focused on constrained \n\\textit{adversaries}\\cite{gs17adversarial,su19one}, who are bound in their \ncapabilities (e.g., how many total features can be perturbed or which features \nare manipulable), we differentiate in that we consider constrained \\textit{domains}.\nSpecifically, these constraints describe the kinds of inputs that are \n\\textit{permissible} in a domain (For example, network packets that do not \nobey the TCP\/IP protocol would not be permissible). In this work, we consider \nthe union of both adversary and domain constraints, unlike previous work. \n\n\\looseness=-1\nIn this paper, we test the hypothesis that constrained domains are less vulnerable\nto the techniques of adversarial machine learning from two perspectives: from the\nperspective of traditional adversarial algorithms, and from universal adversarial\nperturbations. This leads to two approaches in our evaluation. Prior to our\nevaluation, we identify and extract constraints from a dataset. Then, we develop\nan augmented algorithm, the \\textsc{Adaptive JSMA} (\\textsc{AJSMA}{}), to construct\nadversarial examples that obey domain constraints. Next, we design a second\nalgorithm, the \\textsc{Histogram Sketch Generation} (\\textsc{HSG}{}), the first attack to\ncompute \\textit{adversarial sketches}: universal perturbations used to craft\nadversarial examples en masse that comply with domain constraints. Other\nalgorithms that produce universal perturbations neither have a mechanism to comply\nwith constraints, nor are they immediately suitable for attacking constrained\ndomains (as described in \\S\\ref{sec:methodology}). With both algorithms, we\nmeasure the success rate of crafting adversarial examples by attacking models\ndirectly in both constrained and unconstrained domains, and find that we were\ncomparably successful in both domains (greater than 95\\%). Furthermore, we use\nadversarial examples crafted from a model to attack different learning techniques\ntrained on similar, but not identical, data, demonstrating that adversarial\nexamples can \\textit{transfer} in constrained domains (reaching up to 93\\% for the\n\\textsc{AJSMA}{} and 100\\% for the \\textsc{HSG}{}). Finally, we show that even if an adversary\nmaintains attack behavior \\textit{and} cannot arbitrarily control certain features\n\\textit{and} must obey the TCP\/IP protocol, there is still a surprising amount of\nexploitable attack surface to craft adversarial examples. We make four\ncontributions:\n\n\\begin{enumerate}\n\n \\item We introduce formalism to express constraints and design an algorithm that\n is able to extract them systematically in the domains we consider. Learning\n constraints codifies the space of permissible adversarial examples.\n\n \\item We introduce two algorithms. The \\textsc{Adaptive JSMA}, which crafts\n adversarial examples that obey domain constraints. The \\textsc{Histogram\n Sketch Generation}, which produces adversarial sketches: universal adversarial\n perturbations that obey domain constraints.\n\n \\item We perform experiments where a we impose an extreme amount of\n constraints, to the point where an adversary can only control five random\n features, and show that adversarial examples can still be crafted with a\n $\\mathtt{\\sim}50\\%$ success rate within our studied constrained domains. \n\n \\item We demonstrate promising results for both algorithms, reaching greater\n than \\results{} misclassification rates across the datasets used in our\n experiments. This suggests that our studied constrained domains are as\n vulnerable as their unconstrained counterparts.\n\n\\end{enumerate}\n\n\\section{Methodology} \\label{sec:methodology} In this section, we explain the\nintuition behind generating adversarial examples in constrained domains.\n\n\\subsection{AML in Constrained Domains} Throughout, both in this section for\nexamples and in the evaluation, we will focus principally on network intrusion\ndetection data (specifically, the use of TCP\/IP) due to the constrained nature of\nthe domain. Here, we assume a network intrusion detection system classifies\nfeature vectors (representing network traffic flows) as benign or\nmalicious\\footnotemark. These feature vectors are created through a feature\nextraction algorithm, which accepts network packets as input. Therefore, the\nadversary seeks to bypass the network intrusion detection system by tweaking\nmalicious network traffic (guided by adversarial machine learning techniques), so\nthat it is subsequently classified as benign. Naturally, the adversary must obey\nthe TCP\/IP protocol so that the network attacks can be realizable. Any feature\nvector that violates the TCP\/IP protocol (such as negative port numbers) would be\nmanifestly adversarial. \n\n\\footnotetext{\\looseness=-1 Without loss of generality, the models we use in the\nevaluation discriminate between different kinds of network attacks (as well as\nbenign traffic). Nonetheless, the goal of the adversary remains unchanged.}\n\n\\subsection{Challenges in AML with Constraints} Crafting adversarial examples in\nconstrained domains is a necessarily different process from crafting adversarial\nexamples in unconstrained domains. Not all features represent the same kind of\ninformation (pixels vs packet information), nor do they describe the same kind of\nstatistical data (discrete vs a blend of categorical, continuous, and discrete).\nThese differences change the threat surface and the underlying assumptions\nsurrounding the capabilities of an adversary in constrained domains. Existing\nalgorithms are unsuitable for attacking constrained domains for these reasons:\n\n\\looseness=-1\n\\vspace{2pt}\\noindent\\emph{(1) Existing algorithms are largely optimized for human\nperception.} While there is an open discussion on the amount and kinds of\ndistortion that are appropriate~\\cite{cl17towards}, existing algorithms have been\ntuned for image domains~\\cite{gf15explain,cl17towards,ms16deep}. That is, these\nalgorithms try to minimize human perception of the distortion introduced in\nadversarial examples. However, such metrics have no meaning for many\nconstrained domains because they are not perceived by humans. Thus, using\nalgorithms optimized for human perception offers us limited utility.\n\n\\looseness=-1\n\\vspace{2pt}\\noindent\\emph{(2) Existing algorithms assume adversaries have full\ncontrol over the feature space.} Most algorithms perturb the entire feature space\nto minimize an $l_p$ norm as a surrogate for estimating a measure of human\nperception~\\cite{gf15explain,cl17towards,ms16deep}. This is likely an unreasonable\nassumption in constrained domains. For example, in network intrusion detection,\nfeatures can represent broad network behaviors that exist outside the control of\nan adversary, e.g., round-trip times.\n\n\\vspace{2pt}\\noindent\\emph{(3) Existing algorithms do not consider domain\nconstraints.} Crafting adversarial examples that obey domain constraints is\nnecessary to mount practical attacks. However, prior works strategically applied\nperturbations in order to avoid constraints~\\cite{gs17adversarial,kj18adversarial}\n(such as only adding bytes at the end of a binary). While a reasonable strategy\nfor the studied domain (malware), it may not be effective if the adversary can\nonly perturb features that are constrained. As an example of a constraint, network\nintrusion detection datasets commonly have protocol and service (port number) as\nfeatures and certain services are exclusive to certain protocols. Therefore, to\nproduce an adversarial example that is representative of a legitimate traffic\nflow, algorithms need to enforce these constraints.\n\n\\subsection{Crafting with Constraints} In this subsection, we describe how we\nlearn constraints and incorporate them into crafting adversarial examples.\n\n\\looseness=-1 \n\\vspace{3pt}\\noindent\\textbf{Learning Constraints.} If the constraints are not\nexplicitly given, then they must be inferred. One of the first requirements for\nproperly modeling the constraints of any modeled domain is to codify the specific\nrelationships between features. One of the most important of these relationships\nis the notion that we introduce as a \\textit{primary feature.} Intuitively, a\nprimary feature is a feature that, when set to particular value, limits the range\nof permissible values for other features. Conversely, a \\textit{secondary feature}\ndoes not impose any limitations for other features. Therefore, the first task is\nto identify primary features and the relationships from these primary features to\nsecondary features. Interestingly, primary features may have relationships amongst\nthemselves, as well as secondary features.\n\n\\looseness=-1 As an example, many popular network intrusion detection datasets\ninclude features that represent services, packet flags, and other protocol-related\ninformation. Since these features share a causal relation with protocols, we\ndesignate transport layer protocols to be primary features and the others as\nsecondary features\\footnotemark. As another example, we could designate using SSL\nas a primary feature, which would imply using TCP as the transport protocol, which\nfinally dictates the values for other secondary features. Table\n\\ref{tab:nsl-constraints} in the \\hyperref[sec:appendix]{Appendix} demonstrates\nthe constraints for one of our evaluated datasets.\n\n\\footnotetext{\\looseness=-1 We identified our primary features based on our\nunderstanding of the domain and observations in the data. We explore methods to\nsystematically discover primary features when domain expertise is unavailable in\n\\S\\ref{sec:discussion}.}\n\n\\looseness=-1\nGiven that constraints are restrictions on where and how features can be\nperturbed, it is intuitive to model these constraints as a simple form of\nfirst-order logic. Here, primary features are the predicates in logical\nexpressions that determine the properties (e.g., values) that a collection of\nvariables (e.g., secondary features) can have. Said alternatively, the set of\nprimary features are the conditionals on the values of other secondary features.\nFormally, we can understand constraints to have the following form:\n\\begin{equation*} \\forall \\vect{x} \\in \\mathbb{X}: \\vect{x}_k \\Rightarrow\n(\\vect{x}_1 \\in \\mathbb{Y}_1) \\land (\\vect{x}_2 \\in \\mathbb{Y}_2) \\land \\dots\n\\land (\\vect{x}_n \\in \\mathbb{Y}_n) \\end{equation*} \n\\noindent where $\\vect{x}$ represents an input in a dataset $\\mathbb{X}$, $k$ is a\nprimary feature, and $\\mathbb{Y}_n$ represents the values permissible by the\nsemantics of feature $n$ (e.g., $\\{0,1\\} \\in \\mathbb{Y}$ if the feature is binary,\nor $\\{\\mathbb{R}: 0 \\leq y \\leq 1\\} \\in \\mathbb{Y}$ if the feature is continuous).\nFor example, we can represent a simple TCP constraint as follows:\n\\begin{equation*}\n \\forall \\vect{x} \\in \\mathbb{X}: \\vect{x}_{TCP} \\Rightarrow \\vect{x}_{port} \\in\n [1,\\dots, 65535]\n\\end{equation*}\n\\noindent There are many existing algorithms that can learn constraints from\nlogic~\\cite{dr18learning}, as inputs can be modeled as simple instantiations of\nexpressions (i.e., \\textit{propositions}). Here, we use primary features to design\nan algorithm to learn these constraints. \n\nAfter primary features have been identified, we begin to learn constraints based\non the following heuristic: a constraint exists between a primary feature $k$, and\nany other feature $p$, if there exists at least one input in the training set\nwhere both $k$ and $p$ are seen together. For example, features that describe TCP\npacket flags (i.e., $p$) would have the value 1.0 for TCP traffic flows (i.e.,\n$k$) and 0.0 for non-TCP traffic flows. Therefore, TCP packet flag features are\n\\textit{constrained} to TCP flows. Conceptually, these constraints encode the\nmaneuvers that are possible (and probable) for an adversary\\footnotemark.\n\n\\footnotetext{In \\S\\ref{sec:uncontrollable}, we also perform experiments where we\nfurther add additional constraints that an adversary would have to obey.}\n\n\\looseness=-1 \\vspace{3pt}\\noindent\\textbf{Addressing Constraints.} To address the\nchallenges discussed prior, we integrate \\textit{constraint resolution} into the\ncrafting process. This guarantees that generating an adversarial example obeys not\nonly the semantic constraints of the domain, but also the probabilistic\nconstraints of the dataset as well. As an artifact of our constraint learning\nprocess, we may also learn constraints that are technically allowable within the\nsemantics of the domain, but are simply never observed in the dataset. As a second\ntechnique, we also simultaneously minimize the total number of features perturbed\n(i.e., $l_0$) to obey constraints and comply with the limited control over\nfeatures an adversary may have. \n\n\\vspace{3pt}\\noindent\\textbf{Measuring Distance.} The distance between an\nadversarial example and its original counterpart is a measurement of the\ndistortion introduced by an algorithm, commonly represented by an $l_p$ norm. Most\nalgorithms have either limits on the maximum allowable distortion or explicit\ntermination conditions when a particular amount of distortion is introduced. There\nis debate on the most appropriate distance measure (i.e., the choice of $l_p$\nnorm) to use for modeling levels of human perception~\\cite{cl17towards}. Our study\nof constrained domains departs from this debate, as most of these domains are not\ninherently visual. Therefore, we argue that measuring distance under the $l_0$\nnorm is appropriate for our purposes. In particular, we choose $l_0$ to model how\nan adversary may be limited to controlling certain features, to enforce domain\nconstraints, and because we are not optimizing for human perception.\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Augmenting the JSMA.} Before we can use the \\textsc{JSMA}{}\nin constrained domains, we make a slight adjustment to the algorithm so that it\ncan better search the space of possible adversarial examples. By default, the\n\\textsc{JSMA}{} parameter $\\theta$ determines the magnitude and direction of a selected\nperturbation. A positive $\\theta$ will increase feature values and a negative\n$\\theta$ will decrease them. To allow an adversary more freedom when crafting\nadversarial examples, we modified the \\textsc{JSMA}{} to perturb in either direction\ndynamically. This directly improved the success rate of the \\textsc{JSMA}{}. We refer to\nthis modified version as the \\textsc{A(daptive)} \\textsc{JSMA}{}. \n\nTo allow the \\textsc{AJSMA}{} to perturb in either direction, we simply evaluate both masks\nused in~\\cite{pp16limit} that are applied to the saliency map when $\\theta$ is\npositive or negative. Formally, for any feature $i$ to be a perturbation\ncandidate, $i$ must satisfy:\n\n\\footnotesize\n\\[\n \\begin{split}\n \\left(\\frac{\\partial f_t(\\vect{x})}{\\partial \\vect{x}_i} > 0 \\text{ and }\n \\sum_{j\\neq t} \\frac{\\partial f_j(\\vect{x})}{\\partial \\vect{x}_i} < 0\\right)\n \\text{ or } \\left(\\frac{\\partial f_t(\\vect{x})}{\\partial \\vect{x}_i} < 0\n \\text{ and } \\sum_{j\\neq t} \\frac{\\partial f_t(\\vect{x})}{\\partial\n \\vect{x}_i} > 0\\right)\n \\end{split}\n\\]\n\\normalsize\n\n\\noindent where $\\frac{\\partial f_t(\\vect{x})}{\\partial \\vect{x}_i}$ represents\nthe forward derivative for a model $f$ and target class $t$ with respect to\nfeature $i$ in an input $\\vect{x}$. Conceptually, the \\textsc{AJSMA}{} only considers\nfeatures to be perturbable if the target gradient and sum of non-target gradients\nare in opposing directions. Intuitively, this simply means that any perturbation\nreduces the distance to the target class or increases the distance to non-target\nclasses.\n\nThis modification enables us to determine the optimal perturbation direction for\nincreasing (the left half of the mask) or decreasing (the right half of the mask)\nfeatures. Afterwards, we use the scoring metric found in~\\cite{pp16limit} and\nreturn the most influential feature with the optimal perturbation direction. \n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Integrating Constraints into the \\textsc{AJSMA}{}.}\nIntegrating constraint resolution distills to preventing the \\textsc{AJSMA}{} from\nselecting features that violate constraints. Once a feature is\nselected for perturbation, we check if this feature is constrained to a primary\nfeature\\footnotemark. This check is described by Algorithm \\ref{alg:constraints}.\n\n\\footnotetext{We note that any additional perturbations made by our constraint \nresolution algorithm are included when we measure the distance between an\noriginal input and an adversarial example. To this effect, adversarial examples\nthat obey domain constraints do not violate $l_0$ norms.}\n\nConsider the following example using the Algorithm \\ref{alg:constraints}. A UDP\ntraffic flow is given to the \\textsc{AJSMA}{}. After analyzing the saliency map, the\n\\textsc{AJSMA}{} suggests that the current service, \\textsc{tftp\\_u}, should be switched\nto \\textsc{ftp}, which is a service constrained to TCP. After the service switch\nis made, the perturbed input is presented to Algorithm \\ref{alg:constraints} to\ncheck whether or not any constraint is violated. The first condition determines if\nthe perturbed feature $p$ is a primary feature (i.e., TCP, UDP, or ICMP). In this\nexample, it is not, so we move to the second condition and evaluate if $p$ is\nconstrained to exclusively one primary feature. In this case, $p$ is exclusively\nassociated with TCP. Thus, the search domain is further restricted to\nTCP-compliant features and the transport protocol of the input is switched from\nUDP to TCP. Since the input has switched primary features from UDP to TCP, we\nfinally set all non-TCP features to 0. Once the \\textsc{AJSMA}{} terminates, the produced\nadversarial example will be representative of a permissible traffic flow, i.e., it\nobeys the domain constraints.\n\n\\begin{spacing}{0.1}\n\\begin{algorithm}[!t]\n\n \\footnotesize\n \\DontPrintSemicolon\n \\SetAlgoNoLine\n \\SetKwInOut{Input}{Input}\n \\Input{$p$, $\\Gamma$, $\\matrx{S}$, $\\vect{X}$, $h$}\n \\tcp{p is a primary feature}\n \\If{$p\\in \\mathbb{K}$} {\n $\\Gamma = \\Gamma\\cap h(p)$ \\\\\n $\\vect{x}_p\\leftarrow$ switch primary feature to $p$\\\\\n }\n \\tcp{p is constrained to exactly one primary feature}\n \\ElseIf{$\\exists! k\\in \\mathbb{K} \\textrm{ \\upshape s.t. } p\\in h(k)$}{\n $\\Gamma = \\Gamma\\cap h(k)\\setminus\\{p\\}$\\\\\n $\\vect{x}_k\\leftarrow$ switch primary feature to $k$\\\\\n }\n \\tcp{p is constrained to multiple primary features}\n \\ElseIf{$\\exists k\\in \\mathbb{K} \\textrm{ \\upshape where } p\\in h_k$}{\n $\\Gamma = \\Gamma\\setminus\\{p\\}$\\\\\n $k'\\leftarrow$ the current primary feature in $\\vect{x}$\\\\\n \\tcp{x is using an illegal primary feature wrt p}\n \\If{$p\\notin h(k')$}\n {\n $k= \\textrm{arg max}_{\\{k\\in\\mathbb{K} \\rvert p\\in h(k)\\}} \\, S_k$\\\\\n $\\Gamma = \\Gamma\\cap h(k)$\\\\\n $\\vect{x}_k\\leftarrow$ switch primary feature to $k$\\\\\n }\n }\n \\tcp{p constrains all primary features}\n \\ElseIf{$\\forall k \\in \\mathbb{K}, p\\in h_k$}{\n $\\Gamma = \\Gamma\\setminus\\{p\\}$\\\n }\n \\If{\\textrm{ \\upshape switched primary feature to }$k$}{\n \\tcp{ensure x is not using illegal features}\n $\\forall i \\notin h(k),\\, \\vect{x}_i\\leftarrow 0$\n }\n \\textbf{return} $\\Gamma$, $\\vect{x}$\n \\caption{\\textsc{Resolving Constraints\\newline}\n \\label{alg:constraints}\n \\footnotesize\n \\textrm{$p$} is a candidate feature, \\textrm{$\\Gamma$} is the search domain,\n $\\matrx{s}$ is the saliency map for the current input $\\vect{x}$,\n $h:\\mathbb{K}\\mapsto\\mathbb{V}$ is an associative array containing\n constraints.}\n\n\\end{algorithm}\n\\end{spacing}\n\n\\subsection{Creating Adversarial Sketches} In this subsection, we describe how we\ncreate adversarial sketches\\footnotemark: universal \nperturbations that obey constraints.\n\n\\footnotetext{\\looseness=-1 The concept of ``sketching,'' also known as\n\\textit{approximate query processing}~\\cite{cm11sketch}, was first introduced by\nFlajolet et al. Sketching refers to a class of streaming algorithms that seek to\nextract information from a data stream in a single pass~\\cite{fj85probabilistic}.\nCommonly deployed in memory-constrained environments, these algorithms approximate\nor summarize the information in a given data stream. Adversarial sketches are\nsimilar as they are an approximation of a universal perturbation and computed\nthrough one pass of inputs.}\n\nFor the same reasons described at the beginning of \\S\\ref{sec:methodology},\nexisting algorithms to generate universal adversarial\nperturbations~\\cite{hy17service,df16universal,bn17adversarial} are not immediately\nusable in constrained domains: the algorithms are largely optimized for human\nperception, they assume adversaries have full control over the feature space, and\nthey do not consider domain constraints.\n\n\\looseness=-1\nIn our search for adversarial sketches in constrained domains, we take a\nprincipled approach using two tools: adversarial examples generated from the\n\\textsc{AJSMA}{} and a \\textit{perturbation histogram}. Prior to the algorithms mentioned\nearlier for computing universal adversarial perturbations, encountering these\nperturbations was a matter of chance: an adversary would be required to apply a\nperturbation generated from an attack on other unperturbed inputs and simply\nobserve the universality of the perturbation, i.e., brute-forcing the universal\nperturbation~\\cite{gf15explain}.\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Adversarial Examples from the AJSMA.} Initially, we\nfollowed the same brute-force approach in~\\cite{gf15explain}: we used adversarial\nexamples crafted from the \\textsc{AJSMA}{} to glean insights for universal perturbations.\nThis brute-force approach is feasible (in terms of computational complexity) for\nnetwork intrusion detection datasets as their dimensionality and cardinality is\nsmall. After evaluating every perturbation generated by the \\textsc{AJSMA}{} on every\ninput in the test set, we discovered a handful of universal perturbations.\n\n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Perturbation Histograms.} The perturbation histogram\nencodes how perturbations are distributed en masse (an example is shown later in\n\\S\\ref{sec:evaluation}). To produce the histogram, we enumerate over all of the\nperturbations generated by the \\textsc{AJSMA}{} (including any additional perturbations\nmade to resolve constraints) and record the perturbed features and directions. Our\ninsight for the histogram is rooted in how the \\textsc{AJSMA}{} scoring metric ranks\ninfluential features; we hypothesized that features commonly perturbed across\ninputs from different classes would be optimal candidates for building an\nadversarial sketch. This hypothesis was reinforced by our observation that the\nperturbation histogram was (relatively) static: random shuffling of partitioned\ntraining sets, unique training parameters, and random subsets of analyzed\nadversarial examples yielded minor changes to the perturbation histogram. These\nsubstantial adjustments to our experiment workflow demonstrated little change\nbetween perturbation histograms. These observations suggest that the perturbation\nhistogram is a combined representation of class-based saliency \\textit{and} domain\nconstraints.\n\n\\looseness=-1\nWith the universal perturbations discovered through the \\textsc{AJSMA}{} and the static\nnature of the perturbation histogram, we made an observation: \\textbf{the majority\nof the perturbed features (and their associated directions) in the most successful\nuniversal perturbations generated from the \\textsc{AJSMA}{} mapped \\textit{directly} to\nthe most perturbed features in the perturbation histogram. This key observation\nled us to the creation of the \\textit{Histogram Sketch Generation}}.\n \n\\looseness=-1\n\\vspace{3pt}\\noindent\\textbf{Histogram Sketch Generation.} The \\textsc{Histogram\nSketch Generation} accepts a perturbation histogram $\\matrx{h}$ and integer $n$ as\nparameters and returns a adversarial sketch $\\vect{a}$, which consists of the top\n$n$ most frequently perturbed features and optimal directions\\footnotemark. We\nobserved that the most successful universal perturbations generated by the\n\\textsc{AJSMA}{} had a subset of features that directly mapped to the most frequently\nperturbed features in the perturbation histogram. Intuitively, if we consider\nthese successful universal perturbations as the optimal solution, then the \\textsc{HSG}{}\napproximates the optimal solution by returning a subset of those features (and\nassociated directions). Figure \\ref{fig:graphics} in the\n\\hyperref[sec:appendix]{Appendix} demonstrates a handful of sketches.\n\n\\footnotetext{While the \\textsc{HSG}{} does not take a target class as a parameter, it\nuses information directly from the perturbation histogram to create an adversarial\nsketch. As a consequence, the \\textsc{HSG}{} will return a targeted adversarial sketch if\nthe histogram is built from targeted adversarial examples.}\n\n\\looseness=-1\nIt is interesting to note that the \\textsc{HSG}{} is essentially the problem of variable\nselection in classical statistics. Variable selection involves the selection of a\nsubset of relevant variables (or features) in the model, such that a ``minimal''\namount of information is lost (e.g., minimal impact on model loss). Many common\nprocedures in classical statistics, such as LASSO or stepwise regression, aim to\nbalance model fit with a penalty on the $l_0$ norm (or a relaxation of this norm\nto $l_1$, in the case of LASSO)\\cite{tr96lasso}. Our approach of greedily\nselecting the top $n$ most perturbed features is directly analogous to stepwise\nregression's (greedy) selection of the $n$ features which best explain the data.\nAs the complexity grows combinatorially with the number of features, greedy\nmethods are necessarily resorted to, and our approach here is no exception.\n\n\n\\section{Uncontrollable Features} \\label{sec:uncontrollable} Throughout this\npaper, we have discussed some of the differences between adversarial machine\nlearning in constrained domains versus unconstrained domains. One of the most\nfundamental questions within this community is: \\textit{what precisely is an\n``adversarial example?''} There are varying definitions with different objectives\nused throughout AML research. In the image space, it has been generally agreed\nthat if an attack algorithm produces perturbations that are undetectable by a\nhuman observer, then it is an adversarial example. However, it is not clear how to\ntranslate this objective in other domains.\n\n\\looseness=-1 Research outside of image space (usually) provides their own\ndefinitions: perturbed malware must maintain its properties of\nmalware~\\cite{gs17adversarial,kj18adversarial}, perturbed audio must be nearly\ninaudible~\\cite{cl18audio}, perturbed text must preserve its\nsemantics~\\cite{eh18hotflip,ji17adversarial}, among other definitions. For our\nwork in network intrusion detection, we follow an intuitive definition: perturbed\nnetwork flows must maintain their attack behavior. For example, a DoS attack must\nstill be a DoS attack post-perturbation.\n\nHowever, validating attack behaviors is a nontrivial task as security is\ncontextual: a DoS attack on a government network has different behaviors than an\nattack on a family business. Therefore, any sort of simulation to observe attack\nbehavior must be in a similar context to the one in which the dataset was built\nfrom. This is particularly challenging for old datasets like the NSL-KDD, as even\na similar network setup would have different behavior given the modern hardware\nand software on the systems that would constitute that network. Even if all these\nfactors could be accounted for, there are certain classes of attacks whose\n``success'' is challenging to measure. For example, in reconnaissance attacks it\nis difficult to know if the information gathered by an adversary is useful.\n\n\\looseness=-1\nRegardless, all of these points of contention are driven by a single hypothesis:\n\\textit{Perhaps adversarial examples cannot be crafted if features which represent\nthe semantics of the attack cannot be perturbed}. Instead of attempting to justify\nwhy any set of features is critical to the semantics of the attack, we take a\ndifferent stance on addressing this hypothesis: even if some reasonably sized\nsubset of features could not be perturbed (as to not invalidate the attack), we\nargue that adversarial examples can still be successfully crafted. With the\nlessons learned through this research, we set out to investigate this hypothesis\nwith a simple experiment.\n\n\\looseness=-1\nTo test this hypothesis, we iterate over multiple sets of randomly selected\nfeatures and make them unperturable by the adversary. We repeat this random\nselection for sets of varying cardinalities from 1 to 41 features (where 41\nrepresents the entire feature space). We train a new model on the full NSL-KDD\ntraining set and use the 100 most representative\\footnotemark\\footnotetext{We find\nthe most representative inputs by maximizing the difference (via the softmax\nlayer) between the output component that corresponds to the label and the sum of\nthe components that represent all non-label classes.} inputs from each\nclass\\footnotemark\\footnotetext{The ``R2L'' class only had 17 inputs that were\ncorrectly classified. Thus, we crafted from 317 inputs as opposed to 400.} from\nthe test set. Next, we iterate over all of the possible combinations of features\nmade unperturbable to an adversary, i.e., ${\\binom{41}{k}}$\\footnotemark for\n\\footnotetext{The NSL-KDD has 41 features before expanding categorical features to\none-hot vectors. If a particular combination contains a categorical feature, we\neliminate all possible values associated with the feature from the search domain.}\n$k \\in \\{1, 2,\\dotsc, 41\\}$\\footnotemark. Once we have identified a set of\nunperturable features, we simply eliminate that set from initial search domain of\nthe \\textsc{AJSMA}{}. In this experiment, we crafted a total of 17,664,191 adversarial\nexamples. Figure \\ref{fig:fixed} demonstrates the success rate of the \\textsc{AJSMA}{} as\na function of the number of controllable (i.e., perturbable) features.\n\n\\footnotetext{To prevent combinatoric explosion, we randomly sampled 1,500 unique\ncombinations if the total number of possible combinations for a particular value\nof $k$ exceeded 1,500. In total, we evaluated 55,723 unique combinations of\nunperturbable features.}\n\n\\begin{figure}\n\n \\includegraphics[width=\\columnwidth]{figs\/controllable-features.png}\n\n \\caption{\\textsc{AJSMA}{} Success Rate with unperturbable features - The overall success\n rate starts to decrease significantly when the adversary is restricted to controlling \n $\\mathtt{\\sim}12$ features.}\n \n \\label{fig:fixed}\n\\end{figure}\n\n\\looseness=-1 The results support our argument: the success rate of \\textsc{AJSMA}{}\nbegins to decline when the adversary can only control $\\mathtt{\\sim}12$ features.\nEven when an adversary has control over only \\textit{five random} features (which\nrepresents around $\\mathtt{\\sim}10\\%$ of the feature space), the success rate of\ncrafting adversarial examples (with the most representative forms of an attack) is\nslightly less than 50\\%. Furthermore, this result demonstrates that applying more\nrestrictive constraints would have little impact on the success of the adversary. \n\nFinally, we would also like to highlight how this experiment demonstrates a type\nof constraint not covered in the evaluation section: features that the adversary\nsimply does not have control over. Throughout this work, our constraints were\ndefined via the semantics of the domain, i.e. the TCP\/IP protocol. However, this\nexperiment to preserve the semantics of the attack also serves as a demonstration\nof the efficacy of an adversary under this second type of constraints. These\nresults suggest that even if an adversary maintains attack behavior \\textit{and}\ncannot arbitrarily control certain features \\textit{and} must obey the TCP\/IP\nprotocol, there is still a surprising amount of exploitable attack surface to\ncraft legal adversarial examples.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $p , q \\in \\mathbb{N}-\\{0\\}$;\na $p \\times q$ interval matrix is a $p \\times q$ matrix \nwhose entries are closed bounded nonempty intervals in $\\mathbb{R}$.\nWe say that a $p \\times q$ matrix $A$ with entries in $\\mathbb{R}$ is contained in $\\mu$ if and only if, for every $i,j$, we have that $A_{i,j} \\in \\mu_{i,j}$. \n \n On the other side, for any field \n $K$, a partial matrix over $K$ is defined to be a \nmatrix where only some of the entries are given and they are elements of \n$K$;\na completion of a partial matrix is a specification in $K$ of the unspecified entries. \nWe say that a submatrix of a partial matrix is specified if all its entries are specified.\n\nThere are several papers both on interval matrices and\non partial matrices. \nOn partial matrices, there is a wide literature about\nthe problem of determining the maximal and the minimal rank of the completions of a partial matrix.\n We quote, for instance, the papers \\cite{CD}, \\cite{Woe} and \n\\cite{CJRW}. In the last, Cohen, Johnson, Rodman and Woerdeman determined \nthe maximal rank of the completions of a partial\nmatrix in terms of the ranks and the sizes of its maximal \nspecified submatrices; see also \\cite{CD}\nfor the proof.\n The problem of determining the minimal rank of the completions of a partial matrix seems more difficult and it has been solved only in some particular cases,\n see for example \\cite{Woe1} for the case of triangular matrices. \n \n Also interval matrices have been widely studied and in particular\nthere are several papers studying when a $p \\times q $ interval matrix $\\mu$ has full rank, that is when all the matrices contained in $\\mu$ have rank equal to $\\min\\{p,q\\}$. \nFor any $p \\times q $ interval matrix $\\mu = ( [m_{i,j}, M_{i,j}])_{i,j}$ with $m_{i,j} \\leq M_{i,j}$,\nlet $C^{\\mu}$,$\\Delta^{\\mu}$ and $|\\mu|$ be \n the $p \\times q$ matrices such that $$ C^{\\mu}_{i,j}= \\frac{m_{i,j}+ M_{i,j}}{2}, \\hspace*{1cm} \\Delta^{\\mu}_{i,j}= \\frac{M_{i,j}- m_{i,j}}{2}, \\hspace*{1cm} \n|\\mu|_{i,j} = \\max\\{|m_{i,j}|,| M_{i,j}|\\} $$ for any $i,j$. \nThe following theorem\ncharacterizes full-rank square interval matrices:\n\n\\begin{thm} {\\bf (Rohn, \\cite{Rohn})} \nLet $\\mu =( [m_{i,j}, M_{i,j}])_{i,j}$ be a $p \\times p$ interval matrix, where\n $m_{i,j} \\leq M_{i,j}$ for any $i,j$.\nLet $Y_p=\\{-1,1\\}^p$ and, for any $x\n \\in Y_p$, denote by $T_x$ the diagonal matrix whose diagonal is $x$; moreover\n define\n $$C^{\\mu}_{x,y} = C^{\\mu} - T_x \\, \\Delta^{\\mu} \\, T_y.$$\n\nThen $\\mu$ is a full-rank interval matrix \nif and only if, for each\n$x,y ,x',y'\\in Y_p$, $$det(C^{\\mu}_{x,y}) \\,det(C^{\\mu}_{x',y'})>0. $$\n\\end{thm}\n\n\nSee \\cite{Rohn} and \\cite{Rohn2} for other characterizations. Finally, as to interval matrices, we want to quote also the paper \\cite{JR}, where \n an algorithm for checking regularity of interval matrices is presented, and \nthe following theorem characterizing \nfull-rank $p \\times q$ interval matrices, see\n\\cite{Rohn3}, \\cite{Rohn4}, \\cite{Shary}:\n\n\\begin{thm} {\\bf (Rohn)} \nA $p \\times q$ interval matrix $\\mu$ with $p \\geq q$ has full rank if and only if the system\nof inequalities $$\\hspace*{2cm} |C^{\\mu} x| \\leq \\Delta^{\\mu}\n|x|, \\hspace*{1.5cm} x \\in \\mathbb{R}^q$$ has only the trivial solution $x=0$. \n\\end{thm}\n\n \n Obviously the problem of partial matrices in the real case and the one of interval matrices are connected; in fact we can consider matrices whose entries\n are closed connected subsets of $\\mathbb{R}$; these matrices generalize both the interval matrices and the partial matrices.\n We call a matrix \n whose entries are closed connected subsets of $\\mathbb{R}$\n a ``general closed interval matrix''.\n \n In this paper we generalize Rhon's result \n on full-rank square interval matrices\n to \ngeneral closed interval matrices, see Theorem \\ref{gen} for the precise statement. \n\n \n\\section{Notation and some recalls}\n\n$\\bullet$ Let $\\mathbb{R}_{>0}$ be the set $\\{x \\in \\mathbb{R} | \\; x >0\\}$ and\nlet $\\mathbb{R}_{\\geq 0}$ be the set $\\{x \\in \\mathbb{R} | \\; x \\geq 0\\}$; we define analogously $\\mathbb{R}_{<0}$ and $\\mathbb{R}_{ \\leq 0}$.\n\n$\\bullet $ Throughout the paper let $p , q \\in \\mathbb{N}-\\{0\\}$. \n\n$\\bullet $ Let $\\Sigma_p$ be the set of the permutations on $\\{1,....,p\\}$.\n\n$\\bullet$ Let $M(p \\times q, \\mathbb{R}) $ denote the set of the $p \\times q $ matrices\nwith entries in $\\mathbb{R}$. For any $A\n\\in M(p \\times q, \\mathbb{R})$, let $rk(A) $ denote the rank of $A$\nand let $A^{(j)}$ be the $j$-th column of $A$.\n\n$\\bullet$ For any $p \\times q$ general closed interval matrix $\\mu$, any $\\{i_1, \\ldots, i_s\\} \\subset \\{1, \\ldots , p\\}$ and any $\\{j_1, \\ldots, j_r\\} \\subset \\{1, \\ldots , q\\}$, we denote by $\\mu_{\\widehat{i_1,\\ldots , i_s}, \\widehat{j_1, \\ldots , j_s}} $ the matrix obtained from $\\mu$ by deleting the rows $i_1, \\ldots , i_s$ and the columns $j_1, \\ldots , j_r$. \n\n\n\\begin{defin} A {\\bf general interval matrix} is a matrix \n whose entries are connected nonempty subsets of $\\mathbb{R}$.\n \n A {\\bf general closed interval matrix} is a matrix \n whose entries are closed connected nonempty subsets of $\\mathbb{R}$.\n \n An {\\bf interval matrix} is a matrix \n whose entries are closed bounded nonempty intervals of $\\mathbb{R}$.\n\n\n\n\n Let $\\mu $ be a $p \\times q $ general interval matrix.\n As we have already said, \n given a matrix $A \\in M(p \\times q, \\mathbb{R}) $, we say that $ A \\in \\mu $ if and only if $a_{i,j} \\in \\mu_{i,j} $ for any $i,j$.\n \n We define\n$$mrk(\\mu) = min\\{rk(A) | \\; A \\in \\mu \\},$$\n$$Mrk(\\mu) = max\\{rk(A) | \\; A \\in \\mu \\}.$$\nWe call them respectively {\\bf minimal rank} and {\\bf maximal rank} of $\\mu$.\nMoreover, we define \n$$rkRange (\\mu) = \\{ rk(A) | \\; A \\in \\mu \\};$$ \nwe call the set above the {\\bf rank range} of $\\mu$.\n\nWe say that the entry $i,j$ of $\\mu$ is a {\\bf constant}\n if $ \\mu_{i,j}$ is a subset of $\\mathbb{R}$ given by only one element. \n\nWe say that the entry $i,j$ of $\\mu$ is {\\bf bounded}\n if $ \\mu_{i,j}=[a,b]$ for some $a,b \\in \\mathbb{R}$; we say that the entry $i,j$ of $\\mu$ is {\\bf half-bounded}\n if either $ \\mu_{i,j}=[a,+ \\infty ) $ ({\\bf left-bounded}) or \n $ \\mu_{i,j}=(- \\infty ,a] $ ({\\bf right-bounded})\n for some $a \\in \\mathbb{R}$.\n\\end{defin}\n\n\\begin{rem}\nLet $\\mu$ be an interval matrix. Observe that $$rkRange(\\mu) =\n [mrk(\\mu) , Mrk(\\mu)] \\cap \\mathbb{N}.$$\n\\end{rem}\n\nSee \\cite{Rub} for a proof.\n\n\n\\begin{defin}\nGiven a $p \\times p $ interval matrix, $\\nu$, a {\\bf partial generalized\ndiagonal} ({\\bf pg-diagonal} for short) of length $k$ of $\\nu$ is a $k$-uple of the kind $$\n(\\nu_{i_1, j_1},\\dots, \\nu_{i_k, j_k})$$ \nfor some $\\{i_1, \\dots i_k\\}$ and $ \\{j_1, \\dots, j_k\\} $ subsets of $ \\{1,\\dots ,p\\}$.\n\nIts {\\bf complementary matrix} is defined to be the submatrix of $\\mu$ given by the rows and columns whose indices are respectively in $\\{1,\\ldots , p\\}- \\{i_1, \\ldots , i_k\\}$ and \nin $\\{1,\\ldots , p\\}- \\{j_1, \\ldots , j_k\\}$. \n\nWe say that a pg-diagonal is {\\bf \ntotally nonconstant} if and only if all its entries are not constant.\n\nWe define $det^c(\\mu) $ as follows:\n$$ det^c(\\mu) = \\sum_{\\sigma \\in \\Sigma_p \\; s.t. \\; \\mu_{1, \\sigma(1)}, \\dots, \\mu_{p, \\sigma(p)} \\; are \\; constant} \\epsilon (\\sigma) \\,\n\\mu_{1, \\sigma(1)} \\cdot \\ldots \\cdot\\mu_{p, \\sigma(p)} ,$$ \nwhere $ \\epsilon (\\sigma)$ denotes the sign of the permutation $ \\sigma$.\n\\end{defin}\n\n\n\nThe following theorem and corollary were proved in \\cite{Rub} for interval matrices; the same \nproof yields the results for general closed interval matrices:\n\n\n\\begin{thm} \\label{Mrkquadrate}\nLet $\\mu $ be a $p \\times p $ general closed interval matrix. Then $Mrk(\\mu) < p$ if and only if the following conditions hold:\n\n(1) in $\\mu$ there is no totally nonconstant pg-diagonal of length $p$,\n\n\n(2) the complementary matrix of every totally nonconstant pg-diagonal of length between $0$ and $p-1$ has $det^c $ equal to $0$.\n\n\\end{thm}\n\n\\begin{cor} \\label{Mrk}\nLet $\\mu $ be a general closed interval matrix. Then $Mrk(\\mu)$\nis the maximum of the natural numbers $t$ such that there is a $ t \\times t $ submatrix of\n $\\mu$ either with a totally nonconstant pg-diagonal of length \n $t$ or with a totally nonconstant\n pg-diagonal of length between $0$ and $t-1$ whose complementary matrix has $det^c \\neq 0$.\n \\end{cor}\n\n \n \\begin{nota} Let $\\mu$ be a general closed interval matrix.\n\n$\\bullet $ \nWe denote by $\\tilde{\\mu}$ the matrix obtained from $\\mu$ by replacing the entries $(-\\infty , + \\infty)$ with $0$.\n \n $\\bullet$ \n We denote by $ \\overline{\\mu}$ \n the matrix obtained from $\\mu$ by replacing every entry of kind $[a, + \\infty)$, for some $a \\in \\mathbb{R}$, with $a$ and every entry of kind $(-\\infty, b]$, for some $b \\in \\mathbb{R}$, with $b$.\n \n $\\bullet $ \n We denote by $\\mu_l$ \n the matrix obtained from $\\mu$ by replacing every entry of kind $[a, b] $, for some $a ,b \\in \\mathbb{R}$ with $a \\leq b$, with $a$.\n We denote by $\\mu_r$ \n the matrix obtained from $\\mu$ by replacing every entry of kind $[a, b] $, for some $a ,b \\in \\mathbb{R}$ with $a \\leq b$, with $b$.\n \n \n \n\\end{nota}\n\n\n \\begin{defin} \\label{vertexmatrix}\nLet $\\mu $ be a general closed interval matrix.\n \nWe say that $\\gamma $ is a {\\bf vertex matrix} of $\\mu$ if $\\gamma_{i,j} \\in \\{m_{i,j}, M_{i,j}\\}$ for any $i,j$ such that $\\mu_{i,j}$ is a bounded interval and $\\gamma_{i,j}= \\mu_{i,j}$ otherwise.\n\n\n\n\nWe say that a vertex matrix $\\gamma$ of $\\mu$ is {\\bf of even type} if,\nfor every $2 \\times 2$ submatrix of $\\mu$ such that all its entries are bounded intervals,\neither the number of the entries of the corresponding submatrix of $\\gamma$ that are equal to the minimum of the corresponding entries of $\\mu$ is even or some of its entries are constant. \nWe say that it is {\\bf of odd type} if it is not of even type. \\end{defin}\n \n {\\bf Example.} Let $$\\mu=\\begin{pmatrix}\n[1,2] & [2,3] & [2,+\\infty] \\\\\n[-3,4] & [-1, 5] & [1,4]\n \\end{pmatrix}.$$ Then $\\gamma=\\begin{pmatrix}\n1 & 2 & [2,+\\infty] \\\\\n4 & 5 & 1\n \\end{pmatrix}$ is a vertex matrix of $\\mu $ of even type, while $\\delta=\\begin{pmatrix}\n1 & 3 & [2 , -\\infty)\\\\\n4 & 5 & 1\n \\end{pmatrix}$ is a vertex matrix of $\\mu$ of odd type. \n\n\\bigskip\n\nObserve that, by using Definition \\ref{vertexmatrix}, we can restate Rohn's theorem as follows: \n\n\n\\begin{thm} {\\bf (Rohn, \\cite{Rohn})} \\label{Rohnbis}\nLet $\\mu$ be a $p \\times p$ interval matrix.\nThen $\\mu$ is a full-rank interval matrix \nif and only if, for any vertex matrices $A_1, A_2$ of even type of $\\mu$, $$det(A_1) \\,det(A_2)>0. $$\n\\end{thm}\n \n \n\\section{The main result}\n\n\\begin{rem} \\label{polin}\nLet $p(x_1,\\ldots, x_n)$ be a polynomial\nwith coefficients in $\\mathbb{R}$ of degree $1$ in every variable. Let $\\overline{x} \\in \\mathbb{R}^n$.\nThen $p(x) \\geq 0 $ (respectively $p(x)>0$) for every $x \\in \\overline{x} + \\mathbb{R}^n_{\\geq 0}$ if and only if \n$p(\\overline{x}) \\geq 0$ (respectively $p(\\overline{x})>0$)\nand $\\frac{ \\partial \\,p }{\\partial x_i} (x) \\geq 0 $ for every $x \n \\in \\overline{x} + \\mathbb{R}^n_{\\geq 0}$ and for every $i=1, \\ldots, n$.\n\n\\end{rem}\n\n\\begin{lem} \\label{lem1}\nLet $\\mu $ be a general closed interval $p \\times p$ matrix. \nThen $\\mu $ is full-rank if and only if\n$\\tilde{\\mu}$ is full-rank and for every \n$i,j$ such that $ \\mu_{i,j}= (-\\infty, + \\infty)$ \nwe have that $ Mrk( \\mu_{\\hat{i}, \\hat{j}})\n < p-1$.\n\\end{lem}\n\n\n\\begin{proof} Suppose \n $\\mu $ is full-rank. Then obviously \n$\\tilde{\\mu}$ is full-rank. Moreover, let \n$i,j$ be such that $ \\mu_{i,j}= (-\\infty, + \\infty)$. Then the determinant of every $A \\in \\mu_{\\hat{i}, \\hat{j}}$ must be zero; otherwise, suppose there exists $A \\in \\mu_{\\hat{i}, \\hat{j}}$ with $det(A) \\neq 0$; then,\nfor any choice of\n $x_{i,s}$ in $\\mu_{i,s}$ for any $s \\in \\{1, \\ldots , p\\}-\\{j\\}$\n and $x_{r,j}$ in $\\mu_{r,j}$ for any $r \\in \\{1, \\ldots , p\\}-\\{i\\}$,\nwe can choose $x_{i,j} \\in \\mu_{i,j}$ such that, if we define $X$ to be the matrix \nsuch that $X_{r,s}= x_{r,s}$ for $(r,s)$ such that \neither $r=i$ or $s=j$ and\n$X_{\\hat{i}, \\hat{j}}=A$, we have that \nthe determinant of $X$ is zero, which is absurd since \n$\\mu$ is full-rank.\nSo $ Mrk( \\mu_{\\hat{i}, \\hat{j}})\n < p-1$ and we have proved the right-handed implication.\n \n On the other side, let $(i_1,j_1), \\ldots, \n (i_s, j_s) $ be the indices of the entries of $\\mu $ equal to $ (-\\infty, + \\infty)$ and suppose that $\\tilde{\\mu}$ is full-rank and that $ Mrk( \\mu_{\\hat{i_k}, \\hat{j_k}})< p-1$\n for every $k=1, \\ldots ,s$. \n Let $B \\in M(p \\times p, \\mathbb{R})$ be such that \n$B \\in \\mu$. For every $k=1, \\ldots, s$, let $B_k$ be the matrix obtained from $B$ by \n replacing the entries $(i_1,j_1), \\ldots , (i_k, j_k) $ with $0$. Then \n $det (B)= det (B_1)= \\ldots =det(B_s) $ \n since $ Mrk( \\mu_{\\hat{i_k}, \\hat{j_k}})< p-1$ for every $k=1, \\ldots ,s$.\nObviously $B_s \\in \\tilde{\\mu}$, so $det (B_s)$ is nonzero since $\\tilde{\\mu}$ is full-rank; hence $det(B)$ is nonzero and\nwe conclude.\n\\end{proof}\n\n\n\\begin{lem} \\label{lem2}\nLet $\\nu $ be a general closed interval $p \\times p $ matrix with only bounded or half-bounded entries. Then $\\nu $ is full-rank\n if and only if, for any $\\gamma $ vertex matrix of $\\nu$ of even type, we have that $\\gamma $ is full-rank and $det(\\overline{\\gamma})$ has the same sign as $det( \\overline{\\nu}_l)$. \n\\end{lem}\n\n\n\\begin{proof} $\\Longrightarrow$ \nObviously if $\\nu$ has full-rank, then the determinant of all\nthe matrices contained in $\\nu $ must have the same sign (since the determinant is a continous function and the image by a continous function of a connected subset is connected), in particular \n $det(\\overline{\\gamma})$ must have the same sign as $det( \\overline{\\nu}_l)$.\n \n $\\Longleftarrow$ \nLet $A \\in M(p \\times p, \\mathbb{R}) $ be such that $A \\in \\nu$; define \n$\\alpha $ to be the interval matrix such that \n$\\alpha_{i,j} = \\nu_{i,j}$ if $\\nu_{i,j}$ is bounded and $\\alpha_{i,j} = a_{i,j}$ if $\\nu_{i,j}$ is half-bounded.\n\n \n To prove that $A$ has full-rank, obviously it is sufficient to prove that $\\alpha$ is full-rank.\n By Rohn's theorem (see Theorem \\ref{Rohnbis}), to prove that $\\alpha$ is full-rank, it is sufficient to prove that any two\n vertex matrices $A_1,A_2$ of even type of $\\alpha$ are full-rank and the sign of their determinant is the same.\nObviously, for $i=1,2$, the matrix $A_i$ is contained in a vertex matrix $\\gamma_i$ of even type of $\\nu$; by assumption, for $i=1,2$, the matrix $\\gamma_i$ is full-rank, so the matrix $A_i$ is full-rank\nand its determinant has the same sign\nas $det(\\overline{\\gamma}_i)$.\nMoreover, by assumption the sign of \n$det(\\overline{\\gamma}_i)$ is equal to the \nsign of $det( \\overline{\\nu}_l)$ for $i=1,2$;\n in particular $det(\\overline{\\gamma}_1)$\n and $det(\\overline{\\gamma}_2)$ have the same sign, so $det(A_1)$ and $det (A_2)$\n have the same sign, as we wanted to prove.\n\\end{proof}\n\n\\begin{nota}\nLet $\\rho$ be a square general closed interval matrix. We say that $DET(\\rho)$ is \ngreater than $0$ (respectively less than $0$, equal to $0$...) if and only if, for any $A \\in \\rho$, we have that \n$det(A)$ is greater than $0$ (respectively less than $0$, equal to $0$...).\n\\end{nota}\n\n\\begin{lem} \\label{lem3}\nLet $\\rho $ be a general closed interval $p \\times p $ matrix with only constant or half-bounded entries. Then $\\rho $ is full-rank\n if and only if $det(\\overline{\\rho}) \\neq 0$ and, \n for any $(i_1,j_1), \\ldots, \n (i_s,j_s) \\in \\{1,\\ldots , p \\} \\times \\{1,\\ldots , p \\} $ such that $i_1, \\ldots , i_s$ are distinct and $j_1, \\ldots , j_s$ are distinct and $\\rho_{i_1,j_1}, \\ldots , \\rho_{i_s,j_s} $ are half-bounded, \n$$ (-1)^{i_1+j_1+ \\widetilde{i_2} + \\widetilde{j_2}+ \\ldots + \\widetilde{i_s}+ \\widetilde{j_s}+ \\chi((i_1,j_1),\n\\ldots, (i_s, j_s) ) } det (\\overline{\\rho}) \\,det (\\overline{\\rho}_{\\widehat{i_1,\\ldots, i_s}, \\widehat{j_1, \\ldots, j_s}} ) \\geq 0, $$\nwhere: \n\n$\\bullet $ the determinant of a $0 \\times 0 $ matrix is defined to be $1$,\n\n$\\bullet$ \n$\\chi((i_1,j_1),\n\\ldots, (i_s, j_s) )$ is defined to be the number of the right-bounded intervals in \n$\\rho_{i_1,j_1}, \\ldots, \\rho_{i_s,j_s}$, \n\n\n$\\bullet$ \n for $t=2, \\ldots, s$, we define \n$\\widetilde{i_t}$ to be $i_t$ minus the number of the elements among $i_1, \\ldots i_{t-1}$ smaller than \n$i_t$ and $\\widetilde{j_t}$ to be $j_t$ minus the number of the elements among $j_1, \\ldots j_{t-1}$ smaller than $j_t$.\n\n\\end{lem}\n\n\\begin{proof}\nBy Remark \\ref{polin}, the matrix $\\rho $ is full-rank if and only if $det(\\overline{\\rho}) \\neq 0$ and,\nfor any $i_1, j_1$ such that $\\rho_{i_1,j_1}$ is half-bounded,\n$$ (-1)^{i_1+ j_1 + \\chi(i_1,j_1)} \\, det (\\overline{\\rho} ) \\,DET (\\rho_{\\hat{i_1},\\hat{j_1}} ) \\geq 0 $$ \nand, again by Remark \\ref{polin}, the last condition holds if and only if, \nfor any $i_1, j_1$ such that $\\rho_{i_1,j_1}$ is half-bounded, we have that\n$$ (-1)^{i_1+ j_1 + \\chi(i_1,j_1)} \\, det (\\overline{\\rho} ) \\,det (\\overline{\\rho}_{\\hat{i_1},\\hat{j_1}} ) \\geq 0 $$ \nand, for any $i_2, j_2$ such that $i_1 \\neq i_2$ and $j_1\n\\neq j_2$ and \n$\\rho_{i_2,j_2}$ is half-bounded, we have that\n$$ (-1)^{i_1+ j_1 +\\widetilde{i_2} + \\widetilde{j_2} + \\chi((i_1,j_1), (i_2,j_2))} \\, det (\\overline{\\rho} ) \\,DET (\\rho_{\\widehat{i_1,i_2},\\widehat{j_1,j_2}} ) \\geq 0 $$ \nand so on.\n\\end{proof}\n\n\n\\begin{thm} \\label{gen}\nLet $\\mu $ be a general closed interval matrix. \nThen $\\mu $ is full-rank if and only if the following conditions hold:\n\n(1) for every $i,j$ such that $\\mu_{i,j} = (- \\infty, +\\infty) $ we have that in $\\mu_{\\hat{i}, \\hat{j}} $ \n there are no totally nonconstant pg-diagonal of length \n $p-1$ and the complementary matrix of every totally nonconstant\n pg-diagonal in $\\mu_{\\hat{i}, \\hat{j}} $ of length between $0$ and $p-2$ has $det^c $ equal to $0$.\n\n(2) the product of the determinant every vertex matrix of $\\overline{\\tilde{\\mu}}$ of even type and the determinant of $\\overline{\\tilde{\\mu}}_l$ \nis positive, \n\n(3) for every vertex matrix $\\gamma$ of $\\tilde{\\mu}$ of even type and every $(i_1,j_1),\\ldots, (i_s, j_s) \n\\in \\{1,\\ldots , p \\} \\times \\{1,\\ldots , p \\} $ such that $i_1, \\ldots , i_s$ are distinct and $j_1, \\ldots , j_s$ are distinct and \n$\\mu_{i_1,j_1}, \\ldots , \\mu_{i_s,j_s} $ are half-bounded, we have:\n$$ (-1)^{i_1+j_1+ \\widetilde{i_2}+ \\widetilde{j_2}+\\ldots + \\widetilde{i_s}+ \\widetilde{j_s}+ \\chi((i_1,j_1),\n\\ldots, (i_s, j_s) ) } \ndet (\\overline{\\tilde{\\mu}}_l) \\,\ndet (\\overline{\\gamma}_{\\widehat{i_1,\\ldots, i_s}, \\widehat{j_1, \\ldots, j_s}} ) \\geq 0 ,$$\nwhere:\n\n\n$\\bullet $ the determinant of a $0 \\times 0 $ matrix is defined to be $1$, \n\n\n$\\bullet$ the number $\\chi((i_1,j_1),\n\\ldots, (i_s, j_s) )$ is defined to be the number of right-bounded \nintervals in \n$\\mu_{i_1,j_1}, \\ldots, \\mu_{i_s,j_s}$,\n\n$\\bullet $ \n for $t=2, \\ldots, s$, we define \n$$\\widetilde{i_t} := i_t - \\sharp \\{i_r\\; \\mbox{\\em for} \\; r=1, \\dots, t-1 | \\; i_r z_x 1_x,\\\\\n\\\\\n0& {\\rm otherwise}, \\end{array} \\right. \n\\;\\;\\;\\;\\;\\;\\\nY_x(t) = \\left\\{ \\begin{array}{ll}\n1\\;\\;\\;\\; & z_x \\geq t,\\;\\; z_{(x)}> t 1_x,\\\\\n\\\\\n0 & {\\rm otherwise}, \\end{array} \\right. \n\\label{eqn:counts}\n\\end{equation*}\nwhere $1_x$ is a unit vector of the same length as $z_{(x)}$. Define $N(t)=\\sum_xN_x(t)$ and $Y(t)=\\sum_xY_x(t)$. Now let\n\\begin{equation}\n\\hat{A}(t) = \\int_{-\\infty}^t \\frac{dN(u)}{Y(u)}.\n\\label{eqn:nelaal}\n\\end{equation}\nThis is an analogue of the Nelson-Aalen estimator of the cumulative hazard for a survival time variable \\citep{aal08}. If preferred, we can redefine the level scale $t$ to $t^\\ast=\\exp(t)$ so as to be a more familiar non-negative argument. Like the Nelson-Aalen estimator, $\\hat{A}(t)$ is defined \nfor all $t$ and has a finite number of discontinuities at the observed birth times. It is non-negative and uniformly bounded for fixed $\\mathcal{X}$. It is a cadlag process, meaning continuous to the right with left limits. However, the \nstandard Nelson-Aalen variance estimator\n\\begin{equation}\n {\\hat{\\rm{var}}\\{\\hat{A}(t)}\\} = \\int_{-\\infty}^t \\frac{dN(u)}{Y^2(u)} \n\\label{eqn:var}\n\\end{equation}\ndoes not in general apply, for a variety of reasons. One is possible non-independence between counting processes at different locations given non-independence of the underlying field values. Another, related, reason is informative censoring: knowledge that location $x$ falls out of the risk set at level $u$ provides information about other locations. And a third reason is that we have not constructed a martingale with respect to the observed data filtration.\n\nWe propose such Nelson-Aalen estimators for topological features as tools for comparing properties of fields beyond marginal and correlation structures, for comparing observed field data or model residuals with an assumed parametric model, and as a diagnostic for assumed or fitted correlation structures. We allow either a single field $\\mathcal{Z}$ or independent and identically distributed replicates $\\mathcal{Z}_1, \\mathcal{Z}_2, \\ldots,\\mathcal{Z}_N$ defined on a common space $\\mathcal{X}$.\n\n\n\\subsection{Asymptotics}\n\nIf $\\mathcal{X}$ is fixed and there are $N$ independent replicates $\\hat{A}_1(t), \\ldots, \\hat{A}_N(t)$, then as each $\\hat{A}_i(t)$ is non-negative and bounded above, for each level $t$ the variance of $\\hat{A}_i(t)$ is finite and the classic central limit theorem applies. Hence pointwise inference for large $N$ is straightforward, based on the sample mean $\\bar{A}(t)$ and associated sample standard deviation. \n\nFor simultaneous confidence bounds we need to be a little careful because of the discontinuities in $\\hat{A}_i(t)$. \\cite{hahn77} for instance points out that there can be uniformly bounded cadlag processes for which no central limit theorem applies, depending upon properties of their jumps. \\cite{bloz94} provide sufficient conditions under which $\\surd N\\bar{A}(t)$ converges weakly to a Gaussian process with finite variance and smooth sample paths in $(0,\\tau)$, for any arbitrarily large $\\tau$ for which pr$\\{Y(\\tau)>0\\}>0$. These conditions are verified in Supplementary Material. Hence a functional central limit theorem applies.\n\n\n\n\nFurther, for fixed $\\mathcal{X}$ we can show that $\\bar{A}(t)$ is consistent in $N$ for \n\\begin{equation}\n A(t) = \\displaystyle\\int_{-\\infty}^t J(u)\\frac{ \\sum_x {\\rm pr}(z_{(x)}> u 1_x \\mid z_x=u) f_x(u)}{\\sum_x {\\rm pr}(z_x>u, \\;z_{(x)}> u 1_x)} du,\n\\label{eqn:limit}\n\\end{equation}\nwhere $f_x(.)$ is the marginal density of $z_x$ and $J(u)=1$ if $E\\{Y(u)\\}>0$, zero otherwise. If $N=1$ and there is just a single realisation, the estimator $\\hat{A}(t)$ is also consistent for (\\ref{eqn:limit}), but this time as the cardinality of $\\mathcal{X}$ increases, provided cor$(z_x, z_{x^\\prime})$ approaches zero as the distance between $x$ and $x^\\prime$ increases.\n\n\n\n\\subsection{Inference}\n\nFor inference, we need to distinguish between the single and multiple replication situations. We will begin by assuming multiple replications: $\\hat{A}_1(t), \\ldots, \\hat{A}_N(t)$.\n\nWe can consider the replicates $\\hat{A}_i(t)$ as functional data. It is helpful to discretise the time\/level scale of the estimator (\\ref{eqn:nelaal}) to a grid of $M$ distinct levels $\\tau_1=t_1,\\ldots,t_M=\\tau_2$. This is a common assumption in functional data analysis and deals effectively with the discontinuities in our estimators, which with probability one will not occur at measurement points. \nThere are then a huge array of analysis techniques immediately available: see for example the excellent review by \\cite{wang16} for general approaches, and \n\\cite{chiou09} for a specific example of treating discretised hazards as functional data. With our applications in mind, we will mention only the construction of simultaneous confidence bands. This problem has had attention for functional data that are either sparse \\citep{yao05, ma12} or dense \\citep{degras11, cao12}. Given that our discretisation is by design, a dense functional data approach is appropriate.\n\nLet $\\hat{A}_{ij}$ be the Nelson-Aalen estimator for field $\\mathcal{Z}_i$ at level $t_j$. Let $\\bar{A}_j$ and $\\hat{\\sigma}_j^2$ be the sample mean and sample variance of the estimators at $t_j$, and let $\\hat{\\rho}_{jk}$ be the estimated correlation between $\\hat{A}_{.j}$ and $\\hat{A}_{.k}$. \\cite{degras11} \nestablishes that for fixed $\\tau_1$ and $\\tau_2$, as both $N$ and $M$ increase \n\\begin{equation*}\n\\bar{A}_j\\pm c_\\gamma\\hat{\\sigma}_j\/\\surd N\n\\label{eqn:ci}\n\\end{equation*}\nhas asymptotic coverage $1-\\alpha$, where $c_\\alpha$ is the upper $\\alpha$-quantile \nof the maximum absolute value over $(\\tau_2-\\tau_1)\/\\tau_2$ of a Gaussian process with standard margins and with correlation function equal to an appropriately scaled limit of $\\hat{\\rho}_{jk}$. The threshold $c_\\gamma$ needs to be obtained numerically: software is available in the R package {\\tt SCBmeanfd} \\citep{scb16}.\n\nTurning to the case of a single replication only, we assume a parametric model $F_\\theta$ is available for the random field $\\mathcal{Z}$ and we propose a parametric bootstrap approach. Parametric and more general bootstrap methods for functional data have been studied empirically and theoretically by, for example, \\cite{cuevas06}. We require \na consistent estimator $\\hat{\\theta}$ of the parameter vector $\\theta$. We then generate $B$ simulated realisations of $\\mathcal{Z}$ on $\\mathcal{X}$ from $F_{\\hat{\\theta}}$, and obtain the bootstrap Nelson-Aalen estimators $\\hat{A}_b(t)$ for $b=1,\\ldots,B$. Pointwise intervals that have correct coverage asymptotically in $B$ and $\\mid \\mathcal{X} \\mid$ can be obtained from the bootstrap quantiles or a Normal approximation justified by the central limit theorem. \n\nFor simultaneous confidence bands we suggest a Monte Carlo method used by \\cite{crain11} that avoids explicit estimation of the correlation function. We again discretise the original and bootstrap estimators $\\hat{A}_b(t)$ to a dense grid of $M$ points over a fixed range of levels $(\\tau_1, \\tau_2)$. Let $\\hat{A}_j$ be the Nelson-Aalen estimate (\\ref{eqn:nelaal}) at level $t_j$ obtained from the original data, and let $\\hat{A}_{bj}$ be corresponding value from bootstrap replicate $b$. Let $\\bar{A}_j$ and $\\hat{\\sigma}_j$ be the bootstrap mean and estimated standard deviation at $t_j$. Define \n\\[ G_b= \\max_j\\{ \\mid \\hat{A}_{bj}- \\bar{A}_j \\mid\/\\hat{\\sigma}_j \\},\\]\nand take $\\hat{d}_\\alpha$ to be the upper $\\alpha$-quantile of the empirical distribution of $G_1, \\ldots, G_B$. Then for large $M$ an approximate simultaneous confidence band for $A(t)$ over $(\\tau_1, \\tau_2)$ is\n\\[ \\hat{A}_j\\pm \\hat{d}_\\alpha \\hat{\\sigma}_j \\hspace*{1cm} j=1,\\ldots, M.\\]\n\n\n\n\n\n\\subsection{Simulation studies}\n\n\n\nWe consider random fields on $d\\times d$ lattices and investigate three models. The first,\nM1, is a stationary and isotropic Gaussian random field with standard N(0,1) marginals and Mat\\'{e}rn correlation function. If locations $x$ and $x'$ are separated by Euclidean distance $u$ then \n\\[ \\mbox{cor}\\left(z_x,z_{x'}\\right)= \\frac{2^{1-\\nu}}{\\Gamma(\\nu)}\\left(\\sqrt{2\\nu}u\/\\eta\\right)^\\nu K_\\nu\\left(\\sqrt{2\\nu} u\/\\eta\\right), \\]\nwhere $K_\\nu(.)$ is a modified Bessel function of the third kind. Let $\\theta_1=(\\eta, \\nu)$. \n\nThe second model, M2, is a marginally transformed $\\chi^2_1$ field. We simulate from the Gaussian random field model M1, with parameters $\\theta_2$. Values are then squared to give $\\chi^2_1$ margins. Let $y_x$ be the value at location $x$. We next marginally transform to N(0,1) using $z_x=\\Phi^{-1}\\{F_1(y_x)\\}$, where $\\Phi(.)$ and $F_1(.)$ are the N(0,1) and $\\chi^2_1$ cumulative distribution functions, respectively. \n\n\nThe third model, M3, is a marginally transformed $F_{3,3}$ field. We begin by constructing two independent $\\chi^2_3$ fields, each from the sum of three squared independent\nGaussian random fields M1 with Mat\\'{e}rn parameters $\\theta_3$. The ratio of these $\\chi^2_3$ fields has $F_{3,3}$ distribution and we again marginally transform back to N(0,1).\n\n\n\nAll simulated fields are mean-corrected to help with the comparisons. The purpose of the marginal transformations is to generate fields that are not Gaussian random fields but which have Gaussian N(0,1) marginal distributions. This is so that we can assess the added value of topological methods over and above simple comparisons of marginal distributions. Similarly, in our simulations, for each choice of correlation $\\theta_1$ for \nmodel M1 we choose $\\theta_2$ and $\\theta_3$ so as to match as closely as we can the correlation of the final M2 and M3 fields to Mat\\'{e}rn with parameters $\\theta_1$. Details of this are presented in Supplementary Material. Again, the point is to show that differences between fields are not explained by differences in correlation functions. We have no particular interest in M2 and M3 other than as easily generated non-Gaussian random fields to be used as comparators.\n\n\nThe previously-presented Fig. \\ref{fig:intro1} gives in panel (a) an example simulation from M1 on a $60 \\times 60$ lattice with $\\theta=(5,1).$ Panels (b) and (c) provide two examples of M3, and panel (d) an example of M2, with matched correlations. The topological Nelson-Aalen plots for these fields are in Fig. \\ref{fig:intro2}. The solid line is the limiting value \\eqref{eqn:limit}, which we obtained using the multivariate Gaussian distribution function routine in the R package {\\tt mvtnorm}.\n\n\n\n\n\n\n\nTable \\ref{tab:cov1} shows coverage of nominal 95\\% confidence intervals and bands based on Nelson-Aalen plots for connected components from Gaussian random fields M1 simulated on a 60$\\times$60 lattice. Pointwise coverage is evaluated at selected percentiles of the at-risk distributions, ie levels $t$ satisfying $E\\{Y_x(t)\\}=p$ for five different values of $p$. The simultaneous confidence bands cover the central 90\\% of the at-risk distributions, discretised to a grid of 200 equally spaced points. For each parameter combination we give results for three methods. The first, for reference, is a naive approach that assumes standard methods based on the variance estimator (\\ref{eqn:var}) can be applied and using \\cite{borgan87} for simultaneous confidence bands. The second method assumes we have\na single replication and uses a parametric bootstrap with $B=200$ simulated Gaussian random fields, generated with Mat\\'{e}rn correlation parameters taken as the maximum likelihood estimates from the sole replicate.\nThe third method assumes we have replicates. The standard and parametric bootstrap results are based on 1000 single simulations, the results with replications are from 1000 batches of $N=40$ replicates, which matches an application in the next subsection. \nThere is severe undercoverage at times \nfor the standard confidence intervals, but the other methods both produce good results.\n\nSimulation results for M2 and M3 are provided in Supplementary Material, along with further simulations to assess size and power when using our method to test for a Gaussian random field.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Coverage of nominal 95\\% pointwise confidence intervals and 95\\% simultaneous confidence band (SCB) around Nelson-Aalen component plots for Gaussian random fields. Results from 1000 simulations on $60 \\times 60$ lattices, with Mat\\'{e}rn correlation function with parameters $\\eta$ and $\\nu$ }\n\\label{tab:cov1} \n\\begin{tabular}{clrrrrrr}\n & & \\multicolumn{5}{c}{Percentile}\\\\\n$(\\eta, \\nu)$ & Method & 0.9 & 0.7 & 0.5 & 0.3 & 0.1 & SCB \\\\ \n\\\\\n(5,1) & Standard & 84.0& 92.2& 93.5& 88.9& 82.2 & 73.9\\\\\n&Par. bootstrap& 95.3 &94.4 & 95.8& 95.3& 96.2& 94.1\\\\\n&Replications & 94.2 &94.7 &95.7 &93.9 &94.7 &94.4\\\\\n\n\\\\ \n(10,1) & Standard & 67.8 & 82.7 & 83.3 & 73.7 & 65.5 & 39.1\\\\\n&Par. bootstrap& 95.3 &95.1 & 95.6 & 95.7 & 94.7 & 93.6 \\\\\n&Replications & 95.1 &94.0& 94.9 & 95.9 & 94.0 & 96.4\\\\\n \\\\\n\n\\\\\n(5,2) & Standard & 93.4 & 94.6 & 95.0 & 93.2 & 90.8 & 82.9\\\\ \n&Par. bootstrap & 95.4 &95.2& 95.7& 95.5 &95.8& 95.9\\\\\n&Replications & 94.0 & 94.6 &95.0& 94.3& 93.7 & 93.4\\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\nIn Table \\ref{tab:class} we provide a brief comparison between using the proposed Nelson-Aalen estimator and existing topological data analysis techniques when discriminating between random fields, all on 60$\\times$60 lattices. The statistical properties of topological summaries such as persistence diagrams and landscape functions are not understood, which means we cannot use these methods to assess the adequacy of an assumed model such as a Gaussian random field, or for single-sample problems. However, if we have two or more groups of replicated data we can compare classification performance. \n\nWe took training data to be batches of 40 random fields simulated from models M1, M2 and M3, for each of the three parameter choices. We then used a support vector machine to develop pairwise probabilistic classifiers, using the default parameter choices for the {\\tt svm} routine in the R package {\\tt e1071}. We built classifiers based on the proposed Nelson-Aalen estimator $\\hat{A}(t)$ defined at (\\ref{eqn:nelaal}) and on the first 10 landscape functions \\citep{bubenik2015} for each of components and holes, choosing the best of these to compare with $\\hat{A}(t)$. Landscape functions are popular summaries of persistence diagrams and are available through the R package {\\tt TDA}. We replicated the training data 20 times and assessed classifier performance on separate groups of 200 simulated random fields for each model. \n\nWe assume test data would be allocated to the group with the highest class probability. In Table \\ref{tab:class} we provide the accuracy of the classifier, which is the proportion of test data that are allocated to the correct group, and as a calibration measure we give the mean class probabilities for the correct groups.\n For example, a classifier that estimates the probability of the correct group to be 0.51 gives accurate allocation but is poorly calibrated, whereas one that estimates the correct class probability to be 0.99 would be both accurate and well calibrated. The table shows that classification based on the Nelson-Aalen estimator consistently outperforms classification based on the landscape functions in both accuracy and calibration. Additional simulation results provided in Supplementary Material support this conclusion.\n\n \n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Classification of random fields. Accuracy (Acc.) measures the proportion of random fields that were correctly allocated and calibration (Cal.) gives the mean estimated class probabilities for the correct classes. Values are multiplied by 100 for presentation}\n\\label{tab:class} \n\\begin{tabular}{clrrrrrr}\nMat\\'{e}rn &Method & \\multicolumn{2}{c}{M1 v M2}& \\multicolumn{2}{c}{M1 v M3}& \\multicolumn{2}{c}{M2 v M3}\\\\\n $(\\eta, \\nu)$ & & Acc. & Cal. & Acc. & Cal. & Acc. & Cal. \\\\\n\\\\\n(5,1) & Nelson-Aalen & 100.0 &97.7 & 99.4 & 96.6 & 100.0 & 97.5\\\\\n & Landscape &96.5& 92.5& 91.7& 85.5& 95.4& 91.6\\\\\n\\\\\n(10,1) & Nelson-Aalen &99.7 &97.2 &95.2 &91.0 &100.0 &97.3\\\\\n & Landscape &99.7& 97.0& 80.7& 72.0& 95.8& 90.1\\\\\n\\\\\n(5,2) & Nelson-Aalen &100.0 &97.6 &99.7 &97.1 &100.0& 97.4\\\\\n & Landscape & 92.3& 87.7& 79.4& 71.0& 87.5 &80.1\\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Applications: climate modelling, interstellar medium}\n\nThe Community Earth System Model Large Ensemble project \\citep{kay2015community} is a publicly available ensemble of 40 high resolution climate model simulations. Although the models used to generate the data are deterministic, spatial and spatio-temporal statistical modelling is often applied as a means to both explore and concisely summarise the output \\citep{cast16assess, edwards19}. Such models often assume that the modelled residuals form a Gaussian random field. We can use the topological Nelson-Aalen approach to assess this assumption, at least in part, and as a quick and easy method to compare ensemble members and to examine the suitability of assumed correlation structures.\n\nTo illustrate our method we considered annual wind intensities at $10$m for the year 2020. Each ensemble member has data on a 192$\\times$288 grid over the surface of a sphere and at each data point we standardised by the mean and standard deviation across members. To keep the correlation matrices manageable for this illustration we used\nevery third lattice point in both the latitudinal and longitudinal directions. Following \n\\cite{castruccio2014beyond} and \\cite{castruccio2016compressing} we also trimmed the regions near the poles, leaving data on a $51 \\times 96$ lattice for each ensemble member. \n\nTopological Nelson-Aalen plots for connected components are shown as functional boxplots \\citep{sun11} in the left panel of Fig. \\ref{fig:naexamples} together with a simultaneous confidence band for the mean. There is reassurance that the standard assumption of a Gaussian random field for these climate residuals is reasonable: the band almost uniformly includes the expected value for a Gaussian random field\n with correlation function taken to be the empirical correlation found in the data. To illustrate the potential of our method as a diagnostic for fitted correlation models, the figure also includes expected values under three modelled correlation structures. These are the three dotted lines in the plot. In order from the top, the first line corresponds to a stationary \nMat\\'{e}rn model. It is clearly poor for these data, and the same is true for the second dotted line, which corresponds to a regional block model with separate Mat\\'ern correlation functions fitted to land, ocean and coastal regions. In contrast, as seen by the lowest dotted line, a semiparametric non-stationary model proposed by \n\\cite{konzen2019} fits these data well.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics{naexamplesfig.pdf}}\n\\caption{Nelson-Aalen plots for connected components. In (a) the grey region and lines represent functional boxplots of 40 separate realisations of climate data residuals. The long dashed lines represent the mean and associated 95\\% simultaneous confidence band, and the other lines are limiting values for Gaussian random fields with different correlation matrices. The solid line uses the empirical correlation matrix, the others are explained in the text. Panel (b) considers the neutral hydrogen data: the short dashed lines are Region 1, the dotted lines Region 2 and the long dashed lines Region 3. The shaded regions indicate $\\pm$ two standard deviations. The solid lines \nshow the limiting values for Gaussian random fields with matched correlations. The upper lines, which are almost indistinguishable, are for Regions 1 and 2, the lower is for Region 3. }\n\\label{fig:naexamples}\n\\end{figure}\n\n\n\n\nOur second application is into the distribution in the Milky Way of neutral atomic hydrogen, \\textup{H}\\,{\\sc i}, using data obtained by the Galactic All-Sky Survey using the Parkes 64 m radio telescope\n\\citep{mclure09, kalberla10}. Neutral atomic hydrogen makes up around 90\\% of the atoms in the interstellar medium, which embeds stars and galaxies. Turbulence and shocks lead to a highly heterogeneous random pattern to the local density of \\textup{H}\\,{\\sc i}, and a common assumption in astrophysics is that after the removal of trends and large-scale features, the distribution of the log-density of \\textup{H}\\,{\\sc i} is that of a Gaussian random field \\citep{elmeg04, monin07}. This assumption was investigated by \\cite{hend2020}, who considered topological features of three distinct regions of the southern sky. Based on counts of components and holes, but not event history, the authors concluded that two regions were consistent, after monotonic transformation, with Gaussian random fields but the third was not.\n\nThe right panel of Fig. \\ref{fig:naexamples} shows Nelson-Aalen plots for connected components for these data. Each region consists of density measurements over a 256$\\times$256 lattice and following \\cite{hend2020} we considered the residuals after trend removal using a thin plate smoothing spline. The data were again marginally transformed to N(0,1) to preempt questions as to whether simple monotonic transformations could suffice to convert the data to Gaussian random fields. Region 3 is clearly distinct from the others and not consistent with the fitted Gaussian random field. Regions 1 and 2 have very similar Nelson-Aalen plots and almost indistinguishable limiting values for Gaussian random fields with Mat\\'ern correlations fitted to the data. There is evidence however that at the higher levels these regions are not consistent with the fitted Gaussian random fields.\n\n\n\n\n\n\\section{COX PROPORTIONAL HAZARDS FOR EMBEDDED METRIC TREES}\n\\subsection{Topology and event history for tree data}\n\nWe now turn to trees $\\mathcal{T}=\\{\\mathcal{N},\\mathcal{E}\\}$ made up of nodes $\\mathcal{N}$ and associated edges $\\mathcal{E}$ embedded in $\\mathbb{R}^2$ or $\\mathbb{R}^3$, and with a single common ancestor. A simple example was shown in panel (a) of Fig. \\ref{fig:trees}. \n\n\nA number of approaches for the comparison of tree structures have been inspired by persistent homology. \\cite{kanari2018} considered persistence of one-dimensional features on contours equidistant from the root, whether path length or radial distance, as previously illustrated in Panel (a) of Fig. \\ref{fig:trees}. \\cite{LI2017} independently proposed a similar approach in the comparison of neuronal tree structures. \\cite{BENDICH2016} \napplied persistent homology methods to brain artery trees in $\\mathbb{R}^3$, \\cite{MAOLI2017} used a geodesic distance function to derive persistence diagrams for the branching architectures of plants, and \n \\cite{BRODZKI2018} used persistent homology to classify bronchial trees.\n\n\nNone of the methods of which we are aware can deal with the possibility of censored or incomplete trees. For example, an image may capture only part of a tree, in which case edges that cross the image boundary would not be fully observed. Edges that reach the boundary are not true first order nodes or leaves, and should\nnot be considered as birth or death locations when applying any of the persistent homological techniques. Equally they do contain information and these edges should not be ignored or discarded.\n\nThis type of problem is of course fundamental to event history and survival analysis methodology, where methods for censored or truncated data are very well established. Hence our proposal is straightforward: construct a filtration as above but consider as an event the occurrence of either a leaf or a branching point. More specifically we propose the following.\n\n\\begin{enumerate}\n\\item Filter outwards by radial distance $r$.\n\\item Consider a leaf event to occur at $r$ if a known leaf has that radial distance from the root. Consider a branch event similarly. If desired, we might distinguish bifurcations from trifurcations and so on. Censored points cannot be events.\n\\item Construct a risk set consisting of all edges in $\\mathcal{E}$ that intersect a circle of radius $r$ centered at the root, as in panel (a) of Fig. \\ref{fig:trees}.\n\\item Use standard event history methods to contrast events with risk sets.\n\\end{enumerate}\n\nLet $\\mathcal{C}_r$ be a disc of radius $r$ centred on the root and define $\\mathcal{T}_r=\\mathcal{T} \\cap \\mathcal{C}_r$. Covariates can be included provided they are exogenous or dynamically defined on \n$\\mathcal{C}_r$. Thus spatial location, branch order, edge weights, proximity to other branches in $\\mathcal{C}_r$ and so on can be handled easily using a Cox or other event time regression model. Further, we can construct risk sets from multiple trees and use fixed or random effects (frailties) to include tree-to-tree comparisons.\n\nFor this method to work we need three common event history assumptions. First, that events occur independently given the risk sets and covariates. Second, that censoring is independent. Third, that whichever hazard or intensity model is assumed is appropriate. \nUnder these assumptions standard results apply, including asymptotics for Cox or other model estimates.\n\n\n\n\n\n\\subsection{Application: vascular patterns in the eye}\n\nThe left panel of Fig. \\ref{fig:eyefig} shows the vascular pattern of a human eye, taken \nfrom a high dimensional image provided by \\cite{budai13}. The right panel of the figure shows a tree representation, with vertices joined by linear edges, a simplification of the optic nerve area to provide a root, and with the omission of vessels that appear to originate outwith the plot region. Vessels that reach the boundary of the plot region are marked as censored points. This is a two-dimensional projection of a three-dimensional tree and branches can cross. There are no cycles.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics{eyefig.pdf}}\n\\caption{Left panel: vascular patterns in a healthy human eye, with data taken from \n\\cite{budai13}. Right panel: tree representation. Censored edges are marked by points. }\n\\label{fig:eyefig}\n\\end{figure}\n\n\n\nThe data in Fig. \\ref{fig:eyefig} are from one of the 15 fundus images of eyes of healthy patients provided by \\cite{budai13}, who also give fundus images of 15 patients with diabetic retinopathy. We use our methods to compare diabetic retinopathy and healthy patients, and also to investigate the effect of a number of covariates. These are the width of the vessel, the Euclidean distance between nodes, the ratio of path length to Euclidean distance between nodes as a measure of curvature, the number of other nodes within radius 200 units of the current node (about 10\\% of the image size), the generation order, and the azimuth to the root measured from the vertical. We assumed log-linear effects for all covariates except azimuth, for which we assumed a smooth effect via a $P$-spline with four degrees of freedom. We also included patient-level terms as either fixed effects or frailties. In total over the 30 patients there were 12590 edges. At this large sample size the fixed effect and frailty model fits were almost identical.\n\n\nTable \\ref{tab:cox} summarises some of the results. In the upper part we consider an event to be the termination of a vessel at a leaf, and in the lower part we consider branching points. Given the large sample, statistical significance is almost irrelevant and so to compare the magnitudes of effects we take the hazard ratio (hr) between \nthe 20\\% and 80\\% points of the covariate distributions. \n\nAs might be expected, large vessels are highly unlikely to terminate in a leaf but are more likely to branch. The hazard functions for both types of event fall as Euclidean distance from an earlier node increases. There is no real effect of either curvature of an edge or proximity to other nodes, but there is some evidence that branching becomes less likely as the generation of an edge increases. There is a modest azimuth effect on branching, with a plot of the smooth fit (not shown here) indicating a decrease in branching hazard as aziumth moves away from the vertical, fairly symmetrically in both directions.\n\nOf most interest is the difference between patients. The estimated frailty terms for both leaf and branch events are shown in Fig. \\ref{fig:frailfig}. There are relatively small differences between patients in vessel branching frequency, but large differences in the hazard for terminating in a leaf, given covariate effects. Interestingly, we have complete separation between healthy and diabetic retinopathy patients in these frailties. \n\n\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Cox proportional hazards results for eye trees}\n\\label{tab:cox}\n\\begin{tabular}{lrrrrr} \n & \\multicolumn{5}{c}{Event = leaf}\\\\\n & Coefficient & SE & $\\chi^2$ & df & hr\\\\ \nWidth & -0.3849 & 0.0079 & 2369.51 & 1 & 0.02\\\\ \nEuclid & -0.0051 & 0.0002 & 797.16 & 1 & 0.41\\\\ \nPath length ratio & 1.0142 & 0.1260 & 64.76 & 1 & 1.05\\\\ \nNodes within 200 & 0.0037 & 0.0037 & 1.02 & 1 & 1.02\\\\ \nOrder & -0.0147 & 0.0049 & 8.90 & 1 & 0.86\\\\ \nAzimuth & & & 34.66 & 4 & 1.18\\\\ \nPatient effect & & & 823.98 & 29 & 2.29\\\\ \n\\\\\n & \\multicolumn{5}{c}{Event = branch}\\\\\n & Coefficient & SE & $\\chi^2$ & df & hr\\\\ \nWidth & 0.0591 & 0.0016 & 1420.28 & 1 & 1.88\\\\ \nEuclid & -0.0058 & 0.0001 & 1874.37 & 1 & 0.37\\\\ \nPath length ratio & -0.9058 & 0.2406 & 14.18 & 1 & 0.95\\\\ \nNodes within 200 & 0.0375 & 0.0033 & 128.54 & 1 & 1.21\\\\ \nOrder & -0.0622 & 0.0051 & 149.35 & 1 & 0.54\\\\ \nAzimuth & & & 119.30 & 4 & 1.41\\\\ \nPatient effect & & & 103.60 & 29 & 1.37\\\\ \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[height=3in,width=3in]{treefrail.pdf}}\n\\caption{Subject-level frailties for 15 diabetic retinopathy patients (triangles) and 15 healthy patients (circles).}\n\\label{fig:frailfig}\n\\end{figure}\n\n\n\n\\section{DISCUSSION}\n\nThere are a number of areas for further work. In Section 3 we suggested empirical variances for the Nelson-Aalen estimator when there are repeated samples of random fields, as in the climate example, and parametric bootstrap estimators when there was interest in a particular model, such as a Gaussian random field. In Supplementary Material, an adjusted parametric bootstrap assuming local misspecification as in \\cite{Copas2005} is suggested and\nprovides a conservative approach for the case of a single sample and no model. However, it would be useful formally to develop a non-parametric variance estimator. \n\nIn both Section 3 and Section 4 we concentrated on the birth times of connected components. We might also consider death times, though the identification of risk sets is less straightforward in this case, which in turn means that limiting properties will be harder to derive. For complexes in two or higher dimensions, including the random fields of Section 3, we could consider higher order features. In two-dimensions it will be particularly simple to use our methods to investigate death times of holes, which as a result of Alexander duality \\citep{edel12} can be analysed in the same manner as births of components simply by reversing the direction of the filtration. \n\n\\section{ACKNOWLEDGEMENT}\n\nWe thank Evandro Konzen and Jian Shi for providing software to fit their non-stationary correlation model to the climate data.\n\n\n\n\n\\clearpage\n\n\\bibliographystyle{biometrika}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\nThe intracluster medium (ICM), the intragroup medium (IGrM) and the circumgalactic medium (CGM) refer to the gaseous atmosphere pervading the halos of galaxy clusters, groups and individual galaxies, respectively. These media comprise of gas in multiple temperature phases, ranging from $10$--$10^7~\\mathrm{K}$ \\citep[for example, see][]{Bonamente2001ApJ,Fabian2003MNRAS,Salome2006A&A} and play a critical role in the evolution of their member galaxies through the cosmic baryon cycle. Feedback from supernovae and active galactic nuclei (AGNs) jets, interaction between member galaxies and with external galaxies, infall of galaxies into the group\/cluster, all drive turbulence in the halo gas. This driven turbulence plays a number of key roles in the thermodynamics of these media, to name a few--(i) it seeds density fluctuations \\citep{Mohapatra2021MNRAS} in the gas, which can become thermally unstable \\citep{sharma2010thermal}, since denser gas cools faster (cooling rate $\\propto\\rho^2$, where $\\rho$ is density); (ii) it converts kinetic energy from feedback processes and galaxy--galaxy interactions into thermal energy through viscous dissipation and heats the gas; and (iii) it mixes hot ($10^6$--$10^8$~$\\mathrm{K}$) and cold ($10$--$10^4$~$\\mathrm{K}$) components of the gas together to form intermediate-temperature gas with a short cooling time, which generates more cold gas \\citep{Armillotta2016,Gronke2018,Kanjilal2021MNRAS}. Thus, understanding the properties of turbulence, such as the driving scale, the nature of the driving, in particular the fraction of solenoidal and compressive components, and the amplitude of turbulent velocities induced by the driving, is important to understand the physics of these systems. \n\nMany recent theoretical and numerical studies of turbulence in the CGM\/ICM \\cite[for example,][]{Mohapatra2019,Shi2019,Grete2020ApJ,Mohapatra2022VSF} are limited to solenoidal (divergence-free) turbulence, and mostly ignored the effects of compressive modes. The assumption of solenoidal driving is influenced by studies of incompressible\/subsonic turbulence in the earth's atmosphere and oceans, because air\/water are nearly incompressible, and the Boussinesq assumption ($\\div{\\bm{v}}=0$, where $\\bm{v}$ is velocity) holds. Although the gas motions in the ICM have been observed to be subsonic \\citep{hitomi2016}, the ICM\/CGM is quite compressible, i.e., Mach numbers are of the order of $0.1$--$0.5$. Moreover, we observe strong converging and diverging motions in these media - such as expanding X-ray cavities around AGN-inflated bubbles \\citep{Wise2007ApJ,Biava2021A&A,Tiwari2022MNRAS}, compressive waves during galaxy infall and passages \\citep{Churazov2003,Churazov2021A&A}, gas sloshing and cold fronts \\citep{Botteon2021A&A,Gendron-Marsolais2021ApJ,Ichinohe2021MNRAS, Ubertosi2021MNRAS,Ueda2021ApJ,Brienza2022A&A}, which indicate density contrasts and compressive motions in the ICM. Similarly, galactic outflows consisting of radially expanding motions driven by diverging flows such as stellar winds, supernova bubbles, super-bubbles and AGNs can also drive compressive motions in the disk-halo interface and the CGM \\citep{Mulcahy2017A&A,Das2019ApJ,Tejos2021MNRAS}. The evolution of these compressive motions and their role in the thermodynamics of the halo gas could be very different from the divergence-free forcing studied in current literature.\n\nMany recent studies of turbulence in the interstellar medium (ISM) have investigated the effects of compressive driving and its effects on star formation \\citep{Federrath2010A&A,konstandin2012}. They find that compressive forcing leads to stronger density fluctuations (up to $3$ times larger in their standard deviation) which significantly increases the star formation efficiency, since a larger fraction of the gas becomes gravitationally unstable \\citep{Federrath2012}. Compressively driven turbulence also shows a steeper velocity power spectrum \\citep{Federrath2013}, which has a slope closer to Burgers turbulence with $P_v(k)\\propto k^{-2}$ \\citep{Burgers1948171}, rather than Kolmogorov turbulence with $P_v(k)\\propto k^{-5\/3}$ \\citep{kolmogorov1941dissipation}, where $k$ is the wavenumber. Although these studies model supersonic turbulence in star-forming molecular clouds (which are assumed to be isothermal) and are not directly applicable to the subsonic hot plasma in the CGM\/IGrM\/ICM, these effects of compressive forcing could affect both the physics of these media and our interpretations of their observations. Similar to increasing the fraction of gas that becomes Jeans-unstable in molecular clouds, compressive forcing could lead to larger fractions of the gas becoming thermally unstable in the case of halo gas, since the cooling time $\\propto\\rho^{-1}$. These larger density fluctuations in the hot-phase gas could also affect the relation between X-ray surface brightness fluctuations and turbulent velocities, which is used to indirectly measure turbulent gas velocities in several nearby galaxy clusters \\citep{zhuravleva2014turbulent,zhuravleva2014relation,zhuravleva2018}. The steep slopes of the velocity structure functions of H$\\alpha$ filaments in nearby galaxy clusters studied in \\cite{Li2020ApJ} could also be affected by the nature of turbulence forcing. Hence it is important to study the effects of different types of turbulence forcing in the context of the CGM, IGrM and ICM.\n\nRecent numerical and theoretical studies such as \\cite{Shi2018MNRAS,Zhang2019MNRASa,Zhang2019MNRASb,Zhang2020MNRAS,Shi2020MNRAS} have investigated the evolution of shocks and turbulence driven by galaxy infall and cluster mergers, and their effects on the ICM. However, they have mainly focused on more global, large-scale effects of mergers, such as changes to the ICM entropy profile, contribution of turbulent pressure to non-thermal pressure in the ICM, properties of bow shocks, accretion shocks, merger shocks, etc. The current literature lacks an in-depth investigation of the effects of the turbulence driving on the multi-phase gas physics of galaxy halos.\nStudies such as \\citet{Ryu2008Sci} and \\citet{Iapichino2012MNRAS}, using simulations and analytical modelling, respectively, have explored the role of large-scale shocks and discontinuities in driving turbulence in the ICM and the magnification of magnetic fields through the small-scale dynamo mechanism \\citep[also see][]{Federrath2016JPlPh}. \\cite{Porter2015ApJ} studied the effect of the turbulence driving in an isothermal ICM setup and studied the evolution of shocks, vorticity and magnetic fields. More recently, \\citet{Vazza2017MNRAS}, \\citet{Wittor2017MNRAS}, \\citet{Wittor2020MNRAS}, and \\citet{VallesPerez2021MNRAS} have studied the evolution of enstrophy, the squared amplitude of the vorticity, and its sources and sinks using cluster-scale numerical simulations.\n\nIn this study, we conduct a set of high-resolution ($1008^3$ resolution elements) local idealised simulations in a box of size $10-40~\\mathrm{kpc}$. We scan the parameter space corresponding to the turbulence driving parameter $\\zeta$, which quantifies the fraction of solenoidal driving. We include radiative cooling in our model. We study the effects of the driving parameter on the morphology of the gas, the density and temperature distributions and the amplitude of density fluctuations. We also determine the different sources of vorticity and the fraction of solenoidal and compressive motions at different temperatures. In order to compare with observations, we construct mock observations of the X-ray emission from hot-phase gas ($T\\gtrsim10^7~\\mathrm{K}$) and H$\\alpha$ emission from filaments ($T\\sim10^4~\\mathrm{K}$). We also compare the kinematics of the two temperature phases and their dependence on the driving parameter.\n\nThis paper is organised as follows. In \\cref{sec:Methods}, we introduce our model and setup. In \\cref{sec:results-discussion} we present the results from our analysis and discuss their implications for the CGM\/ICM. We present the shortcomings of our model and their possible effects on our results, as well as the future prospects of our work in \\cref{sec:caveats-future}, and summarise and conclude in \\cref{sec:Conclusion}.\n\n\\section{Methods}\\label{sec:Methods}\n\\subsection{Model equations}\\label{subsec:ModEq}\nWe model the CGM\/ICM as a fluid using the compressible hydrodynamic (HD) equations and the ideal gas equation of state. We evolve the following equations:\n\\begin{subequations}\n\t\\begin{align}\n\t\\label{eq:continuity}\n\t&\\frac{\\partial\\rho}{\\partial t}+\\nabla\\cdot (\\rho \\bm{v})=0,\\\\\n\t\\label{eq:momentum}\n\t&\\frac{\\partial(\\rho\\bm{v})}{\\partial t}+\\nabla\\cdot (\\rho\\bm{v}\\otimes\\bm{v})+ \\nabla P=\\rho\\bm{a},\\\\\n\t\\label{eq:energy}\n\t&\\frac{\\partial E}{\\partial t}+\\nabla\\cdot ((E+P)\\bm{v})=\\rho\\bm{a}\\cdot\\bm{v}+Q-\\mathcal{L},\\\\\n\t\\label{eq:tot_energy}\n\t&E=\\frac{\\rho\\bm{v}\\cdot\\bm{v}}{2} + \\frac{P}{\\gamma-1},\n\t\\end{align}\n\\end{subequations}\nwhere $\\rho$ is the gas mass density, $\\bm{v}$ is the velocity, $P=\\rho k_B T\/(\\mu m_p)$ is the thermal pressure, $\\bm{a}$ is the turbulent acceleration field that we apply, $E$ is the total energy density, $\\mu$ is the mean molecular mass, $m_p$ is the proton mass, $k_B$ is the Boltzmann constant, $T$ is the temperature, $Q(\\bm{x},t)$ and $\\mathcal{L}(\\rho,T)$ are the thermal heating and cooling rate densities respectively, and $\\gamma=5\/3$ is the adiabatic index. The cooling rate density $\\mathcal{L}$ is given by\n\\begin{equation}\\label{eq:cooling_function}\n\\mathcal{L} = n_en_i\\Lambda(T),\n\\end{equation}\nwhere $n_e$ and $n_i$ \nare electron and ion number densities, respectively and $\\Lambda(T)$ is the temperature-dependent cooling function of \\cite{Sutherland1993} corresponding to $Z_{\\odot}\/3$ solar metallicity. \n\n\\subsection{Numerical methods}\\label{subsec:numerical_methods}\nWe use a modified version of the FLASH code \\citep{Fryxell2000,Dubey2008}, version~4, for our simulations. We evolve our model equations using the HLL5R Riemann solver \\citep{Bouchut2007,Bouchut2010,Waagan2011}. For time integration, we use the MUSCL-Hancock scheme \\citep{van1984SIAM,Waagan2009JCoPh} and a second-order reconstruction method that uses primitive variables and ensures positive values of density and internal energy. We use a uniformly spaced 3D Cartesian grid with equal size and periodic boundary conditions along all three directions. For our ICM simulations, we use a box of size $40~\\mathrm{kpc}$. For our CGM simulations, we use a box of size $10~\\mathrm{kpc}$. \nWe use $1008^3$ resolution elements for our parameter scan, but also include lower resolution runs with $504^3$ resolution elements for checking convergence of our results. \n\nWe do not explicitly include viscosity in our model, instead, the viscosity is due to numerical dissipation \\citep[see][for a discussion]{Benzi2008PhRvL,Federrath2011PhRvL}. The numerical viscous scale in our simulations is expected to be around $0.6~\\mathrm{kpc}$, roughly at $15\\Delta x$ (where $\\Delta x$ is the numerical cell size), which we obtain from fitting the power spectrum model in \\cite{Kriel2022MNRAS} to our velocity power spectrum. For the hot ICM, we compare the numerical viscosity with the Spitzer viscosity values from \\cite{Spitzer1962pfigbook} and find it to be suppressed by a factor of $\\sim600$. This is in line with expectations from recent ICM observations such as \\cite{Zhuravleva2019NatAs}, who constrain the ICM viscosity to be suppressed to around $10$--$1000$ times smaller than the Spitzer viscosity.\n\n\\subsubsection{Turbulent forcing}\\label{subsubsec:Turb_forcing}\nWe drive turbulence by exciting only large-scale modes in spectral space, and letting turbulence develop self-consistently on smaller scales. We control the amplitude of the turbulent acceleration field $\\bm{a}$ such that the power is a parabolic function of $k$ and peaks at $k=2$ (for simplicity, we have dropped the wavenumber unit $2\\pi\/L$). We only drive modes with $1\\leq\\abs{\\bm{k}}L\/2\\pi\\leq3$. We use the stochastic Ornstein-Uhlenbeck (OU) process to model $\\bm{a}$ with a finite autocorrelation time-scale $t_{\\mathrm{turb}}$, which we have fixed to be $260~\\mathrm{Myr}$ (approximately $3$ times the eddy turnover time on the driving scale) across all our ICM simulations and $100~\\mathrm{Myr}$ for CGM simulations \\citep{eswaran1988examination,schmidt2006numerical,Federrath2010A&A}. In steady state, the gas reaches an rms velocity of $\\sim250~\\mathrm{km\/s}$ for our ICM-like runs \\citep[consistent with observations by the][]{hitomi2016} and $\\sim80~\\mathrm{km\/s}$ for our CGM-like runs \\citep[consistent with][]{Werk2016ApJ,Faerman2017ApJ}.\n\nTo control the contribution of solenoidal and compressive driving, we introduce a parameter $\\zeta$ which denotes the fraction of contribution from solenoidal driving.\nThe Fourier transform of the acceleration $\\bm{a}$ is given by:\n\\begin{subequations}\n\\begin{align}\n \\label{eq:zeta_forcing}\n &\\bm{a}(\\bm{k})=(1-\\zeta) \\Tilde{\\bm{a}}_{\\mathrm{comp}}(\\bm{k})+\\zeta \\Tilde{\\bm{a}}_{\\mathrm{sol}}(\\bm{k})\\text{, where}\\\\\n &\\Tilde{\\bm{a}}_{\\mathrm{comp}}(\\bm{k}) = \\left(\\bm{k}\\cdot\\Tilde{\\bm{a}}(\\bm{k})\\right)\\bm{k}\/k^2 \\label{eq:Forc_comp}\\text{, }\\\\\n &\\Tilde{\\bm{a}}_{\\mathrm{sol}}(\\bm{k}) = \\Tilde{\\bm{a}}(\\bm{k})-\\Tilde{\\bm{a}}_{\\mathrm{comp}}(\\bm{k}) \\label{eq:Forc_sol}\\text{, }\n\\end{align}\n\\end{subequations}\nwhere $\\zeta$ is the turbulence driving parameter, $\\Tilde{\\bm{a}}_{\\mathrm{comp}}(\\bm{k})$ and $\\Tilde{\\bm{a}}_{\\mathrm{sol}}(\\bm{k})$ are the components of $\\Tilde{\\bm{a}}(\\bm{k})$ parallel and perpendicular to the wave vector $\\bm{k}$, respectively. The quantity $\\Tilde{\\bm{a}}(\\bm{k})$ is the non-decomposed forcing term which is evolved in Fourier space using the stochastic OU process. For further details of the forcing method, we refer the reader to section~2.1 of \\cite{Federrath2010A&A}. For our analysis, we have chosen three values of the driving parameter---$\\zeta=0.0$ which corresponds to purely compressive (or curl-free) forcing, $\\zeta=0.5$ which corresponds to natural mixture or equal contributions from the solenoidal and compressive components \\footnote{Note that the ratio of compressive power to total power for $\\zeta=0.5$ is $1\/3$ (and not $1\/2$), since in 3D we have one longitudinal (compressive) mode and two transverse (solenoidal) modes.}, and $\\zeta=1.0$, which corresponds to purely solenoidal or divergence-free forcing. \n\n\\subsubsection{Thermal heating rate and global energy balance}\\label{subsubsec:therm_heating}\nIn all our simulations, we maintain global energy balance. This prevents the gas from undergoing runaway cooling, which is also motivated by the rarity of cooling flows in cluster observations. In order to achieve this, we inject thermal energy into the domain at a rate $Q$, such that the net thermal+kinetic energy of the system as a whole does not decrease. We also ensure the positivity of $Q$ by setting it to zero whenever the instantaneous turbulence energy injection rate is larger than the radiative cooling rate (which is rare). Mathematically, we impose the following condition at every time step:\n\\begin{subequations}\n\\begin{equation}\n Q=\\max\\left(0,\\frac{\\rho(\\bm{x},t)\\left(\\int\\mathcal{L}\\mathrm{d}V-\\int \\rho\\bm{a}\\cdot\\bm{v}\\mathrm{d}V\\right)}{\\int\\rho\\mathrm{d}V}\\right).\\label{eq:thermal_heating_rate_Q}\n\\end{equation}\nThis heating rate density is proportional to the gas density of each cell, which is motivated by the several gas-density dependent heating processes, such as heating by photons and cosmic rays. We introduce the fraction $f_{\\mathrm{turb}}$ which denotes the ratio of the net turbulence energy injection rate to the net cooling rate\\footnote{The ratio $f_{\\mathrm{turb}}$ varies as a function of time, unlike our previous studies (\\citealt{Mohapatra2019,Mohapatra2022characterising}), where we fixed $f_{\\mathrm{turb}}$ to a constant value and scaled $\\bm{a}$ according to \\cref{eq:f_turb}. Here we maintain a fixed rms velocity and not a fixed $f_{\\mathrm{turb}}$.}, given as: \n\\begin{equation}\nf_{\\mathrm{turb}}=\\frac{\\int\\rho\\bm{a}\\cdot\\bm{v}\\mathrm{d}V}{\\int\\mathcal{L}\\mathrm{d}V}.\\label{eq:f_turb}\n\\end{equation}\n\\end{subequations}\n\n\n\\subsubsection{Modifications to the cooling module}\\label{subsubsec:temp_ceiling_and_cool_cutoff}\n\\paragraph{Temperature ceiling.}\\label{par:temp_ceiling} \nSince we use a periodic box for our setup and do not let the energy escape, the thermal heat that we inject to impose global energy balance and the dissipation of turbulence kinetic energy can heat the hot phase ($T\\gtrsim10^7~\\mathrm{K}$ for the ICM and $T\\gtrsim10^6~\\mathrm{K}$ for the CGM) to unrealistic high temperatures ($T\\gtrsim10^8~\\mathrm{K}$). As we do not consider gravity, this hot, low-density gas does not buoyantly rise up, expand and cool as it would in a realistic stratified atmosphere. Very hot grid cells lead to an increase in the maximum speed of sound ($c_s$) and a decrease in the code time step ($dt_{\\mathrm{cs}}=\\Delta x\n\/c_s$, where $c_s=\\sqrt{\\gamma k_BT\/\\mu m_p}$, and $\\Delta x$ is the size of a cell). Thus, in order to prevent the gas from reaching unrealistic hot temperatures and to control the code time step, we introduce a temperature ceiling to our cooling module, which limits any gas with $T>T_{\\mathrm{ceiling}}$ to $T_{\\mathrm{ceiling}}$. For all our simulations, we set $T_{\\mathrm{ceiling}}=3\\times10^8~\\mathrm{K}$ for our ICM-like runs and $10^8~\\mathrm{K}$ for our CGM-like runs. We have ensured that adding a temperature ceiling does not significantly affect the temperature distribution of the gas, but only cuts off its high temperature tail at $T_{\\mathrm{ceiling}}$.\n\n\\paragraph{Pressure and temperature cutoffs on cooling.}\\label{par:cool_cutoffs}\nWe set the cooling function to zero below a temperature $T_{\\mathrm{cutoff}}$ and a pressure $P_{\\mathrm{cutoff}}$. The cooling function from \\cite{Sutherland1993} that we use is truncated at $10^4\\,\\mathrm{K}$, which we set to be our $T_{\\mathrm{cutoff}}$. In order to limit numerical instability issues associated with low-temperature gas dropping to extremely low pressures, we set $P_{\\mathrm{cutoff}}=P_0\/1000$, where $P_0$ is the initial pressure of the gas. This cutoff only limits the gas pressure of a small portion of the cold gas from decreasing below $P_{\\mathrm{cutoff}}$. We have verified that it does not significantly affect any other properties of the gas, such as density, temperature and velocity distributions.\n\n\n\\paragraph{Cooling subcycling.}\\label{par:cooling_subcycling}\nThe cooling time of the gas is given by:\n\\begin{equation}\\label{eq:tcool}\n t_{\\mathrm{cool}}=\\frac{E_{\\mathrm{int}}}{\\mathcal{L}}=\\frac{P}{(\\gamma-1)n_e n_i\\Lambda(T)}.\n\\end{equation}\nThe cooling time can be quite short for gas at intermediate temperatures ($10^5~\\mathrm{K}\\lesssim T\\lesssim10^6~\\mathrm{K}$) where $\\Lambda(T)$ peaks. This makes our multiphase turbulence runs numerically expensive. In order to achieve a reasonable run-time, we evolve the internal energy with operator splitting in the short $t_{\\mathrm{cool}}$ grid cells using smaller time steps. We reduce the global timestep if it exceeds the shortest cooling time by a fixed factor $\\mathrm{sub}_{\\mathrm{factor}}$. We have explained this module in detail in section~2.2.5 of \\cite{Mohapatra2022characterising} (hereafter \\citetalias{Mohapatra2022characterising}). In this setup, we set the hydrodynamical time step to\n\\begin{equation}\\label{eq:cool_subcycling}\n \\mathrm{dt}_{\\mathrm{code}}=\\mathrm{min}(0.5\\times \\mathrm{sub}_{\\mathrm{factor}} \\times t_{\\mathrm{cool,min}},\\mathrm{dt}_{\\mathrm{CFL}}), \n\\end{equation}\nwhere $t_{\\mathrm{cool,min}}$ is the minimum value of $t_{\\mathrm{cool}}$ in the entire domain and $\\mathrm{dt}_{\\mathrm{CFL}}$ is the time step set by the CFL (Courant-Friedrichs-Lewy) criterion. We use $\\mathrm{sub}_{\\mathrm{factor}}=25$ for this study. We have tested the effect of different values of $\\mathrm{sub}_{\\mathrm{factor}}$ on the temperature distribution of the gas and discuss this further in Appendix~C.\n\n\\paragraph{Density ceiling on cooling.}\\label{par:dens_ceiling_cooling}\nThe cooling time $t_{\\mathrm{cool}}\\propto\\rho^{-1}$. For compressive driving runs, the cold phase can reach high densities due to the combined effects of cooling and convergent driving. In order to control the code time step, we set the cooling function to zero for gas with density $\\rho>\\rho_{\\mathrm{ceiling}}$. In this study, we use $\\rho_{\\mathrm{ceiling}}=500\\times\\rho_0$, where $\\rho_0$ is the initial density. We have checked the effect of introducing a density ceiling on the density distribution of the gas and find that it only affects a small part of the high-density tail (c.f. fig. \\ref{fig:dens-PDF}).\n\nThus, the complete cooling function is given by\n\\begin{equation}\n \\mathcal{L}=n_en_i\\Lambda(T)\\mathcal{H}(T-T_{\\mathrm{cutoff}})\\mathcal{H}(P-P_{\\mathrm{cutoff}})\\mathcal{H}(\\rho_{\\mathrm{ceiling}}-\\rho), \\label{eq:cooling_modified}\n\\end{equation}\nwhere $\\mathcal{H}$ is the Heaviside function.\n\n\n\\subsection{Initial conditions}\\label{subsec:init_conditions}\nFor our ICM-like runs, we use the same initial conditions as in our setup in \\citetalias{Mohapatra2022characterising}. We initialise the gas with a temperature $T_0=4\\times10^6~\\mathrm{K}$, $n_e=0.086~\\mathrm{cm}^{-3}$ and with an initial sound speed of $300~\\mathrm{km\/s}$. These conditions mimic the cool central regions of groups and clusters. For our CGM setup, we initialise the gas with $T_0=10^5~\\mathrm{K}$, but with the same initial density as the ICM-like runs.\nWe drive turbulence in the box using the same forcing module (with the same seed for the OU process) for all three runs. We only vary $\\zeta$ across the three runs and control the amplitude of the driving, such that in steady state the gas has similar root-mean-square (rms) velocities across our different $\\zeta$ runs. \nThis driven turbulence creates density fluctuations in the gas, which can cool in a runaway fashion.\nThe gas separates into hot and cold phases, with the cold phase at $T_{\\mathrm{cutoff}}=10^4~\\mathrm{K}$ and the hot phase at $10^7$--$10^8~\\mathrm{K}$, which mimics typical ICM conditions.\n\n\\subsection{List of simulations}\\label{subsec:list_of_models}\n\\begin{table*}\n\t\\centering\n\t\\def1.2{1.2}\n\t\\caption{Simulation parameters for different runs}\n\t\\label{tab:sim_params}\n\t\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{lcccccccc} \n\t\t\t\\hline\n\t\t\tLabel & Driving & Resolution & $\\mathrm{sub}_{\\mathrm{factor}}$ & $\\mathcal{M}_{\\mathrm{hot}}$ & $v\\ (\\mathrm{km\/s})$ & $v_{\\mathrm{comp}}\\ (\\mathrm{km\/s})$ & $\\sigma_{s,\\mathrm{hot}}^2$ & $\\sigma_{\\ln(\\bar{P}),\\mathrm{hot}}^2$ \\\\\n\t\t\t(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\\\\\n\t\t\t\\hline\n\t\t\t$\\zeta0.0$ & Compressive & $1008^3$ & $25$ & $0.20\\pm0.01$ & $250\\pm30$ & $100\\pm30$ & $0.36\\pm0.04$ & $0.012\\pm0.005$ \\\\\n\t\t\t$\\zeta0.5$ & Natural & $1008^3$ & $25$ & $0.23\\pm0.01$ & $240\\pm30$ & $90\\pm20$ & $0.23\\pm0.01$ & $0.006\\pm0.001$ \\\\\n\t\t\t$\\zeta1.0$ & Solenoidal & $1008^3$ & $25$ & $0.24\\pm0.01$ & $240\\pm30$ & $90\\pm20$ & $0.19\\pm0.01$ & $0.006\\pm0.002$ \\\\\n\t\t \\hline\n\t\t\t$\\zeta0.0$LR & Compressive & $504^3$ & $25$ & $0.18\\pm0.01$ & $240\\pm20$ & $80\\pm20$ & $0.34\\pm0.02$ & $0.008^{+0.003}_{-0.002}$\\\\\n\t\t\t$\\zeta0.5$LR & Natural & $504^3$ & $25$ & $0.23\\pm0.01$ & $240\\pm30$ & $70\\pm10$ & $0.22\\pm0.01$ & $0.0045\\pm0.0008$\\\\\n\t\t\t$\\zeta0.5$LR$10$ & Natural & $504^3$ & $10$ & $0.27\\pm0.02$ & $250\\pm30$ & $80\\pm10$ & $0.19\\pm0.01$ & $0.010^{+0.004}_{-0.003}$ \\\\\n\t\t\t$\\zeta0.5$LR$1$ & Natural & $504^3$ & $1$ & $0.32\\pm0.02$ & $250\\pm30$ & $90\\pm20$ & $0.15\\pm0.02$ & $0.025^{+0.013}_{-0.008}$ \\\\\n\t\t\t$\\zeta1.0$LR & Solenoidal & $504^3$ & $25$ & $0.23\\pm0.01$ & $230\\pm30$ & $60\\pm10$ & $0.18\\pm0.01$ & $0.0033^{+0.008}_{-0.006}$\\\\\n\t\t\t\\hline\n\t\t\t$\\zeta0.0$LR-CGM & Compressive & $504^3$ & $25$ & $0.14\\pm0.02$ & $80\\pm10$ & $32\\pm10$ & $0.57\\pm0.09$ & $0.003^{+0.004}_{-0.001}$\\\\\n\t\t\t$\\zeta0.5$LR-CGM & Natural & $504^3$ & $25$ & $0.28\\pm0.04$ & $80\\pm10$ & $33\\pm7$ & $0.26^{+0.14}_{-0.09}$ & $0.02^{+0.03}_{-0.01}$\\\\\n\t\t\t$\\zeta1.0$LR-CGM & Solenoidal & $504^3$ & $25$ & $0.24\\pm0.02$ & $70\\pm10$ & $27\\pm4$ & $0.34\\pm0.05$ & $0.008\\pm0.002$\\\\\n\t\t \\hline\n\t\t\t$\\zeta0.0$HR-nocool & Compressive & $1536^3$ & N\/A & $0.20\\pm0.01$ & $270\\pm30$ & $140\\pm50$ & $0.059\\pm0.005$ & $0.066\\pm0.005$ \\\\\n\t\t\t$\\zeta0.5$HR-nocool & Natural & $1536^3$ & N\/A & $0.25\\pm0.01$ & $270\\pm20$ & $16\\pm4$ & $0.0011\\pm0.0001$ & $0.0026\\pm0.0004$ \\\\\n\t\t\t$\\zeta1.0$HR-nocool & Solenoidal & $1536^3$ & N\/A & $0.24\\pm0.01$ & $240\\pm30$ & $5\\pm1$ & $0.0005\\pm0.0001$ & $0.0010\\pm0.0002$ \\\\\n\t\t \\hline\n\t\t\t\n\t\\end{tabular}}\n\t\\justifying \\\\ \\begin{footnotesize} Notes: Column 1 shows the simulation label. The number following $\\zeta$ denotes the fraction of power in solenoidal modes in our turbulent driving. We denote the type of driving in column 2. In column 3, we show the resolution of the simulations. We use $1008^3$ cells for our main parameter study, but we have also performed simulations at lower resolution ($504^3$, indicated by `LR' in the label) for checking convergence. The fourth column shows the subcycle factor for a simulation, defined in \\cref{eq:cool_subcycling}. The default value of $\\mathrm{sub}_{\\mathrm{factor}}$ is $25$, unless indicated in the simulation label (following LR). In column 5, we show the volume-weighted rms Mach number of the hot phase ($T>10^7~\\mathrm{K}$for ICM-like runs and $T>10^6~\\mathrm{k}$ for CGM-like runs). In columns 6 and 7, we show the volume-weighted standard deviations of velocity $v$ and of the longitudinal component of velocity $v_{\\mathrm{comp}}$, respectively ($v^2=v_{\\mathrm{comp}}^2+v_{\\mathrm{sol}}^2$). Finally, in columns 8 and 9, we show $\\sigma^2_{s,\\mathrm{hot}}$ and $\\sigma^2_{\\ln{\\bar{P}},\\mathrm{hot}}$, the squares of the standard deviations of the logarithms of density and pressure of the hot phase, respectively.\\end{footnotesize} \n\t\n\\end{table*}\n\nWe scan the parameter space of the driving parameter $\\zeta$ using three high-resolution simulations ($1008^3$ resolution elements) with $\\zeta$ values $0.0$ (compressive forcing), $0.5$ (natural mixture) and $1.0$ (solenoidal forcing). These simulations use ICM-like initial conditions and form our fiducial set, which we study in most of our analysis. To extend the applicability of our results to smaller galactic halos, we have conducted three simulations with $\\zeta$ values $0.0$, $0.5$ and $1.0$ and CGM-like initial conditions, and present them in \\cref{subsec:CGM case}. We further conduct three non-radiative simulations with $\\zeta$ values $0.0$, $0.5$ and $1.0$, and present their results in Appendix~A, which helps to distinguish between the effects of driving and the effects of radiative cooling and thermal instability. In order to check convergence with numerical resolution, we have conducted three more ICM-like simulations at a resolution of $504^3$ grid cells. We discuss the dependence of our results on resolution in Appendix~B. We also test the dependence on the sub-cycling factor $\\mathrm{sub}_{\\mathrm{factor}}$ in Appendix~C, and we have conducted two more simulations with $\\mathrm{sub}_{\\mathrm{factor}}=10\\text{ and }1$. These simulations, along with some of their key setup parameters and statistical properties are listed in \\cref{tab:sim_params}. \n\n\\section{Results and discussion}\\label{sec:results-discussion}\nIn this section, we present the key results of our work and discuss their implications to the physics of the halo gas and our interpretation of their observations. We start with the results from our high-resolution ICM-like runs and study the dependence of their gas properties on the driving parameter. We then discuss the effect of the solenoidal driving fraction $\\zeta$ on interpreting X-ray and H$\\alpha$ observations of the ICM. Finally, we present the effect of $\\zeta$ on the gas properties of our CGM-like runs.\n\n\\subsection{2D projection and slice}\\label{subsec:2D_proj}\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figures\/Turb_0400_yz_dens_temp_proj_plot.png}\t\n\t\\caption[density-temp-projection plots]{First column: snapshots of normalised density projected along the $x$ direction at $t=1.042$~$\\mathrm{Gyr}$ for three representative simulations: $\\zeta0.0$, $\\zeta0.5$ and $\\zeta1.0$. Second column: mass-weighted projections of temperature at this snapshot along the $x$ direction for these runs. Third column: A slice of temperature in the $yz$ plane, at $x=0$. All colorbars are in log scale, and are shown at the bottom of each column. These plots show strong large-scale variations in the gas density and temperature for the $\\zeta0.0$ run and more small-scale, weaker variations for the $\\zeta0.5$ and $\\zeta1.0$ runs.}\n\t\\label{fig:dens-temp-proj-2d}\n\\end{figure*}\n\nIn \\cref{fig:dens-temp-proj-2d} we show projections of the logarithm of volume-weighted gas density and mass-weighted temperature integrated along the $x$-direction ($\\Sigma_x$ and $T_x$, respectively) and a slice of temperature along the $yz$-plane at $x=0$ at $t=1.042~\\mathrm{Gyr}$. Across all three runs, we find a clear correlation between the density and the temperature of the gas---dense regions correspond to cooler gas at $\\sim10^4~\\mathrm{K}$ and rarefied regions correspond to hot gas with $T\\sim10^7$--$10^8~\\mathrm{K}$. \n\nWith increasing $\\zeta$, we observe a change in the morphology of the gas. The contrast between high- and low-density regions (or corresponding cold- and hot-phase regions), denoted by $\\chi=\\rho_{\\mathrm{cold}}\/\\rho_{\\mathrm{hot}}$ (where $\\rho_{\\mathrm{cold}}$ and $\\rho_{\\mathrm{hot}}$ denote the characteristic gas densities of the cold and hot phases, respectively), is much stronger for the compressive forcing run ($\\zeta0.0$) and this decreases for the natural ($\\zeta0.5$) and solenoidal forcing ($\\zeta1.0$) runs. This is a result of strong converging and diverging motions associated with compressive driving.\nFor the $\\zeta0.0$ run, the cold phase has a filamentary structure and it surrounds the low-density hot-phase regions, seen in both the temperature projection and slice. The cold clouds are large in size, almost equivalent to the driving scale ($L\/2=20~\\mathrm{kpc}$). This is also the scale on which we expect the density fluctuations to be the strongest. However, these structures are transient, lasting around a few $10\\mathrm{s}$--$100~\\mathrm{Myr}$ (see movie of projected temperature in supplementary data). Although the cold-phase gas does shatter to small scales, we observe a lack of misty small-scale structures. The turbulent mixing time scale $t_{\\mathrm{mix}}\\propto\\ell\/u_\\ell$, decreases with decreasing $\\ell$ (for both compressive and solenoidal turbulence). The lack of small-scale clouds for compressive forcing indicates that large-scale modes dominate the evolution of the medium. Turbulent mixing on small scales is relatively weak, which is expected due to the larger $\\chi$ (density contrast) for this run \\citep[also see section~4.4 of][]{Gronke2022MNRAS}. The small-scale clouds are swept up to form high-density filaments due to our continuous compressive forcing. As we increase $\\zeta$, $\\chi$ decreases due to the lack of strong compressive modes. The number of high-density large-scale cold clouds decreases. This also leads to enhanced turbulent mixing on small scales and the production of a large number of misty small-scale cold clouds. Thus, large-scale cold structures can shatter and be mixed with the ambient hot phase and we require continuous compressive forcing to generate and maintain them. \n\n\n\\subsection{Evolution of cold gas mass, rms Mach number and longitudinal velocities}\\label{subsec:time-evolution}\n\n\\begin{figure}\n\t\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/time-evolution.pdf}\t\n\t\\caption[time-evolution plots]{Time evolution of mass fraction of gas in the cold phase (first panel $m_\\mathrm{cold}\/m_{\\mathrm{tot}}$), volume-weighted rms Mach number (second panel $\\mathcal{M}$), the volume-weighted standard deviation of the longitudinal component of velocity (third panel, $v_{\\mathrm{comp}}$) and the turbulence power fraction (fourth panel, $f_{\\mathrm{turb}}$, defined in \\cref{eq:f_turb}). In general, the $\\zeta0.5$ run shows properties closer to the $\\zeta1.0$ run. The $\\zeta0.0$ run shows the strongest temporal variation in all the above quantities.\n\t}\n\t\\label{fig:time-evolution}\n\\end{figure}\n\nIn this subsection, we discuss the evolution of some statistical properties of our multiphase setup. In the first panel of \\cref{fig:time-evolution}, we show the evolution of net mass fraction of cold gas $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$, where $m_{\\mathrm{cold}}$ refers to the total mass of gas with temperature $T<2\\times10^4~\\mathrm{K}$ (the results are insensitive to the exact choice of the cut-off temperature as long as the PDF peak at $10^4$ K is captured). Initially, the gas is at $T_0=4\\times10^6~\\mathrm{K}$, so $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}=0$. As we drive turbulence, we seed density fluctuations in the gas, where the denser regions cool faster than the lower-density regions. All three runs start forming multiphase gas around $100~\\mathrm{Myr}$ and then the fraction $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$ reaches a steady state value of $\\sim0.8$--$0.9$ at around $\\sim200$--$300~\\mathrm{Myr}$. We expect compressive forcing to generate stronger density fluctuations compared to solenoidal forcing, where the densest regions have a shorter cooling time. Due to these fast cooling seed over-densities, the $\\zeta0.0$ run forms multiphase gas the fastest among the three. The $\\zeta0.5$ run forms multiphase gas at almost the same time as the $\\zeta0.0$ run, and its later evolution closely follows that of the $\\zeta1.0$ run. The fraction $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$ shows a sharp increase once the initial seed over-densities start cooling. It reaches the steady state value faster for the $\\zeta0.5$ and $\\zeta1.0$ runs compared to the $\\zeta0.0$ run. The later evolution of $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$ is controlled by small-scale solenoidal modes, which form cold gas through turbulent mixing of the hot and cold phases. The solenoidal modes saturate (i.e., reach a balance between generation and dissipation) later for the lower $\\zeta$ runs, which we discuss later in \\cref{subsec:evol_enstrophy_sources}. \nIn steady state, the fraction $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$ decreases with increasing $\\zeta$. The larger value of this fraction is associated with the high-density cold gas structures in \\cref{fig:dens-temp-proj-2d}, which become less abundant with increasing $\\zeta$. \n\nIn the second and third panels of \\cref{fig:time-evolution}, we show the evolution of the volume-weighted rms Mach number $\\mathcal{M}$ and the volume-weighted standard deviation of the longitudinal component of velocity $v_{\\mathrm{comp}}$. Both $\\mathcal{M}$ and $v_{\\mathrm{comp}}$ for the $\\zeta0.0$ run increase rapidly initially and reach a steady state value. We do not find any strong correlations between the evolution of $\\mathcal{M}$ or $v_{\\mathrm{comp}}$ with $m_{\\mathrm{cold}}\/m_{\\mathrm{tot}}$, especially around the time when the cold gas forms. In contrast, for the $\\zeta0.5$ and $\\zeta1.0$ runs, $\\mathcal{M}$ and $v_{\\mathrm{comp}}$ increase initially, then show a steep increase around $t\\sim100~\\mathrm{Myr}$, marked by a change in the slope. \nThis is followed by a decrease in the values of $\\mathcal{M}$ and $v_{\\mathrm{comp}}$ which reach their steady state values by $300~\\mathrm{Myr}$. The sharp initial increase corresponds to strong compressive flows associated with the condensation of the dense cold-phase gas and the decrease later corresponds to the dissipation of these compressive modes. \nIn steady state, the value of $v_{\\mathrm{comp}}$ decreases slightly with increasing $\\zeta$ (see column~7 of \\cref{tab:sim_params} for its temporally averaged value in steady state). This is expected, since compressive velocities in the $\\zeta0.0$ run are driven directly in addition to the converging flows associated with gas condensation.\n\nIn the fourth panel of \\cref{fig:time-evolution}, we show the evolution of $f_{\\mathrm{turb}}$, which denotes the ratio of the kinetic energy injection rate due to external turbulence forcing and the net radiative cooling rate, defined in \\cref{eq:f_turb}. For all three runs, $f_{\\mathrm{turb}}$ is positive and in steady state it has a value of around $1\\%$. This is in line with our results from \\citetalias{Mohapatra2022characterising}, where we found that weaker turbulence driving runs best reproduced ICM-like gas temperature and Mach number distributions.\n\nThe average value of $f_{\\mathrm{turb}}$ is the largest for the $\\zeta0.0$ run. This run also shows the strongest variations in $f_{\\mathrm{turb}}$. In general, the injected energy dissipates faster for compressive modes\\footnote{The turbulence energy injection rate for the $\\zeta0.0$ run is $8$ times larger than the rate for the $\\zeta1.0$ run, while they produce similar $v$. In steady state, assuming that the injection rate is balanced by the dissipation rate, we infer that the compressive modes dissipate faster than the solenoidal modes. This inference is corroborated by the quicker saturation of $v_{\\mathrm{comp}}$ in the third row of \\cref{fig:time-evolution}.} (at around the sound crossing time on the driving scale $\\sim L\/2c_s$) compared to solenoidal modes (which dissipate over a few $t_{\\mathrm{eddy}}\\sim L\/2u_L$ on the driving scale). Although the value of $\\mathcal{M}_{\\mathrm{hot}}$ is smaller for the $\\zeta0.0$ run, $f_{\\mathrm{turb}}$ is larger because we inject kinetic energy at a faster rate for this run compared to the $\\zeta0.5$ and $\\zeta1.0$ runs for similar turbulent velocity amplitudes. \n\n\\subsection{Density and temperature probability distribution functions}\\label{subsec:dens_temp_PDFs}\n\\begin{figure}\n\t\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/temp_pdf.pdf}\t\n\t\\caption[temperature PDF]{Upper panel: Volume-weighted PDF of temperature for our three representative runs. Lower panel: Mass-weighted PDFs of temperature for the same runs. With increasing $\\zeta$, there is more gas (by mass) at $T\\gtrsim10^{5.5}\\,\\mathrm{K}$ and the temperature of the hot-phase peak decreases.\n\t\\label{fig:temp-PDF}\n\\end{figure}\n\nHere we present the volume-weighted probability distribution functions (PDF) of gas temperature and density for our fiducial runs and discuss their implications. We average the PDFs temporally from $t=0.65~\\mathrm{Gyr}$ to $1.3~\\mathrm{Gyr}$, which is when turbulence is fully developed and is roughly in a steady state.\n\n\\subsubsection{Temperature distribution}\\label{subsubsec:temp_distribution}\nWe show the volume-weighted temperature PDF in the upper panel of \\cref{fig:temp-PDF}. For all three runs, the gas has two distinct peaks, one close to $T_{\\mathrm{cutoff}}~(10^4~\\mathrm{K})$ and another between $10^7$--$10^8~\\mathrm{K}$. There is a clear lack of gas at intermediate temperatures between the two peaks. The small bumps at these intermediate temperatures correspond to the shape of the cooling curve. With increasing $\\zeta$ (the solenoidal driving parameter), the temperature of the hot phase decreases slightly. In the lower panel of \\cref{fig:temp-PDF}, we show the mass-weighted temperature PDF. The amplitude of the cold-phase peak is much larger than that of the hot-phase peak. The mass of gas with $10^{5.5}\\lesssim T\/\\mathrm{K}\\lesssim 10^8$ increases With increasing $\\zeta$. This implies that for the run with compressive driving, the hot gas has lower density compared to the runs with solenoidal or natural driving. This feature is also seen in \\cref{fig:dens-PDF}, which we discuss below.\n\n\\begin{figure}\n\t\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/dens_pdf.pdf}\t\n\t\\caption[density PDF]{Volume-weighted PDF of density for our three representative runs with different fractions of solenoidal and compressive forcing. We average the PDFs in steady state from $t=0.65~\\mathrm{Gyr}$ to $1.3~\\mathrm{Gyr}$. The shaded regions around the PDFs show the $1-\\sigma$ temporal variation in the PDF. The lighter coloured lines show the PDFs of gas in the hot phase ($T>10^7$~$\\mathrm{K}$). The density bimodality decreases with increasing solenoidal fraction $\\zeta$.}\n\t\\label{fig:dens-PDF}\n\\end{figure}\n\n\\subsubsection{Density distribution}\\label{subsubsec:dens_distribution}\nIn \\cref{fig:dens-PDF} we show the PDF of the logarithm of density ($\\log_{10}(\\bar{\\rho})$, where $\\bar{\\rho}=\\rho\/\\rho_0$). The shaded region shows the $1\\sigma$ variation in time. The lighter coloured lines show the PDF when we only consider the hot-phase gas ($T>10^7~\\mathrm{K}$). In all three cases, the density PDF has two peaks, where the low-density peak corresponds to the hot phase ($\\rho_{\\mathrm{hot}}$) and the high-density peak corresponding to the cold phase ($\\rho_{\\mathrm{cold}}$). The value of $\\chi$ decreases with increasing $\\zeta$. This is explained by strong converging and diverging motions associated with compressive forcing, which lead to large variations in gas density. The $\\zeta0.0$ run also shows a high-density tail, which extends till $\\bar{\\rho}\\approx10^{2.5}$, due to compressive motions further compressing the cold dense regions and increasing its density. These high-density power-law tails are also seen in the density distribution of expanding HII region shocks \\cite[][see their fig.~16]{Tremblin2012A&A} as well as compressively forced turbulence simulations of the warm and cold neutral media in \\cite{Seifried2011A&A}. The amount of gas at intermediate densities, in between $\\rho_{\\mathrm{cold}}$ and $\\rho_{\\mathrm{hot}}$ increases with increasing $\\zeta$. This can we explained by weaker mixing between the hot and cold phases due to the larger $\\chi$ values associated with the compressive forcing run. \n\n\\subsection{Phase diagram of density and temperature fluctuations}\\label{subsec:dens_temp_joint_PDF}\n\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figures\/dens_temp_map.png}\t\n\t\\caption[Temperature vs density 2D PDF]{The two-dimensional mass-weighted PDF of the logarithm of temperature versus logarithm of density for our three fiducial runs averaged over $2t_{\\mathrm{eddy}}$, from $t=0.78~\\mathrm{Gyr}$ to $t=0.94~\\mathrm{Gyr}$. The straight lines show the different fluctuation modes--adiabatic, isobaric, isothermal, and isochoric. The gas with $T\\gtrsim10^{5.5}~\\mathrm{K}$ is mostly isobaric, and isochoric just below $\\sim10^{5.5}~\\mathrm{K}$--which corresponds to the peak of cooling function that we use, where $t_{\\mathrm{cool}}\\ll t_{\\mathrm{cs}}$ across the cooling blob. The cold gas accumulates just above the isothermal track at $T_{\\mathrm{cutoff}}=10^4$~$\\mathrm{K}$. We do not observe any strong trends with varying $\\zeta$.\n\t\\label{fig:temp-dens-mw-2D-PDF}\n\t}\n\\end{figure*}\n\nWe show the joint mass distribution of $\\log_{10}{T}$ and $\\log_{10}{\\bar{\\rho}}$ in \\cref{fig:temp-dens-mw-2D-PDF}, averaged from $t=0.78~\\mathrm{Gyr}$ to $t=0.94~\\mathrm{Gyr}$. The straight lines show the different fluctuation modes--adiabatic, isobaric, isochoric and isothermal. The gas fluctuations are isobaric for $T\\gtrsim10^{5.5}~\\mathrm{K}$ and below this temperature, the gas cooling to lower temperatures is mainly isochoric for our three fiducial runs. This temperature corresponds to the peak of the \\cite{Sutherland1993} cooling curve that we use, where $t_{\\mathrm{cool}}\\ll t_{\\mathrm{cs}}$ across the cooling blob. The trends that we observe in \\cref{fig:temp-dens-mw-2D-PDF} agree with the weak turbulent heating run (f0.001) in \\citetalias{Mohapatra2022characterising} (see their fig.~8), since $f_{\\mathrm{turb}}\\sim10^{-2}$ for all three runs (fourth panel of \\cref{fig:time-evolution}). We do not observe any strong effects of the nature of driving on these phase diagrams. Our results are in agreement with the X-ray observations by \\cite{Zhuravleva2016MNRAS,zhuravleva2018}, who find perturbations in the hot gas to be mostly isobaric.\n\n\n\\subsection{Density and pressure fluctuations in the hot phase gas} \\label{subsec:dens_pres_fluc_hot}\n\n\nThe hot-phase density follows a log-normal distribution for all three runs, as seen in \\cref{fig:dens-PDF}. We list the values of $\\sigma_{s,\\mathrm{hot}}^2$ (where $s=\\ln(\\bar{\\rho})=\\log_{10}(\\bar{\\rho})\/\\ln(10)$) and $\\sigma_{\\ln{\\bar{P}},\\mathrm{hot}}^2$ (the width of the pressure PDF\\footnote{We multiply $\\sigma_{\\ln{\\bar{P}}}^2$ by $1\/\\gamma^2$ so that for subsonic and non-stratified turbulence without cooling, it has similar scaling with $\\mathcal{M}$ as $\\sigma_s^2$ \\citep[see][]{Mohapatra2021MNRAS}.}) in columns~8 and~9 of \\cref{tab:sim_params}, respectively. In \\cref{fig:sig-mach-hot}, we show $\\sigma_s^2$ and $\\sigma_{\\ln{\\bar{P}}}^2\/\\gamma^2$ as a function of the hot-phase rms Mach number $\\mathcal{M}_{\\mathrm{hot}}$ (left panel) and the hot-phase compressive rms Mach number $\\mathcal{M}_{\\mathrm{comp,hot}}$ (right panel).\nThe amplitude of the density and pressure fluctuations in the hot phase are used to obtain indirect estimates of turbulent gas velocities using both X-ray surface brightness \\citep{zhuravleva2014turbulent,zhuravleva2014relation,Eckert2017ApJ,Bonafede2018MNRAS} and thermal Sunyaev-Zeldovich effect (tSZ) observations \\citealt{khatri2016}, respectively \\citep[see][for a review]{Simionescu2019SSRv}.\n\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=1.5\\columnwidth]{figures\/sig_mach_hot.pdf}\t\n\t\\caption[hot gas density and pressure fluctuations]{\\underline{Left panel:} The variance of $s$ (filled data points) and $\\ln(\\bar{P})$ (unfilled), the logarithm of normalised density and pressure, respectively of the hot-phase gas (defined as $T>10^6$~$\\mathrm{K}$ for CGM-like runs, $T>10^7$~$\\mathrm{K}$ for ICM-like runs) plotted as a function of the rms Mach number of the hot phase $\\mathcal{M}_{\\mathrm{hot}}$ (left panel) . The solid and dashed line shows the predicted scaling relation for subsonic turbulence with solenoidal forcing (coefficient $b=0.33$) without radiative cooling (see eq.~13f in \\citealt{Mohapatra2021MNRAS} with the Froude number $\\mathrm{Fr}\\rightarrow\\infty$). We choose $b=1.5$ for compressive forcing, which seems to agree well with the scaling of the pressure fluctuations. In general, the pressure fluctuations agree better with the scaling relation as compared to density fluctuations, which are much larger in the presence of radiative cooling due to the associated thermal instability rather than purely driven by turbulence. \\underline{Right panel:} Similar to the left panel, except we plot the fluctuations (squared) against the compressive rms Mach number of the hot phase $\\mathcal{M}_{\\mathrm{comp,hot}}$ along the $x$-axis. The dotted line shows the scaling relation from \\cite{konstandin2012}, which agrees relatively well with the amplitude of density and pressure fluctuations in our non-radiative (nocool) runs. The hot phase pressure fluctuations in radiative runs show a much steeper scaling, varying as $\\mathcal{M}_{\\mathrm{comp,hot}}^{2.5}$, where as the density fluctuations are larger than the scaling relation and do not show any dependence on $\\mathcal{M}_{\\mathrm{comp,hot}}$. \n\t\\label{fig:sig-mach-hot}\n\t}\n\\end{figure*}\n\nWe observe that the amplitude of the density fluctuations is clearly much larger than the scaling relation between $\\sigma_s^2$ and $\\mathcal{M}_{\\mathrm{hot}}$ from \\cite{Mohapatra2021MNRAS} (for unstratified, non-radiative sims, shown as solid and dashed lines in the left panel). The value of $\\sigma_{s,\\mathrm{hot}}^2$ decreases with increasing $\\mathcal{M}_{\\mathrm{hot}}$ which may be due to enhanced turbulent mixing of large-density fluctuations in these runs. We also compare the value of $\\sigma_{s,\\mathrm{hot}}^2$ to the scaling relation for subsonic isothermal turbulence in \\cite{konstandin2012} in the right panel, where the relation of $\\sigma_{s,\\mathrm{hot}}^2$ with $\\mathcal{M}_{\\mathrm{comp,hot}}$ is shown using the dotted line. We find that density fluctuations for the non-radiative runs agree with this scaling relation and for the radiative runs, they are slightly larger than the prediction for the corresponding $\\mathcal{M}_{\\mathrm{comp,hot}}$. \n\n\\cite{Simonte2022A&A} find the density fluctuations to be independent of the degree of stratification of the ICM in their sample of galaxy clusters, with $\\sigma_s^2\\sim0.05$--$0.5$. This is consistent with the \\cite{konstandin2012} scaling relation for $\\mathcal{M}_{\\mathrm{comp,hot}}$ in the range $0.1$--$0.4$. The amplitude of the density fluctuations are also slightly larger than those predicted by \\cite{Mohapatra2020,Mohapatra2021MNRAS}, who only considered solenoidal driving. Since the fluctuations generated by compressive motions are much larger, it explains the lack of dependence of the fluctuations on the degree of stratification in \\cite{Simonte2022A&A}.\n\nThus, the density structure in radiative runs is shaped mainly by thermal instability of the medium (and not the turbulence driving), which generates large compressive velocities and contact discontinuities leading to larger fluctuations.\nX-ray surface brightness fluctuations, which measure the amplitude of gas density fluctuations may overestimate the turbulent gas velocities in thermally unstable regions of the clusters and galactic halos, if one uses the $\\sigma_s$--$\\mathcal{M}$ scaling relation calibrated using non-radiative simulations. However, they can be useful to obtain an upper limit on the compressive component of the velocity field.\n \nIn comparison to the density fluctuations, the pressure fluctuations are much smaller in amplitude and appear to scale with the $\\mathcal{M}_{\\mathrm{hot}}$ as $\\sigma_{\\ln{\\bar{P}}}\\propto\\mathcal{M}_{\\mathrm{hot}}^2$ for constant $\\zeta$ (left panel of \\cref{fig:sig-mach-hot}), for both radiative and non-radiative (nocool) runs. But the constant of proportionality is larger for smaller value of $\\zeta$ ($\\sim1.5^2$ for compressive forcing, $\\sim0.33^2$ for solenoidal), as expected for stronger compressive forcing. This dependence on $\\zeta$ is quantified better in the plot on the right panel of \\cref{fig:sig-mach-hot}, where we find that $\\sigma_{\\ln{\\bar{P}}}^2\\propto\\mathcal{M}_{\\mathrm{comp,hot}}^{2.5}$ for the radiative runs, where as non-radiative runs do not show this scaling. The exact scaling of $\\sigma_{\\ln{\\bar{P}}}$ is likely to depend on both $\\mathcal{M}_{\\mathrm{hot}}$ and $\\mathcal{M}_{\\mathrm{comp,hot}}$, but this is not the focus of our current study and we leave a detailed analysis of the scaling relation for future work. \n\nThus, in thermally unstable regions of the halo gas, the large X-ray brightness fluctuations could be due to contact discontinuities and compressive motions associated with thermal instability, and not be associated with turbulence. In these regions, tSZ fluctuations are more robust probes of both turbulent gas velocities and their compressive component. One could separate the effects of the two through a Helmholtz decomposition of the observed quantity. \n\n\n\\subsection{Compressive to solenoidal velocity ratio}\\label{subsec:comp_sol_ratio}\n\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figures\/comp_temp_pdf.png}\t\n\t\\caption[comp-temp-2d-pdf]{Two-dimensional volume-weighted probability distribution functions of the ratio between compressive to solenoidal velocity magnitudes $v_{\\mathrm{comp}}\/v_{\\mathrm{sol}}$. The solid cyan line shows the median distribution ($50\\mathrm{th}$ percentile) of this ratio as a function of the temperature, and the dotted lines show the $16\\mathrm{th}$ and $84\\mathrm{th}$ percentiles. Although the ratio increases with decreasing $\\zeta$, its median value is $\\sim0.6$ for $\\zeta0.0$, indicating the dominance of solenoidal modes for all the runs.}\n\t\\label{fig:comp-temp-pdf-2d}\n\\end{figure*}\n\nWe calculate the compressive ($\\bm{v}_{\\mathrm{comp}}$) and solenoidal ($\\bm{v}_{\\mathrm{sol}}$) components of the velocity field by performing a Helmholtz decomposition of the velocity field ($\\bm{v}$) in Fourier space, followed by an inverse Fourier transform of the decomposed $\\bm{v}$-components. This methodology for the Helmholtz decomposition of the acceleration field is described in \\cref{eq:Forc_comp} and \\cref{eq:Forc_sol}. In \\cref{fig:comp-temp-pdf-2d}, we present the ratio between the compressive and solenoidal velocity amplitudes $v_{\\mathrm{comp}}\/v_{\\mathrm{sol}}$ and its dependence on the temperature of the gas in a 2D volume-weighted PDF. \nFor all three runs, the hot phase ($T>10^7~\\mathrm{K}$) has the strongest peak in the PDF since it is the volume-filling phase. It shows the broadest distribution in the value of $v_{\\mathrm{comp}}\/v_{\\mathrm{sol}}$. Although the velocities are predominantly solenoidal, a significant fraction of the hot gas also has a stronger compressive velocity component. At intermediate temperatures, we have very less gas and the gas motions are predominantly solenoidal with a small spread. The cold-phase gas ($10^4~\\mathrm{K}10^7$~$\\mathrm{K}$ at $t=1.042$~$\\mathrm{Gyr}$ for three representative simulations: $\\zeta0.0$, $\\zeta0.5$ and $\\zeta1.0$. Second column: Fluctuations in the hot-phase surface brightness from the mean value.\n\tThird column: same as first column, but for cold-phase gas with $10^4~\\mathrm{K}10^7~\\mathrm{K}$).\n\\end{subequations}\nWe discuss these scale-dependent turbulent statistics for our fiducial runs and their implications for the gas in galactic halos below.\n\n\\subsubsection{Second-order velocity structure function}\\label{subsubsec:VSF2}\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figures\/vsf2_hot_cold_3d.pdf}\t\n\t\\caption[density PDF]{Second-order structure function of hot ($T > 10^7 K$, unfilled data points) and cold phase $T < 2 \\times 10^4 K$, filled data points) gas, showing the transverse and longitudinal components, for our three representative runs. The $\\mathrm{VSF}_2$ are normalised by $v^2$. The error bars denote the temporal variation in the $\\mathrm{VSF}_2$. We fit separate power-law scaling relations to the $\\mathrm{VSF}_2$ of the two phases, as well as to the transverse and longitudinal components. We denote the scaling exponents by $m$ and the phases by the subscripts $h$ and $c$, and the components by $\\mathrm{trsv}$ and $\\mathrm{comp}$. These slopes are listed in the bottom right part of each panel.\n\tThe transverse and the longitudinal components have similar scaling with separation $r$, but the transverse component has a larger magnitude for all our runs. The cold-phase $\\mathrm{VSF}$ is steeper than the hot-phase $\\mathrm{VSF}$ for $\\zeta=0.0$ but it becomes parallel to the hot phase for $\\zeta=0.5$ and shallower for $\\zeta=1.0$.}\n\t\\label{fig:vsf2_hc_3d}\n\\end{figure*}\n\nIn \\cref{fig:vsf2_hc_3d}, we show the second-order velocity structure function for the hot and cold phases for our fiducial set of runs. The velocity structure functions represent the velocity difference between two points as a function of separation and are useful for determining turbulence statistics such as (i) the driving scale(s) of turbulence--indicated by a flattening of $\\mathrm{VSF}$ at the corresponding scale(s); (ii) the distribution of turbulence kinetic energy across scales and comparison with theoretical predictions such as Burgers turbulence \\citep{Burgers1948171} and Kolmogorov turbulence \\citep{kolmogorov1941dissipation}. Recently, integral field spectroscopy observations of atomic filaments \\citep{Gendron-Marsolais2018MNRAS,Sarzi2018MNRAS,Tremblay2018ApJ,Boselli2019A&A} and Atacama Large Millimeter Array (ALMA) observations of molecular gas filaments \\citep{Simionescu2018MNRAS,Tremblay2018ApJ} have been used to determine the $\\mathrm{VSF}$ for the cold phase of the ICM \\citep{Li2020ApJ}. Both observations and recent numerical simulations such as \\cite{Hillel2020ApJ,Wang2021MNRAS,Gronke2022MNRAS,Mohapatra2022VSF,Hu2022arXiv,Zhang2022arXiv} have attempted to interpret the slopes and amplitude of the cold phase $\\mathrm{VSF}$ and its implications for the turbulence kinematics of the hot phase.\nUsing our multiphase turbulence simulations, we study the $\\mathrm{VSF}_2$ of the hot and cold phases, as well as the distribution of the kinetic energy into compressive and solenoidal modes. \n\nWe normalise the $\\mathrm{VSF}_2$ by $v^2$ and decompose it into its transverse and longitudinal components. For comparison with non-radiative runs with the same $\\zeta$, we present the decomposed $\\mathrm{VSF}_2$ for our three non-radiative runs in \\cref{fig:app_vsf2_nocool}. For all runs and both phases, both the longitudinal and transverse components of the $\\mathrm{VSF}_2$ increase as a function of $r$ and peak close to the driving scale. The transverse component has a larger amplitude compared to the longitudinal component for all runs and both phases, as expected from the results in \\cref{subsec:comp_sol_ratio}.\n\n\nFor the $\\zeta0.0$ run, both the longitudinal and transverse components of the cold-phase $\\mathrm{VSF}_2$ are larger than the corresponding hot phase $\\mathrm{VSF}_2$ at the driving scale ($L\/2$). However, they are much steeper than the hot-phase $\\mathrm{VSF}_2$, and on small scales, the hot-phase $\\mathrm{VSF}_2$ is much larger. The large cold-phase velocities close to the driving scale could be due to strong compressive motions associated with multiphase condensation of high-density regions (notice that $\\mathrm{VSF}_2^{\\mathrm{cold}}\/\\mathrm{VSF}_2^{\\mathrm{hot}}$ is much larger for the longitudinal component), in addition to the external turbulence driving. \nThe steepness of the structure function could be due to Burgers turbulence ($\\mathrm{VSF}_2\\propto\\ell$, \\citealt{Burgers1948171}) in the large, supersonic cold clouds. The $\\mathrm{VSF}_2$ could also be affected by weak coupling between the hot and cold phases due to large $\\chi$ (see \\cref{fig:dens-PDF}), which in turn is caused by the large-scale compressive driving.\n\nWith increasing $\\zeta$, both components of the hot-phase $\\mathrm{VSF_2}$ become steeper, whereas the cold-phase $\\mathrm{VSF_2}$ becomes flatter. For the hot phase, this trend is the opposite of what we observe in the non-radiative runs (see appendix A, subsection 2), where the $\\mathrm{VSF}_2$ flattens upon increasing $\\zeta$, as is expected from turbulence scaling changing from compressible Burgers-like to incompressible Kolmogorov-like with increasing $\\zeta$. The flatter $\\mathrm{VSF}_2$ is likely due to the contribution of velocities due to thermal instability, which are expected to scale as $r^0$ since both small- and large-scale isobaric modes are expected to grow at the same rate, as we discussed in section~4.3.1.2 in \\cite{Mohapatra2022VSF} \\citep[also see fig.~2 of ][which shows similar growth rate of small- and large-scale isobaric modes]{sharma2010thermal}. The flattening of velocity scaling due to thermal instability is also observed in the velocity power spectra of \\cite{Gazol2010ApJ}, where the velocity power spectrum for their non-isothermal run with thermal instability has a flatter scaling compared to their isothermal run. The velocities due to thermal instability are expected to be larger for the compressive driving run, since the driving generates stronger density fluctuations, which act as seeds for the isobaric modes to grow from. The flattening of the cold-phase $\\mathrm{VSF}_2$ with increasing $\\zeta$ is expected due to two reasons-- (i) smaller velocities on large scales, due to weaker contribution from velocities associated with large-scale gas condensation and (ii) larger cold-phase velocities at intermediate scales ($0.05\\lesssim r\/L\\lesssim0.1$) due to stronger coupling between the hot and cold phases with decreasing $\\chi$. \n\n\n\n\\subsubsection{Velocity cross-correlation function}\\label{subsubsec:VCF}\n\n\\begin{figure}\n\t\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/vcorrf2_hot_cold_3d.pdf}\t\n\t\\caption[hot-cold correlation function]{Cross correlation function between hot- and cold-phase velocities, for our three representative runs. The correlation between the velocities decreases with increasing $r$ and increases with increasing $\\zeta$.}\n\t\\label{fig:vcorrf2_hc_3d}\n\\end{figure}\n\nIn \\cref{fig:vcorrf2_hc_3d}, we show the spatial cross-correlation function between the hot- and cold-phase velocities $\\mathrm{VCF}_{\\mathrm{hc}}(r)$, as defined in \\cref{eq:vcorrfunc_cold_hot}. The $\\mathrm{VCF}_{\\mathrm{hc}}$ is a useful statistical tool to check the correlation between gas velocities as a function of separation and it gives us a sense of coupling between the hot and cold phases. We have normalised $\\mathrm{VCF}_{\\mathrm{hc}}(r)$ by dividing out the standard deviations of hot- and cold-phase velocities $v_{\\mathrm{hot}}$ and $v_{\\mathrm{cold}}$, respectively. For all three fiducial runs, the $\\mathrm{VCF}_{\\mathrm{hc}}$ peaks on small scales and decreases as a function of separation $r$. With increasing $\\zeta$, the value of $\\mathrm{VCF}_{\\mathrm{hc}}$ increases, due to stronger coupling between the phases. The coupling becomes stronger due to smaller density contrast ($\\chi$) in larger $\\zeta$ runs, since large-scale compressive motions are weaker. The dependence on $\\zeta$ in these runs are in line with our results from \\cite{Mohapatra2022VSF}, where we found the amplitude of the $\\mathrm{VCF}_{\\mathrm{hc}}$ to depend inversely on $\\chi$.\n\n\\subsection{CGM case}\\label{subsec:CGM case}\n\\begin{figure*}\n\t\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figures\/mach_temp_dens_CGM_pdf.pdf}\t\n\t\\caption[PDF CGM case]{The volume-weighted PDFs of the logarithms of $\\mathcal{M}$ (first column), temperature (second column) and density (third column) for three runs with CGM-like initial conditions. The PDFs are temporally averaged over 26 snapshots from $t=0.98$~$\\mathrm{Gyr}$ to $t=1.95$~$\\mathrm{Gyr}$, and the shaded region shows the $1\\sigma$ variation. With decreasing $\\zeta$, the PDFs show more mixing between the hot and cold phases. For the $\\zeta0.5$LR-CGM and $\\zeta1.0$LR-CGM runs, we observe a strong third peak at intemediate temperatures and densities.}\n\t\\label{fig:CGM_PDFs}\n\\end{figure*}\n\nIn this subsection, we discuss the effects of the turbulence driving on our simulations with CGM-like initial conditions. We focus on its effects on the volume-weighted PDFs of $\\mathcal{M}$, temperature (T) and normalised density ($\\bar{\\rho}$), shown in the first, second and third panels of \\cref{fig:CGM_PDFs}, respectively. For all three runs, the hot phase is subsonic, with $10^6~\\mathrm{K}\\lesssim T\\lesssim10^7~\\mathrm{K}$ and density $\\sim10^{-4}$--$10^{-2}~\\mathrm{cm}^{-3}$ ($\\bar{\\rho}=\\rho\/\\rho_0$, where $\\rho_0=0.086~\\mathrm{cm^{-3}}$). \n\nFor the $\\zeta0.0$-LR-CGM run, the gas distribution is bimodal with strong and distinct peaks corresponding to the two temperature phases, similar to our ICM-like runs in \\cref{fig:temp-PDF}. The hot phase is at somewhat higher temperatures due to the strong heating by fast dissipation of compressive modes. The compressive and expansive gas motions also lead to high-density cold-phase gas and lower-density hot-phase gas compared to the higher $\\zeta$ runs.\n\nWith increasing $\\zeta$, the density contrast between hot and cold phases becomes weaker in the absence of large-scale compressive modes. The gas distribution is more continuous rather than bimodal, with the hot and cold peaks moving closer to each other. Since $\\chi$ is much smaller in the CGM compared to the ICM case, mixing due to turbulence continuously produces gas at intermediate temperatures $T_{\\mathrm{mix}}\\sim\\sqrt{T_{\\mathrm{hot}}T_{\\mathrm{cold}}}\\sim10^5~\\mathrm{K}$ \\citep{Begelman1990MNRAS}. Since $T_{\\mathrm{mix}}$ is close to the peak of the \\cite{Sutherland1993} cooling curve, this gas has a short cooling time and thus cools rapidly and accumulates just below the peak of the cooling curve, at $\\sim10^{4.7}~\\mathrm{K}$. This third peak also shows up in the $\\mathcal{M}$ and density distributions, as a strong peak at intermediate $\\mathcal{M}$ and $\\bar{\\rho}$. The effect of increasing $\\zeta$ on the PDFs is much stronger for the CGM-like runs, since the value of $\\chi$ is an order of magnitude smaller than in our ICM-like runs, leading to stronger mixing. \\cite{Gronke2022MNRAS} also study the effect of $\\chi$ on mixing between hot and cold phases and find turbulent mixing to become stronger with decreasing $\\chi$.\n\nIn section~3.3.2 of \\citetalias{Mohapatra2022characterising}, we highlighted that the amount of gas with temperatures in between the hot and cold phase depends on the turbulent heating fraction $f_{\\mathrm{turb}}$. Since we only used solenoidal forcing in the previous study, increasing $f_{\\mathrm{turb}}$ led to both a smaller $\\chi$ and increased amplitude of solenoidal velocities, both of which are associated with increased mixing between the hot and cold phases and larger amounts of gas at intermediate temperatures. In this study, the effect of increasing $\\zeta$ decreases $\\chi$ and as a consequence produces larger amounts of gas at intermediate temperatures, even though $v_{\\mathrm{sol}}$ is similar across all runs. \n\nThe gas at these temperatures is traced by ions such as Mg\\texttt{II} at $\\sim10^4~\\mathrm{K}$ \\citep{Anand2021MNRAS,Anand2022arXiv}, Si\\texttt{IV} at$\\sim10^{5}~\\mathrm{K}$ \\citep{Zheng2017ApJ} and O\\texttt{VI} at $\\sim10^{5.5}~\\mathrm{K}$ \\citep{Tumlinson2011ApJ}. We refer the reader to fig.~6 of \\cite{Tumlinson2017review} for a larger list of ions that are abundant at these temperatures. These ions are detected as absorption features in the spectra of background quasars as well as emission from nearby bright sources. By comparing the relative abundance of these ions and using our estimate of $\\chi$, we can constrain the amplitude of the turbulent velocity. We can use this information further to infer the nature of the driving (compressive vs solenoidal) by the different sources of turbulence, such as stellar winds, supernovae, outflows, mergers and tidal interactions.\n\n\\section{Caveats and future work}\\label{sec:caveats-future}\nIn this section, we discuss some of the assumptions and shortcomings of the present study. We also present some future prospects.\n\nWe model the ICM as a fluid in a periodic box with a fixed injection scale for turbulence, which allows us to perform a controlled parameter scan and study the dependence of gas properties and kinematics on the turbulence driving. However, the driving due to supernovae, AGN jets, galaxy infall and mergers, etc., can occur across multiple scales and contribute to both solenoidal and compressive modes during the course of a galaxy\/cluster's evolution, which are not captured by our simple model. \n\nIn order to run our simulations with the available computing resources, we have used subcycling for the radiative cooling, as explained in paragraph \\ref{par:cooling_subcycling}. We set $\\mathrm{sub}_{\\mathrm{factor}}=25$ for our fiducial set of runs. Implementing subcycling affects the gas density, temperature and pressure distributions, as we discuss in \\cref{app:mach_temp_dens_dist_subc}. Larger values of $\\mathrm{sub}_{\\mathrm{factor}}$ lead to less amount of gas at intermediate temperatures (see \\cref{fig:app_subc_dependence}), since this gas has a short cooling time and is cooled all the way to $T_{\\mathrm{cutoff}}$ when we evolve the cooling module independent of the hydro evolution. The density cooling cutoff ($\\rho_{\\mathrm{cutoff}}$, see paragraph~\\ref{par:cool_cutoffs}) affects the high-density tail of the density distribution for the compressive driving run. For this reason, we have excluded regions with $\\rho>\\rho_{\\mathrm{cutoff}}$ from our analysis.\n\nWe have ignored important physics such as magnetic fields, thermal conduction, gravity and density stratification of the ICM. All of these can affect the physics of gas interactions as well as the evolution of thermal instabilities in the ICM. \n\nTurbulence kinetic energy can be converted into magnetic energy \\citep{Federrath2016JPlPh,DiGennaro2021NatAs}, increase the coupling between hot and cold phases and also affect the slopes of the $\\mathrm{VSF}_2$ \\citep{Mohapatra2022VSF}. Magnetic fields can also affect the growth rate of enstrophy, as seen in studies by \\citet{Porter2015ApJ} and \\citet{Seta2022arXiv}. On sub-$\\mathrm{kpc}$ scales, the ICM is expected to be weakly collisional. Kinetic plasma instabilities such as the firehose and mirror instabilities can affect the evolution of magnetic fields and other gas properties \\citep[see e.g.,][]{Santos-Lima2014ApJ,Santos-Lima2016MNRAS}. \n\nThermal conduction has been shown to affect the structure functions\/power spectra on small scales \\citep{gaspari2013constraining,gaspari2014} and can smear out some of the strong temperature gradients that we observe in the $\\zeta0.0$ run. \n\nGravity and density stratification can affect the growth of thermal instability depending on the ratio between $t_{\\mathrm{cool}}$ and the free fall time scale \\citep{sharma2012thermal,choudhury2016,Voit2021ApJ}. It can provide another channel of energy transport, from turbulence kinetic energy into buoyancy potential energy \\citep{Mohapatra2020}. In the presence of gravity and the absence of pressure support, the cold-phase filaments may fall towards the cluster centre, as seen in \\cite{Wang2021MNRAS}. We plan to include the effects of gravity and magnetic fields in a future study.\n\nWe have modelled external, non-turbulent sources of energy by distributing the heat throughout the volume as a function of the local gas density. In future works, we plan to implement heating due to other feedback sources such as cosmic rays, which are expected to have a different distribution of energy and momentum compared to our simple model \\citep{Ji2020MNRAS,Butsky2020ApJ,Butsky2021arXiv}.\n\nOur simulations have a resolution of $\\sim40~\\mathrm{pc}$ for the ICM-like runs and $\\sim10~\\mathrm{pc}$ for the CGM-like runs. The minimum cooling length $\\ell_{\\mathrm{cool}}=\\min(c_st_{\\mathrm{cool}})\\approx0.1~\\mathrm{pc}$, which is one of the\nsmallest length scale in the problem and perhaps necessary for resolving the turbulent mixing layers between the hot and cold phases (\\citealt{mccourt2018}; but bottom right panel in Fig. 17 of \\citealt{Das2021MNRAS} shows that clouds even smaller than this length scale condense in 1D). This scale is still two orders of magnitude smaller than our current resolution. We refer the reader to \\cref{app:mach_temp_dens_dist_res} for a convergence study. With increasing resolution, we expect the cold gas to form more small-scale misty clouds, especially for the higher $\\zeta$ runs. We also expect $\\chi$ to decrease and turbulent mixing to be stronger as a consequence. \n\n\\section{Summary and conclusions}\\label{sec:Conclusion}\nIn this work, we have studied the effects of the turbulence driving--compressive ($\\zeta0.0$, curl-free), natural mixture ($\\zeta0.5$) and solenoidal ($\\zeta1.0$, divergence-free) on the different statistical properties of multiphase turbulence and discussed their implications for the ICM and the CGM. Here we present some of the main takeaway points of this work:\n\n\\begin{itemize}\n \\item Turbulence driving directly affects gas properties on large scales. We observe larger contrasts between the density and temperature of the hot and cold phases with compressive driving compared to solenoidal driving. \n \n \\item The density contrast ($\\chi$) between the cold and hot phases plays a key role in several other statistical properties of the gas. A smaller ratio of the cold-to-hot-phase gas density, $\\chi=\\rho_{\\mathrm{cold}}\/\\rho_{\\mathrm{hot}}$, in solenoidal driving leads to more efficient turbulent mixing between the two phases, resulting in larger amounts of gas at intermediate temperatures. This effect is stronger for the CGM-like runs, where $\\chi$ is an order of magnitude smaller than in the ICM-like runs.\n \n \\item Gas density and pressure fluctuations, which are important observational tools to estimate turbulent gas velocities of the ICM depend on $\\zeta$. The density fluctuations in the hot X-ray-emitting phase are orders of magnitude larger than known scaling relations due to additional compressive velocities and contact discontinuities induced by thermal instability of the medium. Gas-pressure fluctuations scale as $\\mathcal{M}^2$, and their amplitudes decrease with increasing $\\zeta$. \n \n \\item The gas velocities are dominated by solenoidal modes at all temperatures, even for compressive driving. Baroclinicity efficiently generates enstrophy (vorticity squared), especially for purely compressive driving, which is further amplified by vortex stretching. In steady state, the enstrophy is nearly independent of the driving mode.\n \n \\item Compressive driving affects the appearance of the gas in X-rays and optical mock-emission maps, with the X-ray emission showing cavities surrounded by strong emission regions. The H$\\alpha$ filaments coincide with the peaks in X-ray emission. The emission maps are relatively uniform for solenoidal driving and show no strong large-scale features.\n \n \\item The coupling between the velocities of the hot and cold phases increases with decreasing density contrast ($\\chi$), which decreases with increasing fraction of solenoidal driving. For the compressive driving run, the cold-phase 2nd-order velocity structure function, $\\mathrm{VSF}_2$, has larger velocities, but has a much steeper scaling with separation (Burgers-like) compared to the hot phase. For all runs, the transverse component of $\\mathrm{VSF}_2$ is much larger than the longitudinal component, implying the dominance of solenoidal velocities at all scales for both hot and cold phases.\n\\end{itemize}\n\n\\section*{Acknowledgements}\nThis work was carried out during the ongoing COVID-19 pandemic. The authors would like to acknowledge the health workers all over the world for their role in fighting in the frontline of this crisis. \nThe authors would like to thank the anonymous referee for a constructive report, which helped to improve this work.\nCF acknowledges funding provided by the Australian Research Council (Future Fellowship FT180100495), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD).\nPS acknowledges a Swarnajayanti Fellowship (DST\/SJF\/PSA-03\/2016-17) and a National Supercomputing Mission (NSM) grant from the Department of Science and Technology, India.\nWe further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants~pr32lo, pr48pi and GCS Large-scale project~10391), the Australian National Computational Infrastructure (grant~ek9) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software, \\texttt{FLASH}, was in part developed by the Flash Centre for Computational Science at the Department of Physics and Astronomy of the University of Rochester.\n\nThis work used the following software\/packages:\n\\texttt{FLASH} \\citep{Fryxell2000,Dubey2008}, \\texttt{matplotlib} \\citep{Hunter4160265}, \\texttt{cmasher} \\citep{Ellert2020JOSS}, \\texttt{scipy} \\citep{Virtanen2020}, \\texttt{NumPy} \\citep{Harris2020}, \\texttt{h5py} \\citep{collette_python_hdf5_2014}, \\texttt{LMfit} \\citep{Newville2016ascl} and \\texttt{astropy} \\citep{astropy2018}.\n\n\\section{Data Availability}\nAll relevant data associated with this article is available upon reasonable request to the corresponding author.\n\n\\section{Additional Links}\nMovies of projected density and temperature of different simulations are available as online supplementary material, as well as at the following links:\n\\begin{enumerate}\n \\item \\href{https:\/\/youtu.be\/qBsJti2R0HY}{Movie} of mass-weighted projection of temperature for the fiducial runs.\n \\item \\href{https:\/\/youtu.be\/JvTMXYU4m_U}{Movie} of mass-weighted projection of temperature for the three CGM runs.\n \\item \\href{https:\/\/youtu.be\/YXWYFtM0P94}{Movie} of the $\\zeta0.0$ simulation.\n \\item \\href{https:\/\/youtu.be\/RBjN6RD0k78}{Movie} of the $\\zeta0.5$ simulation.\n \\item \\href{https:\/\/youtu.be\/aGn7_EX8JvI}{Movie} of the $\\zeta1.0$ simulation.\n\\end{enumerate}\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThis paper is a sequel of sorts to the first author's paper \\cite{kedlaya-dri}, \nalthough there is no logical dependence between the two. Both papers concern themselves with\nanalogues of ``Drinfeld's lemma'' in \\'etale cohomology, and in particular with corresponding statements in $p$-adic cohomology; however, there are some differences in scope and methodology which we highlight below.\n\n\\begin{secnumber} \\label{sss:intro setting}\nDrinfeld's lemma was first introduced by Drinfeld in his proof of the Langlands correspondence for $\\GL_2$ over global function fields of characteristic $p>0$ \\cite{Dri78}. \nLet us briefly review the result. Let $X_1,X_2$ be two connected schemes over $k=\\mathbb{F}_p$. The scheme $X=X_1\\times_k X_2$ is equipped with two endomorphisms $F_{1}, F_{2}$, obtained by base changes of the absolute Frobenius on $X_1,X_2$, respectively. \nWe consider the category $\\mathcal{C}(X,\\Phi)$ of objects $(T,F_{ \\{1\\}},F_{ \\{2\\}})$ consisting of a finite \\'etale morphism $T\\to X$ and isomorphisms $F_{ \\{i\\}}:T\\times_{X,F_i}X \\xrightarrow{\\sim} T$ commuting with each other, whose composition is the relative Frobenius morphism $F_{T\/X}$ of $T$ over $X$. \nThis category is a Galois category and we denote by $\\pi_1(X,\\Phi,\\overline{x})$ the Galois group defined by a geometric point $\\overline{x}$ of $X$. \nThen Drinfeld's lemma says that the projection maps $X\\to X_i$ induce an isomorphism of profinite groups:\n\\begin{equation} \\label{eq:origin Drinfeld lemma}\n\t\\pi_1(X,\\Phi,\\overline{x}) \\xrightarrow{\\sim} \\pi_1(X_1,\\overline{x})\\times \\pi_1(X_2,\\overline{x}).\n\\end{equation}\n(See \\cite[Theorem~8.1.4]{lau} or \\cite[Theorem~4.2.12]{kedlaya-aws}. For the key case where $X_2$ is a geometric point, see also \\cite[Lemme~8.11]{lafforgue-v}.)\n\nA closely related result is that any quasicompact \nopen immersion which is stable under the $F_i$ is covered by products of open immersions into the $X_i$. \n(See \\cite[Lemma~9.2.1]{lau} or \\cite[Theorem~4.3.6]{kedlaya-aws}. For the key case where $X_2$ is a geometric point, see also \\cite[Lemme~8.12]{lafforgue-v}.)\nThese two results together allow us to view lisse $\\ell$-adic sheaves with ``partial Frobenius'' on $X$ as $\\ell$-adic representations of $\\pi_1(X_1,\\overline{x})\\times \\pi_1(X_2,\\overline{x})$, and to work with constructible sheaves with partial Frobenius on $X$ via stratifications coming from $X_1$ and $X_2$. \nWe refer to \\cite{lau} and \\cite[Lecture 4]{kedlaya-aws} for more detailed expositions. \n\\end{secnumber}\n\n\\begin{secnumber}\n\tDrinfeld's lemma also plays an essential role in V. Lafforgue's work \\cite{lafforgue-v} on the automorphic-to-Galois direction of the Langlands correspondence for reductive groups over a global function field $F$. \n\tRoughly speaking, V. Lafforgue showed that the space of cuspidal automorphic functions (with values in $\\overline{\\mathbb{Q}}_{\\ell}$) of a reductive group $G$ over $F$ admits a decomposition indexed by certain $\\ell$-adic Langlands parameters. \n\tThis decomposition is obtained by investigating the $\\ell$-adic cohomology of certain moduli stacks of shtukas for $G$. \n\tMoreover, he conjectured that this decomposition should be $\\ell$-independent and indexed by certain motivic Langlands parameters. \n\t\n\tIn particular, we expect that there is a variant of V. Lafforgue's result in terms of $p$-adic Langlands parameters, corresponding to Abe's adaptation of the work of L. Lafforgue from $\\ell$-adic to $p$-adic coefficients \\cite{abe-companion}.\n\tIn this adaptation, the $p$-adic analogues of of lisse $\\ell$-adic sheaves are \\textit{overconvergent $F$-isocrystals}. \n\tBesides those objects, we may also consider the larger category of \\textit{convergent $F$-isocrystals}, which admit no $\\ell$-adic analogue but play an important role in the $p$-adic setup.\n\t\n\tRecently, Drinfeld proposed an unconditional definition of motivic Langlands parameters \\cite{Dri18}. \n\tInspired by this work, we consider $p$-adic Langlands parameters as homomorphisms of the Tannakian group of the category of overconvergent $F$-isocrystals over a curve to the Langlands dual group of $G$. \n\\end{secnumber}\t\n\n\\begin{secnumber}\n\tFrom this perspective, we prove a Tannakian form of Drinfeld's lemma for overconvergent\/convergent $F$-isocrystals, which aims to establish the aforementioned result for $p$-adic Langlands parameters. \n\tNote that another ingredient of V. Lafforgue's approach, the \\emph{geometric Satake equivalence}, was established for $F$-isocrystals by the second author and X. Zhu \\cite{XZ}. \n\n\tKeep the notations of \\S~\\ref{sss:intro setting}. \n\tGiven an overconvergent (resp. convergent) isocrystal $\\mathscr{E}$ on $X$, a \\emph{partial Frobenius structure} on $\\mathscr{E}$ consists of two isomorphisms $\\varphi_i:F_i^*(\\mathscr{E})\\xrightarrow{\\sim} \\mathscr{E}$ such that $\\varphi_1\\circ F_1^*(\\varphi_2)= \\varphi_2\\circ F_2^*(\\varphi_1)$; this composition in particular provides $\n\\mathscr{E}$ with the structure of an $F$-isocrystal.\n\tThe category $\\PhiIsocd(X)$ (resp. $\\PhiIsoc(X)$) of overconvergent (resp. convergent) isocrystals with a partial Frobenius structure is a Tannakian category and we denote by $\\pi_1^{\\PhiIsocd}(X)$ (resp. $\\pi_1^{\\PhiIsoc}(X)$) the associated Tannakian group (with respect to a fiber functor). \n\\end{secnumber}\n\n\\begin{theorem}[\\ref{t:main thm}, \\ref{t:Drinfeld lemma origin}, \\ref{t:Drinfeld lemma conv}] \\label{t:thm intro}\n\t\\textnormal{(i)} \n\tSuppose each $X_i$ is geometrically connected (resp. smooth and geometrically connected) over $k$.\n\tThe pullback functors of projections $p_i:X\\to X_i$ induce a canonical isomorphism of Tannakian groups:\n\t\\begin{equation} \\label{eq:iso intro}\n\tp_1^{\\circ}\\times p_2^{\\circ}: \n\t\\pi_1^{\\PhiIsocd}(X)\\xrightarrow{\\sim} \\pi_1^{\\FIsocd}(X_1)\\times \\pi_1^{\\FIsocd}(X_2) \\quad\n\t\\textnormal{(resp. } \\pi_1^{\\PhiIsoc}(X)\\xrightarrow{\\sim} \\pi_1^{\\FIsoc}(X_1)\\times \\pi_1^{\\FIsoc}(X_2)).\n\\end{equation}\n\twhere $\\pi_1^{\\FIsocd}$ (resp. $\\pi_1^{\\FIsoc}$) denotes the Tannakian group of the category of overconvergent (resp. convergent) $F$-isocrystals over $\\overline{\\mathbb{Q}}_p$, and similarly with $F$ replaced by $\\Phi$. \n\n\t\\textnormal{(ii)}\n\tSuppose each $X_i$ is smooth and geometrically connected over $k$. \n\tBy taking connected components in the above isomorphism, we recover the isomorphism \\eqref{eq:origin Drinfeld lemma}. \n\\end{theorem}\n\nIn the overconvergent case, we also establish a similar isomorphism for overconvergent isocrystals which can be equipped with a Frobenius structure but do not carry a specified one. This roughly corresponds to passing from arithmetic to geometric fundamental groups in the $\\ell$-adic setting. \n\n\\begin{secnumber}\n\tIn very recent work \\cite{kedlaya-mono-rel}, the first author has established a relative version of the $p$-adic local monodromy theorem for differential modules with a Frobenius structure over an annulus \\cite{And02II,Meb02,kedlaya-mono}. \n\tAn application is the local monodromy theorem for modules with an integrable connection and a partial Frobenius structure over polyannuli \\cite[Theorem 3.3.6]{kedlaya-mono-rel}. \n\tWe formulate these results in terms of a Tannakian form of local Drinfeld's lemma (Theorem \\ref{t:local Drinfeld lemma}) and discuss some related constructions. \n\n\tWe remark that variants of local Drinfeld's lemma for $\\ell$-adic sheaves are key ingredients of the local Langlands correspondence of Genestier--V. Lafforgue \\cite{GL} and of Fargues--Scholze \\cite{FS}. \n\n\tInspired by Berger's thesis \\cite{Berger}, we expect to deduce a ``de Rham implies potentially semistable'' result for $p$-adic representations of powers of Galois groups from local Drinfeld's lemma. \n\tThere are some related results in this direction: (i) the overconvergence of multivariate $(\\varphi,\\Gamma)$-modules has been proved by the first author, A. Carter, and G. Z\\'abr\\'adi \\cite{CKZ21}; (ii) multivariable de Rham representations and the associated $p$-adic differential equations are studied by Brinon, Chiarellotto, and Mazzari \\cite{BCM21}. \n\\end{secnumber}\n\n\\begin{secnumber}\n We now describe the structure of the paper.\n\t\n The proof of Theorem~\\ref{t:thm intro} in the overconvergent case is given in \\S\\ref{sec-overconvergent}.\tA key ingredient is Proposition \\ref{P:pushforward},\n which says that the pushforward functor (for arithmetic $\\mathscr{D}$-modules) of the projection $p_i:X\\to X_i$ ($i=1,2$) sends overconvergent isocrystals on $X$, which can be equipped with a partial Frobenius structure, to overconvergent isocrystals on $X_i$. \n\tCombined with a criterion of Esnault--Hai--Sun \\cite{EPS} on exact sequences of Tannakian groups, we conclude Theorem \\ref{t:thm intro}(i) in the overconvergent case. \n\n\tThe proof in the convergent case is contained in section \\ref{s:conv case} and follows a similar line as in \\cite{kedlaya-dri} (see below). \n\tWe study unit-root and diagonally unit-root convergent isocrystals with a partial Frobenius structure (Propositions \\ref{P:Crew diagonal unitroot}, \\ref{P:Crew partial Frob}), and the diagonal (resp. partial) Frobenius slope filtrations (Theorems \\ref{t:slope filtration}, \\ref{T:partial slope filtration}).\n\tThese tools allow us to define a pushforward functor along the projection $p_i:X\\to X_i$ from $\\PhiIsoc(X)$ to $\\FIsoc(X_i)$. \n\tThen we deduce Theorem \\ref{t:thm intro}(i) in the convergent case by a similar argument as in the overconvergent case.\n\n\tWe upgrade the isomorphism \\eqref{eq:iso intro} in an $\\ell$-independent form \u00e0 la Drinfeld \\cite{Dri18} in section \\ref{s:motivic} (see Proposition \\ref{P:l-independence} and \\eqref{eq:Drinfeld lemma mot}). \n\n\tThe last section is devoted to the local Drinfeld's lemma. \n\n\\end{secnumber}\n\n\\begin{secnumber}\nWe now compare more carefully the results of this paper with those of \\cite{kedlaya-dri}.\nIn \\cite{kedlaya-dri}, each $X_i$ is required to be smooth over some perfect field $k_i$, but it is not required that $k_i$ be either finite or independent of $i$. The more restricted situation considered here (in which $k_i = \\mathbb{F}_p$ for all $i$) is sufficient for the application to geometric Langlands; moreover, it is not immediately apparent how to reproduce the Tannakian formulation in the setting of \\cite{kedlaya-dri}. \n\nAnother important difference with \\cite{kedlaya-dri} is in the overall structure of the arguments. Therein, the convergent case is treated first, using the isomorphism \\eqref{eq:origin Drinfeld lemma}\n as input, and then the overconvergent case is deduced as a corollary; herein, we obtain the overconvergent case directly using cohomological methods, deduce \\eqref{eq:origin Drinfeld lemma}\n as a corollary, and finally recover the convergent case.\n\\end{secnumber}\n\n\\textbf{Notations}:\n\tLet $k$ be the finite field with $q=p^s$ elements where $s$ a positive integer, and $K$ a finite extension of $\\mathbb{Q}_p$ with residue field $k$. \n\tWe equip $K$ with the lift $\\id$ of the $s$-th Frobenius structure on $k$. \n\tWe fix an algebraic closure $\\overline{\\mathbb{Q}}_p$ of $K$. \n\tIn the following, a \\textit{$k$-scheme} $X$ means a separated scheme of finite type over $k$.\n\tWe denote the $s$-th Frobenius morphism on $X$ by $F_X$. \n\n\\section{Drinfeld's lemma for overconvergent $F$-isocrystals} \\label{sec-overconvergent}\n\\subsection{Generalities on Tannakian categories}\n\tIn the following, we study some general constructions in the Tannakian formalism. \n\tLet $E$ be a field of characteristic $0$.\n\tLet $\\Vect_E$ denote the category of finite-dimensional $E$-vector spaces. \n\tFor any neutral Tannakian category $\\mathscr{C}$ over $E$ with respect to a fiber functor $\\omega$, we use $\\pi_1(\\mathscr{C},\\omega)$ to denote its Tannakian group (i.e., the group of natural isomorphisms of $\\omega$). \n\tWe omit $\\omega$ from the notation if there is no risk of confusion. \n\n\t\\begin{secnumber}\\label{setting}\n\t\tLet $\\widetilde{\\mathscr{C}}$ be a Tannakian category over $E$, neutralized by a fiber functor $\\omega:\\widetilde{\\mathscr{C}}\\to \\Vect_F$, where $F$ is a finite extension of $E$. Suppose there exist $E$-linear tensor equivalences $\\tau_i:\\widetilde{\\mathscr{C}}\\to \\widetilde{\\mathscr{C}}$ and isomorphisms of tensor functors $\\eta_i:\\omega\\circ \\tau_i\\xrightarrow{\\sim} \\omega$ for $i=1,2,\\dots,n$. \n\tMoreover, we assume that there exist natural isomorphisms $\\sigma_{ij}: \\tau_i\\circ \\tau_j \\simeq \\tau_{j}\\circ \\tau_i$ for $i,j$ such that the following diagram commutes:\n\t\\begin{equation} \\label{sigma ij}\n\t\t\\xymatrix{\n\t\t\\omega\\circ \\tau_i\\circ \\tau_j \\ar[r]^{\\eta_i\\circ \\id} \\ar[d]_{\\omega(\\sigma_{ij})} & \\omega\\circ \\tau_j \\ar[r]^{\\eta_j} & \\omega \\\\\n\t\t\\omega \\circ \\tau_j\\circ \\tau_i \\ar[r]^{\\eta_j\\circ \\id} & \\omega\\circ \\tau_i \\ar[ru]_{\\eta_i}\n\t\t}\n\t\\end{equation}\n\tSince $\\omega$ is faithful, such an isomorphism $\\sigma_{ij}$ is unique.\n\tIn the following, for every $i=1,\\dots,n$, we fix a quasi-inverse $\\tau_i^{-1}$ of $\\tau_i$. For $m\\in \\mathbb{Z}$, we set $\\tau_i^m$ to be the $|m|$-th composition of $\\tau_i$ (or $\\tau_i^{-1}$ if $m<0$) and define $\\eta_i^m: \\omega\\circ \\tau_i^m \\xrightarrow{\\sim} \\omega$ by the composition of $\\eta_i$ (or the inverse of $\\eta_i$ if $m<0$). \n\n\tWe define a category $\\mathscr{C}_0$ as follows: an object $(\\mathscr{E},\\varphi_1,\\dots,\\varphi_{n})$ consists of an object $\\mathscr{E}$ of $\\widetilde{\\mathscr{C}}$ and isomorphisms $\\varphi_i:\\tau_i(\\mathscr{E})\\xrightarrow{\\sim} \\mathscr{E}$ such that $\\varphi_j\\circ \\tau_j(\\varphi_i)=\\varphi_i\\circ \\tau_i(\\varphi_j)$ via $\\sigma_{ij}$. Morphisms of $\\mathscr{C}_0$ are morphisms of $\\widetilde{\\mathscr{C}}$ compatible with the $\\varphi_i$'s. \n\tWe have a canonical functor \n\\begin{displaymath}\n\t\\mathscr{C}_0\\to \\widetilde{\\mathscr{C}}, \\quad (\\mathscr{E},\\varphi_i)\\mapsto \\mathscr{E}.\n\\end{displaymath} \n\nBy \\cite[\\S~2.5]{Del90} and a similar argument to \\cite[Proposition 3.4.5]{XZ}, we can show that $\\mathscr{C}_0$ is a Tannakian category over $E$ neutralized by $\\omega$ over $\\Vect_F$. \n\t\n\tWe say an object of $\\mathscr{C}_0$ is \\textit{constant} if its image in $\\widetilde{\\mathscr{C}}$ is isomorphic to a finite direct sum of the unit object.\n\n\tRecall \\cite[\\S~2]{Ber08} that a Tannakian subcategory of $\\widetilde{\\mathscr{C}}$ is a strictly full abelian subcategory, closed under $\\otimes$, duals, subobjects (and thus quotients). \n\tThe constant objects of $\\mathscr{C}_0$ forms a Tannakian subcategory $\\mathscr{C}_{0,\\cst}$ of $\\mathscr{C}_0$ (\\textit{loc.cit}).\n\n\n\tLet $\\mathscr{C}$ be the smallest Tannakian subcategory of $\\widetilde{\\mathscr{C}}$ containing the essential image of $\\mathscr{C}_0$ (i.e. generated by the subquotients of the essential image of $\\mathscr{C}_0$). \n\n\\end{secnumber}\n\n\\begin{secnumber}\n\tIn the following, we assume that $\\widetilde{\\mathscr{C}}$ is neutral over $E$ by $\\omega\\colon \\widetilde{\\mathscr{C}}\\to \\Vect_E$. \n\tThen so are $\\mathscr{C},\\mathscr{C}_0$ and $\\mathscr{C}_{0,\\cst}$. \n\tWe prove the following results by a similar approach of \\cite[Appendix]{DZ20}, where D'Addezio treated the case $n=1$. A similar discussion appeared in \\cite[Appendix 2]{HNY} and \\cite[\\S~3.4]{XZ}. \n\\end{secnumber}\n\n\\begin{prop} \\label{ess of pione} \n\t\\textnormal{(i)} \n\tThe canonical functors $\\mathscr{C}_{0,\\cst}\\to \\mathscr{C}_0\\to \\mathscr{C}$ induce a short exact sequence \n\t\\begin{displaymath}\n\t\t1\\to \\pi_1(\\mathscr{C}) \\to \\pi_1(\\mathscr{C}_0)\\to \\pi_1(\\mathscr{C}_{0,\\cst})\\to 1.\n\t\\end{displaymath}\n\n\t\\textnormal{(ii)} The category $\\mathscr{C}_{0,\\cst}$ is equivalent to the category of representations of $\\mathbb{Z}^n$ over $E$ and the Tannakian group $\\pi_1(\\mathscr{C}_{0,\\cst})$ is isomorphic to the pro-algebraic completion of $\\mathbb{Z}^n$. \n\\end{prop}\n\t\nIn view of \\cite[Proposition 2.21]{DM} and definition of $\\mathscr{C}$, the morphism $\\pi_1(\\mathscr{C}) \\to \\pi_1(\\mathscr{C}_0)$ (resp. $\\pi_1(\\mathscr{C}_0)\\to \\pi_1(\\mathscr{C}_{0,\\cst})$) is a closed immersion (resp. faithfully flat). \n\tWe use the following criterion to prove the exactness.\n\n\t\\begin{theorem}[\\cite{DM} Proposition 2.21, \\cite{EPS} Theorem A.1] \\label{exact criterion}\n\t\tWe consider a sequence of affine group schemes over $E$:\n\t\t\\begin{equation} \\label{eq:exact criterion group sequence}\n\t\tL\\xrightarrow{q} G\\xrightarrow{p} A \n\t\t\\end{equation}\n\t\t and the associated functors:\n\t\t\\begin{equation}\n\t\t\\mathbf{Rep}_E(A)\\xrightarrow{p^*}\\mathbf{Rep}_E(G)\\xrightarrow{q^*} \n\t\t\\mathbf{Rep}_E(L).\n\t\t\\end{equation}\n\n\t\t\\textnormal{(i)} The map $p:G\\to A$ is faithfully flat if and only if $p^*(\\mathbf{Rep}_E(A))$ is a full subcategory of $\\mathbf{Rep}_E(G)$, closed under taking subquotients. \n\n\t\t\\textnormal{(ii)} The map $q:L\\to G$ is a closed immersion if and only if any object of $\\mathbf{Rep}_E(L)$ is a subquotient of an object $q^*(V)$ for some $V\\in \\mathbf{Rep}_E(G)$. \n\n\t\t\\textnormal{(iii)} Assume that $q$ is a closed immersion and $p$ is faithfully flat. Then the sequence \\eqref{eq:exact criterion group sequence} is exact if and only if the following conditions are satisfied.\n\n\t\t(a) For an object $V\\in \\mathbf{Rep}_E(G)$, $q^*(V)$ in $\\mathbf{Rep}_E(L)$ is trivial if and only if $V\\simeq p^*U$ for some object $U$ in $\\mathbf{Rep}_E(A)$. \n\n\t\t(b) Let $W_0$ be the maximal trivial subobject of $q^*(V)$ in $\\mathbf{Rep}_E(L)$. Then there exists a subobject $V_0$ of $V$ in $\\mathbf{Rep}_E(G)$ such that $q^*(V_0)\\simeq W_0$. \n\n\t\t(c) Any object of $\\mathbf{Rep}_E(L)$ is a subobject of an object in the essential image of $q^*$.\n\t\\end{theorem}\n\n\tFor any $E$-algebra $R$ and $1\\le i \\le n$, the above structure defines a homomorphism:\n\t\\begin{equation} \\label{def Z action}\n\t\tu_i:\\mathbb{Z} \\to \\Aut(\\pi_1(\\mathscr{C})(R)),\\quad k \\mapsto (h:\\omega\\to \\omega ~\\mapsto~ \\omega \\xrightarrow{\\eta_i^{-k}} \\omega\\circ \\tau_i^k \\xrightarrow{ h\\circ \\id} \\omega\\circ \\tau_i^k \\xrightarrow{\\eta_i^k} \\omega). \n\t\\end{equation}\n\tIn view of \\eqref{sigma ij}, the images of $u_i,u_j$ commute with each other. \n\tWe thus obtain an action of $\\mathbb{Z}^n$ on $\\pi_1(\\mathscr{C})$ and this allows us to define a group scheme $\\pi_1(\\mathscr{C})\\rtimes \\mathbb{Z}^n$ over $E$, which we denote by $W(\\mathscr{C}_0)$.\n\n\\begin{lemma} \\label{lemma Weil gp}\n\t\\textnormal{(i)} There exist a canonical equivalence of categories $\\mathscr{C}_0\\xrightarrow{\\sim} \\mathbf{Rep}_{E}(W(\\mathscr{C}_0))$ and a canonical morphism of group schemes $\\iota:W(\\mathscr{C}_0)\\to \\pi_1(\\mathscr{C}_0)$ such that the following diagram is $2$-commutative:\n\t\\begin{displaymath}\n\t\t\\xymatrix{\n\t\t& \\mathscr{C}_0 \\ar[ld]^{\\sim} \\ar[rd]^{\\sim} & \\\\\n\t\t\\mathbf{Rep}_E(\\pi_1(\\mathscr{C}_0)) \\ar[rr]^{\\iota^*} & & \\mathbf{Rep}_E(W(\\mathscr{C}_0))\n\t\t}\n\t\\end{displaymath}\n\tMoreover, the image of $\\iota$ is Zariski-dense in $\\pi_1(\\mathscr{C}_0)$. \n\t\n\t\\textnormal{(ii)} The subgroup $\\pi_1(\\mathscr{C})$ of $\\pi_1(\\mathscr{C}_0)$ is normal. In particular, every object of $\\mathscr{C}$ is a subobject of an object in the essential image of $\\mathscr{C}_0$. \n\\end{lemma}\n\n\\begin{proof}\n\t(i) We construct a functor\n\t\\begin{displaymath}\n\t\t\\mathscr{C}_0\\to \\mathbf{Rep}_E(W(\\mathscr{C}_0)).\n\t\\end{displaymath}\n\tGiven an object $(\\mathscr{E},\\varphi_i)$ of $\\mathscr{C}_0$, we construct a representation $\\rho$ of $W(\\mathscr{C}_0)(E)$ on $\\omega(\\mathscr{E})$. \n\tFor any element $(g,m_1,\\cdots,m_n)\\in \\pi_1(\\mathscr{C})(E)\\rtimes \\mathbb{Z}^n$, we define $\\rho(g,m_i)$ as the composition:\n\t\\begin{displaymath}\n\t\t\\omega (\\mathscr{E})\\xrightarrow{\\prod \\eta_{i}^{-m_i}} \n\t\t\\omega(\\prod\\tau_{i}^{m_i}(\\mathscr{E})) \\xrightarrow{g} \n\t\t\\omega(\\prod\\tau_{i}^{m_i}(\\mathscr{E}))\\xrightarrow{\\prod \\eta_{i}^{m_i}} \n\t\t\\omega(\\mathscr{E}).\n\t\\end{displaymath}\n\tIn view the definition of $u_i$ \\eqref{def Z action}, one checks that the above formula defines a representation. Then we obtain the above functor and we can check that it is an equivalence. \n\n\tBy the Tannakian reconstruction theorem \\cite[Theorem 2.11]{DM},\n\t$\\pi_1(\\mathscr{C}_0)$ is the pro-algebraic completion of $W(\\mathscr{C}_0)$ and the image of $\\iota$ is therefore Zariski-dense. \n\n\t(ii) The second assertion follows from the first one by condition \\ref{exact criterion}(iii.c). \n\tLet $H$ denote the normalizer of $\\pi_1(\\mathscr{C})$ in $\\pi_1(\\mathscr{C}_0)$. \n\tAs $\\pi_1(\\mathscr{C})$ is normal in $W(\\mathscr{C}_0)$, the image of $\\iota:W(\\mathscr{C}_0)\\to \\pi_1(\\mathscr{C}_0)$ is contained in $H$. \n\tAs $\\iota$ has Zariski-dense image by (i), this implies that $H=\\pi_1(\\mathscr{C}_0)$ and the assertion follows. \n\\end{proof}\n\n\\textit{Proof of Proposition \\ref{ess of pione}.}\n(i) We know the exactness at the left and the exactness on the right follows from \\cite[Proposition 2.21]{DM}. We use Theorem \\ref{exact criterion} to verify the exactness. Condition (a) follows from the definition. \nGiven an object $(V,\\varphi_1,\\dots,\\varphi_n)$ of $\\mathscr{C}_0$, the maximal trivial subobject $W_0$ of $V$ in $\\mathscr{C}$ is preserved by the $\\varphi_i$'s. Therefore condition (b) is verified. Condition (c) is proved in Lemma \\ref{lemma Weil gp}(ii).\n\n(ii) The assertion follows from Lemma \\ref{lemma Weil gp} applied to the category $\\Vect_E$.\n\\hfill \\qed\n\n\n\n\n\\subsection{Tannakian form of Drinfeld's lemma for overconvergent $F$-isocrystals}\n\n\\begin{secnumber} \\label{sss:setting}\n\tLet $X$ be a $k$-scheme. \n\tWe denote by $\\Isocd(X\/K)$ (resp. $\\Isoc(X\/K)$) the category of overconvergent (resp. convergent) isocrystals over $X$ relative to $K$. \n\tIf we apply the construction of \\ref{setting} to the Tannakian category $\\widetilde{\\mathscr{C}}=\\Isocd(X\/K)$ (resp. $\\Isoc(X\/K)$) over $k$ and the ($s$-th) Frobenius pull-back functor $\\tau=F_X^*$, then $\\mathscr{C}_0$ corresponds to the category $\\FIsocd(X\/K)$ (resp. $\\FIsoc(X\/K)$) of overconvergent (resp. convergent) $F$-isocrystals over $X\/K$.\n\tWe set $\\mathbf{Isoc}^{\\dagger\\dagger}(X\/K)=\\mathscr{C}$, the full Tannakian subcategory of $\\Isocd(X\/K)$ generated by subquotients of overconvergent isocrystals which admit a ($s$-th) Frobenius structure, considered in \\cite{abe-companion}. \n\\end{secnumber}\n\n\\begin{secnumber}\n\tThe above construction can be generalized as follows. \n\tFor $i=1,\\ldots,n$, let $X_i$ be a $k$-scheme and set $X=X_1\\times_k \\cdots \\times_k X_n$. \n\tWe define a morphism $F_i:X\\to X$ by the ($s$-th) Frobenius morphism $F_{X_i}$ of $X_i$ on the component $X_i$ and the identity map on other components.\n\tThe morphisms $F_i$ commute with each other and their composition is equal to $F_X$. \n\tThe morphisms $F_i$ induce tensor equivalences \n\t\\[\n\tF_i^*:\\Isocd(X\/K)\\to \\Isocd(X\/K)\\quad \\textnormal{(resp. } \\Isoc(X\/K)\\to \\Isoc(X\/K)),\\quad i=1,\\ldots,n,\n\t\\]\n\tcommuting with each other and their composition is equal to $F^*_X$. \n\n\tGiven an overconvergent (resp. convergent) isocrystal $\\mathscr{E}$ over $X\/K$, \\textit{a partial ($s$-th) Frobenius structure on $\\mathscr{E}$} consists of isomorphisms $\\phi_i:F_i^{*}(\\mathscr{E})\\xrightarrow{\\sim} \\mathscr{E}$ for $i=1,\\dots, n$ such that $\\phi_i\\circ F_i^{*}(\\phi_j)=\\phi_j\\circ F_j^{*}(\\phi_i)$. \n\tFor $i=1,\\dots,n$, $\\phi_i$ is called the \\textit{$i$-th partial Frobenius structure} (on $\\mathscr{E}$). \n\tNote that the composition of the $\\phi_i$ (in any order) forms a Frobenius structure on $\\mathscr{E}$. \n\n\tIf we apply the construction of \\ref{setting} to $(\\widetilde{\\mathscr{C}}=\\Isocd(X\/K)$ (resp. $\\Isoc(X\/K)$); $F_1^{*},\\ldots,F_n^{*})$, then $\\mathscr{C}_0$ corresponds to the category of overconvergent (convergent) isocrystals over $X$ with a partial Frobenius structure, which we denote by $\\PhiIsocd(X\/K)$ (resp. $\\PhiIsoc(X\/K)$). \n\tWe set $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/K)=\\mathscr{C}$, the full subcategory of $\\Isocd(X\/K)$, generated by subquotients of overconvergent isocrystals which admits a $s$-th partial Frobenius structure.\n\n\tWe denote by $F_i\\textnormal{-}\\Isocd(X\/K)$ (resp. $F_i\\textnormal{-}\\Isoc(X\/K)$) the category of pairs $(\\mathscr{E},\\varphi_i)$ consisting of an overconvergent (resp. convergent) isocrystal with an $i$-th partial Frobenius structure. \n\n\tLet $k'\/k$ be a finite extension of degree $a$, $K'=\\rW(k')\\otimes_{\\rW(k)}K$. \n (We denote the Witt vector functor by $\\rW$ to reduce confusion with the notation $W(\\mathscr{C}_0)$ from Lemma \\ref{lemma Weil gp}).\n\tSet $X_{k'}=X\\otimes_k k'$. Then we have a canonical functor of extension of scalars:\n\t\\begin{equation}\\label{eq:ext sca}\n\t\\PhiIsoc(X\/K)\\to \\PhiIsoc(X_{k'}\/K'),\\quad (\\mathscr{E},\\varphi_i)\\mapsto (\\mathscr{E}\\otimes_K K',\\varphi_i^{a}\\otimes_K K').\n\\end{equation}\n\\end{secnumber}\n\n\\begin{secnumber} \\label{sss:Tannakian gps}\n\tThe category $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/K)$ defined above is a Tannakian category over $K$ and may not be neutral. \n\tFor any algebraic extension $L$ of $K$ in $\\overline{\\mathbb{Q}}_p$, we set $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/L)=\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/K)\\otimes_K L$. \n\tWhen $L=\\overline{\\mathbb{Q}}_p$, this category is a neutral Tannakian category over $\\overline{\\mathbb{Q}}_p$ (with respect to a fiber functor) and we omit $\/\\overline{\\mathbb{Q}}_p$ from the notation. \n\tWe denote by $\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)$ (resp. $\\pi_1^{\\PhiIsocd}(X)$, resp. $\\pi_1^{\\PhiIsoc}(X)$) the Tannakian group of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ (resp. $\\PhiIsocd(X)$, $\\PhiIsoc(X)$) over $\\overline{\\mathbb{Q}}_p$.\n\tWhen $n=1$ (i.e. partial Frobenius structures reduce to a Frobenius structure), we omit $\\Phi$ from the notation or replace it with $F$.\n\t\n\tThe pullback functor $p_{i}^*$ induces a canonical tensor functor\n\t\\begin{eqnarray} \\label{functor p_1*} \\\\\n\t\\nonumber\n\t\tp_i^{*}: \\FIsocd(X_i) &\\to& \\PhiIsocd(X) \\quad \n\t\t\\textnormal{(resp. } \\mathbf{Isoc}^{\\dagger\\dagger}(X_i)\\to \\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi), \\textnormal{resp.} \\FIsoc(X_i)\\to \\PhiIsoc(X)) \\\\\n\t\t\\quad (\\mathscr{E}_i,\\varphi_i) &\\mapsto& (p_i^{*}\\mathscr{E}_i, \\id,\\ldots,p_i^{*}(\\varphi_i),\\ldots,\\id). \\nonumber \n\t\\end{eqnarray}\n\tBy the K\u00fcnneth formula, the functor $p_i^*:\\mathbf{Isoc}^{\\dagger\\dagger}(X_i)\\to \\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ is fully faithful.\n\tIt induces a canonical $\\overline{\\mathbb{Q}}_p$-homomorphism\n\t\\begin{displaymath}\n\t\tp_i^{\\circ}: \\pi_1^{\\PhiIsocd}(X) \\to \\pi_1^{\\FIsocd}(X_i) \\quad \\textnormal{(resp. } \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)\\to \\pi_{1}^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_i) , ~ \\textnormal{resp. } \\pi_1^{\\PhiIsoc}(X) \\to \\pi_1^{\\FIsoc}(X_i)).\n\t\\end{displaymath}\n\\end{secnumber}\n\nThe Tannakian form of Drinfeld's lemma can be summarized as follows. Its proof will occupy most of the remainder of \\S\\ref{sss:Tannakian gps}.\n\n\\begin{theorem} \\label{t:main thm}\n\tAssume each $X_i$ is a geometrically connected $k$-scheme. \n\tThe following canonical homomorphisms are isomorphisms:\n\t\\begin{equation} \\label{Tannakian Drinfeld lemma}\n\t\t\\prod_{i=1}^n p_i^{\\circ}:\n\t\t\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)\\xrightarrow{\\sim} \\prod_{i=1}^n \\pi_{1}^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_i), \\quad \n\t\t\\prod_{i=1}^n p_i^{\\circ}:\n\t\t\\pi_1^{\\PhiIsocd}(X) \\xrightarrow{\\sim} \\prod_{i=1}^n \\pi_1^{\\FIsocd}(X_i) . \n\t\\end{equation}\n\\end{theorem}\n\nWe first establish some basic properties of the category $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$. \n\n\t\\begin{prop}\n\t\t\\label{embedding of partial Frobenius}\t\n\t\t\\textnormal{(i)} An irreducible object of $\\Isocd(X)$ belongs to $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ if and only if it is equipped with a $s'$-th partial Frobenius structure for some $s|s'$. \n\n\t\t\\textnormal{(ii) [D'Addezio--Esnault \\cite{DE20}]}\n\tThe category $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ is closed under extension. Every object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ can be embedded into an object of $\\PhiIsocd(X)$. \n\n\t\t\\textnormal{(iii)} The category $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ is equivalent to the thick full subcategory of $\\Isocd(X)$ generated by those objects which can be equipped with a $s'$-th partial Frobenius structure for some $s|s'$. \n\t\\end{prop}\n\t\\begin{proof}\n\t(i) Given an object $\\mathscr{E}$ of $\\PhiIsocd(X)$, the partial Frobenius pullbacks permute the isomorphism classes of the irreducible constituents of $\\mathscr{E}$ in $\\mathbf{Isoc}^{\\dagger\\dagger}(X)$. We thus conclude that an irreducible object $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ admits $s'$-th partial Frobenius structure for some $s|s'$. \n\n\tOn the other hand, if an object $\\mathscr{E}$ of $\\Isocd(X)$ is irreducible equipped with an $s'(=st)$-th partial Frobenius structure for some $t\\in \\mathbb{N}$,\n\tthen $\\mathscr{E}$ is a subobject of \\[\n\t\\mathscr{F}=\\bigoplus_{j_1=0}^{t-1} \\cdots \\bigoplus_{j_n=0}^{t-1} F_1^{j_1*}\\circ\\cdots\\circ F_{n}^{j_n*}(\\mathscr{E}),\n\t\\]\n\t which admits an $s$-th Frobenius structure. \n\tTherefore $\\mathscr{E}$ belongs to $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$. \n\n\t(ii) When $n=1$, assertion (ii) is proved in \\cite[Theorem 5.4]{DE20}. \n\tThe general case can be showed in a similar way as in \\textit{loc.cit.}\n\n\t(iii) Assertion (iii) follows from assertions (i--ii). \n\t\\end{proof}\n\n\t\\begin{secnumber}\n\tWe will use the theory of holonomic (arithmetic) $\\mathscr{D}$-modules and their six functors formalism developed in \\cite{abe-companion}. \n\tLet $L$ be an extension of $K$ in $\\overline{\\mathbb{Q}}_p$, $\\mathfrak{T}=\\{k,\\mathscr{O}_K,K,L\\}$ the associated geometric base tuple and $\\mathfrak{T}_F=\\{k,\\mathscr{O}_K,K,L,s,\\id_L\\}$ the associated arithmetic base tuple \\cite[1.4.10, 2.4.14]{abe-companion}. \n\n\tLet $X$ be a $k$-scheme. There exists an $L$-linear triangulated category $\\rD(X\/L)$ (resp. $\\rD(X\/L_F)$) relative to the geometric base tuple $\\mathfrak{T}$ (resp. arithmetic base tuple $\\mathfrak{T}_F$). \n\tThis category is denoted by $\\rD^{\\rb}_{\\hol}(X\/\\mathfrak{T})$ or $\\rD^{\\rb}_{\\hol}(X\/L)$ (resp. $\\rD^{\\rb}_{\\hol}(X\/\\mathfrak{T}_F)$ or $\\rD^{\\rb}_{\\hol}(X\/L_F)$) in \\cite[1.1.1, 2.1.16]{abe-companion}. \n\tFor $\\blacktriangle\\in \\{\\emptyset,F\\}$, there exists a \\textit{holonomic t-structure} on $\\rD(X\/L_{\\blacktriangle})$, whose heart is denoted by $\\Hol(X\/L_{\\blacktriangle})$, called the \\textit{category of holonomic modules}.\n\tWe denote by $\\mathcal{H}^*$ the cohomological functor for holonomic t-structure. \n\t\n\tThe six functors formalism for $\\rD(X\/L)$ (resp. $\\rD(X\/L_F)$) has been established recently. We refer to \\cite[\\S~2.3]{abe-companion} for details and to \\cite[1.1.3]{abe-companion} for a summary.\n\t\\end{secnumber}\n\n\\begin{secnumber} \\label{rem thick}\n\tLet $X$ be a smooth and quasiprojective geometrically connected $k$-scheme of dimension $d_X$. \t\n\tThe category $\\mathbf{Isoc}^{\\dagger\\dagger}(X)$ (resp. $\\FIsocd(X)$) is equivalent to the full subcategory $\\Sm(X\/\\overline{\\mathbb{Q}}_p)$ (resp. $\\Sm(X\/\\overline{\\mathbb{Q}}_{p,F})$) consisting of smooth objects of the category $\\Hol(X\/\\overline{\\mathbb{Q}}_p)[-d_X]$ (resp. $\\Hol(X\/\\overline{\\mathbb{Q}}_{p,F})[-d_X]$) of holonomic arithmetic $\\mathscr{D}$-modules shifted by the dimension $-d_X$ \\cite[1.1.3 (12)]{abe-companion}. \n\n\tWe briefly review the pushforward and pullback functors for a smooth morphism $f:X\\to Y$ of relative dimension $d$ between quasiprojective geometrically connected $k$-schemes following \\cite[1.2.8]{abe-companion}. \n\tWe have an adjoint pair $(f_+,f^+[2d])$\n\t\\[\n\tf_+:\\rD(X\/L_{\\blacktriangle}) \\rightleftarrows \\rD(Y\/L_{\\blacktriangle}): f^+[2d]. \n\t\\]\n\tThe functor $f^+[d]$ is exact and induces a fully faithful functor \\cite[Proposition 2.1.6]{XZ}:\n\t\\begin{equation} \\label{eq:fullyfaithful}\n\tf^+[d]:\\Hol(Y\/L_{\\blacktriangle})\\to \\Hol(X\/L_{\\blacktriangle}).\n\\end{equation}\n\tIf $d_X$ (resp. $d_Y$) denotes the dimension of $X$ (resp. $Y$), the functor \n\t\\begin{equation} \\label{eq:pi pushforward Hol}\nf_{*}:=\\mathcal{H}^{d}(f_{+}(-))[-d]:\\Hol(X\/L_{\\blacktriangle})[-d_X]\\to \\Hol(Y\/L_{\\blacktriangle})[-d_Y]\n\\end{equation}\nis a right adjoint of $f^*:=f^+$ and is left exact. \nIts right derived functor $\\rR f_*$ is compatible with $f_+$. \n\nMoreover, there exists a canonical isomorphism (Poincar\\'e duality \\cite[1.5.13]{abe-companion}): \n\t\\[\n\tf^+(d)[2d]\\xrightarrow{\\sim} f^!.\n\t\\]\n\\end{secnumber}\n\n\\begin{prop} \\label{P:pushforward}\n\tAssume each $X_i$ is smooth and quasiprojective over $k$.\n\n\t\t\\textnormal{(i)} The functor $p_{i,*}$ sends objects of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ to $\\mathbf{Isoc}^{\\dagger\\dagger}(X_i)$ (resp. $\\PhiIsocd(X)$ to $\\FIsocd(X_i)$). \n\n\t\t\\textnormal{(ii)} The functor $p_{i,*}:\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)\\to \\mathbf{Isoc}^{\\dagger\\dagger}(X_i)$ is a right adjoint of $p_i^*$ \\eqref{functor p_1*}.\n\t\tThe adjoint morphism $\\id\\to p_{i,*}p_i^*$ is an isomorphism and $p_i^*p_{i,*}\\to \\id$ is injective. \n\t\\end{prop}\n\n\n\\begin{secnumber}\n\tLet $Y=Y_1\\times_k Y_2$ be a product of two geometrically connected $k$-schemes, $k'$ a perfect field over $k$, $K'=\\rW(k')\\otimes_{\\rW(k)}K$.\n\tWe denote by $F_1\\textnormal{-}\\Isocd(Y_{1,k'}\/K')$ (resp. $F_1\\textnormal{-}\\Isoc(Y_{1,k'}\/K')$) the category of pairs $(\\mathscr{E},\\phi)$ consisting of an object $\\mathscr{E}$ of $\\Isocd(Y_{1,k'}\/K')$ (resp. $\\Isoc(Y_{1,k'}\/K')$) and an isomorphism $\\phi:(F_{Y_1}\\otimes \\id_{k'})^*(\\mathscr{E})\\xrightarrow{\\sim} \\mathscr{E}$. \n\t\n\tA point $i:\\Spec(k')\\to Y_2$ induces a natural functor: \n\t\\begin{eqnarray} \\label{eq:partial Frob to FIsocd}\n\t\\quad\\quad\t\\iota:F_1\\textnormal{-}\\Isocd(Y\/K) &\\to& F_1\\textnormal{-}\\Isocd(Y_{1,k'}\/K'), \\quad \n\t\\textnormal{(resp. } \n\tF_1\\textnormal{-}\\Isoc(Y\/K)\\to F_1\\textnormal{-}\\Isoc(Y_{1,k'}\/K') ),\\\\\n\t(\\mathscr{E},\\varphi)&\\mapsto& ((\\id_{Y_1}\\times i)^*(\\mathscr{E}),(\\id_{Y_1}\\times i)^*\\varphi). \\nonumber \n\\end{eqnarray}\nWhen $k'$ is a finite extension of $k$ of degree $a$, we associate an $F$-isocrystal over $Y_{1,k'}\/K'$ to an object of $F_1\\textnormal{-}\\Isocd(Y_{1,k'}\/K')$ by composing its Frobenius structure $a$ times. \n\t\n\tWhen $Y_2=\\Spec(k)$, we regard the above functor as the functor of extension of scalars:\n\t\\[\n\t\\iota_{k\/k'}:\\FIsocd(Y\/K) \\to F_1\\textnormal{-}\\Isocd(Y_{1,k'}\/K').\n\t\\]\n\\end{secnumber}\n\n\\begin{secnumber}\n\t\\textit{Proof of Proposition \\ref{P:pushforward}}. \n\t(i)\n\tIt suffices to prove the assertion for objects of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/K)$. We may assume $i=1$. We set $X'=\\prod_{i=2}^n X_i$. \n\t\n\t(a) We first prove the assertion for an object $\\mathscr{E}$ of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi\/K)$ equipped with a partial Frobenius structure $(\\varphi_1,\\ldots,\\varphi_n)$. \n\tThere exists an open subscheme $U_1$ of $X_1$ such that $p_{1,*}(\\mathscr{E})|_{U_1}$ is smooth. In view of their fibers, the adjoint morphism of overconvergent isocrystals:\n\t\\[\n\t\tp_1^*(p_{1,*}(\\mathscr{E})|_{U_1})\\to \\mathscr{E}|_{U_1\\times X'}\n\t\\]\n\t\tis injective. \n\t\tThe first partial Frobenius $\\varphi_1$ on $\\mathscr{E}$ induces a Frobenius structure $\\phi_1$ on $p_{1,*}(\\mathscr{E})|_{U_1}$ \t\tand the above morphism is compatible with the first partial Frobenius structures $p_1^*(\\phi_1)$ and $\\varphi_1$. \n\t\tLet $k'$ be a separable closure of $k(X')$ and $K':=W(k')[\\frac{1}{p}]$. By \\eqref{eq:partial Frob to FIsocd}, we obtain an injection in $F_1\\textnormal{-}(U_{1,k'}\/K')$:\n\t\t\\[\n\t\t\\iota_{k'\/k}(p_{1,*}(\\mathscr{E})|_{U_1}) \\hookrightarrow \\iota(\\mathscr{E}|_{U_{1}\\times X'}).\n\t\t\\]\n\t\tBy \\cite[Proposition 5.3.1]{Ked07}, the left hand side extends to a subobject $\\mathscr{F}'$ of $\\iota(\\mathscr{E})$ of $\\Isocd(X_{1,k'}\/K')$. \n\n\t\tWe claim that there exists an overconvergent isocrystal $\\mathscr{F}$ of $\\Isocd(X_1\/K)$ extending $p_{1,*}(\\mathscr{E})|_{U_1}$ such that $\\iota_{k'\/k}(\\mathscr{F})\\simeq \\mathscr{F}'$. \n\tBy Kedlaya--Shiho's purity theorem \\cite[Theorem 5.1]{kedlaya-isocrystals}, we may assume that the boundary $D=X_1-U_1$ is a smooth divisor. \n\tSince the local monodromy of $\\mathscr{F}'$ around $D_{k'}$ is constant \\cite[Theorem 5.2.1]{Ked07}, then so is the local monodromy of $p_{1,*}(\\mathscr{E})|_{U_1}$ around $D$. Then the claim follows. \n\t\t\n\tMoreover, the Frobenius structure on $p_{1,*}(\\mathscr{E})|_{U_1}$ extends to $\\mathscr{F}$. \n\tFinally, by full faithfullness of pullback for holonomic $\\mathscr{D}$-modules along $U_1\\to X_1$ \\cite[Proposition 2.1.6]{XZ}, we deduce that $\\mathscr{F}$ is isomorphic to $p_{1,*}(\\mathscr{E})$.\n\tThen the assertion in this case follows. \n\t\n\t\t(b) An object $\\mathscr{E}$ of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ can be embedded into an object $\\mathscr{F}$, which can be equipped with a partial Frobenius structure (Proposition \\ref{embedding of partial Frobenius}(ii)). \n\t\tAs $p_{1,*}$ is left exact, we have an injection to a smooth module $\\mathscr{N}$ of $\\FIsocd(X_1)$: \n\t\t\\begin{displaymath}\n\t\t\t\\mathscr{M}:=p_{1,*}(\\mathscr{E})\\hookrightarrow \\mathscr{N}:=p_{1,*}(\\mathscr{F}).\n\t\t\\end{displaymath}\n\n\t\tIt remains to show that $\\mathscr{M}$ is also smooth. \n\t\tWe first consider the case where $X_1$ is a curve. \n\t\tLet $U_1$ be an open subset of $X_1$ on which $\\mathscr{M}$ is smooth.\n\t\tBy applying \\cite[Corollary 2.3.4]{XZ} to $\\mathscr{M}|_{U_1}$, the exactness of pullback on smooth modules and of nearby cycle functor, we deduce the smoothness of $\\mathscr{M}$ from that of $\\mathscr{N}$. \n\n\n\tIn general, there is a dense open subscheme $j:U_1\\to X_1$ and a smooth object $\\mathscr{L}$ on $U_1$ such that $\\mathscr{M}|_{U_1}\\simeq \\mathscr{L}$. \nLet $c:C\\to X_1$ be a morphism from a smooth curve to $X$ such that $c(C)\\cap U_1$ is non-empty and $p_c:C\\times X'\\to C$ the projection. \nBy the above argument, $p_{c,*}( (c\\times\\id)^*\\mathscr{E})$ is smooth and the pullback $c^*\\mathscr{L}$ extends to an overconvergent isocrystal on $C$. \nBy Shiho's cut-by-curves theorem \\cite{Shiho11} and Kedlaya--Shiho's purity theorem, $\\mathscr{L}$ can be extended to an overconvergent isocrystal over $X$. \nBy full faithfullness of pullback along $U_1\\to X_1$, we deduce that $\\mathscr{M}$ is isomorphic to the extension of $\\mathscr{L}$ to $X_1$. This finishes the proof. \n\n\t(ii) The isomorphism $\\id\\to p_{i,*}p_i^*$ follows from the K\u00fcnneth formula \\cite[Proposition 1.1.7]{abe-companion}. \n\tIn view of fibers at closed points, the injectivity of $p_i^*p_{i,*}\\to \\id$ follows. \\hfill \\qed\n\\end{secnumber}\n\n\n\\begin{secnumber} \n\\label{pf main thm}\nWe now turn to the proof of Theorem \\ref{t:main thm}. \nWe first prove Theorem \\ref{t:main thm} for $\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}$ under the additional hypothesis that each $X_i$ is \\textit{smooth and quasiprojective} over $k$. \n\n\\textit{Proof.} As we work with arithmetic $\\mathscr{D}$-modules with coefficients in $\\overline{\\mathbb{Q}}_p$, we may enlarge the base field $k$ as in \\cite[1.4.11]{abe-companion}. \n\t\tTherefore we may assume there exists a $k$-point $x$ of $X_1$. \n\t\tWe consider the following diagram\n\t\\begin{displaymath}\n\t\t\\xymatrix{\n\t\t\tX'\\ar[r]^-{u=(x,\\id)} \\ar[d]_{g} & X=X_1\\times X' \\ar[d]_{p_1} \\\\\n\t\t\t\\Spec(k)\\ar[r]^-{x} & X_1\n\t\t}\n\t\\end{displaymath}\t\n\n\tWe denote by $\\PhipIsocd(X')$ the category of overconvergent isocrystals over $X'$ with a partial Frobenius structure and $\\mathbf{Isoc}^{\\dagger\\dagger}(X',\\Phi')$ the Tannakian subcategory of $\\Isocd(X')$ generated by those can be equipped with some partial Frobenius structure. \n\tThe morphism $u$ induces a tensor functor \n\t\\begin{equation} \\label{functor u*}\n\t\tu^{*}: \\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)\\to \\mathbf{Isoc}^{\\dagger\\dagger}(X',\\Phi')\\quad \\textnormal{(resp. } \\PhiIsocd(X) \\to \\PhipIsocd(X')) \\quad \\mathscr{E}\\mapsto u^{*}(\\mathscr{E})\n\t\\end{equation}\n\tand a homomorphism \n\t$u^{\\circ}: \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X',\\Phi')\\to \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)$.\n\tConsider the following commutative diagram:\n\t\\begin{equation} \\label{eq:compare tannakian exact sequences}\n\t\t\\xymatrix{\n\t\t1\\ar[r] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X',\\Phi') \\ar[r]^{u^{\\circ}} \\ar[d]_{\\prod_{i=2}^n p_i^{\\circ}} & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi) \\ar[r]^{p_1^{\\circ}} \\ar[d]_{\\prod_{i=1}^n p_i^{\\circ}} & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_1) \\ar[r] \\ar@{=}[d] & 1 \\\\\n\t\t1\\ar[r] & \\prod_{i=2}^n \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_i)\\ar[r] & \\prod_{i=1}^n \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_i) \\ar[r] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_1) \\ar[r] & 1\n\t\t}\n\t\\end{equation}\n\tHere the upper sequence is defined as above and the lower sequence is exact. \n\tBy induction, it suffices to show the exactness of the upper sequence. \n\tIn the following, we will do it by checking the conditions of Theorem \\ref{exact criterion}\n\tfor the functors:\n\t\\begin{displaymath}\n\t\t\\mathbf{Isoc}^{\\dagger\\dagger}(X_1)\\xrightarrow{p_1^{*}} \\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi) \\xrightarrow{u^{*}} \\mathbf{Isoc}^{\\dagger\\dagger}(X',\\Phi').\n\t\\end{displaymath}\n\twhere $p_1^{*}$ is defined as in \\eqref{functor p_1*}.\n\tWe use the functor $p_{i,*}$ defined in Proposition~\\ref{P:pushforward}(i).\n\n\t(i)\n\tLet $\\mathscr{E}$ be an object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X_1)$ and $\\mathscr{F}$ a subobject of $p_1^*(\\mathscr{E})$. We need to show that $\\mathscr{F}$ is the image under $p_1^*$ of a subobject of $\\mathscr{E}$. If $\\mathscr{G}$ denotes the quotient $p_1^*(\\mathscr{E})\/\\mathscr{F}$, by applying $p_1^*p_{1,*}$ we obtain a commutative diagram:\n\t\\begin{displaymath}\n\t\t\\xymatrix{\n\t\t0\\ar[r] & p_1^*p_{1,*}\\mathscr{F}\\ar[r] \\ar[d]_{\\alpha}& p_1^*\\mathscr{E} \\ar[r]^{\\delta} \\ar[d]_{\\beta} & p_1^*p_{1,*}\\mathscr{G} \\ar[d]_{\\gamma} &\\\\\n\t\t0\\ar[r] & \\mathscr{F}\\ar[r] & p_1^*\\mathscr{E} \\ar[r] & \\mathscr{G}\\ar[r] &0 \n\t\t}\n\t\\end{displaymath}\n\tHere $\\beta$ is the identity, while $\\alpha,\\gamma$ are injective by Proposition~\\ref{P:pushforward}(ii). Then $\\delta$ is surjective. We deduce that $\\alpha$ is an isomorphism. \n\tThen we conclude that $\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi) \\to \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_1)$ is faithfully flat by \\cite[Proposition 2.21]{DM}. \n\n\t(ii) and (iii.c) Since $u$ is a section of $p':X=X_1\\times X'\\to X'$, we deduce condition (iii.c) of Theorem \\ref{exact criterion} for $u^{\\circ}:\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X',\\Phi')\\to \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)$. Then $u^{\\circ}$ is a closed immersion by \\cite[Proposition 2.21]{DM}. \n\n\t(iii.b) \n\tGiven an object $\\mathscr{E}$ of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$, the maximal trivial subobject of $u^{*}\\mathscr{E}$ in the category $\\mathbf{Isoc}^{\\dagger\\dagger}(X',\\Phi')$ is $g^{*}g_{*}u^{*}\\mathscr{E}$. \n\tSince $u^+(\\mathscr{E})[2d]\\simeq u^!(\\mathscr{E})$, we have $g^{*}g_{*}u^{*}\\mathscr{E}\\simeq u^* p_1^*p_{1,*}\\mathscr{E}$ by smooth base change. \n\tThen condition (b) follows from the fact that $p_1^*p_{1,*}\\to \\id$ is injective by Proposition~\\ref{P:pushforward}(ii).\n\n\t(iii.a) If an object $\\mathscr{E}$ of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ comes from the essential image of $p_1^{*}$, then $u^*(\\mathscr{E})$ is trivial. \n\tConversely, in view of (iii.b), if $u^*(\\mathscr{E})$ is trivial, then $\\mathscr{E}\\simeq p_1^*p_{1,*}(\\mathscr{E})$. This verifies condition (a). \\hfill\\qed\n\\end{secnumber}\n\nWe now upgrade the previous argument to eliminate the smooth quasiprojective hypothesis.\n\n\\begin{lemma} \\label{lem:descent}\nThe categories $\\Isocd(X), \\PhiIsocd(X), \\mathbf{Isoc}^{\\dagger\\dagger}(X, \\Phi)$ admit descent with respect to any proper hypercoverings of the $X_i$.\n\\end{lemma}\n\\begin{proof}\nDescent for $\\Isocd(X)$ is established in \\cite[Theorem~5.1]{lazda}.\nThis then implies descent for $\\PhiIsocd(X)$; we deduce descent for $\\mathbf{Isoc}^{\\dagger\\dagger}(X, \\Phi)$ using Proposition~\\ref{embedding of partial Frobenius}.\n\\end{proof}\n\n\\begin{secnumber}\n\\textit{Proof of Theorem \\ref{t:main thm} for $\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}$ with general $X_i$}. \t\nBy de Jong's alterations theorem \\cite{dejong-alterations}, there exist smooth and quasiprojective $k$-schemes $Y_i$ and proper surjective, generic \\'etale maps $\\pi_i:Y_i\\to X_i$ for $i=1,2,\\ldots,n$.\nWe use the hypercovering produced by the $\\pi_i$'s to show that the \nupper sequence in \\eqref{eq:compare tannakian exact sequences} is exact, by again checking the conditions of Theorem \\ref{exact criterion}.\n\nWe set $Y=\\prod Y_i$ and $\\pi:Y\\to X$ the projection. \nBy induction, it suffices to treat the case where $Y_i=X_i$ for $i=2,\\cdots,n$. Consider the following diagram:\n\\[\n\\xymatrix{\n\t1\\ar[r] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X') \\ar[r]^{u^{\\circ}} \\ar@{=}[d] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi) \\ar[r]^{p_1^{\\circ}} \\ar[d]_{\\pi^{\\circ}} & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_1) \\ar[r] \\ar[d]_{\\pi_1^{\\circ}} & 1 \\\\\n\t1\\ar[r] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X') \\ar[r]^{u^{\\circ}} & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(Y,\\Phi) \\ar[r]^{p_1^{\\circ}} & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(Y_1) \\ar[r] & 1 \n}\n\\]\nBy induction hypotheses, we may assume the exactness of the second line. \nWe will deduce the exactness of the first line from the second one. \nThe above diagram corresponds to the functor:\n\\[\n\\xymatrix{\n\\mathbf{Isoc}^{\\dagger\\dagger}(X_1) \\ar[r]^{p_1^*} \\ar[d]_{\\pi_1^*} & \\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi) \\ar[r]^{u^*} \\ar[d]_{\\pi^*} & \\mathbf{Isoc}^{\\dagger\\dagger}(X') \\ar@{=}[d] \\\\ \n\t\\mathbf{Isoc}^{\\dagger\\dagger}(Y_1) \\ar[r]^{p_1^*} \\ar[d] & \\mathbf{Isoc}^{\\dagger\\dagger}(Y,\\Phi) \\ar[r]^{u^*} \\ar[d] & \\mathbf{Isoc}^{\\dagger\\dagger}(X') \\\\\n\t\\mathbf{Isoc}^{\\dagger\\dagger}(Y_1^{(2)}) \\ar[r] & \\mathbf{Isoc}^{\\dagger\\dagger}(Y^{(2)}, \\Phi) & \n}\n\\]\nHere $Y_1^{(2)}=Y_1\\times_{X_1}Y_1$ and $Y^{(2)}=Y\\times_X Y$. \nWe check the conditions of \\ref{exact criterion} for the first line. \n\n(i) \nLet $U$ be an object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X_1)$ and $V_0$ a subobject of $p_1^*(U)$. \nBy exactness of the second line, we deduce that there exists a subobject $U_{0,Y}$ of $\\pi_1^*(U)$ over $Y_1$ such that $p_1^*(U_{0,Y})\\simeq \\pi^*(V_0)$ as \nsubobjects of $\\pi^*(p_1^*(U))$.\nAs the functor $p_1^*:\\mathbf{Isoc}^{\\dagger\\dagger}(Y_1^{(2)})\\to \\mathbf{Isoc}^{\\dagger\\dagger}(Y^{(2)},\\Phi)$ is fully faithful, the descent data on $\\pi^*(V_0)$ gives rise to a descent data on $U_0$. \nIn this way, we obtain a subobject $U_{0}$ of $U$ over $X_1$ sending to $V_0\\to p_1^*(U)$, via $p_1^*$. \nHence the map $p_1^\\circ: \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi)\\to \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_1)$ is faithfully flat.\n\nConditions (ii) and (iii.c) follow from the same argument as in the smooth and quasiprojective case. \n\n(iii.a) Let $V$ be an object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ such that $u^*(V)$ is trivial. Then there exists an object $U_Y$ over $Y_1$ such that $p_1^*(U_Y)\\simeq \\pi^*(V)$. \nThe descent data on $\\pi^*(V)$ induces a descent data on $U_Y$, which gives rise to an object $U$ on $X_1$, sending to $V$ via $p_1^*$. \n\n(iii.b) Let $V$ be an object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$, $W_0$ be the maximal trivial subobject of $u^*(V)$. \nBy exactness of second line, there exists a subobject $V_0$ of $\\pi^*(V)$ on $Y$, sending to $W_0$ via $u^*$. \nSince $u^*$ is faithful, we deduce that the descent data on $\\pi^*(V)$ preserves $V_0$. \nThis gives rise to a subobject of $V$, sending to the trivial object $W_0$.\n\\hfill \\qed \n\\end{secnumber}\n\\begin{secnumber}\n\t\\textit{Proof of Theorem \\ref{t:main thm} for $\\pi_1^{\\PhiIsocd}$}. \n\tBy \\ref{ess of pione}(i), we have a commutative diagram: \n\t\\begin{displaymath}\n\t\t\\xymatrix{\n\t\t1\\ar[r] & \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi) \\ar[r] \\ar[d]_{\\prod p_i^{\\circ}} & \\pi_1^{\\PhiIsocd}(X) \\ar[r] \\ar[d]_{ \\prod p_i^{\\circ}} & \\pi_1^{\\PhiIsocd}(X)^{\\cst} \\ar[r] \\ar[d] & 1 \\\\\n\t\t1\\ar[r] & \\prod_{i=1}^n \\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X_i) \\ar[r] & \\prod_{i=1}^n \\pi_1^{\\FIsocd}(X_i) \\ar[r] & \\prod_{i=1}^n \\pi_1^{\\FIsocd}(X_i)^{\\cst} \\ar[r] & 1\n\t\t}\n\t\\end{displaymath}\n\twhere the first and second lines are exact. \n\tBy \\ref{ess of pione}(ii), the right vertical arrow identifies with the projection of the pro-algebraic completion of $\\mathbb{Z}^n$ to the pro-algebraic completion of $\\mathbb{Z}$ and is therefore an isomorphism. \n\tThe assertion follows from that the left vertical arrow is an isomorphism. \\hfill \\qed\n\\end{secnumber}\n\n\n\\section{Drinfeld's lemma for convergent $\\Phi$-isocrystals}\n\\label{s:conv case}\nIn this section, $X_i$ denotes a $k$-scheme for $i=1,\\ldots,n$ and $X:=\\prod_{i=1}^n X_i$. \nExcept in \\S~\\ref{sss:partial Frob slope}--\\ref{def:unit-root}, we assume each $X_i$ is smooth and geometrically connected over $k$. \n\n\\subsection{Unit-root and diagonally unit-root convergent $\\Phi$-isocrystals}\n\\begin{secnumber} \\label{sss:partial Frob slope}\n\tWe first define the \\textit{partial Frobenius slopes} at a closed point of $X$. \n\n\tLet $(\\mathscr{E},\\varphi_i)$ be an object of $\\PhiIsoc(X\/K)$, $x\\in |X|$ a closed point, $K_x:=\\rW(k_x)\\otimes_{\\rW(k)}K$ and $a:=[k_x:k]$. \n\tWe take the extension of scalars to $k_x$ \\eqref{eq:ext sca} and then take its fiber $(\\mathscr{E}_x,\\varphi_1^a|_x,\\ldots,\\varphi_n^a|_x)$ at $x$, which is an object of $\\PhiIsoc(\\Spec(k_x)\/K_x)$ with respect to the $sa$-th power of Frobenius, that is a $K_x$-vector space together with commuting linear automorphisms $\\varphi_i^a|_x$. \n\tThis allows us to define the \\textit{$i$-th partial Frobenius slopes of $(\\mathscr{E},\\varphi_i)$ at $x$} for $i=1,\\ldots,n$ by Dieudonn\\'e--Manin theory. \n\n\tWe have a surjection from points of $X$ to products of points of the $X_i$:\n\t\\[\n\t\\pi: |X|\\twoheadrightarrow \\prod_{i=1}^n |X_i|.\n\t\\]\n\n\tLet $(x_i)_{i=1}^n\\in \\prod |X_i|$ be a product of closed points. To simplify notation, assume that there exists a finite extension $k'$ of $k$ such that $k_{x_i}\\simeq k'$ (this can always be enforced by enlarging $k$). \n\tWe have a decomposition \n\t\\begin{equation} \\label{eq:decomposition points}\n\t\t\\prod x_i\\simeq \\bigsqcup_G \\Spec(k'),\n\t\\end{equation}\n\tindexed by $G=\\left( \\prod \\Gal(k'\/k) \\right)\/\\Gal(k'\/k)$, for the diagonal action. \n\tFor an object $(\\mathscr{E},\\varphi_i)$ of $\\PhiIsoc(\\prod x_i\/K)$, $\\mathscr{E}$ corresponds to a direct sum of vector spaces over $K'$, indexed by $G$. \n\tIf $a=[k':k]$, then the $a$-th power of each partial Frobenius $\\varphi_{i}^a$ preserves each component. \n\\end{secnumber}\n\n\n\t\\begin{lemma}\n\t\t\\label{l:partial Frob}\n\t\tThe partial Frobenius slope at a point $x\\in |X|$ depend only on its image $\\pi(x)=(x_1,\\ldots,x_n)\\in \\prod |X_i|$. \n\t\\end{lemma}\n\t\\begin{proof}\n\t\tWe keep the above notation. Since $G$ acts transitively on each component of \\eqref{eq:decomposition points}, the action induces isomorphisms between the pullback of $(\\mathscr{E}_x,\\varphi_{i,x}^a)$ to each component and the claim follows. \n\\end{proof}\n\n\\begin{defn} \\label{def:unit-root}\n\tLet $(\\mathscr{E},\\varphi_1,\\ldots,\\varphi_n)$ be an object of $\\PhiIsoc(X)$. \n\n\t(i) We say the $i$-th partial Frobenius structure $\\varphi_i:F_i^*(\\mathscr{E})\\to \\mathscr{E}$ is \\textit{unit-root}, if its slope at each closed point is zero. \n\t\n\t(ii) We say $\\mathscr{E}$ is \\textit{unit-root}, if every partial Frobenius structure $\\varphi_i$ is unit-root. \n\n\t(iii) We say $\\mathscr{E}$ is \\textit{diagonally unit-root} if the associated convergent $F$-isocrystal is unit-root. \n\\end{defn}\n\n\\begin{secnumber} \\label{sss:crew equiv}\n\tIn the following, we assume each $X_i$ is a smooth geometrically connected $k$-scheme. Then so is $X$. \n\n\tLet $x$ be a closed point of $X$ and $\\overline{x}$ the associated geometric point. \n\tRecall \\cite{crew-f} that there exists a canonical equivalence between the category of continuous $K$-representations of $\\pi_1^{\\et}(X,\\overline{x})$ and the category of unit-root convergent $F$-isocrystals over $X\/K$:\n\t\\begin{equation}\\label{eq:crew}\n\t\t\\mathbf{Rep}^{\\cont}_{K}(\\pi_1^{\\et}(X,\\overline{x}))\\xrightarrow{\\sim} \\FIsoc^{\\ur}(X\/K).\n\t\\end{equation}\n\tVia the equivalence between the left hand side and the category $\\LS(X,K)$ of lisse $K$-sheaves over $X$, we have:\n\t\\begin{equation} \\label{eq:crew LS}\n\t\t\\LS(X,K)\\xrightarrow{\\sim} \\FIsoc^{\\ur}(X\/K).\n\t\\end{equation}\n\tThe above equivalence is compatible with the following operations: \n\t\\begin{itemize}\n\t\\item[(i)] The pullback functoriality along a morphism between smooth connected $k$-schemes. \n\t\\item[(ii)] Extensions of scalars. \n\t\\end{itemize}\n\n\tRecall that a lisse $K$-sheaf $\\mathbb{L}$ over $X$ is equipped with the Frobenius correspondence \\cite[XIV=XV]{SGAV}:\n\t\\begin{equation} \\label{eq:phiV}\n\t\t\\phi_{\\mathbb{L}}:F_X^*(\\mathbb{L})\\xrightarrow{\\sim} \\mathbb{L}. \n\t\\end{equation}\n\\end{secnumber}\n\n\\begin{lemma}\n\tLet $(\\mathscr{E},\\varphi)$ be a unit-root convergent $F$-isocrystal over $X$ and $\\mathbb{L}$ the associated sheaf. \n\tThen the Frobenius structure $\\varphi:(F_X^*\\mathscr{E},F_X^*\\varphi)\\xrightarrow{\\sim} (\\mathscr{E},\\varphi)$ gives rise to the above isomorphism $\\phi_{\\mathbb{L}}$. \n\\end{lemma}\n\\begin{proof}\n\tIn view of the construction of \\eqref{eq:crew}, we may reduce to the case where $\\mathbb{L}$ is a lisse $\\mathscr{O}_K\/p^n \\mathscr{O}_K$-sheaf for some $n$, and we may assume $X$ is affine and admits a smooth lifting $\\mathfrak{X}_n$ to $\\mathscr{O}_K\/p^n\\mathscr{O}_K$, equipped with a Frobenius lift. \n\tAfter taking pullback along a finite \\'etale morphism trivializing $\\mathbb{L}$, we may moreover assume that $\\mathbb{L}$ is a trivial $\\mathscr{O}_K\/p^n\\mathscr{O}_K$-module. \n\tIn this case, the assertion is clear. \n\\end{proof}\n\\begin{remark}\n\tWhen $X=x=\\Spec(k)$, the endomorphism $\\phi_{\\mathbb{L}}$ on the geometric fiber $\\mathbb{L}_{\\overline{x}}$ is nothing but the action of the geometric Frobenius $F_k\\in \\Gal(\\overline{k}\/k)$ on $\\mathbb{L}_{\\overline{x}}$ in view of the construction of \\eqref{eq:crew}. \t\n\\end{remark}\n\n\\begin{secnumber} \\label{sss:Crew diagonal unitroot}\n\tLet $i$ be an integer $\\in [1,n]$. The $i$-th partial Frobenius $F_i$ of $X$ is a homeomorphism and induces an equivalence of \\'etale topoi of $X$. \n\tLet $(\\mathscr{E},\\varphi_1,\\ldots,\\varphi_n)$ be a diagonally unit-root convergent $\\Phi$-isocrystal over $X$ and $\\mathbb{L}$ the lisse $K$-sheaf over $X$ associated to $(\\mathscr{E},\\varphi)$. \n\tThe $i$-th partial Frobenius structure $\\varphi_i$ induces an isomorphism of sheaves:\n\t\\[\n\t\\phi_i:F_i^*(\\mathbb{L})\\xrightarrow{\\sim} \\mathbb{L}. \n\t\\]\n\tThe isomorphisms $\\phi_i$ commute with each other in the following sense: for any $1\\le i,j \\le n$, the identifications\n\t\\[\n\tF_i^* (F_j^* (\\mathbb{L})) \\cong (F_i \\circ F_j)^* (\\mathbb{L}) = (F_j \\circ F_i)^* (\\mathbb{L}) \\cong F_j^*( F_i^* (\\mathbb{L}))\n\t\\]\n\tinduce an equality $\\phi_j\\circ F_j^*(\\phi_i)=\\phi_i\\circ F_i^*(\\phi_j)$.\n\tThe composition of $\\phi_i$ coincides with the isomorphism $\\phi_{\\mathbb{L}}$ \\eqref{eq:phiV}. \n\tThe above construction is clearly functorial. \n\\end{secnumber}\n\n\\begin{prop} \\label{P:Crew diagonal unitroot}\n\tThe category of diagonally unit-root $\\Phi$-isocrystals over $X$ is equivalent to the category of pairs $(\\mathbb{L},\\{\\phi_i\\}_{i=1}^n)$ consisting of a lisse $K$-sheaf $\\mathbb{L}$ over $X$ together with isomorphisms $\\phi_i:F_i^*(\\mathbb{L})\\xrightarrow{\\sim} \\mathbb{L}$ commuting to each other and whose composition is $\\phi_{\\mathbb{L}}$ \\eqref{eq:phiV}. \n\tThe morphisms in the later category are morphisms of $\\LS(X,K)$ compatible with $\\phi_i$'s. \n\\end{prop}\n\n\\begin{proof}\n\tWe construct a quasi-inverse of the functor in \\S~\\ref{sss:Crew diagonal unitroot}. \n\tLet $(\\mathbb{L},\\{\\varphi_i\\}_{i=1}^n)$ be a collection of data as above and $(\\mathscr{E},\\varphi)$ the unit-root $F$-isocrystal over $X$ associated to $\\mathbb{L}$. \n\tBy functoriality of \\eqref{eq:crew LS}, $\\phi_i$ induces an $i$-th partial Frobenius structure $\\varphi_i$ on $\\mathscr{E}$. \n\tThe commutativity of $\\varphi_i$, and the fact that the composition of the $\\varphi_i$ equals $\\varphi$, follow from the corresponding properties of $\\phi_i$. \n\tThis construction is clearly functorial and provide a quasi-inverse of \\S~\\ref{sss:Crew diagonal unitroot}. \n\\end{proof}\n\n\\begin{cor} \\label{c:constant partial Frob}\n\tLet $(\\mathscr{E},\\varphi_i)$ be a diagonally unit-root $\\Phi$-isocrystal over $X$. \n\tThen for $i=1,2,\\ldots,n$, the $i$-th partial Frobenius slopes of $\\mathscr{E}$ are constant on $|X|$. \t\n\\end{cor}\n\\begin{proof}\n\tLet $(\\mathbb{L},\\{\\phi_i\\}_{i=1}^n)$ be the associated data in above proposition. \n\tThen the $i$-th partial Frobenius slope of $\\mathscr{E}$ at $x\\in X(k)$ can be calculated by that of $\\phi_{i,\\overline{x}}$ on $\\mathbb{L}_{\\overline{x}}$. \n\tSince $\\mathbb{L}$ admits a lisse $\\mathscr{O}_K$-sheaf as an integral model, the slopes of $\\phi_i$ at each fibers $\\mathbb{L}_{\\overline{x}}$ are constant as function on $x\\in X(k)$.\n\n\tThe corollary follows from applying the previous argument to extensions of scalars of $(\\mathscr{E},\\varphi_i)$ (\\S~\\ref{sss:crew equiv}). \n\\end{proof}\n\n\\begin{theorem}\n\t\\label{t:slope filtration}\n\tLet $(\\mathscr{E},\\varphi_i)$ be a convergent $\\Phi$-isocrystal over $X$. \n\tSuppose that the diagonal Newton polygon of $(\\mathscr{E},\\varphi_i)$ is constant as a function on $|X|$. \n\t\n\t\\textnormal{(i)} \n\tThere exists a filtration\n\t\\begin{equation} \\label{eq:d slope filtration}\n0 = \\mathscr{E}_0 \\subset \\cdots \\subset \\mathscr{E}_l = \\mathscr{E}\n\\end{equation}\n\tof $\\PhiIsoc(X)$ and an increasing sequence $\\mu_1<\\mu_2<\\cdots<\\mu_{\\ell}$ of rational numbers such that for $j=1,\\cdots,\\ell$, the diagonal Newton polygon of $\\mathscr{E}_j\/\\mathscr{E}_{j-1}$ is constant with slope $\\mu_j$. \n\tMoreover, the filtration and sequence are both uniquely determined by this condition. (We call it \\emph{diagonal slope filtration} of $\\mathscr{E}$.)\n\t\n\t\\textnormal{(ii)} \n\tFor each partial Frobenius structure $\\varphi_i$ of $\\mathscr{E}$, its Newton polygon is also constant on $|X|$. \n\\end{theorem}\n\n\\begin{proof}\n\t(i) There exists a slope filtration \\eqref{eq:d slope filtration} as convergent $F$-isocrystals \\cite[Corollary~4.2]{kedlaya-isocrystals}. \n\tIt suffices to show that each partial Frobenius $\\varphi_i$ preserves this filtration, that is the composition $F_i^*(\\mathscr{E}_j)\\xrightarrow{\\varphi_i} \\mathscr{E} \\to \\mathscr{E}\/\\mathscr{E}_j$ vanishes. \n\tThen uniqueness follows from that of the slope filtration for $F$-isocrystals. \n\t\n\tWe reduce to checking the above claim at the fiber of each closed point $x\\in |X|$. \n\tWe may assume there exists a finite extension $k'\/k$ of degree $a$ such that $x\\simeq \\Spec(k')$ as in \\S~\\ref{sss:partial Frob slope}. \n\tThen the fiber of the Frobenius structure $\\varphi_x^a$ is a linear automorphism of $\\mathscr{E}_x$, and its generalized eigenspaces decomposition is a refinement of the filtration $\\mathscr{E}_{0,x}\\subset\\cdots \\subset \\mathscr{E}_{\\ell,x}$. \n\tSince each partial Frobenius structure $\\varphi_{i,x}$ commutes with $\\varphi_{x}^a$, $\\varphi_{i,x}$ preserves each generalized eigenspace of $\\varphi_x^a$. \n\tThen the assertion follows. \n\n\t(ii) By assertion (i), we may reduce to the case where $\\mathscr{E}$ is diagonally unit-root after twisting. \n\tIn this case, assertion (ii) follows from Corollary \\ref{c:constant partial Frob}. \n\\end{proof}\n\n\\subsection{A variant of Crew's theorem for unit-root convergent $\\Phi$-isocrystals}\n\\begin{secnumber} \\label{sss:CXPhi}\n\tWe denote by $\\mathcal{C}(X,\\Phi)$ the category of objects $(T,\\{F_{ \\{i\\}}\\}_{i=1}^n)$ consisting of a finite \\'etale morphism $T\\to X=X_1\\times_k\\cdots \\times_k X_n$ and isomorphisms $F_{ \\{i\\}}:T\\times_{X,F_i}X \\xrightarrow{\\sim} T$ commuting with each other, i.e. $F_{ \\{i\\}}\\circ F_{ \\{j\\}}\\times_{X,F_j}X= F_{ \\{j\\}}\\circ F_{ \\{i\\}}\\times_{X,F_i}X$, whose composition is the relative Frobenius morphism $F_{T\/X}$ of $T$ over $X$.\n\tA morphism in this category is a morphism above $X$ compatible with $F_{ \\{i\\}}$'s. \n\tNote that an object of $\\mathcal{C}(X,\\Phi)$ is equivalent to the data of a pair $(T, \\{\\varphi_{T,i}\\}_{i=1}^n)$ consisting of a finite \\'etale morphism $T\\to X$ together with morphisms $\\varphi_{T,i}:T\\to T$ above $F_i$, commuting with each other, whose composition is the Frobenius morphism $F_T$. \n\n\tThis category is a Galois category and we denote by $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$ the Galois group defined by the fiber functor associated to $\\overline{x}$.\n\tWe have the following equivalence:\n\n\t\\begin{itemize}\n\t\t\\item the category of continuous actions of $\\pi_1(X,\\Phi,\\overline{x})$ on finite sets;\n\n\t\t\\item the category of locally constant constructible sheaves $\\mathbb{L}$ of $X_{\\et}$, equipped with a partial Frobenius structure $\\phi_i:F_i^*(\\mathbb{L})\\xrightarrow{\\sim} \\mathbb{L}$, commuting with each other, whose composition is the Frobenius correspondence $\\phi_{\\mathbb{L}}$. \n\t\\end{itemize}\n\\end{secnumber}\n\n\\begin{prop} \\label{P:Crew partial Frob}\n\tThere is a canonical equivalence between the category of continuous representations of $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$ on finite-dimensional $K$-vector spaces and the full subcategory $\\PhiIsoc^{\\ur}(X\/K)$ of $\\PhiIsoc(X\/K)$ consisting of unit-root convergent $\\Phi$-isocrystals. \n\\end{prop}\n\n\\begin{secnumber}\n\tLet $\\pi$ be a uniformizer of $\\mathscr{O}_K$. \n\tWe set $R:=\\mathscr{O}_K$ and $R_n:=\\mathscr{O}_K\/\\pi^n \\mathscr{O}_K$ for $n\\ge 1$. \n\tWe may reduce to the case where each $X_i$ is affine and admits a smooth formal lifting $\\mathfrak{X}_i$ to $R$ and a lifting $F_{\\mathfrak{X}_i}:\\mathfrak{X}_i\\to \\mathfrak{X}_i$ of the Frobenius $F_{X_i}$. We set $\\mathfrak{X}:=\\prod \\mathfrak{X}_i$ and denote by $F_{i,\\mathfrak{X}}:\\mathfrak{X}\\to \\mathfrak{X}$ the product of $F_{\\mathfrak{X}_i}$ and the identity maps on other components. \n\t\n\tA \\textit{unit-root $\\Phi$-lattice} on $\\mathfrak{X}\/R$ is a locally free $\\mathscr{O}_{\\mathfrak{X}}$-module of finite rank together with isomorphisms $\\varphi_i: F_{i,\\mathfrak{X}}^*(M)\\xrightarrow{\\sim} M$ commuting with each other. \n\tWe first establish an equivalence following \\cite{crew-f}:\n\t\\begin{equation}\n\t\t\\iota: \\mathbf{Rep}_{R}^{\\cont}(\\pi_1(X,\\Phi,\\overline{x})) \\xrightarrow{\\sim} \\{\\textnormal{Unit-root $\\Phi$-lattices on $\\mathfrak{X}\/R$}\\}.\n\t\t\\label{eq:equiv integral rep}\n\t\\end{equation}\n\n\tLet $V$ be a continuous $R$-linear representation of $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$. \n\tFor $n\\ge 1$, let $G_n$ be the image of $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$ in $\\GL(V\/\\pi^nV)$, which corresponds to a Galois cover $Y^{(n)}\\to X$ and $\\pi_n:\\mathfrak{Y}_n^{(n)}\\to \\mathfrak{X}_n$ its lifting to $R_n$. \n\tMoreover, each isomorphism $F_{ \\{i\\}}$ uniquely lifts to an isomorphism $F_{ \\{i\\}}:F_{i,\\mathfrak{X}}^*(\\mathfrak{Y}_n^{(n)})\\xrightarrow{\\sim} \\mathfrak{Y}_n^{(n)}$, which commute with each other. As in \\cite{crew-f}, we consider a locally free $\\mathscr{O}_{\\mathfrak{X}_n}$-module (resp. $\\mathscr{O}_{\\mathfrak{X}}$-module)\n\t\\[\n\tM_n= \\pi_{n,*}(\\mathscr{O}_{\\mathfrak{Y}_n^{(n)}})\\otimes_{R_n[G_n]}V\/\\pi^n V, \\quad M=\\varprojlim M_n. \n\t\\]\n\tThen the isomorphism $F_{ \\{i\\}}$ induces an isomorphism $F_{i,\\mathfrak{X}}^*(M_n)\\xrightarrow{\\sim} M_n$ and gives rise to a unit-root parital Frobenius structure\n\t\\[\n\\varphi_i:F_{i,\\mathfrak{X}}^*(M)\\xrightarrow{\\sim} M.\n\t\\]\n\tIn this way, we obtain the functor $\\iota$ as in \\eqref{eq:equiv integral rep}.\n\t\n\tConversely, given a unit-root $\\Phi$-lattice $(\\mathscr{E},\\varphi_i)$ on $\\mathfrak{X}\/R$, the associated unit-root $F$-lattice $(\\mathscr{E},\\varphi)$ defines a lisse $R$-sheaf $\\mathbb{L}$ on $X$ by \\cite[Theorem 2.2]{crew-f}.\n\tThe partial Frobenius structures $\\varphi_i$ define isomorphisms $\\phi_i:F_i^*(\\mathbb{L})\\xrightarrow{\\sim} \\mathbb{L}$ commuting with each other. Then we obtain a continuous $R$-representation of $\\pi_1(X,\\Phi,\\overline{x})$ on $\\mathbb{L}_{\\overline{x}}$. \n\tThe above construction is functorial and defines a quasi-inverse of $\\iota$. \n\\end{secnumber}\n\\begin{lemma}\n The isomorphism $\\varphi_i$ is horizontal with respect to the canonical convergent connection $\\nabla$ on $M^{\\rig}$ \\cite[Proposition 2.3]{crew-f}.\n\\end{lemma}\n\\begin{proof}\n\tLet $\\Delta$ be the formal completion of the diagonal map $\\mathfrak{X}\\to \\mathfrak{X}\\times_{R}\\mathfrak{X}$ and $p_1,p_2:\\Delta\\to \\mathfrak{X}$ the natural projections. \n\tFor $n\\ge 1$, we set $\\Delta_n:=\\Delta\\otimes_{R}R_n$. \n\tLet $p_i^*\\mathfrak{Y}_{n}^{(n)}$ be the fiber product $\\mathfrak{Y}_{n}^{(n)}\\times_{\\mathfrak{X}_n,p_i}\\Delta_n$. \n\tThe map $\\mathfrak{Y}_n^{(n)}\\to \\mathfrak{X}_n \\xrightarrow{diag} \\Delta_n$ gives rise to formal thickenings $\\mathfrak{Y}_n^{(n)}\\to p_i^*\\mathfrak{Y}_{n}^{(n)}$. \n\tRecall \\textit{loc.cit} that, since $\\mathfrak{Y}_n^{(n)}\\to \\mathfrak{X}_n$ is \\'etale, there exists a unique isomorphism $p_1^*\\mathfrak{Y}_n^{(n)}\\xrightarrow{\\sim} p_2^*\\mathfrak{Y}_n^{(n)}$ which fits into the following diagram:\n\t\\[\n\t\\xymatrix{\n\tp_1^* \\mathfrak{Y}_n^{(n)} \\ar[d] \\ar[rd]^{\\sim} & \\mathfrak{Y}_n^{(n)} \\ar[l] \\ar[d] \\\\\n\t\\Delta_n & p_2^*\\mathfrak{Y}_n^{(n)} \\ar[l]\n\t}\n\t\\]\n\tBy uniqueness, this isomorphism is compatible with the $G_n$-action, the partial Frobenius maps on $\\mathfrak{Y}_n^{(n)}$, and the canonical connection on $M$. Then the lemma follows. \n\\end{proof}\n\\begin{secnumber} \\textit{Proof of Proposition \\ref{P:Crew partial Frob}.}\n\tBy the above lemma, we have a fully faithful functor \n\t\\[\n\t\\{\\textnormal{Unit-root $\\Phi$-lattices on $\\mathfrak{X}\/R$}\\}\\otimes_{R}K \\to\n\t\\{\\textnormal{Unit-root $\\Phi$-isocrystals on $X\/K$}\\}\n\t\\]\n\tIt suffices to show that the functor in Proposition \\ref{P:Crew partial Frob} is essentially surjective. \n\n\tLet $(\\mathscr{M},\\varphi_i)$ be a unit-root $\\Phi$-isocrystal on $X\/K$ and $(\\mathbb{L},\\phi_i)$ the associated data in Proposition \\ref{P:Crew diagonal unitroot}. \n\tThere exists a $R$-lattice $\\mathbb{L}^{\\circ}$ of $\\mathbb{L}$ such that $\\phi_{\\mathbb{L}}$ induces an isomorphism $F_X^*(\\mathbb{L}^{\\circ})\\xrightarrow{\\sim} \\mathbb{L}^{\\circ}$. \n\tAs $\\phi_i$ is unit-root, we claim there exist integers $r,s$ such that for every integer $m\\ge 0$, we have \n\t\t\\[\n\t\tp^r \\mathbb{L}^{\\circ} \\subset \\phi_i^m(F_i^{m,*}\\mathbb{L}^{\\circ}) \\subset p^s \\mathbb{L}^{\\circ}.\n\t\t\\]\n\tIndeed, let $x\\in|X|$ be a closed point with degree $[k_x:k]=a$. \n\tThe linear action of $\\phi_{i,x}^a$ on the fiber $\\mathbb{L}^{\\circ}_{\\overline{x}}$ is unit-root and satisfies the above properties \\cite[Proposition 1.11]{crew-f}. \n\tThen the claim follows.\n\n\tFollowing \\cite{crew-f}, we consider $\\mathbb{L}':=\\sum_{m\\ge 0} \\IM \\phi_i^m: \\mathbb{L}^{\\circ}\\to p^s \\mathbb{L}$ and $\\mathbb{L}_i:= \\cap_{m\\ge 1} \\IM \\phi_i^m: \\mathbb{L}'\\to \\mathbb{L}'$. \n\tThen $\\phi_i$ induces an isomorphism $F_i^*\\mathbb{L}_i\\xrightarrow{\\sim} \\mathbb{L}_i$. \n\n\tBy repeating the above argument to each $\\phi_i$, we obtain a $R$-lattice $\\widetilde{\\mathbb{L}}$ of $\\mathbb{L}$ such that $\\phi_i$ induces an isomorphism $F_i^*(\\widetilde{\\mathbb{L}})\\xrightarrow{\\sim} \\widetilde{\\mathbb{L}}$ for every $i$. \n\tThen the data $(\\widetilde{\\mathbb{L}},\\phi_i)$ descent to a representation of $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$.\n\tIn view of Proposition \\ref{P:Crew diagonal unitroot}, this defines a quasi-inverse of previous construction. \n\tThe proposition follows. \\hfill \\qed\n\\end{secnumber}\n\n\n\n\\subsection{Original Drinfeld's lemma} \n\tFollowing an argument of Drinfeld--Kedlaya \\cite[Appendix B]{DK17}, we recover the original Drinfeld's lemma from Theorem \\ref{t:main thm} and Proposition \\ref{P:Crew partial Frob}.\n\n\\begin{secnumber}\n\tWe say a representation of $\\mathbf{Rep}^{\\cont}_K(\\pi_1^{\\et}(X,\\Phi,\\overline{x}))$ is \\textit{smooth} if the action of $\\pi_1^{\\et}(X,\\Phi,\\overline{x})$ factors through a finite quotient. \n\tIn view of \\'etale descent for overconvergent $F$-isocrystals, the canonical functor \n\t\\begin{equation}\n\t\t\\mathbf{Rep}^{\\smooth}_{K}(\\pi_1^{\\et}(X,\\Phi,\\overline{x})) \\to \\PhiIsoc(X)\n\t\t\\label{Crew smooth}\n\t\\end{equation}\n\tfactors through $\\PhiIsocd(X)$. \n\tIt induces canonical homomorphisms: \n\t\\begin{equation}\n\t\t\\pi_1^{\\PhiIsoc}(X)\\to \\pi_1^{\\PhiIsocd}(X)\\to \\pi_1^{\\et}(X,\\Phi,\\overline{x}).\n\t\t\\label{composition pi}\n\t\\end{equation}\n\tSince the functor \\eqref{Crew smooth} is fully faithful, the composition is an epimorphism. \n\t\n\tIf $\\mathbf{Fet}(X_i)$ denotes the category of finite \\'etale schemes over $X_i$, the canonical functor\n\t\\[\n\t\\mathbf{Fet}(X_i)\\to \\mathcal{C}(X,\\Phi), ~ T_i\\mapsto \\prod_{j\\neq i}X_j \\times T_i\n\t\\]\n\tinduces a canonical homomorphism $\\widetilde{p}_i:\\pi_1^{\\et}(X,\\Phi,\\overline{x})\\to \\pi_1^{\\et}(X_i,\\overline{x}_i)$, where $\\overline{x}_i=p_i(\\overline{x})$. \n\tIn view of the construction, the above morphism is compatible with the projection, that is the following diagram commutes:\n\t\\begin{equation}\n\t\t\\label{commutative projection}\n\t\\xymatrix{\n\t\\pi_1^{\\PhiIsocd}(X) \\ar[r] \\ar[d]_{p_i^{\\circ}} & \\pi_1^{\\et}(X,\\Phi,\\overline{x}) \\ar[d]_{\\widetilde{p}_i} \\\\\n\t\\pi_1^{\\FIsocd}(X_i) \\ar[r] & \\pi_1^{\\et}(X_i,\\overline{x}_i)\n\t}\n\t\\end{equation}\n\\end{secnumber}\n\n\n\\begin{theorem} \\label{t:Drinfeld lemma origin}\n\t\\textnormal{(i)} The homomorphisms \\eqref{composition pi} induce isomorphisms\n\t\\[\n\t\\pi_0(\\pi_1^{\\PhiIsoc}(X))\\xrightarrow{\\sim} \\pi_0(\\pi_1^{\\PhiIsocd}(X))\\xrightarrow{\\sim} \\pi_1(X,\\Phi,\\overline{x}).\t\t\n\t\\]\n\n\t\\textnormal{(ii)}\n\tIf we apply $\\pi_0$ to the isomorphism \\eqref{Tannakian Drinfeld lemma}, we obtain the original Drinfeld's lemma:\n\t\\[\n\t\\pi_1(X,\\Phi,\\overline{x})\\xrightarrow{\\sim} \\prod_{i=1}^n \\pi_1^{\\et}(X_i,\\overline{x}_i).\n\t\\]\n\\end{theorem}\n\n\\begin{proof}\n\t(i) \n\tWe first show that for any finite group $\\Gamma$ and any tensor functor\n\t\\begin{equation} \\label{eq:Drinfeld lemma pi1}\n\t\\mathbf{Rep}_{\\overline{\\mathbb{Q}}_p}(\\Gamma) \\to \\PhiIsoc(X)\n\\end{equation}\n\tfactors through $\\mathbf{Rep}_{\\overline{\\mathbb{Q}}_p}^{\\cont}(\\pi_1(X,\\Phi,\\overline{x}))$. \n\tBy Proposition \\ref{P:Crew partial Frob}, it suffices to show that objects in the essential image of the above functor are unit-root at each point of $|X|$. \n\tThis can be shown by a similar argument to \\cite[Proposition~B.4.1]{DK17}. \n\t\n\tMoreover, we deduce that the above functor factors through $\\mathbf{Rep}_{\\overline{\\mathbb{Q}}_p}^{\\smooth}(\\pi_1(X,\\Phi,\\overline{x}))$ as in \\textit{loc.cit.}. \n\n\tHence, the kernel of the canonical epimorphism $\\pi_1^{\\PhiIsoc}(X)\\twoheadrightarrow \\pi_1(X,\\Phi,\\overline{x})$ is the connected component of $\\pi_1^{\\PhiIsoc}(X)$. \n\tWe obtain the isomorphism $\\pi_0(\\pi_1^{\\PhiIsoc}(X))\\xrightarrow{\\sim} \\pi_1(X,\\Phi,\\overline{x})$.\n\n\tBy a full faithfulness result \u00e0 la Kedlaya (Proposition \\ref{P:fully faithful}), $\\pi_1^{\\PhiIsocd}(X)\\to \\pi_1(X,\\Phi,\\overline{x})$ is an epimorphism. \n\tThen by \\cite[Lemma~B.7.4]{DK17}, we deduce the isomorphism $\\pi_0(\\pi_1^{\\PhiIsocd}(X))\\xrightarrow{\\sim} \\pi_1(X,\\Phi,\\overline{x})$. \n\n\t(ii) \n\tAssertion (ii) follows from assertion (i), Theorem \\ref{t:main thm}, and the commutative diagram \\eqref{commutative projection}. \t\n\\end{proof}\n\t\n\t\\begin{prop}\\label{P:fully faithful}\n\t\t\\textnormal{(i)} The canonical functor $\\iota_{\\Phi}:\\PhiIsocd(X\/K)\\to \\PhiIsoc(X\/K)$ is fully faithful.\n\n\t\t\\textnormal{(ii)} \n\t\t\tFor $i=1,\\dots,n$, let $U_i\\subset X_i$ be an open immersion with dense image and set $U:= \\prod_{i=1}^n U_i$. \n\tThen the following restriction functors are fully faithful:\n\t\\[\n\t\\PhiIsoc(X\/K)\\to \\PhiIsoc(U\/K),\\quad \\PhiIsocd(X\/K)\\to \\PhiIsocd(U\/K).\n\t\\]\n\t\\end{prop}\n\\begin{proof}\n\t(i) Consider the following diagram:\n\t\\[\n\t\\xymatrix{\n\t\\PhiIsocd(X)\\ar[r]^{\\iota_{\\Phi}} \\ar[d]& \\PhiIsoc(X) \\ar[d] \\\\\n\t\\FIsocd(X)\\ar[r]^{\\iota_F} & \\FIsoc(X) }\n\t\\]\n\tThe vertical arrows are faithful and $\\iota_F$ is fully faithful \\cite{Ked04}. \n\tHence it remains to show the fullness of $\\iota_{\\Phi}$. \n\t\n\tGiven two objects $(\\mathscr{E},\\phi_i),(\\mathscr{E}',\\phi'_i)$ of $\\PhiIsocd(X)$ and a morphism $f:\\mathscr{E}\\to \\mathscr{E}'$ of $\\PhiIsoc(X)$,\n\t $f$ extends to a morphism $f^{\\dagger}:\\mathscr{E}\\to \\mathscr{E}'$ of overconvergent $F$-isocrystals by the full faithfulness of $\\iota_F$. \n\tSince the canonical functor $\\Isocd(X)\\to \\Isoc(X)$ is faithful, we deduce that $f^{\\dagger}$ is compatible with partial Frobenius structures. \n\n\t(ii) By assertion (i), it suffices to prove the assertion for convergent $\\Phi$-isocrystals. \n\tBy \\cite[Theorem 5.3]{kedlaya-isocrystals} and a similar argument as in (i), it suffices to show its fullness. \n\t\n\tGiven two objects $(\\mathscr{E},\\phi_i),(\\mathscr{E}',\\phi'_i)$ of $\\PhiIsoc(X\/K)$ and a morphism $f:\\mathscr{E}\\to \\mathscr{E}'$ of $\\PhiIsoc(U\/K)$. \n\tThen $f$ extends to a morphism $g:\\mathscr{E}\\to \\mathscr{E}'$ of convergent $F$-isocrystals on $X$. \n\tThe compatibility between $f$ and partial Frobenius structures follows from \\textit{loc.cit}. \n\\end{proof}\n\n\n\\subsection{Partial Frobenius slope filtrations}\n\n\\begin{theorem} \\label{T:partial slope filtration}\nSuppose that an object $\\mathscr{E}$ of $\\PhiIsoc(X\/K)$ has a constant diagonal Newton polygon on $|X|$. Then for $i=1,\\dots,n$, $\\mathscr{E}$ admits a filtration\n\\[\n0 = \\mathscr{E}_0^{(i)} \\subset \\cdots \\subset \\mathscr{E}_l^{(i)} = \\mathscr{E}\n\\]\nin $\\PhiIsoc(X\/K)$ and an ascending sequence $\\mu_1 < \\cdots < \\mu_l$ of rational numbers such that for $j=1,\\dots,l$, the $i$-th partial Frobenius slope of $\\mathscr{E}^{(i)}_j\/\\mathscr{E}^{(i)}_{j-1}$ equals to $\\mu_j$. Moreover, the filtration and sequence are both uniquely determined by this condition. (We call it the \\emph{$i$-th partial slope filtration} of $\\mathscr{E}$.)\n\\end{theorem}\n\\begin{lemma} \\label{L:diagonally unit-root decomposition}\n\tSuppose that $\\mathscr{E} \\in \\PhiIsoc(X\/K)$ is diagonally unit-root.\n\t\n\t\\textnormal{(i)} Then there exists a decomposition in $\\PhiIsoc(X\/K)$\n\\[\n\\mathscr{E}\\simeq \\bigoplus_{d_1,\\dots,d_n} \\mathscr{E}_{d_1,\\dots,d_n}, \n\\]\nindexed by tuples $(d_1,\\dots,d_n) \\in \\mathbb{Q}^n$ with $d_1 + \\cdots + d_n =\n0$, in which\n$\\mathscr{E}_{d_1,\\dots,d_n}$ has the constant partial Frobenius slopes $(d_1,\\ldots,d_n)$ on $|X|$. \n\n\\textnormal{(ii)} After taking extension of scalars, each $\\Phi$-isocrystal $\\mathscr{E}_{d_1,\\ldots,d_n}$ is isomorphic to successive extensions of $\\Phi$-isocrystals $\\mathscr{E}_1\\boxtimes \\mathscr{E}_2\\boxtimes\\cdots\\boxtimes \\mathscr{E}_n$, where $\\mathscr{E}_i\\in \\FIsoc(X_i\/K)$ is isoclinic of slope $d_i$. \n\\end{lemma}\n\n\\begin{proof} \n\t(i) Let $(\\mathbb{L},\\phi_i)$ be the data associated to $\\mathscr{E}$ in Proposition \\ref{P:Crew diagonal unitroot}.\n\tLet $x\\in |X|$ be a closed point of degree $a$ and $(V=\\mathbb{L}_{\\overline{x}},\\rho)$ the associated $\\pi_1(X,\\overline{x})$-representation. \n\tSince $\\phi_{i,\\overline{x}}$ commute with each other, $V$ admits a decomposition \n\t\\[\n\tV\\simeq \\bigoplus_{(d_1,\\ldots,d_n)} V_{d_1,\\ldots,d_n},\n\t\\]\n\taccording to partial Frobenius slopes of $(\\phi_{1,\\overline{x}},\\ldots,\\phi_{n,\\overline{x}})$. \n\tIt remains to show that each component is invariant under the $\\pi_1(X,\\overline{x})$-action. \n\tNote that each $\\phi_{i,\\overline{x}}$ preserves $V_{d_1,\\ldots,d_n}$, and then does their composition $\\phi_{\\mathbb{L},\\overline{x}}$. \n\n\tSince $F_X^a$ induces an identity on \\'etale fundamental groups $F_X^a:\\pi_1(X,\\overline{x})=\\pi_1(X,\\overline{x})$, $\\phi_{\\mathbb{L},\\overline{x}}^a$ is an automorphism of the representation $(V,\\rho)$. \n\tSince $\\phi_{\\mathbb{L},\\overline{x}}^a$ commutes with the action of $\\pi_1(X,\\overline{x})$ on $V$, we deduce that the $\\pi_1(X,\\overline{x})$-action also preserves each component $V_{d_1,\\ldots,d_n}$. Then assertion (i) follows. \n\n\t(ii) After taking extension of scalars and twisting, we may assume $\\mathscr{F}=\\mathscr{E}_{d_1,\\ldots,d_n}$ is unit-root. \n\tWe prove the assertion by induction on $n$. \n\tLet $\\overline{x}$ be a geometric point of $X$. \n\tBy Proposition \\ref{P:Crew partial Frob} and Theorem \\ref{t:Drinfeld lemma origin}, $\\mathscr{E}$ corresponds to a continuous $K$-linear representation $V$ of $\\pi_1(X_1,\\overline{x})\\times\\cdots \\times \\pi_1(X_n,\\overline{x})$. \n\tLet $W$ be an irreducible sub-$\\pi_1(X_n,\\overline{x})$-representation of $V$. \n\tThen $W$ corresponds to a unit-root $F$-isocrystal $\\mathscr{F}_n$ on $X_n$, and $\\Hom_{\\pi_1(X_n,\\overline{x})}(W,V)$, viewed as a $\\pi_1(X_1,\\overline{x})\\times\\cdots\\times \\pi_1(X_{n-1},\\overline{x})$-representation, corresponds to a unit-root $\\Phi$-isocrystal $\\mathscr{E}'$ on $X_1\\times\\cdots\\times X_{n-1}$. \n\tWe obtain a monomorphism $\\mathscr{E}'\\boxtimes \\mathscr{F}_n\\to \\mathscr{F}$. \n\tApplying the induction hypothesis to $\\mathscr{E}'$, then assertion (ii) follows. \n\\end{proof}\n\n\\begin{lemma}\n\t\\label{L:split by slope}\n\tLet $\\mathscr{E},\\mathscr{E}'$ be two objects of $\\PhiIsoc(X\/K)$ with constant partial Frobenius slopes $(\\mu_i)_{i=1}^n,(\\mu_i')_{i=1}^n$ respectively. \n\n\t\\textnormal{(i)} If $\\Hom_{\\PhiIsoc(X\/K)}(\\mathscr{E}',\\mathscr{E})\\neq 0$, then $\\mu_i=\\mu_i'$ for every $i=1,\\cdots,n$. \n\n\t\\textnormal{(ii)} If $\\Ext_{\\PhiIsoc(X\/K)}(\\mathscr{E}',\\mathscr{E})\\neq 0$, then $\\mu_i\\ge \\mu_i'$ for every $i=1,\\cdots,n$. \n\\end{lemma}\n\\begin{proof}\n\t(i) Let $i$ be an integer, $X'=\\prod_{j\\neq i} X_j$ and $k'$ a closed point of $X'$.\n\tWe apply the exact and faithful functor \\eqref{eq:partial Frob to FIsocd} to the pullback along $\\Spec(k')\\to X'$ to obtain convergent $F$-isocrystals on $X_{i,k'}\/K'$. \n\tThen the assertion follows from the corresponding assertion for convergent $F$-isocrystals. \n\n\t(ii) We will deduce this vanishing result of $\\Ext$ from the case of $F$-isocrystals, which is known (it follows from the existence of the usual slope filtration). \n\tWe set $\\mathscr{F}=\\mathscr{H}om(\\mathscr{E}',\\mathscr{E})$. \n\tBy Lemma \\ref{L:diagonally unit-root decomposition}(ii), we may reduce to the case where $\\mathscr{F}\\simeq \\mathscr{F}_1\\boxtimes\\cdots\\boxtimes \\mathscr{F}_n$, with $\\mathscr{F}_i\\in \\FIsoc(X_i\/K)$ after taking extension of scalars. \n\tSuppose $\\mu_i'>\\mu_i$ for some integer $i\\in [1,n]$. \n\tWe consider the extension group $\\Ext^1_{F_i\\textnormal{-}\\Isoc(X\/K)}(\\mathscr{E}',\\mathscr{E})$, which fits into the following diagram:\n\t\\begin{equation} \\label{eq:ext}\n\t0\\to \\rH^0(X,\\mathscr{F})\/(\\phi_i-\\id)\\to \\Ext^1_{F_i\\textnormal{-}\\Isoc(X\/K)}(\\mathscr{E}',\\mathscr{E}) \\to \\rH^1(X,\\mathscr{F})^{\\phi_i=\\id}\\to 0.\n\\end{equation}\n\n\tSince the slope of $\\mathscr{F}_i$ is $>0$, we obtain the vanishing of $\\rH^0(X_i,\\mathscr{F}_i)^{\\phi_i=\\id},\\rH^0(X_i,\\mathscr{F}_i)\/(\\phi_i-\\id)$, and $\\rH^1(X_i,\\mathscr{F}_i)^{\\phi_i=\\id}$; note that the last case follows from the assertion (ii) for convergent $F$-isocrystals. \n\tWe conclude that the middle term of \\eqref{eq:ext} vanishes by the K\u00fcnneth formula \\cite[Lemma 4.5]{Abe14}. Then the assertion follows. \n\\end{proof}\n\\begin{secnumber} \\textit{Proof of Theorem \\ref{T:partial slope filtration}.} \n\tBy Theorem \\ref{t:slope filtration}, each partial Newton polygon of $\\mathscr{E}$ is also constant. \n\tThen we apply Lemma \\ref{L:diagonally unit-root decomposition} to the successive quotients of the diagonal slope filtration; this yields a filtration in which each successive quotient has the property that every $i$-th partial Frobenius slope has a fixed value, but these values may not occur in ascending order. However, using Lemma \\ref{L:split by slope} we can reorder the successive quotients to enforce this condition. \\hfill \\qed\n\\end{secnumber}\n\n\n\\subsection{Drinfeld's lemma for convergent $\\Phi$-isocrystals}\n\nRecall that the category $\\FIsoc(X\/K)\\otimes_K \\overline{\\mathbb{Q}}_p$ (resp. $\\PhiIsoc(X\/K)\\otimes_K\\overline{\\mathbb{Q}}_p$) is denoted by $\\FIsoc(X)$ (resp. $\\PhiIsoc(X)$) and is neutral Tannakian over $\\overline{\\mathbb{Q}}_p$. \nWith notations of \\S~\\ref{sss:Tannakian gps}, we show a version of Drinfeld's lemma for convergent $F$-isocrystals as in Theorem \\ref{t:main thm}. \n\n\\begin{theorem}\n\t\\label{t:Drinfeld lemma conv}\n\tAssume $X_i$ is a smooth geometrically connected $k$-scheme. \n\tThe following canonical homomorphisms are isomorphisms:\n\t\\begin{equation} \\label{eq:Drinfeld lemma conv}\n\t\t\\prod_{i=1}^n p_i^{\\circ}:\n\t\t\\pi_1^{\\PhiIsoc}(X) \\xrightarrow{\\sim} \\prod_{i=1}^n \\pi_1^{\\FIsoc}(X_i) . \n\t\\end{equation}\t\n\\end{theorem}\n\n\\begin{prop}\n\t\\label{p:pullback subquotients}\n\tThe pullback functor:\n\t\\[\n\t\tp_i^*:\\FIsoc(X_i)\\to \\PhiIsoc(X) \n\t\\]\n\tis fully faithful and its essential image is closed under subquotients.\n\\end{prop}\n\n\\begin{proof}\n\tWe may assume $i=n$ and set $X':=\\prod_{i=1}^{n-1}X_i$.\n\n\t(i) We first prove the full faithfulness. \n\tBy Proposition \\ref{P:fully faithful}, we may assume each $X_i$ is affine and admits a smooth formal lifting $\\mathfrak{X}_i$ over $\\mathscr{O}_K$. \n\tLet $(\\mathscr{E},\\varphi)$ be an object of $\\FIsoc(X_n)$. Since each $X_i$ is smooth and geometrically connected, the canonical morphism\n\t\\[\n\t\\rH^0(X_n,\\mathscr{E})\\xrightarrow{\\sim} \\rH^0(X,p_n^*(\\mathscr{E}))\n\t\\]\nis an isomorphism by K\u00fcnneth formula for the sheaf of differential operators \\cite[Lemma 4.5]{Abe14} and for quasi-coherent modules. \n\tThen we deduce a canonical isomorphism: \n\t\\[\n\t\\rH^0(X_n,\\mathscr{E})^{\\varphi=\\id}\\to \\rH^0(X,p_n^*(\\mathscr{E}))^{\\Phi=\\id}.\n\t\\]\n\tThe full faithfulness follows from the above isomorphism applied to internal homomorphisms.\t\n\n\t(ii) Next we treat the second assertion. \n\tLet $\\mathscr{E}$ be an object of $\\FIsoc(X_n)$ and $p_n^*(\\mathscr{E})\\to \\mathscr{F}$ a surjection in $\\PhiIsoc(X)$. \n\tBy enlarging $k$ \\eqref{eq:ext sca}, we may assume there exists a $k$-point $x$ of $X'$. \n\tWe will show that the surjection $\\mathscr{E}\\to \\mathscr{G}=(x\\times \\id_{X_n})^*(\\mathscr{F})$ over $X_n$ is isomorphic to $p_n^*(\\mathscr{E})\\to \\mathscr{F}$ after pullback via $p_n^*$. \n\n\tLet $U$ be a dense open subset of $X$ such that the diagonal Newton polygon of $\\mathscr{F}$ is constant, which is preserved by partial Frobenius morphisms. \n\tBy Drinfeld's lemma for open immersions \\cite[Lemma~9.2.1]{lau}, \\cite[Theorem~4.3.6]{kedlaya-aws} and Proposition \\ref{P:fully faithful}, we may assume that the diagonal Newton polygons of $p_n^*(\\mathscr{E})$ and $\\mathscr{F}$ are constant on $|X|$ after shrinking $X_i$.\n\tThen the set of each partial Newton polygon of $\\mathscr{F}$ is also constant on $|X|$. \n\tTo show the claim, we may reduce to the case where the Frobenius slopes of $\\mathscr{E}$ are constant on $X_n$ by applying Theorem \\ref{t:slope filtration} to $p_n^*(\\mathscr{E})\\to \\mathscr{F}$. \n\tMoreover, we may assume $\\mathscr{E}$ is unit-root by twisting. \n\tThen $\\mathscr{F}$ is also a unit-root $\\Phi$-isocrystal. \n\tBy Proposition \\ref{P:Crew partial Frob} and Theorem \\ref{t:Drinfeld lemma origin}, for a geometric point $\\overline{y}$ of $X$, the action of $\\prod_{i=1}^{n-1}\\pi_1(X_i,\\overline{y})$ on $p_n^*(\\mathscr{E})_{\\overline{y}}$ is trivial. \n\tThen the same holds for $\\mathscr{F}$ and the assertion follows. \n\\end{proof}\n\n\\begin{prop} \\label{p:pushforward conv}\n\t\\textnormal{(i)} The functor $p_i^*:\\FIsoc(X_i)\\to \\PhiIsoc(X)$ admits a right adjoint $p_{i,*}$. \n\tMoreover, the canonical morphism $\\id\\to p_{i,*}p_i^*$ is an isomorphism and $p_i^*p_{i,*}\\to \\id$ is injective. \n\n\t\\textnormal{(ii)} The functor $p_{i,*}$ commutes with the base change of $g: Y\\to X_i$. \n\\end{prop}\n\n\\begin{proof}\n\t(i) Let $\\mathscr{E}$ be an object of $\\PhiIsoc(X)$. By Proposition \\ref{p:pullback subquotients}, the collection of subobjects of $\\mathscr{E}$ which belong to the essential image of $p^*_i$ has a maximal element $\\mathscr{F}$. \n\tThen we define $p_{i,*}(\\mathscr{E})$ to be the object of $\\FIsoc(X_i)$ of which $\\mathscr{F}$ is a pullback. \n\n\tLet $f:\\mathscr{E}_1\\to \\mathscr{E}_2$ be a morphism of $\\PhiIsoc(X)$. \n\tLet $\\mathscr{G}$ be the image of $p_i^*(p_{i,*}(\\mathscr{E}_1))$ in $\\mathscr{E}_2$. \n\tBy Proposition \\ref{p:pullback subquotients}, $\\mathscr{G}$ belongs to the essential image of $p_i^*$ and the functoriality of $p_{i,*}$ follows. \n\tThe second assertion is clear. \n\n\t(ii) By Lemma \\ref{P:fully faithful}, we may assume that the diagonal Newton polygon of $\\mathscr{E}$ is constant on $|X|$. \n\tBy Theorem \\ref{t:slope filtration}, we may moreover assume that $\\mathscr{E}$ is diagonally unit-root after twisting the $i$-th partial Frobenius structure. \n\tThen $p_{i,*}(\\mathscr{E})$ is unit-root and $p_i^*(p_{i,*}(\\mathscr{E}))$ lies in the unit-root component $\\mathscr{E}_{(0,\\ldots,0)}$ of $\\mathscr{E}$ (c.f. Lemma \\ref{L:diagonally unit-root decomposition}). \n\tMoreover, the unit-root $F$-isocrystal $p_{i,*}(\\mathscr{E})$ over $X_i$ can be identified with the $\\prod_{j\\neq i}\\pi_1(X_j,\\overline{x})$-invariant part of the $\\pi_1(X,\\Phi,\\overline{x})$-representation associated to $\\mathscr{E}_{(0,\\ldots,0)}$ via Proposition \\ref{P:Crew partial Frob}. \n\tThe base change property follows from this identification in view of Theorem \\ref{t:Drinfeld lemma origin}. \n\\end{proof}\n\n\\begin{secnumber}\n\t\\textit{Proof of Theorem \\ref{t:Drinfeld lemma conv}.}\n\tThe proof is similar to that of \\S~\\ref{pf main thm}(a). \n\tWe keep the notation of \\textit{loc.cit}. \n\tIt suffices to show the exactness of the following sequence:\n\t\\[\n\t\\pi_1^{\\PhiIsoc}(X',\\Phi') \\xrightarrow{u^{\\circ}}\n\t\\pi_1^{\\PhiIsoc}(X,\\Phi) \\xrightarrow{p_1^{\\circ}}\n\t\\pi_1^{\\FIsoc}(X_1).\n\t\\]\n\tWe verify the exactness using Theorem \\ref{exact criterion}. \n\n\tCondition (i) follows from Proposition \\ref{p:pullback subquotients}. \n\tConditions (ii) and (iii.c) can be verified in the same way as in \\S~\\ref{pf main thm}(a).\n\tFinally, as in \\textit{loc.cit}, we verify conditions (iii.a) and (iii.b) using functor $p_{i,*}$ and its base change property Proposition \\ref{p:pushforward conv}. This finishes the proof. \\hfill \\qed \n\\end{secnumber}\n\n\\begin{remark}\n\tThe functor $\\PhiIsocd(X)\\to \\PhiIsoc(X)$ induces a canonical morphism $\\pi_1^{\\PhiIsoc}(X)\\to \\pi_1^{\\PhiIsocd}(X)$, which fits into the following diagram:\n\t\\[\n\t\\xymatrix{\n\t\\pi_1^{\\PhiIsoc}(X) \\ar[r]^-{\\prod_{i=1}^n p_i^{\\circ}}_-{\\sim}\\ar[d] & \\prod_{i=1}^n \\pi_1^{\\FIsoc}(X_i) \\ar[d] \\\\\n\t\\pi_1^{\\PhiIsocd}(X) \\ar[r]^-{\\prod_{i=1}^n p_i^{\\circ}}_-{\\sim}& \\prod_{i=1}^n \\pi_1^{\\FIsocd}(X_i)\n\t}\n\t\\]\n\tIn view of the above diagram, the pushforward functor for convergent $\\Phi$-isocrystals is compatible with the one for overconvergent $\\Phi$-isocrystals.\n\\end{remark}\n\n\\begin{cor}\n\tAny object $\\mathscr{E}$ of $\\mathbf{Isoc}^{\\dagger\\dagger}(X,\\Phi)$ (resp. $\\PhiIsocd(X)$, resp. $\\PhiIsoc(X)$) is a subobject (or quotient) of an object of the form $\\boxtimes_{i=1}^n \\mathscr{E}_i$, where each $\\mathscr{E}_i$ is an object of $\\mathbf{Isoc}^{\\dagger\\dagger}(X_i)$ (resp. $\\FIsocd(X_i)$, resp. $\\FIsoc(X)$).\n\\end{cor}\n\\begin{proof}\n\tIt suffices to show the assertion about the quotient. By induction, we can reduce to the case where $n=2$. \n\tWe set $G=\\pi_1^{\\mathbf{Isoc}^{\\dagger\\dagger}}(X,\\Phi), G_i=\\pi_1^{\\Isocd}(X_i)$.\n\tLet $V$ be a representation of $G\\simeq G_1\\times G_2$ corresponding to $\\mathscr{E}$. We have morphisms of $G_1\\times G_2$-representations:\n\t\\[\n\t(V\\otimes V^{\\vee})^{G_2}\\otimes V \\hookrightarrow V\\otimes V^{\\vee}\\otimes V \\xrightarrow{\\id_V\\otimes \\ev} V,\n\t\\]\n\twhere $G_1$ only acts at the first component of $V\\otimes V^{\\vee}\\otimes V$ and $G_2$ acts diagonally. \n\tBy composing with $V\\xrightarrow{\\coev\\otimes \\id} V\\otimes V^{\\vee}\\otimes V$, we see that the above composition is surjective. This finishes the proof. \t\n\\end{proof}\n\n\\section{A variant for $\\widehat{\\Pi}^{\\mot}$} \\label{s:motivic}\n\nIn \\cite{Dri18}, Drinfeld considered the $\\ell$-adic pro-semisimple completion of the \\'etale fundamental group and showed this object is independent of $\\ell$ up to conjugation by elements of the neutral connected component. \nIn this section, we incorporate Drinfeld's lemma in that setting.\n\n\tThroughout \\S\\ref{s:motivic}, let $X_i$ be smooth geometrically connected $k$-schemes for $i=1,\\ldots,n$ and $X:=\\prod_{i=1}^n X_i$. \n\n\t\\subsection{On semisimplicity of the monodromy group}\nIn this subsection, we discuss the semisimplicity of the monodormy group for an overconvergent $\\Phi$-isocrystal following \\cite{WeilII,Cr92}. \n\nA \\textit{$n$-twist} is a $\\Phi$-isocrystal on $\\Spec(k)\/K$ of rank 1.\nLet $\\chi=(\\chi_i)_{i=1}^n$ be a $n$-twist and $\\mathscr{E}$ an overconvergent $\\Phi$-isocrystal. \nWe denote by $\\mathscr{E}(\\chi)$ the tensor product $\\mathscr{E}\\otimes f^*(\\chi)$, where $f:X\\to \\Spec(k)$ is the structure morphism. \nWe first prove a result in the rank one case, generalizing a result of Abe \\cite[Lemma 6.1]{Abe15}.\n\n\n\\begin{lemma} \\label{p:rankone}\n\tLet $\\mathscr{E}$ be an overconvergent $\\Phi$-isocrystal of rank one over $X$. \n\tThen there exists a $n$-twist $\\chi$ such that $\\mathscr{E}(\\chi)$ has finite order. \n\\end{lemma}\n\n\\begin{proof}\n\tAfter twisting, we may assume $\\mathscr{E}$ is unit-root.\n\tLet $\\rho$ be the character of $\\pi_1(X,\\Phi,\\overline{x})$ associated to $\\mathscr{E}$ (Proposition \\ref{P:Crew partial Frob}) and $\\rho_i$ the restriction of $\\rho$ to $\\pi_1(X_i,\\overline{x})$ (Theorem \\ref{t:Drinfeld lemma origin}). \n\tWe may shrink each $X_i$ and in particular, we may assume that $X_i$ admits a smooth compactification $X_i\\to \\overline{X}_i$ such that the complement is a simple normal crossing divisor. \n\tUsing \\cite[Theorem 4.3]{Shi11}, \\cite[Theorem 2]{KL} and the same argument of \\cite[Corollary 4.13]{crew-f}, one can show that some power $\\mathscr{E}^{\\otimes N}$ is geometrically trivial, i.e. its underlying overconvergent isocrystal is trivial. \n\tThen $\\rho^{\\otimes N}$ factors through the quotient $\\pi_1(X,\\Phi,\\overline{x})\\to \\widehat{\\mathbb{Z}}^n$ and each $\\rho_i^{\\otimes N}$ is also geometrically trivial.\n\tHence the unit-root convergent $F$-isocrystal $\\mathscr{E}_i$ over $X_i$ associated to $\\rho_i$ is overconvergent. \n\tWe may take suitable twists $\\chi_i$ such that each $\\mathscr{E}_i(\\chi_i)$ has finite order \\cite[Lemma 6.1]{Abe15}. Then the assertion follows. \n\\end{proof}\n\nThis allows us to conclude the following corollary.\n\\begin{cor}\n\tLet $\\mathscr{E}$ be an overconvergent $\\Phi$-isocrystal over $X$. \n\tAfter taking a finite extension of $K$, there exist an integer $m\\ge 1$, $n$-twists $\\chi_i$ for $1\\le i\\le m$, and a decomposition of the semisimplification of $\\mathscr{E}$:\n\t\\[\n\t\\mathscr{E}^{ss}\\simeq \\bigoplus_{i=1}^m \\mathscr{F}_i(\\chi_i),\n\t\\]\n\twhere $\\mathscr{F}_i$ is an irreducible overconvergent $\\Phi$-isocrystal of finite order determinant for each $i$. \n\\end{cor}\n\n\\begin{defn}\nWe say $\\mathscr{E}$ is \\textit{untwisted}, if we can choose each $\\chi_i$ to be the trivial $\\Phi$-isocrystal on $\\Spec(k)$ in the above corollary. \n\\end{defn}\n\\begin{secnumber}\n\tLet $\\mathscr{E}$ be an object of $\\PhiIsocd(X)$. \n\tWe denote by $G_{\\geo}(\\mathscr{E})$ (resp. $G_{\\arith}(\\mathscr{E})$) its geometric (resp. arithmetic) monodromy group over $\\overline{\\mathbb{Q}}_p$, which is defined by the Tannakian full subcategory of $\\PhiIsocd(X)$ (resp. $\\Isocd(X)$) whose objects are subquotients of $\\mathscr{E}^{\\otimes m}\\otimes \\mathscr{E}^{\\vee, \\otimes n}$. \n\tRecall that the radical of $G_{\\geo}(\\mathscr{E})$ is unipotent \\cite[Theorem 4.9]{Cr92}. \n\n\tGiven a field $E$, by a \\textit{semisimple} group (resp. \\textit{reductive} group) over $E$, we mean an algebraic group of finite type over $E$ whose neutral component is semisimple (resp. reductive). \n\t\n\tBy a similar argument to \\cite[Theorem 3.4.7]{DZ20}, we conclude the following result for $\\Phi$-isocrystals. \n\\end{secnumber}\n\n\\begin{prop} \\label{p:radical unipotent}\n\tThe following properties are equivalent:\n\n\t\\begin{itemize}\n\t\t\\item[(i)] The neutral component of $G_{\\arith}(\\mathscr{E})\/G_{\\geo}(\\mathscr{E})$ is unipotent. \n\n\t\t\\item[(ii)] The radical of $G_{\\arith}(\\mathscr{E})$ is unipotent. \n\n\t\t\\item[(iii)] The object $\\mathscr{E}$ is untwisted. \n\t\\end{itemize}\n\\end{prop}\n\n\\begin{cor} \\label{c:semisimple monodromy}\n\tLet $\\mathscr{E}$ be a semisimple object of $\\PhiIsocd(X)$.\n\n\t\\textnormal{(i)} The geometric monodromy group $G_{\\geo}(\\mathscr{E})$ is semisimple. \n\n\t\\textnormal{(ii)} If the determinant of each irreducible component of $\\mathscr{E}$ has finite order, then $G_{\\arith}(\\mathscr{E})$ is semisimple. \t\n\\end{cor}\n\n\\begin{proof}\n\tAssertion (i) follows from the fact that the radical of $G_{\\geo}(\\mathscr{E})$ is unipotent and the argument of \\cite[Corollaire 1.3.9]{WeilII}.\n\tAssertion (ii) can be shown in a similar way using Proposition \\ref{p:radical unipotent}. \n\\end{proof}\n\n\\subsection{Pro-semisimple completion of the fundamental group of a smooth variety with partial Frobenius}\n\n\\begin{secnumber}\n\tLet $E$ be an algebraically closed field. \n\tWe refer to \\cite[\\S 2.1.1]{Dri18} for the notion of pro-semisimple groups and pro-reductive groups. \n\tFollowing \\cite[\\S 1.2.3]{Dri18}, $\\Pred(E)$ denotes the groupoid whose objects are pro-reductive groups over $E$ and whose morphisms are as follows: a morphism $G_1\\to G_2$ is an isomorphism of group schemes $G_1\\xrightarrow{\\sim} G_2$ defined up to composing with automorphisms of $G_2$ of the form $x\\mapsto gxg^{-1}$, $g\\in G_2^{\\circ}$. \n\tLet $\\Pss(E)\\subset \\Pred(E)$ be the full subcategory formed by pro-semisimple groups. \n\nFor any pro-algebraic group $G$ over $E$, we denote by $G^{\\red}$ (resp. $G^{\\semis}$) its pro-reductive (resp. pro-semi-simple) quotient. \n\\end{secnumber}\n\n\\begin{secnumber}\n\tFix an algebraic closure $\\overline{\\mathbb{Q}}$ of $\\mathbb{Q}$. Let $\\lambda$ be a non-Archimedean place of $\\overline{\\mathbb{Q}}$, $\\ell$ the prime that $\\lambda$ divides $\\ell$, $\\overline{\\mathbb{Q}}_{\\lambda}$ the direct limit of $E_{\\lambda}$ for subfields $E\\subset \\overline{\\mathbb{Q}}$ finite over $\\mathbb{Q}$. \n\tLet $(\\widetilde{X},\\Phi)$ (resp. $\\widetilde{X}_i$) be the universal cover of the category $\\mathcal{C}(X,\\Phi)$ \\eqref{sss:CXPhi} (resp. the universal \\'etale cover of $X_i$) and $\\Pi(X,\\Phi)$ the automorphism group of $(\\widetilde{X},\\Phi)\/X$, which is isomorphic to $\\pi_1(X,\\Phi,\\overline{x})$ after choosing a base point. \n\tWhen $n=1$, we write $\\Pi(X)$ for $\\Pi(X,\\Phi)$.\n\t\n\tFollowing Drinfeld, we define a pro-semisimple group $\\widehat{\\Pi}_{\\lambda}(X,\\Phi)$ over $\\overline{\\mathbb{Q}}_{\\lambda}$ as follows:\n\t\\begin{itemize}\n\t\t\\item[(i)] When $\\ell=p$, we denote by $\\mathcal{T}_{\\lambda}(X,\\Phi)$ the full subcategory of $\\PhiIsocd(X)$ of semisimple objects $M$ such that the determinant of each irreducible component of $M$ has finite order. \n\tThis category is a Tannakian full subcategory of $\\PhiIsocd(X)$ and we denote by $\\widehat{\\Pi}_{\\lambda}(X,\\Phi)$ its Tannakian group. \n\tBy Corollary \\ref{c:semisimple monodromy}, the quotient $\\pi_1^{\\PhiIsocd}(X)\\to \\widehat{\\Pi}_{\\lambda}(X,\\Phi)$, induced by the inclusion functor, identifies with the pro-semisimple quotient of $\\pi_1^{\\PhiIsocd}(X)$. \n\n\\item[(ii)] When $\\ell\\neq p$, we define $\\widehat{\\Pi}_{\\ell}(X,\\Phi)$ to be the $\\ell$-adic pro-semisimple completion of $\\Pi(X,\\Phi)$ and set $\\widehat{\\Pi}_{\\lambda}(X,\\Phi)=\\widehat{\\Pi}_{\\ell}(X,\\Phi)\\otimes_{\\mathbb{Q}_{\\ell}}\\overline{\\mathbb{Q}}_{\\lambda}$ \\cite[\\S 1.2.1]{Dri18}. \n\t\n\tThere is an equivalent definition: we define a full subcategory $\\mathcal{T}_{\\lambda}(X,\\Phi)$ of the category of lisse $\\overline{\\mathbb{Q}}_{\\lambda}$-sheaves on $X$ with a partial Frobenius structure, formed by semisimple objects $M$ such that the determinant of each irreducible component of $M$ has finite order as in (i). \n\tIt is a Tannakian category over $\\overline{\\mathbb{Q}}_{\\lambda}$ and the associated Tannakian group is isomorphic to $\\widehat{\\Pi}_{\\lambda}(X,\\Phi)$ (c.f. \\cite[\\S 3.6]{Dri18} in the case $n=1$). \n\\end{itemize}\n\nThe embedding $\\overline{\\mathbb{Q}}\\to \\overline{\\mathbb{Q}}_{\\lambda}$ induces an equivalence \\cite[Proposition 2.2.5]{Dri18}:\n\t\\begin{equation}\n\t\t\\Pss(\\overline{\\mathbb{Q}})\\xrightarrow{\\sim} \\Pss(\\overline{\\mathbb{Q}}_{\\lambda}).\n\t\t\\label{eq:equi alg closed}\n\t\\end{equation}\n\tWe denote by $\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)$ the object of $\\Pss(\\overline{\\mathbb{Q}})$ associated to $\\widehat{\\Pi}_{\\lambda}(X,\\Phi)$ by above equivalence.\n\tWhen $n=1$, we ignore $\\Phi$ from the above notations. \n\n\tBy Drinfeld's lemma, there exists a canonical isomorphism over $\\overline{\\mathbb{Q}}_{\\lambda}$:\n\t\\[\n\\widehat{\\Pi}_{\\lambda}(X,\\Phi)\\xrightarrow{\\sim} \\prod_{i=1}^n \\widehat{\\Pi}_{\\lambda}(X_i).\n\t\\]\n\tWhen $\\lambda$ divides $p$, it follows from taking pro-semisimple quotient of \\eqref{Tannakian Drinfeld lemma}; when $\\lambda$ does not divide $p$, it follows from \\eqref{eq:Drinfeld lemma pi1} and taking $\\ell$-adic pro-semisimple completion. \n\tVia the equivalence \\eqref{eq:equi alg closed}, we obtain an isomorphism over $\\overline{\\mathbb{Q}}$: \n\t\\begin{equation}\n\t\t\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)\\xrightarrow{\\sim} \\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda)}(X_i). \n\t\t\\label{eq:HPi prod}\n\t\\end{equation}\n\\end{secnumber}\n\n\\begin{secnumber} \\label{sss:tildePi}\n\tWe review some constructions in \\cite{Dri18}. \n\tFor each $i$, let $|\\widetilde{X}_i|^\\circ$ be the set of closed points of $\\widetilde{X}_i$. We have a canonical $\\Pi(X_i)$-equivariant map \\cite[(1.1)]{Dri18}:\n\t\\[\n\t|\\widetilde{X}_i|^\\circ \\to \\Pi(X_i),\\quad \\widetilde{x}\\mapsto F_{\\tilde{x}},\n\t\\]\n\twhere $F_{\\tilde{x}}$ denotes the geometric Frobenius automorphism at $\\widetilde{x}$.\n\n\tThe set $\\Pi(X_i)$ contains a dense subset $\\Pi_{F\\textnormal{-}}(X_i)$ formed by the elements $F_{\\tilde{x}}^n$ for $\\tilde{x}\\in |\\widetilde{X}_i|^\\circ$ and $n\\in \\mathbb{Z}_{\\ge 0}$. \n\tThe group $\\Pi(X_i)$ acts on $\\Pi_{F\\textnormal{-}}(X_i)$ by conjugation. \n\tWe denote by $\\widetilde{\\Pi}_{F\\textnormal{-}}(X_i):=\\mathbb{Z}_{\\ge 0}\\times |\\widetilde{X}|^\\circ$. One has the canonical $\\Pi(X_i)$-equivariant surjection:\n\t\\begin{equation} \\label{eq:Pi tilde}\n\t\\widetilde{\\Pi}_{F\\textnormal{-}}(X_i)\\twoheadrightarrow \\Pi_{F\\textnormal{-}}(X_i),\\quad (n,\\tilde{x}) \\mapsto F_{\\tilde{x}}^n.\n\\end{equation}\n\t\n\tFor any pro-reductive group $G$, let $[G]$ denote the GIT quotient of $G$ by the conjugation action of the neutral connected component $G^{\\circ}$. \n\tBy Theorem \\ref{t:Drinfeld lemma origin}, we have an adjoint action of $\\Pi(X,\\Phi)$ on $[\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)]$ and a canonical $\\Pi(X,\\Phi)$-equivariant map $[\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)]\\twoheadrightarrow \\Pi(X,\\Phi)$. \n\n\tThe map \\eqref{eq:Pi tilde} has a canonical lift to a $\\Pi(X_i)$-equivariant map (c.f. \\cite[\\S 7.2.4, 7.3.5]{Dri18})\n\t\\begin{equation}\n\t\t\\widetilde{\\Pi}_{F\\textnormal{-}}(X_i)\\to [\\widehat{\\Pi}_{(\\lambda)}(X_i)](\\overline{\\mathbb{Q}}_{\\lambda}),\\quad (n,\\tilde{x})\\mapsto F_{\\tilde{x}}^n. \n\t\t\\label{eq:Pitilde Hat}\n\t\\end{equation}\n\tBy the companion theorem \\cite{abe-companion, AE16, kedlaya-companionI, kedlaya-companionII}, the above map factors as \\cite[Corollary 7.4.2]{Dri18}\n\t\\[\n\t\\widetilde{\\Pi}_{F\\textnormal{-}}(X_i)\\twoheadrightarrow \\Pi_{F\\textnormal{-}}(X_i) \\to [\\widehat{\\Pi}_{(\\lambda)}(X_i)](\\overline{\\mathbb{Q}}) \\hookrightarrow [\\widehat{\\Pi}_{(\\lambda)}(X_i)](\\overline{\\mathbb{Q}}_{\\lambda}).\n\t\\]\n\tFor any non-Archimedean place $\\lambda$ of $\\overline{\\mathbb{Q}}$, we have a diagram of sets:\n\t\\[\n\t\\Pi_{F\\textnormal{-}}(X_i)\\to [\\widehat{\\Pi}_{(\\lambda)}(X_i)](\\overline{\\mathbb{Q}}) \\twoheadrightarrow \\Pi(X_i).\n\t\\]\n\t\n\tVia the isomorphism \\eqref{eq:HPi prod}, we obtain a diagram of sets:\n\t\\begin{equation}\n\t\t\\prod_{i=1}^n \\Pi_{F\\textnormal{-}}(X_i) \\to [\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)](\\overline{\\mathbb{Q}})\\twoheadrightarrow \\Pi(X,\\Phi). \n\t\t\\label{eq:Frob seq}\n\t\\end{equation}\n\t\nThe result of \\cite[Theorem 1.4.1]{Dri18} can be generalized as follow. \n\\end{secnumber}\n\n\\begin{prop} \\label{P:l-independence}\n\tLet $\\lambda, \\lambda'$ be non-Archimedean places of $\\overline{\\mathbb{Q}}$. \n\tThere exists a unique isomorphism \n\t\\[\n\t\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)\\xrightarrow{\\sim} \\widehat{\\Pi}_{(\\lambda')}(X,\\Phi)\n\t\\]\n\tin the category $\\Pss(\\overline{\\mathbb{Q}})$ which sends the diagram \\eqref{eq:Frob seq} to the corresponding diagram for $\\widehat{\\Pi}_{(\\lambda')}$. \n\t\\label{t:HPi unique}\n\\end{prop}\n\n\\begin{proof}\n\tWhen $n=1$ (i.e. we don't consider partial Frobenius structures), Drinfeld proved the above result for higher-dimensional smooth geometrically connected $k$-schemes if $\\lambda,\\lambda'$ do not divide $p$ and the result in the curve case when $\\lambda$ or $\\lambda'$ divides $p$ \\cite[Theorem 1.4.1, 5.2.1]{Dri18}. \n\tNow we can obtain the full generality of the theorem in the case $n=1$ using the recent breakthrough in the companion theorem for $p$-adic coefficients \\cite{AE16, kedlaya-companionI, kedlaya-companionII} over smooth $k$-schemes and the same argument of \\cite{Dri18}. \n\n\tIn general, the isomorphism $\\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda)}(X_i)\\xrightarrow{\\sim} \\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda')}(X_i)$, obtained in the case $n=1$, induces an isomorphism $\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)\\xrightarrow{\\sim} \\widehat{\\Pi}_{(\\lambda')}(X,\\Phi)$, which fits into the following diagram:\n\t\\[\n\t\\xymatrix{\n\t\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)\\ar[r]^-{\\sim} \\ar[d]_{\\sim} & \\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda)}(X_i) \\ar[d]^{\\sim}\\\\\n\t\t\\widehat{\\Pi}_{(\\lambda')}(X,\\Phi) \\ar[r]^-{\\sim}& \\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda')}(X_i)\n\t}\n\t\\]\n\tIn view of the definition of \\eqref{eq:Frob seq}, the required properties and the uniqueness follows from the case $n=1$. \n\\end{proof}\n\n\\begin{cor}\n\tThe neutral component $\\widehat{\\Pi}^{\\circ}_{(\\lambda)}(X,\\Phi)$ is simply connected. \n\\end{cor}\n\n\\begin{proof}\n\tWe may assume that $\\lambda$ is coprime to $p$. Then the statement follows from isomorphism \\eqref{eq:HPi prod} and \\cite[Proposition 3.3.4]{Dri18}. \n\\end{proof}\n\n\\begin{secnumber}\n\tLet $\\lambda$ be a non-Archimedean place of $\\overline{\\mathbb{Q}}$. \n\tIn \\cite[\\S~6]{Dri18}, Drinfeld defined a pro-reductive group $\\widehat{\\Pi}^{\\mot}_{(\\lambda)}$ over $\\overline{\\mathbb{Q}}$ (independent of $\\lambda$ up to unique isomorphisms) and introduced an unconditional definition of motivic Langlands parameters proposed by V. Lafforgue \\cite{lafforgue-v}. \n\t\n\tNow we introduce its variant $\\widehat{\\Pi}^{\\mot}_{(\\lambda)}(X,\\Phi)$ with partial Frobenius. \n\tLet $\\mathcal{W}_p\\subset \\overline{\\mathbb{Q}}^{\\times}$ denote the subgroup of $p$-Weil numbers. \n\tFor every $n\\ge 1$, we set $D_n=\\Hom(\\mathcal{W}_p,\\mathbb{G}_m^n)$ and $D_{n,\\lambda}=D_n\\otimes_{\\overline{\\mathbb{Q}}}\\overline{\\mathbb{Q}}_{\\lambda}$ \\cite[\\S~6.1]{Dri18}. \n\tNote that the group $\\pi_0(D_n)$ identifies with the group $\\Hom(\\mu_{\\infty}(\\overline{\\mathbb{Q}}),\\mathbb{G}_m^n)\\simeq \\widehat{\\mathbb{Z}}^n$. \n\n\tOn the other hand, we have a canonical surjection:\n\t\\[\n\t\\Pi(X,\\Phi)\\twoheadrightarrow \\Gal(\\overline{k}\/k)^n \\simeq \\widehat{\\mathbb{Z}}^n.\n\t\\]\n\n\tFollowing \\cite[\\S 6.1]{Dri18}, we define $\\widehat{\\Pi}_{(\\lambda)}^{\\mot}(X,\\Phi)$ as the fiber product of $\\widehat{\\Pi}_{(\\lambda)}(X,\\Phi)$ and $D_{n}$ over $\\widehat{\\mathbb{Z}}^{n}$. \n\tThen we can upgrade the isomorphism \\eqref{eq:HPi prod} to be an isomorphism in $\\Pred(\\overline{\\mathbb{Q}})$:\n\t\\begin{equation}\n\t\t\\widehat{\\Pi}_{(\\lambda)}^{\\mot}(X,\\Phi)\\xrightarrow{\\sim} \\prod_{i=1}^n \\widehat{\\Pi}_{(\\lambda)}^{\\mot}(X_i).\n\t\t\\label{eq:Drinfeld lemma mot}\n\t\\end{equation}\n\tNote that the above isomorphism is independent of $\\lambda$ by Proposition \\ref{t:HPi unique}. \n\\end{secnumber}\n\n\n\\section{Local Drinfeld's lemma over polyannuli}\n\nFor a connected subinterval $I$ of $(0,\\infty)$, we denote by $A_K[I]$ the rigid annulus $|t|\\in I$ over $K$. \nRecall that the Robba ring $\\mathcal{R}$ over $K$ is defined as\n\\[\n\\mathcal{R}:=\\lim_{\\varepsilon\\to 1} \\mathscr{O}(A_K[\\varepsilon,1[).\n\\]\nLet $n$ be a positive integer. We consider the $n$-fold Robba ring $\\mathcal{R}^n$ over $K$:\n\\begin{eqnarray}\n\t\\mathcal{R}^n &:=& \\lim_{\\varepsilon_1,\\ldots,\\varepsilon_n\\to 1} \\mathscr{O}(\\prod_{i=1}^n A_K[\\varepsilon_i,1[),\\nonumber \\label{eq:Robba-n}\\\\\n\t&=& \\{\\sum_{I=(i_1,\\ldots,i_n)} a_I \\underline{t}^I ;~ |a_I|\\underline{\\rho}^I \\to 0 ~(|i_1|+\\cdots+|i_n|\\to \\infty), \\forall \\rho_i\\in [\\varepsilon_i,1[ \\textnormal{ for some } \\varepsilon_i\\in ]0,1[ \\}, \\nonumber \n\\end{eqnarray}\nwhere $\\underline{t}^I=t_1^{i_1}\\cdots t_n^{i_n}$ and $\\rho^I=\\rho_1^{i_1}\\cdots\\rho_n^{i_n}$. \nFor any algebraic extension $L\/K$, we set $\\mathcal{R}^n_{L}:=\\mathcal{R}^n\\otimes_{K}L$. \n\nLet $h$ be an integer $\\ge 1$, and $\\sigma$ a continuous automorphism of $\\overline{\\mathbb{Q}}_p$ lifting the $h$-th Frobenius on $\\overline{k}$ and preserving each finite extension $L$ of $K$. \nWe denote by $\\phi_i:\\mathcal{R}^n\\to \\mathcal{R}^n$ the $K$-linear endomorphism defined by $t_i\\mapsto t_i^{p^h}$ and $t_j\\mapsto t_j$ if $j\\neq i$. \n\nWe consider the category $\\MIC(\\mathcal{R}^n\/K)$ of free $\\mathcal{R}^n$-modules of finite rank equipped with an integrable $K$-linear connection and we denote by $\\MIC^{\\uni}(\\mathcal{R}^n\/K)$ its full subcategory consisting of unipotent objects, which is isomorphic to a successive extension of the trivial connection over $\\mathcal{R}^n$. \n\n\\textit{A partial Frobenius structure of order $h$ (resp. a partial Frobenius structure)} on an object $(M,\\nabla)$ of $\\MIC(\\mathcal{R}^n\/K)$ consists of isomorphisms $\\varphi_i:(\\sigma\\circ\\phi_i)^*(M,\\nabla)\\simeq (M,\\nabla)$ of $\\MIC(\\mathcal{R}^n\/K)$ commuting with each other (resp. without specifying order $h$). \nWe denote by $\\MIC(\\mathcal{R}^n\/K,\\Phi)$ the full subcategory of $\\MIC(\\mathcal{R}^n\/K)$ consisting of objects whose irreducible sub-quotients can be equipped with a partial Frobenius structure (of order $h'$ with $h|h'$). \nNote that a unipotent object lies in $\\MIC(\\mathcal{R}^n\/K,\\Phi)$. \n\nLet $K^{\\ur}$ be the maximal unramified extension of $K$ in $\\overline{\\mathbb{Q}}_p$. \nWe can extend above definition to free $\\mathcal{R}^n_{K^{\\ur}}$-modules of finite rank with an integrable $K^{\\ur}$-linear connection. \nNote that an object of $\\MIC(\\mathcal{R}^n_{K^{\\ur}}\/K^{\\ur})$ comes from the extension of scalars of an object of $\\MIC(\\mathcal{R}^n_{L}\/L)$ for an unramified finite extension $L\/K$. \n\nLet $\\mathcal{K}=k( (t))$, $G_{\\mathcal{K}}=\\Gal(\\mathcal{K}^{\\sep}\/\\mathcal{K})$ and $I_{\\mathcal{K}}$ the inertia subgroup of the Galois group $G_{\\mathcal{K}}$. \nWe reformulate the local monodromy theorem for $\\MIC(\\mathcal{R}^n\/K,\\Phi)$ \\cite[Theorem 3.3.6]{kedlaya-mono-rel} as follows; this generalizes Andr\\'e's result \\cite[Th\\'eor\\`eme 7.1.1]{And02II} to polyannuli. \n\n\\begin{theorem}\n\t\\label{t:local Drinfeld lemma}\n\tThe category $\\MIC(\\mathcal{R}^n_{K^{\\ur}}\/K^{\\ur},\\Phi)$ is a neutral Tannakian category over $K^{\\ur}$ and its Tannakian group is isomorphic to \n\t\\[\n\t(I_{\\mathcal{K}}\\times \\mathbb{G}_{a})^n.\n\t\\]\n\\end{theorem}\n\n\\subsection{Unipotent connections on polyannuli}\n\n\\begin{prop} \\label{p:unipotent}\n\tThe functor $(V,N_1,\\ldots,N_n) \\mapsto (V\\otimes_K \\mathcal{R},\\nabla_{\\mathcal{N}})$, where the connection $\\nabla_{\\mathcal{N}}$ is defined by:\n\t\\[\n\t\\nabla_{\\mathcal{N}}(v \\otimes 1)= \\sum_{i=1}^n N_iv\\otimes \\frac{dt_i}{t_i},\n\t\\]\n\tinduces an equivalence of tensor categories between the category of finite dimensional $K$-vector spaces with $n$ commuting nilpotent operators $\\mathcal{N}=(N_1,\\ldots,N_n)$ and the category $\\MIC^{\\uni}(\\mathcal{R}^n\/K)$. \n\\end{prop}\n\n\\begin{proof}\n\tWe construct a quasi-inverse of the above functor. Let $(M,\\nabla)$ be a unipotent connection on a polyannulus $\\prod_{i=1}^n A_K[\\varepsilon_i,1[$. \n\tThe operators $\\partial_i=\\nabla(t_i \\frac{d}{d t_i})$ on $M$ commute with each other; set $\\partial:=\\partial_1\\circ\\cdots \\circ \\partial_n$. \n\tSince $(M,\\nabla)$ is unipotent, the $K$-vector space $V:=\\cup_{n\\ge 1} (\\Ker \\partial)^n$ has the same rank as $M$. \n\tWe claim that the operator $\\partial_i$ is nilpotent on $V$. \n\tIndeed, if $x$ is a point of the polyannulus $\\prod_{j\\neq i} A_K[\\varepsilon_j,1[$, then its fiber $(M_x,\\nabla_x)$ is a unipotent connection over the annulus $A_{K(x)}[\\varepsilon_i,1[$. \n\tWe deduce that $\\partial_i$ is nilpotent on $V\\otimes_K K(x)$ and the claim follows. \n\tThen\n\t\\[\n\t(M,\\nabla)\\mapsto (\\cup_{n\\ge 1} (\\Ker \\partial)^n, \\partial_1,\\ldots,\\partial_n) \n\t\\]\n\tdefines a quasi-inverse functor. The proposition follows. \n\\end{proof}\n\n\\begin{remark}\n\tUnlike the case of an annulus \\cite[Lemma 4.3]{Mat02}, a unipotent object on a polyannulus may not admit a Frobenius structure. \n\\end{remark}\n\n\\begin{secnumber} \\textit{Proof of Theorem \\ref{t:local Drinfeld lemma}.}\n\tLet $(M,\\nabla)$ be an object of $\\MIC(\\mathcal{R}^n_{K^{\\ur}}\/K^{\\ur},\\Phi)$ defined on a polyannulus $\\prod_{i=1}^n A_{K^{\\ur}}[\\varepsilon_i,1[$. \n\tBy \\cite[Theorem 3.3.6]{kedlaya-mono-rel}, there exist eligible \\'etale covers $\\{Y_i\\to A_{K^{\\ur}}[\\varepsilon_i,1[\\}_{i=1}^n$ such that the pullback of $(M,\\nabla)$ to $\\prod_{i=1}^n Y_i$ is unipotent. \n\tThen by Proposition \\ref{p:unipotent}, we obtain a fiber functor:\n\t\\[\n\t\\omega: \\MIC(\\mathcal{R}^n_{K^{\\ur}}\/K^{\\ur},\\Phi)\\to \\Vect_{K^{\\ur}},\n\t\\]\n\tby forgetting nilpotent operators. \n\tThis makes $\\MIC(\\mathcal{R}^n_{K^{\\ur}}\/K^{\\ur},\\Phi)$ into a neutral Tannakian category over $K^{\\ur}$. \n\tIn this way, we obtain a continuous action of $I_{\\mathcal{K}}^n$ on $\\omega(M,\\nabla)$ via $I_{\\mathcal{K}}^n\\to \\prod \\Aut(Y_i\/A_{K^{\\ur}}[I])$, which commutes with the action of nilpotent operators. \n\tThen the theorem follows. \\hfill \\qed \n\\end{secnumber}\n\n\\subsection{Construction of Weil--Deligne representations}\n\nIn this subsection, we briefly review representations of a self-product of the Weil group of $\\mathcal{K}$ associated to differential modules with a partial Frobenius structure on polyannuli, following \\cite{Mar08}. \n\n\\begin{secnumber}\n\tRecall that $q=p^s$ is the cardinality of $k$. \n\tWe denote by $\\Phi\\textnormal{-}\\mathbf{MIC}(\\mathcal{R}^n\/K)$ the category of free modules over $\\mathcal{R}^n$ equipped with an integrable $K$-linear connection and a partial Frobenius structure of order $s$. \n\n\tWe denote by $\\Del_{K^{\\ur}}(G_{\\mathcal{K}}^n)$ the category of triples $(V,\\varphi_1,\\ldots,\\varphi_n,N_1,\\ldots,N_n)$ consisting of a continuous semilinear representation of $G_{\\mathcal{K}}^n$ on a finite dimensional $K^{\\ur}$-vector space $V$ (with discrete topology), $\\sigma$-semilinear equivariant Frobenius isomorphisms $\\{\\varphi_i\\}_{i=1}^n$ on $V$ commuting with each other, and equivariant monodromy operators $\\{N_i:V\\to V\\}_{i=1}^n$ commuting with each other satisfying $N_i\\varphi_j=q^{\\delta_{ij}}\\varphi_jN_i$ for $i,j=1,\\ldots,n$. \n\n\tThe proof of Theorem \\ref{t:local Drinfeld lemma} shows that there exists a canonical tensor functor:\n\t\\[\n\t\t\\Phi\\textnormal{-}\\mathbf{MIC}(\\mathcal{R}^n\/K) \\to \\Del_{K^{\\ur}}(G_{\\mathcal{K}}^n).\n\t\\]\n\\end{secnumber}\n\n\\begin{secnumber}\n\tLet $W_{\\mathcal{K}}$ be the Weil group of $\\mathcal{K}$ and $v_i:W_{\\mathcal{K}}^n\\to \\mathbb{Z}$ the projection on the $i$-th component. \n\tWe consider the category $\\mathbf{Rep}_{K^{\\ur}}(\\WD^n_{\\mathcal{K}})$ of pairs $(V,N_1,\\cdots,N_n)$ consisting of a continuous linear representation $\\rho$ of $W_{\\mathcal{K}}^n$ on a finite dimensional $K^{\\ur}$-vector space $V$ (with discrete topology), monodromy operators $\\{N_i\\}_{i=1}^n$ commuting with each other satisfying $N_i\\rho(g)=q^{v_i(g)}\\rho(g)N_i$ for $i=1,\\ldots,n$, and $g\\in W_{\\mathcal{K}}^n$.\n\tWhen $n=1$, this is the category of Weil--Deligne representations of $W_{\\mathcal{K}}$. \n\n\tWe have a Frobenius linearization functor\n\t\\[\n\t\tL: \\Del_{K^{\\ur}}(G_{\\mathcal{K}}^n)\\to \\mathbf{Rep}_{K^{\\ur}}(\\WD^n_{\\mathcal{K}}),\n\t\\]\n\tsending $(V,\\varphi_i,N_i)$ to a continuous linear representation $\\rho$ of $W_{\\mathcal{K}}^n$ on $V$ defined by \n\t\\[\n\t\\rho(g)(m)=g(\\prod_{i=1}^n \\varphi^{v_i(g)}_i(m)),\n\t\\]\n\ttogether with monodromy operators $\\{N_i\\}_{i=1}^n$. \n\tIn summary, we obtain a tensor functor:\n\t\\[\n\t\\Phi\\textnormal{-}\\mathbf{MIC}(\\mathcal{R}^n\/K) \\to \\mathbf{Rep}_{K^{\\ur}}(\\WD^n_{\\mathcal{K}}).\n\t\\]\n\\end{secnumber}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzeise b/data_all_eng_slimpj/shuffled/split2/finalzzeise new file mode 100644 index 0000000000000000000000000000000000000000..3ad6dea04cb7116be26d89a44b37a06404a71569 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzeise @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nWithin a fixed class of dark energy models, such as the cosmological constant\nor scalar field quintessence, various cosmological observables are all\ninterrelated by the properties of the class itself.\nThe narrower the class,\nthe higher the expected correlation between measurements of different\nobservables. Therefore, given a class of dark energy models, constraints from\none set of cosmic acceleration observables make predictions for other observables.\nFor example, it is well known that since the first release of WMAP data\n\\cite{Spergel_2003}, the Hubble constant in a spatially flat universe with a\ncosmological constant and cold dark matter ($\\Lambda$CDM) has been predicted\nto a precision better than it has yet been measured. Predictions like this one\ntherefore offer the opportunity for more precise measurements to falsify\nthe dark energy model (in this case, flat $\\Lambda$CDM)~\\cite{Hu_standards}.\n\nIn a previous paper (hereafter MHH) \\cite{PaperI}, we showed how this idea\ncan be generalized to additional acceleration observables and wider classes of\ndark energy models. Other observables include the expansion rate $H(z)$, the \ncomoving angular diameter distance $D(z)$, and the linear growth function\n$G(z)$. The model classes we considered include a cosmological constant, \nwith and without\nspatial curvature, and scalar field quintessence models, with and without \nearly dark energy and spatial curvature components. \nUsing forecasts for a Stage IV \\cite{DETF} SN sample and Planck CMB data, we found \nthat future data sets will provide numerous strong predictions that we \nmay use to attempt to falsify various acceleration paradigms.\n\n\\vspace{1cm}\nIn this paper, we evaluate the predictive power of {\\it current} measurements\nto constrain the expansion rate, distance, and growth as a function of\nredshift. Specifically, we consider current measurements of supernovae (SN),\nthe cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and\nthe Hubble constant ($H_0$). These predictions \ntarget the redshift ranges and required precision for future measurements\nseeking to rule out whole classes of models for cosmic acceleration.\n\nOur approach complements studies that seek to constrain an ever expanding set\nof parameters of the dark energy. The most ambitious analyses currently\nutilize $\\sim 5$ parameters to describe the dark energy equation of state\n$w(z)$ \\cite{Huterer_Cooray,Wang_Tegmark_2005,Riess_2006,Zunckel_Trotta,\n Sullivan_Cooray_Holz,Zhao_Huterer_Zhang,Zhao_Zhang:2009,Serra:2009}. We\ntake these studies in a new direction: rather than constraining parameters\nassociated with the equation of state, we propagate constraints from the data\ninto allowed ranges for $H(z)$, $D(z)$, $G(z)$, and auxiliary observables that\ncan be constructed from them through a principal component representation of\n$w(z)$ that is complete in these observables for $z<1.7$. This work goes\nbeyond previous studies that are similar in spirit\n(e.g.~\\cite{Kujat,Sahlen05,Huterer_Peiris,Chongchitnan_Efstathiou,ZhaKnoTys08})\nby directly applying constraints from current data sets\n to complete representations of several dark energy model classes and making\n concrete predictions for a number of observable quantities.\n\nThis paper is organized as follows.\nWe begin in Sec.~\\ref{sec:methods} with a discussion of the methodology of\npredicting observables within classes of dark energy models, including \ndescriptions of each of the acceleration observables, cosmological data sets, \nand model classes. We present our\npredictions from current data in Sec.~\\ref{sec:predict} \nand discuss the results in Sec.~\\ref{sec:discussion}.\n\n\n\\section{Methodology}\n\\label{sec:methods}\n\n\n\n\\subsection{Acceleration Observables}\n\\label{sec:obs}\n\nThere are two general types of acceleration observables: those related\nto the expansion history and geometry of the universe, and those related to the growth of structure.\nIn terms of a general evolution for the dark energy \nequation of state $w(z)$, \nthe expansion history observables are the Hubble expansion rate\n\\begin{eqnarray}\nH(z) &=& H_0 \\left[ \\Omega_{\\rm m} (1+z)^3 + \\Omega_{\\rm DE} f(z) + \\Omega_{\\rm K} (1+z)^2 \\right]^{1\/2},\n\\nonumber \\\\\n&& f(z) = \\exp\\left[3 \\int_0^z dz' \\frac{1+w(z')}{1+z'}\\right], \\label{eq:hz} \n\\end{eqnarray}\nwhere $\\Omega_{\\rm m}$ and $\\Omega_{\\rm DE}$ are the present matter and dark energy densities, \nrespectively, as fractions of the critical density for flatness, \nspatial curvature is parametrized by $\\Omega_{\\rm K}\\equiv 1-\\Omega_{\\rm m}-\\Omega_{\\rm DE}$, \nand the small contribution of radiation at $z\\sim 1$ is neglected; \nand the comoving angular diameter distance\n\\begin{equation}\nD(z) = \\frac{1}{(|\\Omega_{\\rm K}|H_0^2)^{1\/2}} S_{\\rm K}\\left[(|\\Omega_{\\rm K}|H_0^2)^{1\/2} \n\\int_0^z \\frac{dz'}{H(z')} \\right],\n\\label{eq:dist}\n\\end{equation}\nwhere the function $S_{\\rm K}(x)$ is equal to $x$ in a flat universe ($\\Omega_{\\rm K}=0$), \n$\\sinh x$ in an open universe ($\\Omega_{\\rm K}>0$), and $\\sin x$ in a closed universe ($\\Omega_{\\rm K}<0$).\nThe growth of\nlinear density perturbations $\\delta \\propto G a$\nis given by\n\\begin{equation}\nG'' + \\left(4+\\frac{H'}{H}\\right)G' + \\left[\n3+\\frac{H'}{H}-\\frac{3}{2}\\Omega_{\\rm m}(z)\\right]G = 0,\n\\label{eq:growth}\n\\end{equation}\nwhere primes denote derivatives with respect to $\\ln a$ and $\\Omega_{\\rm m}(z) = \\Omega_{\\rm m}\nH_0^2(1+z)^3\/H^2(z)$. We assume scales for which the dark energy density is\nspatially smooth compared with the matter and normalize $G(z)=1$ at $z=10^3$.\n\n\nThere are several auxiliary quantities related to the growth function that are also\ninteresting to examine. Since growth measurements like the evolution of the\ncluster abundance often compare the change in growth relative to the present, we\nalso consider a different normalization for the growth function,\n\\begin{equation}\nG_0(z)\\equiv \\frac{G(z)}{G(0)}.\n\\end{equation}\nVelocity field measurements, on the other hand, are sensitive to the growth {\\it rate}\n\\begin{equation}\nf(z) \\equiv 1 + \\frac{G'}{G}.\n\\end{equation}\nSpecifically, the amplitude of the velocity power spectrum can be measured\nfrom redshift space distortions and constrains $f(z)G(z)$ independently of galaxy bias (e.g.\\ see~\\cite{Percival_White}).\nFinally, given that the growth rate is approximately related to expansion\nhistory observables by $f(z) = [\\Omega_{\\rm m}(z)]^{\\gamma}$ where the growth index is\n$\\gamma \\approx 0.55$ for flat $\\Lambda$CDM\\ \\cite{WangSteinhardt,Linder_gamma} we\nalso consider predictions for\n\\begin{equation}\n\\gamma(z)\\equiv {\\ln[f(z)] \\over \\ln[\\Omega_{\\rm m}(z)]} \\,.\n\\label{eq:gamma_z}\n\\end{equation}\nNote however that $\\gamma(z)$ is not a direct observable but rather \nmust be inferred from a combination of measurements in a \nspecific dark energy context.\n\n\nWe ignore the influence of massive neutrinos throughout this study. \nThe effect of massive neutrinos on the growth of structure is\nsignificantly scale-dependent, \nbut on present linear scales well below the horizon, \n$k\\sim 0.01-0.1~h$~Mpc$^{-1}$, the growth suppression from a \nnormal neutrino mass hierarchy with $\\sum m_{\\nu}\\sim 0.05$~eV \\cite{nu_oscill}\nis $\\lesssim 1\\%$ in $G(z)$ and $f(z)G(z)$ and smaller for other \nobservables. The maximum decrement in growth from nearly-degenerate \nneutrinos with $\\sum m_{\\nu}\\sim 0.5$~eV (e.g.~\\cite{Reid_SDSSDR7}) \nis $\\sim 1-10\\%$ on these scales. In the predictions we present here, \nthese effects would appear as an additional ``early'' dark energy \ncomponent with $w\\approx 0$. Future precise measurements of $\\sum m_{\\nu}$ \nfrom independent data could be used to correct the growth predictions here \nby scaling them by the appropriate suppression factor.\n\n\n\n\n\\subsection{Constraints from Current Data}\n\\label{sec:data}\n\nThe main observational constraints we consider when making predictions for\nacceleration observables include relative distances at $z \\lesssim 1.5$ from\nType Ia SNe and absolute distances at $z_*=1090$ from the CMB, $z_{\\rm BAO}\\approx\n0.35$ from BAO, and $z_h\\approx 0.04$ from low-redshift SNe calibrated with\nmaser and Cepheid distances. Since low-$z$ distances mainly probe the Hubble\nconstant for smoothly varying $w(z)$, we refer to the low-$z$ SN\ncalibration as an $H_0$ constraint. The CMB data additionally constrain\nparameters that impact dark energy models such as the matter density $\\Omega_{\\rm m} h^2$\nand the fraction of dark energy density at recombination.\n\nIn the simplest classes of models, the SN and CMB data suffice to make\naccurate predictions for expansion and growth observables. In more complex\nclasses, BAO and $H_0$ constraints on distances are necessary. Even\nin these cases, predictive power is still retained in that measured distances\nto a few specific redshifts constrain $H(z)$, $D(z)$, and $G(z)$ at all redshifts.\nWe now describe each of these data sets in more detail.\n\nThe Type Ia SN sample we use is the Union compilation~\\cite{SCP_Union}. These\nSN observations measure relative distances, $D(z_1)\/D(z_2)$, over a range of\nredshifts spanning $0.015 \\leq z \\leq 1.551$, with most SNe at $z \\lesssim\n1$. We add the SN constraints using the likelihood code for the Union data\nsets \\cite{Union_like}, which includes estimated systematic errors for the SN\ndata~\\cite{SCP_Union}.\n\nFor the CMB, we use the most recent, 5-year release of data from the WMAP\nsatellite \\cite{Komatsu_2008,Nolta_2008,Dunkley_2008} employing the likelihood\ncode available at the LAMBDA web site \\cite{WMAP_like}. Unlike the CMB\ndistance priors on $D(z_*)$ and $\\Omega_{\\rm m} h^2$ used for the forecasts in MHH, the\nlikelihood used here contains the full information from the CMB angular power\nspectra; in particular this provides sensitivity to large fractions of\nearly dark energy at recombination as well as information about late-time dark\nenergy and spatial curvature from the ISW effect without necessitating\nadditional priors. We compute the CMB angular power spectra using the code\nCAMB \\cite{Lewis:1999bs,camb_url} modified with the parametrized\npost-Friedmann (PPF) dark energy module \\cite{PPF,ppf_url} to include \n models with general dark energy equation of state evolution\nwhere $w(z)$ may cross $w=-1$. Note that while our {\\it predictions} for\ngrowth observables apply to scales on which dark energy is smooth relative to\nmatter, the CAMB+PPF code self-consistently accounts for the effects of \nscale-dependent dark energy perturbations on the CMB anisotropies.\n\n\nThe BAO constraint we use is based on the measurement of the correlation\nfunction of SDSS Luminous Red Galaxies (LRGs) \\cite{Eisenstein}, which\ndetermines the distance and expansion rate at $z_{\\rm BAO}\\approx 0.35$ through the\ncombination $D_V(z) \\equiv [z D^2(z)\/H(z)]^{1\/3}$. We implement this\nconstraint by taking the volume average of this quantity, $\\langle D_V \\rangle$, over\nthe LRG redshifts, $0.16z_{\\rm max}$, we adopt a simple parametrization by \nassuming a constant equation of state, $w(z>z_{\\rm max})=w_{\\infty}$, \nrestricted to $-1\\leq w_{\\infty}\\leq 1$. The dark energy density at $z>z_{\\rm max}$ can be\nextrapolated from its value at $z_{\\rm max}$ as\n\\begin{equation}\n\\rho_{\\rm DE}(z) = \\rho_{\\rm DE}(z_{\\rm max})\\left(\\frac{1+z}{1+z_{\\rm max}}\\right)^{3(1+w_{\\infty})}.\n\\end{equation}\nFor more restricted model classes where we assume that there is \nno significant early dark energy, \nwe fix $w_{\\infty}=-1$ since a constant dark energy density \nrapidly becomes negligible relative to the matter density \nat increasing redshift.\nNote that the possibility of early dark energy is automatically \nincluded in the smooth $w_0-w_a$\nmodel class where the equation of state at high redshift is $w\\approx w_0+w_a$.\n\n\nIn addition to the dark energy parameters described above ($\\bm{\\theta}_{\\rm DE}$), we include\ncosmological parameters that affect the CMB angular power spectra but \nnot the acceleration observables ($\\bm{\\theta}_{\\rm nuis}$): the physical baryon density $\\Omega_{\\rm b} h^2$, \nthe normalization and tilt of the primordial curvature\nspectrum $\\Delta_{\\cal R}^2 = A_s (k\/k_0)^{n_s-1}$ with $k_0 = 0.05$~Mpc$^{-1}$,\nand the optical depth to reionization $\\tau$.\nThis brings our full set of parameters for $\\Lambda$CDM\\ to $\\bm{\\theta}_{\\Lambda}=\\bm{\\theta}_{{\\rm DE},\\Lambda}+\\bm{\\theta}_{\\rm nuis}$,\nand for quintessence and smooth $w_0-w_a$ dark energy\nwe define the analogous parameter sets with\n\\begin{eqnarray}\n\\bm{\\theta}_{{\\rm DE},\\Lambda} &=&\\{\\Omega_{\\rm m} h^2, \\Omega_{\\rm m}, \\Omega_{\\rm K}\\} \\,,\\nonumber\\\\\n\\bm{\\theta}_{\\rm DE,Q} &=& \\bm{\\theta}_{{\\rm DE},\\Lambda} + \\{\\alpha_1,\\ldots, \\alpha_{N_{\\rm max}}, w_{\\infty}\\} \\,,\\nonumber\\\\\n\\bm{\\theta}_{\\rm DE,S} &=& \\bm{\\theta}_{{\\rm DE},\\Lambda} + \\{w_0, w_a\\} \\,,\\nonumber\\\\\n\\bm{\\theta}_{\\rm nuis} &=& \\{ \\Omega_{\\rm b} h^2, n_s, A_s, \\tau \\} \\,,\n\\label{eq:parametersfull}\n\\end{eqnarray}\nwhere we count $\\Omega_{\\rm m}$ and $\\Omega_{\\rm K}$ as dark energy parameters\nsince $\\Omega_{\\rm DE}= 1-\\Omega_{\\rm m} - \\Omega_{\\rm K}$.\nNote that the Hubble constant is a derived parameter, $h=H_0\/(100~{\\rm km~s}^{-1}{\\rm\n Mpc}^{-1}) = (\\Omega_{\\rm m} h^2\/\\Omega_{\\rm m})^{1\/2}$. \nAlthough the observable predictions mainly depend on constraints on the \ndark energy parameters $\\bm{\\theta}_{\\rm DE}$, we include the additional ``nuisance'' \nparameters $\\bm{\\theta}_{\\rm nuis}$ due to degeneracies between \n$\\bm{\\theta}_{\\rm DE}$ and $\\bm{\\theta}_{\\rm nuis}$ parameters in current CMB data; \nthese nuisance parameters are marginalized over in our predictions \nfor acceleration observables.\nThe parameter sets and priors on the parameters for each model \nclass are summarized in Table~\\ref{tab:modelclasses}.\n\n\n\n\n\\subsection{MCMC Predictions}\n\\label{sec:mcmc}\n\n\nTo make predictions for the acceleration observables using constraints from current data, we\nuse a Markov Chain Monte Carlo (MCMC) likelihood analysis. Given a dark energy\nmodel class parametrized by $\\bm{\\theta}_{\\Lambda}$, $\\bm{\\theta}_{\\rm Q}$, or $\\bm{\\theta}_{\\rm S}$, the MCMC algorithm\nestimates the joint posterior distribution of cosmological parameters and\npredicted observables by sampling the parameter space and evaluating the\nlikelihood of each proposed model compared with the data described in\nSec.~\\ref{sec:data}\n(e.g.\\ see~\\cite{Christensen:2001gj,Kosowsky:2002zt,Dunetal05}). We use the\ncode CosmoMC \\cite{Lewis:2002ah,cosmomc_url} for the MCMC analysis.\n\nThe posterior distribution is obtained using Bayes' Theorem,\n\\begin{equation}\n{\\cal P}(\\bm{\\theta}|{\\bf x})=\n\\frac{{\\cal L}({\\bf x}|\\bm{\\theta}){\\cal P}(\\bm{\\theta})}{\\int d\\bm{\\theta}~\n{\\cal L}({\\bf x}|\\bm{\\theta}){\\cal P}(\\bm{\\theta})},\n\\label{eq:bayes}\n\\end{equation}\nwhere ${\\cal L}({\\bf x}|\\bm{\\theta})$ is the likelihood of the data ${\\bf x}$\ngiven the model parameters $\\bm{\\theta}$ and ${\\cal P}(\\bm{\\theta})$ is the\nprior probability density. The MCMC algorithm generates random draws from the\nposterior distribution that are fair samples of the likelihood surface.\nWe test convergence of the samples to a stationary distribution that\napproximates the joint posterior density ${\\cal P}(\\bm{\\theta}|{\\bf x})$ \nby applying a conservative Gelman-Rubin criterion \\cite{gelman\/rubin}\nof $R-1\\lesssim 0.01$ across a minimum of four chains for each model class.\n\nAs described in MHH, the MCMC approach allows us to straightforwardly\ncalculate confidence regions for the acceleration observables by computing\n$H(z)$, $D(z)$, $G(z)$ and the auxiliary observables \n$G_0(z)$, $f(z)G(z)$, and $\\gamma(z)$ for each MCMC sample\nusing Eqs.~(\\ref{eq:hz})$-$(\\ref{eq:gamma_z}). The posterior distribution of\nthe model parameters $\\bm{\\theta}$ thus maps onto a distribution of each\nacceleration observable at each redshift. These redshift-dependent\ndistributions of the expansion and growth observables form the predictions\nthat we describe in the next section.\n\n\n\\section{Dark Energy Model Predictions}\n\\label{sec:predict}\n\n\nIn this section, we show the predictions for growth and expansion\nobservables from the combined current CMB, SN, BAO, and $H_0$\nconstraints. Since plotting full distributions for the six observables \ndefine in Sec.~\\ref{sec:obs} at\nseveral different redshifts is impractical, we instead plot only the regions\nenclosing 68\\% and 95\\% of the models at each redshift, defined such that the number density of\nmodels is equal at the upper and lower limit of each region. \n(When describing the predictions, we will typically quote the 68\\% CL limits.)\nTo provide examples of\nfeatures of individual models that may not be apparent from the 68\\% and 95\\% CL limits,\nwe also plot the evolution of observables for the maximum\nlikelihood (ML) MCMC model within each model class. We caution, however, that the\nMCMC algorithm is designed to approximate the overall shape of the likelihood\nand is not optimized for precisely computing the ML\nparameters, so the ``best fit'' models shown here may be slightly displaced\nfrom the true ML points.\n\nIn most figures in this section, we compare the predictions for two model\nclasses, one of which is a subclass of the second, more general class (for\nexample, $\\Lambda$CDM\\ and quintessence). The potential to falsify the simpler class\nin favor of the more complex one is greatest where the two sets of predictions\ndiffer most, i.e.~where one class gives strong predictions and the other does\nnot.\n\n\n\\subsection{$\\Lambda$CDM}\n\nWe begin with the simplest and most predictive model class: flat $\\Lambda$CDM.\nSince $\\Omega_{\\rm K}=0$, this model has only two free dark energy parameters in\nEq.~(\\ref{eq:parametersfull}), $\\Omega_m$ and $\\Omega_{\\rm m} h^2$ (or $H_0$), \nproviding very little freedom to alter the acceleration observables at {\\it any} \nredshift as shown in Fig.~\\ref{fig:flcdm}:\n$H(z)$, $D(z)$, and $G(z)$ are currently predicted with a precision of $\\sim 2\\%$\n(68\\% CL) or better everywhere. The velocity observable $f(z)G(z)$ is predicted to better than 5\\% and the growth index $\\gamma$ to $0.1\\%$.\nThese predictions are more precise than\ncurrent measurements of the acceleration observables at any redshift.\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=flcdmpredict4.eps, width=2.5in}}\n\\caption{Flat $\\Lambda$CDM\\ predictions for growth and \nexpansion observables, showing the 68\\% CL (shading) and 95\\% CL (curves) \nregions allowed by current CMB, SN, BAO, and $H_0$ data. \nObservables include the linear growth function normalized in two \ndifferent ways, $G(z)$ equal to unity at high redshift and \n$G_0(z)=G(z)\/G(0)$; the product of the differential growth rate \nand the growth function $f(z)G(z)$; \nthe growth index $\\gamma(z)$ which relates $f(z)$ and $\\Omega_{\\rm m}(z)$; \nthe expansion rate $H(z)$; and the comoving distance $D(z)$ \n(scaled by a factor of 1\/10 in the lower panel).\nNote that the separation between the 68\\% and 95\\% CL regions \nis not visible where the observables are extremely well predicted, \ne.g. in the $\\gamma(z)$ predictions in the middle panel.\n}\n\\label{fig:flcdm}\n\\end{figure}\n\n\nThe strong predictions for flat $\\Lambda$CDM\\ arise largely due to CMB constraints: the two parameters\n$\\Omega_{\\rm m}$ and $H_0$ are tied together by the measurement of $\\Omega_{\\rm m} h^2$,\nand the remaining freedom in $H_0$ or the extragalactic distance scale is\nfixed by the measurement of the distance to $z_*$. \nHowever, given the present uncertainties in $\\Omega_{\\rm m} h^2$ and $D(z_*)$, \nthe addition of the other data (SN, BAO, and $H_0$) increases the \nprecision of the predictions by almost a factor of 2 \nrelative to WMAP constraints alone.\n\n \nThe flat $\\Lambda$CDM model is therefore highly falsifiable in that future measurements\nmay find that these quantities deviate substantially from the predictions. \nFor example, an $H_0$ measurement with $\\lesssim 2\\%$ accuracy would match the precision of the\npredictions and hence provide a sharp test of flat $\\Lambda$CDM. \nThese predictions are only a factor of $2-3$ weaker\nthan the Stage IV SN and CMB forecasts from MHH. Since flat $\\Lambda$CDM is the current\nstandard model of the cosmic expansion history and structure formation, falsifying it would represent\nthe most important observational breakthrough since the discovery of cosmic acceleration and would require revision of basic assumptions about the nature \nof dark energy, spatial curvature, or the theory of gravity. \n\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_lcdm_fg2.eps, width=3.5in}}\n\\caption{Predicted growth and expansion observables for \nnon-flat (dark blue) and flat (light gray) $\\Lambda$CDM, plotted \nrelative to the reference cosmology (the best fit model for flat $\\Lambda$CDM).\nHere and in subsequent figures, \n68\\% CL regions are marked by shading, 95\\% CL regions are bounded \nby solid curves, and red curves outlined in white show the \nbest fit model of the more general (dark blue) model class \n(in this case, non-flat $\\Lambda$CDM). \n}\n\\label{fig:lcdm}\n\\end{figure}\n\n\nGeneralizing the model to $\\Lambda$CDM with curvature increases the range of\npredictions by less than a factor of~2. In Fig.~\\ref{fig:lcdm}, we plot the\npredictions for flat and non-flat $\\Lambda$CDM\\ relative to the ML flat $\\Lambda$CDM\nmodel with $\\Omega_{\\rm m}=0.268$, $h=0.711$. Curvature opens up the ability to free the\nextragalactic distance scale from the constraints imposed by the CMB acoustic\npeak measurements. The tight constraints on SN, $H_0$, and BAO distances\nlimit this freedom. Since the forecasts from MHH used only the current BAO\nmeasurement and a weaker $H_0$ constraint as priors, the relative impact of\ncurvature here is substantially smaller. In particular, predictions of the\ngrowth function are nearly unchanged by curvature and still vary by less than\n$2\\%$.\nLikewise, $fG$ is nearly unaffected by curvature. Although\nthe growth index, $\\gamma(z)$, is not as perfectly determined for non-flat\n$\\Lambda$CDM, especially at high redshift, it is still predicted to better than 1\\%\nat $z \\lesssim 3$, and both $D(z)$ and $H(z)$ are predicted to better than\n$3\\%$. Any measurement that deviates by significantly more than these amounts\nwould prove that the dark energy is not a \ncosmological constant.\\footnote{A substantial decrement in \n growth from high redshifts,\n which in the context of our treatment would be interpreted as \n evidence for early dark energy thus falsifying $\\Lambda$CDM,\n could alternately indicate neutrinos with more than the minimal \n allowed masses.}\n\n\n\n\\subsection{Quintessence}\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_q_f_fg2.eps, width=3.5in}}\n\\caption{Flat quintessence models without early dark energy (dark blue) \nvs.\\ flat $\\Lambda$CDM\\ (light gray). Other aspects here and in later figures follow\nFig.~\\ref{fig:lcdm}.\n}\n\\label{fig:quint}\n\\end{figure}\n\nIf $\\Lambda$CDM is falsified, then in the context of dark energy \nwe must consider models with $w(z) \\ne -1$. Our\nnext class of models are therefore flat quintessence models \nwith $w(z)$ parametrized by 10 principal components at $z<1.7$, assuming\nno early dark energy (``$w_{\\infty}=-1$''). The predictions for acceleration observables \nwithin this model class are compared with the flat $\\Lambda$CDM\\ predictions\nin Fig.~\\ref{fig:quint}. \n\nInterestingly, the quintessence predictions are no longer centered on the\nflat $\\Lambda$CDM ML model. From the $H(z)$ predictions which mainly reflect\nvariation in evolution of the dark energy density, we see that on average the\ndata favor a smaller low-redshift ($z\\lesssim 0.5$) and larger\nintermediate-redshift ($0.5\\lesssim z\\lesssim 2$) dark energy density.\nCorrespondingly, the best fit growth function $G(z)$ of $\\Lambda$CDM is higher\nthan that of $\\sim 85\\%$ of the quintessence models in the chain.\nTherefore a measurement of the growth relative to high redshift that is\nsmaller than the $\\Lambda$CDM prediction by more than a few percent not only\nrules out a cosmological constant but actually favors these quintessence models.\nThe additional freedom in growth opens up predictions for $\\gamma$\nto include $2-3\\%$ deviations at $z\\lesssim 1$.\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=data_bestfit_hd.eps, width=3.5in}}\n\\caption{Upper panel: Comparison of distance constraints from SN data and best\n fit models, plotted relative to the best fit $H_0D(z)$ for flat $\\Lambda$CDM\\ (dotted\n line). Blue points with error bars show the Union SN data in redshift\n bins of width $\\Delta \\log z = 0.05$. \nThe best fit model for flat quintessence without\n early dark energy is plotted as a dashed curve, and the solid curve \nshows how the relative distances are affected by smoothing $w(z)$ \nfor this model by a Gaussian of width $\\sigma_z=0.1$. \n The full distribution of relative distance predictions for this quintessence \n model class is also\n shown with light gray shading (68\\% CL) and curves (95\\% CL). \nLower panel: $w(z)$ for each of the models from the upper panel. }\n\\label{fig:databf}\n\\end{figure}\n\n\n\nMany of the shifts in the predictions relative to flat $\\Lambda$CDM\\ are \nreflected in the evolution of $w(z)$ in the maximum likelihood model for \nflat quintessence without early dark energy. The ML model in this class marginally \nimproves the fit to the current data sets relative to the \n$\\Lambda$CDM\\ ML model, largely due to\nvariations in the SN data with redshift that are fit \nmarginally better by dynamical dark energy than by a cosmological constant.\nFigure~\\ref{fig:databf} compares ML models, quintessence \npredictions, and relative distance constraints from the Union SN data sets \nat $z\\lesssim 1$.\nFreedom in $w(z)$ at these redshifts allows changes in the dark energy \ndensity to improve the fit to SN distances by $-2\\Delta\\ln\\mathcal{L}\\sim 4.5$.\nHowever, some of this improvement is due to the large oscillations in \nthe equation of state at $z\\sim 0.1$, which are allowed to violate \nthe $-1\\leq w\\leq 1$ bound due to the conservative implementation \nof the quintessence prior on PC amplitudes described in Sec.~\\ref{sec:pcs}.\nSmoothing the ML $w(z)$ by a Gaussian with width $\\sigma_z\\sim 0.1$ \nor requiring $w(z)$ to satisfy stricter quintessence bounds\nreduces the improvement relative to $\\Lambda$CDM\\ to $-2\\Delta\\ln\\mathcal{L}\\sim 2$, \nbut has little effect on the overall distributions of the predicted \nobservables.\n\nAlthough differences in the ML models cause quintessence to not be centered\naround $\\Lambda$CDM, the allowed {\\it width} of quintessence predictions around\nthe maximum likelihood relative to $\\Lambda$CDM\\ follows the expectations of the Stage IV predictions\nfrom MHH except for being weaker by a factor of $2-3$. The PCs allow for\noscillatory variations in $H(z)$, $f(z)G(z)$, and $\\gamma(z)$ at $z<1$ that would\nnot be readily observable with expansion history or growth measures due to\nlimited resolution in redshift. On the other hand, $G(z)$, $G_0(z)$, and\n$D(z)$ are still predicted with $\\sim 2-3\\%$ precision, so the class of flat\nquintessence models without early dark energy remains highly falsifiable.\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_q_fe_fg2.eps, width=3.5in}}\n\\caption{Flat quintessence models with (dark blue) and without (light gray) \nearly dark energy.\n}\n\\label{fig:ede}\n\\end{figure}\n\nAdding early dark energy to flat quintessence (Fig.~\\ref{fig:ede}) \nhas very little impact on the 68\\% CL predictions of most observables due to the\nrestriction that $w\\ge -1$ for a canonical scalar field. To satisfy CMB\ndistance constraints, any increase in the expansion rate due to early dark\nenergy must be compensated by a lower expansion rate at intermediate redshift relative to\n$z=0$, i.e. a dark energy density that decreases with increasing redshift\nrequiring $w<-1$. While adding early dark energy does allow a larger\nsuppression of growth at high redshift (which is also a possible sign of \nmassive neutrinos given current upper limits), a measurement of a $\\gtrsim 10-15\\%$\ndecrement or $\\gtrsim 2\\%$ increment in the growth relative to high redshift\nwould still suggest that a broader class of models is necessary. This freedom in\ngrowth leaves the amplitude relative to $z=0$ practically unchanged as the\n$G_{0}(z)$ predictions show. The only qualitative change with early dark energy\nis to open up the allowed range in $\\gamma(z)$ so that the high redshift end has as much\nfreedom as the low redshift end. All of these trends for early dark energy\nwithout curvature reflect those of the forecasts in MHH.\n\n\n\nIncluding curvature in the quintessence class, but not early dark energy, opens up more\nfreedom as shown in Fig.~\\ref{fig:curv}. Now $z>2$ deviations in $D(z)$ are\nallowed at the $\\sim 5\\%$ level relative to $\\Lambda$CDM. \nThus a BAO distance measurement at $z>2$\n could falsify flat quintessence in favor of quintessence with curvature.\nAs discussed in MHH, because of the $w \\ge -1$ quintessence bound, this additional freedom skews\n to smaller distances and lower growth relative to high redshift.\n \n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_q_c_fg2.eps, width=3.5in}}\n\\caption{Non-flat (dark blue) and flat (light gray) \nquintessence models without early dark energy.\n}\n\\label{fig:curv}\n\\end{figure}\n\n\nPredictions from the most general quintessence class which includes \nboth curvature and early dark energy, shown in Fig.~\\ref{fig:curvede}, \ncombine features of the previous quintessence classes in ways that \nare similar to the Stage IV predictions in MHH. \nThe ML model in this class improves the fit to the combined data by \n$-2\\Delta\\ln\\mathcal{L}\\sim 4$, mostly due to changing the SN likelihood by $-2\\Delta\\ln\\mathcal{L}\\sim 5$;\nhowever, removing the large low-$z$ oscillations by smoothing $w(z)$ reduces \nthe improvement in the SN fit to $-2\\Delta\\ln\\mathcal{L}\\sim 2-3$.\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_q_ce_fg2.eps, width=3.5in}}\n\\caption{Non-flat quintessence models with (dark blue)\nand without (light gray) early dark energy.\n}\n\\label{fig:curvede}\n\\end{figure}\n\n\nThe predictions for $G_0(z)$, $D(z)$, and $H(z)$, which \nwere affected little by early dark energy alone, \nare nearly the same as those for non-flat \nquintessence without early dark energy.\nThe other observables show a mixture of the effects of\ncurvature at low $z$ and early dark energy at high $z$. Large suppression\n($\\gtrsim 20\\%$) of $G(z)$ (and similarly $fG$) relative to $\\Lambda$CDM\\ is allowed, but enhancement of\nthe growth function over the $\\Lambda$CDM\\ best fit is still limited at the $\\sim\n2\\%$ level. Note that this upper limit on $G(z)$ is robust to neutrino mass \nuncertainties. Likewise, low-redshift distances (including $z_h\nH_0^{-1}$) cannot be smaller than in $\\Lambda$CDM by substantially more than\n$\\sim 2\\%$. As in Fig.~\\ref{fig:ede}, \nthe high redshift predictions for $\\gamma(z)$ in Fig.~\\ref{fig:curvede} weaken\nsubstantially but only in the positive direction. Indeed, all of the\nobservables display similar asymmetric weakening of the predictions with the\naddition of curvature and early dark energy, which can be understood in terms\nof the $w\\geq -1$ quintessence bound.\n\n\nThe existence of an upper or lower bound on each observable that is robust to\nfreedom in curvature and early dark energy provides the possibility of\nfalsifying the entire quintessence model class. In fact, in this most general\nclass, the statistical predictions from current SN and CMB bounds are already\ncomparable to those that can be achieved by a Stage IV version of these\nprobes, which can be understood from the fact\n that the forecasts from MHH used current BAO and\n$H_0$ measurements.\n\n\nThe comparable predictions in large part reflect the fact that curvature is\nalready well constrained through the BAO and $H_0$ measurements. \nThe constraint in this most\ngeneral class of quintessence models is $-0.006<\\Omega_K<0.033$ (95\\%~CL), \na factor of $\\sim 2$ weaker than for non-flat $\\Lambda$CDM\\ and skewed\ntoward open models due to the quintessence prior on $w(z)$.\n\n\nFinally, as an example of the use of the asymmetric quintessence predictions,\nwe consider the application of these results \nto observables which measure some combination of $\\sigma_8$ and $\\Omega_{\\rm m}$. \nTo compute predictions for $\\sigma_8$ given our predictions for the raw acceleration observables, \nwe use the fitting formula \\cite{Hu_Jain}\n\\begin{eqnarray}\n\\sigma_8 &=& \\frac{G(z=0)}{0.76}\\left[\\frac{A_s(k=0.05~{\\rm Mpc}^{-1})}{3.12\\times 10^{-9}}\\right]^{1\/2} \\left(\\frac{\\Omega_{\\rm b} h^2}{0.024}\\right)^{-1\/3} \\nonumber \\\\\n&& \\times \\left(\\frac{\\Omega_{\\rm m} h^2}{0.14}\\right)^{0.563}\n\\left(\\frac{h}{0.72}\\right)^{0.693}(3.123h)^{(n_s-1)\/2}\n\\label{eq:sigma8}\n\\end{eqnarray}\nfor each model sampled in the MCMC likelihood analysis. \nNote that on top of allowed\nvariations in $G(z=0)$, $\\sigma_{8}$ predictions include uncertainties in the\nreionization optical depth $\\tau$ through its covariance with $A_{s}$. While\nthis analysis assumes instantaneous reionization, the uncertainty introduced\nby more general ionization histories is small \\cite{MorHu08}.\nWe have checked that the $\\sigma_8$ distributions obtained using \nEq.~(\\ref{eq:sigma8}) closely match those \nfrom the more accurate computation of $\\sigma_8$ using CAMB.\nThe joint predictions for $\\sigma_8$ and $\\Omega_{\\rm m}$ from the current SN, CMB, \nBAO, and $H_0$ constraints are shown in Fig.~\\ref{fig:sigma8} for \nflat $\\Lambda$CDM\\ and two quintessence model classes.\n\n\nIn particular, in the context of flat $\\Lambda$CDM\\ the current SN, CMB, BAO, and\n$H_0$ data predict the combination best measured by the local abundance of\nmassive galaxy clusters to be $0.394 < \\sigma_8 \\Omega_{\\rm m}^{0.5} < 0.441$ (68\\% CL).\nFlat quintessence without early dark energy weakens the lower end somewhat but\nleaves the upper limit nearly unchanged: $0.358 < \\sigma_8 \\Omega_{\\rm m}^{0.5} < 0.419$.\nQuintessence with both early dark energy and curvature yields $0.306 < \\sigma_8\n\\Omega_{\\rm m}^{0.5} < 0.396$. Therefore a measurement of a local cluster abundance in\nsignificant excess of the flat $\\Lambda$CDM\\ predictions rules out the whole quintessence class,\nwhereas a measurement that is substantially lower would remain consistent with\nquintessence but would rule out a cosmological constant (see also \\cite{Kunz_sigma8}).\nA measurement below the flat $\\Lambda$CDM\\ prediction by $\\lesssim 10\\%$ could \nalso indicate large neutrino masses, but an excess cluster abundance could \nnot be alternately explained by massive neutrinos.\nCurrent cluster surveys, with $\\sim 5\\%$ measurements of similar \ncombinations of $\\sigma_8$ and $\\Omega_{\\rm m}$ \\cite{Vikhlinin,Rozo,Mantz},\nare beginning to reach the precision necessary to test these predictions. \nIn fact, the\nlack of an observed excess already places strong constraints on modified\ngravity explanations of cosmic acceleration \\cite{Schmidt:2009am}.\n\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=sigma8_om_2.eps, width=3.5in}}\n\\caption{Predictions for $\\sigma_8$ and $\\Omega_{\\rm m}$ for flat $\\Lambda$CDM\\ \n(gray contours, top), flat quintessence without early dark energy\n(red contours, middle), \n and non-flat quintessence with early dark energy \n(blue contours, bottom), showing 68\\% CL (light) and 95\\% CL (dark) regions.\n}\n\\label{fig:sigma8}\n\\end{figure}\n\n\n\n\n\\subsection{Smooth $w_0-w_a$ Dark Energy}\n\nAs a final case we consider the class of models defined by an equation of\nstate $w(z)= w_{0} +(1-a)w_{a}$ \\cite{Chevallier_Polarski,Linder_wa} under the\nassumption that dark energy is smooth relative to matter.\nUnlike our previous cases, this class does not define a physical candidate for\ndark energy such as the cosmological constant or a scalar field but rather\nrepresents a simple but illustrative phenomenological parametrization. Note\nthat early dark energy is included in this parametrization since \n$\\lim_{z\\to\\infty} w(z) = w_0 + w_a$.\n \nThe predictions for the $w_0-w_a$ model class serve two purposes. First, the\ncomparison of predictions for smooth, monotonic $w_0-w_a$ models with \nthose for PC quintessence models test the dependence of the\npredictions on rapid transitions and non-monotonic evolution of the\nequation of state. The second use of the $w_0-w_a$ predictions is to\nillustrate how predictions are affected by the $-1\\leq w(z) \\leq 1$\nquintessence bound. Unlike the model classes where $w(z)$ is parametrized by\nprincipal components, it is simple to impose a strict \nquintessence prior on $w_0-w_a$ models\nby requiring $-1\\leq w_0 \\leq 1$ and $-1\\leq w_0+w_a\\leq 1$. We compare\npredictions using this prior with the more general case, where the priors are\nweak enough that constraints on $w_0$ and $w_a$ are determined solely \nby the data (``no $w$ prior'').\n\n\nA fair comparison can be made between the predictions for flat and non-flat\n$w_{0}-w_{a}$ models with the $-1\\leq w\\leq 1$ prior (light gray contours in\nFigs.~\\ref{fig:w0waf} and \\ref{fig:w0wac}) and PC quintessence models with\nearly dark energy (dark blue contours in Figs.~\\ref{fig:ede} and\n\\ref{fig:curvede}). In particular, observables relatively insensitive to both\nthe amount of early dark energy and large changes in the PC equation of\nstate at low redshift, such as $G_0(z)$ and $D(z)$, \nare generally in good agreement.\nThe expansion rate and growth rate are more sensitive to sudden changes in\n$w(z)$ than the distances and the integrated growth function. Therefore, the\nimpact of large, low-$z$ oscillations in the PCs is greatest for $H(z)$, \n$f(z)G(z)$, and $\\gamma(z)$ at $z\\lesssim 1$, increasing the width of those predictions\nrelative to the corresponding predictions for the smooth $w_0-w_a$ models. The\nPC quintessence models also have more freedom in early dark energy than\n$w_0-w_a$ models since $w_{\\infty}$, unlike $w_0+w_a$, is completely free from the\nlow-redshift SN, BAO, and $H_0$ constraints. As a result, $w_0-w_a$\npredictions for $G(z)$ and the high-redshift values of $\\gamma(z)$ \nand $f(z)G(z)$ are stronger than, but still qualitatively similar to, \nthose for PC quintessence with early dark energy.\n\n\nLike the PC quintessence predictions, the predictions for $w_0-w_a$ models \nbounded by $-1\\leq w\\leq 1$ are shifted relative to flat $\\Lambda$CDM\\ \ndue to marginal improvements in the fit to SN data ($-2\\Delta\\ln\\mathcal{L}\\sim 0.5$ for \nthe ML model) enabled by an evolving equation of state.\nThis is a somewhat smaller change in the likelihood than \nfor PC quintessence models, but the \nmagnitude of the ML model shift in the observables is similar for \n$w_0-w_a$ and PC quintessence, at least for those observables that \ndepend little on early dark energy.\n\n\nComparing the two sets of predictions in Figs.~\\ref{fig:w0waf}\nand~\\ref{fig:w0wac} (no $w$ prior vs.\\ the $-1\\leq w\\leq 1$ prior) shows the\neffect on the $w_0-w_a$ predictions of allowing freedom in $w(z)$ beyond that\nallowed by the quintessence bounds. As discussed in MHH, eliminating these\nbounds makes the range in predictions for observables such as growth more\nsymmetric around the best fit for flat $\\Lambda$CDM\\ since $w(z)$ is allowed to cross\nbelow $w=-1$. In particular, growth in excess of flat\n $\\Lambda$CDM is now allowed. Based on the analysis of MHH, we expect the\n amount of the remaining skewness in the predictions around flat $\\Lambda$CDM\n to be affected by the available volume of parameter space as determined by\n how priors on dark energy parameters weight models with $w<-1$ relative to\n those with $w>-1$.\n\nRemoving the quintessence bounds also allows models with greater amounts of\nearly dark energy, and (for non-flat $w_0-w_a$) more closed models, to fit the\ndata. A notable consequence for models with nonzero curvature is that the\npredictions for $\\gamma(z)$ at 95\\% CL diverge at $z>1$. \nThis is the same effect noted in MHH for $\\gamma(z)$ forecasts\nin the non-flat smooth dark energy model class.\nThe divergence in the tails of the high-redshift $\\gamma(z)$ distribution \nis caused by the appearance of\na singularity in $\\gamma(z)$ for closed models where $\\Omega_{\\rm K}$ is sufficiently\nnegative so that $\\Omega_{\\rm m}(z)$ crosses unity at some redshift; when $\\Omega_{\\rm m}(z)=1$,\n$\\gamma(z)$ is no longer well defined by Eq.~(\\ref{eq:gamma_z}). Such caveats\nmust be kept in mind when using $\\gamma$ as a test of not only quintessence\nbut of all smooth dark energy models.\n\n\n\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_w0wa_f_fg2.eps, width=3.5in}}\n\\caption{Flat $w_0-w_a$ without priors on $w(z)$ (dark blue) and with\n quintessence priors ($-1\\leq w_0\\leq 1$, $-1\\leq w_0+w_a\\leq 1$; light\n gray). \n}\n\\vskip 0.25cm\n\\label{fig:w0waf}\n\\end{figure}\n\n\n\\begin{figure}[tp]\n\\centerline{\\psfig{file=rp_w0wa_c_fg2.eps, width=3.5in}}\n\\caption{Non-flat $w_0-w_a$ without priors on $w(z)$ (dark blue) and \nwith quintessence priors ($-1\\leq w_0\\leq 1$, $-1\\leq w_0+w_a\\leq 1$; \nlight gray).\n}\n\\vskip 0.25cm\n\\label{fig:w0wac}\n\\end{figure}\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nAny given class of dark energy models makes concrete predictions for the\nrelationship between the expansion history, geometry, and growth \nof structure as a function\nof redshift. Therefore, current distance-based measurements, though limited in\nredshift, make predictions for other dark energy observables that can be used\nto test and potentially rule out whole classes of dark energy models.\n\n\nIn this paper we present the allowed ranges for the expansion rate $H(z)$,\ndistances $D(z)$, the linear growth rate $G(z)$, \nand several auxiliary growth observables from the current\ncombination of cosmological measurements of supernovae, the cosmic microwave\nbackground, baryon acoustic oscillations, and the Hubble constant. In\nparticular, growth at any redshift or a Hubble constant in significant excess\nof $2\\%$ ($68\\%$ CL range) of the current best fit $\\Lambda$CDM model would\nfalsify both a cosmological constant and more general quintessence \nmodels with or without curvature and early dark energy. On the\nother hand, comparable measurements of a decrement in these quantities would\nrule out a cosmological constant but would be fully consistent\nwith quintessence. Alternately, a substantial reduction in growth \nrelative to the expectation for $\\Lambda$CDM\\ could indicate neutrinos with \nlarge masses ($\\sum m_{\\nu}>0.05$~eV).\n\nRemarkably, predictions for the main acceleration observables, $H(z)$, $D(z)$,\nand $G(z)$, are only weaker than Stage IV SN and CMB predictions (MHH) by a\nfactor of $\\sim 2-3$. However, this improvement applies across a wide range\nof redshifts, indicating that multiple phenomenological parameters may each be\nimproved by this factor. For example, parameter-based figures of merit\neffectively involve products of individual parameters (e.g.\\ area in the\n$w_0-w_a$ plane \\cite{Huterer_Turner,DETF} or volume of the principal\ncomponent parameter error ellipsoid \\cite{Albrecht_Bernstein,FoMSWG}), and in\nsuch figures of merit the total improvement with future data can be\nsignificant. \nIf novel dark energy physics affects small pockets of these high-dimensional\nparameter spaces --- that is, if only specific dark energy parameter combinations\nare sensitive to new physics --- then these multiparameter figures of merit\nwill justly indicate a much more significant improvement with future cosmological\ndata.\n\nIn this work we have considered only known and quantifiable sources of error\nin the current data. Recent analyses of supernova data (e.g.~\\cite{Constitution,SDSS_SN,Kelly09}) indicate that unknown systematic\nerrors remain and can significantly affect cosmological constraints.\nFurthermore, the systematic error estimates used here for the SN data were\noptimized for models with a cosmological constant and therefore may be\nunderestimated for dynamical dark energy \\cite{SCP_Union}. We intend to\nexplore the implications of SN systematics for dark energy predictions in\nfuture work. Our predictive methodology can alternately be viewed as a means\nof ferreting out unknown systematics by looking for inconsistencies between\nthe predictions from one set of observations and data from another.\n\n\nOver the course of this study, new data have become available that could\nimprove the predictions for acceleration observables or begin to test\npredictions within the various classes. In particular, BAO measurements from\nSDSS DR7 and 2dFGRS provide a 2.7\\% constraint on $D_V(z=0.275)$ and a 3.7\\%\nconstraint on $D_V(z=0.35)\/D_V(z=0.2)$ \\cite{Percival09}. We have estimated the\nimpact of these new measurements on our predictions by using the updated BAO\nlikelihood to modify the weighting of MCMC samples for each model class. For\nall quintessence model classes, the effect of updating the BAO data is\nnegligible for most observables except for $D(z\\lesssim 1)$ and (to a lesser\nextent) $H(z\\lesssim 0.5)$, reflecting the improved BAO constraint on\nlow-redshift $D$ and $H$.\n\nThe impact of the newer BAO measurements on $\\Lambda$CDM\\ models is greater than\nfor quintessence since the reduced freedom in dark energy evolution ties\nlow-redshift measurements to high-redshift predictions. The updated BAO\nconstraints exclude models on one side of the predicted observable\ndistributions in Fig.~\\ref{fig:lcdm}, reducing their width by $10-30\\%$ and\nshifting the distributions by an equal amount. However, these changes appear\nto be mainly due to a slight tension between the new BAO constraints and the\nother data sets used for $\\Lambda$CDM\\ predictions. Note that the BAO constraints of\nRef.~\\cite{Percival09} are still less precise than the flat $\\Lambda$CDM\\ predictions\nin Fig.~\\ref{fig:lcdm} and comparable to the non-flat $\\Lambda$CDM\\ predictions, so\nthey do not yet represent a significant additional test of the cosmological\nconstant.\n\n\nFalsifiable predictions from current data reveal many opportunities for sharp\nobservational tests of paradigms for cosmic acceleration by requiring \nconsistency within a given theoretical framework between observables \nthat depend on the expansion history, geometry, and growth of structure \nin the universe. These predictions can be used to \ninform future surveys as to the optimal choice of observables, redshifts, and\nrequired measurement accuracies for testing whole\nclasses of dark energy models.\nFalsification of even the simplest model, flat $\\Lambda$CDM, would have \nrevolutionary consequences for cosmology and fundamental physics.\n\n\n\\vspace{1cm}\n{\\it Acknowledgments:} We thank David Weinberg for useful conversations about this\nwork. MM and WH were supported by the KICP under NSF contract PHY-0114422. MM\nwas additionally supported by the NSF GRFP and CCAPP at Ohio State; WH by DOE\ncontract DE-FG02-90ER-40560 and the Packard Foundation; DH by the DOE OJI\ngrant under contract DE-FG02-95ER40899, NSF under contract AST-0807564, and\nNASA under contract NNX09AC89G.\n\n\n\\vfill\n\\bibliographystyle{arxiv_physrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nData classification -- e.g. object recognition -- is a fundamental computing problem in machine learning and artificial intelligence. Large-scale classification competitions such as the annual ImageNet challenge \\cite{krizhevsky2012imagenet, russakovsky2015imagenet}, where a super-human accuracy of 95\\% has been achieved within about 5 years of steady progress, have contributed greatly to the general popularity of machine learning. Understandably, ImageNet is mostly discussed in the context of technical improvements regarding the classification methods which enabled this drastic boost of performance. But it also illustrates some fundamental problems that arise when computers are to create models of human-defined data categories:\nFor example, the fact that classification accuracies are typically leveling off at values below 100\\% does not necessarily reflect a limitation of the algorithms, but instead may reveal the classification limits of the humans who provided the ground truth data. Indeed, in the case of ImageNet, the massive work of annotating millions of images had been crowd-sourced using Amazon Mechanical Turk, and so a large number of individuals were involved in the labeling process, individuals who may place certain ambiguous images into different categories. This problem of ambiguity due to non-rigorously defined object categories is most pronounced in biological and medical data, where sample-to-sample variations are notoriously large. \n\nIn this work, we use artificially generated surrogate data, as well as real-world bio-medical data, to explore the implications of this inevitable data ambiguity. We demonstrate that the overlap of data classes leads to a theoretical upper limit of classification accuracy, a limit that can be mathematically computed in low-dimensional examples and which depends in a systematic way on the statistical properties of the data set. We find that sufficiently powerful classifier models of different kinds all perform at this same upper limit of accuracy, even if they are based on completely different operating principles. Interestingly, this accuracy limit is not affected by applying certain non-linear transformations to the data, even if these transformations are non-reversible and drastically reduce the information content (entropy) of the input data. \n\nIn a next step, the same three models that reached the common classification limit for artificial data are now applied to human EEG data measured during sleep. In a pre-processing step, two kinds of features are extracted from raw EEG signals, yielding different marginal distributions and mutual correlations. It turns out that a more complex Bayesian model, based on correlated multi-variate Gaussian likelihoods (CMVG), performs worse than two other models (naive Bayes, perceptron), because the statistical properties of the pre-processed features do not match those of the likelihoods. In contrast, the perceptron and the naive Bayes model still show very similar classification accuracies, indicating that both reach the theoretical accuracy limit for sleep stage classification.\n\nFinally, we address the question whether typical human-defined object categories can also be considered as 'natural kinds', that is, whether the data vectors in input space have a built-in cluster structure that can be detected by objective machine-learning models even in non-labeled data. For this purpose, we use as real-world examples the MNIST data set \\cite{deng2012mnist}, as well as the above EEG sleep data. We find that a simple visualization by multi-dimensional scaling (MDS) \\cite{torgerson1952multidimensional, kruskal1964nonmetric,kruskal1978multidimensional,cox2008multidimensional} already reveals an inherent cluster structure of the data in both cases. Interestingly, the degree of clustering, quantified by the general discrimination value (GDV) \\cite{krauss2018statistical, schilling2021quantifying}, can be enhanced by a step-wise dimensionality reduction of the data, using an autoencoder that is trained in an unsupervised manner. A perceptron classifier with a layer design comparable to the autoencoder, trained on the same data in a supervised fashion, achieves as expected a much stronger cluster separation. However, the enhancement of clustering by unsupervised data compression, combined with automatic labeling methods, could be a promising way to automatically detect 'natural kinds' in non-labeled data. \n\n\n\\clearpage\n\\section{Methods}\n\n\\subsection*{Part 1: Accuracy limit}\n\n\\subsubsection*{Derivation of theoretical accuracy limit}\n\nClassification is the general problem of assigning a discrete class label $i=1\\ldots K$ to each given input data $\\vec{x}$, where the latter is considered as a vector with $N$ real-valued components $x_{n=1\\ldots N}$. Such a discrimination is possible when the conditional probability distributions $p_{gen}(\\vec{x}\\;|\\;i)$ of data vectors, here called {\\bf 'generation densities'}, are different for each of the possible data classes $i$. In the simple case of a two- or three-dimensional data space, each data class can be visualized as a 'point cloud' (See Fig.\\ref{figure_1}(a,b) for examples), and either the shapes or the center positions of these point clouds must vary sufficiently in order to facilitate a reliable classification. However, since the data generation process typically involves not only the system of interest (which might indeed have $K$ well-distinguished modes of operation), but also some measurement or data transmission equipment (which introduces noise into the data), a certain 'overlap' of the different data classes is usually not avoidable.\n\n\\vspace{0.2cm}\\noindent A classifier is receiving the data vectors $\\vec{x}$ as input and computes a set of $K$ {\\bf 'classification probabilities'} $q_{cla}(j\\;|\\;\\vec{x})$, quantifying the belief that $\\vec{x}$ belongs to $j$. They are normalized to one over all possible classes, so that $\\sum_{j=1}^K q_{cla}(j\\;|\\;\\vec{x}) = 1\\; \\forall\\; \\vec{x}$.\n\n\\vspace{0.2cm}\\noindent We can now define a {\\bf 'confusion density'} as the product\n\\begin{equation}\nC_{ji}(\\vec{x}) = q_{cla}(j\\;|\\;\\vec{x})\\;p_{gen}(\\vec{x}\\;|\\;i).\n\\end{equation}\nIt can be interpreted as the probability density that the generator is producing data vector $\\vec{x}$ under class $i$, which is then assigned to class $j$ by the classifier. Because there is usually a very small but non-zero probability density that {\\em any} vector $\\vec{x}$ can occur under {\\em any} class $i$, we expect that the non-diagonal elements $C_{j\\!\\neq\\!i}(\\vec{x})$ are larger than zero as well. These non-diagonal confusion densities will have their largest values in regions of data space where the classes $i$ and $j$ overlap (See Fig.\\ref{figure_1}(e,f) for examples).\n\n\\vspace{0.2cm}\\noindent By integrating the confusion density over all possible data vectors $\\vec{x}$,\n\\begin{equation}\nC_{ji} = \\int C_{ji}(\\vec{x})\\; d\\vec{x},\n\\label{cij}\n\\end{equation}\nwe obtain the {\\bf 'confusion matrix'} of the classifier, which comes out properly normalized, so that $\\sum_{j=1}^K C_{ji} = 1\\; \\forall\\; i$. The confusion matrix $C_{ji}$ therefore is the probability that a data point originating from class $i$ is assigned to class $j$.\n\n\\vspace{0.2cm}\\noindent Assuming for simplicity that all data classes appear equally often, we can compute the {\\bf accuracy} $A$ of the classifier as the average over all diagonal elements of the confusion matrix:\n\\begin{equation}\nA = \\frac{1}{K}\\sum_{i=1}^K C_{ii}.\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent In the following, we are particularly interested in the {\\bf theoretical limit of the classification accuracy}, denoted by $A_{max}$. We therefore consider an ideal classifier that has learned the exact generation densities $p_{gen}(\\vec{x}\\;|\\;i)$. In this case, the {\\bf 'ideal classification probability'} corresponds to the Bayesian posterior\n\\begin{equation}\nq_{cla}(\\;j\\;|\\;\\vec{x}\\;) = \\frac{p_{gen}(\\;\\vec{x} \\;|\\; j\\;)}{\\sum_k p_{gen}(\\;\\vec{x} \\;|\\; k\\;)}. \n\\end{equation}\n\n\\vspace{0.2cm}\\noindent In our numerical experiments, we will use classifiers that output a definite class label $j$ for each given data vector $\\vec{x}$, corresponding to most probably class. To compute the theoretical accuracy maximum for such a model, we replace $q_{cla}$ by the binary {\\bf 'class indicator function'}\n\\begin{equation}\n\\hat{q}_{cla}(\\;j\\;|\\;\\vec{x}\\;) = \\delta_{jk}\\;\\;\\mbox{with}\\;\\;k = \\mbox{argmax}_c\\; q_{cla}(\\;c\\;|\\;\\vec{x}\\;).\n\\end{equation}\nIt has the value $1$ for all data points $\\vec{x}$ assigned to class $j$, and the value $0$ for all other data points (See Fig.\\ref{figure_1}(c,d) for examples). When the ideal accuracy $A$ is evaluated using $\\hat{q}_{cla}$ instead of $q_{cla}$, the result can be directly compared with numerical accuracies based on one-hot classifier outputs.\n\n\\subsubsection*{Numerical evaluation of $A_{max}$}\n\nIn Fig.\\ref{figure_1}, the above quantities have been numerically evaluated for a simple Gaussian test data set. For this purpose, the two-dimensional integral \\ref{cij} has been evaluated numerically on a regular grid of linear spacing 0.01, ranging from -8 to +8 in each feature dimension. \n\n\n\\subsubsection*{Classifiers and input data}\n\n\\vspace{0.2cm}\\noindent In the following subsections, we provide the implementation details for the different classifier models that are compared in this work. The input data for these models is given as lists of $D$-dimensional feature vectors $\\vec{u} = (u_1,u_2,\\ldots,u_f,\\ldots,u_D)$, each belonging to one of $K$ possible classes $c$. In the case of artificially generated data, these lists contain 10000 feature vectors distributed equally over the data classes. They are split randomly into training (80\\%) and test (20\\%) data sets.\n\n\\subsubsection*{Perceptron model}\n\nThe perceptron model is implemented using Keras\/Tensorflow. It has one hidden layer, containing $N_{neu}=100$ neurons with RELU activation function. The output layer has $N_{out}$ neurons with softmax activation function, where $N_{out}=D$ corresponds to the number of data classes. The loss function is categorical crossentropy. We optimize the perceptron on each training data set using the Adams optimizer over at least 10 epochs with a batch size of 128 and a validation split of 0.2. After training, the accuracy of the perceptron is evaluated with the independent test data set.\n\n\\subsubsection*{Naive Bayesian model}\n\nThe naive Bayesian model is implemented using the Python libraries Numpy and Scipy.\n\n\\vspace{0.2cm}\\noindent In the training phase, the training data set is sorted according to the $K$ class labels $c$. Then an individual Gaussian Kernel Density (KDE) approximation (Scott method) is computed for each feature $f$ and class label $c$, corresponding to the empirical marginalized probability densities $p_{f,c}(u_f)$. \n\n\\vspace{0.2cm}\\noindent In the testing phase, the accuracy of the model is evaluated with the independent test data set as follows: According to the naive Bayes approach, the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}=(u_1,u_2,\\ldots,u_D)$ under class $c$ is approximated by a product of the marginalized probabilities, so that\n\\begin{equation}\nL(\\vec{u}\\;|\\;c) = \\prod_{f=1\\ldots D} p_{f,c}(u_f).\n\\end{equation}\nSince we assume a flat prior probability ($P_{prior}(c)=1\/K$) over the data classes, the posterior probability of data class $c$, given the input data vector $\\vec{u}$, is given by\n\\begin{equation}\nP_{post}(c\\;|\\;\\vec{u}) = \\frac{L(\\vec{u}\\;|\\;c)}{\\sum_{i\\!=\\!1}^K\\;L(\\vec{u}\\;|\\;i)}\n\\end{equation}\n\n\\subsubsection*{Naive Bayesian model with Random Dimensionality Expansion (RDE)}\n\nSince the naive Bayesian model takes into account only the marginal feature distributions $p_{f,c}(u_f)$, it cannot distinguish data classes which accidentally have identical $p_{f,c}(u_f)$ distributions, but differ in the correlations between the features. In principle, this problem can be fixed by multiplying the $D$-dimensional input vectors $\\vec{u}$ by a random $D_2 \\times D$ matrix $\\textbf{M}$, for example with normally distributed entries $M_{ij}\\propto N(\\mu=0,\\sigma=1)$, which yields transformed vectors $\\vec{v} = \\textbf{M} \\vec{u}$. Provided that $D_2\\gg D$, at least some of the new feature linear combinations $v_f$ will have marginal distributions that vary between the data classes. \n\n\\subsubsection*{CMVG Bayesian model}\n\nThe Correlated Multi-Variate Gaussian (CMVG) Bayesian model is also implemented using the Python libraries Numpy and Scipy. \n\n\\vspace{0.2cm}\\noindent In the training phase, the training data set is sorted according to the two class labels $c$. Then, for each class label $c$, we compute the mean values $\\mu_f^{(c)}$ of the features $f=1\\ldots D$, as well as the covariances $\\Sigma_{fg}^{(c)}$ between features $f$ and $g$. These quantities are packed as one vector ${\\bf \\mu}^{(c)}$ and one matrix ${\\bf \\Sigma}^{(c)}$ for each class $c$.\n\n\\vspace{0.2cm}\\noindent In the testing phase, the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}=(u_1,u_2,\\ldots,u_D)$ under class $c$ is computed as the correlated, multi-variate Gaussian probability density\n\n\\begin{equation}\nL(\\vec{u}\\;|\\;c) = p_{cmvg}\\left( \\vec{u} \\;,\\; {\\bf\\mu}\\!=\\!{\\bf\\mu}^{(c)} \\;,\\;\n{\\bf \\Sigma}\\!=\\!{\\bf \\Sigma}^{(c)}\n\\right).\n\\end{equation}\nSince we assume a flat prior probability ($P_{prior}(c)=1\/2$) for the two data classes, the posterior probability of data class $c$, given the input data vector $\\vec{u}$, is given by\n\\begin{equation}\nP_{post}(c\\;|\\;\\vec{u}) = \\frac{L(\\vec{u}\\;|\\;c)}{\\sum_{i\\!=\\!1}^K\\;L(\\vec{u}\\;|\\;i)}\n\\end{equation}\n\n\n\\subsection*{Part 2: The DSC data model}\n\nWe consider an artificial classification problem with two multivariate Gaussian data classes $c\\in\\left\\{0,1\\right\\}$ and with statistical properties that can be tuned by {\\bf three control quantities}: the {\\bf dimensionality} $D$ of the feature space, the {\\bf separation} $S$ between the centers of the point clouds, and the {\\bf correlation} $C$ between features (within the same class), which is associated with the shape of the point cloud. The generation of artificial data within this DSC model works as follows:\n\n\\vspace{0.2cm}\\noindent Starting from a given triple $D,S,C$ of control quantities, we first generate $N_{rep}$ independent {\\bf parameter sets} $[\\mu_f^{(c)}, \\Sigma_{fg}^{(c)}]$ that describe the statistical properties of the two classes $c\\in\\left\\{0,1\\right\\}$. Here, $\\mu_f^{(c)}$ is the mean value of feature $f$ in class $c$, and $\\Sigma_{fg}^{(c)}$ is the covariance of features $f$ and $g$ in class $c$.\n\n\\vspace{0.2cm}\\noindent The mean values $\\mu_f^{(c=0)}$ in class $0$ are always set to zero, whereas the mean values $\\mu_f^{(c=1)}$ in class $1$ are random numbers, drawn from a uniform distribution with values in the range from 0 to $S$. The separation quantity $S$ is therefore the maximum distance between corresponding feature mean values in each dimension $f$.\n\n\\vspace{0.2cm}\\noindent The diagonal elements $\\Sigma_{ff}^{(c)}$ of the symmetric covariance matrix are set to 1 in both classes. The off-diagonal elements $\\Sigma_{f\\neq g}$ are assigned independent, continuous random numbers $x$, drawn from a box-shaped probability density distribution $q(x,C)$ that depends on the correlation quantity $C$ as follows:\n\n\\begin{eqnarray}\n q(x,C) &=& \\mbox{uniform}[0\\;,\\;C]\\;\\;\\;\\;\\;\\;\\mbox{for}\\;C\\le1 \\nonumber\\\\\n q(x,C) &=& \\mbox{uniform}[C\\!-\\!1\\;,\\;1]\\;\\mbox{for}\\;C>1\n\\end{eqnarray}\n\n\\vspace{0.2cm}\\noindent For $C=0$, the distribution $q(x,C)$ peaks at $x=0$, so that $\\Sigma_{ij}$ becomes a diagonal unit matrix. For $C=1$, the distribution $q(x,C)$ is uniform in the range $[0,1]$, and for $C=2$ it peaks at $x=1$, leading to $\\Sigma_{ij}=1$. A plot of the distribution is shown in \\ref{figure_2}(b).\n\n\\vspace{0.2cm}\\noindent According to the parameter set $[\\mu_f^{(c)},\\Sigma_{fg}^{(c)}]$, we then generate for each of the two classes $c$ a number $N_{vec}\/2$ of random, Gaussian data vectors $\\vec{u}(t)$, in which the $D$ components (features) are correlated to a degree controlled by quantity $C$. In the limiting case $C=0$, the $D$ time series $u_{f}(t)$ become statistically independent, whereas for $C=1$, the time series become fully correlated and thus identical. The total number of $N_{vec}$ data vectors is combined to a complete data set, in which vectors from the two classes (with corresponding labels $c$) appear in random order. By this way, we obtain for each triple $D,S,C$ of control quantities a total number of $N_{rep}$ independent data sets, each consisting of $N_{vec}$ data vectors. Since each data set obeys its own random parameters $[\\mu_f^{(c)},\\Sigma_{fg}^{(c)}]$, the DSC model reflects some of the heterogeneity of typical real world data. Finally, we split each data set into a training set (80\\%) and a test set (20\\%).\n\n\\vspace{0.2cm}\\noindent Before applying different types of classifiers to the DSC data sets, we test that the feature correlations and the class separation can be controlled reliably and over a sufficiently large range, using the quantities $C$ and $S$ (Fig.\\ref{figure_2}(d)).\n\n\\subsubsection*{Control of feature correlations by quantity $C$}\n\nTo evaluate correlation control, we fix the quantities $D=10$ and $S=1.0$ (Note that the separation has no effect on the correlations) and vary $C$ over the complete available range of supported values from 0 to 2. For each $C$, we generate $N_{rep}=100$ independent data sets, each consisting of $N_{vec}=10000$ data vectors. For each data set, we estimate the empirical covariance matrix $\\Sigma_{ij}^{(0)}$ of class 0. Because the matrix is symmetric, we compute the root-mean-square (RMS) average of all matrix elements above the diagonal. The blue line in (Fig.\\ref{figure_2}(d)) shows for each $C$ the mean RMS, averaged over the $N_{rep}=100$ repetitions (The latter are shown as gray dots). We find an almost linear relation between $C$ and the mean RMS. In particular, we can realize the full range of correlations, including the limiting cases of independently fluctuating features (for $C=0$) and identically fluctuating features (for $C=2$).\n\n\\subsubsection*{Control of class separation by quantity $S$}\n\nTo evaluate separation control, we fix the quantities $D=10$ and $C=0.5$ and vary $S$ between 0 and 10. For each $S$, we generate $N_{rep}=100$ independent data sets, each consisting of $N_{vec}=10000$ data vectors. For each (labeled) data set, we compute the general discrimination value (GDV), a quantity that has been specifically designed to quantify the separation between classes in high dimensional data sets \\cite{krauss2018statistical, schilling2021quantifying}. The orange line in Fig.\\ref{figure_2}(d) is the mean negative GDV, averaged over the $N_{rep}=100$ repetitions (The latter are shown as gray dots). \n\n\\vspace{0.2cm}\\noindent The GDV is computed as follows: We consider $N$ points $\\mathbf{x_{n=1..N}}=(x_{n,1},\\cdots,x_{n,D})$, distributed within $D$-dimensional space. A label $l_n$ assigns each point to one of $L$ distinct classes $C_{l=1..L}$. In order to become invariant against scaling and translation, each dimension is separately z-scored and, for later convenience, multiplied with $\\frac{1}{2}$:\n\\begin{align}\ns_{n,d}=\\frac{1}{2}\\cdot\\frac{x_{n,d}-\\mu_d}{\\sigma_d}.\n\\end{align}\nHere, $\\mu_d=\\frac{1}{N}\\sum_{n=1}^{N}x_{n,d}\\;$ denotes the mean, and $\\sigma_d=\\sqrt{\\frac{1}{N}\\sum_{n=1}^{N}(x_{n,d}-\\mu_d)^2}$ the standard deviation of dimension $d$.\nBased on the re-scaled data points $\\mathbf{s_n}=(s_{n,1},\\cdots,s_{n,D})$, we calculate the {\\em mean intra-class distances} for each class $C_l$ \n\\begin{align}\n\\bar{d}(C_l)=\\frac{2}{N_l (N_l\\!-\\!1)}\\sum_{i=1}^{N_l-1}\\sum_{j=i+1}^{N_l}{d(\\textbf{s}_{i}^{(l)},\\textbf{s}_{j}^{(l)})},\n\\end{align}\nand the {\\em mean inter-class distances} for each pair of classes $C_l$ and $C_m$\n\\begin{align}\n\\bar{d}(C_l,C_m)=\\frac{1}{N_l N_m}\\sum_{i=1}^{N_l}\\sum_{j=1}^{N_m}{d(\\textbf{s}_{i}^{(l)},\\textbf{s}_{j}^{(m)})}.\n\\end{align}\nHere, $N_k$ is the number of points in class $k$, and $\\textbf{s}_{i}^{(k)}$ is the $i^{th}$ point of class $k$.\nThe quantity $d(\\textbf{a},\\textbf{b})$ is the euclidean distance between $\\textbf{a}$ and $\\textbf{b}$. Finally, the Generalized Discrimination Value (GDV) is calculated from the mean intra-class and inter-class distances as follows:\n\\begin{align}\n\\mbox{GDV}=\\frac{1}{\\sqrt{D}}\\left[\\frac{1}{L}\\sum_{l=1}^L{\\bar{d}(C_l)}\\;-\\;\\frac{2}{L(L\\!-\\!1)}\\sum_{l=1}^{L-1}\\sum_{m=l+1}^{L}\\bar{d}(C_l,C_m)\\right]\n \\label{GDVEq}\n\\end{align}\n\n\\noindent whereas the factor $\\frac{1}{\\sqrt{D}}$ is introduced for dimensionality invariance of the GDV with $D$ as the number of dimensions.\nIn the case of two Gaussian distributed point clusters, the resulting discrimination value becomes $-1.0$ if the clusters are located such that the mean inter cluster distance is two times the standard deviation of the clusters.\n\n\\subsection*{Part 3: Comparing classifiers}\n\nIn Fig.\\ref{figure_3}, we determine the average accuracy of the three classifier types (See part 1 of the Methods section) for different combinations of the DSC control parameters. For each parameter combination, 100 data sets are sampled from the superstatistical distribution. For every data set, consisting of 8000 training vectors and 2000 test vectors, the three classifiers are trained from the scratch and then evaluated. This results in 100 accuracies for each classifier and each parameter combination. We then compute the mean value of these 100 accuracies, and this is the average accuracy plotted as colored lines in Fig.\\ref{figure_3}(c-f). The individual, non-averaged accuracies are plotted as gray points.\n\n\\subsection*{Part 4: Feature transformations}\n\nIn Fig.\\ref{figure_4}, we return to a much simpler test data set, consisting of two 'spherical' Gaussian data clusters in a two-dimensional feature space, which are centered at $\\vec{x}=(-\\frac{1}{2},0)$ and $\\vec{x}=(+\\frac{1}{2},0)$. respectively. All three classifier types reach the theoretical accuracy limit of about 0.69 in this case.\n\n\\vspace{0.2cm}\\noindent In this part we explore how certain non-linear transformations of the original features (that is, $(x_1,x_2) \\longrightarrow (f(x_1),f(x_2))$) affect the classification accuracy. In particular, we investigate the cases $f(x)=\\sin(x)$, $f(x)=\\cos(x)$ and $f(x)=\\mbox{sgn}(x)$. The signum function yields -1 for negative arguments and +1 for positive arguments. For the special case $x=0$ it would return zero, but this practically does not happen, as the features $x$ are continually distributed random variables. \n\n\\subsection*{Part 5: Sleep EEG data}\n\nFor a real-world evaluation of classifier performance, we are using 68 multi-channel EEG data sets from our sleep laboratory, each corresponding to a full-night recording of brain signals from a different human subject. The data were recorded with a sampling rate of 256 Hz, using three separate channels F4-M1, C4-M1, O2-M1. In this work, however, the signals from these channels are pooled, effectively treating them as data sets of their own.\n\n\\vspace{0.2cm}\\noindent The participants of the study included 46 males and 22 females, with an age range between 21 and 80 years. Exclusion criteria were a positive history of misuse of sedatives, alcohol or addictive drugs, as well as untreated sleep disorders. The study was conducted in the Department of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander University Erlangen-N\u00fcrnberg (FAU), following approval by the local Ethics Committee (323\u201316 Bc). Written informed consent was obtained from the participants before the cardiorespiratory polysomnography (PSG). \n\n\\vspace{0.2cm}\\noindent After recording, the raw EEG data were analyzed by a sleep specialist accredited by the German Sleep Society (DGSM), who removed typical artifacts\n\\cite{tatum2011artifact} from the data and visually identified the sleep stages in subsequent 30-second epochs, according to the AASM criteria (Version 2.1, 2014) \\cite{iber2007aasm,american2012aasm}. The resulting, labeled raw data were then used as a ground truth for testing the accuracy of the different classifier types.\n\n\\vspace{0.2cm}\\noindent In this work, we are primarily testing the ability of the classifiers to assign the correct sleep label $s$\n(Wake, REM, N1, N2, N3) independently to each epoch, without providing further context information. Such a single-channel epoch consists of $30\\times256=7680$ subsequent raw EEG amplitudes $x_{d,e}(t_n)$, where $d$ is the data set, $e$ the number of the epoch within the data set, and $t_n$ the $n$th recording time within the epoch. \n\n\\vspace{0.2cm}\\noindent In order to facilitate classification of these 7680-dimensional input vectors $\\vec{x}_{d,e}$ by a simple Bayesian model, or by a flat two-layer perceptron with relatively few neurons, the vectors have to be suitably pre-processed and compressed down to feature vectors $\\vec{u}_{d,e}$ of much smaller dimensionality $D\\ll 7680$. \n\\vspace{0.2cm}\\noindent Instead of relying on self-organized (and thus 'black-box') features, we are using mathematically well-defined features with a simple interpretation. In particular, we are interested in the case where all $D$ components $u_f$ of a feature vector $\\vec{u}$ are fundamentally of the same kind and only differ by some tunable parameter. \n\n\\subsubsection*{Fourier features}\n\nOur first type of feature estimates the momentary Fourier component of the raw EEG signal $x_{d,e}(t_n)$ at a certain, tunable frequency $\\nu_f$: \n\n\\begin{equation}\nu_f = \n\\sqrt{\\left(\\;\n\\sum_{n=1}^{7680} x_{d,e}(t_n)\\cdot \\cos(2\\pi \\nu_f t_n)\n\\right)^2 + \n\\left(\n\\sum_{n=1}^{7680} x_{d,e}(t_n)\\cdot \\sin(2\\pi \\nu_f t_n)\n\\right)^2}. \n\\end{equation}\n\n\\vspace{0.2cm}\\noindent The set of frequencies $\\nu_{f=1}\\ldots\\nu_{f=D}$ is in our case chosen as an equidistant grid between 0 Hz and 30 Hz, because our EEG system is filtering out the higher-frequency components of the raw signals above about 30 Hz. \n\n\\subsubsection*{Correlation features}\n\nOur second type of feature is the normalized auto-correlation coefficient of the raw EEG signal $x_{d,e}(t_n)$ at a certain, tunable lag-time $\\Delta t_f$: \n\n\\begin{equation}\nu_f = \\frac{\\left\\langle \\left( x_{d,e}(t_n) - \\overline{x}_{d,e} \\right)\\cdot\\left( x_{d,e}(t_n\\!+\\!\\Delta t_f) - \\overline{x}_{d,e} \\right) \\right\\rangle_n}{\\sigma_{d,e}^2}.\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent Here, $ \\overline{x}_{d,e}$ is the mean and $\\sigma_{d,e}$ the standard deviation of the raw EEG signal within the epoch. The symbol $\\left\\langle \\right\\rangle_n$ stands for averaging over all time steps within the epoch. The set of lag-times $\\Delta t_{f=1} \\ldots \\Delta t_{f=D}$ must be integer multiples of the recording time interval $\\delta t = 1\/256$ sec.\n\n\\subsection*{Part 6: Sleep stage detection}\n\nIn Fig.\\ref{figure_6}, we investigate the performance of the three classifier types described in part 1 in the real-world scenario of personalized sleep-stage detection. For this purpose, the classifiers are trained and tested individually on each of our 68 full-night sleep recordings, using as inputs the same 6-dimensional Fourier- or correlation features as in Fig.\\ref{figure_5} (Note that the aggregated distribution functions and covariance matrices in Fig.\\ref{figure_5} have been computed by pooling over all data sets and therefore show a much more regular behavior than the individual ones).\n\n\\vspace{0.2cm}\\noindent As a result, we obtain 68 accuracies for each combination of classifier type (Fig.\\ref{figure_6}, rows) and used input feature (Fig.\\ref{figure_6}, columns). The distributions of these accuracies are presented as histograms in the figure.\n\n\\subsection*{Part 7: Natural data clustering}\n\nIn Fig.\\ref{figure_7}, we address the question whether typical real-world data sets have a built-in clustering structure that can be detected (and possibly enhanced) by unsupervised methods of data analysis. For this purpose, we visualize the clustering structure.\nA frequently used method to generate low-dimensional embeddings of high-dimensional data is t-distributed stochastic neighbor embedding (t-SNE) \\cite{van2008visualizing}. However, in t-SNE the resulting low-dimensional projections can be highly dependent on the detailed parameter settings \\cite{wattenberg2016use}, sensitive to noise, and may not preserve, but rather often scramble the global structure in data \\cite{vallejos2019exploring, moon2019visualizing}.\nIn contrats, multi-Dimensional-Scaling (MDS) \\cite{torgerson1952multidimensional, kruskal1964nonmetric,kruskal1978multidimensional,cox2008multidimensional} is an efficient embedding technique to visualize high-dimensional point clouds by projecting them onto a 2-dimensional plane. Furthermore, MDS has the decisive advantage that it is parameter-free and all mutual distances of the points are preserved, thereby conserving both the global and local structure of the underlying data. \nWhen interpreting patterns as points in high-dimensional space and dissimilarities between patterns as distances between corresponding points, MDS is an elegant method to visualize high-dimensional data. By color-coding each projected data point of a data set according to its label, the representation of the data can be visualized as a set of point clusters. For instance, MDS has already been applied to visualize for instance word class distributions of different linguistic corpora \\cite{schilling2021analysis}, hidden layer representations (embeddings) of artificial neural networks \\cite{schilling2021quantifying,krauss2021analysis}, structure and dynamics of recurrent neural networks \\cite{krauss2019analysis, krauss2019recurrence, krauss2019weight}, or brain activity patterns assessed during e.g. pure tone or speech perception \\cite{krauss2018statistical,schilling2021analysis}, or even during sleep \\cite{krauss2018analysis,traxdorf2019microstructure}. \nIn all these cases the apparent compactness and mutual overlap of the point clusters permits a qualitative assessment of how well the different classes separate.\n\nIn addition, we measure the degree of clustering objectively by calculating the general discrimination value (GDV) \\cite{krauss2018statistical,schilling2021quantifying}, described in part 2.\n\n\\vspace{0.2cm}\\noindent For the clustering analysis we analyze two examples of 'natural data': One is the MNIST data \\cite{deng2012mnist} set with 10 classes of handwritten digits, in which the input vectors are 784-dimensional (28x28 pixels) and have continuous positive values (between 0 and 1 after normalization). \n\n\\vspace{0.2cm}\\noindent As the second example we use, again, our full-night EEG recordings with the 5 data classes corresponding to the sleep stages Wake, REM, N1, N2, and N3. In order to reduce setup-differences between measurements, we first perform a z-transform over each individual full-night EEG recording, so that the one-channel EEG signal of each participant has now zero mean and unit variance. Next, in order to make the EEG data more comparable with MNIST, we produce one 784-dimensional input vector from each 30-second epoch of the EEG recordings in the following way: The 7680 subsequent one-channel EEG signals of the epoch are first transformed to the frequency domain using Fast Fourier Transform (FFT), yielding 3840 complex amplitudes. Since the phases of the amplitudes change in a highly irregular way between epochs, we discard this information by computing (the square roots of) the magnitudes of the amplitudes. We keep only the first 784 values of the resulting real-valued frequency spectrum, corresponding to the lowest frequencies. By pooling over all epochs and participants, we obtain a long list of these 784-dimensional input vectors. They are globally normalized, so that the components in the list range between 0 and 1, just as in the MNIST case. Finally, the list is randomly split into train (fraction 0.8) and test (fraction 0.2) data sets. \n\n\\vspace{0.2cm}\\noindent It is possible to directly compute the MDS projection of the uncompressed 784-dimensional test data vectors into two dimensions, and also to calculate the corresponding GDV value that quantifies the degree of class separation (using the known sleep stage labeling). In Fig.\\ref{figure_7}, these uncompressed data distributions are always shown in the left upper scatter plot of each two-by-two block.\n\n\\vspace{0.2cm}\\noindent In this context, we also test if step-wise dimensionality reduction in an autoencoder leads to an enhanced clustering. The used autoencoder has RELU activation functions and 7 fully connected layers with the following numbers of neurons: 784,128,64,16,64,128,728. The mean squared error between input vectors and reconstructed vectors is minimized using the Adams optimizer. We also compute the MDS projections and GDV values for layers 2, 3 and 4 (the 16-dimensional bottleneck) of the autoencoder. In Fig.\\ref{figure_7}, these three compressed data distributions are shown within the two-by-two blocks of scatter plots.\n\n\\vspace{0.2cm}\\noindent As a reference for the resulting MDS projections and GDV values in the unsupervised autoencoder, we also process the two kinds of natural data with a perceptron that is trained in a supervised manner, so that it separates the known classes as far as possible. To make the perceptron comparable to the autoencoder, the first 4 layers (from the input to the bottleneck) are identical: Fully connected, RELU activations, and layer sizes 784,128,64,16. However, the decoder-part of the autoencoder is replaced by a softmax layer in the perceptron, which has either 10 (MNIST) or 5 (sleep) neurons. The perceptron is trained by back-propagation to minimize categorical cross-entropy between the true and predicted labels, using the Adams optimizer. Just as in the autoencoder, we compute MDS projections and GDV values for the first 4 perceptron layers. \n\n\n\n\n\n\\clearpage\n\\section{Results}\n\n\\subsection*{Part 1: Accuracy limit}\n\nIn order to demonstrate the existence of an accuracy limit in classification tasks, we assume a statistical process is generating data vectors $\\vec{x}$ which are distributed in the input space (subsequently also called feature space) according to given {\\bf generation densities} $p_{gen}(\\vec{x}\\;|\\;i)$ that depend on the class $i$. For reasons of mathematical tractability and visual clarity, we start with a simple problem of two Gaussian data classes in a two-dimensional feature-space. We assume that class $i=0$ is centered at $\\vec{x}=(x_1,x_2)=(0,0)$, whereas class $i=1$ is centered a distance $d$ away, at $\\vec{x}=(d,0)$. As another discriminating property, the two class-dependent distributions are assumed to have different correlations between the features $x_1$ and $x_2$ (Compare Fig.\\ref{figure_1}(a,b)). \n\n\\vspace{0.2cm}\\noindent As derived in the Methods section, an ideal classifier would divide the feature-space $\\left\\{\\vec{x}\\right\\}$ among the two classes in a way that is perfectly consistent with the true generation densities $p_{gen}(\\vec{x}\\;|\\;i)$. The resulting ideal assignment of a discrete class $j=0$ or $j=1$ to each data vector $\\vec{x}$ can be described by binary {\\bf class indicator functions} $\\hat{q}_{cla}(\\;j\\;|\\;\\vec{x}\\;)$ (Compare Fig.\\ref{figure_1}(c,d)). \n\n\\vspace{0.2cm}\\noindent The latter two quantities can be combined to the {\\bf confusion densities} $\\hat{q}_{cla}(j|\\vec{x}) \\; p_{gen}(\\vec{x}|i)$, which give the probability density that data point $\\vec{x}$ is generated in class $i$ but assigned to class $j$ by the ideal classifier (Compare Fig.\\ref{figure_1}(e,f)). The parts of feature space where the confusion density is large for $i\\neq j$ correspond to the overlap regions of the data classes, and it is this overlap that makes the theoretical limit of the classification accuracy smaller than one.\n\n\\vspace{0.2cm}\\noindent It is possible to compute the {\\bf confusion matrix} of the ideal classifier by integrating the confusion densities over the entire feature space, which is feasible only in very low-dimension spaces. The confusion matrix, in turn, yields the {\\bf theoretical accuracy limit} $A_{max}$ of the ideal classifier.\nIn our simple example, $A_{max}$ is expected to increase with the distance $d$ between the two data classes, as this separation reduces the class overlap. By numerically computing the integral over the two-dimensional feature space of our Gaussian test example, we indeed find a monotonous increase of $A_{max}=A_{max}(d)$ from about 0.62 at $d=0$ to nearly one at $d=5$ (Compare Fig.\\ref{figure_1}(g, black line)).\n\n\\vspace{0.2cm}\\noindent Our next goal is to apply different types of classifier models to data drawn from the generation densities $p_{gen}(\\vec{x}\\;|\\;i)$ of the Gaussian test example above.\n\n\\vspace{0.2cm}\\noindent As an example for a 'black box' classifier, we consider a {\\bf perceptron} with one hidden layer (See Methods section for details). In the training phase, the connection weights of this neural network are optimized using the back-propagation algorithm. \n\n\\vspace{0.2cm}\\noindent As an example of a mathematically transparent, but simple classifier type, we consider a {\\bf Naive Bayesian} model. Here, correlations between the input features are neglected, and so the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}$, given the data class $c$, is approximated as the product of the marginal likelihood factors for each individual feature $f$ (See Methods section for details). In the 'training phase', the naive Bayesian classifier is simply estimating the distribution functions of these marginal likelihood factors, using Kernel Density Approximation (KDE). \n\n\\vspace{0.2cm}\\noindent Finally, we consider a {\\bf Correlated Multi-Variate Gaussian (CMVG) Bayesian} model as an example of a mathematically transparent classifier that can also account for correlations in the data, but which assumes that all features are normally distributed (See Methods section for details). In the training phase, the CMVG Bayesian classifier has to estimate the mean values and covariances of the data vectors.\n\n\\vspace{0.2cm}\\noindent When applying these three classifiers to the Gaussian test data, we indeed find that {\\bf all models reach the same theoretical classification limit, even though their operating principles are very different} (Compare Fig.\\ref{figure_1}(g)). The only exception is the Naive Bayes classifier at small class distances $d$ (Compare Fig.\\ref{figure_1}(g, orange line)). This model fails because it can only use the marginal feature distributions, which happen to be identical for both classes in the case $d=0$. However, the problem can be easily fixed by multiplying the original two-dimensional feature vectors with a random, non-quadratic matrix (See Methods section for details) and thereby creating many new linear feature combinations, some of which usually have significantly different marginal distributions. Such a {\\bf Random Dimensionality Expansion (RDE)}, as proposed in Yang et al. \\cite{yang2021neural}, allows even the Naive Bayes model to reach the accuracy limit in strongly overlapping data classes (Compare Fig.\\ref{figure_1}(g, olive line)).\n\n\n\\subsection*{Part 2: The DSC data model}\n\nIn order to investigate how the performance of different classifiers depends on the statistical properties of the data, we generate large numbers of artificial data sets with two labeled classes $c \\in \\left\\{0,1\\right\\}$, in which the dimensionality $D$ of the individual data vectors $\\vec{u}$, the degree of correlations $C$ between their components $u_{f=1\\ldots D}$ (here also called features), and the separation $S$ between the two classes in feature space can be independently adjusted (See Figs.\\ref{figure_2}(b,c) for an illustration of $C$ and $S$). To replicate some of the heterogeneity of real world data, we design our data generator as a two-level superstatistical model \\cite{metzner2015superstatistical,mark2018bayesian}: The mean values $\\bf \\mu^{(c)}$ and covariances $\\bf \\Sigma^{(c)}$ of the multi-variate probability distributions $p_c(\\vec{u})$ in each of the data classes $c$ are themselves random variables. They are drawn from certain meta-distributions, which are in turn controlled by the three quantities $D,S,C$ (See Methods for details, as well as Fig.\\ref{figure_2}(a)).\n\n\\vspace{0.2cm}\\noindent Using the General Discrimination Value (GDV), a measure designed to quantify the separability of labeled point sets (data classes) in high-dimensional spaces \\cite{schilling2021quantifying}, we show that the mean separability of data classes in the DSC-model is indeed monotonously increasing with the control quantity $S$ (Orange line in Fig.\\ref{figure_2}(d)), whereas the separability of individual data sets is fluctuating heavily around this mean value (Grey dots in Fig.\\ref{figure_2}(d)).\n\n\\vspace{0.2cm}\\noindent Moreover, we quantify the degree of correlation between the $D$ features of the data vectors in each class $c$ by the root-mean-square average of the upper triangular matrix elements in the covariance matrix $\\bf \\Sigma^{(c)}$. We show that this RMS-average is an almost linear function of $C$ (Blue line in Fig.\\ref{figure_2}(d)) and can be varied between zero (Corresponding to independently fluctuating features, or statistical independence) and one (Corresponding to identically fluctuating features, or perfect correlations).\n\n\\subsection*{Part 3: Comparing classifiers}\n\nNext, we apply the three classifier types to artificial data, with statistical properties controlled by the quantities $D$, $S$ and $C$. We first investigate the {\\bf accuracy of the classifiers as a function of data dimensionality $D$} (Fig.\\ref{figure_3}(c,d)), considering correlated data ($C=1.0$). \n\n\\vspace{0.2cm}\\noindent When the separation of the data classes in feature space is small ($S=0.1$, panel (c)), the classification accuracy for one-dimensional data ($D=1$) is very close to the minimum possible value of 0.5 (corresponding to a purely random assignment of the two class labels) in all three models. As data dimensionality $D$ increases, all three models monotonically increase their average accuracies (colored lines), whereas the accuracies of individual cases show a large fluctuation (gray dots). However, the Naive Bayes classifier (orange line) does not perform well even for large data dimensionality, because the point clouds corresponding to the two classes are strongly overlapping in feature space. By contrast, the CMVG Bayes classifier (red line) and the Perceptron (blue line) eventually achieve a very good performance, because they can exploit the correlations in the data. The similarity of the latter two accuracy-versus-$D$ plots is remarkable, considering that these two classifiers work in completely different ways (the Bayesian model performing theory-based mathematical operations with estimated probability distributions, the neural network computing quite arbitrary non-linear transformations of weighted sums). We therefore conclude that the latter two models approach the theoretical optimum of accuracy for each combination of the control quantities $D,S,C$. \n\n\\vspace{0.2cm}\\noindent As the separation of the data classes in feature space gets larger ($S=1.0$, panel (d)), the accuracy-versus-$D$ plots are qualitatively similar to panel (c), but for one-dimensional data ($D=1$) the common accuracy is now slightly above the random baseline, at 0.6. By comparing panels (c) and (d) we note that Naive Bayes is profiting from the larger class separation, but the other two classifiers reach the theoretical performance maximum even without this extra separation.\n\n\\vspace{0.2cm}\\noindent Next, we investigate the {\\bf accuracy of the classifiers as a function of class separation $S$} (Fig.\\ref{figure_3}(e,f)), considering five-dimensional data ($D=5$). Without correlations ($C=0$, panel (e)), all three models show exactly the same monotonous increase of accuracy with separation $S$, starting at the random baseline of 0.5 and finally approaching perfect accuracy of 1.0. \n\n\\vspace{0.2cm}\\noindent With feature correlations present ($C=1.0$, panel (f)), the Naive Bayes classifier shows the same behavior as in panel (e), whereas the other two correlation-sensitive models now already start with a respectable accuracy of 0.8 at zero class separation.\n\n\\vspace{0.2cm}\\noindent Finally, we investigate the {\\bf accuracy of the classifiers as a function of the feature correlations $C$} (Fig.\\ref{figure_3}(g,h)), considering again five-dimensional data ($D=5$). For strongly overlapping data classes ($S=0.1$, panel (g)), Naive Bayes cannot exceed an accuracy of about 0.55, whereas the two correlation-sensitive models show a super-linear increase of accuracy with increasing feature correlations. However, this decrease is ending rather abruptly at about $C\\approx 0.7$. Above this transition point, both models stay at a plateau accuracy of about 0.8, independent of the correlation quantity. Note that this discontinuity of the slope of the accuracy-versus-$D$ plots is likely not an artifact of the DSC data, since the RMS-average of empirical correlations versus $C$ (Fig.\\ref{figure_3}(d)) did not show such an effect at $C\\approx 0.7$. Moreover, the fact that functionally distinct classifiers such as CMVG Bayes and Perceptron produce an almost identical behaviour here suggests that the accuracy plateau in the strong correlation regime indeed reflects the theoretical performance maximum.\n\n\\vspace{0.2cm}\\noindent As the class separation is increased ($S=1.0$, panel (h)), all three models start at a larger accuracy of about 0.75 in the uncorrelated case. Now the performance of Naive Bayes is even declining with increasing $C$, because this model wrongly assumes uncorrelated data. The other two models show again the super-linear increase up to $C\\approx 0.7$. However, now a further improvement of performance is possible with increasing correlations.\n\n\\subsection*{Part 4: Feature transformations}\n\nThe accuracy limit is determined by the overlap of data classes, that is, by the possibility that different classes $i \\neq j$ produce exactly the same data vector $\\vec{x}^{\\ast}$. Transformations $\\vec{x} \\rightarrow \\vec{f}(\\vec{x})$ of the input features can drastically change the distributions of data points (As an example, compare the rows in Fig.\\ref{figure_4}). However, they cannot be expected to reduce the fundamental amount of class overlap, because transformations are just redirecting the common points $\\vec{x}^{\\ast}$ to new locations in feature space. In particular, invertible transformations can be viewed as variable substitutions in the integral Eq.\\ref{cij} for the confusion matrix. They do not affect the resulting matrix values and thus leave the accuracy invariant.\n\n\\vspace{0.2cm}\\noindent In order to test this expectation, we start with two overlapping Gaussian data classes in a two-dimensional feature space (Fig.\\ref{figure_4}, top row), resulting in an accuracy limit of $\\approx 0.69$. All three classifiers actually reach this limit with the original data as input.\n\n\\vspace{0.2cm}\\noindent Next we perform simple non-linear transformations on the input data, by replacing each of the two features $x_1$ and $x_2$ with a function of themselves (in particular: $\\sin$, $\\mbox{sgn}$, and $\\cos$). We find that the application of the $\\sin$-transformation (second row in Fig.\\ref{figure_4}) has indeed no effect on the accuracy of the three classifiers, even though the joint (first column) and marginal distributions (second and third column) are now strongly distorted. Even the application of the $\\mbox{sgn}$-transformation (third row), which collapses all data onto just 4 possible points in feature space, leaves the accuracies invariant. This works because the two classes in our simple example can be distinguished by the sign of the $x_1$-feature, and both the $\\sin$- as well as the $\\mbox{sgn}$-transformation leave this information intact. By contrast, the application of the $\\cos$-transformation destroys this crucial information, and consequently all accuracies drop to the random baseline of 0.5.\n\n\\vspace{0.2cm}\\noindent The above numerical experiments illustrate that transformations of the input-data can reduce (by destroying information that is essential for class-discrimination), but never increase the theoretical accuracy limit, which is an inherent property of the data. Of course, the subsequent data transformations which are taking place in the layers of deep neural networks are still useful, because they re-shape data distributions until classes can be linearly separated in the final layer of the network. \n\n\n\n\\subsection*{Part 5: Sleep EEG data}\n\nIn our artificial data sets, all feature distributions were normally distributed. Moreover, it was possible to introduce extremely strong correlations between these features, which could then be exploited by two of the three classifier models. It is however unclear if the ability of a classifier to detect correlations is always crucial in real-world problems.\n\n\\vspace{0.2cm}\\noindent We therefore turn in a next step to actually measured EEG data, recorded over-night from 68 different sleeping human subjects. In this case, our final goal is to assign to each 30-second epoch of a raw one-channel EEG signal one of the five sleep stages (Wake, REM, N1, N2, N3). \n\n\\vspace{0.2cm}\\noindent At our sample rate, a single epoch of EEG data corresponds to 7680 subsequent amplitudes. Such high-dimensional data vectors $\\vec{x}$ are however not suitable as direct input for a Bayesian classifier, nor for a flat neural network with only $\\approx 100$ neurons. For this reason, we first compress the raw data vectors $\\vec{x}=(x_1,\\ldots x_{7680})$ into suitable feature vectors $\\vec{u}=(u_1,\\ldots u_D)$ of strongly reduced dimensionality $D\\approx 10$. Since we aim to develop a fully transparent classifier system, we use mathematically well-defined, human-interpretable features $u_f = G(\\vec{x},\\alpha_f)$, which depend on a freely tunable parameter $\\alpha$. The dimensionality $D$ of the feature space is then determined by how many of these parameters $\\alpha_{f=1\\ldots D}$ are chosen.\n\n\\vspace{0.2cm}\\noindent The huge literature on brain waves suggests that the momentary {\\bf Fourier components} of the EEG signal are suitable features for the classification of sleep stages. The parameter $\\alpha$ is then naturally given by the frequency $\\nu$ of the Fourier component (For details see methods). In a first experiment, we use a set of six equally spaced frequencies ($\\nu_1=$5 Hz, $\\nu_2=$10 Hz,$\\ldots$ $\\nu_6=$30 Hz). Based on training data sets that have been manually labeled by a sleep specialist, we then compute the marginal probability density functions of these Fourier features, as well as their covariance matrices, for each of the 5 sleep stages $s$ (Fig.\\ref{figure_5}, left two columns). We find that within each sleep stage, the Fourier features have unimodal distributions, with peak positions and widths depending quite systematically on the frequency $\\nu$. There are characteristic differences between the sleep stages (in particular the distributions are wider in the wake stage), but they are not very pronounced. In the covariance matrices, we find that the off-diagonal elements are significantly smaller than the diagonal elements (The latter have been set to zero in Fig.\\ref{figure_5} to emphasize the actual inter-feature correlations), with the exception of the wake state. Also the N1 state has slightly larger inter-feature correlations compared to the REM, N2 and N3 states. \n\n\\vspace{0.2cm}\\noindent As an alternative or complement to the Fourier features, we also consider the normalized (Pearson) {\\bf auto-correlation coefficients} of the raw EEG signal (Fig.\\ref{figure_5}, right two columns. For details see methods). The feature parameter $\\alpha$ is in this case given by the lag-time $\\Delta t$, for which we choose six equally spaced values ($1,3,\\ldots,11$ in units of the EEG sampling period). Since these correlation features cannot exceed the value of one by definition, the marginal distributions are highly non-Gaussian with pronounced tails towards small values. These tails show relatively strong differences between some of the sleep stages, but also surprising similarities, in particular for REM and N2. In the covariance matrices, we find the strongest inter-feature correlations in the wake and N1 stages. Again, the covariance matrices are very similar in REM and N2.\n\n\\subsection*{Part 6: Sleep stage detection}\n\nNext, we apply our three classifier models to the above sleep EEG data. However, while the feature distributions and correlations in Fig-\\ref{figure_5} were based on the global data, pooled over all 68 full-night EEG recordings, we are considering here the task of personalized sleep-stage recording. That is, the classifiers are trained and evaluated individually on each of the 68 data sets. Because the amount of training data is severely limited in this task, classification accuracies are expected to be rather low and strongly dependent on the participant. We therefore compute the distributions of accuracies over the 68 personalized data sets (histograms in Fig.\\ref{figure_6}) for all three classifiers and for the two types of pre-processed features. \n\n\\vspace{0.2cm}\\noindent We find that the CMVG Bayes model is performing very poorly in this task, presumably because the feature distributions are non-Gaussian and only weakly correlated except in the wake stage. In particular, for some participants the classification accuracy is less then the random baseline of about 0.2, corresponding to consistent miss-classifications. This can happen in Bayesian classifiers when the likelihood distributions learned from the training data set do not match the actual distributions in the test data set. \n\n\\vspace{0.2cm}\\noindent By contrast, the Naive Bayes model can properly represent the non-Gaussian feature distributions by KDE approximations, and it furthermore profits from the lack of correlations. The performance of the Perceptron is comparable to that of the Naive Bayes model. Both for Fourier- and correlation-features, these two models show accuracies well above the baseline, roughly in the range from 0.3 to 0.6.\n\n\\subsection*{Part 7: Natural data clustering}\n\nBoth the ten digits in MNIST, as well as the five sleep stages in overnight EEG recordings, are human-defined classes. It is therefore unclear whether these classes can also be considered as 'natural kinds'. \n\n\\vspace{0.2cm}\\noindent After a suitable pre-processing that brings both data sets into the same format of 784-dimensional, normalized feature vectors (for details see Methods sections), we address this question by computing two-dimensional MDS projections, coloring the data points according to the known, human-assigned labels (In Fig.\\ref{figure_7}, see the upper left scatter plot in each 2-by-2 block). Indeed, the projected data distributions show a small degree of clustering, which is also quantitatively confirmed by the corresponding GDV values (-0.061 for MNIST and -0.035 for sleep EEG data). Note that in the sleep data, a large number of extreme outliers are found which might not correspond to any of the standard classes.\n\n\\vspace{0.2cm}\\noindent The purpose of classifiers is to transform and re-shape the data distribution in such a way that the final network layer (often a softmax layer with one neuron for each data class) can separate the classes easily from each other. Although, as we have shown above, these re-shaping transformations cannot reduce the natural overlap of classes (which would push the accuracy beyond the data-inherent limit), they might as a side-effect lead to a larger 'centrality' of the clusters associated with each class. This would show up quantitatively as a decrease of the General Discrimination Value (GDV) in the higher network layers of the classifier, as compared to the original input data. In order to test this hypothesis, we have trained a four-layer perceptron (see Methods section for details) in a supervised manner on both the MNIST and sleep EEG data. In the case of MNIST, we indeed observe a systematic decrease of the GDV in subsequent network layers: GDV(L0)=-0.061, GDV(L1)=-0.174, GDV(L2)=-0.250, and GDV(L3)=-0.300 (See Fig.\\ref{figure_7}(b)). An analogous layer-wise decrease is found for the sleep EEG data: GDV(L0)=-0.035, GDV(L1)=-0.096, GDV(L2)=-0.122, and GDV(L3)=-0.181 (See Fig.\\ref{figure_7}(d)).\n\n\\vspace{0.2cm}\\noindent We finally address the question whether a natural clustering in novel, unlabeled data sets can be automatically detected, and possibly enhanced, in an unsupervised manner. For this purpose, we consider an autoencoder that performs a layer-wise dimensionality reduction of the data, and then re-expands these low-dimensional embeddings back to the original number of dimensions. During this process of 'compression' and 're-expansion', fine details of the data have to be discarded, and it appears reasonable that this might go hand in hand with a 'sharpening' of the clusters. Again, in our test case where the labels of the data points are actually known, this enhancement of cluster centrality can be quantitatively measured by the GDV. For comparability, we have used an autoencoder that has the same design as the perceptron for the first four network layers. In the case of MNIST, we indeed find that the unsupervised compression enhances cluster centrality: GDV(L0)=-0.061, GDV(L1)=-0.115, GDV(L2)=-0.122, and GDV(L3)=-0.137 (See Fig.\\ref{figure_7}(a)). The behavior is similar with the sleep EEG data, except for the last layer: GDV(L0)=-0.035, GDV(L1)=-0.037, GDV(L2)=-0.041, and GDV(L3)=-0.036 (See Fig.\\ref{figure_7}(c)).\n\n\n\\vspace{0.2cm}\\noindent \n\n\n\\clearpage\n\\section{Discussion and Outlook}\n\nIn this work, we have addressed various aspects of data ambiguity: the fact that multi-dimensional data spaces usually contain vectors that cannot be unequivocally assigned to any particular class. \nThe probability of encountering such ambiguous vectors is easily underestimated in machine learning, because the data sets used to train classifiers - rather than being sampled randomly from the entire space of possible data - typically represent just a tiny, pre-selected subset of 'reasonable' examples. For instance, the space of monochrome images with full HD resolution and 256 gray values contains $256^{1920 \\times 1080} \\approx 10^{4993726}$ possible vectors. The fraction of these images that resemble any human-recognizable objects is virtually zero, whereas the largest part would be described as noise by human observers. One may argue that these 'structure-less' images should not play any role in real-world applications. However, it is conceivable that sensors in autonomous intelligent systems, such as self-driving cars, can produce untypical data under severe environmental conditions, such as snow storms. How to deal with data ambiguity is therefore a practically relevant problem. Moreover, as we have tried to illustrate in this paper, data ambiguity has interesting consequences from a theoretical point of view.\n\n\\vspace{0.2cm}\\noindent In part one, we have derived the theoretical limit $A_{max}$ of accuracy that can be achieved by a perfect classifier, given a data set with partially overlapping classes. By generating artificial data classes with Gaussian probability distributions in a two-dimensional feature space and with a controllable distance $d$ between the maxima, we verified that different types of classifiers (The CMVG Bayesian model with multi-variate Gaussian likelihoods and a perceptron) exactly follow the predicted accuracy limit $A_{max}(d)$ (Fig.\\ref{figure_1}(g)). The naive Bayesian model, which cannot exploit correlations to distinguish between data classes, originally yields sub-optimal accuracies for small distances $d$, but this problem can be fixed by applying a random dimensionality expansion to the data as a trivial pre-processing step \\cite{yang2021neural}. We have restricted ourselves to only two features (dimensions) for this test, because predicting the accuracy limit involves the exact computation of the confusion matrix, which in turn is an integral over the entire data space. Note, however, that for high-dimensional data with known class-dependent generation densities $p_{gen}(\\vec{x}\\;|\\;i)$, the integral could be approximated by Monte Carlo sampling. In this case, the element $C_{ji}$ of the confusion matrix would be computed by drawing random vectors $\\vec{x}$ from class $i$. The class indicator function $\\hat{q}_{cla}(\\;k\\;|\\;\\vec{x}\\;)$ of the perfect classifier, which is fully determined by the generation densities, yields the corresponding predicted classes $k$ for these data vectors. The matrix element $C_{ji}$ is then given by the fraction of cases where $k=j$. \n\n\\vspace{0.2cm}\\noindent In part two, we have constructed a two-level model to generate artificial test data (Fig.\\ref{figure_2}). The model has high-level parameters $D$, $S$ and $C$ which control the number of dimensions (features), the average separation of the two classes in feature space, as well as the average correlation between the features. For each triple of high-level parameters $D,S,C$, a large number of low-level parameters $\\mu, \\Sigma$ are randomly drawn according to specified distributions, which are in turn used to generate the final test data sets. The super-statistical nature of the model allows us to prescribe the essential statistical features of dimensionality, separation and correlation, while at the same time ensuring a large variability of the test data. By using the General Discrimination Value (GDV), a quantitative measure of class separability (centrality), we have confirmed that the high-level parameter $S$ controls the class separability as intended. Moreover, the proper action of parameter $C$ was confirmed by computing the root-mean-square average over the elements of the data's covariance matrix.\n\n\\vspace{0.2cm}\\noindent In part three, we have applied our three types of classifiers to the test data generated with the DSC-model. Without intra-class feature correlations ($C=0$), we find that all three models show with growing separation parameter $S$ exactly the same monotonically increasing average accuracy (Fig.\\ref{figure_3}(e)). Although the exact computation of $A_{max}$ is not possible in this five-dimensional data space, the perfect agreement of the three different classifiers indicates that they all have reached the accuracy limit. When intra-class feature correlations are present ($C\\neq 0$), we find by systematically varying the parameters $D$, $S$ and $C$ that the resulting accuracies of the CMVG-Bayes classifier and of the perceptron are extremely similar in all considered cases, indicating again that they have reached the theoretical accuracy limit. As expected, the naive Bayesian classifier shows sub-optimal accuracies in all cases where feature correlations are required to distinguish between the classes. In general, this analysis shows that the accuracy of classification can be systematically enhanced by providing more features (larger data dimensionality $D$) as input. Extra features that do not provide additional useful information are 'automatically ignored' by the classifiers and never reduce the achievable accuracy. Moreover, accuracy can be enhanced by providing features that are correlated with each other (larger parameter $C$), but differently in each data class. Such class-specific feature correlations can be exploited for discrimination by models such as CMVG Bayes and the perceptron, but not by the naive Bayes model. Moreover, we find that the theoretical accuracy maximum as a function of the correlation parameter $C$ shows an interesting abrupt change of slope at around $C\\approx 0.8$ (Fig.\\ref{figure_3}(g,h)). The origin of this effect is at present unclear, but will be explored in follow-up studies.\n\n\\vspace{0.2cm}\\noindent In part four, we have investigated the effect of non-linear feature transformations, applied as a pre-processing step, on classification accuracy (Fig.\\ref{figure_4}). Since the achievable accuracy in a classification task is limited by the degree of overlap between the data classes, feature transformations can certainly reduce the accuracy to below the limit $A_{max}$ (when they destroy information that is essential for discrimination), but they can never push the accuracy to above $A_{max}$. This is indeed confirmed in a simple test case where all three classifier types perform at the accuracy maximum with the non-transformed data: Applying a feature-wise sine-transformation drastically changes the data distributions $p_{gen}(\\vec{x}\\;|\\;i)$, but leaves the accuracies unchanged at $A_{max}$. The accuracy remains invariant even under a signum-transformation, although this non-invertible operation reduces the data distributions to only four possible points in feature space. In this extreme case, most of the detailed information about the input data vectors is lost, but the part that is essential for class discrimination, namely the sign of the feature $x_1$, is retained. This example demonstrates that classification is a type of lossy data processing where irrelevant information can be safely discarded. For this reason, neural-network based classifiers usually project the input data vectors into spaces of ever smaller dimensions, up to the final discrimination layer which needs only as many neural units as there are data classes. In this context, it is interesting that biological organisms with nervous systems, relying on an efficient classification of objects in their environment for survival, have probably evolved sensory organs and filters that only transmit the small class-discriminating part of the available information to the higher stages of the neural processing chain. As a consequence, our human perception is almost certainly not a veridical representation of the world \\cite{mark2010natural,hoffman2014objects, hoffman2018interface}.\n\n\\vspace{0.2cm}\\noindent In part five, we have analyzed full-night EEG recordings of sleeping humans, divided into epochs of 30 seconds that have been labeled by a specialist according to the five sleep stages. Such recordings can been used as training data for automatic sleep stage classifiers - an application of machine learning that could in the future remove a large work load from clinical sleep laboratories. In our context of data ambiguity, sleep EEG is an interesting case because different human specialists agree about individual sleep-label assignments only in 70\\% - 80\\% of the cases, even if multiple EEG channels and other bio-signals (such as electro-oculograms or electro-myograms) are provided \\cite{fiorillo2019automated}. This low inter-rater reliability suggests that a considerable fraction of the 30-second epochs is actually ambiguous with respect to sleep stage classification, in particular when only the time-dependent signal of a single EEG channel is available as input-data. Our first goal is a suitable dimensionality reduction of the raw data, which (at a sample rate of 256 Hz) consist of 7680 subsequent EEG values in each epoch. As a pre-processing step, we map each 7680-dimensional raw data vector onto an only 6-dimensional feature vector, so that our Bayesian classifiers (Naive and CMVG) can be efficiently used. We consider as features the real-valued Fourier amplitudes at different frequencies, as well as the auto-correlation coefficients at different lag-times (Fig.\\ref{figure_5}). The Fourier features are expected to be particularly useful, as it is well-known that the activity in different EEG frequency bands varies in characteristic ways over the five sleep stages. The correlation features have been successfully applied for Bayesian classification in a former study \\cite{metzner2021sleep}. In our present study, we are using either Fourier or correlation features, but no combinations of those. By performing a statistical analysis of the features, we find that within the same sleep stage, the six features have significantly different marginal probability distributions. However, these distributions are quite similar in all sleep stages, so that their value for the classification task is limited. Moreover, the correlations between features, which could be exploited by the CMVG Bayes classifier and by the perceptron, turn out to be very weak, except for the Fourier features in the wake stage. Another problem is the strongly non-Gaussian shape of the marginal probability distributions in the case of the correlation features, which cannot be properly represented by the CMVG Bayes model. \n\n\\vspace{0.2cm}\\noindent In part six, we have used our three classifier models, based on the above Fourier- and correlation features, for personalized sleep stage detection. In this very hard task, the classifiers are trained and tested, independently, on the full-night EEG data set of a single individual only. Since an individual data set contains typically less than 1000 epochs (each corresponding to one feature vector), random deviations from the 'typical' sleeping patterns are likely to be picked up during the training phase. We consequently find that the accuracies vary widely between the individual data sets. As expected, the CMVG Bayes model performs badly in this task, because there are almost no inter-feature correlations present that could be exploited for sleep stage discrimination, and because feature distributions are non-Gaussian. Interestingly, both the Naive Bayesian classifier and the perceptron achieve relatively good accuracies, mainly in the range from 0.3 to 0.6. However, these accuracies may be further increased by using more sophisticated neural network architectures \\cite{stephansen2018neural, krauss2021analysis}, and hence do not represent the accuracy limit.\n\n\\vspace{0.2cm}\\noindent In the final part seven, we have started to explore whether the distinct classes in typical real-world data sets are defined arbitrarily (and therefore can only be detected after supervised learning), or if the differences between these classes are so prominent that even unsupervised machine learning methods can recognize them as distinct clusters in feature space. Besides the (pooled) sleep EEG data, we have used the MNIST data set to test for any inherent clustering structure. For this investigation, the individual data points, corresponding to respectively one epoch of EEG signal or one handwritten digit, have been brought into the same format of 784-dimensional, normalized vectors. Computing directly the General Discrimination Value (GDV) of the MNIST data, based on the known labels, has indeed revealed a small amount of 'natural clustering', even in this raw data distribution. This quantitative result was qualitatively confirmed by a two-dimensional visualization using multi-dimensional scaling (MDS), however the cluster structure would hardly be visible without the class-specific coloring (left upper scatter plots in Fig.\\ref{figure_7}(a,b)). By contrast, no natural clustering was found for the raw sleep EEG data when the 7680 values in each epoch were simply down-sampled in the time-domain to 784 values (data not shown). This presumably fails because the relevant class-specific signatures appear randomly at different temporal positions within each epoch, and so the Euclidean distance between two data vectors is not a good measure of their dissimilarity. However, when we instead used as data vectors the magnitudes of the 784 Fourier amplitudes with lowest frequencies, a weak natural clustering was found also in the sleep data (left upper scatter plots in Fig.\\ref{figure_7}(c,d)). We have furthermore demonstrated that the degree of clustering (for both data sets) is systematically increasing in the higher layers of a perceptron that has been trained to discriminate the classes in a supervised manner (Fig.\\ref{figure_7}, right column). Finally, we have used a multi-layer autoencoder to produce embeddings of the data distributions with reduced dimensionality in an unsupervised setting. It has turned out that the degree of clustering (with respect to the known data classes) tends to increase systematically with the degree of dimensional compression (Fig.\\ref{figure_7}, left column). This interesting finding, previously reported in Schilling et al. \\cite{schilling2021quantifying}, suggests that unsupervised dimensionality reduction could be used to automatically detect and enhance natural clustering in unlabeled data. In combination with automatic labeling methods, such as Gaussian Mixture Models, this may provide an objective way to define 'natural kinds' in arbitrary data sets. \n\n\n\n\\clearpage\n\\section{Additional information}\n\n\\noindent{\\bf Author contributions statement:}\nCM has conceived of the project, implemented the methods, evaluated the data, and wrote the paper, PK co-designed the study, discussed the results and wrote the paper, AS discussed the results, MT provided access to resources and wrote the paper, HS provided access to resources and wrote the paper.\n \\vspace{0.5cm} \n\n\\noindent{\\bf Funding:}\nThis work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation): grant SCHU\\,1272\/16-1 (AOBJ 675050) to HS, grant TR\\,1793\/2-1 (AOBJ 675049) to MT, grant SCHI\\,1482\/3-1 (project number 451810794) to AS, and grant KR\\,5148\/2-1 (project number 436456810) to PK. \\vspace{0.5cm}\n\n\\noindent{\\bf Competing interests statement:}\nThe authors declare no competing interests. \\vspace{0.5cm}\n\n\\noindent{\\bf Data availability statement:}\nData and analysis programs will be made available upon reasonable request.\n\\vspace{0.5cm}\n\n\\noindent{\\bf Ethical approval and informed consent:} The study was conducted in the Department of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander University Erlangen-N\u00fcrnberg (FAU), following approval by the local Ethics Committee (323 \u2013 16 Bc). Written informed consent was obtained from the participants before the cardiorespiratory poly-somnography (PSG).\n\n\\vspace{0.5cm}\n\n\\noindent{\\bf Third party rights:}\nAll material used in the paper are the intellectual property of the authors. \\vspace{0.5cm}\n\n\\clearpage\n\\bibliographystyle{unsrt}\n\n\\section{Introduction}\n\nData classification -- e.g. object recognition -- is a fundamental computing problem in machine learning and artificial intelligence. Large-scale classification competitions such as the annual ImageNet challenge \\cite{krizhevsky2012imagenet, russakovsky2015imagenet}, where a super-human accuracy of 95\\% has been achieved within about 5 years of steady progress, have contributed greatly to the general popularity of machine learning. Understandably, ImageNet is mostly discussed in the context of technical improvements regarding the classification methods which enabled this drastic boost of performance. But it also illustrates some fundamental problems that arise when computers are to create models of human-defined data categories:\nFor example, the fact that classification accuracies are typically leveling off at values below 100\\% does not necessarily reflect a limitation of the algorithms, but instead may reveal the classification limits of the humans who provided the ground truth data. Indeed, in the case of ImageNet, the massive work of annotating millions of images had been crowd-sourced using Amazon Mechanical Turk, and so a large number of individuals were involved in the labeling process, individuals who may place certain ambiguous images into different categories. This problem of ambiguity due to non-rigorously defined object categories is most pronounced in biological and medical data, where sample-to-sample variations are notoriously large. \n\nIn this work, we use artificially generated surrogate data, as well as real-world bio-medical data, to explore the implications of this inevitable data ambiguity. We demonstrate that the overlap of data classes leads to a theoretical upper limit of classification accuracy, a limit that can be mathematically computed in low-dimensional examples and which depends in a systematic way on the statistical properties of the data set. We find that sufficiently powerful classifier models of different kinds all perform at this same upper limit of accuracy, even if they are based on completely different operating principles. Interestingly, this accuracy limit is not affected by applying certain non-linear transformations to the data, even if these transformations are non-reversible and drastically reduce the information content (entropy) of the input data. \n\nIn a next step, the same three models that reached the common classification limit for artificial data are now applied to human EEG data measured during sleep. In a pre-processing step, two kinds of features are extracted from raw EEG signals, yielding different marginal distributions and mutual correlations. It turns out that a more complex Bayesian model, based on correlated multi-variate Gaussian likelihoods (CMVG), performs worse than two other models (naive Bayes, perceptron), because the statistical properties of the pre-processed features do not match those of the likelihoods. In contrast, the perceptron and the naive Bayes model still show very similar classification accuracies, indicating that both reach the theoretical accuracy limit for sleep stage classification.\n\nFinally, we address the question whether typical human-defined object categories can also be considered as 'natural kinds', that is, whether the data vectors in input space have a built-in cluster structure that can be detected by objective machine-learning models even in non-labeled data. For this purpose, we use as real-world examples the MNIST data set \\cite{deng2012mnist}, as well as the above EEG sleep data. We find that a simple visualization by multi-dimensional scaling (MDS) \\cite{torgerson1952multidimensional, kruskal1964nonmetric,kruskal1978multidimensional,cox2008multidimensional} already reveals an inherent cluster structure of the data in both cases. Interestingly, the degree of clustering, quantified by the general discrimination value (GDV) \\cite{krauss2018statistical, schilling2021quantifying}, can be enhanced by a step-wise dimensionality reduction of the data, using an autoencoder that is trained in an unsupervised manner. A perceptron classifier with a layer design comparable to the autoencoder, trained on the same data in a supervised fashion, achieves as expected a much stronger cluster separation. However, the enhancement of clustering by unsupervised data compression, combined with automatic labeling methods, could be a promising way to automatically detect 'natural kinds' in non-labeled data. \n\n\n\\clearpage\n\\section{Methods}\n\n\\subsection*{Part 1: Accuracy limit}\n\n\\subsubsection*{Derivation of theoretical accuracy limit}\n\nClassification is the general problem of assigning a discrete class label $i=1\\ldots K$ to each given input data $\\vec{x}$, where the latter is considered as a vector with $N$ real-valued components $x_{n=1\\ldots N}$. Such a discrimination is possible when the conditional probability distributions $p_{gen}(\\vec{x}\\;|\\;i)$ of data vectors, here called {\\bf 'generation densities'}, are different for each of the possible data classes $i$. In the simple case of a two- or three-dimensional data space, each data class can be visualized as a 'point cloud' (See Fig.\\ref{figure_1}(a,b) for examples), and either the shapes or the center positions of these point clouds must vary sufficiently in order to facilitate a reliable classification. However, since the data generation process typically involves not only the system of interest (which might indeed have $K$ well-distinguished modes of operation), but also some measurement or data transmission equipment (which introduces noise into the data), a certain 'overlap' of the different data classes is usually not avoidable.\n\n\\vspace{0.2cm}\\noindent A classifier is receiving the data vectors $\\vec{x}$ as input and computes a set of $K$ {\\bf 'classification probabilities'} $q_{cla}(j\\;|\\;\\vec{x})$, quantifying the belief that $\\vec{x}$ belongs to $j$. They are normalized to one over all possible classes, so that $\\sum_{j=1}^K q_{cla}(j\\;|\\;\\vec{x}) = 1\\; \\forall\\; \\vec{x}$.\n\n\\vspace{0.2cm}\\noindent We can now define a {\\bf 'confusion density'} as the product\n\\begin{equation}\nC_{ji}(\\vec{x}) = q_{cla}(j\\;|\\;\\vec{x})\\;p_{gen}(\\vec{x}\\;|\\;i).\n\\end{equation}\nIt can be interpreted as the probability density that the generator is producing data vector $\\vec{x}$ under class $i$, which is then assigned to class $j$ by the classifier. Because there is usually a very small but non-zero probability density that {\\em any} vector $\\vec{x}$ can occur under {\\em any} class $i$, we expect that the non-diagonal elements $C_{j\\!\\neq\\!i}(\\vec{x})$ are larger than zero as well. These non-diagonal confusion densities will have their largest values in regions of data space where the classes $i$ and $j$ overlap (See Fig.\\ref{figure_1}(e,f) for examples).\n\n\\vspace{0.2cm}\\noindent By integrating the confusion density over all possible data vectors $\\vec{x}$,\n\\begin{equation}\nC_{ji} = \\int C_{ji}(\\vec{x})\\; d\\vec{x},\n\\label{cij}\n\\end{equation}\nwe obtain the {\\bf 'confusion matrix'} of the classifier, which comes out properly normalized, so that $\\sum_{j=1}^K C_{ji} = 1\\; \\forall\\; i$. The confusion matrix $C_{ji}$ therefore is the probability that a data point originating from class $i$ is assigned to class $j$.\n\n\\vspace{0.2cm}\\noindent Assuming for simplicity that all data classes appear equally often, we can compute the {\\bf accuracy} $A$ of the classifier as the average over all diagonal elements of the confusion matrix:\n\\begin{equation}\nA = \\frac{1}{K}\\sum_{i=1}^K C_{ii}.\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent In the following, we are particularly interested in the {\\bf theoretical limit of the classification accuracy}, denoted by $A_{max}$. We therefore consider an ideal classifier that has learned the exact generation densities $p_{gen}(\\vec{x}\\;|\\;i)$. In this case, the {\\bf 'ideal classification probability'} corresponds to the Bayesian posterior\n\\begin{equation}\nq_{cla}(\\;j\\;|\\;\\vec{x}\\;) = \\frac{p_{gen}(\\;\\vec{x} \\;|\\; j\\;)}{\\sum_k p_{gen}(\\;\\vec{x} \\;|\\; k\\;)}. \n\\end{equation}\n\n\\vspace{0.2cm}\\noindent In our numerical experiments, we will use classifiers that output a definite class label $j$ for each given data vector $\\vec{x}$, corresponding to most probably class. To compute the theoretical accuracy maximum for such a model, we replace $q_{cla}$ by the binary {\\bf 'class indicator function'}\n\\begin{equation}\n\\hat{q}_{cla}(\\;j\\;|\\;\\vec{x}\\;) = \\delta_{jk}\\;\\;\\mbox{with}\\;\\;k = \\mbox{argmax}_c\\; q_{cla}(\\;c\\;|\\;\\vec{x}\\;).\n\\end{equation}\nIt has the value $1$ for all data points $\\vec{x}$ assigned to class $j$, and the value $0$ for all other data points (See Fig.\\ref{figure_1}(c,d) for examples). When the ideal accuracy $A$ is evaluated using $\\hat{q}_{cla}$ instead of $q_{cla}$, the result can be directly compared with numerical accuracies based on one-hot classifier outputs.\n\n\\subsubsection*{Numerical evaluation of $A_{max}$}\n\nIn Fig.\\ref{figure_1}, the above quantities have been numerically evaluated for a simple Gaussian test data set. For this purpose, the two-dimensional integral \\ref{cij} has been evaluated numerically on a regular grid of linear spacing 0.01, ranging from -8 to +8 in each feature dimension. \n\n\n\\subsubsection*{Classifiers and input data}\n\n\\vspace{0.2cm}\\noindent In the following subsections, we provide the implementation details for the different classifier models that are compared in this work. The input data for these models is given as lists of $D$-dimensional feature vectors $\\vec{u} = (u_1,u_2,\\ldots,u_f,\\ldots,u_D)$, each belonging to one of $K$ possible classes $c$. In the case of artificially generated data, these lists contain 10000 feature vectors distributed equally over the data classes. They are split randomly into training (80\\%) and test (20\\%) data sets.\n\n\\subsubsection*{Perceptron model}\n\nThe perceptron model is implemented using Keras\/Tensorflow. It has one hidden layer, containing $N_{neu}=100$ neurons with RELU activation function. The output layer has $N_{out}$ neurons with softmax activation function, where $N_{out}=D$ corresponds to the number of data classes. The loss function is categorical crossentropy. We optimize the perceptron on each training data set using the Adams optimizer over at least 10 epochs with a batch size of 128 and a validation split of 0.2. After training, the accuracy of the perceptron is evaluated with the independent test data set.\n\n\\subsubsection*{Naive Bayesian model}\n\nThe naive Bayesian model is implemented using the Python libraries Numpy and Scipy.\n\n\\vspace{0.2cm}\\noindent In the training phase, the training data set is sorted according to the $K$ class labels $c$. Then an individual Gaussian Kernel Density (KDE) approximation (Scott method) is computed for each feature $f$ and class label $c$, corresponding to the empirical marginalized probability densities $p_{f,c}(u_f)$. \n\n\\vspace{0.2cm}\\noindent In the testing phase, the accuracy of the model is evaluated with the independent test data set as follows: According to the naive Bayes approach, the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}=(u_1,u_2,\\ldots,u_D)$ under class $c$ is approximated by a product of the marginalized probabilities, so that\n\\begin{equation}\nL(\\vec{u}\\;|\\;c) = \\prod_{f=1\\ldots D} p_{f,c}(u_f).\n\\end{equation}\nSince we assume a flat prior probability ($P_{prior}(c)=1\/K$) over the data classes, the posterior probability of data class $c$, given the input data vector $\\vec{u}$, is given by\n\\begin{equation}\nP_{post}(c\\;|\\;\\vec{u}) = \\frac{L(\\vec{u}\\;|\\;c)}{\\sum_{i\\!=\\!1}^K\\;L(\\vec{u}\\;|\\;i)}\n\\end{equation}\n\n\\subsubsection*{Naive Bayesian model with Random Dimensionality Expansion (RDE)}\n\nSince the naive Bayesian model takes into account only the marginal feature distributions $p_{f,c}(u_f)$, it cannot distinguish data classes which accidentally have identical $p_{f,c}(u_f)$ distributions, but differ in the correlations between the features. In principle, this problem can be fixed by multiplying the $D$-dimensional input vectors $\\vec{u}$ by a random $D_2 \\times D$ matrix $\\textbf{M}$, for example with normally distributed entries $M_{ij}\\propto N(\\mu=0,\\sigma=1)$, which yields transformed vectors $\\vec{v} = \\textbf{M} \\vec{u}$. Provided that $D_2\\gg D$, at least some of the new feature linear combinations $v_f$ will have marginal distributions that vary between the data classes. \n\n\\subsubsection*{CMVG Bayesian model}\n\nThe Correlated Multi-Variate Gaussian (CMVG) Bayesian model is also implemented using the Python libraries Numpy and Scipy. \n\n\\vspace{0.2cm}\\noindent In the training phase, the training data set is sorted according to the two class labels $c$. Then, for each class label $c$, we compute the mean values $\\mu_f^{(c)}$ of the features $f=1\\ldots D$, as well as the covariances $\\Sigma_{fg}^{(c)}$ between features $f$ and $g$. These quantities are packed as one vector ${\\bf \\mu}^{(c)}$ and one matrix ${\\bf \\Sigma}^{(c)}$ for each class $c$.\n\n\\vspace{0.2cm}\\noindent In the testing phase, the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}=(u_1,u_2,\\ldots,u_D)$ under class $c$ is computed as the correlated, multi-variate Gaussian probability density\n\n\\begin{equation}\nL(\\vec{u}\\;|\\;c) = p_{cmvg}\\left( \\vec{u} \\;,\\; {\\bf\\mu}\\!=\\!{\\bf\\mu}^{(c)} \\;,\\;\n{\\bf \\Sigma}\\!=\\!{\\bf \\Sigma}^{(c)}\n\\right).\n\\end{equation}\nSince we assume a flat prior probability ($P_{prior}(c)=1\/2$) for the two data classes, the posterior probability of data class $c$, given the input data vector $\\vec{u}$, is given by\n\\begin{equation}\nP_{post}(c\\;|\\;\\vec{u}) = \\frac{L(\\vec{u}\\;|\\;c)}{\\sum_{i\\!=\\!1}^K\\;L(\\vec{u}\\;|\\;i)}\n\\end{equation}\n\n\n\\subsection*{Part 2: The DSC data model}\n\nWe consider an artificial classification problem with two multivariate Gaussian data classes $c\\in\\left\\{0,1\\right\\}$ and with statistical properties that can be tuned by {\\bf three control quantities}: the {\\bf dimensionality} $D$ of the feature space, the {\\bf separation} $S$ between the centers of the point clouds, and the {\\bf correlation} $C$ between features (within the same class), which is associated with the shape of the point cloud. The generation of artificial data within this DSC model works as follows:\n\n\\vspace{0.2cm}\\noindent Starting from a given triple $D,S,C$ of control quantities, we first generate $N_{rep}$ independent {\\bf parameter sets} $[\\mu_f^{(c)}, \\Sigma_{fg}^{(c)}]$ that describe the statistical properties of the two classes $c\\in\\left\\{0,1\\right\\}$. Here, $\\mu_f^{(c)}$ is the mean value of feature $f$ in class $c$, and $\\Sigma_{fg}^{(c)}$ is the covariance of features $f$ and $g$ in class $c$.\n\n\\vspace{0.2cm}\\noindent The mean values $\\mu_f^{(c=0)}$ in class $0$ are always set to zero, whereas the mean values $\\mu_f^{(c=1)}$ in class $1$ are random numbers, drawn from a uniform distribution with values in the range from 0 to $S$. The separation quantity $S$ is therefore the maximum distance between corresponding feature mean values in each dimension $f$.\n\n\\vspace{0.2cm}\\noindent The diagonal elements $\\Sigma_{ff}^{(c)}$ of the symmetric covariance matrix are set to 1 in both classes. The off-diagonal elements $\\Sigma_{f\\neq g}$ are assigned independent, continuous random numbers $x$, drawn from a box-shaped probability density distribution $q(x,C)$ that depends on the correlation quantity $C$ as follows:\n\n\\begin{eqnarray}\n q(x,C) &=& \\mbox{uniform}[0\\;,\\;C]\\;\\;\\;\\;\\;\\;\\mbox{for}\\;C\\le1 \\nonumber\\\\\n q(x,C) &=& \\mbox{uniform}[C\\!-\\!1\\;,\\;1]\\;\\mbox{for}\\;C>1\n\\end{eqnarray}\n\n\\vspace{0.2cm}\\noindent For $C=0$, the distribution $q(x,C)$ peaks at $x=0$, so that $\\Sigma_{ij}$ becomes a diagonal unit matrix. For $C=1$, the distribution $q(x,C)$ is uniform in the range $[0,1]$, and for $C=2$ it peaks at $x=1$, leading to $\\Sigma_{ij}=1$. A plot of the distribution is shown in \\ref{figure_2}(b).\n\n\\vspace{0.2cm}\\noindent According to the parameter set $[\\mu_f^{(c)},\\Sigma_{fg}^{(c)}]$, we then generate for each of the two classes $c$ a number $N_{vec}\/2$ of random, Gaussian data vectors $\\vec{u}(t)$, in which the $D$ components (features) are correlated to a degree controlled by quantity $C$. In the limiting case $C=0$, the $D$ time series $u_{f}(t)$ become statistically independent, whereas for $C=1$, the time series become fully correlated and thus identical. The total number of $N_{vec}$ data vectors is combined to a complete data set, in which vectors from the two classes (with corresponding labels $c$) appear in random order. By this way, we obtain for each triple $D,S,C$ of control quantities a total number of $N_{rep}$ independent data sets, each consisting of $N_{vec}$ data vectors. Since each data set obeys its own random parameters $[\\mu_f^{(c)},\\Sigma_{fg}^{(c)}]$, the DSC model reflects some of the heterogeneity of typical real world data. Finally, we split each data set into a training set (80\\%) and a test set (20\\%).\n\n\\vspace{0.2cm}\\noindent Before applying different types of classifiers to the DSC data sets, we test that the feature correlations and the class separation can be controlled reliably and over a sufficiently large range, using the quantities $C$ and $S$ (Fig.\\ref{figure_2}(d)).\n\n\\subsubsection*{Control of feature correlations by quantity $C$}\n\nTo evaluate correlation control, we fix the quantities $D=10$ and $S=1.0$ (Note that the separation has no effect on the correlations) and vary $C$ over the complete available range of supported values from 0 to 2. For each $C$, we generate $N_{rep}=100$ independent data sets, each consisting of $N_{vec}=10000$ data vectors. For each data set, we estimate the empirical covariance matrix $\\Sigma_{ij}^{(0)}$ of class 0. Because the matrix is symmetric, we compute the root-mean-square (RMS) average of all matrix elements above the diagonal. The blue line in (Fig.\\ref{figure_2}(d)) shows for each $C$ the mean RMS, averaged over the $N_{rep}=100$ repetitions (The latter are shown as gray dots). We find an almost linear relation between $C$ and the mean RMS. In particular, we can realize the full range of correlations, including the limiting cases of independently fluctuating features (for $C=0$) and identically fluctuating features (for $C=2$).\n\n\\subsubsection*{Control of class separation by quantity $S$}\n\nTo evaluate separation control, we fix the quantities $D=10$ and $C=0.5$ and vary $S$ between 0 and 10. For each $S$, we generate $N_{rep}=100$ independent data sets, each consisting of $N_{vec}=10000$ data vectors. For each (labeled) data set, we compute the general discrimination value (GDV), a quantity that has been specifically designed to quantify the separation between classes in high dimensional data sets \\cite{krauss2018statistical, schilling2021quantifying}. The orange line in Fig.\\ref{figure_2}(d) is the mean negative GDV, averaged over the $N_{rep}=100$ repetitions (The latter are shown as gray dots). \n\n\\vspace{0.2cm}\\noindent The GDV is computed as follows: We consider $N$ points $\\mathbf{x_{n=1..N}}=(x_{n,1},\\cdots,x_{n,D})$, distributed within $D$-dimensional space. A label $l_n$ assigns each point to one of $L$ distinct classes $C_{l=1..L}$. In order to become invariant against scaling and translation, each dimension is separately z-scored and, for later convenience, multiplied with $\\frac{1}{2}$:\n\\begin{align}\ns_{n,d}=\\frac{1}{2}\\cdot\\frac{x_{n,d}-\\mu_d}{\\sigma_d}.\n\\end{align}\nHere, $\\mu_d=\\frac{1}{N}\\sum_{n=1}^{N}x_{n,d}\\;$ denotes the mean, and $\\sigma_d=\\sqrt{\\frac{1}{N}\\sum_{n=1}^{N}(x_{n,d}-\\mu_d)^2}$ the standard deviation of dimension $d$.\nBased on the re-scaled data points $\\mathbf{s_n}=(s_{n,1},\\cdots,s_{n,D})$, we calculate the {\\em mean intra-class distances} for each class $C_l$ \n\\begin{align}\n\\bar{d}(C_l)=\\frac{2}{N_l (N_l\\!-\\!1)}\\sum_{i=1}^{N_l-1}\\sum_{j=i+1}^{N_l}{d(\\textbf{s}_{i}^{(l)},\\textbf{s}_{j}^{(l)})},\n\\end{align}\nand the {\\em mean inter-class distances} for each pair of classes $C_l$ and $C_m$\n\\begin{align}\n\\bar{d}(C_l,C_m)=\\frac{1}{N_l N_m}\\sum_{i=1}^{N_l}\\sum_{j=1}^{N_m}{d(\\textbf{s}_{i}^{(l)},\\textbf{s}_{j}^{(m)})}.\n\\end{align}\nHere, $N_k$ is the number of points in class $k$, and $\\textbf{s}_{i}^{(k)}$ is the $i^{th}$ point of class $k$.\nThe quantity $d(\\textbf{a},\\textbf{b})$ is the euclidean distance between $\\textbf{a}$ and $\\textbf{b}$. Finally, the Generalized Discrimination Value (GDV) is calculated from the mean intra-class and inter-class distances as follows:\n\\begin{align}\n\\mbox{GDV}=\\frac{1}{\\sqrt{D}}\\left[\\frac{1}{L}\\sum_{l=1}^L{\\bar{d}(C_l)}\\;-\\;\\frac{2}{L(L\\!-\\!1)}\\sum_{l=1}^{L-1}\\sum_{m=l+1}^{L}\\bar{d}(C_l,C_m)\\right]\n \\label{GDVEq}\n\\end{align}\n\n\\noindent whereas the factor $\\frac{1}{\\sqrt{D}}$ is introduced for dimensionality invariance of the GDV with $D$ as the number of dimensions.\nIn the case of two Gaussian distributed point clusters, the resulting discrimination value becomes $-1.0$ if the clusters are located such that the mean inter cluster distance is two times the standard deviation of the clusters.\n\n\\subsection*{Part 3: Comparing classifiers}\n\nIn Fig.\\ref{figure_3}, we determine the average accuracy of the three classifier types (See part 1 of the Methods section) for different combinations of the DSC control parameters. For each parameter combination, 100 data sets are sampled from the superstatistical distribution. For every data set, consisting of 8000 training vectors and 2000 test vectors, the three classifiers are trained from the scratch and then evaluated. This results in 100 accuracies for each classifier and each parameter combination. We then compute the mean value of these 100 accuracies, and this is the average accuracy plotted as colored lines in Fig.\\ref{figure_3}(c-f). The individual, non-averaged accuracies are plotted as gray points.\n\n\\subsection*{Part 4: Feature transformations}\n\nIn Fig.\\ref{figure_4}, we return to a much simpler test data set, consisting of two 'spherical' Gaussian data clusters in a two-dimensional feature space, which are centered at $\\vec{x}=(-\\frac{1}{2},0)$ and $\\vec{x}=(+\\frac{1}{2},0)$. respectively. All three classifier types reach the theoretical accuracy limit of about 0.69 in this case.\n\n\\vspace{0.2cm}\\noindent In this part we explore how certain non-linear transformations of the original features (that is, $(x_1,x_2) \\longrightarrow (f(x_1),f(x_2))$) affect the classification accuracy. In particular, we investigate the cases $f(x)=\\sin(x)$, $f(x)=\\cos(x)$ and $f(x)=\\mbox{sgn}(x)$. The signum function yields -1 for negative arguments and +1 for positive arguments. For the special case $x=0$ it would return zero, but this practically does not happen, as the features $x$ are continually distributed random variables. \n\n\\subsection*{Part 5: Sleep EEG data}\n\nFor a real-world evaluation of classifier performance, we are using 68 multi-channel EEG data sets from our sleep laboratory, each corresponding to a full-night recording of brain signals from a different human subject. The data were recorded with a sampling rate of 256 Hz, using three separate channels F4-M1, C4-M1, O2-M1. In this work, however, the signals from these channels are pooled, effectively treating them as data sets of their own.\n\n\\vspace{0.2cm}\\noindent The participants of the study included 46 males and 22 females, with an age range between 21 and 80 years. Exclusion criteria were a positive history of misuse of sedatives, alcohol or addictive drugs, as well as untreated sleep disorders. The study was conducted in the Department of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander University Erlangen-N\u00fcrnberg (FAU), following approval by the local Ethics Committee (323\u201316 Bc). Written informed consent was obtained from the participants before the cardiorespiratory polysomnography (PSG). \n\n\\vspace{0.2cm}\\noindent After recording, the raw EEG data were analyzed by a sleep specialist accredited by the German Sleep Society (DGSM), who removed typical artifacts\n\\cite{tatum2011artifact} from the data and visually identified the sleep stages in subsequent 30-second epochs, according to the AASM criteria (Version 2.1, 2014) \\cite{iber2007aasm,american2012aasm}. The resulting, labeled raw data were then used as a ground truth for testing the accuracy of the different classifier types.\n\n\\vspace{0.2cm}\\noindent In this work, we are primarily testing the ability of the classifiers to assign the correct sleep label $s$\n(Wake, REM, N1, N2, N3) independently to each epoch, without providing further context information. Such a single-channel epoch consists of $30\\times256=7680$ subsequent raw EEG amplitudes $x_{d,e}(t_n)$, where $d$ is the data set, $e$ the number of the epoch within the data set, and $t_n$ the $n$th recording time within the epoch. \n\n\\vspace{0.2cm}\\noindent In order to facilitate classification of these 7680-dimensional input vectors $\\vec{x}_{d,e}$ by a simple Bayesian model, or by a flat two-layer perceptron with relatively few neurons, the vectors have to be suitably pre-processed and compressed down to feature vectors $\\vec{u}_{d,e}$ of much smaller dimensionality $D\\ll 7680$. \n\\vspace{0.2cm}\\noindent Instead of relying on self-organized (and thus 'black-box') features, we are using mathematically well-defined features with a simple interpretation. In particular, we are interested in the case where all $D$ components $u_f$ of a feature vector $\\vec{u}$ are fundamentally of the same kind and only differ by some tunable parameter. \n\n\\subsubsection*{Fourier features}\n\nOur first type of feature estimates the momentary Fourier component of the raw EEG signal $x_{d,e}(t_n)$ at a certain, tunable frequency $\\nu_f$: \n\n\\begin{equation}\nu_f = \n\\sqrt{\\left(\\;\n\\sum_{n=1}^{7680} x_{d,e}(t_n)\\cdot \\cos(2\\pi \\nu_f t_n)\n\\right)^2 + \n\\left(\n\\sum_{n=1}^{7680} x_{d,e}(t_n)\\cdot \\sin(2\\pi \\nu_f t_n)\n\\right)^2}. \n\\end{equation}\n\n\\vspace{0.2cm}\\noindent The set of frequencies $\\nu_{f=1}\\ldots\\nu_{f=D}$ is in our case chosen as an equidistant grid between 0 Hz and 30 Hz, because our EEG system is filtering out the higher-frequency components of the raw signals above about 30 Hz. \n\n\\subsubsection*{Correlation features}\n\nOur second type of feature is the normalized auto-correlation coefficient of the raw EEG signal $x_{d,e}(t_n)$ at a certain, tunable lag-time $\\Delta t_f$: \n\n\\begin{equation}\nu_f = \\frac{\\left\\langle \\left( x_{d,e}(t_n) - \\overline{x}_{d,e} \\right)\\cdot\\left( x_{d,e}(t_n\\!+\\!\\Delta t_f) - \\overline{x}_{d,e} \\right) \\right\\rangle_n}{\\sigma_{d,e}^2}.\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent Here, $ \\overline{x}_{d,e}$ is the mean and $\\sigma_{d,e}$ the standard deviation of the raw EEG signal within the epoch. The symbol $\\left\\langle \\right\\rangle_n$ stands for averaging over all time steps within the epoch. The set of lag-times $\\Delta t_{f=1} \\ldots \\Delta t_{f=D}$ must be integer multiples of the recording time interval $\\delta t = 1\/256$ sec.\n\n\\subsection*{Part 6: Sleep stage detection}\n\nIn Fig.\\ref{figure_6}, we investigate the performance of the three classifier types described in part 1 in the real-world scenario of personalized sleep-stage detection. For this purpose, the classifiers are trained and tested individually on each of our 68 full-night sleep recordings, using as inputs the same 6-dimensional Fourier- or correlation features as in Fig.\\ref{figure_5} (Note that the aggregated distribution functions and covariance matrices in Fig.\\ref{figure_5} have been computed by pooling over all data sets and therefore show a much more regular behavior than the individual ones).\n\n\\vspace{0.2cm}\\noindent As a result, we obtain 68 accuracies for each combination of classifier type (Fig.\\ref{figure_6}, rows) and used input feature (Fig.\\ref{figure_6}, columns). The distributions of these accuracies are presented as histograms in the figure.\n\n\\subsection*{Part 7: Natural data clustering}\n\nIn Fig.\\ref{figure_7}, we address the question whether typical real-world data sets have a built-in clustering structure that can be detected (and possibly enhanced) by unsupervised methods of data analysis. For this purpose, we visualize the clustering structure.\nA frequently used method to generate low-dimensional embeddings of high-dimensional data is t-distributed stochastic neighbor embedding (t-SNE) \\cite{van2008visualizing}. However, in t-SNE the resulting low-dimensional projections can be highly dependent on the detailed parameter settings \\cite{wattenberg2016use}, sensitive to noise, and may not preserve, but rather often scramble the global structure in data \\cite{vallejos2019exploring, moon2019visualizing}.\nIn contrats, multi-Dimensional-Scaling (MDS) \\cite{torgerson1952multidimensional, kruskal1964nonmetric,kruskal1978multidimensional,cox2008multidimensional} is an efficient embedding technique to visualize high-dimensional point clouds by projecting them onto a 2-dimensional plane. Furthermore, MDS has the decisive advantage that it is parameter-free and all mutual distances of the points are preserved, thereby conserving both the global and local structure of the underlying data. \nWhen interpreting patterns as points in high-dimensional space and dissimilarities between patterns as distances between corresponding points, MDS is an elegant method to visualize high-dimensional data. By color-coding each projected data point of a data set according to its label, the representation of the data can be visualized as a set of point clusters. For instance, MDS has already been applied to visualize for instance word class distributions of different linguistic corpora \\cite{schilling2021analysis}, hidden layer representations (embeddings) of artificial neural networks \\cite{schilling2021quantifying,krauss2021analysis}, structure and dynamics of recurrent neural networks \\cite{krauss2019analysis, krauss2019recurrence, krauss2019weight}, or brain activity patterns assessed during e.g. pure tone or speech perception \\cite{krauss2018statistical,schilling2021analysis}, or even during sleep \\cite{krauss2018analysis,traxdorf2019microstructure}. \nIn all these cases the apparent compactness and mutual overlap of the point clusters permits a qualitative assessment of how well the different classes separate.\n\nIn addition, we measure the degree of clustering objectively by calculating the general discrimination value (GDV) \\cite{krauss2018statistical,schilling2021quantifying}, described in part 2.\n\n\\vspace{0.2cm}\\noindent For the clustering analysis we analyze two examples of 'natural data': One is the MNIST data \\cite{deng2012mnist} set with 10 classes of handwritten digits, in which the input vectors are 784-dimensional (28x28 pixels) and have continuous positive values (between 0 and 1 after normalization). \n\n\\vspace{0.2cm}\\noindent As the second example we use, again, our full-night EEG recordings with the 5 data classes corresponding to the sleep stages Wake, REM, N1, N2, and N3. In order to reduce setup-differences between measurements, we first perform a z-transform over each individual full-night EEG recording, so that the one-channel EEG signal of each participant has now zero mean and unit variance. Next, in order to make the EEG data more comparable with MNIST, we produce one 784-dimensional input vector from each 30-second epoch of the EEG recordings in the following way: The 7680 subsequent one-channel EEG signals of the epoch are first transformed to the frequency domain using Fast Fourier Transform (FFT), yielding 3840 complex amplitudes. Since the phases of the amplitudes change in a highly irregular way between epochs, we discard this information by computing (the square roots of) the magnitudes of the amplitudes. We keep only the first 784 values of the resulting real-valued frequency spectrum, corresponding to the lowest frequencies. By pooling over all epochs and participants, we obtain a long list of these 784-dimensional input vectors. They are globally normalized, so that the components in the list range between 0 and 1, just as in the MNIST case. Finally, the list is randomly split into train (fraction 0.8) and test (fraction 0.2) data sets. \n\n\\vspace{0.2cm}\\noindent It is possible to directly compute the MDS projection of the uncompressed 784-dimensional test data vectors into two dimensions, and also to calculate the corresponding GDV value that quantifies the degree of class separation (using the known sleep stage labeling). In Fig.\\ref{figure_7}, these uncompressed data distributions are always shown in the left upper scatter plot of each two-by-two block.\n\n\\vspace{0.2cm}\\noindent In this context, we also test if step-wise dimensionality reduction in an autoencoder leads to an enhanced clustering. The used autoencoder has RELU activation functions and 7 fully connected layers with the following numbers of neurons: 784,128,64,16,64,128,728. The mean squared error between input vectors and reconstructed vectors is minimized using the Adams optimizer. We also compute the MDS projections and GDV values for layers 2, 3 and 4 (the 16-dimensional bottleneck) of the autoencoder. In Fig.\\ref{figure_7}, these three compressed data distributions are shown within the two-by-two blocks of scatter plots.\n\n\\vspace{0.2cm}\\noindent As a reference for the resulting MDS projections and GDV values in the unsupervised autoencoder, we also process the two kinds of natural data with a perceptron that is trained in a supervised manner, so that it separates the known classes as far as possible. To make the perceptron comparable to the autoencoder, the first 4 layers (from the input to the bottleneck) are identical: Fully connected, RELU activations, and layer sizes 784,128,64,16. However, the decoder-part of the autoencoder is replaced by a softmax layer in the perceptron, which has either 10 (MNIST) or 5 (sleep) neurons. The perceptron is trained by back-propagation to minimize categorical cross-entropy between the true and predicted labels, using the Adams optimizer. Just as in the autoencoder, we compute MDS projections and GDV values for the first 4 perceptron layers. \n\n\n\n\n\n\\clearpage\n\\section{Results}\n\n\\subsection*{Part 1: Accuracy limit}\n\nIn order to demonstrate the existence of an accuracy limit in classification tasks, we assume a statistical process is generating data vectors $\\vec{x}$ which are distributed in the input space (subsequently also called feature space) according to given {\\bf generation densities} $p_{gen}(\\vec{x}\\;|\\;i)$ that depend on the class $i$. For reasons of mathematical tractability and visual clarity, we start with a simple problem of two Gaussian data classes in a two-dimensional feature-space. We assume that class $i=0$ is centered at $\\vec{x}=(x_1,x_2)=(0,0)$, whereas class $i=1$ is centered a distance $d$ away, at $\\vec{x}=(d,0)$. As another discriminating property, the two class-dependent distributions are assumed to have different correlations between the features $x_1$ and $x_2$ (Compare Fig.\\ref{figure_1}(a,b)). \n\n\\vspace{0.2cm}\\noindent As derived in the Methods section, an ideal classifier would divide the feature-space $\\left\\{\\vec{x}\\right\\}$ among the two classes in a way that is perfectly consistent with the true generation densities $p_{gen}(\\vec{x}\\;|\\;i)$. The resulting ideal assignment of a discrete class $j=0$ or $j=1$ to each data vector $\\vec{x}$ can be described by binary {\\bf class indicator functions} $\\hat{q}_{cla}(\\;j\\;|\\;\\vec{x}\\;)$ (Compare Fig.\\ref{figure_1}(c,d)). \n\n\\vspace{0.2cm}\\noindent The latter two quantities can be combined to the {\\bf confusion densities} $\\hat{q}_{cla}(j|\\vec{x}) \\; p_{gen}(\\vec{x}|i)$, which give the probability density that data point $\\vec{x}$ is generated in class $i$ but assigned to class $j$ by the ideal classifier (Compare Fig.\\ref{figure_1}(e,f)). The parts of feature space where the confusion density is large for $i\\neq j$ correspond to the overlap regions of the data classes, and it is this overlap that makes the theoretical limit of the classification accuracy smaller than one.\n\n\\vspace{0.2cm}\\noindent It is possible to compute the {\\bf confusion matrix} of the ideal classifier by integrating the confusion densities over the entire feature space, which is feasible only in very low-dimension spaces. The confusion matrix, in turn, yields the {\\bf theoretical accuracy limit} $A_{max}$ of the ideal classifier.\nIn our simple example, $A_{max}$ is expected to increase with the distance $d$ between the two data classes, as this separation reduces the class overlap. By numerically computing the integral over the two-dimensional feature space of our Gaussian test example, we indeed find a monotonous increase of $A_{max}=A_{max}(d)$ from about 0.62 at $d=0$ to nearly one at $d=5$ (Compare Fig.\\ref{figure_1}(g, black line)).\n\n\\vspace{0.2cm}\\noindent Our next goal is to apply different types of classifier models to data drawn from the generation densities $p_{gen}(\\vec{x}\\;|\\;i)$ of the Gaussian test example above.\n\n\\vspace{0.2cm}\\noindent As an example for a 'black box' classifier, we consider a {\\bf perceptron} with one hidden layer (See Methods section for details). In the training phase, the connection weights of this neural network are optimized using the back-propagation algorithm. \n\n\\vspace{0.2cm}\\noindent As an example of a mathematically transparent, but simple classifier type, we consider a {\\bf Naive Bayesian} model. Here, correlations between the input features are neglected, and so the global likelihood $L(\\vec{u}\\;|\\;c)$ of a data vector $\\vec{u}$, given the data class $c$, is approximated as the product of the marginal likelihood factors for each individual feature $f$ (See Methods section for details). In the 'training phase', the naive Bayesian classifier is simply estimating the distribution functions of these marginal likelihood factors, using Kernel Density Approximation (KDE). \n\n\\vspace{0.2cm}\\noindent Finally, we consider a {\\bf Correlated Multi-Variate Gaussian (CMVG) Bayesian} model as an example of a mathematically transparent classifier that can also account for correlations in the data, but which assumes that all features are normally distributed (See Methods section for details). In the training phase, the CMVG Bayesian classifier has to estimate the mean values and covariances of the data vectors.\n\n\\vspace{0.2cm}\\noindent When applying these three classifiers to the Gaussian test data, we indeed find that {\\bf all models reach the same theoretical classification limit, even though their operating principles are very different} (Compare Fig.\\ref{figure_1}(g)). The only exception is the Naive Bayes classifier at small class distances $d$ (Compare Fig.\\ref{figure_1}(g, orange line)). This model fails because it can only use the marginal feature distributions, which happen to be identical for both classes in the case $d=0$. However, the problem can be easily fixed by multiplying the original two-dimensional feature vectors with a random, non-quadratic matrix (See Methods section for details) and thereby creating many new linear feature combinations, some of which usually have significantly different marginal distributions. Such a {\\bf Random Dimensionality Expansion (RDE)}, as proposed in Yang et al. \\cite{yang2021neural}, allows even the Naive Bayes model to reach the accuracy limit in strongly overlapping data classes (Compare Fig.\\ref{figure_1}(g, olive line)).\n\n\n\\subsection*{Part 2: The DSC data model}\n\nIn order to investigate how the performance of different classifiers depends on the statistical properties of the data, we generate large numbers of artificial data sets with two labeled classes $c \\in \\left\\{0,1\\right\\}$, in which the dimensionality $D$ of the individual data vectors $\\vec{u}$, the degree of correlations $C$ between their components $u_{f=1\\ldots D}$ (here also called features), and the separation $S$ between the two classes in feature space can be independently adjusted (See Figs.\\ref{figure_2}(b,c) for an illustration of $C$ and $S$). To replicate some of the heterogeneity of real world data, we design our data generator as a two-level superstatistical model \\cite{metzner2015superstatistical,mark2018bayesian}: The mean values $\\bf \\mu^{(c)}$ and covariances $\\bf \\Sigma^{(c)}$ of the multi-variate probability distributions $p_c(\\vec{u})$ in each of the data classes $c$ are themselves random variables. They are drawn from certain meta-distributions, which are in turn controlled by the three quantities $D,S,C$ (See Methods for details, as well as Fig.\\ref{figure_2}(a)).\n\n\\vspace{0.2cm}\\noindent Using the General Discrimination Value (GDV), a measure designed to quantify the separability of labeled point sets (data classes) in high-dimensional spaces \\cite{schilling2021quantifying}, we show that the mean separability of data classes in the DSC-model is indeed monotonously increasing with the control quantity $S$ (Orange line in Fig.\\ref{figure_2}(d)), whereas the separability of individual data sets is fluctuating heavily around this mean value (Grey dots in Fig.\\ref{figure_2}(d)).\n\n\\vspace{0.2cm}\\noindent Moreover, we quantify the degree of correlation between the $D$ features of the data vectors in each class $c$ by the root-mean-square average of the upper triangular matrix elements in the covariance matrix $\\bf \\Sigma^{(c)}$. We show that this RMS-average is an almost linear function of $C$ (Blue line in Fig.\\ref{figure_2}(d)) and can be varied between zero (Corresponding to independently fluctuating features, or statistical independence) and one (Corresponding to identically fluctuating features, or perfect correlations).\n\n\\subsection*{Part 3: Comparing classifiers}\n\nNext, we apply the three classifier types to artificial data, with statistical properties controlled by the quantities $D$, $S$ and $C$. We first investigate the {\\bf accuracy of the classifiers as a function of data dimensionality $D$} (Fig.\\ref{figure_3}(c,d)), considering correlated data ($C=1.0$). \n\n\\vspace{0.2cm}\\noindent When the separation of the data classes in feature space is small ($S=0.1$, panel (c)), the classification accuracy for one-dimensional data ($D=1$) is very close to the minimum possible value of 0.5 (corresponding to a purely random assignment of the two class labels) in all three models. As data dimensionality $D$ increases, all three models monotonically increase their average accuracies (colored lines), whereas the accuracies of individual cases show a large fluctuation (gray dots). However, the Naive Bayes classifier (orange line) does not perform well even for large data dimensionality, because the point clouds corresponding to the two classes are strongly overlapping in feature space. By contrast, the CMVG Bayes classifier (red line) and the Perceptron (blue line) eventually achieve a very good performance, because they can exploit the correlations in the data. The similarity of the latter two accuracy-versus-$D$ plots is remarkable, considering that these two classifiers work in completely different ways (the Bayesian model performing theory-based mathematical operations with estimated probability distributions, the neural network computing quite arbitrary non-linear transformations of weighted sums). We therefore conclude that the latter two models approach the theoretical optimum of accuracy for each combination of the control quantities $D,S,C$. \n\n\\vspace{0.2cm}\\noindent As the separation of the data classes in feature space gets larger ($S=1.0$, panel (d)), the accuracy-versus-$D$ plots are qualitatively similar to panel (c), but for one-dimensional data ($D=1$) the common accuracy is now slightly above the random baseline, at 0.6. By comparing panels (c) and (d) we note that Naive Bayes is profiting from the larger class separation, but the other two classifiers reach the theoretical performance maximum even without this extra separation.\n\n\\vspace{0.2cm}\\noindent Next, we investigate the {\\bf accuracy of the classifiers as a function of class separation $S$} (Fig.\\ref{figure_3}(e,f)), considering five-dimensional data ($D=5$). Without correlations ($C=0$, panel (e)), all three models show exactly the same monotonous increase of accuracy with separation $S$, starting at the random baseline of 0.5 and finally approaching perfect accuracy of 1.0. \n\n\\vspace{0.2cm}\\noindent With feature correlations present ($C=1.0$, panel (f)), the Naive Bayes classifier shows the same behavior as in panel (e), whereas the other two correlation-sensitive models now already start with a respectable accuracy of 0.8 at zero class separation.\n\n\\vspace{0.2cm}\\noindent Finally, we investigate the {\\bf accuracy of the classifiers as a function of the feature correlations $C$} (Fig.\\ref{figure_3}(g,h)), considering again five-dimensional data ($D=5$). For strongly overlapping data classes ($S=0.1$, panel (g)), Naive Bayes cannot exceed an accuracy of about 0.55, whereas the two correlation-sensitive models show a super-linear increase of accuracy with increasing feature correlations. However, this decrease is ending rather abruptly at about $C\\approx 0.7$. Above this transition point, both models stay at a plateau accuracy of about 0.8, independent of the correlation quantity. Note that this discontinuity of the slope of the accuracy-versus-$D$ plots is likely not an artifact of the DSC data, since the RMS-average of empirical correlations versus $C$ (Fig.\\ref{figure_3}(d)) did not show such an effect at $C\\approx 0.7$. Moreover, the fact that functionally distinct classifiers such as CMVG Bayes and Perceptron produce an almost identical behaviour here suggests that the accuracy plateau in the strong correlation regime indeed reflects the theoretical performance maximum.\n\n\\vspace{0.2cm}\\noindent As the class separation is increased ($S=1.0$, panel (h)), all three models start at a larger accuracy of about 0.75 in the uncorrelated case. Now the performance of Naive Bayes is even declining with increasing $C$, because this model wrongly assumes uncorrelated data. The other two models show again the super-linear increase up to $C\\approx 0.7$. However, now a further improvement of performance is possible with increasing correlations.\n\n\\subsection*{Part 4: Feature transformations}\n\nThe accuracy limit is determined by the overlap of data classes, that is, by the possibility that different classes $i \\neq j$ produce exactly the same data vector $\\vec{x}^{\\ast}$. Transformations $\\vec{x} \\rightarrow \\vec{f}(\\vec{x})$ of the input features can drastically change the distributions of data points (As an example, compare the rows in Fig.\\ref{figure_4}). However, they cannot be expected to reduce the fundamental amount of class overlap, because transformations are just redirecting the common points $\\vec{x}^{\\ast}$ to new locations in feature space. In particular, invertible transformations can be viewed as variable substitutions in the integral Eq.\\ref{cij} for the confusion matrix. They do not affect the resulting matrix values and thus leave the accuracy invariant.\n\n\\vspace{0.2cm}\\noindent In order to test this expectation, we start with two overlapping Gaussian data classes in a two-dimensional feature space (Fig.\\ref{figure_4}, top row), resulting in an accuracy limit of $\\approx 0.69$. All three classifiers actually reach this limit with the original data as input.\n\n\\vspace{0.2cm}\\noindent Next we perform simple non-linear transformations on the input data, by replacing each of the two features $x_1$ and $x_2$ with a function of themselves (in particular: $\\sin$, $\\mbox{sgn}$, and $\\cos$). We find that the application of the $\\sin$-transformation (second row in Fig.\\ref{figure_4}) has indeed no effect on the accuracy of the three classifiers, even though the joint (first column) and marginal distributions (second and third column) are now strongly distorted. Even the application of the $\\mbox{sgn}$-transformation (third row), which collapses all data onto just 4 possible points in feature space, leaves the accuracies invariant. This works because the two classes in our simple example can be distinguished by the sign of the $x_1$-feature, and both the $\\sin$- as well as the $\\mbox{sgn}$-transformation leave this information intact. By contrast, the application of the $\\cos$-transformation destroys this crucial information, and consequently all accuracies drop to the random baseline of 0.5.\n\n\\vspace{0.2cm}\\noindent The above numerical experiments illustrate that transformations of the input-data can reduce (by destroying information that is essential for class-discrimination), but never increase the theoretical accuracy limit, which is an inherent property of the data. Of course, the subsequent data transformations which are taking place in the layers of deep neural networks are still useful, because they re-shape data distributions until classes can be linearly separated in the final layer of the network. \n\n\n\n\\subsection*{Part 5: Sleep EEG data}\n\nIn our artificial data sets, all feature distributions were normally distributed. Moreover, it was possible to introduce extremely strong correlations between these features, which could then be exploited by two of the three classifier models. It is however unclear if the ability of a classifier to detect correlations is always crucial in real-world problems.\n\n\\vspace{0.2cm}\\noindent We therefore turn in a next step to actually measured EEG data, recorded over-night from 68 different sleeping human subjects. In this case, our final goal is to assign to each 30-second epoch of a raw one-channel EEG signal one of the five sleep stages (Wake, REM, N1, N2, N3). \n\n\\vspace{0.2cm}\\noindent At our sample rate, a single epoch of EEG data corresponds to 7680 subsequent amplitudes. Such high-dimensional data vectors $\\vec{x}$ are however not suitable as direct input for a Bayesian classifier, nor for a flat neural network with only $\\approx 100$ neurons. For this reason, we first compress the raw data vectors $\\vec{x}=(x_1,\\ldots x_{7680})$ into suitable feature vectors $\\vec{u}=(u_1,\\ldots u_D)$ of strongly reduced dimensionality $D\\approx 10$. Since we aim to develop a fully transparent classifier system, we use mathematically well-defined, human-interpretable features $u_f = G(\\vec{x},\\alpha_f)$, which depend on a freely tunable parameter $\\alpha$. The dimensionality $D$ of the feature space is then determined by how many of these parameters $\\alpha_{f=1\\ldots D}$ are chosen.\n\n\\vspace{0.2cm}\\noindent The huge literature on brain waves suggests that the momentary {\\bf Fourier components} of the EEG signal are suitable features for the classification of sleep stages. The parameter $\\alpha$ is then naturally given by the frequency $\\nu$ of the Fourier component (For details see methods). In a first experiment, we use a set of six equally spaced frequencies ($\\nu_1=$5 Hz, $\\nu_2=$10 Hz,$\\ldots$ $\\nu_6=$30 Hz). Based on training data sets that have been manually labeled by a sleep specialist, we then compute the marginal probability density functions of these Fourier features, as well as their covariance matrices, for each of the 5 sleep stages $s$ (Fig.\\ref{figure_5}, left two columns). We find that within each sleep stage, the Fourier features have unimodal distributions, with peak positions and widths depending quite systematically on the frequency $\\nu$. There are characteristic differences between the sleep stages (in particular the distributions are wider in the wake stage), but they are not very pronounced. In the covariance matrices, we find that the off-diagonal elements are significantly smaller than the diagonal elements (The latter have been set to zero in Fig.\\ref{figure_5} to emphasize the actual inter-feature correlations), with the exception of the wake state. Also the N1 state has slightly larger inter-feature correlations compared to the REM, N2 and N3 states. \n\n\\vspace{0.2cm}\\noindent As an alternative or complement to the Fourier features, we also consider the normalized (Pearson) {\\bf auto-correlation coefficients} of the raw EEG signal (Fig.\\ref{figure_5}, right two columns. For details see methods). The feature parameter $\\alpha$ is in this case given by the lag-time $\\Delta t$, for which we choose six equally spaced values ($1,3,\\ldots,11$ in units of the EEG sampling period). Since these correlation features cannot exceed the value of one by definition, the marginal distributions are highly non-Gaussian with pronounced tails towards small values. These tails show relatively strong differences between some of the sleep stages, but also surprising similarities, in particular for REM and N2. In the covariance matrices, we find the strongest inter-feature correlations in the wake and N1 stages. Again, the covariance matrices are very similar in REM and N2.\n\n\\subsection*{Part 6: Sleep stage detection}\n\nNext, we apply our three classifier models to the above sleep EEG data. However, while the feature distributions and correlations in Fig-\\ref{figure_5} were based on the global data, pooled over all 68 full-night EEG recordings, we are considering here the task of personalized sleep-stage recording. That is, the classifiers are trained and evaluated individually on each of the 68 data sets. Because the amount of training data is severely limited in this task, classification accuracies are expected to be rather low and strongly dependent on the participant. We therefore compute the distributions of accuracies over the 68 personalized data sets (histograms in Fig.\\ref{figure_6}) for all three classifiers and for the two types of pre-processed features. \n\n\\vspace{0.2cm}\\noindent We find that the CMVG Bayes model is performing very poorly in this task, presumably because the feature distributions are non-Gaussian and only weakly correlated except in the wake stage. In particular, for some participants the classification accuracy is less then the random baseline of about 0.2, corresponding to consistent miss-classifications. This can happen in Bayesian classifiers when the likelihood distributions learned from the training data set do not match the actual distributions in the test data set. \n\n\\vspace{0.2cm}\\noindent By contrast, the Naive Bayes model can properly represent the non-Gaussian feature distributions by KDE approximations, and it furthermore profits from the lack of correlations. The performance of the Perceptron is comparable to that of the Naive Bayes model. Both for Fourier- and correlation-features, these two models show accuracies well above the baseline, roughly in the range from 0.3 to 0.6.\n\n\\subsection*{Part 7: Natural data clustering}\n\nBoth the ten digits in MNIST, as well as the five sleep stages in overnight EEG recordings, are human-defined classes. It is therefore unclear whether these classes can also be considered as 'natural kinds'. \n\n\\vspace{0.2cm}\\noindent After a suitable pre-processing that brings both data sets into the same format of 784-dimensional, normalized feature vectors (for details see Methods sections), we address this question by computing two-dimensional MDS projections, coloring the data points according to the known, human-assigned labels (In Fig.\\ref{figure_7}, see the upper left scatter plot in each 2-by-2 block). Indeed, the projected data distributions show a small degree of clustering, which is also quantitatively confirmed by the corresponding GDV values (-0.061 for MNIST and -0.035 for sleep EEG data). Note that in the sleep data, a large number of extreme outliers are found which might not correspond to any of the standard classes.\n\n\\vspace{0.2cm}\\noindent The purpose of classifiers is to transform and re-shape the data distribution in such a way that the final network layer (often a softmax layer with one neuron for each data class) can separate the classes easily from each other. Although, as we have shown above, these re-shaping transformations cannot reduce the natural overlap of classes (which would push the accuracy beyond the data-inherent limit), they might as a side-effect lead to a larger 'centrality' of the clusters associated with each class. This would show up quantitatively as a decrease of the General Discrimination Value (GDV) in the higher network layers of the classifier, as compared to the original input data. In order to test this hypothesis, we have trained a four-layer perceptron (see Methods section for details) in a supervised manner on both the MNIST and sleep EEG data. In the case of MNIST, we indeed observe a systematic decrease of the GDV in subsequent network layers: GDV(L0)=-0.061, GDV(L1)=-0.174, GDV(L2)=-0.250, and GDV(L3)=-0.300 (See Fig.\\ref{figure_7}(b)). An analogous layer-wise decrease is found for the sleep EEG data: GDV(L0)=-0.035, GDV(L1)=-0.096, GDV(L2)=-0.122, and GDV(L3)=-0.181 (See Fig.\\ref{figure_7}(d)).\n\n\\vspace{0.2cm}\\noindent We finally address the question whether a natural clustering in novel, unlabeled data sets can be automatically detected, and possibly enhanced, in an unsupervised manner. For this purpose, we consider an autoencoder that performs a layer-wise dimensionality reduction of the data, and then re-expands these low-dimensional embeddings back to the original number of dimensions. During this process of 'compression' and 're-expansion', fine details of the data have to be discarded, and it appears reasonable that this might go hand in hand with a 'sharpening' of the clusters. Again, in our test case where the labels of the data points are actually known, this enhancement of cluster centrality can be quantitatively measured by the GDV. For comparability, we have used an autoencoder that has the same design as the perceptron for the first four network layers. In the case of MNIST, we indeed find that the unsupervised compression enhances cluster centrality: GDV(L0)=-0.061, GDV(L1)=-0.115, GDV(L2)=-0.122, and GDV(L3)=-0.137 (See Fig.\\ref{figure_7}(a)). The behavior is similar with the sleep EEG data, except for the last layer: GDV(L0)=-0.035, GDV(L1)=-0.037, GDV(L2)=-0.041, and GDV(L3)=-0.036 (See Fig.\\ref{figure_7}(c)).\n\n\n\\vspace{0.2cm}\\noindent \n\n\n\\clearpage\n\\section{Discussion and Outlook}\n\nIn this work, we have addressed various aspects of data ambiguity: the fact that multi-dimensional data spaces usually contain vectors that cannot be unequivocally assigned to any particular class. \nThe probability of encountering such ambiguous vectors is easily underestimated in machine learning, because the data sets used to train classifiers - rather than being sampled randomly from the entire space of possible data - typically represent just a tiny, pre-selected subset of 'reasonable' examples. For instance, the space of monochrome images with full HD resolution and 256 gray values contains $256^{1920 \\times 1080} \\approx 10^{4993726}$ possible vectors. The fraction of these images that resemble any human-recognizable objects is virtually zero, whereas the largest part would be described as noise by human observers. One may argue that these 'structure-less' images should not play any role in real-world applications. However, it is conceivable that sensors in autonomous intelligent systems, such as self-driving cars, can produce untypical data under severe environmental conditions, such as snow storms. How to deal with data ambiguity is therefore a practically relevant problem. Moreover, as we have tried to illustrate in this paper, data ambiguity has interesting consequences from a theoretical point of view.\n\n\\vspace{0.2cm}\\noindent In part one, we have derived the theoretical limit $A_{max}$ of accuracy that can be achieved by a perfect classifier, given a data set with partially overlapping classes. By generating artificial data classes with Gaussian probability distributions in a two-dimensional feature space and with a controllable distance $d$ between the maxima, we verified that different types of classifiers (The CMVG Bayesian model with multi-variate Gaussian likelihoods and a perceptron) exactly follow the predicted accuracy limit $A_{max}(d)$ (Fig.\\ref{figure_1}(g)). The naive Bayesian model, which cannot exploit correlations to distinguish between data classes, originally yields sub-optimal accuracies for small distances $d$, but this problem can be fixed by applying a random dimensionality expansion to the data as a trivial pre-processing step \\cite{yang2021neural}. We have restricted ourselves to only two features (dimensions) for this test, because predicting the accuracy limit involves the exact computation of the confusion matrix, which in turn is an integral over the entire data space. Note, however, that for high-dimensional data with known class-dependent generation densities $p_{gen}(\\vec{x}\\;|\\;i)$, the integral could be approximated by Monte Carlo sampling. In this case, the element $C_{ji}$ of the confusion matrix would be computed by drawing random vectors $\\vec{x}$ from class $i$. The class indicator function $\\hat{q}_{cla}(\\;k\\;|\\;\\vec{x}\\;)$ of the perfect classifier, which is fully determined by the generation densities, yields the corresponding predicted classes $k$ for these data vectors. The matrix element $C_{ji}$ is then given by the fraction of cases where $k=j$. \n\n\\vspace{0.2cm}\\noindent In part two, we have constructed a two-level model to generate artificial test data (Fig.\\ref{figure_2}). The model has high-level parameters $D$, $S$ and $C$ which control the number of dimensions (features), the average separation of the two classes in feature space, as well as the average correlation between the features. For each triple of high-level parameters $D,S,C$, a large number of low-level parameters $\\mu, \\Sigma$ are randomly drawn according to specified distributions, which are in turn used to generate the final test data sets. The super-statistical nature of the model allows us to prescribe the essential statistical features of dimensionality, separation and correlation, while at the same time ensuring a large variability of the test data. By using the General Discrimination Value (GDV), a quantitative measure of class separability (centrality), we have confirmed that the high-level parameter $S$ controls the class separability as intended. Moreover, the proper action of parameter $C$ was confirmed by computing the root-mean-square average over the elements of the data's covariance matrix.\n\n\\vspace{0.2cm}\\noindent In part three, we have applied our three types of classifiers to the test data generated with the DSC-model. Without intra-class feature correlations ($C=0$), we find that all three models show with growing separation parameter $S$ exactly the same monotonically increasing average accuracy (Fig.\\ref{figure_3}(e)). Although the exact computation of $A_{max}$ is not possible in this five-dimensional data space, the perfect agreement of the three different classifiers indicates that they all have reached the accuracy limit. When intra-class feature correlations are present ($C\\neq 0$), we find by systematically varying the parameters $D$, $S$ and $C$ that the resulting accuracies of the CMVG-Bayes classifier and of the perceptron are extremely similar in all considered cases, indicating again that they have reached the theoretical accuracy limit. As expected, the naive Bayesian classifier shows sub-optimal accuracies in all cases where feature correlations are required to distinguish between the classes. In general, this analysis shows that the accuracy of classification can be systematically enhanced by providing more features (larger data dimensionality $D$) as input. Extra features that do not provide additional useful information are 'automatically ignored' by the classifiers and never reduce the achievable accuracy. Moreover, accuracy can be enhanced by providing features that are correlated with each other (larger parameter $C$), but differently in each data class. Such class-specific feature correlations can be exploited for discrimination by models such as CMVG Bayes and the perceptron, but not by the naive Bayes model. Moreover, we find that the theoretical accuracy maximum as a function of the correlation parameter $C$ shows an interesting abrupt change of slope at around $C\\approx 0.8$ (Fig.\\ref{figure_3}(g,h)). The origin of this effect is at present unclear, but will be explored in follow-up studies.\n\n\\vspace{0.2cm}\\noindent In part four, we have investigated the effect of non-linear feature transformations, applied as a pre-processing step, on classification accuracy (Fig.\\ref{figure_4}). Since the achievable accuracy in a classification task is limited by the degree of overlap between the data classes, feature transformations can certainly reduce the accuracy to below the limit $A_{max}$ (when they destroy information that is essential for discrimination), but they can never push the accuracy to above $A_{max}$. This is indeed confirmed in a simple test case where all three classifier types perform at the accuracy maximum with the non-transformed data: Applying a feature-wise sine-transformation drastically changes the data distributions $p_{gen}(\\vec{x}\\;|\\;i)$, but leaves the accuracies unchanged at $A_{max}$. The accuracy remains invariant even under a signum-transformation, although this non-invertible operation reduces the data distributions to only four possible points in feature space. In this extreme case, most of the detailed information about the input data vectors is lost, but the part that is essential for class discrimination, namely the sign of the feature $x_1$, is retained. This example demonstrates that classification is a type of lossy data processing where irrelevant information can be safely discarded. For this reason, neural-network based classifiers usually project the input data vectors into spaces of ever smaller dimensions, up to the final discrimination layer which needs only as many neural units as there are data classes. In this context, it is interesting that biological organisms with nervous systems, relying on an efficient classification of objects in their environment for survival, have probably evolved sensory organs and filters that only transmit the small class-discriminating part of the available information to the higher stages of the neural processing chain. As a consequence, our human perception is almost certainly not a veridical representation of the world \\cite{mark2010natural,hoffman2014objects, hoffman2018interface}.\n\n\\vspace{0.2cm}\\noindent In part five, we have analyzed full-night EEG recordings of sleeping humans, divided into epochs of 30 seconds that have been labeled by a specialist according to the five sleep stages. Such recordings can been used as training data for automatic sleep stage classifiers - an application of machine learning that could in the future remove a large work load from clinical sleep laboratories. In our context of data ambiguity, sleep EEG is an interesting case because different human specialists agree about individual sleep-label assignments only in 70\\% - 80\\% of the cases, even if multiple EEG channels and other bio-signals (such as electro-oculograms or electro-myograms) are provided \\cite{fiorillo2019automated}. This low inter-rater reliability suggests that a considerable fraction of the 30-second epochs is actually ambiguous with respect to sleep stage classification, in particular when only the time-dependent signal of a single EEG channel is available as input-data. Our first goal is a suitable dimensionality reduction of the raw data, which (at a sample rate of 256 Hz) consist of 7680 subsequent EEG values in each epoch. As a pre-processing step, we map each 7680-dimensional raw data vector onto an only 6-dimensional feature vector, so that our Bayesian classifiers (Naive and CMVG) can be efficiently used. We consider as features the real-valued Fourier amplitudes at different frequencies, as well as the auto-correlation coefficients at different lag-times (Fig.\\ref{figure_5}). The Fourier features are expected to be particularly useful, as it is well-known that the activity in different EEG frequency bands varies in characteristic ways over the five sleep stages. The correlation features have been successfully applied for Bayesian classification in a former study \\cite{metzner2021sleep}. In our present study, we are using either Fourier or correlation features, but no combinations of those. By performing a statistical analysis of the features, we find that within the same sleep stage, the six features have significantly different marginal probability distributions. However, these distributions are quite similar in all sleep stages, so that their value for the classification task is limited. Moreover, the correlations between features, which could be exploited by the CMVG Bayes classifier and by the perceptron, turn out to be very weak, except for the Fourier features in the wake stage. Another problem is the strongly non-Gaussian shape of the marginal probability distributions in the case of the correlation features, which cannot be properly represented by the CMVG Bayes model. \n\n\\vspace{0.2cm}\\noindent In part six, we have used our three classifier models, based on the above Fourier- and correlation features, for personalized sleep stage detection. In this very hard task, the classifiers are trained and tested, independently, on the full-night EEG data set of a single individual only. Since an individual data set contains typically less than 1000 epochs (each corresponding to one feature vector), random deviations from the 'typical' sleeping patterns are likely to be picked up during the training phase. We consequently find that the accuracies vary widely between the individual data sets. As expected, the CMVG Bayes model performs badly in this task, because there are almost no inter-feature correlations present that could be exploited for sleep stage discrimination, and because feature distributions are non-Gaussian. Interestingly, both the Naive Bayesian classifier and the perceptron achieve relatively good accuracies, mainly in the range from 0.3 to 0.6. However, these accuracies may be further increased by using more sophisticated neural network architectures \\cite{stephansen2018neural, krauss2021analysis}, and hence do not represent the accuracy limit.\n\n\\vspace{0.2cm}\\noindent In the final part seven, we have started to explore whether the distinct classes in typical real-world data sets are defined arbitrarily (and therefore can only be detected after supervised learning), or if the differences between these classes are so prominent that even unsupervised machine learning methods can recognize them as distinct clusters in feature space. Besides the (pooled) sleep EEG data, we have used the MNIST data set to test for any inherent clustering structure. For this investigation, the individual data points, corresponding to respectively one epoch of EEG signal or one handwritten digit, have been brought into the same format of 784-dimensional, normalized vectors. Computing directly the General Discrimination Value (GDV) of the MNIST data, based on the known labels, has indeed revealed a small amount of 'natural clustering', even in this raw data distribution. This quantitative result was qualitatively confirmed by a two-dimensional visualization using multi-dimensional scaling (MDS), however the cluster structure would hardly be visible without the class-specific coloring (left upper scatter plots in Fig.\\ref{figure_7}(a,b)). By contrast, no natural clustering was found for the raw sleep EEG data when the 7680 values in each epoch were simply down-sampled in the time-domain to 784 values (data not shown). This presumably fails because the relevant class-specific signatures appear randomly at different temporal positions within each epoch, and so the Euclidean distance between two data vectors is not a good measure of their dissimilarity. However, when we instead used as data vectors the magnitudes of the 784 Fourier amplitudes with lowest frequencies, a weak natural clustering was found also in the sleep data (left upper scatter plots in Fig.\\ref{figure_7}(c,d)). We have furthermore demonstrated that the degree of clustering (for both data sets) is systematically increasing in the higher layers of a perceptron that has been trained to discriminate the classes in a supervised manner (Fig.\\ref{figure_7}, right column). Finally, we have used a multi-layer autoencoder to produce embeddings of the data distributions with reduced dimensionality in an unsupervised setting. It has turned out that the degree of clustering (with respect to the known data classes) tends to increase systematically with the degree of dimensional compression (Fig.\\ref{figure_7}, left column). This interesting finding, previously reported in Schilling et al. \\cite{schilling2021quantifying}, suggests that unsupervised dimensionality reduction could be used to automatically detect and enhance natural clustering in unlabeled data. In combination with automatic labeling methods, such as Gaussian Mixture Models, this may provide an objective way to define 'natural kinds' in arbitrary data sets. \n\n\n\n\\clearpage\n\\section{Additional information}\n\n\\noindent{\\bf Author contributions statement:}\nCM has conceived of the project, implemented the methods, evaluated the data, and wrote the paper, PK co-designed the study, discussed the results and wrote the paper, AS discussed the results, MT provided access to resources and wrote the paper, HS provided access to resources and wrote the paper.\n \\vspace{0.5cm} \n\n\\noindent{\\bf Funding:}\nThis work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation): grant SCHU\\,1272\/16-1 (AOBJ 675050) to HS, grant TR\\,1793\/2-1 (AOBJ 675049) to MT, grant SCHI\\,1482\/3-1 (project number 451810794) to AS, and grant KR\\,5148\/2-1 (project number 436456810) to PK. \\vspace{0.5cm}\n\n\\noindent{\\bf Competing interests statement:}\nThe authors declare no competing interests. \\vspace{0.5cm}\n\n\\noindent{\\bf Data availability statement:}\nData and analysis programs will be made available upon reasonable request.\n\\vspace{0.5cm}\n\n\\noindent{\\bf Ethical approval and informed consent:} The study was conducted in the Department of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander University Erlangen-N\u00fcrnberg (FAU), following approval by the local Ethics Committee (323 \u2013 16 Bc). Written informed consent was obtained from the participants before the cardiorespiratory poly-somnography (PSG).\n\n\\vspace{0.5cm}\n\n\\noindent{\\bf Third party rights:}\nAll material used in the paper are the intellectual property of the authors. \\vspace{0.5cm}\n\n\\clearpage\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\@startsection{section}{1}{\\z@{INTRODUCTION}\n\nTwo Galactic X-ray sources known to produce relativistic radio jets\nare GRS 1915+105 \\cite{Mir94} and GRO J1655-40\n\\cite{Tin95,Hje95}. Optical observations of GRO J1655-40 have provided\ndynamical evidence for a 7 $M_{\\odot}$ black hole \\cite{Oro97} in a 2.6 day\nbinary orbit with a $\\sim$F4 IV companion star. GRS1915+105 is\npresumed to be a black hole binary, based on its X-ray high luminosity\nand similarities with GRO J1655-40. However, a direct measurement of\nthe motion of its companion star has been prevented by interstellar\nextinction, which limits optical\/IR studies of GRS1915+105 to\nwavelengths $> 1$ micron \\cite{Mir94}. While each source was active\nat radio frequencies, H I absorption measurements were combined with\nGalactic rotation models to derive distance measurements of 12.5 kpc\nand 3.2 kpc for GRS 1915+105 \\cite{Mir94} and GRO J1655-40\n\\cite{Tin95}, respectively.\n\nGRS 1915+105 is a transient X-ray source, and the BATSE light curve\n(20--100 keV) indicates that bright X-ray emission began during May\n1992 \\cite{Har97}. Before the launch of the $Rossi ~X$-$ray ~Timing\n~Explorer$ ($RXTE$), observations in soft X-rays were sporadic, and\nGRS1915+105 may have persisted as a bright source in soft X-rays since\n1992. When the All Sky Monitor (ASM) on $RXTE$ established regular\ncoverage on 1996 Feb 22, the source was bright and highly variable,\nand it has remained so throughout 1996 and 1997. The ASM light curve,\nwhich is shown in Figure~\\ref{fig:asm19}, illustrates both the extent\nof the intensity variations and also the repetitive character of\nparticular variability patterns. The early ASM light curve was used to\ninitiate $RXTE$ pointed observations (PCA and HEXTE instruments),\nwhich began on 1996 April 6. Since then the source has been observed\nonce or twice per week, and most of the data are available in a public\narchive. At the higher time resolution provided by PCA light curves,\nthere are again dramatic and repetitive patterns of variations\n\\cite{Gre96}. These results are one of the extraordinary chapters in\nthe history of high-energy astronomy.\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=asm1915.ps,width=16cm,height=16cm} }\n\\caption{ASM light curve (2--12 keV) of GRS1915+105 for 1996 and\n1997. The Crab Nebula, for reference, yields 75.5 ASM c\/s. The ASM\nhardness ratio, $HR2$ is defined as the count rate in the 5--12 keV\nband relative to the rate in the 3--5 keV band. The time intervals\nthat correspond with our groups of combined X-ray power spectra (see\nTable 1) are shown above the light curve.}\n\\label{fig:asm19}\n\\end{figure*}\n\nFourier analyses of the first 31 PCA observations \\cite{Mor97} of\nGRS1915+105 revealed 3 different types of oscillations: a\nquasi-periodic oscillation (QPO) with a constant frequency of 67 Hz;\ndynamic, low-frequency (0.05 to 10 Hz) QPO with a large variety\nof amplitudes and widths; and complex, high-amplitude dip cycles\n($10^{-3}$ to $10^{-1}$ Hz) that are related to the extreme X-ray\nvariations noted above. The combined characteristics of the power\nspectra, light curves, and energy spectra were interpreted as\nrepresenting four different emission states \\cite{Mor97}, none of\nwhich resemble the canonical states of black hole binaries\n\\cite{Van95}.\n\nThe other microquasar, GRO J1655-40, was first detected with BATSE on\n1994 July 27, and the correlation between hard X-ray activity and the\nejections of relativistic radio jets \\cite{Har95} was an important\nstep in establishing the relationship between accretion changes and\nthe formation of jets. During late 1995 and early 1996, GRO J1655-40\nentered a quiescent accretion state, permitting optical spectroscopy\nof the companion star, which led to our knowledge of the binary\nconstituents and mass of the black hole~\\cite{Oro97}, as noted above.\n\nThe ASM recorded a renewed outburst from GRO J1655-40 \\cite{Lev96}\nthat began on 1996 April 25. The ASM light curve is shown in Figure\n~\\ref{fig:asm16}. With great fortune a concurrent optical campaign\nwas in progress, and it was determined that optical brightening\npreceded the X-ray turn-on by 6 days, beginning first in the I band\nand then accelerating rapidly in the B and V bands. These results\nprovide concrete evidence favoring the accretion disk instability as\nthe cause of the X-ray nova episode.\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=asm1655.ps,width=16cm,height=16cm} }\n\\caption{(top) ASM light curve (1.5--12 keV) of GRO J1655-40 for 1996\nand 1997. The tick marks above the light curve show the times of RXTE\npointed observations, either from the public archive (1997) or our\nguest observer program (1996). (bottom) The ASM hardness ratio, $HR2$\nas defined previously.}\n\\label{fig:asm16}\n\\end{figure*}\n\nThe $RXTE$ observations of GRO J1655-40 indicate a more stable form of\naccretion. X-ray spectral variations (see Fig.~\\ref{fig:asm16})\nresemble the canonical ``soft\/high'' and ``very high'' states in black\nhole binaries \\cite{Rem98,Van95}. There are X-ray QPOs in the range of\n8--22 Hz, and there is also a transient, high-frequency QPO at 300\nHz~\\cite{Rem98}. This QPO is detected only when the X-ray power-law\ncomponent reaches its maximum strength.\n\nThe efforts to explain the 67 Hz QPO in GRS1915+105 and the 300 Hz QPO\nin GRO J1655-40 commonly invoke effects rooted in General Relativity\n(GR). There are at least 4 proposed mechanisms that relate the QPO\nfrequency to a natural time scale of the inner accretion disk in a\nblack hole binary. These are: the last stable orbit\n\\cite{Sha83,Mor97}, diskoseismic oscillations \\cite{Per97,Now97},\nframe dragging \\cite{Cui98}, and an oscillation in the centrifugal\nbarrier \\cite{Tit98}. The physics of all of these phenomena invokes GR\neffects in the inner accretion disk. It has also been proposed that\nthe high frequency QPOs may be caused by an inertial-acoustic\ninstability in the disk \\cite{Che95} (with non-GR origin), although\nthe oscillation in GRO J1655-40 would extend this application to\nhigher frequencies than had been argued previously.\n\nIn this paper we advertise some recent work that associates jet\nformation in GRS1915+105 with features in the X-ray light curve. We\nthen turn to the topic of X-ray QPOs. New results are presented on the\nreappearance of 67 Hz oscillations in GRS1915+105. Finally we\ndescribe the various QPO tracks that appear in GRO J1655-40, and we\nexplain how they behave in response to the strength of the power-law\ncomponent in the X-ray spectrum.\n\n\\@startsection{section}{1}{\\z@{CLUES FOR THE ORIGIN OF JETS IN GRS1915+105}\n\nSeveral groups have combined X-ray, radio, and\/or infrared\nobservations of GRS 1915+105 to probe the properties of jet formation\nand relate the ejection events to features in the X-ray light curves.\nInfrared jets were discovered \\cite{Sam96}, and infrared flares were\nseen to occur after radio flares\\cite{Fen97,Mir97}. These\ninvestigations provide solid evidence that the infrared flares\nrepresent synchrotron emission from rapidly evolving jets.\n\nIt has been further demonstrated that the radio, infrared, and X-ray\nbands occasionally show strong oscillations with a quasiperiodic time\nscale of 20--40 min \\cite{Rod97,Fen97,Eik98,Poo98}. In perhaps the\nmost impressive of these studies to date, there were a series of\ninfrared flares (with 20 min recurrence time), and in six of six\npossible cases the flares were seen to follow dramatic dipping cycles\nin the X-ray light curve. Since these dips have been analyzed as\nrepresenting the disappearance of the thermal X-ray emission from the\ninner disk \\cite{Bel97a,Bel97b}, the infrared\/X-ray correlation shows\nthat the jet material originates in the inner accretion\ndisk\\cite{Eik98}. Another conclusion drawn from the recent\nX-ray\/radio\/infrared studies is that there is a wide distribution of\n``baby jets'' in which quantized impulses appear at $\\sim30$ min\nintervals. The radio strength of these events is one to three orders\nof magnitude below the levels of the superluminal outbursts of 1994\n\\cite{Poo98,Mir94}.\n\nWe expect that $RXTE$ will continue to support multifrequency\nobservations of GRS1915+105 during 1998. There are opportunities for\nfurther analysis to characterize the distribution and expansion times\nof the jets, analyze the infrared and radio spectra of these events,\nand study the details of the X-ray light curve in the effort to\nconstrain the physics of the trigger mechanism.\n\n\\@startsection{section}{1}{\\z@{67 HZ OSCILLATIONS IN GRS1915+105}\n\nThere have been many observations of GRS1915+105 with $RXTE$ since the\nsix (1996 April 6--June 11) that provided detections of QPO at 67 Hz\n\\cite{Mor97}. Given the importance of this QPO and also the variety of\nemission states recorded for GRS1915+105 (see Figure~\\ref{fig:asm19}),\nwe investigated the data archive for new detections of this QPO. We\nadopted a global perspective, and we divided the $RXTE$ observations\ninto a sequence of X-ray state intervals, which we label as groups\n``g1'' through ``g10'' in Figure~\\ref{fig:asm19}. The groups were\nselected with consideration of both the ASM light curve and the\ncharacteristics of the PCA power spectra, and some observations\nbetween the group boundaries were ignored as representing transition\nstates.\n\nIn Table~\\ref{tab:67hz} we list the time intervals (cols. 2, 3) the\nnumber of observations (col. 4), the X-ray state (col. 5), and the\naverage X-ray flux (in Crab units) for each group. The typical\nobservation has an exposure time of 10 ks. The X-ray state\ndescription follows the convention of Morgan et al. \\cite{Mor97},\nwhich describes GRS1915+105 as being relatively steady and bright (B),\nflaring (FL), chaotic (CH), or low-hard (LH).\n\n\\begin{table*}\n\n\n\\newlength{\\digitwidth} \\settowidth{\\digitwidth}{\\rm 0}\n\\catcode`?=\\active \\def?{\\kern\\digitwidth}\n\\caption{The 67 Hz QPO in GRS1915+105}\n\\label{tab:67hz}\n\\begin{tabular*}{\\textwidth}{@{}l@{\\extracolsep{\\fill}}llrccccc}\n\\hline\ngroup & start & end & obs & state & flux & freq. & FWHM & ampl. \\\\\n\\hline\n\n1 & 1996 Apr 06 & 1996 May 14 & 7 & B & 1.06 & 64.5 & 4.0 & 0.0069 \\\\\n2 & 1996 May 21 & 1996 Jul 06 & 14 & FL & 1.00 & 65.7 & 2.3 & 0.0022 \\\\\n3 & 1996 Jul 14 & 1996 Aug 10 & 6 & LH & 0.58 & & & \\\\\n4 & 1996 Sep 16 & 1996 Oct 15 & 8 & B & 1.01 & 67.6 & 1.5 & 0.0016 \\\\\n5 & 1996 Nov 28 & 1997 May 08 & 28 & LH & 0.31 & 68.3 & 2.3 & 0.0023 \\\\\n6 & 1997 May 13 & 1997 Jun 30 & 18 & CH\/B & 0.64 & & & \\\\\n7 & 1997 Jul 07 & 1997 Aug 21 & 17 & B & 1.33 & 66.9 & 4.3 & 0.0039 \\\\\n8 & 1997 Aug 24 & 1997 Sep 29 & 15 & CH\/FL & 1.17 & & & \\\\\n9 & 1997 Oct 09 & 1997 Oct 25 & 4 & LH & 0.47 & & & \\\\\n10 & 1997 Oct 30 & 1997 Dec 22 & 15 & FL & 1.41 & 67.4 & 4.2 & 0.0035 \\\\\n\\hline\n\\end{tabular*}\n\\end{table*}\n\nWe then combined the power spectra in each group, using the full\nenergy coverage of the PCA instrument. We fit the results for a power\ncontinuum (with a power-law function) and a QPO feature (with a\nLorentian profile) over the range of 40--120 Hz. We emphasize that the\nlocation of the central QPO frequency is free to wander within this\nfrequency interval. The average power spectra for the 10 groups\n(linear units) and the QPO fits for 6 cases are shown in\nFigure~\\ref{fig:fit67hz}.\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=hmult67.ps,width=16cm,height=16cm} }\n\\caption{Average power density spectra in the range of 20--120 Hz for RXTE PCA observations of 1996 and 1997, combined in 10 groups. For the 6 cases in which a QPO is detected (see Table 1), the QPO fit is shown with a solid line.}\n\\label{fig:fit67hz}\n\\end{figure*}\n\nThe results derived from this analysis are listed in\nTable~\\ref{tab:67hz}. The central QPO frequency is given in col. 7,\nand there is a narrow distribution of $66.7 \\pm 1.4$ Hz. The QPO FWHM\nvalues (col. 8) have a mean value of $3.4 \\pm 1.0$ Hz. Comparing these\nobserving intervals, we conclude that the average X-ray luminosity of\nGRS1915+105 may vary by a factor of 4 with no significant change in\nthe characteristics of the 67 Hz QPO.\n\nThe integrated QPO amplitude is given in col. 9. The amplitudes (like\nthe power spectra in Figure~\\ref{fig:fit67hz}) are normalized by the\nmean X-ray count rate for GRS1915+105. The integrated power in the 67\nHz QPO is in the range of 0.2\\%--0.7\\% of the mean X-ray flux.\n\nThe results for group 5 are particularly noteworthy. During this\nperiod the source was in the low-hard state for a long time (see\nFigure~\\ref{fig:asm19}. The PCA light curves in 1 s time bins show\nvariations limited to moderate flickering, with rms variations $\\sim\n10$\\%. However the continuum power at 40--120 Hz is relatively high\nduring this interval (see Figure~\\ref{fig:fit67hz}). The large number\nof observations in group 5 partially compensates for the losses in\nstatistical sesitivity to QPO detection due to lower count rate and\nelevated continuum power. Nevertheless the QPO search does find a\nsmall feature that is consistent in frequency (68.3 Hz), width (2.3\nHz), and amplitude (0.23\\%) with the other detections. We estimate that\nthe uncertainty in the amplitude is 0.09\\%, so that the detection of\nthe 67 Hz QPO in group 5 has a signigicance of 2.6 $\\sigma$. For the\n4 groups that do not yield QPOs in the range of 40--120 Hz, the\nuncertainties are slightly larger, and we cannot exclude the\npossibility that GRS1915+105 is $always$ emitting X-ray QPOs at 67 Hz\nwith amplitudes in the range of 0.1\\% or larger.\n\nThere are yet many avenues for further investigation of this QPO,\ne.g. time lags at 67 Hz, analysis of the energy spectrum for the\ngroups with positive QPO detection, and segregation of data with\nalternative schemes such as the phases of jet-related dipping cycles.\nAll of these topics will be pursued during the next several months.\n\n\\@startsection{section}{1}{\\z@{QPOs in GRO J1655-40}\n\nWe have conducted similar analyses of PCA power spectra for individual\nobservations of GRO J1655-40. As reported previously \\cite{Rem98},\nthere are transient QPOs in the range of 8--30 Hz and there is a high\nfrequency QPO near 300 Hz. All of these QPOs are associated with the\nstrength of the power-law component. With respect to\nFigure~\\ref{fig:asm16}, the QPOs at 8--30 Hz appear when observations have\nhard spectra that correspond with ASM HR2 values above 0.8, while\nthe 300 Hz QPO is significant only when the combine the power spectra\nfor the 7 ``hardest'' observations made with the PCA (1996 August and\nOctober). \n\nWe fit the individual PCA power spectra for power continuum and QPOs,\nas described above, using frequency windows of 0.02--2 Hz and 5--50\nHz. In Figure~\\ref{fig:qpo16} we show the central QPO frequencies as a\nfunction of the source count rate in the PCA energy channels above 13\nkeV (or above channel 35). We use on open triangle for narrow\nQPOs ($\\nu \/ \\delta\\nu > 5$) and the ``*'' symbol for broad QPOs\n($\\nu \/ \\delta\\nu < 4$). In some observations, both narrow and broad\nQPOs appear in the same power spectrum (i.e. one 10 ks observation).\nThe ``x'' symbol shows a narrow and weak QPO derived from the average\npower spectrum obtained during the 1997 PCA observations (MJD interval\n50500--50650).\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=pub_fxqpo.ps,width=10cm,height=5.5cm} }\n\\caption{The central frequency of X-ray QPOs in GRO J1655-40 as a function of the PCA count rate above 13 keV. The open triangles represent broad QPOs, while the solid triangles represent narrow ones.}\n\\label{fig:qpo16}\n\\end{figure*}\n\nThe results in Figure~\\ref{fig:qpo16} show that the low-frequency QPOs\nin GRO J1655-40 are organized in three tracks. A broad QPO appears to\nbe stationary near 8 Hz, while the narrow QPOs shifts to lower\nfrequency as the hard X-ray flux increases. The QPO derived from the\nsum of 1997 observations appears to be a simple extension of this\nnarrow QPO track, occurring when the X-ray flux above 13 keV is nearly\nzero. Very low frequency QPO (0.085 and 0.11 Hz) are seen on two\noccasions when the hard X-ray flux is near maximum. These QPO coexist\nwith 300 Hz QPO, and they are reminiscent of the 0.067 Hz QPOs in\nGRS1915+105. We speculate that the 0.1 Hz QPOs appear near the\nthreshold of the chaotic light curves manifest in GRS1915+105.\nGRO1655-40 approaches this threshhold but does not cross the line into\nunstable light curves during the 1996-1997 outburst.\n\nIn Figure~\\ref{fig:asm16} we see that GRO J1655-40 fades below 20\nmCrab on 1997 Aug 17. Whether there will be a renewed outburst in\n1998 is anyone's guess, but the ASM will surely be monitoring this\nsource for any signs of X-ray activity.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $M \\subset \\R^n$ be an open, bounded and connected set with a smooth boundary, and consider the wave equation\non $M$,\n\\begin{align}\n\\label{eq:wave_isotropic}\n&\\p_t^2 u(t,x) - c(x)^2 \\Delta u(t,x) = 0, \n\\quad &(t,x) \\in (0, \\infty) \\times M,\n\\\\&u(0,x) = 0,\\ \\p_t u(0,x) = 0,\n\\quad &x \\in M, \\nonumber\n\\\\&\\p_\\nu u(t,x) = f(t,x), \n\\quad &(t,x) \\in (0, \\infty) \\times \\p M, \\nonumber\n\\end{align}\nwhere $c$ is a smooth stricty positive function on $\\bar M$, and $\\p_\\nu$ is the normal derivative on \nthe boundary $\\p M$.\n\nDenote the solution of (\\ref{eq:wave_isotropic}) by $u^f(t,x) = u(t,x)$, let $T > 0$, and define the operator \n\\begin{equation}\n\\label{eq:dtn}\n\\Lambda_{2T} : f \\mapsto u^f|_{(0,2T) \\times \\p M}.\n\\end{equation}\nOperator $\\Lambda_{2T}$ models boundary measurements and is called the Neumann-to-Dirichlet operator. \nLet us assume that $c|_{\\p M}$ is known but $c|_M$ is unknown.\nThe inverse problem for the wave equation is to reconstruct the wave speed $c(x)$, $x \\in M$,\nusing the operator $\\Lambda_{2T}$.\n\nLet $\\Gamma \\subset \\p M$ be open, $\\tau \\in C(\\bar \\Gamma)$, and\nconsider a wave source \n$f$ in $L^2((0,\\infty) \\times \\p M)$ \nsatisfying the support condition\n\\begin{equation}\n\\label{eq:source_supp_condition}\n\\supp(f) \\subset \\{ (t, y) \\in [0, T] \\times \\bar \\Gamma;\\ t \\in [T - \\tau(y), T] \\}.\n\\end{equation}\nBy the finite speed of progation for the wave equation \\cite{Ga, Mi},\nthe solution $u^f$ satisfies then the support condition, \n\\begin{equation}\n\\label{eq:sol_supp_condition}\n\\supp(u^f(T)) \\subset \\{x \\in M;\\ \\text{there is $y \\in \\bar \\Gamma$ such that $d(x,y) \\le \\tau(y)$}\\},\n\\end{equation}\nwhere $d(x,y)$ is the travel time between points $x$ and $y$, see (\\ref{domain_of_influence}) below. Let us denote the set in (\\ref{eq:sol_supp_condition}) by $M(\\Gamma, \\tau)$ and call it the domain of influence. \n\nThe contribution of this paper is twofold.\nFirst, we present a method to compute the volume of $M(\\Gamma, \\min(\\tau, T))$ using the operator $\\Lambda_{2T}$.\nThe method works even when the wave speed is anisotropic, that is, when the wave speed is given by a Riemannian metric tensor $g(x) = (g_{jk}(x))_{j,k}^n$, $x \\in \\bar M$. \nIn the case of the isotropic wave equation (\\ref{eq:wave_isotropic}) we have $g(x) = (c(x)^{-2} \\delta_{jk})_{j,k}^n$.\n\nSecond, assuming that the Riemannian manifold $(\\bar M, g)$ is simple, we show that the volumes of $M(\\Gamma, \\tau)$\nfor $\\tau \\in C(\\p M)$ contain enough information to \ndetermine the metric tensor $g$ up to a change of coordinates in $M$.\nWe recall the definition of a simple compact manifold below,\nsee Definition \\ref{def:simple}.\nIn the case of the isotropic wave equation (\\ref{eq:wave_isotropic}) we can\ndetermine the wave speed $c$\nin the Cartesian coordinates of $M$. \n\nOur method to compute the volume of \nthe domain of influence\nis a quadratic minimization scheme in $L^2((0, 2T) \\times \\p M)$ for the source \n$f$ satisfying the support condition (\\ref{eq:source_supp_condition}).\nAfter a finite dimensional discretization,\nan approximate minimizer \ncan be computed by solving\na positive definite system of linear equations. \nWe show that the system can be solved very efficiently if \nwe use an iterative method, such as the conjugate gradient method,\nand intertwine measurements with computation.\nIn particular, instead of solving the equation (\\ref{eq:wave_isotropic}) computationally in an iteration step, we measure $\\Lambda_{2T} f$ \nfor two sources $f$:\none is the approximate minimizer given by the previous iteration step and the other is related to the time-reversed version of the approximate minimizer, see (\\ref{eq:cg_step_measurements}) below.\nWe believe that our intertwined algorithm is more robust against noise than an algorithm where noise is propagated by simulation of the wave equation.\n\nLet us consider next the problem to determine the metric tensor $g$ given the volumes of $M(\\p M, \\tau)$ for all \n$\\tau \\in C(\\p M)$. Our approach exploits the fact that\n$C(\\p M)$ is a lattice with the natural partial order \n\\begin{equation}\n\\tau \\le \\sigma \n\\quad \\text{if and only if} \\quad \n\\tau(y) \\le \\sigma(y)\\ \\text{for all $y \\in \\p M$}.\n\\end{equation}\nLet us define the greatest lower bound of $\\tau$ and $\\sigma$ in $C(M)$ as their pointwise minimum and\ndenote it by $\\tau \\wedge \\sigma$.\nWe recall that a subset of $C(M)$ is a meet-semilattice if it is closed under the binary operation $\\wedge$.\n\nLet us define the {\\em boundary distance functions},\n\\begin{equation}\n\\label{eq:boundary_distance_functions}\nr_x : \\p M \\to [0, \\infty), \\quad r_x(y) := d(x,y),\n\\end{equation}\nfor $x \\in \\overline M$.\nWe show that the volumes of \n$M(\\p M, \\tau)$, $\\tau \\in C(\\p M)$, determine the meet-semilattice, \n\\begin{equation}\n\\label{eq:semilattice_QM_intro}\n\\overline{Q(M)} = \\bigcup_{x \\in \\overline M}\n\\{ \\tau \\in C(M);\\ \\tau \\le r_x \\}.\n\\end{equation}\nMoreover, we show that if $(\\bar M, g)$ is simple \nthen the boundary distance functions are the maximal elements of $\\overline{Q(M)}$.\nThe set of boundary distance functions\ndetermines the Riemannian manifold $(M,g)$ \\cite{Ku_proc, KKL}. Thus the volumes of $M(\\p M, \\tau)$, $\\tau \\in C(\\p M)$, determine $(M,g)$ if it is simple.\n\n\nOur results give a new uniqueness proof for the \ninverse problem for the wave equation in the case of\na simple geometry. Belishev and Kurylev have proved the uniqueness even when the geometry is not simple \\cite{BeKu}. \nTheir proof is based on the boundary control method\n\\cite{AKKLT, Be3, KK, KKLima, Pestov}, originally developed for the isotropic wave equation \\cite{Be}. \nOur uniqueness proof might be the first systematic use of lattice structures in the context of inverse boundary value problems.\n\nIn previous literature, $M(\\Gamma, \\tau)$ has been defined in the case of a constant function $\\tau$, see e.g. \\cite{KKL} and the references therein.\nIn this paper we establish some properties of $M(\\Gamma, \\tau)$ when $\\Gamma \\subset \\p M$ is open and $\\tau \\in C(\\bar \\Gamma)$.\nIn particular, we show that its boundary is of measure zero. \nThis important detail seems to be neglected in previous literature also in the case of a constant function $\\tau$.\n\nOur method to compute the volume of a domain of influence is related to the iterative time-reversal control method \nby Bingham, Kurylev, Lassas and Siltanen \\cite{ITRC}.\nTheir method produces a certain kind of focused waves, \nand they also prove uniqueness for the \ninverse problem for the wave equation using these waves. Moreover, they give a review of \nmethods that use time-reversed measurements\n\\cite{Bardos, Bardos2, Papa1, CIL, FinkD, FinkMain, Kliba}. \nA modification of the iterative time-reversal control method is presented in \\cite{DKL}. \n\n\\section{Main results}\n\nLet $(M, g)$ be a $C^\\infty$-smooth, \ncompact and connected Riemannian manifold \nof dimension $n \\ge 2$ with nonempty boundary $\\p M$.\nWe consider the wave equation \n\\begin{align}\\label{eq:wave}\n&\\p_t^2 u(t,x) + a(x,D_x) u(t, x) = 0, \\quad (t,x) \\in (0,\\infty) \\times M,\n\\\\\\nonumber& u|_{t=0} = 0, \\quad \\p_t u|_{t=0}=0, \n\\\\\\nonumber& b(x, D_x) u(t,x) = f(t,x), \\quad (t,x) \\in (0,\\infty) \\times \\p M,\n\\end{align}\nwhere $a(x, D_x)$ is a weighted Laplace-Beltrami operator and $b(x, D_x)$ is the \ncorresponding normal derivative. \nIn coordinates, $(g^{jk}(x))_{j,k=1}^n$ denotes the inverse of $g(x)$ and $|g(x)|$ the determinant of $g(x)$.\nThen\n\\begin{align*}\na(x,D_x) u\n&:= -\\sum_{j,k=1}^n \\mu(x)^{-1}|g(x)|^{-\\frac 12}\\frac {\\p}{\\p x^j} \n\\ll( \\mu(x)|g(x)|^{\\frac 12}g^{jk}(x)\\frac {\\p u}{\\p x^k} \\rr),\n\\\\b(x, D_x) u \n&:= \\sum_{j,k=1}^n \\mu(x)g^{jk}(x) \\nu_k(x) \\frac{\\p u}{\\p x^j},\n\\end{align*}\nwhere $\\mu$ is a $C^\\infty$-smooth strictly positive weight function and \n$\\nu = (\\nu_1, \\dots, \\nu_n)$ is the exterior co-normal vector of $\\p M$\nnormalized with respect to $g$, that is $\\sum_{j,k=1}^m g^{jk}\\nu_j\\nu_k=1$.\nThe isotropic wave equation (\\ref{eq:wave_isotropic}) is a special case of (\\ref{eq:wave})\nwith $g(x) := (c(x)^{-2} \\delta_{jk})_{j,k=1}^n$ and $\\mu(x) = c(x)^{n-2}$.\n\nWe denote the indicator function of a set $A$ by $1_A$,\nthat is, $1_A(x) = 1$ if $x \\in A$ and $1_A(x) = 0$\notherwise. \nMoreover, we denote\n\\begin{equation} \\label{eq:integration_triangle_L}\nL := \\{ (t,s) \\in \\R^2; t + s \\le 2 T,\\ s > t > 0 \\},\n\\end{equation}\nand define the operators\n\\begin{align*}\n&J f(t) := \\frac{1}{2} \\int_0^{2 T} 1_L(t,s) f(s) ds, \\quad\nR f(t) := f(2 T - t), \\quad\n\\\\&K := J \\Lambda_{2T} - R \\Lambda_{2T} R J, \\quad\nI f (t) := 1_{(0,T)}(t) \\int_0^t f(s) ds,\n\\end{align*}\nwhere $\\Lambda_{2T}$ is the operator defined by (\\ref{eq:dtn}) $u^f$ being the solution of (\\ref{eq:wave}).\nWe denote by $dS_g$ the Riemannian volume measure of the manifold $(\\p M, g|_{\\p M})$.\nFurthermore, we denote by $(\\cdot, \\cdot)$ and $\\norm{\\cdot}$ \nthe inner product and the norm of $L^2((0, 2T) \\times \\p M; dt \\otimes dS_g)$.\nWe study the regularized minimization problem\n\\begin{equation}\n\\label{eq:minimization_regularized}\n\\argmin_{f \\in S} \\ll ((f, K f) - 2(I f, 1) + \\alpha \\norm{f}^2 \\rr),\n\\end{equation}\nwhere the regularization parameter $\\alpha$ is strictly positive and $S$\nis a closed subspace of $L^2((0, 2T) \\times \\p M)$.\n\nOperator $a(x, D_x)$ with the domain $H^2(M) \\cap H^1_0(M)$ is self-adjoint on\nthe space $L^2(M; dV_\\mu)$, where $dV_\\mu = \\mu |g|^{1\/2} dx$ in coordinates.\nThus we call $dV_\\mu$ the natural measure corresponding to $a(x, D_x)$\nand denote it also by $m$.\nIn \\cite{ITRC} it is shown that\n\\begin{equation}\n\\label{eq:inner_products}\n(u^f(T), u^h(T))_{L^2(M; dV_\\mu)} \n= (f, K h).\n\\end{equation}\nThis is a reformulation of the Blagovestchenskii identity \\cite{Bl}.\nIn Lemma \\ref{lem:cross_term} we show the following identity\n\\begin{equation}\n\\label{eq:inner_product_with_1}\n(u^f(T), 1)_{L^2(M; dV_\\mu)} = (I f, 1).\n\\end{equation}\nThis is well known at least in the isotropic case, see e.g. \\cite{Be2}.\nThe equations (\\ref{eq:inner_products}) and (\\ref{eq:inner_product_with_1}) imply that\n\\begin{align}\n\\label{eq:minimization}\n(f, K f) - 2(I f, 1)\n&= \\norm{u^{f}(T)}_{L^2(M; dV_\\mu)}^2 - 2 (u^{f}(T), 1)_{L^2(M; dV_\\mu)} \n\\\\\\nonumber&=\n\\norm{u^{f}(T) - 1}_{L^2(M; dV_\\mu)}^2 + C,\n\\end{align}\nwhere $C = -\\norm{1}_{L^2(M; dV_\\mu)}^2$ does not depend on the source $f$.\nThus the minimization problem (\\ref{eq:minimization_regularized}) is equivalent with the \nminimization problem \n\\begin{equation*}\n\\argmin_{f \\in S} \\ll( \\norm{u^{f}(T) - 1}_{L^2(M; dV_\\mu)}^2 + \\alpha \\norm{f}^2 \\rr).\n\\end{equation*}\n\nFor $\\Gamma \\subset M$ and $\\tau : \\bar \\Gamma \\to \\R$, we define {\\em the domain of influence},\n\\begin{equation}\n\\label{domain_of_influence}\nM(\\Gamma, \\tau) := \\{x \\in M;\\ \\text{there is $y \\in \\bar \\Gamma$ such that $d(x,y) \\le \\tau(y)$}\\},\n\\end{equation}\nwhere $d$ is the distance on the Riemannian manifold $(M,g)$.\nIn Section \\ref{sec:regularization} we show the following two theorems.\n\\begin{theorem}\n\\label{thm:minimization_on_subspace}\nLet $\\alpha > 0$ and let $S \\subset L^2( (0,2T) \\times \\p M)$ be a closed subspace.\nDenote by $P$ the orthogonal projection\n\\begin{equation*}\nP : L^2( (0,2T) \\times \\p M) \\to S.\n\\end{equation*}\nThen the regularized minimization (\\ref{eq:minimization_regularized}) has unique minimizer $f_\\alpha \\in S$, \nand $f_\\alpha$ is the unique $f \\in S$ solving\n\\begin{equation}\n\\label{eq:normal}\n(PKP + \\alpha) f = P I^+ 1,\n\\end{equation}\nwhere $I^+$ is the adjoint of $I$ in $L^2( (0,2T) \\times \\p M)$.\nMoreover, $PKP + \\alpha$ is positive definite on $S$.\n\\end{theorem}\n\\begin{theorem}\n\\label{thm:indicator_functions}\nLet $\\Gamma \\subset \\p M$ be open, $\\tau \\in C(\\bar \\Gamma)$ and define\n\\begin{equation*}\nS = \\{f \\in L^2((0, 2T) \\times \\p M);\\ \\text{$\\supp(f)$ satisfies (\\ref{eq:source_supp_condition})}\\}.\n\\end{equation*}\nLet $f_\\alpha$, $\\alpha > 0$, \nbe the minimizer in Theorem \\ref{thm:minimization_on_subspace}.\nThen in $L^2(M)$\n\\begin{equation*}\n\\lim_{\\alpha \\to 0} u^{f_\\alpha}(T) \n= 1_{M(\\Gamma, \\tau \\wedge T)}.\n\\end{equation*}\n\\end{theorem}\n\nWe denote $m_\\tau := m(M(\\p M, \\tau))$, for $\\tau \\in C(\\p M)$, and $m_\\infty := m(M)$.\nMoreover, we define\n\\begin{align}\n Q(M) &:= \\{ \\tau \\in C(\\p M);\\ m_\\tau < m_\\infty \\}, \\label{eq:meet_semilattice_QM}\n\\\\R(M) &:= \\{ r_x \\in C(\\p M);\\ x \\in M \\}, \\nonumber\n\\end{align}\nwhere $r_x$ is the boundary distance function defined by \n(\\ref{eq:boundary_distance_functions}) $d$ being the distance on the Riemannian manifold $(M,g)$.\nWe denote by $\\overline{Q(M)}$ the closure of $Q(M)$ in $C(M)$.\nIn Section \\ref{sec:maximal_elements} we prove the \nequation (\\ref{eq:semilattice_QM_intro}) and show\nthe following theorem.\n\\begin{theorem}\n\\label{thm:maximal_elements}\nIf $(M,g)$ satisfies the condition \n\\begin{itemize}\n\\item[(G)] $x_1, x_2 \\in M$ and $r_{x_1} \\le r_{x_2}$ imply $x_1 = x_2$,\n\\end{itemize}\nthen $R(M)$ is the set of maximal elements of $\\overline{Q(M)}$.\n\\end{theorem}\n\nLet $T \\ge \\max\\{ d(x, y) ;\\ x \\in M,\\ y \\in \\p M \\}$.\nThen the set of volumes,\n\\begin{equation}\n\\label{eq:volumes_for_uniqueness}\n\\mathcal V := \\{m_\\tau;\\ \\tau \\in C(\\p M)\\ \\text{and $0 \\le \\tau \\le T$} \\},\n\\end{equation}\ndetermines the set $Q(M)$. Note that $r_x(y) \\le T$, for all $x \\in M$ and all $y \\in \\p M$, and that $m_\\infty = \\max \\mathcal V$.\nMoreover, by Theorem \\ref{thm:indicator_functions} and equation (\\ref{eq:inner_products}) we can compute the volume $m_\\tau \\in \\mathcal V$ as the limit \n\\begin{equation}\n\\label{eq:volumes_via_minimizers}\nm_\\tau = \\lim_{\\alpha \\to 0} (f_\\alpha, K f_\\alpha).\n\\end{equation}\nThe set $R(M)$ determines the manifold $(M,g)$ up to an isometry \\cite{Ku_proc, KKL}.\nHence the volumes (\\ref{eq:volumes_for_uniqueness}) contain enough information to \ndetermine the manifold $(M,g)$ in the class of manifolds satisfying (G).\nIn section \\ref{sec:maximal_elements}, we show that simple manifolds satisfy (G).\n\\begin{definition}\n\\label{def:simple}\nA compact Riemannian manifold $(M, g)$ with boundary is {\\em simple}\nif it is simply connected, any geodesic has no conjugate points and\n$\\p M$ is strictly convex with respect to the metric $g$.\n\\end{definition}\n\nLet us discuss Theorem \\ref{thm:minimization_on_subspace} from the point of view of practical computations.\nWhen the subspace $S$ is finite-dimensional, the positive definite system of linear equations (\\ref{eq:normal})\ncan be solved using the conjugate gradient method.\nIn each iteration step of the conjugate gradient method we must evaluate one matrix-vector product. \nIn our case, the product can be realized by two measurements \n\\begin{equation}\n\\label{eq:cg_step_measurements}\n\\Lambda_{2 T}f, \\quad \\Lambda_{2 T} R J f, \n\\end{equation}\nwhere $f$ is the approximate solution given by the previous iteration step. \nThe remaining computational part of the iteration step consists of a few inexpensive vector-vector operations. \nThus if we intertwine computation of a conjugate gradient steps with measurements (\\ref{eq:cg_step_measurements}), \nthe computational cost of our method is very low. \n\n\n\\section{The open and the closed domain of influence}\n\\label{sec:domains_of_influence}\n\nLet us recall that the domain of influence $M(\\Gamma, \\tau)$ \nis defined in (\\ref{domain_of_influence}) for $\\Gamma \\subset M$ and $\\tau : \\bar \\Gamma \\to \\R$.\nWe call $M(\\Gamma, \\tau)$ also the {\\em closed} domain of influence and\ndefine the {\\em open} domain of influence\n\\begin{equation*}\nM^0(\\Gamma, \\tau) := \\{ x \\in M;\\ \\text{there is $y \\in \\Gamma$ s.t. $d(x,y) < \\tau(y)$} \\}.\n\\end{equation*}\n\nLet us consider the closed domain of influence $M(\\Gamma, \\tau)$ when $\\Gamma \\subset \\p M$ is open and $\\tau$ is a constant.\nFinite speed of propagation for the wave equation guarantees that the solution $u^f$ at time $T$\nis supported on $M(\\Gamma, \\tau)$ whenever the source $f$ satisfies the support condition \n(\\ref{eq:source_supp_condition}).\nMoreover, using Tataru's unique continuation result \\cite{Ta1, Ta2}, it is possible to show that the set of functions,\n\\begin{equation*}\n\\{ u^f(T);\\ \\text{$f \\in L^2((0,2T) \\times \\p M)$ and $\\supp(f)$ satisfies (\\ref{eq:source_supp_condition})}\\},\n\\end{equation*}\nis dense in $L^2(M^0(\\Gamma, \\tau))$, see e.g. the proof of Theorem 3.16 and the orthogonality argument of Theorem 3.10 in \\cite{KKL}.\nIt is easy generalize this for $\\tau$ of form\n\\begin{equation*}\n\\tau(y) = \\sum_{j=1}^N T_j 1_{\\Gamma_j}(y), \\quad y \\in \\p M,\n\\end{equation*}\nwhere $N \\in \\N$, $T_j \\in \\R$ and $\\Gamma_j \\subset \\p M$ are open, see \\cite{ITRC}.\nHowever, the fact that $M(\\Gamma, \\tau) \\setminus M^0(\\Gamma, \\tau)$ is of measure zero, seems to go unproven in the literature.\nIn this section we show that this is indeed the case even for $\\tau \\in C(\\bar \\Gamma)$.\n\nTo our knowledge, this can not be proven just by considering the boundaries of the balls $B(y, \\tau(y))$, \n$y \\in \\Gamma$.\nIn fact, we give below an example showing that the union of the boundaries $\\p B(y, \\tau(y))$, for $y \\in \\p \\Gamma$,\ncan have positive measure.\n\\begin{example}\nLet $\\mathcal C$ be the fat Cantor set, $M \\subset \\R^2$ be open, $g$ be the Euclidean metric, \n$(0,1) \\times \\{0\\} \\subset \\p M$ and $\\Gamma = \\ll( (0,1) \\setminus \\mathcal C \\rr) \\times \\{0\\}$.\nThen the union $B := \\bigcup_{y \\in \\p \\Gamma} \\p B(y, 1)$ has positive measure. \n\\end{example}\n\\begin{proof}\nThe fat Cantor set $\\mathcal C$ is an example of a closed subset of $[0,1]$ whose boundary has positive measure, see e.g. \\cite{Pugh}.\nThe map\n\\begin{equation*}\n\\Phi : (s, \\alpha) \\mapsto (s + \\cos \\alpha, s + \\sin \\alpha)\n\\end{equation*}\nis a diffeomorphism from $\\R \\times (0, \\pi\/2)$ onto its image in $\\R^2$.\nThe image of $H := \\p \\mathcal C \\times (0, \\pi\/2)$ under $\\Phi$\nlies in $B$.\nAs $H$ has positive measure so has $B$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:characterization_of_domi}\nLet $\\Gamma \\subset \\p M$ be open and let $\\tau \\in C(\\bar \\Gamma)$. \nThen the function \n\\begin{equation*}\nr_{\\Gamma, \\tau}(x) := \\inf_{y \\in \\Gamma} (d(x, y) - \\tau(y))\n\\end{equation*}\nis Lipschitz continuous and \n\\begin{align}\nM(\\Gamma, \\tau) &= \\{ x \\in M;\\ r_{\\Gamma, \\tau}(x) \\le 0 \\}, \\label{eq:closed_domi_r}\n\\\\M^0(\\Gamma, \\tau) &= \\{ x \\in M;\\ r_{\\Gamma, \\tau}(x) < 0 \\}. \\label{eq:open_domi_r}\n\\end{align}\nIn particular, $M(\\Gamma, \\tau)$ is closed and $M^0(\\Gamma, \\tau)$ is open.\n\\end{lemma}\n\\begin{proof}\nLet us define\n\\begin{equation*}\nr(x) := r_{\\Gamma, \\tau}(x), \\quad \\tilde r(x) := \\min_{y \\in \\bar \\Gamma} (d(x, y) - \\tau(y)),\n\\end{equation*}\nand show that $\\tilde r = r$. Clearly $\\tilde r\\le r$.\nLet $x \\in M$. The minimum in the definition of $\\tilde r(x)$ is attained at a point $y_0 \\in \\bar \\Gamma$.\nWe may choose a sequence $(y_j)_{j=1}^\\infty \\subset \\Gamma$ such that $y_j \\to y_0$ as $j \\to \\infty$.\nThen\n\\begin{equation*}\nr(x) \\le d(x, y_j) - \\tau(y_j) \\to \\tilde r(x)\n\\quad \\text{as $j \\to \\infty$}.\n\\end{equation*}\nHence $\\tilde r = r$.\n\nLet us show that $\\tilde r$ is Lipschitz. \nLet $x \\in M$, and let $y_0$ be as before. \nLet $x' \\in M$. Then \n\\begin{equation*}\n\\tilde r(x') - \\tilde r(x) \n\\le d(x', y_0) - \\tau(y_0) - \\ll( d(x, y_0) - \\tau(y_0) \\rr) \\le d(x, x').\n\\end{equation*}\nBy symmetry with respect to $x'$ and $x$, $\\tilde r$ is Lipschitz.\n\nLet us show (\\ref{eq:closed_domi_r}). \nClearly $r(x) \\le 0$ for $x \\in M(\\Gamma, \\tau)$.\nLet $x \\in M$ satisfy $r(x) \\le 0$, and let $y_0$ be as before.\nThen \n\\begin{equation*}\nd(x, y_0) - \\tau(y_0) = \\tilde r(x) = r(x) \\le 0,\n\\end{equation*}\nand $x \\in M(\\Gamma, \\tau)$. Hence (\\ref{eq:closed_domi_r}) holds. \nThe equation (\\ref{eq:open_domi_r}) can be proven in a similar way.\n\\end{proof}\n\nIf $\\Gamma \\subset \\p M$ and $\\tau$ is a constant function, then\n\\begin{align*}\nM(\\Gamma, \\tau) &= \\{ x \\in M;\\ r_{\\Gamma, \\tau}(x) \\le 0 \\} \n= \\{ x \\in M;\\ \\inf_{y \\in \\Gamma} d(x,y) \\le \\tau \\}\n\\\\&= \\{ x \\in M;\\ d(x,\\Gamma) \\le \\tau \\}.\n\\end{align*}\nThus for a constant $\\tau$, our definition of $M(\\Gamma, \\tau)$ coincides with the definition \nof the domain of influence in \\cite{KKL}. \n\n\\begin{lemma}\n\\label{lem:level_set_is_null}\nLet $A \\subset M$ be compact and let $\\tau : A \\to \\R$ be continuous. \nWe define\n\\begin{equation*}\nr(x) := \\inf_{y \\in A} (d(x,y) - \\tau(y)), \\quad x \\in M.\n\\end{equation*}\nIf $\\tau$ is strictly positive on $A$ or $A$ is a null set, then $\\{x \\in M;\\ r(x) = 0\\}$\nis a null set.\nWe mean by a null set a set of measure zero with respect to the Riemannian volume measure.\n\\end{lemma}\n\\begin{proof}\nDenote by $V_g$ the volume measure of $M$ and define\n\\begin{equation*}\nZ := \\{p \\in M;\\ r(p) = 0\\}.\n\\end{equation*}\nLet us show that \n\\begin{equation} \\label{eq:A_does_not_matter}\nV_g(Z) = V_g(Z \\setminus A).\n\\end{equation}\nIf $V_g(A) = 0$, then (\\ref{eq:A_does_not_matter}) is immediate. \nIf $\\tau > 0$, then \n\\begin{equation*}\nr(q) \\le d(q,q) - \\tau(q) = -\\tau(q) < 0, \\quad q \\in A.\n\\end{equation*}\nHence $Z \\cap A = \\emptyset$ and (\\ref{eq:A_does_not_matter}) holds.\n\nLet $p \\in M^\\text{int}$. There is a chart $(U, \\phi)$ of $M^\\text{int}$ such that\n$\\phi(p) = 0$ and that the closure of the open Euclidean unit ball $B$ of $\\R^n$\nis contained in $\\phi(U)$.\nWe denote $U_p := \\phi^{-1}(B)$.\n\nThe sets $U_p$, $p \\in M^\\text{int}$, form an open cover for $M^\\text{int}$,\nand as $M^\\text{int}$ is second countable, \nthere is a countable cover $U_{p_j}$, $j = 1, 2, \\dots$, of $M^\\text{int}$.\nHence\n\\begin{equation*}\nV_g(Z) = V_g( (Z \\setminus A) \\cap (\\p M \\cup \\bigcup_{j=1}^\\infty U_{p_j}))\n\\le \\sum_{j=1}^\\infty V_g( (Z \\setminus A) \\cap U_{p_j}).\n\\end{equation*}\nIt is enough to show that $\\phi((Z \\setminus A) \\cap U_p)$\nis a null set with respect to the Lebesgue measure on $B$.\n\nWe define for $v = (v^1, \\dots, v^n) \\in \\R^n$ and $x \\in B$,\n\\begin{equation*}\n|v|_{g(x)}^2 := \\sum_{j,k = 1}^n v^j g_{jk}(x) v^k, \n\\quad |v|^2 := \\sum_{j = 1}^n (v^j)^2,\n\\end{equation*}\nwhere $(g_{jk})_{j,k=1}^n$ is the metric $g$ in the local coordinates on $\\phi(U)$.\nAs $\\overline B$ is compact in $\\phi(U)$, there is $c_p > 0$ such that\nfor all $v \\in \\R^n$ and $x \\in B$\n\\begin{equation*}\nc_p |v|_{g(x)} \\le |v| \\le \\frac{1}{c_p} |v|_{g(x)}.\n\\end{equation*}\n\nAs in the proof of Lemma \\ref{lem:characterization_of_domi}\nwe see that $r$ is Lipschitz continuous on $M$.\nThus by Rademacher's theorem there is a null set $N \\subset B$ such that \n$r$ is differentiable in the local coordinates in $B \\setminus N$.\nWe denote $Z_p := \\phi((Z \\setminus A) \\cap U_p) \\setminus N$.\n\nLet $x \\in Z_p$ and denote $p_x := \\phi^{-1}(x)$.\nAs $A$ is compact and $q \\mapsto d(p_x, q) - \\tau(q)$ \nis continuous, there is $q_x \\in A$ such that\n\\begin{equation*}\nd(p_x, q_x) - \\tau(q_x) = r(p_x) = 0.\n\\end{equation*}\nWe denote $s := d(p_x, q_x) = \\tau(q_x)$.\nAs $p_x \\notin A$ and $q_x \\in A$, we have that $0 < s$.\n\nAs $M$ is connected and complete as a metric space, Hopf-Rinow theorem gives\na shortest path $\\gamma : [0, s] \\to M$ \nparametrized by arclength and joining $q_x = \\gamma(0)$ and $p_x = \\gamma(s)$.\nFor a study of shortest paths on Riemannian manifolds with boundary\nsee \\cite{Ax2}.\nAs $p_x \\in U_p$, there is $a \\in (0, s)$ such that $\\gamma|_{[a , s]}$\nis a unit speed geodesic of $U_p \\subset M^\\text{int}$.\nAs $\\gamma$ is parametrized by arclength,\n\\begin{equation*}\nr(\\gamma(t)) \\le d(\\gamma(t), q_x) - \\tau(q_x) = t - s, \\quad t \\in [a, s],\n\\end{equation*}\nand as $r(\\gamma(s)) = r(p_x) = 0$,\n\\begin{equation*}\n\\frac{r(\\gamma(t)) - r(\\gamma(s))}{t - s} \\ge \\frac{t - s - 0}{t - s} = 1, \\quad t \\in (a,s).\n\\end{equation*}\n\nThe function $r \\circ \\gamma$ is differentiable at $s$ by the chain rule, and\n\\begin{equation*}\n\\p_t (r \\circ \\gamma)(s)\n= \\lim_{t \\to s^-} \\frac{r(\\gamma(t)) - r(\\gamma(s))}{t - s} \\ge 1.\n\\end{equation*}\nAs $\\gamma$ is a unit speed geodesic near $s$,\n\\begin{equation*}\nc_p = c_p |\\p_t \\gamma(s)|_{g(x)} \\le |\\p_t \\gamma(s)| \n\\le \\frac{1}{c_p}|\\p_t \\gamma(s)|_{g(x)} = \\frac{1}{c_p}.\n\\end{equation*}\nHence in the local coordinates in $B$\n\\begin{equation*}\n|D r(x)| \\ge D r(x) \\cdot \\frac{\\p_t \\gamma(s)}{|\\p_t \\gamma(s)|}\n\\ge c_p \\p_t (r \\circ \\gamma)(s) \\ge c_p.\n\\end{equation*}\n\nLet $\\epsilon > 0$. \nThere is $\\delta(x) > 0$ such that \n\\begin{equation}\n\\label{eq:r_derivative_approximation}\n|r(y) - r(x) - D r(x) \\cdot (y - x)| \\le c_p \\epsilon |y - x|, \\quad y \\in B(x, \\delta(x)),\n\\end{equation}\nand $B(x, \\delta(x)) \\subset B$.\nHere $B(x, \\delta)$ is the open Euclidean ball with center $x$ and radius $\\delta$.\n\nThe sets $B(x, \\delta(x)\/5)$, $x \\in Z_p$, form an open cover for $Z_p$,\nand as $Z_p$ is second countable, \nthere is $(x_j)_{j=1}^\\infty \\subset Z_p$ such that the sets \n\\begin{equation*}\nB_j' := B(x_j, \\delta(x_j)\/5)\n\\end{equation*}\nform an open cover for $Z_p$.\nBy Vitali covering lemma there is an index set $J \\subset \\N$ such that\nthe sets $B_j'$, $j \\in J$, are disjoint and the sets\n\\begin{equation*}\nB_j := B(x_j, \\delta(x_j)), \\quad j \\in J,\n\\end{equation*}\nform an open cover for $Z_p$.\n\nWe denote \n\\begin{equation*}\nv_j := \\frac{D r(x_j)}{|D r(x_j)|}, \\quad \\delta_j := \\delta(x_j).\n\\end{equation*}\nIf $y \\in Z_p \\cap B_j$, then $r(y) = 0 = r(x_j)$ and by (\\ref{eq:r_derivative_approximation})\n\\begin{equation*}\n|v_j \\cdot (y - x_j)| \\le \\frac{1}{|D r(x_j)|} c_p \\epsilon |y - x_j| \\le \\epsilon \\delta_j.\n\\end{equation*}\n\nWe denote by $\\alpha_m$ the volume of the open Euclidean unit ball in $\\R^m$\nand by $V$ the Lebesgue measure on $B$.\nLet $j \\in J$.\nUsing a translation and a rotation we get such \ncoordinates that $x_j = 0$ and $v_j = (1, 0, \\dots, 0)$.\nIn these coordinates \n\\begin{equation*}\nZ_p \\cap B_j \\subset \\{ (y^1, y') \\in \\R \\times \\R^{n-1};\\ |y^1| \\le \\epsilon \\delta_j, |y'| \\le \\delta_j \\}.\n\\end{equation*}\nHence $V(Z_p \\cap B_j) \\le 2 \\epsilon \\delta_j \\alpha_{n-1} \\delta_j^{n-1}$.\nParticularly,\n\\begin{equation*}\nV(Z_p \\cap B_j) \n\\le \\epsilon \\frac{2 \\alpha_{n-1}}{\\alpha_{n}} V(B_j) \n= \\epsilon c_n V(B_j'),\n\\end{equation*}\nwhere $c_n := 2 \\cdot 5^n \\alpha_{n-1} \/ \\alpha_{n}$.\nThen\n\\begin{align*}\nV(\\phi((Z \\setminus A) \\cap U_p))\n&= V(Z_p) = V(Z_p \\cap \\bigcup_{j \\in J} B_j) \n\\le \\sum_{j \\in J} V(Z_p \\cap B_j)\n\\\\&\\le \\epsilon c_n \\sum_{j \\in J} V(B_j') \n= \\epsilon c_n V(\\bigcup_{j \\in J} B_j') \n\\le \\epsilon c_n V(B).\n\\end{align*}\nAs $\\epsilon > 0$ is arbitrary, $V(\\phi((Z \\setminus A) \\cap U_p)) = 0$ and the claim is proved.\n\\end{proof}\n\n\\section{Approximately constant wave fields on a domain of influence}\n\\label{sec:regularization}\n\n\\begin{lemma}\n\\label{lem:cross_term}\nLet $f \\in L^2((0, 2T) \\times \\p M)$. Then the equation (\\ref{eq:inner_product_with_1}) holds.\n\\end{lemma}\n\\begin{proof}\nThe map $h \\mapsto u^h(T)$ is bounded $L^2((0, 2T) \\times \\p M) \\to L^2(M)$, see e.g. \\cite{LaTr}.\nThus is it enough to prove the equation (\\ref{eq:inner_product_with_1}) for $f \\in C_c^\\infty( (0, 2T) \\times \\p M)$.\nLet us denote\n\\begin{equation*}\nv(t) := (u^f(t), 1)_{L^2(M; dV_\\mu)}.\n\\end{equation*}\nAs $a(x,D_x) 1 = 0$ and $b(x,D_x)1 = 0$,\nwe may integrate by parts\n\\begin{align*}\n\\p_t^2 v(t) &= - (a(x, D_x) u^f(t), 1)_{L^2(M; dV_\\mu)}\n\\\\&= - \\ll( (a(x, D_x) u^f(t), 1)_{L^2(M; dV_\\mu)} - (u^f(t), a(x, D_x) 1)_{L^2(M; dV_\\mu)} \\rr)\n\\\\&= (b(x, D_x) u^f(t), 1)_{L^2(\\p M; dS_g)} - (u^f(t), b(x, D_x) 1)_{L^2(\\p M; dS_g)}\n\\\\&= (f(t), 1)_{L^2(\\p M; dS_g)}.\n\\end{align*}\nAs $\\p_t^j v(0) = 0$ for $j=0,1$,\n\\begin{align*}\nv(T) &= \\int_0^T \\int_0^t \\int_{\\p M} f(s,x) dS_g(x) ds dt\n\\\\&= \\int_0^{2T} \\int_{\\p M} 1_{(0,T)}(t) \\int_0^t f(s,x) ds dS_g(x) dt.\n\\end{align*}\n\\end{proof}\n\nThe proof of Theorem \\ref{thm:minimization_on_subspace} \nis similar to the proof of the corresponding result in \\cite{ITRC}.\nWe give the proof for the sake of completeness.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:minimization_on_subspace}.]\nWe define\n\\begin{equation*}\nE(f) := (f, Kf) - 2 (If, 1) + \\alpha \\norm{f}^2.\n\\end{equation*}\nThen \n\\begin{equation}\n\\label{eq:energy_functional}\nE(f) = \\norm{u^{f}(T) - 1}_{L^2(M; dV_\\mu)}^2 - \\norm{1}_{L^2(M; dV_\\mu)}^2 + \\alpha \\norm{f}^2.\n\\end{equation}\nLet $(f_j)_{j=1}^\\infty \\subset S$ be such that \n\\begin{equation*}\n\\lim_{j \\to \\infty} E(f_j) = \\inf_{f \\in S} E(f).\n\\end{equation*}\nThen \n\\begin{equation*}\n\\alpha \\norm{f_j} \\le E(f_j) + \\norm{1}_{L^2(M; dV_\\mu)}^2,\n\\end{equation*}\nand $(f_j)_{j=1}^\\infty$ is bounded in $S$.\nAs $S$ is a Hilbert space,\nthere is a subsequence of $(f_j)_{j=1}^\\infty$ converging weakly in $S$.\nLet us denote the limit by $f_\\infty \\in S$ and the subsequence still by $(f_j)_{j=1}^\\infty$.\n\nThe map $h \\mapsto u^h(T)$ is bounded\n\\begin{equation*}\nL( (0, 2T) \\times \\p M) \\to H^{5\/6 - \\epsilon}(M)\n\\end{equation*}\nfor $\\epsilon > 0$, see \\cite{LaTr}.\nHence $h \\mapsto u^h(T)$ is a compact operator\n\\begin{equation*}\nL^2( (0, 2T) \\times \\p M) \\to L^2(M),\n\\end{equation*}\nand $u^{f_j}(T) \\to u^{f_\\infty}(T)$ in $L^2(M)$ as $j \\to \\infty$.\nMoreover, the weak convergence implies \n\\begin{equation*}\n\\norm{f_\\infty} \\le \\liminf_{j \\to \\infty} \\norm{f_j}.\n\\end{equation*}\nHence\n\\begin{align*}\nE(f_\\infty) &= \\lim_{j \\to \\infty} \\norm{u^{f_j}(T) - 1}_{L^2(M; dV_\\mu)}^2 - \\norm{1}_{L^2(M; dV_\\mu)}^2 + \\alpha \\norm{f_\\infty}^2\n\\\\&\\le \\lim_{j \\to \\infty} \\norm{u^{f_j}(T) - 1}_{L^2(M; dV_\\mu)}^2 - \\norm{1}_{L^2(M; dV_\\mu)}^2 + \\alpha \n\\liminf_{j \\to \\infty} \\norm{f_j}^2\n\\\\&= \\liminf_{j \\to \\infty} E(f_j) = \\inf_{f \\in S} E(f),\n\\end{align*}\nand $f_\\infty \\in S$ is a minimizer.\n\nLet $f_\\alpha$ be a minimizer and $h \\in S$. \nBy orthogonality of the projection $P$ and identity (\\ref{eq:inner_products}),\nit is clear that $PKP$ is self-adjoint and positive semidefinite.\nDenote by $D_h$ the Fr\\'echet derivative to direction $h$. \nAs $f_\\alpha = Pf_\\alpha$ and $h = Ph$\n\\begin{equation*}\n0 = D_h E(f_\\alpha) = 2 (h, PKP f_\\alpha) - 2 (h, PI^+ 1) + 2 \\alpha (h, f_\\alpha).\n\\end{equation*}\nHence $f_\\alpha$ satisfies (\\ref{eq:normal}).\nAs $PKP$ is positive semidefinite, $PKP + \\alpha$ is positive definite\nand solution of (\\ref{eq:normal}) is unique.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:approximation_by_simple}\nLet $\\Gamma \\subset \\p M$ be open, $\\tau \\in C(\\bar \\Gamma)$ and $\\epsilon > 0$.\nThen there is a simple function\n\\begin{equation*}\n\\tau_\\epsilon(y) = \\sum_{j=1}^N T_j 1_{\\Gamma_j}(y), \\quad y \\in \\p M,\n\\end{equation*}\nwhere $N \\in \\N$, $T_j \\in \\R$ and $\\Gamma_j \\subset \\Gamma$ are open,\nsuch that \n\\begin{align*}\n\\tau - \\epsilon &< \\tau_\\epsilon \\quad \\text{almost everywhere on $\\Gamma$ and}\n\\\\\\tau_\\epsilon &< \\tau \\quad \\text{on $\\bar \\Gamma$}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAs $\\p M$ is compact, there is a finite set of coordinate charts covering $\\p M$.\nUsing partition of unity, we see that it is enough to prove the claim in the case \nwhen $\\Gamma \\subset \\R^{n-1}$ is an open set. \nBut then $\\tau$ is a continuous function on a compact set $\\overline \\Gamma \\subset \\R^{n-1}$,\nand it is clear that there is a simple function with the required properties. \n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:measures_by_approximation}\nLet $\\Gamma \\subset \\p M$ be open, $\\tau \\in C(\\bar \\Gamma)$ and let $\\tau_\\epsilon$, $\\epsilon > 0$, satisfy\n\\begin{align*}\n\\tau - \\epsilon &< \\tau_\\epsilon \\quad \\text{almost everywhere on $\\Gamma$ and}\n\\\\\\tau_\\epsilon &< \\tau \\quad \\text{on $\\bar \\Gamma$}.\n\\end{align*}\nThen\n\\begin{equation*}\n\\lim_{\\epsilon \\to 0} m(M(\\Gamma, \\tau_\\epsilon)) = m(M(\\Gamma, \\tau)).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nLet $\\epsilon > 0$ and denote by $N \\subset \\Gamma$ \nthe set of measure zero where $\\tau - \\epsilon \\ge \\tau_\\epsilon$ as functions on $\\Gamma$.\nLet us show that $M^0(\\Gamma, \\tau - \\epsilon) \\subset M(\\Gamma, \\tau_\\epsilon)$.\nLet $x \\in M^0(\\Gamma, \\tau - \\epsilon)$. Then there is $y_0 \\in \\Gamma$\nsuch that \n\\begin{equation*}\nd(x,y_0) < \\tau(y_0) - \\epsilon.\n\\end{equation*}\nAs $\\tau$ and the function $y \\mapsto d(x,y)$ are continuous\nand $\\Gamma \\setminus N$ is dense in $\\Gamma$, there is $y \\in \\Gamma \\setminus N$ such that\n\\begin{equation*}\nd(x,y) < \\tau(y) - \\epsilon < \\tau_\\epsilon(y).\n\\end{equation*}\nHence $x \\in M(\\Gamma, \\tau_\\epsilon)$.\nA similar argument shows that $M(\\Gamma, \\tau_\\epsilon) \\subset M^0(\\Gamma, \\tau)$.\n\nClearly $M^0(\\Gamma, \\tau - \\epsilon_1) \\subset M^0(\\Gamma, \\tau - \\epsilon_2)$ for $\\epsilon_1 \\ge \\epsilon_2 > 0$,\nand\n\\begin{equation*}\n\\bigcup_{\\epsilon > 0} M^0(\\Gamma, \\tau - \\epsilon) = M^0(\\Gamma, \\tau).\n\\end{equation*}\nHence $m(M^0(\\Gamma, \\tau - \\epsilon)) \\to m(M^0(\\Gamma, \\tau))$ as $\\epsilon \\to 0$, and\n\\begin{align*}\n0 \\le m(M^0(\\Gamma, \\tau)) - m(M(\\Gamma, \\tau_\\epsilon)) &\\le m(M^0(\\Gamma, \\tau)) - m(M^0(\\Gamma, \\tau - \\epsilon))\n\\\\&\\to 0, \\quad \\text{as $\\epsilon \\to 0$}.\n\\end{align*}\nMoreover, by Lemmas \\ref{lem:characterization_of_domi} and \\ref{lem:level_set_is_null}\n\\begin{equation*}\nm(M(\\Gamma, \\tau)) = m(M^0(\\Gamma, \\tau)) = \\lim_{\\epsilon \\to 0} m(M(\\Gamma, \\tau_\\epsilon)).\n\\end{equation*}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:indicator_functions}.]\nWe may assume without loss of generality that $\\tau \\le T$, as\nwe may replace $\\tau$ by $\\tau \\wedge T$ in what follows.\nLet us denote\n\\begin{equation*}\nS(\\Gamma, \\tau) := \\{ f \\in L^2((0,2T) \\times \\p M);\\ \\text{$\\supp(f)$ satisfies (\\ref{eq:source_supp_condition})}\\}.\n\\end{equation*}\nBy the finite speed of propagation for the wave equation,\nwe have that $\\supp(u^f(T)) \\subset M(\\Gamma, \\tau)$ whenever $f \\in S(\\Gamma, \\tau)$.\nHence for $f \\in S(\\Gamma, \\tau)$,\n\\begin{align}\n\\label{eq:splitting_the_difference_1}\n\\norm{u^f(T) - 1}_{L^2(M; dV_\\mu)}^2\n&= \\int_{M(\\Gamma, \\tau)} (u^f(T) - 1)^2 dV_\\mu + \\int_{M \\setminus M(\\Gamma, \\tau)} 1 dV_\\mu\n\\\\\\nonumber&= \\norm{u^f(T) - 1_{M(\\Gamma, \\tau)}}_{L^2(M; dV_\\mu)}^2 \n + m(M \\setminus M(\\Gamma, \\tau)).\n\\end{align}\n\nLet $\\epsilon > 0$.\nBy Lemmas \\ref{lem:approximation_by_simple} and \\ref{lem:measures_by_approximation}\nthere is a simple function $\\tau_\\delta$ satisfying\n\\begin{equation*}\n\\tau_\\delta < \\tau, \\quad m(M(\\Gamma, \\tau)) - m(M(\\Gamma, \\tau_\\delta)) < \\epsilon.\n\\end{equation*}\nBy the discussion in the beginning of Section \\ref{sec:domains_of_influence}, the set \n\\begin{equation*}\n\\{ u^f(T) \\in L^2(M(\\Gamma, \\tau_\\delta));\\ f \\in S(\\Gamma, \\tau_\\delta) \\}\n\\end{equation*}\nis dense in $L^2(M(\\Gamma, \\tau_\\delta))$.\nThus there is $f \\in S(\\Gamma, \\tau_\\delta) \\subset S(\\Gamma, \\tau)$ such that\n\\begin{equation*}\n\\norm{u^f(T) - 1_{M(\\Gamma, \\tau_\\delta)}}_{L^2(M; dV_\\mu)}^2 \\le \\epsilon.\n\\end{equation*}\nThen \n\\begin{equation*}\n\\norm{u^f(T) - 1_{M(\\Gamma, \\tau)}}_{L^2(M; dV_\\mu)}^2 \\le \n\\epsilon + \\norm{1_{M(\\Gamma, \\tau_\\delta)} - 1_{M(\\Gamma, \\tau)}}_{L^2(M; dV_\\mu)}^2\n\\le 2 \\epsilon.\n\\end{equation*}\n\nMoreover, $E(f_\\alpha) \\le E(f)$ and equations (\\ref{eq:splitting_the_difference_1}) and (\\ref{eq:energy_functional}) give\n\\begin{align*}\n&\\norm{u^{f_\\alpha}(T) - 1_{M(\\Gamma, \\tau)}}_{L^2(M; dV_\\mu)}^2 \n\\\\&\\quad= \\norm{u^{f_\\alpha}(T) - 1}_{L^2(M; dV_\\mu)}^2 - m(M \\setminus M(\\Gamma, \\tau))\n\\\\&\\quad\\le E(f_\\alpha) + \\norm{1}_{L^2(M; dV_\\mu)}^2 - m(M \\setminus M(\\Gamma, \\tau))\n\\\\&\\quad\\le \\norm{u^f(T) - 1}_{L^2(M; dV_\\mu)}^2 - m(M \\setminus M(\\Gamma, \\tau)) + \\alpha \\norm{f}^2\n\\\\&\\quad= \\norm{u^f(T) - 1_{M(\\Gamma, \\tau)}}_{L^2(M; dV_\\mu)}^2 + \\alpha \\norm{f}^2 \n\\le 2 \\epsilon + \\alpha \\norm{f}^2.\n\\end{align*}\nWe may choose first small $\\epsilon > 0$ and then small $\\alpha > 0$ to \nget $u^{f_\\alpha}(T)$ arbitrarily close to $1_{M(\\Gamma, \\tau)}$ in $L^2(M)$. \n\\end{proof}\n\n\\section{The boundary distance functions as maximal elements}\n\\label{sec:maximal_elements}\n\n\\def\\tilde{\\widetilde}\nWe denote $M^0(\\tau) := M^0(\\p M, \\tau)$, for $\\tau \\in C(\\p M)$, and define\n\\begin{equation*}\n\\tilde Q := \\{ \\tau \\in C(M);\\ M \\setminus M^0(\\tau) \\ne \\emptyset \\}.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{lem:maximal_elements}\nIf $\\tau$ is a maximal element of $\\tilde Q$, then $\\tau = r_x$ \nfor some $x \\in M$.\nMoreover, if the manifold $(M,g)$ satisfies (G),\nthen $R(M)$ is the set of the maximal elements of $\\tilde Q$.\n\\end{lemma}\n\\begin{proof}\nLet $x \\in M$ and $\\tau \\in C(\\p M)$. Then $\\tau \\le r_x$ if and only if $x \\notin M^0(\\tau)$.\nIn other words,\n\\begin{equation}\n\\label{eq:characterization_of_Q}\n\\tilde Q = \\{ \\tau \\in C(\\p M);\\ \\text{there is $x \\in M$ such that $\\tau \\le r_x$} \\}.\n\\end{equation}\nMoreover, $r_x \\in \\tilde Q$ for all $x \\in M$.\nIndeed, $r_x$ is continuous and trivially $r_x \\le r_x$.\n\nSuppose that $\\tau$ is a maximal element of $\\tilde Q$. \nBy (\\ref{eq:characterization_of_Q}) there is $x \\in M$ such that $\\tau \\le r_x$,\nbut $r_x \\in \\tilde Q$ and maximality of $\\tau$ yields $\\tau = r_x$\n\nLet us now suppose that $(M,g)$ satisfies (G) and show that $r_x$ \nis a maximal element of $\\tilde Q$.\nSuppose that $\\tau \\in \\tilde Q$ satisfies $r_x \\le \\tau$.\nBy (\\ref{eq:characterization_of_Q}) there is $x' \\in M$ such that $\\tau \\le r_x'$.\nHence \n\\begin{equation*}\nr_x \\le \\tau \\le r_{x'},\n\\end{equation*}\nand (G) yields that $x = x'$. \nThus $\\tau = r_x$ and $r_x$ is a maximal element of $\\tilde Q$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:closure}\nThe set $\\tilde Q$ is the closure of $Q(M)$ in $C(M)$.\n\\end{lemma}\n\\begin{proof}\nLet us first show that $\\tilde Q$ is closed.\nLet $(\\tau_j)_{j=1}^\\infty \\subset \\tilde Q$ satisfy $\\tau_j \\to \\tau$ in $C(\\p M)$ as $j \\to \\infty$.\nBy (\\ref{eq:characterization_of_Q}) there is $(x_j)_{j=1}^\\infty \\subset M$ such that $\\tau_j \\le r_{x_j}$.\nAs $M$ is compact there is a converging subsequence $(x_{j_k})_{k=1}^\\infty \\subset (x_j)_{j=1}^\\infty$.\nLet us denote the limit by $x$, that is, $x_{j_k} \\to x$ as $k \\to \\infty$.\nBy continuity of the distance function,\n\\begin{equation*}\n\\tau(y) = \\lim_{k \\to \\infty} \\tau_{j_k}(y) \\le \\lim_{k \\to \\infty} r_{x_{j_k}}(y) = r_x(y), \\quad y \\in \\p M.\n\\end{equation*}\nHence $\\tau \\in \\tilde Q$ and $\\tilde Q$ is closed.\n\nClearly $Q(M) \\subset \\tilde Q$ and it is enough to show that $Q(M)$ is dense in $\\tilde Q$.\nSuppose that $\\tau \\in \\tilde Q$. Then there is $x_0 \\in M$ such that $\\tau \\le r_{x_0}$.\nLet $\\epsilon > 0$. As $M \\times \\p M$ is compact and the distance function is continuous,\nthere is $r > 0$ such that\n\\begin{equation*}\n\\sup_{y \\in \\p M} |d(x, y) - d(x_0, y)| < \\epsilon, \\quad \\text{when $d(x, x_0) < r$ and $x \\in M$}.\n\\end{equation*}\nHence $\\tau(y) - \\epsilon \\le r_{x_0}(y) - \\epsilon < r_x(y)$ for all $y \\in \\p M$ and all $x \\in B(x_0, r)$.\nIn other words, \n\\begin{equation*}\nB(x_0, r) \\subset (M \\setminus M(\\tau - \\epsilon)),\n\\end{equation*}\nand this yields that $\\tau - \\epsilon \\in Q(M)$. \nFunctions $\\tau - \\epsilon$ converge to $\\tau$ in $C(\\p M)$ as $\\epsilon \\to 0$. Thus $\\tau$ is in the closure of $A$.\n\\end{proof}\n\nLemmas \\ref{lem:maximal_elements} and \\ref{lem:closure} together prove Theorem \\ref{thm:maximal_elements}.\nMoreover, (\\ref{eq:characterization_of_Q})\nyields the equation (\\ref{eq:semilattice_QM_intro})\nin the introduction.\n\n\\begin{lemma}\n\\label{lem:simple_manifold}\nIf $(M,g)$ is simple or the closed half sphere, then (G) holds.\n\\end{lemma}\n\\begin{proof}\nLet $x_1, x_2 \\in M$ satisfy $x_1 \\ne x_2$, and let us show that $r_{x_1} \\nleq r_{x_2}$.\nFirst, if $x_2 \\in \\p M$ then $r_{x_1}(x_2) > 0 = r_{x_2}(x_2)$.\nSecond, if $x_2 \\in M^{\\text{int}}$, then there\nis the unique unit speed geodesic $\\gamma$ and the unique point $y \\in \\p M$ such that \n$\\gamma(0) = x_1$, $\\gamma(s) = x_2$ and $\\gamma(s') = y$,\nwhere $0 < s < s'$.\nAs $\\gamma$ is a shortest path\nfrom $x_1$ to $y$ (the shortest path if $(M,g)$ is simple) and the shortest path from $x_2$ to $y$,\n\\begin{equation*}\nr_{x_1}(y) = s' > s = r_{x_2}(y).\n\\end{equation*}\n\\end{proof}\n\nThe closed half sphere is not simple, since for a point on the boundary the corresponding \nantipodal point is a conjugate point.\nHence the manifolds satisfying (G) form a strictly larger class than the simple manifolds.\n\n\\vspace{1cm}\n{\\em Acknowledgements.}\nThe author would like to thank Y. Kurylev for useful discussions. \nThe research was partly supported by Finnish Centre of Excellence in Inverse Problems Research,\nAcademy of Finland COE 213476,\nand partly by Finnish Graduate School in Computational Sciences.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAn old question of Lusin asked whether there exists a Borel\nfunction which cannot be decomposed into countably many\ncontinuous functions. By now several examples have been\ngiven, by Keldi\\v{s}, Adyan and Novikov among others. A\nparticularly simple example, the function\n$P:(\\omega+1)^\\omega\\rightarrow\\omega^\\omega$, has been found by\nPawlikowski (cf. \\cite{CMPS}). By definition,\n\\begin{displaymath}\n P(x)(n) = \\left\\{ \n \\begin{array}{ll}\n x(n)+1 & \\mbox{if}\\quad x(n)<\\omega,\\\\\n 0 & \\mbox{if}\\quad x(n)=\\omega.\n \\end{array} \\right.\n\\end{displaymath}\nIt is proved in \\cite{CMPS} that if $A\\subseteq(\\omega+1)^\\omega$ is such\nthat $P\\!\\!\\upharpoonright\\!\\! A$ is continuous then\n$P[A]\\subseteq\\omega^\\omega$ is nowhere dense. Since $P$ is\na surjection, it is not $\\sigma$-continuous.\n\nIn \\cite{Sol} Solecki showed that the above function is, in\na sense, the only such example, at least among Baire class 1\nfunctions (in other words, it is the initial object in a\ncertain category).\n\n\\begin{theorem}[Solecki, \\cite{Sol}]\\label{pmin}\n For any Baire class 1 function $f:X\\rightarrow Y$, where\n $X,Y$ are Polish spaces, either $f$ is $\\sigma$-continuous\n or there exist topological embeddings $\\varphi$ and $\\psi$\n such that the following diagram commutes:\n $$\n \\begin{CD}\n \\omega^\\omega @ >\\psi >> Y\\\\\n @AAP A @AAfA\\\\\n (\\omega+1)^\\omega @> \\varphi >> X\n \\end{CD}\n $$\n\\end{theorem}\n\nIn \\cite{Zpl:DSTDF} Zapletal generalized Solecki's dichotomy\nto all Borel functions by proving the following theorem.\n\n\\begin{theorem}[Zapletal, \\cite{Zpl:DSTDF}]\n If $f:X\\rightarrow Y$ is a Borel function which is not\n $\\sigma$-continuous then there is a compact set\n $C\\subseteq X$ such that $f\\!\\!\\upharpoonright\\!\\! C$ is not\n $\\sigma$-continuous and of Baire class 1.\n\\end{theorem}\n\nIn this paper we give a new proof of the above dichotomy for\nall Borel functions, which is direct, shorter and more\ngeneral than the original proof from \\cite{Sol}.\n\n\\section{Notation}\n\nWe say that a Borel function $f:X\\rightarrow Y$, where $X,Y$\nare Polish spaces, is $\\sigma$-continuous if there exist a\ncountable cover of the space $X=\\bigcup_n X_n$ (with\narbitrary sets $X_n$) such that $f\\!\\!\\upharpoonright\\!\\! X_n$ is continuous\nfor each $n$. It follows from the Kuratowski extension\ntheorem that we may require that the sets $X_n$ be Borel. If\n$f$ is a Borel function which is not $\\sigma$-continuous\nthen the family of sets on which it is $\\sigma$-continuous\nis a proper $\\sigma$-ideal in $X$. We denote this\n$\\sigma$-ideal by $I_f$.\n\nIn a metric space $(X,d)$ for $A,B\\subseteq X$ let us denote\nby $h(A,B)$ the Hausdorff distance between $A$ and $B$.\n\nThe spaces $(\\omega+1)^\\omega$ and $\\oplus^n$ for $n<\\omega$ are endowed\nwith the product topology of order topologies on $\\omega+1$.\n\n\\section{The Zapletal's game}\n\nIn \\cite{Zpl:DSTDF} Zapletal introduced a two-player game,\nwhich turnes out to be very useful in examining\n$\\sigma$-continuity of Borel functions. Let\n$B\\subseteq\\omega^\\omega$ be a Borel set and $f:B\\rightarrow\n2^\\omega$ be a Borel function. Let\n$\\rho:\\omega\\rightarrow\\omega\\times 2^{<\\omega}\\times\\omega$\nbe a bijection. The game $G_f(B)$ is played by Adam and Eve.\nThey take turns playing natural numbers. In his $n$-th move,\nAdam picks $x_n\\in\\omega$. In her $n$-th move, Eve chooses\n$y_n\\in 2$. At the end of the game we have\n$x\\in\\omega^\\omega$ and $y\\in 2^\\omega$ formed by the\nnumbers picked by Adam and Eve, respectively. Next, $y\\in\n2^\\omega$ is used to define a sequence of partial continuous\nfunctions (with domains of type $G_\\delta$ in\n$\\omega^\\omega$) in the following way. For $n<\\omega$ let\n$f_n$ be a partial function from $\\omega^\\omega$ to\n$2^\\omega$ such that for $t\\in\\omega^\\omega$ and $\\sigma\\in\n2^{<\\omega}$\n$$ f_n(t)\\supseteq\\sigma\\quad\\iff\\quad\\exists\nk\\in\\omega\\,\\,\\, y(\\rho(n,\\sigma,k))=1$$ and\n$\\mbox{dom}(f_n)=\\{t\\in\\omega^\\omega: \\forall\nn<\\omega\\,\\exists!\\sigma\\in 2^n\\,\\,\nf_n(t)\\supseteq\\sigma\\}$. Eve wins the game $G_f(B)$ if\n$x\\not\\in B$ or $\\exists n\\, f(x)=f_n(x).$ Otherwise Adam\nwins the game.\n\nIt is easy to see that if $f$ is a Borel function then $G_f$\nis a Borel game. The key feature of the game $G_f$ is that\nit detects $\\sigma$-continuity of the function $f$.\n\n\\begin{theorem}[Zapletal,\\cite{Zpl:DSTDF}]\n For $B\\subseteq\\omega^\\omega$ and $f:B\\rightarrow\n 2^\\omega$ Eve has a winning strategy in the game $G_f(B)$\n if and only if $f$ is $\\sigma$-continuous on $B$.\n\\end{theorem}\n\nNote that if Adam has a winning strategy then the image of\nhis strategy (treated as a continuous function from\n$2^\\omega$ to $B$) is a compact set on which $f$ is also not\n$\\sigma$-continuous. This observation and the Borel\ndeterminacy gives the following corollary.\n\n\\begin{corollary}[Zapletal,\\cite{Zpl:DSTDF}]\\label{cldense}\n If $B$ is a Borel set and $f:B\\rightarrow 2^\\omega$ is a\n Borel function which is not $\\sigma$-continuous then there\n is a compact set $C\\subseteq B$ such that $f\\!\\!\\upharpoonright\\!\\! C$ is\n also not $\\sigma$-continuous.\n\\end{corollary}\n\n\n\\section{Proof of the dichotomy}\n\nIn the statement of Theorem \\ref{pmin} both functions\n$\\varphi$ and $\\psi$ are to be topological embeddings.\nHowever, as we will see below, for the dichotomy it is\nenough that they both are injective, $\\varphi$ continuous\nand $\\psi$ open. We are going to prove first this version of\nthe dichotomy.\n\n\\begin{theorem}\\label{dichotomy}\n Let $X$ be a Polish space and $f:X\\rightarrow 2^\\omega$ be\n a Borel function. Then precisely one of the following\n conditions holds:\n \\begin{enumerate}\n \\item either $f$ is $\\sigma$-continuous\n \\item\\label{fact} or there are an open injection $\\psi$ and\n a continuous injection $\\varphi$ such that the following\n diagram commutes:\n $$\n \\begin{CD}\n \\omega^\\omega @ >\\psi >> 2^\\omega\\\\\n @AAP A @AAf A\\\\\n (\\omega+1)^\\omega @> \\varphi >> X\n \\end{CD}\n $$ \n \\end{enumerate}\n\\end{theorem}\n\\noindent Notice that compactness of $(\\omega+1)^\\omega$ implies that the\n$\\psi$ above must be a topological embedding.\n\\begin{proof}\n \n It is straightforward that (\\ref{fact}) implies that $f$\n is not $\\sigma$-continuous. Let us assume that $f$ is not\n $\\sigma$-continuous and prove that (\\ref{fact}) holds. By\n Corollary \\ref{cldense} we may assume that $X$ is compact.\n\n\n \\subsection*{Notation.}\n\n First we introduce some notation. For a fixed $n$ and\n $0\\leq k\\leq n$ let $S^n_k$ be the set of points in\n $\\oplus^n$ of Cantor-Bendixson rank $\\geq n-k$. For each\n $n<\\omega$ and $1\\leq k\\leq n$ let us pick a function\n $\\pi^n_k:S^n_k\\rightarrow S^n_{k-1}$ such that\n \\begin{itemize}\n \\item on $S^n_{k-1}$ $\\pi^n_k$ is the identity,\n \\item if $\\tau\\in S^n_k\\setminus S^n_{k-1}$ then we pick\n one $i\\in n$ such that $\\tau(i)<\\omega$ and $\\tau(i)$ is\n maximal such and define\n $$\\pi^n_k(\\tau)(i)=\\omega, \\quad\n \\pi^n_k(\\tau)(j)=\\tau(j) \\ \\ \\ \\mbox{for}\\ j\\not= i.$$\n \\end{itemize} \n This definition clearly depends on the choice of the index\n $i$ above. Note, however, that we may pick the functions\n $\\pi^n_k$ so that they are coherent, in the sense that for\n $\\tau\\in\\oplus^{n+1}$, unless $\\tau(n)$ is the biggest\n finite value of $\\tau$, we have\n $\\pi^{n+1}_{k+1}(\\tau)=\\pi^n_k(\\tau\\!\\!\\upharpoonright\\!\\!\n n)^\\smallfrown\\tau(n)$. In particular\n $\\pi^{n+1}_{k+1}(\\sigma^\\smallfrown\\omega)=\\pi^n_k(\\sigma)^\\smallfrown\\omega$\n for any $\\sigma\\in\\oplus^n$. The functions $\\pi^n_k$ will\n be called projections.\n\n \\begin{lemma}\n For each $n$ and $1\\leq k\\leq n$ the projection\n $\\pi^n_k: S^n_k\\rightarrow S^n_{k-1}$ is continuous.\n \\end{lemma}\n \\begin{proof}\n Note that any point in $S^n_k$ except\n $(\\omega,\\ldots,\\omega)$ ($k$ times $\\omega$) has a\n neighborhood in which projection is unambigous and hence\n continuous. But it is easy to see that at the point\n $(\\omega,\\ldots,\\omega)$ any projection is continuous.\n \\end{proof}\n\n For each $n<\\omega$ let us also introduce the function\n $r_n:\\oplus^n\\rightarrow\\oplus^{n}$ defined as\n $r_n(\\tau^\\smallfrown a)=\\tau^\\smallfrown\\omega$.\n\n To make the above notation more readable we will usually\n drop subscripts and superscripts in $\\pi^n_k$ and $r_n$.\n\n We pick a well-ordering $\\leq$ of $(\\omega+1)^{<\\omega}$ into type $\\omega$\n such that for each point $\\tau\\in(\\omega+1)^{<\\omega}$ all elements of the\n transitive closure of $\\tau$ with respect to $\\pi$, $r$\n and restrictions (i.e. functions of the form\n $\\oplus^n\\ni\\tau\\mapsto\\tau\\!\\!\\upharpoonright\\!\\! m\\in\\oplus^m$ for $m\\tau$, as can be seen in\n the first condition above. The last condition, as we will\n see later, will be used to guarantee ``continuity'' of the\n family of sets $X_\\tau$. For technical reasons we will\n also make sure that $X^n_\\sigma=(X^n_\\sigma)^*$.\n\n We are going to ensure disjointness of $C_\\tau$'s by\n satisfying the following conditions:\n \\begin{itemize}\n \\item $C_{\\tau^\\smallfrown a}\\subseteq C_\\tau$,\n \\item $C_{\\tau^\\smallfrown a}\\cap C_{\\tau^\\smallfrown\n b}=\\emptyset$ for $a\\not=b$.\n \\end{itemize}\n\n The fact that $\\mbox{diam}(X_\\tau)<1\\slash|\\tau|$ will follow\n from the following inductive conditions (recall that\n $\\pi(\\tau)\\leq\\tau$ for any $\\tau$):\n \\begin{itemize}\n \\item $\\mbox{diam}(X_\\tau)< 3\\,\\mbox{diam}(X_{\\pi(\\tau)})$,\n \\item $\\mbox{diam}(X_{\\tau^\\smallfrown\\omega}) <\n 1\\slash(3^{|\\tau|+1}(|\\tau|+1))$,\n \\end{itemize}\n because iterating projections in $\\oplus^n$ stabilizes\n before $n+1$ steps.\n\n The crucial feature of the sets $X_\\tau$ is that this\n family should be ``continuous''. Namely, we will require\n that if $\\tau$ and $\\pi(\\tau)$ occur by the $n$-th step\n then\n \\begin{equation}\\label{ineq}\n h(X^n_\\tau,X^n_{\\pi(\\tau)})<3^{|\\tau|}\\,d(\\tau,\\pi(\\tau))\n \\end{equation}\n\n This condition is the most diffucult. To fulfill it we\n will construct yet another kind of objects. Notice first\n that if $h(A,B)<\\varepsilon$ for two nonempty sets in $X$\n then there are two finite families (we will refer to them\n as to ``anchors'') $A_i$ and $B_i$ ($i\\in I_0$) of subsets\n of $A$ and $B$ respectively such that for any\n $A_i'\\subseteq A_i$, $B_i'\\subseteq B_i$ still\n $h(\\bigcup_i A_i',\\bigcup_i B_i')<\\varepsilon$. Similarly,\n if $h(A,B)<\\varepsilon$ and $C\\subseteq A$ then there\n exist a finite family $D_i$ ($i\\in I_0$) of subsets of $B$\n such that for any $D_i'\\subseteq D_i$ $h(\\bigcup_i D_i',\n C)<\\varepsilon$.\n\n At each step $n$ if $\\tau$ is the $n$-the element of $(\\omega+1)^{<\\omega}$\n we will additionally construct anchors \n \\begin{itemize}\n \\item for each pair $X^n_\\sigma$ and $X^n_{\\pi(\\sigma)}$\n such that $\\sigma,\\pi(\\sigma)\\leq\\tau$\n \\item and for each tripple\n $X^n_\\sigma,X^n_{\\pi(\\sigma)},X^n_{\\sigma^\\smallfrown\n a}$ such that $a\\in\\omega+1$, $\\pi(\\sigma^\\smallfrown\n a)\\subseteq\\pi(\\sigma)$ and\n $\\sigma,\\pi(\\sigma),\\sigma^\\smallfrown a\\leq\\tau$.\n \\end{itemize}\n \n\n \\subsection*{Completing the diagram.}\n\n As we now have a clear picture of what should be constructed\n let us argue that this is enough to finish the proof. For\n each $t\\in(\\omega+1)^\\omega$ the intersection $\\bigcap_n\n X_{t\\upharpoonright n}$ has precisely one point so let us\n define $\\varphi(t)$ to be this point. The other function,\n $\\psi$ is defined as $f\\circ\\varphi\\circ P^{-1}$. Let us\n check that this works. Both functions $\\psi$ and $\\varphi$\n are injective thanks to the disjoitness of the sets\n $C_\\tau$ and to the fact that $X_\\tau\\subseteq\n f^{-1}[C_\\tau]$. The function $\\psi$ is open because\n $C_\\tau$ are clopens.\n\n To see continuity of $\\varphi$ notice first that since the\n sets $X_\\tau$ have diameters vanishing to $0$, it suffices\n to check that $\\varphi$ is continuous on each $\\oplus^n$\n (which are treated as subsets of $(\\omega+1)^\\omega$ via the embedding\n $e: \\tau\\mapsto\\tau^\\smallfrown(\\omega,\\omega,\\ldots)$).\n Continuity on $\\oplus^n$ is checked inductively on the\n sets $S^n_k$ for $0\\leq k\\leq n$.\n\n The set $S^n_0$ consists of one point, so there is nothing\n to check. Suppose that $\\tau_i\\rightarrow \\tau$,\n $\\tau,\\tau_i\\in S^n_k,i\\in\\omega$. Then either the\n sequence is eventually constant or $\\tau\\in S^n_{k-1}$.\n Let us assume the latter. By the inductive assumption and\n continuity of projection\n $\\varphi(\\pi(\\tau_i))\\rightarrow\\varphi(\\tau)$. Now pick\n any $\\varepsilon>0$. Let $m$ be such that\n $\\mbox{diam}(X_\\sigma)<\\varepsilon$ for $\\sigma\\in\\oplus^m$ and\n $j\\in\\omega$ such that\n $d(\\tau_j,\\pi(\\tau_j))<3^{-m}\\varepsilon$. Let us write\n $\\rho^\\smallfrown\\omega^l$ for $\\rho$ extended by $l$ many\n $\\omega$'s. By (\\ref{ineq}) and coherence of projections\n we have\n $$h(X_{{\\tau_j}^\\smallfrown\\omega^{m-n}},X_{\\pi(\\tau_j)^\\smallfrown\\omega^{m-n}})<\\varepsilon,$$\n which implies that $\\varphi(\\tau_j)$ and\n $\\varphi(\\pi(\\tau_j))$ are closer than $3\\varepsilon$.\n This shows that $\\varphi(\\tau_j)\\rightarrow\\varphi(\\tau)$\n and proves continuity of $\\varphi$.\n\n \\subsection*{Key lemma.}\n\n Now we state the key lemma, which will be used to\n guarantee ``continuity'' of the family of sets $X_\\tau$.\n\n \\begin{lemma}\\label{limit}\n Let $X$ be a Borel set, $f:X\\rightarrow \\omega^\\omega$ a\n Borel, not $\\sigma$-continuous function. There exist\n a basic clopen $C_\\omega\\subseteq f[X]$\n \n and a compact set $X_\\omega\\subseteq\n f^{-1}[C_\\omega]$ such that\n \\begin{itemize}\n \\item $f\\!\\!\\upharpoonright\\!\\! X_\\omega$ is not $\\sigma$-continuous,\n \\item\n $X_\\omega\\subseteq\\mbox{cl}\\big((f^{-1}[\\omega^\\omega\\setminus\n C_\\omega])^*\\big)$.\n \\end{itemize}\n The compact set $X_\\omega$ can be chosen of arbitrarily\n small diameter.\n \\end{lemma}\n \\begin{proof} Without loss of generality assume that\n $f^{-1}[C]=(f^{-1}[C])^*$ for all clopen sets\n $C\\subseteq\\omega^\\omega$. Let us consider the\n following tree of open sets, indexed by $\\omega^{<\\omega}$\n $$U_\\tau=\\int\\big(f^{-1}[[\\tau]]\\big).$$ Let\n $G=\\bigcap_n\\bigcup_{|\\tau|=n} U_\\tau$ and\n $Z_\\tau=f^{-1}[[\\tau]]\\setminus U_\\tau$. Notice that\n $f\\!\\!\\upharpoonright\\!\\! G$ is continuous and since $X=G\\cup\\bigcup_\\tau\n Z_\\tau$ there is $\\tau\\in\\omega^{<\\omega}$ such that $Z_\\tau\\not\\in\n I_f$. Observe that\n $Z_\\tau\\subseteq\\mbox{cl}\\big(\\bigcup_{\\tau'\\not=\\tau,|\\tau'|=|\\tau|}\n f^{-1}[[\\tau']]\\big)$ because if an open set $U\\subseteq\n f^{-1}[[\\tau]]$ is disjoint from\n $\\bigcup_{\\tau'\\not=\\tau,|\\tau'|=|\\tau|}\n f^{-1}[[\\tau']]$ then $U\\subseteq U_\\tau$. Now put\n $C_\\omega=[\\tau]$ and pick any compact set with small\n diameter $X_\\omega\\subseteq Z_\\tau$ such that\n $X_\\omega\\not\\in I_f$.\n \\end{proof}\n\n \\subsection*{The construction.}\n\n We begin with $X_\\emptyset=X^0_\\emptyset=X$ and\n $C_\\emptyset=\\omega^\\omega$. Without loss of generality\n assume that $X=X^*$. Suppose we have done $n-1$ steps of\n the inductive construction up $\\tau\\in(\\omega+1)^{<\\omega}$. Let $|\\tau|=l$\n and $\\sigma=\\tau\\!\\!\\upharpoonright\\!\\!(l-1)$. There are three cases.\n\n \\textbf{Case 1.} The four points $\\tau$, $\\pi(\\tau)$,\n $r(\\tau)$ and $r(\\pi(\\tau))$ are equal. So\n $\\tau=(\\omega,\\ldots,\\omega)$ and $C_{\\tau\\upharpoonright\n n-1}$ and $X^{n-1}_\\sigma$ are already constructed. In\n this case we use Lemma \\ref{limit} to find a clopen set\n $C_\\tau$ and a compact set $X_\\tau\\subseteq\n X^{n-1}_\\sigma$ of diameter $<|\\tau|\\slash 3^{n+1}$ small\n enough so that no element of the anchors constructed so\n far is contained in $X_\\tau$. We put $X^n_\\tau=X_\\tau^*$,\n $X^n_\\sigma=(X^{n-1}_\\sigma\\setminus f^{-1}[C_\\tau])^*$\n and $X^n_\\rho=X^{n-1}_\\rho$ for other $\\rho<\\tau$. By the\n assertion of Lemma \\ref{limit} we still have\n $X^n_\\tau\\subseteq\\mbox{cl}(X^n_\\sigma)$. In this case we do not\n need to construct any new anchors.\n\n \\textbf{Case 2.} The two points $\\pi(\\tau)$ and $r(\\tau)$\n are equal but distinct from $\\tau$. Let\n $\\delta=d(\\tau,r(\\tau))$. Since\n $X^{n-1}_{r(\\tau)}\\subseteq \\mbox{cl} (X^{n-1}_\\sigma)$ by the\n inductive assumption, we may find finitely many sets\n $B_i\\subseteq X^{n-1}_\\sigma,i\\leq k$ such that\n \\begin{itemize}\n \\item $h(\\bigcup_i B'_i, X_{r(\\tau)})<\\delta$ for any\n $B'_i\\subseteq B_i$,\n \\item $B_i\\not\\in I_f$.\n \\end{itemize}\n The second condition follows from\n $X^{n-1}_\\sigma=(X^{n-1}_\\sigma)^*$. We may assume that\n for each clopen set $C\\subseteq 2^\\omega$ the set $B_i\\cap\n f^{-1}[C]$ is either empty or outside of the ideal $I_f$.\n\n We are going to find clopens $C_i\\subseteq C_\\sigma$, for\n $i\\leq k$ such that $C_i\\cap C_{r(\\tau)}=\\emptyset$ and\n then put $C_\\tau=\\bigcup_{i\\leq k} C_i$,\n $X^n_\\sigma=(X^{n-1}_\\sigma\\setminus\\bigcup_i\n f^{-1}[C_i])^*$ and find $X_\\tau\\subseteq\\bigcup_{i\\leq k}\n B_i\\cap f^{-1}[C_i]$. We will have to carefully define\n $X^n_{r(\\tau)}$ so that $X^n_{r(\\tau)}\\subseteq\n \\mbox{cl}(X^n_\\sigma)$.\n\n It is easy to see that for any $A\\subseteq X^{n-1}_\\sigma$\n $$X^{n-1}_{r(\\tau)}=X^{n-1}_{r(\\tau)}\\cap\\mbox{cl}\\big((X^{n-1}_\\sigma\\cap A)^*\\big)\\ \\cup\\\n X^{n-1}_{r(\\tau)}\\cap\\mbox{cl}\\big((X^{n-1}_\\sigma\\cap\n A^{c})^*\\big)$$ so (putting\n $A=f^{-1}[C_\\sigma\\cap[(m,0)]]$ for $m<\\omega$) we may\n inductively on $m$ pick binary sequences $\\beta^m_i\\in\n 2^m,i\\leq k$ such that $f^{-1}[[\\beta^m_i]]\\cap\n B_i\\not=\\emptyset$ and\n \\begin{displaymath}\n X^{n-1}_{r(\\tau)}\\cap\\mbox{cl}\\big((X^{n-1}_\\sigma\\setminus f^{-1}[\\bigcup_{i\\leq k} [\\beta^m_i]])^*\\big)\\not\\in I_f.\n \\end{displaymath}\n\n We are going to carry on this construction up to some\n $m<\\omega$ and put\n $X^n_\\rho=\\big(X^{n-1}_\\rho\\setminus\\bigcup_{i\\leq k}\n f^{-1}[[\\beta^m_i]]\\big)^*$ for\n $\\rho<\\tau,\\rho\\not\\supseteq r(\\tau)$ and\n $X^n_{\\rho}=\\big(X^{n-1}_{\\rho}\\cap\\mbox{cl}(X^n_\\sigma)\\big)^*$\n for $\\rho<\\tau,\\rho\\supseteq r(\\tau)$. We must, however,\n take care that this does not destroy the existing anchors.\n\n Since $f^{-1}[\\{x\\}]\\in I_f$ for any $x\\in 2^\\omega$ and\n there are only finitely many elements of the existing\n anchors, we may pick $m<\\omega$ and construct the\n sequences $\\beta^m_i$ so that for any element $A$ of an\n anchor ``below'' $X^{n-1}_{r(\\tau)}$ it is the case that\n $A\\cap\\mbox{cl}(f^{-1}[C_\\sigma\\setminus\\bigcup_{i\\leq k}\n [\\beta^m_i]]\\cap X^{n-1}_\\sigma)\\not\\in I_f$ and for any\n element $A$ of other anchors\n $A\\setminus\\big(\\bigcup_{i\\leq k}\n f^{-1}[\\beta^m_i]\\big)\\not\\in I_f$.\n\n Once we have constructed the sequences $\\beta^m_i$ for\n $i\\leq k$ we put $C_i=[\\beta^m_i]$ and $C_\\tau=\\bigcup_i\n [\\beta^m_i]$. Next we find $I_f$-positive compact sets\n $X_i$ inside $B_i\\cap f^{-1}[[\\beta_i]]$, each of diameter\n $<1\\slash(3^{n+1}|\\tau|)$.\n\n If $\\delta>\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})$ then we can pick\n one $X_i$ as $X_\\tau$ and then\n $h(X_\\tau,X^{n-1}_{\\pi(\\tau)})\\leq 3\\,\n h(X^{n-1}_\\sigma,X^{n-1}_{\\pi(\\sigma)})<3^{|\\tau|}\\,\\delta$.\n Otherwise, let $X_\\tau=\\bigcup_{i\\leq k}X_i$ and then\n $\\mbox{diam}(X_\\tau)<3\\,\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})\\leq\n 3\\,\\mbox{diam}(X_{\\pi(\\tau)})$. Define $X^n_\\tau=X_\\tau^*$.\n\n At this step we create anchors for the pair $X^n_\\tau$ and\n $X^n_{r(\\tau)}$ as well as for the tripples $X^n_\\sigma$,\n $X^n_\\rho$, $X^n_\\tau$ for $\\rho<\\tau$.\n\n \\textbf{Case 3.} The two points $\\pi(\\tau)$, $r(\\tau)$ are\n distinct. Let $\\delta=d(\\tau,\\pi(\\tau))$. By coherence of\n the projections $\\pi(\\tau)\\supseteq\\pi(\\sigma)$. By the\n inductive assumption we have\n $h(X^{n-1}_\\sigma,X^{n-1}_{\\pi(\\sigma)})<3^{|\\sigma|}\\,\\delta$.\n Using the existing anchor for the tripple\n $X^{n-1}_\\sigma$, $X^{n-1}_{\\pi(\\sigma)}$,\n $X^{n-1}_{\\pi(\\tau)}$ let us find finitely many sets\n $B_i,i\\leq k$ in $X_\\sigma$ such that\n \\begin{itemize}\n \\item $h(\\bigcup_i B'_i,\n X_{\\pi(\\tau)})<3^{|\\sigma|}\\,\\delta$ for any\n $B'_i\\subseteq B_i$,\n \\item $B_i\\not\\in I_f$.\n \\end{itemize}\n As before, we assume assume that for each clopen set\n $C\\subseteq 2^\\omega$ if $B_i\\cap f^{-1}[C]\\in I_f$ then\n it is empty. We have now two subcases, in analogy to the\n two previous cases.\n\n \\textbf{Subcase 3.1.} Suppose $\\tau=r(\\tau)$. Similarly as\n in Case 1, we use Lemma \\ref{limit} to find $X_i\\subseteq\n B_i$ and $C_i$ for $i\\leq k$. Put $C_\\tau=\\bigcup_{i\\leq\n k} C_i$. If $\\delta>\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})$ then we\n can pick one $X_i$ as $X_\\tau$ and then\n $h(X_\\tau,X^{n-1}_{\\pi(\\tau)})\\leq 3\\,\n h(X^{n-1}_\\sigma,X^{n-1}_{\\pi(\\sigma)})<3^{|\\tau|}\\,\\delta$.\n Otherwise, let $X_\\tau=\\bigcup_{i\\leq k}X_i$ and then\n $\\mbox{diam}(X_\\tau)<3\\,\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})\\leq\n 3\\,\\mbox{diam}(X_{\\pi(\\tau)})$. Again, similarly as in Case 1,\n we put $X^n_\\tau=(X^n_\\tau)^*$,\n $X^n_\\sigma=(X^{n-1}_\\sigma\\setminus f^{-1}[C_\\tau])^*$,\n $X^n_\\rho=X^{n-1}_\\rho$ for other $\\rho<\\tau$.\n\n\n \\textbf{Subcase 3.2.} Suppose $\\tau\\not=r(\\tau)$.\n Similarly as in Case 2, we find clopens $C_i$ in\n $\\omega^\\omega$ such that $X^{n-1}_{r(\\tau)}\\cap\n \\mbox{cl}\\big((f^{-1}[C_\\sigma\\setminus\\bigcup_{i\\leq\n k}C_i])^*\\big)\\not\\in I_f$ and no existing anchor is\n destroyed when we put\n $X^n_\\rho=(X^{n-1}_\\rho\\setminus\\bigcup_{i\\leq\n k}f^{-1}[C_i])^*$ for $\\rho<\\tau,\\rho\\not\\supseteq\n r(\\tau)$ and $X^n_\\rho=X^{n-1}_\\rho\\cap\\mbox{cl}(X^n_\\sigma)$\n for $\\rho<\\tau,\\rho\\supseteq r(\\tau)$.\n\n Next we find $I_f$-positive compact sets $X_i\\subseteq\n B_i\\cap f^{-1}[C_i]$ each of diameter\n $<1\\slash(3^{|\\tau|+1}|\\tau|)$. As previously, if\n $\\delta>\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})$ then we can pick one\n $X_i$ as $X_\\tau$ and then\n $h(X_\\tau,X^{n-1}_{\\pi(\\tau)})\\leq 3\\,\n h(X^{n-1}_\\sigma,X^{n-1}_{\\pi(\\sigma)})<3^{|\\tau|}\\,\\delta$.\n Otherwise, let $X_\\tau=\\bigcup_{i\\leq k}X_i$ and then\n $\\mbox{diam}(X_\\tau)<3\\,\\mbox{diam}(X^{n-1}_{\\pi(\\tau)})\\leq\n 3\\,\\mbox{diam}(X_{\\pi(\\tau)})$. Again, we put\n $X^n_\\tau=(X^n_\\tau)^*$,\n\n In Case 3 we construct the same anchors as in Case 2.\n\n This ends the construction and the entire proof.\n\\end{proof}\n\n\\begin{theorem}\n If $f:X\\rightarrow\\omega^\\omega$ is not\n $\\sigma$-continuous then there exist topological\n embeddings $\\varphi$ and $\\psi$ such that the following\n diagram commutes:\n $$\n \\begin{CD}\n \\omega^\\omega @ >\\psi >> \\omega^\\omega\\\\\n @AAP A @AAf A\\\\\n (\\omega+1)^\\omega @> \\varphi >> X\n \\end{CD}\n $$ \n\\end{theorem}\n\\begin{proof}\n By Theorem \\ref{dichotomy} we have $\\psi$ and $\\varphi$\n such that $\\psi$ is $1$-$1$ open. But as a Borel function\n it continuous on a dense $G_\\delta$ set\n $G\\subseteq\\omega^\\omega$. On the other hand by the\n properties of the function $P$ $X\\in I_P$ implies $P[X]$\n is meager. So $P^{-1}[G]\\not\\in I_P$ and the problem\n reduces to the restriction of the function $P$. This,\n however, has been proved in \\cite{MS.for} (Corollary 2).\n So we get the following diagram:\n $$\n \\begin{CD}\n \\omega^\\omega @>\\psi'>> G @>\\psi>> \\omega^\\omega\\\\\n @AAP A @AAP\\upharpoonright G A @AAf A\\\\\n (\\omega+1)^\\omega@>\\varphi'>> P^{-1}[G] @>\\varphi>> X\n \\end{CD}\n $$ \n which ends the proof.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzejns b/data_all_eng_slimpj/shuffled/split2/finalzzejns new file mode 100644 index 0000000000000000000000000000000000000000..158eda1de3fc76010d667ce9068f062444691fad --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzejns @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\n\nLarge pre-trained Transformers have yielded remarkable performance on abstractive summarization~\\cite{liu-lapata-2019-text,lewis-etal-2020-bart,zhang2020pegasus} with impeccable fluency, yet their summaries often contain factually inconsistent content~\\cite{maynez-etal-2020-faithfulness,zhang-etal-2020-optimizing,goyal-durrett-2020-evaluating}, even for state-of-the-art models.\nThree types of remedies have been proposed: running a separately learned error correction component~\\cite{dong-etal-2020-multi}, removing noisy training samples~\\cite{nan-etal-2021-entity,goyal-durrett-2021-annotating}, and modifying the Transformer architecture~\\cite{huang-etal-2020-knowledge,zhu-etal-2021-enhancing}. \nYet they either rely on heuristically created data for error handling, falling short of generalization, or require learning a large number of new parameters, and summary informativeness is often sacrificed. \n\n\n\\begin{figure}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{3pt}\n \\begin{tabular}{p{0.47\\textwidth}}\n \\toprule\n \\textbf{XSum Article}: The Fermanagh MLA Phil Flanagan tweeted after Tom Elliott appeared on a BBC radio programme in May 2014. $\\ldots$\n ``I wonder if he will reveal how many people he harassed and shot as a member of the UDR.''$\\ldots$ \\\\\n \\midrule\n \\textbf{Contrastive learning} (our method): A Sinn F\u00e9in MLA has been ordered to apologise and pay compensation to a former member of the \\hlc[yellow!70]{ Ulster Defence Regiment} (UDR). \\\\\n \\midrule\n \\textbf{Cross-entropy}: A Sinn F\u00e9in MLA has agreed to pay compensation to a former \\hlc[cyan!40]{Ulster Unionist Party} (UDR) MP after he tweeted that he had harassed and shot people as a member of the party. \\\\\n \\midrule\n \\textbf{Entailment reranking}: A Sinn F\u00e9in MLA has agreed to pay compensation to a former \\hlc[cyan!40]{Ulster Unionist Party} (UDR) councillor for a tweet he sent about him. \\\\\n \\midrule\n \\textbf{Unlikelihood}: An MLA has been ordered to apologise and pay compensation to a former \\hlc[cyan!40]{loyalist} MP for a remark he made about him while serving in \\hlc[cyan!40]{the Ministry of Defence}. \\\\\n \n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Sample article and system summaries by different methods. \n Our contrastive learning model trained on low confidence system outputs correctly generates the \\hlc[yellow!70]{full name}.\n Comparisons using cross-entropy loss,\n beam reranking by entailment scores~\\cite{kryscinski-etal-2020-evaluating}, and unlikelihood objective~\\cite{Welleck2020Neural} over negative samples all produce \\hlc[cyan!40]{unfaithful content}.\n }\n \\label{fig:intro_sample}\n\\end{figure}\n\nOur goal is to train abstractive summarization systems that generate both faithful and informative summaries in an end-to-end fashion.\nWe observe that, while the commonly used maximum likelihood training optimizes over references, there is no guarantee for the model to distinguish references from incorrect generations~\\cite{Holtzman2020The,Welleck2020Neural}. \nTherefore, potential solutions reside in designing new learning objectives that can effectively inform preferences of factual summaries over incorrect ones. \n\nConcretely, we hypothesize that including factually inconsistent summaries (i.e., \\textit{negative samples}) for training, in addition to references (i.e., \\textit{positive samples}), let models become better at differentiating these two types of summaries. \nAlthough using negative samples has been effective at text representation learning, e.g., word2vec~\\cite{mikolov2013distributed} and BERT~\\cite{devlin-etal-2019-bert}, there exist two major challenges for it to succeed in concrete language tasks. \nFirst, \\textit{a suitable training objective} is critical to avoid performance degradation~\\cite{saunshi2019theoretical}. \nSecond, it is nontrivial to \\textit{construct ``natural'' samples} that mimic the diverse errors made by state-of-the-art systems that vary in words and syntax~\\cite{goyal-durrett-2021-annotating}. \n\nTo address both challenges, we first propose a new framework, \\textbf{\\textsc{CLIFF}}, that uses \\underline{c}ontrastive \\underline{l}earning for \\underline{i}mproving \\underline{f}aithfulness and \\underline{f}actuality of the generated summaries.\\footnote{Our code and annotated data are available at \\url{https:\/\/shuyangcao.github.io\/projects\/cliff_summ}.} \nContrastive learning (CL) has obtained impressive results on many visual processing tasks, such as image classification~\\cite{khosla2020supervised,pmlr-v119-chen20j} and synthesis~\\cite{park2020contrastive,zhang2021cross}. \nIntuitively, CL improves representation learning by compacting positive samples while contrasting them with negative samples.\nHere, we design a task-specific CL formulation that teaches a summarizer to expand the margin between factually consistent summaries and their incorrect peers.\n\nMoreover, we design four types of strategies with different variants to construct negative samples by \\textit{editing reference summaries} via rewriting entity-\/relation-anchored text, and using \\textit{system generated summaries} that may contain unfaithful errors. \nImportantly, these strategies are inspired by our new annotation study on errors made by state-of-the-art summarizers---models fine-tuned from BART~\\cite{lewis-etal-2020-bart} and PEGASUS~\\cite{zhang2020pegasus}---on two benchmarks: XSum~\\cite{narayan-etal-2018-dont} and CNN\/DailyMail (CNN\/DM)~\\cite{NIPS2015_5945}. \n\nWe fine-tune pre-trained large models with our contrastive learning objective on XSum and CNN\/DM. \nResults based on QuestEval~\\cite{scialom2020QuestEval}, a QA-based factuality metric of high correlation with human judgments,\nshow that our models trained with different types of negative samples uniformly outperform strong comparisons, including using a summarizer with post error correction and reranking beams based on entailment scores to the source.\nMoreover, compared with unlikelihood training method that penalizes the same negative samples~\\cite{Welleck2020Neural}, our summaries also obtain consistently better QuestEval scores. \nHuman evaluation further confirms that our models consistently reduce both extrinsic and intrinsic errors over baseline across datasets. \n\n\\section{Related Work}\n\\label{sec:related_work}\n\n\n\\paragraph{Factuality Improvement and Evaluation.} \nNeural abstractive summaries often contain unfaithful content with regard to the source~\\cite{falke-etal-2019-ranking}. To improve summary factuality, three major types of approaches are proposed. First, a separate correction model is learned to fix errors made by the summarizers~\\cite{zhao-etal-2020-reducing,chen-etal-2021-improving}, including replacing entities absent from the source~\\cite{dong-etal-2020-multi} or revising all possible errors~\\cite{cao-etal-2020-factual}. \nThe second type targets at modifying the sequence-to-sequence architecture to incorporate relation triplets~\\cite{cao2018faithful}, knowledge graphs~\\cite{zhu-etal-2021-enhancing}, and topics~\\cite{aralikatte-etal-2021-focus} to inform the summarizers of article facts. Yet additional engineering efforts and model retraining are often needed. \nFinally, discarding noisy samples from model training has also been investigated~\\cite{nan-etal-2021-entity,goyal-durrett-2021-annotating}, however, it often leads to degraded summary informativeness. \nIn comparison, our contrastive learning framework allows the model to be end-to-end trained and does not require model modification, thus providing a general solution for learning summarization systems. \n\n\n\n\nAlongside improving factuality, we have also witnessed growing interests in automated factuality evaluation, since popular word-matching-based metrics, e.g., ROUGE, correlate poorly with human-rated factual consistency levels~\\cite{gabriel-etal-2021-go,fabbri2021summeval}. \nEntailment-based scorers are designed at summary level~\\cite{kryscinski-etal-2020-evaluating} and finer-grained dependency relation level~\\cite{goyal-durrett-2020-evaluating}. \nQA models are employed to measure content consistency by reading the articles to answer questions generated from the summaries~\\cite{wang-etal-2020-asking,durmus-etal-2020-feqa}, or considering the summaries for addressing questions derived from the source~\\cite{scialom-etal-2019-answers}.\nThough not focusing on evaluation, our work highlights that models can produce a significant amount of world knowledge which should be evaluated differently instead of as extrinsic hallucination~\\cite{maynez-etal-2020-faithfulness}. We also show that world knowledge can possibly be distinguished from errors via model behavior understanding.\n\n\n\\medskip\n\\noindent \\textbf{Training with negative samples} has been investigated in several classic NLP tasks, such as grammatical error detection~\\cite{foster-andersen-2009-generrate} and dialogue systems~\\cite{li-etal-2019-sampling}. \nNotably, negative sampling plays a key role in word representation learning~\\cite{mikolov2013distributed} and training large masked language models, such as BERT and ALBERT, to induce better contextual representations~\\cite{devlin-etal-2019-bert,Lan2020ALBERT:}. \nFor text generation tasks, unlikelihood training is proposed to penalize the generation of negative tokens (e.g., repeated words) and sentences (e.g., contradictory responses in a dialogue system)~\\cite{Welleck2020Neural,li-etal-2020-dont,he-glass-2020-negative}. \nWe use contrastive learning that drives enhanced representation learning to better distinguish between factual and incorrect summaries, which encourages more faithful summary generation. \n\n\n\\paragraph{Contrastive Learning (CL) for NLP.}\nCL has been a popular method for representation learning, especially for vision understanding~\\cite{hjelm2018learning,pmlr-v119-chen20j}. \nOnly recently has CL been used for training language models with self-supervision~\\cite{fang2020cert}, learning sentence representations~\\cite{gao2021simcse}, and improving document clustering~\\cite{zhang2021supporting}. \nWith a supervised setup, \\citet{gunel2021supervised} adopt the contrastive objective to fine-tune pre-trained models on benchmark language understanding datasets. Using a similar idea, \\citet{liu-liu-2021-simcls} enlarge the distances among summaries of different quality as measured by ROUGE scores. \n\n\\section{\\textsc{CLIFF}: Contrastive Learning Framework for Summarization}\n\\label{sec:method}\n\n\nWe design a contrastive learning (CL)-based training objective that drives the summarization model to learn a preference of faithful summaries over summaries with factual errors.\nIt is then used for fine-tuning BART~\\cite{lewis-etal-2020-bart} and PEGASUS~\\cite{zhang2020pegasus} for training summarization models.\nFormally, let an article $x$ have a set of reference summaries $P$ (henceforth \\textit{positive samples}) and another set of erroneous summaries $N$ (\\textit{negative samples}). \nThe contrastive learning objective is~\\cite{khosla2020supervised,gunel2021supervised}:\n\n\\begin{equation}\n \\fontsize{10}{11}\\selectfont\n l_{cl}^{x} = - \\frac{1}{\\binom{|P|}{2}} \\sum_{\\substack{y_i, y_j \\in P\\\\y_j \\ne y_i}}\n \\log \\frac{\\exp(\\mathrm{sim} (\\bm{h}_i, \\bm{h}_j) \/ \\tau)}{\\sum\\limits_{\\substack{y_k \\in P \\cup N\\\\y_k \\ne y_i}} \\exp(\\mathrm{sim} (\\bm{h}_i, \\bm{h}_k) \/ \\tau)}\n \\label{eq:contrast}\n\\end{equation}\nwhere $\\bm{h}_i$, $\\bm{h}_j$, and $\\bm{h}_k$ are representations for summaries $y_i$, $y_j$, and $y_k$.\n$\\mathrm{sim}(\\cdot, \\cdot)$ calculates the cosine similarity between summary representations. $\\tau$ is a temperature and is set to $1.0$.\n\n\nImportantly, summaries in $P$ and $N$ are included in the \\textit{same batch} during training, so that the model acquires better representations to differentiate correct summaries from those with errors by comparing the two types of samples, thus maximizing the probabilities of the positive samples and minimizing the likelihoods of the corresponding negative samples.\nThe \\textbf{CL objective} on the full training set, denoted as $\\mathcal{L}_{CL}$, is the sum of losses $l_{cl}^{x}$ over all samples. \n\nTo effectively employ CL in summarization, we need to address two challenges: (1) how to automatically construct both positive and negative samples, which are critical for CL efficacy~\\cite{pmlr-v119-chen20j}, and (2) how to represent the summaries (i.e., $\\bm{h}_{\\ast}$).\nBelow we describe positive sample generation and options for $\\bm{h}_{\\ast}$, leaving the strategies for negative samples to \\S~\\ref{sec:sample_construct}. \n\n\n\\paragraph{Positive Sample Construction ($P$).}\nSummarization datasets often contain a single reference for each article. To create multiple positive samples, in our pilot study,\nwe experiment with paraphrasing with synonym substitution~\\cite{ren-etal-2019-generating}, randomly replacing words based on the prediction of masked language models~\\cite{kobayashi-2018-contextual}, and back-translation~\\cite{mallinson-etal-2017-paraphrasing}. We find back-translation to be best at preserving meaning and offering language variation, and thus use NLPAug\\footnote{\\url{https:\/\/github.com\/makcedward\/nlpaug}} to translate each reference to German and back to English. Together with the reference, the best translation is kept and added to $P$, if no new named entity is introduced.\n\n\n\\paragraph{Summary Representation ($\\bm{h}_{\\ast}$).}\nWe use the outputs of the decoder's last layer, and investigate three options that average over \\textit{all tokens}, \\textit{named entity tokens}, and the \\textit{last token} of the decoded summary. Entities and other parsing results are obtained by spaCy~\\cite{spacy}. \nWe further consider adding a multi-layer perceptron (MLP) with one hidden layer to calculate the final $\\bm{h}_{\\ast}$.\n\n\\medskip\n\\noindent \\textbf{The final training objective} combines the typical cross-entropy loss $\\mathcal{L}_{CE}$ and our contrastive learning objective: $\\mathcal{L} = \\mathcal{L}_{CE} + \\lambda \\mathcal{L}_{CL}$, where $\\lambda$ is a scalar and set to $1.0$ for all experiments. \n\n\\section{Summary Error Annotation and Model Behavior Analysis}\n\\label{sec:error_annotation}\n\nWe first describe annotating unfaithfulness errors by state-of-the-arts, i.e., models fine-tuned from BART and PEGASUS\non XSum and CNN\/DM.\nWe then probe into the model generation behavior that is indicative of errors, which guides the design of negative sample construction strategies.\n\n600 ($150\\times2\\times2$) summaries are annotated to demonstrate how often do the models ``hallucinate\", i.e., generating content not grounded by the source.\nTo characterize errors, we annotate \\textit{text spans} in summaries with \n(i) \\textbf{intrinsic errors} caused by misconstructing phrases or clauses from the source; \nand (ii) \\textbf{extrinsic errors} which include words not in the source that are either unverifiable or cannot be verified by Wikipedia. \nContent not covered by the article but can be validated by Wikipedia is annotated as \\textbf{world knowledge}, and the models' behavior pattern when generating them differs from when they generate errors.\n\nTwo fluent English speakers with extensive experience in summary evaluation and error labeling are hired. For each sample, they are shown the article and two system summaries, and instructed to annotate text spans with the aforementioned errors and world knowledge. After labeling every 50 samples, the annotators discuss and resolve any disagreement.\nThe Fleiss's Kappas on XSum and CNN\/DM are $0.35$ and $0.45$.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/error_type_dist.pdf}\n \n \\caption{Percentage of samples with intrinsic and extrinsic error spans for models fine-tuned from BART and PEGASUS on XSum and CNN\/DM.}\n \\label{fig:error_type_dist}\n \n\\end{figure}\n\n\\medskip\n\\noindent \\textbf{Error statistics} are displayed in Fig.~\\ref{fig:error_type_dist}. \nExtrinsic errors dominate both datasets, especially on XSum. $58.7\\%$ of summaries by BART (and $44.0\\%$ by PEGASUS) contain at least one extrinsic error.\nNoticeably, PEGASUS is a newer model pre-trained with a larger amount of data, thus contains less errors than BART and other older models studied for error annotations by~\\newcite{maynez-etal-2020-faithfulness}. This observation also highlights the usage of our annotations for future development and evaluation of summarization models. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.46\\textwidth]{figures\/prob_both.pdf}\n \\caption{\n Probability distributions of generating the first tokens of proper nouns and numbers, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens.}\n \\label{fig:output_prob_propnum}\n\\end{figure}\n\n\\paragraph{Low confidence generation is indicative of extrinsic errors.}\nInspired by recent work that studies model prediction confidence~\\cite{liu2021tokenlevel}, we examine generation probabilities for tokens of \\textit{different part-of-speech (POS) tags}.\nFig.~\\ref{fig:output_prob_propnum} shows salient results on the generation probabilities of the first token of a proper noun or a number (with additional analysis provided in Appendix~\\ref{appendix:error_annotation}). \nAs observed, model confidence tends to be lower for the first tokens of proper nouns and numbers if they are part of spans with \\textit{extrinsic errors}.\nAlso note that world knowledge, which cannot be inferred from the source either, often has higher generation probability than extrinsic errors. \nTake this snippet generated by a fine-tuned BART as an example: \\textit{``Manchester United captain Wayne Rooney's testimonial game against Manchester City$\\ldots$''}. \\textit{``Manchester City\"} is an extrinsic error and \\textit{``Wayne\"} is produced as world knowledge. The model assigns a low probability of $0.10$ to the first token of \\textit{``Manchester City''} and a high probability of $0.92$ to token \\textit{``Wayne''}. \nThis implies that model confidence can be a useful indicator for negative sample collection. \n\n\n\\section{Negative Sample Construction}\n\\label{sec:sample_construct}\n\nHere we describe four strategies for constructing negative samples that \n\\textit{modify the references} (\\S~\\ref{subsec:swapping}-\\ref{subsec:conditional}) or use \\textit{system generated summaries} (\\ref{subsec:modelgeneration}). \n\n\\begin{table}[t]\n \n \\fontsize{8.5}{9}\\selectfont\n \\setlength{\\tabcolsep}{2pt}\n \n \\hspace{-1mm}\n \\begin{tabular}{p{76mm}}\n \\toprule\n \\textbf{\\textsc{Reference}}: A ``rare'' short-eared owl found emaciated in Flintshire is now recuperating well, the RSPCA have said. \\\\\n \\midrule \\midrule\n \\textbf{\\textsc{SwapEnt}}: Flintshire $\\rightarrow$ Bettisfield \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in \\textcolor{red!80!black}{\\textbf{Bettisfield}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{MaskEnt}}: A ``rare'' short-eared owl found emaciated in \\texttt{[MASK]} is now recuperating well, the RSPCA have said. \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in a field in \\textcolor{red!80!black}{\\textbf{South Yorkshire}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{MaskRel}}: A ``rare'' short-eared owl found \\texttt{[MASK]} in \\texttt{[MASK]} is now recuperating well, the RSPCA have said. \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found \\textcolor{red!80!black}{\\textbf{dead} in \\textbf{London}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{RegenEnt}}: A ``rare'' short-eared owl found emaciated in \\rule{6mm}{0.15mm} \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in \\textcolor{red!80!black}{\\textbf{Nottinghamshire}} \\textbf{is now at a wildlife centre to recover.} \\\\\n \\midrule\n \\textbf{\\textsc{RegenRel}}: A ``rare'' short-eared owl found \\rule{6mm}{0.15mm} \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found \\textbf{in the grounds of a former coal mine is being cared for \\textcolor{red!80!black}{by the RSPCA in Somerset.}} \\\\\n \\midrule\n \\textbf{\\textsc{SysLowCon}}: An injured \\textcolor{red!80!black}{golden} owl found in a former coal mine in \\textcolor{red!80!black}{Lancashire} is being cared for \\textcolor{red!80!black}{by the RSPCA}. \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Negative sample construction strategies (\\S~\\ref{sec:sample_construct}). For summaries edited from the reference, their differences are \\textbf{bolded}. Introduced errors are in \\textcolor{red!80!black}{\\textbf{red}}. Text before \\rule{6mm}{0.15mm} is the prefix for regeneration.\n }\n \\label{tab:construction_example}\n \n\\end{table}\n\n\n\\subsection{Entity Swap}\n\\label{subsec:swapping}\n\nEntity swap imitates intrinsic errors, as over $55\\%$ of intrinsic errors in our annotations are found to contain named entities.\nWe construct negative samples by swapping named entities in the references with other randomly selected entities of the same entity type in the source (\\textbf{\\textsc{SwapEnt}}). One sample is constructed for each entity in the reference. \nThough this idea has been studied by \\citet{kryscinski-etal-2020-evaluating}, they allow entities of different types to be used, e.g., a PERSON can be replaced by a LOCATION. \nExamples are displayed in Table~\\ref{tab:construction_example}. \n\n\\textsc{SwapEnt} has the advantage of not depending on any trained model. Yet it only introduces intrinsic errors and lacks the coverage for extrinsic errors, which is addressed by the following generation-based methods. \n\n\n\\subsection{Mask-and-fill with BART}\n\\label{subsec:unconditional}\nTo simulate extrinsic errors, we leverage large unconditional language models' capability of converting a sequence with masked tokens into a fluent and appropriate sequence. Specifically, we replace each named entity in a reference with a \\texttt{[MASK]} token and encode it with BART (without any fine-tuning). BART then fills this partially masked summary with newly generated entities (\\textbf{\\textsc{MaskEnt}}). BART is chosen since it can fill \\texttt{[MASK]} with varying number of tokens.\nFor each entity in the reference, we sample three summaries and only retain the ones containing at least one entity that is absent from both the source and the reference.\n\nUp to now, the two introduced strategies both focus on incorrect named entities. To cover more diverse extrinsic and intrinsic errors~\\cite{goyal-durrett-2020-evaluating}, we extend \\textsc{MaskEnt} to contain relations (\\textbf{\\textsc{MaskRel}}). \nWe first obtain dependency relations using Stanza~\\cite{qi-etal-2020-stanza}, with each relation denoted as $<$\\texttt{gov}, \\texttt{rel}, \\texttt{dep}$>$. \nTo incorporate more context, we consider noun phrase spans enclosing the token of \\texttt{gov} or \\texttt{dep} if it is a content word and the noun phrase contains a named entity. \nSimilar to \\textsc{MaskEnt}, three negative samples are generated by BART based on the input with both \\texttt{gov} and \\texttt{dep} spans masked in the reference. Only the samples that introduce any new dependency relation that is not contained in the source nor the reference are kept. Specifically, we consider a match of a dependency relation as the same form or synonyms of its \\texttt{gov} and and \\texttt{dep} is found in the source or the reference with the same relation. \n\n\nBoth \\textsc{MaskEnt} and \\textsc{MaskRel} can create more extrinsic errors compared to other strategies introduced in this section, since negative samples are generated without being grounded on the source articles. However, their constructed negative samples may contained drifted topics that can be easily detected by a summarization model, resulting with less efficient training signals. \n\n\n\\subsection{Source-conditioned Regeneration}\n\\label{subsec:conditional}\n\nTo ground negative sample generation with the article, we further design a regeneration strategy based on conditional generation. For each named entity in the reference, we treat the text before it as a \\textit{prompt}.\nA summarizer, e.g., fine-tuned from BART or PEGASUS, first reads in the source using the encoder, then receives the prompt as the first part of the decoder output, and finally decodes the rest of the content based on nucleus sampling~\\cite{Holtzman2020The} with a cumulative probability threshold of $0.7$. \nThe prompt and the regenerated text comprise the final negative sample. This method is denoted as \\textbf{\\textsc{RegenEnt}}. \n\nWe also extend entities to relations with expanded governor and dependent spans (\\textbf{\\textsc{RegenRel}}). Here, we consider a prompt as the text before the \\texttt{gov} or \\texttt{dep} span, whichever occurs first. \nFor both \\textsc{RegenEnt} and \\textsc{RegenRel}, three negative samples are generated for each prompt, and a sample is kept if it introduces any new entity (for \\textsc{RegenEnt}) or dependency relation (for \\textsc{RegenRel}) with regard to the source and the reference. \n\nNegative samples generated by both methods are more relevant to the article than the mask-and-fill strategy, yet they may still miss certain types of errors and differ from real model outputs, since they are modified from the reference summaries. \n\n\n\\subsection{System Generation}\n\\label{subsec:modelgeneration}\n\nMotivated by the model confidence analysis in \\S~\\ref{sec:error_annotation}, we explore using system generated summaries as negative samples. We first run fine-tuned BART or PEGASUS on the same training set to decode summaries. For each summary, we check the model confidence on the first token of each proper noun and number span. If the probability is below a threshold, we keep it as a negative sample (\\textbf{\\textsc{SysLowCon}}). The threshold is tuned by maximizing F1 based on our error annotations. \n\nWe consider all beams at the last decoding step as candidates. We use beam sizes of $6$ and $4$ for XSum and CNN\/DM.\nStatistics of negative samples constructed by different strategies are in Appendix~\\ref{appendix:dataset_stat}.\n\n\n\\section{Experiment Setup}\n\\label{sec:exp_setup}\n\n\\paragraph{Evaluation Metrics.} \nQuestEval~\\cite{scialom2020QuestEval} is used as the main metric to evaluate summaries' factual consistency. \nGiven an article and a summary, QuestEval first generates natural language questions for entities and nouns from both. \nA QA model then consumes the article to answer questions derived from the summary, producing a score. Another score is obtained from a QA model addressing article-based questions after reading the summary.\nThe final QuestEval score is the harmonic mean of the two. We use the version with learned weights for questions, which has shown high correlation with human judged consistency and relevance.\n\n\nWe further use FactCC~\\cite{kryscinski-etal-2020-evaluating}, trained based on their negative sample construction method, to measure if the summary can be entailed by the source.\nWe also report ROUGE-L~\\cite{lin-2004-rouge}. \nBoth FactCC and ROUGE-L reasonably correlate with summary factuality as judged by human~\\cite{pagnoni2021understanding}. \n\nBased on our error annotations, we report the correlations between each metric and the error rate---percentage of tokens being part of an error span, and the raw number of errors (Table~\\ref{tab:metric_correlation}).\nQuestEval correlates better on both aspects than other metrics. \n\n\n\\begin{table}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{4.5pt}\n \\begin{tabular}{lcccc}\n \\toprule\n \\textbf{Metric} & \\multicolumn{2}{c}{\\textbf{XSum}} & \\multicolumn{2}{c}{\\textbf{CNN\/DM}} \\\\\n & \\textbf{\\% of Err} & \\textbf{\\# of Err} & \\textbf{\\% of Err} & \\textbf{\\# of Err} \\\\\n \\midrule\n QuestEval & \\textbf{-0.43}\\rlap{$^\\ast$} & \\textbf{-0.25}\\rlap{$^\\ast$} & \\textbf{-0.33}\\rlap{$^\\ast$} & \\textbf{-0.29}\\rlap{$^\\ast$} \\\\\n FactCC & -0.02 & -0.15\\rlap{$^\\ast$} & -0.13\\rlap{$^\\ast$} & -0.12\\rlap{$^\\ast$} \\\\\n ROUGE-1 & -0.16\\rlap{$^\\ast$} & -0.02 & -0.03 & -0.06 \\\\\n ROUGE-2 & -0.11\\rlap{$^\\ast$} & -0.05 & -0.02 & -0.04 \\\\\n ROUGE-L & -0.13\\rlap{$^\\ast$} & -0.03 & -0.06 & -0.08 \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Pearson correlation between metrics and error rates and numbers of errors. \n $\\ast$: p-value $< 0.05$.\n %\n }\n \\label{tab:metric_correlation}\n \n\\end{table}\n\n\\paragraph{Comparisons.}\nIn addition to the models fine-tuned with cross-entropy loss (\\textbf{\\textsc{CrsEntropy}}), we consider reranking beams based on FactCC score (also our metric) at the last decoding step (\\textbf{\\textsc{EntailRank}}). \nWe also include three common methods of improving factuality: \n(1) (\\textbf{\\textsc{Correction}}) fine-tunes BART to fix summary errors as a separate step~\\cite{cao-etal-2020-factual}. \n(2) \\textbf{\\textsc{SubsetFT}} fine-tunes large models based on training samples without any dependency relation error~\\cite{goyal-durrett-2021-annotating}, with released checkpoint only available for XSum. \n(3) \\textbf{\\textsc{FASum}} modifies Transformer by incorporating knowledge graphs for factual consistency~\\cite{zhu-etal-2021-enhancing}, with model outputs only on CNN\/DM. \n\n\nMoreover, we compare with \\textbf{unlikelihood training} that penalizes the probabilities of all tokens in a negative sample~\\cite{li-etal-2020-dont}. Given a negative sample $y'$, the loss is defined as $-\\sum_{t=1}^{| y' |} \\log (1 - p(y'_t | y'_{1:t-1}, x))$, where $p(y'_t | y'_{1:t-1}, x)$ is the output probability at the $t$-th step. We combine the unlikelihood training objective with cross-entropy loss with equal weights for fine-tuning. \n\n\nLastly, we compare our negative sample strategies with negative samples constructed for training the FactCC scorer, denoted as \\textbf{\\textsc{FCSample}}. \nFor CL only, we compare with using other samples in the same batch as negative samples (\\textbf{\\textsc{Batch}}), a common practice for CL-based representation learning~\\cite{gao2021simcse, zhang2021supporting}.\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Automatic Evaluation}\n\\label{subsec:autoeval}\nWe report results by models fine-tuned from BART and PEGASUS with different objectives and negative samples on XSum and CNN\/DM in Tables~\\ref{tab:main_result} and~\\ref{tab:pegasus_result}. \n\\textsc{CLIFF} models use a summary representation of averaging over all tokens with MLP projection, with other variants discussed in \\S~\\ref{subsec:ablaton}. Unless explicitly stated, comparison models are fine-tuned from the same large model used by \\textsc{CLIFF}. \n\n\nFirst, comparing with other factuality improvement models (top of the tables), \\textit{almost all \\textsc{CLIFF} models trained with different negative samples uniformly produce higher QuestEval scores across datasets with both large models}, with the improvements more pronounced on XSum.\nImportantly, ROUGE scores for \\textsc{CLIFF} models are comparable or better than baselines trained with cross-entropy, e.g., on CNN\/DM as in Table~\\ref{tab:main_result}. \nA similar trend is observed with the FactCC metric, especially when using PEGASUS as the base model (Table~\\ref{tab:pegasus_result}). \nNote that \\textsc{EntailRank} tends to yield significantly higher FactCC scores, though it obtains lower QuestEval scores than the cross-entropy baseline. Human inspection finds that \\textsc{EntailRank} can pick up beams with peculiar words of high FactCC scores, without improving factuality. \nMoreover, other comparisons based on post \\textsc{Correction} and model engineering (\\textsc{FASum}) only offer incremental gains. \nThe sample selection-based method, \\textsc{SubsetFT}, sacrifices ROUGE scores significantly. \nOverall, \\textsc{CLIFF} demonstrates stronger generalizability.\n\n\n\\begin{table}[!t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{2.5pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n \\textbf{Model} & \\multicolumn{3}{c}{\\textbf{XSum}} & \\multicolumn{3}{c}{\\textbf{CNN\/DM}} \\\\\n \\cmidrule(lr){2-4} \\cmidrule(lr){5-7}\n & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} \\\\\n \\midrule\n \\multicolumn{7}{l}{\\textit{Comparisons without Negative Samples}}\\\\\n \\textsc{CrsEntropy} & 32.50 & 25.48 & \\textbf{39.07} & 50.21 & 44.44 & 40.39 \\\\\n \\textsc{EntailRank} & 32.42 & \\textbf{41.90} & 38.47 & 50.15 & \\textbf{61.04} & \\textbf{40.67} \\\\\n \\textsc{Correction} & 32.55 & 25.15 & 39.02 & 49.48 & 32.96 & 39.79 \\\\\n \\midrule\n \\midrule\n \\multicolumn{7}{l}{\\textit{Comparisons with Unlikelihood Training}} \\\\\n \\textsc{FCSample} & 32.79 & \\hlc[xgreen]{25.37} & 38.46 & 50.63 & 45.45 & 39.28 \\\\\n \n \\textsc{SwapEnt} & 32.88 & 24.76 & 37.91 & 50.43 & 43.02 & \\hlc[xgreen]{38.96} \\\\\n \\textsc{MaskEnt} & 33.04 & \\hlc[xgreen]{26.30} & 37.51 & 51.11 & 52.19 & \\hlc[xgreen]{39.34} \\\\\n \\textsc{MaskRel} & \\hlc[xgreen]{33.08} & 24.38 & 38.05 & 51.14 & 52.93 & 39.31 \\\\\n \\textsc{RegenEnt} & 32.89 & 24.46 & \\hlc[xgreen]{38.47} & \\hlc[xgreen]{51.11} & \\hlc[xgreen]{52.90} & 39.23 \\\\\n \\textsc{RegenRel} & 32.91 & 24.80 & \\hlc[xgreen]{38.46} & 51.07 & \\hlc[xgreen]{53.68} & \\hlc[xgreen]{39.45} \\\\\n \\textsc{SysLowCon} & 31.66 & \\hlc[xgreen]{26.06} & 34.03 & \\hlc[xgreen]{50.92} & 51.08 & 39.19 \\\\\n \\midrule\n \\multicolumn{7}{l}{\\textit{Our Method:} \\textsc{CLIFF}} \\\\\n \\textsc{Batch} & 32.64 & 24.96 & 38.42 & 51.03\\rlap{$^\\ast$} & 51.81\\rlap{$^\\ast$} & 39.38 \\\\\n \\hdashline\n \\textsc{FCSample} & \\hlc[xgreen]{32.96}\\rlap{$^\\ast$} & 25.28 & \\hlc[xgreen]{38.58} & \\hlc[xgreen]{51.00}\\rlap{$^\\ast$} & \\hlc[xgreen]{51.80}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.37} \\\\\n \\textsc{SwapEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & \\hlc[xgreen]{25.09} & \\hlc[xgreen]{38.58} & \\hlc[xgreen]{51.16}\\rlap{$^\\ast$} & \\hlc[xgreen]{52.97}\\rlap{$^\\ast$} & 38.95 \\\\\n \\textsc{MaskEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & 25.75 & \\hlc[xgreen]{38.12} & \\hlc[xgreen]{51.13}\\rlap{$^\\ast$} & \\hlc[xgreen]{53.60}\\rlap{$^\\ast$} & 39.24 \\\\\n \\textsc{MaskRel} & 33.06\\rlap{$^\\ast$} & \\hlc[xgreen]{25.28} & \\hlc[xgreen]{38.37} & \\hlc[xgreen]{\\textbf{51.17}}\\rlap{$^\\ast$} & \\hlc[xgreen]{53.34}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.36} \\\\\n \\textsc{RegenEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & \\hlc[xgreen]{24.48} & 38.33 & 50.99\\rlap{$^\\ast$} & 52.18\\rlap{$^\\ast$} & \\hlc[xgreen]{39.28} \\\\\n \\textsc{RegenRel} & \\hlc[xgreen]{33.16}\\rlap{$^\\ast$} & \\hlc[xgreen]{24.82} & 38.30 & \\hlc[xgreen]{51.16}\\rlap{$^\\ast$} & 53.21\\rlap{$^\\ast$} & 39.25 \\\\\n \\textsc{SysLowCon} & \\hlc[xgreen]{\\textbf{33.21}}\\rlap{$^\\ast$} & 25.18 & \\hlc[xgreen]{38.18} & 50.85\\rlap{$^\\ast$} & \\hlc[xgreen]{53.73}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.30} \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Results of models fine-tuned from PEGASUS on XSum and CNN\/DM. We report results on $5,000$ randomly selected samples on CNN\/DM, due to long running time of QuestEval. \n \n \n For models of unlikelihood training and CLIFF that use the same negative samples, the better of the two is highlighted with \\hlc[xgreen]{green}. \n $\\ast$: our model is significantly better than \\textsc{CrsEntropy} (approximation randomization test, $p < 0.005$).}\n \\label{tab:pegasus_result}\n\\end{table}\n\n\nSecond, \\textit{\\textsc{CLIFF} is more effective and robust than unlikelihood training} with the same negative samples. \nAccording to Table~\\ref{tab:main_result}, using 7 negative sample construction strategies on two datasets, \\textsc{CLIFF} obtains higher QuestEval scores than unlikelihood training in 12 out of the 14 comparisons. Using PEGASUS, \\textsc{CLIFF} also outperforms in 11 setups as listed in Table~\\ref{tab:pegasus_result}. \nSimilar trends are found on FactCC and ROUGE-L.\nAnother noteworthy piece is that \\textsc{CLIFF}'s improvements over the cross-entropy baseline are more consistent, whereas unlikelihood training occasionally hurts factuality or ROUGE scores significantly. \nWe believe the key advantage of \\textsc{CLIFF} resides in its measure of representation similarities between positive and negative samples in the same batch, allowing models to better differentiate between correct and erroneous summaries. \n\n\nFinally, among all variants, \\textit{\\textsc{CLIFF} trained with low confidence summaries as negative samples obtains the best QuestEval scores on the more abstractive dataset}. As seen in Table~\\ref{tab:main_result}, using low confidence summaries also improves FactCC scores on both datasets, and enhances ROUGE-L on the more extractive dataset CNN\/DM. \nThis indicates that \\textit{system generated summaries contribute more diverse errors made by existing models organically}, which are particularly suitable for our CL framework. As we use summaries generated by the same model for \\textsc{CLIFF} training, one future direction is to use outputs by different models. \nFor our mask-and-fill and source-conditioned regeneration strategies, we find that relation-anchored construction often beats their entity-anchored counterparts. This calls for efforts that steer the entity-driven methods to a more relation-focused direction. \n\n\n\\paragraph{Combining Strategies.}\nWe further show results by fine-tuning BARTs using samples based on combined negative sample construction strategies in Table~\\ref{tab:combine_strategy}. As can be seen, \\textit{combining \\textsc{SysLowCon} and other strategies yields better QuestEval scores than models trained with negative samples by any single strategy}, except for \\textsc{MaskEnt} and \\textsc{RegenEnt} on XSum. This signifies the importance of covering diverse types of errors in negative samples. \n\n\n\\begin{table}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{2.5pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n \\textbf{Strategy} & \\multicolumn{3}{c}{\\textbf{XSum}} & \\multicolumn{3}{c}{\\textbf{CNN\/DM}} \\\\\n \\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \n & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} \\\\\n \\midrule\n \\textsc{SysLowCon} & 33.35 & 25.47 & \\textbf{36.19} & 51.05 & 50.05 & \\textbf{41.01} \\\\\n + \\textsc{SwapEnt} & \\textbf{33.40} & \\textbf{25.50} & 35.50 & \\textbf{51.32} & \\textbf{53.95} & 40.57 \\\\\n + \\textsc{MaskEnt} & 33.21 & 25.47 & 35.91 & 51.16 & 51.90 & 40.66 \\\\\n + \\textsc{MaskRel} & 33.39 & 25.20 & 35.70 & 51.24 & 52.48 & 40.80 \\\\\n + \\textsc{RegenEnt} & 33.31 & 25.07 & 35.94 & 51.21 & 51.86 & 40.91 \\\\\n + \\textsc{RegenRel} & 33.38 & 24.97 & 36.03 & 51.13 & 50.85 & 40.97 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Results of fine-tuned BART with combinations of negative sample construction strategies.}\n \\label{tab:combine_strategy}\n\\end{table}\n\n\n\\subsection{Human Evaluation}\n\\label{subsec:humaneval}\n\n\\begin{table}[t]\n \\begin{subtable}[h]{0.48\\textwidth}\n \\small\n \\setlength{\\tabcolsep}{2pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n & \\multicolumn{3}{c}{\\textbf{Inform.}} & \\multicolumn{3}{c}{\\textbf{Factual.}} \\\\\n \\textbf{Model} & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ \\\\\n \\midrule\n \\textsc{EntailRank} & 3.3 & 84.3 & \\textbf{12.3} & 23.7 & 71.7 & \\textbf{4.7} \\\\\n \\textsc{ULL. MaskEnt} & 6.3 & 80.7 & 13.0 & 26.3 & 62.0 & 11.7 \\\\\n \\textsc{CL. Batch} & 5.3 & 80.0 & 14.7 & 21.7 & 68.0 & 10.3 \\\\\n \\textsc{CL. SysLowCon} & \\textbf{8.7} & 78.3 & 13.0 & \\textbf{31.3} & 61.7 & 7.0 \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-1mm}\n \\caption{XSum}\n \n \\end{subtable}\n \n \\begin{subtable}[h]{0.48\\textwidth}\n \\small\n \\setlength{\\tabcolsep}{2pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n & \\multicolumn{3}{c}{\\textbf{Inform.}} & \\multicolumn{3}{c}{\\textbf{Factual.}} \\\\\n \\textbf{Model} & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ \\\\\n \\midrule\n \\textsc{EntailRank} & 2.3 & 86.3 & 11.3 & 4.7 & 94.7 & \\textbf{0.7} \\\\\n \\textsc{ULL. MaskEnt} & \\textbf{18.0} & 71.0 & 11.0 & 17.3 & 79.7 & 3.0 \\\\\n \\textsc{CL. Batch} & 17.3 & 74.7 & \\textbf{8.0} & \\textbf{20.7} & 77.0 & 2.3 \\\\\n \\textsc{CL. SysLowCon} & 15.7 & 75.7 & 8.7 & 20.0 & 77.7 & 2.3 \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-1mm}\n \\caption{CNN\/DM}\n \\end{subtable}\n \n \\caption{\n Percentages of summaries that are better than, tied with, or worse than \\textsc{CrsEntropy}, in informativeness (Inform.) and factual consistency (Factual.) \n \n The Krippendorff's $\\alpha$s are $0.33$ and $0.62$ for the two aspects on XSum, and $0.34$ and $0.89$ on CNN\/DM. Our CL method using low confidence summaries is more frequently rated as better for informativeness and factuality on the more abstractive dataset XSum.\n }\n \\label{tab:human_eval}\n \n\\end{table}\n\n\n\\paragraph{Pairwise Comparison with Cross-entropy.} \nWe recruit the two human annotators for our summary error study, as well as another experienced annotator, to evaluate summary \\textbf{informativeness} and \\textbf{factual consistency}. For each article, the judges are shown summaries generated by the \\textsc{CrsEntropy} model and four other systems. They then rate each system summary against the \\textsc{CrsEntropy} summary.\nAll four summaries generated by different factuality-improved models are shown in random order without system names shown, ensuring the fair comparison among them.\n\nWe randomly pick $100$ articles from each dataset used in our error analysis study in \\S~\\ref{sec:error_annotation}, and evaluate summaries generated by \\textsc{EntailRank}, unlikelihood training (\\textsc{ULL}) with negative samples constructed by \\textsc{MaskEnt}, and \\textsc{CLIFF} models trained with \\textsc{Batch} and \\textsc{SysLowCon} negative samples. All are fine-tuned from BART. Detailed evaluation guidelines are in Appendix~\\ref{appendix:human_eval}. \n\n\nTable~\\ref{tab:human_eval} shows that on the more abstractive XSum data \\textit{CL trained with low confidence samples are more frequently rated as being more informative and more factual} than \\textsc{CrsEntropy} summaries. This echos our automatic evaluations with QuestEval in \\S~\\ref{subsec:autoeval}.\nOn CNN\/DM, all models trained with negative samples produce summaries with better informativeness and faithfulness.\nIn contrast, \\textsc{EntailRank} summaries are less distinguishable from outputs by \\textsc{CrsEntropy} on both datasets, as more ties are found. \nWe show sample outputs in Fig.~\\ref{fig:intro_sample}, with additional examples in Appendix~\\ref{appendix:outputs}. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{figures\/human_eval_error_type_dist.pdf}\n \\vspace{-2mm}\n \\caption{Portions of summaries with errors. CL models consistently reduce both types of errors.\n }\n \\label{fig:human_eval_error_dist}\n \\vspace{-3mm}\n\\end{figure}\n\n\\paragraph{Intrinsic vs. Extrinsic Errors.}\nNext, the annotators are asked to label text spans with intrinsic and extrinsic errors as done in \\S~\\ref{sec:error_annotation}. \nFig.~\\ref{fig:human_eval_error_dist} shows that \\textit{CL is more effective at reducing extrinsic errors than unlikelihood training can} on both datasets.\nWe also observe slight decreases of world knowledge in the summaries (figure attached in Appendix~\\ref{appendix:human_eval}). \n\n\\paragraph{Error Correction Operations.}\nFinally, with reference to \\textsc{CrsEntropy} summaries, human judges are instructed to label each system summary as whether it corrects any error by \\textsc{CrsEntropy} using \\textbf{deletion} of the incorrect content, \\textbf{substitution} with factual information, or \\textbf{both}. \nAs seen in Fig.~\\ref{fig:error_correct_technique}, CL-based models restore factually consistent information, e.g., by replacing erroneous names and numbers with correct ones, more frequently than unlikelihood training or entailment reranking. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{figures\/error_correct_dist.pdf}\n \\vspace{-2mm}\n \\caption{Summaries use different portions of error correction operations. \n Contrastive learning with \\textsc{SysLowCon} (CL.SLC) and \\textsc{Batch} (CL.B) substitute errors with correct content more often than unlikelihood training with \\textsc{MaskEnt} and \\textsc{EntailRank}. \n \n \n }\n \\label{fig:error_correct_technique}\n \\vspace{-3mm}\n\\end{figure}\n\n\n\n\\subsection{Variants of Summary Representation}\n\\label{subsec:ablaton}\n\nSample representation is critical for CL to be effective. Here we investigate summary representation variants as discussed in \\S~\\ref{sec:method}. \nThere are two major considerations: (1) Should we consider all tokens in a summary or only representative ones (e.g., entities or last token)? (2) Should additional transformation, i.e., an MLP, be used?\n\nExperiments on XSum using three negative sample construction strategies demonstrate that \\textit{averaging the decoder outputs of all tokens and adding an MLP projection yield the best overall performance}, as shown in Table~\\ref{tab:ablation_representation}. The implications are at least two-fold. \nFirst, even for entity- or relation-triggered sample modifications, using more global context helps with CL training. \nSecond, additional transformation can help avoid model degeneration. For instance, more nonsensical and repetitive content is produced by variants without MLP. \n\n\n\\begin{table}[t]\n \\centering\n \\setlength{\\tabcolsep}{2.5pt}\n \\fontsize{8.5}{10}\\selectfont\n \\begin{tabular}{lccccccc}\n \\toprule\n \\multicolumn{2}{l}{} & \\multicolumn{2}{c}{\\textbf{\\textsc{SwapEnt}}} & \\multicolumn{2}{c}{\\textbf{\\textsc{MaskRel}}} & \\multicolumn{2}{c}{\\textbf{\\textsc{SysLowCon}}} \\\\\n \\textbf{Rep.} & \\textbf{MLP} & \\textbf{QEval} & \\textbf{FC} & \\textbf{QEval} & \\textbf{FC} & \\textbf{QEval} & \\textbf{FC} \\\\\n \\midrule\n \\multicolumn{8}{l}{\\textit{BART}} \\\\\n Last & \\checkmark & 33.15 & 25.10 & 33.20 & 25.29 & 33.10 & 24.85 \\\\\n Last & & \\textcolor{red!53}{+0.13} & \\textcolor{red!42}{+0.02} & \\textcolor{blue!21}{--0.01} & \\textcolor{blue!72}{--0.32} & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!50}{--0.10} \\\\\n Entity & \\checkmark & \\hlc[xgreen]{\\textbf{33.35}} & 25.41 & 33.34 & 25.44 & 33.32 & 24.46 \\\\\n Entity & & \\textcolor{blue!50}{--0.13} & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!41}{--0.14} & \\textcolor{blue!45}{--0.05} & \\textcolor{blue!69}{--0.29} & \\textcolor{red}{+0.72} \\\\\n All & \\checkmark & 33.30 & \\hlc[xgreen]{\\textbf{25.67}} & \\hlc[xgreen]{\\textbf{33.35}} & \\hlc[xgreen]{\\textbf{25.69}} & \\hlc[xgreen]{\\textbf{33.35}} & \\hlc[xgreen]{\\textbf{25.47}} \\\\\n All & & \\textcolor{blue!43}{--0.23} & \\textcolor{blue!90}{--0.80} & \\textcolor{blue!44}{--0.04} & \\textcolor{blue!88}{--0.48} & \\textcolor{blue!46}{--0.26} & \\textcolor{blue!80}{--0.40} \\\\\n \\midrule\n \\multicolumn{8}{l}{\\textit{PEGASUS}} \\\\\n Last & \\checkmark & 33.07 & \\hlc[xgreen]{\\textbf{25.45}} & 32.99 & 25.09 & 33.18 & 24.94 \\\\\n Last & & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!96}{--0.56} & \\textcolor{red!41}{+0.01} & \\textcolor{blue!41}{-0.01} & \\textcolor{blue!47}{--0.02} & \\textcolor{blue!44}{--0.04} \\\\\n Entity & \\checkmark & 33.03 & 25.43 & 33.05 & 24.77 & 33.20 & 24.59 \\\\\n Entity & & \\textcolor{red!41}{+0.01} & \\textcolor{blue!74}{--0.34} & \\textcolor{blue!42}{--0.05} & \\textcolor{red}{\\textbf{\\hlc[xgreen]{+0.64}}} & \\textcolor{blue!70}{--0.30} & \\textcolor{red!45}{+0.05} \\\\\n All & \\checkmark & \\hlc[xgreen]{\\textbf{33.09}} & 25.09 & 33.06 & 25.28 & \\hlc[xgreen]{\\textbf{33.21}} & \\hlc[xgreen]{\\textbf{25.18}} \\\\\n All & & \\textcolor{blue!51}{--0.11} & \\textcolor{red!65}{+0.25} & \\textcolor{red!43}{\\hlc[xgreen]{\\textbf{+0.03}}} & \\textcolor{blue!59}{-0.19} & \\textcolor{blue!42}{--0.02} & \\textcolor{blue}{--0.80} \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-2mm}\n \\caption{\n Comparing different formulations of summary representation in CL. \n For models without MLP, we display score changes from their counterparts. \n Overall, using all tokens with MLP produces better summaries. \n }\n \\label{tab:ablation_representation}\n \\vspace{-2mm }\n\\end{table}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe present \\textsc{CLIFF}, a contrastive learning-based framework to promote faithfulness and factuality of abstractive summaries. \\textsc{CLIFF} uses both references and summaries that are factually inconsistent with the articles to train systems to be better at discriminating errors from factual and salient content. \nWe further study strategies that automatically create erroneous summaries by editing from references or leveraging systems outputs, inspired by our new summary error analysis on state-of-the-art models. \nBoth automatic evaluation and human ratings show that \\textsc{CLIFF} achieves consistent improvements over competitive comparison methods, and is generalizable across datasets with systems fine-tuned from different large models. \n\n\n\\section*{Acknowledgements}\nThis research is supported in part by Oracle for Research Cloud Credits, National Science Foundation through a CAREER award IIS-2046016, and the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via contract \\# FA8650-17-C-9116. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. \nWe thank three anonymous reviewers for their valuable suggestions. \n\n\n\\section{Additional Analysis for Summary Error Annotation}\n\\label{appendix:error_annotation}\n\nWe hire two fluent English speakers to annotate summary errors on XSum and CNN\/DailyMail (CNN\/DM). They annotate a common batch of 100 summaries generated by summarizers fine-tuned from BART and PEGASUS, with 50 articles in each batch. The two annotators are shown 50 HTML pages in a batch, each of which contains an article and two summaries generated by the two models. \nThe detailed annotation guideline is given in Fig.~\\ref{fig:annotation_guideline}.\n\nFor our analysis on token generation probabilities, we additionally show the distributions of the first token's probability for nouns and verbs in Fig.~\\ref{fig:first_token}. We also report the distributions of the non-first token's probability for proper nouns, numbers, nouns, and verbs in Fig.~\\ref{fig:nonfirst_token}. As can be seen, tokens within extrinsic and intrinsic errors have high generation probabilities when they are non-first tokens.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/prob_other_first.pdf}\n \\caption{Probability distributions of generating the \\textit{first} tokens of nouns and verbs, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens. No verb is annotated as world knowledge.}\n \\label{fig:first_token}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/prob_all_nonfirst.pdf}\n \\caption{Probability distributions of generating the \\textit{non-first} tokens of proper nouns, numbers, nouns, and verbs, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens. Non-first tokens do not exist for numbers and verbs, as they only contain single tokens.}\n \\label{fig:nonfirst_token}\n\\end{figure}\n\n\n\n\\section{Statistics for Datasets and Training Samples}\n\\label{appendix:dataset_stat}\n\n\\paragraph{Summarization Datasets.}\n\nWe follow the official data splits for the two datasets, with the number of samples in each split listed in Table~\\ref{tab:data_split}.\n\n\\begin{table}[t]\n \\centering\n \\small\n \\begin{tabular}{lccc}\n \\toprule\n \\textbf{Dataset} & \\textbf{Train} & \\textbf{Validation} & \\textbf{Test} \\\\\n \\midrule\n XSum & 204{,}045 & 11{,}332 & 11{,}334 \\\\\n CNN\/DM & 287{,}227 & 13{,}368 & 11{,}490 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Numbers of samples in train\/validation\/test splits of XSum and CNN\/DM.}\n \\label{tab:data_split}\n\\end{table}\n\n\\paragraph{Positive Samples.}\n\nWe observe unfaithful paraphrases by back-translation for some reference summaries, which are mainly due to the introduction of new entities and the rewriting of quoted text. Thus, we discard samples generated by back-translation that contain new entities and inconsistent quoted text. Finally, we obtain $182{,}114$ and $91{,}468$ positive samples by back-translation on XSum and CNN\/DM.\n\n\\paragraph{Negative Samples.}\n\nFor consistency, we use the summarizer fine-tuned from BART in \\textsc{RegenEnt}, \\textsc{RegenRel} (\\S~\\ref{subsec:conditional}), and \\textsc{SysLowCon} (\\S~\\ref{subsec:modelgeneration}) strategies. We tune a threshold to select negative samples from model generations in our \\textsc{SysLowCon} strategy. The threshold is set to $0.21$, with F1 scores of $73.99$ and $40.49$ on XSum and CNN\/DM annotated samples generated by BART.\n\nThe numbers of negative samples constructed by each strategy for training on XSum and CNN\/DM are shown in Table~\\ref{tab:num_neg_samples}.\n\\textsc{SysLowCon} constructs the least negative samples in total, while it achieves the best results as reported in our main paper (\\S~\\ref{subsec:autoeval}), indicating that its negative samples are more effective for training.\n\n\\begin{table}[ht]\n \\centering\n \\small\n \\begin{tabular}{lcc}\n \\toprule\n \\textbf{Strategy} & \\textbf{XSum} & \\textbf{CNN\/DM} \\\\\n \\midrule\n \\textsc{FCSample} & 936{,}164 & 1{,}291{,}710 \\\\\n \\textsc{SwapEnt} & 438{,}003 & 1{,}617{,}764 \\\\\n \\textsc{MaskEnt} & 360{,}795 & 1{,}050{,}200 \\\\\n \\textsc{MaskRel} & 391{,}224 & 1{,}345{,}317 \\\\\n \\textsc{RegenEnt} & 732{,}986 & 1{,}941{,}886 \\\\\n \\textsc{RegenRel} & 993{,}694 & 1{,}453{,}044 \\\\\n \\textsc{SysLowCon} & 401{,}112 & 502{,}768 \\\\\n \n \\bottomrule\n \\end{tabular}\n \\caption{Numbers of negative samples constructed by different strategies on XSum and CNN\/DM.}\n \\label{tab:num_neg_samples}\n\\end{table}\n\n\n\n\\section{Implementation Details}\n\\label{appendix:implementation}\n\nWe use Fairseq~\\cite{ott2019fairseq} and Huggingface Transformers~\\cite{wolf-etal-2020-transformers} for our experiments with BART and PEGASUS. Our experiments are conducted on the RTX 8000 GPU with 48GB memory and the A100 GPU with 40GB memory.\n\n\\paragraph{Training Settings.}\n\nFor hyperparameters, we follow \\citet{lewis-etal-2020-bart} for BART and \\citet{zhang2020pegasus} for PEGASUS.\nDuring training, we randomly select 5 and 4 negative samples for each input article in XSum and CNN\/DM. Mixed-precision training is not supported by the PEGASUS implementation and is utilized on BART only.\n\n\\paragraph{Decoding Settings.}\n\nWe use the beam search algorithm to decode summaries.\nFor BART, we set the beam sizes as 6 and 4 on XSum and CNN\/DM. A beam size of 8 is used for PEGASUS on both datasets.\n\n\\paragraph{Running Time and Model Sizes.}\n\nThe BART-based models take 6 and 13 hours for training on XSum and CNN\/DM, and it takes 1.5 hour to decode on the two datasets.\nMeanwhile, training the PEGASUS-based models takes 8 and 25 hours for XSum and CNN\/DM, and the decoding takes 1 hour.\n\nAs for model sizes, our BART-based models and PEGASUS-based models have 400M and 568M parameters.\n\n\n\\section{Human Evaluation}\n\\label{appendix:human_eval}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/human_eval_world_dist.pdf}\n \\caption{Percentages of samples containing world knowledge as labeled by human on the outputs of XSum and CNN\/DM.}\n \\label{fig:percent_world}\n\\end{figure}\n\nIn \\S~\\ref{subsec:humaneval}, we demonstrate the percentages of samples containing intrinsic errors and extrinsic errors for each model evaluated by human judges. \nHere, we report the percentages of samples containing world knowledge in Fig.~\\ref{fig:percent_world}. On XSum, all models produce less world knowledge compared to the model trained with cross-entropy loss, while generating similar or greater amounts of samples with world knowledge on CNN\/DM.\n\nOur human evaluation guideline is shown in Fig.~\\ref{fig:human_eval_guideline}.\n\n\n\n\\section{Sample Outputs}\n\\label{appendix:outputs}\n\nWe include more sample outputs in Fig.~\\ref{fig:generation_examples}.\n\n\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\fontsize{10}{12}\\selectfont\n \\begin{tabular}{p{0.88\\textwidth}}\n \\toprule\n In this study, you will first read article-summary pairs and then identify three types of text spans in the summaries. These spans include content that is contradicted by or cannot be implied from the article. The description for each type is described below: \\\\\n \\begin{itemize}\n \\item \\textbf{Intrinsic errors:} Text spans that misconstruct phrases or clauses from the article. \n \\item \\textbf{Extrinsic errors:} Text spans that include words that are not in the article and are not verifiable or cannot be verified by Wikipedia. \n \\item \\textbf{world knowledge:} Text spans that contain information that is not covered by the article but can be validated by Wikipedia. \n \\end{itemize} \\\\\n When selecting spans, you should always make sure the spans are complete words. \\\\\n \\\\\n In practice, you should follow the these steps carefully: (1) read the article and summaries carefully; (2) figure out if there is content contradicted by or not presented in the article; (3) label the span as an intrinsic error if it misconstructs phrases or clauses from the article; (4) if the span does not belong to intrinsic errors, search within Wikipedia and determine whether the content in the span can be verified; (5) label it as world knowledge if the it can be verified by Wikipedia, otherwise label it as an extrinsic error. \\\\\n \\midrule\n \\textbf{Example annotations 1} \\\\\n \\textbf{Article:} Isis Academy in Oxford said it had rebranded as ``Iffley Academy'' to protect its ``reputation, integrity and image''. \\textbf{The name `Isis' was originally chosen as the school is near to the section of the River Thames of the same name}. Formerly Iffley Mead School, it became Isis Academy in 2013. A statement issued by the school said it had changed name following ``the unforeseen rise of ISIS (also known as ISIL and the Islamic State) and related global media coverage of the activities of the group''. ``Our priority is to remove the detrimental impact which the name `Isis' had on pupils, their families and our staff.'' Last year a language school in the city removed Isis from its name for the same reason. The Isis is the name given to the part of the River Thames above Iffley Lock in Oxford. It is also the name of the goddess wife of the god Osiris in Egyptian beliefs. \\\\\n \n \\textbf{Summary:} A school that \\hlc[crimsonglory!40]{was named after} the Islamic State (IS) militant group has changed its name. \\\\\n \\textbf{Explanation:} \\textit{``was name after''} is an intrinsic error contradicted by the article sentence in \\textbf{bold}. \\\\%The name is not chosen because of the Islamic State. \\\\\n \\midrule\n \\textbf{Example annotations 2} \\\\\n \\textbf{Article:} Khalil Dale, 60, was abducted in \\textbf{Quetta} in January 2012 and was found dead on a roadside a few months later. He had been beheaded. A note next to his body said he was killed because a ransom had not been paid. Mr Dale was born in York but lived in Dumfries. He spent 30 years working in countries including Somalia, Afghanistan and Iraq. An inquest into his death was held at Chesterfield Coroners Court because he is buried in Derbyshire. The court heard that the Muslim convert, who was formerly known as Kenneth, worked as a humanitarian assistance relief worker. Following his abduction, negotiations were undertaken by the International Committee of the Red Cross with the help of the UK government. His body was found on 29 April 2012. The inquest was told that he died as a result of decapitation. Senior coroner Dr Robert Hunter concluded that Mr Dale was unlawfully killed while providing international humanitarian assistance. \\\\\n \\textbf{Summary:} A British aid worker was unlawfully killed by \\hlc[bleudefrance!40]{Islamist militants} in \\hlc[yellow!60]{Pakistan}, an inquest has heard. \\\\\n \\textbf{Explanation:} \\textit{``Islamist militant''} is an extrinsic error as it can not be found in or inferred from the article. The information is also not verifiable by Wikipedia. \\textit{``Pakistan''} is world knowledge as \\textbf{Quetta} in the article is a city in Pakistan according to Wikipedia. \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Guideline for our summary error annotation (\\S~\\ref{sec:error_annotation}).}\n \\label{fig:annotation_guideline}\n\\end{figure*}\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\fontsize{10}{12}\\selectfont\n \\begin{tabular}{p{0.92\\textwidth}}\n \\toprule\n In this study, you will evaluate 100 sets of summaries produced by four systems. For each set, its corresponding article and a baseline summary are shown before the four system summaries. The errors in the baseline summary are highlighted. \\\\\n Please \\textit{first read the article and the baseline summary} and then \\textit{compare each system summary against the baseline summary} based on \\textbf{informativeness} and \\textbf{factual consistency}. In addition, please decide the \\textbf{operations} made by the system to achieve better factual consistency. \\\\\n For informativeness and factual consistency, you need to label whether the system summary is better or worse than the baseline summary. You can also label the system summary as tying with the baseline summary. \\\\\n You need to consider two types of operations: \\textbf{deletions} and \\textbf{substitutions}. Please label the system summary as making deletions, substitutions, or both operations. Examples for the aspects and the operations are as follows. \\\\\n \\midrule\n \n \\textbf{Article:} Alexys Brown, also known as Lexi, died at her home in Emmadale Close, Weymouth, on Thursday. An investigation is under way to discover how she became trapped. A post-mortem examination is due to be carried out this week. It was originally hoped the appeal would raise \u00a32,000. Alison Record, who started the Just Giving appeal, said she was \"heart broken\" over the death. ``Everybody by now has heard of the terrible tragedy the Brown family have suffered with the loss of their beautiful and beloved little girl Lexi,'' the appeal page reads. Many other comments have been posted on the appeal page. Steph Harris said: ``Thinking of you all at this devastating time, fly high beautiful princess. Love Steph and family xxx'' Lesley Andrews added: ``No amount of money will take away the pain, but so much love comes with every penny. Take care. xx'' Aster Group, the housing association responsible for managing the home, is assisting with the police investigation. The Health and Safety Executive (HSE) is also investigating. Dorset County Council said it had not installed the disabled lift at the property. \\\\\n \n \\textbf{Baseline Summary:} An appeal to raise \\hlc[crimsonglory!40]{10,000 pounds} for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}. \\\\\n \n \\smallskip\n \\textbf{Informativeness:} Whether the summary captures salient content from the input article. Note that incorrect content should be considered as invalid. \\\\\n \\textbf{Win.} \\textit{An appeal to raise money for the family of a three-year-old girl who died after getting stuck in a lift was originally hoped for raising \u00a32,000.} The target money of the appeal is a salient information. \\\\\n \\textbf{Tie.} \\textit{An appeal to raise money for the family of a girl who died after getting stuck in a lift has raised more than \u00a320,000.} Compared to the baseline, missing incorrect information does not affect the informativeness. \\\\\n \\textbf{Lose.} \\textit{An appeal to raise money for the family of a three-year-old girl has raised more than \u00a320,000.} This system summary does not mention the death of the girl, which is a salient content of the article. \\\\\n \n \\smallskip\n \\textbf{Factual Consistency:} Whether the summary is factually correctly based on the article and knowledge from Wikipedia. \\\\\n \\textbf{Win.} \\textit{An appeal has been set up for the family of an \\hlc[bleudefrance!40]{eight-year-old} girl who died after becoming trapped in a lift at her Dorset home.} This system summary does not generate the incorrect numbers of money. \\\\\n \\textbf{Tie.} \\textit{An appeal to raise \\hlc[crimsonglory!40]{5,000 pounds} for the family of a \\hlc[bleudefrance!40]{seven-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} This system summary makes similar errors to the baseline. \\\\\n \\textbf{Lose.} \\textit{\\hlc[crimsonglory!40]{The family} of an \\hlc[bleudefrance!40]{eight-year-old} girl who died after becoming trapped in a lift at her Dorset home \\hlc[crimsonglory!40]{have set a fundraising target} of \\hlc[bleudefrance!40]{10,000 pounds}.} This system summary fabricates an event \\textit{The family have set a fundraising target}, which is more severe than errors of modifiers. \\\\\n \n \\smallskip\n \\textbf{Deletion:} The incorrect content in the baseline summary is deleted. \\\\\n - \\textit{An appeal for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} The error \\textit{``10{,}000 pounds''} is deleted. \\\\\n \n \\textbf{Substitution:} The incorrect content in the baseline summary is replaced with correct one. \\\\\n - \\textit{An appeal to raise 2,000 pounds for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} The error \\textit{``10{,}000 pounds''} is substituted with \\textit{``2,000 pounds''}, which is the correct information. \\\\\n \n \\bottomrule\n \\end{tabular}\n \\caption{Guideline for our human evaluation (\\S~\\ref{subsec:humaneval}).}\n \\label{fig:human_eval_guideline}\n\\end{figure*}\n\n\n\\begin{figure*}[th]\n \\centering\n \\small\n \\begin{tabular}{p{0.95\\textwidth}}\n \\toprule\n \\textbf{Example 1} \\\\\n \\midrule\n \\textbf{CNN\/DM Article:} At the grand old age of 75, Jack Nicklaus is still capable of hitting aces. The Golden Bear added another magic moment to his storied career at Augusta National in the Par-3 Contest. Stepping up to the tee on the 130-yard fourth, the greatest golfer of all time saw his shot sail beyond the flag before spinning back into the hole. Jack Nicklaus gave the crowd something to cheer with a hole in one on the fourth during the Par-3 Contest. Nicklaus holds up his ball to an adoring crowd as Gary Player (left) and Ben Crenshaw salute the great. Crenshaw and Nicklaus fist pump following his ace on the 130-yard hole at Augusta National. Nicklaus was playing alongside Gary Player and Ben Crenshaw as the first roar of Masters week filled the skies in a sunswept Georgia. 'I had an interview this morning, and I said \"well, all I've got to do is go out and win the Par-3 and make a hole and one\", and I make a hole in one,' Nicklaus said afterwards. . The ace was Nicklaus' first on the hallowed turf of Augusta, but the veteran saw his achievement eclipsed by that of Camilo Villegas on a day which saw five hole-in-ones, levelling the record from 2002. The Colombian followed Nicklaus with an ace on the fourth and also managed one on the eighth, but he couldn't quite land the title, losing out to world No 74 Kevin Streelman in a play-off. Nicklaus became the youngest person to wear a green jacket in 1963, and collected his sixth in 1986. He is one of five men to complete the career grand slam, an accolade which favourite Rory McIlroy can achieve if he wins his third major in succession. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas and \\hlc[crimsonglory!40]{Kevin Streelman} also hit hole-in-ones on the day. \\\\\n \\textbf{Entailment reranking:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas and \\hlc[crimsonglory!40]{Kevin Streelman} also hit hole-in-ones on day one. \\\\\n \\textbf{Unlikelihood training:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas followed Nicklaus with a hole-in-one on the same hole. Nicklaus is one of five men to complete the career grand slam. \\\\\n \\textbf{Contrastive learning:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas followed Nicklaus with a hole-in-one on the eighth. \\\\\n \\midrule\n \\textbf{Example 2} \\\\\n \\midrule\n \\textbf{XSum Article:} The British supermarket giant, the world's third largest retailer, has been looking to dispose of its loss-making US food chain for some time. Around 4,000 of Fresh \\& Easy's 5,000 employees are to transfer to the new business as part of the deal. Tesco said in a statement that this is an \"orderly and efficient exit from the US market\". Under the terms of the deal, Tesco will loan the new business around \u00a380m. Fresh \\& Easy, which operates 200 stores across California, Nevada and Arizona, has lost money since opening in 2007. Tesco did not specify which stores are being transferred to Yucaipa, but it is understood the remaining 50 shops will be closed. It will cost Tesco \u00a3150m to dispose of the Fresh \\& Easy brand, but part of that is a loan which may be paid back. Also, if Yucaipa is sucessful and the business becomes profitable, Tesco has the option to buy a stake. Yucaipa is a Los Angeles based private equity company focused on supermaket and restaurant businesses and is run by billionaire investor Ron Burkle. In April of this year, Tesco reported its first fall in annual profits for 20 years. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Tesco has agreed to sell its US food chain Fresh \\& Easy to Yucaipa in a deal \\hlc[bleudefrance!40]{worth about 150m}, the BBC understands. \\\\\n \\textbf{Entailment reranking:} Tesco is to sell its US food chain Fresh \\& Easy to Yucaipa \\hlc[bleudefrance!40]{for an undisclosed sum}. \\\\\n \\textbf{Unlikelihood training:} Tesco has agreed to sell its US food business, Fresh \\& Easy, to a private equity firm called Yucaipa \\hlc[bleudefrance!40]{for an undisclosed sum}. \\\\\n \\textbf{Contrastive learning:} Tesco has agreed to sell its US food chain, Fresh \\& Easy, to Yucaipa. \\\\\n \\midrule\n \\textbf{Example 3} \\\\\n \\midrule\n \\textbf{XSum Article:} Neil Aspin's promotion-chasing hosts have not lost in nine National League matches while Adam Lockwood's side are unbeaten in five. Guiseley went ahead on 15 minutes against the run of play when a throw-in found James Hurst who squared to Jake Lawlor to stroke into an empty net. Gateshead defender Liam Hogan superbly blocked Jordan Preston's effort and Guiseley keeper Jonny Maxted then saved well from Wesley York's shot just before the break. The hosts, who started the second half well, levelled on 62 minutes when a slip by half-time substitute Derek Asamoah let York curl sweetly into the top-right corner from the edge of the box. Match report supplied by the Press Association. Match ends, Gateshead 1, Guiseley 1. Second Half ends, Gateshead 1, Guiseley 1. Substitution, Guiseley. Michael Rankine replaces Jordan Preston. Substitution, Gateshead. Luke Hannant replaces Gus Mafuta. Gus Mafuta (Gateshead) is shown the yellow card. Substitution, Guiseley. Adam Boyes replaces Jake Cassidy. Goal! Gateshead 1, Guiseley 1. Wes York (Gateshead). Substitution, Guiseley. Derek Asamoah replaces Kevan Hurst. Second Half begins Gateshead 0, Guiseley 1. First Half ends, Gateshead 0, Guiseley 1. Simon Walton (Guiseley) is shown the yellow card. Goal! Gateshead 0, Guiseley 1. Jake Lawlor (Guiseley). First Half begins. Lineups are announced and players are warming up. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Gateshead and Guiseley maintained their \\hlc[crimsonglory!40]{unbeaten starts to the season} with a 1-1 draw at the International Stadium. \\\\\n \\textbf{Entailment reranking:} Gateshead and Guiseley shared the spoils after a 1-1 draw at the International Stadium. \\\\\n \\textbf{Unlikelihood training:} Gateshead and Guiseley shared the spoils after a \\hlc[crimsonglory!40]{goalless} draw in the National League. \\\\\n \\textbf{Contrastive learning:} Gateshead and Guiseley shared the spoils after a 1-1 draw at Gateshead. \\\\\n \\bottomrule\n \n \\end{tabular}\n \\caption{Sample generated summaries by fine-tuned BART models. Intrinsic errors are highlighted in \\hlc[crimsonglory!40]{red} and extrinsic errors are in \\hlc[bleudefrance!40]{blue}. \n \\textsc{MaskEnt} and \\textsc{SysLowCon} are used for negative sample construction with unlikelihood training and contrastive learning.}\n \\label{fig:generation_examples}\n\\end{figure*}\n\n\\section{Summary Error Annotation and Model Behavior Analysis}\n\\label{sec:error_annotation}\n\nWe first describe annotating unfaithfulness errors by state-of-the-arts, i.e., models fine-tuned from BART and PEGASUS\non XSum and CNN\/DM.\nWe then probe into the model generation behavior that is indicative of errors, which guides the design of negative sample construction strategies.\n\n600 ($150\\times2\\times2$) summaries are annotated to demonstrate how often do the models ``hallucinate\", i.e., generating content not grounded by the source.\nTo characterize errors, we annotate \\textit{text spans} in summaries with \n(i) \\textbf{intrinsic errors} caused by misconstructing phrases or clauses from the source; \nand (ii) \\textbf{extrinsic errors} which include words not in the source that are either unverifiable or cannot be verified by Wikipedia. \nContent not covered by the article but can be validated by Wikipedia is annotated as \\textbf{world knowledge}, and the models' behavior pattern when generating them differs from when they generate errors.\n\nTwo fluent English speakers with extensive experience in summary evaluation and error labeling are hired. For each sample, they are shown the article and two system summaries, and instructed to annotate text spans with the aforementioned errors and world knowledge. After labeling every 50 samples, the annotators discuss and resolve any disagreement.\nThe Fleiss's Kappas on XSum and CNN\/DM are $0.35$ and $0.45$.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/error_type_dist.pdf}\n \n \\caption{Percentage of samples with intrinsic and extrinsic error spans for models fine-tuned from BART and PEGASUS on XSum and CNN\/DM.}\n \\label{fig:error_type_dist}\n \n\\end{figure}\n\n\\medskip\n\\noindent \\textbf{Error statistics} are displayed in Fig.~\\ref{fig:error_type_dist}. \nExtrinsic errors dominate both datasets, especially on XSum. $58.7\\%$ of summaries by BART (and $44.0\\%$ by PEGASUS) contain at least one extrinsic error.\nNoticeably, PEGASUS is a newer model pre-trained with a larger amount of data, thus contains less errors than BART and other older models studied for error annotations by~\\newcite{maynez-etal-2020-faithfulness}. This observation also highlights the usage of our annotations for future development and evaluation of summarization models. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.46\\textwidth]{figures\/prob_both.pdf}\n \\caption{\n Probability distributions of generating the first tokens of proper nouns and numbers, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens.}\n \\label{fig:output_prob_propnum}\n\\end{figure}\n\n\\paragraph{Low confidence generation is indicative of extrinsic errors.}\nInspired by recent work that studies model prediction confidence~\\cite{liu2021tokenlevel}, we examine generation probabilities for tokens of \\textit{different part-of-speech (POS) tags}.\nFig.~\\ref{fig:output_prob_propnum} shows salient results on the generation probabilities of the first token of a proper noun or a number (with additional analysis provided in Appendix~\\ref{appendix:error_annotation}). \nAs observed, model confidence tends to be lower for the first tokens of proper nouns and numbers if they are part of spans with \\textit{extrinsic errors}.\nAlso note that world knowledge, which cannot be inferred from the source either, often has higher generation probability than extrinsic errors. \nTake this snippet generated by a fine-tuned BART as an example: \\textit{``Manchester United captain Wayne Rooney's testimonial game against Manchester City$\\ldots$''}. \\textit{``Manchester City\"} is an extrinsic error and \\textit{``Wayne\"} is produced as world knowledge. The model assigns a low probability of $0.10$ to the first token of \\textit{``Manchester City''} and a high probability of $0.92$ to token \\textit{``Wayne''}. \nThis implies that model confidence can be a useful indicator for negative sample collection. \n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe present \\textsc{CLIFF}, a contrastive learning-based framework to promote faithfulness and factuality of abstractive summaries. \\textsc{CLIFF} uses both references and summaries that are factually inconsistent with the articles to train systems to be better at discriminating errors from factual and salient content. \nWe further study strategies that automatically create erroneous summaries by editing from references or leveraging systems outputs, inspired by our new summary error analysis on state-of-the-art models. \nBoth automatic evaluation and human ratings show that \\textsc{CLIFF} achieves consistent improvements over competitive comparison methods, and is generalizable across datasets with systems fine-tuned from different large models. \n\n\n\\section{Introduction}\n\n\nLarge pre-trained Transformers have yielded remarkable performance on abstractive summarization~\\cite{liu-lapata-2019-text,lewis-etal-2020-bart,zhang2020pegasus} with impeccable fluency, yet their summaries often contain factually inconsistent content~\\cite{maynez-etal-2020-faithfulness,zhang-etal-2020-optimizing,goyal-durrett-2020-evaluating}, even for state-of-the-art models.\nThree types of remedies have been proposed: running a separately learned error correction component~\\cite{dong-etal-2020-multi}, removing noisy training samples~\\cite{nan-etal-2021-entity,goyal-durrett-2021-annotating}, and modifying the Transformer architecture~\\cite{huang-etal-2020-knowledge,zhu-etal-2021-enhancing}. \nYet they either rely on heuristically created data for error handling, falling short of generalization, or require learning a large number of new parameters, and summary informativeness is often sacrificed. \n\n\n\\begin{figure}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{3pt}\n \\begin{tabular}{p{0.47\\textwidth}}\n \\toprule\n \\textbf{XSum Article}: The Fermanagh MLA Phil Flanagan tweeted after Tom Elliott appeared on a BBC radio programme in May 2014. $\\ldots$\n ``I wonder if he will reveal how many people he harassed and shot as a member of the UDR.''$\\ldots$ \\\\\n \\midrule\n \\textbf{Contrastive learning} (our method): A Sinn F\u00e9in MLA has been ordered to apologise and pay compensation to a former member of the \\hlc[yellow!70]{ Ulster Defence Regiment} (UDR). \\\\\n \\midrule\n \\textbf{Cross-entropy}: A Sinn F\u00e9in MLA has agreed to pay compensation to a former \\hlc[cyan!40]{Ulster Unionist Party} (UDR) MP after he tweeted that he had harassed and shot people as a member of the party. \\\\\n \\midrule\n \\textbf{Entailment reranking}: A Sinn F\u00e9in MLA has agreed to pay compensation to a former \\hlc[cyan!40]{Ulster Unionist Party} (UDR) councillor for a tweet he sent about him. \\\\\n \\midrule\n \\textbf{Unlikelihood}: An MLA has been ordered to apologise and pay compensation to a former \\hlc[cyan!40]{loyalist} MP for a remark he made about him while serving in \\hlc[cyan!40]{the Ministry of Defence}. \\\\\n \n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Sample article and system summaries by different methods. \n Our contrastive learning model trained on low confidence system outputs correctly generates the \\hlc[yellow!70]{full name}.\n Comparisons using cross-entropy loss,\n beam reranking by entailment scores~\\cite{kryscinski-etal-2020-evaluating}, and unlikelihood objective~\\cite{Welleck2020Neural} over negative samples all produce \\hlc[cyan!40]{unfaithful content}.\n }\n \\label{fig:intro_sample}\n\\end{figure}\n\nOur goal is to train abstractive summarization systems that generate both faithful and informative summaries in an end-to-end fashion.\nWe observe that, while the commonly used maximum likelihood training optimizes over references, there is no guarantee for the model to distinguish references from incorrect generations~\\cite{Holtzman2020The,Welleck2020Neural}. \nTherefore, potential solutions reside in designing new learning objectives that can effectively inform preferences of factual summaries over incorrect ones. \n\nConcretely, we hypothesize that including factually inconsistent summaries (i.e., \\textit{negative samples}) for training, in addition to references (i.e., \\textit{positive samples}), let models become better at differentiating these two types of summaries. \nAlthough using negative samples has been effective at text representation learning, e.g., word2vec~\\cite{mikolov2013distributed} and BERT~\\cite{devlin-etal-2019-bert}, there exist two major challenges for it to succeed in concrete language tasks. \nFirst, \\textit{a suitable training objective} is critical to avoid performance degradation~\\cite{saunshi2019theoretical}. \nSecond, it is nontrivial to \\textit{construct ``natural'' samples} that mimic the diverse errors made by state-of-the-art systems that vary in words and syntax~\\cite{goyal-durrett-2021-annotating}. \n\nTo address both challenges, we first propose a new framework, \\textbf{\\textsc{CLIFF}}, that uses \\underline{c}ontrastive \\underline{l}earning for \\underline{i}mproving \\underline{f}aithfulness and \\underline{f}actuality of the generated summaries.\\footnote{Our code and annotated data are available at \\url{https:\/\/shuyangcao.github.io\/projects\/cliff_summ}.} \nContrastive learning (CL) has obtained impressive results on many visual processing tasks, such as image classification~\\cite{khosla2020supervised,pmlr-v119-chen20j} and synthesis~\\cite{park2020contrastive,zhang2021cross}. \nIntuitively, CL improves representation learning by compacting positive samples while contrasting them with negative samples.\nHere, we design a task-specific CL formulation that teaches a summarizer to expand the margin between factually consistent summaries and their incorrect peers.\n\nMoreover, we design four types of strategies with different variants to construct negative samples by \\textit{editing reference summaries} via rewriting entity-\/relation-anchored text, and using \\textit{system generated summaries} that may contain unfaithful errors. \nImportantly, these strategies are inspired by our new annotation study on errors made by state-of-the-art summarizers---models fine-tuned from BART~\\cite{lewis-etal-2020-bart} and PEGASUS~\\cite{zhang2020pegasus}---on two benchmarks: XSum~\\cite{narayan-etal-2018-dont} and CNN\/DailyMail (CNN\/DM)~\\cite{NIPS2015_5945}. \n\nWe fine-tune pre-trained large models with our contrastive learning objective on XSum and CNN\/DM. \nResults based on QuestEval~\\cite{scialom2020QuestEval}, a QA-based factuality metric of high correlation with human judgments,\nshow that our models trained with different types of negative samples uniformly outperform strong comparisons, including using a summarizer with post error correction and reranking beams based on entailment scores to the source.\nMoreover, compared with unlikelihood training method that penalizes the same negative samples~\\cite{Welleck2020Neural}, our summaries also obtain consistently better QuestEval scores. \nHuman evaluation further confirms that our models consistently reduce both extrinsic and intrinsic errors over baseline across datasets. \n\n\\section{Example Appendix}\n\n\n\\end{document}\n\n\\section*{Acknowledgements}\nThis research is supported in part by Oracle for Research Cloud Credits, National Science Foundation through a CAREER award IIS-2046016, and the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via contract \\# FA8650-17-C-9116. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. \nWe thank three anonymous reviewers for their valuable suggestions. \n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Automatic Evaluation}\n\\label{subsec:autoeval}\nWe report results by models fine-tuned from BART and PEGASUS with different objectives and negative samples on XSum and CNN\/DM in Tables~\\ref{tab:main_result} and~\\ref{tab:pegasus_result}. \n\\textsc{CLIFF} models use a summary representation of averaging over all tokens with MLP projection, with other variants discussed in \\S~\\ref{subsec:ablaton}. Unless explicitly stated, comparison models are fine-tuned from the same large model used by \\textsc{CLIFF}. \n\n\nFirst, comparing with other factuality improvement models (top of the tables), \\textit{almost all \\textsc{CLIFF} models trained with different negative samples uniformly produce higher QuestEval scores across datasets with both large models}, with the improvements more pronounced on XSum.\nImportantly, ROUGE scores for \\textsc{CLIFF} models are comparable or better than baselines trained with cross-entropy, e.g., on CNN\/DM as in Table~\\ref{tab:main_result}. \nA similar trend is observed with the FactCC metric, especially when using PEGASUS as the base model (Table~\\ref{tab:pegasus_result}). \nNote that \\textsc{EntailRank} tends to yield significantly higher FactCC scores, though it obtains lower QuestEval scores than the cross-entropy baseline. Human inspection finds that \\textsc{EntailRank} can pick up beams with peculiar words of high FactCC scores, without improving factuality. \nMoreover, other comparisons based on post \\textsc{Correction} and model engineering (\\textsc{FASum}) only offer incremental gains. \nThe sample selection-based method, \\textsc{SubsetFT}, sacrifices ROUGE scores significantly. \nOverall, \\textsc{CLIFF} demonstrates stronger generalizability.\n\n\n\\begin{table}[!t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{2.5pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n \\textbf{Model} & \\multicolumn{3}{c}{\\textbf{XSum}} & \\multicolumn{3}{c}{\\textbf{CNN\/DM}} \\\\\n \\cmidrule(lr){2-4} \\cmidrule(lr){5-7}\n & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} \\\\\n \\midrule\n \\multicolumn{7}{l}{\\textit{Comparisons without Negative Samples}}\\\\\n \\textsc{CrsEntropy} & 32.50 & 25.48 & \\textbf{39.07} & 50.21 & 44.44 & 40.39 \\\\\n \\textsc{EntailRank} & 32.42 & \\textbf{41.90} & 38.47 & 50.15 & \\textbf{61.04} & \\textbf{40.67} \\\\\n \\textsc{Correction} & 32.55 & 25.15 & 39.02 & 49.48 & 32.96 & 39.79 \\\\\n \\midrule\n \\midrule\n \\multicolumn{7}{l}{\\textit{Comparisons with Unlikelihood Training}} \\\\\n \\textsc{FCSample} & 32.79 & \\hlc[xgreen]{25.37} & 38.46 & 50.63 & 45.45 & 39.28 \\\\\n \n \\textsc{SwapEnt} & 32.88 & 24.76 & 37.91 & 50.43 & 43.02 & \\hlc[xgreen]{38.96} \\\\\n \\textsc{MaskEnt} & 33.04 & \\hlc[xgreen]{26.30} & 37.51 & 51.11 & 52.19 & \\hlc[xgreen]{39.34} \\\\\n \\textsc{MaskRel} & \\hlc[xgreen]{33.08} & 24.38 & 38.05 & 51.14 & 52.93 & 39.31 \\\\\n \\textsc{RegenEnt} & 32.89 & 24.46 & \\hlc[xgreen]{38.47} & \\hlc[xgreen]{51.11} & \\hlc[xgreen]{52.90} & 39.23 \\\\\n \\textsc{RegenRel} & 32.91 & 24.80 & \\hlc[xgreen]{38.46} & 51.07 & \\hlc[xgreen]{53.68} & \\hlc[xgreen]{39.45} \\\\\n \\textsc{SysLowCon} & 31.66 & \\hlc[xgreen]{26.06} & 34.03 & \\hlc[xgreen]{50.92} & 51.08 & 39.19 \\\\\n \\midrule\n \\multicolumn{7}{l}{\\textit{Our Method:} \\textsc{CLIFF}} \\\\\n \\textsc{Batch} & 32.64 & 24.96 & 38.42 & 51.03\\rlap{$^\\ast$} & 51.81\\rlap{$^\\ast$} & 39.38 \\\\\n \\hdashline\n \\textsc{FCSample} & \\hlc[xgreen]{32.96}\\rlap{$^\\ast$} & 25.28 & \\hlc[xgreen]{38.58} & \\hlc[xgreen]{51.00}\\rlap{$^\\ast$} & \\hlc[xgreen]{51.80}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.37} \\\\\n \\textsc{SwapEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & \\hlc[xgreen]{25.09} & \\hlc[xgreen]{38.58} & \\hlc[xgreen]{51.16}\\rlap{$^\\ast$} & \\hlc[xgreen]{52.97}\\rlap{$^\\ast$} & 38.95 \\\\\n \\textsc{MaskEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & 25.75 & \\hlc[xgreen]{38.12} & \\hlc[xgreen]{51.13}\\rlap{$^\\ast$} & \\hlc[xgreen]{53.60}\\rlap{$^\\ast$} & 39.24 \\\\\n \\textsc{MaskRel} & 33.06\\rlap{$^\\ast$} & \\hlc[xgreen]{25.28} & \\hlc[xgreen]{38.37} & \\hlc[xgreen]{\\textbf{51.17}}\\rlap{$^\\ast$} & \\hlc[xgreen]{53.34}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.36} \\\\\n \\textsc{RegenEnt} & \\hlc[xgreen]{33.09}\\rlap{$^\\ast$} & \\hlc[xgreen]{24.48} & 38.33 & 50.99\\rlap{$^\\ast$} & 52.18\\rlap{$^\\ast$} & \\hlc[xgreen]{39.28} \\\\\n \\textsc{RegenRel} & \\hlc[xgreen]{33.16}\\rlap{$^\\ast$} & \\hlc[xgreen]{24.82} & 38.30 & \\hlc[xgreen]{51.16}\\rlap{$^\\ast$} & 53.21\\rlap{$^\\ast$} & 39.25 \\\\\n \\textsc{SysLowCon} & \\hlc[xgreen]{\\textbf{33.21}}\\rlap{$^\\ast$} & 25.18 & \\hlc[xgreen]{38.18} & 50.85\\rlap{$^\\ast$} & \\hlc[xgreen]{53.73}\\rlap{$^\\ast$} & \\hlc[xgreen]{39.30} \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Results of models fine-tuned from PEGASUS on XSum and CNN\/DM. We report results on $5,000$ randomly selected samples on CNN\/DM, due to long running time of QuestEval. \n \n \n For models of unlikelihood training and CLIFF that use the same negative samples, the better of the two is highlighted with \\hlc[xgreen]{green}. \n $\\ast$: our model is significantly better than \\textsc{CrsEntropy} (approximation randomization test, $p < 0.005$).}\n \\label{tab:pegasus_result}\n\\end{table}\n\n\nSecond, \\textit{\\textsc{CLIFF} is more effective and robust than unlikelihood training} with the same negative samples. \nAccording to Table~\\ref{tab:main_result}, using 7 negative sample construction strategies on two datasets, \\textsc{CLIFF} obtains higher QuestEval scores than unlikelihood training in 12 out of the 14 comparisons. Using PEGASUS, \\textsc{CLIFF} also outperforms in 11 setups as listed in Table~\\ref{tab:pegasus_result}. \nSimilar trends are found on FactCC and ROUGE-L.\nAnother noteworthy piece is that \\textsc{CLIFF}'s improvements over the cross-entropy baseline are more consistent, whereas unlikelihood training occasionally hurts factuality or ROUGE scores significantly. \nWe believe the key advantage of \\textsc{CLIFF} resides in its measure of representation similarities between positive and negative samples in the same batch, allowing models to better differentiate between correct and erroneous summaries. \n\n\nFinally, among all variants, \\textit{\\textsc{CLIFF} trained with low confidence summaries as negative samples obtains the best QuestEval scores on the more abstractive dataset}. As seen in Table~\\ref{tab:main_result}, using low confidence summaries also improves FactCC scores on both datasets, and enhances ROUGE-L on the more extractive dataset CNN\/DM. \nThis indicates that \\textit{system generated summaries contribute more diverse errors made by existing models organically}, which are particularly suitable for our CL framework. As we use summaries generated by the same model for \\textsc{CLIFF} training, one future direction is to use outputs by different models. \nFor our mask-and-fill and source-conditioned regeneration strategies, we find that relation-anchored construction often beats their entity-anchored counterparts. This calls for efforts that steer the entity-driven methods to a more relation-focused direction. \n\n\n\\paragraph{Combining Strategies.}\nWe further show results by fine-tuning BARTs using samples based on combined negative sample construction strategies in Table~\\ref{tab:combine_strategy}. As can be seen, \\textit{combining \\textsc{SysLowCon} and other strategies yields better QuestEval scores than models trained with negative samples by any single strategy}, except for \\textsc{MaskEnt} and \\textsc{RegenEnt} on XSum. This signifies the importance of covering diverse types of errors in negative samples. \n\n\n\\begin{table}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{2.5pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n \\textbf{Strategy} & \\multicolumn{3}{c}{\\textbf{XSum}} & \\multicolumn{3}{c}{\\textbf{CNN\/DM}} \\\\\n \\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \n & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} & \\textbf{QEval} & \\textbf{FC} & \\textbf{R-L} \\\\\n \\midrule\n \\textsc{SysLowCon} & 33.35 & 25.47 & \\textbf{36.19} & 51.05 & 50.05 & \\textbf{41.01} \\\\\n + \\textsc{SwapEnt} & \\textbf{33.40} & \\textbf{25.50} & 35.50 & \\textbf{51.32} & \\textbf{53.95} & 40.57 \\\\\n + \\textsc{MaskEnt} & 33.21 & 25.47 & 35.91 & 51.16 & 51.90 & 40.66 \\\\\n + \\textsc{MaskRel} & 33.39 & 25.20 & 35.70 & 51.24 & 52.48 & 40.80 \\\\\n + \\textsc{RegenEnt} & 33.31 & 25.07 & 35.94 & 51.21 & 51.86 & 40.91 \\\\\n + \\textsc{RegenRel} & 33.38 & 24.97 & 36.03 & 51.13 & 50.85 & 40.97 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Results of fine-tuned BART with combinations of negative sample construction strategies.}\n \\label{tab:combine_strategy}\n\\end{table}\n\n\n\\subsection{Human Evaluation}\n\\label{subsec:humaneval}\n\n\\begin{table}[t]\n \\begin{subtable}[h]{0.48\\textwidth}\n \\small\n \\setlength{\\tabcolsep}{2pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n & \\multicolumn{3}{c}{\\textbf{Inform.}} & \\multicolumn{3}{c}{\\textbf{Factual.}} \\\\\n \\textbf{Model} & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ \\\\\n \\midrule\n \\textsc{EntailRank} & 3.3 & 84.3 & \\textbf{12.3} & 23.7 & 71.7 & \\textbf{4.7} \\\\\n \\textsc{ULL. MaskEnt} & 6.3 & 80.7 & 13.0 & 26.3 & 62.0 & 11.7 \\\\\n \\textsc{CL. Batch} & 5.3 & 80.0 & 14.7 & 21.7 & 68.0 & 10.3 \\\\\n \\textsc{CL. SysLowCon} & \\textbf{8.7} & 78.3 & 13.0 & \\textbf{31.3} & 61.7 & 7.0 \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-1mm}\n \\caption{XSum}\n \n \\end{subtable}\n \n \\begin{subtable}[h]{0.48\\textwidth}\n \\small\n \\setlength{\\tabcolsep}{2pt}\n \\begin{tabular}{lcccccc}\n \\toprule\n & \\multicolumn{3}{c}{\\textbf{Inform.}} & \\multicolumn{3}{c}{\\textbf{Factual.}} \\\\\n \\textbf{Model} & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ & \\textbf{Win}$\\uparrow$ & \\textbf{Tie} & \\textbf{Lose}$\\downarrow$ \\\\\n \\midrule\n \\textsc{EntailRank} & 2.3 & 86.3 & 11.3 & 4.7 & 94.7 & \\textbf{0.7} \\\\\n \\textsc{ULL. MaskEnt} & \\textbf{18.0} & 71.0 & 11.0 & 17.3 & 79.7 & 3.0 \\\\\n \\textsc{CL. Batch} & 17.3 & 74.7 & \\textbf{8.0} & \\textbf{20.7} & 77.0 & 2.3 \\\\\n \\textsc{CL. SysLowCon} & 15.7 & 75.7 & 8.7 & 20.0 & 77.7 & 2.3 \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-1mm}\n \\caption{CNN\/DM}\n \\end{subtable}\n \n \\caption{\n Percentages of summaries that are better than, tied with, or worse than \\textsc{CrsEntropy}, in informativeness (Inform.) and factual consistency (Factual.) \n \n The Krippendorff's $\\alpha$s are $0.33$ and $0.62$ for the two aspects on XSum, and $0.34$ and $0.89$ on CNN\/DM. Our CL method using low confidence summaries is more frequently rated as better for informativeness and factuality on the more abstractive dataset XSum.\n }\n \\label{tab:human_eval}\n \n\\end{table}\n\n\n\\paragraph{Pairwise Comparison with Cross-entropy.} \nWe recruit the two human annotators for our summary error study, as well as another experienced annotator, to evaluate summary \\textbf{informativeness} and \\textbf{factual consistency}. For each article, the judges are shown summaries generated by the \\textsc{CrsEntropy} model and four other systems. They then rate each system summary against the \\textsc{CrsEntropy} summary.\nAll four summaries generated by different factuality-improved models are shown in random order without system names shown, ensuring the fair comparison among them.\n\nWe randomly pick $100$ articles from each dataset used in our error analysis study in \\S~\\ref{sec:error_annotation}, and evaluate summaries generated by \\textsc{EntailRank}, unlikelihood training (\\textsc{ULL}) with negative samples constructed by \\textsc{MaskEnt}, and \\textsc{CLIFF} models trained with \\textsc{Batch} and \\textsc{SysLowCon} negative samples. All are fine-tuned from BART. Detailed evaluation guidelines are in Appendix~\\ref{appendix:human_eval}. \n\n\nTable~\\ref{tab:human_eval} shows that on the more abstractive XSum data \\textit{CL trained with low confidence samples are more frequently rated as being more informative and more factual} than \\textsc{CrsEntropy} summaries. This echos our automatic evaluations with QuestEval in \\S~\\ref{subsec:autoeval}.\nOn CNN\/DM, all models trained with negative samples produce summaries with better informativeness and faithfulness.\nIn contrast, \\textsc{EntailRank} summaries are less distinguishable from outputs by \\textsc{CrsEntropy} on both datasets, as more ties are found. \nWe show sample outputs in Fig.~\\ref{fig:intro_sample}, with additional examples in Appendix~\\ref{appendix:outputs}. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{figures\/human_eval_error_type_dist.pdf}\n \\vspace{-2mm}\n \\caption{Portions of summaries with errors. CL models consistently reduce both types of errors.\n }\n \\label{fig:human_eval_error_dist}\n \\vspace{-3mm}\n\\end{figure}\n\n\\paragraph{Intrinsic vs. Extrinsic Errors.}\nNext, the annotators are asked to label text spans with intrinsic and extrinsic errors as done in \\S~\\ref{sec:error_annotation}. \nFig.~\\ref{fig:human_eval_error_dist} shows that \\textit{CL is more effective at reducing extrinsic errors than unlikelihood training can} on both datasets.\nWe also observe slight decreases of world knowledge in the summaries (figure attached in Appendix~\\ref{appendix:human_eval}). \n\n\\paragraph{Error Correction Operations.}\nFinally, with reference to \\textsc{CrsEntropy} summaries, human judges are instructed to label each system summary as whether it corrects any error by \\textsc{CrsEntropy} using \\textbf{deletion} of the incorrect content, \\textbf{substitution} with factual information, or \\textbf{both}. \nAs seen in Fig.~\\ref{fig:error_correct_technique}, CL-based models restore factually consistent information, e.g., by replacing erroneous names and numbers with correct ones, more frequently than unlikelihood training or entailment reranking. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{figures\/error_correct_dist.pdf}\n \\vspace{-2mm}\n \\caption{Summaries use different portions of error correction operations. \n Contrastive learning with \\textsc{SysLowCon} (CL.SLC) and \\textsc{Batch} (CL.B) substitute errors with correct content more often than unlikelihood training with \\textsc{MaskEnt} and \\textsc{EntailRank}. \n \n \n }\n \\label{fig:error_correct_technique}\n \\vspace{-3mm}\n\\end{figure}\n\n\n\n\\subsection{Variants of Summary Representation}\n\\label{subsec:ablaton}\n\nSample representation is critical for CL to be effective. Here we investigate summary representation variants as discussed in \\S~\\ref{sec:method}. \nThere are two major considerations: (1) Should we consider all tokens in a summary or only representative ones (e.g., entities or last token)? (2) Should additional transformation, i.e., an MLP, be used?\n\nExperiments on XSum using three negative sample construction strategies demonstrate that \\textit{averaging the decoder outputs of all tokens and adding an MLP projection yield the best overall performance}, as shown in Table~\\ref{tab:ablation_representation}. The implications are at least two-fold. \nFirst, even for entity- or relation-triggered sample modifications, using more global context helps with CL training. \nSecond, additional transformation can help avoid model degeneration. For instance, more nonsensical and repetitive content is produced by variants without MLP. \n\n\n\\begin{table}[t]\n \\centering\n \\setlength{\\tabcolsep}{2.5pt}\n \\fontsize{8.5}{10}\\selectfont\n \\begin{tabular}{lccccccc}\n \\toprule\n \\multicolumn{2}{l}{} & \\multicolumn{2}{c}{\\textbf{\\textsc{SwapEnt}}} & \\multicolumn{2}{c}{\\textbf{\\textsc{MaskRel}}} & \\multicolumn{2}{c}{\\textbf{\\textsc{SysLowCon}}} \\\\\n \\textbf{Rep.} & \\textbf{MLP} & \\textbf{QEval} & \\textbf{FC} & \\textbf{QEval} & \\textbf{FC} & \\textbf{QEval} & \\textbf{FC} \\\\\n \\midrule\n \\multicolumn{8}{l}{\\textit{BART}} \\\\\n Last & \\checkmark & 33.15 & 25.10 & 33.20 & 25.29 & 33.10 & 24.85 \\\\\n Last & & \\textcolor{red!53}{+0.13} & \\textcolor{red!42}{+0.02} & \\textcolor{blue!21}{--0.01} & \\textcolor{blue!72}{--0.32} & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!50}{--0.10} \\\\\n Entity & \\checkmark & \\hlc[xgreen]{\\textbf{33.35}} & 25.41 & 33.34 & 25.44 & 33.32 & 24.46 \\\\\n Entity & & \\textcolor{blue!50}{--0.13} & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!41}{--0.14} & \\textcolor{blue!45}{--0.05} & \\textcolor{blue!69}{--0.29} & \\textcolor{red}{+0.72} \\\\\n All & \\checkmark & 33.30 & \\hlc[xgreen]{\\textbf{25.67}} & \\hlc[xgreen]{\\textbf{33.35}} & \\hlc[xgreen]{\\textbf{25.69}} & \\hlc[xgreen]{\\textbf{33.35}} & \\hlc[xgreen]{\\textbf{25.47}} \\\\\n All & & \\textcolor{blue!43}{--0.23} & \\textcolor{blue!90}{--0.80} & \\textcolor{blue!44}{--0.04} & \\textcolor{blue!88}{--0.48} & \\textcolor{blue!46}{--0.26} & \\textcolor{blue!80}{--0.40} \\\\\n \\midrule\n \\multicolumn{8}{l}{\\textit{PEGASUS}} \\\\\n Last & \\checkmark & 33.07 & \\hlc[xgreen]{\\textbf{25.45}} & 32.99 & 25.09 & 33.18 & 24.94 \\\\\n Last & & \\textcolor{blue!47}{--0.07} & \\textcolor{blue!96}{--0.56} & \\textcolor{red!41}{+0.01} & \\textcolor{blue!41}{-0.01} & \\textcolor{blue!47}{--0.02} & \\textcolor{blue!44}{--0.04} \\\\\n Entity & \\checkmark & 33.03 & 25.43 & 33.05 & 24.77 & 33.20 & 24.59 \\\\\n Entity & & \\textcolor{red!41}{+0.01} & \\textcolor{blue!74}{--0.34} & \\textcolor{blue!42}{--0.05} & \\textcolor{red}{\\textbf{\\hlc[xgreen]{+0.64}}} & \\textcolor{blue!70}{--0.30} & \\textcolor{red!45}{+0.05} \\\\\n All & \\checkmark & \\hlc[xgreen]{\\textbf{33.09}} & 25.09 & 33.06 & 25.28 & \\hlc[xgreen]{\\textbf{33.21}} & \\hlc[xgreen]{\\textbf{25.18}} \\\\\n All & & \\textcolor{blue!51}{--0.11} & \\textcolor{red!65}{+0.25} & \\textcolor{red!43}{\\hlc[xgreen]{\\textbf{+0.03}}} & \\textcolor{blue!59}{-0.19} & \\textcolor{blue!42}{--0.02} & \\textcolor{blue}{--0.80} \\\\\n \\bottomrule\n \\end{tabular}\n \\vspace{-2mm}\n \\caption{\n Comparing different formulations of summary representation in CL. \n For models without MLP, we display score changes from their counterparts. \n Overall, using all tokens with MLP produces better summaries. \n }\n \\label{tab:ablation_representation}\n \\vspace{-2mm }\n\\end{table}\n\n\\section{Related Work}\n\\label{sec:related_work}\n\n\n\\paragraph{Factuality Improvement and Evaluation.} \nNeural abstractive summaries often contain unfaithful content with regard to the source~\\cite{falke-etal-2019-ranking}. To improve summary factuality, three major types of approaches are proposed. First, a separate correction model is learned to fix errors made by the summarizers~\\cite{zhao-etal-2020-reducing,chen-etal-2021-improving}, including replacing entities absent from the source~\\cite{dong-etal-2020-multi} or revising all possible errors~\\cite{cao-etal-2020-factual}. \nThe second type targets at modifying the sequence-to-sequence architecture to incorporate relation triplets~\\cite{cao2018faithful}, knowledge graphs~\\cite{zhu-etal-2021-enhancing}, and topics~\\cite{aralikatte-etal-2021-focus} to inform the summarizers of article facts. Yet additional engineering efforts and model retraining are often needed. \nFinally, discarding noisy samples from model training has also been investigated~\\cite{nan-etal-2021-entity,goyal-durrett-2021-annotating}, however, it often leads to degraded summary informativeness. \nIn comparison, our contrastive learning framework allows the model to be end-to-end trained and does not require model modification, thus providing a general solution for learning summarization systems. \n\n\n\n\nAlongside improving factuality, we have also witnessed growing interests in automated factuality evaluation, since popular word-matching-based metrics, e.g., ROUGE, correlate poorly with human-rated factual consistency levels~\\cite{gabriel-etal-2021-go,fabbri2021summeval}. \nEntailment-based scorers are designed at summary level~\\cite{kryscinski-etal-2020-evaluating} and finer-grained dependency relation level~\\cite{goyal-durrett-2020-evaluating}. \nQA models are employed to measure content consistency by reading the articles to answer questions generated from the summaries~\\cite{wang-etal-2020-asking,durmus-etal-2020-feqa}, or considering the summaries for addressing questions derived from the source~\\cite{scialom-etal-2019-answers}.\nThough not focusing on evaluation, our work highlights that models can produce a significant amount of world knowledge which should be evaluated differently instead of as extrinsic hallucination~\\cite{maynez-etal-2020-faithfulness}. We also show that world knowledge can possibly be distinguished from errors via model behavior understanding.\n\n\n\\medskip\n\\noindent \\textbf{Training with negative samples} has been investigated in several classic NLP tasks, such as grammatical error detection~\\cite{foster-andersen-2009-generrate} and dialogue systems~\\cite{li-etal-2019-sampling}. \nNotably, negative sampling plays a key role in word representation learning~\\cite{mikolov2013distributed} and training large masked language models, such as BERT and ALBERT, to induce better contextual representations~\\cite{devlin-etal-2019-bert,Lan2020ALBERT:}. \nFor text generation tasks, unlikelihood training is proposed to penalize the generation of negative tokens (e.g., repeated words) and sentences (e.g., contradictory responses in a dialogue system)~\\cite{Welleck2020Neural,li-etal-2020-dont,he-glass-2020-negative}. \nWe use contrastive learning that drives enhanced representation learning to better distinguish between factual and incorrect summaries, which encourages more faithful summary generation. \n\n\n\\paragraph{Contrastive Learning (CL) for NLP.}\nCL has been a popular method for representation learning, especially for vision understanding~\\cite{hjelm2018learning,pmlr-v119-chen20j}. \nOnly recently has CL been used for training language models with self-supervision~\\cite{fang2020cert}, learning sentence representations~\\cite{gao2021simcse}, and improving document clustering~\\cite{zhang2021supporting}. \nWith a supervised setup, \\citet{gunel2021supervised} adopt the contrastive objective to fine-tune pre-trained models on benchmark language understanding datasets. Using a similar idea, \\citet{liu-liu-2021-simcls} enlarge the distances among summaries of different quality as measured by ROUGE scores. \n\n\\section{\\textsc{CLIFF}: Contrastive Learning Framework for Summarization}\n\\label{sec:method}\n\n\nWe design a contrastive learning (CL)-based training objective that drives the summarization model to learn a preference of faithful summaries over summaries with factual errors.\nIt is then used for fine-tuning BART~\\cite{lewis-etal-2020-bart} and PEGASUS~\\cite{zhang2020pegasus} for training summarization models.\nFormally, let an article $x$ have a set of reference summaries $P$ (henceforth \\textit{positive samples}) and another set of erroneous summaries $N$ (\\textit{negative samples}). \nThe contrastive learning objective is~\\cite{khosla2020supervised,gunel2021supervised}:\n\n\\begin{equation}\n \\fontsize{10}{11}\\selectfont\n l_{cl}^{x} = - \\frac{1}{\\binom{|P|}{2}} \\sum_{\\substack{y_i, y_j \\in P\\\\y_j \\ne y_i}}\n \\log \\frac{\\exp(\\mathrm{sim} (\\bm{h}_i, \\bm{h}_j) \/ \\tau)}{\\sum\\limits_{\\substack{y_k \\in P \\cup N\\\\y_k \\ne y_i}} \\exp(\\mathrm{sim} (\\bm{h}_i, \\bm{h}_k) \/ \\tau)}\n \\label{eq:contrast}\n\\end{equation}\nwhere $\\bm{h}_i$, $\\bm{h}_j$, and $\\bm{h}_k$ are representations for summaries $y_i$, $y_j$, and $y_k$.\n$\\mathrm{sim}(\\cdot, \\cdot)$ calculates the cosine similarity between summary representations. $\\tau$ is a temperature and is set to $1.0$.\n\n\nImportantly, summaries in $P$ and $N$ are included in the \\textit{same batch} during training, so that the model acquires better representations to differentiate correct summaries from those with errors by comparing the two types of samples, thus maximizing the probabilities of the positive samples and minimizing the likelihoods of the corresponding negative samples.\nThe \\textbf{CL objective} on the full training set, denoted as $\\mathcal{L}_{CL}$, is the sum of losses $l_{cl}^{x}$ over all samples. \n\nTo effectively employ CL in summarization, we need to address two challenges: (1) how to automatically construct both positive and negative samples, which are critical for CL efficacy~\\cite{pmlr-v119-chen20j}, and (2) how to represent the summaries (i.e., $\\bm{h}_{\\ast}$).\nBelow we describe positive sample generation and options for $\\bm{h}_{\\ast}$, leaving the strategies for negative samples to \\S~\\ref{sec:sample_construct}. \n\n\n\\paragraph{Positive Sample Construction ($P$).}\nSummarization datasets often contain a single reference for each article. To create multiple positive samples, in our pilot study,\nwe experiment with paraphrasing with synonym substitution~\\cite{ren-etal-2019-generating}, randomly replacing words based on the prediction of masked language models~\\cite{kobayashi-2018-contextual}, and back-translation~\\cite{mallinson-etal-2017-paraphrasing}. We find back-translation to be best at preserving meaning and offering language variation, and thus use NLPAug\\footnote{\\url{https:\/\/github.com\/makcedward\/nlpaug}} to translate each reference to German and back to English. Together with the reference, the best translation is kept and added to $P$, if no new named entity is introduced.\n\n\n\\paragraph{Summary Representation ($\\bm{h}_{\\ast}$).}\nWe use the outputs of the decoder's last layer, and investigate three options that average over \\textit{all tokens}, \\textit{named entity tokens}, and the \\textit{last token} of the decoded summary. Entities and other parsing results are obtained by spaCy~\\cite{spacy}. \nWe further consider adding a multi-layer perceptron (MLP) with one hidden layer to calculate the final $\\bm{h}_{\\ast}$.\n\n\\medskip\n\\noindent \\textbf{The final training objective} combines the typical cross-entropy loss $\\mathcal{L}_{CE}$ and our contrastive learning objective: $\\mathcal{L} = \\mathcal{L}_{CE} + \\lambda \\mathcal{L}_{CL}$, where $\\lambda$ is a scalar and set to $1.0$ for all experiments. \n\n\n\\section{Negative Sample Construction}\n\\label{sec:sample_construct}\n\nHere we describe four strategies for constructing negative samples that \n\\textit{modify the references} (\\S~\\ref{subsec:swapping}-\\ref{subsec:conditional}) or use \\textit{system generated summaries} (\\ref{subsec:modelgeneration}). \n\n\\begin{table}[t]\n \n \\fontsize{8.5}{9}\\selectfont\n \\setlength{\\tabcolsep}{2pt}\n \n \\hspace{-1mm}\n \\begin{tabular}{p{76mm}}\n \\toprule\n \\textbf{\\textsc{Reference}}: A ``rare'' short-eared owl found emaciated in Flintshire is now recuperating well, the RSPCA have said. \\\\\n \\midrule \\midrule\n \\textbf{\\textsc{SwapEnt}}: Flintshire $\\rightarrow$ Bettisfield \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in \\textcolor{red!80!black}{\\textbf{Bettisfield}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{MaskEnt}}: A ``rare'' short-eared owl found emaciated in \\texttt{[MASK]} is now recuperating well, the RSPCA have said. \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in a field in \\textcolor{red!80!black}{\\textbf{South Yorkshire}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{MaskRel}}: A ``rare'' short-eared owl found \\texttt{[MASK]} in \\texttt{[MASK]} is now recuperating well, the RSPCA have said. \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found \\textcolor{red!80!black}{\\textbf{dead} in \\textbf{London}} is now recuperating well, the RSPCA have said. \\\\\n \\midrule\n \\textbf{\\textsc{RegenEnt}}: A ``rare'' short-eared owl found emaciated in \\rule{6mm}{0.15mm} \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found emaciated in \\textcolor{red!80!black}{\\textbf{Nottinghamshire}} \\textbf{is now at a wildlife centre to recover.} \\\\\n \\midrule\n \\textbf{\\textsc{RegenRel}}: A ``rare'' short-eared owl found \\rule{6mm}{0.15mm} \\\\\n \n $\\Rightarrow$ A ``rare'' short-eared owl found \\textbf{in the grounds of a former coal mine is being cared for \\textcolor{red!80!black}{by the RSPCA in Somerset.}} \\\\\n \\midrule\n \\textbf{\\textsc{SysLowCon}}: An injured \\textcolor{red!80!black}{golden} owl found in a former coal mine in \\textcolor{red!80!black}{Lancashire} is being cared for \\textcolor{red!80!black}{by the RSPCA}. \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Negative sample construction strategies (\\S~\\ref{sec:sample_construct}). For summaries edited from the reference, their differences are \\textbf{bolded}. Introduced errors are in \\textcolor{red!80!black}{\\textbf{red}}. Text before \\rule{6mm}{0.15mm} is the prefix for regeneration.\n }\n \\label{tab:construction_example}\n \n\\end{table}\n\n\n\\subsection{Entity Swap}\n\\label{subsec:swapping}\n\nEntity swap imitates intrinsic errors, as over $55\\%$ of intrinsic errors in our annotations are found to contain named entities.\nWe construct negative samples by swapping named entities in the references with other randomly selected entities of the same entity type in the source (\\textbf{\\textsc{SwapEnt}}). One sample is constructed for each entity in the reference. \nThough this idea has been studied by \\citet{kryscinski-etal-2020-evaluating}, they allow entities of different types to be used, e.g., a PERSON can be replaced by a LOCATION. \nExamples are displayed in Table~\\ref{tab:construction_example}. \n\n\\textsc{SwapEnt} has the advantage of not depending on any trained model. Yet it only introduces intrinsic errors and lacks the coverage for extrinsic errors, which is addressed by the following generation-based methods. \n\n\n\\subsection{Mask-and-fill with BART}\n\\label{subsec:unconditional}\nTo simulate extrinsic errors, we leverage large unconditional language models' capability of converting a sequence with masked tokens into a fluent and appropriate sequence. Specifically, we replace each named entity in a reference with a \\texttt{[MASK]} token and encode it with BART (without any fine-tuning). BART then fills this partially masked summary with newly generated entities (\\textbf{\\textsc{MaskEnt}}). BART is chosen since it can fill \\texttt{[MASK]} with varying number of tokens.\nFor each entity in the reference, we sample three summaries and only retain the ones containing at least one entity that is absent from both the source and the reference.\n\nUp to now, the two introduced strategies both focus on incorrect named entities. To cover more diverse extrinsic and intrinsic errors~\\cite{goyal-durrett-2020-evaluating}, we extend \\textsc{MaskEnt} to contain relations (\\textbf{\\textsc{MaskRel}}). \nWe first obtain dependency relations using Stanza~\\cite{qi-etal-2020-stanza}, with each relation denoted as $<$\\texttt{gov}, \\texttt{rel}, \\texttt{dep}$>$. \nTo incorporate more context, we consider noun phrase spans enclosing the token of \\texttt{gov} or \\texttt{dep} if it is a content word and the noun phrase contains a named entity. \nSimilar to \\textsc{MaskEnt}, three negative samples are generated by BART based on the input with both \\texttt{gov} and \\texttt{dep} spans masked in the reference. Only the samples that introduce any new dependency relation that is not contained in the source nor the reference are kept. Specifically, we consider a match of a dependency relation as the same form or synonyms of its \\texttt{gov} and and \\texttt{dep} is found in the source or the reference with the same relation. \n\n\nBoth \\textsc{MaskEnt} and \\textsc{MaskRel} can create more extrinsic errors compared to other strategies introduced in this section, since negative samples are generated without being grounded on the source articles. However, their constructed negative samples may contained drifted topics that can be easily detected by a summarization model, resulting with less efficient training signals. \n\n\n\\subsection{Source-conditioned Regeneration}\n\\label{subsec:conditional}\n\nTo ground negative sample generation with the article, we further design a regeneration strategy based on conditional generation. For each named entity in the reference, we treat the text before it as a \\textit{prompt}.\nA summarizer, e.g., fine-tuned from BART or PEGASUS, first reads in the source using the encoder, then receives the prompt as the first part of the decoder output, and finally decodes the rest of the content based on nucleus sampling~\\cite{Holtzman2020The} with a cumulative probability threshold of $0.7$. \nThe prompt and the regenerated text comprise the final negative sample. This method is denoted as \\textbf{\\textsc{RegenEnt}}. \n\nWe also extend entities to relations with expanded governor and dependent spans (\\textbf{\\textsc{RegenRel}}). Here, we consider a prompt as the text before the \\texttt{gov} or \\texttt{dep} span, whichever occurs first. \nFor both \\textsc{RegenEnt} and \\textsc{RegenRel}, three negative samples are generated for each prompt, and a sample is kept if it introduces any new entity (for \\textsc{RegenEnt}) or dependency relation (for \\textsc{RegenRel}) with regard to the source and the reference. \n\nNegative samples generated by both methods are more relevant to the article than the mask-and-fill strategy, yet they may still miss certain types of errors and differ from real model outputs, since they are modified from the reference summaries. \n\n\n\\subsection{System Generation}\n\\label{subsec:modelgeneration}\n\nMotivated by the model confidence analysis in \\S~\\ref{sec:error_annotation}, we explore using system generated summaries as negative samples. We first run fine-tuned BART or PEGASUS on the same training set to decode summaries. For each summary, we check the model confidence on the first token of each proper noun and number span. If the probability is below a threshold, we keep it as a negative sample (\\textbf{\\textsc{SysLowCon}}). The threshold is tuned by maximizing F1 based on our error annotations. \n\nWe consider all beams at the last decoding step as candidates. We use beam sizes of $6$ and $4$ for XSum and CNN\/DM.\nStatistics of negative samples constructed by different strategies are in Appendix~\\ref{appendix:dataset_stat}.\n\n\n\\section{Experiment Setup}\n\\label{sec:exp_setup}\n\n\\paragraph{Evaluation Metrics.} \nQuestEval~\\cite{scialom2020QuestEval} is used as the main metric to evaluate summaries' factual consistency. \nGiven an article and a summary, QuestEval first generates natural language questions for entities and nouns from both. \nA QA model then consumes the article to answer questions derived from the summary, producing a score. Another score is obtained from a QA model addressing article-based questions after reading the summary.\nThe final QuestEval score is the harmonic mean of the two. We use the version with learned weights for questions, which has shown high correlation with human judged consistency and relevance.\n\n\nWe further use FactCC~\\cite{kryscinski-etal-2020-evaluating}, trained based on their negative sample construction method, to measure if the summary can be entailed by the source.\nWe also report ROUGE-L~\\cite{lin-2004-rouge}. \nBoth FactCC and ROUGE-L reasonably correlate with summary factuality as judged by human~\\cite{pagnoni2021understanding}. \n\nBased on our error annotations, we report the correlations between each metric and the error rate---percentage of tokens being part of an error span, and the raw number of errors (Table~\\ref{tab:metric_correlation}).\nQuestEval correlates better on both aspects than other metrics. \n\n\n\\begin{table}[t]\n \\centering\n \\small\n \\setlength{\\tabcolsep}{4.5pt}\n \\begin{tabular}{lcccc}\n \\toprule\n \\textbf{Metric} & \\multicolumn{2}{c}{\\textbf{XSum}} & \\multicolumn{2}{c}{\\textbf{CNN\/DM}} \\\\\n & \\textbf{\\% of Err} & \\textbf{\\# of Err} & \\textbf{\\% of Err} & \\textbf{\\# of Err} \\\\\n \\midrule\n QuestEval & \\textbf{-0.43}\\rlap{$^\\ast$} & \\textbf{-0.25}\\rlap{$^\\ast$} & \\textbf{-0.33}\\rlap{$^\\ast$} & \\textbf{-0.29}\\rlap{$^\\ast$} \\\\\n FactCC & -0.02 & -0.15\\rlap{$^\\ast$} & -0.13\\rlap{$^\\ast$} & -0.12\\rlap{$^\\ast$} \\\\\n ROUGE-1 & -0.16\\rlap{$^\\ast$} & -0.02 & -0.03 & -0.06 \\\\\n ROUGE-2 & -0.11\\rlap{$^\\ast$} & -0.05 & -0.02 & -0.04 \\\\\n ROUGE-L & -0.13\\rlap{$^\\ast$} & -0.03 & -0.06 & -0.08 \\\\\n \\bottomrule\n \\end{tabular}\n \n \\caption{\n Pearson correlation between metrics and error rates and numbers of errors. \n $\\ast$: p-value $< 0.05$.\n %\n }\n \\label{tab:metric_correlation}\n \n\\end{table}\n\n\\paragraph{Comparisons.}\nIn addition to the models fine-tuned with cross-entropy loss (\\textbf{\\textsc{CrsEntropy}}), we consider reranking beams based on FactCC score (also our metric) at the last decoding step (\\textbf{\\textsc{EntailRank}}). \nWe also include three common methods of improving factuality: \n(1) (\\textbf{\\textsc{Correction}}) fine-tunes BART to fix summary errors as a separate step~\\cite{cao-etal-2020-factual}. \n(2) \\textbf{\\textsc{SubsetFT}} fine-tunes large models based on training samples without any dependency relation error~\\cite{goyal-durrett-2021-annotating}, with released checkpoint only available for XSum. \n(3) \\textbf{\\textsc{FASum}} modifies Transformer by incorporating knowledge graphs for factual consistency~\\cite{zhu-etal-2021-enhancing}, with model outputs only on CNN\/DM. \n\n\nMoreover, we compare with \\textbf{unlikelihood training} that penalizes the probabilities of all tokens in a negative sample~\\cite{li-etal-2020-dont}. Given a negative sample $y'$, the loss is defined as $-\\sum_{t=1}^{| y' |} \\log (1 - p(y'_t | y'_{1:t-1}, x))$, where $p(y'_t | y'_{1:t-1}, x)$ is the output probability at the $t$-th step. We combine the unlikelihood training objective with cross-entropy loss with equal weights for fine-tuning. \n\n\nLastly, we compare our negative sample strategies with negative samples constructed for training the FactCC scorer, denoted as \\textbf{\\textsc{FCSample}}. \nFor CL only, we compare with using other samples in the same batch as negative samples (\\textbf{\\textsc{Batch}}), a common practice for CL-based representation learning~\\cite{gao2021simcse, zhang2021supporting}.\n\n\\section{Additional Analysis for Summary Error Annotation}\n\\label{appendix:error_annotation}\n\nWe hire two fluent English speakers to annotate summary errors on XSum and CNN\/DailyMail (CNN\/DM). They annotate a common batch of 100 summaries generated by summarizers fine-tuned from BART and PEGASUS, with 50 articles in each batch. The two annotators are shown 50 HTML pages in a batch, each of which contains an article and two summaries generated by the two models. \nThe detailed annotation guideline is given in Fig.~\\ref{fig:annotation_guideline}.\n\nFor our analysis on token generation probabilities, we additionally show the distributions of the first token's probability for nouns and verbs in Fig.~\\ref{fig:first_token}. We also report the distributions of the non-first token's probability for proper nouns, numbers, nouns, and verbs in Fig.~\\ref{fig:nonfirst_token}. As can be seen, tokens within extrinsic and intrinsic errors have high generation probabilities when they are non-first tokens.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/prob_other_first.pdf}\n \\caption{Probability distributions of generating the \\textit{first} tokens of nouns and verbs, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens. No verb is annotated as world knowledge.}\n \\label{fig:first_token}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/prob_all_nonfirst.pdf}\n \\caption{Probability distributions of generating the \\textit{non-first} tokens of proper nouns, numbers, nouns, and verbs, grouped by extrinsic errors, intrinsic errors, world knowledge, and other correct tokens. Non-first tokens do not exist for numbers and verbs, as they only contain single tokens.}\n \\label{fig:nonfirst_token}\n\\end{figure}\n\n\n\n\\section{Statistics for Datasets and Training Samples}\n\\label{appendix:dataset_stat}\n\n\\paragraph{Summarization Datasets.}\n\nWe follow the official data splits for the two datasets, with the number of samples in each split listed in Table~\\ref{tab:data_split}.\n\n\\begin{table}[t]\n \\centering\n \\small\n \\begin{tabular}{lccc}\n \\toprule\n \\textbf{Dataset} & \\textbf{Train} & \\textbf{Validation} & \\textbf{Test} \\\\\n \\midrule\n XSum & 204{,}045 & 11{,}332 & 11{,}334 \\\\\n CNN\/DM & 287{,}227 & 13{,}368 & 11{,}490 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Numbers of samples in train\/validation\/test splits of XSum and CNN\/DM.}\n \\label{tab:data_split}\n\\end{table}\n\n\\paragraph{Positive Samples.}\n\nWe observe unfaithful paraphrases by back-translation for some reference summaries, which are mainly due to the introduction of new entities and the rewriting of quoted text. Thus, we discard samples generated by back-translation that contain new entities and inconsistent quoted text. Finally, we obtain $182{,}114$ and $91{,}468$ positive samples by back-translation on XSum and CNN\/DM.\n\n\\paragraph{Negative Samples.}\n\nFor consistency, we use the summarizer fine-tuned from BART in \\textsc{RegenEnt}, \\textsc{RegenRel} (\\S~\\ref{subsec:conditional}), and \\textsc{SysLowCon} (\\S~\\ref{subsec:modelgeneration}) strategies. We tune a threshold to select negative samples from model generations in our \\textsc{SysLowCon} strategy. The threshold is set to $0.21$, with F1 scores of $73.99$ and $40.49$ on XSum and CNN\/DM annotated samples generated by BART.\n\nThe numbers of negative samples constructed by each strategy for training on XSum and CNN\/DM are shown in Table~\\ref{tab:num_neg_samples}.\n\\textsc{SysLowCon} constructs the least negative samples in total, while it achieves the best results as reported in our main paper (\\S~\\ref{subsec:autoeval}), indicating that its negative samples are more effective for training.\n\n\\begin{table}[ht]\n \\centering\n \\small\n \\begin{tabular}{lcc}\n \\toprule\n \\textbf{Strategy} & \\textbf{XSum} & \\textbf{CNN\/DM} \\\\\n \\midrule\n \\textsc{FCSample} & 936{,}164 & 1{,}291{,}710 \\\\\n \\textsc{SwapEnt} & 438{,}003 & 1{,}617{,}764 \\\\\n \\textsc{MaskEnt} & 360{,}795 & 1{,}050{,}200 \\\\\n \\textsc{MaskRel} & 391{,}224 & 1{,}345{,}317 \\\\\n \\textsc{RegenEnt} & 732{,}986 & 1{,}941{,}886 \\\\\n \\textsc{RegenRel} & 993{,}694 & 1{,}453{,}044 \\\\\n \\textsc{SysLowCon} & 401{,}112 & 502{,}768 \\\\\n \n \\bottomrule\n \\end{tabular}\n \\caption{Numbers of negative samples constructed by different strategies on XSum and CNN\/DM.}\n \\label{tab:num_neg_samples}\n\\end{table}\n\n\n\n\\section{Implementation Details}\n\\label{appendix:implementation}\n\nWe use Fairseq~\\cite{ott2019fairseq} and Huggingface Transformers~\\cite{wolf-etal-2020-transformers} for our experiments with BART and PEGASUS. Our experiments are conducted on the RTX 8000 GPU with 48GB memory and the A100 GPU with 40GB memory.\n\n\\paragraph{Training Settings.}\n\nFor hyperparameters, we follow \\citet{lewis-etal-2020-bart} for BART and \\citet{zhang2020pegasus} for PEGASUS.\nDuring training, we randomly select 5 and 4 negative samples for each input article in XSum and CNN\/DM. Mixed-precision training is not supported by the PEGASUS implementation and is utilized on BART only.\n\n\\paragraph{Decoding Settings.}\n\nWe use the beam search algorithm to decode summaries.\nFor BART, we set the beam sizes as 6 and 4 on XSum and CNN\/DM. A beam size of 8 is used for PEGASUS on both datasets.\n\n\\paragraph{Running Time and Model Sizes.}\n\nThe BART-based models take 6 and 13 hours for training on XSum and CNN\/DM, and it takes 1.5 hour to decode on the two datasets.\nMeanwhile, training the PEGASUS-based models takes 8 and 25 hours for XSum and CNN\/DM, and the decoding takes 1 hour.\n\nAs for model sizes, our BART-based models and PEGASUS-based models have 400M and 568M parameters.\n\n\n\\section{Human Evaluation}\n\\label{appendix:human_eval}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/human_eval_world_dist.pdf}\n \\caption{Percentages of samples containing world knowledge as labeled by human on the outputs of XSum and CNN\/DM.}\n \\label{fig:percent_world}\n\\end{figure}\n\nIn \\S~\\ref{subsec:humaneval}, we demonstrate the percentages of samples containing intrinsic errors and extrinsic errors for each model evaluated by human judges. \nHere, we report the percentages of samples containing world knowledge in Fig.~\\ref{fig:percent_world}. On XSum, all models produce less world knowledge compared to the model trained with cross-entropy loss, while generating similar or greater amounts of samples with world knowledge on CNN\/DM.\n\nOur human evaluation guideline is shown in Fig.~\\ref{fig:human_eval_guideline}.\n\n\n\n\\section{Sample Outputs}\n\\label{appendix:outputs}\n\nWe include more sample outputs in Fig.~\\ref{fig:generation_examples}.\n\n\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\fontsize{10}{12}\\selectfont\n \\begin{tabular}{p{0.88\\textwidth}}\n \\toprule\n In this study, you will first read article-summary pairs and then identify three types of text spans in the summaries. These spans include content that is contradicted by or cannot be implied from the article. The description for each type is described below: \\\\\n \\begin{itemize}\n \\item \\textbf{Intrinsic errors:} Text spans that misconstruct phrases or clauses from the article. \n \\item \\textbf{Extrinsic errors:} Text spans that include words that are not in the article and are not verifiable or cannot be verified by Wikipedia. \n \\item \\textbf{world knowledge:} Text spans that contain information that is not covered by the article but can be validated by Wikipedia. \n \\end{itemize} \\\\\n When selecting spans, you should always make sure the spans are complete words. \\\\\n \\\\\n In practice, you should follow the these steps carefully: (1) read the article and summaries carefully; (2) figure out if there is content contradicted by or not presented in the article; (3) label the span as an intrinsic error if it misconstructs phrases or clauses from the article; (4) if the span does not belong to intrinsic errors, search within Wikipedia and determine whether the content in the span can be verified; (5) label it as world knowledge if the it can be verified by Wikipedia, otherwise label it as an extrinsic error. \\\\\n \\midrule\n \\textbf{Example annotations 1} \\\\\n \\textbf{Article:} Isis Academy in Oxford said it had rebranded as ``Iffley Academy'' to protect its ``reputation, integrity and image''. \\textbf{The name `Isis' was originally chosen as the school is near to the section of the River Thames of the same name}. Formerly Iffley Mead School, it became Isis Academy in 2013. A statement issued by the school said it had changed name following ``the unforeseen rise of ISIS (also known as ISIL and the Islamic State) and related global media coverage of the activities of the group''. ``Our priority is to remove the detrimental impact which the name `Isis' had on pupils, their families and our staff.'' Last year a language school in the city removed Isis from its name for the same reason. The Isis is the name given to the part of the River Thames above Iffley Lock in Oxford. It is also the name of the goddess wife of the god Osiris in Egyptian beliefs. \\\\\n \n \\textbf{Summary:} A school that \\hlc[crimsonglory!40]{was named after} the Islamic State (IS) militant group has changed its name. \\\\\n \\textbf{Explanation:} \\textit{``was name after''} is an intrinsic error contradicted by the article sentence in \\textbf{bold}. \\\\%The name is not chosen because of the Islamic State. \\\\\n \\midrule\n \\textbf{Example annotations 2} \\\\\n \\textbf{Article:} Khalil Dale, 60, was abducted in \\textbf{Quetta} in January 2012 and was found dead on a roadside a few months later. He had been beheaded. A note next to his body said he was killed because a ransom had not been paid. Mr Dale was born in York but lived in Dumfries. He spent 30 years working in countries including Somalia, Afghanistan and Iraq. An inquest into his death was held at Chesterfield Coroners Court because he is buried in Derbyshire. The court heard that the Muslim convert, who was formerly known as Kenneth, worked as a humanitarian assistance relief worker. Following his abduction, negotiations were undertaken by the International Committee of the Red Cross with the help of the UK government. His body was found on 29 April 2012. The inquest was told that he died as a result of decapitation. Senior coroner Dr Robert Hunter concluded that Mr Dale was unlawfully killed while providing international humanitarian assistance. \\\\\n \\textbf{Summary:} A British aid worker was unlawfully killed by \\hlc[bleudefrance!40]{Islamist militants} in \\hlc[yellow!60]{Pakistan}, an inquest has heard. \\\\\n \\textbf{Explanation:} \\textit{``Islamist militant''} is an extrinsic error as it can not be found in or inferred from the article. The information is also not verifiable by Wikipedia. \\textit{``Pakistan''} is world knowledge as \\textbf{Quetta} in the article is a city in Pakistan according to Wikipedia. \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Guideline for our summary error annotation (\\S~\\ref{sec:error_annotation}).}\n \\label{fig:annotation_guideline}\n\\end{figure*}\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\fontsize{10}{12}\\selectfont\n \\begin{tabular}{p{0.92\\textwidth}}\n \\toprule\n In this study, you will evaluate 100 sets of summaries produced by four systems. For each set, its corresponding article and a baseline summary are shown before the four system summaries. The errors in the baseline summary are highlighted. \\\\\n Please \\textit{first read the article and the baseline summary} and then \\textit{compare each system summary against the baseline summary} based on \\textbf{informativeness} and \\textbf{factual consistency}. In addition, please decide the \\textbf{operations} made by the system to achieve better factual consistency. \\\\\n For informativeness and factual consistency, you need to label whether the system summary is better or worse than the baseline summary. You can also label the system summary as tying with the baseline summary. \\\\\n You need to consider two types of operations: \\textbf{deletions} and \\textbf{substitutions}. Please label the system summary as making deletions, substitutions, or both operations. Examples for the aspects and the operations are as follows. \\\\\n \\midrule\n \n \\textbf{Article:} Alexys Brown, also known as Lexi, died at her home in Emmadale Close, Weymouth, on Thursday. An investigation is under way to discover how she became trapped. A post-mortem examination is due to be carried out this week. It was originally hoped the appeal would raise \u00a32,000. Alison Record, who started the Just Giving appeal, said she was \"heart broken\" over the death. ``Everybody by now has heard of the terrible tragedy the Brown family have suffered with the loss of their beautiful and beloved little girl Lexi,'' the appeal page reads. Many other comments have been posted on the appeal page. Steph Harris said: ``Thinking of you all at this devastating time, fly high beautiful princess. Love Steph and family xxx'' Lesley Andrews added: ``No amount of money will take away the pain, but so much love comes with every penny. Take care. xx'' Aster Group, the housing association responsible for managing the home, is assisting with the police investigation. The Health and Safety Executive (HSE) is also investigating. Dorset County Council said it had not installed the disabled lift at the property. \\\\\n \n \\textbf{Baseline Summary:} An appeal to raise \\hlc[crimsonglory!40]{10,000 pounds} for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}. \\\\\n \n \\smallskip\n \\textbf{Informativeness:} Whether the summary captures salient content from the input article. Note that incorrect content should be considered as invalid. \\\\\n \\textbf{Win.} \\textit{An appeal to raise money for the family of a three-year-old girl who died after getting stuck in a lift was originally hoped for raising \u00a32,000.} The target money of the appeal is a salient information. \\\\\n \\textbf{Tie.} \\textit{An appeal to raise money for the family of a girl who died after getting stuck in a lift has raised more than \u00a320,000.} Compared to the baseline, missing incorrect information does not affect the informativeness. \\\\\n \\textbf{Lose.} \\textit{An appeal to raise money for the family of a three-year-old girl has raised more than \u00a320,000.} This system summary does not mention the death of the girl, which is a salient content of the article. \\\\\n \n \\smallskip\n \\textbf{Factual Consistency:} Whether the summary is factually correctly based on the article and knowledge from Wikipedia. \\\\\n \\textbf{Win.} \\textit{An appeal has been set up for the family of an \\hlc[bleudefrance!40]{eight-year-old} girl who died after becoming trapped in a lift at her Dorset home.} This system summary does not generate the incorrect numbers of money. \\\\\n \\textbf{Tie.} \\textit{An appeal to raise \\hlc[crimsonglory!40]{5,000 pounds} for the family of a \\hlc[bleudefrance!40]{seven-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} This system summary makes similar errors to the baseline. \\\\\n \\textbf{Lose.} \\textit{\\hlc[crimsonglory!40]{The family} of an \\hlc[bleudefrance!40]{eight-year-old} girl who died after becoming trapped in a lift at her Dorset home \\hlc[crimsonglory!40]{have set a fundraising target} of \\hlc[bleudefrance!40]{10,000 pounds}.} This system summary fabricates an event \\textit{The family have set a fundraising target}, which is more severe than errors of modifiers. \\\\\n \n \\smallskip\n \\textbf{Deletion:} The incorrect content in the baseline summary is deleted. \\\\\n - \\textit{An appeal for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} The error \\textit{``10{,}000 pounds''} is deleted. \\\\\n \n \\textbf{Substitution:} The incorrect content in the baseline summary is replaced with correct one. \\\\\n - \\textit{An appeal to raise 2,000 pounds for the family of a \\hlc[bleudefrance!40]{three-year-old} girl who died after becoming trapped in a lift has raised \\hlc[bleudefrance!40]{more than 20,000 pounds}.} The error \\textit{``10{,}000 pounds''} is substituted with \\textit{``2,000 pounds''}, which is the correct information. \\\\\n \n \\bottomrule\n \\end{tabular}\n \\caption{Guideline for our human evaluation (\\S~\\ref{subsec:humaneval}).}\n \\label{fig:human_eval_guideline}\n\\end{figure*}\n\n\n\\begin{figure*}[th]\n \\centering\n \\small\n \\begin{tabular}{p{0.95\\textwidth}}\n \\toprule\n \\textbf{Example 1} \\\\\n \\midrule\n \\textbf{CNN\/DM Article:} At the grand old age of 75, Jack Nicklaus is still capable of hitting aces. The Golden Bear added another magic moment to his storied career at Augusta National in the Par-3 Contest. Stepping up to the tee on the 130-yard fourth, the greatest golfer of all time saw his shot sail beyond the flag before spinning back into the hole. Jack Nicklaus gave the crowd something to cheer with a hole in one on the fourth during the Par-3 Contest. Nicklaus holds up his ball to an adoring crowd as Gary Player (left) and Ben Crenshaw salute the great. Crenshaw and Nicklaus fist pump following his ace on the 130-yard hole at Augusta National. Nicklaus was playing alongside Gary Player and Ben Crenshaw as the first roar of Masters week filled the skies in a sunswept Georgia. 'I had an interview this morning, and I said \"well, all I've got to do is go out and win the Par-3 and make a hole and one\", and I make a hole in one,' Nicklaus said afterwards. . The ace was Nicklaus' first on the hallowed turf of Augusta, but the veteran saw his achievement eclipsed by that of Camilo Villegas on a day which saw five hole-in-ones, levelling the record from 2002. The Colombian followed Nicklaus with an ace on the fourth and also managed one on the eighth, but he couldn't quite land the title, losing out to world No 74 Kevin Streelman in a play-off. Nicklaus became the youngest person to wear a green jacket in 1963, and collected his sixth in 1986. He is one of five men to complete the career grand slam, an accolade which favourite Rory McIlroy can achieve if he wins his third major in succession. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas and \\hlc[crimsonglory!40]{Kevin Streelman} also hit hole-in-ones on the day. \\\\\n \\textbf{Entailment reranking:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas and \\hlc[crimsonglory!40]{Kevin Streelman} also hit hole-in-ones on day one. \\\\\n \\textbf{Unlikelihood training:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas followed Nicklaus with a hole-in-one on the same hole. Nicklaus is one of five men to complete the career grand slam. \\\\\n \\textbf{Contrastive learning:} Jack Nicklaus hit a hole in one on the fourth during the Par-3 Contest. The 75-year-old was playing alongside Gary Player and Ben Crenshaw. Camilo Villegas followed Nicklaus with a hole-in-one on the eighth. \\\\\n \\midrule\n \\textbf{Example 2} \\\\\n \\midrule\n \\textbf{XSum Article:} The British supermarket giant, the world's third largest retailer, has been looking to dispose of its loss-making US food chain for some time. Around 4,000 of Fresh \\& Easy's 5,000 employees are to transfer to the new business as part of the deal. Tesco said in a statement that this is an \"orderly and efficient exit from the US market\". Under the terms of the deal, Tesco will loan the new business around \u00a380m. Fresh \\& Easy, which operates 200 stores across California, Nevada and Arizona, has lost money since opening in 2007. Tesco did not specify which stores are being transferred to Yucaipa, but it is understood the remaining 50 shops will be closed. It will cost Tesco \u00a3150m to dispose of the Fresh \\& Easy brand, but part of that is a loan which may be paid back. Also, if Yucaipa is sucessful and the business becomes profitable, Tesco has the option to buy a stake. Yucaipa is a Los Angeles based private equity company focused on supermaket and restaurant businesses and is run by billionaire investor Ron Burkle. In April of this year, Tesco reported its first fall in annual profits for 20 years. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Tesco has agreed to sell its US food chain Fresh \\& Easy to Yucaipa in a deal \\hlc[bleudefrance!40]{worth about 150m}, the BBC understands. \\\\\n \\textbf{Entailment reranking:} Tesco is to sell its US food chain Fresh \\& Easy to Yucaipa \\hlc[bleudefrance!40]{for an undisclosed sum}. \\\\\n \\textbf{Unlikelihood training:} Tesco has agreed to sell its US food business, Fresh \\& Easy, to a private equity firm called Yucaipa \\hlc[bleudefrance!40]{for an undisclosed sum}. \\\\\n \\textbf{Contrastive learning:} Tesco has agreed to sell its US food chain, Fresh \\& Easy, to Yucaipa. \\\\\n \\midrule\n \\textbf{Example 3} \\\\\n \\midrule\n \\textbf{XSum Article:} Neil Aspin's promotion-chasing hosts have not lost in nine National League matches while Adam Lockwood's side are unbeaten in five. Guiseley went ahead on 15 minutes against the run of play when a throw-in found James Hurst who squared to Jake Lawlor to stroke into an empty net. Gateshead defender Liam Hogan superbly blocked Jordan Preston's effort and Guiseley keeper Jonny Maxted then saved well from Wesley York's shot just before the break. The hosts, who started the second half well, levelled on 62 minutes when a slip by half-time substitute Derek Asamoah let York curl sweetly into the top-right corner from the edge of the box. Match report supplied by the Press Association. Match ends, Gateshead 1, Guiseley 1. Second Half ends, Gateshead 1, Guiseley 1. Substitution, Guiseley. Michael Rankine replaces Jordan Preston. Substitution, Gateshead. Luke Hannant replaces Gus Mafuta. Gus Mafuta (Gateshead) is shown the yellow card. Substitution, Guiseley. Adam Boyes replaces Jake Cassidy. Goal! Gateshead 1, Guiseley 1. Wes York (Gateshead). Substitution, Guiseley. Derek Asamoah replaces Kevan Hurst. Second Half begins Gateshead 0, Guiseley 1. First Half ends, Gateshead 0, Guiseley 1. Simon Walton (Guiseley) is shown the yellow card. Goal! Gateshead 0, Guiseley 1. Jake Lawlor (Guiseley). First Half begins. Lineups are announced and players are warming up. \\\\\n \\midrule\n \\textbf{Cross-entropy:} Gateshead and Guiseley maintained their \\hlc[crimsonglory!40]{unbeaten starts to the season} with a 1-1 draw at the International Stadium. \\\\\n \\textbf{Entailment reranking:} Gateshead and Guiseley shared the spoils after a 1-1 draw at the International Stadium. \\\\\n \\textbf{Unlikelihood training:} Gateshead and Guiseley shared the spoils after a \\hlc[crimsonglory!40]{goalless} draw in the National League. \\\\\n \\textbf{Contrastive learning:} Gateshead and Guiseley shared the spoils after a 1-1 draw at Gateshead. \\\\\n \\bottomrule\n \n \\end{tabular}\n \\caption{Sample generated summaries by fine-tuned BART models. Intrinsic errors are highlighted in \\hlc[crimsonglory!40]{red} and extrinsic errors are in \\hlc[bleudefrance!40]{blue}. \n \\textsc{MaskEnt} and \\textsc{SysLowCon} are used for negative sample construction with unlikelihood training and contrastive learning.}\n \\label{fig:generation_examples}\n\\end{figure*}\n\n\\section{Example Appendix}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\input{section1}\n\n\\section{Problem Statement and the Main Result}\\label{sec:statement}\n\\input{section2}\n\n\\section{Proof of the Reconstruction: Non-Symmetric Paths}\n\\label{sec:proof_nonsymmetric}\n\\input{section3}\n\n\\section{Symmetric Paths}\n\\label{sec:proof_symmetric}\n\\input{section4}\n\n\\section{Conclusion and Future Work}\n\\label{sec:conclusion_futurework}\n\\input{section5}\n\n\n\\bibliographystyle{plain}\n\n\\input{exact_reconstruction.bbl}\n\n\\end{document}\n\\subsection{Background and Motivation}\n\n\\newcommand{\\vspace{0.02in}}{\\vspace{0.02in}}\n\\newcommand{\\parab}[1]{\\vspace{0.02in}\\noindent{\\textsl{#1}}}\n\\def\\mathcal G{\\mathcal G}\n\\def\\mathcal W{\\mathcal W}\n\\def\\mathcal D{\\mathcal D}\n\\def\\mathcal P{\\mathcal P}\n\n\\parab{Graphs and Communication Networks.}\nThe problem that we study originates in performance measurement of\npacket communications networks. These are modeled as a a weighted\ndirected graph $\\mathcal G=(V,E,\\mathcal W)$ in which the vertex set $V$ represents\nrouters, the edges $E$ represent directed links between routers, and\nthe edge weights $\\mathcal W$ characterize performance metrics of the\nassociated links. A subset $V^B\\subset V$ of vertices represents the\n``boundary'' of the network where the measurements are performed. We\nassume that there is a fixed directed path $\\Path(b,b')$ connecting each\nboundary node pair $b,b' \\in V^B$ and that one can measure the common\nportion of any two paths.\n\nMore specifically, we will assume one can measure the length of the\nfollowing: the path between any two boundary nodes, the common part of\nthe two paths from a boundary node to any two other boundary nodes, as\nwell as the common part of the paths to a boundary node from any\ntwo other boundary nodes. This set of measurements we will call the\nPath Correlation Data (PCD). We establish necessary and sufficient\nconditions --- which turn out to be rather natural --- for the\nreconstruction of a graph from its PCD and present an algorithm to\nachieve it. In the case when the underlying graph violates the\nreconstructibility conditions, we describe the result of the algorithm\n--- it turns out to be the ``simplest'' routing network that produces\nthe observed PCD.\n\n\\parab{Network Tomography and the Inversion Problem.}\nThe form of our model and the assumptions we make originate from a\nbody of work developed under the term \\textsl{Network Tomography}\n\\cite{10.2307\/2291416} that seeks to infer link metrics and even the\nunderlying network topology from the measured metric values on paths\ntraversing the network between a set of routers at the network\nboundary, represented by $V^B$. This setting is similar to other\ngraph reconstruction problems, such as tomography of electrical\nresistance networks (see, e.g., \\cite{Cheney99,MR2719770}), optical\nnetworks \\cite{MR3634446}, and graph reconstruction from\nSchr\\\"odinger-type spectral data (see, e.g.,\n\\cite{MR2353313,MR2545980}). However, in a communication network\nmodel there is a single path between given origin and destination, in\ncontrast to the electrical current flowing between two points a\nresistive medium via all possible paths. In this sense, our model\nis more similar to combinatorial reconstruction problems\n\\cite{MR1047783,EvaLan_aam18}.\n\nIn many practical cases the communication network metrics are additive\nin the sense that the sum of metric values over links in a path\ncorresponds to the same performance metric for the path. Examples of\nadditive metrics include mean packet delay, log packet transmission\nprobability, and variances in an independent link metric model. For\nadditive metrics, a putative solution to the network tomography\nproblem attempts to invert a linear relation expression between the\nset of path metrics $\\mathcal{D}$ and the link metrics $\\mathcal{W}$ in the form\n\\begin{equation}\n \\label{eq:lin} \n \\mathcal{D} = \\mathcal{A} \\mathcal{W}\n\\end{equation}\nHere $\\mathcal{A}$ is the incidence matrix of links over paths,\n$\\mathcal{A}_{\\mathcal{P},\\ell}$ is 1 if path $\\mathcal{P}$ traverses link\n$\\ell$, and zero otherwise. The linear system (\\ref{eq:lin}) is\ngenerally underconstrained in real-life networks, and hence does not\nadmit a unique solution \\cite{4016134}. To overcome this deficiency,\none approach has been to impose conditions on the possible solutions,\ntypically through sparseness, effectively to find the ``simplest''\nexplanation of the observed path metrics; see\n\\cite{DBLP:journals\/tit\/Duffield06,10.1007\/978-3-642-30045-5_22}. A\ndifferent approach in the similar problem of traffic matrix tomography\nhas been to reinterpret (\\ref{eq:lin}) as applying to bi-modal\nmeasurements of packet and bytes counts \\cite{SinMic_ip07}, or \nof empirical means and variances then imposing constraints between these based on\nempirical models \\cite{10.2307\/2291416,866369}. However, the high\ncomputational complexity of this approach makes it infeasible for\nreal-world communications networks \\cite{Zhang:2003:FAC:781027.781053}, although\nquasi-likelihood methods offer some reduction in\ncomplexity \\cite{1212664}. A related approach known as Network\nKriging seeks to reduce dimensionality in the path set by assumption\non prior covariances \\cite{4016134}.\n\n\\parab{Correlations and Trees.}\nMore relevant for the work of this paper has been the idea to \nexploit correlations between metrics on different paths that occur due\nto common experience of packets on their intersection. One variant\nuses multicast probing \\cite{796384} or emulations thereof\n\\cite{DBLP:journals\/ton\/DuffieldPPT06}. Another variant exploits the\nfact that variances of some measurable packet statistics are both\nadditive and independent over links. If $\\mathcal D$ is such a\nvariance-based metric, then\n$ \\text{Cov}(\\mathcal D_{P_1}, \\mathcal{D}_{P_2}) =\n\\text{Var}(\\mathcal D_{P_1\\cap P_2})$\nwhere $\\mathcal D_P$ denotes the metric value across path $P$; see\n\\cite{DBLP:journals\/ton\/DuffieldP04}.\n\nWe abstract both these cases into a unified data model in which for\nevery triple $b$, $b_1$ and $b_2$ of boundary vertices, we can\nmeasure, via covariances of the packet statistics, the metric of the\nintersection $P=\\Path(b,b_1)\\cap \\Path(b,b_2)$ as well as to the metric of\nthe intersection $P=\\Path(b_1,b)\\cap \\Path(b_2,b)$. The results of such\nmeasurements we will denote by $\\PCD(b \\prec b_1,b_2)$ and\n$\\PCD(b_1,b_2 \\succ b)$ correspondingly. We remark that we use the\nsymbols $\\prec$ and $\\succ$ here not as binary comparison operators\nbut as pictograms meant to evoke the topological structure of the\ncorresponding pair of paths. The totality of such measurements we\ncall the \\textbf{path correlation data}. Note that we will not, in\ngeneral, assume that the paths are symmetric; the path $\\Path(b,b_1)$\nmay be different topologically from the path $\\Path(b_1,b)$. As a\nconsequence, the values $\\PCD(b \\prec b_1,b_2)$ and\n$\\PCD(b_1,b_2 \\succ b)$ are in general different. We will assume that\nthe function $\\PCD$ is measured exactly; see below for a brief\ndiscussion of possible sources of error and the techniques for\nerror-correction. \n\nUnder fairly general conditions that will be in force in this paper,\nthe problem has a natural formulation in terms of trees. First assume,\nthat for each $b,b_1,b_2\\in V^B$, the intersection\n$\\Path(b,b_1)\\cap \\Path(b,b_2)$ is connected. Second, assume that the\nmetric $\\mathcal D$ is path increasing, i.e., $\\mathcal D(\\Path)<\\mathcal D(\\Path')$ for\n$\\Path\\subsetneq \\Path'$. As shown in \\cite{971737}, the quantities\n$\\{\\PCD(b \\prec b_1,b_2): b_1, b_2\\in V^B\\}$ give rise to an embedded\nlogical weighted tree rooted at $b$ (called ``source tree''). The tree\nis computed iteratively by finding node pairs $(b_1,b_2)$ of maximal\n$\\PCD(b \\prec b_1,b_2)$ and identifying each such pair with a branch\npoint in the logical tree. Each logical link is assigned a weight\nequal to the difference between the values of\n$\\text{PCD}(b \\prec \\cdot,\\cdot)$ associated with its end points.\nSimilarly, the quantities $\\PCD(b_1,b_2 \\succ b)$ give rise to an\nembedded logical tree with a single receiver $b$ and the sources\n$V^B \\setminus \\{b\\}$. Such trees are called ``receiver trees''.\nUnder the assumed conditions, our problem can be restated as how to\nrecover the underlying weighted graph from the set of logical source\nand receiver trees rooted at every $b\\in V^B$.\n\n\n\\parab{Summary of the Results}. The main contribution of this paper,\nTheorem~\\ref{thm:main}, is to show that under natural conditions, a\nweighted directed graph $\\mathcal G$ can be recovered knowing only the\ngraph's Path Correlation Data (PCD). The conditions under which\nTheorem~\\ref{thm:main} holds are (i) each edge is traversed by at\nleast one path in $\\Path$; (ii) each non-boundary node is\n\\textsl{nontrivial} in the sense that in-degree and out-degree are not\nboth equal $1$; and (iii) each non-boundary node $x$ is\n\\textsl{nonseparable} in the sense that the set of paths\n$\\Path(x)\\subset \\Path$ that pass through $x$ cannot be partitioned\ninto two or more subsets with non intersecting end point sets. Our\nresult holds without the assumption of weight symmetry (defined as\nrequiring the existence of the reverse of any edge in $G$, having the\nsame weight) or path symmetry (defined as the paths in either\ndirection between two boundary nodes traversing the same set of\nedges). We prove the correctness of our reconstruction algorithm\n(Algorithm~\\ref{algorithm:non_symmetric}) under the stated\nassumptions.\n\nOur solution does not assume that link weights are symmetric. Neither\ndo we assume that that paths in either direction between two endpoints\nare symmetric: they are not required to traverse the same set of\ninternal (i.e. non-boundary) vertices. This level of generality\nreflects networking practice, in which non-symmetric routing is\nemployed for policy reasons including performance and revenue\noptimization \\cite{Teixeira:2004:DHR:1012888.1005723}. However, in\nthe cases where symmetric paths can be assumed {\\it a priori}, this\nknowledge enlarges the set of reconstructible networks. We therefore\npay special attention to this case, providing alternative definitions\nof the \\emph{nontrivial} and \\emph{nonseparable} vertices and a\nseparate proof of the correctness of\nAlgorithm~\\ref{algorithm:non_symmetric} (which requires a one-line\nchange). We also establish the correctness of a second, more\ncustomized Algorithm~\\ref{algorithm:symmetric} which applies only in\nthe case of symmetric weights and paths, and which is computationally\nless expensive than Algorithm~\\ref{algorithm:non_symmetric} applied to\nthis case.\n\n\\parab{Inference from Inconsistent Data.}\nBefore describing our approach in more detail, we note that practical\nnetwork measurement may provision data with imperfect consistency. For\nexample, input data may be provided in the form of weighted trees\ncomputed from packet measurement over time intervals that are not\nperfectly aligned, so that the metric of a path $\\Path(b,b')$ may be\nreported differently in the source tree from $b$ and the receiver tree\nto $b'$. Even with aligned intervals, deviations from the model and\nstatistical fluctuations due to finitely many probe packets may result\nin inconsistency. To enable our proposed algorithm to operate with\nsuch data, we propose to compute a $\\text{PCD}$ that is a\nleast-squares fit to inconsistent tree data. This extension and\nassociated error sensitivity analysis is described in the forthcoming\ncompanion paper \\cite{PrepUs2018}.\n\n\n\\subsection{An example of a communication network}\n\nTo illustrate the information available to an observer in our model,\nconsider the network graph schematically shown in\nFig.~\\ref{fig:AltSource}. It is assumed that the end-to-end\nmeasurements are possible among the boundary vertices\n$V^B = \\{b_1, ... , b_6\\}$. The three versions of the same graph\nshown contain information about the paths between the given source\n($b_1$, $b_2$ and $b_3$, correspondingly) and the\ncorresponding set of receivers $V^B \\setminus \\{b_i\\}$; the links belonging to\nthese paths are highlighted in thicker lines.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.7]{f_AltRouting0}\n \\caption{Alternative selections of the source and the corresponding\n routing paths (bold edges).} \n \\label{fig:AltSource}\n\\end{figure}\n\nFrom the point of view of an external observer, the routing paths on\nthe graph are hidden but can be reconstructed, to a certain extent, by\nthe measurements with a fixed source and alternating receivers,\nrepresented in our setting by queries to the $\\PCD$ function. As\nFig.~\\ref{fig:AltSourceLogic} shows, the trees reconstructed from PCD\nare \\emph{logical} trees where the edges represent the amalgamated\nversions of the actual physical edges. For example, the logical tree\nlabeled $(SLb_3)$ has a direct edge from $b_3$ to $b_6$, whereas the\nactual route, shown in $(Sb_3)$ passes through an internal node.\nSince this node does not feature as a junction in the tree $(Sb_3)$,\nit will not be detected from the PCD. Moreover each internal vertex\nin the original graph has multiple appearances in the logical trees\nwith no identifying information attached to them. Correctly\nidentifying multiple representation of the same internal node will be\nthe central challenge of this work.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.8]{f_AltRouting_LogicT}\n \\caption{Representation of PCD on the network graph through the set of\n observed logical source trees for sources $b_1$ to $b_6$.}\n \\label{fig:AltSourceLogic}\n\\end{figure}\n\nThe information form logical source trees\n(Fig.~\\ref{fig:AltSourceLogic}) is only enough for reconstruction of\nspecial class of network graph, namely the symmetric one where both\nthe routing and edge weights are symmetric. When this is not the\ncase, the\nmeasurements of the form $\\text{PCD}(b_1,b_2 \\succ b)$ will be essential to\nreconstruct \\emph{logical receiver trees} shown in \nFig.~\\ref{fig:AltRoutingLogicalT2}. Those contain information about\nthe paths with\nthe selected receiver $b$ and with the source set $V^B \\setminus\n\\{b\\}$.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.8]{f_AltRouting_LogicT2}\n \\caption{Representation of PCD on the network graph through the set of\n observed logical receiver trees for receivers $b_1$ to $b_6$.} \n \\label{fig:AltRoutingLogicalT2}\n\\end{figure}\n\nTo summarize, the main goal in this paper is to develop a practical\nalgorithm to reconstruct the original graph from set of measurements\ninformation of the form $\\text{PCD}(b_1 \\prec b_2, b_3)$ and\n$\\text{PCD}(b_2, b_3 \\succ b_1)$ for $b_1,b_2,b_3 \\in V^B$ and to\nestablish necessary and sufficient conditions for the successful\nreconstruction.\n\n\n\\subsection{Outline of paper}\n\nThe outline of the paper is as follows. In Section~\\ref{sec:statement}\nwe state the definitions and assumptions concerning the network\ntopology and path measurements. We then state our main\nTheorem~\\ref{thm:main} concerning the reconstruction of the general\nnon-symmetric weighted graph under these assumptions. The proof of\nTheorem~\\ref{thm:main} will proceed by establishing that a computation\ncodified as Algorithm~\\ref{algorithm:non_symmetric} reconstructs the\ntopology under the stated assumptions, as established in\nSection~\\ref{sec:proof_nonsymmetric} for non-symmetric routing. We\nalso discuss the operation of Algorithm~\\ref{algorithm:non_symmetric}\non graphs that do not satisfy some assumptions of\nTheorem~\\ref{thm:main}, and the relation of the output to the true\ngraph. The flow of Algorithm~\\ref{algorithm:non_symmetric} is\nslightly modified if the routing is symmetric. This case is\nconsidered in Section~\\ref{sec:proof_symmetric}. We also present a\nmodified Algorithm~\\ref{algorithm:symmetric}, suitable \\emph{only} for\nthe case of symmetric paths. We conclude in\nSection~\\ref{sec:conclusion_futurework} with a discussion of possible\nfuture work.\n\n\n\\subsection{Problem Setup}\n\\label{sec:setup}\n\n\n\n\\begin{defn}\n\tA network graph $\\mathcal{N} = (\\Graph, V^B, \\Paths)$ is an edge-weighted graph $\\mathcal{G} =\n\t(V,E,\\mathcal{W})$ together with a set $V^B\\subset V$ of\n\t\\emph{boundary vertices} and the paths $\\Paths$ between\n\tthem. In detail, \n\t\\begin{itemize}\n\t\t\\item $V$ is a finite set of vertices,\n\t\t\\item $E \\subset V\\times V$ is the set of directed edges (no loops\n\t\tor multiple edges are allowed),\n\t\t\\item $\\mathcal{W} : E \\to \\mathbb{R}_+$ are the edge weights,\n\t\t\\item $V^B$ is an arbitrary subset of $V$\n\t\t\\item there is a path $\\Path(b,b')$ between any pair of boundary\n\t\tvertices $b \\neq b'$; each path $\\Path(b,b')$ is simple and\n\t\tassumed to be fixed for the duration of observation of the\n\t\tnetwork. The paths are assumed to have the \\emph{tree consistency\n\t\t\tproperty}: for any $b_1$, $b_1'$, $b_2$ and $b_2'$ in $V^B$ the\n\t\tintersection $\\Path(b_1,b_1') \\cap \\Path(b_2,b_2')$ is\n\t\tconnected.\n\t\\end{itemize}\n\\end{defn}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.375\\textwidth]{f_TreeConst}\n\t\\caption{An example of the violation of the tree consistency\n\t\tproperty. The paths $\\Path(b,b_1)$ and $\\Path(b,b_2)$ first\n\t\tdiverge and then meet again.} \n\t\\label{fig:TreeConst}\n\\end{figure}\n\n\nWe will assume that a path $\\Path(b,b')$ does not pass through any\nother boundary vertices. This is done for convenience only; a graph\ncan be easily made to satisfy this condition by ``drawing out'' the\nboundary vertices from the bulk of the graph as shown in\nFig.~\\ref{fig:BvRemove}. The vertices that do not belong to the\nboundary we will call \\emph{internal vertices} and use the notation\n$V^I = V \\setminus V^B$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{f_BvRemove}\n\t\\caption{A graph (square with no diagonals) with the path\n\t\t$\\Path(b_1,b_3)$ going through $b_2$. The same graph with the\n\t\tboundary vertices drawn out.}\n\t\\label{fig:BvRemove}\n\\end{figure}\n\nWe remark that we do not assume that the weights are symmetric:\n$\\mathcal{W}_{x,y}$ is generally different from $\\mathcal{W}_{y,x}$. We also\ndo not need to assume that the paths are symmetric. However, since\nthe latter case is important in applications and allows for a\nsimplified reconstruction algorithm, we will devote some time to its\nseparate treatment, in particular in Definition~\\ref{def:symm_routing}\nand Section~\\ref{sec:proof_symmetric}.\n\nLet us consider some implications of the tree consistency property.\nConsider two paths, $\\Path(b,b_1)$ and $\\Path(b,b_2)$ with some\ndistinct $b,b_1,b_2 \\in V^B$. The tree property implies that the paths\ncan be written as\n\\begin{align}\n\\label{eq:path1}\n\\Path(b,b_1) &= [b, x_1, x_2, ... , x_j, y_1, y_2, ..., b_1] \\\\\n\\label{eq:path2}\n\\Path(b,b_2) &= [b, x_1, x_2, ... , x_j, z_1, z_2, ..., b_2],\n\\end{align} \nwhere the vertex sets $\\{y_1, y_2, ...\\}$ and $\\{z_1, z_2, ... \\}$ are\ndisjoint, see Fig.~\\ref{fig:Junction}(a).\n\nSimilarly tree consistency property applied to paths \n$\\Path(b_1,b)$ and $\\Path(b_2,b)$ implies that\n\\begin{align}\n\\label{eq:path_b1b}\n\\Path(b_1,b) &= [b_1, y_1, y_2, ... ,y_{i}, x_1, x_2, ..., b] \\\\\n\\label{eq:path_b2b}\n\\Path(b_2,b) &= [b_2, z_1, z_2, ... ,z_{j}, x_1, x_2, ..., b],\n\\end{align} \nwith disjoint $\\{y_1, y_2, ... , y_{m}\\}$ and $\\{z_1, z_2, ... ,\nz_n\\}$, see Fig.~\\ref{fig:Junction}(b).\n\n\\begin{defn}\n\tThe vertex $x_j$ in equations (\\ref{eq:path1})--(\\ref{eq:path2}) is\n\tcalled the $(b \\prec b_1,b_2)$-junction. Note that the set\n\t$\\{x_1, ... , x_j\\}$ is allowed to be empty in which case $b$ acts\n\tas the junction. The vertex $x_1$ in equations\n\t(\\ref{eq:path_b1b})--(\\ref{eq:path_b2b}) is called the\n\t$(b_1,b_2 \\succ b)$-junction.\n\\end{defn}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{f_Junction}\n\t\\caption{(a) $x_j = (b \\prec b_1,b_2)$-junction, \n\t\t(b) $x_1 = (b_1,b_2 \\succ b)$-junction.} \n\t\\label{fig:Junction}\n\\end{figure}\n\nWe remark that we use symbols $\\prec$ and $\\succ$ not as relational\noperators but as separators in the list of 3 vertices which are\npictorially similar to the path configurations in Figure\n\\ref{fig:Junction}.\n\nTo specify the graph reconstruction problem we will be solving we need\nto define the set of \\emph{measurements} available to us.\nThe length of a path is defined as the sum of the weights $\\mathcal{W}$ of\nits edges; we will denote the length by $|\\cdot|$. We consider a\nsingle vertex as a zero-length path; the length of an empty set is\nalso set to be zero. This allows us to assign length to an intersection\nof two paths between boundary vertices which is either empty or a single\nvertex or a connected subpath.\n\nThe totality of the measurements we can make will be called the Path\nCorrelation Data (PCD). In includes, for all $b, b_1, b_2 \\in V^B$,\n\\begin{itemize}\n\t\\item the length $|\\Path(b,b_1)|$,\n\t\n\t\\item the length $\\left| \\Path(b,b_1) \\cap \\Path(b,b_2)\n \\right|$, which we will denote by $\\PCD(b \\prec b_1,b_2)$,\n\t\n\t\\item the length $\\left| \\Path(b_1,b) \\cap \\Path(b_2,b)\n \\right|$, which we will denote by $\\PCD(b_1,b_2 \\succ b)$.\n\\end{itemize}\n\nThus we can directly measure the distance from $b \\in V^B$ to any $(b\n\\prec b_1,b_2)$-junction, or from the $(b_1,b_2 \\succ\nb)$-junction to $b$. We can also infer the distance from the $(b\n\\prec b_1,b_2)$-junction to $b_1$, or from $b_1$ to the $(b_1,b_2\\succ\nb)$-junction, see Fig.~\\ref{fig:JunctionMeas}. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{f_JunctionMeas}\n\t\\caption{Various distances we can measure from Path\n Correlation Data (PCD). Here $\\delta = \\PCD(b, \\prec\n b_1,b_2)$ and $\\gamma = \\PCD(b_1,b_2 \\succ b)$.} \n\t\\label{fig:JunctionMeas}\n\\end{figure}\n\nOur principal question is thus: \\emph{ Which network graphs\n $(\\mathcal{G},V^B, \\Paths)$ can be reconstructed from their path\n correlation data and how does one accomplish this?}\n\n\n\\subsection{Some obvious necessary conditions}\n\\label{sec:necessary_examples}\n\nBefore we state our result and the associated reconstruction\nalgorithm, let us consider examples that show some obvious\nnecessary conditions we need to impose on the network graph\n$\\mathcal{N}=(\\mathcal{G},V^B,\\Paths)$ in order to be able to reconstruct it.\n\n\\begin{exmp}\n\tConsider the network graphs in Fig.~\\ref{fig:Example_1}, with $V^B =\n\t\\{u,v,w\\}$ and with the routing\n\tpaths indicated by dashed lines. \n\tNone of the routing paths pass through the\n\tedge $e=(u,w)$ therefore the length of this edge cannot influence\n\tthe Path Correlation Data in any way. Conversely, the length\n of the edge $e$ (and even its existence) cannot be inferred\n from the PCD.\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{f_Example_1}\n\t\t\\caption{Failure to recover the edge $e=(u,w)$: the graphs (a) and\n\t\t\t(b) will produce the same PCD since none of the paths pass\n\t\t\tthrough the extra edge.}.\n\t\t\\label{fig:Example_1}\n\t\\end{figure}\n\\end{exmp}\n\n\\begin{exmp}\n\t\\label{ex:bad_degree2}\n\tConsider the network graphs in Fig.~\\ref{fig:Example_2} with\n boundary vertex set $V^B = \\{u,w\\}$. In the left graph the\n length of the edge $(x,u)$ will never appear in the PCD on its\n own, without being added to the length of the edge $(x,w)$.\n Therefore, it will be impossible to reconstruct the location\n of the vertex $x$, and even detect it at all. This will be\n the case for any internal vertex of degree 2.\n\t\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{f_Example_2}\n\t\t\\caption{Failure to recover the internal vertex $x$: the graphs\n\t\t\t(a) and (b) will produce the same PCD as long as the sum of the\n\t\t\tlengths of $(u,x)$ and $(x,w)$ in the graph (a) is equal to the\n\t\t\tlength of the edge $(u,w)$ in the graph (b).}\n\t\t\\label{fig:Example_2}\n\t\\end{figure}\n\\end{exmp}\n\n\\begin{exmp}\n\tConsider the network graphs in Fig.~\\ref{fig:Example_3} with the\n\tboundary vertex set\n\t\\begin{equation}\n\tV^B = \\{u_1,v_1,u_2,v_2\\}. \n\t\\end{equation}\n\tIn Fig.~\\ref{fig:Example_3}(a) the paths $\\Path(u_1,v_1)$ and\n $\\Path(u_2,v_2)$ intersect at an internal vertex $x$, while in\n Fig.~\\ref{fig:Example_3}(b) they do not have any vertices in\n common. However, the two graphs will produce the same PCD and\n will be indistinguishable.\n\t\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{f_Example_3}\n\t\t\\caption{Failure to recover the integrity of the\n vertex $x$: the graphs (a), (b) and (c) will produce\n the same PCD since the path between $u_1$ and $v_1$\n is in no way correlated to the path between $u_2$\n and $v_2$.}\n\t\t\\label{fig:Example_3}\n\t\\end{figure}\n\t\n\tThe reader will undoubtedly observe that the vertices $x_1$ and $x_2$\n\tin Fig.~\\ref{fig:Example_3}(b) will not be detected, and the\n graph in Fig.~\\ref{fig:Example_3}(c) is the ``minimal'' graph\n which will have the same PCD. By making the graph\n\tstructure more complicated one can easily construct an example\n\twhere $x$, $x_1$ and $x_2$ will act as junctions for some pairs of\n\tpaths and thus will be detectable, see\n Fig.~\\ref{fig:Example_3_Sep}, but the two graphs are still not\n distinguishable from their PCD.\n\t\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{f_Example_3_Sep}\n\t\t\\caption{A more complicated example of failure to recover the\n\t\t\tintegrity of the vertex $x$.}\n\t\t\\label{fig:Example_3_Sep}\n\t\\end{figure}\n\t\n\tThus the real problem is that in the left graph in\n\tFig.~\\ref{fig:Example_3_Sep} there are two families of paths going\n\tthrough the internal vertex $x$ that do not interact in any way.\n\\end{exmp}\n\n\n\\subsection{Statement of the main result}\n\\label{sec:main_result}\n\nThe main result of this paper is that the necessary conditions\nillustrated by examples in Section~\\ref{sec:necessary_examples} are in\nfact sufficient for the reconstruction! We start by formalizing\n(and naming) the conditions we observed.\n\n\\begin{defn}\n\tAn edge is called \\emph{unused} if there is no path in $\\Paths$\n\tcontaining it.\n\\end{defn}\n\nWe remark that if there are no unused edges in a network graph, each\ninternal vertex has at least one incoming and at least one outgoing edge.\n\n\\begin{defn}\n\tAn internal vertex $x$ is called \\emph{trivial} if it has only one\n\tincoming and only one outgoing edge (i.e.\\ edges of the form\n\t$(y_1,x)$ and $(x,y_2)$ correspondingly).\n\\end{defn}\n\nWe remark that if there are no unused edges, then there are at least\ntwo paths through every non-trivial internal vertex.\n\n\\begin{defn}\n\tFor an internal vertex $x \\in V^I$, let $S_x \\subset V^B$ to be the set\n\tof the sources and $R_x \\subset V^B$ be the set of the receivers\n\twhose paths pass through $x$. More precisely, \n\t\\begin{equation*}\n\t\\begin{split}\n\tS_x &= \\big \\{ b \\in V^B : \\exists \\hat{b} \\in V^B,\\ \n\tx \\in \\Path(b,\\hat{b}) \\big \\} \\\\\n\tR_x &= \\big \\{ \\hat{b} \\in V^B : \\exists b \\in V^B,\\ \n\tx \\in \\Path(b,\\hat{b}) \\big \\}.\n\t\\end{split}\n\t\\end{equation*}\n\\end{defn}\n\n\\begin{defn}\n \\label{def:separable}\n A vertex $x \\in V^I$ is called \\emph{separable} if there are\n disjoint non-empty partitions $S_x = S_x^1 \\cup S_x^2$ and\n $R_x = R_x^1 \\cup R_x^2$ with the property that\n\t\\begin{equation}\n\t\\label{eq:separable_def}\n\tb\\in S_x^j,\\ \\hat{b}\\in R_x^{j'} \\mbox{ with }j\\neq j'\n \\quad\\Rightarrow\\quad x \\notin \\Path(b,\\hat{b}).\n\t\\end{equation}\n\\end{defn}\n\nAn example of a separable vertex is shown in Fig.~\\ref{fig:SeparateExam}. The\npartition sets here are $S^1 = \\{b_1\\}$, $S^2 = \\{b_2\\}$, $R^1 =\n\\{b_3,b_4\\}$ and $R^2 = \\{b_5,b_6\\}$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{f_SeparateExam}\n\t\\caption{Paths through a separable vertex $x$ and its separation\n\t\tinto $x_1$ and $x_2$.}\n\t\\label{fig:SeparateExam}\n\\end{figure}\n\nFinally, if the graph has symmetric routing we need to modify the\nconditions slightly but the resulting reconstructibility theorem will\nstay the same. Naturally, symmetric routing is an extra piece of\ninformation and more graphs are reconstructable in this case. The\nnatural setting for the problem with symmetric routing is a\nnon-directed graph, but since we still allow non-symmetric edge\nweights, we will keep the edges directed. As a result, edges come in\npairs which correspond to undirected edges splitting into two directed\nones. This is formalized in the definition of a ``network graph with\nsymmetric routing'' below.\n\n\\begin{defn}\n\t\\label{def:symm_routing}\n\tWe will say that the network graph $\\mathcal{N}=(\\Graph, V^B, \\Paths)$ has\n\t\\emph{symmetric routing} if\n\t\\begin{itemize}\n\t\t\\item for all $x,y\\in V$, $(x,y) \\in E$ implies $(y,x) \\in E$ and\n\t\t\\item for all $b,b'\\in V^B$, the path $\\Path(b, b')$ is the\n\t\treversal of the path $\\Path(b',b)$, namely\n\t\t\\begin{equation}\n\t\t\\label{eq:reversal}\n\t\t\\Path(b, b') = [b, x_1, x_2, \\ldots, x_j, b']\n\t\t\\quad \\Rightarrow \\quad\n\t\t\\Path(b', b) = [b', x_j, x_{j-1}, \\ldots, x_1, b].\n\t\t\\end{equation}\n\t\\end{itemize}\n\\end{defn}\n\t\n\\begin{defn}\n \\label{def:symm_vertex}\n\tA vertex $x \\in V^I$ in a network graph with symmetric routing is\n\t\\emph{trivial} if it has two (or less) adjacent vertices. \n\n\n\tA vertex $x \\in V^I$ in a network graph with symmetric routing is\n\t\\emph{separable} if there is a disjoint partition $S_x = S_x^1 \\cup S_x^2$\n\tso that\n\t\\begin{equation}\n\t\\label{eq:separable_def_symm}\n\tb_1 \\in S_x^1,\\ b_2 \\in S_x^2 \n\t\\quad \\Rightarrow \\quad\n\tx \\notin \\Path(b_1,b_2).\n\t\\end{equation}\n\t\n\\end{defn}\n\nWe can now state our Main Theorem. We stress that the statement of\nthe theorem applies uniformly to network graphs with or without\nsymmetric routing, the differences being absorbed by the definitions\nabove. We will still need to provide two separate (but similar!)\nproofs.\n\n\\begin{thm}[Main Theorem]\n\t\\label{thm:main}\n\tLet $(\\mathcal{G},V^B,\\Paths)$ be a network graph. If\n\t\\begin{enumerate}\n\t\t\\item \\label{cond:edges}\n\t\tno edge $e \\in E$ is unused,\n\t\t\\item \\label{cond:degrees}\n\t\tno $x \\in V^I$ is trivial,\n\t\t\\item \\label{cond:nonsep} \n\t\tno $x \\in V^I$ is separable,\n\t\\end{enumerate}\n\tthen $(\\mathcal{G},V^B,\\Paths)$ is uniquely reconstructable from\n\tits Path Correlation Data (PCD).\n\t\n\tTo put it more generally, in every class of network graphs with the\n\tsame PCD, there is a unique network graph which satisfies the above\n\tconditions.\n\\end{thm}\n\nThe theorem will be proved constructively, by presenting a\nreconstruction algorithm and verifying its result. The second part,\nwhich posits not only uniqueness but also the existence of the\nreconstructed graph, means, in practical terms, that even when given\nPCD from a graph that does not satisfy the conditions, the algorithm\nwill terminate and produce a ``nearby'' result which does.\n\n\n\\subsection{Comments on the algorithm for non-symmetric routing}\n\nThe algorithm for the case of non-symmetric routing\n(Algorithm~\\ref{algorithm:non_symmetric}) works by discovering the\ninternal vertices and reconstructing the routing paths in the format\n\\begin{equation}\n\\label{eq:reconstructed_path_format}\n\\mathcal{R}(b, \\hat{b}) = [(b,0),\\,(x_1,\\delta_1),\\,(x_2,\\delta_2),\n\\ldots,\\, (\\hat{b},\\delta)],\n\\end{equation}\nwhere $\\delta_i$ is the cumulative distance from $b$ to $x_j$ along the\npath (naturally, $\\delta = |\\Path(b, \\hat{b})|$). Once every path is\ncomplete, the edges can be read off as pairs of consecutive vertices\nappearing in a path. The internal vertices\nare discovered as junctions. The main difficulty lies in identifying\ndifferent junctions that correspond to the same vertex. This is done\nby a depth-first search in the function {\\sc UpdatePath}.\n\n\\algnewcommand{\\IfThen}[2]{%\n \\State \\algorithmicif\\ #1\\ \\algorithmicthen\\ #2}\n\\algnewcommand{\\IfThenElse}[3]\n \\State \\algorithmicif\\ #1\\ \\algorithmicthen\\ #2\\ \\algorithmicelse\\ #3}\n\n\n\\begin{algorithm}[t]\n\t\\caption{Reconstruction of the network graph\n\t\t$(\\Graph, V^B, \\Paths)$}\n\t\\label{algorithm:non_symmetric}\n\t\\begin{algorithmic}[1]\n\t\t\\For{$b_1,b_2 \\in V^B$} \\Comment{{\\bf Initialization}}\n\t\t\\State $\\mathcal{R}(b_1,b_2) = [ (b_1,0),(b_2, |\\Path(b_1,b_2)|) ]$\n\t\t\\EndFor\n\t\t\n\t\n\t\t\\For{$b_1,b_2,b_3 \\in V^B$} \\Comment{{\\bf Main Loop}}\n\t\t\\State create label $a$ \\label{algo:create_x}\n\t\t\\State $\\delta = \\PCD(b_1 \\prec b_2,b_3)$ \\label{algo:distance_x}\n\t\t\\State {\\sc UpdatePath}($\\mathcal{R}(b_1,b_2), a,\\delta$) \\label{algo:first_call}\n \\IfThenElse {``symmetric routing''} {$a'=a$} {create label $a'$} \\label{algo_create_y}\n\t\t\\State $\\delta' = |\\Path(b_2,b_1)| - \\PCD(b_2,b_3 \\succ b_1)$ \\label{algo:dist_y} \n\t\t\\State {\\sc UpdatePath}($\\mathcal{R}(b_2,b_1), a',\\delta'$) \\label{algo:second_call}\n\t\t\\EndFor\n\t\t\n\t\n\t\t\\State \\textbf{Read off} the graph from reconstructed paths\n\t\t$\\mathcal{R}$. \\Comment{{\\bf Return the result}}\n\t\t\n\t\t\\Statex\n\t\t\\Function{UpdatePath}{$\\mathcal{R}(u,v),a,\\delta$}\n\t\t\\Comment{{\\bf Recursive Function}} \\label{algo:function_start}\n\t\t\\IfThen {$\\exists\\,(\\cdot,\\delta) \\in \\mathcal{R}(u,v)$} {return}\n\t\t\\label{algo:check_vertex}\n\t\t\\State insert $(a,\\delta)$ into $\\mathcal{R}(u,v)$\n \\State $\\delta' = |\\Path(u,v)| - \\delta$\n\t\t\\For {$z \\in V^B$} \\label{algo:discovery_loop}\n\t\t\\IfThen {$\\PCD(u \\prec v,z) \\geq \\delta$} \n\t\t\\label{algo:transfer_x_tail}\n\t\t{\\sc UpdatePath}($\\mathcal{R}(u,z),a,\\delta$)\n\t\n\t\t\\IfThen {$\\PCD(u,z \\succ v) \\geq \\delta'$} \n\t\t\\label{algo:transfer_x_head} \n {\\sc UpdatePath}($\\mathcal{R}(z,v),a,|\\Path(z,v)| - \\delta'$)\n\t\n\t\t\\EndFor\n\t\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\nThe following comments might be in order. In\nlines~\\ref{algo:create_x}--\\ref{algo:distance_x} $a$ is the label for\nthe vertex that is the $(b_1 \\prec b_2,b_3)$-junction and $\\delta$ is the distance from $b_1$ to $a$ along the path $\\Path(b_1,b_2)$. In\nlines~\\ref{algo_create_y}--\\ref{algo:dist_y} $a'$ is the label for the\n$(b_2,b_3 \\succ b_1)$-junction and $\\delta'$ is the distance from $b_2$ to $a'$ along the path $\\Path(b_2,b_1)$. If we know {\\it a priori\\\/} that the routing is symmetric, the $(b_1 \\prec b_2,b_3)$-junction and the $(b_2,b_3 \\succ b_1)$-junction are the same vertex and can receive the same label. \n\nLine~\\ref{algo:check_vertex} checks if there is already a vertex\ndistance $\\delta$ from $b_1$ (if there is, label $a$ is unused).\nFinally, the loop starting on line~\\ref{algo:discovery_loop} looks for\nany other paths that the vertex with label $a$ must belong to. Here we rely heavily on the tree consistency property.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{f_Algorithm1}\n\t\\caption{Insertion of internal vertex $x$ in the reconstructed paths, (a) $\\mathcal{R}(u,z)$, and (b) $\\mathcal{R}(z,v)$ by fulfillment of conditions~\\ref{algo:transfer_x_tail} and~\\ref{algo:transfer_x_head} respectively called by line~\\ref{algo:first_call} of Algorithm~\\ref{algorithm:non_symmetric}.}\n\t\\label{fig:VertexInsertion}\n\\end{figure}\n\nSome further code improvements are possible. Creating and then\ndiscarding unused labels can be avoided either by performing a check\nsimilar to line~\\ref{algo:check_vertex} in the main loop or, more\nelegantly, by making $a$ an optional argument to the function {\\sc\nUpdatePath} and creating a label after line~\\ref{algo:check_vertex}\nif no $a$ was supplied. Additionally, if the edge weights are\nsymmetric, the call to $\\PCD$ in line \\ref{algo:dist_y} can be avoided\nby using $\\PCD(b_2,b_3 \\succ b_1) = \\PCD(b_1 \\prec b_2,b_3) = \\delta$.\n\nFinally, a crude upper bound on complexity of the algorithm (in terms\nof label insertions into various $\\mathcal{R}$) is $|V^I| \\times |V^B|^2$\ni.e. the product of number of internal vertices and the square number\nof boundary vertices of the graph.\n\n\n\n\n\\subsection{Specialized algorithm for symmetric routing}\t\n\nWe have shown that the reconstruction\nAlgorithm~\\ref{algorithm:non_symmetric} is universal: it covers both\nthe general non-symmetric network and also graphs with symmetric\nrouting. However, having the prior information that the routing on the\nnetwork is symmetric makes it possible to call the recursive function\n{\\sc UpdatePath} less. This is due to the symmetric routing property\nthat if an internal vertex $x$ is inserted in reconstruction path\n$\\Path(u,v)$ for $u,v \\in V^B$ then this vertex should also inserted\nin the reverse path $\\Path(v,u)$ with appropriate distance from root\n$v$ (this is not generally true for non-symmetric routing case). To\ntake advantage of this feature we propose\nAlgorithm~\\ref{algorithm:symmetric} as a specialized version of the\nreconstruction algorithm for the graphs with symmetric routing.\n\nCompared to the Algorithm~\\ref{algorithm:non_symmetric}, the new\nalgorithm calls the recursive function twice less (as can be seen by\ncomparing the main loops of the two algorithms). The change in the number\nof calls of reconstruction function is compensated by adding line 14\nin the new algorithm where the label of internal vertex which is\ninserted in the given path will be inserted in the reverse one as well.\n\nIt should be pointed out that although less calls of main\nreconstruction function makes the algorithm computationally more\neffective, from the point of view of mathematical complexity (in terms\nof insertions of a vertex into a reconstructed path), the two\nalgorithms can be considered the same.\n\n\n\\begin{algorithm}[h]\n\t\\caption{Reconstruction of a network graph with symmetric routing}\n\t\\label{algorithm:symmetric}\n\t\\begin{algorithmic}[1]\n\t\t\\For{$b_1,b_2 \\in V^B$} \\Comment{{\\bf Initialization}}\n\t\t\\State $\\mathcal{R}(b_1,b_2) = [ (b_1,0),(b_2, |\\Path(b_1,b_2)|) ]$\n\t\t\\EndFor\n\t\t\n\t\t\\For{$b_1,b_2,b_3 \\in V^B$} \\Comment{{\\bf Main Loop}}\n\t\t\\State create label $a$ \\label{algo:create_x_sym}\n\t\t\\State $\\delta = \\text{PCD}(b_1 \\prec b_2,b_3)$ \\label{algo:distance_x_sym_f}\n\t\t\\State $\\delta' = \\text{PCD}(b_2,b_3 \\succ b_1)$ \\label{algo:distance_x_sym_r}\n\t\t\\State {\\sc UpdatePath}($\\mathcal{R}(b_1,b_2),a,\\delta, \\delta'$)\n\t\t\\EndFor\n\t\t\n\t\n\\State \\textbf{Read off} the graph from reconstructed paths\n$\\mathcal{R}$. \\Comment{{\\bf Return the result}}\n\n\\Statex\n\\Function{UpdatePath}{$\\mathcal{R}(u,v),a,\\delta, \\delta'$}\n\\Comment{{\\bf Recursive Function}} \\label{algo:function_start_sym}\n\\IfThen {$\\exists\\,(\\cdot,\\delta) \\in \\mathcal{R}(u,v)$} {return}\n\\label{algo:check_vertex_sym}\n\\State insert $(a,\\delta)$ into $\\mathcal{R}(u,v)$\n\\State insert $(a,|\\Path(v,u)| - \\delta')$ into $\\mathcal{R}(v,u)$\n\n\\For {$z \\in V^B$} \\label{algo:discovery_loop_sym}\n\\IfThen {$\\PCD(u \\prec v,z) \\geq \\delta$} {{\\sc UpdatePath}($\\mathcal{R}(u,z),a,\\delta,\\delta'$)}\n\\label{algo:transfer_x_tail_sym}\n\\IfThen {$\\PCD(v \\prec u,z) \\geq \\gamma'$} {{\\sc UpdatePath}($\\mathcal{R}(v,z),a,|\\Path(v,u)| - \\delta',|\\Path(u,v)| - \\delta $)}\n\\label{algo:transfer_x_head_sym} \n\\EndFor\n\n\n\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Reconstruction example}\n\nIn the following example, we will show how having the prior\ninformation of symmetric routing will help to uniquely reconstruct the\nnetwork graph shown in Fig.~\\ref{fig:Delta}(a). The selected routing\namong the boundary vertices $V^B = \\{b_1,b_2,b_3\\}$ follows the\nsequence $\\Path(b_i,b_j) = [b_i,x_i,x_j,b_j]$ for $i,j = 1,2,3$. From\nthe measurement point of view, the PCD on the graph is represented as\nthe set of observed logical trees in Fig.~\\ref{fig:DeltaReconst}.\n\nIf there is no information on the symmetry of routing, then we can\nnot conclude that $a_1 = a_4$, $a_2=a_5$ and $a_3=a_6$. The resulting\nreconstruction will be the graph appearing in Fig.~\\ref{fig:Delta}(b).\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.9]{f_Delta_Underlying}\n\t\\caption{(a) An example of a network graph. To avoid clutter, the edges\n\t\twith no specified direction are assumed to go in both directions\n\t\twith the same weight, (b) reconstructed network}\n\t\\label{fig:Delta}\n\\end{figure}\n\nWe remark that such geometry is fairly realistic if we, for example,\nconsider the vertices $b_j$ to be internet service providers (ISPs)\nwho are eager to push the traffic addressed outside their network to\nother ISPs as soon as possible. Still, the reconstruction does not\nmatch the graph we had originally.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[scale=0.9]{f_DeltaReconst}\n\t\\caption{Representation of PCD on the network example through set of logical source and receiver trees.}\n\t\\label{fig:DeltaReconst}\n\\end{figure}\n\nHowever, the reconstruction \\emph{is possible} if we know a priori\nthat the routes are symmetric, as they are in this example. Then the\npaths going out of a given source $b$ and the paths going to $b$\nacting as the receiver have exactly the same topology. This\nadditional information allows us to identify $a_1 = a_4$, $a_2=a_5$\nand $a_3=a_6$ in Fig.~\\ref{fig:Delta}(b) and thus recover the\noriginal graph. Since symmetric routing networks appear in\napplications, we expect the optimized\nAlgorithm~\\ref{algorithm:symmetric} to be a practical value.\n\n\n\\subsection{The effect of symmetric edge weights}\n\nSymmetric network graph for which both the routing on the graph and\nedge weights are symmetric can be considered as a special class of\ngeneral networks. For this class of networks, similar to the graphs\nwith symmetric routing (see eq.~\\ref{eq:reversal}), same set of edges\nare passed in the two reverse paths $\\Path_1(u,v)$ and\n$\\Path_2(v,u)$. Additionally, the symmetric edge weights property\nimplies that for any vertex $x \\in \\Path_1$, then\n$|\\Path(v,x)| = |\\Path(u,v)| - |\\Path(u,x)|$. In other words, path\n$\\Path_2$ is fully identified by the available information form the\npath $\\Path_1$. This property of symmetric networks implies that\n$\\text{PCD}(v,w \\succ u) = \\text{PCD}(u \\prec v,w)$, and as a result\nthe information from \\textit{path correlation data} encoded in the form\nof $(u \\prec v,w)$-junctions can be applied to identify the\ninformation for the $(v,w \\succ u)$-junctions. From the\n\\textbf{reconstruction} point of view, same conditions are required\nfor exact reconstruction of graphs with symmetric edge weights as\ndiscussed in previous sections and thereby\nAlgorithm~\\ref{algorithm:symmetric} can be applied for reconstruction\npurposes.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n \n Methyl chloride has been proposed as an observable biosignature gas in the search for life outside the Solar System.~\\cite{05SeKaMe.CH3Cl} A model of different hypothetical Earth-like planets orbiting a range of M stars predicted that a higher concentration of CH$_{3}$Cl would exist than on Earth, and with stronger spectral features. Seager et al. have since gone on to classify CH$_3$Cl as a Type III biomarker - a molecule produced from a secondary metabolic process - and estimated the required concentration needed for a realistic detection in a generalized oxidized atmosphere,~\\cite{13SeBaHu.CH3Cl} and for an exoplanet with a thin H$_2$ rich atmosphere and a habitable surface temperature.~\\cite{13aSeBaHu.CH3Cl} The rotation-vibration spectrum of CH$_3$Cl has received increased interest as a result.\n \n A highly accurate and comprehensive line list is lacking for methyl chloride, with varying coverage in the spectroscopic databases.~\\cite{HITRAN,PNNL,GEISA,JPL} The HITRAN database~\\cite{HITRAN} is the most extensive, containing over $100{\\,}000$ transitions for each of the two main isotopologues, $^{12}$CH$_{3}{}^{35}$Cl and $^{12}$CH$_{3}{}^{37}$Cl (henceforth labelled as CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl), in the range $0$ to $3200{\\,}$cm$^{-1}$. Although the latest update HITRAN2012 has seen improvements, notably around $3000{\\,}$cm$^{-1}$,~\\cite{11BrPeJa.CH3Cl} there are still deficiencies with certain line positions and intensities taken from an empirically refined theoretical anharmonic force field.~\\cite{85KoKoNa.CH3Cl}\n \n Due to its prominent role in depletion of the ozone layer, levels of methyl chloride are being closely monitored by satellite missions such as the Atmospheric Chemistry Experiment~\\cite{ACE_Bern06,06NaBeBo.CH3Cl,11BrChMa.CH3Cl,13BrVoSc.CH3Cl} and the Microwave Limb Sounder.~\\cite{13SaLiMa.CH3Cl} A number of recent publications focusing on line shapes and broadening coefficients,~\\cite{12BrJaBu.CH3Cl,13BrJaLa.CH3Cl,13aBrJaLa.CH3Cl,12GuRoBu.CH3Cl,\n 11BuGuEl.CH3Cl,12BuRoxx.CH3Cl,13Buxxxx.CH3Cl,13BuMaMo.CH3Cl,\n 13RaJaDh.CH3Cl,14RaJaDh.CH3Cl,14aRaJaDh.CH3Cl,13DuLaBu.CH3Cl,14DuLaBu.CH3Cl} needed for a realistic modelling of atmospheric spectra, confirm its terrestrial importance. The $3.4{\\,}\\mu$m region is particularly relevant for atmospheric remote sensing due to a relatively transparent window and strong spectral features of the $\\nu_1$ band of CH$_{3}$Cl. A high-resolution study of the $\\nu_1$, $\\nu_4$ and $3\\nu_6$ bands in this region produced a line list for the range $2920$ to $3100{\\,}$cm$^{-1}$.~\\cite{11BrPeJa.CH3Cl} The $6.9{\\,}\\mu$m region has seen line positions, intensities, and self-broadening coefficients determined for more than 900 rovibrational transitions of the $\\nu_{5}$ band.~\\cite{13RaJaDh.CH3Cl} Nikitin et al. have also measured, modelled and assigned over $20{\\,}000$ transitions for each isotopologue in the region $0$ to $2600{\\,}$cm$^{-1}$.~\\cite{03NiFeCh.CH3Cl,04NiChBu.CH3Cl,05NiChxx.CH3Cl,05NiChBu.CH3Cl} An effective Hamiltonian model adapted to the polyad structure of methyl chloride reproduced observed transitions involving the ground state and $13$ vibrational states with an overall standard deviation of $0.0003{\\,}$cm$^{-1}$.\n \n There is a large body of experimental work on the rovibrational spectrum of methyl chloride. We refer the reader to the most recent publications~\\cite{11BrPeJa.CH3Cl,12BrTrJa.CH3Cl,12BrJaBu.CH3Cl,13BrJaLa.CH3Cl,13aBrJaLa.CH3Cl,12GuRoBu.CH3Cl,11BuGuEl.CH3Cl,12BuRoxx.CH3Cl,13Buxxxx.CH3Cl,13BuMaMo.CH3Cl,13RaJaDh.CH3Cl,14RaJaDh.CH3Cl,14aRaJaDh.CH3Cl,13DuLaBu.CH3Cl,14DuLaBu.CH3Cl,03NiFeCh.CH3Cl,04NiChBu.CH3Cl,05NiChxx.CH3Cl,05NiChBu.CH3Cl} (and references therein) for a more complete overview.\n \n Theoretically there has been a consistent effort over the years to characterize the spectrum of CH$_3$Cl. Much attention has been given to a description of harmonic~\\cite{70DuAlMc.CH3Cl,72DuMcSp.CH3Cl,76Duxxxx.CH3Cl,79Wixxxx.CH3Cl,80LaTaBe.CH3Cl,87ScThxx.CH3Cl,95Haxxxx.CH3Cl,01BlLaxx.CH3Cl} and anharmonic~\\cite{77ScWoBe.CH3Cl,83BeAlxx.CH3Cl,83KoKoNa.CH3Cl,84KoKoNa.CH3Cl,85KoKoNa.CH3Cl,90DuLaxx.CH3Cl,92ScThxx.CH3Cl} force fields, both empirically and using \\textit{ab initio} methods. The latest work by Black and Law~\\cite{01BlLaxx.CH3Cl} employed spectroscopic data from ten isotopomers of methyl chloride to produce an empirical harmonic force field incorporating the most up to date treatment of anharmonic corrections. These were largely based on a complete set of empirical anharmonicity constants derived from a joint local mode and normal mode analysis of 66 vibrational energy levels in the region $700$ to $16{\\,}500{\\,}$cm$^{-1}$,~\\cite{90DuLaxx.CH3Cl} and follow-up work in a similar vein by Law.~\\cite{99Laxxxx.CH3Cl}\n \n From a purely \\textit{ab initio} standpoint, Nikitin has computed global nine-dimensional potential energy surfaces (PESs) for vibrational energy level calculations, considering both CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl in the region $0$ to $3500{\\,}$cm$^{-1}$.~\\cite{08Nixxxx.CH3Cl} Using fourth-order M{\\o}ller-Plesset perturbation theory MP4 and a correlation-consistent quadruple-zeta basis set, as well as coupled cluster theory CCSD(T) with a triple-zeta basis set, a combined total of $7241$ points with energies up to $h c \\cdot 40{\\,}000{\\,}$cm$^{-1}$ were employed to generate and fit the PESs ($h$ is the Planck constant and $c$ is the speed of light). Vibrational energies were calculated variationally using a finite basis representation and an exact kinetic energy operator, reproducing the fundamental term values with a root-mean-square error of $1.97$ and $1.71{\\,}$cm$^{-1}$ for CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl respectively.\n\n The potential energy surface is the foundation of rovibrational energy level calculations. Its quality not only dictates the accuracy of line positions, but is also crucial for achieving significant improvements in calculated band intensities.~\\cite{OvThYu08a.PH3} Achieving ``spectroscopic accuracy'' (better than $\\pm 1{\\,}$cm$^{-1}$) in a purely \\textit{ab initio} fashion is extremely challenging due to the limitations of electronic structure methods. To do so one must account for higher-level (HL) electron correlation beyond the initial coupled cluster method when generating the PES, and use a one-particle basis set near the complete basis set (CBS) limit. Core-valence (CV) electron correlation, scalar relativistic (SR) effects, higher-order (HO) electron correlation, and the diagonal Born-Oppenheimer correction (DBOC) are considered to be the leading HL contributions.~\\cite{Peterson12}\n \n The goal of this study is to use state-of-the-art electronic structure calculations to construct a global PES for each of the two main isotopologues of methyl chloride, CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl. This requires inclusion of the leading HL corrections and extrapolation to the CBS limit. The quality of the respective PESs will be assessed by variational calculations of the vibrational energy levels using the computer program TROVE.~\\cite{TROVE2007} To obtain fully converged term values we will exploit the smooth convergence of computed energies with respect to vibrational basis set size, and perform a complete vibrational basis set (CVBS) extrapolation.~\\cite{OvThYu08.PH3} By using a range of theoretical techniques we aim to find out exactly what accuracy is possible for a molecule such as methyl chloride.\n\n The paper is structured as follows: In Sec.~\\ref{sec:PES} the \\textit{ab initio} calculations and analytic representation of the PES are detailed. The variational calculations will be discussed in Sec.~\\ref{sec:variational}, where we assess the combined effect of the HL corrections and CBS extrapolation on the vibrational term values and equilibrium geometry. We offer concluding remarks in Sec.~\\ref{sec:conc}.\n \n\\section{Potential Energy Surface}\n\\label{sec:PES} \n\n\\subsection{Electronic structure calculations}\n \n We take a focal-point approach~\\cite{Csaszar98} to represent the total electronic energy,\n\\begin{equation}\\label{eq:tot_en}\nE_{\\mathrm{tot}} = E_{\\mathrm{CBS}}+\\Delta E_{\\mathrm{CV}}+\\Delta E_{\\mathrm{HO}}+\\Delta E_{\\mathrm{SR}}+\\Delta E_{\\mathrm{DBOC}}\n\\end{equation}\n\n\\noindent which allows for greater control over the PES. To compute $E_{\\mathrm{CBS}}$ we employed the explicitly correlated F12 coupled cluster method CCSD(T)-F12b~(Ref.~\\onlinecite{Adler07} - for a detailed review of this method see Refs.~\\onlinecite{F12_Tew2010,F12_Werner2010}) in conjunction with the F12-optimized correlation consistent polarized valence basis sets, cc-pVTZ-F12 and cc-pVQZ-F12,~\\cite{Peterson08} in the frozen core approximation. The diagonal fixed amplitude ansatz 3C(FIX),~\\cite{TenNo04} and a Slater geminal exponent value of $\\beta=1.0$~$a_0^{-1}$ as recommended by Hill et al.~\\cite{Hill09} were used. To evaluate the many electron integrals in F12 theory additional auxiliary basis sets (ABS) are required. For the resolution of the identity (RI) basis, and the two density fitting (DF) basis sets, we utilized the corresponding OptRI,~\\cite{Yousaf08} cc-pV5Z\/JKFIT,~\\cite{Weigend02} and aug-cc-pwV5Z\/MP2FIT~\\cite{Hattig05} ABS, respectively. All calculations were carried out using MOLPRO2012~\\cite{Werner2012} unless stated otherwise.\n \n To extrapolate to the CBS limit we used a parameterized two-point, Schwenke-style~\\cite{Schwenke05} formula,\n\\begin{equation}\\label{eq:cbs_extrap}\nE^{C}_{\\mathrm{CBS}} = (E_{n+1} - E_{n})F^{C}_{n+1} + E_{n}\n\\end{equation}\n\n\\noindent originally proposed by Hill et al.~\\cite{Hill09} The coefficients $F^{C}_{n+1}$ are specific to the $\\mathrm{CCSD-F12b}$ and $\\mathrm{(T)}$ components of the total CCSD(T)-F12b energy and we use values of $F^{\\mathrm{CCSD-F12b}}=1.363388$ and $F^{\\mathrm{(T)}}=1.769474$ as recommended in Ref.~\\onlinecite{Hill09}. No extrapolation was applied to the Hartree-Fock (HF) energy, rather the HF+CABS singles correction~\\cite{Adler07} calculated in the larger basis set was used.\n\n The energy correction from core-valence electron correlation $\\Delta E_{\\mathrm{CV}}$ was calculated at the CCSD(T)-F12b level of theory with the F12-optimized correlation consistent core-valence basis set cc-pCVQZ-F12.~\\cite{Hill10} The same ansatz and ABS as in the frozen core approximation computations were used, however we set $\\beta=1.5$~$a_0^{-1}$. All-electron calculations kept the (1\\textit{s}) orbital of Cl frozen with all other electrons correlated due to the inability of the basis set to adequately describe this orbital.\n \n Core-valence and higher-order electron correlation often contribute to the electronic energy with opposing signs and should thus be considered jointly. We use the hierarchy of coupled cluster methods to estimate the HO correction as $\\Delta E_{\\mathrm{HO}} = \\Delta E_{\\mathrm{T}} + \\Delta E_{\\mathrm{(Q)}}$, including the full triples contribution $\\Delta E_{\\mathrm{T}} = \\left[E_{\\mathrm{CCSDT}}-E_{\\mathrm{CCSD(T)}}\\right]$, and the perturbative quadruples contribution $\\Delta E_{\\mathrm{(Q)}} = \\left[E_{\\mathrm{CCSDT(Q)}}-E_{\\mathrm{CCSDT}}\\right]$. Calculations were carried out in the frozen core approximation at the CCSD(T), CCSDT and CCSDT(Q) levels of theory using the general coupled cluster approach~\\cite{Kallay05,Kallay08} as implemented in the MRCC code~\\cite{mrcc} interfaced to CFOUR.~\\cite{cfour} For the full triples and the perturbative quadruples calculations, we employed the augmented correlation consistent triple zeta basis set, aug-cc-pVTZ(+d for Cl),~\\cite{Dunning89,Kendall92,Woon93,Dunning01} and the double zeta basis set, aug-cc-pVDZ(+d for Cl), respectively. Note that for HO coupled cluster corrections, it is possible to use successively smaller basis sets at each step up in excitation level due to faster convergence.~\\cite{Feller06}\n \n In exploratory calculations, the contributions from the full quadruples $\\left[E_{\\mathrm{CCSDTQ}}-E_{\\mathrm{CCSDT(Q)}}\\right]$ and from the perturbative pentuples $\\left[E_{\\mathrm{CCSDTQ(P)}}-E_{\\mathrm{CCSDTQ}}\\right]$ were found to largely cancel each other out. Thus to reduce the computational expense, only $\\Delta E_{\\mathrm{T}}$ and $\\Delta E_{\\mathrm{(Q)}}$ were deemed necessary for an adequate representation of HO electron correlation.\n \n Scalar relativistic effects $\\Delta E_{\\mathrm{SR}}$ were included through the one-electron mass velocity and Darwin terms (MVD1) from the Breit-Pauli Hamiltonian in first-order perturbation theory.~\\cite{Cowan76} Calculations were performed with all electrons correlated (except for the (1\\textit{s}) of Cl) at the CCSD(T)\/aug-cc-pCVTZ(+d for Cl)~\\cite{Woon95,Peterson02} level of theory using the MVD1 approach~\\cite{Klopper97} implemented in CFOUR. The contribution from the two-electron Darwin term is expected to be small enough to be neglected.~\\cite{Gauss07}\n \n The diagonal Born-Oppenheimer correction $\\Delta E_{\\mathrm{DBOC}}$ was computed again with the (1\\textit{s}) orbital of Cl frozen and all other electrons correlated. Calculations employed the CCSD method~\\cite{Gauss06} as implemented in CFOUR with the aug-cc-pCVTZ(+d for Cl) basis set. The DBOC is the contribution from the nuclear kinetic energy operator acting on the ground state electronic wavefunction. It is mass dependent, so separate contributions were generated for CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl.\n\n The spin-orbit interaction was not considered as it can be safely neglected in spectroscopic calculations on light closed-shell molecules.~\\cite{Tarczay01} A simple estimate of the Lamb shift was also calculated from the MVD1 contribution,~\\cite{Pyykko01} but its effect on the vibrational energy levels was negligible. The differing levels of theory and basis set size reflect the fact that different HL energy corrections converge at different rates.\n\n Grid points were generated using a random energy-weighted sampling algorithm of Monte Carlo type, in terms of nine internal coordinates: the C-Cl bond length $r_0$; three C-H bond lengths $r_1$, $r_2$ and $r_3$; three $\\angle(\\mathrm{H}_i\\mathrm{CCl})$ interbond angles $\\beta_1$, $\\beta_2$ and $\\beta_3$; and two dihedral angles $\\tau_{12}$ and $\\tau_{13}$ between adjacent planes containing H$_i$CCl and H$_j$CCl (see Figure~\\eqref{fig:geometry}). This led to a global grid of $44{\\,}820$ geometries with energies up to $h c \\cdot 50{\\,}000{\\,}$cm$^{-1}$, which included geometries in the range $1.3\\leq r_0 \\leq 2.95{\\,}\\mathrm{\\AA}$, $0.7\\leq r_i \\leq 2.45{\\,}\\mathrm{\\AA}$, $65\\leq \\beta_i \\leq 165^{\\circ}$ for $i=1,2,3$ and $55\\leq \\tau_{jk} \\leq 185^{\\circ}$ with $jk=12,13$. To ensure an adequate description of the equilibrium region, around $1000$ carefully chosen low-energy points were also incorporated into the data set. At each grid point, the computed coupled cluster energies were extrapolated to the CBS limit using Eq.\\eqref{eq:cbs_extrap}.\n\n \\begin{figure}\n \\includegraphics{new_ch3cl_draw_int_coord}\n \\caption{\\label{fig:geometry}Definition of internal coordinates used for CH$_3$Cl.}\n \\end{figure}\n\n The HL energy corrections are generally small in magnitude and vary in a smooth manner,~\\cite{YaYuRi11.H2CS} displaying a straightforward polynomial-type dependence as can be seen in Figures \\eqref{fig:1d_cv_ho} and \\eqref{fig:1d_mvd1_dboc}. For each of the HL terms, a reduced grid was carefully designed to obtain a satisfactory description of the correction with minimum computational effort. Reduced grids of $9377$, $3526$, $12{\\,}296$ and $3679$ points with energies up to $h c \\cdot 50{\\,}000{\\,}$cm$^{-1}$ were used for the CV, HO, SR and DBOC corrections, respectively.\n \n \\begin{figure*}\n \\includegraphics{cv_ho_1d_ch3cl}\n \\caption{\\label{fig:1d_cv_ho}One-dimensional cuts of the core-valence (CV) and higher-order (HO) corrections with all other coordinates held at their equilibrium values.}\n \\end{figure*}\n \n \\begin{figure*}\n \\includegraphics{mvd1_dboc_1d_ch3cl}\n \\caption{\\label{fig:1d_mvd1_dboc} One-dimensional cuts of the scalar relativistic (MVD1) and diagonal Born-Oppenheimer (DBOC) corrections with all other coordinates held at their equilibrium values.}\n \\end{figure*}\n\n\\subsection{Analytic representation}\n\n Methyl chloride is a prolate symmetric top molecule of the $\\bm{C}_{3\\mathrm{v}}\\mathrm{(M)}$ symmetry group.~\\cite{MolSym_BuJe98} Of the six symmetry operations $\\left\\lbrace E,(123),(132),(12)^{*},(23)^{*},(13)^{*}\\right\\rbrace$ which make up $\\bm{C}_{3\\mathrm{v}}$(M), the cyclic permutation $(123)$ replaces nucleus 1 with nucleus 2, nucleus 2 with nucleus 3, and nucleus 3 with nucleus 1. The permutation-inversion operation $(12)^{*}$ interchanges nuclei 1 and 2 and inverts all particles (including electrons) in the molecular centre of mass. The identity operation $E$ leaves the molecule unchanged.\n \n To represent the PES analytically, an on-the-fly symmetrization procedure has been implemented. We first introduce the coordinates\n\\begin{equation}\\label{eq:stretch1}\n\\xi_1=1-\\exp\\left(-a(r_0 - r_0^{\\mathrm{eq}})\\right)\n\\end{equation}\n\\begin{equation}\\label{eq:stretch2}\n\\xi_j=1-\\exp\\left(-b(r_i - r_1^{\\mathrm{eq}})\\right){\\,};\\hspace{2mm}j=2,3,4{\\,}, \\hspace{2mm} i=j-1\n\\end{equation}\n\n\\noindent where $a=1.65{\\,}\\mathrm{\\AA}^{-1}$ for the C-Cl internal coordinate $r_0$, and $b=1.75{\\,}\\mathrm{\\AA}^{-1}$ for the three C-H internal coordinates $r_1,r_2$ and $r_3$. For the angular terms\n\\begin{equation}\\label{eq:angular1}\n\\xi_k = (\\beta_i - \\beta^{\\mathrm{eq}}){\\,};\\hspace{2mm}k=5,6,7{\\,}, \\hspace{2mm} i=k-4\n\\end{equation}\n\\begin{equation}\\label{eq:angular2}\n\\xi_8 = \\frac{1}{\\sqrt{6}}\\left(2\\tau_{23}-\\tau_{13}-\\tau_{12}\\right)\n\\end{equation}\n\\begin{equation}\\label{eq:angular3}\n\\xi_9 = \\frac{1}{\\sqrt{2}}\\left(\\tau_{13}-\\tau_{12}\\right)\n\\end{equation}\n\n\\noindent Here $\\tau_{23}=2\\pi-\\tau_{12}-\\tau_{13}$, and $r_0^{\\mathrm{eq}}$, $r_1^{\\mathrm{eq}}$ and $\\beta^{\\mathrm{eq}}$ are the reference equilibrium structural parameters of CH$_3$Cl. \n\n Taking an initial potential term of the form\n\\begin{equation}\\label{eq:V_i}\nV_{ijk\\ldots}^{\\mathrm{initial}}=\\xi_{1}^{\\,i}\\xi_{2}^{\\,j}\\xi_{3}^{\\,k}\\xi_{4}^{\\,l}\\xi_{5}^{\\,m}\\xi_{6}^{\\,n}\\xi_{7}^{\\,p}\\xi_{8}^{\\,q}\\xi_{9}^{\\,r}\n\\end{equation}\n\n\\noindent with maximum expansion order $i+j+k+l+m+n+p+q+r=6$, each symmetry operation of $\\bm{C}_{3\\mathrm{v}}$(M) is independently applied to $V_{ijk\\ldots}^{\\mathrm{initial}}$, i.e.\n\\begin{equation}\\label{eq:V_op}\nV_{ijk\\ldots}^{\\mathbf{X}}=\\mathbf{X}{\\,}V_{ijk\\ldots}^{\\mathrm{initial}}=\\mathbf{X}\\left(\\xi_{1}^{\\,i}\\xi_{2}^{\\,j}\\xi_{3}^{\\,k}\\xi_{4}^{\\,l}\\xi_{5}^{\\,m}\\xi_{6}^{\\,n}\\xi_{7}^{\\,p}\\xi_{8}^{\\,q}\\xi_{9}^{\\,r}\\right)\n\\end{equation}\n\n\\noindent where $\\mathbf{X}=\\lbrace E,(123),(132),(12)^{*},(23)^{*},(13)^{*}\\rbrace$, to create six new terms. The results are summed up to produce a final term,\n\\begin{equation}\\label{eq:V_f}\nV_{ijk\\ldots}^{\\mathrm{final}}=V_{ijk\\ldots}^{E}+V_{ijk\\ldots}^{(123)}+V_{ijk\\ldots}^{(132)}+V_{ijk\\ldots}^{(12)^*}+V_{ijk\\ldots}^{(23)^*}+V_{ijk\\ldots}^{(13)^*}\n\\end{equation}\n\n\\noindent which is itself subjected to the six $\\bm{C}_{3\\mathrm{v}}$(M) symmetry operations to check its invariance. The total potential function is then given by the expression\n\\begin{equation}\\label{eq:pot_f}\nV_{\\mathrm{total}}(\\xi_{1},\\xi_{2},\\xi_{3},\\xi_{4},\\xi_{5},\\xi_{6},\\xi_{7},\\xi_{8},\\xi_{9})={\\sum_{ijk\\ldots}}{\\,}\\mathrm{f}_{ijk\\ldots}V_{ijk\\ldots}^{\\mathrm{final}}\n\\end{equation}\n\n\\noindent where $\\mathrm{f}_{ijk\\ldots}$ are the corresponding expansion coefficients, determined through a least-squares fitting to the \\textit{ab initio} data. Weight factors of the form suggested by Partridge and Schwenke,~\\cite{Schwenke97}\n\\begin{equation}\\label{eq:weights}\nw_i=\\left(\\frac{\\tanh\\left[-0.0006\\times(\\tilde{E}_i - 15{\\,}000)\\right]+1.002002002}{2.002002002}\\right)\\times\\frac{1}{N\\tilde{E}_i^{(w)}}\n\\end{equation}\n\n\\noindent were used in the fitting, with normalization constant $N=0.0001$ and $\\tilde{E}_i^{(w)}=\\max(\\tilde{E}_i, 10{\\,}000)$, where $\\tilde{E}_i$ is the potential energy at the $i$th geometry above equilibrium (all values in cm$^{-1}$). In our fitting, energies below $15{\\,}000{\\,}$cm$^{-1}$ are favoured by the weight factors. For geometries where $r_0\\geq 2.35{\\,}\\mathrm{\\AA}$ and $r_i\\geq 2.00{\\,}\\mathrm{\\AA}$ for $i=1,2,3$, the weights were reduced by several orders of magnitude. At such large stretch coordinates, the coupled cluster method is known to become unreliable, as indicated by a T1 diagnostic value $>0.02$.~\\cite{T1_Lee89} Although energies at these points may not be wholly accurate, they are still useful and ensure that the PES maintains a reasonable shape towards dissociation.\n\n The same form of potential function, Eq.\\eqref{eq:pot_f}, and the same procedure, Eqs.\\eqref{eq:V_i} to \\eqref{eq:V_f}, were used to fit the higher-level correction surfaces. The stretching coordinates however were replaced with linear expansion variables; $\\xi_1=(r_0-r_0^{\\mathrm{eq}})$ and $\\xi_j=(r_i-r_1^{\\mathrm{eq}})$ where $j=2,3,4$ and $i=j-1$. The angular terms, Eqs.\\eqref{eq:angular1} to \\eqref{eq:angular3}, remained the same as before. Each HL correction was fitted independently and the parameters $r_0^{\\mathrm{eq}}$, $r_1^{\\mathrm{eq}}$ and $\\beta^{\\mathrm{eq}}$ were optimized for each surface. The four HL corrections were applied at each of the $44{\\,}820$ grid points, either from a directly calculated value at that geometry, or by interpolation using the corresponding analytic representation. Two final data sets were produced, one for each isotopologue of CH$_3$Cl, the only difference being the contribution from the DBOC.\n \n Two separate fits were carried out and in each instance we could usefully vary 414 expansion parameters to give a weighted root-mean-square (rms) error of $0.82{\\,}$cm$^{-1}$ for energies up to $50{\\,}000{\\,}$cm$^{-1}$. The fit employed Watson's robust fitting scheme,~\\cite{Watson03} the idea of which is to reduce the weight of outliers and lessen their influence in determining the final set of parameters. The Watson scheme improves the fit at energies below $10{\\,}000{\\,}$cm$^{-1}$ which is preferable for our purposes. When comparing the expansion parameters for CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl, only very slight differences were observed in the determined values. We refer to these two PESs as CBS-35$^{\\,\\mathrm{HL}}$ and CBS-37$^{\\,\\mathrm{HL}}$ in subsequent calculations.\n \n To assess the combined effect of the HL corrections and CBS extrapolation on the vibrational energy levels and equilibrium geometry of CH$_3$Cl, we fit a reference PES to the raw CCSD(T)-F12b\/cc-pVQZ-F12 energies. Again we used Watson's robust fitting scheme and 414 parameters to give a weighted rms error of $0.82{\\,}$cm$^{-1}$ for energies up to $50{\\,}000{\\,}$cm$^{-1}$. We refer to this PES as VQZ-F12 in subsequent calculations. Note that the CBS-(35\/37)$^{\\,\\mathrm{HL}}$ and VQZ-F12 PESs contain only slightly different parameter sets.\n \n The choice of reference equilibrium structural parameters in our PES expansion is to some extent arbitrary due to the inclusion of linear expansion terms in the parameter set. For this reason, values of $r_0^{\\mathrm{eq}}=1.7775{\\,}\\mathrm{\\AA}$, $r_1^{\\mathrm{eq}}=1.0837{\\,}\\mathrm{\\AA}$, and $\\beta^{\\mathrm{eq}}=108.445^{\\circ}$, used for the CBS-35$^{\\,\\mathrm{HL}}$ PES, were also employed for the CBS-37$^{\\,\\mathrm{HL}}$ and VQZ-F12 PESs. Note that these are not the actual equilibrium parameters which define the minimum of the PES, they are simply parameters of a function. The true equilibrium values will be determined and discussed in Sec.~\\ref{sec:variational}.\n \n Generating a PES on-the-fly is advantageous when it comes to variational calculations as its implementation requires only a short amount of code. Alternatively, one can derive the full analytic expression for the potential and incorporate this into the nuclear motion computations, but this method is cumbersome. The CBS-35$^{\\,\\mathrm{HL}}$ and CBS-37$^{\\,\\mathrm{HL}}$ expansion parameter sets are provided in the supplementary material along with a FORTRAN routine to construct the PESs.~\\cite{EPAPSCH3CL} \n\n\\section{Results}\n\\label{sec:variational}\n\n\\subsection{Extrapolation to the Complete Vibrational Basis Set Limit}\n\\label{sec:cvbs}\n\n The nuclear motion code TROVE (Theoretical ROVibrational Energies)~\\cite{TROVE2007} is designed to calculate the rotation-vibration energy levels and corresponding transition intensities for a polyatomic molecule of arbitrary structure in an isolated electronic state. The flexibility of TROVE has allowed a range of molecular systems to be treated,~\\cite{OvThYu08a.PH3,OvThYu08.PH3,YaYuRi11.H2CS,YuBaYa09.NH3,YuYaTh09.HSOH,YaYuJe10.HSOH,YuCaYa10.SBH3,\n YaYuJe11.H2CO,UnTeYu13.SO3,PoKoOv13.H2O2,YuTeBa13.CH4} and for the present study the functionality has been extended to handle molecules of the form XY$_3$Z.\n\n In TROVE, solution of the rotation-vibration Schr\\\"{o}dinger equation is achieved by numerical diagonalization of the corresponding Hamiltonian constructed in terms of a symmetry adapted basis set. The rovibrational Hamiltonian is represented as a power series expansion around a reference geometry, taken presently at the equilibrium configuration. For the present work we take advantage of recent developments in TROVE, in particular the implementation of a novel method of constructing the rovibrational Hamiltonian in terms of curvilinear internal coordinates.~\\cite{YaYu15.ADF} By employing this new approach, our variational results show much faster convergence with respect to vibrational basis set size. We will see the importance of this later on. For CH$_3$Cl, we truncate the kinetic and potential energy operators at 6th and 8th order respectively in all calculations. This level of truncation is sufficient for our purposes, however we refer the reader to Ref.~\\onlinecite{TROVE2007} for a detailed discussion of the associated errors of such a scheme. Note that atomic mass values have been used in the subsequent TROVE computations.\n \n A multi-step contraction scheme is used to generate the vibrational basis set, the size of which is controlled by the polyad number \n\\begin{equation}\n P = \\sum_{k=1}^{9} a_k n_k\n\\end{equation}\n\n\\noindent The quantum numbers $n_k$ correspond to primitive basis functions $\\phi_{n_k}$, which are obtained from solving one-dimensional Schr\\\"{o}dinger equations for each of the nine vibrational modes by means of the Numerov-Cooley method.~\\cite{Numerov1924,Cooley1961} Using the definition of the polyad coefficient $a_k=\\omega_k\/\\min(\\omega_1\\ldots\\omega_9)$, where $\\omega_k$ denotes the harmonic frequency of the $k$th mode, the polyad number for CH$_{3}$Cl is\n\\begin{equation}\\label{eq:polyad}\nP = n_1+2(n_2+n_3+n_4)+n_5+n_6+n_7+n_8+n_9 \\leq P_{\\mathrm{max}}\n\\end{equation}\n\n\\noindent which does not exceed a predefined maximum value $P_{\\mathrm{max}}$.\n\n Fully converged energies in variational calculations are usually obtained with the use of an extended basis set. We have only been able to compute $J=0$ vibrational energies up to a polyad truncation number of $P_{\\mathrm{max}}=14$ for CH$_3$Cl. As shown in Figure~\\eqref{fig:dimension}, this requires the diagonalization of a Hamiltonian matrix of dimension close to $128{\\,}000$, which in turn equals the number of primitive basis functions generated. The extension to $P_{\\mathrm{max}}=16$ using TROVE would be an arduous computational task. \n \n \\begin{figure}\n \\includegraphics{dim_matrix_poly}\n \\caption{\\label{fig:dimension}Size of the $J=0$ Hamiltonian matrix with respect to the polyad truncation number $P_{\\mathrm{max}}$. Computations were feasible up to $P_{\\mathrm{max}}=14$.}\n \\end{figure}\n \n One means of achieving converged vibrational energy levels without having to diagonalize increasingly large matrices, is the use of a complete vibrational basis set (CVBS) extrapolation.~\\cite{OvThYu08.PH3} In analogy to the common basis set extrapolation techniques of electronic structure theory,~\\cite{Petersson88,Petersson91} the same principles can be applied to TROVE calculations with respect to $P_{\\mathrm{max}}$. We adopt the exponential decay expression\n\\begin{equation}\n E_i(P_{\\mathrm{max}}) = E_i^{\\mathrm{CVBS}}+A_i\\exp(-\\lambda_i P_{\\mathrm{max}})\n\\end{equation}\n\n\\noindent where $E_i$ is the energy of the $i$th level, $E_i^{\\mathrm{CVBS}}$ is the respective energy at the CVBS limit, $A_i$ is a fitting parameter, and $\\lambda_i$ can be found from\n\\begin{equation}\n\\lambda_i=-\\frac{1}{2}\\ln\\left(\\frac{E_i(P_{\\mathrm{max}}+2)-E_i(P_{\\mathrm{max}})}{E_i(P_{\\mathrm{max}})-E_i(P_{\\mathrm{max}}-2)}\\right)\n\\end{equation}\n\n Values of $P_{\\mathrm{max}}=\\lbrace 10,12,14 \\rbrace$ were employed for a CVBS extrapolation of all vibrational term values up to $5000{\\,}$cm$^{-1}$, and for selected higher energies to compare with experiment. This was done for the CBS-35$^{{\\,}\\mathrm{HL}}$, CBS-37$^{{\\,}\\mathrm{HL}}$, and VQZ-F12 PESs. In Figure~\\eqref{fig:converge}, the convergence of the vibrational energy levels up to $5000{\\,}$cm$^{-1}$ for the CBS-35$^{{\\,}\\mathrm{HL}}$ PES can be seen with respect to the final $E_i^{\\mathrm{CVBS}}$ extrapolated values. Below $4000{\\,}$cm$^{-1}$ the computed $P_{\\mathrm{max}}=14$ term values are already reasonably well converged. Only five levels in this range possess a residual $\\Delta E(P_{\\mathrm{max}}-P_{\\mathrm{CVBS}})$ larger than $0.1{\\,}$cm$^{-1}$, none of which is greater than $0.3{\\,}$cm$^{-1}$. As expected, levels involving highly excited modes benefit the most from extrapolation as these converge at a much slower rate. \n\n \\begin{figure}\n \\includegraphics{pcvbs_term_converge}\n \\caption{\\label{fig:converge}Convergence of vibrational term values of CH$_{3}{}^{35}$Cl up to $5000{\\,}$cm$^{-1}$ with respect to $P_{\\mathrm{max}}=P_{\\mathrm{CVBS}}$. For illustrative purposes we restrict the range of $\\Delta E(P_{\\mathrm{max}}-P_{\\mathrm{CVBS}})$ to $10{\\,}$cm$^{-1}$.}\n \\end{figure}\n \n The limiting factor of a CVBS extrapolation is the correct identification of the energy levels at each step up in basis set size. TROVE automatically assigns quantum numbers to the eigenvalues and corresponding eigenvectors by analysing the contribution of the basis functions. Due to the increased density of states above $5000{\\,}$cm$^{-1}$ for higher values of $P_{\\mathrm{max}}$, it quickly becomes difficult to consistently identify and match levels, except for highly excited individual modes. \n\n\\subsection{Vibrational $J=0$ Energies}\n\\label{sec:vib_enr}\n\n The normal modes of methyl chloride are classified by two symmetry species, $A_{1}$ and $E$. Of $A_{1}$ symmetry are the three non-degenerate modes; the symmetric CH$_{3}$ stretching mode $\\nu_{1}$ ($2967.77\/2967.75{\\,}$cm$^{-1}$), the symmetric CH$_{3}$ deformation mode $\\nu_{2}$ ($1354.88\/1354.69{\\,}$cm$^{-1}$), and the C{--}Cl stretching mode $\\nu_{3}$ ($732.84\/727.03{\\,}$cm$^{-1}$). Of $E$ symmetry are the three degenerate modes; the CH$_{3}$ stretching mode $\\nu_{4}^{l_{4}}$ ($3039.26\/3039.63{\\,}$cm$^{-1}$), the CH$_{3}$ deformation mode $\\nu_{5}^{l_{5}}$ ($1452.18\/1452.16{\\,}$cm$^{-1}$), and the CH$_{3}$ rocking mode $\\nu_{6}^{l_{6}}$ ($1018.07\/1017.68{\\,}$cm$^{-1}$). The values in parentheses are the experimentally determined values for CH$_{3}{}^{35}$Cl\/CH$_{3}{}^{37}$Cl from Refs.~\\onlinecite{11BrPeJa.CH3Cl} and \\onlinecite{05NiChBu.CH3Cl}. The additional vibrational angular momentum quantum numbers $l_{4}$, $l_{5}$, and $l_{6}$ are needed to resolve the degeneracy of their respective modes. To be of spectroscopic use we map the vibrational quantum numbers $n_k$ of TROVE to the normal mode quantum numbers $\\mathrm{v}_k$ commonly used. For CH$_3$Cl, the vibrational states are labelled as $\\mathrm{v_1}\\nu_1+\\mathrm{v_2}\\nu_2+\\mathrm{v_3}\\nu_3+\\mathrm{v_4}\\nu_4+\\mathrm{v_5}\\nu_5+\\mathrm{v_6}\\nu_6$ where $\\mathrm{v_i}$ counts the level of excitation.\n\n The calculated $J=0$ energy levels for CH$_{3}{}^{35}$Cl using the CBS-35$^{{\\,}\\mathrm{HL}}$ and VQZ-F12 PESs are listed in Table~\\ref{tab:j0_cbs_35cl}. We compare against all available experimental data taken from Refs.~\\onlinecite{11BrPeJa.CH3Cl}, \\onlinecite{05NiChBu.CH3Cl}, \\onlinecite{90DuLaxx.CH3Cl}, and \\onlinecite{99Laxxxx.CH3Cl}. A small number of levels from Refs.~\\onlinecite{90DuLaxx.CH3Cl} and \\onlinecite{99Laxxxx.CH3Cl} have not been included as we were unable to confidently identify the corresponding values in TROVE.\n \n\\setlength\\LTleft{0pt}\n\\setlength\\LTright{0pt}\n\\LTcapwidth=\\textwidth\n\\begin{longtable*}[ht]{@{\\extracolsep{\\fill}} l c c c c c c c}\n\\caption{\\label{tab:j0_cbs_35cl} Comparison of calculated and experimental $J=0$ vibrational term values (in cm$^{-1}$) for CH$_{3}{}^{35}$Cl. The zero-point energy was computed to be $8219.661\\,$cm$^{-1}$ at the CVBS limit.}\\\\ \\hline\\hline\nMode & Sym. & VQZ-F12{\\,}(A) & CBS-35$^{{\\,}\\mathrm{HL}}${\\,}(B) & Exp. & Obs$-$calc{\\,}(A) & Obs$-$calc{\\,}(B) & Ref.\\\\ \\hline\n\\endfirsthead\n\\caption{(\\textit{Continued})}\\\\ \\hline \nMode & Sym. & VQZ-F12{\\,}(A) & CBS-35$^{{\\,}\\mathrm{HL}}${\\,}(B) & Exp. & Obs$-$calc{\\,}(A) & Obs$-$calc{\\,}(B) & Ref.\\\\ \\hline\n\\endhead\n$\\nu_3$ & $A_1$ & 734.37 & 733.22 & 732.8422 & -1.53 & -0.38 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_6$ & $E$ & 1018.16 & 1018.05 & 1018.0709 & -0.09 & 0.02 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_2$ & $A_1$ & 1355.11 & 1355.01 & 1354.8811 & -0.23 & -0.13 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_5$ & $E$ & 1451.57 & 1452.56 & 1452.1784 & 0.61 & -0.38 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$2\\nu_3$ & $A_1$ & 1459.94 & 1457.54 & 1456.7626 & -3.18 & -0.78 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_3+\\nu_6$ & $E$ & 1747.10 & 1745.78 & 1745.3711 & -1.73 & -0.41 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$2\\nu_6$ & $A_1$ & 2029.67 & 2029.46 & 2029.3753 & -0.29 & -0.09 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$2\\nu_6$ & $E$ & 2038.58 & 2038.37 & 2038.3262 & -0.25 & -0.04 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_2+\\nu_3$ & $A_1$ & 2082.35 & 2080.98 & 2080.5357 & -1.82 & -0.45 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$3\\nu_3$ & $A_1$ & 2176.84 & 2173.09 & 2171.8875 & -4.95 & -1.20 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_3+\\nu_5$ & $E$ & 2183.51 & 2183.30 & 2182.5717 & -0.94 & -0.73 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_2+\\nu_6$ & $E$ & 2368.08 & 2367.90 & 2367.7222 & -0.35 & -0.18 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_5+\\nu_6$ & $E$ & 2461.19 & 2461.98 & 2461.6482 & 0.46 & -0.33 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$2\\nu_3+\\nu_6$& $E$ & 2467.19 & 2464.65 & 2463.8182 & -3.37 & -0.83 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_5+\\nu_6$ & $A_1$ & 2464.50 & 2465.28 & 2464.9025 & 0.40 & -0.38 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$\\nu_5+\\nu_6$ & $A_2$ & 2466.85 & 2467.85 & 2467.6694 & 0.82 & -0.18 & \\onlinecite{05NiChBu.CH3Cl}\\\\\n$2\\nu_2$ & $A_1$ & 2694.69 & 2694.61 & 2693.00 & -1.69 & -1.61 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_3+2\\nu_6$& $A_1$ & 2753.23 & 2751.74 & 2751.18 & -2.05 & -0.56 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_2+2\\nu_3$& $A_1$ & 2800.39 & 2797.64 & 2796.81 & -3.58 & -0.83 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_2+\\nu_5$ & $E$ & 2803.10 & 2803.96 & 2803.26 & 0.16 & -0.70 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$4\\nu_3$ & $A_1$ & 2885.23 & 2880.47 & 2878.00 & -7.23 & -2.47 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_5$ & $A_1$ & 2877.75 & 2879.31 & 2879.25 & 1.50 & -0.06 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_5$ & $E$ & 2896.27 & 2898.22 & 2895.566 & -0.71 & -2.65 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$2\\nu_3+\\nu_5$& $E$ & 2906.64 & 2905.14 & 2907.903 & 1.26 & 2.77 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$\\nu_1$ & $A_1$ & 2965.78 & 2969.16 & 2967.7691 & 1.99 & -1.39 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$\\nu_4$ & $E$ & 3035.50 & 3038.19 & 3039.2635 & 3.76 & 1.07 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$3\\nu_6$ & $E$ & 3045.08 & 3045.76 & 3042.8944 & -2.18 & -2.87 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$3\\nu_6$ & $A_1$ & 3060.95 & 3060.62 & 3060.0064 & -0.95 & -0.62 & \\onlinecite{11BrPeJa.CH3Cl}\\\\\n$\\nu_2+2\\nu_6$& $A_1$ & 3373.81 & 3373.57 & 3373.5 & -0.31 & -0.07 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_2+\\nu_3$& $A_1$ & 3415.53 & 3414.02 & 3413.0 & -2.53 & -1.02 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_3+2\\nu_5$& $A_1$ & 3607.98 & 3608.77 & 3607.70 & -0.28 & -1.07 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_1+\\nu_3$ & $A_1$ & 3700.18 & 3702.43 & 3700.67 & 0.49 & -1.76 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_2+\\nu_6$& $E$ & 3702.92 & 3702.80 & 3702.69 & -0.23 & -0.11 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_3+3\\nu_6$& $E$ & 3760.53 & 3759.07 & 3756.6 & -3.93 & -2.47 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_3+\\nu_4$ & $E$ & 3773.98 & 3776.04 & 3773.52 & -0.46 & -2.52 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_5+\\nu_6$& $E$ & 3884.88 & 3886.75 & 3886.05 & 1.17 & -0.70 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_1+\\nu_6$ & $E$ & 3977.68 & 3980.97 & 3979.66 & 1.98 & -1.31 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_4+\\nu_6$ & $E$ & 4047.37 & 4049.83 & 4051.22 & 3.85 & 1.39 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_2+\\nu_5$& $E$ & 4137.96 & 4138.86 & 4138.29 & 0.33 & -0.57 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_2+2\\nu_5$& $A_1$ & 4229.37 & 4231.18 & 4230.34 & 0.97 & -0.84 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_2+3\\nu_6$& $E$ & 4378.39 & 4379.54 & 4380.52 & 2.13 & 0.98 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_2+\\nu_4$ & $E$ & 4384.03 & 4385.90 & 4382.64 & -1.39 & -3.26 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_1+\\nu_5$ & $E$ & 4412.81 & 4416.81 & 4415.4 & 2.59 & -1.41 & \\onlinecite{99Laxxxx.CH3Cl}\\\\\n$\\nu_1+2\\nu_6$& $A_1$ & 4982.68 & 4985.90 & 4984.0 & 1.32 & -1.90 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_1+2\\nu_2$& $A_1$ & 5655.67 & 5658.93 & 5657.0 & 1.33 & -1.93 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_2+\\nu_4$& $E$ & 5708.85$^a$& 5711.12$^a$&5713 & 4.15 & 1.88 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$\\nu_1+\\nu_4$ & $E$ & 5870.30 & 5875.98 & 5873.8 & 3.50 & -2.18 & \\onlinecite{99Laxxxx.CH3Cl}\\\\\n$2\\nu_1$ & $A_1$ & 5875.28 & 5881.04 & 5878 & 2.72 & -3.04 & \\onlinecite{99Laxxxx.CH3Cl}\\\\\n$\\nu_4+2\\nu_5$& $E$ & 5918.20$^a$&5923.37$^a$&5923.4& 5.20 & 0.03 & \\onlinecite{99Laxxxx.CH3Cl}\\\\\n$2\\nu_4$ & $A_1$ & 6011.38 & 6018.47 & 6015.3 & 3.92 & -3.17 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_1+\\nu_5$& $E$ & 7303.96 & 7311.08 & 7313.2 & 9.24 & 2.12 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_4+\\nu_5$& $E$ & 7437.80 & 7445.86 & 7443.2 & 5.40 & -2.66 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$2\\nu_4+2\\nu_5$& $A_1$& 8870.15 & 8877.25 & 8874.3 & 4.15 & -2.95 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n$3\\nu_4$ & $A_1$ & 9069.53 & 9079.28 & 9076.9 & 7.37 & -2.38 & \\onlinecite{90DuLaxx.CH3Cl}\\\\\n\\hline\\hline\n$^a$ $P_{\\mathrm{max}}=14$ value. \\\\[-2mm]\n\\end{longtable*}\n \n The CBS-35$^{{\\,}\\mathrm{HL}}$ PES reproduces the six fundamental term values with a root-mean-square (rms) error of $0.75{\\,}$cm$^{-1}$ and a mean-absolute-deviation (mad) of $0.56{\\,}$cm$^{-1}$. This is a considerable improvement over the results of the VQZ-F12 PES, which reproduces the fundamentals with a rms error of $1.86{\\,}$cm$^{-1}$ and a mad of $1.37{\\,}$cm$^{-1}$. Inspection of all computed CH$_{3}{}^{35}$Cl energy levels shows that on the whole, the CBS-35$^{{\\,}\\mathrm{HL}}$ PES is more reliable. This is gratifying as the effort required to generate the CBS-35$^{{\\,}\\mathrm{HL}}$ PES is far greater than that of the VQZ-F12 PES. Unlike other instances,~\\cite{YaYuRi11.H2CS} the VQZ-F12 results do not benefit from an extensive cancellation of errors. Note that the PES reported in Ref.~\\onlinecite{08Nixxxx.CH3Cl}, which did not treat any additional HL energy corrections, produces results with errors similar to those of the VQZ-F12 PES.\n \n The accuracy achieved at lower energies with the CBS-35$^{{\\,}\\mathrm{HL}}$ PES is quite remarkable, with residuals larger than $2{\\,}$cm$^{-1}$ starting to appear around $3000{\\,}$cm$^{-1}$. This is a notoriously difficult region of CH$_{3}$Cl with strong resonances, but the experimental values we compare against are from a recent high-resolution study and should thus be trustworthy.~\\cite{11BrPeJa.CH3Cl} In the comparison against values reported in Ref.~\\onlinecite{90DuLaxx.CH3Cl}, and subsequently used in Ref.~\\onlinecite{99Laxxxx.CH3Cl}, we exercise some caution. The $2\\nu_5(E)$ and $2\\nu_3+\\nu_5(E)$ levels presented in Ref.~\\onlinecite{90DuLaxx.CH3Cl} are lower by around $3$ and $5{\\,}$cm$^{-1}$ respectively when compared with new values measured in Ref.~\\onlinecite{11BrPeJa.CH3Cl}. However the agreement for the $\\nu_1$, $\\nu_4$ and $3\\nu_6(E)$ levels is excellent. The residual for the $2\\nu_2$ level seems large given the residual for the $\\nu_2$ term value, and we suspect that the experimental value is incorrect. At higher energies the quality of the CBS-35$^{{\\,}\\mathrm{HL}}$ PES does not appear to deteriorate significantly.\n \n For the $^{37}$Cl isotopologue of methyl chloride, the $J=0$ term values calculated from the CBS-37$^{{\\,}\\mathrm{HL}}$ PES are compared with all available experimental data in Table~\\ref{tab:j0_cbs_37cl}. The CBS-37$^{{\\,}\\mathrm{HL}}$ PES reproduces the six fundamental term values with a rms error of $1.00{\\,}$cm$^{-1}$ and a mad of $0.70{\\,}$cm$^{-1}$. The reduction in accuracy when compared to the CBS-35$^{{\\,}\\mathrm{HL}}$ PES is primarily due to the $\\nu_4$ mode, whose residual has gone from $1.07{\\,}$cm$^{-1}$ for CH$_{3}{}^{35}$Cl to $1.92{\\,}$cm$^{-1}$ for CH$_{3}{}^{37}$Cl. The accuracy of the $3\\nu_6(E)$ level has also declined, but for energies leading up to $3000{\\,}$cm$^{-1}$ the agreement with experiment is excellent. Despite being unable to compare against higher energies we expect the CBS-37$^{{\\,}\\mathrm{HL}}$ PES to perform as well as its $^{35}$Cl counterpart.\n \n\\begin{table}[!ht]\n\\tabcolsep=5pt\n\\caption{\\label{tab:j0_cbs_37cl} Comparison of calculated and experimental $J=0$ vibrational term values (in cm$^{-1}$) for CH$_{3}{}^{37}$Cl. The zero-point energy was computed to be $8216.197\\,$cm$^{-1}$ at the CVBS limit.}\n\\begin{center}\n\t\\begin{tabular}{lcccc}\n\t\\hline\\hline\n \tMode & Sym. & CBS-37$^{{\\,}\\mathrm{HL}}$ & Exp.$^a$ & Obs$-$calc \\\\\n\t\\hline\n\t\n\t$\\nu_3$ & $A_1$ & 727.40 & 727.0295 & -0.37 \\\\\n\t$\\nu_6$ & $E$ & 1017.66 & 1017.6824 & 0.02 \\\\\n\t$\\nu_2$ & $A_1$ & 1354.82 & 1354.6908 & -0.13 \\\\\n\t$2\\nu_3$ & $A_1$ & 1446.12 & 1445.3509 & -0.77 \\\\\n\t$\\nu_5$ & $E$ & 1452.53 & 1452.1552 & -0.38 \\\\\n\t$\\nu_3+\\nu_6$ & $E$ & 1739.64 & 1739.2357 & -0.41 \\\\\n\t$2\\nu_6$ & $A_1$ & 2028.68 & 2028.5929 & -0.09 \\\\\n\t$2\\nu_6$ & $E$ & 2037.59 & 2037.5552 & -0.04 \\\\\n\t$\\nu_2+\\nu_3$ & $A_1$ & 2074.90 & 2074.4526 & -0.45 \\\\\n\t$3\\nu_3$ & $A_1$ & 2156.31 & 2155.1179 & -1.19 \\\\\n\t$\\nu_3+\\nu_5$ & $E$ & 2177.47 & 2176.7504 & -0.72 \\\\\n\t$\\nu_2+\\nu_6$ & $E$ & 2367.32 & 2367.1394 & -0.18 \\\\\n\t$2\\nu_3+\\nu_6$& $E$ & 2452.76 & 2451.9048 & -0.85 \\\\\n\t$\\nu_5+\\nu_6$ & $E$ & 2461.78 & 2461.4849 & -0.29 \\\\\n\t$\\nu_5+\\nu_6$ & $A_1$ & 2464.85 & 2464.4690 & -0.38 \\\\\n\t$\\nu_5+\\nu_6$ & $A_2$ & 2467.43 & 2467.2469 & -0.18 \\\\\n\t$\\nu_2+\\nu_5$ & $E$ & 2803.73 & 2803.2$\\,^b$ & -0.53 \\\\\n $2\\nu_5$ & $A_1$ & 2879.81 & 2879.0$\\,^b$ & -0.81 \\\\\n\t$2\\nu_3+\\nu_5$& $E$ & 2893.71 & 2893.7394$\\,^c$ & 0.03 \\\\\n\t$2\\nu_5$ & $E$ & 2898.19 & 2895.449$\\,^c$ & -2.74 \\\\\n\t$\\nu_1$ & $A_1$ & 2969.14 & 2967.7469$\\,^c$ & -1.39 \\\\\n\t$\\nu_4$ & $E$ & 3037.71 & 3039.6311$\\,^c$ & 1.92 \\\\\n\t$3\\nu_6$ & $E$ & 3044.97 & 3041.2568$\\,^c$ & -3.72 \\\\\n\t$3\\nu_6$ & $A_1$ & 3059.47 & 3058.6913$\\,^c$ & -0.78 \\\\\n\t\\hline\\hline\n \\end{tabular}\n\\end{center}\n $^a$ Values from Ref.~\\onlinecite{05NiChBu.CH3Cl} unless stated otherwise.\\\\[-2mm]\n $^b$ Ref.~\\onlinecite{82BeAlGu.CH3Cl}.\n $^c$ Ref.~\\onlinecite{11BrPeJa.CH3Cl}.\\\\[-2mm]\n\\end{table}\n \n We have not computed term values for CH$_{3}{}^{37}$Cl using the VQZ-F12 PES but we expect errors similar to those reported for CH$_{3}{}^{35}$Cl. It is evident that for methyl chloride, the inclusion of additional HL corrections and a CBS extrapolation in the PES lead to considerable improvements in computed $J=0$ energies. We have been able to identify and assign over $100$ new energy levels for both CH$_{3}{}^{35}$Cl and CH$_{3}{}^{37}$Cl which we provide as supplementary material.~\\cite{EPAPSCH3CL} We recommend the CBS-35$^{{\\,}\\mathrm{HL}}$ and CBS-37$^{{\\,}\\mathrm{HL}}$ PESs for future use.\n \n\\subsection{Equilibrium Geometry of CH$_3$Cl}\n\n The equilibrium geometry of methyl chloride determined empirically by Jensen et al.~\\cite{81JeBrGu.CH3Cl} has often served as reference. However the reliability of the axial rotational constants used in their analysis has been questioned.~\\cite{97DeWlRu.CH3Cl} The C-H bond length reported in Ref.~\\onlinecite{81JeBrGu.CH3Cl} also appears too large to be consistent with \\textit{ab initio} calculations, and also with the isolated C-H bond stretching frequency.~\\cite{94DeWlxx.CH3Cl} A combined empirical and \\textit{ab initio} structure has later been determined based on $^{12}$CH$_{3}{}^{35}$Cl, $^{12}$CH$_{3}{}^{37}$Cl, $^{12}$CD$_{3}{}^{35}$Cl and $^{12}$CD$_{3}{}^{37}$Cl experimental data.~\\cite{97DeWlRu.CH3Cl} We compare against this as well as another high-level \\textit{ab initio} study.~\\cite{03DeMaBo.CH3Cl}\n\n The equilibrium structural parameters calculated from the CBS-(35\/37)$^{\\,\\mathrm{HL}}$ PES and the VQZ-F12 PES are listed in Table~\\ref{tab:eq_ref}. The CBS-(35\/37)$^{\\,\\mathrm{HL}}$ bond lengths are shorter than the VQZ-F12 values, which is to be expected due to the inclusion of core-valence electron correlation.~\\cite{Coriani:2005} There is good agreement with the values from Refs.~\\onlinecite{97DeWlRu.CH3Cl} and \\onlinecite{03DeMaBo.CH3Cl}. The largest discrepancy concerns the bond angle determined in Ref.~\\onlinecite{97DeWlRu.CH3Cl} which is around $0.3$ degrees larger than all \\textit{ab initio} computed values.\n \n\\begin{table}\n\\tabcolsep=5pt\n\\caption{\\label{tab:eq_ref}Equilibrium structural parameters of CH$_3$Cl}\n\\begin{center}\n\\begin{tabular}{l c c c}\n\\hline\\hline\n& $r$(C-Cl)\/$\\mathrm{\\AA}$ & $r$(C-H)\/$\\mathrm{\\AA}$ & $\\beta$(HCCl)\/deg \\\\[0.5mm]\n\\hline \\\\[-2.5mm]\nCBS-(35\/37)$^{\\,\\mathrm{HL}}$ & 1.7777 & 1.0834 & 108.38 \\\\\nVQZ-F12 & 1.7805 & 1.0849 & 108.39 \\\\ \nRef.~\\onlinecite{97DeWlRu.CH3Cl}$\\,^a$ & 1.7768 & 1.0842 & 108.72 \\\\\nRef.~\\onlinecite{03DeMaBo.CH3Cl}$\\,^b$ & 1.7772 & 1.0838 & 108.45 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n$^a$ Value determined from empirical data and CCSD(T) calculations.\\\\[-2mm]\n$^b$ CCSD(T)(\\textit{fc})\/cc-pV(Q,5)Z + MP2(\\textit{ae})\/cc-pwCVQZ - MP2(\\textit{fc})\/cc-pwCVQZ.\n\\end{table}\n \n For further validation we studied the pure rotational spectrum as rotational energies are highly dependent on the molecular geometry through the moments of inertia. In Table~\\ref{tab:rotational} we present the calculated $J\\leq5$ rotational energies in the ground vibrational state for CH$_{3}{}^{35}$Cl using the CBS-35$^{\\,\\mathrm{HL}}$ PES. The computed values reproduce the experimental levels with a rms error of $0.0018{\\,}$cm$^{-1}$. The CBS-(35\/37)$^{\\,\\mathrm{HL}}$ \\textit{ab initio} structural parameters reported in Table~\\ref{tab:eq_ref} can thus be regarded as reliable, and we expect the true equilibrium geometry of methyl chloride to be close to these values.\n \n\\begin{table}[!h]\n\\tabcolsep=5pt\n\\caption{\\label{tab:rotational} Comparison of calculated and experimental $J\\leq5$ pure rotational term values (in cm$^{-1}$) for CH$_{3}{}^{35}$Cl. The observed ground state energy levels are from Ref.~\\onlinecite{05NiChxx.CH3Cl}.}\n\\begin{center}\n\t\\begin{tabular}{cccrrr}\n\t\\hline\\hline\n \t $J$ & $K$ & Sym. & Exp. & CBS-35$^{{\\,}\\mathrm{HL}}$ & Obs$-$calc \\\\ \n\t\\hline\t\n\t0 & 0 & $A_1$ & 0.0000 & 0.0000 & 0.0000 \\\\\n\t1 & 0 & $A_2$ & 0.8868 & 0.8868 & 0.0000 \\\\\n\t1 & 1 & $E$ & 5.6486 & 5.6489 & -0.0003 \\\\\n\t2 & 0 & $A_1$ & 2.6604 & 2.6603 & 0.0001 \\\\\n\t2 & 1 & $E$ & 7.4222 & 7.4223 & -0.0001 \\\\\n\t2 & 2 & $E$ & 21.7067 & 21.7075 & -0.0008 \\\\\n\t3 & 0 & $A_2$ & 5.3208 & 5.3205 & 0.0003 \\\\\n\t3 & 1 & $E$ & 10.0825 & 10.0826 & -0.0001 \\\\\n\t3 & 2 & $E$ & 24.3668 & 24.3676 & -0.0008 \\\\\n\t3 & 3 & $A_1$ & 48.1707 & 48.1727 & -0.0020 \\\\\n\t3 & 3 & $A_2$ & 48.1707 & 48.1727 & -0.0020 \\\\\n\t4 & 0 & $A_1$ & 8.8678 & 8.8675 & 0.0003 \\\\\n\t4 & 1 & $E$ & 13.6295 & 13.6294 & 0.0001 \\\\\n\t4 & 2 & $E$ & 27.9137 & 27.9143 & -0.0006 \\\\\n\t4 & 3 & $A_1$ & 51.7173 & 51.7191 & -0.0018 \\\\\n\t4 & 3 & $A_2$ & 51.7173 & 51.7191 & -0.0018 \\\\\n\t4 & 4 & $E$ & 85.0354 & 85.0389 & -0.0035 \\\\\n\t5 & 0 & $A_2$ & 13.3015 & 13.3010 & 0.0005 \\\\\n 5 & 1 & $E$ & 18.0632 & 18.0629 & 0.0003 \\\\\n 5 & 2 & $E$ & 32.3472 & 32.3476 & -0.0004 \\\\\n 5 & 3 & $A_1$ & 56.1505 & 56.1521 & -0.0016 \\\\\n 5 & 3 & $A_2$ & 56.1505 & 56.1521 & -0.0016 \\\\\n 5 & 4 & $E$ & 89.4681 & 89.4714 & -0.0033 \\\\\n 5 & 5 & $E$ & 132.2931 & 132.2985 & -0.0054 \\\\\n \\hline\\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n \n\\section{Conclusions}\n\\label{sec:conc}\n\n Using state-of-the-art electronic structure calculations, we have generated two global \\textit{ab initio} PESs for the two main isotopologues of methyl chloride. We believe that the accuracy achieved by these PESs is at the limit of what is currently possible using solely \\textit{ab initio} methods, and that it would be extremely challenging to go beyond this without empirical refinement of the respective PESs. Considering that the PESs are purely \\textit{ab initio} constructs, the computed vibrational wavenumbers show remarkable accuracy when compared with experiment for both isotopologues. It is evident that higher-level energy corrections and an extrapolation to the complete basis set limit should be included to obtain accurate theoretical vibrational energies for CH$_3$Cl. The same applies to the determination of equilibrium structural parameters.\n \n For the requirements of high-resolution spectroscopy, the \\textit{ab initio} surfaces presented here will no doubt need to be refined to empirical data.~\\cite{YuBaTe11.NH3} The resulting ``spectroscopic PESs'' can then be used to achieve unprecedented accuracy in the simulation of rotation-vibration spectra, and it is at this stage that the predictive power of the variational approach is fully realised. The PESs presented in this work provide an excellent starting point for this procedure. Once the refinement is complete, a comprehensive rovibration line list applicable for elevated temperatures will be generated for methyl chloride as part of the ExoMol project.~\\cite{ExoMol2012}\n\n\\begin{acknowledgments}\nThis work was supported by ERC Advanced Investigator Project 267219, and FP7-MC-IEF project 629237. A.O. is grateful to UCL for a studentship under their Impact Student Scheme and thanks J\\\"{u}rgen Breidung for valuable discussions.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\t\n\t\\label{sec:introduction}\n\tLicense Plate (LP) detection and recognition are the key parts of intelligent transportation systems because it is the unique identification of vehicles. The relevant methods are widely used on electronic toll payment, parking managing, and traffic monitoring systems. In order to achieve effective detection and recognition, researchers proposed a variety of techniques that are capable of handling the task in most conditions. Before the deep learning era, most of the methods were on the basis of artificial designs~\\cite{tradm1,tradm2,alpr}. They utilized hand-crafted features such as colors, shadows and textures, and integrated them by a cascaded strategy with license plate detection, segmentation, and recognition. Although they reached promising performance, the robustness may not be enough for some uncontrolled circumstances like weather, illumination, and rotation since the scheme relies on manually designed features. Then with the explosion of deep learning methods in recent years, most researchers turned their attention to this framework that is able to learn features automatically. The convenient and efficient technique quickly became popular, and networks for detection and recognition also sprung up~\\cite{pvw, vlpd, lprec1}. In these works, great successes have been achieved. However, for Chinese LPs, the problem of data scarcity gradually emerges. Existing annotated Chinese LP data that is representative of most scenarios cannot meet the huge demands. Thus, to alleviate the issue, we present our Chinese Road Plate Dataset (CRPD).\n\t\n\tAdmittedly, there are already some excellent public datasets with LPs~\\cite{pvw,reid, caltech,zemris,ccpd,aolp,ssig,ufpr,pku,clpd}. These public benchmarks lay the foundation of various LP processing methods, and our CRPD is an effective supplement to existing Chinese LP datasets, which is more challenging. Images of CRPD are collected from electronic monitoring systems in most provinces of mainland China in different periods and weather conditions. The images contain cars with different statuses and types, and quite a part of the data contains more than one LP in one image. Each image has annotations of (i) LP content. (ii) Locations of four vertices. (iii) LP type. More details will be introduced in Section~\\ref{sec:1}.\n\t\n\n\t\n\tAs for detection and recognition tasks, most prestigious methods designed the two branches separately. Zhou \\textsl{et al.}~\\cite{pvw} proposed a scheme for LP detection with Principal Visual Word (PVW) generation and applied bag-of-words in partial-duplicate image search. Chen \\textsl{et al.}~\\cite{vlpd} put forward a method to detect the vehicles and the LPs simultaneously, where the results can be used for further recognition. The two-stage framework is effective, but the error accumulation problems hinder further progresses. Therefore, end-to-end frameworks are increasingly prevalent. In this paper, we propose an end-to-end trainable Chinese LP detection and recognition network with both high efficiency and satisfactory performance as the baseline of our CRPD. Our method is a unified network that consists of two branches. The branch for detection is based on STELA~\\cite{stela}, which is a learned anchor-based detector. It only associates one reference box at each spatial position, which highly reduces the computation to reach a fast running speed. In the recognition branch, we abandon the recognition by segmentation pipeline and utilize a sequence-to-sequence method to accomplish the recognition task. The region-wise features are extracted by the RRoIAlign~\\cite{fots} operator and then will be fed into components for recognition. This scheme blurs the line between detection and recognition, which strongly alleviates the error accumulation issues. \n\t\n\n\t\n\tIn summary, there are three main contributions in this work:\n\t\n\t\\begin{itemize}\n\t\t\\item We publish a new Chinese LP dataset with more than 30k images, which covers more scenes and administrative regions of mainland China. We argue that this new dataset is more difficult than existing datasets and is also a supplement to the Chinese LP research field.\n\t\t\\item We propose an end-to-end trainable network for Chinese LP detection and recognition, which almost reaches a trade-off between accuracy and efficiency as a baseline. Through utilizing the common feature extraction branch and the RRoIAlign~\\cite{fots} operator, end-to-end training is achieved, and the error accumulation problems are alleviated with a real-time efficiency kept.\n\t\t\\item Our code and dataset will be publicly available soon. To facilitate the reference of researchers and get more progress, we will upload related materials.\n\t\\end{itemize}\n\t\n\t\\section{Related Work}\n\t\\label{sec:related}\n\t\\subsection{LP Datasets}\n\tDue to the importance of LP detection and recognition, researchers built and published a number of LP datasets. ReId~\\cite{reid} is a dataset for license plate recognition with 76k images gathered from surveillance cameras on highway toll gates. Caltech~\\cite{caltech} and Zemris~\\cite{zemris} collected over 600 images from the road and freeways with high-resolution cameras. Hsu~\\textsl{et al.}~\\cite{aolp} presented a dataset for applications of access control, traffic law enforcement and road patrol. Gon{\\c{c}}alves~\\textsl{et al.}~\\cite{ssig} proposed Sense SegPlate Database to evaluate license plate character segmentation problem. Laroca~\\textsl{et al.}~\\cite{ufpr} provided a dataset that includes 4,500 fully annotated images from 150 vehicles in real-world scenarios. These datasets strongly support the researches of LP detection and recognition methods.\n\t\n\tHowever, LPs in different countries and regions are usually not the same. The mentioned datasets mostly contain LPs that only include alphanumeric characters. In some countries or regions, such as Chinese mainland regions, Japan and South Korea, the LPs contain some special characters. Therefore, it is still significant to build new datasets for these LPs. Among them, mainland Chinese LP detection and recognition are one of the important tasks which require a large amount of data. To meet the demand, Chinese LP datasets were proposed. Zhou~\\textsl{et al.}~\\cite{pvw} collected 220 LP images where the LPs were of little affine distortion. Yuan~\\textsl{et al.}~\\cite{pku} presented a dataset that contains vehicle images captured from various scenes under diverse conditions. Zhang~\\textsl{et al.}~\\cite{clpd} proposed a dataset that contains 1,200 LP images from all 31 provinces in mainland China. CCPD~\\cite{ccpd} is a large and comprehensive dataset with about 290k images that contain plates with various angles, distances, and illuminations. Though they made important contributions to the progress of detection and recognition methods, there are still not enough large and representative datasets.\n\t\n\t\\subsection{LP Detection and Recognition}\n\tOwing to the successes of text detection and recognition, LP processing is also well developed.\n\tThere are methods with end-to-end frameworks, which achieved excellent performance. Zhang~\\textsl{et al.}~\\cite{vlpdr} integrated LP detection, tracking, and recognition into a unified framework via deep learning. Silva~\\textsl{et al.}~\\cite{cnnlpdr} proposed to identify the vehicle and the LP region using two passes on the same CNN and then to recognize the characters using a second CNN. Kessentini~\\textsl{et al.}~\\cite{twostage} presented a two-stage network to achieve the detection and recognition of LPs. In~\\cite{mtlpr}, a light CNN was proposed for detection and recognition, which achieved real-time efficiency.\n\t\n\tAs for Chinese LPs, there are also excellent end-to-end frameworks. Laroca~\\textsl{et al.}~\\cite{yoloalpr} proposed a unified approach for LP detection and layout classification to improve the recognition results. Qin~\\textsl{et al.}~\\cite{eulpr} proposed a unified method that can recognize both single-line and double-line LPs in an end-to-end way without line segmentation and character segmentation. However, Chinese LP processing is still challenging due to the large number of categories of Chinese characters. Meanwhile, LP processing under unconstrained scenarios also faces many problems. Therefore, it is still valuable to propose a new end-to-end framework for Chinese LP detection and recognition.\n\t\n\t\\section{CRPD Overview}\n\t\\label{sec:1}\n\t\\subsection{Constitution of Data}\n\t\\label{sec:2}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=2.7in]{Fig1.eps}\n\t\t\\caption{The constitution of CRPD dataset.}\n\t\t\\label{cons}\n\t\\end{figure}\n\t\n\tBecause CRPD is presented as a supplement for existing datasets, special attention is paid to the diversity of data. The images are mainly captured on electronic monitoring systems, including vehicles that are running, turning, parked, or far away which may cause blur and rotation. The scene includes day and night and different weathers. Quite a part of the images contains more than one LP from different provinces and cities, and there are LPs of special vehicles involved, such as coach cars, police cars, and trailers, whose LPs will contain some special characters. The dataset includes three sub-datasets according to the numbers of LPs: CRPD-single, CRPD-double, and CRPD-multi, as shown in Figure~\\ref{cons}. CRPD-single contains images with only one LP, CRPD-double contains images with two LPs, and CRPD-multi contains images with three or more LPs.\n\t\n\tThere are totally about 25k images for training, 6.25k for validating, and 2.3k for testing. In CRPD-single, there are 20k images for training, 5k for validating, and 1k for testing. In CRPD-double, there are 4k images for training, 1k for validating, and 1k for testing. In CRPD-multi, there are 1k images for training, 0.25k for validating, and 0.3k for testing.\n\t\n\tThe annotations consist of three parts. The first is LP content which includes numbers, Chinese and English characters. There are some LPs that are too small or seriously blurred whose content is unidentifiable, and they are also annotated while the unrecognizable characters are replaced with a special one. The second is the coordinate of four vertices of the LPs. The last is the LP type, including blue (small cars), yellow and single line (front of large cars), yellow and double lines (back of large cars), and white (police cars).\n\t\n\t\\subsection{Data Analysis}\n\tCRPD provides more than 30k LP images with annotations. Though it is not the largest dataset in current frequently used Chinese datasets, some characteristics of CRPD will be helpful for training a robust model. Existing datasets mostly contain single and focused LP in one image, as shown in Figure~\\ref{public_datasets}. In some cases, such as electronic toll payment or parking managing systems, the data is highly effective. But when dealing with data for traffic monitoring systems, suspect car tracking, or vehicle flow measuring, images with more LPs will be required. CRPD aims to fill up this deficiency, so we paid special attention to the number of LPs, as shown in Figure~\\ref{crpd}.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=3.5in]{Fig2.eps}%\n\t\t\\caption{Examples of LP images in existing public datasets.}\n\t\t\\label{public_datasets}\n\t\\end{figure}\n\t\n\tAlso, CRPD has some other advantages for building a robust model. To illustrate them, we compare with CCPD~\\cite{ccpd}, EasyPR~\\cite{easypr}, ChineseLP~\\cite{pvw} and CLPD~\\cite{clpd} in some aspects, which are shown in Tables~\\ref{LP_number},~\\ref{status} and~\\ref{type}.\n\t\n\tThe first is the number of LPs. Images in CCPD~\\cite{ccpd} and CLPD~\\cite{clpd} contain one LP, which can better indicate the detection Precision of a network. But as noted above, this may restrict the scenarios where it can be used. In comparison, EasyPR~\\cite{easypr}, ChineseLP~\\cite{pvw} and our CRPD have better compatibility in this aspect.\n\t\n\tThe second is the status of vehicles. CCPD~\\cite{ccpd} contains images captured from parking lots, and they are more interested in LPs in various circumstances with different illumination, rotation, and blur. Therefore, the vehicle status is not focused. Our CRPD concentrates on the capability to deal with a variety of vehicles, so there is better coverage on vehicle status.\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=3.5in]{Fig3.eps}%\n\t\t\\caption{Examples of LP images in CRPD-single (the first row), CRPD-double (the second row), and CRPD-multi (the third row).}\n\t\t\\label{crpd}\n\t\\end{figure}\n\t\n\t\\begin{table}[!t]\n\t\n\t\t\\caption{A comparison on the number of images with different numbers of LPs between current public Chinese LP datasets.}\n\t\t\\label{LP_number}\n\t\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tLP Number & CCPD & EasyPR & ChineseLP & CLPD & CRPD \\\\\n\t\t\t\\midrule[0.5pt]\t\n\t\t\t$=1$ & 224001 & 225 & 392& 1200& 26659 \\\\\n\t\t\t$=2$ & 0 & 21 & 16& 0& 6242 \\\\\n\t\t\t$=3$ & 0 & 10 & 1& 0& 1232 \\\\\n\t\t\t$\\geq4$ & 0 & 0 & 2& 0& 371 \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{table}[t]\n\t\n\t\t\\caption{A comparison of the coverage of different vehicle statuses between current public Chinese LP datasets.}\n\t\t\\label{status}\n\t\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tStatus of Vehicles & CCPD & EasyPR & ChineseLP& CLPD& CRPD \\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\tParked & \\ding{52} &\\ding{52} &\\ding{52} &\\ding{52} &\\ding{52} \\\\\n\t\t\tRunning & \\ding{55} &\\ding{52} &\\ding{52} &\\ding{52} &\\ding{52} \\\\\n\t\t\tTurning & \\ding{55} &\\ding{52} &\\ding{52} &\\ding{52} &\\ding{52} \\\\\n\t\t\tFar away & \\ding{55} &\\ding{55} &\\ding{55} &\\ding{52} &\\ding{52} \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\tThe last is the vehicle types. Though there are not a number of special vehicles on the road, the detection and recognition of them are still of great importance. Thus, we paid attention to the LPs of special vehicles to ensure the capability of the network trained by CRPD to deal with these LPs. Some images of them are shown in Figure~\\ref{special}. Altogether, our CRPD covers most a variety of common scenes. It is a strong supplement to Chinese LP datasets either used as training or evaluating data.\n\t\n\t\\begin{table}[t]\n\t\n\t\t\\caption{A comparison of the coverage of different vehicle types between current public Chinese LP datasets.}\n\t\t\\label{type}\n\t\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tType of Vehicles & CCPD & EasyPR & ChineseLP& CLPD& CRPD \\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\tCoach Vehicles & \\ding{55} & \\ding{52} & \\ding{55}& \\ding{52}& \\ding{52} \\\\\n\t\t\tPolice Vehicles & \\ding{55} & \\ding{55} & \\ding{55}& \\ding{52}& \\ding{52} \\\\\n\t\t\n\t\t\tTrailers & \\ding{55} & \\ding{55} & \\ding{55}& \\ding{55}& \\ding{52} \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{figure}[!t]\n\t\t\\centering\n\t\t\\includegraphics[width=3.5in]{Fig4.eps}%\n\t\t\\caption{LPs of special vehicles with their annotations in CRPD. The green rectangles are the annotated boxes and the text on the top left corner of the zooming rectangle is the annotated LP content.}\n\t\t\\label{special}\n\t\\end{figure}\n\t\n\tHowever, there are also some limitations of CRPD. First, the location of each LP character is not annotated, which restricts the applications of object detection and data augmentation. Meanwhile, it is also difficult to detect each character because it is very small in the perspective of electronic monitoring systems. Second, the LPs are all on the vehicles. The actual LPs can be held in hand or placed on the ground in some circumstances, and the detection and recognition of these LPs are also useful.\n\t\n\t\\begin{figure*}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=5in]{Fig5.eps}%\n\t\t\\caption{The framework of our network. Conv, DeConv, Bn, MaxPooling, and ResBlock stand for convolution layer, deformable convolution layer, batch normalization layer, max pooling layer, and residual block, respectively. k, s, p, and c stand for kernel size, stride, padding size, and output channel number, respectively, with the size behind each of them.}\n\t\t\\label{pipeline}\n\t\\end{figure*}\n\t\n\t\\section{Methodology}\n\t\n\t\\label{methodology}\n\t\n\tIn this section, we will describe the proposed method in detail, and the pipeline is shown in Figure~\\ref{pipeline}.\n\t\n\t\\subsection{Detection Branch}\n\t\\label{methodology_det}\n\t\n\tThe first stage is the LP detection step. As noted above, there are some obstacles for LP detection networks to reach a balance between accuracy and efficiency. Considering the trade-off, we utilize our previous STELA~\\cite{stela}, a totally real-time detector, as the basis of our detection branch. The detection network is implemented on RetinaNet~\\cite{retinanet} and utilizes Feature Pyramid Network (FPN)~\\cite{fpn} to construct a rich, multi-scale feature pyramid from a single resolution input image. It consists of three portions: anchor classification, rotated bounding box regression, and anchor refining.\n\t\n\t\\subsubsection{Anchor Classification}\n\t\\label{methodology_det_cls}\n\t\n\tAs we do not generate region proposals, class imbalance problems still exist in our scheme. That means there are only a few anchors that are annotated as positive (the object), while the others are negative. Therefore, Focal Loss~\\cite{retinanet} are utilized to calculate the loss of classification, as it is designed to deal with the problem. Firstly, we define $p_t$ with\n\t\\begin{equation}\n\t\tp_t=\n\t\t\\begin{cases}\n\t\t\tp& if~y=1\\\\\n\t\t\t1-p& otherwise\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere $p\\in [0,1]$ is the predicted probability and $y=1$ specifies the ground-truth class. Then the loss is defined as \n\t\\begin{equation}\n\t\tL_{cls} = FL(p_t)=-\\alpha_{t}(1-p_{t})^{\\gamma}log(p_t)\n\t\\end{equation}\n\t$\\alpha_{t}$ is a balanced weighting factor and $\\gamma$ is a focusing parameter. They are set to 0.25 and 2.0 respectively, which is the same as the original Focal Loss~\\cite{retinanet}.\n\t\n\t\\begin{figure}[b]\n\t\t\\centering\n\t\t\\includegraphics[width=2.3in]{Fig6.eps}%\n\t\t\\caption{Illustration of the parameters. The red dotted box represents $b$ which is a box and the green solid box represents $g$ which is the groundtruth.}\n\t\t\\label{regression}\n\t\\end{figure}\n\t\n\t\\subsubsection{Rotated Bounding Box Regression}\n\t\\label{methodology_det_reg}\n\t\n\tFor the detection of tilted LP, we utilize rotated bounding boxes to match the instances. The box can be represented by a five tuple $(x,y,w,h,\\theta)$, in which $x$ and $y$ are the coordinate of the center point, $w$ and $h$ are the width and height of the box, and $\\theta$ is the angle to horizontal, as shown in Figure~\\ref{regression}. For the regression operation, the distance vector $\\Delta=(\\delta_x,\\delta_y,\\delta_w,\\delta_h,\\delta_{\\theta})$ is defined as\n\t\\begin{equation}\n\t\t\\delta_x = (g_x -b_x )\/b_w , ~\\delta_y = (g_y -b_y )\/b_h\n\t\\end{equation}\n\t\\begin{equation}\n\t\t\\delta_w = log(g_w \/b_w ), ~\\delta_h = log(g_h \/ b_h)\n\t\\end{equation}\n\t\\begin{equation}\n\t\t\\delta_{\\theta}=tan(g_{\\theta})-tan(b_{\\theta})\n\t\\end{equation}\n\twhere $b$ and $g$ represent a bounding box and the corresponding target groundtruth respectively. The loss of the regression can be calculated by\n\t\\begin{equation}\n\t\tL_{loc} = smooth_{L_1}(\\Delta_{t}-\\Delta_{p})\n\t\\end{equation}\n\twhere $smooth_{L_1}$ is the smooth L1 loss~\\cite{fastrcnn}, $\\Delta_t$ is the target and $\\Delta_p$ is the predicted tuple.\n\t\n\t\\subsubsection{Learned Anchor}\n\t\\label{methodology_det_ref}\n\t\n\tAs depicted in~\\cite{stela}, the most important part of the proposed scheme in two-stage is that the selected proposals are chosen by learning. The manually-defined original anchors with fixed scales and aspect ratios may not be the optimal designs, so an extra regression branch for anchor refining is added. The final classification and regression task will be reached on the learned anchors, which brings an improvement in accuracy with a little increment of computation. The original anchor, learned anchor, and output boxes are illustrated in Figure~\\ref{anchors}. It is obvious that the center should be well aligned with the pixel in feature maps. Thus, the offsets are only regressed within $\\Delta'=(\\delta'_{w},\\delta'_{h},\\delta'_{\\theta})$, which means that only the shapes are adjusted. The loss can be calculated with\n\t\\begin{equation}\n\t\tL_{ref} = smooth_{L_1}(\\Delta'_{t}-\\Delta'_{p})\n\t\\end{equation}\n\tin which $\\Delta'_t$ is the target and $\\Delta'_p$ is the predicted tuple. And finally, the total loss is\n\t\\begin{equation}\n\t\tL_{det} = \\lambda_{ref}L_{ref}+\\lambda_{loc}L_{loc}+\\lambda_{cls}L_{cls}\n\t\\end{equation}\n\tin which $\\lambda_{ref}, \\lambda_{loc}$ and $\\lambda_{cls}$ are the weights which are set to 0.5, 0.5, and 1 which have been proven to be effective.\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=3.5in]{Fig7.eps}%\n\t\t\\caption{The illustration of anchors and boxes. The red, yellow, and green boxes represent the original anchor, the learned anchor, and the final output boxes respectively. The original anchor (red) and learned anchor (yellow) have the same center point.}\n\t\t\\label{anchors}\n\t\\end{figure}\n\t\n\t\\subsection{Recognition Branch}\n\t\\label{methodology_rec}\n\t\n\tThe recognition step is the second stage of the LP processing tasks. Because most of the networks for recognition achieved high efficiency, we simply utilize modules based on CRNN~\\cite{crnn} to achieve the recognition. There are three parts in the modules: the convolutional layers, the recurrent layers, and the transcription layer.\n\t\n\tIn consideration that the backbone of our network has already extracted critical features of the input images, to avoid redundant computation, we utilize the processed feature maps as the input of the recognition branch. In order to effectively deal with rotated plates, RRoIAlign~\\cite{fots} that can crop the feature map with a rotated box, is applied to crop the maps according to the groundtruth boxes of LPs, and the cropped size is $8\\times 25$. While training, we consider that the feature maps may not be processed well when the predicted boxes are not accurate, so the feature maps will also be cropped with both groundtruth boxes and predicted boxes with a score higher than 0.9.\n\t\n\tAnd because the input is feature maps, we remove the first three convolutions of the original convolutional layers of CRNN~\\cite{crnn} to avoid the over-fitting problems. And in order to reach better accuracy, we refer to our another previous work~\\cite{drnn}, and replace two convolution layers with deformable convolution layers and add four residual blocks in this branch. The deformable convolution layers have an adaptive receptive field that can better cover the text area, as shown in Figure~\\ref{deformable}.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=2.7in]{Fig8.eps}\n\t\t\\caption{Indication of fixed receptive fields in standard convolution (the first row) and adaptive receptive fields in deformable convolution (the second row). In each image triplet, the left shows the sampling locations of two levels of $3\\times3$ filters on the preceding feature map, the middle shows the sampling locations of a $3\\times3$ filter, and the right shows two activation units. Two sets of locations are highlighted according to the activation units.}\n\t\t\\label{deformable}\n\t\\end{figure}\n\t\n\tThe architecture of the convolutional layers is shown in Figure~\\ref{pipeline}. The recurrent layers are based on bidirectional LSTM~\\cite{lstm}, and the transcription component is based on CTCLoss~\\cite{ctc}. The total loss function is a weighted loss\n\t\n\t\\begin{equation}\n\t\tL=\\lambda_{det} L_{det} +\\lambda_{rec} L_{rec}\n\t\\end{equation}\n\t\n\twhere $\\lambda_{det}$ and $\\lambda_{rec}$ are constants that indicate the strength of the detection and recognition modules. And in our training, the value of $L_{rec}$ is one-magnitude-order larger than $L_{det}$, so to keep a balance, we set them to 1 and 0.1, respectively.\n\t\n\t\n\t\n\t\\section{Experiments}\n\t\\label{experiments}\n\t\\subsection{Training Details}\n\t\\label{experiments_details}\n\t\n\tThe training and testing datasets are from CCPD~\\cite{ccpd}, EasyPR~\\cite{easypr}, and our CRPD. The input images are resized to $640\\times 640$ with three channels. In the training stage, the optimizer of the network is Adam [18], the batch size is set to 32, and the learning rate is 1e-4. The network is trained for 35000 iterations which consume about 10 hours. The proposed method is implemented by PyTorch~\\cite{pytorch}. The experiments are carried on a platform with Intel Xeon(R) E5-2630 v3 CPU and a single NVIDIA TITAN RTX GPU.\n\t\n\t\\subsection{Evaluation Metrics}\n\t\\label{experiments_metrics}\n\t\n\tTo demonstrate the effectiveness of the methods, we utilize the protocols described in~\\cite{ccpd} to evaluate the models. The detection will be considered as a match if it overlaps a ground truth bounding box by more than 60\\% and the words match exactly. Then Recall, Precision, and F-score are calculated with\n\t\\begin{equation}\n\t\tRecall=\\frac{TP}{TP+FN}\n\t\\end{equation}\n\t\\begin{equation}\n\t\tPrecision=\\frac{TP}{TP+FP}\n\t\\end{equation}\n\t\\begin{equation}\n\t\tF-score=\\frac{2\\times Precision \\times Recall}{Precision+Recall}\n\t\\end{equation}\n\twhere $TP$ represents the number of positive objects that are predicted as positive and the words match, $FP$ represents the number of negative objects that are predicted as positive, and $FN$ represents the number of negative objects that are predicted as negative. \n\t\n\t\\subsection{Ablations}\n\t\\label{experiments_ablations}\n\t\n\tTo demonstrate that the unified architecture brings some improvements and our scheme reaches the best performance, we evaluate the proposed components. Firstly, we test the effectiveness of our unified network. As comparisons, cascaded STELA~\\cite{stela} and CRNN~\\cite{crnn} models are utilized. The two networks are trained respectively, and the CRNN~\\cite{crnn} model will process the original images cropped in the light of the boxes predicted by STELA~\\cite{stela}. All the models are trained on CRPD and evaluated on CCPD~\\cite{ccpd}, EasyPR~\\cite{easypr}, and CRPD. From Table~\\ref{unified_expr}, we see that our network achieves a few improvements. A better result is reached on CRPD because the training and testing data have the same distribution. The LPs in CCPD~\\cite{ccpd} have a larger size, so the model trained on our CRPD, which contains LPs with small sizes, is not able to reach the best performance.\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\caption{Ablations of the unified scheme on different datasets.}\n\t\t\\label{unified_expr}\n\t\t\\begin{threeparttable}\n\t\t\\begin{tabular}{p{1.3cm}<\\centering p{0.55cm}<\\centering p{0.55cm}<\\centering p{0.55cm}<\\centering c p{0.55cm}<\\centering p{0.55cm}<\\centering p{0.55cm}<\\centering}\n\t\t\t\\toprule[0.75pt]\n\t\t\t\\multirow{2}{*}{Dataset} & \\multicolumn{3}{c}{STELA+CRNN} & & \\multicolumn{3}{c}{Our Method} \\\\\n\t\t\t\\cmidrule[0.5pt]{2-4}\\cmidrule[0.5pt]{6-8}\n\t\t\t& R & P & F & & R & P & F \\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\tCCPD & 79.1 & 67.8 & 73.0 & & 75.6 & 72.1 & 73.8 \\\\\n\t\t\tEasyPR & \\textbf{90.2} & 72.8 & 80.6 & & 89.9 & 73.0 & 80.6 \\\\\n\t\t\tCRPD & 88.3 & \\textbf{82.9} & \\textbf{85.5} & & \\textbf{95.4} & \\textbf{84.1} & \\textbf{89.4} \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\t\\begin{tablenotes}\n\t\t\t\\item R: Recall; P: Precision; F: F-score\n\t\t\\end{tablenotes}\n\t\t\\end{threeparttable}\n\t\\end{table}\n\t\n\tThen as we utilize the predicted boxes of the detection modules to crop the feature maps in the training stage, we evaluate the effectiveness of them with different scores. In Table~\\ref{ablations_predbox}, it is obvious that utilizing predicted boxes will bring an improvement on the Recall because more incomplete LPs will be detected, with a little descend on Precision. Considering the recognition branch will also become more robust with this mechanism, it will make the network able to better deal with inaccurate boxes. But because Chinese LPs usually have seven characters, a box with a score lower than 0.86 may cut a whole character off. Thus, there must be a deterioration of the performance when using boxes with scores only higher than 0.85. And the number of boxes with scores higher than 0.95 is less, which causes a smaller batch size for recognition modules, so we choose 0.9 as the threshold finally.\n\t\n\t\\begin{table}[h]\n\t\n\t\t\\caption{Ablations on CRPD-all between different thresholds of scores of the predicted boxes utilized to train the model.}\n\t\t\\label{ablations_predbox}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tThreshold & Recall & Precision & F-score \\\\\n\t\t\t\\midrule[0.5pt] \n\t\t\tNone & 91.2 & \\textbf{85.4} & 88.2 \\\\\n\t\t\t\\textgreater0.95 & 91.7 & 84.6 & 88.0 \\\\\n\t\t\t\\textgreater0.90 & \\textbf{95.4} & 84.1 & \\textbf{89.4} \\\\\n\t\t\t\\textgreater0.85 & 94.5 & 82.5 & 88.1 \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\tThe way to crop the feature maps may also bring different results of the recognition modules. We compare RoIPool~\\cite{fastrcnn}, RoIAlign~\\cite{maskrcnn}, and RRoIAlign~\\cite{fots}, and the results are shown in Table~\\ref{ablations_crop}. There is no obvious difference between the first two methods, and we think the reason is that LPs are not small objects, so some slight errors will not influence much. RRoIAlign~\\cite{fots} can better handle the work because the recognition branch will have great progress with consideration of the rotation degrees.\n\t\n\t\\begin{table}[t]\n\t\n\t\t\\caption{Ablations on CRPD-all between different region feature extracting approaches.}\n\t\t\\label{ablations_crop}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tApproach & Recall & Precision & F-score \\\\\n\t\t\t\\midrule[0.5pt] \n\t\t\tRoIPooling & 95.0 & 80.2 & 87.0 \\\\\n\t\t\tRoIAlign & \\textbf{95.7} & 80.1 & 87.2 \\\\\n\t\t\tRRoIAlign & 95.4 & \\textbf{84.1} & \\textbf{89.4} \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\tIn the recognition branch, we crop the feature maps yield by different layers of the FPN~\\cite{fpn} components to explain why we utilize those from the third layer. Though it seems to utilize the feature maps from the third layer is optimal in Table~\\ref{ablations_fpnlayer}, but we consider it is because most of the LPs in the images of CRPD have sizes which match the anchors in the third layer. In the fifth or deeper layers, the anchors are mappings of some huge boxes in the original images, but most of the LPs have a small size.\n\t\n\t\\begin{table}[h]\n\t\n\t\t\\caption{Ablations on CRPD-all with feature maps from different FPN layers to train the model.}\n\t\t\\label{ablations_fpnlayer}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tFeature Map Layer & Recall & Precision & F-score \\\\\n\t\t\t\\midrule[0.5pt] \n\t\t\tP3 & \\textbf{95.4} & \\textbf{84.1} & \\textbf{89.4} \\\\\n\t\t\tP4 & 82.4 & 70.1 & 75.8 \\\\\n\t\t\tP5 & 2.6 & 2.9 & 2.7 \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{table}[h]\n\t\n\t\t\\caption{Ablations on CRPD-all with feature maps from different FPN layers to train the model.}\n\t\t\\label{ablations_deformable}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\toprule[0.75pt]\n\t\t\tDeformable Conv & Recall & Precision & F-score \\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\t\\ding{55} & 93.0 & 83.7 & 88.1 \\\\\n\t\t\t\\ding{52} & \\textbf{95.4} & \\textbf{84.1} & \\textbf{89.4} \\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\tFinally, to demonstrate that the deformable layers and residual blocks in the recognition branch are effective, ablations are involved, as shown in Table~\\ref{ablations_deformable}. In our metrics, only when the content matches exactly, the result will be regarded as correct. Therefore, deformable convolution brings an improvement in recognition, and the Recall and Precision are both improved.\n\t\n\t\\subsection{Comparisons}\n\t\\label{experiments_comp}\n\t\n\t\\begin{table*}[h]\n\t\n\t\t\\scriptsize\n\t\t\\caption{Comparisons on CRPD between our method and other methods. The methods that are based on deep-learning are trained on CRPD.}\n\t\t\\label{table_crpd_comp}\n\t\t\\centering\n\t\t\\begin{threeparttable}\n\t\t\\begin{tabular}{m{2.9cm}<{\\centering}m{0.5cm}<{\\centering}m{0.8cm}<{\\centering}m{0.9cm}<{\\centering}m{0.3cm}<{\\centering}cm{0.5cm}<{\\centering}m{0.8cm}<{\\centering}m{0.9cm}<{\\centering}m{0.3cm}<{\\centering}}\n\t\t\t\\toprule[0.75pt]\n\t\t\t\\multirow{2}{*}{Method}& \\multicolumn{4}{c}{CRPD-all} & & \\multicolumn{4}{c}{CRPD-single} \\\\\n\t\t\t\\cmidrule[0.5pt]{2-5}\\cmidrule[0.5pt]{7-10}\n\t\t\t& Recall& Precision& F-score& FPS& & Recall& Precision& F-score& FPS\\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\tEasyPR& 2.0& 1.3& 1.6& 6& & 2.3& 1.3& 1.7& 6\\\\\n\t\t\tSSD512 + CRNN& \\textbf{97.8}& 27.2& 42.6& \\textbf{66}& & \\textbf{98.9}& 28.7& 44.4& \\textbf{71}\\\\\n\t\t\t YOLOv3 + CRNN& 73.0& 61.0& 66.5& 18& & 73.7& 59.4& 65.8& 18\\\\\n\t\t\t YOLOv4 + CRNN& 84.4& 60.5& 70.5& 40& & 87.3& 68.4& 76.7& 40\\\\\n\t\t\t SYOLOv4 + CRNN& 86.8& 71.0& 78.2& 35& & 90.1& 72.4& 80.3& 35\\\\\n\t\t\tFaster-RCNN + CRNN& 79.9& 73.7& 76.7& 19& & 81.4& 71.7& 76.3& 20\\\\\n\t\t\t STELA + CRNN& 88.3& 82.9& 85.5& 36& & 83.1& 73.3& 77.9& 36\\\\\n\t\t\tOurs& 95.4& \\textbf{84.1}& \\textbf{89.4}& 30& & 96.3& \\textbf{83.6}& \\textbf{89.5}& 35\\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Method} & \\multicolumn{4}{c}{CRPD-double} & & \\multicolumn{4}{c}{CRPD-multi} \\\\\n\t\t\t\\cmidrule[0.5pt]{2-5}\\cmidrule[0.5pt]{7-10}\n\t\t\t& Recall& Precision& F-score& FPS& & Recall& Precision& F-score& FPS\\\\\n\t\t\t\\hline\n\t\t\tEasyPR& 1.5& 1.2& 1.3& 6& & 1.8& 1.7& 1.8& 6\\\\\n\t\t\tSSD512 + CRNN& \\textbf{97.5}& 27.5& 42.9& \\textbf{66}& & \\textbf{93.5}& 21.1& 34.5& \\textbf{63}\\\\\n\t\t\t YOLOv3 + CRNN& 74.6& 64.4& 69.1& 17& & 66.2& 61.3& 63.6& 17\\\\\n\t\t\t YOLOv4 + CRNN& 90.4& 41.2& 56.6& 40& & 88.9& 36.8& 52.0& 39\\\\\n\t\t\t SYOLOv4 + CRNN& 91.5& 75.9& 83.0& 35& & 91.5& 75.2& 82.5& 35\\\\\n\t\t\tFaster-RCNN + CRNN& 81.1& 77.9& 79.5& 19& & 69.3& 75.2& 72.1& 17\\\\\n\t\t\t STELA + CRNN& 84.0& 80.6& 82.3& 34& & 77.6& 82.8& 80.1& 33\\\\\n\t\t\tOurs& 95.8& \\textbf{84.5}& \\textbf{89.8}& 30& & 90.8& \\textbf{85.0}& \\textbf{87.7}& 26\\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\t\\begin{tablenotes}\n\t\t\t\\item SYOLOv4: Scaled YOLOv4\n\t\t\\end{tablenotes}\n\t\t\\end{threeparttable}\n\t\\end{table*}\n\t\n\tTo demonstrate that our method has reached a satisfactory performance, we make some comparisons with other methods. We involve EasyPR~\\cite{easypr}, SSD~\\cite{ssd}, YOLO-v3~\\cite{yolov3}, YOLO-v4~\\cite{yolov4}, Scaled YOLO-v4~\\cite{scaledyolov4} and Faster-RCNN~\\cite{fasterrcnn}. A CRNN~\\cite{crnn} model is appended to achieve recognition. We utilize cascaded STELA~\\cite{stela} and CRNN~\\cite{crnn} as our baseline in experiments. From Table~\\ref{table_crpd_comp}, it can be observed that EasyPR~\\cite{easypr} cannot treat LPs in CRPD well because it depends on manually designed features. SSD~\\cite{ssd}, YOLOv3~\\cite{yolov3}, YOLO-v4~\\cite{yolov4}, Scaled YOLO-v4~\\cite{scaledyolov4} and Faster-RCNN~\\cite{fasterrcnn} are not specially designed for LPs or text, but the results are still competitive. Our method reaches the best on all the sub-datasets, which proves the effectiveness.\n\t\n\tBecause Xu~\\textsl{et al.}~\\cite{ccpd} evaluated their method on different circumstances, for fair comparisons, experiments of our method on these datasets are also utilized. And as the other methods are trained on CCPD~\\cite{ccpd}, we also utilize it as the training data of our method in this comparison. Because images in CCPD~\\cite{ccpd} only contain one LP in one image, only Precision is involved when the Recall is not considered. Following the experiments of Xu~\\textsl{et al.}~\\cite{ccpd}, Cascade classifier~\\cite{wang2007cascade}, SSD300~\\cite{ssd}, YOLO9000~\\cite{yolov2}, Faster-RCNN~\\cite{fasterrcnn} are involved as the detector with Holistic-CNN~\\cite{reid} as the recognition model. And end-to-end methods TE2E~\\cite{te2e} and RPNet~\\cite{ccpd} are utilized for comparisons. We also involve YOLO-v4~\\cite{yolov4} and Scaled YOLO-v4~\\cite{scaledyolov4} with CRNN~\\cite{crnn} to report more results. Our baseline which consists of STELA~\\cite{stela} and CRNN~\\cite{crnn} are also involved. From Table~\\ref{table_ccpd_comp}, it can be observed that our method has reached the best in most circumstances, and the efficiency is also competitive.\n\t\n\t\\begin{table*}[h]\n\t\n\t\t\\caption{Comparisons of Precision on CCPD between our method and others.}\n\t\t\\scriptsize\n\t\t\\label{table_ccpd_comp}\n\t\t\\centering\n\t\t\\begin{threeparttable}\n\t\t\\begin{tabular}{m{2.95cm}<{\\centering}m{0.3cm}<{\\centering}m{0.3cm}<{\\centering}m{0.4cm}<{\\centering}m{0.3cm}<{\\centering}m{0.4cm}<{\\centering}m{0.6cm}<{\\centering}m{0.4cm}<{\\centering}m{0.8cm}<{\\centering}m{0.9cm}<{\\centering}m{0.3cm}<{\\centering}} \n\t\t\t\\toprule[0.75pt]\n\t\t\tMethod \n\t\t\t& Size& AP& Base& DB& FN& Rotate& Tilt& Weather& Challenge& FPS\\\\\n\t\t\t\\midrule[0.5pt]\n\t\t\tCascade classifier + HC\n\t\t\t& 480& 58.9& 69.7& 67.2& 69.7& 0.1& 3.1& 52.3& 30.9& 29\\\\\n\t\t\tSSD300 + HC\n\t\t\t& 300& 95.2& 98.3& 96.6& 95.9& 88.4& 91.5& 87.3& 83.8& 35\\\\\n\t\t\tYOLO9000 + HC\n\t\t\t& 480& 93.7& 98.1& 96.0& 88.2& 84.5& 88.5& 87.0& 80.5& 36\\\\\n\t\t\tYOLOv4 + CRNN\n\t\t\t& 512& 94.7& 97.8& 94.6& 87.3& 82.9& 89.9& 83.3& 75.7& 40\\\\\n\t\t\tSYOLOv4 + CRNN\n\t\t\t& 640& 95.3& 97.8& 95.0& 88.9& 84.9& 91.5& 90.4& 77.1& 34\\\\\n\t\t\tFaster-RCNN + HC\n\t\t\t& 600& 92.8& 97.2& 94.4& 90.9& 82.9& 87.3& 85.5& 76.3& 13\\\\\n\t\t\tTE2E\n\t\t\t& 600& 94.4& 97.8& 94.8& 94.5& 87.9& 92.1& 86.8& 81.2& 3\\\\\n\t\t\tRPNet\n\t\t\t& 480& 95.5& \\textbf{98.5}& 96.9& 94.3& 90.8& 92.5& 87.9& 85.1& \\textbf{61}\\\\\n\t\t\tSTELA + CRNN\n\t\t\t& 640& 97.8& 97.9& \\textbf{98.3}& 94.5& 90.1& 91.3& 89.5& 83.6& 41\\\\\n\t\t\tOurs\n\t\t\t& 640& \\textbf{97.9}& 98.3& 98.0& \\textbf{97.2}& \\textbf{92.5}& \\textbf{93.7}& \\textbf{90.7}& \\textbf{87.9}& 30\\\\\n\t\t\t\\bottomrule[0.75pt]\n\t\t\\end{tabular}\n\t\t\\begin{tablenotes}\n\t\t\t\\item SYOLOv4: Scaled YOLOv4; HC: Holistic-CNN; AP: average percent of all the circumstances\n\t\t\\end{tablenotes}\n\t\t\\end{threeparttable}\n\t\\end{table*}\n\t\n\tFinally, some detection and recognition results on CRPD are shown in Figure~\\ref{results} and~\\ref{results_wrong}. In the illustrations, we see that most LPs that are recognized incorrectly are also hard to distinguish by humans. Our proposed method is still effective in most cases, and the performance is competitive.\n\t\n\t\\begin{figure*}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=4.5in]{Fig9.eps}%\n\t\t\\caption{Spotting results of our method on CRPD. The green rectangles are the predicted rotated bounding boxes and the text on the top left corner of the zooming rectangle is the predicted text.}\n\t\t\\label{results}\n\t\\end{figure*}\n\t\n\t\\begin{figure*}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=4.5in]{Fig10.eps}%\n\t\t\\caption{Examples of failed recognition on CRPD. The green rectangles are the predicted rotated bounding boxes and the text on the top left corner of the zooming rectangle is the description of the LP, and the red characters are the ones that are recognized incorrectly. For simplicity, only the failed examples are zoomed in.}\n\t\t\\label{results_wrong}\n\t\\end{figure*}\n\t\n\t\\section{Discussion and Conclusion}\n\t\\label{conclusion}\n\t\n\tIn this paper, we present a dataset with Chinese LP images, which is named CRPD. As a supplement to multi-LP datasets, CRPD includes three sub-datasets, CRPD-single, CRPD-double, and CRPD-multi, which are able to deal with a variety of application scenarios. And CRPD covers many kinds of vehicles and a number of environments that will be helpful to build a robust model. We also propose an end-to-end trainable network to detect and recognize LPs with high efficiency as the baseline of the dataset. The experiments demonstrate the effectiveness of our proposed components, and the performance of the network is satisfactory. In the future, we hope CRPD will become a new benchmark on multi-LP detection and recognition tasks. We also consider utilizing the network for end-to-end scene text spotting and integrating more advanced techniques to achieve better portability and adaptation capability.\n\t\n\t\\section*{Acknowledgments}\n\tThis work was partly supported by the National\u00a0Key\u00a0Research\u00a0and\u00a0Development\u00a0Program\u00a0of\u00a0China\u00a0with\u00a0ID\u00a02018AAA0103203.\n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe lunar Cherenkov technique \\cite{Dagkesamanskii1989} provides a\npromising method of UHE neutrino detection since it utilises the\nlunar regolith as a detector; which has a far greater volume than\ncurrent ground-based detectors. This technique makes use of\nEarth-based radio telescopes to detect the coherent Cherenkov\nradiation emitted when a UHE neutrino interacts in the outer\nlayers of the Moon. It was first applied by Hankins, Ekers and\nO'Sullivan using the 64-m Parkes radio telescope\n\\cite{Hankins1996Asearch} and significant limits have been already\nbeen placed on the UHE neutrino flux by several collaborations\n\\citep{Hankins1996Asearch, Gorham2004Experimental, Beresnyak2005,\nJames2010MNRAS, NuMoon2010}.\n\nElectromagnetic pulses originating in the lunar surface will be\ndispersed when they arrive at Earth-based receivers due to\npropagation through the ionosphere which introduces a\nfrequency-dependent time delay. This dispersion reduces the peak\namplitude of a pulse, however, dedispersion techniques can be used\nto recover the full pulse amplitude and consequently increase the\nchances of detection. Accurate dedispersion requires an\nunderstanding of the ionospheric dispersion characteristic and\nit's effect on radio-wave propagation.\n\n\\section{Effects of Ionospheric Dispersion}\n\nThe ionosphere is a weakly ionized plasma which is formed by\nultraviolet ionizing radiation from the sun. Due to its\nrelationship with the sun, the ionosphere's electron density\nexperiences a strong diurnal cycle and is also dependent on the\nseason of the year, the current phase of the 11-year solar cycle\nand the geometric latitude of observation. The differential\nadditive delay, caused by pulse dispersion, is parameterised by\nthe ionospheric TEC (see Equation \\ref{eq:Derived3})\n\n\\begin{equation} \\Delta t= 0.00134 \\times\nSTEC \\times (\\nu_{\\text{lo}}^{-2}-\\nu_{\\text{hi}}^{-2}),\n\\label{eq:Derived3}\n\\end{equation}\nwhere $\\Delta t$ is the duration of the dispersed pulse in\nseconds, $STEC$ is the Slant Total Electron Content in electrons\nper cm$^{2}$ and $\\nu_{\\text{lo}}$ and $\\nu_{\\text{hi}}$ are the\nreceiver bandwidth band edges in Hz.\n\nCherenkov emission produces a sub-nanosecond pulse and therefore\ndetection requires gigahertz bandwidths to achieve the high time\nresolution needed to optimse the signal to noise. Due to\nexcessive data storage requirements, the only way to exploit such\nhigh data rates is to implement real-time dedispersion and\ndetection algorithms and to store potential events at full\nbandwidth for later processing. This requires an accurate\nknowledge of the real-time ionospheric TEC.\n\nIonospheric dispersion also offers some potential experimental\nadvantages, particularly for single dish experiments which can not\nuse array timing information to discriminate against RFI. A lunar\npulse will travel a one-way path through the ionosphere and be\ndispersed according to the current ionospheric TEC. Conversely,\nterrestrial RFI will not be dispersed at all and any Moon-bounce\nRFI will travel a two-way path through the ionosphere and be\ndispersed according to twice the current ionospheric TEC.\nTherefore performing real-time ionospheric dedispersion will\noptimise detection for lunar pulses and provide discrimination\nagainst RFI. Dispersion may also be seen to offer an increase in\ndynamic range. If triggering is performed on a dedispersed data\nstream while the raw data is buffered, any pulse clipping that\noccurs in the triggering hardware can be recovered during offline\nprocessing by reconstructing the pulse from the raw, undispersed\ndata.\n\n\\section{Dedispersion Hardware}\n\nPulse dispersion can be corrected using matched filtering\ntechniques implemented in either analog or digital technology.\nEarly LUNASKA experiments made use of the Australia Telescope\nCompact Array (ATCA) which consists of six 22-m dish antennas.\nThree antennas were fitted with custom-designed hardware for the\nneutrino detection experiments and pulse dedispersion was achieved\nthrough the use of innovative new analog dedispersion filters that\nemploy a method of planar microwave filter design based on inverse\nscattering \\cite{Roberts1995}.\n\nAs the microwave dedispersion filters have a fixed dedispersion\ncharacteristic, an estimate had to be obtained for the TEC which\nwould minimise errors introduced by temporal ionospheric\nfluctuations. The ATCA detection experiments were performed in\n2007 and 2008 during solar minimum and therefore relatively stable\nionospheric conditions. Initial observations were during the\nnights of May 5, 6 and 7, 2007 and these dates were chosen to\nensure that the Moon was at high elevation (particularly during\nthe night-time hours of ionospheric stability) and positioned such\nthat the ATCA would be sensitive to UHE particles from the\ngalactic center. The filter design was based on predictions made\nusing dual-frequency GPS data and assumed a differential delay of\n5 ns across the 1.2--1.8 GHz bandwidth. Data available post\nexperiment revealed that the average differential delay for these\nnights was actually 4.39 ns, with a standard deviation of 1.52 ns.\n\nThe ionosphere experiences both temporal and spatial fluctuations\nin TEC and therefore some signal loss is expected with a fixed\ndedispersion filter. A promising digital solution to overcome\nthese losses lies in the use of high speed Field Programmable Gate\nArrays (FPGAs). An FPGA implementation allows the dedispersion\ncharacteristic to be tuned in real time to reflect temporal\nchanges in the ionospheric TEC. A fully coherent or predetection\ndedispsersion method was pioneered by Hankins and Rickett\n\\cite{Hankins1971, Hankins1975} which completely eliminates the\neffect of dispersive smearing. This is achieved by passing the\npredetected signal through an inverse ionosphere filter which can\nbe implemented in either the time domain, as an FIR filter, or in\nthe frequency domain.\n\nIn 2009, the LUNASKA collaboration started a series of UHE\nneutrino detection experiments using the 64-m Parkes radio\ntelescope. For these experiments, dedispersion was achieved via a\nsuite of FIR filters implemented on a Vertex 4 FPGA. As GPS TEC\nestimates are currently not available in real-time, near real-time\nTEC measurements were derived from foF2 ionosonde measurements.\nIonosondes probe the peak transmission frequency (fo) through the\nF2-layer of the ionosphere which is related to the ionospheric TEC\nsquared. A comparison to GPS data available post-experiment\nrevealed that the foF2-derived TEC data consistently\nunderestimated the GPS TEC measurements. This is attributed to the\nground-based ionosondes probing mainly the lower ionospheric\nlayers and not properly measuring TEC contribution from the\nplasmasphere \\cite{Titheridge1972determination}.\n\n\\section{Monitoring the Ionospheric TEC}\n\nCoherent pulse dedispersion requires an accurate knowledge of the\nionospheric dispersion characteristic which is parameterised by\nthe instantaneous ionospheric TEC.\n\nTEC Measurements can be derived from dual-frequency GPS signals\nand are available online from NASA's CDDIS \\cite{CDDIS}, however,\nthese values are not available in real time. The CDDIS TEC data is\nsampled at two-hour intervals and is currently published after at\nleast a few days delay. Estimates derived from foF2 ionosonde\nmeasurements are available hourly from the Australian Ionospheric\nPrediction Service \\cite{IPS}. However, as discussed, there are\nknown inaccuracies in the derivation of the foF2-based TEC\nestimates.\n\nBoth of these products are published as vertical TEC (VTEC) maps\nwhich must be converted to Slant TEC (STEC) estimates to obtain\nthe true total electron content through the slant angle\nline-of-sight to the Moon. To perform this conversion, the\nionosphere can be modeled as a Single Layer Model (SLM)\n\\cite{Todorova2008} which assumes all free electrons are\nconcentrated in an infinitesimally thin shell and removes the need\nfor integration through the ionosphere. Slant and vertical TEC are\nrelated via\n\n\\begin{equation}\nSTEC=F(z)VTEC. \\label{Eq:slant}\n\\end{equation}\nwhere $F(z)$ is a slant angle factor defined as\n\n\\begin{align}\nF(z)&=\\frac{1}{\\cos(z^{\\prime})}\\\\\n&=\\frac{1}{\\sqrt{1-\\left(\\frac{R_e}{R_e+H}\\sin(z)\\right)^2}},\n\\end{align}\n$R_e$ is the radius of the Earth, $z$ is the zenith angle to the\nsource and $H$ is the height of the idealised layer above the\nEarth's surface (see Figure \\ref{Iono_model}). The CDDIS also use\nan SLM ionosphere for GPS interpolation algorithms and assume a\nmean ionospheric height of 450 km.\n\n\\begin{figure}[!tbph]\n\\begin{center}\n\\includegraphics [width=0.4\\textwidth,keepaspectratio]{fig1.eps}\n\\caption{Parameters of the Ionospheric Single Layer\nModel.}\\label{Iono_model}\n\\end{center}\n\\end{figure}\n\n\\section{A New Method of Ionospheric Calibration}\n\nAs the solar cycle enters a more active phase accurate pulse\ndedispersion is becoming a more important experimental concern for\nthe lunar Cherenkov technique. This requires methods of obtaining\nmore accurate measurements of the ionospheric TEC.\n\nA new technique has been formulated to obtain TEC measurements\nthat are both instantaneous and line-of-sight to the Moon. The\nionospheric TEC can be deduced from Faraday rotation measurements\nof a polarised source combined with geomagnetic field models,\nwhich are more stable than ionospheric models (the CCDIS\n\\cite{CDDIS} states that ionospheric TEC values are accurate to\n$\\sim$20\\% while the IGRF \\cite{IGRF} magnetic field values are\naccurate to better than 0.01\\%). Lunar thermal emission can be\nused as the polarised source since Brewster angle effects produce\na nett polarisation excess in the emission from the lunar limb\n\\cite{Heiles1963}. This provides a method for calibrating the\nionosphere directly line-of-sight to the Moon and makes the lunar\nCherenkov technique extremely attractive for UHE cosmic ray and\nneutrino astronomy as it allows the characteristic dispersion to\nbe used as a powerful discriminant against terrestrial RFI whilst\nremoving the need to search in dispersion-space.\n\nThe unique constraints of an UHE neutrino detection experiment\nusing the lunar Cherenkov technique conflict with traditional\nmethods of planetary synthesis imaging and polarimetry which\nrequires a complete set of spacings and enough observing time for\nearth rotation. Therefore to apply this method of ionospheric\ncalibration to the ATCA detection experiments, innovations in the\nanalysis of lunar polarisation observations were required. In\nparticular, a method of obtaining lunar Faraday rotation estimates\nin the visibility domain (i. e. without Fourier inversion to the\nimage plane) had to be developed.\n\nWorking in the visibility domain removes both the imaging\nrequirement of a compact array configuration, which would increase\nthe amount of correlated lunar noise between receivers, and also\nremoves the need for earth rotation allowing measurements to\nobtained in real time. This technique makes use of the angular\nsymmetry in planetary polarisation distribution. The intrinsic\nthermal radiation of a planetary object appears increasingly\npolarised toward the limb, when viewed from a distant point such\nas Earth \\cite{Heiles1963, SPORT2002}. The polarised emission is\nradially aligned and is due to the changing angle of the planetary\nsurface toward the limb combined with Brewster angle effects. The\nangular symmetry of this distribution can be exploited by an\ninterferometer so that an angular spatial filtering technique may\nbe used to obtain real-time position angle measurements directly\nin the visibility domain. The measured position angles are\nuniquely related to the corresponding $uv$ angle at the time of\nthe observation. Comparison with the expected radial position\nangles, given the current $uv$ angle of the observation, gives an\nestimate of the Faraday rotation induced on the Moon's polarised\nemission. Faraday rotation estimates can be combined with\ngeomagnetic field models to determine the associated ionospheric\nTEC and subsequently provide a method of calibrating the current\natmospheric effects on potential Cherenkov pulses.\n\nObservations of the Moon were taken using the 22-m telescopes of\nthe Australia Telescope Compact Array with a center frequency of\n1384 MHz. At this frequency the Moon is in the near field of the\narray, however, investigation of the Fresnel factor in polar\ncoordinates showed that it has no dependence on the spatial\nparameter, which determines the polarisation distribution of a\nplanetary body.\n\nUsing the angular spatial filtering technique, position angle\nestimates were calculated directly in the visibility domain of the\nlunar observational data. The Faraday rotation estimates were\nobtained by comparing these angles to the instantaneous $uv$ angle\nand the resultant Faraday rotation estimates were averaged over\nsmall time increments to smooth out noise-like fluctuations. Since\nthe polarised lunar emission received on each baseline varied in\nintensity over time, there were nulls during which the obtained\nposition angle information was not meaningful. A threshold was\napplied to remove position angle measurements taken during these\nperiods of low polarised intensity and baseline averaging was\nconsidered necessary as the results on each baseline were slightly\ndifferent and each affected differently by intensity nulls. The\nFaraday rotation estimates were converted to estimates of\nionospheric TEC via\n\n\\begin{equation}\n\\Omega = 2.36 \\times 10^4 \\nu^{-2}\\int_{\\text{path}}N(s)B(s)\\cos\n(\\theta) ds, \\label{eq:Faraday_rotation}\n\\end{equation} where $\\Omega$ is the rotation angle in radians, $f$ is the signal\nfrequency in Hz, $N$ is the electron density in m$^3$, $B$ is the\ngeomagnetic field strength in T, $\\theta$ is the angle between the\ndirection of propagation and the magnetic field vector and $ds$ is\na path element in m.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics [width=0.5\\textwidth,keepaspectratio]{fig2.eps}\n\\caption{Lunar Faraday rotation estimates converted to\n(\\emph{left}) ionospheric TEC values and (\\emph{right}) the\ndifferential delay across 1.2--1.8 GHz.}\\label{TECdelay}\n\\end{center}\n\\end{figure}\n\nTo evaluate the effectiveness of this new ionospheric calibration\ntechnique, the TEC results were compared against ionospheric TEC\nestimates derived from dual-frequency GPS data (Figure\n\\ref{TECdelay}) . Slant angle factors were used to convert the GPS\nVTEC estimates to STEC toward the Moon for comparison with the\nATCA data. Both data sets exhibited a similar general trend of\nsymmetry around the Moon's transit. However, the ATCA data often\nunderestimated the GPS data, particularly around 14:30--17:00 UT\nwhere the STEC estimates may have been influenced by bad data from\nthe shorter baselines or due to TEC contribution from the\nplasmasphere which is not in the presence of a magnetic field\n\\cite{Titheridge1972determination}. These observations were taken\nwhen the TEC was very low and therefore the relative error in the\nTEC estimates is large.\n\n\\section{Conclusions}\n\nAs the sun enters a more active phase, accurate ionospheric pulse\ndispersion is becoming a more important experimental concern for\nUHE neutrino detection using the lunar Cherenkov technique.\nHardware dedispersion options rely on the accuracy of real-time\nionospheric TEC measurements and, while there are a few options\navailable for obtaining these measurements, they are not currently\navailable in real time nor directly line-of-sight to the Moon. A\nnew ionospheric calibration technique has been developed. This\ntechnique uses Faraday rotation measurements of the polarised\nthermal radio emission from the lunar limb combined with\ngeomagnetic field models to obtain estimates of the ionospheric\nTEC which are both instantaneous and line-of-sight to the Moon.\nSTEC estimates obtained using this technique have been compared to\ndual-frequency GPS data. Both data sets exhibited similar features\nwhich can be attributed to ionospheric events, however, more\nobservations are required to investigate this technique further.\n\n\\section{Acknowledgements}\n\nThis research was supported as a Discovery Project by the\nAustralian Research Council. The Compact Array and Parkes\nObservatory are part of the Australia Telescope which is funded by\nthe Commonwealth of Australia for operation as a National Facility\nmanaged by CSIRO.\n\n\\biboptions{square,sort&compress}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzevgo b/data_all_eng_slimpj/shuffled/split2/finalzzevgo new file mode 100644 index 0000000000000000000000000000000000000000..74cd4273479dadfd406108becd3ea03bdf9536cf --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzevgo @@ -0,0 +1,5 @@ +{"text":"\\section*{Experimental setup}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.85\\textwidth]{.\/Figs\/fig_1_setup.pdf}\n \\caption{\n \\textbf{Experimental setup. a,}\n Alice and Bob generate light beams from their local continuous-wave laser sources (LSs) and send them onto their beam splitters.\n From each beam splitter, one output beam goes to the Encoders and is used to perform TF-QKD.\n The other is used to lock the users' LSs through a service fibre, depicted in orange.\n Alice sends part of her light to Bob through the service fibre.\n Bob interferes it with his own, after shifting its frequency by 80~MHz through an acousto-optic modulator (AOM).\n The beating signal resulting from the interference is detected with a phase-sensitive detector (PSD) whose current is proportional to the phase difference between Alice's and Bob's light beams.\n An electronic feedback is then given to Bob's LS based on the detected difference to lock its phase to Alice's one.\n This constitutes a heterodyne optical phase-locked loop (OPLL).\n \\textbf{b,}\n In the Encoder modules, the continuous-wave light prepared locally by Alice's and Bob's LSs, seeds a gain-switched laser diode (LD) that carves it into pulses.\n The optical pulses are either rapidly modulated or finely controlled in phase by the phase modulators (PMs).\n After crossing the electrical polarisation controller (EPC) and the intensity controller (INT), part of the pulses is directed to the power detector (PD) for monitoring the intensity and the other part travels through the quantum channel towards Charlie's beam splitter (BS).\n Here, they interfere with the other user's pulses.\n The outcome of the interference is registered by the SNSPD detectors D1, D2 and D3.\n Variable optical attenuators (VOAs) add losses to the quantum channel.\n C, circulator; SF, spectral filter; FA, fixed attenuator; PL, polariser; PBS, polarising beam splitter.\n }\n \\label{fig:setup}\n\\end{figure*}\n\nIn our realisation of TF-QKD, we consider a generalised protocol (see Methods) that can be modified to encompass various TF-QKD protocols based on coherent states~\\cite{LYDS18,Tamaki.,Ma.,Wang.2018,CYW+18,CAL18}.\nTo experimentally validate these protocols and overcome the SKC$_0$ bound, Alice and Bob should use two separate lasers to prepare coherent states in a given phase and polarisation state, with various intensities.\nThe two separate lasers should be phase-locked to let the users reconcile their phase values.\nWe represent Alice's states as $\\ket{\\sqrt{\\mu_a} e^{i\\varphi_a}}$, where $\\mu_a$ is the intensity and $\\varphi_a \\in [0,2\\pi)$ is the phase.\nBob prepares similar states with the subscript $a$ replaced by $b$.\nThe phases $\\varphi_{a,b}$ include both the bit information and the random values needed in coherent-state TF-QKD.\nThe optical pulses emitted by the users should interfere with high visibility in the intermediate station after having travelled through a pair of highly lossy channels.\nHigh loss is needed to overcome the SKC$_0$~\\cite{LYDS18}.\nThe optical phase should remain stable in time, which is challenging when the channel loss reduces the amount of detected counts.\n\nWe implement these features using the experimental setup shown in Fig.~1.\nEach user is endowed with a continuous-wave laser source (LS).\nAlice's LS acts as the phase reference.\nIts light is split in two at a first beam splitter (BS).\nOne part is sent to Bob through a service fibre, depicted in orange in Fig.~\na, and is used to lock Bob's LS via a heterodyne optical phase-locked loop (OPLL~\\cite{Bordonalli.1999}, see Supplementary Section~1 for further details).\nIn Bob's module, the reference light interferes with another light beam prepared by Bob and shifted by 80~MHz by an acousto optic modulator (AOM).\nA photodiode acts as a phase-sensitive detector (PSD), whose intensity is mapped onto a phase difference $\\delta\\varphi$ between the reference light and Bob's local light.\nA feedback is then given to Bob's laser based on the value of $\\delta\\varphi$.\nWith this OPLL, Bob's laser is locked to Alice's with a phase error less than $5^{\\circ}$, which includes a potential phase fluctuation in the fibre connecting Alice to Bob.\nAn attacker could modify the reference light while it travels from Alice to Bob, but that would not affect the security of the scheme.\nAny modification would translate into a different value of $\\delta\\varphi$, which is equivalent to Eve introducing phase noise on the main channels going from the users to Charlie (see also~\\cite{Koa04} for a similar argument applied to QKD).\nHowever, we do not claim here the robustness of Alice's and Bob's modules to side-channel attacks, which requires more scrutiny, similarly to the one ongoing for the MDI-QKD sending modules.\n\nThe fraction of each user's light not involved in the phase-locking mechanism is directed to the Encoder, depicted in Fig.~1b.\nHere it enters the cavity of a slave laser diode (LD) that is periodically gain-switched to produce a pulse train at 2~GHz.\nThis ensures that each pulse will inherit the phase of the injected optical field, which is locked to the reference light.\nMoreover, Alice and Bob's LDs will emit pulses as narrow as \\SI{70}{ps} at 1548.92~nm, with high extinction ratio and constant intensity due to the strength of optical injection into the slave laser being 1,400 times weaker than the electrical injection, as we measured. After the LD, the optical pulses pass through an in-line phase modulator (PM), which applies fast modulation from an RF signal to encode the phase values required by the specific TF-QKD protocol, and a slow correction from a DC signal to compensate the phase noise on the paths linking to Charlie.\nAfter setting the optical pulses' polarisation and intensity, the pulses pass through 15~GHz filters that clean their spectral mode~\\cite{CLF+16a}, thus ensuring high visibility interference between the twin-fields. Then they are sent to variable optical attenuators (VOAs) that vary the loss of the channel connecting the users to Charlie.\n\nAlice and Bob's optical fields interfere on Charlie's BS and are eventually detected by superconducting nanowire single photon detectors (SNSPDs, Single Quantum EOS 410 CS) cooled at 3.2~K, featuring 22~Hz dark count rate and 44\\% detection efficiency.\nDetector D1 is associated with a 100~ps resolution time tagger and is used to extract the raw key rate.\nD2 monitors the optical field leakage into the non-intended polarisation, which is minimised by Alice and Bob through their polarisation controllers.\nD3 is sampled by a photon counter at a minimum interval of 10~ms to stabilise the overall phase.\n\n\\section*{Results}\n\nThe first task is providing the users with weak coherent pulses that are locked to a common phase reference and capable of interfering on Charlie's BS. Then part of these pulses have to be phase randomised with respect to the phase reference. In some TF-QKD protocols~\\cite{LYDS18,Tamaki.,Ma.,Wang.2018}, it is necessary that the users know the values of the random phases, whereas in others~\\cite{CYW+18,CAL18,LL18} this is not mandatory.\n\nIn the current setup we randomise the phase in an active way and obtain a first-order interference visibility at Charlie of 96.4\\% when the OPLL is active and the two PMs in Alice and Bob encode equal phases.\nWe encode a pseudo-random pattern containing $2^{10}$ symbols having $2^5$ modulation levels through the PMs driven by high-speed 12~GSa\/s digital-to-analogue converters (DACs) with 8-bit amplitude resolution.\nThe number of phases we chose is sufficiently close to a phase randomisation with infinite random phases~\\cite{CZLM15}.\nHowever, to further demonstrate a full phase randomisation, we performed a parallel experiment using a continuous phase randomisation from a master gain-switched laser~\\cite{JCS+11,YLD+14} (see Supplementary Section~2).\nFor that, we removed the OPLL and active phase randomisation from the setup, while leaving active phase encoding.\nThen we locked Alice and Bob's lasers to the main master laser by optical injection locking~\\cite{YPJ+03,CLF+16a} and obtained a visibility of 97.5\\%, 1.1\\% higher than in the previous case.\nWe attribute this difference to the absence of errors from the OPLL and the active phase randomisation.\nThis result shows that the overall visibility is not affected by the absolute number of encoded phases (it is higher with more phases than less) as much as by the components used to implement it.\nThe base quantum bit error rate (QBER) of the system remains in all cases smaller than 1.8\\% and there is no in-principle limitation to increasing the number of the encoded phases.\n\n\nThe ambient temperature fluctuations cause the experimentally obtained interference to drift.\nEven with 40~metres of optical fibre, the environmental fluctuations cause a relative phase drift in our setup of \\SI{0.7}{rad\/s}.\nThis requires a feedback control every 10 -- 100~ms to avoid detrimental effects on the QBER.\nTo implement this phase control, we doubled the pulse pattern to $2048$ bits and temporally interleaved phase-encoded pulses and unmodulated reference pulses, with equal duty cycles and intensities.\nThis is done by clocking the phase modulators in Alice and Bob's setup at 2~GHz and by actively switching between reference pulses and encoding pulses.\nThis reduces the effective clock rate of the TF-QKD protocol to 1~GHz.\nThe photon detection clicks are recorded by Charlie with time-tagging electronics and are grouped in post-processing according to their phase values.\nFrom these events we retrieve the gain and the QBER of the system (Supplementary Section~4).\nThe phase correction from reference pulses is designed to keep Alice and Bob's optical fields locked on Charlie's BS at a constant $\\pi\/2$ phase difference.\nThis is the most efficient solution as it exploits the linear part of the response function (see Supplementary Section~3 for further discussion).\nThe phase offset is continuously monitored by detector D3 and corrected by acting on the DC level of one of the PMs in the transmitting modules.\nThis is equivalent to having an extra PM in Charlie's station, as proposed in ref.~\\cite{LYDS18}.\n\nA main advantage of TF-QKD is the scaling property of the secret key rate with the square-root of the channel transmission, $\\eta^{1\/2}$.\nThis would be impossible without correspondingly having the square-root scaling of the detection rate.\nWe verified this essential feature of TF-QKD directly and summarised the result in Fig.~2\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{.\/Figs\/fig_2_gain_qber.pdf}\n \\caption{\n \\textbf{Gain and QBER of TF-QKD.}\n The gain is the detection probability per encoding gate.\n Gain and QBER are plotted against the channel loss $1-\\eta$.\n The equivalent fibre length on the top axis pertains to an ultra-low-loss (ULL) fibre with attenuation coefficient 0.16~dB\/km.\n The different scaling laws of the gain for a QKD-like single-path quantum transmission and for a TF-QKD-like double-path quantum transmission are apparent.\n The upward and downward triangular (square) points on the dotted (dashed) line are the single-path (double-path) experimental detection rate recorded for different channel loss.\n The circle points on the solid line are the double-path experimental QBER of the interfering pulses.\n All the experimental data agrees well with the theoretical curves.\n }\n \\label{fig:gain_TFQKDvsQKD}\n\\end{figure}\nThe data corresponding to a direct-link quantum transmission were taken by shutting off one arm of our experimental setup, thus allowing a single user at a time to signal to Charlie's station.\nThe data for double-path transmission, on the other hand, were taken with both arms open.\nAs is apparent from the figure, the single-path gain (triangular points on the dotted line) scales linearly with the loss, $1-\\eta$, whereas the double-path gain (square points on the dashed line) scales with the square-root.\nAt any given gain, the double-path TF-QKD can tolerate channel loss twice as large than single-path direct-link QKD.\nIn the same figure, we also report the experimental QBER of TF-QKD, which is composed of three main contributions: the quantum state preparation, detectors' dark counts and feedback routine.\nIn our setup, the last two terms significantly affect the overall QBER only at losses higher than 70~dB.\n\nOur experimental results are independent of the specific security analysis adopted to extract a key rate.\nHence they can be used as a reference to test the performance of any TF-QKD-like protocol.\nHere we analyse the data for three TF-QKD protocols, two~\\cite{LYDS18,Wang.2018} over the whole loss range and one~\\cite{CAL18} at a specific loss around 70~dB.\nThe protocol in ref.~\\cite{LYDS18} is the original TF-QKD scheme and acts as a reference.\nThe protocols in refs.~\\cite{Wang.2018,CAL18}, on the other hand, have been conceived to be unconditionally secure.\nWe have implemented them using three intensities, which is practical as compared with infinite intensities in their initial proposals.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{.\/Figs\/fig_3_SKR.pdf}\n \\caption{\n \\textbf{TF-QKD key rates.}\n Secret key rates are plotted against the channel loss (lower horizontal axis) and the corresponding ULL fibre distance (upper horizontal axis).\n The markers show the acquired experimental data whereas the solid lines follow from simulations.\n The ideal repeaterless SKC$_0$~\\cite{PLOB17} (dashed line) and the realistic one (dotted line) are plotted along with the key rates of the original TF-QKD protocol~\\cite{LYDS18} (red circles), the protocol in ref.~\\cite{Wang.2018} (blue triangles) and the protocol in ref.~\\cite{CAL18} (yellow square).\n The TF-QKD supremacy region is in pink shades.\n %\n The simulations assume 1~GHz effective clock rate.\n The realistic repeaterless bound assumes a total detection efficiency of 35\\% plus 3~dB loss due to having one detector in Charlie's module.\n Other parameters are:\n $\\alpha=0.16$~dB\/km, ULL fibre attenuation;\n $f_{EC}=1.15$, error correction coefficient.\n $\\eta_{C} = 30.8\\%$, total transmission of Charlie's module, resulting from $\\eta_{det}=44\\%$ and $\\eta_{\\textrm{coupling}}=70\\%$;\n $P_{dc} = 22$~Hz, dark count rate.\n Charlie is assumed to be at equal distance from Alice and Bob.\n The photon fluxes are specified in the Methods.\n }\n \\label{fig:SKR}\n\\end{figure*}\n\nIn Fig.~3\nwe plot the secure key rates (SKRs) versus the channel loss for the protocols analysed.\nWe also plot two lines for the repeaterless secret key capacity, which we call \\textit{ideal} and \\textit{realistic} SKC$_0$, respectively.\nThe former is the expression given in~\\cite{PLOB17}, $\\log_{2}[1\/(1-\\eta)]$, with detectors implicitly assumed to be 100\\% efficient.\nThis SKC$_0$ is impossible to overcome without an intermediate repeater.\nThe latter represents a direct comparison with our experiment and assumes a QKD performed with one detector and efficiency slightly larger than in our setup.\n\nThe darker-pink (lighter-pink) shaded area is the supremacy region where the SKR of the protocol in~\\cite{Wang.2018} (the protocol in~\\cite{LYDS18}) surpasses the realistic SKC$_0$.\nThis region extends from about 50~dB to 83~dB, limited only by detectors' dark counts.\nIn this range, our TF-QKD scheme provides more SKR than a QKD scheme with the same components.\nThis is remarkable in light of the fact that TF-QKD is more secure than QKD, as it protects against attacks directed at the detection devices.\nEven more interestingly, there are experimental points that fall beyond the \\textit{ideal} SKC$_0$.\nAt 71.1~dB, for instance, the SKRs of the protocols in refs.~\\cite{Wang.2018} and \\cite{CAL18} are $213.0$~bit\/s and $270.7$~bit\/s, respectively, i.e.~1.90 times and 2.42 times larger than the corresponding ideal SKC$_0$ ($112.0$~bit\/s).\nThis is the first time that this fundamental limit has been overcome experimentally.\nIt is worth mentioning that all the reported SKRs are quite conservative as they include the penalty due to an imperfect error-correction ($f_{EC}=1.15$).\nThe maximum channel loss over which we can stabilise the phase and obtain a positive key rate is 90.8~dB (rightmost red circle in Fig.~\n).\nThis is equivalent to 454~km and 567~km of standard and ultralow-loss (ULL, 0.16~dB\/km) single-mode optical fibre, respectively, connecting the users.\nWe incidentally notice that values up to 0.1419~dB\/km are currently achievable in fibres at 1560~nm~\\cite{TSM+18}.\n\nIt is interesting to compare these results with the current record distances~\\cite{BBR+18,YCY+16} obtained in QKD and MDI-QKD experiments over long-haul fibres in the finite-size scenario.\nThe QKD record distance is 421.1~km in ULL fibre, with a key rate of 0.25~bit\/s over a total channel loss of 71.9~dB~\\cite{BBR+18}.\nThis was made possible by a 2.5~GHz clock rate and a detector dark count rate of 0.1~Hz.\nFor a similar channel loss, with our 1~GHz effective clock rate, the SKR of any of the TF-QKD protocols analysed is three orders of magnitude higher.\nThe longest demonstration of MDI-QKD is on 404~km of ULL optical fibre obtained with the protocol in~\\cite{ZYW16}.\nWith a clock rate of 75~MHz, it provided an SKR of \\num{3.2e-4}~bit\/s over a total channel loss of 64.64~dB~\\cite{YCY+16}, which is six orders of magnitude smaller than the TF-QKD key rates at 71.1~dB.\nAlthough our results have been achieved in the asymptotic regime and do not include finite-size effects or long-haul real fibres, as the experiments in~\\cite{YCY+16,BBR+18}, the improvement they entail appears to be substantial.\n\n\\section*{Conclusions}\n\nIn TF-QKD, quantum information is carried by the optical fields prepared by Alice and Bob.\nFields can tolerate larger loss than photons and can potentially increase the rate and range of quantum communications~\\cite{LYDS18}.\nIn the present work, we have provided the first experimental evidence of this potential, surpassing, for the first time, the rate-loss limit of direct-link quantum communications~\\cite{PLOB17} with an intermediate untrusted node.\n\nTo achieve this goal, the users have to prepare light pulses that are phase stabilised and at the same time phase randomised respect to a shared phase reference. The phase stabilisation between the lasers was achieved by an optical phase-locked loop, which also guarantees that the reference light emitted by Alice and possibly manipulated by Eve is securely transferred to Bob.\nThe phase stabilisation across the channels linking Alice and Bob to Charlie was achieved using efficient feedback based on unmodulated reference optical pulses of the same intensity as the quantum pulses used for key generation.\nThis maintains interference stability to overcome the rate-loss theoretical limit and extract a positive key rate over a 90~dB loss link, which is about 20~dB larger than in any other previous quantum communication test.\n\nOur proof-of-concept experiment shows that TF-QKD can greatly enhance the range and rate of quantum communications using presently available technology.\n\n\\section*{Methods}\n\\smallskip\n\n{\\small\n\\noindent\\textbf{Generalised TF-QKD protocol}\n\\smallskip\n\n\\textit{State Preparation} -- Alice randomly selects: the bit value $\\alpha_a=\\{0,1\\}$, with probability $p_{\\alpha_a}$; the basis value $\\beta_a=\\{0,1\\}=\\{Z,X\\}$, with probability $p_{\\beta_a}$; the global phase value $\\phi_a\\in [0,2\\pi)$, with uniform probability $p_{\\phi_a}$; the intensity value $\\mu_a=\\{u_a,v_a,w_a\\}$, with probability $p_{\\mu_a}$. She uses the setup in Fig.~\nto prepare a coherent state $\\ket{\\sqrt{\\mu_a}e^{i\\varphi_a}}$, where $\\varphi_a=\\phi_a+\\alpha_a\\pi+\\beta_a\\pi\/2$. Bob does the same, with subscripts $a$ replaced by $b$ and same values for the parameters, unless explicitly stated.\n\nThis step represents the state preparation of the original TF-QKD protocol~\\cite{LYDS18} and of the one in ref.~\\cite{Tamaki.}.\nTo increase the asymptotic key rate, the probability of the majority basis, $p_Z$, can be set arbitrarily close to 1, similarly to the efficient version of the BB84 protocol~\\cite{LCA05}.\nAlong the same line, the state preparation of the TF-QKD protocol in ref.~\\cite{Ma.} can be obtained by simply setting the probability of the minority basis, $p_{X}$, equal to 0.\nIn ref.~\\cite{CYW+18} and in the Protocol~3 of ref.~\\cite{CAL18}, which relies on coherent states, the global phase is randomised only in the `test' basis, which we choose here equal to $X$, while it is constant in the `encoding' basis, which we choose here equal to $Z$.\nFor the state preparation of the protocol in ref.~\\cite{Wang.2018}, we can treat the $X$ basis as in the original TF-QKD protocol whereas for the $Z$ basis we set $w_{a,b}=0$ and $p_{u_{a,b}}=1-p_{w_{a,b}}=\\epsilon$, $p_{v_{a,b}}=0$.\nA close-to-optimal value for $\\epsilon$ is $10\\%$. In our simulations, we set it equal to 7.8\\%. We then relate the bit value $0$ ($1$) to the instances of the $Z$ basis where Alice encoded $w_a$ ($u_a$) and Bob encoded $u_b$ ($w_b$).\n\nIn the above preparation of the intensity, we considered for simplicity only three values, as opposed to the infinite values considered in most of the TF-QKD proposals.\nOn one hand, this setting can easily be generalised to any number of intensities, even infinite, like in the original decoy-state QKD~\\cite{LMC05}.\nOn the other hand, the possibility to import the decoy-state technique into TF-QKD is the main motivation for having multiple intensities in the protocol.\nThe only exception is ref.~\\cite{LL18}, which we could not include in the above description as it does not resort to phase randomisation and decoy states to extrapolate a key rate from the acquired sample.\nFor the other protocols, we describe below how to apply the decoy-state technique with three intensities.\n\n\\smallskip\n\n\\textit{Measurement} -- Alice and Bob send their optical pulses to the central relay station, Charlie, who does not need to be honest.\nA honest Charlie would send the incoming pulses on his beam splitter, measure the output pulses and report which of his two detectors clicked.\nA dishonest Charlie, however, could use any detection scheme he pleases.\nThis would not affect the security of the protocol as TF-QKD's security, similarly to MDI-QKD, is independent of the detection scheme.\nThis implies that the secret key rate extracted by the users when Charlie is dishonest is always lower than or equal to the one they would extract if Charlie is honest.\nIn our experiment, Charlie announces counts from the detector D1 only.\n\n\\smallskip\n\n\\textit{Announcement} -- After repeating the above steps many times, the honest Charlie announces over a public authenticated channel the events where one and only one detector clicked. Alice and Bob announce their intensity values $\\mu_{a,b}$, their basis values $\\beta_{a,b}$ and their global phases $\\phi_{a,b}$.\n\nThis step holds for protocols in~\\cite{LYDS18,Ma.}, even if the announcement of the basis is redundant in~\\cite{Ma.} because $p_X=0$. For the protocols in~\\cite{Wang.2018,CYW+18,CAL18}, the intensity values are announced only in the $X$ basis, whereas the global phases are announced only in the $X$ basis for \\cite{Wang.2018} and never announced in \\cite{CYW+18,CAL18}. This latter feature is remarkable and can entail a great experimental simplification. For the protocol in \\cite{Tamaki.}, Alice selects two modes of execution. In the `test mode' the global phases are never disclosed, thus allowing a rigorous application of the decoy-state method; in the `code mode', the global phases are announced to let the users reconcile their bit values. The bases are always announced unless a particular event occurs in the code mode.\n\\smallskip\n\n\\textit{Sifting} -- Among the announced successful detections, Alice and Bob keep the events that have matching values.\nIn all cases, either for a specific basis or for both bases, they keep those events whose phases are `twins', i.e., no more different than a certain tolerance level $\\Delta$ modulo $\\pi$, due to the symmetry of TF-QKD with respect to the addition of $\\pi$ to the phase values~\\cite{Ma.}.\nAfter sifting their data, the users keep $\\alpha_a$ and $\\alpha_b$ ($\\alpha_b \\oplus 1$) as their raw key bits if Charlie announced a detection related to a $0$ ($\\pi$) phase difference between Alice's and Bob's phases.\n\nThis holds for protocols in~\\cite{LYDS18,Ma.,Tamaki.} with minor differences. For the protocol in \\cite{Wang.2018}, it holds in the $X$ basis. The raw key bit, however, is obtained from the $Z$ basis when single clicks are announced by Charlie, irrespective of which detector clicked.\nFor the protocols in \\cite{CYW+18,CAL18}, it holds with $\\Delta=0$ in the encoding basis.\n\n\\smallskip\n\n\\textit{Parameter Estimation} -- A raw key is formed by concatenating the raw key bits obtained in the previous step. All the remaining data unrelated to the key bits can be fully disclosed to estimate the decoy-state parameters related to security.\n\nUp to minor differences, this step is the same in all TF-QKD protocols.\nIn fact, they all use the decoy-state technique~\\cite{Hwang.2003,Wan05,LMC05} to estimate the single-photon quantities related to security, with the exception of \\cite{LL18}, which was already kept out of our description.\nEven in~\\cite{Ma.} decoy states are extensively used to estimate the photon-number dependent quantities appearing in the phase error rate of the protocol. In this case, having only three intensities for $\\mu_{a,b}$ might be insufficient to obtain a tight estimation of the phase error rate.\nA similar argument applies to ref.~\\cite{CYW+18}, where four intensity levels were used to obtain a good key rate.\nHere we find that three intensity levels are sufficient to extract good key rates from the protocols in Refs.~\\cite{LYDS18,Wang.2018} and \\cite{CAL18}.\n\n\\smallskip\n\n\\textit{Key distillation} -- The users run classical post-processing procedures such as error correction and privacy amplification to distil the final secure key from the raw key.\n\nThe amount of privacy amplification in this step is specific to each TF-QKD protocol as it depends on the detailed security analysis. In the present work, we only consider two specific key distillation rates for exemplificative purposes, given in Refs.~\\cite{LYDS18,Wang.2018}. For both protocols we consider the standard decoy state equations in the asymptotic scenario, providing\nthe lower bound for the 0-photon yield, $\\underline{y}_0 = (v Q_w e^w - w Q_v e^v)\/(v-w)$;\nthe lower bound for the 1-photon yield, $\\underline{y}_1 = [u^2 Q_v e^v - u^2 Q_w e^w - (v^2-w^2) ( Q_u e^u - \\underline{y}_0 )] \/ [u(u v-u w-v^2+w^2)]$;\nand the upper bound for the 1-photon error rate, $\\overline{e}_1 = (E_v Q_v e^v - E_w Q_w e^w)\/[(v-w)\\underline{y}_1]$.\nIn these equations, $u=u_a+u_b$, $v=v_a+v_b$ and $w=w_a+w_b$ are the total intensity values for the signal state, the decoy state and the vacuum state, respectively.\nFor the total intensity, we also use the symbol $\\mu=\\mu_a+\\mu_b=\\{u,v,w\\}$.\nIn our experiment, we set $u=0.4$, $v=0.16$ and $w=10^{-5}$ for the protocols in \\cite{LYDS18,Wang.2018} and $u_a=u_b=0.02$, $v_a=v_b=0.2$, $w=10^{-5}$ for the protocol in \\cite{CAL18}.\nThe parameters $\\underline{y}_1$ ($\\underline{y}_0$) and $\\overline{e}_1$ are, respectively, the lower bound for the single-photon (zero-photon) yield and the upper bound for the single-photon phase error rate; $Q_\\mu$ and $E_\\mu$ are the gain and QBER measured by detector D1 in Fig.~\na.\nThen for the original TF-QKD key rate we use\n\\begin{equation}\\label{SKR_TFQKD}\n R^{\\prime}=\\{\\underline{Q}_1 [1-h(\\overline{e}_1)] - f_{EC} Q_{u} h(E_u) \\}\/M^{\\prime}.\n\\end{equation}\nHere, $\\underline{Q}_1 = \\mu e^{-\\mu} \\underline{y}_1$ is the lower bound for the single-photon gain;\n$f_{EC}$ is the error correction factor, set equal to 1.15 in our simulations;\n$h$ the binary entropy function;\n$M^{\\prime}=M\/2$ and $M$ is the number of phase slices used to reconcile the global random phase, set equal to 16 in our simulations.\nIn Eq.~\\eqref{SKR_TFQKD}, $E_u$ includes $E_M$, the intrinsic misalignment error of TF-QKD, which is equal to 1.275\\% for $M=16$~\\cite{LYDS18}.\nThe key rate in Eq.~\\eqref{SKR_TFQKD} is secure under the conditions clarified in ref.~\\cite{LYDS18}. When these conditions are not met, Eve can perform other attacks like the `collective beam splitting' (CBS) attack~\\cite{LYDS18} or the one described in \\cite{WHY18}.\n\nFor the `Send-Not Send' TF-QKD protocol by Wang \\textit{et al.}~\\cite{Wang.2018} we use the key rate equation\n\\begin{equation}\\label{SKR_SNSTFQKD}\n R^{\\prime\\prime}=\\underline{Q}_0^z + \\underline{Q}_1^z [1-h(\\overline{e}_1)] - f_{EC} Q^z h(E^z),\n\\end{equation}\nwith $Q^z = \\epsilon^2 Q_{u} + \\epsilon (1 - \\epsilon) (Q_{u_a} + Q_{u_b}) + (1 - \\epsilon)^2 Q_0$, $ E^z = [\\epsilon^2 Q_{u} + (1 - \\epsilon)^2 Q_0]\/{Q^z}$.\nThe parameter $\\epsilon$ has been defined in the state preparation step of the protocol; $Q_{u}$, $Q_{u_a}$, $Q_{u_b}$, $Q_{0}$ are the gains (i.e. ratio of successfully detected events to sent optical pulses) of the protocol, measured in the experiment when both of the users, only Alice, only Bob, none of the users, respectively, send out optical pulses.\nThe values for these quantities at each attenuation level are reported the Supplementary Information.\nThe single-photon gain in the $Z$ basis assumes the form $\\underline{Q}_1^z = \\[\\epsilon (1 - \\epsilon) (u_a e^{-u_a}+u_b e^{-u_b}) + \\epsilon^2 (u e^{-u})\\] \\underline{y}_1$ and $\\underline{Q}_0=\\[ (1-\\epsilon)^2 + \\epsilon (1-\\epsilon) (e^{-u_a}+e^{-u_b}) + \\epsilon^2 e^{-u} \\]\\underline{y}_0$.\nThe single-photon quantities $\\underline{y}_0$, $\\underline{y}_1$ and $\\overline{e}_1$ are drawn from the $X$ basis of the protocol using the equations written above.\nIn this specific protocol, the number of phase slices is large, leading to no misalignment error in the $X$ basis.\nAlso, unlike the original send-not send TF-QKD~\\cite{Wang.2018}, we consider here three intensities to implement the decoy-state technique, which is practical.\nFinally, we include an extra term $\\underline{Q}_0^z$ in the key rate equation \\eqref{SKR_SNSTFQKD}, accounting for the fact that Eve cannot extract any useful information from the vacuum pulses prepared by the users~\\cite{Lo05,Koa06}.\n\nFor the protocol by Curty \\textit{et al.}~\\cite{CAL18} we use the key rate equation written for their Protocol~3, after adding the error correction factor $f_{EC}$, to make the proposal more practical:\n\\begin{equation}\\label{SKR_Curty}\n R^{\\prime\\prime\\prime} = Q^z [1-h(\\overline{e}^x_{1})] - f_{EC} Q^z h(E^z).\n\\end{equation}\nThis extra term makes the key rate smaller, so it is even more difficult to overcome the SKC$_0$ when it is taken into account.\nThe counts for the raw key come from detector D1 in the setup of Fig.~\na, as for the other protocols.\nMost quantities in Eq.~\\eqref{SKR_Curty} are similar to those already introduced for the other protocols, with the exception of the phase error rate $\\overline{e}^x_{1}$, which deserves a specific discussion.\nIt has been taken from Eq.~(15) of ref.~\\cite{CAL18} and amounts to\n\\begin{align}\\label{e1Cur}\n\\nonumber \\overline{e}^x_{1} &=\\frac{1}{Q^z}\\sum_{j=0,1}\\[\\sum_{m,n=0}^{\\infty} c_{m}^{(j)} c_{n}^{(j)}\\sqrt{\\overline{Y}^x_{mn}}\\]^2 \\\\\n\\nonumber &\\leq \\frac{1}{Q^z}\\sum_{j=0,1}\\[\\sum_{m,n=0}^{\\infty} c_{m}^{(j)} c_{n}^{(j)}\\sqrt{g_{mn}(\\overline{Y}^x_{mn},Y_{\\textrm{cut}})}\\]^2\\\\\n &\\simeq \\frac{1}{Q^z}\\sum_{j=0,1}\\[\\sum_{m,n=0}^{N_{\\textrm{cut}}} c_{m}^{(j)} c_{n}^{(j)}\\sqrt{g_{mn}(\\overline{Y}^x_{mn},Y_{\\textrm{cut}})}\\]^2.\n\\end{align}\nIn Eq.~\\eqref{e1Cur}, the coefficient $c_{k}^{(0)}$ ($c_{k}^{(1)}$) is defined as $c_{k}^{(0)}=e^{-\\mu\/2} \\mu^{k\/2}\/\\sqrt{k!}$ when the integer $k$ is even (odd) and $0$ otherwise~\\cite{CAL18};\n$g_{mn}(\\overline{Y}^x_{mn},Y_{\\textrm{cut}})$ is a function equal to $\\overline{Y}^x_{mn}$ if $m+n1$. In the vertical direction, the\nupward component of the aerodynamic drag force $F_{d,\\perp}$ is\ncounterbalanced by the excess of the gravitational attraction over the\nair buoyancy force \n\\begin{equation}\n F_g = \\frac{1}{6} \\ \\pi \\ D_V^3 \\ (\\rho_{\\rm H_2O} - \\rho_{\\rm air})\n \\ g \\, ,\n\\end{equation}\nwhere $\\rho_{\\rm H_2O} \\simeq 997~{\\rm kg}\/{\\rm m}^3$ and $g \\simeq\n9.8~{\\rm m\/s}^2$ is the acceleration of gravity. Since $\\rho_{\\rm air}\n\\ll \\rho_{\\rm H_2O}$ the air buoyancy force\nbecomes negligible, and so $F_g \\approx M_V g$, with $M_V$ the aerosol\nmass. When the upward\naerodynamic drag force\nequals the gravitational attraction the droplet reaches mechanical\nequilibrium and starts falling with a terminal speed\n\\begin{equation}\nv_{V,f,\\perp} \\approx \\frac{M_V \\ g \\ \\varkappa}{3 \\pi \\ \\eta \\ D_V} \\, .\n\\end{equation}\nThe terminal speed is $\\propto D_V^2$ (due to the diameter dependence of the mass), and hence larger droplets\nwould have larger terminal velocities thereby reaching the ground\nfaster. The terminal speed for various particle sizes is given in\nTable~\\ref{tabla1}. The time $t_f$ it will take the virus to fall to the\nground is simply given by the distance to the ground divided by\n$v_{V,f,\\perp}$. For an initial height, $h \\sim 2~{\\rm m}$, we find\nthat for $D_V = 2~\\mu{\\rm m}$,\n\\be\nt_f = \\frac{h}{v_{V,f,\\perp}} \\sim 4~{\\rm hr} \\, .\n\\ee\nThe time scale as a function of the droplet size and heigth is shown in Fig.~\\ref{fig:1}.\n\nThe aerodynamic drag force holds for rigid spherical particles moving\nat constant velocity relative to the gas flow. To determine the\nstopping range, in the next section we model the elastic scattering of\nthe turbulent puff cloud with the air molecules.\n\n\n\\begin{table}\n\\caption{Cunningham slip correction factor and terminal speed. \\label{tabla1}}\n \\begin{tabular}{ccc}\n \\hline\n \\hline\n $D_V~(\\mu{\\rm m})$ & $\\varkappa$ & $v_{V,f,\\perp}~({\\rm m\/s})$ \\\\ \n\\hline\n $\\phantom{1}0.001$ & $215.3$ & $6.51 \\times 10^{-9}$ \\\\\n $\\phantom{1}0.010$ & $\\phantom{2}22.05$ & $6.67 \\times 10^{-8}$ \\\\\n $\\phantom{1}0.100$ & $\\phantom{22}2.851$ & $8.62 \\times 10^{-7}$ \\\\\n $\\phantom{1}0.500$ & $\\phantom{22}1.327$ & $1.00 \\times 10^{-5}$ \\\\\n $\\phantom{1}1.000$ & $\\phantom{22}1.163$ & $3.52 \\times 10^{-5}$ \\\\\n $\\phantom{1}1.500$ & $\\phantom{22}1.109$ & $7.54 \\times 10^{-5}$ \\\\\n $\\phantom{1}2.000$ & $\\phantom{22}1.081$ & $1.31 \\times 10^{-4}$ \\\\\n $\\phantom{1}3.000$ & $\\phantom{22}1.054$ & $2.87 \\times 10^{-4}$ \\\\\n $\\phantom{1}5.000$ & $\\phantom{22}1.033$ & $7.81 \\times 10^{-4}$ \\\\\n $\\phantom{1}7.000$ & $\\phantom{22}1.023$ & $1.52 \\times 10^{-3}$ \\\\\n ~~~~~~~~~$10.000$~~~~~~~~~ & ~~~~~~~~~$\\phantom{22}1.016$~~~~~~~~~ & ~~~~~~~~~$3.07 \\times 10^{-3}$~~~~~~~~~ \\\\\n \\hline\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}[tb]\n\\postscript{height-vs-diameter.pdf}{0.9}\n\\caption{Contours of the time $t_f$ in minutes in the\n $h-D_V$ plane. \\label{fig:1}}\n\\end{figure}\n\n \n\\section{Stopping Range}\n\\label{sec:3}\n\nRespiratory particles of saliva and mucus are expelled together\nwith a warm and humid air, which generates a convective current. The\naerosols and droplets are initially transported as part of a \ncoherent gas puff of buoyant fluid. The ejected puff of air remains\ncoherent in a volume that varies from $0.00025$ to $0.0025~{\\rm m}^3$~\\cite{Balachandar}.\nThis corresponds to a puff size $0.78 \\lesssim D_P\/{\\rm m} \\lesssim\n1.68$, where follwoing~\\cite{Balachandar} we have taken an entrainment\ncoefficient~\\cite{Morton} of $\\alpha = 0.1$. \nThe puff is ejected with $1 \\lesssim v_{V,0,\\parallel}\/{\\rm\n (m\/s)} \\lesssim 10$~\\cite{Balachandar}. \nThe turbulent puff cloud consists of an admixture of moist exhaled air and\nmucosalivary filaments. Next, in line with our stated plan, we use the\nexperimental data to calculate the range of the average\ndensity of the buoyant fluid in the turbulent cloud.\n\nThe mass ratio of the average air molecule compared to the aerosol, $m_{\\rm{air}}\/ M_V$, is roughly $10^{-12} $\n(since the size of the aerosol and the mass for its chief constituent,\nH$_2$O, compared to the air molecule are $10^4$ and $10^{3}$), though\nthere is an obvious variation with aerosol size at constant\ndensity. If we consider instead the mass inside the puff $M_P$ the ratio $R \\equiv\nm_{\\rm{air}}\/ M_{P}$ is even smaller. Due to the enormous mass ratio, the virions inside\nthe puff will not undergo large angular deflections, so we will treat the virions as having the same direction for its initial and final velocities (since we are looking at a stopping distance, this is a reasonable assumption). Starting with the non-relativistic one-dimensional equation for the virus velocity $\\beta$ we have in the lowest nontrivial order (in $R \\ll 1$) and any frame\n\\bea \n\\left(\n\\begin{array}{c}\n\\beta_1\\\\\nv_{{\\rm air},f}\n\\end{array}\n\\right)\n= \\mathbb{M} \\left(\\begin{array}{c} \\beta_0 \\\\ v_{{\\rm air},0} \\end{array} \\right) \\,,\n\\eea\nwhere the matrix $\\mathbb{M}$ is derived by imposing conservation of energy and momentum, and\nis given by\n\\bea\n\\mathbb{M} = \\left(\n\\begin{array}{cc}\n~~1-2R~~ & ~~2R~~ \\\\\n~~2~~ & ~~ -1~~\n\\end{array}\n\\right) \\,,\n\\eea\nwith $\\beta_0 = v_{V,0,\\parallel}$, and $v_{{\\rm air},0}$ and $v_{{\\rm air},f}$ the initial and final velocities of the air molecule, respectively. \nAs the velocity $ \\beta $ falls with each interaction, the velocity\nloss remains constant; the target particle is a new air molecule at\neach interaction.\n\n\nThough individual air molecules are traveling at an average speed of a few hundred meters per\nsecond, throughout we assume the medium to be stationary. In analogy\nwith the description of the slowing down of alpha particles in matter (which assumes the\nelectronic cloud is at rest), we can describe the\nscattering of the puff in the frame in which the air molecule is at rest, i.e.,\n$v_{{\\rm air},0} = 0$ (in essence, adopting a stationary medium on average). The stopping power is given by the velocity-loss equation\n\\bea\n d\\beta\/dx = \\Delta\\beta\/\\lambda^V_{\\rm mfp} =\n2R \\beta\/\\lambda^V_{\\rm mfp} \\,,\n\\eea\nwith solution \n$ \\ln\\beta= ( 2R \/ \\lambda^V_{\\rm mfp} ) \\int dx $. Finally, we have\nfor the stopping distance\n\\bea \nL=\\lambda^V_{\\rm mfp} \\ \\frac{1}{2R} \\ \\ln\\left(\\frac{\\beta_0}{\\beta_f} \\right) \\,,\n\\label{stopdis}\n\\eea\nwith $\\beta_f \\equiv v_{V,f,\\parallel}$. Note that $L\/\\lambda_{\\rm mfp}^V$ is not only the number of mean free paths traversed by the fiducial virus, but is also the number of interactions of the virus with air molecules; of course, there is a one-to-one correlation between the number of mean free paths traveled and interactions. \n\nSince $\\beta$ is homogeneous and the mass ratio $R$ is a constant for\na given puff size $D_P$, we\nhave the above simple equation. The mass ratio $R$ is very small, and\n$(2R)^{-1}$ is correspondingly very large. There are a tremendous number of mean free paths\/interactions involved as the virions bowling ball rolls over the air molecule.\n\nFinally, we must calculate \n$ \\lambda^V_{\\rm mfp} = 1\/ (n_{\\rm air} \\sigma) $. The air molecules\nact collectively as a fluid, so the volume $V$ over the air density is\ngiven by the ideal gas law as $ k_{\\rm B}T\/P $, where $P$ is the\npressure, $T$ the temperature, and $k_{\\rm B}$ is the Boltzmann\nconstant. We assume a contact interaction equal to the cross-sectional hard-sphere size of the puff, \ni.e. $ \\sigma = \\pi (D_P\/2)^2 $. Substituting into Eq.(\\ref{stopdis}) we\nobtain the final result for the stopping distance\n\\begin{equation}\nL= \\frac{k_{\\rm B}T}{P} \\ \\frac{1}{\\pi (D_P\/2)^2} \\frac{1}{2R} \\\n\\ln \\left(\\frac{\\beta_{0}}{\\beta_{f}} \\right) \\, .\n\\label{buga}\n\\end{equation}\nWe take the sneeze or cough which causes the droplets expulsion to be\nat a standard ambient air pressure of $P = 101$~kPa and\na temperature of $T \\sim 293~{\\rm K}$. It is important to stress that {\\it temperature variation could cause an $\\mathcal{O}(\\lesssim \\pm8 \\%)$ effect in $L$ for extreme ambient cold or warmth}. We now proceed to fit the experimental data. For $L \\sim 8~{\\rm m}$ and\ntaking $v_{V,f,\\parallel} \\sim 3~{\\rm mm\/s}$~\\cite{Bourouiba},\n we obtain $1.8 <\n\\rho_P\/\\rho_{\\rm air} < 4.0$ for $0.78 \\lesssim D_P\/{\\rm m} \\lesssim\n1.68$, \nwhere\n$\\rho_P$ is the average density of the fluid in the puff. \n\nA point worth noting at this juncture is that our model provides an\neffective description of the turbulent puff cloud. Note that\nindependently of their size and their initial velocity all respiratory\nparticles in the cloud experience both gravitational settling and\nevaporation. Aerosols and droplets of all sizes are subject to\ncontinuous settling, but those with settling speed smaller than the\nfluctuating velocity of the surrounding puff would remain trapped\nlonger within the puff. Actually, because of evaporation the water\ncontent of the respiratory particles is monotonically decreasing. At\nthe point of almost complete evaporation the settling velocity of the\naerosols is sufficiently small that they can remain trapped in the puff\nand get advected by ambient air currents and dispersed by ambient\nturbulence. The size of the puff then continuously grows in\ntime~\\cite{Balachandar}. Our result can equivalently be interpreted in\nterms of the effective coherence length of the turbulent cloud\nassuming $\\rho_P \\sim \\rho_{\\rm air}$. The effective size of the puff\nand its effective density are entangled in Eq.~(\\ref{buga}). Numerical\nsimulations show that during propagation\nthe puff edge grows $\\propto t^{1\/4}$~\\cite{Bourouiba}. After a 100~s the puff would grow by\na factor of 3 (see Fig.~7 in~\\cite{Balachandar}), in agreement with\nour analytical estimates. In\nclosing, we note that if we ignore the motion of the air puff carrying\nthe aerosols, as in the analysis of~\\cite{Wells}, it is\nstraightforward to see substituting $R$ by $m_{\\rm air}\/M_V \\sim\n10^{-12}$ into Eq.~(\\ref{buga}) that the individual aerosols would not travel\nmore than a few cm away from the exhaler, even under conditions of\nfast ejections, such as in a sneeze. This emphasizes the relevance of\nincorporating the complete multiphase flow physics in the modeling of\nrespiratory emissions when ascertaining the risk of SARS-CoV-2 airborne\ninfection.\n\n\n\n\n\\section{Conclusions}\n\\label{sec:4}\n \nWe have carried out a physics modeling study for SARS-CoV-2 transport\nin air. We have developed a nuclear physics analogy-based modeling of\nthe complex gas cloud and its payload of pathogen-virions. Using our\npuff model we estimated the average density of the fluid in the\nturbulent cloud is in the range $1.8 < \\rho_P\/\\rho_{\\rm air} < 4.0$. We have also shown that aerosols and droplets can remain suspended\nfor hours in the air. Therefore, once the puff slows down sufficiently,\nand its coherence is lost, the eventual spreading of the infected\naerosols becomes dependent on the ambient air currents and\nturbulence. De facto, as it was first pointed out in~\\cite{Anchordoqui} and later developed in~\\cite{Augenbraun,Evans} airflow\nconditions strongly influence the distribution of viral particles in\nindoor spaces, cultivating a health threat from COVID-19 airborne\ninfection. \n\n\nAltogether, it seems reasonable to adopt additional infection-control\nmeasures for airborne transmission in high-risk settings, such as the\nuse of face masks when in public. \nIf the results of this study - $t_f$ of ${\\cal O} ({\\rm hr})$ for aerosols, for example - are borne out by experiment, then these findings should be taken into account in policy decisions going forward as we continue to grapple with this pandemic.\n\n\n\n\\section*{Appendix}\n\nThere are important considerations in the development of Stokes' law,\nincluding the hypothesis that the gas at particle\nsurface has zero velocity relative to the particle. This hypothesis\nholds well when the diameter of the particle is much larger than the mean\nfree path of gas molecules. The mean free path $\\lambda_{\\rm mfp}^{\\rm\n air}$ is the average distance\ntraveled by a gas molecule between two successive collisions. In\nanalyses of the interaction between gas molecules and particles, it is\nconvenient to use the Knudsen number ${\\rm Kn} = 2 \\lambda_{\\rm mfp}^{\\rm\n air}\/D_V$, a dimensionless number\ndefined as the ratio of the mean free path to particle radius. For ${\\rm Kn}\n\\agt 1$, the drag force is smaller than\npredicted by Stokes' law. Conventionally this condition is described as\na result of slip on the particle surface. The so-called slip\ncorrection is estimated to be~\\cite{Crowder}\n\\begin{equation}\n\\varkappa = 1 + {\\rm Kn} \\left[1.257 + 0.4 \\ \\exp(-1.1\/{\\rm Kn}) \\right] \\, .\n\\end{equation}\nIn our calculations we take \n\\bea\n\\lambda_{\\rm mfp}^{\\rm air} = \\frac{\\eta_{\\rm air}}{\\rho_{\\rm\n air}}\\left(\\frac{\\pi m_{\\rm air}}{2 \\, k_{\\rm B} \\, T}\\right)^{1\/2} \\,,\n\\eea\nwhere $k_{\\rm B}$ is the Boltzmann constant, $T$ is the temperature in\nKelvin, and the density of air is given by\n\\bea\n\\rho_{\\rm air} = \\frac{P}{R_gT} \\,,\n\\eea\nwith $P = 101$ kPa, and where $R_g = 287.058$~J\/(kg~$\\cdot$~K) is the ideal gas constant. The molar mass of air is\n$m_{\\rm mol} = 29\\; {\\rm g}\/{\\rm mol}$, which leads to $m_{\\rm air} = 4.8\\times10^{-26}\\;{\\rm kg}\/{\\rm molecule}$.\\\\\n\n\n\\noindent{\\bf Funding\/Support:} The research of L.A.A. is supported by the U.S. National Science\nFoundation (NSF Grant PHY-1620661). J.B.D. acknowledges support from\nthe National Science Foundation under Grant No. NSF PHY182080. The\nwork of T.J.W. was supported in part by the U.S. Department of Energy\n(DoE grant No. DE-SC0011981).\n\n\\noindent {\\bf Role of the Funder\/Sponsor:} The sponsors had no role in\nthe preparation, review or approval of the manuscript and decision to\nsubmit the manuscript for publication. Any opinions,\nfindings, and conclusions or recommendations expressed in this\narticle are those of the authors and do not necessarily reflect the\nviews of the NSF or DOE.\n\n\\noindent{\\bf Declaration of Competing Interest:} The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\\noindent{\\bf Ethical Approval:} The manuscript does not contain experiments on animals and humans; hence ethical permission not required.\n\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nMostow rigidity implies that for hyperbolic $3$-manifolds, the\nhyperbolic metric is a topological invariant, so one might hope that\nthe topological and metric complexities are related. We shall show\nthat this is indeed the case for certain definitions of topological\nand metric complexity. We first describe the notions of complexity we\nshall use, and then give a brief outline of the arguments used to\nrelate topological and metric complexity in the subsequent sections.\nIn \\cite{hoffoss-maher} we considered the linear version of these\ninvariants, while in this paper we consider the more general case of\ninvariants constructed from maps to graphs. It will be convenient to\nwork with the collection of hyperbolic $3$-manifolds which are\ncomplete, but do not contain cusps, and are not necessarily of finite\nvolume. The reason for not considering manifolds with cusps is that\nin the cusped case the surfaces separating the $3$-dimensional regions\nin the topological decomposition we construct from the metric might\nhave essential intersection with the cusps. In other words, if the\ncusps are truncated to form a hyperbolic manifold with torus boundary\ncomponents, then the dividing surfaces may be surfaces with essential\nboundary components on the boundary tori. However, the currently\navailable versions of the topological decomposition results we use,\ndue to Scharlemann-Thompson and Saito-Scharlemann-Schultens, assume\nthat the dividing surfaces are closed.\n\nThis paper is not entirely self-contained, and relies on the results\nof \\cite{hoffoss-maher}, however we review the main definitions and\nresults from \\cite{hoffoss-maher} for the convenience of the reader.\n\n\n\\subsection{Metric complexity}\n\n\n\nIn \\cite{hoffoss-maher}, we considered the following definition of\nmetric complexity. Let $M$ be a closed Riemannian $3$-manifold, and\nlet $f \\colon M \\to \\mathbb{R}$ be a Morse function, i.e. $f$ is a smooth\nfunction, all critical points are non-degenerate, and distinct\ncritical points have distinct images in $\\mathbb{R}$. We define the\n\\emph{area} of $f$ to the maximum area of any level set\n$F_t = f^{-1}(t)$ over all points $t \\in \\mathbb{R}$. We define the\n\\emph{Morse area} of $M$ to be the infimum of the area of all Morse\nfunctions $f \\colon M \\to \\mathbb{R}$.\n\nMore generally, we may consider maps $f \\colon M \\to X$, where $X$ is\na trivalent graph. Recall that for a Morse function $f \\colon M \\to\n\\mathbb{R}$ there are singularities of index $0, 1, 2$ and $3$. The\nsingularities of index $0$ and $3$ are known as birth or death\nsingularities respectively, and the level set foliation near the\nsingular point in $M$ is locally homeomorphic to the level sets of the\nfunction $x^2 + y^2 + z^2$ close to the origin in $\\mathbb{R}^3$. For\nsingularities of index $2$ and $3$, the level sets near the singular\npoint in $M$ are locally homeomorphic to the level sets of the\nfunction $x^2 + y^2 - z^2$ close to the origin in $\\mathbb{R}^3$. \n\nIn the case of index $2$ or $3$, there is a map from a small open ball\ncontaining the singular point to the leaf space of the level set\nfoliation. As the singular leaf divides a small ball about the\nsingular point into three connected components, the leaf space is a\ntrivalent graph with a single vertex and three edges, and we call such\na map a \\emph{trivalent singularity}. If $X$ is a trivalent tree, we\nsay a map $f \\colon M \\to X$ is \\emph{Morse} if it is a Morse function\non the interior of each edge of $X$, and at each trivalent vertex $v$\nof $X$ the pre-image under $f$ is locally homeomorphic to a trivalent\nsingularity. We say that the area of $f$ is the maximum area of\n$F_t$, as $t$ runs over all points $t \\in X$. The \\emph{Gromov area}\nof $M$ is the infimum of the area of $f \\colon M \\to X$ over all\ntrivalent graphs $X$, and all Morse functions $f \\colon M \\to X$.\n\n\nThis definition of metric complexity is a variant of Uryson width,\nstudied by Gromov in \\cite{gromov}, though we consider the area of the\nlevel sets instead of the diameter. Alternatively, one may consider\nit to be a variant of the definition of the waist of a manifold, but\nwe prefer to call it area, as the dimension of our spaces is fixed,\nand the fibers have dimension two.\n\n\n\n\\subsection{Topological complexity}\n\nWe now describe the notions of topological complexity we shall\nconsider. A \\emph{handlebody} is a compact $3$-manifold with boundary,\nhomeomorphic to a regular neighborhood of a graph in\n$\\mathbb{R}^3$. Up to homeomorphism, a handlebody is determined by the\ngenus $g$ of its boundary surface. Every $3$-manifold $M$ has a\n\\emph{Heegaard splitting}, which is a decomposition of the manifold\ninto two handlebodies. This immediately gives a notion of complexity\nfor a $3$-manifold, called the \\emph{Heegaard genus}, which is the\nsmallest genus of any Heegaard splitting of the $3$-manifold.\n\nThere is a refinement of this, due to Scharlemann and Thompson\n\\cite{st}, which we now describe. A \\emph{compression body} $C$ is a\ncompact $3$-manifold with boundary, constructed by gluing some number\nof $2$-handles to one side of a compact (but not necessarily\nconnected) surface cross interval and capping off any resulting\n$2$-sphere components with $3$-balls. The side of the surface cross\ninterval with no attached $2$-handles is called the \\emph{top\n boundary} of the compression body and denoted by $\\partial_+ C$, and\nany other boundary components are called the \\emph{lower boundary} of\nthe compression body, and denoted by $\\partial_- C$. A \\emph{linear\n generalized Heegaard splitting},\\footnote{We warn the reader that\n these are often referred to as generalized Heegaard splittings in\n the literature; however we wish to distinguish them from a more\n general notion described subsequently, which is also occasionally\n referred to in the literature as a generalized Heegaard splitting.}\nwhich we shall abbreviate to \\emph{linear splitting}, is a\ndecomposition of a $3$-manifold $M$ into a linearly ordered sequence\nof (not necessarily connected) compression bodies\n$C_1, \\ldots C_{2n}$, such that the top boundary of an odd numbered\ncompression body $C_{2i+1}$ is equal to the top boundary of the\nsubsequent compression body $C_{2i+2}$, and the lower boundary of\n$C_{2i+1}$ is equal to the lower boundary of the previous compression\nbody $C_{2i}$. Let $H_i$ be the sequence of surfaces consisting of\nthe top boundaries of the compression bodies $C_{2i-1}$ and\n$C_{2i}$. The complexity $c(H_i)$ of the surface $H_i$ is the sum of\nthe genera of each connected component, and the complexity of the\nlinear splitting is the collection of integers $\\{c(H_i)\\}$, arranged\nin decreasing order. We order these complexities with the\nlexicographic ordering. The \\emph{width} of the linear splitting is\nthe maximum value (i.e. the first value) of $c(H_i)$ in the collection\n$\\{c(H_i)\\}$. The \\emph{linear width} of a $3$-manifold $M$ is the\nminimum width over all possible linear generalized Heegaard\nsplittings. As a Heegaard splitting is a special case of a linear\nsplitting, the Heegaard genus of $M$ is an upper bound for the linear\nwidth of $M$. A linear splitting which gives the minimum complexity\nof all possible linear splittings is called the \\emph{thin position}\nlinear splitting.\n\n\nThere is a further refinement of this, described in Saito, Scharlemann\nand Schultens \\cite{sss}. A \\emph{graph generalized Heegaard\n splitting}, which we shall abbreviate to \\emph{graph splitting}, and\nis called a \\emph{fork complex} in \\cite{sss}, is a decomposition of a\ncompact $3$-manifold $M$ into compression bodies $\\{ C_i \\}$, such\nthat for each compression body $C_i$, there is a compression body\n$C_j$ such that the top boundary of $C_i$ is equal to the top boundary\nof $C_j$. Furthermore, for each component of the lower boundary of\n$C_i$, there is a compression body $C_k$, such that that component of\nthe lower boundary of $C_i$ is equal to a component of the lower\nboundary of $C_k$. We emphasize that different components of the lower\nboundary of $C_i$ may be attached to lower boundary components of\ndifferent compression bodies. Let $\\{ H_i \\}$ be the collection of top\nboundary surfaces. The complexity of the graph splitting is the\ncollection of integers $\\{c(H_i)\\}$, arranged in decreasing\norder. Again, we put the lexicographic ordering on these\ncomplexities. A graph splitting which realizes the minimum complexity\nis called a \\emph{thin position graph splitting}. The \\emph{width} of\nthe graph splitting is the maximum integer (i.e the first integer)\nthat appears in the complexity. The \\emph{graph width} of a\n$3$-manifold $M$ is the minimum width over all possible graph\nsplittings of $M$. As a linear splitting is a special case of a graph\nsplitting, the linear width of $M$ is an upper bound for the graph\nwidth of $M$. The graph corresponding to the graph splitting is the\ngraph whose vertices are compression bodies, with edges connecting\npairs of compression bodies with common boundary components.\n\n\n\n\n\n\\subsection{Results}\n\n\nIn order to bound metric complexity in terms of topological complexity\nwe shall assume the following result announced by Pitts and Rubinstein\n\\cite{pr} (see also Rubinstein \\cite{rubinstein}).\n\n\\begin{theorem} \\cites{pr,rubinstein} \\label{conjecture:pr} %\nLet $M$ be a Riemannian $3$-manifold with a strongly irreducible\nHeegaard splitting. Then the Heegaard surface is isotopic to a\nminimal surface, or to the boundary of a regular neighborhood of a\nnon-orientable minimal surface with a small tube attached vertically\nin the I-bundle structure.\n\\end{theorem}\n\nA full proof of this result has not yet appeared in the literature,\nthough recent progress has been made by Colding and De Lellis\n\\cite{cdl}, De Lellis and Pellandrini \\cite{dlp}, and Ketover\n\\cite{ketover}.\n\nIn \\cite{hoffoss-maher} we showed:\n\n\\begin{theorem} \\label{theorem:linear} %\nThere is a constant $K > 0$, such that for any closed hyperbolic\n$3$-manifold,\n\\begin{equation} \nK ( \\text{linear width}(M) ) \\leqslant \\text{Morse area}(M) \\leqslant 4 \\pi\n(\\text{linear width}(M)), \n\\end{equation}\nwhere the right hand bounds hold assuming Theorem\n\\ref{conjecture:pr}.\n\\end{theorem}\n\nIn this paper we show:\n\n\\begin{theorem} \\label{theorem:graph} %\nThere is a constant $K > 0$, such that for any closed hyperbolic\n$3$-manifold,\n\\begin{equation}\n K ( \\text{graph width}(M) ) \\leqslant \\text{Gromov area}(M) \\leqslant 4 \\pi\n(\\text{graph width}(M)), \n\\end{equation}\nwhere the right hand bounds hold assuming Theorem\n\\ref{conjecture:pr}.\n\\end{theorem}\n\nWe also expect there to be upper and lower bounds on topological\ncomplexity in terms of Uryson width, i.e. using diameter instead of\narea, but we do not expect them to be linear. \n\n\n\n\\subsection{Related work in $3$-manifolds}\n\n\n\n\n\nIt may be of interest to compare our results with recent work of\nBrock, Minsky, Namazi and Souto \\cite{bmns} on manifolds with bounded\ncombinatorics. Let $C_1, \\ldots C_n$ be a finite collection of\nhomeomorphism types of compact $3$-manifolds with marked boundary,\nwhich we shall refer to as \\emph{model pieces}, and fix a metric on\neach one. A $3$-manifold $M$ is said to have \\emph{bounded\n combinatorics} if it is a union of (possibly infinitely many) model\npieces glued together by homeomorphisms along their boundaries, with\ncertain restrictions on the gluing maps, which we do not describe in\ndetail here. In particular, a manifold with bounded combinatorics is a\nmanifold of bounded topological width. They show that such a manifold\n$M$ is hyperbolic, with a lower bound on the injectivity radius, and\nthe hyperbolic metric is $K$-bilipshitz homeomorphic to the induced\nmetric on $M$ arising from the metrics on the model pieces. A choice\nof foliation with compact leaves, containing the boundary leaves, on\neach model piece then shows that the metric complexity is linearly\nrelated to the topological complexity for this class of manifolds,\nwhere the linear constants depend on the collection of model pieces.\n\nNote that in our context, a bound on the topological width of the\nmanifold implies that the manifold is a union of compression bodies of\nbounded genus, and there are finitely many of these up to\nhomeomorphism. Their result assumes restrictions on the gluing maps,\nbut then shows the resulting manifold is hyperbolic, but the\nbilipshitz constant $K$ depends on the width of $M$, i.e the genus of\nthe compression bodies. We assume that the manifold $M$ is compact and\nhyperbolic, and make no restriction on the gluing maps between the\ncompression bodies, but we show that the linear constants relating\ntopological and metric complexities are independent of the genus of\nthe compression bodies.\n\n\n\n\\subsection{Outline}\n\nIn \\cite{hoffoss-maher} we considered the linear case, in which the\nrange of the Morse function $f \\colon M \\to \\mathbb{R}$ is $\\mathbb{R}$. Such a Morse\nfunction has the property that for each $t \\in \\mathbb{R}$, the pre-image\n$f^{-1}(t)$ is compact and separating. For the case in which the\nrange of the Morse function $f \\colon M \\to X$ is a graph, one may\nconsider the lifted Morse function\n$\\widetilde f \\colon \\widetilde M \\to \\widetilde X$, where\n$\\widetilde X$ is the universal cover of $X$, and $\\widetilde M$ is\nthe corresponding cover of $M$. This lifted Morse function has the\nproperty that for each $t \\in \\widetilde X$, each pre-image\n$\\widetilde f^{-1}(x)$ is compact and separating, and so many of the\narguments from \\cite{hoffoss-maher} go through directly in this case.\nIn particular, we construct polyhedral approximations to the level\nsets of $\\widetilde f$, and show that they have bounded topological\ncomplexity, as we now describe.\n\nA choice of Margulis constant $\\mu$ determines a thick-thin\ndecomposition for $M$, in which the thin part is a disjoint union of\nMargulis tubes. We also choose a Voronoi decomposition determined by\na maximal $\\epsilon$-separated collection of points in $M$. This implies\nthat every Voronoi cell has diameter at most $\\epsilon$, and, given $\\mu$,\nwe may choose $\\epsilon$ small enough such that every Voronoi cell that\nintersects the thick part contains an embedded ball of radius $\\epsilon\/2$.\nThe thick-thin decomposition of $M$, and the Voronoi decomposition of\n$M$, lift to thick-thin decompositions and Voronoi decompositions of\nthe cover $\\widetilde M$. We give the details of this construction in\nSections \\ref{section:tree}, \\ref{section:voronoi} and\n\\ref{section:thickthin}.\n\nA separating surface $F$ in $\\widetilde M$ determines a partition of\nthe Voronoi cells, depending on which side of the surface the majority\nof the volume of the (metric) ball of radius $\\epsilon\/2$ inside the Voronoi\ncell lies. We will call the boundary between these two sets of\nVoronoi cells a \\emph{polyhedral surface} $S$, which is a union of\nfaces of Voronoi cells, and we can think of this as a combinatorial\napproximation to the original surface $F$.\n\nA key observation from \\cite{hoffoss-maher} is that the number of\nfaces of the polyhedral surface in the thick part is bounded by the\narea of $F$. This is because in the thick part of $M$, the metric\nball of radius $\\epsilon\/2$ in each Voronoi cell is embedded, so moving the\nball along a geodesic connecting the centers of the two Voronoi\nproduces at some point a metric ball whose volume is divided exactly\nin two, giving a lower bound to the area of $F$ near that point.\nThere are bounds on the number of vertices and edges of any Voronoi\ncell in terms of $\\epsilon$, so a bound on the number of faces of $S$ in the\nthick part gives a bound on the Euler characteristic of $S$. We are\nunable to control the number of faces in the thin part, so we cap off\nthe part of $S$ in the thick part with surfaces of bounded Euler\ncharacteristic contained in the thin part. This produces surfaces of\nbounded genus, which we call \\emph{capped surfaces}.\n\nIn this way, the lift of a Morse function\n$\\widetilde f \\colon \\widetilde M \\to \\widetilde X$ gives rise to a\ncollection of polyhedral surfaces in $\\widetilde M$ of bounded genus.\nThese surfaces are constant except at finitely many points of\n$\\widetilde X$, which we call \\emph{cell splitters}, where a level set\ndivides the ball contained in a Voronoi cell exactly in half. We give\nthe details of the construction of the capped surfaces and the\nproperties of the cell splitters in Sections \\ref{section:splitter}\nand \\ref{section:capped}.\n\nThe key step, in Section \\ref{section:equivariant}, is to show that we\nmay construct these surfaces equivariantly in $\\widetilde M$, so they\nproject down to embedded surfaces in $M$, with the same bounds on\ntheir topological complexity.\n\nFinally, in Section \\ref{section:bounded}, by considering the local\nconfiguration near a cell splitter, we show that the regions between\nthe capped surfaces may be constructed using a number of handles\nbounded in terms of the area of the level sets $\\widetilde f^{-1}(t)$,\nand so this the bounds topological complexity of the decomposition of\n$M$ given by the capped surfaces in terms of metric complexity of $M$.\n\nThe bound in the other direction is a direct consequence of the bound\nfrom \\cite{hoffoss-maher}, though we review the argument in the\nSection \\ref{section:sss} for the convenience of the reader.\n\n\n\n\n\n\\subsection{Acknowledgements}\n\nThe authors would like to thank Dick Canary, David Futer, David Gabai,\nJoel Hass, Daniel Ketover, Sadayoshi Kojima, Yair Minksy, Yo'av Rieck\nand Dylan Thurston for helpful conversations, and the Tokyo Institute\nof Technology for its hospitality and support. The second author was\nsupported by the Simons Foundation and PSC-CUNY. This material is\nbased upon work supported by the National Science Foundation under\nGrant No.~DMS-1440140 while the second author was in residence at the\nMathematical Sciences Research Institute in Berkeley, California,\nduring the Fall 2016 semester.\n\n\n\n\\section{Gromov area bounds graph width} \n\\label{section:metric bounds topology}\n\nIn this section we show that we can bound the topological complexity of\nthe manifold in terms of its metric complexity, i.e. we show that\ngraph width is bounded in terms of Gromov area.\n\n\\begin{theorem}\nThere is a constant $K$, such that for any closed hyperbolic\n$3$-manifold $M$,\n\\[ \\text{graph width}(M) \\leqslant K ( \\text{Gromov area}(M) ). \\]\n\\end{theorem}\n\nLet $f \\colon M \\to X$ be a Morse function onto a graph $X$, such that\nthe Gromov area of $f$ is arbitrarily close to the Gromov area of $M$.\nAny metric graph is arbitrarily close to a trivalent metric graph, so\nwe may assume the graph is trivalent. We now show that we may assume\nthe level sets of $f$ are connected.\n\n\\begin{proposition}\nLet $M$ be a Riemannian manifold, and let $f \\colon M \\to X$ be a\nMorse function onto a trivalent graph $X$. Then there is a trivalent\ngraph $X'$, and a Morse function $f' \\colon M \\to X'$ with connected\nlevel sets, with $\\text{Gromov area}(f') \\leqslant \\text{Gromov area}(f)$.\n\\end{proposition}\n\n\\begin{proof}\nThe level sets of the function $f$ give a singular foliation of $M$\nwith compact leaves, which we shall call the \\emph{level set\n foliation}, and the leaves of this foliation are precisely the\nconnected components of the pre-images of points in $M$. Consider the\nleaf space $L$ of the level set foliation, i.e. the space obtained\nfrom $M$ by identifying points in the same leaf. As all leaves are\ncompact, the leaf space is Hausdorff. The leaf space is a trivalent\ngraph, with vertices corresponding to vertex singularities, and the\nmaximum area of the pre-images of the quotient map is less than or\nequal to the maximum area of the pre-images of $f$. Therefore, we may\nchoose $f'$ to be the leaf space quotient map $f' \\colon M \\to L$,\nwhich is a Morse function onto a trivalent graph, and has connected\nlevel sets, with the property that the area of the level sets of $f'$\nis bounded by the area of the level sets of $f$.\n\\end{proof}\n\nIn particular, this means that the vertices of $X$ are precisely the\ncritical points of the Morse function $f$ in which a connected level\nset splits into two connected components.\n\n\n\n\\subsection{Morse functions to trees} \n\\label{section:tree}\n\nWe would like to work in the cover $\\widetilde M$ of $M$ corresponding\nto the universal cover $\\widetilde X$ of the graph $X$, which will\nhave the key advantage that all pre-image surfaces are separating in\n$\\widetilde M$. In fact, the induced map on fundamental groups $f_*\n\\colon \\pi_1 M \\to \\pi_1 X$ is surjective, but as we do not use this\nproperty, we omit the proof.\n\nLet $p \\colon \\twid{M} \\rightarrow M$ be the cover of $M$\ncorresponding to the kernel of the induced map $f_* \\colon \\pi_1 M\n\\rightarrow \\pi_1 X $, and let $c:\\twid{X}\\rightarrow X$ be the\nuniversal cover of $X$, so $\\twid{X}$ is a tree. Then the map $f \\circ\np \\colon \\twid{M} \\rightarrow X$ lifts to a map $h = \\twid{f \\circ p}\n: \\twid{M} \\rightarrow \\twid{X}$. Since each leaf $F_t$ in $M$ maps\nto a single point in $X$, the fundamental group of each leaf is\ncontained in $\\ker(f)$. Therefore, each leaf in $M$ lifts to a leaf\nin $\\twid{M}$, and as the cover is regular, the pre-image of a point\n$t \\in \\widetilde{X}$ is a disjoint union of homeomorphic copies of\n$F_{c(t)}$. In particular, the area bound for the leaves $F_t$ in\n$M$ is also an area bound for the leaves $H_t = h^{-1}(t)$ in\n$\\twid{M}$.\n\n\\begin{center}\n\\begin{tikzpicture}[node distance=2cm, auto]\n \\node (A) {$\\twid{M}$};\n \\node (B) [right of=A] {$\\twid{X}$};\n \\node (C) [below of=A] {$M$};\n \\node (D) [below of=B] {$X$};\n \\draw[->] (A) to node {$h = \\twid{f \\circ p}$} (B);\n \\draw[->] (A) to node {$p$} (C);\n \\draw[->] (B) to node {$c$} (D);\n \\draw[->] (C) to node {$f$} (D);\n\\end{tikzpicture}\n\\end{center}\n\nAs $\\widetilde{X}$ is a tree, every point is separating, and so every pre-image\nsurface $H_t = h^{-1}(t)$ is also separating. \n\n\n\n\n\n\\subsection{Voronoi cells} \\label{section:voronoi}\n\nWe will approximate the level sets of $f$ by surfaces consisting of\nfaces of Voronoi cells. We now describe in detail the Voronoi cell\ndecompositions we shall use, and their properties. The definitions in\nthis section are taken verbatim from \\cite{hoffoss-maher}, but we\ninclude them in this section for the convenience of the reader.\n\nA \\emph{polygon} in $\\mathbb{H}^3$ is a bounded subset of a hyperbolic plane\nwhose boundary consists of a finite number of geodesic segments. A\n\\emph{polyhedron} in $\\mathbb{H}^3$ is a convex topological $3$-ball in $\\mathbb{H}^3$\nwhose boundary consists of a finite collection of polygons. A\n\\emph{polyhedral cell decomposition} of $\\mathbb{H}^3$ is a cell decomposition\nin which which every $3$-cell is a polyhedron, each $2$-cell is a\npolygon, and the edges are all geodesic segments. We say a cell\ndecomposition of a complete hyperbolic manifold $M$ is\n\\emph{polyhedral} if its pre-image in the universal cover $\\mathbb{H}^3$ is\npolyhedral.\n\nLet $X = \\{ x_i \\}$ be a discrete collection of points in\n$3$-dimensional hyperbolic space $\\mathbb{H}^3$. The Voronoi cell $V_i$\ndetermined by $x_i \\in X$ consists of all points of $M$ which are\ncloser to $x_i$ than any other $x_j \\in X$, i.e.\n\\[ V_i = \\{ x \\in \\mathbb{H}^3 \\mid d(x, x_i) \\leqslant d(x, x_j) \\text{ for all }\nx_j \\in \\widetilde{X} \\}. \\]\nWe shall call $x_i$ the \\emph{center} of the Voronoi cell $V_i$, and\nwe shall write ${\\cal{V}} = \\{ V_i \\}$ for the collection of Voronoi cells\ndetermined by $X$. Voronoi cells are convex sets in $\\mathbb{H}^3$, and hence\ntopological balls. The set of points equidistant from both $x_i$ and\n$x_j$ is a totally geodesic hyperbolic plane in $\\mathbb{H}^3$. A \\emph{face}\n$\\Phi$ of the Voronoi decomposition consists of all points which lie\nin two distinct Voronoi cells $V_i$ and $V_j$, so $\\Phi$ is contained\nin a geodesic plane. An \\emph{edge} $e$ of the Voronoi decomposition\nconsists of all points which lie in three distinct Voronoi cells\n$V_i, V_j$ and $V_k$, which is a geodesic segment, and a \\emph{vertex}\n$v$ is a point lying in four distinct Voronoi cells $V_i, V_j, V_k$\nand $V_l$. By general position, we may assume that all edges of the\nVoronoi decomposition are contained in exactly three distinct faces,\nthe collection of vertices is a discrete set, and there are no points\nwhich lie in more than four distinct Voronoi cells. We shall call such\na Voronoi decomposition a \\emph{regular} Voronoi decomposition, and it\nis a polyhedral decomposition of $\\mathbb{H}^3$. As each edge is $3$-valent,\nand each vertex is $4$-valent, this implies that the dual cell\nstructure is a simplicial triangulation of $\\mathbb{H}^3$, which we shall\nrefer to as the \\emph{dual triangulation}. The dual triangulation may\nbe realised in $\\mathbb{H}^3$ by choosing the vertices to be the centers $x_i$\nof the Voronoi cells and the edges to be geodesic segments connecting\nthe vertices, and we shall always assume that we have done this. In\nthis case the triangles and tetrahedra are geodesic triangles and\ntetrahedra in $\\mathbb{H}^3$.\n\nGiven a collection of points $X = \\{ x_i \\}$ in a hyperbolic\n$3$-manifold $M$, let $\\widetilde{X}$ be the pre-image of $X$ in the universal\ncover of $M$, which is isometric to $\\mathbb{H}^3$. As $\\widetilde{X}$ is equivariant,\nthe corresponding Voronoi cell decomposition ${\\cal{V}}$ of $\\mathbb{H}^3$ is also\nequivariant. The distance condition implies that the interior of each\nVoronoi cell $V$ is mapped down homeomorphically by the covering\nprojection, though the covering projection may identify faces, edges\nor vertices of $V_i$ under projection into $M$. By abuse of notation,\nwe shall refer to the resulting polyhedral decomposition of $M$ as the\nVoronoi decomposition ${\\cal{V}}$ of $M$. By general position, we may assume\nthat ${\\cal{V}}$ is regular. The dual triangulation is also equivariant, and\nprojects down to a triangulation of $M$, which we will also refer to\nas the dual triangulation, though this triangulation may no longer be\nsimplicial.\n\nWe shall write $B(x, r)$ for the closed metric ball of radius $r$\nabout $x$ in $M$, i.e.\n\\[ B(x, r) = \\{ y \\in M \\mid d(x, y) \\leqslant r \\}. \\]\nA metric ball in $M$ need not be a topological ball in general. We\nshall write $\\text{inj}_M(x)$ for the injectivity radius of $M$ at $x$,\ni.e. the radius of the largest embedded ball in $M$ centered at $x$.\nThen the injectivity radius of $M$, denoted $\\text{inj}(M)$, is defined to\nbe\n\\[ \\text{inj}(M) = \\inf_{x \\in M} \\text{inj}_M(x). \\]\n\nWe say a collection $\\{ x_i \\}$ of points in $M$ is\n\\emph{$\\epsilon$-separated} if the distance between any pair of points\nis at least $\\epsilon$, i.e. $d(x_i, x_j) \\geqslant \\epsilon$, for all $i\n\\not = j$. Let $\\{ x_i \\}$ be a maximal collection of\n$\\epsilon$-separated points in $M$, and let ${\\cal{V}}$ be the corresponding\nVoronoi cell division of $M$. Since the collection $\\{ x_i \\}$ is\nmaximal, each Voronoi cell is contained in a metric ball of radius\n$\\epsilon$ about its center. Furthermore, if the injectivity radius at the\ncenter $x_i$ is at least $2\\epsilon$, then as the points $x_i$ are distance\nat least $\\epsilon$ apart, each Voronoi cell contains a topological ball of\nradius $\\epsilon\/2$ about its center, i.e.\n\\[ B(x_i, \\epsilon\/2 ) \\subset V_i \\subset B(x_i, \\epsilon). \\]\n\n\\begin{definition}\nLet $M$ be a complete hyperbolic $3$-manifold. We say a Voronoi\ndecomposition ${\\cal{V}}$ is $\\epsilon$-regular, if it is regular, and it arises\nfrom a maximal collection of $\\epsilon$-separated points.\n\\end{definition}\n\nA \\emph{simple arc} in the boundary of a tetrahedron is a properly\nembedded arc in a face of the tetrahedron with endpoints in distinct\nedges. A \\emph{triangle} in a tetrahedron is a properly embedded disc\nwhose boundary is a union of three simple arcs, and a\n\\emph{quadrilateral} is a properly embedded disc whose boundary is the\nunion of four simple arcs. A \\emph{normal surface} in a triangulated\n$3$-manifold is a surface that intersects each tetrahedron in a union\nof normal triangles and quadrilaterals.\n\nOne useful property of $\\epsilon$-regular Voronoi decompositions is that the\nboundary of any union of Voronoi cells is an embedded surface, in fact\nan embedded normal surface in the dual triangulation.\n\n\\begin{proposition} \\cite{hoffoss-maher}*{Proposition 2.2} %\nLet $M$ be a complete hyperbolic manifold without cusps, and let ${\\cal{V}}$\nbe an $\\epsilon$-regular Voronoi decomposition. Let $P$ be a union of\nVoronoi cells in ${\\cal{V}}$, and let $S$ be the boundary of $P$. Then $S$ is\nan embedded surface in $M$.\n\\end{proposition}\n\nIn \\cite{hoffoss-maher} this result is stated for compact hyperbolic\n$3$-manifolds, but the proof works for complete hyperbolic\n$3$-manifolds without cusps.\n\nWe shall say a Voronoi cell $V_i$ with center $x_i$ is an\n\\emph{$\\epsilon$-deep} Voronoi cell if the injectivity radius at $x_i$ is at\nleast $4\\epsilon$, i.e. $\\text{inj}_M(x_i) \\geqslant 4\\epsilon$, and in particular this\nimplies that the metric ball $B(x_i, 3\\epsilon)$ is a topological ball. We\nshall also call centers, faces, edges and vertices of $\\epsilon$-deep\nVoronoi cells $\\epsilon$-deep.\nIn the next section we will choose a fixed $\\epsilon$ independent of the\nmanifold $M$, and we will just say \\emph{deep} instead of\n$\\epsilon$-deep. We shall write $\\mathcal{W}$ for the subset of ${\\cal{V}}$ consisting of\ndeep Voronoi cells. If $\\epsilon < \\tfrac{1}{4}\\text{inj}(M)$, then ${\\cal{V}} = \\mathcal{W}$ and\nall Voronoi cells are deep.\n\n\n\nFinally, we recall that there are bounds, which only depend on\n$\\epsilon$, on the number of vertices, edges and faces of a deep\nVoronoi cell.\n\n\\begin{proposition} \\cite{hoffoss-maher}*{Proposition 2.3}\\label{prop:bound} %\nLet $M$ be a complete hyperbolic $3$-manifold with an $\\epsilon$-regular\nVoronoi decomposition ${\\cal{V}}$, and let $\\mathcal{W}$ be the collection of deep\nVoronoi cells. Then there is a number $J$, which only depends on\n$\\epsilon$, such that each deep Voronoi cell $W_i \\in \\mathcal{W}$ has at most $J$\nfaces, edges and vertices.\n\\end{proposition}\n\nAgain, in \\cite{hoffoss-maher}, these results are stated for compact\nhyperbolic $3$-manifolds, but the proofs work for complete hyperbolic\n$3$-manifolds without cusps.\n\n\n\n\n\n\n\\subsection{Margulis tubes} \\label{section:thickthin}\n\n\n\n\n\nWe will use the Margulis Lemma and the \\emph{thick-thin} decomposition\nfor finite volume hyperbolic $3$-manifolds, and we now review these\nresults.\n\nGiven a number $\\mu > 0$, let $X_\\mu = M_{[\\mu, \\infty)}$ be the\n\\emph{thick part} of $M$, i.e. the union of all points $x$ of $M$ with\n$\\text{inj}_M(x) \\geqslant \\mu$. We shall refer to the closure of the complement\nof the thick part as the \\emph{thin part} and denote it by $T_\\mu =\n\\overline{M \\setminus X}$.\n\nThe Margulis Lemma states that there is a constant $\\mu_0 > 0$, such\nthat for any compact hyperbolic $3$-manifold, the thin part is a\ndisjoint union of solid tori, called \\emph{Margulis tubes}, and each\nof these solid tori is a regular metric neighborhood of an embedded\nclosed geodesic of length less than $\\mu_0$. In the case in which $M$\nis complete without cusps, there is an extra possibility, as a\ncomponent of the thin part may also be the universal cover of such a\nsolid torus, and we shall refer to such a component as an\n\\emph{infinite Margulis tube}. We shall call a number $\\mu_0$ for\nwhich this result holds a \\emph{Margulis constant} for\n$\\mathbb{H}^3$. If $\\mu_0$ is a Margulis constant for $\\mathbb{H}^3$,\nthen so is $\\mu$ for any $0 < \\mu < \\mu_0$, and furthermore, given\n$\\mu$ and $\\mu_0$ there is a number $\\delta > 0$ such that\n$N_{\\delta}(T_{\\mu}) \\subseteq T_{\\mu_0}$. For the remainder for this\nsection we shall fix a pair of numbers $(\\mu, \\epsilon)$ such that there are\nMargulis constants $0 < \\mu_1 < \\mu < \\mu_2$, a number $\\delta$ such\nthat $N_{\\delta}(T_{\\mu}) \\subseteq T_{\\mu_2} \\setminus T_{\\mu_1}$,\nand $\\epsilon = \\tfrac{1}{4} \\min \\{ \\mu_1, \\delta \\}$. We shall call $(\\mu,\n\\epsilon)$ a choice of \\emph{MV}-constants for $\\mathbb{H}^3$.\n\nLet $(\\mu, \\epsilon)$ be a choice of $MV$-constants, and consider an\n$\\epsilon$-regular Voronoi decomposition of $M$. The fact that\n$N_{\\delta}(T_{\\mu}) \\subseteq T_{\\mu_2} \\setminus T_{\\mu_1}$ means\nthat we may adjust the boundary of $T_{\\mu}$ by an arbitrarily small\nisotopy so that it is transverse to the Voronoi cells, and we will\nassume that we have done this for the remainder of this section. Our\nchoice of $\\epsilon$ implies that the thick part $X_\\mu$ is contained in the\nVoronoi cells in the deep part, i.e. $X_\\mu \\subset \\bigcup_{W_i \\in\n \\mathcal{W}} W_i$, so in particular $\\partial X_\\mu = \\partial T_\\mu$ is\ncontained in the deep part. Furthermore, each deep Voronoi cell hits\nat most one component of $T_\\mu$.\n\n\n\n\\subsection{Cell splitters}\\label{section:splitter}\n\nThe polyhedral surfaces we construct will be constant, except for a\ndiscrete collection of points in $Y$, which roughly speaking\ncorrespond to points $t \\in Y$ for which the level set $f^{-1}(t)$\ndivide a Voronoi cell in half. For technical reasons, we use points\nwhich divide a ball of fixed size in the Voronoi cell in half, as we\nnow describe.\n\nLet $t$ be a point in a trivalent tree $Y$. We shall write $Y_t^{c_i}$\nfor the closures of the connected components of $Y \\setminus t$, and\nwe shall call these the \\emph{complements} of $t$. If $t$ lies in the\ninterior of an edge, then there are precisely two complements, while\nif $t$ is a vertex, there are precisely three complements.\n\nLet $M$ be a complete hyperbolic $3$-manifold without cusps, and let\n$h \\colon M \\rightarrow Y$ be a Morse function onto a trivalent tree\n$Y$. Given $t \\in Y$, let $H_t^{c_i} = h^{-1}(Y_t^{c_i})$, and we\nshall refer to these as the \\emph{complements} of $H_t$ in $M$. As\nbefore, there are either two or three complementary regions depending\non whether $t$ lies in the interior of an edge, or is a vertex in $Y$.\n\n\\begin{definition} \\label{definition:splitter} %\nLet $M$ be a complete hyperbolic $3$-manifold without cusps, with an\n$\\epsilon$-regular Voronoi decomposition ${\\cal{V}}$. Let $h \\colon M \\to Y$ be a\nMorse function to a tree $Y$, and let $V$ be a Voronoi cell with\ncenter $x$. Suppose that a point $t \\in Y$ has the property that for\neach complementary region $H_t^{c_i}$, the volume of\n$H_t^{c_i} \\cap B(x, \\epsilon\/2) \\cap V$ is at most half the volume of\nthe topological ball $B(x, \\epsilon\/2) \\cap V$. Then we say that $t$\nis a \\emph{cell splitter} for the Voronoi cell $V$.\n\\end{definition}\n\n\\begin{proposition}\nLet $M$ be a complete hyperbolic $3$-manifold without cusps, with an\n$\\epsilon$-regular Voronoi decomposition ${\\cal{V}}$. Let $h \\colon M \\to Y$ be a\nMorse function to a tree, and let $V$ be a Voronoi cell with center\n$x$. Then there is a unique cell splitter $t \\in Y$ for $V$.\n\\end{proposition}\n\n\\begin{proof}\nWe first show existence. Let $B$ be the topological ball $B(x, \\epsilon\/2)\n\\cap V$, and let $v$ be the volume of this ball. Consider $h(B)\n\\subset Y$. If there is a vertex of $Y$ which is a cell splitter,\nthen we are done. Otherwise, suppose no vertex of $h(B)$ is a cell\nsplitter. If $t$ is a vertex in $h(B)$ which is not a cell splitter,\nthen there is at least one complementary region $Y_t^{c_i}$ such that\n$H_t^{c_i} \\cap B(x, \\epsilon\/2) \\cap V$ has volume more than\n$\\tfrac{1}{2}v$, and $Y_t^{c_i} \\cap h(B)$ has at least one fewer\nvertex. So proceeding by induction, we may reduce to the case in which\n$h(B)$ contains an interval $I$ with no vertices such that $h^{-1}(I)\n\\cap B(x, \\epsilon\/2) \\cap V$ has volume at least $\\tfrac{1}{2} v$. In this\ncase, let $t_0$ and $t_1$ be the endpoints of $I$, and consider\n$h^{-1}([t_0, s])$, for $s \\in I$. When $s = t_0$, this has volume\nless than $\\tfrac{1}{2} v$, and has volume greater than\n$\\tfrac{1}{2}v$ when $s = t_1$. As the volume changes continuously\nwith $s$, there is a point $t'$ such that $H_{t'}$ divides $B$ into\ntwo regions, each of which has volume exactly $\\tfrac{1}{2}v$, so $t'$\nis a cell splitter for $V$.\n\nWe now show uniqueness. First suppose $t$ is a cell splitter which is\nnot a vertex. Then there are precisely two complementary regions\n$H_t^{c_1}$ and $H_t^{c_2}$, each of which must have exactly half the\nvolume of $B(x, \\epsilon\/2) \\cap V$, and we shall denote this volume by\n$v$. Any other point $t'$ has a complementary region which contains at\nleast one of these complements, and so has volume greater than\n$\\tfrac{1}{2} v$, and so can not be a cell splitter.\n\nFinally suppose $t$ is a cell splitter which is a vertex. Then there\nare three complements $H_t^{c_1}, H_t^{c_2}$ and $H_t^{c_3}$, each of\nwhich has volume at most $\\tfrac{1}{2}v$. As each region has volume at\nmost $\\tfrac{1}{2}v$, any two regions must have total volume at least\n$\\tfrac{1}{2}v$. Any other point $t' \\in Y$ must have a complementary\nregion which contains at least two of the complements of $H_t$, and so\nhas a complement with volume strictly greater than $\\tfrac{1}{2}v$,\nand so can not be a cell splitter.\n\\end{proof}\n\n\\begin{definition} \\label{definition:generic} %\nWe say that a Morse function $f \\colon M \\to Y$ to a tree $Y$ is\n\\emph{generic} with respect to a Voronoi decomposition $\\mathcal{V}$\nif the cell splitters for distinct Voronoi cells $V_i$ correspond to\ndistinct points $t_i \\in Y$. We say a point $t \\in Y$ is\n\\emph{generic} if it is not a critical point for the Morse function,\nand is not a cell splitter.\n\\end{definition}\n\nWe may assume that $f$ is generic by an arbitrarily small perturbation\nof $f$, and we shall always assume that $f$ is generic from now on.\nFinally, we remark that a trivalent vertex in $Y$ is not necessarily a\ncell splitter.\n\n\n\n\n\\subsection{Polyhedral and capped surfaces} \\label{section:capped}\n\n\nLet $Q$ be a $3$-dimensional submanifold of a complete hyperbolic\n$3$-manifold $M$ without cusps, with boundary an embedded separating\nsurface $F$. In this section we show how to approximate $Q$ by a union\nof Voronoi cells, which in turn gives an approximation to $F$ by an\nembedded surface $S$ which is a union of faces of Voronoi cells.\n\nWe say a region $R$ is \\emph{generic} if for every Voronoi cell $V_i$\nwith center $x_i$, the region consisting of the intersection of\n$B(x_i, \\epsilon\/2)$ with the interior of $V_i$ does not have exactly half\nits volume lying in $R$. We say a separating surface $F$ in $M$ is\n\\emph{generic} if it bounds a generic region.\n\nLet $P$ be the collection of Voronoi cells for which at least half of\nthe volume of $B(x_i, \\epsilon\/2) \\cap \\text{interior}(V_i)$ lies in $Q$. We\nsay the $P$ is the \\emph{polyhedral region} determined by $Q$. The\npolyhedral region $P$ may be empty, even if $Q$ is non-empty. The\nboundary of $Q$ is a polyhedral surface $S$, which we shall call the\n\\emph{polyhedral surface} associated to $F = \\partial Q$, and is a\nnormal surface in the dual triangulation. We will use the following\nbound on the number of faces and boundary components of the\nintersection of the polyhedral surface $S$ with the thick part of the\nmanifold, in terms of the area of the corresponding surface $F$. If\n$S$ is a surface, we will write $\\norm{\\partial S}$ for the number of\nboundary components of $S$, and if $S'$ is a subset of a polyhedral\nsurface $S$, we will write $\\| S' \\|$ for the number of faces of $S$\nwhich intersect $S'$.\n\n\n\n\\begin{proposition} \n\\cite{hoffoss-maher}*{Proposition 2.10, 2.13} \\label{cor:bound} %\nLet $(\\mu, \\epsilon)$ be $MV$-constants, and let $M$ be a complete\nhyperbolic $3$-manifold without cusps, with an $\\epsilon$-regular Voronoi\ndecomposition ${\\cal{V}}$ with deep part $\\mathcal{W}$ and thick part $X_\\mu$. Then\nthere is a constant $K$, which only depends on the $MV$-constants,\nsuch that for any generic embedded separating surface $F$ in $M$, the\ncorresponding polyhedral surface $S$ satisfies:\n\\[ \\| S \\cap X_\\mu \\| \\leqslant K \\text{area}(F), \\]\nand\n\\[ \\norm{\\partial ( S \\cap X_\\mu ) } \\leqslant K \\text{area}(F). \\]\n\\end{proposition}\n\nIn \\cite{hoffoss-maher}, this result is stated for the level set of a\nMorse function $F$ on a compact hyperbolic manifold, and one may then\nobserve that every separating surface is the level set of some Morse\nfunction, though in fact, the proof only uses the fact that $F$ is\nseparating. In \\cite{hoffoss-maher}*{Proposition 2.10} the bound is\nstated in terms of $S \\cap \\mathcal{W}$. However, as $S \\cap X_\\mu \\subset S\n\\cap \\mathcal{W}$, the stated bound follows immediately.\n\n\nFor a polyhedral surface $S$, each boundary component of the surface\n$S \\cap X_\\mu$ is contained in $T_\\mu$, so $S \\cap X_\\mu$ is a\nproperly embedded surface in $X_\\mu$. We now wish to cap off the\nproperly embedded surfaces $S \\cap X_\\mu$ with properly embedded\nsurfaces in $T_\\mu$ to form closed surfaces. We warn the reader that\nthe following definition differs slightly from the definition in\n\\cite{hoffoss-maher}, as we extend the definition to include the case\nin which $T_\\mu$ has infinite components.\n\n\\begin{definition}\nA separating surface $F$ in $M$ gives rise to a polyhedral surface\n$S$, which meets $\\partial T_\\mu$ transversely, and intersects\n$\\partial T_\\mu$ in a collection of simple closed curves which is\nseparating in $\\partial T_\\mu$. We replace $S$ inside the thin part\nby surfaces $\\{ U_i \\}$ which we now describe. For each torus\ncomponent $T_i$ in $\\partial T_\\mu$ choose a subsurface $U_i$ bounded\nby $S \\cap \\partial T_i$. For each infinite component $T_i$, choose a\nnot necessarily connected surface $U_i$ as follows: for each essential\ncurve in the annulus $\\partial T_i$ choose a disc it bounds in $T_i$,\nand then let $U_i$ be the union of these discs with the planar surface\nbounded by the remaining inessential curves. We call the resulting\nsurface a \\emph{capped surface} $S^+ = (S \\cap X_\\mu) \\cup \\bigcup_i\nU_i$.\n\\end{definition}\n\n\nWe will use the following property of the capped surfaces.\n\n\\begin{proposition} \\label{prop:capped}\nLet $(\\mu, \\epsilon)$ be $MV$-constants, and let $M$ be a complete\nhyperbolic $3$-manifold without cusps, with thin part $T_\\mu$, and\nwith with an $\\epsilon$-regular Voronoi decomposition ${\\cal{V}}$. Then there is a\nconstant $K$, which only depends on $\\epsilon$, such that for any generic\nembedded separating surface $F$ in $M$, the corresponding capped\nsurface $S^+$ satisfies:\n\\[ \\text{genus}(S^+) \\leqslant K \\text{area}(F). \\]\n\\end{proposition}\n\nThe proof of this result is essentially the same as the proof of\n\\cite{hoffoss-maher}*{Proposition 2.14}, and instead of repeating the\nentire argument, we explain the minor extension needed. The only\ndifference is that \\cite{hoffoss-maher}*{Proposition 2.14} is stated\nfor closed hyperbolic manifolds, whereas Proposition \\ref{prop:capped}\nis stated for complete hyperbolic manifolds without cusps, so the thin\npart of $M$ may have infinite Margulis tubes. This makes no\ndifference to the estimates of the number of faces and boundary\ncomponents of the resulting polyhedral surface in terms of the area of\nthe original surface. The extension of the definition of capped\nsurface to the infinite case only involves capping off with planar\nsurfaces, so the same genus bounds hold.\n\n\n\n\n\n\n\\subsection{Disjoint equivariant surfaces}\\label{section:equivariant}\n\nEach collection of points $t_i$ in $Y$ corresponds to a collection\n$S^+_i$ of capped surfaces. In this section we show that if the\ncollection of points is equivariant, then we may arrange for the\ncapped surfaces to be disjoint and equivariant.\n\nLet $M$ be a $3$-manifold which admits a group of covering\ntranslations $G$. We say a subset $U \\subset M$ is \\emph{equivariant}\nif it is preserved by $G$. We say a Voronoi decomposition ${\\cal{V}}$ of $M$\nis \\emph{equivariant} if the centers of the Voronoi cells form an\nequivariant set in $M$.\n\nLet $W$ be an equivariant collection of points in $\\widetilde X$, none\nof which are either cell splitters or critical points of the Morse\nfunction $h$. We say two points $t_i$, $t_j$ in $W$ are\n\\emph{adjacent} if the geodesic connecting them in the tree $\\widetilde{X}$ does\nnot contain any other point of $W$. We may choose $W$ such that the\ngeodesic in $\\widetilde{X}$ connecting any pair of adjacent points in $W$\ncontains either a single cell splitter, a single trivalent trivalent\nvertex of $\\widetilde{X}$, or neither of these two types of points.\n\nConsider the collection $S$ of polyhedral surfaces $S_t$, as $t$ runs\nover $W$. As the collection $W$ is equivariant, $S$ is also\nequivariant. Note that although each surface in $S$ is individually\nembedded, each surface in $S$ will share many common faces with other\nsurfaces in $S$. We will now make this collection simultaneously\nequivariantly disjoint, so that we may push them down to $M$ to obtain\na collection of disjoint surfaces which will act as our splitting\nsurfaces in a graph splitting of $M$.\n\n\\begin{proposition}\nLet $M$ be a closed hyperbolic 3-manifold of injectivity radius at\nleast $2\\epsilon$, with an $\\epsilon$-regular Voronoi decomposition ${\\cal{V}}$, and a\ngeneric Morse function $f : M \\rightarrow X$ onto a trivalent graph\n$X$ with connected level sets. Let $p \\colon \\twid{M} \\rightarrow M$\nbe the cover of $M$ corresponding to the kernel of the induced map\n$f_* \\colon \\pi_1 M \\rightarrow \\pi_1 X $, and let\n$c:\\twid{X}\\rightarrow X$ be the universal cover of $X$. Let $W$ be a\ndiscrete equivariant collection of points in $\\widetilde X$. Then the\ncollection of polyhedral surfaces $\\{ S_w \\mid w \\in W \\}$ in\n$\\twid{M}$ is equivariantly isotopic to a disjoint collection of\nsurfaces $\\{ \\Sigma_w \\mid w \\in W \\}$, and furthermore this equivariant\nisotopy may be chosen to be supported in a neighborhood of the\n2-skeleton of the induced Voronoi decomposition of $\\widetilde M$.\n\\end{proposition}\n\n\n\\begin{proof}\nWe now give a recipe for constructing surfaces $\\Sigma_t$, for $t \\in\nW$. Each individual surface $\\Sigma_t$ will be isotopic to the original\n$S_t$, but the union of the surfaces $\\Sigma_t$ will be equivariantly\ndisjointly embedded in $\\twid{M}$.\n\nWe first show that there is a canonical ordering of the polyhedral\nsurfaces $\\Sigma_t$ which share a common face. Let $\\Phi$ be a face of a\nVoronoi cell in $\\widetilde M$, and let $V(x_1)$ and $V(x_2)$ be the\nadjacent Voronoi cells. Let $t_1$ and $t_2$ be cell splitters for\n$V(x_1)$ and $V(x_2)$, so that $H_{t_i} = h^{-1}(t_i)$ is the surface\nwhich divides $B_{\\epsilon\/2}(x_i)$ precisely in half, for $i = 1,2$.\n\nWe say a point in $\\widetilde{X}$ is \\emph{regular} if it is not a\ncell splitter, and not a critical point for the Morse function $h$.\n\n\\begin{claim}\nThe collection of regular points in $\\widetilde X$ corresponding to\npolyhedral surfaces $\\Sigma_t$ which contain the face $\\Phi$ is precisely\nthe regular points contained in the geodesic in $\\widetilde{X}$ from $t_1$ to\n$t_2$.\n\\end{claim}\n\n\\begin{proof}\nThe two embedded surfaces $H_{t_1}$ and $H_{t_2}$ divide $\\widetilde M$ into\nthree parts; call them $A, B$ and $C$, with $A$ the part only hitting\n$H_{t_1}$, and $B$ the part hitting both $H_{t_1}$ and $H_{t_2}$.\n\nLet $\\gamma$ be the geodesic in $\\widetilde{X}$ from $t_1$ to $t_2$. Each point\n$t$ in $\\gamma$ corresponds to a surface $H_t$ dividing $\\widetilde M$ at most\n3 parts, one of which contains $A$, and another containing $C$. Let\n$P_t$ be the part containing $A$. Then, writing $\\norm{A}$ for the\nvolume of a region $A$, \n\\[ \\norm{ B_{\\epsilon\/2}(x_1) \\cap P_t} \\geqslant \\norm{B_{\\epsilon\/2}(x_1)\n \\cap A} \\geqslant \\frac{1}{2} \\norm{ B_{\\epsilon\/2}(x_1) }\n\\]\nand \n\\[ \\norm{ B_{\\epsilon\/2}(x_2)\n \\setminus P_t} \\geqslant \\norm{B_{\\epsilon\/2}(x_2) \\setminus C} \\geqslant \\frac{1}{2} \\norm{\n B_{\\epsilon\/2}(x_2) }.\n\\]\nTherefore the two Voronoi cells $V(x_1)$ and $V(x_2)$ lie in different\npartitions of the Voronoi cells determined by $t$, and so $\\Phi$ lies\nin the polyhedral surface $\\Sigma_t$.\n\nConversely, suppose $t$ does not lie on the path $\\gamma$, then $t$\ndivides $\\widetilde{X}$ into at most three parts, and $\\gamma$ is contained in\nexactly one of these parts. This means that $H_{t_1}$ and $H_{t_2}$\nare contained in the same complementary component of $H_t$, and so $\\Phi$\ncannot be a face of $\\Sigma_t$.\n\\end{proof}\n\nIt suffices to show that we can isotope the normal surfaces,\npreserving the fact that they are normal, so that they have disjoint\nintersection in the $2$-skeleton of the dual triangulation.\n\nLet $e$ be an edge of the dual triangulation, with vertices $x_1$ and\n$x_2$, with corresponding cell splitters $t_1$ and $t_2$. A normal\nsurface $S_i$ intersects $e$ if and only if the corresponding point\n$w_i$ lies in the geodesic $[t_1, t_2]$ in $\\widetilde{X}$ connecting $t_1$ and\n$t_2$. The points $w_i$ in $e$ therefore inherit an order from $[t_1,\nt_2]$, and we may isotope the normal surfaces by a normal isotopy so\nthat they intersect the edge $e$ in the same order. As the interiors\nof each edge have disjoint images under the covering translations, and\nthe collection of edges is equivariant, we may do this normal isotopy\nequivariantly.\n\nLet $\\Phi$ be a triangle in the dual triangulation, with vertices\n$x_1, x_2$ and $x_3$, and corresponding cell splitters $t_1, t_2$ and\n$t_3$. As above, the collection of normal surfaces which intersect an\nedge $[x_i, x_j]$ of $\\Phi$ corresponds to those $w_i$ lying in the\ngeodesic $[t_i, t_j]$ in $\\widetilde{X}$. The union of the three geodesics\n$[t_i, t_j]$ forms a minimal spanning tree for the three cell\nsplitters in $\\widetilde{X}$. Let $t_0$ be the center of this tree, i.e. the\nunique point that lies in all three geodesics. Note that the tree may\nbe degenerate, so $t_0$ may be equal to one of the other vertices.\n\n\\begin{figure}[H]\n\\begin{center} \n\\begin{tikzpicture}\n\n\\draw (0, 0) node [below] {$x_1$} -- \n (1, 4) node [right] {$x_3$} node [midway, right] {$\\Phi$}-- \n (-2, 2) node [left] {$x_2$}-- cycle;\n\n\n\n\\draw [thick] ( $(0,0)!0.833!(1,4)$ ) -- node [midway, below] {$S_1$}\n($(-2,2)!0.833!(1,4)$ );\n\n\\draw [thick] ( $(0,0)!0.666!(1,4)$ ) -- node [midway, below] {$S_2$}\n($(-2,2)!0.666!(1,4)$ );\n\n\\draw [thick] ( $(-2,2)!0.25!(1,4)$ ) -- node\n[midway, right] {$S_3$} ($(0,0)!0.75!(-2,2)$ );\n\n\\draw [thick] ( $(0,0)!0.25!(1,4)$ ) -- node [midway, above] {$S_4$}\n($(0,0)!0.25!(-2,2)$ );\n\n\\begin{scope}[xshift=1cm, yshift=0.5cm, scale=1.4]\n\\filldraw[black] (3,0) circle (0.05cm) node [right]{$t_1$};\n\\filldraw[black] (2.25,2) circle (0.05cm) node [left]{$t_2$};\n\\filldraw[black] (3.75,2) circle (0.05cm) node [right]{$t_3$};\n\\filldraw[black] (3,1) circle (0.05cm) node [right]{$t_0$};\n\\filldraw[black] ( $(3.75,2)!0.333!(3, 1)$ ) circle (0.05cm) node [right] {$w_1$};\n\\filldraw[black] ( $(3.75,2)!0.666!(3, 1)$ ) circle (0.05cm) node [right] {$w_2$};\n\\filldraw[black] ( $(2.25,2)!0.5!(3, 1)$ ) circle (0.05cm) node [left] {$w_3$};\n\\filldraw[black] (3,0.5) circle (0.05cm) node [right]{$w_4$};\n\n\\draw (3,0) -- (3, 1) -- (3.75, 2);\n\n\\draw (2.25, 2) -- (3, 1); \n\n\\end{scope}\n\n\\end{tikzpicture}%\n\\end{center} \n\\caption{Example of normal surfaces intersecting a face of the dual\n triangulation.}\n\\label{pic:normal tree} \n\\end{figure}\n\nNormal arcs parallel to the edge $[x_2, x_3]$ correspond to surfaces\nwhich hit both of the edges $[x_1, x_2]$ and $[x_1, x_3]$, so\ncorrespond points $w_i$ which lie in both $[t_1, t_2]$ and $[t_1,\nt_3]$, and similarly for the other two cases. The intersection of\nthese two geodesics in $\\widetilde{X}$ is $[t_1, t_0]$, and so the corresponding\nsurfaces appear in the same order on each of the edges in $\\Phi$, and so\nthe arcs are disjoint. The same argument applies to each vertex of\n$\\Phi$.\n\\end{proof}\n\nAs the resulting surfaces in $\\twid{M}$ are disjoint and equivariant,\nthey project down to disjoint surfaces in $M$.\n\nWe now show that the polyhedral surfaces, and their complements,\nproject down homeomorphically into $M$. As the level set surfaces lift\nhomeomorphically to $\\twid{M}$, the area bound for the level sets of\n$f$ is also an area bound for the level sets of $h$. Therefore, each\npolyhedral surface contains a bounded number of faces. The deck\ntransformation group of the universal cover of a graph is equal to the\nfundamental group of the graph, which is a free group, so the orbit of\nany face consists of infinitely many disjoint translates. If two lie\nin the same connected component of a polyhedral surface, then that\npath corresponds to a covering translation, which has infinite order,\nso in fact the connected component contains infinitely many faces,\nwhich contradicts the fact that there is a bound on the number of\nfaces in each component.\n\nEach complementary region is compact, so the same argument applied to\nthe complementary regions shows that they are all mapped down\nhomeomorphically as well.\n\n\n\n\n\\subsection{Bounded handles}\\label{section:bounded}\n\nWe now bound the number of handles in a complementary region of the\ncapped surfaces, which contains a single cell splitter. The following\nresult will complete the proof of the left hand inequality in Theorem\n\\ref{theorem:graph}.\n\n\\begin{proposition}\\label{bounded_handles} \nLet $(\\mu, \\epsilon)$ be $MV$-constants, and let $M$ be a complete\nhyperbolic $3$-manifold without cusps, with an $\\epsilon$-regular Voronoi\ndecomposition ${\\cal{V}}$, and a generic Morse function $h \\colon M \\to Y$,\nwhere $Y$ is a tree. Let $\\{ u_i \\}$ be a collection of points in $Y$,\nwhich separate the cell splitters in $Y$, and let $\\{ S^+_i \\}$ be the\ncorresponding collection of capped surfaces. If $P$ is a\ncomplementary component of the capped surfaces in $M$, the region $P$\nhas at most three boundary components, $S^+_{i_1}, S^+_{i_2}$ and\n$S^+_{i_3}$ say, where the final surface may be empty. Then $P$ is\nhomeomorphic to a manifold with a handle decomposition containing at\nmost\n\\[ K \\text{Gromov area} ( M ) \\]\nhandles, where $K$ depends only on the $MV$-constants.\n\\end{proposition}\n\nWe start with the observation that attaching a compression body $P$ to\na $3$-manifold $Q$ by a subsurface $S$ of the upper boundary component\nof $P$, requires a number of handles which is bounded in terms of the\nHeegaard genus of $P$, and the number of boundary components of the\nattaching surface.\n\n\\begin{proposition}\\label{bounded_handles3}\\cite{hoffoss-maher}*{Proposition 2.16}\nLet $Q$ be a compact $3$-manifold with boundary, and let $R = Q \\cup\nP$, where $P$ is a compression body of genus $g$, attached to $Q$ by a\nhomeomorphism along a (possibly disconnected) subsurface $S$ contained\nin the upper boundary component of $P$ of genus $g$. Then $R$ is\nhomeomorphic to a $3$-manifold obtained from $Q$ by the addition of at\nmost $(4g + 2 \\norm{\\partial S})$ $1$-and $2$-handles, where\n$\\norm{\\partial S}$ is the number of boundary components of $S$.\n\\end{proposition}\n\n\\begin{proof}[Proof (of Proposition \\ref{bounded_handles}).]\nIf $P$ has two boundary components, then the argument is exactly the\nsame as \\cite{hoffoss-maher}*{Proposition 2.15}, so we now consider the\ncase in which $P$ has three boundary components, which, without loss\nof generality we may relabel $S^+_{1}, S^+_{2}$ and $S^+_3$. Let $t$\nbe the cell splitter corresponding to the region $P$, and let $V$ be\nthe corresponding Voronoi cell. As $P$ has three boundary components,\n$t$ must be a vertex of $Y$.\n\nWe first consider the case in which the Voronoi region $V$\ncorresponding to the cell splitter $t$ in $h(P)$ is disjoint from the\nthin part $T_\\mu$. Consider the three polyhedral surfaces $S_1, S_2$\nand $S_3$, corresponding to the three capped surfaces, and let\n$\\Sigma = \\cup S_i \\cup V$ be the union of the polyhedral surfaces,\ntogether with the Voronoi cell $V$. By Proposition \\ref{cor:bound},\nthere is a constant $K$, which only depends on the $MV$-constants,\nsuch that the number of faces of $\\Sigma$ in the thick part is at most\n$3 K_1 \\text{Gromov area}(M)$, i.e.\n\\[ \\| \\Sigma \\cap X_\\mu \\| \\leqslant 3 K_1 \\text{Gromov area}(M), \\]\nwhere $K_1$ is the constant from Proposition \\ref{cor:bound}. The\nnumber of boundary components of each surface $S_i \\cap X_\\mu$ is also\nbounded by Proposition \\ref{cor:bound}, and by Proposition\n\\ref{prop:bound}, the Voronoi cell $V$ has a bounded number $J$ of\nvertices, edges and faces, where $J$ depends only on the\n$MV$-constants. In particular, there is a constant $A$, depending\nonly on the $MV$-constants, such that $P \\cap X_\\mu$ has a handle\nstructure with at most $A ( \\text{Gromov area}(M) )$ handles.\n\nTo bound the number of handles contained in $P$, we observe that $P$\nis a regular neighbourhood of the $3$-complex obtained from capping\noff the boundary components of $\\Sigma \\cap X_\\mu$, using the parts of the\ncapped surfaces in the thin part, i.e. the union of the components of\n$S^+_i \\cap T_\\mu$ over all three capped surfaces. Each component of\n$S^+_i \\cap T_\\mu$ has genus at most one, and the number of boundary\ncomponents of $\\Sigma \\cap X_\\mu$ is bounded linearly in terms of\n$\\text{Gromov area}(M)$, therefore, there is a constant $B$, depending\nonly on the $MV$-constants, such that the number of handles in $P$ is\nat most $B ( \\text{Gromov area}(M) )$, as required. \n\nWe now consider the case in which the region $P$ has image $h(P)$ in\n$Y$ which contains the cell splitter $t$, and the corresponding\nVoronoi cell $V$ intersects $T_\\mu$. In this case, the connected\ncomponents of $V \\cap X_\\mu$ need not be topological balls, and there\nmay be connected components of $P \\cap T_\\mu$ whose boundary components\nare not parallel.\n\nThe connected components of $V \\cap X_\\mu$ are handlebodies of bounded\ngenus, as show in the following result of Kobayashi and Rieck\n\\cite{kobayashi-rieck}. We state a simplified version of their result\nwhich suffices for our purposes, see \\cite{hoffoss-maher} for further\ndetails.\n\n\\begin{proposition}\\cite{kobayashi-rieck}\nLet $\\mu$ be a Margulis constant for $\\mathbb{H}^3$, $M$ be a finite\nvolume hyperbolic $3$-manifold, let $0 < \\epsilon < \\mu$, and let ${\\cal{V}}$ be a\nregular Voronoi decomposition of $M$ arising from a maximal collection\nof $\\epsilon$-separated points. Then there is a number $G$, depending only\non $\\mu$ and $\\epsilon$, such that for any Voronoi cell $V_i$, there are at\nmost $G$ connected components of $V_i \\cap X_\\mu$, each of which is a\nhandlebody of genus at most $G$, attached to $T_\\mu$ by a surface with\nat most $G$ boundary components.\n\\end{proposition}\n\nRecall that attaching a handlebody of genus $G$ to a $3$-manifold\nalong a subsurface of the boundary with at most $G$ boundary\ncomponents requires at most $6G$ handles:\n\n\\begin{proposition}\\cite{hoffoss-maher}*{Proposition 2.16}\nLet $Q$ be a compact $3$-manifold with boundary and let $R = Q \\cup P$\n, where $P$ is a compression body of genus $g$, attached to $Q$ by a\nhomeomorphism along a (possibly disconnected) subsurface $S$ contained\nin the upper boundary component of $P$ of genus $g$. Then $R$ is\nhomeomorphic to a $3$-manifold obtained from $Q$ by the addition of at\nmost $(4 \\text{genus} + 2 \\norm{ \\partial S})$ $1$- and $2$-handles,\nwhere $\\norm{\\partial S}$ is the number of boundary components of $S$.\n\\end{proposition}\n\nTherefore, adding a Voronoi cell which intersects $\\partial T_\\mu$ may\nbe realized by at most $6 G^2$ handles.\n\nIf the Voronoi cell intersects $T_\\mu$, then there may be components\nof $P \\cap T_\\mu$ whose boundary surfaces are not parallel. This case\nis considered in the proof of \\cite{hoffoss-maher}*{Proposition 2.15},\nwhen the manifold has no infinite Margulis tubes, so it suffices to\nconsider the case of a component of $P$ contained in an infinite\nMargulis tube. However, the case of an infinite Margulis tube in\nwhich neither surface is an essential disc is the same as the ordinary\nMargulis tube case, and if both surfaces essential discs then they are\nparallel. Finally, if exactly one surface is an essential disc, then\nthe other surface lies in the same homology class, via the component\nof $P$ in the infinite Margulis tube, and so, after surgering\ninessential boundary components, is also an essential disc. However,\nthe number of boundary components is at most $K_1 \\text{Gromov\n area}(M)$, and so the total number of extra handles over all\ncomponents of $P$ in the infinite Margulis tubes is also bounded by\n$K_1 \\text{Gromov area}(M)$.\n\nWe may choose the constant $K$ to be the maximum of the constants\narising from the two cases considered above, thus completing the proof\nof Proposition \\ref{bounded_handles3}.\n\\end{proof}\n\n\n\n\n\n\\section{Topological complexity bounds metric complexity}\\label{section:sss}\n\n\nIn this section we will show bounds for metric complexity in terms of\ntopological complexity, i.e. the right hand inequality in Theorem\n\\ref{theorem:graph}, assuming the Pitts and Rubinstein result, Theorem\n\\ref{conjecture:pr}.\n\nWe start by reminding the reader of the topological properties of thin\nposition for generalized Heegaard splittings, as shown by Scharlemann\nand Thompson \\cite{st} for the linear case and Saito, Scharlemann and\nSchultens \\cite{sss} for the graph case.\n\n\\begin{theorem} \\cites{st, sss} \\label{theorem:sss} %\nLet $H$ be a graph splitting that is in thin position. Then every even\nsurface is incompressible in $M$ and the odd surfaces form strongly\nirreducible Heegaard surfaces for the components of $M$ cut along the\neven surfaces.\n\\end{theorem}\n\nWe will use the following result due to Gabai and Colding\n\\cite{colding-gabai}*{Appendix A}, building on recent work of Colding\nand Minicozzi \\cite{colding-minicozzi}. It is not stated explicitly\nin their paper, but see \\cite{hoffoss-maher}*{Theorem 3.2} for further\ndetails.\n\n\\begin{theorem} \\cite{colding-gabai} \\label{theorem:minimal}\nLet $M$ be a hyperbolic manifold, with (possibly empty) least area\nboundary, with a minimal Heegaard splitting $H$ of genus $g$. Then,\nassuming Theorem \\ref{conjecture:pr}, the manifold $M$ has a\n(possibly singular) foliation by compact leaves, containing the\nboundary surfaces as leaves, such that each leaf has area at most $4\n\\pi g$.\n\\end{theorem}\n\nBy Theorem \\ref{theorem:sss}, we may consider the compression bodies in the\ngraph splitting in pairs, glued along strongly irreducible Heegaard\nsplittings, and then Theorem \\ref{theorem:minimal} guarantees that\neach pair has a foliation with each leaf having area at most $4 \\pi\ng$. These foliations contain the boundary surfaces as leaves, and so\nthe foliations on each pair extend to foliations of the entire\nmanifold, as required.\n\n\n\n\n\n\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{bmns}{article}{\n author={Brock, Jeffrey},\n author={Minsky, Yair},\n author={Namazi, Hossein},\n author={Souto, Juan},\n title={Bounded combinatorics and uniform models for hyperbolic\n 3-manifolds},\n journal={J. Topol.},\n volume={9},\n date={2016},\n number={2},\n pages={451--501},\n issn={1753-8416},\n}\n\n\\bib{cdl}{article}{\n author={Colding, Tobias H.},\n author={De Lellis, Camillo},\n title={The min-max construction of minimal surfaces},\n conference={\n title={Surveys in differential geometry, Vol.\\ VIII},\n address={Boston, MA},\n date={2002},\n },\n book={\n series={Surv. Differ. Geom., VIII},\n publisher={Int. Press, Somerville, MA},\n },\n date={2003},\n pages={75--107},\n}\n\n\\bib{colding-gabai}{article}{\n author={Colding, Tobias H.},\n author={Gabai, David},\n title={Effective Finiteness of irreducible Heegaard splittings of non Haken 3-manifolds},\n eprint={arXiv:1411.2509},\n date={2014},\n}\n\n\\bib{colding-minicozzi}{article}{\n author={Colding, Tobias Holck},\n author={Minicozzi, William P., II},\n title={The singular set of mean curvature flow with generic\n singularities},\n journal={Invent. Math.},\n volume={204},\n date={2016},\n number={2},\n pages={443--471},\n issn={0020-9910},\n}\n\n\\bib{dlp}{article}{\n author={De Lellis, Camillo},\n author={Pellandini, Filippo},\n title={Genus bounds for minimal surfaces arising from min-max\n constructions},\n journal={J. Reine Angew. Math.},\n volume={644},\n date={2010},\n pages={47--99},\n issn={0075-4102},\n}\n\n\n\\bib{gromov}{article}{\n author={Gromov, M.},\n title={Width and related invariants of Riemannian manifolds},\n language={English, with French summary},\n note={On the geometry of differentiable manifolds (Rome, 1986)},\n journal={Ast\\'erisque},\n number={163-164},\n date={1988},\n pages={6, 93--109, 282 (1989)},\n issn={0303-1179},\n}\n\n\n\\bib{hoffoss-maher}{article}{\n author={Hoffoss, Diane},\n author={Maher, Joseph},\n title={Morse area and Scharlemann-Thompson width for hyperbolic\n 3-manifolds},\n journal={Pacific J. Math.},\n volume={281},\n date={2016},\n number={1},\n pages={83--102},\n issn={0030-8730},\n}\n\n\n\n\n\\bib{kobayashi-rieck}{article}{\n author={Kobayashi, Tsuyoshi},\n author={Rieck, Yo'av},\n title={A linear bound on the tetrahedral number of manifolds of bounded\n volume (after J\\o rgensen and Thurston)},\n conference={\n title={Topology and geometry in dimension three},\n },\n book={\n series={Contemp. Math.},\n volume={560},\n publisher={Amer. Math. Soc., Providence, RI},\n },\n date={2011},\n pages={27--42},\n}\n\n\n\\bib{ketover}{article}{\n\tauthor={Ketover, Daniel},\n\ttitle={Degeneration of Min-Max Sequences in 3-manifolds},\n\tdate={2013},\n\teprint={arXiv:1312.2666},\n}\n\n\n\\bib{pr}{article}{\n author={Pitts, Jon T.},\n author={Rubinstein, J. H.},\n title={Existence of minimal surfaces of bounded topological type in\n three-manifolds},\n conference={\n title={},\n address={Canberra},\n date={1985},\n },\n book={\n series={Proc. Centre Math. Anal. Austral. Nat. Univ.},\n volume={10},\n publisher={Austral. Nat. Univ.},\n place={Canberra},\n },\n date={1986},\n pages={163--176},\n}\n\n\n\\bib{rubinstein}{article}{\n author={Rubinstein, J. Hyam},\n title={Minimal surfaces in geometric 3-manifolds},\n conference={\n title={Global theory of minimal surfaces},\n },\n book={\n series={Clay Math. Proc.},\n volume={2},\n publisher={Amer. Math. Soc., Providence, RI},\n },\n date={2005},\n pages={725--746},\n}\n\n\n\\bib{sss}{book}{\n author={Scharlemann, Martin},\n author={Schultens, Jennifer},\n author={Saito, Toshio},\n title={Lecture notes on generalized Heegaard splittings},\n note={Three lectures on low-dimensional topology in Kyoto},\n publisher={World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ},\n date={2016},\n pages={viii+130},\n isbn={978-981-3109-11-7},\n}\n\n\n\n\\bib{st}{article}{\n author={Scharlemann, Martin},\n author={Thompson, Abigail},\n title={Thin position for $3$-manifolds},\n conference={\n title={Geometric topology},\n address={Haifa},\n date={1992},\n },\n book={\n series={Contemp. Math.},\n volume={164},\n publisher={Amer. Math. Soc.},\n place={Providence, RI},\n },\n date={1994},\n pages={231--238},\n}\n\n\n\n\n\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\n\\bigskip\n\n\\noindent Diane Hoffoss \\\\\nUniversity of San Diego \\\\\n\\url{dhoffoss@sandiego.edu} \\\\\n\n\\noindent Joseph Maher \\\\\nCUNY College of Staten Island and CUNY Graduate Center \\\\\n\\url{joseph.maher@csi.cuny.edu} \\\\\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{EXPERIMENTAL DETAILS} \\label{appx:experimental_details}\n\\subsection{Workflow}\n\nInput: Reference data $\\mathcal{D}_{ref}$ , synthetic data $\\mathcal{D}_{syn}$ and test data $\\mathcal{D}_{test}$.\\\\\nOutput: $\\hat{m}$ for all $x\\in \\mathcal{D}_{test}$.\\\\\nSteps:\n\\begin{enumerate}\n\\item Train density model $p_R(X)$ on $\\mathcal{D}_{ref}$.\n\\item Train density model $p_G(X)$ on $\\mathcal{D}_{syn}$.\n\\item Compute $A_{DOMIAS}(x)=\\frac{p_G(x)}{p_R(x)}$ for all $x\\in\\mathcal{D}_{test}$\n\\item Choose threshold $\\tau$, e.g. $\\tau = median \\{A_{DOMIAS}(x)|x\\in\\mathcal{D}_{test}\\}$\n\\item Infer \n\\begin{equation*}\n \\hat{m} =\\begin{cases}1, &\\text{if } A_{DOMIAS}(x)>\\tau,\\\\ 0, &\\text{ otherwise},\n \\end{cases}\n\\end{equation*}\nfor all $x\\in\\mathcal{D}_{test}$.\n\\end{enumerate}\n\n\\subsection{Data}\nWe use the California housing \\citep{Pace1997SparseAutoregressions} (license: CC0 public domain) and Heart Failure (private) datasets, see Table \\ref{tab:datasets} and Figure \\ref{fig:data_correlation} for statistics. All data is standardised.\n\n\\begin{table}[hbt]\n \\centering\n \\caption{Dataset statistics}\n \\label{tab:datasets}\n \\begin{tabular}{l|c|c} \\toprule\n & California Housing & Heart Failure \\\\ \\midrule\n Number of samples & 20640 & 40300\\\\\n Number of features & 8 & 35\\\\\n - binary & 0 & 25\\\\\n - continuous & 8 & 10\\\\ \\bottomrule\n \\end{tabular}\n\n\\end{table}\n\n\n\\begin{figure*}[hbt]\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/correlation_housing.png}\n \\caption{Housing}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/correlation_maggic.png}\n \\caption{Heart Failure}\n \\end{subfigure}\n \\caption{Correlation matrices of features within Housing and Heart Failure datasets. The first feature of the Heart Failure dataset is used for defining the minority group in Section 5.3.}\n \\label{fig:data_correlation}\n\\vspace{-0.25cm}\\end{figure*}\n\n\n\\subsection{Experimental settings}\n\nAll results reported in our paper are based on $8$ repeated runs, with shaded area denoting standard deviations. We experiment on a machine with 8 Tesla K80 GPUs and 32 Intel(R) E5-2640 CPUs. We shuffle the dataset and split the dataset into training set, test set, and reference set. The attack performance is computed over a test set consisting of $50\\%$ training data (i.e. samples from $\\mathcal{D}_{mem})$ and $50\\%$ non-training data. Choices of sizes for those sets are elaborated below\n\n\\paragraph{Experimental Details for Section 5.1}\nIn this section, we experimented on the California Housing Dataset to compare different MIA performance with DOMIAS. For the experiment varying the number of members in the training dataset (i.e. left panel of Figure 3), we use a fixed training epoch $2000$, a fixed number of reference example $|\\mathcal{D}_{ref}|=10000$ and a fixed number of generated example $|\\mathcal{D}_{syn}|=10000$. For the experiment varying the number of training epochs of TVAE (i.e. the right panel of Figure 3), we use a fixed training set size $|\\mathcal{D}_{mem}|=500$, a fixed number of reference example $|\\mathcal{D}_{ref}|=10000$ and a fixed number of generated example $|\\mathcal{D}_{syn}|=10000$. Training with a single seed takes $2$ hours to run in our machine with BNAF as the density estimator. \n\nIn BNAF density estimation, the hyper-parameters we use are listed in Table~\\ref{tab:hyper-param-bnaf}. Our implementation of TVAE is based on the source code provided by ~\\citep{Xu2019ModelingGAN}.\n\\begin{table}[hbt]\n \\centering\n \\caption{Hyperparameters for BNAF}\n \\label{tab:hyper-param-bnaf}\n \\begin{tabular}{c|c}\n \\toprule\n batch-dim & 50 \\cr\n n-layer & 3 \\\\\n hidden-dim & 32 \\\\\n flows & 5 \\\\\n learning rate &$0.01$ \\\\\n epochs & 50 \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\paragraph{Experimental Details for Section 5.2}\nIn our experiments varying the number of reference data $n_{ref}$, i.e. results reported in the left panel of Figure 4, we fix the training epoch to be $2000$, set $n_{syn}=10000$ and $n_{M}=500$. In the experiments varying the number of generated data $n_{syn}$, i.e. results reported in the right panel of Figure 4, we set $n_{ref}=10000$, training epoch to be $2000$, and $n_{mem}=500$. Our implementation of the kernel density estimation is based on \\textit{sklearn} with an automated adjusted bandwidth. Training with a single seed takes $0.5$ hours to finish in our machine with the kernel density estimator.\n\n\\paragraph{Experimental Details for Section 5.3} Based on results of Section 5.2, the attacking performance on different subgroups can be immediately calculated by adopting appropriate sample weights.\n\n\\paragraph{Experimental Details for Section 5.4}\nIn the Additive-Noise baseline curve, results are generated with the following noise values: \n$[0.7,\n0.9,\n1.1,\n1.3,\n1.5,\n1.7,\n1.9,\n2.3,\n2.5,\n2.9,\n3.5,\n3.9]$. In the ADS-GAN curve, results are generated with the following privacy parameter $\\lambda = [0.2, 0.5, 0.7, 1.0, 1.1, 1.3, 1.5]$. \nIn the WGAN-GP we use a gradient penalty coefficient $10.0$. All the other methods are implemented with recommended hyper-parameter settings. Training different generative models are not computational expensive and take no more than $10$ minutes to finish in our machine. Using a kernel density estimator and evaluating all baseline methods take another $20$ minutes, while using a BNAF estimator takes around $1.5$ more hours.\n\n\n\n\n\\section{ADDITIONAL EXPERIMENTS} \\label{appx:additional_results}\n\n\\subsection{Experiment 5.1 and 5.2 on Heart Failure dataset}\nWe repeat the experiments of Section 5.1 and 5.2 on the Heart Failure dataset, see Figures \\ref{fig:domias_vs_baselines_heart_failure} and \\ref{fig:ablation_results_heart_failure}. Results are noisier, but we observe the same trends as in Sections 5.1 and 5.2\n\n\n\\begin{figure*}[hbt]\n\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/MAGGIC_n_M.jpg}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/MAGGIC_t_epochs.jpg}\n \n \\end{subfigure}\n \\caption{\\textit{DOMIAS outperforms baselines on Heart Failure dataset.} MIA performance of DOMIAS and baselines versus the generative model training set size $|\\mathcal{D}_{mem}|$ and training time $t_{epochs}$, evaluated on Heart Failure datasets. The same trends are observed as in Section 5.1.}\n \\label{fig:domias_vs_baselines_heart_failure}\n\\vspace{-0.25cm}\n\\end{figure*}\n\n\\begin{figure*}[hbt]\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\includegraphics[width=0.9\\textwidth]{figures\/MAGGIC_n_ref.jpg}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/MAGGIC_n_G.jpg}\n \n \\end{subfigure}\n \\caption{\\emph{DOMIAS source of gain.} Ablation study of DOMIAS on Heart Failure dataset, with attack performance as a function of the reference dataset size (left) and the synthetic dataset size (right). Similar to Section 5.2, we see that the MIA performance of DOMIAS is largely due to assumption Eq.2 vs Eq. 1, i.e. the value of the reference dataset.}\n \\label{fig:ablation_results_heart_failure}\n\\vspace{-0.25cm}\n\\end{figure*}\n\n\n\n\\subsection{Experiment 5.4: Results other attackers}\nIn Figure \\ref{fig:other_attackers_against_gen} we include the results of experiment 5.4 for all attacks, including error bars. Indeed, we see that DOMIAS outperforms all baselines against most generative models. This motivates using DOMIAS for quantifying worst-case MIA vulnerability.\n\n\\begin{figure*}[hbt]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/5.4_other_attackers.jpg}\n \\caption{DOMIAS consistently outperforms baseline attackers at attacking the different generative models.}\n \\label{fig:other_attackers_against_gen}\n\\end{figure*}\n\n\n\\subsection{CelebA image data} \\label{sec:celeba}\nWe include additional results for membership inference attacks against the image dataset CelebA. Results indicate DOMIAS is significantly better at attacking this high-dimensional data than baseline methods.\n\n\\paragraph{Set-up} We use \\href{https:\/\/mmlab.ie.cuhk.edu.hk\/projects\/CelebA.html}{CelebA}~\\citep{Liu2015DeepWild}, a large-scale face attributes dataset with more than 200K celebrity images. We generate a synthetic dataset with 10k examples using a convolutional VAE with a training set containing the first 1k examples, and use the following 1k examples as test set. Then the following 10k examples are used as reference dataset. As training the BNAF density estimator is computational expensive (especially when using deeper models), we conduct dimensionality reduction with a convolutional auto-encoder with $128$ hidden units in the latent representation space (i.e. output of the encoder) and apply BNAF in such a representation space. The hyper-parameters and network details we use in VAE are listed in Table~\\ref{tab:hyper-param-vae} and Table \\ref{tab:vae_architecture}.\n\n\n\n\n\\begin{table}[hbt]\n \\centering\n \n \\caption{Hyperparameters for VAE}\n \\label{tab:hyper-param-vae}\n \\begin{tabular}{c|c}\n \\toprule\n batch size & 128 \\cr\n n-layer & 5 \\cr\n Optimizer & Adam \\cr\n learning rate &$0.002$ \\cr\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\begin{table*}\n \\centering\n \\caption{Architecture of VAE}\n \\label{tab:vae_architecture}\n \\begin{subtable}[t]{0.48\\textwidth}\n \n \\caption{Network Structure for Encoder}\n \\label{tab:stru-encoder}\n \\begin{tabular}{c|c}\n \\toprule\n Layer & Params (PyTorch-Style) \\cr\n \\hline\n Conv1 & $(3,64,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n Conv2 & $(64,128,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n Conv3 & $(128,256,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n Conv4 & $(256,256,4,2,1)$ \\cr\n ReLU &$\\cdot$ \\cr\n Linear1 & $(256*4*4,256)$ \\cr\n ReLU & $\\cdot$\\cr\n Linear2 & $(256,256)$ \\cr\n ReLU & $\\cdot$\\cr\n Linear3 & $(256,128*2)$ \\cr\n \\bottomrule\n \\end{tabular}\n \\end{subtable}\n \\hspace{\\fill}\n \\begin{subtable}[t]{0.48\\textwidth}\n \n \\caption{Network Structure for Decoder}\n \\label{tab:stru-decoder}\n \\begin{tabular}{c|c}\n \\toprule\n Layer & Params (PyTorch-Style) \\cr\n \\hline\n Linear1 & $(128,256)$ \\cr\n ReLU & $\\cdot$\\cr\n Linear2 & $(256,256)$ \\cr\n ReLU & $\\cdot$\\cr\n Linear3 & $(256,256*4*4)$ \\cr\n ReLU & $\\cdot$\\cr\n ConvTranspose1 & $(256,256,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n ConvTranspose2 & $(256,128,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n ConvTranspose3 & $(128,64,4,2,1)$ \\cr\n ReLU & $\\cdot$\\cr\n ConvTranspose4 & $(64,3,4,2,1)$ \\cr\n Tanh & $\\cdot$\\cr\n \\bottomrule\n \\end{tabular}\n \\end{subtable}\n\\end{table*}\n\n\\paragraph{Results}\nFigure \\ref{fig:celeba} includes the attacking AUC of DOMIAS and baselines of 8 runs. DOMIAS consistently outperforms other MIA methods, most of which score not much better than random guessing. These methods fail to attack the 128-dimensional representations of the data (originally $64\\times 64$ pixel images), due to most of them using nearest neighbour or KDE-based approaches. On the other hand, DOMIAS is based on the flow-based density estimator BNAF \\citep{deCao2019BlockFlow}, which is a deeper model that is more apt at handling the high-dimensional data.\n\\begin{figure*}[hbt]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/CelebA_results.png}\n \\caption{\\emph{Attacking performance on CelebA.} DOMIAS scores significantly better at attacking image data compared to baselines.}\n \\label{fig:celeba}\n\\end{figure*}\n\n\n\n \n\\section{HIGH-LEVEL PRIOR KNOWLEDGE} \\label{appx:gaussian_prior}\nIf we have no reference data at all, we can still perform more successful attacks compared to baselines if we have high-level statistics of the underlying distribution. Effectively, any informed prior can improve upon methods that use Eq. \\ref{eq:assumption_prev}; this being a special case of Eq. \\ref{eq:assumption_domias}, where one assumes a uniform prior on $p_R$. In this Appendix, we use the Housing dataset and we assume that we only know the mean and standard deviation of the first variable, median income. This is a very realistic setting in practice, since an adversary can relatively easily acquire population statistics for individual features. We subsequently model the reference dataset distribution $p_{ref}$ as a normal distribution of only the age higher-level statistics---i.e. not making any assumptions on any of the other variables, implicitly putting a uniform prior on these when modelling $p_{ref}$. Otherwise, we use the same training settings as in Experiment 5.1 (left panel Figure 3). In Figure \\ref{fig:high-level statistics}. We see that even with this minimal assumption, we still outperform its ablated versions. These results indicate that a relatively weak prior on the underlying distribution without any reference data, can still provide a relatively good attacker model.\n\n\\begin{figure}[hbt]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{figures\/appx_prior_statistics.png}\n \\caption{\\textit{Using DOMIAS with no reference data but high-level statistics of the underlying data.} Using just the mean and standard deviation of the population's median income, DOMIAS outperforms its ablated counterparts that are based on Eq. \\ref{eq:assumption_prev}. }\n \\label{fig:high-level statistics}\n\\end{figure}\n\n\n\\section{HIGH-PRECISION ATTACKS} \\label{appx:high_precision attacks}\n\\citet{Hu2021MembershipRegions} focus on high-precision membership attacks, i.e. can we attack a small set of samples with high certainty. This is an interesting question, since the risk of high-precision attacks may be hidden if one only looks at overall attacking performance. Their work is not applicable to our setting, e.g. they assume full generator and discriminator access. In this section, we show that even in the full black-box setting high-precision MIAs are a serious risk.\n\n\\subsection{Tabular data}\n\n\\paragraph{Set-up} We assume the same dataset and generative model set-up as in Section 5.3. We study which samples the different methods give the highest score, i.e. mark as most likely to be in $\\mathcal{D}_{mem}$. Let $\\mathcal{D}_{test}$ be a test set consisting for 50\\% of samples $x^i$ in $\\mathcal{D}_{mem}$ and 50\\% samples not in $\\mathcal{D}_{mem}$, respectively denoted by $m=1$ and $m=0$. Let $\\hat{m} = A(x)$ be the attacker's prediction, and let $S(A, \\mathcal{D}_{test},q) = \\{x\\in\\mathcal{D}_{test}|\\hat{m}>Quantile(\\{\\hat{m}^i\\}_i,1-q)\\}$ be the set of samples that are given the $q$-quantile's highest score by attacker $A$. We are interested in the mean membership of this set, i.e. the precision if threshold $Quantile(\\{\\hat{m}^i|x^i\\in\\mathcal{D}_{test}\\},1-q)$ is chosen. We include results for DOMIAS and all baselines. Results are averaged over 8 runs.\n\n\\paragraph{Results} In Figure \\ref{fig:high-precision} we plot the top-score precision-quantile curve for each method for each MIA method, i.e. $P(A, \\mathcal{D}_{test}, q) = \\text{mean}(\\{m|x\\in S(A, \\mathcal{D}_{test}, q)\\})$ as a function of $q$. These figures show the accuracy of a high-precision attacker, if this attacker would choose to attack only the top $q$-quantile of samples. We see that unlike other methods, the precision of DOMIAS goes down almost linearly and more gradually. Though MC and GAN-Leaks are able to find the most overfitted examples, they do not find all---resulting from their flawed underlying assumption Eq. 1 that prohibits them from finding overfitted examples in low-density regions. \n\n\n\\begin{figure*}[hbt]\n \\centering\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_DOMIAS.png}\n \\caption{DOMIAS}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_Eq.1_BNAF.jpg}\n \\caption{Eq. 1 (BNAF)}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_hayes-0.png}\n \\caption{LOGAN 0}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_hayes-D1.png}\n \\caption{LOGAN D1}\n \\end{subfigure} \n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_gan-leaks.png}\n \\caption{GAN-leaks 0}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_gan-leaks-cal.png}\n \\caption{GAN-leaks CAL}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_mc.png}\n \\caption{MC}\n \\end{subfigure}\n \\caption{\\emph{DOMIAS is better at high-precision attacks than baselines on heart failure dataset.} Plotting the top-quantile precision $P(A, \\mathcal{D}_{test}, q)$ versus $q$. For example, if the attacker decides to attack only the $20\\%$ highest samples, we get DOMIAS is significantly more precise ($86.2\\pm 5.5\\%$) compared to baselines---LOGAN D0 ($51.0\\pm 3.9\\%$), LOGAN D1 ($72.6\\pm 5.3\\%$), MC ($74.2\\pm 3.0\\%$), GAN-leaks ($74.9\\pm 3.1\\%$), GAN-Leaks CAL ($57.0\\pm 4.1\\%$). Additionally included is Eq. 1 (BNAF), the ablation attacker that does not make use of the reference data. We see that the reference data helps DOMIAS attack a a larger group with high precision.}\n \\label{fig:high-precision}\n\\end{figure*}\n\n\\subsection{Image data}\nLet us run the same high-precision attack on the CelebA dataset---see Appendix \\ref{sec:celeba}, including settings. Again, we see that high-precision attacks are more successful when using DOMIAS, see Figure \\ref{fig:high-precision-celeba}\n\n\\begin{figure*}[hbt]\n \\centering\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_DOMIAS_celeba.jpg}\n \\caption{DOMIAS}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_Eq.1_BNAF_celeba.jpg}\n \\caption{Eq. 1 (BNAF)}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_LOGAN_0_celeba.jpg}\n \\caption{LOGAN 0}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_LOGAN_D1_celeba.jpg}\n \\caption{LOGAN D1}\n \\end{subfigure} \n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_gan-leaks_celeba.jpg}\n \\caption{GAN-leaks 0}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_GAN-Leaks_CAL_celeba.jpg}\n \\caption{GAN-leaks CAL}\n \\end{subfigure}\n \\begin{subfigure}{0.33\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/high_precision_mc_celeba.jpg}\n \\caption{MC}\n \\end{subfigure}\n \\caption{\\emph{DOMIAS is better at high-precision attacks than baselines on CelebA image data.} For example, an attacker could attack only the examples with top 2\\% scores, and get a precision of $P=65.7\\pm11.6\\%$---much higher than the second-best method LOGAN 0, scoring $P=54.8\\pm6.5\\%$.}\n \\label{fig:high-precision-celeba}\n\\end{figure*}\n\n\n\\section{DISTRIBUTION SHIFT $\\mathcal{D}_{ref}$ AND $\\mathcal{D}_{mem}$} \\label{appx:distributional_shift}\nThere may exist a distributional shift between reference and training data. Because DOMIAS is primarily intended as a tool for data publishers to test their own synthetic data vulnerability, it is recommended that testing is conducted with a reference dataset from the same distribution (e.g. a hold-out set): this effectively tests the worst-case vulnerability. Hence, our work focused on the case where there is no shift.\n\nNonetheless, reference data may not always come from the same target distribution. For example, reference data may come from a different country, or synthetic data may be created by intentionally changing some part of the real data distribution, e.g. to include fairness guarantees \\citep{xu2019achieving,vanBreugel2021DECAF:Networks}. Thus, let us assume there is a shift and that the reference data $\\mathcal{D}_{ref}$ comes from $\\tilde{p}_R$, a shifted version of $p_R$ (i.e. the distribution from which $\\mathcal{D}_{mem}$ is drawn). We give a specific example and run an experiment to explore how this could affect DOMIAS attacking performance.\n\nLet us assume there is a healthcare provider that publishes $\\mathcal{D}_{syn}$, a synthetic dataset of patients suffering from diabetes, based on underlying data $\\mathcal{D}_{mem}\\sim p_R$. Let us assume there is an attacker that has their own data $\\mathcal{D}_{ref}\\sim \\tilde{p}_R$, for which some samples have diabetes ($A=1$), but others do not ($A=0$). We assume that $A$ itself is latent and unobserved (s.t. the attacker cannot just train a classification model) and that there is a shift in the distribution of $A$ (i.e. with a slight abuse of notation $\\tilde{p}_R(A=1)<1$). Diabetes is strongly correlated with other features $X$ in the data, additionally we assume the actual condition distribution $p_R(X|A)$ is fixed across datasets. This implies the reference and membership set distributions can be written respectively as:\n\\begin{align}\n \\tilde{p}_R(X) &= \\tilde{p}_R(A=1)p(X|A=1) + \\tilde{p}_R(A=0)p(X|A=0) \\\\\n p_R(X) &= p(X|A=1)\n\\end{align}\nSince $p_R(X|A=1)\\neq p_R(X|A=0)$ and $\\tilde{p}_R(A=1)\\neq 1$, there is a distributional shift between $\\tilde{p}_R$ and $p_R$.\n\nNow let us see how different attackers perform in this setting as a function of the amount of shift. Evidently, since some of the baselines do not use reference data, some attackers will be unaffected, but we should expect DOMIAS performance to degrade. We take the Heart Failure dataset, which indeed has a feature denoting diabetes,. We vary the amount of shift of $\\tilde{p}_R$ w.r.t. $p_R$, from $\\tilde{p}(A=0)=0$ (no shift), to $\\tilde{p}(A=0)=0.8$ (a large shift and the original Heart Failure non-diabetes prevalence). Let us assume test data follows the attacker's existing dataset, i.e. $\\tilde{p}_R$. This gives Figure \\ref{fig:distributional_shift}.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figures\/covariate_shift1_non-avg.png}\n \\caption{\\textit{Effect of distributional shift on DOMIAS performance.} A distributional shift between $\\mathcal{D}_{mem}$ and $\\mathcal{D}_{ref}$ degrades attacking performance, but preliminary experiments show that for small to moderate shifts it is still preferable to use reference data even though it is slightly shifted.}\n \\label{fig:distributional_shift}\n\\end{figure*}\n\nWe see performance of DOMIAS degrades with increasing shift, due to it approximating $p_R$ with $\\tilde{p}_R$, affecting its scores (Eq. 2). However, we see that for low amounts of shift this degradation is minimal and we still perform beter than not using the reference dataset (baseline Eq. 1 (BNAF)). This aligns well with the results from 5.2, Figure 4, that showed that an inaccurate approximation of $p_R$ due to few samples is still preferable over not using any reference data.\n\n\\subsubsection*{\\bibname}}\n\n\n\n\n\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc} \n\\usepackage{hyperref} \n\\usepackage{url} \n\\usepackage{booktabs} \n\\usepackage{enumerate}\n\n\\usepackage{graphicx}\n\\usepackage{caption\n\\usepackage{subcaption}\n\\usepackage[table]{xcolor}\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{tablefootnote}\n\n\\def\\do\\\/\\do-{\\do\\\/\\do-}\n\n\n\\usepackage{dsfont}\n\n\\newcommand{\\mathds{1}}{\\mathds{1}}\n\\newcommand{\\mathds{E}}{\\mathds{E}}\n\n\\newcommand{\\mathbf{x}}{\\mathbf{x}}\n\\newcommand{\\mathds{R}}{\\mathds{R}}\n\\newcommand{\\mathcal{D}}{\\mathcal{D}}\n\\newcommand{\\cmark}{\\checkmark}%\n\\newcommand{\\xmark}{$\\times$}%\n\n\n\\newtheorem{assumption}{Assumption}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{remark}{Remark}\n\\newtheorem{example}{Example}\n\n\n\\begin{document}\n\n\n\n\\twocolumn[\n\n\\aistatstitle{Membership Inference Attacks against Synthetic Data through Overfitting Detection}\n\n\\aistatsauthor{ Boris van Breugel \\And Hao Sun \\And Zhaozhi Qian \\And Mihaela van der Schaar }\n\n\\aistatsaddress{ University of Cambridge \\And University of Cambridge \\And University of Cambridge \\And University of Cambridge \\\\Alan Turing Institute } ]\n\n\\begin{abstract}\nData is the foundation of most science. Unfortunately, sharing data can be obstructed by the risk of violating data privacy, impeding research in fields like healthcare. Synthetic data is a potential solution. It aims to generate data that has the same distribution as the original data, but that does not disclose information about individuals. Membership Inference Attacks (MIAs) are a common privacy attack, in which the attacker attempts to determine whether a particular real sample was used for training of the model. Previous works that propose MIAs against generative models either display low performance---giving the false impression that data is highly private---or need to assume access to internal generative model parameters---a relatively low-risk scenario, as the data publisher often only releases synthetic data, not the model. In this work we argue for a realistic MIA setting that assumes the attacker has some knowledge of the underlying data distribution. We propose DOMIAS, a density-based MIA model that aims to infer membership by targeting local overfitting of the generative model. Experimentally we show that DOMIAS is significantly more successful at MIA than previous work, especially at attacking uncommon samples. The latter is disconcerting since these samples may correspond to underrepresented groups. We also demonstrate how DOMIAS' MIA performance score provides an interpretable metric for privacy, giving data publishers a new tool for achieving the desired privacy-utility trade-off in their synthetic data.\n\\end{abstract}\n\n\n\\section{INTRODUCTION}\nReal data may be privacy-sensitive, prohibiting open sharing of data and in turn hindering new scientific research, reproducibility, and the development of machine learning itself. Recent advances in generative modelling provide a promising solution, by replacing the \\textit{real} dataset with a \\textit{synthetic} dataset---which retains most of the distributional information, but does not violate privacy requirements. \n\n\\textbf{Motivation} The motivation behind synthetic data is that data is generated \\emph{from scratch}, such that no synthetic sample can be linked back to any single real sample. However, how do we verify that samples indeed cannot be traced back to a single individual? Some generative methods have been shown to memorise samples during the training procedure, which means the synthetic data samples---which are thought to be genuine---may actually reveal highly private information \\citep{Carlini2018TheNetworks}. To mitigate this, we require good metrics for evaluating privacy, and this is currently one of the major challenges in synthetic data \\citep{Jordon2021Hide-and-SeekRe-identification, Alaa2022HowModels}. Differential privacy (DP) \\citep{Dwork2014ThePrivacy} is a popular privacy definition and used in several generative modelling works \\citep{Ho2021DP-GAN:Nets,Torkzadehmahani2020DP-CGAN:Generation,Chen2020GS-WGAN:Generators,Jordon2019PATE-GAN:Guarantees,Long2019G-PATE:Discriminators,Wang2021DataLens:Aggregation,Cao2021DontDivergence}. However, even though DP is theoretically sound, its guarantees are difficult to interpret and many works \\citep{Rahman2018MembershipModel,Jayaraman2019EvaluatingPractice,Jordon2019PATE-GAN:Guarantees,Ho2021DP-GAN:Nets} reveal that for many settings, either the theoretical privacy constraint becomes meaningless ($\\epsilon$ becomes too big), or utility is severely impacted. This has motivated more lenient privacy definitions for synthetic data, e.g. see \\citep{Yoon2020Anonymizationads-gan}. We take an adversarial approach by developing a privacy attacker model---usable as synthetic data evaluation metric that quantifies the practical privacy risk. \n\n\\textbf{Aim} Developing and understanding privacy attacks against generative models are essential steps in creating better private synthetic data. There exist different privacy attacks in machine learning literature---see e.g. \\citep{Rigaki2020ALearning}---but in this work we focus on Membership Inference Attacks (MIAs) \\citep{Shokri2017MembershipModels}. The general idea is that the attacker aims to determine whether a particular sample they possess was used for training the machine learning model. Successful MIA poses a privacy breach, since mere membership to a dataset can be highly informative. For example, an insurance company may possess a local hospital's synthetic cancer dataset, and be interested to know whether some applicant was used for generating this dataset---disclosing that this person likely has cancer \\citep{Hu2022MembershipSurvey}. Additionally, MIAs can be a first step towards other privacy breaches, like profiling or property inference \\citep{DeCristofaro2021ALearning}. \n\nPrevious work in MIA attacks against generative models is inadequate, conveying a false pretense of privacy. In the NeurIPS 2020 Synthetic Data competition \\citep{Jordon2021Hide-and-SeekRe-identification}, none of the attackers were successful at MIA.\\footnote{Specifically, none performed better than random guessing in at least half of the datasets.} Similar negative results were found in the black-box results of \\citep{Liu2019PerformingModels,Hayes2019LOGAN:Models,Hilprecht2019MonteModels, Chen2019GAN-Leaks:Models}, where additional assumptions were explored to create more successful MIAs. Most of these assumptions (see Sec. \\ref{sec:related}) rely on some access to the generator, which we deem relatively risk-less since direct access is often avoidable in practice. Nonetheless, we show that even in the black-box setting---in which we only have access to the synthetic data---MIA can be significantly more successful than appears in previous work, when we assume the attacker has some independent data from the underlying distribution. In Sec. \\ref{sec:MIA_formalism} we elaborate further on why this is a realistic assumption. Notably, it also allows an attacker to perform significantly better attacks against underrepresented groups in the population (Sec. \\ref{sec:underrepresented}).\n\n\\textbf{Contributions} This paper's main contributions are the following.\n\\begin{enumerate}\n \\item We propose DOMIAS: a membership inference attacker model against synthetic data, that incorporates density estimation to detect generative model overfitting. DOMIAS improves upon prior MIA work by i) leveraging access to an independent reference dataset and ii) incorporating recent advances in deep density estimation.\n \\item We compare the MIA vulnerability of a range of generative models, showcasing how DOMIAS can be used as a metric that enables generative model design choices\n \\item We find that DOMIAS is more successful than previous MIA works at attacking underrepresented groups in synthetic data. This is disconcerting and strongly motivates further research into the privacy protection of these groups when generating synthetic data.\n \n\\end{enumerate}\n\n\n\n\\section{MEMBERSHIP INFERENCE: FORMALISM AND ASSUMPTIONS} \\label{sec:MIA_formalism\n\\textbf{Formalism for synthetic data MIA}\nMembership inference aims to determine whether a given sample comes from the training data of some model \\citep{Shokri2017MembershipModels}. Let us formalise this for the generative setting. Let random variable $X$ be defined on $\\mathcal{X}$, with distribution $p_R(X)$. Let $\\mathcal{D}_{mem}\\overset{iid}{\\sim} p_R(X)$ be a training set of independently sampled points from distribution $p_R(X)$. Now let $G:\\mathcal{Z}\\rightarrow \\mathcal{X}$ be a generator that generates data given some random (e.g. Gaussian) noise $Z$. Generator $G$ is trained on $\\mathcal{D}_{mem}$, and is subsequently used to generate synthetic dataset $\\mathcal{D}_{syn}$. Finally, let $A:\\mathcal{X}\\rightarrow [0,1]$ be the attacker model, that possesses the synthetic dataset $\\mathcal{D}_{syn}$, some test point $x^*$, with $X^*\\sim p_R(X)$, and possibly other knowledge---see below. Attacker $A$ aims to determine whether some $x^*\\sim p_R(X)$ they possess, belonged to $\\mathcal{D}_{mem}$, hence the perfect attacker outputs $A(x^*)=\\mathds{1}[x^*\\in\\mathcal{D}_{mem}]$. The MIA performance of an attacker can be measured using any classification metric. \n\n\\textbf{Assumptions on attacker access} The strictest black-box MI setting assumes the attacker only has access to the synthetic dataset $\\mathcal{D}_{syn}$ and test point $x^*$. In this work we assume access to a real data set that is independently sampled from $p_R(X)$, which we will call the reference dataset and denote by $\\mathcal{D}_{ref}$. The main motivation of this assumption is that an attacker needs some understanding of what real data looks like to infer MI---in Sec. \\ref{sec:method} we will elaborate further on this assumption's benefits. Similar assumptions have been made in the supervised learning MI literature, see e.g. \\citep{Shokri2017MembershipModels, Ye2021EnhancedModels}.\nThis is a realistic scenario to consider for data publishers: though they can control the sharing of their own data, they cannot control whether attackers acquires similar data from the general population. A cautious data publisher would assume the attacker has access to a sufficiently large $\\mathcal{D}_{ref}$ to approximate $p_R(X)$ accurately, since this informally bounds the MIA risk from above. Related MI works \\citep{Liu2019PerformingModels,Hayes2019LOGAN:Models,Hilprecht2019MonteModels, Chen2019GAN-Leaks:Models} consider other assumptions that all require access to the synthetic data's generative model.\\footnote{Though with varying extents, see \\citep{Chen2019GAN-Leaks:Models}} These settings are much less dangerous to the data publisher, since these can be avoided by only publishing the synthetic data. Individual assumptions of related works are discussed further in Sec. \\ref{sec:related}.\n\n\n\\begin{figure*}[t]\n \\centering\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/example_transform_dep.png}\n \n \\caption{Generative distribution in original space}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/example_transform_dep_2.png}\n \n \\caption{Distribution in log-transformed space}\n \\end{subfigure}\n \\caption{Should we infer membership $m=1$ for point $A$? Consider the generative distribution for two representations of $X$, optimal methods based on Eq. \\ref{eq:assumption_prev} will infer $m=1$ for green and $m=0$ for red areas. This is problematic; it implies inference of these methods is dependent on the (possibly arbitrary) representation of variable $X$. \\emph{Conclusion: it does not make sense to focus on mere density, MIA needs to target local overfitting directly}. This requires data from (or assumptions on) the underlying distribution.}\n \\label{fig:toy_example_eq1}\n\\end{figure*}\n\n\\section{DOMIAS} \\label{sec:method\n\\subsection{Rethinking the black-box setting: why $\\mathcal{D}_{syn}$ alone is insufficient} \nThe most popular black-box setting assumes only access to $\\mathcal{D}_{syn}$. This gives little information, which is why previous black-box works \\citep{Hayes2019LOGAN:Models, Hilprecht2019MonteModels, Chen2019GAN-Leaks:Models} implicitly assume: \n\\begin{equation}\n\\label{eq:assumption_prev}\n A_{prev}(x^*) = f(p_G(x^*)),\n\\end{equation}\nwhere $A$ indicates the attacker's MIA scoring function, $p_G(\\cdot)$ indicates the generator's output distribution and $f:\\mathds{R}\\rightarrow [0,1]$ is some monotonically increasing function.\nThere are two reasons why Eq. \\ref{eq:assumption_prev} is insufficient. First, the score does not account for the intrinsic distribution of the data. Consider the toy example in Figure \\ref{fig:toy_example_eq2}a. There is a local density peak at $x=4$, but without further knowledge we cannot determine whether this corresponds to an overfitted example or a genuine peak in the real distribution. \\textbf{It is thus naive to think we can do MI without background knowledge}. \n\nSecond, the RHS of Eq. \\ref{eq:assumption_prev} is not invariant w.r.t. bijective transformations of the domain. Consider the left and right plot in Figure \\ref{fig:toy_example_eq1}. Given the original representation, we would infer $M=0$ for any point around $x=4$, whereas in the right plot we would infer $M=1$ for the same points. This dependence on the representation is highly undesirable, as any invertible transformation of the representation should contain the same information. \n\nHow do we fix this? We create the following two desiderata: i) the MI score should target overfitting \\textit{w.r.t. the real distribution}, and ii) it should be independent of representation.\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/example_Eq2_with_baseline.png}\n \n \n \\caption{Original space}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/example_Eq2_transformed_with_baseline.png}\n \n \n \\caption{Log-transformed space}\n \\end{subfigure}\n \\caption{\\emph{DOMIAS scores are not dependent on the feature representation.} This is the same toy example as in Figure \\ref{fig:toy_example_eq1}, where we now assume the bump at $x=4$ has been caused by overfitting in the generator, s.t. this part of the space has become overrepresented w.r.t. the original distribution. DOMIAS infers MI by weighting the generative and real distribution, inferring $m=1$ ($m=0$) for green (red) areas. Note the difference with Figure \\ref{fig:toy_example_eq1}: whereas MI predictions of previous works that use Eq. \\ref{eq:assumption_domias} are dependent on the representation, DOMIAS scores are the same in both domains (Theorem \\ref{theorem:representation}).}\n \\label{fig:toy_example_eq2}\n\\end{figure*}\n\n\n\\subsection{DOMIAS: adding knowledge of the real data.}\nWe need to target overfitting directly. We propose the DOMIAS framework: Detecting Overfitting for Membership Inference Attacks against Synthetic Data. \n\nLet us assume we know the true data distribution $p_R(X)$. We change Eq. \\ref{eq:assumption_prev} to:\n\\begin{equation}\n\\label{eq:assumption_domias}\n A_{\\mathrm{DOMIAS}}(x^*) = f(\\frac{p_G(x^*)}{p_R(x^*)}),\n\\end{equation}\nthat is, we weight Eq. \\ref{eq:assumption_prev} by the real data distribution $p_R(X)$.\\footnote{This work focuses on relative scores, hence we ignore choosing $f$---see Sec. \\ref{sec:discussion}.} Figure \\ref{fig:toy_example_eq2} shows the difference between DOMIAS and previous work using Eq. \\ref{eq:assumption_prev}, by considering the same toy example as in Figure \\ref{fig:toy_example_eq1}. Effectively, Eq. \\ref{eq:assumption_domias} distinguishes between the real and generative distribution, similar in vain to global two-sample tests (e.g. see \\cite{Gretton2012ATest,Arora2019ALearning, Gulrajani2019TowardsGeneralization}). The probability ratio has the advantage that (cf. e.g. probability difference) it is independent of the specific representation of the data:\n\\begin{theorem} \\label{theorem:representation}\nLet $X_G$ and $X_R$ be two random variables defined on $\\mathcal{X}$, with distributions $p_G(X)$ and $p_R(X)$, s.t. $p_G\\ll p_R$, i.e. $p_R$ dominates $p_G$. Let $g:\\mathcal{X}\\rightarrow \\tilde{\\mathcal{X}}, x\\mapsto g(x)$ be some invertible function, and define representations $\\tilde{X}_G = g(X_G)$ and $\\tilde{X}_R=g(X_R)$ with respective distribution $\\tilde{p}_G(\\tilde{X})$ and $\\tilde{p}_R(\\tilde{X})$. Then $\\frac{p_G(X)}{p_R(X)} = \\frac{\\tilde{p}_G(g(X))}{\\tilde{p}_R(g(X))}$, i.e. the same score is obtained for either data representations.\n\\end{theorem}\n\\begin{proof}\nWithout loss of generalisation let us assume continuous variables and almost everywhere continuous $g$. Using the chain rule, we have $\\tilde{p}_{\\cdot}(g(x)) = \\frac{p_\\cdot(x)}{|J(x)|}$ with Jacobian $J(x) = \\frac{dg}{dx}(x)$. Hence we see:\n\\begin{equation*}\n \\frac{\\tilde{p}_G(g(x))}{\\tilde{p}_R(g(x))}= \\frac{p_G(x)\/|J(x)|}{p_R(x)\/|J(x)|} = \\frac{p_G(x)}{p_R(x)}, a.e.\n\\end{equation*}\nas desired.\n\\end{proof}\n\n\n\\textbf{DOMIAS does not purport false privacy safety for underrepresented groups} Figure \\ref{fig:toy_example_eq1}a pinpoints a problem with previous works: methods that rely on assumption Eq. \\ref{eq:assumption_prev} cannot attack low-density regions. As a result, one might conclude that samples in these regions are safer. Exactly the opposite is true: in Figure \\ref{fig:toy_example_eq2} we see DOMIAS infers MI successfully for these samples, whatever the representation. This is distressing, as low-density regions may correspond to underrepresented groups in the population, e.g. ethnic minorities. We will explore this further in the experimental section. \n\n\n\\subsection{Illustrative attacker examples}\nAny density estimator can be used for approximating $p_G(X)$ and $p_R(X)$---e.g. fitting of some parametric family, training a generative model with Monte Carlo Integration, or a deep density estimator. The choice of density estimator should largely depend whether prior knowledge is available---e.g. $p_R$ falls in some parametric family---and on the size of the datasets---for a large dataset a more powerful and more flexible density estimator can be used, whereas for little data this is not suitable as it might lead to overfitting. In the experimental section, we illustrate DOMIAS using the flow-based BNAF \\citep{deCao2019BlockFlow} density estimator, chosen for its training efficiency. For the ablation study in Sec. \\ref{sec:ablation} we also include a Gaussian KDE-based method as a non-parametric alternative.\n\n\\section{RELATED WORK} \\label{sec:related}\n\\textbf{MIAs against generative models} Most of the literature on privacy attacks is focused on discriminative models, not generative models. The few works that are concerned with generative models all focus on membership inference (MIA) \\citep{Shokri2017MembershipModels}. Here we focus on works that can be applied to our attacker setting, see Table \\ref{tab:attacks}.\n\n\\citet{Hayes2019LOGAN:Models} propose LOGAN, a range of MIA attacks for both white-box and black-box access to the generative model, including possible auxiliary information. Two attacks can be applied to our setting. They propose a full black-box attack without auxiliary knowledge (i.e. no reference dataset). This model trains a GAN model on the synthetic data, after which the GAN's discriminator is used to compute the score for test examples. They also propose an attack that assumes an independent test set, similar to DOMIAS' $\\mathcal{D}_{ref}$---see Section 4.1 \\citep{Hayes2019LOGAN:Models}, discriminative setting 1 (D1). Their attacker is a simple classifier that is trained to distinguish between synthetic and test samples. \\citet{Hilprecht2019MonteModels} introduce a number of attacks that focus on approximating the generator distribution at each test point. \nImplicitly, they make assumption \\ref{eq:assumption_prev}, and approximate the probability by using Monte Carlo integration, i.e. counting the proportion of generated points that fall in a given neighbourhood. They do not consider the possible attacker access to a reference dataset. Choosing a suitable distance metric for determining neighbourhoods is non-trivial, however this is somewhat alleviated by choosing a better space in which to compute metrics, e.g. \\citeauthor{Hilprecht2019MonteModels} show that using the Euclidean distance is much more effective when used in conjunction with Principal Component Analysis (PCA). We refer to their method as MC, for Monte Carlo integration.\n\n\\citet{Chen2019GAN-Leaks:Models} give a taxonomy of MIAs against GANs and propose new MIA method GAN-leaks that relies on Eq. \\ref{eq:assumption_prev}. For each test point $x^*$ and some $k\\in\\mathbb{N}$, they sample $S^k_G = \\{x_i\\}_{i=1}^k$ from generator $G$ and use score $A(x^*;G) = \\min_{x_i\\in S^k_G} L_2(x^*, x_i)$ as an unnormalised surrogate for $p_G(x^*)$. \nThey also introduce a calibrated method that uses a reference dataset $\\mathcal{D}_{ref}$ to train a generative reference model $G_{ref}$, giving calibrated score $A(x^*;G,k)-A(x^*;G_{ref},k)$. \nThis can be interpreted as a special case of DOMIAS---Eq. \\ref{eq:assumption_domias}---that approximates $p_R$ and $p_G$ with Gaussian KDEs with infinitesimal kernel width, trained on a random subset of $k$ samples from $\\mathcal{D}_{ref}$ and $\\mathcal{D}_{syn}$. At last, we emphasise that though \\citep{Hayes2019LOGAN:Models, Chen2019GAN-Leaks:Models} consider $\\mathcal{D}_{ref}$ too, they (i) assume this implicitly and just for one of their many models, (ii) do not properly motivate or explain the need for having $\\mathcal{D}_{ref}$, nor explore the effect of $n_{ref}$, and (iii) their MIAs are technically weak and perform poorly as a result, leading to incorrect conclusions on the danger of this scenario (e.g. \\citet{Hayes2019LOGAN:Models} note in their experiments that their D1 model performs no better than random guessing).\n\n\\begin{table*}[bt]\n \\centering\n \\caption{Membership Inference attacks on generative models. (1) Underlying ML method (GAN: generative adversarial network, NN: (weighted) Nearest neighbour, KDE: kernel density estimation, MLP: multi-layer perceptron, DE: density estimator); (2) uses $\\mathcal{D}_{ref}$; (3) approximates Eq. \\ref{eq:assumption_prev} or \\ref{eq:assumption_domias}; (4) by default does not need generation access to generative model---only synthetic data itself. \\textit{\\textsuperscript{\\textdagger}GAN-leaks calibrated is a heuristic correction to GAN-leaks, but implicitly a special case of Eq. \\ref{eq:assumption_domias}.}} \n \\label{tab:attacks}\n \\begin{tabular}{lcccc} \\toprule\n Name & (1) & (2) & (3) & (4)\\\\ \\midrule\n LOGAN 0\\citep{Hayes2019LOGAN:Models} & GAN & \\xmark & Eq. \\ref{eq:assumption_prev}& \\cmark\\\\\n LOGAN D1 \\citep{Hayes2019LOGAN:Models} & MLP & \\cmark & N\/A (heuristic) & \\cmark\\\\\n MC \\citep{Hilprecht2019MonteModels}& NN\/KDE & \\xmark & Eq. \\ref{eq:assumption_prev}& \\xmark\\\\\n GAN-leaks 0 \\citep{Chen2019GAN-Leaks:Models} & NN\/KDE & \\xmark & Eq. \\ref{eq:assumption_prev}& \\xmark\\\\\n GAN-leaks CAL \\citep{Chen2019GAN-Leaks:Models} & NN\/KDE & \\cmark & \\ \\ Eq. \\ref{eq:assumption_domias}\\textsuperscript{\\textdagger} &\\xmark\\\\ \\hline\n DOMIAS (Us) & any DE & \\cmark & Eq. \\ref{eq:assumption_domias}& \\cmark\\\\ \\bottomrule\n \\end{tabular}\n\\end{table*}\n\n\n\\textbf{Stronger attacker access assumptions} Other methods in \\citep{Hayes2019LOGAN:Models, Hilprecht2019MonteModels, Chen2019GAN-Leaks:Models} make much stronger assumptions on attacker access. \\citep{Hayes2019LOGAN:Models} propose multiple attacks with a subset of the training set known, which implies that there has already been a privacy breach---this is beyond the scope of this work. They also propose an attack against GANs that uses the GANs discriminator to directly compute the MIA score, but discriminators are usually not published. \\citet{Chen2019GAN-Leaks:Models} propose attacks with white-box access to the generator or its latent code, but this scenario too can be easily avoided by not publishing the generative model itself. All methods in \\citep{Hilprecht2019MonteModels, Chen2019GAN-Leaks:Models} assume unlimited generation access to the generator (i.e. infinitely-sized $\\mathcal{D}_{syn}$), which is unrealistic for a real attacker---either on-demand generation is unavailable or there is a cost associated to it that effectively limits the generation size \\citep{DeCristofaro2021ALearning}. These methods can still be applied to our setting by sampling from the synthetic data directly.\n\n\n\\textbf{Tangential work}\nThe following MIA work is not compared against. \\citet{Liu2019PerformingModels,Hilprecht2019MonteModels} introduce \\textit{co-membership} \\citep{Liu2019PerformingModels} or \\textit{set MIA} \\citep{Hilprecht2019MonteModels} attacks, in which the aim is to determine for a whole set of examples whether either all or none is used for training. Generally, this is an easier attack and subsumes the task of single attacks (by letting the set size be 1).\n\\citet{Webster2021ThisFaces} define the \\textit{identity} membership inference attack against face generation models, which aims to infer whether some person was used in the generative model (but not necessarily a specific picture of that person). This requires additional knowledge for identifying people in the first place, and does not apply to our tabular data setting. \\citet{Hu2021MembershipRegions} focus on performing high-precision attacks, i.e. determining MIA for a small number of samples with high confidence. Similar to us they look at overrepresented regions in the generator output space, but their work assumes full model access (generator and discriminator) and requires a preset partitioning of the input space into regions. \\citep{Zhang2022MembershipData} is similar to \\citep{Hilprecht2019MonteModels}, but uses contrastive learning to embed data prior to computing distances. In higher dimensions, this can be an improvement over plain data or simpler embeddings like PCA---something already considered by \\citep{Hilprecht2019MonteModels}. However, the application of contrastive learning is limited when there is no \\textit{a priori} knowledge for performing augmentations, e.g. in the unstructured tabular domain. \n \nOn a final note, we like to highlight the relation between MIA and the evaluation of overfitting, memorisation and generalisation of generative models. The latter is a non-trivial task, e.g. see \\citep{Gretton2012ATest,Lopez-Paz2016RevisitingTests,Arora2017GeneralizationGANs,Webster2019DetectingRecovery, Gulrajani2019TowardsGeneralization}. DOMIAS targets overfitting directly and locally through Eq. \\ref{eq:assumption_domias}, a high score indicating local overfitting. \nDOMIAS differs from this line of work by focusing on MIA, requiring sample-based scores. DOMIAS scores can be used for interpreting overfitting of generative models, especially in the non-image domain where visual evaluation does not work. \n \n\\section{EXPERIMENTS} \\label{sec:experiments}\nWe perform experiments showing DOMIAS' value and use cases. In Sec. \\ref{sec:domias_vs_baselines} we show how DOMIAS outperforms prior work, in Sec. \\ref{sec:ablation} we explore why. Sec. \\ref{sec:underrepresented} demonstrates how underrepresented groups in the population are most vulnerable to DOMIAS attack, whilst Sec. \\ref{sec:generative_model_comparison} explores the vulnerability of different generative models---showcasing how DOMIAS can be used as a metric to inform synthetic data generation. For fair evaluation, the same experimental settings are used across MIA models (including $n_{ref}$). Details on experimental settings can be found in Appendix \\ref{appx:experimental_details}.\\footnote{Code is available at \\\\ \\href{https:\/\/github.com\/vanderschaarlab\/DOMIAS}{https:\/\/github.com\/vanderschaarlab\/DOMIAS}} \n\n\\subsection{DOMIAS outperforms prior MIA methods} \\label{sec:domias_vs_baselines}\n\\begin{figure*}[hbt]\n\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/n_M.png}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/t_epochs.png}\n \n \\end{subfigure}\n \\caption{\\emph{DOMIAS outperforms baselines.} MIA performance of DOMIAS and baselines versus the generative model training set size $|\\mathcal{D}_{mem}|$ and training time $t_{epochs}$ on the California Housing dataset. We observe how MIA AUC goes up for fewer training samples and long generative model training time, as both promote overfitting.}\n \\label{fig:domias_vs_baselines}\n\\end{figure*}\n\n\\textbf{Set-up} We use the California Housing Dataset \\citep{Pace1997SparseAutoregressions} and use TVAE \\citep{Xu2019ModelingGAN} to generate synthetic data. In this experiment we vary the number of TVAE training samples $|\\mathcal{D}_{mem}|$ and TVAE number of training epochs. We compare DOMIAS against LOGAN 0 and LOGAN D1 \\citep{Hayes2019LOGAN:Models}, MC \\citep{Hilprecht2019MonteModels}, and GAN-Leaks 0 and GAN-Leaks CAL \\citep{Chen2019GAN-Leaks:Models}---see Table \\ref{tab:attacks}.\n\n\\textbf{DOMIAS consistently outperforms baselines}\nFigure \\ref{fig:domias_vs_baselines}(a) shows the MIA accuracy of DOMIAS and baselines against TVAE's synthetic dataset, as a function of the number of training samples TVAE $n_{mem}$. For small $n_{mem}$ TVAE is more likely to overfit to the data, which is reflected in the overall higher MIA accuracy. Figure \\ref{fig:domias_vs_baselines}(b) shows the MIA accuracy as a function of TVAE training epochs. Again, we see TVAE starts overfitting, leading to higher MIA for large number of epochs. \n\nIn both plots, we see DOMIAS consistently outperforms baseline methods. Similar results are seen on other datasets and generative models, see Appendix \\ref{appx:additional_results}. Trivially, DOMIAS should be expected to do better than GAN-Leaks 0 and LOGAN 0, since these baseline methods do not have access to the reference dataset and are founded on the flawed assumption of Eq. \\ref{eq:assumption_prev}---which exposes the privacy risk of attacker access to a reference dataset.\n\n\\begin{figure*}[hbt]\n \\centering\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/n_ref.png}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figures\/n_G.png}\n \n \\end{subfigure}\n \\caption{\\emph{DOMIAS source of gain.} Ablation study of DOMIAS on the California Housing dataset, with attack performance as a function of the reference dataset size (left) and the synthetic dataset size (right). We see that the MIA performance of DOMIAS is largely due to assumption Eq. \\ref{eq:assumption_domias} vs. Eq. \\ref{eq:assumption_prev}, i.e. the value of the reference dataset. The deep flow-based density estimator delivers gains over the simpler KDE approach when enough samples are available.}\n \\label{fig:results_ablation}\n\\end{figure*}\n\n\\subsection{Source of gain} \\label{sec:ablation}\nUsing the same set-up as before, we perform an ablation study on the value of i) DOMIAS' use of the reference set, and ii) the deep density estimator. For the first, we compare using the DOMIAS assumption (Eq. \\ref{eq:assumption_domias}) vs the assumption employed in many previous works (Eq. \\ref{eq:assumption_prev}). For the latter, we compare the results for density estimation based on the flow-based BNAF \\citep{deCao2019BlockFlow} versus a Gaussian kernel density estimator---kernel width given by the heuristic from \\citep{Scott1992MultivariateEstimation}. \n\nFigure \\ref{fig:results_ablation} shows the MIA performance as a function of $n_{syn}$ and $n_{ref}$. Evidently, the source of the largest gain is the use of Eq. \\ref{eq:assumption_domias} over Eq. \\ref{eq:assumption_prev}. As expected, the deep density estimator gives further gains when enough data is available. For lower amounts of data, the KDE approach is more suitable. This is especially true for the approximation of $p_R$ (the denominator of Eq. \\ref{eq:assumption_domias})---small noise in the approximated $p_R$ can lead to large noise in MIA scores. Also note in the right plot that MIA performance goes up with $|\\mathcal{D}_{syn}|$ across methods due to the better $p_G$ approximation; this motivates careful consideration for the amount of synthetic data published.\n\n\\subsection{Underrepresented group MIA vulnerability} \\label{sec:underrepresented}\n\\textbf{Set-up} We use a private medical dataset on heart failure, containing around $40,000$ samples with $35$ mixed-type features (see Appendix \\ref{appx:experimental_details}). We generate synthetic data using TVAE \\citep{Xu2019ModelingGAN}.\n\n\\begin{figure*}[bt]\n\n \\centering\n \n \n \\includegraphics[width=\\textwidth]{figures\/experiment_53.png}\n \\caption{\\emph{DOMIAS is more successful at attacking patients taking high-blood pressure medication. }(left) T-SNE plot of Heart Failure test dataset. There is a cluster of points visible in the top right corner, which upon closer inspection corresponds to subjects who take ARB medication. (right, bottom) Attacking accuracy of DOMIAS and baselines on majority and minority group (averaged over 8 runs). DOMIAS is significantly better at attacking the minority group than the general population. Except for GAN-leaks CAL, baselines fail to capture the excess privacy risk to the patients with blood pressure medication. Comparing DOMIAS with Eq. 1 (BNAF) (see Sec. \\ref{sec:ablation}), we see that the minority vulnerability is largely due to the availability of the reference data. (right, top) Single run attacking success of different MIA methods on these underrepresented samples; correctly inferred membership in green, incorrectly inferred in red.}\n \\label{fig:vulnerable}\n\\end{figure*}\n\n\n\\textbf{Minority groups are most vulnerable to DOMIAS attack} As seen in Sec. \\ref{sec:method}, the assumption underlying previous work (Eq. \\ref{eq:assumption_prev}) will cause these methods to never infer membership for low-density regions. This is problematic, as it gives a false sense of security for these groups---which are likely to correspond to underrepresented groups.\n\nThe left side of Figure \\ref{fig:vulnerable} displays a T-SNE embedding of the Heart Failure dataset, showing one clear minority group, drawn in blue, which corresponds to patients that are on high-blood pressure medication---specifically, Angiotensin II receptor blockers. The right side of Figure \\ref{fig:vulnerable} shows the performance of different MIA models. DOMIAS is significantly better at attacking this vulnerable group compared to the overall population, as well as compared to other baselines. This is not entirely surprising; generative models are prone to overfitting regions with few samples. Moreover, this aligns well with supervised learning literature that finds additional vulnerability of low-density regions, e.g. \\citep{Kulynych2019DisparateAttacks, Bagdasaryan2019DifferentialAccuracy}. Importantly, most MIA baselines give the false pretense that this minority group is \\textit{less vulnerable}. Due to the correspondence of low-density regions and underrepresented groups, \\emph{these results strongly urge further research into privacy protection of low-density regions when generating synthetic data.} \n\n\\begin{figure}[hbt]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/Eval_GANs.png}\n \\caption{\\emph{DOMIAS can be used to quantify synthetic data MIA vulnerability.} We plot the synthetic data quality versus DOMIAS AUC for different generative models on the California Housing dataset. There is a clear trade-off: depending on the tolerated MIA vulnerability, different synthetic datasets are best.}\n \\label{fig:generative_model_comparison}\n \n\\end{figure}\n\n\\subsection{DOMIAS informs generative modelling decisions} \\label{sec:generative_model_comparison} \n\\textbf{Set-up} Again we use the California Housing dataset, this time generating synthetic data using different generative models. We evaluate the quality and MIA vulnerability of GAN, \\citep{Goodfellow2014GenerativeNetworks}, WGAN-GP \\citep{Arjovsky2017WassersteinNetworks,Gulrajani2017ImprovedGANs}, CTGAN and TVAE \\citep{Xu2019ModelingGAN},\nNFlow \\citep{Durkan2019NeuralFlows}, PATE-GAN \\citep{Jordon2019PATE-GAN:Guarantees}, PrivBayes \\citep{Zhang2017Privbayes:Networks}, and ADS-GAN \\citep{Yoon2020Anonymizationads-gan}. As a baseline, we also include the anonymization method of sampling from training data and adding Gaussian noise. For ADS-GAN and the additive noise model, we vary the privacy level by raising the hyperparameter $\\lambda$ and noise variance, respectively. Results for other attackers are found in Appendix \\ref{appx:additional_results}.\n\n\\textbf{DOMIAS quantifies MIA vulnerability}\nFigure \\ref{fig:generative_model_comparison} presents the DOMIAS MIA AUC against the data quality (in terms of Wasserstein Distance to an independent hold-out set), averaged over eight runs. We see a clear privacy-utility trade-off, with the additive noise model giving a clean baseline. The NeurIPS 2020 Synthetic Data competition \\citep{Jordon2021Hide-and-SeekRe-identification} concluded that disappointingly, adding noise usually outperformed generative models in terms of the privacy-utility trade-off. Though we find this is true for WGAN-GP, PATE-GAN and CTGAN---which fall on the right side of the additive noise curve---other methods do yield better synthetic datasets. \n\nADS-GAN is based on WGAN-GP, hence for small $\\lambda$ (the privacy regularizer) it gets a similar score. Increasing $\\lambda$ promotes a higher distance between generated and training data, hence this reduces vulnerability. At first, it also leads to an increase in quality---raising $\\lambda$ leads to lower overfitting---but when $\\lambda$ increases further the generative distribution is distorted to the point that quality is significantly reduced. In contrast to \\citep{Hilprecht2019MonteModels}, we do not find evidence that VAEs are more vulnerable to MIAs than GANs. The Pareto frontier is given by the additive noise method, TVAE, NFlow and PrivBayes, hence the best synthetic data model will be one of these, depending on the privacy requirements.\n\n\\section{DISCUSSION} \\label{sec:discussion}\n\\textbf{DOMIAS use cases} DOMIAS is primarily a tool for evaluating and interpreting generative model privacy. The overall DOMIAS attacking success is a metric for MIA vulnerability, and may hence guide generative model design choices---e.g. choosing privacy parameters---or aid evaluation---including for competitions like \\citep{Jordon2021Hide-and-SeekRe-identification}. Since DOMIAS provides a sample-wise metric, its scores can also provide insight into privacy and overfitting of specific samples or regions in space---as seen in Sec. \\ref{sec:underrepresented}. Future work may adopt DOMIAS for active privacy protection, e.g. as a loss during training or as an auditing method post-training---removing samples that are likely overfitted.\n\n\\textbf{Underrepresented groups are more vulnerable to MIA attacks} Generative models are more likely to overfit low-density regions, and we have seen DOMIAS is indeed more successful at attacking these samples. This is distressing, since these regions can correspond to underrepresented groups in the population. Similar results have been found in supervised learning literature, e.g. \\citep{Kulynych2019DisparateAttacks, Bagdasaryan2019DifferentialAccuracy}. Protecting against this vulnerability is a trade-off, as outliers in data can often be of interest to downstream research. It is advisable data publishers quantify the excess MIA risk to specific subgroups.\n\n\\textbf{Attacker calibration} In practice, it will often be unknown how much of the test data was used for training. Just like related works, we have ignored this. This challenge is equivalent to choosing a suitable threshold, or suitable $f$ in Eq. \\ref{eq:assumption_domias} and relates closely to calibration of the attacker model, which is challenging for MIA since---to an attacker---usually no ground-truth labels are available. Future work can explore assumptions or settings that could enable calibrated attacks. In Appendix \\ref{appx:high_precision attacks} we include results for high-precision attacks.\n\n\\textbf{High-dimensionality and image data} Traditional density estimation methods (e.g. KDE) perform notoriously poorly in high dimensions. Recent years have seen a rise in density estimation methods that challenge this conception. Domain-specific density estimators, e.g. that define density on lower-dimensional embeddings, can be readily used in DOMIAS. We include preliminary results for the high-dimensional CelebA image dataset in Appendix \\ref{sec:celeba}.\n\n\\textbf{Training data size} We have seen that for large number of training samples, the performance of all attackers goes down to almost 0.5. The same is observed for large generative image models, Appendix \\ref{sec:celeba}. This is reassuring for synthetic data publishers, for whom this indicates a relatively low privacy risk globally. However, global metrics may hide potential high-precision attacks on a small number of individuals, see Appendix \\ref{appx:high_precision attacks}.\n\n\\textbf{Availability of reference dataset} DOMIAS assumes the presence of a reference dataset that enables approximating the true distribution $p_R(X)$. In case there is not sufficient data for the latter, more prior knowledge can be included in the parametrisation of $p_R$; e.g. choose $p_R(X)$ to lie in a more restrictive parametric family. Even in the absence of any data $\\mathcal{D}_{ref}$, an informed prior (e.g. Gaussian) based on high-level statistics can already improve upon related works that rely on assumption Eq. \\ref{eq:assumption_prev}---see Appendix \\ref{appx:gaussian_prior} for results. In Appendix \\ref{appx:distributional_shift} we include further experiments with distributional shifts between the $\\mathcal{D}_{ref}$ and $\\mathcal{D}_{mem}$, in which we find that even with moderate shifts the use of a reference dataset is beneficial.\n\n\n\n\\textbf{Publishing guidelines} Synthetic data does not guarantee privacy, however the risk of MIA attacks can be lessened when synthetic data is published considerately. Publishing just the synthetic data---and not the generative model---will in most cases be sufficient for downstream research, while avoiding more specialised attacks that use additional knowledge. Further consideration is required with the amount of data published: increasing the amount of synthetic data leads to higher privacy vulnerability (Figure \\ref{fig:results_ablation}b and see \\citep{Gretton2012ATest}). Though the amount of required synthetic data is entirely dependent on the application, DOMIAS can aid in finding the right privacy-utility trade-off.\n\n\\textbf{Societal impact} We believe DOMIAS can provide significant benefits to the future privacy of synthetic data, and that these benefits outweigh the risk DOMIAS poses as a more successful MIA method. On a different note, we highlight that success of DOMIAS implies privacy is not preserved, but not vice versa. Specifically, DOMIAS should not be used as a certificate for data privacy. Finally, we hope the availability of a reference dataset is a setting that will be considered in more ML privacy work, as we believe this is more realistic in practice than many more popular MIA assumptions (e.g. white-box generator), yet still poses significant privacy risks. \n\n\n\n\n\\subsubsection*{Acknowledgements}\nWe would like to thank the Office of Navel Research UK, who funded this research.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec1}}\n\n\nThis paper is motivated by a class of problems in graph deep learning, where the\nprimary task is either graph classification or graph regression. \nIn either case, the result should be invariant to arbitrary permutations of graph nodes.\n\nAs we explain below, the mathematical problem analyzed in this paper is a special case \nof the permutation invariance issue described above. To set the notations consider the\nvector space ${\\mathbb{R}}^{n\\times d}$ of $n\\times d$ matrices endowed with the Frobenius norm \n $\\norm{X}=\\left(trace(XX^T)\\right)^{1\/2}$\nand its associated Hilbert-Schmidt scalar product, $\\ip{X}{Y}=trace(XY^T)$.\n Let ${\\mathcal S}_n$ denote the symmetric group of $n\\times n$ permutation matrices. \n ${\\mathcal S}_n$ is a finite group of size $|{\\mathcal S}_n|=n!$.\n\nOn ${\\mathbb{R}}^{n\\times d}$ we consider the equivalence relation $\\sim$ \ninduced by the symmetric group of permutation matrices ${\\mathcal S}_n$ as follows. Let $X,Y\\in{\\mathbb{R}}^{n\\times d}$. \nThen we say $X\\sim Y$ if there is $P\\in{\\mathcal S}_n$ so that $Y=PX$. In other words, two matrices are equivalent if one is a row permutation of the other. \nThe equivalence relation induces a natural distance on the quotient space \n${\\widehat{\\Rnd}}:={\\mathbb{R}}^{n\\times d}\/\\sim$,\n\\begin{equation}\n \\label{eq:1.1}\nd: {\\widehat{\\Rnd}}\\times {\\widehat{\\Rnd}} \\rightarrow\\mathbb{R} ~~,~~d(\\hat{X},\\hat{Y})=\\min_{\\Pi\\in{\\mathcal S}_n}\\norm{X-\\Pi Y} \n\\end{equation}\nThis makes $({\\widehat{\\Rnd}},d)$ a complete metric space.\n\nOur main problem can now be stated as follows:\n\\begin{prob}\\label{prob1}\nGiven $n,d\\geq 1$ positive integers, find $m$ and a bi-Lipschitz map\n $\\hat{\\alpha}:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^m,\\norm{\\cdot}_2)$.\n\\end{prob}\nExplicitly the problem can be restated as follows. One is asked to construct a \nmap $\\alpha:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m$ that satisfies the following conditions:\n\\begin{enumerate}\n \\item If $X,Y\\in{\\mathbb{R}}^{n\\times d}$ so that $X\\sim Y$ then $\\alpha(X)=\\alpha(Y)$\n \\item If $X,Y\\in{\\mathbb{R}}^{n\\times d}$ so that $\\alpha(X)=\\alpha(Y)$ then $X\\sim Y$\n \\item There are constants $01$. \n \\item {\\em Sorting Embedding}.\n For $x\\in\\mathbb{R}^n$, consider the sorting map\n\\begin{equation}\\label{eq:ord}\n \\downarrow:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n ~~,~~\n \\downarrow(x)=(x_{\\pi(1)},x_{\\pi(2)},\\ldots,\n x_{\\pi(n)})^T \n\\end{equation}\n where the permutation $\\pi$ is so that\n $x_{\\pi(1)}\\geq x_{\\pi(2)}\\geq\\cdots\\geq x_{\\pi(n)}$. It is obvious that $\\downarrow$ satisfies Conditions (1) and (2) and therefore lifts to an injective map on ${\\widehat{\\Rnd}}$. As we see in Section \\ref{sec3}, the map $\\downarrow$ is bi-Lipschitz. In fact it is isometric, and hence produces an ideal embedding. Our work in Section \\ref{sec3} is to extend such construction to the more general \n case $d>1$.\n\\end{enumerate}\nThe algebraic embedding is a special case of the more general {\\em kernel method} that can be thought of as a projection of the measure \n$a_{\\infty}(X)$ onto a finite dimensional space, e.g., the space of polynomials spanned by $\\{X,X^2,\\cdots,X^n\\}$. In applications such kernel method is known as a ``Readout Map\" \\cite{deepsets}, based on ``Sum Pooling\".\n\nThe sorting embedding has been used in applications under the name of ``Pooling Map\" \\cite{deepsets}, based on ``Max Pooling\". A na\\\"{\\i}ve extension of the unidimensional map (\\ref{eq:ord}) to the case $d>1$ might employ the lexicographic order: order monotone decreasing the rows according to the first column, and break the tie by going to the next column. While this gives rise to an injective map, it is easy to see it is not even continuous, let alone Lipschitz. The main work in this paper is to extend the sorting embedding to the case $d>1$ using a three-step procedure, first embed ${\\mathbb{R}}^{n\\times d}$ into a larger vector space $\\mathbb{R}^{n\\times D}$, then apply $\\downarrow$ in each column independently, and then perform a dimension reduction by a linear map into $\\mathbb{R}^{2nd}$. Similar to the phase retrieval problem (\\cite{bcmn,bod,balan16}), the redundancy introduced in the first step counterbalances the loss of information (here, relative order of one column with respect to another) in the second step. \n\nA summary of main results presented in this paper is contained in the following result.\n\\begin{thm}\\label{t1}\nConsider the metric space $({\\widehat{\\Rnd}},d)$.\n\\begin{enumerate}\n\\item (Polynomial Embedding) There exists a Lipschitz injective map\n\\[ {\\hat{\\alpha}}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m \\]\nwith $m=\\left( \\begin{array}{c}\n\\mbox{$d+n$} \\\\\n\\mbox{$d$}\n\\end{array} \\right)$. Two explicit constructions of this map are given in (\\ref{eq:alpha1}) and (\\ref{eq:alpha2}).\n\\item (Sorting based Embedding) There exists a class of bi-Lipschitz maps \n\\[ {\\hat{\\beta}}_{A,B}:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^m,\\norm{\\cdot}) ~,~ {\\hat{\\beta}}_{A,B}(\\hat{X})=B\\left({\\hat{\\beta}}_A(\\hat{X})\\right) \\]\nwith $m=2nd$, where each map ${\\hat{\\beta}}_{A,B}$ is the composition of two bi-Lipschitz maps: a full-rank linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow \\mathbb{R}^m$, with the nonlinear bi-Lipschitz map ${\\hat{\\beta}}_A:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{n\\times D}$\n parametrized by a matrix $A\\in\\mathbb{R}^{d\\times D}$ called \"key\". \n Explicitly,\n${\\hat{\\beta}}(\\hat{X})=\\downarrow(XA)$, where $\\downarrow$ acts column-wise.\n These maps are characterized by the following properties:\n \\begin{enumerate}\n\\item For $D=1+(d-1)n!$, any\n$A\\in\\mathbb{R}^{d\\times (1+(d-1)n!)}$ whose columns form a full spark frame defines a bi-Lipschitz map ${\\hat{\\beta}}_A$ on ${\\widehat{\\Rnd}}$. \nFurthermore, a lower Lipschitz constant is given by the smallest $d^{th}$ singular value among all $d\\times d$ sub-matrices of $A$,\n $\\min_{J\\subset[D],|J|=d}s_d(A[J])$.\n\\item For any matrix (``key\") $A\\in\\mathbb{R}^{d\\times D}$ such that the map \n${\\hat{\\beta}}_A$ is injective, then ${\\hat{\\beta}}_A:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^{n\\times D},\\norm{\\cdot})$ is bi-Lipschitz. Furthermore, an upper Lipschitz constant is given by $s_1(A)$, the largest singular value of $A$.\n\\item Assume $A\\in\\mathbb{R}^{d\\times D}$ is such that the map \n${\\hat{\\beta}}_A$ is injective (i.e., a \"universal key\"). Then for almost any linear map $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$ the map ${\\hat{\\beta}}_{A,B}=B\\circ{\\hat{\\beta}}_A$ is\nbi-Lipschitz.\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\nAn immediate consequence of this result is the following corollary whose proof is included in subsection \\ref{subsec4.4}:\n\\begin{cor}\n\\label{c0}\nLet $\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}^m$ induce a bi-Lipschitz embedding ${\\hat{\\beta}}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m$ of the\nmetric space $({\\widehat{\\Rnd}},d)$ into $(\\mathbb{R}^m,\\norm{\\cdot}_2)$. \n\\begin{enumerate}\n\\item For any continuous function $f:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ \ninvariant to row-permutation (i.e., $f(PX)=f(X)$ for every \n$X\\in\\mathbb{R}^{n\\times d}$ and $P\\in{\\mathcal S}_n$) there exists a continuous\nfunction $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ such that $f=g\\circ\\beta$.\nConversely, for any $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ continuous function, the \nfunction $f=g\\circ\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ is continuous\nand row-permutation invariant.\n\\item For any Lipschitz continuous function $f:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ \ninvariant to row-permutation (i.e., $f(PX)=f(X)$ for every \n$X\\in\\mathbb{R}^{n\\times d}$ and $P\\in{\\mathcal S}_n$) there exists a Lipschitz continuous\nfunction $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ such that $f=g\\circ\\beta$.\nConversely, for any $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ Lipschitz continuous function, the \nfunction $f=g\\circ\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ is Lipschitz continuous\nand row-permutation invariant.\n\\end{enumerate}\n\\end{cor}\n\\vspace{5mm}\n\n\nThe structure of the paper is as follows. Section \\ref{sec2} contains the algebraic embedding method and encoders $\\alpha$ described at part (1) of Theorem \\ref{t1}. Corollary \\ref{cor2} contains part (1) of the main result stated above. Section \\ref{sec3} introduces the sorting based embedding procedure and describes the key-based encoder $\\beta$. Necessary and sufficient conditions for key universality are presented in Proposition\n\\ref{prop3.8}; the injectivity of the encoder described at part (2.a) of Theorem \\ref{t1} is proved in Theorem \\ref{t4}; the bi-Lipschitz property of any universal key described at part (2.b) of Theorem \\ref{t1} is shown in Theorem \\ref{t5}; the dimension reduction statement (2.c) of Theorem \\ref{t1}\nis included in Theorem \\ref{t6}. Proof of Corollary \\ref{c0} is presented in subsection \\ref{subsec4.4}. Section \\ref{sec4} contains applications to graph deep learning. These application use Graph Convolution Networks and the numerical experiments are carried out on two graph data sets: a chemical compound data set (QM9) and a protein data set (PROTEINS\\_FULL). \n\nWhile the motivation of this analysis is provided by graph deep learning applications,\nthis is primarily a mathematical paper. Accordingly the formal theory is presented first, and then is followed by the machine learning application. Those interested in the application (or motivation) can skip directly to Section \\ref{sec4}. \n\n{\\bf Notations}. For an integer $d\\geq 1$, $[d]=\\{1,2,\\ldots,d\\}$. For a matrix $X\\in{\\mathbb{R}}^{n\\times d}$,\n $x_1,\\ldots x_d\\in\\mathbb{R}^n$ denote its columns, $X=[x_1\\vert\\cdots\\vert x_d]$. All norms are Euclidean; for a matrix $X$, $\\norm{X}=\\sqrt{trace(X^TX)}=\\sqrt{\\sum_{k,j}|X_{k,j}|^2}$ denotes the Frobenius norm; for vectors $x$, $\\norm{x}=\\norm{x}_2=\\sqrt{\\sum_{j} |x_j|^2}$. \n \n\n\\subsection{Prior Works}\n\nSeveral methods for representing orbits of vector spaces under the action of permutation (sub)groups have been studied in literature. Here we describe some of these results, without claiming an exhaustive literature survey.\n\nA rich body of literature emanated from the early works on \nsymmetric polynomials and group invariant representations of \nHilbert, Noether, Klein and Frobenius. They are part of standard\ncommutative algebra and finite group representation theory. \n\nPrior works on permutation invariant mappings have predominantly employed some form of summing procedure, though some have alternatively employed some form of sorting procedure.\n\nThe idea of summing over the output nodes of an equivariant network has been well studied. \nThe algebraic invariant theory goes back to Hilbert and Noether (for finite groups) and then continuing with the continuous invariant function theory of \nWeyl and Wigner (for compact groups), \nwho posited that a generator function $\\psi:X\\rightarrow\\mathbb{R}$ gives rise to a function $E:X\\rightarrow\\mathbb{R}$ invariant to the action of a finite group $G$ on $X$, $(g,x)\\mapsto g.x$, via the averaging formula $E(x)=\\frac{1}{|G|}\\sum_{g\\in G} \\psi(g.x)$.\n\nMore recently, this approach provided the framework for universal approximation results of $G$-invariant functions. \\cite{maron2018invariant} showed that invariant or equivariant networks must satisfy a fixed point condition. The equivariant condition is naturally realized by GNNs. The invariance condition is realized by GNNs when followed by summation on the output layer, as was further shown in \\cite{keriven2019universal}, \\cite{pmlr-v97-maron19a} and \\cite{lipman2022}. Subsequently, \\cite{yarotsky2021universal} proved universal approximation results over compact sets for continuous functions invariant to the action of finite or continuous groups. In \\cite{geerts2022}, the authors\nobtained bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests by tensorizing the input-output mapping. \n\\cite{sannai2020universal} studied approximations of equivariant maps, while \\cite{NEURIPS2019_71ee911d} showed that if a GNN with sufficient expressivity is well trained, it can solve the graph isomorphism problem.\n\nThe authors of \\cite{OrderMatters_2015arXiv151106391V} designed an algorithm for processing sets with no natural orderings. The algorithm applies an attention mechanism to achieve permutation invariance with the attention keys being generated by a Long-Short Term Memory (LSTM) network. Attention mechanisms amount to a weighted summing and therefore can be considered to fall within the domain of summing based procedures.\n\nIn \\cite{GGsNN_2015arXiv151105493L}, the authors designed a permutation invariant mapping for graph embeddings. The mapping employs two separate neural networks, both applied over the feature set for each node. One neural network produces a set of new embeddings, the other serves as an attention mechanism to produce a weighed sum of those new embeddings.\n\n\n\n\nSorting based procedures for producing permutation invariant mappings over single dimensional inputs have been addressed and used by \\cite{deepsets}, notably in their {\\it max pooling} procedure.\n\nThe authors of \\cite{qi2017pointnet} developed a permutation\ninvariant mapping \n$pointnet$ for point sets that is based on a $max$ function. The mapping takes in a set of vectors, processes each vector through a neural network followed by an scalar output function, and takes the maximum of the resultant set of scalars.\n\nThe paper \\cite{zhang2018end} introduced {\\it SortPooling}. {\\it SortPooling} orders the latent embeddings of a graph according to the values in a specific, predetermined column. All rows of the latent embeddings are sorted according to the values in that column. While this gives rise to an injective map, it is easy to see it is not even continuous, let alone Lipschitz. The same issue\narises with any lexicographic ordering, including the well-known Weisfeiler-Leman embedding \\cite{wl}.\nOur paper introduces a novel method that bypasses this issue.\n\nAs shown in \\cite{pmlr-v97-maron19a}, the sum pooling-based GNNs provides universal approximations for of any permutation invariant continuous function but only on \\emph{compacts}. Our sorting based embedding removes the compactness restriction as well as it extends to all Lipschitz maps.\n\nWhile this paper is primarily mathematical in nature, methods developed here are applied to two graph data sets, QM9 and PROTEINS\\_FULL. Researchers have applied various graph deep learning techniques to both data sets. In particular, \\cite{Gilmer_2017arXiv170401212G} studied extensively the QM9 data set, and compared their method with many other algorithms\nproposed by that time.\n\n\\section{Algebraic Embeddings\\label{sec2}}\n\nThe algebraic embedding presented in this section can be thought of a special kernel to project equation (\\ref{eq:measure}) onto.\n\n\\subsection{Kernel Methods}\nThe kernel method employs a family of continuous kernels (test) functions, $\\{K(x;y)~;~x\\in\\mathbb{R}^d~,~y\\in Y\\}$ parametrized\/indexed by a set $Y$. \nThe measure representation $\\mu=a_{\\infty}(X)$ in (\\ref{eq:measure}) yields a nonlinear map\n\\[ \\alpha:\\mathbb{R}^{n\\times d} \\rightarrow C(Y)\n~~,~~X \\mapsto F(y)=\\int_{R^d} K(x;y)d\\mu \\]\ngiven by\n\\[ \\alpha(X)(y)= \\frac{1}{n}\\sum_{k=1}^n K(x_k;y) \\]\nThe embedding problem \\ref{prob1}) can be restated as follows. One is asked\nto find a finite family of kernels $\\{K(x;y)~;~x\\in\\mathbb{R}^d~,~y\\in Y\\}$, \n $m=|Y|$ so that\n\\begin{equation}\n\\label{eq:kernel}\n{\\hat{\\alpha}}:({\\widehat{\\Rnd}},d) \\rightarrow l^2(Y)\\sim (\\mathbb{R}^m,\\norm{\\cdot}_2) ~~,~~ ({\\hat{\\alpha}}(\\hat{X}))_y = \\frac{1}{n} \\sum_{k=1}^n K(x_k;y)\n\\end{equation}\nis injective, Lipschitz or bi-Lipschitz. \n\nTwo natural choices for the kernel $K$ are the Gaussian kernel and the complex exponential (or, the Fourier) kernel:\n\\[ K_{G}(x,y) = e^{-\\norm{x-y}^2\/\\sigma^2} ~~,\nK_{F}(x,y) = e^{2\\pi i \\ip{x}{y}}\n\\]\nwhere in both cases $Y\\subset\\mathbb{R}^d$. \nIn this paper we analyze a different kernel, namely the polynomial kernel $K_P(x,y)=x_1^{y_1}x_2^{y_2}\\cdots x_d^{y_d}$, $Y\\subset\\{0,1,2,\\ldots,n\\}^d$. \n\n\\subsection{The Polynomial Embedding}\n\nSince the polynomial representation is intimately related to the Hilbert-Noether algebraic invariants theory \\cite{compuinvar} and the Hilbert-Weyl theorem, it is advantageous to start our construction from a different perspective. \n\nThe linear space ${\\mathbb{R}}^{n\\times d}$ is isomorphic to $\\mathbb{R}^{nd}$ by stacking the columns one on top of each other. In this case, the action of the permutation group $S_n$ can be recast as the action of the subgroup $I_d\\otimes S_n$ of the bigger group $S_{nd}$ on $\\mathbb{R}^{nd}$. Specifically, let us denote by $\\sim_G$ the equivalence relation\n\\[ x,y\\in\\mathbb{R}^{nd}~~,~~x\\sim_G y \\Longleftrightarrow y=\\Pi x~,~{\\rm for ~ some}~\\Pi\\in G \\]\ninduced by a subgroup $G$ of $S_{nd}$. In the case\n $G=I_d\\otimes S_n=\\{diag_d(P)~,~P\\in S_n\\}$ of block diagonal permutation obtained by repeating $d$ times the same $P\\in S_n$ permutation along the main diagonal, two vectors $x,y\\in\\mathbb{R}^{nd}$ are $\\sim_G$ equivalent iff there is a permutation matrix $P\\in S_n$ so that $y(1+(k-1)n:kn) = Px(1+(k-1)n:kn)$ for each $1\\leq k\\leq d$. In other words, each disjoint $n$-subvectors in $y$ and $x$ are related by the same permutation. In this framework, the Hilbert-Weyl theorem (Theorem 4.2, Chapter XII, in \\cite{BifTheory2}) states that the ring of invariant polynomials is finitely generated. The G\\\"{o}bel's algorithm (Section 3.10.2 in \\cite{compuinvar}) provides a recipe to find a complete set of invariant polynomials. In the following we provide a direct approach to construct a complete set of polynomial invariants. \n \n Let $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,...,{\\bf x}_d]$ denote the algebra of polynomials in $d$-variables with real coefficients. \nLet us denote $X\\in{\\mathbb{R}}^{n\\times d}$ a generic data matrix.\nEach row of this matrix defines a \nlinear form over ${\\bf x}_1,...{\\bf x}_d$,\n $\\lambda_k = X_{k,1}{\\bf x}_1+\\cdots + X_{k,d}{\\bf x}_d$.\n Let us denote by $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d][{\\bf t}]$ the algebra of polynomials in variable ${\\bf t}$ with coefficients in the ring $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]$. Notice $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d][{\\bf t}]=\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$ \n by rearranging the terms according to degree in ${\\bf t}$. \n Thus $\\lambda_k\\in\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]\\subset\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d][{\\bf t}]$ can be encoded as zeros of a polynomial $P_X$ of degree $n$ in variable ${\\bf t}$ with coefficients in $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]$:\n \\begin{equation}\n \\label{eq:polyencoding}\n P_X({\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d) = \\prod_{k=1}^n ({\\bf t}-\\lambda_k({\\bf x}_1,\\ldots,{\\bf x}_d))\n =\\prod_{k=1}^n ({\\bf t}-X_{k,1}{\\bf x}_1-\\ldots -X_{k,d}{\\bf x}_d)\n \\end{equation}\n Due to identification $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d][{\\bf t}]=\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$,\n we obtain that \\\\\n $P_X\\in \\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$ is a homogeneous polynomial of degree $n$ in $d+1$ variables. Let $\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ denote the vector space of homogeneous polynomials in $d+1$ variables of degree $n$ with real coefficients. Notice the real dimension of this vector space is \n \\begin{equation}\n \\label{eq:dimRn}\n \\dim_\\mathbb{R} \\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d] = \\left( \n \\begin{array}{c}\n n+d \\\\\n d\n \\end{array}\n \\right) = \\left(\n \\begin{array}{c}\n n+d \\\\\n n\n \\end{array}\n \\right).\n \\end{equation}\nBy noting that $P_X$ is monic in ${\\bf t}$ (the coefficient of ${\\bf t}^n$ is always 1) we obtain an injective embedding of ${\\widehat{\\Rnd}}$ into $\\mathbb{R}^m$ with \n$m=\\dim_\\mathbb{R} \\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]-1$ via the coefficients of $P_X$ similar to (\\ref{eq:poly}). This is summarized in the following theorem:\n\\begin{thm}\n\\label{t2}\nThe map $\\alpha_0:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^{m-1}$ with $m=\\left(\\begin{array}{c}n+d \\\\ d \\end{array} \\right)$ given by the (non-trivial) coefficients of polynomial $P_X\\in\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ lifts to an analytic embedding ${\\hat{\\alpha}}_0$ of $({\\widehat{\\Rnd}},d)$ into $\\mathbb{R}^m$. Specifically, for $X\\in{\\mathbb{R}}^{n\\times d}$ expand the polynomial\n\\begin{equation} \\label{eq:PA}\nP_X({\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d) = \\prod_{k=1}^n ({\\bf t}-X_{k,1}{\\bf x}_1-\\ldots -X_{k,d}{\\bf x}_d)\n = {\\bf t}^n + \\hspace{-10mm}\\sum_{\\begin{array}{c}\n \\mbox{$p_0,p_1,...,p_d\\geq 0$} \\\\\n \\mbox{$p_0+\\cdots+p_d=n$} \\\\ \n \\mbox{$p_0 1$}\n \\end{array}\\right.\n\\end{equation}\nbe a Lipschitz monotone decreasing function with Lipschitz constant 1.\n\\begin{cor}\\label{cor2}\nConsider the map:\n\\begin{equation}\\label{eq:alpha1}\n\\alpha_1:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m~~,~~\n\\alpha_1(X) = \\left( \\begin{array}{c}\n\\mbox{$\\alpha_0\\bigg (\\varphi_0(\\norm{X})X \\bigg )$} \\\\\n\\mbox{$\\norm{X}$}\n\\end{array}\\right),\n\\end{equation}\nwith $m=\\left(\\begin{array}{c}n+d \\\\ d \\end{array} \\right)$.\nThe map $\\alpha_1$ lifts to an injective and globally Lipschitz map ${\\hat{\\alpha}}_1:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m$ with Lipschitz constant $Lip({\\hat{\\alpha}}_1) \\leq \\sqrt{1+L_0^2}$.\n\\end{cor}\n{\\bf Proof}\n\nClearly $\\alpha_1(\\Pi X)=\\alpha_1(X)$ for any $\\Pi\\in{\\mathcal S}_n$ and $X\\in{\\mathbb{R}}^{n\\times d}$. Assume now that $\\alpha_1(X)=\\alpha_1(Y)$. Then $\\norm{X}=\\norm{Y}$ and since ${\\hat{\\alpha}}_0$ is injective on ${\\widehat{\\Rnd}}$ it follows $\\varphi(\\norm{X})X = \\Pi \\varphi(\\norm{Y})Y$ for some $\\Pi\\in{\\mathcal S}_n$. Thus $X\\sim Y$ which proves $\\alpha_1$ lifts to an injective map on ${\\widehat{\\Rnd}}$. \n\nNow we show ${\\hat{\\alpha}}_1$ is Lipschitz on $({\\widehat{\\Rnd}},d)$ of appropriate Lipschitz constant. Let $X,Y'\\in{\\mathbb{R}}^{n\\times d}$ and $\\Pi_0\\in{\\mathcal S}_n$ so that $d(\\hat{X},\\hat{Y'})=\\norm{X-\\Pi_0 Y'}$. Let $Y=\\Pi_0 Y'$ so that $d(\\hat{X},\\hat{Y})=\\norm{X-Y}$. \n\nChoose two matrices $X,Y\\in{\\mathbb{R}}^{n\\times d}$. We claim $\\norm{\\alpha_1(X)-\\alpha_1(Y)}\\leq \\sqrt{1+L_0^2}\n\\norm{X-Y}$.\nThis follows from two observations: \n\n(i) The map\n\\[ X \\mapsto \\rho(X):=\\varphi_0(\\norm{X})X \\]\nis the nearest-point map to (or, the metric projection map onto) the convex closed set $B_1({\\mathbb{R}}^{n\\times d})$. This means $\\norm{\\varphi_0(\\norm{X})X - Z}\\leq \\norm{X-Z}$ for any $Z\\in B_1({\\mathbb{R}}^{n\\times d})$. \n\n(ii) The nearest-point map to a convex closed subset of a Hilbert space is Lipschitz with constant 1, i.e. it shrinks distances, see \\cite{phelps56}.\n\nThese two observations yield:\n\\begin{multline*} \n\\norm{\\alpha_1(X)-\\alpha_1(Y)}^2 = \\norm{\\alpha_0(\\rho(Y))\n- \\alpha_0(\\rho(Y) )}^2 + |\\norm{X}-\\norm{Y}|^2 \\\\\n \\leq \nL_0^2 \\norm{\\rho(X)-\\rho(Y) }^2 + \\norm{X-Y}^2 \\leq (1+L_0^2)\\norm{X-Y}^2 .\n\\end{multline*}\nThis concludes the proof of this result. $\\qed$\n\\vspace{5mm}\n\nA simple modification of $\\phi_0$ can produce a $C^\\infty$ map by smoothing it out around $x=1$.\n\nOn the other hand the lower Lipschitz constant of ${\\hat{\\alpha}}_1$ is 0 due to terms of the form $X_{i,j}^k$ with $k\\geq 2$. \nIn \\cite{Cahill19}, the authors built a Lipschitz map by a retraction to the unit sphere instead of unit ball. \nInspired by their construction, a modification of $\\alpha_0$ in their spirit reads:\n\\begin{equation}\n \\label{eq:alpha2}\n\\alpha_2:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m~~,~~\n\\alpha_2(X)=\\left( \\begin{array}{c}\n\\mbox{$\\norm{X}\\alpha_0\\bigg ( \\frac{X}{\\norm{X}} \\bigg )$} \\\\\n\\mbox{$\\norm{X}$}\n\\end{array}\\right)~,~if~X\\neq 0~~,~and~~\\alpha_2(0)=0.\n\\end{equation}\nIt is easy to see that $\\alpha_2$ satisfies the non-parallel property in \\cite{Cahill19} and is Lipschitz with a slightly better constant than $\\alpha_1$ (the constant is determined by the tangential derivatives of $\\alpha_0$). \nBut, for the same reasons as in \\cite{Cahill19} this map is not bi-Lipschitz. \n\n\\subsection{Dimension reduction in the case $d=2$ and consequences}\n\nIn this subsection we analyze the case $d=2$. \nThe embedding dimension for $\\alpha_0$ is $\\left( \\begin{array}{c} n \\\\ 2 \\end{array}\\right)-1=\\frac{n(n-1)}{2}-1$. \nOn the other hand, consider the following approach. \nEach row of $X$ defines a complex number $z_1=X_{1,1}+i\\,X_{1,2}$, ... , $z_n=X_{n,1}+i\\,X_{n,2}$ that\ncan be encoded by one polynomial of degree $n$ with complex coefficients $Q\\in\\mathbb{C}_n[t]$,\n\\[ Q({\\bf t}) = \\prod_{k=1}^n ({\\bf t}-z_k) = {\\bf t}^n + \\sum_{k=0}^{n-1}\n{\\bf t}^k q_k \\]\nThe coefficients of $Q$ provide a $2n$-dimensional real embedding $\\zeta_0$,\n\\[ \\zeta_0:\\mathbb{R}^{n\\times 2}\\rightarrow\\mathbb{R}^{2n}~~,~~\\zeta_0(X)=(Re(q_{n-1}),Im(q_{n-1}),\\ldots,Re(q_{0}),Im(q_0)) \\]\nwith properties similar to those of $\\alpha_0$. \nOne can similarly modify this embedding to obtain a globally Lipschitz embedding $\\hat{\\zeta}_1$ of $\\hat{R_{n,2}}$ \ninto $\\mathbb{R}^{2n+1}$. \n\nIt is instructive to recast this embedding in the framework of commutative algebras. Indeed, let $\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle$ denote\nthe ideal generated by polynomials ${\\bf x}_1-1$ and ${\\bf x}_2^2+1$\nin the algebra $\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]$. Consider the quotient space\n $\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle$ and the quotient map\n $\\sigma:\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\\mapsto \\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle{\\bf x}_1-1,{\\bf x}_2^2+1\\rangle$.\n In particular, let $S=\\sigma(\\mathbb{R}_n[{\\bf t},{\\bf x}_1,{\\bf x}_2])$ denote the vector space projected through this quotient map.\nThen a basis for $S$ is given by $\\{1,{\\bf t},\\ldots,{\\bf t}^n,{\\bf x}_2,{\\bf x}_2 {\\bf t},\\ldots,{\\bf x}_2 {\\bf t}^{n-1},{\\bf x}_2 {\\bf t}^n\\}$. Thus $\\dim S=2n+2$. \nLet \n$\\mathfrak{S}=\\{P_X~,~X\\in\\mathbb{R}^{n\\times 2} \\}\\subset\\mathbb{R}_2[{\\bf t},{\\bf x}_1,{\\bf x}_2]$ \ndenote the set of polynomials realizable as in (\\ref{eq:PA}).\nThen the fact that $\\hat{\\zeta}_0:\\mathbb{R}^{n\\times 2}\\rightarrow\\mathbb{R}^{2n}$ \nis injective is equivalent to the fact that $\\sigma{\\vert}_{\\mathfrak{S}}:\\mathfrak{S}\\rightarrow S$ is injective.\nOn the other hand note \n\\[\n\\sigma(\\mathfrak{S})\\subset {\\bf t}^n+\nspan_\\mathbb{R}\\{1,{\\bf t},\\ldots,{\\bf t}^{n-1},{\\bf x}_2,{\\bf x}_2 {\\bf t}, \\ldots,{\\bf x}_2 {\\bf t}^{n-1} \\} \\]\nwhere the last linear subspace is of dimension $2n$. \n\nIn the case $d=2$ we obtain the identification\n$\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle = \\mathbb{C}[{\\bf t}]$ due to uniqueness of polynomial factorization.\n\nThis observation raises the following {\\em open problem}:\n\nFor $d>2$, is there a non-trivial ideal \n$I=\\langle Q_1,\\ldots,Q_r \\rangle \\subset\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$\nso that the restriction $\\sigma{\\vert}_{\\mathfrak{S}}$\nof the quotient map $\\sigma:\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]\\rightarrow\n\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]\/I$ is injective? Here $\\mathfrak{S}$ denote the set of polynomials in $\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ realizable via (\\ref{eq:PA}).\n\\begin{rmk}\nOne may ask the question whether the quaternions can be\nutilized in the case $d=4$. While the quaternions form an associative division algebra, unfortunately polynomials have in general an infinite number of factorization. This prevents an immediate extension of the previous construction to the case $d=4$. \n\\end{rmk}\n\n\\begin{rmk}\nSimilar to the construction in \\cite{Cahill19}, a linear dimension reduction technique may be applicable here (which, in fact, may answer the open problem above) which would reduce the embedding dimension to $m=2nd+1$ (twice the intrinsec dimension plus one for the homogenization variable). \nHowever we did not explore this approach since, even if possible, it would not produce a bi-Lipschitz embedding. \nInstead we analyze the linear dimension reduction technique in the next section in the context of sorting based embeddings. \n\\end{rmk}\n\n\\section{Sorting based Embedding\\label{sec3}}\n\nIn this section we present the extension of the sorting embedding (\\ref{eq:ord}) to the case $d>1$.\n\nThe embedding is performed by a linear-nonlinear transformation that resembles the phase retrieval problem. \nConsider a matrix $A\\in\\mathbb{R}^{d\\times D}$ and the induced nonlinear \ntransformation:\n\n\\begin{equation}\n\\label{eq:qA}\n\\beta_A:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^{n\\times D}~~,~~\\beta_A(X)=\\downarrow (XA)\n\\end{equation}\nwhere $\\downarrow$ is the monotone decreasing sorting operator acting in each column independently. Specifically, let \n$Y=XA\\in\\mathbb{R}^{n\\times D}$ and note its column vectors\n$Y=[y_1,y_2,\\ldots,y_D]$. Then \n\\[ \\beta_A(X)=\\left[ \\begin{array}{cccc}\n\\mbox{$\\Pi_1 y_1$} & \\mbox{$\\Pi_2 y_1$} & \\cdots & \\mbox{$\\Pi_D y_D$}\n\\end{array} \\right] \\]\nfor some $\\Pi_1,\\Pi_2,\\ldots,\\Pi_D\\in{\\mathcal S}_n$ so that each column is sorted monotonically decreasing:\n\\[ (\\Pi_k y_k)_1\\geq (\\Pi_k y_k)_2\\geq \\cdots\\geq (\\Pi_k y_k)_n. \\]\nNote the obvious invariance $\\beta_A(\\Pi X)=\\beta_A(X)$ for any $\\Pi\\in{\\mathcal S}_n$ and $X\\in{\\mathbb{R}}^{n\\times d}$. Hence $\\beta_A$ \nlifts to a map $\\hat{\\beta_A}$ on ${\\widehat{\\Rnd}}$. \n\\begin{rmk}\nNotice the similarity to the phase retrieval problem, e.g., \\cite{balan16}, where the data is obtained via a linear transformation of the \ninput signal followed by the nonlinear operation of taking the absolute value of the frame coefficients. Here the nonlinear transformation is implemented by sorting the coefficients. \nIn both cases it represents the action of a particular\nsubgroup of the unitary group. \n\\end{rmk}\n\n\nIn this section we analyze necessary and sufficient conditions so that maps of type (\\ref{eq:qA}) are injective, or injective almost everywhere. \nFirst a few definitions.\n\n\\begin{defn}\nA matrix $A\\in\\mathbb{R}^{d\\times D}$ is called a \\emph{universal key} (for ${\\mathbb{R}}^{n\\times d}$) if $\\hat{\\beta_A}$ is injective\non ${\\widehat{\\Rnd}}$.\n\\end{defn}\nIn general we refer to $A$ as a {\\em key} for encoder $\\beta_A$. \n\\begin{defn}\nFix a matrix $X\\in{\\mathbb{R}}^{n\\times d}$. A matrix $A\\in\\mathbb{R}^{d\\times D}$ is said \\emph{admissible} (or an {\\em admissible key}) for $X$ if for any $Y\\in{\\mathbb{R}}^{n\\times d}$ so that $\\beta_A(X)=\\beta_A(Y)$ then $Y=\\Pi X$ for some $\\Pi\\in{\\mathcal S}_n$. \n\\end{defn}\nIn other words, $\\hat{\\beta_A}^{-1}(\\hat{\\beta_A}(\\hat{X}))=\\{\\hat{X}\\}$.\nWe let ${\\mathcal{A}}_{D}(X)$, or simply ${\\mathcal{A}}(X)$, denote the set of admissible keys for $X$. \n\\begin{defn}\nFix $A\\in\\mathbb{R}^{d\\times D}$. A matrix $X\\in{\\mathbb{R}}^{n\\times d}$ is said to be {\\em separated} by $A$ if $A\\in{\\mathcal{A}}(X)$.\n\\end{defn}\nFor a key $A$, we let $\\mathfrak{S}_{n}(A)$, or simply $\\mathfrak{S}(A)$, denote the set of {\\em matrices separated by $A$}. Thus a matrix $X\\in\\mathfrak{S}_n(A)$ if and only if, for any matrix $Y\\in\\mathbb{R}^{n\\times d}$, if $\\beta_A(X)=\\beta_A(Y)$ then $X\\sim Y$.\n\nThus a key $A$ is universal if and only if $\\mathfrak{S}_n(A)={\\mathbb{R}}^{n\\times d}$.\n\nOur goal is to produce keys that are admissible for all matrices in ${\\mathbb{R}}^{n\\times d}$, or at least for almost every data matrix.\nAs we show in Proposition \\ref{prop3.6} below this requires that $D\\geq d$ and $A$ is full rank. In particular this means that the columns of $A$ form a frame for $\\mathbb{R}^d$. \n\n\\subsection{Characterizations of ${\\mathcal{A}}(X)$ and $\\mathfrak{S}(A)$}\n\nWe start off with simple linear manipulations of sets of admissible keys and separated data matrices.\n\n\\begin{prop}\\label{prop3.5}\nFix $A\\in\\mathbb{R}^{d\\times D}$ and $X\\in{\\mathbb{R}}^{n\\times d}$.\n\\begin{enumerate}\n \\item For an invertible $d\\times d$ matrix $T\\in\\mathbb{R}^{d\\times d}$,\n \\begin{equation}\\label{eq:TA}\n \\mathfrak{S}_n(TA) = \\mathfrak{S}_n(A)T^{-1}.\n \\end{equation}\n In other words, if $X$ is separated by $A$ then $XT^{-1}$ is separated by $TA$.\n \n \\item For any permutation matrix $L\\in\\SS_D$ and diagonal invertible matrix $\\Lambda\\in\\mathbb{R}^{D\\times D}$,\n \\begin{equation}\\label{eq:AL}\n \\mathfrak{S}_n(AL\\Lambda)=\\mathfrak{S}_n(A\\Lambda L) = \\mathfrak{S}_n(A).\n \\end{equation}\n In other words, if $X$ is separated by $A$ then $X$ is separated also by $AL\\Lambda$ as well as by $A\\Lambda L$.\n \n \\item Assume $T\\in\\mathbb{R}^{d\\times d}$ is a $d\\times d$ invertible matrix. Then\n \\begin{equation}\\label{eq:XT}\n {\\mathcal{A}}_D(XT)=T^{-1}{\\mathcal{A}}_D(X).\n \\end{equation}\n In other words, if $A$ is an admissible key for $X$ then $T^{-1}A$ is an admissible key for $XT$.\n\\end{enumerate}\n\\end{prop}\n{\\bf Proof}\n\nThe proof is immediate, but we include it here for convenience of the reader. \n\n(1) Denote $B=TA$. Let $Y\\in\\mathbb{R}^{n\\times d}$. Then\n\\[ \\beta_B(Y)=\\beta_B(X) \\Longleftrightarrow \\downarrow(XB)=\\downarrow(YB) \\Longleftrightarrow \\downarrow(XTA)=\\downarrow(YTA)\n\\Longleftrightarrow\\beta_A(XT)=\\beta_A(YT). \\]\nThus, if $X\\in\\mathfrak{S}_n(A)$ and $Y'\\in{\\mathbb{R}}^{n\\times d}$ so that $\\beta_B(Y')=\\beta_B(X')$ with $X'=XT^{-1}$, then $\\beta_A(Y'T)=\\beta_A(X)$. Therefore there exists $\\Pi\\in{\\mathcal S}_n$ so that $Y'T=\\Pi X$. Thus $Y'\\sim X'$. Hence $X'\\in\\mathfrak{S}_n(B)$.\nThis shows $\\mathfrak{S}_n(A)T^{-1}\\subset \\mathfrak{S}_n(TA)$. The reverse include follows by replacing $A$ with $TA$ and $T$ with $T^{-1}$. Together they prove (\\ref{eq:TA}).\n\n(2) Let $Y\\in{\\mathbb{R}}^{n\\times d}$ such that $\\beta_{AL\\Lambda}(X)=\\beta_{AL\\Lambda}(Y)$. \nFor every $1\\leq j\\leq D$ let $k\\in [D]$ be so that $L_{jk}=1$. \n\nIf $\\Lambda_{kk}>0$ then $\\downarrow((XA)_j)=\\downarrow((YA)_j)$. \n\nIf $\\Lambda_{kk}<0$ then $\\downarrow(-(XA)_j)=\\downarrow(-(YA)_j)$.\nBut this implies also $\\downarrow((XA)_j)=\\downarrow((YA)_j)$ since\n$\\downarrow(-z)=L_0\\downarrow(z)$ where $L_0$ is the permutation matrix that has 1 on its main antidiagonal.\n\nEither way, $\\downarrow((XA)_j)=\\downarrow((YA)_j)$. Hence \n$\\downarrow(XA)=\\downarrow(YA)$. Therefore $X\\sim Y$ and thus $X\\in \\mathfrak{S}_n(AL\\Lambda)$. This shows $\\mathfrak{S}_n(A)\\subset\\mathfrak{S}_n(AL\\Lambda)$.\nthe reverse inclusion follows by a similar argument.\nFinally, notice $\\{L\\Lambda\\}$ forms a group since $L^{-1}\\Lambda L$ is also a diagonal matrix. This shows $\\mathfrak{S}_n(A\\Lambda L)=\\mathfrak{S}(AL\\Lambda')$ for some diagonal matrix $\\Lambda'$, and the conclusion (\\ref{eq:AL}) then follows.\n\n(3) The relation (\\ref{eq:XT}) follows from noticing $\\beta_{T^{-1}A}(Y)=\\beta_A(YT)$. $\\qed$\n\nRelation (\\ref{eq:AL}) shows that, since $A$ is assumed full rank, without loss of generality we can assume the first $d$ columns are linearly independent. Let $V$ denote the first $d$ columns of $A$ so that\n\\begin{equation}\\label{eq:AA}\nA = V\\left[ \\begin{array} {ccc}\n\\mbox{$I$} & \\mbox{$\\vert$} & \\mbox{$\\tilde{A}$} \n\\end{array} \\right] \n\\end{equation}\nwhere $\\tilde{A}\\in\\mathbb{R}^{d\\times (D-d)}$. \nThe following result shows that, unsurprisingly, when $D=d>1$, almost every matrix $X$ is not separated by $A$. By Proposition \\ref{prop3.5} we can reduce the analysis to the case $A=I$ by a change of coordinates.\n\\begin{prop}\\label{prop3.6}\nAssume $D=d>1$, $n>1$. Then\n\\begin{enumerate}\n \\item The set of data matrices not separated by $I_d$ includes:\n\\begin{equation}\\label{eq:SA1}\n \\mathbb{B}:=\\{ X\\in\\mathbb{R}^{n\\times d}~,~\\exists i,j,k,l~,~ 1\\leq i0$ and $b_0>0$ so that\n for all $X,Y\\in{\\mathbb{R}}^{n\\times d}$,\n \\begin{equation}\n \\label{eq:Lipbeta2}\n a_0\\, d(\\hat{X},\\hat{Y})\\leq \\norm{\\beta_A(X)-\\beta_A(Y)}\\leq b_0\\,\n d(\\hat{X},\\hat{Y})\n \\end{equation}\n where all are Frobenius norms.\n Furthermore, an estimate for $b_0$ is provided by the largest singular value of $A$, $b_0= s_1(A)$.\n\\end{thm}\n\n{\\bf Proof}\n\nThe upper bound in (\\ref{eq:Lipbeta2}) follows as in the proof of Theorem \\ref{t4}, from equations (\\ref{eq:betaA}) and (\\ref{eq:betaAA}). Notice that\nno property is assumed in order to obtain the upper Lipschitz bound.\n\nThe lower bound in (\\ref{eq:Lipbeta2}) is more difficult. \nIt is shown by contradiction following the strategy \nutilized in the \nComplex Phase Retrieval problem \\cite{balazou}.\n\nAssume $\\inf_{X\\not\\sim Y}\\frac{\\norm{\\beta_A(X)-\\beta_A(Y)}_2^2}{d(\\hat{X},\\hat{Y})^2}=0$. \n\n{\\em Step 1: Reduction to local analysis.} \nSince $d(\\hat{tX},\\hat{tY})=t\\,d(\\hat{X},\\hat{Y})$ for all $t>0$, the \nquotient $\\frac{\\norm{\\beta_A(X)-\\beta_A(Y)}_2}{d(\\hat{X},\\hat{Y})}$ \nis scale invariant. Therefore, there are sequences $(X^t)_t,(Y^t)_t$\nwith $\\norm{Y^t}\\leq\\norm{X^t}=1$ and $d(\\hat{X^t},\\hat{Y^t})>0$ so that\n$\\lim_{t\\rightarrow\\infty} \\frac{\\norm{\\beta_A(X^t)-\\beta_A(Y^t)}_2}{d(\\hat{X^t},\\hat{Y^t})} = 0$.\nBy compactness of the closed unit ball, one can extract convergence subsequences. For easiness of notation, assume $(X^t)_t,(Y^t)_t$ are\nthese subsequences. Let ${X^{\\infty}}=\\lim_t X^t$ and ${Y^{\\infty}} = \\lim_t Y^t$ denote their limits. Notice $\\lim_t \\norm{\\beta_A(X^t)-\\beta_A(Y^t)}_2=0$.\nThis implies $\\norm{\\beta_A({X^{\\infty}})-\\beta_A({Y^{\\infty}})}=0$ and thus $\\beta_A({X^{\\infty}})=\\beta_A({Y^{\\infty}})$. Since $\\widehat{\\beta_A}$ is assumed injective, it follows that $\\widehat{{X^{\\infty}}}=\\widehat{{Y^{\\infty}}}$. \n\nThis means that, if the lower Lipschitz bound vanishes, then this \nis achieved by vanishing of a local lower Lipschitz bound. To follow the terminology in \\cite{balazou}, the type I local lower Lipschitz bound vanishes at some \n$Z_0\\in{\\mathbb{R}}^{n\\times d}$, with $\\norm{Z_0}=1$:\n\\begin{equation}\n \\label{eq:lb}\n{A}(Z_0):= \\lim_{r\\rightarrow 0} \\inf_{\n\\begin{array}{c}\n\\hat{X}\\neq\\hat{Y} \\\\\nd(\\hat{X},\\hat{Z_0})0$ by the definition of $G$.\n\nConsider $X=Z_0+U$ and $Y=Z_0+V$ where $U,V\\in{\\mathbb{R}}^{n\\times d}$ are ``aligned\" in the sense that $d(\\hat{X},\\hat{Y})=\\norm{U-V}$. This property requires that $\\norm{U-V}\\leq\\norm{PX-Y}$, for every $P\\in{\\mathcal S}_n$. \nNext result replaces equivalently this condition \nby requirements involving $(U,V)$ and the group $G$ only.\n\\begin{lem}\n\\label{l3.1}\nAssume $\\norm{U},\\norm{V}<\\frac{1}{4}\\delta_0$, where $\\delta_0=\\min_{P\\in{\\mathcal S}_n\\setminus G} \\norm{(I_n-P)Z_0}$. Let $X=Z_0+U$, $Y=Z_0+V$. \nThen:\n\\begin{enumerate}\n \\item $d(\\hat{X},\\hat{Z_0})=\\norm{U}$ and $d(\\hat{Y},\\hat{Z_0})=\\norm{V}$.\n \\item $d(\\hat{X},\\hat{Y})=\\min_{P\\in G}\\norm{U-PV}=\\min_{P\\in G}\\norm{PU-V}$\n \\item The following\nare equivalent:\n\\begin{enumerate}\n \\item $d(\\hat{X},\\hat{Y})=\\norm{U-V}$.\n \\item For every $P\\in G$, $\\norm{U-V}\\leq \\norm{PU-V}$.\n \\item For every $P\\in G$, $\\ip{U}{V}\\geq \\ip{PU}{V}$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{lem}\n{\\bf Proof of Lemma \\ref{l3.1}}.\n(1)\n\nNote that is $U=0$ then the claim follows. Assume $U\\neq 0$. Then\n\\[ d(\\hat{X},\\hat{Z_0})=\\min_{P\\in{\\mathcal S}_n} \\norm{X-PZ_0}\n= \\min_{P\\in {\\mathcal S}_n}\\norm{(I_n-P)Z_0 + U}\\leq \\norm{U} \\]\nOn the other hand, assume the minimum is achieved for a permutation $P_0\\in{\\mathcal S}_n$. If $P_0\\in G$ then\n$d(\\hat{X},\\hat{Z_0})=\\norm{(I_n-P_0)Z_0+U}=\\norm{U}$. If $P_0\\not\\in G$ then \n\\[ d(\\hat{X},\\hat{Z_0})\\geq \\norm{(I_n-P_0)Z_0}-\\norm{U}>\\frac{3\\delta_0}{4}>\\norm{U}\\geq d(\\hat{X},\\hat{Z_0}) \\]\nwhich yields a contradiction.\nHence $d(\\hat{X},\\hat{Z_0})=\\norm{U}$. Similarly, one shows $d(\\hat{X},\\hat{Z_0})=\\norm{V}$. \n\n(2) Clearly\n\\[ d(\\hat{X},\\hat{Y})=\\min_{P\\in{\\mathcal S}_n}\\norm{PX-Y}\\leq \\min_{P\\in G}\\norm{PX-Y}=\\min_{P\\in G}\\norm{PU-V} \\]\nOn the other hand, for $P\\in{\\mathcal S}_n\\setminus G$ and $Q\\in G$,\n\\[ \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\\geq \\norm{(I_n-P)Z_0} - \\norm{U}-\\norm{V}\\geq \\]\n\\[ \\geq \\delta_0\n-2\\norm{U}-2\\norm{V}+\\norm{QU-V}\\geq \\min_{Q\\in G}\\norm{QU-V}\\geq d(\\hat{X},\\hat{Y}). \\]\n\n(3)\n\n(a)$\\Rightarrow$(b).\n\nIf $d(\\hat{X},\\hat{Y})=\\norm{U-V}$ then\n\\[ \\norm{U-V}\\leq \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\n~~,~~\\forall P\\in {\\mathcal S}_n. \\]\nIn particular, for $P\\in G$, $(P-I_n)Z_0=0$ and\nthe above inequality reduces to (b).\n\n(b)$\\Rightarrow$(a).\n\nAssume (b). For $P\\in G$,\n\\[ \\norm{U-V}=\\norm{X-Y}\\leq\\norm{PU-V}=\\norm{PX-Y}. \\]\nFor $P\\in{\\mathcal S}_n\\setminus G$,\n\\[ \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\\geq \\norm{(I_n-P)Z_0} - \\norm{U}-\\norm{V}\\geq \\]\n\\[ \\geq \\delta_0\n-2\\norm{U}-2\\norm{V}+\\norm{U-V}\\geq \\norm{U-V}=\\norm{X-Y}. \\]\nThis shows $d(\\hat{X},\\hat{Y})=\\norm{X-Y}=\\norm{U-V}$.\n\n(b)$\\Longleftrightarrow$(c). This is immediate after squaring (b) and simplifying the terms.\n\n$\\Box$\n\\vspace{3mm}\n\nConsider now sequences $(\\hat{X^t})_t,(\\hat{Y^t})_t$ that converge to $\\hat{Z_0}$ \nand achieve lower bound 0 as in (\\ref{eq:lb}).\nChoose\nrepresentatives $X_t$ and $Y_t$ in their equivalence classes that satisfy the hypothesis of Lemma \\ref{l3.1} so that $X_t=Z_0+U_t$, $Y_t=Z_0+V_t$, $\\norm{U_t},\\norm{V_y}<\\frac{1}{4}\\delta_0$,\n $d(\\hat{X_t},\\hat{Z_0})=\\norm{U_t}$, $d(\\hat{Y_t},\\hat{Z_0})=\\norm{V_t}$ and $d(\\hat{X_t},\\hat{Y_t})=\\norm{U_t-V_t}>0$.\n With $A=[a_1 |\\cdots|a_D]$ we obtain:\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 = \\sum_{j=1}^D \\norm{\\downarrow(X_t a_j)-\\downarrow(Y_t a_j)}_2^2 =\n\\sum_{j=1}^D \\norm{(Z_0+U_t)a_j-\\Pi_{j,t}(Z_0+V_t)a_j}_2^2 \\]\nfor some $\\Pi_{j,t}\\in{\\mathcal S}_n$. In fact $\\Pi_{j,t}\\in argmin_{\\Pi\\in H_j}\\norm{U_t-\\Pi V_t)a_j}_2$. \nPass to sub-sequences (that will be indexed by $t$ for an easier notation) so that $\\Pi_{j,t}=\\Pi_j$ for some $\\Pi_j\\in{\\mathcal S}_n$. Thus\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 =\n\\sum_{j=1}^D \\norm{(I_n-\\Pi_j)Z_0a_j + (U_t-\\Pi_j V_t)a_j}_2^2 \\]\nSince the above sequence must converge to $0$ as $t\\rightarrow\\infty$, while $U_t,V_t\\rightarrow 0$, it follows that necessarily $\\Pi_j\\in H_j$ and the\nexpressions simplify to\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 =\n\\sum_{j=1}^D \\norm{(U_t-\\Pi_j V_t)a_j}_2^2 \\]\nThus equation (\\ref{eq:lb}) implies that for\nevery $j\\in[D]$,\n\\begin{equation}\n\\label{eq:lb2}\n\\lim_{t\\rightarrow\\infty} \\frac{\\norm{(U_t-\\Pi_j V_t)a_j}_2^2}{\\norm{U_t-V_t}^2} = 0\n\\end{equation} \nwhere $\\Pi_j\\in H_j$, $\\norm{U_t},\\norm{V_t}\\rightarrow 0$, and $U_t,V_t$ are aligned so that $\\ip{U_t}{V_t}\\geq \\ip{PU_t}{V_t}$ for every $P\\in G$.\nEquivalently, relation (\\ref{eq:lb}) can be restated as:\n\\begin{equation}\n \\label{eq:opt2}\n \\inf_{\\begin{array}{c} U,V\\in{\\mathbb{R}}^{n\\times d} \\\\ s.t. \\\\\n U\\neq V \\\\\n \\ip{U}{V}\\geq \\ip{PU}{V} , \\forall P\\in G\n \\end{array} } \\frac{\\sum_{j=1}^D \\norm{(U-\\Pi_j V)a_j}_2^2}{\\norm{U-V}^2} = 0\n\\end{equation}\nfor some permutations $\\Pi_j\\in H_j$, $j\\in[D]$.\nBy Lemma \\ref{l3.1} the constraint in the optimization problem above implies $\\norm{U-V}=\\min_{P\\in G}\\norm{U-PV}$. Hence (\\ref{eq:opt2}) implies:\n\\begin{equation}\n \\label{eq:opt3}\n \\inf_{\\begin{array}{c} U,V\\in{\\mathbb{R}}^{n\\times d} \\\\ s.t. \\\\\n U\\neq P V , \\forall P\\in G\n \\end{array} }\n \\max_{P\\in G} \\frac{\\sum_{j=1}^D \\norm{(U-\\Pi_j V)a_j}_2^2}{\\norm{U-P V}^2} = 0\n\\end{equation}\nfor same permutation matrices $\\Pi_j$'s.\nWhile the above optimization problem seems a relaxation of (\\ref{eq:opt2}), in fact (\\ref{eq:opt3}) implies (\\ref{eq:opt2})\nwith a possibly change of permutation matrices $\\Pi_j$, but \nremaining still in $H_j$.\n\\vspace{5mm}\n\n\n{\\em Step 3. Existence of a Minimizer.} \n \n\n\nThe optimization problem (\\ref{eq:opt2}) is a Quadratically Constrained Ratio of Quadratics (QCRQ) optimization problem. A significant number of papers \nhave been published on this topic \\cite{teb06,teb10}. \nIn particular, \\cite{QCRQbook} presents\na formal setup for analysis of QCRQ problems. \nOur interest is to utilize some of these techniques in order to establish the existence of a minimizer for (\\ref{eq:opt2}) or (\\ref{eq:opt3}). Specifically we show:\n\\begin{lem}\\label{l3.2}\nAssume the key $A$ has linearly independent rows (equivalently, the columns of $A$ form a frame for $\\mathbb{R}^d$) and the lower Lipschitz bound of ${\\hat{\\beta}}_A$ is $0$. Then there are $\\tilde{U},\\tilde{V}\\in{\\mathbb{R}}^{n\\times d}$ so that:\n\\begin{enumerate}\n \\item $\\tilde{U}\\neq P \\tilde{V}$, for every $P\\in G$;\n \\item For every $j\\in[D]$, $(\\tilde{U}-\\Pi_j \\tilde{V})a_j=0$.\n\\end{enumerate}\n\\end{lem}\n{\\bf Proof of Lemma \\ref{l3.2}}\n\n\n\nWe start with the formulation (\\ref{eq:opt3}). Therefore there are sequences \n$(U_t,V_t)_{t\\geq 1}$ so that $U_t\\neq PV_t$ for any $P\\in G, t\\geq 1$, and yet for any $P\\in G$,\n\\[ \\lim_{t\\rightarrow\\infty} \\frac{\\sum_{j=1}^D\\norm{(U_t-\\Pi_j V_t)a_j}_2^2}{\\norm{U_t-P V_t}^2} = 0. \\]\nLet $E=\\{(U,V)\\in{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}~,~(U-\\Pi_j)V)a_j=0~,~\\forall j\\in[D]\\}$ denote the null space of the linear operator \n\\[ T:{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}\\rightarrow \\mathbb{R}^D~,~(U,V)\\mapsto \\left[\\begin{array}{ccccc}\n(U-\\Pi_1 V)a_1 & \\vert & \\cdots & \\vert & (U-\\Pi_D V)a_D\n\\end{array}\\right],\n\\]\nassociated to the numerator of the above quotient. Let $F_P=\\{(U,V)\\in{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}~,~U-PV=0\\}$ be the null space of the linear operator \n\\[ R_P:{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}\\rightarrow {\\mathbb{R}}^{n\\times d}~,~(U,V)\\mapsto U-P V. \\]\nA consequence of (\\ref{eq:opt3}) is that for every $P\\in G$, $E\\setminus F_P\\neq\\emptyset$. \nIn particular, $F_p\\cap E$ is a subspace of $E$ of positive codimension. Using the \nBaire category theorem (or more elementary linear algebra arguments), we conclude that\n\\[ E\\setminus \\left(\\cup_{P\\in G} F_P\\right) \\neq \\emptyset. \\]\nLet $(\\tilde{U},\\tilde{V})\\in E\\setminus\\left(\\cup_{P\\in G}F_P\\right)$. This pair satisfies the\nconclusions of Lemma \\ref{l3.2}.\n\n\n\\ignore{\nFirst we rewrite (\\ref{eq:opt2}) in terms of new matrices. Let $S_1=S_1({\\mathbb{R}}^{n\\times d})$ denote the unit sphere of ${\\mathbb{R}}^{n\\times d}$. Let $W=U-V$. Since $W\\neq 0$, let $W_0=\\frac{1}{\\norm{W}}W$ with $\\norm{W_0}=1$.\nLet also $V_0\\in S_1$ be so that $V=t\\norm{W}V_0$\n for some $t\\geq 0$. Then the objective function in (\\ref{eq:opt2}), i.e., the quotient of the two quadratics, simplifies to\n \\[ \\sum_{j=1}^D \\norm{(W_0 + t(I-\\Pi_j)V_0)a_j}_2^2. \\]\n Let $\\Gamma_t$ define the constraints set:\n \\[ \\Gamma_t =\\cap_{P\\in G} \n \\{ (W_0,V_0)\\in S_1\\times S_1~:~\n t^2 \\ip{V_0}{(I-P)V_0}+t\\ip{W_0}{(I-P)V_0}\\geq 0 \\}.\n \\]\nNotice that for each $t\\geq 0$, $\\Gamma_t$ is a closed and hence a compact subset of $S_1\\times S_1$. It may be an empty set for some values of $t$. The problem (\\ref{eq:opt2}) is equivalent to:\n\\[ \\inf_{t\\geq 0} \\inf_{(W_0,V_0)\\in\\Gamma_t}\n\\sum_{j=1}^D \\norm{W_0 a_j + t(I-\\Pi_j)V_0a_j}_2^2 = 0 \\]\nLet $(W(t_k),V(t_k),t_k)\\in S_1\\times S_1\\times [0,\\infty)$, $k\\geq 1$, be a sequence that achieves the lower bound $0$. Extract a subsequence indexed again by $k$ so that\n $\\lim_{k\\rightarrow\\infty} W(t_k)=W_\\infty\\in S_1$ and $\\lim_{k\\rightarrow\\infty}V(t_k)=V_\\infty$. Thus, for all $j\\in [D]$, $\\lim_{k\\rightarrow\\infty} \\norm{W_\\infty a_j + t_k(I-\\Pi_j)V(t_k) a_j}_2 = 0$, which implies\n \\[ \\lim_{k\\rightarrow\\infty} t_k(I-\\Pi_j)V(t_k) a_j = -W\\infty a_j. \\]\n \n Case 1. $\\liminf_{k\\rightarrow\\infty} t_k<\\infty$. In this case extract a subsequence, say\n $(t_{k_l})_l$, so that\n $\\lim_{l\\rightarrow\\infty} t_{k_l}=t_\\infty\\in[0,\\infty)$. \nThis implies\n \\[ W_\\infty a_j +t_\\infty(I-\\Pi_j)V_\\infty a_j = 0~,~\\forall j\\in[D]. \\]\nNotice $(W_\\infty,V_\\infty)\\in \\Gamma_{t_\\infty}$.\nTherefore $\\tilde{U}=W_\\infty + t_\\infty V_\\infty$ and $\\tilde{V}=t_\\infty V_\\infty$ satisfy the conclusions (1),(2), and (3) and lemma \\ref{l3.2} is proved.\n\n\nCase 2. $\\liminf_{k\\rightarrow\\infty} t_k=\\infty$.\n\nIn the rest of the proof of this lemma, we construct an inductive process which ends with a scenario that either satisfies Case 1, or produces a (geo)metric contradiction. \n\nTo simplify notation we shall reuse the index $k$ at each stage.\n\n{\\em Initialization:} Set $p=1$. \nLet $V^{(1)}_{\\infty}=V_\\infty$, $t^{(1)}_k=t_k$, and $R^{(1)}_k=V(t_k)-V_\\infty$.\n\n{\\em Preamble:} Sequences $(t^{(p)}_k,R^{(p)}_k)$ satisfy, for every $j\\in[D]$:\n\\begin{equation}\n\\label{eq:sequences}\n\\lim_{k\\rightarrow\\infty}t^{(p)}_k = +\\infty, \\lim_{k\\rightarrow\\infty}R^{(p)}_k = 0, \\norm{R^{(p)}_k+V^{(p)}_\\infty}=1=\\norm{V^{(p)}_\\infty} , \\lim_{k\\rightarrow\\infty}t^{(p)}_k(I-\\Pi_j)R^{(p)}_k a_j = -W_\\infty a_j \n\\end{equation}\n\n{\\em Refinement:} Extract a subsequence \nindexed again by $k$ that\nsatisfies additionally:\n\\begin{equation}\n\\label{eq:seq2}\n\\norm{R^{(p)}_k}\\leq \\frac{1}{p}~,~ \\lim_{k\\rightarrow\\infty} \\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}}\\in S_1\n\\end{equation}\n\n{\\em Setting up the next iteration:} Set\n\\[ t^{(p+1)}_k=t^{(p)}_k\\norm{R^{(p)}_k} ~,~V^{(p+1)}_\\infty = \\lim_{k\\rightarrow\\infty} \\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}} ~,~ \nR^{(p+1)}_k=\\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}} - V^{(p+1)}_\\infty \\]\n\n{\\em Testing:} If $\\liminf_{k\\rightarrow\\infty}t^{(p+1)}_k<\\infty$ then proceed with Case 1 above, which ends the proof of this lemma.\n\nOtherwise $\\lim_{k\\rightarrow\\infty}t^{(p+1)}_k=\\infty$. Thus $(I-\\Pi_j)V^{(p+1)}_\\infty a_j=0$ for all $j\\in[D]$. \nSet $p\\leftarrow p+1$. The {\\em preamble} conditions (\\ref{eq:sequences}) are again satisfied \nfor all $j\\in[D]$. Then proceed by going to the {\\em refinement} step and iterate. \n\nIf the iterative process described above does not end at some finite $p$, then we construct sequences doubly indexed $(t^{(p)}_k,R^{(p)}_k)_{p,k}$ that satisfy (\\ref{eq:sequences}) and (\\ref{eq:seq2}). \n}\n\n $\\Box$\n\n\n\n\\vspace{5mm}\n\n{\\em Step 4. Contradiction with the universality property of the key.}\n\nSo far we obtained that if the lower Lipschitz bound of ${\\hat{\\beta}}_A$ vanishes than there are $Z_0,\\tilde{U},\\tilde{V}\\in{\\mathbb{R}}^{n\\times d}$ with $Z_0\\neq 0$ and $\\tilde{U}\\neq P \\tilde{V}$, for all $P\\in G$ that satisfy the conclusions of Lemma \\ref{l3.2}. Notice $\\ip{Z_0}{Z_0}=\\ip{PZ_0}{Z_0}$\n for all $P\\in G$ and $(Z_0-\\Pi_j Z_0)a_j=0$ for all $j\\in[D]$. Choose $s>0$ but small enough so that $s\\norm{\\tilde{U}},s\\norm{\\tilde{V}}<\\frac{1}{4}\\delta_0$ with $\\delta_0=\\min_{P\\in{\\mathcal S}_n\\setminus G} \\norm{(I_n-P)Z_0}$.\n Let $X=Z_0+s \\tilde{U}$ and $Y=Z_0+s\\tilde{V}$.\n Then Lemma \\ref{l3.1} implies $d(\\hat{X},\\hat{Y})=\\min_{P\\in G}\\norm{\\tilde{U}-P\\tilde{V}}>0$. \n Hence $\\hat{X}\\neq\\hat{Y}$. On the other hand,\n for every $j\\in[D]$, $Xa_j = \\Pi_j Ya_j$. Thus\n ${\\hat{\\beta}}_A(\\hat{X})={\\hat{\\beta}}_A(\\hat{Y})$. \n Contradiction with the assumption that ${\\hat{\\beta}}_A$ is injective.\n \n This ends the proof of Theorem \\ref{t5}.\n\n$\\Box$\n\\ignore{\n\\begin{rem}\nThe proof of the previous theorem provides estimates for \nboth type I and type II local lower and upper Lipschitz bounds.\n\\end{rem}\n}\n\n\\subsection{Dimension Reduction}\n\nTheorem \\ref{t4} provides an Euclidean bi-Lipschitz embedding of very high dimension, $D=1+(d-1)n!$. On the other hand, Theorem \\ref{t5} shows that any universal key $A\\in\\mathbb{R}^{d\\times D}$ for ${\\widehat{\\Rnd}}$, \nand hence any injective map $\\hat{\\beta}_A$ is bi-Lipschitz. In this subsection we show that \nany bi-Lipschitz Euclidean embedding $\\hat{\\beta}_A:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{n\\times D}$ with $D>2d$ \ncan be further compressed to a smaller dimension space $\\mathbb{R}^m$ with $m=2nd$ thus yielding\nbi-Lipschitz Euclidean embeddings of redundancy 2. This is shown in the next result.\n\n\\begin{thm}\n \\label{t6} Assume $A\\in\\mathbb{R}^{d\\times D}$ is a universal key for ${\\widehat{\\Rnd}}$ with $D\\geq 2d$. \n Then, for $m\\geq 2nd$, a generic linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{m}$ with respect to Zariski topology on\n $\\mathbb{R}^{n\\times D\\times m}$, the map\n \\begin{equation}\n \\label{eq:AB1}\n \\hat{\\beta}_{A,B}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{2nd}~,~ \\hat{\\beta}_{A,B}(\\hat{X})=B\\left(\\hat{\\beta}_A(\\hat{X})\\right)\n \\end{equation}\n is bi-Lipschitz. In particular, almost every full-rank linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$ produces such a \n bi-Lipschitz map.\n\\end{thm}\n\n\\begin{rmk}\nThe proof shows that, in fact, the complement set of linear operators $B$ that produce bi-Lipschitz embeddings is included \nin the zero-set of a polynomial. \n\\end{rmk}\n\n\\begin{rmk}\nPutting together Theorems \\ref{t4}, \\ref{t5}, \\ref{t6} we obtain that the metric space ${\\widehat{\\Rnd}}$ admits\na global bi-Lipschitz embedding in the Euclidean space $\\mathbb{R}^{2nd}$. This result is compatible\nwith a Whitney embedding theorem (see \\S 1.3 in \\cite{hirsh}) with the important caveat that the Whitney embedding result\napplies to smooth manifolds, whereas here ${\\widehat{\\Rnd}}$ is merely a non-smooth algebraic variety.\n\\end{rmk}\n\n\\begin{rmk}\nThese three theorems are summarized in part two of the \nTheorem \\ref{t2} presented in \nthe first section.\n\\end{rmk}\n\n\\begin{rmk}\nWhile the embedding dimension grows linearly in $nd$, in fact $m=2nd$, the computational complexity of constructing ${\\hat{\\beta}}_{A,B}$ is NP due to the $1+(d-1)n!$ intermediary dimension.\n\\end{rmk}\n\n\\begin{rmk}\nAs the proofs show, for $D\\geq 1+(d-1)n!$, a generic $(A,B)$ with\nrespect to Zariski topology, $A\\in\\mathbb{R}^{d\\times D}$ and linear map $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$, produces a bi-Lipschitz embedding $({\\hat{\\beta}}_{A,B},d)$ of ${\\widehat{\\Rnd}}$ into $(\\mathbb{R}^{2nd},\\norm{\\cdot}_2)$. \n\\end{rmk}\n{\\bf Proof of Theorem \\ref{t6} }\n\nThe proof follows a similar approach as in Theorem 3 of \\cite{Cahill19}.\nSee also \\cite{DUFRESNE20091979}.\n\nWithout loss of generality, assume $m0$ so that for every $X,Y\\in\\mathbb{R}^{n\\times d}$, \n$\\norm{B(L_{\\gamma}(X,Y))}\\geq a_\\gamma \\norm{L_{\\gamma}(X,Y)}$.\nLet $a_\\infty = \\min_{\\gamma}a_\\gamma >0$. Thus\n\\[ \\norm{\\beta_{A,B}(X) - \\beta_{A,B}(Y)} =\\norm{B(L_{\\gamma_0}(X,Y))}\n\\geq a_\\infty \\norm{L_{\\gamma_0}(X,Y)}=a_\\infty \\norm{\\beta_A(X)-\\beta_A(Y)} \\]\nwhere $\\gamma_0\\in(S_n)^{2D}$ is a particular $2D$-tuple of permutations. This shows that \n$B{\\vert}_{\\beta_A(\\mathbb{R}^{n\\times d})}:\\beta_A(\\mathbb{R}^{n\\times d})\\rightarrow\\mathbb{R}^m$ is bi-Lipschitz.\nBy Theorem \\ref{t5}, the map ${\\hat{\\beta}}_A$ is bi-Lipschitz. Therefore\nwe get ${\\hat{\\beta}}_{A,B}$ is bi-Lipschitz as well.\n$\\Box$\n\n\n\\subsection{Proof of Corollary \\ref{c0}\\label{subsec4.4}}\n\n(1) It is clear that any continuous $f$ induces a continuous $\\varphi:\\beta(\\mathbb{R}^{n\\times d})\\rightarrow\\mathbb{R}$ via $\\varphi(\\beta(X))=f(X)$. Furthermore, \n$F:=\\beta(\\mathbb{R}^{n\\times d})={\\hat{\\beta}}({\\widehat{\\Rnd}})$\nis a closed subset of $\\mathbb{R}^m$ since ${\\hat{\\beta}}$ is bi-Lipschitz. \nThen a consequence of Tietze extension theorem \n(see problem 8 in \\S 12.1 of \\cite{roydenfitzpatrick})\nimplies that $\\varphi$ admits a continuous extension $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$. Thus $g(\\beta(X))=f(X)$ \nfor all $X\\in\\mathbb{R}^{n\\times d}$. The converse is trivial.\n\n(2) As at part (1), the Lipschitz continuous function $f$ induces a Lipschitz continuous function $\\varphi:F\\rightarrow\\mathbb{R}$. Since $F\\subset\\mathbb{R}^m$ is a subset of a Hilbert space, by Kirszbraun\nextension theorem (see \\cite{WelWil75}), $\\varphi$ \nadmits a Lipschitz continuous extension \n(even with the same Lipschitz constant!)\n $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ so that $g(\\beta(X))=f(X)$ for every $X\\in\\mathbb{R}^{n\\times d}$. The converse is trivial. $\\Box$\n\n\n\\section{Applications to Graph Deep Learning\\label{sec4}}\n\nIn this section we take an empirical look at the permutation invariant mappings presented in this paper. We focus on the problems of graph classification, for which we employ the PROTEINS\\_FULL dataset \\cite{DobsonDoing_proteins}, and graph regression, for which we employ the quantum chemistry QM9 dataset\n\\cite{ramakrishnan2014quantum}. In both problems we want to estimate a function $F: (A,Z) \\rightarrow p$, where $(A,Z)$ characterizes a graph where $A \\in \\mathbb{R}^{n\\times n}$ is an adjacency matrix and $Z \\in \\mathbb{R}^{n\\times r}$ is an associated feature matrix where the $i^{th}$\nrow encodes an array of $r$ features associated with the $i^{th}$ node. $p$ is a scalar output where we have $p \\in \\{0,1\\}$ for binary classification and $p \\in \\mathbb{R}_+$ for regression.\n\nWe estimate $F$ using a deep network that is trained in a supervised manor. The network is comprised of three successive components applied in series: $\\Gamma$, $\\phi$, and $\\zeta$. $\\Gamma$ represents a graph deep network \\cite{GCN},\nwhich produces a set of embeddings $X \\in \\mathbb{R}^{N\\times d}$ across the nodes in the graph. Here $N\\geq n$ is chosen to accommodate the graph with the largest number of nodes. In this case, the last $N-n$ rows of $Y$ are filled with 0's. $\\phi: \\mathbb{R}^{N\\times d} \\rightarrow \\mathbb{R}^{m}$ represents a permutation invariant mapping such as those proposed in this paper. $\\zeta: \\mathbb{R}^{m} \\rightarrow \\mathbb{R}$ is a fully connected neural network. The entire end-to-end network is shown in Figure \\ref{fig:gcn_end2end}.\n\nIn this paper, we model $\\Gamma$ using a Graph Convolutional Network (GCN) outlined in \\cite{GCN}. \nLet ${\\bf D} \\in \\mathbb{R}^{n \\times n}$ be the associated degree matrix for our graph $\\mathcal{G}$. Also let $\\tilde{A}$ be the associated adjacency matrix of $\\mathcal{G}$ with added self connection: $\\tilde{A}=I+A$, where $I$ is the $n \\times n$ identity matrix, and $\\tilde{{\\bf D}}={\\bf D}+I$. Finally, we define the modified adjacency matrix $\\hat{A}=\\tilde{{\\bf D}}^{-1\/2} \\tilde{A} \\tilde{{\\bf D}}^{-1\/2}$. A GCN layer is defined as $H^{(l+1)}=\\sigma (\\hat{A}H^{(l-1)}W^{(l)})$.\nHere $H^{(l-1)}$ represents the GCN state coming into the $l^{th}$ layer, $\\sigma$ represents a chosen nonlinear element-by-element operation such as ReLU, and $W^{(l)}$ represents a matrix of trainable weights assigned to the $l^{th}$ layer whose number of rows match the number of columns in $H^{l}$ and number of columns is set to the size of the embeddings at the (l+1)'th layer. The initial state $H^{(0)}$ of the network is set to the feature set of the nodes of the graph $H^{(0)}=Z$. \n\n\nFor $\\phi$ we employ seven (7) different methods that are described next.\n\\begin{enumerate}\n \\item ordering: For the ordering method, we set $D=d+1$, $\\phi_{ordering}(X)=\\beta_A(X)=\\downarrow(XA)$ with $A=[I~1]$ the identity matrix followed by a column of ones. The ordering and identity-based mappings have the notable disadvantage of not producing the same output embedding size for different sized graphs. To accommodate this and have consistently sized inputs for $\\eta$, we choose to zero-pad $\\phi(X)$ for these methods to produce a vector in $\\mathbb{R}^{m}$, where $m=ND=N(d+1)$ and $N$ is the size of the largest graph in the dataset.\n \\item kernels: For the kernels method, \n $$(\\phi_{kernel}(X))_j=\\sum_{k=1}^n K_G(x_k,a_j)=\\sum_{k=1}^n exp(-\\norm{x_k-a_j}^2),\n ~~j\\in[m],$$ \n for $X=[x_1|\\cdots|x_n]^T$, where kernel vectors $a_1,\\ldots,a_m\\in\\mathbb{R}^d$ \n are generated randomly, each element of each vector is drawn from a standard normal distribution. Each resultant vector is then normalized to produce a kernel vector of magnitude one. When inputting the embedding $X$ to the kernels mapping, we first normalized the embedding for each respective node.\n \\item identity: In this case $\\phi_{id}(X)=X$, which is obviously not a permutation invariant map.\n \\item data augmentation: In this case $\\phi_{data\\;augment}(X)=X$ but data augmentation is used. Our data augmentation scheme works as follows. We take the training set and create multiple permutations of the adjacency and associated feature matrix for each graph in the training set. We add each permuted graph to the training set to be included with the original graphs. In our experiments we use four added permutations for each graph when employing data augmentation.\n \\item sum pooling: The sum pooling method sums the feature values across the set of nodes: $\\phi_{sum\\;pooling}(X)=\\mathbf{1}_{n\\times 1}^T X$.\n \\item sort pooling: The sort pooling method flips entire rows of $X$ so that the last column is ordered descendingly, $\\phi_{sort\\;pool}(X)=\\Pi X$ where $\\Pi\\in{\\mathcal S}_n$ so that $\\Pi\\,X(:,d)=\\downarrow(X(:,d))$. \n \\item set-2-set: This method employs a recurrent neural network\n that achieves permutation invariance through attention-based weighted summations. It has been introduced in \\cite{OrderMatters_2015arXiv151106391V}.\n\\end{enumerate}\n\nFor our deep neural network $\\eta$ we use a simple multilayer perceptron of size described below.\n\nSize parameters related to $\\Gamma$ and $\\zeta$ components are largely held constant across the different implementations.\nHowever the network parameters are trained independently for each method.\n\n\\begin{figure}[!htbp]\n \\centering\n\t\\includegraphics[width=.8\\linewidth]{Results\/GCN_end2end_2.png}\n\t\\caption[.]{.}\n\t\\label{fig:gcn_end2end}\n\\end{figure}\n\n\n\\subsection{Graph Classification}\n\n\\subsubsection{Methodology}\nFor our experiments in graph classification we consider the PROTEINS\\_FULL dataset obtained from \\cite{KKMMN2016} and originally introduced in \\cite{DobsonDoing_proteins}. \nThe dataset consists of 1113 proteins falling into one of two classes: those that function as enzymes and those that do not. Across the dataset there are 450 enzymes in total. \nThe graph for each protein is constructed such that the nodes represent amino acids and the edges represent the bonds between them. The number of amino acids (nodes) vary from around 20 to a maximum of 620 per protein with an average of 39.06. \nEach protein comes with a set of features for each node. \nThe features represent characteristics of the associated amino acid represented by the node. The number of features is $r=29$.\nWe run the end-to-end model with three GCN layers in $\\Gamma$, each with 50 hidden units. \n$\\zeta$ consists of three dense multi-layer perceptron layers, each with 150 hidden units. \nWe set d equal to 1, 10, 50 and 100.\n\nFor each method and embedding size we train for 300 epochs. Note though that the data augmentation method will have experienced five times as many training steps due to the increased size of its training set. We use a batch size of 128 graphs. The loss function minimized during training is the binary cross entropy loss (BCE) defined as\n\\begin{equation}\n\\label{eq:BCE}\nBCE = -\\frac{1}{B}\\sum_{t=1}^B p_t log(\\sigma(\\eta(\\phi(X^{(t)}))))+(1-p_t)log(1-\\sigma(\\eta(\\phi(X^{(t)})))) \n\\end{equation} \nwhere $B=128$ is the batch size, $p_t=1$ when the $t^{th}$ graph\n(protein) is an enzyme and $p_t=0$ otherwise, $\\sigma(x)=\\frac{1}{1+e^{-x}}$ is the sigmoid function that maps the output $\\eta(\\phi(X^{(t)})$ of the 3-layer fully connected network $\\eta$ to $[0,1]$. Three performance metrics were computed: accuracy (ACC), area under the receiver operating characteristic curve (AUC), and average precision (AP) as area under the precision-recall curve from precision scores. These measure are defined as follows (see sklearn.metrics module documentation in pytorch, or \\cite{roc}).\n\nFor a threshold $\\tau\\in[0,1]$, the classification decision $\\hat{p}_t(\\tau)$ is given by:\n\\begin{equation}\n \\hat{p}_t(\\tau) = \\left\\{\n \\begin{array}{rcl}\n 1 & \\mbox{if} & \\mbox{$\\sigma(\\eta(\\phi(X^{(t)}))\\geq \\tau$} \\\\\n 0 & \\mbox{if} & \\mbox{otherwise}\n \\end{array}\\right. .\n\\end{equation}\nBy default $\\tau=\\frac{1}{2}$. For a given threshold, one computes the four scores, true positive (TP), false positive (FP), true negative (TN) and false negative (FN):\n\\begin{equation}\n TP(\\tau) = \\frac{1}{B_1}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 1}1_{p_t = 1} ~~, ~~\n TN(\\tau) = \\frac{1}{B_0} 1_{\\hat{p}_t(\\tau) = 0}1_{p_t = 0}\n\\end{equation}\n\\begin{equation}\n FP(\\tau) = \\frac{1}{B_0}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 1}1_{p_t = 0} = 1-TN(\\tau) ~~,~~\n FN(\\tau) = \\frac{1}{B_1}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 0}1_{p_t = 1} = 1-TP(\\tau)\n\\end{equation}\nwhere $B_0=\\sum_{t=1}^B 1_{p_t = 0}$ and $B_1=\\sum_{t=1}^B 1_{p_t = 1}=B-B_0$.\n\nThese four statistics predict Precision $P(\\tau)$, Recall $R(\\tau)$ (also known as sensitivity or true positive rate), and Specificity $S(\\tau)$ (also known as true negative rate)\n\\begin{equation} P(\\tau) = \\frac{TP(\\tau)}{TP(\\tau)+FP(\\tau)}\n~~,~~R(\\tau) = \\frac{TP(\\tau)}{TP(\\tau)+FN(\\tau)}\n~~,~~S(\\tau) = \\frac{TN(\\tau)}{TN(\\tau)+FP(\\tau)}\n\\end{equation}\n\nAccuracy (ACC) is defined as the fraction of correct classification for default threshold $\\tau=\\frac{1}{2}$ over the set of batch samples:\n\\begin{equation}\n\\label{eq:ACC}\n ACC = \\frac{1}{B}\\sum_{t=1}^B 1_{p_t = \\hat{p}_t(\\frac{1}{2})} =\\frac{B_0}{B} TN(\\frac{1}{2}) + \\frac{B_1}{B} TP(\\frac{1}{2}) \n\\end{equation}\nArea under the receiver operating characteristic curve (AUC) is computed from prediction scores as the area under true positive rate (TPR) vs. false positive rate (FPR) curve, i.e. the recall vs. 1-specificity curve\n\\begin{equation}\n\\label{eq:AUC}\n AUC = \\frac{1}{2}\\sum_{k=1}^K (S(\\tau_{k-1})-S(\\tau_k))(R(\\tau_{k-1})+R(\\tau_k))\n\\end{equation}\nwhere $K$ is the number of thresholds.\nAverage precision (AP) summarizes a precision-recall curve as the weighted mean of precision achieved at each threshold, with the increase in recall from the previous thresholds used as the weight:\n\\begin{equation}\n\\label{eq:AP}\n AP = \\sum_{k=1}^K (R(\\tau_k) - R(\\tau_{k-1}))P(\\tau_k).\n\\end{equation}\n\nWe track the binary cross entropy (BCE) through training and we compute it on the holdout set and a random node permutation of the holdout set (see Figures \\ref{fig:prot1} and \\ref{fig:prot2}). The lower the value the better.\n\nWe look at the three performance metrics on the training set, the holdout set, and a random node permutation of the holdout set: see Figures \\ref{fig:prot3}, and \\ref{fig:prot4} for accuracy (ACC); see Figures \\ref{fig:prot5}, and \\ref{fig:prot6} for area under the receiver operating characteristic curve (AUC); and see Figures \\ref{fig:prot7}, and \\ref{fig:prot8} for average precision (AP). For all these performance metrics, the higher the score the better.\n\n\\subsubsection{Discussion}\n\nTables \\ref{table:t1}-\\ref{table:t12} list values of the three performance metrics (ACC, AUC, AP) at the end of training (after 300 epochs). \nPerformances over the course of training are plotted in Figures \\ref{fig:prot1} through \\ref{fig:prot8}.\n\nThe authors of \\cite{KKMMN2016} utilized a Support Vector Machine (1-layer perceptron) for classification and \nobtained an accuracy (ACC) of 77\\% on the entire data set \nusing 52 features, and an accuracy of 80\\% on a smaller set of 36 features. By comparison, our data augmentation method for $d=100$ achieved an accuracy of 97.5\\% on training data set,\nbut dropped dramatically to 73\\% on holdout data, and 72\\% on \nholdout data set with randomly permuted nodes. \nOn the other hand, both the kernels method and the sum-pooling\nmethod with $d=50$ achieved an accuracy of around 79\\% on\ntraining data set, while dropping accuracy performance by \nonly 2\\% to around\n77\\% on holdout data (as well as holdout data with nodes permuted).\n\nFor $d=1$, data augmentation performed the best on the training set with an area under the receiver operating characteristic (AUC) of 0.896, followed closely by the identity method with an AUC of 0.886. On the permuted holdout set however, sort-pooling performed the best with an AUC of 0.803.\n\nFor $d=10$, sum-pooling, ordering, and kernels performed well on the permuted holdout set with AUC's of 0.821, 0.820, and 0.818 respectively. The high performance of the identity method, data augmentation, and sort-pooling on the training set did not translate to the permuted holdout set at $d=10$. By $d=100$, sum-pooling still performed the best on the permuted holdout set with an AUC of 0.817. This was followed by the kernels method which achieved an AUC of 0.801 on the permuted holdout set.\n\nFor experiments where $d>1$, the identity method and data augmentation show a notable drop in performance from the training set to the holdout set. This trend is also, to a lesser extent, visible in the sort pooling and ordering methods. In the holdout permuted set we see significant oscillations in the performance of both the identity and data augmentation methods.\n\n\\subsection{Graph Regression}\n\n\\subsubsection{Methodology}\nFor our experiments in graph regression we consider the qm9 dataset \\cite{ramakrishnan2014quantum}. This dataset consists of 134 thousand molecules represented as graphs, where the nodes represent atoms and edges represent the bonds between them. \n\nEach graph has between 3 and 29 nodes, $3\\leq n\\leq 29$. Each node has 11 features, $r=11$. We hold out 20 thousand of these molecules for evaluation purposes. The dataset includes 19 quantitative features for each molecule.\n\nFor the purposes of our study, we focus on electron energy gap (units $eV$), which is $\\Delta\\varepsilon$ in \\cite{DFTpaper} whose chemical accuracy is $0.043 eV$ and whose prediction performance of any machine learning technique\nis worse than any other feature.\nThe best existing estimator for this feature is enn-s2s-ens5 from \\cite{Gilmer_2017arXiv170401212G}\n and has a mean absolute error (MAE) of $0.0529eV$ which is $1.23$ larger than the chemical accuracy. \n We run the end to end model with three GCN layers in $\\Gamma$, each with 50 hidden units. $\\eta$ consists of three multi-layer perceptron layers, each with 150 hidden units. We use rectified linear units as our nonlinear activation function. Finally, we vary $d$, the size of the node embeddings that are outputted by $\\Gamma$. We set $d$ equal to 1, 10, 50 and 100.\n\nFor each method and embedding size we train for 300 epochs. Note though that the data augmentation method will have experienced five times as many training steps due to the increased size of its training set. We use a batch size of 128 graphs. The loss function minimized during training is the mean square error (MSE) between the ground truth and the network output \n (see Figures \\ref{fig:a1}, \\ref{fig:a2}) \n\\begin{equation}\n\\label{eq:MSE}\nMSE = \\frac{1}{B}\\sum_{t=1}^B |\\Delta\\varepsilon_t -\\eta(\\phi(X^{(t)}))))|^2\n\\end{equation}\nwhere $B=128$ is the batch size of 128 graphs and $\\Delta\\varepsilon_t$ is the electron energy gap of the $t^{th}$ graph (molecule). The performance\nmetric is Mean Absolute Error (MAE)\n\\begin{equation}\n\\label{eq:MAE}\nMAE = \\frac{1}{B}\\sum_{t=1}^B |\\Delta\\varepsilon_t -\\eta(\\phi(X^{(t)}))))|.\n\\end{equation}\nWe track the mean absolute error through the course of training. We look at this performance metric on the training set, the holdout set, and a random node permutation of the holdout set (see Figures \\ref{fig:b1}, and \\ref{fig:b2}). \n\n\n\\subsubsection{Discussion}\n\nNumerical results at the end of training (after 300 epochs) are included in Tables \\ref{table:a}, \\ref{table:b}, \\ref{table:c} and \\ref{table:d}.\nFrom the results we see that the ordering method performed best for $d=100$\nfollowed closely by the data augmentation method, while both the ordering method and the kernels method performed well for $d=10$, though both fell slightly short of data augmentation which performed marginally better on both the training data and the holdout data, though with significantly more training iterations. For $d=1$, the kernels method failed to train adequately. The identity mapping performed relatively well on training data (for $d=100$ it achieved the smallest MAE among all methods and all parameters) and even the holdout data, however it lost its performance on the permuted holdout data. The identity mapping's failure to generalize across permutations of the holdout set is likely exacerbated by the fact that the QM9 data as presented to the network comes ordered in its node positions from heaviest atom to lightest. Data augmentation notably kept its performance despite this due to training on many permutations of the data. \n\nFor $d=100$, our ordering method achieved a MAE of $0.155eV$ on training data set and $0.187eV$ on holdout data set, which are $3.6$ and $4.35$ times larger than the chemical accuracy ($0.043eV$\\ignore{ cf. Supplementary material of \\cite{Gilmer_2017arXiv170401212G}}), respectively. This is worse than the enn-s2s-ens5 method in \\cite{Gilmer_2017arXiv170401212G} (current best method) that achieved a MAE $0.0529$ (eV), $1.23$ larger than the chemical accuracy, \nbut better than the Coulomb Matrix (CM) representation in \\cite{PhysRevLett.108.058301} that achieved a MAE $5.32$ larger than the chemical accuracy whose features were optimized for this task.\n\n\n\\bibliographystyle{amsplain}\n\n\\section{Introduction\\label{sec1}}\n\n\nThis paper is motivated by a class of problems in graph deep learning, where the\nprimary task is either graph classification or graph regression. \nIn either case, the result should be invariant to arbitrary permutations of graph nodes.\n\nAs we explain below, the mathematical problem analyzed in this paper is a special case \nof the permutation invariance issue described above. To set the notations consider the\nvector space ${\\mathbb{R}}^{n\\times d}$ of $n\\times d$ matrices endowed with the Frobenius norm \n $\\norm{X}=\\left(trace(XX^T)\\right)^{1\/2}$\nand its associated Hilbert-Schmidt scalar product, $\\ip{X}{Y}=trace(XY^T)$.\n Let ${\\mathcal S}_n$ denote the symmetric group of $n\\times n$ permutation matrices. \n ${\\mathcal S}_n$ is a finite group of size $|{\\mathcal S}_n|=n!$.\n\nOn ${\\mathbb{R}}^{n\\times d}$ we consider the equivalence relation $\\sim$ \ninduced by the symmetric group of permutation matrices ${\\mathcal S}_n$ as follows. Let $X,Y\\in{\\mathbb{R}}^{n\\times d}$. \nThen we say $X\\sim Y$ if there is $P\\in{\\mathcal S}_n$ so that $Y=PX$. In other words, two matrices are equivalent if one is a row permutation of the other. \nThe equivalence relation induces a natural distance on the quotient space \n${\\widehat{\\Rnd}}:={\\mathbb{R}}^{n\\times d}\/\\sim$,\n\\begin{equation}\n \\label{eq:1.1}\nd: {\\widehat{\\Rnd}}\\times {\\widehat{\\Rnd}} \\rightarrow\\mathbb{R} ~~,~~d(\\hat{X},\\hat{Y})=\\min_{\\Pi\\in{\\mathcal S}_n}\\norm{X-\\Pi Y} \n\\end{equation}\nThis makes $({\\widehat{\\Rnd}},d)$ a complete metric space.\n\nOur main problem can now be stated as follows:\n\\begin{prob}\\label{prob1}\nGiven $n,d\\geq 1$ positive integers, find $m$ and a bi-Lipschitz map\n $\\hat{\\alpha}:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^m,\\norm{\\cdot}_2)$.\n\\end{prob}\nExplicitly the problem can be restated as follows. One is asked to construct a \nmap $\\alpha:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m$ that satisfies the following conditions:\n\\begin{enumerate}\n \\item If $X,Y\\in{\\mathbb{R}}^{n\\times d}$ so that $X\\sim Y$ then $\\alpha(X)=\\alpha(Y)$\n \\item If $X,Y\\in{\\mathbb{R}}^{n\\times d}$ so that $\\alpha(X)=\\alpha(Y)$ then $X\\sim Y$\n \\item There are constants $01$. \n \\item {\\em Sorting Embedding}.\n For $x\\in\\mathbb{R}^n$, consider the sorting map\n\\begin{equation}\\label{eq:ord}\n \\downarrow:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n ~~,~~\n \\downarrow(x)=(x_{\\pi(1)},x_{\\pi(2)},\\ldots,\n x_{\\pi(n)})^T \n\\end{equation}\n where the permutation $\\pi$ is so that\n $x_{\\pi(1)}\\geq x_{\\pi(2)}\\geq\\cdots\\geq x_{\\pi(n)}$. It is obvious that $\\downarrow$ satisfies Conditions (1) and (2) and therefore lifts to an injective map on ${\\widehat{\\Rnd}}$. As we see in Section \\ref{sec3}, the map $\\downarrow$ is bi-Lipschitz. In fact it is isometric, and hence produces an ideal embedding. Our work in Section \\ref{sec3} is to extend such construction to the more general \n case $d>1$.\n\\end{enumerate}\nThe algebraic embedding is a special case of the more general {\\em kernel method} that can be thought of as a projection of the measure \n$a_{\\infty}(X)$ onto a finite dimensional space, e.g., the space of polynomials spanned by $\\{X,X^2,\\cdots,X^n\\}$. In applications such kernel method is known as a ``Readout Map\" \\cite{deepsets}, based on ``Sum Pooling\".\n\nThe sorting embedding has been used in applications under the name of ``Pooling Map\" \\cite{deepsets}, based on ``Max Pooling\". A na\\\"{\\i}ve extension of the unidimensional map (\\ref{eq:ord}) to the case $d>1$ might employ the lexicographic order: order monotone decreasing the rows according to the first column, and break the tie by going to the next column. While this gives rise to an injective map, it is easy to see it is not even continuous, let alone Lipschitz. The main work in this paper is to extend the sorting embedding to the case $d>1$ using a three-step procedure, first embed ${\\mathbb{R}}^{n\\times d}$ into a larger vector space $\\mathbb{R}^{n\\times D}$, then apply $\\downarrow$ in each column independently, and then perform a dimension reduction by a linear map into $\\mathbb{R}^{2nd}$. Similar to the phase retrieval problem (\\cite{bcmn,bod,balan16}), the redundancy introduced in the first step counterbalances the loss of information (here, relative order of one column with respect to another) in the second step. \n\nA summary of main results presented in this paper is contained in the following result.\n\\begin{thm}\\label{t1}\nConsider the metric space $({\\widehat{\\Rnd}},d)$.\n\\begin{enumerate}\n\\item (Polynomial Embedding) There exists a Lipschitz injective map\n\\[ {\\hat{\\alpha}}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m \\]\nwith $m=\\left( \\begin{array}{c}\n\\mbox{$d+n$} \\\\\n\\mbox{$d$}\n\\end{array} \\right)$. Two explicit constructions of this map are given in (\\ref{eq:alpha1}) and (\\ref{eq:alpha2}).\n\\item (Sorting based Embedding) There exists a class of bi-Lipschitz maps \n\\[ {\\hat{\\beta}}_{A,B}:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^m,\\norm{\\cdot}) ~,~ {\\hat{\\beta}}_{A,B}(\\hat{X})=B\\left({\\hat{\\beta}}_A(\\hat{X})\\right) \\]\nwith $m=2nd$, where each map ${\\hat{\\beta}}_{A,B}$ is the composition of two bi-Lipschitz maps: a full-rank linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow \\mathbb{R}^m$, with the nonlinear bi-Lipschitz map ${\\hat{\\beta}}_A:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{n\\times D}$\n parametrized by a matrix $A\\in\\mathbb{R}^{d\\times D}$ called \"key\". \n Explicitly,\n${\\hat{\\beta}}(\\hat{X})=\\downarrow(XA)$, where $\\downarrow$ acts column-wise.\n These maps are characterized by the following properties:\n \\begin{enumerate}\n\\item For $D=1+(d-1)n!$, any\n$A\\in\\mathbb{R}^{d\\times (1+(d-1)n!)}$ whose columns form a full spark frame defines a bi-Lipschitz map ${\\hat{\\beta}}_A$ on ${\\widehat{\\Rnd}}$. \nFurthermore, a lower Lipschitz constant is given by the smallest $d^{th}$ singular value among all $d\\times d$ sub-matrices of $A$,\n $\\min_{J\\subset[D],|J|=d}s_d(A[J])$.\n\\item For any matrix (``key\") $A\\in\\mathbb{R}^{d\\times D}$ such that the map \n${\\hat{\\beta}}_A$ is injective, then ${\\hat{\\beta}}_A:({\\widehat{\\Rnd}},d)\\rightarrow(\\mathbb{R}^{n\\times D},\\norm{\\cdot})$ is bi-Lipschitz. Furthermore, an upper Lipschitz constant is given by $s_1(A)$, the largest singular value of $A$.\n\\item Assume $A\\in\\mathbb{R}^{d\\times D}$ is such that the map \n${\\hat{\\beta}}_A$ is injective (i.e., a \"universal key\"). Then for almost any linear map $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$ the map ${\\hat{\\beta}}_{A,B}=B\\circ{\\hat{\\beta}}_A$ is\nbi-Lipschitz.\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\nAn immediate consequence of this result is the following corollary whose proof is included in subsection \\ref{subsec4.4}:\n\\begin{cor}\n\\label{c0}\nLet $\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}^m$ induce a bi-Lipschitz embedding ${\\hat{\\beta}}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m$ of the\nmetric space $({\\widehat{\\Rnd}},d)$ into $(\\mathbb{R}^m,\\norm{\\cdot}_2)$. \n\\begin{enumerate}\n\\item For any continuous function $f:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ \ninvariant to row-permutation (i.e., $f(PX)=f(X)$ for every \n$X\\in\\mathbb{R}^{n\\times d}$ and $P\\in{\\mathcal S}_n$) there exists a continuous\nfunction $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ such that $f=g\\circ\\beta$.\nConversely, for any $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ continuous function, the \nfunction $f=g\\circ\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ is continuous\nand row-permutation invariant.\n\\item For any Lipschitz continuous function $f:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ \ninvariant to row-permutation (i.e., $f(PX)=f(X)$ for every \n$X\\in\\mathbb{R}^{n\\times d}$ and $P\\in{\\mathcal S}_n$) there exists a Lipschitz continuous\nfunction $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ such that $f=g\\circ\\beta$.\nConversely, for any $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ Lipschitz continuous function, the \nfunction $f=g\\circ\\beta:\\mathbb{R}^{n\\times d}\\rightarrow\\mathbb{R}$ is Lipschitz continuous\nand row-permutation invariant.\n\\end{enumerate}\n\\end{cor}\n\\vspace{5mm}\n\n\nThe structure of the paper is as follows. Section \\ref{sec2} contains the algebraic embedding method and encoders $\\alpha$ described at part (1) of Theorem \\ref{t1}. Corollary \\ref{cor2} contains part (1) of the main result stated above. Section \\ref{sec3} introduces the sorting based embedding procedure and describes the key-based encoder $\\beta$. Necessary and sufficient conditions for key universality are presented in Proposition\n\\ref{prop3.8}; the injectivity of the encoder described at part (2.a) of Theorem \\ref{t1} is proved in Theorem \\ref{t4}; the bi-Lipschitz property of any universal key described at part (2.b) of Theorem \\ref{t1} is shown in Theorem \\ref{t5}; the dimension reduction statement (2.c) of Theorem \\ref{t1}\nis included in Theorem \\ref{t6}. Proof of Corollary \\ref{c0} is presented in subsection \\ref{subsec4.4}. Section \\ref{sec4} contains applications to graph deep learning. These application use Graph Convolution Networks and the numerical experiments are carried out on two graph data sets: a chemical compound data set (QM9) and a protein data set (PROTEINS\\_FULL). \n\nWhile the motivation of this analysis is provided by graph deep learning applications,\nthis is primarily a mathematical paper. Accordingly the formal theory is presented first, and then is followed by the machine learning application. Those interested in the application (or motivation) can skip directly to Section \\ref{sec4}. \n\n{\\bf Notations}. For an integer $d\\geq 1$, $[d]=\\{1,2,\\ldots,d\\}$. For a matrix $X\\in{\\mathbb{R}}^{n\\times d}$,\n $x_1,\\ldots x_d\\in\\mathbb{R}^n$ denote its columns, $X=[x_1\\vert\\cdots\\vert x_d]$. All norms are Euclidean; for a matrix $X$, $\\norm{X}=\\sqrt{trace(X^TX)}=\\sqrt{\\sum_{k,j}|X_{k,j}|^2}$ denotes the Frobenius norm; for vectors $x$, $\\norm{x}=\\norm{x}_2=\\sqrt{\\sum_{j} |x_j|^2}$. \n \n\n\\subsection{Prior Works}\n\nSeveral methods for representing orbits of vector spaces under the action of permutation (sub)groups have been studied in literature. Here we describe some of these results, without claiming an exhaustive literature survey.\n\nA rich body of literature emanated from the early works on \nsymmetric polynomials and group invariant representations of \nHilbert, Noether, Klein and Frobenius. They are part of standard\ncommutative algebra and finite group representation theory. \n\nPrior works on permutation invariant mappings have predominantly employed some form of summing procedure, though some have alternatively employed some form of sorting procedure.\n\nThe idea of summing over the output nodes of an equivariant network has been well studied. \nThe algebraic invariant theory goes back to Hilbert and Noether (for finite groups) and then continuing with the continuous invariant function theory of \nWeyl and Wigner (for compact groups), \nwho posited that a generator function $\\psi:X\\rightarrow\\mathbb{R}$ gives rise to a function $E:X\\rightarrow\\mathbb{R}$ invariant to the action of a finite group $G$ on $X$, $(g,x)\\mapsto g.x$, via the averaging formula $E(x)=\\frac{1}{|G|}\\sum_{g\\in G} \\psi(g.x)$.\n\nMore recently, this approach provided the framework for universal approximation results of $G$-invariant functions. \\cite{maron2018invariant} showed that invariant or equivariant networks must satisfy a fixed point condition. The equivariant condition is naturally realized by GNNs. The invariance condition is realized by GNNs when followed by summation on the output layer, as was further shown in \\cite{keriven2019universal}, \\cite{pmlr-v97-maron19a} and \\cite{lipman2022}. Subsequently, \\cite{yarotsky2021universal} proved universal approximation results over compact sets for continuous functions invariant to the action of finite or continuous groups. In \\cite{geerts2022}, the authors\nobtained bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests by tensorizing the input-output mapping. \n\\cite{sannai2020universal} studied approximations of equivariant maps, while \\cite{NEURIPS2019_71ee911d} showed that if a GNN with sufficient expressivity is well trained, it can solve the graph isomorphism problem.\n\nThe authors of \\cite{OrderMatters_2015arXiv151106391V} designed an algorithm for processing sets with no natural orderings. The algorithm applies an attention mechanism to achieve permutation invariance with the attention keys being generated by a Long-Short Term Memory (LSTM) network. Attention mechanisms amount to a weighted summing and therefore can be considered to fall within the domain of summing based procedures.\n\nIn \\cite{GGsNN_2015arXiv151105493L}, the authors designed a permutation invariant mapping for graph embeddings. The mapping employs two separate neural networks, both applied over the feature set for each node. One neural network produces a set of new embeddings, the other serves as an attention mechanism to produce a weighed sum of those new embeddings.\n\n\n\n\nSorting based procedures for producing permutation invariant mappings over single dimensional inputs have been addressed and used by \\cite{deepsets}, notably in their {\\it max pooling} procedure.\n\nThe authors of \\cite{qi2017pointnet} developed a permutation\ninvariant mapping \n$pointnet$ for point sets that is based on a $max$ function. The mapping takes in a set of vectors, processes each vector through a neural network followed by an scalar output function, and takes the maximum of the resultant set of scalars.\n\nThe paper \\cite{zhang2018end} introduced {\\it SortPooling}. {\\it SortPooling} orders the latent embeddings of a graph according to the values in a specific, predetermined column. All rows of the latent embeddings are sorted according to the values in that column. While this gives rise to an injective map, it is easy to see it is not even continuous, let alone Lipschitz. The same issue\narises with any lexicographic ordering, including the well-known Weisfeiler-Leman embedding \\cite{wl}.\nOur paper introduces a novel method that bypasses this issue.\n\nAs shown in \\cite{pmlr-v97-maron19a}, the sum pooling-based GNNs provides universal approximations for of any permutation invariant continuous function but only on \\emph{compacts}. Our sorting based embedding removes the compactness restriction as well as it extends to all Lipschitz maps.\n\nWhile this paper is primarily mathematical in nature, methods developed here are applied to two graph data sets, QM9 and PROTEINS\\_FULL. Researchers have applied various graph deep learning techniques to both data sets. In particular, \\cite{Gilmer_2017arXiv170401212G} studied extensively the QM9 data set, and compared their method with many other algorithms\nproposed by that time.\n\n\\section{Algebraic Embeddings\\label{sec2}}\n\nThe algebraic embedding presented in this section can be thought of a special kernel to project equation (\\ref{eq:measure}) onto.\n\n\\subsection{Kernel Methods}\nThe kernel method employs a family of continuous kernels (test) functions, $\\{K(x;y)~;~x\\in\\mathbb{R}^d~,~y\\in Y\\}$ parametrized\/indexed by a set $Y$. \nThe measure representation $\\mu=a_{\\infty}(X)$ in (\\ref{eq:measure}) yields a nonlinear map\n\\[ \\alpha:\\mathbb{R}^{n\\times d} \\rightarrow C(Y)\n~~,~~X \\mapsto F(y)=\\int_{R^d} K(x;y)d\\mu \\]\ngiven by\n\\[ \\alpha(X)(y)= \\frac{1}{n}\\sum_{k=1}^n K(x_k;y) \\]\nThe embedding problem \\ref{prob1}) can be restated as follows. One is asked\nto find a finite family of kernels $\\{K(x;y)~;~x\\in\\mathbb{R}^d~,~y\\in Y\\}$, \n $m=|Y|$ so that\n\\begin{equation}\n\\label{eq:kernel}\n{\\hat{\\alpha}}:({\\widehat{\\Rnd}},d) \\rightarrow l^2(Y)\\sim (\\mathbb{R}^m,\\norm{\\cdot}_2) ~~,~~ ({\\hat{\\alpha}}(\\hat{X}))_y = \\frac{1}{n} \\sum_{k=1}^n K(x_k;y)\n\\end{equation}\nis injective, Lipschitz or bi-Lipschitz. \n\nTwo natural choices for the kernel $K$ are the Gaussian kernel and the complex exponential (or, the Fourier) kernel:\n\\[ K_{G}(x,y) = e^{-\\norm{x-y}^2\/\\sigma^2} ~~,\nK_{F}(x,y) = e^{2\\pi i \\ip{x}{y}}\n\\]\nwhere in both cases $Y\\subset\\mathbb{R}^d$. \nIn this paper we analyze a different kernel, namely the polynomial kernel $K_P(x,y)=x_1^{y_1}x_2^{y_2}\\cdots x_d^{y_d}$, $Y\\subset\\{0,1,2,\\ldots,n\\}^d$. \n\n\\subsection{The Polynomial Embedding}\n\nSince the polynomial representation is intimately related to the Hilbert-Noether algebraic invariants theory \\cite{compuinvar} and the Hilbert-Weyl theorem, it is advantageous to start our construction from a different perspective. \n\nThe linear space ${\\mathbb{R}}^{n\\times d}$ is isomorphic to $\\mathbb{R}^{nd}$ by stacking the columns one on top of each other. In this case, the action of the permutation group $S_n$ can be recast as the action of the subgroup $I_d\\otimes S_n$ of the bigger group $S_{nd}$ on $\\mathbb{R}^{nd}$. Specifically, let us denote by $\\sim_G$ the equivalence relation\n\\[ x,y\\in\\mathbb{R}^{nd}~~,~~x\\sim_G y \\Longleftrightarrow y=\\Pi x~,~{\\rm for ~ some}~\\Pi\\in G \\]\ninduced by a subgroup $G$ of $S_{nd}$. In the case\n $G=I_d\\otimes S_n=\\{diag_d(P)~,~P\\in S_n\\}$ of block diagonal permutation obtained by repeating $d$ times the same $P\\in S_n$ permutation along the main diagonal, two vectors $x,y\\in\\mathbb{R}^{nd}$ are $\\sim_G$ equivalent iff there is a permutation matrix $P\\in S_n$ so that $y(1+(k-1)n:kn) = Px(1+(k-1)n:kn)$ for each $1\\leq k\\leq d$. In other words, each disjoint $n$-subvectors in $y$ and $x$ are related by the same permutation. In this framework, the Hilbert-Weyl theorem (Theorem 4.2, Chapter XII, in \\cite{BifTheory2}) states that the ring of invariant polynomials is finitely generated. The G\\\"{o}bel's algorithm (Section 3.10.2 in \\cite{compuinvar}) provides a recipe to find a complete set of invariant polynomials. In the following we provide a direct approach to construct a complete set of polynomial invariants. \n \n Let $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,...,{\\bf x}_d]$ denote the algebra of polynomials in $d$-variables with real coefficients. \nLet us denote $X\\in{\\mathbb{R}}^{n\\times d}$ a generic data matrix.\nEach row of this matrix defines a \nlinear form over ${\\bf x}_1,...{\\bf x}_d$,\n $\\lambda_k = X_{k,1}{\\bf x}_1+\\cdots + X_{k,d}{\\bf x}_d$.\n Let us denote by $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d][{\\bf t}]$ the algebra of polynomials in variable ${\\bf t}$ with coefficients in the ring $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]$. Notice $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d][{\\bf t}]=\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$ \n by rearranging the terms according to degree in ${\\bf t}$. \n Thus $\\lambda_k\\in\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]\\subset\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d][{\\bf t}]$ can be encoded as zeros of a polynomial $P_X$ of degree $n$ in variable ${\\bf t}$ with coefficients in $\\mathbb{R}[{\\bf x}_1,\\ldots,{\\bf x}_d]$:\n \\begin{equation}\n \\label{eq:polyencoding}\n P_X({\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d) = \\prod_{k=1}^n ({\\bf t}-\\lambda_k({\\bf x}_1,\\ldots,{\\bf x}_d))\n =\\prod_{k=1}^n ({\\bf t}-X_{k,1}{\\bf x}_1-\\ldots -X_{k,d}{\\bf x}_d)\n \\end{equation}\n Due to identification $\\mathbb{R}[{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d][{\\bf t}]=\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$,\n we obtain that \\\\\n $P_X\\in \\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2,\\ldots,{\\bf x}_d]$ is a homogeneous polynomial of degree $n$ in $d+1$ variables. Let $\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ denote the vector space of homogeneous polynomials in $d+1$ variables of degree $n$ with real coefficients. Notice the real dimension of this vector space is \n \\begin{equation}\n \\label{eq:dimRn}\n \\dim_\\mathbb{R} \\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d] = \\left( \n \\begin{array}{c}\n n+d \\\\\n d\n \\end{array}\n \\right) = \\left(\n \\begin{array}{c}\n n+d \\\\\n n\n \\end{array}\n \\right).\n \\end{equation}\nBy noting that $P_X$ is monic in ${\\bf t}$ (the coefficient of ${\\bf t}^n$ is always 1) we obtain an injective embedding of ${\\widehat{\\Rnd}}$ into $\\mathbb{R}^m$ with \n$m=\\dim_\\mathbb{R} \\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]-1$ via the coefficients of $P_X$ similar to (\\ref{eq:poly}). This is summarized in the following theorem:\n\\begin{thm}\n\\label{t2}\nThe map $\\alpha_0:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^{m-1}$ with $m=\\left(\\begin{array}{c}n+d \\\\ d \\end{array} \\right)$ given by the (non-trivial) coefficients of polynomial $P_X\\in\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ lifts to an analytic embedding ${\\hat{\\alpha}}_0$ of $({\\widehat{\\Rnd}},d)$ into $\\mathbb{R}^m$. Specifically, for $X\\in{\\mathbb{R}}^{n\\times d}$ expand the polynomial\n\\begin{equation} \\label{eq:PA}\nP_X({\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d) = \\prod_{k=1}^n ({\\bf t}-X_{k,1}{\\bf x}_1-\\ldots -X_{k,d}{\\bf x}_d)\n = {\\bf t}^n + \\hspace{-10mm}\\sum_{\\begin{array}{c}\n \\mbox{$p_0,p_1,...,p_d\\geq 0$} \\\\\n \\mbox{$p_0+\\cdots+p_d=n$} \\\\ \n \\mbox{$p_0 1$}\n \\end{array}\\right.\n\\end{equation}\nbe a Lipschitz monotone decreasing function with Lipschitz constant 1.\n\\begin{cor}\\label{cor2}\nConsider the map:\n\\begin{equation}\\label{eq:alpha1}\n\\alpha_1:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m~~,~~\n\\alpha_1(X) = \\left( \\begin{array}{c}\n\\mbox{$\\alpha_0\\bigg (\\varphi_0(\\norm{X})X \\bigg )$} \\\\\n\\mbox{$\\norm{X}$}\n\\end{array}\\right),\n\\end{equation}\nwith $m=\\left(\\begin{array}{c}n+d \\\\ d \\end{array} \\right)$.\nThe map $\\alpha_1$ lifts to an injective and globally Lipschitz map ${\\hat{\\alpha}}_1:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^m$ with Lipschitz constant $Lip({\\hat{\\alpha}}_1) \\leq \\sqrt{1+L_0^2}$.\n\\end{cor}\n{\\bf Proof}\n\nClearly $\\alpha_1(\\Pi X)=\\alpha_1(X)$ for any $\\Pi\\in{\\mathcal S}_n$ and $X\\in{\\mathbb{R}}^{n\\times d}$. Assume now that $\\alpha_1(X)=\\alpha_1(Y)$. Then $\\norm{X}=\\norm{Y}$ and since ${\\hat{\\alpha}}_0$ is injective on ${\\widehat{\\Rnd}}$ it follows $\\varphi(\\norm{X})X = \\Pi \\varphi(\\norm{Y})Y$ for some $\\Pi\\in{\\mathcal S}_n$. Thus $X\\sim Y$ which proves $\\alpha_1$ lifts to an injective map on ${\\widehat{\\Rnd}}$. \n\nNow we show ${\\hat{\\alpha}}_1$ is Lipschitz on $({\\widehat{\\Rnd}},d)$ of appropriate Lipschitz constant. Let $X,Y'\\in{\\mathbb{R}}^{n\\times d}$ and $\\Pi_0\\in{\\mathcal S}_n$ so that $d(\\hat{X},\\hat{Y'})=\\norm{X-\\Pi_0 Y'}$. Let $Y=\\Pi_0 Y'$ so that $d(\\hat{X},\\hat{Y})=\\norm{X-Y}$. \n\nChoose two matrices $X,Y\\in{\\mathbb{R}}^{n\\times d}$. We claim $\\norm{\\alpha_1(X)-\\alpha_1(Y)}\\leq \\sqrt{1+L_0^2}\n\\norm{X-Y}$.\nThis follows from two observations: \n\n(i) The map\n\\[ X \\mapsto \\rho(X):=\\varphi_0(\\norm{X})X \\]\nis the nearest-point map to (or, the metric projection map onto) the convex closed set $B_1({\\mathbb{R}}^{n\\times d})$. This means $\\norm{\\varphi_0(\\norm{X})X - Z}\\leq \\norm{X-Z}$ for any $Z\\in B_1({\\mathbb{R}}^{n\\times d})$. \n\n(ii) The nearest-point map to a convex closed subset of a Hilbert space is Lipschitz with constant 1, i.e. it shrinks distances, see \\cite{phelps56}.\n\nThese two observations yield:\n\\begin{multline*} \n\\norm{\\alpha_1(X)-\\alpha_1(Y)}^2 = \\norm{\\alpha_0(\\rho(Y))\n- \\alpha_0(\\rho(Y) )}^2 + |\\norm{X}-\\norm{Y}|^2 \\\\\n \\leq \nL_0^2 \\norm{\\rho(X)-\\rho(Y) }^2 + \\norm{X-Y}^2 \\leq (1+L_0^2)\\norm{X-Y}^2 .\n\\end{multline*}\nThis concludes the proof of this result. $\\qed$\n\\vspace{5mm}\n\nA simple modification of $\\phi_0$ can produce a $C^\\infty$ map by smoothing it out around $x=1$.\n\nOn the other hand the lower Lipschitz constant of ${\\hat{\\alpha}}_1$ is 0 due to terms of the form $X_{i,j}^k$ with $k\\geq 2$. \nIn \\cite{Cahill19}, the authors built a Lipschitz map by a retraction to the unit sphere instead of unit ball. \nInspired by their construction, a modification of $\\alpha_0$ in their spirit reads:\n\\begin{equation}\n \\label{eq:alpha2}\n\\alpha_2:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^m~~,~~\n\\alpha_2(X)=\\left( \\begin{array}{c}\n\\mbox{$\\norm{X}\\alpha_0\\bigg ( \\frac{X}{\\norm{X}} \\bigg )$} \\\\\n\\mbox{$\\norm{X}$}\n\\end{array}\\right)~,~if~X\\neq 0~~,~and~~\\alpha_2(0)=0.\n\\end{equation}\nIt is easy to see that $\\alpha_2$ satisfies the non-parallel property in \\cite{Cahill19} and is Lipschitz with a slightly better constant than $\\alpha_1$ (the constant is determined by the tangential derivatives of $\\alpha_0$). \nBut, for the same reasons as in \\cite{Cahill19} this map is not bi-Lipschitz. \n\n\\subsection{Dimension reduction in the case $d=2$ and consequences}\n\nIn this subsection we analyze the case $d=2$. \nThe embedding dimension for $\\alpha_0$ is $\\left( \\begin{array}{c} n \\\\ 2 \\end{array}\\right)-1=\\frac{n(n-1)}{2}-1$. \nOn the other hand, consider the following approach. \nEach row of $X$ defines a complex number $z_1=X_{1,1}+i\\,X_{1,2}$, ... , $z_n=X_{n,1}+i\\,X_{n,2}$ that\ncan be encoded by one polynomial of degree $n$ with complex coefficients $Q\\in\\mathbb{C}_n[t]$,\n\\[ Q({\\bf t}) = \\prod_{k=1}^n ({\\bf t}-z_k) = {\\bf t}^n + \\sum_{k=0}^{n-1}\n{\\bf t}^k q_k \\]\nThe coefficients of $Q$ provide a $2n$-dimensional real embedding $\\zeta_0$,\n\\[ \\zeta_0:\\mathbb{R}^{n\\times 2}\\rightarrow\\mathbb{R}^{2n}~~,~~\\zeta_0(X)=(Re(q_{n-1}),Im(q_{n-1}),\\ldots,Re(q_{0}),Im(q_0)) \\]\nwith properties similar to those of $\\alpha_0$. \nOne can similarly modify this embedding to obtain a globally Lipschitz embedding $\\hat{\\zeta}_1$ of $\\hat{R_{n,2}}$ \ninto $\\mathbb{R}^{2n+1}$. \n\nIt is instructive to recast this embedding in the framework of commutative algebras. Indeed, let $\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle$ denote\nthe ideal generated by polynomials ${\\bf x}_1-1$ and ${\\bf x}_2^2+1$\nin the algebra $\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]$. Consider the quotient space\n $\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle$ and the quotient map\n $\\sigma:\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\\mapsto \\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle{\\bf x}_1-1,{\\bf x}_2^2+1\\rangle$.\n In particular, let $S=\\sigma(\\mathbb{R}_n[{\\bf t},{\\bf x}_1,{\\bf x}_2])$ denote the vector space projected through this quotient map.\nThen a basis for $S$ is given by $\\{1,{\\bf t},\\ldots,{\\bf t}^n,{\\bf x}_2,{\\bf x}_2 {\\bf t},\\ldots,{\\bf x}_2 {\\bf t}^{n-1},{\\bf x}_2 {\\bf t}^n\\}$. Thus $\\dim S=2n+2$. \nLet \n$\\mathfrak{S}=\\{P_X~,~X\\in\\mathbb{R}^{n\\times 2} \\}\\subset\\mathbb{R}_2[{\\bf t},{\\bf x}_1,{\\bf x}_2]$ \ndenote the set of polynomials realizable as in (\\ref{eq:PA}).\nThen the fact that $\\hat{\\zeta}_0:\\mathbb{R}^{n\\times 2}\\rightarrow\\mathbb{R}^{2n}$ \nis injective is equivalent to the fact that $\\sigma{\\vert}_{\\mathfrak{S}}:\\mathfrak{S}\\rightarrow S$ is injective.\nOn the other hand note \n\\[\n\\sigma(\\mathfrak{S})\\subset {\\bf t}^n+\nspan_\\mathbb{R}\\{1,{\\bf t},\\ldots,{\\bf t}^{n-1},{\\bf x}_2,{\\bf x}_2 {\\bf t}, \\ldots,{\\bf x}_2 {\\bf t}^{n-1} \\} \\]\nwhere the last linear subspace is of dimension $2n$. \n\nIn the case $d=2$ we obtain the identification\n$\\mathbb{R}[{\\bf t},{\\bf x}_1,{\\bf x}_2]\/\\langle {\\bf x}_1-1,{\\bf x}_2^2+1 \\rangle = \\mathbb{C}[{\\bf t}]$ due to uniqueness of polynomial factorization.\n\nThis observation raises the following {\\em open problem}:\n\nFor $d>2$, is there a non-trivial ideal \n$I=\\langle Q_1,\\ldots,Q_r \\rangle \\subset\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$\nso that the restriction $\\sigma{\\vert}_{\\mathfrak{S}}$\nof the quotient map $\\sigma:\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]\\rightarrow\n\\mathbb{R}[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]\/I$ is injective? Here $\\mathfrak{S}$ denote the set of polynomials in $\\mathbb{R}_n[{\\bf t},{\\bf x}_1,\\ldots,{\\bf x}_d]$ realizable via (\\ref{eq:PA}).\n\\begin{rmk}\nOne may ask the question whether the quaternions can be\nutilized in the case $d=4$. While the quaternions form an associative division algebra, unfortunately polynomials have in general an infinite number of factorization. This prevents an immediate extension of the previous construction to the case $d=4$. \n\\end{rmk}\n\n\\begin{rmk}\nSimilar to the construction in \\cite{Cahill19}, a linear dimension reduction technique may be applicable here (which, in fact, may answer the open problem above) which would reduce the embedding dimension to $m=2nd+1$ (twice the intrinsec dimension plus one for the homogenization variable). \nHowever we did not explore this approach since, even if possible, it would not produce a bi-Lipschitz embedding. \nInstead we analyze the linear dimension reduction technique in the next section in the context of sorting based embeddings. \n\\end{rmk}\n\n\\section{Sorting based Embedding\\label{sec3}}\n\nIn this section we present the extension of the sorting embedding (\\ref{eq:ord}) to the case $d>1$.\n\nThe embedding is performed by a linear-nonlinear transformation that resembles the phase retrieval problem. \nConsider a matrix $A\\in\\mathbb{R}^{d\\times D}$ and the induced nonlinear \ntransformation:\n\n\\begin{equation}\n\\label{eq:qA}\n\\beta_A:{\\mathbb{R}}^{n\\times d}\\rightarrow\\mathbb{R}^{n\\times D}~~,~~\\beta_A(X)=\\downarrow (XA)\n\\end{equation}\nwhere $\\downarrow$ is the monotone decreasing sorting operator acting in each column independently. Specifically, let \n$Y=XA\\in\\mathbb{R}^{n\\times D}$ and note its column vectors\n$Y=[y_1,y_2,\\ldots,y_D]$. Then \n\\[ \\beta_A(X)=\\left[ \\begin{array}{cccc}\n\\mbox{$\\Pi_1 y_1$} & \\mbox{$\\Pi_2 y_1$} & \\cdots & \\mbox{$\\Pi_D y_D$}\n\\end{array} \\right] \\]\nfor some $\\Pi_1,\\Pi_2,\\ldots,\\Pi_D\\in{\\mathcal S}_n$ so that each column is sorted monotonically decreasing:\n\\[ (\\Pi_k y_k)_1\\geq (\\Pi_k y_k)_2\\geq \\cdots\\geq (\\Pi_k y_k)_n. \\]\nNote the obvious invariance $\\beta_A(\\Pi X)=\\beta_A(X)$ for any $\\Pi\\in{\\mathcal S}_n$ and $X\\in{\\mathbb{R}}^{n\\times d}$. Hence $\\beta_A$ \nlifts to a map $\\hat{\\beta_A}$ on ${\\widehat{\\Rnd}}$. \n\\begin{rmk}\nNotice the similarity to the phase retrieval problem, e.g., \\cite{balan16}, where the data is obtained via a linear transformation of the \ninput signal followed by the nonlinear operation of taking the absolute value of the frame coefficients. Here the nonlinear transformation is implemented by sorting the coefficients. \nIn both cases it represents the action of a particular\nsubgroup of the unitary group. \n\\end{rmk}\n\n\nIn this section we analyze necessary and sufficient conditions so that maps of type (\\ref{eq:qA}) are injective, or injective almost everywhere. \nFirst a few definitions.\n\n\\begin{defn}\nA matrix $A\\in\\mathbb{R}^{d\\times D}$ is called a \\emph{universal key} (for ${\\mathbb{R}}^{n\\times d}$) if $\\hat{\\beta_A}$ is injective\non ${\\widehat{\\Rnd}}$.\n\\end{defn}\nIn general we refer to $A$ as a {\\em key} for encoder $\\beta_A$. \n\\begin{defn}\nFix a matrix $X\\in{\\mathbb{R}}^{n\\times d}$. A matrix $A\\in\\mathbb{R}^{d\\times D}$ is said \\emph{admissible} (or an {\\em admissible key}) for $X$ if for any $Y\\in{\\mathbb{R}}^{n\\times d}$ so that $\\beta_A(X)=\\beta_A(Y)$ then $Y=\\Pi X$ for some $\\Pi\\in{\\mathcal S}_n$. \n\\end{defn}\nIn other words, $\\hat{\\beta_A}^{-1}(\\hat{\\beta_A}(\\hat{X}))=\\{\\hat{X}\\}$.\nWe let ${\\mathcal{A}}_{D}(X)$, or simply ${\\mathcal{A}}(X)$, denote the set of admissible keys for $X$. \n\\begin{defn}\nFix $A\\in\\mathbb{R}^{d\\times D}$. A matrix $X\\in{\\mathbb{R}}^{n\\times d}$ is said to be {\\em separated} by $A$ if $A\\in{\\mathcal{A}}(X)$.\n\\end{defn}\nFor a key $A$, we let $\\mathfrak{S}_{n}(A)$, or simply $\\mathfrak{S}(A)$, denote the set of {\\em matrices separated by $A$}. Thus a matrix $X\\in\\mathfrak{S}_n(A)$ if and only if, for any matrix $Y\\in\\mathbb{R}^{n\\times d}$, if $\\beta_A(X)=\\beta_A(Y)$ then $X\\sim Y$.\n\nThus a key $A$ is universal if and only if $\\mathfrak{S}_n(A)={\\mathbb{R}}^{n\\times d}$.\n\nOur goal is to produce keys that are admissible for all matrices in ${\\mathbb{R}}^{n\\times d}$, or at least for almost every data matrix.\nAs we show in Proposition \\ref{prop3.6} below this requires that $D\\geq d$ and $A$ is full rank. In particular this means that the columns of $A$ form a frame for $\\mathbb{R}^d$. \n\n\\subsection{Characterizations of ${\\mathcal{A}}(X)$ and $\\mathfrak{S}(A)$}\n\nWe start off with simple linear manipulations of sets of admissible keys and separated data matrices.\n\n\\begin{prop}\\label{prop3.5}\nFix $A\\in\\mathbb{R}^{d\\times D}$ and $X\\in{\\mathbb{R}}^{n\\times d}$.\n\\begin{enumerate}\n \\item For an invertible $d\\times d$ matrix $T\\in\\mathbb{R}^{d\\times d}$,\n \\begin{equation}\\label{eq:TA}\n \\mathfrak{S}_n(TA) = \\mathfrak{S}_n(A)T^{-1}.\n \\end{equation}\n In other words, if $X$ is separated by $A$ then $XT^{-1}$ is separated by $TA$.\n \n \\item For any permutation matrix $L\\in\\SS_D$ and diagonal invertible matrix $\\Lambda\\in\\mathbb{R}^{D\\times D}$,\n \\begin{equation}\\label{eq:AL}\n \\mathfrak{S}_n(AL\\Lambda)=\\mathfrak{S}_n(A\\Lambda L) = \\mathfrak{S}_n(A).\n \\end{equation}\n In other words, if $X$ is separated by $A$ then $X$ is separated also by $AL\\Lambda$ as well as by $A\\Lambda L$.\n \n \\item Assume $T\\in\\mathbb{R}^{d\\times d}$ is a $d\\times d$ invertible matrix. Then\n \\begin{equation}\\label{eq:XT}\n {\\mathcal{A}}_D(XT)=T^{-1}{\\mathcal{A}}_D(X).\n \\end{equation}\n In other words, if $A$ is an admissible key for $X$ then $T^{-1}A$ is an admissible key for $XT$.\n\\end{enumerate}\n\\end{prop}\n{\\bf Proof}\n\nThe proof is immediate, but we include it here for convenience of the reader. \n\n(1) Denote $B=TA$. Let $Y\\in\\mathbb{R}^{n\\times d}$. Then\n\\[ \\beta_B(Y)=\\beta_B(X) \\Longleftrightarrow \\downarrow(XB)=\\downarrow(YB) \\Longleftrightarrow \\downarrow(XTA)=\\downarrow(YTA)\n\\Longleftrightarrow\\beta_A(XT)=\\beta_A(YT). \\]\nThus, if $X\\in\\mathfrak{S}_n(A)$ and $Y'\\in{\\mathbb{R}}^{n\\times d}$ so that $\\beta_B(Y')=\\beta_B(X')$ with $X'=XT^{-1}$, then $\\beta_A(Y'T)=\\beta_A(X)$. Therefore there exists $\\Pi\\in{\\mathcal S}_n$ so that $Y'T=\\Pi X$. Thus $Y'\\sim X'$. Hence $X'\\in\\mathfrak{S}_n(B)$.\nThis shows $\\mathfrak{S}_n(A)T^{-1}\\subset \\mathfrak{S}_n(TA)$. The reverse include follows by replacing $A$ with $TA$ and $T$ with $T^{-1}$. Together they prove (\\ref{eq:TA}).\n\n(2) Let $Y\\in{\\mathbb{R}}^{n\\times d}$ such that $\\beta_{AL\\Lambda}(X)=\\beta_{AL\\Lambda}(Y)$. \nFor every $1\\leq j\\leq D$ let $k\\in [D]$ be so that $L_{jk}=1$. \n\nIf $\\Lambda_{kk}>0$ then $\\downarrow((XA)_j)=\\downarrow((YA)_j)$. \n\nIf $\\Lambda_{kk}<0$ then $\\downarrow(-(XA)_j)=\\downarrow(-(YA)_j)$.\nBut this implies also $\\downarrow((XA)_j)=\\downarrow((YA)_j)$ since\n$\\downarrow(-z)=L_0\\downarrow(z)$ where $L_0$ is the permutation matrix that has 1 on its main antidiagonal.\n\nEither way, $\\downarrow((XA)_j)=\\downarrow((YA)_j)$. Hence \n$\\downarrow(XA)=\\downarrow(YA)$. Therefore $X\\sim Y$ and thus $X\\in \\mathfrak{S}_n(AL\\Lambda)$. This shows $\\mathfrak{S}_n(A)\\subset\\mathfrak{S}_n(AL\\Lambda)$.\nthe reverse inclusion follows by a similar argument.\nFinally, notice $\\{L\\Lambda\\}$ forms a group since $L^{-1}\\Lambda L$ is also a diagonal matrix. This shows $\\mathfrak{S}_n(A\\Lambda L)=\\mathfrak{S}(AL\\Lambda')$ for some diagonal matrix $\\Lambda'$, and the conclusion (\\ref{eq:AL}) then follows.\n\n(3) The relation (\\ref{eq:XT}) follows from noticing $\\beta_{T^{-1}A}(Y)=\\beta_A(YT)$. $\\qed$\n\nRelation (\\ref{eq:AL}) shows that, since $A$ is assumed full rank, without loss of generality we can assume the first $d$ columns are linearly independent. Let $V$ denote the first $d$ columns of $A$ so that\n\\begin{equation}\\label{eq:AA}\nA = V\\left[ \\begin{array} {ccc}\n\\mbox{$I$} & \\mbox{$\\vert$} & \\mbox{$\\tilde{A}$} \n\\end{array} \\right] \n\\end{equation}\nwhere $\\tilde{A}\\in\\mathbb{R}^{d\\times (D-d)}$. \nThe following result shows that, unsurprisingly, when $D=d>1$, almost every matrix $X$ is not separated by $A$. By Proposition \\ref{prop3.5} we can reduce the analysis to the case $A=I$ by a change of coordinates.\n\\begin{prop}\\label{prop3.6}\nAssume $D=d>1$, $n>1$. Then\n\\begin{enumerate}\n \\item The set of data matrices not separated by $I_d$ includes:\n\\begin{equation}\\label{eq:SA1}\n \\mathbb{B}:=\\{ X\\in\\mathbb{R}^{n\\times d}~,~\\exists i,j,k,l~,~ 1\\leq i0$ and $b_0>0$ so that\n for all $X,Y\\in{\\mathbb{R}}^{n\\times d}$,\n \\begin{equation}\n \\label{eq:Lipbeta2}\n a_0\\, d(\\hat{X},\\hat{Y})\\leq \\norm{\\beta_A(X)-\\beta_A(Y)}\\leq b_0\\,\n d(\\hat{X},\\hat{Y})\n \\end{equation}\n where all are Frobenius norms.\n Furthermore, an estimate for $b_0$ is provided by the largest singular value of $A$, $b_0= s_1(A)$.\n\\end{thm}\n\n{\\bf Proof}\n\nThe upper bound in (\\ref{eq:Lipbeta2}) follows as in the proof of Theorem \\ref{t4}, from equations (\\ref{eq:betaA}) and (\\ref{eq:betaAA}). Notice that\nno property is assumed in order to obtain the upper Lipschitz bound.\n\nThe lower bound in (\\ref{eq:Lipbeta2}) is more difficult. \nIt is shown by contradiction following the strategy \nutilized in the \nComplex Phase Retrieval problem \\cite{balazou}.\n\nAssume $\\inf_{X\\not\\sim Y}\\frac{\\norm{\\beta_A(X)-\\beta_A(Y)}_2^2}{d(\\hat{X},\\hat{Y})^2}=0$. \n\n{\\em Step 1: Reduction to local analysis.} \nSince $d(\\hat{tX},\\hat{tY})=t\\,d(\\hat{X},\\hat{Y})$ for all $t>0$, the \nquotient $\\frac{\\norm{\\beta_A(X)-\\beta_A(Y)}_2}{d(\\hat{X},\\hat{Y})}$ \nis scale invariant. Therefore, there are sequences $(X^t)_t,(Y^t)_t$\nwith $\\norm{Y^t}\\leq\\norm{X^t}=1$ and $d(\\hat{X^t},\\hat{Y^t})>0$ so that\n$\\lim_{t\\rightarrow\\infty} \\frac{\\norm{\\beta_A(X^t)-\\beta_A(Y^t)}_2}{d(\\hat{X^t},\\hat{Y^t})} = 0$.\nBy compactness of the closed unit ball, one can extract convergence subsequences. For easiness of notation, assume $(X^t)_t,(Y^t)_t$ are\nthese subsequences. Let ${X^{\\infty}}=\\lim_t X^t$ and ${Y^{\\infty}} = \\lim_t Y^t$ denote their limits. Notice $\\lim_t \\norm{\\beta_A(X^t)-\\beta_A(Y^t)}_2=0$.\nThis implies $\\norm{\\beta_A({X^{\\infty}})-\\beta_A({Y^{\\infty}})}=0$ and thus $\\beta_A({X^{\\infty}})=\\beta_A({Y^{\\infty}})$. Since $\\widehat{\\beta_A}$ is assumed injective, it follows that $\\widehat{{X^{\\infty}}}=\\widehat{{Y^{\\infty}}}$. \n\nThis means that, if the lower Lipschitz bound vanishes, then this \nis achieved by vanishing of a local lower Lipschitz bound. To follow the terminology in \\cite{balazou}, the type I local lower Lipschitz bound vanishes at some \n$Z_0\\in{\\mathbb{R}}^{n\\times d}$, with $\\norm{Z_0}=1$:\n\\begin{equation}\n \\label{eq:lb}\n{A}(Z_0):= \\lim_{r\\rightarrow 0} \\inf_{\n\\begin{array}{c}\n\\hat{X}\\neq\\hat{Y} \\\\\nd(\\hat{X},\\hat{Z_0})0$ by the definition of $G$.\n\nConsider $X=Z_0+U$ and $Y=Z_0+V$ where $U,V\\in{\\mathbb{R}}^{n\\times d}$ are ``aligned\" in the sense that $d(\\hat{X},\\hat{Y})=\\norm{U-V}$. This property requires that $\\norm{U-V}\\leq\\norm{PX-Y}$, for every $P\\in{\\mathcal S}_n$. \nNext result replaces equivalently this condition \nby requirements involving $(U,V)$ and the group $G$ only.\n\\begin{lem}\n\\label{l3.1}\nAssume $\\norm{U},\\norm{V}<\\frac{1}{4}\\delta_0$, where $\\delta_0=\\min_{P\\in{\\mathcal S}_n\\setminus G} \\norm{(I_n-P)Z_0}$. Let $X=Z_0+U$, $Y=Z_0+V$. \nThen:\n\\begin{enumerate}\n \\item $d(\\hat{X},\\hat{Z_0})=\\norm{U}$ and $d(\\hat{Y},\\hat{Z_0})=\\norm{V}$.\n \\item $d(\\hat{X},\\hat{Y})=\\min_{P\\in G}\\norm{U-PV}=\\min_{P\\in G}\\norm{PU-V}$\n \\item The following\nare equivalent:\n\\begin{enumerate}\n \\item $d(\\hat{X},\\hat{Y})=\\norm{U-V}$.\n \\item For every $P\\in G$, $\\norm{U-V}\\leq \\norm{PU-V}$.\n \\item For every $P\\in G$, $\\ip{U}{V}\\geq \\ip{PU}{V}$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{lem}\n{\\bf Proof of Lemma \\ref{l3.1}}.\n(1)\n\nNote that is $U=0$ then the claim follows. Assume $U\\neq 0$. Then\n\\[ d(\\hat{X},\\hat{Z_0})=\\min_{P\\in{\\mathcal S}_n} \\norm{X-PZ_0}\n= \\min_{P\\in {\\mathcal S}_n}\\norm{(I_n-P)Z_0 + U}\\leq \\norm{U} \\]\nOn the other hand, assume the minimum is achieved for a permutation $P_0\\in{\\mathcal S}_n$. If $P_0\\in G$ then\n$d(\\hat{X},\\hat{Z_0})=\\norm{(I_n-P_0)Z_0+U}=\\norm{U}$. If $P_0\\not\\in G$ then \n\\[ d(\\hat{X},\\hat{Z_0})\\geq \\norm{(I_n-P_0)Z_0}-\\norm{U}>\\frac{3\\delta_0}{4}>\\norm{U}\\geq d(\\hat{X},\\hat{Z_0}) \\]\nwhich yields a contradiction.\nHence $d(\\hat{X},\\hat{Z_0})=\\norm{U}$. Similarly, one shows $d(\\hat{X},\\hat{Z_0})=\\norm{V}$. \n\n(2) Clearly\n\\[ d(\\hat{X},\\hat{Y})=\\min_{P\\in{\\mathcal S}_n}\\norm{PX-Y}\\leq \\min_{P\\in G}\\norm{PX-Y}=\\min_{P\\in G}\\norm{PU-V} \\]\nOn the other hand, for $P\\in{\\mathcal S}_n\\setminus G$ and $Q\\in G$,\n\\[ \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\\geq \\norm{(I_n-P)Z_0} - \\norm{U}-\\norm{V}\\geq \\]\n\\[ \\geq \\delta_0\n-2\\norm{U}-2\\norm{V}+\\norm{QU-V}\\geq \\min_{Q\\in G}\\norm{QU-V}\\geq d(\\hat{X},\\hat{Y}). \\]\n\n(3)\n\n(a)$\\Rightarrow$(b).\n\nIf $d(\\hat{X},\\hat{Y})=\\norm{U-V}$ then\n\\[ \\norm{U-V}\\leq \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\n~~,~~\\forall P\\in {\\mathcal S}_n. \\]\nIn particular, for $P\\in G$, $(P-I_n)Z_0=0$ and\nthe above inequality reduces to (b).\n\n(b)$\\Rightarrow$(a).\n\nAssume (b). For $P\\in G$,\n\\[ \\norm{U-V}=\\norm{X-Y}\\leq\\norm{PU-V}=\\norm{PX-Y}. \\]\nFor $P\\in{\\mathcal S}_n\\setminus G$,\n\\[ \\norm{PX-Y}=\\norm{(P-I_n)Z_0 + PU-V}\\geq \\norm{(I_n-P)Z_0} - \\norm{U}-\\norm{V}\\geq \\]\n\\[ \\geq \\delta_0\n-2\\norm{U}-2\\norm{V}+\\norm{U-V}\\geq \\norm{U-V}=\\norm{X-Y}. \\]\nThis shows $d(\\hat{X},\\hat{Y})=\\norm{X-Y}=\\norm{U-V}$.\n\n(b)$\\Longleftrightarrow$(c). This is immediate after squaring (b) and simplifying the terms.\n\n$\\Box$\n\\vspace{3mm}\n\nConsider now sequences $(\\hat{X^t})_t,(\\hat{Y^t})_t$ that converge to $\\hat{Z_0}$ \nand achieve lower bound 0 as in (\\ref{eq:lb}).\nChoose\nrepresentatives $X_t$ and $Y_t$ in their equivalence classes that satisfy the hypothesis of Lemma \\ref{l3.1} so that $X_t=Z_0+U_t$, $Y_t=Z_0+V_t$, $\\norm{U_t},\\norm{V_y}<\\frac{1}{4}\\delta_0$,\n $d(\\hat{X_t},\\hat{Z_0})=\\norm{U_t}$, $d(\\hat{Y_t},\\hat{Z_0})=\\norm{V_t}$ and $d(\\hat{X_t},\\hat{Y_t})=\\norm{U_t-V_t}>0$.\n With $A=[a_1 |\\cdots|a_D]$ we obtain:\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 = \\sum_{j=1}^D \\norm{\\downarrow(X_t a_j)-\\downarrow(Y_t a_j)}_2^2 =\n\\sum_{j=1}^D \\norm{(Z_0+U_t)a_j-\\Pi_{j,t}(Z_0+V_t)a_j}_2^2 \\]\nfor some $\\Pi_{j,t}\\in{\\mathcal S}_n$. In fact $\\Pi_{j,t}\\in argmin_{\\Pi\\in H_j}\\norm{U_t-\\Pi V_t)a_j}_2$. \nPass to sub-sequences (that will be indexed by $t$ for an easier notation) so that $\\Pi_{j,t}=\\Pi_j$ for some $\\Pi_j\\in{\\mathcal S}_n$. Thus\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 =\n\\sum_{j=1}^D \\norm{(I_n-\\Pi_j)Z_0a_j + (U_t-\\Pi_j V_t)a_j}_2^2 \\]\nSince the above sequence must converge to $0$ as $t\\rightarrow\\infty$, while $U_t,V_t\\rightarrow 0$, it follows that necessarily $\\Pi_j\\in H_j$ and the\nexpressions simplify to\n\\[ \\norm{\\beta_A(X_t)-\\beta_A(Y_t)}_2^2 =\n\\sum_{j=1}^D \\norm{(U_t-\\Pi_j V_t)a_j}_2^2 \\]\nThus equation (\\ref{eq:lb}) implies that for\nevery $j\\in[D]$,\n\\begin{equation}\n\\label{eq:lb2}\n\\lim_{t\\rightarrow\\infty} \\frac{\\norm{(U_t-\\Pi_j V_t)a_j}_2^2}{\\norm{U_t-V_t}^2} = 0\n\\end{equation} \nwhere $\\Pi_j\\in H_j$, $\\norm{U_t},\\norm{V_t}\\rightarrow 0$, and $U_t,V_t$ are aligned so that $\\ip{U_t}{V_t}\\geq \\ip{PU_t}{V_t}$ for every $P\\in G$.\nEquivalently, relation (\\ref{eq:lb}) can be restated as:\n\\begin{equation}\n \\label{eq:opt2}\n \\inf_{\\begin{array}{c} U,V\\in{\\mathbb{R}}^{n\\times d} \\\\ s.t. \\\\\n U\\neq V \\\\\n \\ip{U}{V}\\geq \\ip{PU}{V} , \\forall P\\in G\n \\end{array} } \\frac{\\sum_{j=1}^D \\norm{(U-\\Pi_j V)a_j}_2^2}{\\norm{U-V}^2} = 0\n\\end{equation}\nfor some permutations $\\Pi_j\\in H_j$, $j\\in[D]$.\nBy Lemma \\ref{l3.1} the constraint in the optimization problem above implies $\\norm{U-V}=\\min_{P\\in G}\\norm{U-PV}$. Hence (\\ref{eq:opt2}) implies:\n\\begin{equation}\n \\label{eq:opt3}\n \\inf_{\\begin{array}{c} U,V\\in{\\mathbb{R}}^{n\\times d} \\\\ s.t. \\\\\n U\\neq P V , \\forall P\\in G\n \\end{array} }\n \\max_{P\\in G} \\frac{\\sum_{j=1}^D \\norm{(U-\\Pi_j V)a_j}_2^2}{\\norm{U-P V}^2} = 0\n\\end{equation}\nfor same permutation matrices $\\Pi_j$'s.\nWhile the above optimization problem seems a relaxation of (\\ref{eq:opt2}), in fact (\\ref{eq:opt3}) implies (\\ref{eq:opt2})\nwith a possibly change of permutation matrices $\\Pi_j$, but \nremaining still in $H_j$.\n\\vspace{5mm}\n\n\n{\\em Step 3. Existence of a Minimizer.} \n \n\n\nThe optimization problem (\\ref{eq:opt2}) is a Quadratically Constrained Ratio of Quadratics (QCRQ) optimization problem. A significant number of papers \nhave been published on this topic \\cite{teb06,teb10}. \nIn particular, \\cite{QCRQbook} presents\na formal setup for analysis of QCRQ problems. \nOur interest is to utilize some of these techniques in order to establish the existence of a minimizer for (\\ref{eq:opt2}) or (\\ref{eq:opt3}). Specifically we show:\n\\begin{lem}\\label{l3.2}\nAssume the key $A$ has linearly independent rows (equivalently, the columns of $A$ form a frame for $\\mathbb{R}^d$) and the lower Lipschitz bound of ${\\hat{\\beta}}_A$ is $0$. Then there are $\\tilde{U},\\tilde{V}\\in{\\mathbb{R}}^{n\\times d}$ so that:\n\\begin{enumerate}\n \\item $\\tilde{U}\\neq P \\tilde{V}$, for every $P\\in G$;\n \\item For every $j\\in[D]$, $(\\tilde{U}-\\Pi_j \\tilde{V})a_j=0$.\n\\end{enumerate}\n\\end{lem}\n{\\bf Proof of Lemma \\ref{l3.2}}\n\n\n\nWe start with the formulation (\\ref{eq:opt3}). Therefore there are sequences \n$(U_t,V_t)_{t\\geq 1}$ so that $U_t\\neq PV_t$ for any $P\\in G, t\\geq 1$, and yet for any $P\\in G$,\n\\[ \\lim_{t\\rightarrow\\infty} \\frac{\\sum_{j=1}^D\\norm{(U_t-\\Pi_j V_t)a_j}_2^2}{\\norm{U_t-P V_t}^2} = 0. \\]\nLet $E=\\{(U,V)\\in{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}~,~(U-\\Pi_j)V)a_j=0~,~\\forall j\\in[D]\\}$ denote the null space of the linear operator \n\\[ T:{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}\\rightarrow \\mathbb{R}^D~,~(U,V)\\mapsto \\left[\\begin{array}{ccccc}\n(U-\\Pi_1 V)a_1 & \\vert & \\cdots & \\vert & (U-\\Pi_D V)a_D\n\\end{array}\\right],\n\\]\nassociated to the numerator of the above quotient. Let $F_P=\\{(U,V)\\in{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}~,~U-PV=0\\}$ be the null space of the linear operator \n\\[ R_P:{\\mathbb{R}}^{n\\times d}\\times{\\mathbb{R}}^{n\\times d}\\rightarrow {\\mathbb{R}}^{n\\times d}~,~(U,V)\\mapsto U-P V. \\]\nA consequence of (\\ref{eq:opt3}) is that for every $P\\in G$, $E\\setminus F_P\\neq\\emptyset$. \nIn particular, $F_p\\cap E$ is a subspace of $E$ of positive codimension. Using the \nBaire category theorem (or more elementary linear algebra arguments), we conclude that\n\\[ E\\setminus \\left(\\cup_{P\\in G} F_P\\right) \\neq \\emptyset. \\]\nLet $(\\tilde{U},\\tilde{V})\\in E\\setminus\\left(\\cup_{P\\in G}F_P\\right)$. This pair satisfies the\nconclusions of Lemma \\ref{l3.2}.\n\n\n\\ignore{\nFirst we rewrite (\\ref{eq:opt2}) in terms of new matrices. Let $S_1=S_1({\\mathbb{R}}^{n\\times d})$ denote the unit sphere of ${\\mathbb{R}}^{n\\times d}$. Let $W=U-V$. Since $W\\neq 0$, let $W_0=\\frac{1}{\\norm{W}}W$ with $\\norm{W_0}=1$.\nLet also $V_0\\in S_1$ be so that $V=t\\norm{W}V_0$\n for some $t\\geq 0$. Then the objective function in (\\ref{eq:opt2}), i.e., the quotient of the two quadratics, simplifies to\n \\[ \\sum_{j=1}^D \\norm{(W_0 + t(I-\\Pi_j)V_0)a_j}_2^2. \\]\n Let $\\Gamma_t$ define the constraints set:\n \\[ \\Gamma_t =\\cap_{P\\in G} \n \\{ (W_0,V_0)\\in S_1\\times S_1~:~\n t^2 \\ip{V_0}{(I-P)V_0}+t\\ip{W_0}{(I-P)V_0}\\geq 0 \\}.\n \\]\nNotice that for each $t\\geq 0$, $\\Gamma_t$ is a closed and hence a compact subset of $S_1\\times S_1$. It may be an empty set for some values of $t$. The problem (\\ref{eq:opt2}) is equivalent to:\n\\[ \\inf_{t\\geq 0} \\inf_{(W_0,V_0)\\in\\Gamma_t}\n\\sum_{j=1}^D \\norm{W_0 a_j + t(I-\\Pi_j)V_0a_j}_2^2 = 0 \\]\nLet $(W(t_k),V(t_k),t_k)\\in S_1\\times S_1\\times [0,\\infty)$, $k\\geq 1$, be a sequence that achieves the lower bound $0$. Extract a subsequence indexed again by $k$ so that\n $\\lim_{k\\rightarrow\\infty} W(t_k)=W_\\infty\\in S_1$ and $\\lim_{k\\rightarrow\\infty}V(t_k)=V_\\infty$. Thus, for all $j\\in [D]$, $\\lim_{k\\rightarrow\\infty} \\norm{W_\\infty a_j + t_k(I-\\Pi_j)V(t_k) a_j}_2 = 0$, which implies\n \\[ \\lim_{k\\rightarrow\\infty} t_k(I-\\Pi_j)V(t_k) a_j = -W\\infty a_j. \\]\n \n Case 1. $\\liminf_{k\\rightarrow\\infty} t_k<\\infty$. In this case extract a subsequence, say\n $(t_{k_l})_l$, so that\n $\\lim_{l\\rightarrow\\infty} t_{k_l}=t_\\infty\\in[0,\\infty)$. \nThis implies\n \\[ W_\\infty a_j +t_\\infty(I-\\Pi_j)V_\\infty a_j = 0~,~\\forall j\\in[D]. \\]\nNotice $(W_\\infty,V_\\infty)\\in \\Gamma_{t_\\infty}$.\nTherefore $\\tilde{U}=W_\\infty + t_\\infty V_\\infty$ and $\\tilde{V}=t_\\infty V_\\infty$ satisfy the conclusions (1),(2), and (3) and lemma \\ref{l3.2} is proved.\n\n\nCase 2. $\\liminf_{k\\rightarrow\\infty} t_k=\\infty$.\n\nIn the rest of the proof of this lemma, we construct an inductive process which ends with a scenario that either satisfies Case 1, or produces a (geo)metric contradiction. \n\nTo simplify notation we shall reuse the index $k$ at each stage.\n\n{\\em Initialization:} Set $p=1$. \nLet $V^{(1)}_{\\infty}=V_\\infty$, $t^{(1)}_k=t_k$, and $R^{(1)}_k=V(t_k)-V_\\infty$.\n\n{\\em Preamble:} Sequences $(t^{(p)}_k,R^{(p)}_k)$ satisfy, for every $j\\in[D]$:\n\\begin{equation}\n\\label{eq:sequences}\n\\lim_{k\\rightarrow\\infty}t^{(p)}_k = +\\infty, \\lim_{k\\rightarrow\\infty}R^{(p)}_k = 0, \\norm{R^{(p)}_k+V^{(p)}_\\infty}=1=\\norm{V^{(p)}_\\infty} , \\lim_{k\\rightarrow\\infty}t^{(p)}_k(I-\\Pi_j)R^{(p)}_k a_j = -W_\\infty a_j \n\\end{equation}\n\n{\\em Refinement:} Extract a subsequence \nindexed again by $k$ that\nsatisfies additionally:\n\\begin{equation}\n\\label{eq:seq2}\n\\norm{R^{(p)}_k}\\leq \\frac{1}{p}~,~ \\lim_{k\\rightarrow\\infty} \\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}}\\in S_1\n\\end{equation}\n\n{\\em Setting up the next iteration:} Set\n\\[ t^{(p+1)}_k=t^{(p)}_k\\norm{R^{(p)}_k} ~,~V^{(p+1)}_\\infty = \\lim_{k\\rightarrow\\infty} \\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}} ~,~ \nR^{(p+1)}_k=\\frac{R^{(p)}_k}{\\norm{R^{(p)}_k}} - V^{(p+1)}_\\infty \\]\n\n{\\em Testing:} If $\\liminf_{k\\rightarrow\\infty}t^{(p+1)}_k<\\infty$ then proceed with Case 1 above, which ends the proof of this lemma.\n\nOtherwise $\\lim_{k\\rightarrow\\infty}t^{(p+1)}_k=\\infty$. Thus $(I-\\Pi_j)V^{(p+1)}_\\infty a_j=0$ for all $j\\in[D]$. \nSet $p\\leftarrow p+1$. The {\\em preamble} conditions (\\ref{eq:sequences}) are again satisfied \nfor all $j\\in[D]$. Then proceed by going to the {\\em refinement} step and iterate. \n\nIf the iterative process described above does not end at some finite $p$, then we construct sequences doubly indexed $(t^{(p)}_k,R^{(p)}_k)_{p,k}$ that satisfy (\\ref{eq:sequences}) and (\\ref{eq:seq2}). \n}\n\n $\\Box$\n\n\n\n\\vspace{5mm}\n\n{\\em Step 4. Contradiction with the universality property of the key.}\n\nSo far we obtained that if the lower Lipschitz bound of ${\\hat{\\beta}}_A$ vanishes than there are $Z_0,\\tilde{U},\\tilde{V}\\in{\\mathbb{R}}^{n\\times d}$ with $Z_0\\neq 0$ and $\\tilde{U}\\neq P \\tilde{V}$, for all $P\\in G$ that satisfy the conclusions of Lemma \\ref{l3.2}. Notice $\\ip{Z_0}{Z_0}=\\ip{PZ_0}{Z_0}$\n for all $P\\in G$ and $(Z_0-\\Pi_j Z_0)a_j=0$ for all $j\\in[D]$. Choose $s>0$ but small enough so that $s\\norm{\\tilde{U}},s\\norm{\\tilde{V}}<\\frac{1}{4}\\delta_0$ with $\\delta_0=\\min_{P\\in{\\mathcal S}_n\\setminus G} \\norm{(I_n-P)Z_0}$.\n Let $X=Z_0+s \\tilde{U}$ and $Y=Z_0+s\\tilde{V}$.\n Then Lemma \\ref{l3.1} implies $d(\\hat{X},\\hat{Y})=\\min_{P\\in G}\\norm{\\tilde{U}-P\\tilde{V}}>0$. \n Hence $\\hat{X}\\neq\\hat{Y}$. On the other hand,\n for every $j\\in[D]$, $Xa_j = \\Pi_j Ya_j$. Thus\n ${\\hat{\\beta}}_A(\\hat{X})={\\hat{\\beta}}_A(\\hat{Y})$. \n Contradiction with the assumption that ${\\hat{\\beta}}_A$ is injective.\n \n This ends the proof of Theorem \\ref{t5}.\n\n$\\Box$\n\\ignore{\n\\begin{rem}\nThe proof of the previous theorem provides estimates for \nboth type I and type II local lower and upper Lipschitz bounds.\n\\end{rem}\n}\n\n\\subsection{Dimension Reduction}\n\nTheorem \\ref{t4} provides an Euclidean bi-Lipschitz embedding of very high dimension, $D=1+(d-1)n!$. On the other hand, Theorem \\ref{t5} shows that any universal key $A\\in\\mathbb{R}^{d\\times D}$ for ${\\widehat{\\Rnd}}$, \nand hence any injective map $\\hat{\\beta}_A$ is bi-Lipschitz. In this subsection we show that \nany bi-Lipschitz Euclidean embedding $\\hat{\\beta}_A:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{n\\times D}$ with $D>2d$ \ncan be further compressed to a smaller dimension space $\\mathbb{R}^m$ with $m=2nd$ thus yielding\nbi-Lipschitz Euclidean embeddings of redundancy 2. This is shown in the next result.\n\n\\begin{thm}\n \\label{t6} Assume $A\\in\\mathbb{R}^{d\\times D}$ is a universal key for ${\\widehat{\\Rnd}}$ with $D\\geq 2d$. \n Then, for $m\\geq 2nd$, a generic linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{m}$ with respect to Zariski topology on\n $\\mathbb{R}^{n\\times D\\times m}$, the map\n \\begin{equation}\n \\label{eq:AB1}\n \\hat{\\beta}_{A,B}:{\\widehat{\\Rnd}}\\rightarrow\\mathbb{R}^{2nd}~,~ \\hat{\\beta}_{A,B}(\\hat{X})=B\\left(\\hat{\\beta}_A(\\hat{X})\\right)\n \\end{equation}\n is bi-Lipschitz. In particular, almost every full-rank linear operator $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$ produces such a \n bi-Lipschitz map.\n\\end{thm}\n\n\\begin{rmk}\nThe proof shows that, in fact, the complement set of linear operators $B$ that produce bi-Lipschitz embeddings is included \nin the zero-set of a polynomial. \n\\end{rmk}\n\n\\begin{rmk}\nPutting together Theorems \\ref{t4}, \\ref{t5}, \\ref{t6} we obtain that the metric space ${\\widehat{\\Rnd}}$ admits\na global bi-Lipschitz embedding in the Euclidean space $\\mathbb{R}^{2nd}$. This result is compatible\nwith a Whitney embedding theorem (see \\S 1.3 in \\cite{hirsh}) with the important caveat that the Whitney embedding result\napplies to smooth manifolds, whereas here ${\\widehat{\\Rnd}}$ is merely a non-smooth algebraic variety.\n\\end{rmk}\n\n\\begin{rmk}\nThese three theorems are summarized in part two of the \nTheorem \\ref{t2} presented in \nthe first section.\n\\end{rmk}\n\n\\begin{rmk}\nWhile the embedding dimension grows linearly in $nd$, in fact $m=2nd$, the computational complexity of constructing ${\\hat{\\beta}}_{A,B}$ is NP due to the $1+(d-1)n!$ intermediary dimension.\n\\end{rmk}\n\n\\begin{rmk}\nAs the proofs show, for $D\\geq 1+(d-1)n!$, a generic $(A,B)$ with\nrespect to Zariski topology, $A\\in\\mathbb{R}^{d\\times D}$ and linear map $B:\\mathbb{R}^{n\\times D}\\rightarrow\\mathbb{R}^{2nd}$, produces a bi-Lipschitz embedding $({\\hat{\\beta}}_{A,B},d)$ of ${\\widehat{\\Rnd}}$ into $(\\mathbb{R}^{2nd},\\norm{\\cdot}_2)$. \n\\end{rmk}\n{\\bf Proof of Theorem \\ref{t6} }\n\nThe proof follows a similar approach as in Theorem 3 of \\cite{Cahill19}.\nSee also \\cite{DUFRESNE20091979}.\n\nWithout loss of generality, assume $m0$ so that for every $X,Y\\in\\mathbb{R}^{n\\times d}$, \n$\\norm{B(L_{\\gamma}(X,Y))}\\geq a_\\gamma \\norm{L_{\\gamma}(X,Y)}$.\nLet $a_\\infty = \\min_{\\gamma}a_\\gamma >0$. Thus\n\\[ \\norm{\\beta_{A,B}(X) - \\beta_{A,B}(Y)} =\\norm{B(L_{\\gamma_0}(X,Y))}\n\\geq a_\\infty \\norm{L_{\\gamma_0}(X,Y)}=a_\\infty \\norm{\\beta_A(X)-\\beta_A(Y)} \\]\nwhere $\\gamma_0\\in(S_n)^{2D}$ is a particular $2D$-tuple of permutations. This shows that \n$B{\\vert}_{\\beta_A(\\mathbb{R}^{n\\times d})}:\\beta_A(\\mathbb{R}^{n\\times d})\\rightarrow\\mathbb{R}^m$ is bi-Lipschitz.\nBy Theorem \\ref{t5}, the map ${\\hat{\\beta}}_A$ is bi-Lipschitz. Therefore\nwe get ${\\hat{\\beta}}_{A,B}$ is bi-Lipschitz as well.\n$\\Box$\n\n\n\\subsection{Proof of Corollary \\ref{c0}\\label{subsec4.4}}\n\n(1) It is clear that any continuous $f$ induces a continuous $\\varphi:\\beta(\\mathbb{R}^{n\\times d})\\rightarrow\\mathbb{R}$ via $\\varphi(\\beta(X))=f(X)$. Furthermore, \n$F:=\\beta(\\mathbb{R}^{n\\times d})={\\hat{\\beta}}({\\widehat{\\Rnd}})$\nis a closed subset of $\\mathbb{R}^m$ since ${\\hat{\\beta}}$ is bi-Lipschitz. \nThen a consequence of Tietze extension theorem \n(see problem 8 in \\S 12.1 of \\cite{roydenfitzpatrick})\nimplies that $\\varphi$ admits a continuous extension $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$. Thus $g(\\beta(X))=f(X)$ \nfor all $X\\in\\mathbb{R}^{n\\times d}$. The converse is trivial.\n\n(2) As at part (1), the Lipschitz continuous function $f$ induces a Lipschitz continuous function $\\varphi:F\\rightarrow\\mathbb{R}$. Since $F\\subset\\mathbb{R}^m$ is a subset of a Hilbert space, by Kirszbraun\nextension theorem (see \\cite{WelWil75}), $\\varphi$ \nadmits a Lipschitz continuous extension \n(even with the same Lipschitz constant!)\n $g:\\mathbb{R}^m\\rightarrow\\mathbb{R}$ so that $g(\\beta(X))=f(X)$ for every $X\\in\\mathbb{R}^{n\\times d}$. The converse is trivial. $\\Box$\n\n\n\\section{Applications to Graph Deep Learning\\label{sec4}}\n\nIn this section we take an empirical look at the permutation invariant mappings presented in this paper. We focus on the problems of graph classification, for which we employ the PROTEINS\\_FULL dataset \\cite{DobsonDoing_proteins}, and graph regression, for which we employ the quantum chemistry QM9 dataset\n\\cite{ramakrishnan2014quantum}. In both problems we want to estimate a function $F: (A,Z) \\rightarrow p$, where $(A,Z)$ characterizes a graph where $A \\in \\mathbb{R}^{n\\times n}$ is an adjacency matrix and $Z \\in \\mathbb{R}^{n\\times r}$ is an associated feature matrix where the $i^{th}$\nrow encodes an array of $r$ features associated with the $i^{th}$ node. $p$ is a scalar output where we have $p \\in \\{0,1\\}$ for binary classification and $p \\in \\mathbb{R}_+$ for regression.\n\nWe estimate $F$ using a deep network that is trained in a supervised manor. The network is comprised of three successive components applied in series: $\\Gamma$, $\\phi$, and $\\zeta$. $\\Gamma$ represents a graph deep network \\cite{GCN},\nwhich produces a set of embeddings $X \\in \\mathbb{R}^{N\\times d}$ across the nodes in the graph. Here $N\\geq n$ is chosen to accommodate the graph with the largest number of nodes. In this case, the last $N-n$ rows of $Y$ are filled with 0's. $\\phi: \\mathbb{R}^{N\\times d} \\rightarrow \\mathbb{R}^{m}$ represents a permutation invariant mapping such as those proposed in this paper. $\\zeta: \\mathbb{R}^{m} \\rightarrow \\mathbb{R}$ is a fully connected neural network. The entire end-to-end network is shown in Figure \\ref{fig:gcn_end2end}.\n\nIn this paper, we model $\\Gamma$ using a Graph Convolutional Network (GCN) outlined in \\cite{GCN}. \nLet ${\\bf D} \\in \\mathbb{R}^{n \\times n}$ be the associated degree matrix for our graph $\\mathcal{G}$. Also let $\\tilde{A}$ be the associated adjacency matrix of $\\mathcal{G}$ with added self connection: $\\tilde{A}=I+A$, where $I$ is the $n \\times n$ identity matrix, and $\\tilde{{\\bf D}}={\\bf D}+I$. Finally, we define the modified adjacency matrix $\\hat{A}=\\tilde{{\\bf D}}^{-1\/2} \\tilde{A} \\tilde{{\\bf D}}^{-1\/2}$. A GCN layer is defined as $H^{(l+1)}=\\sigma (\\hat{A}H^{(l-1)}W^{(l)})$.\nHere $H^{(l-1)}$ represents the GCN state coming into the $l^{th}$ layer, $\\sigma$ represents a chosen nonlinear element-by-element operation such as ReLU, and $W^{(l)}$ represents a matrix of trainable weights assigned to the $l^{th}$ layer whose number of rows match the number of columns in $H^{l}$ and number of columns is set to the size of the embeddings at the (l+1)'th layer. The initial state $H^{(0)}$ of the network is set to the feature set of the nodes of the graph $H^{(0)}=Z$. \n\n\nFor $\\phi$ we employ seven (7) different methods that are described next.\n\\begin{enumerate}\n \\item ordering: For the ordering method, we set $D=d+1$, $\\phi_{ordering}(X)=\\beta_A(X)=\\downarrow(XA)$ with $A=[I~1]$ the identity matrix followed by a column of ones. The ordering and identity-based mappings have the notable disadvantage of not producing the same output embedding size for different sized graphs. To accommodate this and have consistently sized inputs for $\\eta$, we choose to zero-pad $\\phi(X)$ for these methods to produce a vector in $\\mathbb{R}^{m}$, where $m=ND=N(d+1)$ and $N$ is the size of the largest graph in the dataset.\n \\item kernels: For the kernels method, \n $$(\\phi_{kernel}(X))_j=\\sum_{k=1}^n K_G(x_k,a_j)=\\sum_{k=1}^n exp(-\\norm{x_k-a_j}^2),\n ~~j\\in[m],$$ \n for $X=[x_1|\\cdots|x_n]^T$, where kernel vectors $a_1,\\ldots,a_m\\in\\mathbb{R}^d$ \n are generated randomly, each element of each vector is drawn from a standard normal distribution. Each resultant vector is then normalized to produce a kernel vector of magnitude one. When inputting the embedding $X$ to the kernels mapping, we first normalized the embedding for each respective node.\n \\item identity: In this case $\\phi_{id}(X)=X$, which is obviously not a permutation invariant map.\n \\item data augmentation: In this case $\\phi_{data\\;augment}(X)=X$ but data augmentation is used. Our data augmentation scheme works as follows. We take the training set and create multiple permutations of the adjacency and associated feature matrix for each graph in the training set. We add each permuted graph to the training set to be included with the original graphs. In our experiments we use four added permutations for each graph when employing data augmentation.\n \\item sum pooling: The sum pooling method sums the feature values across the set of nodes: $\\phi_{sum\\;pooling}(X)=\\mathbf{1}_{n\\times 1}^T X$.\n \\item sort pooling: The sort pooling method flips entire rows of $X$ so that the last column is ordered descendingly, $\\phi_{sort\\;pool}(X)=\\Pi X$ where $\\Pi\\in{\\mathcal S}_n$ so that $\\Pi\\,X(:,d)=\\downarrow(X(:,d))$. \n \\item set-2-set: This method employs a recurrent neural network\n that achieves permutation invariance through attention-based weighted summations. It has been introduced in \\cite{OrderMatters_2015arXiv151106391V}.\n\\end{enumerate}\n\nFor our deep neural network $\\eta$ we use a simple multilayer perceptron of size described below.\n\nSize parameters related to $\\Gamma$ and $\\zeta$ components are largely held constant across the different implementations.\nHowever the network parameters are trained independently for each method.\n\n\\begin{figure}[!htbp]\n \\centering\n\t\\includegraphics[width=.8\\linewidth]{Results\/GCN_end2end_2.png}\n\t\\caption[.]{.}\n\t\\label{fig:gcn_end2end}\n\\end{figure}\n\n\n\\subsection{Graph Classification}\n\n\\subsubsection{Methodology}\nFor our experiments in graph classification we consider the PROTEINS\\_FULL dataset obtained from \\cite{KKMMN2016} and originally introduced in \\cite{DobsonDoing_proteins}. \nThe dataset consists of 1113 proteins falling into one of two classes: those that function as enzymes and those that do not. Across the dataset there are 450 enzymes in total. \nThe graph for each protein is constructed such that the nodes represent amino acids and the edges represent the bonds between them. The number of amino acids (nodes) vary from around 20 to a maximum of 620 per protein with an average of 39.06. \nEach protein comes with a set of features for each node. \nThe features represent characteristics of the associated amino acid represented by the node. The number of features is $r=29$.\nWe run the end-to-end model with three GCN layers in $\\Gamma$, each with 50 hidden units. \n$\\zeta$ consists of three dense multi-layer perceptron layers, each with 150 hidden units. \nWe set d equal to 1, 10, 50 and 100.\n\nFor each method and embedding size we train for 300 epochs. Note though that the data augmentation method will have experienced five times as many training steps due to the increased size of its training set. We use a batch size of 128 graphs. The loss function minimized during training is the binary cross entropy loss (BCE) defined as\n\\begin{equation}\n\\label{eq:BCE}\nBCE = -\\frac{1}{B}\\sum_{t=1}^B p_t log(\\sigma(\\eta(\\phi(X^{(t)}))))+(1-p_t)log(1-\\sigma(\\eta(\\phi(X^{(t)})))) \n\\end{equation} \nwhere $B=128$ is the batch size, $p_t=1$ when the $t^{th}$ graph\n(protein) is an enzyme and $p_t=0$ otherwise, $\\sigma(x)=\\frac{1}{1+e^{-x}}$ is the sigmoid function that maps the output $\\eta(\\phi(X^{(t)})$ of the 3-layer fully connected network $\\eta$ to $[0,1]$. Three performance metrics were computed: accuracy (ACC), area under the receiver operating characteristic curve (AUC), and average precision (AP) as area under the precision-recall curve from precision scores. These measure are defined as follows (see sklearn.metrics module documentation in pytorch, or \\cite{roc}).\n\nFor a threshold $\\tau\\in[0,1]$, the classification decision $\\hat{p}_t(\\tau)$ is given by:\n\\begin{equation}\n \\hat{p}_t(\\tau) = \\left\\{\n \\begin{array}{rcl}\n 1 & \\mbox{if} & \\mbox{$\\sigma(\\eta(\\phi(X^{(t)}))\\geq \\tau$} \\\\\n 0 & \\mbox{if} & \\mbox{otherwise}\n \\end{array}\\right. .\n\\end{equation}\nBy default $\\tau=\\frac{1}{2}$. For a given threshold, one computes the four scores, true positive (TP), false positive (FP), true negative (TN) and false negative (FN):\n\\begin{equation}\n TP(\\tau) = \\frac{1}{B_1}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 1}1_{p_t = 1} ~~, ~~\n TN(\\tau) = \\frac{1}{B_0} 1_{\\hat{p}_t(\\tau) = 0}1_{p_t = 0}\n\\end{equation}\n\\begin{equation}\n FP(\\tau) = \\frac{1}{B_0}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 1}1_{p_t = 0} = 1-TN(\\tau) ~~,~~\n FN(\\tau) = \\frac{1}{B_1}\\sum_{t=1}^B 1_{\\hat{p}_t(\\tau) = 0}1_{p_t = 1} = 1-TP(\\tau)\n\\end{equation}\nwhere $B_0=\\sum_{t=1}^B 1_{p_t = 0}$ and $B_1=\\sum_{t=1}^B 1_{p_t = 1}=B-B_0$.\n\nThese four statistics predict Precision $P(\\tau)$, Recall $R(\\tau)$ (also known as sensitivity or true positive rate), and Specificity $S(\\tau)$ (also known as true negative rate)\n\\begin{equation} P(\\tau) = \\frac{TP(\\tau)}{TP(\\tau)+FP(\\tau)}\n~~,~~R(\\tau) = \\frac{TP(\\tau)}{TP(\\tau)+FN(\\tau)}\n~~,~~S(\\tau) = \\frac{TN(\\tau)}{TN(\\tau)+FP(\\tau)}\n\\end{equation}\n\nAccuracy (ACC) is defined as the fraction of correct classification for default threshold $\\tau=\\frac{1}{2}$ over the set of batch samples:\n\\begin{equation}\n\\label{eq:ACC}\n ACC = \\frac{1}{B}\\sum_{t=1}^B 1_{p_t = \\hat{p}_t(\\frac{1}{2})} =\\frac{B_0}{B} TN(\\frac{1}{2}) + \\frac{B_1}{B} TP(\\frac{1}{2}) \n\\end{equation}\nArea under the receiver operating characteristic curve (AUC) is computed from prediction scores as the area under true positive rate (TPR) vs. false positive rate (FPR) curve, i.e. the recall vs. 1-specificity curve\n\\begin{equation}\n\\label{eq:AUC}\n AUC = \\frac{1}{2}\\sum_{k=1}^K (S(\\tau_{k-1})-S(\\tau_k))(R(\\tau_{k-1})+R(\\tau_k))\n\\end{equation}\nwhere $K$ is the number of thresholds.\nAverage precision (AP) summarizes a precision-recall curve as the weighted mean of precision achieved at each threshold, with the increase in recall from the previous thresholds used as the weight:\n\\begin{equation}\n\\label{eq:AP}\n AP = \\sum_{k=1}^K (R(\\tau_k) - R(\\tau_{k-1}))P(\\tau_k).\n\\end{equation}\n\nWe track the binary cross entropy (BCE) through training and we compute it on the holdout set and a random node permutation of the holdout set (see Figures \\ref{fig:prot1} and \\ref{fig:prot2}). The lower the value the better.\n\nWe look at the three performance metrics on the training set, the holdout set, and a random node permutation of the holdout set: see Figures \\ref{fig:prot3}, and \\ref{fig:prot4} for accuracy (ACC); see Figures \\ref{fig:prot5}, and \\ref{fig:prot6} for area under the receiver operating characteristic curve (AUC); and see Figures \\ref{fig:prot7}, and \\ref{fig:prot8} for average precision (AP). For all these performance metrics, the higher the score the better.\n\n\\subsubsection{Discussion}\n\nTables \\ref{table:t1}-\\ref{table:t12} list values of the three performance metrics (ACC, AUC, AP) at the end of training (after 300 epochs). \nPerformances over the course of training are plotted in Figures \\ref{fig:prot1} through \\ref{fig:prot8}.\n\nThe authors of \\cite{KKMMN2016} utilized a Support Vector Machine (1-layer perceptron) for classification and \nobtained an accuracy (ACC) of 77\\% on the entire data set \nusing 52 features, and an accuracy of 80\\% on a smaller set of 36 features. By comparison, our data augmentation method for $d=100$ achieved an accuracy of 97.5\\% on training data set,\nbut dropped dramatically to 73\\% on holdout data, and 72\\% on \nholdout data set with randomly permuted nodes. \nOn the other hand, both the kernels method and the sum-pooling\nmethod with $d=50$ achieved an accuracy of around 79\\% on\ntraining data set, while dropping accuracy performance by \nonly 2\\% to around\n77\\% on holdout data (as well as holdout data with nodes permuted).\n\nFor $d=1$, data augmentation performed the best on the training set with an area under the receiver operating characteristic (AUC) of 0.896, followed closely by the identity method with an AUC of 0.886. On the permuted holdout set however, sort-pooling performed the best with an AUC of 0.803.\n\nFor $d=10$, sum-pooling, ordering, and kernels performed well on the permuted holdout set with AUC's of 0.821, 0.820, and 0.818 respectively. The high performance of the identity method, data augmentation, and sort-pooling on the training set did not translate to the permuted holdout set at $d=10$. By $d=100$, sum-pooling still performed the best on the permuted holdout set with an AUC of 0.817. This was followed by the kernels method which achieved an AUC of 0.801 on the permuted holdout set.\n\nFor experiments where $d>1$, the identity method and data augmentation show a notable drop in performance from the training set to the holdout set. This trend is also, to a lesser extent, visible in the sort pooling and ordering methods. In the holdout permuted set we see significant oscillations in the performance of both the identity and data augmentation methods.\n\n\\subsection{Graph Regression}\n\n\\subsubsection{Methodology}\nFor our experiments in graph regression we consider the qm9 dataset \\cite{ramakrishnan2014quantum}. This dataset consists of 134 thousand molecules represented as graphs, where the nodes represent atoms and edges represent the bonds between them. \n\nEach graph has between 3 and 29 nodes, $3\\leq n\\leq 29$. Each node has 11 features, $r=11$. We hold out 20 thousand of these molecules for evaluation purposes. The dataset includes 19 quantitative features for each molecule.\n\nFor the purposes of our study, we focus on electron energy gap (units $eV$), which is $\\Delta\\varepsilon$ in \\cite{DFTpaper} whose chemical accuracy is $0.043 eV$ and whose prediction performance of any machine learning technique\nis worse than any other feature.\nThe best existing estimator for this feature is enn-s2s-ens5 from \\cite{Gilmer_2017arXiv170401212G}\n and has a mean absolute error (MAE) of $0.0529eV$ which is $1.23$ larger than the chemical accuracy. \n We run the end to end model with three GCN layers in $\\Gamma$, each with 50 hidden units. $\\eta$ consists of three multi-layer perceptron layers, each with 150 hidden units. We use rectified linear units as our nonlinear activation function. Finally, we vary $d$, the size of the node embeddings that are outputted by $\\Gamma$. We set $d$ equal to 1, 10, 50 and 100.\n\nFor each method and embedding size we train for 300 epochs. Note though that the data augmentation method will have experienced five times as many training steps due to the increased size of its training set. We use a batch size of 128 graphs. The loss function minimized during training is the mean square error (MSE) between the ground truth and the network output \n (see Figures \\ref{fig:a1}, \\ref{fig:a2}) \n\\begin{equation}\n\\label{eq:MSE}\nMSE = \\frac{1}{B}\\sum_{t=1}^B |\\Delta\\varepsilon_t -\\eta(\\phi(X^{(t)}))))|^2\n\\end{equation}\nwhere $B=128$ is the batch size of 128 graphs and $\\Delta\\varepsilon_t$ is the electron energy gap of the $t^{th}$ graph (molecule). The performance\nmetric is Mean Absolute Error (MAE)\n\\begin{equation}\n\\label{eq:MAE}\nMAE = \\frac{1}{B}\\sum_{t=1}^B |\\Delta\\varepsilon_t -\\eta(\\phi(X^{(t)}))))|.\n\\end{equation}\nWe track the mean absolute error through the course of training. We look at this performance metric on the training set, the holdout set, and a random node permutation of the holdout set (see Figures \\ref{fig:b1}, and \\ref{fig:b2}). \n\n\n\\subsubsection{Discussion}\n\nNumerical results at the end of training (after 300 epochs) are included in Tables \\ref{table:a}, \\ref{table:b}, \\ref{table:c} and \\ref{table:d}.\nFrom the results we see that the ordering method performed best for $d=100$\nfollowed closely by the data augmentation method, while both the ordering method and the kernels method performed well for $d=10$, though both fell slightly short of data augmentation which performed marginally better on both the training data and the holdout data, though with significantly more training iterations. For $d=1$, the kernels method failed to train adequately. The identity mapping performed relatively well on training data (for $d=100$ it achieved the smallest MAE among all methods and all parameters) and even the holdout data, however it lost its performance on the permuted holdout data. The identity mapping's failure to generalize across permutations of the holdout set is likely exacerbated by the fact that the QM9 data as presented to the network comes ordered in its node positions from heaviest atom to lightest. Data augmentation notably kept its performance despite this due to training on many permutations of the data. \n\nFor $d=100$, our ordering method achieved a MAE of $0.155eV$ on training data set and $0.187eV$ on holdout data set, which are $3.6$ and $4.35$ times larger than the chemical accuracy ($0.043eV$\\ignore{ cf. Supplementary material of \\cite{Gilmer_2017arXiv170401212G}}), respectively. This is worse than the enn-s2s-ens5 method in \\cite{Gilmer_2017arXiv170401212G} (current best method) that achieved a MAE $0.0529$ (eV), $1.23$ larger than the chemical accuracy, \nbut better than the Coulomb Matrix (CM) representation in \\cite{PhysRevLett.108.058301} that achieved a MAE $5.32$ larger than the chemical accuracy whose features were optimized for this task.\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzflxk b/data_all_eng_slimpj/shuffled/split2/finalzzflxk new file mode 100644 index 0000000000000000000000000000000000000000..2c80307932e1fdc81683de40676bbbb550a65b5e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzflxk @@ -0,0 +1,5 @@ +{"text":"\\section*{Introduction}\n\nLet $f\\colon(\\C^2,0)\\to (\\C^2,0)$ be a germ of a holomorphic map that fixes the origin $0\\in \\C^2$, and which is finite-to-one near $0$. Suppose that $C$ and $D$ are two germs of holomorphic curves passing through $0$. In this article, we will study the sequence $\\mu(n) := C\\cdot f^n(D)$ of local intersection multiplicities at the origin, for $n\\geq 0$. Specifically, we will address the question: \\emph{how fast can the sequence $\\mu(n)$ grow?} This and related questions were posed and studied by V.\\ ~I.\\ ~Arnold, who conjectured that, if $\\mu(n)<\\infty$ for every $n$, the sequence $\\mu(n)$ grows at most exponentially fast, see \\cite[\\S5]{MR1215971}, \\cite[p.\\ 215]{MR1350971}, and \\cite[problems 1994-49 and 1994-50]{MR2078115}. Arnold proved this conjecture in the case when $f$ is a local biholomorphism and in some cases when the complex derivative $f'(0)$ has exactly one zero eigenvalue \\cite[\\S5]{MR1215971}, but the general case appears to be unknown. A new proof of the conjecture in the case when $f$ is a local biholomorphism has recently been obtained by Seigal and Yakovenko \\cite{SY}.\n\nIn this article, we will show by explicit construction that Arnold's conjecture is false in general, and that in fact the sequence $\\mu(n)$ can grow arbitrarily fast. More precisely, we will prove the following theorem.\n\n\\begin{letthm}\\label{thmA} Let $f\\colon \\C^2\\to \\C^2$ be the polynomial map $f(x,y) = (x^2 - y^4,y^4)$ and $\\nu\\colon \\N\\to \\R$ be any function. Then there exist germs of holomorphic curves $C$ and $D$ through the origin such that the local intersection multiplicities $\\mu(n) = C\\cdot f^n(D)$ are always finite, and such that $\\mu(n)>\\nu(n)$ for infinitely many $n$.\n\\end{letthm}\n\nNote that the complex derivative $f'(0) $ for this map is $0$, so $f$ defines a \\emph{superattracting} germ at $0$ (in general, $f$ is superattracting if $f'(0)$ is nilpotent). The dynamics of superattracting germs is an active area of research in holomorphic dynamics in several variables, see for instance \\cite{MR1275463, MR1759437, MR2339287, MR2904007, Rug2, MR2853790, BEK, GR} and the notes \\cite{Mattias}.\n\nIt has long been known that intersection multiplicities in \\emph{smooth} dynamical systems (local or global) can grow arbitrarily fast \\cite{MR1039340, MR1167716, MR1199715}. On the other hand, it is also known that intersection multiplicities ``generically\" grow at most exponentially (see \\cite{MR1139553} for a precise formulation). Our second theorem is a holomorphic version of this principle.\n\n\\begin{letthm}\\label{thmB} Let $f\\colon (\\C^d, 0)\\to (\\C^d,0)$ be a holomorphic fixed point germ at the origin $0\\in \\C^d$, where $d\\geq 2$, such that $f$ is finite-to-one near $0$. Fix holomorphic function germs $\\psi_1,\\ldots, \\psi_m$ at the origin such that $\\{\\psi_1 = 0\\}\\cap \\cdots\\cap \\{\\psi_m = 0\\} = \\{0\\}$. For each $z\\in \\C^m$, let $D_z$ denote the hypersurface germ $D_z = \\{z_1\\psi_1 + \\cdots + z_m\\psi_m = 0\\}$ through the origin. Fix an integer $k\\in \\{1,\\ldots, d-1\\}$. For each $a = (a_1,\\ldots, a_k)\\in (\\C^m)^k$ and $b = (b_1,\\ldots, b_{d-k})\\in (\\C^m)^{d-k}$, let $V_a$ and $W_b$ be the local intersection cycles $V_a := D_{a_1}\\cdot\\ldots \\cdot D_{a_k}$ and $W_b := D_{b_1}\\cdot\\ldots\\cdot D_{b_{d-k}}$. Then there exists a dense set $U\\subseteq (\\C^m)^k\\times (\\C^m)^{d-k}$, given as the complement of a countable union of algebraic subsets, with the property that for all $(a,b)\\in U$, the cycles $V_a$ and $W_b$ are of codimension $k$ and $d-k$, respectively, and the sequence of local intersection multiplicities $\\mu(n) := V_a\\cdot f^n_*W_b$ grows subexponentially, that is to say, $\\mu(n)\\leq AB^n$ for some $A,B>0$.\n\\end{letthm}\n\nIn the special case when $d = 2$ and $f$ is superattracting, we will in fact be able to prove much more about the sequence $\\mu(n)$.\n\n\\begin{letthm}\\label{thmD} With the same setup and notations as \\hyperref[thmB]{Theorem~\\ref*{thmB}} but with the additional assumptions that $d = 2$ and $f$ is superattracting, we have for each $(a,b)\\in U$ that the sequence of local intersection multiplicities $\\mu(n) := V_a\\cdot f^n_*W_b$ at the origin eventually satisfies an integral linear recursion relation. Moreover, there exist constants $A_1, A_2>0$ such that $A_1c_\\infty^n \\leq \\mu(n)\\leq A_2 c_\\infty^n$ as $n\\to \\infty$, where here $c_\\infty>1$ denotes the asymptotic attraction rate of $f$ (see \\S5 for the definition of $c_\\infty$). If one replaces $f$ by $f^2$, then in fact there is a constant $A>0$ such that $\\mu(n)\\sim Ac_\\infty^n$.\n\\end{letthm}\n\nThe proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}}, which will be given in \\S4, is a rather easy application of Teissier's theory of mixed multiplicities. \\hyperref[thmD]{Theorem~\\ref*{thmD}}, which we will prove in \\S5, relies on recent non-elementary results of the author and M.\\ Ruggiero \\cite{GR} within the subject of dynamics on valuation spaces. Unlike Theorems \\ref{thmB} and \\ref{thmD}, our proof of \\hyperref[thmA]{Theorem~\\ref*{thmA}} requires no high-powered techniques, and lends itself to an easy overview, which we give now.\n\n\nLet $S$ be the space of binary sequences $S = \\{0,1\\}^\\N$, and let $\\sigma\\colon S\\to S$ denote the left-shift map on $S$. For any two sequences $s,t\\in S$, set $M(s,t)$ to be the smallest index $m$ such that $s_m\\neq t_m$, with $M(s,t) = \\infty$ if $s = t$. To prove \\hyperref[thmA]{Theorem~\\ref*{thmA}}, we will construct a family $\\{C_s\\}_{s\\in S}$ of holomorphic curve germs through the origin with the properties that \\begin{enumerate}\n\\item[1.] $f(C_s) = C_{\\sigma(s)}$ for each $s\\in S$, and\n\\item[2.] for any $s,t\\in S$, the local intersection multiplicity $C_s\\cdot C_t$ is $\\asymp 4^{M(s,t)}$.\n\\end{enumerate} The theorem then follows easily from the following simple proposition, the proof of which is left to the reader.\n\n\\begin{nonumprop} Let $\\nu\\colon \\N\\to \\R$ be any function. Then there exist sequences $s,t\\in S$ such that $M(s,\\sigma^n(t))$ is finite for all $n\\geq 0$, and such that $M(s,\\sigma^n(t))>\\nu(n)$ for infinitely many $n$.\n\\end{nonumprop}\n\nIn \\S1, we will construct the $C_s$ as \\emph{formal} curves, that is, as curves defined by irreducible formal power series $\\varphi_s\\in \\C\\llbracket x, y\\rrbracket$. The coefficients of the power series $\\varphi_s$ will be determined via a recursive procedure that guarantees properties 1.\\ and 2.\\ are satisfied. In \\S2, we will prove that each formal power series $\\varphi_s$ is actually convergent, and hence that the curve germs $C_s$ are \\emph{holomorphic}. It should be noted that the construction of the power series $\\varphi_s$ in \\S1 is purely algebraic, and that if we replace the word \\emph{holomorphic} with \\emph{formal}, \\hyperref[thmA]{Theorem~\\ref*{thmA}} holds when $\\C$ is replaced by any field of characteristic $\\neq 2$. \n\nIn \\S3, we will sketch an alternative, geometric construction of the curves $C_s$, valid when working over $\\C$. The construction realizes the curves as a \\emph{Cantor bouquet} of holomorphic stable manifolds, similar to the construction carried out first in \\cite{MR1808626} for rational maps; see also the related works \\cite{Yam2, MR2195140, MR2307152, Tomoko, MR2629648}. Finally, it is worth mentioning that this counterexample is by no means isolated; one can construct similar Cantor bouquets for many other superattracting germs.\n\n\\subsection*{Acknowledgements} I would like to wholeheartedly thank Mattias Jonsson and Charles Favre for their support and guidance during the course of this project. I would also like to thank the referee for very useful commentary, and especially for pointing out the work of Yamagishi \\cite{MR1808626}. This work was supported by the grants DMS-1001740 and DMS-1045119, as well as the ERC-Starting grant ``Nonarcomp\" no. 307856.\n\n\n\\section{The formal construction of the curves $C_s$}\n\nIn this and the next two sections, we let $f\\colon \\C^2\\to \\C^2$ be the polynomial map $f(x,y) = (x^2 - y^4, y^4)$. We will write $S$ to denote the space of binary sequences $S = \\{0,1\\}^\\N$, and write $\\sigma$ to denote the left-shift map $\\sigma\\colon S\\to S$.\n\nWe now define a family $\\{\\varphi_s\\}_{s\\in S}$ of irreducible formal power series $\\varphi_s\\in \\C\\llbracket x,y\\rrbracket$ of the form \\begin{equation}\\label{eqn1} \\varphi_s(x,y) = x + a_0^sy^2 + a_1^sy^6 + \\cdots + a_n^sy^{2+4n} + \\cdots,\\end{equation} by recursively defining the coefficients $a_n^s$, in the following manner. First, we set $a_0^s = (-1)^{s_0}$; then, assuming $a_0^s,\\ldots, a_n^s$ have been defined for all $s\\in S$, we set\\begin{equation}\\label{recursion}\na_{n+1}^s = \\begin{cases}\\displaystyle-\\frac{1}{2a_0^s}\\sum_{\\substack{i+j = n+1\\\\i,j\\geq 1}} a_i^sa_j^s & \\mbox{ if }4\\nmid n.\\bigskip\\\\\n\\displaystyle-\\frac{a_{n\/4}^{\\sigma(s)}}{2a_0^s} - \\frac{1}{2a_0^s}\\sum_{\\substack{i+j = n+1\\\\i,j\\geq 1}} a_i^sa_j^s & \\mbox{ if }4\\mid n.\\end{cases}\\end{equation} Let $C_s$ denote the formal curve through the origin in $\\C^2$ defined by $\\varphi_s$. As the next proposition shows, the curves $\\{C_s\\}_{s\\in S}$ have the properties discussed in the introduction\n\n\\begin{prop}\\label{formalprop} The formal curves $\\{C_s\\}_{s\\in S}$ satisfy \\begin{enumerate}\n\\item[$1.$] $f(C_s) = C_{\\sigma(s)}$ for all $s\\in S$, and\n\\item[$2.$] the local intersection multiplicity $C_s\\cdot C_t$ is $\\frac{1}{3}(4^{m+1} + 2)$, where $m$ is the smallest index such that $s_m\\neq t_m$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof} To prove 1., we must show that $\\varphi_s\\mid (\\varphi_{\\sigma(s)}\\circ f)$ in the ring $\\C\\llbracket x,y\\rrbracket$. Indeed, we show that $\\varphi_{\\sigma(s)}\\circ f = (x + a_0^sy^2 + a_1^sy^6 + \\cdots)(x - a_0^sy^2 - a_1^sy^6 - \\cdots)$. To see this, observe that \\begin{equation}\\label{eqn3}\n(x + a_0^sy^2 + a_1^sy^6 + \\cdots)(x - a_0^sy^2 - a_1^sy^6 - \\cdots) = x^2 - y^4 - \\sum_{n\\geq 0}\\sum_{\\substack{i+j = n+1\\\\i,j\\geq 0}}a_i^sa_j^sy^{4(n+2)}.\\end{equation} The recursion formula (\\ref{recursion}) then gives that the coefficient of $y^{4(n+2)}$ in this expression is $0$ when $4\\nmid n$ and is $a_{n\/4}^{\\sigma(s)}$ when $4\\mid n$, so the right hand side of (\\ref{eqn3}) is \\[x^2 - y^4 + \\sum_{k\\geq 0} a_k^{\\sigma(s)}y^{4(4k + 2)} = x^2 - y^4 + \\sum_{k\\geq 0}a_k^{\\sigma(s)}y^{8 + 16k} = \\varphi_{\\sigma(s)}\\circ f.\\] This completes the proof of 1. \n\nTo prove 2., we first make the easy observation that the intersection multiplicity $C_s \\cdot C_t$ is precisely the smallest integer $k$ such that the coefficients of $y^k$ in the power series $\\varphi_s$ and $\\varphi_t$ differ. From equation (\\ref{eqn1}), it then follows that $C_s\\cdot C_t = 2 + 4n$, where $n$ is the smallest integer such that $a_n^s\\neq a_n^t$. We will prove 2.\\ by induction on $m\\geq 0$, where $m$ is the smallest index such that $s_m\\neq t_m$. If $m = 0$, then $a_0^s\\neq a_0^t$, and hence $C_s\\cdot C_t = 2$, establishing the base case of the induction. Now assume that $m>0$ is the smallest index index such that $s_m\\neq t_m$. Then, by induction, $C_{\\sigma(s)}\\cdot C_{\\sigma(t)} = \\frac{1}{3}(4^m + 2)$, from which it follows that the first index $n$ for which $a_n^{\\sigma(s)}\\neq a_n^{\\sigma(t)}$ is $n = \\frac{1}{3}(4^{m-1}-1)$. Using the recursion formula (\\ref{recursion}), we can then conclude that the first index $n$ such that $a_n^s\\neq a_n^t$ is \\[\nn = 1 + \\frac{4}{3}(4^{m-1} - 1) = \\frac{1}{3}(4^m-1).\\] Thus $C_s\\cdot C_t = 2 + \\frac{4}{3}(4^m-1) = \\frac{1}{3}(4^{m+1}+2),$ completing the induction, and the proof.\n\\end{proof}\n\n\\section{Analyticity}\n\nIn this section, we complete the proof of \\hyperref[thmA]{Theorem~\\ref*{thmA}} by proving that each of the power series $\\varphi_s$ constructed in \\S1 are convergent. Indeed, using very crude estimates, we will prove the following proposition.\n\n\\begin{prop}\\label{analyticity} Let $C = 1\/20$ and $R = 10$. Then $|a_n^s|\\leq CR^n\/n^2$ for each $n\\geq 1$ and each $s\\in S$. In particular, $\\varphi_s$ converges on the set $\\{(x,y)\\in \\C^2 : |y|< 1\/10\\}$.\n\\end{prop}\n\nTo prove the proposition, we will make use of the following lemma.\n\n\\begin{lem}\\label{lemma} Let $n\\geq 1$ be an integer. Then \\[\\sum_{k=1}^n\\frac{1}{k^2(n-k+1)^2}\\leq \\frac{20}{(n+1)^2}.\\]\n\\end{lem}\n\\begin{proof} The symmetry in the terms of the left hand sum implies that \\[\n\\sum_{k=1}^n\\frac{1}{k^2(n-k+1)^2}\\leq 2\\sum_{k=1}^{\\lfloor\\frac{n+1}{2}\\rfloor} \\frac{1}{k^2(n-k+1)^2}.\\] Multiplying both sides of this inequality by $(n+1)^2$ yields \\[\\sum_{k=1}^n\\frac{(n+1)^2}{k^2(n-k+1)^2}\\leq 2\\sum_{k=1}^{\\lfloor\\frac{n+1}{2}\\rfloor} \\frac{(n+1)^2}{k^2(n-k+1)^2} = 2\\sum_{k=1}^{\\lfloor\\frac{n+1}{2}\\rfloor}\\frac{1}{k^2(1 - \\frac{k}{n+1})^2}\\leq 8\\sum_{k=1}^{\\lfloor\\frac{n+1}{2}\\rfloor}\\frac{1}{k^2}<\\frac{8\\pi^2}{6}.\\] Since $8\\pi^2\/6<20$, the proof is complete.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{analyticity}] We will prove the proposition by induction on $n\\geq 1$. When $n = 1$, the recursion formula (\\ref{recursion}) gives $a_1^s = -a_0^{\\sigma(s)}\/2a_0^s =\\pm\\frac{1}{2}$ for each $s\\in S$, and hence $|a_1^s| = \\frac{1}{2} = CR$, establishing the base case of the induction. Now assume that the proposition holds for $a_k^s$ when $k\\leq n$. If $4\\nmid n$, then (\\ref{recursion}), the triangle inequality, and \\hyperref[lemma]{Lemma~\\ref*{lemma}} give that\\[\n|a_{n+1}^s|\\leq \\frac{1}{2} \\sum_{k=1}^n\\frac{C^2R^{n+1}}{k^2(n-k+1)^2}\\leq \\frac{20C^2R^{n+1}}{2(n+1)^2} = \\frac{CR^{n+1}}{2(n+1)^2}<\\frac{CR^{n+1}}{(n+1)^2},\\] establishing the proposition in this case. If $4\\mid n$, then (\\ref{recursion}) gives \\begin{equation}\\label{eqn4}|a_{n+1}^s|\\leq \\frac{CR^{n\/4}}{2(n\/4)^2} + \\frac{1}{2}\\sum_{k=1}^n\\frac{C^2R^{n+1}}{k^2(n-k+1)^2}\\leq \\frac{CR^{n\/4}}{2(n\/4)^2} + \\frac{CR^{n+1}}{2(n+1)^2}.\\end{equation}Since $4\\mid n$, and in particular $n\\geq 4$, the inequality $\\frac{n}{4}\\leq (n+1) - 4$ is valid, and hence \\[\\frac{CR^{n\/4}}{2(n\/4)^2} = \\frac{8CR^{n\/4}}{n^2}\\leq \\frac{8CR^{n+1}}{n^2R^4}<\\frac{8CR^{n+1}}{(n+1)^2R^4}.\\] Putting this estimate into (\\ref{eqn4}), we see that \\[\n|a_{n+1}^s|\\leq \\left(\\frac{8}{R^4} + \\frac{1}{2}\\right)\\frac{CR^{n+1}}{(n+1)^2}<\\frac{CR^{n+1}}{(n+1)^2}.\\] This completes the proof.\n\\end{proof}\n\nWe have thus shown that the curves $C_s$ are holomorphic. On the other hand, it should be noted that they cannot all be algebraic. This is because if $C$ and $D$ are algebraic plane curves passing through $0$, then, as a simple consequence of Bezout's theorem, the local intersection multiplicities $C\\cdot f^n_*D$ grow subexponentially in $n$. We are then led to the following interesting, but possibly difficult, questions.\n\n\\begin{Quest} Which, if any, of the curves $C_s$ just constructed are (local irreducible components of) germs of algebraic curves? More specifically, if a sequence $s\\in S$ is not eventually periodic, is it possible for $C_s$ to be a germ of an algebraic curve?\n\\end{Quest}\n\n\n\\section{Realization as a Cantor bouquet}\n\nIn this section, we reconstruct the curves $C_s$ from \\S1 as a \\emph{Cantor bouquet} of holomorphic stable manifolds using a geometric procedure given first by Yamagishi in \\cite{MR1808626}, see also \\cite{Yam2, MR2195140, MR2307152, Tomoko, MR2629648}. We only sketch this construction here; for details see \\cite[\\S2]{MR1808626}.\n\nLet $\\pi\\colon X\\to \\C^2$ denote the blowup of the origin in $\\C^2$, and let $E$ denote the exceptional divisor of $\\pi$. It is easy to check that the lift $f_X\\colon X\\dashrightarrow X$ of $f$ has exactly one indeterminacy point $q$, given by $z = w = 0$ in the local coordinates $z = x\/y$ and $w = y$. \n\nNow let $\\pi'\\colon X'\\to X$ denote the blowup of the point $q$. It is a straightforward computation to check that $f$ lifts to an (everywhere defined) holomorphic map $F\\colon X'\\to X$. Moreover, with respect to the local coordinates $z,w$ on $X$ and $u = z\/w$, $v = w$ on $X'$, the map $F$ is given simply as $F(u,v) = (u^2 - 1, v^4)$. The preimage $F^{-1}(q)$ then consists of two points, $q_0 = (1,0)$ and $q_1 = (-1,0)$. Because $F$ is not a local biholomorphism near either $q_0$ or $q_1$, we are not in an identical situation to the one considered in \\cite[\\S2]{MR1808626}, but nonetheless the map $(\\pi')^{-1}\\circ F$, defined away from the $q_i$, exhibits similar local dynamics around the $q_i$ as the maps studied in \\cite[\\S2]{MR1808626}; specifically, $(\\pi')^{-1}\\circ F$ is contracting in the $v$-direction and expanding in the $u$-direction near the $q_i$. It is this behavior that allows us to consider holomorphic stable manifolds of $q$. Let $U$ be a small neighborhood of $q$ in $X$, so that $F^{-1}(U)$ is a disjoint union $U_0\\sqcup U_1$ of neighborhoods of $q_0$ and $q_1$, respectively. Let \\begin{align*}\nB & = \\{p\\in X\\smallsetminus E : f_X^n(p)\\in U \\mbox{ for all } n\\geq 0\\}\\\\\n& = \\{p\\in X\\smallsetminus E : f_X^n(p)\\in \\pi'(U_0\\sqcup U_1)\\mbox{ for all }n\\geq0\\}.\n\\end{align*} Near $q$, the set $B$ will be the union of the (strict transforms in $X$ of the) curves $C_s$ constructed in \\S1. To be precise, if $s\\in S := \\{0,1\\}^\\N$, then the set $\\wt{C}_s := \\{p\\in X\\smallsetminus E : f_X^n(p)\\in \\pi'(U_{s_n})$ for all $n\\geq 0\\}$ is a curve transverse to $E$ at $q$, and the family $\\{\\wt{C}_s\\}_{s\\in S}$ consists of the strict transforms in $X$ of the curves $C_s$ from \\S1. The $\\wt{C}_s$ are \\emph{local stable manifolds} of $q$ in the sense that they form an invariant family for $f_X$, and $f_X^n(p)\\to q$ as $n\\to\\infty$ for all $p\\in B = \\bigcup_s \\wt{C}_s$ near $q$. Moreover, it is clear from this construction that $f(\\wt{C}_s) = \\wt{C}_{\\sigma(s)}$, where $\\sigma\\colon S\\to S$ is the left shift map.\n\nIt is also possible to use the geometry in this construction to compute the local intersection multiplicites of the curves $C_s$ at the origin, rederiving \\hyperref[formalprop]{Proposition~\\ref*{formalprop}(2)}. To see how, first observe that because the $\\wt{C}_s$ are transverse to $E$ at $q$, the projection formula gives \\[\nC_s\\cdot C_t = (\\pi^*C_s \\cdot \\pi^*C_t) = (\\wt{C}_s + E)\\cdot (\\wt{C}_t + E) = \\wt{C}_s\\cdot \\wt{C}_t + 1.\\] Similarly, if $D_s$ denotes the strict transform of $C_s$ in $X'$, then $\\wt{C}_s\\cdot \\wt{C}_t = D_s\\cdot D_t + 1$, and thus $C_s\\cdot C_t = D_s\\cdot D_t + 2$. If $s_0\\neq t_0$, then the germs $D_s$ and $D_t$ lie in different open sets $U_0$ and $U_1$, and hence do not intersect, proving that $C_s\\cdot C_t = 2$, as previously derived. Suppose, on the other hand, that $s_0 = t_0$, say without loss of generality $s_0 = t_0 = 0$. If $F_0$ denotes the restriction $F_0 = F|_{U_0}$, then $D_s = F_0^*\\wt{C}_{\\sigma(s)}$ and $D_t = F_0^*\\wt{C}_{\\sigma(t)}$. Because $F_0$ has local topological degree $4$ at $q_0$, it follows that \\begin{align*}\nC_s\\cdot C_t & = 2 + D_s\\cdot D_t = 2 + F_0^*\\wt{C}_{\\sigma(s)}\\cdot F_0^*\\wt{C}_{\\sigma(t)} = 2 + 4(\\wt{C}_{\\sigma(s)}\\cdot \\wt{C}_{\\sigma(t)})\\\\\n& = 4(C_{\\sigma(s)}\\cdot C_{\\sigma(t)}) - 2\\end{align*} when $s_0= t_0$. Using this identity and the fact that $C_s\\cdot C_t = 2$ when $s_0\\neq t_0$, one easily rederives \\hyperref[formalprop]{Proposition~\\ref*{formalprop}(2)}.\n\nFinally, it is worth pointing out that this argument applies equally well to any other superattracting germ with similar geometry. For instance, using either the methods in this section or in \\S1, one can show that the maps $f(x,y) = (x^p - y^q, y^r)$ where $2\\leq p < r\\leq q$ all have Cantor bouquets of curves.\n\n\\section{Mixed multiplicities and the proof of Theorem B}\n\nIn this section, we give a proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}} using Teissier's theory of \\emph{mixed multiplicities}. In fact, we will prove a slightly stronger theorem, namely \\hyperref[thmC]{Theorem~\\ref*{thmC}} below. For ease of notation, let $R$ denote the formal power series ring $\\C\\llbracket x_1,\\ldots, x_d\\rrbracket$, where $d\\geq 2$ is a fixed integer. Recall that $R$ is a local ring with maximal ideal $\\mf{m} = (x_1,\\ldots, x_d)$. An ideal $\\mf{a}$ of $R$ is said to be $\\mf{m}$-primary if $\\mf{a}$ contains some power of $\\mf{m}$, or equivalently if $\\mf{a}$ defines the origin $0\\in \\C^d$.\n\n\\begin{thm}\\label{thmC} Let $f\\colon (\\C^d,0)\\to (\\C^d,0)$ be a holomorphic fixed point germ at the origin $0\\in \\C^d$ that is finite-to-one near $0$. Let $\\mf{a}_1,\\ldots, \\mf{a}_d$ be $\\mf{m}$-primary ideals of $R$. Choose generators $\\psi_1^{(i)},\\ldots, \\psi_{m_i}^{(i)}$ for each of the ideals $\\mf{a}_i$. For every point $z\\in \\C^{m_i}$, let $D_{z}^{(i)}$ denote the formal hypersurface germ $D_{z}^{(i)} = \\{z_{1}\\psi_1^{(i)} + \\cdots + z_{m_i}\\psi_{m_i}^{(i)} = 0\\}$ through the origin. Fix an integer $k\\in \\{1,\\ldots, d-1\\}$. For each point $a = (a_1,\\ldots, a_k)\\in \\C^{m_1}\\times\\cdots\\times \\C^{m_k}$ and $b = (b_1,\\ldots, b_{d-k})\\in \\C^{m_{k+1}}\\times\\cdots\\times \\C^{m_d}$, let $V_a$ and $W_b$ be the local intersection cycles $V_a := D_{a_1}^{(1)}\\cdot\\ldots\\cdot D_{a_k}^{(k)}$ and $W_b := D_{b_1}^{(k+1)}\\cdot\\ldots \\cdot D_{b_{d-k}}^{(d)}$. Then there is a dense subset $U\\subseteq (\\C^{m_1}\\times\\cdots\\times \\C^{m_k})\\times(\\C^{m_{k+1}}\\times\\cdots\\times \\C^{m_d})$, given as the complement of a countable union of algebraic subsets, such that for all $(a,b)\\in U$, the cycles $V_a$ and $W_b$ are of codimension $k$ and $d - k$, respectively, and the sequence of local intersection multiplicities $\\mu(n) = V_a\\cdot f^n_*W_b$ grows subexponentially.\n\\end{thm}\n\nObserve that \\hyperref[thmC]{Theorem~\\ref*{thmC}} implies \\hyperref[thmB]{Theorem~\\ref*{thmB}} by simply taking each of the ideals $\\mf{a}_i$ of \\hyperref[thmC]{Theorem~\\ref*{thmC}} to be the same ideal $(\\psi_1,\\ldots, \\psi_m)$.\n\nBefore beginning the proof of \\hyperref[thmC]{Theorem~\\ref*{thmC}}, we recall some basic facts from the theory of mixed multiplicities developed by B.\\ Teissier in the 70s \\cite{MR0374482, MR0467800, MR645731, MR708342, MR518229}. A concise and clear overview to the topic that suffices for our purposes can be found in \\cite[\\S1.6.8]{MR2095471}. For us, the relevant results are the following.\n\n\\begin{thm}[Teissier]\\label{teissier1} Let $\\mf{b}_1,\\ldots, \\mf{b}_d$ be $\\mf{m}$-primary ideals of $R$. Fix generators $\\varphi_1^{(i)},\\ldots, \\varphi_{m_i}^{(i)}$ of each of the ideals $\\mf{b}_i$. For a point $z\\in \\C^{m_i}$, let $D_{z}^{(i)} = \\{z_{1}\\varphi_1^{(i)} + \\cdots + z_{m_i}\\varphi_{m_i}^{(i)} = 0\\}$. Then there is an integer $e(\\mf{b}_1;\\cdots; \\mf{b}_d)\\geq 1$ and a nonempty Zariski open subset $U\\subseteq \\C^{m_1}\\times\\cdots\\times \\C^{m_d}$ such that if $ (a_1,\\ldots, a_d)\\in U$, then the hypersurface germs $D_{a_1}^{(1)},\\ldots, D_{a_d}^{(d)}$ intersect properly at the origin, and $e(\\mf{b}_1; \\cdots; \\mf{b}_d)$ is exactly the local intersection multiplicity $ D_{a_1}^{(1)}\\cdot D_{a_2}^{(2)}\\cdot\\ldots\\cdot D_{a_d}^{(d)}$. Moreover, one has the inequality \\[\ne(\\mf{b}_1; \\cdots; \\mf{b}_d) \\leq e(\\mf{b}_1)^{1\/d}\\cdots e(\\mf{b}_d)^{1\/d},\\] where here $e(\\mf{b}_i)$ denotes the standard Samuel multiplicity of $\\mf{b}_i$, that is \\[\ne(\\mf{b}_i) := \\lim_{n\\to \\infty} \\frac{d!}{n^d}\\,\\mathrm{length}_R(R\/\\mf{a}^{n+1}) \\in \\N.\\] The integer $e(\\mf{b}_1;\\cdots; \\mf{b}_d)$ is called the \\emph{mixed multiplicity} of the ideals $\\mf{b}_i$.\n\\end{thm}\n\nWith these facts at our disposal, we can now prove \\hyperref[thmC]{Theorem~\\ref*{thmC}}.\n\n\\begin{proof}[Proof of {\\hyperref[thmC]{Theorem~\\ref*{thmC}}}] The projection formula says precisely that \\begin{equation}\\label{proj}\n\\mu(n) = V_a\\cdot f^n_*W_b = f^{n*}D_{a_1}^{(1)}\\cdot \\ldots \\cdot f^{n*}D_{a_k}^{(k)}\\cdot D_{b_1}^{(k+1)}\\cdot\\ldots\\cdot D_{b_{d-k}}^{(d)}.\\end{equation} By \\hyperref[teissier1]{Theorem~\\ref*{teissier1}}, for each $n$ there is a nonempty Zariski open subset $U_n\\subseteq (\\C^{m_1}\\times\\cdots\\times \\C^{m_k})\\times(\\C^{m_{k+1}}\\times\\cdots\\times \\C^{m_d})$ such that if $(a,b)\\in U_n$, then the right hand side of equation (\\ref{proj}) is exactly the mixed multiplicity $e(f^{n*}\\mf{a}_1; \\cdots; f^{n*}\\mf{a}_k; \\mf{a}_{k+1}; \\cdots; \\mf{a}_d)$, where $f^{n*}\\mf{a}_i$ is the ideal $(\\psi_1^{(i)}\\circ f^n, \\ldots, \\psi_{m_i}^{(i)}\\circ f^n)$. We point out that $f^{n*}\\mf{a}_i$ is an $\\mf{m}$-primary ideal by our assumption that $f$ is finite-to-one near $0$. Let $U = \\bigcap_n U_n$. For $(a,b)\\in U$, this proves that $\\mu(n) = e(f^{n*}\\mf{a}_1; \\cdots; f^{n*}\\mf{a}_k; \\mf{a}_{k+1}; \\cdots; \\mf{a}_d)$ for all $n$.\n\nThe problem is now reduced to showing that the sequence $e(f^{n*}\\mf{a}_1; \\cdots; f^{n*}\\mf{a}_k; \\mf{a}_{k+1}; \\cdots; \\mf{a}_d)$ grows subexponentially. Again using \\hyperref[teissier1]{Theorem~\\ref*{teissier1}}, we see \\[\ne(f^{n*}\\mf{a}_1; \\cdots; f^{n*}\\mf{a}_k; \\mf{a}_{k+1}; \\cdots; \\mf{a}_d)\\leq e(f^{n*}\\mf{a}_1)^{1\/d}\\cdots e(f^{n*}\\mf{a}_k)^{1\/d}e(\\mf{a}_{k+1})^{1\/d}\\cdots e(\\mf{a}_d)^{1\/d},\\] so it suffices to show that $e(f^{n*}\\mf{a}_i)$ grows subexponentially. Let $r\\geq 1$ be an integer such that $\\mf{m}^r\\subseteq \\mf{a}_i$ for each $i$, and let $s\\geq 1$ be an integer such that $\\mf{m}^s\\subseteq f^*\\mf{m}$. Then one has inclusions\\[\nf^{n*}\\mf{a}_i\\supseteq f^{n*}\\mf{m}^r\\supseteq f^{(n-1)*}\\mf{m}^{sr}\\supseteq f^{(n-2)*}\\mf{m}^{s^2r}\\supseteq\\cdots\\supseteq \\mf{m}^{s^nr}.\\] It follows that \\[\ne(f^{n*}\\mf{a}_i) \\leq e(\\mf{m}^{s^nr}) = (s^nr)^d e(\\mf{m}) = r^ds^{dn},\\] which grows subexponentially, completing the proof. \n\\end{proof}\n\n\\begin{rem} With very little extra work, one can show that the exponential growth rate of the sequence $e(f^{n*}\\mf{a}_i)^{1\/d}$ can be bounded above in the following manner:\\[ \n\\lim_{n\\to \\infty} \\frac{1}{n} \\log e(f^{n*}\\mf{a}_i)^{1\/d} \\leq \\lim_{n\\to \\infty}\\frac{1}{n} \\log\\min\\{s\\geq 1 : \\mf{m}^s\\subseteq f^{n*}\\mf{m}\\}.\\] We mention this because quantities such as that on the right hand side have been recently studied by Majidi-Zolbanin, Miasnikov, and Szpiro \\cite{MZMS}. In their notation, the right hand side of this inequality is exactly $w_h(f)$.\n\\end{rem}\n\n\n\n\n\\section{Valuative dynamics and the proof of Theorem C}\n\nIn this final section, we will prove \\hyperref[thmD]{Theorem~\\ref*{thmD}} using techniques from valuation theory and dynamics on valuation spaces. Let us begin by recalling the setup of the theorem. We fix a superattracting holomorphic fixed point germ $f\\colon (\\C^2,0)\\to (\\C^2,0)$, which we assume to be finite-to-one near $0$. Let $\\mf{a} = (\\psi_1,\\ldots, \\psi_m)$ be an $\\mf{m}$-primary ideal in the formal power series ring $\\C\\llbracket x,y\\rrbracket$. For any $z\\in \\C^m$, we set $D_z = \\{z_1\\psi_1 + \\cdots + z_m\\psi_m = 0\\}$. We aim to show that there is a dense subset $U\\subseteq \\C^m\\times \\C^m$, given as the complement of a countable union of algebraic subsets, such that for all $(z,w)\\in U$, the sequence $\\mu(n) := D_z\\cdot f^n_*D_w$ eventually satisfies an integral linear recursion relation. We have already seen in \\S4 that we can find such a set $U$ for which one has $\\mu(n) = e(f^{n*}\\mf{a}; \\mf{a})$ for all $(z,w)\\in U$ and all $n\\geq 1$. Our starting point in this section is the following alternate characterization of mixed multiplicities, which can be found in \\cite[\\S1.6.8]{MR2095471} and \\cite{MR0354663}.\n\n\\begin{thm}\\label{teissier2} Let $\\mf{b}_1$ and $\\mf{b}_2$ be two $\\mf{m}$-primary ideals of $\\C\\llbracket x,y\\rrbracket$. Let $\\pi\\colon X\\to (\\C^2,0)$ be a modification over $0$ which dominates the normalized blowup of each of the ideals $\\mf{b}_i$. Then there exist divisors $Z_1$ and $Z_2$ of $X$, both supported within the exceptional locus $\\pi^{-1}(0)$ of $\\pi$, for which $\\mf{b}_i\\cdot \\mc{O}_X = \\mc{O}_X(Z_i)$. The mixed multiplicity $e(\\mf{b}_1; \\mf{b}_2)$ is given by the intersection number $-(Z_1\\cdot Z_2)$. Finally, the divisors $Z_i$ are \\emph{relatively nef}, which is to say that for every irreducible component $E$ of $\\pi^{-1}(0)$, one has $Z_i\\cdot E \\geq 0$.\n\\end{thm}\n\nHere, and for the rest of the article, a \\emph{modification} $\\pi\\colon X\\to (\\C^2,0)$ over $0$ is defined to be a proper birational morphism $\\pi\\colon X\\to \\C^2$ from a normal variety $X$ to $\\C^2$ that is an isomorphism over $\\C^2\\smallsetminus \\{0\\}$. Such a modification will be called a \\emph{blowup} if it is obtained as a composition of point blowups. If $\\pi\\colon X\\to (\\C^2,0)$ is any modification, and $E_1,\\ldots, E_r$ denote the irreducible components of the exceptional locus $\\pi^{-1}(0)$, we define $\\Div(\\pi)$ to be the vector space of $\\R$-divisors $\\Div(\\pi) = \\bigoplus_{i=1}^r \\R E_i$. The intersection pairing on $\\Div(\\pi)$ is nondegenerate by the Hodge index theorem \\cite[Theorem V.1.9]{MR0463157}, and thus there is a dual basis $\\check{E}_1,\\ldots, \\check{E}_r\\in \\Div(\\pi)$, i.e., a basis satisfying the relation $\\check{E}_i\\cdot E_j = \\delta_{ij}$.\n\nBefore beginning the proof of \\hyperref[thmD]{Theorem~\\ref*{thmD}}, let us give a (very) brief overview of the \\emph{valuative tree} $\\mc{V}$ at the origin $0\\in \\C^2$ of Favre-Jonsson, which is the relevant valuation space for us. A full-on introduction to the valuative tree would take us too far afield here; detailed references can be found in \\cite{MR2097722, Mattias}, and more concise introductions can be found in \\cite{MR2339287, GR}. \n\n\n\nThe valuative tree $\\mc{V}$ at $0\\in \\C^2$ is defined as the set of all semivaluations $\\nu\\colon \\C\\llbracket x,y\\rrbracket\\to \\R\\cup\\{+\\infty\\}$ with the properties that $\\nu|_{\\C^\\times} \\equiv 0$ and $\\min\\{\\nu(x), \\nu(y)\\} = 1$. For us, the most important example is that of a \\emph{divisorial valuation}. A valuation $\\nu\\in \\mc{V}$ is divisorial if there is a blowup $\\pi\\colon X\\to (\\C^2,0)$, an irreducible component $E$ of the exceptional locus $\\pi^{-1}(0)$, and a constant $\\lambda\\in \\R$ such that $\\nu(P) = \\lambda\\ord_E(P\\circ \\pi)$ for all $P\\in \\C\\llbracket x,y\\rrbracket$. In this case, the constant $\\lambda$ is exactly $\\lambda = b_E^{-1}$, where $b_E = \\min\\{\\ord_E(x\\circ \\pi), \\ord_E(y\\circ \\pi)\\}\\in \\N$. The constant $b_E$ is sometimes called the \\emph{generic multiplicity} of $E$. If $\\nu$ is a divisorial valuation of the above form, we will denote it simply as $\\nu_E$.\n\nSuppose that $\\pi\\colon X\\to (\\C^2,0)$ is a blowup, and $E_1,\\ldots, E_r$ are the irreducible components of $\\pi^{-1}(0)$. Then any divisorial valuation $\\nu\\in \\mc{V}$ defines a linear functional $\\nu\\colon\\Div(\\pi)\\to \\R$; essentially, $\\nu(E_i)$ is the $\\nu$-valuation of a local defining equation of $E_i$ at the \\emph{center} of $\\nu$ in $\\pi$, see \\cite[\\S1.2]{favre:holoselfmapssingratsurf} for details. Since the intersection pairing is nondegenerate, it follows that there is a divisor $Z_{\\nu, \\pi}\\in \\Div(\\pi)$ such that $\\nu(D) = Z_{\\nu, \\pi}\\cdot D$ for all $D\\in \\Div(\\pi)$. If $\\pi'\\colon X'\\to (\\C^2, 0)$ is a blowup dominating $\\pi$, say $\\eta\\colon X'\\to X$ is such that $\\pi' = \\pi\\eta$, then $Z_{\\nu, \\pi'} = \\eta^*Z_{\\nu, \\pi}$ and $Z_{\\nu, \\pi} = \\eta_*Z_{\\nu, \\pi'}$. Finally, if $\\nu = b_{E_i}^{-1}\\ord_{E_i}$ for one of the exceptional components $E_i$ of $\\pi^{-1}(0)$, then one easily checks that $Z_{\\nu, \\pi} = b_{E_i}^{-1}\\check{E}_i$.\n\nThe valuative tree $\\mc{V}$ has a natural topology and a natural poset structure $(\\mc{V}, \\leq)$. With respect to these structures, $\\mc{V}$ is a \\emph{rooted tree} (see \\cite[\\S2]{Mattias} for a precise definition). For any two elements $\\nu_1,\\nu_2\\in \\mc{V}$, there is a unique greatest element $\\nu_1\\wedge \\nu_2$ that is both $\\leq \\nu_1$ and $\\leq \\nu_2$. In addition, there is defined on $\\mc{V}$ an increasing function $\\alpha\\colon \\mc{V}\\to [1,+\\infty]$, called the \\emph{skewness} function, which is finite on divisorial valuations and has the following geometric property: if $\\pi\\colon X\\to (\\C^2,0)$ is a blowup and $E_1$ and $E_2$ are two irreducible components of the exceptional locus $\\pi^{-1}(0)$, then \n\\begin{equation}\\label{Beqn3}\\alpha(\\nu_{E_1}\\wedge \\nu_{E_2}) = -(Z_{\\nu_{E_1}, \\pi}\\cdot Z_{\\nu_{E_2}, \\pi}).\\end{equation}\n\nFinally, we note that $f$ induces in a natural way a dynamical system $f_\\bullet \\colon \\mc{V}\\to \\mc{V}$. Indeed, if $\\nu\\in \\mc{V}$, then we obtain a semivaluation $f_*\\nu$ defined by $(f_*\\nu)(P) = \\nu(P\\circ f)$. In general the value $c(f,\\nu):= \\min\\{(f_*\\nu)(x), (f_*\\nu)(y)\\}$ is greater than $1$, so $f_*\\nu$ is not an element of $\\mc{V}$, but by appropriately normalizing we obtain a semivaluation $f_\\bullet\\nu = c(f,\\nu)^{-1}f_*\\nu\\in \\mc{V}$. The quantity $c(f,\\nu)$ is called the \\emph{attraction rate} of $f$ along $\\nu$, and is the primary object of study in the paper \\cite{GR}, the results of which we will use shortly.\n\n\\begin{proof}[Proof of {\\hyperref[thmD]{Theorem~\\ref*{thmD}}}] We begin the proof by deriving an alternate expression for the mixed multiplicity $e(f^*\\mf{a}; \\mf{a})$ using the valuative language just discussed. Let $\\pi_1\\colon X_1\\to (\\C^2,0)$ and $\\pi_2\\colon X_2\\to (\\C^2,0)$ be the normalized blowups of the ideals $f^*\\mf{a}$ and $\\mf{a}$, respectively, and let $Z_i$ be the divisors on $X_i$ for $i = 1,2$ such that $f^*\\mf{a}\\cdot \\mc{O}_{X_1} = \\mc{O}_{X_1}(Z_1)$ and $\\mf{a}\\cdot \\mc{O}_{X_2} = \\mc{O}_{X_2}(Z_2)$. Let $E_1,\\ldots, E_r$ be the irreducible components of the exceptional locus $\\pi_2^{-1}(0)$ of $\\pi_2$. For each $i = 1,\\ldots, r$, let $a_i\\in \\Z$ be the integer $Z_2\\cdot E_i$, so that $Z_2$ can be written $Z_2 = \\sum_{i=1}^r a_i\\check{E_i}$. By \\hyperref[teissier2]{Theorem~\\ref*{teissier2}}, these integers $a_i$ are nonnegative.\n\nAs a consequence of Hironaka's theorem on resolution of singularities, it is possible to find blowups $\\eta_1\\colon Y_1\\to (\\C^2,0)$ and $\\eta_2\\colon Y_2\\to (\\C^2,0)$ over $0$ with the following properties:\\begin{enumerate}\n\\item[1.] Each $\\eta_i$ is a \\emph{log resolution} of both of the ideals $\\mf{a}$ and $f^*\\mf{a}$, that is to say, the ideals $\\mf{a}\\cdot \\mc{O}_{Y_i}$ and $f^*\\mf{a}\\cdot \\mc{O}_{Y_i}$ are locally principal. In particular, the $\\eta_i$ dominate both $\\pi_1$ and $\\pi_2$, so there exist proper birational morphisms $\\sigma_i\\colon Y_1\\to X_i$ and $\\gamma_i\\colon Y_2\\to X_i$ for $i = 1,2$ such that one has $\\eta_1 = \\pi_i\\sigma_i$ and $\\eta_2 = \\pi_i\\gamma_i$.\n\\item[2.] The map $f$ lifts to a holomorphic map $F \\colon Y_1\\to Y_2$, or in other words, $F = \\eta_2^{-1}f\\eta_1$ has no indeterminacy points.\n\\item[3.] If $\\wt{E}_1,\\ldots, \\wt{E}_r$ denote the strict transforms of the $E_i$ in $Y_1$ under $\\sigma_2$, then $F$ does not contract any of the $\\wt{E}_i$ to a point.\n\\end{enumerate} The ideal $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$ is obtained on the one hand by first pulling back the ideal $\\mf{a}$ by $f$ to get $f^*\\mf{a}$, and then by pulling back $f^*\\mf{a}$ by $\\eta_1$ to get $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$. On the other hand, because $f\\eta_1 = \\eta_2F$, we may also obtain $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$ by first pulling back $\\mf{a}$ by $\\eta_2$ to get $\\mf{a}\\cdot \\mc{O}_{Y_2} = \\mc{O}_{Y_2}(\\gamma_2^*Z_2)$, and then pulling this back by $F$ to get $f^*\\mf{a}\\cdot \\mc{O}_{Y_1} = \\mc{O}_{Y_1}(F^*\\gamma_2^*Z_2)$. Using this and the fact that $\\mf{a}\\cdot \\mc{O}_{Y_1} = \\mc{O}_{Y_1}(\\sigma_2^*Z_2)$, \\hyperref[teissier2]{Theorem~\\ref*{teissier2}} implies that \\[\ne(f^*\\mf{a};\\mf{a}) = -( \\sigma_2^*Z_2\\cdot F^*\\gamma_2^*Z_2) = -(F_*\\sigma_2^*Z_2\\cdot \\gamma_2^*Z_2).\\] Using our previously derived expression $Z_2 = \\sum_{i=1}^r a_i\\check{E}_i$, we can express this as \\begin{equation}\\label{Beqn1}\ne(f^*\\mf{a}; \\mf{a}) = -\\sum_{i,j = 1}^r a_ia_j(F_*\\sigma_2^*\\check{E}_i \\cdot \\gamma_2^*\\check{E}_j) = -\\sum_{i,j=1}^r a_ia_jb_{E_i}b_{E_j}(F_*Z_{\\nu_{E_i},\\eta_1} \\cdot Z_{\\nu_{E_j},\\eta_2}).\\end{equation} Because of our assumption that $F$ does not contract any of the $\\wt{E}_i$ to a point, we may now apply \\cite[Lemma 1.10]{favre:holoselfmapssingratsurf} to conclude that \\begin{align*}\ne(f^*\\mf{a}; \\mf{a}) & = -\\sum_{i,j=1}^r a_ia_jb_{E_i}b_{E_i}c(f, \\nu_{E_i})(Z_{f_\\bullet\\nu_{E_i}, \\eta_2}\\cdot Z_{\\nu_{E_j}, \\eta_2})\\\\\n& = \\sum_{i,j =1}^r a_ia_jb_{E_j}b_{E_j}\\alpha(f_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j})c(f, \\nu_{E_i}).\n\\end{align*} Of course, this identity is equally valid for any iterate of $f$, leading us to our final equation for the local intersection multiplicities $\\mu(n)$. Namely, if $(z,w)\\in U$, then \\begin{equation}\\label{Beqn4}\n\\mu(n) = \\sum_{i,j = 1}^r a_ia_jb_{E_i}b_{E_j}\\alpha(f_\\bullet^n\\nu_{E_i}\\wedge \\nu_{E_j})c(f^n, \\nu_{E_i})\\,\\,\\,\\,\\,\\,\\mbox{ for all $n\\geq 1$}.\\end{equation}\n\nTo prove that $\\mu(n)$ eventually satisfies an integral linear recursion relation, it therefore suffices to show that for each $i$ and $j$, the sequence $\\alpha(f^n_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j})c(f^n, \\nu_{E_i})$ eventually satisfies an integral linear recursion relation. More generally, we will prove the following: if $\\nu\\in \\mc{V}$ is any divisorial valuation, then $\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})c(f^n, \\nu)$ eventually satisfies an integral linear recursion relation. \n\nWe first observe that we may without loss of generality prove this for any iterate $f^p$ of $f$. Indeed, the sequence $\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})c(f^n,\\nu)$ is obtained by joining the sequences \\[\n\\{\\alpha((f^{np}_\\bullet f^k_\\bullet\\nu)\\wedge \\nu_{E_j})c(f^{np}, f^k_\\bullet\\nu)\\}_{n=1}^\\infty\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,k = 0,1,\\ldots, p-1\\] in alternating fashion. If each of these sequences eventually satisfies an integral linear recursion relation, then so does the combined sequence.\n\nImmediately let us replace $f$ by $f^2$. By doing so, we may apply \\cite[Theorem 3.1]{GR} to conclude that there is a fixed point $\\nu_\\star\\in \\mc{V}$ for $f_\\bullet$ such that $f^n_\\bullet\\nu\\to \\nu_\\star$ in a strong sense as $n\\to \\infty$. In particular, $\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})\\to \\alpha(\\nu_\\star\\wedge \\nu_{E_j})<+\\infty$. If the sequence $\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})$ is eventually constant, then we are done by \\cite[Theorem 6.1]{GR}, which says that $c(f^n,\\nu)$ eventually satisfies an integral linear recursion relation.\n\n\n\nWe may assume, therefore, that the sequence $\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})$ is not eventually constant; this implies, in particular, that $f^n_\\bullet \\nu \\leq \\nu_{E_j}$ for infinitely many $n$. Such a condition imposes strong restrictions on the possible asymptotic behavior of the sequence $f^n_\\bullet\\nu$. Indeed, the work done in \\cite{GR} shows that in this case, $\\nu_\\star\\leq \\nu_{E_j}$, and $f^n_\\bullet\\nu\\to \\nu_\\star$ along a periodic cycle of tangent directions $\\vec{v}_1,\\ldots, \\vec{v}_p$ at $\\nu_\\star$ (see \\cite[\\S2]{Mattias} for the notion of tangent directions at a valuation). Replacing $f$ by $f^p$, we can assume without loss of generality that $f^n_\\bullet \\nu\\to \\nu_\\star$ along a fixed tangent direction $\\vec{v}$.\n\n\n\nIn this situation, it is a rather non-trivial fact (see \\cite[\\S5.2]{MR2339287}) that one can find a blowup $\\pi\\colon X\\to (\\C^2,0)$ and irreducible components $V, W$ of $\\pi^{-1}(0)$ which intersect transversely such that the following hold: \\begin{enumerate}\n\\item[1.] The interval $I:= [\\nu_{V}, \\nu_{W}]\\subset \\mc{V}$ is invariant for $f_\\bullet$, that is, $f_\\bullet (I)\\subseteq I$.\n\\item[2.] The interval $I$ contains $\\nu_\\star$ and intersects the tangent direction $\\vec{v}$.\n\\item[3.] The divisorial valuation $\\nu_{E_j}$ corresponds to an irreducible component of $\\pi^{-1}(0)$.\n\\end{enumerate} We now proceed similarly to the proof of \\cite[Lemma 6.2]{GR}. For any valuation $\\lambda\\in I$ and any $n\\geq 1$, one has that $Z_{f^{np}_* \\lambda, \\pi} = r_n\\check{V} + s_n\\check{W}$ for some constants $r_n, s_n\\geq 1$, and so \\[\nc(f^{np}, \\lambda)\\alpha(f^{np}_\\bullet \\lambda\\wedge \\nu_{E_j}) = -(Z_{f^{np}_*\\lambda, \\pi}\\cdot Z_{\\nu_{E_j}, \\pi}) = -r_n(\\check{V}\\cdot Z_{\\nu_{E_j}, \\pi}) - s_n(\\check{W}\\cdot Z_{\\nu_{E_j}, \\pi}).\\] Just as in the proof of \\cite[Lemma 6.2]{GR}, there is a $2\\times 2$ integer matrix $M$ for which one has the identity $(r_n, s_n) = (r_{n-1}, s_{n-1})M$. We conclude that $c(f^{np}, \\lambda)\\alpha(f^{np}_\\bullet\\lambda\\wedge \\nu_{E_j})$ satisfies the integral linear recursion relation with characteristic polynomial $t^2 - \\mathrm{tr}(M)t + \\det(M)$. If $N\\geq 1$ is large enough that $f^N_\\bullet\\nu\\in I$, this proves that the sequence $\\{(\\alpha(f^n_\\bullet\\nu\\wedge \\nu_{E_j})c(f^n, \\nu)\\}_{n = N}^\\infty$ satisfies an integral linear recursion relation, completing the proof.\n\\end{proof}\n\n\\begin{rem} One consequence of the study of the sequences $c(f^n, \\nu)$ in \\cite{GR} is that for any divisorial valuation $\\nu$, there is a constant $B = B(\\nu)$ such that $c(f^n,\\nu)\\sim Bc_\\infty^n$, where $c_\\infty>1$ is the \\emph{asymptotic attraction rate} of $f$, that is, \\[\nc_\\infty := \\lim_{n\\to \\infty}\\left(\\max\\{s\\geq 1 : f^{n*}\\mf{m}\\subseteq \\mf{m}^s\\}\\right)^{1\/n}.\\] Since one has $\\alpha(f^n_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j}) \\leq \\alpha(\\nu_{E_j})<+\\infty$ for all $n$, we can conclude from equation (\\ref{Beqn4}) that there exist constants $A_1, A_2>0$ such that $A_1c_\\infty^n\\leq \\mu(n)\\leq A_2c_\\infty^n$ for all $n$. Moreover, as we saw in the proof of \\hyperref[thmD]{Theorem~\\ref*{thmD}}, if we replace $f$ by $f^2$, then $f^n_\\bullet \\nu_{E_i}\\to \\nu_\\star$ for some $\\nu_\\star\\in \\mc{V}$, and thus $\\alpha(f^n_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j})\\to \\alpha(\\nu_\\star\\wedge\\nu_{E_j})<+\\infty$. Therefore in this case equation (\\ref{Beqn4}) implies that $\\mu(n)\\sim Ac_\\infty^n$ for some constant $A>0$.\n\\end{rem}\n\n\\begin{rem} For any irreducible curve germ $C$ through the origin in $\\C^2$, there is a corresponding \\emph{curve valuation} $\\nu_C\\in \\mc{V}$, and one has $f_\\bullet \\nu_C = \\nu_{f(C)}$. Thus the dynamics of $f$ on curves $C$ through the origin is reflected in the dynamics of $f_\\bullet$ on curve valuations $\\nu_C$. With this in mind, it should not come as a surprise that one can study Arnold's conjecture by examining the dynamics of $f_\\bullet$ on $\\mc{V}$. In brief, the outline of our proof of \\hyperref[thmD]{Theorem~\\ref*{thmD}} can be expressed as follows: for ``general\" enough curves $C$, the dynamics of $f_\\bullet$ on the curve valuations $\\nu_C$ is reflected in the dynamics of $f_\\bullet$ on certain associated divisorial valuations. The dynamics of $f_\\bullet$ on divisorial valuations is very regular: \\cite[Theorem 3.1]{GR} says that there is a set $K$ of fixed valuations of $f_\\bullet$ that attract all $\\nu\\in \\mc{V}$ with $\\alpha(\\nu)<+\\infty$, which includes all divisorial valuations.\n\nTo emphasize this point further, when $f(x,y) = (x^2-y^4, y^4)$ is the example studied in \\S\\S1-3, the attracting set $K\\subset \\mc{V}$ consists of a single point $\\nu_\\star\\in \\mc{V}$, and the points $\\nu\\in \\mc{V}$ that are \\emph{not} attracted to $\\nu_\\star$ are exactly the curve valuations $\\nu_{C_s}$ associated to the curves $C_s$ constructed in \\S1. That is, the curve valuations $\\nu_{C_s}$ are exactly the points of $\\mc{V}$ where the dynamics of $f_\\bullet$ is \\emph{not} regular.\n\nIn the case when $f\\colon (\\C^2,0)\\to (\\C^2,0)$ is a finite germ such that the derivative $f'(0)$ has exactly one nonzero eigenvalue, M.\\ Ruggiero \\cite{MR2904007} has studied the dynamics of $f_\\bullet\\colon \\mc{V}\\to \\mc{V}$, and found that there exist fixed curve valuations $\\nu_1,\\nu_2\\in \\mc{V}$ such that $f^n_\\bullet\\nu\\to \\nu_1$ as $n\\to \\infty$ for all $\\nu\\in \\mc{V}\\smallsetminus\\{\\nu_2\\}$. From this one can conclude that the local intersection multiplicities $\\mu(n) = C\\cdot f^n(D)$ grow at most exponentially fast for all curve germs $C,D$ through $0$, provided these numbers are always finite, confirming \\cite[Theorem 4]{MR1215971}.\n\\end{rem}\n\n\n\n\n\\begin{comment}\n\n\\section{D}\n\nIn this final section, we will prove \\hyperref[thmB]{Theorem~\\ref*{thmB}}. Let us start by recalling the setup of the theorem. Let $f\\colon (\\C^2, 0)\\to (\\C^2,0)$ be a holomorphic superattracting fixed point germ at the origin $0\\in \\C^2$, which we assume to be finite-to-one near $0$. Let $\\psi_1,\\ldots, \\psi_m$ be germs of holomorphic functions at $0$ such that $\\{\\psi_1 = 0\\}\\cap \\cdots\\cap \\{\\psi_m = 0\\} = \\{0\\}$, or, in other words, such that the ideal $\\mf{a} = (\\psi_1,\\ldots, \\psi_m)\\subseteq \\C\\llbracket x, y\\rrbracket$ is primary for the maximal ideal $\\mf{m}$ of $\\C\\llbracket x,y\\rrbracket$. We will consider the holomorphic curve germs $D_w$ through the origin of the form $D_w := \\{w_1\\psi_1 + \\cdots + w_m\\psi_m = 0\\}$, where $w\\in \\C^m$. We aim to show that if $(z,w)\\in \\C^m\\times \\C^m$ is chosen outside of a certain countable union of proper algebraic subsets, then the sequence $\\mu(n) := D_z\\cdot f^n(D_w)$ of local intersection multiplicities grows at most exponentially in $n$.\n\nBefore beginning the proof, we first establish some notation. Suppose $\\pi\\colon X\\to (\\C^2,0)$ is a \\emph{modification}, by which we mean a proper birational morphism $\\pi$ from a normal variety $X$ to $\\C^2$ that is an isomorphism over $\\C^2\\smallsetminus\\{0\\}$. If the \\emph{exceptional locus} $\\pi^{-1}(0)$ of $\\pi$ has irreducible components $E_1,\\ldots, E_r$, then the Hodge index theorem \\cite[Theorem V.1.9]{MR0463157} gives that the intersection product restricted to the vector space $\\bigoplus_i \\R E_i$ is negative definite. In particular, there exists a dual basis $\\check{E}_1,\\ldots, \\check{E}_r\\in \\bigoplus_i \\R E_i$, that is, a basis satisfying the relation $\\check{E}_i\\cdot \\check{E}_j = \\delta_{ij}$.\n\nThe proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}} will use some basic facts from the theory of \\emph{mixed multiplicities}, developed by B.\\ Teissier in the 70s [\\color{red}CITES\\color{black}]. A concise introduction to the topic that suffices for our purposes can be found in [\\color{red}CITES, \\S1.6.8\\color{black}]. We collect the relevant results for us in the following theorem.\n\n\\begin{thm}[Teissier]\\label{teissier} Let $\\mf{b}_1 = (g_1,\\ldots, g_r)$ and $\\mf{b}_2 = (h_1,\\ldots, h_s)$ be two $\\mf{m}$-primary ideals of the ring $\\C\\llbracket x,y\\rrbracket$. Then there is an integer $e(\\mf{b}_1; \\mf{b}_2)\\in \\N$, called the \\emph{mixed multiplicity} of $\\mf{b}_1$ and $\\mf{b}_2$, which can be computed in the following two ways:\\begin{enumerate}\n\\item[$1.$] Let $A_\\alpha$ be the curve $A_\\alpha = \\{\\alpha_1g_1 + \\cdots + \\alpha_rg_r = 0\\}$ for $\\alpha\\in \\C^r$ and $B_\\beta$ be the curve $B_\\beta = \\{\\beta_1h_1 + \\cdots + \\beta_sh_s = 0\\}$ for $\\beta\\in \\C^s$. Then we have for $(\\alpha,\\beta)\\in \\C^r\\times \\C^s$ outside of a certain proper algebraic subset that the mixed multiplicity $e(\\mf{b}_1; \\mf{b}_2)$ is exactly the local intersection number $A_\\alpha\\cdot B_\\beta$ at the origin.\n\\item[$2.$] Let $\\pi\\colon X\\to (\\C^2,0)$ be any modification that dominates the blowups of the ideals $\\mf{b}_1$ and $\\mf{b}_2$. Then $\\mf{b}_1\\cdot\\mc{O}_X = \\mc{O}_X(Z_1)$ and $\\mf{b}_2\\cdot\\mc{O}_X = \\mc{O}_X(Z_2)$ for some divisors $Z_1$ and $Z_2$ supported on the exceptional locus $\\pi^{-1}(0)$ of $\\pi$, and the mixed multiplicity $e(\\mf{b}_1; \\mf{b}_2)$ is given by the intersection number $-(Z_1\\cdot Z_2)$.\n\\end{enumerate} Moreover, the divisors $Z_1$ and $Z_2$ in statement $2.$ are \\emph{relatively nef} in the sense that for any irreducible component $E$ of $\\pi^{-1}(0)$, one has $Z_i\\cdot E \\geq 0$ for $i = 1,2$.\n\\end{thm}\n\nWe are now ready to prove \\hyperref[thmB]{Theorem~\\ref*{thmB}}.\n\n\\begin{proof}[Proof of {\\hyperref[thmB]{Theorem~\\ref*{thmB}}}] Let $z,w\\in \\C^m$. We begin by estimating the local intersection number $D_z\\cdot f(D_w)$ at the origin. Certainly one has the inequality $D_z\\cdot f(D_w) \\leq D_z\\cdot f_*D_w = f^*D_z\\cdot D_w$, so we will proceed by computing $f^*D_z\\cdot D_w$. Observe that $f^*D_z$ is the curve \\[\nf^*D_z = \\{z_1(\\psi_1\\circ f) + \\cdots + z_m(\\psi_m\\circ f) = 0\\}.\\] Applying \\hyperref[teissier]{Theorem~\\ref*{teissier}(1)} to the $\\mf{m}$-primary ideals $\\mf{a}$ and $f^*\\mf{a} = (\\psi_1\\circ f, \\ldots, \\psi_m\\circ f)$, we conclude that for $(z,w)\\in \\C^m\\times \\C^m$ chosen outside of a certain proper algebraic subset, the local intersection number $f^*D_z\\cdot D_w$ is exactly the mixed multiplicity $e(\\mf{a}; f^*\\mf{a})$.\n\nLet $\\pi_1\\colon X_1\\to (\\C^2,0)$ and $\\pi_2\\colon X_2\\to (\\C^2,0)$ be the normalized blowups of the ideals $f^*\\mf{a}$ and $\\mf{a}$, respectively, and let $Z_i$ be the divisors on $X_i$ for $i = 1,2$ such that $f^*\\mf{a}\\cdot \\mc{O}_{X_1} = \\mc{O}_{X_1}(Z_1)$ and $\\mf{a}\\cdot \\mc{O}_{X_2} = \\mc{O}_{X_2}(Z_2)$. Let $E_1,\\ldots, E_r$ be the irreducible components of the exceptional locus $\\pi_2^{-1}(0)$ of $\\pi_2$. For each $i = 1,\\ldots, r$, let $a_i\\in \\Z$ be the integer $Z_2\\cdot E_i$, so that $Z_2$ can be written $Z_2 = \\sum_{i=1}^r a_i\\check{E_i}$. By \\hyperref[teissier]{Theorem~\\ref*{teissier}}, these integers $a_i$ are nonnegative.\n\nAs a consequence of Hironaka's theorem on resolution of singularities, it is possible to find modifications $\\eta_1\\colon Y_1\\to (\\C^2,0)$ and $\\eta_2\\colon Y_2\\to (\\C^2,0)$ with the following properties:\\begin{enumerate}\n\\item[1.] Each $\\eta_i$ is a \\emph{log resolution} of both of the ideals $\\mf{a}$ and $f^*\\mf{a}$. In other words, each $\\eta_i$ is a composition of point blowups over the origin, and the ideals $\\mf{a}\\cdot \\mc{O}_{Y_i}$ and $f^*\\mf{a}\\cdot \\mc{O}_{Y_i}$ are locally principal. In particular, the $\\eta_i$ dominate both $\\pi_1$ and $\\pi_2$, so there exist proper birational morphisms $\\sigma_i\\colon Y_1\\to X_i$ and $\\gamma_i\\colon Y_2\\to X_i$ for $i = 1,2$ such that one has $\\eta_1 = \\pi_i\\sigma_i$ and $\\eta_2 = \\pi_i\\gamma_i$.\n\\item[2.] The map $f$ lifts to a holomorphic map $F \\colon Y_1\\to Y_2$, or in other words, $F = \\eta_2^{-1}f\\eta_1$ has no indeterminacy points.\n\\item[3.] If $\\wt{E}_1,\\ldots, \\wt{E}_r$ denote the strict transforms of the $E_i$ in $Y_1$ under $\\sigma_2$, then $F$ does not contract any of the $\\wt{E}_i$ to a point.\n\\end{enumerate} We will now use these modifications and \\hyperref[teissier]{Theorem~\\ref*{teissier}(2)} to obtain a convenient formula for the mixed multiplicity $e(\\mf{a}; f^*\\mf{a})$. \n\nThe ideal $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$ is obtained on the one hand by first pulling back the ideal $\\mf{a}$ by $f$ to get $f^*\\mf{a}$, and then by pulling back $f^*\\mf{a}$ by $\\eta_1$ to get $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$. On the other hand, because $f\\eta_1 = \\eta_2F$, we may also obtain $f^*\\mf{a}\\cdot \\mc{O}_{Y_1}$ by first pulling back $\\mf{a}$ by $\\eta_2$ to get $\\mf{a}\\cdot \\mc{O}_{Y_2} = \\mc{O}_{Y_2}(\\gamma_2^*Z_2)$, and then pulling this back by $F$ to get $f^*\\mf{a}\\cdot \\mc{O}_{Y_1} = \\mc{O}_{Y_1}(F^*\\gamma_2^*Z_2)$. Since $\\mf{a}\\cdot \\mc{O}_{Y_1} = \\mc{O}_{Y_1}(\\sigma_2^*Z_2)$, we conclude that the mixed multiplicity $e(\\mf{a};f^*\\mf{a})$ is exactly the intersection number \\[\ne(\\mf{a};f^*\\mf{a}) = -( \\sigma_2^*Z_2\\cdot F^*\\gamma_2^*Z_2) = -(F_*\\sigma_2^*Z_2\\cdot \\gamma_2^*Z_2).\\] Using our previously derived expression $Z_2 = \\sum_{i=1}^r a_i\\check{E}_i$, we can express this as \\begin{equation}\\label{Beqn1}\ne(\\mf{a}; f^*\\mf{a}) = -\\sum_{i,j = 1}^r a_ia_j(F_*\\sigma_2^*\\check{E}_i \\cdot \\gamma_2^*\\check{E}_j).\\end{equation} \n\nLet $G_1,\\ldots, G_s\\subset Y_2$ be the irreducible components of the exceptional locus $\\eta_2^{-1}(0)$ of $\\eta_2$. Recall that $F$ does not contract any of the $\\wt{E}_i$ to a point, and thus for each $i=1,\\ldots, r$, there is an index $\\theta(i)\\in \\{1,\\ldots, s\\}$ such that $F$ maps $\\wt{E}_i$ onto $G_{\\theta(i)}$. If $k\\in \\{1,\\ldots, s\\}$ is any index, then the coefficient of $\\wt{E}_i$ in the divisor $F^*G_k$ is $0$ if $k\\neq \\theta(i)$, and is some positive integer $\\lambda_i$ if $k = \\theta(i)$. It follows that \\[\nF_*\\sigma_2^*\\check{E}_i\\cdot G_k = \\check{E}_i\\cdot \\sigma_{2*}F^*G_k = \\begin{cases} \\lambda_i & k = \\theta(i).\\\\ 0 & k\\neq \\theta(i).\\end{cases}\\] In other words, $F_*\\sigma_2^*\\check{E}_i = \\lambda_i\\check{G}_{\\theta(i)}$. Putting this into (\\ref{Beqn1}) then yields \\begin{equation}\\label{Beqn2} e(\\mf{a}; f^*\\mf{a}) = -\\sum_{i,j=1}^r a_ia_j\\lambda_i (\\check{G}_{\\theta(i)}\\cdot \\gamma_2^*\\check{E}_j).\n\\end{equation} \n\nExpressions such as (\\ref{Beqn2}) are common in the study of dynamics on valuation spaces [\\color{red}CITES\\color{black}], and in order to proceed we will translate (\\ref{Beqn2}) into this language. The relevant valuation space here is the \\emph{valuative tree} $\\mc{V}$ at $0\\in \\C^2$ of Favre-Jonsson. A full-on introduction to the valuative tree would take us too far afield here; detailed references can be found in [\\color{red}CITES\\color{black}], and more concise introductions can be found in [\\color{red}CITES\\color{black}]. For us, the following discussion should suffice.\n\nThe valuative tree $\\mc{V}$ at the origin $0\\in \\C^2$ is defined as the set of all semivaluations $\\nu\\colon \\C\\llbracket x,y\\rrbracket\\to \\R\\cup\\{+\\infty\\}$ with the properties that $\\nu|_{\\C^\\times} \\equiv 0$ and $\\min\\{\\nu(x), \\nu(y)\\} = 1$. For us, the most important example of is that of a \\emph{divisorial valuation}. A valuation $\\nu\\in \\mc{V}$ is divisorial if there is a composition of point blowups $\\pi\\colon X\\to (\\C^2,0)$ over the origin, an irreducible component $E$ of the exceptional locus $\\pi^{-1}(0)$, and a constant $\\lambda\\in \\R$ such that $\\nu(P) = \\lambda\\ord_E(P\\circ \\pi)$ for all $P\\in \\C\\llbracket x,y\\rrbracket$. In this case, the constant $\\lambda$ is exactly $\\lambda = b_E^{-1}$, where $b_E = \\min\\{\\ord_E(x\\circ \\pi), \\ord_E(y\\circ \\pi)\\}\\in \\N$. The constant $b_E$ is sometimes called the \\emph{generic multiplicity} of $E$. If $\\nu$ is a divisorial valuation of the above form, we will denote it simply as $\\nu_E$.\n\nThe valuative tree $\\mc{V}$ has a natural topology and a natural poset structure $(\\mc{V}, \\leq)$. With respect to the poset structure, $\\mc{V}$ is a \\emph{rooted tree} (see [\\color{red}CITE\\color{black}]) for a precise definition. For any two elements $\\nu_1,\\nu_2\\in \\mc{V}$, there is a unique greatest element $\\nu_1\\wedge \\nu_2$ that is both $\\leq \\nu_1$ and $\\leq \\nu_2$. In addition, there is defined on $\\mc{V}$ an increasing function $\\alpha\\colon \\mc{V}\\to [1,+\\infty]$, called the \\emph{skewness} function, which is finite on divisorial valuations and has the following geometric property: if $\\pi\\colon X\\to (\\C^2,0)$ is a composition of point blowups over the origin and $E_1$ and $E_2$ are two irreducible components of the exceptional locus $\\pi^{-1}(0)$, then \n\\begin{equation}\\label{Beqn3}b_{E_1}b_{E_2}\\alpha(\\nu_{E_1}\\wedge \\nu_{E_2}) = -(\\check{E}_1\\cdot \\check{E}_2).\\end{equation}\n\nFinally, we note that $f$ induces in a natural way a dynamical system $f_\\bullet \\colon \\mc{V}\\to \\mc{V}$. Indeed, if $\\nu\\in \\mc{V}$, then we obtain a semivaluation $f_*\\nu$ defined by $(f_*\\nu)(P) = \\nu(P\\circ f)$. In general the value $c(f,\\nu):= \\min\\{(f_*\\nu)(x), (f_*\\nu)(y)\\}$ is greater than $1$, so $f_*\\nu$ is not an element of $\\mc{V}$, but by appropriately normalizing we obtain a semivaluation $f_\\bullet\\nu = c(f,\\nu)^{-1}f_*\\nu\\in \\mc{V}$. The quantity $c(f,\\nu)$ is called the \\emph{attraction rate} of $f$ along $\\nu$, and is the primary object of study in the paper [\\color{red}CITE\\color{black}], the results of which we will use shortly.\n\nWe now continue with the proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}}. The divisors $E_1,\\ldots, E_r$ and $G_1,\\ldots, G_s$ define divisorial valuations $\\nu_{E_1},\\ldots, \\nu_{E_r}$ and $\\nu_{G_1},\\ldots, \\nu_{G_s}$ in $\\mc{V}$. Moreover, the fact that $F$ maps $\\wt{E}_i$ onto $G_{\\theta(i)}$ implies that $f_\\bullet \\nu_{E_i} = \\nu_{G_{\\theta(i)}}$. Using (\\ref{Beqn3}), we may then rewrite (\\ref{Beqn2}) as \\[\ne(\\mf{a};f^*\\mf{a}) = \\sum_{i,j=1}^r a_ia_j\\lambda_ib_{G_{\\theta(i)}}b_{E_j}\\alpha(f_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j}).\\] By [\\color{red}CITE, Proposition 2.5\\color{black}], we have the equality $\\lambda_ib_{G_{\\theta(i)}} = c(f, \\nu_{E_i})b_{E_i}$, and thus \\begin{equation}\\label{Beqn4}\ne(\\mf{a}; f^*\\mf{a}) = \\sum_{i,j = 1}^r a_ia_jb_{E_i}b_{E_j}\\alpha(f_\\bullet\\nu_{E_i}\\wedge \\nu_{E_j})c(f, \\nu_{E_i}).\\end{equation} In this expression, the quantities $a_i$, $a_j$, $b_{E_i}$, and $b_{E_j}$ depend only on the ideal $\\mf{a}$, and not on the map $f$. While $\\alpha(f_\\bullet \\nu_{E_i}\\wedge \\nu_{E_j})$ does depend on $f$, it is bounded above by $\\alpha(\\nu_{E_j})$, which depends only on $E_j$ and not on $f$. Thus there is a positive constant $K = K(\\mf{a})$ such that \\[\ne(\\mf{a}; f^*\\mf{a}) \\leq K\\sum_{j=1}^r c(f, \\nu_{E_j}).\\] Therefore, if $(z,w)\\in \\C^m\\times \\C^m$ is chosen outside of a certain proper algebraic subset, we obtain the inequality \\[\nf^*D_z\\cdot D_w = e(\\mf{a};f^*\\mf{a}) \\leq K\\sum_{j=1}^r c(f,\\nu_{E_j}).\\]\n\nIf we now repeat this argument for each of the iterates $f^n$, we see that for $(z,w)\\in \\C^m\\times \\C^m$ chosen outside of a \\emph{countable} union of proper algebraic subsets that \\[\n\\mu(n) = D_z\\cdot f^n(D_w) \\leq f^{n*}D_z\\cdot D_w = e(\\mf{a}; f^{n*}\\mf{a}) \\leq K\\sum_{j=1}^r c(f^n, \\nu_{E_j}),\\] where again, the constant $K$ does not depend on $n$. To complete the proof, it suffices to know that for any $j = 1,\\ldots, r$, the sequence $c(f^n, \\nu_{E_j})$ grows subexponentially. Indeed, in [\\color{red}CITE, Theorem 6.1\\color{black}], it was shown that this sequence eventually satisfies an integral linear recursion relation, and thus in particular grows subexponentially. This completes the proof.\n\\end{proof}\n\n\\begin{rem} One has more information about the sequence $c(f^n, \\nu_{E_j})$ from [\\color{red}CITE\\color{black}] than simply that it satisfies an integral linear recursion relation. In fact, $c(f^n, \\nu_{E_j})\\sim Ac_\\infty^n$ for some constant $A>0$, where $c_\\infty>1$ is the \\emph{asymptotic attraction rate} of $f$, defined as follows. For each $n\\geq 1$, express the iterate $f^n$ as $f^n(x,y) = g_c(x, y) + g_{c+1}(x,y) + \\cdots$, where each $g_i$ is a homogeneous polynomial of degree $i$, and $g_c\\neq 0$. Then $c = c(f^n)$ is called the \\emph{attraction rate} of $f^n$. The asymptotic attraction rate $c_\\infty$ is defined to be the exponential growth rate $c_\\infty = \\lim_{n\\to \\infty} c(f^n)^{1\/n}$ of the sequence $c(f^n)$. We can therefore conclude that for $(z,w)\\in \\C^m\\times\\C^m$ outside of a certain countable union of proper algebraic subsets that $\\mu(n) = D_z\\cdot f^n(D_w) = O(c_\\infty^n)$.\n\\end{rem}\n\n\\begin{rem} The main tool used in the proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}} was the result of [\\color{red}CITE\\color{black}] that the sequence $c(f^n, \\nu_{E_j})$ eventually satisfies an integral linear recursion relation. But this result is itself a consequence of a theorem [\\color{red}CITE, Theorem 3.1\\color{black}] about the dynamics of $f_\\bullet$ on the valuative tree $\\mc{V}$, which very roughly says the following: there is a subset $A\\subset \\mc{V}$ of fixed points of $f_\\bullet$ which is attracting in the sense that for every $\\nu\\in \\mc{V}$ with $\\alpha(\\nu)<+\\infty$, the sequence $f^n_\\bullet \\nu$ converges to $A$. For the example $f(x,y) = (x^2-y^4, y^4)$ discussed in $\\S\\S1-3$, the attracting set $A$ consists of a single point $\\nu_\\star$ (that is a curve valuation), and the semivaluations $\\nu\\in \\mc{V}$ that are \\emph{not} attracted to $\\nu_\\star$ under the dynamics of $f_\\bullet$ are exactly the curve valuations $\\nu_{C_s}\\in \\mc{V}$ for the curves $C_s$ previously constructed. In this way, we see that the validity of Arnold's conjecture for a given germ $f$ is closely connected to the basin of attraction of the attracting set $A$.\n\\end{rem}\n\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\nBefore beginning the proof, we first establish some notation. Suppose $\\pi\\colon X\\to (\\C^2,0)$ is a \\emph{modification}, by which we mean a proper birational morphism that is an isomorphism over $\\C^2\\smallsetminus\\{0\\}$. If the \\emph{exceptional locus} $\\pi^{-1}(0)$ of $\\pi$ has irreducible components $E_1,\\ldots, E_r$, the Hodge index theorem \\cite[Theorem V.1.9]{MR0463157} gives that the intersection product restricted to the vector space $\\bigoplus_i \\R E_i$ is negative definite. In particular, there exists a dual basis $\\check{E}_1,\\ldots, \\check{E}_r\\in \\bigoplus_i \\R E_i$, that is, a basis satisfying the relation $\\check{E}_i\\cdot \\check{E}_j = \\delta_{ij}$. In the proof of \\hyperref[thmB]{Theorem~\\ref*{thmB}}, we will make use of the following algebro-geometric result of Teissier:\n\n\\begin{thm}[\\color{red}CITE\\color{black}] Let $\\pi\\colon Y\\to (\\C^2,0)$ be the normalized blowup of the ideal $\\mf{a}$, and let $E_1,\\ldots, E_r$ denote the irreducible components of the exceptional locus $\\pi^{-1}(0)$. Then there exist positive integers $a_1,\\ldots, a_r$ such that for all $w\\in \\C^m$ chosen outside of a proper algebraic subset, one has that \\[\n\\pi^*D_w = -\\sum_{i=1}^r a_i\\check{E}_i + \\sum_{i=1}^r\\sum_{j=1}^{a_i} \\wt{D}_w^{ij}\\] as divisors, where the $\\wt{D}^{ij}_w$ are pairwise disjoint irreducible smooth formal curves with the property that each $\\wt{D}^{ij}_w$ intersects $E_i$ transversely, and does not intersect $E_k$ for any $k\\neq i$. Moreover, the divisors $\\sum_{i,j}\\wt{D}_w^{ij}$ have no common point of intersection on the exceptional locus $\\pi^{-1}(0)$ as $w\\in \\C^m$ varies.\n\\end{thm}\n\nWe are now in a position to prove \\hyperref[thmB]{Theorem~\\ref*{thmB}}.\n\n\n\\begin{proof}[Proof of {\\hyperref[thmB]{Theorem~\\ref*{thmB}}}] Let $z,w\\in \\C^m$. We start by estimating the local intersection number $D_z\\cdot f(D_w)$. Certainly one has the inequality $D_z\\cdot f(D_w) \\leq D_z\\cdot f_*D_w = f^*D_z\\cdot D_w$, so we will proceed by computing $f^*D_z\\cdot D_w$.\n\nLet $\\pi_1\\colon X_1\\to (\\C^2,0)$ and $\\pi_2\\colon X_2\\to (\\C^2,0)$ be two smooth modifications dominating $\\pi$, say with $\\eta_i\\colon X_i\\to Y$ satisfying $\\pi_i = \\pi\\circ \\eta_i$. Suppose, moreover, that the $\\pi_i$ are chosen with the following two properties: \\begin{enumerate}\n\\item[(a)] $f$ lifts to a holomorphic map $F\\colon X_1\\to X_2$, and \n\\item[(b)] if $\\wt{E}_i$ denotes the strict transform of $E_i$ in $X_1$ for $i = 1,\\ldots, r$, then $F$ does not contract any of the $\\wt{E}_i$ to a point. \n\\end{enumerate} One can always find such modifications. To compute $f^*D_z\\cdot D_w$, we now note that by the preceding theorem, if $z,w\\in \\C^m$ are chosen outside a suitable proper algebraic subset, then \\begin{align*}\nf^*D_z\\cdot D_w & = \\pi_1^*f^*D_z\\cdot \\pi_1^*D_w = F^*\\pi_2^*D_z\\cdot \\pi_1^*D_w\\\\\n& = F^*\\left(-\\sum_{i=1}^r a_i\\eta_2^*\\check{E}_i + \\sum_{i=1}^r\\sum_{j=1}^{a_i} \\eta_2^*\\wt{D}^{ij}_z\\right)\\cdot \\left(-\\sum_{i=1}^ra_i\\eta_1^*\\check{E}_i + \\sum_{i=1}^r\\sum_{j=1}^{a_i}\\eta_1^*\\wt{D}_w^{ij}\\right),\n\\end{align*} where the $\\eta_2^*\\wt{D}^{ij}_z$ and $\\eta_1^*\\wt{D}^{ij}_w$ are smooth irreducible formal curves intersecting the exceptional locus transversely at exactly one component, namely the component that is the strict transform of $E_i$ in $X_2$ and $X_1$, respectively. Moreover, because the divisors $\\sum_{i,j} \\eta_2^*\\wt{D}_z^{ij}$ and $\\sum_{i,j} \\eta_1^*\\wt{D}^{ij}_w$ have no common point of intersection on the exceptional loci $\\pi_2^{-1}(0)$ and $\\pi_1^{-1}(0)$, respectively, as $z$ and $w$ vary, we can conclude that for $(z,w)\\in \\C^m\\times \\C^m$ chosen outside of a proper algebraic subset, the divisors $F^*\\sum_{i,j}\\eta_2^*\\wt{D}_z^{ij}$ and $\\sum_{i,j}\\eta_1^*\\wt{D}^{ij}_w$ will have disjoint supports; in particular, these two divisors yield $0$ when intersected. For such pairs $(z,w)$, we therefore have\\begin{align*}\nf^*D_z\\cdot D_w & = F^*\\left(\\sum_{i=1}^ra_i\\eta_2^*\\check{E}_i\\right)\\cdot\\sum_{i=1}^r a_i\\eta_1^*\\check{E}_i - F^*\\left(\\sum_{i=1}^ra_i\\eta_2^*\\check{E}_i\\right)\\cdot\\sum_{i=1}^r\\sum_{j=1}^{a_i}\\eta_1^*\\wt{D}_w^{ij}\\\\\n& \\hspace{1 cm} - F^*\\left(\\sum_{i=1}^r\\sum_{j=1}^{a_i}\\eta_2^*\\wt{D}^{ij}_z\\right)\\cdot \\sum_{i=1}^r a_i\\eta_1^*\\check{E}_i\n\\end{align*} If, for simplicity, we let $V = \\sum_i a_i\\check{E}_i$, this can be rewritten \\begin{align*}\nf^*D_z\\cdot D_w & = (F^*\\eta_2^*V)\\cdot(\\eta_1^*V) - (F^*\\eta_2^*V)\\cdot(\\pi_1^*D_w + \\eta_1^*V) - F^*(\\pi_2^*D_z + \\eta_2^*V)\\cdot (\\eta_1^*V)\n\\end{align*} By the projection formula, $(F^*\\eta_2^*V)\\cdot (\\pi_1^*D_w) = 0 = (F^*\\pi_2^*D_z)\\cdot (\\eta_1^*V)$, thus giving \\[\nf^*D_z\\cdot D_w = - (F^*\\eta_2^*V)\\cdot(\\eta_1^*V) = -(\\eta_2^*V)\\cdot(F_*\\eta_1^*V).\\] Quantities such as the right hand side have been studied in the context of dynamics on valuation spaces, see for instance \\color{red}[CITES]\\color{black}. In the language of these works, the right hand side can be expressed as \\[\nf^*D_z\\cdot D_w = \\sum_{i=1}^r\\sum_{j=1}^r a_ia_jb_{E_i}b_{E_j}\\alpha(\\nu_{E_i}\\wedge f_\\bullet\\nu_{E_j})c(f, \\nu_{E_j}).\\] We will not precisely define each of the terms appearing in this sum, but instead simply say the following: \\begin{enumerate}\n\\item[1.] $b_{E_i}$ is sometimes called the \\emph{generic multiplicity} of the divisor $E_i$, and is a positive constant depending on $E_i$, but not on $f$. \\color{red}[CITES]\\color{black}\n\\item[2.] $\\alpha(\\nu_{E_i}\\wedge f_\\bullet\\nu_{E_j})$ denotes the so-called \\emph{skewness} of the \\emph{valuation} $\\nu_{E_i}\\wedge f_\\bullet \\nu_{E_j}$. It is a positive real number that is bounded above by $\\alpha(\\nu_{E_i})$, which itself is a positive constant depending on $E_i$, but not on $f$. \\color{red}[CITES]\\color{black}\n\\item [3.] $c(f,\\nu_{E_j})$, sometimes called the \\emph{attraction rate} of $f$ along $E_j$, will be the most relevant term for us here. It is positive, and depends on both $E_j$ and $f$. \\color{red}[CITES]\\color{black}\n\\end{enumerate} We have therefore derived that there is a constant $K>0$ depending only on $\\mf{a}$ such that for $(z,w)\\in \\C^m\\times \\C^m$ chosen in a suitable nonempty Zariski open set, \\[\nf^*D_z\\cdot D_w \\leq K\\sum_{j=1}^r c(f,\\nu_{E_j}).\\] This reasoning applies equally to any iterate $f^n$ of $f$, and therefore for $(z,w)\\in \\C^m\\times \\C^m$ chosen away from a suitable \\emph{countable} union of proper algebraic subsets, \\[\n\\mu(n) = D_z\\cdot f^n(D_w) \\leq f^{n*}D_z\\cdot D_w \\leq K\\sum_{j=1}^r c(f^n, \\nu_{E_j})\\,\\,\\,\\,\\,\\mbox{ for all $n$}.\\] To complete the proof, then, it suffices to know that the sequence $c(f^n, \\nu_{E_j})$ grows subexponentially in $n$. Indeed, it was shown in \\color{red}[CITE] \\color{black} that in fact this sequence satisfies an integral linear recursion relation, and hence in particular grows subexponentially, completing the proof.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and motivation}\nPDE models for the propagation of ultrasound waves -- more specifically, high intensity ultrasound propagation (HIUP) -- are relevant to a number of medical and industrial applications.\nTo name but a few, lithotripsy, thermoterapy, (ultrasound) welding, sonochemistry; \ncf., e.g., \\cite{dreyer}. \nThe excitation of induced acoustic fields in order to attain a given task, such as destroying certain `obstacles' (stones in kidneys or deposits resulting from chemical reactions), renders the presence of control functions within the model well-founded. \n \nThe subject of the present investigation is an optimal control problem for a third order in time PDE,\nreferred to in the literature as the Moore-Gibson-Thompson equation, which is the linearization of \nthe Jordan-Moore-Gibson-Thompson (JMGT) equation, arising in the modeling\nof ultrasound waves; see \\cite{jordan,jordan1}, \\cite{kalten-eect}, \\cite{straughan}.\nIn contrast with the renowned Westervelt (\\cite{westervelt}) and Kuznetsov equations, the JMGT equation displays a {\\em finite} speed of propagation of acoustic waves,\nthereby providing a solution to the infinite speed of propagation paradox.\nThis is achieved by replacing the Fourier's law of heat conduction by the Cattaneo law\n(\\cite{cattaneo_1958}); the distinct constitutive law brings about an additional time\nderivative of the acoustic velocity field (or acoustic pressure).\n\nRestricting the analysis to the relevant spatial dimensions $n=2,3$, a Neumann boundary control will be acting as a force on a manifold $\\Gamma_0$ of dimension $n-1$; $\\Gamma_0$ will eventually represent a boundary portion of a bounded domain $\\Omega \\subset \\mathbb{R}^n$.\n(It is an established procedure to reduce the analysis of wave processes on {\\em unbounded} domains to boundary or initial\/boundary value problems (IBVP) on {\\em bounded} domains via the introduction of artificial boundaries.)\nThus, {\\em absorbing} boundary conditions (BC) will be taken on a complementary part of the boundary\n$\\Gamma_1 =\\partial \\Omega \\setminus \\Gamma_0$; see section~\\ref{ss:setting}.\nWe shall assume that the two parts of the boundary do not intersect.\nThe optimal control problem arises from the minimization of the acoustic pressure in $\\Omega$.\nThis setup, which is motivated by significant applications and technologies, has been already adopted\nin the literature in connection with the said nonlinear PDE's; \nsee \\cite{kalten-eect}, \\cite{clason-etal_2009}, \\cite{vania}, \\cite{vania1},\\cite{clason-k} and references therein. \n\nFrom the mathematical point of view, two main challenges appear.\nThe first one is due to the presence of {\\em boundary} controls, which naturally bring about\nunbounded input operators $B$ into the (linear) abstract state equation $y'=Ay+Bg$; see \\cite{bddm}, \\cite{redbook}. \nIt is well known that this issue can be dealt with by exploiting the additional regularity of the\nPDE dynamics: this occurs in the case of parabolic-like dynamics, plainly governed by analytic semigroups $e^{At}$.\nThe reader is referred to the classical texts \\cite{bddm} and \\cite[Vol.~I]{redbook} for a thorough study of the Linear-Quadratic (LQ) problem for parabolic-like PDE's, along with the related differential\nand algebraic Riccati equations.\n\\\\\n(We note that the same is actually valid in the case of PDE problems whose corresponding abstract control systems satisfy the so called {\\em singular estimates} for $e^{At}B$, even if the semigroup \n$e^{At}$ is not analytic (\\cite{las-cbms}).\nAnd, further, appropriate regularity properties can be displayed by certain coupled systems of hyperbolic-parabolic PDE's subject to boundary control -- including thermoelastic systems, acoustic-structure and fluid-structure interactions --, which ensure the solvability of the associated optimal control problems (with quadratic functionals), along with well-posed Riccati equations.\nThe ultimate finite and infinite time horizon theories, as well as references to the motivating\nPDE systems, are found respectively in \\cite{abl-2005} and \\cite{abl_2013}.)\n\nReturning to the PDE under investigation, as we know from \\cite{marchand} and \\cite{kalten-las-mar_2011}, the dynamics of the (uncontrolled) SMGT equation, with classical Dirichlet or Neumann\nBC, is described by a {\\em group} of operators, displaying an intrinsic hyperbolic character, and hence a lack of regularity of its dynamics. \nIn addition, a major challenge is brought about by the presence -- that cannot be eluded -- of the time derivative of the control function $g(t,x)$ within the control system, which becomes\n\\begin{equation} \\label{e:g_t}\ny'=Ay+B_0g+B_1g_t\\,, \n\\end{equation}\nwhereas on the other hand, penalization involves only the $L^2$ (in time) norm of the controls. \nThis means that the cost functional is not coercive with respect to $g_t$. \nThe resulting linear-quadratic problem becomes {\\em singular}.\nIt must be recalled that these features have been already encountered and dealt with\nin the study of optimal boundary control of (second-order in time) wave equations with structural damping; see the former study \\cite{bucci_1992} and the subsequent analysis and solutions proposed in \\cite{LLP} and \\cite{LPT}.\nBecause of the strong damping, in the aforementioned case the free dynamics yields an analytic semigroup, along with an enhanced regularity of the control-to-state map; this feature has been exploited in the studies \\cite{bucci_1992}, \\cite{trig_1994}, \\cite{LLP}, \\cite{LPT}. \nInstead, the present PDE problem is of hyperbolic type.\n\nThe goal of the present paper is to provide a framework for such class of singular control problems, in the case of a hyperbolic-like dynamics which intrinsically does not exhibit \nregularizing effects on its evolution.\nIt is important to emphasize that while the singularity of the control is reflected in difficulties when treating {\\em time} dependence, unbounded inputs affect the analysis of {\\em space} dependence. \nSo, the infinite-dimensional aspect of evolution is at the heart of the problem studied. \nTo the authors' best knowledge this is a first investigation where a singular control \nproblem and the control system \\eqref{e:g_t} appear simultaneously, \nin an infinite dimensional context and with a general semigroup governing the free dynamics. \n\n\\subsection{The nonlinear model and its linearization}\nThe Jordan-Moore-Gibson-Thompson (JMGT) equation is one of the fundamental equations in nonlinear acoustics\nwhich describes wave propagation in viscous thermally relaxing fluids.\nIts linearization is found in the literature as the Moore-Gibson-Thompson (MGT) equation.\n(In recognition of the original work on it by Stokes (\\cite{stokes}), it might rather be termed \nStokes-Moore-Gibson-Thompson equation, as Pedro Jordan himself suggested; \nhence the acronym SMGT (in place of MGT) will be utilized throughout the paper.)\nThe fully nonlinear PDE, that is the JMGT equation, is the following one:\n\\begin{equation}\\label{e:jmgt}\n\\tau \\psi_{ttt} + \\psi_{tt}-c^2\\Delta \\psi - b\\Delta \\psi_t=\n\\frac{\\partial}{\\partial t}\\Big(\\frac1{c^2}\\frac{B}{2A}\\psi^2_t+ |\\nabla \\psi|^2\\Big)\n\\end{equation}\nwhere $\\tau > 0 $ is a time relaxation parameter, the unknown $\\psi=\\psi(t,x)$ is the {\\em acoustic velocity potential}, the space variable $x$ varies in a bounded domain $\\Omega\\subset \\mathbb{R}^n$, $c$ is the speed of sound, parameter $b$ stands for diffusivity, $\\alpha >0 $ is a damping parameter and $A, B$ are suitable nonlinearity constants; then, $-\\nabla \\psi$ is the acoustic {\\em particle velocity}. \n\nWhen $\\tau=0$ the model becomes the Kuznetsov equation, that is \n\\begin{equation}\\label{e:kuzn}\n\\psi_{tt}-c^2\\Delta \\psi - b\\Delta \\psi_t=\n\\frac{\\partial}{\\partial t}\\Big(\\frac1{c^2}\\frac{B}{2A}\\psi^2_t+ |\\nabla \\psi|^2\\Big)\\,,\n\\end{equation} \na (second order in time) quasilinear PDE characterized by an infinite speed of propagation.\nThe positive diffusivity coefficient $b$ provides a regularizing effect on its evolution;\nthe corresponding linearized equation is of parabolic type, as its dynamics is governed by\nan analytic semigroup. \nInstead, as found out in the former works \\cite{kalten-eect} and \\cite{marchand}, in the case\n$\\tau > 0$ the PDE turns into a finite speed of propagation and to a hyperbolic character. \n\nOptimal control problems with quadratic functional for both the Kuznetsov and Westervelt\nequations have been studied first in \\cite{clason-etal_2009} and \\cite{clason-k}; \nsee also \\cite{kalten-eect}. \nThe latter reads as \n\\begin{equation*}\nu_{tt}-c^2\\Delta u - b\\Delta u_t= \\beta \\frac{\\partial^2}{\\partial t^2}\\big(u^2\\big)\n\\end{equation*}\nin terms of the acoustic pressure $u$, where $\\beta > 0$ is a suitable parameter of nonlinearity.\n\\\\\n(The relation $u=\\rho \\psi_t$ between the acoustic pressure and velocity potential \n-- $\\rho(x)$ being the mass density -- allows another formulation of the Kuznetsov equation, \nwith the pressure as the unknown variable.) \nThen, the ultrasound excitation on a certain manifold $\\Gamma_0$ (of dimension~$n-1$)\ncan be represented by means of the Neumann boundary condition \n$\\frac{\\partial u}{\\partial \\nu}=g$ on $\\Gamma_0$, where $g$ is the control function.\nA question which arises is to minimize appropriate cost functionals associated with the controlled PDE. \n\nIn the works \\cite{clason-etal_2009} and \\cite{clason-k} quadratic functionals of tracking type are taken into consideration, such as \n\\begin{equation*}\nJ(g)= \\frac{1}{2}\\int_{\\Omega} |u(T,x)-u^d(x)|^2\\, dx \n+ \\frac{\\alpha}{2}\\int_0^T\\!\\!\\!\\int_{\\Gamma_0} |g|^2 \\,d \\sigma\\,dt\n\\end{equation*}\nand\n\\begin{equation*}\nJ(g)= \\frac{1}{2}\\int_0^T \\!\\!\\!\\int_{\\Omega} |u - u^d|^2\\, dx\\,dt \n+ \\frac{\\alpha}{2}\\int_0^T\\!\\!\\!\\int_{\\Gamma_0} |g|^2 \\,d \\sigma\\,dt\\,,\n\\end{equation*}\nrespectively, where $u^d$ is a given reference pressure;\nthe class of admissible controls $G^{ad}$ is a suitably chosen space \nwhose topology is induced by \n\\begin{equation} \\label{e:smooth-space}\nH^1(0,T; H^{1\/2}(\\Gamma_0)) \\cap H^2(0,T;H^{-1\/2}(\\Gamma_0))\\,.\n\\end{equation}\nA critical role in these studies was played by (i) the assumption that $G^{ad}$ represents a space of smooth controls -- more precisely, differentiable in time and subject to appropriate compatibility conditions (with respect to initial data) --, as well as \n(ii) the control constructed is an open-loop one, rather than a feedback one;\n(iii) the solutions considered are suitably small and the state equation is of parabolic type.\n\\\\ \nFor such class of controls existence, uniqueness of solutions for {\\it small data} (due to quasilinearity) has\nbeen derived (\\cite{kalten-las-pos_2012}, \\cite{kalten-eect}). \nThe optimal control is characterized via the Pontryagin Maximum Principle; see \\cite{clason-etal_2009}. \n\nThe present study, although focused on a simpler linear equation, \ndeparts from the avenues (i)--(iii), guided by two major goals.\nOn one hand, we aim at minimizing a quadratic functional that penalizes controls functions in the $L^2$\n(in time and space) norm, with (state) solutions under consideration not necessarily smooth (in space). \nA set of admissible controls that possess a low regularity is consistent with physical and engineering applications; see, e.g., \\cite{dreyer}. \nIn addition, feedback or closed-loop controls\nare of particular interest. \n\nOn the other hand, as already apparent in the case of the Westervelt equation -- as well\nas in the case of its linearization, that is the strongly damped wave equation (\\cite{bucci_1992}) --, the\nmodeling of boundary control actions naturally brings about the time derivative of the control function,\nwhich is somehow ``hidden'' within the PDE problem. \nThis intrinsic analytical aspect will be made clear later -- once we derive the \ninput-to-state solution formula.\nIf one were to pursue such a study in the case of the JMGT equation, a natural choice would\nbe to begin with the linear dynamics: it is already there where non-smoothness of controls will provide sufficient challenge.\nIn fact, the minimization problem overall $L^2$ controls may not ensure an optimal solution even in the linear case, as already noted in \\cite{LPT}.\nWe shall confirm this finding in the case of the problem under consideration.\n\nThe above suggests that appropriate adjustments in the formulation of the problem and its modeling need to be made.\nWe shall show that by enlarging slightly the class of controls resolves the issue of existence of optimal solution.\nHaving established this, we shall proceed with the optimality analysis and the construction of a\nfeedback control for the PDE which will still display `rough' states. \nHowever, the feedback solution will be shown to generate sufficiently regular outputs which can be used to control the system on-line -- via the solution to a {\\em non-standard} differential Riccati Equation (RE).\nThe well-posedness of these corresponding non-standard Riccati equations provides a contribution of independent interest. \nIn fact, the construction of solutions to the RE requires the extension of the dynamics to extrapolation spaces with very low regularity. \nThis is needed in order to make the dynamics invariant. \n\n\\smallskip\nTo recapitulate: \nthe novel contribution of the present work pertains to optimal feedback control of the acoustic \nSMGT equation; the closed-loop control will be generated by an appropriate non-standard Riccati equation.\n(The non-standard structure is due to the singular nature of the optimization problem.)\nFocus is placed on the linearized version of the model, which already provides significant challenges in terms of the underlying analysis and constitutes a necessary step for a further treatment of nonlinear problems.\nThe expectation\nis that once a solution is given for the optimal feedback control of the linearized dynamics, such control may be used\nfor the nonlinear problem, which then will have to be considered with small initial data.\nA similar approach has been pursued successfully in the case of the Navier-Stokes equations;\ncf. \\cite{barbu}, \\cite{barbu1}. \n\n\n\\subsection{Mathematical setting} \\label{ss:setting}\nWe consider the problem of controlling the acoustic excitation on a certain closed region \n$\\Gamma_0$ while maintaining the acoustic pressure below a certain threshold; $\\Gamma_0$ will be\nsubsequently identified as a part of the boundary of an introduced bounded domain $\\Omega$. \nThen, as usually done in the study of wave propagation phenomena in unbounded spatial domains, an artificial boundary $\\Gamma_1$ is introduced in order to limit the area of observation\/computation.\nThe {\\em absorbing} boundary conditions (BC) on $\\Gamma_1$ are then used to avoid reflections:\nroughly, no waves can `come back'.\nAccordingly, and consistently with the analysis carried out in \\cite{clason-etal_2009} \n(on a classical nonlinear model for ultrasound wave propagation like the Westervelt equation), \nwe will complement the SMGT equation with the BC which are the most pertinent: namely, \n\\begin{itemize}\n\\item\nNeumann {\\em boundary control} acting on\n$\\Gamma_0$ (the so called excited boundary); $g$ below represents a surface force;\n\\item\nabsorbing BC on the complement $\\Gamma_1=\\partial\\Omega\\setminus \\Gamma_0$ \n(the so called {\\em absorbing boundary}).\n\\end{itemize}\nThus, the boundary value problem (BVP) is as follows:\n\\begin{equation} \\label{e:bvp-0}\n\\begin{cases}\n\\tau u_{ttt} + \\alpha u_{tt} -c^2 \\Delta u -b \\Delta u_t =0 & \\textrm{on $(0,T)\\times\\Omega$}\n\\\\\n\\frac{\\partial u}{\\partial \\nu}=g & \\textrm{on $(0,T)\\times\\Gamma_0$}\n\\\\\n\\frac{\\partial u}{\\partial \\nu}+\\frac{1}{c}u_t=0 & \\textrm{on $(0,T)\\times\\Gamma_1$}\n\\end{cases}\n\\end{equation}\nto be supplemented with initial conditions.\n\n\\noindent\nAiming at studying optimal control problems with quadratic functionals associated with the IBVP \\eqref{e:bvp-0}, the following features need to be taken into account: \n\\\\\n(i) {\\em finite} time horizon problems, in the absence of penalization of the final time are the most pertinent ones (e.g., in lithotripsy);\n\\\\\n(ii) with $u$ representing the acoustic pressure, the quantity to be minimized (under the action of the surface force $g$) is\n$\\|u-u^d\\|_{L^2(0,T;L^2(\\Omega))}^2$, where $u^d$ is a reference pressure;\n\\\\\n(iii)\nlonger times (i.e. $T=+\\infty$) might be taken into consideration (e.g., in connection with thermotherapy).\n\nDepending on the applications, different cost functionals may be considered.\nIn what follows we shall focus on the physically significant minimization of the following \ncost functional (of tracking type): \n\\begin{equation} \\label{e:cost-funct}\nJ(g) = \\int_0^T\\!\\!\\!\\int_{\\Omega} |u-u^d|^2 \\,dx\\, dt + \\int_0^T\\!\\!\\! \\int_{\\Gamma_0} |g|^2 \\,d\\sigma\\, dt\\,.\n\\end{equation}\n\n\\begin{remark}\n\\begin{rm}\nThe fact that the functional cost penalizes the control $g$ only in the $L^2$ norm renders the optimization problem a singular one. \nIndeed, if one penalizes also the velocity $g_t$ of the control, then we would obtain a standard boundary control problem with coercive cost functional.\n\\end{rm} \n\\end{remark}\n\nControl problems associated with acoustic equations (Westervelt, Kuznetsov, JMGT ones) have\nbeen recently studied in the literature; see the review paper \\cite{kalten-eect}. \nHowever, the principal difference is that the present minimization involves control functions which belong to $L^2(\\Sigma)$, $\\Sigma := (0,T)\\times \\Gamma_0$, \nrather than more regular -- time-space differentiable -- controls (see \\eqref{e:smooth-space}, \nand the optimal control problems studied in \\cite{clason-etal_2009} and \\cite{vania}). \nIn addition, control laws provided in the past literature were {\\it open loop} controls. \nOur goal is to construct {\\it feedback control} with controls of limited regularity and control gains represented by solutions to Riccati equation. \nThis last aspect is the main trait of our contribution. \nA brief outline-guide to the paper follows below. \n\n\\smallskip\nIn order to state our results and to explain ramifications of the low regularity of the control, \nit is necessary to derive an abstract input-to-state formula of the IBVP problem, within the realm of classical control theory.\nThis means we will seek an explicit representation for the map \n\\begin{equation}\\label{gu}\ng \\longrightarrow (u,u_t,u_{tt})\n\\end{equation}\nThis will be accomplished in the next Section~\\ref{s:semigroup-perspective} by using semigroup theory. \nStarting with uncontrolled dynamics and its representation via generator of a strongly continuous semigroup, we shall then proceed introducing boundary controls into the ``variation of parameters formula'' which will provide an explicit map \\eqref{gu} -- singular and defined on appropriately selected extrapolation spaces, though. \n\nIn the next step we shall formulate control problems associated with the input-state dynamics and we shall discuss existence and non-existence of optimal solutions. \nThe final result pertaining to well-posedness of Riccati equations and to the feedback synthesis \nof the optimal control is presented in Section~\\ref{s:auxiliary-and-riccati}. \nIt is important to notice here that in spite of the singularity of input-state dynamics, the feedback synthesis and the resulting Riccati equations are defined and well-posed on the basic state and control spaces. \nThis is due to the effects of the observation. \n\nThe proofs of the auxiliary and main results are deferred to Sections~\\ref{s:proofs_1} \nand \\ref{s:proofs_2}. \nThe proofs will rely on techniques introduced in the study of the LQ problem for hyperbolic-like equations with unbounded inputs, where the dynamics does not provide beneficial regularizing effects. \nTo handle this issue, we establish appropriate bounds by exploiting structural properties of the observation; see \\cite[Vol.~II]{redbook}.\n\n\n\\section{Input-to-state formulation of the PDE problem} \\label{s:semigroup-perspective}\nA prerequisite step for the understanding of the control-theoretic properties of the \ninitial\/boundary value problem (IBVP)\n\\begin{equation} \\label{e:ibvp-1}\n\\begin{cases}\n\\tau u_{ttt} + \\alpha u_{tt} -c^2 \\Delta u -b \\Delta u_t =0 & \\textrm{on $(0,T)\\times\\Omega$}\n\\\\\n\\frac{\\partial u}{\\partial \\nu}=g & \\textrm{on $(0,T)\\times\\Gamma_0$}\n\\\\\n\\frac{\\partial u}{\\partial \\nu}+\\frac{1}{c} u_t= 0& \\textrm{on $(0,T)\\times\\Gamma_1$}\n\\\\\nu(0,x)=u_0(x)\\,,\\; u_t(0,x)=u_1(x)\\,; u_{tt}(0,x)=u_2(x) & \\textrm{on $\\Omega$}\n\\end{cases}\n\\end{equation}\nfor the SMGT equation is to introduce the corresponding abstract operator model in \nan appropriate function spaces.\n\n\\subsection{Abstract setup. Preliminary analysis}\nIn order to incorporate into the equation the boundary control action on $\\Gamma_0$,\nalong with the absorbing BC on $\\Gamma_1$, we follow a well-established method. \n\nLet ${\\mathcal A}$ be the realization of $-\\Delta$ in $L^2(\\Omega)$ with Neumann BC: namely,\n\\begin{equation*\n{\\mathcal A}=-\\Delta\\,, \\quad \n{\\mathcal D}({\\mathcal A}) =\\Big\\{ f\\in H^2(\\Omega): \\; \\frac{\\partial f}{\\partial \\nu}\\Big|_{\\partial \\Omega}\n=0\\Big\\}\\,.\n\\end{equation*}\nIt is well known that ${\\mathcal A}$ is not boundedly invertible on $L^2(\\Omega)$; it has\nbounded inverse on\n\\begin{equation*}\nL^2_0(\\Omega):=L^2(\\Omega)\/\\ker({\\mathcal A})\n=\\Big\\{f\\in L^2(\\Omega)\\colon \\int_\\Omega f\\,d\\Omega=0\\Big\\}\\,,\n\\end{equation*}\nwhere $\\ker({\\mathcal A})$ is the null space of ${\\mathcal A}$ spanned by the normalized constant functions.\nThen, introduce the Green maps $N_i$, $i=0,1$, which define appropriate harmonic extensions into $\\Omega$ of data defined on $\\partial\\Omega$. \nMore precisely, for $\\varphi\\in L^2(\\Gamma_0)$, $N_i$ will be defined as follows: \n\\begin{equation}\\label{e:N_i}\nN_i\\colon\\varphi\\longmapsto N_i\\varphi=:v \\;\n\\Longleftrightarrow \\;\n\\begin{cases}\n\\Delta v -v =0 & \\textrm{on $\\Omega$}\n\\\\[1mm]\n\\frac{\\partial v}{\\partial \\nu}=\\varphi & \\textrm{on $\\Gamma_i$}\n\\\\[1mm]\n\\frac{\\partial v}{\\partial \\nu}=0 & \\textrm{on $ \\partial \\Omega \\setminus \\Gamma_i$.}\n\\end{cases}\n\\end{equation}\nEither elliptic problem that defines the operator $N_i$ in \\eqref{e:N_i} admits a unique\nsolution $v_i\\in H^{3\/2}(\\Omega)$, for (respective) boundary data $\\varphi\\in L^2(\\Gamma_i)$,\n$i=0,1$.\nThen, by elliptic theory one has for each $i=0,1$ and any positive $\\sigma<3\/4$ \n\\begin{equation} \\label{e:neumann-regularity}\nN_i \\; \\textrm{continuous}\\colon L^2(\\Gamma_i) \\longrightarrow H^{3\/2}(\\Omega)\n\\subset H^{3\/2-2\\sigma}(\\Omega)\\equiv {\\mathcal D}((I+{\\mathcal A})^{3\/4-\\sigma})\\,,\n\\end{equation}\nwith identification of the Sobolev spaces $H^s(\\Omega)$ with the fractional powers of the operator $(I+{\\mathcal A})$, and equivalent norms, that will be especially useful in the sequel. \n\nIf now $N_i^*$ denote the respective adjoint operators of $N_i$, $i=0,1$ -- defined by \n$(N_i\\phi,w)_{L^2(\\Omega)}=(\\phi,N_i^*w)_{L^2(\\Gamma_i)}$ --, it then follows\nfor each $i=0,1$ and any $\\sigma\\in (0,3\/4)$,\n\\begin{equation*}\n(I+{\\mathcal A})^{3\/4-\\sigma}N_i\\in {\\mathcal L}(L^2(\\Gamma_i),L^2(\\Omega))\\,,\nN^*_i(I+{\\mathcal A})^{3\/4-\\sigma}\\in {\\mathcal L}(L^2(\\Omega),L^2(\\Gamma_i))\\,.\n\\end{equation*} \nAs in \\cite{trig_1994}, a computation which utilizes the (second) Green Theorem yields, for $f\\in {\\mathcal D}({\\mathcal A})$, \nthe following fundamental trace results:\n\\begin{equation} \\label{e:trace-result-ni}\nN_i^*({\\mathcal A}+I) f =f|_{\\Gamma_i} \\qquad \\textrm{$i=0,1$.}\n\\end{equation}\n(For the reader's convenience: take $v\\in {\\mathcal D}({\\mathcal A})$, $\\varphi\\in L^2(\\Gamma_0)$,\nand compute\n\\begin{equation*}\n\\begin{split}\n& -\\big(N_0^*({\\mathcal A}+I) v,\\varphi\\big)_{\\Gamma_0}= \\big(-({\\mathcal A}+I) v,N_0\\varphi\\big)_\\Omega\n= (\\Delta v,N_0\\varphi)_\\Omega-(v,N_0\\varphi)_\\Omega=\n\\\\\n& \\qquad = \\big(v,\\Delta (N_0\\varphi)\\big)_\\Omega\n+ \\cancel{\\Big(\\frac{\\partial v}{\\partial\\nu},N_0\\varphi\\Big)_{\\partial\\Omega}}\n- \\Big(v,\\frac{\\partial N_0\\varphi}{\\partial\\nu}\\Big)_{\\partial\\Omega}\n-(v,N_0\\varphi)_\\Omega=\n\\\\\n& \\qquad = (v,N_0\\varphi)_\\Omega\n-(v,\\varphi)_{\\Gamma_0}-(v,N_0\\varphi)_{\\Omega}\n=-(v,\\varphi)_{\\Gamma_0}\\,.\n\\end{split}\n\\end{equation*}\nThe above shows that \\eqref{e:trace-result-ni} holds true when $i=0$; the case $i=1$ is proved in the same way.\nWe note that it has been used that since $v$ belongs to ${\\mathcal D}({\\mathcal A})$,\nthen $\\frac{\\partial v}{\\partial\\nu}=0$ on $\\partial\\Omega$; in addition, the definition\nof $N_0\\varphi$ in \\eqref{e:N_i} -- as the solution of an elliptic problem -- gives\nin particular $\\Delta (N_0\\varphi)=N_0\\varphi$.)\n\n\\smallskip\nIn view of the definition of the introduced operators $N_i$, $i=0,1$, we see that\n\\begin{equation*}\n\\begin{cases}\n(\\Delta -I)\\big(u+\\frac1{c}N_1 u_t|_{\\Gamma_1}-N_0g\\big)=(\\Delta -I)u & \n\\textrm{on $\\Omega\\times (0,T)$}\n\\\\[1mm]\n\\frac{\\partial}{\\partial \\nu}(u+\\frac1{c}N_1 u_t-N_0g)=0 \n& \\textrm{on $\\Gamma_0\\times (0,T)$}\n\\\\[1mm]\n\\frac{\\partial}{\\partial \\nu}(u+\\frac1{c}N_1 u_t-N_0g)=0\\,; \n& \\textrm{on $\\Gamma_1\\times (0,T)$}\n\\end{cases}\n\\end{equation*}\nproceeding formally we get\n\\begin{equation*}\n\\begin{split}\n\\Delta u &= (\\Delta -I)\\big(u+\\frac1{c_1}N_1 u_t|_{\\Gamma_1}-N_0g\\big) +u\\,,\n\\\\\n\\Delta u_t &= (\\Delta -I)\\big(u_t+\\frac1{c_1}N_1 u_{tt}|_{\\Gamma_1}-N_0 g_t\\big) +u_t\\,, \n\\end{split}\n\\end{equation*}\nwhich enable us to rewrite the SMGT equation as \n\\begin{equation*}\n\\begin{split}\n& \\tau u_{ttt}+\\alpha u_{tt} -c^2 (\\Delta - I)\\big(u+\\frac1{c}N_1 u_t|_{\\Gamma_1}-N_0g\\big)-c^2 u-\n\\\\\n& \\qquad\\qquad\\qquad - b(\\Delta -I)\\big(u_t+\\frac1{c}N_1 u_{tt}|_{\\Gamma_1}-N_0 g_t\\big) - b u_t=0\\,,\n\\end{split}\n\\end{equation*}\nwhere $\\frac{\\partial}{\\partial \\nu}\\big(u+\\frac1{c_1}N_1 u_t|_{\\Gamma_1}-N_0g\\big)\\big|_\\Gamma=0$.\nThus, by using the trace results \\eqref{e:trace-result-ni}, the BVP \\eqref{e:bvp-0} for the SMGT equation\ntranslates to the following abstract equation, where both the absorbing BC on $\\Gamma_1$ and the boundary control action on $\\Gamma_0$ are incorporated:\n\\begin{equation*}\n\\begin{split}\n& \\tau u_{ttt}+\\alpha u_{tt} +c^2 ({\\mathcal A}+I)\\Big[u+\\frac1{c}N_1 N_1^*({\\mathcal A}+I)u_t-N_0 g\\Big]\n-c^2 u+\n\\\\[1mm]\n& \\qquad\\qquad\\qquad + b({\\mathcal A}+I)\\Big[u_t+\\frac1{c}N_1N_1^*({\\mathcal A}+I) u_{tt}-N_0 g_t\\Big] - b u_t=0\\,,\n\\end{split}\n\\end{equation*}\nthat is \n\\begin{equation}\\label{e:controlled-eq}\n\\begin{split}\n& \\tau u_{ttt}+\\alpha u_{tt} +c^2 {\\mathcal A} u +c({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)u_t + b {\\mathcal A} u_t +\n\\\\[1mm]\n& \\qquad\\qquad\\qquad +\\frac{b}{c} ({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I) u_{tt} = c^2 ({\\mathcal A}+I)N_0 g + b ({\\mathcal A}+I)N_0 g_t\\,;\n\\end{split}\n\\end{equation} \nthe equality is understood with respect to the duality pairing, i.e. in $[{\\mathcal D}({\\mathcal A})]'$. \n\nThe third order abstract equation \\eqref{e:controlled-eq} gives rise readily to a first order control system,\ninitially defined on an extended space $L^2(\\Omega) \\times L^2(\\Omega) \\times [{\\mathcal D}({\\mathcal A})]'$:\n\\begin{equation}\\label{e:1order-system}\n\\frac{d}{dt}\\begin{pmatrix}\nu\\\\\nu_t\\\\\nu_{tt}\n\\end{pmatrix} = A \\begin{pmatrix}\nu\\\\\nu_t\\\\\nu_{tt}\n\\end{pmatrix} + B_0 g + B_1 g_t\\,,\n\\end{equation}\nwhere the operator describing the {\\em free} dynamics is\n{\\small\n\\begin{equation}\\label{e:generator}\nA= \n\\begin{pmatrix}\n0 & I & 0\n\\\\[1mm]\n0 & 0 & I\n\\\\[1mm]\n- \\tau^{-1} c^2{\\mathcal A} & -\\tau^{-1} \\big[b {\\mathcal A}+c({\\mathcal A}+I)N_1N_1^*({\\mathcal A}+I)\\big] \n& - \\tau^{-1} \\big[\\alpha I +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)\\big]\n\\end{pmatrix}\n\\end{equation}\n}\nwhile the input operators $B_i\\in {\\mathcal L}(L^2(\\Gamma_0),[{\\mathcal D}({\\mathcal A})]'), i=0,1$, are given by\n\\begin{equation}\\label{e:input-operators}\nB_0=\n\\begin{pmatrix}\n0 \n\\\\[1mm]\n0 \n\\\\[1mm]\n\\tau^{-1}c^2 ({\\mathcal A}+I)N_0\n\\end{pmatrix}\\,,\n\\qquad \nB_1=\n\\begin{pmatrix}\n0 \n\\\\[1mm]\n0 \n\\\\[1mm]\n\\tau^{-1}b ({\\mathcal A}+I)N_0\n\\end{pmatrix} \n=\\frac{b}{c^2}B_0\\,.\n\\end{equation}\nThe (free dynamics) operator $A$ in \\eqref{e:generator} will be shown to generate a $C_0$-semigroup on the space $Y = H^1(\\Omega) \\times H^1(\\Omega) \\times L^2(\\Omega)$. \n\n\\begin{remark}\n\\begin{rm}\nThe first order equation \\eqref{e:1order-system} is a control system in (extended to) the dual space\n$[{\\mathcal D}(A^*)]'$;\nthis is due to the fact that $A^{-1} B_i \\in {\\mathcal L}(U,Y)$, $i=0,1$, as it will be verified later.\nHowever, the given formulation involves the time derivative of control, which is not supported by the\ncost functional; as a consequence, the minimization problem lacks coercivity.\nTo cope with this, we will follow \\cite{LLP}: \nintegration by parts in the input-to-state formula enables to `eliminate' the time derivative of the control function, however with the drawback that the states will become `rougher'.\nThe smoothing properties of the observation operator $R$ -- here, {\\em intrinsic} -- will play a major role\nin the entire subsequent analysis, which will eventually bring about the solution of the optimization problem.\n\\end{rm}\n\\end{remark}\n \nBefore we proceed, let us consider the uncontrolled equation first. \nThis step is necessary in order to formulate a correct notion of duality -- which is always with respect to\nthe generator of the semigroup underlying the dynamics. \n\n\n\\subsection{The uncontrolled equation. Semigroup well-posedness}\nIn order to pinpoint the control-theoretic properties of the abstract system \n\\eqref{e:1order-system} -- an ineludible preliminary step for the analysis \nof the optimal control problem --, we consider first the uncontrolled equation, that is\nequation \\eqref{e:controlled-eq} in the absence of the boundary action $g$.\nWith $g\\equiv 0$, the equation \\eqref{e:controlled-eq} reads as\n\\begin{equation}\\label{e:free-eq}\n\\begin{split}\n& \\tau u_{ttt}+\\alpha u_{tt} +c^2 {\\mathcal A} u +c({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)u_t + b {\\mathcal A} u_t +\n\\\\\n& \\qquad\\qquad\\qquad + \\frac{b}{c} ({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I) u_{tt}=0\\,.\n\\end{split}\n\\end{equation} \nWe follow an idea introduced\nand utilized in \\cite{kalten-las-mar_2011} and \\cite{marchand}.\nCalculations below might appear formal: however, they are fully justified with respect to the duality in\n$[{\\mathcal D}(A^*)]'$.\nAfter having set $\\tau=1$ for the sake of simplicity, the rewriting of equation \\eqref{e:free-eq} as\n\\begin{equation}\\label{e:free-eq-1}\n(u_t+\\alpha u)_{tt}+b {\\mathcal A}\\Big(u_t+\\frac{c^2}{b} u\\Big) \n+\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)\\Big(u_{tt}+ \\frac{c^2}{b} u_t\\Big)=0\\,,\n\\end{equation} \nsuggests the introduction of the auxiliary variable \n\\begin{equation} \\label{e:def-of-z}\nz:= u_t+\\frac{c^2}{b} u\\,.\n\\end{equation} \nThe new variable $z$ plays a major role in deriving well-posedness\nresults for the third order equation \\eqref{e:free-eq} in the unknown $u$;\nthis is because it allows to connect the (free) equation under investigation with\nthe following system in the unknowns $(u,z)$:\n\\begin{equation}\\label{e:auxiliary-free-sys}\n\\begin{cases}\nu_t=-\\frac{c^2}{b} u+z\n\\\\[1mm]\nz_{tt}=-b {\\mathcal A} z -\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)z_t -\\gamma z_t+\\gamma \\frac{c^2}b z \n- \\gamma \\Big(\\frac{c^2}b\\Big)^2 u\n\\end{cases}\n\\end{equation}\nwhere $\\gamma:= \\alpha - \\frac{c^2}{b}$ will be assumed to be positive.\nThe explicit statement and proof of this claim, that is an immediate generalization of \nwhat done in \\cite{kalten-las-mar_2011}, is given below for the reader's convenience\nand the sake of completeness.\n \n\\begin{lemma}\nThe uncontrolled third order (in time) equation \\eqref{e:free-eq} is equivalent to \nthe coupled ODE-PDE system \\eqref{e:auxiliary-free-sys}, with \n$\\gamma=\\alpha - \\frac{c^2}{b}$.\n\\end{lemma}\n\n\\begin{proof}\nThe starting point is equation \\eqref{e:free-eq-1} that is nothing but a rewriting of\n\\eqref{e:free-eq}.\nWith the new variable $z= u_t+c^2\/b u$, the term $u_t+\\alpha u$ in \\eqref{e:free-eq-1}\nis rewritten in terms of $z$ and $u$ as follows:\n\\begin{equation*}\nu_t+\\alpha u= z+\\gamma u\\,, \\quad \\gamma:= \\alpha - \\frac{c^2}{b}\\,,\n\\end{equation*} \nso that \\eqref{e:free-eq-1} becomes\n\\begin{equation}\\label{e:free-eq-2}\nz_{tt}+b {\\mathcal A} z +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)z_t + \\gamma u_{tt}=0\\,.\n\\end{equation} \nOn the other hand, using once again the definition of $z$\nwe see that $u_t=z-c^2\/b \\,u$, which gives \n\\begin{equation}\\label{e:u_tt}\nu_{tt}=z_t-\\frac{c^2}{b} u_t=z_t-\\frac{c^2}{b} z+\\Big(\\frac{c^2}{b}\\Big)^2 u\\,;\n\\end{equation} \nthe above, inserted in \\eqref{e:free-eq-2} yields the following equation:\n\\begin{equation*\nz_{tt}+b {\\mathcal A} z +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)z_t + \\gamma z_t-\\gamma \\frac{c^2}b z + \n\\Big(\\frac{c^2}b\\Big)^2 u=0\\,.\n\\end{equation*}\n\nThe latter second order in time equation for $z$, combined with\n\\eqref{e:u_tt}\nleads to the following coupled system of (second-order in time) equations in the \nunknowns $(u,z)$\n\\begin{equation*\n\\begin{cases}\nu_{tt}=z_t-\\frac{c^2}{b} u_t\n\\\\[1mm]\nz_{tt}+b {\\mathcal A} z +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)z_t + \\gamma z_t-\\gamma \\frac{c^2}b u_t =0\\,;\n\\end{cases}\n\\end{equation*}\nor, equivalently, to the coupled ODE-PDE system \\eqref{e:auxiliary-free-sys}.\n\\end{proof}\n\nWe establish semigroup well-posedness of the Cauchy problems associated with\nsystem \\eqref{e:auxiliary-free-sys} in three different function spaces.\n\n\\begin{theorem}[Equivalent system. Well-posedness, I] \\label{t:first}\nThe (first order in time) system in the unknown $(u,z,z_t)$ corresponding to\nsystem \\eqref{e:auxiliary-free-sys} is well-posed in the space\n\\begin{equation*}\nY= \\underbrace{H^1(\\Omega)}_{u}\\times \\underbrace{H^1(\\Omega)\\times L^2(\\Omega)}_{(z,z_t)}\\,.\n\\end{equation*}\nIts dynamics is described by a closed operator $\\tilde{A}: {\\mathcal D}(\\tilde{A})\\subset Y\\to Y$ which is the generator of a $C_0$-semigroup $e^{\\tilde{A} t}$ on $Y$, $t\\ge 0$. \n\\end{theorem}\n\n\\begin{proof}\nThe second-order system \\eqref{e:auxiliary-free-sys} is rewritten as \na first-order system\n\\begin{equation*}\n\\begin{pmatrix}\nu\\\\\nz\\\\\nz_t\n\\end{pmatrix}_t= \\tilde{A}\\begin{pmatrix}\nu\\\\\nz\\\\\nz_t\n\\end{pmatrix}\\,,\n\\end{equation*}\nwith dynamics operator\n\\begin{equation*}\n\\tilde{A}=\n\\begin{pmatrix}\n-\\frac{c^2}{b}\\,I & I & 0\n\\\\[1mm]\n0 & 0 & I\n\\\\[1mm]\n- \\gamma\\big(\\frac{c^2}b\\big)^2 \\,I & -b {\\mathcal A}+\\gamma \\frac{c^2}b\\,I & -\\gamma I \n- \\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I) \n\\end{pmatrix}\\,.\n\\end{equation*}\nIt is then natural to observe that the decomposition \n\\begin{equation*}\n\\tilde{A}= \\tilde{A}_1+ C_1+K_1\n\\end{equation*}\nholds true, where we set\n\\begin{equation*}\n\\tilde{A}_1=\n\\begin{pmatrix}\n-\\frac{c^2}{b}\\,I & 0 & 0\n\\\\\n0 & 0 & I\n\\\\\n0 & -b ({\\mathcal A}+I) & -\\gamma I - \\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I) \n\\end{pmatrix}\\,,\n\\end{equation*}\n\\begin{equation*}\nC_1=\\begin{pmatrix}\n0 & I & 0\n\\\\\n0 & 0 & 0 \n\\\\\n-\\gamma \\big(\\frac{c^2}{b}\\big)^2\\,I & 0 & 0 \n\\end{pmatrix}\\,,\n\\qquad \nK_1=\\begin{pmatrix}\n0 & 0 & 0\n\\\\\n0 & 0 & 0\n\\\\\n0 & (\\gamma \\frac{c^2}{b}+b)\\,I & 0\n\\end{pmatrix}\\,.\n\\end{equation*}\nIt is enough to single out the following respective features: \n\\\\\n(i) the operator $\\tilde{A}_1: {\\mathcal D}(\\tilde{A}_1)\\subset Y\\longrightarrow Y$ is a \n(maximally) dissipative operator on \n\\begin{equation*}\n\\underbrace{H^1(\\Omega)}_{u}\\times \\underbrace{{\\mathcal D}(({\\mathcal A}+I)^{1\/2})\\times L^2(\\Omega)}_{(z,z_t)}\n\\end{equation*}\nand hence it is the generator of a $C_0$-semigroup of {\\em contractions} \n$e^{\\tilde{A}_1 t}$ on $Y$ (which, however, is {\\em not} analytic); \n\\\\\n(ii)\n$C_1$ is a {\\em bounded} operator from $Y$ into itself;\n\\\\\n(iii)\n$K_1$ is a {\\em compact} operator: in fact, with $f\\in {\\mathcal D}(({\\mathcal A}+I)^{1\/2})$ one has \n\\begin{equation*}\n\\gamma \\Big(\\frac{c^2}{b}+b\\Big)\\,f\n=\\gamma \\Big(\\frac{c^2}{b}+b\\Big)({\\mathcal A}+I)^{-1\/2}[({\\mathcal A}+I)^{1\/2}f]\\,.\n\\end{equation*}\nThe generation of a $C_0$-semigroup $e^{\\tilde{A} t}$ on $Y$ follows\nby semigroup theory.\n\\end{proof}\n\n\n\\begin{remark}\n\\begin{rm}\nThe space $Y$ will provide an appropriate functional setting where the original uncontrolled\nsystem is well-posed, and a state space for the optimal control problem under investigation.\nIt is however important to add that well-posedness remains valid in distinct functional spaces;\nthe corresponding results are stated below for the sake of completeness, while the relative proofs are\nomitted.\n\\end{rm}\n\\end{remark}\n\n\n\\begin{corollary}[Equivalent system. Well-posedness, II]\nThe uncontrolled problem is well-posed in\n\\begin{equation*}\nY_2= \\underbrace{H^2(\\Omega)}_{u}\\times \\underbrace{H^1(\\Omega)\\times L^2(\\Omega)}_{(z,z_t)}\\,.\n\\end{equation*}\n\\end{corollary}\n\nThus, in view of the definition of the domain of the generator $\\tilde{A}$, that is\n\\begin{equation*}\n\\begin{split}\n{\\mathcal D}(\\tilde{A}) &= \\big\\{ (u,z,z_t)\\in [H^1(\\Omega)]^3\\colon\nz+\\frac{1}{c}N_1\\,N_1^*({\\mathcal A}+I)z_t\\in {\\mathcal D}({\\mathcal A})\\big\\}=\n\\\\[1mm]\n& = \\Big\\{ (u,z,z_t)\\in H^1(\\Omega)\\times H^2(\\Omega)\\times H^1(\\Omega)\\,\\colon\n\\; \\frac{\\partial z}{\\partial \\nu}\\Big|_{\\Gamma_0} =0 \\,, \\;\n\\Big[c\\frac{\\partial z}{\\partial \\nu}+z_t\\Big]_{\\Gamma_1} =0 \\Big\\}\\,,\n\\end{split}\n\\end{equation*}\ntaking the dual $[{\\mathcal D}(\\tilde{A})]'$ (duality with respect to $Y_2$), we are able to infer the following result.\n\\begin{corollary}[Equivalent system. Well-posedness, III]\nThe uncontrolled problem is well-posed in\n\\begin{equation}\nY_0 \\sim \\underbrace{H^1(\\Omega)}_{u}\\times \\underbrace{L^2(\\Omega)\\times [H^1(\\Omega)]'}_{(z,z_t)}\\,,\n\\end{equation}\nwhere $\\sim$ indicates topological equivalence.\n\\end{corollary}\n\nThe next Theorem~\\ref{t:first} summarizes relevant wellposedness results which will be used throughout . \n\n\\begin{theorem}[The uncontrolled equation. Well-posedness and stability] \\label{t:wellposed_1}\nWith reference to the third order abstract equation \\eqref{e:free-eq} describing the free dynamics,\nthe following statements hold true.\n\\begin{enumerate}\n\\item[i)]\nThe boundary value problem \\eqref{e:bvp-0} with $g\\equiv 0$ admits the abstract formulation\n\\eqref{e:free-eq} as a third order equation; \nequivalently, it is rewritten as a first order abstract system $y'=Ay$, where $y$ denotes the\nstate variable $(u,u_t,u_{tt})$.\n\n\\item[ii)]\nThe operator $A$ which governs the free dynamics, detailed in \\eqref{e:generator},\nis the generator of a $C_0$-semigroup $\\{e^{At}\\}_{t\\ge 0}$ on the function space\n$Y=H^1(\\Omega)\\times H^1(\\Omega)\\times L^2(\\Omega)$.\n\n\\item[iii)] \nThe semigroup $e^{At}$ is exponentially stable when $\\gamma > 0$. \n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\begin{rm}\nIn the critical case, when $\\gamma =0$, it is expected that with $\\Gamma_0$ subject to \nthe ``star-shaped'' Geometric Condition (cf.~\\cite{lagnese}) the resulting semigroup is exponentially stable.\n\\end{rm} \n\\end{remark}\n\n\\begin{remarks}\n\\begin{rm}\nThe first assertion in Theorem \\ref{t:wellposed_1} establishes the existence of a linear semigroup defined\non $Y$ which describes the original uncontrolled dynamics. \nIt is worth noting that if the SMGT equation is complemented with either Dirichlet or Neumann\nBC the same result holds true, as it was first proved in \\cite{marchand} and \\cite{kalten-las-mar_2011};\nin that case the semigroup is actually a {\\em group} on $Y$. \nInstead, the group property is not valid any more in the presence of absorbing BC on $\\Gamma_1$. \n\nThe studies \\cite{kalten-las-mar_2011} and \\cite{marchand} -- the latter, providing a clarifying \nspectral analysis -- obtain that (still in the case of Dirichlet or Neumann BC)\nthe semigroup $e^{tA}$ is exponentially stable on the factor space $Y\/{\\ker(A)}$, \nprovided $\\gamma > 0$; it is marginally stable when $\\gamma =0$ and unstable when $\\gamma < 0$.\nIn the present case, assuming appropriate geometric conditions on $\\Gamma_0$, the absorbing boundary conditions\nturn marginal stability ($\\gamma =0$) to stability. \nThis issue has not been fully investigated so far, yet it is expected that the multipliers' method combined\nwith a background on wave equations would provide the tools. \n\\end{rm}\n\\end{remarks}\n\n\\begin{remark}\n\\begin{rm}\n({\\em A distinct perspective) }\nThe connection between the MGT equation with wave equations with memory has been pointed out in \nthe recent independent works \\cite{pata1} and \\cite{bucci-pan_arxiv2017}.\nThe critical role of $\\gamma$ as a threshold for uniform stability is revisited and recovered in \n\\cite{pata1} via the analysis of a corresponding viscoelastic equation. \nIt is apparent that appropriate compatibility conditions on initial data must be assumed,\nin order to study the third order (in time) equation by using theories pertaining to\nwave equations with a non-local term.\n\nAnd yet, the perspective of equations with memory opens a distinct avenue of investigation\nof the (interior and trace) regularity properties of the corresponding solutions, fruitfully explored in \\cite{bucci-pan_arxiv2017} -- as well as, possibly, of other control-theoretic properties.\nIn this connection, we mention the paper \\cite{pandolfi-TAC_2018}, which provides an analysis\nof the LQ problem and Riccati equations for finite dimensional systems with memory. \n\\end{rm} \n\\end{remark}\n\n\n\n\\subsection{Domain of the generator} \\label{ss:domain-generator}\nWe give an explicit description of the natural domain of the the (free) dynamics generator $A$ introduced in\n\\eqref{e:generator}: given the state space $Y = H^1(\\Omega) \\times H^1(\\Omega) \\times L^2(\\Omega)$, \none has\n\\begin{equation*}\n\\begin{split}\ny\\in {\\mathcal D}(A) \\Longleftrightarrow \\; & y\\in \\Big\\{y =(y_1,y_2,y_3) \\in Y\\colon y_3 \\in H^1(\\Omega)\\,,\n\\\\[1mm]\n& \\qquad\\qquad\\qquad \nc^2 y_1 + b y_2 + N_1 (c N_1^* {\\mathcal A} y_2 + \\frac{b}{c} N_1^* {\\mathcal A} y_3)\\in {\\mathcal D}({\\mathcal A})\\Big\\}\\,. \n\\end{split}\n\\end{equation*}\nFrom the PDE vriewpoint the above corresponds to \n\\begin{equation*}\n\\begin{split}\ny\\in {\\mathcal D}(A) \\Longleftrightarrow \\; & y\\in \\Big\\{y \\in [H^1(\\Omega)]^3\\colon \\Delta (c^2 y_1 + b y_2) \\in L^2(\\Omega)\\,, \n\\;{ \\frac{\\partial}{\\partial \\nu } }(c^2 y_1 + b y_2 ) =0 \\;\\textrm{on $\\Gamma_0$}\\,,\n\\\\[1mm]\n& \\qquad\\qquad\\qquad \nc{ \\frac{\\partial}{\\partial \\nu } } (c^2 y_1 + b y_2 )\\Big|_{\\Gamma_1} = - \\Big[c^2y_2+ b y_3\\Big]\\Big|_{\\Gamma_1}\n\\;\\textrm{on $\\Gamma_1$}\\Big\\}\\,.\n\\end{split}\n\\end{equation*}\nNotice that by a standard variational argument the normal derivatives are first well defined on \n$H^{-1\/2}(\\Gamma)$. \nThen, the $H^{1\/2}(\\Gamma)$-regularity of $y_i$, $i =1,2,3$, along with elliptic theory gives\n\\begin{equation}\\label{e:domain-pde}\n\\begin{split}\n& {\\mathcal D}(A) = \\Big\\{y \\in [H^1(\\Omega)]^3\\colon (c^2 y_1 + b y_2) \\in H^2(\\Omega)\\,, \n\\;{ \\frac{\\partial}{\\partial \\nu } }(c^2 y_1 + b y_2 ) =0 \\;\\textrm{on $\\Gamma_0$}\\,,\n\\\\[1mm]\n& \\qquad\\qquad\\qquad \nc{ \\frac{\\partial}{\\partial \\nu } } (c^2 y_1 + b y_2 )\\Big|_{\\Gamma_1} = - \\Big[c^2y_2+ b y_3\\Big]\\Big|_{\\Gamma_1}\n\\;\\textrm{on $\\Gamma_1$}\\Big\\}\\,.\n\\end{split}\n\\end{equation}\nWe also note that the resolvent of $A$ is not compact, which is important to be pointed out.\n\n\n\\subsection{The SMGT equation subject to smooth controls}\nNow let us turn our attention to the controlled (abstract) equation \\eqref{e:controlled-eq} corresponding to the BVP \\eqref{e:bvp-0} and to its reformulation as the first-order control system \\eqref{e:1order-system}.\nThis system produces readily a solution formula, assuming that $g\\in H^1(0,T;L^2(\\Gamma_0))$:\nthe following Proposition provides a rigorous justification.\n \n\\begin{proposition} \\label{p:firstorder-controlled-wellposed}\nAssume that $g\\in H^1(0,T;L^2(\\Gamma_0))$.\nThe boundary value problem \\eqref{e:bvp-0} for the SMGT equation can be recast as the (third order in time) abstract equation \\eqref{e:controlled-eq}; equivalently, it is rewritten as a first order abstract system \\eqref{e:1order-system} that is\n\\begin{equation} \\label{e:state-eq_0}\ny'=Ay+B_0g+B_1g_t\\,;\n\\end{equation}\n$y$ denotes the state variable $(u,u_t,u_{tt})$ and $g$ is the control variable,\nwhile the linear operators $A$ and $B_i$ satisfy the following analytical properties.\n\\begin{enumerate}\n\n\\item[i)] \nthe operator $A$ which describes the free dynamics, detailed in \\eqref{e:generator},\nis the generator of a $C_0$-semigroup $\\{e^{At}\\}_{t\\ge 0}$ on the function space\n$Y=H^1(\\Omega)\\times H^1(\\Omega)\\times L^2(\\Omega)$, with domain \n${\\mathcal D}(A)$ given in \\eqref{e:domain-pde};\n\n\\item[ii)]\nthe control operators $B_i$, $i=0,1$ defined in \\eqref{e:input-operators} satisfy \n$B_i\\in {\\mathcal L}(U,[{\\mathcal D}(A^*)]')$. \n\\end{enumerate}\nThen, the third order equation \\eqref{e:controlled-eq} is understood on the \nextrapolation space $[{\\mathcal D}({\\mathcal A})]'$.\n\n\\end{proposition}\n\n\\begin{proof}\nSince the Neumann maps $N_i$ defined in \\eqref{e:N_i} enjoy the regularity in \\eqref{e:neumann-regularity}, \nthat is $N_i \\in {\\mathcal L}(L^2(\\Gamma_i),{\\mathcal D}({\\mathcal A}^{3\/4-\\sigma}))$, we accordingly have \nthat the distributional range of the control maps $B_i$ is such that \n\\begin{equation*}\n{\\mathcal R}(B_i) \\subset \\{0\\}\\times \\{0\\} \\times [{\\mathcal D}({\\mathcal A}^{1\/4 + \\sigma})]'\\,.\n\\end{equation*}\nTo see this, just recall the explicit form of the input operators $B_0$ in \\eqref{e:input-operators},\nwhich gives \n\\begin{equation*}\n\\begin{split}\n|(B_0 g,y)|_Y&=|(c^2 ({\\mathcal A}+I)N_0 g,y)|_Y\n=c^2 |(g,y_3|_{\\Gamma_0})_{L^2(\\Gamma_0)}\n=c^2 \\,|y_3|_{H^{1\/2+2\\sigma}(\\Omega)} |g|_{L^2(\\Gamma_0)}\n\\\\[1mm]\n& =c^2\\,|{\\mathcal A}^{1\/4+\\sigma} y_3 |_{L^2(\\Omega)} |g|_{L^2(\\Gamma_0)}\n\\end{split}\n\\end{equation*}\nwhich proves that there exists a positive constant $C$ such that \n\\begin{equation*}\n|(B_i g,y)|_Y|\\le C\\,|{\\mathcal A}^{1\/4+\\sigma} y_3 |_{L^2(\\Omega)} |g|_{L^2(\\Gamma_0)}\n\\le C\\,|A y|_Y \\,|g|_{L^2(\\Gamma_0)}\\,, \\qquad i=0,1\\,,\n\\end{equation*}\nsince $B_1=b\/{c^2} B_0$.\n\nBy using interpolation trace results, a stronger inequality is obtained:\nfor any $\\epsilon > 0 $ one has \n\\begin{equation*}\n(B_i g, y)_Y\\le C \\,|Ay|_Y^{1\/2}|y|_Y^{1\/2} |g|_{L^2(\\Gamma_i)}\n\\le \\big(\\epsilon |Ay|_Y + C_\\epsilon |y|_Y\\big)\\,|g|_{L^2(\\Gamma_i)}\n\\end{equation*}\nwhich gives \n\\begin{equation*}\n|B_i^*y|_{L_2(\\Gamma_i)} \\le \\epsilon |Ay|_Y + C_{\\epsilon} |y|_Y \\qquad \\forall \\epsilon > 0\\,.\n\\end{equation*}\n\\end{proof}\n\nIn view of Proposition~\\ref{p:firstorder-controlled-wellposed} (hence, still under the assumption\n$g\\in H^1(0,T,U)$), semigroup theory yields a first input-to-state formula in the extrapolation space $[{\\mathcal D}(A^*)]'$.\n\n\\begin{corollary}\\label{c:state-in-extrapolation}\nFor any initial state $y_0 \\in [{\\mathcal D}(A^*)]'$ and any control $g\\in H^1(0,T,U)$,\nthe control system \\eqref{e:state-eq_0} has a unique mild solution $y\\in C([0,T];[{\\mathcal D}(A^*)]')$\ngiven by \n\\begin{equation}\\label{e:sln_0}\n\\begin{split}\ny(t) &= e^{At} y(0) + \\int_0^t e^{A(t-s)} \\big(B_0 g(s)+B_1 g_t(s)\\big)\\,ds=\n\\\\[1mm]\n&= e^{At} y(0) + \\int_0^t e^{A(t-s)} B_0 \\Big(g(s) + \\frac{b}{c^2}g_t(s)\\Big)\\,ds\\,.\n\\end{split}\n\\end{equation}\n\n\\end{corollary}\n\n\n\\section{The control problem. Main results} \\label{s:auxiliary-and-riccati}\nIf the cost functional \\eqref{e:cost-funct} penalized (quadratically) the time derivative of\nthe control function, we might choose as space of admissible controls \n${\\mathcal U}=H^1(0,T;L^2(\\Gamma_0))$, and the obtained semigroup solution formula \\eqref{e:sln_0}\nas the state equation.\nRemember however that we seek to minimize the functional \\eqref{e:cost-funct} over all controls $g$ which belong to $L^2(0,T;L^2(\\Gamma_0))$, where the acoustic pressure $u$ satisfies the IBVP \\eqref{e:ibvp-1}.\nHence, in this Section we first derive from \\eqref{e:sln_0} a solution formula which requires\ncontrols which just belong to $L^2(0,T;L^2(\\Gamma_0))$ and are continuous at time $t=0$;\nthis is done by an elementary integration (in time) by parts.\nThen, following an idea proposed in \\cite{LLP} and \\cite{LPT}, we introduce an (auxiliary) optimal control\nproblem associated to an equation depending on a parameter $g_0\\in L^2(\\Gamma_0)=:U$.\nThe main result pertaining to the auxiliary problem and the connection with the original one \nare stated collectively in the section. The respective proofs are the subject of the\nsubsequent two sections.\n\n\\subsection{Control problem with the observation} \nOur next step is to provide a representation formula for the solutions to the controlled dynamics \nby assuming that controls belong to $L^2(0,T;U)$. \nThis is done, as usual, integrating by parts (in a dual space) and exploiting the structure of the domain\nof the generator. \n\n\\begin{lemma}\\label{l:rep}\nGiven an initial state $y_0\\in [{\\mathcal D}({A^2}^*)]'$ and any control function $g\\in C([0,T;U)$, the solution\nto the original control system \\eqref{e:1order-system}, represented via the\ninput-to-state formula \\eqref{e:sln_0}, is equivalently given by \n\\begin{equation}\\label{e:eq-for-U}\ny(t) = e^{At}[y_0 - B_1g(0)] + Lg(t)\\,,\n\\end{equation}\nwith \n\\begin{equation}\\label{e:input-to-state-map}\n\\begin{split}\n(Lg)(t) &= B_1 g(t) + (L_0 g)(t)\\,,\n\\\\[1mm]\n(L_0 g)(t) &= \\int_0^t e^{A(t-s)} B_0 g(s) ds + A\\int_0^t e^{A(t-s)} B_1 g(s) ds\\,.\n\\end{split}\n\\end{equation}\nThe map $(y_0,g) \\rightarrow y(\\cdot)$ is bounded from \n$[{\\mathcal D}({A^2}^*)]' \\times C([0,T; U) \\rightarrow C([0,T;[{\\mathcal D}({A^2}^*)]')$.\n\\end{lemma}\n\n\\begin{proof}\nThe novel representation formula \\eqref{e:eq-for-U} is easily established integrating by parts in \n\\eqref{e:sln_0}; what we need to justify rigorously is the claimed regularity. \nWe know already that $e^{At}$ generates a $C_0$-semigroup on $[{\\mathcal D}({A^2}^*)]'$, and that \n$A^{-1} B_i \\in L(U,Y)$.\nThen, it suffices to analyze the regularity of the operator $L_0$ in \\eqref{e:input-to-state-map},\nwhich depends on the one of the operator $AB_1$. \nRecalling the definitions of $A$ and $B_1$, it is easily seen that \n\\begin{equation*}\nAB_1 =b\n\\begin{pmatrix}\n0 \n\\\\[1mm]\n({\\mathcal A}+I)N_0\n\\\\[1mm]\n- \\alpha ({\\mathcal A}+I)N_0\n\\end{pmatrix} \n\\end{equation*}\nwhere we have used that the distributions on $\\Gamma_0$ and $\\Gamma_1$ have disjoint support.\nThis implies that the contribution of the operator $N_1$ in the definition of $A$, when applied to $B_1$, produces the zero element. \nAs a consequence, we obtain that \n\\begin{equation*}\n{\\mathcal R}(AB_1) \\subset \\{0\\} \\times [{\\mathcal D}({\\mathcal A}^{1\/4+\\epsilon}]' \\times [{\\mathcal D}({\\mathcal A}^{1\/4+\\epsilon}]' \n\\subset [{\\mathcal D}(A^{2*})]'\\,,\n\\end{equation*}\nwhich gives the desired conclusion.\n\\end{proof}\n\nObserve that -- just like in the works \\cite{LLP} and \\cite{LPT} -- the drawback of the chosen approach\nis that the space regularity of the state function gets worse.\nMoreover, in contrast with the dynamics under investigation therein, whose underlying semigroup is \nanalytic, we are dealing with a purely hyperbolic problem.\n\\\\\nOn the other hand, recall that the goal is to minimize the $L^2(\\Omega)$-norm of the acoustic pressure,\ndescribed by the state variable $u$, that is the first component of the state variabile $y$.\nBy setting $u^d=0$ in \\eqref{e:cost-funct} just for the sake of simplicity, \nthe cost functional is abstractly rewritten as\n\\begin{equation}\\label{J}\nJ(g) = \\int_0^T \\|R y\\|_Y^2 \\,dt + \\int_0^T \\|g\\|_U^2\\,dt\\,,\n\\end{equation}\nwhere $U$ denotes the control space, i.e. $U=L^2(\\Gamma_0)$, and the observation operator\n$R$ is acting as follows: for any $y = [y_1,y_2,y_3]^T$, it holds \n\\begin{equation}\\label{e:def-observation-op}\nR y = \\begin{pmatrix}\n{\\mathcal A}^{-1\/2}y_1 \n\\\\[1mm]\n0\n\\\\[1mm]\n0\n\\end{pmatrix}\\,.\n\\end{equation}\nIn fact, after identifying $H^1(\\Omega)$ with ${\\mathcal D}({\\mathcal A}^{1\/2})$, we see that\n\\begin{equation*}\n\\|Ry\\|_Y = \\|{\\mathcal A}^{1\/2} {\\mathcal A}^{-1\/2} y_1\\|_{L^2(\\Omega)} = |y_1|_{L^2(\\Omega)}\\,.\n\\end{equation*}\nThus, the simple -- and yet natural -- quadratic functional taken into consideration,\nattributes to the observation operator $R$ a very special structure and an {\\em intrinsic} strong smoothing effect.\nThe improved regularity of the observed states enables us to pursue an adaptation of \nthe theory developed in \\cite[Vol.~II]{redbook} in the study of hyperbolic-like PDE's\nwith boundary or point control actions and ``smoothing'' observations. \n\n\\medskip\n\n\n\\subsection{Main Results}\nIn this subsection we shall formulate the main results, while the proofs are relegated to the next section. \nWe shall begin with a negative result.\n\nConsider the following minimization problem.\n\n\\begin{problem} \\label{p:pbm_0}\nFor any $y_0 \\in Y$, minimize the cost functional \\eqref{J} over all controls $L^2((0,T)\\times \\Gamma_0)$, \nwhere $y(\\cdot)$ satisfies the controlled equation \\eqref{e:eq-for-U}.\n\\end{problem}\n\n\\begin{theorem} \\label{l:neg} \nIf the initial state $y_0$ belongs to ${\\mathcal R}(B_1)$, then Problem \\ref{p:pbm_0} does not have a solution. \n\\end{theorem}\n\n\nGiven this negative result, one might wonder what are the additional constraints which render the problem solvable. \nThe proof of the negative result (cf.~\\cite{LPT}) reveals that the issue is in singularity of control, as\nthe ``candidate'' to be the optimal control is no longer in the space $L^2(0,T; U)$.\n(This depends upon the appearance of a (time-)trace operator -- intrisincally uncloseable -- in the definition of the state.) \n\nIn view of the above, we shall consider an input-to-state formula depending on a given parameter $g_0 \\in U$, that is \n\\begin{equation}\\label{g0}\ny_{g_0}(t) = e^{At}(y_0 - B_1g_0) + Lg(t)\\,,\n\\end{equation}\nwith $L$ defined in \\eqref{e:input-to-state-map}. This idea has been developed in \\cite{LLP,LPT}.\nWhen $g(0) =g_0$ the above controlled dynamics coincides with the one given by \\eqref{e:eq-for-U}.\nWith \\eqref{g0} we associate the same cost functional \\eqref{J}.\nA new (`extended') optimal control problem is formulated as follows.\n\n\\begin{problem} \\label{p:pbm_1-parameter}\nFor any $y_0 \\in [{\\mathcal D}({A^*}^2)]'$, $g_0\\in U$, minimize the cost functional \\eqref{J} overall controls $g\\in L^2((0,T)\\times \\Gamma_0)$, with $y$ subject to \\eqref{g0}.\n\\end{problem}\nFor this problem the following results holds true.\n\n\\begin{theorem}\\label{l:pos} \nThe optimization Problem \\ref{p:pbm_1-parameter} has a unique solution \n$\\hat{g}_{g_0} \\in L^2(0,T;U)$.\nIts corresponding optimal trajectory satisfies\n\\begin{equation}\\label{e:memberships} \n\\hat{y}_{g_0} \\in C([0,T];[{\\mathcal D}({A^*}^2]')\\,, \n\\quad\nR\\hat{y}_{g_0} \\in C([0,T];Y)\\,.\n\\end{equation} \n\\end{theorem}\n\nThe first main result of the paper establishes the feedback synthesis of the optimal control referred to in Theorem~\\ref{l:pos}. \nFor clarity of the exposition, we shall take $u_d =0$. \n\n\\begin{theorem}\\label{T0}\nWith reference to the minimization Problem \\ref{p:pbm_1-parameter}, the following \nstatements are valid.\n\n\\begin{enumerate}\n\n\\item[i)] (Partial regularity)\nFor any $y_0 \\in [{\\mathcal D}({A^2}^*)]'$, and any $g_0 \\in U$, the unique optimal control \n$\\hat{g}_{g_0}$ belongs to $C([0,T];U]$, and produces the output \n$R\\hat{y}_{g_0} \\in C([0,T];Y)$.\n \n\\item[ii)] (Riccati Equation) \nFor every $t \\in [0,T]$, there exists a self-adjoint positive operator $P(t)$ on $L(Y)$, \nwhose regularity is as follows,\n\\begin{equation*}\nA^* P(t) \\in {\\mathcal L}(Y)\\,, \\quad B_1^* A^* P(t) \\in {\\mathcal L}(Y,U) \\; \\textrm{continuously in time,}\n\\end{equation*}\nand which satisfies the following (non-standard) Riccati equation: \n\\begin{equation}\\label{e:RE-0}\n\\begin{split}\n& \\frac{d}{dt}(P(t) y, w)_Y +(Ay, P(t) w)_Y + (P(t) y, Aw)_Y + (Ry, Rw)_Y = \n\\\\[1mm] \n& \\qquad =((B_0^* + B_1^* A^*) P(t)y,([B_0^* + B_1 A^*) P(t) w)_{U} \n\\quad \\textrm{for all $y, w\\in {\\mathcal D}(A)$}\n\\end{split}\n\\end{equation}\nwith terminal condition $P(T) =0$.\nThe equation \\eqref{e:RE-0} actually extends to all $y, w \\in Y$.\n\n\\item[iii)] (Feedback synthesis) \nThe optimal control $\\hat{g}_{g_0}(\\cdot)$ has the following feedback representation:\n\\begin{equation*} \n\\hat{g}_{g_0}(t) = - \\big(I - [B_0^* + B_1^* A^*] P(t) B_1 \\big)^{-1} \n[B_0^* + B_1^* A^* ] P(t) \\hat{y}_{g_0}(t)\\,,\n\\end{equation*}\nwhere the operator $G(t)=I - [B_0^* + B_1^* A^*] P(t) B_1$ is boundedly invertible on $U$ for each \n$t \\in [0,T]$.\n\n\\end{enumerate}\n\n\\end{theorem}\nFrom the structure of the Riccati equation \\eqref{e:RE-0}, along with the space regularity of the operator \n$P(t)$ asserted in Theorem \\ref{T0}, some additional regularity of the operator $P(t)$ follows. \n\\begin{corollary}\nThe Riccati operator $P(t)$ is time differentiable from $Y$ into itself. \nMore precisely, the operator $ \\frac{d}{dt} P(t)\\colon Y \\rightarrow C([0,T];Y)$ is bounded. \n\\end{corollary}\n\n\n\\begin{remark}\n\\begin{rm}\nWe note that the Riccati equation \\eqref{e:RE-0} is termed {\\em non-standard} \n(already in \\cite{LPT}) because of the special structure of its quadratic term. \nThis feature results from the lack of coercivity in the functional cost, a cause for singularity of the minimization problem.\nThen, the feedback formula which allows the synthesis of the optimal control of Problem~\\ref{p:pbm_1-parameter}\ninvolves the inverse of certain operator defined on the control space $U$.\nInvertibility of the said operator is an issue already encountered in \\cite{LLP} and \\cite{LPT}: however, differently from those studies, in the present case we cannot appeal to the analyticity of the semigroup underlying the controlled dynamics.\n\\end{rm}\n\\end{remark}\n\nTheorem \\ref{T0} provides the optimal control and the optimal synthesis for the input-state dynamics\n\\eqref{g0}, given $y_0$ and the parameter $g_0$. \nOne aims then at exploring the relation between the parameter $g_0$ with the optimal control $\\hat{g}$,\nwhich is known from Theorem~\\ref{T0} to be continuous on $[0,T]$. \nThus, a question of major concern is whether the parameter $g_0\\in U$ can be selected \nin order that $\\hat{g}(0) = g_0$. \nThe validity of this property will prove the equivalence of the state description in \\eqref{e:eq-for-U} with the one in \\eqref{g0}, thereby ensuring that the latter system corresponds to the original PDE model. \nThe answer to this question is positive, as asserted by the Theorem below. \n\n\\begin{theorem}\\label{T:1}\nThe operator $[I + G(0)B_1]$ is bounded invertible on $U$; in particular, $[I + G(0)B_1]^{-1} \\in {\\mathcal L}(U)$. \nBy choosing $g_0 = [I + G(0) B_1]^{-1} G(0) y_0$, one obtains that\n\\begin{equation*}\n\\hat{y}(t) = e^{At }[ y_0 -B_1 \\hat{g}(0) ] + (L \\hat{g})(t)\\,,\n\\end{equation*} \nso that the original dynamics \\eqref{e:eq-for-U} coincides with the one in \\eqref{g0}. \nMoreover, the obtained $\\hat{g}$ is continuous in time, i.e. $\\hat{g}\\in C([0,T];U)$. \n\\end{theorem}\n\nForcing the original model with continuity of the control at the origin may compromise the optimality. \nInstead, the additional `player' $g_0 \\in U$ is advantageous from the optimality point of view. \nWhile we know that in general there is no optimal control in the class of $L^2(0,T;U)$ functions \n(cf.~Theorem~\\ref{l:neg}), reformulating the solution formula as in \\eqref{g0}, with an additional\nparameter, gives additional possibilities for optimization with respect to the parameter. \n\n\\begin{theorem}\\label{T:2}\nLet $U_0 \\subset U$ be a bounded and weakly closed set in $U$. \nThen, there exists a $g^* \\in U_0$ such that the resulting control $\\hat{g}_{g^*}$ attains the infimum of the functional $J(g)$ with respect to $g_0 \\in U_0$, $g \\in L^2(0,T;U)$ and $y$ satisfying (\\ref{g0}).\nMoreover, the following characterization holds true: either $g^*$ is such that \n$y_0 - B_1 g^* \\in \\ker(B_1^* P(0))$, or $g^* \\in \\partial U_0$.\n\\end{theorem}\n\n\n\\begin{remark}\n\\begin{rm}\nNote that the optimal control of Theorem~\\ref{T:2} provides control which is in a larger space than just\n$L^2(0,T; U)$. \nThis is singular control.\nThe corresponding state is described by \\eqref{g0} and it satisfies $R\\hat{y}_{\\hat{g}_{g^*}} \\in C([0,T]; Y)$.\n\\end{rm}\n\\end{remark}\n\nIt is important to note that from both the point of view of applications as well as of mathematical developments, it is significant to have two versions of optimal solutions corresponding to two different formulations of the input-state map. \nIf one is to develop nonlinear versions of the problem, where regularity of controls and of the states is of paramount importance, the first version in Theorem~\\ref{T:1} is the most relevant. \nHowever, from the point of view of automatic control -- where discontinuous inputs are feasible and lead to `better' optimization solutions --, Theorem~\\ref{T:2} becomes more relevant. \nClearly, further discussion of the topic along with relevant examples is appropriate and desirable. \n\n\\begin{remark}\n\\begin{rm}\nThere are several open problems sparked off by the present work. We name but a few.\n \n\\begin{enumerate}\n\\item[i)]\nExtension of the theory to more general observation operators $R$. \nHowever, it is clear that $R$ should display some kind of smoothing effect. \nMoreover, the structure of the problem -- namely, an appropriate interplay between control and observation\noperators -- will need to be carefully chosen, in order that the optimal ($L^2$) solution does exist. \n\n\\item[ii)]\nThe infinite horizon LQ problem in both the stable and the critical case. \nIt is expected that under suitable geometric conditions imposed on $\\Gamma_1$ one could guarantee \nsolvability of the optimization problem, along with a feedback synthesis of the optimal control. \n\n\\item[iii)]\nApplication of the previous result to the feedback control of the nonlinear equation. \nA local theory for small initial data should emerge, while the feedback control should provide a stabilizing effect on the nonlinear dynamics. \n\n\\end{enumerate}\n\\end{rm}\n\\end{remark}\n\nThe remaining parts of the paper are devoted to proofs of four Theorems. \n\n\\section{Proofs of Theorems \\ref{l:neg}, \\ref{l:pos}} \\label{s:proofs_1}\nWe point out at the outset that the main challenge in proving the stated results is to be able to `run' the dynamics on much larger dual spaces, still preserving the invariance \nof the said dynamics. \nThe following Proposition singles out some basic regularity and structural properties \npertaining to the observation operator $R$.\n\n\\begin{proposition}\\label{p:R}\nThe observation $R$ satisfies the following properties. \n\\begin{itemize}\n\\item\n$R \\colon Y \\rightarrow {\\mathcal D}({\\mathcal A}) \\times \\{0\\} \\times \\{0\\}$ is bounded;\n\\item\n$R \\in {\\mathcal L}(Y,{\\mathcal D}(A))$; \n\\item\n$R = R^* $ on $Y$, hence $R \\in {\\mathcal L}([{\\mathcal D}(A^*)]',Y)$.\n\\end{itemize}\n\n\\end{proposition}\n\n\\begin{proof}\nFor the first statement, take $y\\in Y$: then $y_1 \\in {\\mathcal D}({\\mathcal A}^{1\/2})$, and since\n$Ry =({\\mathcal A}^{-1\/2} y_1, 0, 0)^T$ we obtain ${\\mathcal A}^{-1\/2} y_1 \\in {\\mathcal D}({\\mathcal A})$. \n\nThe second statement follows from the calculation with $y \\in Y$\n\\begin{equation*}\nA R y = [0,0,-\\tau^{-1} c^2 {\\mathcal A}^{1\/2} y_1]^T \\in Y \n\\end{equation*}\nWe also note that $A^{-1} \\in {\\mathcal L}(Y,[{\\mathcal D}({\\mathcal A}^{1\/2})]^3)$. \nThe third statement follows from direct calculations using the inner product in $Y$. \n\nThe fourth statement follows combining the third with the second one. \n\\end{proof}\n\n\\subsection{Properties of the input-to-output map}\nThe following Lemma captures \na set of functional-analytic properties pertaining to appropriate combination of the involved abstract operators -- namely, the dynamics, control and observation operators --, which will play a major role in the proof of well-posedness of generalized differential\/integral Riccati equations, eventually leading to solvability of the optimal control problem.\n\n\\begin{lemma} \\label{l:abstract-basics}\nLet $A$, $B_i$ and $R$ the dynamics, control, observation operators defined by\n\\eqref{e:generator}, \\eqref{e:input-operators}, \\eqref{e:def-observation-op}, respectively.\nThen,\n\\begin{enumerate}\n\\item[i)]\n$RA^2$ can be extended to a {\\em bounded} operator on the state space $Y$;\n\\item[ii)]\n$RB_1=0$;\n\\item[iii)]\n$(I+A)^{-1}B_i$ are bounded and compact operators from $L^2(\\Gamma_i)$ into $Y$, $i=0,1$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\ni) We take an element $y=(y_1,y_2,y_3)$ initially assumed in ${\\mathcal D}(A^2)$, and compute\n\\begin{equation*}\n\\begin{split}\n& A^2y = A(Ay)= \n\\\\[1mm]\n& \\quad= A\n\\begin{pmatrix}\ny_2\n\\\\[1mm]\ny_3\n\\\\[1mm]\n- c^2{\\mathcal A} y_1- [b {\\mathcal A}+c({\\mathcal A}+I)N_1N_1^*({\\mathcal A}+I)]y_2\n-\\big[\\alpha I +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)\\big]y_3\n\\end{pmatrix}\n\\\\[1mm]\n& \\quad = \\begin{pmatrix}\ny_3\n\\\\[1mm]\n\\ldots\\ldots\\ldots\n\\\\[1mm]\n\\ldots\\ldots\\ldots\\ldots\\ldots\\ldots\n\\end{pmatrix}\n\\end{split}\n\\end{equation*}\nwhere the second and third component of $A^2y$ are unspecified, owing to the \nstructure of the observation operator $R$ to be applied.\nConsequently, \n\\begin{equation*}\nR\\, A^2y = \\begin{pmatrix}\n(I+{\\mathcal A})^{-1\/2} y_3 \\\\ 0 \\\\ 0\n\\end{pmatrix}\n\\end{equation*}\nand \n\\begin{equation*}\n\\|R\\, A^2y\\|_Y = \\left\\| \\begin{pmatrix} (I+{\\mathcal A})^{-1\/2} y_3 \\\\ 0 \\\\ 0 \\end{pmatrix}\\right\\|_Y\n= \\big\\|(I+{\\mathcal A})^{1\/2}(I+{\\mathcal A})^{-1\/2}y_3\\big\\|=\\|y_3\\|_{L^2(\\Omega)}\n\\end{equation*}\n\n\\smallskip\n\\noindent\nii) It is immediately verified that for any $h\\in L^2(0,T;L^2(\\Gamma_1))$ \n\\begin{equation*}\nR\\,B_1 h=R \\begin{pmatrix}\n0 \n\\\\[1mm]\n0 \n\\\\[1mm]\nb ({\\mathcal A}+I)N_1\\,h\n\\end{pmatrix}\n=(I+{\\mathcal A})^{-1\/2}\\,0=0\\,.\n\\end{equation*}\n\n\\smallskip\n\\noindent\niii) It is clear that the resolvent $(I+A)^{-1}$ is not compact.\nHowever, we have\n\\begin{equation}\n(I+A)^{-1} B_0 = c^2 \\begin{pmatrix} N_0 \\\\ 0 \\\\ 0 \\end{pmatrix}\\,, \n\\qquad\n(I+A)^{-1} B_1 = \\frac{b}{c^2} (I+A)^{-1} B_0=\nbc^{-2}\\begin{pmatrix} N_0 \\\\ 0 \\\\ 0 \\end{pmatrix}\n\\end{equation}\nand because ${\\mathcal R}(N_0) \\subset H^{3\/2}(\\Omega)$, the operators $A^{-1} B_i$\nare not only bounded from $L_2(\\Gamma_0) \\rightarrow Y$, but also compact. \n\n\\end{proof}\n\nThe following Lemma pertains to the regularity of the map $RL_0$.\n\n\\begin{lemma}\\label{l:L}\nLet $L_0$ be the operator defined by \\eqref{e:input-to-state-map}.\nThen \n\\begin{itemize}\n\\item\n$RL_0$ is a compact operator from $L^2(0,T;L^2(\\Gamma_0))$ into $C([0,T];Y)$.\n\\item\n$R e^{A\\cdot} B_i \\colon L^2(0,T;L^2(\\Gamma_0)) \\rightarrow C([0,T];Y)$, $i=0,1$,\nare compact. \n\\end{itemize}\n\n\\end{lemma}\n\n\\begin{proof}\nThe first statement follows computing \n\\begin{equation}\n\\begin{split}\nR L_0\\colon g\\longmapsto R L_0 g \n& = R \\int_0^t e^{A(t-s)} B_0 g(s) ds - R A\\int_0^t e^{A(t-s)} B_1 g(s) ds\n\\\\[1mm]\n& = R A^2 \\int_0^t e^{A(t-s)} A^{-2} B_0 g(s) - R A^2 \\int_0^t e^{A(t-s)} A^{-1} B_1 g(s)ds\n\\end{split}\n\\end{equation}\nin view of Lemma~\\ref{l:abstract-basics}, combined with Aubin-Simon compactness criterion. \n\nThe second statement follows rewriting $R e^{At } B_0$ as follows,\n\\begin{equation*}\nR e^{At } B_0 = R A e^{At} A^{-1} B_0\\,,\n\\end{equation*}\nwhere $RA\\in {\\mathcal L}(Y)$ and $A^{-1}B_0 \\colon U \\rightarrow Y$ compactly. \nThe strong additional regularity $R A^2 \\in {\\mathcal L}(Y) $ allows to handle \nthe time derivative \n\\begin{equation*}\n\\frac{d}{dt} R Ae^{At } A^{-1} B_0 = RA^2 e^{At} A^{-1} B_0 \\in {\\mathcal L}(U,Y)\\,,\n\\end{equation*}\nas needed for the applicability of the Aubin-Simon compactness criterion. \n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{l:neg}} \nWe will denote by $J(g)$ the cost functional $J(g,y)$, where $y(\\cdot)=y(\\cdot;g)$ corresponds to the state variable given by \\eqref{e:eq-for-U}. \nTake $y_0 \\in {\\mathcal R}(B_1)\\in [{\\mathcal D}(A^*)]'$ and select a sequence of controls $g_n \\in H^1(0,T; U )$\nsuch that \n\\begin{itemize}\n\\item[i)]\n$B_1 g_n(0) = y_0$, \n\\item[ii)]\n$g_n \\rightarrow 0$ in $L^2(0,Y;U)$. \n\\end{itemize}\nThen, with $y_n(t) =y_n(t,g_n) = e^{At} \\big(y_0 - B_1 g_n(0)\\big) + (L_0 g_n )(t) \n+ B_1 g_n (t)$ \nwe have \n\\begin{equation*}\nRy_n = R L_0 g_n \\longrightarrow 0 \\quad \\textrm{in $L^2(0,T;Y)$,} \n\\end{equation*}\non the strength of Lemma~\\ref{l:L}. \nConsequently, $J(g_n) \\rightarrow 0$.\n\nSince $g_n \\rightarrow 0$ in $L^2(0,T;U )$, we turn to\n$J(0) = \\int_0^T |R e^{At} y_0 |_Y^2 dt > 0$, \nwhich combined with $g_n \\rightarrow 0$ contradicts the existence of a minimizer. \n\n\\subsection{Proof of Theorem \\ref{l:pos}} \nThe argument is in principle standard, as it is based on proving weak lower semicontinuity\nof the cost functional. \nThus, the challenge is to establish appropriate regularity of the input-to-state map,\nwhich is not obvious in view of the high unboundedness of the control input operators. \nHowever, this is possible exploiting the smoothing effect of the observation operator\nas well as the properties specifically established for the input-to-output map (cf.~Lemma~\\ref{l:L}). \nTo wit: for a given $g_0 \\in U$ consider a minimizing sequence $g_n \\in L^2(0,T;U)$,\nso that $J(g_n) \\rightarrow d = \\inf_{g_n \\in L^2(0,T;U)} J(g_n)$.\nThen, coercivity of the cost in $L^2(0,T;U)$ gives the bound $\\|g_n\\|_{L_2(0,T;U)} \\le M$\nwhich implies that\n\\begin{equation} \ng_n \\rightarrow g \\quad \\text{weakly in $L^2(0,T;U)$.} \n\\end{equation}\nWe also have \n\\begin{equation*}\nR y_n(t) = Re^{At} (y_0 - B_1 g_0) + (RL_0 g_n)(t)\\,.\n\\end{equation*}\nOn the strength of Lemma~\\ref{l:abstract-basics} and Lemma \\ref{l:L}, for a subsequence\n-- denoted by the same symbol -- it follows $R L_0 g_n \\rightarrow R L_0 g$ in $L^2(0,T;Y)$.\nIn addition, $R e^{At} B_1 = R A e^{At}A^{-1} B_1$ is bounded from $L^2(0,T;U)$ into\n$L^2(0,T;Y)$. \nThis implies the weak lower semicontinuity of $J(g)$, along with $J(g) \\le d $, which proves optimality.\nThe regularity in \\eqref{e:memberships} pertaining to the observed optimal state,\nfollows in view of the obtained regularity of the three summands in \n\\begin{equation*}\nR y(t) = Re^{At} (y_0 - B_1 g_0) + (RL_0 g)(t)\\,,\n\\end{equation*}\nwhere in particular $Re^{At} y_0=RA^2 e^{At} A^{-2}y_0\\in C([0,T];Y)$ for any \n$y_0\\in [{\\mathcal D}((A^2)^*)]'$, \nthanks to the property i) of Lemma~\\ref{l:abstract-basics}. \n\n\\section{Proof of Theorem \\ref{T0}} \\label{s:proofs_2}\nGiven the the solution formula \\eqref{g0}, with the input-to-state map $L$ defined\nin \\eqref{e:input-to-state-map}, let us consider the dynamics \n\\begin{equation}\\label{alpha}\ny_{\\alpha}(t) = e^{At} \\alpha + (Lg)(t) \n\\end{equation}\ndepending on the parameter $\\alpha \\in [{\\mathcal D}(A^*)]'$.\nThis choice is justified by $B_1 g_0 \\in [{\\mathcal D}(A^*)]'$ for $g_0\\in U$.\n(We note that $y_{g_0}(\\cdot)$ has been used to denote the function in \\eqref{g0}, with emphasis on the dependence of $y$ on $g_0\\in U$, beside to $y_0$. \nIn the present section, although with a certain abuse of notation, with $y_{\\alpha}(\\cdot)$\nwe shall be always referring to the ``full'' parameter $\\alpha$, rather than to its component $g_0$.) \nRecall that \n\\begin{equation}\\label{A-1B}\nA^{-1} B_0 g= [{\\mathcal A}^{-1} ({\\mathcal A} + I ) N_0g, 0, 0 ]^T \\in \nH^{3\/2}(\\Omega) \\times \\{0\\} \\times \\{0\\} \\subset Y\\,. \n\\end{equation}\nWe add that on the strength of \\eqref{A-1B} and $A^{-2} AB_1 = A^{-1} B_1$,\none gets \n\\begin{equation}\\label{L}\nL\\in {\\mathcal L}(L^2(0,T;U),C([0,T];[{\\mathcal D}(A^*)^2]'))\\,.\n\\end{equation}\n\nThe following auxiliary control problem is naturally associated to \\eqref{alpha}.\n\n\\begin{problem}[\\bf Problem ${\\mathcal P}_\\alpha$] \\label{p:alfa}\nFor any $\\alpha \\in [{\\mathcal D}(A^{2*})]'$, minimize the functional \n\\begin{equation}\\label{Jx}\nJ(g,y_{\\alpha}) = \\int_0^T \\|R y_{\\alpha}\\|_Y^2 \\,dt + \\int_0^T \\|g\\|^2_{U}\\,dt\\,,\n\\end{equation}\noverall controls $g\\in L^2(0,T;U)$, with $y_{\\alpha}(\\cdot)$ solution to \\eqref{alpha}.\n\\end{problem}\n\nOf course, our goal is to obtain the results in the topology of the original spaces $Y$ and $U$. While this is not possible for the entire control system, \nit turns out that the optimal solution displays an additional regularity that will make it\npossible the return to the original state space. \nThe corresponding result is formulated below. \nFor simplicity of notation we shall set $C(Y) = C([0,T];Y)$ and $L^2(Y) = L_2(0,T;Y)$;\na similar notation will be adopted with $Y$ replaced by $U$. \n\n\\begin{proposition}\\label{T}\nWith reference to the parametrized control Problem~\\ref{p:alfa}, the following statements\nare valid. \n\\begin{enumerate}\n\n\\item[i)]\nFor any $\\alpha \\in [{\\mathcal D}({A^*}^2)]'$, there exists a unique optimal control\n$g^0 (\\cdot)\\in L^2(0,T;U)$, which additionally satisfies $g^0 \\in C([0,T];U)$. \nMoreover, $R y_{\\alpha}^0 \\in C[[0,T];Y)$. \n\n\\item[ii)]\nThere exists a selfadjoint, positive operator $P(t)$ on ${\\mathcal L}(Y)$ with the\nfollowing regularity,\n\\begin{equation*}\nA^* P(t)A \\in {\\mathcal L}(Y,C(Y))\\,, \\quad B_1^* A^* P(t) \\in {\\mathcal L}(Y,C(U))\\,,\n\\quad P_t \\in {\\mathcal L}(Y,C(Y))\\,;\n\\end{equation*}\n$P(t)$ satisfies the following (non-standard) Riccati equation, valid for any $y, w \\in {\\mathcal D}(A)$:\n\\begin{equation}\\label{Ric}\n\\begin{split}\n& \\frac{d}{dt}(P(t)y,w)_Y +(Ay, P(t)w)_Y + (P(t) y, Aw)_Y + (Ry, R\\hat{y} )_Y = \n\\\\[1mm] \n& \\; \\big( (B_0^* + B_1^* A^*) P y, [I + B_1^* R^* R B_1]^{-1} \n[ (B_0^* + B_1 A^*) P(t)w] \\big)_U\\,,\n\\end{split}\n\\end{equation}\nwith terminal condition $P(T) =0$.\n\n\\item[iii)]\nFor every $\\alpha \\in {\\mathcal D}({A^2}^*)]'$, the optimal cost \n$J(g^0) = \\min_{g \\in L_2(0,T; U )} J(g,y_{\\alpha})$ is given by \n$J(g^0) = (P(0)\\alpha, \\alpha)_Y$. \n\n\\item[iv)]\nThe optimal control has the following feedback representation:\n\\begin{equation*} \ng^0(t) = - \\big[I - (B_0^* + B_1^* A^*) P(t) B_1\\big]^{-1} \n\\big[ (B_0^* + B_1^* A^*)P(t)\\big] y_{\\alpha}^0(t)\\,,\n\\end{equation*}\nwhere the operator $I - (B_0^* + B_1^* A^*)P(t)B_1$ is boundedly invertible on $U$ \nfor each $t \\ge 0$. \n\n\\end{enumerate}\n\\end{proposition}\n\n\\ifdefined\\xxxxxx\n\\subsection{Matching the initial condition}. \nWe note that for any $x= y_0 - B_1g_0 $ with $y_0 \\in [{\\mathcal D}(A^*)]'$ and $g_0 \\in U$, \nthe corresponding optimal control $g^0 \\in C([0,T];U)$. \nThe latter follows from Part 1 of Theorem~\\ref{T}. \nTherefore, in order to comply with the original model one is asking for the following selection of $g_0$ $g_0 = g^0(0) $. This amounts to \n$$ g^0_{x}(t=0)=g_0, ~x = y_0 - B_1g_0 $$\nThe above implicit relation is always uniquely solvable for $g_0 \\in U $ as shown in \\cite{T}. \nIn fact, the matching condition amounts to solving \n$$ g_0 = G x = G (y_0 - B_1 g_0 ) $$\n$$[I + G B_1] g_0 = Gy_0 $$\nwhere $G\\equiv - [ I + B_1^* R^* R B_1]^{-1} [ B_1^* R^* R + ( B_0^* + B_1^* A^* ) P(0) ]$\n\nWith the key properties $G \\in L([D(A^*)]') \\rightarrow U )$ \nand $ [ I + GB_1 ]^{-1}\\in L(U ) $ to be shown later. \nThus we obtain \n\\begin{corollary}\nLet $y_0 \\in D(A^*)]' $ be given. Consider Problem $\\mathcal{P}_x $ with $x = y_0 -B_1g_0$ and $g_0 \\in U $ is given by\n\\begin{equation}\\label{g00}\n g_0 = [ I + GB_1 ]^{-1} G y_0 \n \\end{equation}\n Then there exists unique optimal control $g^0\\in L_2(0,T; U) $ with the \n corresponding trajectory (\\ref{e:eq-for-U}) and initial condition $y^0(0) = y_0 $, such that \n the results of Theorem \\ref{T} holds with $x = y_0 - B_1 g_0 $ .\n \\end{corollary}\n \nIn other words, by solving parametrized optimal control with a given $x = y_0 - B_1 g_0 $ and a parameter p $g_0 \\in U $ we solve a family of parametrized optimal control problems, which always has a unique solution. \n The original dynamics is included in tis family. \n By selecting $g_0$ according to the matching condition, we make a selection of a problem whose dynamics coincides with the original one. However, the above does not imply that the constructed optimal control for parametrized control problem is also optimal for the original problem -when considered within $L_2(U)$ framework for optimal controls. In fact, the latter may not have optimal solution at all when \n$y_0 \\in {\\mathcal R}(B_1) $ -as shown in \\cite{LPT}. Thus, the constructed control is suboptimal -but it corresponds \n the original dynamics. \n However, if the original problem does have $L_2(U) $ optimal control, then such control coincides with \n a parametrized control where $g_0$ is selected according to the matching condition. \n\\fi\n\n\\subsection{Proof of Proposition \\ref{T} }\n\\ifdefined\\xxx\n\n\\subsection{Properties of the input-output map}\nThe following Lemma captures\na set of functional-analytic properties pertaining to appropriate combination of the involved abstract operators -- namely, the dynamics, control and observation operators --, which will play a major role in the proof of well-posedness of generalized differential\/integral Riccati equations, eventually leading to solvability of the optimal control problem.\n\n\\begin{lemma} \\label{l:abstract-basics}\nLet $A$, $B_i$ and $R$ the dynamics, control, observation operators defined by\n\\eqref{e:generator}, \\eqref{e:input-operators}, \\eqref{e:def-observation-op} respectively.\nThen, we have\n\\begin{enumerate}\n\\item[i)]\n$RA^2$ can be extended to a {\\em bounded} operator on the state space $Y$;\n\\item[ii)]\n$RB_1=0$;\n\\item[iii)]\n$(I+A)^{-1}B_i$ are bounded and compact operators from $L^2(\\Gamma_i)$ into $Y$, $i=0,1$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\ni) We take an element $y=(y_1,y_2,y_3)$ initially assumed in ${\\mathcal D}(A^2)$, and compute\n\\begin{equation*}\n\\begin{split}\n& A^2y = A(Ay)= \n\\\\[1mm]\n& \\quad= A\n\\begin{pmatrix}\ny_2\n\\\\[1mm]\ny_3\n\\\\[1mm]\n- c^2{\\mathcal A} y_1- [b {\\mathcal A}+c({\\mathcal A}+I)N_1N_1^*({\\mathcal A}+I)]y_2\n-\\big[\\alpha I +\\frac{b}{c}({\\mathcal A}+I)N_1 N_1^*({\\mathcal A}+I)\\big]y_3\n\\end{pmatrix}\n\\\\[1mm]\n& \\quad = \\begin{pmatrix}\ny_3\n\\\\[1mm]\n\\ldots\\ldots\\ldots\n\\\\[1mm]\n\\ldots\\ldots\\ldots\\ldots\\ldots\\ldots\n\\end{pmatrix}\n\\end{split}\n\\end{equation*}\nwhere the second and third component of $A^2y$ are unspecified, owing to the \nstructure of the observation operator $R$ to be applied.\nConsequently, \n\\begin{equation*}\nR\\, A^2y = \\begin{pmatrix}\n(I+{\\mathcal A})^{-1\/2} y_3 \\\\ 0 \\\\ 0\n\\end{pmatrix}\n\\end{equation*}\nand \n\\begin{equation*}\n\\|R\\, A^2y\\|_Y = \\left\\| \\begin{pmatrix} (I+{\\mathcal A})^{-1\/2} y_3 \\\\ 0 \\\\ 0 \\end{pmatrix}\\right\\|_Y\n= \\big\\|(I+{\\mathcal A})^{1\/2}(I+{\\mathcal A})^{-1\/2}y_3\\big\\|=\\|y_3\\|_{L^2(\\Omega)}\n\\end{equation*}\n\n\\smallskip\n\\noindent\nii) It is immediatly verified that for any $h\\in L^2(0,T;L^2(\\Gamma_1))$ \n\\begin{equation*}\nR\\,B_1 h=R \\begin{pmatrix}\n0 \n\\\\[1mm]\n0 \n\\\\[1mm]\nb ({\\mathcal A}+I)N_1\\,h\n\\end{pmatrix}\n=(I+{\\mathcal A})^{-1\/2}\\,0=0\\,.\n\\end{equation*}\n\n\\smallskip\n\\noindent\niii) It is clear that the resolvent $(I+A)^{-1}$ is not compact.\nHowever, we have\n\\begin{equation}\n(I+A)^{-1} B_0 = c^2 \\begin{pmatrix} N_0 \\\\ 0 \\\\ 0 \\end{pmatrix}\\,, \n\\qquad\n(I+A)^{-1} B_1 = \\frac{b}{c^2} (I+A)^{-1} B_0=\nbc^{-2}\\begin{pmatrix} N_0 \\\\ 0 \\\\ 0 \\end{pmatrix}\n\\end{equation}\nand because ${\\mathcal R}(N_0) \\subset H^{3\/2}(\\Omega)$, the operators $A^{-1} B_i$\nare not only bounded from $L_2(\\Gamma_0) \\rightarrow Y$, but also compact. \n\n\\end{proof}\n\nThe following Lemma pertains to regularity of the map $RL_0$, that is\n\n\\begin{lemma}\nLet $L_0$ be the operator defined by \\eqref{e:input-to-state-map}.\nThen $RL_0$ is a compact operator from $L^2(0,T;L^2(\\Gamma_0))$ into $C([0,T];Y)$.\n\\end{lemma}\n\n\\begin{proof}\nFollows from Lemma~\\ref{l:abstract-basics} \nfollowed by\n\\begin{equation}\n\\begin{split}\nR L_0\\colon g\\longmapsto R L_0 g \n& = R \\int_0^t e^{A(t-s)} B_0 g(s) ds - R A\\int_0^t e^{A(t-s)} B_1 g(s) ds\n\\\\[1mm]\n& = R A^2 \\int_0^t e^{A(t-s)} A^{-2} B_0 g(s) - R A^2 \\int_0^t e^{A(t-s)} A^{-1} B_1 g(s)ds\n\\end{split}\n\\end{equation}\nand combined with Aubin-Simon compactness criterion. \n\\end{proof}\n\\fi\n\n\\subsubsection{The parametrized LQ-problem} \nThe starting point is the semigroup solution $y(t)= e^{At} \\alpha + Lg(t)$.\nIn order to derive the synthesis for the ``enlarged'' problem by introducing a parameter \n$\\alpha \\in Y$ and later considering the family of control problems depending on a parameter \n$\\alpha \\in Y \\oplus {\\mathcal R}(B_1)$, one needs to develop a dynamics that is invariant on the space compatible with initial data. \n\nIn view of the above, it is essential to extend the action of the semigroup $e^{At}$, originally defined on $Y$, to a larger space which contains $Y \\oplus {\\mathcal R} (B_1)$. \nThis can be done on the strength of the extended regularity of the operator $B_i$ as acting \ninto dual spaces of ${\\mathcal D}(A^*)$. This will be seen below. \nThe low regularity of the input-to-state mapping $L$ will force us to run the dynamics written below on an even larger space which is $[{\\mathcal D}({A^2}^*)]'$. \n\\begin{equation}\\label{e:eq-in-U-with-alpha}\ny(t) = e^{At} \\alpha + Lg(t) = e^{At} \\alpha + B_1 g(t) +[ L_0 g](t) \\,,\n\\end{equation}\nIt is important to emphasize that $y(0)\\ne \\alpha$, whereas $y(0)= \\alpha+B_1g(0)$. \nSince\n\\begin{equation*}\nA^{-1} B_1g = \n\\begin{pmatrix}b c^{-2} {\\mathcal A}^{-1} ({\\mathcal A} + I ) N_0g\n\\\\\n0\n\\\\\n0\n\\end{pmatrix}\\,,\n\\end{equation*}\nwe have $A^{-1} B_1 \\in {\\mathcal L}(U,Y)$ (in fact, compactly). \nThis follows from the regularity of the Neumann map where \n$N_0 \\in {\\mathcal L}(L^2(\\Gamma_0),H^{3\/2}(\\Omega))$, where $H^{3\/2}(\\Omega) \\subset {\\mathcal D}({\\mathcal A}^{1\/2})$ \n(the latter being a compact embedding). \nWe can thus take $\\alpha$ in $[{\\mathcal D}({A^2}^*)]'$.\nSo the dynamics operator with $g \\in C([0,T];U)$ will have values in the dual space \n$[{\\mathcal D}({A^2}^*)]'$.\n\nBy the same arguments as these used for the proof of Theorem \\ref{l:pos} we obtain\n\n\\begin{lemma}[Auxiliary optimal control problem] \\label{p:auxiliary}\nGiven $\\alpha \\in [{\\mathcal D}({A^*}^2]'$, there exists a control function \n$g^0\\in L^2(0,T;U)$ which minimizes the cost functional \\eqref{Jx}, where \n$y(\\cdot)$ is the solution to \\eqref{e:eq-in-U-with-alpha} corresponding to\nthe control $g(\\cdot)$.\n\\end{lemma}\nOur main goal is to provide a feedback synthesis of the optimal control $g^0$. \n\nWhile the existence of optimal solution for the parametrized problem follows from\nLemma~\\ref{p:auxiliary}, in order to provide a (pointwise in time) feedback representation\nof the optimal control -- via the optimal cost operator $P(t)$ -- one needs to introduce, for any $s\\in [0,T)$, the dynamics described by the equation \n\\begin{equation}\\label{e:s-eq-in-U-with-alpha}\ny(t,s;\\alpha) = e^{A(t-s)} \\alpha + L_sg(t)\\,, \\qquad s\\le t\\le T\\,,\n\\end{equation}\nas well as the cost functional\n\\begin{equation}\\label{e:s-cost}\nJ_{s,T}(g) \\equiv \\int_s^T \\big(\\|Ry(t)\\|^2_{Y} + \\|g(t)\\|^2_U \\big)\\,dt\\,, \n\\end{equation}\nwhere as before $y=(u,u_t,u_t)$ and $L_{s,T}$ -- $L_s$, in short -- is the operator defined by \n\\begin{equation}\\label{e:s-input-to-state-operator}\n\\{L_sg\\}(t)= \\int_s^t e^{A(t-\\tau)} B_0 g(\\tau)\\,d\\tau \n+ A\\int_s^t e^{A(t-\\tau)} B_1 g(\\tau)\\,d\\tau +B_1 g(t) \\qquad \\forall t\\in [s,T]\\,. \n\\end{equation}\n(Note that the subscript {\\em s} refers to initial time: in order to avoid confusion, the former operator $L_0=L-B_1$ is written $L^0$.)\n\n\\begin{lemma}\\label{l:regularity-input-to-state}\nOne has the following basic regularity of the input-to-state map:\n\\begin{equation*}\nL^0_s \\; \\text{is continuous}\\colon L^2(s,T;U) \\longrightarrow C([s,T];[{\\mathcal D}({A^*}^2)]')\\,,\n\\end{equation*}\n\\begin{equation*}\nL_s \\; \\text{is continuous}\\colon L^2(s,T;U) \\longrightarrow \nL^2(s,T;[{\\mathcal D}(A^*)]')\\oplus C([s,T];[{\\mathcal D}({A^*}^2)]')\\,,\n\\end{equation*}\nThe above regularity improves when input-to-state map is combined with the observation operator\n$R$; indeed, for the operator $RL$ and its adjoint it holds\n\\begin{equation*}\n\\begin{split}\n& RL_s \\; \\text{continuous}\\colon L^1(s,T;U) \\longrightarrow C([s,T];Y)\\,;\n\\\\[1mm]\n& L_s^*R^* \\; \\text{continuous}\\colon L^1(s,T;Y) \\longrightarrow C([s,T];U)\\,.\n\\end{split}\n\\end{equation*}\nIn addition, the operator $L_s^*R^*$ satisfies \n\\begin{equation*}\nL_s^*R^* \\; \\text{continuous}\\colon L^2(s,T;Y) \\longrightarrow C([s,T];U)\n\\end{equation*}\nuniformly with respect to $s\\in [0,T)$. \n\\end{lemma}\n\n\\begin{proof}\nThe regularity of the control-state map follows from the quantified regularity of $B_1$ map which takes boundedly $U$ into $[{\\mathcal D}((A^*)]'$. \nThen the first statement in the Lemma follows from the structure of $L$ operator.\nThe key in the regularity control $\\rightarrow$ observation operator is the combination of the three properties $A^{-2}B_i\\in{\\mathcal L}(U,Y)$, $i=1,2$, $RA^2\\in {\\mathcal L}(Y)$, $RB_1=0$.\n\\end{proof}\n\n\\begin{lemma}\nWith reference to the optimal control problem \\eqref{e:s-eq-in-U-with-alpha}-\\eqref{e:s-cost},\nthe following statements are valid:\n\\begin{enumerate}\n\\item[i)]\n{\\bf (Optimal pair). }\nGiven $\\alpha\\in [{\\mathcal D}({A^*}^2)]'$, there exists a unique optimal pair\n\\begin{equation*}\n(\\hat{y}(t,s;\\alpha),\\hat{g}(t,s;\\alpha)) \n\\end{equation*}\nfor Problem~\\ref{p:auxiliary}, with \n\\begin{subequations}\n\\begin{align}\n& \\hat{g}(t,s;\\alpha)=[I+L_s^*R^*RL_s]^{-1}L_s^*R^*Re^{A(\\cdot-s)}\\alpha\\in C([s,T];U)\\,,\n\\\\[2mm]\n& \\hat{y}(t,s;\\alpha)= \ne^{A(t-s)}\\alpha + \\{L_s\\hat{g}(\\cdot,s;\\alpha)\\}(t)\\in C([s,T];[{\\mathcal D}({A^*}^2)]')\\,,\n\\label{e:basicregularity}\\\\[2mm]\n& R\\hat{y}(t,s;\\alpha)= [I+RL_sL_s^*R^*]^{-1}Re^{A(\\cdot-s)}\\alpha \\in C([s,T];Y)\\,.\n\\end{align}\n\\end{subequations}\n\\item[ii)]\n{\\bf (Riccati operator). }\nThe operator $P(t)\\in {\\mathcal L}(Y)$, $t\\in [s,T]$, is \ngiven by\n\\begin{equation}\\label{e:riccati-operator-2}\nP(t) \\alpha = \\int_t^T e^{A^*(\\tau-t)}R^*R \\hat{y}(\\tau,t;\\alpha)\\,d\\tau\\,, \n\\end{equation} \nThe operator $P(t)$ is positive selfadjoint on $Y$, and represents the optimal cost (or Riccati) operator; its regularity properties are detailed separately (cf.~Proposition~\\ref{p:Riccati-operator} below).\n\\item[iii)]\n{\\bf (Implicit feedback formula). }\nThe optimal control satisfies\n\\begin{equation*}\n\\hat{g}(t,s;\\alpha)= -[B_0^*P(t) +B_1^*A^*P(t)]\\Phi(t,s)\\alpha\\,,\n\\end{equation*}\nthat is the following implicit equation\n\\begin{equation*}\n\\hat{g}(t,s;\\alpha)= -[B_0^*P(t) +B_1^*A^*P(t)]\\hat{y}(t,s;\\alpha)\n+[B_0^*P(t) +B_1^*A^*P(t)]B_1\\hat{g}(t,s;\\alpha)\\,,\n\\end{equation*}\nwhere the operator $\\Phi(t,s)$ is defined in \\eqref{e:transition}.\n\\item[iv)]\n{\\bf (Optimal cost). }\nThe optimal cost for Problem~\\ref{p:auxiliary} is given by\n\\begin{equation*}\nJ_s(\\hat{g}) =\\int_s^T \\big(\\|R\\hat{y}\\|^2_Y + |\\hat{g}(t)|^2_U \\big)\\,dt\n= \\|[I+RL_sL_s^*R^*]^{-1\/2}\\, Re^{A(\\cdot-s)}\\alpha\\|_{L^2(s,T;Y)}^2 \n\\end{equation*}\nwhich is rewritten in terms of the optimal cost (or Riccati) operator as follows\n\\begin{equation}\n\\begin{split}\nJ_s(\\hat{g}) &=(P(s)\\alpha,\\alpha)=\n\\\\\n&= \\big([I+RL_sL_s^*R^*]^{-1}\\, Re^{A(\\cdot-s)}\\alpha,Re^{A(\\cdot-s)}\\alpha\\big)_{L^2(s,T;Y)}\\,,\n\\end{split}\n\\end{equation}\nthereby providing \n\\begin{equation}\nP(s)\\alpha=e^{A^*(\\cdot-s)}R^*\\,[I+RL_sL^*_sR^*]^{-1}\\,Re^{A(\\cdot-s)}\\alpha \n\\quad \\forall \\alpha \\in [{\\mathcal D}({A^*}^{2}]'\\,.\n\\end{equation}\n\n\\end{enumerate}\n\n\\end{lemma}\n\n\\begin{proof}\n1. The first statement follows by standard variational arguments applied to the \nLQ-problem (cf.~\\cite{redbook}), \nafter taking into consideration the regularity of input-output map stated\nin the preceding Lemma. \nThe formulas for the optimal control, optimal state, observed state are derived\nas usual from the optimality conditions. \nThe regularity of the optimal quantities follows from the regularity of the map $L$. \nIn fact $A^{-2}\\alpha \\in Y$ gives $R e^{At} \\alpha = RA^2 e^{At} A^{-2}\\alpha \\in C([0,T];Y)$ and by Lemma~\\ref{l:regularity-input-to-state}\n\\begin{equation*}\nL_s^* R^* R e^{A\\cdot} \\alpha\\in C([0,T];U)\\,.\n\\end{equation*}\n\nWe note that the invertibility of the operator $I + L_s^* R^* R L_s$ on $C([s,T];U)$ follows\ncombining the self-adjointness and positivity of $L_s^* R^* R L_s$ -- which guarantees the invertibility on $L_2(U)$ -- with boundedness of the latter operator on $C([s,T];U)$. \nA classical bootstrap argument yields the claimed regularity: \none starts from \n\\begin{equation*}\nv=[I + L_s^* R^* R L_s ]^{-1} g\\,,\n\\end{equation*}\n with $g \\in C(U)$, obtaining first \n$v \\in L^2(U)$; then, since $v = - L_s^* R^* R L_s v +g$, the regularity improves to \n$v \\in C(U)$. \n\nA word of caution: while $RL_0 $ is compact on $L^2(U)$, this is no longer the case for $RL$,\nowing to the presence of the summand $R B_1$, which is not time compact. \n\nThe regularity of $R\\hat{y}(t,s;\\alpha)$ is a consequence of the regularity of the operator \n$RL$ in Lemma~\\ref{l:regularity-input-to-state}. \nThen, by the optimality condition \n\\begin{equation}\\label{e:optimal-control-from-optimality}\n\\hat{g}(t,s;\\alpha)=-\\{L_s^*R^*[R\\hat{y}(\\cdot,s;\\alpha)]\\}(t)\\,,\n\\end{equation}\nwhich combined with the regularity of the operator $L_s^*R^*$ yields continuity (in time)\nof the optimal control. \n\n\\noindent\n\\smallskip\n2. All the statements ii)-iv) follow by variational arguments, \nby using the structure of the optimal quantities, once several properties \nthat specifically pertain the operators $\\Phi(\\cdot,\\cdot)$ and $P(\\cdot)$ are proved. \nThese technical results are given in the Propositions which follow next. \n\\end{proof}\n\n\\begin{remark}\n\\begin{rm}\nA peculiarity of the parametrized minimization problem is that the optimal trajectory\ndoes not satisfy the evolution property. \n(For this reason the Riccati operator and the resulting synthesis cannnot be standard, as certain cancellations do not occur.)\nIn the next section we study the evolution operator, defined only on a dual (extrapolation) space. \nThis is a consequence of the low regularity of the control-to-state map. \n\\end{rm}\n\\end{remark}\n\n\\subsubsection{The operator $\\Phi(t,s)$}\nOne of the most critical ingredients of Riccati theory is the evolution operator which\ndescribes controlled dynamics. \nWhile in the standard theory the evolution operator is constructed directly from the optimal trajectory, this is not the case in singular theory. \nThe reason is that such operator will not display the evolution property -- the most fundamental feature. \nFor this reason we define evolution differently, as in the formula below. \n \nFor any couple $(t,s)$ such that $0\\le s\\le t\\le T$, let \n$\\Phi(t,s)\\colon [{\\mathcal D}({A^*}^2)]'\\rightarrow [{\\mathcal D}({A^*}^2)]'$ defined by\n\\begin{equation}\\label{e:transition}\n\\Phi(t,s)\\alpha := \\hat{y}(t,s;\\alpha)-B_1\\hat{g}(t,s;\\alpha)=\ne^{A(t-s)} \\alpha + \\{L^0_s \\hat{g}(\\cdot,s;\\alpha)\\}(t)\\,.\n\\end{equation}\nThe regularity properties of the operator $\\Phi(\\cdot,\\cdot)$, which a priori\nbelongs to ${\\mathcal L}([{\\mathcal D}({A^*}^2)]')$ (for $(t,s)$ given), are collected in the following Proposition.\n\n\\begin{proposition}\\label{p:Phi}\nFor the operator $\\Phi(\\cdot,\\cdot)$ defined in \\eqref{e:transition} \nthe following properties are valid:\n\\begin{enumerate}\n\\item[i)]\n$\\Phi(t,t)\\alpha =\\alpha$ for all $\\alpha\\in [{\\mathcal D}({A^*}^2)]'$;\n\\item[ii)]\nfor any $s, \\tau$ with $0\\le s\\le \\tau\\le T$, it holds\n\\begin{equation}\\label{e:transition-item2}\nRe^{A(\\cdot-\\tau)}\\Phi(\\tau,s)\\alpha\\in C([\\tau,T];Y) \\qquad \\forall \\alpha\\in [{\\mathcal D}({A^*}^2)]'\n\\end{equation}\ncontinuously with respect to $\\alpha$ and uniformly in $s$ and $\\tau$;\n\\item[iii)]\nfor any $s, \\tau$ with $0\\le s\\le \\tau\\le T$, it holds\n\\begin{equation*}\nR\\Phi(\\cdot,\\tau)\\,\\Phi(\\tau,s)\\alpha\\in C([\\tau,T];Y) \\qquad \\forall \\alpha\\in [{\\mathcal D}({A^*}^2)]'\n\\end{equation*} \ncontinuously with respect to $\\alpha$ and uniformly in $s$ and $\\tau$;\n\\item[iv)]\nfor any $s,\\tau, t$ with $0\\le s\\le \\tau\\le t\\le T$, it holds in $Y$ \n\\begin{equation*}\nR\\Phi(t,\\tau)\\,\\Phi(\\tau,s)\\alpha= R \\Phi(t,s)\\alpha \\qquad \\forall \\alpha\\in [{\\mathcal D}({A^*}^2)]'\n\\end{equation*} \n\\end{enumerate}\n\n\\end{proposition}\n\n\\begin{proof}\nSince the operator $\\Phi(t,s)$ -- as defined above -- has the same algebraic structure as in\nthe classical LQ-theory, we can treat this operator as an evolution on the dual space to \n${\\mathcal D}({A^*}^2)$. \nThe needed regularity is established by referring to preceding Lemmas: in particular,\nto Lemma~\\ref{l:regularity-input-to-state}.\nThe proof of the above properties can be produced along the lines of Lemma~8.3.2.3\nand Lemma~8.3.2.4 in \\cite{redbook}, on the basis of the powerful facts\n$RB_1=0$, $R A^2\\in {\\mathcal L}(Y)$, beside $A^{-2}B_i\\in{\\mathcal L}(U,Y)$, $i=1,2$. \n\\end{proof}\n\n\n\\subsubsection{The optimal cost operator} \\label{s:riccati-operator}\nWe note that the Riccati Operator defined via optimal trajectory coincides with \n\\begin{equation}\\label{e:riccati-operator-1}\nP(t) \\alpha = \\int_t^T e^{A^*(\\tau-t)}R^*R \\Phi(\\tau,t)\\alpha\\,d\\tau\\,,\n\\qquad 0\\le t\\le T\\,, \\; \\alpha\\in [{\\mathcal D}({A^*}^2)]'\\,,\n\\end{equation} \nwhere $\\Phi(\\tau,t)$ is defined in \\eqref{e:transition}.\nIt is readily seen that, combining\n$\\Phi(\\tau, t)\\alpha=\\hat{y}(\\tau,t;\\alpha)-B_1 \\hat{g}(\\tau,t;\\alpha)$ with $RB_1=0$, \n\\eqref{e:riccati-operator-1} is actually equivalently rewritten as follows\n\\begin{equation*} \nP(t) \\alpha = \\int_t^T e^{A^*(\\tau-t)}R^*R \\hat{y}(\\tau,t;\\alpha)\\,d\\tau\\,,\n\\qquad 0\\le t\\le T\\,, \\; \\alpha\\in [{\\mathcal D}({A^*}^2)]' \n\\end{equation*} \nwhich confirms the equivalent relation \\eqref{e:riccati-operator-2}.\n\n\\begin{proposition} \\label{p:Riccati-operator}\nThe optimal cost operator $P(t)$ defined by \\eqref{e:riccati-operator-1} (equivalently,\nby \\eqref{e:riccati-operator-2}) satisfies the following (enhanced) regularity properties:\n\n\\begin{enumerate}\n\\item\n{\\rm (\\bf Space regularity)}\nFor any given $t\\in [0,T]$, one has \n\\begin{equation}\\label{e:riccati-spaceregularity}\n{A^*}^2 P(t) A^2 \\in {\\mathcal L}(Y)\\,;\n\\end{equation}\nequivalently,\n\\begin{equation\nP(t)\\in {\\mathcal L}([{\\mathcal D}({A^*}^{\\gamma_1})]',{\\mathcal D}({A^*}^{\\gamma_2})) \n\\qquad \\forall \\gamma_1, \\gamma_2\\le 2\\,.\n\\end{equation}\n\nAs a consequence, $B_i^*P(\\cdot)A^2\\in {\\mathcal L}(Y,U)$, $i=1,2$ and\nthe gain operator $B^*P(t)\\equiv B_0^*P(t)+B_1^*A^*P(t)$ satisfies $B^*P(t) A^2 \\in {\\mathcal L}(Y,U)$;\nnamely,\n\\begin{equation}\\label{gain-spaceregularity}\nB_i^*P(t)\\in {\\mathcal L}([{\\mathcal D}({A^*}^2)]',U))\\,; \\qquad i =0,1\\,.\n\\end{equation}\n\\item\n{\\bf (Time regularity)} As for the regularity in time of the optimal cost operator -- then,\nof the value function -- one has\n\\begin{equation}\\label{e:riccati-timeregularity}\nP(\\cdot) \\;\\textrm{continuous} \\colon \n[{\\mathcal D}({A^*}^2)]' \\longrightarrow C(0,T;{\\mathcal D}({A^*}^2))\\,.\n\\end{equation}\n\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n1. Let $\\alpha\\in [{\\mathcal D}({A^*}^2)]'$ be given.\nWe write down and compute, with $0\\le t\\le T$, \n\\begin{equation*}\n\\begin{split}\n(-A^*)^2P(t)\\alpha &= (-A^*)^2\\int_t^T e^{A^*(\\tau-t)}R^*R \\Phi(\\tau,t\n\\alpha\\,d\\tau \n\\\\[1mm]\n& = \\int_t^T e^{A^*(\\tau-t)}[(-A^*)^2R^*]\\,R \\Phi(\\tau,t)\\alpha\\,d\\tau \n\\end{split}\n\\end{equation*}\nwhere the application of the operator $(-A^*)^2$ commutes with the integration\nin time on the extrapolation space.\n \nThen, the conclusion in \\eqref{e:riccati-spaceregularity} follows recalling that\nthe function $R \\Phi(\\cdot,t)\\alpha$ takes values in $Y$ (cf.~\\eqref{e:transition-item2}), whilst $(-A^*)^2R^*$ is a bounded operator on $Y$.\n\\\\\nAs for gain operator, on the basis of \\eqref{e:riccati-spaceregularity}, we next obtain \n\\begin{equation*}\nB_i^*P(\\cdot)A^2=[B_i^*(-A^*)^{-\\gamma_0}]\\,(-A^*)^{\\gamma_0}P(\\cdot)A^2 \\in {\\mathcal L}(Y,U)\\,,\n\\qquad i=1,2\\,,\n\\end{equation*}\nowing to $B_i^*(-A^*)^{-\\gamma_0}\\in {\\mathcal L}(Y,U)$, $ \\gamma_0 =1 $ thereby confirming the exceptional \nboundedness and smoothing effect of the gain operator in \\eqref{gain-spaceregularity}.\n\n\\smallskip\n\\noindent\n2. Finally, the regularity in time of \\eqref{e:riccati-timeregularity}\nfollows combining the continuity in time of the function $R \\Phi(\\cdot,t)\\alpha$ \n(see, once again, \\eqref{e:transition-item2}) with more standard semigroup properties;\nsee the proof in \\cite[p.~697]{redbook}.\n\\end{proof}\n\n\\subsubsection{The Riccati equation}\nIn this section we shall provide several key relations which lead to a characterization of the Riccati operator via Differential Riccati equation. \nOne of the fundamental properties is time evolution (of the evolution operator) with respect to the initial time, that is the second argument. \nIn the case of semigroups both evolutions are the same. \nHowever, in the case of time dependent evolutions -- as in the present case -- proving differentiability with respect to the initial time is challenging.\nThe challenge is due to compromised regularity and the intrinsic lack of invariance. \n\n\\begin{lemma}[Differentiability of the evolution with respect to initial time]\\label{Right}\n\nThe evolution operator $\\Phi(\\tau,t)$ defined in \\eqref{e:transition} satisfies\n\\begin{equation*}\n\\frac{d}{dt} \\big (R\\Phi(\\tau,t)\\alpha\\big)=- R\\Phi(\\tau,t)\\big[A-BB^*P(t)\\big]\\alpha\n\\qquad \\forall \\alpha \\in [{\\mathcal D}({A^*})]'\\,, \\quad \\textrm{a.e. in $t$,}\n\\end{equation*}\nwhere $B$ denotes $B_0 + A B_1$.\n\\end{lemma}\n\n\\begin{proof}\nWe will sketch the major steps of the proof.\n\n\\noindent\n1. We have seen that $R\\Phi(t,s)$ may be defined on the extrapolation space \n$[{\\mathcal D}({A^*}^2)]'$.\nIn particular, it makes sense $R\\Phi(t,s)Bu$ and it holds\n\\begin{equation*}\n\\sup_{0\\le t\\le T}\\|R\\Phi(\\cdot,t)Bu\\|_{L^1(t,T;Y)}\\le c_T\\|u\\|_U\\,.\n\\end{equation*}\nTo justify the above assertion: we recall that \n\\begin{equation*}\nR\\Phi(\\cdot,t)\\alpha=Re^{A(\\tau-t)}\\alpha+R\\{L_t \\hat{g}(\\cdot,t;\\alpha)\\}(\\tau) \n\\end{equation*}\nwhich combined with \\eqref{e:optimal-control-from-optimality} gives \n\\begin{equation}\\label{e:rphi}\nR\\Phi(\\tau,t)\\alpha = \\big\\{ \\big[I+RL_tL_t^*R^*\\big]^{-1} \\, Re^{A(\\tau-t)}\\alpha\\big\\}(\\tau)\\,,\n\\quad \\alpha \\in [{\\mathcal D}({A^*}^2)]'\n\\end{equation}\nInsertion of $Bu \\in [D(A^{2*})]'$ in place of $\\alpha$ brings about the estimate \n\\begin{equation*}\n\\sup_{0\\le t\\le T}\\|R\\Phi(\\cdot,t)Bu\\|_{L^1(t,T;Y)}\\le \\dots \\le \n\\|Re^{A(\\tau-t)}\\alpha\\|_{L^?(t,T;Y)}\n\\le c_T\\|u\\|_U\\,.\n\\end{equation*}\n\n\\smallskip\n\\noindent\n2. A major step is to show existence (as well as to pinpoint the regularity) of the derivative\nof $R\\Phi(\\tau,t)\\alpha$ with respect to $t$, with $\\alpha$ belonging to the largest possible space. \nThe arguments here owe to \\cite[Vol.~II, Lemma~8.3.4.2]{redbook}. \n\\\\\nRewrite\n\\begin{equation}\\label{e:implicit-eq-for-rphi}\nR\\Phi(\\tau,t)\\alpha+ \\big\\{RL_tL_t^*R^*\\, R \\Phi(\\cdot,t)\\alpha\\big\\}(\\tau)\n= Re^{A(\\tau-t)}\\alpha\n\\end{equation}\nand notice that if $\\alpha\\in [{\\mathcal D}(A^*)]'$\n(please note that here it is {\\bf not} $\\alpha\\in [{\\mathcal D}({A^*}^2)]'$), \nthen \n\\begin{equation*}\nRe^{A(\\tau-t)} x= RA^2\\,A^{-1}e^{A(\\tau-t)}A^{-1}x\\,,\n\\end{equation*}\nwhich gives \n\\begin{equation*}\n\\frac{d}{dt} Re^{A(\\tau-t)} x= -[RA^2]e^{A(\\tau-t)} \\underbrace{A^{-1}x}_{\\in H}\\,.\n\\end{equation*}\nRewrite next \\eqref{e:implicit-eq-for-rphi} explicitly:\n\\begin{equation*}\n\\begin{split}\n& R\\Phi(\\tau,t)\\alpha + R\\int_t^\\tau e^{A(\\tau-\\sigma)}B\\int_\\sigma^T B^*e^{A^*(r-\\sigma)}\nR^*R \\Phi(r,t)\\alpha\\, dr\\, d\\sigma=\n\\\\[1mm]\n& \\qquad\\qquad\\qquad = Re^{A(\\tau-t)}\\alpha\n\\end{split}\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n\\begin{split}\n& \\frac{d}{dt} \\big(R\\Phi(\\tau,t)\\alpha\\big)- Re^{A(\\tau-t)}B\\int_t^T B^*e^{A^*(r-t)}\nR^*R \\Phi(r,t)\\alpha\\, dr \\,+\n\\\\[1mm]\n& \\qquad +\\,R\\int_t^\\tau e^{A(\\tau-\\sigma)}B\\int_\\sigma^T B^*e^{A^*(r-\\sigma)}\nR^*R \\frac{d}{dt} \\big(R\\Phi(\\tau,t)\\alpha\\big)\\, dr\\, d\\sigma\n\\\\[1mm]\n& \\qquad\\qquad\\qquad =- Re^{A(\\tau-t)}A\\alpha\\,.\n\\end{split}\n\\end{equation*}\nThe above implicit equation is rewritten as\n\\begin{equation*}\n\\big[I+RL_tL_t^*R^*\\big]\\frac{d}{dt}\\big(R\\Phi(\\cdot,t)\\alpha\\big) =\n-\\underbrace{Re^{A(\\tau-t)}A\\alpha}_{T_1(\\tau,t)}+ \\underbrace{Re^{A(\\tau-t)}BB^*P(t)\\alpha}_{T_2(\\tau,t)}\n\\end{equation*}\nwhich makes sense at least in $H^{-1}(0,T;Y)$.\n\nThen, noting that \n\\begin{equation*}\nT_1(\\cdot,t)\\in C([t,T];Y)\\,, \\qquad T_2(\\cdot,t)\\in L^\\infty(t,T;Y)\n\\end{equation*}\nwe get \n\\begin{equation*}\n\\frac{d}{dt} \\big(R\\Phi(\\tau,t)\\alpha\\big)=\\big[I+RL_tL_t^*R^*\\big]^{-1}\n\\Big\\{- Re^{A(\\tau-t)}A\\alpha+Re^{A(\\tau-t)}BB^*P(t)\\alpha\\Big\\}\\in L^2(t,T;HY\\,.\n\\end{equation*}\nRecalling \\eqref{e:rphi} we finally obtain\n\\begin{equation*}\n\\frac{d}{dt} \\big(R\\Phi(\\tau,t)\\alpha\\big)=- R\\Phi(\\tau,t)A\\alpha+R\\Phi(\\tau,t)BB^*P(t)\\alpha\n\\end{equation*}\n(cf.~\\cite[Vol.~II, \\S~8.3.4, p.~701]{redbook}), thereby providing with \n\\begin{equation*}\n\\begin{split}\n& \\frac{d}{dt} (\\big(R\\Phi(\\tau,t)x\\big),y)_Y=\n\\\\[1mm]\n& \\qquad\n=- (R\\Phi(\\tau,t)\\,[A-(B_0 + A B_1 )\\,(B_0^* + B_1^* A^*)P(t)]x, y)_Y\\,,\n\\quad x\\in [{\\mathcal D}(A^*)]', y\\in Y\\,.\n\\end{split}\n\\end{equation*} \n\\end{proof}\n\n\n\\begin{lemma}[\\bf First Feedback Synthesis] \\label{l:F}\nThe optimal control $\\hat{g}$ admits the representation\n\\begin{equation*}\n\\hat{g}(\\tau,t;\\alpha) =- [B_0^* + B_1^* A^*] P(\\tau)\\Phi(\\tau,t)\\alpha\n\\qquad \\forall \\alpha \\in [{\\mathcal D}({A^*}^2)]'\\,.\n\\end{equation*}\n\n\\end{lemma}\n\n\\begin{proof}\nFrom the optimality conditions we know that \n\\begin{equation*}\n\\hat{g}(\\tau,t;\\alpha) =-\\{L_t^*R^*R\\hat{y}(\\cdot,t;\\alpha)\\}(\\tau)\\,.\n\\end{equation*}\nBecause $RB_1 =0$, and exploiting the evolution property enjoyed by $\\Phi$, it follows\n\\begin{equation*}\n\\hat{g}(\\tau,t;\\alpha) =-L_t^*R^*R \\Phi(\\cdot,t) \\alpha\\,.\n\\end{equation*}\nObserving that for any $\\alpha \\in [{\\mathcal D}({A^*})]'$ one has $R\\Phi(t,s) \\alpha \\in Y$ and \n$L_t^* R^* \\colon L^1(Y) \\rightarrow C(U)$, makes the above composition of operators \nmeaningful -- as acting on appropriate domains. \nThis concludes the optimal synthesis as stated in the Lemma. \n\\end{proof}\n\n\\begin{lemma}[\\bf Riccati Equation]\\label{RIC}\nFor all $x, y \\in {\\mathcal D}(A)$ the Riccati operator $P(\\cdot)$ satisfies \n\\begin{equation*}\n\\begin{split}\n& \\big(\\frac{d}{dt} P(t)x,y\\big)_Y= -(R^*Rx,y)_H-(A^* P(t)x,y)_Y -\n\\\\[1mm]\n& \\qquad\\qquad\\qquad - (P(t)Ax,y)_Y- ([B_0^*+B_1^*A^*]P(t)x,[B_0^*+B_1^*A^*]P(t)y)_Y\\,,\n\\end{split}\n\\end{equation*}\nwith \n\\begin{equation*}\n\\begin{cases}\nA^*P_t(t) A \\in {\\mathcal L}(Y)\\,, \n\\\\[1mm]\n\\textrm{$A^*P_t(t) A$ continuous $\\colon Y\\longrightarrow L^\\infty(0,T;Y)$.}\n\\end{cases}\n\\end{equation*}\n\n\\end{lemma}\n\n\\begin{proof}\nIn order to derive the Riccati equation, we follow the so called direct approach (cf.~\\cite{redbook}).\nDifferentiation (in a weak sense) of the Riccati operator requires the characterization of the left derivative (with respect to the initial time) of the evolution.\nHowever, in the present case, Proposition~\\ref{p:Phi} provides the needed regularity for the evolution when acted upon by the observation. \nThis allows to obtain the critical representation for the right evolutionary derivative which is given by Lemma~\\ref{Right}. \nThe said representation, when combined with the ``first feedback synthesis'' in Lemma~\\ref{l:F}\ngives the final conclusion.\n\nCalculations are justified by the already proved regularity of the quantities involved. \nIn particular, the compromised regularity of the derivative of the evolution (which requires \n$\\alpha \\in [{\\mathcal D}(A^*)]'$, is sufficient to obtain the final conclusion. \n\\end{proof}\n\nWe note that the feedback synthesis given in Lemma \\ref{l:F} is in terms of the evolution operator $\\Phi(t,s)$. \nWhat is needed, instead, is the feedback synthesis in terms of the actual trajectory $\\hat{y}$.\nThis is attained below. \n\n\\begin{lemma}[\\bf Feedback Synthesis] \\label{l:feed} \nFor any $\\alpha \\in [{\\mathcal D}({A^{2*}}]'$, the following feedback representation of the\noptimal control $\\hat{g}(t;\\alpha)$ holds true: \n\\begin{equation*}\n\\hat{g}(t;\\alpha) =- \\big[I - [B_0^* + B_1^* A^*] P(t) B_1\\big]^{-1} \n[B_0^* + B_1^* A^*]P(t)\\hat{y}(t,\\alpha)\\,; \n\\end{equation*}\nthe formula provides an ``on line'' optimal control $\\hat{g}(\\cdot, \\alpha ) \\in L_2(U)$\nfor the $\\alpha$-parametrized problem. \n\\end{lemma}\n\n\\begin{proof}\nFor the feedback synthesis of the optimal control it remains to discuss the invertibility of the operator \n\\begin{equation*}\nI-[{B_0}^*+{B_1}^*A^*]P(t)B_1\\,.\n\\end{equation*}\n\n\\begin{proposition}\\label{p:feed}\nThe operator $I-[{B_0}^*+{B_1}^*A^*]P(t)B_1$ is boundedly invertible on $U$ for each \n$t \\in [0, T]$. \n\\end{proposition}\n\n\\begin{proof}\n{\\sl Step 1.} \nWe shall first prove the injectivity of the operator \n$I-[{B_0}^*+{B_1}^*A^*]P(t)B_1$ for $t =0$. \nThen, the dynamic programming argument extends the argument to all $t\\in [0,T]$.\n\nBy contradiction, let $v \\in U$ be such that $v \\ne 0$, and \n\\begin{equation}\\label{1}\nv=[{B_0}^*+{B_1}^*A^*]P(t)B_1v\\,.\n\\end{equation}\nConsider then the optimal control problem with $y_0=0$, and $\\alpha =-B_1v$.\nThe (implicit) optimal synthesis gives \n\\begin{equation}\\label{2}\n\\hat{g}_{\\alpha}(0) = - [B_0^* + B_1^* A^*] P(0) \n\\big(\\hat{ y}_{\\alpha}(0) - B_1 \\hat{g}_{\\alpha} (0)\\big)\\,.\n\\end{equation}\nBut from the continuity of optimal control, we also have \n$\\hat{y}_{\\alpha}(0) = \\alpha + B_1 \\hat{g}_{\\alpha}(0)$. \nThis, combined with \\eqref{2} give\n\\begin{equation}\\label{3}\n\\hat{g}_{\\alpha}(0) =\n- [B_0^* + B_1^* A^*] P(0) [\\hat{ y}_{\\alpha}(0) - B_1 \\hat{g}_{\\alpha} (0) ]\n= [B_0^* + B_1^* A^*] P(0) B_1v\\,. \n\\end{equation}\nFrom the contradiction argument \\eqref{1} it follows that $g^0_{\\alpha}(0) =v$.\nOn the other hand, the optimal control problem with $y_0 =0$ produces only one solution \nwhich is equal identically to zero. \nTherefore, the optimal control $g^0$ should be zero as well. \nThis contradicts the fact that $v \\ne 0$. \n\nThe same argument applied to the dynamics originating at the time $t$ yields\ninjectivity of $I - [B_0^* + B_1^* P(t) ]B_1$ on $U$, for any $t \\in [0,T]$. \n\n\\smallskip\n\\noindent\n{\\sl Step 2}. \nCompactness of the operator $[B_0^* + B_1^* P(t)]B_1$. \nThis follows from regularity properties of $P(t)$ which asserts that \n$P(t)\\colon {\\mathcal D}({A^*}^2)]' \\rightarrow {\\mathcal D}({A^*}^2)$ is bounded. \nHowever, the injection $B_1 \\colon U \\rightarrow {\\mathcal D}({A^*}^2)$ is compact.\nThe latter follows from the fact \n$A^{-1} B_1g = [ b c^{-2} {\\mathcal A}^{-1} ( {\\mathcal A} + I ) N_0 g, 0, 0]$ and elliptic theory giving \n$N_0 : L_2(\\Gamma_0 ) \\rightarrow H^1(\\Omega) $ is compact. \n\nThus, the final conclusion follows from spectral theory of compact operator. \n\\end{proof}\n\nNow, the conclusion in Lemma \\ref{l:feed} follows from the Proposition \\ref{p:feed} and the representation in Lemma \\ref{l:F} supported by definition of evolution operator $\\Phi$. \n\\end{proof}\nCompletion of the proof of Proposition \\ref{T}: combine the results stated in Proposition \\ref{p:Riccati-operator}, Lemma \\ref{RIC} and Lemma \\ref{l:feed}. \n\nCompletion of the proof of Theorem \\ref{T0}: setting $\\alpha = y_0 - B_1 g_0$ provides the conclusions stated in Theorem \\ref{T0}. \n \n\\subsection{Proof of Theorem \\ref{T:1}}\nIt remains to be shown that $\\hat{g}(0)$ coincides with the parameter $g_0$.\nThis is done below. \n\nLet $y_0 \\in [{\\mathcal D}({A^2}^*)]'$ and $g_0 \\in U$ be given.\nWith $\\alpha= y_0 - B_1g_0$, we know from from Part 1 of Theorem \\ref{T0} that the optimal control $g^0$ belongs $C([0,T];U)$. \nTherefore, in order to comply with the original model one is asking for the following selection of the parameter $g_0$: $g_0 = g^0(0)$. \nThis amounts to \n\\begin{equation*}\ng^0_\\alpha(t=0)=g_0\\,, \\qquad \\alpha = y_0 - B_1g_0\\,.\n\\end{equation*}\nThe above implicit relation is always uniquely solvable for some $g_0 \\in U$.\nIn fact, the matching condition amounts to solving \n$g_0 = F \\alpha = F (y_0 - B_1 g_0 )$, that is $(I - F B_1) g_0 = Fy_0$,\nwhere $F\\equiv [B_0^* + B_1^* A^*] P(0)$.\n\nWith the key properties $F \\in {\\mathcal L}([{\\mathcal D}(A^*)]',U)$ and $(I - FB_1)^{-1}\\in {\\mathcal L}(U)$.\nHowever, we recognize that $I - F B_1$ coincides with the operator $G(0)$, for which the requisite boundeness and invertibility have been shown in Proposition \\ref{p:feed}. \n\nThus we obtain \n\\begin{corollary}\nLet $y_0 \\in {\\mathcal D}({A^2}^*)]'$ be given. \nConsider Problem $\\mathcal{P}_{\\alpha}$ with $\\alpha = y_0 -B_1g_0$ and $g_0\\in U$\ngiven by\n\\begin{equation}\\label{g00}\ng_0 = (I - FB_1 )^{-1} F y_0\\,. \n \\end{equation}\nThen, there exists a unique optimal control $g^0\\in C([0,T]; U)$ and a \ncorresponding trajectory \\eqref{e:eq-for-U}, with $y^0(0) = y_0$, \nsuch that the results of Proposition \\ref{T} hold with $\\alpha= y_0 - B_1 g_0$ and \n$g_0$ given by \\eqref{g00}.\n\\end{corollary}\n \nIn other words, by solving the parametrized optimal control problem with a given \n$\\alpha = y_0 - B_1 g_0$ and a parameter $g_0 \\in U $ we solve a family of parametrized optimal control problems, which always has a unique solution. \nThe original dynamics is included in this family. \nBy selecting $g_0\\in U$ according to the matching condition, we make a selection of a problem whose dynamics coincides with the original one.\nHowever, the above does not imply that the constructed optimal control for the parametrized control problem is also optimal for the original problem -- when considered within the $L_2(U)$ framework for optimal controls.\nIn fact, the latter may not have an optimal solution at all when $y_0 \\in {\\mathcal R}(B_1)$, as shown in Theorem \\ref{l:neg}; see also \\cite{LPT}.\nThus, the constructed control is suboptimal, yet it corresponds to the original dynamics. \nHowever, if the original problem does have an $L_2(U)$ optimal control, then such control coincides with a parametrized control where $g_0$ is selected according to the matching condition.\n \n\\subsection{Proof of Theorem \\ref{T:2}}\nTheorem \\ref{T:2} follows from Theorem \\ref{T:1} by using a rather standard argument in calculus of variations. \nTo wit: we recall from Proposition \\ref{T} that the optimal value for the parametrized problem equals \n\\begin{equation*} \nJ(\\hat{g},\\hat{y}_{g_0}) = (P(0) \\alpha, \\alpha )_Y \n= (P(0) (y_0 - B_1 g_0), y_0 - B_1 g_0 )_Y\\,. \n\\end{equation*}\nOn the strength of positivity and selfadjointness of $P(0)$ we can write the above as \n\\begin{equation*} \nJ(\\hat{g},\\hat{y}_{g_0}) = || P^{1\/2} (0) (y_0 - B_1 g_0 ) ||^2_Y\\,.\n\\end{equation*}\nAppealing to the regularity properies of $P(0)$ listed in Theorem \\ref{T:1} we obtain that \n$J(g_0) \\equiv J(\\hat{g},\\hat{y}_{g_0})$ is weakly lower semicontinuous on $U$. \nIndeed, the latter follows from \n\\begin{eqnarray}\\label{opt}\nJ(\\hat{g},\\hat{y}_{g_0})= (P(0) (y_0 - B_1 g_0), y_0 - B_1 g_0 )_Y = (P(0) y_0, y_0 )_Y \n\\notag \\\\ \n- 2 (P(0) y_0, B_1 g_0 )_Y + (P(0) B_1 g_0, B_1 g_0 )_Y\\,,\n\\end{eqnarray}\nwhere $A^{-1} B_1 \\colon U \\rightarrow Y$ is compact and $A^*P(0) A \\colon Y \\rightarrow Y$\nis bounded. \nThis gives compactness of the map $g \\rightarrow P^{1\/2}(0) B_1 g$ from $U$ to $Y$,\nadressing the convergence of the last quadratic term in \\eqref{opt}.\n \nAs for the first term, we simply recall Proposition \\ref{p:Riccati-operator} which states $ {A^*}^2P(0) A^2 :Y \\rightarrow Y $ is also bounded.\nStrong continuity of the second term (linear in $g_0$ ) follows now from $ A^{-1} B_1 \\in L(Y) $ and $A^* P(0) A^2 \\in L(Y) $. \nThus the regularity of the Riccati operator $ P(0)$ along with $ A^{-1} B_1 \\in L(Y) $ implies weal lower-semicontinuity of the functional. Since $U_0$ is weakly compact, we obtain a minimizing sequence $ g_n \\in U_0$ such that $J(g_n) \\rightarrow d = \\inf_ {g_0\\in {U_0}} J(g_0) $ \nand $g_n \\rightarrow g^* \\in U_0 $ weakly in $U$. \n Weak lower semicontinuity of $ J(g_0)$ gives an existence of a minimizer. The characterization of the minimizer follows now from a standard argument in calculus of variations,\nafter taking into consideration the representation of the functional via Riccati operator. This leads to the final conclusion stated in Theorem \\ref{T:2}.\n \n\\ifdefined\\xxx\n1. Notice that unlike the more general framework of Lasiecka-Lukes-Pandolfi and \nLasiecka-Pandolfi-Triggiani, in the present case there is no need of an analysis \nof a non-standard LQ-problem, along with the invocation of a Dissipation Inequality\nsatisfied by the value function. \nIndeed, in view of $RB_1\\equiv 0$, the dynamics $y(t) -B_1g(t)$ is such that \n\\begin{equation}\n\\|R(y(t) -B_1g(t))\\|^2\\equiv \\|Ry(t)\\|_H^2 \n\\end{equation}\nand the cost functional is the same for both dynamics.\n\n\n\\smallskip\n\\noindent\n2. I have analyzed the case $d>0$ and $d_0=0$: we have once more $RA^2$ bounded and $RB_1=0$;\nit must be computed the value of $\\lambda\\ne 0$ for wich $(\\lambda-A)^{-1}B_i$ is bounded. \nWe will return on this after your first feedback.\n\\end{remarks}\n\\fi\n\n\n\\section*{Acknowledgements}\nThe authors are grateful to Barbara Kaltenbacher, whose work has provided motivation for studying \ncontrol problems associated with the SMGT acoustic model. \nInspiring and illuminating mathematical conversations of both authors with Barbara are gratefully\nacknowledged. \n\n\\smallskip\n\\noindent\nThe research of F.B. was partially supported by the Universit\\`a degli Studi di Firenze under the Project \n{\\em Analisi e controllo di sistemi di Equazioni a Derivate Parziali di evoluzione}, and by the GDRE (Groupement de Recherche Europ\\'een) ConEDP (Control of PDEs). \nF.B. is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), whose occasional support is acknowledged. \n\\\\\nThe research of I.L. was partially supported by the NSF Grant DMS-1713506.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this note, we consider the question of determining the number of\ncovers between projective lines in positive characteristic with\nspecified ramification data and fixed branch points. The ramification\ndata considered are the degree of the cover, together with a list of\nthe ramification indices in the fibers of the branch points. Over an\nalgebraically closed field of characteristic zero, it is in principal\n possible to solve this problem by Riemann's Existence Theorem. Namely, the number of covers can be expressed as the cardinality of a finite set,\nwhich can be explicitly constructed in concrete cases. In particular, this approach\nshows that the number of covers is finite and does not depend on the\nposition of the branch points. \n\nIn positive characteristic, the situation is drastically different. For example, the\nnumber of covers with fixed ramification depends on the position\nof the branch points. Moreover, if the characteristic $p$ divides one\nof the ramification indices, the number of covers is in general\ninfinite. There are only few general results on the number of covers\nin this situation (we refer to \\cite{BO} for an overview).\n\n The work of Osserman (\\cite{Osserman1}, \\cite{Osserman2}, \\cite{LO})\n suggests that a particularly nice case to look at is that of covers\n $f:\\mathbb{P}^1\\to \\mathbb{P}^1$ of degree $d$ which are ramified at $r$ points\n $x_1, \\ldots, x_r$ with $f(x_i)$ pairwise distinct (the\n so-called {\\sl single-cycle} case). We write $h(d; e_1, e_2, e_3,\n \\ldots, e_r)$ for the number of single-cycle covers with fixed branch\n locus over $\\mathbb{C}$, where $e_i$ is the ramification index of\n $x_i$; this number is called the {\\sl Hurwitz number}.\n\nLet $k$ be an algebraically closed field of positive characteristic $p$. We\nonly consider covers $f:\\mathbb{P}^1_k\\to \\mathbb{P}^1_k$ in the tame and single-cycle\ncase. We denote by $h_p(d; e_1, e_2, e_3, \\ldots,\ne_r)$ the maximal number of covers with fixed branch locus, where\nthe maximum is taken over all possible branch loci. This number is called the\n$p$-{\\sl Hurwitz number}. Since $p\\nmid e_i$ for all $i$, this number\nis finite and does not depend on $k$. It can be shown that there the maximum is attained if the branch locus belongs to a dense open subset $U\\subset (\\mathbb{P}^1_k)^r\\setminus \\Delta$. Here $\\Delta$\nis the fat diagonal.\n\n\nWe start by summarizing the results on the number of covers with fixed\nbranch locus in the single-cycle case for $r\\in \\{3,\n4\\}$. In \\cite{LO}, F. Liu and B. Osserman give a closed formula for the number\nof such covers in characteristic zero. In \\cite{Osserman1} and \n\\cite{Osserman2}, B. Osserman determines the $p$-Hurwitz number $h_p(d;\ne_1, e_2, e_3)$ using linear series. In \\cite{BO} the number $h_p(p;\ne_1, e_2, e_3, e_4)$ is computed. This last case is substantially more\ndifficult and he proof relies on the theory of stable reduction of covers. \n\nIn this note, we also consider covers $f:\\mathbb{P}^1_k\\to \\mathbb{P}^1_k$ of\nramification type $(d; e_1, e_2, e_3, e_4)$. In contrast with the\nsituation in \\cite{BO}, the degree $d$ is not fixed. We consider two\nelementary constructions, which yield previously unknown results on\nsome $p$-Hurwitz numbers $h_p(d; e_1, e_2, e_3, e_4)$. Both\nconstructions were known before and can be found for example in\n\\cite{Osserman1}. However, the implications for the $p$-Hurwitz\nnumbers have not been fully exploited. As an additional result, we\nobtain rather complete information on the structure of the Hurwitz\ncurve, parameterizing covers of the type considered, in positive\ncharacteristic. These are the first such results.\n\nThe first result deals with the case $1< e_i< p$ and $e_4=p-1$. In\nthis situation, we compute the $p$-Hurwitz number $h_p(d; e_1, e_2, e_3, e_4)$. We can even obtain\nsomething stronger, namely an explicit description of the Hurwitz curve\n$\\mathcal{H}_p(d; e_1, e_2, e_3, e_4)$ parameterizing all covers of type $(d;\ne_1, e_2, e_3, e_4)$. This yields, in particular, not only a formula for\nall covers with generic branch locus, but also exactly describes the values for which the number of covers drops. As far as we know, this is the first nontrivial example of a complete description of Hurwitz curves in positive characteristic. Other papers on Hurwitz curves in positive characteristic (eg\n\\cite{crelle}, \\cite{meta} and \\cite{BO}) do not yield such description.\nWe refer to \\S\\ \\ref{multconstsec} for the precise\nstatement of the result.\n\nThe second result considers the case $e_1>p$ and $20$.\n\\begin{itemize}\n\\item[(a)] The $p$-Hurwitz number $h_p({\\boldsymbol C})$ only depends\n on $p$, and not on the field $k$.\n\\item[(b)] We have $h_p({\\boldsymbol C})\\leq h({\\boldsymbol C})$ with\n equality if $d0$.\n\nLet $k$ be an algebraically closed field of characteristic $p>0$. We\nfix a genus-$0$ ramification type ${\\boldsymbol C}=(d; e_1, e_2, e_3,\np-1)$ and a branch locus ${\\boldsymbol x}=(x_1=0, x_2=1, x_3=\\infty,\nx_4=:\\lambda)$. We assume that $11,\\\\\ne_3&0$.\n\nComputing $p$-Hurwitz numbers is in general more difficult than\ncounting covers with fixed ramification. Beside the classical result\nfrom Lemma \\ref{p-hurwitzlem}.(b), the only general result on\n$p$-Hurwitz numbers is the main result of \\cite{BO}, which\ncomputes $h_p(p; e_1, e_2, e_3, e_4)$. That result relies on subtle\nand deep results on the stable reduction of Galois covers. \n\\end{rem}\n\n\nThe following corollary translates the statement of Proposition\n\\ref{hurwitznrprop} into a statement on the Hurwitz curve\n$\\mathcal{H}_p({\\boldsymbol C})$.\n\n\\begin{cor}\\label{hurwitznrcor} Let ${\\boldsymbol C}=(d; e_1, e_2, e_3, p-1)$\n be as in the \nstatement of Proposition \\ref{hurwitznrprop}. The Hurwitz curve\n$\\mathcal{H}_p({\\boldsymbol C})$ is connected.\n\\end{cor}\n\n{\\bf Proof:\\ } The statement immediately follows from the proposition. Let\n$\\pi_p:\\mathcal{H}_p({\\boldsymbol C})\\to \\mathbb{P}^1_\\lambda$ be the natural map which\nsends a cover of type ${\\boldsymbol C}$ to the branch point\n$\\lambda$. Then $\\pi$ is birationally equivalent to the map\n$\\mu\\mapsto \\lambda$ described in the prof of that proposition.\n\\hspace*{\\fill} $\\Box$ \\vspace{1ex} \\noindent \n\n\n\nLet ${\\boldsymbol C}:=(d; e_1, e_2, e_3, p-1)$ be a\nramification type satisfying the equivalent conditions of Proposition\n\\ref{hurwitznrprop}.(a). Put $\\tilde{{\\boldsymbol C}}=(\\tilde{d}; e_1,\ne_2, p-e_3)$ and let $h:\\mathbb{P}^1_k\\to \\mathbb{P}^1_k$ be the unique cover of type\n$\\tilde{{\\boldsymbol C}}$ (compare with the proof of Proposition\n\\ref{hurwitznrprop}.(a)). We may write $h(y)=h_1(y)\/h_2(y)$, where the\n$h_i\\in k[y]$ are relatively prime and satisfy the relations\n$\\deg(h_1)=\\tilde{d}-e_1$ and\n$\\deg(h_2)=\\tilde{d}-(p-e_3)=(e_1+e_2+e_3-p-1)\/2$. \n\nIt follows from Lemma \\ref{p-hurwitzlem} that there exist finitely many values $\\lambda\\in \\mathbb{P}^1_\\lambda\\setminus\\{ 0,1,\\infty\\}$ for\nwhich the number of covers of ramification type ${\\boldsymbol C}$ and\nbranch locus $(0, 1, \\infty, \\lambda)$ is strictly less than\n$h_p({\\boldsymbol C})$. We let $\\Sigma({\\boldsymbol C})\\subset\n\\mathbb{P}^1_\\lambda\\setminus\\{0,1,\\infty\\}$ be this exceptional set and\ncall it the {\\sl supersingular locus}.\n\n\n\\begin{cor}\\label{supersingcor}\nWith the above notation, we have\n\\[\n\\Sigma({\\boldsymbol C})=\\{y\\in \\mathbb{P}^1_k\\setminus\\{0,1,\\infty\\}\\mid h_2(y)=0\\}.\n\\]\n\\end{cor}\n\n{\\bf Proof:\\ } We recall that the construction of Lemma\n\\ref{multconstlem}.(b) works if and only if $h(\\mu)\\neq 0,1, \\infty,\n\\mu^p$. Moreover, equation (\\ref{lambdaeq}) gives an expression of the fourth branch point $\\lambda$ of $f$ as function of $\\mu$. \n\nAssume that $h(\\mu)=0$. Then (\\ref{lambdaeq}) implies that either\n$\\mu=0$ or $h(\\mu)=1$. By definition $0, 1\\not\\in \\Sigma({\\boldsymbol\n C})$. Therefore it suffices to consider the solutions of $h(\\mu)=1$\nwith $\\mu\\neq 1$. We may write $h(\\mu)-1=\\mu^{e_2}\\varphi(\\mu)$, where\n$\\varphi(1)\\neq 1$. Substituting this in (\\ref{lambdaeq}) yields\n\\[\n\\lambda(\\mu)=\\frac{\\mu^p\\varphi(\\mu)}{(\\mu-1)^{p-e_2}-\\varphi(\\mu)}.\n\\]\nIn particular, it follows that $\\lambda(\\mu)=0$ if $\\mu$ is a zero of\n$\\varphi$. Hence these zeroes are not contained in $\\Sigma({\\boldsymbol\n C})$. Similarly it follows that the solutions of $h(\\mu)=1$ don't belong to $\\Sigma({\\boldsymbol C})$.\n\nAssume that $h(\\mu)=\\infty$ and $\\mu\\neq \\infty$, i.e.\\ $h_2(\\mu)=0$ according to the notation introduced above the statement of the corollary. We then have the identity $\\lambda(\\mu)=\\mu^p$. Therefore $\\mu\\in\n\\Sigma({\\boldsymbol C})$, since $\\mu\\neq 0,1,\\infty$.\n\nFinally, assume that $\\mu=\\mu^p$ and $\\mu\\not\\in\\{0,1,\\infty\\}$. Then\n$\\lambda=\\infty$, hence this does not yields any new value.\n\\hspace*{\\fill} $\\Box$ \\vspace{1ex} \\noindent \n\n\n\n\\begin{exa} We illustrate the results of this section with two concrete \nexamples.\n\n(a) Let $p\\geq 5$ be a prime, and consider the genus-$0$ ramification\ntype ${\\boldsymbol C}=(d; 2,2,p-3, p-1)$. Note that the condition of\nProposition \\ref{hurwitznrprop}.(a) is satisfied. Hence Proposition\n\\ref{hurwitznrprop}.(b) implies that $h_p({\\boldsymbol C})=p-1$ and\nProposition \\ref{4ptprop}.(a) leads to $h({\\boldsymbol\n C})=\\min(3(p-3), 2(p-2),p-1)=p-1$. We therefore find the equality $h({\\boldsymbol\n C})=h_p({\\boldsymbol C})$, so that all covers of this type with generic\nbranch locus have good reduction.\n\nThe unique normalized cover $h:\\mathbb{P}^1\\to \\mathbb{P}^1$ of\ntype $(3; 2,2,3)$ is given by\n\\[\nh(y)=3y^3-2y^2.\n\\]\nTherefore \n\\[\n\\lambda(\\mu)=\\frac{\\mu^{p}(1+2\\mu^3-3\\mu^2)}{\\mu^p+2\\mu^3-3\\mu^2}=\n\\frac{\\mu^{p-2}(2\\mu-1)}{\\sum_{i=1}^{p-4}i\n \\mu^{p-3-i}}.\n\\]\nThis confirms that the degree $\\deg(\\lambda)$ equals\n$(3p-(e_1+e_2+e_3))\/2=p-1$.\n\nWe have already remarked that all covers with generic branch locus\nhave good reduction to characteristic $p>0$. Arguing as in the proof\nof \\cite[Theorem 4.2]{meta}, one may deduce from this observation that\nthe map $\\pi_p:\\mathcal{H}_p({\\boldsymbol C})\\to\n\\mathbb{P}^1_\\lambda\\setminus\\{0,1,\\infty\\}$ is finite. However, in this\nconcrete example the finiteness of $\\pi_p$ immediately follows from\nCorollary \\ref{supersingcor}.\n\n(b) Next, we consider the case ${\\boldsymbol C}=(p; 3,2,p-2, p-1)$\nand ${\\tilde{\\boldsymbol C}}=(3; 3,2,2)$, again assuming that $p\\geq\n5$. The unique cover $h:\\mathbb{P}^1_k\\to \\mathbb{P}^1_k$ of type\n${\\tilde{\\boldsymbol C}}$ is given by $h(y)=y^3\/(3y-2)$ and a direct computation leads to the expression\n\\[\n\\lambda(\\mu)=\\frac{\\mu^{p-3}(-\\mu^3+3\\mu-2)}{3\\mu^{p-2}-2\\mu^{p-3}-1}.\n\\]\nDividing the numerator and the denominator by $(\\mu-1)^2$, we find\n$\\deg(\\lambda)=p-2$, which confirms Proposition\n\\ref{hurwitznrprop}.(b). As in Corollary \\ref{supersingcor}, the supersingular values are the poles of $h$ different from $\\infty$. In this concrete example, we find a unique value, namely $\\mu=2\/3$. The case $d=p$ has been considered in \\cite{BO}. This result may also be deduced from\n\\cite[Remark 8.3]{BO} (note, however, that the proof relies on subtle\narguments involving stable reduction, which are only sketched in that\npaper). As in the previous example, Proposition \\ref{4ptprop} implies that $h({\\boldsymbol\n C})=2(p-1)$ and Proposition \\ref{hurwitznrprop}.(b) asserts that\n$h_p({\\boldsymbol C})=p-2$ We conclude that $h({\\boldsymbol\n C})-h_p({\\boldsymbol C})=p$, which confirms the main result of \\cite{BO}.\n\\end{exa}\n\n\n\nFix a genus-$0$ ramification type ${\\boldsymbol C}=(d; e_1, e_2, e_3,\np-1)$ which is $p$-tame, i.e.\\ $p\\nmid e_i$. Recall that\n$h({\\boldsymbol C})-h_p({\\boldsymbol C})$ denotes the ``bad degree''\nof the ramification type. This is the number of covers with generic\nbranch locus which have bad reduction to characteristic $p$. \n\n\\begin{prop}\\label{baddegprop}\nThe notation being as above, assume that the minimum\\\\ $\\min_{i\\in\n \\{1,2,3\\}} e_i(d+1-e_i)$ is attained for $e_1$. This is not a\nrestriction, since\\\\ we may permute the branch points. Then, the bad degree\n$h(\\boldsymbol{C})-h_p(\\boldsymbol{C})$ is\\\\ given by\n\\[\n\\begin{cases} \n0& \\text{ if }d\\leq p-1,\\\\\np(d+1-p)& \\text{ if }p\\leq d\\leq p-2+e_1,\\\\\nh({\\boldsymbol C})=e_1(d+1-e_1)&\\text{ otherwise}.\n\\end{cases}\n\\]\n\\end{prop}\n\n{\\bf Proof:\\ } The first case immediately follows from Lemma\n\\ref{p-hurwitzlem}. Assume that $p\\leq d\\leq p-2+e_1$. In this case, Propositions \\ref{4ptprop}.(a) and\n\\ref{hurwitznrprop}.(a) assert that $h({\\boldsymbol C})=(p-1)(d+2-p)$ and\n$h_p({\\boldsymbol C})\\neq 0$. Statement (b) therefore follows from\nProposition \\ref{hurwitznrprop}.(b).\n\nFor $d> p-2+e_1$, Proposition \\ref{4ptprop}.(a) implies\nthat $h({\\boldsymbol C})=e_1(d+1-e_1)$. Since $d>p-2+\\min_i\ne_i=p-2+e_1$ by assumption, we conclude from Proposition\n\\ref{hurwitznrprop}.(a) that $h_p({\\boldsymbol C})=0$ and statement (c)\nfollows. \\hspace*{\\fill} $\\Box$ \\vspace{1ex} \\noindent \n\nIn the second case of Proposition \\ref{baddegprop}, some covers have\ngood reduction while others have bad reduction. In the first\n(resp.\\ third) case all covers have good (resp.\\ bad) reduction to\ncharacteristic $p>0$. The following corollary therefore follows\nfrom Proposition \\ref{baddegprop} and its proof. A similar phenomenon\noccurs in the situation of \\cite[Section 4]{meta}.\n\n\\begin{cor}\\label{baddegcor}\nLet ${\\boldsymbol C}$ be as in Proposition \\ref{baddegprop}, and\nassume that $h({\\boldsymbol C})\\neq h_p({\\boldsymbol C})\\neq 0$. Then\nthe bad degree $h({\\boldsymbol C})-h_p({\\boldsymbol C})$ is divisible\nby $p$.\n\\end{cor}\n\n\\section{A variant}\\label{addconstsec}\nIn this section, we present a variant of the construction of Section\n\\ref{multconstsec}. This construction and the idea of the proof of the\nfollowing lemma has been taken from \\cite[Prop. 5.4]{Osserman1}. We\nfix integers $e_1>p$ and $1p$ it follows also that the ramification index of $f_c$ in\n$x=\\infty$ is $e_1$. Similarly, the ramification\nindices of $f_c$ in $x=1, \\rho$ are $e_2, e_3$, respectively.\n\nThe equality\n\\[\n\\frac{\\partial f_c}{\\partial x}=\\frac{\\partial f}{\\partial x}\n\\]\nimplies that $f_c$ is unramified outside $x=0, 1, \\rho,\\infty$. The\nassumption on $c$ implies that the image of $x=0, 1, \\rho,\\infty$\nunder $f_c$ are all distinct and the statement of the lemma follows.\n\n(b) Let $g$ be as in the statement of the lemma. Define $g_c=g+cx^p.$\nSince $\\partial g_c\/\\partial x=\\partial g\/\\partial x\\neq 0$ it follows\nthat $g_c$ is separable. Moreover, for all $c$ such that the image\nunder $g_c$ of the ramification points are pairwise distinct, the\nramification type of $g_c$ is still $\\boldsymbol{C}=(d; e_1, e_2, e_3,\ne_4)$. Assume that two of the ramification points, for example $x_3$\nand $x_4$, have the same image under $g_c$. \nThen the ramification type is $\\tilde{\\boldsymbol{C}}=(d; e_1,\ne_2, e_3\\text{-}e_4)$. The connectedness of the Hurwitz curve\n$\\mathcal{H}_p(\\boldsymbol{C})$ (Proposition \\ref{4ptprop}.(b)) implies that\nthere exists a $c$ such that $g_c(x_3)=g_c(x_4)$, which proves (b).\n\\hspace*{\\fill} $\\Box$ \\vspace{1ex} \\noindent \n\n\nThe following proposition is a direct consequence of Lemma \\ref{addconstlem}.\n\n\n\\begin{prop}\\label{addconstprop} The assumptions being as above, assume additionally that $e_3\\neq\ne_4$. \n\\begin{itemize}\n\\item[(a)]\nWe then have the equality\n\\[\nh_p(d, e_1, e_2, e_3, e_4)=h_p(d; e_1, e_2, e_3\\text{-}e_4).\n\\]\n\\item[(b)] If $h_p(d; e_1, e_2, e_3\\text{-}e_4)>0$ then the Hurwitz\n curve $\\mathcal{H}_p(d, e_1, e_2, e_3, e_4)$ contains $h_p(d; e_1, e_2,\n e_3\\text{-}e_4)$ irreducible components of genus $0$. Moreover, the restriction of\n the natural map $\\pi: \\mathcal{H}_p(d, e_1, e_2, e_3, e_4)\\to \\mathbb{P}^1_\\lambda$\n which sends $[f]$ to its fourth branch point has degree $1$ on each\n of these components.\n\\end{itemize}\n\\end{prop}\n\n\n{\\bf Proof:\\ } To prove (a), it is sufficient to show that nonisomorphic covers $f_i$ of\ntype $(d, e_1, e_2, e_3, e_4)$ give rise to nonisomorphic covers\nunder the construction of Lemma \\ref{addconstlem}.\n\nLet $f^i:\\mathbb{P}^1_k\\to \\mathbb{P}^1_k$ be two nonisomorphic covers of type $(d,\ne_1, e_2, e_3$-$e_4)$, and assume they are normalized as in the\nstatement of Lemma \\ref{addconstlem}. The branch points of $f^i_c$ are\n$\\infty, 0, 1+c, 1+c\\rho^p$. Normalizing the third branch point to $1$\nyields the normalized cover $g^i_c(x):=f^i_c(x)\/(1+c)$ with branch\npoints $\\infty, 0, 1, (1+c\\rho^p)\/(1+c)=:\\lambda_i$. The assertion\nthat the $g^i_c$ are nonisomorphic follows immediately from the\nassumption that $e_3\\neq e_4$. \n\nStatement (b) follows immediately from the explicit expression\nfor the cover $f_c$ given in the proof of Lemma \\ref{addconstlem}.\n\\hspace*{\\fill} $\\Box$ \\vspace{1ex} \\noindent \n\nIn the rest of this section, we discuss a concrete application of this\nresult to Hurwitz curves in positive characteristic.\n\n\n\\begin{lem}\\label{3ptaddlem}\nLet $p>3$ be a prime, and choose $e_1=p+2, e_2=3, 2\\leq e_3< e_43$ be a prime, and choose $e_1=p+2, e_2=3, 2\\leq e_3< e_4 0}|\\langle n|\\hat{D}|0\\rangle|^2\\delta(E-(E_n-E_0)) ,\n\\end{equation} \nwhere $E_n$ are the excitation energies of the states $|n\\rangle$ while\n$E_0$ is the energy of the ground state $\\displaystyle |0\\rangle=|\\Phi_{0} \\rangle$. In our approach this is obtained \nfrom the imaginary part of the Fourier transform of the time-dependent expectation value of \nthe dipole momentum $ \\displaystyle D(t) = \\frac{NZ}{A} X(t)= \\langle \\Phi (t) |\\hat{D}| \\Phi (t) \\rangle $ extracted\nfrom our simulations (see Fig. \\ref{dip}). We have:\n\\begin{equation}\n S(E) =\\frac{Im(D(\\omega))}{\\pi \\eta \\hbar}~~,\n\\label{stre}\n\\end{equation}\nwhere $\\displaystyle D(\\omega) =\\int_{t_0}^{t_{max}} D(t) e^{i\\omega t} dt$.\nWe consider the initial perturbation along the z-axis and integrate numerically\nthe Vlasov equations (\\ref{vlaprot}, \\ref{vlaneut}) until $t_{max}=1830fm\/c$.\n$\\eta$ was determined from the numerical value of the collective momentum at $t=t_0=30fm\/c$. \n\\begin{figure}\n\\begin{center}\n\\includegraphics*[scale=0.36]{stre_sn132_148.eps}\n\\end{center}\n\\caption{(Color online) The strength function for $^{132}Sn$ [the blue (solid) lines] and $^{148}Sn$ [the red (dashed) lines].\nAsystiff EOS.}\n\\label{stre} \n\\end{figure} \nIn order to eliminate the artifacts resulting from a finite time domain analysis of the signal a filtering procedure, as described\nin \\cite{reiPRE2006}, was considered. A smooth cut-off function was introduced such\nthat $D(t) \\rightarrow D(t)cos^{2}(\\frac{\\pi t}{2 t_{max}}) $. \nThe E1 strength functions of $^{132}Sn$ and $^{148}Sn$ \nare represented in Fig. \\ref{stre}. A test of the quality of our method is the comparison of the numerically estimated\nvalue of the first moment $\\displaystyle m_1=\\int_0^\\infty E S(E) dE$ with the value predicted by the\nThomas-Reiche-Kuhn (TRK) sum rule $\\displaystyle m_1= \\frac{\\hbar^2}{2m} \\frac{N Z}{A}$. In all cases \nthe difference was below $5\\%$. \n\nBefore discussing the dipole response below GDR region let us observe that from the strength function one can determine the \nnuclear dipole polarizability:\n\\begin{equation}\n\\alpha_D = 2 e^2 \\int_0^{\\infty}\\frac{S(E)}{E} dE ~.\n\\end{equation}\nFor $^{68}Ni$ the experimental value of $\\displaystyle \\alpha_D$ reported recently, \\cite{rosPRL2013} is $3.14 fm^3$ while we\nobtained values from $4.1 fm^3$ from $5.7 fm^3$ when we pass from asysoft to asysuperstiff EOS \\cite{barPRC2013}. In the case of $^{208}Pb$ the \nexperimental value of $\\displaystyle \\alpha_D$ is around $20.1 fm^3$ \\cite{tamPRL2011}. In our approach it changes from \n$21.1 fm^3$ for asysoft EOS to $28.6 fm^3$ for asysuperstiff.\nHere we want to explore the mass dependence of this quantity for the three asy-EOS. In order to accomplish this goal we\nconsider the systems $^{48}Ca, $$^{68}Ni$, $^{86}Kr$, $^{208}Pb$ as well as the mentioned isotopic chain of Sn.\nThe Migdal estimation of polarizability \\cite{migJP1944}, valid for large systems, provides a $A^{5\/3}$ dependence with mass: \n\\begin{equation}\n\\alpha_D = \\frac{e^2 A }{24 \\epsilon_{sym}}=\\frac{1.44 e^2}{40 \\epsilon_{sym}} A^{\\frac{5}{3}} ~,\n\\end{equation}\nconsidering that $ \\displaystyle = \\frac{3}{5} R^2$ and $\\displaystyle R=1.2A^{\\frac{1}{3}}$.\nSince $\\epsilon_{sym}$, at saturation, has similar values for the three asy-EOS one also expect, in this situation, close values for the polarizability. \nIn Fig. \\ref{polar} we show the polarizability $\\displaystyle \\alpha_D$ as a function of $A^{5\/3}$. The linear correlation is quite well\nverified. Nevertheless a clear dependence of the slope with asy-EOS is evidenced. This can be related to the surface effects and the interplay\nbetween surface and volume symmetry energy, expected to manifest in finite systems \\cite{lipPLB1982} and which will be influenced by the\nsymmetry energy slope parameter L. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics*[scale=0.36]{polariz_mass.eps}\n\\end{center}\n\\caption{(Color online) The dipole polarizability as a function of $A^{5\/3}$\nfor asysuperstiff (blue squares) asystiff (red circles) and asysoft (green diamonds) EOS. \nThe corresponding dashed lines provide the best linear fit of $\\alpha_D$ with $A^{5\/3}$.\nThe correlation coefficients $r_{fit}$ are $97\\%$, $98\\%$ and $99\\%$ respectively.}\n\\label{polar} \n\\end{figure}\n\nReturning to the strength function, one can identify the appearance of a resonant response below GDR, more important when the number\nof neutrons in excess is larger. In the present model the energy centroid \nis very well described by the parametrization $\\displaystyle 41 A^{- \\frac{1}{3}}$ \\cite{barPRC2013}, in nice agreement with several experimental data.\nThis new mode we associate with Pygmy Dipole Resonance (PDR)\nand notice that other studies based on Vlasov equations\narrived at similar conclusions \\cite{barPRC2012,barRJP2012,abrJU2009,urbPRC2012}. \nHere our purpose is to investigate the dependence of PDR response on the isospin parameter $I$.\nWe calculate EWSR exhausted by this mode by integrating over the low-energy resonance region:\n\\begin{equation}\n m_{1,y} = \\int_{PDR} E S(E) dE ~.\n\\end{equation}\nand plot its dependence on $\\displaystyle I=\\frac{N-Z}{A}$ in Fig. \\ref{m1y}. \nFrom our calculations, for Sn isotopes, a quadratic correlation appears to describe quite well the observed dependence of $\\displaystyle m_{1y}$\nwith the isospin parameter $I$. We remark that, as in the case of polarization, the linear correlation between\n$ m_{1,y}$ and $I^2$ is influenced by the symmetry energy slope parameter $L$. We can therefore conclude that\npolarization effects in the isovector density play an important role in the dynamics of Pygmy Dipole Resonance. \n\\begin{figure}\n\\begin{center}\n\\includegraphics*[scale=0.36]{m1y_iso2.eps}\n\\end{center}\n\\caption{(Color online) The EWSR exhausted by PDR as a function of $I$ square, for asysuperstiff (blue squares),\nasystiff (red circles) and asysoft (green diamonds) EOS. Were\n considered the systems $^{108}Sn$, $^{116}Sn$, $^{124}Sn$, $^{132}Sn$, $^{140}Sn$ and $^{148}Sn$.\nThe dashed lines correspond to the best fit of $ m_{1,y}$ with $I^2$. The correlation coefficients\n$r_{fit}$ are $99.3\\%$, $99.6\\%$ and $99.1\\%$ respectively.}\n\\label{m1y} \n\\end{figure}\n\nThe PDR was observed experimentally for several systems \\cite{aumPS2013,savPPNP2013} and discussed in different theoretical models \\cite{paaJPG2010} for various nuclei, including\nthe Sn isotopic chain \\cite{paaPLB2005,tsoPRC2008,artPRC2009,daoPRC2012,papPRC2014}. Concerning the features of this mode, in literature still exists\nan intense debate about the collective character of this mode, about the role of symmetry energy as well as about its isovector\/isoscalar structure.\nWhile the relativistic quasiparticle RPA (RQRPA) \\cite{paaRPP2007,paaPRL2009} provides evidences about collectivity of PDR, from amplitudes and transition matrix elements, the nonrelativistic Hartree-Fock-Bogoliubov treatment within quasiparticle-phonon model \\cite{tsoPLB2004}, assign to the resonant\nstructures noncollective properties. The calculations based on relativistic time-blocking \\cite{litPRC2009} also shows in the dipole spectra\nof even-even $\\displaystyle ^{130}Sn$-$^{140}Sn$ nuclei two well separated collective structures, the lower lying one, having a specific\nbehavior of the transition densities of states, being ascribed to PDR. For $\\displaystyle ^{34}Mg$, from the time-dependent density plots obtained within TDHF calculations\nwith Skyrme interaction, was identified a superimposed surface mode, not fully coupled to the bulk dynamics. This was related to the pygmy-like peak, obtained around 10 MeV, in the dipole response strength \\cite{briIJMPE2006}.\n\n\\section{Conclusions}\n\\label{concl}\n\nSummarizing, the main task in this paper was to present new results regarding the collective dipole response\nin connection with the properties of the symmetry energy below saturation.\nOur investigation was performed in a microscopic transport model based on a system of two coupled \nVlasov equations for protons and neutrons. \n\n For all studied asy-EOS our model predicts that the energy weighted sum-rule exhausted by the\nPygmy Dipole Resonance manifests a linear dependence with the square of isospin parameter $I=(N-Z)\/A$, with a slope which\nis influenced by the variation rate with density of the symmetry energy around saturation. \nEven if were considered asy-EOS providing similar values of symmetry energy at saturation\nwas also observed that the slope of dipole polarizability as a function of $\\displaystyle A^{5\/3}$\nchanges with the symmetry energy slope parameter L. \nWe interpret these results as an indication of the surface effects associated to the polarization of \nisovector density in finite nuclei. \n\nIn conclusion, the models based on Vlasov equation prove to be appropriate tools for the study of several\naspects of nuclear dynamics, including the development of quite feeble modes as is Pygmy Dipole Resonance, \nfor which provides qualitative insights but also quantitative information regarding its dependence on the\nsymmetry energy or its evolution with the isospin parameter and mass number. \n\n\\section{Acknowledgments}\nThis work for V. Baran and A. Croitoru was supported by a grant of the Romanian National\nAuthority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2011-3-0972.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzharc b/data_all_eng_slimpj/shuffled/split2/finalzzharc new file mode 100644 index 0000000000000000000000000000000000000000..de0fd55be3d2eaa8ec9e0f37835115303ef63762 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzharc @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \nLocalization is from a mathematical and physical point of view considered to be an unresolved issue in a relativistic context, see for example \\cite{NW49}, \\cite{F1}, \\cite{H2}, \\cite{H1} and references therein. The concept is well-defined from a non-relativistic quantum-mechanical (QM) point of view. However, by combining concepts of QM and relativity, from which the frame-work of quantum field theory (QFT) emerged, the localization issue was never completely resolved. This is not merely an issue of construction but points to a more fundamental question about the existence of localizable particles.\n\\\\\\\\\nThe authors in \\cite{NW49} tackled the problem by demanding certain requirements from the states of a coordinate operator, which are used to localize elementary systems. Independently, among other operators the author in \\cite{PR48} gave as well a definition of an appropriate coordinate operator, which is equivalent to the one obtained by \\cite{NW49}. Although the operator is unique for massive particles with arbitrary spin and for every massless system with spin $0,\\,1\/2$, it has two major physical setbacks. First, eigenstates of the Newton-Wigner-Pryce (NWP) operator propagate superluminally and moreover the states are not covariant w.r.t. general Lorentz-transformations. Hence, there are two different opinions on this subject; either such localizable particles do not exist in a relativistic context or one can still make physical sense of the Newton-Wigner-Pryce operator and its respective states. \n\\\\\\\\\nIn this work, we take the second point of view and give, in our opinion, convincing arguments why the NWP sates are of physical interest. In particular, the importance of these states relies on the fact that they can be used to calculate the probability amplitude of finding a particle at a certain spatial-position at time $t$. Moreover, the second quantization of this operator allows us to calculate the probability amplitude of finding $k$-particles at certain spatial-positions at time $t$.\n\\\\\\\\\nIn order to have a richer example of the NWP-space in a quantum field theoretical context (than the massive scalar field \\cite{Muc7}), we work with the free \\textbf{massless} scalar field. In particular, we show that the localizability of particles, in the massless case, is not a lost cause. In general, the covariance group in the massless case is more than solely the Poincar\\'e group, it is given by the conformal group. Hence, we investigate the covariant transformation property of NWP-states under conformal transformations. The importance therein has a physical reasoning. It is first of all done in order to prove that although the NWP-states are non-covariant w.r.t. to the whole conformal group there exist transformations of a subgroup of the conformal group that preserve their covariant character. Furthermore, given two observers measuring, for a given covariant state, the probability of finding $k$-particles at $k$-spatial-positions. If these two observers are in two relatively to each other transformed frames, for example one is rotated, space-time translated or dilated to the other then they can agree on the observation of the probability of spatial positions, by using the covariant transformation rules.\n\\\\\\\\\nBesides calculations of position-probability amplitudes of particles, the NWP-operator has an additional physical relevance. In this work we prove that by solely using the coordinate and relativistic momentum operator, one is able to define all generators of the conformal group. Hence, in some sense, the NWP-operator can be seen as essential for implementing relativistic symmetries. \n\\\\\\\\\nAnother motivation besides understanding localization to translate the conformal group into the coordinate space, is non-commutative (NC) geometry applied to quantum field theory (QFT). In particular the geometry that is quantized in NCQFT is the geometry of the NWP-space. Hence, in order to understand the outcome and the action of a deformation of QFT using, for example (see \\cite{MUc1}) the conformal group, we need a better understanding of the commutative case.\\\\\\\\\nThe paper is organized as follows; Section two gives a quick introduction to the tools needed in this paper, where we define the Fock space of the free massless scalar field, the Newton-Wigner-Pryce operator and the algebra of the Fourier-transformed ladder operators. In section four, the conformal algebra for a massless scalar field is transformed into the NWP-space.\nSection five investigates the covariance of the NWP-space under the Conformal group.\n\n\n\n\n \\begin{convention}\n \tWe use $d=n+1$, for $n\\in\\mathbb{N}$ and the Greek letters are split into $\\mu, \\,\\nu=0,\\dots,n$. Moreover, we use Latin letters for the spatial components which run from $1,\\dots,n$ and we choose the following convention for the Minkowski scalar product of $d $-dimensional vectors, $a\\cdot b=a_0b^0+a_kb^k=a_0b^0- \\vec{a}\\cdot\\vec{b}$. Furthermore, we use the common symbol $\\mathscr{S}(\\mathbb{R}^n)$ for the Schwartz-space.\n \\end{convention} \n\\section{Preliminaries}\n\\subsection{Bosonic Fock-space and Fourier-transformation}\nWe briefly define the Bosonic Fock \nspace for a free scalar field \n $\\phi$ with mass $m=0$ on the $n+1$-dimensional Minkowski spacetime. In particular a particle with momentum $\\mathbf{p} \\in \\mathbb{R}^n$ has \n the energy $\\omega_{\\mathbf{p} }$ given by $\\omega_{\\mathbf{p} }=|\\mathbf{p}|$. Another quantity needed for the definition of the Fock-space is the Lorentz-invariant measure: $d^n\\mu(\\mathbf{p} )=d^n\\mathbf{p}\/( {2\\omega_{\\mathbf{p}}})$.\n \\begin{definition} The Bosonic Fock space $\\mathscr{H}^{+}$ is defined \n \tas in \\cite{Fr,S}:\n \t\\begin{equation*}\n \t\\mathscr{H}^{+}=\\bigoplus_{k=0}^{\\infty}\\mathscr{H}_{k}^{+}\n \t\\end{equation*}\n \twhere the $k$ particle subspaces are given as\n \t\\begin{align*}\n \t\\mathscr{H}_{k}^{+}&=\\{\\Psi_{k}: \\partial V_{+} \\times \\dots \\times \n \t\\partial V_{+} \\rightarrow \\mathbb{C}\\quad \\mathrm{symmetric}\n \t|\\\\ &\\left\\Vert \\Psi_k \\right\\Vert^2 =\\int \n \td^n\\mu(\\mathbf{p_1})\\dots\\int d^n\\mu(\\mathbf{p_k})\n \t|\\Psi_{k}(\\mathbf{p_1},\\dots,\\mathbf{p_k})|^2<\\infty\\},\n \t\\end{align*}\n \twith\n \t\\begin{equation*}\n \t\\partial V_{+}:=\\{p\\in \\mathbb{R}^{n+1}|p^2=0,p_0>0\\}.\n \t\\end{equation*}\n \\end{definition}\n The particle annihilation and creation operators can be defined by their \n action on $k$-particle wave functions\n \\begin{align*}\n (a_c(f)\\Psi)_k(\\mathbf{p_1},\\dots,\\mathbf{p_k})&=\\sqrt{k+1}\\int \n d^n\\mu(\\mathbf{p})\\overline{f(\\mathbf{p})}\n \\Psi_{k+1}(\\mathbf{p},\\mathbf{p_1},\\dots,\\mathbf{p_k})\\\\\n (a_c(f)^{*}\\Psi)_k( \\mathbf{p },\\mathbf{p_1},\\dots,\\mathbf{p_k})&= \\left\\{\n \\begin{array} {cc}\n 0, \\qquad &k=0 \\\\ \\frac{1}{\\sqrt{k}}\\sum\\limits_{l=1}^{k} f(\\mathbf{p_l})\n \\Psi_{k-1}(\\mathbf{p_1},\\dots,\\mathbf{p_{l-1}},\\mathbf{p_{l+1}},\\dots,\\mathbf{p_k}),\\quad \n &k>0\n \\end{array} \\right.\n \\end{align*}\n with $f \\in \\mathscr{H}_{1} $ and $\\Psi_k \\in \\mathscr{H}_{k}^{+}$ . The commutation relations of the smeared annihilation and creation operators $a_c(f), \n a_c(f)^{*}$ follow from their action on the functions $\\Psi$ and are given as follows\n \\begin{align*}\n [a_c(f), a_c(g)^{*}]=\\langle f,g\\rangle=\\int d^n\\mu(\\mathbf{p}) \\overline{f(\\mathbf{p})} \n g(\\mathbf{p}), \\qquad\n [a_c(f), a_c(g)]=0=[a_c^{*}(f), a_c^{*}(g)].\n \\end{align*}\nThe ladder operators with sharp momentum are \n introduced as operator valued distributions in the following\n \\begin{align*}\n a_c(f)=\\int d^n\\mu(\\mathbf{p}) \\overline{f(\\mathbf{p})}a_c(\\mathbf{p}), \n \\qquad a_c(f)^{*}=\\int d^n\\mu(\\mathbf{p}) {f(\\mathbf{p})}a_c^{*}(\\mathbf{p}),\n \\end{align*}\n where the ladder operators with sharp \n momentum satisfy the well-known commutator relations\n \\begin{align*}\n [a_c(\\mathbf{p}), a_c(\\mathbf{q})^{*}]=2\\omega_{\\mathbf{p} \n }\\delta^n(\\mathbf{p}-\\mathbf{q}), \\qquad\n [a_c(\\mathbf{p}), a_c(\\mathbf{q})]=0=[a_c^{*}(\\mathbf{p}), a_c^{*}(\\mathbf{q})].\n \\end{align*}\n In the following sections, we use the non-covariant normalization given by, \n \n \\begin{align}\n {a}(\\mathbf{p})= \\frac{{a}_c(\\mathbf{p})}{\\sqrt{2\\omega_{\\mathbf{p} \n \t}}},\\qquad {a}^{*}(\\mathbf{p})= \\frac{{a}^{*}_c(\\mathbf{p})}{\\sqrt{2\\omega_{\\mathbf{p} \n }}}.\n \\end{align}\n Next, we define the Fourier-transformation of the ladder operators that were given in the former expressions. These transformations are of physical importance in regards to the following sections. \n \\begin{definition}\\label{sec2.2}\\textbf{Fourier-transformation}\\\\\\\\\n \tIn order to change from momentum space to the NWP-space we use explicit expressions for the Fourier-transformed creation and annihilation operators which are given by,\n \t\\begin{align*}\n \t{a}(\\mathbf{p})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}(\\mathbf{x}),\\qquad {a}^{*}(\\mathbf{p})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{x}\\, e^{-ip_kx^k} \\tilde{a}^*(\\mathbf{x}).\n \t\\end{align*}\n \tThe commutation relation between the ladder operators in momentum space gives us directly the relations for ladder operators of the NWP space, \n \t\\begin{align*}\n \t\\delta^n(\\mathbf{p}-\\mathbf{q})&=[{a}(\\mathbf{p}),{a}^*(\\mathbf{q})]=(2\\pi)^{-n} \n \t\\iint\n \td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, e^{ip_kx^k}\\, e^{-iq_ky^k}\\underbrace{ [\\tilde{a}(\\mathbf{x}),\\tilde{a}^*(\\mathbf{y})]}_{=\\delta^n(\\mathbf{x}-\\mathbf{y})}.\n \t\\end{align*} \n Inverse Fourier-transformations of momentum space operators to the NWP-space ladder operators are given by,\n \t\\begin{align}\\label{inf}\n \t\\tilde{a}(\\mathbf{x})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{p}\\, e^{-ip_kx^k}{a}(\\mathbf{p}),\\qquad \n \t\\tilde{a}^*(\\mathbf{x})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{p}\\, e^{ ip_kx^k}{a}^*(\\mathbf{p}).\n \t\\end{align}\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \\end{definition} \n \n\\subsection{Massless NWP-operator}\nThe (spatial)-QFT-position operator for a massive free scalar field is defined by the Newton-Wigner-Pryce operator, \\cite{PR48} and \\cite{NW49}. On a one particle wave-function it acts as follows, \\cite[Chapter 3c, Equation 35]{Sch}\n \\begin{equation}\\label{NWP}\n (X_{j} \\varphi)(\\mathbf{p})=-i \\left( \\frac{p_j}{2\\omega_{m,\\mathbf{p}}^2}\n + \\frac{\\partial}{\\partial p^j } \n \\right)\\varphi(\\mathbf{p}),\n \\end{equation}\n where $\\omega_{m,\\mathbf{p}}$ is the relativistic-energy for a particle with mass $m$. \n In order to prove that the NWP-operator is in the massless case equivalently represented as in the massive case, we can proceed in different ways. First, we can take Equation (\\ref{NWP}) and perform the massless limit. Since, the energy $\\omega_{m,\\mathbf{p}}$ is the only term that is affected by the limit, the massless NWP-operator is given by exchanging in Formula (\\ref{NWP}) the massive energy with $\\omega_{\\mathbf{p}}$. Therefore, the coordinate operator for a massless QFT has the same form as in the massive case. Another possible path to proceed is to define the coordinate operator as unitary equivalent to the second-quantized spatial-component of the relativistic momentum operator. In terms of the Fock space operators the relativistic momentum operator is given as,\n \\begin{align}\\label{mop}\n P_{\\mu}&= \\int d^n \\mathbf{p}\\, p_{ \\mu}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\n .\n \\end{align} \n The unitary equivalence of the coordinate to the spatial momentum operator is given by the unitary map represented by the Fourier-transformation. In particular, it was proven that the NWP-operator can be represented as a self-adjoint operator on the domain, $\\bigotimes_{i=1}^k \\mathscr{S}(\\mathbb{R}^n)$, with $\\mathscr{S}(\\mathbb{R}^n)$ denoting the Schwartz space, for details see \\cite{Muc2} and \\cite{Muc3}. The explicit Fock-space representation of the spatial-coordinate operator is given by,\n \\begin{align*}\n X_j&=-i\\int d^n \\mathbf{p}\\, {a}^*(\\textbf{p}) \\frac{\\partial}{\\partial p^j} {a}(\\textbf{p}).\n \\end{align*} \n Since the spatial-momentum operator has the same form in the massless case as in the massive one, the NWP-operator, defined through the unitary equivalence, takes the same form as in the massive case. This is stems from the fact that ladder operators do not explicitly depend on the mass. The commutation relations between the spatial-momentum operator and the NWP-operator are given by, see \\cite{SS} or \\cite{Muc3},\n \\begin{align}\\label{ccr}\n [X_j,P_k]&=-i\\eta_{jk}N,\n \\end{align} \n where $N$ is the particle-number operator represented in Fock-space as\n \\begin{align}\\label{pn}\tN= \\int\n d^n \\mathbf{p}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p}).\n \\end{align}\n Next, we want to give the physical interpretation and relevance of this operator. The eigenfunctions of the Newton-Wigner-Pryce operator, simultaneously representing the localized wave functions at time $x_{0}=0$, are given by, \\cite[Chapter 3, Equation 38]{Sch}\n\\begin{align*} \n\\Psi_{\\mathbf{x},0}(\\mathbf{p})=(2\\pi)^{-n\/2}\\,e^{-i\\mathbf{p} \\cdot \\mathbf{x}}\\,(2\\omega_{p})^{1\/2}.\n\\end{align*} \n Let a free massless scalar field be in a state $\\Phi(\\mathbf{p})$ at time $t=0$. Then, the probability amplitude of finding a particle at the position $\\mathbf{x}$ is given by, \\cite[Chapter 3, Equation 44]{Sch}\n\\begin{align*} \n\\langle \\overline{\\Psi}_{\\mathbf{x}}, \\Phi\\rangle=\\int d^n\\mu(\\mathbf{p})\\, \\Psi_{\\mathbf{x},0}(\\mathbf{p}) \\Phi(\\mathbf{p}).\n\\end{align*} \nIn order to extend this quantity to $k$-particles to calculate the probability amplitude for finding $k$-particle at positions $\\mathbf{x}_{1} \\cdots \\mathbf{x}_{k}$ at time $x_{0}={x}_{10}= \\cdots={x}_{k0}$, the following operator is introduced, \\cite[Chapter 7, Equation 99]{Sch},\n \\begin{align}\\label{copes}\n \\phi_1( {x}) =\\int d^n\\mu(\\mathbf{p}) \\overline{\\Psi_{\\mathbf{x},x_{0}}(\\mathbf{p})} a_c(\\mathbf{p}) =(2\\pi)^{-n\/2}\\,\n \\int d^n \\mathbf{p} \\, e^{-i {p} \\cdot {x}}\\, a (\\mathbf{p}),\n \\end{align}\n and we apply $k$-times $\\phi_1$ on $\\vert\\Phi^1 \\rangle$ and the vacuum as follows,\n \\begin{align*} \n \\Phi^1( x_1 ,\\cdots, x_k ) = (k!)^{-1\/2} \\langle 0\\vert \\phi_1(x_{1})\\cdots\\phi_1(x_{k})\n \\vert\\Phi^1 \\rangle.\n \\end{align*}\n It is important to point out that the operator $\\phi_1(x)$ at time $x^{0}=0$ is nothing else than the Fourier-transformed annihilation operator, see Equation (\\ref{inf}). Moreover, \tthe Newton-Wigner-Pryce operator has the following coordinate space representation (for proof see \\cite[Lemma 3.1]{Muc7}),\n \\begin{align*}\n X_j&= \\int d^n \\mathbf{x}\\, x_j \\, \\tilde{a}^*(\\textbf{x}) \\tilde{a}(\\textbf{x})\n .\n \\end{align*} \n\\subsection{Constantly Used Integrals}\nTo make this work self-contained, we give in this section a general formula for certain Fourier-transformed functions. In particular, these formulas can be found in \\cite[Chapter III, Section 2.6-2.8]{GS1},\n\\begin{align}\\label{f1}\n\\tilde{P}^{\\lambda}&= \\int\\,d^n\\mathbf{p}\\, \\vert\\vec{p}\\vert^{2\\lambda} \\exp({-i\\,p_k z^k})\\\\\\nonumber\n&= 2^{2\\lambda+n}\\pi^{\\frac{1}{2}n}\\,\\frac{\\Gamma(\\lambda+\\frac{1}{2}n)}{\\Gamma(-\\lambda)}\\left(z_1^2+\\cdots +z_n^2\\right)^{-\\lambda-\\frac{1}{2}n},\n\\end{align}\nwhere $\\Gamma$ denotes the Gamma-function.\n \n\\section{Conformal Group in the NWP-space}\nIn this section we change the basis of the conformal group, which is given in terms of Fock-space operators in the momentum space, to the coordinate space. The base changes are mathematically well-defined on the domain $\\bigotimes_{i=1}^k \\mathscr{S}(\\mathbb{R}^n)$. This conclusion follows from the fact that we use the Fourier-transformation for the base change and this transformation acts as a linear automorphism on the Schwartz-space. Hence, we conclude that the base change is well-defined.\n\\\\\\\\However, an exception to the rule is given by the Lorentz generators generating boosts. They do not map the Schwartz-space to itself since the coefficient functions of the differential operators are\nnot differentiable at $p=0$. Hence, if one intends to calculate the adjoint action w.r.t. the Lorentz-boosts one has to choose the space of analytic functions on $S^2 \\times \\R$, more specifically of analytic sections, which is the light-cone without the tip, (see \\cite[ Chapter 8.6]{ND}).\\footnote[1]{I am indebted to Prof. N. Dragon for this remark.}\n\\\\\\\\\nFrom a quantum field theoretical point of view, the conformal group is of great interest. \nIn particular, the group is defined as the set of all conformal transformations. The definition of a conformal transformation is an \ninvertible mapping $\\mathbf{x}'\\rightarrow \\mathbf{x}$, leaving a\n$d$-dimensional metric $g$ invariant. The invariance holds, modulo a scale factor, \\cite{DMS}:\n\\begin{equation} \\label{cond1}\ng'_{\\mu\\nu}(x')=F(x)g_{\\mu\\nu}(x).\n\\end{equation}\nConformal mappings can be essentially summarized as Lorentz \ntransformations, translations, dilations and special \nconformal transformations. The generators of these transformations are given by the \noperators $M_{\\mu\\nu}$, $P_{\\rho}$, $D$, $K_{\\sigma}$.\n\\\\\\\\\nThe conformal algebra is defined by the commutation relations of the \ngenerators and is given as follows:\n\\begin{equation}\n[M_{\\mu\\nu},M_{\\rho\\sigma}]\n=i\\left(\\eta_{\\mu\\sigma}M_{\\nu\\rho}+\\eta_{\\nu\\rho}M_{\\mu\\sigma}-\\eta_{\\mu\\rho}M_{\\nu\\sigma}-\n\\eta_{\\nu\\sigma}M_{\\mu\\rho},\n\\right)\n\\end{equation}\n\\begin{equation}\n[P_{\\rho},M_{\\mu\\nu}]=i\\left(\\eta_{\\rho\\mu}P_{\\nu}-\\eta_{\\rho\\nu}P_{\\mu}\\right),\n\\qquad \n[K_{\\rho},M_{\\mu\\nu}]=i\\left(\\eta_{\\rho\\mu}K_{\\nu}-\\eta_{\\rho\\nu}K_{\\mu}\\right),\n\\end{equation}\n\\begin{equation}\\label{f11}\n[P_{\\rho},D]=iP_{\\rho}, \\qquad [K_{\\rho},D]=-iK_{\\rho},\n\\end{equation}\n\\begin{equation}\\label{f12}\n[P_{\\rho},K_{\\mu}]= 2i\\left(\\eta_{\\rho\\mu}D-M_{\\rho\\mu}\\right),\n\\end{equation}\nwhere all other commutators are equal to zero.\n\\\\\\\\ \nTo transform the momentum and Lorentz operators in to the NWP-space we can take the massive expressions (see \\cite{Muc7}) and perform a massless limit, i.e. $m\\rightarrow0$. However, in order to make this work self-contained we calculate most expressions explicitly. \n\\begin{lemma}\n\tThe zero component in the massless case is given by,\n\t\\begin{align*} P_{0}\n\t= \n\t\\int\n\td^n \\mathbf{x}\\,\n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega} \\ast\n\t\\tilde{a} \\right)(\\mathbf{x} ),\n\t\\end{align*} \n\twith function $\\tilde{\\omega}(x):=-\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2}) |\\mathbf{x} |^{- ( n+1)}$ and $\\ast$ denoting the convolution.\\\\\\\\\n\tThe massless spatial momentum operator takes the same form in coordinate space as in the massive case, \n\t\\begin{align*} \n\tP_j&=\n\t- i\n\t\\int\n\td^n \\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\frac{\\partial}{\\partial x^{j}}\\tilde{a} (\\mathbf{x} ).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tFor the proof we take the expression of the massless energy operator (see Equation (\\ref{mop})) and transform, by explicit Fourier-transformation, into the coordinate space,\n\t\\begin{align*} \n\tP_{0}&= \\int\n\td^n \\mathbf{p}\\, \\omega_{\\mathbf{p}} \\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\\\\&=(2\\pi)^{-n} \n\t\\int\n\td^n \\mathbf{p}\\, |\\mathbf{p}| \\int\n\td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^n \\mathbf{y} \\,e^{-ip_ly^l} \\tilde{a} (\\mathbf{y} ) \\\\&=(2\\pi)^{-n} \n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, \\left( \\int\n\td^n \\mathbf{p}\\, {|\\mathbf{p}| } \\, e^{ip_k(x-y)^k} \\right)\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\t \\\\ &= -\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2})\n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, \n\t\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ {n+1}} } \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere the Fourier-transformation can be found in \\cite[Chapter III, Section 2.6]{GS1} or see Equation (\\ref{f1}). The spatial momentum operator does not depend explicitly on the mass term, hence its representation in the massless is equivalent to the massive case, (see \\cite{Muc7}).\n\\end{proof}\nOther important expressions in the NWP-context are the velocity and the particle number operator. One way to calculate the velocity operator is by using the Heisenberg equation of motion, \n\\begin{align}\\label{heq}\n[P_{0},X_{j}]=-iV_j.\n\\end{align}\nSince, we represented the relativistic energy and the NWP-operator in coordinate space, the calculation of the commutator gives us the velocity directly in coordinate space. In order to represent the particle number operator in the NWP-space we apply a Fourier-transformation on the expression in momentum space (see Equation (\\ref{pn})).\n\\begin{lemma}\n\tThe velocity operator is given, in the massless case, as follows\n\t\\begin{align}\n\tV_j= -i\\int d^n \\mathbf{x} \\, \n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega} _j\\ast\n\t\\tilde{a}\\right) (\\mathbf{x} ),\n\t\\end{align}\n\twith function $\\tilde{\\omega} _j(\\mathbf{x}):=- \\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2}) \\, |\\mathbf{x} |^{- ( n+1)}\\,x_j$ and $\\ast$ denoting the convolution. \\\\\\\\ Moreover, the particle number operator $N$ is in the coordinate space given by,\n\t\\begin{align}\n\tN= \\int d^n \\mathbf{x} \\, \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{x} ).\n\t\\end{align}\n\t\n\n\\end{lemma}\n\\begin{proof}\nWe calculate the velocity by taking the commutator of the spatial coordinate space and the zero component of the momentum (see Heisenberg-Equation (\\ref{heq})), which is given in the former theorem, i.e. \n\t\\begin{align*}\n\t[P_0,X_j]&=\\iint d^n \\mathbf{x} \\, d^n \\mathbf{z} \\, z_j\\, [\n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega}\\ast\n\t\\tilde{a}\\right) (\\mathbf{x} ),\\tilde{a}^*(\\mathbf{z}) \\tilde{a} (\\mathbf{z}) ]\\\\&=\n\t\\iiint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\,d^n \\mathbf{z}\\,z_j \\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y})\\underbrace{[\n\t\t\\tilde{a}^*(\\mathbf{x}) \n\t\t\\tilde{a} (\\mathbf{y} ),\\tilde{a}^*(\\mathbf{z}) \\tilde{a} (\\mathbf{z}) ]}_{-\\delta(\\mathbf{x}-\\mathbf{z}) \\tilde{a}^*(\\mathbf{z})\\tilde{a} (\\mathbf{y})+\\delta(\\mathbf{y}-\\mathbf{z}) \\tilde{a}^*(\\mathbf{x})\\tilde{a} (\\mathbf{z})}\\\\&=\n\t-\\iint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\, \\,(x-y)_j \\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y}) \\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}).\n\t\\end{align*}\n\tFor the particle number operator we use the Fourier-transformation as in the proof of the former lemma,\n\t\t\\begin{align*} \n\tN&= \\int\n\t\td^n \\mathbf{p}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\\\\&=(2\\pi)^{-n} \n\t\t\\int\n\t\td^n \\mathbf{p}\\, \\int\n\t\td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}^*(\\mathbf{x}) \\int\n\t\td^n \\mathbf{y} \\,e^{-ip_ly^l} \\tilde{a} (\\mathbf{y} ) \\\\&=(2\\pi)^{-n} \n\t\t\\iint\n\t\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\,\\underbrace{ \\left( \\int\n\t\td^n \\mathbf{p}\\, e^{ip_k(x-y)^k} \\right)}_{(2\\pi)^{n}\\delta(\\mathbf{x}-\\mathbf{y}) }\\tilde{a}^*(\\mathbf{x}) \n\t\t\\tilde{a} (\\mathbf{y} ) .\n\t\\end{align*}\n\\end{proof}\nNext, we turn our attention to the expressions of Lorentz generators. They are given as in the massive case by (see \\cite[Equation\n3.54]{IZ} and in this context see also \\cite[Appendix]{SS}),\n\\begin{align}\\label{lbcaop1}\nM_{j0}&= i\\int d^n\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \n\\left( \\frac{p_j}{2\\omega_{\\textbf{p}}}-\\omega_{\\textbf{p}}\\frac{\\partial}{\\partial p^j } \\right) a(\\textbf{p}),\n\\\\ \\label{lbcaop2}\nM_{ik}&=i \\int d^n\\mathbf{p}\\, {a}^{*}(\\textbf{p}) \n\\left(p_i \\frac{\\partial}{\\partial p^k }-p_k\\frac{\\partial}{\\partial p^i }\\right)\na(\\textbf{p}).\n\\end{align}\n\tIt is interesting and important to note that operators of boost and rotations can be given by the second quantization (denoted by $d\\Gamma(\\cdot)$ see \\cite[Chapter X.7]{RS2}) of symmetric or skew-symmetric products of the momentum and coordinate operator.\n\\begin{theorem}For the massless scalar field the generators of the proper orthochronous Lorentz-group $\\mathscr{L}^{\\uparrow}_{+}$ can be written in terms of the second quantization of products of the NWP-operator with relativistic four-momentum as,\n\t\\begin{align}\\label{p1}\n\tM_{0j}=\\frac{1}{2}\\big( d\\Gamma(X_jP_0)+d\\Gamma(P_0X_j)\\big) ,\n\\qquad \n\t\tM_{ik} = d\\Gamma(X_iP_k)-d\\Gamma(X_{k}P_i).\n\t\t\\end{align}\n\t\\end{theorem}\n\t\\begin{proof}\n\tFor the boosts we have,\n\t\\begin{align*}\n\tM_{0j}&=\\frac{1}{2}\\big( d\\Gamma(X_jP_0)+d\\Gamma(P_0X_j)\\big) =\\frac{1}{2} d\\Gamma([X_j,P_0])+d\\Gamma(P_0X_j) \n\t \\\\ &=\\frac{i}{2} d\\Gamma(V_j)+d\\Gamma(P_0X_j) ,\n\t\\end{align*}\n\twhere in the last lines we used the Heisenberg equation of motion. By comparing the last line with Formula (\\ref{lbcaop1}) the proof follows. \t For rotations, the representation follows trivially. \n\t\\end{proof}\n\tThe interesting fact about the former Theorem is that in principle one can define the generators of the Lorentz-group in a $k$-particle space by solely using the translation group and the NWP-operator. This resembles the proof of the Maxwell-equations by using the canonical commutation relations of Feynman (published by Dyson, \\cite{FD}). In particular, the existence of the NWP-operator and the translation group are sufficient to build the Poincar\\'e group, i.e. the group responsible for the implications of special relativity.\n\tHence, the commutation relations with the addition of relativistic energy can be used to implement relativistic principles. This fact, is, in our opinion, an additional argument for the physical sense of the NWP-operator. \\\\\\\\\nNext, we turn our attention to the explicit expressions of Lorentz generators in the coordinate space.\n\\begin{lemma}\n\tThe boost generators of the Lorentz group expressed in the terms of ladder operators of the free massless scalar field are given in the coordinate space as\n\t\\begin{align*}\n\tM_{0j}=\\frac{1}{2}\t\\iint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\,(x+y)_j\\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y}) \\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}) .\n\t\\end{align*}\n\tThe operator of rotations takes the, from a quantum mechanical point of view, familiar and to the massive case equivalent form, \n\t\\begin{align*}\n\tM_{ik}=\t i \\int\n\td^n \\mathbf{x} \\,\n\t\\tilde{a}^*(\\mathbf{x})\n\t\\left(x_i \\frac{\\partial}{\\partial x^k }-x_k\\frac{\\partial}{\\partial x^i }\\right) \\tilde{a} (\\mathbf{x}).\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}Since in the proof of \\cite[Lemma 4.3]{Muc7} we did not explicitly encounter the mass when we calculated the representation of rotations in the coordinate space, we conclude that they have the same form.\tNext, we turn to the proof of the representation of Lorentz boosts in coordinate space. The first term is simply the massless velocity operator times $\\frac{i}{2}$ (see Equation (\\ref{p1})). Hence, we focus here only on the second part,\n\t\\begin{align*}\n\t&- i\\int d^n\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\, \\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^j} a(\\textbf{p})\\\\&=\n\t- i\\int d^n\\mathbf{p}\\,\\int\n\td^n \\mathbf{x}\\, e^{-ip_rx^r} \\,\\tilde{a}^*(\\mathbf{x}) \\, \\omega_{\\textbf{p}}\\frac{\\partial}{\\partial p^j} \\int\n\td^n \\mathbf{y}\\, e^{ip_sy^s} \\tilde{a}(\\mathbf{y}) \n\t\\\\&= (2\\pi)^{-n}\\iint\td^n \\mathbf{x}\\,\td^n \\mathbf{y}\n\t\\left( \\int d^n\\mathbf{p}\\, |\\mathbf{p}| \\, e^{-ip_k(x-y)^k} \\right)\\,y_j\\,\\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}) \n\t\\\\&=-\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2})\n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, y_j\\, \n\t\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ {n+1}} } \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\n\t\\\\&= \n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, y_j\\,\\tilde{\\omega}(\\mathbf{x}-\\mathbf{y})\n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\n\t,\n\t\\end{align*} \n\\end{proof}\nIn order to translate the special conformal and the dilatation operator into the coordinate space, we first write the momentum space representation, see \\cite{SS}. Moreover, we restrict this part to the physical space-time dimension four. The dilatation operator is given in momentum space as \n\\begin{align}\nD=-i\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\n{a}(\\textbf{p}),\n\\end{align}\nand for the special conformal operators we have,\n\\begin{align}\nK_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{4 \\,\\omega_{\\mathbf{p}}}+\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}-\\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\na(\\textbf{p}),\\\\ \nK_{j}&= -\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\\frac{p_j}{4\\,\\omega_{\\mathbf{p}}^2}+3 \\frac{\\partial}{\\partial p^{j}}+2p^l\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p^{j}}-p_{j}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\na(\\textbf{p}).\n\\end{align}\n\\begin{lemma}\n\tThe four-dimensional dilatation operator is given in the coordinate representation as follows,\n\t\\begin{align}\n\tD=i\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+x^{l}\\frac{\\partial}{\\partial x^{l}}\n\t\\right)\\tilde{a}(\\mathbf{x}).\n\t\\end{align}\n\\end{lemma}\n\\begin{proof}\n\tThe Fourier-transformation is straight-forward in this case, e.g.\n\t\\begin{align*}\n\tD&=-i\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\n\t{a}(\\textbf{p})\n\t\\\\&=-i(2\\pi)^{-3} \n\t\\int\n\td^3 \\mathbf{p}\\, \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\\int\n\td^3 \\mathbf{y} \\,e^{ ip_ky^k} \\tilde{a} (\\mathbf{y} ) \t\\\\&=-\n\t\\frac{3i}{2}\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x})\n\t-i(2\\pi)^{-3} \n\t\\iint\n\td^3 \\mathbf{p}\\, \n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^3 \\mathbf{y} \\,\\left(y^{l}\\frac{\\partial}{\\partial y^{l}}e^{ ip_ky^k}\\right) \\tilde{a} (\\mathbf{y} ) \\\\&=\n\t \t\\frac{3i}{2}\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x})\n\t+i(2\\pi)^{-3} \n\t\\iint\n\td^3 \\mathbf{p}\\, \n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^3 \\mathbf{y} \\,e^{ ip_ky^k} y^{l}\\frac{\\partial}{\\partial y^{l}}\\tilde{a} (\\mathbf{y} ) \\\\&=\n\ti\t\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+ x^{l}\\frac{\\partial}{\\partial x^{l}}\\right)\\tilde{a}(\\mathbf{x}) .\n\t\\end{align*}\n\\end{proof}\nThe outcome of the dilatation operator is interesting. In essence, it is equivalent to its momentum representation modulo a sign. This is what one might expect, since this operator performs dilatations in a complimentary manner, with regard to the momentum and coordinate space. \\\\\\\\\nNext, we transform the special conformal operators into the coordinate space. From a calculational point of view these operators are the most difficult ones. \n\\begin{lemma}\n\tThe zero component of the special conformal operators has the following form in coordinate space \n\t\\begin{align*}\n\tK_{0}&= \\pi^{-2}\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x}) \\,\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\left( \\frac{9}{8}-\n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2}\\vert\\mathbf{y}\\vert^2\n\t+ \\frac{y^{l}}{2} \\frac{\\partial}{\\partial y^{l}}\n\t\\right) \\tilde{a} (\\mathbf{y} ).\n\t\\end{align*}\n\t\n\t\n\\end{lemma}\n\n\n\n\\begin{proof}\n\tIn order to make this calculation more readable we separate the zero component into three parts as follows,\n\t\\begin{align*}\n\tK_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{4 \\,\\omega_{\\mathbf{p}}}+\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}-\\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\n\ta(\\textbf{p})\\\\&=:\n\tK_{0}^1+K_{0}^2+K_{0}^3.\n\t\\end{align*}\n\tLet us take a look at the first part,\n\t\\begin{align*}\n\tK^1_{0}&=-\\frac{3}{4}\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\,\\omega_{\\mathbf{p}}^{-1}\n\ta(\\textbf{p})\n\t\\\\&=-\\frac{3}{4}(2\\pi)^{-3} \n\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left( \\int d^3 \\mathbf{p} \n\t\\, e^{-ip_k(x-y)^k } \\,\\omega_{\\mathbf{p}}^{-1}\\right)\n\t\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= -\\frac{3}{8} \\pi^{-2}\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x}) \\,\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\tilde{a} (\\mathbf{y} ) .\n\t\\end{align*}\n\tNext we turn our attention to the second term of the zero component of the special conformal operator, \n\t\\begin{align*}\n\tK^2_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\,\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}\n\ta(\\textbf{p})\n\t\\\\&=- (2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}e^{ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= - (2\\pi)^{-3} \\int\n\td^3 \\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}} e^{-ip_rx^r}\n\t\\int d^3 \\mathbf{y} \\,\\left( y^{l}\\frac{\\partial}{\\partial y^{l}}e^{ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= (2\\pi)^{-3} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}} e^{-ip_k(x-y)^k}\\right)\n\t\\, \\left(3+y^{l} \\frac{\\partial}{\\partial y^{l}}\\right)\\tilde{a} (\\mathbf{y} )\n\t\\\\&= (2)^{-1} \\pi^{-2} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left( \\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\right)\n\t\\, \\left(3+y^{l} \\frac{\\partial}{\\partial y^{l}}\\right)\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere in the last lines we performed a partial integration and used Formula (\\ref{f1}). The last term of the zero component is given by\n\t\\begin{align*}\n\tK^3_{0}&=\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\, \\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\n\ta(\\textbf{p})\\\\&\n\t=(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left( \n\t\\omega_{\\mathbf{p}}\\,\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\n\te^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= (2\\pi)^{-3}\\iint\n\td^3 \\mathbf{x}\\,\t d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\\left(\n\t\\int d^3\\mathbf{p}\\, {\\omega_{\\mathbf{p}}}\\, e^{-ip_k(x-y)^k}\\right) \\,\\vert\\mathbf{y}\\vert^2 \\tilde{a} (\\mathbf{y} ) \n\t\\\\&= -( \\pi)^{-2}\\iint\n\td^3 \\mathbf{x}\\, d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x})\\left(\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-4} \n\t\\right)\t\\vert\\mathbf{y}\\vert^2 \\tilde{a} (\\mathbf{y} ) , \n\t\\end{align*}\n\twhere in the last lines we used the action of the differential operators w.r.t. $\\mathbf{p}$ on the exponential function and used as before Equation (\\ref{f1}).\n\t\n\\end{proof}\nNext, we give the base change of the spatial part of the special conformal operator. \n\\begin{lemma}\n\tThe spatial components of the special conformal operators have the following form in coordinate space,\n\t\\begin{align}\n\tK_{j} = -\\frac{i}{16}\\pi^{-1}\\iint&\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-1} \n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} )\\\\&\\nonumber+\n\ti \\int\n\td^3 \\mathbf{x} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(3x_j+ 2x_{j}x^{l} \\frac{\\partial}{\\partial x^{l}}- x_{l}x^{l} \\frac{\\partial}{\\partial x^{j}} \\right) \\tilde{a} (\\mathbf{x} )\n\t\\end{align}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\tLet us first recall and define the expression of the special conformal operators in the momentum space, i.e. \n\t\\begin{align*}\n\tK_{j}&= -\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\t\\frac{p_j}{4\\,\\omega_{\\mathbf{p}}^2}+3 \\frac{\\partial}{\\partial p^{j}}+2p_l\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}-p_{j}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\n\ta(\\textbf{p})\\\\&=\n\tK^{1}_{j}-3i\tX_{j}+\tK^{2}_{j}+\tK^{3}_{j}.\n\t\\end{align*}\n\tThe second term in the spatial conformal operator is simply the coordinate operator times a constant. This fact was used in \\cite{SE}. Hence, the remaining terms of interest are $K^{1}_{j},\\,K^{2}_{j},\\,K^{3}_{j}$. Let us start with first object, \n\t\\begin{align*}\n\tK^{1}_{j}&= -\n\t\\frac{1}{4}\n\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\t\\frac{p_j}{\\,\\omega_{\\mathbf{p}}^2} \\right)\n\ta(\\textbf{p})\\\\&= -\n\t\\frac{1}{4}\n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\t\\frac{p_j}{\\,\\omega_{\\mathbf{p}}^2} e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \\\\&= -\n\t\\frac{i}{4}\n\t(2\\pi)^{-3} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}^2} e^{-ip_k(x-y)^k}\\right)\n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} )\n\t\\\\&= - \\frac{i}{16}\\pi^{-1}\\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-1} \n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere in the last lines we expressed the momentum as a derivative, performed a partial integration and used the Fourier-transformation given in Formula (\\ref{f1}) for $\\lambda=-1$. A general term which includes, after taking the corresponding indices, the second and third operator of the spatial special conformal object is given by,\n\t\\begin{align*}\n\tO_{j}&= \n\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(p_{r}\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}\\right)\n\ta(\\textbf{p})\\\\&= \n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_sx^s} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\tp_r\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} )\n\t\\\\&= i\n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_sx^s } \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(y_{j}y^{l} \\frac{\\partial}{\\partial y^{r}} e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} )\\\\&= - i \\int\n\td^3 \\mathbf{x} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\eta_{rj}x^l+\\eta_{r}^{\\,\\,l}x_{j}+ x_{j}x^{l} \\frac{\\partial}{\\partial x^ {r}} \\right) \\tilde{a} (\\mathbf{x} ),\n\t\\end{align*}\n\twhere in the last lines we applied the derivatives to the exponential, expressed the coordinate $y$\tas a derivative and performed a partial integration. \n\\end{proof}\n \\begin{remark}\n \tAs for Lorentz generators, dilatation and special conformal operators can be written in terms of symmetric second-quantized products of the momentum and NWP-operator. For the dilatation operator it is straight forward,\n \t\\begin{align*}\n \tD=\\frac{1}{2}\\big(d\\Gamma(P_jX^j) +d\\Gamma(X^jP_j)\\big).\n \t\\end{align*}\n \twhile the special-conformal operators are a bit more involved. For the zero component we have, \n \t\t\\begin{align*}\n \t\tK_0&=-\\frac{3}{4}d\\Gamma(P_0^{-1})-id\\Gamma(V^{l}X_l)+d\\Gamma(P_0X_lX^l) \t\\end{align*}\t\nwhile the spatial components in terms of the NWP and the momentum operator read, \n\\begin{align*}\n \t\tK_j&=-\\frac{1}{4}d\\Gamma(P_0^{-1}V_j)-3id\\Gamma(X_j)+2d\\Gamma(P^lX_lX_j)-d\\Gamma(P_jX_lX^l) .\n \t\t\\end{align*}\n \t\t \n \\end{remark}\n \n In this section we expressed all generators of the conformal group in terms of the ladder operators of the NWP space for $x_0=0$. However, the operator $\\phi_1$ (see Equation (\\ref{copes})) responsible for calculating the probability of finding $k$-particle at a certain position has an explicit time dependence. Hence, in order to connect our results with this time-dependency we give the following proposition.\n \\begin{proposition}\\label{sf}\nThe operator $\\phi_1$ that has the following form,\n \t\\begin{align*} \n \t\\phi_1( {x}) =(2\\pi)^{-n\/2}\\,\n \t\\int d^n \\mathbf{p}\\, e^{-i {p} \\cdot {x}}\\,a(\\mathbf{p}),\n \t\\end{align*}\n can be obtained by performing a time-translation on the annihilation operator $\\tilde{a}$ of the NWP-space\n \tas follows\n \t\\begin{align*} \n \t\\phi_1( x^{0}, \\mathbf{x}) = e^{ix^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-ix^{0}P_{0}}, \\qquad x^0\\in\\mathbb{R}.\n \t\\end{align*}\n \t\n \\end{proposition}\t\\begin{proof}\n By using the inverse Fourier-transformation of $\\tilde{a}$ and the transformation of the annihilation operator in momentum space under time-translation, the proof is completed, i.e. \n \\begin{align*} \n e^{ix^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-ix^{0}P_{0}}&=(2\\pi)^{-n\/2} \\int\n d^n \\mathbf{p}\\, e^{-ip_kx^k}e^{ix^{0}P_{0}}{a}(\\mathbf{p})e^{-iy^{0}P_{0}}\\\\&=(2\\pi)^{-n\/2} \\int\n d^n \\mathbf{p}\\, e^{-ip_kx^k}\\,e^{-ip_{0}x^{0} }{a}(\\mathbf{p}) .\n \\end{align*}\n\\end{proof}Hence, we can simply time-translate all the operators of the conformal group in the NWP-space, by using the same time $x^0$, in order to obtain the expressions for a general time and in terms of the operator $\\phi_1$. Since the zero component of the momentum commutes with the spatial momentum and the rotations, we only have to calculate the time-translation for the remaining generators by using the algebra.\n\n\n\n\\begin{lemma}\n\tThe dilatation transforms under time-translations as follows,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}D e^{-ix^{0}P_{0}} \t= D-x^{0}P_{0}, \n\t\\end{align*}\n\twhere $x^0\\in\\mathbb{R}$. Under time-translations the special conformal operator transforms as,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}K_{\\mu} e^{-ix^{0}P_{0}}&=\n\tK_{\\mu}-2x^{0}\\left(\n\t\\eta_{0\\mu}D-M_{0\\mu}\n\t\\right)-(x^{0})^2\n\t\\left( P_{\\mu}\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tThe proof solely uses the conformal algebra (see Formulas (\\ref{f11}) and (\\ref{f12})). For the dilatations we have,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}D e^{-ix^{0}P_{0}} =\n\tD-x^{0}P_{0} +\\frac{i^2}{2!}(x^{0})^2\n\t\\underbrace{[P_{0},[P_{0},D]]+ \\cdots}_{=0}= D-x^{0}P_{0},\n\t\\end{align*}\n\twhere in the last lines we used the Backer-Hausdorff formula.\tA more complex expression is the transformation of the special-conformal operator, i.e. \n\t\\begin{align*}\n\te^{ix^{0}P_{0}}K_{\\mu} e^{-ix^{0}P_{0}}&=K_{\\mu}+ix^{0}[P_{0},K_{\\mu}]+\\frac{i^2}{2!}(x^{0})^2\n\t[P_{0},[P_{0},K_{\\mu}]]+\\underbrace{\\cdots}_{=0}\n\t\\\\&= K_{\\mu}-2x^{0}\\left(\n\t\\eta_{0\\mu}D-M_{0\\mu}\n\t\\right)-(x^{0})^2\n\t\\left( \\eta_{00}P_{\\mu}\\right) ,\n\t\\end{align*}\n\twhere we used the Baker-Campbell-Hausdorff formula and the specific commutator relations of the conformal algebra.\n\t\n\\end{proof}\nSince, we have all expressions in terms of generators of the NWP-space we can equivalently give the conformal algebra expressed by operators $\\phi^*_1(x_0,\\mathbf{x})$ and $\\phi_1(x_0,\\mathbf{y})$. Note that neither the coordinate operator nor its respective eigenstates are covariant w.r.t. general Lorentz-transformations. Nevertheless, the generators of the conformal group obey the covariant transformation property independent of their representation. \tHence, even though we use non-covariant eigenstates, operators of the conformal group represented in the NWP-space respect the Poincar\\'e symmetry. \n\\section{Conformal-Transformations of NWP-States} \nIn this section we calculate the explicit adjoint action of the conformal group on the Fourier-transformed ladder operators. In some cases this is done by taking the massless limit of the massive theory. \\\\\\\\\n\t The main physical motivation to calculate the adjoint action w.r.t. conformal transformations, in the coordinate space, is given by the fact that we use operator $\\phi_1(x)$ to calculate the probability of localizing $k$-particles at $k$-spatial positions. Therefore, we intend to investigate under which subgroup of the conformal group the probability amplitudes of localization are covariant. \n\t\\\\\\\\ In order find the covariance group, we first define the unitary operator generating transformations of the orthochronous proper Poincar\\'e group $\\mathscr{P}^{\\uparrow}_{+}=\\mathscr{L}^{\\uparrow}_{+}\\ltimes\\mathbb{R}^4$ denoted by $U(y,\\Lambda)$. It transforms creation and annihilation operators in the following fashion, \\cite[Chapter 7]{Sch},\n\t\\begin{align}\\label{traf}\n\tU(y,\\Lambda) a(\\mathbf{p})U(y,\\Lambda)^{-1}&= \\sqrt{\\frac{\t\\omega_{\\Lambda \\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}e^{-i(\\Lambda p)_{\\mu}y^{\\mu}}{a}(\\Lambda\\mathbf{p}) ,\\qquad (y,\\Lambda) \\in \\mathscr{P}^{\\uparrow}_{+},\\\\\\label{traf1}\n\tU(y,\\Lambda) a^{*}(\\mathbf{p})U(y,\\Lambda)^{-1}&= \\sqrt{\\frac{\t\\omega_{\\Lambda \\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}e^{ i(\\Lambda p)_{\\mu}y^{\\mu}}{a}^{*}(\\Lambda\\mathbf{p}) ,\\qquad (y,\\Lambda) \\in \\mathscr{P}^{\\uparrow}_{+}\n\t.\n\t\\end{align} \n\tBy using the former equations we first calculate the adjoint action of translations on the NWP-ladder operators.\n\t\\begin{lemma}\\label{l63}\n\t\tThe Fourier-transformed ladder operator $\\tilde{a}$ of the massless field transforms under translations as follows, \n\t\t\\begin{align}\\label{eq51}\n\t\tU({y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U({y},\\mathbb{I}) ^{-1} = c_{y } \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{x+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{x+y}),\n\t\t\\end{align} \n\t\twhere $c_{y }= \\frac{i y^{0}}{ \\pi^2}$, $\\ast$ denotes the convolution and $y\\in\\mathbb{R}^d$.\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tThe proof for spatial translations can be done as in the massive case (see \\cite[Lemma 5.1]{Muc7}) since for this proof there is no explicit mass dependency. In particular they act accordingly as spatial translations in coordinate space, i.e.\n\t\t\t\\begin{align*}\t\tU( \\mathbf{y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( \\mathbf{y},\\mathbb{I}) ^{-1} \n\t\t\t &=\n\t\t\t(2\\pi)^{3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} U(\\mathbf{y},\\mathbb{I}) {a}(\\mathbf{p}) U(\\mathbf{y},\\mathbb{I}) ^{-1}\\\\\n\t\t\t&=(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_k(x+y)^k} {a}(\\mathbf{p})= \\tilde{a}(\t \\mathbf{x}+\\mathbf{y}).\n\t\t\t\t\\end{align*} \n\t\t For time translations, we take the massless limit of the massive case (see \\cite[Lemma 5]{Muc7}) where we had the integral, \n\t\t\t\\begin{align*}(2\\pi)^{-3}\\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} e^{-iy^{0} \\omega_{m,\\mathbf{p}}} =\n\t\t\\frac{i\ty^{0}}{ 2\\pi^2} \\frac{ m^2K_{2}(m\\sqrt{\\vert \\mathbf{x}-\\mathbf{z}\\vert^{2}-(y^{0})^2})}{\\vert \\mathbf{x}-\\mathbf{z}\\vert^{2}-(y^{0})^2},\\end{align*} \n\t\t with $\\omega_{m,\\mathbf{p}}$ being the energy for a particle with mass $m$. By using the following limit, \n\t\t\\begin{align}\\label{ml0}\n\t\t\\lim\\limits_{m\\rightarrow 0} m^2\\, K_{2}(m f)=\\frac{2}{f^2},\n\t\t\\end{align} \n\t the explicit form of the time translation can be given in the massless case as follows, \n\t\t\t \\begin{align*} U(y^{0},\\mathbb{I}) \\tilde{a}(\t \\mathbf{x}) U(y^{0},\\mathbb{I})^{-1} &=e^{iy^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-iy^{0}P_{0}}\\\\\n\t\t\t &=(2\\pi)^{-3\/2} \\int\n\t\t\t d^{3} \\mathbf{p}\\, e^{-ip_kx^k} e^{-iy^{0} \\omega_{\\mathbf{p}}} {a}(\\mathbf{p}) \\\\\n\t\t\t &=(2\\pi)^{-3 }\\int\n\t\t\t d^{3} \\mathbf{z} {\\left(\\int\n\t\t\t \td^{3} \\mathbf{p}\\, e^{-ip_k(x-z)^k} e^{-i \\omega_{\\mathbf{p}}y^{0}}\\right)}_{\n\t\t\t } \\tilde{a}(\\mathbf{z})\\\\&=\\frac{i\ty^{0}}{ \\pi^2}\n\t\t\t \\int\n\t\t\t d^{3} \\mathbf{z} \\left(\n\t\t\t \\frac{\t 1}{\\vert \\mathbf{x-z} \\vert^{2}-(y^{0})^2 } \\right)^2\\tilde{a}(\\mathbf{z}).\n\t\t\t \\end{align*}\n\t\t\t By taking the commutativity of the components of the momentum operator into account, we obtain the general transformation behavior under translations by using the former two results, i.e.\n\t\t\t \t\\begin{align} \tU( {y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( {y},\\mathbb{I}) ^{-1} =\n\t\t U(y^{0},\\mathbb{I}) \tU( \\mathbf{y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( \\mathbf{y},\\mathbb{I}) ^{-1} U(y^{0},\\mathbb{I})^{-1}.\n\t\t\t \t\\end{align} \n\t\t\t \n\t\\end{proof}\nBefore we turn to the Lorentz transformations in the massless case, we define the rotation group. \tIn particular we denote matrices of the Lorentz group representing pure rotations as $\\Lambda_R$. They are given by matrices of the following form,\n\\[\\Lambda_R=\\left(\n\\begin{matrix} \n1 & 0\\\\\n0 & R\t \n\\end{matrix}\\right), \\qquad R\\in SO(3).\n\\]\n\t\\begin{lemma}\\label{l622}Under pure rotations the Fourier-transformed ladder operators $\\tilde{a}$ transforms in a covariant manner as follows, \n\t\t\\begin{align*}\n\t\tU(0,\\Lambda_R) \\tilde{a} (\t \\mathbf{x}) U(0,\\Lambda_R) ^{-1} =\n\t\t\\tilde{a} (\t \\mathbf{R x }) .\n\t\t\\end{align*}\n\t \n\t\\end{lemma}\n\t\\begin{proof}\n\t\tThe proof is done analog to the proof of the former Lemma. However, we could take the results of \\cite[Lemma 5.2]{Muc7} and perform the massless limit by using Equation (\\ref{ml0}). First, we calculate the action of pure spatial rotations on the Fourier-transformed annihilation operator, \n\t\t\t\\begin{align*}\n\t\t\tU(0,\\Lambda_R) \\tilde{a}^{\\,}(\t \\mathbf{x}) \tU(0,\\Lambda_R) ^{-1} &=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} U(0,\\Lambda_R){a}(\\mathbf{p}) U(0,\\Lambda_R)^{-1}\\\\&=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} \\sqrt{\\frac{\t\\omega_{R\\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}{a}(R\\mathbf{p}) \n\t\t\t\\\\&=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-i p_k(R x)^k} {a}( \\mathbf{p}),\n\t\t\t\\end{align*}\n\t\t\tin the last lines we used the transformation behavior of the non-covariant momentum ladder operators (see \\cite[Chapter 7, Equation 57]{Sch}) and the orthogonality of $\\Lambda_{R}$ for pure spatial rotations. \n\t\\end{proof}\t\nThe NWP ladder operators transform non-covariantly w.r.t. the Lorentz-boosts. This is to be expected, since the NWP-operator is non-covariant w.r.t. the boosts. However, we see in the next theorem that the operator $\\phi_1(x)$ is at least covariant w.r.t. space-time translations and pure rotations.\n\\begin{theorem}\\label{t2}\n\tThe NWP-ladder operators transform under space-time translations and pure rotations, i.e. \twith $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$, as follows,\n\\begin{align}\n\tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}&=\n\t \\frac{i y^{0}}{ \\pi^2} \\left(\n\t \\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{Rx+y}) \\\\\n\t \tU(y,\\Lambda_R) \\tilde{a}^{*}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}&=-\n\t \t\\frac{i y^{0}}{ \\pi^2} \\left(\n\t \t\\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a}^{*} (\\mathbf{Rx+y}).\n\\end{align}\nMoreover, the operator used to calculate the probabilities, i.e. $\\phi_1(x)$ transforms covariantly,\n\\begin{align}\\label{t2e2}\nU(y,\\Lambda_R) \\phi_1(x) U(y,\\Lambda_R)^{-1}= \\phi_1(\\Lambda_Rx+y)\\qquad (y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}.\n\\end{align}\n\t\t\n\n\\end{theorem}\n\t\n \n\\begin{proof}\n\tBy using Lemmas \\ref{l63} and \\ref{l622} we deduce,\n\t\t\\begin{align*} \tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1} &= \tU(y,\\mathbb{I})\tU(0,\\Lambda_R)\\tilde{a}(\\mathbf{x})U(0,\\Lambda_R)^{-1}\tU(y,\\mathbb{I})^{-1}\t\t\n\t\\\\&=\t\t \t \tU(y,\\mathbb{I}) \\tilde{a}(R\\mathbf{x})\n\t\tU(y,\\mathbb{I})^{-1}\t \n\t\t\\\\&= \\frac{i y^{0}}{ \\pi^2} \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{Rx+y}) ,\t\n\t\t\\end{align*} \n\t\twhere in the last lines we used the fact that a general Poincar\\'e transformation can be split in first Lorentz-transformation and then space-time translation. An analog proof can be done for the creation operator $\\tilde{a}^{*}$ or we can use the fact that we have unitary operators generating the transformations and take the adjoint of the former equality. \\\\\\\\ In order to calculate the adjoint action of the operator $\\phi_1(x)$, we use the fact that time translations commute with the unitary operator of the Poincar\\'e group for pure rotations, i.e.\n\t\t\\begin{align*}\n\t\t\tU(y,\\Lambda_R) \\phi_1(x) U(y,\\Lambda_R)^{-1}& =\n\t\t\t\tU(y,\\Lambda_R)U(x_0,\\mathbb{I}) \\tilde{a}(\\mathbf{x}) U(x_0,\\mathbb{I})^{-1}U(y,\\Lambda_R)^{-1}\\\\&=U(x_0,\\mathbb{I})\n\t\t\t\tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}U(x_0,\\mathbb{I})^{-1}\\\\&=\n\t\t\t\tU(x_0+y_0,\\mathbb{I})\n\t\t\t\tU(\\mathbf{y},\\mathbb{I}) \\tilde{a}(R\\mathbf{x}) U(\\mathbf{y},\\mathbb{I})^{-1}U(x_0+y_0,\\mathbb{I})^{-1}\n\t\t\t\t\\\\&=\n\t\t\t\tU(x_0+y_0,\\mathbb{I}) \\tilde{a}(R\\mathbf{x+y}) U(x_0+y_0,\\mathbb{I})^{-1}= \\phi_1(\\Lambda_Rx+y).\n\t\t\\end{align*}\n\t\t\n\t\t\n\n\t\t\n\\end{proof}\nNext, we investigate the adjoint action w.r.t. the dilatations. \n\tIn order to calculate the adjoint action of dilatations on the NWP-space we first calculate the commutator of the dilatation operator with the coordinate operator. From that we can deduce the adjoint action of the dilatations acting on the coordinate ladder operators. This construction is analog to the one in \\cite[Chapter 9.4]{SU}.\n\t\\begin{lemma}\nThe adjoint action w.r.t. the dilatations acting on the NWP-operator is given as follows,\n\t\\begin{align*}\n\te^{i\\alpha D} X_j e^{-i\\alpha D}=e^{-\\alpha}X_j,\n\t\\end{align*}\nwhere in order to perform the calculations of the adjoint action the following commutator relation was used,\n\t\\begin{align*}\n\t[D,X_{j}]=iX_{j}.\n\t\\end{align*} \nFrom the former equation it follows that $X_{j}e^{-i\\alpha D}|\\mathbf{x}\\rangle$ is an eigenvector of \n$X_{j}$ with eigenvalue $e^{-\\alpha}x_{j}$. Therefore, one concludes that the transformation behavior of the operator $\\tilde{a}(\\mathbf{x})$ under dilatations is\n\t\\begin{align*}\n\te^{i\\alpha D}\\tilde{a}(\\mathbf{x})\te^{-i\\alpha D}=\ne^{ \\frac{3}{2}\\alpha}\\,\\tilde{a}(e^{ \\alpha}\\mathbf{x}).\n\t\\end{align*}\n \n\t\\end{lemma}\n\t\n\t\n\t\\begin{proof}\n\t\tIn order to proof the transformational behavior we first calculate the commutator of the dilatation operator with the NWP-operator, \n\t\t\\begin{align*}\n\t\t[D,X_{j}]&=\n\t\ti\t\\iint d^3\\mathbf{x}\\,d^3\\mathbf{y}\\,y_j\\,[\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+ x^{l}\\frac{\\partial}{\\partial x^{l}}\\right)\\tilde{a}(\\mathbf{x}) , \\tilde{a}^*(\\mathbf{y}) \\tilde{a}(\\mathbf{y})]\\\\&=\n\t\ti\t\\iint d^3\\mathbf{x}\\,d^3\\mathbf{y}\\,y_j\\,x^{l}\\,\\underbrace{[\\tilde{a}^*(\\mathbf{x}) \\frac{\\partial}{\\partial x^{l}} \\tilde{a}(\\mathbf{x}) , \\tilde{a}^*(\\mathbf{y}) \\tilde{a}(\\mathbf{y})]}_{-\\delta(\\mathbf{x}-\\mathbf{y})\\tilde{a}^*(\\mathbf{y})\\frac{\\partial}{\\partial x^{l}} \\tilde{a}(\\mathbf{x})+\\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y})\\frac{\\partial}{\\partial x^{l}}\\delta(\\mathbf{x}-\\mathbf{y})} \t \n\t\t\t\\\\&\t=-i\\int d^3x \\left(x_{j}\\,x^l \\tilde{a}^*(\\mathbf{x})\\frac{\\partial}{\\partial x^{l}}\\tilde{a}(\\mathbf{x})+\n\t\t\t \\,x_j\\,\\left(\\frac{\\partial}{\\partial x^{l}}\\left(x^l\\tilde{a}^*(\\mathbf{x})\\right) \\right) \\tilde{a}(\\mathbf{x})\\right)\n\t\t\t\t\\\\&=\t\t-i\\int d^3x \\left(x_{j}\\,x^l \\tilde{a}^*(\\mathbf{x})\\frac{\\partial}{\\partial x^{l}}\\tilde{a}(\\mathbf{x})-x^l\\tilde{a}^*(\\mathbf{x})\n\t\t\t\\left(\\frac{\\partial}{\\partial x^{l}}\\left(\t x_j\\, \\tilde{a}(\\mathbf{x})\\right) \\right)\\right)\n\t\t\t\t\\\\&=\t i\\int d^3x \\,x_j\\, \t\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x}),\n\t\t\\end{align*}\n\t\twhere in the last lines we used the vanishing commutator of the particle number operator with the coordinate operator. Moreover, we integrated over delta distributions and performed partial integrations. \\\\\\\\\t\tFrom the former commutator relation we obtain for the adjoint action of the NWP-operator w.r.t. dilatations the following, \n\t\t\\begin{align*}\n\t\t e^{i\\alpha D}X_{j}e^{-i\\alpha D}&=X_{j}+i\\alpha [D,X_{j}]+\\frac{i^2}{2!}\\alpha^2 [D,[D,X_{j}]]+\\cdots\\\\\n\t\t&= X_{j}+i^2\\alpha\\,X_{j}+\\frac{i^4}{2!}\\alpha^2X_{j}+\\cdots\n\t\t\\\\&= \\sum\\limits_{n=0}^{\\infty}\\frac{(i^2\\alpha)^n}{n!}X_{j}=e^{-\\alpha}X_j,\n\t\t\\end{align*}\n\t\twhere in the last lines we used the commutator relation between the dilatation and the NWP-operator and the Baker-Campbell-Hausdorff formula. As in \\cite[Chapter 9.4]{SU} we look at the following expression,\n\t\t\t\\begin{align*}\n\t\t\t X_{j}\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle &=e^{-i\\alpha D}e^{i\\alpha D} X_{j}\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle\\\\&= e^{-i\\alpha D}e^{-\\alpha}X_j|\\mathbf{x}\\rangle=e^{-\\alpha}x_j\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle.\n\t\t\t\\end{align*}\n\t\tTherefore $X_{j}e^{-i\\alpha D}|\\vec{x}\\rangle$ is an eigenvector of $X_{j}$ for\n\t\tthe eigenvalue $e^{-\\alpha}x_j$, \n\t\tthus contained in the\n\t\teigenspace $\\mathcal{H}_{e^{-\\alpha}\\vec{x} }$ of $X_{j}$. Hence, the transformation on the creation operator of the NWP-space has to have the following form, \n\t\t\\begin{align}\\label{daf}\n\t\te^{i\\alpha D}\\tilde{a}^*(\\mathbf{x})\te^{-i\\alpha D}=\n\t\tf(\\alpha)\\,\\tilde{a}^*(e^{ \\alpha}\\mathbf{x}),\n\t\t\\end{align}\n\t\twhere $f(\\alpha)$ is a real-valued function of the transformation parameter $\\alpha$. To find the concrete form of the function $f$ we calculate the adjoint action of the coordinate operator\n\t\tw.r.t. the dilatations but in the Fock representation of the NWP-space, i.e. \n\t\\begin{align*}\n\te^{i\\alpha D}X_j\te^{-i\\alpha D}&= e^{i\\alpha D}\\,\\int d^3 \\mathbf{x}\\,x_j\\,\n\t\\tilde{a}^*( \\mathbf{x})\t\\tilde{a} ( \\mathbf{x})\te^{-i\\alpha D}\\\\&=\n\t\\int d^3 \\mathbf{x}\\,x_j\\,\n\tf(\\alpha)^2\\tilde{a}^*(e^{ \\alpha}\\mathbf{x})\\tilde{a} (e^{ \\alpha}\\mathbf{x})\\\\&=e^{-\\alpha}\n\t\\int d^3 \\mathbf{x}\\,e^{ -3\\alpha }\\,x_j\\,\n\tf(\\alpha)^2\\tilde{a}^*( \\mathbf{x})\\tilde{a} ( \\mathbf{x}), \n\t\\end{align*}\n\t\twhere in order for the NWP-operator to transform correctly the function $f$ has to be chosen as $f(\\alpha)= e^{ \\frac{3}{2}\\alpha }$. In order to solidify our approach we apply the adjoint action on the particle number operator in the coordinate space, \n\t\t\\begin{align*}\n e^{i\\alpha D}N\te^{-i\\alpha D}&= e^{i\\alpha D}\\,\\int d^3 \\mathbf{x}\\,\n\t\\tilde{a}^*( \\mathbf{x})\t\\tilde{a} ( \\mathbf{x})\te^{-i\\alpha D}\\\\&=\n\t \\int d^3 \\mathbf{x}\\,\n\t\tf(\\alpha)^2\\tilde{a}^*(e^{ \\alpha}\\mathbf{x})\\tilde{a} (e^{ \\alpha}\\mathbf{x})\\\\&=\n\t\t\\int d^3 \\mathbf{x}\\,e^{ -3\\alpha }\n\t\tf(\\alpha)^2\\tilde{a}^*( \\mathbf{x})\\tilde{a} ( \\mathbf{x})=N, \n\t\t\\end{align*}\n\twhere in the last lines a variable substitution was performed and the commutator relation $[D,N]=0$ is used to argue that the adjoint action leaves the particle number operator invariant and therefore the Ansatz (\\ref{daf}) with explicit form of $f$ gives us the right transformational behavior. \n\t\\end{proof}\n\t To calculate the adjoint action of dilatations on the ladder operators of the NWP-space we could have, as well, taken the well-known transformation property of the ladder operators in momentum space. After doing so we could have performed an inverse Fourier-transformation. However, one of the purposes of this work is to demonstrate to the reader that all calculations can as well be done in the NWP-space. In particular, the NWP-space is on an equal footing with the momentum space. \\\\\\\\\nSince we are interested in transformations of the time translated expression of the ladder operators we need to take the commutation relations between the momentum and dilatation operators into account. \n\\begin{theorem}\\label{t3}\n\tThe operator $\\phi_1(x)$ that is responsible for calculating probabilities of finding $k$-particles in $k$-spatial positions transforms under dilatations as follows, \n\t\\begin{align*}\ne^{i\\alpha D}\\phi_1(x)e^{-i\\alpha D}=e^{\\frac{3}{2} \\alpha}\\phi_1(e^{\\alpha}x).\n\t\\end{align*}\n\\end{theorem}\n\\begin{proof}\nWe can rewrite the operator $\\phi_1(x)$ as \n\t\\begin{align*}\n\\phi_1(x)= \tU(x^{0},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U(-x^{0},\\mathbb{I}).\n\t\\end{align*}\nSince we know the adjoint action on the ladder operators in the NWP-space we can directly calculate the transformation on all expressions. First we calculate the adjoint action of the dilatations on the zero component of the momentum operator,\n\t\\begin{align*}\ne^{i\\alpha D}\\,P_0\\,e^{-i\\alpha D}&= -\\pi^{-2}\\Gamma(2)\n\\iint\nd^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\, \n\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ 4} } \ne^{i\\alpha D}\\,\\tilde{a}^*(\\mathbf{x}) \\, e^{i\\alpha D}e^{-i\\alpha D}\\,\n\\tilde{a} (\\mathbf{y} )\\,e^{-i\\alpha D}\\\\&\n= -\\pi^{-2}\\Gamma(2)\n\\iint\nd^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\, \n\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ 4} } \\, e^{ 3\\alpha}\\,\n \\tilde{a}^*(e^{ \\alpha}\\mathbf{x}) \\,\n\\tilde{a} (e^{ \\alpha}\\mathbf{y} ) \\\\&\n= e^{\\alpha} P_{0},\n\t\\end{align*}\nwhere in the last lines we used the explicit transformation given in Equation (\\ref{daf}) and performed a variable substitution. Next, we apply the adjoint action of dilatations on this operator and use the fact that we are dealing with unitary transformations, i.e.\n\\begin{align*}\ne^{i\\alpha D}U(x^{0},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U(-x^{0},\\mathbb{I})e^{-i\\alpha D} &=e^{\\frac{3}{2} \\alpha}\nU(e^{\\alpha}x^{0},\\mathbb{I}) \\tilde{a} (e^{ \\alpha} \\mathbf{x}) U(-e^{\\alpha}x^{0},\\mathbb{I})\\\\&= \\frac{i\te^{\\frac{5}{2} \\alpha} x^{0}}{2\\pi^2}\n\\int\nd^{3} \\mathbf{z} \\left(\n\\frac{\t 1}{\\vert e^{ \\alpha}\\mathbf{x-z} \\vert^{2}-(e^{\\alpha}x^{0})^2 } \\right)^2\\tilde{a}(\\mathbf{z}).\n\\end{align*}\n\\end{proof}\nBy using the former lemmas we calculate the adjoint action of translations, rotations and dilatations on $\\phi_1(x)$. This is in particular important since the operator $\\phi_1(x)$ is used to calculate probability amplitudes of finding $k$-particles in $k$-spatial positions. Hence, by knowing the transformational behavior, two frames that are connected via translations, rotations and\/or dilatations can calculate the same probability by taking the proper transformation into account.\n\t\\begin{theorem}\n\t\tThe operator\t $\\phi_1(x)$ transforms covariantly under space-\\textbf{time} translations, pure rotations and dilatations, i.e. \n\t\t\\begin{align*}&\n\te^{i\\alpha D}\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\n\t\t= e^{\\frac{3}{2} \\alpha}\\phi_1(\te^{\\alpha}(\\Lambda_R x+ y))\n ,\n\t\t\\end{align*} \n\t\twith $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$ and $\\alpha\\in\\mathbb{R}$.\n\t\tMoreover, the explicit transformation of the Fourier-transformed operator $\\tilde{a}$ under the action of the subgroup $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$ is in coordinate space given as, \n\t\t\\begin{align*}&\n\t\te^{i\\alpha D}\t\tU(y,\\Lambda_R)\\tilde{a}(\\mathbf{x})U(y,\\Lambda_R)^{-1}e^{-i\\alpha D}\n\t\t= c_{y' } \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{Rx}+ \\mathbf{y} \\vert^{2}-( y^{0})^2 } \\right)^2\\ast\\tilde{a} (e^{\\alpha}(\\mathbf{Rx}+ \\mathbf{y}) ),\n\t\t\\end{align*} \n\t\twhere $c_{y' }:= \\frac{i e^{-\\frac{3}{2}\\alpha} y^{0}}{ \\pi^2}$, $\\ast$ denotes the convolution and $y\\in\\mathbb{R}^4$.\n \n\t\\end{theorem}\n\\begin{proof}\nIn order to prove the first part of the theorem we use \tthe former Lemmas and the algebra of the Poincar\\'e group. First, we examine how the operator $\\phi_1(x)$ transforms under space-time translations and pure rotations,\n\t\\begin{align*}\n\t& e^{i\\alpha D}\t\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\\\\ & =\te^{i\\alpha D}\t\\phi_1(\\Lambda_R x+y )\te^{-i\\alpha D} \\\\& =\n\te^{\\frac{3}{2}\\alpha}\\phi_1(e^{\\alpha}\\Lambda_R x+e^{\\alpha}y )\n\\\\&=\t \ne^{\\frac{3}{2}\\alpha}U(e^{\\alpha}(x_0+ y_0),\\mathbb{I}) \\tilde{a}(e^{\\alpha}(\\mathbf{Rx+ y}))U(-e^{\\alpha}(x_0+ y_0),\\mathbb{I}) \\\\&=e^{\\frac{3}{2}\\alpha}\nc_{y^{''} } \\left(\n\\frac{\t 1}{\\vert e^{\\alpha}(\\mathbf{Rx+ y}) \\vert^{2}-(e^{\\alpha}(x_0+ y_0))^2 } \\right)^2\\ast\\tilde{a} (e^{\\alpha}(\\mathbf{Rx+ y})),\n\\end{align*}\nwhere $c_{y^{''} }:=\\frac{i e^{\\alpha}(x_0+ y_0)}{ \\pi^2}$ and \n\twhere for the calculation we used the transformation property given in Theorem \\ref{t3}, the dilatation action Equation (\\ref{t2e2})\n and the explicit time translation given in Equation (\\ref{eq51}).\n\\end{proof}\n \n\\section{Conclusion and Outlook}\nIn this work we supported the point of view, that the NWP operator and its respective states have a physical meaning. In particular, we defined the second-quantized version of the NWP-operator for a massless scalar field and wrote the operator in its respective eigenbase. The meaning of the second-quantized operator is given by its respective eigenfunctions. They are used to calculate the probability amplitude of finding particles at certain spatial positions for a fixed time. Furthermore, by transforming the whole conformal group into the NWP-eigenbase we showed that this space can be treated and worked on at equal footing as the momentum space. \n\\\\\\\\\nAlthough eigenstates of this operator are non-covariant w.r.t. the Poincar\\'e group we showed indirectly that combinations of these states are covariant. This is proven by the fact that covariant operators remain covariant, independently of their respective representation. Hence, even though we work with non-covariant eigenstates, symmetries of the theory remain untouched. This is, in our opinion, another strong argument why the coordinate space, i.e. the NWP-space, is equally entitled from a physical point of view as the momentum space. \n\\\\\\\\\nMoreover, we showed that the operator generating the NWP-space, i.e. $\\phi_1(x)$ transforms in a covariant manner under the subgroup of space-time translations, rotations and dilatations of the conformal group, namely \n\t\\begin{align*}&\n\te^{i\\alpha D}\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\n\t= e^{\\frac{3}{2} \\alpha}\\phi_1(\te^{\\alpha}(\\Lambda_R x+ y))\n\t.\n\t\\end{align*}\nHence, given two observers that are related by space-time translations, rotations or\/and dilatations, they both measure the same probability amplitude, by taking the respective covariant transformation into account. \\\\\\\\\t\nFurthermore, we were able to supply an additional argument to point out the physical relevance of the NWP-operator. Namely, it is possible, even in a second-quantized context, to express the generators \nof the conformal group by using the NWP-operator and the relativistic momentum. In our opinion this supplies a strong argument for the physical sense of the NWP-operator, since the conformal group can be written in terms of products of the coordinate and momentum operators. \\\\\\\\\n\tSince we formulated the whole framework of a free massless scalar field, its respective transformations and symmetries into the NWP-space, the next step in progress is to understand localization from a non-commutative point of view. In particular, by imposing a space-space non-commutativity the subject of interest is to be able to calculate the probability of finding a particle at a certain mean value of a spatial position. In our opinion, the ground work for such investigations was presented in this paper. \n\n\t\\section*{Aknowledgments}\n\tThe author is indebted to Prof. K. Sibold for initiating deep conceptual questions of this work. Furthermore, we would like to thank Prof. K. Sibold and S. Pottel for many fruitful discussions during different stages of this paper. With regards to the domains of the Lorentz-boosts am indebted to Prof. Norbert Dragon. The corrections of Dr. Z. Much are thankfully acknowledged.\n\t\n \n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \n If $k$ and $N$ are positive integers, let $S_k(N)$ be the rational vector space of cusp forms of weight $k$ on $\\Gamma_0(N)$ with rational Fourier coefficients.\n These forms have a Fourier expansion at $\\infty$ of the form\n\\[\n f(z)= \\sum_{n=n_0}^{\\infty} a(n) q^{n} \\text{ }\\text{ with }\\text{ }a(n_0) \\neq 0,\n\\]\nand we define $\\operatorname{ord}_{\\infty}(f) := n_0$. Let $g(N)=\\dim(S_2(N))$ be the genus of $X_0(N)$. We say that $\\infty$ is a Weierstrass point on the modular curve $X_0(N)$ if there exists $0 \\neq f \\in S_2(N)$ such that $\\operatorname{ord}_{\\infty}(f) > g(N)$. Ogg \\cite{Ogg} proved the following theorem.\n\\begin{theorem}\\label{thm:Ogg}\n If $p$ is a prime such that $p \\nmid N$, and if $g(N)=0$, then $ \\infty $ is not a Weierstrass point on $X_0(pN)$.\n\\end{theorem}\n A non-geometric proof of Theorem 1.1 was given in [AMR09] (previously, certain cases of level $p\\ell$ for distinct primes $p$ and $\\ell$ were considered in \\cite{Kohnen}, \\cite{Kilger}). To state our first result, when $N$ is a positive integer and $p$ is a prime such that $p \\nmid N$, we require the Atkin-Lehner operator $W_{p}^{pN}$ on $S_{k}(pN)$ defined in \\eqref{eqn:Atkin-Lehner}. Furthermore, if \n\\[\nf(z)= \\sum_{n=n_0}^{\\infty} a(n) q^{n} \\in S_k(N),\n\\] \ndefine \n\\[\n v_p(f):=\\inf\\{v_p(a(n))\\}.\n\\]\n With this notation, we prove the following theorem.\n\\begin{theorem}\\label{thm:main}\n Let $N$ be a positive integer and $k$ be a positive even integer. Let \n $p$ be a prime with $p \\geq \\max(5,k+1)$ and $p \\nmid N$. Suppose that $0 \\neq f \\in S_k(pN)$ satisfies \n\\[\n v_p(f)=0, \\text{ }\\text{ }v_p(f|_{k}W^{pN}_{p}) \\geq 1-k\/2.\n\\]\n Then \n\\[\n \\operatorname{ord}_{\\infty}(f) \\leq \\dim(S_k(pN)).\n\\]\n\\end{theorem} \n As a corollary, we prove an analogue of Ogg's theorem.\n\\begin{corollary}\\label{thm:analogue}\n Suppose that $N$ is a positive integer, that $k$ is a positive even integer, and that $p$ is a prime with $p \\geq \\max(5,k+1)$ and $p \\nmid N$. If $S_{k}(N)= \\{0\\}$, then for $0 \\neq f \\in S_{k}(pN)$, we have \n\\[\n \\operatorname{ord}_{\\infty}(f) \\leq \\dim(S_k(pN)).\n\\]\n\\end{corollary}\n There is a finite list of $N$ and $k$ for which $S_k(N)=\\{0\\}$. For $k=2$, there are $15$ such values of $N$ \\cite[pg. 110]{Ono}. For $k \\geq 4$, the rest are\n\\begin{align*}\n k&=4: N= 1, 2, 3, 4 \\\\\n k&=6: N= 1, 2 \\\\\n k&=8,10,14: N= 1. \n\\end{align*}\n \n It is natural to seek to understand the subspace of forms $f \\in S_{k}(N)$ which vanish to order greater than the dimension.\n If $N$ is a positive integer and $k$ is a positive even integer, define the subspace\n\\[\n W_{k}(N):=\\{f \\in S_k(N): \\operatorname{ord}_{\\infty}(f) > \\dim(S_k(N))\\}.\n\\]\n With this notation, we have $W_{2}(N)=\\{0\\}$ if and only if $\\infty$ is not a Weierstrass point on $X_0(N)$. As a corollary of Theorem~\\ref{thm:main}, we obtain a bound for $\\dim(W_{k}(pN))$.\n\\begin{corollary}\\label{thm:Subspace}\n Suppose that $p \\geq \\max(5,k+1)$ is a prime satisfying $p \\nmid N$. Then we have \n\\[\n\\dim(W_{k}(pN)) \\leq \\dim(S_k(N)).\n\\]\n \\end{corollary}\nNote that this implies Theorem~\\ref{thm:Ogg} in the case $k=2$. It is interesting to note that the bound in Corollary~\\ref{thm:Subspace} is independent of $p$. Thus, for fixed $N$, the spaces $W_{k}(pN)$ have uniformly bounded dimension as $p \\rightarrow \\infty$.\n\\begin{remark}\nFor squarefree $N$, Arakawa and B\\\"{o}cherer \\cite{Arakawa-Bocherer} study the space\n\\[\nS_k(N)^{*}:= \\{ f \\in S_k(N): f\\big|_k W_{p}^{pN}+p^{1-\\frac{k}{2}}\\sl U_p=0 \\ \\ \\text{for all $p \\mid N$} \\}.\n\\]\nWe will use a similar subspace to prove Corollary~\\ref{thm:Subspace}.\n\\end{remark}\nThe following examples, which we computed with Magma, illustrate Corollary~\\ref{thm:Subspace} for small values of $N$.\n\\begin{example}\n For an example which is sharp, set $N=1$, $p=19$, and $k=16$. Here, we have $\\dim(S_k(pN))=24$ and $\\dim(S_k(N))=1$. In this case, there is a form $f \\in S_k(pN)$ with $f=q^{25}+\\cdots$.\n \\end{example}\n \\begin{example}\n To get an example which is sharp and for which $pN$ is not prime, set $N=2$, $p=23$, and $k=12$. Here, $\\dim(S_k(pN))=64$ and $\\dim(S_k(N))=2$. In this case, there are forms $f$ and $g$ with $f=q^{67}+\\cdots$ and $g=q^{68}+\\cdots$.\n \\end{example}\n \\begin{example}\n Corollary~\\ref{thm:Subspace} is not always sharp. For example, set $N=1$, $p=29$, and $k=28$. Here, $\\dim(S_k(pN))=67$ and $\\dim(S_k(N))=3$. In this case, there is no non-zero $f \\in S_k(N)$ satisfying $\\operatorname{ord}(f) > 67$.\n \\end{example}\n The paper is organized as follows. Section $2$ contains the background necessary to prove these results. Section $3$ contains the proof of Theorem~\\ref{thm:main}, which uses results from \\cite{Ahlgren-Masri-Rouse}. Finally, Section $4$ contains the proofs of Corollary~\\ref{thm:analogue} and Corollary~\\ref{thm:Subspace}. \n\\section{Preliminaries on Modular Forms}\nThe definitions and facts given here can be found in \\cite{Diamond-Shurman} and \\cite{Ahlgren-Masri-Rouse}. Let $N$ and $k$ be positive integers.\nLet $\\varepsilon_{\\infty}(N)$ denote the number of cusps on $X_0(N)$, let $g(N)$ denote its genus, and let $\\varepsilon_{2}(N)$, $\\varepsilon_{3}(N)$ denote the numbers of elliptic points of orders $2$ and $3$, respectively. Then we have\n\\begin{equation}\\label{genus}\ng(N)=\\mfrac{[\\operatorname{SL}_2(\\mathbb{Z}):\\Gamma_0(N)]}{12}-\\mfrac12\\varepsilon_{\\infty}(N)-\\mfrac14\\varepsilon_{2}(N)-\\mfrac13\\varepsilon_{3}(N)+1,\n\\end{equation}\n\\[ \\varepsilon_{2}(N)= \\begin{cases} \n0 & \\text{if } 4 \\mid N, \\\\\n\\displaystyle\\prod_{p \\mid N} \\(1+\\(\\mfrac{-4}{p}\\)\\) & \\text{otherwise,} \n\\end{cases}\n\\] \n\\[ \\varepsilon_{3}(N)= \\begin{cases} \n0 & \\text{if } 9 \\mid N, \\\\\n\\displaystyle\\prod_{p \\mid N} \\(1+\\(\\mfrac{-3}{p}\\)\\) & \\text{otherwise.} \n\\end{cases}\n\\]\nWe have the well-known formula\n\\[\n[\\operatorname{SL}_{2}(\\mathbb{Z}):\\Gamma_0(N)]=N\\prod_{p \\mid N}\\(1+\\mfrac{1}{p}\\).\n\\] \nFor weights $k \\geq 4$, we have\n\\begin{equation}\\label{*}\n\\dim(S_k(N))=(k-1)(g(N)-1)+\\floor{\\mfrac{k}{4}}\\varepsilon_{2}(N)+\\floor{\\mfrac{k}{3}}\\varepsilon_{3}(N)+\\(\\mfrac{k}{2}-1\\)\\varepsilon_{\\infty}(N).\n\\end{equation} \n \n A form in $S_k(N)$ may have forced vanishing at the elliptic points. As in \\cite{Ahlgren-Masri-Rouse}, let $\\alpha_{2}(N,k)$ and $\\alpha_{3}(N,k)$ count the number of forced complex zeroes of a form $f \\in S_k(N)$ at the elliptic points of order $2$ and $3$, respectively. These are given by \n\\begin{equation}\\label{eq:table}\n(\\alpha_{2}(N,k),\\alpha_{3}(N,k))= \\begin{cases} \n(\\varepsilon_{2}(N),2\\varepsilon_{3}(N)) & \\text{if } k \\equiv 2 \\pmod{12}, \\\\\n(0,\\varepsilon_{3}(N)) & \\text{if } k \\equiv 4 \\pmod{12}, \\\\\n(\\varepsilon_{2}(N),0) & \\text{if } k \\equiv 6 \\pmod{12}, \\\\\n(0,2\\varepsilon_{3}(N)) & \\text{if } k \\equiv 8 \\pmod{12}, \\\\\n(\\varepsilon_{2}(N),\\varepsilon_{3}(N)) & \\text{if } k \\equiv 10 \\pmod{12}, \\\\\n(0,0) & \\text{if } k \\equiv 0 \\pmod{12}.\n\\end{cases}\n\\end{equation} \n\n If $d=\\dim(S_k(N)),$ then $S_k(N)$ has a basis $\\{f_1,...,f_d\\}$ with integer coefficients with the property \n\\begin{equation}\\label{eqn:INTEGRALITY}\nf_i(z)=a_iq^{c_i}+O(q^{c_{i}+1}), \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1 \\leq i \\leq d,\n\\end{equation}\nwhere $a_i \\neq 0$ and $c_10$. \nThe equations for $\\nu$ are \n\\begin{equation}\n\\begin{split}\n\\frac{2}{\\rho}\\,\\partial_z \\nu \\= & \\sum_{\\Lambda=0}^3 \\left[ \\partial_\\rho \\log W_\\Lambda \\,\\partial_z \\log W_\\Lambda + a_\\Lambda^2\\, \\partial_\\rho L_\\Lambda \\,\\partial_z L_\\Lambda \\right], \\\\\n\\frac{4}{\\rho}\\,\\partial_\\rho \\nu \\=& \\sum_{\\Lambda=0}^3 \\left[ \\left(\\partial_\\rho \\log W_\\Lambda\\right)^2 -\\left( \\partial_z \\log W_\\Lambda\\right)^2+ a_\\Lambda^2\\, \\left( \\left(\\partial_\\rho L_\\Lambda\\right)^2 - \\left(\\partial_z L_\\Lambda\\right)^2 \\right)\\right]\\,.\\label{eq:nuEq}\n\\end{split}\n\\end{equation}\nThe integrability is guaranteed by the harmonicity of the functions on the right-hand sides. These equations can be integrated in a case-by-case manner depending on the choice of sources for the harmonic functions.\n\\footnotetext{ More generically, the scalars are given by\n\\begin{equation}\nZ_\\Lambda = {\\cal G}^{(\\Lambda)}_\\ell\\left(L_\\Lambda\\right) \\,,\\qquad \\star_3 d\\left(H_0\\,d\\phi\\right) = dL_0\\,, \\qquad dT_I = \\frac{dL_I}{ {\\cal G}^{(I)}_\\ell(L_I)^2} \\,,\n\\label{eq:WGF&Hfoot}\n\\end{equation}\nwhere ${\\cal G}^{(\\Lambda)}_\\ell$ can be freely chosen among the following five generating functions of one variable:\n\\begin{align}\n{\\cal G}^{(\\Lambda)}_1 (x) &= \\frac{\\sinh(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda} \\,,\\qquad {\\cal G}^{(\\Lambda)}_2 (x) = i\\,\\frac{\\cosh(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda}\\,,\\qquad {\\cal G}^{(\\Lambda)}_3 (x) = x +b_\\Lambda\\,,\\nonumber \\\\\n{\\cal G}^{(\\Lambda)}_4 (x) &= \\frac{\\sin(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda} \\,,\\qquad {\\cal G}^{(\\Lambda)}_5(x) =\\frac{\\cos(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda}\\,,\\qquad a_\\Lambda \\in \\mathbb{R}_+, \\quad b_\\Lambda\\in \\mathbb{R} \\,.\n\\label{eq:DefFi}\n\\end{align}\nHowever, we did not find regular solutions for the branches $\\ell=2,4,5$ (see \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki} for more details). Therefore, we consider only the branch $\\ell=1$ from which the function $\\ell=3$ can be obtained with \\eqref{eq:BPSlimit}.}\n\nIt is important to point out that $W_\\Lambda$ are ``vacuum'' warp factors, as they are not coupled to any gauge potentials, whereas $Z_\\Lambda$ are generated and induced by the gauge potentials. However, one can take a neutral limit by sending all gauge potentials to zero while keeping $Z_\\Lambda$ non-trivial, which then satisfy the vacuum equation $\\Delta \\log Z_\\Lambda =0$. The neutral limit is given by\n\\begin{equation}\n(b_\\Lambda, a_\\Lambda) \\to \\infty \\qquad\\text{with}\\quad e^{b_\\Lambda}\/a_\\Lambda \\quad\\text{and}\\quad a_\\Lambda L_\\Lambda\\quad \\text{held fixed.}\n\\label{eq:neutrallimit}\n\\end{equation}\nIt shows that the present ansatz is generically in the non-BPS regime of the M2-M2-M2-KKm system.\n\nMoreover, at the other side of the parameter space,\n\\begin{equation}\n(b_\\Lambda, a_\\Lambda) \\to 0 \\qquad\\text{with}\\quad b_\\Lambda\/a_\\Lambda \\quad \\text{held fixed,}\n\\label{eq:BPSlimit}\n\\end{equation}\nthe solutions converge to the linear branch $Z_\\Lambda = L_\\Lambda+b_\\Lambda\/a_\\Lambda$. As we will see in the next section, this leads to the BPS equations for the M2-M2-M2-KKm system. Therefore, the ansatz admits a BPS limit considering the above transformation.\n\nIt is remarkable that one can deviate significantly from the BPS regime while maintaining the linearity of the Einstein-Maxwell equations. Moreover, it only changes the warp factors from a linear function (BPS regime) to a $\\sinh$ function (non-BPS regime) in terms of harmonic functions. As we will see later, the price to pay for having a linear ansatz will be to use non-BPS sources that have a fixed charge-to-mass ratio, but not necessarily 1.\n\n\n\\subsection{BPS regime}\n\n\nWe consider that the two-tori and six-torus are rigid, $W_\\Lambda=1$. Moreover, we take the limit \\eqref{eq:BPSlimit} for the scalars $(Z_\\Lambda, H_0,T_I)$ \\eqref{eq:WGF&H}, which leads to $L_\\Lambda \\= Z_\\Lambda - b_\\Lambda\/a_\\Lambda$. Therefore, the solutions can be better written such that $Z_\\Lambda$ are harmonic functions and the gauge potentials are determined by\n\\begin{equation}\n\\Delta Z_\\Lambda \\= 0\\,,\\qquad \\star_3 d (H_0\\,d\\phi) \\= dZ_0\\,,\\qquad T_I \\= - \\frac{1}{Z_I}\\,.\n\\end{equation}\nWe recognize the BPS equations of motion one can obtain from supersymmetric static M2-M2-M2-KKm solutions. With such a choice, $\\nu=0$ \\eqref{eq:nuEq}, and the base is flat $\\mathbb{R}^3$ as expected. Physical solutions are necessarily sourced by point particles on the $z$-axis:\n\\begin{equation}\nZ_\\Lambda \\= \\sum_i^n \\frac{q_i^{(\\Lambda)}}{\\sqrt{\\rho^2+(z-z_i)^2}}\\+ c_\\Lambda\\,, \\qquad H_0 \\= \\sum_i^n \\frac{q_i^{(0)}\\,(z-z_i)}{\\sqrt{\\rho^2+(z-z_i)^2}}\\+ \\bar{c}_0\\,.\n\\end{equation}\nThe sources at $(\\rho=0,z=z_i)$ are either static extremal four-charge black holes if $q_i^{(\\Lambda)} \\neq 0$ or smooth Gibbons-Hawking centers if $q_i^{(I)}=0$. \n\nTherefore, the linear non-BPS floating-brane ansatz contains the BPS floating brane ansatz for the static M2-M2-M2-KKm system.\n\n\n\n\\subsection{Non-BPS regime}\n\\label{sec:GenWeylSol}\n\nGeneric non-BPS solutions are constructed for non-zero $(a_\\Lambda,b_\\Lambda)$ in \\eqref{eq:WGF&H}. One can check that sourcing the harmonic functions by point sources lead to naked singularities. We therefore consider that the harmonic functions are sourced by \\emph{rods}, that is segment sources. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.23\\textwidth]{Rods.pdf}\n\\caption{Schematic description of axisymmetric rod sources.}\n\\label{fig:RodsSources}\n\\end{figure}\n\nIn this section, we establish our formalism and highlight the basic ingredients that can be added linearly through the ansatz. We save the construction of the solutions for later. \n\n\n We consider $n$ distinct rods of length $M_i>0$ along the $z$-axis centered around $z=z_i$. Without loss of generality, we can order them as $z_i < z_j$ for $i 0$ \\eqref{eq:WarpfactorsRods}, which requires, if $b_\\Lambda \\neq +\\infty$, \n\\begin{equation}\nb_\\Lambda \\geq 0 \\,,\\quad a_\\Lambda \\,P_i^{(\\Lambda)}\\, \\geq\\, 0 \\,,\\qquad \\Lambda=0,...,3, \\,\\, i=1,...,n.\n\\label{eq:RegPositivity}\n\\end{equation}\nWith this condition, the solutions are regular out of the $z$-axis. \n\\item[\u2022] \\textbf{Regularity at the rods on the $z$-axis:}\n\n\nWe consider the local spherical coordinates centered around the $i^\\text{th}$ rod, \\begin{equation}\n\\rho= \\sqrt{r_{i}\\left(r_{i}+M_{i}\\right)} \\sin \\theta_i,\\, \\qquad z\\= \\left(r_{i}+\\frac{M_{i}}{2}\\right) \\cos \\theta_{i}+z_{i}. \n\\label{eq:localcoordrod}\n\\end{equation}\n At the rod, $r_i \\to 0$, we have\n\\begin{equation}\n\\frac{R_+^{(i)}}{R_-^{(i)}} \\propto \\frac{1}{r_i}\\,,\\qquad \\frac{E_{+-}^{(i,i)}E_{-+}^{(i,i)}}{E_{++}^{(i,i)}E_{--}^{(i,i)}} \\propto r_i^2,\n\\end{equation}\nwhile all other quantities are well-behaved. From \\eqref{eq:WarpfactorsRods}, we have\\footnote{Since $a_\\Lambda P_i^{(\\Lambda)}\\geq 0$ is required if $b_\\Lambda \\neq \\infty$, $Z_\\Lambda$ can vanish at the rod only if one takes $b_\\Lambda = \\infty$, that is the neutral limit \\eqref{eq:neutrallimit} for the pair $(Z_\\Lambda,T_\\Lambda)$. In that case, $Z_\\Lambda \\,\\propto\\, r_i^{- a_\\Lambda P_i^{(\\Lambda)}}$ where $a_\\Lambda P_i^{(\\Lambda)}$ can be taken negative.}\n\\begin{equation}\nZ_\\Lambda \\,\\propto\\, r_i^{- a_\\Lambda P_i^{(\\Lambda)} }\\,, \\qquad W_\\Lambda \\,\\propto\\, r_i^{-G_i^{(\\Lambda)}}\\,,\\qquad e^{2\\nu} \\,\\propto \\, r_i^{\\alpha_{ii}}\\,,\n\\end{equation}\nand therefore, the metric \\eqref{eq:nonBPSfloatingRemind} is singular at the rod, except for specific choices of weights that will characterize the physical rods of the non-BPS floating brane ansatz in M-theory. It will be convenient to use the following aspect ratios, $d_i$,\n\\begin{equation}\n\\begin{split}\nd_1 &\\,\\equiv\\, 1\\,,\\qquad d_i \\,\\equiv\\, \\prod_{j=1}^{i-1} \\prod_{k=i}^n \\left(\\dfrac{(z_k^- - z_j^+)(z_k^+ - z_j^-)}{(z_k^+ - z_j^+)(z_k^- - z_j^-)} \\right)^{\\alpha_{jk}}\\quad \\text{when } i=2,\\ldots n\\,.\n\\end{split}\n\\label{eq:dialphaDefcharged}\n\\end{equation}\nThe regularity at each rod requires that the sources are in one of these categories (see appendix \\ref{App:AttheRod} for more details):\n\\begin{itemize}\n\\item[-] \\underline{\\emph{Black rods:}} the $y_a$-fibers and the $\\phi$-circle have a finite size at the $i^\\text{th}$ rod if\n\\begin{equation}\nG^{(\\Lambda)}_i \\= 0 \\,,\\qquad P^{(\\Lambda)}_i \\= \\frac{1}{2 \\, a_\\Lambda}\\,.\n\\label{eq:blackrodWeight}\n\\end{equation}\nOne can check that $\\alpha_{ii}=1$ and the timelike Killing vector $\\partial_t$ shrinks at the rod which therefore defines the locus of a horizon. The topology of the horizon is either T$^7\\times$S$^2$ or T$^6\\times$S$^3$ depending on the close environment around the rod (see Fig.\\ref{fig:RodCategories} and \\ref{fig:TouchingRods}).\\footnote{\\label{footnote1}The horizon topology depends on the close environment of the black rod. If the rod is connected on one side as in Fig.\\ref{fig:TouchingRods}, it corresponds to a T$^6\\times$S$^3$ horizon. Otherwise, it is a T$^7\\times$S$^2$ horizon as in Fig.\\ref{fig:RodCategories}.} \n\nMoreover, the surface gravity and horizon area associated to the black hole give\\footnote{The surface gravity can be read from the local geometry such as $ds^2 \\propto d\\rho_i^2 - \\rho_i^2 \\kappa_i^2 dt^2 + \\ldots$ with $\\rho_i\\to 0$, and is associated to the temperature of the black hole, ${\\cal T}_i = \\kappa_i\/(2\\pi )$. Moreover, note that if the rod is connected to another one, for instance $z_{i-1}^+=z_i^-$, the surface gravity is still finite since $(z_{i-1}^+-z_i^-)\/d_i=\\text{cst}$.}\n\\begin{equation}\n\\begin{aligned}\n\\kappa_{i} & \\equiv \\frac{2}{d_{i} M_{i}} \\prod_{\\Lambda=0}^4 \\sqrt{\\frac{a_{\\Lambda}}{e^{b_{\\Lambda}}}}\\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(i-j) \\alpha_{i j}},\\\\\nA_{i} &=\\frac{(2 \\pi)^8\\,M_{i} }{\\kappa_{i}}\\,\\prod_{a=1}^7 R_{y_a}.\n\\end{aligned}\n\\label{eq:Area&SurfaceGrav}\n\\end{equation}\nThe rod on its own corresponds to the static limit of a four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.\\footnote{Since all $P_i^{(\\Lambda)}$ are non-zero, the rod carries non-zero M2-M2-M2-KKm charges given by \\eqref{eq:individualMcharges}.} However, in the present ansatz, one can stack linearly multiple black holes on a line and their interactions can be studied.\n\\item[-] \\underline{\\emph{Bubble rods:}} One of the T$^6$ directions, for instance $y_1$ or $y_2$, shrinks at the $i^\\text{th}$ rod while all other fibers are finite by taking\n\\begin{equation}\n\\pm G_i^{(1)} =-G_i^{(0)} = a_0 P_i^{(0)} = a_1 P_i^{(1)} =\\frac{1}{2} , \\quad G_i^{(2)}=G_i^{(3)}=P_i^{(2)}=P_i^{(3)}= 0,\n\\label{eq:bubblerodWeight1}\n\\end{equation}\nwhere the ``$+$'' corresponds to a shrinking $y_1$-circle and the ``$-$'' corresponds to a shrinking $y_2$-circle. The weights for which another T$^6$ direction degenerates can be obtained by permuting the $I=1,2,3$ indexes. Similarly, the $y_7$ circle shrinks while all other fibers remain finite by considering\n\\begin{equation}\na_0\\,P_i^{(0)}\\= 1 , \\qquad G_i^{(\\Lambda)} \\= P_i^{(I)} \\= 0\\,.\n\\label{eq:bubblerodWeight2}\n\\end{equation}\nFor each possibility, the $(r_i, y_a)$-subspace corresponds to an origin of an $\\mathbb{R}^2$ defining the locus of a bubble with a T$^6\\times$S$^2$ or T$^5\\times$S$^3$ (see Fig.\\ref{fig:RodCategories} and \\ref{fig:TouchingRods}).\\footnote{\\label{footnote2}As for a black rod, the local topology of the bubble depends if the bubble rod is disconnected (Fig.\\ref{fig:RodCategories}) or connected (Fig.\\ref{fig:TouchingRods}).} It corresponds to a smooth coordinate degeneracy if an algebraic equation, which we can denote as a \\emph{bubble equation}, is satisfied. For a rod where the $y_1$ or $y_2$ circle degenerates smoothly, the bubble equation relates the asymptotic radius to the internal parameters as follows\\footnote{If the rod is connected to another one, e.g. $z_{i-1}^+=z_i^-$, the constraint is always finite since $(z_{i-1}^+-z_i^-)\/d_i=\\text{cst}$.}\n\\begin{equation}\nk_{i}\\,R_{y_{a}} \\= d_{i} M_{i}\\,\\sqrt{\\frac{e^{b_{1}+b_0}}{a_{1}a_{0}}}\\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(j-i) \\alpha_{i j}}, \\quad a=1 \\text{ or }2,\n\\label{eq:RegCondBu}\n\\end{equation}\nwhere $k_i$ is an orbifold parameter $k_i\\in \\mathbb{N}$ that corresponds to the order of the conical deficit on the $\\mathbb{R}^2$. The constraint corresponding to the degeneracy of another T$^6$ direction or the $y_7$-circle is obtained by permuting $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_I}}{a_I}$ or $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_0}}{a_0}$ respectively. Moreover, one can check that the gauge field is regular at the rod (see Appendix \\ref{App:AttheRod}).\n\nThe smooth bubble where one of the T$^6$ directions degenerates has nonzero $P_i^{(0)}$ and $P_i^{(I)}$. Therefore, it carries only a M2 brane charge and a KKm charge, given by \\eqref{eq:individualMcharges}. Similarly, the smooth bubble that shrinks the $y_7$ direction carries only a KKm charge. Moreover, these charges can be freely reduced to zero without changing the topology of the bubble by taking the neutral limit \\eqref{eq:neutrallimit}, and the resulting objects will be smooth vacuum bubbles in M-theory.\n\nTherefore, \\emph{seven types of physical rod sources} correspond to a smooth coordinate degeneracy of one the T$^6\\times$S$^1$ directions defined by an origin of an $\\mathbb{R}^2$ where a smooth bubble is sitting.\n\\end{itemize}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{RodSpecies.pdf}\n\\caption{Schematic description of three physical rods and the behavior of the circles on the $z$-axis when the rod are disconnected. The black rod sources a static M2-M2-M2-KKm black hole with a T$^7\\times$S$^2$ horizon. The species-1 bubble rod corresponds to the degeneracy of the $y_1$-circle inducing a T$^6\\times$S$^2$ M2-KKm bubble The species-7 bubble rod induces the degeneracy of the $y_7$-circle and a T$^6\\times$S$^2$ KKm bubble. Each section in between the sources has a conical excess, defining a strut.}\n\\label{fig:RodCategories}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{ConnectedRods.pdf}\n\\caption{Schematic description of connected rods by making the rods of Fig.\\ref{fig:RodCategories} touch. The $\\phi$-circle does not shrink along the rod configuration, and the solution is strut-free.}\n\\label{fig:TouchingRods}\n\\end{figure}\n\n\nThere are 8 categories of sources that can be used to generate physical asymptotically-$\\mathbb{R}^{1,3}$ solutions in M-theory: a non-extremal M2-M2-M2-KKm black hole, 6 non-BPS M2-KKm bubbles and a non-BPS KKm bubble. Remarkably, we have for any choices:\n\\begin{equation}\n\\begin{split}\n\\alpha_{jk}& \\= \\begin{cases} \n1 \\qquad &\\text{if the }j^\\text{th}\\text{ and }k^\\text{th}\\text{ rods are of the same nature,} \\\\\n\\frac{1}{2} \\qquad &\\text{otherwise,}\n\\end{cases} \n\\end{split}\n\\label{eq:dialphaDefcharged2}\n\\end{equation}\nwhich means that the three-dimensional base does not depend on the specific nature of the sources.\n\nFinally, one can think of other types of sources that are singular in M-theory but regular in other duality frameworks. This will be clarified when discussing the ansatz in other string theory frameworks in the section \\ref{sec:dualFrame}. \n\n\nThe effective mass induced by the $i^\\text{th}$ rod, ${\\cal M}_i$, can be derived from \\eqref{eq:ADMmass4d} by isolating the contribution of the rod. We find\n\\begin{equation}\n{\\cal M}_i \\= \\frac{1}{4 G_4} \\left(a_0 \\coth b_0\\, Q_{\\text{KKm},i}+ \\sum_{I=1}^3 a_I \\coth b_I\\, Q_{\\text{M2},i}^I\\right)\\,,\n\\end{equation}\nwhere the brane charges carried by the rod are given in \\eqref{eq:individualMcharges}. Therefore, the charge-to-mass ratios induced by the branes are determined by $a_\\Lambda \\coth b_\\Lambda$, and are the same for all the rods. It is a consequence for having a linear ansatz. If it still allows to remarkably deviate from the BPS regime, one cannot have two rods with opposite charges for instance.\n\n\\newpage\n\\item[\u2022] \\textbf{Regularity on the $z$-axis out of the rods:}\n\nAt $\\rho=0$ and $z \\notin [ z_i^-,z_i^+]$, the functions $R_+^{(i)}\/R_-^{(i)}$ and $E_{\\pm \\pm }^{(i,j)}$ are finite and non-zero. Therefore, the warp factors are well-defined and the $\\phi$-circle degenerates as the cylindrical coordinate singularity. The analysis reduces to the regularity of the three-dimensional base, $ds_3^2 = e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2$, and the warp factor $e^{2\\nu}$ can induce conical singularities. More precisely, we have\n\\begin{equation}\ne^{2 \\nu} \\sim \\begin{cases}\n~~ 1 &\\text{if } z < z_{1}^- \\text{ and } z > z_{n}^+,\\\\\n~~ d_i^2\\qquad &\\text{if } z \\in \\, ]z_{i-1}^+, z_i^- [\\,,\\quad i=2,..,n,\n\\end{cases}\n\\end{equation}\nwhere we remind that $d_i$ are the aspect ratios \\eqref{eq:dialphaDefcharged}. Therefore, the base metric above and below the rod configuration, $z < z_{1}^-$ and $z > z_{n}^+$, has no conical singularity and is smooth there. However, in between the $(i-1)^\\text{th}$ and $i^\\text{th}$ rods, we have $ds_3^2 \\sim d_i^2 \\left(d\\rho^2 + dz^2 +\\frac{\\rho^2}{d_i^2} d\\phi^2 \\right)$. Since $d_i <1$, the segment has a conical excess and describes a strut, i.e. a string with a negative tension (see Fig.\\ref{fig:RodCategories}). The strut is a singularity that accounts for the lack of repulsion between the $(i-1)^\\text{th}$ and $i^\\text{th}$ sources. It can be associated with a curvature singularity and an energy as done in \\cite{Costa:2000kf} and reviewed in \\cite{Bah:2021owp}.\n\nThe existence of a strut between two rods requires that they are disconnected. Therefore, by connecting the rods, which implies that they are of different nature, the solutions are free of struts and conical excesses \\cite{Elvang:2002br,Bah:2021owp,Bah:2021rki}. These solutions consist in connecting bubbles and possibly non-extremal black holes on a line (see Fig.\\ref{fig:TouchingRods}). For such configurations, the sources do not require a singular mechanism to be prevented from collapsing and are kept apart by topological bubbles \\cite{Bah:2021owp}. This provides a new paradigm in the microstate geometry program for constructing smooth bubbling geometries without the aid of supersymmetry. The standard lore of supersymmetry is that gravitational attraction is counterbalanced by electromagnetic repulsion. In the present ansatz, the solutions are generically non-BPS with arbitrarily low electromagnetic potentials, and the lack of repulsion is balanced by the inherent pressure of squeezed bubbles \\cite{Bah:2021owp}. \n\\end{itemize}\n\nIn conclusion, the non-BPS floating brane ansatz contains a large phase space of axisymmetric regular static solutions that are asymptotic to four-dimensional flat space plus compactification circles. Typical solutions consist of non-extremal four-charge black holes and 7 types of smooth bubbles on a line, which are kept apart by struts if disconnected. In addition, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section \\ref{sec:Sol4d}.\n\n\n\n\n\n\\subsection{Five-dimensional solutions}\n\\label{sec:5dNonBPSGen}\n\nTo construct five-dimensional non-BPS solutions, the four-dimensional base, $(\\rho,z,\\phi,y_7)$, must be considered as a four-dimensional infinite space. That is, we take $y_7$ to be an angle of periodicity $2\\pi$. As detailed in the appendix \\ref{App:Reg5d}, one must also consider the neutral limit \\eqref{eq:neutrallimit} for the pair $(Z_0,H_0)$ and source $Z_0$ by a semi-infinite rod.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.22\\textwidth]{Rods5d.pdf}\n\\caption{Schematic description of axisymmetric rod sources with a semi-infinite rod in $Z_0$.}\n\\label{fig:RodsSources5d}\n\\end{figure}\n\nGeneric five-dimensional solutions are therefore given by \\eqref{eq:nonBPSfloatingRemind}, sourced by $n$ finite rod of length $M_i$ and a $(n+1)^\\text{th}$ semi-infinite rod (see Fig.\\ref{fig:RodsSources5d}). We will consider that the $n^\\text{th}$ and $(n+1)^\\text{th}$ rods are connected to avoid a conical excess between them, $z_n^+=z_{n+1}^-$. The warp factors and gauge potentials are similar to the ones given for four-dimensional solutions \\eqref{eq:WarpfactorsRods} with the neutral limit \\eqref{eq:neutrallimit} for $(Z_0,H_0)$ and additional terms corresponding to the semi-infinite rod, that is\n\\begin{align}\n&Z_I= \\frac{1}{2a_I} \\left[e^{b_I} \\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{a_I P^{(I)}_i}-e^{-b_I} \\prod_{i=1}^n \\left( \\frac{R_-^{(i)}}{R_+^{(i)}}\\right)^{a_I P^{(I)}_i} \\right]\\,,\\quad T_I\\=-\\frac{1}{Z_I}\\,\\sqrt{1+a_I^2\\,Z_I^2} \\,,\\nonumber\\\\\n& Z_0 \\= \\left(r_+^{(n)}-(z-z_n^+) \\right)^{-G_{n+1}}\\,\\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{G_i} \\,,\\quad H_0 \\=0\\,,\\quad W_\\Lambda = \\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{G^{(\\Lambda)}_i} \\, , \\label{eq:WarpfactorsRods5d} \n\\end{align}\n\\begin{align}\n&e^{2\\nu} \\= \\frac{1}{2} \\left(\\frac{r_+^{(n)}-(z-z_n^+)}{r_+^{(n)}} \\right)^{{G_{n+1}}^2}\\,\\prod_{i=1}^n \\left(\\frac{E_{++}^{(n,i)}\\,R_-^{(i)}}{E_{+-}^{(n,i)}\\,R_+^{(i)}} \\right)^{G_{n+1}G_i }\\,\\prod_{i,j=1}^n\\, \\left( \\frac{E_{+-}^{(i,j)}E_{-+}^{(i,j)}}{E_{++}^{(i,j)}E_{--}^{(i,j)}}\\right)^{\\frac{1}{2}\\,\\alpha_{ij}} \\,, \\nonumber\n\\end{align}\nwhere the main functions are given in \\eqref{eq:RpmdefRemind} and the exponents, $\\alpha_{ij}$, are now given by\\footnote{The exponents are slightly different from \\eqref{eq:Defalpha} due to the neutral limit \\eqref{eq:neutrallimit} taken for the pair $(Z_0,H_0)$. More precisely, we have taken $(a_0,b_0) \\to \\infty$ with $G_i = a_0 P^{(0)}_i$ finite in \\eqref{eq:HarmFuncGen} and \\eqref{eq:HarmFuncGen2}.}\n\\begin{align}\n\\alpha_{ij} &\\,\\equiv\\, G_i G_j\\+ \\sum_{\\Lambda =0}^3 G_i^{(\\Lambda)}G_j^{(\\Lambda)} \\+ \\sum_{I=1}^3 a_I^2\\,P_i^{(I)}P_j^{(I)}\\,. \\label{eq:alpha5d}\n\\end{align}\n\nNote that the solutions have no KKm charges along $y_7$ because $H_0=0$. We have therefore defined a family of non-BPS M2-M2-M2 solutions given by $10 n + 8$ parameters. \n\n\\subsubsection{Regularity constraints}\n\\label{sec:regM5d}\n\nWe will be brief in the regularity analysis since it is similar to the discussion for four-dimensional solutions. We refer to the appendix \\ref{App:Reg5d} for more details.\n\nThe five-dimensional metric \\eqref{eq:5dframework} is asymptotically flat if \n\\begin{equation}\n \\prod_{I=1}^3 \\frac{\\sinh b_I}{a_I} \\= 1\\,.\n\\label{eq:condonasymp5d}\n\\end{equation}\nThe solution is regular everywhere out of the finite rods, especially at the semi-infinite rod where the $y_7$ degenerates smoothly as an origin of a $\\mathbb{R}^2$ if and only if\n\\begin{equation}\nG_{n+1} \\= 1\\,.\n\\label{eq:CondGn+1}\n\\end{equation}\nMoreover, the solution is regular at the finite $i^\\text{th}$ rod if its weights fall into one of these eight categories (see appendix \\ref{App:AttheRod2} for more details):\n\\begin{itemize}\n\\item[-] \\underline{\\emph{Black rods:}} \nWe consider that\n\\begin{equation}\nG^{(\\Lambda)}_i \\=0\\,,\\qquad G_i\\= a_I P^{(I)}_i\\= \\frac{1}{2},\n\\label{eq:blackrodWeight5d}\n\\end{equation}\nand the rod corresponds to a horizon of a black hole where the timelike Killing vector $\\partial_t$ shrinks. The surface gravity, temperature and area of the black hole are given by\n\\begin{align}\n\\kappa_i &\\,\\equiv\\, \\frac{2 \\sqrt{z_n^+-z_i^+}}{d_i\\,M_i}\\, \\sqrt{\\frac{a_1 a_2 a_3}{e^{b_1+b_2+b_3}}}\\,\\prod_{j=1}^i \\left( \\frac{z_n^+ - z_j^-}{z_n^+ - z_j^+}\\right)^{ G_j} \\,\\prod_{j\\neq i} \\left(\\frac{z_j^+ - z_i^-}{z_j^- -z_i^-} \\right)^{\\text{sign}(i-j)\\, \\alpha_{ij}}\\,, \\nonumber\\\\\n{\\cal T}_i &\\= \\frac{\\kappa_i}{2\\pi}\\,,\\qquad A_i \\= \\frac{(2\\pi)^8\\,M_i}{\\kappa_i}\\,\\prod_{a=1}^6 R_{y_a}\\,,\n\\label{eq:surfaceGrav5d}\n\\end{align}\nwhere $d_i$ is still given by \\eqref{eq:dialphaDefcharged}. Because the three $P_i^{(I)}$ are finite, the black hole carries generically three M2 charges given by \\eqref{eq:individualMcharges}. Each charge can be taken to be zero by considering the neutral limit \\eqref{eq:neutrallimit}, which corresponds to $a_I = \\sinh b_I\\to \\infty$. \n\nMore precisely, the rod corresponds to the horizon of a three-charge static non-extremal black hole for which the single-center solutions have been derived in \\cite{Cvetic:1996xz,Cvetic:1997uw,Cvetic:1998xh}. \n\n\\item[-] \\underline{\\emph{Bubble rods:}} The $y_1$ or $y_2$ circle shrinks at the $i^\\text{th}$ rod while all other fibers are finite if we consider\n\\begin{equation}\n \\pm G_i^{(1)} = -G_i^{(0)} =G_i = a_1 P_i^{(1)} =\\frac{1}{2} , \\quad G_i^{(2)}=G_i^{(3)}=P_i^{(2)}=P_i^{(3)}= 0,\n\\label{eq:bubblerodWeight15d}\n\\end{equation}\nwhere the ``$+$'' corresponds to a shrinking $y_1$-circle and the ``$-$'' corresponds to a shrinking $y_2$-circle. One can obtain the weights for another degenerating T$^6$ direction by permuting the $I=1,2,3$ indexes. Similarly, on can make the $y_7$ circle degenerate by considering\n\\begin{equation}\nG_i \\= 1 , \\qquad G_i^{(\\Lambda)} \\= P_i^{(I)} \\= 0\\,.\n\\label{eq:bubblerodWeight25d}\n\\end{equation}\nFor each possibility, the local geometry corresponds to a bubble sitting at a smooth origin of a $\\mathbb{R}^2$ if a bubble equation is satisfied. For a rod that makes the circle $y_1$ or $y_2$ degenerate smoothly, the bubble equation constrains the internal parameters as a function of the extra-dimension radius such as\n\\begin{equation}\nk_i \\,R_{y_a} \\= \\frac{d_{i} M_{i}}{\\sqrt{z_n^+-z_i^+}}\\,\\sqrt{\\frac{e^{b_{1}}}{a_{1}}}\\, \\prod_{j=1}^i \\left( \\frac{z_n^+ - z_j^+}{z_n^+ - z_j^-}\\right)^{ G_j} \\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(j-i) \\alpha_{i j}}, \\quad a=1 \\text{ or }2,\n\\label{eq:RegCondBu5d}\n\\end{equation}\nwhere $k_i$ is an orbifold parameter, $k_i\\in \\mathbb{N}$, that corresponds to the order of the conical deficit on the $\\mathbb{R}^2$. The bubble equation corresponding to another degenerating T$^6$ direction or the $y_7$-circle is obtained by permuting $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_I}}{a_I}$ or $\\frac{e^{b_1}}{a_1} \\to 1$ respectively. \n\nThe smooth bubble where one of the T$^6$ direction degenerates carries only a M2 brane charge, given by \\eqref{eq:individualMcharges}, because only one $P^{(I)}_i$ is nonzero \\eqref{eq:bubblerodWeight15d}. Similarly, the smooth bubble where the $y_7$ direction shrinks is purely topological and carries no charge. We have therefore six types of M2 bubbles and one type of vacuum bubble that can smoothly source the solutions.\n\\end{itemize}\n\nThe mass induced by each rod, ${\\cal M}_i$, can be derived from \\eqref{eq:ADMmass5d} and related to the brane charges \\eqref{eq:individualMcharges} such as\n\\begin{equation}\n{\\cal M}_i \\= \\frac{\\pi}{2 G_5} \\,\\sum_{I=1}^3 a_I \\coth b_I \\,Q_{\\text{M2},i}^I\\,.\n\\end{equation}\nThe charge-to-mass ratios for each stack of M2 branes are fixed by $a_I \\coth b_I$ for all rods.\n\nAs for four-dimensional solutions, a strut is separating each pair of disconnected rods. More precisely, if the $(i-1)^\\text{th}$ and $i^\\text{th}$ rods are not connected, the $\\phi$-circle degenerates in between them with a conical excess of order $d_i^{-1}>1$ \\eqref{eq:dialphaDefcharged}. This singularity can be bypassed by considering connected rods. For such configurations, the struts disappear and the non-BPS sources are held apart by pure topology. \n\n\nTo conclude, the family of solutions, given by \\eqref{eq:WarpfactorsRods5d}, contains a large phase space of regular non-BPS axisymmetric static solutions that are asymptotic to five-dimensional flat space plus compactification circles. Typical solutions consist of non-BPS non-extremal three-charge black holes and 7 types of smooth bubbles on a line, which are held apart by struts if disconnected. Moreover, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section \\ref{sec:Sol5d}.\n\n\\subsection{Other duality frames}\n\\label{sec:dualFrame}\n\nSo far we have limited the discussion to the M-theory frame, but the construction can be dualized to different string theory frames through dimensional reduction and T-dualities. Each frame will have new types of regular rod sources that are singular in M-theory. To illustrate this property, we will focus on dualization in the D1-D5-P-KKm frame, but we refer the interested reader to the appendix \\ref{App:dualFrame} where we have dualized the solutions in six different frames, from the D2-D2-D2-D6 to the P-M5-M2-KKm frame. In addition, the study of the D1-D5-P-KKm frame will provide a link to the author's previous constructions of smooth bubbling geometries \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki}.\n\n\n\nAfter reduction along $y_5$ and a series of T-dualities along $y_1$, $y_2$ and $y_6$, the M2-M2-M2-KKm solutions \\eqref{eq:nonBPSfloatingRemind} transform to D1-D5-P-KKm solutions where the common direction for the momentum P charge, the D1 and D5 branes is $y_6$. The metric, the R-R gauge fields $C^{(p)}$, the NS-NS two-form gauge field $B_2$ and the dilaton in type IIB are given by \n\\begin{align}\nds_{10}^2 \\= &- \\frac{dt^2}{Z_3 \\sqrt{W_3W_0 Z_1 Z_2}} \\+ \\sqrt{\\frac{ Z_1 Z_2}{W_3 W_0}} \\left[\\frac{1}{Z_0} \\left(dy_7 +H_0 \\,d\\phi\\right)^2 \\+ Z_0\\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\right] \\nonumber\\\\\n& +\\sqrt{\\frac{Z_1}{Z_2}}\\left[\\sqrt{\\frac{W_3}{W_0}} \\,\\left(W_1\\,dy_1^2+ \\frac{dy_2^2}{W_1}\\right) +\\sqrt{\\frac{W_0}{W_3}} \\left(\\frac{dy_3^2}{W_2} + W_2 \\,dy_4^2 \\right) \\right] \\label{eq:D1D5PKKm} \\\\\n& + \\frac{Z_3}{\\sqrt{W_3 W_0 Z_1 Z_2}}\\,\\left(dy_6+T_3\\, dt \\right)^2\\,, \\qquad \\Phi \\= \\frac{1}{2} \\log \\frac{Z_1}{W_0 W_3 Z_2}\\,,\\qquad B_2 \\= 0\\,, \\nonumber \\\\\nC^{(0)} \\=& 0 \\,,\\qquad C^{(2)}\\= H_2 \\,d\\phi \\wedge dy_7 \\- T_1 \\,dt \\wedge dy_6\\,,\\qquad C^{(4)} \\= 0\\,,\\nonumber \n\\end{align}\nwhere $H_2$ is the electromagnetic dual of $T_2$, and is given by the same expression as $H_0$ in \\eqref{eq:WarpfactorsRods} but with $P_i^{(0)} \\to P_i^{(2)}$ (see appendix \\ref{App:dualFrame}).\n\nOne can recognize the type IIB frame of topological stars and five-dimensional charged Weyl solutions of \\cite{Bah:2020pdz} by taking\n\\begin{equation}\nW_0 \\= W_1\\= W_2\\= W_3 \\= 1 \\,,\\qquad Z_0 \\=Z_1 \\=Z_2 \\= Z \\,,\\qquad T_3 \\= 0 \\,,\\qquad Z_3 \\= W\\,,\n\\end{equation}\nTaking $T_3=0$ while keeping a non-trivial $Z_3=W$ is possible by considering the neutral limit for the pair $(Z_3,T_3)$ \\eqref{eq:neutrallimit}, that is no P-charges. If we now consider $Z_0$ to be different from $Z_1=Z_2$, we retrieve the framework of \\cite{Bah:2021owp,Bah:2021rki} where six-dimensional smooth bubbling solutions, as bubble bag ends \\cite{Bah:2021rki}, have been constructed.\n\n\n\nThe warp factors and gauge potentials are governed by the linear system of equations obtained from the non-BPS floating brane ansatz of section \\ref{sec:WeylAns}.\n\n\nFor the non-BPS solutions, the physical rod sources are different from the eighth found in sections \\ref{sec:regM4d} and \\ref{sec:regM5d}. For instance, the $y_6$-circle degenerates at the $i^\\text{th}$ rod for a solutions given by \\eqref{eq:WarpfactorsRods} defining the locus of a bubble if we consider its associated weights to be \n\\begin{equation}\nG^{(\\Lambda)}_i \\= 0 \\,,\\qquad a_0 P^{(0)}_i \\= a_1 P^{(1)}_i \\= a_2 P^{(2)}_i \\= -a_3 P^{(3)}_i \\=\\frac{1}{2}\\,.\n\\label{eq:bubblerodWeight4}\n\\end{equation}\nAs a black rod \\eqref{eq:blackrodWeight}, they do not require to turn on the weights of the torus warp factors $W_\\Lambda$, and therefore the T$^4$ is rigid as for BPS solutions. The only difference is that $a_3 P_i^{(3)}=-\\frac{1}{2}$. However, taking $a_3 P_i^{(3)}$ negative is compatible with the regularity condition \\eqref{eq:RegPositivity} only if one takes the neutral limit, $(a_3,b_3)\\to \\infty$ \\eqref{eq:neutrallimit}. It requires to take the gauge potential for the P charge to be zero and the solutions have no momentum charges as in \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki}. A rod given by \\eqref{eq:bubblerodWeight4} corresponds to a smooth D1-D5-KKm bubble in type IIB.\\footnote{Smoothness requires a bubble equation of a similar type to those obtained previously.} However, it is singular in the M-theory frame \\eqref{eq:nonBPSfloatingRemind}. Therefore, there are specific species of bubble rods that correspond to smooth loci in a unique string theory frame.\n\n\\section{Explicit four-dimensional solutions}\n\\label{sec:Sol4d}\n\nIn this section, we derive explicit solutions that are asymptotic to $\\mathbb{R}^{1,3}$ with extra compact dimensions using the non-BPS floating brane ansatz in different string theory frames. There is a very large number of configurations that one can think about. First, we will construct chain of non-extremal four-charge black holes, either separated by struts or smooth bubbles, and discuss their physics. Second, we will construct smooth bubbling geometries in the manner of \\cite{Bah:2021owp,Bah:2021rki}, that have the same mass and charges as non-extremal four-charge black holes.\n\n\n\n\\subsection{Chain of static non-extremal black holes}\n\nWe first study solutions for which the main ingredients are non-extremal four-charge black holes on a line.\n\n\\subsubsection{Black holes separated by struts}\n\nWe consider $n$ finite rods of length $M_i$ on the $z$-axis with weights corresponding to black rods given by \\eqref{eq:blackrodWeight} (see Fig.\\ref{fig:ChainofBHStruts}). It is convenient to divide the warp factors, $Z_\\Lambda$ \\eqref{eq:WarpfactorsRods}, into two parts: one will encode changes in topology and the other will correspond to a well-behaved flux decoration such as\n\\begin{equation}\nZ_\\Lambda \\= \\frac{{\\cal Z}_\\Lambda}{\\sqrt{U_t}}\\,,\\qquad U_t \\,\\equiv\\, \\prod_{i=1}^n\n\\left(1-\\frac{2 M_i}{R_+^{(i)}} \\right)\\,,\\qquad {\\cal Z}_\\Lambda \\,\\equiv\\, \\frac{e^{b_\\Lambda}-e^{-b_\\Lambda}\\,U_t}{2 a_\\Lambda}\n\\end{equation}\nThe remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_\\Lambda = 1,\\quad e^{2\\nu} \\=\\prod_{i,j=1}^n\\, \\sqrt{ \\frac{E_{+-}^{(i,j)}E_{-+}^{(i,j)}}{E_{++}^{(i,j)}E_{--}^{(i,j)}}},\\quad H_0 \\= \\frac{1}{2 a_0}\\sum_{i=1}^n \\left(r_-^{(i)}-r_+^{(i)} \\right),\\quad T_I\\=-\\sqrt{a_I^2+\\frac{U_t}{{\\cal Z}_I^2}}, \\nonumber \n\\end{align}\nwhere the generating functions $R_\\pm^{(i)}$ and $E_{\\pm \\pm}^{(i,j)}$ are given in \\eqref{eq:RpmdefRemind}. The metric and gauge potential in M-theory are given in \\eqref{eq:nonBPSfloatingRemind}, while the four-dimensional reduction along T$^6\\times$S$^1$ is given by \\eqref{eq:4dSol}:\n\\begin{align}\nds_4^2 &\\= -\\frac{U_t\\,dt^2}{\\sqrt{{\\cal Z}_0 {\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3}} \\+ \\frac{\\sqrt{{\\cal Z}_0 {\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3}}{U_t} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right), \\nonumber \\\\\nz^I &\\= i\\,\\frac{|\\epsilon_{IJK}|}{2} \\sqrt{\\frac{{\\cal Z}_J {\\cal Z}_K}{{\\cal Z}_0\\,{\\cal Z}_I} },\\quad \\Phi_\\Lambda \\=0 \\,,\\quad \\bar{F}^{0} \\= -dH_0 \\wedge d\\phi \\,,\\quad \\bar{F}^{I} \\= dT_I \\wedge dt\\,.\n\\label{eq:4dprofilechainofBH}\n\\end{align}\nThe function $U_t$ behaves as a product of ``Schwarzschild factors'' that vanishes at each rod and makes the timelike Killing vector shrink. It induces the topology of the solutions and forms the horizons. The ${\\cal Z}_\\Lambda$ depend only on the gauge field parameters, and are always positive and finite. In the neutral limit \\eqref{eq:neutrallimit}, all ${\\cal Z}_\\lambda$ converge to 1 while $U_t$ remains unchanged. We retrieved the solutions of Schwarzschild black holes on a line \\cite{Israel1964,Gibbons:1979nf}.\n\n\nAt large distance $r$, with $(\\rho,z)=(r\\sin\\theta,r\\cos\\theta)$, the geometry is asymptotically flat if \\eqref{eq:condonasymp} is satisfied, and one can read the ADM mass from \\eqref{eq:ADMmass4d}:\n\\begin{equation}\n{\\cal M} \\= \\frac{\\sum_{\\Lambda=0}^3 \\coth b_\\Lambda}{8 G_4}\\,\\sum_{i=1}^n M_i\\,.\n\\label{eq:MassBHchain}\n\\end{equation}\nMoreover, the solutions carry four charges that are M2-M2-M2-KKm charges in M-theory. They are given by the sum of the rod charges carried by each rod \\eqref{eq:individualMcharges}: \n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=1}^n M_i \\,,\\qquad Q_{\\text{KKm}} \\= \\frac{1}{2a_0} \\,\\sum_{i=1}^n M_i.\n\\end{equation}\n\\begin{itemize}\n\\item[\u2022] \\underline{A unique rod: map to the four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.}\n\\end{itemize} \nWe consider a unique rod, $n=1$, and the spherical coordinates $$(\\rho,\\,z)= \\left(\\sqrt{r \\left(r -M_{1}\\right)} \\sin \\theta,\\, \\left(r-\\frac{M_{1}}{2}\\right) \\cos \\theta +z_{1}\\right). $$ The four-dimensional metric, gauge fields and scalars \\eqref{eq:4dprofilechainofBH} simplify to\n\\begin{align}\nds_4^2 &\\= -\\frac{r(r-M_1)\\,dt^2}{\\sqrt{r_0 r_1 r_2 r_3}} \\+ \\sqrt{r_0 r_1 r_2 r_3} \\left(\\frac{dr^2}{r(r-M_1)}+d\\theta^2 +\\sin^2\\theta d\\phi^2\\right),\\quad \\Phi_\\Lambda \\=0, \\label{eq:4chargeBH} \\\\\nz^I &\\= i\\,\\frac{a_0 a_I\\,|\\epsilon_{IJK}|}{2\\sinh b_0 \\sinh b_I}\\sqrt{\\frac{r_J r_K}{r_0\\,r_I} } \\,,\\quad \\bar{F}^{0} \\= \\frac{M_1}{2 \\sinh b_0}\\sin \\theta \\,d\\theta \\wedge d\\phi \\,,\\quad \\bar{F}^{I} \\= \\frac{M_1}{2 \\sinh b_I\\,r_I^2}dt\\wedge dr\\,, \\nonumber\n\\end{align}\nwhere we have defined $r_\\Lambda \\,\\equiv\\, r+ \\frac{e^{-b_\\Lambda}}{2 \\sinh b_\\Lambda}\\,M_1$.\\footnote{Note that the metric no longer depend on $a_\\Lambda$. In four dimensions, the dependence in $a_\\Lambda$ for regular solutions is a function of $\\prod_\\Lambda a_\\Lambda$ which is fixed in terms of $b_\\Lambda$ to have asymptotically flat solutions \\eqref{eq:condonasymp}.} We retrieve the static solutions of a four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca} by relating the boost parameters in these papers, $\\delta_\\Lambda$, to our parameters, $b_\\Lambda$, such as $\\cosh 2\\delta_\\Lambda = \\coth b_\\Lambda$.\n\\begin{itemize}\n\\item[\u2022] \\underline{Multiple four-charge black holes on a line.}\n\\end{itemize}\nBy considering $n>1$, we perform the multicentric generalization of the static solutions in \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.\nIn the IR, the geometry is made of a chain of four-charge static non-extremal black holes on the $z$-axis held apart by struts (see Fig.\\ref{fig:ChainofBHStruts}). At each rod, the timelike Killing vector $\\partial_t$ vanishes defining the locus of a horizon with area \\eqref{eq:Area&SurfaceGrav}. If one wants the black holes to be in thermal equilibrium, their surface gravity must be equal \\eqref{eq:Area&SurfaceGrav}, which non-trivially constrains their positions and sizes. Moreover, in between each rod the $\\phi$-circle shrinks with a conical excess of order $d_i^{-1} >1$ \\eqref{eq:dialphaDefcharged} defining the loci of the struts.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.38\\textwidth]{ChainofBHStruts.pdf}\n\\caption{Description of a chain of static four-charge non-extremal black holes in the four-dimensional framework \\eqref{eq:4dprofilechainofBH} and the behavior of the $\\phi$-circle on the $z$-axis.}\n\\label{fig:ChainofBHStruts}\n\\end{figure}\n\nThe struts carry an energy that counterbalances the attraction between the black holes. This energy corresponds to the interaction between the black holes in string theory. It can be derived using the method of \\cite{Costa:2000kf,Bah:2021owp}. More precisely, the energy of the strut that separates the $(i-1)^\\text{th}$ and $i^\\text{th}$ black holes on the chain carries an energy given by\n\\begin{equation}\nE_i \\= - \\frac{1-d_i}{4 G_4}\\,(z_i^- - z_{i-1}^+)\\,, \\qquad G_4 = \\frac{G_{11}}{(2\\pi)^{7} \\prod_{i=1}^{7} R_{y_i}}.\n\\end{equation}\nFor a configuration of two identical black holes separated by a distance $\\ell$, the energy of the strut in between them is given by\n\\begin{equation}\nE \\= - \\frac{M^2 \\ell}{4 G_4 (M+\\ell)^2},\n\\end{equation}\nwhere $M$ is the length of each rod. When the separation is large, $\\ell \\gg M$, the energy of the strut approximates the Newtonian potential between two particles of mass, $\\frac{M}{2G_4}$, in four dimensions. From this perspective, the strut measures the binding energy between the two black holes, or rather the potential energy needed to keep the two black holes from collapsing on each other. An important observation is that the effective ADM masses of the black holes \\eqref{eq:MassBHchain}, from the Newtonian point of view, depend on $b_\\Lambda$, which are associated to the four charges of the black holes \\eqref{eq:4chargeBH} with the regularity condition \\eqref{eq:condonasymp}. This implies that the binding energy as measured by the strut also accounts for effects due to the electromagnetic fields of the non-extremal black holes.\n\nFor finite separation between the two black holes, the gravitational potential between them deviates significantly from that of the Newtonian limit. In particular, the gravitational potential between two black holes vanish when $\\ell \\to 0$. In this limit, the two rod sources merge and the two-body configuration becomes a single non-extremal black hole. \n\n\n\n\n\\subsubsection{Black holes separated by smooth bubbles}\n\\label{sec:chainBHbu}\n\nStruts are singularities that do not have a consistent UV description in string theory as they correspond to cosmic strings with negative tension. Therefore, it is crucial to find a mechanism to resolve them and to obtain more relevant configurations in string theory. The non-BPS regime will require a new mechanism for this task. In the analysis of \\cite{Costa:2000kf} and continued in \\cite{Bah:2021owp}, it was shown that struts can be classically resolved into smooth bubbles. More precisely, one can consider the same black hole systems as before but sourcing each segment between them by a bubble rod. The resulting configuration will consist in a chain of connected rods which would be a succession of black holes and bubbles without struts. The condition for having connected rods fixes their centers in terms of their size:\n\\begin{equation}\nz_i^+ \\= z_{i+1}^- \\,,\\quad \\forall i \\qquad \\Rightarrow \\qquad z_i \\= z_n^+ + \\frac{M_i}{2} - \\sum_{j=i}^n M_j\\,.\n\\label{eq:connectedcond}\n\\end{equation}\nOne can also freely choose the origin of the $z$-axis so that $z_n^+=0$. \n\nFor simplicity, we consider the bubble rods to be smooth KKm bubbles given by the weights \\eqref{eq:bubblerodWeight2}, but one could a priori take any of the seven species of bubble rods detailed in section \\ref{sec:regM4d}. This allows a faithful description of the solutions in the five-dimensional framework \\eqref{eq:5dframework}. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{ChainofBHBub.pdf}\n\\caption{Description of a chain of static four-charge non-extremal black holes and KKm bubbles in the five-dimensional framework \\eqref{eq:5dprofilechainofBH} and the behavior of the $\\phi$ and $y_7$ circles on the $z$-axis.}\n\\label{fig:ChainofBHBub}\n\\end{figure}\nTherefore, we consider $n=2N+1$ rods of lengths $M_i$ where the $N+1$ odd rods correspond to black rods \\eqref{eq:blackrodWeight} and the $N$ even rods correspond to KKm bubble rods \\eqref{eq:bubblerodWeight2} (see Fig.\\ref{fig:ChainofBHBub}). As in the previous section, we divide the warp factors $Z_\\Lambda$ \\eqref{eq:WarpfactorsRods} in meaningful pieces such as\n\\begin{equation}\n\\begin{split}\nZ_0 &\\= \\frac{{\\cal Z}_0}{U_{y_7}\\sqrt{U_t}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_t}}\\,,\\\\\nU_t &\\,\\equiv\\, \\prod_{i=1}^{N+1} \\left(1-\\frac{2 M_{2i-1}}{R_+^{(2i-1)}} \\right)\\,, \\qquad U_{y_7} \\,\\equiv\\, \\prod_{i=1}^N \\left(1-\\frac{2 M_{2i}}{R_+^{(2i)}} \\right)\\,, \\\\\n{\\cal Z}_0 &\\,\\equiv\\, \\frac{e^{b_0}-e^{-b_0}\\,U_t U_{y_7}^2}{2 a_0} \\,,\\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_t}{2 a_I}.\n\\end{split}\n\\end{equation}\n\nThe remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_\\Lambda = 1,\\qquad H_0 \\= \\frac{1}{2 a_0} \\left[r_-^{(1)}- r_+^{(n)}+\\sum_{i=1}^N \\left(r_-^{(2i)}-r_+^{(2i)} \\right) \\right],\\qquad T_I\\=-\\sqrt{a_I^2+\\frac{U_t}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{E_{-+}^{(1, n)}}{\\sqrt{E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i=1}^{N} \\prod_{j=1}^{N+1} \\sqrt{\\frac{E_{--}^{(2 i, 2 j-1)} E_{++}^{(2 i, 2 j-1)}}{E_{-+}^{(2 i, 2 j-1)} E_{+-}^{(2 i, 2 j-1)}}}\\,,\n\\end{align}\nwhere $R_\\pm^{(i)}$ and $E_{\\pm \\pm}^{(i,j)}$ are given in \\eqref{eq:RpmdefRemind}. The metric and gauge potential in M-theory are given in \\eqref{eq:nonBPSfloatingRemind}, while the five-dimensional reduction along T$^6$ is given by \\eqref{eq:5dframework}\n\\begin{align}\nds_5^2 &\\= - \\frac{U_t\\,dt^2}{\\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{2}{3}}} \\+ \\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{1}{3}} \\left[\\frac{U_{y_7}}{{\\cal Z}_0} \\left(dy_7 +H_0 \\,d\\phi\\right)^2 \\+ \\frac{{\\cal Z}_0}{U_t U_{y_7}} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\right], \\nonumber \\\\\nX^I &\\= \\frac{|\\epsilon_{IJK}|}{2} \\left(\\frac{{\\cal Z}_J {\\cal Z}_K}{{\\cal Z}_I^2} \\right)^\\frac{1}{3},\\qquad \\Phi_\\Lambda \\= 0 \\,,\\qquad F^{(I)} \\= dT_I \\wedge dt\\,.\n\\label{eq:5dprofilechainofBH}\n\\end{align}\nThe warp factors $U_t$ and $U_{y_7}$ force the degeneracy of the $t$ and $y_7$ fibers at the rods. In the regions where $U_{y_7}$ vanishes, that is at the even rods, the $y_7$ direction shrinks forming a smooth origin of an $\\mathbb{R}^2$ (with potential conical defects) if we have \\eqref{eq:RegCondBu}\n\\begin{equation}\nk_{i}\\,R_{y_7} \\,\\equiv\\, \\,\\frac{d_{2i} M_{2i}\\,e^{b_{0}}}{a_{0}}\\, \\prod_{j \\neq i}^N\\left(\\frac{z_{2j}^{+}-z_{2i}^{-}}{z_{2j}^{-}-z_{2i}^{-}}\\right)^{\\operatorname{sign}(j-i)} \\,\\prod_{j =1}^{N+1}\\left(\\frac{z_{2j-1}^{+}-z_{2i}^{-}}{z_{2j-1}^{-}-z_{2i}^{-}}\\right)^{\\frac{\\operatorname{sign}(j-i-1\/2)}{2}}, \\quad \\forall i=1,...,N,\n\\end{equation}\nwhere we remind that $z_j^\\pm$ are the rod endpoint coordinates on the $z$-axis \\eqref{eq:RpmdefRemind} and $d_j$ are the aspect ratios \\eqref{eq:dialphaDefcharged}. These $N$ ``bubble equations'' fix the lengths of the bubble rods, $M_{2i}$, according to the other parameters. Moreover, if one wants the black holes to be in thermal equilibrium, one needs in addition to impose that their surface gravities \\eqref{eq:Area&SurfaceGrav} are equal, which also constrains the length of the black rods. \n\nThe $\\phi$-circle does not shrink anymore along the rod configuration and the four-charge non-extremal black holes are held apart by KKm bubbles. The struts have been then successfully replaced by smooth bubbles. This mechanism has been explored in \\cite{Costa:2000kf,Bah:2021owp}. In one word, vacuum bubbles are reluctant to be squeezed as it wants to expand \\cite{Witten:1981gj}, and provide the necessary pressure between two non-BPS objects \\cite{Costa:2000kf}. If the electromagnetic fluxes stabilize the bubbles \\cite{Stotyn:2011tv}, their reluctance to be squeezed remains \\cite{Bah:2021owp}.\n\nFinally, the solutions are asymptotic to $\\mathbb{R}^{1,3}\\times$S$^1$ if \\eqref{eq:condonasymp} is satisfied. The ADM mass is given by \\eqref{eq:ADMmass4d}\n\\begin{equation}\n{\\cal M} \\= \\frac{\\sum_{\\Lambda=0}^3 \\coth b_\\Lambda\\,\\sum_{i=1}^{N+1} M_{2i-1}+2 \\coth b_0\\,\\sum_{i=1}^{N} M_{2i}}{8 G_4}\\,.\n\\end{equation}\nMoreover, the M2-M2-M2-KKm charges of the solutions in M-theory can be obtained from the sum of the charges carried by each rod \\eqref{eq:individualMcharges}: \n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=1}^{N+1} M_{2i-1}\\,,\\qquad Q_{\\text{KKm}} \\= \\frac{\\sum_{i=1}^{2N+1} M_{i}+\\sum_{i=1}^{N} M_{2i}}{2a_0}.\n\\end{equation}\n\n\\subsection{Smooth bubbling geometries}\n\nWe now seek to construct smooth non-BPS bubbling solutions without black hole sources and struts. This consists in building a chain of connected bubble rods of different species. In M-theory alone, there are 7 species of smooth bubbles that can be used, and there are several others in different duality frameworks as described in section \\ref{sec:dualFrame}. This induces a wide variety of configurations possible. In this section, we will construct an explicit M2-M2-M2-KKm configuration and a D1-D5-KKm configuration in type IIB. \n\n\\subsubsection{In M-theory}\n\\label{sec:BuSolMtheory}\n\nAs described in section \\ref{sec:regM4d}, among the seven species of physical bubble rods in M-theory, six correspond to the smooth degeneracy of one of the T$^6$ direction and carry a unique M2 charge and a KKm charge. To construct a smooth configuration with M2-M2-M2-KKm charges, one needs at least three species of these kinds. \n\n\nTherefore, we will consider a chain of $n=3N$ connected bubble rods of length $M_i$ which will successively make the $y_1$, $y_3$ and $y_5$ circles shrink (see Fig.\\ref{fig:MtheoryBubbles}). Having connected rods constrains the rod centers in terms of their sizes as in \\eqref{eq:connectedcond}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{ChainofBuMtheory.pdf}\n\\caption{Description of a chain of smooth M2-KKm bubbles in M-theory and the behavior of the $\\phi$, $y_1$, $y_3$ and $y_5$ circles on the $z$-axis (the other fibers have finite size everywhere). The species-$k$ bubble rod corresponds to the degeneracy of the $y_k$-circle as an origin of a $\\mathbb{R}^2$.}\n\\label{fig:MtheoryBubbles}\n\\end{figure}\n\nThe expression of the warp factors and gauge potentials are given by \\eqref{eq:WarpfactorsRods} with the following choices of weights, obtained from \\eqref{eq:bubblerodWeight1},\n\\begin{equation}\na_0 P_i^{(0)}=-G_i^{(0)}= \\frac{1}{2}\\,,\\quad \\begin{array}{l} G_{3i-2}^{(1)} = a_1 P_{3i-2}^{(1)} =\\frac{1}{2} , \\quad G_{3i-2}^{(2)}=G_{3i-2}^{(3)}=P_{3i-2}^{(2)}=P_{3i-2}^{(3)}= 0, \\\\\nG_{3i-1}^{(2)} = a_2 P_{3i-1}^{(2)} =\\frac{1}{2} , \\quad G_{3i-1}^{(1)}=G_{3i-1}^{(3)}=P_{3i-1}^{(1)}=P_{3i-1}^{(3)}= 0, \\\\\nG_{3i}^{(3)} = a_3 P_{3i}^{(3)} =\\frac{1}{2} , \\quad G_{3i}^{(1)}=G_{3i}^{(2)}=P_{3i}^{(1)}=P_{3i}^{(2)}= 0.\n\\end{array}\n\\end{equation}\nAs in the previous constructions, it is convenient to split the warp factors $Z_\\Lambda$ such as\n\\begin{align}\nZ_0 &\\= \\frac{{\\cal Z}_0}{\\sqrt{U_1 U_2 U_3}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_I}}\\,,\\label{eq:WarpfactorMtheoryBu1} \\\\\nU_I &\\,\\equiv\\, \\prod_{i=0}^{N-1} \\left(1-\\frac{2 M_{3i+I}}{R_+^{(3i+I)}} \\right)\\,, \\qquad {\\cal Z}_0 \\,\\equiv\\, \\frac{e^{b_0}-e^{-b_0}\\,U_1 U_2 U_3}{2 a_0} \\,,\\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_I}{2 a_I}. \\nonumber\n\\end{align}\nThe functions $U_I$ are products of ``Schwarzschild factors'' that vanish at all the $(3i+I)^\\text{th}$ rods while the ${\\cal Z}_\\Lambda$ are non-zero and finite. The remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_0 = \\left( U_1 U_2 U_3\\right)^{\\frac{1}{2}},\\quad W_I \\= U_I^{-\\frac{1}{2}} ,\\quad H_0 \\= \\frac{1}{2 a_0} \\left( r_-^{(1)}- r_+^{(n)} \\right),\\quad T_I\\=-\\sqrt{a_I^2+\\frac{U_I}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{E_{-+}^{(1, n)}}{\\sqrt{E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i,j=0}^{N-1} \\,\\prod_{\\substack{I,J=1\\\\I1$ consists therefore in a chain of static three-charge black holes in five dimensions. The black holes are separated by struts, that are segments where the $\\phi$-circle degenerates with a conical excess of order $d_i^{-1} >1$ \\eqref{eq:dialphaDefcharged}. The struts account for the lack of repulsion between the non-extremal black holes and encodes the binding energy of the system.\n\n\nThe singular struts can be replaced by regular bubbles in a manner similar to that used for the chain of four-dimensional black holes in the section \\ref{sec:chainBHbu}. Specifically, one can source the M-theory solutions with bubble rods at each segment between the black holes. The bubble rods can either be vacuum bubbles that make the $y_7$ circle degenerate smoothly \\eqref{eq:bubblerodWeight25d}, or M2 bubbles that correspond to the degeneracy of one of the T$^6$ directions \\eqref{eq:bubblerodWeight15d}.\n\n\n\\subsection{Smooth bubbling geometries}\n\nWe now construct smooth non-BPS bubbling solutions that consist in a chain of connected bubble rods of different species. We will restrict to the five-dimensional analogs of the M-theory solutions constructed in section \\ref{sec:BuSolMtheory}. One could have also built asymptotically-$\\mathbb{R}^{1,4}$ D1-D5 non-BPS bubbling solutions by taking the equivalent of the four-dimensional solutions of section \\ref{sec:BuSolIIB}.\n\nWe consider a chain of $n=3N$ connected bubble rods of length $M_i$ where the circles $y_1$, $y_3$ and $y_5$ successively degenerate. Having connected rods constrains the rod centers in terms of their sizes as in \\eqref{eq:connectedcond}, and we also assume that the origin of the $z$-axis is at $z_n^+=0$. The warp factors and gauge potential are given by \\eqref{eq:WarpfactorsRods5d} with the following choices of weights, obtained from \\eqref{eq:bubblerodWeight15d},\n\\begin{equation}\nG_i =-G_i^{(0)}= \\frac{1}{2}\\,,\\quad \\begin{array}{l} G_{3i-2}^{(1)} = a_1 P_{3i-2}^{(1)} =\\frac{1}{2} , \\quad G_{3i-2}^{(2)}=G_{3i-2}^{(3)}=P_{3i-2}^{(2)}=P_{3i-2}^{(3)}= 0, \\\\\nG_{3i-1}^{(2)} = a_2 P_{3i-1}^{(2)} =\\frac{1}{2} , \\quad G_{3i-1}^{(1)}=G_{3i-1}^{(3)}=P_{3i-1}^{(1)}=P_{3i-1}^{(3)}= 0, \\\\\nG_{3i}^{(3)} = a_3 P_{3i}^{(3)} =\\frac{1}{2} , \\quad G_{3i}^{(1)}=G_{3i}^{(2)}=P_{3i}^{(1)}=P_{3i}^{(2)}= 0.\n\\end{array}\n\\end{equation}\nIt is also convenient to divide the warp factors $Z_\\Lambda$ such as\n\\begin{align}\nZ_0 &\\= \\frac{1}{\\left(r_+^{(n)}-z\\right)\\,\\sqrt{U_1 U_2 U_3}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_I}}\\,,\\label{eq:WarpfactorMtheoryBu15d} \\\\\nU_I &\\,\\equiv\\, \\prod_{i=0}^{N-1} \\left(1-\\frac{2 M_{3i+I}}{R_+^{(3i+I)}} \\right)\\,, \\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_I}{2 a_I}\\,. \\nonumber\n\\end{align}\nThe remaining functions \\eqref{eq:WarpfactorsRods5d} give\n\\begin{align}\n& W_0 = \\left( U_1 U_2 U_3\\right)^{\\frac{1}{2}},\\qquad W_I \\= U_I^{-\\frac{1}{2}} ,\\qquad H_0 \\= 0 ,\\qquad T_I\\=-\\sqrt{a_I^2+\\frac{U_I}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{1}{Z_0}\\,\\sqrt{\\frac{E_{-+}^{(1, n)}}{2E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i,j=0}^{N-1} \\,\\prod_{\\substack{I,J=1\\\\Iz_n^+=0$, corresponds to a coordinate degeneracy since $r_+^{(n)}-z=0$. The $y_7$-circle reduces to zero size as a smooth origin of $\\mathbb{R}^2$ (see appendix \\ref{App:AttheRod2} for more details).\n\nAt the finite rods, the $y_1$, $y_3$ and $y_5$ fibers shrink successively to zero size since $U_1$, $U_2$ and $U_3$ vanish alternatively. These loci correspond to smooth origin of $\\mathbb{R}^2$ with potential conical defects if $3N$ bubble equations are satisfied (see Appendix \\ref{App:AttheRod2} for more details). These bubble equations arise from the regularity condition \\eqref{eq:RegCondBu5d} at each rod. They are non-trivial multivariate polynomials that fix all rod lengths, $M_i$. \n\nEach species of bubble rod carries a M2 charge as discussed in section \\ref{sec:regM5d}. Therefore, by combining the three species, the solutions have M2-M2-M2 charges. They are given by\n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=0}^N M_{3i+I}.\n\\label{eq:ChargeMtheoBu5d}\n\\end{equation}\nIn the five-dimensional frame after reduction on the T$^6$ \\eqref{eq:5dframework}, the solutions are \n\\begin{align}\nds_5^2 &\\= \\left(U_1 U_2 U_3 \\right)^{\\frac{1}{3}} \\left[- \\frac{dt^2}{\\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{2}{3}}} \\+ \\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{1}{3}}\\, \\Biggl[\\left(r_+^{(n)}-z \\right) \\,dy_7^2 \\right. \\\\\n& \\left. \\hspace{5.5cm}\\+ \\frac{1}{\\left(r_+^{(n)}-z \\right)U_1 U_2 U_3} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\Biggr] \\right] \\nonumber \\\\\n\\Phi_0 &= \\frac{1}{2}\\log (U_1 U_2 U_3) \\,,\\quad \\Phi_I = -\\frac{1}{2}\\log U_I \\,,\\quad X^I =\\frac{|\\epsilon_{IJK}|}{2} \\left(\\frac{U_I\\,{\\cal Z}_J {\\cal Z}_K}{\\sqrt{U_J U_K}\\,{\\cal Z}_I} \\right)^\\frac{1}{3},\\quad F^{I} = dT_I \\wedge dt\\,. \\nonumber\n\\end{align}\nTherefore, they are singular at the rods in five dimensions, and these singularities are resolved in M-theory as the degeneracy of the extra dimensions. Moreover, the solutions are asymptotic to $\\mathbb{R}^{1,4}$ if \\eqref{eq:condonasymp5d} is satisfied. The ADM mass is given by \\eqref{eq:ADMmass5d}\n\\begin{equation}\n{\\cal M} \\= \\frac{\\pi}{4G_5} \\sum_{I=1}^3 \\left( \\coth b_I \\sum_{i=0}^{N-1} M_{3i+I}\\right)\\,.\n\\label{eq:MassMtheoBu5d}\n\\end{equation}\nFurthermore, since the extra scalars, $\\Phi_\\Lambda$, are turned on, the five-dimensional solutions are not solutions of the STU Lagrangian as detailed in section \\ref{sec:5dreduction}. Nevertheless, they have the same conserved charges as a non-extremal three-charge static black hole in five dimensions, but they are horizonless and terminate the spacetime as a chain of non-BPS bubbles in M-theory.\n\n\\begin{itemize}\n\\item[\u2022] \\underline{BPS limit:}\n\\end{itemize}\n\nAs for the four-dimensional bubbling geometries constructed in section \\ref{sec:BuSolMtheory}, one can derive the BPS limit of the present five-dimensional solutions. This consists of taking the gauge field parameters to zero, $(b_I, a_I) \\to 0 $. For simplicity, we consider $b_I= a_I=\\lambda$ with $\\lambda \\to 0$. First, from the bubble equations \\eqref{eq:RegCondBu5d}, we have \n\\begin{equation}\nM_{i} \\underset{\\lambda\\to 0}{\\sim} \\lambda \\,q_{i},\\qquad i=0,...,3N,\n\\end{equation}\nwhere $q_i$ are finite constants as $\\lambda \\to 0$, and the whole structure reduces to a point. However, it is clear that the charges \\eqref{eq:ChargeMtheoBu5d} and the mass \\eqref{eq:MassMtheoBu5d} do not vanish:\n\\begin{equation}\nQ^I_{\\text{M2}} \\sim \\frac{1}{2}\\,\\sum_{i=0}^N q_{3i+I} , \\qquad {\\cal M} \\sim \\frac{\\pi}{4 G_5}\\left(Q^1_{\\text{M2}}+Q^2_{\\text{M2}}+Q^3_{\\text{M2}} \\right),\n\\end{equation}\nThe warp factors and gauge potentials, \\eqref{eq:WarpfactorMtheoryBu15d} and \\eqref{eq:WarpfactorMtheoryBu25d}, behave as\n\\begin{equation}\n\\begin{split}\nU_I &\\sim1\\,, \\qquad {\\cal Z}_I \\sim 1+ \\frac{Q_\\text{M2}^I}{r^2}, \\qquad r_+^{(n)}-z \\sim r^2 \\sin^2 \\theta \\,, \\qquad T_I \\sim - \\frac{1}{{\\cal Z}_I}\\,,\\qquad e^{2\\nu} \\sim 1\\,.\n\\end{split}\n\\end{equation}\nwhere $(r,\\theta)$ are the five-dimensional spherical coordinates $(\\rho,z)=\\frac{1}{2}(r^2\\sin2\\theta,r^2\\cos 2 \\theta)$. Therefore, the solutions approach a three-charge static BPS black hole in five dimensions.\n\nThe BPS black hole horizon is thus smoothly resolved into smooth non-BPS bubbles wrapped by flux by adding a small amount of non-extremity. At small $(a_I,b_I)$, the solutions are almost indistinguishable from the BPS black hole. They must develop an AdS$_2$ throat that does not end in a horizon but as non-BPS bubbles.\n\n\n\\section{Discussion}\n\n\nIn this paper, we have shown that non-BPS linear ansatz that allow non-trivial matter fields and topology can be directly derived from Maxwell-Einstein equations in string theory. In M-theory on T$^6\\times$S$^1$, our ansatz enables four gauge potentials corresponding to three stacks of M2 branes and a KKm vector, and it can be dualized to other string frames such as the D1-D5-P-KKm frame. Focusing on solutions that are asymptotic to four- and five-dimensional flat space, we derived families of four-charge non-extreme black holes on a line and non-BPS bubbling geometries. The latter can have the same charges and mass as non-extremal black holes but are horizonless and smooth. We have highlighted examples that are almost indistinguishable from the BPS black hole when they are slightly non-BPS but terminate the spacetime smoothly as a chain of non-BPS bubbles.\n\n\n\nWhile the present non-BPS floating brane ansatz opens a new door to the study of non-BPS solitons in string theory, several questions need to be explored in future work. First, although the smooth charged bubble is known to be a meta-stable vacuum \\cite{Stotyn:2011tv,StabilityPaper}, the stability of chains of such objects remains to be studied. Second, one can think of constructing new families of solutions with different boundaries than the flat asymptotic. Third, it will also be interesting to add brane degrees of freedom to the ansatz in M-theory, such as M5 or P brane charges. This will enable Chern-Simons interactions to be turned on as in the BPS floating brane ansatz for multicenter solutions. In addition, to construct more astrophysically-interesting solutions, a rotational degree of freedom will need to be added. \n\nFurthermore, understanding the origin of the solutions as bound states of strings and branes will require careful analysis of the geometric transition that occurs. For BPS solutions, for example, the geometric transition that gives rise to smooth geometries is well understood \\cite{Bena:2013dka}. The present one uses very different mechanisms because our solutions are far from being the non-BPS extensions of known BPS smooth bubbling solutions as their BPS limit suggests, and they are based on non-trivial topology changes of compact tori. They should therefore be very illuminating on constructions of string and brane boundary states in the non-supersymmetric regime, but one should not expect these bound states to use transitions similar to those of BPS bound states.\n\nFinally, it would be interesting to have a better geometrical and physical understanding of the smooth bubbling geometries. One can compare them with the non-extremal black hole with the same mass and charges as in \\cite{Bah:2021rki}. Moreover, one can derive their multipole moments, probe them by light geodesics or compute their quasi-normal modes using technologies developed for similar geometries \\cite{Bena:2020uup,Bianchi:2020miz,Bah:2021jno,Mayerson:2020tpn,Bacchini:2021fig,Ikeda:2021uvc,Bena:2020yii}. One could also being interested in performing M2-brane probe computation in the smooth bubbling solutions in M-theory. If we do not expect the branes to feel an entirely flat potential that allows them to ``float'' everywhere in space, our non-linear ansatz suggest special flat regions where probe can be trapped.\n\n\\section*{Acknowledgments}\nI am grateful to Ibrahima Bah for his rich advice, and to Iosif Bena, Nejc Ceplak, Anthony Houppe and Nick Warner for interesting discussions. This work is supported in part by NSF grant PHY-1820784.\n\n\n\\vspace{1cm}\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\nExploring various phases of quark and nuclear matter\nis one of the primary goals in high-energy heavy-ion collision physics.\nCompared to the QCD phase transition at high $T$,\nless is known for the cold and dense matter,\npartly because the lattice Monte-Carlo simulations are difficult\nat large $\\mu$.\nOne of the most instructive\napproaches to compressed baryonic matter is to consider\nthe strong coupling limit (SCL) of lattice \nQCD~\\cite{Kawamoto,Faldt1986,Bilic1992,Nishida2004}.\nIn fact, effective free energies at finite $T$ and $\\mu$\nhave been analytically derived in SCL,\nand it is successfully applied to nuclear many-body problems~\\cite{Tsubakihara}.\nWhile baryon effects would be important in dense matter,\nthey have been ignored in finite $T$ treatments\nthrough the $1\/d$ expansion ($d$ is the spatial dimension).\n\nIn this proceedings, first we derive an expression of the effective free energy\nat finite $T$ and $\\mu$ including baryon effects~\\cite{KMOO_2007}.\nWe find the second and first order phase transition at high and low $T$,\nseparated by the tricritical point (TCP) in the chiral limit. \nWith finite quark masses,\nthese second order phase boundary and TCP\nbecome the cross over and the critical end point,\nthus the obtained phase diagram seems to show essential qualitative features\nof that in the real world.\n\nOne of the problems in SCL\nis in the ratio, $R_{\\mu T} = \\mu_c(T=0)\/T_c(\\mu=0)$;\nthis ratio would be larger than two in the real world,\nbut it is less than one half in all of the works based on SCL.\nIn the second part of this proceedings, \nwe demonstrate that\nthe ratio increases to $R_{\\mu T} \\sim 1.8$ at $g \\sim 1$\nwith $1\/g^2$ corrections.\nWith finite bare quark mass and baryon effects,\nit may be possible to understand the shape of the actual phase diagram.\n\n\\section{Effective free energy in the strong coupling limit of lattice QCD}\n\\label{Sec:Model}\n\nIn SCL, we ignore pure gluonic action ($\\propto 1\/g^2$),\nand we obtain the following staggered fermion effective action\nafter integrating spatial links,\n\\begin{equation}\n{\\cal Z}=\\int {\\cal D}[\\chi,{\\bar{\\chi}},U_0]\n\\exp\\left[-S_F^{(0)}\n\t+\\frac12 (M, V_M M)\n\t+(\\bar{B},V_B B)\n\\right]\\,,\n\\label{Eq:ActionA}\n\\end{equation}\nwhere\n$(A,B)=\\sum_x A_xB_x$,\nand \nthe mesonic and baryonic composites and their propagators are\n$M_x={\\bar{\\chi}}^a_x\\chi^a_x$, \n$B_x=\\varepsilon_{abc}\\chi^a_x\\chi^b_x\\chi^c_x\/6$,\n$V_M(x,y)$\nand\n$V_B(x,y)$.\nWe keep the timelike link and bare quark mass terms\nunexpanded,\n$S_F^{(0)}=\\sum_{x}[{\\bar{\\chi}}_xe^{\\mu}U_0(x)\\chi_{x+\\hat{0}}\n\t-{\\bar{\\chi}}_{x+\\hat{0}}e^{-\\mu}U^\\dagger_0(x)\\chi_x]\/2\n\t+m_0({\\bar{\\chi}},\\chi)$.\n\nWe decompose the effective action in Eq.~(\\ref{Eq:ActionA}),\ncontaining six fermion terms,\nto the bilinear form in $\\chi$\nin order to perform the quark integral in a finite $T$ treatment.\nDecomposition has been carried out in three steps,\n\\numparts\n\\begin{eqnarray}\n&&e^{(\\bar{B},V_B,B)}\n=\\int {\\cal D}[\\bar{b},b,\\phi,\\phi^\\dagger]\n\te^{\n\t\t-(\\bar{b},V_B^{-1}b)-\\phi_a^\\dagger\\phi_a\n\t\t+\\phi_a^\\dagger D_a+D^\\dagger_a\\phi_a\n\t\t- \\gamma^2 M^2\/2+M\\bar{b}b\/9\\gamma^2\n\t}\n\t\\ ,\n\\label{Eq:DecompA}\n\\\\\n&&e^{M \\bar{b}b\/9\\gamma^2}\n=\\int d[\\omega]\\,e^{\n\t\t-\\omega^2\/2\n\t\t-\\omega(\\alpha M+\\bar{b}b\/9\\alpha\\gamma^2)\n\t\t-\\alpha^2 M^2\/2\n\t\t}\n\\ ,\n\\label{Eq:DecompB}\n\\\\\n&&e^{\\frac12(M,\\widetilde{V}_M M)}\n=\\int d[\\sigma]\\,e^{\n\t\t-(\\sigma,\\widetilde{V}_M\\sigma)\/2\n\t\t-(\\widetilde{V}_M\\sigma,M)\n\t\t}\n\\label{Eq:DecompC}\n\\ .\n\\end{eqnarray}\n\\endnumparts\nIn the first step (\\ref{Eq:DecompA}),\nthe baryonic action is decomposed \nusing the diquark field $\\phi_a$~\\cite{Azcoiti2003},\nwhose expectation value is $\\VEV{\\phi_a}=\\VEV{D_a}$,\nwhere $D_a=\\gamma\\varepsilon_{abc}\\chi^b\\chi^c\/2+\\bar{\\chi}^a b\/3\\gamma$.\nIn the second step (\\ref{Eq:DecompB}),\nthe four fermi interaction term, $M\\bar{b}b$,\nis decomposed utilizing the null potency, $(\\bar{b}b)^2=0$.\nIn the third step (\\ref{Eq:DecompC}),\nchiral condensate $\\sigma$ is introduced\nafter including the remaining four quark interaction terms\nin the propagator,\n$\\widetilde{V}_M=V_M+(\\alpha^2+\\gamma^2)\\delta_{x,y}$.\nThe effective action is now in a bilinear and pfaffian form of $\\chi$,\nand quarks are separated in each spatial point\ncoupling to the auxiliary fields,\n$m_q=\\widetilde{V}_M\\sigma+\\alpha\\omega$,\n\\begin{eqnarray}\n\\fl~~~~~~~~~~~~\nS_F&=&\nS_F^{(0)}\n+({\\bar{\\chi}},m_q\\chi)\n+\\frac12(\\sigma,\\widetilde{V}_M\\sigma)\n+(\\bar{b},\\widetilde{V}_B^{-1} b)\n+\\frac12 (\\omega,\\omega)\n\\nonumber\\\\\n\\fl\n&+&(\\phi^\\dagger,\\phi)\n+{1\\over3\\gamma}\\left[\n\t ({\\bar{\\chi}}^a, \\phi_a^\\dagger b)\n\t+(\\bar{b}\\phi_a,\\chi^a)\n\t\\right]\n+\\frac{\\gamma}{2}\\varepsilon_{cab}\\left[\n\t (\\phi_c^\\dagger, \\chi^a \\chi^b)\n\t+({\\bar{\\chi}}^b{\\bar{\\chi}}^a,\\phi_c)\n\t\\right]\n\\,.\n\\label{Eq:ActionB}\n\\end{eqnarray}\n\nWe can analytically carry out\nMatsubara frequency product of quarks and temporal link integral\nin the mean field approximation\nat zero diquark condensate.\nAt equilibrium, \nan approximate linear relation $\\omega \\propto \\sigma$ holds,\nand the effective free energy is found to be\n\\begin{eqnarray}\n{\\cal F}_\\mathrm{eff}(\\sigma)\n&=&\\frac12b_\\sigma\\sigma^2\n+{\\cal F}_\\mathrm{eff}^{(q)}(b_\\sigma\\sigma)\n+{\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}(g_\\sigma\\sigma)\n\\,,\n\\label{Eq:Feff}\n\\\\\n{\\cal F}_\\mathrm{eff}^{(q)}(m_q)\n&=&-T\\log\\left[C_\\sigma^3-\\frac12C_\\sigma\n+\\frac14\\cosh\\left(\\frac{3\\mu}{T}\\right)\\right]\n\\,,\n\\\\\n{\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}(m_b)\n&\\simeq&-f^{(b)}\\left({\\pi m_b\\over8}\\right)\n\\,, \n\\end{eqnarray}\nwhere $C_\\sigma=\\cosh(\\mathrm{arcsinh}\\,(m_q)\/T)$\nand \n$f^{(b)}(x)=\\log(1+x^2)\/2-[\\arctan{x}-x+x^3\/3]\/x^3-3x^2\/10$.\nIn this effective free energy ${\\cal F}_\\mathrm{eff}$,\nwe have two parameters, $b_\\sigma$ and $g_\\sigma$,\nwhich are related to the decomposition parameters,\n$\\alpha$ and $\\gamma$.\nWithout baryon effects,\nthe first two terms remain in the effective free energy (\\ref{Eq:Feff})\nwith a fixed coefficient $b_\\sigma=d\/2N_c$~\\cite{Nishida2004}.\n\nBaryon effects appear\nin the modification of the coefficient $b_\\sigma$\nand in the additional term ${\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}$\ncoming from the auxiliary baryon determinant.\nWhen we adopt the parameter $\\alpha=0.2$\nwhich approximately maximize the ratio\n$R_{\\mu T}=\\mu_c\/T_c$,\nbaryons are found to have effects\nof the effective free energy gain with respect to $T_c$\nand the extension of the hadronic phase in the larger $\\mu$ direction,\nas shown in Fig.~\\ref{Fig:SCL}~\\cite{KMOO_2007}.\n\n\\begin{figure}\n\\begin{minipage}{\\textwidth}\n\\Psfig{8cm}{QM06-Ohnishi_Fig1a.eps}{}\n\\Psfig{8cm}{QM06-Ohnishi_Fig1b.eps}{}\n\\end{minipage}\n\\caption{Effective free energy as a function of $\\sigma$ (left panel),\nand the phase diagram in the strong coupling limit (right panel).\n}\\label{Fig:SCL}\n\\end{figure}\n\n\\section{Finite coupling correction}\n\\label{Sec:Finite_g}\n\nWhile the SCL seems to show\nqualitative features of the phase diagram in the real world,\nthere are several problems,\nsuch as\nthe parameter dependence and scale modifications~\\cite{KMOO_2007},\nno stable color superconductor (CSC) phase,\nand too small ratio $R_{\\mu T} < 0.5$.\nFinite coupling corrections may solve\nsome of these problems~\\cite{Faldt1986,Bilic1992,Bilic1995,Ipp}.\n\nThe $1\/g^2$ correction on the effective action was\nderived by Faldt and Petersson~\\cite{Faldt1986},\nand we can bosonize the plaquett contributions\nin the mean field approximation as,\n\\begin{eqnarray}\n\\fl~~~~~~~~\n{\\it \\Delta}S_F&=&\n \t\\frac{1}{4N_c^2g^2}\\sum_{x,j>0} \n\t\t(\n\t\t V^\\dagger_x V_{x+\\hat{j}}\n\t\t+V^\\dagger_x V_{x-\\hat{j}}\n\t\t)\n \t- \\frac{1}{8N_c^4g^2}\\sum_{x,k>j>0} \n\t\tM_{x}\n\t\tM_{x+\\hat{j}}\n\t\tM_{x+\\hat{k}}\n\t\tM_{x+\\hat{k}+\\hat{j}}\n\\nonumber\\\\\n\\fl\n&\\simeq&\n\tN_s^3N_\\tau\\left(\n\t\t \\frac{\\beta_t}{4}\\varphi_t^2\n\t\t+\\frac{\\beta_sd}{4}\\varphi_s^2\n\t\\right)\n\t+\\frac{\\beta_t\\varphi_t}{4}\n\t\\sum_x(V_x-V_x^\\dagger)\n\t-\\beta_s\\varphi_s\n\t\\sum_{x,j>0} M_x M_{x+\\hat{j}}\n\t\\,.\n\\label{Eq:ActionC}\n\\end{eqnarray}\nWe have defined $V_x \\equiv {\\bar{\\chi}}_x U_0(x) \\chi_{x+\\hat{0}}$,\n$\\beta_s=(d-1)\/8N_c^4g^2$, and $\\beta_t=d\/2N_c^2g^2$,\nand the auxiliary fields have expectation values of\n$\\VEV{\\varphi_t}=\\VEV{V^\\dagger-V}$\nand \n$\\VEV{\\varphi_s}=2\\VEV{M_xM_{x+\\hat{j}}}$.\nHere we have omitted baryon effects (${\\cal O}(1\/d^{1\/2})$)\nand higher order terms (${\\cal O}(1\/d)$) \nin the $1\/d$ expansion.\n\nThese corrections have a similar structure\nto the SCL effective action (\\ref{Eq:ActionA}),\nand they lead to\nthe modifications of the quark mass and effective chemical potential as \n$\\widetilde{m}_q=\\sigma d(1+4N_c\\beta_s\\varphi_s-\\beta_t\\varphi_t\\cosh\\mu)\/2N_c$\nand\n$\\widetilde{\\mu}=\\mu-\\beta_t\\varphi_t\\sinh\\mu$.\nAt equilibrium, \nwe can put $\\varphi_s=2\\sigma^2 + {\\cal O}(1\/g^2)$,\nand the effective free energy up to ${\\cal O}(1\/g^2)$ is obtained as,\n\\begin{eqnarray}\n{\\cal F}_\\mathrm{eff}=\\frac{d}{4N_c}\\sigma^2+3d\\beta_s\\sigma^4\n\t+\\frac{\\beta_t}{4}\\varphi_t^2-N_c\\beta_t\\varphi_t\\cosh\\mu\n\t+{\\cal F}_\\mathrm{eff}^{(q)}(\\widetilde{m}_q;T,\\widetilde{\\mu})\n\t\\,.\n\t\\label{Eq:FeffCorr}\n\\end{eqnarray}\n\nWith this effective free energy,\n$T_c$ is found to decrease as $g$ decreases,\nwhile $\\mu_c$ stays almost constant.\nAs a result, $R_{\\mu T}$ grows to around 1.8 at $g \\sim 1$\nas shown in Fig.~\\ref{Fig:Finite_g}.\nThe present results are not fully consistent with the previous \nfindings ~\\cite{Bilic1995}.\nThe difference in the bosonization scheme and lattice\nanisotropy~\\cite{Bilic1992,Bilic1995} has to be investigated further.\n\n\\begin{figure}\n\\begin{minipage}{\\textwidth}\n\\Psfig{6cm}{QM06-Ohnishi_Fig2a.eps}{}\n\\Psfig{8cm}{QM06-Ohnishi_Fig2b.eps}{}\n\\end{minipage}\n\\caption{Evolution of the phase diagram with $\\beta=6\/g^2$ (left panel),\nand the $\\beta$ dependence of $T_c$ and $\\mu_c$ (right panel).}\n\\label{Fig:Finite_g}\n\\end{figure}\n\n\n\\section{Summary}\n\nIn this proceedings,\nwe have investigated the phase diagram\nin the strong coupling region of lattice QCD.\nBaryons are found to have effects of extending the hadron phase\nin the larger $\\mu$ direction with respect to $T_c$ up to around\n30 \\%~\\cite{KMOO_2007} in the strong coupling limit.\nWith finite coupling corrections,\nwe have found that the ratio $R_{\\mu T}=\\mu_c\/T_c$ becomes\ncloser to that expected in the real world.\nIt would be interesting to evaluate both of the finite coupling correction\nand baryon effects simultaneously.\n\nThis work is supported in part by the Ministry of Education,\nScience, Sports and Culture,\nGrant-in-Aid for Scientific Research\nunder the grant numbers,\n 13135201,\t\n 15540243,\t\nand 1707005.\t\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction}\nThe search for periodic gravitational wave signals is a stimulating challenge\nfor data analists because of the considerable amount of computing time\nrequired.\n\nFor blind searches, i.e. without any {\\it a priori} knowledge about the source,\na fully coherent analysis can not handle more than a few days of data because\nof the steep dependence of the size of the parameter space on the\nfrequency resolution.\n\nIn \\cite{Astone:2005fn} three data sets, each two days long, from the\nExplorer 1991 run have been coherently studied by means of the \n{\\cal F} statistics method \\cite{Jaranowski:1998qm}\nwhich led to and an upper limit of $1\\cdot 10^{-22}$ on $h$ in\nthe narrow band $921.00$-$921.76$Hz.\n\nA similar technique, applied in \\cite{Abbott:2006vg}\nto the most sensitive 10 hours of the the LIGO S2 run, led\nto an upper limit of $6.6\\cdot 10^{-23}$ for isolated\nneutron stars in the band between $160$ and $728.8$Hz.\n\nWith the widening of the frequency band, due to the advent of\ninterferometers as well as to improvements in the readout of resonant\ndetectors, several incoherent and semi-coherent methods have been\nconceived and employed.\n\nIn \\cite{Abbott:2005pu} the Hough transform technique has been applied to\nthe LIGO S2 data to perform a blind search for isolated neutron stars\non a set of narrow frequency bands in the range $200$-$400$Hz, and a best\nupper limit of $4.43\\cdot 10^{-23}$ has been set.\n\nIn the present work, the simple technique of adding power spectra has been\napplied to the most sensitive $40$Hz band of the 2005 run of the Explorer bar\n\\cite{Astone:2006uf}, resulting in a further improvement on the best upper\nlimit on $h$, which is set to $3.1\\cdot 10^{-23}$ at $920.14$Hz.\n\nAccording to the results reported at the recent Amaldi7 conference,\nthe analysis of the LIGO S4 run \\cite{Abbott:2007td} is leading to a sensible\nimprovement in this direction (about an order of magnitude);\nremarkably, the limit set in the present work is still competitive with\nthe one coming from the S4 data in the same frequency band.\n\n\\subsection{The data set}\nAt the end of April 2005, after a short commissioning break, the resonant\nantenna Explorer was again\n{\\it on air} and operated with the usual, remarkable duty cycle ($86\\%$\nfrom April to December 2005) and a good stability.\n\nThe data stream taken by the bar until the end of 2005\n(after which the sensitivity curve has been modified)\nhas been divided into $25161$ segments, each\nabout 14 minutes long.\nThe most sensitive band of the Fourier transform of these segments,\nnamely $N_f=32178$ frequency bins in the range $885$-$925$Hz,\nhas been selected for the analysis.\n \n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=-90]{noiseexp05.ps} \n \\caption{The noise level of Explorer during the 2005 run.\nThe solid horizontal line is the cut applied to select the best spectra.}\n \\label{fignoise}\n\\end{figure}\n\nA noise cut has been applied to the total power contained in each \nspectrum, with the purpose of discarding the noisy ones thus creating an\nhomogeneous\nset of spectra. This allowed us to apply for the subsequent analysis\nthe simple power addition method, without weighting each spectrum with the\ncorresponding noise level.\n\nThis selection led to the creation of a data set $\\left\\{S_i\\right\\}$\nmade of the $N_1=11749$ cleanest spectra (corresponding to an effective data\ntime of 114 days), and of a second set\ncontaining just the best $N_2=3875$ (used to deal with the critical zones\nof the spectrum, corresponding to the resonant modes of the bar, around 888Hz\nand 920Hz).\n\n\nThe average sensitivity of the second data set is shown in Fig. \\ref{figsensi}\n(the first set is similar except around the resonant modes);\na few noisy lines may be noted, including small 1Hz harmonics on the left part.\n\\begin{figure}\n \\centering\n \\includegraphics[width=6in,angle=0]{spenoshift.eps} \n \\caption{The square root of the average spectrum represents the typical\nExplorer sensitivity curve during the 2005 run.}\n \\label{figsensi}\n\\end{figure}\n\n\n\n\\subsection{The analysis method}\n\nFor a given direction $\\hat{r_j}$ in the sky, the selected spectra have\nbeen deformed according to the Doppler shift formula\n\\begin{eqnarray}\\label{doppler}\nf^{\\rm true}_j\\simeq f_{\\rm exp}\\left(1-\\frac{\\vec{v}_{rel}\n\\cdot\\hat{r_j}}{c}\\right)\\nonumber\n\\end{eqnarray}\nand then summed and renormalized dividing by $N_1$ or by $N_2$:\n\\begin{eqnarray}\\label{sumspe}\n{\\cal S}_j (f^{\\rm true}_j)=\\frac{1}{N_{1,2}}\\sum_{i=1}^{N_{1,2}}\nS_i(f^{\\rm true}_j)\\, .\\nonumber\n\\end{eqnarray}\n\n\nThe speed of the detector relative to the Solar System Baricenter has been\ncomputed thanks to the JPL ephemerides \\cite{JPL}.\nAs the speed of the source have not been taken into account, the search is\nsensitive to isolated neutron stars, and not to those which are part of\nbinary systems.\n\nSpindown has also been neglected: given the frequency resolution of\n$1.2\\cdot 10^{-3}$Hz, this means being sensitive to an average frequency\ndrift up to $6\\cdot10^{-11}$Hz\/s during the observation\nperiod.\\\\\nThe procedure has been repeated for any point of an optimized sky grid\nmade of $N_{\\rm sky}=23927$ possible directions, thus leading to the creation\nof a set $\\left\\{{\\cal S}_j\\right\\}$ containing\n$N_{\\rm sky}$ ``deformed and summed'' spectra.\n\nThe variance of the noise is obtained calculating, for each value of the\nfrequency, the variance of the distribution of the $N_{\\rm sky}$ deformed\nspectra.\nThe result, whose square root is shown in Fig. \\ref{figsigma}, agrees with\nthe general expectation, based on the central limit theorem, that\n\\begin{equation}\n\\sigma\\simeq\\frac{S_h}{\\sqrt{N_{1,2}}}\\, .\n\\end{equation}\n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=0]{sigsky.eps} \n \\caption{Noise variance of the set of deformed and summed spectra.\nTo help comparison with Fig. \\ref{figsensi}, the square root of $\\sigma$ is\nactually shown.}\n \\label{figsigma}\n\\end{figure}\n\nAs expected, the plot shows an anomalous behavior of the noise\nvariance in correspondence of the disturbances of the initial data set.\nThese anomalous zones have not been taken into account for the candidate\nsearch, but have been included in the upper limit determination.\n\nThe detection threshold is fixed by the requirement that the false alarm rate\nshould be less than $1\\%$. According to Poisson statistics one has to impose\n\n\\begin{eqnarray}\\label{poisson}\nP(0,\\lambda)={\\rm e}^{-\\lambda}>.99\\, ,\\nonumber\n\\end{eqnarray}\nwhere the expected number of threshold crossings in absence of signal is\n\\begin{eqnarray}\\label{lambda}\n\\lambda=p\\cdot N_f\\cdot N_{\\rm sky}\\, ,\\nonumber\n\\end{eqnarray}\nbeing $p$ the probability of false detection in a single frequency bin\nand for a single direction in the sky, and $N_f\\cdot N_{\\rm sky}$\nthe trial factor.\n\nOne thus finds the condition $p<1.3\\cdot 10^{-11}$\nwhich, assuming that the $N_{\\rm sky}$ values of the shifted spectra at a\ngiven frequency bin are gaussian distributed, translates to a $7\\sigma$\nthreshold.\n\nIn other words, a detection is claimed if, for some value of\nthe frequency $f$, a given deformed spectrum ${\\cal S}_j$\nsatisfies the condition\n\\begin{eqnarray}\\label{detcond}\n{\\cal S}_j(f)-\\bar{\\cal S}(f)>7 \\sigma(f)\\, ,\n\\end{eqnarray}\nbeing $\\bar{\\cal S}$ the average of the deformed spectra\n$\\left\\{{\\cal S}_j\\right\\}$ as $j$ spans over the $N_{\\rm sky}$\nsky grid points. \n\nSince $h\\propto\\sqrt{S}$, to translate the detection threshold\nin $h$ units one has to multiply the values shown in Fig. \\ref{figsigma}\nby the factor\n\\begin{eqnarray}\\label{factor}\n\\sqrt{\\frac{7}{T}}\\cdot\\sqrt{\\frac{15}{4}}\\, ,\\nonumber\n\\end{eqnarray}\nwhere $T$ is the length of each data segment and the factor $4\/15$\n(the average angular sensitivity of the bar over the solid angle) is\nintroduced to compensate the fact that amplitude modulation has not been taken\ninto account when summing the deformed spectra.\n\n\\subsection{Results}\n\\subsubsection{Cadidate search}\n\nThe threshold is shown in red in Figure \\ref{figthremax},\nwhile the blue line is the maximum value of $h$ found, at any given\nfrequency bin, among the set of the $N_{\\rm sky}$ spectra according to\nthe following formula:\n\\begin{eqnarray}\\label{geth}\nh_{\\rm max}(f)={\\rm max}_j \\left[\\sqrt{\\frac{15}{4}\\frac{{\\cal S}_j(f)-\\bar{\\cal S}(f)}{T}}\\right]\\, .\\nonumber\n\\end{eqnarray}\n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=0]{thremax.eps} \n \\caption{Detection threshold (red, upper curve) and maximal triggers\n (blue, lower). The $1$Hz disturbances on the left have been left only\n for illustrative purpose and cannot be considered as genuine threshold\n crossings.}\n \\label{figthremax}\n\\end{figure}\n\n The anomalous regions of the spectrum have been cut out, since it is not\nreasonable to assume that they follow a gaussian distribution. To illustrate\nthe point, the $1$Hz disturbances have been keft in Figure \\ref{figthremax},\nto show that they would have\nproduced fake candidates, if they had been included in the analysis.\n\nWith these specifications, the maxima are never above the threshold,\nand thus no candidates have been found.\n\n\\subsubsection{Software injections and upper limit}\n\nTo find an upper limit on $h$, a variation of the loudest event method\n\\cite {Brady:2004gt} has been applied.\nThis method allows to determine an upper limit starting from the\nloudest event present in a data stream, irrespectively of the fact that\nsuch an event may be due to noise or to a real signal.\nThe idea is basically that if a strong real signal would have been present\nduring the data taking, it would have produced an event louder than the\nloudest event actually recorded. This idea can be made quantitative by\nstudying the detection efficiency of the experiment, for example by performing\nsoftware injections at various SNR's.\n\nIn our case, we have a loudest event for any frequency bin, and all these\nevents form precisely the curve of maxima depicted in Figure \\ref{figthremax}.\nAs a preliminary step, we had to determine our detection efficiency by\ninjecting fake periodic signals in the Explorer data stream and\nfinding out how hey did loook like after the analysis chain: the left plot of\nFigure \\ref{figinjupp} shows for instance the result of 16 injections\nof signals with $h=3\\cdot 10^{-23}$, equally spaced in frequency by\n$0.1$Hz starting from $919.4$Hz, and coming from randomly chosen directions\nin the sky.\n\\begin{figure}\n \\centering\n \\includegraphics[width=6cm,angle=0]{inje.eps}\n \\includegraphics[width=6cm,angle=0]{upper.eps}\n \\caption{On the left, the outcome of the 16 injections,\n with initial amplitude given by the dashed red line.\n On the right, the upper limt curve.}\n \\label{figinjupp} \n\\end{figure}\nThen, the upper limit at $95\\%$ c.l. at has been determined for each\nfrequency bin as the lowest injected amplitude which had produced a\nsignal larger than the actual maximum at least in $95\\%$ of the cases.\n\nThe right plot on Figure \\ref{figinjupp} shows the result, i.e. the curve\nof $h$ upper limit at $95\\%$ confidence level: the minimum is\n$3.1\\cdot 10^{-23}$ at $920.14$ Hz.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhrzv b/data_all_eng_slimpj/shuffled/split2/finalzzhrzv new file mode 100644 index 0000000000000000000000000000000000000000..0ee718c24164ca5e535d0636d174c57b7e412de2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhrzv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nRare-earth orthorhombic perovskites, orthoferrites \\textit{R}FeO$_3$ and orthochromites \\textit{R}CrO$_3$ (where \\textit{R} is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon of the spontaneous spin reorientation. The spin reorientation (SR) is one of their unique properties that have attracted a lot of attention back in the 70s of the last century\\,\\cite{BZKL,KP}, though their exact microscopic origin is still a challenge to theorists and experimentalists.\n\nThe revival of interest in the mechanism of the spontaneous spin reorientation and magnetic compensation in rare-earth perovskites in recent years is related with the discovery of the magnetoelectric and the exchange bias effect, which can have a direct application in magnetoelectronics. Along with the emergence of new experimental studies (see, e.g., Refs.\\,\\cite{PRB102,PRB103}), there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous spin-reorientation transitions\\,\\cite{Bazaliy} or to scrutinize the microscopic mechanism responsible for spin reorientations and magnetization reversal\\,\\cite{Luxem}. In fact, these results are not directly related to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and orthochromites. For instance, the authors of the most recent paper\\,\\cite{Luxem} did not take into account a number of interactions, such as the fourth-order anisotropy for the $ 3d $ sublattice of orthoferrites and the crystal field for $ R $-ions, which play a fundamental role in determining the spontaneous spin reorientation. The spin anisotropy of the second order in the $3d$ sublattice of orthorhombic orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in Ref.\\,\\cite{Luxem}. Furthermore, the density functional theory does not allow in principle to give an adequate description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric exchange\\,\\cite{CM-2019}.\n\nIn this paper, we present the results of a simple but realistic microscopic model of the spontaneous spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main relevant interactions. This model was developed back in the 80s of the last century\\,\\cite{thesis}, but has not been published until now.\n\n\n\\section{Model formulation}\n\nThe most popular examples of systems with the spontaneous SR transitions are magnets based on $3d$ and $4f$ elements such as rare-earth orthoferrites \\textit{R}FeO$_3$, orthochromites \\textit{R}CrO$_3$, intermetallic compounds \\textit{R}Co$_5$, \\textit{R}Fe$_2$ etc. In all cases, an important cause of the spontaneous SR is the $4f-3d$ interaction. Usually this interaction is taken into account by introducing an effective field of the magnetically ordered $3d$ sublattice acting on the $4f$ ions.\n\nTo consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we are developing a model which takes into account either the well isolated lower Kramers doublet of the $4f$ ions (with an odd number of the $4f$ electrons) or the well isolated two lower Stark sublevels with close energies that form a quasi-doublet.\n\nWithin the framework of such ``single-doublet'' approximation we consider the spontaneous SR transition in orthorhombic weak ferromagnets \\textit{R}FeO$_3$ and \\textit{R}CrO$_3$, where the free energy per ion can be represented as follows\n\\begin{equation}\\label{phiRFEO3}\n \\Phi (\\theta) = K_1 \\cos 2 \\theta + K_2 \\cos 4 \\theta - k T \\ln 2 \\, \\cosh \\dfrac{\\Delta (\\theta)}{2 k T},\n\\end{equation}\nwhere $K_1$, $K_2$ are the first and second anisotropy constants of the $3d$ sublattice, which are temperature independent (at least in the SR region), $\\theta$ is the orientation angle of the antiferromagnetic, or N\\'{e}el vector $\\bf G$ of the $3d$ sublattice (e.g. in the $ac$ plane), and $\\Delta (\\theta)$ is the lower doublet (quasi-doublet) splitting of the $4f$ ion in a magnetic field induced by the $3d$ sublattice. \n\nTheoretical estimations\\,\\cite{thesis,M_2021,JETP-1981} of the different\ncontributions to the first constants of the magnetic anisotropy for orthoferrites \\textit{R}FeO$_3$ point to a competition of several main mechanisms with relatively regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion anisotropy, SIA) dependence on the type of \\textit{R}-ion. For instance, the microscopic theory predicts an unexpectedly strong increase in values of the constant $K_1(ac)$ for LuFeO$_3$ as compared with YFeO$_3$.\nThe SIA contribution to $K_1(ac)$ partially compensates for the large contribution of the DM interaction in YFeO$_3$, whereas in LuFeO$_3$, they add up. \nThis result is confirmed by experimental data on the measurement of the threshold field $H_{SR}$ of spin reorientation $\\Gamma_4\\rightarrow\\Gamma_2$ ($G_x\\rightarrow G_z$) in the orthoferrite Lu$_{0.5}$Y$_{0.5}$FeO$_3$, in which $H_{SR}$\\,=\\,15\\,T as compared to $H_{SR}$\\,=\\,7.5\\,T in YFeO$_3$\\,\\cite{JETP-1981}.\nThus, one can estimate $K_1(ac)$ in LuFeO$_3$ as around three times as much as $K_1(ac)$ in YFeO$_3$.\n\nLet us pay attention to recent works on the determination of the parameters of the spin Hamiltonian in YFeO$_3$ from measurements of the spin-wave spectrum by the inelastic neutron scattering\\,\\cite{Hahn,Park} and terahertz absorption spectroscopy\\,\\cite{Amelin}. However, these authors started with a simplified spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-ion anisotropy. Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the ``single-ion anisotropy'' constants found by the authors are some effective quantities that are not directly related to the SIA.\n\nUnfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of various anisotropy mechanisms.\n\nAs shown by theoretical calculations\\,\\cite{thesis,M_2021,cubic} the constants $K_2$ of the fourth order spin anisotropy rather smoothly decrease in absolute value, changing by no more than two times on going from La to Lu. But the most interesting was the conclusion about the different signs of these constants, positive for the $ac$ and $bc$ planes and negative for the $ab$ plane, thus indicating a different character of spin-reorientation transitions in the corresponding planes, i.e., second-order transitions in the $ac$ and $bc$ planes and first-order transitions in the $ab$ plane\\,\\cite{KP}. \nIndeed, all currently known spin-reorientation transitions of the $\\Gamma_4 - \\Gamma_2$ ($G_x - G_z$) type\nin orthoferrites \\textit{R}FeO$_3$ (\\textit{R} = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb) \nare smooth, with two characteristic temperatures of the second-order phase transitions to be a start and finish of the spin reorientation, and the only known jump-like first order SR transition for these crystals is the SR transition $\\Gamma_4 - \\Gamma_1$ ($G_x - G_y$) in the $ab$ plane in DyFeO$_3$\\,\\cite{KP}. \nA unique example that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite Ho$_{0.5}$Dy$_{0.5}$FeO$_3$\\,\\cite{KP}\nin which two spin-reorientation\ntransitions $G_x - G_y$ ($T$\\,=\\,46\\,K) and $G_y - G_z$ (18\\,$\\div$\\,24\\,K) are realized through one phase transition of the first order in the $ab$ plane and two phase transitions of the second order in the $bc$ plane, respectively.\n\n\nThe splitting value $\\Delta (\\theta)$ for the Kramers doublet in a magnetic field $\\bf H$ has the well-known form\n\\begin{equation}\\label{DeltaKramers}\n \\Delta (\\theta) = \\mu_{B} \\left[ \\left( g_{xx} H_x + g_{xy} H_y \\right)^2 + \\left( g_{xy} H_x + g_{yy} H_y \\right)^2 + g_{zz}^2 H_z^2 \\right]^{1\/2},\n\\end{equation}\nwhere it is taken into account that for the $4f$ ions in \\textit{R}FeO$_3$ the $\\hat{g}$-tensor (with the local symmetry $C_s$) has the form \n\\begin{equation}\\label{g-tensor}\n \\hat{g} = \n \\begin{pmatrix}\n g_{xx} & g_{xy} & 0\\\\\n g_{xy} & g_{yy} & 0\\\\\n 0 & 0 & g_{zz}\n\\end{pmatrix}.\n\\end{equation}\n\nThe effective field $\\bf H$ for the SR transition $G_x\\rightarrow G_z$ in the $ac$ plane can be represented as follows\n\\begin{equation}\\label{MagnField}\n H_x = H_x^{(0)} \\cos \\theta, \\,\\, H_y = H_y^{(0)} \\cos \\theta, \\,\\, H_z = H_z^{(0)} \\sin \\theta,\n\\end{equation}\nso in the absence of an external magnetic field, for $\\Delta (\\theta)$ we have the rather simple expression:\n\\begin{equation}\\label{DeltaKramers2}\n \\Delta (\\theta) = \\left( \\dfrac{\\Delta_a^2 - \\Delta_c^2}{2} \\cos 2 \\theta + \\dfrac{\\Delta_a^2 + \\Delta_c^2}{2} \\right)^{1\/2},\n\\end{equation}\nwhere $\\Delta_{a,c}$ are the doublet splitting for the cases of $\\theta = 0$ ($G_z$-phase) and $\\theta = \\pi \/2$ ($G_x$-phase) respectively. The dependence $\\Delta (\\theta)$ from \\eqref{DeltaKramers2} is also valid in the case of quasi-doublet.\n\nA contribution of splitting $\\Delta$ to the free energy $\\Phi (\\theta)$ for the rare-earth sublattice is usually considered in the ``high-temperature'' approximation, when $kT \\gg \\Delta$ and the influence of the $4f$ sublattice are reduced only to renormalization of the first anisotropy constant $K_1$:\n\\begin{equation}\\label{Ku*}\n K_1^* = K_1 \\left( 1 - \\dfrac{1}{\\tau} \\right),\n\\end{equation}\nwhere $\\tau = T\/T_{SR}$ is the reduced temperature and $T_{SR} = (\\Delta_a^2 - \\Delta_c^2)\/ 16k K_1$ is the characteristic transition temperature.\n\nBelow we will consider a specific situation when $K_1 > 0$ and $\\Delta_a > \\Delta_c$, i.e. when the configuration $G_x$ ($\\theta = \\pi\/2$) is realized at high temperatures and a decrease in temperature can lead to the spin reorientation $G_x \\rightarrow G_z$ or $G_x \\rightarrow G_{xz}$ (transition to an angular spin structure). The type of the phase transition of the spin reorientation in the ``high-temperature'' approximation is determined by the sign of the second constant $K_2$: at $K_2 < 0$ it will be realized by one first-order phase transition at $T = T_{SR}$, i.e. $\\tau = 1$, or at $K_2 > 0$ by two second-order phase transitions at $\\tau_s = (1 + \\gamma)^{-1}$ and $\\tau_f = (1 - \\gamma)^{-1}$, where $\\tau_s$ and $\\tau_f$ are the reduced temperatures of the beginning and end of the SR phase transition and $\\gamma = 4 K_2 \/ K_1$.\n\n\n\n\\section{Analysis of the ``single-doublet'' model}\n\nA behavior of a system described by the free energy \\eqref{phiRFEO3} can be analyzed rigorously. The condition $\\partial \\Phi \/ \\partial \\theta = 0$ reduces in this case to two equations:\n\\begin{equation}\\label{SinEq}\n \\sin 2 \\theta = 0,\n\\end{equation}\n\\begin{equation}\\label{mainEq}\n \\alpha \\mu + \\beta \\mu^3 = \\tanh \\dfrac{\\mu}{\\tau};\n\\end{equation}\nwhere the following notations are introduced:\n\\begin{equation}\\label{mainEq_params}\n\\alpha = 1 - \\gamma \\dfrac{\\Delta_a^2 + \\Delta_c^2}{\\Delta_a^2 - \\Delta_c^2},\\ \\beta = \\dfrac{2 \\gamma}{\\mu_f^2 - \\mu_{s}^2},\\ \\mu = \\dfrac{\\Delta (\\theta)}{2k T_{SR}},\\ \\mu_{s} = \\dfrac{\\Delta_c}{2k T_{SR}},\\ \\mu_f = \\dfrac{\\Delta_a}{2k T_{SR}}.\n\\end{equation}\nThis corresponds to three possible magnetic configurations:\n\\begin{itemize}\n\\item The configuration $G_x$: $\\theta = \\pm \\pi\/2$, stable at $\\tanh {\\mu_s}\/{\\tau} \\leq \\alpha \\mu_s + \\beta \\mu_s^3$ .\n\\item The configuration $G_z$: $\\theta = 0, \\ \\pi$, stable at $\\tanh {\\mu_f}\/{\\tau} \\geq \\alpha \\mu_f + \\beta \\mu_f^3$ .\n\\item The angular configuration $G_{xz}$: the temperature dependence of $\\theta (\\tau)$ is determined by solving the equation \\eqref{mainEq} (see Figure\\,\\ref{fig1}), the state is stable at $\\partial \\mu \/ \\partial \\tau \\leq 0$.\n\\end{itemize}\nThe peculiar $\\mu$-$\\tau$ phase diagram which represents solutions of the master equation \\eqref{mainEq} given a fixed value of the $\\alpha$ parameter and different value of the $\\beta$ parameter is shown in Figure\\,\\ref{fig1}, where areas with different character of the SR transition are highlighted in different colors. For the solutions in the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one or two second-order phase transitions, in the MO$_{1,2}$ regions we arrive at a \"mixture\" of the first and second-order phase transitions. All the lines $\\mu (\\tau)$ on the right side converge to $\\sqrt{|\\alpha\/\\beta|}$ at $\\tau \\rightarrow \\infty$; on the left side, when $\\tau \\rightarrow 0$ the branch point $\\mu = \\frac{3}{2\\alpha}$ is obtained at $\\beta = - \\frac{4}{27}\\alpha^3$, and the point $\\mu = 1\/\\alpha$ at $\\beta = 0$; all the solutions, where $\\mu$ can reach zero, converge to $\\tau = 1\/\\alpha$. \n\n\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=0.7\\textwidth]{classes.pdf}\n\\caption{(Color online) The peculiar $\\mu$-$\\tau$ phase diagram which represents solutions of the master equation \\eqref{mainEq} given a fixed value of the $\\alpha$ parameter and different value of the $\\beta$ parameter (see text for detail). }\n\\label{fig1}\n\\end{figure}\n\nThe character of the SR transition will be determined by the form of the solution of the equation \\eqref{mainEq} in the region $\\mu_s \\leq \\mu \\leq \\mu_f$. Let us analyze this equation starting with the simplest case $K_2 = 0$, i.e. $\\alpha = 1$, $\\beta = 0$. In this case, the main equation transforms into the molecular field equation well known in the basic theory of ferromagnetism:\n\\begin{equation}\\label{MolFieldEq}\n \\mu = \\tanh \\dfrac{\\mu}{\\tau} = B_{\\frac{1}{2}} \\left( \\dfrac{\\mu}{\\tau} \\right) ,\n\\end{equation}\nwhere $B_{1\/2}(x)$ is the Brillouin function. The equation has only one non-trivial solution at $0 \\leq \\tau \\leq 1$, $0 \\leq \\mu \\leq 1$, and the function $\\mu (\\tau)$ has the usual ``Weiss'' form. Thus, with the absence of the cubic anisotropy ($K_2 = 0$) in the ``single-doublet'' model the SR will be realized either through two second-order phase transitions at $\\mu_f \\leq 1$ (the complete spin-reorientation $G_x \\rightarrow G_z$), or through one second-order phase transition at $\\mu_f > 1$, but in this case the SR will be incomplete, i.e. it will end with a transition to the angular spin structure $G_{xz}$. The spin reorientation will begin at a temperature $T_s \\leq T_{SR}$ and $T_s$ is equal to $T_{SR}$ only in the case $\\mu_s = 0$ ($\\Delta_c = 0$), which can be realized in the general case only for Ising ions (e.g. Dy$^{3+}$ in DyFeO$_3$). For this type of ions, the temperature dependence of the ``order parameter'' $\\mu$ (in fact the splitting $\\Delta (\\theta)$ of the doublet) in a close range of $T_{SR}$ will be very sharp: $\\mu (T) \\sim (T - T_{SR})^{-1\/2}$. Nevertheless, the SR will be continuous and the temperature range of the SR $\\Delta T = T_s - T_f$ at $\\mu \\ll 1$ can theoretically reach arbitrarily small values.\n\nThus, the results of the rigorous analysis of the ``single-doublet'' model are fundamentally different from the conclusions of the simplified model (the ``high-temperature'' approximation), according to which for $K_2 = 0$ the spin reorientation always occurs as the first-order phase transition at $T = T_{SR}$.\n\nFor a positive second anisotropy constant ($K_2 > 0$, $\\beta > 0$), the main equation \\eqref{mainEq} has one non-trivial solution in the region $0 \\leq \\tau \\leq 1\/\\alpha$, $0 \\leq \\mu \\leq \\mu_0$ at $\\alpha > 0$, and one in the region $0 \\leq \\tau \\leq \\infty$, $ \\sqrt{|\\alpha\/\\beta|} \\leq \\mu \\leq \\mu_0$ at $\\alpha \\leq 0$, where $\\mu_0$ is determined from the solution of the equation $\\alpha \\mu_0 + \\beta \\mu_0^3 = 1$. The situation in this case is very similar to the previous one, i.e. the beginning of the SR will always be a second-order phase transition, and the reorientation will be complete ($G_x \\rightarrow G_z$) or incomplete ($G_x \\rightarrow G_{xz}$). Note that under the condition $(\\mu_f^2 - \\mu_s^2)\/(\\mu_f^2 + \\mu_s^2) \\geq \\gamma$, i.e. $\\alpha \\leq 0$, the width of the reorientation region becomes very large, even if $\\mu_s$ differs slightly from $\\mu_f$.\n \nFor Ising ions at $\\Delta_c = 0$, the SR beginning temperature is determined in exactly the same way as in the ``high-temperature'' approximation $T_s = T_{SR} \/ (1 + \\gamma)$.\n\nFor a negative second anisotropy constant ($K_2 < 0$, $\\beta < 0$), the several fundamentally different solutions of the main equation \\eqref{mainEq} are possible. For $K_2^* \\geq K_2$, where $K_2^*$ is determined from the condition $\\beta = -\\frac{1}{3} \\alpha^3$, i.e.\n\\begin{equation}\\label{ConditionForK*}\n \\dfrac{2 \\gamma}{\\mu_f^2 - \\mu_{s}^2} = -\\dfrac{1}{3} \\left( 1 - \\gamma \\dfrac{\\mu_f^2 + \\mu_{s}^2}{\\mu_f^2 - \\mu_{s}^2} \\right)^3,\n\\end{equation}\nthere is one non-trivial solution of the equation \\eqref{mainEq} in the region $1\/\\alpha \\leq \\tau < \\infty$, $\\mu \\leq \\sqrt{\\alpha \/ \\beta}$, but here $\\mu (T)$ decreases with decreasing temperature, i.e. $\\partial \\mu \/ \\partial \\tau > 0$. This solution is unstable and there is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the first-order phase transition.\n\nIn the intermediate range of values $K_2$ ($K_2^* < K_2 < 0$ or $-\\frac{1}{3} \\alpha^3 < \\beta < 0$) the main equation has two non-trivial solutions, and for one of them $\\partial \\mu \/ \\partial \\tau > 0$ (corresponding to bigger values of $\\mu$), and for the second $\\partial \\mu \/ \\partial \\tau < 0$ (corresponding to smaller values of $\\mu$). It is convenient to consider separately three areas of variation $\\beta$.\n\n1. $-\\frac{4}{27} \\alpha^3 < \\beta < 0$:\\\\\na) the first solution: $0 \\leq \\tau < \\infty, \\,\\, \\mu_\\text{\\scriptsize >} \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution: $0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq \\mu_\\text{\\scriptsize <}$,\\\\\nwhere $\\mu_\\text{\\scriptsize >}$, $\\mu_\\text{\\scriptsize <}$ are the bigger and smaller positive solution of the equation $\\alpha \\mu + \\beta \\mu^3 = 1$.\n\n2. $\\beta = -\\frac{4}{27} \\alpha^3$:\\\\\na) the first solution: $0 \\leq \\tau < \\infty, \\,\\, 3\/(2 \\alpha) \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution: $ 0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq 3\/(2 \\alpha),$\\\\\nmoreover, in this case we have a branch point of the main equation solution at $\\tau = 0$, $\\mu = 1$.\n\n3. $-\\frac{1}{3} \\alpha^3 < \\beta < -\\frac{4}{27} \\alpha^3$:\\\\\na) the first solution: $\\tau_0 \\leq \\tau < \\infty, \\,\\, \\mu_0 \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution: $\\tau_0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq \\mu_0$,\\\\\nwhere the quantities $\\mu_0$, $\\tau_0$ correspond to the branch points of the main equation solutions.\n\n\nIllustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple (quasi)doublet model are shown in Figure\\,\\ref{fig2}. The Figure\\,\\ref{fig2}a, built with $K_1 = 1,\\ \\gamma = 0.05,\\ \\Delta_a = 30.84,\\ \\Delta_c = 14.82$, which corresponds to $T_{SR} = 45.73,\\ \\mu_s = 0.162,\\ \\mu_f = 0.337,\\ \\tau_s = 1.04,\\ \\tau_f = 0.91$, describes a typical smooth SR transition\n with two second-order phase transitions $G_x-G_{xz}$ at the beginning ($\\tau_s$) and $G_{xz}-G_{z}$ at the end ($\\tau_f$) of the spin reorientation.\n \nThe Figure\\,\\ref{fig2}b, built with $K_1 = 1,\\ \\gamma = -0.1,\\ \\Delta_a = 33.19,\\ \\Delta_c = 27.1$, which corresponds to $T_{SR} = 22.95,\\ \\mu_s = 0.59,\\ \\mu_f = 0.72,\\ \\tau_s = 0.762,\\ \\tau_f = 0.93$, describes an abrupt first-order SR transition. For $\\tau > \\tau_f$ there is the $G_x$-phase, which can remain stable up to $\\tau_s$ when cooled. For $\\tau < \\tau_s$ there is the $G_z$-phase, which can remain stable up to $\\tau_f$ when heated. The point $A$ marks a phase transition point when the phases $G_x$ and $G_z$ have equal energies.\n\n\nThe Figure\\,\\ref{fig2}c, built with $K_1 = 1,\\ \\gamma = -0.222,\\ \\Delta_a = 6.72,\\ \\Delta_c = 1.63$, which corresponds to $T_{SR} = 2.65,\\ \\mu_s = 0.307,\\ \\mu_f = 1.266,\\ \\tau_s = 0.778,\\ \\tau_f = 0.523$ and the Figure\\,\\ref{fig2}d, built with $K_1 = 1,\\ \\gamma = -0.25,\\ \\Delta_a = 6.71,\\ \\Delta_c = 2.02$, which corresponds to $T_{SR} = 2.56,\\ \\mu_s = 0.396,\\ \\mu_f = 1.31,\\ \\tau_s = 0.73,\\ \\tau_f = 0.545$ describe unconventional \"mixed\" SR transitions. At $\\tau_s$ there is the smooth second-order phase transition $G_x-G_{xz}$. At $\\tau \\leq \\tau_f$ we have two stable phases $G_z$ and $G_{xz}$: at those temperatures the sharp first-order phase transition $G_{xz}-G_{z}$ can happen, or the system could stay in the angular $G_{xz}$-phase.\n\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=1.0\\textwidth]{4cases.pdf}\n\\caption{Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple (quasi)doublet model (see text for detail). The arrows indicate the direction of the antiferromagnetic vector $\\bf G$ in the $ac$ plane. The insets in panel (b) show the $\\theta$-dependence of the free energy.}\n\\label{fig2}\n\\end{figure}\n\nThus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range of intermediate values $K_2$ is the fundamental possibility of the existence of ``mixed'' SR transitions, in which the spins first smoothly rotate through a certain angle and then jump to the position with $\\theta = 0$. For this, it is sufficient that $\\mu_f$ corresponds to a point on the upper branch of solutions, and $\\mu_s$ to a point on the lower branch of solutions at $\\tau_f < \\tau_s$. In this case, the spin reorientation begins with the single second-order transition $G_x \\rightarrow G_{xz}$ and then ends with the first-order phase transition $G_{xz} \\rightarrow G_z$.\nIn contrast to the ``high-temperature'' approximation, the ``single-doublet'' model claims the nature of the phase transition is determined not simply by the sign of the second anisotropy constant, but also it depends on the ratio between $K_1$, $K_2$ and the doublet splitting in both phases. Nevertheless, if we apply the simplified model to describe the SR transition, we have to renormalize both the first and the second anisotropy constant, giving the last one sometimes a rather complicated temperature dependence, in particular with a change in sign when considering transitions of the ``mixed'' type. Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second constant.\n\n\n\\section{Conclusion}\nThe model of the spin-reorientation transitions induced by the $4f-3d$ interaction in rare-earth orthoferrites and orthochromites has been investigated. It is shown that both the temperature and the character of the spin-reorientation transition following from the solution of the transcendental equation \\eqref{mainEq} are the result of competition between the second and fourth order spin anisotropy of the $3d$ sublattice, the crystal field for 4f ions, and the $4f-3d$ interaction. At variance with the ``high-temperature'' approximation, the ``single-doublet'' model, along with typical smooth and abrupt SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order transition and a final abrupt first-order transition.\n\n\\ \\\\\n\\textbf{Funding:} The research was supported by the Ministry of Education and Science of the Russian Federation, project \u2116 FEUZ-2020-0054, and by Russian Science Foundation, project \u2116 22-22-00682.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\n\nCombinatorics is often of great importance in the study of the moduli space of stable curves of genus $g$, $\\overline M_g$.\nRecent examples are the combinatorial computation of its Euler characteristic \\cite{bodo}; \nthe combinatorial aspects in the study of the Nef cone of $\\overline M_g$ (see in particular question 0.13 of \\cite{GKM}). \nThe importance of combinatorics is evident also in the study of spin curves \\cite{CC} \\cite{CCC}.\n\nThe main result relating the geometry of $\\overline M_g$ to the combinatoric of stable curves is \n is the stratification of $\\overline M_g$ by topological type, \nwhich is governed by the {\\em weighted dual graph} associated to any stable curve \n (Definition \\ref{dwg}).\nTo any (multi)graph, \nit is naturally associated a group,\nwhich we will call {\\em complexity group} of the graph (Definition \\ref{cg}).\nIn particular, given a stable (or, more generally, nodal) curve $C$ we call $\\Delta_C$ the complexity group of the\ndual graph of the curve.\nThis group has been extensively studied as an invariant of graphs, with applications to\n Physics, Chemistry, and many other areas, and it goes under many different names, \n such as critical group \\cite{biggschip} \\cite{C-R}, determinant group \\cite{tatiana}, Picard group \\cite{tatiana}, Jacobian group \\cite{B-N}, \n abelian sandpiles group \\cite{C-E}.\nFrom the point of view of geometry, the complexity group was introduced in \\cite{cap}, with the name of {\\em degree class group},\nin order to describe and handle the fibres of the compactification of the universal Picard variety $\\overline P_{d,g}$ \nover $\\overline M_g$ (also constructed in the same article). \nMoreover, this group arises naturally in the study of the \ncompactified Jacobians of families of curves.\nMore precisely, let $R$ be a discrete valuation ring with fraction field $K$ and residue field ${\\mbox{\\itshape {\\textbf k}}}$ and denote by $B$ the spectrum of $R$. \nConsider a flat and proper regular curve $X\\longrightarrow B$, such that the closed fiber $X_{\\mbox{\\small \\itshape {\\textbf k}}}$\nis geometrically irreducible.\nUnder some technical assumptions, it is defined the group $\\Phi$ of connected\ncomponents of the the N\\'eron model of the Jacobian $J_K$ of the generic fiber $X_K$\n(see \\cite{BLR} sec. 9.6).\nNotice that in this definition the closed fiber doesn't need to be nodal;\nwhen $X_{\\mbox{\\small \\itshape {\\textbf k}}}$ is nodal, $\\Phi$ is precisely $\\Delta_{X_{\\mbox{\\small \\itshape {\\textbf k}}}}$.\nCaporaso in \\cite{capneron} gave a geometric counterpart of this construction, showing that, under the assumption that $(d-g+1, 2g-2)=1$, there exists a space over $\\overline M_g$ such that, for every regular family of stable curves over $B$, the N\\'eron model of the Picard variety of degree $d$ of $X_K$ is obtained by base change via the moduli map $B\\longrightarrow \\overline M_g$. A yet another geometric interpretation of $\\Phi$ is given by Chiodo in \\cite{chiodo}, where the $r$-torsion points of $\\Phi$ are described as N\\'eron models of $r$-torsion line bundles on $ X_K$.\n\n\n\nThe relationship between the structure of this group and the structure of the graph has been studied, among others, by Lorenzini\nin several papers (see for instance \\cite{lorgraphs} and \\cite{lorjac}).\nOther important references are \\cite{biggschip}, \\cite{tatiana} and \\cite{C-R}.\nWe addressed this problem in \\cite{BMS}, where in particular we constructed\na family of graphs with cyclic complexity group.\nIt seems that the question of finding a relation between the structure of the complexity group and the geometry of the curve \nis extremely difficult and intricate.\n\n\n\nRather that in the structure, we are interested here in the cardinality of the complexity group.\nKirkoff's Matrix Tree Theorem says that this \ninteger is the number of spanning trees of the dual graph of the curve, usually called the {\\em complexity}; \nthis is the reason for our notation, also suggested by L. Caporaso in \\cite{prag}. \nMoreover, it can be calculated \nas the determinant of a certain matrix (Theorem \\ref{MTT}, Remarks \\ref{spanningtrees} and \\ref{bourbaki}).\n\nWe shall call complexity of a curve the corresponding complexity of its dual graph.\nThe complexity being an upper semicontinuous function over $\\overline M_g$ (Lemma \\ref{usc}), \nit defines a weak stratification on $\\overline M_g$.\nIn Section \\ref{pre} we investigate the relationship between this stratification and the (strong) one given by topological type.\nThis relation is also enlighted using the list of possible graphs for curves of genus $3$ given in Section \\ref{lista}.\n\n\nIn particular, we are interested in the classes of curves with {\\em maximal complexity}. Define the function\n$\\psi(g):=\\mbox{max}\\{ |\\Delta_C|,\\,\\, C \\mbox{ stable curve of genus } g\\}$;\nour guiding problems are the following.\\\\\n\n1. Give a characterization of the curves $C$ such that $c(C)=\\psi (g)$, or at least with ``big'' complexity.\\\\\n\n2. Find bounds for $\\psi$, depending only on $g$.\\\\\n\nIn Section \\ref{MAX}, we give partial answers to problem 1, \nfinding necessary conditions for curves to have maximal complexity.\nWe show that these curves are all graph curves in the sense of \\cite{B-E} i.e., they have simple trivalent dual graph \n$\\Gamma$ with $b_1(\\Gamma)=g$; moreover, they have no disconnecting nodes (Theorem \\ref{maxconf} and its combinatorial version Theorem \\ref{combinmaxconf}).\nUsing this reduction, we can apply some results of the immense literature regarding the complexity of \nregular graphs.\n\nIn Section \\ref{ub} we give an answer to our second problem, \n using results of Biggs \\cite{biggs}, McKay \\cite{McK} and Chung-Yau \\cite{C-Y}.\nIn particular, we obtain an asymptotically sharp upper bound for $g\\gg 0$.\nBy what observed in the beginning, these bounds limit the number of connected components of the \nN\\'eron model of the generalized Jacobian of stable curves. \nNote that this result is different from the one given by Lorenzini in \\cite{lorner}: see Remark \\ref{confrontoL}.\nIn Section \\ref{lower}, we provide an example of families of graphs corresponding to stable curves with increasing genus,\nand compute explicitly the complexity depending on the genus, thus giving an explicit lower bound.\n\nIn the last section we discuss the conjectural behavior of the curves with maximal complexity.\nIn particular, it seems that the graph of such a curve should have maximal connectivity (which is 3 in this case).\nMoreover, it seems that the girth has to be big (Conjecture \\ref{connectivity} and \\ref{girth}).\nWe prove in this section some partial results that seem to support the conjectures (Proposition \\ref{bah} and Corollary \\ref{g<10}).\nMoreover, we prove a uniform necessary condition holding for any sequence of graphs of curves with maximal complexity\n(Theorem \\ref{cicli}), using again a result of McKay.\n\nEven though our point of view is irreparably geometrical, our main results are proven with (very simple) combinatorics methods, \nand can be rewritten in purely combinatorial terms (see in particular \\ref{combi}).\nWe would like to express the hope that our geometric approach does not discourage non-geometers from the reading of the article;\nand in particular from considering the list of open question given in section \\ref{congetture}.\n\n\n\\medskip\n\n\\noindent {\\bf Acknowledgments}\n\nWe would like to express our gratitude to L. Caporaso for introducing us the problems we are dealing with in the present paper and for following this work very closely\nand to C. Casagrande for the important corrections and suggestions she gave us.\n\nWe also thank A. Machi for transmitting us much encouragement after reading a preliminary version.\n\n\n\n\\section{Combinatorial invariants of nodal curves}\\label{pre}\n\n\nWe work over an algebraically closed field $\\mbox{\\itshape {\\textbf k}} =\\overline \\mbox{\\itshape {\\textbf k}}$.\nA {\\em nodal} curve is a reduced curve which has only ordinary double points as singularities.\n\n\\subsubsection*{The dual graph of a curve}\n\\begin{defi}\\label{dwg}\nTo a nodal curve $C$ we can associate a graph $\\Gamma_C$,\ncalled the \\textit{weighted dual graph}, \ngiven by a triple $(V,E,g)$, where \n$V$ is the set of vertices, $E$ the set of edges, and $g$ a function on the set $V$ with non-negative integer values.\nThis triple is defined in the following way\n\\begin{itemize}\n\\item to each irreducible component $A$ corresponds a vertex\\footnote{\nSometimes in the literature the vertices of a graph are denoted ``nodes''; of course we will never adopt this notation\nwhich is extremely confusing in our context. A node will always be for us an ordinary double point of a curve!} \n$v_A$; \n\\item to each node intersecting the components $A$ and $B$ (where $A$ and $B$ can coincide) \ncorresponds an edge connecting the vertices $v_A$ and $v_B$;\n\\item $g: V\\longrightarrow \\mathbb Z_{\\geq 0}$ is \n the function that associates to any vertex $v$ the geometric genus of the corresponding component of $C$.\n\\end{itemize}\n\\end{defi}\nCall $\\gamma$ the number of irreducible components of $C$ and \n$\\delta$ the number of nodes.\nThus $\\Gamma_C$ has $\\gamma$ vertices, $\\delta$ edges, \nand among the edges there is a loop \nfor every node lying on a single irreducible component of $C$.\nThe weighted graph encodes all the topological \ninformation about the curve.\nOf course, conversely, any weighted graph can be realized as the dual graph of a nodal curve.\n\n\n\n\n\nIt is important to stress that dual graphs of nodal curves can have more than one edge connecting two nodes, and can also have loops.\nThese are usually called {\\em multigraphs}. \nIn this paper, by graph we will always mean multigraph, while a graph without loops and multiple edges will be called {\\em simple}.\n\nGiven a graph $\\Gamma$ with $\\delta$ edges, $\\gamma$ vertices and $c$ connected components, its first Betti number is\n$b_1(\\Gamma):= \\delta-\\gamma+c$; it corresponds to the number of independent cycles on $\\Gamma$.\n\nLet $\\{C_1,\\ldots,C_{\\gamma}\\}$ be the set of irreducible components of a nodal curve $C$ \nwith $\\delta$ nodes and $c$ connected components.\nRecall that the arithmetic genus of $C$ can be computed by the following formula\n\\begin{equation}\\label{genere}\ng(C)=\\sum_{i=1}^{\\gamma}g(C_i)+\\delta-\\gamma+c=\\sum_{v\\in V(\\Gamma_C)}g(v)+b_1(\\Gamma_C).\n\\end{equation}\nBy analogy, for any weighted graph $\\Gamma$ given by the triple $(V,E,g)$, we will call \n$g(\\Gamma):=\\sum_{v\\in V}g(v)+b_1(\\Gamma)$ the {\\em (arithmetic) genus of the graph}.\n\n\n\n\n\\subsubsection*{Complexity group} \n\nLet us consider a connected nodal curve $C$.\nLet $\\left\\{C_i\\right\\}_{i=1,...,\\gamma}$ be the irreducible components of $C$. \nDefine \n$$\n\\begin{matrix}k_{ij}:= & \\left\\{\\begin{array}{l}\\;\\;\\, \\sharp(C_i\\cap C_j) \\, \\mbox{ if }i\\not= j\\\\ \\\\\n-\\sharp(C_i\\cap \\overline{C\\setminus C_i}) \\, \\mbox{ if } i=j\\\\\\end{array}\\right.\\end{matrix}\n$$\nAs $C_i\\cap \\overline{C\\setminus C_i}=\\bigcup_{j\\not= i} C_i\\cap C_j$, we have that, for fixed $i$, $\\sum_j k_{ij}=0$.\nFor every $i$ set \n$$\\underline{c}_i:=(k_{1i},\\ldots,k_{\\gamma i}) \\in \\mathbb Z^\\gamma.$$ \nCall $Z:=\\{\\underline{z}\\in\\mathbb{Z}^\\gamma : |\\underline z|=0\\}$.\nAs observed before, $\\underline c_i\\in Z$. Let us call $\\Lambda_C$ the sublattice of $Z$ spanned by \n$\\{\\underline c_1,\\ldots,\\underline c_\\gamma\\}$. In fact, $\\Lambda_C$ is a lattice in $Z$ (it\nhas rank $\\gamma -1$) as we will show in a moment (see \\cite{capneron} for a geometric proof of this fact).\n\n\\begin{defi}\\label{cg}\nThe complexity group of $C$ is the finite abelian group $\\Delta_C:=Z\/\\Lambda_C$.\n\\end{defi}\nIt is important to notice that this group depends {\\em only} on the dual (non-weighted) graph\nof the curve; clearly it is defined for any connected\\footnote{The definition could be easily extended to \nnon connected graphs\/curves.} graph.\nAs noted in the Introduction, this group arises in many contexts where graphs are used,\nand it is known with many other names.\n\n\n\nLet $M$ be the $\\gamma\\times\\gamma$ matrix whose columns are the $\\underline c_i$'s. \nWe will call $M$ the {\\em intersection matrix}, the name clearly deriving from its geometrical meaning. \nHowever, in literature, $M$ is known as the {\\em (combinatorial) Laplacian} matrix (cf. e.g. \\cite{bollobas} and \\cite{lorfinite}). \nIt is obtained from the so-called adjacency matrix of $\\Gamma_C$ subtracting the vertex degrees on the diagonal.\n\n\n\n\n\\subsubsection*{Complexity of a graph}\n\n\nA tree is a connected graph $G$ with $b_1(G)=0$.\nLet $\\Gamma$ be a graph. A {\\it spanning tree} of $\\Gamma$ is a subgraph of $\\Gamma$ which is a tree having \nthe same vertices as $\\Gamma$. \n\\begin{defi}\nThe complexity of $\\Gamma$, indicated by the symbol $c(\\Gamma)$, is the number of spanning trees contained \nin $\\Gamma$ (see e.g. \\cite{biggs}, sec 6, \\cite{berge}, cap.3 $\\natural$ 5, \\cite{west}, sec 2.2).\n\\end{defi}\n\n\nObserve that $c(\\Gamma)=0$ if and only if $\\Gamma$ is not connected, \nand that if $\\Gamma$ is a connected tree $c(\\Gamma)=1$.\nFor the complexity of the dual graph associated to a curve $C$, we will often use the symbol $c(C)$, instead of\n$c(\\Gamma_C)$.\nThe following theorem, known as Kirkoff's Matrix Tree Theorem, will be a key ingredient for our work. \nThere are at least three different proofs of this result; see \\cite{BMS} for a proof and for the references.\n\n\\begin{teo}{\\upshape (Matrix Tree Theorem)}\\label{MTT}\nLet $s, t \\in \\{1,\\ldots \\gamma\\}$.\nUsing the above notations, if $M^\\star$ is obtained from $M$ by deleting the $t$-th column and the $s$-th row, then \n$$c(\\Gamma)=(-1)^{s+t+\\gamma -1}\\mbox{det}(M^\\star).$$\n\\end{teo}\n\n\nIn particular, the Matrix Tree Theorem assures that, in the case of the \ndual graph of a {\\em connected} curve $C$, the matrix $M$ has rank $\\gamma-1$\ni.e., $\\Lambda_C$ is indeed a lattice.\n\n\\begin{rem}\\label{spanningtrees} \\upshape{\nFor $r\\in\\{1,\\ldots,\\gamma\\}$, consider the isomorphism $\\alpha_r:Z\\stackrel{\\sim}\\longrightarrow\\mathbb{Z}^{\\gamma - 1}$\nwhich consists of deleting the $r$-th component.\nThe group $\\Delta_C$ is the quotient of $\\mathbb{Z}^{\\gamma - 1}$ by the lattice generated by \n$$\\underline c_i^\\prime:=(k_{1i}\n,\\ldots,\\widehat{k_{ri}},\\ldots,k_{\\gamma i}).$$\nObserve that again $\\sum_i \\underline c_i^\\prime = \\underline 0 \\in \\mathbb{Z}^{\\gamma -1}$.\nTherefore $\\Delta_C$ is presented by the matrix $M^\\star$ obtained from $M$ by deleting a column and the $r$-th row.\nHence, we can compute its cardinality via Theorem \\ref{MTT}\n$$c(\\Gamma_C)=|\\Delta_C|=|\\mbox{det}(M^\\star)|.$$\nSo, we can conclude that {\\em the cardinality of the complexity group of a curve $C$ is the complexity of the dual graph $\\Gamma_C$.}}\n\\end{rem}\n\n\\begin{rem}\\label{bourbaki}\\upshape{\nBy the diagonalization theorem of integer matrices (i.e. the structure theorem for finite abelian groups, \nsee \\cite{ar}), $M$ is equivalent over $\\mathbb Z$ to a diagonal matrix\n$diag (d_1, d_2,\\ldots, d_\\gamma)$, where the $d_i's$ are the invariant factors, i.e. the \nnon-negative integers obtained in the following way:\nLet $D_i$ be the greatest common divisor of the $i\\times i$ minors of $M$. \nThen $d_i=D_i\/D_{i-1}$\n(cf. also \\cite{lorarithmetical}, Theorem 1.5). \nNote that $d_\\gamma=0$ and $d_i>0$ for $i\\not = \\gamma$, so \n$\\Delta_C=\\oplus _{i=0}^{\\gamma-1}\\mathbb Z\/d_i \\mathbb Z $.\nIn particular, $ |\\Delta_C|$ is equal to the greatest common divisor of all the $(\\gamma-1)\\times(\\gamma-1)$ \nminors of $M$. \n\nOn the other hand, if we diagonalize $M$ over the real numbers, we get real eigenvalues\n$0=\\lambda_1>\\lambda_2\\geq \\ldots\\geq \\lambda_\\gamma$. \nThese eigenvalues are deeply studied in Combinatorics.\nNote that in general the $\\lambda_i$'s have no relation with the invariant factors, not even if they happen to be integers; \na nice counterexample can be found in Section 9.2 of \\cite{tatiana}.\nAn attempt to construct the invariant factors from the $\\lambda_i's$, for a particular class of graphs, can be find in \\cite{C-R}.\n\nLet us note that in particular $ | \\Delta_C|=\\gamma^{-1}\\lambda_2\\lambda_3\\ldots\\lambda_\\gamma$ \n(cf. \\cite{biggs} cor.6.5).\n}\n\\end{rem}\n\n\n\n\n\\subsubsection*{Stable curves and their graphs}\n\n\\begin{defi}\\label{stable}\nA curve $C$ of genus $g\\geq 2$ over $\\mbox{\\itshape {\\textbf k}}$ is stable (resp. semistable) if it is nodal, connected and such that if \n$D\\subset C$ is a smooth rational component, then $|D\\cap\\overline{C\\setminus D}|\\geq 3$ (resp. $\\geq 2$).\n\\end{defi}\n\nThe moduli space of stable curves of genus $g$, $\\overline M_g$, is a projective variety of dimension $3g-3$, the so-called Deligne-Mumford compactification of the moduli space of smooth curves of genus $g$, $M_g$.\nThe theory of stable curves was first introduced by A. Mayer and D. Mumford and was first developed in \\cite{DM} in order to prove the irreducibility of $M_g$ in any characteristic. In that paper, the authors prove the main properties of stable curves used later by D. Gieseker in \\cite{gie} to establish the existence of $\\overline M_g$.\n\n\\medskip\n\nGiven a graph $\\Gamma$ and a vertex $v$, the degree $d(v)$ of $v$ is the number of half edges touching $v$.\nThe combinatorial version of the definition of a stable curve is the following.\n\n\\begin{defi}\\label{cstable}\nA weighted graph $\\Gamma$ of arithmetic genus $g\\geq 2$ is stable if\n\\begin{equation}\\label{stability}\n2g(v)-2+d(v)>0 \\,\\,\\,\\,\\,\\,\\mbox{ for any }\\,\\,\\,\\,\\,v\\in V.\n\\end{equation}\n$\\Gamma$ is said to be semistable if the inequality above holds with $\\geq$.\n\\end{defi}\n\n\n\n\\subsubsection*{Topological stratification of $\\overline M_g$}\n\nThere is a natural stratification on $\\overline M_g$ given by topological type.\nEach stratum of codimension $k$ is the subset of $\\overline M_g$ \nconsisting of classes of stable curves with weighted graphs of genus $g$ having exactly $k$ edges.\nIn particular, the stratum of codimension $0$ is the open set $M_g\\subset \\overline M_g$ of smooth curves.\nThe strata of codimension $1$ are $[g\/2]+1$, and the corresponding graphs are of the following types.\n\n\\bigskip\n\n\\centerline{\n\\xymatrix@=.2pc{\n*{\\bullet} \\ar@{-}^<{g-1} @(ur,dr)\\\\{}\n}\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n}\n\n\\bigskip\n\\centerline{\n\\xymatrix@R=.1pc{\n*{\\bullet} \\ar@{-}^<{i}^>{g-i}[r] & *{\\bullet}\n\\\\{}\n}\n\\, for $i=1,\\ldots, [g\/2]$.\n}\n\n\\bigskip\n\n\\noindent The closure of these strata are effective divisors on $\\overline M_g$, the so-called boundary divisors,\nusually denoted by $\\Delta_i$, for $i=0,\\ldots, [g\/2]$, according to the preceding list.\nFor instance, $\\Delta_0$ has as generic point corresponding to an irreducible curve of geometric genus $g-1$ \nwith exactly one node.\nIt can be described as the locus \nof curves having at least one non-disconnecting node.\nThe boundary of $\\overline M_g$ is $\\partial \\overline M_g=\\overline M_g\\setminus M_g=\\cup_{i=0}^{[g\/2]}\\Delta_i$.\n\nThe union of the strata of codimension $k$ is the variety of isomorphism classes of stable curves having exactly $k$ nodes; \nits closure is the locus of curves with at least $k$ nodes.\n\n\n\\subsubsection*{Degeneration of nodal curves}\n\nThe following well-known result describes the possible transformations on a dual graph corresponding to the degenerations of a nodal curve.\n\n\\begin{prop}\\label{degenerations}\nLet $\\Gamma=(V,E,g)$ be a weighted graph. \nLet $C$ be a nodal curve having $\\Gamma$ as dual weighted graph.\nThe dual graphs of the possible nodal curves obtained by degenerations of $C$ \nare obtained from $\\Gamma$ via a sequence of the following\ntransformations:\n\\begin{itemize}\n\\item[(I)] given a vertex $v$ such that $g(v)\\geq 1$ add a loop on $v$ and decrease its weight by 1;\n\\item[(II)] given a vertex $v$, and given two nonnegative integers $a$ and $b$ such that $a+b=g(v)$, substitute it with \ntwo vertices $v_a$ and $v_b$ with weights respectively $a$ and $b$ and one edge $l$ connecting them.\n\\end{itemize}\n\\end{prop}\n\\noindent The figure below illustrates operation (II).\n\n\n\\vspace{0.5cm}\n\\centerline{\n\\begin{tabular}{cc}\n{}\n&\n\\xymatrix@=1pc{\n& &\\ar@{-}[dl]_>v &&&& &&&&\\ar@{-}[dl]\\\\\n& *{\\bullet} && \\ar@{:>}[rr] && & &*{\\bullet} \\ar@{-}[rr]^l_<{v_a}_>{v_b}&& *{\\bullet}& \\\\\n\\ar@{-}[ur]& &\\ar@{-}[ul] &&&& \\ar@{-}[ur] &&&&\\ar@{-}[ul]\n}\n\\end{tabular}}\n\\vspace{0.5cm}\n\n\n\n\\noindent Of course, there are many ways to perform operation (II) (see also Definition \\ref{operazione} below).\nNotice that this is the opposite operation of contracting an edge, in the sense that if we contract the \nnew edge $l$ we get the original graph.\n\nNote also that if our given curve $C$ is stable, degenerations of $C$ obtained by performing operation (I) are still stable, while if we perform operation (II) this is the case only if\n$2a-2+d(v_a)\\geq 1$ and $2b-2+d(v_b)\\geq 1$.\n\n\n\n\n\n\n\\subsubsection*{Polygonal curves}\n\nWe present here an elementary combinatorial proof of the following well-known fact for stable curves. \nA geometric proof can be obtained using the above results about the topological stratification of $\\overline M_g$.\n\\begin{lem} \\label{lemcomb}\nLet $C$ be a stable curve of genus $g\\ge 2$. \nThen\n\\begin{enumerate}\n\\item $C$ has at most $3g-3$ nodes and $2g-2$ irreducible components.\n\\item Assume that $C$ has $3g-3$ nodes. Then $C$ has $2g-2$ components $C_1,\\ldots,C_{2g -2}$ and, \nif $\\nu_i:C_i^\\nu\\longrightarrow C_i$ is the normalization of $C_i$, then \n$C_i^\\nu\\simeq \\mathbb{P}^1$ and $|\\nu_i^{-1}(C_i\\cap C_{sing})|=3$ for all $i$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nLet $\\Gamma=(V,E,g)$ be the weighted graph associated to $C$.\nNote that $\\sum_{v\\in V}d(v)=2\\delta$, hence we can rewrite formula (\\ref{genere}) as\n\\begin{equation}\\label{genere2}\ng=\\sum_{v\\in V}\\left(g(v)+\\frac{d(v)}{2}\\right)-\\gamma +1.\n\\end{equation}\nBy the connectedness of $C$, $d(v)\\geq 1$ for any $v$. \nUsing also the stability condition, we have that \n$\ng\\geq \\sum_{v\\in V}\\frac{3}{2}-\\gamma +1=\\frac{\\gamma}{2}+1,\n$\nhence, $\\gamma \\leq 2g-2$. Now, using (\\ref{genere}) again, we have that $\\delta \\leq 3g-3$, and (1) follows.\n\nNow, if $\\delta =3g-3$, again by (\\ref{genere}), we see that necessarily $\\gamma =2g-2$ and \n$g(v)=0$ for any $v\\in V$.\nMoreover, using again (\\ref{genere2}), we see that $d(v)=3$ for any vertex $v$, so also (2) is proved.\n\\end{proof}\n\nStable curves with $3g-3$ nodes (and $2g-2$ components), \nare the $0$-strata of the topological stratification of $\\overline M_g$; \nthey are rigid, in the sense that \nany deformation of such curves in a family of stable curves must have necessarily at least one node smoothed. \nIndeed, from Proposition \\ref{degenerations} we see that both operations (I) and (II) cannot be performed, \ni.e. there is no possible further degeneration for such curves.\nWe will call such curves {\\em polygonal curves}\n(also called large limit curves, see \\cite{tyurin}).\nPolygonal curves with simple graph are called graph curves (\\cite{B-E}).\n\nWe refer to \\cite{B-E} for an ample discussion on the importance and the properties of these curves.\nLet us just make a couple of remarks. \nThe associated dual graphs are trivalent (multi)graphs such that the weight function $g$ is $0$ on every vertex.\nDue to the fact that $\\mathbb P^1$ with $3$ fixed points has no non-trivial automorphisms, \nthe automorphism group of such curves coincides with the automorphism group of the graph.\nArtamkin in \\cite{artamkingen} has given a recursive rule that computes the number\n$$\n\\sum_{\\begin{array}{c}\\Gamma \\,\\,\\,trivalent,\\\\\nb_1(\\Gamma)=g\n\\end{array}}\n\\frac{1}{|Aut(\\Gamma)|} \\mbox{ ,}\n$$\nwhich is the ``stacky top self-intersection'' of the boundary divisor $\\partial \\overline M_g$.\n\n\n\\subsubsection*{Stratification by complexity of $\\overline M_g$}\n\n\nWe can define a function $c\\colon \\overline M_g\\longrightarrow \\mathbb Z_{\\geq 0}$ \nassociating to every $[C]\\in \\overline M_g$ its complexity $c(C)$.\n\nOf course, $c$ is bounded from above, i.e., there is a bound for the complexity of a stable curve of given genus;\nindeed, by Lemma \\ref{lemcomb}, there is only a finite number of possible graphs.\nIn section \\ref{ub} we provide upper bounds, which are asymptotically sharp for $g\\gg 0$.\n\\begin{rem}\\upshape{\nClearly, this wouldn't make sense for nodal curves (not even for semistable ones). \nIndeed, blowing up a node an arbitrary number of times doesn't change the genus of the curve, \nbut it can increase arbitrarily the complexity of a nodal curve (see for instance \\cite{BMS} Proposition 3.3.).}\n\\end{rem}\n\n\n\\begin{lem}\\label{usc}\nThe function $c\\colon \\overline M_g\\longrightarrow \\mathbb Z_{\\geq 0}$ is upper semi-continuous.\n\\end{lem}\n\\begin{proof}\nFollows from Proposition \\ref{increasing} below.\n\\end{proof}\nThis result implies in particular that we could define a stratification ``by complexity'' of $\\overline M_g$.\nThis is clearly a much rougher stratification than the one by topological type; \n{\\em the set of curves in $\\overline{M}_g$ with given complexity is a (maybe empty) union of components of different codimension strata \nof the topological type stratification, of different codimension} (see the case of genus $3$ in the next section).\n\n\nFor instance, the set $\\overline M_g^{c=1}$ of curves with complexity one is the set of curves whose dual graph is a tree with loops.\nIt contains the curves of compact type $\\overline M_g^{ct}=\\overline M_g\\setminus \\Delta_0$, but also the interior of $\\Delta_0$,\nand other strata of bigger codimension; it also contains $0$-strata, i.e. isolated points (see next section).\nThe complement $\\overline M_g\\setminus \\overline M_g^{c=1}$ of curves with complexity greater than one is a closed \nsubset of codimension $2$.\n\n\n\n\n\n\\section{List of graphs for $\\overline M_3$}\\label{lista}\n\nIn this section, we list all the possible weighted graphs for stable curves of genus 3, \nas well as their complexity and their complexity group. \nThis list (with one error, now corrected, which was kindly pointed out to us by A. Chiodo) appeared in \\cite{BMS}.\nWe will use $\\mathbb{Z}_n$ to denote the quotient group $\\mathbb{Z}\/n\\mathbb{Z}$. \nThe graphs are ordered by increasing the number of nodes (i.e. according to the codimension of corresponding \nstratum of the stratification by topological type).\nIn the graphs we will indicate the weight of each vertex only if it is not zero.\nRecall that if $C$ is a stable curve of genus 3, then $C$ has at most $6$ nodes and $4$ components.\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline\n\\xymatrix@=.2pc{\n{}\\\\\n*{\\bullet}\\\\{}\n\\scriptstyle{3}}\n&\n\\xymatrix@=.001pc{\n\\\\\n0}\n&\n\\xymatrix@=.001pc{\n\\\\\n1}\n&\n\\xymatrix@=.001pc{\n\\\\\n1}\n&\n\\xymatrix@=.001pc{\n\\\\\n0}\\\\\n\n\\xymatrix{\n*{2 \\,\\bullet} \\ar@{-}@(ur,dr) & {}\n}\n&\n1\n&\n1\n&\n1\n&\n0 \\\\\n&&&&\\\\\n\n\\xymatrix{\n*{2\\, \\bullet} \\ar@{-}[r] & *{\\bullet \\, 1}\n}\n&\n1&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}_<{1}@(ul,dl) \\ar@{-}@(dr,ur)\n}\n&\n2&1&\n1\n&\n0\n\\\\\n&&&&\\\\\n\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]^<{1}^>{1}& *{\\bullet}\n}\n&\n2&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]^>{2}& *{\\bullet}\n}\n&\n2&2&\n1\n&\n0\n\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_\/[r]_>{1} & *{\\bullet} \\ar@{-} @\/_\/[l]_>{1}\n}\n&\n2&2&\n2\n&\n$\\mathbb{Z}_2$\\\\\n&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}[r]^<{1}^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n2&3&\n1\n&\n0\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline \n{}&&&&\\\\\n{}&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}@(ur,dr) \\ar@{-}@(ur,ul)\\\\\n{}\n}\n&\n3&1\n&\n1&\n0\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}@(dr,dl) \\ar@{-}[r]^>{1}& *{\\bullet}\\\\\n{}\n}\n&\n3&2&\n1\n&\n0\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]_<{1} & *{\\bullet} \\ar@{-} @(ur,dr)\n}\n&\n3&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}_<{1}@\/_\/[l]\n}\n&\n3&2&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}_>{1} @\/_\/[l]\n}\n&\n3&2&\n3\n&\n$\\mathbb Z_3$\\\\\n&&&&\\\\\n\\xymatrix@R=.5pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r] & *{\\bullet}\\ar@{-}[r]^<{1}^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n3&3\n&\n1\n&\n0\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r]^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n3&3&\n1&\n0\\\\\n\\xymatrix{\n *{\\bullet} \\ar@{-}^<{1}[r]& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}_<{1}@\/_\/[l]\n}\n&\n3&3&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}@(dr,dl) \\ar@{-}[r]& *{\\bullet} \\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n4&2&\n1&\n0\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)\n}\n&\n4&2&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}@\/_\/[l]\n}\n&\n4&2&\n3\n&\n$\\mathbb Z_3$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_.3pc\/[r] \\ar @{-}@\/_.8pc\/[r]\/& *{\\bullet} \\ar@{-}@\/_.3pc\/[l] \\ar@{-}@\/_.8pc\/[l]\n}\n&\n4&2&\n4\n&\n$\\mathbb Z_4$\\\\\n&&&&\\\\\n\\xymatrix@R=.5pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r] & *{\\bullet}\\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n4&3\n&\n1\n&\n0\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r]^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n4&3&\n1&\n0\\\\\n\\xymatrix@R=.1pc{\n{} & *{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}_<{1}@\/_\/[l]\\\\\n{}\n}\n&4&3\n&\n2\n&\n$\\mathbb Z_2$\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}^<{1}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n4}\n&\n\\xymatrix@=.3pc{\n\\\\\n3}\n&\n\\xymatrix@=.3pc{\n\\\\\n5\\\\\n{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n{\\mathbb Z_5}\n}\n\\\\\n\\xymatrix@R=.8pc{\n{} & *{\\bullet} \\ar@{-}^<{1}[r]& *{\\bullet} \\ar@{-}[r] \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l]&\n}\n&\n4&3&\n3\n&\n$\\mathbb Z_3$\n\\\\\n\\xymatrix@=.5pc{\n*{\\bullet}\\ar@{-}^<{1}[dr]&&&\\\\\n&*{\\bullet}\\ar@{-}[rr]&&*{\\bullet}\\ar@{-}@(ur,dr)\\\\\n*{\\bullet} \\ar@{-}_<{1}[ur]&&\n}&\n\\xymatrix@=.3pc{\n\\\\\n4\n\\\\\n{}\n\\\\\n{}}\n&\\xymatrix@=.3pc{\n\\\\\n4}\n&\n\\xymatrix@=.3pc{\n\\\\\n1}\n&\n\\xymatrix@=.3pc{\n\\\\\n0}\n\\\\\n\\xymatrix@R=.3pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r] & *{\\bullet}\n\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n5&3&\n1&\n0\\\\\n\\xymatrix@R=.3pc{\n{}&&\\\\\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet} \\ar@{-}@\/^\/[r] \\ar@{-}@\/_\/[r] & *{\\bullet}\n\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n5}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n2}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n\\mathbb{Z}_2}\\\\\n\n\\hline \\hline\n\\end{tabular}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline \n{}&&&&\\\\\n\\xymatrix@R=.2pc{\n{}\\\\\n{} & *{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar@{-}[r] \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l]&\\\\&&&&\n}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n5}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n\\mathbb Z_3}\n\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}@\/_\/[dl] \\ar@{-}@\/^\/[dr] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] & & *{\\bullet}\\\\{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n5}\n&\n\\xymatrix@=.3pc{\n\\\\\n3}\n&\n\\xymatrix@=.3pc{\n\\\\8\n\\\\\n{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n{\\mathbb Z_8}}\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr] \\ar@{-}@(ul,ur)& \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\n\\xymatrix@=.5pc{\n\\\\5}\n&\n\\xymatrix@=.5pc{\n\\\\3\n\\\\\n{}\n}\n&\n\\xymatrix@=.5pc{\n\\\\5\n\\\\\n{}\n}\n&\n\\xymatrix@=.5pc{\n\\\\\n{\\mathbb Z_5}}\n\\\\\n\\xymatrix@=1.2pc{\n&*{\\bullet} \\ar@{-}^<{1}[d]&\\\\\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr] & \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n5\n}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n4}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n5\n\\\\\n{}\\\\{}\n}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n{\\mathbb Z_5}}\\\\\n\\xymatrix@R=.2pc{\n{} & *{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l] \\ar@{-}^>{1}[r]& *{\\bullet}\n\\\\&&&&\n}\n&5&4&\n2\n&\n$\\mathbb Z_2$\n\\\\\n\\xymatrix@=.5pc{\n*{\\bullet}\\ar@{-}^<{1}[ddd]&&&\\\\\n&&*{\\bullet}\\ar@{-}@(ur,dr)&\\\\\n&&&\\\\\n*{\\bullet}\\ar@{-}[rrr]\\ar@{-}[rruu]&& &*{\\bullet}\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\\xymatrix{\n\\\\5}\n&\\xymatrix{\n\\\\4}\n&\n\\xymatrix{\n\\\\1}\n&\n\\xymatrix{\n\\\\\n0}\n\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l] \\ar@{-}[r]& *{\\bullet} \\ar@{-}@(ur,dr)\n\\\\{}\n}\n&6&4&\n2\n&\n$\\mathbb Z_2$\n\\\\\n\\xymatrix@=1pc{\n*{\\bullet} \\ar@{-}[dd]\\ar@{-}@\/_.5pc\/[dd]\\ar@{-}[drr]&&&\\\\\n&&*{\\bullet}\\ar@{-}[r]&*{\\bullet} \\ar@{-}@(ur,dr)\\\\\n*{\\bullet} \\ar@{-}[urr]&&&\n}\n&\n\\xymatrix@=.6pc{\n\\\\6}\n&\\xymatrix@=.6pc{\n\\\\4}\n&\n\\xymatrix@=.6pc{\n{}\\\\\n5\n\\\\\n{}\\\\{}\n}\n&\n\\xymatrix@=.6pc{\n{}\\\\\n{\\mathbb Z_5}}\\\\\n\n\\xymatrix@=.5pc{\n&*{\\bullet}\\ar@{-}[ddd] \\ar@{-}@(ur,ul)&\\\\\n*{\\bullet} \\ar@{-}@(ul,dl)&&*{\\bullet}\\ar@{-}@(ur,dr)&\\\\\n&&&\\\\\n&*{\\bullet}\\ar@{-}[ruu]\\ar@{-}[luu] & \\\\\n{}\n}\n&\\xymatrix@=.5pc{\n\\\\6}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\1}\n&\n\\xymatrix@=.5pc{\n\\\\\n0}\n\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}[d] \\ar@{-}@\/_\/[d] & *{\\bullet} \\ar@{-}[d] \\ar@{-}@\/^\/[d]\\\\\n*{\\bullet} \\ar@{-}[r] & *{\\bullet}\n}\n&\\xymatrix@=.5pc{\n\\\\6}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\12}\n&\n\\xymatrix@=.5pc{\n\\\\\n\\mathbb Z_{12}\n}\n\\\\\n\\xymatrix@=1.2pc{\n&*{\\bullet}\\ar@{-}[dd] \\ar@{-}[dl] \\ar@{-}[dr] &\\\\\n*{\\bullet} \\ar@{-}[rr] |!{[r]}\\hole & &*{\\bullet}\\\\\n&*{\\bullet}\\ar@{-}[ru]\\ar@{-}[lu] & \n}\n&\\xymatrix@=.5pc{\n\\\\6\\\\\n{}}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\16}\n&\n\\xymatrix@=.5pc{\n\\\\\n\\mathbb Z_4\\times\\mathbb Z_4}\n\\\\\n\n\\hline \\hline\n\\end{tabular}\n\\end{equation*}\n\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\noindent Note that the set of curves of genus $3$ with complexity $1$ contains at least one stratum \nof the topological stratification of any codimension.\nThe set of curves with complexity $2$ contains at least one stratum among the ones of codimension\ngreater or equal to $2$.\nThe set of curves with complexity $6$ is empty.\n\n\n\n\n\n\n\n\\section{Stable curves with maximal complexity}\\label{MAX}\n\nWe will now focus our interest in curves with maximal complexity.\nWe can see that in case $g=3$ above, the curves reaching the maximal complexity are polygonal curves. \nHowever, as already observed, there are also polygonal curves with small, even $0$, complexity.\nIn what follows we prove that, indeed, any curve with maximal complexity is necessarily a polygonal curve\nand, moreover, it is a graph curve without disconnecting nodes.\nIn Section \\ref{congetture} we shall make some remarks and conjectures on sufficient conditions.\n\nWe shall need the following well-known result, whose proof is elementary.\n\n\\begin{prop}\\label{comb}\nLet $\\Gamma$ be a graph. If $e$ is an edge of $\\Gamma$ which is not a loop,\ncall $\\Gamma -e$ the graph obtained from $\\Gamma$ by removing $e$, \nand $\\Gamma\\cdot e$ the one obtained by contracting $e$. \nThen, we have the following relation between the complexities of these three graphs:\n\\begin{equation}\\label{fond}\nc(\\Gamma)=c(\\Gamma - e)+c(\\Gamma\\cdot e).\n\\end{equation}\n\\end{prop}\n\\noindent See \\cite{BMS} for a geometric interpretation and discussion of this proposition.\\\\\n\n\n\n\nThe main result of this section is the following\n\n\\begin{teo} \\label{maxconf}\nLet $C$ be a stable curve of genus $g\\geq 3$ with maximal complexity. \nThen, $C$ is a curve without disconnecting nodes and with trivalent simple dual graph.\nIn particular, $C$ is a graph curve.\n\\end{teo}\n\n\n\nFrom a geometrical point of view, this result implies in particular that the curves with maximal complexity \nlie on those $0$-strata of the topological stratification\nwhich are contained in $\\partial \\overline M_g\\setminus \\cup_{i=1}^{[g\/2]} \\Delta_i$.\n\nSee the end of this section for a purely combinatorial statement.\n\n\\bigskip\n\nOur strategy is the following: we consider a generic stable curve $C$ of genus $g$ and its weighted dual graph, $\\Gamma_C$;\nthen, if $\\Gamma_C$ is not as stated in Theorem \\ref{maxconf}, we modify the graph obtaining a new weighted graph $\\Gamma'$, which is the dual graph of another stable curve \nof genus $g$ with the desired properties, and we prove that $c(\\Gamma_C)c(\\Gamma_{K})$.\n\n\\end{prop}\n\\begin{proof}\nClearly, the weights on the graphs do not interfere with the complexity, as well as adding loops.\n\nWe now prove that, applying operation (II) the complexity remains the same only in case (II)a, and it increases strictly in case (II)b.\nCall $\\Gamma'$ the graph obtained by applying operation (II) to a vertex $v$ of degree $d$, and, accordingly to the notation of Proposition \\ref{degenerations}, call \n$d_a$ and $d_b$ the vertex degrees of $v_a$ and $v_b$, respectively, in $\\Gamma'$. \nClearly \n$$\nd=d_a+d_b -2,\n$$\nSince we are adding a vertex and a component, the first Betti number of the graph, $b_1$, remains the same. \nHowever, the complexity can increase, but not decrease. \nIndeed, as observed above, we have that $\\Gamma=\\Gamma'\\cdot l$. \nSo, by formula (\\ref{fond}),\n$$\nc(\\Gamma')=c(\\Gamma)+c(\\Gamma'-l)\\geq c(\\Gamma).\n$$\nMoreover, we see from the above formula that the complexity remains the same if and only if $c(\\Gamma'-l)=0$, \ni.e., if and only if $l$ \nis a disconnecting edge for $\\Gamma'$. \n\\end{proof}\n\n\n\n\n\\subsubsection*{The switching of two edges}\nAs we will often make use of another operation on graphs, it is convenient to describe it separately.\nLet us consider a graph $\\Gamma$, and fix two distinct edges $l$ and $m$, with associated vertices $v$, $v'$ and $w$, $w'$ respectively. Let us suppose that $v\\not = w$ (but we do not ask that $v\\not = v'$ or $w\\not = w'$, i.e. $l$ and $m$ can be loops). \nWe shall construct a new graph $\\Gamma'$ which has the same vertices as $\\Gamma$, \nsuch that $\\Gamma\\setminus\\{l,m\\}$ is a subgraph of $\\Gamma'$, and the edges $l$, $m$, are substituted by \n$l'$ and $m'$ as illustrated in figure below.\n\n\n\n\\bigskip\n\n\\centerline{\n\\begin{tabular}{rcl}\n\\xymatrix@R=2pc{\n*{\\bullet} & &*{\\bullet} \\\\\n& &\\\\\n *{\\bullet} \\ar @{-}[uu]^{l}^>{v}^<{v'} & & *{\\bullet} \\ar @{-}[uu]_{m}_>{w}_<{w'}\\\\}\n&\n\\xymatrix@R=2pc{\n\\\\\n=\\Rightarrow\\\\\n\\\\ \n\\\\}\n&\n\\xymatrix@R=2pc{\n*{\\bullet} \\ar @{-}[ddrr]^{l^\\prime } & & *{\\bullet} \\\\\n& &\\\\\n *{\\bullet} \\ar @{.}[uu]^{l}^>{v}^<{v'} \\ar @{-}[uurr]^{m^\\prime} & & *{\\bullet} \\ar @{.}[uu]_{m}_>{w}_<{w'}\\\\}\n\\end{tabular}}\n\n\\bigskip\n\n\n\\noindent So, the new edges $l'$ and $m'$ connect $v$ and $w'$, $v'$ and $w$ respectively. \nWe will call this process {\\em the switching of $l$ and $m$ with respect to $v$ and $w$\\footnote{Note that this operation\n is the $\\widetilde X$-transformation described in \\cite{tsukui}.}.}\nNote that the vertex degrees of $\\Gamma$ and $\\Gamma'$ are the same.\n\nIn general, this operation doesn't increase the complexity; counterexamples are easy to construct.\nHowever, we shall prove that, for certain graph configurations, it does.\n\n\n\n\\begin{prop}\\label{1}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen $C$ has no disconnecting nodes.\n\\end{prop}\n\\begin{proof}\nSuppose that $C$ has a disconnecting node, which corresponds in $\\Gamma=\\Gamma_C$ to a disconnecting edge $r$.\nConsider $l$, $m$ two edges adjacent to $r$. \nThe only case in which we cannot find two different edges, is if one of the vertex joined by $r$, say $v$, has degree one.\nIn this case, as the curve $C$ is stable, necessarily $ g(v)\\geq 1$; we therefore modify the graph by adding a loop on $v$, and \ndecreasing by one the weight on $v$.\nWe still have a graph associated to a stable curve of genus $g$ with the same complexity, which we will call again $\\Gamma$, and we can choose $l$ and $m$ as above.\nNow, we perform the switching of $l$ and $m$ w.r.t. $v$ and $w$:\\footnote{this operation is the \n``slide transformation of $l$ and $m$ at $v$ and $w$ along $r$'' with the terminology of \\cite{tsukui}.}\n\n\\bigskip\n\n\\centerline{\n\\xymatrix@R=1pc{\n\\ar @{}[r]|->{\\ldots} &*{\\bullet} \\ar @{-}[dr]^>{v} & & & & *{\\bullet} \\ar@{-}[dl]_>{w}& \\ar @{}[l]|->{\\ldots} &\\ar @{}[r]|->{\\ldots} &*{\\bullet} \\ar @{-}[dr]^>{v} & & & & *{\\bullet} \\ar@{-}[dl]_>{w}& \\ar @{}[l]|->{\\ldots} \\\\\n&& *{\\bullet} \\ar@{-}[rr]^{r} & & *{\\bullet} & &\\ar@{:>}[r] &&& *{\\bullet} \\ar@{-}[rr]^{r} & & *{\\bullet} & \\\\\n*{} \\ar @{}[r]_>{}|->{\\ldots}& *{\\bullet} \\ar @{-}[ur]|-{l} & & & & *{\\bullet} \\ar @{-}[ul]|-{m} & \\ar @{}[l]^>{}|->{\\ldots}&\n*{} \\ar @{}[r]_>{}|->{\\ldots}& *{\\bullet} \\ar @{.}[ur]|-{l} \\ar @{-}[urrr]|-{l^\\prime} & & & & *{\\bullet} \\ar @{.}[ul]|-{m} \\ar @{-}[ulll]|-{m^\\prime} & \\ar @{}[l]^>{}|->{\\ldots}\\\\}\n}\n\n\\bigskip\n\n\n\nNote that $\\Gamma\\cdot r=\\Gamma' \\cdot r$.\nNow, $\\Gamma'$ is the dual graph of a stable curve (indeed, we can think of the modification made as if we have \nresolved two nodes of the curve, and attached the components where they belong in a different way).\n\nNow, if the edge $r$ is still a disconnecting edge for $\\Gamma'$, then both $l$ and $m$ should be also disconnecting edges in $\\Gamma$. In this case, we must have $g(v)\\geq 1$ and we perform operation (I) in $v$. Then, if we choose $l$ to be the new loop, after the switching of $l$ and $m$ with respect to $v$, the edge $r$ will not be diconnecting anymore.\n\nSo, we can suppose that $r$ is no longer a disconnecting edge for $\\Gamma'$, and it is immediate to check that we have \nintroduced no new disconnecting edges, i.e. if $f$ is a disconnecting edge for $\\Gamma'$, then it is also a disconnecting edge for $\\Gamma$.\n\nNow, applying Proposition \\ref{comb} and observing that $c(\\Gamma-r)=0$, while $c(\\Gamma'-r)>0$, we get:\n$$\nc(\\Gamma')=c(\\Gamma'\\cdot r)+c(\\Gamma'-r)=c(\\Gamma\\cdot r)+c(\\Gamma'-r)>c(\\Gamma\\cdot r)=c(\\Gamma).\n$$\nThis proves the statement.\n\\end{proof}\n\n\\begin{rem}\n\\upshape{In \\cite{DJ-IR}, sec.3, it is observed that a general graph (not necessarily regular) has to be\nfree of disconnecting edges in order to have maximal complexity. \nProposition \\ref{1}, as stated, could be indeed proven in the same way.\nHowever, the operation used by the two authors is the switching \nof {\\em one} edge, which modifies the vertex degrees; as we will see, it is necessary for us to remain in the class \nof cubic graphs, that's why we apply our switching.}\n\\end{rem}\n\n\n\\begin{prop}\\label{2}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen $C$ is a loopless graph curve.\n\\end{prop}\n\\begin{proof}\nFrom Proposition \\ref{1} we can suppose that $C$ has no disconnecting nodes.\nLet $\\Gamma$ be the weighted graph associated to $C$ and suppose $\\Gamma$ is not a loopless trivalent graph.\nUsing the degeneration operations described in Definition \\ref{degenerations},\nwe will describe an algorithm in two steps that, given $\\Gamma$ with no disconnecting nodes,\nproduces a loopless graph $\\Gamma_3$ \nwith strictly bigger complexity than $\\Gamma$, such that $b_1(\\Gamma_3)=g$ and each vertex has degree $3$ and weight $0$. \nTherefore $\\Gamma_3$ is the dual graph associated to a loopless graph curve of genus $g$.\n\n\\medskip\n\n\\noindent FIRST STEP (Reduction to a curve having only rational components): \nWe replace any vertex $v_i$ with strictly positive weight $g(v_i)$ with a bouquet given by a vertex of weight $0$ and $g(v_{i})$ \nloops attached to it. \nThis is a reiterate application of operation (I) and does not change the arithmetic genus, nor the complexity. \nCall $\\Gamma_1$ the graph obtained in this way.\n\n\\medskip\n\n\\noindent SECOND STEP (Reduction to a loopless graph curve):\nsuppose that in $\\Gamma_1$ there is a vertex $v$ with degree $d$ greater or equal to $4$. \nAs $\\Gamma_1$ is obtained from $\\Gamma$ by adding loops, it has no disconnecting edges. \nWe can therefore perform operation (II)b on $v$, with the request that $\\deg(v_a)\\geq 3$, and $\\deg(v_b)\\geq 3$\n(i.e. remaining in the class of graphs of stable curves).\nAs proved in Proposition \\ref{increasing}, this operation increases strictly the complexity.\nMoreover, the degrees of $v_a$ and $v_b$ are strictly smaller than the degree of $v$.\nWe perform this transformation until all the vertices of the new graph have exactly degree 3. \nCall $\\Gamma_2$ the resulting graph. \nSo by the genus formula $\\Gamma_2$ has $2g-2$ vertices of weight 0, $3g-3$ edges, and $b_1(\\Gamma_2)=g$; \nequivalently, $\\Gamma_2$ is the graph associated to a graph curve.\nMoreover, $\\Gamma_2$ has no disconnecting edges, and no loops. \nIndeed, if there were a loop, $\\Gamma_2$ would contain a subgraph of the form:\n\n\\bigskip\n\n\\begin{equation*}\n\\xymatrix@=1.2pc{\n&*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}[d]\\\\\n&*{\\bullet} & \\\\\n\\ar@{-}[ur] && \\ar@{-}[ul] & .\n}\n\\end{equation*}\nhence with a disconnecting node.\nMoreover, by what observed above, $c(\\Gamma_2)> c(\\Gamma_1)$.\n\\end{proof}\n\n\n\n\\begin{ex}\n\\upshape{\nLet us consider the dual graph of a smooth genus $3$ curve; this is just one vertex with weight $3$.\nLet us apply the above algorithm to this graph.\nWith the first step, we obtain the following `` bouquet\" of $3$ loops,\n\n\\medskip\n\n\\begin{equation*}\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}@(ur,dr) \\ar@{-}@(ur,ul)\n}\n\\end{equation*}\n\n\\medskip\n\nwhich is the graph of a stable curve with a single irreducible rational component with $3$ nodes. \nAs the irreducible component has geometric genus $0$, we apply the second step. \nWe have several choices. \nSo, we can obtain either one of the following configurations.\n\n\\begin{equation*}\n\\begin{tabular}{cccccccccc}\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_.3pc\/[r] \\ar @{-}@\/_.8pc\/[r]\/& *{\\bullet} \\ar@{-}@\/_.3pc\/[l] \\ar@{-}@\/_.8pc\/[l]\n}\n&{}&{}&\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}@\/_\/[l]\n}\n&{}&{}&\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)\n}\n\\end{tabular}\n\\end{equation*}\n}\n\n\nNotice that all these graphs have still vertices with degree greater then 3. So, we therefore must go on applying step 2. \nFor example, one of the possibilities for the first graph would be the following.\n\n\\begin{equation*}\n\\begin{tabular}{lcr}\n\\xymatrix{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}@\/_\/[dl] \\ar@{-}@\/^\/[dr] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] & & *{\\bullet} & \\ar@{:>}[r] &\n}\n&\n&\\xymatrix@=1.2pc{\n&*{\\bullet}\\ar@{-}[dd] \\ar@{-}[dl] \\ar@{-}[dr] &\\\\\n*{\\bullet} \\ar@{-}[rr] |!{[r]}\\hole & &*{\\bullet}\\\\\n&*{\\bullet}\\ar@{-}[ru]\\ar@{-}[lu] & \n}\n\\end{tabular}\n\\end{equation*}\n\nIf we start from the third graph, we can perform the following chain of transformations\n(indeed, in this case, these operations are forced).\n\n\\begin{equation*}\n\\begin{tabular}{lcccr}\n\\xymatrix@R=1pc{\n{}\\\\\n *{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)&\\ar@{:>}[r]&\n}\n&&\n\\xymatrix@=1pc{\n*{\\bullet} \\ar@{-}[dd]\\ar@{-}@\/_.5pc\/[dd]\\ar@{-}[drr]&&&&\\\\\n&&*{\\bullet} \\ar@{-}@(ur,dr)&&\\ar@{:>}[r]&\\\\\n*{\\bullet} \\ar@{-}[urr]&& \n}\n&\n\\xymatrix{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}[d] \\ar@{-}@\/_\/[d] & *{\\bullet} \\ar@{-}[d] \\ar@{-}@\/^\/[d]\\\\\n*{\\bullet} \\ar@{-}[r] & *{\\bullet}\n}\n\\end{tabular}\n\\end{equation*}\n\\end{ex}\n\nAs also the simple example above shows, the algorithm of Proposition \\ref{2} doesn't give a unique output.\nOn the contrary, if we start from a bouquet of $g$ loops we can obtain \\emph{all} the possible cubic graphs without disconnecting edges using step 2, as one can see in the following way. \nConsider a cubic graph without disconnecting edges $\\Sigma$ with $b_1(\\Sigma)=g$. \nLet $T$ be a spanning tree for $\\Sigma$. To contract all the edges of $T$ is the reverse operation to step 2 of the algorithm, and the result is the bouquet of $g$ loops.\n\n\n\n\\begin{proof} ({\\it of Theorem \\ref{maxconf}})\nBy Proposition \\ref{1} and \\ref{2} we can suppose that $\\Gamma=\\Gamma_C$ is trivalent,\nloopless and without disconnecting edges.\nHence, the only possible multiple edges are of the form\n\n\\medskip\n\n\\centerline{\n\\xymatrix{ &&&&\\ar @{-}[dl]\\\\\n*{\\bullet} \\ar @{-}[r]_s^>{v_2}^<{v_1}&*{\\bullet} \\ar @{-}@\/_\/[r]_l& *{\\bullet} \\ar @{-}@\/_\/[l]_f & *{\\bullet} \\ar @{-}[l]^r_>{v_3}_<{v_4}&\\\\\n&&&& *{\\bullet}\\ar @{-}[ul]^m^<{v_5}\\\\\n}\n}\n\n\\bigskip\n\nNote that $l$ and $m$ have to be distinct as well as $v_1$ and $v_4$ (otherwise $\\Gamma$ would have disconnecting edges).\nWith the notations adopted in the figure, we apply the switching of $l$ and $m$ w.r.t. $v_3$ and $v_4$:\n\n\n\n\\centerline{\n\\xymatrix{ &&&&\\ar @{-}[dl]\\\\\n& & *{\\bullet} \\ar @{-}@\/_\/[dl]_{f} & *{\\bullet} \\ar @{-}[l]_r_>{v_3}_<{v_4}&\\\\\n*{\\bullet} \\ar @{-}[r]_s^<{v_1}^>{v_2}&*{\\bullet} \\ar @{.}@\/_\/[ur]_{l^\\prime}\\ar @{-}[urr]&&&*{\\bullet} \\ar @{.}[ul]\\ar @{-}[ull]^{m^\\prime}^<{v_5}\\\\\n}\n}\n\n\\bigskip\n\n\\bigskip\n\nWe have constructed a new trivalent graph $\\Gamma'$ which has one less couple of multiple edges: indeed, \nit is immediate to check that we have not created any other multiple edges.\n\n\nLet us prove that $c(\\Gamma ')>c(\\Gamma)$.\nWe prove it by giving an injective map from the spanning trees of $\\Gamma$ to the spanning trees of $\\Gamma'$, as follows.\n\nLet $T$ be a spanning tree of $\\Gamma$. If $r\\in T$, the switching of $\\Gamma$ we performed transforms $T$ into a spanning tree $T^\\prime$ of $\\Gamma^\\prime$, with $r\\in \\Gamma^\\prime$.\n\nNow, suppose $r \\not\\in T$. We shall distinguish between $4$ different situations.\n\\begin{itemize}\n\\item $l,m\\not\\in T$ ($\\Rightarrow f\\in T$). $T$ is transformed into a tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\not\\in T^\\prime,m^\\prime\\not\\in T^\\prime,r\\not\\in T^\\prime$ and $f\\in T^\\prime$. \n\\item $l\\in T$, $m\\not\\in T$ ($\\Rightarrow f\\not\\in T$). Then, in $T$, the path connecting $v_2$ and $v_4$ passes by $l$. So, the switching of $l$ and $m$ along $r$ restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$ since $l^\\prime$ would create a cycle containing $l^\\prime$ and $v_3$ and $v_5$ get disconnected. So, we consider the subgraph of $\\Gamma^\\prime$ obtained by the union of $m^\\prime$ with the switching of $T$ minus $l^\\prime$. It is easy to see that this is a spanning tree $T^\\prime$ of $\\Gamma^\\prime$, with $l^\\prime\\not\\in T^\\prime$, $m^\\prime\\in T$, $r\\not\\in T^\\prime$ and $f\\not\\in T^\\prime$.\n\\item $l\\not\\in T$, $m\\in T$ ($\\Rightarrow f\\in T$). In this case we must do a further distinction. \n\\begin{itemize}\n\\item In $T$ the path connecting $v_2$ and $v_4$ does not contain $m$. \n\nIn this case $T$ is transformed into a spanning tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\not\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\item In $T$ the path connecting $v_2$ and $v_4$ contains $m$.\n\nIn this case the switching we performed restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$, since $m^\\prime$ creates a cycle and $v_4$ and $v_2$ get disconnected. So, we consider $T^\\prime$ as the union of $l^\\prime$ with the switching of $\\Gamma$ restricted to $T$ minus $m^\\prime$. Then $T^\\prime$ is a spanning tree of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\not\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\end{itemize}\n\\item $l,m\\in T$ ($\\Rightarrow f\\not\\in T$). Also in this case we have to distinguish between the following situations:\n\\begin{itemize}\n\\item In $T$ the path connecting $v_2$ and $v_4$ does not contain $m$.\n\nThen the switching of $l$ and $m$ along $r$ restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$ since it creates a cycle containing $l^\\prime$ and $v_4$ and $v_5$ get disconnected. So if we consider $T^\\prime$ as the union of $f$ with the switching of $T$ minus $s$, then $T^\\prime$ is a spanning tree of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\item In $T$ the path connecting $v_2$ and $v_4$ contains $m$.\n\nIn this case the switching we performed restricted to $T$ is a spanning tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\not\\in T^\\prime$.\n\\end{itemize}\n\\end{itemize}\n\nSo, the association of each spanning tree $T$ of $\\Gamma$ with $T^\\prime$ gives an injective map from the spanning trees of $\\Gamma$ into the spanning trees of $\\Gamma^\\prime$.\n\nNow, to prove that, indeed strict inequality holds, we observe that $\\Gamma^\\prime$ has at least one spanning tree $T^\\prime$ such that $l'$, $f$ and $r$ are not in $T^\\prime$ ($\\Rightarrow m^\\prime\\in T^\\prime$), and this kind of spanning trees are not in the image of the map constructed above. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Combinatorial version}\\label{combi}\nWe can give a purely combinatorial translation of the above results, as follows.\nRecall that a graph $\\Gamma$ is said to be {$k$-connected} if for any set $S$ of $k-1$ edges of \n $\\Gamma$, $\\Gamma\\setminus S$ is still connected. \n $\\Gamma$ is said to be {\\it strictly} $k$-connected if it is $k$-connected but not $k+1$-connected. \nSo, a graph is $1$-connected if and only if it is connected; it is strictly $1$-connected if and only if it is connected and \n it has at least one disconnecting node.\n Of course, a trivalent graph can never be $4$-connected (any triple of edges incident in one vertex is a disconnecting set).\n \n Given a positive integer $g$, let $\\mathcal G_g$ be the set of all weighted multigraphs of genus $g$ satisfying \n condition (\\ref{stability}). Theorem \\ref{maxconf} can now be written in the following way.\n \n \\begin{teo}\\label{combinmaxconf}\n The graphs $\\Gamma\\in \\mathcal G_g$ reaching the maximal complexity\n are trivalent, simple and $2$-connected.\n \\end{teo}\n \n \nWe shall call a simple trivalent graph a {\\em cubic graph}. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Bounds on the maximal complexity of stable curves}\\label{ub}\n\nThe question of an upper bound on the complexity depending on the genus has several geometrical meanings: for instance, it implies that there is a bound for the irreducible components of the compactified Jacobian of a stable curve, and a bound for the group of components of the N\\'eron model of the relative Jacobian of a family of curves having $C$ as special fiber; see also Remark \\ref{confrontoL} .\nLet us define \n$$\\psi(g):=\\mbox{max}\\{c(C), C \\mbox{ stable curve of genus } g\\}.$$\n\n\\begin{rem}\\upshape{\nNotice that $\\psi$ is strictly increasing with respect to the genus $g$. \nIn fact, if $C$ is a curve of genus $g$ such that $\\psi(g)=|\\Delta_C|$, let $C^\\prime$ be the stable curve obtained from $C$ by adding an extra node connecting two of the components of $C$. Then, using the above formula for the genus of $C^\\prime$, one gets $g_{C^\\prime}=g+1$ and, clearly, $c(C^\\prime)>c(C)$.}\n\\end{rem}\n\n\n\nAccording to the results of the preceding section, we know that curves achieving maximal complexity are in particular graph curves. \nUsing this fact, we can find a first rough bound:\n$$\\psi(g)\\le \\binom{3g-3}{g}\\le 2^{3g-4}.$$ \nFor the proof, see \\cite{lorjac}, Lemma 2.7.\nNote that the first inequality follows immediately from the fact that any spanning tree in a cubic\ngraph of order $2g-2$ is obtained by removing $g$ edges. \n\n\nApplying the results on the surjectivity of the ``Abel-Jacobi map'' in \\cite{B-N},\nwe could derive a better bound: $$\\psi(g)\\le \\binom{2g-2}{g}.$$\n\nBoth these bounds are not sharp, even for low genus.\nUsing a result of Biggs \\cite{biggs}, we can prove the following.\n\\begin{equation}\\label{disbiggs}\nc(\\Gamma)\\leq \\frac{1}{2g-2}\\left(\\frac{6g-6}{2g-3}\\right)^{2g-3}=:\\alpha(g).\n\\end{equation}\n\nFor $g=3$, this bound is optimal.\nHowever, the bound is not asymptotically sharp.\n\nThe complexity of $k$-regular graphs has been studied also by McKay \\cite{McK}, and Chung-Yau \\cite{C-Y}.\nWe can apply their results obtaining a sharper bound. \n\\begin{teo}\\label{bound}\nLet $C$ be a stable curve of genus g, and let $\\Gamma$ be its dual graph. \nThen \n\\begin{equation}\\label{disCY}\nc(\\Gamma)\\leq \\frac{2 \\ln{(2g-2)}}{3(2g-2) \\ln{(9\/8)}}\n\\exp{\\left(\\frac{12}{\\sqrt\\pi}\\left(\\frac{\\ln{9\/8}}{\\ln{(2g-2)}}\\right)^{5\/2}\\right)}\n\\left(\\frac{4}{\\sqrt{3}}\\right)^{2g-2}=:\\beta(g).\n\\end{equation}\nMoreover, this bound is asymptotically sharp for $g\\gg 0$.\n\\end{teo}\n\\begin{proof}\nBy making explicit the computations in Theorem 4 of \\cite{C-Y}, we obtain the bound (\\ref{disCY})\nfor $3$-regular graphs with $2g-2$ vertices. The thesis is now straightforward because of the reduction to\ngraph curves of Theorem \\ref{maxconf}.\nMcKay proves in \\cite{McK} that there is a sequence of cubic graphs \n$\\{\\Gamma_i\\}_{i\\in\\mathbb N}$ with increasing orders $n_i$'s, such that, if we set \n\\begin{equation}\\label{tau}\n\\tau(\\Gamma_i):=\\left(n_ic(\\Gamma_i)\\right)^{\\frac{1}{n_i}},\n\\end{equation}\nthen $\\tau(\\Gamma_i)\\longrightarrow \\frac{4}{\\sqrt 3}$ for $i\\rightarrow \\infty$.\nIndeed, he proves much more than the existence, but that a ``random'' sequence of cubic\ngraphs satisfies this property, see also \\cite{lyons}.\nHence, the constant $ \\frac{4}{\\sqrt 3}$ in the bound (\\ref{disCY}) is the best possible.\n\\end{proof}\n\n\n\nIn the next picture we describe the graphs with maximal complexity for $g=4$, $g=5$ and $g=6$, respectively.\n\n\\bigskip\n\n\\centerline{\n\\begin{tabular}{ccc}\n\\xymatrix@C=1.7pc@R=3.3pc{\n& *{\\bullet} \\ar@{-}[rr] \\ar@{-}[ddrr] & & *{\\bullet} \\ar@{-}[ddll]|!{[ll];[dd]}\\hole &\\\\\n*{\\bullet} \\ar@{-}[rrrr]|!{[rr]}\\hole \\ar@{-}[ur] \\ar@{-}[dr] & & & & *{\\bullet} \\ar@{-}[ul] \\ar@{-}[dl] \\\\\n& *{\\bullet} \\ar@{-}[rr] & & *{\\bullet}\n}\n&\n\\xymatrix@=.8pc{\n& & & *{\\bullet} \\ar@{-}[dll] \\ar@{-}[drr] \\ar@{-}[dddddd]|!{[ddd]}\\hole & & & \\\\\n& *{\\bullet} \\ar@{-}[ddl] \\ar@{-}[ddddrrrr] & & & & *{\\bullet} \\ar@{-}[ddr]\n\\ar@{-}[ddddllll]|!{[llll];[dddd]}\\hole & \\\\\n&&&&&&\\\\\n*{\\bullet} \\ar@{-}[rrrrrr]|!{[rrr]}\\hole & & &&& & *{\\bullet} \\\\\n&&&&&&\\\\\n& *{\\bullet} \\ar@{-}[uul] & & & & *{\\bullet} \\ar@{-}[uur] & \\\\\n& & & *{\\bullet} \\ar@{-}[ull] \\ar@{-}[urr]\n}\n&\n\\xymatrix@C=.9pc@R=.8pc{\n& *{\\bullet} \\ar@{-}[rrrr] \\ar@{-}[dddl] \\ar@{-}[ddddr] &&&& *{\\bullet} \\ar@{-}[dddr] \\ar@{-}[ddddl]|!{[ddd];[lll]}\\hole & \\\\\n&&&&&&\\\\\n& & & *{\\bullet} \\ar@{-}[d] \\ar@{-}[dlll]|!{[d];[lll]}\\hole \\ar@{-}[drrr]&&&\\\\\n*{\\bullet} \\ar@{-}[dddr] && &*{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& && *{\\bullet} \\ar@{-}[dddl] \\\\\n&&*{\\bullet} \\ar@{-}[ddrrr]|!{[d];[rrr]}\\hole &&*{\\bullet} \\ar@{-}[ddlll]&&\\\\\n&&&&&&\\\\\n& *{\\bullet} \\ar@{-}[rrrr] &&&&*{\\bullet}\n}\\\\\n$\\Gamma_1 , g=4$\n&\n$\\Gamma_2 , g=5$\n&\n$\\Gamma_3 , g=6$\n\\end{tabular}\n}\n\n\\bigskip\n\nNote that $\\Gamma_3$ is the famous Peterssen graph. \nThese graphs are proved to be of maximal complexity among simple cubic graphs of their respective orders in \\cite{DJ-IR}, sec.5. \n\n\n\n\\begin{rem}\\label{confrontoL}\n\\upshape{As already noticed, the geometric meaning of this result is that it gives a bound on the group of connected components of the N\\'eron model of the degree-$d$ Picard variety for families of {\\em stable}\ncurves (\\cite{BLR}, Theorem 1, sec.9.6), and as well on the number of irreducible components of the fibres of the scheme\n$P_g^d$ constructed in \\cite{capneron} and of $\\overline{P_{d,g}}$ of \\cite{cap}. \nOf course this is also a bound on the cardinality of the group of components of the N\\'eron models of the Jacobians of such families.\nHowever, notice that this bound is different from the ones found by Lorenzini in \\cite{lorner} (see also \\cite{BLR}, \nTheorem 9 sec.6.9). \nIndeed, given any strictly henselian discrete valuation ring $R$, with algebraically closed residue field $\\mbox{\\itshape {\\textbf k}}$ and field of fractions $K$, any regular family of curves $f\\colon X\\longrightarrow \\mbox{spec }R$ such that the {\\em general} fiber $X_K$ is of genus $g$, and the Jacobian $J_K$ has {\\em potential good reduction}, Lorenzini finds a bound, depending only on \n$g$, for the group of components of the N\\'eron model of $f$.\nWe obtain instead a bound for the group of components of the N\\'eron model of \\emph{any} smooth family $f\\colon X\\rightarrow \\mbox{spec }R$ such that the \\emph{closed} fiber $X_{\\mbox{\\small \\itshape {\\textbf k}}}$ is a stable curve.\nSo one could say that his approach is dynamic, in the sense that it depends on the generic fiber, while our approach is static, as it starts from a given curve.\nMoreover, the boundedness comes on the one hand from the assumption of potential good reduction on the general fiber, on the other from the assumption of stability of the special one.\nAs it is proven in \\cite{lorgraphs}, a family has potential good reduction if and only if the closed fiber has a {\\em tree} as dual graph, \nhence complexity 1.\n}\n\\end{rem}\n\n\n\n\n\n\\subsection{Example and lower bound}\\label{lower}\n\n\n\n\n\n\n\n\n\n\nWe present here an example of a family of cubic graphs of increasing order and increasing complexity.\nIn particular, this gives explicit lower bounds on $\\psi(g)$ (even though not sharp, given the above result).\n\nLet $\\Gamma_m$ be the graph with $4m$ vertices of valency $3$ formed from $m$ \npairwise disjoint graphs $G_i$ of the following form:\n\n\\centerline{\n\\begin{tabular}{c}\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n\\end{tabular}\n}\n\n\\noindent by adding $m$ edges $l_1,\\ldots ,l_m$ to link them as a ring, as shown in the figure below for $m=6$. \nClearly, $\\Gamma_m$ is the dual graph of a graph curve of genus $g=2m+1$.\n\n\\centerline{\n\\xymatrix@=.2pc{\n&&&&&*{\\bullet} \\ar@{-}[drr] \\ar@{-}[ddl] \\ar@{-}[dddr] &&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[ddr] \\ar@{-}[dddl]&&& \\\\\n&&&&&&&*{\\bullet} \\ar@{-}[ddl] \\ar@{-}[rrr] &&& *{\\bullet} \\ar@{-}[ddr] &&&&& \\\\\n&&&&*{\\bullet} \\ar@{-}[drr] \\ar@{-}[dddll] &&&&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[dddrr]\\\\\n&&&&&&*{\\bullet} &&&&& *{\\bullet} &&&& \\\\\n\\\\\n&&*{\\bullet} \\ar@{-}[ddll] \\ar@{-}[ddrr] &&&&&&&&&&&&& *{\\bullet} \\ar@{-}[ddrr]\\ar@{-}[ddll]& \\\\\n\\\\\n*{\\bullet} \\ar@{-}[rrrr] \\ar@{-}[ddrr] &&&& *{\\bullet} \\ar@{-}[ddll] &&&&&&&&& *{\\bullet} \\ar@{-}[ddrr] \\ar@{-}[rrrr] &&&& *{\\bullet} \\ar@{-}[ddll]\\\\\n\\\\\n&&*{\\bullet} \\ar@{-}[dddrr] &&&&&&&&&&&&& *{\\bullet} \\ar@{-}[dddll] & \\\\\n\\\\\n&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[ddr] \\ar@{-}[dddl] &&&&& *{\\bullet} \\ar@{-}[drr] \\ar@{-}[ddl] \\ar@{-}[dddr]\\\\\n&&&& *{\\bullet} \\ar@{-}[ddr] &&&&&&&&& *{\\bullet} \\ar@{-}[ddl]\\\\\n&&&&&&&*{\\bullet} \\ar@{-}[dll] &&& *{\\bullet} \\ar@{-}[lll] \\ar@{-}[drr]\\\\\n&&&&& *{\\bullet} &&&&&&& *{\\bullet}\n}\n}\n\n\\vspace{.1cm}\n\n\n\\begin{prop}\n$\nc(\\Gamma_m)=2m8^m.\n$\n\\end{prop}\n\\noindent {\\bf Proof}: To prove this formula we will proceed by hands counting the number of spanning trees for $\\Gamma_m$.\nNotice that the complexity of the subgraphs $G_i$'s is $8$. \n\nChoosing one of the edges $l_i$, the number of spanning trees not containing it is $8^m$. \nIndeed, all the others $l_j$'s have to be included in any spanning tree, while for any $G_k$, we have to count the $8$ possibilities for the spanning trees. So, we have $m8^m$ of these.\n\nOn the other hand, if $T$ is a spanning tree that contains all the $l_i$, then there is one and only one $j$ such that $T\\cap G_j$ is disconnected, and it has to be of one of the $8$ following forms:\n\n\\vspace{.2cm}\n\\begin{tabular}{cccccccc}\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& \\\\\n*{\\bullet} && *{\\bullet} \\\\\n& *{\\bullet}\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet}& \\\\\n*{\\bullet} && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] & \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} & \\\\\n*{\\bullet} \\ar@{-}[rr]&& *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} & \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] & \\\\\n*{\\bullet} && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dr]& \\\\\n*{\\bullet} && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \n}\n\\end{tabular}\n\n\\vspace{.2cm}\n\nFor $i\\not =j$, $T\\cap G_i$ is a spanning tree for $G_i$ as above.\nTherefore, we have $8m8^{m-1}=m8^m$ possible spanning trees of this form.\n\nSo, summing all up, we have $2m8^m$ spanning trees, as required.\\hfill $\\Box$\n\n\\bigskip\n\nNotice that $$c(\\Gamma_m) =(g-1)8^{\\frac{g-1}{2}}=(g-1)(2\\sqrt{2})^{g-1}.$$\nSo, for odd $g$, we have this lower bound on $\\psi$. \nFor even $g$, remembering that $\\psi$ is an increasing function of the genus, we have\n$\\psi(g)>\\psi(g-1)$. \nHence, $\\psi$ is bounded from below by $(g-2)(2\\sqrt{2})^{g-2}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\\section{Further results and conjectures}\\label{congetture}\n\n\\subsubsection*{3-connectedeness}\n\nNote that a trivalent graph with loops, or with double edges is strictly $2$-connected. \nWith Theorem \\ref{MAX}, we have therefore excluded the strictly $1$-connected case, \nand some of the cases of strict $2$-connectedness. \n>From this, and from the observation of graphs with maximal complexity for low genus, \nit is natural to ask\n\n\\begin{conj}\\label{connectivity}\nA cubic graph with maximal complexity is $3$-connected.\n\\end{conj}\nThis seems to be generally believed, also for bigger classes of graphs (see e.g.\\cite{DJ-IR} sec.3), \nbut no proof is known.\nLet us note that $3$-connected graph curves have also interesting geometrical properties; \nindeed, they are precisely those graph curves for which the canonical bundle is very ample, yielding an embedding morphism \\cite{B-E}.\n\nWith similar techniques to the ones used in Section \\ref{MAX}, we can prove a partial result.\n\\begin{prop}\\label{bah}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen \n\\begin{enumerate}\n\\item $\\Gamma_C$ has no couple of disconnecting edges that lay on a cycle of length $\\leq 6$.\n\\item $\\Gamma_C$ has no couple of disconnecting edges such that at least one of them is adjacent to a cycle of length $\\leq 4$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{cor}\\label{g<10}\nConjecture \\ref{connectivity} holds for $g\\leq 8$.\n\\end{cor}\n\n\n>From a result on the so-called ``Abel-Jacobi map'' for graphs proved on \\cite{B-N} (Theorem 1.8),\nwe can derive the following \n\\begin{prop}[Baker-Norine]\nIf $\\Gamma$ is any $k$-connected graph of order $n$, \n$$\nc(\\Gamma)\\geq \\binom{n}{k-1}.\n$$\n\\end{prop}\nThis proves that the complexity grows with the connectivity.\nIn the case of cubic graphs, however, the bound obtained for $3$-connected graphs is just \nquadratic in the order.\n\n\n\\subsubsection*{Maximal girth}\nThe explicit sequences of cubic graphs with large complexity \nhave all {\\em large girth}. \nLet us explain the terminology.\nThe {\\em girth} of a graph is the length of the shortest cycle.\nA simple argument shows that in a cubic graph the girth cannot exceed $2\\ln n\/\\ln2$ ($n$\nbeing the order of the graph).\nHence, the girth of a family of cubic graphs can grow at most as the rate of the logarithm of \nthe number of vertices.\n\n\\begin{defi}\\label{largegirth}\nA sequence $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$ of cubic graphs with increasing \norders $n_i$ is called {\\em sequence of large girth} if\n$$\n\\lim_{i\\to \\infty}\\frac{\\ln(n_i)}{\\ln2 \\,\\, girth(\\Gamma_i)} \\,\\,\\,\\mbox{ is finite. }\n$$\n\\end{defi}\nMcKay proves in \\cite{McK} that sequences with large girth satisfy condition (\\ref{tau}).\n\n\\begin{conj}\\label{girth}\nA sequence $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$ of cubic graphs with maximal complexity has large girth. \n\\end{conj}\nSee \\cite{DJ-IR}, sec.3 for an heuristic argument supporting this conjecture.\n \n\n\nFrom a result of McKay we can derive the following property of sequences of trivalent graphs of maximal complexity.\nGiven a graph $\\Gamma$, and a positive integer $m$, let $C_{\\Gamma}(m)$ be the number of \ncycles of length $\\leq m$ in $\\Gamma$.\n\n\\begin{teo}\\label{cicli}\nLet $m$ be a positive integer. \nFor any $\\epsilon>0$ there exist a positive integer $j$ such that: \ngiven any infinite sequence of trivalent graphs\n $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$ such that \n$\\Gamma_i$ has $2i$ vertices, and reaches the maximal complexity between cubic graphs of order $2i$,\nthe following inequality holds\n$$\n\\frac{C_i(m)}{2i}\\leq \\epsilon \\,\\,\\mbox{ for any }i\\geq j,\n$$\nwhere $C_i(m)=C_{\\Gamma_i}(m)$.\n\\end{teo}\n\\begin{proof}\nTheorem 4.5 of \\cite{McK} states that, given a sequence of cubic graphs $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$ \n satisfying (\\ref{tau}), then, for each fixed $m$,\n $$\n\\frac{C_i(m)}{n_i}\\rightarrow 0 \\,\\,\\mbox{ for } i\\to \\infty.\n$$\nThis does not directly imply the statement, which is uniform, in the sense that the constant $j$ depends only on $m$ and not \non the chosen sequence.\nHowever, let us argue as follows: given $m$, consider a sequence of cubic graphs with maximal complexity {\\em and} with\nmaximal cardinality of $C(m)$.\nClearly this sequence satisfies condition (\\ref{tau}).\nNow, by assumption, the statement holds uniformly on every sequence of graphs of maximal complexity.\n\\end{proof}\nNote that a sequence of graphs of large girth satisfies trivially the condition of Theorem \\ref{cicli}.\n\n\n\n\n\n\\subsubsection*{Further speculations}\n\nOther open questions are: \n\\begin{itemize}\n\\item Are cubic graphs of maximal complexity \n{\\em strongly regular}?\n\\item Do cubic graphs of maximal complexity have {\\em maximal automorphism group}? \n\\item Is a cubic graph with maximal complexity and given genus {\\em unique}?\n\\item Let $C$ be a graph curve with maximal complexity; suppose that it is indeed $3$-connected. \nThen the canonical map is an embedding realizing $C$ as a configurations of lines in $\\mathbb{P}^{g-1}$.\nWe conjecture that these lines are in ``general position'', in a sense that can be made precise.\nThis property seems to be connected with the large girth property.\n\\item Let $\\Gamma$ be a cubic graph with maximal complexity; suppose that it is indeed $3$-connected. \nWe think that the ``Clifford index'' of this graph, as defined for instance by Bayer and Eisenbud, has to be maximal as well. \n\\item Let $\\Gamma$ be a cubic graph with maximal complexity; suppose that it is indeed $3$-connected. \nWe wonder if any cutset of $3$ edges is made of adjacent edges.\n\\end{itemize}\n\n\n\\bibliographystyle{amsalpha}\n \\nocite{*}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA majority of modern Silicon Photomultipliers (SiPMs) is designed as an array of independent Single-Photon Avalanche Diode (SPAD) cells with individual quenching resistors connected to a common electrode. An active photosensitive part of the cell is formed by planar pn-junctions. The inactive cell periphery is typically formed by guard rings or trenches to provide an independent operation of the cells in the Geiger breakdown processes. This design has been initially developed in Russia \\cite{A,B,1}. Since the mid-2000s, the planar SiPM design concept has been utilized with some modifications by Hamamatsu \\cite{2}, SensL \\cite{C}, STM \\cite{3}, FBK \\cite{4}, Excelitas \\cite{5}, and KETEK \\cite{6}. It was worldwide recognized as the SiPM, a new photon-number-resolving avalanche detector of outstanding performance \\cite{7}.\n\nNowadays, the planar SiPMs are approaching the physical limits of the design performance with its main inherent trade-off between photon detection efficiency (PDE) and dynamic range (DR) because of the dead space at the cell periphery \\cite{Vinogradov2011}. \nThis trade-off becomes challenging when developing a SiPM which should be sensitive for a wide spectral range, especially for red and near-infrared (NIR) wavelengths.\nThe development of a \"NIR-SiPM\" is anticipated to be a breakthrough in a number of applications \u2013 first and foremost in LIDARs \u2013 as well as in medicine and health sciences, biology and physics, environmental monitoring and quantum telecommunications. To achieve efficient absorption of photons and a high PDE for the red part of the light spectrum, a thicker photosensitive layer is required due to the low absorption coefficient of silicon at these wavelengths. But, the available increase of the thickness is eliminated by the so-called \"border effect\". \nThe thicker the active layer of a SiPM for efficient absorption of NIR photons, the higher the losses of a sensitive volume adjacent to trenches \\cite{9}. Further, the breakdown voltage increases proportionally to the depletion depth for planar technologies. \nThis leads to high operating voltages in the Geiger-Mode with high temperature coefficients. Simultaneously, the lateral distance of the active area to the micro-cell edge has to be increased in order to prevent edge breakdowns at high voltages. \nDespite these limitations, further progress in planar SiPM technology has been made with respect to the sensitivity at \\SI{905}{\\nano m}. For example, FBK started from \\SI{11}{\\%} for \\SI{35}{\\micro m} cells in 2017 \\cite{9} and Broadcom improved this result to \\SI{18}{\\%} (cell size is claimed as \"smallest\") in 2020 \\cite{Broadcom2020}.\n\\IEEEpubidadjcol\n\\\\\nIn contrast, there are also several photoelectric devices based on non-planar configurations.\nThe most relevant examples are Geiger-Mode APDs developed as predecessors or alternatives to the planar SiPMs. Spherical avalanche diodes with a radius of the pn-junction of \\SI{2}{\\micro m} and a breakdown voltage of \\SI{50}{V} seems to be the earliest device of this type operated in a photon counting mode \\cite{12}. Geiger-Mode APDs with negative feedback, also known as Metal-Resistor-Semiconductor (MRS) APDs, were designed as an array of avalanche micro-channels. A few micrometer-size n+ diffusion dots or \"needle\" junctions on p-Si wafer are covered by a thin resistive SiC layer as quenching resistor \\cite{G,13}. The first NIR-GM-APD with negative feedback has also been developed with a similar design \\cite{14}. Another kind of the microchannel GM-APD, the Micropixel APD (MAPD), is based on deep n-type micro-dots buried into a p-type epi-layer \\cite{H,15}. The MAPDs demonstrated a unique pixel-density of up to $\\SI{4e4}{mm^{-2}}$ with the highest geometric efficiency of almost \\SI{100}{\\%}. However, the MAPD design is also associated with a long pixel recovery time eliminating its dynamic range for continuous illumination and long light pulse detection.\n\nFor years, KETEK has been providing state-of-the-art SiPMs optimized for a blue range of the light spectrum, which is commonly used in medical and high energy physics applications.\nTo overcome the above mentioned limitations of the planar technology, we developed a new SiPM concept according to the patent application \\cite{Patent_TAPD}, the \"Tip Avalanche Photodiode\" (TAPD).\nThe concept is based on utilizing the properties of tip-like electrodes to focus and enhance the electric field, to reduce the breakdown voltage and cell capacitance and to eliminate the needs in a peripheral separation of the SiPM cells (avalanche regions). \n\nIn this article, we give an in-depth overview of the physical model of our new SiPM design and explain the working principle of the device. Further, we present the metrological characterization of our prototype samples and put them in relation to our simulations.\n\n\n\\section{Concept and Physical Model}\nThe simplified cross section of the TAPD is shown in Fig.~\\ref{fig:schematic_concept}.\nA spherical tip which consists of high doped n-silicon is placed in a low doped epitaxial layer.\nThe tip is connected to the bias supply through a quenching resistor $R_q$ placed on the surface.\nThe pn-junction on the surface of the tip causes a depletion and therefore an electric field around the sphere.\n\n\\subsection{Analytic Description}\n\nIn this article, the basic properties of the new SiPM concept are derived using a simplified model of an n-doped sphere inside an infinite p-doped bulk. The transition from n-doped to p-doped region is first approximated as an abrupt junction where the depleted charge has a box profile (in 3D spherical shells).\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\columnwidth]{Model_with_electricfield_plot.pdf}\n\\caption{Schematic of the spherical SiPM concept}\n\\label{fig:schematic_concept}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{Model_with_electricfield_plot3.pdf}\n\\caption{Dimensions and depletion layers of the spherical pn-junction model.}\n\\label{fig:concept_model} \n\\end{figure}\n\nIn thermal equilibrium, the charge carrier currents of drift and diffusion cancel out and the Fermi level $E_F$ is constant throughout the junction:\n\\begin{equation}\n\\begin{split}\nJ_{n} = & q \\mu_n \\left( n \\mathcal{E} + \\frac{kT}{q} \\frac{dn}{dr} \\right) = 0 \\\\\nJ_p = & q \\mu_p \\left( p \\mathcal{E} + \\frac{kT}{q} \\frac{dp}{dr} \\right) = 0\n\\end{split}\n\\label{eq:zero_net_current}\n\\end{equation}\nThe depletion approximation considers complete impurity ionization in the n- and p-region. At thermal equilibrium (subscript '$o$') the free charge carrier concentration can be simplified to $n_{no} \\approx N_D$ in the n-region and $p_{po} \\approx N_A $ in the p-region. \nThe condition of Eq.(\\ref{eq:zero_net_current}) requires a constant Fermi level and therefore a built-in voltage is present in the pn-junction. The built-in potential $\\Psi_{bi}$ can be written as: \n\\begin{equation}\n\\begin{split}\n\\Psi_{bi}\t& =\t\\frac{kT}{q} \\ln \\left( \\frac{n_{no}}{n_i} \\right) + \\frac{kT}{q} \\ln \\left( \\frac{p_{po}}{n_i} \\right) \\\\\n\t\t\t& \\approx \\frac{k_B T}{q} \\ln \\left(\\frac{N_D N_A}{n_i^2}\\right)\n\\end{split}\n\\end{equation}\nThis expression of the built-in potential is valid for planar and spherical junctions.\nThe electric field and the potential of the space charge region can be obtained solving the Poisson equation.\nThe one dimensional Poisson equation in spherical coordinates for any arbitrary charge density $\\rho(r)$ is \\cite{JayantBaliga1976}:\n\\begin{equation}\n\t\\frac{1}{r^2} \\frac{d}{d r} \\left( r^2 \\frac{d \\Psi_i(r)}{d r} \\right) = - \\frac{\\rho(r)}{\\varepsilon \\varepsilon_0}\n\\label{eq:poisson}\n\\end{equation}\n\n\nIn the first case of an abrupt junction, the charge carrier densities are given by the completely ionized acceptor and donor impurities, $N_A$ and $N_D$. \nThe ionized regions and the notation of the dimensional variables are shown in Fig.~\\ref{fig:concept_model}. \nThe analytic solution for the electric field in the n-region and p-region ($\\mathcal{E}_n$ and $\\mathcal{E}_p$) can be obtained from integration of Eq.(\\ref{eq:poisson}) with the boundary condition of zero electric field outside the depletion region: \n\\begin{align}\n\\begin{split}\n\\mathcal{E}_n(r)& = \\frac{ e N_D}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r^3 - r_D^3\\right)}{r^2},\\\\\n\\mathcal{E}_p(r) &= \\frac{ e N_D}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r_j^3 - r_D^3\\right)}{r^2} - \\frac{e N_A}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r^3 - r_j^3\\right)}{r^2}\n\\end{split}\n\\label{eq:Efield_ana}\n\\end{align}\nThe electric field in the n-region is zero at the inner depletion edge $r_D$ and will increase linear if the term $r_D^3 \/ r^2$ is small.\nIn the p-region, the expression of the electric field has two terms.\nThe first term is dominant if $N_D \\gg N_A$ and the electric field will decrease proportional to $1\/r^2$. \nThe second term becomes dominant if the first term is small enough with increasing radius and the electric field will equal zero at the outer depletion edge $r_A$.\n\nIn this concept, the donor concentration in the sphere is much higher than the acceptor concentration in the epitaxial layer.\nA exemplary electric field distribution according to Eq.(\\ref{eq:Efield_ana}) for different metallurgical junction radii is presented in Fig.~\\ref{fig:Efield_ana}. Here, the donor and acceptor concentrations were arbitrarily set to $N_A = \\SI{1e14}{\\per \\cm}$, $N_D = \\SI{1e18}{\\per \\cm}$.\nThe depletion width of each sphere size was adapted to create a depletion potential of \\SI{40}{\\V}.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{ElectricField_AnalyticSolution.pdf}\n\\caption{Electric field distribution in respect with the analytic solution for different sphere radii at a depletion potential of \\SI{40}{\\V}.}\n\\label{fig:Efield_ana}\n\\end{figure}\n\n\n\\subsection{Numerical Simulation}\n\nThe electric field in the previous section is derived using an approximation of the depletion regions as box profiles.\nIn a processed device, the dopant concentration will vary along the radius of the sphere.\nA dopant distribution is typically created by ion implantation and thermal annealing during different process steps. \nThe diffusion of impurities is thermally activated and occurs in direction of the concentration gradient.\nIn the case of the TAPD, especially the donor impurities of the tip will start to diffuse into the p-region.\nThree different structure sizes were processed and tested. The nominal junction radii of these devices are \\SI{0.6}{\\micro \\m}, \\SI{0.8}{\\micro \\m} and \\SI{1.0}{\\micro \\m}.\nThe measured dopant profile of these devices is used for the following simulations.\n\nThe complete impurity ionization (as assumed in Eq.(\\ref{eq:Efield_ana})) is prevented by diffusion of free charge carriers into the depleted regions \\cite{Sze2007}.\nWhile Eq.(\\ref{eq:zero_net_current}) deals with the steady-state, the continuity equations describe the net current flowing in and out of a region of interest. Here, the generation and recombination in the depletion region is neglected. The divergence operator has to be adapted for one dimensional spherical coordinates. The simplified continuity equations and the differential equation of the electric field from the system of coupled equation which has to be solved:\n\\begin{equation}\n\\frac{\\partial n}{\\partial t}\t= \\frac{1}{r^2} \\frac{\\partial}{\\partial r} r^2 \\left( \\mu_n \\mathcal{E}\tn + D_n \\frac{\\partial n}{\\partial r} \\right)\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial p}{\\partial t} = \\frac{1}{r^2} \\frac{\\partial }{\\partial r} r^2 \\left( - \\mu_p \\mathcal{E} p + D_p \\frac{\\partial p}{\\partial r} \\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:depl}\n\t\\frac{1}{r^2} \\frac{d}{d r} r^2 \\mathcal{E}(r) = \\frac{q}{\\varepsilon_s} \\left( p(r) - n(r) + D(r) \\right)\n\\end{equation}\nwhere $\\mu_n$, $\\mu_p$ are the electron and hole mobility and $D_n$, $D_p$ are the electron and hole diffusivities.\nThe spatial dependent doping profile $D(r)$ defines the p and n-region as:\n\\begin{equation}\nD(r) = \n\\begin{cases}\nr \\leq r_j; &N_D(r) \\\\\nr \\geq r_j; &-N_A(r)\n\\end{cases}\n\\end{equation}\n\nThe presented governing system of equations is considered in a spherical domain.\nThe boundary conditions in the center and on the spherical shell of the region of interest were of the Dirichlet type. \nWe used the finite difference method with a centred difference stencil for the spatial numerical approximation. \nThe approximation in time of the continuity equations was done using the Crank Nicolson method \\cite{crank_nicolson_1947}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Model_with_electricfield_plot2.pdf}\n\\caption{Schematic 2D electric field distribution due to the spherical depletion around the tip.}\n\\label{fig:2D-field}\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\columnwidth]{ElectricField_Ion_at_Breakdown.pdf} \n\\caption{Spatial distribution of the electric field and the ionization rates of electrons and holes for the smallest structure S06.}\n\\label{fig:plot_Efield} \n\\end{figure}\n\n\n\n\\begin{figure*}[bp]\n\\centering\n\\includegraphics[width=14cm]{TAPD_layout.pdf}\n\\caption{High density layout and bias supply of a TAPD array.}\n\\label{fig:layout}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:2D-field}, a schematic 2D distribution of the normalized electric field around the tip is illustrated. The highest field strength is located close to tip surface and decreases with increasing distance to the junction.\nThe electric field is present up to the passivated surface and a spherical active volume is available for electron attraction in direction of the n-region.\nHowever, impact ionization multiplication due to a high electric field occurs just close to tip surface.\nThe solution of the numerical approximation at the breakdown voltage (here \\SI{43.4}{V}) is presented in Fig.~\\ref{fig:plot_Efield} for a nominal junction radius of \\SI{0.6}{\\micro \\m}.\nThe ionization rates $\\alpha_n$ and $\\alpha_p$ for electrons and holes depend on the electric field:\n\\begin{equation}\n\\alpha(\\mathcal{E}) = \\alpha_\\infty \\exp\\left(\\frac{-b}{\\mathcal{E}} \\right)\n\\end{equation}\nwith $\\alpha_\\infty$ and $b$ at room temperature taken from \\cite{VanOverstraeten1970} for electrons and holes, respectively.\nIn order to estimate the breakdown voltage of different sphere sizes, the multiplication of charge carriers due to impact ionization has to be evaluated using the spatial dependent ionization rates at different bias voltages.\nA breakdown occurs if the impact ionization multiplication becomes infinite which is equivalent to the condition that the ionization integral \\cite{McINTYRE1966} equals one:\n\\begin{equation}\n\\int_{r_D}^{r_A} \\alpha_n \\exp\\left[ - \\int_{r}^{r_A} \\left( \\alpha_n - \\alpha_p \\right) dr' \\right] dr = 1\n\\end{equation}\nThe results of this evaluation are presented in Tab.~\\ref{tab:structures}. The breakdown voltage increases with the size of the tip due to a lower electric field at the same bias voltage.\n\\begin{table}[h]\n\\caption{Tested device structures and nominal radii of the metallurgical junction.}\n\\label{tab:structures} \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\nStructure Name & Nominal Radius ($r_j$) & Breakdown Voltage \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nS06 & \\SI{0.6}{\\micro \\m} & \\SI{43.4}{V} \\\\\nS08 & \\SI{0.8}{\\micro \\m} & \\SI{50.7}{V} \\\\\nS10 & \\SI{1.0}{\\micro \\m} & \\SI{53.9}{V} \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\nThe range of the electric field directly affects the photon detection capability regarding light with increasing wavelength.\nThe absorption of a photon inside of a material at a certain depth $x$ can be described by the Beer-Lambert law.\nThe probability for the generation of charge carriers at a certain depth inside the epitaxial layer decreases exponentially with increasing distance to the surface.\nAdditionally, in the range of visible light, the absorption coefficient decreases with increasing wavelength.\nA SiPM with a high charge collection efficiency aims to absorb as much of the incoming photon flux as possible. \nConsequently, the electric field has to be present as deep as possible.\nPhotoelectrons generated at a distance to the tip are first accelerated due the drift field in direction of the multiplication region (high electric field) and a Geiger discharge is triggered.\n\n\nThe simulated range of the electric field into the epitaxial layer for the three structure sizes at different bias voltages is shown in Fig.~\\ref{fig:depletion_width}.\nAll structures reach at least \\SI{8}{\\micro \\m} at their respective breakdown voltage.\nThe maximal active volume is reached placing the center of the sphere at a depth of the maximal depletion range. In this configuration, photoelectrons which are generated close to the surface and up to the maximal active depth are detected. This advantage of the new concept allows for a high photon detection efficiency in a wide spectral range. Theoretically, it is possible to reach a total active depth of \\SI{20}{\\micro m} with the current technology (see Fig.~\\ref{fig:depletion_width}) and consequently \na photon detection efficiency above \\SI{30}{\\%} at \\SI{905}{nm}.\nOur prototypes were processed in an epitaxial layer of \\SI{12}{\\micro m}, which limits the maximal depleted volume for all structure types. \n\n\n\n\\subsection{Cell Placement}\n\nThe operation of a SiPM above the breakdown voltage in Geiger-Mode requires a serial connected quenching resistor to keep the device in a quasi-stable state. \nThe potential of the epitaxial layer during operation is set to ground while the tip is biased through the resistor.\nThe spherical shape of the active volume allows a high density placement of multiple single cells in an array.\nIn the top view, the cells are placed in a hexagonal lattice achieving a theoretical packing density of $\\eta \\approx \\SI{90.7}{\\percent}$.\nThe bias voltage is supplied through a metal grid connected to the quenching resistor of each cell.\nThe schematic layout is shown in Fig.~\\ref{fig:layout}.\nThe aim of the layout is to cover the smallest possible area with metal. \nThe metal on the surface blocks visible light therefore reducing the active area of the SiPM for light detection.\nThe presented placement offers the advantage to use just one metal line for two rows of SiPM cells.\n\nThe relation of uncovered area to total area is called geometric efficiency (see also Sec.~\\ref{sec:PDE}).\nCompared to planar devices, the new concept offers the advantage of a frameless layout. The TAPD cells are biased through the center and can be placed as close as possible next to each other without losing active area.\n\n\nA typical parameter for SiPMs is the cell pitch which equals the cell size for planar devices. We defined the cell pitch for the TAPD as the distance of pillars (see Fig.~\\ref{fig:layout}).\nA small cell pitch offers the advantage of a higher dynamic range \\cite{Vinogradov2011}.\nThe absorption of a photoelectron in the active volume triggers a Geiger-discharge of a single cell. During the discharge and the following recharge of the cell, the cell is partially inactive for incoming light \\cite{Oide2007,Renker2009}.\nConsequently, the SiPM array shown in Fig.~\\ref{fig:layout} could only detect a maximum of 24 photons during a fast light pulse (e.g. shorter than recharge time).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{DepletionRadius_vs_Bias_Graph.pdf}\n\\caption{Range of the electric field for the three structure sizes.}\n\\label{fig:depletion_width}\n\\end{figure}\n\nThe aim of a high dynamic range and a high geometric efficiency is in general contradictory.\nRegarding the results of Fig.~\\ref{fig:depletion_width}, we chose a cell pitch of \\SI{15}{\\micro \\m} for the processed devices.\nThe active areas of two neighbouring cells for the smallest structure S06 are just in contact.\nThe geometric efficiency of our device is only determined by the opaque materials like metal lines and semi-transparent materials like the quenching resistors, which are located on top of the active area. \nFor the prototypes with a \\SI{15}{\\micro m} pitch, the nominal geometric efficiency of \\SI{83}{\\%} was realized. The value was calculated from the layout, accounting metal lines and quenching resistors as opaque.\n\n\n\\subsection{Depletion Layer Capacitance}\n\nThe depletion layer around the tip creates a certain amount of charge on each side of the junction whereas the total charge is zero.\nThe incremental charge $dQ_D$ of one side of the junction upon an incremental change of the bias voltage defines the depletion layer capacitance $C_D = dQ_D \/ dV$.\nThe spatial depletion of the respective structures was obtained with the solution of the continuity equations (see Eq.(\\ref{eq:depl})).\nThe simulated depletion layer capacitance dependent on the bias voltage is presented in Fig.~\\ref{fig:depletion_cap}.\nAll three structures show a similar curve progression while the increased tip size leads to a higher capacitance.\n\nThe simulated depletion layer capacitance is part of the total cell capacitance $C_{\\mathrm{cell}}$, which includes additionally the parasitic capacitance due to the pillar and the connection to the quenching resistor.\nThe gain of a single cell is proportional to the cell capacitance and can be expressed as: \n\\begin{equation}\nG = \\frac{Q}{e} = \\frac{\\Delta V \\cdot C_{\\mathrm{cell}} }{e}\n\\end{equation}\nwhere $\\Delta V$ stands for the excess bias voltage.\nThe main goal for the presented new SiPM design was to a achieve a high photon detection efficiency over a wide spectral range.\nThe recovery time (recharge time) should be short to provide a fast device with a high dynamic range. Consequently, a low cell capacitance is beneficial.\nIn respect with the simulations of this section, we chose the smallest device S06 as the most promising structure. \n\n\n\\section{Metrological Characterization}\n\\subsection{Single electron response}\n\\label{sec:SER}\nTo achieve a high photon count rate and a strong ambient light immunity, a fast recovery of the micro-cells is required. In Fig.~\\ref{fig:SER}, the single electron response (SER) of the 3D-SiPM is shown at $\\Delta V=4\\text{\\,V}$.\nIt was measured as the voltage drop across a \\SI{25}{\\Omega} load resistor. \nThe decay part of the pulse consists of two exponential components with time constants of $\\tau_{1}\\approx \\SI{0.5}{\\nano s}$ and $\\tau_{2}\\approx \\SI{4.3}{ns}$.\nAfter approximately \\SI{9.5}{\\nano s}, the micro-cells are recovered to \\SI{90}{\\%} of their maximum charge. Using the double-light-pulse method, as proposed in \\cite{Popova2012}, comparable results were achieved.\nThis confirms that the SER shape reflects the true recovery process in this case. With this result, the TAPD has the fastest recovery time with respect to state-of-the-art planar SiPMs with enhanced red-sensitivity \\cite{Broadcom2020,RB_Series2020}. \n\\begin{figure}[h!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{SER.pdf}}\n\\caption{Normalized pulseshape of the S06 structure at an excess bias voltage of \\SI{4}{V}. }\n\\label{fig:SER} \n\\end{figure}\n\nFig.~\\ref{fig:SPE_Spectrum} provides an example of a single photoelectron charge spectrum. For this spectrum, we used a pulsed laser illumination with a pulse width of \\SI{70}{ps}. The acquisition was synchronous with the light pulses. The charge was integrated within a time-window of \\SI{10}{\\nano s}. The peaks up to 4 photoelectrons are well separated, which makes precise single photon counting possible.\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{Spectrum_EPJC.pdf}}\n\\caption{Single photoelectron charge spectrum of the S06 structure at an excess bias voltage of \\SI{4}{V}.}\n\\label{fig:SPE_Spectrum} \n\\end{figure}\n\n\\subsection{Photon detection efficiency}\n\\label{sec:PDE}\nThe photon detection efficiency (PDE) describes the capability of the sensor to detect light as the ratio of the average number of incident and the average number of detected photons. In Eq.(\\ref{eq:PDE_Def}), the PDE is described as a product of three quantities:\n\n\\begin{equation}\n\\label{eq:PDE_Def}\nPDE=\\varepsilon \\cdot QE \\cdot P_{trigg}\n\\end{equation}\n\n(i) The geometric efficiency $\\varepsilon$ describes the fraction of the SiPM area, which is able to detect photons. The area of the SiPM which is not sensitive to light is mainly due to the metal lines for signal readout, the quenching resistors, guard rings for electric field attenuation towards the micro-cell edges and trenches for the suppression of optical crosstalk.\n\n(ii) The quantum efficiency $QE$ describes the efficiency to collect a fraction of charge carriers that a photon generated within the active volume of a micro-cell.\nTo maximize the PDE for blue light, the depletion region has to be extended as close to the surface as possible. Lower energetic photons are also absorbed at larger depths. To reach an enhanced detection of red and near-infrared light, the active region has to reach deeper inside the silicon. With our devices, we reach a spherical depletion volumes with radii of about \\SI{8}{\\micro m} to \\SI{9}{\\micro m} (see Fig.~\\ref{fig:depletion_width}).\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PDE_vs_Overvoltage_EPJC.pdf}}\n\\caption{Photon detection efficiency vs. the excess bias voltage for the S06 structure. The lines are drawn as eye guides.}\n\\label{fig:PDE_vs_EBV} \n\\end{figure}\n\n\n(iii) The avalanche triggering probability $P_{trigg}$ describes the probability that a generated e-h pair will successfully initiate an avalanche breakdown by impact ionization. \n \nIn this work, the PDE was measured by using the continuous low-level light method, as reported in \\cite{Piemonte2012,Vachon2018,Vinogradov2016,Engelmann2018}. The incident photon rate was determined by a calibrated reference PIN-diode \\cite{Thorlabs2019}. Both, the SiPM and the reference diode were homogeneously illuminated trough a lens. The homogeneous part of the light spot was significantly larger than the active area of the photosensors. The photon rate was determined as the difference of the SiPM pulse count rate during illumination and at dark conditions. The count rates were measured according to the method described in Sec.~\\ref{sec:DCR_PCP}.\nIn Fig.~\\ref{fig:PDE_vs_EBV}, the PDE is shown as a function of the excess bias voltage for several wavelengths from \\SI{460}{\\nano m} to \\SI{905}{\\nano m}. In this spectral range, the PDE reaches \\SI{90}{\\%} of its saturation value at excess bias voltages between \\SI{4}{V} and \\SI{5}{V}. This is comparable to state-of-the-art planar SiPMs which are optimized for the detection of blue light and hence have smaller depletion volumes \\cite{Otte2017}.\n\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PDE_vs_WL_EPJC.pdf}}\n\\caption{Photon detection efficiency vs. wavelength for the S06 structure at an excess bias voltage of \\SI{5}{V}.}\n\\label{fig:PDE_vs_WL} \n\\end{figure}\n\n\n\nIn Fig.~\\ref{fig:PDE_vs_WL}, the PDE is shown at different wavelengths. This result was obtained by converting the spectral response measurement with a monochromator into an absolute PDE measurement as described in \\cite{Otte2017}.\nThe TAPD demonstrates the highest peak PDE value of \\SI{73}{\\%} at \\SI{600}{\\nano m} with respect to state of the art devices with a \\SI{15}{\\micro m} pitch size \\cite{Gola2019,Yamamoto2018}. \nAdditionally, the PDE curve does not show the typical fast decrease with increasing wavelengths and remains above a value of \\SI{45}{\\%} up to a wavelength of $800\\text{\\,nm}$. The measured PDE is in good agreement with the expected values when assuming a Geiger-efficiency of \\SI{90}{\\%} (see Eq.(\\ref{eq:PDE_Def}) and dashed curve in Fig.~\\ref{fig:PDE_vs_WL}).\nFor wavelengths in the NIR-regime, we expect the geometric efficiency to increase approximately to \\SI{90}{\\%} due to the decreasing absorption by the semi-transparent quenching resistors.\nThe oscillations in the PDE curve are caused by destructive interference due to the protective $Si\/SiO_2$ stack on top of the entrance window of the prototype devices.\nThe results of the applied method are in agreement with the well-known method based on pulsed laser illumination \\cite{Eckert2010,Otte2017}.\nThe applied method offers the possibility to directly measure the absolute PDE for a larger number of wavelengths due to the easy access to LEDs of different wavelengths. \n\n\n\\subsection{Dark count rate and delayed correlated pulses}\n\\label{sec:DCR_PCP}\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{DCR_vs_EBV_EPJC.pdf}}\n\\caption{Dark count rate vs. the excess bias voltage for the S06 structure. The error bars represent the standard deviation of 5 samples. The line is drawn as an eye guide.}\n\\label{fig:DCR_vs_EBV} \n\\end{figure}\n\nIn this work, the dark count rate and the probability of delayed correlated pulses is determined by using the method proposed in \\cite{Vinogradov2016}. The method is based on the analysis of the complementary cumulative distribution function (CCDF) of pulses subsequent to a primary dark pulse. Contrary to the pulse counting approach, the applied method provides the benefit that delayed correlated pulses which exceed the detection threshold do not contribute to the dark count rate of the device \\cite{Engelmann2018}. This advantage is of special significance for SiPMs with a fast recovery time, like the TAPD, where delayed correlated pulses reach the maximum charge after a few ns.\nThe CCDF-method is also suited to measure the count rate of a continuous Poissonian photon output from a light source. In this way the photon detection efficiency can be measured (cf. Sec.~\\ref{sec:PDE}).\n\nAs delayed correlated pulses, we understand pulses which may be caused by two kinds of effects: \n\\\\\nThe first one is afterpulsing. Here, trapping centres present as energy states within the bandgap may capture electrons or holes from the conduction or valence band and re-emit them after a certain delay-time $\\Delta t$ into the same band. If the trapping centre is located in the active region of a micro-cell, the re-emitted charge carrier has a finite probability to trigger a subsequent avalanche in the same micro-cell. The delay-times depend on the respective trap type and may vary by many orders of magnitude.\n\\\\\nThe second effect is the so called delayed optical crosstalk.\nHere, one or more photons which are emitted during the avalanche breakdown are absorbed outside the high field region. The generated minority charge carriers then diffuse towards the active region and are able to trigger consecutive breakdowns of the original or a neighbouring micro-cell. This process is significantly slower compared to the prompt optical crosstalk.\n\nThe applied method is only valid under the assumption that the time constants of the delayed correlated effects are smaller with respect to the reciprocal of the dark count rate. In \\cite{Engelmann_PhD_2018}, the applied measurement procedure is described in detail.\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PCP_T0P6_EPJC.pdf}}\n\\caption{Probability of delayed correlated pulses vs. the excess bias voltage for the S06 structure. The error bars represent the standard deviation of 5 samples.}\n\\label{fig:PCP_T0P6} \n\\end{figure}\n\nIn Fig.~\\ref{fig:DCR_vs_EBV}, the dark count rate is shown as a function of the excess bias voltage.\nAt recommended excess bias voltages between $\\Delta V=\\SI{4}{V}$ and $\\Delta V=\\SI{5}{V}$, the prototypes show a DCR between $\\SI{700}{kHz\/mm^2}$ and $\\SI{1.3}{MHz\/mm^2}$. In comparison, KETEK's planar \\SI{15}{\\micro m} pitch SiPM shows a DCR of typically $125\\text{\\,kHz}\/\\text{mm}^2$ at $\\Delta V=\\SI{5}{V}$ \\cite{KETEK2019}.\n\nThis result matches our expectations, since the active volume of the TAPD is about a factor 10 larger with respect to the planar structures. The state of the art red-sensitive SiPMs from other manufacturers show dark count rates between $\\SI{600}{kHz\/mm^2}$ \\cite{Broadcom2020} and $\\SI{3.5}{MHz\/mm^2}$ \\cite{RB_Series2020} at the recommended operation voltages.\nIn Fig.~\\ref{fig:PCP_T0P6}, the probability of delayed correlated pulses is plotted as a function of the excess bias voltage. It is below \\SI{3}{\\%} at excess bias voltages up to $\\Delta V=\\SI{5}{V}$.\n\n\n\n\\subsection{Prompt optical crosstalk}\nDuring an avalanche breakdown, optical photons are generated by a variety of processes. These photons are able to propagate to neighbouring micro-cells and initiate further avalanche breakdowns. The propagation may occur by a direct path or by several reflections at the top and bottom side of the device. In either case, the time difference between the first and the consecutive pulse is not sufficient for a distinction of the two pulses. \nFor this reason, only one pulse with an amplitude of multiple p.e. (photoelectron equivalent) is registered. This effect is called \"prompt optical crosstalk\" (CT). The prompt optical crosstalk probability scales with the number of generated photons during an avalanche breakdown, the geometric cross-section for the interaction between two micro-cells and the avalanche triggering probability.\nThe prompt optical crosstalk probability $P_{CT}$ is estimated as shown in Eq.(\\ref{eq:P_CT_estimate}). Here, $DCR_{n}$ is the dark count rate measured by the pulse counting method with a discriminator threshold of $n\\text{ p.e.}$ \\cite{Renker2009,Eckert2010,ONeill2015,Otte2017}:\n\\begin{equation}\n\\label{eq:P_CT_estimate}\nP_{CT}\\approx \\frac{DCR_{1.5}}{DCR_{0.5}}\n\\end{equation}\nAt typical excess bias voltages between \\SI{4}{V} and \\SI{5}{V}, we measure crosstalk probabilities between \\SI{27}{\\%} and \\SI{35}{\\%}. For comparison: The reported values from other state-of-the-art devices with a red-enhanced sensitivity are between \\SI{20}{\\%} and \\SI{43}{\\%} \\cite{Broadcom2020,RB_Series2020}.\n\n\\begin{figure}[h!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{CTP_vs_EBV_EPJC.pdf}}\n\\caption{Prompt optical crosstalk probability of the S06 structure.}\n\\label{CTP_vs_EBV} \n\\end{figure}\n\n\n\\FloatBarrier\n\\subsection{Gain}\n\\label{sec:Gain}\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{GAIN.pdf}}\n\\caption{Gain vs. the excess bias voltage for the S06, S08 and the S10 structure.}\n\\label{fig:Gain_vs_EBV} \n\\end{figure}\n\nThe fast recovery time, described in Sec.~\\ref{sec:SER}, is realized by a strongly reduced micro-cell capacity. As a consequence, the intrinsic gain is about a factor of 20 lower with respect to the KETEK's planar SiPM products \\cite{KETEK2019}. In Fig.~\\ref{fig:Gain_vs_EBV}, the absolute gain is shown as a function of the excess bias voltage for two different structures.\nThe higher gain of the S10 structure can be attributed to the larger ball radius and the larger contribution from the parasitic capacitance of the quenching resistor.\nTo measure the gain, we applied a combination of the single photoelectron charge spectrum and the dark current \\cite{Pagano2012,Engelmann_PhD_2018}. \nIn this procedure, the single photoelectron spectrum was recorded in dark environment with a trigger threshold set to \\SI{0.5}{p.e.}.\nTo receive the probability density function (PDF) of the primary and secondary events, the spectrum was normalized to the total number of detected events.\nThe expected number of firing micro-cells per initial photoelectron is described by the mean value of the PDF which is equivalent to the excess charge factor (ECF) \\cite{Klanner2019}. The dark count rate was determined as described in Sec.~\\ref{sec:DCR_PCP}. In this case, the contribution from afterpulses is neglected. Since the investigated devices have a low afterpulsing probability (see Fig.~\\ref{fig:PCP_T0P6}), this approach is reasonable. The gain $G$ is then determined by using Eq.(\\ref{eq:Gain}). $I_{dark}$ is the dark current at the respective operation voltage:\n\\begin{equation}\n\\label{eq:Gain}\nG(V)=\\frac{I_{dark}(V)}{q\\cdot DCR(V) \\cdot ECF(V)}\n\\end{equation}\nFrom the slopes of the linear fits in Fig.~\\ref{fig:Gain_vs_EBV}, we extract the micro-cell capacitances. The experimentally determined micro-cell capacities differ from the simulated ones by about \\SI{3.3}{\\femto F} to \\SI{3.8}{\\femto F} (see Fig.~\\ref{fig:depletion_cap}). We attribute this discrepancy to the parasitic capacitance of the quenching resistor and variation of the real geometry from the simulated spherical one. Here, we would like to point out that we expect the parasitic capacitance to be larger than the micro-cell capacitance and hence strongly contribute to the charge output of the TAPD.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Capacitance_vs_Bias_Graph.pdf}\n\\caption{Simulated depletion layer capacitance and measured total cell capacitance (plotted at $V_{bd}$).}\n\\label{fig:depletion_cap}\n\\end{figure}\n\\subsection{Temperature dependence of the breakdown voltage}\nEspecially in automotive applications, the systems must operate in a wide temperature range. Independent of whether the photosensor has a temperature stabilization\/compensation or not, a low temperature coefficient of the breakdown voltage ($V_{bd}$) is beneficial. For our TAPD, we measure a linear decrease of $V_{bd}$ with temperature (see Fig.~\\ref{fig:VBD_vs_Temp}).\nThe slopes increase with increasing pillar diameter from $\\SI{26}{mV\/^\\circ C}$ for the S06 structure to $\\SI{31.5}{mV\/^\\circ C}$ for the S10 structure. These values are comparable with respect to $\\SI{22}{mV\/^\\circ C}$ for planar KETEK SiPMs \\cite{KETEK2019}, despite the clearly larger depletion width and the increased breakdown voltage.\nThe breakdown voltage was determined from the inverse logarithmic derivative (ILD) of the reverse current-voltage-characteristic with low-level light illumination, as described in \\cite{Klanner2017}.\nThe simulated breakdown voltages (see Tab.~\\ref{tab:structures}) overestimate the experimental values at room temperature. The discrepancy decreases from \\SI{5}{V} for the S10, to \\SI{4.7}{V} for the S08 and to \\SI{1}{V} for the S06 structure.\nWe attribute this observation to the fact that the deviation of the processed structures from a perfect sphere increases with increasing tip size.\nFor the S10 structure, the shape of the tip is closer to the one of an ellipse than a sphere. Here, the breakdown voltage is defined by the point with the lowest curvature.\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{VBD_vs_Temp_EPJC.pdf}}\n\\caption{Breakdown voltage vs. temperature. The error bars are within the symbols.}\n\\label{fig:VBD_vs_Temp} \n\\end{figure}\n\n\\newpage\n\\section{Conclusion}\n\\label{sec_Conclusion}\nIn this work, we presented a new Silicon Photomultiplier, the Tip Avalanche Photodiode. The TAPD concept is based on quasi-spherical electrodes (tips) which are placed inside an epitaxial layer.\nWe gave a theoretical overview of the physical models and presented the metrological characterization of the existing prototypes.\nOur simulations are in a good agreement with the obtained experimental data.\nWith the TAPD we achieved a record photon detection efficiency over a wide spectral range from \\SI{400}{\\nano m} to \\SI{905}{\\nano m} with a small micro-cell pitch of 15 \u00b5m. In combination with a high dynamic range, a fast recovery time (\\SI{4}{\\nano s}), and a low temperature coefficient of the breakdown voltage (\\SI{26}{mV\/K}), this SiPM is a promising detector for a large variety of applications.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent Modulational instability of high-frequency nonlinear \nwaves is a common process in a variety of circumstances involving \nwave propagation in continuous systems. Modulational processes can be seen \nto occur in a wide range of physical situations, from nonlinear waves in \nplasmas \\cite{dter78} to nonlinear electromagnetic waves propagating in optical \nfibers \\cite{mal97}. What usually happens in all those cases is that due to \ngeneric nonlinear interactions, the amplitude of a high-frequency carrier \ndevelops slow modulations in space and time. If the modulations are indeed much \nslower than the high-frequencies involved, one can obtain simplified equations \ndescribing the dynamics of the slowly varying amplitudes solely, the amplitude \nequations \\cite{lili91,novo99,olig96,riz98,erich98}. In the present analysis \nwe consider systems that become integrable in this modulational limit, a \nfeature often displayed. Should this be indeed the case, \nno spatiotemporal chaos would be observed there. The basic interest \nthen would be to see what happens when the approximations leading to \nmodulational approximations cease to be satisfied. The paper is organized as \nfollows: in \\S 2 we introduce our model equation and discuss \nhow and when it can be approximated by appropriate \namplitude equations; in \\S 3 we investigate the modulational process from the \npoint of view of nonlinear dynamics; in \\S4 we perform full spatiotemporal \nsimulations and compare the results with those obtained in \\S 3; and in \\S 5 \nwe conclude the work. \n\n\\section{Model equation, modulational approximations, and amplitude equations}\n\nIn the present paper we \nfocus attention on a nonlinear variant of the Klein-Gordon equation (NKGE) to \ninvestigate the breakdown of modulational approximations in the context of \nnonlinear wave fields. The NKGE used here reads\n\\begin{equation}\n\\partial_t^2 A(x,t) - \\partial_x^2 A(x,t) + {\\partial \\Phi \\over \\partial A}=0,\n\\label{eqzero}\n\\end{equation}\n($\\partial_t \\equiv \\partial \/ \\partial t, \\> \n\\partial_x \\equiv \\partial \/ \\partial x$) \nwhere we write the generalized nonlinear potential as \n\\begin{equation}\n\\Phi(A) = \\omega^2 \\> {A(x,t)^2 \\over 2} - {A(x,t)^4 \\over 4} \n+ {A(x,t)^6 \\over 6},\n\\label{phi}\n\\end{equation}\n$\\omega$ playing the role of a linear frequency which will set the \nfast time scale. The remaining coefficients on the right-hand-side \nwere chosen to \nallow for modulational instability and saturation; we shall see that \nwhile the negative sign of the second fulfills the condition for \nmodulational instability, the positive sign of the third provides saturation. \nThe choice of their numerical values is arbitrary, but our results are \nnevertheless generic. The NKGE is known to describe wave propagation in \nnonlinear media and the idea here is to see how the dynamics changes as a \nfunction of the parameters of the theory: wave amplitude, and time and length \nscales. \n\nLet us first derive the conditions for slow modulations. We start by supposing \nthat the field $A(x,t)$ be expressed in the form \n\\begin{equation}\nA(x,t) = \\tilde A (x,t) e^{i \\omega t} + {\\rm complex\\>\\>conjugate}. \n\\label{fatora}\n\\end{equation}\nThen, if one assumes slow modulations and discards terms like \n$\\partial_t^2 \\tilde A$ and the highest-order power of the potential \n$\\Phi$, one obtains \n\\begin{equation}\n2 i \\omega \\partial_t \\tilde A(x,t) - \\partial_x^2 \\tilde A(x,t) - \n3 |\\tilde A(x,t)|^2 \\tilde A(x,t) = 0, \n\\label{schroclassica}\n\\end{equation}\nwhich is, apart from some rescalings, the Nonlinear Schr\\\"odinger Equation - \nwe shall refer to it as NLSE here - an integrable equation. \n\nThis is no novelty; it is known that the modulational approximation is \nobtainable \nwhen there is a great disparity between the time scales of the high-frequency \n$\\omega$ and the modulational frequency, we call it $\\Omega$, such that \nterms of order $\\Omega^2 \\tilde A$, when compared to $\\omega^2 \\tilde A$, can be \ndropped from the governing equation. The magnitude of the modulational frequency \ncan be estimated as follows. Consider Eq. (\\ref{schroclassica}) and suppose a \ndemocratic balance among the magnitude of its various terms; \n$\\omega \\partial_t \\tilde A \\sim \\partial_x^2 \\tilde A \\sim {\\tilde A}^3$. Then one \nobtains for $\\partial_t \\rightarrow \\Omega$,\n\\begin{equation}\n{\\Omega \\over \\omega} \\sim ({\\tilde A \\over \\omega})^2.\n\\label{condition}\n\\end{equation}\nIt is thus clear that the modulational approximation is valid only \nwhen $\\tilde A \\ll \\omega$, since this condition slows down the modulational \nprocess causing $\\Omega \\ll \\omega$. The next question would be on what is to \nbe expected when the modulational approach ceases to be valid. Before \nproceeding along this line, let us mention that a stability analysis \ncan be performed on Eq. (\\ref{schroclassica}). One perturbs an homogeneous \nself-sustained state with small fluctuations of a given wavevector $k>0$ (we \nchoose $k>0$ here, but the theory is invariant when $k \\rightarrow - k$) and \nafter some algebra one concludes that: (i) the field in the homogeneous state, \nlet us call this field $A_h$, is given by \n\\begin{equation}\nA_h = a_o e^{- i {3 a_o^2 \\over 2 \\omega} t}\n\\label{comp1}\n\\end{equation}\nwhere $a_o$ is an arbitrary \namplitude parameter and where the exponential term should be seen as providing a \nsmall nonlinear correction to the linear frequency $\\omega$; and (ii) the \nperturbation is unstable when \n\\begin{equation}\nk < k_{tr} \\equiv \\sqrt{6} a_o.\n\\label{modinst}\n\\end{equation}\nwith maximum growth rate at\n\\begin{equation}\nk_{max} \\equiv {k_{tr} \\over \\sqrt{2}}.\n\\label{maxgrowth}\n\\end{equation}\nWhen unstable, the homogeneous state typically evolves towards a state \npopulated by regular structures which can be formed precisely because \nthe underlying governing NLSE, Eq. (\\ref{schroclassica}), is of the \nintegrable type as mentioned earlier. \n\n\\section{Beyond the modulational approximation}\n\nTo advance the analysis beyond the modulational regimes we start from \nthe basic equation, Eq. (\\ref{eqzero}), but do not use approximation \n$\\partial_t^2 \\tilde A \\ll \\omega^2 \\tilde A$ leading to Eq. (\\ref{schroclassica}) \nand eventually to condition $\\tilde A \\ll \\omega$. The idea is precisely to \nexamine what happens as the ratio $\\tilde A \/ \\omega$ of Eq. (\\ref{condition}) \ngrows from values much smaller, up to values comparable to the unit.\n\nOur first task is to examine how the regular and well known modulational \ninstability analyzed in the previous section comes directly from Eq. (\\ref{eqzero}). \nTo do that, let us write a truncated solution as the sum of an \nhomogeneous term plus fluctuations with wavevector $k$, \n$A = A_h(t) + A_1(t) (e^{i k x} + e^{-i k x})$, $A_h$ and $A_1$ real. The \ntruncation, that discards higher harmonics, is legitimate within linear \nregimes, but fortunately we shall see that it is not as restrictive as it might \nappear even in nonlinear regimes. The general idea favouring truncation here \nis that in the reasonable situation where modes with the fastest growth rates \nare more strongly excited, relation (\\ref{maxgrowth}) indicates that second \nharmonics are already outside the instability band since \n$2 k_{max} = \\sqrt{2} k_{tr} > k_{tr}$. Under these circumstance \none would be led to think that the most important modes would be the \nhomogeneous and those at the fundamental spatial harmonic. We shall actually \nsee that the truncation provides a nice and representative approach to the \ncase of regular regimes. \n\nNote that we are already considering the amplitudes of the exponential functions as \nequal. This results from a simplified symmetrical choice of initial conditions and is \ntotally consistent with the real character of Eq. (\\ref{eqzero}). After some lengthy \nalgebra, one finds out that the coupled nonlinear dynamics of the fields $A_h$ and \n$A_1$ is governed by the Hamiltonian\n$$\nH={p_h^2 \\over 2} + {\\omega^2 q_h^2 \\over 2} - {q_h^4 \\over 2} + \n{2 q_h^6 \\over 3} + \n{p_1^2 \\over 2} + {\\chi^2 q_1^2 \\over 2} - {3 q_1^4 \\over 4} + \n{5 q_1^6 \\over 3} - \n$$\n\\begin{equation}\n3 q_h^2 q_1^2 + 10 q_h^4 q_1^2 + 15 q_h^2 q_1^4,\n\\label{hamiltonian}\n\\end{equation}\nwhere $\\chi^2 \\equiv \\omega^2+k^2$, $q_h = A_h\/\\sqrt{2}$, $q_1 = A_1$, and \nwhere the $p$'s denote the two momenta conjugate to the respective \n$q$-coordinates.\n\nThe Hamiltonian (\\ref{hamiltonian}) can be informative. As a first \ninstance it can be used to determine the stability properties of the homogeneous \npump, as mentioned before. To see this, assume that in average \n$q_h \\gg q_1$ and solve the dynamics perturbatively. In zeroth order, \none would have the following Hamiltonian governing the $(p_h,q_h)$ \ndynamics:\n\\begin{equation}\nh_o = {p_h^2 \\over 2} + {\\omega^2 q_h^2 \\over 2} - {q_h^4 \\over 2} + \n{2 q_h^6 \\over 3}.\n\\label{ho}\n\\end{equation}\nWith help of action-angle variables ($\\rho,\\Theta$) for the \nzeroth-order part and conventional perturbative techniques \nthe solution reads \n\\begin{equation}\nq_h = \\sqrt{2 \\rho \\over \\omega} \\cos(\\omega t - {3 \\over 2} \n{\\rho \\over \\omega^2} t), \n\\label{ordemzero}\n\\end{equation}\nif $\\rho$, the amplitude parameter, is not too large. Note that the \noscillatory frequency undergoes a small nonlinear correction which \nis determined by the quartic term in $q_h$ of the Hamiltonian $h_o$. \n\nNext we consider the driven Hamiltonian controlling the dynamics of \nthe canonical pair $(p_1,q_1)$\n\\begin{equation}\nh_1 = {p_1^2 \\over 2} + {\\chi^2 q_1^2 \\over 2} + 3\\>q_h^2 q_1^2,\n\\label{h1}\n\\end{equation}\nwhere we recall that the pair $(p_1,q_1)$ describes the inhomogeneity \nof the system. One again introduces action-angle variables $(I,\\theta)$ \nto rewrite the linear Hamiltonian (\\ref{h1}) in the form\n\\begin{equation}\nh_1 = \\chi I + 12 {\\rho \\over \\omega} {I \\over \\chi} \\cos^2 \\theta \n\\cos^2 (\\omega t - {3 \\over 2} {\\rho \\over \\omega^2} t),\n\\label{h11}\n\\end{equation}\nfrom which we obtain the resonant form\n\\begin{equation}\nh_{1,r} = {1 \\over 2 \\omega^2}(k^2 - 3\\>\\rho) I + \n{3\\>\\rho I \\over 2 \\omega^2} \\cos (2 \\phi),\n\\label{hres}\n\\end{equation}\nwith $\\phi = \\theta -(\\omega - 3\\rho \/ 2\\omega^2)t$, and where use is made of the \napproximation $\\chi \\approx \\omega + k^2\/(2 \\omega)$ valid when \n$k \\ll \\omega$. Now consider the stability of a \nsmall perturbation $I \\sim 0$. For this initial condition \n$h_{1,r} \\rightarrow 0$. \nSince $h_{1,r}$ is a constant of motion, solutions for arbitrarily large \nvalues of $I$, what would indicate instability, are possible only \nwhen $|\\cos(2 \\phi)| \\le 1$. \nFrom the resonant Hamiltonian (\\ref{hres}), \nthis demands \n\\begin{equation}\nk < \\sqrt{6 \\rho \\over \\omega},\n\\label{cond2}\n\\end{equation}\nand also indicates that maximum growth rate for $I$ occurs at \n$k_{max}=\\sqrt{3 \\rho \/ \\omega}$. Comparisons of temporal dependence \nof Eqs. (\\ref{fatora}), (\\ref{comp1}) and (\\ref{ordemzero}) shows that \n$\\rho \/ \\omega = a_o^2$, and that conditions (\\ref{modinst}) and (\\ref{cond2}) \nare therefore one and the same as they should be. In other words, starting \nfrom the full nonlinear wave equation, we recover the typical results \nnaturally yielded by the NLSE.\n\nBut the Hamiltonian (\\ref{hamiltonian}) gives further information because \napart the truncation, it is not an adiabatic approximation like \nEq. (\\ref{schroclassica}); it can therefore tell us whether the reduced \ndynamics, if unstable, is of the regular or chaotic type. The interest on this \nissue comes from the fact that the reduced dynamics usually helps to determine \nthe spatiotemporal patterns of the full system: while regular reduced dynamics \nis associated with regular structures - frequently a collection of \nsolitons or soliton-like structures, chaotic reduced \ndynamics is associated with spatiotemporal chaos. The correlation has its roots \non the so called stochastic pump model \\cite{lili91}. The model states that \nintense chaos in a low-dimensional subsystem of a multidimensional environment can \nmake the subsystem act like a thermal source, irreversibly delivering energy into \nothers degrees-of-freedom. In the limit of a predominantly regular dynamics \nundergoing in the reduced system, \nirreversibility is greatly reduced and energy tends to remain confined \nwithin the subsystem. On the other hand, in the limit of deeply chaotic \ndynamics with no periodicity at all, energy flow out of the subsystem is fast \nand there is not even much sense in defining the subsystem as an approximately \nisolated entity. The intermediary cases are those more amenable to a \ndescription in terms of the stochastic pump \\cite{lili91,novo99}. We point out, \nhowever, that in all cases, even in the chaotic one, analysis of the reduced \nsubsystem serves as an orientation on what to be expected in the full \nspatiotemporal dynamics. In our case, the appropriate subsystem is precisely \nthe one we have been using. This is so because it is the smallest subsystem \ncomprising the most important ingredients of the dynamics: the homogeneous \nand the only linearly unstable modes.\n\nWe now examine these points in some detail. Let us consider $\\omega = 0.1$. When\n$A_{h,o} \\ll \\omega$, $A_{h,o} \\equiv A_h(t=0)$, the modulational\napproximation can be used to determine the instability range, $k < k_{tr}$. \nA surface of section plot based on the two-degrees-of-freedom \nHamiltonian (\\ref{hamiltonian}) produces Fig. (\\ref{fig1.ps}), where we record \nthe values of $(q_1,p_1)$ whenever $p_h = 0$ with $dp_h\/dt > 0$, and \nwhere we take $q_1 = 0.01 q_h \\ll q_h$ and $p_h = p_1 = 0$ to determine the \ntotal unique energy of the various initial conditions we launch in the \nsimulations. Note that the particular ``seed'' initial condition introduced \nabove is included in the ensemble of initial conditions launched and represents \nsmall perturbations to an homogeneous background.\n\\begin{figure}[h]\n\\vspace{4in}\n\\special{psfile=fig1.ser.eps angle=-0 voffset=0 hoffset=40 hscale=30 vscale=40}\n\\caption{Low-dimensional Poincar\\'e plots on the projected\nphase-space $(p_1,q_1)$. $\\omega=0.1$ in all cases, and $k=k_{max}$ in panels \n(b)-(d). Initial conditions discussed in the text.}\n\\label{fig1.ps}\n\\end{figure}\nIn Fig. (\\ref{fig1.ps}a) \nwe take $A_{h,o}\/\\omega = 0.5$ and $k = 2 k_{max} > k_{tr}$. One lies outside \nthe instability range and the figure reveals that the origin, which \nrepresents the purely homogeneous state $q_1=A_1=0$, is indeed stable. \nThe remaining panels are all made for $k=k_{max}$ and increasing values of \nthe ratio $A_{h,o}\/\\omega$. One sees that for such values of $k$ and \n$A_{h,o}$ not only the origin is rendered unstable, as it also becomes \nprogressively surrounded by chaotic activity. The inner chaotic trajectory \nissuing from the origin - i.e., the trajectory representing the \nmodulational instability - is encircled by invariant curves, but the last panel \nalready shows that some external chaos, here represented by scattered points \naround the invariant curves, is also present. At some amplitude \n$A_{h,o} = A_{critical}$ slightly larger than the one used in panel (d), invariant \ncurves are completely destroyed. Above the critical field, inner orbits are no \nlonger restricted to move within confined regions of phase-space - these \norbits are in fact engulfed by the external chaos seen in panel (d). External \nchaos here is a result of the hyperbolic point positioned at the local \nmaximum of the generalized potential $\\Phi(A)$, at $A_1 \\sim 0.1$.\nOne gross determination of the critical field based on simple \nnumerical observation yields $A_{critical}\/\\omega \\sim 1\/1.62$, although \nthe corresponding transition from order to disorder in the full simulations \nmay not be so sharply defined. Yet, one may expect that in both \nlow-dimensional and full simulations, the transition should occur at \ncomparable amplitudes. \n\n\\section{Full spatiotemporal simulations}\n\nTo make the appropriate comparisons, we now look into the full simulation of \nEq. (\\ref{eqzero}). Full simulations are made through a discretization of the \nspatial domain via a finite difference method. The dynamics is resolved temporally \nby means of a sympletic integrator, as was the purely temporal Hamiltonian. The \nresults are quite robust and energy is conserved up to $10^{-6}$ parts in one. \nIn Fig. (\\ref{fig2.ps}) we display the spacetime history of the quantity \n$Q = \\sqrt{\\omega^2 A(x,t)^2 + \\dot A(x,t)^2}$ - we plot this quantity \nbecause it becomes a constant of motion in the limit where we discard nonlinearities \nand inhomogeneities. Initial conditions are the same as the seed initial \ncondition used in Fig. (\\ref{fig1.ps}), but the control parameters differ. In \nthe case of Fig. (\\ref{fig2.ps}a) where we choose $A_{h,o}\/\\omega = 0.1$ so as \nto safely satisfy $A_{h,o} \\ll A_{critical}$ and $A_{h,o} \\ll \\omega$, \nit is seen that the \nspatiotemporal dynamics is very regular. The homogeneous state is unstable, but \nonly periodic spatiotemporal spikes can be devised. This is the regular \nspatiotemporal dynamics so typical of the integrable NLSE and this kind of \ndynamics agrees very well with the low-dimensional predictions of the reduced \nHamiltonian (\\ref{hamiltonian}). The fact that only one single structure can be \nfound along the spatial axis at any given time, indicates that the dynamics is \nsingly periodic (period-1) along this axis and thus basically understood in \nterms of the reduced number of active modes (homogeneous plus fundamental \nharmonics) of Hamiltonian (\\ref{hamiltonian}). We now move into the vicinity of \nthe critical amplitude $A_{critical}$ discussed earlier. Under such conditions, \none may expect to see the effects of spatiotemporal chaos. The value \n$A_{h,o}\/\\omega = 1\/1.625$ is chosen in Fig. (\\ref{fig2.ps}b), where full \nsimulations indeed display a highly disorganized state after a short regular \ntransient. Regularly interspersed \nspikes can no longer be seen and spatial and temporal periodicities are lost, \nwhich characterizes spatiotemporal chaos. In this regime many modes become \nactive (we shall return to this point later) and the reduced Hamiltonian \n(\\ref{hamiltonian}) fails to provide an accurate description of the full \ndynamics. However it still provides a good estimate on the point of transition. \nA more throughful examination of the transition in the full simulations \nsuggests that the critical field there is a bit smaller - a value close to \n$\\omega \/ 1.95$; for smaller values we have not observed noticeable \nsignals of spatiotemporal chaos even for much longer runs than those \npresented here. The much larger oscillations \nexecuted by $Q$ in chaotic cases (see the legend of Fig. (\\ref{fig2.ps})) \nis a direct result of the destruction of the invariant curves seen in \nFig. (\\ref{fig1.ps}). In the absence of invariant curves, initial conditions \nare no longer restricted to move near the origin. \n\\begin{figure}[h]\n\\vspace{4.25in}\n\\special{psfile=fig2.mat.eps angle=-0 voffset=-55 hoffset=0 hscale=50 vscale=50\n}\n\\caption{Full spatio-temporal simulations for $\\omega = 0.1$ and $k=k_{max}$.\n$A_{h,o}\/\\omega=0.1$ in (a) and $A_{h,o}\/\\omega=1\/1.625$ in (b).\nThe intensity plots are made for the quantity\n$Q = \\sqrt{\\omega^2 A(x,t)^2 + {\\dot A}(x,t)^2}$; lighter shades are associated\nwith larger values of $Q$, which varies within the range $0 < Q < 0.0014$ in\n(a) and $0 < Q < 0.8$ in (b). $\\tau \\equiv \\omega t \/ 2 \\pi$ here and in all\nthe remaining simulations}\n\\label{fig2.ps}\n\\end{figure}\nIt is thus seen that there are limits to an integrable modulational \ndescription of the dynamics of a wave field. The limits are essentially \nset by the parameter $A_{h,o} \/ \\omega$. If it is much smaller than \nthe unit, the modulational description is valid and one can \nexpect to see a collection of spatiotemporal periodic structures \nbeing formed as asymptotic states of the dynamics. On the other hand, as the \nparameter approaches the unit, nonintegrable features are likely to be \nseen. In particular, regularity does not survive for very long, and \nfluctuations with various length scales appear in the system. This \nis a regime of spatiotemporal chaos which fundamentally involves the \npresence of nonlinear resonances between the frequency of the \ncarrier, $\\omega$, and the intrinsic nonlinear modulational frequency, \n$\\Omega$.\n\nDue to the presence of chaos, one is suggested that the transition involves an \nirreversible energy flow out of the reduced subsystem. If one computes the average \nnumber of active modes\n\\begin{equation}\n \\equiv {\\sum_n n^2 |A_n|^2 \\over \\sum_n |A_n|^2},\n\\label{soma}\n\\end{equation}\nwhere the amplitudes are defined in the form\n\\begin{equation}\nA_n = \\sum_j A(x_j,t) e^{i n k x_j},\n\\label{aene}\n\\end{equation}\none obtains the plots shown in Fig. (\\ref{fig3.ps}). \n\\begin{figure}[b]\n\\vspace{4.25in}\n\\special{psfile=fig3.kg.eps angle=-0 voffset=-0 hoffset=0 hscale=37 vscale=37}\n\\caption{Average number of active modes as a function of time. Parameters\nrespectively equal to those of the previous figure with exception of\npanel (d) where we take $A_{h,o}\/\\omega = 1\/1.95$.}\n\\label{fig3.ps}\n\\end{figure}\n\\noindent In Eq. (\\ref{aene}), \n\\lq\\lq$j$'' is the discretization index. In Fig. (\\ref{fig3.ps}a) we use the \nsame conditions as in Fig. (\\ref{fig2.ps}a). This panel \nshows that in rough terms, energy keeps periodically migrating between the \nhomogeneous mode (when $\\sqrt{} \\sim 0$) and the fundamental harmonic \n(when $\\sqrt{} \\sim 1$). Conditions of Fig. (\\ref{fig3.ps}b) are the same \nas those of Fig. (\\ref{fig2.ps}b); one sees that as the ratio \n$A_{h,o}\/\\omega$ grows, periodicity is lost, and that energy flow out of the \ninitial subsystem into other modes becomes clearly irreversible. In \nFig. (\\ref{fig3.ps}c) we use the same previous conditions with exception \nof $A_{h,o}$ which we take $A_{h,o} = \\omega \/ 1.95$. This slightly smaller, \nbut not too small, value of the initial amplitude allows to observe the \nslow diffusive transit of energy during the initial stages of the \ncorresponding simulations. During this stage one can actually look at \nthe subsystem as an energy source adiabatically delivering energy into other \nmodes - the concept of the stochastic pump applies more appropriately in those \nsituations.\n\n\\section{Final conclusions}\n\nTo summarize, in this paper we have studied the breakdown of modulational \napproximations in nonlinear wave interactions. We have analyzed \na nonlinear Klein-Gordon equation to draw the following \nconclusions. Adiabatic or modulational approximations are accurate while the \nhigh-frequency of the carrier wave keeps much larger than the modulational \nfrequency. Under these circumstances the \nfull spatiotemporal patterns are regular as is the dynamics \nin the reduced subsystem where the system energy is initially injected. There \nis no net flow of energy out of the reduced subsystem into the remaining \nmodes.\n\nOn the other hand, \nwhen both frequencies become of the same order of magnitude, the reduced \nsubsystem undergoes a transition to chaos. Correspondingly, the spatiotemporal \npatterns of the full system become highly disordered and energy spreads \nout over many modes. The correlation between the low-dimensional and high-dimensional \nspatiotemporal chaos has its roots on the stochastic pump model \\cite{lili91}. \nAccording to the model, a low-dimensional chaotic subsystem may act like a thermal \nsource, delivering energy in a irreversible fashion to other degrees-of-freedom \nof the entire system. Spectral simulations performed here indicates \nthat this seems to be the case with the present setting. \n\nThe transition to \nchaos involves a noticeable increase in terms of wave amplitude. This \ntakes place when the invariant curves of Fig. (\\ref{fig1.ps}) are destroyed, \nallowing for the merge of the external and internal chaotic bands into one \nextended chaotic sea. \nThis merging of chaotic bands is actually a result of reconnections involving the \nmanifolds of the unstable fixed point at the origin, and other hyperbolic points \nassociated with the curvature of the generalized potential $\\Phi$ \\cite{corso98}. \nWhen both manifolds reconnect, inner trajectories issuing from the origin start to \nexecute the large and irregular oscillations seen in Fig. (\\ref{fig2.ps}). More \ndetailed studied of the process is under current analysis. \n\n\\acknowledgments\nThis work was partially supported by \nFinanciadora de Estudos e Projetos (FINEP), Conselho Nacional de \nDesenvolvimento Cient\\'{\\i}fico e Tecnol\\'ogico (CNPq), and \nFunda\\c{c}\\~ao da Universidade Federal do Paran\\'a - FUNPAR, Brazil. S.R. Lopes \nwishes to express his thanks for the hospitality at the Plasma Research \nLaboratory, University of Maryland. Part of the numerical work was performed on \nthe CRAY Y-MP2E at the Supercomputing Center of the Universidade Federal do \nRio Grande do Sul. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:1}Introduction}\nThe physics of low-dimensional spin systems has attracted much \ninterest for past decades.\nAlthough the quantum fluctuation is expected to be large in two dimensions,\nthere exist strong evidences that the quantum Heisenberg antiferromagnet\n(QHAF) with nearest-neighbor coupling in a square lattice\nexhibits the N\\'eel long-range order at zero temperature.\\cite{Chakravarty89}\nUnder the presence of the long-range order, the linear spin-wave (LSW) \ntheory works rather well.\\cite{Anderson52,Kubo52}\nA natural way to include quantum fluctuations is\nan expansion in terms of $1\/S$, where $S$ is the magnitude of spin,\nbecause the LSW theory is made up of a leading-order in the $1\/S$ expansion.\nSuch attempts have been done and turned out to be \nuseful.\\cite{Oguchi60,Igarashi92-1,Igarashi92-2,Igarashi93,Canali92,Hamer92}\n\nIn our previous paper,\\cite{Igarashi92-1} basing on the Holstein-Primakoff \ntransformation,\\cite{Holstein40} \nwe calculated corrections up to the second order \nof $1\/S$ in various physical quantities such as the spin-wave dispersion,\nthe sublattice magnetization, the perpendicular susceptibility,\nand the spin-stiffness constant.\nWe also calculated the dynamical structure\nfactors of the transverse and the longitudinal components up to the \nsecond order of $1\/S$.\\cite{Igarashi92-2}\nAt that time, however, available experimental data of inelastic \nneutron scattering (INS) were limited to the momentum transfer \nin a narrow region of the Brillouin zone (BZ).\nTherefore, our study was just a demonstration of usefulness\nof the $1\/S$ expansion.\nNow that INS experiments provide us the information of the excitation \nspectra in the whole BZ, it may be interesting to calculate the excitation\nspectra in the whole BZ in comparison with recent experiments.\nWe show that the second-order correction makes the spin wave energy\nat momentum $(\\pi,0)$ about $2\\%$ smaller than that at $(\\pi\/2,\\pi\/2)$.\nThis difference is smaller than the value $7\\sim 9\\%$ obtained by\nthe series expansion\\cite{Singh95,Zheng04} \nand the Monte Carlo simulation.\\cite{Sandvik01}\nWe have so far not been able to find why the $1\/S$ expansion \nwithin the second order gives different results, since the higher-order \ncorrections is expected to be quite small. \nNote that the spin-wave dispersion was recently measured by\nthe INS experiment for Cu(DCOO)$_2\\cdot$4D$_2$O (CFTD),\nrevealing the $6\\%$ difference.\\cite{Ronnow01,Christensen04}\nThis material is believed to be well described by the $S=1\/2$\nHeisenberg model within the nearest-neighbor coupling.\\cite{Com1}\nIn addition to the spin-wave energy, we calculate the transverse dynamical \nstructure factor up to the second order of $1\/S$. \nIt consists of the $\\delta$-function-like peak\nof one spin-wave excitation and the continuum of three spin-wave excitations.\nThe second-order correction is found quite small in the spin-wave-peak\nintensity due to a cancellation of various second-order processes.\nThe result is compared with the recent experiment\\cite{Christensen04} \nas well as other $1\/S$-expansion study\\cite{Canali93} based on the \nDyson-Maleev transformation and the series expansion \nstudies.\\cite{Singh95,Zheng04}\nIn the three-spin-wave continuum, such a cancellation in the second-order\nprocesses is not severe, and the substantial intensities come out.\nThis is consistent with our previous study\\cite{Igarashi92-2}\nand others.\\cite{Canali93,Singh95,Zheng04}\n\nAnother purpose of this paper is to study the crossover behavior in\nthe QHAF from two dimensions to one dimension with weakening \nexchange coupling in one direction. In purely one dimension,\nof course, the antiferromagnetic long-range order\ndisappears due to quantum fluctuation \nand therefore the concept of spin waves breaks down.\nCarrying out the $1\/S$ expansion, we demonstrate that\nthe second-order corrections increase with approaching\nthe quasi-one dimensional situation.\nThe second-order correction works to increase considerably \nthe sublattice magnetization, although the first-order correction makes\nit decrease, in the quasi-one dimensional situation. \nInterestingly, the spin-wave dispersion is found to approach to the curve\nknown as the des Cloizaux-Pearson boundary in the $S=1\/2$ QHAF \nchain.\\cite{desCloizeaux62}\nAt the same time, for the spin-wave peak in the transverse dynamical \nstructure factor, the first-order correction makes the intensity\ndecrease but the second-order correction makes it increase.\nIn the quasi-one dimensional situation, the former is much larger than\nthe latter, and the net intensity is strongly reduced from the LSW value.\nOn the other hand, the intensity of three-spin-wave continuum \nby the second-order correction increases and exceeds the spin-wave-peak\nintensities in the quasi-one dimensional situation. \nThis contrasts with the description of using spinon\\cite{Ho01}\nto describe the large intensity of the spectral continuum.\nThe above three characteristics in the quasi-one dimension, \n(a) the spin-wave energy approaches to the des Cloizaux-Pearson boundary,\n(b) the spin-wave-peak intensity decreases, and (c) the intensity of\nthree-spin-wave continuum increases, are suggesting a close relation to\nthe purely-one dimensional behavior that the spectra are described by\ncontinuum of two spinons.\n\nThe present paper is organized as follows.\nIn Sec. \\ref{sect.2}, the Heisenberg Hamiltonian is expressed in terms of \nthe 1\/S-expansion. The Green's functions for spin waves are introduced\nin Sec. \\ref{sect.3}. The sublattice magnetization is calculated \nwith the help of the Green's functions in Sec. \\ref{sect.4}.\nThe spin-wave dispersion is calculated in Sec. \\ref{sect.5},\nand the transverse dynamical structure factor is calculated\nin Sec. \\ref{sect.6}.\nSection \\ref{sect.7} is devoted to the concluding remarks.\n\n\\section{\\label{sect.2}Hamiltonian}\n\nWe consider the Heisenberg Hamiltonian on the square lattice\nwith directional anisotropy of exchange couplings:\n\\begin{equation}\n H= J\\sum_{\\bf\\ell}{\\bf S}_{\\bf\\ell}\\cdot{\\bf S}_{{\\bf\\ell}+{\\bf a}}\n + J'\\sum_{\\bf\\ell}{\\bf S}_{\\bf\\ell}\\cdot{\\bf S}_{{\\bf\\ell}+{\\bf b}},\n\\label{eq.heis}\n\\end{equation}\nwhere ${\\bf\\ell}$ runs over all lattice sites and\n${\\bf\\ell}+{\\bf a}$ and ${\\bf\\ell}+{\\bf b}$ indicate\nthe nearest neighbors to the ${\\bf\\ell}$th site in the positive $x$ and $y$\ndirections, respectively. \nQuasi-one dimensional situations are realized by weakening the exchange \ncoupling $J'$ in the $y$ direction.\n\nIntroducing the Holstein-Primakoff transformation,\\cite{Holstein40}\nwe express the spin operators in terms of boson annihilation operators \n$a_i$ and $b_j$ (and their Hermite conjugates),\n\\begin{eqnarray}\n S_i^z &=& S - a_i^\\dagger a_i , \n \\label{eq.boson1}\\\\\n S_i^+ &=& (S_i^-)^\\dagger = \\sqrt{2S}f_i(S)a_i ,\\\\\n S_j^z &=& -S + b_j^\\dagger b_j ,\\\\\n S_j^+ &=& (S_j^-)^\\dagger = \\sqrt{2S}b_j^\\dagger f_j(S) ,\n \\label{eq.boson2}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n f_\\ell (S) = \\left(1 - \\frac{n_\\ell}{2S}\\right)^{1\/2}\\\\\n = 1 - \\frac{1}{2}\\frac{n_\\ell}{2S} -\n \\frac{1}{8}\\left(\\frac{n_\\ell}{2S}\\right)^2 + \\cdots .\n\\end{equation}\nwith $n_\\ell=a_i^\\dagger a_i$ and $b_j^\\dagger b_j$.\nIndices $i$ and $j$ refer to sites on the \"up\" and \"down\" sublattices, \nrespectively.\nSubstituting Eqs.~(\\ref{eq.boson1})-(\\ref{eq.boson2}) into Eq.~(\\ref{eq.heis})\nwe expand the Hamiltonian in powers of $1\/S$ as\n\\begin{equation}\n H = -S^2N(J+J')+H_0 + H_1 + H_2 + \\cdots, \n\\end{equation}\nwith $N$ the number of lattice sites.\n\nThe leading term $H_0$ is expressed as\n\\begin{eqnarray}\n H_0 &=& JS\\sum_{i}(2 a_i^\\dagger a_i \n + 2 b_{i+{\\bf a}} b_{i+{\\bf a}}\n + a_i b_{i+{\\bf a}}\n + a_i b_{i-{\\bf a}}\n + a_i^\\dagger b_{i+{\\bf a}}^\\dagger\n + a_i^\\dagger b_{i-{\\bf a}}^\\dagger ) \\nonumber \\\\\n &+& J'S\\sum_{i} (2 a_i^\\dagger a_i \n + 2 b_{i+{\\bf b}} b_{i+{\\bf b}}\n + a_i b_{i+{\\bf b}}\n + a_i b_{i-{\\bf b}}\n + a_i^\\dagger b_{i+{\\bf b}}^\\dagger\n + a_i^\\dagger b_{i-{\\bf b}}^\\dagger ).\n\\label{eq.h0}\n\\end{eqnarray} \nWe diagonalize $H_0$ by rewriting the boson operators in the momentum \nspace as\n\\begin{eqnarray}\n a_i &=& \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_{\\bf k} a_{\\bf k}\n \\exp(i{\\bf k\\cdot r}_i), \\\\\n b_j &=& \\left(\\frac{2}{N}\\right)^{1\/2}\n \\sum_{\\bf k} b_{\\bf k} \\exp(i{\\bf k\\cdot r}_j),\n\\end{eqnarray}\nand by introducing the Bogoliubov transformation,\n\\begin{equation}\n a_{\\bf k}^\\dagger = \\ell_{\\bf k}\\alpha_{\\bf k}^\\dagger\n + m_{\\bf k}\\beta_{-{\\bf k}}, \\quad\n b_{-{\\bf k}} = m_{\\bf k}\\alpha_{\\bf k}^\\dagger\n + \\ell_{\\bf k}\\beta_{-{\\bf k}}, \n\\end{equation}\nwhere\n\\begin{equation}\n \\ell_{\\bf k} = \\Bigl[\\frac{1+\\epsilon_{\\bf k}}\n {2\\epsilon_{\\bf k}}\\Bigr]^{1\/2},\\quad\n m_{\\bf k} = -{\\rm sgn}(\\gamma_{\\bf k})\\Bigl[\\frac{1-\\epsilon_{\\bf k}}\n {2\\epsilon_{\\bf k}}\\Bigr]^{1\/2} \\equiv - x_{\\bf k}\\ell_{\\bf k},\n\\end{equation}\nwith\n\\begin{eqnarray}\n \\epsilon_{\\bf k} &=& \\left(1-\\gamma_{\\bf k}^2\\right)^{1\/2},\\\\\n \\gamma_{\\bf k} &=& (\\cos k_x+\\zeta\\cos k_y)\/(1+\\zeta), \\\\\n \\zeta &=& J'\/J. \n\\end{eqnarray}\nMomentum ${\\textbf k}$ is defined in the first magnetic Brillouin zone\n(BZ). The ${\\rm sgn}(\\gamma_{\\textbf k})$ denotes the sign of \n$\\gamma_{\\textbf k}$,\nwhich is absorbed into the definition of $x_{\\bf k}$.\nFor the study of the isotropic exchange coupling ($\\zeta=1$), \nwe have neglected this factor \nbecause $\\gamma_{\\bf k}$ is always positive in the first BZ.\nFor the anisotropic coupling, however, this redefinition of $x_{\\bf k}$ \nis necessary, because $\\gamma_{\\bf k}$ is negative in a certain region \nof the first BZ. After this transformation, we have\n\\begin{eqnarray}\n H_0 &=& 2JS(1+\\zeta)\\sum_{\\bf k}(\\epsilon_{\\bf k}-1) \\nonumber \\\\\n &+& 2JS(1+\\zeta)\\sum_{\\bf k} \\epsilon_{\\bf k}\n (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k}\n + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k}). \n\\end{eqnarray}\nThis expression is the same as that for the isotropic coupling,\nexcept for the first factor $2JS(1+\\zeta)$.\n\nThe first-order term $H_1$ can be expressed in terms of \nspin-wave operators through the same procedure as above.\nThe result for the anisotropic coupling is given by\nthe previous expression in Ref.~\\onlinecite{Igarashi92-1} with \nsimply replacing \n$JSz$ by $2JS(1+\\eta)$:\n\\begin{eqnarray}\n H_1 &=& \\frac{2JS(1+\\zeta)}{2S} A\\sum_{\\bf k}\\epsilon_{\\bf k}\n (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k} + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k})\n \\nonumber \\\\\n &+&\\frac{-2JS(1+\\zeta)}{2SN}\\sum_{1234}\\delta_{\\bf G}(1+2-3-4)\n \\ell_1\\ell_2\\ell_3\\ell_4 \\nonumber \\\\\n &\\times& \\biggl[\\alpha_1^\\dagger\\alpha_2^\\dagger\\alpha_3\\alpha_4 \n B_{1234}^{(1)}+\\beta_{-3}^\\dagger\\beta_{-4}^\\dagger\\beta_{-1}\\beta_{-2} \n B_{1234}^{(2)}+4\\alpha_1^\\dagger\\beta_{-4}^\\dagger\\beta_{-2}\\alpha_3 \n B_{1234}^{(3)} \\nonumber \\\\\n &+&\\bigl(2\\alpha_1^\\dagger\\beta_{-2}\\alpha_3\\alpha_4 B_{1234}^{(4)}\n +2\\beta_{-4}^\\dagger\\beta_{-1}\\beta_{-2}\\alpha_3 B_{1234}^{(5)}\n +\\alpha_1^\\dagger\\alpha_2^\\dagger\\beta_{-3}^\\dagger\\beta_{-4}^\\dagger\n B_{1234}^{(6)} + {\\rm H.c.}\\bigr)\\biggr],\n\\label{eq.intham}\n\\end{eqnarray}\nwith \n\\begin{equation}\n A=\\frac{2}{N}\\sum_{\\bf k}(1-\\epsilon_{\\bf k}) .\n \\label{eq.A}\n\\end{equation}\nMomenta ${\\bf k}_1$, ${\\bf k}_2$, ${\\bf k}_3$, $\\cdots$ are abbreviated\nas $1,2,3,\\cdots$.\nThe first term arises from setting the products of four boson operators \ninto normal product forms with respect to spin-wave operators.\nThe second term in Eq.~(\\ref{eq.intham}) represents the scattering\nof spin waves. The Kronecker delta $\\delta_{\\bf G}(1+2-3-4)$ represents\nthe conservation of momenta within a reciprocal lattice vector ${\\bf G}$.\nThe vertex functions $B^{(i)}$'s in a symmetric\nparameterization are the same as those given by Eqs.~(2.16)-(2.20) \nin Ref.~\\onlinecite{Igarashi92-1}, so that they are omitted here.\n\nThe second-order term $H_2$ is composed of products of six boson operators.\nWriting it in a normal product form with respect to spin-wave operators,\nwe have \n\\begin{equation}\n H_2 = \\frac{2JS(1+\\zeta)}{(2S)^2}\\sum_{\\bf k}\\left[ C_1({\\bf k})\n (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k} + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k})\n + C_2({\\bf k})(\\alpha_{\\bf k}^\\dagger\\beta_{\\bf -k}^\\dagger\n + \\beta_{\\bf -k}\\alpha_{\\bf k})+\\cdots \\right].\n\\end{equation}\nNeglected terms are unnecessary for calculating corrections up to the\nsecond order. The explicit forms of $C_1({\\bf k})$ and $C_2({\\bf k})$,\nare given by Eqs.~(2.22) and (2.23) in Ref.~\\onlinecite{Igarashi92-1}.\nNote that $C_1({\\bf k})$ and $C_2({\\bf k})$ diverge as $1\/\\epsilon_{\\bf k}$ \nwith $|{\\bf k}|\\to 0$. \n \n\\section{\\label{sect.3}Green's Function} \n\nFor systematically carrying out the 1\/S-expansion,\nit is convenient to introduce the Green's functions for spin-waves,\n\\begin{eqnarray}\n G_{\\alpha\\alpha}({\\bf k},t) &=& -i \\langle T(\\alpha_{\\bf k}(t)\n \\alpha_{\\bf k}^\\dagger (0)) \\rangle, \\\\\n G_{\\alpha\\beta}({\\bf k},t) &=& -i \\langle T(\\alpha_{\\bf k}(t)\n \\beta_{-{\\bf k}}(0)) \\rangle, \\\\\n G_{\\beta\\alpha}({\\bf k},t) &=& -i \\langle T(\\beta_{-{\\bf k}}^\\dagger(t)\n \\alpha_{\\bf k}^\\dagger (0)) \\rangle, \\\\\n G_{\\beta\\beta}({\\bf k},t) &=& -i \\langle T(\\beta_{-{\\bf k}}^\\dagger(t)\n \\beta_{-{\\bf k}}(0)) \\rangle,\n\\end{eqnarray}\nwhere $ \\langle \\cdots \\rangle$ denotes the average over the ground state, and \n{\\textit T} is the time-ordering operator.\n\nIn this paper, we measure energies in units of $2JS(1+\\zeta)$.\nThe unperturbed propagators corresponding to $H_0$ are given by\n\\begin{eqnarray}\n G_{\\alpha\\alpha}^0({\\bf k},\\omega) &=& [\\omega \n - \\epsilon_{\\bf k} + i\\delta]^{-1}, \\\\\n G_{\\alpha\\beta}^0({\\bf k},\\omega) &=& G_{\\beta\\alpha}^0({\\bf k},\\omega) = 0, \n \\\\\n G_{\\beta\\beta}^0({\\bf k},\\omega) &=& [-\\omega \n - \\epsilon_{\\bf k} + i\\delta]^{-1}.\n\\end{eqnarray}\nThe self-energy is defined by\n\\begin{equation}\n G_{\\mu\\nu}({\\bf k},\\omega) = G_{\\mu\\nu}^0({\\bf k},\\omega)\n + \\sum_{\\mu'\\nu'} G_{\\mu\\mu'}^0({\\bf k},\\omega)\n \\Sigma_{\\mu'\\nu'}({\\bf k},\\omega)G_{\\nu'\\nu}({\\bf k},\\omega). \n\\end{equation}\nIt is expanded in powers of $1\/(2S)$,\n\\begin{equation}\n \\Sigma_{\\mu\\nu}({\\bf k},\\omega) = \\frac{1}{2S}\n \\Sigma_{\\mu\\nu}^{(1)}({\\bf k},\\omega)\n + \\frac{1}{(2S)^2}\\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega) + \\cdots. \n\\end{equation}\n>From $H_1$ we have the first-order correction as\n\\begin{equation}\n \\Sigma_{\\alpha\\alpha}^{(1)}({\\bf k},\\omega)\n = \\Sigma_{\\beta\\beta}^{(1)}({\\bf k},\\omega) = A\\epsilon_{\\bf k}, \\quad\n \\Sigma_{\\alpha\\beta}^{(1)}({\\bf k},\\omega)\n = \\Sigma_{\\beta\\alpha}^{(1)}({\\bf k},\\omega)= 0.\n\\end{equation}\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.1.eps}%\n\\caption{\\label{fig.diagram1}\nSecond-order diagrams for the self-energy, \n(a) $\\Sigma_{\\alpha\\alpha}({\\bf k},\\omega)$ \nand (b) $\\Sigma_{\\alpha\\beta}({\\bf k},\\omega)$.\nSolid lines represent the unperturbed Green's functions.\n}\n\\end{figure}\n\nThe second-order term $\\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega)$ is obtained from\nthe second-order perturbation, whose diagrams are shown \nin Fig.~\\ref{fig.diagram1}.\nWe obtain formally the same expression for the self-energy as \nin our previous paper:\\cite{Igarashi92-1}\n\\begin{eqnarray}\n \\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\omega)\n &=& \\Sigma_{\\beta\\beta}^{(2)}(-{\\bf k},-\\omega) \\nonumber \\\\\n &=& C_1({\\bf k}) + \\left(\\frac{2}{N}\\right)^2\n \\sum_{{\\bf p}{\\bf q}}2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\n \\ell_{\\bf q}^2\\ell_{{\\bf k}+{\\bf p}-{\\bf q}}^2 \\nonumber \\\\\n &\\times& \\left[\\frac{\\mid B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n \\mid^2}{\\omega-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}+i\\delta}\n -\\frac{\\mid B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\\mid^2}\n{\\omega+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}-i\\delta}\\right], \n\\label{eq.self1}\\\\\n \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\omega)\n &=& \\Sigma_{\\beta\\alpha}^{(2)}(-{\\bf k},-\\omega) \\nonumber \\\\\n &=& C_2({\\bf k}) + \\left(\\frac{2}{N}\\right)^2\n \\sum_{{\\bf p}{\\bf q}}2\n \\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{{\\bf k}+\n {\\bf p}-{\\bf q}}^2 {\\rm sgn}(\\gamma_{\\bf G}) \\nonumber \\\\\n &\\times& B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\n \\frac{2(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{{\\bf k}+{\\bf p}-{\\bf q}})}\n {\\omega^2-(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2+i\\delta},\n\\label{eq.self2}\n\\end{eqnarray}\nwhere $\\delta\\to 0$, and $[{\\bf k+p-q}]$ is the vector ${\\bf k+p-q}$ reduced to\nthe 1st BZ by a reciprocal vector ${\\bf G}$.\nWe have used the relations\n\\begin{equation}\n\\begin{array}{l}\nB_{[{\\bf k+p-q}],{\\bf q},{\\bf p},{\\bf k}}^{(5)} =\n{\\rm sgn}(\\gamma_{\\bf G})B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)},\\\\\nB_{{\\bf q},[{\\bf k+p-q}],{\\bf k},{\\bf p}}^{(6)} =\n{\\rm sgn}(\\gamma_{\\bf G})B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}. \\\\\n\\end{array}\n\\end{equation}\nThe terms divergent with ${\\bf k}\\to 0$ in $C_1({\\bf k})$ and $C_2{\\bf k})$\nare canceled by the second-order perturbation terms in Eqs.~(\\ref{eq.self1}) \nand (\\ref{eq.self2}).\nOne can prove $\\Sigma_{\\mu\\nu}^{(2)}({\\bf k}\\to 0,\\omega=0)\\to 0$\nfrom these equations.\n\n\\section{\\label{sect.4}Sublattice Magnetization} \n\nOnce the Green's function is known, the sublattice magnetization is \ncalculated from the relation,\n\\begin{eqnarray}\n M &\\equiv& S-\\langle a_i^\\dagger a_i\\rangle \\nonumber \\\\\n &=& S - \\frac{2}{N}\\sum_{\\bf k}\\lim_{\\eta\\to 0^+}\\int_{-\\infty}^{+\\infty}\n\\frac{{\\rm d}\\omega}{2\\pi}i{\\rm e}^{i\\omega\\eta}\n\\Bigl\\{\\ell_{\\bf k}^2G_{\\alpha\\alpha}({\\bf k},\\omega) \\nonumber\\\\\n&+& \\ell_{\\bf k}m_{\\bf k}\\bigl[G_{\\alpha\\beta}({\\bf k},\\omega)\n+G_{\\beta\\alpha}({\\bf k},\\omega)\\bigr]\n+ m_{\\bf k}^2G_{\\beta\\beta}({\\bf k},\\omega)\\Bigr\\},\n\\end{eqnarray}\nwith $\\eta\\to 0^+$.\nAfter carrying out the integration with respect to $\\omega$,\nwe obtains\n\\begin{equation}\n M = S -\\Delta S + \\frac{M_2}{(2S)^2},\n\\end{equation}\nwith\n\\begin{eqnarray}\n \\Delta S &=& \\frac{2}{N}\\sum_{\\bf k}\\frac{1}{2}(\\epsilon_{\\bf k}^{-1}-1), \\\\\n M_2 &=& \\frac{2}{N}\\sum_{\\bf k}\\Biggl\\{\n \\frac{\\ell_{\\bf k}m_{\\bf k}}{\\epsilon_{\\bf k}}\\Sigma_{\\alpha\\beta}^{(2)}\n({\\bf k},-\\epsilon_{\\bf k}) \\nonumber\\\\\n &-& \\left(\\frac{2}{N}\\right)^2\\sum_{{\\bf pq}}\n 2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n \\biggl[\\frac{(\\ell_{\\bf k}^2+m_{\\bf k}^2)\\mid\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\\mid^2}\n {(\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2} \n \\nonumber\\\\\n &+&\\frac{2\\ell_{\\bf k} m_{\\bf k}{\\rm sgn}(\\gamma_{\\bf G})\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}}\n {\\epsilon_{\\bf k}^2-(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}\n +\\epsilon_{\\bf k+p-q})^2}\\biggr]\\Biggr\\}.\n\\label{eq.zero}\n\\end{eqnarray}\nHere $[{\\bf k+p-q}]$ was defined before, and \n${\\bf G}={\\bf k+p-q}-[{\\bf k+p-q}]$. The zeroth-order correction\n$\\Delta S$ represents the well-known ``zero-point\" reduction in the LSW\ntheory. \nTo evaluate Eq.~(\\ref{eq.zero}), we sum up the values of $N_L^2\/4$ points\nof ${\\bf k}$ in the 1\/4 part of the first BZ and $N_L^2$ points of\n${\\bf p}$ and ${\\bf q}$ in the first BZ, with $N_L=20,48$.\nFor $J'\/J=1$, the convergence is very good; $M_2=0.0035059$ for $N_L=20$,\nand $M_2=0.0035065$ for $N_L=48$.\n\n\\begin{table}\n\\caption{\\label{table2}\nSublattice magnetization}\n\\begin{ruledtabular}\n\\begin{tabular}{llllll}\n $J'\/J$ & $1$ & $0.5$ & $0.1$ & $0.075$ & $0.05$ \\\\\n\\hline\n $\\Delta S$ & $0.196$ & $0.213$ & $0.355$ & $0.391$ & $0.445$ \\\\ \n $M_2$ & $0.0035$ & $0.024$ & $0.323$ & $0.481$ & $0.818$ \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nTable \\ref{table2} lists the values of $\\Delta S$ and $M_2$ for several \nvalues of $J'\/J$. These values are evaluated for $N_L=48$.\nFor the isotropic coupling, we reproduce the values obtained previously.\n\\cite{Igarashi92-1} \nThe zero-point reduction $\\Delta S$ increases with decreasing values of $J'\/J$.\nOn the other hand, $M_2$ is found always positive, tending to cancel\nthe zero-point reduction. The value increases with decreasing values of\n$J'\/J$, and becomes comparable to $\\Delta S$ around \n$J'\/J=0.1$, suggesting the applicability limit of the expansion for \nthe case of $S=1\/2$. \n\n\\section{\\label{sect.5}Spin-Wave Dispersion} \n\nWithin the second order in $1\/S$,\nthe renormalized spin-wave energy\n${\\tilde \\epsilon_{\\bf k}}$ in units of $2JS(1+\\zeta)$\nis obtained from \n\\begin{equation}\n {\\tilde\\epsilon}_{\\bf k} = \\epsilon_{\\bf k} + \\frac{1}{2S}A\\epsilon_{\\bf k} \n + \\frac{1}{(2S)^2}\\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\epsilon_{\\bf k}).\n\\label{eq.spin-wave}\n\\end{equation}\nFrom this equation, we define the renormalized spin-wave velocity $V_x$ \nalong the $x$ direction by $V_x \\equiv \\lim_{k_x\\to 0}\n2JS(1+\\zeta)\\tilde \\epsilon_{\\bf k}\/(\\hbar k_x)$ with $k_y=0$.\nThus the renormalization factor is expressed as\n\\begin{equation}\n Z_v\\equiv\\frac{V_x}{2JS(1+\\zeta)^{1\/2}}= 1+\\frac{v_1}{2S}+\\frac{v_2}{(2S)^2}, \n\\end{equation}\nwith $v_1=A$ given by Eq.~(\\ref{eq.A}).\nIn the following numerical evaluation, we divide\nthe first BZ into $N_L^2$ meshes with $N_L=64$.\n\n\\subsection{Isotropic case}\n\nFigure \\ref{fig.iso.disp} shows the spin-wave energy \n$2JS(1+\\zeta)\\tilde\\epsilon_{\\bf k}$\nas a function of momentum for the isotropic coupling ($J'\/J=1$)\nwith $S=1\/2$, in comparison with the experimental data\ntaken from the INS for \nCu(DCOO)$_2\\cdot$4D$_2$O (CFTD).\\cite{Ronnow01,Christensen04}\nMomentum is measured in units of (nearest neighbor distance)$^{-1}$.\nIn the whole BZ, both the first and second order corrections make\nthe spin-wave energy larger.\nThe curve along the line $(0,0)-(\\pi,0)$ has already been reported\nin our previous paper.\\cite{Igarashi92-1}\nThe dispersion along $(\\pi\/2,\\pi\/2)-(\\pi,0)$ is completely \nflat within the first-order correction.\nThe second-order correction makes the excitation energy \nat $(\\pi,0)$ about $2\\%$ smaller than the energy at $(\\pi\/2,\\pi\/2)$.\nExplicitly they are \n$\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.196$ and $\\tilde\\epsilon_{(\\pi,0)}=1.179$. \nA previous series expansion study predicted the energy difference \nabout $7\\%$,\\cite{Singh95} and a recent study gave about $9\\%$ difference,\nthat is, $\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.192$ and \n$\\tilde\\epsilon_{(\\pi,0)}=1.09$.\\cite{Zheng04}\nA Monte Carlo simulation has given \n$\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.195$ and \n$\\tilde\\epsilon_{(\\pi,0)}=1.08$.\\cite{Sandvik01}\nThese values at $(\\pi\/2,\\pi\/2)$ agree well with our value, while\nthe values at $(\\pi,0)$ is rather different from our estimate.\nThe experimental data indicate that the excitation energy at $(\\pi,0)$ is \n$6\\%$ smaller than that at $(\\pi\/2,\\pi\/2)$.\\cite{Ronnow01,Christensen04}\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.2.eps}%\n\\caption{\\label{fig.iso.disp}\nSpin-wave energy as a function of momentum for the isotropic coupling\n($J'\/J=1$). $S=1\/2$.\nThe dotted, broken and solid lines represent the values calculated \nwithin the LSW theory, up to the first-order correction, \nand up to the second-order correction, respectively. \nExperimental data are taken from Ref.~\\onlinecite{Christensen04}.\nInset indicates high symmetry lines which momentum varies along.\n}\n\\end{figure}\n\n\\subsection{Anisotropic case}\n\nFigure \\ref{fig.aniso.disp} shows the renormalized spin-wave energy \nas a function of momentum along $(0,0)-(\\pi,0)$ for $J'\/J<1$.\nAs the same as the isotropic coupling,\nboth the first-order and the second-order \ncorrections are found to be positive, making the energy larger. \nBoth corrections increase with decreasing values of $J'\/J$.\nIn quite small interchain couplings ($J'\/J < 0.1$),\nthe excitation energy seems approaching \nthe {\\em des Cloizeaux-Pearson boundary} \nin one dimension.\\cite{desCloizeaux62}\nAs shown in Table \\ref{table1}, the renormalization constant\n$Z_v$ seems approaching $\\pi\/2$, corresponding to the value of\nthe boundary.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.3.eps}%\n\\caption{\\label{fig.aniso.disp}\nSpin-wave energy as a function of momentum for anisotropic couplings,\n(a) $\\zeta=0.5$, (b) $\\zeta=0.1$, (c) $\\zeta=0.075$, and (d) $\\zeta=0.05$.\nThe dotted, broken and solid lines represent the values calculated \nwithin the LSW theory, up to the first-order correction, and up to\nthe second-order correction, respectively. \nThe thin solid line labelled ``1D Exact\" represents the des Cloizeaux-Pearson \nboundary.}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{table1}\nRenormalization of spin-wave velocity}\n\\begin{ruledtabular}\n\\begin{tabular}{llllll}\n $J'\/J$& $1$ & $0.5$ & $0.1$ & $0.075$ & $0.05$ \\\\\n\\hline\n $v_1$ & $0.158$ & $0.174$ & $0.272$ & $0.287$ & $0.306$ \\\\ \n $v_2$ & $0.021$ & $0.053$ & $0.130$ & $0.141$ & $0.155$ \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\section{\\label{sect.6}Dynamical Structure Factor} \n\nThe dynamical structure factor is an important quantity, \nsince it is directly related to the INS spectra.\nWe have already reported the expression within the second order \nof $1\/S$ in the isotropic coupling situation.\\cite{Igarashi92-2}\nAs is evident from the forms of $H_0$ and $H_1$, the\nformulas for the anisotropic coupling are\nformally the same as those for\nthe isotropic coupling.\n\nWe consider only the transverse component, \nwhich is defined by\n\\begin{equation}\n S^{+-}_{u(s)}({\\bf k}, \\omega) = \\frac{1}{2\\pi}\n \\int {\\rm d}t {\\rm e}^{i\\omega t}\n \\langle Q_{u(s)}({\\bf k},t)Q_{u(s)}({\\bf k},0)^{\\dagger}\\rangle,\n\\end{equation}\nwhere\n\\begin{equation}\n Q_{u(s)}({\\bf k})=S^{+}_a({\\bf k})\\pm S^{+}_b({\\bf k}),\n \\label{eq.Q}\n\\end{equation}\nwith \n\\begin{eqnarray}\n S^{+}_a({\\bf k})&=&[S^{-}_a({\\bf k})]^{\\dagger}=\n \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_i S^{+}_i \\exp(-i{\\bf k}\\cdot{\\bf r}_i),\n \\label{eq.sa}\\\\\n S^{+}_b({\\bf k})&=&[S^{-}_b({\\bf k})]^{\\dagger}=\n \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_j S^{+}_j \\exp(-i{\\bf k}\\cdot{\\bf r}_j).\n \\label{eq.sb}\n\\end{eqnarray}\nWe need the ``uniform\" and ``staggered\" parts because the momentum \nis defined inside the first BZ.\nThey are labelled as the suffix ``u\" and ``s\", \nand correspond to upper and lower signs in Eq.~(\\ref{eq.Q}), respectively.\n\nWe start by introducing the operators,\n\\begin{eqnarray}\n Y^{+}_{\\alpha}({\\bf k})&=&[Y^{-}_{\\alpha}({\\bf k})]^{\\dagger}\n = [\\ell_{\\bf k}S^{+}_{a}({\\bf k})-m_{\\bf k}S^{+}_{b}({\\bf k})]\/(2S)^{1\/2},\\\\\n Y^{+}_{\\beta}({\\bf k})&=&[Y^{-}_{\\beta}({\\bf k})]^{\\dagger}\n = [-m_{\\bf k}S^{+}_{a}({\\bf k})+\\ell_{\\bf k}S^{+}_{b}({\\bf k})]\/(2S)^{1\/2},\n\\end{eqnarray}\nand the associated Green's functions,\n\\begin{equation}\n F_{\\mu\\nu}({\\bf k},\\omega)=-i\\int_{-\\infty}^{\\infty}dt {\\rm e}^{i\\omega t}\n \\langle T[Y^{+}_{\\mu}({\\bf k},t)Y^{-}_{\\nu}({\\bf k},0)]\\rangle.\n\\end{equation}\nThen, with the help of the fluctuation-dissipation theorem,\nwe have\n\\begin{eqnarray}\n& & S^{+-}_{u(s)}({\\bf k},\\omega) = \n2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2 \\left(- \\frac{1}{\\pi} \\right) \\nonumber\\\\\n& &\\times {\\rm Im}[F_{\\alpha\\alpha}({\\bf k},\\omega) \n \\pm F_{\\alpha\\beta}({\\bf k},\\omega)\n \\pm F_{\\beta\\alpha}({\\bf k},\\omega)\n +F_{\\beta\\beta}({\\bf k},\\omega)],\n\\nonumber \\\\\n \\label{eq.dyna}\n\\end{eqnarray}\nwhere the upper (lower) signs correspond to the uniform (staggered) part.\n\nFor calculating $F_{\\mu\\nu}({\\bf k},\\omega)$, we expand the operator \n$Y^{+}_{\\mu}({\\bf k})$ in terms of spin-wave operators with the help of\nthe HP transformation and the Bogoliubov transformation.\nAfter lengthy calculations, we have\n\\begin{eqnarray}\n Y^{+}_{\\alpha}({\\bf k}) &=&\n D\\alpha_{\\bf k}-\\frac{1}{2S}\\frac{2}{N}\\sum_{234}\n \\delta_{\\bf G}({\\bf k}+2-3-4)\\frac{1}{2}\\ell_{\\bf k}\\ell_2\\ell_3\\ell_4\n \\nonumber\\\\\n &\\times& (M^{(1)}_{{\\bf k}234}\\beta_{-2}\\alpha_3\\alpha_4\n +M^{(2)}_{{\\bf k}234}\\alpha^{\\dagger}_{2}\\beta^{\\dagger}_{-3}\n \\beta^{\\dagger}_{-4}+ \\cdots),\n \\label{eq.y1}\\\\\n Y^{+}_{\\beta}({\\bf k}) &=&\n D\\beta^{\\dagger}_{-\\bf k} \\nonumber \\\\\n&-&\\frac{1}{2S}\\frac{2}{N}\\sum_{234}\n \\delta_{\\bf G}({\\bf k}+2-3-4)\\frac{1}{2}\\ell_{\\bf k}\\ell_2\\ell_3\\ell_4\n {\\rm sgn}(\\gamma_{\\bf G})\\nonumber\\\\\n &\\times& (M^{(2)}_{{\\bf k}234}\\beta_{-2}\\alpha_3\\alpha_4\n +M^{(1)}_{{\\bf k}234}\\alpha^{\\dagger}_{2}\\beta^{\\dagger}_{-3}\n \\beta^{\\dagger}_{-4}+ \\cdots),\n \\label{eq.y2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n D&=&1-\\frac{\\Delta S}{2S}-\\frac{1}{4}\\frac{\\Delta S(1+3\\Delta S)}{(2S)^2},\n \\label{eq.D}\\\\\n M^{(1)}_{{\\bf k}234} &=& -x_2+ {\\rm sgn}(\\gamma_{\\bf G})x_{\\bf k}x_3x_4, \\\\\n M^{(2)}_{{\\bf k}234} &=& x_3x_4 - {\\rm sgn}(\\gamma_{\\bf G})x_{\\bf k}x_2.\n\\end{eqnarray}\nwith ${\\bf G}={\\bf k}+2-3-4$. \nThe first-order and second-order corrections in Eq.~(\\ref{eq.D}) arises from \nsetting four and six boson operators in the HP transformation \ninto the normal product forms with spin-wave operators, respectively.\nThereby, the second terms in Eqs. (\\ref{eq.y1}) and (\\ref{eq.y2})\nare normally ordered.\nNote that ${\\rm sgn}(\\gamma_{\\bf G})$ arises from the phase difference\nin the definitions, Eqs.~(\\ref{eq.sa}) and (\\ref{eq.sb}).\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.4.eps}%\n\\caption{\\label{fig.struc.diagram}\nDiagrams for $F_{\\mu\\nu}({\\bf k},\\omega)$.\nSolid lines represent the unperturbed Green's functions \n$G_{\\mu\\nu}^0({\\bf k},\\omega)$. \n}\n\\end{figure}\n\nWith the use of Eqs.~(\\ref{eq.y1}) and (\\ref{eq.y2}),\n$F_{\\mu\\nu}({\\bf k},\\omega)$ is expanded up to the second order by\nthe diagrams shown in Fig.~\\ref{fig.struc.diagram}.\nExplicitly, it is given by\n\\begin{eqnarray}\n F_{\\mu\\nu}({\\bf k},\\omega) &=& D^2G_{\\mu\\nu}^0({\\bf k},\\omega)\\delta_{\\mu\\nu}\n + G_{\\mu\\mu}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n \\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega)\\nonumber\\\\\n &+& I_{\\mu\\nu}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega) \n + G_{\\mu\\mu}^0({\\bf k},\\omega)\\tilde I_{\\mu\\nu}({\\bf k},\\omega)\n + J_{\\mu\\nu}({\\bf k},\\omega),\n \\label{eq.f}\n\\end{eqnarray}\nwhere each term in Eq.~(\\ref{eq.f}) corresponds to the diagrams \n(a), (b), (c), (d), and (e), respectively.\nExplicit expressions for $I_{\\mu\\nu}({\\bf k},\\omega)$,\n$\\tilde I_{\\mu\\nu}({\\bf k},\\omega)$, and $J_{\\mu\\nu}({\\bf k},\\omega)$ \nare given by Eqs.~(4.5)-(4.22) in Ref.~\\onlinecite{Igarashi92-1}.\n\nThe dynamical structure factor is obtained by substituting Eq.~(\\ref{eq.f}) \ninto Eq.~(\\ref{eq.dyna}). \nIt consists of the $\\delta$-function-like peak of the one spin-wave excitation\nand the continuum of three spin-wave excitations:\n\\begin{equation}\n S_{u(s)}^{+-}({\\bf k},\\omega) = \n \\rho_{u(s)}^{(1)}({\\bf k})\\delta(\\omega-\\epsilon_{\\bf k})\n + \\rho_{u(s)}^{(2)}({\\bf k},\\omega).\n \\label{eq.s+-}\n\\end{equation}\nFor the term of one spin-wave excitation, the bare energy $\\epsilon_{\\bf k}$\nis to be replaced by the renormalized value $\\tilde\\epsilon_{\\bf k}$\ngiven by Eq.~(\\ref{eq.spin-wave}).\nHowever, the spectral weight $\\rho^{(1)}({\\bf k})$ \nwithin the second order of $1\/S$ is safely evaluated \nby putting $\\omega=\\epsilon_{\\bf k}$ in Eq.~(\\ref{eq.f}).\nIt is expressed as\n\\begin{equation}\n \\rho_{u(s)}^{(1)}({\\bf k}) = 2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2\n \\left(1+\\frac{d_{u(s),1}}{2S}+\\frac{d_{u(s),2}}{(2S)^2}\\right),\n\\end{equation}\nwith\n\\begin{eqnarray}\n d_{u(s),1} &=& -2\\Delta S, \n \\label{eq.dyna1}\\\\\n d_{u(s),2} &=& -\\frac{1}{2}\\Delta S(1+\\Delta S) \n \\mp\\frac{1}{\\epsilon_{\\bf k}} \n \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\epsilon_{\\bf k}) \\nonumber\\\\\n &+& \\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n 2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\\nonumber\\\\\n &\\times&\\left[\\frac{-|B^{(4)}_{\\bf k,p,q,[k+p-q]}|^2}\n {(\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})^2}\n +\\frac{|B^{(6)}_{\\bf k,p,q,[k+p-q]}|^2}\n {(\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2}\n \\right] \\nonumber\\\\\n &+& \\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n 2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n (M_{\\bf k,p,q,[k+p-q]}^{(1)}\n \\pm{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)})\\nonumber\\\\\n &\\times& \\left[\\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n {(\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})}\n \\mp\\frac{{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n \\right].\n \\label{eq.dyna2}\n\\end{eqnarray}\nThe upper (lower) signs correspond to the uniform (staggered) part.\nThe first-order correction Eq.~(\\ref{eq.dyna1}) arises from the first term of\nEq.~(\\ref{eq.f}).\nIn the second-order correction given by Eq.~(\\ref{eq.dyna2}), \nthe first term arises\nfrom the first term of Eq.~(\\ref{eq.f}), and the second term arises from \nthe second term of Eq.~(\\ref{eq.f}),\n\\[ G_{\\alpha\\alpha}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\omega)G_{\\beta\\beta}^0({\\bf k},\\omega)\n +G_{\\beta\\beta}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n \\Sigma_{\\beta\\alpha}^{(2)}({\\bf k},\\omega)G_{\\alpha\\alpha}^0({\\bf k},\\omega).\n\\]\nThe third term of Eq.~(\\ref{eq.dyna2}) is equivalent to\n\\[ \\left.\n \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}({\\bf k},\\omega)}\n {\\partial\\omega}\\right|_{\\omega=\\epsilon_{\\bf k}}, \\]\nand arises from the second term of Eq.~(\\ref{eq.f}),\n\\[ G_{\\alpha\\alpha}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n \\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\omega)\n G_{\\alpha\\alpha}^0({\\bf k},\\omega).\n\\]\nThis is related to the second-order correction to the residue of \nthe spin-wave pole in $G_{\\alpha\\alpha}({\\bf k},\\omega)$,\n\\[ \\frac{1}{1- \\left.\n \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}\n ({\\bf k},\\omega)}\n {\\partial\\omega}\\right|_{\\omega=\\epsilon_{\\bf k}}}\n \\approx \n 1+ \\left.\n \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}({\\bf k},\\omega)}\n {\\partial\\omega}\n \\right|_{\\omega=\\epsilon_{\\bf k}} . \\]\nThe fourth term arises from the third and fourth terms of Eq.~(\\ref{eq.f}),\n\\[ I_{\\mu\\nu}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega) \n + G_{\\mu\\mu}^0({\\bf k},\\omega)\\tilde I_{\\mu\\nu}({\\bf k},\\omega).\n\\]\nNo contribution arises from the last term of Eq.~(\\ref{eq.f}),\n$J_{\\mu\\nu}({\\bf k},\\omega)$.\nNote that the main momentum dependence around ${\\bf k}=(0,0)$ arises from\nthe prefactors, $(\\ell_{\\bf k}+m_{\\bf k})^2\\propto \\epsilon_{\\bf k}$\nand $(\\ell_{\\bf k}-m_{\\bf k})^2\\propto 1\/\\epsilon_{\\bf k}$.\n\nThe three-spin-wave continuum arises only from the second-order corrections.\nNote that the first term of Eq.~(\\ref{eq.f}) has no contribution. \nAfter careful evaluation of other terms in Eq.~(\\ref{eq.f}), we obtain\n\\begin{eqnarray}\n \\rho_{u(s)}^{(2)}({\\bf k},\\omega) &=& 2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2\n \\frac{1}{(2S)^2}\\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n \\delta(\\omega-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})\n \\nonumber\\\\\n &\\times& \\frac{1}{2}\n \\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n \\Biggl[M_{\\bf k,p,q,[k+p-q]}^{(1)}\n \\pm{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)}\\nonumber\\\\\n &-& \\frac{2B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}}\n \\mp\\frac{2{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n \\Biggr]^2.\n \\label{eq.conti}\n\\end{eqnarray}\nThe spectral shape may be modified by the renormalization of spin-wave\nenergies and by taking account of scattering spin waves due to mutual\ninteraction, which terms are present in $H_1$.\nTherefore, it may be difficult to determine the spectral shape\nin a consistent way with the $1\/S$ expansion.\nHowever, the total intensity, which is given by\n\\begin{equation}\n I_{u(s)}^{(2)}({\\bf k})=\\int_0^{\\infty}{\\rm d}\\omega\n \\rho_{u(s)}^{(2)}({\\bf k},\\omega),\n\\end{equation}\nmay be safely evaluated from Eq.~(\\ref{eq.conti}).\nNote that around ${\\bf k}=(0,0)$ \n\\begin{eqnarray}\n M_{\\bf k,p,q,[k+p-q]}^{(1)}\n &\\pm&{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)} \\propto\n \\epsilon_{\\bf k}, \\\\\n -\\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}}\n &\\mp&\\frac{{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n \\nonumber \\\\\n &\\approx&\n \\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]} \n \\mp{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n {\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}} \n \\propto\\epsilon_{\\bf k},\n\\end{eqnarray}\nand $(\\ell_{\\bf k}+m_{\\bf k})^2\\ell_{\\bf k}^2\\propto\nconst.$, $(\\ell_{\\bf k}-m_{\\bf k})^2\\ell_{\\bf k}^2\n\\propto 1\/\\epsilon_{\\bf k}^2$, we notice the dependences\naround ${\\bf k}=(0,0)$ as\n$I^{(2)}_u({\\bf k})\\propto \\epsilon_{\\bf k}^2$,\n$I^{(2)}_s({\\bf k})\\propto const.$ \n\nIn the numerical evaluation of Eqs.~(\\ref{eq.dyna2}) and (\\ref{eq.conti}),\nwe sum up the values on $N_L^2$ points of ${\\bf p}$ and ${\\bf q}$ in the\nfirst BZ, with $N_L=64$.\n\n\\subsection{Isotropic case}\n\nFigure \\ref{fig.iso.struc} shows the spin-wave-peak intensity \nand the intensity of three-spin-wave continuum as a function of momentum \nalong high-symmetry lines for $S=1\/2$. \nUsing the extended zone scheme, we assign the staggered part for line\n$(0,0)-(\\pi\/2,\\pi\/2)$ to the values for line $(\\pi,\\pi)-(\\pi\/2,\\pi\/2)$ \nand also the staggered part for line $(0,0)-(\\pi,0)$ to\nthe vales for line $(\\pi,\\pi)-(\\pi,0)$.\nThe uniform part is assigned inside the first BZ.\nAt the zone boundary of the reduced BZ, the uniform and staggered parts\ncoincide with each other. \nThe second-order corrections to the spin-wave-peak intensity becomes\none order of magnitude smaller than the first-order correction,\ndue to a cancellation among contributions of four terms \nin Eq.~(\\ref{eq.dyna2}).\nThe intensity is almost determined within the first-order correction.\nThus the correction relative to the zero-th order value is independent \nof momentum. We obtain around ${\\bf k}=(0,0)$, \n\\begin{equation}\n \\rho^{(1)}_u({\\bf k})=0.215 |{\\bf k}|, \\quad\n \\rho^{(1)}_s({\\bf k})= 1.72\/|{\\bf k}| \\quad (S=1\/2).\n\\end{equation}\nThese values should be compared with the $1\/S$-expansion analysis\nbased on the Dyson-Maleev transformation,\\cite{Canali93} 0.202 and 1.86.\nA series expansion analysis by Singh\\cite{Singh93} \ngives the values, 0.246 and 2.10, while a recent analysis by Zheng \n\\textit{et al}.\\cite{Zheng04} gives the values, 0.216 and 1.86.\n\nThe small second-order correction to $\\rho^{(1)}_{u(s)}({\\bf k})$\ndoes not necessarily mean small three-spin-wave continuum.\nAt the zone boundary $(\\pi,0)$, for example, we have \n$I^{(2)}(\\pi,0)=0.143$ in addition to $\\rho^{(1)}(\\pi,0)=0.618$.\nSuch a considerable intensity of three spin-wave continuum has been\npredicted in our previous paper\\cite{Igarashi92-2} and \nothers.\\cite{Canali93,Zheng04}\nIt varies as proportional to $\\epsilon_{\\bf k}^2$ at the zone center,\nand converges to a constant value at $(\\pi,\\pi)$.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.5.eps}%\n\\caption{\\label{fig.iso.struc}\nTransverse dynamical structure factor as a function of momentum \nfor the isotropic coupling. $S=1\/2$.\nThe dotted, broken and solid lines represent the intensity of one spin-wave\nexcitation calculated within the LSW theory, the first-order correction, \nand the second-order correction, respectively. \nThin solid line represents the intensity of three spin-wave excitations.\nExperimental data are taken from Ref.~\\onlinecite{Christensen04}.\nInset indicates high symmetry lines which momentum varies along.} \n\\end{figure}\n\n\\subsection{Anisotropic case}\n\nFigure \\ref{fig.aniso.struc} shows the spin-wave-peak intensity \nand the intensity of three-spin-wave continuum along the symmetry line\n$(0,0)-(\\pi,0)$ for several anisotropic couplings. $S=1\/2$.\n\nThe spin-wave-peak intensity is reduced by the first-order correction,\nbut is increased by the second-order correction.\nNote that, although the residue of the spin-wave pole in the Green's function\nis reduced by the self-energy, the other terms of the second-order\ncorrection work to increase the intensity. The second-order correction \nincreases with decreasing values of $J'\/J$, but\nthe reduction due to the first-order correction is much larger than\nthe gain due to the second-order correction. As a result, the spin-wave-peak\nintensity is strongly reduced. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.6.eps}%\n\\caption{\\label{fig.aniso.struc}\nTransverse dynamical structure factor \nas a function of momentum along the symmetric line $(0,0)-(\\pi,0)$\nfor anisotropic couplings;\n(a)$J'\/J=0.5$, (b)$J'\/J=0.1$, (c)$J'\/J=0.075$, and (d)$J'\/J=0.05$. $S=1\/2$.\nThe dotted, broken and solid lines represent the spin-wave-peak intensity\ncalculated within the LSW theory, up to the first-order correction, \nand up to the second-order correction, respectively. \nThe thin solid line represents the three-spin-wave continuum intensity.}\n\\end{figure}\n\nOn the other hand, the intensity of three-spin-wave continuum increases\nwith decreasing values of $J'\/J$.\nIt exceeds the spin-wave-peak intensity in the quasi-one dimensional situation.\nSuch large intensities of continuum spectra have been observed \nin the recent INS experiments on the quasi-one dimensional QHAF\nsuch as KCuF$_3$\\cite{Lake00} and BaCu$_2$Si$_2$O$_7$.\\cite{Zheludev02}\n\n\\section{\\label{sect.7} Concluding remarks}\n\nWe have systematically carried out the $1\/S$ expansion up to the second order\non the basis of the HP transformation in the two-dimensional QHAF.\nWe have calculated the spin-wave energy in the whole BZ,\nin comparison with the recent INS experiment \nfor CFTD.\\cite{Ronnow01,Christensen04}\nWe have found that the spin-wave energy at $(\\pi,0)$ is about $2\\%$\nsmaller than that at $(\\pi\/2,\\pi\/2)$ due to the second-order correction.\nThis is a correct tendency, but the value is somewhat smaller than\nthe experimental value $6\\%$ and other theoretical estimates \n$7-9\\%$.\\cite{Singh95,Sandvik01,Zheng04}\nWe have so far not been able to find the reason for the difference, \nsince the corrections higher than the second order of $1\/S$ is expected\nto be quite small. We have also calculated the transverse dynamical\nstructure factor. The second-order correction is found extremely small\nin the one-spin-wave-peak intensity due to the cancellation\nin the second-order terms, while it gives rise to substantial intensities \nof three-spin-wave continuum. This is consistent with previous \nstudies.\\cite{Igarashi92-2,Canali93,Singh95,Zheng04}\n\nCanali and Wallin\\cite{Canali93} reported that the value of the perpendicular \nsusceptibility $\\chi_{\\perp}$ is different from the value \nin our previous paper\\cite{Igarashi92-1} (for $J'\/J=1$),\nalthough they confirmed the same values for the second-order correction\nto the sublattice magnetization $M_2$ and for the spin-wave velocity $V_x$.\nThe difference is not due to numerical errors, since the convergence \nwith respect to $N_L$ has been carefully checked with changing \n$N_L=160, 320, 480$ (see Ref.~\\onlinecite{Igarashi92-1}). \nAs already discussed there, the value in Ref.~\\onlinecite{Igarashi92-1}\nsatisfies the hydrodynamic relation, $V_x=(\\rho_s\/\\chi)^{1\/2}$ in an \nappropriate unit, with independently-evaluated spin-stiffness constant \n$\\rho_s$.\nAs regards the dynamical structure factor, the difference\nbetween the values by Canali and Wallin and the present values\nmight have the same origin as the difference in the perpendicular \nsusceptibility, because the diagrams for the perpendicular susceptibility \nare closely related to those for the transverse dynamical structure factor.\n\nThe second-order correction is expected to become more important in \nquasi-one dimensional systems.\nWe have studied the crossover from two dimensions to one dimension\nby weakening the exchange coupling in one direction.\nAll formulas are found formally the same as those for the isotropic coupling\nwith replacing $JSz$ by $2JS(1+\\zeta)$.\nIt is found that the excitation energy is pushed up by the first-order and \nsecond-order corrections. With approaching the quasi-one dimensional situation,\nthe corrections make the excitation energy close to the des Cloizeaux-Pearson \nboundary in the one-dimensional QHAF for $S=1\/2$. \nIn the transverse dynamical structure factor,\nthe spin-wave-peak intensity is reduced by the first-order\ncorrection, but is increased by the second-order correction.\nThe former exceeds the latter in the quasi-one dimensional situation, \nand thereby the peak intensity is strongly reduced from the LSW value.\nOn the other hand, the intensity of three-spin-wave continuum is found to \nbecome larger and exceeds the spin-wave-peak intensity.\n\nIn purely one-dimension, spin-one excitations are considered excitations\nof two spinons of a spin-one-half excitation.\nIn this respect, the spin wave might be considered \nas a bound state of two spinons. Our finding that the weight \n$\\rho_1({\\bf k})$ of the spin-wave peak decreases with $J'\/J\\to 0$\nis consistent with this picture. Large intensities\nof three-spin-wave continuum might be replaced by two-spinon continuum\nin the one-dimensional limit. \nRecently, INS experiments have been carried out at low temperatures\nin the quasi-one dimensional systems such as KCuF$_3$\\cite{Lake00} \nand BaCu$_2$Si$_2$O$_7$,\\cite{Zheludev02} and large broad spectra \nhave been observed in addition to a peak in the transverse dynamical structure \nfactors. This behavior as well as the behavior of the longitudinal component\nhave been analyzed by the chain-mean-field and random phase \napproximation.\\cite{Essler97,Zheludev03}\nIt may be interesting to analyze these data in terms of the $1\/S$ expansion\nby starting from a detailed three-dimensional model \nwith directional anisotropy.\n\n\\begin{acknowledgments}\nJ.I. thanks MPI-PKS at Dresden for hospitality during his stay, \nwhere this work started.\nThis work was partially supported by a Grant-in-Aid for Scientific Research \nfrom the Ministry of Education, Science, Sports and Culture, Japan.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzibuq b/data_all_eng_slimpj/shuffled/split2/finalzzibuq new file mode 100644 index 0000000000000000000000000000000000000000..1735583ffbe1f0cdbf4f7fa0cad09320d7201f01 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzibuq @@ -0,0 +1,5 @@ +{"text":"\\section{Discussion on the time complexity of NetEmd}\n\\label{app:time_complexity}\n\nCalculating the NetEmd measure is a process that can be separated in two phases: acquiring the distributions of the network statistics of interest and comparing the statistics using the EMD. In the case of the algorithm we propose in this work, obtaining the distributions of network statistics involves calculating the graphlet degree matrix and performing PCA or ICA on this matrix.\n\nObtaining the graphlet degree matrix is the most computationally expensive step of this process, with a time complexity of $O(Nd^{m-1}$), where $N$ is the number of nodes in the network, $d$ the maximum degree of any node in the network and $m$ the size of the graphlets being enumerated. In undirected networks, we use the combinatorial algorithm \\emph{ORCA}~\\cite{hovcevar2014combinatorial} that relies on an analytical approach to set up a system of linear equations that relates different orbit frequencies. In directed networks, no such approaches are known~\\cite{ribeiro2019survey}, so we rely on \\emph{G-Tries}~\\cite{ribeiro2014g,aparicio2016extending}, a data structure that supports a graphlet enumeration algorithm by representing subgraphs in a prefix tree, the state of the art in enumerating directed graphlets, to our knowledge~\\cite{ribeiro2019survey,aparicio2016extending}.\n\nThe complexity of PCA is split in two parts, computing the covariance matrix is $O(Np^2)$ and doing the eigenvalue decomposition is $O(p^3)$, where $N$ is the number of nodes in the network and $p$ the number of orbits being considered. This leads to an overall complexity of $O(Np^2 + p^3)$.\n\nThe complexity of ICA is harder to characterize. Firstly, the FastICA algorithm assumes that the data has been centered and whitened~\\cite{hyvarinen2000independent}. The implementation we use, from scikit-learn~\\cite{scikit-learn}, uses PCA to do the whitening preprocessing, so its complexity is at least as high as PCA. The iterative algorithm to find the weight matrix $\\mathbf{W}$ is repeated until the matrix has converged, but there is no prior assurance that the algorithm converges or how many iterations it takes to do so. The scikit-learn implementation defines a maximum number of iterations to stop execution, we set this value to 1000. Each iteration involves calculating $\\log\\cosh(\\mathbf{w}^\\intercal\\mathbf{f}_i)$ for each component and each node, which the scikit-learn implementation does in $O(Nc^2)$, where $c$ is the number of components to be calculated, and the decorrelation step, which is done in $O(c^2)$. The overall complexity of ICA then becomes $O(Np^2 + p^3 + INc^2)$, where $I$ is the number of iterations.\n\nWegner et al.~\\cite{wegner2017identifying} calculate the complexity of comparing two graphlet distributions using $EMD^*$ to be $O(k(N+N')\\log(N+N'))$, where $N$ and $N'$ are the number of nodes of each network and $k$ is the maximum number of function calls to the optimization algorithm used to align the distributions.\n\n\n\\section{Results for Task 1 in directed networks}\n\\label{app:task1_dir}\n\n\\afterpage{\n\\begin{landscape}\n\\begin{table}\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|c|c|ccccc|}\n \n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{5}{c|}{Reciprocity} \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% \\\\\n \\hline \\multirow{2}{*}{All Orbits} & G3D & 0.989 $\\pm$ 0.002 & 0.990 $\\pm$ 0.002 & 0.989 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & 0.968 $\\pm$ 0.003 \\\\\n \\cline{2-7} & G4D & 0.992 $\\pm$ 0.002 & 0.993 $\\pm$ 0.001 & 0.993 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\textit{0.989 $\\pm$ 0.001} \\\\\n \\hline \\multirow{2}{*}{$Weighted\\_NetEmd$} & G3D & 0.989 $\\pm$ 0.002 & 0.990 $\\pm$ 0.002 & 0.989 $\\pm$ 0.002 & 0.989 $\\pm$ 0.002 & 0.969 $\\pm$ 0.003 \\\\\n \\cline{2-7} & G4D & 0.992 $\\pm$ 0.002 & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\textit{0.989 $\\pm$ 0.001} \\\\\n \\hline \\multirow{5}{*}{$PCA\\_NetEmd$} & 50\\% Variance & 0.989 $\\pm$ 0.003 & 0.993 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & \\textit{0.993 $\\pm$ 0.001} & 0.978 $\\pm$ 0.004 \\\\\n \\cline{2-7} & 80\\% Variance & 0.990 $\\pm$ 0.003 & 0.993 $\\pm$ 0.001 & 0.990 $\\pm$ 0.002 & 0.992 $\\pm$ 0.001 & 0.982 $\\pm$ 0.003 \\\\\n \\cline{2-7} & 90\\% Variance & \\textit{0.993 $\\pm$ 0.002} & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\bf{0.990 $\\pm$ 0.002} \\\\\n \\cline{2-7} & 95\\% Variance & 0.992 $\\pm$ 0.002 & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\textit{0.988 $\\pm$ 0.001} \\\\\n \\cline{2-7} & 99\\% Variance & \\textit{0.992 $\\pm$ 0.002} & 0.992 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & 0.986 $\\pm$ 0.002 \\\\\n \\hline \\multirow{3}{*}{$ICA\\_NetEmd$} & 2 components & 0.990 $\\pm$ 0.003 & 0.993 $\\pm$ 0.002 & 0.992 $\\pm$ 0.002 & \\textit{0.993 $\\pm$ 0.001} & 0.987 $\\pm$ 0.002 \\\\\n \\cline{2-7} & 10 components & \\textit{0.992 $\\pm$ 0.002} & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.002 & \\textit{0.993 $\\pm$ 0.001} & 0.985 $\\pm$ 0.002 \\\\\n \\cline{2-7} & 15 components & \\textit{0.993 $\\pm$ 0.002} & 0.994 $\\pm$ 0.001 & 0.993 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & 0.980 $\\pm$ 0.001 \\\\ \n \\hline\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & 0.992 $\\pm$ 0.001 & 0.991 $\\pm$ 0.001 & 0.990 $\\pm$ 0.002 & 0.992 $\\pm$ 0.002 & 0.970 $\\pm$ 0.003 \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.986 $\\pm$ 0.003 & 0.992 $\\pm$ 0.002 & 0.990 $\\pm$ 0.002 & 0.988 $\\pm$ 0.002 & 0.968 $\\pm$ 0.004 \\\\\n \\cline{2-7} & 129 orbits & \\bf{0.995 $\\pm$ 0.002} & \\bf{0.996 $\\pm$ 0.001} & \\bf{0.995 $\\pm$ 0.001} & \\bf{0.995 $\\pm$ 0.002} & 0.987 $\\pm$ 0.002 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.988 $\\pm$ 0.003 & 0.990 $\\pm$ 0.002 & 0.988 $\\pm$ 0.003 & 0.992 $\\pm$ 0.002 & 0.959 $\\pm$ 0.005 \\\\\n \\cline{2-7} & G4D & 0.992 $\\pm$ 0.003 & 0.992 $\\pm$ 0.002 & 0.991 $\\pm$ 0.003 & \\textit{0.993 $\\pm$ 0.002} & 0.977 $\\pm$ 0.007 \\\\\n \\hline\n\n \\end{tabular}\n \\caption{Results for Task 1 in directed networks, for each level of reciprocity. The metric used for evaluation is the mean (and standard error of the mean) of the 16 values for $\\overline{P}$, in each combination of number of nodes, network density and reciprocity. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each reciprocity level. Italic values are within one standard error of the maximum performance.}\n \\label{tab:scores_dir_task1}\n\\end{table}\n\\end{landscape}\n}\n\n\\section{Discussion on orbits from smaller graphlet sizes}\n\\label{app:smaller_orbits}\n\nWegner et al.~\\cite{wegner2017identifying} have shown that reducing the graphlet size as input to NetEmd can improve the quality of the clusters under certain conditions. A more common reason to use smaller graphlets for comparison is a computational load argument, as we show in Appendix~\\ref{app:time_complexity}, increasing the graphlet size leads to more computation time in each step of the NetEmd framework. In the first step, computing orbit frequencies of graphlets of size 4 in directed networks or size 5 in undirected networks can be prohibitive for very large graphs. Another consideration is the time taken to compute the EMD between orbit distributions, particularly in directed networks with graphlets of size 4 requiring 22 times more calls to the $EMD^*$ function than using size 3. Finally, computing the principal or independent components also contain a complexity component dependent on the number of orbits we consider as features to these methods, which also contribute to greater computational load when increasing the graphlet sizes as input to NetEmd. Therefore, it is important to understand in which situations decreasing the graphlet size as input to NetEmd leads to similar or better performance to avoid paying the high computational cost.\n\nWe previously presented the performance of $ICA\\_NetEmd$ under smaller graphlet sizes, for multiple components used, in Figures~\\ref{fig:undir_task1_pbar} to~\\ref{fig:real_nets_pbar}. The results for Task 1, both in directed and undirected networks, demonstrate that when comparing networks of similar sizes and densities using bigger graphlet sizes yields a more accurate comparison. The same holds for Task 2 in the undirected case and in the extreme cases of reciprocity (0\\% and 100\\%) in Task 2 of the directed experiments, where ICA achieves the top performance with size 5 undirected and size 4 directed graphlets, respectively. On the other hand, in the datasets of real world networks and in directed Task 2 with reciprocity of 25\\%, 50\\% and 75\\%, size 5 graphlets in undirected networks and size 4 in directed networks perform worse than size 4 and size 3, respectively. \nFor the datasets of real networks in particular, this difference in performance can be explained by noticing that these datasets contain very small networks such that some graphlets have no matches in them. This issue is amplified the larger the graphlet size considered is and it adds a confounding factor to the network comparison, similar to the example of Figure~\\ref{fig:weight_by_orbit_example}. Incidentally, this is the only dataset where $Weighted\\_NetEmd$ achieves better performance than following the original formulation. \n\nFigures~\\ref{fig:undir_task1_pbar} to~\\ref{fig:real_nets_pbar} also show the performance results of using the full set of size $k$ orbits to denoise the distributions of $k-1$ or $k-2$ orbits. In this second way of using ICA, the only time saved compared to using the full set of orbits is the time to compute the EMD between orbit distributions and we find that in no case it leads to the best clustering performance. However, Figure~\\ref{fig:undir_task2_pbar} shows a breakpoint at 30 components, where the performance using size 3 and 4 orbits after denoising with size 5 succeeds in performing better than size 5. The reason for this is that, as more components are added, the smaller the reconstruction error $||\\mathbf{F}_G - \\hat{\\mathbf{F}}_G||_2$ is, so the performance with those parameters tends to approximate the performance of the original NetEmd with those graphlets sizes.\n\nThe performance of $PCA\\_NetEmd$ under smaller graphlet sizes follows a similar set of rules to $ICA\\_NetEmd$, with better performances in Task 1 measured when using larger graphlets and in Task 2 and real world networks when using smaller graphlet sizes. Tables~\\ref{tab:pca_undir_small} to~\\ref{tab:pca_dir_task2_small} detail the results for multiple combinations of graphlet size and percentage of variance explained. Unlike $ICA\\_NetEmd$, our experimental setup does not allow us to claim that after a certain value of explained variance performance degrades or stabilizes. Instead, results suggest that the optimal threshold depends on both the dataset and the size of graphlet chosen. Like with $ICA\\_NetEmd$, choosing a low value for percentage of explained variance for Task 2 in directed networks yields the best performance for 0, 25, 50 and 75\\% reciprocity, but otherwise choosing 90 or 95\\% explained variance seems to be a good rule of thumb for best performance with $PCA\\_NetEmd$.\n\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{|c|>{\\centering\\arraybackslash}m{60pt}>{\\centering\\arraybackslash} m{60pt}|>{\\centering\\arraybackslash} m{60pt}|}\n \\hline \\multirow{2}{*}{Parameters} & \\multicolumn{2}{c|}{Synthetic} & Real \\\\\n \\cline{2-4} & Task 1 & Task 2 & Onnela et al.\\\\\n \\hline 3 - 50\\% Variance & 0.981 $\\pm$ 0.004 & 0.912 & 0.872\\\\\n \\hline 3 - 80\\% Variance & 0.980 $\\pm$ 0.003 & 0.924 & 0.874 \\\\\n \\hline 3 - 90\\% Variance & 0.984 $\\pm$ 0.002 & 0.929 & 0.881\\\\\n \\hline 3 - 95\\% Variance & 0.982 $\\pm$ 0.002 & 0.911 & 0.891 \\\\\n \\hline 3 - 99\\% Variance & 0.986 $\\pm$ 0.002 & 0.928 & 0.889\\\\\n \\hline 4 - 50\\% Variance & 0.986 $\\pm$ 0.003 & 0.917 & 0.873\\\\\n \\hline 4 - 80\\% Variance & 0.987 $\\pm$ 0.003 & 0.926 & 0.885 \\\\\n \\hline 4 - 90\\% Variance & 0.992 $\\pm$ 0.001 & 0.931 & \\textbf{0.898}\\\\\n \\hline 4 - 95\\% Variance & 0.990 $\\pm$ 0.001 & 0.921 & 0.897 \\\\\n \\hline 4 - 99\\% Variance & 0.991 $\\pm$ 0.001 & 0.931 & 0.897\\\\\n \\hline 5 - 50\\% Variance & 0.986 $\\pm$ 0.003 & 0.915 & 0.863\\\\\n \\hline 5 - 80\\% Variance & 0.985 $\\pm$ 0.002 & 0.918 & 0.893 \\\\\n \\hline 5 - 90\\% Variance & \\textit{0.995 $\\pm$ 0.001} & 0.922 & 0.891\\\\\n \\hline 5 - 95\\% Variance & 0.993 $\\pm$ 0.001 & 0.919 & 0.892 \\\\\n \\hline 5 - 99\\% Variance & \\textit{0.995 $\\pm$ 0.001} & 0.921 & 0.891\\\\\n \\hline Other best & \\textbf{0.996 $\\pm$ 0.001} (ICA G5 13 components) & \\textbf{0.933} (ICA G5 7 components) & \\textbf{0.898} (Original NetEmd G4)\\\\\n \\hline\n \\end{tabular}\n \\caption{Results for $PCA\\_NetEmd$ with different combinations of graphlet size and percentage of variance explained in Task 1, Task 2 and the Onnela et al. dataset in the undirected case. The metric used for Task 1 is the mean (and standard error of the mean) of the 16 values for $\\overline{P}$ in each combination of number of nodes with network density. The metric used for Task 2 is the sole value of $\\overline{P}$ after comparing the 1280 networks. The bolded values are the maximum in each task. Italic values are within one standard error of the maximum performance.}\n \\label{tab:pca_undir_small}\n\\end{table}\n\n\n\\begin{landscape}\n\\begin{table}\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \n \\begin{tabular}{|>{\\centering\\arraybackslash}m{50pt}|>{\\centering\\arraybackslash}m{60pt}>{\\centering\\arraybackslash} m{60pt}>{\\centering\\arraybackslash} m{60pt}>{\\centering\\arraybackslash} m{60pt}>{\\centering\\arraybackslash} m{60pt}|}\n \\hline \\multirow{2}{*}{Parameter} & \\multicolumn{5}{c|}{Reciprocity} \\\\\n \\cline{2-6} & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% \\\\\n \\hline 3 - 50\\% Variance & 0.984 $\\pm$ 0.003 & 0.987 $\\pm$ 0.003 & 0.985 $\\pm$ 0.003 & 0.987 $\\pm$ 0.003 & 0.928 $\\pm$ 0.004\\\\\n \\hline 3 - 80\\% Variance & 0.985 $\\pm$ 0.003 & 0.991 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & 0.992 $\\pm$ 0.002 & 0.932 $\\pm$ 0.004 \\\\\n \\hline 3 - 90\\% Variance & 0.989 $\\pm$ 0.003 & 0.991 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & 0.990 $\\pm$ 0.002 & 0.929 $\\pm$ 0.005\\\\\n \\hline 3 - 95\\% Variance & 0.992 $\\pm$ 0.002 & 0.992 $\\pm$ 0.001 & 0.990 $\\pm$ 0.002 & 0.992 $\\pm$ 0.002 & 0.940 $\\pm$ 0.003 \\\\\n \\hline 3 - 99\\% Variance & 0.990 $\\pm$ 0.002 & 0.991 $\\pm$ 0.001 & 0.990 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & 0.938 $\\pm$ 0.004\\\\\n \\hline 4 - 50\\% Variance & 0.989 $\\pm$ 0.003 & 0.993 $\\pm$ 0.002 & 0.991 $\\pm$ 0.002 & \\textit{0.993 $\\pm$ 0.001} & 0.978 $\\pm$ 0.004\\\\\n \\hline 4 - 80\\% Variance & 0.990 $\\pm$ 0.003 & 0.993 $\\pm$ 0.001 & 0.990 $\\pm$ 0.002 & 0.992 $\\pm$ 0.001 & 0.982 $\\pm$ 0.003 \\\\\n \\hline 4 - 90\\% Variance & \\textit{0.993 $\\pm$ 0.002} & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\textbf{0.990 $\\pm$ 0.002}\\\\\n \\hline 4 - 95\\% Variance & 0.992 $\\pm$ 0.002 & 0.993 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & \\textit{0.993 $\\pm$ 0.001} & \\textit{0.988 $\\pm$ 0.001} \\\\\n \\hline 4 - 99\\% Variance & 0.992 $\\pm$ 0.002 & 0.992 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & 0.992 $\\pm$ 0.001 & 0.986 $\\pm$ 0.002\\\\\n \\hline\n \\hline Other best & \\textbf{0.995 $\\pm$ 0.002} (DGCD129) & \\textbf{0.996 $\\pm$ 0.001} (DGCD129) & \\textbf{0.995 $\\pm$ 0.001} (DGCD129) & \\textbf{0.995 $\\pm$ 0.002} (DGCD129) & \\textit{0.989 $\\pm$ 0.001} (Weighted G4D)\\\\\n \\hline\n\n \\end{tabular}\n \\caption{Results for $PCA\\_NetEmd$ with different combinations of graphlet size and percentage of variance explained in Task 1 of directed networks, for each level of reciprocity. The metric used for evaluation is the mean (and standard error of the mean) of the 16 values for $\\overline{P}$, in each combination of number of nodes, network density and reciprocity. The bolded values are the maximum for each reciprocity level. Italic values are within one standard error of the maximum performance.}\n \\label{tab:pca_dir_task1_small}\n\\end{table}\n\\end{landscape}\n\n\\begin{table}[h]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|>{\\centering\\arraybackslash}m{31pt}|>{\\centering\\arraybackslash} m{28pt}>{\\centering\\arraybackslash} m{28pt}>{\\centering\\arraybackslash} m{28pt}>{\\centering\\arraybackslash} m{28pt}>{\\centering\\arraybackslash} m{28pt}|>{\\centering\\arraybackslash} m{28pt}| }\n \\hline \\multirow{3}{*}{Parameters} & \\multicolumn{5}{c|}{Synthetic} & \\multirow{3}{*}{\\shortstack{Real\\\\Networks}}\\\\\n \\cline{2-6} & \\multicolumn{5}{c|}{Reciprocity} & \\\\\n & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% & \\\\\n \\hline 3 - 50\\% Variance & 0.915 & 0.919 & 0.918 & 0.918& 0.857 & 0.825\\\\\n \\hline 3 - 80\\% Variance & 0.925 & 0.909 & 0.905 & 0.907 & 0.865 & 0.858 \\\\\n \\hline 3 - 90\\% Variance & 0.933 & 0.909 & 0.905 & 0.906 & 0.866 & 0.861\\\\\n \\hline 3 - 95\\% Variance & 0.934 & 0.908 & 0.905 & 0.904 & 0.867 & 0.868 \\\\\n \\hline 3 - 99\\% Variance & 0.921 & 0.903 & 0.900 & 0.903 & 0.879 & 0.867\\\\\n \\hline 4 - 50\\% Variance & 0.924 & 0.897 & 0.900 & 0.894 & 0.908 & 0.821\\\\\n \\hline 4 - 80\\% Variance & 0.924 & 0.885 & 0.892 & 0.881 & 0.920 & 0.825\\\\\n \\hline 4 - 90\\% Variance & 0.921 & 0.882 & 0.889 & 0.878 & \\textbf{0.927} & 0.824\\\\\n \\hline 4 - 95\\% Variance & 0.915 & 0.880 & 0.887 & 0.876 & 0.922 & 0.822\\\\\n \\hline 4 - 99\\% Variance & 0.914 & 0.878 & 0.885 & 0.875 & 0.924 & 0.818\\\\\n \\hline\n \\hline Other best & \\textbf{0.941} (ICA G4D 2 comps) & \\textbf{0.926} (ICA G3D 2 comps) & \\textbf{0.925} (ICA G3D 2 comps) & \\textbf{0.928} (ICA G3D 2 comps) & \\textbf{0.927} (ICA G4D 4 comps) & \\textbf{0.870} (ICA G3D 10 comps) \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for $PCA\\_NetEmd$ with different combinations of graphlet size and percentage of variance explained in Task 2 in directed networks, for each level of reciprocity, and for the real world networks dataset. The metric used for evaluation is the mean (and standard error of the mean) of the 16 values for $\\overline{P}$, in each combination of number of nodes, network density and reciprocity. The bolded values are the maximum for each reciprocity level.}\n \\label{tab:pca_dir_task2_small}\n\\end{table}\n\n\n\\section{Different metrics}\n\\label{app:diff_metrics}\n\n\\subsection{Area Under Precision-Recall (AUPR)}\n\nSarajlic et al.~\\cite{sarajlic2016graphlet} evaluate the performance of their graph comparison tool using the Area Under Precision - Recall curve (AUPR), after the framework proposed by Yavero{\\u{g}}lu et al.~\\cite{yaverouglu2015proper}. The metric is calculated as follows: for each value of a parameter $\\epsilon \\ge 0$, if the distance between two networks is less than $\\epsilon$, then the networks belong to the same cluster. The parameter $\\epsilon$ ranges from 0 to 1, with values incrementing by $\\num{5e-3}$. For each value of $\\epsilon$, we compute the number of \\emph{true positives}, \\emph{false positives}, \\emph{true negatives} and \\emph{false negatives}. In this context, we define these concepts as:\n\n\\begin{itemize}\n \\item \\emph{True positive (TP)}: the distance between the networks is smaller than $\\epsilon$ and they were generated by the same random network model.\n \\item \\emph{False positive (FP)}: the distance between the networks is smaller than $\\epsilon$ but they were generated by different random network models.\n \\item \\emph{True negative (TN)}: the distance between the networks is greater than $\\epsilon$ and they were generated by different random network models.\n \\item \\emph{False negative (FN)}: the distance between the networks is greater than $\\epsilon$ but they were generated by the same random network model.\n\\end{itemize}\n\nFrom these quantities, we calculate \\emph{precision} as $\\frac{TP}{TP + FP}$ and \\emph{recall} as $\\frac{TP}{TP + FN}$, obtaining 200 tuples of (precision, recall), from which we are able to plot the precision-recall curve. The area under this curve is a metric relevant to our problem as it puts an emphasis on the positive predictive value of the model, disregarding the true negatives that compose the majority of datasets with imbalanced labels such as this one (with 8 random network models, positive labels comprise only $12.5\\%$ of the dataset).\n\n\\subsubsection{Undirected Results}\n\nTable~\\ref{tab:aupr_undir} shows the AUPR for Task 1 and Task 2 in the undirected case and the Onnela et al. dataset.\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|c|c|cc|c|}\n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{2}{c|}{Synthetic} & Real \\\\\n \\cline{3-5} & & Task 1 & Task 2 & Onnela et al.\\\\\n \\hline \\multirow{3}{*}{Original NetEmd} & G3 & 0.91 $\\pm$ 0.01 & 0.670 & 0.762 \\\\\n \\cline{2-5} & G4 & \\textit{0.95 $\\pm$ 0.01} & 0.705 & \\textbf{0.790} \\\\\n \\cline{2-5} & G5 & \\textbf{0.96 $\\pm$ 0.01} & 0.712 & 0.761 \\\\\n \\hline \\multirow{5}{*}{PCA} & 50\\% Variance & 0.93 $\\pm$ 0.01 & 0.668 & 0.702\\\\\n \\cline{2-5} & 80\\% Variance & 0.92 $\\pm$ 0.01 & 0.701 & 0.759 \\\\\n \\cline{2-5} & 90\\% Variance & \\textit{0.96 $\\pm$ 0.01} & 0.717 & 0.756 \\\\\n \\cline{2-5} & 95\\% Variance & \\textit{0.96 $\\pm$ 0.01} & 0.716 & 0.760 \\\\\n \\cline{2-5} & 99\\% Variance & \\textbf{0.96 $\\pm$ 0.01} & 0.718 & 0.760 \\\\\n \\hline \\multirow{3}{*}{ICA} & 2 components & 0.93 $\\pm$ 0.01 & \\textbf{0.742} & 0.717 \\\\\n \\cline{2-5} & 10 components & 0.94 $\\pm$ 0.01 & 0.728 & 0.750\\\\\n \\cline{2-5} & 15 components & \\textbf{0.96 $\\pm$ 0.01} & 0.729 & 0.754 \\\\\n \\hline\n \\hline \\multirow{2}{*}{GCD} & 11 orbits & 0.89 $\\pm$ 0.02 & 0.516 & 0.698\\\\\n \\cline{2-5} & 73 orbits & 0.94 $\\pm$ 0.02 & 0.600 & 0.747 \\\\\n \\hline \\multirow{3}{*}{GDA} & G3 & 0.92 $\\pm$ 0.02 & 0.605 & 0.783 \\\\\n \\cline{2-5} & G4 & 0.80 $\\pm$ 0.04 & 0.535 & 0.692 \\\\\n \\cline{2-5} & G5 & 0.74 $\\pm$ 0.04 & - & 0.638 \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 1, Task 2 and the Onnela et al.~\\cite{onnela2012taxonomies} datasets in undirected networks. The metric used for Task 1 is the mean (and standard error of the mean) of the 16 values for AUPR in each combination of number of nodes with network density. The metric used for Task 2 and Onnela et al. dataset is the sole value of AUPR after comparing the 1280 and 151 networks, respectively. The parameter column indicates: graphlet size used in original NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 5); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 5; the number of orbits used by GCD; graphlet sizes used in GDA. The bolded values are the maximum for each task. Italic values are within one standard error of the maximum performance. Note that results for GDA with graphlet size 5 took longer than a week to return results for Task 2, at which point we stopped the computation.}\n \\label{tab:aupr_undir}\n\\end{table}\n\nIn Task 1, according to AUPR, using PCA or ICA does not lead to a gain in performance, unlike what we observed when using $\\overline{P}$. This is not surprising as the improvement we displayed with $\\overline{P}$ was within a standard error and both methodologies are able to separate networks between these random models with a very high degree of accuracy. Task 1 serves as a proof of concept that our proposed measure is able to perform the most simple task.\n\nThe results for Task 2 are aligned with what we observe when using $\\overline{P}$, coupling ICA with NetEmd leads to the best separation between models when different network sizes and densities are present. In the Onnela et al. dataset, the results with AUPR also agree with $\\overline{P}$, both performance metrics consider the original NetEmd with size 4 orbits to be one of the top performers with this dataset. The difference between the two metrics is that when measuring with $\\overline{P}$, we observe that using PCA with 90\\% explained variance leads to similar performance (Table~\\ref{tab:pca_undir_small}), but the top performer in this dataset according to AUPR is PCA with 99\\% explained variance, achieving an AUPR score of 0.791.\n\n\\afterpage{\n\\begin{landscape}\n\\begin{table}\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|c|c|ccccc|}\n \n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{5}{c|}{Reciprocity} \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% \\\\\n \\hline \\multirow{2}{*}{All Orbits} & G3D & 0.89 $\\pm$ 0.01 & 0.90 $\\pm$ 0.01 & 0.89 $\\pm$ 0.01 & 0.90 $\\pm$ 0.01 & 0.914 $\\pm$ 0.007 \\\\\n \\cline{2-7} & G4D & \\textit{0.961 $\\pm$ 0.007} & 0.945 $\\pm$ 0.005 & \\textbf{0.941 $\\pm$ 0.005} & 0.944 $\\pm$ 0.005 & \\textbf{0.992 $\\pm$ 0.002} \\\\\n \\hline \\multirow{2}{*}{$Weighted\\_NetEmd$} & G3D & 0.87 $\\pm$ 0.01 & 0.88 $\\pm$ 0.01 & 0.87 $\\pm$ 0.02 & 0.89 $\\pm$ 0.02 & 0.875 $\\pm$ 0.009 \\\\\n \\cline{2-7} & G4D & 0.93 $\\pm$ 0.01 & 0.930 $\\pm$ 0.007 & 0.925 $\\pm$ 0.009 & 0.928 $\\pm$ 0.007 & 0.923 $\\pm$ 0.009 \\\\\n \\hline \\multirow{5}{*}{$PCA\\_NetEmd$} & 50\\% Variance & 0.94 $\\pm$ 0.01 & 0.929 $\\pm$ 0.009 & 0.928 $\\pm$ 0.008 & 0.927 $\\pm$ 0.008 & 0.957 $\\pm$ 0.005 \\\\\n \\cline{2-7} & 80\\% Variance & 0.94 $\\pm$ 0.01 & 0.935 $\\pm$ 0.006 & 0.932 $\\pm$ 0.008 & 0.932 $\\pm$ 0.006 & 0.958 $\\pm$ 0.006 \\\\\n \\cline{2-7} & 90\\% Variance & \\textit{0.958 $\\pm$ 0.008} & 0.938 $\\pm$ 0.005 & \\textit{0.936 $\\pm$ 0.005} & 0.936 $\\pm$ 0.005 & 0.974 $\\pm$ 0.004 \\\\\n \\cline{2-7} & 95\\% Variance & \\textit{0.958 $\\pm$ 0.007} & 0.940 $\\pm$ 0.005 & \\textit{0.937 $\\pm$ 0.005} & 0.939 $\\pm$ 0.005 & 0.971 $\\pm$ 0.003 \\\\\n \\cline{2-7} & 99\\% Variance & \\textit{0.958 $\\pm$ 0.007} & 0.941 $\\pm$ 0.005 & \\textit{0.936 $\\pm$ 0.005} & 0.940 $\\pm$ 0.005 & 0.977 $\\pm$ 0.002 \\\\\n \\hline \\multirow{3}{*}{$ICA\\_NetEmd$} & 2 components & 0.941 $\\pm$ 0.007 & 0.938 $\\pm$ 0.007 & 0.934 $\\pm$ 0.007 & 0.940 $\\pm$ 0.007 & 0.982 $\\pm$ 0.003 \\\\\n \\cline{2-7} & 10 components & 0.936 $\\pm$ 0.009 & 0.941 $\\pm$ 0.005 & \\textit{0.939 $\\pm$ 0.007} & 0.937 $\\pm$ 0.006 & 0.936 $\\pm$ 0.005 \\\\\n \\cline{2-7} & 15 components & 0.946 $\\pm$ 0.007 & 0.938 $\\pm$ 0.005 & \\textit{0.938 $\\pm$ 0.006} & 0.937 $\\pm$ 0.005 & 0.92 $\\pm$ 0.01 \\\\ \n \\hline\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & 0.938 $\\pm$ 0.006 & 0.936 $\\pm$ 0.008 & 0.93 $\\pm$ 0.01 & 0.94 $\\pm$ 0.01 & 0.936 $\\pm$ 0.005 \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.92 $\\pm$ 0.02 & 0.93 $\\pm$ 0.02 & 0.91 $\\pm$ 0.02 & 0.89 $\\pm$ 0.02 & 0.87 $\\pm$ 0.02 \\\\\n \\cline{2-7} & 129 orbits & \\textbf{0.97 $\\pm$ 0.01} & \\textbf{0.96 $\\pm$ 0.01} & \\textit{0.94 $\\pm$ 0.02} & 0.92 $\\pm$ 0.02 & 0.90 $\\pm$ 0.02 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.90 $\\pm$ 0.03 & \\textbf{0.96 $\\pm$ 0.01} & \\textit{0.94 $\\pm$ 0.01} & \\textbf{0.96 $\\pm$ 0.01} & 0.82 $\\pm$ 0.04 \\\\\n \\cline{2-7} & G4D & 0.88 $\\pm$ 0.03 & 0.84 $\\pm$ 0.03 & 0.86 $\\pm$ 0.03 & 0.84 $\\pm$ 0.03 & 0.80 $\\pm$ 0.04 \\\\\n \\hline\n\n \\end{tabular}\n \\caption{Results for Task 1 in directed networks, for each level of reciprocity. The metric used for evaluation is the mean (and standard error of the mean) of the 16 values for AUPR, in each combination of number of nodes, network density and reciprocity. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each reciprocity level. Italic values are within one standard error of the maximum performance.}\n \n \\label{tab:aupr_dir_task1}\n\\end{table}\n\\end{landscape}\n}\n\n\\subsubsection{Directed Results}\n\nThe results using AUPR for Task 1 in directed networks are shown in Table \\ref{tab:aupr_dir_task1} and for Task 2 and the dataset of real world directed networks in Table~\\ref{tab:aupr_dir_task2}.\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|>{\\centering\\arraybackslash}m{29pt}|>{\\centering\\arraybackslash}m{43pt}|>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}|>{\\centering\\arraybackslash} m{28pt}| }\n \\hline \\multirow{3}{*}{Algorithm} & \\multirow{3}{*}{Parameter} & \\multicolumn{5}{c|}{Synthetic} & \\multirow{3}{*}{\\shortstack{Real\\\\Networks}}\\\\\n \\cline{3-7} & & \\multicolumn{5}{c|}{Reciprocity} & \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% & \\\\\n \\hline \\multirow{2}{*}{\\shortstack{All\\\\Orbits}} & G3D & 0.644 & 0.594 & 0.583 & 0.598 & 0.721 & \\textbf{0.883} \\\\\n \\cline{2-8} & G4D & 0.702 & 0.630 & 0.632 & 0.628 & \\textbf{0.953} & 0.820 \\\\\n \\hline \\multirow{2}{*}{Weighted} & G3D & 0.605 & 0.554 & 0.545 & 0.559 & 0.639 & \\textbf{0.883} \\\\\n \\cline{2-8} & G4D & 0.608 & 0.567 & 0.563 & 0.565 & 0.660 & 0.847 \\\\\n \\hline \\multirow{5}{*}{PCA} & 50\\% Variance & 0.696 & 0.638 & 0.644 & 0.631 & 0.855 & 0.829 \\\\\n \\cline{2-8} & 80\\% Variance & 0.700 & 0.626 & 0.634 & 0.621 & 0.845 & 0.824 \\\\\n \\cline{2-8} & 90\\% Variance & 0.718 & 0.628 & 0.639 & 0.623 & 0.876 & 0.822 \\\\\n \\cline{2-8} & 95\\% Variance & 0.705 & 0.628 & 0.636 & 0.623 & 0.872 & 0.822 \\\\\n \\cline{2-8} & 99\\% Variance & 0.703 & 0.628 & 0.637 & 0.623 & 0.892 & 0.819 \\\\\n \\hline \\multirow{3}{*}{ICA} & 2 components & \\textbf{0.767} & \\textbf{0.688} & \\textbf{0.691} & \\textbf{0.683} & 0.926 & 0.726 \\\\\n \\cline{2-8} & 10 components & 0.725 & 0.669 & 0.679 & 0.664 & 0.820 & 0.837 \\\\\n \\cline{2-8} & 15 components & 0.729 & 0.660 & 0.680 & 0.658 & 0.768 & 0.841 \\\\\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & 0.664 & 0.629 & 0.623 & 0.629 & 0.738 & 0.881 \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.631 & 0.625 & 0.600 & 0.581 & 0.567 & 0.872 \\\\\n \\cline{2-8} & 129 orbits & 0.653 & 0.643 & 0.620 & 0.593 & 0.558 & 0.860 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.584 & 0.552 & 0.539 & 0.551 & 0.549 & 0.760 \\\\\n \\cline{2-8} & G4D & - & - & - & - & 0.535 & - \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 2 in directed networks, for each level of reciprocity, and for the real world networks dataset. The metric used for this task is the sole value of AUPR after comparing the 1280 and 1231 networks, respectively. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each dataset. Note that results for GDA with graphlet size 4 took longer than a week to return results for Task 2 and in the real world networks dataset, at which point we stopped the computation.}\n \\label{tab:aupr_dir_task2}\n\\end{table}\n\n\nIn Task 1, the main difference we find is that GDA with size 3 orbits shows an improvement over the other algorithms, becoming the top performer with 25\\% (tied with DGCD with 129 orbits) and 75\\% reciprocity. DGCD with 129 orbits still shows the best performance with 0\\% reciprocity, but PCA with 90, 95 and 99\\% explained variance is within one standard error, similarly to the results measured with $\\overline{P}$. At a 100\\% reciprocity, NetEmd with size 4 orbits achieves the best performance, but PCA with 90, 95 and 99\\% explained variance and ICA with 2 components obtain similar AUPR scores. As before, we highlight that the high scores achieved by the 6 algorithms in Table~\\ref{tab:aupr_dir_task1} serve as a proof of concept that they are able to distinguish the random network models for different levels of reciprocity, even if the AUPR score is not within a standard error of the best score.\n\nIn Task 2, similarly to the undirected version, we find an agreement between the AUPR score and $\\overline{P}$, with ICA with 2 components highlighted as the highest performing algorithm for 0, 25, 50 and 75\\% reciprocity datasets and NetEmd with size 4 orbits for 100\\% reciprocity. The same agreement between AUPR and $\\overline{P}$ is found on the dataset of real world directed networks, with the top performer being ICA with 6 components with an AUPR of 0.891 but not highlighted in Table~\\ref{tab:aupr_dir_task2}.\n\nThe conclusions regarding the difference between our proposed NetEmd methods and TriadEMD also hold when using AUPR instead of $\\overline{P}$; in the synthetic datasets TriadEMD achieves better AUPR scores than both NetEmd and $Weighted\\_NetEmd$ with orbits from size up to 3 graphlets, but worse results than NetEmd with size 4 graphlets. In the real world directed networks dataset, both NetEmd and $Weighted\\_NetEmd$ with size 3 orbits obtain better AUPR scores than TriadEMD.\n\n\\subsection{Adjusted Rand Index}\n\nThe adjusted Rand index~\\cite{rand1971objective,hubert1985comparing} is a metric to evaluate the similarity of data clusterings; a corrected for chance version of the Rand index~\\cite{rand1971objective} was proposed by Hubert and Arabie~\\cite{hubert1985comparing}. Let $\\mathcal{G} = \\{G_1, G_2, \\ldots, G_N\\}$ be the set of networks we wish to partition into clusters. For the synthetic datasets, the ground truth partition of this set is known, we have 8 models of random graphs and we evaluate a network comparison method by its ability to reconstruct these groups. For the Onnela et al.~\\cite{onnela2012taxonomies} dataset, the ground truth partition used is the one shown in the supplementary material of~\\cite{ali2014alignment}. Let $\\mathcal{P}^* = \\{P_1^*, P_2^*, \\ldots, P_c^*\\}$ be the ground truth partitions and $\\mathcal{P} = \\{P_1, P_2, \\ldots, P_c\\}$ be the clusters produced by the network comparison algorithms. These clusters are generated by applying a hierarchical clustering algorithm to the matrix of pair-wise distances calculated by the network comparison; in our case we used the implementation by \\emph{scikit-learn}~\\cite{scikit-learn}, the \\emph{AgglomerativeClustering} class with complete linkage.\n\nGiven the two set of clusters, we define the following four quantities:\n\n\\begin{itemize}\n \\item $a$: the number of pairs of elements of $\\mathcal{G}$ that are in the same cluster in $\\mathcal{P}^*$ and in the same cluster in $\\mathcal{P}$.\n \\item $b$: the number of pairs of elements of $\\mathcal{G}$ that are in different clusters in $\\mathcal{P}^*$ and also in different clusters in $\\mathcal{P}$.\n \\item $c$: the number of pairs of elements of $\\mathcal{G}$ that are in the same cluster in $\\mathcal{P}^*$ but in different clusters in $\\mathcal{P}$.\n \\item $d$: the number of pairs of elements of $\\mathcal{G}$ that are in different clusters in $\\mathcal{P}^*$ but in the same cluster in $\\mathcal{P}$.\n\\end{itemize}\n\nThese quantities can be interpreted as the number of true positives, true negatives, false negatives and false positives, respectively, if we consider the attribution of a pair of networks to a cluster as a decision problem. The Rand index is then defined as:\n\n\\begin{equation*}\n RI = \\frac{a+b}{a+b+c+d} = \\frac{a+b}{\\binom{N}{2}}.\n\\end{equation*}\nThe general form for correcting a index for chance is:\n\n\\begin{equation*}\n \\frac{RI - \\mathbb{E}(RI)}{\\text{max}(RI) - \\mathbb{E}(RI)}.\n\\end{equation*}\n\\label{eq:general_correction}\nHubert and Arabie~\\cite{hubert1985comparing} show how to calculate $\\mathbb{E}(RI)$, which we omit for brevity, leading to the formula for the adjusted Rand index (assuming a maximum value of 1 for the Rand index when $c$ and $d$ are 0):\n\n\\begin{equation*}\n ARI = \\frac{\\sum_{ij}\\binom{n_{ij}}{2} - \\frac{\\sum_i\\binom{a_i}{2}\\sum_j\\binom{b_j}{2}}{\\binom{N}{2}}}{\\frac{1}{2} \\left[ \\sum_i\\binom{a_i}{2} + \\sum_j\\binom{b_j}{2} \\right] - \\frac{\\sum_i\\binom{a_i}{2}\\sum_j\\binom{b_j}{2}}{\\binom{N}{2}}},\n\\end{equation*}\nwhere $n_{ij}$ is the number of networks in common between $P_i^*$ and $P_j$, i.e., $n_{ij} = |P_i^* \\cap P_j|$, $a_i = \\sum_{j=1}^c n_{ij}$ and $b_j = \\sum_{i=1}^c n_{ij}$.\n\n\\subsubsection{Undirected Results}\n\nTable~\\ref{tab:ari_undir} shows the ARI for Task 1, Task 2 and real world networks in the undirected case.\n\nAs with AUPR, we find that using ARI as a performance metric does not change the conclusions drawn from the results with $\\overline{P}$. Recalling that the ARI metric measures the quality of hierarchical clusters generated from a distance matrix, for Task 1, the output of original NetEmd with size 5 orbits leads the best score using this metric. More evidence of the quality of the clusters produced by the original NetEmd with size 5 orbits is to notice that increasing the number of components in ICA or the percentage of variance explained in PCA (and therefore approximating the results of original NetEmd) leads to higher ARI score.\n\nFor Task 2, the gain measured by $\\overline{P}$ and AUPR when using $ICA\\_NetEmd$ with 2 components is also reflected in the ARI score. For the Onnela et al. dataset, the best performance we measure not included in Table~\\ref{tab:ari_undir} is $ICA\\_NetEmd$ with size 4 graphlets and 8 components, with the components calculated with size 5 graphlets, with a score of 0.713.\n\n\\begin{table}[ht]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|c|c|cc|c|}\n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{2}{c|}{Synthetic} & Real \\\\\n \\cline{3-5} & & Task 1 & Task 2 & Onnela et al.\\\\\n \\hline \\multirow{3}{*}{Original NetEmd} & G3 & 0.63 $\\pm$ 0.03 & 0.472 & 0.592 \\\\\n \\cline{2-5} & G4 & 0.69 $\\pm$ 0.02 & 0.427 & \\textbf{0.675} \\\\\n \\cline{2-5} & G5 & \\textbf{0.81 $\\pm$ 0.02} & 0.445 & 0.576 \\\\\n \\hline \\multirow{5}{*}{PCA} & 50\\% Variance & 0.72 $\\pm$ 0.03 & 0.439 & 0.558 \\\\\n \\cline{2-5} & 80\\% Variance & 0.71 $\\pm$ 0.03 & 0.447 & 0.619 \\\\\n \\cline{2-5} & 90\\% Variance & 0.78 $\\pm$ 0.02 & 0.436 & 0.612 \\\\\n \\cline{2-5} & 95\\% Variance & \\textit{0.80 $\\pm$ 0.02} & 0.414 & 0.610 \\\\\n \\cline{2-5} & 99\\% Variance & \\textbf{0.81 $\\pm$ 0.02} & 0.396 & 0.587 \\\\\n \\hline \\multirow{3}{*}{ICA} & 2 components & 0.67 $\\pm$ 0.02 & \\textbf{0.477} & 0.521 \\\\\n \\cline{2-5} & 10 components & 0.73 $\\pm$ 0.03 & 0.452 & 0.593\\\\\n \\cline{2-5} & 15 components & 0.76 $\\pm$ 0.03 & 0.469 & 0.602 \\\\\n \\hline\n \\hline \\multirow{2}{*}{GCD} & 11 orbits & 0.64 $\\pm$ 0.07 & 0.309 & 0.435\\\\\n \\cline{2-5} & 73 orbits & 0.77 $\\pm$ 0.05 & 0.430 & 0.488 \\\\\n \\hline \\multirow{3}{*}{GDA} & G3 & 0.74 $\\pm$ 0.05 & 0.361 & 0.456 \\\\\n \\cline{2-5} & G4 & 0.62 $\\pm$ 0.07 & 0.332 & 0.197 \\\\\n \\cline{2-5} & G5 & 0.41 $\\pm$ 0.05 & - & 0.207 \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 1, Task 2 and the Onnela et al.~\\cite{onnela2012taxonomies} datasets in undirected networks. The metric used for Task 1 is the mean (and standard error of the mean) of the 16 values for ARI in each combination of number of nodes with network density. The metric used for Task 2 and Onnela et al. dataset is the sole value of ARI after comparing the 1280 and 151 networks, respectively. The parameter column indicates: graphlet size used in original NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 5); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 5; the number of orbits used by GCD; graphlet sizes used in GDA. The bolded values are the maximum for each task. Italic values are within one standard error of the maximum performance. Note that results for GDA with graphlet size 5 took longer than a week to return results for Task 2, at which point we stopped the computation.}\n \\label{tab:ari_undir}\n\\end{table}\n\n\\subsubsection{Directed Results}\n\n\nThe results using ARI for Task 1 in directed networks are shown in Table \\ref{tab:ari_dir_task1} and for Task 2 and the dataset of real world directed networks in Table~\\ref{tab:ari_dir_task2}.\n\nIn Task 1, we find that, according to ARI, the performance of all NetEmd variants is significantly below the performance of DGCD with 129 orbits for 0, 25 and 50\\% reciprocity and of GDA with size 3 graphlets for 75\\% reciprocity. At 100\\% reciprocity, NetEmd, $PCA\\_NetEmd$ with 80\\% variance and GDA, all with size 4 graphlets, share the top performance, but with a smaller difference to other algorithms and parameters. These results are a stark contrast to the ones we observe using AUPR and $\\overline{P}$, where scores for this task did not differ significantly between all the algorithms.\n\nIn Task 2, unlike in the other metrics, $ICA\\_NetEmd$ with 2 components is no longer considered the best performer for 0\\% reciprocity, with TriadEMD achieving the best result instead. On the other hand, according to ARI, $ICA\\_NetEmd$ with 2 components is the top performer for 100\\% reciprocity. For 25, 50 and 75\\% reciprocity, $ICA\\_NetEmd$ with 2 components performs significantly worse when comparing to the results using AUPR and $\\overline{P}$. ARI scores indicate that the best performers in these levels of reciprocity are $PCA\\_NetEmd$ with 99\\% variance, NetEmd with size 3 graphlets and $PCA\\_NetEmd$ with 95\\% variance. As we mention previously, using a smaller graphlet size usually leads to better performance when different network sizes and densities are involved. We find partial agreement to this claim when using ARI as the performance metric. At 0\\% reciprocity, we find that $ICA\\_NetEmd$ with 2 components and size 3 orbits achieves an ARI of 0.509 (higher than the score of $ICA\\_NetEmd$ with 2 components and size 4 orbits in Table~\\ref{tab:ari_dir_task2}, but lower than the score of TriadEMD); at 75\\% reciprocity, $PCA\\_NetEmd$ with 90\\% variance explained and size 3 orbits achieves an ARI of 0.495 (higher than the score of $PCA\\_NetEmd$ with 95\\% variance explained and size 4 orbits in Table~\\ref{tab:ari_dir_task2}).\n\nFor the dataset of real world directed networks, ARI disagrees with AUPR and $\\overline{P}$, assigning the best performance to GDA with size 3 orbits. This task also favours using smaller graphlets for the comparison, so the $PCA\\_NetEmd$ and $ICA\\_NetEmd$ results are at a disadvantage compared to the other methods as we only report results with size 4 in Table~\\ref{tab:ari_dir_task2}. For these two algorithms with size 3 graphlets, the best ARI scores on this task are 0.524, when using $PCA\\_NetEmd$ with 90\\% explained variance, and 0.489, when using $ICA\\_NetEmd$ with 19 components.\n\nOn the overall, in spite of some agreements between ARI and AUPR or $\\overline{P}$, we find ARI to be an unreliable metric to gauge the performance of network distance measures, as it relies on an intermediate step of clustering that is unrelated to the measures themselves. Using a different clustering method or even a different linkage leads to disparate results compared to the ones we present in this section, which adds a confounding factor when measuring the performance of the network comparison itself. An example of this unreliability is the ARI score of 0 given to GDA with size 3 orbits and to TriadEMD at 100\\% reciprocity, when the other metrics showed a performance comparable to DGCD.\n\n\\afterpage{\n\\begin{landscape}\n\\begin{table}\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|c|c|ccccc|}\n \n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{5}{c|}{Reciprocity} \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% \\\\\n \\hline \\multirow{2}{*}{All Orbits} & G3D & 0.56 $\\pm$ 0.02 & 0.61 $\\pm$ 0.03 & 0.62 $\\pm$ 0.03 & 0.60 $\\pm$ 0.03 & 0.61 $\\pm$ 0.02 \\\\\n \\cline{2-7} & G4D & 0.65 $\\pm$ 0.03 & 0.70 $\\pm$ 0.02 & 0.69 $\\pm$ 0.03 & 0.71 $\\pm$ 0.02 & \\textit{0.64 $\\pm$ 0.02} \\\\\n \\hline \\multirow{2}{*}{$Weighted\\_NetEmd$} & G3D & 0.55 $\\pm$ 0.02 & 0.60 $\\pm$ 0.03 & 0.60 $\\pm$ 0.03 & 0.59 $\\pm$ 0.03 & 0.60 $\\pm$ 0.03 \\\\\n \\cline{2-7} & G4D & 0.63 $\\pm$ 0.03 & 0.68 $\\pm$ 0.02 & 0.68 $\\pm$ 0.03 & 0.69 $\\pm$ 0.02 & \\textit{0.63 $\\pm$ 0.03} \\\\\n \\hline \\multirow{5}{*}{$PCA\\_NetEmd$} & 50\\% Variance & 0.67 $\\pm$ 0.03 & 0.71 $\\pm$ 0.03 & 0.72 $\\pm$ 0.03 & 0.71 $\\pm$ 0.02 & 0.56 $\\pm$ 0.01 \\\\\n \\cline{2-7} & 80\\% Variance & 0.67 $\\pm$ 0.03 & 0.69 $\\pm$ 0.03 & 0.67 $\\pm$ 0.02 & 0.71 $\\pm$ 0.02 & \\textit{0.64 $\\pm$ 0.02} \\\\\n \\cline{2-7} & 90\\% Variance & 0.69 $\\pm$ 0.03 & 0.69 $\\pm$ 0.02 & 0.69 $\\pm$ 0.03 & 0.70 $\\pm$ 0.02 & \\textit{0.63 $\\pm$ 0.02} \\\\\n \\cline{2-7} & 95\\% Variance & 0.65 $\\pm$ 0.03 & 0.70 $\\pm$ 0.02 & 0.68 $\\pm$ 0.02 & 0.71 $\\pm$ 0.02 & 0.60 $\\pm$ 0.01 \\\\\n \\cline{2-7} & 99\\% Variance & 0.66 $\\pm$ 0.03 & 0.69 $\\pm$ 0.03 & 0.68 $\\pm$ 0.02 & 0.70 $\\pm$ 0.02 & 0.58 $\\pm$ 0.02 \\\\\n \\hline \\multirow{3}{*}{$ICA\\_NetEmd$} & 2 components & 0.63 $\\pm$ 0.03 & 0.67 $\\pm$ 0.03 & 0.68 $\\pm$ 0.03 & 0.70 $\\pm$ 0.03 & 0.57 $\\pm$ 0.02 \\\\\n \\cline{2-7} & 10 components & 0.64 $\\pm$ 0.02 & 0.69 $\\pm$ 0.03 & 0.70 $\\pm$ 0.03 & 0.68 $\\pm$ 0.02 & 0.56 $\\pm$ 0.01 \\\\\n \\cline{2-7} & 15 components & 0.64 $\\pm$ 0.02 & 0.70 $\\pm$ 0.02 & 0.68 $\\pm$ 0.02 & 0.70 $\\pm$ 0.02 & 0.58 $\\pm$ 0.02 \\\\ \n \\hline\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & 0.69 $\\pm$ 0.03 & 0.62 $\\pm$ 0.03 & 0.65 $\\pm$ 0.03 & 0.64 $\\pm$ 0.03 & \\textbf{0.65 $\\pm$ 0.02} \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.70 $\\pm$ 0.03 & \\textit{0.79 $\\pm$ 0.03} & \\textit{0.77 $\\pm$ 0.04} & 0.69 $\\pm$ 0.05 & 0.59 $\\pm$ 0.04 \\\\\n \\cline{2-7} & 129 orbits & \\textbf{0.84 $\\pm$ 0.03} & \\textbf{0.80 $\\pm$ 0.03} & \\textbf{0.78 $\\pm$ 0.04} & 0.74 $\\pm$ 0.05 & 0.58 $\\pm$ 0.05 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.66 $\\pm$ 0.04 & \\textit{0.79 $\\pm$ 0.03} & 0.73 $\\pm$ 0.03 & \\textbf{0.81 $\\pm$ 0.03} & 0.56 $\\pm$ 0.05 \\\\\n \\cline{2-7} & G4D & 0.66 $\\pm$ 0.06 & 0.63 $\\pm$ 0.05 & 0.62 $\\pm$ 0.07 & 0.61 $\\pm$ 0.06 & \\textit{0.64 $\\pm$ 0.06}\\\\\n \\hline\n\n \\end{tabular}\n \\caption{Results for Task 1 in directed networks, for each level of reciprocity. The metric used for evaluation is the mean (and standard error of the mean) of the 16 values for ARI, in each combination of number of nodes, network density and reciprocity. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each reciprocity level.}\n \n \\label{tab:ari_dir_task1}\n\\end{table}\n\\end{landscape}\n}\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.7}\n \\begin{tabular}{|>{\\centering\\arraybackslash}m{29pt}|>{\\centering\\arraybackslash}m{43pt}|>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}|>{\\centering\\arraybackslash} m{28pt}| }\n \\hline \\multirow{3}{*}{Algorithm} & \\multirow{3}{*}{Parameter} & \\multicolumn{5}{c|}{Synthetic} & \\multirow{3}{*}{\\shortstack{Real\\\\Networks}}\\\\\n \\cline{3-7} & & \\multicolumn{5}{c|}{Reciprocity} & \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% & \\\\\n \\hline \\multirow{2}{*}{\\shortstack{All\\\\Orbits}} & G3D & 0.432 & 0.423 & \\textbf{0.464} & 0.387 & 0.188 & 0.167 \\\\\n \\cline{2-8} & G4D & 0.395 & 0.409 & 0.348 & 0.433 & 0.439 & 0.308\\\\\n \\hline \\multirow{2}{*}{Weighted} & G3D & 0.393 & 0.361 & 0.343 & 0.319 & 0.311 & 0.449 \\\\\n \\cline{2-8} & G4D & 0.362 & 0.324 & 0.319 & 0.378 & 0.422 & 0.373 \\\\\n \\hline \\multirow{5}{*}{PCA} & 50\\% Variance & 0.448 & 0.274 & 0.456 & 0.439 & 0.429 & 0.153 \\\\\n \\cline{2-8} & 80\\% Variance & 0.425 & 0.371 & 0.338 & 0.396 & 0.424 & 0.124 \\\\\n \\cline{2-8} & 90\\% Variance & 0.415 & 0.460 & 0.350 & 0.392 & 0.448 & 0.192 \\\\\n \\cline{2-8} & 95\\% Variance & 0.427 & 0.468 & 0.440 & \\textbf{0.475} & 0.445 & 0.151 \\\\\n \\cline{2-8} & 99\\% Variance & 0.431 & \\textbf{0.481} & 0.394 & 0.451 & 0.434 & 0.159 \\\\\n \\hline \\multirow{3}{*}{ICA} & 2 components & 0.451 & 0.261 & 0.336 & 0.403 & \\textbf{0.484} & 0.118 \\\\\n \\cline{2-8} & 10 components & 0.399 & 0.419 & 0.397 & 0.284 & 0.417 & 0.282 \\\\\n \\cline{2-8} & 15 components & 0.418 & 0.441 & 0.431 & 0.438 & 0.394 & 0.172 \\\\\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & \\textbf{0.539} & 0.467 & 0.436 & 0.392 & 0.001 & 0.297 \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.428 & 0.445 & 0.420 & 0.264 & 0.154 & 0.436 \\\\\n \\cline{2-8} & 129 orbits & 0.432 & 0.402 & 0.421 & 0.386 & 0.373 & 0.203 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.354 & 0.255 & 0.178 & 0.270 & 0.000 & \\textbf{0.542} \\\\\n \\cline{2-8} & G4D & - & - & - & - & 0.455 & - \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 2 in directed networks, for each level of reciprocity, and for the real world networks dataset. The metric used for this task is the sole value of ARI after comparing the 1280 and 1231 networks, respectively. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each dataset. Note that results for GDA with graphlet size 4 took longer than a week to return results for Task 2 and in the real world networks dataset, at which point we stopped the computation.}\n \\label{tab:ari_dir_task2}\n\\end{table}\n\n\\section{Details of datasets}\n\\label{app:network_models}\n\n\\subsection{Synthetic datasets}\n\nWe generate 16 datasets using the combinations of number of nodes $N \\in \\left\\{1250, 2500, 5000, 10000 \\right\\}$ and average degree $d \\in \\left\\{ 10, 20, 40, 80\\right\\}$. Each dataset contains 10 realizations of each model for a total of 80 networks.\n\n\\begin{itemize}\n\\item Erd\\H{o}s-R{\\'e}nyi (ER) model~\\cite{erdHos1960evolution}. A random graph with $n$ is generated by picking $m$ unique edges chosen at random from the $\\binom{n}{2}$ possible edges. $m$ was chosen to generate networks with the appropriate density.\n\\item Barabasi-Albert (BA) preferential attachment model~\\cite{barabasi1999emergence}. An initial graph is created with $m$ nodes and new nodes are added iteratively to the network, each new node is connected to $m$ existing nodes, picked randomly with probability proportional to their degree. $m$ was chosen to generate networks with the appropriate density.\n\\item Geometric random graphs~\\cite{gilbert1961random}, using a 3-dimensional square ($D = 3$). Nodes are randomly embedded in a $D$-dimensional space and are connected if the Euclidean distance between them is smaller than a threshold $r$. This threshold parameter was determined by grid search to generate networks with the desired number of edges.\n\\item Geometric gene duplication model~\\cite{higham2008fitting}. Starting with an initial network of 5 nodes embedded in a 3-dimensional space, on each iteration a node is chosen randomly to be duplicated. The duplicated node is placed randomly in the 3-dimensional space at an Euclidean distance of at most 2 from the original node. This process is repeated until the desired number of nodes, and nodes at a distance of $r$ or less at connected. This distance $r$ is chosen to generate a network with the appropriate number of edges.\n\\item Duplication divergence model of Vazquez et al.~\\cite{vazquez2003modeling}. The network grows in two stages. In the first stage, a node $v$ is chosen randomly to be duplicated into a node $v'$, that keeps all the edges from $v$. The nodes $v$ and $v'$ are connected with probability 0.05. In the second stage, one of the duplicated edges ($(v,u)$ or $(v',u)$) is chosen randomly and deleted with probability $q$. This process is repeated until the network grows to the desired number of nodes and $q$ is chosen to generate networks with the appropriate density.\n\\item Duplication divergence model of Ispolatov et al.~\\cite{ispolatov2005duplication}. Starting from a network with a single edge, a node is chosen randomly and duplicated, with the new node keeping each of the original's neighbours with probability $p$. This parameter is chosen to generate networks with the desired number of edges.\n\\item Configuration model, using the Duplication divergence model of Vazquez et al.~\\cite{vazquez2003modeling} as the generator for the graphical degree sequence.\n\\item Watts-Strogatz (WS) model~\\cite{watts1998collective}. Nodes are placed in a ring and connected to their $k$ nearest neighbours in both sides of the ring. Each edge is then rewired with probability 0.05 to a new node selected at random. The parameter $q$ is chosen to generate networks with the appropriate average degree.\n\\end{itemize}\n\n\\subsection{Real world networks}\n\nThe summary statistics of the Onnela et al.~\\cite{onnela2012taxonomies} and the real world directed networks are given in Table~\\ref{tab:summary_stats}.\n\n\n\\begin{table}[!ht]\n \\centering\n \n \\small\n \\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{|c|c|c|c|}\n \n \\hline \\multicolumn{2}{|c|}{} & Onnela et al. & Real directed \\\\\n \\hline \\multicolumn{2}{|c|}{\\# Networks} & 151 & 1283 \\\\\n \\hline \\multirow{3}{*}{Number of Nodes} & Min. & 30 & 9\\\\\n \\cline{2-4} & Median & 918 & 160\\\\\n \\cline{2-4} & Max & 11586 & 62586 \\\\\n \\hline \\multirow{3}{*}{Number of Edges} & Min. & 62 & 21 \\\\\n \\cline{2-4} & Median & 2436 & 2428\\\\\n \\cline{2-4} & Max & 232794 & 1614977\\\\\n \\hline \\multirow{3}{*}{Density} & Min. & \\num{4.26e-5} & \\num{3.78e-5} \\\\\n \\cline{2-4} & Median & 0.015 & 0.095\\\\\n \\cline{2-4} & Max & 0.499 & 0.75 \\\\\n \\hline \\multirow{3}{*}{Average degree} & Min. & 1.70 & 2.76 \\\\\n \\cline{2-4} & Median & 5.46 & 25.6\\\\\n \\cline{2-4} & Max & 455 & 726 \\\\\n \\hline \\multirow{3}{*}{Reciprocity} & Min. & - & 0.0 \\\\\n \\cline{2-4} & Median & - & 0.533 \\\\\n \\cline{2-4} & Max & - & 1.0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Summary statistics for the Onnela et al. and real world directed networks datasets.}\n \\label{tab:summary_stats}\n\\end{table}\n\n\\section{Background}\n\n\\subsection{Graph theoretical concepts}\n\nA \\emph{graph} $G$ is composed of a set of \\emph{vertices} (or \\emph{nodes}) $V(G)$ and a set of edges $E(G) \\subseteq V(G)\\times V(G)$, represented by pairs $(a,b) \\in E(G)$ for $a, b \\in V(G)$. A graph can be either $directed$, when the order of the vertices in the pairs expresses direction, meaning that $(a,b)\\in E(G)$ does not imply that $(b,a)\\in E(G)$, or \\textit{undirected} otherwise (i.e.\\ when $(a,b)\\in E(G)$ if and only if $(b,a)\\in E(G)$). The \\emph{size} of a graph is the number of vertices in the graph, written as $|V(G)|$, and the \\emph{density} is the portion of edges present in the graph over the total potential ones ($|E(G)|\/\\binom{|V(G)|}{2}$ in undirected networks). For a directed network, the \\textit{reciprocity} $\\rho$ is the probability that, given an edge $(a,b)$ selected uniformly at random from $E(G)$, $(b,a)$ is also a member of $E(G)$. A graph is called \\emph{simple} if it does not contain multiple edges (two or more edges connecting the same pair of vertices) or self-loops (an edge of the form $(a,a)$ that connects a vertex to itself). In this work, we consider only simple graphs.\n\nThe \\emph{neighbourhood} of a vertex $u \\in V(G)$ is defined as $N(u) = \\{ v : (v,u) \\in E(G) \\lor (u,v) \\in E(G) \\}$. All nodes are assigned consecutive integer numbers starting from $0$ and running to $|V(G)|-1$. The \\emph{degree} of a vertex is the number of edges it participates in, which is equivalent to the size of the node's neighbourhood. In directed networks, the degree of a vertex can be split into the \\emph{in-degree} and \\emph{out-degree}, counting the number of incoming and outgoing edges, respectively, from that node. In directed networks, the \\emph{degree} of a vertex is the sum of its \\emph{in-degree} and \\emph{out-degree}. In undirected networks, the \\emph{degree} of a node $u$ is simply $|N(u)|$. The distribution of degrees in a network is called the \\emph{degree sequence}.\n\nA \\emph{subgraph} of size $k$, $G_k$, of a graph $G$ is a graph with $k$ vertices, such that $V(G_k) \\subseteq V(G)$ and $E(G) \\supseteq E(G_k) \\subseteq V(G_k) \\times V(G_k)$. A subgraph is \\emph{induced} if $\\forall u,v \\in V(G_k), \\; (u,v) \\in E(G_k) \\leftrightarrow (u,v) \\in E(G)$ and is said to be \\emph{connected} when all pairs of vertices have a sequence of edges connecting them. Two graphs $G$ and $H$ are \\emph{isomorphic}, written as $G\\sim H$, if there is a bijection between $V(G)$ and $V(H)$ such that two vertices are adjacent in $G$ if and only if their correspondent vertices in $H$ are adjacent. A \\emph{match} of a graph $H$ in a larger graph $G$ is a set of nodes that induce the respective subgraph $H$. In other words, it is a subgraph $G_k$ of $G$ that is isomorphic to $H$. The frequency of a subgraph $G_k$ is then the number of different matches of $G_k$ in $G$.\n\n\\emph{Orbits} are unique positions of a graph, calculated by partitioning the set of vertices into equivalence classes where two vertices belong to the same class if there is an automorphism that maps one into the other~\\cite{ribeiro2019survey}.\n\n\nSmall, connected, non-isomorphic, induced subgraphs are commonly called \\emph{graphlets}~\\cite{prvzulj2007biological}. The smallest graphlet considered is a single edge, which can be seen as a size-2 subgraph. An undirected edge has a single orbit (three in the directed case) and the frequency of a node in this orbit is equivalent to the degree of the node. The \\emph{graphlet degree vector} of a node is an extension of the definition of degree to size-$k$ graphs, representing how many times the node occurs in each orbit. The \\emph{graphlet degree matrix} of a graph is the collection of graphlet degree vectors of each node in the graph.\n\nFigure~\\ref{fig:graphlets} shows the graphlets of size 2, 3 and 4 in undirected networks and size 2 and 3 in directed networks, alongside the respective orbits.\n\n\\begin{figure}\n \n \\includegraphics[width=0.95\\textwidth]{graphlets_g4_g3d.png}\n \\caption{Undirected graphlets of size 2, 3 and 4 and directed graphlets of size 2 and 3. Nodes are numbered according to their orbit, counted by type (directed or undirected) from the size 2 graphlets, and nodes with the same number represent the same orbit.}\n \\label{fig:graphlets}\n\\end{figure}\n\n\\subsection{NetEmd}\n\nNetEmd~\\cite{wegner2017identifying} is a network comparison measure that relies on structural features of the network, mainly the distribution of orbit frequencies. The core idea is formalizing the intuition that the shape of the degree distribution is indicative of the network's generation mechanisms, for instance, a network with a power law degree distribution is generated by a process distinct from a network with a uniform degree distribution. As the graphlet degree vector is a generalization of the degree distribution for graphlets of size $k \\geq 3$, the shapes of the distributions of each orbit also carry information about the topology of the network. Note that because the shape of a distribution is invariant under linear transformations such as translations, using the shape as the focus of the comparison is well-suited to comparing networks of different sizes and densities. \n\nWegner et al.~\\cite{wegner2017identifying} postulate that ``any metric that aims to capture the similarity of shapes should be invariant under linear transformations of its inputs.\" Thus, they define a measure of similarity between distributions $p$ and $q$, with non-zero and finite variances, using the Earth Mover's Distance (EMD)~\\cite{rubner1998metric}:\n\n\\begin{equation*}\\label{eq:emd}\nEMD^{*}(p,q) = \\text{inf}_{c \\in \\mathbb{R}} (EMD(\\tilde{p}(\\cdot + c), \\tilde{q}(\\cdot))),\n\\end{equation*}\nwhere $\\tilde{p}$ and $\\tilde{q}$ are the distributions resulting of scaling $p$ and $q$ to variance 1. Any distance metric $d$ can be used to generate $d^*$; EMD was used by the authors as it has been shown to be an appropriate metric to compare shapes of distribution in domains such as information retrieval and it produced better results than other distance metrics, like the Kolmogorov or $L^1$ distances.\n\nGiven two networks $G$ and $H$ and a set of $m$ orbits $\\mathcal{O} = \\{o_1, o_2, \\ldots, o_m\\}$, the \\emph{NetEmd} measure is defined as:\n\n\\begin{equation}\n\\label{eq:netemd}\nNetEmd_{\\mathcal{O}}(G,H) = \\frac{1}{m} \\sum\\limits_{i=1}^{m} EMD^*(p_{o_i}(G),p_{o_i}(H)),\n\\end{equation}\nwhere $p_{o_i}(G)$ and $p_{o_i}(H)$ are the distributions of orbit $i$ in graphs $G$ and $H$, respectively. Note that $\\mathcal{O}$ may be replaced by any set of network features; in this work we focus on orbits only. \n\n\\section{Conclusion}\n\nWe presented two extensions of NetEmd to directed networks, one a direct extension that uses all orbits of graphlets up to size 4 and another that compares networks based only on the orbits that exist in at least one of the networks. We showed that our methodology achieves state of the art performance in large datasets of synthetic networks with heterogeneous network sizes and in datasets of real world networks.\n\nWe also proposed to add dimensionality reduction techniques, namely PCA and ICA, as a preprocessing step to the network comparison. The goal of this step is to use dimensionality reduction as a process to attenuate noise within orbit frequencies, that can be introduced by random number generation (in synthetic networks) or data collection\/representation (in real world networks). Results show that this preprocessing improves the performance of NetEmd not only when comparing networks with the same number of nodes and edges, but especially in large datasets containing networks of different sizes. \n\nFrom an end-user perspective, our extensive testing allows us to recommend guidelines for which version of NetEmd to use in which situation. If the set of networks is homogeneous in number of nodes and average degree, using the largest graphlet size available (size 5 in undirected, size 4 in directed) coupled with ICA (10 to 15 components) leads to the best results. In the cases where there is a large variety in number of nodes or average degree between the networks in the dataset, using smaller graphlets is generally advised. The exception is in directed networks, if the connections within the networks have no reciprocity, then using graphlets of size 4 (with ICA) is more likely to yield a more accurate comparison, due to having a larger set of orbits to with non-zero frequency. On the other hand, if the average reciprocity allows using the whole range of graphlets of size 3 and 4, then it is more likely that using graphlets of size 3 coupled with ICA will lead to a more accurate distinction between networks. In this case, 2 components might be enough if the size of the networks are all within one order of magnitude from each other but 5 to 10 components if there is a wider difference.\n\\section{NetEmd with Dimension Reduction}\n\\label{sec:dimred}\n\n\\subsection{Motivation}\n\nIn this section, we describe how we couple dimensionality reduction techniques with NetEmd, using them as noise reduction techniques. The rationale is that processes that generate networks are inherently noisy due to the complexity of the systems they represent. This is particularly evident when we consider random networks generated by the same model; we expect them to have similar structure and that similarity to be reflected in the distribution of subgraph and orbit frequencies, but the stochastic nature of the generation process introduces noise in these distributions that makes them harder to identify.\n\nConsidering the graphlets and orbits shown in Figure~\\ref{fig:graphlets}, it is clear that (for example) directed graphlet $G_2$ cannot exist if $G_0$ does not, and similarly directed graphlet $G_{12}$ cannot exist if $G_1$ does not. More generally, it is clear that the different orbits do not represent independent degrees of freedom, and are expected to be constrained by the requirement of combinatorial consistency of the underlying full graph $G$ as well as correlations induced by the generation process.\n\nIn practice, both combinatorial enumeration of graphlets and \\textit{a priori} determination of statistical relationships between them (beyond comparison to simple random graphs as in \\cite{milo2004superfamilies}) are computationally infeasible. While this does not matter in practice for the undirected case, for directed graphs as we consider here, the relatively large number and complexity of orbits makes these much more important. We therefore seek an approach that can adjust for the non-independence of orbit counts.\n\nProceeding somewhat formally, let $\\mathbf{F}_G$ denote the graphlet degree matrix of graph $G$ for a set of orbits $\\mathcal{O}$, a $n \\times m$ matrix where $n = |V(G)|$, $m = |\\mathcal{O}|$, such that $[\\mathbf{F}_{G}]_{i,j}$ represent the frequency of orbit $j$ for node $i$. Further, let $\\mathbf{f}_i$ be the vector of frequencies of node $i$, i.e.\\ $(\\mathbf{f}_{i})_{j} = [\\mathbf{F}_{G}]_{i,j}$ meaning $\\mathbf{f}_i$ is the $i$-th row of $\\mathbf{F}_G$. Now note that we are only interested into networks up to isomorphism; in fact, our distance measures depend on empirical histograms of orbit counts, deriving the distribution $p_{o_i}$ in \\eqref{eq:netemd} from histogram heights like\n\\begin{equation}\n h_{o_j,y} = \\frac{1}{n}\\sum_{i=1}^{n} \n \\mathbf{1}_{\\{(\\mathbf{f}_{i})_{j} = y\\}} ,\n \\label{hoiy}\n\\end{equation}\nwhich is the proportion of nodes with $y$ counts of orbit $o_j$, where $\\mathbf{1}$ is an indicator function taking the value 1 if its argument is true and the value 0 otherwise. The right-hand side of \\eqref{hoiy} is a sum of random variables, one for each node, and from \\eqref{eq:netemd} we will further sum over orbits for a function of two such histogram heights.\n\nIn general, we will expect a random graph model to assign a probability to each graph in some finite set, and according to this measure we expect a probability distribution to be induced on $\\mathbf{h}_{G} = [h_{o_j,y}]$. As discussed above, we expect both combinatorial constraints and correlations between orbit counts, meaning that a sum over all elements of $\\mathbf{h}_{G}$ as implied by \\eqref{hoiy} and \\eqref{eq:netemd} will inflate, due to the additivity of variances of random variables, the amount of noise in EMD realisations compared to the amount that is absolutely necessary under the random graph model.\n\nTherefore, a natural approach is to denoise using techniques that do not require explicit solution of or simulation from random graph models. In particular, we will use the linear techniques of principal component analysis (PCA) and independent component analysis (ICA), noting their computational efficiency compared to potentially more general nonlinear methods \\cite{Hinton:2006}. The main idea of our methodology is therefore to project the orbit frequencies to a lower dimension, training the dimension reduction model to lose minimal information while removing noise.\n\nSince it is not guaranteed that each graph will dimensionally reduce to the same size, the approaches we use allow for expansion of the dimensionally reduced features back to the original (high) dimension. The NetEmd comparison from Equation~\\ref{eq:netemd} is consequently applied to the reconstructed frequencies. By using the first $L$ components of the linear methods, which contain the most signal from the dataset, to reconstruct the original counts, this can be intuitively understood as decreasing noise within the orbit frequencies when we do not have a good model for that noise.\n\n\n\n\\subsection{Principal Component Analysis}\n\nPrincipal component analysis (PCA) is a technique for dimensionality reduction that preserves as much variability of the data as possible, by computing principal components of the data. Principal components are sequences of orthogonal unit vectors that form an orthonormal basis, in which the original dimensions of the data are linearly uncorrelated. When projecting the data onto this new basis, these vectors can be seen as the directions that maximize the variance of the projected data, with the first principal component representing the maximum variance.\n\nLet $\\mathbf{F}_G$ denote the graphlet degree matrix of graph $G$ for a set of orbits $\\mathcal{O}$, a $n \\times m$ matrix where $n = |V(G)|$, $m = |\\mathcal{O}|$ and $[\\mathbf{F}_{G}]_{i,j}$ represents the frequency of orbit $j$ for node $i$. We assume that the frequencies have been normalized, a preprocessing step advised before applying PCA (to prevent notation overload, we use the same $\\mathbf{F}_G$ to denote the normalized version of the graphlet degree matrix). The principal components of this matrix are defined by $\\mathbf{V} = \\mathbf{F}_G\\mathbf{W}$, where $\\mathbf{W}$ is a $m \\times m$ matrix whose columns are the eigenvectors of $\\mathbf{F}_G^\\intercal\\mathbf{F}_G$. To allow for only the first L components, the matrix $\\mathbf{W}$ can be truncated to $\\mathbf{W}_L$, with dimensions $m \\times L$, leading to a transformation $\\mathbf{V}_L = \\mathbf{F}_G\\mathbf{W}_L$. This truncation is done such that the $L$ eigenvectors chosen to correspond to the largest eigenvalues of $\\mathbf{W}$. The original graphlet degree matrix can be reconstructed as $\\hat{\\mathbf{F}}_G = \\mathbf{V}_L\\mathbf{W}_L^\\intercal = \\mathbf{F}_G\\mathbf{W}_L\\mathbf{W}_L^\\intercal$. The goal of PCA is to learn $\\mathbf{W}_L$ such that the variance of the original data preserved is maximized, while also minimizing the total squared reconstruction error $||\\mathbf{F}_G - \\hat{\\mathbf{F}}_G||_2^2 = ||\\mathbf{V}\\mathbf{W}^\\intercal - \\mathbf{V}_L\\mathbf{W}_L^\\intercal||_2^2$.\n\nWe use the reconstructed graphlet degree matrix to compare networks with NetEmd, keeping Equation~\\ref{eq:netemd} virtually unchanged from the original formulation:\n\n\\begin{equation}\n\\label{eq:pca_netemd}\nPCA\\_NetEmd_{\\mathcal{O}}(G,H) = \\frac{1}{m} \\sum\\limits_{i=1}^{m} EMD^*(\\hat{p}_{o_i}(G),\\hat{p}_{o_i}(H)),\n\\end{equation}\nwhere $\\hat{p}_i(G)$ and $\\hat{p}_i(H)$ are obtained from $\\hat{\\mathbf{F}}_G$ and $\\hat{\\mathbf{F}}_H$ respectively, instead of\n$\\mathbf{F}_G$ and $\\mathbf{F}_H$.\n\n\\subsubsection{Choosing the number of components.}\n\\label{subsec:num_comps}\n\nChoosing an appropriate number of components to project the data down affects the reconstruction error and therefore the amount of noise reduced. If we take $L = m$, then clearly we are able to reconstruct the original frequencies perfectly and no noise has been removed from the data. On the other hand, picking a value for L too low, leads to foregoing descriptiveness of the data. A common strategy for picking the number of components is through the amount of variance explained by each component, a strategy that allows us to adapt the number of components to suit the networks we are comparing.\n \nThe sample covariance matrix of each orbit, i.e. $F_{G_{i,1}}, \\, F_{G_{i,2}}, \\, \\ldots, \\, F_{G_{i,m}}$, is proportional to $\\mathbf{F}_G^\\intercal\\mathbf{F}_G$ (the derivation for this result can be found in~\\cite{jolliffe2002springer}, pp.~30-31) and contains the variance of the frequency of each orbit in its diagonal. As a square matrix, according to the spectral theorem, the covariance matrix can be diagonalized by its eigenvectors and the values of the resulting diagonal matrix are the corresponding eigenvalues. These eigenvalues represent variability of each axis in the projected space. Therefore, by taking the sum of the $L$ highest eigenvalues, the ones corresponding to the first $L$ components, we get the variance explained by the first $L$ principal components. The ratio $\\sum_1^L \\lambda_i \/ \\sum_1^m \\lambda_i$, where $\\lambda_i$ is the eigenvalue corresponding to the $i$th eigenvector, measures the proportion of variance explained by the first $L$ principal components. Finally, to condition the number of components on the proportion of variance explained, the smallest $L$ is calculated such that $\\sum_1^L \\lambda_i \/ \\sum_1^m \\lambda_i \\geq r$ where $r \\times 100\\%$ is the percentage of variance explained.\n\n\n\n\\subsection{Independent Component Analysis}\n\nIndependent component analysis (ICA) is a statistical technique in which an observed vector of random variables is thought to be the linear combination of unknown independent components. These components are assumed to be mutually statistically independent and with zero mean. The classic example of an ICA application is a party where a microphone is picking up multiple conversations and the goal is to separate the source signal from each conversation from the mixed one picked up by the microphone.\n\nTransposing ICA to the domain of network comparison, we can draw an analogy from the above application by considering that the voices in the conversation are the nodes in the network and the orbit frequencies are the data points captured by the microphone. Our goal is then to search for the source signals, i.e., network characteristics stemming from its generation mechanism that generate such frequency distribution. We conjecture that by using these source signals to reduce noise in the orbit frequency distributions, our network comparison measure becomes able to more accurately distinguish networks with different generation mechanisms.\n\nAs before, let $\\mathbf{F}_G$ denote the graphlet degree matrix of graph $G$ for a set of orbits $\\mathcal{O}$ and let $\\mathbf{f}_i$ be the vector of frequencies of node $i$, i.e. $f_{i,j} = \\mathbf{F}_{G_{i,j}}$. We can write $\\mathbf{f}_i$ as $\\mathbf{f}_i = \\mathbf{A}\\mathbf{s}_i$, where $\\mathbf{A}$ is called the mixing matrix and $\\mathbf{s}_i$ are the $L$ independent components. The goal of ICA is to estimate $\\mathbf{A}$ and $\\mathbf{s}_i$ from $\\mathbf{f}_i$ only. In practice, algorithms to calculate independent components compute a weight matrix $\\mathbf{W}$, a pseudo-inverse of $\\mathbf{A}$ and obtain the independent components through $\\mathbf{s}_i = \\mathbf{W}\\mathbf{f}_i$. Using the FastICA algorithm~\\cite{hyvarinen2000independent,hyvarinen1999fixed}, $\\mathbf{W}$ is constructed iteratively by finding unit vectors $\\mathbf{w}$ such that the projection $\\mathbf{w}^\\intercal\\mathbf{f}_i$ maximizes non-gaussianity, measured by the approximation of negentropy $J(\\mathbf{w}^\\intercal\\mathbf{f}_i) \\propto \\left[ E(G(\\mathbf{w}^\\intercal\\mathbf{f}_i)) - E(G(\\mathbf{\\nu})) \\right]^2$, where $G(x) = \\log\\cosh x$ and $\\mathbf{\\nu}$ is a standard Gaussian random variable with mean 0 and variance 1. These unit vectors $\\mathbf{w}$ are combined into the weight matrix $\\mathbf{W}$ and decorrelated to prevent convergence to the same maxima by $\\mathbf{W} = \\mathbf{W} \/ \\sqrt{\\| \\mathbf{W}\\mathbf{W}^\\intercal \\|}$ (the matrix norm is calculated using the $l^2$ norm); this procedure is then repeated with $\\mathbf{W} = \\frac{3}{2}\\mathbf{W} - \\frac{1}{2} \\mathbf{W} \\mathbf{W}^\\intercal\\mathbf{W}$ until convergence.\n\nUpon obtaining the weight matrix, we calculate its pseudo-inverse to obtain the mixing matrix $\\mathbf{A}$ and calculate $\\hat{\\mathbf{F}}_G$ similarly to what was done with PCA: we project $\\mathbf{F}_G$ to a lower dimensional space using $\\mathbf{W}$ and apply the mixing matrix $\\mathbf{A}$ to retrieve the original data. The final NetEmd formula then takes a similar form as for $PCA\\_NetEmd$:\n\n\\begin{equation}\n\\label{eq:ica_netemd}\nICA\\_NetEmd_{\\mathcal{O}}(G,H) = \\frac{1}{m} \\sum\\limits_{i=1}^{m} EMD^*(\\hat{p}_{o_i}(G),\\hat{p}_{o_i}(H)),\n\\end{equation}\nwith $\\hat{p}_i(G)$ and $\\hat{p}_i(H)$ obtained from $\\hat{\\mathbf{F}}_G$ and $\\hat{\\mathbf{F}}_H$ respectively, as before.\n\n\\section{Directed NetEmd}\n\nDirected networks pose challenges unlike the ones observed in undirected networks, due to the combinatorial explosion in number of orbits introduced by distinguishing $(u,v)$ and $(v,u)$ as different edges between $u$ and $v$. There are 730 orbits when considering directed graphlets of size up 4, compared to 15 in the undirected case, and scaling up to size 5 in directed networks becomes unfeasible as the number of orbits rises to $45,637$. This sharp increase makes the task of counting orbit frequencies for each node even harder, with no known analytical approaches like ORCA~\\cite{hovcevar2014combinatorial} that rely on crafting sets of equations that exploit combinatorial relationships between smaller graphlets to compute orbit counts. Instead, enumeration-based approaches are required in the directed case, which are known to be at least an order of magnitude slower than analytical approaches in undirected networks. In order to adapt NetEmd to directed networks, we use the G-Trie~\\cite{ribeiro2014g} data structure and the counting algorithm proposed by Aparicio et al.~\\cite{aparicio2016extending}, publicly available at~\\cite{gtscanner}, modified to return the graphlet degree matrix instead of the graphlet degree distribution.\n\nHow to handle this increased number of orbits in the comparison is also a challenging problem, especially when comparing networks that contain only a small subset of orbits and differences within those orbits can get diluted when taking the average over the whole set. Aparicio et al.~\\cite{aparicio2016extending} argue that the comparison should only be done over the orbits present in at least one of the networks, using networks $G$ and $H$ in Figure~\\ref{fig:weight_by_orbit_example} as an example. In this case, the authors show that using the original formulation of the Graphlet Degree Distribution Agreement (GDA) (described in Section~\\ref{sec:gda}) inflates the similarity score between the two networks far beyond what would be expected from two so distinct looking networks, with an agreement score of 0.92 against 0.32 when using their modified version that compares only orbits in at least one of the networks. This modification to GDA is well-founded because GDA is a measure of \\emph{agreement}, bounded between 0 and 1 and meant to be interpretable; a score close to 1 means that the networks are similar and a score close to 0 the opposite, so the magnitude of the value that the measure outputs is of interest. However, NetEmd is a measure of distance between networks and the relative difference between distances is more informative than the absolute value of that distance, so including information that an orbit is missing in both networks improves our ability to tell if they are less distant from each other when comparing against another pair of networks.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{weight_by_orbit_example}\n \\caption{Example to illustrate the usefulness of only using orbits that appear in at least one of the networks being compared. Graphs $G$ and $H$ are the example provided by Aparicio et al.~\\cite{aparicio2016extending}; we adapt them to directed networks $G'$ and $H'$, keeping the same number of orbits in each network. We measure $NetEmd(G',H') = \\num{5e-5}$ and $Weighted\\_NetEmd(G',H') = 0.12$.}\n \\label{fig:weight_by_orbit_example}\n\\end{figure}\n\nWith the above in mind, we propose two versions of directed NetEmd. Given two networks $G$ and $H$ and a set of $m$ orbits $\\mathcal{O} = \\{o_1, o_2, \\ldots, o_m\\}$, the first uses all $m$ orbits in this set, which in practice are all the orbits in graphlets of size up to 3 or 4 and the formulation is the same as the original NetEmd in Equation~\\ref{eq:netemd}. The second is using the same idea of Aparicio et al.~\\cite{aparicio2016extending}, which we refer to as \\emph{weighted NetEmd}, where we restrict the set of orbits to those that occur in at least one of the networks $G$ or $H$. This leads to a new set of $m'$ orbits $\\mathcal{O'} = \\{o_1', o_2', \\ldots, o_m'\\}$, with $m' \\leq m$. The average in Equation~\\ref{eq:netemd} is done over this set $\\mathcal{O'}$ instead of the original $\\mathcal{O}$, but the rest of the formula is the same:\n\n\\begin{equation}\n\\label{eq:weighted_netemd}\nWeighted\\_NetEmd_{\\mathcal{O'}}(G,H) = \\frac{1}{m'} \\sum\\limits_{i=1}^{m'} EMD^*(p_{o_i}(G),p_{o_i}(H)).\n\\end{equation}\n\nGraphs $G'$ and $H'$ in Figure~\\ref{fig:weight_by_orbit_example} show an analogous example to the one shown by Aparicio et al.~\\cite{aparicio2016extending} to demonstrate the difference between weighted and original NetEmd applied to directed networks. Although it is usually impractical to scale up to size 5 in directed networks, in a small example like this one it becomes feasible to do so. In this context, we calculate a distance of \\num{5e-5} using all orbits and $0.12$ when using the weighted version.\n\n\n\\section{Experimental Setup}\n\nExperiments were performed on an AMD Opteron Processor 6380 with 1.4 GHz and 2 MB of cache memory, using Python version 3.5.4. The implementation of our methods is available online at \\url{https:\/\/github.com\/migueleps\/denoise-dir-netemd}.\n\n\\subsection{Measure of Cluster Performance}\n\nGiven a set of networks $\\mathcal{G} = \\{G_1, G_2, \\ldots, G_N\\}$, divided in disjoint classes $C = \\{c_1, c_2, \\ldots, c_m\\}$, we use the performance measure proposed by Wegner et al.~\\cite{wegner2017identifying} that captures the idea that networks of the same class should be nearer to each other than to networks of other classes. It is defined as an empirical probability $P(G) = d(G,G_1) < d(G,G_2)$, where $G_1$ is a network selected randomly from the same class as $G$ and $G_2$ is randomly selected from a different class, and $d$ is the network comparison statistic. The performance over the whole dataset is the average $P(G)$ over all the networks in $\\mathcal{G}$, and we write this as $\\overline{P} = P(\\mathcal{G}) = \\frac{1}{|\\mathcal{G}|} \\sum _{G \\in \\mathcal{G}} P(G)$. In Appendix~\\ref{app:diff_metrics}, we present results using area under precision-recall curve (AUPR) and adjusted Rand index (ARI).\n\n\\subsection{Synthetic datasets}\n\nWe reproduce the experimental setup of Wegner et al.~\\cite{wegner2017identifying}, testing our proposed modifications to NetEmd on the same eight random network models. These models are Erd\\H{o}s-R{\\'e}nyi~\\cite{erdHos1960evolution}, Barabasi-Albert preferential attachment~\\cite{barabasi1999emergence}, configuration model, geometric random graphs~\\cite{gilbert1961random}, geometric gene duplication model~\\cite{higham2008fitting}, duplication divergence of Vazquez et al.~\\cite{vazquez2003modeling}, duplication divergence of Ispolatov et al.~\\cite{ispolatov2005duplication} and Watts-Strogatz~\\cite{watts1998collective}. Details of parameters for each model are described in Appendix~\\ref{app:network_models}.\n\nWe generate 10 networks per model per combination of number of nodes and average degree. The set of number of nodes used is $N \\in \\{1250, 2500, 5000, 10000\\}$ and the set of average degrees is $k \\in \\{10, 20, 40, 80\\}$. This leads to a total of 160 networks per model and 1280 networks in total. \n\nFor the directed datasets, we add varying levels of reciprocity $\\rho \\in \\{0, 0.25, 0.5, 0.75, 1\\}$. To generate a directed network with reciprocity $\\rho$, we take the undirected version, we duplicate and invert all edges (if $(u,v)$ is in the undirected network, we add $(v,u)$ to the directed version) and we then take a proportion $1-\\rho$ of these pairs and choose randomly a direction to remove (either $(u,v)$ or $(v,u)$). This is done incrementally such that if $(u,v)$ is removed from the dataset with 75\\% reciprocity, then it is also removed from the datasets with 50\\%, 25\\% and 0\\% reciprocity.\n\nThese datasets of synthetic networks give rise to two tasks aimed at gauging how well a network comparison measure separates clusters according to their generation mechanism, with the ground truth given by the random network model used to generate the networks in each cluster. The first, more simple task is to separate networks with the same number of nodes and the same density. To this end, we create 16 groups based on the combinations of $N$ and $k$, each group containing 10 realizations of each random network model, for a total of 80 networks. This task is equivalent to $RG_1$ in the original NetEmd paper~\\cite{wegner2017identifying}. The second task, equivalent to $RG_3$ in the original NetEmd paper, is to compare all 1280 networks simultaneously, finding the 8 clusters of 160 networks. This task measures how sensitive the comparison measure is to differences in an order of magnitude in number of nodes and edges, making the separation by model type more difficult. In directed networks, we repeat Task 1 and 2 for each level of reciprocity, determining how sensitive the network comparison is to this third parameter that impacts the set of orbits available for comparison (the set of orbits with 0\\% reciprocity is disjoint from the set of orbits with 100\\% reciprocity).\n\n\n\\vspace{-5pt}\n\\subsection{Real world datasets}\n\nWe use the dataset of Onnela et al.~\\cite{onnela2012taxonomies} to validate our methodology on a mix of real world and synthetic networks. The multiple sources that make up this dataset lead to a heterogeneous set of networks; however, the ability to separate these networks according to their domain is desired from a network comparison method. We were unable to find sources for the original 746 networks, so we use the reduced set proposed by Ali et al.~\\cite{ali2014alignment} with 151 unweighted and undirected networks. There is no ground truth for the true clusters in this dataset and the dendograms constructed through the methods proposed by Onnela et al.~\\cite{onnela2012taxonomies} and Ali et al.~\\cite{ali2014alignment} disagree on the composition of each cluster. Therefore, we aim to reconstruct clusters according to the type of data, which can be visualized in the supplementary material of Ali et al.~\\cite{ali2014alignment}. \n\nFor directed networks, we download four datasets from the SNAP library~\\cite{snapnets}: Gnutella peer-to-peer file sharing network from August 2002~\\cite{ripeanu2002mapping,leskovec2007graph} (9 networks with an average reciprocity of 0\\%), CAIDA autonomous systems relationships datasets from January 2004 to November 2007~\\cite{leskovec2005graphs} (122 networks with an average reciprocity of 100\\%), ego networks of \\emph{circles} from Google+~\\cite{mcauley2012learning} (132 networks with an average reciprocity of 28\\%) and ego networks of \\emph{lists} from Twitter~\\cite{mcauley2012learning} (973 networks with an average reciprocity of 54\\%). \n\n\\subsection{Other network comparison methods}\n\n\\subsubsection{GCD.}\n\nYavero{\\u{g}}lu et al.~\\cite{yaverouglu2014revealing} note that orbit counts have dependencies between them, making them redundant since they can be expressed as a linear combination of other orbits. The authors identify 11 out of 15 orbits in graphlets of size up to 4 and 56 out of 73 orbits in graphlets of size up to 5 as non-redundant. The authors construct the graphlet degree vector of each node in the network using this reduced number of orbits (although the full set of orbits may also be used), making up a matrix where each row is the graphlet degree vector of each node, and compute the Spearman's correlation coefficient between each pair of orbits, i.e., each pair of columns in this matrix. The pairwise Spearman's correlation coefficients are aggregated in a square $m \\times m$ matrix called the \\emph{Graphlet Correlation Matrix} (GCM), where $m$ is the number of orbits used (e.g., 11 if using non-redundant orbits of graphlets up to size 4).\n\nTo compare two networks, the authors propose the \\emph{Graphlet Correlation Degree} (GCD), which is the Euclidean distance between the upper triangle of the GCM of each network.\n\nSarajli{\\'c} et al.~\\cite{sarajlic2016graphlet} extend GCD to directed networks, finding 13 non-redundant orbits in graphlets of size up to 3 and 129 in graphlets of size up to 4.\n\n\\subsubsection{GDA.}\n\\label{sec:gda}\n\nPrzulj et al.~\\cite{prvzulj2007biological} introduced the Graphlet Degree Distribution (GDD), calculated as follows. Let $d^o_G(k)$ be the number of nodes of graph $G$ that participate $k$ times in orbit $o$, i.e., $d^o_G(k)$ is graphlet degree distribution of orbit $o$. The authors scale the distribution to decrease the contributions of larger orbits by calculating $S^o_G(k) = d^o_G(k)\/k$ and then normalize the distribution as $N_G^o(k) = S^o_G(k) \/ \\sum_{k=1}^{\\infty}S_G^o(k)$. To compare the GDD distributions of the same orbit in different networks $G$ and $H$, the authors propose the GDD-agreement (GDA) metric defined as:\n\n\\begin{equation*}\n \\label{eq:gda}\n A^o(G,H) = 1 - \\left(\\sum\\limits_{k=1}^{\\infty} \\left[ N_G^o(k) - N_H^o(k)\\right]^2\\right)^{1\/2}.\n\\end{equation*}\n\nAs a measure of \\emph{agreement}, the output of GDA is 1 if the distributions are identical, as opposed to NetEmd which is a measure of \\emph{distance} meaning it will output 0 for identical distributions. To aggregate the agreements of the multiple orbits, the authors propose either a arithmetic or a geometric mean. We compare against the implementation of Aparicio et al.~\\cite{aparicio2016extending}, which uses the arithmetic mean and is available at~\\cite{gtscanner}. This implementation uses the idea that if a graphlet has a frequency of 0 in both networks, then the orbits from that graphlet are excluded from the arithmetic mean, for both in the directed and undirected versions.\n\n\\section{Results}\n\n\\subsection{Undirected Results}\n\nWe present the results for Tasks 1 and 2 in undirected networks and the results for the Onnela et al.~\\cite{onnela2012taxonomies} dataset in Table~\\ref{tab:undir_scores}. The results for Task 1 show the average and standard error for the 16 values of $\\overline{P}$, one for each combination of $N$ and $k$. The results for Task 2 and Onnela et al. dataset show the single value of $\\overline{P}$ after comparing the 1280 and 151 networks, respectively.\n\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \n \\renewcommand{\\arraystretch}{1.6}\n \\begin{tabular}{|c|c|cc|c|}\n \\hline \\multirow{2}{*}{Algorithm} & \\multirow{2}{*}{Parameter} & \\multicolumn{2}{c|}{Synthetic} & Real \\\\\n \\cline{3-5} & & Task 1 & Task 2 & Onnela et al.\\\\\n \\hline Original NetEmd & G5 & \\textit{0.995 $\\pm$ 0.003} & 0.918 & 0.890\\\\\n \\hline \\multirow{5}{*}{$PCA\\_NetEmd$} & 50\\% Variance & 0.98 $\\pm$ 0.01 & 0.915 & 0.864 \\\\\n \\cline{2-5} & 80\\% Variance & 0.98 $\\pm$ 0.01 & 0.918 & \\textbf{0.893} \\\\\n \\cline{2-5} & 90\\% Variance & \\textit{0.995 $\\pm$ 0.004} & 0.922 & 0.891 \\\\\n \\cline{2-5} & 95\\% Variance & \\textit{0.993 $\\pm$ 0.004} & 0.919 & 0.892\\\\\n \\cline{2-5} & 99\\% Variance & \\textit{0.995 $\\pm$ 0.004} & 0.921 & 0.891 \\\\\n \\hline \\multirow{3}{*}{$ICA\\_NetEmd$} & 2 components & 0.985 $\\pm$ 0.002 & 0.926 & 0.869 \\\\\n \\cline{2-5} & 10 components & \\textit{0.994 $\\pm$ 0.004} & \\bf{0.930} & 0.887 \\\\\n \\cline{2-5} & 15 components & \\bf{0.996 $\\pm$ 0.003} & 0.929 & 0.890 \\\\\n \\hline\n \\hline \\multirow{2}{*}{GCD} & 11 orbits & \\textit{0.99 $\\pm$ 0.01} & 0.888 & 0.789\\\\\n \\cline{2-5} & 73 orbits & \\textit{0.994 $\\pm$ 0.008} & 0.885 & 0.819\\\\\n \\hline \\multirow{3}{*}{GDA} & G3 & 0.986 $\\pm$ 0.003 & 0.868 & 0.876 \\\\\n \\cline{2-5} & G4 & 0.977 $\\pm$ 0.007 & 0.886 & 0.855 \\\\\n \\cline{2-5} & G5 & 0.95 $\\pm$ 0.01 & - & 0.832 \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 1, Task 2 and the Onnela et al.~\\cite{onnela2012taxonomies} datasets in undirected networks. The metric used for Task 1 is the mean (and standard error of the mean) of the 16 values for $\\overline{P}$ in each combination of number of nodes with network density. The metric used for Task 2 and Onnela et al. dataset is the sole value of $\\overline{P}$ after comparing the 1280 and 151 networks, respectively. The parameter column indicates: graphlet size used in original NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 5); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 5; the number of orbits used by GCD; graphlet sizes used in GDA. The bolded values are the maximum for each task. Italic values are within one standard error of the maximum performance. Note that results for GDA with graphlet size 5 took longer than a week to return results for Task 2, at which point we stopped the computation.}\n \\label{tab:undir_scores}\n\\end{table}\n\nWe find that our proposed modifications to NetEmd achieve an increase in performance, with $ICA\\_NetEmd$ outperforming original NetEmd in synthetic datasets and $PCA\\_NetEmd$ in the Onnela et al. dataset. The performance gain in Task 1 is not significant as the results are within a standard error of each other, but on Task 2, the more difficult task, the difference is more pronounced. This performance improvement in synthetic datasets is not reflected in the Onnela et al. dataset, where 15 components are necessary to equalize the performance of original NetEmd. The best performance in the Onnela et al. dataset is $PCA\\_NetEmd$ with 80\\% explained variance, which contrasts with the results in the synthetic dataset, where 80\\% explained variance achieves worse performances than the original NetEmd. Using higher values for explained variance in $PCA\\_NetEmd$ also improves performance over original NetEmd in Task 2 and Onnela et al. dataset, but not in Task 1.\n\n\n\\subsection{Directed Results}\n\nWe show the results for Task 1 in directed networks in Table~\\ref{tab:scores_dir_task1} and the results for Task 2 and the dataset of real world directed networks in Table~\\ref{tab:scores_dir_task2}. The results for Task 1 show the average and standard error for the 16 values of $\\overline{P}$, one for each combination of $N$ and $k$, for each level of reciprocity. The results for Task 2 and real world directed networks dataset show the single value of $\\overline{P}$ after comparing the 1280, for each level of reciprocity, and 1232 networks, respectively.\n\nSimilarly to the undirected case, we find that the results for each algorithm in Task 1 are similar across different levels of reciprocity, meaning that the various different versions of NetEmd and DGCD are able to distinguish the generation mechanism regardless of how many edges are reciprocated in a directed network. Results show that DGCD with 129 orbits achieves best performance for 0\\%, 25\\%, 50\\% and 75\\% and $PCA\\_NetEmd$ with 90\\% explained variance for 100\\% reciprocity. We observe no significant difference between using NetEmd with all orbits, with weighted orbits or coupled with dimensionality reduction techniques. Results with TriadEMD are non-significantly better than our NetEmd version with orbits from size 3 graphlets in this task, indicating that using graphlets of size 2 for the comparison in this task does not help differentiating between different models. This is likely due to the configuration model, which shares the same degree distribution as the duplication divergence model of Vazquez et al.~\\cite{vazquez2003modeling}.\n\n\n\\begin{table}[t]\n\\centering\n \\scriptsize\n \\renewcommand{\\arraystretch}{1.6}\n \\begin{tabular}{|>{\\centering\\arraybackslash}m{29pt}|>{\\centering\\arraybackslash}m{43pt}|>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}>{\\centering\\arraybackslash} m{16pt}|>{\\centering\\arraybackslash} m{28pt}| }\n \\hline \\multirow{3}{*}{Algorithm} & \\multirow{3}{*}{Parameter} & \\multicolumn{5}{c|}{Synthetic} & \\multirow{3}{*}{\\shortstack{Real\\\\Networks}}\\\\\n \\cline{3-7} & & \\multicolumn{5}{c|}{Reciprocity} & \\\\\n & & 0\\% & 25\\% & 50\\% & 75\\% & 100\\% & \\\\\n \\hline \\multirow{2}{*}{\\shortstack{All\\\\Orbits}} & G3D & 0.923 & 0.905 & 0.907 & 0.905 & 0.915 & 0.864 \\\\\n \\cline{2-8} & G4D & 0.908 & 0.879 & 0.881 & 0.878 & \\textbf{0.927} & 0.813 \\\\\n \\hline \\multirow{2}{*}{Weighted} & G3D & 0.918 & 0.899 & 0.902 & 0.898 & 0.913 & \\textbf{0.868} \\\\\n \\cline{2-8} & G4D & 0.902 & 0.876 & 0.878 & 0.875 & 0.923 & 0.827 \\\\\n \\hline \\multirow{5}{*}{PCA} & 50\\% Variance & 0.924 & 0.897 & 0.900 & 0.894 & 0.908 & 0.821 \\\\\n \\cline{2-8} & 80\\% Variance & 0.924 & 0.885 & 0.892 & 0.881 & 0.920 & 0.825 \\\\\n \\cline{2-8} & 90\\% Variance & 0.921 & 0.882 & 0.889 & 0.878 & \\textbf{0.927} & 0.824 \\\\\n \\cline{2-8} & 95\\% Variance & 0.915 & 0.880 & 0.887 & 0.876 & 0.922 & 0.822 \\\\\n \\cline{2-8} & 99\\% Variance & 0.914 & 0.878 & 0.885 & 0.875 & 0.924 & 0.818 \\\\\n \\hline \\multirow{3}{*}{ICA} & 2 components & \\textbf{0.941} & \\textbf{0.916} & \\textbf{0.914} & \\textbf{0.911} & \\textbf{0.927} & 0.712 \\\\\n \\cline{2-8} & 10 components & 0.934 & 0.900 & 0.903 & 0.896 & 0.923 & 0.843 \\\\\n \\cline{2-8} & 15 components & 0.931 & 0.896 & 0.903 & 0.892 & 0.917 & 0.844 \\\\\n \\hline \\multicolumn{2}{|c|}{TriadEMD} & 0.926 & 0.903 & 0.906 & 0.906 & 0.916 & 0.859 \\\\\n \\hline \\multirow{2}{*}{DGCD} & 13 orbits & 0.905 & 0.904 & 0.895 & 0.885 & 0.871 & 0.812 \\\\\n \\cline{2-8} & 129 orbits & 0.887 & 0.884 & 0.880 & 0.880 & 0.885 & 0.799 \\\\\n \\hline \\multirow{2}{*}{GDA} & G3D & 0.842 & 0.802 & 0.802 & 0.800 & 0.858 & 0.736 \\\\\n \\cline{2-8} & G4D & - & - & - & - & 0.886 & - \\\\\n \\hline\n \\end{tabular}\n \\caption{Results for Task 2 in directed networks, for each level of reciprocity, and for the real world networks dataset. The metric used for this task is the sole value of $\\overline{P}$ after comparing the 1280 and 1231 networks, respectively. The parameter column indicates: graphlet size used in directed and weighted NetEmd; percentage of variance explained to determine the number of components in $PCA\\_NetEmd$ (using orbits in graphlets of size up to 4); the number of components used in $ICA\\_NetEmd$, using orbits in graphlets of size up to 4; the number of orbits used by DGCD; graphlet sizes used in GDA. The bolded values are the maximum for each dataset. Note that results for GDA with graphlet size 4 took longer than a week to return results for Task 2 and in the real world networks dataset, at which point we stopped the computation.}\n \\label{tab:scores_dir_task2}\n\\end{table}\n\nIn Task 2, our results show a noticeable gain in performance when using $ICA\\_NetEmd$ over other NetEmd versions and DGCD, in particular when using only 2 components and in reciprocity levels smaller than 100\\%. We find that using orbits in graphlets of only size 2 and 3 orbits achieves better results for this task than using ones with size up to 4, with the exception of the two extremities in reciprocity (0\\% and 100\\%), where having more available orbits for comparison helps performance. In this task, we also find that using $Weighted\\_NetEmd$ degrades performance compared to using all orbits. Results also show that our versions of NetEmd perform very similarly to TriadEMD, with a slight advantage to TriadEMD at 0, 75\\% and 100\\% reciprocity, which is perhaps expected again due to the configuration model adding a confounding factor when using graphlets $G_0$ and $G_1$ from Figure~\\ref{fig:graphlets} in the comparison. \n\nOn the other hand, $Weighted\\_NetEmd$ reaches the best performance in the real directed networks dataset, where, similarly to Task 2 in synthetic networks, using smaller graphlets yields better results. Table~\\ref{tab:scores_dir_task2} shows that $ICA\\_NetEmd$ with 10 and 15 components obtains better performance than NetEmd without dimensionality reduction and with size 4 graphlets. This comparison is unfair to $ICA\\_NetEmd$, as results suggest that when there is a large discrepancy between the sizes of the networks, using smaller graphlet sizes improves the performance of NetEmd. Therefore, although $Weighted\\_NetEmd$ achieves the best performance out of the parameters we include in Table~\\ref{tab:scores_dir_task2}, the top performance in the real world directed networks dataset is achieved by $ICA\\_NetEmd$ with size 3 graphlets and 10 components. All four versions of NetEmd that we propose achieve better results than DGCD and GDA in this dataset. This is also true when comparing our versions with size 3 graphlets against TriadEMD, indicating that in real world networks the degree distribution is able to contribute in a positive manner towards being able to discriminate networks of different sources\n\n\\vspace{-5pt}\n\\subsection{Performance by number of ICA components}\n\nWe examine the impact on performance of varying the number of components used for $ICA\\_NetEmd$, with the caveat that increasing the number of components leads to a greater computational complexity as the FastICA algorithm takes more iterations to converge. We also explore the performance of $ICA\\_NetEmd$ under smaller graphlet sizes, namely graphlets sizes 3 and 4 for undirected networks and size 3 for directed. There are two different ways of using smaller graphlets on $ICA\\_NetEmd$, the more natural way in which we compute the independent components using the set of orbits related to the desired graphlet size, or an alternative way in which the components are computed using a larger graphlet size and the reconstructed graphlet degree matrix is posteriorly truncated to include only the orbits of the desired graphlet size. For example, when calculating network distances using undirected graphlets of size up to 4, there are 15 orbits. In the first method, the graphlet degree matrix $\\mathbf{F}_G$ is a $|V(G)| \\times 15$ matrix and we simply calculate $\\hat{\\mathbf{F}}_G$ as described in Section~\\ref{sec:dimred}, constraining the number of components used to a maximum of 14 components. In the second method, $\\mathbf{F}_G$ is a $|V(G)| \\times 73$ matrix, allowing a maximum of 72 components to compute $\\hat{\\mathbf{F}}_G$, which is then truncated to 15 columns.\n\nFigures~\\ref{fig:undir_task1_pbar} to~\\ref{fig:real_nets_pbar} show, in order, performance with varying number of components in undirected networks Task 1, undirected networks Task 2, directed networks Task 1, directed networks Task 2, the Onnela et al. dataset and directed real world networks.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{undir_task1_pbar.png}\n \\caption{Undirected networks - Task 1}\n \\label{fig:undir_task1_pbar}\n \n \n \\includegraphics[width=0.75\\textwidth]{undir_task2_pbar.png}\n \\caption{Undirected networks - Task 2}\n \\label{fig:undir_task2_pbar}\n \n \\centering\n \\includegraphics[width=0.95\\textwidth]{dir_task1_pbar.png}\n \\caption{Directed networks - Task 1}\n \\label{fig:dir_task1_pbar}\n \\end{figure}\n \\begin{figure}\n \\centering\n \\includegraphics[width=0.81\\textwidth]{dir_task2_pbar.png}\n \\caption{Directed networks - Task 2}\n \\label{fig:dir_task2_pbar}\n \n \\includegraphics[width=0.75\\textwidth]{onnela_pbar.png}\n \\caption{Onnela et al. dataset}\n \\label{fig:onnela_pbar}\n \n \\includegraphics[width=0.75\\textwidth]{real_nets_pbar.png}\n \\caption{Real world directed networks dataset}\n \\label{fig:real_nets_pbar}\n\n\\end{figure}\n\nWe find that adding more components to ICA does not always translate to a better performance, instead we find that there is a performance maximum obtained with a low number of components that depends on the task. Adding more components after this maximum leads to a decline in performance, either an immediately visible decline (Figures~\\ref{fig:undir_task2_pbar},~\\ref{fig:dir_task2_pbar} and~\\ref{fig:onnela_pbar}) or after plateauing at that maximum value (Figures~\\ref{fig:undir_task1_pbar},~\\ref{fig:dir_task1_pbar} and ~\\ref{fig:real_nets_pbar}). This behaviour indicates that our ICA algorithm is working as expected, as we find a suitable number of components that maximizes the noise reduced in the orbit frequency distributions, so adding more components reduces the difference between $\\mathbf{F}_G$ and $\\hat{\\mathbf{F}}_G$ which negates the positive impact on that noise reduction.\n\nFigures~\\ref{fig:dir_task1_pbar} and~\\ref{fig:dir_task2_pbar} show a sharp decline in performance when using graphlets of size up to 3 with reciprocity 0\\% and 100\\%, after 13 and 4 components respectively. This is caused by these extreme levels of reciprocity limiting the orbits that can occur in the networks we compare. With 0\\% reciprocity, only graphlets with no reciprocal edges can occur in these networks, so all orbits in graphlets with reciprocal edges have a frequency of 0. As there are only 13 orbits belonging to graphlets with no reciprocal edges, when we try to compute more than 13 independent components the performance degrades as more components are added. With 100\\% reciprocity, we find a similar situation but there are only 4 orbits in graphlets of size 2 and 3 for which all edges are reciprocal (orbits 2, 27, 28 and 32 in Figure~\\ref{fig:graphlets}).\n\n\n\\section{Introduction}\n\nComplex networks represent relationships between actors of systems with non-trivial structural properties and are ubiquitous in a myriad of domains. Studying the networks underlying these complex systems is then a step towards gaining an understanding about the systems themselves. Comparing different objects is a fundamental part of human cognition, therefore it is only natural that analysis of networks encompasses a comparison element. It is sometimes straightforward to show two networks are different, through finding a mismatch between the set of nodes and edges that compose the networks, which would and claim a difference if these sets do not match. On the other hand, knowing if two networks are exactly the same is known as the graph isomorphism problem, which has been shown to be in the NP complexity class~\\cite{cook1971complexity}. Between these two extremes, \\emph{network comparison} has been established as an area of study within network analysis that combines network statistics and computable properties of a network to tell how similar or dissimilar two networks are. Network comparison has been the subject of an increasing amount of research, see for example~\\cite{wegner2017identifying, sarajlic2016graphlet, aparicio2016extending, ali2014alignment,tantardini2019comparing}, and its applications are widespread, most notably in biological areas like protein-protein interaction networks~\\cite{ali2014alignment} and metabolic networks~\\cite{aparicio2016extending}, but also in other domains, like tracking dynamics of world trade networks~\\cite{yaverouglu2014revealing}.\n\nNetwork comparison methods can be broadly split in two categories: node alignment methods~\\cite{kuchaiev2010topological,mamano2017sana,gu2018homogeneous}, that attempt to create a mapping between the nodes of the networks being compared, and alignment-free methods~\\cite{milo2004superfamilies,prvzulj2007biological,onnela2012taxonomies,koutra2013deltacon}, that use global features of the networks to determine similarity or dissimilarity. Among the latter, methods that use distributions of small connected subgraphs, known as \\emph{graphlets}, have emerged as the state of the art in the area~\\cite{prvzulj2007biological,yaverouglu2014revealing,ali2014alignment,aparicio2016extending,sarajlic2016graphlet}. In particular, Wegner et al.~\\cite{wegner2017identifying} recently introduced the NetEmd measure that achieves state of the art performance in undirected networks, by comparing the shape of graphlet distributions using the \\emph{Earth Mover's Distance} (EMD).\n\nUndirected networks represent relationships between entities under the assumption that the relationship is symmetrical. For example, a Facebook friendship is mutual because both users mutually agree to become friends and thus there is no difference between source and target in the friendship. This abstraction is often insufficient to capture more nuanced relationships. For instance, in Twitter there is an asymmetry in the follower-followed relationship because the action of following another user is not necessarily reciprocated. Directed networks allow us to model such relationships, but this increased wealth of information leads to greater complexity when analyzing them. Such is the case when comparing directed networks, where network comparison metrics designed for undirected networks have been shown to be unsuitable for the task~\\cite{sarajlic2016graphlet,aparicio2016extending}.\n\nThe first contribution of our work is an extension of NetEmd to directed networks, using directed graphlets of size up to 4. Xu and Reinert~\\cite{xu2018triad} previously proposed a similar extension, named \\emph{TriadEMD}, that is limited to graphlets of size 3. The main obstacle when going from size 3 to size 4 graphlets in directed networks is the combinatorial explosion in number of orbits due to edge direction, there are 33 orbits in graphlets of size up to 3 but 730 when including size 4 as well. In the undirected case, scaling up from size 3 to size 4 is a comparatively much smaller jump, from 4 to 15. We propose two methodologies for our extension to size 4: the first is a simple extension, where all orbits are used in the NetEmd calculation; in the second, we adapt the idea of Aparicio et al.~\\cite{aparicio2016extending} and use only orbits that have non-zero frequency in both networks for the comparison. We test systematically the performance difference between using size 3 and size 4 graphlets to understand under which circumstances the added computational load of size 4 graphlets leads to gains in comparison performance. It is also worth noting that the previous methods that use graphlets for comparing directed networks~\\cite{sarajlic2016graphlet,aparicio2016extending} also use size 4 graphlets as input to their comparison measures.\n\nAnother difference between our work and \\emph{TriadEMD} is the inclusion of orbits from size 2 graphlets in the comparison, which is tantamount to including a comparison of the degree distribution in the network comparison measure. Degree distributions are the most commonly studied property to measure the structure of networks and are able to distinguish between networks created with different models~\\cite{ravasz2003hierarchical,newman2006structure,prvzulj2004modeling}, but often they are insufficient alone as a heuristic for network comparison as it is possible to craft networks with the same degree distribution but widely different structures~\\cite{prvzulj2004modeling,li2005towards}. The inclusion of orbits from size 2 graphlets is also supported by the undirected version of NetEmd, where these orbits are also included~\\cite{wegner2017identifying}.\n\nComplex networks are heterogeneous in the amount of data they represent, with famous examples ranging from Zachary's karate club~\\cite{zachary1977information} with only 34 individuals to gigantic networks like the Friendster social network~\\cite{yang2015defining}, which contains more than one billion relationships between its users. Such disparity makes the task of comparing networks even harder, as networks that differ in size by multiple orders of magnitude may still be organized according to similar topography, for instance when they are generated by the same process. Identifying such ``common organizational principles\"~\\cite{wegner2017identifying} is the basis for the NetEmd comparison measure. However, networks representing real world phenomena are often inaccurate representations of that phenomena due to unseen data, measurement errors or simply because the systems they represent are too complex to fully describe. These inaccuracies lead to noise in the statistics used to describe the network and similar topographies become harder to recognize, especially when networks differ greatly in size.\n\nThe distributions of graphlets are not immune to these errors, so our second contribution in this work is using denoising methods to reduce the impact of these errors so that NetEmd is able to more accurately distinguish networks according to those core structures that compose each network. To this end, we propose a framework that employs Principal Component Analysis (PCA)~\\cite{hotelling1933analysis} as the most commonly used denoising method before comparing the graphlet distributions with EMD. PCA finds uncorrelated directions of ordered by contribution to variance, which is sufficient for independence under the assumption of normality~\\cite{shlens2014tutorial}. Graphlet distributions for complex networks, however, break this assumption as, although hard to characterize exactly, they are thought to be similar to degree distributions, which, whether or not they are approximately power-law as often claimed~\\cite{barabasi1999emergence}, are at least often far from normal due to being heavy-tailed~\\cite{broido_scale-free_2019}. Therefore, we hypothesize that PCA struggles to create an appropriate model for these distributions, which can make the denoising process ineffective, so we also propose an alternative intermediate step that uses Independent Component Analysis (ICA)~\\cite{comon1994independent}, a method that is explicitly designed to work with heavy-tailed distributions \\cite{mackay2003information}.\n\nWe test our contributions in clustering tasks involving synthetic and real world networks, following the same experimental procedure as the original NetEmd paper~\\cite{wegner2017identifying}, which adheres to the framework of graph comparison introduced by Yavero{\\u{g}}lu et al.~\\cite{yaverouglu2015proper}. We find that our proposed method of denoising using component analysis techniques is able to improve NetEmd's performance on clustering tasks, both in undirected and directed networks. This improvement is particularly significant when comparing networks of different sizes and densities, where NetEmd is already the state of the art.\n\n\n\n\n\n\n\n\\input{Background}\n\n\\input{DimRed}\n\n\\input{Directed}\n\n\\input{Results}\n\n\\input{Conclusion}\n\n\\section*{Acknowledgments}\n\nMEPS is funded by Engineering and Physical Sciences Research Council Manchester Centre for Doctoral Training in Computer Science (grant number EP\/I028099\/1). TH is supported by the UKRI through the JUNIPER modelling consortium (grant no. MR\/V038613\/1) and the Engineering and Physical Sciences COVID-19 scheme (grant number EP\/V027468\/1), the Royal Society (grant number INF\/R2\/180067), and the Alan Turing Institute for Data Science and Artificial Intelligence.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nIn the authors' previous works\n\\cite{F}--\\cite{FRUYY2}, \\cite{UY}--\\cite{Y}\non zero mean curvature surfaces \nin the Lorentz-Minkowski $3$-space ${\\mathbf R}^3_1$\nand constant mean curvature one surfaces\n(i.e. CMC-$1$ surfaces) in de \nSitter 3-space $S^3_1$, a number of \nconcrete examples were constructed.\nSuch surfaces in ${\\mathbf R}^3_1$ or\n$S^3_1$ have singularities, in general.\nMoreover, some of them\nhave non-trivial analytic extensions\n(cf.\\ Definition \\ref{def:r-A6}).\nTo find such extensions,\none sometimes needs \nfurther techniques beyond\nthe usual reflection principle. \n\n\n\\begin{figure}[hbt]%\n \\begin{center}\n \\begin{tabular}{c@{\\hspace{2em}}c@{\\hspace{2em}}c@{\\hspace{2em}}c}\n \\includegraphics[width=5.0cm]{k1.eps} & \n \\includegraphics[width=5.0cm]{k2.eps} \n \\end{tabular}\n \\end{center}\n\\caption{ The image of $f_K$ (left) and its analytic extension to\na zero mean curvature graph (right)\n(the curves where the surface changes type are also indicated).\n}%\n\\label{fig1c}\n\\end{figure}\nThinking of ${\\mathbf R}^3_1$ with signature $(-++)$,\nthe analytic map\n$f_K\\colon{}{\\mathbf R}\\times T^1\\to ({\\mathbf R}^3_1;t,x,y)$\ndefined by\n\\begin{equation}\\label{eq:A-cat-max}\nf_K(u, v) = (\\sinh u \\cos v, u, \\cosh u \\cos v)\n\\qquad (u\\in {\\mathbf R}, \\,\\, 0\\le v < 2\\pi),\n\\end{equation}\ngives a typical such example \n(the subscript ``K'' stands for ``Kobayashi'' \\cite{K}), \nwhere\n\\begin{equation}\\label{eq:T1}\nT^1:={\\mathbf R}\/2\\pi {\\mathbf Z}\n\\end{equation}\nis a $1$-dimensional torus.\nThis map $f_K$ gives a parametrization of a maximal surface\nderived from the Weierstrass type representation formula\nand has two disjoint \nsingular sets along the lines $v=0,\\pi$.\nThe map is a proper (i.e. the inverse image of a compact set is\ncompact) analytic map, but\nis a two-to-one mapping away from its singular set \n(consisting of\nfold singular points).\nOsamu Kobayashi \\cite{K} found that \nthe image of $f_K$ is a proper subset of the \nentire (zero-mean curvature) \ngraph of the function\n$\nt=y \\tanh x\n$.\nThe complement of the image of\n$f_K$ within this graph contains two disjoint\ntime-like zero mean curvature surfaces,\nthat is, the graph of the function $t=y \\tanh x$ can be considered as\nan analytic extension of the image of $f_K$ \n(see Example \\ref{ex:intro},\nthe precise definition of analytic extension is given in \nDefinition \\ref{def:r-A6}.)\n\nUp to now several space-like maximal surfaces\nadmitting analytic extensions have been found\nby several geometers, \nincluding the authors, see\n\\cite{A, A3, CR, FKKRSUYY2, FKKRUY4, HK, UY2},\nwhich have motivated the authors to \nfind criteria that ensure the resulting \nsurfaces satisfy \\lq\\lq analytic completeness\\rq\\rq,\nnamely, they have no further non-trivial \nanalytic extensions (cf. Proposition \\ref{prop;Aext}). \n\nFor example, space-like maximal surfaces in ${\\mathbf R}^3_1$\nwhich are invariant under a one-parameter family of \nisometries with a common fixed point (i.e. $G$-catenoids, see below)\nare congruent to one of the \nfollowing three surfaces up to a homothety:\n\\begin{itemize}\n \\item the elliptic \\emph{$G$-catenoid}\n \\begin{equation}\\label{eq:e-catenoid}\nf_E(u,v)=(\\sinh u, u \\cos v,u \\sin v) \\qquad\n (u,v)\\in {\\mathbf R}\\times T^1,\n \\end{equation}\n \\item the parabolic $G$-catenoid\n \\begin{equation}\\label{eq:p-catenoid}\n f_P(u,v):=\n \\left(v-\\frac{v^3}3+u^2v, v+\\frac{v^3}3-u^2v,2uv\n \\right)\\qquad (u,v)\\in {\\mathbf R}^2,\n \\end{equation}\n\\item the hyperbolic $G$-catenoid\n \\begin{equation}\\label{eq:h-catenoid}\n f_H(u,v)=(\\cosh u\\sin v, v, \\sinh u \\sin v)\\qquad\n(u,v)\\in {\\mathbf R}^2.\n \\end{equation}\n\\end{itemize}\nThe image of $f_E$ is rotationally symmetric with\nrespect to the time-like axis, and has a cone-like singular point.\nMoreover,\n$f_E$ has no analytic extensions (in fact,\nthe map is proper and admits only one cone-like singular point, \nand so we can apply Proposition~\\ref{prop:first} \nand conclude that $f_E$ is analytically complete). \n\nAs for $f_P$ in \\eqref{eq:p-catenoid}, its image \nis not closed.\nIf we set\n\\begin{equation}\\label{eq:p2}\n \\mc P:=\\Big\\{(t,x,y)\\in{\\mathbf R}^3_1\\,;\\,\n 12 (x^2 - t^2) = (x + t)^4 - 12 y^2\\Big\\},\n\\end{equation}\nthen it has a cone-like singular point\nand satisfies\n$$\n\\mc P=f_P({\\mathbf R}^2)\\cup L,\\qquad L:=\\Big\\{(t,-t,0)\\,;\\,t\\in {\\mathbf R}\\Big\\}.\n$$\nThe inclusion map associated with $\\mc P$ can be considered as an analytic \nextension of the map $f_P$ (see Examples \\ref{ex:P0} and \\ref{ex:P2}).\nSimilarly, $f_H$ also\nhas an analytic extension.\nIn fact, by a suitable symmetry $T$ in ${\\mathbf R}^3_1$, \nwe can write\n\\begin{equation}\\label{eq:S}\n \\mc H:=\\Big\\{(t,x,y)\\in {\\mathbf R}^3_1\\,;\\, \\sin^2 x+y^2-t^2=0\\Big\\}\n=\\overline{f_H({\\mathbf R}^2)\\cup T\\circ f_H({\\mathbf R}^2)},\n\\end{equation}\nwhich is a singly-periodic surface\nwith a countably infinite number of\ncone-like singular points, giving an analytic\nextension of $f_H$ (see Examples \\ref{ex:H0} and \\ref{ex:H2}). \n\nThe figures of $\\mc P$ and $\\mc H$ are found \nin \\cite{FKKRUY4}, and\nthey can be expressed as the zero sets of \nthe single analytic functions\n\\begin{align*}\n&F_P(x,y,t):= 12 (x^2 - t^2) - (x + t)^4 + 12 y^2, \\\\\n&F_H(x,y,t):=\\sin^2 x+y^2-t^2\n\\end{align*}\nrespectively, and by this fact, we can show that they are\nanalytically complete\n(cf. Proposition~\\ref{prop:first}).\nHowever, the converse is not true, in general.\nIn other words,\nanalytically complete subsets\nmight not be expressed as the zero set of \na certain analytic function:\nFor example, the image of the map\n\\begin{equation}\nf_0(t):=(t,e^{-1\/t})\\,\\, (t>0)\n\\end{equation}\nis an analytically complete subset of ${\\mathbf R}^2$\n(cf. Example \\ref{ex:Kok})\nalthough it cannot be characterized as\nthe zero set of any real analytic function.\nIn fact, \n$f_0((0,\\infty))$ is not a closed subset of ${\\mathbf R}^2$. \n\nIn this paper, we define\nanalytic completeness\nfor subsets of real analytic manifolds \nwhich is essentially the same concept as\nthe arc-symmetric property \nintroduced by Kurdyka \\cite{arc} (see also Nash \\cite{N}\nfor arc-structures).\nMoreover, inspired by \\cite{arc},\nwe introduce \n\\lq\\lq arc-properness\\rq\\rq\\ for \ncontinuous maps to an analytic manifold\n(cf.~Definition~\\ref{def:ap}).\nOne may regard arc-properness as a\nvery weak version of properness of maps (cf. Proposition \\ref{prop:P}). \nIn fact, proper maps are arc-proper, but the\nconverse is not true, in general.\nThis concept plays an important\nrole in describing our criterion\nfor analytic completeness (cf. Theorem~\\ref{thm:first}).\nIn the remainder of this paper,\nwe shall demonstrate that this criterion \nis actually applicable to concrete examples. \nIn fact, in the last two sections of this paper, \nit is applied \nto show the analytic completeness \nof analytic extensions of constant\nmean curvature one catenoids in de Sitter $3$-space,\nas follows:\nLet ${\\mathbf R}^4_1$ be the Lorentz-Minkowski $4$-space\nwith the metric $\\inner{~}{~}$ of \nsignature $(-++\\,+)$.\nThen \n\\[\n S^3_1=\n \\Big\\{X:=(x_0,x_1,x_2,x_3) \\in{\\mathbf R}^4_1\\, ; \n\\, (\\inner{X}{X}=)\n-(x_0)^2+\\sum_{j=1}^3 (x_j)^2=1\\Big\\},\n\\]\nwith the metric induced from ${\\mathbf R}^4_1$, \nis a simply-connected Lorentzian \n$3$-manifold with constant sectional curvature $1$,\ncalled the {\\it de Sitter 3-space}.\n\nSpace-like constant mean curvature one surfaces in $S^3_1$ \nhave a Weierstrass-type representation formula\nlike space-like maximal\nsurfaces in Lorentz-Minkowski space ${\\mathbf R}^3_1$\n(see \\cite{AA}, \\cite{F} and \\cite{FRUYY}).\nIn this paper, we consider constant mean curvature one surfaces\nin $S^3_1$\nand zero mean curvature surfaces in ${\\mathbf R}^3_1$ with either of the\nfollowing properties:\n\n\\begin{itemize}\n\\item[(G)]\nThe surface is invariant under a one-parameter family of\nisometries with a common fixed point.\n\\item[(W)]\nThe surface is weakly complete with Gauss map of degree $1$ and Hopf differential having a pole of order $2$ at each of two ends.\n\\end{itemize}\nWe call them geometric catenoids (G-catenoids in short) if they satisfy\n(G), and catenoids with the same holomorphic data as the\nEuclidean catenoid (W-catenoids in short) if they satisfy (W).\nHere the subscript ``W'' stands for ``Weierstrass-type representation formulas''\nfor space-like constant mean curvature one surfaces\nin $S^3_1$ and zero mean curvature surfaces in ${\\mathbf R}^3_1$ ).\n\nAs mentioned above, $G$-catenoids in ${\\mathbf R}^3_1$\nare space-like maximal surfaces \nthat are congruent to one of $f_E,\\, f_P,\\, f_H$ up to a homothety,\nand the\nlast two maps have analytic extensions as mentioned above. \nW-catenoids in ${\\mathbf R}^3_1$\nare congruent to either $f_E$ or $f_K$ up to homotheties. \n(We have already seen that $f_K$ can be\nextended as an entire graph.)\n\nOn the other hand, in \\cite{Y} and \\cite{FKKRUY2}, \nall G-catenoids and W-catenoids in $S^3_1$ \nare classified.\nSpace-like constant mean curvature one surfaces in $S^3_1$ \nhave a Weierstrass-type representation formula\nlike as in the case of space-like maximal \nsurfaces in Lorentz-Minkowski space ${\\mathbf R}^3_1$.\n\nIn light of this, it is interesting that\nthere are so many different types of\nG-catenoids and W-catenoids in $S^3_1$.\nG-catenoids in $S^3_1$ admit only cone-like singular points\nlike as in the case of catenoids in ${\\mathbf R}^3_1$.\nOn the other hand, W-catenoids in $S^3_1$ admit \nvarious kind of singular points.\nIn this paper, we focus on $G$-catenoids.\nSuch a family of isometries fixes a geodesic of $S^3_1$.\nBy the last author's classification \\cite{Y},\ngeometric catenoids are divided into \nthe following eight classes:\n\\begin{itemize}\n\\item G-catenoids of type TE, TP or TH\n(which are special cases of W-catenoids with elliptic, parabolic or\nhyperbolic monodromies, respectively), \n\\item G-catenoids of type SE, SP or SH \n(exceptional W-catenoids of type II belong\nto a special class of G-catenoids of type SE),\n\\item G-catenoids of type LE or LH, \n\\end{itemize}\nwhere $T,S$ and $L$ denote whether the geodesic\nfixed by the 1-parameter group of isometries \nis time-like, space-like or light-like,\nand $E,P$ and $H$ stand for elliptic, parabolic and hyperbolic, respectively.\nWe show that,\nexcept for G-catenoids of type TE, TP and TH which are \nalready analytically complete, \nthe other five types of G-catenoids have\nanalytic extensions which are analytically complete.\n\nTo formulate the notion of analytic extension of analytic maps,\nwe define the concept \n\\lq\\lq double-cone manifold\". \nIn fact, all analytic extensions of G-catenoids \nadmit only cone-like singular points \nas their singular points, and can be parametrized\nby analytic maps defined on double-cone manifolds.\n\nIn Section 1, we define analytic completeness. \nIn Section 2, we define analytic maps on double-cone manifolds\nand show their fundamental properties.\nAll analytic extensions of G-catenoids\ncan be interpreted as DC-immersions (i.e. double-cone immersions, \nthat is immersions of DC-manifolds). \nIn Section 3, we give a criterion for analytic completeness of\nreal analytic DC-immersions.\nIn Section 4, we prove that G-catenoids belonging to the above \nfive types SE, SP, SH, LE or LH can be extended as \nreal analytic DC-immersions from certain \nDC-manifolds of dimension 2.\n\nThe analytic completeness of maximal catenoids or their particular\nanalytic extensions in ${\\mathbf R}^3_1$ is also discussed in the appendix.\n\n\\section{Preliminaries}\n\nThroughout this paper,\nwe fix a real analytic manifold $N^n$ of dimension $n(\\ge 1)$.\nWe also fix a positive integer $m(\\le n)$.\nWe begin by defining analytic completeness as follows:\n\n\\begin{Definition}\nA non-empty subset $S$ of $N^n$\nis said to be {\\it analytically complete}\nif it satisfies the following \nproperty: \n\\begin{quote}\nAny real analytic map $\\Gamma : [0,1] \\to N^n$ satisfying \n$\\Gamma([0,\\varepsilon)) \\subset S$ for some $\\varepsilon \\in (0,1)$\nsatisfies $\\Gamma([0,1]) \\subset S$.\n\\end{quote}\n\\end{Definition}\n\nAnalytic completeness is the same concept as \nthe \\lq\\lq arc-symmetric property\\rq\\rq\\\ngiven in \\cite{arc}. \nFor example:\n \\begin{itemize}\n \\item \nAny affine subspace $A^m$ in ${\\mathbf R}^n$ is an analytically complete \nsubset of ${\\mathbf R}^n$ (cf. Example \\ref{ex:Rn}). \n\\item Any subinterval $I \\subsetneq {\\mathbf R}$ \nis not an analytically complete subset of ${\\mathbf R}$.\nIn fact, if we set $\\Gamma(t):=t$, then\n$\\Gamma(I)\\subset I$ but $\\Gamma(J)\\not \\subset I$\nif $J$ is an interval containing $I$ as a proper subset.\n\\item Let $S = \\{ (t, e^{-1\/t}) \\in {\\mathbf R}^2 \\mid 0 < t < \\infty \\}$. \nThen $S$ is an analytically complete subset of ${\\mathbf R}^2$,\nalthough $S$ has a $C^\\infty$-extension\n(cf. Example \\ref{ex:Kok}).\n \\end{itemize}\n\n\\begin{Definition}\nA subset $S$ of $N^n$ is called {\\it globally analytic} \nif there exists a positive integer $l$ and a real analytic map\n$F:N^n \\to {\\mathbf R}^l$ such that the zero set\n$$\nZ(F):=\\Big\\{P\\in N^n\\,;\\, F(P)=(0,\\dots,0)\\Big\\}\n$$\nof $F$ coincides with $S$. \n\\end{Definition}\n\n\n\\begin{Exa}\\label{ex:Rn}\nIf $S$ is an $m$-dimensional linear subspace \nin ${\\mathbf R}^n$, then $S$ is a globally analytic subset,\nas there is a linear map $F:{\\mathbf R}^n\\to {\\mathbf R}^{l}$ ($l\\ge 1$)\nsuch that $S$ is the kernel of $F$.\n\\end{Exa}\n\n\n\\begin{Exa}\nFor $m\\ge 2$, we set \n\\begin{align*}\n\\mc C^m_1:=\\left\\{(x_1,\\dots,x_{m+1})\\in {\\mathbf R}^{m+1}\\,;\\, \n\\sum_{j=1}^m (x_j)^2=(x_{m+1})^2,\\,\\,\\, x_{m+1}\\ge 0\\right\\},\n\\end{align*}\nwhich is called the {\\it $m$-dimensional standard cone}, and the\norigin is called the {\\it standard cone point}.\n$\\mc C^m_1$ is not globally analytic. \nIt should be remarked that $\\mc C^m_1$ is the image of the analytic map\n\\begin{equation}\\label{eq:h}\nh_C:{\\mathbf R}\\times S^{m-1}\\ni (t,P)\\mapsto (t^2 P, t^2)\\in {\\mathbf R}^{m+1}.\n\\end{equation}\nOn the other hand, the {\\it $m$-dimensional standard double cone}\n\\begin{equation}\\label{eq:cone}\n\\mc C_2^m:=\\left\\{(x_1,\\dots,x_{m+1})\\in {\\mathbf R}^{m+1}\\,;\\, \n\\sum_{j=1}^m (x_j)^2=(x_{m+1})^2\\right\\}\n\\end{equation}\nis a globally analytic subset of ${\\mathbf R}^{m+1}$.\nMoreover, $\\mc C^m_2$ is the image of the analytic map\n\\begin{equation}\\label{eq:fdc}\nf_{DC}:{\\mathbf R}\\times S^{m-1}\\ni (t,P)\\mapsto (t P, t)\\in {\\mathbf R}^{m+1}.\n\\end{equation}\nWe call the origin $\\mb 0\\in \\mc C_2^m$ the \n{\\it standard double-cone point}.\n\\end{Exa}\n\n\nAs pointed out in the introduction,\nthe two subsets $\\mc P$ and $\\mc H$ of ${\\mathbf R}^3_1$\ngiven by \\eqref{eq:p2} and \\eqref{eq:S} are globally analytic.\nThe following assertion holds:\n\n\\begin{Proposition}\\label{prop:first}\nLet $S$ be a subset of $N^n$.\nIf $S$ is a globally analytic subset of $N^n$, then it\n is analytically complete.\n\\end{Proposition}\n\n\n\\begin{proof}\nSuppose that $S$ is globally analytic.\nThen there exist a positive integer \n$l$ and a real analytic map $F:N^n\\to {\\mathbf R}^l$\nsuch that $S=Z(F)$. Let $\\Gamma:[0,1]\\to N^n$ be a real analytic curve\nsuch that $\\Gamma([0,\\epsilon))\\subset S(\\subset Z(F))$ \nfor some sufficiently small $\\epsilon>0$.\nThen the real analyticity of $\\Gamma$ and $F$\nimplies $\\Gamma([0,1])\\subset S$.\nSo $S$ is analytically complete.\n\\end{proof}\n\n\\section{Analytic maps on double-cone manifolds (DC-manifolds)}\n\nIn the previous section, we defined\nthe standard cone $\\mc C^m_1$ and\nthe standard double-cone $\\mc C^m_2$, respectively. \nUsing them, we define\nreal analytic cone manifolds and double-cone manifolds, as\nfollows (we fix an integer $m\\ge 2$): \n\n\\begin{Definition}\\label{def:CR}\nLet $U$ be an open subset of $\\mc C_i^m$ ($i=1,2$).\nA map $\\mu:U \\to \\mc C_i^m$ is said to be {\\it real analytic} if\nthere exists an open subset $\\Omega$ of ${\\mathbf R}^{m+1}$ \nand a real analytic map $\\tilde \\mu:\\Omega\\to {\\mathbf R}^{m+1}$\nsuch that $\\mc C_i^m\\cap \\Omega=U$ and the restriction of \n$\\tilde \\mu$ to $U$ coincides with $\\mu$.\n\\end{Definition}\n\n\\begin{Definition}\nLet $X^m$ be a locally compact \nHausdorff space with the second axiom of countability.\nThen $X^m$ is called an {\\it $m$-dimensional real analytic\ndouble-cone manifold} (i.e. real analytic DC-manifold) \n(resp. a {\\it real analytic cone manifold})\nif there exists a family of maps\n$\\phi_\\lambda:U_\\lambda\\to \\mc C_2^m$\n(resp. $\\phi_\\lambda:U_\\lambda\\to \\mc C_1^m$)\nindexed by a set $\\Lambda$ satisfying the following\nproperties: \n\\begin{itemize}\n\\item $\\{U_\\lambda\\}_{\\lambda\\in \\Lambda}$\nis an open covering of $X^m$,\n\\item for each $\\lambda\\in \\Lambda$, \n$\\phi_\\lambda(U_\\lambda)$ is a connected open subset of $\\mc C^m_2$,\n\\item \neach $\\phi_\\lambda$ is a homeomorphism\nbetween $U_\\lambda$ and $\\phi_\\lambda(U_\\lambda)$, and\n\\item for two indices $\\lambda_i$ ($i=1,2$),\nthe map \n$$\n\\phi_{\\lambda_2}\\circ \\phi_{\\lambda_1}^{-1}:\n\\phi_{\\lambda_1}(U_{\\lambda_1}\\cap U_{\\lambda_2})\n\\to \\phi_{\\lambda_2}(U_{\\lambda_1}\\cap U_{\\lambda_2})\n$$\nis a real analytic map in the sense of Definition \\ref{def:CR}.\n\\end{itemize}\nWe call $\\{(U_\\lambda,\\phi_\\lambda)\\}_{\\lambda\\in \\Lambda}$\nthe {\\it DC-structure} (resp. {\\it cone structure})\nof $X^m$.\n\nWe let $X^m$ be a real analytic DC-manifold of dimension $m$.\nA pair $(V,\\psi)$ consisting of a map $\\psi:V\\to \\mc C^m_{2}$ defined\non an open subset $V$ of $X^m$ is called a {\\it local DC-coordinate system} of $X^m$\nif \n\\begin{itemize}\n\\item $\\psi$ is a homeomorphism\nbetween $V$ and $\\psi(V)$, and\n\\item for each index $\\lambda\\in \\Lambda$,\nthe maps \n$$\n\\phi_{\\lambda}\\circ \\psi^{-1}:\n\\psi(U_{\\lambda}\\cap V)\n\\to \\phi_{\\lambda}(U_{\\lambda}\\cap V)\n\\,\\,\\, \\text{and}\\,\\,\\,\n\\psi\\circ \\phi_{\\lambda}^{-1}:\n\\phi_\\lambda(U_{\\lambda}\\cap V)\n\\to \\psi(U_{\\lambda}\\cap V)\n$$\nare real analytic in the sense of Definition \\ref{def:CR}.\n\\end{itemize}\n\\end{Definition}\n\nLet $X^m$ be a real analytic DC-manifold as above.\nThen $p\\in X^m$ is called a {\\it DC-point}\n(resp. a {\\it non-DC-point})\nif there exists a local coordinate system $(V,\\psi)$\nsuch that \n$\\psi(p)$ is (resp. is not) \nthe standard double-cone point of $\\mc C^m_2$.\nIf $p$ is a DC-point (resp. non-DC-point)\nand $(U,\\phi)$ is another local coordinate system\nsuch that $p\\in U\\cap V$, then\nit can be easily checked that\n$\\phi(p)$ is (resp. is not)\nthe standard double-cone point of $\\mc C^m_2$\nas well as $\\psi(p)$.\n\nBy definition, usual manifolds can be considered as \nDC-manifolds without DC-points.\n \nOne can easily see that $\\mc C_2^m$ (resp. $\\mc C_1^m$)\nis a typical example of a real analytic DC-manifold\n (resp. cone manifold). \nHere, we show that\nthe images $\\mc P$ and $\\mc H$ of geometric catenoids in ${\\mathbf R}^3_1$\ngiven in the introduction can be considered as real analytic DC-manifolds,\nas follows:\n\n\n\\begin{Exa}\\label{ex:H0}\nWe consider the subset $\\mathcal H$ given by \\eqref{eq:S} \nwhich is the zero-level \nset of the real analytic function $F_H = \\sin^2 x + y^2 -t^2$. \nThe exterior derivative $dF_{H}=-2t\\, dt+\\sin{2x}\\, dx+2y\\, dy$ \nvanishes on the set \n$$\n\\Sigma =\\{ (0, n\\pi, 0) \\in \\mathcal H ; n \\in \\mathbf Z \\}.\n$$\nSo $\\mathcal H \\setminus \\Sigma$ has the structure of a \nreal analytic $2$-manifold. \nWe consider the map \n\\begin{equation*}\n \\Phi_H \\colon \\mathbf R^3_1 \\ni (t,x,y) \\mapsto \n(t, \\sin x, y) \\in \\mathbf R^3. \n\\end{equation*}\nFor every $n \\in \\mathbf Z$, the map $\\Phi_H$ gives a \nreal analytic diffeomorphism \nfrom a neighborhood $\\Omega_n$ of $(0, n \\pi, 0)$ to a neighborhood of \n$\\boldsymbol 0 \\in \\mathbf R^3$. The image $\\Phi_H ( \\Omega_n \\cap \\mathcal H )$ \ncoincides with $\\Phi_H(\\Omega_n) \\cap \\mathcal C^2_2$ and the image \n$\\Phi_H((0, n \\pi, 0))$ is the standard double-cone point $\\boldsymbol 0$. \nHence, we can give $\\Omega_n \\cap \\mathcal H$ the structure of a real analytic \nDC-manifold \nso that \nthe pair $(\\Phi_H |_{\\Omega_n \\cap \\mathcal H}, \\Omega_n \\cap \\mathcal H)$ \nis a (local) DC-coordinate system around the DC-point $(0,n\\pi,0)$. \nIt is obvious that the DC-structure on $\\mc H\\setminus \\Sigma$\nand on $\\Omega_n\\cap \\mc H$ ($n\\in {\\mathbf Z}$) are compatible. Therefore, $\\mc H$\nhas the structure of a real analytic DC-manifold whose set of DC-points\nis $\\Sigma$.\n\\end{Exa}\n\n\\begin{Definition}\\label{def:mDC-map}\nLet $X^m$ and $Y^n$ be two real analytic DC-manifolds whose \nDC-structures are\n$\\{(U_\\alpha,\\phi_\\alpha)\\}_{\\alpha\\in A}$\nand\n$\\{(V_\\beta,\\psi_\\beta)\\}_{\\beta\\in B}$, respectively.\nA map $f:X^m \\to Y^n$ is said to be {\\it DC-analytic}\nif $\\psi_\\beta\\circ f\\circ \\phi_{\\alpha}^{-1}$\nis a real analytic map.\nMoreover, if $f$ admits a non-DC-point at which $f$ is an immersion,\nthen $f$ is called an {\\it $m$-dimensional DC-analytic map}.\n\\end{Definition}\n\nGiven an arbitrary DC-point of a DC-manifold, there exists a blow-up for it. \n\n\\begin{Lemma}\\label{lem:B}\nLet $X^m$ be a connected $m$-dimensional real analytic DC-manifold\nsuch that $p$ is a DC-point of $X^m$.\nThen there exists a connected real analytic DC-manifold $\\tilde X^m$\nand a surjective DC-analytic map $\\pi:\\tilde X^m\\to X^m$\nsuch that \n\\begin{enumerate}\n\\item $\\pi^{-1}(p)$ consists of non-DC-points, \n\\item there exists a neighborhood $U$ of $p$\nsuch that $\\pi^{-1}(q)$ $(q\\in U\\setminus \\{p\\})$\nconsists of a single non-DC-point.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nWe take a local analytic DC-coordinate \nneighborhood $(U,\\phi)$ of $p$ in $X^m$ \nsuch that $\\phi(p)=\\mb 0$.\nThen $g:=\\phi^{-1}\\circ f_{DC}$ (cf. \\eqref{eq:fdc})\nis a DC-analytic map. In particular, $g$ is continuous, and\n$\nV:=g^{-1}(U)\n$\nis an open submanifold of ${\\mathbf R}\\times S^{m-1}$.\nWe then consider the disjoint union\n$$\n\\textstyle \\mc W:=(X^m\\setminus \\{p\\})\\, \\coprod\\, V\n$$\nof the two DC-manifolds $X^m\\setminus \\{p\\}$ and $V$.\nConsider the injection\n$$\n\\hat g:=g|_{V\\setminus g^{-1}(p)}:V\\setminus g^{-1}(p)\\to X^m\\setminus \\{p\\},\n$$\nwhich gives a local DC-analytic diffeomorphism.\nHere we regard this map $\\hat g$ as a gluing map, by which\nwe can identify each point of $V\\setminus g^{-1}(p)$\nwith its image via $\\hat g$ and obtain\nthe DC-manifold $\\tilde X^m$ of dimension $m$.\nWe let\n$\n\\tilde \\pi:\\mc W \\to X^m\n$\nbe the canonical projection, that is,\nit is defined by \n$$\n\\tilde \\pi(x):=\n\\begin{cases}\nx & \\text{if $x\\in X^m \\setminus \\{p\\}$}, \\\\\ng(x) & \\text{if $x\\in V$}.\n\\end{cases}\n$$\nIn fact, $\\tilde \\pi$ induces a DC-analytic projection\n$\n\\pi: \\tilde X^m\\to X^m\n$\nsuch that $\\pi^{-1}(p)$ consists of non-DC-points.\nBy this construction, it is obvious that $\\tilde X^m$\nis connected.\nHence, the DC-manifold $\\tilde X^m$ has the desired properties.\n\\end{proof}\n\nWe call this $\\tilde X^m$ the {\\it blowup} of $X^m$ at \nthe DC-point $p$ and call the map $\\pi:\\tilde X^m\\to X^m$\nthe {\\it canonical projection}.\nThe following assertion holds:\n\n\\begin{Proposition}\\label{prop:Long}\nLet $f:X^m\\to Y^n$ be an $m$-dimensional DC-analytic map\n$($Definition \\ref{def:mDC-map}$)$.\nIf $X^m$ is connected,\nthen there exists an open dense subset $O$ of $X^m$ consisting\nonly of non-DC-points so that $f|_O$ is an immersion.\n\\end{Proposition}\n\nTo prove this, we prepare a lemma:\n\n\\begin{Lemma}\\label{lem:DC}\nLet $f:X^m\\to Y^m$ be an $m$-dimensional analytic DC-map\nand $p\\in X^m$ a non-DC-point at which $f$ is an immersion.\nLet $\\Sigma$ be the set of DC-points of $X^{m}$\nand suppose that $X_0^m$ is a connected component\nof $X^m\\setminus \\Sigma$ containing $p$.\nThen there exists an open dense subset $O$ of $X^m_0$\n so that $f|_O$ is an immersion.\n\\end{Lemma}\n\n\\begin{proof}\nSince $X^m_0$ has no DC-point, \nit is a (usual) connected real analytic $m$-manifold. \nWe denote by $J$ the Jacobian of the map $f$ on $ X^m_0$.\nSince $J$ is a real analytic function, the fact $J(p)\\ne 0$\nimplies that the set of non-zero points of $J$ is an open \ndense subset of $X^m_0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:Long}]\nLet $p\\in X^m$ be\na non-DC-point at which $f$ is an immersion.\nWe denote by $\\Sigma$ the set of DC-points of $X^{m}$\nand suppose that $X_0^m$ is a connected component\nof $X^m\\setminus \\Sigma$ containing $p$.\nWe also fix a point $q\\in X^m\\setminus (\\Sigma\\cup X_0^m)$ arbitrarily,\nand denote by $X_1^m$ the connected component of \n$X^m\\setminus \\Sigma$ which contains $q$. \nSince $X^m$ is connected, there exists a continuous curve\n$\\gamma:[0,1]\\to X^m$ such that $\\gamma(0)=p$\nand $\\gamma(1)=q$. \nSince the set of DC-points are discrete, \nthe curve $\\gamma$ passes through only finitely many\nDC-points $x_1,\\dots,x_k$. So it is sufficient to \nconsider the case that $k=1$. Let $\\tilde X^m$ be the blowup of $X^m$\nat $x_1$ and denote by $\\pi:\\tilde X^m\\to X^m$ the canonical projection.\nWe let $\\tilde p$ and $\\tilde q$ be points on\n$\\tilde X^m$ so that $\\pi(\\tilde p)=p$ and\n$\\pi(\\tilde q)=q$. Since $\\pi^{-1}(x_1)$ consists of \nnon-DC-points in $\\tilde X^m$,\n$\\tilde p$ and $\\tilde q$ belong to the same\nconnected component $\\tilde X^m_0$ in $\\tilde X^m\\setminus \\tilde \\Sigma$,\nwhere $\\tilde \\Sigma$ is the set of DC-points in $\\tilde X^m$.\nSince $\\tilde f:=f\\circ \\pi$ is an $m$-dimensional\nanalytic DC-map, Lemma \\ref{lem:DC} implies that\nthere exists an open dense subset $\\tilde O$ of $\\tilde X^m_0$ such\nthat $\\tilde f$ is an immersion on $\\tilde X^m_0$. \nThen $f$ is an immersion on $O:=\\pi(\\tilde O)$\nand $O\\cap X^m$ is dense in $X^m_0\\cup X_1^m$.\nSince $q$ is arbitrarily chosen, we obtain the assertion.\n\\end{proof}\n\nWe next define DC-immersions as follows:\n\n\\begin{Definition}\\label{def:XN}\nLet $f:X^{m}\\to N^{n}$ ($n>m$) be a DC-analytic map. \nA point $p\\in X^m$ is called a {\\it DC-regular point} if there exist\n\\begin{itemize}\n\\item a DC-coordinate neighborhood $(U,\\phi)$ \nof $p$ and \n\\item a real analytic diffeomorphism $\\Phi$ \ndefined on a neighborhood $\\Omega$ of $f(p)$ \ninto a neighborhood $\\Omega_0$ of the origin $\\mb 0$\nof ${\\mathbf R}^{n}$ such that $$\n\\Phi\\circ f\\circ \\phi^{-1}(a)=(a,\\mb 0_{n-m-1}) \n\\qquad (a\\in \\phi(U)\\subset \\mc C^m_2),\n$$\nwhere $\\mb 0_{n-m-1}$ is the zero vector of ${\\mathbf R}^{n-m-1}$.\n\\end{itemize}\nBy definition $\\Phi\\circ f(b)=\\mb 0$ holds\nif and only if $b$ is a DC-point.\nIf $b$ is not a DC-point, then $f$ is an immersion at $b$.\nIf $f$ admits only DC-regular points,\nwe call $f$ a {\\it real analytic DC-immersion}.\n\nMoreover, a point $p\\in X^m$ is called a {\\it cone-like singular point} of $f$\nif $p$ is a DC-point of $X^m$ and is a DC-regular point of $f$.\nIf a real analytic DC-immersion has an inverse map which is also\na DC-immersion, it is called a {\\it real analytic DC-diffeomorphism}.\n\\end{Definition}\t\n\nBy definition, the following assertion is obvious:\n\n\\begin{Proposition}\\label{prop:DC-N}\nA real analytic DC-immersion $f:X^{m}\\to N^{n}$ $(n>m)$\nis an $m$-dimensional DC-analytic map.\n\\end{Proposition}\n\nHere, we give two examples of real analytic DC-diffeomorphisms:\n\n\n\\begin{Exa}\\label{ex:E0}\nThe image $\\mc E$ of the elliptic catenoid $f_E$ given in\n\\eqref{eq:e-catenoid} can be characterized as \nthe zero set of the function\n$\nF_E:=x^2+y^2-(\\sinh^{-1}t)^2\n$, that is, we have\n\\begin{equation}\\label{eq:E0}\n\\mc E=\\{(t,x,y)\\in {\\mathbf R}^3_1\\,;\\, x^2+y^2=(\\sinh^{-1}t)^2\\},\n\\end{equation}\nwhere $\\sinh^{-1}$ denotes the inverse function of \n$\\sinh.$ The exterior derivative of $F_E$ \nvanishes only at the origin $\\mb 0$ of ${\\mathbf R}^3_{1}$. \nSo $\\mc E \\setminus \\{\\mb 0\\}$ has the structure of \na (usual) real analytic $2$-manifold.\nThe map defined by\n$$\n\\tilde \\Phi_E:{\\mathbf R}^3\\ni (x,y,t)\\to (x,y,\\sinh^{-1}t)\\in {\\mathbf R}^3\n$$\ngives a real analytic diffeomorphism on ${\\mathbf R}^3$.\nSo the inverse map of the restriction $\\Phi_E:=\\tilde \\Phi_E$ \ngives a local DC-coordinate system of $\\mc E$ \nwhich covers the point $\\mb 0\\in \\mc E$.\nSo $\\mc E$ has the structure of a $2$-dimensional DC-manifold.\nMoreover, the map $\\Phi_E$\ngives a real analytic DC-diffeomorphism between\n$\\mc C^2_2$ and $\\mc E$.\n\\end{Exa}\n\n\\begin{Exa}\\label{ex:P0}\nWe consider the subset $\\mc P$ given by \\eqref{eq:p2}\nof ${\\mathbf R}^3_1$ which is the zero-level set of the real analytic function \n$$\nF_P= 12 (x^2 - t^2) - (x + t)^4 + 12 y^2.\n$$\nThe exterior derivative of $F_P$\nvanishes only at the origin $\\mb 0$ of ${\\mathbf R}^3_{1}$. \nSo $\\mc P\\setminus \\{\\mb 0\\}$ has the structure of \na real analytic $2$-manifold.\nWe set\n$$\n\\xi+\\zeta:=x+t,\\qquad \\xi-\\zeta:=12(x-t)-(x+t)^3,\\quad \\eta:=2\\sqrt{3}y,\n$$\nand consider the map defined by\n$$\n\\Phi_P:\\mc P\\ni (t,x,y) \\mapsto (\\zeta,\\xi,\\eta)\\in \\mc C^2_2.\n$$\nSince the Jacobian matrix of the map\n${\\mathbf R}^3\\ni (t, x, y) \\mapsto (\\zeta,\\xi,\\eta)\\in {\\mathbf R}^3$ is of full rank \nat $\\mb 0$,\nwe can consider the inverse map $\\Phi^{-1}_P$\nwhich is defined on a neighborhood of the origin in ${\\mathbf R}^3$.\nThe restriction of this to ${\\mc C}^2_2$\ngives a local DC-coordinate system of $\\mc P$ centered at the origin $\\mb 0$.\nSo $\\mc P$ has the structure of a real analytic $2$-dimensional \nDC-manifold whose set of DC-points is $\\{\\mb 0\\}$. \nMoreover, the map $\\Phi_P$\nis a real analytic DC-diffeomorphism between\n$\\mc P$ and $\\mc C^2_2$.\n\\end{Exa}\n\n\n\\begin{Definition}\\label{def:r-A6}\nLet $f_i:X^m_i\\to N^n$ ($i=1,2$)\nbe two real DC-analytic maps. \nThen $f_2$ is called an {\\it analytic extension} \nof $f_1$ if there exists\na DC-analytic map $\\phi:X^m_1\\to X^m_2$\nsuch that $f_1=f_2\\circ \\phi$\nand $f_2(X^m_2)$ is a proper subset of $f_1(X^m_1)$. \n\\end{Definition}\n\n\n\\begin{Example}\\label{ex:intro}\nLet $f_K\\colon{}{\\mathbf R}\\times T^1\\to {\\mathbf R}^3_1$\nbe the real analytic map defined by\n\\eqref{eq:A-cat-max} in the introduction (cf.\n\\eqref{eq:T1} for the definition of $T^1$).\nWe set\n$$\ng:{\\mathbf R}^2\\ni (u,s)\\mapsto (s \\sinh u,u,s \\cosh u)\\in {\\mathbf R}^3_1.\n$$\nIt can be easily seen that $g$ is an analytic\nextension of $f_K$. \nBut if we set $y=s \\cosh u$ and $x=u$, \nthen $g$ can be written as\n$$\nh:{\\mathbf R}^2\\ni (x,y)\\mapsto (y \\tanh x,x,y)\\in {\\mathbf R}^3_1,\n$$\nwhich parametrizes the entire graph of the function\n$\nt=y \\tanh x.\n$\nThis implies the image of $h$ is globally analytic\n(cf. Proposition \\ref{prop:first}),\nand so $h$ is an analytic extension of $f_K$, which is\nanalytically complete.\n\\end{Example}\n\n\\begin{Example}\\label{ex:P2}\nIn Example \\ref{ex:P0},\nwe gave the structure of a $2$-dimensional DC-manifold\non \n\\begin{equation}\\label{eq:FP}\n\\mc P:=\\{(t,x,y)\\in {\\mathbf R}^3_1\\,;\\, \n12 (x^2 - t^2) - (x + t)^4 + 12 y^2=0\\}.\n\\end{equation}\nIt can be easily checked that\nthe inclusion map $\\Phi_P$ of $\\mc P$ into ${\\mathbf R}^3_1$ \nis a real analytic DC-immersion.\nMoreover, $\\Phi_P$ can be considered as an analytic extension of the\nmap $f_P$ given in \\eqref{eq:p-catenoid}.\nSince $\\mc P$ is globally analytic, it is analytically complete\n(cf. Proposition \\ref{prop:first}).\n\\end{Example}\n\n\n\\begin{Example}\\label{ex:H2}\nSimilarly, \n\\begin{equation}\\label{eq:FH}\n\\mc H:=\\{(t,x,y)\\in {\\mathbf R}^3_1\\,;\\,\n\\sin^2 x+y^2-t^2=0 \\}\n\\end{equation}\nhas\nthe structure of $2$-dimensional DC-manifold\n(cf. Example \\ref{ex:H0}) and one can easily check that\nthe inclusion map $\\Phi_H$ of $\\mc H$ into ${\\mathbf R}^3_1$ \nis a real analytic DC-immersion, which is\nconsidered as an analytic extension of the\nmap $f_H$ given in \\eqref{eq:h-catenoid}.\nSince $\\mc H$ is globally analytic, it is analytically complete\n(cf. Proposition \\ref{prop:first}).\n\\end{Example}\n\n\\begin{Proposition}\\label{prop;Aext}\nLet $f:X^m\\to N^n$ be an $m$-dimensional DC-analytic map. \nIf $f(X^m)$ is analytically complete, then any \n$m$-dimensional DC-analytic map\ncannot be an analytic extension of $f$.\n\\end{Proposition}\n\n\\begin{proof}\nSuppose that $g:Y^m\\to N^n$ is an $m$-dimensional DC-analytic map\n which is an\nanalytic extension of $f$. Then there exists an $m$-dimensional DC-analytic map \n$\\phi:X^m\\to Y^m$ such that $f=g\\circ \\phi$.\nSince $f$ is an $m$-dimensional DC-analytic map,\nthere exists a non-DC-point $p_0$\nsuch that $f$ is an immersion at $p_0$.\nThen there exists a neighborhood $U$ of $p_0$\nso that $f|_U$ is an immersion defined on\nthe $C^\\omega$-differentiable $m$-manifold $U$.\nIf the image $\\phi(U)$ consists only of DC-points of $Y^m$, \nit must be a single point set of $Y^m$,\nwhich contradicts the fact that $f|_U$ is an immersion.\nSo replacing $p$ by a suitable point on $U$ if necessary, \nwe may assume $p:=\\phi(p_0)$ is a non-DC-point of $Y^m$. \nThen replacing $U$ by a smaller neighborhood\nof $p_0$ if necessary, we may also assume that $\\phi(U)$\ndoes not contain any DC-points. In particular, $\\phi|_U$ is an immersion \nbetween two $m$-dimensional $C^\\omega$-differentiable \nmanifolds $U$ and $\\phi(U)$.\nBy the inverse function theorem, \nwe may assume that $\\phi$ is an analytic diffeomorphism \nbetween $U$ and $\\phi(U)$.\n\nSince $g$ is an analytic extension of $f$,\nwe can find a point $q\\in Y^m$ such that\n$g(q)\\not\\in f(X^m)$.\nSince $Y^m$ is connected, there exists a continuous curve\n$\\gamma:[0,1]\\to Y^m$ such that $\\gamma(0)=p$ and $\\gamma(1)=q$. \nHere $\\Sigma$ is the set of DC-points of $Y^{m}$.\nSince the set of DC-points are discrete, \nthe curve $\\gamma$ passes through only finitely many\nDC-points $y_1,\\dots,y_k$. \nWe can take a connected open subset $U_0(\\subset Y_m)$\nsuch that the image of $\\gamma$ lies in $U_0$\nand there are no DC-points of $U_0$ other than\n$y_1,\\dots,y_k$.\nWe then inductively define $U_j$\n($j=1,...,k$) such that $U_j$ is a blowup of $U_{j-1}$ at $y_j$. \nThen $U_k$ has no DC-points and \nthe canonical projection $\\pi:U_k\\to U_0$ is induced.\nTake points $\\tilde p$ and $\\tilde q$ on $U_k$\nso that $\\pi(\\tilde p)=p$ and\n$\\pi(\\tilde q)=q$. \nSince $U_k$ is a connected real analytic manifold,\nthere exists a real analytic curve $\\sigma:[0,1]\\to U_k$\nsuch that $\\sigma(0)=\\tilde p$ and $\\sigma(1)=\\tilde q$\n(cf. \\cite[the footnote on page 402]{HP}).\nSince $p$ is an interior point of $\\phi(U)$, the image of the curve\n$g\\circ \\pi\\circ \\sigma([0,\\epsilon))$ is\ncontained in $f(X^m)$ for sufficiently small $\\epsilon>0$.\nSince $f$ is analytically complete, we can conclude that\n$$\ng\\circ \\pi\\circ \\sigma(1)=g\\circ \\pi(\\tilde q)=g(q)\n$$\nbelongs to $f(X^m)$,\na contradiction.\n\\end{proof}\n\n\n\\section{Analytic completeness for the image of DC-immersions}\\label{sec:ap}\n\nIn this section, we give a criterion which we can apply \nto showing the analytic completeness of G-catenoids in Section 4. \nWe first prepare the following lemma:\n\n\\begin{Lemma}\\label{lem:X}\nLet $f:X^m\\to N^n$ be a real analytic DC-immersion.\nThen for each $p \\in X^m$, there exist \n\\begin{itemize}\n\\item a neighborhood $U(\\subset X^m)$\nof $p$,\n\\item a neighborhood $\\Omega(\\subset N^n)$ \nof $f(p)$,\nand \n\\item a real anaytic function\n$F:\\Omega\\to {\\mathbf R}^k$ $(k\\ge n-m)$\n\\end{itemize}\nsuch that $Z(F)=f(U)$ and\n$f|_{U}$ is a homeomorphism between\n$U$ and $f(U)$.\n\\end{Lemma}\n\n\\begin{proof}\nWe fix a point $p\\in X^m$ arbitrarily.\nSince $p$ is a DC-regular point,\nthere exist\na DC-coordinate neighborhood $(U,\\phi)$ \nof $p$ and \na real analytic diffeomorphism $\\Phi$ \ndefined on a neighborhood $\\Omega$ of $f(p)$ \ninto a neighborhood $\\Omega_0$ of the origin $\\mb 0$\nof ${\\mathbf R}^{n}$ such that \n\\begin{equation}\\label{eq:phiDC22}\n\\Phi\\circ f\\circ \\phi^{-1}(a)=(a,\\mb 0_{n-m-1})\n\\qquad (a\\in \\phi(U)\\subset \\mc C^m_2).\n\\end{equation}\nWe let\n$\n\\pi:{\\mathbf R}^{m}\\to {\\mathbf R}^{m+1}\n$\nbe the orthogonal projection. Then\nwe have\n\\begin{equation}\\label{eq:phiDC}\n\\Psi\\circ f\\circ \\phi^{-1}(a)=a\n\\qquad (a\\in \\phi(U)\\subset \\mc C^m_2),\n\\end{equation}\nwhere $\\Psi:=\\pi\\circ \\Phi|_{f(U)}$.\nThen this implies\n\\begin{equation}\\label{eq:phiDC3}\n\\Psi\\circ f(b)=\\phi(b)\n\\qquad (b\\in U).\n\\end{equation}\nThus $\\phi^{-1}\\circ \\Psi$ gives the inverse map of $f|_U$.\nSo $f|_U$ is a homeomorphism.\n\nWe next define a map $G:{\\mathbf R}^{n}\\to {\\mathbf R}^{n-m}$ by\n$$\nG (x_{1}, \\ldots , x_{n}) =\n (x^{2}_{1}+\\cdots +x^{2}_{m}-x^{2}_{m+1}, x_{m+2}, \\ldots, x_{n}). \n$$\nThen we have\n$$\n\\Phi \\circ f(U) = \\{ (\\varphi (b), \\mb 0_{n-m-1})\\in {\\mathbf R}^{n} \\, ; \\, b\\in U \\} \n= Z(G) \\cap \\Omega_{0} = Z(G |_{\\Omega_{0}}). \n$$\nSince $\\Psi$ is a diffeomorphism, we have\n$$\nf(U)=\\Phi^{-1}(Z(G|_{\\Omega_{0}}))=Z(G\\circ \\Phi).\n$$\nSo, by setting $F:=G\\circ \\Phi$, we obtain\nthe relation $Z(F)=f(U)$.\n\\end{proof}\n\n\\begin{Lemma}\\label{prop:X}\nLet $f:X^m\\to N^n$ be a real analytic DC-immersion.\nSuppose that $I$ is an open interval of ${\\mathbf R}$ and that \n$\\Gamma:I\\to N^n$ is a real analytic map satisfying \n$\\Gamma(I) \\subset f(X)$.\nThen there exists a continuous map \n$\\gamma:J\\to X^m$ \ndefined on\nan open subinterval $J$ of $I$\nsuch that\n$\\Gamma=f\\circ \\gamma$ on $J$.\n\\end{Lemma}\n\n\\begin{proof}\nWe let $\\{(U_p,\\Omega_p,F_p)\\}_{p\\in X^m}$\nbe a family of triples determined by the previous lemma.\nSince $X^m$ \nsatisfies the second axiom of countability,\nthere exists a sequence $\\{p_j\\}_{j=1}^\\infty$ \nof points in $X^m$ \nsuch that $\\{U_{p_j}\\}_{j=1}^\\infty$ is\nan open covering of $X^m$. \nWe set\n$$\nW_j:=\\Gamma^{-1}(\\Omega_{p_j})\\,\\, (\\subset I)\n$$\nand consider the map\n$$\n\\lambda_j:=F_{p_j}\\circ \\Gamma:W_j\\to {\\mathbf R}^l\\qquad (j=1,2,3,\\dots). \n$$\nWe denote by $Z_j$ the zero set of $\\lambda_j$\non $W_j$. Since $\\Gamma(I) \\subset f(X^m)$\nand $f(X^m)\\subset \\bigcup_{j=1}^\\infty Z(F_{p_j})$,\nwe have\n$\nI=\\bigcup_{j=1}^\\infty Z_j.\n$\nBy Baire's category theorem, there exists a number $j_0$\nsuch that $Z_{j_0}$ has an interior point.\nSo there exists an open subinterval $J$ of $I$\nsuch that $J\\subset Z_{j_0}$, that is,\n$\\gamma(J)\\subset Z(F_{p_{j_0}})=f(U_{p_{j_0}})$.\nSince $f|_{U_p}$ is a homeomorphism,\nwe can set\n$\n\\phi:=(f|_{U_p})^{-1}\\circ \\Gamma,\n$\nand $f\\circ \\phi=\\Gamma$ holds on $J$.\n\\end{proof}\n\n\\begin{Lemma}\\label{prop:Y}\nLet $f:X^m\\to N^n$ be a real analytic DC-immersion.\nSuppose that $I$ is an open interval of ${\\mathbf R}$ and\n$\\gamma_{i}:I\\to X^m$ $(i=1,2)$\nare two real analytic maps so that \n$f\\circ \\gamma_1(t)=f\\circ \\gamma_2(t)$\nfor all $t\\in I$.\nIf $\\gamma_1(0)=\\gamma_2(0)$,\nthen $\\gamma_1$ coincides with $\\gamma_2$\nas maps on the interval $I$.\n\\end{Lemma}\n\n\\begin{proof}\nWe consider the subset $A$ of $I$ given by\n$$\nA:=\\{t\\in I\\,;\\, \\gamma_1(t)=\\gamma_2(t)\\}.\n$$\nSince $\\gamma_1(0)=\\gamma_2(0)$, we have $0\\in A$, that is,\n$A$ is a non-empty set.\nObviously, $A$ is a closed subset.\nWe fix $t_0\\in A$ arbitrarily.\nBy Lemma \\ref{lem:X},\nthere exist a neighborhood $U(\\subset X^m)$\nof $\\gamma(t_0)$, a neighborhood $\\Omega(\\subset N^n)$ \nof $f\\circ \\gamma(t_0)$\nand a real analytic function\n$F:\\Omega\\to {\\mathbf R}^k$ ($k\\ge n-m$)\nsuch that $Z(F)=f(U)$ and\n$f|_{U}$ gives a homeomorphism between\n$U$ and $f(U)$.\nBy the continuity of $\\gamma_2$, \nthere exists $\\delta(>0)$ such that\n$\\gamma_2(t)\\in U$ for $|t-t_0|<\\delta$.\nBy Lemma \\ref{lem:X},\nthe inverse map $g$ of $f|_U$ is defined\nand $\\gamma_1(t)=g^{-1}\\circ f\\circ \\gamma_2(t)$\nholds when $|t-t_0|<\\delta$.\nSo $(t_0-\\delta,t_0+\\delta)$ is a subset of $A$.\nThus $A$ is an open subset of $I$.\nSince $I$ is connected, we have $A=I$, proving the\nassertion.\n\\end{proof}\n\nWe prepare one more lemma:\n\n\\begin{Lemma}\\label{prop:Z}\nLet $f:X^m\\to N^n$ be a real analytic DC-immersion, and let\n$\\Gamma:[a,b]\\to N^n$ $(a0$.\nSince $f|_{U}$ is a homeomorphism,\nthe curve\n$$\n(f|_{U})^{-1}\\circ \\Gamma(t)\\qquad (t\\in (c-\\epsilon,c+\\epsilon))\n$$ \nis a continuous map, which coincides with $\\gamma$\non $(c-\\epsilon,c)$, and so $\\gamma$ can be\nextended on $[a,c+\\epsilon)$.\n\\end{proof}\n\n\nTo define arc-properness, we give the following definition:\n\n\\begin{Def}\nLet $X$ be a topological space,\nand let $f:X\\rightarrow N^n$ \nbe a continuous map.\nA continuous map \n$\\sigma:[0,1)\\to X$ is said to be {\\it $(C^r,f)$-extendable}\nif $f\\circ \\sigma(t)$ can be \nextended to a $C^r$-map defined on $[0,1]$,\nwhere $r$ is a non-negative \ninteger or $\\infty$ or $r=\\omega$\n(the notion $C^\\omega$ means real analyticity). \n\\end{Def}\n\nWe now define the concept \\lq\\lq arc-properness\" as follows:\n\n\\begin{Def}\\label{def:ap}\nLet $X$ be a locally compact Hausdorff space satisfying the\nsecond axiom of countability (as a consequence,\n$X$ is metrizable, affecting convergence of sequences of points on $X$). \nWe let $f:X\\rightarrow N^n$ \nbe a continuous map.\nWe fix $r$ as a non-negative \ninteger or $\\infty$ or $r=\\omega$. \nThen $f$ is called \\textit{$C^r$-arc-proper}\nif \nfor each $(C^r,f)$-extendable\ncontinuous curve $\\sigma:[0,1) \\rightarrow X$, \nthere exists a sequence $\\{t_k\\}_{k=1}^\\infty$ \non $[0,1)$\nconverging to $1$ such that the limit\n$\n\\lim_{k\\to \\infty}\\sigma(t_k)\n$\nexists.\n\\end{Def}\n\n\n\\begin{Prop}\\label{prop:P}\nProperness implies $C^r$-arc-properness\nfor $r\\in \\{0,1,\\dots\\}\\cup \\{\\infty, \\omega\\}$. \n\\end{Prop}\n\n\\begin{proof}\nLet \n$f:X\\rightarrow N^n$ be a proper \ncontinuous map. We fix a \n$(C^r,f)$-extendable\ncontinuous curve $\\sigma:[0,1) \\rightarrow X$. \nThen $f\\circ \\sigma(t)$ can be \nextended to a $C^r$-curve $\\Gamma:[0,1]\\to N^n$.\nSince $K:=\\Gamma([0,1])$ is compact,\nthe assumption that $f$ is proper implies\n$f^{-1}(K)$\nis compact.\nWe let $\\{t_k\\}_{k=1}^\\infty$ be a sequence on\n$[0,1)$ converging to $1$.\nSince $f^{-1}(K)$ is compact, \nthe sequence \n$\\{\\sigma(t_k)\\}_{k=1}^\\infty$ \nhas a convergent subsequence. \n\\end{proof}\n\n\\begin{Example}\\label{ex:T-comp}\nWe consider the image $f({\\mathbf R})$ of the real analytic immersion \n\\begin{equation}\\label{eq:T}\n f:{\\mathbf R}\\ni t\\mapsto (t, \\sqrt{2}t)\\in T^2\n :={\\mathbf R}^2\/{\\mathbf Z}^2.\n\\end{equation}\nAlthough $f$ is not a proper map,\nas will be seen below,\n$f$ is $C^0$-arc-proper.\nLet $\\Gamma:[0,1]\\to T^2$ be a \ncontinuous map satisfying $\\Gamma([0,1))\\subset f({\\mathbf R})$. \nWe denote by\n $\\pi:{\\mathbf R}^2\\to T^2$ the canonical covering projection.\nThen by the homotopy lifting property on covering spaces,\nthere exists a continuous map\n$\n\\tilde \\Gamma:I\\to {\\mathbf R}^2\n$\nsuch that $\\pi\\circ \\tilde \\Gamma=\\Gamma$ on $I$\nand \n$$\n\\tilde \\Gamma(0)\\in A,\\quad\nA:=\\{(x,y)\\in {\\mathbf R}^2\\,;\\, y=\\sqrt{2}x\\}.\n$$\nSince $A$ is a closed subset of ${\\mathbf R}^2$,\nwe have $\\tilde\\Gamma(1)\\in A$\nand so\n$$\n\\Gamma(1)=\\pi\\circ \\tilde\\Gamma(1)\\in \\pi(A)=f({\\mathbf R}).\n$$\nThus, $f$ is $C^0$-arc-proper. \n\\end{Example}\n\n\\begin{Exa}\\label{ex:Kok}\nConsider the analytic curve\n$\nf(t):=(t,e^{-1\/t})\\,\\, (t>0)\n$\nin the $xy$-plane.\nAs will be shown below, $f$ is a $C^\\omega$-arc-proper immersion.\nWe set $S:=f((0,\\infty))$.\nLet $\\sigma:[0,1) \\rightarrow (0,\\infty)$ be\na $(C^\\omega,f)$-extendable continuous curve. \nThen there exists a real analytic curve $\\Gamma:[0,1]\\to {\\mathbf R}^2$\nsuch that $\\Gamma(s)=f\\circ \\sigma(s)$ holds for $s\\in [0,1)$.\nWe can write\n$$\n\\Gamma(s)=(x(s),y(s)) \\qquad (s\\in [0,1]).\n$$\nSuppose that $\\Gamma(1)\\not \\in S$.\nSince $\\overline{S}\\setminus S=\\{(0,0)\\}$,\nwe have\n$\n\\Gamma(1)=(0,0).\n$\nMoreover, since $\\Gamma([0,1))\\subset S$,\nwe have\n$$\ny(s)=e^{-1\/x(s)},\\quad x(s)>0 \\qquad (s\\in [0,1)).\n$$\nThen one can inductively check that \n$$\n\\lim_{s\\to 1-0}\\frac{d^k y}{ds^k}(s)=0\\qquad (k=0,1,2,\\dots)\n$$\nholds. \nThis contradicts the fact that $y(s)$ is\na real analytic function at $s=1$.\nThus, we have $\\Gamma(1)\\in S$.\nThen $x(1)>0$ and\n$$\n(x(s),y(s))=\\Gamma(s)=f\\circ \\sigma(s)=(\\sigma(s),e^{-1\/\\sigma(s)})\n$$\nhold. In particular,\t\n$\\sigma(s)=x(s)$ converges to $x(1)$ as $s$ tends to $1$.\nThus, $f$ is a $C^\\omega$-arc proper map.\n\nBy the reflection with respect to\nthe $y$-axis, the map $f$ can be\nextended as a $C^\\infty$-map defined by\n$$\nF(t):=\n\\begin{cases}\nf(t)\n&\\mbox{($t>0$)},\\\\\n(0,0) \\qquad & \\mbox{($t=0$)},\\\\\n(t,e^{1\/t})&\\mbox{($t<0$)}.\n\\end{cases} \n$$\nThen the curve $F(t)$ itself is a smooth curve on\n${\\mathbf R}^2$, and $\\sigma(t):=t$ ($t>0$) satisfies\n$f\\circ \\sigma(t)=F(t)$ for $t>0$.\nSince $\\lim_{t\\to 0}\\sigma(0)=0$ holds and\nthe origin $0$ does not\nbelong to $ (0,\\infty) $,\n$f$ is not $C^\\infty$-arc proper.\nSo $f$ gives an example which is\nnot $C^\\infty$-arc-proper \nbut is $C^\\omega$-arc-proper.\n\\end{Exa}\n\n\\begin{Thm}\\label{thm:first}\nLet $f:X^m\\to N^n$ be a real analytic DC-immersion which is\n$C^\\omega$-arc-proper as a continuous map. \nThen $f(X^m)$ is analytically complete.\nIn particular, if $f$ is a proper map $($cf. Proposition \\ref{prop:P}$)$, \nthen $f(X^m)$ is analytically complete.\n\\end{Thm}\n\nWe will apply this to prove the analytic completeness of\nseveral $G$-catenoids in Section \\ref{app:Gcatenoids}.\n\n\\begin{proof}\nLet $\\Gamma : [0, 1] \\to N^{n}$ be an analytic curve such\nthat $\\Gamma ([0, \\varepsilon ))\\subset f(X^{m})$ \nfor some $\\varepsilon \\in (0, 1]$. It is sufficient to show that \n$\\Gamma (1) \\in f(X^{m})$.\n\nGiven $0\\leq a< b\\leq 1$, a continuous curve \n$\\sigma : [a, b) \\to X^{m}$ is called a {\\it lift} of $\\Gamma$ \non the interval [a, b) if \n$f\\circ \\sigma (t) = \\Gamma (t)$ for all $t\\in [a, b)$. \nBy Lemma \\ref{prop:Y}, if two lifts ${\\sigma}_{1}$ \nand $\\sigma_{2}$ of $\\Gamma$ on $[a, b)$ satisfy \n$\\sigma_{1} (a) = \\sigma_{2} (a)$, \nthen $\\sigma_{1} (t)= \\sigma_{2} (t)$ for all $t\\in [a, b)$. \n\nBy Lemma \\ref{prop:X}, there exists $t_{0}$, $c_{0}$ with \n$0< t_{0}< c_{0}< \\varepsilon $ and a lift \n$\\sigma_{0}: [t_{0}, c_{0}) \\to X^{m}$ of $\\Gamma$ on $[t_{0}, c_{0})$.\nNow let \n$$\n\\mc C :=\\{ c_{1}\\in (t_{0}, 1] \\, ; \\, \\text{There exists a lift $\\sigma_{1}$ \nof $\\Gamma$ on $[t_{0}, c_{1})$ with \n$\\sigma_{1} (t_{0}) = \\sigma_{0} (t_{0})$.}\\}. \n$$\n$\\mc C$ is nonempty since $c_{0} \\in \\mc C$. Let $c$ be the supremum \nof $\\mc C$. \nThen in fact $c\\in \\mc C$. \nLet $\\sigma : [t_{0}, c) \\to X^{m}$ be the corresponding (maximal) \nlift of $\\Gamma$ on $[t_{0}, c)$. \n\nSince $f$ is $C^{\\omega}$-arc-proper, there exists a sequence \n$\\{ t_{k} \\}_{k=1}^{\\infty}$ in $[t_{0}, c)$ converging \nto $c$ such that $\\lim_{k\\to \\infty}\\sigma(t_k)$\nexists. If $c< 1$, then by Lemma \\ref{prop:Z}, $\\sigma$ \ncan be continuously \nextended on $[t_{0}, c+{\\varepsilon}_{1} )$ \nfor some ${\\varepsilon}_{1} >0$ such that $f\\circ \\sigma = \\Gamma$, \nwhich contradicts the maximality of $c$. \nSo $c=1$. Then we have \n$$\n\\Gamma(1)=\\lim_{k\\to \\infty}\\Gamma(t_k)\n=f(\\lim_{k\\to \\infty}\\sigma(t_k))\\in f(X^m),\n$$\nproving the assertion.\n\\end{proof}\n\nAs a consequence,\nwe get the following: \n\n\\begin{Corollary}\\label{cor:deep}\n{\\it \nLet $X^m$ be a connected real analytic manifold\nand $f:X^m\\to N^n$ a real analytic immersion.\nIf $f$ is a $C^\\omega$-arc-proper map, \nthen $f(X^m)$ is analytically complete.}\n\\end{Corollary}\n\nMoreover, the following assertion holds:\n\n\\begin{Corollary}\\label{cor:deep2}\n{\\it \nLet $X^m$ be a connected real analytic DC-manifold of dimension $m$\nand $f:X^m\\to N^n$ a real analytic DC-immersion.\nSuppose that $N^n$ is a properly $C^\\omega$-embedded in\na real analytic manifold $\\tilde N^k$ $(k\\ge n)$, and\ndenote by $\\iota:N^n\\hookrightarrow \\tilde N^k$ the inclusion map.\nThen $f(X^m)$ is analytically complete in $N^n$\nif and only if it is analytically complete in $\\tilde N^k$.}\n\\end{Corollary}\n\n\\begin{proof}\nThe \\lq\\lq if \" part is obvious.\nWe suppose that $f(X^m)$ is an analytically complete in $N^n$.\nWe consider a real analytic curve\n$\n\\Gamma:[0,1]\\to \\tilde N^k\n$\nsuch that $\\Gamma ([0,\\epsilon) )\\subset f(X^m)$\nfor sufficiently small $\\epsilon>0$.\nThen we have\n$$\n\\Gamma ([0,\\epsilon)) \\subset f(X^m)\\subset N^n.\n$$\nSince $N^n$ is analytically complete in $\\tilde N^k$,\nwe can conclude $\\Gamma([0,1])\\subset N^n$.\nSince $f(X^m)$ is an analytically complete subset of $N^n$,\nwe can also conclude $\\Gamma([0,1])\\subset f(X^m)$.\nThus $f(X^m)$ is analytically complete in $\\tilde N^k$.\n\\end{proof}\n\n\n\\section{Analytic extensions of catenoids in $S^3_1$}\n\\label{app:Gcatenoids}\n\nIn this section, we introduce a class of \n\\lq\\lq real analytic space-like constant mean curvature one DC-immersions\\rq\\rq\\ \nand show that every G-catenoid or its appropriate analytic \nextension in the de Sitter 3-space $S^3_1$ \nbelongs to this class.\nWe will further show that all of these maps satisfy one of \nthe criteria for analytic completeness provided so far, \nhence prove that their images are analytically complete. \n\n\\subsection{Space-like CMC-1 DC-immersions}\n\nWe give the following definition.\n\n\\begin{Definition}\\label{def:CMC-1}\nLet $f:X^2\\to S^3_1$ be a\nreal analytic DC-immersion \ndefined on a connected $2$-dimensional real analytic\nDC-manifold $X^2$.\nThen \n$f$ is called \na {\\it real analytic space-like CMC-1 DC-immersion\nif\nthe restriction $f|_{X^2\\setminus \\Sigma}$\nis a space-like CMC-1 immersion to $S^3_1$,\nwhere $\\Sigma$ is the set of DC-points in $X^2$.}\n\\end{Definition}\n\nSimilarly, real analytic ZMC DC-maps\nin ${\\mathbf R}^3_1$ can be defined. See Appendix A for details.\nThe following assertion holds.\n\n\\begin{Proposition}\\label{prop:AB-1}\nLet $f:X^2\\to S^3_1$ \nbe a real analytic DC-immersion\ndefined on a connected $2$-dimensional real analytic\nDC-manifold $X^2$.\nSuppose that there exist a point $p$ and a non-empty open subset\n$U$ of $p$ such that the restriction\n$f|_U:U\\to S^3_1$ gives a space-like CMC-1 immersion.\nThen $f$ is a real analytic space-like CMC-1 \nDC-immersion.\n\\end{Proposition}\n\n\\begin{proof}\nWe let $\\Sigma$ be the set of DC points in $X^2$.\nSince $\\Sigma$ consists of isolated points,\n$X^2\\setminus \\Sigma$ is a connected $2$-manifold.\nWe fix a point $q\\in X^2\\setminus \\Sigma$.\nThen there exists a continuous curve\n$\\gamma:[0,1]\\to X^2\\setminus \\Sigma$ such that\n$\\gamma(0)\\in U$ and $\\gamma(1)=p$.\nSince $\\gamma([0,1])$ is compact,\nwe can take a partition\n$$\n0=t_00$) such that \n\\begin{itemize}\n\\item $\\sigma(0)=p$,\n\\item $\\sigma(t)$ ($|t|<\\epsilon$) are light-like points of $f$, and\n\\item $f\\circ \\sigma$ is a light-like geodesic segment in $S^3_1$.\n\\end{itemize}\nThis is the reason why the following analytic extensions of\nCMC-1 catenoids in $S^3_1$ often contain light-like geodesic segments.\n\\end{Remark}\n\n\\subsection{The classification of G-catenoids}\n\nA \\emph{geometric catenoid} (or a \\emph{G-catenoid} for short)\nis a space-like CMC-1 surface in de Sitter $3$-space $S^3_1$ \nwhich is invariant\nunder the action of a one-parameter subgroup $G$ of isometries \nof $S^3_1$ with a common fixed point.\nSuch a subgroup $G$ fixes a geodesic, which is called the \\emph{axis}\nof the subgroup $G$.\nWe say that a G-catenoid is of \\emph{type T} (resp.\\ \\emph{L}, \\emph{S})\nif the axis is a time-like (resp.\\ light-like, space-like) geodesic.\nThe analytic part of a G-catenoid can be considered as a CMC-1 face\nin the sense of \\cite{FKKRUY2}, that is, it can be expressed as \n\\[\n f = (x_0,x_1,x_2,x_3),\\qquad \\begin{pmatrix}\n\t\t\t\t x_0+x_3 & x_1+\\op{i} x_2 \\\\\n\t\t\t\t x_1-\\op{i} x_2 & x_0-x_3\n\t\t\t\t \\end{pmatrix} = F\\begin{pmatrix}\n\t\t\t\t\t\t 1 & \\hphantom{-}0 \\\\\n\t\t\t\t\t\t 0 & -1\n\t\t\t\t\t\t \\end{pmatrix}\n \t\t\t\t\t F^*,\n\\]\nwhere $F$ is a holomorphic immersion into $\\op{SL}(2,{\\mathbf C})$\ndefined on a certain Riemann surface.\nSuch an $F$ is called the \\emph{holomorphic lift} of the surface\n(cf.\\ \\cite{FKKRUY2} for details).\nG-catenoids were completely classified \nby the last author \\cite{Y}\nas in Table~\\ref{tab:G-catenoids}.\n(A surface of type LE (resp.\\ LH)\n with holomorphic lift $F_L(\\alpha,z)$ is congruent\n to $F_L(1,z)$ (resp.\\ $F_L(-1,z)$).)\n\n\\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c||c|c|c|}\n \\hline\n axis & E-type & P-type & H-type\\\\ \\hline \\hline\n \\begin{minipage}[t]{1.6cm}\n time-like \\\\\n (type T)\n \\end{minipage}\n &\n (TE)\n \\begin{minipage}[t]{2.3cm}\n $F_T\\left(\\frac{1-\\mu^2\\mathstrut}{2},z\\right)$\\newline\n \\hfill $\\bigl(\\mu\\in{\\mathbf R}_+\\setminus\\{1\\}\\bigr)$\n \\end{minipage}\n & (TP) $F_T(0,z)$ \n & (TH) \n \\begin{minipage}[t]{2.3cm}\n $F_T\\left(\\frac{1-\\mu^2\\mathstrut}{2},z\\right)$\\newline\n \\hfill $\\bigl(\\mu\\in\\op{i}{\\mathbf R}_+\\bigr)$\n \\end{minipage}\\\\\n \\hline\n \\begin{minipage}[t]{1.6cm}\n light-like \\\\\n (type L)\n \\end{minipage}&\n (LE) \n \\begin{minipage}[t]{2.3cm}\n $F_L\\bigl(\\alpha, z\\bigr)$\\newline\n \\hfill $\\bigl(\\alpha>0\\bigr)$\n \\end{minipage}& ---\n & (LH) \n \\begin{minipage}[t]{2.3cm}\n $F_L\\bigl(\\alpha, z\\bigr)$\\newline\n \\hfill $\\bigl(\\alpha<0\\bigr)$\n \\end{minipage}\n \\\\ \\hline\n \\begin{minipage}[t]{1.6cm}\n space-like \\\\\n (type S)\n \\end{minipage}\n &\n (SE)\n \\begin{minipage}[t]{2.3cm}\n $F_S\\left(\\frac{1-\\mu^2\\mathstrut}{2},z\\right)$\\newline\n \\hfill $\\bigl(\\mu\\in{\\mathbf R}\\bigr)$\n \\end{minipage}\n & (SP) $F_S(0,z)$ \n & (SH) \n \\begin{minipage}[t]{2.3cm}\n $F_S\\left(\\frac{1-\\mu^2\\mathstrut}{2},z\\right)$\\newline\n \\hfill $\\bigl(\\mu\\in\\op{i}{\\mathbf R}_+\\bigr)$\n \\end{minipage}\\\\\n \\hline\n \\end{tabular}\n \\end{center}\n \\caption{The classification of G-catenoids in \\cite{Y}.\n $F_T$, $F_L$ and $F_S$ denote the holomorphic lifts\n of surfaces, which are defined in \n \\cite[Lemma 4.2, Section 4]{Y}.}\\label{tab:G-catenoids}\n\\end{table}\n\nFrom now on, we show that each $G$-catenoid \nis analytically complete in $S^3_1$ or\nits suitable analytic extension is analytically complete in $S^3_1$.\nSince $S^3_1$ is a properly embedded real analytic submanifold\nof ${\\mathbf R}^4$, all those surfaces are also analytically complete in ${\\mathbf R}^4$\n(cf. Corollary \\ref{cor:deep2}).\n\n\n\\subsection{G-catenoids of type T}\n\n\\subsubsection{G-catenoids of type TE}\nAs mentioned in the introduction,\nthere are three subclasses of G-catenoids of type T.\nIn this subsection, we show that the images of all of \nthem are analytically complete. \n\nWe set $T^1:={\\mathbf R}\/2\\pi {\\mathbf Z}$.\nFor each \n$\\mu\\in {\\mathbf R}_+ \\setminus \\{1\\}$,\nwe consider a real analytic map\n$$\n f_{\\op{TE}}^\\mu \n\\colon{} {\\mathbf R}\\times T^1 \\ni\n(s,\\theta) \\mapsto (x_0(s),x_1(s,\\theta),x_2(s,\\theta),x_3(s))\\in \nS^3_1,\n$$\nwhich \ngives a G-catenoid of type {\\rm TE}, \nwhere\n\\begin{align*}\nf^{\\mu}_{\\mathrm{TE}}(s, \\theta) =\n \\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\ 0 & \\cos \\theta & -\\sin \\theta & 0 \\\\ \n 0 & \\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 0 & 1\n \\end{pmatrix}\n\\Gamma^{\\mu}_{\\mathrm{TE}}(s),\\quad\n\\Gamma^{\\mu}_{\\mathrm{TE}}(s):=\\begin{pmatrix}\n x_0(s) \\\\ \n\\frac{1- \\mu^2}{2 \\mu} \\sinh \\mu s \\\\ \n0 \\\\ \nx_3(s) \n\\end{pmatrix}, \n\\end{align*}\nand\n\\begin{equation}\\label{eq:x0x3-TE} \n\\begin{aligned}\nx_0(s)&=\\sinh s \\cosh \\mu s-\\frac{\\left(\\mu^2+1\\right) \\cosh s \\sinh \\mu s}{2 \\mu},\\\\\nx_3(s)&=\\cosh s\\cosh \\mu s-\\frac{\\left(\\mu^2+1\\right) \\sinh s \\sinh \\mu s}{2 \\mu}.\n \\end{aligned}\n\\end{equation}\nHere $\\Gamma_{\\mathrm{TE}}^\\mu(s)$ ($s\\in {\\mathbf R}$) is a\nprofile curve of $f^{\\mu}_{\\mathrm{TE}}$\nlying on $S^2_1 = S^3_1 \\cap \\{ x_2=0 \\}$. \nThe axes of $f_{\\op{TE}}^\\mu$ \nare $\\{(\\sinh t,0,0,\\pm\\cosh t)\\;;\\;t\\in{\\mathbf R}\\}$.\nThe singular point set of $f_{\\op{TE}}^\\mu$ \nis $\\{(0,\\theta)\\in {\\mathbf R}\\times T^1\\}$,\nwhose image consists of one point $\\{(0,0,0,1)\\}$\n(see Figure \\ref{fig:typeTE}).\n\n\\begin{figure}[hbt]%\n \\begin{center}\n \\includegraphics[width=3.5cm]{desittercat-TE1.eps} \\hskip 2cm\n \\includegraphics[width=3.5cm]{desittercat-TE2.eps} \n\\end{center}\n \\caption{G-catenoids of type TE with $0<\\mu<1$ (left) and $\\mu>1$ (right)}%\n\\label{fig:typeTE}\n\\end{figure}\n\nThe following assertion can be checked easily:\n\n\\begin{Lemma}\\label{lem:5-1}\nFor each \n$\\mu\\in {\\mathbf R}_+ \\setminus \\{1\\}$,\nthe plane curve $s\\mapsto (x_0(s),x_3(s))$ is a regular curve on ${\\mathbf R}$.\n\\end{Lemma}\n\n\\begin{proof}\nIn fact, we have\n$$\n\\pmt{x_0'(s) \\\\ x'_3(s)}=\n\\frac{\\mu^2-1}{2\\mu}\n\\pmt{ \\mu \\cosh s & \\sinh s \\\\\n \\mu \\sinh s &\\cosh s \n}\n\\pmt{\n -\\mu \\cosh \\mu s \\\\\n \\sinh \\mu s\n},\n$$\nwhich implies the assertion.\n\\end{proof}\n\n\\begin{Proposition}\nThe map\n$f_{\\op{TE}}^\\mu$ is a real analytic space-like CMC-1\n DC-immersion, and\nits image is analytically complete.\n\\end{Proposition}\n\nNote that $f_{\\op{TE}}^\\mu$ is a W-catenoid with \nelliptic monodromy as in \\cite{FKKRUY2}.\n\n\\begin{proof}\nWe set\n$$\n\\mc X_{\\op{TE}}^\\mu:=\n\\left\n\\{(\\xi,\\eta,s)\\in {\\mathbf R}^3\\,;\\, \n\\xi^2+\\eta^2=\\left(\\frac{1-\\mu^2}{2\\mu}\\right)^2\\sinh^2\n\\mu s\n\\right\\},\n$$\nwhere $\\mu\\in {\\mathbf R}_+\\setminus \\{1\\}$.\nIt can be easily checked that\n$\\mc X_{\\op{TE}}^\\mu$ has the structure of \na $2$-dimensional real analytic DC-manifold \nso that $(0,0,0)\\in \\mc X_{\\op{TE}}^\\mu$ \nis a DC-point. On the other hand, \nusing the functions \\eqref{eq:x0x3-TE},\nconsider the continuous map\n$$\n\\tilde f_{\\op{TE}}^\\mu:\\mc X_{\\op{TE}}^\\mu\\ni (\\xi,\\eta,s)\n\\mapsto (x_0(s),\\xi,\\eta,x_3(s))\\in S^3_1, \n$$\nwhich is obviously a real analytic map from\nthe DC-manifold $\\mc X_{\\op{TE}}^\\mu$.\nBy Lemma \\ref{lem:5-1},\nthe map\n$$\n\\Psi:{\\mathbf R}^3 \\ni (\\xi,\\eta,s)\n\\mapsto (x_0(s),\\xi,\\eta,x_3(s))\\in S^3_1\n$$\nis an immersion.\nSince the inclusion map $\\iota:\\mc X_{\\op{TE}}^\\mu\\hookrightarrow {\\mathbf R}^3$\nis a real analytic DC-immersion, the composition\n$\\tilde f_{\\op{TE}}=\\Psi\\circ \\iota$\nis also a real analytic DC-immersion.\nSince\n$$\nf_{\\op{TE}}^\\mu(s,\\theta)=\\tilde f_{\\op{TE}}^\\mu(x_1(s,\\theta),x_2(s,\\theta),s),\n$$\nthe image of $\\tilde f_{\\op{TE}}^\\mu$ coincides with that of $f_{\\op{TE}}^\\mu$.\nThe map $\\tilde f_{\\op{TE}}^\\mu$ is \na real analytic space-like CMC-1\nDC-immersion. \nIt can be easily checked that $\\tilde f_{\\op{TE}}^\\mu$ is a proper map.\nSo, Theorem \\ref{thm:first}\nyields that\nthe image of $\\tilde f_{\\op{TE}}^\\mu$ is analytically complete.\n\\end{proof}\n\n\\subsubsection{G-catenoids of type TP}\nA G-catenoid of type {\\rm TP} \ncan be expressed as\n \\begin{align*}\n f_{\\op{TP}}\n \\colon{} {\\mathbf R}\\times T^1\\ni\n(s,\\theta) \\mapsto (x_0(s),x_1(s,\\theta),x_2(s,\\theta),x_3(s))\\in S^3_1,\n \\end{align*}\nwhere\n\\begin{align*}\nf_{\\op{TP}}(s, \\theta)=\n \\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\ 0 & \\cos \\theta & -\\sin \\theta & 0 \\\\ \n 0 & \\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 0 & 1\n \\end{pmatrix}\n\\Gamma_{\\op{TP}}(s), \\quad \n\\Gamma_{\\op{TP}}(s) :=\n\\begin{pmatrix}\nx_0(s) \\\\ \ns\/2 \\\\ \n0 \\\\ \nx_3(s) \n\\end{pmatrix}, \n\\end{align*}\nand\n\\begin{equation}\\label{eq:x0x3-TP}\n x_0(s)=\\sinh s - \\frac{s}{2} \\cosh s,\\qquad\nx_3(s)=\\cosh s - \\frac{s}{2} \\sinh s. \n\\end{equation}\nHere\n$\\Gamma_{\\op{TP}}(s)$ ($s\\in {\\mathbf R}$) is the profile curve\nof $f_{\\op{TP}}$ lying on $S^2_1 = S^3_1 \\cap \\{ x_2=0 \\}$.\nThe axes of $f_{\\op{TP}}$ are the same as the axes of $f_{\\op{TE}}^\\mu$.\nThe singular point set of $f_{\\op{TP}}$ is \n$\t \\{(0,\\theta)\\in {\\mathbf R}\\times T^1\\}$,\nwhose image consists of one point $\\{(0,0,0,1)\\}$.\nNote that $f_{\\op{TP}}$ is the limit of $f_{\\op{TE}}^\\mu$ as $\\mu \\to 0$. \nWe set\n$$\n\\mc X_{\\op{TP}}:=\n\\left\\{\n(\\xi,\\eta,s)\\in {\\mathbf R}^3\\,;\\, \\xi^2+\\eta^2=\\frac{s^2}{4}\n\\right\\}.\n$$\nObviously, $\\mc X_{\\op{TP}}$ has the structure of a \nreal analytic $2$-dimensional DC-manifold\nsuch that $(0,0,0)$ is a DC-point.\nUsing the functions \\eqref{eq:x0x3-TP},\nconsider the real analytic map defined on\nthe DC-manifold by\n$$\n\\tilde f_{\\op{TP}}:\\mc X_{\\op{TP}}\\ni (\\xi,\\eta,s)\n\\mapsto (x_0(s),\\xi,\\eta,x_3(s))\\in S^3_1. \n$$\nSince\n$\nf_{\\op{TP}}(s,\\theta)=\\tilde f_{\\op{TP}}(x_1(s,\\theta),x_2(s,\\theta),s),\n$\nthe image of $\\tilde f_{\\op{TP}}$ coincides with that \nof $f_{\\op{TP}}$. \nIt can be easily checked that \n${\\mathbf R}\\ni s\\mapsto (x_0(s),x_3(s))\\in {\\mathbf R}^2$ is a \nregular curve.\nSo \n$$\n\\Psi:{\\mathbf R}^3\\ni (\\xi,\\eta,s)\n\\mapsto (x_0(s),\\xi,\\eta,x_3(s))\\in S^3_1\n$$\nis an immersion.\nSince the inclusion map $\\iota:\\mc X_{\\op{TP}}\\hookrightarrow {\\mathbf R}^3$\nis a real analytic DC-immersion, the composition\n$\\tilde f_{\\op{TP}}=\\Psi\\circ \\iota$\nis also a real analytic DC-immersion.\nOn the other hand, it is easily checked that\n$\\tilde f_{\\op{TP}}$ is a proper map.\nApplying\nTheorem \\ref{thm:first}\nyields the following:\n\n\\begin{Proposition}\nThe map\n$f_{\\op{TP}}$ is a \nreal analytic space-like CMC-1 DC-immersion, and\nits image is analytically complete.\n\\end{Proposition}\n\nNote that $f_{\\op{TP}}$ is\nan W-catenoid with parabolic monodromy as in \\cite{FKKRUY2}.\n\n\\subsubsection{G-catenoids of type TH}\nFor each \n$\\nu\\in {\\mathbf R}_+ \\setminus \\{1\\}$,\nwe consider the following real analytic map\n\\begin{equation*}\n f_{\\op{TH}}^\\nu \\colon{} {\\mathbf R}\\times T^1\n\\ni (s,\\theta) \\mapsto (x_0(s),x_1(s, \\theta),x_2(s, \\theta),x_3(s))\n \\in S^3_1\n\\end{equation*}\nwhich gives a G-catenoid of type TH, where\n\\begin{align*}\nf^{\\nu}_{\\op{TH}}(s, \\theta):=\n \\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\ 0 & \\cos \\theta & -\\sin \\theta & 0 \\\\ \n 0 & \\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 0 & 1\n \\end{pmatrix}\n\\Gamma^{\\nu}_{\\op{TH}}(s), \\quad \n\\Gamma^{\\nu}_{\\op{TH}}(s) :=\n\\begin{pmatrix}\nx_0(s) \\\\ \n\\frac{\\nu^2+1}{2 \\nu} \\cos s \\\\ \n0 \\\\ \nx_3(s) \n\\end{pmatrix},\n\\end{align*}\nand\n\\begin{equation}\\label{eq:x0x3-TH}\n\\begin{aligned}\nx_0 (s) &:=\\sin s \\sinh \\left(s\/\\nu\\right)\n-\\frac{\\left(\\nu^2-1\\right) \\cos s \\cosh \\left(s\/\\nu\\right)}{2 \\nu}, \\\\\nx_3 (s) &:=\\sin s \\cosh \\left(s\/\\nu\\right)\n-\\frac{\\left(\\nu^2-1\\right) \\cos s \\sinh \\left(s\/\\nu\\right)}{2 \\nu}.\n \\end{aligned}\n\\end{equation}\nHere $\\Gamma^\\nu_{\\op{TH}}(s)$ ($s\\in {\\mathbf R}$) is a profile curve\nof $f^\\nu_{\\op{TH}}$\nlying on $S^2_1 = S^3_1 \\cap \\{ x_2=0 \\}$.\nThe axes of $f_{\\op{TH}}^\\nu$ are the same as the axes of $f_{\\op{TE}}^\\mu$.\n\nThe singular point set of $f_{\\op{TH}}^\\nu$\nis given by\n \\[\n\t \\Sigma:=\\bigcup_{k\\in{\\mathbf Z}}\\Sigma_k,\\qquad\n \\Sigma_k := \\left\\{\n \\left(\\left(\\frac{1}{2}+k\\right)\\pi,\\theta\\right)\\,;\n \\theta\\in T^1\n \\right\\},\n \\]\n and the image of each connected component $\\Sigma_k$ consists\n of one point\n \\[\n \\left(\n (-1)^{k+1}\\sinh\\left(\\left(\\frac{1}{2}+k\\right)\\frac{\\pi}{\\nu}\\right),\n0, 0,\n\t (-1)^{k+1}\\cosh\\left(\\left(\\frac{1}{2}+k\\right)\n\\frac{\\pi}{\\nu}\\right)\\right).\n \\]\nUnlike the case of $f_{\\op{TE}}^\\mu$ and $f_{\\op{TP}}$,\nthe map $f_{\\op{TH}}^\\nu\\colon{}{\\mathbf R}\\times T^1\\to S^3_1$ is not\na proper map. In fact, the image of the map accumulates to the set\n \\[\n\t \\mc L = \\mc L_+\\cup \\mc L_-, \\qquad \n \\mc L_\\pm := \\{(x_0,x_1,x_2,x_3)\\in S^3_1\\,;\\, x_0\\mp x_3=0\\}.\n \\]\nFigures for the image of $f_{\\op{TH}}^\\nu$ are given\nin \\cite[Fig. 3 (page 35)]{FKKRUY2}.\nWe set\n$$\n\\mc X_{\\op{TH}}^\\nu:=\n\\left\\{\n(\\xi,\\eta,s)\\in {\\mathbf R}^2\\times T^1\\,;\\, \\xi^2+\\eta^2=\\frac{(1+\\nu^2)^2\\cos^2 s}{4\\nu^2}\n\\right\\},\n$$\nwhich has the structure of a real analytic \n$2$-dimensional DC-manifold whose DC-points are\n $(0,0,\\pi\/2+\\pi k)$ \n($k\\in {\\mathbf Z}$).\nUsing the functions \\eqref{eq:x0x3-TH}, \ndefine the following map on the DC-manifold \n$\\mc X_{\\op{TH}}^\\nu$:\n$$\n\\tilde f_{\\op{TH}}^\\nu:\\mc X_{\\op{TH}}^\\nu\\ni (\\xi,\\eta,s)\n\\mapsto (x_0(s),\\xi,\\eta,x_3(s))\\in S^3_1.\n$$\n\n\\begin{Lemma}\\label{lem:5-2}\nFor each $\\nu\\in{\\mathbf R}_+\\setminus\\{1\\}$, \nthe plane curve $s\\mapsto (x_0(s),x_3(s))$ is a regular curve on ${\\mathbf R}$.\n\\end{Lemma}\n\n\\begin{proof}\nIn fact, the assertion follows from the following identity\n$$\n\\pmt{x_0'(s) \\\\ x'_3(s)}=\n\\frac{\\nu^2+1}{2\\nu^2}\n\\pmt{ \\sinh(s\/\\nu) & \\cosh(s\/\\nu) \\\\\n \\cosh(s\/\\nu) & \\sinh(s\/\\nu) \n}\n\\pmt{\n \\cos s \\\\\n \\nu\\sin s\n}.\n$$\n\\end{proof}\n\nBy this lemma, $\\tilde f_{\\op{TH}}^\\nu$\nis a real analytic real analytic DC-immersion. \nSince\n$$\nf_{\\op{TH}}^\\nu(s,\\theta)\n=\\tilde f_{\\op{TH}}^\\nu(x_1(s,\\theta),x_2(s,\\theta),s),\n$$\nthe image of $\\tilde f_{\\op{TH}}^\\nu$ coincides \nwith that of $f_{\\op{TH}}^\\nu$.\nWe can prove the following:\n\n\\begin{Proposition}\nThe map\n$f_{\\op{TH}}^\\nu$ is a \nreal analytic space-like CMC-1 DC-immersion,\nand its image is analytically complete.\n\\end{Proposition}\n\n\\begin{proof}\nAlthough $\\tilde f_{\\op{TH}}^\\nu$ is not a proper map,\n$\\tilde f_{\\op{TH}}^\\nu$ is a $C^0$-arc-proper map:\nIn fact, let $\\hat\\gamma:[0,1]\\to S^3_1$ be a continuous\nmap, and suppose that\n$\n\\gamma(t)=(s(t),\\xi(t),\\eta(t))\n$\n($t\\in [0,1)$) satisfies \n$\\hat\\gamma(t)=\\tilde f_{\\op{TH}}^\\nu\\circ \\gamma(t)$ for each\n$t\\in [0,1)$.\nSince the second and the third components of\n$\\hat\\gamma(t)$ are $\\xi(t),\\eta(t)$,\nthe limits $\\lim_{t\\to 1-0}\\xi(t)$ and $\\lim_{t\\to 1-0}\\eta(t)$\nexist. Since\n$$\n\\xi(t)^2+\\eta(t)^2=\\frac{(1+\\nu^2)^2\\cos^2 s(t)}{4\\nu^2}\n\\qquad (t\\in [0,1)),\n$$\n$\\lim_{t\\to 1-0}\\cos s(t)$ exists.\nBy Lemma~\\ref{lem:C}, $\\lim_{t\\to 1-0}s(t)$ also exists.\nThus, we can show the existence of\nthe limit $\\lim_{t\\to 1-0} \\gamma(t)$, proving the\n$C^0$-arc-properness of $\\tilde f_{\\op{TH}}^\\nu$.\nSo the assertion follows from Theorem~\\ref{thm:first}, \nsince $\\tilde f_{\\op{TH}}^\\nu$ is real analytic.\n\\end{proof}\n\n\\begin{Remark}\\label{rmk:infty}\nIf $\\nu=1$, we have that\n$$\nx_0=\\sin s \\sinh s,\\quad x_1=\\cos \\theta \\cos s,\n\\quad x_2=\\sin \\theta \\cos s,\n\\quad x_3= \\sin s \\cosh s.\n$$\nThen $f_{\\op{TH}}^\\nu(k\\pi,\\theta)=\n(0,(-1)^k\\cos \\theta, (-1)^k\\sin \\theta,0)$ \n($\\nu=1$) for each $k\\in {\\mathbf Z}$. \nIn particular, $f_{\\op{TH}}^\\nu$ ($\\nu=1$)\ntakes the same value infinitely many times.\n\\end{Remark}\n\n\nFrom now on, we consider G-catenoids of type $S$ and $P$,\nall of which have non-trivial analytic extensions:\n\n\\subsection{G-catenoids of type S}\\par\n\n\\subsubsection{G-catenoids of type SE}\nFor each $\\mu\\in {\\mathbf R}_+\\setminus \\{1\\}$,\nwe consider a map\n\\begin{equation*}\n f_{\\op{SE}}^\\mu \\colon {\\mathbf R}^2\n\\ni (s,\\theta) \\mapsto (x_0(s,\\theta),x_1(\\theta),\nx_2(\\theta),x_3(s,\\theta)) \n\\in S^3_1\n\\end{equation*}\ngiven by\n\\begin{align*}\nf_{\\op{SE}}^\\mu (s,\\theta) \n=\n\\begin{pmatrix}\n \\cosh s & 0 & 0 & \\sinh s \\\\ \n 0 & 1 & 0 & 0 \\\\ \n 0 & 0 & 1 & 0 \\\\ \n \\sinh s & 0 & 0 & \\cosh s\n\\end{pmatrix}\n\\Gamma^\\mu_{\\op{SE}}(\\theta), \n \\quad \n\\Gamma^\\mu_{\\op{SE}}(\\theta) :=\n\\begin{pmatrix}\n -\\frac{\\mu ^2-1}{2 \\mu} \\cos \\mu \\theta \\\\ \nx_1(\\theta) \\\\\nx_2(\\theta) \\\\\n0\n\\end{pmatrix},\n\\end{align*}\nwhere\n\\begin{equation}\\label{eq:f-SE}\n \\begin{aligned}\nx_1(\\theta)&=-\\frac{(\\mu^2+1) \\cos \\theta \n\\cos \\mu\\theta}{2 \\mu}-\\sin\\theta \\sin \\mu\\theta, \\\\\nx_2(\\theta)&=-\\frac{(\\mu^2+1) \\sin \\theta \n\\cos \\mu \\theta}{2\\mu}+\\cos \\theta \\sin \\mu \\theta.\n\\end{aligned}\n\\end{equation}\nThen this gives\na G-catenoid of type {\\rm SE}.\nHere, $\\Gamma^{\\mu}_{\\op{SE}}(\\theta)$ ($\\theta \\in {\\mathbf R}$) \nis the profile curve of\n$f^{\\mu}_{\\op{SE}}$\nlying on $S^2_1 = S^3_1 \\cap \\{ x_3=0 \\}$.\nThe axis of $f_{\\op{SE}}^\\mu$ is $\\{(0,\\cos t,\\sin t,0)\\;;\\;t\\in T^1\\}$.\nThe singular point set of $f_{\\op{SE}}^\\mu$ is \n \\[\n\t \\Sigma:=\\{(s,\\theta)\\,;\\, \\cos \\mu \\theta=0\\}\n = \\bigcup_{k\\in{\\mathbf Z}} \\Sigma_k,\\qquad\n \\Sigma_k:=\\left\\{\n \\left(s,\\frac{1}{\\mu}\n \\left(\\frac{\\pi}{2}+k\\pi\\right)\\right)\\,;\\, s\\in{\\mathbf R}\\right\\},\n \\]\n and the image of each $\\Sigma_k$ consists of one point\n \\begin{equation}\\label{eq:SE-sing}\n\t \\pt{P}_k:=f_{\\op{SE}}^\\mu(\\Sigma_k) =\n \\left(0,\\gamma_{\\op{SE}}^\\mu \\left(\\frac{1}{\\mu}\n \\left(\\frac{\\pi}{2}+k\\pi\\right)\\right),\n 0\\right),\n \\end{equation}\nwhere \n$ \\gamma^{\\mu}_{\\op{SE}} (\\theta) := (x_1(\\theta), x_2(\\theta)) \\in {\\mathbf R}^2$. \nThe limit point set of $f_{\\op{SE}}^\\mu$ is \n \\[\n\t \\mc L:= \\bigcup_{k\\in{\\mathbf Z}} \\mc L_k,\\quad\n \\mc L_k:=\\left\\{\\left(\n u,\\gamma_{\\op{SE}}^{\\mu}\\left(\\frac{\\pi}{2\\mu}\n \\left(2k+1\\right)\\right),\\pm u\\right)\\in S^3_1\\,;\\,\n u\\in{\\mathbf R}\\right\\}.\n \\]\nWe prove that the image of this map $f^{\\mu}_{\\op{SE}}$ \nhas an analytic extension:\nFor any $\\mu \\in {\\mathbf R}_+ \\setminus \\{ 1\\}$, \ndefine a subset $\\mc X_{\\op{SE}}^\\mu\\subset{\\mathbf R}^3$ \nby \n\\begin{equation}\\label{eq:X-SE} \n\\mc X_{\\op{SE}}^\\mu:=\\left\\{(\\xi,\\eta,\\theta)\\in {\\mathbf R}^3\\,;\\,\\xi\\eta=\n\\left(\\frac{(\\mu^2-1)\\cos \\mu\\theta}{2\\mu}\\right)^2\n\\right\\}.\n\\end{equation}\n By \\eqref{eq:f-SE}, we have\n\\begin{equation}\\label{eq:cosSE}\n |\\gamma^\\mu(\\theta)|^2\n= 1+\\left(\\frac{\\mu^2-1}{2\\mu}\\right)^2\\cos^2 \\mu \\theta.\n\\end{equation}\nThus, if we set \n$$\\xi:=x-y,\\quad\n\\eta:=x+y,\\quad\nz:=\\left(\\frac{\\mu^2-1}{2\\mu}\\right)^2\\cos^2 \\mu \\theta,\n$$\nthen $-x^2+y^2+z^2=0$ holds.\nUsing this expression, one can show that \n$\\mc X_{\\op{SE}}^\\mu$ \nhas the structure of a real analytic $2$-dimensional\nDC-manifold with a countably infinite number of DC-points\nso that the inclusion map\n$\\mc X_{\\op{SE}}^\\mu\\hookrightarrow {\\mathbf R}^3$\nis real analytic.\n\nUsing the functions \\eqref{eq:f-SE}, we set \n$$\n\t \\tilde f_{\\op{SE}}^\\mu\\colon{}\\mc X_{\\op{SE}}^\\mu \n\\ni (\\xi,\\eta,\\theta)\n \\mapsto \\left(\\frac{\\xi+\\eta}{2},x_1(\\theta),x_2(\\theta),\n \\frac{\\xi-\\eta}{2}\\right)\\in S^3_1.\n$$\nIt holds that\n$$\nf^\\mu_{\\op{SE}}(s,\\theta)=\\tilde f_{\\op{SE}}^\\mu\n\\left(-\\frac{(\\mu ^2-1) e^s \\cos \\mu \\theta}{2 \\mu},\n-\\frac{(\\mu ^2-1) e^s \\cos \\mu \\theta}{2 \\mu },\\theta\\right).\n$$\nIf we set\n$$\n\\check f^\\mu_{\\op{SE}}(s,\\theta)=\\tilde f_{\\op{SE}}^\\mu\n\\left(\\frac{(\\mu ^2-1) e^s \\cos \\mu \\theta}{2 \\mu },\n\\frac{(\\mu ^2-1) e^s \\cos \\mu \\theta}{2 \\mu },\\theta\\right),\n$$\nthen we have\n$$\n\\check f_{\\op{SE}}^\\mu=\n\\pmt{\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -1 \n}f^\\mu_{\\op{SE}},\n$$\nwhich implies the image of $\\check f_{\\op{SE}}^\\mu$ is congruent to\nthat of $\\check f_{\\op{SE}}^\\mu$ in $S^3_1$.\nIt can be easily checked that\n\\begin{equation}\\label{eq:SE-E}\n\\tilde f_{\\op{SE}}^\\mu(\\mc X_{\\op{SE}}^\\mu)=\nf_{\\op{SE}}^\\mu({\\mathbf R}^2)\\cup \\mc L\\cup \\check f_{\\op{SE}}^\\mu({\\mathbf R}^2).\n\\end{equation}\nWe prove the following: \n\n\\begin{Lemma}\\label{lem:5-3}\nThe plane curve $\\theta\\mapsto (x_1(\\theta),x_2(\\theta))$ \nis a regular curve on ${\\mathbf R}$.\n\\end{Lemma}\n\n\\begin{proof}\nIn fact, the assertion follows from the following identity\n$$\n\\pmt{x_0'(\\theta) \\\\ x'_3(\\theta)}=\n\\frac{\\mu^2-1}{2\\mu}\n\\pmt{ \\cos \\theta & -\\sin \\theta \\\\\n \\sin \\theta & \\cos \\theta\n}\n\\pmt{\n\\mu \\sin\\mu \\theta \\\\ \\cos\\mu \\theta\n}.\n$$\n\\end{proof}\n\n\n\\begin{Proposition}\\label{prop:SE-noext}\nThe map $\\tilde f_{\\op{SE}}^\\mu$ \nis a \nreal analytic space-like CMC-1 DC-immersion\nwhose\nimage is analytically complete.\n\\end{Proposition}\n\n\\begin{proof}\nBy Lemma \\ref{lem:5-3},\nthe curve $\\gamma_{\\op{SE}}^\\mu$\nis regular, hence \nthe map\n$$\n\\Psi:{\\mathbf R}^3\\ni (\\xi,\\eta,\\theta)\\mapsto \n\\Big (\\frac{\\xi+\\eta}2, x_1(\\theta),x_2(\\theta),\\frac{\\xi-\\eta}2\n\\Big)\n\\in S^3_1\n$$\nis an immersion.\nSince the inclusion map $\\iota:\\mc X_{\\op{SE}}^\\mu\\hookrightarrow {\\mathbf R}^3$\nis a real analytic DC-immersion, the composition\n$\\tilde f_{\\op{SE}}=\\Psi\\circ \\iota$\nis also a real analytic DC-immersion.\nThen \\eqref{eq:SE-E} implies that\n$\\tilde f_{\\op{SE}}^\\mu$ is an analytic extension of $f^\\mu_{\\op{SE}}$.\nMoreover, by Proposition~\\ref{prop:AB-1},\n$\\tilde f_{\\op{SE}}^\\mu$ is a \nreal analytic space-like CMC-1 DC-immersion. \n\nTo obtain the analytic completeness of $\\tilde f_{\\op{SE}}^\\mu$, \nit is sufficient to show that\nthe map $\\tilde f_{\\op{SE}}^\\mu$ is $C^0$-arc-proper, \nsince $\\tilde f_{\\op{SE}}^\\mu$ is real analytic.\nIn fact, let $\\Gamma:[0,1]\\to S^3_1$ be a continuous\nmap, and suppose that\n$$\n\\sigma(t):=(\\xi(t),\\eta(t),\\theta(t))\\qquad (t\\in [0,1))\n$$\nsatisfies $\\Gamma(t)=f\\circ \\sigma(t)$ for each\n$t\\in [0,1)$.\nSince the second and the third components of\n$\\Gamma(t)$ are $\\xi(t)\\pm \\eta(t)$,\nthe limits\n$$\n\\lim_{t\\to 1-0}\\xi(t),\\qquad\n\\lim_{t\\to 1-0}\\eta(t)\n$$\nexist. Since\n$$\n\\xi(t)\\eta(t)+1=|\\gamma_{\\op{SE}}^\\mu(\\theta(t))|^2\n=1+\\left(\\frac{\\mu^2-1}{2\\mu}\\right)^2\\cos^2 \\mu \\theta(t)\n\\qquad (t\\in [0,1)),\n$$\nthe limit $\\lim_{t\\to 1-0}\\cos \\mu \\theta(t)$ exists,\nand so \n$\\lim_{t\\to 1-0}\\theta (t)$ also exists (applying Lemma B.1 in the appendix).\nThus $\\lim_{t\\to 1-0} \\sigma(t)$ exists, which implies\nthe weak $C^0$-arc-properness of $f$.\nSo the conclusion follows from\nTheorem \\ref{thm:first}.\n\\end{proof}\n\nThe map $\\tilde f_{\\op{SE}}^\\mu$ is not a proper map,\nsince the curve $\\gamma_{\\op{SE}}^\\mu$ is bounded.\nHowever, if $\\mu\\in {\\mathbf R}_+\\setminus\\{1\\}$ is \nan integer, \n$\\tilde f_{\\op{SE}}^\\mu$\ninduces a map \n$\n\\hat f_{\\op{SE}}^\\mu:\\hat {\\mc X}_{\\op{SE}}^{\\mu}\\to S^3_1,\n$\nwhere\n$$\n\\hat {\\mc X}_{\\op{SE}}^\\mu:=\\{(\\xi,\\eta,\\theta)\\in {\\mathbf R}^2\\times T^1\n\\,;\\,-\\xi\\eta+|\\gamma_{\\op{SE}}^\\mu(\\theta)|^2=1\\}\n$$\nis a real analytic $2$-dimensional DC-manifold with finitely many DC-points. \nIt can be easily checked that\n$\\hat f_{\\op{SE}}^\\mu$ is a proper map, and its image\nis congruent to the image of the exceptional W-catenoid \n$f^{\\mbox{\\tiny II}}_{\\mu}$ given\nin \\cite{FKKRUY3}.\n\n\\subsubsection{G-catenoids of type SH}\nFor each\n $\\nu\\in {\\mathbf R}_+$, we consider\na map\n\\begin{equation*}\n f_{\\op{SH}}^\\nu :\n{\\mathbf R}^2\\ni (s,\\theta) \\mapsto \n(x_0(s,\\theta),x_1(\\theta),x_2(\\theta),x_3(s,\\theta))\n\\in S^3_1, \n\\end{equation*}\ndefined by\n\\begin{align*}\nf^{\\nu}_{\\mathrm{SH}} (s, \\theta)\n=\n\\begin{pmatrix}\n \\cosh s & 0 & 0 & \\sinh s \\\\ \n 0 & 1 & 0 & 0 \\\\ \n 0 & 0 & 1 & 0 \\\\ \n \\sinh s & 0 & 0 & \\cosh s\n\\end{pmatrix}\n\\Gamma^{\\nu}_{\\op{SH}}(\\theta), \\quad \n\\Gamma^{\\nu}_{\\op{SH}}(\\theta) :=\n\\begin{pmatrix}\n \\frac{\\nu ^2+1}{2 \\nu} \\sinh \\nu \\theta \\\\ \nx_1(\\theta) \\\\\nx_2(\\theta) \\\\\n0\n\\end{pmatrix}, \n\\end{align*}\nwhere $\\nu\\in {\\mathbf R}_+$ and \n\\begin{equation}\\label{eq:x1x2-SH}\n \\begin{aligned}\nx_1(\\theta)&=\\frac{(\\nu^2-1) \\cos \\theta \\sinh \\nu\\theta}{2\\nu}+\\sin \\theta \\cosh \\nu\\theta,\\\\\nx_2(\\theta)&=\\frac{(\\nu^2-1) \\sin \\theta \\sinh \\nu\\theta}{2 \\nu}-\\cos \\theta \\cosh \\nu\\theta.\n \\end{aligned}\n\\end{equation}\nHere $\\Gamma^{\\nu}_{\\op{SH}}(\\theta)$ ($\\theta \\in {\\mathbf R}$) is the profile curve\nof\n$f^{\\nu}_{\\op{SH}}$\nlying on $S^2_1 = S^3_1 \\cap \\{ x_3=0 \\}$.\nThen $f^{\\nu}_{\\mathrm{SH}}$ gives \na G-catenoid of type {\\rm SH}. \nThe axis of $f_{\\op{SH}}^\\nu$ is the same as the axis of $f_{\\op{SE}}^\\mu$.\n\nThe singular point set of $f_{\\op{SH}}^\\nu$ is \n$\\{\\theta=0\\}$, and its image consists of one point \n$(0,0,-1,0)$.\nThe limit point set of $f_{\\op{SH}}^\\nu$ is \n$\n\t \\mc L:= \\{(u,0,-1,\\pm u)\\,;\\,u\\in{\\mathbf R}\\}\\subset S^3_1.\n$\n Let\n \\begin{equation}\\label{eq:X-SH}\n\t \\mc X_{\\op{SH}}^\\nu:=\\left\\{(\\xi,\\eta,\\theta)\\in {\\mathbf R}^3\n\\,;\\,\\xi\\eta=\\left(\\frac{(\\nu^2+1) \n\\sinh \\nu\\theta}{2\\nu}\\right)^2\\right\\}\\subset{\\mathbf R}^3,\n \\end{equation}\nwhich has the structure of a real analytic $2$-dimensional DC-manifold\nwhose DC-point is $(0,0,0)$.\nUsing the functions in \\eqref{eq:x1x2-SH}, we set\n$$\n\t \\tilde f_{\\op{SH}}^\\nu\\colon{}\\mc X_{\\op{SH}}^\\nu\\ni \n(\\xi,\\eta,\\theta)\n \\mapsto \\left(\\frac{\\xi+\\eta}{2},x_1(\\theta),x_2(\\theta),\n \\frac{\\xi-\\eta}{2}\\right)\\in S^3_1.\n$$\nThen we have \n$$\nf_{\\op{SH}}^\\nu(s,\\theta)=\\tilde f_{\\op{SH}}^\\nu\\left(\\frac{(\\nu^2+1)e^s}{\\nu}\\sinh(\\nu \\theta),\\frac{(\\nu^2+1)e^{-s}}{\\nu}\n\\sinh(\\nu \\theta),\\theta\\right).\n$$\nBy setting\n$$\n\\check f_{\\op{SH}}^\\nu(s,\\theta):\n=\\tilde f_{\\op{SH}}^\\nu\\left(\n-\\frac{(\\nu^2+1)e^s}{\\nu}\\sinh(\\nu \\theta),-\\frac{(\\nu^2+1)e^{-s}}{\\nu}\n\\sinh(\\nu \\theta),\\theta\\right),\n$$\nwe have\n$$\n\\check f_{\\op{SH}}^\\nu(s,\\theta)=\n\\pmt{\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -1 \n}f^\\nu_{\\op{SH}} (s, \\theta),\n$$\nwhich implies the image of $\\check f_{\\op{SH}}^\\nu$ is \ncongruent to that of $f_{\\op{SH}}^\\nu$. \nIt holds that \n\\begin{equation}\\label{eq:SH-I}\n\\tilde f_{\\op{SH}}^\\nu(\\mc X_{\\op{SH}}^\\nu)=\nf^\\nu_{\\op{SH}}({\\mathbf R}^2)\\cup \\mc L\\cup \\check f_{\\op{SH}}^\\nu({\\mathbf R}^2),\n\\end{equation}\nWe prove the following: \n\n\\begin{Lemma}\\label{lem:5-4}\nThe plane curve $\\theta\\mapsto (x_1(\\theta),x_2(\\theta))$ is a \nregular curve on ${\\mathbf R}$.\n\\end{Lemma}\n\n\\begin{proof}\nIn fact, the assertion follows from the following identity\n$$\n\\pmt{x_0'(\\theta) \\\\ x'_3(\\theta)}=\n\\frac{\\nu^2+1}{2\\nu}\n\\pmt{ \\cos \\theta & -\\sin \\theta \\\\\n \\sin \\theta & \\cos \\theta\n}\n\\pmt{\n\\nu \\cosh \\nu \\theta \\\\ -\\sinh \\nu \\theta\n}.\n$$\n\\end{proof}\nBy this lemma, we can conclude that\n$$\n\\Psi:{\\mathbf R}^3\\ni (\\xi,\\eta,\\theta)\n \\mapsto \\left(\\frac{\\xi+\\eta}{2},x_1(\\theta),x_2(\\theta)\\right)\n\\in S^3_1\n$$\nis an immersion.\nSince the inclusion map $\\iota:\\mc X_{\\op{SH}}^\\nu\\hookrightarrow {\\mathbf R}^3$\nis a real analytic DC-immersion, the composition\n$\\tilde f_{\\op{SH}}=\\Psi\\circ \\iota$\nis also a real analytic DC-immersion. \nThe relation \\eqref{eq:SH-I} implies that\n$\\tilde f_{\\op{SH}}^\\nu$ is an analytic extension of $f_{\\op{SH}}^\\nu$.\nMoreover, by Proposition \\ref{prop:AB-1},\n$\\tilde f_{\\op{SH}}^\\nu$\nis a real analytic space-like CMC-1 DC-immersion. \n\nWe set $\\gamma_{\\op{SH}}^\\nu(\\theta)=(x_1(\\theta), x_2(\\theta))$. \nSince\n \\[\n |\\gamma_{\\op{SH}}^\\nu(\\theta)|^2= 1+\\left(\\frac{\\nu^2+1}{2\\nu}\n\\right)^2\\sinh^2 \\nu \\theta,\n \\]\n$|\\gamma_{\\op{SH}}^\\nu(\\theta)|$ diverges as $\\theta\\to \\pm \\infty$.\nSince the curve $\\gamma_{\\op{SH}}^\\nu$ is a \nproper map, $\\tilde f_{\\op{SH}}^\\nu$ is also a proper map.\nSo we can apply \nTheorem \\ref{thm:first} and obtain\nthe following:\n\n\n\\begin{Proposition}\\label{prop:SH-noext}\nThe map\n$\\tilde f_{\\op{SH}}^\\nu$\nis a \nreal analytic space-like CMC-1 DC-immersion\nwhose\nimage is analytically complete.\n\\end{Proposition}\n\n\\subsubsection{G-catenoids of type SP}\n A G-catenoid of type {\\rm SP} \ncan be expressed as\n\\begin{equation*}\n f_{\\op{SP}} \\colon {\\mathbf R}^2 \\ni (s,t) \\mapsto \n(x_0(s,t),x_1(t),x_2(t),x_3(s,t)) \\in S^3_1,\n\\end{equation*}\nwhere\n\\begin{align*}\nf_{\\mathrm{SP}}(s,t)\n=\n\\begin{pmatrix}\n \\cosh s & 0 & 0 & \\sinh s \\\\ \n 0 & 1 & 0 & 0 \\\\ \n 0 & 0 & 1 & 0 \\\\ \n \\sinh s & 0 & 0 & \\cosh s\n\\end{pmatrix}\n\\Gamma_{\\op{SP}}(t), \\quad \n\\Gamma_{\\op{SP}}(t) := \n\\begin{pmatrix}\n -\\frac{t}{2} \\\\ \\frac{1}{2} t \\cos t -\\sin t \\\\ \n\\frac{1}{2} t \\sin t+\\cos t \\\\ 0\n\\end{pmatrix}. \n\\end{align*}\nHere $\\Gamma_{\\op{SP}}(t)$ ($t \\in {\\mathbf R}$) is the profile curve\nof \n$f_{\\op{SP}}$\nlying on $S^2_1 = S^3_1 \\cap \\{ x_3=0 \\}$.\nThe axis of $f_{\\op{SP}}$ is the same as the axis of $f_{\\op{SE}}^\\mu$.\n\nThe singular point set of $f_{\\op{SP}}$ is \n $\\{t=0\\}$, and its image consists of one point \n $(0,0,1,0)$. On the other hand,\nthe limit point set of $f_{\\op{SP}}$ is \n \\[\n\t \\mc L:= \\{(u,0,1,\\pm u)\\,;\\,u\\in{\\mathbf R}\\}\\subset S^3_1.\n \\]\nThis map has an analytic extension as follows:\nLet \n \\begin{equation}\\label{eq:X-SP}\n\t \\mc X_{\\op{SP}}:=\n\\{(\\xi,\\eta,t)\\,;\\,\\xi\\eta=t^2\/4\\}\\subset{\\mathbf R}^3.\n \\end{equation}\nThen one can check that $\\mc X_{\\op{SP}}$ \nhas the structure of a real analytic $2$-dimensional DC-manifold\nadmitting only\none cone point at $(0,0,0)$.\nWe define a real analytic map on the DC-manifold \n$\\mc X_{\\op{SP}}$ by\n$$\n\t \\tilde f_{\\op{SP}}\\colon{}\\mc X_{\\op{SP}}\\ni\n (\\xi,\\eta,t)\n \\mapsto \\frac12\\Big(\\xi+\\eta,t \\cos t-2 \\sin t,\nt \\sin t+2 \\cos t,\n \\xi-\\eta\\Big)\\in S^3_1.\n$$\nThen, we have\n$$\nf_{\\op{SP}}(s,t)=\\tilde f_{\\op{SP}}\n(-\\frac{e^st}2,-\\frac{e^{-s}t}2,t).\n$$\nIf we set\n$$\n\\check f_{\\op{SP}}(s,t):=\n\\tilde f_{\\op{SP}}\n(\\frac{e^st}2,\\frac{e^{-s}t}2,t),\n$$\nthen\n\\begin{equation}\\label{eq:SP-S}\n\\check f_{\\op{SP}}:=\n\\pmt{\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -1 \n}\nf_{\\op{SP}}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:SP-I}\n\\tilde f_{\\op{SP}}(\\mc X_{\\op{SP}})=f_{\\op{SP}}({\\mathbf R}^2)\\cup \\mc L\\cup\n\\check f_{\\op{SP}}({\\mathbf R}^2).\n\\end{equation}\nThe relation \\eqref{eq:SP-S}\nimplies that $\\check f_{\\op{SP}}$ is congruent to $f_{\\op{SP}}$. \n\nIt can be easily checked that ${\\mathbf R}\\ni t \\mapsto (x_1(t), x_2(t))\\in {\\mathbf R}^2$\nis an immersion, and so\n$$\n\\Psi:{\\mathbf R}^3 \\ni \n (\\xi,\\eta,t)\n \\mapsto \\frac12\\Big(\\xi+\\eta,t \\cos t-2 \\sin t,\nt \\sin t+2 \\cos t,\n \\xi-\\eta\\Big)\\in S^3_1 \n$$\nalso gives an immersion. \nSince the inclusion map $\\iota:\\mc X_{\\op{SP}}\\hookrightarrow {\\mathbf R}^3$\nis a real analytic DC-immersion, the composition\n$\\tilde f_{\\op{SP}}=\\Psi\\circ \\iota$\nis also a real analytic DC-immersion.\nBy \\eqref{eq:SP-I},\n$\\tilde f_{\\op{SP}}$ is an analytic extension of $\\tilde f_{\\op{SP}}$.\nApplying Proposition \\ref{prop:AB-1},\nwe can also conclude that $\\tilde f_{\\op{SP}}$\nis a real analytic space-like CMC-1 DC-immersion\nin $S^3_1$. \n\nOn the other hand,\nsince \n${\\mathbf R}\\ni t \\mapsto (x_1(t), x_2(t))\\in {\\mathbf R}^2$ \nis a proper map,\nso is $\\tilde f_{\\op{SP}}$.\nIn particular, we obtain the following:\n\n\\begin{Proposition}\\label{prop:SP-noext}\nThe map $\\tilde f_{\\op{SP}}$ \nis a real analytic space-like CMC-1 \nDC-immersion whose\nimage is analytically complete.\n\\end{Proposition}\n\n\n\\subsection{G-catenoids of type L}\n\n\\subsubsection{G-catenoids of type LH}\nA G-catenoid of type {\\rm LH} \ncan be expressed as\n\\begin{equation*}\n f_{\\op{LH}} \\colon{}{\\mathbf R}^2\n\\ni (u,v) \\mapsto P(u)\\Gamma_{\\op{LH}}(v) \\in S^3_1, \n\\end{equation*}\nwhere \n\\begin{equation*}\n P(u)= \n \\begin{pmatrix}\n 1+\\frac{u^2}{2} & u & 0 & -\\frac{u^2}{2} \\\\ \n u & 1 & 0 & -u \\\\ \n 0 & 0 & 1 & 0 \\\\ \n \\frac{u^2}{2} & u & 0 & 1 -\\frac{u^2}{2}\n \\end{pmatrix}, \n\\quad\n\\Gamma_{\\op{LH}}(v) := \n\\begin{pmatrix}\n v \\cos 2v+\\frac{1}{2} v^2 \\sin 2v \\\\ \n0 \\\\ \n\\cos 2v + v \\sin 2v \\\\\n v \\cos 2v+\\frac{1}{2} (v^2-2) \\sin 2v\n\\end{pmatrix}, \n\\end{equation*}\nthat is, $f_{\\op{LH}}=(x_0,x_1,x_2,x_3)$ is given by \n\\begin{equation}\\label{eq:f-LE}\n \\begin{aligned}\nx_0&= v \\cos 2 v + \\frac{u^2+v^2}{2} \\sin 2v, &\nx_1&= u \\sin 2 v, \\\\ \nx_2&= y_{\\op{LH}}(v):=\\cos 2v + v \\sin 2v, &\nx_3&= v \\cos 2 v + \\frac{u^2+v^2-2}{2} \\sin 2v. \n\\end{aligned}\n\\end{equation}\nThis $f_{\\op{LH}}$ is generated by \nthe profile curve $\\Gamma_{\\op{LH}}(v)$ ($v \\in {\\mathbf R}$)\nlying on $S^2_1 = S^3_1 \\cap \\{ x_1=0 \\}$. \nThe singular point set is $\\{2v\\equiv 0\\pmod \\pi\\}$,\n and the image of each connected component of the singular\n set consists of one point.\nThe limit point set of $f_{\\op{LH}}\\colon{}{\\mathbf R}^2\\to S^3_1$ \n consists of two light-like lines\n \\[\n\t \\mc L_{\\pm}:=\\{(t,0,\\pm 1,t)\\in S^3_1\\,;\\, t\\in{\\mathbf R}\\},\n \\]\nwhich are the axes of $f_{\\op{LH}}$.\nThe figure of the image of $f_{\\op{LH}}$ is complicated \n(cf. Figure \\ref{fig:typeL}, left).\n\n\\begin{figure}[hbt]%\n \\begin{center}\n \\includegraphics[width=3.5cm]{desittercat-LH.eps} \\hskip 2cm\n \\includegraphics[width=3.5cm]{desittercat-LE.eps} \n \n\\end{center}\n \\caption{G-catenoids of type LH (left) and LE (right)}%\n\\label{fig:typeL}\n\\end{figure}\n\n\n Let\n \\begin{equation}\\label{eq:X-LH}\n\t \\mc X_{\\op{LH}}:=\\{(\\xi,x,v)\\,;\\,-\\xi\\sin 2v +x^2 \n+(\\cos^2 2v -v \\sin^2 2v)^2=1\\}\\subset{\\mathbf R}^3,\n \\end{equation}\nand let\n$$\n\t \\tilde f_{\\op{LH}}\\colon{}\\mc X_{\\op{LH}}\\ni\n (\\xi,x,v)\n \\mapsto \\left(\\frac{\\xi+\\sin 2v}{2},x,y^{}_{\\op{LH}}(v),\n \\frac{\\xi-\\sin 2v}{2}\\right)\\in S^3_1.\n$$\nThen it holds that\n$$\nf_{\\op{LH}}(u,v)=\\tilde f_{\\op{LH}}\n\\left(\\frac{(u^2+v^2+1)\\sin 2v}2,u \\sin 2v,v\\right)\n$$\nand\n\\begin{equation}\\label{eq:LH-I}\n\\tilde f_{\\op{LH}}(\\mc X_{\\op{LH}})\n= f_{\\op{LH}}({\\mathbf R}^2)\\cup \\mc L_+\\cup \\mc L_-.\n\\end{equation}\nIn particular, $\\tilde f_{\\op{LH}}$ is an analytic extension of\n$f_{\\op{LH}}$, and is a real analytic \nspace-like constant mean curvature one DC-immersion,\ngiving an analytic extension of the image of $f_{\\op{LH}}$.\nIt can be easily checked that\n$$\n{\\mathbf R}^3\\ni \n (\\xi,x,v)\n \\mapsto \\left(\\frac{\\xi+\\sin 2v}{2},x,y^{}_{\\op{LH}}(v),\n \\frac{\\xi-\\sin 2v}{2}\\right)\\in S^3_1\n$$\nis an immersion. Since $\\tilde f_{\\op{LH}}$ is \nthe restriction of this map to $\\mc X_{\\op{LH}}$,\nit is a real analytic DC-immersion.\nThe map $\\tilde f_{\\op{LH}}$ is not proper. \nIn fact, the sequence\n$\\{(0,0,n\\pi)\\}_{n=1}^\\infty$ lies in $\\mc X_{\\op{LH}}$\nand diverges in ${\\mathbf R}^3$, although its image under $\\tilde f_{\\op{LH}}$ is\nthe point set $\\{(0,0,1,0)\\}$.\nWe prepare the following:\n\n\\begin{Lemma}\nThe set $\\mc X_{\\op{LH}}(\\subset {\\mathbf R}^3)$\nhas the structure of a real analytic $2$-dimensional DC-manifold\nwith a countably infinite number of DC-points.\nMoreover, \n$\\tilde f_{\\op{LH}}$ is a real analytic DC-immersion,\ngiving the analytic extension of $f_{\\op{LH}}$. \n\n\\end{Lemma}\n\n\\begin{proof}\nWe set $F:=-\\xi\\sin 2v +x^2 +y_{\\op{LH}}(v)^2-1$.\nIt can be easily checked that\n$F=F_{\\xi}=F_x=F_v=0$ if and only if\n$$\nv=v_0,\\quad \\xi=2\\epsilon_0 v_0, \\quad x=0 \n\\qquad \\left(\\epsilon_0:=\\cos(2v_0)\\in \\{\\pm 1\\},\\,\\,v_0\\in \\frac{\\pi}2{\\mathbf Z}\\right).\n$$\nSo we fix such a $v_0$.\nWe set $\\tilde v:=\\sin 2v$, and\nthen we can write\n$$\nv-v_0=\\tilde v\\phi(\\tilde v), \\qquad \n\\cos 2v=\\epsilon_0 +\\tilde v\\psi(\\tilde v),\n$$\nwhere $\\phi(\\tilde v)$ and $\\psi (\\tilde v)$ are \nfunctions defined on a neighborhood of $\\tilde v=0$.\nSo if we set\n$$\n\\tilde \\xi:=\\xi-\\xi_0+\\tilde v (\\phi(\\tilde v)^2\n+2\\epsilon_0 \\phi(\\tilde v) + 1),\n$$\nthen we can write $F=-\\tilde \\xi \\tilde v +x^2$.\nUsing this expression, one can conclude that \nthe zero set of $F$ has a \ncone-like singular point\nat $(\\xi_0,0,v_0)$\nof $\\tilde f_{\\op{LH}}$.\nSince the map\n$$\nS^3_1\\ni (x_0,x_1,x_2,x_3) \\mapsto \\Big(x_0+x_3, x_1,\n\\frac{\\sin^{-1}(x_0-x_3)}2\\Big ) \\in {\\mathbf R}^3\n$$\ngives a local inverse map of the immersion\n$$\n{\\mathbf R}^3\\ni (\\xi,x,v)\n \\mapsto \\left(\\frac{\\xi+\\sin 2v}{2},x,y^{}_{\\op{LH}}(v),\n \\frac{\\xi-\\sin 2v}{2}\\right)\\in S^3_1,\n$$\n$\\tilde f_{\\op{LH}}$ is a real analytic DC-immersion.\nBy \\eqref{eq:LH-I}, it also gives an analytic extension of\n$f_{\\op{LH}}$. \n\\end{proof}\n\n\\begin{Proposition}\\label{prop:LH-noext}\nThe map \n$\\tilde f_{\\op{LH}}$ is a real analytic space-like constant\nspace-like mean curvature one DC-immersion whose\nimage is analytically complete.\n\\end{Proposition}\n\n\\begin{proof}\nApplying Proposition \\ref{prop:AB-1},\nwe can conclude that\n$\\tilde f_{\\op{LH}}$ is a real analytic \nspace-like CMC-1 DC-immersion. \nThe map $\\tilde f_{\\op{LH}}$ is not a proper map,\nsince $\\tilde f_{\\op{LH}}(\\xi,0,2m\\pi)=(\\xi\/2,0,1,\\xi\/2)$\nfor $m\\in {\\mathbf Z}$. \nBy \nTheorem \\ref{thm:first}, \nit is sufficient to show that \n$\\tilde f_{\\op{LH}}$ is $C^0$-arc-proper.\nWe consider a continuous curve\n$\n\\Gamma:[0,1]\\to S^3_1\n$\nsuch that $\\Gamma([0,1))\\subset \\tilde f_{\\op{LH}}(\\mc X_{\\op{LH}})$.\n Let $\\sigma(t)=(\\xi(t),x(t),v(t))$ ($0\\le t<1$)\nbe a continuous\ncurve on $\\mc X_{\\op{LH}}$\nsuch that \n$$\n\\tilde f_{\\op{LH}}\\circ \\sigma(t)=\\Gamma(t) \\qquad\n(0\\le t<1).\n$$\nSince $\\Gamma(1)$ exists, the three limits\n$$\n\\lim_{t\\to 1-0}\\xi(t), \\quad \\lim_{t\\to 1-0} x(t),\\quad\n\\lim_{t\\to 1-0} \\sin 2v(t)\n$$\nexist. \nThen\n$\\lim_{t\\to 1-0} v(t)$ also exists (cf. Lemma B.1 in the appendix),\nproving the assertion.\n\\end{proof}\n\n\\subsubsection{G-catenoids of type LE}\nOn the other hand,\na G-catenoid of type {\\rm LE} \ncan be expressed as\n\\begin{equation*}\n f_{\\op{LE}} \\colon{}{\\mathbf R}^2\n\\ni (u,v) \\mapsto P(u)\\Gamma_{\\op{LE}}(v) \\in S^3_1, \n\\end{equation*}\nwhere \n\\begin{equation*}\n P(u)= \n \\begin{pmatrix}\n 1+\\frac{u^2}{2} & u & 0 & -\\frac{u^2}{2} \\\\ \n u & 1 & 0 & -u \\\\ \n 0 & 0 & 1 & 0 \\\\ \n \\frac{u^2}{2} & u & 0 & 1 -\\frac{u^2}{2}\n \\end{pmatrix}, \n\\quad\n\\Gamma_{\\op{LE}}(v) := \n\\begin{pmatrix}\n v \\cosh 2v-\\frac{1}{2} (v^2+2) \\sinh 2v \\\\ \n0 \\\\ \n\\cosh 2v - v \\sinh 2v \\\\\n v \\cosh 2v-\\frac{1}{2} v^2 \\sinh 2v\n\\end{pmatrix}, \n\\end{equation*}\nthat is, $f_{\\op{LE}}=(x_0,x_1,x_2,x_3)$ is given by \n\\begin{equation}\\label{eq:f-LH}\n \\begin{aligned}\nx_0&= v \\cosh 2v - \\frac{u^2+v^2+2}{2} \\sinh 2v, &\nx_1&= -u \\sinh 2v, \\\\ \nx_2&= y_{\\op{LE}}(v):=\\cosh 2v - v \\sinh 2v, &\nx_3&= v \\cosh 2 v - \\frac{u^2+v^2}{2} \\sinh 2v. \n\\end{aligned}\n\\end{equation}\nHere $\\Gamma_{\\op{LE}}(v)$ ($v \\in {\\mathbf R}$) is a profile curve\nof\n$f_{\\op{LE}}$\nlying on $S^2_1 := S^3_1 \\cap \\{ x_1=0 \\}$.\nThe axes of $f_{\\op{LE}}$ are the same as the axes of $f_{\\op{LH}}$.\n\nThe singular point set is $\\{v=0\\}$, whose \nimage consists of one point $(0,0,1,0)$.\nThe limit point set of $f_{\\op{LE}}\\colon{}{\\mathbf R}^2\\to S^3_1$ consists\nof a light-like line\n $\n\t \\mc L:=\\{(t,0,1,t)\\in S^3_1\\,;\\, t\\in{\\mathbf R}\\}.\n $\nSet\n\\begin{equation}\\label{eq:X-LE}\n\\mc X_{\\op{LE}}:=\\{(\\xi,x,v)\\,;\\,\\xi\n\\sinh 2 v+x^2 +y_{\\op{LE}}^{}(v)^2=1\\}\\subset{\\mathbf R}^3,\n \\end{equation}\nand let\n$$\n\t \\tilde f_{\\op{LE}}\\colon{}\\mc X_{\\op{LE}}\\ni \n(\\xi,x,v)\n \\mapsto \\left(\\frac{\\xi-\\sinh 2v}{2},x,y_{\\op{LE}}^{}(v),\n \\frac{\\xi+\\sinh 2v}{2}\\right)\\in S^3_1.\n$$\nThen it can be easily checked that $\\tilde f_{\\op{LE}}$\nis a real analytic map defined on the\nDC-manifold $\\mc X_{\\op{LE}}$,\nand is a proper map.\nMoreover, \nwe have\n$$\nf_{\\op{LE}}(u,v)=\\tilde f_{\\op{LE}}(v \\cosh 2v - \\frac{u^2+v^2+1}{2} \\sinh 2v,-u\\sinh v,v)\n$$\nand $\\tilde f_{\\op{LE}}(\\mc X_{\\op{LE}})= f_{\\op{LE}}({\\mathbf R}^2)\\cup \\mc L$.\nIn particular, $\\tilde f_{\\op{LE}}$ \nis an analytic extension of $f_{\\op{LE}}$.\nSince\n$$\n{\\mathbf R}^3\\ni(\\xi,x,v)\n \\mapsto \\left(\\frac{\\xi-\\sinh 2v}{2},x,y_{\\op{LE}}^{}(v),\n \\frac{\\xi+\\sinh 2v}{2}\\right)\\in S^3_1\n$$\nis an immersion,\n$\\tilde f_{\\op{LE}}$ is a real analytic DC-immersion. \nMoreover, by Proposition \\ref{prop:AB-1}, it is \na real analytic space-like CMC-1 DC-immersion giving \nan analytic extension of $f_{\\op{LE}}$. \nThe image of $\\tilde f_{\\op{LE}}$ can be expressed as \nthe intersection of $S^3_1$ and the graph\n\\[\n x_2 = \\sqrt{1+(x_0-x_3)^2} -\\frac12 (x_0-x_3)\\sinh^{-1}(x_0-x_3).\n\\]\nBy Proposition \\ref{prop:first},\nwe obtain the following:\n\n\\begin{Proposition}\\label{prop:LH-noext2}\nThe map $\\tilde f_{\\op{LE}}$ \nis a real analytic space-like CMC-1 DC-immersion whose\nimage is analytically complete.\n\\end{Proposition}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Proof of Theorem \\ref{thm: fedavg_conv}}\nBy Lipschitz smoothness, we have \n\\begin{align}\n f(x_{t+1}) \\leq f(x_t) + \\langle \\nabla f(x_t), x_{t+1} - x_t \\rangle + \\frac{L}{2} \\|x_{t+1} - x_t\\|^2\n\\end{align}\nBefore we proceed, we define following quantities to simplify notation.\n\\begin{align}\n&\\alpha_i^t = \\frac{c}{\\max(c,\\eta_l \\|\\sum_{q=0}^{Q-1} g_i^{t,q} \\|)}, \\quad \n\\tilde \\alpha_i^t = \\frac{c}{\\max(c,\\eta_l \\| \\mathbb E[ \\sum_{q=0}^{Q-1} g_i^{t,q} ]\\|)}, \\quad \n\\overline \\alpha^t = \\frac{1}{N} \\sum_{i=1}^N \\tilde \\alpha_i^t, \\nonumber \\\\\n&\\Delta_i^t = -\\eta_l \\sum_{q=0}^{Q-1} g_i^{t,q} \\cdot \\alpha_i^t , \\quad\n\\tilde \\Delta_i^t = -\\eta_l \\sum_{q=0}^{Q-1} g_i^{t,q} \\cdot \\tilde \\alpha_i^t, \\nonumber \\\\\n&\\overline \\Delta_i^t = -\\eta_l \\sum_{q=0}^{Q-1} g_i^{t,q} \\cdot \\overline \\alpha^t \\quad \\breve \\Delta_i^t = -\\eta_l \\sum_{q=0}^{Q-1} \\nabla f(x_i^{t,q}) \\cdot \\overline \\alpha^t \n\\end{align}\nwhere the expectation in $\\tilde \\alpha_i^t$ is taken over all possible randomness.\n\nThen we have $x_{t+1} - x_t = \\eta_g \\frac{1}{N} \\sum_{i=1}^N (\\Delta_i^t + z_i^t)$ with $z_i^t \\sim \\mathcal N (0, \\sigma^2 I)$ and conditioned on $x_t$,\n\\begin{align}\\label{eq: lip}\n \\mathbb E [f(x_{t+1})] \\leq f(x_t) + \\eta_g {\\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t\\right] \\right\\rangle} + \\frac{L}{2} \\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t \\right\\|^2\\right] + \\frac{L}{2} \\eta_g^2 \\frac{1}{N} \\sigma^2 d \n\\end{align}\nThe next steps will analyze the bias caused by clipping by looking at the first order term. We have\n\\begin{align}\\label{{eq: bias_split}}\n & \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n= & \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right] \\right\\rangle + \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\tilde \\Delta_i^t - \\overline \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n& + \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\overline \\Delta_i^t\\right] \\right\\rangle \n\\end{align}\nwhere the first two terms of RHS of the above equality can be viewed as bias caused by clipping. The first order predicted descent can be analyzed from the last term by completing the square.\n\\begin{align}\n & \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\overline \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n = & \\mathbb E\\left[ \\left\\langle \\nabla f(x_t), \\frac{1}{N} \\sum_{i=1}^N \\breve \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n =& \\frac{-\\eta_l \\overline \\alpha^t Q}{2} \\|\\nabla f(x_t)\\|^2 - \\frac{\\eta_l\\overline \\alpha^t}{2Q} \\mathbb E \\left[ \\left\\| \\frac{1}{\\eta_l N \\overline \\alpha^t} \\sum_{i=1}^N \\breve \\Delta_i^t\\right\\|^2\\right] \\nonumber \\\\\n &+ \\frac{\\eta_l \\overline \\alpha^t}{2} \\underbrace{ \\mathbb E \\left[\\left\\|\\sqrt{Q} \\nabla f(x_t) - \\frac{1}{\\sqrt{Q}} \\frac{1}{\\eta_l N \\overline \\alpha^t} \\sum_{i=1}^N \\overline \\Delta_i^t\\right \\|^2 \\right]}_{A_1}\n\\end{align}\nWe further upper bound $A_1$ as\n\\begin{align}\\label{eq: A1}\n A_1 \n = & Q \\mathbb E \\left[\\left \\| \\nabla f(x_t) - \\frac{1}{{Q}N} \\sum_{i=1}^N \\sum_{q=0}^{Q-1} \\nabla f_i(x_i^{t,q}) \\right \\|^2\\right] \\nonumber \\\\\n = & Q \\mathbb E \\left[\\left \\| \\frac{1}{{Q}N} \\sum_{i=1}^N \\sum_{q=0}^{Q-1} \\nabla f_i(x^t) - \\nabla f_i(x_i^{t,q}) \\right \\|^2 \\right] \\nonumber \\\\\n \\leq & \\frac{1}{N} \\sum_{i=1}^N \\sum_{q=0}^{Q-1} \\mathbb E [ \\| \\nabla f_i(x^t) - \\nabla f_i(x_i^{t,q}) \\|^2 ] \\nonumber \\\\\n \\leq & \\frac{1}{N} \\sum_{i=1}^N \\sum_{q=0}^{Q-1} L^2 \\mathbb E [ \\| x^t - x_i^{t,q} \\|^2 ] \\nonumber \\\\\n \\leq & L^2 5 Q^2 \\eta_l^2 (\\sigma_l^2 + 6Q \\sigma_g^2) + L^2 30 Q^3 \\eta_l^2 \\|\\nabla f(x_t)\\|^2\n\\end{align}\nwhere the last inequality is due to \\cite{}[a lemma in Yang et al., 2021 and Reddi et al., 2021].\n\nNow we turn to upper bounding the second order term in \\eqref{eq: lip}.\n\\begin{align}\\label{eq: second_term}\n & \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t \\right\\|^2\\right] \\nonumber \\\\ \n \\leq & 3 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right\\|^2\\right] + 3 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\tilde \\Delta_i^t - \\overline \\Delta_i^t \\right\\|^2\\right] + 3 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\overline \\Delta_i^t \\right\\|^2\\right] \\nonumber \\\\\n\n\\end{align}\nFor the last term in \\eqref{eq: second_term}, we have \n\\begin{align}\\label{eq: split_variance}\n &\\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\overline \\Delta_i^t \\right\\|^2\\right] \\nonumber \\\\\n = & \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N -\\eta_l \\sum_{q=0}^{Q-1} g_i^{t,q} \\cdot \\overline \\alpha^t \\right\\|^2\\right] \\nonumber \\\\\n \\leq & \\mathbb E \\left [ 2 \\left \\|\\frac{1}{N} \\sum_{i=1}^N -\\eta_l \\sum_{q=0}^{Q-1} \\nabla f(x_i^{t,q}) \\cdot \\overline \\alpha^t \\right\\|^2 + 2 \\left \\|\\frac{1}{N} \\sum_{i=1}^N -\\eta_l \\sum_{q=0}^{Q-1} (\\nabla f(x_i^{t,q}) -g_i^{t,q}) \\cdot \\overline \\alpha^t \\right\\|^2\\right] \\nonumber \\\\\n = & 2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\breve \\Delta_i^t \\right\\|^2\\right] + \\frac{2}{N}\\eta_l^2\\overline \\alpha^2 Q \\sigma_l^2\n\\end{align}\n\nCombining \\eqref{eq: lip}-\\eqref{eq: second_term}, we have\n\\begin{align}\n \\mathbb E [f(x_{t+1})] \\leq & f(x_t) + \\frac{-\\eta_g\\eta_l \\overline \\alpha^t Q}{2} \\|\\nabla f(x_t)\\|^2 - \\frac{\\eta_g\\eta_l\\overline \\alpha^t}{2Q} \\mathbb E \\left[ \\left\\| \\frac{1}{\\eta_l N \\overline \\alpha^t} \\sum_{i=1}^N \\breve \\Delta_i^t\\right\\|^2\\right] \\nonumber \\\\\n &+ \\frac{\\eta_g\\eta_l \\overline \\alpha^t}{2} (5L^2 Q^2 \\eta_l^2 (\\sigma_l^2 + 6Q \\sigma_g^2) + 30L^2 Q^3 \\eta_l^2 \\|\\nabla f(x_t)\\|^2) \\nonumber \\\\\n &+ \\eta_g \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right] \\right\\rangle + \\eta_g \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\tilde \\Delta_i^t - \\overline \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n & + 3L\\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\breve \\Delta_i^t \\right\\|^2\\right] + \\frac{3L}{N}\\eta_g^2\\eta_l^2(\\overline \\alpha^t)^2 Q \\sigma_l^2 + \\frac{L}{2} \\eta_g^2 \\frac{1}{N} \\sigma^2 d \\nonumber \\\\\n & + \\frac{3L}{2} \\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right\\|^2\\right] + \\frac{3L}{2} \\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\tilde\\Delta_i^t - \\overline\\Delta_i^t \\right\\|^2\\right] \n\\end{align}\nWhen $\\eta_g\\eta_l \\leq \\frac{1}{6QL}$ and $\\eta_l \\leq \\frac{1}{\\sqrt{60}QL}$, the above inequality simplifies to \n\\begin{align}\n \\mathbb E [f(x_{t+1})] \n \\leq & f(x_t) + \\frac{-\\eta_g\\eta_l \\overline \\alpha^t Q}{4} \\|\\nabla f(x_t)\\|^2 \\nonumber \\\\\n &+ \\frac{\\eta_g\\eta_l^3 \\overline \\alpha^t}{2} (5L^2 Q^2 (\\sigma_l^2 + 6Q \\sigma_g^2)) \\nonumber \\\\\n &+ \\eta_g \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right] \\right\\rangle + \\eta_g \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\tilde \\Delta_i^t - \\overline \\Delta_i^t\\right] \\right\\rangle \\nonumber \\\\\n & + \\frac{3L}{N}\\eta_g^2\\eta_l^2(\\overline \\alpha^t)^2 Q \\sigma_l^2 + \\frac{L}{2} \\eta_g^2 \\frac{1}{N} \\sigma^2 d \\nonumber \\\\\n & + \\frac{3L}{2} \\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right\\|^2\\right] + \\frac{3L}{2} \\eta_g^2 \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\tilde\\Delta_i^t - \\overline\\Delta_i^t \\right\\|^2\\right] \n\\end{align}\nSum over $t$ from $1$ to $T$, divide both sides by $T\\eta_g\\eta_lQ\/4$, and rearrange, we have\n\\begin{align}\n &\\frac{1}{T} \\sum_{t=1}^T \\mathbb E[ \\overline \\alpha^t\\|\\nabla f(x_t)\\|^2] \\nonumber \\\\\n \\leq & \\frac{4}{T\\eta_g\\eta_lQ}(\\mathbb E [f(x_{1})] - \\mathbb E [f(x_{T+1})]) \\nonumber \\\\\n &+ {10\\eta_l^2 } L^2 Q (\\sigma_l^2 + 6Q \\sigma_g^2) \\frac{1}{T}\\sum_{t=1}^T \\overline \\alpha^t + \\frac{12L}{N}\\eta_g\\eta_l \\sigma_l^2 \\frac{1}{T}\\sum_{t=1}^T (\\overline \\alpha^t)^2+ 2L\\frac{ \\eta_g}{\\eta_lQN} d \\sigma^2 \\nonumber \\\\\n &+ \\frac{1}{T}\\sum_{t=1}^T \\frac{4}{\\eta_lQ} \\mathbb E\\left[\\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right] \\right\\rangle + \\left\\langle \\nabla f(x_t), \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N \\tilde \\Delta_i^t - \\overline \\Delta_i^t\\right] \\right\\rangle \\right ] \\nonumber \\\\\n & + \\frac{6L}{\\eta_lQ} \\eta_g \\frac{1}{T} \\sum_{t=1}^T \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\Delta_i^t - \\tilde \\Delta_i^t \\right\\|^2\\right] + \\frac{6L}{\\eta_lQ} \\eta_g \\frac{1}{T} \\sum_{t=1}^T \\mathbb E \\left [\\left \\|\\frac{1}{N} \\sum_{i=1}^N \\tilde\\Delta_i^t - \\overline\\Delta_i^t \\right\\|^2\\right]. \n\\end{align}\nUpper-bounding the last four terms using $\\|g_i^{t,q}\\| \\leq G$ yields the desired result. \\qed\n\n\\section{Introduction}\n First proposed by~\\cite{konevcny2016federated}, \\emph{Federated Learning} (FL) is a distributed learning framework that aims to reduce communication complexity and to provide privacy protection during training. The popular FedAvg algorithm \\cite{konevcny2016federated} has been proposed to reduce the communication cost by using periodic averaging and client sampling. {There has been many extensions of this algorithm, mostly by modifying the local update directions \\cite{karimireddy2020scaffold,Zhang_FedPD_Arxiv_2020,liang2019variance}.} Even though FL algorithms \\swedit{have the goal of privacy protection}, recent works have shown that they are vulnerable to inference attacks and leak local information during training~\\cite{zhao2020idlg,zhu2020deep,wei2020framework}. \n\\swedit{As a result, striking a balance between \\emph{formal} privacy guarantees and desirable optimization performance remains one of the fundamental challenges in FL.}\n\n\n\nRecently, various FL algorithms~\\cite{geyer2017differentially,truex2020ldp, truex2019hybrid, wang2020d2p, triastcyn2019federated} have been proposed to provide the formal guarantees of \\emph{differential privacy} (DP)~\\cite{Dwork2006}.\nIn these algorithms, the clients perform multiple local updates between two communication steps, and then perturbation mechanisms are added to aggregate updates across individual clients. In order for the perturbation mechanism to have formal privacy guarantees, each client's model update needs to have a bounded norm, which is ensured by applying a clipping operation\nthat shrinks individual model updates when their norm exceeds a given threshold. While there has been prior work that studies the clipping effects on stochastic gradients~\\cite{bassily2014private,chen2020understanding,song2021evading} in the differentially private SGD \\cite{abadi2016deep}, there has not been any work on providing understanding how clipping the model updates affect the optimization performance of FL subject to DP. Our work provides the first in-depth study on such clipping effects. \n\\swdelete{Notably, the aggregation step is not directly performed on the individual stochastic gradients, which is typically the case for centralized algorithm like differentially private SGD (DP-SGD),\nbut instead a sequence of (stochastic) gradients. On the contrary, classical analysis for differentially private (stochastic) gradient descent directly clips the (stochastic) gradients, and the key in the analysis is {to bound the bias introduced by clipping the (stochastic) gradient} \\cite{abadi2016deep}. Therefore, all such existing analyses can not be directly applied to the same operation in FL.\nHowever, none of the existing work provides a rigorous understanding of how clipping operation affect the convergence of FL algorithms. Therefore it is not clear how these DPFL algorithms can balance between optimization performance and privacy guarantees.}\n\n\n\\noindent{\\bf Contributions.} In this work, we will conduct rigorous theoretical analysis and provide extensive empirical evidence to understand how to best protect client-level DP for FL algorithms. Specifically, we make the following contributions: \n\n\\noindent {\\bf 1)} We analyze the existing model and difference clipping strategies for clipping-enabled FedAvg and prove that difference clipping outperforms model clipping. Our result provides theoretical insight into designing FL algorithms with clipping operation. \n\n\\noindent {\\bf 2)} We empirically show that the performance of the clipping-enabled FedAvg depends on the structure of the neural network being used -- when \\swedit{the structure of the network induces \\emph{concentrated} clients' updates}, and the performance drop becomes negligible.\n\n\\noindent {\\bf 3)} We provide the convergence analysis of the clipping-enabled FedAvg algorithm and highlight the relationship between clipping bias and the distribution of the clients' updates. Our result leads to a natural guarantee of client-level DP for FedAvg. \n\nTo the best of our knowledge, this is the first work that rigorously investigates theoretical and empirical issues regarding the clipping operation in FL algorithms.\n\n\n\\subsection{Preliminaries \\& Related Work}\n\nFederated learning typically considers the following optimization problem:\n\\begin{equation}\\label{eq:FL}\n \\min_{\\xx} \\bigg[f(\\xx) \\triangleq \\sum^N_{i=1}f_i(\\xx)\\bigg], \\;\\mbox{\\rm where } f_i(\\xx) = \\mathbb{E}_{\\xi\\sim \\cD_i} F(\\xx;\\xi),\n\\end{equation}\nwhere $N$ is the number of participating clients; the $i^\\mathrm{th}$ client optimizes a local model $f_i$, which is the expectation of a loss function $F(\\xx;\\xi)$, where the expectation is taken over local data distribution $\\cD_i$. At each communication round $t$, the server samples a subset of clients $\\cP_t$ and broadcasts the global model parameters $\\xx^t$. The sampled clients perform $Q$ steps of SGD updates and compute the total update differences $\\Delta\\xx^t_i$'s, and then the server aggregates the update differences to update the global model. In Algorithm \\ref{alg:FedAvg}, we present a slightly generalized FedAvg algorithm \\cite{karimireddy2020scaffold,yang2021achieving}, in which the server uses a stepsize $\\eta_g$ to perform its update. When $\\eta_g=1$, the algorithm becomes the same as the original FedAvg. \n\n\\begin{algorithm}[b!]\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {Initialize: $\\xx^{0}_i\\triangleq\\xx^{0}, i=1,\\dots,N$}\\\\\n\t\t\\FOR{$t=0,\\dots,T-1$ {\\it (stage)}}\n\t\t\\FOR{$i\\in \\cP_t\\subseteq [N]$ in parallel}\n\t\t\\STATE {Update agents' $\\xx^{t,0}_i =\\xx^{t}$}\n\t\t\\FOR{$q=0,\\dots,Q-1$ {\\it (iteration)}}\n\t\t\\STATE{Compute stochastic gradient $g_i^{t,q}$ with $\\mathbb E [g_i^{t,q}] = \\nabla f_i(x_i^{t,q})$ \n\t\t\\STATE{Local update:} $\\xx^{t,q+1}_i = \\xx^{t,q}_i - \\eta_l g_i^{t,q} $}\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\t\\STATE {Global averaging:} { $\\Delta\\xx^{t}_i = \\xx^{t,Q}_i - \\xx^{t}$, \\quad $\\xx^{t+1} =\\xx^{t}+\\eta_g\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_t\n\t\t\\Delta\\xx^{t}_i$}\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\t\\caption{FedAvg Algorithm}\\label{alg:FedAvg}\n\\end{algorithm}\n\n\\swedit{In this work, we study FL subject to the rigorous privacy guarantees of \\emph{Differential Privacy} (DP) \\cite{Dwork2006}}, whose formal definition is given below.\n\n\\begin{defn}\\label{def:CDP}\\cite{Dwork2006}\n\tAn algorithm $\\cM$ is $(\\epsilon, \\delta)$-differentially private if \n\t\\begin{equation}\n\t P(\\cM(\\cD)\\in\\cS) \\leq e^\\epsilon P(\\cM(\\cD')\\in\\cS) + \\delta,\n\t\\end{equation}\n\twhere $\\cD$ and $\\cD'$ \\swedit{are neighboring datasets, $\\cS$ is an arbitrary subset of outputs of $\\cM$.}\n\\end{defn}\n\n The common mechanism used to protect DP in centralized training is straightforward: 1) clip the stochastic gradient with the so-called clipping operation~\\eqref{eq:clip}; 2) add a random perturbation $\\zz\\sim\\cN(0,\\sigma^2I)$ to the clipped quantity~\\cite{abadi2016deep}. The clipping operation is the key step to guarantee DP as the noise level $\\sigma^2$ is determined by the clipping threshold $c$~\\cite{dwork2014algorithmic}:\n\\begin{equation}\\label{eq:clip}\n\t\\mbox{\\rm clip}(g^{t},c) = g^t\\cdot\\min\\bigg\\{1,\\frac{c}{\\norm{g^t}}\\bigg\\}.\n\\end{equation}\n\nHowever, DP is more complex in FL than that in centralized training. Two key factors distinguish FL from existing DP machine learning framework are:\n\\vspace{-0.2cm}\n\\begin{itemize}[leftmargin=*]\n \\item {\\it Data distribution}: unlike centralized training, in FL the data are naturally distributed on the clients, and the clients can potentially have very different data distributions. In the centralized setting, the recent work \\cite{chen2020understanding} has shown that the distribution of the samples affects the performance of the DP-SGD, but how heterogeneous data distribution affects the design and analysis of FL algorithm that protects DP is unclear. \n \\item {\\it Local updates}: as described in \\algref{alg:FedAvg}, the clients will perform multiple local update steps before sending the model to the server, and it is well-known that when $Q>1$, the data heterogeneity will cause performance degradation in FedAvg even without clipping and perturbation \\cite{khaled2019first}. Although there are multiple alternatives of how the DP mechanism can be applied to FL algorithms, none of those mechanisms has a rigorous theoretical guarantee, and it is not clear how to properly balance the optimization performance and privacy guarantees. \n\\end{itemize}\n\nThese two factors result in different {\\it definitions} and {\\it clipping operations} in FL. \n\n\\underline{DP definitions in FL:} Based on the distribution pattern of the client and local datasets, two DP definitions are commonly considered in FL algorithm design: \n\\vspace{-0.2cm}\n\\begin{itemize}[leftmargin=*]\n \\item Sample-level differential privacy (SL-DP): SL-DP directly follows the centralized DP and protects each local sample so that the server could not identify one sample from the union of all local datasets, i.e., $\\cD = \\bigcup^N_{i=1}\\cD_i$, and $\\cD,\\cD'$ differ by one sample $\\xi$. SL-DP fits in the cross-silo FL scenario that has a relatively small number of clients, each with a large dataset. E.g., SL-DP is used in medical image classification application to protect patients' personal information~ {\\cite{choudhury2019differential}}. However, in the Google Keyboard application~\\cite{hard2018federated} where each client is an application user, SL-DP that only protects one sample (i.e., an input record) will not be sufficient to protect the user's personal information.\n \\item Client-level differential privacy (CL-DP): CL-DP has a stricter privacy guarantee compared with SL-DP. It requires that the server cannot identify the participation of one client by observing the output of the local updates, i.e., $\\cD =\\{\\cD_i\\}^N_{i=1}$, and $\\cD,\\cD'$ differ by one dataset $\\cD_i$. CL-DP is suitable for the cross-device FL scenario such as the Google Keyboard application, which has a large number of distributed clients. \n\\end{itemize}\n\n\\underline{Clipping operation in FL:} Based on different DP requirements and the algorithm structures, a number of FL algorithms have been proposed which protect DP to some extent. \n\nTo protect SL-DP, \\cite{truex2019hybrid} proposes to clip and inject noise to every local update. {That is, some Gaussian noise is added to the stochastic gradients $g^{t,q}_i$ given in \\algref{alg:FedAvg}.} However, as intermediate updates are kept local and private, the clipping and perturbation to the local steps appear to be unnecessary, and such operations result in significant performance degradation. {Moreover, it is not clear how such kind of operation impact other aspects of the algorithm performance (such as algorithm convergence, quality of solutions, etc.)}\n\nTo protect CL-DP, \\cite{wei2020federated} proposes to clip the local models to be transmitted directly. Similarly, \\cite{truex2020ldp} assumes that the model parameters are upper and lower bounded by some constant and directly apply perturbations to the local models. {However, this scheme also significantly reduces the training and test accuracy empirically and has no theoretical convergence guarantee}. Recently, \\cite{geyer2017differentially} proposes to clip the difference between the input model and the output models of the FedAvg algorithm. In particular, one can replace the update directions $\\Delta\\xx^{t}_i$'s of line 8 in Algorithm 1 by their clipped versions as expressed below: \n\\begin{equation}\n\t\\label{eq:diffclip}\n\t\\begin{aligned}\n\t\\mbox{\\rm clip}(\\Delta\\xx^{t}_i,c) = \\Delta\\xx^{t}_i\\cdot\\min\\bigg\\{1,\\frac{c}{\\norm{\\Delta\\xx^{t}_i}}\\bigg\\}, \\\\\n\t\\xx^{t+1} =\\xx^{t}+\\eta_g\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_t}\n\t\t\\mbox{\\rm clip}(\\Delta\\xx^{t}_i,c).\n\t\\end{aligned}\n\\end{equation}\nIt is shown that such a scheme has better numerical performance than model clipping, but no convergence proof for the algorithm is given. {Reference \\cite{triastcyn2019federated} also clips the update difference and proposed Bayesian DP to measure the privacy loss and only demonstrates the numerical performance of the proposed algorithm. D2P-Fed \\cite{wang2020d2p} follows the same clipping strategy and further apply compression and quantization during communication to improve communication efficiency while having DP guarantee, but its convergence guarantee only applies to the non-clipping version.}\n\nIn summary, despite extensive recent research about DP-enabled FL, there are still a number of technical challenges and open research questions in this area. First, it is not clear how various kinds of clipping operations can affect the performance of FL algorithms. Second, it is not clear how to add noise to balance the convergence of FL algorithms and its CL-DP guarantee.\n\\section{Clipping Issues in FL}\n\nAs discussed above, clipping is a key operation in providing DP guarantee for FL algorithms. Therefore, to design algorithms that protect DP in FL, the first step is to understand how clipping affects the convergence performance of a FL algorithm. Towards this end, we start with analyzing two common clipping strategies, and identify their theoretical properties. {Then we provide a series of empirical studies to demonstrate how system parameters such as training models, datasets and data distributions can affect the performance of clipping-enabled FedAvg algorithm.} These empirical studies will be combined with our theoretical analysis in the next section to provide a comprehensive understanding about the optimization performance and CL-DP guarantees in FL.\n\n\\subsection{Model clipping versus Difference Clipping}\n\nThe two major clipping strategies used in protecting CL-DP for FL algorithms are {\\it local model} clipping and {\\it local update difference} clipping, as we describe below.\n\\vspace{-0.2cm}\n\\begin{enumerate}[leftmargin=*]\n \\item {\\bf Model clipping}~\\cite{wei2020federated}: The clients directly clip the models sent to the server. For FedAvg algorithm, this means performing $\\mbox{\\rm clip}(x^{t,Q}_i,c)$. This method appears to be straightforward, but clipping the model directly results in relatively large clipping threshold, so it requires to add larger perturbation. \n \\item {\\bf Difference clipping}~\\cite{geyer2017differentially}: The clients clip the local update difference between the initial model and the output model according to~\\eqref{eq:diffclip}. This method needs to record the initial model and to perform extra computation before clipping, but the update difference typically has smaller magnitudes than the model itself, so the clipping threshold and the perturbation can be smaller than using model clipping. Note that when $Q = 1$, the difference clipping is equivalent to the standard mini-batch gradient clipping (i.e., the DP-SGD), but in the general case where $Q>1$, their behaviors are very different.\n\\end{enumerate}\n\nBelow we analyze how they perform on simple quadratic problems. Our results indicate that the difference clipping strategy is more preferable, because it is less likely to have strong impact on the optimization performance.\n\n\\begin{claim}\\label{eq:claim:model}\n Given any constant clipping threshold $c$, there exists a convex quadratic problem, for which FedAvg with model clipping does not converge to the global optimal solution with any fixed $Q\\geq 1$ and $\\eta_l > 0$.\n\\end{claim}\n\\noindent{\\bf Proof.} Given a fixed clipping threshold $c$, consider the following quadratic problem $$f(x) = \\sum^3_{i=1} \\frac{1}{2}(x-b_i)^2,$$ where we have $N = 3$ clients. By applying model clipping to FedAvg, one round update can be expressed as:\n\n\\begin{equation*}\n \\begin{aligned}\n x^+ = \\frac{1}{3}\\sum^3_{i=1}\\mbox{\\rm clip}(\\lambda x+(1-\\lambda)b_i,c), \\text{ where } \\lambda = (1-\\eta_l)^Q \\in (0,1),\n \\end{aligned}\n\\end{equation*}\n{ where $\\eta_l$ is the local stepsize.}\n\nSuppose that the algorithm converges, then we will have solution $x^+ = x = x^\\infty$. This implies that\n\\begin{align}\\label{eq:updates}\n \\frac{1}{3}\\sum^3_{i=1}\\mbox{\\rm clip}(\\lambda x^\\infty+(1-\\lambda)b_i,c) = x^\\infty.\n\\end{align}\nLet us set $b_1 = b_2 = -0.5c, b_3 = kc$, then it is easy to verify that the optimal solution of the problem is given by $x^\\star = \\frac{(k-1)c}{3}>0$.\nHowever, when $k>4$, from \\eqref{eq:updates} we can see that $x^\\infty \\leq c$ and $x^\\star >c$. Therefore, the only possibility is that $x^\\infty = \\frac{\\lambda}{3-2\\lambda}c\\leq c \\neq x^\\star$, and this holds true for any $\\lambda\\in(0,1)$. So the stationary solution of FedAvg with model clipping to this problem will not converge to the original optimal solution no matter how we choose $Q$ and $\\eta_l$. \\hfill $\\blacksquare$\n\n\\begin{claim}\\label{eq:claim:difference}\n For all linear regression problem with fixed clipping threshold $c$, there exist $\\eta_l$ and local update step $Q\\ge 1$ such that FedAvg with difference clipping converges to the global optimal solution. Furthermore, there exist a linear regression problem such that under the same $c, \\eta_l$ and $Q$, FedAvg with difference clipping converges to a better solution with smaller loss than the original FedAvg.\n\\end{claim}\n\n\\noindent {\\bf Proof.} First, we prove that using difference clipping, FedAvg can converge to global optimal by carefully selecting $Q$ and $\\eta_l$. Consider the following convex quadratic problem $$f(x) = \\sum^N_{i=1} \\frac{1}{2}(A_ix-b_i)^2.$$ By applying FedAvg with update difference clipping, one round of update can be expressed as:\n\n\\begin{equation*}\n \\begin{aligned}\n x^+ &= x - \\frac{1}{N}\\sum^N_{i=1} \\mbox{\\rm clip}(\\Lambda_i \\nabla f_i(x),c), \\text{ where } \\Lambda_i = (I - (I - \\eta_l A^T_iA_i)^Q)(A^T_iA_i)^{-1}.\n \\end{aligned}\n\\end{equation*}\n\nIn order for the problem to converge to the original problem, it is easy to verify that the following condition has to hold: $$\\sum^N_{i=1} \\mbox{\\rm clip}(\\Lambda_i \\nabla f_i(x^\\star),c) = 0.$$ \n\nThe above example can be viewed as using gradient descent to optimize a problem with the following gradient \n\\begin{equation}\n \\nabla f'_i(x) = \\left\\{\\begin{array}{cc}\n \\Lambda_i \\nabla f_i(x) & \\norm{\\Lambda_i \\nabla f_i(x) } \\leq c,\\\\\n \\frac{c \\Lambda_i \\nabla f_i(x) }{\\norm{\\Lambda_i \\nabla f_i(x)}}& \\text{otherwise}.\n \\end{array}\\right.\n\\end{equation}\nNote that in general it is hard to write down the exact local problems $f'_i$ that satisfies the above condition, but when $x\\in \\mathbb{R}$ is a scalar, $f'_i(x)$ is the Huberized loss of $\\Lambda_i f_i(x)$~\\cite{song2021evading} \n\n\\begin{equation}\n f'_i(x) = \\begin{cases}\n \\Lambda_i f_i(x)& \\mbox{if } \\abs{\\Lambda_i A_i(A_ix -b_i)}\\leq c,\\\\\n c\\abs{\\frac{\\Lambda_i}{A_i} f_i(x)} -\\frac{1}{2}c^2& \\mbox{otherwise.}\n \\end{cases}\n\\end{equation}\n\nIn general, the re-weighted problem does not have the same solution as the original problem, but we can select $\\eta_l$ and $Q$ (determined by on $x^\\star$ and $f_i$'s) so that $f'(x)$ has the same solution as $f(x)$. For example, one set of parameters that satisfy the above requirement is $Q = 1, \\eta_l = 1\/\\max_i\\{\\norm{\\nabla f_i(x^\\star)}\\}$. In this case, $\\Lambda_i = I\\eta_l$, and when $\\eta_l$ is small enough, the clipping will not be activate when $x = x^\\star$ and $\\sum^N_{i=1} \\mbox{\\rm clip}(\\Lambda_i \\nabla f_i(x^\\star),c) = \\sum^N_{i=1} \\eta_l\\nabla f_i(x^\\star) = 0$. \n\nNext, we show that Clipping-enabled FedAvg can outperform the non-clipped version. Note that when $Q > 1$, even when $\\eta$ is small such that the clipping is not activated, the algorithm will not converge to the original solution. So in general one cannot draw the conclusion about whether clipping helps or hurts the performance of FedAvg. Consider the following problem: \n\n\\begin{equation}\\label{eq:example_3}\n \\begin{aligned}\n f(x) &= \\sum^3_{i=1} f_i(x),\\\\\n f_1(x) &= \\frac{1}{2}(x-4)^2,\\; f_2(x) = \\frac{1}{2}(2x-1)^2, \\; f_3(x) = \\frac{1}{2}(6x+1)^2.\n \\end{aligned}\n\\end{equation}\n\nAs $\\nabla f(x) = (x-4) + (4x-2) + (36x +6) = 41x$, the optimal solution of this problem is $x^\\star = 0$. Table \\ref{tab:solutions} show the stationary points of FedAvg under different choice of parameters. When $Q = 1$, FedAvg is equivalent to SGD and clipping hurts the performance of FedAvg. However, when $Q$ is large, clipped FedAvg has a better performance than the non-clipped version, in the sense that the stationary solution it obtains are closer to the global optimal solution $x^*=0$. \\hfill $\\blacksquare$\n\n\\begin{table}[tb!]\n \\centering\n \\begin{tabular}{c|cc}\n \\hline\n & $Q = 1$ & $Q = \\infty$ \\\\\n \\hline\n $c = \\infty$ & $x^\\infty = 0$ & $x^\\infty = \\frac{13}{9}$ \\\\\n $c = 1$ & $x^\\infty = \\frac{1}{2}$ & $x^\\infty = \\frac{2}{3}$\\\\\n \\hline\n \\end{tabular}\n \\caption{Stationary points of FedAvg with gradient clipping for \\eqref{eq:example_3} under different parameter settings.}\n \\vspace{-0.8cm}\n \\label{tab:solutions}\n\\end{table}\n\n\\begin{rem}\nTo prove Claim \\ref{eq:claim:model}, we construct a problem whose magnitude of the optimal solution is larger than the clipping threshold. Then FedAvg with model clipping will converge to a stationary point with magnitude bounded by the clipping threshold, therefore the algorithm will not converge to global optimal solution.\n\nThe technique to prove the first part of Claim \\ref{eq:claim:difference} is related to the analysis for centralized gradient clipping algorithms \\cite{song2020characterizing}. The main difference is that our algorithm considers $Q$ steps of local update before clipping. We show that by allowing multiple local updates, FedAvg algorithm with difference clipping optimizes the sum of the Huberzied re-weighted local loss functions. By properly choosing the learning rate $\\eta_l$ for each local loss function, we can balance the re-weighting factors so that the optimal solution to the new loss function matches the solution to the original problem. \\hfill $\\blacksquare$\n\\end{rem}\n\nThe above claims indicate that the difference clipping should outperform the model clipping in terms of convergence guarantees. Therefore, in the subsequent analysis, we will focus on understanding the difference clipping enabled FL algorithms. In particular, we consider the Clipping-Enabled FedAvg (CE-FedAvg) algorithm described in~\\algref{alg:CE-FedAvg}, which combines the difference clipping with the slightly generalized FedAvg algorithm described in \\algref{alg:FedAvg} (which uses two stepsizes $\\eta_l, \\eta_g$, one for local and one for global updates, respectively). The reason to consider such a {\\it bi-level-stepsize} version of FedAvg is that, it has been proved to have superior performance, especially when not all clients participate in each round of communication~\\cite{karimireddy2020scaffold,yang2021achieving}.\n\n\n\\begin{algorithm}[b!]\n\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {Initialize: $\\xx^{0}_i\\triangleq\\xx^{0}, i=1,\\dots,N$}\\\\\n\t\t\\FOR{$t=0,\\dots,T-1$ {\\it (stage)}}\n\t\t\\FOR{$i\\in \\cP_t\\subseteq [N]$ in parallel}\n\t\t\\STATE {Update agents' $\\xx^{t,0}_i =\\xx^{t}$}\n\t\t\\FOR{$q=0,\\dots,Q-1$ {\\it (iteration)}}\n\t\t\\STATE{Compute stochastic gradient $g_i^{t,q}$ with $\\mathbb E [g_i^{t,q}] = \\nabla f_i(x_i^{t,q})$ \n\t\t\\STATE{Local update:} $\\xx^{t,q+1}_i = \\xx^{t,q}_i - \\eta_l g_i^{t,q} $}\n\t\t\\ENDFOR\n\t\t\\STATE{Compute update difference:} $\\Delta\\xx^{t}_i = \\xx^{t,Q}_i - \\xx^{t,0}_i$\n\t\t\\STATE{Clip:} $\\Hat{\\Delta}\\xx^{t}_i = {\\rm clip}(\\Delta\\xx^t_i,c)$, where ${\\rm clip}(\\cdot)$ is defined in \\eqref{eq:clip}\n\t\t\\ENDFOR\n\t\t\\STATE {Global averaging:} { $\\xx^{t+1} =\\xx^{t}+\\eta_g\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_t}\\Hat{\\Delta}\\xx^{t}_i$}\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\t\\caption{Clipping-enabled FedAvg Algorithm (CE-FedAvg)}\\label{alg:CE-FedAvg}\n\\end{algorithm}\n\n\\subsection{Empirical Results}\n\n\\noindent{\\bf Experiment Setting.} To have a thorough understanding about how the difference clipping can impact the FedAvg, we conduct numerical experiments with different models, datasets and local data distributions. We compare the test accuracies between CE-FedAvg and the original FedAvg. Note that in this set of experiments we do not consider the privacy issues yet, so we do not add perturbation\n\nTo have a fair comparison, we set $Q$, $T$, $N$, $\\abs{\\cP_t}$, $\\eta_l$ and $\\eta_g$ to be identical for both FedAvg and CE-FedAvg. We first run the original FedAvg, compute $\\norm{\\Delta\\xx^{t}_i}$ and average over all clients $i$ and iterations $t$ to obtain $\\bar{\\Delta}$ and choose the clipping threshold $c=0.5\\bar{\\Delta}$. \n\nWe run the algorithm using AlexNet~\\cite{krizhevsky2012imagenet} and ResNet-18~\\cite{he2016deep} with EMNIST dataset~\\cite{cohen2017emnist} and Cifar-10 dataset~\\cite{krizhevsky2009learning} for comparison. We split the dataset in two different ways: 1) {\\it IID Data} setting, where the samples are uniformly distributed to each client; 2) {\\it Non-IID Data} setting, where the clients have unbalanced samples. Details are described below. For EMNIST digit classification dataset, each client has 500 samples without overlapping. In the IID case, each client has around 50 samples of each class and in the Non-IID case, there are 8 classes each has around 5 samples and 2 classes each has 230 samples on each client. For the Cifar-10 dataset, in the IID case (resp. Non-IID case), each client also has 500 samples (resp. $50$ samples); these samples can overlap with those on the other clients and {the samples on each client are uniformly distributed in $10$ classes, i.e., each client has $50$ samples (resp. $5$ samples) from each class.\n\n\\noindent{\\bf Performance Degradation.} In \\figtab{tab:clip_drop}, we compare the classification results produced by using AlexNet and ResNet-18 on the two datasets.\n{\\begin{table*}[t!]\n \\centering\\footnotesize\n \\begin{tabular}{l|rrrrr\n \\hline\n Model& dataset& IID(\\%)& IID Clipping (\\% drop) & Non-IID (\\%)& Non-IID Clipping (\\% drop)\\\\\n \\hline\n AlexNet \t& EMNIST\t& 98.20\t& 0.19\t& 95.60 & 3.60\\\\\n\t\t \t\t\t& Cifar-10\t& 66.01\t& 4.83\t& 57.14 & 7.30\\\\\n ResNet-18 \t& EMNIST\t& 99.61 & 0.02 \t& 95.43 & 0.10\\\\\n\t\t \t\t\t& Cifar-10\t& 76.36\t& 0.53\t& 59.46 & 1.55\\\\\n \\hline\n \\end{tabular}\n \n \\caption{\\footnotesize The accuracy drop between a) FedAvg and clipping-enabled FedAvg, used for training AlexNet and ResNet-18, on IID and Non-IID data.}\n \\vspace{-0.8cm}\n \\label{tab:clip_drop}\n\\end{table*}}\n\nThere are three interesting observations: 1) The data distribution will greatly affect the clipping performance in FL. When data are IID across the clients, clipping has far less impact on the final accuracy, otherwise the clipping will introduce some accuracy drop to the trained models; 2) Clipping has quite different impact on different models -- the best accuracy of the models drops $0.10\\%$ and $3.60\\%$ for ResNet-18 and AlexNet on EMNIST, respetively. The drop is $1.55\\%$ for ResNet-18 and $7.30\\%$ {for AlexNet on Cifar-10}, comparing CE-FedAvg with non-clipped version on the Non-IID data; 3) Data complexity also affects the behavior of the CE-FedAvg -- the accuracy drop on Cifar-10 dataset is much larger than that on EMNIST dataset.\n\nThe empirical experiments show that heterogeneous data distribution among the clients is one of the main causes of the different behavior between the clipped and non-clipped algorithms. The data heterogeneity issue is unique in FL cause by periodical communication. It does not happen in centralized optimization where the data are shared among all workers. \n\n\\noindent{\\bf Update Difference Distribution.} To further understand the clipping procedure, we plot in Fig. \\ref{fig:distribution1} and Fig. \\ref{fig:distribution2} the magnitudes of local updates $\\norm{\\Delta\\xx^t_i}$ and the cosine angles between the last iteration's global update and $\\Delta\\xx^t_i$:\n\\xcdelete{of each client:} $\\cos^{-1}\\left({\\lin{\\Delta\\xx^t_i,\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_{t-1}}{\\Delta}\\xx^{t-1}_i}}\/{\\norm{\\Delta\\xx^t_i}\\norm{\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_{t-1}}{\\Delta}\\xx^{t-1}_i}}\\right).$ Due to page limitation, we only put the distribution of communication round $T = 16$. More detailed results are given in Appendix \\ref{app:exp}. {In the plots, we mainly focus on the variance of the magnitudes of the clients' update difference (i.e., the blue dots). Larger variance indicates that the updates made by different clients are more different from each other.}\n\nCompare Fig. \\ref{fig:distribution1} with Fig. \\ref{fig:distribution2} we can see that the update magnitudes on EMNIST dataset are more concentrated than that on Cifar-10 dataset by having smaller mean and variance. Similarly, by comparing Fig. \\ref{fig:alexiid} with Fig. \\ref{fig:alexnoniid} or Fig. \\ref{fig:resnetiid} with Fig. \\ref{fig:resnetnoniid}, it is clear that the local update magnitudes are more concentrated on IID data than on Non-IID data. Moreover, ResNet-18 has a more concentrated distribution of update magnitudes than AlexNet. Importantly, comparing Table \\ref{tab:clip_drop} with Fig. \\ref{fig:distribution1} and Fig. \\ref{fig:distribution2}, one can observe that the drop in final accuracy of a model caused by clipping is correlated with {\\it the degree of concentration} of update magnitudes, as AlexNet with less concentrated update magnitudes suffers more from clipping, while ResNet-18 exhibits the opposite behavior.\n\nThe above results about the update difference distributions match the accuracy results in \\figtab{tab:clip_drop}, in the sense that clipping performs worse when update differences distribution has a larger divergence and vise versa. Inspired by this observation, in the next subsection, we will characterize the impact of clipping based on the degree of concentration in local updates and develop the convergence analysis of CE-FedAvg.\n\n\\begin{figure}[t!]\n \\centering\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\centering \n\t\t \\includegraphics[width=\\textwidth]{figure\/alex_e16_iid.png}\n \\caption{AlexNet, IID}\n \\label{fig:alexiid}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/alex_e16_noniid.png}\n \\caption{AlexNet, Non-IID}\n \\label{fig:alexnoniid}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/fed_emnist_resnet_32_lFalse_noniid00_e16.png}\n \\caption{ResNet-18, IID}\n \\label{fig:resnetiid}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/fed_emnist_resnet_32_lFalse_noniid09_e16.png}\n \\caption{ResNet-18, Non-IID}\n \\label{fig:resnetnoniid}\n \\end{subfigure}\n \\caption{\\footnotesize The distribution of local updates for AlexNet and ResNet-18 on IID and Non-IID data at communication round $16$ for EMNIST dataset. Each blue dot corresponds to the local update from one client. The black dot shows the magnitude and the cosine angle of averaged local update at iteration $t$.}\n \\label{fig:distribution1}\n\\end{figure}\n\n\\begin{figure}[t!]\n\t\t\\centering\n\t\t \\begin{subfigure}[b]{0.24\\textwidth}\n\t\t\t \\centering \n\t\t\t \\includegraphics[width=\\textwidth]{figure\/fed_cifar_alex_32_lFalse_noniid00_e16.png}\n\t\t\t \\caption{AlexNet, IID}\n\t\t\t \\label{fig:alexiidc}\n\t\t \\end{subfigure}\n\t\t \\hfill\n\t\t \\begin{subfigure}[b]{0.24\\textwidth}\n\t\t\t \\centering\n\t\t\t \\includegraphics[width=\\textwidth]{figure\/fed_cifar_alex_32_lFalse_noniid10_e16.png}\n\t\t\t \\caption{AlexNet, Non-IID}\n\t\t\t \\label{fig:alexnoniidc}\n\t\t \\end{subfigure}\n\t\t \\begin{subfigure}[b]{0.24\\textwidth}\n\t\t\t \\centering\n\t\t\t \\includegraphics[width=\\textwidth]{figure\/fed_cifar_resnet_32_lFalse_noniid00_e16.png}\n\t\t\t \\caption{ResNet-18, IID}\n\t\t\t \\label{fig:resnetiidc}\n\t\t \\end{subfigure}\n\t\t \\hfill\n\t\t \\begin{subfigure}[b]{0.24\\textwidth}\n\t\t\t \\centering\n\t\t\t \\includegraphics[width=\\textwidth]{figure\/fed_cifar_resnet_32_lFalse_noniid10_e16.png}\n\t\t\t \\caption{ResNet-18, Non-IID}\n\t\t\t \\label{fig:resnetnoniidc}\n\t\t \\end{subfigure}\n\t\t\\caption{\\footnotesize The distribution of local updates for AlexNet and ResNet-18 on IID and Non-IID data at communication round $16$ for Cifar-10 dataset. Each blue dot corresponds to the local update from one client. The black dot shows the magnitude and the cosine angle of averaged local update at iteration $t$.}\n\t\t\\label{fig:distribution2}\n\t\t\\vspace{-0.7cm}\n\t\\end{figure}\n\n\n\\section{Convergence Analysis of Clipping-Enabled FedAvg}\n\nIn this section, we analyze the theoretical performance of CE-FedAvg as well as its randomly perturbed version, in order to gain a better understanding of our previous empirical observations and the trad-off between the convergence performance of FedAvg and its DP guarantees.\n\nTowards this end, we will provide the convergence analysis and privacy guarantees for the DP-FedAvg algorithm, described in in \\algref{alg:DP-FedAvg}. Compared to CE-FedAvg, this algorithm further adds a random perturbation $\\zz^t_i$ to the locally clipped model differences.\nDuring the communication, we assume that the attacker can only observe the aggregated update $\\sum_{i\\in\\cP_t}\\Tilde\\Delta\\xx^{t}_i$, and this can be guaranteed by using secure aggregation~\\cite{bonawitz2017practical} or assuming the uplink of the clients to the server is secure.\n\nDespite the similar mechanism used in DPSGD and DP-FedAvg, let us point their major differences: in DPSGD, the goal is to protect SL-DP, while DP-FedAvg is to protect CL-DP. The key difference in DP-FedAvg is that the local dataset size is large enough so that after performing multiple local update steps, the resulting model has relatively good performance. By doing so, we can largely reduce the number of communication and the corresponding privacy noise added per communication. Note that DP-FedAvg becomes DPSGD with the following choices of hyperparameters: 1) enlarge the client number to be the same as the size of the dataset, 2) decrease the local dataset size to 1; 3) decrease the number of local update to 1; 4) decrease the privacy noise accordingly.\n\n\\begin{algorithm}[b!]\n\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {Initialize: $\\xx^{0}_i\\triangleq\\xx^{0}, i=1,\\dots,N$}\\\\\n\t\t\\FOR{$t=0,\\dots,T-1$ {\\it (stage)}}\n\t\t\\FOR{$i\\in \\cP_t\\subseteq [N]$ in parallel}\n\t\t\\STATE {Update agents' $\\xx^{t,0}_i =\\xx^{t}$}\n\t\t\\FOR{$q=0,\\dots,Q-1$ {\\it (iteration)}}\n\t\t\\STATE{Compute stochastic gradient $g_i^{t,q}$ with $\\mathbb E [g_i^{t,q}] = \\nabla f_i(x_i^{t,q})$ \n\t\t\\STATE{Local update:} $\\xx^{t,q+1}_i = \\xx^{t,q}_i - \\eta_l g_i^{t,q} $}\n\t\t\\ENDFOR\n\t\t\\STATE{Compute update difference:} $\\Delta\\xx^{t}_i = \\xx^{t,Q}_i - \\xx^{t,0}_i$\n\t\t\\STATE{Clip and perturb:} $\\Tilde{\\Delta}\\xx^{t}_i = {\\rm clip}(\\Delta\\xx^t_i,c)+\\zz^t_i$, where ${\\rm clip}(\\cdot)$ is defined in \\eqref{eq:clip}\n\t\t\\ENDFOR\n\t\t\\STATE {Global averaging:} { $\\xx^{t+1} =\\xx^{t}+\\eta_g\\frac{1}{|\\cP_t|}\\sum_{i\\in\\cP_t}\\Tilde\\Delta\\xx^{t}_i$}\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\t\\caption{DP-FedAvg Algorithm}\\label{alg:DP-FedAvg}\n\\end{algorithm}\n\n\\subsection{Convergence Analysis}\n\n\\begin{theorem}[Convergence of DP-FedAvg]\\label{thm: fedavg_conv}\nFor Algorithm \\ref{alg:DP-FedAvg}, assume\n\\begin{align*}\n \\|\\nabla f_i(x) - \\nabla f_i(y)\\| & \\leq L \\|x-y\\|,\\; \\forall\\; i,x,y,\\quad \\min_x f(x) \\geq f^*;\\\\\n \\mathbb E [\\|g_i^{t,q} - \\nabla f_i(x_i^{t,q})\\|^2] & \\leq \\sigma_l^2, \\;\\quad\\|g_i^{t,q}\\| \\leq G, \\; \\forall\\; t,q,i,\\quad \\|\\nabla f_i(x) - \\nabla f(x)\\|^2 & \\leq \\sigma_g^2, \\; \\forall i,\n\\end{align*}\nwhere $L$ is the Lipschitz constant of gradient, $\\sigma_l^2$ and $\\sigma_g^2$ are intra-client and inter-client gradient variance, $G$ is the bound on stochastic gradient. \n\nBy letting $\\eta_g\\eta_l \\leq \\min\\{\\frac{P}{48Q},\\frac{P}{6QL(P-1)}\\}$ and $\\eta_l \\leq \\frac{1}{\\sqrt{60}QL}$, we have\n{\\small\n\\begin{align*\n\t& \\frac{1}{T} \\sum_{t=1}^T \\mathbb E [\\overline \\alpha^t \\|\\nabla f(x^t)\\|^2] \\\\\n\t&\\leq \\underbrace{\\frac{4(f(x^0) - f^*)}{\\eta_g \\eta_l QT} + \\frac{25}{2}\\eta_l^2LQ(\\sigma_l^2 + 6Q\\sigma_g^2) \\gamma_1(T) +\\frac{6\\eta_g \\eta_l L \\sigma_l^2}{P} \\gamma_2(T) }_{\\text{\\rm standard terms for FedAvg}} +\\underbrace{ \\frac{2\\eta_gLd \\sigma^2}{\\eta_lPQ}}_{\\text{\\rm caused by privacy noise}} \\nonumber \\\\ \n\t& + \\underbrace{G^2\\frac{4}{T} \\sum_{t=1}^T \\mathbb E\\left[\\frac{1}{N} \\sum_{i=1}^N (|\\alpha_i^t - \\tilde \\alpha_i^t| + |\\tilde \\alpha_i^t - \\overline \\alpha^t|)\\right]}_{\\text{\\rm caused by clipping}} + \\underbrace{\\eta_g\\eta_lLQG^2 \\frac{6}{T} \\sum_{t=1}^T \\mathbb E\\left[ \\frac{1}{P} \\sum_{i=1}^N (|\\alpha_i^t - \\tilde \\alpha_i^t|^2 + |\\tilde \\alpha_i^t - \\overline \\alpha^t|^2)\\right]}_{\\text{\\rm caused by clipping}}\n\\end{align*}\n}%\nwhere $P := \\abs{\\cP_t}$, $\\alpha_i^t := \\frac{c}{\\max(c, \\eta_l\\|\\sum_{q=0}^{Q-1} g_i^{t,q}\\|)}$, $\\tilde \\alpha_i^t := \\frac{c}{\\max(c, \\eta_l \\| \\mathbb E [\\sum_{q=0}^{Q-1} g_i^{t,q}]\\|)}$, $\\overline \\alpha^t := \\frac{1}{N} \\sum_{i=1}^N \\tilde \\alpha_i^t$; $d$ is the dimension of $x$, $\\gamma_1(T) =\\frac{1}{T}\\sum_{t=1}^T \\mathbb E [\\overline \\alpha^t] \\leq 1 $, $\\gamma_2(T) =\\frac{1}{T}\\sum_{t=1}^T \\mathbb E [(\\overline \\alpha^t)^2] \\leq 1$. \n\\end{theorem}\n\nIn the bound of Theorem \\ref{thm: fedavg_conv}, the standard terms are inherited from standard FedAvg with two-sided learning rates which can yield a convergence rate of $O(\\frac{1}{\\sqrt{PQT}}+\\frac{1}{T})$ when setting $\\eta_g=\\sqrt{QP}$ and $\\eta_l=\\frac{1}{\\sqrt{T}QL}$. When there is no clipping bias and privacy noise, Theorem \\ref{thm: fedavg_conv} exactly recovers the standard convergence bounds for FedAvg up to a constant, see \nTheorem 1 in \\cite{yang2021achieving}. In addition to the standard terms, we have extra terms caused by the privacy noise $z_i^{t}$ and the clipping operation. We highlight the terms caused by clipping which characterize the \\swdelete{worse-case}\\mhdelete{adversarial effect of} estimation bias caused by clipping. The bias can be decomposed into terms caused by $|\\alpha_i^t - \\tilde \\alpha_i^t|$ and terms caused by $|\\tilde \\alpha_i^t - \\overline \\alpha^t|$. \\xcedit{Notice that $|\\alpha_i^t - \\tilde \\alpha_i^t| \\leq \\eta_l |\\|\\sum_{q=0}^{Q-1} g_i^{t,q}\\| - \\| \\mathbb E [\\sum_{q=0}^{Q-1} g_i^{t,q}]\\||$ } , it is clear \\mhdelete{we will have} $\\mathbb E [|\\alpha_i^t - \\tilde \\alpha_i^t|] $ will be small if the stochastic local updates have similar \\xcedit{variance or} magnitudes in norm, and $\\mathbb E [|\\alpha_i^t - \\tilde \\alpha_i^t|] =0$ if $\\sigma_l=0$. \\xcedit{This term characterizes the bias caused by local update variance.} In addition, $\\mathbb E[|\\tilde \\alpha_i^t - \\overline \\alpha^t|]$ will be small if the expected local model updates have similar magnitudes in norm across clients and $\\mathbb E[|\\tilde \\alpha_i^t - \\overline \\alpha^t|]= 0$ if $ \\|\\mathbb E [ \\Delta x^t_i]\\| = \\|\\mathbb E [\\Delta x^t_j]\\|, \\forall i, j$. \\xcedit{This term shows the bias caused by cross-client update variance.}\n\nIn FL, sometimes each client will have limited amount of data, and the local model updates can be performed with small $\\sigma_l$ or even $\\sigma_l=0$ (full batch update). Thus, the bias caused by $|\\alpha_i^t - \\tilde \\alpha_i^t|$ can be small \\mhedit{and is} avoidable. However, the bias caused by $|\\tilde \\alpha_i^t - \\overline \\alpha^t|$ is {\\it unavoidable} since this term will not diminish even each client updates its local model with full batch gradient. In addition, this term might be large with heterogeneous data distribution since the heterogeneity may induce quite disparate gradient distributions across clients. Thus, it is crucial to investigate the bias caused by $|\\tilde \\alpha_i^t - \\overline \\alpha^t|$ in practice. Note that $|\\tilde \\alpha_i^t - \\overline \\alpha^t|$ is fully controlled by differences in magnitudes of local model updates when $\\sigma_l = 0$ for fixed $c$. Going back to Fig. \\ref{fig:distribution1}, we do see that how such differences in update magnitudes can be affected by both the neural network models and data heterogeneity. \n\n\n\\subsection{Differential Privacy Guarantee}\n\nThe privacy guarantee of DP-FedAvg can be characterized by standard privacy theorems on Gaussian mechanism. We rephrase \\cite[Theorem 1 ]{abadi2016deep} for client privacy in Theorem \\ref{thm: privacy}. \n\\begin{theorem}[Privacy of DP-FedAvg]\\label{thm: privacy}\nThere exist constants $u$ and $v$ so that\ngiven the number of iterations $T$, for any $\\epsilon \\leq u q^2T$ with $q= \\frac{P}{N}$ and $|\\mathcal P_t|= P,\\ \\forall t$, Algorithm \\ref{alg:FedAvg} is $(\\epsilon,\\delta)$-differentially private for any $\\delta >0$ if $\\sigma^2 \\geq v\\frac{c^2 PT\\ln(\\frac{1}{\\delta})}{N^2\\epsilon^2}$.\n\\end{theorem}\nThe privacy-utility trade-off of DP-FedAvg can be analyzed by substituting $\\sigma^2$ from Theorem \\ref{thm: privacy} into Theorem \\ref{thm: fedavg_conv}. To get more insights on how parameters like $T, \\eta_g, \\eta_l$ and $\\epsilon$ affect DP-FedAvg, let us consider simplified Theorem \\ref{thm: fedavg_conv} in Corollary \\ref{corl: nobias} with $c \\geq \\eta_lQG$ and $\\sigma^2$ substituted . If $c' < G$ in Corollary \\ref{corl: nobias}, then there will be extra bias terms inherited from the bound in Theorem \\ref{thm: fedavg_conv}. \\mhdelete{it can be affected by $c'$ and the distribution of update magnitude of different clients.}\n\\begin{corollary}[Convergence with privacy guarantee]\\label{corl: nobias}\nAssume all assumptions in Theorem \\ref{thm: fedavg_conv}, for any clipping threshold $c = \\eta_l Q c' $ with $c' \\geq G$, and set $\\sigma^2$ as in Theorem \\ref{thm: privacy}, for any $(\\epsilon, \\delta)$ satisfying the constraints in Theorem \\ref{thm: privacy}, we have\n{\\small\n\\begin{align}\\label{eq: convergence}\n\t& \\frac{1}{T} \\sum_{t=1}^T \\mathbb E [ \\|\\nabla f(x^t)\\|^2] \\leq \\underbrace{O\\left(\\frac{1}{\\eta_g \\eta_l QT} + \\eta_l^2Q^2 +\\frac{\\eta_g \\eta_l }{P}\\right) }_{\\text{\\rm standard terms for FedAvg}} +\\underbrace{O\\left( \\frac{\\eta_g\\eta_lQTd \\ln(\\frac{1}{\\delta})}{N^2\\epsilon^2}\\right)}_{\\text{\\rm caused by privacy noise}} \n\\end{align}}\nand the best rate one can get from the above bound is $\\tilde O(\\frac{\\sqrt{d}}{N\\epsilon})$ by optimizing $\\eta_g, \\eta_l,Q,T$.\n\\end{corollary}\nA direct implication of Corollary \\ref{corl: nobias} is that the big-$O$ convergence rate of DP-FedAvg is the same as differentially private SGD (DP-SGD) in terms of $d$, $\\epsilon$, and $N$ (note that $N$ which will be number of training samples in DP-SGD).\n\n\n\\section{Numerical Experiments}\n\nIn the experiment, we compare the performance of FedAvg, CE-FedAvg and DP-FedAvg on two datasets. In both experiments, we set client number $N = 1920$, the number of client participates in each round {$\\abs{\\cP_t} = 80,\\ \\forall~t$}, the number of local iterations $Q = 32$ and the mini-batch size $64$. The clipping threshold is set to $50\\%$ of the average \\xcedit{(over clients and iterations)} of local update magnitudes recorded in FedAvg. For DP-FedAvg we set the clipping threshold the same as in CE-FedAvg, we fix the number of communication rounds and privacy budget for the algorithms to obtain the noise variance that needs to be added. Among all the experiments, we fix privacy budget $\\delta = 10^{-5}.$\n\n\\noindent{\\bf EMNIST dataset.} We use the digit part of the EMNIST dataset, which has 240K training samples and 40K testing samples. We distribute the data in the Non-IID way described in Section II and each client has 125 samples. We conduct experiments on a 2-layer MLP with one hidden layer, AlexNet, ModelNetV2~\\cite{sandler2018mobilenetv2} and ResNet-18. The results are listed in Table~\\ref{tab:performance1} and \\figref{fig:test_acc_dp1}.\n\n{\\begin{table}[b!]\n \\vspace{-0.5cm}\n\t\\centering\n\t\\begin{tabular}{l|rrrrr}\n\t\t\\hline\n\t\t\tModel& \\# Parameters & \\# Layers & Accuracy (\\%)& Clipping (\\% drop) & DP (\\% drop)\\\\\n\t\t\t\\hline\n\t\t\tMLP& 159K & 2 & 94.0 & 1.84 & 0.29\\\\\n\t\t\tAlexNet & 3.3M & 7 & 96.4 & 1.47 & 0.16\\\\\n\t\t\tMobileNetV2 & 2.3M & 24 & 97.8 & 0.35 & 1.62\\\\\n\t\t\tResNet-18 & 11.1M & 18 & 95.2 & -0.15 & 3.76$^*$ \\\\\n\t\t\t\\hline\n\t\\end{tabular}\n\t\\caption{\\footnotesize The accuracy drop between a) FedAvg and clip-enabled FedAvg and b) clip-enabled FedAvg and DP-FedAvg. The clipping threshold is $0.5$ of the average magnitude and privacy budget $\\epsilon = 1.5$ for MLP, AlexNet and MobileNetV2 and $\\epsilon = 5$ for ResNet-18.}\n\t\\label{tab:performance1}\n\\vspace{-0.5cm}\n\\end{table}}\n\n\\begin{figure}[t!]\n \\vspace{-0.5cm}\n \\centering\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/mlp_acc.png}\n \\caption{MLP, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/alex_acc.png}\n \\caption{AlexNet, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/mobilenet_acc.png}\n \\caption{MobileNetV2, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/resnet_acc.png}\n \\caption{ResNet-18, $\\epsilon = 5$}\n \\end{subfigure}\n \\caption{\\footnotesize The test accuracy of FedAvg, CE-FedAvg and DP-FedAvg on different models on EMNIST. The privacy budgets for MLP, AlexNet and MobileNet are $\\epsilon = 1.5$ while for ResNet, we set $\\epsilon = 5$.}\n \\label{fig:test_acc_dp1}\n \\vspace{-0.5cm}\n\\end{figure}\n\n\\noindent{\\bf Cifar-10 dataset.} The dataset we use is the Cifar-10 dataset, which has 50K training samples and 10K testing samples. We distribute the data in the IID way described in Section II and each client has 500 samples. We conduct experiments on a 2-layer MLP with one hidden layer, AlexNet and ResNet-18. The results are listed in Table~\\ref{tab:performance2} and \\figref{fig:test_acc_dp2}.\n\n{\\begin{table}[b!]\n\t\\centering\n\t\\begin{tabular}{l|rrrrr}\n\t\t\\hline\n\t\t\tModel& \\# Parameters & \\# Layers & Accuracy (\\%)& Clipping (\\% drop) & DP (\\% drop)\\\\\n\t\t\t\\hline\n\t\t\tMLP& 616K & 2 & 51.90 & 7.39 & 0.90\\\\\n\t\t\tAlexNet & 3.3M & 7 & 66.01 & 4.83 & -0.18\\\\\n\t\t\tResNet-18 & 11.1M & 18 & 76.36 & 0.53 & 5.15 \\\\\n\t\t\t\\hline\n\t\\end{tabular}\n\t\\caption{\\footnotesize The accuracy drop between a) FedAvg and CE-FedAvg and b) CE-FedAvg and DP-FedAvg. The clipping threshold is $0.5$ of the average magnitude and privacy budget $\\epsilon = 1.5$ for MLP, AlexNet and ResNet-18.}\n\t\\label{tab:performance2}\n\\vspace{-0.5cm}\n\\end{table}}\n\n\\begin{figure}[t!]\n \\centering\n \\begin{subfigure}[b]{0.30\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/MLP_Acc_iid.png}\n \\caption{MLP, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.30\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/Alex_Acc_iid.png}\n \\caption{AlexNet, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.30\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{figure\/Resnet_Acc_iid.png}\n \\caption{ResNet-18, $\\epsilon = 1.5$}\n \\end{subfigure}\n \\caption{\\footnotesize The test accuracy of FedAvg, CE-FedAvg and DP-FedAvg on different models on Cifar-10. The privacy budgets for MLP, AlexNet and ResNet are $\\epsilon = 1.5$.}\n \\label{fig:test_acc_dp2}\n \\vspace{-0.5cm}\n\\end{figure}\n\n\\noindent{\\bf Discussion.} Let us discuss the relation between our empirical observations and the theoretical results. \n\n{\\bf 1)} It appears that when the underlying machine learning model is {\\it structured} (e.g., many layers, has convolution layers, skip connections, etc), the update difference of FedAvg becomes {\\it concentrated}, yielding a better clipping performance (as suggested by the terms related to clipping in Theorem \\ref{thm: fedavg_conv});\n\n{\\bf 2)} When the model has too many parameters and\/or layers, they are sensitive to privacy noise. This is reasonable since the error term caused by privacy noise in Theorem \\ref{thm: fedavg_conv} is linearly dependent on the size of the model $d$ and the square of the Lipschitz constant $L$ (note, that $\\eta_\\ell\\propto 1\/L$). From~\\cite[Corollary 3.3]{herrera2020estimating}, we know that $L$ increases exponentially with the number of layers. Therefore, larger and deeper models are potentially more sensitive to privacy noise.\n\n{\\bf 3)} We conjecture that, to ensure good performance of DP-FedAvg, we need to pick a neural network that is structured enough, while not having too many variables and too many number of layers.\n\n\\section{Conclusion}\nThis work provides empirical and theoretical understanding about clipping operation in FL. We show how to properly combine the clipping operation with existing FL algorithms to achieve the desirable trade-off between convergence and differential privacy guarantees. Numerical results corroborate our theory, and suggest that the distribution of the clients' updates is a key factor that affects the performance of the clipping-enabled FL algorithm. \n \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nOrdinal time series analysis is a special type of symbolic analysis~\\cite{hao1989,daw2003} which makes use of \\textit{symbols} that are ordinal patterns (also referred to as order patterns or permutation patterns) of length of at least 2.\nC. Bandt and B. Pompe introduced this concept in 2002 in their seminal paper~\\cite{bandt2002} together with permutation entropy as a natural complexity measure of time series. \nLet $x_i=x(i),i=1,\\ldots,N$, denote a sequence of observations (or time series) from some system $X$. \nFor a given, but otherwise arbitrary $i$, $m$ amplitude values $X_i=\\{x(i),x(i+l),\\ldots,x(i+(m-1)l)\\}$ are arranged in an ascending order $\\{x(i+(k_{i1}-1)l)\\leq x(i+(k_{i2}-1)l)\\leq\\ldots\\leq x(i+(k_{im}-1)l)\\}$, where $l$ and $m$ denote the appropriately chosen~\\cite{staniek2007,Berger2019,Myers2020} time delay and embedding dimension (cf. Takens' embedding theorem~\\cite{takens1981,sauer1991}). \nIn case of equal amplitude values, one can e.g. carry out the rearrangement according to the associated index $k$, i.e., for $x(i+(k_{i1}-1)l)=x(i+(k_{i2}-1)l)$ one can write $x(i+(k_{i1}-1)l)\\leq x(i+(k_{i2}-1)l)$ if $k_{i1}0$. In terms of $x$, the standardized statistic is given by:\n\\begin{eqnarray}\n \\gamma = \\kappa(x) = \\frac{x}{2 \\sqrt{2 + \\frac{1 + \\exp{(x)}}{\\exp{(x\/2)}}}}.\n\\end{eqnarray}\nUsing basic trigonometric identities:\n\\begin{eqnarray*}\n && \\frac{1 + \\exp{(x)}}{\\exp{(x\/2)}} = 2 \\cosh (x\/2), \\\\\n && 2 + \\frac{1 + \\exp{(x)}}{\\exp{(x\/2)}} = 4 ( \\cosh (x\/4) )^{2}, \\quad \\text{and} \\\\\n && \\sqrt{2 + \\frac{1 + \\exp{(x)}}{\\exp{(x\/2)}}} = 2 \\cosh (x\/4), \n\\end{eqnarray*}\nwe can express $\\kappa(x)$ and its derivative in terms of hyperbolic functions as:\n\\begin{eqnarray}\n \\kappa(x) &=& \\frac{x\/4}{\\cosh(x\/4)} = (x\/4) \\, \\text{sech}(x\/4) \\label{eq.kappa} \\\\ \n \\kappa'(x) &=& \\frac{4 - x \\tanh(x\/4) }{16 \\cosh(x\/4) }.\n\\end{eqnarray}\nTo maximize the standardized ln(OR), we set $\\kappa'(x)$=$0$, which is equivalent to solving $(x\/4)\\tanh(x\/4)=1$. The solution is four times the solution to $x\\tanh(x)=1$ equation, which is 1.19967864... This implies maximum ln(OR) $= 4 \\times 1.19967864...=4.7987...$ and by substituting this value into Eq. (\\ref{gamma.max}) we obtain $\\gamma_{\\text{max}} = 0.6627...$, the LLC. The solution to KE\ninvolves the condition equivalent to Eq. (\\ref{eq.kappa}). Namely, the solution can be expressed as the power series in $\\varepsilon$, provided $|\\varepsilon \\sin(E)| < |E-M|$ and that $\\varepsilon < x \/ \\cosh(x), x = |E-M|$, which is the LLC.\\cite{plummer1918introductory} \n\n\nAlthough it appears that the LLC bound is a function of odds ratio alone, this bound can only be attained at the specific values of population parameters (or the respective sample values). Namely, (i) $v_m=w_m=1\/2$ from Eq. (\\ref{eq7}), which implies $\\text{RR}^2= \\left(\\frac{\\Pr(E|D)}{\\Pr(E|\\bar{D})}\\right)^2 = \\text{OR}$; (ii) $\\Pr(D|E)=1-\\Pr(D|\\bar{E})$ from Eqs. (\\ref{eq5} - \\ref{eq6}); and (iii) ln(OR) $= 4.7987...$ Next, by solving\n\\begin{eqnarray*}\n \\text{OR} &=& \\frac{\\Pr(D|E)\/(1-\\Pr(D|E))}{\\Pr(D|\\bar{E})\/(1-\\Pr(D|\\bar{E}))} \\\\\n &=& \\exp(4.7987\\dots) = 121.354\\dots\n\\end{eqnarray*}\nfor $\\Pr(D|E)$, we obtain:\n\\begin{eqnarray}\n \\Pr(D|E) = \\frac{1}{2\\,z} + \\frac{1}{2},\n\\end{eqnarray}\nwhere $z$ is the solution of $ z \\tanh(z)=1$, i.e., $z =1.19967864\\dots$ and $\\Pr(D|E) = \\Pr(E|D) = 0.9167782798\\dots$\nSimilarly, Pearson correlation between two binary variables ranges between -1 and 1, but this range is not free of parameters: these boundary values are possible only in the case when the population (or sample) frequencies of two binary variables are equal to each other. Moreover, these bounds are asymmetric, depending on the sign of the correlation.\\cite{weir1979inferences} \n\nThe range of the standardized statistic, -LLC to LLC, has implications for statistical analysis. For example, several recent publications on P\\nobreakdash-value{} replicability posed the following question: given a small initial P\\nobreakdash-value{}, what is a likely spread of P\\nobreakdash-values{} in subsequent replication studies?\\cite{halsey2015fickle,lai2012subjective,lazzeroni2014p,lazzeroni2016solutions,vrz2017bayesian} P\\nobreakdash-values{} for $\\ln(\\text{OR})$ are explicit functions of $\\gamma$ because they are defined as $P = \\Pr(Z > z_{\\alpha})$, where $Z = \\sqrt{N} \\gamma$ is asymptotically normal. The two-sided P\\nobreakdash-value{} can be similarly defined in terms of chi-square distributed $Z^2$. The prior distribution for the standardized effect size occurs naturally and needs to be specified in order to give probabilistic bounds for the spread of future replication P\\nobreakdash-values{}. In applications where prior distribution for the effect size is modeled in terms of ln(OR),\nour results allow one to specify a reasonable prior range for the standardized value.\n\n It has been suggested that summary association statistics can be converted to approximate posterior (Bayesian) summaries about parameters of interest. For example, one-sided P\\nobreakdash-value{}, $P$ for testing significance of $\\ln(\\text{OR})$ can be transformed to the normal test statistic, $Z = \\Phi^{-1}(1-P)$. This statistic is $Z=\\sqrt{N}\\ln(\\widehat{\\text{OR}})\/\\hat{\\sigma}$. An approximate Bayesian false discovery probability can be computed based only on the summary statistics $\\ln(\\widehat{\\text{OR}})$ and $\\hat{\\sigma}$, the value $N$, and an assumed variance parameter for the zero-mean prior normal distribution for $\\ln(\\text{OR})$.\\cite{wakefield2007bayesian,wakefield2009bayes} For any given value of OR, $\\mu=\\ln(\\text{OR})$, is fixed, but $\\sigma$ can vary as a function of $w$. The normal prior distribution for $\\mu$ can be characterized simply by $\\Pr(\\text{OR} > x) = \\beta$. Considering the standardized effect, we can write $\\beta = \\Pr(\\text{OR} > x) = \\Pr(\\mu \/ \\sigma > \\ln(x) \/ \\sigma)$.\nDenote the normal cumulative distribution function with the mean $a$ and variance $b$, evaluated at $x$ by $\\Phi(x|a,b)$, and its inverse by $\\Phi^{-1}(x|a,b)$. Then,\n$\\ln(x) \/\\sigma = \\Phi^{-1}(1-\\beta \\mid 0,\\sigma_0) = \\sqrt{\\sigma_0} \\Phi^{-1}(1-\\beta \\mid 0,1)$.\nFrom this, we can obtain the flattest possible prior distribution for $\\mu\/\\sigma$ as the zero-mean normal with variance\n\\begin{eqnarray}\n \\sigma_0 = \\left( \\frac{\\ln(x)\/\\sigma_m}{\\Phi^{-1}(1-\\beta \\mid 0,1)} \\right).\n\\end{eqnarray}\nFor $\\sigma_0$ to be as large as possible, $\\sigma_m$ should be equal to $\\ln(x)\/\\gamma_{\\max}$ (from Eq. \\ref{gamma.max}). Alternatively, either the value $\\sigma(w_m)$ or $\\sigma(v_m)$ can be specified with some additional assumptions. For example, for $\\sigma(w_m)$, $\\Pr(D|E)$ needs to be specified. Then,\n\\begin{eqnarray}\n \\Pr(D \\mid \\bar{E}) &=& \\frac{1}{1-\\text{OR} \\left(1 - \\Pr(D \\mid E)^{-1} \\right)}, \\\\\n \\text{RR} &=& \\Pr(D \\mid E) \/ \\Pr(D \\mid \\bar{E}),\n\\end{eqnarray}\nand $\\sigma(w_m)$ is obtained using the value $w_m$ from Eq. (\\ref{wm}). \\\\\n\n\n\\section*{Acknowledgements}\nThis research was supported in part by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n In the previous decade, various exciting investigations have been made on 4d, ${\\mathcal N}=1$ supersymmetric gauge theory.\n In string theory point of view, supersymmetric gauge theory can be realized \n as a low energy effective theory on D-branes.\n The low energy behavior of supersymmetric gauge theory has been widely analyzed by using gauge\/gravity correspondence.\n In particular, it has been known that the effective superpotential of ${\\mathcal N}=1$ supersymmetric gauge theory \n with an adjoint chiral superfield and a tree level superpotential can be evaluated\n from the gravity theory with flux \\cite{Vafa, CIV, CFIKV} and from the matrix model \\cite{DV}.\n These relations have been analyzed in \\cite{DGLVZ, CDSW, Seiberg2, CSW2} field-theoretically.\n \n Recently, some interesting results have been obtained in ${\\mathcal N}=1$ supersymmetric $U(N)$ gauge theory\n with the gauge kinetic term which depends on the adjoint chiral superfield, \n $\\Im \\int d^2 \\theta \\mathop{\\rm Tr} \\tau(\\phi) {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha$.\n In \\cite{IM, Ferrari}, it has been shown that the effective superpotential of such a theory\n is deformed compared to the theory with constant $\\tau$\n (in \\cite{IM}, a specific case where ${\\mathcal N}=2$ supersymmetry is spontaneously broken to ${\\mathcal N}=1$ \\cite{FIS, IMS}\n has been analyzed).\n Since there are non-renormalizable coupling constants, this theory should have UV completion.\n In \\cite{ABSV}, this theory is obtained as a low energy effective theory on D5-branes\n wrapped on $S^2$'s in Calabi-Yau 3-fold.\n The field-dependent gauge kinetic term is introduced by the integrals over $S^2$'s \n of non-trivial $B$-field flux which holomorphically varies on the base space (complex plane) of Calabi-Yau.\n By using geometric transition duality, \n the deformed superpotential \\cite{IM, Ferrari} has been derived from the gravity theory \\cite{ABSV}.\n Also, it has been argued that the deformation of the effective superpotential\n leads to the existence of supersymmetry breaking vacua in some cases of the parameters.\n (See also \\cite{OT} for IIA and M-theory perspectives \n and \\cite{Maruyoshi} for the case with partially broken ${\\mathcal N}=2$ supersymmetry.)\n \n In this paper, we study ${\\mathcal N}=1$ supersymmetric gauge theory with the flavors\n where the gauge kinetic term depends on the adjoint chiral superfields.\n In the first half of the paper, we analyze ${\\mathcal N}=1$ $A$-$D$-$E$ quiver gauge theories.\n As in \\cite{CKV, CFIKV, OTatar}, \n ${\\mathcal N}=1$ $A$-$D$-$E$ quiver gauge theories can be obtained as low energy effective theories\n on D5-branes wrapped on $S^2$'s in Calabi-Yau 3-folds (and D3-branes in affine case) in IIB string theory.\n More precisely, these Calabi-Yau 3-folds are constructed by fibering the non-singular spaces, \n which are obtained from the ALE spaces with $A$-$D$-$E$ singularity, over the complex plane $t$.\n As in \\cite{ABSV} for $A_1$ case, the non-trivial $B$-field flux which depends on $t$ is turned on \n in order to obtain the field-dependent gauge kinetic term.\n \n An interesting result of the string theory construction of $A$-$D$-$E$ quiver gauge theories is that\n the theory which is obtained by Weyl reflection on a node of the Dynkin diagram (or equivalently quiver diagram) \n is equivalent to the original one,\n due to ambiguities from the fact that there is no unique way to blow up the singularity.\n Since the nodes of the Dynkin diagrams correspond to $S^2$'s, Weyl reflection acts on the gauge kinetic term \n as well as the superpotential in the gauge theory. \n In \\cite{KlebanovS, CFIKV}, it has been analyzed, in the constant $\\tau$ case, that\n the theory obtained by Weyl reflection is a dual description\n (by Kutasov duality \\cite{Kutasov, ASY, KS, KSS, EGKRS}),\n after integrating out the meson fields and the flavors.\n (See also \\cite{ABSV0, ABF} for the case with antibranes.)\n \n In the case which we will consider in this paper, the gauge kinetic term is affected by Weyl reflection.\n Therefore, we propose an extension of the Kutasov duality:\n ${\\mathcal N}=1$ supersymmetric $U(N_c)$ gauge theory with an adjoint $\\phi$ and $N_f$ flavors $Q$ and $\\bar{Q}$,\n equipped with the gauge kinetic term where $\\tau(\\phi)$ is\n \\begin{eqnarray}\n \\tau(\\phi)\n = \\sum_{k = 0}^m t_k \\phi^k,\n \\end{eqnarray}\n and a superpotential, \n has a dual description which is $U(n N_f - N_c)$ gauge theory \n with an adjoint $\\tilde{\\phi}$, $N_f$ flavors $q$ and $\\bar{q}$ and meson fields,\n where the gauge kinetic term is \n \\begin{eqnarray}\n \\tilde{\\tau}(\\tilde{\\phi})\n = \\tilde{t}_0 - \\sum_{k = 1}^m t_k \\tilde{\\phi}^k\n \\end{eqnarray}\n and the dual superpotential is the same as the one in \\cite{Kutasov, KS, KSS}.\n In the latter half of this paper, we will analyze this duality\n from the field theoretical point of view.\n \n The dual superpotential and the dual gauge kinetic term can be determined by the consistency of the duality.\n In particular, a strong constraint is \n that both theories should split to decoupled SQCD theories at low energy \n and there exists a corresponding SQCD theory in the dual theory which is Seiberg dual \\cite{Seiberg} \n to each decoupled SQCD theory in the original one.\n The duality map of several operators can also verified \n by using the above argument and the generalized Konishi anomaly equations.\n (See also \\cite{Ferrarif} for a different analysis of this theory.)\n \n The organization of this paper is as follows.\n We introduce ${\\mathcal N}=1$ $A$-$D$-$E$ quiver gauge theories with the gauge kinetic terms \n which depend on the adjoint chiral superfields, promoting the gauge coupling constants $\\tau_i$ \n to the field-dependent functions $\\tau_i (\\phi_i)$, in section \\ref{sec:classical}.\n We will show that the classical equations of motion reduce to the same vacuum equations as those in the theories \n with constant $\\tau_i$.\n In section \\ref{sec:geometric}, we construct such theories in the framework of superstring theory.\n We then consider a duality: Weyl reflection on $A$-$D$-$E$ nodes in section 4.\n We will see that this induces a non-trivial action on the gauge kinetic term as well as the superpotential.\n In section \\ref{sec:fieldtheory}, we consider a non-trivial check of the duality proposal.\n Finally, we will analyze the duality map of the operators, in section \\ref{sec:dualitymap}.\n\n \n\\section{$A$-$D$-$E$ quiver gauge theories}\n\\label{sec:classical}\n In this section, we introduce ${\\mathcal N}=1$ $A$-$D$-$E$ quiver gauge theories.\n Throughout this paper, we consider the case where the gauge kinetic term depends on the adjoint chiral superfields. \n This is an extended version of the quiver gauge theories considered in \\cite{CKV, CFIKV, OTatar}.\n \n These theories are considered as a deformation of ${\\mathcal N}=2$ $A$-$D$-$E$ quiver gauge theories\n whose gauge groups are $\\prod_i U(N_i)$ and each gauge factor corresponds to each node of the quiver diagrams.\n The quiver diagrams are expressed by the Dynkin diagrams of non-affine or affine $G = A, D, E$ groups.\n In terms of ${\\mathcal N}=1$ superfields, ${\\mathcal N}=2$ quiver gauge theory consists of the vector superfields $V_i$ \n (or the field strength superfields ${\\mathcal W}^\\alpha_i$),\n the adjoint chiral superfields $\\phi_i$ and the matter chiral superfields $Q_{ij}$ and $Q_{ji}$ which\n are respectively in the bi-fundamental representations $(N_i, \\bar{N}_j)$ and $(\\bar{N}_i, N_j)$ \n of $U(N_i) \\times U(N_j)$ groups.\n ($i$ label the nodes of the quiver diagram.)\n We consider the case where the prepotential \n which determines the ${\\mathcal N}=2$ classical Lagrangian has higher order terms, that is, \n \\begin{eqnarray}\n {\\mathcal F}_i (\\Psi_i)\n = \\sum_{k=0} \\frac{t_{i, k}}{(k +1)(k +2)} \\Psi_i^{k + 2},\n \\end{eqnarray}\n where $\\Psi_i$ are the ${\\mathcal N}=2$ vector superfields which contain $\\phi_i$ and ${\\mathcal W}_i^\\alpha$ \n and $t_{i, k}$ are complex parameters.\n In ${\\mathcal N}=1$ superspace formalism, this leads to the field-dependent gauge kinetic term\n \\begin{eqnarray}\n \\sum_i \\Im \\int d^2 \\theta \\mathop{\\rm Tr} \\tau_i (\\phi_i) {\\mathcal W}^\\alpha_i {\\mathcal W}_{i \\alpha},\n \\end{eqnarray}\n and also the K\\\"ahler terms.\n Here $\\tau_i$ are related with the prepotentials as $2 \\tau_i (x) = {\\mathcal F}''_i (x)$.\n \n We add the superpotentials $W_i (\\phi_i)$ which break ${\\mathcal N}=2$ supersymmetry to ${\\mathcal N}=1$.\n We will choose these superpotentials to be polynomials of the same degree $n + 1$, for simplicity.\n Therefore, the holomorphic part of the Lagrangian is\n \\begin{eqnarray}\n \\sum_i \\left[ \\Im \\int d^2 \\theta \\mathop{\\rm Tr} \\tau_i (\\phi_i) {\\mathcal W}^\\alpha_i {\\mathcal W}_{i \\alpha}\n + \\int d^2 \\theta \\left( \\mathop{\\rm Tr} \\sum_j s_{ij} Q_{ij} Q_{ji} \\phi_i - \\mathop{\\rm Tr} W_i (\\phi_i) \\right)\n + h.c. \\right],\n \\end{eqnarray}\n where $s_{ij}$ is the intersection matrix of $i$-th and $j$-th nodes, which is zero if the nodes are not linked \n and $\\pm 1$ if linked (and they also satisfy $s_{ij} = - s_{ji}$).\n In the affine case, the following condition for the superpotentials:\n \\begin{eqnarray}\n \\sum_{i = 0}^r d_i W_i (x)\n = 0\n \\end{eqnarray}\n should be satisfied (where $d_i$ are the Dynkin indices), \n if we geometrically engineer this theory \\cite{CKV}.\n In the non-affine case, there is no restriction on the superpotentials.\n Note that in the case where $t_{i, k}= 0$ for $k>0$, i.e. constant $\\tau_i$ case, \n these theories reduce to the ones analyzed in \\cite{CKV, CFIKV, OTatar}.\n \n\\subsection{Classical equations of motion}\n\\label{subsec:eom}\n The solution of the classical equations of motion \n in the case where $t_{i, k}= 0$, for $k>0$ (constant $\\tau_i$ case) has been derived in \\cite{CKV}.\n In non-zero $t_{i, k}$ case, the equations of motion are slightly complicated, \n but reduce to the same conditions as those in \\cite{CKV}.\n Let us see this in this subsection.\n \n First of all, it is easy to see that the F-term equations are not changed \n compared to the case with constant $\\tau_i$.\n In fact, the gauge kinetic term which depends on the adjoint chiral superfields induces an additional term,\n $\\partial_{\\phi_i} \\tau_i (\\phi_i)$ multiplied by the fermion bilinear, \n in the F-term equations with respect to $\\phi_i$.\n However, the fermion does not get an expectation value in the classical vacua which we are interested in. \n Therefore, this term does not contribute to the solution.\n \n On the other hand, the equations of motion with respect to $D^a_i$ \n (where $a = 1, \\ldots, N^2_i$ label the gauge indices of $U(N_i)$ gauge group) lead to\n \\begin{eqnarray}\n D^a_i\n = \\frac{i}{2} (f_i)_{~bc}^a \\bar{\\phi}_i^b \\phi_i^c\n - (\\Im \\tau_i (\\phi_i)^{-1})^{ab} \n \\left[ \\mathop{\\rm Tr} \\sum_{j} s_{ij} (Q_{ij}^\\dag t_b^i Q_{ij} - Q_{ji} t_b^i Q^\\dag_{ji}) + h.c. \\right]\n = 0,\n \\label{D}\n \\end{eqnarray}\n where $(f_i)^{a}_{~bc}$ and $t_a^i$ are the structure constants and the generators in the fundamental of $U(N_i)$.\n Each matrix $(\\Im \\tau_i (\\phi_i)^{-1})^{ab}$ is defined as an inverse of $N^2_i \\times N_i^2$ matrix\n $\\mathop{\\rm Tr} (\\Im \\tau_i (\\phi_i) t_a^i t_b^i)$.\n While we have non-trivial factors $(\\Im \\tau_i (\\phi_i)^{-1})^{ab}$ in (\\ref{D})\n compared to the case with constant $\\tau_i$ (in this case, the factors are proportional to $\\delta^{ab}$),\n (\\ref{D}) results in\n \\begin{eqnarray}\n \\sum_j s_{ij} (Q_{ij} Q_{ij}^\\dag - Q^\\dag_{ji} Q_{ji} ) + h.c.\n = 0,\n \\label{D2}\n \\end{eqnarray}\n which are the same D-term conditions as those in the constant $\\tau_i$ case.\n This can be seen as follows.\n We are interested in the vacua where the scalar fields get the diagonal vev, \n i.e. non-Cartan parts of $\\langle \\phi_i \\rangle$ are zero.\n Under these, the first term of (\\ref{D}) is zero.\n Since $\\det ( \\Im \\tau_i (\\phi_i)^{-1} )^{ab} \\neq 0$, the solution is trivial and we obtain (\\ref{D2}).\n \n We have shown that the vacuum conditions following from the equations of motion \n are the same as those in \\cite{CKV}.\n Therefore, the structure of the classical vacua is also same.\n So, we only explain these here.\n \n For the non-affine case, the solutions of these equations are specified \n in terms of the positive roots $\\rho_K$ \\cite{CKV}, \n where $K=1, \\ldots, R_+$ with $2 R_+ + r = |G|$ and $r$ is the rank of $G$.\n In terms of the simple roots $e_i$, the positive roots can be written as\n \\begin{eqnarray}\n \\rho_K \n = \\sum_{i=1}^r n_K^i e_i,\n \\label{positiveroot}\n \\end{eqnarray}\n where $n_K^i$ are some positive integers.\n The equations of motion reduce to the following equations\n \\begin{eqnarray}\n W'_K (x) \n = \\sum_i n_K^i W'_i (x)\n = 0.\n \\label{vacuumcondition}\n \\end{eqnarray}\n Each of these equations has $n$ roots for each positive root $\\rho_K$ \n because we have chosen every superpotential is polynomial of degree $n+1$.\n We denote these roots as $x = a_{(p, K)}$ where $p =1, \\ldots, n$.\n Then, a supersymmetric vacuum is given by the value of $a_{(p, K)}$ \n with multiplicities $M_{(p, K)} \\geq 0$ which satisfy\n \\begin{eqnarray}\n N_i \n = \\sum_{K = 1}^{R_+} \\sum_{p = 1}^n M_{(p, K)} n_K^i.\n \\end{eqnarray}\n Furthermore, the gauge group is broken as \n \\begin{eqnarray}\n \\sum_i U(N_i) \n \\rightarrow \n \\sum_K \\sum_p U(M_{(p, K)}),\n \\end{eqnarray}\n by Higgsing.\n \n For the affine case, we have one additional node in quiver diagram and the gauge group is $\\sum_{i=0}^r U(N_i)$\n where $U(N_0)$ gauge group corresponds to that node.\n The classical vacua are similarly specified by the positive roots as above \\cite{CKV}.\n \n\\section{Geometric construction}\n\\label{sec:geometric}\n The above gauge theories can be realized as low energy effective theories \n on D5-branes wrapped on 2-cycles of Calabi-Yau 3-folds in the non-affine case \n and additional D3-branes in the affine case.\n These Calabi-Yau 3-folds are constructed by non-trivially fibering the ALE spaces with $A$-$D$-$E$ singularity \n over the complex plane.\n The important difference between the quiver gauge theory constructed above\n and the one in \\cite{CKV, CFIKV} is the gauge kinetic term.\n As considered in \\cite{ABSV} for $A_1$ case, the field dependent gauge kinetic term can be engineered geometrically\n by introducing the non-trivial $B$-field depending on the complex plane which is the base space of Calabi-Yau 3-fold.\n \n We consider the ALE spaces with $A$-$D$-$E$ singularity at the origin, which can be viewed \n as the hypersurfaces $f(x, y, z) = 0$ with, e.g. for $A_{r}$ singularity,\n \\begin{eqnarray}\n f = x^2 + y^2 + z^{r+1},\n \\end{eqnarray}\n where $x, y, z \\in \\textbf{C}$.\n By deforming these by relevant deformations, \n we obtain non-singular spaces, whose defining equations are, in $A_r$ case,\n \\begin{eqnarray}\n f = x^2 + y^2 + \\prod_{i = 1}^{r+1} (z + t_i),\n ~~\n \\sum_{i = 1}^{r + 1} t_i = 0,\n \\label{defining}\n \\end{eqnarray}\n where $t_i$ are deformation parameters and there are $r$ independent classes of non-vanishing $S^2$'s.\n These classes intersect according to the corresponding $A$-$D$-$E$ Dynkin diagrams.\n The holomorphic volumes of $S^2$'s are defined \n by integrals of the holomorphic 2-form $\\omega = dx dy\/z$ as $\\alpha_i = \\int_{S^2_i} \\omega$.\n These $\\alpha_i$ are simply related to $t_i$ in (\\ref{defining}) by, in $A_r$ case,\n \\begin{eqnarray}\n \\alpha_i \n = t_i - t_{i + 1}.\n \\label{alpha}\n \\end{eqnarray}\n For $G = D,E$ cases, the constructions are similar to the above \\cite{CKV}.\n \n We consider the fibrations of these spaces over the complex plane.\n We denote the coordinate of this plane as $t$.\n If there exists D5-branes wrapping on the above $S_i^2$ and occupying the $R^{1,3}$ direction, \n we obtain 4d non-affine $A$-$D$-$E$ quiver gauge theories as low energy effective theories on the D5-branes, \n whose field contents are the same as those in section \\ref{sec:classical}.\n If we include the D3-branes occupying $R^{1,3}$ direction, \n the gauge theory becomes the affine quiver gauge theory \\cite{DM, CKV, CFIKV}.\n Note that since $t$ plane is orthogonal to the $S^2$'s on which D5-branes wrap,\n it parametrizes the positions of the D-branes.\n Thus, $t$ corresponds to the vacuum expectation value of the adjoint chiral superfield on the D-brane world volume.\n \n In type IIB string theory, there are NSNS field $B^{NS}$ and RR field $B^R$.\n Geometrically, the complexified gauge coupling of each gauge factor of the quiver gauge theory\n corresponds to the integral over corresponding $S^2_i$ of Calabi-Yau 3-fold:\n \\begin{eqnarray}\n \\left( \\frac{\\theta}{2 \\pi} + \\frac{4 \\pi i}{g^2_{YM}} \\right)_i\n = \\int_{S^2_i} \\left( B^R + \\frac{i}{g_s} B^{NS} \\right).\n \\end{eqnarray}\n Note that we have set the K\\\"ahler parameters to zero: $r_i \\equiv \\int_{S_i^2} k = 0$ where $k$ is the K\\\"ahler form.\n As in \\cite{CKV, CFIKV}, if the background $B$-fields do not have any $t$-dependence, \n the above quantities are constants and denote the complexified gauge coupling constants.\n On the other hand, if the background $B$-fields depend on $t$ as in \\cite{ABSV}, we obtain\n \\begin{eqnarray}\n \\tau_i (t)\n \\equiv\n \\left( \\frac{\\theta}{2 \\pi} + \\frac{4 \\pi i}{g^2_{YM}} \\right)_i (t)\n = \\int_{S^2_i} \\left( B^R (t) + \\frac{i}{g_s} B^{NS} (t) \\right),\n \\label{tau}\n \\end{eqnarray}\n which produce the field-dependent gauge kinetic term of the effective theory on the D-branes,\n as in section \\ref{sec:classical}.\n The point is that in order not to break the ${\\mathcal N}=2$ supersymmetry, \n $B$-fields should {\\it holomorphically} depend on $t$ \\cite{KN}.\n Indeed, the dual IIB supergravity solution of this brane set-up, which has ${\\mathcal N}=2$ supersymmetry in 4d, \n can be obtained assuming that the dilaton is constant.\n Generically, $t$-dependent $B$-fields induce the source term in the dilaton equation of motion.\n However, such a source term vanishes in the case with holomorphically $t$-dependent $B$-fields \\cite{KN}.\n Therefore, the dilaton remains constant in this case and ${\\mathcal N}=2$ supersymmetry is not broken.\n \n The superpotentials can be turned on \n by considering the non-trivial fibration of the ALE space over $t$-plane,\n promoting $\\alpha_i$ (\\ref{alpha}) to be dependent on $t$: $\\alpha_i = \\alpha_i (t)$.\n These $\\alpha_i$ give the superpotential $W'_i (z) = \\alpha_i (z)$.\n We only consider the non-monodromic fibration where $\\alpha_i$ are the single-valued functions of $t$ \n as it leads to the single trace functions $W_i (\\phi_i)$ in section \\ref{sec:classical}.\n Also, we choose all the degrees of the superpotentials to be $n + 1$.\n In this case, there are $n$ points in $t$ plane for each positive root $\\rho_K$ (\\ref{positiveroot})\n where the holomorphic volume becomes zero\n \\begin{eqnarray}\n W'_K (t)\n \\equiv\n \\sum_{i =1}^r n^i_K W'_i(t)\n = 0.\n \\label{eom}\n \\end{eqnarray}\n These equations correspond to the conditions for the supersymmetric vacua \n and are same as those obtained in the gauge theory (\\ref{vacuumcondition}).\n The roots of (\\ref{eom}) are expressed as $t = a_{(p, K)}$ where $p = 1, \\ldots n$.\n As noted above, these values correspond to the positions of D-branes and, \n therefore, the vacuum expectation values of $\\phi_i$.\n \n Therefore, we have geometrically engineered the ${\\mathcal N}=1$ $A$-$D$-$E$ quiver gauge theories\n with the field-dependent gauge kinetic term, which have been considered in the previous section.\n This construction is a simple generalization of $A_1$ case \\cite{ABSV} to other quiver cases.\n \n Now we will comment on an important point which arises from the non-trivial fields background.\n Note that the background $B$-fields (\\ref{tau}) and the positions of D-branes $a_{(p, K)}$ determine \n the classical gauge coupling constant of each gauge factor $U(M_{(p, K)})$\n \\begin{eqnarray}\n \\left( \\frac{4 \\pi}{g^2} \\right)_{(p, K)}\n = \\sum_{i =1}^r n^i_K \\Im \\tau_i (a_{(p, K)}).\n \\end{eqnarray}\n The crucial point is that, in contrast to the case in \\cite{CFIKV}, these quantities could be negative\n for generic choice of the background fields and the superpotentials.\n This implies that the field theoretical description is ill-defined in that case, \n but from the string theory point of view, the case where some of the squared gauge coupling constants are negative \n arises from antibranes wrapping on the corresponding $S^2$'s\\footnote{\n In $A_1$ case, if all the squared gauge coupling constants are negative, \n a better field theoretical description which is supersymmetry breaking model by spurion fields \\cite{LM, GMT}\n has been proposed in \\cite{ABSV}.}.\n \n\\section{Duality in string theory}\n\\label{sec:weyl}\n As considered in \\cite{CFIKV}, there are two types of duality in the above theories.\n The one is the geometric transition duality \\cite{Vafa} \n and the other one corresponds to Weyl reflection of $A$-$D$-$E$ groups.\n In this paper, we only consider the latter type.\n \n Weyl reflection about the simple root $e_{i_0}$ of $A$-$D$-$E$ group can be viewed as the following action\n on the simple roots:\n \\begin{eqnarray}\n e_i\n \\rightarrow\n e_i - (e_i \\cdot e_{i_0}) e_{i_0},\n \\end{eqnarray}\n where the inner product of the simple roots is normalized as follows:\n $e_i \\cdot e_{i_0}$ are 2 for $i = i_0$, $-1$ for $i$ connected with $i_0$ node and 0 for the other $i$.\n In the Calabi-Yau geometry, this corresponds to the change of $S^2$'s\n and leads to the following action on $\\tau_i$ and the polynomial parts of the superpotential:\n \\begin{eqnarray}\n \\tau_i (\\phi_i)\n \\rightarrow\n \\tau_i (\\phi_i) - (e_i \\cdot e_{i_0}) \\tau_{i_0} (\\phi_i),\n ~~~\n W_i (\\phi_i)\n \\rightarrow\n W_i (\\phi_i) - (e_i \\cdot e_{i_0}) W_{i_0} (\\phi_i).\n \\label{weyl}\n \\end{eqnarray}\n The action of the Weyl reflection on the superpotentials are exactly same as those in \\cite{CFIKV}.\n But, since the gauge couplings $\\tau_i$ are polynomials of $\\phi_i$, \n the higher order terms in $\\phi_i$ are also affected by the Weyl reflection.\n This induces non-trivial action on the coupling constants $t_{i, k}$ in $\\tau_i$.\n In fact, in the case with constant $\\tau_i$, \n this reduces to the action on the gauge coupling constants, as in \\cite{CFIKV}:\n $(1\/g^2)_i \\rightarrow (1\/g^2)_i - (e_i \\cdot e_{i_0}) (1\/g^2)_{i_0}$.\n \n The different looking gauge theory obtained by Weyl reflection \n should be equivalent to the original one from the string theory perspective \\cite{CFIKV}.\n This is due to ambiguities which come from the fact that \n there is no unique way to blow up the singularity and we can determine a quiver gauge theory up to Weyl group action.\n Since the total brane charge must be conserved, \n the ranks of the gauge groups after the transition are related with the original ranks as\n \\begin{eqnarray}\n \\sum_i N'_i e'_i\n = \\sum_i N_i e_i.\n \\end{eqnarray}\n Hence, the ranks of the gauge groups are changed under the Weyl reflection about $e_{i_0}$\n as $N'_{i_0} = N_f - N_{i_0}$ and $N'_i = N_i$ for $i\\neq i_0$\n where $N_f$ is the number of flavors of $U(N_{i_0})$ gauge theory \n when the other gauge symmetries are considered as flavor symmetries\n and $N_f \\equiv \\sum_{i\\neq i_0} (- e_i \\cdot e_{i_0}) N_i$.\n Note that the number of flavors is not changed under the Weyl reflection.\n \n As discussed in \\cite{CFIKV}, this kind of duality can be considered as ${\\mathcal N}=1$ electric-magnetic duality \n \\cite{Seiberg, IS, Kutasov, KS, KSS, ASY} in the framework of the gauge theory.\n (See also \\cite{EGK, OV, BRP, FHHU} for related approaches.)\n However, as seen above, the duality induces the non-trivial action on the field-dependent gauge kinetic term.\n This is a first example for the electric-magnetic duality in the case with the field dependent gauge kinetic term.\n We will call this as extended electric-magnetic duality.\n Fortunately, string theory has suggested that such a duality exists.\n We will check this duality field-theoretically in the subsequent sections.\n \n Before going to next, let us see the action on the superpotentials and on $\\tau_i$ more explicitly.\n First of all, the action on the superpotentials (\\ref{weyl}) can be written as \n \\begin{eqnarray}\n W'_{i} (\\phi_{i})\n = \\left\\{ \n \\begin{array}{ll}\n - W_{i} (\\phi_{i}), & {\\rm for} ~i=i_0,\\\\\n W_{i} (\\phi_i) + W_{i_0} (\\phi_i),~~ & {\\rm for} ~i ~{\\rm connected ~with}~i_0, \\\\\n W_{i} (\\phi_i), & {\\rm for~the ~ other} ~ i.\n \\end{array}\n \\right.\n \\label{weylsuperpot}\n \\end{eqnarray}\n Also, for the coefficients of the gauge kinetic terms, Weyl reflection acts as\n \\begin{eqnarray}\n \\tau'_{i} (\\phi_{i})\n = \\left\\{ \n \\begin{array}{ll}\n - \\tau_{i} (\\phi_{i}), & {\\rm for}~ i=i_0,\\\\\n \\tau_{i} (\\phi_i) + \\tau_{i_0} (\\phi_i), ~~ & {\\rm for} ~i ~{\\rm connected ~with}~i_0, \\\\\n \\tau_{i} (\\phi_i), & {\\rm for~the ~ other} ~ i,\n \\end{array}\n \\right.\n \\label{weylgauge}\n \\end{eqnarray}\n Let us concentrate on the gauge theory on the $i_0$-th node.\n If we treat the gauge symmetries of the linked nodes as the weakly gauged flavor symmetries,\n we obtain $U(N_{i_0})$ gauge theory with a superpotential \n \\begin{eqnarray}\n W\n = \\sum_{k=1}^n \\frac{g_k}{k + 1} \\mathop{\\rm Tr} \\phi^{k+1}\n + \\tr \\bar{Q} \\phi Q + \\tr m \\bar{Q} Q,\n \\label{superpotential}\n \\end{eqnarray}\n where $Q$ and $\\bar{Q}$ are $N_f$ fundamental and anti-fundamental superfields.\n The symbol $\\tr$ denotes the trace over the flavor indices.\n These come from the bi-fundamental superfield connecting $i_0$-th node with the neighboring nodes.\n The mass term for $Q$ and $\\bar{Q}$ is due to $\\phi_{i'} Q_{i' i_0} Q_{i_0 i'}$ term \n of the neighboring nodes by giving a vev of $\\phi_{i'}$.\n Also, let the gauge kinetic term of this theory be\n \\begin{eqnarray}\n \\mathop{\\rm Tr} \\tau_{i_0} (\\phi) {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha\n = \\sum_{k= 0}^n t_{k} \\mathop{\\rm Tr} (\\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha),\n \\end{eqnarray}\n where we have simplified the notation of the coupling in $\\tau_{i_0}$ as $t_{i_0, k} \\equiv t_k$.\n \n The Weyl action changes the gauge group to $U(N_f - N_{i_0})$ ($N_f \\equiv \\sum_{i\\neq i_0} (- e_i \\cdot e_{i_0}) N_i$)\n and the superpotential (\\ref{superpotential}) to \n \\begin{eqnarray}\n \\widetilde{W}\n = - \\sum_{k = 1}^n \\frac{g_k}{k + 1} \\mathop{\\rm Tr} \\tilde{\\phi}^{k + 1}\n + \\sum_{k = 1}^n \\frac{g_k}{k + 1} \\tr m^{k + 1}\n + \\tr \\bar{q} \\tilde{\\phi} q + \\tr m \\bar{q} q,\n \\label{dualsuperpotential}\n \\end{eqnarray}\n where $\\tilde{\\phi}$ is an adjoint field of $U(N_f - N_{i_0})$ gauge group \n and $q$ and $\\bar{q}$ are the $N_f$ fundamentals and anti-fundamentals.\n The minus sign of the first term reflects \n the Weyl action on the superpotential $W_{i_0}$ (\\ref{weylsuperpot}).\n The second term comes from the Weyl action on the superpotentials of the nodes linked to $i_0$ node.\n (The trace of this term is taken over the flavor indices.)\n Furthermore, the gauge kinetic term of the dual theory becomes\n \\begin{eqnarray}\n - \\sum_{k = 0}^n t_{k} \\mathop{\\rm Tr} (\\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha),\n \\label{dualgaugekineticterm}\n \\end{eqnarray}\n where $\\widetilde{{\\mathcal W}}^\\alpha$ is the field strength superfield of the dual theory.\n \n In the theory corresponding to a node connected with $i_0$-th node, \n as noted above, the dual superpotential of this theory contributes to the second term of (\\ref{dualsuperpotential})\n because the Weyl reflection induces an additional term $W_{i_0} (\\tilde{\\phi}_i)$ (\\ref{weylsuperpot})\n where $\\tilde{\\phi}_i$ is an chiral superfield of this dual theory.\n On the other hand, the gauge kinetic term is affected as follows:\n \\begin{eqnarray}\n \\tau_i (\\phi_i)\n \\rightarrow\n \\tau_i (\\tilde{\\phi}_i) + \\tau_{i_0} (\\tilde{\\phi}_i)\n = \\tau_i (\\tilde{\\phi}_i) + \\sum_{k = 0}^n t_k \\tilde{\\phi}_i^k,\n \\end{eqnarray}\n as easily extracted from (\\ref{weylgauge}).\n\n\\section{Extended electric-magnetic duality}\n\\label{sec:fieldtheory}\n We have seen that the string theory construction has suggested an extension of ${\\mathcal N}=1$ electric-magnetic duality\n to the case where the gauge kinetic term depends on the adjoint chiral superfields.\n In what follows, we concentrate on a particular node of the quiver and consider the duality \n from field-theoretical point of view.\n \n \n Let us specify the model.\n Consider ${\\mathcal N}=1$, $U(N_c)$ gauge theory with an adjoint chiral superfield $\\phi$ \n and $N_f$ fundamental and anti-fundamental superfields $Q$ and $\\bar{Q}$, \n and also with a gauge kinetic term which depends on the adjoint chiral superfield:\n \\begin{eqnarray}\n \\int d^2 \\theta \\mathop{\\rm Tr} \\tau (\\phi) {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha,\n ~~~\n \\tau (\\phi)\n = \\sum_{k=0}^{m} t_k \\phi^k,\n \\end{eqnarray}\n and a superpotential (\\ref{superpotential})\n \\begin{eqnarray}\n W\n = \\sum_{k=1}^n \\frac{g_k}{k + 1} \\mathop{\\rm Tr} \\phi^{k+1}\n + \\tr \\bar{Q} \\phi Q + \\tr m \\bar{Q} Q.\n \\label{fullsuperpotential}\n \\end{eqnarray}\n Without loss of the generality, the mass matrix $m$ can be chosen to be diagonal.\n We will use the indices $f = 1, \\ldots, N_f$ to label the flavors.\n In this notation, the diagonal components of the mass matrix are written as $m_f$.\n\n The vacua of this theory can be divided into two types: confining and Higgs vacua.\n Classically, the confining vacua correspond to \n the vacuum expectation values such that \n $\\langle Q \\rangle = \\langle \\bar{Q} \\rangle = 0$ \n and \n \\begin{eqnarray}\n \\langle \\phi \\rangle = \\diag (a_1, a_2, \\ldots, a_N),\n \\label{vevphi}\n \\end{eqnarray} \n where $a_i$ are determined from the solutions of the F-term equation: \n \\begin{eqnarray}\n W'(x)\n \\equiv\n g_n \\prod_{i = 1}^n (x - a_i)\n = 0.\n \\label{F-term}\n \\end{eqnarray}\n Note that the other terms contributing to the F-term equation vanish in these vacua.\n Indeed, as we have seen in the quiver case, \n the gauge kinetic term, $\\tau(\\phi) {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha$, produces an additional term in the F-term equation\n such as $\\partial_\\phi \\tau (\\phi) \\lambda^\\alpha \\lambda_\\alpha$ where $\\lambda^\\alpha$ is the gluino, \n but this term vanishes because we are interested in the vacua where the vacuum expectation values of the fermions are zero.\n \n The Higgs vacua correspond to the case \n where some of the diagonal elements of $\\langle \\phi \\rangle$ are equal to the mass parameters\n and $Q$ and $\\bar{Q}$ have non-zero vacuum expectation values which are determined from the F-term equation:\n \\begin{eqnarray}\n (W'(\\phi))_{ij} + \\sum_f Q_{j}^f \\bar{Q}_{i f} \n = 0,\n \\label{F-termHiggs}\n \\end{eqnarray}\n where $i, j = 1, \\ldots, N$ are the gauge indices.\n As above, the gauge kinetic term does not contribute to the classical equation (\\ref{F-termHiggs}).\n \n In subsection \\ref{subsec:singletrace}, we begin to consider the case without $\\bar{Q} \\phi Q$ and $m \\bar{Q}Q$ terms.\n In this case, the flavors are massless and, after integrating out the adjoint fields,\n the theory splits into a set of the decoupled SQCD theories with the massless flavors.\n Therefore, the stable vacua exist if \\cite{KS}\n \\begin{eqnarray}\n \\frac{N_c}{n} \\leq N_f.\n \\label{stable}\n \\end{eqnarray}\n We will see the dual description of the above theory, after reviewing the constant $\\tau$ case.\n Then, we will turn to the case with full superpotential (\\ref{fullsuperpotential}) in subsection \\ref{subsec:full}.\n \n\\subsection{Single trace superpotential case}\n\\label{subsec:singletrace}\n We consider the case where the superpotential is \n \\begin{eqnarray}\n W\n = \\sum_{k = 1}^n \\frac{g_k}{k + 1} \\mathop{\\rm Tr} \\phi^{k + 1}.\n \\label{electricsuperpotential}\n \\end{eqnarray}\n We first review the case where $\\tau$ is constant, i.e. $t_k=0$ for $k > 0$.\n In this case, the dual description of the theory has been obtained in \\cite{Kutasov, KS, KSS}, \n which is $U(\\tilde{N}_c)$ ($\\tilde{N}_c = n N_f - N_c$) gauge theory with $N_f$ fundamental and anti-fundamental superfields $q$ and $\\bar{q}$,\n gauge singlet superfields $M_i$ ($i = 1, \\ldots, n$) and an adjoint chiral superfield $\\tilde{\\phi}$.\n The singlet fields $M_i$ are identified with the meson superfields in the original theory as\n \\begin{eqnarray}\n M_i \n = \\bar{Q} \\phi^{i - 1} Q, \n ~~~\n i = 1, \\ldots, n.\n \\label{matching}\n \\end{eqnarray}\n It is not necessary to introduce the other meson fields corresponding to $\\bar{Q} \\phi^\\ell Q$ ($\\ell > n$), \n since such fields can be eliminated by the chiral ring relation.\n In addition, the superpotential of the dual theory is \\cite{KSS}\n \\begin{eqnarray}\n \\widetilde{W}\n = - \\sum_{k=1}^n \\frac{g_k}{k + 1} \\mathop{\\rm Tr} \\tilde{\\phi}^{k + 1} \n + \\frac{1}{\\mu^2} \\sum_{k=1}^{n} g_k \\sum_{i = 1}^k M_i \\bar{q} \\tilde{\\phi}^{k - i} q,\n \\label{magneticsuperpotential}\n \\end{eqnarray}\n where a parameter $\\mu$ has been introduced in order for the dimension of the second term to be correct.\n This duality is a generalization of the electric-magnetic duality (Seiberg duality) in ${\\mathcal N}=1$ SQCD \\cite{Seiberg}\n to the case with an adjoint chiral superfield and a tree level single trace superpotential.\n Below we refer to the original and dual theories as electric and magnetic theories respectively.\n \n The global symmetries of both theories are the same:\n there is $SU(N_f) \\times SU(N_f) \\times U(1)_R \\times U(1)_J$ symmetry.\n The $U(1)$ charges of the fields (and the parameters) of the electric theory are in Table \\ref{tab1},\n \\begin{table}[t]\n \\begin{center}\n \\begin{tabular}{c|c|c|c|c|c|c|c}\n & $\\phi$ & ${\\mathcal W}^\\alpha$ & $Q$ (or $\\bar{Q}$) & $g_k$ & $\\Lambda^{2 N_c - N_f}$ \\\\ \\hline\n $U(1)_R$ & $2$ & $1$ & $0$ & $- 2 k$ & $2 (2 N_c - N_f)$ \\\\ \\hline\n $U(1)_J$ & $0$ & $1$ & $1$ & $2$ & $0$ \\\\\n \\end{tabular}\n \\caption{$U(1)$ charges of the electric fields and the parameters.}\n \\label{tab1}\n \\end{center}\n \\end{table}\n where $\\Lambda$ is the dynamical scale of the electric theory\n and the superspace coordinate $\\theta$ also has charge 1.\n Note that we have allowed the coupling constants $g_k$ and $t_k$ transform non-trivially,\n as $U(1)_R$ and $U(1)_J$ become the symmetries with the superpotential and the gauge kinetic term. \n Also, the charges of the fields of the magnetic theory are in Table \\ref{tab2},\n \\begin{table}[t]\n \\begin{center}\n \\begin{tabular}{c|c|c|c|c|c|c|c|c}\n & $\\tilde{\\phi}$ & {\\small $\\widetilde{{\\mathcal W}^\\alpha}$} & $q$ (or $\\bar{q}$) & $M_\\ell$ & $g_k$ & $\\mu$ & $\\tilde{\\Lambda}^{2 \\tilde{N}_c - N_f}$ \\\\ \\hline\n {\\small $U(1)_R$} & $2$ & $1$ & $\\frac{(n - 1)(N_c - \\tilde{N}_c)}{\\tilde{N}_c}$ & {\\small $2 (\\ell - 1)$} & $- 2 k$ & {\\footnotesize $-2+$}$ \\frac{(n - 1)(N_c - \\tilde{N}_c)}{\\tilde{N}_c}$ & {\\footnotesize $4 \\tilde{N_c} - 2 N_f +$}$\\frac{2 N_f (n - 1)(N_c - \\tilde{N}_c)}{\\tilde{N}_c}$ \\\\ \\hline\n {\\small $U(1)_J$} & $0$ & $1$ & $\\frac{N_c}{\\tilde{N}_c}$ & $2$ & $2$ & $1 + \\frac{N_c}{\\tilde{N}_c}$ & $\\frac{2 N_f (N_c - \\tilde{N}_c)}{\\tilde{N}_c}$ \\\\\n \\end{tabular}\n \\caption{$U(1)$ charges of the magnetic fields and the parameters.}\n \\label{tab2}\n \\end{center}\n \\end{table}\n where $\\tilde{\\Lambda}$ is the dynamical scale of the magnetic theory.\n A non-trivial check of this duality is to compare the 't Hooft anomalies of the theories. \n It has been shown that they perfectly match in the case with the truncated superpotential \\cite{KS}.\n \n An important ingredient of ${\\mathcal N}=1$ duality is the matching relation of the dynamical scales.\n In the case here, the relation is\n \\begin{eqnarray}\n \\Lambda^{2 N_c - N_f} \\tilde{\\Lambda}^{2 \\tilde{N}_c - N_f}\n = g_n^{- 2 N_f} \\mu^{2 N_f}.\n \\end{eqnarray}\n One can easily check that this relation is consistent with the above charge assignment.\n Also, in \\cite{KSS}, it has been shown that this is consistent with the deformations of the theory \n by the mass terms of the flavors.\n \n It is worth noting that, on general grounds,\n the coefficients of $\\mathop{\\rm Tr} \\tilde{\\phi}^{k + 1}$ and $M_i \\bar{q} \\tilde{\\phi}^{k - i} q$ \n in the magnetic superpotential are generic functions of $g_k$.\n However, we can fix these coefficients as in (\\ref{magneticsuperpotential}).\n First of all, in the electric theory, according to (\\ref{vevphi}), the gauge symmetry is broken to $\\prod_{i = 1}^n U(r_i)$ where $\\sum_i r_i = N_c$.\n ($r_i$ denote the number of the eigenvalues of $\\langle \\phi \\rangle$ which are equal to $a_i$.)\n Supposing that the underlying $U(N_c)$ gauge theory is weakly coupled at the mass scale \n which is specified by the above superpotential, \n the theory splits in the low energy into a set of decoupled SQCD theories \n with $U(r_i)$ gauge groups and $N_f$ flavors\n \\footnote{This is the case where all the roots of (\\ref{F-term}) are different from each other.\n In the case where some of $a_i$ coincide, i.e. $W' = \\prod_{i = 1}^r (x - a_i)^{n_i}$ ($r,\n \\nonumber \\\\\n T(z)\n &=& \\left< \\mathop{\\rm Tr} \\frac{1}{z - \\phi} \\right>,\n \\nonumber \\\\\n M(z)_f^{f'}\n &=& \\left< \\bar{Q}_f \\frac{1}{z - \\phi} Q^{f'} \\right>,\n \\end{eqnarray}\n where we have ignored the fermionic one-point function.\n The generalized Konishi anomaly equations in terms of these variables are\\footnote{\n These anomaly equations were derived also in \\cite{Ferrarif} recently.}\n :\n \\begin{eqnarray}\n R(z)^2\n &=& \\Bigg[ W'(z) R(z) \\Bigg]_-,\n \\nonumber \\\\\n 2 R(z) T(z)\n &=& \\Bigg[ W'(z) T(z) \\Bigg]_- + 32 \\pi^2 i \\Bigg[ \\tau'(z) R(z) \\Bigg]_- + M(z),\n \\nonumber \\\\\n - \\delta^{f'}_{f} R(z)\n &=& \\Bigg[ M(z)_f^{f'} (z + m_{f'}) \\Bigg]_-,\n \\label{GKAelectric}\n \\end{eqnarray}\n which can be obtained by generalizing the arguments in \\cite{CDSW, Seiberg2, CSW2, IM, Ferrari}.\n In the last equation, the flavor index $f'$ is not contracted.\n \n In the magnetic theory, we can also define\n \\begin{eqnarray}\n \\widetilde{R}(z)\n &=& - \\frac{1}{64 \\pi^2} \\left< \\mathop{\\rm Tr} \\frac{\\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha}{z - \\tilde{\\phi}} \\right>,\n \\nonumber \\\\\n \\widetilde{T}(z)\n &=& \\left< \\mathop{\\rm Tr} \\frac{1}{z - \\tilde{\\phi}} \\right>,\n \\nonumber \\\\\n \\widetilde{M}(z)_f^{f'}\n &=& \\left< \\bar{q}_f \\frac{1}{z - \\tilde{\\phi}} q^{f'} \\right>.\n \\end{eqnarray}\n In terms of these, the anomaly equations can be obtained as\n \\begin{eqnarray}\n \\widetilde{R}(z)^2\n &=& - \\Bigg[ W'(z) \\widetilde{R}(z) \\Bigg]_-,\n \\nonumber \\\\\n 2 \\widetilde{R}(z) \\widetilde{T}(z)\n &=& - \\Bigg[ W'(z) \\widetilde{T}(z) \\Bigg]_- - 32 \\pi^2 i \\Bigg[ \\tau'(z) \\widetilde{R}(z) \\Bigg]_- \n + \\Bigg[ \\widetilde{M}(z) A'(z) \\Bigg]_-,\n \\nonumber \\\\\n - \\delta^{f'}_{f} \\widetilde{R}(z)\n &=& \\Bigg[ \\widetilde{M}(z)_f^{f''} A(z)_{f''}^{f'} \\Bigg]_-,\n \\label{GKAmagnetic}\n \\end{eqnarray}\n where\n \\begin{eqnarray}\n A(z)\n = \\frac{1}{\\mu^2} \\sum_{k = 1}^n g_k \\sum_{i = 1}^k M_i z^{k - i}.\n \\end{eqnarray}\n Note that $M_i \\bar{q} \\phi^{k - i} q$ terms in the magnetic superpotential do not contribute \n to the first equation of (\\ref{GKAmagnetic})\n because the terms with $\\bar{q} {\\mathcal W}^\\alpha$ and ${\\mathcal W}^\\alpha q$ are zero in the chiral ring.\n \n Another important point of these anomaly equations (\\ref{GKAelectric}) and (\\ref{GKAmagnetic}) is that \n the $\\phi$(or $\\tilde{\\phi}$)-dependence of the gauge kinetic term does not affect \n the anomaly equation for $R(z)$ (or $\\widetilde{R}(z)$), as noted in \\cite{IM, Ferrari}.\n In other words, $t_k$ and $\\tilde{t}_k$ do not enter in those equations.\n This is crucial in the analysis in subsequent subsections.\n\n\\subsection{Duality map of $\\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha$ operators}\n\\label{subsec:R}\n In this subsection, we consider the operators $\\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}^\\alpha$\n ($\\mathop{\\rm Tr} \\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}^\\alpha$ in the magnetic theory).\n We expect from the argument in previous section that the following duality map of the operators are satisfied:\n \\begin{eqnarray}\n \\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha\n = - \\mathop{\\rm Tr} \\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha.\n \\label{operatormatching}\n \\end{eqnarray}\n We will check this relation in the vacuum.\n It should be noted that in the case without $\\bar{Q} \\phi Q$ and $m \\bar{Q}Q$ terms, \n the argument in the rest of this section might be invalid.\n More precisely, we obtain $\\langle \\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha \\rangle = 0$, \n as we can see from the anomaly equation for $M(z)$.\n This is because the flavors remain massless at IR.\n Therefore, we will consider the full superpotential (\\ref{fullsuperpotential}) below.\n \n Since $\\langle \\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha \\rangle = 0$ in the classical vacuum,\n the classical analysis cannot be non-trivial check of the duality map (\\ref{operatormatching}).\n However, they could have non-zero expectation values in the quantum vacuum, as can be seen from the anomaly equations.\n Indeed, we can relate $R(z)$ to $\\widetilde{R}(z)$ \n by using the generalized Konishi anomaly equations and this will be a non-trivial check of (\\ref{operatormatching}).\n Let us see this below.\n \n The generalized Konishi anomaly equations (\\ref{GKAelectric}) and (\\ref{GKAmagnetic}) for $R(z)$ and $\\widetilde{R}(z)$,\n can be rewritten as\n \\begin{eqnarray}\n R(z)^2\n = W'(z) R(z) + \\frac{f(z)}{4},\n ~~~\n \\widetilde{R}(z)^2\n = - W'(z) \\widetilde{R}(z) + \\frac{\\tilde{f}(z)}{4},\n \\end{eqnarray}\n where $f(z)$ and $\\tilde{f}(z)$ are the polynomials of degree $n - 1$.\n These equations can be easily solved as\n \\begin{eqnarray}\n R(z)\n = \\frac{1}{2} \\left( W'(z) - \\sqrt{W'(z)^2 + f(z)} \\right),\n ~~~\n \\widetilde{R}(z)\n = \\frac{1}{2} \\left( - W'(z) + \\sqrt{W'(z)^2 + \\tilde{f}(z)} \\right),\n \\end{eqnarray}\n where the signs of the square roots have chosen to be consistent \n with the large $z$ behavior of $R(z)$ and $\\widetilde{R}(z)$.\n From the above forms, we can see that $R(z)$ and $\\widetilde{R}(z)$ have cuts in the complex $z$ plane\n and are, respectively, meromorphic functions on Riemann surfaces $\\Sigma$ and $\\tilde{\\Sigma}$ of genus $n - 1$:\n $y^2 = W'(z)^2 + f(z)$ and $\\tilde{y}^2 = W'(z)^2 + \\tilde{f}(z)$.\n Let us denote by $\\alpha_i$ and $\\tilde{\\alpha}_i$ $\\alpha$-cycles of $\\Sigma$ and $\\tilde{\\Sigma}$ respectively.\n \n The polynomials $f(z)$ and $\\tilde{f}(z)$ are completely fixed \\cite{CDSW} by \n \\begin{eqnarray}\n - \\frac{1}{64 \\pi^2} \\langle {\\rm Tr}_{U(r_i)} {\\mathcal W}^{i \\alpha} {\\mathcal W}^i_\\alpha \\rangle\n &=& \\frac{1}{2 \\pi i} \\oint_{\\alpha_i} R(z) dz,\n \\nonumber \\\\\n - \\frac{1}{64 \\pi^2} \\langle {\\rm Tr}_{U(\\tilde{r}_i)} \\widetilde{{\\mathcal W}}^{i \\alpha} \\widetilde{{\\mathcal W}}^i_\\alpha \\rangle\n &=& \\frac{1}{2 \\pi i} \\oint_{\\tilde{\\alpha}_i} \\widetilde{R}(z) dz.\n \\end{eqnarray}\n It follows from these equations and (\\ref{WW=-WW}) that $f(z) = \\tilde{f}(z)$.\n Therefore, we obtain $R(z) = - \\widetilde{R}(z)$, which implies\n \\begin{eqnarray}\n \\langle \\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha \\rangle\n = - \\langle \\mathop{\\rm Tr} \\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha \\rangle.\n \\label{relation2}\n \\end{eqnarray}\n \n Note that this could be an alternative check of the magnetic gauge kinetic term.\n Indeed, as we have noted above, $M_i \\bar{q} \\phi^{k - 1} q$ terms and the magnetic gauge kinetic term\n in the Lagrangian do not contribute to the anomaly equation for $\\widetilde{R}(z)$.\n What we have assumed in the above argument is \n that the polynomial part of the magnetic superpotential is $- W(\\tilde{\\phi})$ and\n the relations (\\ref{WW=-WW}).\n However, these follows from that the electric and magnetic superpotentials have the same critical points and \n that both theories split into the decoupled SQCD theories at low energy.\n Once we have derived (\\ref{relation2}), \n we then obtain the following relations by taking derivatives of the partition functions \n with respect to $t_k$ and $\\tilde{t}_k$:\n \\begin{eqnarray}\n \\frac{\\partial Z}{\\partial t_k}\n \\sim \\langle \\mathop{\\rm Tr} \\phi^k {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha \\rangle\n = - \\langle \\mathop{\\rm Tr} \\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha \\rangle\n \\sim- \\frac{\\partial \\widetilde{Z}}{\\partial \\tilde{t}_k}\n \\end{eqnarray}\n Since the duality implies $Z = \\widetilde{Z}$ at least for the holomorphic sector, \n thus we can conclude that the magnetic gauge kinetic term is (\\ref{tildet}).\n \n\\subsection{Duality map of $\\mathop{\\rm Tr} \\phi^k$ operators}\n\\label{subsec:T}\n Finally, we analyze the operator relations between $\\mathop{\\rm Tr} \\phi^k$ and $\\mathop{\\rm Tr} \\tilde{\\phi}^k$.\n In the theory with constant $\\tau$, it has been known \\cite{KSS} that the duality map can be written as\n \\begin{eqnarray}\n \\mathop{\\rm Tr} \\phi^k\n = - \\mathop{\\rm Tr} \\tilde{\\phi}^k \n + \\frac{k}{\\mu^2} \\sum_{i = 1}^{k - 1} M_i \\bar{q} \\tilde{\\phi}^{k - 1 - i} q + \\ldots, \n \\label{matchingT}\n \\end{eqnarray}\n where ellipsis denotes the constant term.\n In \\cite{KSS}, these relations have been checked by substituting the classical vacuum expectation values.\n Also, they have been analyzed in \\cite{Mazzucato} by using the generalized Konishi anomaly equations.\n On general grounds, we can expect that these relations can be deformed \n by the term with the operators $\\mathop{\\rm Tr} \\tilde{\\phi}^k \\widetilde{{\\mathcal W}}^\\alpha \\widetilde{{\\mathcal W}}_\\alpha$ \n and the terms involving $t_k$, in the case with $\\tau(\\phi)$.\n Let us show below that such terms do not exist by making use of the generalized Konishi anomaly equations.\n \n The third equations of the generalized Konishi anomaly equations (\\ref{GKAelectric}) and (\\ref{GKAmagnetic}):\n \\begin{eqnarray}\n - \\delta^f_{f'} R(z)\n = \\Bigg[ M(z)_{f'}^{f} (z + m_f) \\Bigg]_-,\n ~~~\n - \\delta^f_{f'} \\widetilde{R}(z)\n = \\Bigg[ \\widetilde{M}(z)_{f'}^{f} A(z) \\Bigg]_-\n \\end{eqnarray}\n imply that the $t_k$-dependence cannot enter in $M(z)$ and $\\widetilde{M}(z)$, \n since $R(z)$ and $\\widetilde{R}(z)$ are independent of $t_k$.\n On the other hand, the second equations of (\\ref{GKAelectric}) and (\\ref{GKAmagnetic}) are\n \\begin{eqnarray}\n 2 R(z) T(z)\n &=& \\Bigg[ W'(z) T(z) \\Bigg]_- + 32 \\pi^2 i \\Bigg[ \\tau'(z) R(z) \\Bigg]_- + M(z),\n \\nonumber \\\\\n - 2 R(z) \\widetilde{T}(z)\n &=& - \\Bigg[ W'(z) \\widetilde{T}(z) \\Bigg]_- + 32 \\pi^2 i \\Bigg[ \\tau'(z) R(z) \\Bigg]_- \n + \\Bigg[ \\widetilde{M}(z) A'(z) \\Bigg]_-,\n \\label{matchingT2}\n \\end{eqnarray}\n where we have substituted $R(z) = - \\widetilde{R}(z)$.\n At this stage, we can see that the field-dependent gauge kinetic term {\\it does} affect \n the quantum chiral ring relation \n \\footnote{The author thanks Ken Intriligator for a useful comment on this point.}\n :\n the second terms in the right hand sides denote that $T(z)$ and $\\widetilde{T}(z)$ are affected \n by the gauge kinetic terms.\n Indeed, in large $z$, the first equation of (\\ref{matchingT2}) becomes\n \\begin{eqnarray}\n \\left< W'(\\Phi) \\right> - \\frac{i}{2} \\left< \\tau'(\\Phi) {\\mathcal W}^\\alpha {\\mathcal W}_\\alpha \\right>\n + \\left< \\bar{Q} Q \\right> \n = 0,\n \\end{eqnarray}\n and this is the usual F-term equation.\n In the classical vacua, the second term does not contribute, but it does in the quantum vacua.\n \n Let us consider the effect of the second terms in (\\ref{matchingT2}).\n In the constant $\\tau$ case, by the duality map (\\ref{matchingT}), \n the first equation of (\\ref{matchingT2}) should reduce to the second equation, as noted above.\n In the case with $\\tau(\\phi)$, \n since the only difference between the second terms in (\\ref{matchingT2}) is the sign, \n they do not change the duality map (\\ref{matchingT}).\n\n\\section*{Acknowledgements}\n The author especially thanks Ken Intriligator for carefully reading the manuscript\n and giving useful comments.\n The author thanks Philip Argyres, Hiroshi Itoyama, Yutaka Ookouchi, Luca Mazzucato, Al Shapere, \n Ta-Sheng Tai, Masato Taki, Seiji Terashima, Cumrun Vafa and Futoshi Yagi\n for useful discussions and helpful comments.\n The author also thanks Brown University, Harvard University, University of California, San Diego, \n University of Cincinnati, University of Kentucky \n and Perimeter Institute for Theoretical Physics for the hospitality during part of this project.\n The research of the author is supported in part by JSPS Research Fellowships for Young Scientists.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nWhen observing a time series, one crucial issue is to predict the (non-observed) first future value using the\nobserved past values. Since the seventies, different model selection procedures have been\nstudied for inferring how many observed past values are needed for predicting the next value. Procedures such as AIC (\\cite{Akaike1973}), BIC (Schwarz \\cite{bic}) and APE (Ing \\cite{Ing2007}) \nare used by practitioners to select a reasonable linear predictor. When the observations satisfy a linear model, those procedures are proved to be asymptotically efficient (see Ing \\cite{Ing2007} for more details).\n\nIn the same time, the progress of statistical learning theory in the iid setting brought new perspectives in\nmodel selection (see Vapnik \\cite{Vapnik} and Massart \\cite{Massart} among others). Typical machine-learning procedures allow to\nchoose a predictor among a family, with the guarantee that this predictor performs almost\nas well as the best possible predictor of the family (called the oracle). Such results are called oracle inequalities; they provide guarantees on the\nquality of the prediction without any parametric assumption on the observations.\n\nFew works have been done in the context of dependent observations. The\nmachine learning theory was used successfully in the time series prediction context by Modha and Masry\n\\cite{modha}. However, their procedure relies on the knowledge of the $\\alpha$-mixing coefficients. To our knowledge, there is no efficient estimation of this coefficients and their procedure seems difficult to use in practice. Baraud {\\it et al.} \\cite{Baraud} use the\nmodel selection point of view to perform regression and auto-regression on dependent observations.\nThey prove powerful oracle inequalities when the observations satisfy an additive auto-regressive\nmodel.\nWhen the observations are Harris\nrecurrent Markov chains, Lacour \\cite{LACOUR} gives also oracle inequalities for a procedure completely\nfree of the dependence properties. An alternative point of view is provided by the theory of individual sequences prediction\n(see Lugosi and Cesa-Bianchi \\cite{Lugosi} or Stoltz \\cite{Stoltz}). In these works, no assumption on the observations - not even a\nstochastic assumption - is done and oracle inequalities are given.\\\\\n\nIn this paper, our objectives are the following:\n\\begin{enumerate}\n\\item to build various predictors of different form and using different number\nof past observations,\n\\item to select one of these predictors {\\it without any assumptions\non the distribution of the observations},\n\\item to prove oracle inequalities under weak assumptions on the observed time series.\n\\end{enumerate} \nIn the end of this introduction, let us briefly present our two-step procedure.\n\nLet us observe $(X_1,\\ldots,X_n)$ from a stationary time series\n$X=(X_{t})_{t\\in\\mathds{Z}}$ distributed as $\\pi_0$ on $\\mathcal{X}^\\mathbb{Z}$ where $\\mathcal{X}$ is\nan Hilbert space equipped with its usual norm $\\|\\cdot\\|$. Let us fix a (possibly large) family of predictors\n$ \\{f_{\\theta},\\theta\\in\\Theta\\} $: for any $\\theta$ and any \n$t$, $f_{\\theta}$ applied to the past values $(X_{t-1},X_{t-2},\\ldots, X_1)$ is a possible prediction\nof $X_{t}$. We discretize the family of predictors by the number $p$ of past values they use for prediction. Thus we assume that\n$$ \\Theta = \\bigcup_{p=1}^{\\left\\lfloor\\frac{n}{2} \\right\\rfloor}\\Theta_{p} $$\nwhere the $\\Theta_{p}$ are disjoint in order that for any $\\theta\\in\\Theta$, there is only one $p$ such that\n$\\theta\\in\\Theta_{p}$. Now, for any\n$\\theta\\in\\Theta_{p}$, $f_{\\theta}$ is a function $\\mathcal{X}^{p}\\rightarrow\\mathcal{X}$ and at\nany time $t$, $f_{\\theta}(X_{t-1},...,X_{t-p})$ is a prediction of $X_{t}$ according\nto $\\theta$ and denoted $\\hat{X}_{t}^{\\theta}$. As the predictor $f_\\theta$ may take different\nforms (linear, neural networks,\\ldots), we write\n$$\n\\Theta_{p} = \\bigcup_{\\ell=1}^{m_{p}}\\Theta_{p,\\ell}\n$$\nfor a given $m_{p}\\in \\{1,\\ldots,n\\}$. Finally, the risk $R(\\theta)$ of the prediction is defined by\n$$\nR(\\theta)=\\pi_0\\left[\\left\\|f_{\\theta}(X_{t-1},...,X_{t-p})-X_{t} \\right\\|\\right]\n = \\pi_0\\left[\\left\\|\\hat{X}_{t}^{\\theta}-X_{t} \\right\\|\\right],\n$$\nwhere here and all along the paper $\\pi[h]=\\int hd\\pi$ for any measure $\\pi$ and any integrable function $h$. Note that $R(\\theta)$ does not depend on $t$ as $X$ is stationary.\n\n\nThe mathematical counterparts of the points (1), (2) and (3) of our objectives are the following. The point (1) corresponds to build, on the basis of the observations, an estimator $\\hat{\\theta} _{p,\\ell}$ in each model $\\Theta_{p,\\ell}$,\nfor $1\\leq p \\leq \\lfloor n\/2 \\rfloor$ and $1\\leq \\ell \\leq m_{p}$. The point (2) consists in defining a procedure to choose a $\\hat{\\theta}$ among all the possible $\\hat{\\theta} _{p,\\ell}$. \nFinally, for point (3), we prove that $R(\\hat{\\theta})$ is close to $\\inf_{\\theta\\in\\Theta}R(\\theta)$. To attain our objectives we use the PAC-Bayesian paradigm (introduced by Shawe-Taylor and Williamson \\cite{STW97} and McAllester \\cite{McAllester}).\nUsing this approach, Catoni \\cite{Classif,Cat7,manuscrit}, Audibert \\cite{AudibertReg}, Alquier \\cite{Alquier2008},\nTsybakov and Dalalyan \\cite{arnak} solve points (1),\n(2) and (3) simultaneously for various regression and classification problems in the iid setting. In this paper we build a procedure that gives a predictor $\\hat \\theta$ satisfying, under general conditions on $X$ and\nwith probability at least $1-\\varepsilon$,\n$$ R(\\hat{\\theta})\\le\n\\inf_{\\Tiny d_{p,\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta ) +\n{\\rm cst} . \\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\} + {\\rm cst} .\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}} $$\nwhere ${\\rm cst} >0$ is an explicit constant and $d_{p,\\ell}$ an estimate of the complexity of $\\Theta_{p,\\ell}$.\n\nTo obtain such oracle inequalities, we use\nsharp estimates (close to the ones in the iid case) on the Laplace transform of partial sums in dependent settings. For bounded observations we use the $\\theta_\\infty$-coefficients\n(see \\cite{Dedecker2007a}), introduced in Rio \\cite{Rio2000a} as the $\\gamma$-mixing coefficients. These coefficients generalize the uniform mixing ones. For unbounded observations we\nuse the causal Bernoulli shifts representation. It includes all classical linear ARMA models and also the more general chains with infinite memory introduced by Doukhan\nand Wintenberger \\cite{Doukhan2008}. These bounded and unbounded dependent frameworks are not comparable with the $\\beta$ or $\\alpha$-mixing ones as they include\nsome dynamical systems that are not mixing, see Andrews \\cite{Andrews1984} and Dedecker and Prieur\n\\cite{Dedecker2005} for details. Finally, it is important to note that our prediction procedure is the same for the two dependence frameworks. It does not depend on any unknown dependence coefficients; we believe that similar oracle inequalities for our predictor can be proved in other weakly dependent settings.\n\n\nThe paper is organized as follows: First, the prediction procedure is detailed in Section\n\\ref{sec::pl}; Second, the assumptions on the observed time series and the corresponding oracle inequalities are given in Section\n\\ref{sec::res}. In Section \\ref{sec::app1} are given some examples of time series for which these oracle inequalities hold. Our procedure applied on some possible prediction models are given in Section \\ref{sec::app2}. Linear predictors (with simulations), neural networks predictors and\nnon-parametric predictors are considered. Finally the complete proofs are collected in Section \\ref{sec::pf}.\n\n\\section{The prediction procedure}\n\\label{sec::pl}\n\nWe observe $(X_1,\\ldots,X_n)$ from a stationary time process $X=(X_{t})_{t\\in\\mathds{Z}}$\ndistributed as $\\pi_0$ on $\\mathcal{X}^\\mathbb{Z}$ where $\\mathcal{X}$ is an Hilbert space equipped with its\nusual norm $\\|\\cdot\\|$. We fix a family of predictors $\\{f_{\\theta},\\theta\\in\\Theta\\}$ with\n$$ \\Theta= \\bigcup_{p=1}^{\\left\\lfloor\\frac{n}{2} \\right\\rfloor}\\Theta_{p}\n = \\bigcup_{p=1}^{\\left\\lfloor\\frac{n}{2} \\right\\rfloor}\n \\left(\\bigcup_{\\ell=1}^{m_{p}}\\Theta_{p,\\ell}\\right) $$\nsuch that $m_p\\ge n$ and $p(\\theta)$ is the only $p$ such that $\\theta\\in\\Theta_{p}$. For any $\\theta\\in\\Theta$, we denote\n$\\hat{X}_{t}^{\\theta}=f_{\\theta}(X_{t-1},...,X_{t-p})$ and $R(\\theta) = \\pi_0\\left[\\left\\|\n\\hat{X}_{t}^{\\theta}-X_{t} \\right\\|\\right]$.\n\n\\subsection{The Lipschitz predictors}\n\nLet $M$ denotes the set of all possible pairs $(p,\\ell)$:\n$$ M = \\bigcup_{p=1}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor} \\{p\\}\\times\\{1,...,m_{p}\\}. $$\nLet $\\mathcal{T}$ be a $\\sigma$-algebra on $\\Theta$ and $\\mathcal{T}_{p,\\ell}$ be its restriction\nto $\\Theta_{p,\\ell}$ for any $(p,\\ell)\\in M$. For any $(p,\\ell)\\in M$, we assume that $\\Theta_{p,\\ell}$ is a compact subset of $\\mathbb{R}^q$ for\nsome $q<\\infty$ ($q$ depends on $(p,\\ell)$) and that there\nexists $(a_{j}(\\theta))_{j\\in\\{1,...,p\\}}$ satisfying, for any\n$(x_{1},...,x_{p}),(y_{1},...,y_{p})\\in\\mathcal{X}^{p}$, the relation\n\\begin{equation}\n\\label{modellip}\n\\left\\|f_{\\theta}(x_{1},...,x_{p}) - f_{\\theta}(y_{1},...,y_{p})\\right\\|\n\\leq\n\\sum_{j=1}^{p} a_{j}(\\theta) \\left\\|x_{j}-y_{j}\\right\\|.\n\\end{equation}\nMoreover, we assume that\n\\begin{equation}\n\\label{modellip2}\nL:=\\sup_{(p,\\ell)\\in M}\n\\sup_{\\theta \\in\\Theta_{p,\\ell}}\\sum_{j=1}^{p} a_{j}(\\theta) \\qquad\\mbox{satisfies}\\qquad L\\leq \\log(n) -1\n\\end{equation}\nin order to bound the volatility of the predictors uniformly on $M$.\n\n\\subsection{The complexity of $\\Theta_{p,\\ell}$}\nTo control the complexity of each $\\Theta_{p,\\ell}$ \n we \nassume that, for all $(p,\\ell)\\in M$, there exist a probability measure $\\pi_{p,\\ell}$ on the measurable space\n$\\left(\\Theta_{p,\\ell},\\mathcal{T}_{p,\\ell}\\right)$ and a constant\n$1\\le d_{p,\\ell}<\\infty $ satisfying\n\\begin{equation}\n\\label{dimension}\n\\sup_{\\gamma>e} \\left\\{\n\\frac{-\\log \\int_{\\Theta_{p,\\ell}}\n\\left[\\exp\\left(-\\gamma \\left(R(\\theta)-R\\left(\\overline{\\theta}_{p,\\ell}\\right)\\right)\\right)\\right]\nd\\pi_{p,\\ell}(\\theta) }\n{\\log(\\gamma)} \\right\\} \\leq d_{p,\\ell}.\n\\end{equation}\nHere $\\overline{\\theta}_{p,\\ell}=\\arg \\min_{\\Theta_{p,\\ell}}R$ for any $(p,\\ell)\\in M$. The parameter $d_{p,\\ell}$ is linked with classical\ndimensions as the Vapnik dimension and entropy measures. In this paper we only investigate the cases where $\\pi_{p,\\ell}$ is the Lebesgue measure on $\\Theta_{p,\\ell}$. We have the following result:\n\\begin{prop}\\label{propdim}\nLet $q\\in\\mathds{N}^{*}$, $x>0$ and $\\mathcal B_{x}^{q}$ be\nthe closed $\\ell^1$-ball in $\\mathds{R}^{q}$ of radius $x>0$ and centered at $0$.\nIf $\\Theta_{p,\\ell}=\\mathcal B_{c_{p,\\ell}}^{q}$ for $c_{p,\\ell}>0$ and $\\theta \\to R(\\theta)$ is a $C$-Lipschitz function then we have:\n\\begin{equation}\\label{eqdim}\nd_{p,\\ell}\\le q\\times\\left(1+\\log\\left(c_{p,\\ell}\\left(\\frac{Ce}{q}\\vee\n\\frac{1}{c_{p,\\ell}-\\|\\overline\\theta_{p,\\ell}\\|}\\right)\n\\right)\\right).\n\\end{equation}\n\\end{prop}\nThe proof of this result is given at the end of Subsection \\ref{prooflems}. Predictive models whom complexity $d_{p,\\ell}$ is estimated are given in Section \\ref{sec::app2}.\n\n\n\n\\subsection{The empirical risk}\n\nAs the risk $R(\\theta)$ cannot be computed, we use its empirical counterpart $r_{n}(\\theta)$:\n$$\nr_n(\\theta) =\n\\frac{1}{n-p\\left(\\theta\\right)} \\sum_{t=p\\left(\\theta\\right)+1}^{n}\n\\left\\|X_{t}-\\widehat{X}^{\\theta}_{t}\\right\\|.\n$$\n\n\\subsection{The randomized estimators}\n\n\nFor any $(p,\\ell)\\in M$,\nour randomized estimators $\\tilde{\\theta}_{p,\\ell}^{\\lambda}$ is drawn randomly through a Gibbs measure\n$$ \\tilde{\\theta}_{p,\\ell}^{\\lambda} \\sim \\pi_{p,\\ell}\\{-\\lambda r_n\\}.$$\nWe recall that for any measure $\\pi$ and any measurable function $h$\nsuch that $\\pi[\\exp(h)]<+\\infty$, the Gibbs measure denoted $\\pi\\{h\\}$ is defined by the relation:\n\\begin{equation}\\label{Gibbsmeasure}\n\\frac{d\\pi\\{h\\}}{d\\pi}(\\theta) = \\frac{\\exp(h(\\theta))}{\\pi[\\exp( h)]}.\n\\end{equation}\nHere the parameter $\\lambda$ is called the temperature (this terminology comes from the statistical thermodynamics). For $n\\ge 8e (1+L)$, $\\lambda$ takes values in a finite grid $\\mathcal{G}_{p,\\ell}$ defined as\n$$ \\mathcal{G}_{p,\\ell} = \\left\\{ g_{1}\\frac{\\sqrt{d_{p,\\ell}n}\\log(d_{p,\\ell}n)}{(1+L)\\log^{3\/2}(n)}, \\dots,\ng_{n_{0}}\\frac{\\sqrt{d_{p,\\ell}n}\\log(d_{p,\\ell}n)}{(1+L)\\log^{3\/2}(n)}\\right\\}\n \\cap \\left[2e,\\frac{n}{4(1+L)}\\right] .$$\nwhere $\\check{c}\\leq g_{1}<...0$ such that for any $n$,\n$\\theta_{\\infty,n}(1)<\\mathcal{C}$ (the short memory condition).\nCausal Bernoulli shifts with bounded innovations, uniform $\\varphi$-mixing sequences and dynamical systems\nare classical $\\theta_\\infty$ weakly-dependent examples, see Section \\ref{sec::app1} for more details. In this context we prove the following oracle inequality\n\\begin{thm}\\label{mainthm}\nUnder {\\bf (WDP)} and condition \\eqref{dimension},\nthere are explicit constants\n$$({\\rm cst}_{1},{\\rm cst}_{2}) = {\\rm cst}(\\check{c},\\hat{c},L,\\mathcal{C},\\|X_{0}\\|_{\\infty})$$\nsuch that for all $n\\ge 8e (1+L)$ with probability at least $1-\\varepsilon$\n\\begin{multline*}\nR(\\hat{\\theta})\\le\n\\inf_{\\Tiny d_{p,\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta ) +\n{\\rm cst}_{1} . \\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\} + {\\rm cst}_{2}.\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\n\\\\\n+ 4(1+L) \\left(\\frac{(\\|X_{0}\\|_{\\infty}+\\mathcal C)^{2}}{2} - \\log^{3}(n)\\right)_{+}.\n\\end{multline*}\n\\end{thm}\nThe proof of this result is given in Subsection \\ref{proofmain} page \\pageref{proofmain}.\n\n\\subsection{Causal Bernoulli shifts {\\bf(CBS)}}\nLet $\\mathcal{X}'$ be some Banach space equipped with a norm also denoted $\\|\\cdot\\|$.\nLet $H:{\\mathcal X'}^{\\mathbb{N}}\\mapsto \\mathcal X$ be a Lipschitzian function, i.e. such that there exists\n$(a_j(H))_{j\\in\\mathds N}$ satisfying, for any $v=(v_j)_{j\\in\\mathds{N}}$, $v'=(v'_j)_{j\\in\\mathds{N}}\\in\n{\\mathcal X'}^{\\mathds N}$, the relations:\n\\begin{gather}\n\\label{condlip1}\n\\left\\|H(v)-H(v')\\right\\|\n\\le \\sum_{j=0}^\\infty a_j(H) \\|v_j-v'_j\\|,\\\\\n\\label{condsum}\n\\mbox{ with } \\sum_{j=0}^\\infty ja_j(H)<+\\infty.\n\\end{gather}\nWe denote $\\sum_{j=0}^{\\infty}\\ a_j(H)\n:=a(H)$, $\\sum_{j=0}^\\infty ja_j(H)=\\tilde a(H)$. The causal Bernoulli shifts are defined by the relation\n$$\nX_t=H(\\xi_{t},\\xi_{t-1},\\xi_{t-2},\\ldots)\\qquad\\forall t\\in\\mathbb{Z}\n$$\nwhere $\\xi_{t}$ for ${t\\in\\mathbb{Z}}$ are iid variables called the\ninnovations and distributed as $\\mu$. We assume that the\ninnovation's norm admits a finite Laplace transform $\\mu[\\exp(c^\\ast\\|\\xi_0\\|)]:=\\Psi(c^\\ast)<+\\infty$ \n(the Cramer condition) for $c^\\ast\\geq a(H)$. Classical examples\nof such processes are causal linear ARMA models and chains with infinite memory with low-tail innovations,\nsee Section \\ref{sec::app1} for more details. In this context we prove the following oracle inequality\n\n\\begin{thm}\\label{mainthm2}\nUnder {\\bf (CBS)} and condition \\eqref{dimension},\nthere are explicit constants\n$$({\\rm cst}'_{1},{\\rm cst}'_{2}) = {\\rm cst}'(\\check{c},\\hat{c},L,a(H),\\tilde{a}(H),\\Psi(1))$$\nsuch that for all $n\\ge 8e (1+L)$ with probability at least $1-\\varepsilon$\n\\begin{multline*}\nR(\\hat{\\theta})\\le\n\\inf_{\\Tiny d_{p,\\ell}\\leq n}\\left\\{ \\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta )\n+\n{\\rm cst}_{1}' . \\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\} + {\\rm cst}_{2}'.\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\n\\\\\n+ \\sqrt{\\frac{d_{\\hat{p},\\hat{\\ell}}}{n}}\\log(d_{\\hat{p},\\hat{\\ell}}n)4(1+L)\\hat c\n\\left(4a(H)\\Psi(a(H)) + 2\\log^{2}(n)\\Big(1+\\frac{\\tilde{a}(H)}{a(H)}\\Big)^{2} - \\log^{3}(n) \\right)_{+}.\n\\end{multline*}\n\\end{thm}\nThe proof of this result is given in Subsection \\ref{proofmain2}\npage \\pageref{proofmain2}.\n\n\n\n\n\n\n\n\\subsection{Comments on the results}\\label{comments}\nThe constants are roughly (but explicitly) estimated in the proofs, see Subsections \\ref{proofmain} and \\ref{proofmain2}. For example, we obtain\n$$\n{\\rm cst}_1\\le (1+L)\\left( \\frac{6}{\\check{c}}\n+ 8 \\hat{c} \\Bigl(1+\\|X_{0}\\|_{\\infty} + \\mathcal C\\Bigr)^{2}\n\\right)\\quad\\mbox{and}\\quad{\\rm cst}_2\\le\\frac{7(1+L)}{\\check c}.\n$$\nFor $n$ sufficiently large, the last terms in the oracle inequalities vanish. Then it exists a constant $C>0$ such that under {\\bf (WDP)} or {\\bf (CBS)} for all $n\\ge 8e (1+L)$ with probability at least $1-\\varepsilon$:\n$$ R(\\hat{\\theta})\\le\n\\inf_{\\Tiny d_{p,\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta ) +\nC\\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\} + C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.$$\nSimilar oracles inequalities have already proved by Modha and Masry \\cite{modha} and Baraud {\\it et al.}\n\\cite{Baraud}. These inequalities are given in expectation while ours\nare true with high probability. Remark that integrating our oracle inequalities\nwith respect to $\\varepsilon$ leads to a result in expectation: there exists a constant \n$C>0$ independent of $n$ such that in both {\\bf (WDP)} and {\\bf (CBS)} cases\n$$ \\pi_{0} \\left[R(\\hat{\\theta})\\right] \\le\n\\inf_{\\Tiny d_{p,\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta ) +\nC\\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\}.$$\nThe converse is not true: results in expectation\ndo not lead to results that hold with high probability.\n\nIt is difficult to compare our oracle inequalities with the ones in \\cite{modha} and \\cite{Baraud}: contrary with our paper, they\ndeal with the quadratic risk and ($\\beta-$ or $\\alpha-$) mixing time series. However, let us remark that we obtain \nthe same additional term $\\sqrt{d_{p,\\ell}\/n}$ (called the rate) than in the iid case, up to a multiplicative $\\log^{5\/2}(n)$ term (called the loss). Baraud {\\it et al.}\n\\cite{Baraud} obtain the same rate than in the iid case associated with the quadratic risk, while Modha and Masry \\cite{modha}\nsuffer a loss $(n\/d_{p,\\ell})^{c}$ for some $c>0$.\n\n\\section{Examples of time series satisfying {\\bf (WDP)} or {\\bf (CBS)}}\n\\label{sec::app1}\n\nWe present several examples of time series satisfying {\\bf (WDP)} or {\\bf (CBS)}.\n\n\\subsection{Causal Bernoulli shifts}\n\nCausal Bernoulli shifts are stationary time series that admit the representation\n\\begin{equation}\\label{cbs}\nX_t=H(\\xi_{t},\\xi_{t-1},\\xi_{t-2},\\ldots)\\qquad\\forall t\\in\\mathbb{Z}\n\\end{equation}\nwhere the $\\xi_t$ are iid variables called innovations. Almost all known stationary and ergodic processes\nhave this form. However we work here under the restrictive assumption \\eqref{condlip}. Remark that under\nthis Lipschitz condition the existence of the stationary time series $(X_t) $ follows from \\eqref{cbs}\nand it satisfies the Cramer condition as soon as the innovations do. Some examples of causal Bernoulli\nshifts are presented below.\n\n\n\\subsubsection{Linear models}\n\nLet $(X_t)$ be a real time series admitting the MA($\\infty$) representation \n$$\nX_t=\\sum_{j=0}^\\infty a_j\\xi_{t-j}\\quad \\mbox{with}\\quad \\sum_{j=0}^\\infty ja_j <+\\infty.\n$$\nThen it satisfies {\\bf (CBS)} if the iid innovations $\\xi_t$ satisfy the Cramer condition. As an example there is any causal AR($\\infty$) model $X_t=\\phi_0+\\sum_{j=1}^\\infty\n\\phi_jX_{t-j}+\\xi_t$ with $\\phi(z)=1-\\sum_{j=1}^\\infty \\phi_j z^j$ that have no root for $|z|\\le 1$ (such that causal ARMA($p$,$q$) models). Indeed, it is\na real analytic function on the unit disc, $1\/\\phi(z)$ is a well defined real analytic function with the\nexpression $\\sum_{j=1}^\\infty \\psi_j z^j$ on the unit disc with the coefficients $ \\psi_j$ that decrease\nexponentially fast (\\eqref{condsum} is automatically satisfied). \n\n\\subsubsection{Chains with infinite memory}\n\nChains with infinite memory is a class of time series $ (X_{t}) $ introduced by Doukhan and\nWintenberger \\cite{Doukhan2008} as the solution of the equation\n\\begin{equation}\\label{eq::rec}\nX_t=F(X_{t-1},X_{t-2},\\ldots;\\xi_t)\\mbox{ almost everywhere, }\n\\end{equation}\nfor some function $F:\\mathcal{X}^{(\\mathds{N}\\setminus\\{0\\})}\\times \\mathcal{X}'\\to \\mathcal{X}$. Assume also that\nthere exits some $u$ satisfying, for all $x=(x_k)_{k\\in\\mathds{N}\\setminus\\{0\\}}$, $x'=(x'_k)_{k\\in\\mathds{N}\n\\setminus\\{0\\}}\\in \\mathcal{X}^{\\mathds{N}\\setminus\\{0\\}}$ such that there exists $N>0$ as $x_k=x'_k=0$ for all\n$k>N$, the condition\n\\begin{gather}\n\\label{condlip}\n\\left\\|F(x;y)-F(x';y')\\right\\|\n\\le \\sum_{j=1}^\\infty a_j(F) \\|x_j-x'_j\\|+u\\|y-y'\\|,\\\\\n\\label{condcontract}\n\\mbox{ with }\\qquad \\sum_{j=1}^{\\infty}a_j(F)\n:=a(F)<1,\n\\end{gather}\n Many non linear econometrics time series are chains with infinite memory. The following Proposition gives sufficient assumptions such that chains with infinite memory satisfy {\\bf (CBS)}:\n\\begin{prop}\\label{unboundedprop}\\label{lemma_ex_1}\nUnder \\eqref{condlip} and \\eqref{condcontract} there exists a unique solution $(X_t)$ of equation \\eqref{eq::rec}\nsatisfying {\\bf (CBS)} if $\\xi_0$ satisfies the Cramer condition.\n\\end{prop}\nThe proof of Proposition \\ref{lemma_ex_1} is given in Subsection \\ref{proofobservations}.\n\n\\subsection{Weakly dependent processes}\n\\label{secwdp}\n\n\\subsubsection{Bounded causal Bernoulli shifts}\n\nBounded causal Bernoulli shifts are examples of time series satisfying {\\bf (WDP)}:\n\\begin{prop}\\label{taucbs}\nUnder condition \\eqref{condlip} and \\eqref{condsum}, any solution of the equation \\eqref{cbs} is bounded by\n$2a(H)\\|\\xi_0\\|_\\infty$ and is weakly dependent {\\bf (WDP)} with \n$\\mathcal C= 2 \\|\\xi_0\\|_\\infty \\tilde a(H). $\n\\end{prop}\nThe proof of this already known result is given in Subsection \\ref{proofobservations} for completeness. Below are\npresented two examples of time series satisfying {\\bf (WDP)} that are not bounded causal Bernoulli shifts.\n\n\\subsubsection{Uniform $\\varphi$-mixing processes}\n\nLet us remind the definition of the $\\varphi$-mixing coefficients introduced in Ibragimov\n\\cite{Ibragimov1962};\n$$\n\\varphi(r)=\\sup_{(A,B)\\in\\,\\mathfrak{S}_0\\times\\mathfrak{F}_r}|\\pi(B\/A)-\\pi(B)|\n$$\nwhere $\\mathfrak{F}_r=\\sigma(Y_t,t\\ge r)$. The class of $\\varphi$-mixing processes gives examples of time series\nthat satisfied {\\bf (WDP)}:\n\\begin{prop}\\label{lemma_ex_2}\nIf $(X_t)$ is a stationary bounded process then it satisfies {\\bf (WDP)} with\n$$\n\\theta_{\\infty,n}(1)\\leq 2\\|X_0\\|_\\infty \\sum_{r=1}^n \\varphi(r).\n$$\n\\end{prop}\nThe proof of this already known result is given in Subsection \\ref{proofobservations} for completeness. Remark that $(X_t)$ satisfies the short memory condition as soon as $(\\varphi(r))$ is summable. All uniform ergodic Markov chains are examples of $\\varphi$-mixing processes with short memory, see Doukhan \\cite{Doukhan1994}.\n\n\\subsubsection{Dynamical systems on $[0,1]$}\nThe AR($1$) process $X_t=2^{-1}(X_{t-1}+\\xi_t)$ with $\\xi_t$ Bernoulli distributed is not mixing, see\n\\cite{Andrews1984} for more details. Through a reversion of the time, it can be viewed as a dynamical\nsystem $X_{t}=T(X_{t+1})$ where $T(x)=2x$ if $0\\le x<1\/2$, $T(x)=2x-1$ if $1\/2\\le x\\le 1$. Dedecker and\nPrieur \\cite{Dedecker2005} extended this counter-example to processes $(X_t)$ such that\n$X_t=T(X_{t+1})$ where $T$ is an expanding map on $[0,1]$, see Section 4.4 of \\cite{Dedecker2005} for a\nproper definition. Then $(X_t)$ satisfies\n{\\bf (WDP)} with $\n\\mathcal C= K\\sigma\/(1-\\sigma)$ where $K>0$, $0\\le \\sigma<1$, see Section 7.2 of \\cite{Dedecker2005}.\n\n\\section{Examples of predictors}\n\\label{sec::app2}\n\nWe give some examples of Lipschitz predictors where we can estimate the complexity of the $\\Theta_{p,\\ell}$ and then apply our main results. In this section $C>0$ is a constant independent of $\\varepsilon$ and $n$ that may be different from one inequality to another.\n\n\\subsection{Linear predictors}\n\n\\label{arex}\nLet $\\mathcal X =\\mathbb{R}$ and we consider predictors of the form:\n$$ f_{\\theta}(X_{t-1},...,X_{t-p}) =\\theta_0+ \\sum_{i=1}^{p}\\theta_{i}X_{t-i}, $$\nwhere $\\theta\\in\\Theta_{p}\\subset\\mathbb{R}^{p+1}$ with by definition, for some $B>0$,\n$$ \\Theta_{p} = \\Theta_{p,1}\n= \\left\\{\\theta\\in\\mathds{R}^{p+1}, \\quad \\|\\theta\\|_{1}=\\sum_{i=0}^{p}|\\theta_{i}|\\leq B \\right\\} $$\n($m_p=1$ for all $p$ and we omit the index $\\ell$). Using\nProposition \\ref{propdim} it follows that\n$$\nd_{p}\\le (p+1)\\log\\left(eB\\left(\\frac{e}{p+1}\\vee \\frac{1}{B-\\|\\overline\\theta_{p}\\|}\\right)\n\\right),$$\nwhere $\\overline\\theta_{p}=\\arg\\min_{\\Theta_{p}}R(\\theta)$. As a consequence of Theorems \\ref{mainthm} and \\ref{mainthm2} we obtain:\n\\begin{cor}\\label{cor1}\nIf $ \\|\\overline{\\theta}_{p}\\|_1 \\leq B-e\/(p+1) $ for all $p\\ge 0$, then, under {\\bf (WDP)} or {\\bf (CBS)}, for all $n\\ge 8e (1+L)$\n with probability at least $1-\\varepsilon$:\n$$\nR(\\hat{\\theta})\\le\n\\inf_{p+1\\leq n\/2} \\left\\{ \\min_{\\theta\\in\\Theta_{p }} R( {\\theta} ) +\nC \\sqrt{\\frac{p}{n}} \\log^{5\/2}(n) \\right\\} + C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.\n$$\n\n\\end{cor}\nAs an application, consider the innovations $\\xi_t$ iid satisfying the Cramer condition and ${\\rm med} (\\xi_0) = 0$. If $(X_{t})$ is a causal AR($p_0$) process ($0\\le p_0<\\infty$) of the form\n$$ X_{t} = a_{0} + \\sum_{j=1}^{p_{0}} a_{j} X_{t-j} + \\xi_{j} \\mbox{ for all }t\\in\\mathbb{Z}.$$\nIf $B\\ge \\sum_{j=0}^{p}|a_{j}|+e\/(p+1)$ for all $0\\le p\\le p_0$, the error of the best\nlinear predictor is $\\mu[|\\varepsilon_{j}|]$. \nCorollary \\ref{cor1} then implies that for any $0<\\varepsilon<1$ and any $n\\ge 2(p_0+1)$, it holds:\n$$ R(\\hat{\\theta})-\\mu[|\\varepsilon_{0}|]\\le\nC\\left( \\sqrt{\\frac{p_{0}}{n}} \\log^{5\/2}(n) + \\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\\right)\\qquad \\mbox{ with probability at least }1-\\varepsilon.$$\nFor $\\varepsilon>0$ fixed independently of $n$, the rate of convergence of the excess risk is estimated by $\\sqrt{ {p_{0}}\/{n}} \\log^{5\/2}(n)$. Note that $\\hat\\theta$ achieves this rate even if $p_0$ is unknown. One says that our procedure is adaptive in $p_0$ and, using the terminology of \\cite{modha}, memory-universal.\\\\\n\nConsider now that $(X_t)$ comes from an AR($\\infty$) model of the form\n\\begin{equation}\\label{arinf}\nX_t=a_0+\\sum_{i=1}^\\infty a_iX_{t-i}+\\xi_t,\\mbox{ for all }t\\in\\mathbb{Z}.\n\\end{equation}\nIf\n$B\\ge \\sum_{j=0}^{p}|a_{j}|+e\/(p+1)$ for all $p\\ge 0$, we have $\\overline{\\theta}_{p}=(a_{0},...,a_{p})$. Then we roughly bound\n$R(\\overline{\\theta}_{p})= \\pi_{0}[|\\sum_{i>p}a_{i}X_{-i}+\\xi_{0}|]\\leq\n\\mu[|\\xi_{0}|]+\\pi_{0}[|X_{-i}|]\\sum_{i>p}|a_{i}|$ and with probability at least $1-\\varepsilon$:\n$$ R(\\hat{\\theta})-\\mu[|\\xi_{0}|]\\le\n\\inf_{p+1\\leq n\/2} \\left[\n\\pi_{0}[|X_{0}|]\\sum_{i>p}|a_{i}| +\nC \\sqrt{\\frac{p}{n}} \\log^{5\/2}(n)\\right] + C \\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.$$\nIn this non-parametric setting, to obtain a rate of convergence for the excess risk we have to specify the decay rate of the $|a_i|$. For example, if\n$$ \\exists \\gamma>0,\\exists \\beta>0, \\forall p:\\quad\n \\sum_{i>p}|a_{i}| \\leq \\frac{\\gamma}{p^{\\beta}} $$\nthen the convergence rate is $({\\log^{5}(n)}\/{n})^{\\frac{\\beta}{2\\beta+1}}$ (consider the optimal $p= n^{1\/(2\\beta+1)}\\log^{5\/(2\\beta+1)}(n)$).\\\\ \n\n\n\\noindent {\\bf Simulations.}\nWe implement our linear prediction procedure using the R software \\cite{R}. We compare the results\nto the one obtained using the standard ARIMA procedure of R with the AIC criterion for model selection.\nOur theoretical penalization terms, driven by <> bounds, are necessarily pessimistic: our procedure systematically over-penalizes large models. Thus, for having an efficient procedure in practice, adjustments have been done. However we aim with these simulations to\nshow that\n\\begin{enumerate}\n\\item our linear prediction procedure is easily implementable;\n\\item its performances are reasonable when the implemented penalization term is smaller than the theoretical one.\n\\end{enumerate}\nWe only consider observations from simulations of AR($p_0$) models of the form\n$$ X_{t} = \\sum_{i=1}^{p_0}a_{i}X_{t-i} + \\xi_{t} $$\nwhere the $\\xi_{t}$ are iid, either $\\mathcal{N}(0,\\sigma^{2})$-distributed, either\n$(\\delta_{0}+\\mathcal{E}(\\lambda))\/2$ distributed, where $\\delta_{0}$ is the Dirac mass\non $0$ and $\\mathcal{E}(\\lambda)$ the exponential distribution with parameter $\\lambda>0$. In both cases the Cramer condition is satisfied and ${\\rm med} (\\xi_0)=0$. In the first case, mean and median are equal whereas this is no longer true in the second case. Thus the minimizers of the $\\ell_1-$ and quadratic risk differ in the second case.\n\nWe use $p_0=3$, $a_{1}=0.2$, $a_{2}=0.3$, $a_{3}=0.2$, $\\sigma^{2}\\in\\{1,3\\}$,\n$\\lambda\\in\\{1,1\/\\sqrt{12}\\}$, and $n=500$, \n$$ \\Theta = \\bigcup_{p=1}^{8} \\Theta_{p} = \\bigcup_{p=1}^{8}\\{\\theta\\in\\mathds{R}^{p}:\\|\\theta\\|_{1} \\leq 1\\} $$\nand\n$$ \\lambda \\in \\mathcal{G} = \\{2,4,8,...,1024\\} .$$\nIn view of our procedure, we compute the simplified penalized criterion\n$$(\\hat{\\lambda},\\hat{p}) = \\arg\\min_{\n\\tiny{\n\\begin{array}{c}\n1\\leq p \\leq 8\n\\\\\n\\lambda\\in\\mathcal{G}\n\\end{array}\n}\n}\n- \\frac{1}{\\lambda}\\log\\int_{\\Theta_{p,\\ell}} \\exp\\left(-\\lambda r_n(\\theta)\\right)\n d\\pi_{p,\\ell}(\\theta)\n + \\lambda\\frac{K^{2}}{n}.\n$$\nThe theoretical value $K=2(\\log n)^{3\/2}\\approx 9$ systematically over-penalizes the large models and always selects the simplest one ($p=1$). Thus, we fix in practice $K=0,1$. To compute the criteria, the integrand term is approximated using an acceptation-reject algorithm with gaussian proposal and 10000 iterations. To compare one simulation of $\\hat{\\theta}\\sim \\pi_{\\hat p}\\{-\\hat \\lambda r_n\\}$ with $\\hat{\\theta}_{AIC}$ obtained by the classical R procedure we simulate independently\n$(X'_{1},\\ldots,X'_{500})$ distributed as $(X_{1},\\ldots, X_{500})$ and we compare\n$$ {\\rm err}_{1}(\\hat{\\theta}) = \\frac{1}{n-8}\\sum_{i=9}^{500} \\left|X'_{i}\n - \\sum_{p=1}^{\\hat{p}} (\\hat{\\theta})_{p} X'_{i-p} \\right| $$\nwith ${\\rm err}_{1}(\\hat{\\theta}_{AIC})$. As the classical R procedure is based\non least square estimators, we also compare the quadratic prevision error\n$$ {\\rm err}_{2}(\\hat{\\theta}) = \\frac{1}{n-8}\\sum_{i=9}^{500} \\left(X'_{i}\n - \\sum_{p=1}^{\\hat{p}} (\\hat{\\theta})_{p} X'_{i-p} \\right)^{2} $$\nwith ${\\rm err}_{2}(\\hat{\\theta}_{AIC})$. The results of 20 experiments are reported in\nTable \\ref{groszizi}.\n\n\\begin{table}[!h]\n\\caption{For each experiment, we report the median, mean and\nstandard deviation of the ${\\rm err}_{i}(.)$ quantities on the 20 experiments realized. The best results, for\nboth ${\\rm err}_{1}(.)$ and ${\\rm err}_{2}(.)$, are highlighted for each serie. }\n\\label{groszizi}\n\\begin{center}\n\\begin{tabular}{|p{1.7cm}|p{1.3cm}||p{1.7cm}|p{1.7cm}||p{1.7cm}|p{1.7cm}|}\n\\hline\n$\\xi_{t}$ & & ${\\rm err}_{1}(\\hat{\\theta})$ & ${\\rm err}_{1}(\\hat{\\theta}_{AIC})$\n & ${\\rm err}_{2}(\\hat{\\theta})$ & ${\\rm err}_{2}(\\hat{\\theta}_{AIC})$ \\\\\n\\hline \\hline\n $\\mathcal{N}(0,1)$ & median & {\\bf 0.790} & 0.792 & {\\bf 0.975} & {\\bf 0.975} \\\\\n & mean & {\\bf 0.797} & 0.798 & {\\bf 0.985} & 0.988 \\\\\n & s.d. & 0.023 & 0.024 & 0.054 & 0.054 \\\\\n\\hline\n $\\mathcal{N}(0,3)$ & median & 2.433 & {\\bf 2.432} & 0.918 & {\\bf 0.916} \\\\\n & mean & {\\bf 2.409} & 2.412 & {\\bf 0.911} & 0.912 \\\\\n & s.d. & 0.078 & 0.065 & 0.496 & 0.412 \\\\\n\\hline\n $\\frac{\\delta_{0}+\\mathcal{E}(1)}{2}$ & median & {\\bf 0.567} & 0.592 & 0.819 & {\\bf 0.813} \\\\\n & mean & {\\bf 0.580} & 0.589 & 0.836 & {\\bf 0.813} \\\\\n & s.d. & 0.047 & 0.043 & 0.153 & 0.150 \\\\\n\\hline\n $\\frac{\\delta_{0}+\\mathcal{E}(1\/\\sqrt{12})}{2}$ & median & {\\bf 1.973} & 2.000 & 9.525 & {\\bf 9.494} \\\\\n & mean & {\\bf 1.955} & 1.997 & 9.733 & {\\bf 9.390} \\\\\n & s.d. & 0.158 & 0.162 & 1.656 & 1.522 \\\\\n\\hline\n\\end{tabular}\n\\vspace*{-6pt}\n\\end{center}\n\\end{table}\nThe results are coherent with the theory: in the gaussian cases, the optimal values of $\\theta$\nfor the $\\ell_{1}$ and the quadratic risks of prediction are the same. Both procedures estimate efficiently the same $\\theta$\nand their prediction risks are the same.\nIn the other cases, the optimal values of $ \\theta$ for the $\\ell_{1}$ and the\nquadratic risks of prediction are not the same. We observe \n${\\rm err}_{1}(\\hat{\\theta})<{\\rm err}_{1}(\\hat{\\theta}_{AIC})$ and\n${\\rm err}_{2}(\\hat{\\theta})>{\\rm err}_{2}(\\hat{\\theta}_{AIC})$. The choice between\nthe two procedures only depends on the prediction risk considered.\n\n\n\\subsection{Neural networks predictors}\\label{app::NN}\nSimilarly than in \\cite{modha}, we present a procedure that approximates the best possible predictor using the best possible number of past values $p$ for the one-step prediction. Given $p$, the best possible predictor for the $\\mathbb L^1$-risk is $\\mbox{med}(X_0|X_{-1},\\ldots,X_{-p})$. We denote $R_p^\\ast$ the corresponding risk. For $\\mathcal X=\\mathbb{R}$ we use the abstract neural networks predictors \ndefined in Barron \\cite{Barron1994} by the relation\n$$\nf_\\theta= c_0+\\sum_{i=1}^\\ell c_i\\, \\phi(a_i\\cdot x+b_i), \\mbox{ for all }x\\in\\mathbb{R}^p,\n$$\nfor $a_i\\in\\mathbb{R}^p$ and $c_i,b_i\\in\\mathbb{R}$ for all $1\\le i\\le \\ell$, the sigmoidal function $\\phi(x)=(1+\\exp(-x))^{-1}$ for all $x\\in\\mathbb{R}$ and $\\theta=(c_0,a_{1,1},\\ldots,a_{1,p},b_{1},c_{1},\\ldots,\na_{\\ell,1},\\ldots,a_{\\ell,p},b_{\\ell},c_{\\ell})$ in $\\mathcal B^q_{c_{p,\\ell}}$ for some $c_{p,\\ell}>0$, $q={\\ell(p+2)+1}$ and $\\ell\\le n $.\nFor any $p\\ge 1$ we denote\n$$\nr_p(x)=\\mbox{med}(X_0|(X_{-1},\\ldots,X_{-p})=x)\\mbox{ for all }x\\in \\mathbb{R}^p.\n$$\nand we assume that there exists a complex-valued function $\\tilde r_p$ on $\\mathbb{R}^p$ satisfying\n$$\n\\forall x\\in \\mathbb{R}^p\\qquad r_p(x)-r_p(0)=\\int_{\\mathbb{R}^p}(e^{iwx}-1)\\tilde r_p(w)dw\n\\quad\\mbox{\nand }\\quad\n\\int_{\\mathbb{R}^p}\\|w\\|_1|\\tilde r_p(w)|dw\\le C'p^c\n$$\nfor some $C',c>0$. Then\n\\begin{cor}\\label{cor::nn}\nUnder {\\bf (WDP)} if for any $(p,\\ell)\\in M$\n\\begin{equation}\\label{cpl}\n\\frac{q}{e}+2\\sqrt \\ell \\|X\\|_\\infty (C'p^c+\\ell\\log \\ell)\\le c_{p,\\ell} \n\\end{equation}\nthen, for all $n\\ge \\max_M c_{p,\\ell}$, with probability at least\n$1-\\varepsilon$, \n$$\nR(\\hat{\\theta})\\leq\n\\inf_{10(1+\\log n)^2p^{1+2c} \\le n } \\Biggl\\{\nR^\\ast_p +\nC\\frac{p^{1\/4+c\/2} \\log^{3} n}{n^{1\/4}} \\Biggr\\}+ C \\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.\n$$\n\n\\end{cor}\nIf $(X_t)$ satisfies the Markov condition of order $p_0$, then $c=0$ and for $n$ sufficiently large \n$$\nR(\\hat{\\theta})-R^\\ast_{p_0}\\leq C\\left(\\frac{ \\log^{3} n}{n^{1\/4}} + \\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\\right).\n$$\nCompared to the iid case, the loss is $\\log^3 n$ and we do not need to know the order $p_0$ (our procedure is memory-universal). It is smaller than the loss of the memory-universal procedure given in \\cite{modha}.\n\n\n\\subsection{Non-parametric auto-regressive predictors}\\label{app::np}\n\nAs in Baraud, Comte and Viennet \\cite{Baraud}, we assume that\n$(X_t)$ is a solution of the equation:\n$$ X_{t} = f_1(X_{t-1}) +\\dots+ f_{p_{0}}(X_{t-p_{0}})+ \\xi_{t},\\mbox{, for all }t\\in\\mathbb{Z} \n$$\nwhere $\\xi_{t}\\sim\\mathcal{N}(0,\\sigma^{2})$, the\n$f_i$ are functions $[-1;1] \\mapsto \\mathds{R}$ in H\\\"older class\n$ H(s_i,L_i) $: $f_i$ is derivable $\\lfloor s_i\\rfloor$ times and\n\\begin{equation}\n\\label{holder2}\n\\exists \\mathcal L_i>0,\\forall (x,x')\\in [-1,1]^{2},\\quad |f_i^{(\\lfloor s_i\\rfloor)}(x)-f_i^{(\\lfloor s_i\\rfloor)}(x')|\n \\leq \\mathcal L_i|x-x'|^{s_i-\\lfloor s_i\\rfloor} .\n\\end{equation}\nConsider the Fourier basis $(\\phi_{j}(.))_{j\\geq 1}$ \non $[-1,1]$ composed by $\\phi_{2k}(x)=\\sqrt{2}\\cos(2\\pi k x)$ and $\\phi_{2k+1}(x)=\\sqrt{2}\\sin(2\\pi k x)$.\nAssumption \\ref{holder2} implies the existence of $\\gamma_i>0$ such that for any $m\\ge 0$ it holds\n$$ \\min_{(\\alpha_{1},...,\\alpha_{m})\\in\\mathds{R}^{m}}\n \\left\\{\\int_{-1}^{1} \\Bigl[f_i(t)-\\sum_{j=1}^{m}\\alpha_{i,j}\\phi_{j}(t)\\Bigr]^{2}ds\\right\\}^{\\frac{1}{2}}\n\\leq \\gamma_{i}m^{- s_i} .\n$$\nThen natural predictors have the form\n$$\n\\widehat X_{n+1}=\\sum_{i=1}^p\\sum_{j=1}^\\ell\\theta_{i,j}\\varphi_j(X_{n-i})=:f_\\theta(X_n,\\ldots,X_{n-p})\n$$\nfor any $p\\in\\{1,...,\\lfloor n\/2\\rfloor\\}$ and any $\\ell\\in\\{1,...,m_{p}=n\\}$. We restrict the procedure on $\\theta_{p,\\ell} $ in the compact set\n$$\\Theta_{p,\\ell}=\\left\\{\\theta\\in\\mathds{R}^{p \\ell},\n \\sum_{i=1}^{p} \\sum_{j=1}^{\\ell} \\theta_{i,j}^{2} (2[j\/2])^{2} \\leq L^{2} \\right\\} $$\nsuch that any $f_{\\theta}$ is an $L$-Lipschitz function. Define also the coefficients $\\overline \\theta_{p,\\ell}\\in\\mathbb{R}^{p\\ell}$ by the relation\n$$\n\\arg\\min_{\\theta\\in\\Theta_{p,\\ell}}\\pi_0\\left[\\left|X_n-\\sum_{i=1}^p\\sum_{j=1}^\\ell\\theta_{i,j}\\varphi_j(X_{n-i})\\right|\\right].\n$$\nWe obtain as a consequence of Theorem \\ref{mainthm}:\n\\begin{cor}\n\\label{fourier}\nUnder {\\bf (CBS)}, if for any $\\ell \\ge 1$ and any $p\\ge 1$\n$$ \\frac{\\ell p}{e} + \\left(\\sum_{i=1}^{p_{0}} \\sum_{j=1}^{\\ell} (\\overline{\\theta}_{p_{0},\\ell})_{i,j}^{2} (2\\lfloor j\/2\\rfloor)^{2}\n\\right)^{\\frac{1}{2}}\\le L,$$\nthen for all $n\\ge 8e (1+L)$ with probability at least $1-\\varepsilon$ $$\nR(\\hat{\\theta})-\\mu[|\\xi_{0}|]\n\\leq C \\left(\\left(\\frac{\\log(n)}{n}\\right)^{\\frac{s}{2s+1}} + \\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\\right)\n$$\nwhere $s$ denotes $\\min\\{ s_{1},..., s_{p_{0}}\\}$.\n\\end{cor}\nThe (iid) minimax rate of convergence with respect to $ s_{1},..., s_{p_{0}}$ is for the $\\mathbb{L}^1$-risk achieved up\nto a logarithmic loss.\nIn \\cite{Baraud}, the (iid) minimax rate of convergence for the quadratic risk\nis achieved for the empirical quadratic risk.\n\n\\section*{Aknowledgments}\n\nWe would like to thank the anonymous referees for the various corrections and improvements\nthey suggested.\n\n\\section{Proofs}\\label{sec::pf}\n\nTo present the proofs in a unified version whether we work under {\\bf\n(CBS)} or {\\bf (WDP)}, we truncate the observations if we are under\n{\\bf (CBS)}: \n$$\n\\overline X_t=H(\\overline \\xi_t,\\overline \\xi_{t-1},\\overline \\xi_{t-2}, \\ldots),\\mbox{ for all }t\\in \\mathbb{Z},\n$$\nwhere $\\overline \\xi_t=(\\xi_t\\wedge C)\\vee\n(-C)$,\nunder {\\bf (WDP)} we just take $\\overline X_t=X_t$.\nWe denote in the sequel $\\overline X=(\\overline X_t)$ and $\\overline r$, $\\overline R$ the risks\nassociated with $\\overline X$ under {\\bf (CBS)} and with $X$ under {\\bf (WDP)}. For shorten the proofs, we denote $K_n=(1+L)\\log^{3\/2} n$ and $w_{p,\\ell}=1\/(m_{p} \\lfloor n\/2\\rfloor)$ in the sequel.\nThe proof of our main Theorem lies on estimates on Laplace transforms.\n\n\\subsection{Preliminary lemmas: estimates on Laplace transforms}\n\nThe proofs of these lemmas are given in Section \\ref{prooflems}.\nThe first Lemma is an estimate of the Laplace transforms of the risk of $\\overline X$; it is\na direct corollary of the result in Rio \\cite{Rio2000a}.\n\\begin{lemma}[Laplace transform of the risk] \\label{lemrisk}\nFor any $\\lambda>0$ and $\\theta\\in\\Theta$ we have:\n\\begin{equation*}\n\\pi_0 [\\exp ( \\lambda (\\overline R(\\theta) - \\overline r_n(\\theta)) )]\n\\leq\n\\exp \\left(\\frac{ \\lambda^{2}k_n^{2}}{ n(1-p \/n)^{2}}\\right),\n\\end{equation*}\nwhere $k_n=\\sqrt 2C(1+L) (a(H)+\\tilde a(H))$ under {\\bf (CBS)} and\n$k_n= (1+L) (\\|X_0\\|_\\infty+\\theta_{\\infty,n}(1))\/\\sqrt 2$ under {\\bf (WDP)}.\n\\end{lemma}\nGiven a measurable space $(E,\\mathcal{E})$ we let\n$\\mathcal{M}_{+}^{1}(E)$ denote the set of all probability measures\non $(E,\\mathcal{E})$. The Kullback divergence is a pseudo-distance on $\\mathcal{M}_{+}^{1}(E)$ defined, for any\n$(\\pi,\\pi')\\in[\\mathcal{M}_{+}^{1}(E)]^{2}$ by the equation\n$$\n\\mathcal{K}(\\pi,\\pi')= \\left\\{\n\\begin{array}{l}\n\\pi[\\log(d \\pi\/d \\pi')] \\quad \\text{ if } \\pi \\ll \\pi',\n\\\\\n\\\\\n+ \\infty \\quad \\text{ otherwise.}\n\\end{array}\n\\right.\n$$\nThe proof of the following Lemma is omitted as it can be found in \\cite{Classif} or \\cite{Cat7}.\n\\begin{lemma}[Legendre transform of the Kullback divergence function] \\label{LEGENDRE}\nFor any $\\pi\\in\\mathcal{M}_{+}^{1}(E)$,\nfor any measurable function $h:E\\rightarrow\\mathds{R}$ such\nthat $ \\pi[\\exp (h)]<+\\infty $ we have:\n\\begin{equation} \\label{lemmacatoni}\n\\pi[\\exp\n(h)]=\\exp\\left(\\sup_{\\rho\\in\\mathcal{M}_{+}^{1}(E)}\\biggl(\\rho\n[h]-\\mathcal{K}(\\rho,\\pi)\\biggr)\\right),\n\\end{equation}\nwith convention $\\infty-\\infty=-\\infty$. Moreover, as soon as $h$ is\nupper-bounded on the support of $\\pi$, the supremum with respect to\n$\\rho$ in the right-hand side is reached for the Gibbs measure\n$\\pi\\{h\\}$ defined in \\eqref{Gibbsmeasure}.\n\\end{lemma}\nUsing Lemmas \\ref{lemrisk} and \\ref{LEGENDRE} we get an upper-bound for\nthe Laplace transform of the mean risk of Gibbs estimators in all sub-models:\n\\begin{lemma} \\label{lemrisk2}\nUnder the\nassumptions of Theorem \\ref{mainthm} we have for any\n$\\lambda>0$ and $(p,\\ell)\\in M$:\n\\begin{equation}\n\\label{equlem1,equlem2}\n\\pi_0\\left[ \\exp \\Biggl( \\sup_{\\rho\\in \\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})}\n\\left\\{ \\lambda \\rho[\\overline R- \\overline r_n]\n-\\mathcal{K}\\left(\\rho,\\pi_{p,\\ell}\\right) \\right\\}\\right.\n\\left. - \\frac{\\lambda^{2}k_n^{2}}{n(1-p\/n)^{2}}\\Biggr)\\right]\\leq 1,\n\\end{equation}\nwhere $k_n$ has the same expression than in Lemma \\ref{lemrisk}.\n\\end{lemma}\nFollowing the technique used by Catoni \\cite{Classif}, we derive from Lemma\n\\ref{lemrisk2} another upper-bound on the Laplace transform of the mean risk\nof any aggregation estimators of all Gibbs estimators:\n\\begin{lemma}\n\\label{lemrisk3}\nFor any measurable functions\n$ \\hat \\rho_{p,\\ell}:\n\\mathcal{X}^{n} \\rightarrow \\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})$ for $(p,\\ell)\\in M$,\nunder the assumptions of Theorem \\ref{mainthm} we have:\n\\begin{gather*}\n\\pi_0\\left[ \\sum_{(p,\\ell)\\in M} \\sum_{\\lambda\\in\\mathcal{G}_{p,\\ell}}\n \\hat \\rho_{p,\\ell}\\left[\\exp\\left(\n \\lambda (\\overline R- \\overline r_n)\n - \\log \\frac{d\\hat \\rho_{p,\\ell}}{d\\pi_{p,\\ell}}-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\n+\\log (w_{p,\\ell}\/n) \\right)\\right]\\right]\\le 1\\\\\n\\mbox{and}\\qquad \\pi_0\\Biggl[\n \\sum_{(p,\\ell)\\in M} \\sum_{\\lambda\\in\\mathcal{G}_{p,\\ell}}\n \\exp\\left(\n \\lambda \\hat \\rho_{p,\\ell} [\\overline r_{n} - \\overline R]\n - \\mathcal{K}(\\hat \\rho_{p,\\ell},\\pi_{p,\\ell})-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\n+\\log (w_{p,\\ell}\/n)\\right)\\Biggr] \\le 1,\n\\end{gather*}\nwhere we remind that $k_n$ is defined in Lemma \\ref{lemrisk}.\n\\end{lemma}\nFinally, we use a Lemma that quantify the error in the risk due to the truncation under {\\bf (CBS)}:\n\\begin{lemma}\n\\label{lembound}\nUnder {\\bf (CBS)}, for any truncation level $C>0$ and any $0\\le \\lambda \\le n\/(4(1+L))$, we have \n$$\n\\pi_0\\Big[\\exp( \\lambda\\sup_{\\theta\\in\\Theta}|r_n(\\theta)-\\overline r_n(\\theta)|\n-\\lambda2 (1+L)\\Psi(a(H))\\left(\\frac{a(H)^2C}{\\exp(a(H)C)-1}\n+\\lambda \\frac{4(1+L)}{n} \\right)\\Big]\\le 1.\n$$\n\\end{lemma}\n\n\n\\subsection{Proof of Theorem \\ref{mainthm}}\n\n\\label{proofmain}\nRemark that {\\bf (WDP)} is satisfied, so $\\overline R=R$ and $\\overline r=r$. We apply the first\ninequality of Lemma \\ref{lemrisk3} to $\\hat\\rho_{p,\\ell}^\\lambda=\\pi_{p,\\ell}\\{-\\lambda r_n\\}$.\nRemembering that $(\\hat p,\\hat \\ell, \\hat \\lambda)= \\arg\\min \\hat{R}\\left(p,\\ell,\\lambda\\right)$,\nwe obtain in particular:\n\\begin{equation}\\label{lastlapl}\n\\pi_0 \\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}\\left[\\exp\\left(\n \\hat \\lambda (R- r_n)\n - \\log \\left(\\frac{d\\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}}\n {d\\pi_{\\hat p,\\hat\\ell}}\\right)-\\frac{\\hat\\lambda^{2} k_n^{2}}{n(1-\\hat p\/n)^{2}}\n+\\log \\left(\\frac{w_{\\hat p,\\hat \\ell}}{n}\\right)\\right)\\right]\\le 1.\n\\end{equation}\nRemark that $\\pi_0 \\, \\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}$ is a well defined probability\nmeasure as $\\hat \\rho$ are defined conditionally on the observations. Remark also that $\\hat \\theta\n\\sim \\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}$ by definition, then using the classical Chernov\nbound we derive that with probability $1-\\varepsilon$ it holds:\n\\begin{equation}\n\\label{equproof1}\n R(\\hat{\\theta})\n\\leq\n r_n (\\hat \\theta)\n+ \\frac{\\hat \\lambda k_n^{2}}{n(1-\\hat p\/n)^{2}}\n+ \\frac{1}{\\hat\\lambda}\\log\\left( \\frac{d\\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}}{d\\pi_{\\hat p,\\hat\\ell}}\\right)\n+ \\frac{1}{\\hat \\lambda} \\log\\left( \\frac{ n}{w_{\\hat p,\\hat\\ell}}\\right)\n+ \\frac{1}{\\hat\\lambda} \\log\\frac{1}{\\varepsilon}.\n\\end{equation}\nIn order that the term $\\hat R$ appears, we notice that \\eqref{equproof1} is equivalent to\n\\begin{align}\n\\nonumber R(\\hat{\\theta})\n& \\leq - \\frac{1}{\\hat \\lambda} \\log \\int_{\\Theta_{\\hat p,\\hat \\ell}}\n\\exp\\left(-\\hat\\lambda r_n(\\theta)\\right)\\pi_{\\hat p,\\hat \\ell}(d\\theta) +\n\\frac{\\hat \\lambda k_n^{2}}{n(1-\\hat p\/n)^{2}} + \\frac{1}{\\hat \\lambda} \\log\n\\left(\\frac{n}{ w_{\\hat p,\\hat \\ell}}\\right) +\n\\frac{1}{\\hat \\lambda} \\log\\frac{1}{\\varepsilon}\n\\\\\n\\nonumber & \\le \\inf_{p,\\ell,\\lambda}\n\\hat{R}\\left(p,\\ell,\\lambda\\right)\n+\\frac{\\hat \\lambda(k_n^2-K_n^2)}{n(1-\\hat p\/n)^2}- \\frac{1}{\\hat\\lambda} \\log \\varepsilon\n\\end{align}\n(remind that $K_n=(1+L)\\log^{3\/2} n$).\nNow, we upper bound the term\n$\\hat{R}\\left(p,\\ell,\\lambda\\right)$, for any $p$, $\\ell$ and $\\lambda$. \nUsing the second inequality of Lemma\n\\ref{lemrisk3} we obtain for any $(p,\\ell)\\in M$, $\\lambda\\in\\mathcal{G}$\nand $\\rho\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})$,\n\\begin{equation}\n\\label{equproof1prime} \\int_{\\Theta_{p,\\ell}} r_n\n\\left(\\theta\\right) \\rho(d\\theta) \\leq \\int_{\\Theta_{p,\\ell}} R \\left(\\theta\\right) \\rho(d\\theta) + \\frac{\\lambda\nk_n^{2}}{n(1-p\/n)^{2}} + \\frac{1}{\\lambda}\n\\mathcal{K}\\left(\\rho,\\pi_{p,\\ell}\\right) + \\frac{1}{\\lambda} \\log\n\\frac{n}{w_{p,\\ell}} + \\frac{1}{\\lambda}\n\\log\\frac{1}{\\varepsilon}.\n\\end{equation}\nFrom \\eqref{equproof1prime} and using Lemma \\ref{LEGENDRE} two times we derive that\n\\begin{multline*}\n- \\frac{1}{\\lambda} \\log \\int_{\\Theta_{p,\\ell}} \\exp\\left(-\\lambda r_n(\\theta)\\right)\\pi_{p,\\ell}(d\\theta)\n=\n\\inf_{\\rho \\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})}\n\\left\\{\\int_{\\Theta_{p,\\ell}} r_n \\left(\\theta\\right) \\rho(d\\theta)\n + \\frac{1}{\\lambda} \\mathcal{K}\\left(\\rho,\\pi_{p,\\ell}\\right) \\right\\}\\\\\n\\leq\n\\inf_{\\rho \\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})}\n\\left\\{\\int_{\\Theta_{p,\\ell}} R \\left(\\theta\\right) \\rho(d\\theta)\n + \\frac{2}{\\lambda} \\mathcal{K}\\left(\\rho,\\pi_{p,\\ell}\\right) \\right\\}\n+ \\frac{\\lambda k_n^{2}}{n(1-p\/n)^{2}}\n+\\frac{1}{\\lambda} \\log \\frac{n}{\\varepsilon w_{p,\\ell}}\n\\\\\n=\n- \\frac{2}{\\lambda} \\log \\int_{\\Theta_{p,\\ell}} \\exp\\left(-\\frac{\\lambda}{2} R(\\theta)\\right)\\pi_{p,\\ell}(d\\theta)\n+\\frac{\\lambda k_n^{2}}{n(1-p\/n)^{2}}\n+\\frac{1}{\\lambda} \\log \\frac{n}{\\varepsilon w_{p,\\ell}}.\n\\end{multline*}\nFinally we obtain:\n\\begin{equation}\\label{equproof5}\n\\hat{R}\\left(p,\\ell,\\lambda\\right)\n\\leq\n- \\frac{2}{\\lambda} \\log \\int_{\\Theta_{p,\\ell}} \\exp\\left(-\\frac{\\lambda}{2} R(\\theta)\\right)\\pi_{p,\\ell}(d\\theta)\n+\\frac{\\lambda( k_n^{2}+K_n^2)}{n(1-p\/n)^{2}}\n+\\frac{1}{\\lambda} \\log \\frac{n}{ \\varepsilon w_{p,\\ell}}.\n\\end{equation}\nUnder Assumption \\eqref{dimension}, as soon as $\\lambda>2e$ it holds\n$$\n- \\log\\pi_{p,\\ell}\\left[\\exp\\left(-\\frac{\\lambda}{2}\n\\left( R- R(\\overline{\\theta}_{p,\\ell})\\right)\\right)\\right]\n\\le d_{p,\\ell}\\log\\frac{\\lambda}{2}\n$$\n and it easily follows that\n$$-\\log\\pi_{p,\\ell}\\left[\\exp\\left(-\\frac{\\lambda}{2}\\overline R\\right)\\right]\n\\leq d_{p,\\ell} \\log\\frac{\\lambda}{2} + \\frac{\\lambda}{2} R(\\overline{\\theta}_{p,\\ell}).\n$$\nWe plug this result into the inequality \\eqref{equproof5} to obtain:\n\\begin{equation}\n\\label{equproof6}\n\\hat{R}\\left(p,\\ell,\\lambda\\right)\n\\leq\nR\\left(\\overline{\\theta}_{p,\\ell}\\right) +\\frac{1}{\\lambda}\\left(2 d_{p,\\ell} \\log\n\\frac{\\lambda}{2}+\\log\\frac{n}{\\varepsilon w_{p,\\ell}}\\right)\n+\\frac{\\lambda (k_n^{2}+K_n^2)}{n(1-p\/n)^{2}}.\n\\end{equation}\nCollecting the inequalities \\eqref{equproof1} and \\eqref{equproof6}, we obtain:\n\\begin{multline}\n\\label{equproof6-2}\nR(\\hat{\\theta})\n\\le\n\\inf_{p,\\ell,\\lambda\\in\\mathcal{G}_{p,\\ell}}\n\\left\\{R\\left(\\overline{\\theta}_{p,\\ell}\\right) + \\frac{1}{\\lambda}\\left( 2 d_{p,\\ell}\n\\log \\frac{\\lambda}{2}+ \\log \\frac{n}{\\varepsilon w_{p,\\ell}}\n\\right)\n+\\frac{\\lambda (k_n^{2}+K_n^2)}{n(1-p\/n)^{2}}\\right\\}\\\\\n+\n\\frac{ \\hat\\lambda(k_n^2-K_n^2)}{n(1-\\hat p\/n)^2}- \\frac{1}{\\hat \\lambda} \\log \\varepsilon.\n\\end{multline}\nAs for $\\lambda\\in\\mathcal{G}_{p,\\ell}$, we have, by definition of $\\mathcal{G}_{p,\\ell}$ that\n$$ \\lambda \\in\\left[\\check{c} \\frac{\\sqrt{d_{p,\\ell}n}\\log(d_{p,\\ell}n)}{K_{n}}, \\dots,\n\\hat{c} \\frac{\\sqrt{d_{p,\\ell}n}\\log(d_{p,\\ell}n)}{K_{n}}\\right] \\cap [2e,n]$$\nthen it holds\n\\begin{multline}\n\\label{equproof6-2-bis}\nR(\\hat{\\theta})\n\\le\n\\inf_{d_{p,\\ell}\\leq n}\n\\Biggl\\{R\\left(\\overline{\\theta}_{p,\\ell}\\right) +\n \\frac{K_{n}}{\\check{c}\\sqrt{d_{p,\\ell}n}\\log(d_{p,\\ell}n)}\n\\left( 2 d_{p,\\ell}\n\\log \\frac{n}{2}+ \\log \\frac{n}{\\varepsilon w_{p,\\ell}}\n\\right)\n\\\\\n+4 \\hat{c} (k_n^{2}+K_n^2) \\sqrt{\\frac{d_{p,\\ell}}{n}} \\frac{\\log(n d_{p,\\ell})}{K_n} \\Biggr\\}\n+\n4 (k_{n}^{2}-K_{n}^{2})_{+} + \\frac{(1+L)\\log\\frac{1}{\\varepsilon}}{\\check{c} \\sqrt{n}}.\n\\end{multline}\nFor the sake of simplicity, we use rough estimates ($1\\leq d_{p,\\ell}$,\n$1\\leq {1}\/{\\varepsilon}$, $m_p\\le n$, \\ldots) to obtain\n\\begin{multline*}\nR(\\hat{\\theta})\n\\le\n\\inf_{d_{p,\\ell}\\leq n}\n\\Biggl\\{R\\left(\\overline{\\theta}_{p,\\ell}\\right) +\n(1+L) \\left( \\frac{6}{\\check{c}}\n+ 8 \\hat{c} \\Bigl(1+\\|X_{0}\\|_{\\infty} + \\theta_{\\infty,n}(1)\\Bigr)^{2}\n\\right) \\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2} (n)\n\\Biggr\\}\n\\\\\n+\n4 (k_{n}^{2}-K_{n}^{2})_{+} + \\frac{7(1+L)\\log\\frac{n}{\\varepsilon}}{\\check{c} \\sqrt{n}}.\n\\end{multline*}\nThis ends the proof as \n$$\nk_{n}^{2}-K_{n}^{2}=(1+L) \\left(\\frac{(\\|X_{0}\\|_{\\infty}+\\theta_{\\infty,n}(1))^{2}}{2} - \\log^{3}(n)\\right).\n$$\n\n\\subsection{Proof of Theorem \\ref{mainthm2}}\n\\label{proofmain2}\n\nAs we work under {\\bf (CBS)}, we have to deal with the error of approximation of $r$ and $R$ by $\\overline R$. To quantify it, we use Lemma \\ref{lembound}. First remark that as $R=\\pi_0[r]$ it holds\n$$\n \\exp\\left( \\lambda\\sup_{\\theta\\in\\Theta}|R(\\theta)-\\overline R(\\theta)|-\\lambda\\phi(C,\\lambda)\\right)\\le 1,\n$$\nwhere \n$$\n\\phi(C,\\lambda)= 2 (1+L)\\Psi(a(H))\\left(\\frac{a(H)^2C}{\\exp(a(H)C)-1}\n+\\lambda \\frac{4(1+L)}{n} \\right)\n$$\nAn immediate consequence is that \n$$\n\\pi_0\\left[\\exp\\left( \\lambda\\sup_{\\theta\\in\\Theta}|\n(r_n-R)(\\theta)-(\\overline r_n-\\overline R)(\\theta)|-2\\lambda \\phi(C,\\lambda)\\right)\\right]\\le 1.\n$$\nAs $R-r_n=\\overline r_n-\\overline R+(r_n-R) -(\\overline r_n-\\overline R)$, for any measurable function\n$ \\rho_{p,\\ell}:\n\\mathcal{X}^{n} \\rightarrow \\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})$ the Cauchy-Schwartz inequality gives\n$$\n\\pi_0\\rho[\\exp(\\lambda\/2(R-r_n))]\\le \\sqrt{\\pi_0\\rho[\\exp(\\lambda(\\overline R-\\overline r_n))]\n\\pi_0\\rho\\left[\\exp\\left(\\lambda\\sup_{\\theta\\in\\Theta}|\n(r_n-R)(\\theta)-(\\overline r_n-\\overline R)(\\theta)|\\right)\\right]}.\n$$\nUsing this remark and the same reasoning than in the proof of Theorem \\ref{mainthm} that gives\n\\eqref{lastlapl} from Lemma \\ref{lemrisk3}, we get the inequality\n\\begin{multline*}\n\\pi_0 \\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}\\left[\\exp\\left(\n \\frac{\\hat \\lambda}{2} (R- r_n)\n - 0,5\\log \\left(\\frac{d\\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}}{d\\pi_{\\hat p,\\hat\\ell}}\\right)-0,5\\frac{\\hat\\lambda^{2} k_n^{2}}{n(1-\\hat p\/n)^{2}}\n+0,5\\log \\left(\\frac{w_{\\hat p,\\hat \\ell}}{n}\\right)\\right.\\right.\\\\\n\\left.\\left. -\\lambda \\phi(C,\\lambda)\\right)\\right]\\le 1.\n\\end{multline*}\nAs in the proof of Theorem \\ref{mainthm}, we derive an equivalent of\n\\eqref{equproof1}, i.e. with probability $1-\\varepsilon$ it holds:\n$$\n R(\\hat{\\theta})\n\\leq\n r_n (\\hat \\theta)\n+ \\frac{\\hat \\lambda k_n^{2}}{n(1-\\hat p\/n)^{2}}\n+ \\frac{1}{\\hat\\lambda}\\log\\left( \\frac{d\\hat \\rho_{\\hat p,\\hat \\ell}^{\\hat\\lambda}}{d\\pi_{\\hat p,\\hat\\ell}}\\right)\n+ \\frac{1}{\\hat \\lambda} \\log\\left( \\frac{ n}{w_{\\hat p,\\hat\\ell}}\\right)\n+2\\phi(C,\\hat\\lambda)+ \\frac{2}{\\hat\\lambda} \\log\\frac{1}{\\varepsilon}.\n$$\nWith similar arguments we derive an equivalent of \\eqref{equproof1prime}:\n\\begin{multline*}\n \\int_{\\Theta_{p,\\ell}} r_n\n\\left(\\theta\\right) \\rho(d\\theta) \\leq \\int_{\\Theta_{p,\\ell}} R \\left(\\theta\\right) \\rho(d\\theta) + \\frac{\\lambda\nk_n^{2}}{n(1-p\/n)^{2}} + \\frac{1}{\\lambda}\n\\mathcal{K}\\left(\\rho,\\pi_{p,\\ell}\\right) + \\frac{1}{\\lambda} \\log\n\\frac{n}{w_{p,\\ell}} \\\\\n+2\\phi(C,\\lambda)+ \\frac{2}{ \\lambda} \\log\\frac{1}{\\varepsilon}\n\\end{multline*}\nand also\n\\begin{multline}\n\\label{preuve2-pierre}\nR(\\hat{\\theta})\n\\le\n\\inf_{p,\\ell,\\lambda}\\left\\{R\\left(\\overline{\\theta}_{p,\\ell}\\right) + \n\\frac{1}{\\lambda}\\left( 2 d_{p,\\ell} \\log \\frac{\\lambda}{2}+ \\log \\frac{n}{\\varepsilon w_{p,\\ell}}\n\\right)\n+\\frac{\\lambda (k_n^2+K_n^{2})}{n(1-p\/n)^{2}}\n+ 2\\phi(C,\\lambda)\\right\\}\\\\\n+\\frac{\\hat{\\lambda} (k_n^2-K_n^{2})}{n(1-p\/n)^{2}}\n+2\\phi(C,\\hat \\lambda)\n - \\frac{2}{\\hat \\lambda} \\log \\varepsilon.\n\\end{multline}\nWe still have\n$$ -\\frac{2}{\\hat \\lambda} \\log \\varepsilon \\leq \\frac{2(1+L)}{\\check{c}\\sqrt{n}} \\log\\frac{1}{\\varepsilon} $$\nso we now have to upper bound $ 2\\phi(C,\\hat \\lambda) $. As $\\hat \\lambda\\leq n\/(4(1+L))$ by definition of the $\\mathcal G_{p,\\ell}$, fixing $C=a(H)^{-1} \\log n $ we obtain:\n$$\n\\phi(C,\\hat\\lambda)\\le \\frac{4a(H)(1+L)\\Psi(a(H)) [2\\hat \\lambda(1+L) +a(H) \\log(n)]}{n}.\n$$\nAs $\\hat \\lambda \\le \\hat c{d_{\\hat{p},\\hat{\\ell}}}\\log(d_{\\hat{p},\\hat{\\ell}}n)\/(1+L)$ by definition of $\\mathcal G_{p,\\ell}$, we obtain\n\\begin{multline*}\n\\frac{\\hat \\lambda (k_n^2-K_n^{2})}{n(1-p\/n)^{2}}\n+2\\phi(C,\\hat \\lambda)\n\\leq\n\\frac{8a(H)^2(1+L)\\Psi(a(H))\\log(n)}{n}\n\\\\\n+\\sqrt{\\frac{d_{\\hat{p},\\hat{\\ell}}}{n}}\\log(d_{\\hat{p},\\hat{\\ell}}n)4(1+L)\\hat c\n\\left(4a(H)\\Psi(a(H)) + 2\\log^{2}(n)(1+\\tilde{a}(H)\/a(H))^{2} - \\log^{3}(n) \\right)_{+}.\n\\end{multline*}\nWe now plug this result into \\eqref{preuve2-pierre} to end the proof.\n\n\\subsection{Proofs of Lemmas \\ref{lemrisk}, \\ref{lemrisk2}, \\ref{lemrisk3}, \\ref{lembound} and of Proposition \\ref{propdim}}\n\\label{prooflems}\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemrisk}]\nThe proof of this Lemma is based on the following result of Rio \\cite{Rio2000a} on $\\overline X$:\n\\begin{thm}\n\\label{RIOthm}\nLet $Y=(Y_t)_{t\\in\\mathbb{Z}}$ be a bounded stationary time series bounded distributed as $\\pi_0$ on $\\mathcal{X}^\\mathbb{Z}$. Let $h$ be a $1$-Lipschitz function of $ \\mathcal{X}^{n} \\rightarrow \\mathds{R} $, i.e.\nsuch that:\n\\begin{equation} \\label{lip}\n\\forall (x_{1},y_{1},...,x_{n},y_{n})\\in\\mathcal{X}^{2n},\\quad | h(x_{1},...,x_{n}) - h(y_{1},...,y_{n}) |\n \\leq \\sum_{i=1}^{n} \\left\\|x_{i}-y_{i}\\right\\|.\n\\end{equation}\nThen for any\n$t\\ge 0$ we have:\n$$\n\\pi_0\\left[\\exp (t(\\pi_0[h(X_{1},...,X_{n})]-h(X_{1},...,X_{n})))\\right]\n\\le \\exp(t^2n(\\|X_0\\|_\\infty+ \\theta_{\\infty,n}(1) )^{2}\/2 ).\n$$\n\\end{thm}\n\\begin{proof}[Proof of Theorem \\ref{RIOthm}]\nThis version of Theorem 1 of \\cite{Rio2000a} comes rewriting the inequality (3) in \\cite{Rio2000a} as, for any $1$-Lipschitz function $g$:\n$$\n\\Gamma(g)=\\|\\mathbb{E}(g(X_{\\ell+1},\\cdots,X_n)|{\\mathcal F}_\\ell)-\\mathbb{E}(g(X_{\\ell+1},\\cdots,X_n))\\|_\\infty\\le \\theta_{\\infty,n-\\ell}(1).\n$$\nThe result is proved as $\\sup_{1\\le r\\le n}\\theta_{\\infty,r}(1)\\le \\theta_{\\infty,n}(1).$\n\\end{proof}\nWe now apply the result of Theorem \\ref{RIOthm} on $Y=\\overline X$ to obtain the result of Lemma \\ref{lemrisk}.\nLet us fix $\\lambda>0$, $(p,\\ell)\\in M$, $\\theta\\in\\Theta_{p,\\ell}$ and $t=(1+L)\\lambda\/\\left[n-p\\left(\\theta\\right)\\right]$ and the function $h$ defined by:\n$$ h(x_{1},...,x_{n})\n = \\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n}\n \\left\\|x_{i} - f_{\\theta}(x_{i-1},...,x_{i-p\\left(\\theta\\right)}) \\right\\|. $$\nWe easily check that $h$ satisfies condition \\eqref{lip}:\n\\begin{align*}\n\\Bigl|h(x_{1},...,x_{n}) &- h(y_{1},...,y_{n})\\Bigr|\n\\\\\n&\\leq\n\\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\Biggl|\n \\left\\|x_{i} - f_{\\theta}(x_{i-1},...,x_{i-p\\left(\\theta\\right)}) \\right\\|\n - \\left\\|y_{i} - f_{\\theta}(y_{i-1},...,y_{i-p\\left(\\theta\\right)}) \\right\\| \\Biggr|\n\\\\\n&\\leq\n\\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n}\n \\left\\|x_{i}-y_{i} - f_{\\theta}(x_{i-1},...,x_{i-p\\left(\\theta\\right)})\n + f_{\\theta}(y_{i-1},...,y_{i-p\\left(\\theta\\right)})\\right\\|\n\\\\\n&\\leq\n\\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\left\\|x_{i}-y_{i}\\right\\|\n+ \\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\left\\| f_{\\theta}(x_{i-1},...,x_{i-p\\left(\\theta\\right)})\n - f_{\\theta}(y_{i-1},...,y_{i-p\\left(\\theta\\right)})\\right\\|\n\\\\\n&\\leq\n\\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\left\\|x_{i}-y_{i}\\right\\|\n+ \\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\sum_{j=1}^{p(\\theta)} a_{j}(\\theta) \\left\\|x_{i-j}-y_{i-j}\\right\\|\n\\\\\n&\\leq\n\\frac{1}{1+L} \\sum_{i=p\\left(\\theta\\right)+1}^{n} \\left\\|x_{i}-y_{i}\\right\\|\n+ \\frac{L}{1+L} \\sum_{i=1}^{n} \\left\\|x_{i}-y_{i}\\right\\|\\\\\n&\\leq \\sum_{i=1}^{n} \\left\\|x_{i}-y_{i}\\right\\|.\n\\end{align*}\nThe direct application of Theorem \\ref{RIOthm} ends the proof under {\\bf (WDP)}. Under {\\bf (CBS)} $k_n$ follows from the estimates of $\\|X_0\\|_\\infty$ and $\\theta_{\\infty,n}(1)$ obtained in Proposition \\ref{lemma_ex_1}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemrisk2}]\nIntegrate the inequality in Lemma \\ref{lemrisk} with respect $\\pi_{p,\\ell}$ on $\\Theta_{p,\\ell}$ (then $p(\\theta)=p$) for any $(p,\\ell)\\in M$ in order to obtain:\n\\begin{equation*}\n\\pi_{p,\\ell}[ \\pi_0[ \\exp (\n\\lambda (\\overline R- \\overline r_n ))]] \\leq\n\\exp\n\\left(\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\\right).\n\\end{equation*}\nFubini's Theorem implies that\n\\begin{equation*}\n\\pi_0\\left[ \\pi_{p,\\ell}\\left[ \\exp \\left(\n\\lambda\\Bigl(\\overline R- \\overline r_n \\Bigr)\n-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\\right)\\right]\\right]\n\\leq 1.\n\\end{equation*}\nApplying Lemma \\ref{LEGENDRE} for $\\pi=\\pi_{p,\\ell}$ and $h=\\lambda(\\overline R- \\overline r_n)-\\lambda^{2} k_n^{2}\/(n(1-p\/n)^{2})$ on $\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})$ leads to the inequality:\n$$\n\\pi_0\\left[ \\exp \\Biggl(\\sup_{\\rho\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})}\n\\left\\{ \\lambda \\rho[\\overline R- \\overline r_n] - \\mathcal{K}(\\rho,\\pi_{p,\\ell}) \\right\\} -\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\\Biggr)\\right]\\leq 1.\n$$\nThis ends the proof.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemrisk3}]\nFirst, let us choose $\\lambda\\in\\Lambda$.\nLet $h_{p,\\ell}^{\\lambda}$ denotes, for any $(p,\\ell)\\in M$:\n$$\nh_{p,\\ell}^{\\lambda}=\\sup_{\\rho_{p,\\ell}\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})} \\left\\{ \\lambda\n\\rho_{p,\\ell}[\\overline R- \\overline r_n]\n - \\mathcal{K}\\left(\\rho_{p,\\ell},\\pi_{p,\\ell}\\right) \\right\\}\n-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}.\n$$\nFrom Lemma \\ref{lemrisk2} applied on the different $\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})$ we have,\nfor any $(p,\\ell)\\in M$:\n$$\n\\pi_0\\left[\\sum_{(p,\\ell)\\in M}w_{p,\\ell} \\exp \\left(h_{p,\\ell}^{\\lambda}\\right)\\right]\\leq 1.\n$$\nNow we apply Inequality \\eqref{lemmacatoni} in\nLemma \\ref{LEGENDRE} for $\\pi=\\sum_{(p,\\ell)\\in M}w_{p,\\ell}\\delta_{(p,\\ell)}$\nand $h=\\sum_{(p,\\ell)\\in M}h_{p,\\ell}^{\\lambda}\\mathbb{1}_{\\Theta_{p,\\ell}}$ and we obtain\n$$\n\\pi_0\\left[\\exp\\left(\\sup_{\\sum_{(p,\\ell)\\in M}w'_{p,\\ell}=1}\\left\\{\n\\sum_{(p,\\ell)\\in M}w'_{p,\\ell} h_{p\\ell}-\\sum_{(p,\\ell)\\in M}w'_{p,\\ell}\\log (w'_{p,\\ell}\/w_{p,\\ell})\\right\\}\\right)\\right]\\le 1\n$$\nand, by Jensen's inequality, and replacing $h_{p,\\ell}^{\\lambda}$ by its definition,\n\\begin{multline}\\label{eq::a}\n\\pi_0\\left[\\sup_{\\sum_{(p,\\ell)\\in M}w'_{p,\\ell}=1}\\left\\{\n \\sum_{(p,\\ell)\\in M}w'_{p,\\ell} \\sup_{\\rho_{p,\\ell}\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})} \\exp\\left(\n \\lambda \\rho_{p,\\ell} \\left[\\lambda(\\overline R- \\overline r_n)\n - \\log \\frac{d\\rho_{p,\\ell}}{d\\pi_{p,\\ell}}\\right]\\right.\\right.\\right.\\\\\n\\left.\\left.\\left.-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\n+\\log \\frac{w_{p,\\ell}}{ w'_{p,\\ell}}\\right)\\right\\}\\right] \\le 1.\n\\end{multline}\nBy Jensen again, we obtain a bound for the first term in the sum bounded in Lemma \\ref{lemrisk3}:\n\\begin{multline*}\n\\pi_0\\left[\\sup_{\\sum_{(p,\\ell)\\in M}w'_{p,\\ell}=1}\\left\\{\n \\sum_{(p,\\ell)\\in M}w'_{p,\\ell}\n \\sup_{\\rho_{p,\\ell}\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})} \\rho_{p,\\ell}\\left[\\exp\\left(\n \\lambda (\\overline R- \\overline r_n)\n - \\log \\frac{d\\rho_{p,\\ell}}{d\\pi_{p,\\ell}}\\right.\\right.\\right.\\right.\\\\\n\\left.\\left.\\left.\\left.-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\n+\\log \\frac{w_{p,\\ell}}{ w'_{p,\\ell}}\\right)\\right]\\right\\}\\right] \\le 1.\n\\end{multline*}\nFinally, we sum this inequality over all $\\lambda\\in\\mathcal G$ to bound the first expectation.\n\nThe second expectation is bounded by choosing specific weights $w'_{p,\\ell}$ in the supremum in inequality \\eqref{eq::a} such that $w'_{p,\\ell}=1$ for $(p,\\ell)= \\arg\\max_M\\{h_{p,\\ell}\\}$:\n$$\n\\pi_0\\Biggl[\n \\sup_{\n\\tiny{\n\\begin{array}{c}\n(p,\\ell)\\in M\n\\\\\n\\rho_{p,\\ell}\\in\\mathcal{M}_{+}^{1}(\\Theta_{p,\\ell})\n\\end{array}\n}}\n\\left\\{\n \\exp\\left(\n \\lambda \\rho_{p,\\ell} [\\overline R- \\overline r_n]\n - \\mathcal{K}(\\rho_{p,\\ell},\\pi_{p,\\ell})-\\frac{\\lambda^{2} k_n^{2}}{n(1-p\/n)^{2}}\n+\\log w_{p,\\ell}\\right)\\right\\}\\Biggr] \\le 1.\n$$\nAgain a summation over all $\\lambda\\in\\mathcal G$ leads to the result. This ends the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{lembound}]\nFrom the proof of the Lemma \\ref{lemrisk}, we already know that $|\\overline r_n(\\theta)-r_n(\\theta)|\\le (1+L)\/(n-p)\\sum_{i=1}^n\\|X_i-\\overline X_i\\|$. This bound holds uniformly on $\\Theta$. As $p\\le n\/2$ it remains to estimate $\\pi_0[\\exp(\\lambda 2 (1+L)\/n\\sum_{i=1}^n\\|X_i-\\overline X_i\\|])$. From the assumption \\eqref{condlip}, the stationarity of $X$ and as the $\\xi_i$s are iid we have:\n\\begin{multline*}\n\\pi_0\\Big[\\exp\\Big( \\lambda 2(1+L)\/n\\sum_{i=1}^n\\|X_i-\\overline X_i\\|\\Big]\\Big)\\le \\pi_0\\Big[\\exp\\Big(\\lambda 2(1+L)\/n\\sum_{i=1}^n\\sum_{j=0}^\\infty a_j(H)\\|\\xi_{i-j}-\\overline \\xi_{i-j}\\|\\Big)\\Big]\\\\\n\\le \\pi_0\\Big[\\exp\\Big( \\lambda 2 (1+L)\/n\\sum_{j=0}^\\infty \\sum_{i=1\\vee(n-j)}^na_{n-i+j}(H)\\|\\xi_{n-j}-\\overline \\xi_{n-j}\\|\\Big)\\Big]\\\\\n\\le\\prod_{j=0}^\\infty\\pi_0\\Big[\\exp( \\lambda 2(1+L)\/n\\sum_{i=1\\vee(n-j)}^na_{n-i+j}(H)\\|\\xi_0\\|\\mathbb{1}_{\\|\\xi_0\\|>C})\\Big] .\n\\end{multline*}\nDenoting $c_j= \\lambda 2(1+L)\\sum_{i=1\\vee(n-j)}^na_{n-i+j}(H)\/n$, we develop for all $j\\ge 0$\n$$\n\\pi_0\\Big[\\exp( c_j\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C})\\Big]=1+c_j\\pi_0\\Big[\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]+\\sum_{k\\ge 2}\\frac{c_j^k\\pi_0\\Big[\\|\\xi_{0}\\|^k\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]}{k!}.$$\nThere exists $\\delta>0$ such that the complex function $ \\Psi(z)=\\pi_0[\\exp(z\\|\\xi_0\\|)]$ is holomorphic on the open disk $D(0, a(H)+\\delta)$. From the Cauchy estimates we obtain:\n$$\n\\pi_0\\Big[\\|\\xi_{0}\\|^k\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]\\le \\pi_0\\Big[\\|\\xi_{0}\\|^k \\Big]\\le\\Psi^{(k)}(0)\\le \\frac{k!\\max_{c\\in \\overline {D(0, a(H))}}\\Psi(\\lambda)}{{a(H)}^k}\\le \\frac{k! \\Psi(a(H))}{{a(H)}^k}\\quad \\forall k\\ge 2\n$$ \nAs $\\lambda< n\/(4(1+L))$ then $2c_j\\le a(H)$ for all $j\\ge 0$ and then we derive that for all $j\\ge 0$:\n\\begin{multline*}\n\\pi_0\\Big[\\exp( c_j\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C})\\Big]\\le 1+c_j\\pi_0\\Big[\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]+\\Psi(a(H))\\sum_{k\\ge 2}(c_j\/a(H))^k\\\\\n\\le 1+c_j\\pi_0\\Big[\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]+\\frac{\\Psi(a(H))c_j^2}{a(H)(a(H)-c_j)}\n\\le 1+c_j\\pi_0\\Big[\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C}\\Big]+c_j^2\\frac{2\\Psi(a(H))}{a(H)^2}.\n\\end{multline*}\nAs $\\phi(x)= (\\exp(x)-1)\/x$ is an increasing function for $x>0$, then $\\mathbb{1}_{\\|\\xi_0\\|>C}\\le \\phi(a(H)\\|\\xi_0\\|)\/\\phi(a(H)C)$ and the Markov formula gives for all $j\\ge 0$\n$$\n\\pi_0\\Big[\\exp( c_j\\|\\xi_{0}\\|\\mathbb{1}_{\\|\\xi_0\\|>C})\\Big]\\le 1+ c_j \\frac{\\Psi(a(H))a(H)C}{\\exp(a(H)C)-1}+c_j^2\\frac{2\\Psi(a(H))}{a(H)^2}.\n$$\nCollecting those bounds we obtain\n$$\n\\pi_0\\Big[\\exp\\Big( \\lambda\\sup_{\\theta\\in\\Theta}|\\overline r_n(\\theta)-r_n(\\theta)|\\Big)\\Big]\\le \\prod_{j=0}^\\infty\\left(1+c_j \\frac{\\Psi(a(H))a(H)C}{\\exp(a(H)C)-1}+c_j^2\\frac{2\\Psi(a(H))}{a(H)^2}\\right).\n$$\nUsing that $\\log(1+x)\\le x$ for all $x>0$, we finally obtain:\n$$\n\\log\\left(\\pi_0\\Big[\\exp\\Big( \\lambda\\sup_{\\theta\\in\\Theta}|\\overline r_n(\\theta)-r_n(\\theta)|\\Big)\\Big]\\right)\\le \\sum_{j=0}^\\infty c_j \\frac{\\Psi(a(H))a(H)C}{\\exp(a(H)C)-1}+\\sum_{j=0}^\\infty c_j^2\\frac{2\\Psi(a(H))}{a(H)^2}.\n$$\nThe desired result follows from the estimates $\\sum_{j=0}^\\infty c_j\\le \\lambda a(H)2(1+L)$ and $\\sum_{j=0}^\\infty c_j^2\\le \\lambda^2 a(H)^24(1+L)^2\/n$.\n\\end{proof}\n\nNow give the proof of the useful Proposition \\ref{propdim}.\n\\begin{proof}[Proof of Proposition \\ref{propdim}]\nLet us introduce a parameter $\\zeta>0$ then we have\n\\begin{multline*}\n-\\frac{1}{\\gamma}\\log\\pi_{p,\\ell}\\left[\\exp\\left( -\\gamma \\left(R - R(\\overline{\\theta}_{p,\\ell})\\right)\\right)\\right]-\\zeta\n=\n-\\frac{1}{\\gamma}\\log\\pi_{p,\\ell} \\left[\\exp\\left(-\\gamma \\left(R -R(\\overline{\\theta}_{p,\\ell})-\\zeta \\right)\\right)\\right]\\\\\n\\leq\n-\\frac{1}{\\gamma}\\log\\pi_{p,\\ell} \\left( R(\\theta) - R(\\overline{\\theta}_{p,\\ell}) \\leq \\zeta \\right)\n\\end{multline*}\nThen we directly derive from the definition of $d_{p,\\ell}$ that\n$$\nd_{p,\\ell}\\le\\sup_{\\gamma>e} \\frac{\\inf_{\\zeta>0}\\{\\zeta\\gamma-\\log\\pi_{p,\\ell} \\left( R(\\theta) -\nR(\\overline{\\theta}_{p,\\ell}) \\leq \\zeta \\right)\\}}{\\log \\gamma}.\n$$\nSo\n$$\n\\zeta \\gamma-q\\log\\frac{\\zeta}{C c_{p,\\ell}}\\le q\\wedge \\gamma C(c_{p,\\ell}-\\|\\overline \\theta_{p,\\ell}\\|)+ q \\log \\left(\\frac{Cc_{p,\\ell}\\gamma}{q}\\vee \\frac{c_{p,\\ell}}{c_{p,\\ell}-\\|\\overline \\theta_{p,\\ell}\\|}\\right).\n$$\nNow if $q\\le \\gamma C(c_{p,\\ell}-\\|\\overline\\theta_{p,\\ell}\\|)$ then we get the estimate\n$q(1+\\log (Cc_{p,\\ell}\\gamma \/ q))\/\\log \\gamma$\nwhich decreases with $\\gamma$. We then get the desired bound when the supremum is established for $\\gamma=e\\vee q\/(C(c_{p,\\ell}-\\|\\overline\\theta_{p,\\ell}\\|))$. If $q\\ge \\gamma C(c_{p,\\ell}-\\|\\overline\\theta_{p,\\ell}\\|)$ then we get the estimate\n$\n(\\gamma C(c_{p,\\ell}-\\|\\overline \\theta_{p,\\ell}\\|)+q\\log (c_{p,\\ell}\/(c_{p,\\ell}-\\|\\overline \\theta_{p,\\ell}\\|)))\/\\log\\gamma\n$ which increases with $\\gamma$. We have to consider $\\gamma$ as large as possible, i.e. when $q = \\gamma C(c_{p,\\ell}-\\|\\overline\\theta_{p,\\ell}\\|)$ and we are going back to the case treated above.\n\\end{proof}\n\n\n\\subsection{Proofs of the results given in Section \\ref{sec::app1}}\n\n\\label{proofobservations}\n\nAfter proving Proposition \\ref{lemma_ex_1}, we give Lemma \\ref{couplinglemma} that introduces a coupling argument used to estimate the coefficients $\\theta_{\\infty,n}(1)$ in Propositions \\ref{taucbs} and \\ref{lemma_ex_2}.\n\n\\begin{proof}[Proof of Proposition \\ref{lemma_ex_1}]\nThe theorem 3.1 of Doukhan and Wintenberger \\cite{Doukhan2008} gives the existence of a unique stationary solution and the existence of an $H$ such that $X_t=H(\\xi_t,\\xi_{t-1},\\xi_{t-2},\\ldots)$. We prove that conditions \\eqref{condlip1} and \\eqref{condsum} are automatically satisfied. Let $(x_i)$ and $(y_i)$ be two sequences such that there exists $j\\in\\mathbb{N}$ with $x_i=y_i$ for all $i\\neq j$. Then $H(x)=u_0^\\infty$ where $u_0^\\infty=\\lim_{k\\to \\infty}u_0^k$ for $(u_{-i}^k)_{i\\in\\mathbb{N}}$ defined recursively by\n$$\nu_{-i}^k=F(u_{-i-1}^k,u_{-i-2}^k,\\ldots, u_{1-k}^k,u_{-k}^k,0,\\ldots;x_i)\\qquad \\forall 0\\le i\\le k.\n$$\nSimilarly, we denote $H(y)=v_0^\\infty$ such that $\\|H(x)-H(y)\\|=\\|u_0^\\infty-v_0^\\infty\\|$. For $j=0$, using \\eqref{condlip} $\\|u_{0}^k-v_{0}^k\\|\\le u\\|x_j-y_j\\|$ for all $k$. For $j\\ge 1$, as $x_i=y_i$ for $i>j$, for $k$ sufficiently large it holds (with the convention $\\sum_{\\ell=1}^{-k}=0$ for $k\\ge0$):\n$$\n\\|u_0^k-v_0^k\\|\\le \\sum_{\\ell_1 = 1}^j a_{\\ell_1}(F)\\sum_{\\ell_2 = 1}^{j-\\ell_1}a_{\\ell_2}(F)\\cdots \\sum_{\\ell_j = 1}^{j-\\ell_1-\\cdots-\\ell_{j-1}}a_{\\ell_j}(F)\\|u_{-j}^k-v_{-j}^k\\|.\n$$\nBy definition $\\|u_{-j}^k-v_{-j}^k\\le u\\|x_j-y_j\\|$ and we obtain\n$\\|u_0^k-v_0^k\\|\\le ua(F)^{j-1}\\|x_j-y_j\\|$ for sufficiently large $k$. As the estimate does not depends on $k$, we derive that \\eqref{condlip1} holds with $a_j(H)=ua(F)^{j-1}$ and that \\eqref{condsum} follows from the condition \\eqref{condcontract}.\\end{proof}\n\n\n\n\nNow we state a useful coupling Lemma; $(X^\\ast_t) $ is said to be a coupling version of $(X_t)$ if it is similarly distributed and such that $(X^\\ast_t)_{t>0}$ is independent of $\\mathfrak{S}_0=\\sigma(X_t,t\\le 0)$. From a version of the Kantorovitch-Rubinstein duality, see Dedecker and Prieur \\cite{Dedecker2005} for more details, we obtain an estimate of $\\theta_{\\infty,n}(1)$:\n\\begin{lemma}\\label{couplinglemma}\nFor any version $(X^\\ast_t)$ we have\n$$\n\\theta_{\\infty,n}(1)\\le \\sum_{i=1}^n\\|\\mathbb{E}(\\|X_i-X_i^\\ast\\|\/\\mathfrak{S}_0)\\|_\\infty.\n$$\n\\end{lemma}\nFor the sake of completeness, we give the proof of this already known result.\\\\\n\n\\noindent{\\em Proof of Lemma \\ref{couplinglemma}.} As we equipped $\\mathcal {X}^n$ with the norm $\\|(x_1,\\ldots,x_n)\\|=\\sum_{i=1}^n \\|x_i\\|$ we immediately get the inequality\n$$\n{\\tau_{\\infty,n}}(1)\\le \\|\\mathds{\\mathds{E}}(\\|(X_1,\\ldots,X_n)-(X_1^\\ast,\\ldots,X_n^\\ast)\\||\\mathfrak{S}_0)\\|_\\infty\n\\le \\sum_{i=1}^u\\|\\mathds{\\mathds{E}}(\\|X_i-X_i^\\ast\\||\\mathfrak{S}_0)\\|_\\infty.\\qquad\\square\n$$\nThe proof of Propositions \\ref{taucbs} and \\ref{lemma_ex_2} are simple applications of this Lemma:\n\\begin{proof}[Proof of Proposition \\ref{taucbs}] \nLet us consider the coupling version of the causal Bernoulli shift $(X_t)$ given by\n$$\nX_t^\\ast=H(\\xi_t,\\xi_{t-1},\\ldots,\\xi_{1},\\xi_0^\\ast,\\xi_{-1}^\\ast,\\ldots),\\qquad \\forall t\\in\\mathbb{Z}\n$$\nwhere $(\\xi_t^\\ast)$ is similarly distributed than $(\\xi_t)$ and the two processes are independent. Then from Lemma \\ref{couplinglemma} and condition \\eqref{condlip1} we obtain:\n$$\n\\theta_{\\infty,n}(1)\\le \\sum_{i=1}^n\\Big\\|\\sum_{j=i}^\\infty a_j(H)\\mathbb{E}(\\|\\xi_{i-j}-\\xi_{i-j}^\\ast\\|\/\\mathfrak{S}_0)\\Big\\|_\\infty\\le \\sum_{j=i}^\\infty j a_j(H)\\|\\mathbb{E}(\\|\\xi_{i-j}-\\xi_{i-j}^\\ast\\|\/\\mathfrak{S}_0)\\Big\\|_\\infty $$\nand the desired result follows.\n\\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{lemma_ex_2}]\nHere we will consider the maximal coupling scheme of Goldstein \\cite{Goldstein1979}: there exists a version $(X_t^\\ast)$ such that\n$$\n\\|\\mathbb{P}(X_t\\neq X_t^\\ast\\mbox{ for some }t\\ge r\/ \\mathfrak{S}_0)\\|_\\infty=\\sup_{(A,B)\\in\\,\\mathfrak{S}_0\\times\\mathfrak{F}_r}|\\mathbb{P}(A\/B)-P(B)|=\\varphi(r).\n$$\nAs $\\|Y-Z\\|\\le 2\\|X_0\\|_\\infty\\mathbb{1}_{Y\\neq Z}$ for any variables $Y,Z$ bounded by $\\|X_0\\|_\\infty$, we have:\n$$\n\\|\\mathds{\\mathds{E}}(\\|X_i-X_i^\\ast\\|\/ \\mathfrak{S}_0)\\|_\\infty\n\\le 2\\|X_0\\|_\\infty\\|\\mathds{\\mathds{E}}(\\mathbb{1}_{X_i\\neq X_i^\\ast}\/ \\mathfrak{S}_0)\\|_\\infty\n\\le 2\\|X_0\\|_\\infty\\|\\mathds{\\mathds{P}}(X_i\\neq X_i^\\ast\/ \\mathfrak{S}_0)\\|_\\infty.\n$$\nAs\n$\n\\mathds{P}(X_i\\neq X_i^\\ast\/ \\mathfrak{S}_0)\\le\\mathds{P}( X_t\\neq X_t^\\ast\\mbox{ for some }t\\ge r\/ \\mathfrak{S}_0)\n$,\nwe conclude using Lemma \\ref{couplinglemma}.\n\\end{proof}\n\n\\subsection{Proofs of the results given in Section \\ref{sec::app2}}\n\n\\label{proof::app}\nWe proof the Corollaries \\ref{cor::nn} and \\ref{fourier} of Theorem \\ref{mainthm} applied in the context of Neural Networks and projection in the Fourier basis predictors.\n\n\\begin{proof}[Proof of Corollary \\ref{cor::nn}]\\label{proofnn}\nLet us check that all the predictors are $L$-Lipschitz functions of the observations. For any $x,y\\in\\mathbb{R}^p$, as the function $\\phi$ is 1-Lipschitz, we have\n\\begin{multline*}\n|f_\\theta(x)-f_\\theta(y)|\\le \\left|\\sum_{k=1}^\\ell c_k(\\phi(a_k\\cdot x+b_k)-\\phi(a_k\\cdot y+b_k))\\right|\\\\\n\\le \\sum_{k=1}^\\ell |c_k||a_k\\cdot(x-y)|\n\\le \\sum_{k=1}^\\ell |c_k|\\|a_k\\|_1\\|x-y\\|_\\infty\n\\le \\|\\|a_k\\|_1\\|_\\infty\\sum_{k=1}^\\ell |c_k|\\sum_{i=1}^p|x_i-y_i|.\n\\end{multline*}\nFor $\\theta\\in\\mathcal B^q_{c_{p,\\ell}}$ then $L=(c_{p,\\ell}\\vee 1)^3 $ is convenient. Next, using Jensen to estimate $\\mathbb L_1$-risk by $\\mathbb L_2$-risk, we obtain from the theorem 1 of Barron \\cite{Barron1994} the existence of $C>0$ such that\n$$\n\\pi_0\\left[\\left|\\mbox{med}(X_0\\,|\\,X_{-1},\\ldots,X_{-p})-f_{\\overline \\theta_{p,\\ell}}(X_{-1},\\ldots,X_{-p})\\right|\\right]\\le C \\frac{ p^c\\|X_0\\|_\\infty}{\\sqrt \\ell}\n$$\nwhere $\\overline \\theta_{p,\\ell}$ belongs to the compact set\n$$\n\\mathcal B'_{p,\\ell}=\\left\\{\n\\theta\\in\\mathbb{R}^{\\ell(p+2)+1} ; \\sum_{i=1}^\\ell |c_i|\\le C'c^p;\n\\max_{1\\le i\\le \\ell}\\|a_i\\|\\le \\sqrt \\ell \\log\\ell; \\right.\n\\left. \\max_{1\\le i\\le \\ell}|b_i|\\le \\|X_0\\|_\\infty \\sqrt \\ell \\log\\ell\\right\\}.\n$$\nRemark that under the assumptions of Corollary \\ref{cor::nn} we have $c_{p,\\ell}- \\|\\overline\\theta_{p,\\ell}\\|\\ge q\/e$. It implies by Proposition \\ref{propdim} that\n$d_{p,\\ell}\\le 3q(1+\\log(c_{p,\\ell}))$ when $c_{p,\\ell}\\ge1$. From Theorem \\ref{mainthm} there exists $C>0$ satisfying\n$$\nR(\\hat{\\theta})]\\le \\inf_{d_{p,\\ell}\\le n} \\left\\{R^\\ast_p\n+C\\left( \\frac{p^c}{\\sqrt \\ell} + \\log^{3} (n) \\sqrt \\frac{p\\ell}{n}\\right)\\right\\}+C\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.\n$$\nThe result follows from considering $\\ell=\\sqrt n p^{c-1\/2}$.\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{fourier}]\\label{prooffourier}\nLet us apply Theorem \\ref{mainthm2}: there exists $C>0$ such that \n\\begin{multline*}\nR(\\hat{\\theta})\\le\n\\inf_{\\Tiny p,\\ell: d_{p,\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p,\\ell}}R (\\theta ) +\nC \\sqrt{\\frac{d_{p,\\ell}}{n}} \\log^{5\/2}(n) \\right\\}\n + C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\n\\\\\n\\leq\n\\inf_{\\Tiny \\ell: d_{p_{0},\\ell}\\leq n} \\left\\{\n\\min_{\\theta\\in\\Theta_{p_{0},\\ell}}R (\\theta ) +\nC \\sqrt{\\frac{d_{p_{0},\\ell}}{n}} \\log^{5\/2}(n) \\right\\}\n + C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}\n.\n\\end{multline*}\nRemarking that\n\\begin{multline*}\nR\\left(\\overline{\\theta}_{p_{0},\\ell}\\right)\n= \\inf_{\\theta\\in\\Theta}{\\pi_{0}}\n\\left[ \\left|X_{p+1}-f_{\\overline{\\theta}_{p_{0},\\ell}}(X_{p},...,X_{1})\\right|\\right]\n\\\\\n\\leq {\\pi_{0}} \\left[ \\Biggl|X_{p+1}-\\sum_{i=1}^{p_{0}}\nf_{i}(X_{p-i}) \\Biggr|\\right) + \\inf_{\\theta\\in\\Theta}\n {\\pi_{0}} \\left[\\Biggl| \\sum_{i=1}^{p_{0}}\nf_{i}(X_{p-i}) - \\sum_{i=1}^{p_{0}} \\sum_{j=1}^{n} \\theta_{i,j}\n\\varphi_{j}(X_{p-i}) \\Biggr|\\right]\n\\\\\n\\leq \\mu[|\\xi_{0}|] + \\inf_{\\theta\\in\\Theta} \\sum_{i=1}^{p_{0}}\n{\\pi_{0}} \\left[ \\Biggl| f_{i}(X_{1}) - \\sum_{j=1}^{n}\n\\theta_{i,j} \\varphi_{j}(X_{1}) \\Biggr|\\right] .\n\\end{multline*}\nNote also that under our hypothesis $X_{1}$\nhas a density upper bounded by $1\/\\sqrt{2\\pi\\sigma^{2}}$. It then holds\n \\begin{multline*}\nR\\left(\\overline{\\theta}_{p_{0},\\ell}\\right) \\leq \\mu[|\\xi_{0}|] +\n\\frac{1}{\\sqrt{2\\pi\\sigma^{2}}} \\inf_{\\theta\\in\\Theta}\n\\sum_{i=1}^{p_{0}} \\int \\biggl| f_{i}(x) - \\sum_{j=1}^{n}\n\\theta_{i,j} \\varphi_{j}(x) \\biggr| dx\n\\\\\n\\leq \\mu[|\\xi_{0}|] + \\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}\n\\inf_{\\theta\\in\\Theta} \\sum_{i=1}^{p_{0}} \\left(\\int \\biggl[\nf_{i}(x) - \\sum_{j=1}^{n} \\theta_{i,j} \\varphi_{j}(x) \\biggr]^{2}\ndx\\right)^{\\frac{1}{2}}\n\\\\\n\\leq\n\\mu[|\\xi_{0}|] + \\frac{1}{\\sqrt{2\\pi\\sigma^{2}}} \\sum_{i=1}^{p_{0}}\n\\gamma_{i} \\ell^{-s_{i}}\n\\leq \\mu[|\\xi_{0}|] + \\frac{\\sum_{i=1}^{p_{0}}\\gamma_{i}}{\\sqrt{2\\pi\\sigma^{2}}}\n\\ell^{-s}.\n\\end{multline*}\nThen we have\n\\begin{equation}\n\\label{kaka}\n \\pi_{0}[R(\\hat{\\theta})]\\le \\mu[|\\xi_{0}|] + \\inf_{\\ell} \\left\\{\n\\ell^{-s} \\frac{\\sum_{i=1}^{p_{0}}\\gamma_{i}}{\\sqrt{2\\pi\\sigma^{2}}}\n+ C \\sqrt{\\frac{d_{p_{0},\\ell}}{n}} \\log^{5\/2}(n)\n \\right\\}\n+C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.\n\\end{equation}\nThe estimate of $d_{p_{0},\\ell}$ from Propostition \\ref{propdim} is plugged into\n\\eqref{kaka} to obtain for some $C>0$ \n$$\n \\pi_{0}[R(\\hat{\\theta})]\\le \\mu[|\\xi_{0}|] + \\inf_{\\ell} \\left\\{\n\\ell^{-s} \\frac{\\sum_{i=1}^{p_{0}}\\gamma_{i}}{\\sqrt{2\\pi\\sigma^{2}}}\n+ C \\sqrt{\\frac{p_{0}\\ell}{n}} \\log^{5\/2}(n)\n \\right\\}\n+ C\n\\frac{\\log\\frac{1}{\\varepsilon}}{\\sqrt{n}}.\n$$\nIn particular, fixing $\\ell$ proportional to $n^{\\frac{1}{2s+1}}$ leads to the result.\n\\end{proof}\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n \\par Only a small fraction ($\\sim$10\\,\\%) of galaxies are expected to present an active galactic nucleus \\citep[AGN,][]{Ho2008}. This phenomenon occurs when the supermassive black hole (SMBH) in the centre of the galaxy captures nearby matter. \nDuring this process, this matter is gradually accreted, turning gravitational potential energy into electromagnetic radiation and kinetic energy, by ejection of particles in the form of winds. The radiation and winds produced in the accretion disk could play an important role in the evolution of the AGN host galaxy -- the so-called AGN feedback mechanisms \\citep{cattaneo09,fabian12,harrison17}. This feedback may influence the evolution of the galaxy, as predicted in cosmological simulations \\citep{croton}. \n\n\\par In the unified scheme of AGN \\citep{antonucci, urry}, a torus surrounding the accretion disk collimates its radiation leading to a bi-conical morphology for the Narrow-Line Region (NLR). A similar structure could be expected for the winds launched from the accretion disc which could also be collimated by the torus. Narrow-band HST images of type 2 QSOs reveal elongated ENLRs and bipolar ionisation cones whose extents increase with $[\\textsc{O iii}]\\lambda5007$ luminosity \\citep{fischer18,thaisa2018}, but in less-luminous Seyfert galaxies the conical morphology is not as common as expected from the unified scheme \\citep{schmitt2003}. \\citet{He_2018} found bipolar structures with an opening angle of $\\sim 80^{\\circ}$ in NLRs from MaNGA AGN by analysing emission lines ratio behaviours.\n\nThe $[\\textsc{Oiii}]\\lambda 5007$ kinematics of the inner kilo-parsec of nearby AGN reveal that although bi-conical outflows are observed, they appear to be more the exception than the rule. For instance, \\citet{mullaney} find that AGN driven winds due to the interaction of radio jets with the ambient gas usually present a linear structure and not a bi-conical morphology. \\citet{fischer2013} has also studied the $[\\textsc{Oiii}]\\lambda 5007$ kinematics in the NLR of 48 Seyfert galaxies and found that in only 12 objects the ionised gas kinematics is consistent with conical outflows, indicating that the gas kinematics of AGN is more complex than predicted by the unified model. \\citet{fischer18} investigated the $[\\textsc{Oiii}]\\lambda 5007$ emission distribution and kinematics in a sample of 12 type 2 nearby ($z < 0.12$) and luminous (log $L_{\\rm [OIII]}\/{\\rm erg s^{-1}} \\gtrsim 42$ quasars, using HST observations. They found that the gas is kinematically disturbed by the AGN up to distances of $\\sim$ 8-15 kpc and determined \nan average maximum outflow radius of $\\sim$600 pc.\n\\cite{wylezalek2020} discussed the ionised gas kinematics of a sample of AGN observed with MaNGA and found that high velocity gas is more prevalent in AGN compared to non-AGN, based on measurements of the line velocity width that encloses 80\\% of the total flux of the $[\\textsc{Oiii}]\\lambda 5007$ emission line.\n\n\n\\par AGN-driven outflows are known to extend from tens to kpc scales, achieving velocities of hundreds of km s$^{-1}$ \\citep{crenshaw2003,ilha2019,wylezalek2020}. \nHowever, the impact of these outflows in the AGN host galaxies remains unclear, and recent studies have focused in this question by studying the gas kinematics on large (kpc) scales. Using spatially resolved long-slit spectroscopy, \\citet{greene2011} investigated the kinematics of a sample of 15 nearby ($z < 0.5$) quasars with [\\ion{O}{III}]$\\lambda$5007 luminosity $>10^{42}$ erg s$^{-1}$ and reported that the AGN is the main ionising agent of the interstellar medium over the entire structure of their host galaxies.\nIn powerful AGN, large-scale outflows are detected, and their extents increase with the AGN luminosity \\citep{greene2012, fischer18,thaisa2018}. By analysing the $[\\textsc{Oiii}]\\lambda 5007$ and H$\\beta$ emitting gas kinematics of a sample of 14 quasars, observed with Gemini IFUs, \\citet{liu2013} found that all objects possess extended ionised gas nebulae outflows with mean diameter of $28$ kpc. Such powerful outflows are able to halt star formation in the host galaxy \\citep[e.g.][]{alatalo15}. However, \\citet{husemann} find that the results from \\citet{liu2013} and also \\citet{liu2014} may be affected by not considering beam-smearing effects which indicates that probably the outflows present smaller extents \\citep{villarmartin,Karouzos_2016, tadhunter}. \n \n\n\\par The role of the AGN feedback in low luminosity AGN can be addressed with the current surveys of integral field spectroscopy of galaxies, such as the Sloan Digital Sky Survey - IV's \\citep[SDSS-IV][]{blanton} Mapping Nearby Galaxies at APO \\citep[MaNGA; ][]{bundy2015} and Calar Alto Legacy Integral Field Area \\citep[CALIFA; ][]{sanchez12}. We have been using MaNGA data to investigate the AGN hosts properties, with results presented in a series of papers, as follows. \\citet{rembold2017} discusses nuclear stellar population properties of the first 62 AGN observed with the MaNGA survey and defined a control sample of inactive galaxies that match the AGN hosts properties in terms of stellar mass, redshift, visual morphology and inclination. It was found that AGN of increasing luminosity exhibit an increasing contribution from the youngest stellar population relative to control galaxies and a decrease in the oldest components. \n\n\\par The sample defined by \\citet{rembold2017} was used in subsequent studies. \\citet{mallmann2018} presented spatially resolved stellar population properties using the MaNGA data. They have shown that the fraction of young stellar population in high-luminosity AGN is higher in the inner ($R\\le 0.5 R_e$, where $R_e$ is the galaxy effective radius) regions when compared with the control sample. The low-luminosity AGN and control galaxies display similar fractions of young stars over the whole MaNGA field-of-view ($\\sim1\\,R_e$). \\citet{nascimento2019} focused on the emission-line flux distributions and gas excitation of this sample and reported that the extent of the region ionised by the AGN is proportional to $L_{[\\textsc{Oiii}]}^{0.5}$, where $L_{[\\textsc{Oiii}]}$ is the luminosity of the $[\\textsc{Oiii}]\\lambda 5007$ emission line. They also found that the star formation rate is higher in AGN than in the control galaxies for the early-type host galaxies. The analysis of the nuclear stellar and gas kinematics, as well as the difference of the orientation of the line of nodes derived from the stellar and gas velocity fields (kinematic position angle [PA] offset), is presented in \\citet{ilha2019}. By comparing AGN host and inactive galaxies, no difference was found in terms of the kinematic PA offsets between gas and stars. But AGN present higher gas velocity dispersion within the inner 2\\farcs5 diameter region, that has been interpreted as due to AGN-driven outflows. \n\n\n\\par In the present study, we use an expanded sample relative to that of the above papers, consisting of all AGN included in the MaNGA Product Launch 8 (MPL-8) \\citep{law2016}, to investigate the large scale gas kinematics of AGN hosts, and compare them to those of a control sample of inactive galaxies, drawn in the same way as in \\citet{rembold2017}. Our AGN sample consists of 170 AGN and two control galaxies for each AGN host. \n\n\\par This paper is organised as follows: Section \\ref{sec:data} presents a brief description of the MaNGA survey and discusses the sample properties. In Sec. \\ref{measurements} we describe our measurements. In Sec.~\\ref{results}, we present radial profiles of the kinematic properties of the sample galaxies, which are discussed in Sec.~\\ref{disc}. The main conclusions of this work are summarised in Sec.~\\ref{conc}. The assumed cosmological parameters in this work are $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_m=0.3$, and $\\Omega_V=0.7$.\n\n\n\\section{Data}\\label{sec:data}\n\\par The MaNGA survey's focus is on understanding the formation and evolution of galaxies by mapping the kinematics and physical properties of $\\sim10\\,000$ nearby galaxies, with redshift ($z$) ranging from 0.01 to 0.15 and stellar masses greater than $10^9$\\,M$_\\odot$ \\citep{law2015}. The integral field unit (IFU) spectroscopic observations have been done with the 2.5-m Sloan telescope of the Apache Point Observatory (APO) \\citep{gunn, smee, wake, yan2016a, yan20162}. The spectral resolving power is $R \\sim 2000$, resulting in an instrumental broadening of $\\sigma_{\\rm inst}=70$ km\\,s$^{-1}$ \\citep{law2021}, and the average IFU's angular resolution is $2\\farcs5$ \\citep{yan20162,law2016}. Details about the survey, instrument, and data processing are given in \\citet{bundy2015} and \\citet{drory2015}. \n\n\\par SDSS-IV Data Release (DR) 14 \\citep{abolfathi2018} includes 2778 MaNGA data cubes. \\citet{rembold2017} presented a sample of all AGN observed by MaNGA in DR14, as obtained from diagnostic diagrams \\citep{bpt,whan}, hereafter called BPT and WHAN diagrams, based on emission-line fluxes and equivalent widths measured using the SDSS-III spectra by \\citet{thomas13}. These diagrams can be used to identify the dominant gas ionisation mechanism in each galaxy and are among the most commonly used diagnostic diagrams to select AGN using optical observations. To be selected as an AGN host, the galaxy must be located in the Seyfert or LINER region on both BPT and WHAN diagrams simultaneously, including the uncertainties. \n In order to analyse the effects of AGN on the host galaxies, \\citet{rembold2017} have also selected non-active galaxies to compose a control sample. Two control galaxies were selected for each AGN. This selection was performed by matching the morphology, stellar mass, absolute magnitude and redshifts of the inactive galaxies to those of the AGN hosts. The initial control sample contains 124 galaxies (twelve of them were paired to more than one AGN host). The initial AGN and control samples, as well as their main properties, are listed in \\citet{rembold2017}. \n\n\n\\par After the release of the MaNGA Product Launch 8 (MPL-8), which contains data-cubes for 6779 galaxies, the number of observed AGN with MaNGA has grown to 173 objects using the same criteria as in \\citet{rembold2017}, including the 62 AGN presented in that work. Following \\citet{rembold2017}, we selected two control galaxies for each new AGN except for three objects: the galaxies identified as MaNGA ID 1-37440, 1-189584, and 31-115. Since these galaxies have low redshifts ( $z\\lesssim$0.01), which are close to the lower limit of MaNGA's redshift range, we were not able to identify control galaxies at similar redshifts. Considering this issue and the fact that the main goal of this paper is to compare the gas kinematics of AGN hosts and their control galaxies, we do not include these objects in the global statistics. The final AGN sample consists of 170 galaxies, with a $[\\textsc{O iii}]\\lambda 5007$ emission line luminosity ranging from $10^{39}$ to $10^{41}$ erg s$^{-1}$, as measured from the SDSS-III spectra. The AGN and their respective control galaxies are listed in Tables 1 and 2 (available in electronic form only), respectively. Fig.~\\ref{fig:diagnostic} presents the location of AGN hosts and control galaxies of our sample in the diagnostic diagrams BPT and WHAN. Fig. \\ref{fig:hist_params} shows the distributions of redshift ($z$), stellar mass ($M^\\bigstar$), \\textit{r}-band ($M_r$) absolute magnitude and $[\\textsc{O iii}]\\lambda 5007$ luminosity ($L[\\textsc{Oiii}]$) of our AGN sample, as compared to that of their associated control galaxies. As in the initial sample \\citep{rembold2017}, the AGN and control samples present similar distributions in all parameters except ($L[\\textsc{Oiii}]$), which is shifted towards larger\nvalues for the AGN hosts. Fig.~\\ref{fig:examples_agn_cs} presents the SDSS-III multicolor images of four representative AGN hosts in our extended sample -- i.e. those not included in \\citet{rembold2017} -- and their associated controls.\n\n\n\\begin{figure*}\n \\centering\n \\subfloat[]{\\label{bptw1}\\includegraphics[width=0.5\\linewidth]{bpt_agn_MPL8.pdf}}\n\\subfloat[]{\\label{bptw2}\\includegraphics[width=0.5\\linewidth]{whan_agn_MPL8.pdf}}\\\\\n\\subfloat[]{\\label{bptw3}\\includegraphics[width=0.5\\linewidth]{bpt_cs_mpl8.pdf}}\n\\subfloat[]{\\label{bptw4}\\includegraphics[width=0.5\\linewidth]{whan_cs_mpl8.pdf}}\n\\caption{BPT (left) and WHAN (right) diagrams for the AGN hosts (top) and control objects (bottom) in our sample. The separating lines in the plots are from \\citet{kauffmann2003}, \\citet{kewley01} and \\citet{whan}. Grey dots represent all the emission-line galaxies in MPL-8. Large black circles represent AGN and control objects.}\n \\label{fig:diagnostic}\n\\end{figure*}\n\n\n\\par In total, the AGN sample contains 57 early-type (33.5$\\%$), 87 late-type (51.2$\\%$), and 3 merger galaxies (1.8$\\%$), according to the Galaxy Zoo database. There is no classification in Galaxy Zoo for 23 objects (13.5$\\%$). Similarly, the control sample is composed of 125 early-type (36.8$\\%$), 182 late-type (53.5$\\%$), 4 merger galaxies (1.2$\\%$), and 29 objects (8.5$\\%$) whose classifications are uncertain. Regarding the AGN classification, 94 (55.3\\%) out of the 170 sources present LINER nuclei, while the other 76 (44.7\\%) galaxies present Seyfert nuclei, as indicated by the BPT diagram (Fig. \\ref{fig:diagnostic}). By visual inspection of the nuclear spectra of all AGN hosts, we conclude that 12 (7.06 $\\%$ of the sample) objects are type 1 AGN (showing broad components in their hydrogen recombination lines). The fraction of type 1 AGN in our sample is smaller than that obtained using SDSS-DR7 spectra, of 10--34 $\\%$ for similar [O\\,{\\sc iii}] luminosities \\citep{oh15}. This difference is likely due to selection effects of the MaNGA sample and by the visual inspection of the observed spectra, which may not be sensitive to faint broad line components. These weak broad components do not affect the determinations of NLR properties, which is the aim of this study. \n\n\nWe have also separated the sample according to the $[\\textsc{O iii}]\\lambda 5007$ emission line luminosity (L$[\\textsc{O iii}]$), which can be used as a proxy for the bolometric luminosity of the AGN \\citep{heckman_2004}. The $[\\textsc{O iii}]\\lambda 5007$ luminosity L$[\\textsc{O iii}]$ is determined within a circular aperture of 2\\farcs5 diameter from the nucleus, based in a single fiber. Following previous works by our group \\citep{rembold2017,mallmann2018,ilha2019,nascimento2019} we separate our sample into strong and weak AGN: strong AGN are defined as those with $L[\\textsc{O\\,iii}] \\ge 3.8 \\times 10^{40}$ erg s$^{-1}$, measured within a nuclear aperture of 3$''$ diameter; weak AGN are those with $L[\\textsc{Oiii}] < 3.8 \\times 10^{40}$ erg s$^{-1}$. We use the same $L[\\textsc{Oiii}]$, originally proposed by \\citet{kauffmann2003} and that roughly divide their sample in Seyfert (defined by the authors as objects with [O {\\sc iii}]5007\/H$\\beta>3$ and [N {\\sc ii}]6583\/H$\\alpha>0.6$) and LINER ([O {\\sc iii}]5007\/H$\\beta<3$ and [N {\\sc ii}]6583\/H$\\alpha>0.6$) nuclei.\nIn the complete sample, 62 galaxies can be classified as strong AGN, which corresponds to 36$\\%$ of our AGN sample; the new sample includes almost four times the number of strong AGN when compared to the 17 of \\citet{rembold2017}. The number of strong AGN is slightly smaller than the number of Seyfert nuclei (76) and for consistency with our previous works, we will separate our sample into strong and weak AGN in this study. The larger number of strong AGN is relevant, as the main differences in terms of physical properties of the gas and stars of AGN and non-AGN hosts in previous works \\citep{mallmann2018,ilha2019,nascimento2019} were seen in strong AGN. \\citet{wylezalek2020} also observed that the ionised gas kinematics is more extreme in more luminous MaNGA AGN. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{MPL8_agncs_histcomb.pdf}\n \\caption{Redshift ($z$), stellar mass ($M^\\bigstar$), integrated \\textit{r}-band ($M_r$) absolute magnitude and $[\\textsc{O iii}]\\lambda 5007$ distributions for AGN (black) and control samples (red). Both samples are composed mostly of low-luminosity galaxies.}\n \\label{fig:hist_params}\n\\end{figure*}\n\n\n\nThe most important difference between our sample and those of previous studies is that our sample is based on MPL-8, which include data for 6779 galaxies, while the previous works were based on the Data Release (DR) 14 \\citep{dr14}, including 2744 galaxies. In addition, this work and that of \\citet{wylezalek2020} used distinct AGN selection criteria. While our selection is based on SDSS-III spectroscopic data from DR 12 \\citep{dr12}, \\citet{wylezalek2020} used MaNGA data, including off centred photo-ionisation signatures to classify a galaxy as an AGN \\citep{wylezalek18}. Finally, an advantage of our work is the selection of a control sample of non-active galaxies with properties matching those of the AGN hosts. This allows us to better investigate the effects of AGN feedback on the host galaxy properties, by making sure that eventual differences are not related to properties such as stellar mass or Hubble type.\n\n \\begin{figure*\n \\centering\n \\begin{tabular}{ccc}\nAGN & Control object 1 & Control object 2 \\\\\n1-174945 & 1-137845 & 1-135372\\\\\n\\includegraphics[height=3.5cm]{ex_1_agn.jpg} & \\includegraphics[height=3.5cm]{ex_1_cs1.jpg} & \\includegraphics[height=3.5cm]{ex_1_cs2.jpg} \\\\\n1-297172 & 1-456656 & 1-395508\\\\\n\\includegraphics[height=3.5cm]{ex_2_agn.jpg} & \\includegraphics[height=3.5cm]{ex_2_cs1.jpg} & \\includegraphics[height=3.5cm]{ex_2_cs2.jpg} \\\\\n1-31788 & 1-379708 & 1-317822\\\\\n\\includegraphics[height=3.5cm]{ex_3_agn.jpg} & \\includegraphics[height=3.5cm]{ex_3_cs1.jpg} & \\includegraphics[height=3.5cm]{ex_3_cs2.jpg} \\\\\n1-232143 & 1-173966 & 1-627397\\\\\n\\includegraphics[height=3.5cm]{ex_4_agn.jpg} & \\includegraphics[height=3.5cm]{ex_4_cs1.jpg} & \\includegraphics[height=3.5cm]{ex_4_cs2.jpg} \\\\\n\\end{tabular}\n \\caption{SDSS-III multicolor images of four representative examples of AGN hosts (left column) and their respective control galaxies (centre and right columns) in our sample. These galaxies were chosen to roughly reproduce the morphological Hubble sequence from ellipticals (top) to late-type spirals (bottom).}\n \\label{fig:examples_agn_cs}\n\\end{figure*}\n\n\n\n\\section{Measurements}\\label{measurements}\n\n\n\\par The measurements of the emission-line fluxes and the gas and stellar kinematics from the MaNGA data cubes were performed using the Gas AND Absorption Line Fitting (\\textsc{gandalf}) code \\citep{sarzi2006}, written in Interactive Data Language (IDL). This algorithm maps the gas emission and kinematics while separating the stellar continuum and the gas contributions to the galaxy's spectra. In order to subtract the continuum spectra and also to measure the stellar kinematics, \\textsc{gandalf} uses the Penalized Pixel-Fitting (\\textsc{pPXF}) code \\citep{capellari, capellari2}. The \\textsc{pPXF} code consists of stellar absorption lines fitting, using templates spectra as bases. Following \\citet{ilha2019}, we used 30 selected Evolutionary Population Synthesis models from\n\\citet{bruzual-charlot} as template spectra. These models cover ages ranging from 5~Myr to 12~Gyr and three metallicities ($0.004\\,Z_{\\odot},0.02\\,Z_{\\odot},0.05\\,Z_{\\odot}$). In addition, we included multiplicative Legendre polynomial of order 3 to account for the different continuum shapes and the line of sight velocity distribution is represented by a Gaussian.\n\n\nThe following emission lines were fitted: H$\\beta$, [O\\,{\\sc iii}]$\\lambda\\lambda$4959,5007, [O\\,{\\sc i}]$\\lambda$6300, [N\\,{\\sc ii}]$\\lambda\\lambda$6548,6583, H$\\alpha$ and [S\\,{\\sc ii}]$\\lambda\\lambda$6716,6731. Each emission line is fitted by a single Gaussian component, except for type 1 objects, where a broad component is included to reproduce the nuclear profiles of H$\\beta$ and H$\\alpha$. The centroid velocity and width of the doublets [O\\,{\\sc iii}]$\\lambda\\lambda4959,5007$, [N\\,{\\sc ii}]$\\lambda\\lambda6548,6583$ and [S\\,{\\sc ii}]$\\lambda\\lambda6716,6731$ emission lines were kept tied, for each pair separately. During the fit, the following intensity ratios were kept fixed to their theoretical values: [O\\,{\\sc iii}]$\\lambda5007$\/[O\\,{\\sc iii}]$\\lambda4959=2.86$ and [N\\,{\\sc ii}]$\\lambda6583$\/[N\\,{\\sc ii}]$\\lambda6548=2.94$ \\citep{osterbrock06}. As discussed in \\citet{ilha2019}, the emission-line profiles are well reproduced by single-Gaussian fits; details about the spectral fitting can be found there. \nIn Figure\\,\\ref{fig:example} we present an example of the resulting maps of the properties we have derived from our measurements for the AGN host with {\\it{mangaid}} 1-48116.\n\n\n\n\\begin{figure*}\n\\begin{center}\nMaNGA 1-48116\n\\end{center}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{1-48116.png}\n\\caption{Example of maps we have obtained from our measurements for the galaxy MaNGA 1-48116. The first row shows, from left to right, a continuum map in units of $10^{-17}\\,{\\rm erg s^{-1} cm^{-2} \\AA^{-1}}$ obtained by collapsing the whole spectral range, the [O\\,{\\sc iii}]5007\\,\\AA\\ and H$\\alpha$ flux maps in units of $10^{-17}\\,{\\rm erg s^{-1} cm^{-2}}$. The second row shows, from left to right, the stellar, [O\\,{\\sc iii}] and H$_\\alpha$ velocity fields in km\\,s$^{-1}$, relative to the systemic velocity of the galaxy. The bottom row shows the velocity dispersion maps for the stars, [O\\,{\\sc iii}], and H$_\\alpha$, from left to right. In all panels, the North points up and East to the left and the $\\Delta X$ and $\\Delta Y$ labels show the distance relative to the peak of the continuum emission. } \n\\label{fig:example}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[scale=0.46, trim=0.9cm 0cm 0cm 0cm, clip]{agns_controles_razao_all_sem_bpt.pdf}\n \\includegraphics[scale=0.46, trim=0.9cm 0cm 0cm 0cm, clip]{agns_strong_weak_razao_all_sem_bpt.pdf}\n \\includegraphics[scale=0.46, trim=0.9cm 0cm 0cm 0cm, clip]{agns_early_late_razao_all_sem_bpt.pdf}\n \\caption{{Comparison of the $[\\textsc{Oiii}]\\lambda 5007\/\\textrm{H}\\beta$ line ratio spatial profiles. Each point represents the azimuthal median value of the quantities in a bin of 0.5 kpc. The error bars present the standard deviation of the quantities, divided by the square root of the number of spaxels. \\textit{Left panel:} AGN hosts (black) and controls (green); \\textit{central panel:} strong (dark blue) and weak (red) AGN; \\textit{right panel:} early-type (light blue) and late-type (pink) AGN.}}\n \\label{fig:OIII_HB}\n\\end{figure*}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.3]{agns_controles_velo_all_sem_bpt.pdf}\n \\includegraphics[scale=0.3]{agns_controles_sig_apenas_all_sem_bpt.pdf}\n \\\\\n \\includegraphics[scale=0.3]{agns_strong_weak_velo_all_sem_bpt.pdf}\n \\includegraphics[scale=0.3]{agns_strong_weak_sig_apenas_all_sem_bpt.pdf}\n \\includegraphics[scale=0.3]{agns_early_late_velo_all_sem_bpt.pdf}\n \\includegraphics[scale=0.3]{agns_early_late_sig_apenas_all_sem_bpt.pdf}\n \\caption{Comparison of median kinematic radial profiles. The symbols and colors are identical to those of Fig. \\ref{fig:OIII_HB}. The first row shows a comparison between AGN and controls, from left to right: residual between the $[\\textsc{Oiii}]\\lambda 5007$ gas and stellar velocities and residual between the corresponding velocity dispersions. Second row: same as in first row but comparing strong and weak AGN hosts. Third row: same as first but comparing early and late-type AGN hosts.}\n \\label{fig:samples_profiles_host}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.55, trim=1cm 0cm 0cm 1cm, clip]{diferenca_sig_apenas_lumin_sem_bpt.pdf}\n \\caption{Median radial profile of the $\\sigma$ difference between AGN and controls. Each AGN was compared to its respective control galaxies. The colored lines indicate different bins of $[\\textsc{O iii}] \\lambda 5007$ luminosity, on a logarithmic scale. Over all the analysed extension, the largest differences between AGN and control velocity dispersions were observed for the most luminous AGN hosts.}\n \\label{fig:samples_lumin_bpt_host}\n\\end{figure}\n\n\n\\par To identify and characterise the kinematically disturbed gas by the AGN, we constructed radial profiles of the gas kinematic properties and emission-line ratios. Due to the MaNGA's spatial resolution and coverage, the radii range that we consider in the present work ranges from $0.5$ to $5.0$ kpc. We preferred not to correct the radial profiles for projection along the azimuthal direction, because this correction is not clear for many galaxies of our sample. Any bias resulting from this is nevertheless compensated by the fact that the control sample galaxies were selected to present the same aspect ratio of the corresponding AGN. For each galaxy, we calculated the azimuthal median of all values whose spaxels are located within a 0.5 kpc bin, for which the uncertainty in [O{\\sc iii}]5007 flux is smaller than 30\\% and uncertainties in velocity dispersion and centroid velocity are lower than 30 km\\,s$^{-1}$. Throughout this article, median profiles for properties of the different sub-samples were constructed by computing the median azimuthal values determined within each radial bin. The residual values ([O{\\sc iii}] $-$ stars) were calculated at each spaxel and then the median-valued in each radial bin. The differences between the kinematic properties of AGN and each control galaxy were computed between their azimuthal median profiles.\n\n\n\\section{Results}\\label{results}\n\n\\subsection{Excitation} \n\n{ \\par Figure \\ref{fig:OIII_HB} shows median radial profiles of emission-line ratios for the AGN and control samples. We also show \nbars corresponding to plus and minus the root mean square standard deviation of the quantities, divided by the square root of the number of spaxels. \nThese profiles were used to compare the radial behaviour of the $[\\textsc{O iii}]\\lambda 5007\/\\textrm{H}\\beta$ line ratio of different sub-samples: besides the comparison between AGN and control sample, the panels of Fig. \\ref{fig:OIII_HB} also include the AGN profiles separated in strong and weak sub-samples, as well as in sub-samples of early and late-type host galaxies. \n\\par Both AGN and control samples' $[\\textsc{O iii}]\\lambda 5007\/\\textrm{H}\\beta$ ratio profiles decrease with the distance from the nucleus, with a steeper gradient seen for AGN. Although a few control galaxies do not show gas emission, most of them show some emission, but the median $[\\textsc{O iii}] \\lambda 5007\/$H$\\beta$ ratio becomes higher than 1 only in the 1 kpc nuclear bin, while AGN show a median nuclear $[\\textsc{O iii}] \\lambda 5007\/$H$\\beta > 3$. Strong AGN display higher $[\\textsc{O iii}]\/$H$\\beta$ ratios for radial distances up to 5\\,kpc than weak AGN, indicating that the AGN has a major role in the gas excitation even at large radii. The comparison between early and late-type galaxies reveals that they show similar $[\\textsc{O iii}]\/$H$\\beta$ values at the nucleus, but at distances of 1.5 to 5.0 kpc there are clear differences. \nWhile the early-type AGN have $[\\textsc{O iii}] \\lambda 5007\/$H$\\beta\\approx3$ over all distances, this ratio decreases for late-type galaxies with increasing distance from the nucleus. Besides probable H\\,{\\sc ii} regions dominating the outer regions of the disc of late-type galaxies, another possible explanation for this difference is that, as the amount of gas in early type galaxies is smaller than in late-type galaxies, the AGN radiation reaches greater distances from the nucleus.}\n\n\n\\subsection{Kinematics}\n\\par Figure \\ref{fig:samples_profiles_host} shows the radial profiles of the kinematic properties of the samples obtained from measurements of the $[\\textsc{Oiii}]\\lambda 5007$ and stellar kinematics. \nWe analyse residual gas velocities and residual velocities dispersion. The residual velocity is defined as the median of the absolute value of $[\\textsc{Oiii}]\\lambda 5007$ velocity minus the velocity of the stars ($v_{res}=|v_{[\\textsc{Oiii}]}-v_{\\textrm{stars}}|$). Likewise, the residual velocity dispersion ($\\sigma_{\\textrm{res}}$) is defined as $\\sigma_{[\\textsc{Oiii}]}-\\sigma_{\\textrm{stars}}$, where $\\sigma$ is the velocity dispersion. Comparing AGN hosts and controls, the AGN present higher values of these properties, especially if we consider only the strong AGN. All along the analysed extension, $v_{\\textrm{res}}$ for AGN hosts is slightly higher than that for the control galaxies; in strong AGN, $v_{\\textrm{res}}$ is about 10 km\\,s$^{-1}$ larger than in weak AGN. \nLate-type AGN display similar kinematic properties to that of the control and weak AGN samples. Conversely, early-type AGN present higher $v_{\\textrm{res}}$ than late-type AGN, increasing with the distance from the nucleus. \n\n\\par The $\\sigma_{\\textrm{res}}$ profiles shown in the central column panels of Fig. \\ref{fig:samples_profiles_host} indicate that the velocity dispersion of the control galaxies' gas is always lower than that of the AGN sample. In the nuclear aperture, the controls display $\\sigma_{\\textrm{res}}\\approx -25.0$ km s$^{-1}$ whilst for AGN the median value is $\\sigma_{\\textrm{res}} \\approx 2.0$ km s$^{-1}$. Separating the AGN in strong\/weak, the highest $\\sigma_{\\textrm{res}}$ values are dominated by strong AGN all along the analysed extension. While weak AGN do not present positive $\\sigma_{\\textrm{res}}$, the strong AGN present it out to 4 kpc from the nucleus. Regarding the early and late-type AGN $\\sigma_{\\textrm{res}}$ profiles, late-type galaxies show the highest residual velocity dispersion, about $8.0$ km s$^{-1}$ in the nuclear aperture. Early-type galaxies yield negative values of $\\sigma_{\\textrm{res}}$ in all the analysed bins of radius. \n\n\\par As seen in Fig.~\\ref{fig:samples_profiles_host}, the main difference in the kinematics of the AGN and control sample is in the velocity dispersion. A direct comparison of the [\\ion{O}{III}]$\\lambda$5007 $\\sigma$ values of AGN and their controls is shown in Fig. \\ref{fig:samples_lumin_bpt_host}. We obtain the median values of the profiles $\\sigma_{\\textrm{AGN}}-\\sigma_{\\textrm{controls}}$ in three different bins of luminosity: $39.0\\le \\log$ L$[\\textsc{O iii}] < 40.0$ (hereafter low), $40.0\\le \\log$ L$[\\textsc{O iii}] < 41.0$ (hereafter intermediate), and $41.0\\le \\log$ L$[\\textsc{O iii}] < 42.0$ (hereafter high). Each AGN was compared to its respective control galaxies. We define $\\sigma_{\\textrm{controls}}=(\\sigma_{\\textrm{control1}}+\\sigma_{\\textrm{control2}})\/2$. Low and intermediate luminosity AGN present similar profiles, reaching values of $\\sigma_{\\textrm{AGN}}-\\sigma_{\\textrm{controls}}\\sim20.0$ km s$^{-1}$ and $\\sim40.0$ km s$^{-1}$ in the bin corresponding to the inner kpc, respectively. The most luminous AGN display the highest values of $\\sigma_{\\textrm{AGN}}-\\sigma_{\\textrm{controls}}$ in all the radial bins. In the inner 1 kpc, this difference reaches $\\approx 65.0$\\,km\\,s$^{-1}$, indicating a relation between the $[\\textsc{O iii}] \\lambda 5007$ luminosity and $\\sigma$. \n\n\n\\section{Discussion}\\label{disc}\nBy comparing the gas kinematics of AGN hosts and control galaxies, we find evidence that the AGN affect the gas motions up to 4-5 kpc, most clearly seen as differences in the gas velocities dispersion of AGN and controls, but also observable in the centroid velocities. We attribute these differences to AGN driven winds interacting with the NLR gas; in this section, we determine the physical extent of these kinematically disturbed regions (KDRs), the outflow powers and discuss their implication for AGN feedback processes. The KDR is defined here as the region where the gas kinematics is disturbed by the AGN. \n\n\\subsection{Sizes of the NLR and KDR}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.44\\textwidth]{manga-9499-6104_CofG.png}\n\\includegraphics[width=0.44\\textwidth]{manga-8945-3703_CofG.png}\n\\caption{Examples of curves-of-growth. The black dashed line corresponds to the PSF and the red line shows the [\\ion{O}{iii}]$\\lambda$5007 flux and curve-of-growth. The shaded regions delineate the 1$\\sigma$ flux uncertainties. The vertical dotted and dashed lines show the observed radii of the NLR and KDR, respectively. The {\\it mangaid} of the galaxies are identified in the title of each panel. }\n\\label{fig:psf}\n\\end{figure}\n\n\n\n\\begin{figure}\n{\\centering\n\\includegraphics[width=0.44\\textwidth]{RNLR.png}\n\\includegraphics[width=0.44\\textwidth]{Routflow.png}\n}\n\\caption{Comparison of the raw estimates of the\nextents of the NLR (left) and KDR (right) with the corrected sizes obtained after deconvolving the [\\ion{O}{iii}] brightness profiles (see text). The dashed lines indicate the locus\nof the 1:1 relation.}\n\\label{fig:deconvolution}\n\\end{figure}\n\n\n\\subsubsection{Estimates of the NLR and KDR extents}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth, trim=1cm 0cm 0cm 0cm, clip]{r_nlr_outflow_new.pdf}\n \\caption{The KDR extents versus the NLR extents on a logarithmic scale. There is a correlation between these extents, with a Spearman correlation coefficient of $CC \\sim 0.53$ and a p-value of $10^{-4}$. The best-fit linear regression (blue line) results in an intercept of $\\sim1.07 \\pm 0.22$ and a slope of $\\sim0.53 \\pm 0.12$. The uncertainties are represented by gray bars corresponding to differences in the value of the properties of 1$\\sigma$.}\n \\label{fig:nlr_outflow}\n\\end{figure}\n\nMeasurements of the extent of the NLR in AGN are commonly reported in the literature, mainly based on imaging and Integral Field Spectroscopy (IFS) observations. The extent of the NLR correlates with the AGN luminosity and ranges from a few hundred parsecs in low luminosity AGN \\citep[e.g.][]{fraquelli2003,schmitt2003,nascimento2019,chen2019} to tens of kpc in the most luminous objects \\citep[e.g.][]{bennert2002,greene2011,liu2013,liu2014,thaisa2018}. The KDR extents and how they relate to the size of the NLR is less clear. So far, studies on individual objects and small samples find ionised gas outflows on scales from tens of parsecs to kiloparsec \\citep{das2005,das2007,thaisa2010,liu2013b,liu2014,barbosa2014,davies2014,cresci2015,rogemar_mrk79,rogemar_n5643,fischer2017,revalski2018,fischer18,diniz2019,munoz-vergara,soto-pinto2019}. Here, we measure both the extent of the KDR and the size of the NLR. \n\n\n\\par We compute the observed extents of the NLR ($r_{\\mathrm{ NLR,o}}$) and KDR ($r_{\\mathrm{KDR,o}}$) of our AGN sample, by comparing emission-line ratios and gas kinematics of AGN hosts and control galaxies in bins of 0.5 kpc, although the minimal region considered has a size of two spaxels, which corresponds on average to $\\sim$ 0.876 kpc in our sample. All the bins take into account only the sources that have measurements in the corresponding region, e. g., some galaxies do not have the first 0.5 kpc resolved and as consequence are not considered in this bin. We define $r_{\\mathrm{NLR,o}}$ as the farthest radius from the nucleus within which both $[\\textsc{O iii}]\/$H$\\beta$ and $[\\textsc{N ii}]\/$H$\\alpha$ emission lines ratios fall in the AGN region of the $[\\textsc{O iii}]\/$H$\\beta$ vs. $[\\textsc{N ii}]\/$H$\\alpha$ BPT diagram \\citep{bpt}. In addition, the H$\\alpha$ equivalent width is required to be larger than 3.0 $\\AA$ to exclude regions ionised by post-AGB stars \\citep{stasinska08,cid11,singh13,belfiore16}.\n\n\nThe observed extent of the KDR $r_{\\textrm{KDR,o}}$ is defined as the largest radius from the nucleus where the residual $\\sigma_{\\textrm{res, AGN}}=\\sigma_{\\textrm{[OIII], AGN}}-\\sigma_{\\textrm{stars, AGN}}$ and becomes equal to $\\sigma_{\\textrm{res, controls}}=\\sigma_{\\textrm{[OIII], controls}}-\\sigma_{\\textrm{stars, controls}}$. To compare these residuals in both AGN and controls, we fit their radial distribution by a second order polynomial and compute the radius where they intercept, which is $r_{\\textrm{KDR,o}}$.\nThe extents $r_{\\textrm{NLR,o}}$ and $r_{\\textrm{KDR,o}}$ could not be determined for the entire AGN sample, since it was not possible to obtain all the necessary measurements (e.g., when one or both line ratios are AGN-like only for the nuclear aperture, or $\\sigma_{\\textrm{res, AGN}}$ and $\\sigma_{\\textrm{res,controls}}$ are already similar at the nucleus).\n\n\n\\subsubsection{Are the NLR and KDRs resolved?}\n\\label{nlrresolved}\n\nThe emission-line flux distributions of an unresolved nuclear source can be smeared by the seeing to distances much larger than the FWHM of the point spread function. \nIn order to verify whether the NLR and KDR radii $r_{\\textrm{NLR,o}}$ and $r_{\\textrm{KDR,o}}$ derived in the previous section are spatially resolved, we follow \\citet{kakkad20} and compute the [\\ion{O}{iii}]5007 emission-line flux curve-of-growth (CofG) for each galaxy. We then compare the [\\ion{O}{iii}]5007 flux and PSF CofGs, with the latter obtained from the reconstructed g-band point source profiles included in the MaNGA data-cubes. The CofG are not used to estimate the NLR or KDR radii, but just to check if they are spatially resolved by MaNGA. The [\\ion{O}{iii}]5007 emission line is chosen for this purpose because it is the best tracer of the gas ionized by the AGN and of the gas in outflow, being one of the strongest optical lines. \nFirst, we normalise the [\\ion{O}{iii}] flux by its the value in the nuclear spaxel -- defined as that corresponding to the peak of the continuum emission -- and then compute the integrated fluxes increasing the radius in steps of 0\\farcs5. The same procedure is adopted to compute the CofG for the PSF.\n\n\nIn Fig.~\\ref{fig:psf} we show examples of the CofGs. We consider that the KDR is spatially resolved if the [\\ion{O}{iii}] flux in the CofG at $r_{\\rm KDR,o}$ is higher than that of the PSF considering the 1$\\sigma$ error (i.e., the shaded red region must be above the PSF CofG profile). Similarly, the NLR is considered resolved if at $r_{\\rm NLR,o}$ the [\\ion{O}{iii}] flux of the CofG is larger than that of the PSF CofG. The left panel of Fig.~\\ref{fig:psf} shows an example in which both $r_{\\rm KDR,o}$ and $r_{\\rm NLR,o}$ are spatially resolved,\nwhile the right panel shows an example in which the KDR is unresolved and the NLR is resolved. We find that $r_{\\rm NLR,o}$ and $r_{\\rm KDR,o}$ are spatially resolved in 55 and 46 AGN hosts of our sample, respectively. Our further analysis will be focused on these objects.\n\n\n\\subsubsection{Correction for the beam smearing effect}\n\n\nHaving established that the NLR and KDR of most AGN in our sample are spatially resolved, we now correct their measured extents for beam smearing. In particular, our estimates of the extent of the KDRs are based on kinematic parameters, which are hard to correct using purely photometric techniques like surface brightness deconvolution. In what follows, we simply assume that beam smearing affects equally the derived linear extents obtained from the kinematic and photometric analyses, $r_{\\textrm{KDR,o}}$ and $r_{\\textrm{NLR,o}}$, respectively. This assumption allows us to perform a rough correction to the estimated radial extents.\n\n\n\nOur methodology for estimating the corrected extents of both the NLR ($r_{\\textrm{NLR,c}}$) and KDR ($r_{\\textrm{KDR,c}}$) is as follows. We start by deconvolving the [O\\,{\\sc iii}]5007 brightness profiles of each galaxy, which were derived by calculating the integrated fluxes in concentric rings around the galaxy centre at fixed steps 0\\farcs5 wide. The same calculation was performed for the PSF associated to each galaxy (see Sec. \\ref{nlrresolved}). We used these profiles to derive 2D circularly symmetric flux distributions for [\\ion{O}{iii}] and the PSF with a cubic spline fit, producing ``pseudo-images'' whose linear scale was lower than the original MaNGA data-cubes by a factor 10. We then performed a 2D Richardson-Lucy deconvolution on the [\\ion{O}{iii}] brightness distributions using the Python package {\\sc{scikit-image}}, with the 2D symmetric PSF as input. Finally, we calculated in the observed profile the fraction of the total [\\ion{O}{iii}] flux contained within the raw, uncorrected radial extent of a given structure (KDR or NLR), and impose that the corrected extent comprises the same flux fraction in the deconvolved profile.\n\nFig.~\\ref{fig:deconvolution} shows the comparison between the raw estimates\nof the extent of the KDR\/NLR and those corrected as explained above. The beam\nsmearing, as expected, produces an overestimation of the true extent of both structures.\nIn some cases, particularly for the most extended ($\\gtrsim 5^{\\prime\\prime}$) sources, the corrected\nsizes are very similar to the raw ones, which is also expected. The mean ratio between the corrected and observed radii are 0.48$\\pm$0.14 and 0.65$\\pm$0.22 for $r_{\\textrm{KDR}}$ and $r_{\\textrm{NLR}}$, respectively. \n\n\n\n\\subsubsection{Comparison between the NLR and KDR extents}\n\nFig. \\ref{fig:nlr_outflow} displays the $r_{\\textrm{KDR,c}}$ against the $r_{\\textrm{NLR,c}}$. The average $$ is $3.00\\pm0.33$ kpc and individual values range between $0.4$ and $10.1$ kpc, which are in good agreement with previous values obtained for MaNGA AGN based on spatially resolved BPT diagrams \\citep{chen2019}. The KDRs present extents of: $=0.96\\pm 0.09$\\,kpc, varying between 0.2 and 2.3 kpc. \\citet{wylezalek2020} found that the average width of the [OIII]$\\lambda5007$ line in weak AGN reaches the same level of non-AGN at distances of $\\sim$8\\,kpc from the nucleus, while for strong AGN similar values are seen at distances of up to 15\\,kpc, leading to a `qualitative' measurement of the extent of the KDR that are on average larger than ours. The p-value of the correlation between $r_{\\textrm{KDR,c}}$ and $r_{\\textrm{NLR,c}}$ of our sources is $10^{-4}$, suggesting that they are correlated.\nWe fit the data by a linear equation and find\n\\begin{equation}\n\\log r_{\\rm KDR,c} = (0.53\\pm 0.12)\\,\\log r_{\\rm NLR,c} +(1.07\\pm 0.22)\\,, \n\\end{equation}\n\\noindent in units of kpc. $r_{\\textrm{KDR,c}}$ and $r_{\\textrm{NLR,c}}$ present a positive correlation and, on average, $r_{\\textrm{KDR,c}}$ is smaller than $r_{\\textrm{NLR,c}}$, with mean value of $\\langle r_{\\textrm{KDR,c}}\/r_{\\textrm{NLR,c}}\\rangle = 0.32\\pm0.27$. Using HST observations of 12 nearby type 2 quasars ($z<0.12$, $L_{\\rm bol}\\sim10^{46}$ erg s$^{-1}$), \\citet{fischer18} reported that the extent of the outflows are on average only 20\\,\\% of the extent of the [O\\,{\\sc iii}] emission (NLR or ENLR) in a sample of nearby type 2 quasars. Other studies on quasars at larger distances and luminosities ($z=0.5-0.6$, $L_{\\rm bol}\\sim10^{47}$ erg s$^{-1}$) using integral field spectroscopy and HST images find similar extents for the outflow and ionised gas emission \\citep{liu2013b,liu2014,wylezalek2016}. As a cautionary note, we point out that the methods used to estimate the size of the KDR and outflows vary widely among all these studies and the comparison between different measurements is not straightforward. \n\n\n\\subsection{Mass outflow rates} \n\\begin{figure*}\n \\centering\n \\includegraphics[scale=0.5, trim= 0.9cm 0cm 0cm 0cm, clip]{taxa_outflow_lumin_02kpc.pdf}\n \\includegraphics[scale=0.5, trim= 0.9cm 0cm 0cm 0cm, clip]{potencia_cinetica_lumin_02kpc.pdf}\n \\caption{$\\dot{M}_{out}$ (left panel) and $\\dot{E}_{kin}$ (right panel) versus the AGN bolometric luminosity. Red circles indicate objects analysed in \\citet{Fiore}, while blue crosses show the objects from \\citet{baron2019b}. Our sample is represented by black dots and characterise AGN with the lowest luminosity in these relations. We were able to estimate these properties for 20 sources. Black lines show the linear fits considering the three samples. The red dashed lines indicate the linear fit presented in \\citet{Fiore}.}\n \\label{fig:estimativas}\n\\end{figure*}\n\n\n\\par In order to examine the outflow impact on their host galaxies, we first must estimate the mass outflow rate. Following previous works (e.g., \\cite{rogemar_mrk79}, \\cite{riffel2}, \\cite{thaisa2010}), we define the mass outflow rate $\\dot{M}_{\\rm out}$ as \n\\begin{equation}\\label{eqmout}\n\\dot{M}_{\\rm out}=1.4m_pN_ev_{\\rm out}fA ,\n\\end{equation}\nin which $m_p$ is the proton mass, $N_e$ is the electron density, $v_{\\rm out}$ is the outflow velocity in the $[\\textsc{O iii}]\\lambda 5007$ emission line, $f$ is the filling factor, $A$ is the area of the outflow cross section and 1.4 factor is due to the estimated contribution from He to the gas mass. The filling factor can be estimated by\n\\begin{equation}\nf\\approx \\frac{L_{\\rm H\\alpha}}{4\\pi j_{\\rm H\\alpha}V}, \n\\end{equation}\nwhere $L_{\\rm H\\alpha}$ is the H$\\alpha$ luminosity emitted by a volume $V$ and $4\\pi\\,j_{\\rm H\\alpha}\/N_e^2=3.534\\times10^{-25}$ erg\\, cm$^{-3}$\\,s$^{-1}$ \\citep{osterbrock06}. Here, we adopt the assumption that all the gas is in outflow, as our data does not allow the separation between gas in the outflow from gas that is not. This assumption may introduce uncertainties that are nevertheless not larger than those introduced by our ignorance of other variables in these calculations, such as the outflow geometry and orientation. Replacing $f$ in Eq.\\,\\ref{eqmout} and assuming a spherical geometry for the outflow, we obtain\n\\begin{equation}\\label{eq_out}\n\\dot{M}_{\\rm out}=\\frac{4.2 m_p L_{\\rm H\\alpha} v_{\\rm out}}{N_e j_{\\rm H\\alpha} r},\n\\end{equation}\nwhere $r$ is the radius of the sphere. \n\nAs pointed out in the previous section, the radius of the outflow is considered to be smaller than $r_{\\textrm{KDR,c}}$, as previous resolved studies have shown that the mass outflow rates (and powers) \\citep[e.g.][]{thaisa2018,fischer2017} usually peak within the inner kpc of the host galaxies and then decrease for larger distances. This ``peak radius\" is the one to be used in calculations $\\dot{M}_{\\rm out}$ and outflow power, although the effect of the outflows (disturbed kinematics) may extend much beyond this radius. Nevertheless, such radii are not resolved in our MaNGA data. We have thus adopted the outflow radius as the smallest value consistent with our observations, adopted to be the distance corresponding to the radius of the seeing disk of the observation of each source. This value at the galaxies approximately correspond to typical values obtained in resolved studies of $\\approx$\\,1\\,kpc.\n\nThe electron density values were calculated for each galaxy from the $[\\textsc{S II}]\\lambda 6717\/[\\textsc{S II}]\\lambda 6731$ ratio and assuming a temperature of 10\\,000\\,K. Recent works have been arguing that the density of the gas in outflow obtained using $[\\textsc{S II}]$ is an underestimation for outflows occurring on hundred of pc scales, as the $[\\textsc{S II}]$ emission originates from a partially ionised region farther out from the nucleus, that has a lower gas density \\citep{baron2019a,baron2019b,davies2020}. But, as the radii of the outflows are of the order of kpc for our sample, the $[\\textsc{S II}]$ emission can be used to obtain the gas density at these distances. \n\nThe $[\\textsc{S II}]$ doublet is not sensitive to $50 > N_e\\gtrsim1\\times10^4$\\,cm$^{-3}$, where the dependence of the $[\\textsc{S II}]$ line ratios with $N_e$ becomes asymptotically flat. Thus, we estimate the outflow properties only in objects with measured densities $50 5 \\times 10^{-3}$, and the AGN feedback becomes more effective when $\\dot{E}_{\\rm kin}$ is at least $\\sim$5\\,\\% of the bolometric luminosity \\citep{harrison18}. In our sample, this occurs only for one source (MaNGA ID 1-258599). Therefore, the majority of outflows detected in our AGN sample are not powerful enough to affect their host galaxies on a large scale. The gas is outflowing from the nucleus and being re-distributed within the galaxies remaining available for further star formation and AGN feeding, comprising the so-called ``maintenance mode\" AGN feedback. However, the kinetic power of the ionised outflows corresponds only to their mechanical effect in a single gas phase and do not correspond to the initial AGN input energy included in theoretical studies, and thus a comparison between the observed kinetic efficiencies in a specific wind phase and theoretical predictions in not straightforward \\citep[e.g.][]{harrison17}.\n\\par We have also measured the escape velocity (V$_{\\textrm{esc}}$) for each galaxy and found that the outflow velocity is lower than the V$_{\\textrm{esc}}$ in every source, which reinforces the conclusion that the AGN impact is low, the feedback being only in ``maintenance mode''.\n\n\n\\section{Conclusions}\\label{conc}\n\\par We have compared the ionised gas kinematics of a sample of 170 active galaxies to those of a control sample of non-active galaxies from the MaNGA survey. Our goal was to look for properties related to the nuclear activity. Using these properties, we estimated the extent of the NLR and of the Kinetically Disturbed Region -- KDR (attributed to the interaction of AGN outflows with ambient gas in the host galaxy), showing average kinematic profiles as a function of the radius (in units of effective radius). We also estimated the average values of the mass outflow rate within the inner kpc and the kinetic power of outflows for the AGN sample. Our main conclusions are:\n\\begin{itemize}\n\n\\item We find spatially resolved NLR's and KDR's in 55 and 46 AGN host galaxies, respectively, among the 170 AGN of our sample;\n \n\\item The main difference in the mean gas excitation of AGN relative to controls was observed in the line ratio $[\\textsc{O iii}]\\lambda 5007\/$H$\\beta$, which was used to obtain the extent of the NLR;\n \n \\item The main difference in the kinematic radial profiles of AGN relative to controls was observed in the residual velocity dispersion $\\sigma_{\\textrm{res}}=\\sigma_{\\textsc{O iii}}-\\sigma_{\\textrm{stars}}$; \n \n \\item Over the whole analysed radial extent, the most luminous AGN ($41.0 \\le \\log L[\\textsc{O iii}] \\le 42.0$) present the highest residual velocities $v_{res}=|v_{[\\textsc{O iii}]}-v_{\\textrm{stars}}|$ and residual velocity dispersions $\\sigma_{res}$ relative to their respective control galaxies;\n\n \\item The NLR extent -- adopted as the region where the AGN is the dominant source of ionisation -- ranges from 0.4 kpc to 10.1 kpc;\n \n \n \\item The KDR extent ranges from 0.2 to 2.3 kpc, defined as the largest distance from the nucleus where the residual between $\\sigma_{\\rm [OIII]}$ and $\\sigma_{\\rm stars}$ for the AGN and controls become similar. The KDR size is on average 32 percent of the mean NLR extent;\n \n \n \\item There is a correlation between the extent of the KDR and the extent of the NLR: the best fit is $r_{\\rm KDR,c}=(0.53\\pm0.12)r_{\\rm NLR,c}+(1.07 \\pm 0.22)$;\n \n \\item Assuming that the kinematic disturbance observed along the NLR is due to outflows from the AGN, we have estimated the values of the mass outflow rate and power adopting as radius of the outflow the angular resolution of the data;\n \n \\item The mass outflow rate ranges between $\\sim 10^{-5}$ and $\\sim 1\\,M_{\\odot}$\\,yr$^{-1}$ and shows a weak correlation with the AGN luminosity;\n \n \\item The power of the outflows ranges between $\\sim 10^{34}$ and $\\sim 10^{40}$ erg\\,s$^{-1}$, and presents a weak correlation with the AGN luminosity -- although being probably underestimated -- with most values below those expected from the extrapolation of the \\citet{Fiore} relation for more luminous AGN;\n \n \\item The ratio between the kinetic power of the outflow and the AGN bolometric luminosity $\\dot{E}_{\\textrm{kin}}\/L_{\\textrm{bol}}$ ranges between $\\sim 10^{-7}$ and $10^{-4}$. Only for the AGN with MaNGA ID 1-258599, which is one of the most luminous sources in our sample, it is higher than the 0.5\\% threshold AGN feedback postulated by models to significantly affect the evolution of the host galaxy.\n\n\\end{itemize}\nAlthough we have found that the kinematic disturbance on the NLR by the AGN extends up to several kpc in some galaxies, its effect is not powerful enough to significantly affect their host galaxies. Nevertheless, other gas phases -- neutral and molecular -- as well as other forms of feedback besides the kinetic one seen in the ionised gas, should also be considered as possible sources of AGN feedback for a fairer comparison with galaxy evolution models.\n\n\\\n\\begin{acknowledgements}\nThe authors thank the anonymous referee for her\/his valu-able suggestions that helped us to significantly improve the present paper. This study was funded in part by the Coordena\\c c\\~ao de\nAperfei\\c coamento de Pessoal de N\\'ivel Superior - Brasil (CAPES) -\nFinance Code 001, Conselho Nacional de Desenvolvimento Cient\\'ifico e Tecnol\\'ogico (CNPq) and Funda\\c c\\~ao de Amparo \\`a pesquisa do Estado do RS (FAPERGS). ADM acknowledges financial support from the Spanish MCIU grant PID2019-106027GB-C41 and from the State Agency for Research of the Spanish MCIU through the ``Center of Excellence Severo Ochoa'' award for the Instituto de Astrof\u00edsica de Andaluc\u00eda (SEV-2017-0709). ADM also acknowledges the support of the INPhINIT fellowship form ``la Caixa'' Foundation (ID 100010434), under the fellowship code LCF\/BQ\/DI19\/11730018. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofisica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\\\"ur Astrophysik Potsdam (AIP), Max-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"ur Astrophysik (MPA Garching), Max-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\\' orio Nacional \/ MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nEllis, Treciokas and Matravers (ETM) \\cite{ETMa,ETMb} introduced a $1+3$ \ncovariant kinetic theory formalism in which an irreducible representation of\nthe rotation group based on Projected Symmetric and Trace-Free (PSTF) tensors \northogonal to a physically definved 4-vrelocity $u^a$ \ngives a covariant representation of the Cosmic Background Radiation (CBR)\nanisotropies, which is gauge-invariant when the geometry is an\nalmost-Robertson Walker (RW) geometry. This $1+3$ Covariant and Gauge-Invariant \n(CGI) formalism has been used in a previous series of papers\n\\cite{SME,MSEa,MSEb,SAG} to look at the local generation of CBR\nanisotropies by matter and spacetime inhomogeneities and anisotropies\nin an almost-Friedmann Lema\\^{i}tre (FL) universe model \\footnote{\nHere `Robertson-Walker' refers to the geometry, whatever the field\nequations; `Friedman-Lemaitre'\\ assumes that the Einstein gravitational\nfield equations with a perfect fluid matter source are imposed on such a\ngeometry.}. By contrast, the present series of papers\\footnote{And a series by\nChallinor and Lasenby \\cite{cl}, \\cite{cl2}\nwhich are similar in method and intent but\ndifferent in detail and focus.} uses this formalism to\ninvestigate CBR anisotropies in the non-local context of emission of\nradiation near the surface of last scattering in the early universe and its\nreception here and now (the Sachs-Wolfe (SW) effect and its further \ndevelopments).\n\nThere is of course a vast literature investigating these anisotropies both\nfrom a photon viewpoint, developing further the methods of the original\nSachs-Wolfe paper \\cite{SW}, and from a kinetic theory viewpoint, so it\nis useful to comment on why the CGI philosophy and programme \\cite{EB} \nmake the present series of papers \nworthwhile. Rather than beginning with a background described in \nparticular coordinates and perturbing away from this\nbackground, this approach centres on $1+3$ covariantly defined geometric\nquantities, and develops exact nonlinear equations for their evolution.\nThese equations are then systematically linearised about a\nFriedmann-Lemaitre (FL) background universe with a Robertson-Walker (RW)\ngeometry resulting in description by gauge-invariant variables and equations \n\\cite{EB}. Because the definitions and equations used are\ncoordinate-independent, one can adopt any suitable coordinate or tetrad\nsystem to specialise the tensor equations to specific circumstances when\ncarrying out detailed calculations; a harmonic or mode analysis can be\ncarried out at that stage, if desired.\n\nThis approach is geometrically transparent (see \\cite{cl}, \n\\cite{cl2}) because of the CGI variable definitions used. \nIn contrast to the various gauge-dependent approaches to \nperturbations in cosmology, the differential equations used are of just the \norder that is needed to describe the true physical degrees of freedom, so\nno non-physical gauge modes occur. When a harmonic decomposition is \nintroduced in the case of linear perturbations, the CGI variables used here\nprovide a description that is equivalent to that obtained by approaches \nbased Bardeen's GI variables \\cite{bardeen}, see \\cite{bde}, but they do \nnot imply linearisation of the equations from the outset, as occurs in \nthat formalism.\n\nThus the benefit of the present formalism is precisely its $1+3$ covariant and\ngauge invariant nature, together with the fact that we are able to write\ndown the exact non-linear equations governing the growth of structure and\nthe propagation of the radiation, and then linearise them in a transparent\nway in an almost-RW situation. This means it can be extended to non-linear\nanalyses in a straightforward way \\cite{MGE}, which will be essential in \ndeveloping the\ntheory of finer CBR anisotropy structure as reliable small-angle\nobservations become available. Achievements of the CGI approach with respect \nto the CMB are the\nalmost-EGS Theorem \\cite{SME}, related model-independent limits on \ninhomogeneity and isotropy \\cite{MSEa,MSEb,MES96,SAG}, and derivation \nof exact anisotropic solutions of the Liouville equation in a RW \ngeometry (\\cite{me}, see also \\cite{emt}).\n\nThis paper, Part I, deals with algebraic issues, developing further the\nformalism of ETM: namely an irreducible representation of radiation\nanisotropies based on PSTF tensors \\cite{T1,P}. The paper considers \nthis irreducible representation and its relation to observable quantities, \nboth generically and in the context of models linearised about RW \ngeometries \\cite{MSEa}. In section 2 and 3, the underlying $1+3$ \ndecomposition is outlined and the basic CGI harmonic formalism for \nanisotropies developed. In section 4, the angular correlation\nfunctions are constructed from CGI variables, assuming that the multipole\ncoefficients are generated by superpositions of homogeneous and isotropic\nGaussian random fields. The multipole expansion is discussed in detail,\nextending the results of ETM, giving the construction of the multipole\ncoefficient mean-square and developing its link to the angular correlation\nfunction. In section 5, the mode coefficients are found following the\nWilson-Silk approach, but derived and dealt with in the CGI form; the\ncovariant and gauge invariant multipole and mode expanded angular \ncorrelation functions are related to the usual treatments used in the \nliterature \\cite{WS,GSS,HS95a,HS95b,EBW}. In this discussion, the\nCGI mode expansion is related to the\ncoordinate approach by linking the Legendre Tensors to the PSTF \nrepresentation, using a covariant addition theorem \nto generate the Legendre Polynomial recursion relation. The key result \nis the construction of the angular correlation functions in the \nCGI variables, and their link to the \n(non-local) GI Mode functions \\cite{GSS}.\n\nThe following papers in the series look at the Boltzmann equation and\nmultipole divergence relations, solution of the resulting mode equations,\nand relation of the kinetic theory approach to the photon based formalism of\nthe original Sachs-Wolfe paper. Exact non-linear equations are obtained and\nthen linearised, allowing a transparent linearisation process from the\nnon-linear equations that is free from ambiguities and gauge modes.\n\n\n\\section{Temperature anisotropies}\n\nA radiation {\\it temperature measurement} is associated with an antenna\ntemperature, $T(x^i,e^a)$, measured by an observer moving with\n4-velocity $u^a$ at position $x^i$ in a direction $e^a$ on the unit\nsphere ($e^ae_a=1$, $e^au_a=0)$. We assume $u^a$can be uniquely define\nin the cosmological situation, corresponding to the motion of\n`fundamental observers' in cosmology \\cite{varenna} \\footnote{In the\nearly universe, when matter and radiation average velocities differ,\nthere may be several competing possibilities for covariant definition\nof $u^a$; however once a choice has been made between these\npossibilities, this vector field is uniquely defined.}. The direction\n$e^a$ can be given in terms of an orthonormal tetrad\nframe\\footnote{The description of such a tetrad frame is briefly\ndiscussed in last appendix}, for example by : \n\\begin{equation}\ne^a(\\theta ,\\phi )=(0,\\sin \\theta \\sin \\phi ,\\sin \\theta \\cos \\phi\n,cos\\theta )\\,.\\label{3.2}\n\\end{equation}\nThe temperature $T(x^i,e^a)$ can be unambiguously decomposed into the\nall-sky {\\it average bolometric temperature } \\footnote{%\nNote that this is {\\it not} the same as the background temperature, for that\nquantity varies only with cosmic time $t$, whereas the true isotropic\ncomponent of the temperature varies with spatial position as well as time.} $%\nT(x^i)$ at position $x^i$, given by \n\\begin{equation}\nT(x^i)=\\frac 1{4\\pi }\\int_{4\\pi }T(x^i,e^a)d\\Omega . \\label{3.6}\n\\end{equation}\nwhere $\\Omega $ is the solid angle on the sky, and the anisotropic {\\it %\ntemperature perturbation} $\\delta T(x^i,e^a)$ (the difference from the\naverage over the unit sphere surrounding $x^i$ \\cite{MSEa}),\ncan be defined : \n\\begin{equation}\nT(x^i,e^a)=T(x^i)+\\delta T(x^i,e^a). \n\\label{3.5}\n\\end{equation}\n{}From the Stefan-Boltzmann law it follows -- if the radiation is almost\nblack-body, which we assume -- that the radiation energy density is given in\nterms of the average bolometric temperature by $\\rho_R\n(x^i)=r T^4(x^i)$.\n($r$ is Stefan-Boltzmann constant). Both the quantities $T(x^i)$ and $\\delta\nT(x^i,e^a)$ are CGI, for $T(x^i)$ is defined\nin a physically unique frame in the real universe \n(because $u^a$ is assumed to be uniquely defined), \nand $\\delta T(x^i,e^a)$ vanishes in any background without temperature\nanisotropies.\n\nWe can define the fractional temperature variation $\\tau (x^i,e^a)$ by\n\\cite{MSEa}\n\\begin{equation}\n\\tau (x^i,e^a):={\\frac{\\delta T(x^i,e^a)}{T(x^i)}}, \\label{3.10}\n\\end{equation}\nand take a covariant (angular) harmonic expansion of this, \n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_{\\ell=1}^\\infty \\tau\n_{a_1a_2a_3...a_\\ell}(x^i)\\,e^{a_1}e^{a_2}e^{a_3}...e^{a_{\\ell-1}}e^{a_\\ell}\n\\equiv \\sum_{\\ell \\geq 1} \\tau_{A_{\\ell}} \\hat e^{A_{\\ell}}.\n\\label{3.11}\n\\end{equation}\nWe introduce the shorthand notation using the compound index \n$A_{\\ell} = a_1 a_2 ... a_{\\ell}$. Here $\\tau _{a_1a_2a_3...a_l}(x^i)$\nare trace-free symmetric tensors orthogonal to $u^a$ :\n\\begin{eqnarray}\n\\tau_{A_{\\ell}}=\\tau_{(A_{\\ell})}\\,,~~\\tau_{A_{\\ell} ab} h^{ab}\n=0\\,,~~\\tau_{A_{\\ell} a} u^a \\label{symmetry}\n\\end{eqnarray}\nRound bracketts ``(..)'' denote the symmetric part of a set of\nindices, angle bracketts ``$\\langle..\\rangle$'' the (orthogonally-) Projected \nSymmetric Trace-Free (PSTF) part of the indices : $\\tau_{A_{\\ell}} =\n\\tau_{\\langle A_{\\ell} \\rangle}$.\n\nBecause of (\\ref{3.2}), this expansion is entirely equivalent to a more\nusual expansion in terms of spherical harmonics:\n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_{l=1}^\\infty A_l^m(x^i)Y_l^m(\\theta ,\\phi )\n\\label{usual}\n\\end{equation}\n(see \\cite{ETMa} for details), but is more closely related to a tensor\ndescription, and so results in more transparent relations to physical\nquantities. \n\nWe wish to measure the temperature in two different directions to find the\ntemperature difference associated with the directions $e^a$ and $e^{\\prime\n}{}^a$ such that (using \\ref{3.6}) : \n\\begin{eqnarray}\n\\Delta T(x^i;e^a,e^{\\prime }{}^a)=T(x^i,e^a)-T(x^i,e^{\\prime }{}^a)\\,\n~\\Rightarrow~ \\Delta T(x^i;e^a,e^{\\prime }{}^a)\n=\\delta T(x^i,e^a)-\\delta T(x^i,e^{\\prime}{}^a). \\label{3.8}\n\\end{eqnarray}\nIt follows from (\\ref{3.8}), (\\ref{3.10}) and (\\ref{3.11}) that \n ${{\\Delta T(x^i;e^a,e^{\\prime }{}^a)}} = T(x^i) \\sum_{\\ell} \n\\tau _{A_{\\ell}}\\left( { e^{A_{\\ell}}-e^{\\prime}{}^{A_{\\ell}}}\\right)$\\,,\nwhere $\\Delta T\/T$ represents the {\\it real fractional temperature\ndifference} on the current sky. Due to the CGI nature\nof $T(x^i)$ we may relate this directly to the {\\it real temperature \nperturbations} (no background model is involved in these definitions).\n\nThe relation between the two directions $e^a$ and $e^{\\prime }{}^a$ at $x^i$\nis characterised by \n\\begin{equation}\ne^ae^{\\prime }{}_a=\\cos (\\beta )=:X. \\label{3.1}\n\\end{equation}\ni.e. they are an angular distance $\\beta $ apart \\footnote{%\nIt should be pointed out that $e^{\\prime}{}^a$ is distinct from $%\ne^{a^{\\prime}{}} $, the first denotes a direction vector different from $e^a$\nin a given tetrad frame, while the second means the same direction vector in\na different tetrad frame.}. If analogous to (\\ref{3.2}) we write \n$e^a(\\theta ^{\\prime }{},\\phi ^{\\prime }{})\\equiv {e^{\\prime }{}}^a=(0,\\sin\n\\theta ^{\\prime }{}\\sin \\phi ^{\\prime }{},\\sin \\theta ^{\\prime }{}\\cos \\phi\n^{\\prime }{},\\cos \\theta ^{\\prime }{})$, then it can be shown from \n(\\ref{3.1}) that \n\\begin{equation}\n\\cos \\beta =\\sin \\theta \\sin \\theta ^{\\prime }{}\\cos (\\phi -\\phi ^{\\prime\n}{})+\\cos \\theta \\cos \\theta ^{\\prime }{}.\n\\end{equation}\nIn later applications, it is important to relate the different terms of the\nharmonic expansion to angular scales in ths sky. A useful approximation is\n$l\\approx {\\frac 1\\theta }$, where $\\theta $ is in radians. \n\n\\subsection{Covariant and gauge invariant angular correlation function}\n\nThe two-point correlations are an indication of the fraction of temperature\nmeasurements, $T(x^i,e^a)$, that are the same for a given angular\nseparation. This corresponds to the correlation between $\\delta T(x^i,e^a)$\nand $\\delta T(x^i,e^{\\prime }{}^a)$ or equivalently between $\\tau (x^i,e^a)$\nand $\\tau (x^i,e^{\\prime }{}^a)$, given by the {\\it angular position\ncorrelation function} \n\\begin{equation}\nC({e^a,e^{\\prime }{}^a})=\\left\\langle {\\tau (x^i,e^a),\\tau (x^i,e^{\\prime\n}{}^a)}\\right\\rangle , \\label{ang_correl}\n\\end{equation}\nwhere the angular brackets representing an angular average over the complete\nsky. Note this is a function in the sky.{} If we write ${\\tau (x^i,e^a)}$\nand ${\\tau (x^i,e^{\\prime }{}^a)}$ in terms of the angular harmonic\nexpansion (\\ref{3.11}), we can also define correlation functions $C_l$\nfor the anisotropy coefficients ${\\tau _{A_l}(e}^a),{\\tau }_{A_l}(e^{\\prime\na})$ by \n\\begin{eqnarray}\nC_\\ell({e^a,e^{\\prime }{}^a})=(2\\ell+1)^{-1}\\Delta _\\ell\\left\\langle {\\tau\n_{A_\\ell}(x^i,e^a) \\tau ^{A_\\ell}(x^i,e}^{\\prime }{^a)}\\right\\rangle .\n\\label{129a}\n\\end{eqnarray}\nHere the right-hand side term in brackets is the all-sky mean-square value\nof the $l-$th temperature coefficient ${\\tau _{A_\\ell}(x^i,e^a)}$, and the\ncoefficient $\\Delta _\\ell$ is defined in (\\ref{Delta-ortho0-01}). The numerical\nfactor $(2\\ell+1)^{-1}\\Delta _\\ell$ is included in order to agree with\ndefinitions normally used in the literature (see later). This can be\nthought of as the momentum space version of (\\ref{ang_correl}), as we\nhave taken an angular fourier series of the quantities in that equation;\nit says, for each choice of ${e^a,e^{\\prime }{}^a,}$ how much power\nthere is in that expression for that angular separation as contributed\nby a particular $\\ell$-th valued multipole moment on average.\n\n\\subsection{The Central-Limit Theorem}\n\nWe consider an ensemble of temperature anisotropies, where a \nsequence of repeated trials is replaced by a\ncomplete ensemble of outcomes. The temperature\nanisotropy $\\tau_{A_\\ell}$ found in a given member of the ensemble is \na realization of the statistical process represented by the ensemble.\nThe physically measured anisotropy is taken to be \none such realization. The variance of the ensemble, for example\n$\\langle \\tau_{A_\\ell} \\tau^{A_\\ell} \\rangle$, is in principle \nfound by averaging over a sufficiently large number of\nexperiments, where we assume the results will approach the\ntrue ensemble variance -- this is the assumption of\nergodicity. \n\nOn Fourier transforming, we make the assumption that to a good \napproximation the phases of the various multipole moments are \nuncorrelated and random. This corresponds to treating the anisotropies \nas a form of random noise. The random phase assumption has a useful \nconsequence: that the sum of a large number of independent random variables \nwill tend to be normally distributed. By the central-limit theorem \n\\cite{Adler81}, this is true for all quantities that are derived from \nlinear sums over waves. The end result is that one ends up with a \nGaussian Random Field (GRF) which is fully characterized by a power \nspectrum. The central limit theorem holds as long as there exists a \nfinite second moment, i.e. a finite variance.\n\nWe will assume that the angular variance \n$\\langle \\tau_{A_\\ell} \\tau^{A_\\ell} \\rangle$ is independent of \nposition; this is the assumption of statistical homogeneity. \nIts plausibility lies in the underlying use of the weak Copernican \nassumption. Additionally it is convenient to assume that the power \nspectrum will have no directional dependence, thus it will be isotropic:\n$P(k^a) = P(|k^a|)$. Together these imply the statistical distribution \nrespects the symmetries of the RW background geometry.\n\nOne needs to be careful in using the central limit theory to motivate\nGaussian random field, particularly in the presence of nonlinearity, which \ncould result in the elements of the ensemble no longer being \nindependent. While the assumptions of primordial homogeneous and \nisotropic GRF's is plausible, because the perturbations\n are made up of a sufficiently large number\nof independent random variables, the key point is to realize that \nthese are assumptions that should be tested if possible.\nThe simplest test of weak non-Gaussianity is \nlooking for a three-point angular or spatial correlation, for\nthe Gaussian assumption ensures that all the odd higher moments \nare zero and that the even ones can be expressed in terms of the \nvariance alone. \n \n \nIf the primordial perturbations are made up of GRF's, \nthen non-Gaussianity of the CBR\nanisotropy spectrum should arise primarily from foreground \ncontamination due to local physical processes.\nIf the non-Gaussian effects due to these later physical processes or \nevolutionary effects are small enough, one can attempt to determine\na cosmological primordial signature. \n \n\n\n\n\n\n\n \n\\subsection{Gaussian perturbations}\n\nA general {\\it Gaussian perturbation} \\cite{LW}, $\\tau (x^i,e^a)$, will\nbe a superposition of functions, $\\tau _{A_\\ell}$, i.e. (\\ref{3.11}) \nis satisfied, where the probability, P, of finding a particular valued\ntemperature coefficient is given by ($\\sigma^2_{\\ell} = \\langle\n\\tau_{A_\\ell} \\tau^{A_{\\ell}} \\rangle$) :\n\\begin{equation}\nP(\\tau _{A_\\ell})={\\frac 1{\\sqrt{2\\pi \\sigma _\\ell^2}}}\\exp \n\\left\\{ {\\frac{-\\tau_{A_\\ell}\\tau ^{A_\\ell}}{2\\sigma _\\ell^2}}\\right\\}\n. \\label{4.1} \n\\end{equation}\nNote that{\\bf \\ }$\\tau _{A_\\ell}${\\bf \\ }is both the amplitude of the\n$\\ell$-th component, and determines the probability of that\namplitude. The probability of a temperature perturbation, $\\tau $, is\ngiven by the sum of the Gaussian probability distributions (\\ref{4.1})\nweighting the various angular scales, given by $\\ell$, of the general\nperturbation (\\ref{3.11}). \n\nConsidering isotropic and homogeneous gaussian random fields, the angular\nposition correlation function $C({e^a,e^{\\prime }{}^a})$ is a function only\nof the angular separation $\\beta $ of the two temperature measurements. We\nthen write (\\ref{ang_correl}) as \n\\begin{eqnarray}\n\\left\\langle {\\tau \\cdot \\tau ^{\\prime }{}}\\right\\rangle =C(\\beta )=W(X)\\,,\n\\label{w}\n\\end{eqnarray}\nwhere the expression on the left is shorthand for $\\left\\langle {\\tau\n(x^i,e^a),\\tau (x^i,e^{\\prime }{}^a)}\\right\\rangle _\\beta $ , the {\\it %\n2-point angular correlation function} for a given angular separation $\\beta $\nbetween the on-sky temperature measurements, and $X$ $=e^ae^{\\prime\n}{}_a=\\cos \\beta $. This expression is now independent of position in the\nsky. Gaussian Fields are completely specified by the angular power spectrum\ncoefficients $C_\\ell$ (\\ref{129a}), which are now just constants, \nbecause $\\ell$ is uniquely related to $\\beta$, so the power spectrum\nis a function of the modulus of the wavenumber only. One thus expects \nthe temperature perturbations in this case to be fully specified by\nthe {\\it mean squares}, $\\left\\langle \\tau _{A_\\ell}\\tau\n^{A_\\ell}\\right\\rangle $, when (\\ref{3.11}) is\nsubstituted in (\\ref{w})$.$ Equivalently they are uniquely determined by\nthe angular Fourier transform of the 2-point angular correlation function .\n\n\n\\section{Multipole expansions}\n\n\nIn this section we examine the anisotropy properties of radiation described\nin terms of the covariant multipole formalism (\\ref{3.11}), which is\nequivalent to the usual angular harmonic formalism but much more directly\nrelated to space-time tensors. Note that the relations in this section hold\nat any {\\it point} in the space-time, and in particular at the event $R$\n('here and now') where observations take place. Here we consider the\nPSTF part of, $e^{A_{\\ell}}$; some useful properties of $e^{A_{\\ell}}$\nare listed in appendix B.\n\n\\subsection{The PSTF part of $e^{A_{\\ell}}$}\n\nBecause the coefficients in (\\ref{3.11}) are symmetric and trace-free,\nthe important directional quantities defined by directions $e^a$ at a\nposition $x^i$ are the PSTF quantities \n\\begin{equation}\nO^{A_\\ell }=e^{\\left\\langle {A_\\ell }\\right\\rangle\n}=e^{\\langle a_1}e^{a_2}e^{a_3}...e^{a_{\\ell -1}}e^{a_\\ell \\rangle}, \\label{defO}\n\\end{equation}\nfor clearly \n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_{\\ell =1}^\\infty \\tau _{A_\\ell }(x^i)e^{A_\\ell\n}(\\theta ,\\phi \n)=\\sum_{\\ell =1}^\\infty \\tau _{A_\\ell }(x^i)O^{A_\\ell }(\\theta ,\\phi\n). \\label{3.11b} \n\\end{equation}\nIndeed the standard spherical harmonic properties are contained in these\nquantities.\n\nNow the symmetric trace-free (STF) part of a 3-tensor is given in general by\nPirani \\cite{P}, \n\\begin{eqnarray}\n\\left[ {F_{A_\\ell }}\\right]\n^{STF}=\\sum_{n=0}^{[\\ell\n\/2]}B^{ln}h_{(a_1a_2}....h_{a_{2n-1}a_{2n}}F_{a_{2n+1}...a_\\ell )}~~ \n\\mbox{with}~~\nB_{\\ell n}={\\frac{{(-1)^n\\ell !(2\\ell -2n-1)!!}}{{(\\ell -2n)!(2\\ell\n-1)!!(2n)!!}}}. \\label{B} \n\\end{eqnarray}\nHere $[\\ell \/2]$ means the largest integer part less than or equal to\n$\\ell \/2$. The following definitions have also been used : $\\ell !\n=\\ell (\\ell -1)(\\ell -2)(\\ell -3)...(1)$, and $\\ell !! =\\ell (\\ell\n-2)(\\ell -4)(\\ell -6)...(2~~or~~1)$. \n\nThe PSTF part of a tensor, \n\\begin{eqnarray}\nT_{\\langle ab \\rangle}=\\left[ {T_{ab}}\\right] ^{PSTF}=\\left[ {\\\n\\left[ {T_{ab}}\\right] ^{STF}}\\right] ^P,\n\\end{eqnarray}\ncan be constructed recursively from a vector basis following EMT and a\nlittle algebra.\n\nWe take the PSTF part of $e^{\\langle A_\\ell \\rangle}$ \\cite\n{P,T1,T2,ETMa} to find\n\\begin{eqnarray}\nO^{A_\\ell }=e^{\\langle A_\\ell \\rangle}=\\sum_{k=0}^{[\\ell \/2]}\nB_{\\ell k}h^{(A_{2k}}e^{A_{\\ell -2k})}, \\label{II-3}\n\\end{eqnarray}\nwhere $h^{(A_{2k}}e^{A_{\\ell-2k})}\\equiv\nh^{(a_1a_2}...h^{a_{2k-1}a_{2k}}e^{a_{2k+1}}e^{a_{2k+2}}...e^{a_\\ell)}$\nand $B_{\\ell k}$ are given by (\\ref{B}).\n\n{}From \\cite{ETMa} we can now construct recursion relations that\nplay a key role later on. First, \n\\begin{eqnarray}\nO^{\\langle A_\\ell }e^{a_{\\ell +1}\\rangle }=O^{(A_\\ell }e^{a_{\\ell +1})}-{\\frac\n\\ell {{2\\ell +1}}}e^ah_{ad}O^{d(A_{\\ell -1}}h^{a_\\ell a_{\\ell +1})}\n\\label{recursion_01} \n\\end{eqnarray}\n{}From (\\ref{recursion_01}), (\\ref{defO}), and using \n\\begin{eqnarray}\ne_{a_1}O^{A_\\ell }={\\frac \\ell {(2\\ell -1)}}O^{A_{\\ell -1}}, \\label{eO-rec}\n\\end{eqnarray}\nit can then be shown that \n\\begin{eqnarray}\nO^{A_{\\ell +1}}=e^{(a_{\\ell +1}}O^{A_\\ell )}-{\\frac{\\ell ^2}{(2\\ell\n+1)(2\\ell -1)}} h^{(a_{\\ell +1}a_\\ell }O^{A_{\\ell -1})} \\label{PSTF_rec}\n\\end{eqnarray}\nrelates the $(\\ell +1)-th$ term to the $\\ell -th$ term and the $(\\ell\n-1)-th$ term. \n\nThe orthogonality, addition theorem and double integral relations\nof $O^{A_{\\ell}}$ are listed in appendix B. Using the orthogonality\nrelations we obtain the inversion of the harmonic expansion :\n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_{\\ell =0}^\\infty \\tau _{A_\\ell }(x^i)\\,O^{A_\\ell\n}~\\Leftrightarrow ~\\tau _{A_\\ell }(x^i)=\\Delta _\\ell ^{-1}\\int_{4\\pi }d\\Omega \n\\,O_{A_\\ell }\\,\\tau(x^i,e^a)\\,. \\label{inverse} \n\\end{equation}\n\nThe polynomial $L_{\\ell}\\equiv O^{A_\\ell }O^{\\prime }{}_{A_\\ell }\n=\\sum_{m=0}^{[\\ell\/2]}B_{\\ell m}X^{\\ell -2m}, $ \n(see (\\ref{add01a})) is the natural polynomial\nthat arises in the PSTF tensor approach (equivalent to the Legendre \npolynomials, see below), where the coefficients $B_{lm}$ are defined by\n(\\ref{B}). It follows from this that \n\\begin{eqnarray}\nL_\\ell (1)=O^{A_\\ell }O_{A_\\ell }=\\sum_{m=0}^{[\\ell \/2]}B_{\\ell\nm}=:\\beta _\\ell\\,~~\\mbox{with}\n~~\\beta _\\ell =\\left( {\\frac{(\\ell !)^22^\\ell }{(2\\ell )!}}\\right)\n={\\frac{\\ell !}{(2\\ell -1)!!}}. \n\\label{add02a_beta}\n\\end{eqnarray}\nThe $\\beta _\\ell $'s satisfy the recursive relations \n\\begin{eqnarray}\n\\beta _\\ell ={\\frac{(2\\ell +1)}{(\\ell +1)}}\\beta _{\\ell +1},~~\\beta\n_\\ell ={\\frac \\ell {(2\\ell -1)}} \\beta _{\\ell -1},~~{\\frac{\\beta\n_{\\ell +1}}{\\beta _{\\ell-1}}}={\\frac{\\ell (\\ell +1)}{(2\\ell +1)(2\\ell\n-1)}}. \\label{rec} \n\\end{eqnarray}\n\nAny function $W(X)$ can be expanded in terms of the polynomials $L_\\ell(X)$ \n\\footnote{We include the dipole, $\\ell=1$, however the monopole is\ndropped as we will only be considering the expansion of the CGI \nperturbations (\\ref{3.11}) where the isotropic\npart has been factored out according to (\\ref{3.10}). We must beware\nthat this does not cause problems later by omitting the spatial\ngradients of the isotropic term.} and then upon combining\n(\\ref{add01a}) and expansion in terms of $L_{\\ell}(X)$ to find \nthe expansion in terms $O^{A_\\ell}$ : \n\\begin{eqnarray}\nW(X)=\\sum_{\\ell=1}^\\infty \\hat{C}_\\ell L_\\ell(X)\\, \n~~\\mbox{and}~~\nW(X)=\\sum_{n=1}^\\infty \\hat{C}_nO^{A_n}O^{\\prime }{}_{A_n}\\,.\n\\label{add02_add03}\n\\end{eqnarray}\nWhen $W(X)$ is the angular correlation function, the $\\hat{C}_\\ell$ are the\ncorresponding angular power spectrum coefficients (see below). \n\n\\subsubsection{Relationship to Legendre polynomials}\n\nA Legendre polynomial $P_\\ell (X)$ is given by renormalising the\npolynomials $L_\\ell (X)$ defined in (\\ref{add01a}) so that $P_\\ell\n(1)=1.$ By (\\ref{add02a_beta}), this implies \n\\begin{equation}\nP_\\ell (X)=(\\beta _\\ell )^{-1}L_\\ell (X)~~\\Rightarrow P_\\ell (1)=1\\,;\n\\label{pm(x)} \n\\end{equation}\nconsequently from (\\ref{add01a}), \n\\begin{eqnarray}\nO^{A_\\ell }O^{\\prime }{}_{A_\\ell }=\\beta _\\ell P_\\ell (X), \\label{add}\n\\end{eqnarray}\nwhere $\\beta _\\ell $ are given by (\\ref{add02a_beta}). \nIt follows from (\\ref{add}) that \n\\begin{eqnarray}\nP_{\\ell}(X) = \\sum_{m=0}^{[\\ell \/2]} A_{\\ell m} X^{\\ell -2m},\n~~\\mbox{with}~~ \nA_{\\ell k} = {\\frac{(-1)^k (2\\ell -2k)! }{2^\\ell k! (\\ell -k)! (\\ell\n-2k)!}} \\label{legdef_II-2} \n\\end{eqnarray}\nare related to the $B_{lk}$ in (\\ref{add1}) by \n\\begin{eqnarray}\nB_{\\ell k} = \\left( {\\frac{(\\ell !)^2 2^\\ell }{(2\\ell )!}} \\right)\nA_{\\ell k} \\label{II-6} \n\\end{eqnarray}\n\nAny function $W(X)$ can be expanded in terms of both sets of polynomials -\nsee (\\ref{add02_add03}) and the corresponding expression \n\\begin{eqnarray}\nW(X) &=& \\sum_{\\ell =1}^{\\infty} {\\frac{2\\ell +1 }{4 \\pi}} C_\\ell P_\\ell (X).\n\\label{add01}\n\\end{eqnarray}\nThese two expansions can then be related as follows: equating\n(\\ref{add02_add03}) and (\\ref{add01}), and using (\\ref{legdef_II-2})\nand ({\\ref{II-3}), gives \n\\begin{eqnarray}\n\\sum_{\\ell =1}^{\\infty} \\sum_{m=0}^{[\\ell \/2]} \\hat C_\\ell B_{\\ell m}\nX^{\\ell -2m} &=& \\sum_{\\ell =1}^{\\infty} \\sum_{m=0}^{[\\ell \/2]}\n{\\frac{(2\\ell +1) }{4 \\pi}} \n C_{\\ell}\n A_{\\ell m} X^{\\ell -2m}\\, ~\\Rightarrow\n~\\hat C_\\ell B_{\\ell m} = {\\frac{(2\\ell +1) }{4 \\pi}} C_\\ell A_{\\ell m}, \n\\end{eqnarray}\nfrom (\\ref{II-6}) this gives the relation between the expansion\ncoefficients :\n\\begin{eqnarray}\n\\hat C_\\ell = \\left[ {\\frac{(2\\ell +1) (2\\ell )! }{4 \\pi 2^\\ell\n(\\ell !)^2}} \\right] C_\\ell = \\Delta_\\ell ^{-1} C_\\ell \\,. \n\\label{II-25}\n\\end{eqnarray}\n\n\\subsection{The mean square of PSTF coefficients :\n $\\langle {F_{A_\\ell} F^{A_\\ell}} \\rangle$}\n\nIt is known as before, from evaluating $\\int d\\Omega f^2$ and constructing\nthe orthogonality conditions on $O_{A_\\ell}$, that inversion \n\\begin{eqnarray}\n{F}_{A_\\ell}=\\Delta _\\ell^{-1}\\int_{4\\pi }d\\Omega\nO_{A_\\ell}f(x^i,e^a)~~~\\Leftrightarrow ~~~~f(x^i,e^a)=\\sum_{\\ell=0}^\\infty \n{F}_{A_\\ell}O^{A_\\ell},\n\\end{eqnarray}\ncan be constructed. From this we can build \n\\begin{eqnarray}\n{F}_{A_\\ell} {F}^{B_m}=\\Delta _\\ell^{-1}\\Delta _m^{-1}\\int_{4\\pi }d\\Omega\nO_{A_\\ell}f(x^i,e^a)\\int_{4\\pi }d\\Omega ^{\\prime }{}O^{\\prime\n}{}^{B_m}f(x^i,e^{\\prime }{}^a),\n\\end{eqnarray}\nto find \n\\begin{eqnarray}\n {F}_{A_\\ell} {F}^{B_m}=\\Delta _\\ell^{-1}\\Delta _m^{-1}\\int \\int d\\Omega\nd\\Omega ^{\\prime }{}O_{A_\\ell}O^{\\prime }{}^{B_m}f(x^i,e^a)f(x^i,e^{\\prime\n}{}^a)\\,. \n\\end{eqnarray}\nTaking the ensemble average \\cite{LW,W83} then gives \n\\begin{eqnarray}\n\\langle { {F}_{A_\\ell} {F}^{B_m}} \\rangle =\\Delta\n_\\ell^{-1}\\Delta _m^{-1}\\int \\int d\\Omega d\\Omega ^{\\prime\n}{}O_{A_\\ell}O^{\\prime}{}^{B_m} \\langle {f\\cdot f^{\\prime\n}{}} \\rangle \\label{II-29} \n\\end{eqnarray}\nwhere $\\langle ... \\rangle $ indicates an ensemble average over\nsufficiently many realizations of the angular correlation function.\n\nIn order to evaluate this further we assume that the correlations between\nthe function $f(x^i,e^a)$, {\\it {\\ i.e. }}, $\\langle\n{f(x^i,e^a)f(x^i,e^{\\prime }{}^a)} \\rangle$ are a function of the angular\nseparation between the two directions only, \n\\begin{eqnarray}\n\\langle {f\\cdot f^{\\prime }{}} \\rangle =W(e^ae^{\\prime\n}{}_a)=W(X)\\,. \n\\end{eqnarray}\nThis is a consequence of the Gaussian assumption (\\ref{w}), which allows\none to evaluate the angular correlation functions (\\ref{ang_correl}) in\na straightforward way \\footnote{Ideally one would prefer to evaluate\nthe angular correlation function without using the Gaussian\nassumption, as this is one of the features one should test rather than\nassume.}. With this assumption (\\ref{II-29}) becomes \n\\begin{eqnarray}\n\\langle { {F}_{A_\\ell} {F}^{B_m}}\\rangle =\\Delta\n_\\ell^{-1}\\Delta _m^{-1}\\int \\int d\\Omega d\\Omega ^{\\prime\n}{}O_{A_\\ell}O^{\\prime}{}^{B_m}W(X)\\,. \\label{II-29a} \n\\end{eqnarray}\nSubstituting (\\ref{add02_add03}) into (\\ref{II-29a}), we find \n\\begin{eqnarray}\n\\langle { {F}_{A_\\ell} {F}^{B_m}} \\rangle =\\Delta\n_\\ell^{-1}\\Delta _m^{-1}\\int \\int d\\Omega d\\Omega ^{\\prime\n}{}O_{A_\\ell}O^{\\prime}{}^{B_m}\\sum_{n=1}^\\infty\n\\hat{C}_nO^{A_n}O^{\\prime }{}_{A_n}\\,. \n\\end{eqnarray}\nRearranging terms, \n\\begin{eqnarray}\n\\langle { {F}_{A_\\ell} {F}^{B_m}} \\rangle =\\Delta\n_\\ell^{-1}\\Delta _m^{-1}\\sum_{n=1}^\\infty \\hat{C}_n\\left\\{ {\\ \\int_{4\\pi\n}d\\Omega O_{A_\\ell}O^{A_n}}\\right\\} \\left\\{ {\\int_{4\\pi }d\\Omega ^{\\prime\n}{}O^{\\prime }{}^{B_m}O^{\\prime }{}_{A_n}}\\right\\} ,\n\\end{eqnarray}\nwhere the integrals can be evaluated using the orthogonality conditions on\nthe $O_{A_\\ell}$'s, (\\ref{Delta-ortho0-01}), \n\\begin{eqnarray}\n\\langle { {F}_{A_\\ell} {F}^{B_m}} \\rangle &=&\\Delta\n_\\ell^{-1}\\Delta _m^{-1}\\sum_{n=1}^\\infty \\hat{C}_n\\left\\{ {\\delta\n^{\\ell n}\\Delta \n_\\ell h_{\\langle A_\\ell \\rangle }{}^{\\langle A_n \\rangle }}\\right\\} \\left\\{ {\\delta ^{mn}\\Delta\n_m^{+1}h^{\\langle B_m \\rangle }{}_{\\langle A_n \\rangle }}\\right\\} \\nonumber \\\\ \n&=&\\sum_{n=1}^\\infty \\hat{C}_n\\delta ^{\\ell n}\\delta\n^{mn}h_{\\langle A_\\ell \\rangle }{}^{\\langle A_n \\rangle }h^{\\langle B_m \\rangle}{}_{\\langle A_n \\rangle}. \n\\end{eqnarray}\nThus \n\\begin{equation}\n\\langle { {F}_{A_\\ell} {F}^{B_m}} \\rangle = \\hat{C}_\\ell\\delta\n^{\\ell m}h_{\\langle A_\\ell\\rangle }{}^{\\langle B_m\\rangle }.\n~~\\mbox{to find}~~\\langle { {F}_{A_\\ell} {F}^{A_\\ell}} \\rangle \n=\\hat{C}_\\ell h_{\\langle A_\\ell\\rangle }{}^{\\langle A_\\ell \\rangle},\n\\end{equation}\nso using $h_{\\langle A_\\ell\\rangle }{}^{\\langle A_\\ell\\rangle}$ (\\ref{pstfh-contract})\nthe mean-square is found to be \n\\begin{equation}\n\\langle { {F}_{A_\\ell} {F}^{A_\\ell}} \\rangle =\\hat{C}_\\ell \n\\,(2\\ell+1) \\label{II-36}\n\\end{equation}\ngiving the angular power spectrum coefficients $\\hat{C}_\\ell$ in terms of the\nensemble-averages of the harmonic coefficients \\footnote{This corrects\nan error in \\cite{SAG}, removing a spurious factor of\n$3^\\ell=h^{A_\\ell}_{A_\\ell}$ which follows from the orthogonality\nconditions in \\cite{ETMa} which are corrected here.}. If we use the\nLegendre expansion (\\ref{add01}) instead of the covariant expansion\ncoefficients (\\ref{add02_add03}), then from (\\ref{II-36}) and (\\ref{II-25})\nthe relation is \n\\begin{equation}\n\\langle {F_{A_\\ell}F^{A_\\ell}} \\rangle =(2\\ell+1)\\Delta\n_\\ell^{-1}C_\\ell,\\label{87} \n\\end{equation}\nwhere $C_\\ell$ are the usual Legendre angular power spectrum coefficients.\n\n\\subsection{The CGI angular correlation function}\n\nWe can now gather the results above in terms of the application we have in\nmind, namely anisotropy of the CBR. Consider on-sky perturbations made of\nGaussian Random Fields : The {\\it angular correlation function\n}$C(\\beta )$ $=W(X)$ is given by (\\ref{w}); the {\\it angular power \nspectrum coefficients} $C_\\ell$ are given by the mean-square of the \n$\\ell-$th temperature coefficient through (\\ref{87}): \n\\begin{equation}\n\\langle {\\tau _{A_\\ell}\\tau ^{A_\\ell}} \\rangle =(2\\ell+1) \\Delta_\\ell^{-1}C_\\ell\n\\label{129}\n\\end{equation}\nwhere the constants $\\Delta _\\ell^{}$ are given by\n(\\ref{Delta-ortho0-01}). These quantities are related by (\\ref{add01}): \n\\begin{equation}\nW(X)=\\sum_{m=1}^\\infty C_m{\\frac{(2m+1)}{4\\pi }}P_m(X) \\label{128}\n\\end{equation}\nwhere the $P_m(X)$ are given by (\\ref{pm(x)}) from (\\ref{add01})\nand (\\ref{add02a_beta}).\n\n\\subsubsection{Cosmic variance}\n\nThe observations are in fact of $a_\\ell ^2$ (which is \n$\\sum_{m=-\\ell }^{+\\ell }|a_{\\ell m}|^2\/4\\pi $ in the usual\nnotation). This is what is effectively found from experiments, such as\nthe COBE-DMR experiment. This is a {\\it single} realization of the\nangular power spectrum $C_\\ell $. The finite sampling of events\ngenerated by random processes (in this case Gaussian random fields)\nleads to an intrinsic uncertainty in the variance even in \nperfect experiments - this is {\\it sample variance}, or in the\ncosmological setting, {\\it cosmic variance}. We are measuring a single\nrealization of a process that is assumed to be random; there is an error\nassociated with how we fit the single realization to the {\\it averaged}\nangular power spectrum.\n\nThe quantities $\\langle \\tau _{A_\\ell } \\tau ^{A_\\ell}\\rangle$ \nrepresent the averaged (over the entire ensemble of\npossible $C_\\ell$'s) angular power spectrum, this is what one is in fact\ndealing with in the theory, as the reductions are done in terms of Gaussian\nRandom Fields where the entire ensemble is considered rather than a single\nexperimental realization. The $a_\\ell^2$ are a sum of the $2\\ell+1$ Gaussian\nRandom Variables $a_{\\ell m}$, this is taken to be $\\chi ^2$\ndistributed with $2 \\ell+1$ degrees of freedom. Each multipole has \n$2 \\ell +1$ samples{\\footnote{The uncertainty in $C_\\ell$ as \n${\\frac{\\Delta C_\\ell }{C_\\ell }} = \\sqrt{\\frac{2 }{(2 \\ell+1)}}$}}.\n\nThe key point here is that\n cosmic variance is proportional to $\\ell^{-1\/2}$ and so is\nless significant for smaller angular scales than larger scales (as is\npopular wisdom), i.e. {\\it cosmic variance} is not an issue on\nsmall scales. \nHowever on small (and perhaps intermediate scales) systematic errors \ncould be underestimated. \n\nPhysical process deviations and instrument noise are expected\nto dominate the small scales rather than non-Gaussian effects in the\nprimordial perturbations,\nbut on large\nscales the uncertainty due to cosmic variance would swamp out a non-Gaussian\nsignature. It then seems plausible that on both large and small scales the\nassumption of Gaussian Perturbations is acceptable; however on intermediate\nscales this is not the case, on these scales the effects of comic variance\nwould be small enough to allow a non-Gaussian signature to be apparent.\n \n\n\\section{Mode expansions}\n\n\nWe now consider spatial harmonic analysis of the angular coefficients\ndiscussed in the previous section. Note that the relations in this section\nhold in space-like {\\it surfaces}, namely the background space-like surfaces\nin an almost-FL model. The application in the following sections will be to\nthe projection into these spacelike surfaces of null cone coordinates\nassociated with the propagation of the CBR down the null cone.\n\nFollowing the Wilson-Silk approach \\cite{W83,GSS,HS95a} we consider\nthe following CGI expansions. Eigenfunctions $Q(x^\\nu )$ are chosen to\nsatisfy the Helmholtz equation \n\\begin{equation}\n\\mbox{D}^a\\mbox{D}_aQ=-k_{phys}^2 Q \\label{eigen}\n\\end{equation}\nin the (background) space sections of the given space-time of interest,\nwhere the $Q$'s are time-independent scalar functions with the physical\nwavenumber $k_{phys}(t)=k\/a(t)$, the wave number $k$ being independent of\ntime \\footnote{The function $Q$ will be associated with a direction\nvector $e^a_{(k)}$ and wave vector $k_a = k e_a^{(k)}$ normal to the\nsurfaces $Q = \\mbox{const}$, see the following subsection.}. These\ndefine tensors $Q_{A_\\ell}(k^\\nu ,x^i)$ that are Projected,\nSymmetric, and Trace-Free, and in the case of scalar perturbations are\nchosen to be given by PSTF covariant derivatives of the eigenfunctions $Q$: \n\\begin{equation}\nQ_{A_\\ell }=(-k_{phys} )^{-\\ell }\\mbox{D}_{\\langle A_\\ell \\rangle }Q \\label{def1}\n\\end{equation}\nUsing these we define functions of direction and position: \n\\begin{eqnarray}\nG_\\ell [Q](x^\\nu ,e^a)=O^{A_\\ell}Q_{A_\\ell}\n\\end{eqnarray}\nwith the $O^{A_\\ell }$ defined by (\\ref{defO}). Here the $G_\\ell $\nare call {\\it mode operators} and the objects $G_\\ell [Q]$ are called\n{\\it mode functions \\footnote{Note these are functions in phase space,\nnot on $M$.}}. \nIt follows that \n\\begin{eqnarray}\nG_\\ell [Q]=(-k_{phys} )^{-\\ell }O^{A_\\ell }\\mbox{D}_{\\langle A_\\ell \\rangle }Q \\label{mf}\n\\end{eqnarray}\nand we can expand a given function $f(x^i,e^a)$ in terms of these functions.\nIn our case this serves as a way of harmonically analysing the\ncoefficients $\\tau _{A_\\ell }(x^i)$ in (\\ref{3.11}) and (\\ref{3.11b}):\nexpanding the temperature anisotropy in terms of the mode functions, \n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_\\ell \\,\\sum_k\\,\\tau _\\ell (t,k)\\,G_\\ell [Q]\n\\label{exp} \n\\end{equation}\nwhere the $\\ell $-summation is the angular harmonic expansion and the\n$k$-summation the spatial harmonic expansion (in fact $k$ will be a\n3-vector because space is 3-dimensional, see below). Using the\nexpansion (\\ref{3.11}) on the left and (\\ref{mf}) on the right, \n\\begin{equation}\n\\sum_\\ell \\,\\tau _{A_\\ell}(x^i)\\,O^{A_\\ell}~=~\\sum_\\ell\n\\,\\sum_k\\,\\tau _\\ell (t,k)\\,(-k_{phys})^{-\\ell }O^{A_\\ell }\\,\\mbox{D}_{\\langle \nA_\\ell \\rangle }Q \n\\end{equation}\nand so \n\\begin{equation}\n\\tau _{A_\\ell }(x^i)~=~\\sum_k\\,\\tau _\\ell (t,k)\\,(-k_{phys} )^{-\\ell }\\mbox{D}_{\\langle\nA_\\ell \\rangle }Q(x^\\nu ) \\label{mode_exp}\n\\end{equation}\nwhich is the spatial harmonic expansion of the radiation anisotropy\ncoefficients in terms of the symmetric, trace-free spatial derivatives of\nthe harmonic function $Q$. The quantities $\\,\\tau _\\ell (t,k)$ are the\ncorresponding mode coefficients. Note that we have not as yet restricted the\ngeometry of the $Q^{\\prime }$s: they could be either spherical or plane-wave\nharmonics, for example.\n\nBy successively applying the background 3-space Ricci identity, \n\\begin{equation}\n\\mbox{D}_{abA_\\ell }Q-\\mbox{D}_{baA_\\ell }Q=+\\sum_{n=1}^\\ell {\\frac K{a^2}}\\left( {\\delta\n_b^{b_n}h_{a_na}-\\delta _a^{b_n}h_{a_nb}}\\right) \\mbox{D}_{\\bar{A_\\ell }}Q,\n\\end{equation}\nwhere $\\bar{A_\\ell }=a_1...a_{n-1}b_na_{n+1}...a_\\ell$, {\\it {\\ i.e. }}, the\nsequence of $\\ell $ indices with the $n$-th one replaced with a\ncontraction. First, the {\\it curvature-modified Helmholtz equation} is found\n\\footnote{This will be necessary in order to switch from the almost-FLRW\nmultipole divergence equations in part II to the mode representation\nwhere we use (\\ref{eO-rec}) to construct $\\mbox{D}^a \\mbox{D}_{\\langle a}Q_{A_{\\ell} \\rangle}$.}\n(using \\ref{rel-004},\\ref{rel-005},\\ref{rel-003}) :\n\\begin{equation}\n\\mbox{D}^a \\mbox{D}_a Q_{\\langle A_{\\ell} \\rangle} = - \\tilde k_{\\ell}^2 Q_{\\langle A_\\ell\n\\rangle}~~\\mbox{with}~~ - \\tilde k_{\\ell}^2 = \\frac{1}{a^2} \\left( {K \\ell\n(\\ell + 2) - k^2} \\right)\\,. \\label{cmhe}\n\\end{equation}\nSecond, we are able to construct the {\\it mode recursion relation} (using\n\\ref{rel-001}, \\ref{rel-002} in \\ref{PSTF_rec}) :\n\\begin{equation}\ne^a\\mbox{D}_a[G_\\ell [Q]]=+k_{phys} \\left[ {{\\frac{\\ell ^2}{(2\\ell +1)(2\\ell\n-1)}}\\left( {1-{\\frac K{k^2}}(\\ell ^2-1)}\\right) G_{\\ell\n-1}[Q]-G_{\\ell +1}[Q]}\\right] , \\label{Gl-recursion}\n\\end{equation}\nThe latter is the basis of the standard derivation of the \nlinear-FRW mode heirarchy for scalar modes. The derivation\nof these are given in appendix C (we consider only scalar \neigenfunctions). It will be seen, in paper II, that this relation can \nbe used in place of the general divergence relations which allow the \nconstruction of generic multipole divergence equations \\cite{ETMa} if \none restricts oneself to constant curvature space-times.\n\nGiven the recursion relation one can immediatly make the connection\nwith the usual Legendre tensor treatment (\\ref{len-PSTF}) this is\nshown in appendix C (\\ref{len-PSTF}-\\ref{len-rec}).\n\n\n\\subsection{$|\\tau _\\ell|^2$ in almost-FLRW universes}\n\nWe now relate the multipole mean-squares $\\left< \\tau_{A_\\ell} \\tau^{A_\\ell}\n\\right>$ of the ensemble average over the multipole moments with that of the\nmode coefficient mean-squares $|\\tau_\\ell(k)|^2$.\n\nIn order to carry this out we relate two separate spatial harmonic\nexpansions (\\ref{mode_exp}) for the same function: the first is one\nassociated with plane wave harmonics ($Q^k$), naturally used in describing\nstructure existing at any time $t$, and the second, one associated with\nradial and multipole harmonics ($O^{A_\\ell}_{(\\chi)}$ $R_{A_\\ell}$), \n{\\it {\\ i.e. }}, a spherical expansion based at the point of\nobservation, naturally arises when we project the null cone angular\nharmonics into a surface of constant time. These are both related to\nthe Mode Function formulation which becomes useful in the non-flat\nconstant curvature cases. \n\n\n\\subsubsection{Plane-waves and Mode functions}\n\n\nConsidering flat FRW universes, each set of eigenfunctions\nsatisfy (\\ref{eigen}). The temperature anisotropy (\\ref{3.10}) \ncan be expressed in terms of its plane-wave spatial Fourier\ntransform :\n\\begin{equation}\n\\tau(x^i,e^a) = \\sum_{k_{\\nu}} \\tau(k,t,e_a^{(k)},e^a) Q_{flat}.\\label{t-pw}\n\\end{equation}\n(For a more detailed treatment see appendix D). It can be\nshown that for the flat case, $K=0$, that is (\\ref{q-defs}) hold\nin (\\ref{q_a}) and (\\ref{q-eigen}) to find from (\\ref{plane-wave}) :\n\\begin{equation}\n\\mbox{D}_{\\langle A_{\\ell} \\rangle} Q_{flat} = (- i k_{phys})^{\\ell}\nO_{A_{\\ell}}^{(k)} Q_{flat} \\label{DQ-pw}\n\\end{equation}\nwhere $O_{A_{\\ell}}^{(k)}$ are the PSTF tensors associated with the\ndirection $e^a_{(k)}$. Then from (\\ref{def1}) we find that :\n\\begin{equation}\nQ_{flat}{}_{A_\\ell} = (-1)^{\\ell} O_{A_\\ell}^{(k)} Q_{flat}, \\label{QAl-pw}\n\\end{equation} \nalong with (\\ref{mode_exp}) to get the temperature multipole :\n\\begin{equation}\n\\tau_{A_{\\ell}} = (-1)^{\\ell} \\sum_k \\tau_{\\ell}(t,k) Q_{flat} \nO_{A_\\ell}^{(k)}. \\label{tAl-pw}\n\\end{equation}\n\n\n\\subsubsection{Radial expansions and Mode functions}\n\n\nUsing the flat, $K=0$, spherical eigenfunctions centred on a\npoint $x^i_0$, and with associated radial direction vector\n$e_a^{(\\chi)}$. The latter is the same as the (spherically symmetric)\nprojection into the constant time surfaces of the tangent vector $e^a$\nof the radial null geodesics, so we need not distinguish it from that\nvector. In this case the $\\ell$-th harmonic is \n\\begin{equation}\nQ_\\ell (x^\\alpha,e^a_{(\\chi)})|_{\\chi} = R_{A_\\ell }(r) O^{A_\\ell\n}_\\chi, ~~\\mbox{D}_a r = e_a, ~~\\mbox{D}^a e_a = {\\frac{2 }{r}} \\label{sph1}\n\\end{equation}\nwhere $e^a = dx^a\/dr$ is the unit radial vector. Cartesian coordinates in\nspace are given by $r$ and $e^a$ through $x^i = r e^i$. Defining the\nprojection tensor \n\\begin{equation}\np_{ab} = h_{ab} - e_a e_b \\Rightarrow p_{ab} u^a =0= p_{ab}\ne^a,~~p_a^{~a}=2,~~p^a_{~b} p^b_{~c} = p^a_{~c},\n\\end{equation}\nthen ($e^a$ is shear and curl free) \n\\begin{eqnarray}\n\\mbox{D}_a e_b = \\frac{1}{r} p_{ab} \\Rightarrow \\mbox{D}_a p_{bc} = - \\frac{1}{r} (p_{ab}\ne_c + p_{ac} e_b), ~~ \\mbox{D}^a p^b_{~a} = - \\frac{2}{r}e^b\n\\end{eqnarray}\nso from (\\ref{eigen}) \n\\begin{equation}\n\\mbox{D}_a \\mbox{D}^a (R_{A_\\ell } O^{A_\\ell }) = - k_{phys}^2 (R_{A_\\ell } O^{A_\\ell }).\n\\end{equation}\nTo work out the {\\it l.h.s.}, we must first calculate \n$\\mbox{D}_a Q_{\\ell}$ (\\ref{DQ-l}) and $\\mbox{D}_a \\mbox{D}^a Q_{\\ell}$\n(\\ref{D2Q-l}). These are then used to calculated :\n$\\mbox{D}_a R_{A_\\ell}$ (\\ref{DRAl}), $\\mbox{D}^a \\mbox{D}_a R_{A_\\ell}$ (\\ref{D2RAl}),\n$\\mbox{D}_a R^{A_\\ell}$ (\\ref{DOAl}) and $\\mbox{D}_a \\mbox{D}^a O^{A_\\ell}$\n(\\ref{D2OAl}) respectively.\n\nPutting these in (\\ref{D2Q-l}) to find (\\ref{proto-bessel}) frow which \nwe get :\n\\begin{eqnarray}\n\\frac{1}{r^2} {\\frac{\\partial }{\\partial r}} \\left( r^2 {\\frac{\\partial R_{A_\\ell }\n}{\\partial r}} \\right) + R_{A_\\ell } \\left({- \\frac{\\ell (\\ell +1)}{r^2}}\n\\right) = - k_{phys}^2 R_{A_\\ell }\\,, \\label{bessel} \n\\end{eqnarray}\nthe spherical Bessel equation. Thus providing the PSTF derivation of\nthe spherical bessel equation in terms of the irreducible representation.\n\nNow consider that we can choose any basis we like for the tensor basis here,\nindependent of the spatial coordinates used. It is convenient to use the\nplane wave decomposition to get a parallel vector basis. We do this by\nwriting \n\\begin{eqnarray}\nR_{A_\\ell }(r) = \\sum_{k^a} R_{A_\\ell }^{(k)} = \\sum_{k^a} R_\\ell\n(k,r) O_{A_\\ell }^{(k)}\\,. \\label{103}\n\\end{eqnarray}\nthis expresses the tensor eigenfunction in terms of the monopole\neigenfunction. When this is substituted into (\\ref{proto-bessel}) we\nobtain the {\\it radial equation} which has solutions that are\n{\\it spherical Bessel functions} in the flat case :\n\\begin{eqnarray}\n{\\frac{\\partial }{\\partial r}} \\left( {r^2 {\\frac{\\partial R_{\\ell } }{\\partial r}}}\n\\right) + \\left[ {k_{phys}^2 r^2- \\ell (\\ell +1)} \\right] R_\\ell = 0\\,,\n~~\\mbox{to find}~~ R_\\ell (r) = \n\\alpha_l\\, j_\\ell (k_{phys} r) \\label{sph-bessel}\n\\end{eqnarray}\nwhere $k^2_{phys} = k^2\/a^2$ and $\\alpha_\\ell $ are integration constants (the\nsecond set of constants for this second order equation vanish because we\nchoose $R_l(0)$ to be finite; the Neumann functions are not finite at $r=0$).\n\n{}From (\\ref{sph1}), (\\ref{103}), and (\\ref{sph-bessel}) we have found\nthat the solutions to the Helmholtz equations give the eigenfunctions \n\\begin{eqnarray}\nQ_\\ell (x^\\nu) &=& \\alpha_\\ell \\, \\sum_{k^{\\nu}} j_\\ell (k_{phys} r)\nO^{A_\\ell }_{(k)} O_{A_\\ell }^{(\\chi)} = \\alpha_\\ell \\, \\sum_{k^{\\nu}}\nj_\\ell (k_{phys}(k,r) r) L_\\ell (X). \n\\end{eqnarray}\nand we can set $\\alpha_\\ell = (\\Delta_\\ell )^{-1}$ so that $\\int\nQ_\\ell d\\Omega = 1.$ \n\nIt is important to notice that the functions $L_\\ell (X)$ depend both \non $k^a$ and on $e^a$, and so for each $k^a$ is a function of \n$(\\theta,\\phi)$, thus $Q$ is indeed a function of all\nspatial coordinates. We can pick any direction $k^a$ to find the particular\neigenfunctions \n\\begin{eqnarray}\nQ_\\ell ^{(k)} = (\\Delta_\\ell )^{-1} j_\\ell (k_{phys} r) O^{A_\\ell\n}_{(k)} O_{A_\\ell }^{(\\chi)}. \n\\end{eqnarray}\nassociated with that direction. The general $\\ell $-th eigenfunction\nis a sum of such eigenfunctions over a basis of directions $k^a$\n\\footnote{Many treatments choose a particular direction for $k$: $k^a\n= \\delta^a_3$ or similar and omit the summation.}: \n\\begin{equation}\nQ_\\ell (x^\\nu) = \\sum_{k^a} Q_\\ell ^{(k)} \\,.\n\\end{equation}\n\\noindent\nNow we find $O^{A_{\\ell}} D_{A_{\\ell}}Q$ (\\ref{QD-Rl}) in terms of $R_{\\ell}$\n(as show in the appendix using PSTF techniques, \\ref{104a}-\\ref{QD-Rl}) :\n\\begin{equation}\nO^{A_\\ell }_{(\\chi)} \\mbox{D}_{\\langle A_\\ell \\rangle } Q = (- k_{phys} )^{\\ell }\nO^{A_\\ell }_{(\\chi)} O_{A_\\ell }^{(k)} R_\\ell ~~\\Rightarrow ~~\nO^{A_\\ell } Q_{A_\\ell } = (- k_{phys})^\\ell Q_\\ell. \n\\end{equation}\nPutting this in the expansion (\\ref{mode_exp}) \n\\begin{eqnarray}\n\\tau_{A_\\ell }(x^i) = \\sum_{k^{\\nu}} \\tau_\\ell (t,k)\n(-k_{phys})^{-\\ell } \\mbox{D}_{\\langle A_\\ell \\rangle } Q(x^{\\nu}) = \\sum_{k^{\\nu}}\n\\sum_n \\tau_\\ell (t,k) (-k_{phys})^{-\\ell } \\mbox{D}_{\\langle A_\\ell \\rangle } Q_n\n\\label{mean-s2} \n\\end{eqnarray}\ngives the present version of (\\ref{exp}) \n\\begin{equation}\n\\tau(x^i,e^a_{(\\chi)}) = \\sum_\\ell \\tau_{A_\\ell } O^{A_\\ell } =\n\\sum_\\ell \\tau_\\ell (k,t) G_\\ell [Q];\n\\end{equation}\ndirectly analysing the coefficients $\\tau_{A_\\ell }$ in terms of these\nfunctions $Q$ - given that we have 3-dimensions worth of variability\nso as to represent arbitrary spatial functions - with purely\ntime-dependent co-efficients (parametrized by a vector $k^a$), \n\n\n\\subsubsection{Radial expansions and plane-waves}\n\n\nNow consider the inversion\\footnote{Contrast with (\\ref{sph1}):\nthere the l.h.s is the spherical eigenfunction; here it is the plane\none, expressed in terms of spherical ones.} \n\\begin{equation}\ne^{(+i k_{\\alpha} x^{\\alpha})} = \\sum_{\\ell } \\tilde{R}_{A_\\ell }(r)\nO^{A_\\ell }_{(\\chi)},~~\\iff~~ \\tilde{R}_{A_\\ell } = \\Delta_\\ell ^{-1} \\int d\n\\Omega_{(\\chi)} e^{+i k_{\\alpha} x^{\\alpha}} O_{A_\\ell }^{(k)}\n\\end{equation}\nby taking a Taylor expansion using $k_{\\alpha} x^{\\alpha} = k_{phys} r\ne_b^{(k)} e^{b}_{(\\chi)}$ we find \n\\begin{eqnarray}\n\\tilde{R}_{A_\\ell } &=& \\Delta_\\ell ^{-1} \\sum_{n=0}^{\\infty}\n{\\frac{(+i)^n (k_{phys} r)^n }{n!}} e_{B_n}^{(k)} \\int d \\Omega\nO_{A_\\ell }^{(\\chi)} e^{B_n}_{(\\chi)} \\nonumber \\\\\n&=& \\Delta_\\ell ^{-1} \\sum_{n=0}^{\\infty} {\\frac{(+i)^n (k_{phys} r)^n\n}{n!}} {\\frac{\\delta_n^{\\ell +2m} n! (n-\\ell +1)!! }{(n-\\ell +1)!\n(n+\\ell +1)!!}} 4 \\pi e_{B_n}^{(k)} h_{\\langle A_\\ell \\rangle }{}^{\\langle B_\\ell\n} h^{B_{n-\\ell }\\rangle }. \\label{p-r003} \n\\end{eqnarray}\non using (\\ref{dint}). Putting (\\ref{p-r004}) into (\\ref{p-r003})\nwe find \n\\begin{eqnarray}\n\\tilde{R}_{A_\\ell } &=& (+i)^\\ell \\Delta_\\ell ^{-1} 4 \\pi\n\\sum_{m=0}^{\\infty} {\\frac{(-1)^m (k_{phys} r)^{\\ell +2m} (2m+1)!!\n}{(2m+1)! (2m+2\\ell +1)!!}} O_{A_\\ell }^{(k)} \n\\nonumber \\\\\n&=& 4 \\pi (+i)^\\ell \\Delta_\\ell ^{-1} \\sum_{m=0}^{\\infty}\n{\\frac{(-1)^m (k_{phys} r)^{\\ell +2m} }{2^m m! (2m+2\\ell +1)!!}}\nO_{A_\\ell }^{(k)}. \n\\end{eqnarray}\nThis can be re-expressed as \n\\begin{equation}\n\\tilde{R}_{A_\\ell } = 4 \\pi (+i)^\\ell \\Delta_\\ell ^{-1} j_\\ell\n(k_{phys} r) O_{A_\\ell }^{(k)}\n= (+i)^\\ell (2\\ell +1) \\beta_\\ell ^{-1} j_\\ell\n(k_{phys} r) O_{A_\\ell }^{(k)}. \n\\end{equation}\nHence \n\\begin{eqnarray}\nQ_{flat} (x^i,e^{(k)}_a)= e^{+i k^{\\alpha} x_{\\alpha}} = 4 \\pi\n\\sum_{n,k^{\\nu}}^{\\infty} (+i)^n j_n(k_{phys} r) O_{(k)}^{A_n}\nO_{A_n}^{(\\chi)} \\Delta_n^{-1} \\label{p-rfinal}\n\\end{eqnarray}\nlinks the plane-waves to the spherical expansion\nThis recovers, from $L_\\ell (e^a e^{\\prime}{}_a) = \\beta_\\ell P_\\ell (e^a\ne^{\\prime}{}_a)$ the more usual \n\\begin{equation}\ne^{+i k_{phys} r X} = \\sum_\\ell \\tilde{R}_{A_\\ell } O^{A_\\ell\n}{}_{(\\chi)} = \\sum_\\ell (+i)^\\ell (2\\ell +1) j_\\ell (k_{phys} r)\nP_\\ell (X) \n\\end{equation}\n\n\n\\subsubsection{$|\\tau _\\ell |^2$ for $K=0$ (flat) almost-FLRW models}\n\n\nWe now return to the relationship between the\n$\\tau_{A_\\ell }$ and $\\tau_\\ell $. Now from (\\ref{plane-wave}), (\\ref{102}),\n(\\ref{p-rfinal}) and (\\ref{mean-s2}) : \n\\begin{eqnarray}\n\\tau_{A_\\ell } &=& (+i)^\\ell \\int {k^2 d k \\over (2 \\pi)^3}\n \\int d \\Omega_k \\tau_{\\ell }(k,t)\nO^{(k)}_{A_\\ell } \\sum_{n=0}^{\\infty} (+i)^n j_n(\\lambda r) O_{(k)}^{B_n}\nO_{B_n}^{(\\chi)} \\beta_n^{-1} (2n+1).\n\\end{eqnarray}\nThis can be reduced further (\\ref{tau_Al_planewv}-\\ref{Atau_Al_radial}), \nwhere upon using (\\ref{Rl_defn}) one finds that \n\\begin{eqnarray}\n\\tau_{A_\\ell } = 4 \\pi O_{A_\\ell }^{(\\chi)} \\int {k^{\\prime}{}^2\ndk^{\\prime}\\over (2 \\pi)^3 }{}\\tau_\\ell (k^{\\prime}{},t) \nR_\\ell (k^{\\prime}{},\\chi),\\label{tau_Al_radial}\n\\end{eqnarray}\nwhere $\\chi=r\/a$ and by using $R_\\ell (k,\\chi)= j_\\ell (\\lambda r)$ we\ncan identify this with (\\ref{tau_Al_planewv}), the equivalent of the\nmultipole moments found using the explicit form of the plane-waves.\n\nIn order to proceed further, from (\\ref{tau_Al_radial}) we construct the\nensemble average\n{\\footnote{Remembering that \n\\[\n\\langle { j_{\\ell}( k' r) j_{\\ell}(k,r) } \\rangle_{x^a} = 4 \\pi \n\\left( {\\pi \\over 2} \\right) {1 \\over k^2} \\delta(k'-k).\n\\]\n}} \n\\begin{eqnarray}\n\\langle {\\tau_{A_\\ell } \\tau^{A_\\ell }} \\rangle &=& (4 \\pi)^2 O_{A_\\ell }^{(\\chi)}\nO^{A_\\ell }_{(\\chi)} \\left< {\\int {k^{\\prime}{}^2 dk^{\\prime}\\over(2 \\pi)^3}\n{}\\tau_\\ell (k^{\\prime}{},t) R_\\ell (k^{\\prime}{},r) \\int {k^2 dk \\over (2\n\\pi)^3} \\tau_\\ell (k,t) R_\\ell (k,r) } \\right> , \\nonumber \\\\ \n&=& {4 \\pi \\over (2 \\pi)^3} \\beta_\\ell \\int k^2\ndk | \\tau_\\ell (k,t)|^2. \n\\end{eqnarray}\nWhat has happened here is that we imagine an ensemble of universes and we\nuse an ensemble average rather than the space average; we have to do this\nbecause $\\tau_\\ell $ is not square-integrable, {\\it i.e.}, we cannot use the\nr.m.s value as we cannot integrate the square over the space in general;\nthis operation is not well defined. In order to deal with the ensemble\naverage over the mode coefficients, {\\it {\\ i.e. }} $\\langle \\tau_\\ell ^k\n\\tau_m^{k^{\\prime}{}} \\rangle $, we assume the perturbations to be fairly\nhomogeneously spread throughout the space and not confined in a particular\nregion, and assume that there are no correlations between perturbations with\ndifferent wavenumbers. Here the Gaussian assumption is useful, we have that \n\\[\n\\left< {\\tau_\\ell (k,t) \\tau_\\ell (k^{\\prime}{},t)} \\right> =\n (2 \\pi)^3 \\delta(k^{\\prime}{}-k) |\\tau_{\\ell}(k,t)|^2. \n\\]\n\n\n\n\\subsubsection{$|\\tau _\\ell |^2$ for almost-FLRW models}\n\nIt is useful to notice that an alternative although\nequivalent avenue of approach is also possible proceeding directly from the\nmode expansion in $G_\\ell [Q]= O^{A_\\ell }_{(\\chi)} Q_{A_\\ell }$; this can be used for\nany $K$.\n\nNotice that as before at the observer at $x_0^i$, \n\\begin{equation}\n\\tau(x_0^i,e^a) = \\sum_\\ell \\tau_{A_\\ell } O^{A_\\ell },~~ \\iff~~\n\\tau_{A_\\ell }(x_0^i) = \\Delta_\\ell ^{-1} \\int d \\Omega O_{A_\\ell }\n\\tau(x_0^i,e^a). \n\\end{equation}\nNow in the spatial section in general we can write \n\\begin{equation}\n\\tau(x^i,e_{(k)}^a) = \\sum_\\ell \\tau_{A_\\ell } O^{A_\\ell }_{(k)}, ~~ \\iff ~~\n\\tau_{A_\\ell }(x^i) = \\Delta_\\ell ^{-1} \\int d \\Omega^{(k)} O_{A_\\ell }^{(k)}\n\\tau(x^i,e^a_{(k)}).\n\\end{equation}\nNow the point $x_0^{\\nu}$ is chosen at some earlier time in a spatial\nsection, with radial direction vector $e^a_{(\\chi)}$, for FRW models we can\nconsider $e^a_{(\\chi)}$ and $e^a$ to be equivalent and write \n\\begin{eqnarray}\n\\tau(x_0^i,e^a) = \\tau(x_0^i,e^a_{(\\chi)}) = \\sum_\\ell \\tau_{A_\\ell }(x_0^i)\nO^{A_\\ell }_{(\\chi)} \\equiv \\sum_\\ell \\tau_{A_\\ell }(x_0^i) O^{A_\\ell\n} \\nonumber \\\\ ~~\\iff ~~ \\tau_{A_\\ell }(x_0^i) = \\Delta_\\ell ^{-1}\n\\int d \\Omega^{(\\chi)} O_{A_\\ell }^{(\\chi)} \\tau(x_0^i,e^a_{(\\chi)}).\n\\end{eqnarray}\n\n\\noindent\nRemember that in some spatial section (where the integral relations\n(\\ref{OGl},\\ref{GlGm},\\ref{Gl-norm1}) are useful)\n\\begin{equation}\n\\tau(x^i,e^a) = \\sum_{\\ell ,k^{\\nu}} \\tau_\\ell (k,t) O^{A_\\ell }_{(\\chi)}\nQ_{A_\\ell }. \n\\end{equation}\n\nNow from \n\\begin{equation}\n\\tau_{A_\\ell }(x_0^i) = \\Delta_\\ell ^{-1} \\int d \\Omega\n\\sum_{m,k^{\\alpha}} \\tau_m O_{A_\\ell } G_\\ell [Q],\n\\end{equation}\nwe identify $O^{A_\\ell}_{(\\chi)}$ with $O^{A_\\ell }$, to find \n\\begin{equation}\n\\tau_{A_\\ell } = \\sum_{k^{\\alpha}} \\tau_\\ell (k,t) Q_{A_\\ell },\n\\end{equation}\nwhich means that for $K=0$ ($R_\\ell \\propto j_\\ell $). \n\\begin{eqnarray}\n\\left< {\\tau_{A_\\ell } \\tau^{A_\\ell }} \\right> &=& \\sum_{k^{\\nu}}\n|\\tau_\\ell |^2 \\langle {Q_{A_\\ell } Q^{A_\\ell} }\\rangle = (4 \\pi) \\beta_\\ell\n\\sum_{k} |\\tau_\\ell |^2. \\label{mean-s3}\n\\end{eqnarray}\n\nWhat is particularly useful about the last method of calculation is that for\nconstant $K$ surfaces \n\nit can be shown on using relation (\\ref{rec_norm_Gn}) recursively that \n\\begin{equation}\n\\int d \\Omega_k \\left< |G_\\ell [Q] G_m[Q]| \\right> = 4 \\pi \\Xi_{\\ell\n}^2 \\delta^m_\\ell\\,, ~~\\mbox{with}~~ \n\\Xi_\\ell ^2 = \\prod_{n=1}^\\ell (\\alpha_n)^2\n\\end{equation}\nand $\\left< |\\ldots | \\right>$ denotes the ensemble average of the\neigenfunctions. What is useful here is to notice that (using the\nnormalisation for $K=0$), \n\\begin{equation}\n\\Xi_\\ell ^2|_{K=0} = \\prod_{n=1}^{\\ell } {\\frac{n^2 }{(2n+1)(2n-1)}} =\n{\\frac{\\beta_\\ell ^2 }{2\\ell +1}} = (4 \\pi)^{-2} (2\\ell +1)\n\\Delta_\\ell ^2. \n\\end{equation}\nThis means that for $K=0$ \n\\begin{equation}\n\\int d \\Omega_{(k)} \\left< | G_\\ell ^k G_m^{k^{\\prime}{}} | \\right> =\n\\delta_\\ell ^m {\\frac{{2\\ell +1} }{4 \\pi}} \\Delta_\\ell ^2. \\label{Gl-norm2}\n\\end{equation}\n\nNow we can extend the above to FL models with $K\\neq 0$. The mean-square for\nconstant $K$ is found by modifying the normalization after identifying the $%\nK=0$ normalizations in (\\ref{Gl-norm1}) and (\\ref{Gl-norm2}) : \n\\begin{equation}\n\\langle {\\tau _{A_\\ell }\\tau ^{A_\\ell }}\\rangle = \\sum_{k^\\nu\n}|\\tau _\\ell |^2 \\langle {Q_{A_\\ell }Q^{A_\\ell}} \\rangle=(4\\pi ) \\beta _\\ell\n\\sum_k|\\tau _\\ell |^2\\Xi _\\ell^2, \n\\end{equation}\nor alternatively keeping the form of the mean-square in (\\ref{mean-s3})\nby redefining the mode function expansion \\cite{WSc} by a\nwavelength-dependent coefficient; then \n\\begin{equation}\n\\tau (x^i,e^a)=\\sum_{\\ell ,k^\\nu }\\tilde{\\tau}_\\ell (k,t)M_\\ell [Q]\n\\,, ~~\\mbox{and}~~ \nM_\\ell [Q]=\\Xi _\\ell (k,K)G_\\ell [Q], \\label{redef}\n\\end{equation}\ndefines the new coefficients $\\tilde{\\tau }_\\ell (k,t)$ so that \n\\begin{equation}\n|\\tilde{\\tau}_\\ell |^2=|\\tau _\\ell |^2\\Xi _\\ell ^2\\,. \\label{prb_208b} \n\\end{equation}\nBecause we have redefined the mode functions in (\\ref{redef}), the form\nof the equations for $K\\neq 0$ is the same as in the case $K=0$. However the\ncoefficients are different because they are from a different expansion.\nUsing the results from the $K=0$ case, either (\\ref{tau_Al_radial}) or\n(\\ref{tau_Al_planewv}), we have that \n\\begin{eqnarray}\n\\langle {\\tau _{A_\\ell }\\tau ^{A_\\ell }}\\rangle = {1 \\over 2 \\pi^2 }\n\\beta _\\ell \\int {dk \\over k}k^3 |\\tau _\\ell (k,t)|^2. \\label{prb_208}\n\\end{eqnarray}\nHence, on using $\\hat{C}_\\ell =\\Delta ^{-1}C_\\ell $ and $\\langle { \\tau\n_{A_\\ell }\\tau ^{A_\\ell }}\\rangle =\\hat{C}_\\ell (2\\ell +1)$, we now have \n\\begin{eqnarray}\n\\langle {\\tau (x_0^i,e^a)\\tau (x_0^i,e^{\\prime }{}^a)}\\rangle \n=\\sum_{\\ell =0}^\\infty (2\\ell +1)^{-1}\\langle {\\tau _{A_\\ell }\\tau\n^{A_\\ell }} \\rangle O_{B_\\ell }O^{\\prime }{}^{B_\\ell }\n\\end{eqnarray}\nwhich can now be written from (\\ref{prb_208}) and (\\ref{prb_208b}) as \n\\begin{eqnarray}\n\\langle {\\tau (x_0^i,e^a)\\tau (x_0^i,e^{\\prime }{}^a)}\\rangle \n&=& \\left( {1 \\over 2 \\pi^2} \\right) \\sum_{\\ell =0}^\\infty \n(2\\ell +1)^{-1}\\beta _\\ell \\int_0^\\infty {dk \\over k} k^3 \n|\\tilde{\\tau}_\\ell (k,t)|^2 O_{A_\\ell }O^{\\prime}{}^{A_\\ell}\n\\nonumber \\\\ &=& \\left( {1 \\over 2 \\pi^2}\\right) \n\\sum_{\\ell=0}^\\infty \\beta _\\ell^2(2\\ell +1)^{-1} \\int {dk \\over k} k^3 \n|\\tilde{\\tau}_\\ell (k,t)|^2 P_\\ell (e^ae^{\\prime }{}_a) \n\\label{prb_208a}\n\\end{eqnarray}\nthus reobtaining the results of White and Wilson :\n\\begin{eqnarray}\nC_{\\ell} = \\frac{2}{\\pi} {\\beta_{\\ell}^2 \\over (2 \\ell +1)^2}\n\\int_{0}^{\\infty} {dk \\over k} k^3 | \\tilde \\tau_{\\ell}(k,t)|^2.\n\\end{eqnarray}\nOne needs to be careful here with the factors of $(2\\ell +1)$.\nEquation (\\ref{prb_208a}) relates the amount of power in a given \nwavenumber, $|\\tau_\\ell(k,\\eta_0)|^2$, given the intersection of the \nnull-cone fixed at the observer $x_0^i$, on a angular scale $\\ell $ given \nthat the angular correlations are found on the scale $X=e^a e^{\\prime}{}_a$, \n{\\it {\\ i.e. }}, X is the separation between measurements.\n\n\n\n\n\\section{Conclusions}\n\nWe have given here a comprehensive survey of the CGI \nrepresentation of CBR anisotropies in almost-FLRW universes, and\nrelated this formalism to the other major formalisms in use for this purpose\nat the present time. \n\nThis paper has been concerned with algebraic relations: specifically\nthe Multipole, ({\\it e.g.} for $\\tau_{A_\\ell}$), and Mode, ({\\it e.g.} for $\\tau_{\\ell}$), \nformalisms and the relationship between these. Where possible the multipole\nmoments have been treated for a spacetime with generic inhomogeneity and\nanisotropy but small temperature anisotropies. The mode moments however\nare only meaningful in the restricted class of almost-FLRW universes.\n\nThe subsequent papers in the series consider the differential relations\nsatisfied by the quantities mentioned here \\cite{MGE}, and will show how \nboth timelike and null integrations are used to lead to the standard results\nin the literature. Taken together, this will be is an \n{\\it ab initio} demonstration of the way the different formalisms in use, \nand their major results, can be obtained from a single CGI \napproach, as well as providing the natural extension of the usual \nresults into the non-linear (exact) theory.\n\n\n\\section*{Acknowledgements}\n\nWe are grateful to Bill Stoeger, Bruce Bassett, Roy Maartens, and \nPeter Dunsby for\nuseful comments and suggestions. This work was supported by the South \nAfrican Foundation for Research and Development.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjusp b/data_all_eng_slimpj/shuffled/split2/finalzzjusp new file mode 100644 index 0000000000000000000000000000000000000000..fdea795870da945c9449820a92fe5ab8f3ea7c30 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzjusp @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{s:intro}\n\nThe formation of low-mass stars occurs in dense cores inside molecular clouds,\nalthough the actual mechanism by which these cores accumulate material is\n still disputed (ambipolar diffusion vs.\\ turbulence: \\citealt{shu:ppiv};\n \\citealt{hartmann01}). In any case, through the loss of turbulent and\/or\nmagnetic support, dense cores contract towards a `critical state',\nafter which gravitational collapse starts and infall occurs onto a central\nobject, a `protostar'.\nObservers distinguish such protostars in Class~0 objects, where\n the circumstellar envelope is much more massive than the star and\n accretion rates are high, and Class~I objects, where star and\n envelope have similar masses and accretion rates have dropped (see\n \\citealt{andre:ppiv} for a review).\nDuring these stages, most ($\\approx$99\\%) of the angular momentum of\nthe core is carried away by bipolar outflows, the formation of binary\nstars, and magnetic fields. The remaining 1\\% leads to the formation\nof the circumstellar disks commonly observed in subsequent stages of\npre-main sequence stars \\citep{mundy00}.\nThis scenario is convincing, but many fundamental questions remain\nunanswered regarding the stages before formation of a central luminous\nobject, the so-called pre-stellar cores. Does the collapse start from\na state of dynamical equilibrium? When does a rotating structure form,\nand how important are the effects of flattening?\n\nDetailed kinematical studies have been carried out for the Class~I\nobject LDN~1489 \\citep{michiel01} and the Class~0 object IRAM 04191\n\\citep{arnaud02}. These authors used the HCO$^+$ and CS molecules,\nrespectively, to trace the velocity field. \nHowever, at the low temperatures (${_<\\atop{^\\sim}}$10~K) of pre-stellar cores,\nmost neutral molecules freeze out onto dust grains. In particular,\nthe CO and CS molecules freeze out at $n$(H$_2$)~${_>\\atop{^\\sim}}$10$^5$~cm$^{-3}$, so\nthat these species (as well as HCO$^+$) trace the outer parts of the\ncores \\citep{bacmann03,tafalla04}.\nThe more volatile N$_2$ molecule remains abundant at these densities,\nmaking N$_2$H$^+$\\ and N$_2$D$^+$\\ better tracers of pre-stellar core nuclei\nthan CO and CS (e.g., \\citealt{bergin97}; \\citealt{crapsi04}).\nHowever, at the very centers of the cores, where ${_>\\atop{^\\sim}}$10$^6$~cm$^{-3}$,\neven N$_2$ freezes out, causing N$_2$H$^+$\\ and N$_2$D$^+$\\ to disappear from the\ngas phase \\citep{bergin02,belloche04}.\nOne of the few molecules that do not suffer from this depletion effect\nis H$_2$D$^+$. This paper explores the use of H$_2$D$^+$\\ as kinematic tracer of\nthe centers of pre-stellar cores. We focus on the `prototypical'\npre-stellar core LDN 1544 in the Taurus molecular cloud ($d$=140~pc)\nwhich has been well characterized by observations and models.\nPreliminary results of this work appeared in \\citet{zermatt}.\n\n The structure of this paper is as follows. After the data and the radiative\n program have been introduced (\\S\\S~\\ref{sec:data}, \\ref{s:mdl}), we test\n models of quasi-static collapse mediated by ambipolar diffusion, which fit\n previous observations of LDN~1544, in spherical (\\S~\\ref{s:sph}) and axial\n (\\S~\\ref{s:disk}) symmetry. More general spherically symmetric free-fall\n models are discussed in \\S~\\ref{s:alto}. Sect.~\\ref{s:co_dep} tests the\n adopted molecular abundance structure of the core against chemical models. The\n possibility of dynamic rather than quasi-static collapse is considered in\n \\S~\\ref{s:hunter}. The sensitivity of the results to the adopted collisional\n cross sections for H$_2$D$^+$\\ is examined in \\S~\\ref{s:crc}. Sect.~\\ref{s:concl}\n lists the conclusions and future prospects of this work.\n\n\\section{Data}\n\\label{sec:data}\n\n\\begin{figure*}[tb]\n\\includegraphics[width=12cm,angle=-90]{2792_f1.ps}\n\\caption{Line profiles of H$_2$D$^+$\\ (left), HC$^{18}$O$^+$\\ (middle) and\n N$_2$H$^+$\\ (right; only the $F_1F$=1,0--1,1 hyperfine component)\n observed towards the `dust peak' of LDN~1544. Superposed are synthetic\n profiles for the best-fit static model (dotted), and infall models\n after \\citet{ciolek00} for times 2.660 (dashed) and 2.684~Myr\n (dash-dotted). The line width (static: $b_D$=0.15~km~s$^{-1}$; infall:\n $b_D$=0.05~km~s$^{-1}$) is set to match the H$_2$D$^+$\\ data.}\n\\label{f:data}\n\\end{figure*}\n\nThe line profile of H$_2$D$^+$\\ observed toward the `dust peak' of LDN~1544\\ in\nOctober 2002 \\citep{h2d+03} suggested a double-peaked shape but\nwas of poor quality with low\nsignal to noise. Therefore the source was included in our December\n2003 CSO observations, which will be presented by Caselli et al.\\ (in prep.).\nThe left panel of Figure~\\ref{f:data} shows the combined data at\nthe central position, which has an rms noise level of $T_{\\rm mb}$=0.11~K per\n0.04~km~s$^{-1}$\\ channel. In this spectrum, the H$_2$D$^+$\\ line profile is\nclearly double-peaked, like the profiles of the HC$^{18}$O$^+$\\ and N$_2$H$^+$\\ 1--0\nlines, observed by \\citet{paola02a} and shown in the central and right\npanels of the figure. To quantify the similarity between these lines,\nwe fitted two Gaussians to the profiles and found that each line is\nwell described by two thermal components at $T_{\\rm kin}$=7--10~K, separated\nby $\\approx$0.26~km~s$^{-1}$\\ (Table~\\ref{t:twopeaks}).\n\n\\begin{table}[tb]\n \\caption{Centroids and widths of the two velocity components.\nNumbers in brackets denote uncertainties in units of the last decimal.}\n\\label{t:twopeaks}\n \\centering\n \\begin{tabular}{lrrr}\n\n \\hline \\noalign{\\smallskip}\n \\multicolumn{1}{c}{Line} & \\multicolumn{1}{c}{$V_{\\rm LSR}$} &\n\\multicolumn{1}{c}{$\\Delta V_{\\rm obs}$} & \n\\multicolumn{1}{c}{$\\Delta V_{\\rm T}^a$} \\cr\n\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{km~s$^{-1}$} & \n\\multicolumn{1}{c}{km~s$^{-1}$} & \\multicolumn{1}{c}{km~s$^{-1}$} \\cr\n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\nH$_2$D$^+$\\ (1$_{10}$--$1_{11}$) & 7.07(3) & 0.29(7) & 0.28--0.34 \\cr\n & 7.38(3) & 0.27(4) & \\cr\nHC$^{18}$O$^+$\\ (1--0) & 7.04(1) & 0.18(3) & 0.10--0.12 \\cr\n & 7.28(1) & 0.23(3) & \\cr \nN$_2$H$^+$\\ (1--0, $F_1F$=10--11)& 7.08(1) & 0.19(1) & 0.11--0.13 \\cr\n & 7.33(1) & 0.20(2) & \\cr\n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\\multicolumn{4}{l}{$^a$ Thermal line width at $T_{\\rm kin}$=7 and 10 K.}\n\\end{tabular}\n\\end{table}\n\nThe H$_2$D$^+$\\ line has also been detected at positions 20$''$ offset from\nthe dust peak, where the line is only half as strong, suggesting a\ncentrally peaked H$_2$D$^+$\\ abundance \\citep{h2d+03}.\nFigure~\\ref{f:off} shows the line profiles of H$_2$D$^+$\\ and of HC$^{18}$O$^+$\\ and\nN$_2$H$^+$\\ (from \\citealt{paola02a}), as averages of observations taken at\n20$''$ North, South, East and West offsets. For H$_2$D$^+$\\ and N$_2$H$^+$,\nneither these `average offset' spectra appear double-peaked, nor the\nindividual spectra.\nIn contrast, the average HC$^{18}$O$^+$\\ spectrum at the offset position\nis clearly double-peaked.\n\nThe H$_2$D$^+$\\ line profile towards the dust peak could be affected by\nabsorption in the outer parts of the core, as seen in the $J$=1--0 lines of\nCS and HCO$^+$ \\citep{tafalla98} and N$_2$H$^+$\\ \\citep{williams:l1544}.\nAbsorption is not seen for DCO$^+$ 2--1 \\citep{paola02a} but this may be an\nexcitation effect.\nAnalogously, the low H$_2$D$^+$\\ intensity and lack of strong central absorption at\nthe offset position may be an excitation effect. To clarify these points,\ndetailed radiative transfer models have been run for H$_2$D$^+$, as described in the\nfollowing sections.\n\n\\section{Model setup}\n\\label{s:mdl}\n\nThe line profiles of H$_2$D$^+$, HC$^{18}$O$^+$\\ and N$_2$H$^+$\\ toward LDN 1544 have\nbeen modeled with the Monte Carlo program by \\citet{hst}\\footnote{\\tt\n http:\/\/www.mpifr-bonn.mpg.de\/staff\/fvandertak\/ratran\/},\nwhich can treat spherical and axisymmetric geometries.\nFor HC$^{18}$O$^+$\\ and N$_2$H$^+$, we use\nmolecular data from the database by \\citet{moldata}\\footnote{\\tt\n http:\/\/www.strw.leidenuniv.nl\/$\\sim$moldata\/}. For H$_2$D$^+$,\ncollisional rate coefficients have not been calculated, so we use\nscaled radiative rates \\citep{black90},\nwhich probably have an uncertainty of a factor of 3--10.\nSection~\\ref{s:crc} investigates the sensitivity of the\n model results to the adopted rate coefficients.\n\nModeling of the N$_2$H$^+$\\ $J$=1$\\to$0 line concentrates on the lowest\nfrequency $F_1F$=10$\\to$11 hyperfine component, which has an optical\ndepth of only 3.7\\% of the total transition. We treat this line as if\nit were an isotopic version of N$_2$H$^+$: molecular excitation and line\nformation are calculated with the N$_2$H$^+$\\ abundance reduced by 27.\nSimilarly, to model the HC$^{18}$O$^+$\\ $J$=1$\\to$0 line, an oxygen isotope\nratio of $^{16}$O\/$^{18}$O=500 was adopted.\nWe do not model H$^{13}$CO$^+$ spectra because the line profile is\ncomplicated by hyperfine structure \\citep{jsb04}.\n\n\\begin{figure*}[tb]\n \\includegraphics[width=12cm,angle=-90]{2792_f2.ps}\n\\caption{As Figure~\\ref{f:data}, at 20 arcseconds offset (average NSEW)}\n\\label{f:off}\n\\end{figure*}\n\n\\section{Spherically symmetric models}\n\\label{s:sph}\n\n\\begin{figure*}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f3.ps}\n\\caption{Radial profiles of dust and gas temperature, and the\n densities of H$_2$, H$_2$D$^+$\\ ($\\times$2$\\times$10$^9$), HCO$^+$\n ($\\times$10$^{12}$) and N$_2$H$^+$\\ ($\\times$10$^{10}$) in our model of\n LDN~1544.}\n\\label{f:model}\n\\end{figure*}\n\nThe spherical models have inner and outer radii of 80 and 13000~AU, on\na 100-point grid, logarithmically spaced.\nFigure~\\ref{f:model} shows the adopted temperature and density\nstructure, taken from \\citet{galli02},\nas well as the adopted molecular abundance profiles.\nFor HCO$^+$ and N$_2$H$^+$,\nwe use abundance profiles from \\citet{paola02b}\n(Model~3 with a low ionization rate and a high sticking\n coefficient, which fits their data best), \nand assumed zero abundance inside $r$=2500~AU. \nThe distribution of HC$^{18}$O$^+$\\ has a central hole because its chemical\nprecursor, CO, freezes out onto dust grains at $n$(H$_2$)${_>\\atop{^\\sim}}$10$^5$~cm$^{-3}$.\nThe N$_2$H$^+$\\ distribution continues further inward and has a hole at\nsmaller radii, where N$_2$ freezes out.\nThe assumption of an N$_2$H$^+$\\ `hole' of 2500~AU was already needed by\n\\citet{h2d+03} to reproduce the large H$_2$D$^+$\\ abundance at the center\nof LDN~1544.\nFor H$_2$D$^+$, we initially assumed a central abundance of\n1$\\times$10$^{-9}$, dropping by a factor of 5 at a radius of 20$''$\n(2800~AU) \\citep{zermatt} as suggested by the observations of\n\\citet{h2d+03}.\n\nFor the velocity field, we first adopt static models, with zero\nvelocity at all radii, and second models with `infall' velocity\nfields. There are several competing theories for the velocity\n field of collapse onto a protostar. Here we consider models by\n \\citet{ciolek00}, which assume that protostellar collapse is\n quasi-static and regulated by magnetic pressure.\nSpecifically, we model their simulations `t3' and `t5' which\ncorrespond to times of 2.660 and 2.684~Myr after\nthe start of core collapse. At earlier times, the infall speed is\nvery low, while later times are implausible due to the short time\nspent in these phases.\nWe consider these specific infall models because they are known to\n reproduce observations of HCO$^+$ and N$_2$H$^+$\\ toward LDN~1544\\ \\citep{paola02a}.\n Sections~\\ref{s:alto} and~\\ref{s:hunter} treat more general collapse models.\nNote that the velocity field in the simulations by Ciolek \\& Basu is not\n spherically but axially symmetric with an axis ratio of $\\approx$0.3\n (see their Figure 4).\nAdopting these velocity fields in spherical symmetry is therefore a\nsimplification; however, the assumed profiles are quite similar to those of\ncollapsing Bonnor-Ebert spheres, as recently shown by \\citet{myers05}.\nSection~\\ref{s:disk} will present fully consistent, two-dimensional\nmodels with the velocity field of Ciolek \\& Basu.\nFor the turbulent broadening, Doppler parameters $b_D$ between 0.05 and\n0.25~km~s$^{-1}$\\ were tried. Smaller values of $b_D$ are overwhelmed by\nthermal broadening; larger values do not fit the data.\nVariations of $b_D$ with radius are considered in Section~\\ref{s:crc}.\n\nFigure~\\ref{f:data} shows that the total widths of the H$_2$D$^+$, HC$^{18}$O$^+$\\\nand N$_2$H$^+$\\ ground-state lines can be\nmatched using either velocity field. The best-fit static model has\n$b_D$=0.15~km~s$^{-1}$, while $b_D$=0.05~km~s$^{-1}$\\ gives the best fits with the\ninfall models. However, the observed double-peaked line shape rules\nout the static model, both at the dust peak and at the 20$''$ offset\nposition. Using $b_D$=0.05~km~s$^{-1}$\\, the infall models can reproduce the\nHC$^{18}$O$^+$\\ and N$_2$H$^+$\\ line profiles at both positions (Figure~\\ref{f:off}). \nThe data do not allow to decide between the `t3' and `t5' solutions.\nHowever, none of these models reproduces the double-peaked H$_2$D$^+$\\ \nprofile seen at the central position.\n\n\\bigskip\n\nOne reason for the lack of a central dip in the synthetic H$_2$D$^+$\\\nspectrum may be that the model has H$_2$D$^+$\\ abundant at the core center,\nwhere infall velocities are low according to \\citet{ciolek00}.\nTherefore we have tested the possibility that H$_2$D$^+$\\ has a shell-type\ndistribution like HC$^{18}$O$^+$\\ and N$_2$H$^+$.\nThe H$_2$ molecule is too light to freeze out on dust grains and cause\ndepletion of H$_2$D$^+$, but instead, conversion into D$_2$H$^+$\\ and D$_3^+$\\ in the\ngas phase \n(e.g., H$_2$D$^+$$+$HD$\\to$D$_2$H$^+$$+$H$_2$)\nmay cause a central H$_2$D$^+$\\ hole. Formation of multiply\ndeuterated H$_3^+$\\ is expected at the temperature and density of the\ncenter of LDN~1544\\ \\citep{roberts03,walmsley04}.\nSupport for these theories comes from the recent detection of D$_2$H$^+$\\ \ntowards the pre-stellar core LDN 1689N \\citep{vastel04}.\n\nWe have run models with the H$_2$D$^+$\\ abundance set to zero inside inner\nradii of 500--2500~AU (in steps of 500~AU), but none of these models\ngives a double-peaked H$_2$D$^+$\\ line profile. The problem is that the\n\\citet{ciolek00} models predict infall velocities of ${_<\\atop{^\\sim}}$0.1~km~s$^{-1}$\\ \nat the radii where H$_2$D$^+$\\ is abundant, much less than the observed\npeak separation of $\\approx$0.26~km~s$^{-1}$\\ (Table~\\ref{t:twopeaks}). \nIncreasing the outer radius of the H$_2$D$^+$\\ shell from 3000~AU to 6000~AU gives\ntoo strong line intensities at the offset position, but does produce\ndouble-peaked H$_2$D$^+$\\ profiles at the center. However, the two peaks are not\n$\\approx$equally bright as observed, but the blueshifted peak is\n$\\approx$twice as strong as the redshifted one, a phenomenon known as\n`infall asymmetry'.\nTherefore it seems that the H$_2$D$^+$\\ line shape cannot be explained with\nthe physical and geometrical structure adopted so far.\nBefore changing the adopted physical structure of the core, we explore\ndeviations from the hitherto assumed spherical symmetry.\n\n\\section{Axisymmetric models}\n\\label{s:disk}\n\nMaps of dust continuum and molecular line emission of LDN~1544\\ show a\nflattened structure \\citep{ward-t99,paola02a}. For example, the\n1300~$\\mu$m\\ dust continuum, which should be representative of the\nsurface density distribution, has a major axis of $\\approx$2\\farcm5\n(21000~AU), an axis ratio of $\\approx$1.7, and a position angle of\n45$^\\circ$ for the major axis. \nTherefore we explore the effects of flattened geometry on the line profiles.\n\nWe have constructed two-dimensional radiative transfer models with\ninner and outer radii of 70 and 20000~AU, on a 30$\\times$30 grid,\nlogarithmically spaced in both $R$ and $z$. The density structure is\nas described by \\citet{ciolek00}: an `infall disk' with a flaring\nangle of 15$^\\circ$ and an inclination of 74$^\\circ$. \nWe adopt a time of 2.660~Myr, which gives the best fit to\nmillimeter-wave dust continuum observations \\citep{ciolek00}.\nThe radial\ninfall velocity is zero at cloud center and edge, and peaks at\n0.14~km~s$^{-1}$\\ at a radius of 0.03~pc. For the gas and dust temperatures,\nwe tried a uniform value of 10~K (as in the model of Ciolek and Basu)\nand the radial distribution by \\citet{galli02} (assumed vertically\nisothermal). The abundance profiles of N$_2$H$^+$, HC$^{18}$O$^+$\\ and DCO$^+$\\ were\nagain taken from \\citet{paola02b}, Model~3, and for H$_2$D$^+$, we used the\nprofile described in \\S~\\ref{s:sph}.\n\nThese models reproduce the observed spatial distribution of the \nDCO$^+$\\ 2--1 and N$_2$H$^+$\\ 1--0 lines \\citep{paola02a}, and also predict\nthe correct line shape for HC$^{18}$O$^+$\\ and N$_2$H$^+$. However, they predict a\nsingle-peaked H$_2$D$^+$\\ line, which is not observed (Fig.~\\ref{f:2d}).\n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f4.ps}\n\\caption{Results of 2D models with $T_{\\rm kin}$$\\equiv$10~K (dotted line) and with the\ntemperature distribution from Galli et al (dashed line), superposed on\nthe observations (histogram).}\n\\label{f:2d}\n\\end{figure}\n\nAnother possible cause of the double peaked H$_2$D$^+$\\ line profile is\nrotational motion of the central part of the core. For example,\n\\citet{arnaud02} found the Class~0 object IRAM 04191 to rotate\nat an angular rate of $\\Omega\n\\sim$3.9$\\times$10$^{-13}$~s$^{-1}$ inside a `centrifugal radius' $r_C$ of\n3500~AU. We have added rotation to the velocity field by \n\\citet{ciolek00}, while keeping the density and temperature structure\nfrom the previous models.\nRotation rates between $\\Omega$=1$\\times$10$^{-13}$ and\n2$\\times$10$^{-12}$~s$^{-1}$ were tried, inside $r_C$=3000 and\n6000~AU. \nLarger values of $r_C$ are ruled out by the lack of a velocity shift\nof the H$_2$D$^+$\\ profiles at the offset positions, and also by\nthe interferometric observations of N$_2$H$^+$\\ 1--0 by \\citet{williams:l1544}.\nWe find that the H$_2$D$^+$\\ line profile is hardly changed from the\nnon-rotating case for $r_C$=3000~AU.\nFor $r_C$=6000~AU, the H$_2$D$^+$\\ line is considerably broadened, and the\ntotal width of the observed H$_2$D$^+$\\ profile limits $\\Omega$ to\n$<$5$\\times$10$^{-13}$~s$^{-1}$. None of these models predict\ndouble-peaked line shapes.\nIn general, both flattening and rotation appear unimportant for LDN\n1544 on the scales probed by single-dish observations.\n\n\\section{Alternative temperature and density structures}\n\\label{s:alto}\n\nThe reason that the models in \\S\\S~\\ref{s:sph} and~\\ref{s:disk} do not\nindicate H$_2$D$^+$\\ self-absorption may be that the H$_2$D$^+$\\ excitation\ntemperature is almost constant in the range of radii (${_<\\atop{^\\sim}}$3000~AU)\nwhere H$_2$D$^+$\\ is abundant (H$_2$D$^+$\/H$_2$~=~10$^{-9}$). \nIn particular, the models from \\S~\\ref{s:sph} with Galli's temperature\ndistribution have $T_{\\rm ex}$=5.8--5.6~K inside $r$=3000~AU\n(Fig.~\\ref{f:txc}, top), while the isothermal models of\n\\S~\\ref{s:disk} have $T_{\\rm ex}$=8.8--8.2~K at these radii.\n\n\\begin{figure}[htb]\n\\includegraphics[width=6cm,angle=-90]{2792_f5a.ps}\n\\includegraphics[width=6cm,angle=-90]{2792_f5b.ps}\n\\includegraphics[width=6cm,angle=-90]{2792_f5c.ps}\n\\caption{Excitation temperature profiles of H$_2$D$^+$\\ $1_{10}$--$1_{11}$\n for the physical structures by Galli et al (top), Young et al\n (middle), and for the isothermal power-law model (bottom). Within the\n radius of high H$_2$D$^+$\\ abundance ($\\sim$3000~AU), $T_{\\rm ex}$\\ is\n practically constant in the Galli case, but has a gradient in the\n other cases.}\n\\label{f:txc}\n\\end{figure}\n\nIn search of an excitation temperature gradient, we went\nback to spherically symmetric models and adopted the\ndensity and dust temperature distributions by \\citet{evans01},\nwith gas temperatures calculated by \\citet{young:h2co}. \nIn these models for LDN~1544, the density continues to rise toward the\ncenter, rather than flattening off as in the model of \\citet{ward-t99}.\nAs a consequence, $T_{\\rm ex}$\\ drops from 6.9 to 5.5~K at radii of\n500-2500~AU (Fig.~\\ref{f:txc}, middle).\n\nCombining this physical structure with velocity field `t3'\nfrom \\citet{ciolek00}, the H$_2$D$^+$\\ profile remains single peaked. This result\nholds also when adopting constant infall speeds up to 0.2~km~s$^{-1}$.\nLarger infall speeds are inconsistent with the HC$^{18}$O$^+$\\ and N$_2$H$^+$\\ data.\nEven an inward increasing velocity profile, $v=v_0(r\/r_0)^{-0.5}$ with\n$r_0$=10$^4$~AU and $v_0$=0.1~km~s$^{-1}$, fails to produce a double peak\n(Fig.~\\ref{f:alt}).\nAdopting $r_0$=10$^3$~AU and $v_0$=0.2~km~s$^{-1}$, which mimics the `t5'\nvelocity field \\citep{paola03} does not change this result appreciably.\nFurthermore, applying central holes in the H$_2$D$^+$\\ abundance distribution,\nwith radii of 500-1500~AU, changes the total intensity of the line,\nbut not its shape.\n\nTo prevent $T_{\\rm ex}$\\ from dropping off at the very center, we took the\nmore drastic step of forcing a uniform $T_{\\rm kin}$, while keeping the density\npower law structure by Evans et al. Values of $T_{\\rm kin}$=6~K, 8~K, 10~K and\n12~K were tried. \nFig.~\\ref{f:txc} (bottom) shows the $T_{\\rm ex}$\\ profile for $T_{\\rm kin}$=8~K.\nThe H$_2$D$^+$\\ profiles for these models show a classic `infall asymmetry'\nfor abundances ${_>\\atop{^\\sim}}$10$^{-9}$, or a broad peak for lower abundances.\nThe observed profile with two peaks of equal strength is not reproduced.\nLeaving the dust temperature unchanged or setting it equal to $T_{\\rm kin}$\\\nmakes no difference to these results.\n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f6.ps}\n\\caption{Results of models with a power-law density distribution and\n $T_{\\rm kin}$$\\equiv$8~K (dotted line) and with the gas temperature calculated by\n Young et al (dashed line), superposed on the observations (histogram).}\n\\label{f:alt}\n\\end{figure}\n\n\\section{Self-consistent chemical model}\n\\label{s:co_dep}\n\nThe rate of formation of H$_2$D$^+$\\ by the reaction of H$_3^+$\\ with HD\ndepends mainly on the cosmic-ray ionization rate which is not expected\nto change within the pre-stellar core. However, the main destruction\nchannel of H$_2$D$^+$\\ is the reaction with CO, the abundance of which is\nexpected to decrease significantly at low temperatures and high\ndensities due to freeze-out on dust grains. The models so far take\ndepletion into account in a simplified way by considering `jumpy'\nH$_2$D$^+$\\ abundance profiles.\nThis section considers models that compute the abundances of H$_3^+$\\ \nand its isotopomers self-consistently across the LDN~1544 core. For\nthat we adapted the model of H$_3^+$\\ chemistry in young protoplanetary\ndisks developed by \\citet{ceccarelli05}. These models ignore the\neffect of dynamics on the chemistry.\nBriefly, the model computes the abundances of H$_3^+$, H$_2$D$^+$, D$_2$H$^+$\\ and\nD$_3^+$\\ by solving the chemical network for these species, which is\nmostly governed by the CO depletion and the cosmic-ray ionization rate\n(\\citealt{roberts03}; \\citealt{walmsley04}; see the detailed\ndiscussion in \\citealt{ceccarelli05}). \nThe model uses the `fast' rate for the H$_3^+$\\ +~HD reaction\n\\citep{roberts03} and 920~K for the binding energy of CO on the grain\nsurface \\citep{bergin97}. The papers by Roberts et al.\\ and\n Walmsley et al.\\ discuss the uncertainties of the other relevant\n chemical reation rates such as the dissociative recombination of\n H$_2$D$^+$, D$_2$H$^+$\\ and D$_3^+$.\nThe role of cosmic rays is twofold: on the one hand, they set the\nionization degree in the gas, and therefore the total abundance of\nH$_3^+$\\ and its isotopomers in the region where CO is depleted.\nThe larger the ionization degree, the larger the absolute abundance\nof H$_3^+$\\ in all its isotopic forms.\nSecond, the cosmic rays regulate the desorption of CO molecules\nfrom icy grain mantles back into the gas phase.\nThe model includes `spot heating' of grains by cosmic ray impacts, \nbut no non-thermal desorption mechanisms which are much less\n effective \\citep{shen04}. \nThe larger the cosmic-ray flux, the higher the CO gas-phase abundance,\nand the lower the abundance of the deuterated forms of H$_3^+$.\n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=0]{2792_f7.ps}\n\\caption{Calculated abundances of H$_2$D$^+$, D$_2$H$^+$\\ and D$_3^+$\\ for an age\n of 0.1~Myr, a cosmic-ray ionization rate of 3$\\times$10$^{-17}$\n s$^{-1}$ and the physical structure by \\citet{evans01}. The dotted\n lines indicate the radii equivalent to one and two CSO beams, for\n comparison with observations.}\n\\label{f:ccmod}\n\\end{figure}\n\nThe model assumes that at time $t$=0, all CO is in the gas phase, so\nthat the CO abundance is equal to the canonical value $9.5 \\times\n10^{-5}$ with respect to H$_2$ (e.g., \\citealt{frerking82}). As time\npasses, the CO molecules condense out onto the grain mantles and\ndisappear from the gas phase. The time scale of the CO depletion is\nset by the density and temperature, which are taken equal to those\ncomputed in the models of Galli et al.\\ and Evans et al., previously\ndescribed. \nThe result is a zone where the increased CO depletion leads to an\nenhanced H$_2$D$^+$\/H$_3^+$\\ ratio, namely an increased H$_2$D$^+$\\ abundance \n(see also \\citealt{h2d+03} and \\citealt{ceccarelli04}).\nOur computed H$_2$D$^+$, D$_2$H$^+$\\ and D$_3^+$\\ abundances coincide with those\ncalculated by \\citet{roberts03} and \\citet{walmsley04} when the same\ndensities are considered.\n\nFigure~\\ref{f:ccmod} shows an example of the computed H$_2$D$^+$, D$_2$H$^+$\\ \nand D$_3^+$\\ abundance profile across LDN~1544. Also shown is the\n electron fraction, which equals the total ionization. In these highly shielded\nregions, most charge comes from molecules, not from metals, which are also\nexpected to be highly depleted \\citep{paola02b}.\nNote that in the very inner region, the H$_2$D$^+$\/H$_3^+$\\ and D$_2$H$^+$\/H$_3^+$\\\nabundance ratios have a `plateau', whereas the D$_3^+$\/H$_3^+$\\ abundance\nratio keeps increasing. The result is a `hole' in the H$_2$D$^+$\\ and\nD$_2$H$^+$\\ abundances, because D$_3^+$\\ becomes the charge carrier when the\nCO depletion is larger than $\\sim$30.\nThe result that D$_3^+$\\ is the dominant ion at the centers of pre-stellar\n cores was found before by \\citet{roberts03} and \\citet{walmsley04}; adopting\n the temperature and density structure from \\citet{young:h2co} for LDN~1544, it\n happens at $R {_<\\atop{^\\sim}} 3000$~AU.\nThe H$_2$D$^+$\\ line profiles computed from these models depend\non two major parameters: the cosmic-ray ionization rate and the time\n(for the CO depletion). We ran models varying both parameters for the\ntwo physical structures of the core, the Galli et al.\\ and Evans et\nal.\\ structure respectively. We considered ages of 0.05, 0.1 and\n1.0~Myr, and fixed the cosmic-ray ionization rate at the canonical\nvalue of 3$\\times$10$^{-17}$ s$^{-1}$ \\citep{zeta}.\nThe model does not consider any shielding of cosmic rays, as in\n\\citet{paola03}. \n\nSince the central CO--free zone grows in time,\nthe strengths of isotopic CO lines constrain the chemical age.\nWith the velocity field of \\citet{ciolek00}, we calculate\nan intensity for the C$^{18}$O 1--0 line in the 22$''$ beam of the\nIRAM 30m telescope of 4.3~K for $t$=0.05~Myr, 3.5~K for 0.1~Myr and\n1.5~K for 1.0~Myr. The observed strength of $\\approx$5~K\n\\citep{paola02a} thus suggests an age of ${_<\\atop{^\\sim}}$0.1~Myr for LDN~1544.\nIt is not clear if these ages are compatible with our assumption\n of a static cloud. Models including both chemistry and dynamics\n (e.g., \\citealt{rodgers:cocoon}; \\citealt{aikawa05}) are needed to\n address this issue.\n\nTo model the H$_2$D$^+$\\ line profile, we assume an\n ortho\/para ratio of unity as appropriate for low temperatures\n and high densities, although the exact amount of molecular\n depletion may make a factor of $\\sim$2 difference\n (\\citealt{pagani:h2d+}; \\citealt{walmsley04}).\nFigure~\\ref{f:ccchem} shows the H$_2$D$^+$\\ line profiles calculated for a\nconstant infall velocity of 0.2~km~s$^{-1}$\\ and for the Ciolek \\& Basu\nvelocity field. The synthetic profiles are double peaked, but show\nstrong infall asymmetry while two peaks of equal strength are\nobserved. The strong self-absorption makes the emission\n$\\approx$50\\% weaker than observed. At the offset position, the\nemission is also predicted to be asymmetric, and also $\\approx$50\\%\nweaker than observed. \n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f8.ps}\n\\caption{Results of models with a power-law density distribution and the\n gas temperature calculated by Young et al, the effect of depletion\n included for a chemical age of 0.1~Myr for either a constant infall\n velocity (dotted line) or the Ciolek \\& Basu velocity field (dashed\n line), superposed on the observations (histogram).}\n\\label{f:ccchem}\n\\end{figure}\n\nThese models seem to overestimate the depletion of CO and the\nabundance of H$_2$D$^+$. Chemical ages $<$0.05~Myr do not seem plausible,\nbut the age is uncertain through the dust opacity coefficient, which\nmakes the density uncertain by a factor of two.\nAlternatively the gas temperature (which controls the emission) is\nhigher than the dust temperature (which controls the depletion). The\nbinding energy of CO to the grain surface may also have been\noverestimated, so that CO does not stick to the surface as well as\nassumed. \nWe have not considered the reduced cosmic-ray ionization rate which\nprevious studies of the LDN~1544 core indicate in this source\n(\\citealt{paola02b}; \\citealt{tafalla04}), since the CO depletion\nwould be even higher, implying a worse match to the data.\n\n\n\\section{Alternative kinematics}\n\\label{s:hunter}\n\nThe infall model\nby \\citet{ciolek00} is a magnetically mediated, `quasi-static' collapse\nwith relatively slow infall velocities. The collapse may instead be\n`dynamic' which means that the gas never reaches equilibrium. The most\nextreme case is the Larson-Penston collapse (e.g., \\citealt{zhou92})\nwhere the velocity reaches many times the sound speed. We modeled the\nH$_2$D$^+$\\ line with such a velocity profile, but find that emission is\npredicted at much higher velocities than observed.\n\nHydrodynamic simulations by \\citet{hunter77} show the existence of a\ncontinuum of solutions between the extreme static and dynamic cases.\nThe solutions for the density $\\rho$ at time $t$ and radius $r$ are\n$$\n\\frac{4\\pi G\\rho r^2}{a^2} = \\sqrt{\\frac{m_0 r}{2at}}\n$$\nand the velocity $u$ (directed inward) follows\n$$\n\\frac{u}{a} = \\sqrt{\\frac{2m_0at}{r}}\n$$\nwhere $a$ is the sound speed, and $m_0$ is a dimensionless parameter.\nThe Larson-Penston collapse has $m_0 = 46.9$ while $m_0 = 0.975$ for\nthe quasi-static (Shu) collapse.\n\nAs examples of infall models `intermediate' between quasi-static\nand fully dynamic collapse,\nwe have run models for Hunter's cases 11b and 11d which have\n$m_0 = 2.577$ and $m_0 = 1.138$ respectively. The gas and dust\ntemperatures in these models are equal at 10~K, corresponding to a\nsound speed of 0.19~km~s$^{-1}$.\nFig.~\\ref{f:hunt} presents synthetic line profiles for Model 11d.\nResults are shown for the case of a constant H$_2$D$^+$\\ abundance and of\nan inner hole in the H$_2$D$^+$\\ distribution. These models have peaks at\nthe observed velocities, but with `infall asymmetry': the blueshifted\npeak is stronger than the redshifted one. The results of Model 11b\nare similar, but with this asymmetry more pronounced.\n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f9.ps}\n\\caption{Results of models with a power-law density distribution,\n $T_{\\rm kin}$$\\equiv$10~K, Hunter's velocity profile `11d', and with the H$_2$D$^+$\\ \n abundance constant (dotted line) and with an inner hole (dashed\n line), superposed on the observations (histogram).}\n\\label{f:hunt}\n\\end{figure}\n\n\\section{Effect of the collisional rate coefficient}\n\\label{s:crc}\n\nAnother potentially important parameter for the H$_2$D$^+$\\ line formation is the\ncollisional rate coefficient (CRC). To quantify the sensitivity of our model\nresults on the CRC, we have run models with the CRC increased and decreased by\nfactors of~3 and~10.\nThese models use the power-law density profile from \\citet{evans01},\nthe temperature profile from \\citet{young:h2co} and the H$_2$D$^+$\\\nabundance profile from \\citet{ceccarelli05}. \nSince the previous sections indicate that the details of the infall\nvelocity field play a minor role for our models, we simplify the\nkinematics to a `step function' in $b_D$, the turbulent line width.\nThe observations suggest that $b_D$ is larger in the (emitting)\ncentral part of the core than in the (absorbing) outer layers, so we\nadopt $b_D$=0.1~km~s$^{-1}$\\ at radii $<$10$^3$~AU and $b_D$=0 outside this\nradius. The model does not include any systematic radial motions, but\nthe increased $b_D$ at the center is presumed to arise in infall.\nThis setup implies the existence of infall motions, while the results\ndo not depend on the specific infall model. Note that thermal\nbroadening is added to the turbulence at all radii.\n\nFigure~\\ref{f:bdj} shows the results of models with the CRC increased\nby factors of 3 and 10 above the value from \\citet{black90}. The line\nstrength is seen to scale with the CRC; this trend continues for the\nmodels with the CRC decreased below `standard' which are not shown. \nThe shape of the H$_2$D$^+$\\ profile is seen not to depend on the CRC: both\nthe emission from the core and the absorption in the outer layers\nscale with the CRC.\nFor the adopted model parameters, the best fit to the observed H$_2$D$^+$\\\nprofile would be obtained by increasing the CRC by a factor of $\\approx$5.\nHowever, this number depends critically on several parameters,\nparticularly $n$(H$_2$), $T_{\\rm kin}$\\ and the H$_2$D$^+$\\ abundance.\nThis is why we refrain from recommending a value for the CRC at this point.\nEven though $n$(H$_2$) and $T_{\\rm kin}$\\ can be obtained from other observations \n(as we have done), the H$_2$D$^+$\\ abundance cannot.\nTherefore, for the time being, H$_2$D$^+$\\ abundances can only be\ndetermined to factors of $\\sim$10 accuracy. We recommend theoretical\ncalculations of the CRC of H$_2$D$^+$\\ to improve this situation.\n\nThe fact that the simplified kinematics give a better fit to the observed\nH$_2$D$^+$\\ profile than the models in Section~\\ref{s:alto} implies that the infall\nmotions toward LDN~1544\\ are very small, in good agreement with the conclusion\nfrom \\citet{williams:l1544} that the infall speed is much smaller than the\nthermal, rotational and gravitational speeds in this source.\nThe observed intensity at the 20$''$ offset position of $T_{\\rm mb}$=0.4~K is well\nreproduced by the same model that fits the central spectrum. The model predicts\na $\\approx$20\\% absorption at the line center, which the present data do not\nrule out. However, better spectra at offset positions are needed to confirm this\nprediction. \n\n\\begin{figure}[htb]\n\\includegraphics[width=8cm,angle=-90]{2792_f10.ps}\n\\caption{Results of models with the temperature density distributions\n from Young et al, a `jump' in $b_D$ at $R$=10$^3$~AU, and with the\n H$_2$D$^+$\\ collisional rate coefficient at the standard value and\n increased by factors of 3 and 10 (curves from bottom to top),\n superposed on the observations (histogram).}\n\\label{f:bdj}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{s:concl}\n\nRadiative transfer modeling of line profiles of H$_2$D$^+$, HC$^{18}$O$^+$\\ and\nN$_2$H$^+$\\ in the pre-stellar core LDN~1544 shows that previous\ndescriptions of the temperature, density and velocity structure, that\nreproduce HC$^{18}$O$^+$\\ and N$_2$H$^+$\\ data, do not fit H$_2$D$^+$\\ data, which probe\nsmaller radii.\nAt least two modifications are required at these radii, which affect\nH$_2$D$^+$, but not HC$^{18}$O$^+$\\ or N$_2$H$^+$.\n\nFirst, to make the excitation temperature of H$_2$D$^+$\\ increase inward,\nthe density cannot flatten off as proposed by \\citet{ward-t99}, but\nmust continue to increase as in the models by \\citet{evans01}. In\naddition, only very small inward decreases of the gas temperature are\nallowed. \nObservations of NH$_3$\\ toward a sample of pre-stellar cores also\nsuggest a constant kinetic temperature with radius (Tafalla et al.\\\n\\citeyear{tafalla02}, \\citeyear{tafalla04}).\nThe same result is obtained from observations of CO isotopes toward\nthe pre-stellar core B~68 \\citep{bergin05}, suggesting that current\nmodels of pre-stellar cores either underestimate the dust temperature\nbecause of changes in the dust opacity, or overestimate the gas-grain\nthermal coupling because of grain growth.\nHowever, the NH$_3$\\ and CO observations probe larger radii than H$_2$D$^+$,\nand interferometric observations of NH$_3$\\ would be valuable.\n\nSecond, the infall velocity needs to increase with radius down to\nsmall radii. If cloud collapse is quasi-static and mediated by\nambipolar diffusion, the central region supported by thermal pressure,\nwhere the infall speed drops to zero, must be smaller than in the\nmodels by \\citet{ciolek00}. Alternatively, the collapse may be\n`mildly' dynamic, indicated by values of 1--3 for the parameter $m_0$\n\\citep{hunter77,foster93}. Highly supersonic infall velocities, as\nwell as large-scale rotation, are ruled out.\n\nAlternatively, the central dip on the H$_2$D$^+$\\ profile is due to absorption\n in the outer parts of the core, as seen for CS, HCO$^+$ and N$_2$H$^+$. However,\n such absorption is not seen for DCO$^+$ 2--1, nor in H$_2$D$^+$\\ at offset\n positions. Also, to produce an absorption as narrow as observed, the outer\n layers of LDN~1544\\ must have essentially zero infall and turbulent motions,\n consistent with evidence from CS, HCO$^+$ and N$_2$H$^+$\\ \\citep{williams:l1544}.\n To test this `absorption' hypothesis, sensitive mapping of the H$_2$D$^+$\\ line and\n of DCO$^+$\\ 1--0 are needed.\n\nIn the future, making full use of H$_2$D$^+$\\ as kinematic probe requires\nmaps at higher sensitivity and spatial and spectral resolution than\nthe CSO can offer.\nOther single-dish telescopes (JCMT, APEX) can fulfill some of these\nrequirements, but not all.\nTherefore, for a real breakthrough, interferometric observations will be\ncrucial. The SMA will enable case studies, but only ALMA will have the\nsensitivity to probe many pre-stellar cores within reasonable times.\nThe interpretation of the data will require good observational\nconstraints of the \\textit{gas} temperature and density.\nIn addition, theoretical calculations of the\n collisional rate coefficients of H$_2$D$^+$\\ are urgently needed.\n\nIf the current observations cannot be matched with any of these\nmodifications, then we speculate that the doubled-peaked H$_2$D$^+$\\ line\nprofile in LDN~1544 is caused by the presence of two protostellar\ncondensations, orbiting each other, in the core center. We may be\nwitnessing the birth of a \\textit{binary} protostellar system. Once\nagain, interferometer observations are required to test this\nhypothesis.\n\n\\begin{acknowledgement}\n The authors thank Malcolm Walmsley, Arnaud Belloche, Neal Evans\n and the anonymous referee for useful comments on the\n manuscript, Glenn Ciolek and Kaisa Young for sending their model\n results in electronic form, and Aurore Bacmann and Antonio Crapsi\n for help with the observations. P.C.\\ acknowledges support from the MIUR\n grant `Dust and molecules in astrophysical environments'.\n\\end{acknowledgement}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nDisk accretion onto magnetic central objects occurs in a variety of\nastrophysical contexts, ranging from classical T Tauri stars (e.g.,\nBouvier et al.~2007) and cataclysmic variables (intermediate polars;\ne.g. Warner 2004), to accretion-powered X-ray pulsars (e.g. Lewin \\&\nvan der Klis 2006). Such accretion has been studied extensively since\nthe discovery of high-mass accreting X-ray pulsars in the 1970s (e.g.,\nPringle \\& Rees 1972). The basic picture of disk-magnetosphere\ninteraction is well known. The stellar magnetic field disrupts the\naccretion flow at the magnetospheric boundary and funnels the plasma\nonto the polar caps of the star or ejects it to infinity (e.g.,\nPringle \\& Rees 1972). The magnetosphere boundary is located where\nthe magnetic and plasma stresses balance,\n\\begin{equation}\nr_m =\\xi \\left({\\mu^4\\over GM\\dot M^2}\\right)^{1\/7}, \n\\end{equation}\nwhere $M$ and $\\mu$ are the mass and magnetic moment of the central\nobject, $\\dot M$ is the mass accretion rate and $\\xi$ is a\ndimensionless constant of order 0.5-1.\nRoughly speaking, the funnel flow occurs when $r_m$ is less than the corotation\nradius $r_c$ (where the disk rotates at the same rate as the\nstar). For $r_m\\go r_c$, centrifugal forces may lead to ejection of\nthe accreting matter (``propeller'' effect; Illarionov \\& Sunyaev 1975).\n\nHowever, beyond this simple ``Astro-101'' picture (with the estimate of\n$r_m$ given above), the situations are far more complicated. Over the last\nseveral decades, numerous theoretical studies have been devoted to\nunderstanding the interaction between accretion disks and magnetized\nstars. Many different models have been developed. Some examples are\n(see, e.g., Ghosh \\& Lamb 1979; Aly 1980; Lipunov \\& Shakura 1980;\nAnzer \\& B\\\"orner 1980,1983; Arons 1987,1993; Wang 1987,1995; Aly \\&\nKuijpers 1990; Spruit \\& Taam 1990,1993; K\\\"onigl 1991; Shu et al.~1994,2000; van\nBallegooijen 1994; Lovelace et al.~1995,1999; Li et al.~1996;\nWickramasinge \\& R\\\"udiger 1996; Campbell 1997; Lai 1998,1999; Terquem\n\\& Papaloizou 2000; Shirakawa \\& Lai 2002a,b; Pfeiffer \\& Lai 2004;\nUzdensky et al.~2002, Uzdensky 2004; Matt \\& Pudritz 2005; D'Angelo \\&\nSpruit 2010):\n\n$\\bullet$ Ghosh \\& Lamb (1979) assumed that the stellar fields invade\nthe disk over a large range of radii. Differential rotation between\nthe star and the disk will cause the field lines to wind up. They\nargued that a steady state can be reached, where the winding-up of the\nfield lines is balanced by dissipation, so that the field lines slip\nacross the disk. A problem with the GL picture is that in order to\navoid too much winding, they need a very large (and unrealistic) \nmagnetic diffusivity.\n\n$\\bullet$ In fact, the accretion disk is a very good conductor. So it\nis possible that the stellar magnetic field cannot penetrate the disk\nexcept near the magnetosphere boundary. This kind of models were first\nexplored by Arons, McKee \\& Pudritz (see Arons 1986). The field lines\nare strongly pinched and a centrifugally driven wind may come out from\nthe magnetosphere boundary. These ideas were worked out and expanded\nin more details in the so-called x-wind model of Shu et al.~(1994),\nwhich has a rich set of phenomenology and applications.\n\n$\\bullet$ Lovelace et al.~(1995) considered a model where the field\nlines connecting the star and disk becomes open because of the winding-up.\nSo there is an inner magnetosphere corotating with the star but also\nan open region where wind may come out.\n\n$\\bullet$ Matt \\& Pudritz (2005) developed a model where the possible\nrole of stellar wind driven by accretion was emphasized..\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12.5cm,clip]{f1.pdf}\n\\vskip -0.8truecm\n\\caption{Different theoretical models of magnetosphere-disk interaction (see text for \ndiscussion).}\n\\label{f1} \n\\end{figure}\n\nIn parallel to these theoretical studies, since 1990s there have been\nmany numerical simulations, with increasing sophistication, on this\nproblem (Hayashi et al.~1996,2000; Miller \\& Stone 1997; Goodson et\nal.~1997; Fendt \\& Elstner 2000; Matt et al.~2002; Romanova et\nal.~2003,2009). These simulations are playing an important role in\nelucidating the physics of magnetosphere-disk interaction in various\nastrophysical situations (see papers by Romanova and by Zanni in this\nvolume).\n\nIn this paper, we will not discuss any particular models or\nsimulations, which are covered by others. Instead, we will focus on\nseveral uncertain issues in the theory of magnetosphere-disk\ninteraction, and will also discuss their possible applications and\nrelevance in various astrophysical situations.\n\n\n\\section{Magnetospheric Boundary and Disk Inner Radius}\n\\label{sec-1}\n\nLet us first consider the problem of dipole magnetic field of a star\nbeing invaded by a conducting disk (Aly 1980). Note that an accretion disk is\nindeed a good conductor. MRI-driven turbulence gives rise to \nboth viscosity $\\nu$ and magnetic diffusivity $\\eta$, with the ratio\n(called magnetic Prantl number) $\\nu\/\\eta$ of order a few \n(see the simulation by Lesur \\& Longaretti 2009). \nWith the $\\alpha$-parameterization, $\\eta\\sim\\nu=\\alpha H c_s$, the diffusion time\nof magnetic field across the disk is $t_{\\rm diff}\\sim H^2\/\\eta\\sim 1\/(\\alpha\n\\Omega)$ (where $\\Omega$ is the disk rotation velocity), much longer than the\ndisk dynamical time. \n\nFor a perfectly conducting disk\/plate (with inner radius $r_{\\rm in}$\nsurrounding a dipole, the exact solution (Aly 1980) shows that the vertical\nfield at the disk inner edge is $B_z(r_{\\rm in})\\simeq B_\\star (r_{\\rm in})\n(r_{\\rm in}\/H)^{1\/2}$, where $B_\\star(r_{\\rm in})\\sim \\mu\/r_{\\rm in}^3$ is the\ndipole field at $r=r_{\\rm in}$ when the disk is absent.\n\nNow, at the inner edge of the disk, various instabilities (including\nmagnetic Kelvin-Helmholtz and Rayleigh-Taylor, and reconnection) can develop,\nleading to the formation of a boundary layer. The stellar dipole field\npenetrates this magnetospheric boundary layer and gets twisted to produce\n$B_{\\phi\\pm}=\\mp\\zeta B_z$, with $\\zeta\\sim 1$ (the upper\/lower sign specifies\nthe field above\/below the disk). In the boundary layer ($r_m0$ and the magnetic torque on\nthe star is positive, while for $r_m>r_c$, we have $T_m<0$.\n\nThe torque on the star is affected by the propeller effect. When the\naccreting plasma penetrates the magnetosphere and gets attached to the\nstellar field lines, it corotates with the star. If $r_m\\Omega_s$\nexceeds the escape speed $v_{\\rm esc}(r_m)= (2GM\/r_m)^{1\/2}$, or\nequivalently $r_m>2^{1\/3}r_c$, the plasma may be ejected due to centrifugal \nforce. The torque on the star should be modified to \n\\begin{equation}\n{dJ_\\star\\over dt}={\\dot M}_{\\rm acc} r_m^2\\Omega(r_m)+T_m-\\dot M_{\\rm eject}\nr_m^2\\Omega_s, \n\\end{equation}\nwhere $\\dot M_{\\rm eject}$ is the mass ejection rate. \nIn steady-state, we may expect $\\dot M_{\\rm acc}+\\dot M_{\\rm eject}=\\dot M$. \nPrecisely how much mass is ejected would depends on details of\ndissipation\/heating near the magnetosphere boundary, magnetic field\ntopology, etc., If the plasma is not ejected, it may accumulate near\nthe magnetosphere boundary, forming a dead disk (D'Angelo \\& Spruit\n2010,2012). Is there any accretion onto the star through interchange\ninstabilities? Determining $\\dot M_{\\rm acc}$ and $\\dot M_{\\rm eject}$\nin the canonical propeller regime is an important problem.\n\nOf course, the reason we are interested in the torque on the star is\nbecause we want to know what physics determines the equilibrium spin of\naccreting magnetic stars. This is relevant to young stars (T tauri stars), \naccreting millisecond pulsars as well as long-period pulsars.\nIn the standard picture, already developed in the 1970s and 1980s,\nthe equilibrium rotation equals, to within a factor of two, the Keplerian\nrotation rate at $r_m$. That is, equilibrium is reached when \n$r_c\\simeq r_m$. Although there may be various factors (accretion torque,\nmagnetic torque, propeller, wind\/outflows, etc) that compete to\ndrive the star's rotation, all these generally lead to the similar\nequilibrium condition ($r_c\\sim r_m$).\n\nMore constraining to the theory of magnetic star-disk interaction\nis the spinup\/spindown behavior of accreting x-ray pulsars.\nMany x-ray pulsars have been observed to exhibit changing spinup and \nspindown behaviors over timescale of years (Bildsten et al.~1997).\nFigure 3 shows two particularly striking examples: 4U1626-67 is an accreting\npulsar with spin period 7.66~s. The clean spinup before 1990.6 was followed\nby a clean spindown, and another spinup phase starting 2008.2. The \nspindown\/spinup transition lasted 150 days. The spinup rate $\\dot\\Omega_s$\nis positively correlated with the x-ray flux (Camero-Arranz et al.~2010,2012).\nIn GX 1+4, it was found that during the spindown phase, the magnitude of the\nspindown torque is positively correlated with the x-ray flux, \n$-\\dot\\Omega_s\\propto F_x^{0.030\\pm 0.07}$ (Chakrabarty et al.~1997;\nGonzalez-Galan et al.~2012). This latter correlation is difficult to explain.\nIn fact, it directly contradicts the prediction of the original Ghosh-Lamb\nmodel. Perhaps some variations of the models based on variable\n$\\dot M_{\\rm acc}$ and $\\dot M_{\\rm eject}$ can do the job, but exactly \nhow is not clear (see Locsei \\& Melatos 2004; Perna, Bozzo \\& Stella 2006;\nDai \\& Li 2006 for related works).\nUnderstanding this spindown\/spinup behavior and its correlation with\nthe accretion rate remains an outstanding unsolved problem.\n\n\n\\section{Outflows}\n\\label{sec-5}\n\nOutflows is a natural consequence of magnetosphere-disk interaction.\nWe have already touched upon two effects that may give rise to\noutflows. \n\n(1) The quasi-cyclic star-disk linkage (see Section 3) and\nassociated field inflation inevitably lead to ejection of plasma from\nthe magnetosphere boundary. This episodic outflows have been seen in\nsimulations (see Romanova et al. 2009; Zanni \\& Ferreira 2013), but may also be\nrelated to the X-wind model of Shu et al. (although the ``canonical''\nX-wind model was developed for the case when the star has already reached\nspin equilibrium, thus $r_m\\simeq r_c$, and the wind is steady).\n\n(2) In the propeller regime, plasma may be ejected from the magnetosphere.\nSuch outflow is distinct from the boundary outflow discussed in (1).\nIt maybe more more collimated and has a faster speed (see Lii et al.~2013\nfor recent simulations).\n\nIn addition to the above two outflow launching mechanisms, there are two other\npossibilities:\n\n(3) Accretion-induced stellar wind (see paper by R. Pudritz).\n\n(4) Outflow from the disk through magneto-centrifugal mechanism\na la Blandford-Payne. Perhaps such outflow is less collimated and\nhas a slow velocity.\n\nIt is possible that some of these mechanisms are related to each other and\ntherefore not distinguishable in practice (both in simulations and observations).\nObviously, it is important to confront all these mechanisms with observations.\nOutflows\/jets from young stellar objects are well known and discussed elsewhere. \nBut there are also observed in accreting neutron stars in binaries. Perhaps\nthe famous is the relativistic jet from Circinus X-1 (Fender et al.~2004;\nArmstrong et al.~2013),\na bright and highly variable x-ray binary containing a neutron star \non an eccentric orbit (17 days) with a normal star companion.\nIt is interesting to note that jets and outflows have been observed\nin different types of neutron star binaries, including Atolls sources \n(such as Aql X-1), Z-sources (Sco X-1) and accreting millisecond\nx-ray pulsars (SAX J1808). It is usually thought that the key difference\nin these sources lies in their magnetic field strengths, with accreting pulsars\nhaving the highest fields ($\\sim 10^9$~G).\n\nIt is also of interest to compare the jet phenomenology in black-hole x-ray\nbinaries and neutron-star x-ray binaries. In the case of black hole, compact steady\njets are produced in the low-hard state, when the innermost disk is \nreplaced by a geometrically thick flow\/corona; episodic\/ballistic jets are\nproduced in the transitional state between the low state and the high, thermal\nstate; no jets are produced in the ``clean'' thermal state. By contrast,\nfor neutron star systems, jets appear to be produced in the soft x-ray thermal\n(called ``banana'') states (for example, Ser X-1, 4U1820-30; see\nMigliari et al. 2011). The same can be said of young stars, which are always\nin the ``thermal disk'' state. Perhaps this difference points to the important\nrole of magnetosphere, which is absent in black-hole systems.\n\n\n\\section{Misaligned Stellar Fields}\n\\label{sec-6}\n\nThe physics and issues we have discussed so far are applicable\nregardless of whether the stellar magnetic dipole is aligned or misaligned\nwith the stellar spin axis. In general, the stellar dipole axis is misaligned\nwith the spin (this is required for accreting x-ray pulsars). Are there any new\neffects? The answer is yes. One obvious effect is that with misaligned magnetic\nfields, the plasma in the magnetosphere can flow to the polar cap more\neasily (the so-called ``funnel flow''), thereby producing pulsed emission \nfrom the stellar surface. With a very aligned stellar field, this may be difficult,\nand magnetosphere accretion may proceed through instabilities, so that the\nplasma may land on the stellar surface more in the equatorial region than \nthe polar caps. This obviously will have different observational consequences\n(see M. Romanova's paper in this proceedings).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12.5cm,clip]{f4.pdf}\n\\vskip -3.6truecm\n\\caption{Left panel: Bending waves in the disk excited by magnetic force from a\nmisaligned stellar dipole. The wave is most visible at the Lindblad\/vertical\nresonance (from Lai \\& Zhang 2008).\nRight panel: Power density spectrum of accreting millisecond pulsar \nSAX J1808.4-3658, showing the spin frequency $\\nu_s=401$~Hz and two\nkHz QPOs, with $\\nu_h-\\nu_l\\simeq \\nu_2\/2$ (from van der Klis 2005).}\n\\label{f4} \n\\end{figure}\n\nAnother effect of misaligned stellar dipole is that it can excite\nnon-axisymmetric waves in the disk through magnetic forcing (Lai \\&\nZhang 2008; see Romanova et al.~2013 for recent simulations).\nFor example, the vertical magnetic force on the disk can\nbe written as\n\\begin{equation}\nF_z(r,\\phi,t)=F_{\\omega}(r)\\exp(im\\phi-i\\omega t),\n\\end{equation}\nwhere $m=1$. The forcing frequency $\\omega$ can be either $\\Omega_s$ or\n$2\\Omega_s$. While $\\omega=\\Omega_s$ is obvious, the $2\\Omega_s$ component \nrequires some explanation. It arises because while the dipole field\nat the given point in the disk varies as $\\cos \\Omega_s t$, the induced\nscreening current in the disk (by virtue of its high conductivity) also varies\nas $\\cos \\Omega_s t$; thus the force has a term proportional $\\cos^2\\Omega_s t$.\n(Note that this term is present only when the disk is misaligned with the\nstellar spin; see below). This periodic vertical force will excite bending waves\nin the disk. These bending waves are launched at the Lindblad\/vertical\nresonance, where the Doppler-shifted forcing frequency\n$\\omega-\\Omega(r)$ matches the intrinsic disk vertical epicyclic\nfrequency $\\Omega_z\\simeq \\Omega$.\nThus, the perturbation is most ``visible'' at $r=r_L$ given by\n$\\Omega(r_L)=\\omega\/2=\\Omega_s\/2,\\Omega_s$. \n\nThese magnetically excited bending waves may have interesting\napplications to kHz QPOs observed in accreting millisecond pulsars\n(van der Klis 2005). In particular, in SAX J1808.4-3658 (with the spin\nfrequency $\\nu_s=401$~Hz), the two kHz QPOs frequencies both vary as\nthe source luminosity changes, but the difference frequency\n$\\nu_h-\\nu_l$ remains constant and equals $\\nu_s\/2$ to within a few\nHz. In XTE J1807.4-294 (with $\\nu_s=191$~Hz), the difference $\\nu_h-\\nu_l\n\\simeq \\nu_s$. It seems that these are not pure coincidences.\n(One should add that for most LMXBs which do not show coherent pulsations, \nthe difference frequency of kHz QPOs do not exhibit similar correlation with\nthe neutron star spin.) Perhaps the beating of high-frequency\nQPO ($\\nu_h$) generated at the magnetosphere (or inside) with the perturbed\ndisk plasma at the Lindblad\/vertical resonance can explain \nthese intriguing observations.\n\n\\section{Stellar Spin - Disk Misalignment}\n\\label{sec-7}\n\n\nIn the standard picture magnetic star-disk interaction, it is usually\nassumed that the stellar spin axis is aligned with the disk axis (the\ndisk normal vector). This seems reasonable since the star may have\ngained substantial angular momentum from the accreting gas in the\ndisk. However, magnetic interaction between the star and the inner\nregion of the disk may (if not always) change this simple picture\n(Lai 1999; Lai et al.~2011).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12.5cm,clip]{f5.pdf}\n\\vskip -5.0truecm\n\\caption{Left panel: A sketch of magnetic field configuration in a\n star-disk system for nonzero $\\beta$ (the angle between the disk\n axis and the stellar spin axis) and $\\theta_\\star$ (the angle\n between the stellar dipole axis and the spin axis). Part of the\n stellar magnetic fields (dashed lines) penetrate the disk in the\n interaction zone between the disc inner radius $r_{\\rm in}$ and\n $r_{\\rm int}$ in a cyclic manner, while other field lines are\n screened out of the disc. The closed field lines are twisted by the\n differential rotation between the star and the disc, which leads to\n a magnetic braking torque and a warping torque. The screening\n current in the disk leads to a precessional torque. Right panel:\n A toy model for understanding the origin of the warping torque. A\n tilted rotating metal plate (with angular momentum ${\\bf L}$) in an\n external magnetic field ${\\bf B}$ experiences a vertical magnetic\n force around region 2 and 4 due to the interaction between the\n induced current ${\\bf J}$ and the external ${\\bf B}_\\parallel$,\n resulting in a torque ${\\bf N}$ which further increases the tilt\n angle $\\beta$.}\n\\label{f5} \n\\end{figure}\n\nBefore being disrupted at $r_{\\rm in}$ by the stellar magnetic field, \nthe disk generally experiences nontrivial\nmagnetic torques from the star. These torques are of two types: (i) A\nwarping torque ${\\bf N}_w$ which acts in a small interaction region\n$r_{\\rm in} < r < r_{\\rm int}$ (with width $\\Delta r$),\nwhere some of the stellar field lines are\nlinked to the disk. These field lines are twisted by the differential\nrotation between the star and the disk, generating a toroidal field\n$\\Delta B_\\phi=\\mp \\zeta B_z^{(s)}$ from the quasi-static vertical\nfield $B_z^{(s)}$ threading the disk, where $\\zeta\\sim 1$ and the\nupper\/lower sign refers to the value above\/below the disk plane. Since\nthe toroidal field from the stellar dipole $B_\\phi^{(\\mu)}$ is\nsymmetric with respect to the disk plane, the net toroidal field\ndiffers above and below the disk plane, giving rise to a vertical\nforce on the disk. While the mean force (averaging over the azimuthal\ndirection) is zero, the uneven distribution of the force induces a net\nwarping torque which tends to push the orientation of the disk angular\nmomentum $\\hat{\\mbox{\\boldmath $l$}}$ away from the stellar spin axis $\\hat{{\\mbox{\\boldmath $\\Omega$}}}_s$.\nThe essential physics of the warping torque can also be understood\nfrom the ``laboratory'' toy model depicted in Fig.~5. (ii) A\nprecession torque ${\\bf N}_p$ which arises from the screening of the\nazimuthal electric current induced in the highly conducting disk. This\nresults in a difference in the radial component of the net magnetic\nfield above and below the disk plane and therefore in a vertical force\non the disk. The resulting precession torque tends to cause $\\hat{\\mbox{\\boldmath $l$}}$\nto precess around $\\hat{\\mbox{\\boldmath $\\Omega$}}_s$.\n\nThe two magnetic torques (per unit area) on the disk can be written as\n\\begin{equation}\n{\\bf N}_w = -(\\Sigma r^2\\Omega)\\cos\\beta\\,\n\\Gamma_w \\,\\hat{\\mbox{\\boldmath $l$}}\\times(\\hat{{\\mbox{\\boldmath $\\Omega$}}}_s\\times\\hat{\\mbox{\\boldmath $l$}}),\\qquad\n{\\bf N}_p=(\\Sigma r^2\\Omega)\\cos\\beta\\,\n\\Omega_p \\,\\hat{{\\mbox{\\boldmath $\\Omega$}}}_s\\times\\hat{\\mbox{\\boldmath $l$}},\n\\label{eq:torque}\\end{equation}\nwhere $\\Sigma(r)$ is the surface density, $\\Omega(r)$ the\nrotation rate of the disk, and $\\beta(r)$ is the disc tilt angle (the angle\nbetween $\\hat{\\mbox{\\boldmath $l$}} (r)$ and the spin axis $\\hat{\\mbox{\\boldmath $\\Omega$}}_s$).\nThe warping rate and precession angular frequency at radius $r$ are\ngiven by\n\\begin{equation}\n\\Gamma_w (r)=\\frac{\\zeta\\mu^2}{4\\pi r^7\\Omega(r)\\Sigma(r)}\\cos^2\\theta_\\star,\n\\qquad\n\\Omega_p (r)=-\\frac{\\mu^2}{\\pi^2 r^7\\Omega(r)\\Sigma(r)\n D(r)}\\sin^2\\theta_\\star,\n\\label{eqn:Omega_w}\n\\end{equation}\nwhere $\\theta_\\star$ is the angle between the magnetic dipole axis and the\nspin axis, and the dimensionless function $D(r)$ is somewhat less than unity.\n\n\\subsection{Applications to QPOs\/Variabilities}\n\nThe magnetically driven warping and precession effects discussed\nabove may be relevant to the low-frequency (10-50~Hz)\nQPOs (the so-called horizontal branch oscillations)\nobserved in neutron star LMXBs. Stella \\& Vietri (1998) suggested that LFQPOs\nare associated with the Lense-Thirring precession of the inner\naccretion disk around the rotating neutron star. If the LFQPO and\nthe kHz QPOs are generated at the same special radius in\nthe disk, this implies a quadratic relation between the\nLFQPO frequency and the kHz QPO frequency, in rough\nagreement with observations (usually for a certain range of\nX-ray fluxes). However, in order to have precession, the inner disk\nmust be tilted relative to the stellar spin. The magnetic effects may induce\nthe required disk warp and contribute the the disk precession\n(Shirakawa \\& Lai 2002a, Pfeiffer \\& Lai 2004).\n\nMagnetically driven warping and precession may also explain\nmillisecond variabilities in high-field accreting X-ray pulsars \n(Shirakawa \\& Lai 2002b) as well as some photometric variabilities observed in\nT Tauri stars (such as AA Tau; see Terquem \\& Papaloizou 2000;\nBouvier et al.~2013).\n\n\\subsection{Possible Connection to (Exo)Planetary Systems}\n\nOne of the recent surprises in exoplanetary sciences is that many hot\nJupiter systems (orbital periods of order a few days) are found to\nhave a misaligned orbital axis relative the spin axis of the host\nstar. Among the several dozens of hot Jupiter systems with\nmeasurements the stellar obliquity, about $60\\%$ have an orbital axis\naligned (in sky projection) with the stellar spin, while the other\nsystems show a significant spin-orbit misalignment, including some\nwith retrograde orbits.\n\nIn our own Solar System, except for Pluto, all planets\noutside 1~AU lie within 2$^\\circ$ of the ecliptic plane, while the\nSun's equatorial plane is inclined by 7$^\\circ$ with respect to the\necliptic. It is not clear whether the difference between 2$^\\circ$ and\n7$^\\circ$ is significant or needs an explanation.\n\nThe observed spin-orbit misalignment in many hot Jupiter systems\nindicates that gravitational interactions \nbetween multiple planets and\/or secular interactions with a distant\nplanet or stellar companion play an important role \nin shaping the architecture of planetary systems.\nNevertheless, there is an alternative possibility: misaligned\nprotostar and protoplanetary disk. Such misaligned disk may explain the\n$7^\\circ$ ``anomaly'' in Solar System and affect the initial condition \nof planetary formation.\n\nAs discussed above, the inner region of the disk around a magnetic protostar \nexperiences a warping torque and a precessional torque.\nIf we imagine dividing\nthe disk into many rings, and if each ring were allowed to behave\nindependent of each other, it would be driven toward a perpendicular state\nand precess around the spin axis\nof the central star. Obviously, real protoplanetary disks do not\nbehave as a collection of non-interacting rings: Hydrodynamic stresses\n(bending waves and viscosity) provide strong couplings between different rings,\nand the disk will resist the inner disk warping. Detailed calculations\nby Foucart \\& Lai (2011) showed that for most reasonable stellar\/disk\nparameters, the steady-state disk warp is rather small because of\nefficient viscous damping or propagation of bending waves. Thus the inner\ndisk direction is approximate the same as the outer disk direction.\n\nWhat is happening to the stellar spin? Is there a secular change to\nthe stellar spin direction? The answer is maybe. The back-reaction\nmagnetic torque from the disk on the stellar spin can push the stellar\nspin axis away from alignment. The magnetic misalignment\ntorque on the star is of the same order as the fiducial accretion torque\n$\\dot M\\sqrt{GM_\\star r_{\\rm in}}$. Assuming the spin angular momentum\n$J_s=0.2M_\\star R_\\star^2\\Omega_s$, we find the spin evolution time to be\n\\begin{equation}\nt_{\\rm spin}=(1.25\\,{\\rm Myr})\\left(\\!{M_\\star\\over 1\\,M_\\odot}\\!\\right)\n\\!\\!\\left({{\\dot M}\\over 10^{-8}{M_\\odot}{\\rm yr}^{-1}}\\right)^{\\!-1}\n\\left(\\!{r_{\\rm in}\\over 4R_\\star}\\!\\right)^{\\!-2}\n\\!\\!{\\Omega_s\\over\\Omega(r_{\\rm in})}.\n\\label{eq:tspin}\\end{equation}\nGiven enough time, the stellar spin axis will evolve towards the\nperpendicular state and even the retrograde state. Unfortunately,\nthere are other torques acting on the star which counter-act the\nwarping torque. In particular, the accretion torque tends to align the\nstellar spin axis with the disk. Since the magnetic torque and the\naccretion torque are of the same order of magnitude, we cannot be\nmake a definitive conclusion. All we can say is that both alignment\nand misalignment between the stellar spin axis and the disk axis are\npossible (see Lai, Foucart \\& Lin 2011 and Foucart \\& Lai 2011 for more details).\n\n{\\bf Acknowledgments:}\nI thank the organizers for the invitation and support for me \nto attend the conference on the Physics at the Magnetosphere Boundary\nin Geneva. This work has been supported in part by NSF grants\nAST-1008245, 1211061 and NASA grant NNX12AF85G.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDust grains are the building blocks of planets, but there are relatively few ways to accurately characterize their properties. Studies of solar system bodies provide the strongest constraints on the constituents of comets or asteroids (e.g., \\citealp{Frattin2019}, \\citealp{Bertini2019}), but do not inform us directly on what is taking place during the planet formation stage. Observations of disks around young ($\\leq 100$\\,Myr old) stars are sensitive to grain sizes that are smaller than typically $100$\\,$\\mu$m or a few mm. Debris disks, disks of second generation dust, are ideal targets to study dust grains. As those disks are optically thin at all wavelengths, we have direct access to the absorption and scattering properties of the grains, without having to account for non trivial optical depth effects or multiple scattering events. There are two possible ways to characterize dust properties in debris disks, first, via their thermal emission by measuring the spectral slope at (sub-) mm wavelengths (\\citealp{Draine2006}, \\citealp{MacGregor2016}, constraining the slope of the grain size distribution or the maximum grain size), or modelling their emission features in the mid-IR (\\citealp{Olofsson2012}, informing about the dust composition ). The second avenue is to study how stellar light is scattered off of the dust grains (in total intensity or polarized light, at optical or near-infrared wavelengths), either by measuring the colour of the disk between different bands (\\citealp{Debes2008}, \\citealp{Rodigas2105}), or studying the phase function (e.g., \\citealp{Olofsson2016}; \\citealp{Milli2017,Milli2019}, \\citealp{Ren2019}). Both approaches can bring constraints on the typical grain sizes as well as their porosity. The phase function informs us how efficiently the light is scattered as a function of the scattering angle (between the star, the dust grain and the observer). This approach requires the disk to be spatially resolved and therefore became more popular in the past years with the availability of high angular resolution instruments such as VLT\/SPHERE \\citep{Beuzit2019} or GPI \\citep{Perrin2015}, but pioneering works were led with \\textit{Hubble Space Telescope} observations as well (e.g., \\citealp{Graham2007}, \\citealp{Stark2014}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\hsize]{opang.pdf}\n \\caption{Convolved synthetic images of a debris disk with increasing opening angle. From left to right, $\\psi = \\mathrm{arctan}(h\/r) = 0.01$, $0.03$, $0.05$, $0.07$ (see text for further details).}\n\\label{fig:opang}\n\\end{figure*}\n\nThere are two different methods to estimate the phase function of debris disks, either from total intensity or polarized light observations, and in this study, we will focus mostly on polarimetric observations. Out of the two approaches, the first one consists of fitting a model to the observations and the phase function is a free parameter of the modelling, either using scattering theory such as Mie or more complex ones (and in that case, the free parameter(s) mostly govern the grain size distribution or the porosity), or using parametrized approximations such as the Henyey-Greenstein one (\\citealp{Henyey1941}). The second approach being to measure the surface brightness of the disk as a function of the scattering angle, without the use of a model. The first approach requires a disk model for the dust density distribution as well as a scattering theory (or an approximation) that is able to reproduce the true phase function, while the second approach does not account for changes in column density at different azimuthal angles (as is the case along the semi-major axis of inclined disks). One should note that in this method, the final phase function is not exactly equal to the surface brightness as several correction factors have to be applied (e.g., illumination effects, point spread function dilution, and aperture shape, \\citealp{Milli2019}). \n\nThe phase function is best measured for disks with a significant inclination (but not perfectly edge-on as most of the azimuthal information is lost), as a wide range of scattering angles can be probed. Inclinations around $75^{\\circ}$ are ideal, but as a consequence, there is an increase in column density along the major axis, if the disk is not infinitely flat. This increase in column density is illustrated in Figure\\,\\ref{fig:opang} showing disk models computed with different opening angles (increasing from left to right, see Section\\,\\ref{sec:model} for how the images are computed). The models shown in Fig.\\,\\ref{fig:opang} all have an isotropic phase function and therefore directly trace the dust column density. One can note that as the opening angle increases, the major axis of the disk becomes brighter compared to the minor axis. Therefore when retrieving the phase function by measuring the surface brightness as a function of the scattering angle, one has to account for the column density variations along the disk. We here present a novel approach to retrieve the phase function in debris disks, in a non-parametric way, but that remains model-dependent on the density distribution throughout the disk.\n\n\n\\section{Determining the phase function}\n\nIn this Section, we describe our new approach to determine the phase function in a non-parametric way, and briefly present the observations used to test it beforehand.\n\n\\subsection{Observations}\n\nBecause of the brightness of the disk around HR\\,4796, we use the SPHERE\/ZIMPOL polarimetric observations presented in \\citet{Milli2019} and \\citet{Olofsson2019} to illustrate our method. HR\\,4796 is a young ($8\\pm2$\\,Myr old, \\citealp{Stauffer1995}) nearby ($71.9\\pm0.7$\\,pc, \\citealp{Gaia2018}) A-type star surrounded by one of the brightest debris disk ($L_\\mathrm{disk}\/L_\\star \\sim 5 \\times 10^{-3}$, \\citealp{Moor2006}). We used the $Q_\\phi$ and $U_\\phi$ images presented in \\citet{Milli2019} as the signal to noise ratio is larger close to the semi-minor axis, at the expense of some small artefacts along the semi-major axis. The observations were obtained without a coronagraph and the reader is referred to \\citet{Milli2019} for more information on the observing sequence and data processing. For all the calculations described below, we mask all the points that are within $0\\farcs35$ from the estimated position of the star and all the pixels that are outside of an elliptical mask with a semi-major axis of $1\\farcs25$ (see left panel of Fig.\\,\\ref{fig:data}) are excluded when computing the goodness of fit.\n\n\\subsection{Synthetic disk images}\\label{sec:model}\n\nWe use an updated version of the code presented in \\citet{Olofsson2016,Olofsson2018} which can quickly produce images of (eccentric) debris disks which are not infinitely flat. To summarize briefly, the density distribution is computed as\n\\begin{equation}\\label{eqn:nr}\nn(r, z) \\propto \\left[\\left(\\frac{r}{r_0}\\right)^{-2\\alpha_{\\mathrm{in}}} + \\left(\\frac{r}{r_0}\\right)^{-2\\alpha_{\\mathrm{out}}}\\right]^{-1\/2} \\times \\mathrm{e}^{-z^2\/2h^2},\n\\end{equation}\nwhere $n$ is the volumetric density, $r_0$ is a reference radius, $\\alpha_\\mathrm{in}$ and $\\alpha_\\mathrm{out}$ are the slopes of the density distribution and $h$ is the vertical height of the disk (parametrized with the opening angle $\\psi$ such as $\\mathrm{tan}\\,\\psi = h\/r$). For eccentric disks, parametrized with two free parameters, the eccentricity $e$ and the argument of pericenter $\\omega$, the reference radius $r_0$ depends on the azimuthal angle $\\gamma$ such as\n\\begin{equation}\\label{eqn:r0}\n r_0 = \\frac{a(1-e^2)}{1 + e \\mathrm{cos}(\\omega + \\gamma)},\n\\end{equation} \nwhere $a$ is the semi-major axis, $\\gamma = \\mathrm{arctan2}(y,x)$ is the azimuthal angle in the midplane\\footnote{We use arctan2 to place the resulting angle in the proper quadrant when both $y$ and $x$ are negative for instance.}, and $x$ and $y$ are the pixels coordinates (with the origin at the centre of the image) after projecting for the inclination $i$ and rotating according to the position angle $\\phi$. To produce images, the code first defines a bounding box, sufficiently large for the volume density to be negligible at the borders. For each pixel of the image, the entry and exit points of the bounding box are calculated, and that ``column'' is divided into $m = 100$ equal parts of the same volume $V$. For each cell, the volume density is computed following Eq.\\,\\ref{eqn:nr}, the scattering angle $\\theta$ is computed from the dot product between the unit vector along the line of sight and the $3$D coordinates at the centre of the cell, and the flux will be the product of the volume density, $V$, and the scattering $S_{11}(\\theta)$ (or polarized $S_{12}(\\theta)$) phase function. For each pixel, the final flux is the sum over the $m$ cells. The user can provide a $2$D array for $S_{12}$ (or $S_{11}$), a $1$D array for $\\theta$ (between $0$ and $\\pi$) and the code will interpolate at the proper scattering angle when computing an image. The array for the phase function is $2$D so that there can be one phase function for the north side and a different one for the south side. Both sides are identified based on the sign of the azimuthal angle (which ranges between $-\\pi$ and $\\pi$). One should note that the code is not flux-calibrated, therefore, the total dust mass or the polarization degree are not mandatory input parameters. \n\nTo compare the synthetic images to the observations, we follow the approach explained in the Appendix of \\citet{Engler2018}. To summarize briefly, the modelled $Q_\\phi$ image is decomposed in $Q$ and $U$ images, according to the polar coordinates on the detector. Then both images are convolved with a 2D normal distribution with a full width at half maximum (FWHM) of $34$\\,mas ($\\sigma = 2$\\,pixels, comparable to the observing conditions as reported in \\citealp{Milli2019}). The convolved $Q$ and $U$ images are then combined to obtain the final $Q_\\phi$ image (see \\citealp{Engler2018} for further details). \n\n\\subsection{Description of the approach}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\hsize]{data_model.pdf}\n \\caption{Observations, best-fit model with an istropic phase function, final best-fit model with the revised phase function, and residuals to the SPHERE\/ZIMPOL observations of HR\\,4796\\,A, with the same linear stretch (from left to right). The goodness of fit is estimated between the inner circle and the elliptical mask shown in the leftmost panel. In the central right panel, the location of the pericenter is marked with a circle.}\n\\label{fig:data}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\hsize]{surface_b.pdf}\n \\caption{Surface brightness of the north side of the disk, as a function of the scattering angle for the model with an isotropic phase function and for the observations (left and right, respectively). Each black circle represents a pixel. The red curves correspond to the median in each bin of $\\theta$.}\n\\label{fig:surface}\n\\end{figure*}\n\nThe motivation of our approach is to decouple the geometric parameters of the disk (e.g., radius, inclination, eccentricity) from the dust properties (the phase function in this case). This is achieved by comparing a first model computed using an isotropic phase function, which solely traces the dust density distribution, to the observations. By computing the ratio of surface brightness between this first model and the observations one can infer a revised phase function that should account for most of the differences. This revised phase function can then be injected in a new model to be compared to the observations. \n\nWe follow a three-step approach; for a given set of parameters we compute a first model with an isotropic phase function for both north and south sides (with the same number of pixels, and the same pixel scale as the observations, $7.2$\\,mas per pixel) to trace the density distribution as a function of the scattering angle. Because the code is not flux-calibrated, we first find the best scaling factor $S_\\mathrm{scale}$ that minimizes the difference between the model and the observations, as \n\\begin{equation}\\label{eqn:scale}\n S_\\mathrm{Scale} = \\frac{\\sum\\frac{I_\\mathrm{obs} \\times I_\\mathrm{model}}{\\sigma^2}}{\\sum\\left(\\frac{I_\\mathrm{model}}{\\sigma}\\right)^2},\n\\end{equation}\nwhere $I_\\mathrm{model}$ and $I_\\mathrm{obs}$ are the images of the model and the $Q_\\phi$ image, while $\\sigma$ are the uncertainties estimated from the $U_\\phi$ image (see next Section, and see Section\\,\\ref{sec:adi} for a discussion on total intensity observations). The purpose of this first scaling is to try to account for most of the (unknown) multiplicative factors that govern the total flux of the disk (e.g., total dust mass, albedo).\n\nThen, for each side of the disk (north and south) , we estimate the surface brightness of both the model and the observations as a function of the scattering angle in the midplane of the disk. Given that the resulting distribution can be quite noisy for the observations we apply a numerical mask, selecting only the pixels where the model is brighter than $0.45$ times its peak brightness. Given that we use an isotropic phase function for this first model, the whole ring is recovered, for a wide range of scattering angles. \n\nThe second step of the approach is to bin the surface brightness as a function of the scattering angle, for both the model and the observations. The binning is performed over $50$ linearly spaced bins, and for each bin we compute the median value. For the observations, we also compute the median absolute deviation $\\sigma_i$ in each bin. Afterwards, we average the resulting distributions using a running mean over $5$ neighbouring bins, to smooth them. For the observed profile, the corresponding uncertainties are estimated as\n\\begin{equation}\\label{eqn:uncer}\n \\sigma_\\mathrm{binned} = \\left( \\sum_{i=0}^N \\frac{1}{\\sigma_i^2} \\right)^{-1\/2},\n\\end{equation}\nwhere $N = 5$ is the number of neighbouring bins.\n\nThis process is illustrated in Figure\\,\\ref{fig:surface} (the left panel shows an asymmetry as the disk model is slightly eccentric). Given the mask that excludes the central $0\\farcs35$, we are able to probe scattering angles in the range $[22^{\\circ}, 163^{\\circ}]$ on both the north and south sides. The polarized phase function is estimated from the ratio between the functions representing the observations and the model (propagating the uncertainties at the same time), in other words, the red curve of the right panel divided by the red curve in the left panel of Figure\\,\\ref{fig:surface}. The scaling factor $S_\\mathrm{scale}$ of the isotropic model accounts for most of the differences with the observations but this model is not necessarily a good match, and as a consequence the $y$-axis of Figure\\,\\ref{fig:surface} may be different. When computing the ratio between the two averaged surface brightness there may still be some contribution from unconstrained ``flux-calibration'' factors and those factors will be incorporated in the resulting phase function, which is therefore a scaled (up or down) version of the true phase function\\footnote{One could normalize the phase function over $4\\pi$ steradians but since we do not probe the full range of scattering angles, this would remain an approximation}. \n\nWe then compute the final model with the newly evaluated phase function. Once the second model is computed we need to find the new best scaling factor $S_\\mathrm{Scale}$ that minimizes the $\\chi^2$ following Eq.\\,\\ref{eqn:scale}. One should note that computing the second model is only necessary to compare the model to the observations and estimate a goodness of fit to find the best parameters of the disk model. \n\nFrom Figure\\,\\ref{fig:surface}, one can see that by estimating the phase function from the observed surface brightness, for a non flat disk, we over-estimate it quite significantly at scattering angles near $90^{\\circ}$ (or under-estimate it a smaller angles), especially for highly inclined disks (e.g., \\citealp{Olofsson2016}, \\citealp{Milli2019}). \n\n\\section{Modelling and results}\n\n\\begin{table}\n\\caption{Free parameters and best-fit results.}\n\\label{tab:grid}\n\\centering\n\\begin{tabular}{lcc}\n\\hline\\hline\nParameters & Prior & Best-fit \\\\\n\\hline\n$a$ [$\\arcsec$] & [$0.90$, $1.15$] & $1.066\\pm0.001$ \\\\\n$i$ [$^{\\circ}$] & [$70$, $80$] & $77.60\\pm0.06$ \\\\\n$\\alpha_\\mathrm{out}$ & [$-15$, $-5$] & $-11.78\\pm0.20$ \\\\\n$e$ & [$0.0$, $0.1$] & $0.026\\pm0.002$ \\\\\n$\\omega$ [$^{\\circ}$] & [$-180$, $-90$] & $-147.8\\pm5.2$ \\\\\n$\\psi$ [$^{\\circ}$] & [$0.005$, $0.06$] & $0.035\\pm0.001$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTo determine most accurately the shape of the polarized phase function, we modelled the observations, trying to constrain the most relevant parameters of the disk. We put a strong emphasis on free parameters that can have an effect on the local increase in column density along the major axis. Therefore, the free parameters are the semi-major axis $a$, the inclination $i$, the outer slope of the density distribution $\\alpha_\\mathrm{out}$ ($\\alpha_\\mathrm{in}$ being fixed to $+25$), the eccentricity $e$, the argument of periapsis $\\omega$, and the opening angle $\\psi$. The position angle of the disk has already been well constrained for this dataset and we therefore use a value of $-152.1^{\\circ}$ following the results presented in \\citet{Olofsson2019}. The uncertainties are estimated from the $U_\\phi$ image, computing the standard deviation in concentric annuli of $2$\\,pixel width. Neither the $Q_\\phi$ nor the $U_\\phi$ images are convolved. Overall, since our model is convolved with a point spread function representative of the observations (following the approach outlined in \\citealp{Engler2018}), that illumination effects are naturally accounted for in the model, and that we do not use an aperture to measure the surface brightness profiles, we do not need to apply correction factors such as the ones mentioned in the introduction and described in \\citet{Milli2019}. \n\nTo identify the most probable solution, we use the \\texttt{MultiNest} algorithm (\\citealp{Feroz2009}) interfaced to \\texttt{Python} via the \\texttt{PyMultiNest} package (\\citealp{Buchner2014}). The probability distributions are plotted using the \\texttt{corner} package \\citep{corner} and are presented in Figure\\,\\ref{fig:corner}. The best-fit values and their uncertainties are reported in Table\\,\\ref{tab:grid}, while the best-fit model and the residuals are presented in the centre right and right panels of Figure\\,\\ref{fig:data} (the model with the isotropic phase function is shown in the centre left panel). One should note that the uncertainties reported in Table\\,\\ref{tab:grid} are most likely under-estimated and should be taken with some caution. This is most likely due to the uncertainties derived from the $U_\\phi$ image used to compute the goodness of fit. By measuring the standard deviation in concentric annulii, a strategy commonly used in direct imaging studies, we may be under-estimating the true uncertainties, yielding larger $\\chi^2$ values. As a consequence, the Monte-Carlo algorithm may explore a narrower range of values, leading to narrow probability distributions. \nFinally, the polarized phase functions for the north and south sides of the disk are shown in Figure\\,\\ref{fig:pfunc}, in black and dashed red, respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{pfunc.pdf}\n \\caption{Polarized phase function for the north and south sides (black and dashed red, respectively) as a function of the scattering angle.}\n\\label{fig:pfunc}\n\\end{figure}\n\nWe find that the semi-major axis is well constrained at $a=1\\farcs066 \\pm 0.001$, the inclination is $i = 77.60^{\\circ}\\pm0.06$, the outer slope of the density distribution is $\\alpha_\\mathrm{out} = -11.8\\pm0.2$, the eccentricity $e = 0.026\\pm0.002$, the argument of periapsis is $\\omega = -147.8^{\\circ}\\pm5.2$ (and its location is shown in the middle panel of Fig.\\,\\ref{fig:data}), and finally, we find that the opening angle is $\\psi = 0.035\\pm0.001$. From Figure\\,\\ref{fig:corner}, the degeneracy between $e$ and $\\omega$ is clearly noticeable, which explains why we find a smaller value for $e$ compared to the results of \\citet{Milli2017,Milli2019}, \\citet{Olofsson2019} who found the argument of periapsis to be closer to the projected semi-minor axis of the disk.\n\n\\section{Discussion}\n\n\\subsection{Residuals and caveats}\n\nThe residuals image overall shows that most of the signal from the disk has been removed, especially in the south side of the disk. There is still some signal left in the northern side, and this implies that the smoothed surface brightness profile that we estimated from the observations may not accurately capture the true surface brightness distribution of the northern side. As pointed out in \\citet{Olofsson2019}, the brightness asymmetry along the two sides cannot be solely explained by pericenter glow (\\citealp{Wyatt1999}) and that there may be an over-density of small dust grains at the north side of the disk. Our modelling approach is blind to local increase in dust density at different azimuthal angles, and there is no easy way around this issue. Nonetheless, the fact that the phase functions are quite similar between the north and south sides is quite reassuring, as one would not expect to have very different dust grains in different places of the disk (i.e., not surviving half an orbit). If indeed, there is an over-density of small dust grains along the north side as suggested in \\citet{Olofsson2019}, then the slight bump at $90^{\\circ}$ may not be real, and the true phase function may be more similar to the one of the south side.\n\n\\subsection{Other attempt at determining the phase function}\n\nThe approach presented in this study relies on some assumptions, for instance, how to estimate the surface brightness profiles of the model and of the observations. We originally attempted at least another approach that would have circumvented some of those issues, and we briefly discuss it here.\n\nWith the parameters of the disk fixed ($a$, $i$, etc), we sampled the phase function over a small number of angles ($10$), and tried to fit the actual values of the phase function, without any prior nor parametrization. The idea being that the code should find the shape of the phase function that minimizes the $\\chi^2$. Unfortunately, the fitting never really converged. We postulate that the main issue with this approach is that we are trying to minimize second order effects. The $\\chi^2$ is mostly dominated by the geometric shape of the ring, and small changes in the shape of the phase function yields very small changes in the final $\\chi^2$ values. One possible work-around could be to work with relative $\\chi^2$ values, but fine-tuning the evaluation of the goodness of fit may not be that trivial, and overall, we deemed this possible solution out of the scope of this paper.\n\nAnother alternative possibility, as mentioned in the introduction, would be to assume a parametric form for the phase function with a handful of free parameters (e.g., weighted sums of several Henyey-Greenstein functions, or a polynomial form). However, the challenge of such an approach would be to estimate when to increase the complexity of the form and when to stop. With our method, for a given set of disk parameters (e.g., $a$, $e$, $i$), the phase function that we retrieve is the best phase function that would minimize the goodness of fit (but it does not necessarily mean that it is a good solution).\n\n\\subsection{When does this matter?}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\hsize]{map.pdf}\n \\caption{Map showing the surface brightness ratio between the major and minor axis of disk models with an isotropic phase function, as a function of the inclination and opening angle of the disk.}\n \\label{fig:map}\n\\end{figure}\n\nTo quantify when the column density variations make a significant difference, we compute a grid of models using an isotropic phase function. We then compute the synthetic surface brightness as a function of the scattering angle similarly to the left panel of Figure\\,\\ref{fig:surface}. For each model we estimate the ratio between the maximum and minimum values of the profile. In the grid, we explore the two parameters that have the most important impact on the density increase; the inclination $i$ and opening angle $\\psi$. The inclination ranges between $30$ and $85^{\\circ}$, and the opening angle ranges between $0.01$ and $0.06$ radians. The semi-major axis, position angle, $\\alpha_\\mathrm{in}$, and $\\alpha_\\mathrm{out}$ are set to the same values as before. To simplify the problem, we set $e=0$ (therefore $\\omega$ no longer matter). Figure\\,\\ref{fig:map} shows the ratio of density enhancement between the major and minor axis of the disk for the grid. For disks that have an opening angle of $0.04$, the effect starts to become significant for inclinations larger than $\\sim 60^{\\circ}$.\n\n\\subsection{Is the disk around HR\\,4796 vertically thin?}\\label{sec:flat}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\hsize]{map_height.pdf}\n \\caption{Map showing the ratio of the FWHM along the semi-major and semi-minor axis of disks models, as a function of the outer slope of dust density distribution and the opening angle of the disk.}\n \\label{fig:height}\n\\end{figure}\n\n\n\\citet{Milli2019} injected the phase function they inferred into a disk model and obtained residuals that are comparable to the ones of Figure\\,\\ref{fig:data}, but assumed a vertically thin disk (vertical height of $1$\\,au at a reference radius of $77.4$\\,au). Following the description laid out in \\citet{Augereau1999}, this would translate into an opening angle of $\\sim 0.009$ for the code used in this work. Therefore, the true shape of the phase function depends on whether the disk is vertically flat or not.\n\n\\citet{Kennedy2018} modelled ALMA observations but could not firmly conclude on the vertical height of the disk. They mentioned that the disk could be vertically resolved, with a typical height of $\\sim 10$\\,au at a radius of $80$ \\,au, but that a flat disk was also consistent with their observations. Given that a dynamical cold disk would be vertically thin (\\citealp{Thebault2008}, but see also \\citealp{Thebault2009}), and would explain its narrowness, the authors remained cautious and the actual vertical thickness of the disk remains a matter of debate. \n\nNonetheless, \\citet{Thebault2009} argued that debris disks should have a minimum aspect ratio $H\/r$ of $0.04\\pm0.02$ (where $H$ is the local half width at half maximum) and that disks are most likely stratified for different grain sizes; the smallest dust grains having a larger aspect ratio compared to larger grains. Therefore, debris disks may appear vertically thicker in scattered light observations than at millimetre wavelengths. They also argue that the disk vertical height cannot directly be related to its dynamical excitation. Converting our best-fit value for the opening angle to the aspect ratio as defined in \\citet{Thebault2009}, we obtain a value of $0.041$, consistent with their results. \n\nFor an inclined disk, the width measured along the major and minor axis should depend on the vertical thickness; if the disk is vertically flat the minor axis should appear narrower than the major axis due to the inclination, as illustrated in Figure\\,\\ref{fig:opang}. To better quantify this, we computed several models, with the same spatial resolution as the observations, varying the following two free parameters, the opening angle $\\psi$ and the outer slope of the dust density distribution $\\alpha_\\mathrm{out}$ (which governs the width of the disk). For those models the semi-major axis and the inclination are the ones of the best fit model, and we used an isotropic phase function (in App.\\,\\ref{sec:app_pf} we repeat the same exercise for a different phase function to test the impact of this choice). Each image is convolved as explained in the previous Section. We measure the FWHM along the major and minor axis of the disk model and compute their ratio. Figure\\,\\ref{fig:height} shows how the ratio varies as a function of the two free parameters. While the slope of the dust density distribution has a small effect on the ratio of FWHM, the opening angle has the strongest impact. Overall, this suggests that the angular resolution of the observations is sufficient to constrain the height of the disk, and that this can in principle be done by measuring the width of the disk as a function of the azimuthal angle. \n\nHowever, this approach cannot be easily applied to our observations. The innermost regions are affected by strong noise, making the determination of the FWHM of the minor axis difficult. Furthermore, the background level is not the same along the minor and major axis of the disk, which may bias the peak values of both profiles, and hence the values of the FWHM. That being said, the test presented in Figure\\,\\ref{fig:height} strongly suggests that the width of the disk at different azimuthal angles informs us about its vertical structure. This supports our findings that the distribution of small dust grains in the disk around HR\\,4796 is most likely vertically extended. \n\n\n\\subsection{Angular differential observations}\\label{sec:adi}\n\nWe presented a new approach to estimate the phase function measured in polarimetric observations, but it would be extremely valuable to also be able to measure the phase function in total intensity from angular differential imaging (ADI) observations. The main challenge when modelling ADI observations is that self-subtraction effects cannot easily be dealt with \\citep{Milli2012}. After median-collapsing the de-rotated cube (after performing principal component analysis, or any other algorithms), one cannot measure the surface brightness of the disk free of biases. Therefore we cannot properly estimate the surface brightness of the disk to correct the phase function for column density effects. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{s11.pdf}\n \\caption{Fit to the total intensity phase function derived in \\citet{Milli2017} using two Henyey-Greenstein functions (black) ``corrected'' for projection effects (dashed red). Both functions are normalized to unity at $90^{\\circ}$.}\n\\label{fig:s11}\n\\end{figure}\n\nOne way to avoid biases induced by self-subtraction is to perform ``forward modelling'' by subtracting a disk model in the cube (and we then need another free parameter to scale up or down the model before subtracting it). Having the scattered light phase function as a free parameter (as mentioned in a previous sub section) runs into the same shortcomings. One possible, but time costly, approach would be to run a disk model with an isotropic total intensity phase function, find the best scaling factor for the given set of disk parameters, measure the residuals in the final image and modify the shape of the phase function accordingly before re-evaluating the scaling factor.\n\nNonetheless, we here attempt to roughly quantify the changes to the phase function measured on total intensity observations of HR\\,4796. \\citet{Milli2017} presented SPHERE\/IRDIS observations of the disk, and measured the phase function from the surface brightness distribution, correcting for self-subtraction effects based on a disk model. This model has parameters that are compatible with our best fit solution, with a slightly different value for $e$ ($0.06$) and therefore for $\\omega$ as well ($-105.7^{\\circ}$ in our reference frame) given the degeneracy between the two parameters. They then fitted two weighted Henyey-Greenstein functions $f_\\mathrm{HG}$ to the measured phase function, as $w \\times f_\\mathrm{HG}(g_0) + (1-w) \\times f_\\mathrm{HG}(g_1)$, where $f_\\mathrm{HG}$ takes the following form\n\\begin{equation}\n f_\\mathrm{HG}(g) = \\frac{1}{4 \\pi} \\frac{1-g^2}{(1 + g^2 - 2g\\mathrm{cos}\\theta)^{3\/2}}.\n\\end{equation}\n\\citet{Milli2017} found that $g_0 = 0.99$, $g_1 = -0.14$, and $w = 0.83$. If the disk is not flat, the phase function they inferred does not take into account column density variations. We therefore used their analytical form, and applied an additional correction factor based on the dust density variation as a function of the scattering angle for a non-flat disk. The revised phase function being the original phase function divided by the surface brightness of the best fit model with an isotropic phase function. Both phase functions are shown in Figure\\,\\ref{fig:s11}, normalized to unity at $90^{\\circ}$. The most notable difference is that backward scattering becomes much more significant when accounting for the column density variations due to the inclination. One should keep in mind however that this result remains model dependent for both the self-subtraction and the density variation corrections (and the discussion of Section\\,\\ref{sec:flat}).\n\nInterestingly, as a side note, \\citet{Ren2019} measured the surface brightness of the disk and halo around HD\\,191089 and measured strong backward scattering for the halo but not for the main ring. If the halo is vertically very thin, extending mostly from the densest regions, i.e., the midplane, then one should not expect significant column density changes due to the inclination of the system ($\\sim 59^{\\circ}$) and therefore the measurement of the phase function from the halo would not suffer from the same biases described in this paper.\n\n\\subsection{Dust properties in the disk around HR\\,4796}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\hsize]{dust_prop.pdf}\n\\caption{Maps of the $\\chi^2$ when trying to reproduce the shape of the phase functions using DHS for the north and south sides (left and right, respectively). The two free parameters are the porosity and the minimum grain sizes. The insets show the observations and the best-fitting model}\n\\label{fig:dustprop}\n\\end{figure*}\n\nWith the revised phase functions, we can attempt to revise the dust properties inferred in \\citet{Milli2019}. We computed a grid of polarized phase functions, using the \\texttt{OpacityTool} (\\citealp{Woitke2016}, \\citealp{Toon1981}). The code can compute absorption and scattering properties, as well as six elements of the scattering matrix, including $S_{11}$ and $S_{12}$, assuming a distribution of hollow spheres (DHS, \\citealp{Min2005}). We assume that the grain size distribution is a differential power-law of the form d$n(s) \\propto s^{-3.5}$d$s$, where $s$ is the grain size, and we set $s_\\mathrm{max} = 1$ \\,mm. We compute the polarized phase function, integrated over the size distribution, varying $s_\\mathrm{min}$ between $0.01$\\, $\\mu$m and $110\\,\\mu$m, and the porosity between $0$ and $99$\\%. The optical constant are taken from \\citet[][amorphous silicates]{Dorschner1995} with a $85$\\% mass fraction and \\citet[][amorphous carbon]{Zubko1996} with a $15$\\% mass fraction. The maximum filling factor is set to $0.8$. To estimate whether we can reproduce the inferred phase functions, we computed a grid of models, and the $\\chi^2$ maps for both the north and south sides are shown in Figure\\,\\ref{fig:dustprop} (left and right, respectively). For both panels, the insets show the inferred phase functions and their best-fit model. For both sides, we find that the best model is obtained for $s_\\mathrm{min} = 0.01$ \\,$\\mu$m. For the north side, we find that the porosity should be $16$\\%, and $0$\\% for the south side, with significant uncertainties as illustrated in Fig.\\,\\ref{fig:dustprop}. In Appendix\\,\\ref{sec:poro_size} we show in more detail the effects of porosity and minimum grain size on the shape of the phase function. \n\nWhile the best-fit model reproduces rather well the phase function for the south side, it fails to capture the shape of the north side. One has to keep in mind that, as discussed before, the northern phase function may be biased by a possible over-density of small dust grains, but overall, we find that the phase functions suggest the presence of very small dust grains, with rather low porosity values.\n\nWith the exercise described above, the slope of the grain size distribution is set to $-3.5$, to minimize the number of free parameters, but which may be a strong hypothesis. The grain size distribution can show some wavy structures (e.g., \\citealp{Thebault2007}) or there can be an over-abundance of small dust grains in bright debris disks (as is the case for HR\\,4796, \\citealp{Thebault2019}). Therefore, we repeated the same exercise as before, but integrating the size distribution between $s$ and $s+\\delta_s$, keeping the slope as $-3.5$. The motivation being to identify the characteristic size that can best explain the phase functions. We use \\texttt{PyMultiNest} to find the best fit model and for the north side, we obtain $s_\\mathrm{min} = 0.03\\pm0.02$\\,$\\mu$m, $\\delta_s = 0.29\\pm0.15$\\,$\\mu$m, and a porosity of $16\\pm1$\\%. For the south side, we find $s_\\mathrm{min} = 0.02\\pm0.01$ \\,$\\mu$m, $\\delta_s = 0.36\\pm0.29$\\,$\\mu$m, and a porosity compatible with $0$\\%. The shape of the best fit model is quite similar to the previous best fit models. The results of this approach are quite similar to the previous ones, the phase functions are best reproduced by very small dust grains. \n\nUsing the \\texttt{OpacityTool}, we can also compute the asymmetry parameter $g_\\mathrm{sca}$ and compute the unitless $\\beta$ ratio between the stellar radiation pressure and the gravitational force for different grain sizes, as\n\\begin{equation}\n \\beta(s) = \\frac{3L_\\star}{16\\pi G c^2 M_\\star}\\frac{Q_\\mathrm{pr}(s)}{\\rho s},\n\\end{equation}\nwhere $L_\\star$ and $M_\\star$ are the stellar luminosity and mass ($25.75$\\,$L_\\odot$ and $1.31$\\,$M_\\odot$, respectively, \\citealp{Olofsson2019}), $G$ the gravitational constant, $c$ the speed of light, $\\rho$ the dust density, and $Q_\\mathrm{pr}$ the radiation pressure efficiency (equal to $Q_\\mathrm{ext}(\\lambda, s) - g_\\mathrm{sca}(s) \\times Q_\\mathrm{sca}(\\lambda, s)$) averaged over the stellar spectrum. For porosity values of $0$ and $16\\%$ we find that the blow-out sizes (for which $\\beta \\leq 0.5$, assuming the parent bodies are on circular orbit) are $\\sim 13.5$ and $17$\\,$\\mu$m (for larger porosity values, the blow-out size increases even more). All the grains that are smaller no longer are bound to the star and would be removed from the system rapidly. We therefore reach the same conclusions as the ones presented in \\citet{Milli2017,Milli2019}, that the Mie or DHS theory cannot adequately explain the full picture. Indeed, \\citet{Augereau1999} found that to reproduce the spectral energy distribution of the disk, the minimum grain size should be close to $10$\\,$\\mu$m, which is rather compatible with the aforementioned blow-out size but would fail to reproduce the measured phase function. Relatively large aggregates composed of sub-$\\mu$m sized monomers may be a viable alternative to explain the observations. As explained in \\citet{Min2016}, the polarization properties of aggregates are intimately related to the size of the individual monomers and not to the overall size of the aggregate itself.\n\n\n\\section{Conclusions}\n\nIn this paper, we presented an alternative approach to estimate the phase function from polarized observations of debris disks with an emphasis on disks that have a non negligible vertical scale height. While our method remains model-dependent it does not require a parametrized form for the phase function (e.g., Henyey-Greenstein). The total flux depends both on the local density and the true phase function, but when the disks are highly inclined, and not infinitely flat, there are variations in column density along the major axis, due to projection effects, and those variations have to be taken into account.\n\nWe presented an approach to account for those column density variation effects, and find that the inferred phase function is quite different from previous estimates. We tested our model to SPHERE\/ZIMPOL observations of HR\\,4796 and derived phase functions for both the north and south sides, both being relatively similar. We reach similar conclusions as the ones outlined in \\citet{Milli2019}, i.e., we cannot fully reconcile all key aspects with a single scattering theory (e.g., phase function and blow-out size). Our modelling results suggest that the disk is not vertically flat, with an opening angle of $\\psi \\sim 0.035$. The vertical scale height can successfully be constrained by the model based on how the width of the disk varies as a function of the azimuthal angle. \n\nWe also note that our modelling approach remains blind to any local increase of the dust density, and that it cannot readily be applied to ADI observations. We remark however that similar biases are probably occurring when deriving the total intensity phase function, which may lead to an under-estimation of backward scattering.\n\n\\begin{acknowledgements}\nWe are grateful to the referee, Kees Dullemond, for providing helpful comments, especially with respect to the determination of the vertical scale height, as well as several pointers to help clarify the paper.\nThis research has made use of the SIMBAD database (operated at CDS, Strasbourg, France). This research made use of Astropy, a community-developed core Python package for Astronomy (\\citealp{Astropy}).\nJ.~O. and A.~B. acknowledge support from the ICM (Iniciativa Cient\\'ifica Milenio) via the Nucleo Milenio de Formaci\u00f3n planetaria grant. J.~O. acknowledges support from the Universidad de Valpara\\'iso and from Fondecyt (grant 1180395). A.~B. acknowledges support from Fondecyt (grant 1190748). \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nMany of the interesting reactions at present-day and future collider \nexperiments involve a multitude of unstable particles during the intermediate \nstages. In view of the high precision of the experiments, the proper treatment \nof these unstable particles has become a demanding exercise, since on-shell \napproximations are simply not good enough anymore. A proper treatment of \nunstable particles requires the resummation of the corresponding self-energies \nto all orders. In this way the singularities originating from the poles in the \non-shell propagators are regularized by the imaginary parts contained in the\nself-energies, which are closely related to the decay widths of the unstable \nparticles. The perturbative resummation itself involves a simple geometric \nseries and is therefore easy to perform. However, this simple procedure \nharbours the serious risk of breaking gauge invariance. Gauge invariance is \nguaranteed order by order in perturbation theory. Unfortunately one takes into \naccount only part of the higher-order terms by resumming the self-energies. \nThis results in a mixing of different orders of perturbation theory and \nthereby jeopardizes gauge invariance, even if the self-energies themselves are \nextracted in a gauge-invarant way.\n\nDuring recent years awareness has been raised regarding the seriousness of the\nproblem. It was shown explicitly how these gauge-breaking effects, which are \nformally of higher order in the expansion parameter, can nevertheless have \nprofound repercussions on physical observables~\\cite{west,lapp-kek,fls1,fls2}. \nThis applies in particular to kinematical situations that \napproach asymptotic limits, like space-like virtual photons close to the \non-shell limit~\\cite{west,lapp-kek,fls1} or longitudinal gauge bosons at high \nenergies~\\cite{fls2}. These asymptotic regimes are characterized by strong \ngauge cancellations, which are governed by the Ward identities of the theory. \nAny small gauge-breaking effect can upset these intricate gauge cancellations \nand can therefore be amplified significantly.\n\nA solution to the problem is provided by the so-called \npole-scheme~\\cite{pole-scheme}, which allows the gauge-invariant calculation \nof matrix elements in the presence of unstable particles. The pole-scheme \namounts to a systematic expansion of the matrix elements around the complex \npoles in the unstable-particle propagators. This can be viewed as a \nprescription for performing an effective expansion in powers of $\\Gamma_i\/M_i$,\nwhere $M_i$ and $\\Gamma_i$ stand for the masses and widths of the unstable \nparticles. The residues in the pole expansion are physically observable and \ntherefore gauge-invariant. In reactions with multiple unstable-particle \nresonances it is rather awkward to perform the complete pole-scheme expansion\nwith all its subtleties in the treatment of the off-shell phase \nspace~\\cite{dpa-ww}. Therefore one usually approximates the expansion by \nretaining only the terms with the highest degree of resonance. \nThis approximation is called the leading-pole approximation. The accuracy of \nthe approximation is typically ${\\cal O}(\\Gamma_i\/M_i)$, making it a suitable tool \nfor calculating radiative corrections, since in that case the errors are \nfurther suppressed by powers of the coupling constant~\\cite{dpa-ww}. \nThe errors induced at the lowest-order level, however, are as large as the \nradiative corrections themselves. In view of the high precision of present-day \ncollider experiments, this is not acceptable. Therefore either the lowest-order\nexpansion has to be performed explicitly or an alternative gauge-invariant \nresummation method should be used. \n\nA few years ago a dedicated method was developed for the gauge-invariant \ntreatment of unstable gauge bosons~\\cite{fls1,fls2,fls-baur,fls-others}. \nThis so-called fermion-loop scheme exploits the fact that the unstable gauge \nbosons decay exclusively into fermions (at lowest order). Based on this \nobservation, it proved natural to resum the fermionic one-loop self-energies \nand include all other possible one-particle-irreducible fermionic one-loop \ncorrections. This resummation of one-particle-irreducible fermionic one-loop \ncorrections involves the closed subset of all ${\\cal O}([N_c^f\\alpha\/\\pi]^n)$ \ncontributions (with $N_c^f$ denoting the colour degeneracy of fermion $f$), \nwhich makes it manifestly gauge-invariant. Unfortunately this method does not \nwork for particles that also have bosonic decay modes. Moreover, the inclusion \nof a full-fledged set of one-loop corrections in a lowest-order amplitude \ntends to overcomplicate things.\n\nA more general and rapidly developing method is the so-called pinch technique\n(PT)~\\cite{pinch}. This method can be viewed as an extension of the \nfermion-loop scheme to the bosonic sector. It amounts to a re-organization of \nthe various one-loop Green's functions in terms of gauge-invariant off-shell \nbuilding blocks, labelled by the kinematical characteristics of the terms that \nare included (e.g~self-energy-like terms, vertex-like terms, etc.). These \nbuilding blocks satisfy ghost-free tree-level Ward identities (like in the \nfermion-loop scheme) and can be combined into gauge-invariant amplitudes. \nAll this is achieved by making explicit use of the full Ward identities of the\ntheory. After having applied the pinching procedure, the one-loop PT \nself-energies can be resummed in the resonant amplitudes~\\cite{pinch-resum}. \nThe gauge invariance of this resummation then follows from the tree-level-like \nWard identities of the non-resummed (vertex-like, box-like, etc.) one-loop \ncorrections (see Ref.~\\cite{bkg} for a formal proof based on the \nbackground-field-method). The PT is therefore a suitable candidate for treating\nlowest-order reactions involving unstable particles, although the lowest-order\namplitudes will be quite complicated in view of the full set of non-resummed \none-loop corrections. The complexity of these mandatory non-resummed one-loop\ncorrections grows strongly with the amount of final-state particles in the \nlowest-order reaction, just like in the fermion-loop scheme. The terminology \n`lowest-order' refers to the fact that resonant amplitudes are dominated by \nthe decay widths in the propagators, which are calculated in lowest order in \nthe PT. In order to go beyond the lowest order, the imaginary parts of the \ntwo-loop self-energies are needed. Gauge invariance of the resummation \nprocedure in turn requires the inclusion of the relevant imaginary parts of \nthe other two-loop corrections. At the moment some first attempts are under \nway to extend the PT beyond the one-loop order~\\cite{pinch-two-loop}. However, \nthere is still a long way to go. Developing a fundamental (non-diagrammatic) \nunderstanding of the PT might be the most important next step in this context.\n\nSo, the need remains for a novel, preferably non-diagrammatic method to solve \nthe full set of Ward identities. Ideally \nspeaking, such an alternative method should be applicable to arbitrary \nreactions, involving all possible unstable particles and an unspecified \namount of stable external particles. At the same time the \ngauge-restoring terms should be kept to a minimum. In this paper we propose \nsuch a novel technique for tree-level processes. By using gauge-invariant \nnon-local effective Lagrangians, it is possible to generate the self-energy\neffects in the propagators as well as the required gauge-restoring terms in \nthe multi-particle interactions. These multi-particle interactions can be \nderived explicitly in a relatively concise form.\nOf course one should generate physically sensible self-energies, so that \nthe tree-level calculations are phenomenologically meaningful.\n\nThe paper is organized as follows. In Sect.~\\ref{sec:fls} we briefly discuss\nthe essence of the gauge-invariance problem. The non-local effective-Lagrangian\nmethod for the resummation of self-energies is introduced in \nSects.~\\ref{sec:top} and \\ref{sec:ww} within the framework of an unbroken \nnon-abelian $SU(N)$ gauge theory with fermions. In Sect.~\\ref{sec:sm} we apply \nthe method to the treatment of unstable particles in the Standard Model and \ngive some simple examples. In the appendices we list some useful non-local \nFeynman rules, thus demonstrating the application of the method.\n\n\n\n\n\\section{The gauge-invariance problem: a simple example}\n\\label{sec:fls}\n\nIn this section we show the origin of the gauge-invariance problem associated\nwith the resummation of self-energies, which is a minimal requirement for \ntreating unstable particles. To this end we consider the simple example of an \nunbroken non-abelian $SU(N)$ gauge theory with fermions and subsequently \nintegrate out these fermions. This example also allows to make a direct link \nto the philosophy behind the fermion-loop scheme.\n\nFirst we fix our notation and introduce some conventions, which will be used \nthroughout this paper. The $SU(N)$ generators in the fundamental representation\nare denoted by $\\mbox{T}^a$ with $\\,a=1,\\cdots,N^2\\!-\\!1$. They are normalized \naccording to $\\mbox{Tr}\\,(\\mbox{T}^a\\mbox{T}^b)=\\delta^{ab}\/2$ and obey the commutation relation\n$\\Bigl[ \\mbox{T}^a,\\mbox{T}^b \\Bigr] = if^{abc}\\,\\mbox{T}^c$. In the adjoint representation the \ngenerators $\\mbox{F}^a$ are given by $(\\mbox{F}^a)^{bc} = -if^{abc}$. The Lagrangian of \nthe unbroken $SU(N)$ gauge theory with fermions can be written as \n\\begin{equation}\n\\label{lagr}\n {\\cal L}(x) \n = \n -\\,\\frac{1}{2}\\,\\mbox{Tr}\\,\\Bigl[ \\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(x) \\Bigr] \n + \\bar{\\psi}(x)\\, (i\\,D\\hspace{-0.63em}\/\\hspace{0.1em} - m)\\, \\psi(x),\n\\end{equation}\nwith\n\\begin{equation}\n \\mbox{\\boldmath $F$}_{\\mu\\nu} \\equiv \\mbox{T}^a F_{\\mu\\nu}^a = \\frac{i}{g}\\,[D_{\\mu},D_{\\nu}], \n \\qquad \n D_{\\mu} = \\partial_{\\mu} - ig\\,\\mbox{T}^a A_{\\mu}^a \n \\equiv \\partial_{\\mu} - ig\\mbox{\\boldmath $A$}_{\\mu}.\n\\end{equation}\nHere $\\psi$ is a fermionic $N$-plet in the fundamental representation of\n$SU(N)$ and $A_{\\mu}^a$ are the ($N^2\\!-\\!1$) non-abelian $SU(N)$ gauge fields,\nwhich form a multiplet in the adjoint representation. \nThe Lagrangian (\\ref{lagr}) is invariant under the \n$SU(N)$ gauge transformations\n$$\n \\psi(x) \\to \\psi'(x) = \\mbox{G}(x)\\,\\psi(x), \n$$\n\\begin{equation}\n\\label{gauge_trans}\n \\mbox{\\boldmath $A$}_{\\mu}(x) \\to \\mbox{\\boldmath $A$}'_{\\mu}(x) = \\mbox{G}(x)\\,\\mbox{\\boldmath $A$}_{\\mu}(x)\\,\\mbox{G}^{-1}(x)\n + \\frac{i}{g}\\,\\mbox{G}(x)\\Bigl[ \\partial_{\\mu}\\mbox{G}^{-1}(x) \\Bigr],\n\\end{equation}\nwith the $SU(N)$ group element defined as $\\mbox{G}(x)=\\exp[ig\\,\\mbox{T}^a\\theta^a(x)]$. \nThe covariant derivative $D_{\\mu}$ and field strength $\\mbox{\\boldmath $F$}_{\\mu\\nu}$ both \ntransform in the adjoint representation\n\\begin{equation}\n D_{\\mu} \\to \\mbox{G}(x)\\,D_{\\mu}\\,\\mbox{G}^{-1}(x), \n \\qquad \n \\mbox{\\boldmath $F$}_{\\mu\\nu}(x) \\to \\mbox{G}(x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\mbox{G}^{-1}(x).\n\\end{equation}\nSince the Lagrangian is quadratic in the fermion fields, one can integrate \nthem out exactly in the functional integral. The resulting effective action is \nthen given by\n\\begin{equation}\n i\\,S_{\\sss{eff}}[J] \n = \n i\\int d^4x\\,\\biggl\\{ \n -\\,\\frac{1}{2}\\,\\mbox{Tr}\\,\\Bigl[ \\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(x) \\Bigr]\n + J_{\\mu}^a(x)\\,A^{a,\\,\\mu}(x) \\biggr\\}\n + \\mbox{Tr}\\,\\Bigl[ \\ln(-\\,D\\hspace{-0.63em}\/\\hspace{0.1em} - i\\,m) \\Bigr], \n\\end{equation}\nwith $J_{\\mu}^a(x)$ denoting the gauge-field sources. The trace on the \nright-hand side has to be taken in group, spinor, and coordinate space.\nAs a next step one can expand the effective action in terms of the coupling \nconstant\n\\begin{eqnarray}\n\\label{fls\/expansion}\n \\mbox{Tr}\\,\\Bigl[ \\ln(-\\,D\\hspace{-0.63em}\/\\hspace{0.1em} - i\\,m) \\Bigr] &=& \n \\mbox{Tr}\\,\\Bigl[ \\ln(-\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em} - i\\,m) \\Bigr] \n + \\mbox{Tr}\\,\\biggl[ \\ln\\Bigl( 1+\\frac{g}{i\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em}-m}\\,\\TA\\hspace{-0.65em}\/\\hspace{0.1em} \n \\Bigr) \\biggr]\n \\nonumber \\\\[1mm]\n &=& \n \\mbox{Tr}\\,\\Bigl[ \\ln(-\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em} - i\\,m) \\Bigr]\n + \\sum\\limits_{n=1}^{\\infty} \\frac{(-1)^{n-1}}{n}\\,\n \\mbox{Tr}\\,\\Biggl[ \\biggl( \\frac{g}{i\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em}-m}\\,\\TA\\hspace{-0.65em}\/\\hspace{0.1em}\n \\biggr)^n \\Biggr].\n\\end{eqnarray}\nNote that the left-hand side of Eq.~(\\ref{fls\/expansion}) is gauge-invariant as\na result of the trace-log operation. In contrast, the separate terms of the \nexpansion on the right-hand side are not gauge-invariant. This is due to the \nfact that, unlike in the abelian case, the non-abelian gauge transformation \n(\\ref{gauge_trans}) mixes different powers of the gauge field $A_{\\mu}$ in \nEq.~(\\ref{fls\/expansion}). \nThus, if one truncates the series on the \nright-hand side of Eq.~(\\ref{fls\/expansion}) one will in general break gauge \ninvariance. From Eq.~(\\ref{fls\/expansion}) it is also clear that the fermionic \npart of the effective action induces higher-order interactions between the \ngauge bosons. \n\nWhat are these higher-order interactions? Let us consider the quadratic\ngauge-field contribution \n\\begin{equation}\n\\label{fls\/AA} \n -\\,\\frac{1}{2}\\,\\mbox{Tr}\\,\\Biggl[ \\biggl( \\frac{g}{i\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em}-m}\\,\\TA\\hspace{-0.65em}\/\\hspace{0.1em}\n \\biggr)^2 \\Biggr]\n =\n -\\,\\frac{1}{2}\\,\\int d^4x\\,d^4y\\,\\mbox{Tr}\\,\\Bigl[ O(x,y)\\,O(y,x) \\Bigr],\n\\end{equation}\nwhere\n\\begin{equation} \n O(x,y) = g\\,S^{(0)}_{\\sss{F}}(x-y)\\,\\TA\\hspace{-0.65em}\/\\hspace{0.1em}(y)\n\\end{equation}\nand $i\\,S^{(0)}_{\\sss{F}}(x-y)=\\,<\\!0\\,|\\,T(\\psi(x)\\,\\bar{\\psi}(y))\\,\n|\\,0\\!>_{\\sss{free}}\\,$ is the free fermion propagator. The trace on the \nright-hand side of Eq.~(\\ref{fls\/AA})\nhas to be taken in group and spinor space. A quick glance at this quadratic \ngauge-field contribution reveals that it is just the one-loop self-energy of \nthe gauge boson induced by a fermion loop. In the same way, the higher-order \nterms $\\sim g^n A^n$ in Eq.~(\\ref{fls\/expansion}) are just the fermion-loop \ncontributions to the $n$-point gauge-boson vertices.\n\nOne can truncate the expansion in Eq.~(\\ref{fls\/expansion}) at $n=2$, thus \ntaking into account only the gauge-boson self-energy term and neglecting the \nfermion-loop contributions to the higher-point gauge-boson vertices. This is\nevidently the simplest procedure for performing the Dyson resummation of the \nfermion-loop self-energies. However, as was pointed out above, truncation of\nEq.~(\\ref{fls\/expansion}) at any finite order in $g$ in general breaks \ngauge invariance. This leads to the important observation that, {\\it although \nthe resummed fermion-loop self-energies are gauge-independent by themselves, \nthe resummation is nevertheless responsible for gauge-breaking effects in the\nhigher-point gauge-boson interactions through its inherent mixed-order nature}.\nAnother way of understanding this is provided by the gauge-boson Ward\nidentities. Since the once-contracted $n$-point gauge-boson vertex can be \nexpressed in terms of $(n\\!-\\!1)$-point vertices (see Sect.~\\ref{sec:ww}), it \nis clear that gauge invariance is violated if the self-energies are resummed \nwithout adding the necessary compensating terms to the higher-point vertices.\n\nAn alternative is to keep all the terms in Eq.~(\\ref{fls\/expansion}). Then the \nmatrix elements derived from the effective action will be gauge-invariant. \nKeeping all the terms means that we will have to take into account not only \nthe fermion-loop self-energy in the propagator, but also all the possible \nfermion-loop contributions to the higher-point gauge-boson vertices. This is \nexactly the prescription of the fermion-loop scheme \n(FLS)~\\cite{fls1,fls2,fls-baur}. \nAlthough the FLS guarantees gauge invariance of the matrix elements, it has \ndisadvantages as well. Its general applicability is limited to those situations\nwhere non-fermionic particles can effectively be discarded in the \nself-energies, as is for instance the case for $\\Gamma_W$ and $\\Gamma_Z$ at \nlowest order. Another disadvantage is that in the FLS one is forced to do the \nloop calculations, even when calculating lowest-order quantities. For example, \nthe calculation of the tree-level matrix element for the process \n$e^+e^- \\to 4f\\gamma\\,$ involves a four-point gauge-boson interaction, which \nhas to be corrected by fermion loops in the FLS. This overcomplicates an \notherwise lowest-order calculation.\n\nIt is clear that the FLS provides more than we actually need. It does not only \nprovide gauge invariance for the Dyson resummed matrix elements at a given \norder in the coupling constant, but it also takes into account all\nfermion-loop corrections at that given order. In the vicinity of unstable\nparticle resonances the imaginary parts of the fermion-loop self-energies are \neffectively enhanced by ${\\cal O}(1\/g^2)$ with respect to the other \nfermion-loop corrections. Therefore, what is really needed is only a minimal \nsubset of the non-enhanced contributions such that gauge invariance\nis restored. In a sense one is looking for a minimal solution of a system of \nWard identities. The FLS provides a solution, but this solution is far from \nminimal and is only practical for particles that decay exclusively into \nfermions. Since the decay of unstable particles is a physical phenomenon,\nit seems likely that there exists a simpler and more natural method for \nconstructing a solution to a system of Ward identities, without an explicit\nreference to fermions. In the following sections we will try to indicate how \nsuch a solution can be constructed.\nAlthough the solution will be valid for arbitrary self-energies, in practical\ncalculations one will take physically sensible self-energies just like the FLS \ndoes.\n\n\n\n\n\\section{An effective-Lagrangian approach for fermions}\n\\label{sec:top}\n\nIn this section we propose a scheme that allows a gauge-invariant resummation\nof fermion self-energies in tree-level processes, without having to resort\nto a complete set of loop corrections. This scheme will form the basis for\nthe treatment of unstable fermions in the SM, like the top-quark. The crucial \ningredient in the scheme is the use of non-local effective Lagrangians.\nWe start off by briefly repeating the non-local Lagrangian formalism of \nRef.~\\cite{nl-terning}, where the concept of non-local Lagrangians was used \nfor a completely different purpose. \n\nThe usual local $SU(N)$ gauge theory with fermions has a Lagrangian given by \nEq.~(\\ref{lagr}), which is invariant under the gauge transformations \n(\\ref{gauge_trans}). For the gauging procedure of the non-local Lagrangians we \nwill need one more ingredient, the {\\it path-ordered exponential}, which is \ndefined as \n\\begin{eqnarray}\n\\label{Pexp}\n U(x,y) &=& U^{\\dagger}(y,x) \\ =\\ \\mbox{P}\\exp\\Biggl[ \n -\\,ig\\int\\limits_x^y\\mbox{\\boldmath $A$}_{\\mu}(\\omega)\\,d\\omega^{\\mu} \\Biggr]\n \\\\\n &=& \\lim_{d\\omega_i \\to 0}\n \\biggl( 1 - ig\\mbox{\\boldmath $A$}_{\\mu}(x)\\,d\\omega_1^{\\mu} \\biggr)\n \\biggl( 1 - ig\\mbox{\\boldmath $A$}_{\\mu}(x+d\\omega_1)\\,d\\omega_2^{\\mu} \\biggr)\n \\dots\n \\biggl( 1 - ig\\mbox{\\boldmath $A$}_{\\mu}(y-d\\omega_n)\\,d\\omega_n^{\\mu} \\biggr).\n \\nonumber\n\\end{eqnarray}\nHere $d\\omega^{\\mu}$ is the element of integration along some path \n$\\Omega(x,y)$ that connects the points $x$ and $y$. In principle we are free\nto choose this particular path. For reasons that will become clear from the \ndiscussion presented below, we will make our choice in such a way that\n\\begin{equation}\n\\label{nullpath}\n \\delta^{(4)}(x-y) \\int \\limits_x^y \\mbox{\\boldmath $A$}_{\\mu}(\\omega)\\,d\\omega^{\\mu} = 0\n\\end{equation}\nand\n\\begin{equation}\n\\label{U\/der}\n \\partial^{\\mu}_y \\int\\limits_x^y f(\\omega)\\,d\\omega^{\\nu}\n = \n f(y)\\,g^{\\mu\\nu}.\n\\end{equation}\nThe first condition implies that the null path $\\Omega(x,x)$ always has zero \nlength, i.e.~it does not involve a closed loop. The second condition fixes the\nproperties of the path-ordered exponentials under differentiation.\nThe so-defined path-ordered exponential transforms as \n\\begin{equation}\n U(x,y) \\to \\mbox{G}(x)\\,U(x,y)\\,\\mbox{G}^{-1}(y)\n\\end{equation}\nunder the $SU(N)$ gauge transformations. It hence carries the gauge \ntransformation from one space-time point to the other. With the help of the \npath-ordered exponential one can rewrite the local action corresponding to the \nfermionic part of the Lagrangian (\\ref{lagr}) according to\n\\begin{equation}\n\\label{lagr\/2}\n {\\cal S}_{\\sss{L}} \n = \n \\int d^4x\\,d^4y\\,\\bar{\\psi}(x)\\,\\delta^{(4)}(x-y)\\,\n (i\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em}_y-m)\\,U(x,y)\\,\\psi(y).\n\\end{equation}\nNote that the action (\\ref{lagr\/2}) and the one obtained from Eq.~(\\ref{lagr})\nare only equivalent because of the condition (\\ref{nullpath}).\n\nUsing this gauging procedure we can now add a gauge-invariant non-local term to\nthe Lagrangian:\n\\begin{equation}\n\\label{lagr\/nl}\n {\\cal L}_{\\sss{NL}}(x,y) \n = \n \\bar{\\psi}(x)\\,\\Sigma_{\\sss{NL}}(x-y)\\,U(x,y)\\,\\psi(y).\n\\end{equation}\nIt contains a non-local coefficient $\\Sigma_{\\sss{NL}}(x-y)$, which will play \nthe role of a self-energy correction to the free fermion propagator. The \nargument of this coefficient is $x-y$ as a result of translational invariance. \nFor calculational simplicity we will assume that $\\Sigma_{\\sss{NL}}(x-y)$ is \nmass-like, i.e.~it is diagonal in spinor space. Consequently it has to be a \nfunction of the scalar invariant $(x-y)^2$. This condition is not essential for\nthe method, but it will suit our purposes later on. With the new non-local \nterm added to the Lagrangian, the total gauge-invariant fermionic action \nbecomes\n\\begin{eqnarray}\n {\\cal S} &=& {\\cal S}_{\\sss{L}} + {\\cal S}_{\\sss{NL}}, \\nonumber \\\\[1mm]\n {\\cal S}_{\\sss{L}} &=& \\int d^4x\\,\\bar{\\psi}(x)\\,\\Bigl[ i\\,\\partial\\hspace{-0.50em}\/\\hspace{0.1em}-m+g\\,\\TA\\hspace{-0.65em}\/\\hspace{0.1em}(x) \n \\Bigr]\\,\\psi(x), \\nonumber \\\\\n {\\cal S}_{\\sss{NL}} &=& \\int d^4x\\,d^4y\\,\\bar{\\psi}(x)\\,\\Sigma_{\\sss{NL}}(x-y)\n \\,U(x,y)\\,\\psi(y).\n\\end{eqnarray}\n\nAs a next step we derive the relevant Feynman rules from the action ${\\cal S}$.\nFirst we verify that the non-local term acts as a self-energy correction to \nthe free fermion propagator:\n\\begin{equation} \n \\begin{picture}(60,15)(0,-2)\n \\ArrowLine(0,0)(30,0)\n \\ArrowLine(30,0)(60,0)\n \\Text(12,7)[lb]{$p$}\n \\Text(42,7)[lb]{$p'$}\n \\GCirc(30,0){3}{0}\n \\end{picture} \n \\ :\\ i\\,\\Sigma(y,z) \n = \n \\frac{i\\,\\delta^2 {\\cal S}}{\\delta\\psi(y)\\,\\delta\\bar{\\psi}(z)}\n \\Biggl.\\Biggr|_{A=\\psi=\\bar{\\psi}=0} \n =\n -\\,(\\partial\\hspace{-0.50em}\/\\hspace{0.1em}_z + i\\,m)\\,\\delta^{(4)}(z-y) + i\\,\\Sigma_{\\sss{NL}}(z-y).\n\\end{equation} \nOr, in momentum representation:\n\\begin{equation}\n\\label{top\/self-energy}\n \\tilde{\\Sigma}(p,-p') \n = \n (2\\pi)^4\\,\\delta^{(4)}(p-p')\\,\\Bigl[ p\\hspace{-0.42em}\/}%\\hspace{0.1em} - m + \\tilde{\\Sigma}_{\\sss{NL}}(p^2)\n \\Bigr], \n\\end{equation}\nwhere we performed the Fourier transforms\n\\begin{eqnarray}\n \\Sigma_{\\sss{NL}}(z-y) &=& \\frac{1}{(2\\pi)^4}\\,\\int d^4l\\,e^{-il\\cdot (z-y)}\n \\,\\tilde{\\Sigma}_{\\sss{NL}}(l^2), \n \\nonumber \\\\[1mm]\n \\Sigma(y,z) &=& \\frac{1}{(2\\pi)^8}\\,\\int d^4p\\,d^4p'\\,\n e^{ip\\cdot y}\\,e^{-ip'\\cdot z}\\,\n \\tilde{\\Sigma}(p,-p').\n\\end{eqnarray}\nBy convention we use a tilde to indicate functions in the momentum\nrepresentation. Upon inversion of Eq.~(\\ref{top\/self-energy}), the non-local \ncoefficient is indeed seen to act as a mass-like correction to the free \nfermion propagator. \n\nNext we investigate how the gauge-boson--fermion--fermion vertex is modified \nby the non-local interaction. This vertex consists of two parts: a local and a\nnon-local one. The local part is standard:\n\\begin{equation}\n \\begin{picture}(40,35)(0,-2)\n \\Photon(0,0)(40,0){2}{5}\n \\ArrowLine(22,-12)(23,-12)\n \\Line(12,-12)(22,-12)\n \\ArrowLine(40,0)(60,30)\n \\ArrowLine(60,-30)(40,0)\n \\GCirc(40,0){3}{0}\n \\Text(0,8)[lb]{$a,\\mu$}\n \\Text(0,-8)[lt]{$q$}\n \\Text(60,20)[lt]{$p'$}\n \\Text(60,-20)[lb]{$p$}\n \\end{picture} \n \\hspace*{7ex} \n :\\ ig\\,\\Gamma_{\\sss{L}}^{a,\\,\\mu}(x,y,z) \n = \n \\frac{i\\,\\delta^3 {\\cal S}_{\\sss{L}}}{\\delta A_{\\mu}^a(x)\\,\\delta\\psi(y)\n \\,\\delta\\bar{\\psi}(z)}\n \\Biggl.\\Biggr|_{A=\\psi=\\bar{\\psi}=0} \n =\n ig\\,\\mbox{T}^a\\,\\delta^{(4)}(x-y)\\,\\delta^{(4)}(x-z)\\,\\gamma^{\\mu}.\n \\vspace*{2ex}\n\\end{equation}\nOr, in momentum representation:\n\\begin{equation}\n ig\\,\\tilde{\\Gamma}_{\\sss{L}}^{a,\\,\\mu}(q,p,-p') \n = \n ig\\,\\mbox{T}^a\\,\\gamma^{\\mu}\\,(2\\pi)^4\\,\\delta^{(4)}(q+p-p'). \n\\end{equation}\nIn order to calculate the non-local contribution to the vertex, it is \nconvenient to take the Fourier transform of $\\Sigma_{\\sss{NL}}$, perform a \nTaylor expansion of $\\tilde{\\Sigma}_{\\sss{NL}}(l^2)$, and finally perform \nintegration by parts. Then ${\\cal S}_{\\sss{NL}}$ can be rewritten as \n\\begin{eqnarray}\n\\label{Snl_n}\n {\\cal S}_{\\sss{NL}} &=& \\sum_{n=0}^{\\infty}\\frac{1}{n!}\\,{\\cal S}_{\\sss{NL}}^{(n)}\\,\n \\biggl( \\frac{d}{d l^2} \\biggr)^n\\,\n \\tilde{\\Sigma}_{\\sss{NL}}(l^2)\\Biggl. \\Biggr|_{l^2=0}\n \\equiv \n \\sum_{n=0}^{\\infty}\\frac{1}{n!}\\,{\\cal S}_{\\sss{NL}}^{(n)}\\,\n \\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0),\n \\nonumber \\\\\n {\\cal S}_{\\sss{NL}}^{(n)} &=& \\int d^4x\\,d^4y\\,\\delta^{(4)}(x-y)\\,\\bar{\\psi}(x)\\,\n (-\\partial_y^2)^n\\,\\mbox{P}\\exp\\Biggl[ \n -\\,ig\\int\\limits_x^y \\mbox{\\boldmath $A$}_{\\mu}(\\omega)\\,d\\omega^{\\mu} \n \\biggr]\\psi(y).\n\\end{eqnarray}\nOf course, this expansion is not always allowed. We assume, however, that the \nnon-local coefficient has analyticity properties that are similar to the ones \nexpected for a normal self-energy function. As such we can perform the \ncalculation in the regime where the above expansion is applicable and \nsubsequently extend the range of validity by means of an analytical \ncontinuation. In fact, we will be able to present the non-local Feynman rules \nin such a way that the Ward identities are fulfilled irrespective of the \nprecise form of the non-local coefficient.\nThe non-local part of the gauge-boson--fermion--fermion vertex now reads\n\\begin{equation}\n\\label{nl\/u1\/4}\n ig\\,\\Gamma_{\\sss{NL}}^{a,\\,\\mu}(x,y,z) \n = \n \\frac{i\\,\\delta^3 {\\cal S}_{\\sss{NL}}}{\\delta A_{\\mu}^a(x)\\,\\delta\\psi(y)\\,\n \\delta\\bar{\\psi}(z)}\\Biggl.\\Biggr|_{A=\\psi=\\bar{\\psi}=0} \n = \n ig\\,\\mbox{T}^a\\,\\sum_{n=0}^{\\infty} \\frac{1}{n!}\\,\n \\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0)\\,{\\cal A}^{\\mu}_n(z,y|x),\n\\end{equation}\nwhere ${\\cal A}_n^{\\mu}(z,y|x)$ is given by\n\\begin{equation}\n {\\cal A}_n^{\\mu}(z,y|x) \n =\n -\\,i\\int d^4\\tau\\,\\delta^{(4)}(z-\\tau)\\,(-\\partial^2_{\\tau})^n\n \\,\\delta^{(4)}(\\tau-y)\\int\\limits_z^{\\tau} \\delta^{(4)}(\\omega-x)\\,\n d\\omega^{\\mu}.\n\\end{equation}\nThe corresponding Fourier transform can be simplified by eliminating some \n$\\delta$-function integrations and performing integration by parts:\n\\begin{equation}\n\\label{nl\/u1\/5}\n \\tilde{{\\cal A}}_n^{\\mu}(-p',p|q) \n = \n -\\,i\\int d^4\\xi\\,d^4\\tau\\,\\delta^{(4)}(\\xi-\\tau)\\,e^{ip'\\cdot\\xi}\\, \n (-\\partial_{\\tau}^2)^n\\,e^{-ip\\cdot\\tau} \n \\int\\limits_{\\xi}^{\\tau} e^{-iq\\cdot\\omega}\\,d\\omega^{\\mu}.\n\\end{equation}\nBy working out one of the $(-\\partial_{\\tau}^2)$ operators with the help of\nEq.~(\\ref{U\/der}), one can derive the recursion relation \n\\begin{equation}\n \\tilde{{\\cal A}}_n^{\\mu}(-p',p|q) \n = \n p^2 \\tilde{{\\cal A}}_{n-1}^{\\mu}(-p',p|q) \n + (2p+q)^{\\mu}\\,(p+q)^{2n-2}\\,(2\\pi)^4\\,\\delta^{(4)}(q+p-p').\n\\end{equation}\n{}From the base of the recursion, $\\tilde{{\\cal A}}_0^{\\mu}(-p',p|q)=0$, it is clear \nthat all terms in the series will be proportional to \n$(2p+q)^{\\mu}\\,(2\\pi)^4\\,\\delta^{(4)}(q+p-p')$. The solution of the recursion \nrelation can be found in App.~\\ref{app:a}:\n\\begin{equation}\n\\label{A_n1}\n \\tilde{{\\cal A}}_n^{\\mu}(-p',p|q) \n = \n (2p+q)^{\\mu}\\,\\frac{(q+p)^{2n}-p^{2n}}{(q+p)^2-p^2}\n \\,(2\\pi)^4\\,\\delta^{(4)}(q+p-p').\n\\end{equation}\nSubstituting this into the definition of the gauge-boson--fermion--fermion \nvertex (\\ref{nl\/u1\/4}), one obtains for the non-local contribution in \nmomentum representation\n\\begin{eqnarray}\n ig\\,\\tilde{\\Gamma}_{\\sss{NL}}^{a,\\,\\mu}(q,p,-p') &=& \n ig\\,\\mbox{T}^a\\,(p+p')^{\\mu}\\sum_{n=0}^{\\infty} \\frac{1}{n!}\\,\n \\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0)\\,\n \\frac{p'^{\\,2n}-p^{2n}}{p'^{\\,2}-p^2}\\,\n (2\\pi)^4\\,\\delta^{(4)}(q+p-p')\n \\nonumber \\\\[1mm]\n &=&\n ig\\,\\mbox{T}^a\\,\\frac{(p+p')^{\\mu}}{p'^{\\,2}-p^2}\\, \n \\biggl[ \\tilde{\\Sigma}_{\\sss{NL}}(p'^{\\,2})\n - \\tilde{\\Sigma}_{\\sss{NL}}(p^2) \\biggr]\\,\n (2\\pi)^4\\,\\delta^{(4)}(q+p-p').\n\\end{eqnarray}\nThis expression exhibits the proper infrared behaviour,\n\\begin{equation}\n \\tilde{\\Gamma}_{\\sss{NL}}^{a,\\,\\mu}(q,p,-p') \n \\stackrel{q\\to 0}{\\longrightarrow}\n \\mbox{T}^a\\,(2\\pi)^4\\,\\delta^{(4)}(q+p-p')\\,\n \\frac{\\partial\\,\\tilde{\\Sigma}_{\\sss{NL}}(p^2)}{\\partial p_{\\mu}},\n\\end{equation}\nrequired for guaranteeing the usual eikonal factorization in the infrared \nlimit. \n\nIt is easy to verify that the so-obtained full gauge-boson--fermion--fermion \nvertex satisfies the Ward identity for dressed fermion propagators:\n\\begin{equation}\n q_{\\mu}\\,\\Bigl[ \\tilde{\\Gamma}_{\\sss{L}}^{a,\\,\\mu}(q,p,-p') \n + \\tilde{\\Gamma}_{\\sss{NL}}^{a,\\,\\mu}(q,p,-p') \\Bigr]\n = \n (2\\pi)^4\\,\\delta^{(4)}(q+p-p')\\,\\mbox{T}^a\\,\n \\Bigl[ \\tilde{S}_F^{-1}(p')-\\tilde{S}_F^{-1}(p) \\Bigr],\n\\end{equation}\nwith $\\tilde{S}_F^{-1}(p) = p\\hspace{-0.42em}\/}%\\hspace{0.1em}-m+\\tilde{\\Sigma}_{\\sss{NL}}(p^2)$. From this \nwe can conclude that the described non-local approach allows a gauge-invariant \nresummation of fermion self-energies, while at the same time reducing the \ncomplexity of the necessary gauge-restoring higher-point interactions to a \nminimum. The general higher-point interaction between two fermions and $k$ \ngauge bosons reads\n\\begin{eqnarray}\n\\label{2fkGB}\n ig^k\\,\\Gamma_{\\sss{NL}}^{a_1..a_k,\\,\\mu_1..\\mu_k} (x_1,..\\,,x_k,y,z) \n &=& \n \\frac{i\\,\\delta^{k+2} {\\cal S}_{\\sss{NL}}}\n {\\delta A_{\\mu_1}^{a_1}(x_1)\\,..\\,\\delta A_{\\mu_k}^{a_k}(x_k)\\,\n \\delta \\psi(y)\\,\\delta\\bar{\\psi}(z)}\n \\Biggl.\\Biggr|_{A=\\psi=\\bar{\\psi}=0}\n \\nonumber \\\\[1mm]\n &=&\n ig^k \\sum_{n=0}^{\\infty} \\frac{1}{n!}\\,\\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0)\\,\n \\sum\\limits_{\\sss{perm}}{\\cal A}_n^{\\nu_1..\\nu_k}(z,y|y_1,..\\,,y_k)\\,\n \\mbox{T}^{b_1}\\cdots\\,\\mbox{T}^{b_k}.\\hspace*{5ex}\n\\end{eqnarray}\nThe latter sum involves all possible permutations \n$\\{(b_1,\\nu_1,y_1),\\ldots ,(b_k,\\nu_k,y_k)\\}$ of the basic set \n$\\{(a_1,\\mu_1,x_1),\\ldots ,(a_k,\\mu_k,x_k)\\}$.\nThe Fourier transform of the path-ordered tensor-function \n${\\cal A}_n^{\\mu_1..\\mu_k}(z,y|x_1,..\\,,x_k)$ is given by\n\\begin{equation}\n\\label{A_nk}\n \\tilde{{\\cal A}}_n^{\\mu_1..\\mu_k}(-p',p|q_1,..\\,,q_k)\n =\n (-i)^k\\!\\int d^4\\xi\\,d^4\\tau\\,\\delta^{(4)}(\\xi-\\tau)\\,e^{ip'\\cdot\\xi}\n (-\\partial^2_{\\tau})^n\\,e^{-ip\\cdot\\tau}\\,\\prod\\limits_{j=1}^k\n \\,\\,\\int\\limits_{\\omega_{j\\!-\\!1}}^{\\tau}d\\omega_j^{\\mu_j}\\,\n e^{-iq_j\\cdot\\omega_j},\n\\end{equation}\nwith $\\omega_0 = \\xi$. In App.~\\ref{app:a} we discuss briefly the recursion \nrelations corresponding to these tensor-functions and present the solution for \ngeneral $k$. Based on the simple `Ward identities' for the tensor-functions \ngiven in App.~\\ref{app:a}, it is easy to verify that the general interaction\n(\\ref{2fkGB}) obeys the Ward identity\n\\begin{eqnarray}\n \\lefteqn{\\hspace*{-2.0cm}\n q_{r,\\,\\mu_r}\\,\\tilde{\\Gamma}_{\\sss{NL}}^{a_1..a_k,\\,\\mu_1..\\mu_k} \n (q_1,..\\,,q_k,p,-p') = \\tilde{\\Gamma}_{\\sss{NL}}^{a_1....a_k,\\,\n \\mu_1..<\\mu_r>..\\mu_k}(q_1,..\\,,<\\!q_r\\!>,..\\,,q_k,p+q_r,-p')\\,\\mbox{T}^{a_r}}\n \\nonumber \\\\[4mm]\n & & -\\,\\mbox{T}^{a_r}\\,\\tilde{\\Gamma}_{\\sss{NL}}^{a_1....a_k,\\,\n \\mu_1..<\\mu_r>..\\mu_k}(q_1,..\\,,<\\!q_r\\!>,..\\,,q_k,p,-p'+q_r) \n \\nonumber \\\\[2mm]\n & & -\\,\\sum\\limits_{j\\neq r} \\,(\\mbox{F}^{a_r})^{a_j d}\\,\\biggl[\n \\tilde{\\Gamma}_{\\sss{NL}}^{a_1....a_k,\\,\n \\mu_1..<\\mu_r>..\\mu_k}(q_1,..\\,,<\\!q_r\\!>,..\\,,q_k,p,-p')\n \\biggl]_{\\begin{array}{l}\n \\\\[-3mm]\n \\!\\!\\scriptstyle{a_j \\to d} \\\\[-1mm]\n \\!\\!\\scriptstyle{q_j \\to q_j+q_r}\n \\end{array}}\n \\hspace*{-1cm}.\n\\end{eqnarray}\nHere $<\\!i_r\\!>$ indicates that the index $i_r$ has been removed. \nThe substitutions in the last term should be applied to the expression \ninside the brackets only. The $SU(N)$ generator in the adjoint representation \n$\\mbox{F}^{a_r}$ has been defined in the previous section.\n\n{}From all this we can conclude that the above-described procedure allows the \ngauge-invariant resummation of fermion self-energies in the context of a \n$SU(N)$ symmetric theory. At the same time the compensating terms in the \nhigher-point interactions are kept to a minimum. It should be\nnoted, however, that this procedure is not sufficient for a gauge-invariant \ndescription of unstable fermions in the Standard Model, since the symmetry is\nexplicitly broken in that case. We will come back to this point in \nSect.~\\ref{sec:sm}. \n\n\n\n\n\\section{An effective-Lagrangian approach for gauge bosons}\n\\label{sec:ww}\n\nThe next step is to extend the non-local method to the gauge-boson sector.\nWe remind the reader that the non-local Lagrangian should allow the Dyson \nresummation of the gauge-boson self-energies, in order to make the link to\nunstable gauge bosons later on, and it should preserve gauge invariance with a \nminimum of additional higher-point interactions. The starting-point of the\nnon-local effective Lagrangian should therefore be a bilinear gauge-boson \ninteraction. In the light of the discussion presented in Sect.~\\ref{sec:fls}, \nthe main idea is to rearrange the series on the right-hand side of \nEq.~(\\ref{fls\/expansion}) in such a way that each term becomes gauge-invariant \nby itself. Subsequent truncation of the series at a given term is then \nallowed. \n\nSince the gauge bosons transform in the adjoint representation \n($\\mbox{\\boldmath $F$} \\to \\mbox{\\boldmath $F$}'=\\mbox{G} \\mbox{\\boldmath $F$} \\mbox{G}^{-1}$), the non-local action for gauge bosons differs\nfrom the one for fermions in the way the path-ordered exponentials occur. \nFor an $SU(N)$ Yang--Mills theory it takes the form\n\\begin{equation}\n\\label{ww\/action}\n {\\cal S}_{\\sss{NL}} \n = \n -\\,\\frac{1}{2} \\int d^4x\\,d^4y\\,\\Sigma_{\\sss{NL}}(x-y)\\, \n \\mbox{Tr}\\,\\Bigl[ U(y,x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,U(x,y)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(y) \\Bigr],\n\\end{equation}\nor, expanding the non-local coefficient $\\Sigma_{\\sss{NL}}$ in terms of \nderivatives,\n\\begin{eqnarray}\n {\\cal S}_{\\sss{NL}} &=& \\sum\\limits_{n=0}^{\\infty} \\frac{1}{n!}\\, \n \\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0)\\,{\\cal S}_{\\sss{NL}}^{(n)},\n \\nonumber \\\\\n {\\cal S}_{\\sss{NL}}^{(n)} &=& -\\,\\frac{1}{2}\\int d^4x\\,d^4y\\,\\delta^{(4)}(x-y)\\,\n (-\\partial_y^2)^n\\,\\mbox{Tr}\\,\\Bigl[ U(y,x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,U(x,y)\\,\n \\mbox{\\boldmath $F$}^{\\mu\\nu}(y) \\Bigr].\n\\end{eqnarray}\nAs required, the action contains bilinear gauge-boson interactions. The induced\ninfinite tower of higher-point gauge-boson interactions, which are also of\nprogressively higher order in the coupling constant $g$, is needed for \nrestoring gauge invariance.\n\nLet us derive the relevant Feynman rules, starting with the two-point \nfunction \n\\begin{equation}\n \\begin{picture}(80,30)(0,-2)\n \\Photon(0,0)(80,0){2}{10}\n \\ArrowLine(25,-12)(26,-12)\n \\Line(15,-12)(25,-12)\n \\ArrowLine(55,-12)(54,-12)\n \\Line(65,-12)(55,-12)\n \\Text(0,8)[lb]{$a_1,\\mu_1$}\n \\Text(0,-8)[lt]{$q_1$}\n \\Text(80,8)[rb]{$a_2,\\mu_2$}\n \\Text(80,-8)[rt]{$q_2$}\n \\GCirc(40,0){3}{0}\n \\end{picture} \n \\hspace*{5ex}\n :\\\n i\\,\\Sigma^{a_1a_2,\\,\\mu_1\\mu_2}(x_1,x_2) \n = \n \\frac{i\\,\\delta^2 ({\\cal S}_{\\sss{L}}+{\\cal S}_{\\sss{NL}})}{\\delta A^{a_1}_{\\mu_1}(x_1)\\,\n \\delta A^{a_2}_{\\mu_2}(x_2)}\\Biggl.\\Biggr|_{A=0}, \n \\vspace*{2ex}\n\\end{equation}\nwhere the local action ${\\cal S}_{\\sss{L}}$ follows from the gauge-boson term in \nEq.~(\\ref{lagr}). The Fourier transform of this two-point function can be \ncalculated in a straightforward way, since the path-ordered exponentials are\neffectively unity. The result reads \n\\begin{equation}\n\\label{ww\/self-energy}\n i\\,\\tilde{\\Sigma}^{a_1a_2,\\,\\mu_1\\mu_2}(q_1,q_2)\n =\n i\\,\\delta^{a_1a_2} \\Bigl( q_1^{\\mu}q_1^{\\nu} - q_1^2 g^{\\mu\\nu} \\Bigr)\\,\n \\Bigl[ 1 + \\tilde{\\Sigma}_{\\sss{NL}}(q_1^2) \\Bigr]\n \\,(2\\pi)^4\\,\\delta^{(4)}(q_1+q_2).\n\\end{equation}\nNote that this two-point interaction is transverse, as it should be for an \nunbroken theory. The non-local coefficient acts as a (dimensionless) \ncorrection to the transverse free gauge-boson propagator. So, exactly what is\nneeded for the Dyson resummation of the gauge-boson self-energies. \n\nThe general non-local interaction between $k$ gauge bosons consists of four \ndistinct contributions, with either two, three or four gauge fields supplied \nby the field strengths in Eq.~(\\ref{ww\/action}). In order to simplify the \nderivation of the Feynman rules, it is convenient to write \nEq.~(\\ref{ww\/action}) in the adjoint representation rather than the \nfundamental representation:\n\\begin{equation}\n {\\cal S}_{\\sss{NL}} \n = \n -\\,\\frac{1}{4} \\int d^4x\\,d^4y\\,\\Sigma_{\\sss{NL}}(x-y)\\, \n F^a_{\\mu\\nu}(x)\\,U^{ab}(x,y)\\,F^{\\mu\\nu,\\,b}(y), \n\\end{equation}\nwith\n\\begin{equation}\n U^{ab}(x,y) \n = \n \\mbox{P}\\exp\\Biggl[ -\\,ig\\int\\limits_x^y \\mbox{F}^c A^c_{\\mu}(\\omega)\\,\n d\\omega^{\\mu} \\Biggr]^{ab}.\n\\end{equation}\nFor the general interaction between $k$ gauge bosons we now obtain\n\\begin{eqnarray}\n \\begin{picture}(75,45)(0,-2)\n \\Photon(0,0)(40,0){2}{5}\n \\ArrowLine(25,-12)(26,-12)\n \\Line(15,-12)(25,-12)\n \\Photon(40,0)(60,30){2}{5}\n \\ArrowLine(57,8)(55,5)\n \\Line(57,8)(62,15.5)\n \\Photon(60,-30)(40,0){2}{5}\n \\ArrowLine(57,-8)(55,-5)\n \\Line(57,-8)(62,-15.5)\n \\GCirc(40,0){3}{0}\n \\Text(0,8)[lb]{$a_1,\\mu_1$}\n \\Text(0,-8)[lt]{$q_1$}\n \\Text(60,35)[cb]{$a_2,\\mu_2$}\n \\Text(65,20)[lb]{$q_2$}\n \\Text(60,-35)[ct]{$a_k,\\mu_k$}\n \\Text(65,-20)[lt]{$q_k$}\n \\GCirc(75,12){0.5}{0}\n \\GCirc(80,4){0.5}{0}\n \\GCirc(80,-4){0.5}{0}\n \\GCirc(75,-12){0.5}{0}\n \\end{picture} \n \\hspace*{7ex} \n &:& ig^{k-2}\\,\\Gamma_{\\sss{NL}}^{a_1..a_k,\\,\\mu_1..\\mu_k}(x_1,..\\,,x_k) \\ =\\\n \\frac{i\\,\\delta^k {\\cal S}_{\\sss{NL}}}{\\delta A_{\\mu_1}^{a_1}(x_1)\\,..\\,\n \\delta A_{\\mu_k}^{a_k}(x_k)}\n \\Biggl.\\Biggr|_{A=0} \\nonumber \\\\[4mm]\n & & =\\ ig^{k-2} \\sum_{n=0}^{\\infty} \\frac{1}{n!}\\,\n \\tilde{\\Sigma}_{\\sss{NL}}^{(n)}(0)\\,\\sum\\limits_{\\sss{perm}}\n (\\mbox{F}^{b_3}\\cdots\\,\\mbox{F}^{b_k})^{b_1b_2}\\,\n V_{\\sss{NL},\\,n}^{\\nu_1..\\nu_k}(y_1,..\\,,y_k),\n\\end{eqnarray}\nwhere the sum involves all possible permutations\n$\\{(b_1,\\nu_1,y_1),\\ldots ,(b_k,\\nu_k,y_k)\\}$ of the basic set \n$\\{(a_1,\\mu_1,x_1),\\ldots ,(a_k,\\mu_k,x_k)\\}$.\nThe Fourier transform of the path-ordered tensor-function \n$V_{\\sss{NL},\\,n}^{\\mu_1..\\mu_k}(x_1,..\\,,x_k)$ can be expressed in terms of \nthe path-ordered tensor-functions introduced in the previous section: \n\\begin{eqnarray}\n\\label{ww\/k-point}\n \\tilde{V}_{\\sss{NL},\\,n}^{\\mu_1..\\mu_k}(q_1,..\\,,q_k) \n &=&\n \\frac{1}{2}\\,T^{\\mu_1\\mu_2}(q_1,q_2)\\,\n \\tilde{{\\cal A}}_n^{\\mu_3..\\mu_k}(q_1,q_2|q_3,..\\,,q_k) \n \\nonumber \\\\\n &+& \n \\frac{1}{4}\\,A^{\\mu_1,\\,\\mu_2\\mu_k}(q_1)\\,\n \\tilde{{\\cal A}}_n^{\\mu_3..\\mu_{k\\!-\\!1}}(q_1,q_2+q_k|q_3,..\\,,q_{k\\!-\\!1})\n \\nonumber \\\\\n &-&\n \\frac{1}{4}\\,A^{\\mu_2,\\,\\mu_1\\mu_3}(q_2)\\,\n \\tilde{{\\cal A}}_n^{\\mu_4..\\mu_k}(q_1+q_3,q_2|q_4,..\\,,q_k)\n \\nonumber \\\\\n &-&\n \\frac{1}{4}\\,g^{\\mu_1\\mu_2}\\,g^{\\mu_3\\mu_k}\\,\n \\tilde{{\\cal A}}_n^{\\mu_4..\\mu_{k\\!-\\!1}}(q_1+q_3,q_2+q_k|q_4,..\\,,q_{k\\!-\\!1}),\n\\end{eqnarray} \nwhere the first term contributes for $k\\ge 2$, the second\/third term for\n$k\\ge 3$, and the fourth term for $k\\ge 4$.\nHere we introduced the transverse tensors\n\\begin{equation}\n T^{\\mu\\nu}(p,q)\n =\n (p \\cdot q)\\,g^{\\mu\\nu} - p^{\\nu} q^{\\mu},\n \\qquad\n A^{\\mu,\\,\\nu\\rho}(q) \n = \n g^{\\mu\\nu}\\,q^{\\rho} - g^{\\mu\\rho}\\,q^{\\nu}.\n\\end{equation}\nThese tensors have the following properties: $p^{\\mu}\\,T^{\\mu\\nu}(p,q)=\nT^{\\mu\\nu}(p,q)\\,q^{\\nu} = q^{\\mu}A^{\\mu,\\,\\nu\\rho}(q) = 0$,\n$p^{\\rho}A^{\\mu,\\,\\nu\\rho}(q) = T^{\\mu\\nu}(q,p)\\,$ and \n$\\,p^{\\nu}A^{\\mu,\\,\\nu\\rho}(q) = - T^{\\mu\\rho}(q,p)$.\nUsing in addition the properties of the tensor-functions $\\tilde{{\\cal A}}_n$ given \nin App.~\\ref{app:a}, one can verify that the general non-local gauge-boson\ninteraction satisfies the (ghost-free) Ward identity \n\\begin{eqnarray}\n q_{r,\\,\\mu_r}\\,\\tilde{\\Gamma}_{\\sss{NL}}^{a_1..a_k,\\,\\mu_1..\\mu_k}\n (q_1,..\\,,q_k) \n = \n -\\,\\sum\\limits_{j\\neq r} \\,(\\mbox{F}^{a_r})^{a_j d}\\,\\biggl[\n \\tilde{\\Gamma}_{\\sss{NL}}^{a_1....a_k,\\,\\mu_1..<\\mu_r>..\\mu_k}\n (q_1,..\\,,<\\!q_r\\!>,..\\,,q_k) \\biggl]_{\\begin{array}{l}\n \\\\[-3mm]\n \\!\\!\\scriptstyle{a_j \\to d} \\\\[-1mm]\n \\!\\!\\scriptstyle{q_j \\to q_j+q_r}\n \\end{array}}\n \\hspace*{-1.0cm}. \\hspace*{1.1ex}\n\\end{eqnarray}\nOn top of that, the non-local three-point interaction exhibits the proper\ninfrared behaviour,\n\\begin{equation}\n \\tilde{\\Gamma}_{\\sss{NL}}^{a_1a_2a_3,\\,\\mu_1\\mu_2\\mu_3}(q_1,q_2,q_3)\n \\stackrel{q_1\\to 0}{\\longrightarrow}\n -\\,(\\mbox{F}^{a_1})^{a_2a_3}\\,(2\\pi)^4\\,\\delta^{(4)}(q_1+q_2+q_3)\\,\n \\frac{\\partial}{\\partial q_{2,\\,\\mu_1}}\n \\biggl\\{ T^{\\mu_2\\mu_3}(q_2,-q_2)\\,\\tilde{\\Sigma}_{\\sss{NL}}(q_2^2) \n \\biggr\\},\n\\end{equation}\nthereby guaranteeing the usual eikonal factorization in the infrared limit. \n\n\nIn addition to the non-local contributions, the three- and four-point \ngauge-boson interactions also receive contributions from the local action.\nWith our conventions these local contributions read \n\\begin{equation}\n ig^{k-2}\\,\\Gamma_{\\sss{L}}^{a_1..a_k,\\,\\mu_1..\\mu_k}(x_1,..\\,,x_k)\n =\n ig^{k-2}\\,\\sum\\limits_{\\sss{perm}}(\\mbox{F}^{b_3}\\cdots\\,\\mbox{F}^{b_k})^{b_1b_2}\\,\n V_{\\sss{L}}^{\\nu_1..\\nu_k}(y_1,..\\,,y_k),\n\\end{equation}\nwith\n\\begin{eqnarray}\n \\tilde{V}_{\\sss{L}}^{\\mu_1\\mu_2\\mu_3}(q_1,q_2,q_3) &=&\n \\frac{1}{2}\\,A^{\\mu_1,\\,\\mu_2\\mu_3}(q_1)\\,\n (2\\pi)^4\\,\\delta^{(4)}(q_1+q_2+q_3),\n \\nonumber \\\\[1mm]\n \\tilde{V}_{\\sss{L}}^{\\mu_1\\mu_2\\mu_3\\mu_4}(q_1,q_2,q_3,q_4) &=&\n -\\,\\frac{1}{4}\\,g^{\\mu_1\\mu_2}g^{\\mu_3\\mu_4}\\,\n (2\\pi)^4\\,\\delta^{(4)}(q_1+q_2+q_3+q_4),\n \\nonumber \\\\[2mm]\n \\tilde{V}_{\\sss{L}}^{\\mu_1..\\mu_k}(q_1,..\\,,q_k) &=& \n 0 \\qquad (k>4).\n\\end{eqnarray}\n\nAlthough the above-described non-local procedure provides a gauge-invariant \nframework for performing the Dyson resummation of the gauge-boson \nself-energies, we want to stress that it is not unique. We have seen in \nSect.~\\ref{sec:fls} that the FLS provides a different solution of the system \nof gauge-boson Ward identities. In the context of non-local effective \nLagrangians it is always possible to add additional towers of gauge-boson \ninteractions that start with three-point interactions and therefore do not \ninfluence the Dyson resummation of the gauge-boson self-energies. For instance,\nthe non-local action\n\\begin{equation}\n {\\cal S}_{\\sss{NL}}' \n =\n g\\,\\int d^4x\\,d^4y\\,d^4z\\,V(x,y,z)\\,\\mbox{Tr}\\,\\Bigl[ U(z,x)\\,\\mbox{\\boldmath $F$}^{\\mu}_{\\ \\nu}(x)\\, \n U(x,y)\\,\\mbox{\\boldmath $F$}^{\\nu}_{\\ \\rho}(y)\\,U(y,z)\\,\\mbox{\\boldmath $F$}^{\\rho}_{\\ \\mu}(z) \\Bigr]\n\\end{equation}\nis gauge-invariant and does not affect the gauge-boson self-energies.\nIt does contribute, however, to the interaction between three or more gauge\nbosons. As such it leads to a zero-mode solution of the system of \ngauge-boson Ward identities. In our quest for minimality we have opted to \nleave out such zero-mode solutions, as they are anyhow immaterial for the\ndiscussion of self-energies. In the light of the discussion presented in \nSect.~\\ref{sec:fls}, we rearrange the series on the right-hand \nside of Eq.~(\\ref{fls\/expansion}) according to gauge-invariant towers of \ngauge-boson interactions labelled by the minimum number of gauge bosons that\nare involved in the non-local interaction. Effectively this constitutes an \nexpansion in powers of the coupling constant $g$, since a higher minimum\nnumber of particles in the interaction is equivalent to a higher minimum \norder in $g$. In order to achieve minimality we have truncated this series at \nthe lowest effective order. \n \n\n\n\n\\section{Unstable particles in the Standard Model}\n\\label{sec:sm}\n\nIn this section we address the case of phenomenological interest: unstable \nparticles in a broken $[SU(3)_C\\times ]SU(2)_L\\times U(1)_Y$ gauge theory. \nFirst we briefly fix the notations. The $SU(2)_L\\times U(1)_Y$ gauge-group \nelement is defined as\n\\begin{equation}\n G(x) = \\exp\\Bigl[ ig_2\\mbox{T}^a\\theta^a(x) - ig_1\\frac{Y}{2}\\,\\theta_Y(x) \\Bigr],\n\\end{equation}\nwhere the $SU(2)$ generators $\\mbox{T}^a$ can be expressed in terms of the standard \nPauli spin matrices $\\sigma^a$ $\\,(a=1,2,3)$ according to $\\mbox{T}^a=\\sigma^a\/2$. \nThe normalization condition and commutation relation read \n$\\mbox{Tr}\\,(\\sigma^a\\sigma^b) = 2\\delta^{ab}$ and \n$\\Bigl[ \\sigma^a,\\sigma^b \\Bigr] = 2i\\epsilon^{abc}\\,\\sigma^c$, with the\n$SU(2)$ structure constant $\\epsilon^{abc}$ given by\n\\begin{equation}\n \\epsilon^{abc} \n =\n \\left\\{ \\begin{array}{cl}\n +1 & \\mbox{\\ if }(a,b,c)=\\mbox{\\ even permutation of }(1,2,3) \\\\\n -1 & \\mbox{\\ if }(a,b,c)=\\mbox{\\ odd permutation of }(1,2,3)\\ \\ .\\\\\n 0 & \\mbox{\\ else}\n \\end{array}\n \\right.\n\\end{equation}\nThe $SU(2)$ generators in the adjoint representation are given by \n$(\\mbox{F}^a)^{bc} = -i\\epsilon^{abc}$.\n\n\n\\subsection{The gauge bosons}\n\\label{sec:sm\/wz}\n\nIn the Standard Model there are four gauge fields, $W^a_{\\mu}$ $\\,(a=1,2,3)$\nand $B_{\\mu}$, with the corresponding field-strength tensors given by\n\\begin{equation}\n \\mbox{\\boldmath $F$}_{\\mu\\nu} = \\partial_\\mu \\mbox{\\boldmath $W$}_\\nu - \\partial_\\nu \\mbox{\\boldmath $W$}_\\mu \n - i g_2\\,\\bigl[ \\mbox{\\boldmath $W$}_\\mu,\\mbox{\\boldmath $W$}_\\nu \\Bigr],\n \\qquad \n B_{\\mu\\nu} = \\partial_\\mu B_\\nu - \\partial_\\nu B_\\mu,\n\\end{equation}\nusing the shorthand notations\n\\begin{equation}\n \\mbox{\\boldmath $F$}_{\\mu\\nu} \\equiv \\mbox{T}^a F_{\\mu\\nu}^a,\n \\qquad\n \\mbox{\\boldmath $W$}_{\\mu} \\equiv \\mbox{T}^a W_{\\mu}^a.\n\\end{equation}\nIn this notation the Yang--Mills Lagrangian reads:\n\\begin{equation}\n\\label{sm\/yang-mills}\n {\\cal L}^{\\sss{YM}}(x) \n = \n -\\,\\frac{1}{2}\\,\\mbox{Tr}\\,\\Bigl[ \\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(x) \\Bigr]\n -\\frac{1}{4}\\,B_{\\mu\\nu}(x)\\,B^{\\mu\\nu}(x).\n\\end{equation}\nThe physically observable gauge-boson states are given by\n\\begin{equation}\n W^{\\pm}_{\\mu} = \\frac{1}{\\sqrt{2}} (W^1_{\\mu}\\mp i\\,W^2_{\\mu}),\n \\quad\n Z_{\\mu} = c_W W^3_{\\mu} + s_W B_{\\mu},\n \\quad \n A_{\\mu} = c_W B_{\\mu} - s_W W^3_{\\mu},\n\\end{equation}\nwhere $\\,c_W=g_2\/\\sqrt{g_1^2+g_2^2}\\,$ and $\\,s_W=\\sqrt{1-c_W^2}\\,$ are the \ncosine and sine of the weak mixing angle. The electric charge is given by \n$e=g_1 g_2\/\\sqrt{g_1^2+g_2^2}$.\n\nFor the gauge-invariant treatment of unstable gauge bosons we can use a \nnon-local Lagrangian that generates the relevant self-energy effects. \nThe corresponding action can be split into two pieces. One piece is already \nknown from the unbroken theory, bearing in mind that we have two \nfield-strength tensors to work with:\n\\begin{eqnarray}\n\\label{sm\/action}\n {\\cal S}_{\\sss{NL}}^{\\sss{YM}} &=& -\\,\\frac{1}{4} \\int d^4x\\,d^4y\\,\\Sigma_1(x-y)\\,\n B_{\\mu\\nu}(x)\\,B^{\\mu\\nu}(y) \\nonumber \\\\[1mm]\n & & -\\,\\frac{1}{2} \\int d^4x\\,d^4y\\,\\Sigma_2(x-y)\\,\\mbox{Tr}\\,\\Bigl[ \n U_2(y,x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,U_2(x,y)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(y) \\Bigr].\n\\end{eqnarray}\nNote that the path-ordered exponentials vanish in the first term. They are also\nnot needed, since $B_{\\mu\\nu}(x)$ is gauge-invariant by itself. In the second\nterm $U_2$ is the path-ordered exponential corresponding to $SU(2)_L$ \n[defined according to Eq.~(\\ref{Pexp})]. Furthermore,\nit is impossible to construct a gauge-invariant non-local operator \nof the form $B\\cdot F$ using only gauge fields. Such interactions require\nsome additional fields with non-zero vacuum expectation value, i.e.~the \nHiggs fields. For the second piece of the non-local Lagrangian we therefore \nexploit the fact that the theory is spontaneously broken:\n\\begin{eqnarray}\n\\label{sm\/action\/higgs}\n {\\cal S}_{\\sss{NL}}^{\\Phi} &=& -\\,\\frac{g_1g_2}{2M_W^2} \\int d^4x\\,d^4y\\,\\Sigma_3(x-y)\\,\n [ \\Phi^{\\dagger}(x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\Phi(x) ]\\,\n B^{\\mu\\nu}(y) \\nonumber \\\\\n & & -\\,\\frac{g_2^4}{4M_W^4} \\int d^4x\\,d^4y\\,\\Sigma_4(x-y)\\,\n [ \\Phi^{\\dagger}(x)\\,\\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\Phi(x) ]\\,\n [ \\Phi^{\\dagger}(y)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(y)\\,\\Phi(y) ],\n\\end{eqnarray}\nwith the ($Y\\!=\\!1$) Higgs doublet $\\Phi$ given by\n\\begin{equation}\n\\label{higgs}\n \\Phi(x)\n = \n \\left( \\begin{array}{c}\n \\phi^+(x) \\\\[1mm]\n [v+H(x)+i\\,\\chi(x)]\/\\sqrt{2}\n \\end{array} \n \\right)\\ .\n\\end{equation}\nThe non-zero vacuum expectation value $v$ is given by $v=2M_W\/g_2$. \nThe field operators contained in this additional effective action are clearly \nof higher dimension than the ones contained in the previously encountered\neffective actions (see the prefactors $1\/M_W^2$ and $1\/M_W^4$). As such these\nhigher-dimensional operators have no local analogue in the Standard Model \nLagrangian. They are required for achieving an explicit breaking of the \n$SU(2)$ symmetry amongst the $SU(2)$ gauge bosons in the transverse sector. \nAfter all, the loop effects in the Standard Model also lead to such explicit \nsymmetry-breaking effects. \n\nFor completeness we now list the two-point gauge-boson interactions induced by \nthe above-specified non-local operators:\n\\begin{equation}\n \\begin{picture}(80,30)(0,-2)\n \\Photon(0,0)(80,0){2}{10}\n \\ArrowLine(25,-12)(26,-12)\n \\Line(15,-12)(25,-12)\n \\ArrowLine(55,-12)(54,-12)\n \\Line(65,-12)(55,-12)\n \\Text(0,8)[lb]{$V_1,\\mu_1$}\n \\Text(0,-8)[lt]{$q_1$}\n \\Text(80,8)[rb]{$V_2,\\mu_2$}\n \\Text(80,-8)[rt]{$q_2$}\n \\GCirc(40,0){3}{0}\n \\end{picture} \n \\hspace*{5ex}\n :\\\n i\\,\\tilde{\\Sigma}^{\\mu\\nu}_{\\sss{NL}}(q_1,q_2)\n =\n i\\,\\Bigl( q_1^{\\mu} q_1^{\\nu} - q_1^2 g^{\\mu\\nu} \\Bigr)\\,\n \\tilde{\\Pi}_{\\sss{NL}}^{V_1V_2}(q_1^2)\\,(2\\pi)^4\\,\\delta^{(4)}(q_1+q_2),\n \\vspace*{2ex}\n\\end{equation}\nwith the transverse (dimensionless) self-energies given by\n\\begin{eqnarray}\n\\label{sm\/self-energies}\n \\tilde{\\Pi}_{\\sss{NL}}^{WW}(q_1^2) \\!&=&\\! \n \\tilde{\\Sigma}_2(q_1^2) \\nonumber \\\\[1mm]\n \\tilde{\\Pi}_{\\sss{NL}}^{ZZ}(q_1^2) \\!&=&\\! \n s_W^2\\,\\tilde{\\Sigma}_1(q_1^2) + c_W^2\\,\\tilde{\\Sigma}_2(q_1^2) \n - 2s_W^2\\,\\tilde{\\Sigma}_3(q_1^2) + c_W^2\\,\\tilde{\\Sigma}_4(q_1^2) \n \\nonumber \\\\[1mm]\n \\tilde{\\Pi}_{\\sss{NL}}^{\\gamma Z}(q_1^2) \\!&=&\\! \n \\tilde{\\Pi}_{\\sss{NL}}^{Z\\gamma}(q_1^2) \\ =\\ \n s_W c_W\\,\\tilde{\\Sigma}_1(q_1^2) - s_W c_W\\,\\tilde{\\Sigma}_2(q_1^2) \n + (s_W^2-c_W^2)\\,\\frac{s_W}{c_W}\\,\\tilde{\\Sigma}_3(q_1^2) \n - s_W c_W\\,\\tilde{\\Sigma}_4(q_1^2) \\nonumber \\\\\n \\tilde{\\Pi}_{\\sss{NL}}^{\\gamma\\gamma}(q_1^2) \\!&=&\\! \n c_W^2\\,\\tilde{\\Sigma}_1(q_1^2) + s_W^2\\,\\tilde{\\Sigma}_2(q_1^2)\n + 2s_W^2\\,\\tilde{\\Sigma}_3(q_1^2) + s_W^2\\,\\tilde{\\Sigma}_4(q_1^2).\n\\end{eqnarray}\nThus four self-energies are parametrized by four independent functions. As such\nall mass effects can be taken into account properly. If the theory would have\nbeen unbroken, only two functions ($\\Sigma_{1,2}$) would be available for \nparametrizing the four self-energies in the massless limit and two relations\namong the self-energies would emerge. These relations hold indeed in the FLS \nif all fermions are massless (including the top-quark)~\\cite{fls2}. \nAt high energies $\\Sigma_3$ and $\\Sigma_4$ should vanish, since effectively \nthe theory becomes unbroken.\n\nAt this point we remind the reader that we have only considered non-local\ncontributions to the transverse gauge-boson self-energies, which can be\nresummed into dressed transverse gauge-boson propagators. In principle one \nshould add also non-local terms that contribute to the longitudinal gauge-boson\nself-energies \n($\\tilde{\\Sigma}_{\\sss{long}}^{\\mu\\nu} \\propto q_1^{\\mu}q_1^{\\nu}$), which\ncan be resummed into dressed longitudinal gauge-boson propagators. Since the\nresummation in the transverse and longitudinal sectors can be performed \nindependently, the longitudinal sector with its close relation to the \ngauge-fixing procedure can be treated separately. In view of minimality we \nrefrain from adding non-local longitudinal terms. In physical matrix elements\nthe longitudinal propagators do not generate resonances and therefore there \nis no strict need for resumming (imaginary parts of) longitudinal \nself-energies. The imaginary parts that appear in the resummed transverse \npropagators of the $W,Z$ bosons are directly linked to the corresponding\ndecay widths $\\Gamma_{W,Z}$ and are hence sufficient for a proper description \nof the resonance effects. In the covariant $R_{\\xi}$ gauge \n\\begin{eqnarray}\n {\\cal L}_{R_{\\xi}}^{\\sss{gauge fix}}(x) &=& - \\,\\frac{1}{2}\\,\\Biggl\\{\n \\frac{1}{\\xi_{\\gamma}}\\,\\Bigl[\\partial^{\\mu} A_{\\mu}(x)\\Bigr]^{2}\n + \\frac{1}{\\xi_{Z}}\\,\\Bigl[\\partial^{\\mu} Z_{\\mu}(x) \n - \\xi_{Z}M_{Z}\\chi(x)\\Bigr]^{2} \n \\nonumber \\\\\n &+& \\frac{2}{\\xi_{W}}\\, \n \\Bigl[\\partial^{\\mu} W_{\\mu}^{+}(x) - i \\xi_{W}M_{W}\\phi^{+}(x)\\Bigr]\n \\Bigl[\\partial^{\\nu} W_{\\nu}^{-}(x) + i \\xi_{W}M_{W}\\phi^{-}(x)\\Bigr]\n \\Biggr\\},\n\\end{eqnarray}\nfor instance, we \nobtain the following dressed gauge-boson propagators ($V=\\gamma,Z,W$):\n\\begin{eqnarray}\n\\label{prop\/W}\n P^{VV}_{\\mu\\nu}(q,\\xi_{_V}) &=& -i\\,D_T^{VV}(q^2)\\,\n \\biggl( g_{\\mu\\nu} - \\frac{q_{\\mu}q_{\\nu}}{q^2} \\biggr)\n - \\frac{i\\xi_{_V}}{q^2-\\xi_{_V}M_V^2}\\,\\frac{q_{\\mu}q_{\\nu}}{q^2},\n \\nonumber \\\\[1mm]\n P^{\\gamma Z}_{\\mu\\nu}(q) &=& P^{Z\\gamma}_{\\mu\\nu}(q)\n \\ =\\ -i\\,D_T^{\\gamma Z}(q^2)\\,\n \\biggl( g_{\\mu\\nu} - \\frac{q_{\\mu}q_{\\nu}}{q^2} \\biggr). \n\\end{eqnarray}\nThe transverse propagator functions $D_T$ are given by\n\\begin{eqnarray}\n D_T^{WW}(q^2) &=& \n \\Bigl\\{ q^2 - M_W^2 + q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{WW}(q^2) \\Bigr\\}^{-1}, \n \\nonumber \\\\[1mm]\n D_T^{\\gamma\\gamma}(q^2) &=&\n \\Bigl[ q^2 - M_Z^2 + q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{ZZ}(q^2) \\Bigr]\/D(q^2),\n \\nonumber \\\\[1mm]\n D_T^{ZZ}(q^2) &=& \n \\Bigl[ q^2+q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma\\gamma}(q^2) \\Bigr]\/D(q^2),\n \\nonumber \\\\[1mm]\n D_T^{\\gamma Z}(q^2) &=& \n - q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma Z}(q^2)\/D(q^2),\n \\nonumber \\\\[1mm]\n D(q^2) &=&\n \\Bigl[ q^2 - M_Z^2 + q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{ZZ}(q^2) \\Bigr]\\,\n \\Bigl[ q^2 + q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma\\gamma}(q^2) \\Bigr]\n - \\Bigl[ q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma Z}(q^2) \\Bigr]^2,\n\\end{eqnarray}\nwith the explicit mass terms originating from the Higgs part of the Standard\nModel Lagrangian [see Eq.~(\\ref{SMHiggs}) below].\nHowever, this is not the complete story. We will have to redefine the \nphoton field and the electromagnetic coupling, which are by definition \nidentified by means of the $ee\\gamma$ interaction in the Thomson limit \n($q_\\gamma^2=0$). Since the $ee\\gamma$ vertex does not receive non-local\ncontributions, only the non-local photonic self-energy contributions have to \nbe adjusted. This results in a `finite renormalization' of the form \n\\begin{eqnarray}\n && \\left. q^2\\,D_T^{\\gamma\\gamma}(q^2) \\right|_{q^2=0} =\n \\frac{1}{1+\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma\\gamma}(0)} \\to 1,\n \\nonumber \\\\[1mm]\n && \\left. q^2\\,\\tilde{\\Pi}_{\\sss{NL}}^{\\gamma Z}(q^2) \\right|_{q^2=0} \\to 0.\n\\end{eqnarray} \n \nIn App.~\\ref{app:b} we list the Feynman rules for the relevant non-local \nthree- and four-point interactions, needed for a gauge-invariant treatment of\nreactions like $2f \\to 4f,\\,4f\\gamma,\\,6f$. \n\n\n\\subsection{The Higgs boson}\n\\label{sec:sm\/higgs}\n\nIn the Standard Model the Higgs part of the Lagrangian is given by\n\\begin{equation}\n\\label{SMHiggs}\n {\\cal L}^H(x) \n =\n \\biggl( D_{\\mu}\\Phi(x) \\biggr)^{\\dagger} \\biggl( D^{\\mu}\\Phi(x) \\biggr)\n + \\mu^{2}\\,\\Bigl[ \\Phi^{\\dagger}(x)\\,\\Phi(x) \\Bigr] \n - \\frac{\\lambda}{4}\\,\\Bigl[ \\Phi^{\\dagger}(x)\\,\\Phi(x) \\Bigr]^2,\n\\end{equation} \nwith the covariant derivative defined as\n\\begin{equation}\n D_{\\mu} \n = \n \\partial_{\\mu} - i g_2 \\mbox{T}^a W^a_{\\mu} + i g_1 \\frac{Y}{2}\\,B_{\\mu}.\n\\end{equation}\nThe ($Y\\!=\\!1$) complex Higgs doublet $\\Phi$ is expanded around its vacuum \nexpectation value \n$\\,<\\!|\\phi|\\!>_0 = v\/\\sqrt{2} = \\sqrt{2}\\,\\mu\/\\sqrt{\\lambda}\\,$ \naccording to Eq.~(\\ref{higgs}). The resulting Lagrangian describes one physical\nscalar particle $H$ with mass $\\,M_H = \\sqrt{2}\\,\\mu\\,$ and three degrees \nof freedom that are absorbed by the gauge bosons and that are hence rendered \nunphysical. Our aim in this subsection is to construct an effective Lagrangian\nthat generates a self-energy for the physical Higgs boson. At the same time we \nwant to avoid generating any self-energies for the unphysical Higgs modes or \nthe corresponding longitudinal gauge-boson modes. This is based on the same \nphilosophy as adopted in the previous subsection. In order to achieve this \naim we are led to a construction with only singlets, i.e.~without path-ordered \nexponentials: \n\\begin{equation}\n\\label{sm\/NL\/higgs}\n {\\cal S}_{\\sss{NL}}^H\n = \n \\frac{1}{2 v^2}\\,\\int d^4x\\,d^4y\\,\\Sigma_H(x-y)\\,\n \\Biggl(\\Bigl|\\Phi(x)\\Bigr|^{2}-\\frac{v^2}{2}\\Biggr)\\,\n \\Biggl(\\Bigl|\\Phi(y)\\Bigr|^{2}-\\frac{v^2}{2}\\Biggr).\n\\end{equation}\nThis Lagrangian induces the required self-energy in the physical Higgs \npropagator, without generating additional self-energy or tadpole contributions.\nThe combined local and non-local contributions to the two-point interaction\nbetween physical Higgs bosons read:\n\\begin{equation}\n \\begin{picture}(80,30)(0,-2)\n \\DashLine(0,0)(80,0){8}\n \\ArrowLine(25,-12)(26,-12)\n \\Line(15,-12)(25,-12)\n \\ArrowLine(55,-12)(54,-12)\n \\Line(65,-12)(55,-12)\n \\Text(0,8)[lb]{$H$}\n \\Text(0,-8)[lt]{$q_1$}\n \\Text(80,8)[rb]{$H$}\n \\Text(80,-8)[rt]{$q_2$}\n \\GCirc(40,0){3}{0}\n \\end{picture} \n \\hspace*{5ex}\n :\\\n i\\,\\tilde{\\Sigma}(q_1,q_2)\n =\n i\\,\\Bigl[ q_1^2-M_H^2+\\tilde{\\Sigma}_H(q_1^2) \\Bigr]\\,\n (2\\pi)^4\\,\\delta^{(4)}(q_1+q_2),\n \\vspace*{2ex}\n\\end{equation}\nwhich can be inverted trivially to give the dressed Higgs-boson propagator\n\\begin{equation} \n\\label{prop\/H}\n P^H(q) = \\frac{i}{q^2-M_H^2+\\tilde{\\Sigma}_H(q^2)}.\n\\end{equation}\nThe remaining non-local scalar interactions can be found in App.~\\ref{app:b}. \n\nThis time we can explicitly check the proposed gauge-invariant resummation \nprocedure by considering reactions like $\\phi^+\\phi^- \\to \\chi\\chi$ in the \nlimit $M_H^2 \\gg q^2 \\gg M_{W,Z}^2$. Indeed, we find that in leading \napproximation the vertex and box graphs are identical to the non-local three- \nand four-point interactions, provided that tadpole renormalization is applied.\nThis is caused by the fact that the scalar three- and four-point functions \nreduce to two-point functions as a result of the exchange of the heavy \n(physical) Higgs bosons. It should be noted that the check only works for the\nleading terms, since the sub-leading contributions will already contain \ninformation on higher-order non-local towers (e.g.~the ones that start at \nthree-point level). \n\n\n\\subsection{The top-quark}\n\\label{sec:sm\/top}\n\nIn the Standard Model the fermions acquire their mass through the Yukawa\ninteractions with the Higgs field. Since all masses are basically different,\nthe $SU(2)$ symmetry is explicitly broken in the fermionic doublets. As such\nthe method described in Sect.~\\ref{sec:top} is not applicable to the \nresummation of fermion self-energies in the Standard Model. In this subsection \nwe concentrate on the top-quark, which has a large decay width and therefore \ncan be described by means of perturbative methods.\nWe start off by writing down the Yukawa interaction for the top-quark:\n\\begin{equation}\n\\label{sm\/top\/yukawa}\n {\\cal L}_{\\sss{Yu}}^t(x) \n = \n -\\,f_t\\,\\overline{Q}_L(x)\\,\\tilde{\\Phi}(x)\\,t_R(x) + \\mbox{h.c.},\n\\end{equation}\nwhere $f_t$ is the top-quark Yukawa coupling. The doublet $\\overline{Q}_L(x)$\nand singlet $t_R(x)$ can be expressed in terms of the top-quark and \nbottom-quark fields $t(x),b(x)$ according to\n\\begin{equation} \n Q_L(x) = \\left( \\begin{array}{c}\n t_L(x) \\\\[1mm]\n b_L(x)\n \\end{array} \\right)\n\\end{equation}\nwith\n\\begin{equation}\n t_L(x) = \\frac{1-\\gamma_5}{2}\\,t(x),\n \\qquad\n t_R(x) = \\frac{1+\\gamma_5}{2}\\,t(x)\n \\quad \\mbox{and} \\quad\n b_L(x) = \\frac{1-\\gamma_5}{2}\\,b(x).\n\\end{equation}\nIn order to give a mass to the top-quark, a Higgs doublet $\\tilde{\\Phi}$ with \nopposite hypercharge ($Y\\!=\\!-1$) is required: \n\\begin{equation} \n \\tilde{\\Phi}(x)\n =\n \\left( \\begin{array}{c}\n [v+H(x)-i\\,\\chi(x)]\/\\sqrt{2}\\\\[1mm]\n -\\,\\phi^-(x) \n \\end{array} \\right)\n \\quad \\mbox{with} \\quad\n \\phi^-(x) = \\Bigl[ \\phi^+(x) \\Bigr]^{\\dagger}.\n\\end{equation}\nThe resulting top-quark mass is given by $m_t=vf_t\/\\sqrt{2}$. Based on this\nYukawa interaction it is not difficult to construct a non-local effective \naction that generates a mass-like top-quark self-energy. There are two ways to \nnon-localize the three fields in Eq.~(\\ref{sm\/top\/yukawa}).\nThe first one involves a non-local interaction between $SU(2)$ singlets:\n\\begin{equation}\n\\label{sm\/NL\/top}\n {\\cal S}_{\\sss{NL}}^t \n = \n \\frac{\\sqrt{2}}{v}\\,\\int d^4x\\,d^4y\\,\\Sigma_t(x-y)\\,\\biggl\\{\n \\Bigl[\\, \\overline{Q}_L(x)\\,\\tilde{\\Phi}(x) \\,\\Bigr]\n \\,U_1(x,y)\\,U_3(x,y)\\,t_R(y) \n + \\mbox{h.c.} \\biggr\\},\n\\end{equation}\nwhere $U_1$ and $U_3$ are the path-ordered exponentials corresponding\nto $U(1)_Y$ (for $Y=4\/3$) and $SU(3)_C$, respectively. The latter path-ordered\nexponential enters as a result of the fact that the top-quark also carries a \ncolour charge. The second way of non-localizing Eq.~(\\ref{sm\/top\/yukawa})\ninvolves a non-local interaction between the $SU(2)$ doublets \n$\\overline{Q}_L(x)$ and $[\\tilde{\\Phi}(y)t_R(y)]$, connected by a string of \npath-ordered exponentials $U_1(x,y)\\,U_2(x,y)\\,U_3(x,y)$. Note, that both \neffective Lagrangians contribute to the top-quark two-point interaction in the \nsame way and therefore both allow a gauge-invariant resummation of the \nself-energy. A particular choice can be made on the basis of either explicit \nphysical requirements (like the properties under parity transformations) or \nminimality considerations. In this paper we consider in detail the simplest of \nthe two effective Lagrangians, given by Eq.~(\\ref{sm\/NL\/top}).\n\nThe combined local and non-local contributions to the top-quark \ntwo-point interaction then read:\n\\begin{equation} \n \\begin{picture}(60,15)(0,-2)\n \\ArrowLine(0,0)(30,0)\n \\ArrowLine(30,0)(60,0)\n \\Text(0,8)[lb]{$t$}\n \\Text(0,-8)[lt]{$p$}\n \\Text(60,8)[rb]{$t$}\n \\Text(60,-5)[rt]{$p'$}\n \\GCirc(30,0){3}{0}\n \\end{picture}\n \\hspace*{5ex}\n :\\ \n i\\,\\tilde{\\Sigma}(p,-p') \n = \n i\\,(2\\pi)^4\\,\\delta^{(4)}(p-p')\\,\n \\Bigl[ p\\hspace{-0.42em}\/}%\\hspace{0.1em} - m_t + \\tilde{\\Sigma}_t(p^2) \\Bigr],\n \\vspace*{2ex} \n\\end{equation}\nwhich results in the following dressed top-quark propagator:\n\\begin{equation} \n\\label{prop\/t}\n P^t(p) = \\frac{i}{p\\hspace{-0.42em}\/}%\\hspace{0.1em} - m_t + \\tilde{\\Sigma}_t(p^2)}.\n\\end{equation}\n\nAt first sight the effective action (\\ref{sm\/NL\/top}) seems to have little in\ncommon with the effective action introduced in Sect.~\\ref{sec:top}. However,\nthe part originating from the non-zero vacuum expectation value of the Higgs,\nwhich hence only involves fermions and gauge bosons, has the familiar form\n\\begin{equation}\n\\label{sm\/NL\/top\/nohiggs}\n {\\cal S}_{\\sss{NL}}^{t,\\,v} \n = \n \\int d^4x\\,d^4y\\,\\Sigma_t(x-y)\\,\\bar{t}(x)\\,U_1(x,y)\\,U_3(x,y)\\,t(y).\n\\end{equation}\nIn App.~\\ref{app:b} we list the Feynman rules for the various non-local \nthree- and four-point interactions. Particularly noteworthy are the mixed\nQCD--electroweak interactions, involving both gluons and electroweak bosons,\nwhich are needed for the construction of gauge-invariant resummed amplitudes \nin certain mixed QCD--electroweak processes (like $e^{+}e^{-}\\to t\\bar{t}g$).\n\n\n\\subsection{Some simple examples}\n\\label{sec:sm\/examples}\n\nA substantial simplification occurs when all non-local coefficients are taken \nto be delta-functions:\n\\begin{equation}\n \\Sigma_j(x-y) = \\Sigma_j\\,\\delta(x-y) \\qquad (j=1,2,3,4,H,t).\n\\end{equation}\nBy choosing appropriate values for the complex constants $\\Sigma_j$, the \nsimplified set of non-local actions can be used to implement the decay widths \nof unstable particles in a concise, gauge-invariant way. Strictly speaking, \nhowever, the proposed simplification is not supported by the actual loop \neffects in gauge theories, where no imaginary parts occur for space-like \nmomenta. Nevertheless, it has become a very popular (ad hoc) procedure.\n\nAccording to Eqs.~(\\ref{prop\/H}) and (\\ref{prop\/t}), the simplification \ncorreponds to constant shifts in the Higgs and top-quark propagators:\n\\begin{equation}\n P^H(q) = \\frac{i}{q^2 - M_H^2 + \\Sigma_H}\n \\qquad \\mbox{and} \\qquad\n P^t(p) = \\frac{i}{p\\hspace{-0.42em}\/}%\\hspace{0.1em} - m_t + \\Sigma_t}.\n\\end{equation}\nAs a result of the delta-functions in the non-local coefficients, the effective\nLagrangians defined by ${\\cal S}_{\\sss{NL}}^H$ in Eq.~(\\ref{sm\/NL\/higgs}) and ${\\cal S}_{\\sss{NL}}^t$ in \nEq.~(\\ref{sm\/NL\/top}) become proportional to the corresponding local \nLagrangians:\n\\begin{equation}\n {\\cal L}_{\\sss{NL}}^H(x) = -\\,\\Sigma_H\\,\\frac{{\\cal L}_{\\sss{pot}}^H(x)}{M_H^2}\n \\qquad \\mbox{and} \\qquad\n {\\cal L}_{\\sss{NL}}^t(x) = -\\,\\Sigma_t\\,\\frac{{\\cal L}_{\\sss{Yu}}^t(x)}{m_t},\n\\end{equation}\nwith ${\\cal L}_{\\sss{pot}}^H$ indicating the part of the local Higgs Lagrangian\nthat corresponds to the Higgs potential. From this it is clear that the \ncombined effect of all non-local interactions amounts to the mere effective\nreplacements \n\\begin{equation}\nM_H^2 \\to M_H^2 - \\Sigma_H \n\\qquad \\mbox{and} \\qquad \nm_t \\to m_t - \\Sigma_t\n\\end{equation}\nin the Standard Model Lagrangian. Note that for imaginary shifts \n(e.g.~$\\Sigma_H=iM_H\\Gamma_H$) this procedure resembles the so-called \nfixed-width scheme.%\n\\footnote{By appropriately choosing the complex constants $\\Sigma_{t,H}$, it is\n also possible to effectively replace the masses by the complex poles\n of the propagators [e.g.~with \n $\\Sigma_H = iM_H^2\\Gamma_H\/(M_H+i\\,\\Gamma_H)$ one obtains the \n effective replacement $M_H^2 \\to M_H^2\/(1+i\\,\\Gamma_H\/M_H)$]. Such a \n complex pole mass is often better suited for the description of \n resonances associated with heavy unstable particles~\\cite{massdef}.}\nHowever, in contrast to our non-local approach, the fixed-width scheme applies \nthe effective replacements only to the propagators. So, in general the \nfixed-width scheme has to be adapted whenever the mass terms in the \nhigher-point interactions play a role.\n\nIn the gauge-boson sector we need a further simplification. \nThe higher-dimensional operators in ${\\cal S}_{\\sss{NL}}^{\\Phi}$ have no Standard Model\nanalogues. Therefore, a large number of compensating higher-point interactions\nremain, even if all non-local coefficients are taken to be delta-functions.\nAt this point we can exploit the fact that we don't strictly need all four\nnon-local gauge-boson coefficients for a gauge-invariant treatment of unstable\n$W$ and $Z$ bosons. In order to properly generate the two corresponding decay\nwidths, it is formally sufficient to have only two independent non-local\ncoefficients. A huge simplification is achieved by setting \n$\\Sigma_{3,4}=0$. This comes at a price, though. As mentioned in \nSect.~\\ref{sec:sm\/wz}, two relations among the gauge-boson \nself-energies will emerge, like in the unbroken theory. This means that the \nself-energies involving photons are not independent anymore. In fact this is\nnot a real problem, since we will anyhow have to redefine the photon field and \nthe electromagnetic coupling according to the $ee\\gamma$ interaction in the\nThomson limit ($q_\\gamma^2=0$).\n\nThe net effect of the simplifications in the gauge-boson sector amounts to a\nrescaling of the $U(1)_Y$ and $SU(2)_L$ terms in the Yang--Mills Lagrangian \n(\\ref{sm\/yang-mills}):\n\\begin{equation}\n\\label{examples\/YM}\n {\\cal L}^{\\sss{YM}}(x) + {\\cal L}_{\\sss{NL}}^{\\sss{YM}}(x) \n = \n -\\,\\frac{1}{2}\\,(1+\\Sigma_2)\\,\n \\mbox{Tr}\\,\\Bigl[ \\mbox{\\boldmath $F$}_{\\mu\\nu}(x)\\,\\mbox{\\boldmath $F$}^{\\mu\\nu}(x) \\Bigr]\n -\\frac{1}{4}\\,(1+\\Sigma_1)\\,B_{\\mu\\nu}(x)\\,B^{\\mu\\nu}(x). \n\\end{equation}\nThese rescaling factors can be absorbed into the gauge-boson fields and the\ncoupling constants according to\n\\begin{equation}\n\\label{redef} \n W^a = \\frac{{W'}^a}{\\sqrt{1+\\Sigma_2}},\n \\quad \n g_2 = g'_2\\,\\sqrt{1+\\Sigma_2}\n \\qquad \\mbox{and} \\qquad\n B = \\frac{B'}{\\sqrt{1+\\Sigma_1}},\n \\quad \n g_1 = g'_1\\,\\sqrt{1+\\Sigma_1}.\n\\end{equation}\nIn terms of the redefined fields and couplings the Lagrangian \n(\\ref{examples\/YM}) retrieves the original Yang--Mills form. At the same time\nthe other Standard Model interactions are not changed by the redefinitions, as\nthe covariant derivatives stay the same. So, the only noticeable \nchanges involve the gauge-boson mass matrix and consequently the $W^3$--$B$ \nmixing, which are both defined in terms of the coupling constants:\n\\begin{eqnarray}\n\\label{examples\/YM\/upshot}\n M_W^2 = \\frac{1}{4}\\,v^2 g_2^2 &\\to & \n {M'_W\\!\\!}^2 = \\frac{M_W^2}{1+\\Sigma_2},\n \\nonumber \\\\[1mm]\n M_Z^2 = \\frac{1}{4}\\,v^2 (g_1^2+g_2^2) &\\to &\n {M'_Z\\!}^2 = M_Z^2\\,\\frac{1+c_w^2\\Sigma_1+s_w^2\\Sigma_2}\n {(1+\\Sigma_1)(1+\\Sigma_2)} \n \\equiv \\frac{M_Z^2}{1+C_Z},\n \\nonumber \\\\[1mm]\n c_w = M_W\/M_Z &\\to & \n c'_w = M'_W\/M'_Z = c_w\\,\\sqrt{\\frac{1+C_Z}{1+\\Sigma_2}}.\n\\end{eqnarray}\nFor an imaginary non-local coefficient $\\Sigma_2 = i\\,\\Gamma_W\/M_W$ the \nredefined $W$ mass is identical to the so-called complex pole mass \n${M'_W\\!\\!}^2 = (M_W^2-iM_W\\Gamma_W)\/(1+\\Gamma_W^2\/M_W^2)$. A similar pole mass\ncan be obtained for the $Z$ boson by choosing $\\Sigma_1$ in such a way that \n$C_Z = i\\,\\Gamma_Z\/M_Z$.%\n\\footnote{By choosing $\\Sigma_2 = i\\,\\Gamma_W\/(M_W-i\\,\\Gamma_W)$ and \n $C_Z = i\\,\\Gamma_Z\/(M_Z-i\\,\\Gamma_Z)$ one can obtain the usual \n fixed-width masses ${M'_V\\!}^2 = M_V^2-iM_V\\Gamma_V$.}\nThe redefined physical states ${W'}^{\\pm},\\,Z'$ and $A'$ are obtained from \n${W'}^a$ and $B'$ in the usual way in terms of the redefined mixing angle.\nFor instance, $Z'=Z\\sqrt{1+C_Z}$. Since the interactions between the gauge \nbosons and fermions are unchanged, the $A'$ field is by definition the photon \nfield and the redefined coupling $e' = g'_1g'_2\/\\sqrt{{g'_1}^2+{g'_2}^2}$ is \nby definition the electromagnetic coupling constant. This is equivalent to\nperforming finite renormalizations in order to absorb the non-local \ncontributions to the photon wave function and the electromagnetic charge. \nSo, the combined effect of all non-local interactions amounts to the effective \nreplacements given in Eq.~(\\ref{examples\/YM\/upshot}). These effective \nreplacements can be extended to the longitudinal sector by simply rescaling the\ngauge parameter in the gauge-fixing part of the Lagrangian. For instance,\nwith the rescaling $\\xi_{_W} = \\xi'_{_W}\/(1+\\Sigma_2)$ the dressed $W$-boson \npropagator in Eq.~(\\ref{prop\/W}) becomes\n\\begin{equation}\n P^{WW}_{\\mu\\nu}(q,\\xi_{_W}) \n = \n \\frac{-i}{[1+\\Sigma_2]\\,[q^2-{M'_W\\!\\!}^2]}\\,\n \\biggl( g_{\\mu\\nu} - \\frac{q_{\\mu}q_{\\nu}}{q^2} \\biggr)\n - \\frac{i\\xi'_{_W}}{[1+\\Sigma_2]\\,[q^2-\\xi'_{_W}{M'_W\\!\\!}^2]}\\,\n \\frac{q_{\\mu}q_{\\nu}}{q^2},\n\\end{equation}\nwhere the factor $1\/(1+\\Sigma_2)$ can be absorbed into the $W$-boson fields\naccording to Eq.~(\\ref{redef}). This simple example for the gauge bosons \ncoincides with the complex-mixing-angle procedure that was adopted in \nRef.~\\cite{DDRrad} for calculating the radiative processes \n$e^+e^- \\to 4f\\gamma$. \n\nThe appeal of the above-discussed special examples lies in the simplicity of \nthe net prescriptions that follow from the effective Lagrangians, allowing a\nstraightforward implementation into the existing Monte Carlo programs. In spite\nof the simplicity, nevertheless a reasonably good description of the \nunstable-particle resonances can be achieved.\nIn case of a more rigorous treatment of unstable particles, involving a proper \nenergy dependence of the absorptive parts of the self-energies, one is forced\nto take into account the full extent of the effective Lagrangians. Or, in other\nwords, one has to properly take into account the relevant sets of \ngauge-restoring multi-particle interactions (see e.g.~the Feynman rules listed\nin App.~\\ref{app:b}). \n\n\n\n\n\\section{Conclusions and outlook}\n\nIn this paper we have introduced a method that offers the\npossibility of performing gauge-invariant tree-level calculations\nwith unstable particles in intermediate states. To this end non-local\ngauge-invariant Lagrangians are introduced, which allow the gauge-invariant\nresummation of self-energies and therefore give rise to dressed (regular)\npropagators for unstable particles. Certainly for practical applications \nthe resummed self-energies will be taken from the underlying gauge theory,\nbut in principle the choice is arbitrary in our approach. For every choice the \ngauge restoring vertices will be different. This leaves open the possibility \nof studying {\\it ad hoc} methods for implementing the decay widths of the \nunstable particles, like the fixed-width scheme. From the non-local Lagrangian \none obtains in general an infinite set of multi-point vertices. \nThese vertices provide an explicit solution of the full set of \nghost-free Ward identities and thereby restore the gauge invariance of the \nresummed amplitudes. For a given multi-particle process only a limited number \nof those vertices contribute. In the paper we have given the derivation of the \nmulti-point vertices from the non-local Lagrangians, and we have explicitly \nlisted all relevant modifications of the Standard Model vertices for up to \nfour external particles. These modified vertices are related to the unstable \ngauge bosons, the Higgs particle and the top-quark, which all occur in the \nelectroweak\/QCD calculations for present and future collider experiments. \nIt should be kept in mind that there are other multi-point vertices that would \nalso lead to gauge-invariant amplitudes. In other words, the vertices are not \nunique, but our prescription gives in a minimal way a set of vertices that \nrestores gauge invariance.\n\nUsually one restricts the final-state particles in a process to stable\nparticles, i.e.~fermions, photons and gluons. The final-state fermions can be \neither massive or massless. In our approach this poses no problem, since the \ncalculation remains gauge-invariant in either case. The vertices given in \nApp.~\\ref{app:b} allow gauge-invariant calculations for unstable-particle \nprocesses like $e^{+}e^{-}\/q\\bar{q}\\,\/gg\\to 4f\\gamma,\\,4fg,\\,6f$. \nMany of the present-day unstable-particle production processes lead to these \nfinal states. Examples are $W^{+}W^{-}\\gamma$ production at LEP2 and $t\\bar{t}$\nproduction at the Tevatron. For the latter process gluon radiation could\nalso be of practical importance. In that case one should\nextend the list in App.~\\ref{app:b} and add the vertices that contribute \nto a final state with six fermions and one gluon.\nFor instance, a 3-gluon--$t\\bar{t}$ vertex would arise.\n\nAlthough this paper was primarily motivated by the phenomenological\nneed to perform sensible tree-level calculations with unstable\nparticles, other applications seem possible.\n\nOne possible application could be the gauge-invariant resummation of gluon \npropagators in QCD calculations. In this way part of the higher-order \ncorrections can be taken into account in a gauge-invariant way. The effect of \nthis resummation on multiparton amplitudes can now be investigated\nusing our method. In a similar way one could study the resummation \nof the electroweak gauge-boson propagators in terms of running (effective)\ncouplings.\n\nAnother intriguing question is whether the non-local Lagrangian technique\ncould be used to construct a gauge-invariant bosonic self-energy\nby adding gauge-restoring parts from vertex and box diagrams \nto a non-gauge-invariant self-energy. \nIn other words, could the method of non-local Lagrangians be used \nto carry out the pinch technique?\n\nAnother issue is whether one could use the propagators and\nvertices derived in the paper to perform quantum loop corrections. \nTo our knowledge this remains an open question.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nEvolutionary algorithms have been used for a wide range of different tasks. This includes solving classical $\\mathcal{NP}$-hard optimization problems, optimizing the design for complex engineering problems~\\cite{FLEMING20021223}, algorithms for machine learning~\\cite{DBLP:journals\/csur\/TelikaniTBG22}, and approaches in the area of algorithm selection and configuration~\\cite{Kerschke2019aas}.\n\nIn this paper, we focus on a crucial problem in the context of algorithm selection, namely the design of problem instances where two algorithms show a significantly different behaviour. We consider instances for the classical Euclidean traveling salesperson problem~(TSP) for which a wide range of algorithms has been introduced over the last 50 years~\\cite{10.5555\/1374811}. The problem is to find a round-trip tour of minimal costs through $n$ cities where the cities are given by coordinates in the 2D Euclidean space and pairwise distances correspond to the Euclidean distances of the points.\n\nThe complementary ability of algorithms for solving the TSP has been subjected to a wide range of studies. Center of these investigations is the characterization of features of TSP instances where a considered set of algorithms shows a significantly different behaviour. Studies have considered simplified variants of local search~\\cite{smithmiles2010}, approximation algorithms~\\cite{nallaperuma2013}, and state-of-the-art inexact TSP solvers such as EAX and LKH~\\cite{BT2016EvolvingInstances,BT2016UnderstandingCharacteristics,kerschke2018}.\n\nEvolutionary diversity optimization~(EDO) has recently been applied using various diversity measures to create diverse sets of images and TSP instances~\\cite{alexander2017evolution,neumann2018discrepancy,neumann2019evolutionary}. For TSP instances, it has been shown that such approaches are able to create TSP instances with desired properties (such as good or bad performance behaviour) with a much wider range in the feature space~\\cite{doi:10.1162\/evcoa00274}.\nFurthermore, recent investigations have used and analyzed EDO methods to create diverse sets of high quality solutions for the TSP and other permutation problems~\\cite{viet2020evolving,DoGN021,NikfarjamBN021a,NikfarjamB0N21b}.\n\nQuality diversity~(QD)~\\cite{PughSS16} follows a similar goal as EDO. Algorithms explore a feature space of possible solutions to a given problem. Simple approaches, e.~g.,Map-Elites, partition the feature space into boxes and store for each box the best solution that has been found for its range of feature combinations.\nQD algorithms have mainly been applied in the context of robotics and games~\\cite{CullyM13,ZardiniZZIF21}. \nIn the context of combinatorial optimization, a QD approach has recently been used for the traveling thief problem and it has been shown that significantly better solutions can be obtained~\\cite{NNN2021qdttp}.\n\n\\changed{In this paper, we use for the first time the quality diversity approach in the context of instance generation}. We use a Map-Elites approach which for a given set of features divides the feature space into boxes and stores for each box TSP instances that show the maximum performance difference for two given algorithms.\n\nUsing this approach, we explore the feature space and optimize instances within each box with respect \\changed{to solver performance}.\nThis allows to get an overview on differences of instances in the feature space in terms of their difficulties for a given set of TSP solvers.\nIn our experiments we evolve instances for the insertion heuristics Farthest-Insertion and Nearest-Insertion as well as for state-of-the-art exact TSP solvers EAX and LKH on different two-dimensional feature spaces. We compare our QD approach against \\changed{classical $(\\mu+1)$-EAs and EDO-tailored $(\\mu+1)$-EAs modified to store their \"footprint\" in a similar way. Our experimental investigations show that with the proposed QD approach (a) in most cases a much wider range of feature combinations is explored during the evolutionary process and (b) the objective scores of the evolved instances are either comparable or even better in all cases}. Put together the QD approach is capable of generating a huge amount of interesting instances in a single run.\n\nThe paper is organized as follows. In Section~\\ref{sec:problem} we describe the task of instance generation in detail, discuss challenges and review approaches from the literature. In Section~\\ref{sec:method} we introduce our quality diversity approach. Experimental studies on simple and state-of-the-art TSP heuristics follow in Section~\\ref{sec:experiments_cheap} and Section~\\ref{sec:experiments_expensive} respectively. We conclude the paper in Section~\\ref{sec:conclusion}.\n\n\\section{Problem formulation and history}\n\\label{sec:problem}\n\nLet $\\mathcal{A}$ be a portfolio of algorithms which show complementary performance on a set of problem instances $\\mathcal{I}$. Further, let $F:\\mathcal{I} \\to \\mathcal{F} \\subseteq \\mathbb{R}^p, p \\geq 1$ be a \\emph{feature-mapping} which maps instances to a $p$-dimensional real-valued numeric vector in the \\emph{feature-space} $\\mathcal{F}$ that ideally captures the characteristics of the instance which makes it easy or hard for the algorithms in the portfolio. In the context of the Euclidean TSP, features are for example the share of points on the convex-hull of the point coordinates or structural properties of transformations of the original instance, e.~g., statistics on the weak\/strong connected components of a $k$-nearest neighbor graph~($k$-NNG). The idea of \\emph{(per-instance) algorithm selection}~(AS) is to learn an algorithm selector $S : \\mathcal{F} \\to \\mathcal{A}$ which -- for a previously unseen instance -- predicts the algorithm which will likely perform best on it based on its cheap-to-evaluate features only~\\cite{Rice1976ASProblem,Kerschke2019aas}.\n\\begin{algorithm}[t]\n\t\\SetKwInOut{Input}{input}\n \\Input{Objective $f$, population size $\\mu$.}\n Initialize population $P$ of $\\mu$ instances at random\\;\n\t\\While{Stopping condition not met}{\n Select random instance $I \\in P$\\;\n Generate $I'$ by mutating $I$\\;\n \\If{$f(I')$ is not worse than $f(I)$}{\n Replace $I$ with $I'$ in $P$\\;\n }\n }\n \\Return{Best instance from $P$}\\;\n \\caption{Classical $(\\mu+1)$~EA for instance generation.}\n \\label{alg:ea_evolver}\n\\end{algorithm}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.95\\columnwidth]{figures\/operators.pdf}\n \\caption{Random uniform Euclidean instance (left) with Gaussian mutation~(center) and implosion mutation~(right) applied to it. Blue points are affected by the mutation.}\n \\label{fig:mutation_operators}\n\\end{figure}\n\nIn order to understand the strengths and weaknesses of algorithms and to be able to learn reasonable selectors, the instance set $\\mathcal{I}$ should ideally be diverse on two different levels: (D1) diverse\/complementary with respect to algorithm performance and (D2) diverse with respect to the structure of the instances which is captured by the coverage of the feature-space. Finding such a set of instances is challenging~\\cite{smithmiles2010}. \\changed{Since the availability of real-world instances for research is limited}, a systematic generation process is desirable. In this paper we focus on the problem of evolving a large set of diverse instances that satisfy both diversity requirements. \nFor the TSP, early approaches focused on diversity in the performance space (D1). To this end Mersmann et al.~\\cite{Mersmann13-TSPmeta} introduced a $(\\mu+1)$~EA to generate instances which are hard\/easy for an algorithm $A$ by minimizing\/maximizing the gap to a known optimum; they evolved small instances for classical approximation algorithms and local-search algorithms. Subsequent works evolved instances which are easy for one algorithm $A_1$ and hard(er) for another algorithm $A_2$ by minimizing the ratio of performances measured by the \\emph{penalized average runtime~(PAR~\\cite{bischl2016})} for state-of-the-art inexact TSP solvers EAX and LKH~\\cite{BT2016EvolvingInstances,BT2016UnderstandingCharacteristics}. The EAs used for these tasks -- leaving apart details -- very much follow the pseudo-code given in Algorithm~\\ref{alg:ea_evolver}. In a nutshell the algorithm initializes a population $P$ with $\\mu$ \\emph{random uniform Euclidean}~(RUE) instances. I.~e., each of the $n$ city coordinates is placed uniformly at random within a bounding-box usually set to $[0,1]^2$ (see left-most plot in Figure~\\ref{fig:mutation_operators}). Within the evolutionary loop one instance is selected to produce an offspring. The classical mutation operators either add some Gaussian noise to a subset of the points of the parent as illustrated in Figure~\\ref{fig:mutation_operators}~(center) or re-locate a subset of the points by sampling new coordinates, again, uniformly at random. An elitist $(\\mu+1)$-strategy assures that the best individuals with respect to the objective function survive. On termination the best instance is usually returned. It turned out that even though goal (D1) was successfully met, (D2) was not; in fact no difference to RUE instances was visible to the naked eye and as a consequence the feature space $\\mathcal{F}$ was not well covered.\nLater, this issue was approached by different groups of researchers. Gao et al.~\\cite{GaoNN16feature} used a bi-level approach which aimed to evolve a feature-diverse population of instances with minimal performance quality requirements (see also the recent journal extension~\\cite{doi:10.1162\/evcoa00274}). To this end they developed a diversity measure based on a weighted sum approach to calculate each individuals contribution to the feature diversity in the population. Though appealing, the bi-level character delayed the search for low objective values. \\changed{Later TSP instance generation was approached in the context of evolutionary diversity optimization~(EDO). In EDO, given a (close-to-)optimal solution OPT and a parameter $\\alpha >1$ the goal is to evolve a population $P$ such that $f(I) \\leq (1+\\alpha)\\cdot\\text{OPT}$ for all $I \\in P$ and a diversity measure $D : P \\to \\mathbb{R}$ is maximized. Promising contributions in this field adopted the star-discrepancy~\\cite{neumann2018discrepancy} or well-known performance indicators from multi-objective optimization~\\cite{neumann2019evolutionary} as feature-space diversity measure $D$.}\nBossek et al.~\\cite{Bossek2019evolv} recently took a different path and introduced a sophisticated set of \"creative\" mutation operators. These operators addressed the short-comings of the aforementioned operators (re-location and Gaussian noise) by strongly affecting the parents' point coordinates. An easy to illustrate example is the \\emph{implosion} mutation where all points within a random radius around a randomly sampled center of implosion are attracted towards the center (see right-most plot in Figure~\\ref{fig:mutation_operators}). The authors showed in their study, adopting a $(5+1)$~EA where in each iteration one of these creative mutation operators was applied to the parent, that this approach -- applied many times to produce a set of instances -- covered a much wider part of the feature space without explicit feature-space diversity preservation at all.\n\nAll approaches discussed so far are very wasteful in the sense that only the best individual of the final population is returned as output or the whole usually small\\footnote{Due to the computational costs of the instance generation process the population is usually small in order to allow for as many generations as possible.} final population for the approaches with diversity preservation. Nevertheless, many potentially interesting intermediate instances \\added{become} extinct in the course of optimization. This is undesirable since the entire process is obviously very time consuming.\\footnote{In recent studies on state-of-the-art inexact TSP solvers, for each generated instance the \\underline{exact} optimum has to be calculated in order to calculate the PAR-score. E.~g., in~\\cite{Bossek2019evolv} the generation of a single instance was given a time budget of 48h CPU time.} Hence, there is a need for a systematic means to memorize instances of interest.\n\n\n\\section{A Quality Diversity Approach}\n\\label{sec:method}\n\nA straight-forward idea is to simply save all intermediate instances, i.~e., the footprint of the EA so to say. This approach is not target-oriented however since instances with large performance gaps are desirable; see diversity requirement (D1). We now present an idea that borrows ideas from quality diversity~(QD). QD is a recently emerged branch of evolutionary computation. In the so-called \\emph{Map-elites} approach the \\emph{target space} is partitioned into \\emph{boxes} where each box stores the best solution found for all target space realizations closest to its box-center. This approach is highly successful, e.~g., in evolving different robot behavior~\\cite{CullyM13} where the target space corresponds to the behavioral space of the robots.\nIn our setting the target space corresponds to the feature space $\\mathcal{F} \\subseteq \\mathbb{R}^p$ we aim to cover. Each box of our QD-algorithm corresponds to one possible feature combination. The algorithm is designed for discrete features where, given a fixed instance size $n$, the number of possible realizations of each feature is countable and bounded. This is indeed the case for many TSP-features of interest, e.~g., $k$-NNG based features or depth-based MST features~\\cite{Pihera2014MLTSP} which are frequently among the top discriminating features in AS-models~\\cite{Pihera2014MLTSP,Kerschke2018leveraging,Seiler2020DeepLearning}.\\footnote{E.~g., given $n$ cities, the number of weak connected components in the $k$-NNG can take values in $\\{1, 2, \\ldots, n-k\\}$ only~\\cite{HBPSTK2021tspnormalization}.} We refer the interested reader to recent work by Heins et al.~\\cite{HBPSTK2021tspnormalization} who derived theoretical lower and upper bounds for a large collection of such features for feature normalization.\n\\begin{algorithm}[t]\n\t\\SetKwInOut{Input}{input}\n \\Input{Objective $f$, feature-mapping $F$.}\n Initialize empty map $M$\\;\n Initialize instance $I$ at random\\;\n Calculate $F(I)$ and store $I$ in $M[F(I)]$\\;\n \\While{Stopping condition not met} {\n Select an instance $I$ from $M$ uniformly at random\\;\n Generate $I'$ by mutating $I$\\;\n \\If{Box $M[F(I')]$ is not covered}{\n Store $I'$ in $M[F(I')]$\\;\n }\\Else\n Let $I''$ be the instance stored in box $M[F(I')]$\\;\n \\If{$f(I')$ is not worse than $f(I'')$}{\n Replace $I''$ with $I'$ in $M[F(I')]$\\;\n }\n }\n }\n \\Return{$M$}\\;\n \\caption{QD approach for instance generation.}\n \\label{alg:qd_evolver}\n\\end{algorithm}\nFor each box our algorithm -- in classical QD-fashion -- stores the TSP instance found over time with the best performance value. More precisely, the algorithm proceeds as shown in Algorithm~\\ref{alg:qd_evolver}. First, an empty map is initialized. The map takes the role of the \"population\" in Algorithm~\\ref{alg:ea_evolver}. Next, an RUE instance $I$ is created at random and stored in the respective box of the map indexed by $I$'s feature-vector $F(I)$.\\footnote{The map can be implemented as a hash-map allowing for very fast look-up operations.} From there on the following steps are performed until a stopping condition is met: an instance $I$ from the subset of \\emph{covered} boxes, i.~e., those that are already assigned an instance, is sampled uniformly at random. Mutation is applied to $I$ which yields another instance $I'$. Next, if the box at $F(I')$ is empty (we say the box is \\emph{hit for the first time} in the following), $I'$ is saved in any case regardless of its objective value. Otherwise, if the box is already covered, the objective function $f$ comes into play. Let $I''$ be the instance assigned to the box, i.~e., $F(I') = F(I'')$. If $f(I')$ is not worse than $f(I'')$ the box is \\emph{updated} by replacing $I''$ with $I'$. The algorithm returns all instances stored in the map upon termination. Hence, the algorithm follows an elitist strategy in each box and thus box-wise the objective function is monotonically decreasing (w.~l.~o.~g., we assume minimization).\nThe approach is simple but nevertheless very generic and powerful and by no means restricted to the TSP domain that we adopt here for a proof-of-concept study in this contribution.\n\n\n\\section{Experiments on Simple Heuristics}\n\\label{sec:experiments_cheap}\n\nIn this and the next section we continue with an extensive empirical evaluation of the proposed approach. Our experimental setup is two-fold: In the \\emph{cheap} setting we evolve small instances for simple TSP heuristics whereas in the \\emph{expensive} setting we evolve larger instances with the goal to maximize performance differences between state-of-the-art heuristics. \nThe cheap experiments -- due to fast evaluation of the objective function -- allow to explore the differences obtained after a huge number of iterations as a proof-of-concept. In the expensive setting we can afford only a fraction of the iterations due to computational limitations, but tackle interesting algorithms.\n\n\n\\subsection{Experimental setup}\n\nIn a first series of experiments we evolve small instances with $n=100$ nodes. We consider two insertion heuristics: \\emph{Farthest-Insertion}~(FI) and \\emph{Nearest-Insertion}~(NI)~\\cite{Rosenkrantz2009tsp}. FI starts with a partial tour consisting of one randomly selected node. In $(n-1)$ iterations the remaining nodes are added until a TSP tour is constructed. In the selection phase the algorithm selects the node furthest away from the currently maintained sub-route. This node is injected into the existing partial route such that the least extension of the previous partial route is created. (NI) works very much alike with a slight difference in the selection phase: NI selects the nearest node. The goal is to evolve instances which span a two-dimensional feature space space nicely and minimize the tour length ratio $FI(I)\/NI(I)$ or $NI(I)\/FI(I)$ respectively. Ratios below~1 are desirable as this indicates a performance advantage for the \"numerator\" algorithm.\nThe feature combinations consist of combinatorial features calculated on basis of a MST of the instance or its $k$-NNG as introduced in~\\cite{Pihera2014MLTSP}; for details we refer the reader to the original papers. More precisely, we consider feature combinations~(FCs) that were frequently selected by AS-models in various studies (see, e.~g., \\cite{Pihera2014MLTSP,Bossek2019evolv}):\n\\begin{enumerate}\n \\item[FC1)] Maximum size of a strong connected component in the 3-NNG (nng\\_3\\_strong\\_components\\_max) and the number of weak connected components in the 3-NNG~(nng\\_3\\_n\\_weak).\n \\item[FC2)] The number of strong connected components in the 5-NNG (nng\\_5\\_n\\_strong) and the median depth of nodes in a MST of the input graph (mst\\_depth\\_median).\n\\end{enumerate} \nIn the following we will make heavy use of the abbreviations FC1 and FC2 for ease of writing.\n\\changed{For each algorithm pair (FI vs. NI) and (NI vs. FI) and each of the two FCs we evolve instances by ten algorithms in total: QD implements Algorithm~\\ref{alg:qd_evolver}. We compare QD against classic evolutionary algorithms $(1+1)$~EA and $(50+1)$~EA as well as EDO-focused EAs $(50+1)$~ED-IGD and $(50+1)$~EA-HV from~\\cite{neumann2019evolutionary} which use inverted generational distance~(IGD) and the dimension-doubling hyper-volume indicator for (feature-)diversity maximization. These algorithms were among the best-performing EAs in~\\cite{neumann2019evolutionary} (see the reference for details). EDO-based algorithms require a good initial population. Hence, we initialize each run with clones of a solution obtained by running $(1+1)$-EA for additional $10\\,000$ iterations.\nAll algorithms are run with either only \\emph{simple} mutation operators (re-location, Gaussian mutation) from Mersmann et al.~\\cite{Mersmann13-TSPmeta} (suffix [simple]) or the full set of disruptive mutation operators introduced in Bossek et al.~\\cite{Bossek2019evolv} (suffix [all]; see also Section~\\ref{sec:problem}).\\footnote{In each mutation step one of the mutation operators is selected uniformly at random.}\nNote that the baseline algorithms return at most 50 instances and thus it is unfair to compare directly against these; the winner would be determined in advance -- QD -- since it stores all novel instances with not yet seen feature-combinations. Hence, for a fair comparison we modify the EAs to store instances the same way the proposed QD approach does (see Algorithm~\\ref{alg:ea_with_archive_evolver}). We stress that this way of saving the footprint of the EAs has also not been done before in the literature.\n\\begin{algorithm}[t]\n\t\\SetKwInOut{Input}{input}\n \\Input{Objective $f$, population size $\\mu$, feature-mapping $F$.}\n Initialize empty map $M$\\;\n Initialize population $P$ of $\\mu$ instances at random\\;\n Store $I \\in P$ at $M[F(I)]$ for all $I \\in P$\\;\n\t\\While{Stopping condition not met}{\n Select random instance $I \\in P$\\;\n Generate $I'$ by mutating $I$\\;\n \\If{Box $M[F(I')]$ is not covered}{\n Store $I'$ in $M[F(I')]$\\;\n }\\Else{\n Let $I''$ be the instance stored in box $M[F(I')]$\\;\n \\If{$f(I')$ is not worse than $f(I'')$}{\n Replace $I''$ with $I'$ in $M[F(I')]$\\;\n }\n }\n \\If{$f(I')$ is not worse than $f(I)$}{\n Replace $I$ with $I'$ in $P$\\;\n }\n }\n \\Return{$M$}\\;\n \\caption{$(\\mu+1)$~EA (see Alg.~\\ref{alg:ea_evolver}) with archive.}\n \\label{alg:ea_with_archive_evolver}\n\\end{algorithm}\nEach experiment was repeated \\changed{30 times} with different random numbers generators seeds and a budget of $1\\,000\\,000$ function evaluations. Within each objective function evaluation the tour length of each TSP heuristic is calculated as the mean over 5 independent runs to account for stochasticity in the choice of the initial node for tour construction. All algorithms were implemented in the R programming language~\\cite{Rlang}. Code and data will be publicly available in a GitHub repository upon acceptance.\\footnote{GitHub repository: \\url{https:\/\/github.com\/jakobbossek\/GECCO2022-QD-TSP}}\n\n\n\\subsection{Results}\n\nWe first study the number of boxes covered, i.~e., the number of instances \/ feature combinations hit by the algorithms over time. To this end we logged the load of the map $M$ over time.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.6\\columnwidth]{figures\/nr_of_instances_over_time.pdf}\n \\caption{Mean number ($\\pm$ standard deviation) of boxes covered (i.~e., instances detected) in the map over the course of the search for FC1 (top row) and FC2 (bottom row).}\n \\label{fig:nr_of_instances_over_time}\n\\end{figure}\n\nFigure~\\ref{fig:nr_of_instances_over_time} shows the progress of the mean number of boxes covered (with one standard-deviation error bars). \\changed{Note that we show $(50+1)$~EA-IGD in all subsequent plots as a representative EDO-EA since it performs very much like the EA-HV version. We observe that both QD approaches are highly superior to the classic EA-baselines with respect to feature space coverage. After termination the mean number of boxes covered by QD~[all] surpasses the best classic EAs' value by a factor of $>2$ for FC1 and up to a factor of $>4$ for FC2 and the algorithm pair (NI vs. FI).\nThe EDO-EAs perform much better than their classic counterpart. However, expect for (FI. vs. NI) on FC2 they produce less instances than QD~[all] and QD-algorithms show a high robustness \/ very low variation. Table~\\ref{tab:statistics_cheap} shows the mean and standard deviation of boxes covered among other statistics that will be discussed later. The observations are confirmed. A Kruskal-Wallis test with Bonferroni-correction rejects the zero hypothesis $H_0: \\text{median}(X) > \\text{median}($QD~[all]$)$ for every competitor evolver $X$ at significance level $\\alpha=0.0001$ for all settings expect for the mentioned (FI. vs. NI) on setting FC2; the results are highly significant.\nAnother important observation is that QD~[all] dominates the whole field of competitors significantly in the first few thousand iterations. This is important for computationally more involved instance generation task (see Section~\\ref{sec:experiments_expensive}).\nAs expected, all algorithms tend to perform much better\n\\begin{table*}\n\\renewcommand{\\tabcolsep}{3pt}\n\\renewcommand{\\arraystretch}{0.8}\n\\begin{scriptsize}\n\\caption{\\label{tab:statistics_cheap}Table of different aggregated values of interest for the experiments on the simple insertion heuristics. Best values (for the statistics of covered or objective values) or maximal values (for box update statistics) are \\colorbox{gray!40}{highlighted}.}\n\\centering\n\\begin{tabular}[t]{llrrrrrrrrrrrrrr}\n\\toprule\n\\multicolumn{2}{c}{ } & \\multicolumn{7}{c}{\\textbf{FI vs. NI}} & \\multicolumn{7}{c}{\\textbf{NI vs. FI}} \\\\\n\\cmidrule(l{3pt}r{3pt}){3-9} \\cmidrule(l{3pt}r{3pt}){10-16}\n\\multicolumn{2}{c}{ } & \\multicolumn{3}{c}{\\textbf{Nr. of cells}} & \\multicolumn{2}{c}{\\textbf{Cell statistics}} & \\multicolumn{2}{c}{\\textbf{Objective}} & \\multicolumn{3}{c}{\\textbf{Nr. of cells}} & \\multicolumn{2}{c}{\\textbf{Cell statistics}} & \\multicolumn{2}{c}{\\textbf{Objective}} \\\\\n\\cmidrule(l{3pt}r{3pt}){3-5} \\cmidrule(l{3pt}r{3pt}){6-7} \\cmidrule(l{3pt}r{3pt}){8-9} \\cmidrule(l{3pt}r{3pt}){10-12} \\cmidrule(l{3pt}r{3pt}){13-14} \\cmidrule(l{3pt}r{3pt}){15-16}\n\\textbf{FC} & \\textbf{Algorithm} & \\textbf{mean} & \\textbf{std} & <\\textbf{med} & \\textbf{upd.} & \\textbf{hits} & \\textbf{best} & \\textbf{median} & \\textbf{mean} & \\textbf{std} & <\\textbf{med} & \\textbf{upd.} & \\textbf{hits} & \\textbf{best} & \\textbf{median}\\\\\n\\midrule\n & (1+1) EA [all] & 371.93 & 52.20 & 180.00 & \\cellcolor{gray!40}{\\textbf{165.00}} & 186079.00 & \\cellcolor{gray!40}{\\textbf{0.53}} & 0.68 & 431.33 & 56.13 & 215.00 & 45.00 & 192185.00 & \\cellcolor{gray!40}{\\textbf{0.77}} & 0.97\\\\\n\n & (1+1) EA [simple] & 392.53 & 41.07 & 193.00 & 70.00 & \\cellcolor{gray!40}{\\textbf{487405.00}} & 0.65 & 0.77 & 397.13 & 40.60 & 199.00 & \\cellcolor{gray!40}{\\textbf{45.00}} & \\cellcolor{gray!40}{\\textbf{430152.00}} & 0.87 & 1.00\\\\\n\n & (50+1) EA [all] & 527.90 & 25.96 & 263.00 & 76.00 & 30129.00 & 0.55 & 0.70 & 522.90 & 20.88 & 260.00 & 27.00 & 30250.00 & 0.83 & 0.98\\\\\n\n & (50+1) EA [simple] & 478.83 & 19.39 & 240.00 & 31.00 & 51435.00 & 0.69 & 0.77 & 485.90 & \\cellcolor{gray!40}{\\textbf{19.60}} & 243.00 & 26.00 & 51992.00 & 0.88 & 1.00\\\\\n\n & (50+1) EA-HV [all] & 920.43 & 113.77 & 475.00 & 25.00 & 12855.00 & 0.57 & 0.71 & 889.77 & 73.86 & 444.00 & 19.00 & 11843.00 & 0.87 & 0.98\\\\\n\n & (50+1) EA-HV [simple] & 870.03 & 47.61 & 434.00 & 21.00 & 10710.00 & 0.71 & 0.78 & 759.33 & 75.32 & 381.00 & 20.00 & 13330.00 & 0.90 & 0.99\\\\\n\n & (50+1) EA-IGD [all] & 976.50 & 90.07 & 503.00 & 27.00 & 20658.00 & 0.63 & 0.73 & 882.87 & 74.75 & 447.50 & 19.00 & 19466.00 & 0.84 & 0.98\\\\\n\n & (50+1) EA-IGD [simple] & 865.10 & 44.73 & 437.50 & 21.00 & 17756.00 & 0.71 & 0.78 & 739.80 & 69.00 & 369.50 & 22.00 & 18890.00 & 0.91 & 1.00\\\\\n\n & QD [all] & \\cellcolor{gray!40}{\\textbf{1042.73}} & \\cellcolor{gray!40}{\\textbf{17.50}} & \\cellcolor{gray!40}{\\textbf{518.00}} & 72.00 & 4610.00 & 0.56 & \\cellcolor{gray!40}{\\textbf{0.64}} & \\cellcolor{gray!40}{\\textbf{1029.80}} & 22.67 & \\cellcolor{gray!40}{\\textbf{516.00}} & 32.00 & 4433.00 & 0.80 & \\cellcolor{gray!40}{\\textbf{0.93}}\\\\\n\n\\multirow{-10}{*}{\\raggedright\\arraybackslash FC1} & QD [simple] & 894.90 & 23.87 & 445.50 & 38.00 & 3981.00 & 0.66 & 0.73 & 891.93 & 33.72 & 446.00 & 27.00 & 4531.00 & 0.87 & 0.96\\\\\n\\cmidrule{1-16}\n & (1+1) EA [all] & 804.43 & 248.08 & 432.00 & \\cellcolor{gray!40}{\\textbf{100.00}} & 116604.00 & \\cellcolor{gray!40}{\\textbf{0.53}} & 0.64 & 267.77 & 32.19 & 133.00 & 39.00 & 310009.00 & \\cellcolor{gray!40}{\\textbf{0.78}} & 1.02\\\\\n\n & (1+1) EA [simple] & 120.83 & 30.30 & 57.00 & 90.00 & \\cellcolor{gray!40}{\\textbf{582995.00}} & 0.62 & 0.76 & 91.07 & 19.40 & 44.00 & \\cellcolor{gray!40}{\\textbf{95.00}} & \\cellcolor{gray!40}{\\textbf{700120.00}} & 0.81 & 1.02\\\\\n\n & (50+1) EA [all] & 824.33 & 84.32 & 402.50 & 44.00 & 34771.00 & 0.55 & 0.64 & 328.80 & 21.66 & 164.00 & 26.00 & 84605.00 & 0.83 & 0.99\\\\\n\n & (50+1) EA [simple] & 156.03 & \\cellcolor{gray!40}{\\textbf{13.86}} & 78.00 & 38.00 & 246284.00 & 0.66 & 0.77 & 127.43 & \\cellcolor{gray!40}{\\textbf{8.83}} & 64.00 & 34.00 & 244907.00 & 0.88 & 1.02\\\\\n\n & (50+1) EA-HV [all] & 2289.90 & 177.68 & 1148.50 & 24.00 & 8302.00 & 0.59 & 0.69 & 1140.17 & 265.98 & 565.00 & 21.00 & 26899.00 & 0.85 & 0.99\\\\\n\n & (50+1) EA-HV [simple] & 892.73 & 130.89 & 462.00 & 22.00 & 31931.00 & 0.69 & 0.76 & 513.83 & 143.77 & 269.00 & 21.00 & 85790.00 & 0.89 & 1.00\\\\\n\n & (50+1) EA-IGD [all] & \\cellcolor{gray!40}{\\textbf{2335.60}} & 162.63 & \\cellcolor{gray!40}{\\textbf{1184.00}} & 28.00 & 20174.00 & 0.58 & 0.70 & 1047.53 & 233.71 & 505.00 & 17.00 & 49680.00 & 0.87 & 1.00\\\\\n\n & (50+1) EA-IGD [simple] & 768.37 & 121.46 & 387.50 & 23.00 & 153460.00 & 0.68 & 0.77 & 412.63 & 123.33 & 196.00 & 23.00 & 302422.00 & 0.90 & 1.01\\\\\n\n & QD [all] & 1924.33 & 76.97 & 969.00 & 51.00 & 3315.00 & 0.55 & \\cellcolor{gray!40}{\\textbf{0.61}} & \\cellcolor{gray!40}{\\textbf{1779.77}} & 79.11 & \\cellcolor{gray!40}{\\textbf{886.00}} & 27.00 & 5240.00 & 0.81 & \\cellcolor{gray!40}{\\textbf{0.97}}\\\\\n\n\\multirow{-10}{*}{\\raggedright\\arraybackslash FC2} & QD [simple] & 841.63 & 67.22 & 419.00 & 40.00 & 11698.00 & 0.64 & 0.71 & 804.87 & 90.07 & 397.50 & 27.00 & 15761.00 & 0.88 & 0.99\\\\\n\\bottomrule\n\\end{tabular}\n\\end{scriptsize}\n\\end{table*}\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.48\\columnwidth]{figures\/FI_NI_nng3_nng3_tile_obj.pdf}\n \\includegraphics[width=0.48\\columnwidth]{figures\/NI_FI_nng3_nng3_tile_obj.pdf}\n \\includegraphics[width=0.48\\columnwidth, trim={0 0 0 30pt}, clip]{figures\/FI_NI_nng5_mst_tile_obj.pdf}\n \\includegraphics[width=0.48\\columnwidth, trim={0 0 0 30pt}, clip]{figures\/NI_FI_nng5_mst_tile_obj.pdf}\n \\vspace{-0.25cm}\n \\caption{Representative feature space coverage obtained by the different algorithms for feature combination FC1 (top row) and FC2 (bottom row). White crosses mark the boxes hit by the initial population for (FI vs. NI; left) and (NI vs. FI; right). The tiles share a common continuous color gradient with the same limits to make the differences in the objective values visible.}\n \\label{fig:feature_space}\n\\end{figure*}\nif all mutation operators are used due to the disruptive nature of the creative mutation~\\cite{Bossek2019evolv}. However, the benefit of disruptive over simple mutation is surprisingly low for FC1 and very strong for FC2. \nFigure~\\ref{fig:feature_space} shows the feature space (normalized to $[0,1]$~\\cite{HBPSTK2021tspnormalization}) coverage by means of exemplary but representative runs for FC1 and FC2.\nThe covered boxes are colored by their objective value (smaller is better) and white crosses represent the boxes hit by the initial solutions. We see a slight benefit for the *[all] algorithms and FC1. However, the feature space coverage is indeed very similar. Explanation: consider the spread of initial solutions in Figure~\\ref{fig:feature_space} for the $(50+1)$~EA variants on FC1. Recall that each initial solution is generated by placing $n$ points uniformly at random within $[0,1]^2$ (RUE; see Section~\\ref{sec:problem}). These initial instances already show a nice distribution of this particular feature space indicating that simple mutation operators (which produce again RUE-like mutants) are sufficient to further explore the FC1-space. The FC2 space coverage in Figure~\\ref{fig:feature_space} shows a different picture. Here, the initial population is very close to the origin. I.~e., for all initial solutions the feature vectors in FC2-space are quite similar. This is plausible since an RUE instance is likely to have a low number of strong connected components in the $5$-NNG and likewise a rather small median depth of nodes in an MST. In the FC2-space, the strong mutation of the *[all] variants allows all algorithms to explore the feature space much more thoroughly. \n\nComparing the *[simple] and *[all] plots in Figure~\\ref{fig:feature_space} for FC2 and also the respective progress over time in Figure~\\ref{fig:nr_of_instances_over_time} gives rise to another question: why do the \\added{classic and EDO-based} $(\\mu+1)$~EA~[all] variants cover much more of the FC2 feature space when the objective is to evolve instances which are easier to solve for FI than for NI? And why is this not the case for their *[simple] counterparts? One might think that this has to be a bug since an instance is always saved if its feature vector was not seen so far; this \\emph{first hit} of a box is independent of the objective value. However, the reason lies in the way the survival selection of the EAs works: the mutant replaces the parent if and only if its objective value is not worse than that of the parent~(see Algorithm~\\ref{alg:ea_with_archive_evolver}) \\added{or -- for the EDO-EAs -- diversity maximization takes place only if the instance satisfies the minimum quality constraint}. Since it is much more challenging to produce instances which are harder for FI than for NI (see Figure~\\ref{fig:boxplot_objective_simple_heuristics} where the objective values barely fall below~1) this is an unlikely event to happen. Consequently, in particular $(1+1)$~EA suffers from few updates of its \"population\" and its only population member serves as the template for the vast majority of the produced offspring. A single mutation (regardless of the type: explosion, implosion etc.) might be insufficient in producing a previously unseen feature combination. This lack of variation leads to the poor performance on (NI vs. FI). In the reverse case, (FI vs. NI), things are different and the population members get updated more often which allows for a more thorough exploration of the feature space since the \"source material\" changes more frequently. This phenomenon is not observed for FC1. As pointed out already the FC1-space can be easily explored even by simple mutation. \nRecall that the identified issue may be a major hindrance for the classical $(\\mu+1)$ -- and in particular the $(1+1)$-approaches -- from the literature to improve the objective since it carries over directly to EAs without footprint archive.\n\\added{We close the discussion of box-coverage by investigating the robustness of the approach. Figure~\\ref{fig:feature_space_frequency} colors boxes by the coverage frequency across all runs. QD explores the feature space much more reliably with little variation. The observations are the same for all other settings.}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\columnwidth]{figures\/NI_FI_nng5_mst_tile_frequency.pdf}\n \\caption{Distribution of the coverage frequency across all 30 independent runs for (NI vs. FI) on FC2.}\n \\label{fig:feature_space_frequency}\n\\end{figure}\n\nSo far we investigated the ability of the algorithms to cover the feature space, one of the two goals of the evolving procedure. We now turn the focus towards the second \\added{equally if not more important} goal: minimizing the tour length ratio. Figure~\\ref{fig:boxplot_objective_simple_heuristics} shows the distribution of the objective values on all instances over all runs produced by the algorithms. The panels split the data by feature combination (rows) and objective \"direction\" ((FI vs. NI) and (NI vs. FI); columns). Recall that the algorithms strive to minimize the tour length ratio. Thus, values in the green shaded region below~1 are desirable. We observe that QD~[all] dominates the field for all combinations of features and target directions with respect to the median objective value. In particular in the (NI vs. FI) setting, where finding instances which are harder to solve for NI seems difficult, the median objective values are clearly located in the greenish area whereas the median objective scores of the competitor algorithms are closer or even above~1. With the exception of (NI vs. FI) on FC2, the upper quartile for QD~[all] is even lower than the median of the other algorithms. This observation highlights the need for stronger exploration of the feature space and shows the superiority of the QD-approach.\n\\added{Note also that the median objective values produced by the EDO-EAs are worse than the respective QD-values. This is particularly visible in (FI. vs. NI) setting and both FC1 and FC2. Recall that QD~[all] covered less boxes in the FC2 scenario than the EDO-EAs. However, it shines when it comes to the median objective values. This is also confirmed by the precise numbers in Table~\\ref{tab:statistics_cheap}.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\columnwidth]{figures\/boxplot_objectives_by_features_and_algopair.pdf}\n \\caption{Distribution of the objective values (ratio of tour lengths) of all instances over all independent runs. Data is split by the direction of the optimization process (columns) and the feature space covered (rows).}\n \\label{fig:boxplot_objective_simple_heuristics}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/FI_NI_scatterplot_nhits_vs_obj.pdf}\n \\caption{Scatter-plot of the number of box-hits on log-scale and the objective value of the instance stored in the box. The plot shows the first run by example.}\n \\label{fig:FI_NI_scatterplot_nhits_vs_obj}\n\\end{figure}\nKeep in mind that the data used for Figure~\\ref{fig:boxplot_objective_simple_heuristics} is the union of all instances of all \\changed{30} algorithm runs. Considering the advantage of QD~[all] with respect to feature space coverage the QD approaches in consequence produced significantly more easy\/hard instances; a clear benefit.\nRegarding the overall \"best\" objective values, in most cases the $(1+1)$~EAs dominates. This is in line with our expectation. Recall that (1) the $(1+1)$~EAs explores a significantly smaller share of the feature space and (2) has the same budget of function evaluations. Hence, the search process is much more focused and boxes are thus hit much more often. \nThis increases the chances to update respective boxes with better instances. Figure~\\ref{fig:FI_NI_scatterplot_nhits_vs_obj} shows the relation between the (log-scaled) number of hits on the abscissa and the objective value of the instance stored in the respective boxes after termination on the ordinate. The plots shows the result of one particular run of the algorithms. However, the patterns are the same for all other runs. We see obvious negative linear relationships. In addition, we see that for a subset of the feature combinations in the $(\\mu+1)$~EAs the number of hits surpasses the maximum number of hits for the respective QD algorithm by orders of magnitude~(recall that the abscissa is on logarithmic scale).\n\\added{An interesting observation is that the number of box-hits of the EDO-EAs is also much higher. Nevertheless, QD remains the winner with respect to the objective. The reason is the survival selection of the EDO-EAs which may lead to frequent box-hits which do not improve upon the objective however.}\nFigure~\\ref{fig:FI_NI_nng5_mst_tile_nupdates_firsthit} in the top row shows this narrowed search by means of example in the FC2-space explored by $(1+1)$~EA~[all], \\added{$(50+1)$~EA-HV~[all]} and QD~[all] where boxes are colored by the number of actual updates. While for QD~[all] the updates are distributed across the whole explored feature space (leave apart the border regions) only a few boxes are updated more that 30 times in the $(1+1)$~EA~[all] run. \\added{The $(50+1)$~EA-HV~[all] updates cells less often which explains its rather poor performance with respect to the objective values discussed before.}\nThe \\emph{box statistics} in Table~\\ref{tab:statistics_cheap} support these observations. They show the maximum number of updates and hits of boxes over all runs. $(1+1)$~EA shows a huge maximum of hits up to $\\approx 52\\%$ of the total number of iterations on (NI vs. FI) and FC2. The values for QD in comparison are minuscule.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\columnwidth]{figures\/FI_NI_nng5_mst_tile_nupdates.pdf}\n \\includegraphics[width=0.75\\columnwidth]{figures\/FI_NI_nng5_mst_tile_firsthit.pdf}\n \\caption{FC2 with boxes colored by the number of updates, i.~e., the number of times the instance stored in the box was replaced with an instance not worse with respect to the objective score~(top row). In the bottom row boxes are colored by the first hitting time \/ (first discovery time).}\n \\label{fig:FI_NI_nng5_mst_tile_nupdates_firsthit}\n\\end{figure}\nThe decreased number of updates at the borders can be explained by the hardness to produce instances with such \"extreme\" feature vectors. The bottom row in Figure~\\ref{fig:FI_NI_nng5_mst_tile_nupdates_firsthit} shows the same data colored by the iteration number of the boxes' first hit. Some of the boxes were only discovered after more than $800\\,000$ iterations. This observation together with the increasing trend in Figure~\\ref{fig:nr_of_instances_over_time} indicates that even longer runs would hit even more boxes.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\columnwidth]{figures\/LKH_EAX_nng5_mst_tile_obj.pdf}\n \\caption{Representative feature space coverage obtained by the different algorithms for feature combination FC2 and state-of-the-art heuristics LKH and EAX.}\n \\label{fig:feature_space_expensive}\n\\end{figure}\n\\section{Experiments on State-of-the-Art Inexact TSP Solvers}\n\\label{sec:experiments_expensive}\n\nWe now consider EAX and LKH and the feature combination FC2 only since the results in Section~\\ref{sec:experiments_cheap} indicate that this space is harder to cover. Both algorithms are state-of-the-art in inexact TSP solving. EAX~\\cite{Nagata1997EAX,Nagata2013powerful} is a sophisticated genetic algorithm based on the edge-assembly-crossover operator and entropy population diversity preservation. LKH~\\cite{Helsgaun2009submoves} is a local-search algorithm whose main driver is the Lin-Kernighan heuristics for $k$-opt moves. Both algorithms are capable of solving even large instances to optimality in reasonable time. Complementary behavior on a wide range of instances motivated algorithm selection studies~\\cite{Kotthoff2015improving,kerschke2018,Bossek2019evolv,HBPSTK2021tspnormalization}.\n\n\\subsection{Experimental setup}\nPreliminary experiments show that EDO-based algorithms perform worse than QD in this computationally demanding domain with respect to both box-coverage and objective value since the number of iterations is severely reduced to $\\approx 2\\,000$. \n\\begin{table*}\n\\renewcommand{\\tabcolsep}{4pt}\n\\renewcommand{\\arraystretch}{0.8}\n\\centering\n\\begin{scriptsize}\n\\caption{\\label{tab:statistics_expensive}Table of aggregated values of interest for the experiments on state-of-the-art TSP solvers EAX and LKH. Best values (for the statistics of covered or objective values) or maximal values (for box update statistics) are \\colorbox{gray!40}{highlighted}.}\n\\centering\n\\begin{tabular}[t]{llrrrrrrrrrrrrrr}\n\\toprule\n\\multicolumn{2}{c}{ } & \\multicolumn{7}{c}{\\textbf{EAX vs. LKH}} & \\multicolumn{7}{c}{\\textbf{LKH vs. EAX}} \\\\\n\\cmidrule(l{3pt}r{3pt}){3-9} \\cmidrule(l{3pt}r{3pt}){10-16}\n\\multicolumn{2}{c}{ } & \\multicolumn{3}{c}{\\textbf{Nr. of boxes}} & \\multicolumn{2}{c}{\\textbf{Box statistics}} & \\multicolumn{2}{c}{\\textbf{Objective}} & \\multicolumn{3}{c}{\\textbf{Nr. of boxes}} & \\multicolumn{2}{c}{\\textbf{Box statistics}} & \\multicolumn{2}{c}{\\textbf{Objective}} \\\\\n\\cmidrule(l{3pt}r{3pt}){3-5} \\cmidrule(l{3pt}r{3pt}){6-7} \\cmidrule(l{3pt}r{3pt}){8-9} \\cmidrule(l{3pt}r{3pt}){10-12} \\cmidrule(l{3pt}r{3pt}){13-14} \\cmidrule(l{3pt}r{3pt}){15-16}\n\\textbf{FC} & \\textbf{Algorithm} & \\textbf{mean} & \\textbf{std} & <\\textbf{med} & \\textbf{upd.} & \\textbf{hits} & \\textbf{best} & \\textbf{median} & \\textbf{mean} & \\textbf{std} & <\\textbf{med} & \\textbf{upd.} & \\textbf{hits} & \\textbf{best} & \\textbf{median}\\\\\n\\midrule\n & (1+1) EA [all] & 390.83 & 100.32 & 199.50 & 7.00 & 141.00 & \\cellcolor{gray!40}{\\textbf{0.00}} & \\cellcolor{gray!40}{\\textbf{0.82}} & 574.38 & 153.52 & 292.00 & 8.00 & 368.00 & \\cellcolor{gray!40}{\\textbf{0.02}} & \\cellcolor{gray!40}{\\textbf{0.33}}\\\\\n\n & (50+1) EA-HV [all] & 442.83 & 240.29 & 247.50 & \\cellcolor{gray!40}{\\textbf{7.00}} & \\cellcolor{gray!40}{\\textbf{277.00}} & 0.00 & 0.99 & 712.00 & 242.13 & 321.00 & \\cellcolor{gray!40}{\\textbf{8.00}} & \\cellcolor{gray!40}{\\textbf{387.00}} & 0.03 & 0.34\\\\\n\n\\multirow{-3}{*}{\\raggedright\\arraybackslash FC1} & QD [all] & \\cellcolor{gray!40}{\\textbf{877.50}} & \\cellcolor{gray!40}{\\textbf{54.98}} & \\cellcolor{gray!40}{\\textbf{430.50}} & 6.00 & 24.00 & 0.00 & 0.85 & \\cellcolor{gray!40}{\\textbf{882.57}} & \\cellcolor{gray!40}{\\textbf{130.68}} & \\cellcolor{gray!40}{\\textbf{441.00}} & 7.00 & 25.00 & 0.03 & 0.36\\\\\n\\cmidrule{1-16}\n & (1+1) EA [all] & 128.57 & 23.75 & 65.00 & 8.00 & 237.00 & 0.00 & 0.38 & 135.38 & \\cellcolor{gray!40}{\\textbf{12.81}} & 67.50 & 11.00 & \\cellcolor{gray!40}{\\textbf{942.00}} & \\cellcolor{gray!40}{\\textbf{0.03}} & \\cellcolor{gray!40}{\\textbf{0.29}}\\\\\n\n & (50+1) EA-HV [all] & 116.60 & \\cellcolor{gray!40}{\\textbf{22.94}} & 59.00 & 8.00 & \\cellcolor{gray!40}{\\textbf{371.00}} & \\cellcolor{gray!40}{\\textbf{0.00}} & \\cellcolor{gray!40}{\\textbf{0.30}} & 137.88 & 24.19 & 73.00 & \\cellcolor{gray!40}{\\textbf{11.00}} & 797.00 & 0.03 & 0.31\\\\\n\n\\multirow{-3}{*}{\\raggedright\\arraybackslash FC2} & QD [all] & \\cellcolor{gray!40}{\\textbf{281.00}} & 62.02 & \\cellcolor{gray!40}{\\textbf{141.00}} & \\cellcolor{gray!40}{\\textbf{8.00}} & 44.00 & 0.00 & 0.42 & \\cellcolor{gray!40}{\\textbf{353.67}} & 60.70 & \\cellcolor{gray!40}{\\textbf{174.00}} & 9.00 & 72.00 & 0.04 & 0.38\\\\\n\\bottomrule\n\\end{tabular}\n\\end{scriptsize}\n\\end{table*}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\columnwidth]{figures\/eax_vs_lkh_parscores.pdf}\n \\includegraphics[width=0.49\\columnwidth]{figures\/lkh_vs_eax_parscores.pdf}\n \\caption{Scatter-plots of PAR10 scores of the two competing algorithms EAX and LKH. Points are colored by the evolving algorithm. The left plot shows the results for the experiments with the goal to produce instances which are easy for EAX and hard for LKH; the right plot shows the vice versa. Points on the diagonal line indicate equal performance of both algorithms on the respective instance. Points above\/below the diagonal indicate a runtime-advantages for one of the algorithms. The gray dashed lines indicate the maximum possible PAR10-score of $10 \\cdot 180$~[s].}\n \\label{fig:scatter_par_scores}\n\\end{figure}\nThis is in-line with observations extracted from Figure~\\ref{fig:nr_of_instances_over_time} and Figure~\\ref{fig:boxplot_objective_simple_heuristics}. Hence, in the following we use $(1+1)$~EA, $(50+1)$~EA and QD.\nIn line with \\cite{Bossek2019evolv} we evolve instances with $n=500$ nodes and minimize the ratio of the penalized average runtime~(PAR10~\\cite{bischl2016}) scores, the mean of the runtimes of $R$ runs of $A$ on $I$ where \\emph{failure runs} are penalized with $10 \\cdot T$, $T$ being the cutoff-time; a run is termed a \\emph{failure} if the algorithm does not find the optimum within time $T$. The objective function evaluation requires a single run of the exact TSP solver \\texttt{concorde}~\\cite{Applegate2009TSP} to calculate an optimal tour. We set $R=3$ and $T=180~[s]$ and run each evolving algorithm with 48h wall-time.\n\n\\subsection{Results}\nThe observations on feature-space coverage in Figure~\\ref{fig:feature_space_expensive} are in par with Figure~\\ref{fig:feature_space} (bottom row) in the cheap setting. However, due to the costly objective function evaluation the algorithms performs significantly less iterations which explains the lower total coverage. QD~[all] dominates the other algorithms with respect to the covered area in all scenarios due to only up to $\\approx$~2000 performed generations. Table~\\ref{tab:statistics_expensive} shows summary statistics for the number of covered boxes, the objective values and box update\/hit statistics. The numbers confirm the good performance of all approaches with respect to the objective values and the advantage of the QD approaches with respect to feature space coverage. One observation worth to mention is that the mean number of boxes covered is higher for all algorithms for (LKH vs. EAX) in comparison to (EAX vs. LKH). The reason is that LKH hits the cutoff-time of $180$ seconds frequently while this is rarely the case for EAX. Put differently, it is easier for the algorithms to evolve an instance where LKH hits the time limit than the vice verse. Figure~\\ref{fig:scatter_par_scores} shows this nicely by plotting the PAR10-scores of EAX and LKH against each other for the evolving algorithms $(1+1)$~EA~[all], $(50+1)$~EA-HV~[all] and QD~[all]. The three clusters in the top left corner of the left plot indicate that on many evolved instances LKH has either one, two or even three failure runs. In contrast, EAX times our rarely if the goal is to produce EAX-hard instances. \n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe generation of TSP problem instances which are diverse with respect to solver performance and instance features is of utmost importance, e.~g., in the field of algorithm selection. We introduced a flexible quality diversity~(QD) approach to evolve instances. Instances are mapped to their feature vectors. In an iterative evolutionary process not yet seen instances \/ feature combinations are stored in any case; instances are overwritten if newly generated instances share the same feature combination, but show a stronger performance difference for two competing TSP algorithms.\nWe compared our approach to classical $(\\mu+1)$~EAs and versions adapted to feature-diversity optimization from the literature modified to store their footprints in a similar way. The results show impressively the capability of the QD approach in covering a wide range of feature combinations -- and thus producing much more instances in a single, less wasteful run -- while competing or even outperforming the classical approaches with respect to objective scores. These results motivate a broad avenue of future work. Here, we only name a few of our most promising ideas: application to $>2$ features and other domains, introducing selection bias towards border regions, or storing multiple instances per box.\n\n\\section{Acknowledgements}\nJ. Bossek acknowledges support by the European Research Center for Information Systems (ERCIS). This work has been supported by the Australian Research Council (ARC) through grants DP190103894 and FT200100536. The authors thank the anonymous reviewers for their immensely valuable feedback.\n\n\\bibliographystyle{unsrt} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjzzb b/data_all_eng_slimpj/shuffled/split2/finalzzjzzb new file mode 100644 index 0000000000000000000000000000000000000000..ba00499d59a0c76037e99733ddb319dd7d2e690b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzjzzb @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe electronic transport in mesoscopic systems has been\nintensively studied during the last decades.\\cite{Mbook1, Mbook2}\nThe conductance of mesoscopic wires is usually computed using the\nB\\\"uttiker-Landauer formalism, \\cite{BLandauer} which accounts for\nthe quantum mechanical nature of transmission and reflection\nthrough impurities and works well for Fermi liquids.\n\nRecent advances in nanotechnology, together with the discovery of\ncarbon nanotubes \\cite{SumioIijima, CeesDekker} have enabled the\nfabrication of extremely narrow wires, which are basically\none-dimensional. Because the electron-electron interactions become\nessential in 1D, these quantum wires exhibit Luttinger liquid\nbehavior, i.e., they have bosonic collective excitations, instead\nof the usual fermionic quasi-particles present in conventional\nFermi liquids.\\cite{indianLL}\n\nThe role played by impurities in a Luttinger liquid depends\ncrucially on the nature of the interactions. In pathbreaking\npapers, Kane and Fisher\\cite{KaneFisher1,KaneFisher2} showed that\nthe impurity barrier is irrelevant if the electrons have\nattractive interactions ($g > 1$), but it cuts the wire into two\npieces if they interact repulsively ($g < 1$). For the\nnoninteracting case ($g = 1$) the barrier is a marginal\nperturbation, i.e., one can have both, transmission and\nreflection.\n\nFor the case of two barriers, contrarily to the expectation, it is\npossible to have transmission even in the presence of repulsive\ninteractions ($g < 1$) because quasi-bound-states can form between\nthe barriers, which lead to resonant\ntransmission.\\cite{KaneFisher3}\n\nThe vast majority of cases discussed in the literature concerning\ntransport in Luttinger liquids involve {\\it static} {\\it\npoint-like} impurity barriers. Recently, the case of {\\it\ndynamical point-like} impurities was addressed.\\cite{Sharma,\nSchmeltzer, FeldmanGefen, ChengZhou, us} Feldman and\nGefen\\cite{FeldmanGefen} have shown that in the presence of one\nimpurity oscillating with frequency $\\Omega$, the backscattering\ncurrent may change sign and the conductance of a single-mode\nLuttinger liquid with strong repulsive interactions ($g < 1\/2$)\nmay become {\\it larger} than the quantum of conductance $G_0 =\ne^2\/h$. We have generalized their results and shown that in the\npresence of several impurities this phenomenon occurs even for\n$1\/2 < g < 1$.\\cite{us} In fact, we have found that the dc-current\ncan always be expressed in terms of the backscattering current off\na static impurity. The effect of the impurity frequency $\\Omega$\nis solely to split the energy modes into $\\omega_0 + \\Omega$ and\n$\\omega_0 - \\Omega$, where $\\omega_0 = eV\/\\hbar$ denotes the\nJosephson frequency, associated with the external voltage $V$.\nHence, within first order perturbation theory, several point-like\nimpurities oscillating in phase may lead to an increase of the\nconductance, in a certain range of parameters, but in the entire\nregime of repulsive interactions $0 < g < 1$.\\cite{us}\n\nAfter having understood the role of {\\it time-dependent point-like\nimpurities}, the remaining task is to investigate transport\nthrough {\\it extended time-dependent} impurities.\\cite{Barci} This\nis precisely the aim of this paper. The coherent transport\nproperties of multimode quantum channels in the presence of\nGaussian-like scatterers was considered recently.\\cite{Bardarson}\nIn addition, the tunneling current between two counterpropagating\nedge modes described by chiral Luttinger liquids was calculated\nfor the case when tunneling takes place along an extended\nregion.\\cite{Aranzana}\n\nIn this work we evaluate the backscattering current off {\\it\nseveral time-dependent extended impurities} in an otherwise\nperfect Luttinger liquid, in the presence of repulsive\nelectron-electron interactions. We start in Section II with a\nsingle static extended impurity in the noninteracting limit ($g =\n1$), which can be solved using the B\\\"uttiker-Landauer\nformalism,\\cite{BLandauer} and then we generalize the results for\nthe case of an arbitrary static potential. In Section III we\ninclude interactions among the electrons and study the problem\nusing the Keldysh formalism. Within first order perturbation\ntheory, we first derive a general expression for the\nbackscattering current in the presence of several time-dependent\nextended impurities and then we concentrate on a typical example,\nnamely, the case of one impurity, to illustrate the main results.\nFinally, we compare our findings obtained within perturbation\ntheory in the limit of noninteracting electrons ($g = 1$) with the\nexact results derived in the framework of the B\\\"uttiker-Landauer\nformalism and show a perfect agreement between them. Our\nconclusions are presented in Section IV.\n\n\\section{Non-interacting electrons ($g=1$) in an extended backscattering potential}\n\\subsection{The Model}\nWe start by studying the simplest case, namely, noninteracting\nelectrons in 1D in the presence of a backscattering potential. The\nHamiltonian of the model reads $H=H_{0}+H_{imp}$, where\n\\begin{eqnarray}\\label{Hamiltonian}\nH_{0} &=& -i\\hbar v_{F}\\int dx (\\psi_{R}^{\\dagger}\n\\partial_{x}\\psi_{R}-\\psi_{L}^{\\dagger}\n\\partial_{x}\\psi_{L}), \\\\\nH_{imp} &=& \\int dx (\\psi_{R}^{\\dagger}\n\\psi_{L}+\\psi_{L}^{\\dagger} \\psi_{R})W(x,t).\\nonumber\n\\end{eqnarray}\nHere $\\psi_{R\/L}$ are fermionic fields corresponding to the\nright\/left moving electrons and $W(x,t)$ stands for the\nbackscattering potential. We also further set the Planck constant\n$\\hbar=1$ and the Fermi velocity $v_{F}=1$. The equations of\nmotion then read\n\\begin{eqnarray}\n (\\partial_{t}+\\partial_{x})\\psi_{R}+i W(x,t)\\psi_{L}&=& 0,\\\\\n (\\partial_{t}-\\partial_{x})\\psi_{L}+i W(x,t)\\psi_{R}&=& 0. \\nonumber\n \\label{EOM}\n\\end{eqnarray}\n For a static backscattering potential, $W(x,t)=W(x)$, the\n solutions of the above equations of motion have a stationary form,\n $\\mathbf{\\psi}_{R\/L}(x,t)=\\mathbf{\\psi}_{R\/L}(x)e^{-i \\omega t}$.\n The fermionic fields $\\psi_{R\/L}$ then obey the following equation\n of motion\n\\begin{equation}\n\\left( \\begin{array}{cc} \\omega+i\\partial_{x} & -W(x)\\\\\n -W(x) & \\omega-i\\partial_{x}\\\\\n\\end{array}\n \\right)\\mathbf{\\psi}(x)=0,\n \\label{SEOM}\n\\end{equation}\nwhere we introduced a ``spinor'' notation\n\\begin{eqnarray}\n\\mathbf{\\psi}\\equiv \\left( \\begin{array}{c} \\psi_{R} \\\\\n\\psi_{L}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\nIn the following we concentrate on some particular examples for\nwhich an analytical solution can be obtained.\n\\subsection{Single static rectangular-like barrier}\n\nLet us begin by considering a static rectangular-like potential\nbarrier of width $L$ and height $W$,\n\\begin{equation}\\label{rectpotential}\nW(x)=W[\\Theta(x+L\/2)-\\Theta(x-L\/2)],\n\\end{equation}\n where $\\Theta(x)$ is the Heavyside theta-function. The solution of the stationary\nequation of motion\\ (\\ref{SEOM}) may be sought in the form\n\\begin{eqnarray}\n \\label{Eansatz}\n \\psi_{R}(x)&=&\\psi_{R}^{-}e^{- i k x}+\\psi_{R}^{0}e^{i k x }, \\\\\n \\psi_{L}(x)&=&\\psi_{L}^{0}e^{- i k x}+\\psi_{L}^{-}e^{i k x }.\\nonumber\n\\end{eqnarray}\nSubstitution of the ansatz\\ (\\ref{Eansatz}) into the equations of\nmotion (\\ref{SEOM}) yields\n\\begin{equation}\n\\psi_{L}^{-}=\\frac{W(x)}{\\omega+k}\\psi_{R}^{0}, \\qquad\n\\psi_{R}^{-}=\\frac{W(x)}{\\omega+k}\\psi_{L}^{0},\n\\end{equation}\nwith $\\omega^{2}=[W(x)]^{2}+k^{2}$. We now consider the region\n$[-L\/2,L\/2]$ and require both functions $\\psi_{L}(x)$ and\n$\\psi_{R}(x)$ to be continuous at the points $x=\\pm L\/2$. For the\npoint $x=-L\/2$ we have\n\\begin{eqnarray}\n \\psi_{R}(-L\/2)&=&\\psi_{R}^{0}e^{-i k L\/2 } +\n\\frac{W}{\\omega+k} \\psi_{L}^{0}e^{i k L\/2 }, \\\\\n \\psi_{L}(-L\/2)&=& \\frac{W}{\\omega+k} \\psi_{R}^{0}e^{-i k L\/2 }\n+\\psi_{L}^{0}e^{i k L\/2 },\\nonumber\n \\end{eqnarray}\nwhere $k=\\sqrt{\\omega^{2}-W^{2}}$, and for the point $x=L\/2$ the\nexpressions are similar. Introducing the matrices\n\\begin{equation}\nB= \\left( \\begin{array}{cc} 1 & \\frac{W}{\\omega+k}\\\\\n\\frac{W}{\\omega+k} & 1\n\\end{array}\n \\right)\n\\end{equation}\nand\n\\begin{equation}\nP= \\left( \\begin{array}{cc} e^{i k L\/2} & 0\\\\\n0 & e^{-i k L\/2}\n\\end{array}\n \\right),\n\\end{equation}\nthe continuity conditions can be written in a simple form\n\\begin{equation}\n\\mathbf{\\psi}(-L\/2)=BP^{\\dagger}\\mathbf{\\psi}^{0}, \\qquad\n\\mathbf{\\psi}(L\/2)= BP\\mathbf{\\psi}^{0}.\n\\end{equation}\nThe above equations then yield\n \\begin{equation}\\label{scattering}\n\\mathbf{\\psi}(L\/2)=M\\mathbf{\\psi}(-L\/2), \\qquad M=BP^2B^{-1},\n\\end{equation}\nwhere $M$ is the transmission matrix, which gives the\ntransformation from the states on the left of the barrier,\n$\\psi_{R\/L} (-L\/2)$, to the states on the right of the barrier\n$\\psi_{R\/L} (L\/2)$,\n\\begin{equation}\nM= \\left( \\begin{array}{cc} \\cos(k L)+i \\frac{\\omega}{k} \\sin(k L)\n&\n -i \\frac{W}{k} \\sin(k L)\\\\\ni \\frac{W}{k} \\sin(k L) & \\cos(k L)-i \\frac{\\omega}{k} \\sin(k L)\n\\end{array}\n \\right). \\label{Mmatrix}\n\\end{equation}\nLet us observe that\n$M^\\dagger\\mathbf{\\sigma}_{3}M=\\mathbf{\\sigma}_{3}$, and therefore\n$\\bar{\\mathbf{\\psi}}\\mathbf{\\psi}$ is a constant, where\n$\\bar{\\mathbf{\\psi}}=\\mathbf{\\psi}^{\\dagger}\\mathbf{\\sigma}_{3}$\nis the Dirac conjugate. This is a consequence of charge\nconservation and holds for any transmission matrix, since an\narbitrary backscattering potential can always be represented as a\nset of (infinitely narrow) rectangular potentials with different\namplitudes. Therefore, the transmission matrix $M\\in SU(1,1)$ can\nbe always written in the form\n\\begin{equation}\nM= \\left( \\begin{array}{cc} s & r^{\\ast}\\\\\nr & s^{\\ast}\n\\end{array}\n \\right). \\label{Melements}\n\\end{equation}\nA more elegant derivation of Eq.\\ (\\ref{Mmatrix}) using results of\nthe Lie group theory is presented in the Appendix. The\ntransmission matrix given by Eq.\\ (\\ref{Melements}) is related to\nthe scattering matrix, $S$, which gives the transformation from\nthe incoming modes $\\psi_{R}(-L\/2),\\psi_{L}(L\/2)$ to the outgoing\nmodes $\\psi_{R}(L\/2),\\psi_{L}(-L\/2)$. In terms of the transmission\nmatrix elements, the latter reads\n\\begin{equation}\nS= \\frac{s}{|s|}\\left( \\begin{array}{cc} 1\/|s| & r^{\\ast}\/|s|\\\\\n-r\/|s| & 1\/|s|\n\\end{array}\n \\right). \\label{Selements}\n\\end{equation}\n\nLet us now calculate the transmission coefficient for the right\nmoving particles using Eq.\\ (\\ref{scattering}). In that case, we\nhave\n\\begin{equation}\n\\left( \\begin{array}{c} \\psi_{R}(L\/2) \\\\\n0\\\\\n\\end{array}\n\\right)=M\\left( \\begin{array}{c} \\psi_{R}(-L\/2) \\\\\n\\psi_L(-L\/2)\\\\\n\\end{array}\n\\right),\n\\end{equation}\nwith the transmission matrix $M$ given by Eq.\\ (\\ref{Mmatrix}).\nThe transmission coefficient may be then promptly found from the\nprevious equation\n\\begin{equation}\nT\\equiv\\left|\\frac{\\psi_R(L\/2)}{\\psi_R(-L\/2)}\\right|^2=\\frac{1}{|s|^2}=\n\\frac{1}{1+|r|^2},\n\\end{equation}\nUsing the explicit form of the transmission matrix, given by Eq.\\\n(\\ref{Mmatrix}), we obtain for $\\omega^{2}\\geq W^{2}$\n\\begin{equation} T(\\omega)=\\left[{1+\n\\frac{W^{2}}{\\omega^{2}-W^{2}} \\sin^{2}(\\sqrt{\\omega^{2}-W^{2}}\nL)}\\right]^{-1}\n \\label{Transm}\n\\end{equation}\nand for $\\omega^{2}< W^{2}$\n\\begin{equation} T(\\omega)=\\left[{1+\n\\frac{W^{2}}{W^{2}-\\omega^{2}} \\sinh^{2}(\\sqrt{W^{2}-\\omega^{2}}\nL)}\\right]^{-1}.\n \\label{Transm2}\n\\end{equation}\nAn analogous calculation shows that the transmission coefficient\nfor the left moving particles is the same as for the right moving\nones. We can now find the backscattering current using the\nLandauer formula\\cite{BLandauer} and choosing the left and right\nreservoirs to be symmetric. We use the subscript ``$st$'' to\ndenote the backscattering current off a {\\it static} impurity to\ndistinguish it from the backscattering current originating from\n{\\it time-dependent} {\\it barriers}, which is considered in the\nnext section. The current then reads\n\\begin{equation}\nI_{st}=e\\int_{-\\infty}^{\\infty} \\frac{d\\omega}{2\\pi}\n[n^{R}(\\omega)-n^{L}(\\omega)][T(\\omega)-1], \\label{BLform}\n\\end{equation}\nwhere $n^{R\/L}(\\omega)$ are the occupation numbers of the\nright\/left movers, given by the Fermi distribution function\n\\begin{equation}\nn^{R\/L}(\\omega)=\\frac{1}{e^{\\beta(\\omega-\\omega_{F}\\mp\n\\omega_{0}\/2)}+1}.\n\\end{equation}\nHere, $\\omega_{0}=eV$ is the Josephson frequency related to the\napplied bias voltage $V$ and $\\omega_{F}$ is the Fermi energy\n(recall that $\\hbar$=1). We have to distinguish the two cases\ngiven by Eqs.\\ (\\ref{Transm}) and (\\ref{Transm2}), since if\n$\\omega^{2}< W^{2}$ the momentum $k$ becomes imaginary and\ntransport occurs then only via tunnelling. In other words, the\npresence of backscattering opens a gap in the energy spectrum of\nthe fermions from $\\omega_F-W$ to $\\omega_F+W$. Substituting Eq.\\\n(\\ref{Transm}) into Eq.\\ (\\ref{BLform}) and keeping only the\nlowest order terms in $W$, we find the ``static'' backscattering\ncurrent\n\\begin{equation}\nI_{st}=-\\frac{eW^{2}}{2\\pi}\\int_{-\\infty}^{\\infty}\nd\\omega\\frac{\\sin^{2}(\\omega\nL)}{\\omega^{2}}[n^{R}(\\omega)-n^{L}(\\omega)]. \\label{Tbscurr}\n\\end{equation}\nAt zero temperature ($T=0$) and $\\omega_{F}=0$ the above\nexpression simplifies to\n\\begin{equation}\nI_{st}=-\\frac{2eW^{2}L}{\\pi}\\int_{0}^{L\\omega_{0}} d y\n\\frac{\\sin^{2}(y\/2)}{y^{2}}. \\label{bscurr}\n\\end{equation}\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=6cm,angle=-90]{Fig1b.eps}\n\\includegraphics[width=6cm,angle=-90]{Fig1a.eps}\n\\end{center}\n\\caption{\\label{fig1}(color online) $I_{st}$ versus $\\omega_0$ for\nnoninteracting electrons ($g=1$) in the presence of a point-like\n(dashed line) or an extended impurity (solid line). The asymptotic\nbehavior for the latter,\n$I_{st}(\\omega\\rightarrow\\infty)\\rightarrow-1\/2$, is represented\nby dotted lines. The current is plotted in units of $eW^{2}L$ and\nthe frequency is in units of $1\/L$. Upper panel: $T=1$; lower\npanel: $T=0$. }\n\\end{figure}\n\nThe backscattering currents at a finite and zero temperature,\ngiven by Eqs.\\ (\\ref{Tbscurr}) and (\\ref{bscurr}), are plotted,\nrespectively, in the upper and the lower panel of Fig.\\ 1. In both\ncases the current $I_{st}$ is represented in units of $eW^2L$ and\nthe Josephson frequency is in units of $1\/L$. Inspection of Fig.\\\n1 shows that in the region $L\\omega_{0}<2$, the current $I_{st}$\nfor scattering off an extended impurity (solid line) is almost\nindistinguishable from the backscattering current of a point-like\nimpurity represented by a barrier ${\\tilde W}(x)=W L\\delta(x)$\n(dashed line). However, in the region $L\\omega_{0}>2$ the\ndifference becomes drastic, since the backscattering current off a\npoint-like impurity continues to decrease linearly, whereas for\nthe extended impurity, the current saturates at the value $I_{bs}=\n-e W^{2}L\/2$.\n\n\\subsection{Arbitrary static potential}\n\n Let us now consider noninteracting electrons backscattered\noff a static potential $W(x)$ localized in the interval $0\\leq\nx\\leq y$ with the corresponding transmission matrix $M(y)$. In\norder to find the dependence of the transmission matrix on the\nscattering potential, let us add to the existing barrier an\ninfinitesimally thin part with the height $W(y+dy)$.\nIn this case the transmission matrix at the point $y+dy$ is\n$M(y+dy)=M(y)+dM$, where $dM=i{\\cal T}M dy$ (see Appendix). After\nperforming a matrix multiplication, we find the following\ndifferential equations\n\\begin{eqnarray}\n d s=i[\\omega s - W(y) r ]dy, \\\\ \\nonumber\n d r=i[W(y) s - \\omega r]dy.\n\\end{eqnarray}\nIf the potential is weak, $W(y) \\ll \\omega$, then to the lowest\norder in perturbation expansion $r={\\cal O} (W\/\\omega)$. After\nneglecting terms ${\\cal O} (r^{3})$, we obtain\n\\begin{equation}\nd r(y)=i[W(y)e^{i \\omega y}- \\omega r(y)]dy.\n\\end{equation}\nThis is a linear inhomogeneous differential equation of the first\norder with solution of the form\n\\begin{equation}\nr(y)=ie^{-i \\omega y}\\int_{0}^{y}dxW(x)e^{i 2\\omega x}.\n\\end{equation}\nThen, it follows straightforwardly that\n\\begin{equation}\n|r|^2=\\int dx_{1}\\int dx_{2}W(x_{1})W(x_{2})\\cos[ 2\\omega\n(x_{1}-x_{2})].\n\\end{equation}\nOn the other hand, since $|r|^2\\ll 1$, the transmission\ncoefficient becomes\n\\begin{equation}\nT(\\omega)-1=\\frac{1}{1+|r|^2} - 1 =-|r|^2,\n\\end{equation}\nand the backscattering current at finite temperatures then reads\n\\begin{equation}\nI_{st}=-e\\int_{-\\infty}^{\\infty} \\frac{d\\omega}{2\\pi}\n|r|^2[n^{R}(\\omega)-n^{L}(\\omega)].\n\\end{equation}\nConsequently, we find that\n\\begin{eqnarray}\n&& \\frac{\\delta^{2}}{\\delta W(x_{1})\\delta\nW(x_{2})}I_{st}{\\Large{\\vert}}_{W=0}=\\\\ \\nonumber &&\n-e\\int_{-\\infty}^{\\infty} \\frac{d\\omega}{\\pi}\\cos[2\\omega\n(x_{1}-x_{2})][n^{R}(\\omega)-n^{L}(\\omega)].\n\\end{eqnarray} Since\n\\begin{equation}\\nonumber\n\\int_{-\\infty}^{\\infty}d y\\cos(y) [n^{R}(y\/2l)-n^{L}(y\/2l)]=4 \\pi\nT l\\frac{\\sin(\\omega_0 l)\\cos(2\\omega_{F}l)}{\\sinh(2 \\pi T l)},\n\\end{equation}\nwe eventually obtain that\n\\begin{eqnarray}\n \\label{twopointnonint} && \\frac{\\delta^{2}I_{st}}{\\delta\nW(x_{1})\\delta W(x_{2})}{\\Large{\\vert}}_{W=0}=\\\\ \\nonumber &&\n-2eT\\frac{\\cos[2\\omega_{F}(x_{1}-x_{2})]\\sin[\\omega_{0}(x_{1}-x_{2})]}{\\sinh[2T\\pi\n(x_{1}-x_{2})]}.\n\\end{eqnarray}\nAs we will see later, Eq.\\ (\\ref{twopointnonint}) will allow us to\nconnect the results obtained within perturbation theory in the\nbosonization formalism with the ones derived in the framework of\nthe B\\\"uttiker-Landauer approach, for an arbitrary (but weak)\nextended impurity barrier, in the limit of noninteracting\nelectrons ($g=1$).\n\n\\section{Interacting electrons}\n\\subsection{Several time-dependent extended\nimpurities} In this section we consider a one-dimensional quantum\nwire with spinless interacting electrons backscattered by a {\\it\ntime-dependent} potential $W(x,t)$. The Hamiltonian of the model,\nincluding the electron-electron interaction, reads\n\\cite{FeldmanGefen,Chamon2,Rao}\n\\begin{eqnarray}\\nonumber\nH &=&\\int dx \\left\\{-i(\\psi_{R}^{\\dagger}\n\\partial_{x}\\psi_{R}-\\psi_{L}^{\\dagger}\\partial_{x}\\psi_{L})\n+ U(\\psi_{R}^{\\dagger}\\psi_{R}+\\psi_{L}^{\\dagger} \\psi_{L})^{2}\n\\right. \\\\ &+& \\left. W(x,t)\\left[\\psi_{L}^{\\dagger}\n\\psi_{R}e^{i\\omega_{0}t+i2k_{F}x}+H.c.\\right]\\right\\},\n\\label{Hmodel}\n\\end{eqnarray}\nwhere $U$ is the interaction strength. In the next step, we\nconsider a discretized backscattering potential $W(x,t)$ composed\nof several point-like time-dependent impurities located at\npositions $x_p$, $p=1,...,N$, such that\n \\begin{equation}\n \\lim_{\\Delta\nx_p\\rightarrow 0+}\\sum_{p=1}^{N}W(x_p,t)\\Delta x_{p} = \\int W(x,t)\ndx.\n\\end{equation}\n\nWe now use the bosonization technique, which relates the fermionic\nfields, $\\psi_{R\/L}$, with the corresponding bosonic fields,\n$\\Phi_{R\/L}$, through $\\psi_{R\/L}=\\alpha \\eta e^{\\pm\ni\\sqrt{g}\\Phi_{R\/L}}$. Here, $\\alpha$ is a normalization factor,\n$\\eta$ is a fermionic operator, and $g=\\sqrt{\\pi \/(\\pi+2U)}$\n(recall $v_{F}=1$) is the coupling constant in the bosonic theory.\nThe action corresponding to the Hamiltonian\\ (\\ref{Hmodel}) then\nreads \\cite{FeldmanGefen,Chamon,Chamon2,Rao,us,Egger}\n\\begin{eqnarray}\nS &=&\\int dt dx \\left\\{\\frac{1}{8\\pi}\\left[(\\partial_{t}\\Phi)^{2}-\n(\\partial_{x}\\Phi)^{2}\\right] \\right. \\\\ \\nonumber &-&\n\\left.\\sum_{p}\\delta(x-x_{p})V_{p}(t)\\left(e^{i\\sqrt{g}\n\\Phi(t,x_{p})} e^{i\\omega_{0}t}+H.c.\\right)\\right\\},\n\\end{eqnarray}\nwhere $V_p(t)\\equiv W(x_p,t)e^{i2k_{F}x_p}\\Delta x_p\/2\\pi$ and\n$\\Phi\\equiv\\Phi_R+\\Phi_L$. Using the approach developed in Refs.\\\n[\\onlinecite{us}] and [\\onlinecite{Chamon2}], we find that, to the\nlowest order in $V$, the backscattering current reads\n \\begin{equation}\n I_{bs}(\\omega_{0},t)=C \\sum_{k,j} \\int_{|x_{k}-x_{j}|}^{\\infty}\n \\frac{d\\tau\\sin(\\omega_{0}\\tau) V^{\\ast}_{k}(t-\\tau)\n V_{j}(t)}{|\\tau^{2}-(x_{k}-x_{j})^{2}|^{g}}+H.c.,\n\\label{Ibs}\n\\end{equation}\nwhere $C = -2 e\\sin (\\pi g)$. The most general case should involve\na spatially varying impurity potential, with an arbitrary shape.\nIn order to account for this more general situation, we take the\ncontinuum limit in Eq.\\ (\\ref{Ibs}) by setting the number of\nimpurities $N \\rightarrow \\infty$ and the distance between the\nimpurities $\\Delta x_p\\rightarrow0$. Eq.\\ (\\ref{Ibs}) then\nacquires the form\n\\begin{eqnarray} \\nonumber\n I_{bs}(\\omega_{0},t)=-e\\frac{\\sin (\\pi g)}{\\pi^{2}} \\int\\int dx_{1}\ndx_{2}\\int_{|x_{1}-x_{2}|}^{\\infty}d\\tau \\\\\n\\frac{\\sin(\\omega_{0}\\tau)\n\\cos[2k_{F}(x_{1}-x_{2})]W(x_{1},t-\\tau)W(x_{2},t)}\n {|\\tau^{2}-(x_{1}-x_{2})^{2}|^{g}}.\n \\label{extimp}\n \\end{eqnarray}\nEq.\\ (\\ref{extimp}) is a very important result of our paper\nbecause it allows us to evaluate the backscattering current for\nany set of time-dependent extended impurities, described by any\narbitrary function $W(x,t)$.\n\nWe can progress further in the analytical calculation of some\nspecific examples by introducing a simplifying assumption, namely,\nthat the impurity potential is a periodic function of time,\n$W(x,t)=W(x)\\cos(\\Omega t)$. We have shown in Ref.\\\n\\onlinecite{us} that the dc component of the backscattering\ncurrent generated by a set of dynamical point-like impurities\noscillating with a frequency $\\Omega$ can be expressed, in first\norder of perturbation theory, in terms of the dc current for a set\nof quasi-static impurities ($\\omega_{0} \\gg \\Omega >\n1\/\\tau_{meas}$, with $\\tau_{meas}$ being the measuring or the\nrelaxation time),\n\\begin{equation}\nI_{dc}=\\frac{1}{2}[I_{qst}(\\omega_{0}+\\Omega)+I_{qst}(\\omega_{0}-\\Omega)],\n\\label{Idc}\n\\end{equation}\nor static impurities ($1\/\\tau_{meas}\\gg\\Omega$)\n\\begin{equation}\nI_{dc}=\\frac{1}{4}[I_{st}(\\omega_{0}+\\Omega)+I_{st}(\\omega_{0}-\\Omega)],\n\\label{Idc2}\n\\end{equation}\nwith\n\\begin{eqnarray} \\nonumber\nI_{st}(\\omega_0)=-e\\int\\int dx_{1}\ndx_{2}W(x_{1})W(x_{2})\\\\\n\\cos[2k_{F}(x_{1}-x_{2})]H_{g}(\\omega_{0},|x_{1}-x_{2}|,T = 0).\n\\label{Idcstatic}\n\\end{eqnarray}\nThe function $H_{g}$ is defined as\n\\begin{equation}\nH_{g}(\\omega,x,T=0) \\equiv \\frac{\\sin (\\pi g)}{\\pi^{2}}\n\\int_{|x|}^{\\infty}d\\tau \\frac{\\sin(\\omega\n\\tau)}{|\\tau^{2}-x^{2}|^{g}} \\label{Ffunction}\n\\end{equation}\nand after explicit evaluation it reads\n\\begin{equation}\nH_{g}(\\omega,x,T=0)={\\rm {\\rm\nsgn}}(\\omega)\\frac{J_{g-1\/2}(x\\omega)}{2^{1\/2+g}\\sqrt{\\pi}\n\\Gamma(g)}\\left|\\frac{\\omega}{x}\\right|^{g-1\/2},\n\\end{equation}\nwhere $J_{\\alpha}(x)$ are Bessel functions of the first kind and\n$\\Gamma(g)$ is the gamma function.\n\nThis result may be readily generalized for the situation where a\nphase shift of the form $W(x,t)=W(x)\\cos(\\Omega t+2Kx)$ is present\nalong an extended impurity. Then we can again write the dc\ncomponent of the current as\n\\begin{equation}\nI_{dc}=\\frac{1}{4}[I_{st,k_{F}+K}(\\omega_{0}+\\Omega)+I_{st,k_{F}-K}(\\omega_{0}-\\Omega)],\n\\end{equation}\nwhere the channels $\\omega_{0}+\\Omega$ and $\\omega_{0}-\\Omega$\nhave an ``effective'' Fermi level at the momentum $k_{F}+K$ and\n$k_{F}-K$, respectively.\nThis means that it is possible to decrease the ``effective'' Fermi\nmomentum for the $\\omega_{0}-\\Omega$ channel. For simplicity we\nassume $K=0$ in the following.\n\nWe discussed until now the $T=0$ case. When the temperature is\nnonzero, the Green's function is modified by a conformal\ntransformation \\cite{Shankar}\n\\begin{equation}\n|\\tau^{2}-x^{2}|^{g}\\rightarrow \\frac{|\\sinh[\\pi\nT(\\tau-x)]\\sinh[\\pi T(\\tau+x)]|^{g}}{(\\pi T)^{2g}}.\n\\end{equation}\n Consequently, the function $H_{g}$ acquires the form\n\\begin{equation}\nH_{g}(\\omega,x,T) \\equiv \\int_{|x|}^{\\infty}\\frac{d\\tau \\sin (\\pi\ng)\\sin(\\omega \\tau)T^{2g}\\pi^{2g-2}}{|\\sinh[\\pi\nT(\\tau-x)]\\sinh[\\pi T(\\tau+x)]|^{g}}. \\label{FfunctionwithT}\n\\end{equation}\n\nIt is worth noting here that the function $I_{st}(\\omega_0)$,\ndefined by Eq.\\ (\\ref{Idcstatic}), can be represented as a product\nof a dimensional factor(which actually becomes dimensional only\nafter including the cutoff of the model) and a function of\ndimensionless variables. The dimensionless part remains invariant\nwith scaling\n\\begin{eqnarray} \\nonumber\n\\label{scaling} \\omega_0\\rightarrow \\omega'_0 = \\omega_0 \\zeta,\n\\quad x\\rightarrow\nx'=x\/\\zeta,\\\\\n\\quad W\\rightarrow W' = W \\zeta, \\quad T\\rightarrow T' = T \\zeta,\n\\end{eqnarray}\nwhereas the currents $I_{st}$ and $I_{dc}$ scale as the\ndimensional factor\n\\begin{eqnarray} \\nonumber\nI_{st}&\\rightarrow& I'_{st}\\\\\nI'_{st}[W'(x')](\\omega_0',T')&=&\\zeta^{2g-1}\nI_{st}[W(x)](\\omega_0,T). \\label{scaling2}\n\\end{eqnarray}\nThese equations set the relation between currents corresponding to\ndifferent potentials, which have the same shape but a different\nlength parameter. For example, all rectangular-like potentials\nhave the same backscattering current dependence up to the scaling\n(\\ref{scaling}). This leads us to the conclusion that it is enough\nto consider some potential with a fixed length parameter to obtain\nthe backscattering current dependence for all potentials of the\nsame form (in lowest order of perturbation theory).\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=6cm,angle=-90]{Fig2a.eps}\n\\includegraphics[width=6cm,angle=-90]{Fig2b.eps}\n\\end{center}\n\\caption{\\label{fig2}(color online) Dimensionless \"static\" current\n$I_{st}$ versus $\\omega_0$ for $k_{F}=0$ and $W=L=1$. Upper panel:\n$g<1\/2$. Lower panel: $g\\geq1\/2$. }\n\\end{figure}\n\nIn the case of a potential $W(x)$ with a rectangular shape, the\nabove integral Eq.\\ (\\ref{Idcstatic}) simplifies to\n\\begin{eqnarray} \\nonumber\n I_{st}(\\omega_0)&=&-eW^{2}\\int_{-L\/2}^{L\/2}\\int_{-L\/2}^{L\/2} dx_{1}\ndx_{2}\\cos[2k_{F}(x_{1}-x_{2})]\\\\ \\nonumber\n&&H_{g}(\\omega_{0},|x_{1}-x_{2}|,T).\\end{eqnarray} By changing the\nvariables $x=x_{1}-x_{2}$ and $y=x_{1}+x_{2}$, one integration can\nbe performed, finally yielding\n\\begin{equation}\\label{Irect}\nI_{st}(\\omega_0)=-2eW^{2}\\int_{0}^{L} dx (L-x)\\cos(2k_{F}\nx)H_{g}(\\omega_{0},x,T).\n\\end{equation}\n\nIn Fig.\\ 2, the dimensionless current in the presence of a static\nimpurity, given by Eq.\\ (\\ref{Irect}), is plotted for $k_{F}=0$,\n$W=L=1$, since, as it was argued before, we can fix the length of\nthe barrier without loss of generality. The upper panel shows that\nfor $g<1\/2$, $I_{st}$ always increases, analogous to the case of a\npoint-like impurity. On the other hand, for $g>1\/2$, a\nnon-monotonic behavior of the current appears. As we will discuss\nlater, this non-monotonic behavior may lead to an increase of the\nconductance, beyond the value $G_0$.\n\nLet us now discuss the dependence of the current on the value of\n$k_F$. To demonstrate the influence of the Fermi momentum, $k_F$,\non the backscattering current off a static impurity, we plot\n$I_{st}$ versus the Josephson frequency for several values of the\nFermi momentum in Fig.\\ 3.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=8cm,angle=0]{Fig3.eps}\n\\end{center}\n\\caption{\\label{fig3}(color online) Dimensionless $I_{st}$ versus\n$\\omega_0$ for $g=0.7$ and $L=1$}\n\\end{figure}\nIt can be promptly observed that the qualitative features are the\nsame for zero and finite $k_{F}$, the only effect of the latter is\nto shift the minimum of the curve to higher values of\n$\\omega_{0}$.\n\nFor future purposes it is worth noting that for the rectangular\npotential with $k_{F}=0$ the following relation holds\n\\begin{equation}\\label{partialst}\n\\frac{\\partial^{2}}{\\partial\nL^{2}}I_{st}=-2eW^{2}H_{g}(\\omega_{0},L,T).\n\\end{equation}\nEq.\\ (\\ref{partialst}) is a consequence of a more general\nrelation, valid for an arbitrary static potential \\cite{Liguori}\n\\begin{eqnarray} \\nonumber\n&&\\frac{\\delta^{2}}{\\delta W(x_{1})\\delta\nW(x_{2})}I_{st}\\vert_{W=0}= \\\\ \\nonumber\n&&-2e\\cos[2k_{F}(x_{1}-x_{2})]H_{g}(\\omega_{0},|x_{1}-x_{2}|,T).\n\\end{eqnarray}\n\n\\subsection{Noninteracting electrons $g=1$ within perturbation theory}\n\nNow we can compare this result with the one obtained in Sec. II B\nusing the B\\\" uttiker-Landauer formalism by taking the limit\n$g\\rightarrow1$ in Eq.\\ (\\ref{Ffunction}). Defining\n$h(\\tau)\\equiv\\sin(\\omega\\tau)$, $f(\\tau)\\equiv|\\tau^2-x^2|$, and\n$g=1-\\epsilon$, we may use the formula\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow 0+}\\int_{x}^{y}d \\tau\nh(\\tau)[f(\\tau)]^{\\epsilon-1}\\epsilon=\\frac{h(x)}{f'(x)}, \\quad\ny>x,\n\\end{equation}\nwhich holds for a function $f$ with the properties $f(x)= 0$ and\n$f'(x)>0$, yielding\n\\begin{equation}\nH_{1}(\\omega_{0},x,T)=T\\frac{\\sin(\\omega_{0}x)}{\\sinh(2T\\pi x)}.\n\\label{H1}\n\\end{equation}\n\nIn the case of a rectangular potential and $k_{F} = 0$, we may\nsubstitute Eq.\\ (\\ref{H1}) into Eq.\\ (\\ref{Irect}) to obtain\n\\begin{equation}\nI_{st}(\\omega_0)=-2eW^{2}T\\int_{0}^{L} dx\n(L-x)\\frac{\\sin(\\omega_{0}x)}{\\sinh(2T\\pi x)}, \\label{TIst}\n\\end{equation}\nwhich, in the limit $T\\rightarrow0$, becomes\n\\begin{equation} I_{st}(\\omega_0)=-\\frac{eW^{2}}{\\pi}\\int_{0}^{L}\ndx (L-x)\\frac{\\sin(\\omega_0 x)}{x}.\n\\end{equation}\nAfter integrating by parts and using that\n$\\frac{d}{dy}[2\\sin^{2}(y\/2)]=\\sin(y)$, we find\n\\begin{equation}\nI_{st}=-\\frac{2eW^{2}L}{\\pi}\\int_{0}^{L\\omega_{0}} d y\n\\frac{\\sin^{2}(y\/2)}{y^{2}},\n\\end{equation}\nwhich exactly coincides with Eq.\\ (\\ref{bscurr}) found in the\nprevious section.\n\nTo demonstrate the equivalence between the results obtained using\ndifferent approaches in the case of nonzero temperature, we note\nthat the second derivative of the currents given by Eqs.\\\n(\\ref{Tbscurr}) and (\\ref{TIst}) actually coincide,\n\\begin{equation}\n\\frac{\\partial^{2}}{\\partial L^{2}}I_{st}=-2e W^{2}T\n\\frac{\\sin(\\omega_0 L)}{\\sinh(2 \\pi T L)}.\n\\end{equation}\nTherefore, these currents can only differ by a function\n$F(L,\\omega_0,T)=C_{1}(\\omega_0,T)L+C_{2}(\\omega_0,T)$, with $C_1$\nand $C_2$ being arbitrary functions of their arguments. It follows\nfrom $\\frac{\\partial}{\\partial\nL}I_{st}(L=0,\\omega_0,T)=I_{st}(L=0,\\omega_0,T)=0$, that\n$F(L,\\omega_0,T)=0$, and the currents given by Eqs.\\\n(\\ref{Tbscurr}) and (\\ref{TIst}) are thus equal. The same\narguments hold also for the case of an arbitrary potential and\n$k_{F} \\neq 0$, where the equations have similar form, only with\n the partial derivative replaced by the functional one,\n$\\frac{\\partial}{\\partial L}\\rightarrow \\frac{\\delta}{\\delta\nW(x)}$.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=6cm,angle=-90]{Fig4.eps}\n\\end{center}\n\\caption{\\label{fig4}$\\omega_{min}L$ versus $g$. }\n\\end{figure}\n\n\\subsection{One oscillating extended impurity}\nWe can now fix the width of the impurity to be $L=1$ without loss\nof generality, since the other cases are obtained from the latter\nby the scaling relation (\\ref{scaling}). The function\n$I_{st}(\\omega)$ is odd, $I_{st}(-\\omega)=-I_{st}(\\omega)$,\nbecause there is no ratchet effect present in the lowest order of\nperturbation expansion.\\cite{Vinokur} The current for the\ndynamical impurity $I_{dc}$, according to Eq.\\ (\\ref{Idc2}), reads\n\\begin{equation}\nI_{dc}=\\frac{1}{4}[I_{st}(\\Omega+\\omega_0)-I_{st}(\\Omega-\\omega_0)].\n\\end{equation}\nSince $I_{st}(\\omega_0)$ is always negative or zero for\n$\\omega_0>0$, the function $I_{dc}$ is also negative if both\n$\\omega_0+\\Omega$ and $\\omega_0-\\Omega$ are positive. To obtain a\npositive backscattering current it is therefore necessary to have\n$\\Omega>\\omega_0$, i.e., the frequency of the impurity should be\nlarger than the external bias frequency. Then for small $\\omega_0$\n\\begin{equation}\nI_{dc}=\\frac{1}{2}I'_{st}(\\Omega)\\omega_0 \\label{deriv}\n\\end{equation}\nand therefore, the $I_{dc}$ current is positive if\n$I'_{st}(\\Omega)>0$, at least for small $\\omega_0$. Let us begin\nby considering the current in the region $1\/20$, we have to choose\n$\\Omega>\\omega_{min}$ in order to obtain an increase of the dc\ncurrent, see Eq.\\ (\\ref{deriv}) and Fig.\\ (\\ref{fig3}). Now,\nconsidering $I_{dc}$ as a function of the applied voltage $V$ for\nsome fixed $\\Omega>\\omega_{min}$, one finds that the dc current is\npositive in a region $0<\\omega_0 < \\omega_{0}^*(\\Omega)$, see\nFig.\\ (\\ref{fig5}). Therefore, the effect of current enhancement\nin the region $1\/2\\omega_{min}$, the\ncontinuous function $I_{dc}$ is then positive within a region $0<\n\\omega_0 < \\omega_{0}^*$.\n\n\n\\section{Conclusions}\nIn this work we calculated the backscattering current off extended\nimpurities in a Luttinger liquid. We started our analysis by\nconsidering noninteracting electrons ($g=1$) scattered by a\nrectungular-like barrier of width $L$ and height $W$. In this\nsimplified case the problem can be exactly solved using the\nB\\\"uttiker-Landauer formalism. We evaluate the backscattering\ncurrent at finite and zero temperatures for the Fermi momentum\n$k_F=0$, and find that for small Josephson frequencies\n$L\\omega_{0}<2$ the result is almost indistinguishable from the\nbackscattering current off a point-like impurity represented by a\nbarrier ${\\tilde W}(x)=W L\\delta(x)$. In the region\n$L\\omega_{0}>2$ these currents differ considerably, since in the\ncase of a point-like impurity the current decreases linearly\nwhereas for the extended barrier it saturates at the value\n$I_{bs}= -e W^{2}L\/2$. We also calculated the backscattering\ncurrent off a weak backscattering potential with an arbitrary\nshape in lowest order of perturbation expansion.\n\nThe effect of a time-dependent extended barrier, $W(x,t)$, in the\npresence of repulsive electron-electron interactions was then\nconsidered. Using the bosonization technique combined with the\nKeldysh formalism, we showed that the backscattering current off\ndynamical impurities can be expressed in terms of the\nbackscattering current off static ones. It actually turns out that\nthe only effect of the barrier oscillating with a frequency\n$\\Omega$ is to split the conduction channel $\\omega_0$ into two\nchannels, $\\omega_0\\pm\\Omega$.\nWe then evaluated the backscattering current off static barriers\n$I_{st}(\\omega_{0})$ and showed that it is a non-monotonic\nfunction of $\\omega_{0}$, which for $1\/2\\omega_{min}$,\nit may occur that the backscattering current changes sign and in a\ncertain region of applied voltage $0< \\omega_0 < \\omega_{0}^*$ the\nconductance will be greater than the quantum of conductance $G_0$.\nThe phenomenon of the conductance enhancement may occur in the\npresence of oscillating impurities as well, but the main\ndifference concerns the region of parameters where it appears,\nwhich depends on the type of the barrier. In the case of a single\npoint-like oscillating barrier it only occurs for strongly\nrepulsive interactions $0i+1$.\nThese are precisely the matrix entries of \n$D_1+E_1$.\n\\end{proof}\n\nAs we will show in Sections \\ref{PermTableaux} and \\ref{Proof} \nthe matrix product $W_1 (\\prod_{i=1}^n (\\tau_i D_1 + (1-\\tau_i)E_1))V_1$\nfrom Theorem \\ref{ansatz}\nhas a combinatorial interpretation as a generating function\nfor permutation tableaux.\n\n\\begin{example}\\label{example1}\nSuppose that we are considering the PASEP model with $3$ sites, and are\ninterested in computing the probability that in the long time limit,\nthe system reaches the configuration which has a black particle in \nthe second site (but the first and third sites are empty). \nThis configuration is represented by the vector \n$\\tau = (0,1,0)$. \nUsing Theorem \\ref{ansatz}, we see that $f_3(0,1,0)$ is\nequal to the matrix\nproduct $W_1 (E_1 D_1 E_1) V_1$;\nthis product is $\\alpha^{-2} + \\alpha^{-1}\\beta^{-1} + \\alpha^{-2}\\beta^{-1}q$.\nAnd by Remark \\ref{partition}, the \npartition function $Z_3$ is equal to \n$W_1 (D_1 +E_1)^3 V_1$, which is in this case\n$\\alpha^{-3}+2\\alpha^{-2}+2\\alpha^{-1}+\\alpha^{-2}\\beta^{-1}+\n2\\alpha^{-1}\\beta^{-1} +2\\beta^{-1}+\\alpha^{-1}\\beta^{-2} + 2\\beta^{-2}+\n\\beta^{-3} +\nq(\\alpha^{-2}+\\alpha^{-2}\\beta^{-1}+4\\alpha^{-1}\\beta^{-1}+\\alpha^{-1}\\beta^{-2}+\\beta^{-2}) +\nq^2(\\alpha^{-2}\\beta^{-1} + \\alpha^{-1} \\beta^{-2}).$\nTherefore\nthe probability that in the long time limit the system is in \nthe configuration $(0,1,0)$ is \n$\\frac{\\alpha^{-2} + \\alpha^{-1}\\beta^{-1} + \\alpha^{-2}\\beta^{-1}q}{Z_3}$.\n\\end{example}\n\n\n\\section{Connection with permutation tableaux}\\label{PermTableaux}\n\nRecall that a {\\em partition} $\\lambda = (\\lambda_1, \\dots,\n\\lambda_k)$ is a weakly decreasing sequence of nonnegative\nintegers. For a partition $\\lambda$, where $\\sum \\lambda_i = m$, the\n{\\em Young diagram} $Y_\\lambda$ of shape $\\lambda$ is a left-justified\ndiagram of $m$ boxes, with $\\lambda_i$ boxes in the $i$-th row.\nWe define the {\\it expanse} of $\\lambda$ or $Y_\\lambda$ to be the sum \nof the number of rows and the number of columns. Note that \nwe will allow a row to have length $0$-- i.e.\\ we allow $\\lambda_i=0$ --\nand we distinguish two partitions that differ in their number of\nempty rows.\n\nWe will often identify\na Young diagram $Y_\\lambda$ with expanse $n$ \nwith the lattice path $p(\\lambda)$ of length $n$\nwhich takes unit steps south and west,\nbeginning at\nthe north-east corner of $Y_\\lambda$ and ending at\nthe south-west corner.\nNote that such a lattice path always begins with a step south.\nSee Figure \\ref{path} for the path corresponding to the \nYoung diagram of shape $(2,1,0)$.\n\n\\begin{figure}[h]\n\\pspicture(0,0)(40,40) \n\\psline[linecolor=black,linewidth=0.5pt]{-}(0,0)(0,36)\n\\tbox(0,2){1} \n\\tbox(0,1){1} \n\\tbox(1,2){1} \n\\endpspicture\n\\pspicture(0,0)(40,40)\n\\psline[linecolor=black,linewidth=0.5pt]{-}(0,0)(0,12)(12,12)(12,24)(24,24)(24,36)\n\\endpspicture\n\\caption{Young diagram and path for $\\lambda=(2,1,0)$}\n\\label{path}\n\\end{figure}\n\n\nIf $\\tau \\in \\{0,1\\}^{n-1}$, then we associate to it a Young diagram \n$\\lambda(\\tau)$ with expanse $n$ as follows. First we define \na path $p = (p_1, \\dots , p_n) \\in \\{S,W\\}^n$ such that $p_1=S$, and \n$p_{i+1}=S$ if and only if $\\tau_i = 1$. We then define\n$\\lambda(\\tau)$ to be the partition associated to this path $p$.\nWe denote by $\\phi$ the inverse of the above map: \nit is a bijection \nfrom the set of Young diagrams\nwith expanse $n$ to the set of $n-1$-tuples in $\\{0,1\\}^{n-1}$.\n\nAs in \\cite{SW}, we\ndefine a {\\em permutation tableau} $\\mathcal{T}$ to be a partition $\\lambda$\ntogether with a filling of the boxes of $Y_\\lambda$ with $0$'s and\n$1$'s such that the following properties hold:\n\\begin{enumerate}\n\\item Each column of the rectangle contains at least one $1$.\n\\item There is no $0$ which has a $1$ above it in the same column\n{\\em and} a $1$ to its left in the same row.\n\\end{enumerate}\n\nWe call such a filling a {\\em valid} filling of $Y_\\lambda$. \n\nNote that the second requirement above can be rephrased in the following\nway. Read the columns of a permutation tableau $\\mathcal{T}$ from right to left. \nIf in any column we have a $0$ which lies beneath some $1$, then all \nentries to the left of $0$ (which are in the same row) must also\nbe $0$'s.\n\nNote that if we forget the requirement (1) above we recover the definition of\na $\\hbox{\\rotatedown{$\\Gamma$}}$-diagram~\\cite{Postnikov, Williams}, \nan object which represents\na cell in the totally nonnegative part of the Grassmannian.\n\nWe will now define a few statistics on permutation tableaux.\nWe define the {\\it weight} $\\wt(\\mathcal{T})$ of a permutation tableau $\\mathcal{T}$ with k \ncolumns to be the total number of $1$'s in the filling minus $k$.\n(We subtract $k$ since there must be at least $k$ $1$'s in \na valid filling of a tableau with $k$ columns.)\nIn other words, we are counting how many extra $1$'s each column contains\nbeyond the requisite one.\n\nWe define $f(\\mathcal{T})$ to be the number of $1$'s in the first\nrow of $\\mathcal{T}$.\n\nWe say that an entry in a column of a permutation tableau is \n{\\it restricted} if \nthat entry is a $0$ which lies below some $1$. \nAnd we say that a row is \n{\\it unrestricted}\nif it does not contain a restricted entry. \nDefine $u(\\mathcal{T})$ to be the number of unrestricted rows of $\\mathcal{T}$ \nminus $1$. (We subtract $1$ since the top row of a tableau\nis always unrestricted.)\n\n\nFigure \\ref{PermTab} gives an example of a permutation tableau $\\mathcal{T}$\nwith weight $19-10=9$ and expanse $17$, such that \n$u(\\mathcal{T}) = 3$ and $f(\\mathcal{T}) = 5$.\n\n\\begin{figure}[h]\n\\pspicture(-95,-12)(230,94)\n\\rput(190,36)\n{$\\begin{array}{l}\n\\lambda=(10,9,9,8,5,2)\n\\end{array}$}\n\\psline[linecolor=black,linewidth=0.5pt]{-}(0,-12)(120,-12)(120,72)(0,72)(0,-12)\n\\tbox(0,0){1}\n\\tbox(1,0){1}\n\\tbox(0,1){0}\n\\tbox(1,1){0}\n\\tbox(2,1){0}\n\\tbox(3,1){1}\n\\tbox(4,1){1}\n\\tbox(0,2){0}\n\\tbox(1,2){0}\n\\tbox(2,2){0}\n\\tbox(3,2){0}\n\\tbox(4,2){0}\n\\tbox(5,2){0}\n\\tbox(6,2){1}\n\\tbox(7,2){1}\n\\tbox(0,3){0}\n\\tbox(1,3){0}\n\\tbox(2,3){0}\n\\tbox(3,3){0}\n\\tbox(4,3){0}\n\\tbox(5,3){0}\n\\tbox(6,3){0}\n\\tbox(7,3){0}\n\\tbox(8,3){0}\n\\tbox(0,4){1}\n\\tbox(1,4){1}\n\\tbox(2,4){1}\n\\tbox(3,4){1}\n\\tbox(4,4){0}\n\\tbox(5,4){1}\n\\tbox(6,4){1}\n\\tbox(7,4){1}\n\\tbox(8,4){1}\n\\tbox(0,5){0}\n\\tbox(1,5){1}\n\\tbox(2,5){1}\n\\tbox(3,5){0}\n\\tbox(4,5){0}\n\\tbox(5,5){1}\n\\tbox(6,5){0}\n\\tbox(7,5){1}\n\\tbox(8,5){0}\n\\tbox(9,5){1}\n\\endpspicture\n\\caption{A permutation tableau}\n\\label{PermTab}\n\\end{figure}\n\nNow we will be interested in the question of enumerating permutation \ntableaux according to their shape, weight, unrestricted rows, and \nfirst row.\nThat is, we are interested in computing the polynomials\n$F_{\\lambda}(q):= \\sum_{\\mathcal{T}} q^{\\wt(\\mathcal{T})} \\alpha^{-f(\\mathcal{T})} \\beta^{-u(\\mathcal{T})}$, \nwhere the \nsum ranges over all permutation tableaux $\\mathcal{T}$ of shape\n$\\lambda$. Let us also define the \npolynomials $F^n(q):= \n\\sum_{\\mathcal{T}} q^{\\wt(\\mathcal{T})} \\alpha^{-f(\\mathcal{T})} \\beta^{-u(\\mathcal{T})}$, \nwhere the sum ranges over all permutation tableaux $\\mathcal{T}$ with expanse $n$.\n\nOur main result is the following.\n\n\\begin{theorem}\\label{Th1}\nFix a partition $\\lambda$ with expanse $n+1$.\nLet $(\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$ be \n$\\phi(\\lambda)$. Then \n\\begin{equation*}\nF_{\\lambda}(q) = W_1 (\\prod_{i=1}^n (\\tau_i D_1 + (1-\\tau_i) E_1)) V_1. \n\\end{equation*}\nMoreover, the generating function $F^{n+1}(q)$ for {\\it all}\npermutation tableaux with expanse $n+1$ is \n$W_1 (D_1+E_1)^n V_1$.\n\\end{theorem}\n\nAn examination of the formula in Theorem \\ref{Th1} reveals that\n$F_{\\lambda}(q)$ is just the sum of the entries\nin the top row of a certain product of $n$ matrices, each one either\n$D_1$ or $E_1$, where this product corresponds via $\\phi$ to the \nshape of the partition $\\lambda$.\n\nWe remark that Theorem \\ref{Th1} is particularly \nnice because it is a {\\it positive}\nformula. In \\cite{SW}, a recurrence was given for \n$F_{\\lambda}(q)$ (in the case where $\\alpha=\\beta=1$), \nbut not an explicit formula.\n\n\\begin{example}\\label{example2}\nWe now illustrate Theorem \\ref{Th1} with an example.\nSuppose that we want to calculate the weight generating function\nfor permutation tableaux of shape $\\lambda = (2,1)$. Note that\n$\\phi(\\lambda)=(0,1,0)$. Therefore we need to evaluate\nthe expression $W_1 E_1 D_1 E_1 V_1$. This is equal to \n$\\alpha^{-2}\\beta^{-1}q +\n\\alpha^{-2}+\\alpha^{-1}\\beta^{-1}$. \nAnd indeed, as shown in Figure \\ref{Hello}, there are three\npermutation tableaux of shape $(2,1)$, whose statistics correspond to \nthe three terms above.\nCompare our results here to \nExample\n \\ref{example1}.\n\n\\begin{figure}[h]\n\\pspicture(0,0)(40,40) \n\\tbox(0,2){1} \n\\tbox(0,1){1} \n\\tbox(1,2){1} \n\\endpspicture\n\\pspicture(0,0)(40,40)\n\\tbox(0,2){1}\n\\tbox(1,2){1}\n\\tbox(0,1){0}\n\\endpspicture\n\\pspicture(0,0)(40,40)\n\\tbox(0,2){0}\n\\tbox(1,2){1}\n\\tbox(0,1){1}\n\\endpspicture\n\n\\caption{Permutation tableaux of shape $(2,1,0)$}\n\\label{Hello}\n\\end{figure}\n\\end{example}\n\n\nWe will defer the proof of Theorem \\ref{Th1} to Section \\ref{Proof},\nin order to first explain various consequences of the result.\n\nTheorem \\ref{Th1} \ntogether with Theorem \\ref{ansatz} implies the following.\n\n\\begin{corollary}\\label{Cor1}\nFix $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$, and \nlet $\\lambda := \\lambda(\\tau)$. (Note that $\\expanse (\\lambda)=n+1$.)\nThe probability of finding the PASEP model in configuration \n$(\\tau_1, \\dots , \\tau_n)$ in the steady state is \n\\begin{equation*}\n\\frac{F_{\\lambda}(q)}{Z_n}.\n\\end{equation*}\nHere, $F_{\\lambda}(q)$ is the weight-generating function for \npermutation tableaux of shape $\\lambda$. \nMoreover, the partition function $Z_n$ for the PASEP model\nis equal to the generating function for all permutation\ntableaux with expanse $n+1$, enumerated according to \nweight, unrestricted rows, and the first row.\n\\end{corollary}\n\nIn the case that $\\alpha=\\beta=1$, \nthis corollary together with results from work \nwith\nSteingr\\'{\\i}msson \\cite{SW} \nwill give us two more combinatorial interpretations\nof the probabilities in the steady state, in terms of \nexcedences and crossings, and descents and patterns in permutations. \nWe now recall \nsome definitions and results.\n\nIf $\\pi \\in S_n$ is a permutation, we say that $\\pi$ has a \n{\\it weak excedence} in position $i$ if $\\pi(i) \\geq i$. \nNote that there is always a weak excedence in position $1$.\nThe {\\it weak excedence set} of $\\pi$ is the set of all indices\n$i$ where there is a weak excedence.\nAdditionally, we say that $\\pi$ has a {\\it crossing} in positions $(i,j)$ \nif either $j < i \\leq \\pi(j) < \\pi(i)$ or \n$\\pi(i) < \\pi(j) < i < j$.\nCrossings were first defined by the first author in \\cite{Corteel}.\n\n\\begin{theorem} \\cite[Theorem 7]{SW} \\label{Bij}\nLet $T(k,n,c)$ be the set of permutation tableaux with \n$k$ rows, $n-k$ columns, and weight $c$. \nLet \n$M(k,n,c)$ be the set of all permutations $\\pi \\in S_n$\nwith $k$ weak excedences and $c$ crossings. \nThen there is a bijection $\\Phi: T(k,n,c) \\rightarrow M(k,n,c)$.\nMoreover, \n$\\Phi$ \nmaps a tableau $\\mathcal{T}$ whose path $\\phi(\\mathcal{T})$ has south steps in positions \n$S \\subset \\{1, \\dots , n\\}$ to a permutation \nwith weak excedence set $S$.\n\\end{theorem}\n\nGiven $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^{n}$,\nwe define $W(\\tau)$ to be the subset of $\\{1, 2, \\dots , n+1\\}$ which\ncontains $1$ and also contains $i+1$ if and only if $\\tau_i = 1$.\nDefine the {\\it excedence weight} $\\wt(\\pi)$ to be the number of crossings \nof $\\pi$, and \ndefine $F'_{\\tau}(q): = \\sum_{\\pi} q^{\\wt(\\pi)}$, \nwhere the sum is over all permutations in $S_{n+1}$ with \nweak excedence set $W(\\tau)$.\n\nBy combining Corollary \\ref{Cor1} and Theorem \\ref{Bij}, we obtain\nthe following result.\n\n\\begin{corollary}\\label{perminterp}\nFix $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$ and suppose\nthat $\\alpha = \\beta = 1$.\nThe probability of finding the PASEP model in configuration \n$(\\tau_1, \\dots , \\tau_n)$ in the steady state is \n\\begin{equation*}\n\\frac{F'_{\\tau}(q)}{Z_n}.\n\\end{equation*}\nHere $F'_{\\tau}(q)$ is the generating function which enumerates \npermutations in $S_{n+1}$ with weak excedence set $W(\\tau)$ according to number of\ncrossings.\n\\end{corollary}\n\nThis corollary can also be obtained in another way.\nNamely, in \\cite{Corteel2}, the authors provide a lattice path interpretation\nfor the probability of find the PASEP model in a particular configuration.\nIf one then applies a certain bijection on lattice paths \n\\cite[Lemma 9]{Corteel} and then the \nbijection of Foata and Zeilberger \\cite{FZ} from lattice paths to \npermutations, one arrives \nat Corollary \\ref{perminterp} in this manner.\n\n\n\\begin{example}\nSuppose that we are interested in $\\tau=(0,1,0)$. If \nwe set $\\alpha = \\beta = 1$ and use the results of Example\n\\ref{example1}, then we find that the probability of finding the PASEP\nmodel in configuration $\\tau$ in the steady state (for $\\alpha = \\beta = 1$)\nis \n$\\frac{q+2}{2q^2+8q+14}$.\nNote that $W(\\tau) = (1,3)$.\nNow observe that there are precisely three permutations in $S_4$\nwith weak excedence set $(1,3)$: $(3,1,4,2)$, and \n$(4,1,3,2)$, and $(2,1,4,3)$. The first of these permutations\nhas one crossing and the second two have none, so the generating function\nfor permutations of type $\\tau$ is $q+2$, in agreement with\nCorollary \\ref{perminterp}.\n\\end{example}\n\nWe now introduce some more definitions concerning permutations.\nIf $\\pi \\in S_n$ is a permutation, we say that $\\pi$ has a \n{\\it descent} in position $i$ if $\\pi(i) > \\pi(i+1)$. \nThe {\\it descent set} of $\\pi$ is the subset of $\\{1, 2, \\dots , n-1\\}$\nwhere $\\pi$ has descents. \nAnd we say that $\\pi$ has an occurrence of a {\\it generalized pattern of type} \n$2-31$ if there is a pair $(i,j)$ such that $1 \\leq i < j < n$ and \n$\\pi(j+1)<\\pi(i)<\\pi(j)$. These patterns were defined by \nBabson and Steingr\\'{\\i}msson \\cite{BS}.\n\nGiven $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^{n-1}$, \nwe define $D(\\tau)$ to be the subset of $\\{1, 2, \\dots , n \\}$ which \ncontains $i$ if and only if $\\tau_i = 1$.\nWe define \nthe {\\it descent weight} $\\wt'(\\pi)$ to be the number of crossings \nof $\\pi$, and \ndefine $F''_{\\tau}(q): = \\sum_{\\pi} q^{\\wt'(\\pi)}$, \nwhere the sum is over all permutations in $S_{n+1}$ with\ndescent set $D(\\tau)$.\n\nWe need the following result from \\cite{SW}, which \nis also proved in\nproved in \\cite[Proposition 6]{Corteel}.\n\n\\begin{theorem}\\cite[Theorem 16]{SW}.\nThere is a bijection $\\Psi$ on permutations in $S_n$, which \nsends a permutation $\\pi$ with descent set $D(\\tau)$ to a permutation\n$\\Psi(\\pi)$ with weak excedence set $W(\\tau)$. Moreover, the \nexcedence weight of $\\Psi(\\pi)$ is equal to the descent weight of \n$\\Psi(\\pi)$, i.e.\\ the number of crossings of $\\Psi(\\pi)$ is equal \nto the number of occurrences of the pattern $2-31$ in $\\pi$.\n\\end{theorem}\n\nBy applying this result to Corollary \\ref{perminterp}, we get the \nfollowing.\n\n\\begin{corollary}\\label{perminterp2}\nFix $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$ \nand suppose $\\alpha = \\beta = 1$.\nThe probability of finding the PASEP model in configuration \n$(\\tau_1, \\dots , \\tau_n)$ in the steady state is \n\\begin{equation*}\n\\frac{F''_{\\tau}(q)}{Z_n}.\n\\end{equation*}\nHere $F''_{\\tau}(q)$ is the generating function which enumerates \npermutations in $S_{n+1}$ with descent set $D(\\tau)$ \naccording to the number of\noccurrences of the pattern $2-31$.\n\\end{corollary}\n\nNow we will show how these results imply a result of \nthe first author \\cite{Corteel}.\n\nFirst we recall the definition of the polynomials\n$\\hat{E}_{k,n}(q)$ which were introduced by the second author \\cite{Williams}. \nDefine \n\\begin{equation*}\n{\\hat{E}}_{k,n}(q) =\n q^{k-k^2} \\sum_{i=0}^{k-1} (-1)^i [k-i]^n q^{ki-k}\n \\left( {n \\choose i} q^{k-i} + {n \\choose i-1}\\right).\n\\end{equation*} \n\nIt was shown (implicitly) there (and more explicitly \nin \\cite{SW}) that \n$\\hat{E}_{k,n}(q)$ enumerates permutation tableaux with \n$k$ rows and $n-k$ columns\naccording to weight. Additionally, it was shown \nin \\cite{Williams} that at $q=-1, 0, 1$, \n$\\hat{E}_{k,n}(q)$ specializes to binomial coefficients, \nNaryana numbers, and Eulerian numbers. \n\nIn \\cite{Corteel}, the following connection was made between\nthese polynomials and the PASEP model.\n\n\\begin{theorem} \\cite{Corteel} \\label{Sylvie}\nLet $\\alpha = \\beta = 1$.\nThen in the steady state,\nthe probability that the PASEP model with \n$n$ sites is in a configuration with precisely $k$ particles is:\n\\begin{equation*}\n\\frac{\\hat{E}_{k+1,n+1}(q)}{Z_n}\n\\end{equation*} \n\\end{theorem}\n\nSince the polynomials $\\hat{E}_{k,n}(q)$ enumerate permutation \ntableaux with $k$ rows according to weight, \nCorollary \\ref{Cor1} implies Theorem \\ref{Sylvie}.\n\n\\section{Proof of Theorem \\ref{Th1}}\\label{Proof}\n\nRecall the following requirement for permutation tableaux.\nWhen we read the columns of a permutation tableau $\\mathcal{T}$ from right to left,\nif in any column \nwe have a $0$ which lies beneath some $1$, then all \nentries to the left of $0$ (which are in the same row) must also\nbe $0$'s.\n\nAlso recall that an entry in a column of a permutation tableau is \n{\\it restricted} if \nthat entry is a $0$ which lies below some $1$. \nAnd a row is \n{\\it unrestricted}\nif it does not contain a restricted entry. \n\n\\begin{proof}\nWe will prove Theorem \\ref{Th1} inductively, by finding\na precise combinatorial interpretation \nfor each of the entries in the top row of a matrix product such as\n\\begin{equation*}\n\\prod_{i=1}^n (\\tau_i D_1 + (1-\\tau_i) E_1).\n\\end{equation*}\n\nLet \n$F^i_{\\lambda}(q):= \\sum_{\\mathcal{T}} q^{\\wt(\\mathcal{T})} \\alpha^{-f(\\mathcal{T})} \\beta^{-u(\\mathcal{T})}\n= \\sum_{\\mathcal{T}} q^{\\wt(\\mathcal{T})} \\alpha^{-f(\\mathcal{T})} \\beta^{-i+1}$, \nwhere the \nsum ranges over all permutation tableaux $\\mathcal{T}$ of shape\n$\\lambda$ which have precisely $i$ unrestricted rows. \nClearly $F_{\\lambda}(q) =\\sum_{i\\geq 1}\nF^i_{\\lambda}(q)$.\nNote that a permutation tableau will always have at least one \nunrestricted row (the top one).\n\nNow let $M_{\\lambda}$ be the matrix\n$\\prod_{i=1}^n (\\tau_i D_1 + (1-\\tau_i) E_1)$, where\n$(\\tau_1, \\dots , \\tau_n) = \\phi(\\lambda)$. \nWe claim that the entry $M_{\\lambda}[1,i]$ in position $(1,i)$ of $M_{\\lambda}$\nis $F^i_{\\lambda}(q)$.\n\nFirst note that this claim holds when $M_{\\lambda}$ is equal to $D_1$\nor $E_1$. $M_{\\lambda}$ is equal to $D_1$ when $\\lambda$ is the \npartition $(0,0)$. In this case there is a single permutation\ntableaux with two rows and no columns, which has weight $0$, \nand two unrestricted rows (hence $f(\\mathcal{T})=0$ and $u(\\mathcal{T})=1$).\nThis corresponds to the fact that the top row of $D_1$ is \n$(0,\\beta^{-1},0,\\dots)$. \nSimilarly, $M_{\\lambda}$ is equal to $E_1$ when $\\lambda$ is the partition\n$(1)$. \nIn this case there is a single permutation tableaux with one\nbox which is filled with a $1$ (hence $f(\\mathcal{T})=1$). \nThis tableaux has weight $0$ and \none unrestricted row (hence $u(\\mathcal{T})=0$), \ncorresponding to the fact that the top row\nof $E_1$ is $(\\alpha^{-1},0,0,\\dots)$.\n\nUsing induction, assume that our claim is true for $\\lambda$ with expanse \nless than or equal to $n$. In other words, we can \ninterpret the $i$th entry of the top row of $M_{\\lambda}$ as \na generating function enumerating permutation tableaux of shape\n$\\lambda$ with $i$\nunrestricted rows, according to weight.\nLet us now consider how these generating functions\nwill change if we instead consider permutation tableaux of \nshape ${\\lambda}'$, where ${\\lambda}'$ is a new shape obtained from \n$\\lambda$ by either adding a new row of length $0$, or a new column \nwhose length is the number of rows (including rows of length $0$) of \n$\\lambda$. The corresponding operation on paths is the following:\nwe take the path corresponding\nto $\\lambda$ and add an additional step from its south-west corner\nwhich is either south or west.\n\nIt is easy to see how adding a step south to the partition path ---\ni.e.\\ adding an empty row to $\\lambda$ --- will affect the generating\nfunctions. Any permutation tableau $\\mathcal{T}$ of the new shape $\\lambda'$ will\nbe a permutation tableau of shape\n$\\lambda$ with one additional unrestricted row\n(the last row). Therefore if $\\lambda'$ is equal to $\\lambda$ union\na row of length $0$, we will have \n$F_{\\lambda'}^{1}(q)=0$, and $F_{\\lambda'}^{i+1}(q)=\n\\beta^{-1} F_{\\lambda}^{i}(q)$ for\n$i\\geq 1$.\nThis corresponds to the fact that in the matrix product\n$M_{\\lambda}D_1$, the new top row will be \n$(0,\\beta^{-1} M_{\\lambda}[1,1], \\beta^{-1} M_{\\lambda}[1,2], \\dots )$.\n\nIt now remains to see how adding a step west to the partition path --- \ni.e.\\ adding an extra column of length $r$ to the left-hand-side of \n$\\lambda=(\\lambda_1, \\dots , \\lambda_r)$ \n--- will affect the generating functions. \nIn this case our new partition $\\lambda'$ is equal to \n$(\\lambda_1 +1, \\dots , \n\\lambda_r +1)$.\n\nLet $h_{a,b}(q)$ denote the polynomial \n$\\beta^{b-a} \\left(\\alpha^{-1} q^{a-1} {b-1 \\choose a-1} +\n\\sum_{j=0}^{a-2} q^j {b-a+j \\choose j} \\right)$ for $a \\leq b$.\nWe claim that \n$F_{\\lambda'}^{a}(q) = \\sum_{b \\geq a} h_{a,b}(q) F_{\\lambda}^{b}(q)$.\n\nTo prove this, it is enough to show the following.\nFix a permutation tableau $\\mathcal{T}$ of shape $\\lambda$ which has precisely\n$b$ unrestricted rows. Consider all ways of adding an additional\n(maximal) column $C$ to the left of $\\mathcal{T}$, in order to build a new permutation\ntableau $\\mathcal{T}'$ of shape $\\lambda'$ which has precisely $a$ unrestricted\nrows. Then the generating function for these new tableaux according \nto weight is precisely $h_{a,b}(q)q^{\\wt(\\mathcal{T})}$.\n\nTo prove this last statement, consider the process of adding the column\n$C$ to $\\mathcal{T}$. Since $\\mathcal{T}$ has $b$ unrestricted rows, there are only $b$ entries\nin $C$ in which we can choose to put either a $1$ or $0$; all other\nentries are forced to be $0$. Let us number those unrestricted entries from top\nto bottom by \n$c_1$ to $c_b$. Since we want our new column $C$ to \nadd an additional $b-a$ restricted positions, in our filling \nof the entries $c_1,\\dots,c_b$, we must have precisely $b-a$ $0$'s below\na $1$. If the top-most $1$ in our filling of $c_1, \\dots ,c_b$ is \nin position $c_i$, then the $i-1$ entries above it are all $0$'s, and\nwe have precisely ${b-i \\choose b-a}$ ways to choose which entries to make\n$0$ below $c_i$. The other $(b-i)-(b-a)=a-i$ entries must be $1$'s.\nThis particular choice of column will therefore contribute the extra weight \n$q^{a-i}$ to the weight of $\\mathcal{T}$ -- if $i \\neq 1$ -- and \nwill contribute the weight $\\alpha^{-1} q^{a-1}$ -- if $i=1$.\nIf we sum over all possible columns\n$C$ which we may add to $\\mathcal{T}$, we get the following:\n$\\alpha^{-1} q^{a-1} {b-1 \\choose b-a} + \n\\sum_{i=2}^a q^{a-i} {b-i \\choose b-a}$ which is equal to \n$\\alpha^{-1} q^{a-1} {b-1 \\choose a-1} + \n\\sum_{j=0}^{a-2} q^j {b-a+j \\choose j}$.\nThis completes the proof that \n$F_{\\lambda'}^{a}(q) = \\sum_{b \\geq a} h_{a,b}(q) F_{\\lambda}^{b}(q)$.\nAnd now note that this corresponds to the fact that in the matrix product\n$M_{\\lambda}E_1$, the entry in the first row and $a$th column \nwill be \n$\\sum_{b \\geq a} h_{a,b}(q) M_{\\lambda}[1,b]$.\n\n\nThis now completes our proof that for any partition $\\lambda$,\nthe entry $M_{\\lambda}[1,i]$ in position $(1,i)$ of $M_{\\lambda}$\nis $F^i_{\\lambda}(q)$. And since $F_{\\lambda}(q) = \\sum_{i \\geq 1}\nF^i_{\\lambda}(q)$, the theorem follows.\n\\end{proof}\n\n\n\\section{Applications of permutation tableaux}\\label{Applications}\n\nIn this section we will give some applications of \nthe connection of permutation tableaux \nto the PASEP. The first application is a partial order \non states of the PASEP and some monotonicity results (with respect to that \npartial order) of probabilities\nof observing these states. Note that some of the results below hold for\ngeneral $\\alpha, \\beta$ ($0 < \\alpha \\leq 1$ and $0 <\\beta \\leq 1$), and for others we need to assume that \n$\\alpha=\\beta=1$.\n\n\\begin{definition}\nLet $\\tau, \\tau' \\in \\{0,1\\}^n$ be two states of the PASEP which \ncontain exactly $k$ particles. We define the \npartial order $\\prec$ by $\\tau \\prec \\tau'$\nif and only if $\\lambda(\\tau) \\subset \\lambda(\\tau')$. \n\\end{definition}\n\nFigure \\ref{Order} illustrates this partial order when $n=5$ and\n$k=2$, and it shows the (unnormalized) probabilities\n$f_n(\\tau)$ that each of these states occurs, for $\\alpha=\\beta=1$.\n\n\\begin{figure}[h]\n \\centering\n\\includegraphics[height=4in]{order.eps}\n\\caption{A partial order for states of the PASEP}\n\\label{Order}\n\\end{figure}\n\nWe now give two simple inequalities relating \n$f_n(\\tau)$ to $f_n(\\tau')$ when $\\tau \\prec \\tau'$.\n\n\\begin{proposition}\\label{qdiff}\nLet $\\alpha$ and $\\beta$ be general. Suppose that $\\tau \\prec \\tau'$ and let \n$d:= |\\lambda(\\tau')| - |\\lambda(\\tau)|$. That is, \n$d$ is the difference between the cardinalities of \nthe Young diagrams $\\lambda(\\tau')$ and $\\lambda(\\tau)$.\nThen $f_n(\\tau') - q^d f_n(\\tau)$ is a non-negative polynomial \nin $q$, $\\alpha^{-1}$, and $\\beta^{-1}$.\n\\end{proposition}\n\n\\begin{proof}\nObserve that any permutation tableau of shape \n$\\tau$ can be naturally extended to a permutation tableau\nof shape $\\tau'$ by filling in all boxes of \n$\\lambda(\\tau') \\setminus \\lambda(\\tau)$ with $1$'s. \nThis new permutation tableau has weight $q^d$ times the weight\nof the old one.\n\\end{proof}\n\n\\begin{proposition}\\label{mono}\nLet $\\alpha=\\beta=1$.\nSuppose that $\\tau \\prec \\tau'$. Then \n$f_n(\\tau') - f_n(\\tau)$ is a non-negative polynomial in $q$.\nIn other words, as one moves up the partial order\n$\\prec$, the coefficients of $f_n(\\tau)$ monotonically\nincrease.\n\\end{proposition} \n\nProposition \\ref{mono} is false for general $\\alpha$ and $\\beta$.\nFor example, $f_4(1,1,0,0)-f_4(1,0,1,0)$ is a Laurent polynomial\nin $q$, $\\alpha$, and $\\beta$ with some negative coefficients.\n\nWe will defer the proof of Proposition \\ref{mono} to the next\nsection because it is most easily proved using Motzkin paths.\n\nWe thank the referee for pointing out that the corollary below\nfollows from our previous results.\n\n\\begin{corollary}\nLet $\\alpha=\\beta=1$. Suppose that $\\tau \\prec \\tau'$. Let \n$d$ be any integer such that \n$0 \\leq d \\leq |\\lambda(\\tau')| - |\\lambda(\\tau)|$. \nThen\n$f_n(\\tau')-q^d f_n(\\tau)$ is a non-negative polynomial in $q$.\n\\end{corollary}\n\n\\begin{proof}\nThis follows from Propositions \\ref{qdiff} and \\ref{mono}. Simply\nchoose a saturated chain of partitions \n$\\tau = \\tau_0 \\prec \\tau_1 \\prec \\dots \\tau_m=\\tau'$\nsuch that $|\\lambda(\\tau_{k+1})|-|\\lambda(\\tau_k)|=1$\nand apply either Proposition \\ref{qdiff} or Proposition \n\\ref{mono} at each step. \n\\end{proof}\n\nThe above results make sense intuitively,\nsince in a model with particles entering from the left and leaving\nto the right, it is more likely that a given particle will be \nfurther to the left than to the right. Also recall that \nthe probability of hopping left ($\\frac{q}{n+1}$ for some \n$q\\leq 1$) is at most the \nprobability of hopping right ($\\frac{1}{n+1}$)\n\nAnother application of permutation tableaux\nis that these objects\nallow one to \nread off the main\nrecurrences for the PASEP \n(\\cite[Theorem 1]{Corteel2}) quite easily.\nThe rest of the results in this section hold for general \n$\\alpha$ and $\\beta$.\n\nFix a partition $\\lambda = (\\lambda_1, \\dots , \\lambda_k)$, \nand let $\\tau = (\\tau_1, \\dots , \\tau_n)$ be the \nunique vector in $\\{0,1\\}^n$\nsuch that $\\lambda = \\lambda (\\tau)$.\nChoose any {\\it corner} of $\\lambda$, i.e.\\ \nthe last box of a row $\\lambda_i$ in $\\lambda$\nsuch that $\\lambda_{i+1} < \\lambda_i$.\n(Equivalently, a pair $(j,j+1)$ of entries in $\\tau$ such that\n$\\tau_j = 1$ and $\\tau_{j+1}=0$.)\nAs Figure \\ref{Recurrence} illustrates, there is a simple\nrecurrence for $F_{\\lambda}(q)$.\n\n\\begin{figure}[h]\n \\centerline{\\epsfig{figure=newrecurrence.eps}}\n\\caption{Recurrence for $F_{\\lambda}(q)$}\n\\label{Recurrence}\n\\end{figure}\n\nExplicitly, any valid filling of $\\lambda$ is obtained in one\nof the following ways:\n\\begin{itemize}\n\\item inserting a column whose bottom entry is $1$ and whose\nother entries are $0$ after the $(\\lambda_i - 1)$st column of\na valid filling of $(\\lambda_1 - 1, \\lambda_2 - 1, \\dots , \n\\lambda_k-1)$;\n\\item adding a $1$ to the end of the $i$th row of a valid filling\nof the shape $(\\lambda_1, \\lambda_2, \\dots , \\lambda_i - 1, \\dots , \\lambda_k)$;\n\\item inserting an all-zero row of length $\\lambda_i$ as the $i$th row\nin a valid filling of $(\\lambda_1, \\dots , \\lambda_{i-1}, \\lambda_{i+1}, \n\\dots , \\lambda_k)$.\n\\end{itemize}\n\nNote that none of the operations above change the number of $1$'s in \nthe first row or the number of unrestricted rows. Therefore, \nfor $1 < i \\leq k$ and $\\lambda_i > \\lambda_{i+1}$, we have that \n\\begin{equation*}\nF_{\\lambda}(q) = F_{(\\lambda_1 - 1, \\dots , \\lambda_i -1, \\lambda_{i+1}, \\dots , \\lambda_k)} + \nF_{(\\lambda_1, \\dots , \\lambda_{i-1}, \\lambda_{i+1}, \\dots , \\lambda_k)} +\nqF_{(\\lambda_1, \\dots , \\lambda_{i-1},\\lambda_i -1, \\lambda_{i+1}, \\dots,\n\\lambda_k)}.\n\\end{equation*}\n\nBy Corollary \\ref{Cor1}, this recurrence translates into a\nrecurrence for the PASEP model.\nRecall our notation $\\frac{f_n(\\tau_1, \\dots , \\tau_n)}{Z_n}$ for \nthe probability of finding the system in \nconfiguration $(\\tau_1, \\dots , \\tau_n)$ in the steady state.\n\n\\begin{corollary}\\cite[Theorem1]{Corteel2}\nLet $\\alpha$ and $\\beta$ be general. Then\n\\begin{align*}\nf_n(\\tau_1, \\tau_2, \\dots , \\tau_{j-1}, 1, 0, \n\\tau_{j+2}, \\dots , \\tau_n) = \n&f_{n-1}(\\tau_1, \\tau_2, \\dots , \\tau_{j-1}, 1,\n\\tau_{j+2}, \\dots , \\tau_n) +\\\\\n&qf_n(\\tau_1, \\tau_2, \\dots , \\tau_{j-1},0,1,\n\\tau_{j+2},\\dots , \\tau_n) +\\\\\n&f_{n-1}(\\tau_1, \\dots , \\tau_{j-1}, 0, \n\\tau_{j+2}, \\dots , \\tau_n).\n\\end{align*}\n\\end{corollary}\n\nIt is also easy to prove the following, by consideration \nof permutation tableaux.\n\n\\begin{corollary}\\cite[Theorem 1]{Corteel2}\nLet $\\alpha$ and $\\beta$ be general. We have that \n$f_n(\\tau_1, \\dots , \\tau_{n-1},1)=\n\\frac{1}{\\beta} f_n(\\tau_1, \\dots , \\tau_{n-1})$\nand also \n$f_n(0, \\tau_1, \\tau_2, \\dots , \\tau_n)=\n\\frac{1}{\\alpha} f_n(\\tau_1, \\tau_2, \\dots , \\tau_n)$.\n\\end{corollary}\n\n\\begin{proof}\nThis also follows from\nCorollary \\ref{Cor1}. The first equality holds because\nthe generating function for permutation tableaux of shape\n$(\\lambda_1, \\lambda_2, \\dots , \\lambda_{n-1},0)$ is clearly equal \nto the generating function for permutation tableaux of shape\n$(\\lambda_1, \\lambda_2, \\dots , \\lambda_{n-1})$ except that\nthere is one additional unrestricted row. The second\nequality holds because \nthe generating function for permutation tableaux of shape\n$(\\lambda_1 + 1, \\lambda_2, \\dots , \\lambda_{n})$ is equal \nto the generating function for permutation tableaux of shape\n$(\\lambda_1, \\lambda_2, \\dots , \\lambda_{n})$ except that there\nis one additional entry in the first row which is a $1$. \n(Recall that each column of a permutation tableaux is required to \ncontain exactly one $1$ and that the weight of a column is equal\nto the number of $1$'s it contains {\\it beyond} that requisite\none.)\n\\end{proof}\n\n\n\n\n\\section{Connection with Motzkin paths}\\label{Motzkin}\n\nWe now give another solution to the matrix ansatz, \nwhich is essentially the one \ndescribed in\nDerrida et al \\cite{Derrida1}, and which \nhas an interpretation in terms of \nMotzkin paths. \n\nNote that in this section we restrict to the \ncase $\\alpha = \\beta = 1$. These results can be extended to the more\ngeneral case, but we restrict ourselves to this case since our main\npurpose here is to obtain the result needed for the\nmonotonicity result of the previous section.\n\nLet $D_0$ be the (infinite) upper triangular\nmatrix $(d_{ij})$ such that \n$d_{i,i}=[i]$ where $[i]$ is the $q$-analog of $i$, \n$d_{i,i+1}=[i+1]$, and $d_{i,j}=0$ for $j\\neq i, i+1$.\n\nThat is, $D_0$ is the matrix\n\\[ \\left( \\begin{array}{ccccc}\n[1] & [2] & 0 & 0 & \\dots \\\\\n0 & [2] & [3] & 0 & \\dots \\\\\n0 & 0 & [3] & [4] & \\dots \\\\\n0 & 0 & 0 & [4] & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\vdots &\n\\end{array} \\right). \\]\n\nLet $E_0$ be the (infinite) lower triangular matrix\n$(e_{i,j})$ such that \n$e_{i,i}=[i]$, \n$e_{i+1,i}=[i]$, and $e_{i,j}=0$ for $j\\neq i,i-1$.\n\nThat is, $E_0$ is the matrix\n\\[ \\left( \\begin{array}{ccccc}\n1 & 0 & 0 & 0 & \\dots \\\\\n1 & [2] & 0 & 0 & \\dots \\\\\n0 & [2] & [3] & 0 & \\dots \\\\\n0 & 0 & [3] & [4] & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\vdots &\n\\end{array} \\right). \\]\n\nLet $W_0$ be the (row) vector $(1,0,0,\\dots )$ and $V_0$ be the (column)\nvector\n$(1,0,0,\\dots)$.\nIt is now easy to check that \n$D_0 E_0 -q E_0 D_0 = D_0 + E_0$, \n$DV_0 = V_0$, and $W_0 E = W_0$.\n\nWe will now give an interpretation of the steady states\nof the PASEP\nin terms of bicolored\nMotzkin paths. This result is very similar to a result obtained in \nBrak et al \\cite{Corteel2}, via a combinatorial derivation. \nNote also that in \\cite{BrakEssam}, Brak and Essam\nconsidered the case $q=0$ of the PASEP and \ngave multiple interpretations for the steady states \nin terms of various weighted lattice paths.\n\nWe define a {\\it bicolored Motzkin path} of length $n$ to be a sequence of \nsteps in the plane\n$c=(c_1, \\dots , c_n)$ such that \n$c_i \\in \\{N, S, E, \\bar{E}\\}$ for $1 \\leq i \\leq n$, \nwhich starts and ends at height $0$, and \nalways stays at or above height $0$. \nThat is, if we define the \nheight at the $i$th step to be \n$h_i = \\{j < i \\ \\vert \\ c_j = N \\} - \\{j < i \\ \\vert \\ c_j=S\\}$,\nthen $h_i \\geq 0$ for $1 \\leq i \\leq n$, and furthermore, \n$h_{n+1}=0$.\n\n\nWe now assign a weight $\\wt(p)$ to a bicolored Motzkin path $p$ as\nfollows: step $c_i$ is assigned weight $[h+1]$ if $c_i$ ends at height\n$h$, and the weight of $c$ is defined to be the product of the weights of\nall the $c_i$'s.\n\nFix a sequence $\\tau=(\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$.\nWe say that a bicolored Motzkin path $c$ has {\\it type} \n$\\tau$ if $c_i=N$ or $E$ whenever $\\tau_i=1$ and \n$c_i=S$ or $\\bar{E}$ whenever $\\tau_i=0$. \n\nWe now define $F'''_{\\tau}(q)$ to be the generating function for all \nbicolored Motzkin paths of type $\\tau$: that is, \n$F'''_{\\tau}(q) :=\\sum_p q^{\\wt(p)}$ where the sum is over all bicolored\nMotzkin paths of type $\\tau$.\n\nThe main result of this section is the following.\n\n\\begin{proposition}\\label{M}\nChoose a sequence $\\tau=(\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$.\nThen \n\\begin{equation*}\nF'''_{\\tau}(q) = W_0 (\\prod_{i=1}^n (\\tau_i D_0 + (1-\\tau_i) E_0)) V_0\n\\end{equation*}\n\\end{proposition}\n\nLet us make the convention in this section that rows and columns \nof our matrices are indexed by non-negative integers (including $0$).\nThen Proposition \\ref{M} says that \nthe generating function $F'''_{\\tau}(q)$ is equal \nto the entry in the $0$th row and $0$th column of the matrix\nproduct \n$\\prod_{i=1}^n (\\tau_i D_0 + (1-\\tau_i) E_0)$. \n\n\\begin{proof}\nUsing the definition of matrix multiplication, we\nexpand $\\prod_{i=1}^n (\\tau_i D_0 + (1-\\tau_i) E_0)$\nas the sum of terms of the form\n$A_1[1,i_1] A_2[i_1, i_2] \\dots A_n[i_{n-1},1]$\nwhere the $A_j$'s are matrices $D_0$ or $E_0$, depending on\nthe sequence $\\tau$. It is obvious that these terms correspond\nto the Motzkin paths of type $\\tau$.\n\\end{proof}\n\nAs before, we can use \nProposition \\ref{M}\ntogether with Theorem \\ref{ansatz} to deduce the following.\n\n\\begin{corollary}\\label{Cor3}\nFix $\\tau = (\\tau_1, \\dots , \\tau_n) \\in \\{0,1\\}^n$. \nThe probability of finding the PASEP model in configuration \n$(\\tau_1, \\dots , \\tau_n)$ in the steady state is \n\\begin{equation*}\n\\frac{F'''_{\\tau}(q)}{Z_n}.\n\\end{equation*}\nHere $F'''_{\\tau}(q)$ is the generating function for \nbicolored Motzkin paths of type $\\tau$.\n\\end{corollary}\n\nWe now use this result to prove Proposition \\ref{mono}\nfrom the previous section:\n\n\n\\begin{proof}\nIt is sufficient to consider two configurations\n$\\tau$ and $\\tau'$ whose corresponding diagrams\n$\\lambda(\\tau)$ and $\\lambda(\\tau')$ differ by one box.\nThat is, there exists some $i$ such that:\\\\\n$\\bullet$ $\\tau_i = 0$, $\\tau_{i+1}=1$\\\\\n$\\bullet$ $\\tau'_i = 1$, $\\tau'_{i+1}=0$\\\\\n$\\bullet$ $\\tau'_j = \\tau_j$ for $j \\neq i, i+1$.\n\nWe now observe that any bicolored Motzkin path $p$ of \ntype $\\tau$ (with weight $w$) can be mapped \nto a bicolored Motzkin path $p'$\nof type $\\tau'$ whose weight $w'$ is coefficient-wise\ngreater than $w$.\n\nIf $p=(c_1, \\dots , c_n)$ then we let \n$p':=(c_1, \\dots, c_{i-1},c_{i+1},c_{i},c_{i+2},\\dots,c_n)$.\nThat is, $p'$ is the path obtained from $p$ by switching the \n$i$th and $i+1$st steps.\nIt is clear that whether we switch $SN$ steps for $NS$,\nor $SE$ for $ES$, or $\\bar{E}N$ for $N\\bar{E}$, or \n$\\bar{E}E$ for $E\\bar{E}$, the resulting path is a valid\nMotzkin path. The weight $w'$ will differ from $w$ by\n(respectively): replacing a factor \n$[h-1][h]$ with $[h+1][h]$; \nreplacing a factor $[h-1]^2$ with $[h][h-1]$;\nreplacing a factor $[h][h+1]$ with $[h+1]^2$;\nnothing.\nUsing the fact that $[n+1]-[n]$ is nonnegative, all of the\ndifferences\n$[h+1][h]-[h-1][h]$, \n$[h][h-1]-[h-1]^2$, $[h+1]^2-[h][h+1]$\nare nonnegative; therefore in all cases,\n$w'-w$ will be a nonnegative polynomial.\nBy Corollary \\ref{Cor3}, we are done.\n\\end{proof}\n\nNote that one can use the same argument to give another proof of\nProposition \\ref{qdiff} in the case that $\\alpha=\\beta=1$. We repeat the\nsame argument but now \n$w'$ differs from $w$ by: \nreplacing $q[h-1][h]$ with $[h+1][h]$; replacing \n$q[h-1]^2$\nwith $[h][h-1]$; and replacing $q[h][h+1]$ with $[h+1]^2$. Then all\ndifferences of such polynomials will be nonnegative, since $[n+1]-q[n]$ is\nnonnegative.\n\n\n\n\\begin{remark}\nWe now summarize our combinatorial interpretations for the \nsteady state probabilities of the PASEP model; these are given by\nour solutions $(D_j, E_j)$ to the matrix \nansatz (where $\\alpha = \\beta = 1$), for $j=0,1$.\nNote that since \n\\begin{equation*}\nW_j (\\prod_{i=1}^n (\\tau_i D_j + (1-\\tau_i) E_j)) V_j \n\\end{equation*}\ndescribes the steady state probability of the PASEP model\nfor both $j=0,1$, these two formulas must be equal. \nIn particular, both of these are formulas for all of the following:\n\\begin{itemize}\n\\item $F_{\\lambda(\\tau)}(q)$, the weight-generating function for \npermutation tableaux of shape $\\lambda(\\tau)$.\n\\item $F'_{\\tau}(q)$, the generating function for permutations of \nexcedence type $\\tau$, enumerated according to crossings.\n\\item $F''_{\\tau}(q)$, the generating function for permutations of \ndescent type $\\tau$, enumerated according to occurrences of $2-31$.\n\\item $F'''_{\\tau}(q)$, the generating function for weighted\nbicolored Motzkin \npaths.\n\\end{itemize}\n\nMoreover, the product $W_j (D_j+E_j)^n V_j$ is the weight-generating \nfunction for all of the following:\n\\begin{itemize}\n\\item permutation tableaux with expanse $n+1$.\n\\item permutations in $S_{n+1}$, enumerated according to crossings. \n\\item permutations in $S_{n+1}$, enumerated according to occurrences of \n$2-31$.\n\\item weighted bicolored Motzkin paths of length $n$.\n\\end{itemize}\n\n\\end{remark}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nEver since Wigner and Huntington's prediction in 1935 that solid molecular hydrogen\nwould dissociate and form an atomic and metallic phase at high pressures~\\cite{wigner},\nthe phase diagram of hydrogen has been the focus of intense experimental and \ntheoretical studies.\nAdvanced experimental techniques, notably diamond anvil cell approaches, mean that it \nis now possible to explore hydrogen at pressures up to about\n360 GPa~\\cite{mao1994,loubeyre2002,eremets,zha}, and new types of diamond anvil cell may be able to \naccess much higher pressures~\\cite{dubrovinsky_2012}.\nThese experiments, along with numerous theoretical studies, have revealed a remarkably \nrich and interesting phase diagram comprising regions of stability for \na molecular solid, a molecular liquid and an atomic liquid, and\nwithin the solid region four distinct phases have been detected~\\cite{howie,goncharov,eremets,zha}.\nFurthermore, metallic hydrogen has been observed in high temperature\nshock wave experiments~\\cite{fortov,weir,nellis,hicks} and it is accepted to be a major component\nof gas giant planets such as Jupiter and Saturn~\\cite{exoplanet}.\nDespite the tremendous and rapid progress, important gaps in our understanding of the phase diagram \nof high pressure hydrogen remain, with arguably the least well understood issue being the solid to liquid melting\ntransition at very high pressures.\nIndeed the melting curve is only established experimentally and theoretically \nup to around 200 GPa~\\cite{datchi,gregoryanz,deemyad,eremetsjetp,bonev2004}.\nHowever, from 65 GPa up to about 200 GPa the slope of the melting \nline is negative (i.e., the melting point drops with increasing pressure), which suggests that\nat yet higher pressures a low temperature liquid state of hydrogen might exist or, \nas suggested by Ashcroft, perhaps even a metallic liquid state at zero K~\\cite{ashcroft2004}.\nFurther interest in hydrogen at pressures well above 200 GPa stems from other remarkable \nsuggestions such as superfluidity~\\cite{ashcroft2000jpcm} and superconductivity at room\ntemperature~\\cite{ashcroft1968,gross2008,mcmahon}, \nall of which imply that hydrogen at extreme pressures could be one of the most interesting \nand exotic materials in all of condensed matter.\n\n\nIn the current study we use computer simulation techniques to probe \nthe low temperature phase diagram of hydrogen in the ultra-high 500-1,200 GPa regime\nto try and find this potential low temperature liquid state of hydrogen.\nWe describe the proton motion using \\textit{ab initio} path-integral\nmolecular dynamics (PIMD)\nmethods~\\cite{marxpimd1,marxpimd2,tuckermanpimd1,tuckermanpimd2,qfzhang,xzliprl,xzlipnas},\nwhich are based on forces computed ``on the fly'' as the dynamics of\nthe system evolves, and can account for bond formation and breaking in\na seamless manner.\nTo compute the melting line we simulate solid and liquid phases in\ncoexistence~\\cite{morris,ogitsu,alfe,bonev2004}. \nThe coexistence approach minimises hysteresis effects arising from super-heating or\nsuper-cooling during the phase transition, and\nalthough it has been used with model potentials and \\textit{ab initio} approaches before, the\ncurrent study is, to the best of our knowledge, the first to combine\nit with the \\textit{ab initio} PIMD method. \nWith this combination of \napproaches we have found a low-temperature metallic atomic liquid phase at\npressures of 900~GPa and above, down to the lowest temperature we can\nsimulate reliably of 50~K.\nThe existence of this low temperature metallic atomic liquid is\nassociated with a negative slope of the melting line between atomic\nliquid and solid phases at pressures between 500 and 800~GPa.\nThis low temperature metallic atomic liquid is strongly quantum in\nnature since treating the nuclei as classical particles significantly\nraises the melting line of the atomic solid to $\\sim$300~K over the\nwhole pressure range. \nIt also destroys the negative slope of the melting line,\nand consequently it does not predict a low temperature liquid phase.\nOur study provides a finite temperature phase diagram of hydrogen\nunder ultrahigh pressures and gives direct numerical support for\nAshcroft's low temperature metallic liquid.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.70\\linewidth]{.\/figure1.eps}\n\\caption{\\label{figure1}\n\\textit{Ab initio} PIMD simulations of solid-liquid coexistence and melting. Snapshots \nof the PIMD simulations at 700~GPa showing (a) the starting structure, \n(b) the final state at 100~K, and (c) the final state at 125~K.\n32 beads~(green balls) were used to represent the imaginary time path-integral for each atom.\nThe grey balls in the insets of (b) and (c) correspond to the centroid of each atom.\n(d) Radial distribution functions (RDFs) for the same two simulations at 100 and 125 K. \nAt 100~K (black solid line) the solid state persists, whereas at 125~K (red dashed line)\nthe RDF is characteristic of a liquid. \n(e) Mean square displacements \n(MSD) as a function of time from separate adiabatic centroid MD simulations within\nthe path-integral framework.\nThe MSD for the 100~K solid phase saturates rapidly, whereas for the liquid phase at 125~K \nit rises approximately linearly with time, resulting in a finite diffusion\ncoefficient. }\n\\end{figure}\n\nExtensive computational searches for low enthalpy solid structures of\nhydrogen have been performed using density functional theory (DFT)\nmethods~\\cite{pickard2007,ceperley2011,pickard2012,pickard2012b,yanming} and recently a systematic\nanalysis of the evolution of the low-enthalpy phases has been provided\nby Labet \\textit{et al.}~\\cite{labet1,labet3,labet4}.\nThese searches found a metallic phase of $I4_1\/amd$ space group\nsymmetry to be stable from about 500 GPa to 1,200~GPa~\\cite{ceperley2011},\nwhen quasi-harmonic proton zero-point motion was included.\nWe use this phase, therefore, as the starting point for our finite temperature exploration\nof the phase diagram and melting line. \nWith the coexistence method we have performed a series of \ntwo phase solid-liquid simulations at different temperatures (from 50~K to 300~K) which \nare then used to bracket the melting \ntemperature from above and below. \nWe begin by considering the 500--800~GPa pressure regime and show an example of the data \nobtained from the coexistence simulations at 700~GPa in Fig.~\\ref{figure1}.\nAt this pressure we find that for $T\\geq$125~K the system transforms in to a liquid state, while for\n$T\\leq$100~K, it ends up as solid.\nThe phases on either side of the melting line (at this and other pressures) were characterized by \ntheir radial distribution function (RDF) and as can be seen in Fig.~\\ref{figure1} (d) upon moving from\n100 to 125 K the system clearly transforms from solid to liquid.\nThe phases were also characterised by the variations in mean\nsquare displacement (MSD) of the nuclei of the particles over time.\nSince PIMD rigorously provides only thermally averaged\ninformation we used the adiabatic centroid molecular dynamics (MD) approach within the\npath-integral scheme to obtain real-time quantum dynamical\ninformation~\\cite{adcmd}. \nAgain, as shown in Fig.~\\ref{figure1} (e), the distinction\nbetween the solid phase at 100 K and liquid phase at 125 K is clear.\n\nThe same coexistence procedure was used to locate the melting point at 500 and 800 GPa, leading to the\nmelting line shown in Fig.~\\ref{figure2}. \nThe up~(down) triangles indicate the highest (lowest) temperatures at \nwhich the solid (liquid) phases are stable, bracketing the melting temperatures within a 25~K \nwindow. \nFrom this we see that the melting temperature is 150--175~K\nat 500~GPa and that it drops rapidly with increasing pressure yielding a melting \ntemperature of only 75--100~K at 800~GPa.\nThus the melting curve has a substantial negative slope\n($dP$\/$dT<0$) in this pressure range.\nAcross this entire pressure range the molten liquid state is atomic \nand the solid phase which grows is the original atomic $I4_1\/amd$ phase that\nwas used as the starting structure.\nGiven that molecular phases have been observed at pressures lower than \n360~GPa in both experimental and theoretical studies~\\cite{zha,loubeyre2002,kitamura},\nwe suggest that a molecular-to-atomic solid-solid phase transition should occur between\n360 and the lowest pressure of 500~GPa considered here.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{.\/figure2.eps}\n\\caption{\\label{figure2}\nPhase diagram of hydrogen with regions of stability for the molecular solid (yellow), \nmolecular liquid (purple), atomic solid (blue), and atomic liquid (pink) indicated\nby the various colours.\nThe dashed line separating the molecular and atomic liquid phases is taken from \nquantum Monte Carlo calculations~\\cite{ceperley2010}.\nThe solid line separating the molecular solid and molecular liquid phases\nis taken from \\textit{ab initio} MD simulations~\\cite{bonev2010}, whose negative\nslope has been confirmed by several experiments~\\cite{datchi,gregoryanz,deemyad,eremetsjetp}.\nThe thick black line is the melting curve obtained in this study from the\n\\textit{ab initio} PIMD coexistence simulations.\nThe solid lines separating phases I, II, III and IV are from Refs.~\\cite{goncharov,howie}.\nThe inset shows how the high pressure melting curve (dashed lines) are established \nin our simulations.\nThe black and red triangles (inset) correspond to the PIMD and MD results, respectively.\nThe up triangles give the highest temperatures for solidification and\nthe down triangles show the lowest temperatures for liquefaction. }\n\\end{figure}\n\nThe negative slope of the melting line up to\n800 GPa suggests that at even higher pressures a lower temperature liquid phase might\nexist.\nMotivated by this, we carried out simulations at 900 and 1,200~GPa.\nHowever, in this pressure range we have to consider nuclear exchange\neffects, which are neglected in the PIMD simulations performed here,\nbut which could potentially become significant.\nIndeed, analysis of our simulations reveals that at these pressures the dispersion of the beads\nin the path integral ring polymer becomes comparable to the smallest inter-atomic separations when the\ntemperature is below $\\sim$40~K (see Fig.~S6). \nThis is the so-called quantum degeneracy temperature,\nbelow which the exchange of nuclei will be important.\nWith this in mind we performed all simulations in this very high pressure regime at \n$T \\geq 50$~K.\nInterestingly we find that at 50~K at both 900 and 1,200 GPa the systems are \nalready in the liquid state, revealing that the melting temperate at these pressures\nis $<$ 50 K. \nWhether the liquid phase is the 0~K ground state of hydrogen at these pressures is\nnot something we can establish at this stage.\nHowever, the large negative slope of the melting line at lower pressures and\nthe observation of a liquid phase at temperatures as low as 50~K present strong support for\nAshcroft's low temperature liquid metallic state of hydrogen~\\cite{ashcroft2004},\nand it implies that any room temperature superconductor in this regime would have to\nbe a liquid.\n\nIt is instructive to compare the results of the \\textit{ab initio} \nPIMD simulations with those obtained from the \\textit{ab initio} MD\napproach in which the nuclei are approximated by classical point-like particles. \nTo this end we performed a second complete set of coexistence simulations with \\textit{ab initio} MD across the\nentire 500--1,200 GPa range. \nThe \\textit{ab initio} MD melting line is shown by the red data in the inset of Fig.~\\ref{figure2}, \nwhere it can be seen that the melting temperatures obtained from the MD simulations are much higher than\nthose from the fully quantum PIMD simulations. \nThe \\textit{ab initio} MD melting temperature is well above 200 K\nacross the pressure range 500--1,200 GPa, and the slope of the melting\nline is small.\nThese remarkably large differences between the \\textit{ab initio} MD and PIMD melting lines \nclearly demonstrate that quantum nuclear effects play a crucial role in the stability\nof the low temperature liquid.\n\nIn conclusion, our results are consistent with a melting curve with a negative slope\nbetween an atomic solid and a low temperature (50~K or\nbelow) metallic atomic phase.\nA comparison of the MD and PIMD results shows that the quantum nature\nof the nuclei is responsible for the large negative slope of the\nmelting line and the consequent existence of the low temperature metallic liquid. \nA natural conclusion from this is that the phase diagram for hydrogen in the pressure regime\nconsidered should exhibit rather large isotope effects. \nIt is not clear from our studies whether the liquid phase survives\ndown to 0~K, and this needs to be studied using other methods.\nThe occurrence of the $I4_1\/amd$ solid above\nabout 500~GPa indicates that a molecular-to-atomic solid-solid phase\ntransition must occur at lower pressures, which could be accessible to\nexperiments in the near future.\n\n\\section*{Methods}\n\nDensity-functional theory~(DFT) calculations \nwith the PBE exchange correlation functional \nwere performed with the VASP code~\\cite{pbe, VASP1, VASP2, VASP3, VASP4}.\nProjector augmented wave potentials were used along with a 500~eV plane wave \ncutoff energy~\\cite{paw}. \nA Monkhorst-Pack $k$-point mesh of\nspacing $2\\pi \\times 0.05\\text{\\AA}^{-1}$ was used along with a unit cell\nthat typically contained 200 atoms.\nConvergence tests with larger unit cells and higher plane wave cutoff energies are reported\nin the SI.\n\nMD and PIMD simulations were of the ``Born-Oppenheimer'' type within the NVT ensemble, \nusing a Nos\\'e-Hoover chain thermostat~\\cite{nose-hoover}.\nA 0.5~fs time step was used and all simulations were run for at least 20,000 steps (10~ps). \nThe PIMD simulations made use of a recent implementation \nof this method in VASP.\n32 beads were used to sample the imaginary-time path-integral at each temperature, \nas this was found to be the minimum number of beads needed to obtain reliable \nestimates of the melting temperature (SI).\nWhen calculating real-time atomic diffusion\nin the liquid phase using the adiabatic centroid MD\nmethod~(Fig.~\\ref{figure1} (e)), a smaller time step of 0.05~fs was used\nafter thermalization and 20,000 steps were used to characterize the atomic diffusion.\n\nIn the coexistence simulations, each run was initiated from a\nstructure comprising a solid $I4_1\/amd$ phase and a liquid phase\nobtained by selecting samples from earlier simulations at higher\ntemperatures.\nTo test the sensitivity of the results to the chosen starting structures we performed\nsimulations with several different uncorrelated liquid phase\nstructures, and found that the results were independent\nof the starting liquid structure.\nThe system was normally found to liquefy or solidify after about\n1--2~ps and equilibrium was reached soon afterwards~(Fig.~S5).\nFor more computational details, please read the supporting online\nmaterials.\n\n\\begin{acknowledgements}\n J.C., X.Z.L., and E.W.\\ are supported by NSFC. A.M.\\ is supported by\n the European Research Council and the Royal Society through a Royal\n Society Wolfson Research Merit Award.\n We are grateful for computational resources provided by the\n supercomputer TianHe-1A in Tianjin, China.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{{\\normalsize \\bf #2}}\\list\n {[\\arabic{enumi}]}{\\settowidth\\labelwidth{[#1]}\\leftmargin\\labelwidth\n \\advance\\leftmargin\\labelsep\n \\usecounter{enumi}}\n \\def\\newblock{\\hskip .11em plus .33em minus -.07em}\n \\sloppy\n \\sfcode`\\.=1000\\relax}\n\\let\\endthebibliograph=\\endlist\n\\newcommand{\\lcorr}{{\\Large \\bf ![}\\,}\n\\newcommand{\\rcorr}{{\\Large \\bf ]!}\\,}\n\\newcommand{\\dist}{{\\mathcal D}'({\\mathbb{R}}^n)}\n\\newcommand{\\sign}{{\\rm sign}\\, }\n\\newcommand{\\test}{{\\mathcal D}({\\mathbb{R}}^n)}\n\\newcommand{\\tombstone}{\\hspace{0cm}\\hspace*{\\fill} \\rule{3mm}{3mm}\n\\\\[2mm]}\n\\newcommand{\\euclid}{{\\mathbb{R}}^n}\n\\newcommand{\\re}{{\\mathbb{R}}}\n\\newcommand{\\R}{{\\mathbb{R}}^n}\n\\newcommand{\\cj}{{\\mathcal{J}}}\n\\newcommand{\\cs}{{\\mathcal{S}}}\n\\newcommand{\\cl}{{\\mathcal{L}}}\n\\newcommand{\\cf}{{\\mathcal{F}}}\n\\newcommand{\\cfi}{{\\mathcal{F}}^{-1}}\n\\newcommand{\\besovn}[3]{B_{{#2},{#3}}^{#1}({\\mathbb R}^n)}\n\\newcommand{\\besov}[3]{B_{{#2},{#3}}^{#1}({\\mathbb R})}\n\\newcommand{\\besovhom}[3]{\\dot{B}_{{#2},{#3}}^{#1}({\\mathbb R})}\n\\newcommand{\\besovhomn}[3]{\\dot{B}_{{#2},{#3}}^{#1}({\\mathbb R}^n)}\n\\newcommand{\\vp}{{\\mathcal V}_p}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{prop}{Proposition}\n\\newtheorem{definition}{Definition}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{remark}{Remark}\n\\input{tcilatex}\n\\begin{document}\n\n\\title{Sobolev embeddings for Herz-type Triebel-Lizorkin spaces}\n\\author{ Douadi Drihem \\ \\thanks\nM'sila University, Department of Mathematics, Laboratory of Functional\nAnalysis and Geometry of Spaces , P.O. Box 166, M'sila 28000, Algeria,\ne-mail: \\texttt{\\ douadidr@yahoo.fr}}}\n\\date{\\today }\n\\maketitle\n\n\\begin{abstract}\nIn this paper we prove the Sobolev embeddings for Herz-type Triebel-Lizorkin\nspaces, \n\\begin{equation*}\n\\dot{K}_{q}^{\\alpha _{2},r}F_{\\theta }^{s_{2}}\\hookrightarrow \\dot{K\n_{s}^{\\alpha _{1},p}F_{\\beta }^{s_{1}}\n\\end{equation*\nwhere the parameters $\\alpha _{1},\\alpha _{2},s_{1},s_{2},s,q,r,p,\\beta $\nand $\\theta $ satisfy some suitable conditions. An application we obtain new\nembeddings between Herz and Triebel-Lizorkin spaces. Moreover, we present\nthe Sobolev embeddings for Triebel-Lizorkin spaces equipped with power\nweights. All these results cover the results on classical Triebel-Lizorkin\nspaces. \\medskip\n\n\\textit{MSC 2010\\\/}: Primary 46E35: Secondary 42B25.\n\n\\textit{Key Words and Phrases}: Triebel-Lizorkin spaces, Herz spaces,\nSobolev embedding.\n\\end{abstract}\n\n\\section{Introduction}\n\n\nFunction spaces have been widely used in various areas of analysis such as\nharmonic analysis and partial differential equations. In recent years, there\nhas been increasing interest in a new family of function spaces which\ngeneralize the Besov spaces and Triebel-Lizorkin spaces. Some example of\nthese spaces can be mentioned such as Herz-type Triebel-Lizorkin spaces, \n\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$, that initially appeared in the\npapers of J. Xu and D. Yang \\cite{XuYang03} , \\cite{XuYang05} and \\cite{Xu03\n. Several basic properties were established, such as the Fourier analytical\ncharacterisation and\\ lifting properties. When $\\alpha =0$ and$\\ p=q$ they\ncoincide with the usual function spaces $F_{p,q}^{s}$.\\vskip5pt\n\nThe interest in these spaces comes not only from theoretical reasons but\nalso from their applications to several classical problems in analysis. In \n\\cite{LuYang97}, Lu and Yang introduced the Herz-type Sobolev and Bessel\npotential spaces. They gave some applications to partial differential\nequations. Also in \\cite{T11}, Y. Tsutsui, studied the Cauchy problem for\nNavier-Stokes equations on Herz spaces\\ and weak Herz spaces.\\vskip5pt\n\nSince the Sobolev embedding plays an important role in theory of function\nspaces and PDE's, the main aim of this paper is to prove the Sobolev\nembedding of $\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$ spaces. First we shall\nprove the Sobolev embeddings of associated sequence spaces. Then, from the\nso-called $\\varphi $-transform characterization in the sense of Frazier and\nJawerth, we deduce the main result of this paper. As a consequence, we\nobtain new Jawerth-Franke-type embeddings, the Sobolev embeddings for\nTriebel-Lizorkin spaces equipped with power weights, new embeddings between\nHerz and Triebel-Lizorkin spaces, and we present some remarks about the\nwavelet caracterization of Herz-Triebel-Lizorkin spaces. All these results\ngeneralize the existing classical results on Triebel-Lizorkin spaces.\n\nTo recall the definition of these function spaces, we need some notation.\nFor any $u>0,k\\in \\mathbb{Z}$ we set $C\\left( u\\right) =\\{x\\in \\mathbb{R\n^{n}:u\/2\\leq \\left\\vert x\\right\\vert 0$ we denote by $B(x,r)$ the open ball in $\\mathbb{R\n^{n}$ with center $x$ and radius $r$. Let $\\chi _{k}$, for $k\\in \\mathbb{Z}\n, denote the characteristic function of the set $C_{k}$. The expression \nf\\approx g$ means that $C$ $g\\leq f\\leq c\\,g$ for some independent constants \n$c,C$ and non-negative functions $f$ and $g$. \\vskip5pt\n\n\\noindent We denote by $\\left\\vert \\Omega \\right\\vert $ the $n$-dimensional\nLebesgue measure of $\\Omega \\subseteq \\mathbb{R}^{n}$. For any measurable\nsubset $\\Omega \\subseteq \\mathbb{R}^{n}$ the Lebesgue space $L^{p}(\\Omega )\n, $00}\\frac{1}{\\left\\vert B(x,r)\\right\\vert \n\\int_{B(x,r)}\\left\\vert f\\left( y\\right) \\right\\vert dy\n\\end{equation*\nand $\\mathcal{M}_{t}f=\\left( \\mathcal{M}|f|^{t}\\right) ^{1\/t}$ for any \n0-n\/q$\\textrm{. }$\\mathrm{The}$ $\\mathrm{spaces}\n\\textrm{\\ }$\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$\\textrm{\\ }$\\mathrm{are}$ \n\\mathrm{independent}$ $\\mathrm{of}$ $\\mathrm{the}$ $\\mathrm{particular}$ \n\\mathrm{choice}$ $\\mathrm{of}$ $\\mathrm{the}$ $\\mathrm{smooth}$ $\\mathrm\ndyadic}$ $\\mathrm{resolution}$ $\\mathrm{of}$ $\\mathrm{unity}$\\textrm{\\ }\n\\{\\phi _{j}\\}_{j\\in \\mathbb{N}_{0}}$ $\\mathrm{(in}$ $\\mathrm{the}$ $\\mathrm\nsense}$ $\\mathrm{of\\ equivalent}$ $\\mathrm{quasi}$\\textrm{-}$\\mathrm{norms)}\n\\textrm{. }$\\mathrm{In}$ $\\mathrm{particular}$ $\\dot{K}_{q}^{\\alpha\n,p}F_{\\beta }^{s}$\\textrm{\\ }$\\mathrm{are}$ $\\mathrm{quasi}$\\textrm{-}\n\\mathrm{Banach}$ $\\mathrm{spaces}$ $\\mathrm{and}$ $\\mathrm{if}$\\textrm{\\ }\np,q,\\beta \\geq 1\\mathrm{,}$ $\\mathrm{then}$ $\\dot{K}_{q}^{\\alpha ,p}F_{\\beta\n}^{s}$ $\\mathrm{are}$ $\\mathrm{Banach}$ $\\mathrm{spaces}$\\textrm{. }$\\mathrm\nFurther}$ $\\mathrm{results,concerning,}$ $\\mathrm{for}$ $\\mathrm{instance,}$ \n$\\mathrm{lifting}$ $\\mathrm{properties,}$ $\\mathrm{Fourier}$ $\\mathrm\nmultiplier}$ $\\mathrm{and}$ $\\mathrm{local}$ $\\mathrm{means}$ $\\mathrm\ncharacterizations}$ $\\mathrm{can}$ $\\mathrm{be}$ $\\mathrm{found}$ $\\mathrm{i\n}$ $\\mathrm{\\cite{XuYang05},}$ $\\mathrm{\\cite{Xu03},}$ $\\mathrm{\\cite{Xu04}}$\n$\\mathrm{and}$ $\\mathrm{\\cite{Xu09}.}$\n\\end{remark}\n\nNow we give the definitions of the spaces $B_{p,\\beta }^{s}$ and $F_{p,\\beta\n}^{s}$.\\vskip5pt\n\n\\begin{definition}\n$\\mathrm{(i)}$\\textit{\\ Let }$s\\in \\mathbb{R}$\\textit{\\ and }$00\n\\begin{align}\n\\left\\vert \\mathcal{F}k_{0}(\\xi )\\right\\vert & >0\\text{\\quad for\\quad \n\\left\\vert \\xi \\right\\vert <2\\varepsilon \\label{T-cond1} \\\\\n\\left\\vert \\mathcal{F}k(\\xi )\\right\\vert & >0\\text{\\quad for\\quad }\\frac\n\\varepsilon }{2}<\\left\\vert \\xi \\right\\vert <2\\varepsilon \\label{T-cond2}\n\\end{align\nan\n\\begin{equation}\n\\int_{\\mathbb{R}^{n}}x^{\\alpha }k(x)dx=0\\text{\\quad for any\\quad }\\left\\vert\n\\alpha \\right\\vert \\leq S. \\label{moment-cond}\n\\end{equation\nHere $\\mathrm{\\eqref{T-cond1}}$ and $\\mathrm{\\eqref{T-cond2}}$\\ are\nTauberian conditions, while $\\mathrm{\\eqref{moment-cond}}$ are moment\nconditions on $k$. We recall the notatio\n\\begin{equation*}\nk_{t}(x)=t^{-n}k(t^{-1}x)\\text{,}\\quad k_{j}(x)=k_{2^{-j}}(x)\\text{,\\quad fo\n}\\quad t>0\\quad \\text{and}\\quad j\\in \\mathbb{N}.\n\\end{equation*\nUsually $k_{j}\\ast f$ is called local mean. The following result is from \n\\cite[Theorem 1]{Xu05}.\n\n\\begin{theorem}\n\\label{loc-mean-char}\\textit{Let }$\\alpha ,s\\in \\mathbb{R},0-n\/q$ and $s0$. Recall that the $\\varphi $-transform $S_{\\varphi }$ is defined\nby setting $(S_{\\varphi }f)_{0,m}=\\langle f,\\Phi _{m}\\rangle $ where $\\Phi\n_{m}(x)=\\Phi (x-m)$ and $(S_{\\varphi }f)_{v,m}=\\langle f,\\varphi\n_{v,m}\\rangle $ where $\\varphi _{v,m}(x)=2^{vn\/2}\\varphi (2^{v}x-m)$ and \nv\\in \\mathbb{N}$. The inverse $\\varphi $-transform $T_{\\psi }$ is defined by \n\\begin{equation*}\nT_{\\psi }\\lambda =\\sum_{m\\in \\mathbb{Z}^{n}}\\lambda _{0,m}\\Psi\n_{m}+\\sum_{v=1}^{\\infty }\\sum_{m\\in \\mathbb{Z}^{n}}\\lambda _{v,m}\\psi _{v,m},\n\\end{equation*\nwhere $\\lambda =\\{\\lambda _{v,m}\\in \\mathbb{C}:v\\in \\mathbb{N}_{0},m\\in \n\\mathbb{Z}^{n}\\}$, see \\cite{FJ90}.\n\nFor a sequence $\\lambda =\\{\\lambda _{v,m}\\in \\mathbb{C}:v\\in \\mathbb{N\n_{0},m\\in \\mathbb{Z}^{n}\\},00$, se\n\\begin{equation*}\n\\lambda _{v,m,r,d}^{\\ast }=\\Big(\\sum_{h\\in \\mathbb{Z}^{n}}\\frac{|\\lambda\n_{v,h}|^{r}}{(1+2^{v}|2^{-v}h-2^{-v}m|)^{d}}\\Big)^{1\/r}\n\\end{equation*\nand $\\lambda _{r,d}^{\\ast }=\\{\\lambda _{v,m,r,d}^{\\ast }\\in \\mathbb{C}:v\\in \n\\mathbb{N}_{0},m\\in \\mathbb{Z}^{n}\\}$.\n\n\\begin{lemma}\n\\label{lamda-equi}\\textit{Let }$\\alpha ,s\\in \\mathbb{R},0n$ \\textit{and }$\\alpha >-n\/q$. The\n\\begin{equation*}\n\\Big\\|\\lambda _{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta ),d}^{\\ast }\\mid \n\\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\Big\\|\\approx \\big\\|\\lambda \\mid \\dot{K\n_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|\n\\end{equation*}\n\\end{lemma}\n\n\\textbf{Proof.} Obviously, \n\\begin{equation*}\n\\left\\Vert \\lambda \\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\right\\Vert\n\\leq \\Big\\|\\lambda _{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta ),d}^{\\ast\n}\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\Big\\|.\n\\end{equation*\nFrom Lemma A.2 of \\cite{FJ90}, we obtai\n\\begin{equation*}\n\\lambda _{v,m,\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta ),d}^{\\ast }\\leq \n\\mathcal{M}_{a}\\Big(\\sum_{h\\in \\mathbb{Z}^{n}}|\\lambda _{v,h}|\\chi _{v,h\n\\Big)(x),\\quad x\\in Q_{v,m},\n\\end{equation*\nwhere, \n\\begin{equation*}\n0nr\n\\end{equation*\nand $c>0$ depend only $n$ and $d$. Let $\\varepsilon =\\frac{d}{n}-1>0$ and $a\n\\tfrac{r}{1+\\varepsilon \/2}$, then $0nr$. Hence \n\\begin{eqnarray*}\n&&\\Big\\|\\lambda _{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta ),d}^{\\ast\n}\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\Big\\| \\\\\n&\\leq &c\\Big\\|\\Big(\\sum_{v=0}^{\\infty }\\mathcal{M}_{a\/\\beta }\\Big(\\sum_{h\\in \n\\mathbb{Z}^{n}}2^{vs\\beta }|\\lambda _{v,h}|^{\\beta }\\chi _{v,h}\\Big)\\Big\n^{a\/\\beta }\\mid \\dot{K}_{q\/a}^{\\alpha a,p\/a}\\Big\\|^{1\/a}.\n\\end{eqnarray*\nObserve that $\\frac{\\beta }{a}>1$, $\\frac{q}{a}>1$ and $\\frac{-na}{q}<\\alpha\na-n\/q$. \\textit{Suppose that }\n\\varphi $ and $\\Phi $ satisfy \\eqref{Ass1}-\\eqref{Ass4}. The operators \nS_{\\varphi }:\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}\\rightarrow \\dot{K\n_{q}^{\\alpha ,p}f_{\\beta }^{s}$ and $T_{\\psi }:\\dot{K}_{q}^{\\alpha\n,p}f_{\\beta }^{s}\\rightarrow \\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$ are\nbounded. Furthermore, $T_{\\psi }\\circ S_{\\varphi }$ is the identity on $\\dot\nK}_{q}^{\\alpha ,p}F_{\\beta }^{s}$.\n\\end{theorem}\n\n\\textbf{Proof.} We use the same arguments of \\cite[Theorem 2.2]{FJ90}, see\nalso \\cite[Theorem 2.1]{SiYY10} and \\cite[Theorem 3.12]{Bow07}. For any \nf\\in \\mathcal{S}^{\\prime }(\\mathbb{R}^{n})$ we put $\\sup\n(f)=\\{\\sup_{v,m}(f):v\\in \\mathbb{N}_{0},m\\in \\mathbb{Z}^{n}\\}$ where \n\\begin{equation*}\n\\sup_{v,m}(f)=\\sup_{y\\in Q_{v,m}}|\\widetilde{\\varphi _{v}}\\ast f(y)|\n\\end{equation*\nif $v\\in \\mathbb{N},m\\in \\mathbb{Z}^{n}$ and \n\\begin{equation*}\n\\sup_{0,m}(f)=\\sup_{y\\in Q_{0,m}}|\\widetilde{\\Phi }\\ast f(y)|\n\\end{equation*\nif $m\\in \\mathbb{Z}^{n}$. For any $\\gamma \\in \\mathbb{Z}_{+}$, we define the\nsequence $\\inf_{\\gamma }(f)=\\{\\inf_{v,m,\\gamma }(f):v\\in \\mathbb{N}_{0},m\\in \n\\mathbb{Z}^{n}\\}$ by setting \n\\begin{equation*}\n\\inf_{v,m,\\gamma }(f)=\\sup_{h\\in \\mathbb{Z}^{n}}\\{\\inf_{y\\in Q_{v+\\gamma\n,h}}|\\widetilde{\\varphi _{v}}\\ast f(y)|:Q_{v+\\gamma ,h}\\cap Q_{v,m}\\neq\n\\emptyset \\}\n\\end{equation*\nif $v\\in \\mathbb{N},m\\in \\mathbb{Z}^{n}$ and \n\\begin{equation*}\n\\inf_{0,m,\\gamma }(f)=\\sup_{h\\in \\mathbb{Z}^{n}}\\{\\inf_{y\\in Q_{\\gamma ,h}}\n\\widetilde{\\Phi }\\ast f(y)|:Q_{\\gamma ,h}\\cap Q_{0,m}\\neq \\emptyset \\}\n\\end{equation*\nif $m\\in \\mathbb{Z}^{n}$. Here $\\widetilde{\\varphi _{j}}(x)=2^{jn}\\overline\n\\varphi (-2^{j}x)}$ and $\\widetilde{\\Phi }(x)=\\overline{\\Phi (-x)}$. As in\nLemma A.5 of \\cite{FJ90}, see also \\cite{Bow07} and \\cite{SiYY10} we obtain \n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\\n\\leq c\\big\\|f\\mid \\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}\\big\\|\n\\end{equation*\nfor any $s\\in \\mathbb{R},0-n\/q$\nand $\\gamma >0$ sufficiently large. Indeed, we hav\n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\\n=c\\Big\\|\\Big(\\sum\\limits_{m\\in \\mathbb{Z}^{n}}2^{js}\\inf_{j-\\gamma ,m,\\gamma\n}(f)\\chi _{j-\\gamma ,m}\\Big)_{j\\geq \\gamma }\\mid \\dot{K}_{q}^{\\alpha\n,p}(\\ell _{\\beta })\\Big\\|\n\\end{equation*\nDefine a sequence $\\{\\lambda _{i,k}\\}_{i\\in \\mathbb{N}_{0},k\\in \\mathbb{Z\n^{n}}$ by setting $\\lambda _{i,k}=\\inf_{y\\in Q_{i,k}}|\\widetilde{\\varphi\n_{i-\\gamma }}\\ast f(y)|$ and $\\lambda _{0,k}=\\inf_{y\\in Q_{\\gamma ,k}}\n\\widetilde{\\Phi }\\ast f(y)|$. We hav\n\\begin{equation*}\n\\inf_{j-\\gamma ,m,\\gamma }(f)=\\sup_{h\\in \\mathbb{Z}^{n}}\\{\\lambda\n_{j,h}:Q_{j,h}\\cap Q_{j-\\gamma ,m}\\neq \\emptyset \\}\n\\end{equation*\nand \n\\begin{equation*}\n\\inf_{0,m,\\gamma }(f)=\\sup_{h\\in \\mathbb{Z}^{n}}\\{\\lambda _{0,h}:Q_{\\gamma\n,h}\\cap Q_{0,m}\\neq \\emptyset \\}.\n\\end{equation*\nLet $h\\in \\mathbb{Z}^{n}$ with $Q_{j,h}\\cap Q_{j-\\gamma ,m}\\neq \\emptyset $\nand $j\\geq \\gamma $. Then \n\\begin{equation}\n\\lambda _{j,h}\\leq c2^{\\gamma d\/r}\\lambda _{j,z,r,\\gamma }^{\\ast }\\text{, \nj>\\gamma \\text{ \\ \\ and \\ \\ }\\lambda _{0,h}\\leq c2^{\\gamma d\/r}\\lambda\n_{0,z,r,\\gamma }^{\\ast }\\text{, \\ \\ }j=\\gamma \\label{est1lamda}\n\\end{equation\nfor any $z\\in \\mathbb{Z}^{n}$ with $Q_{j,z}\\cap Q_{j-\\gamma ,m}\\neq\n\\emptyset $, where the constant $c>0$ does not depend on $j$, $h$ and $z$.\nIndeed, we observ\n\\begin{equation*}\n\\lambda _{j,h}=\\frac{\\lambda _{j,h}}{(1+2^{j}|2^{-j}h-2^{-j}z|)^{d}\n(1+2^{j}|2^{-j}h-2^{-j}z|)^{d}.\n\\end{equation*\nLet $x\\in Q_{j,h}\\cap Q_{j-\\gamma ,m}$ and $y\\in Q_{j,z}\\cap Q_{j-\\gamma ,m}\n. We hav\n\\begin{equation*}\n|2^{-j}h-2^{-j}z|\\leq |2^{-j}h-x|+|x-y|+|y-2^{-j}z|\\lesssim\n2^{2-j}+2^{\\gamma -j}.\n\\end{equation*\nThis implies \\eqref{est1lamda}. Henc\n\\begin{equation*}\n\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\inf_{j-\\gamma ,m,\\gamma }(f)\\chi\n_{j-\\gamma ,m}\\leq c\\sum\\limits_{k\\in \\mathbb{Z}^{n}}\\lambda\n_{j,k,r,d}^{\\ast }\\chi _{j,k},\\text{ \\ \\ }j>\\gamma \n\\end{equation*\nan\n\\begin{equation*}\n\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\inf_{0,m,\\gamma }(f)\\chi _{0,m}\\leq\nc\\sum\\limits_{k\\in \\mathbb{Z}^{n}}\\lambda _{0,k,r,d}^{\\ast }\\chi _{\\gamma\n,k},\\text{ \\ \\ }j=\\gamma ,\n\\end{equation*\nwith $r=\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta )$ and $d>n$. Therefore\n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\\n\\leq c\\Big\\|\\Big(\\sum\\limits_{k\\in \\mathbb{Z}^{n}}2^{js}\\lambda\n_{j,k,r,d}^{\\ast }\\chi _{j,k}\\Big)_{j\\geq \\gamma }\\mid \\dot{K}_{q}^{\\alpha\n,p}(\\ell _{\\beta })\\Big\\|.\n\\end{equation*\nNotice that if $j=\\gamma $ we replace $\\lambda _{j,k,r,d}^{\\ast }\\chi _{j,k}$\nby $\\lambda _{0,k,r,d}^{\\ast }\\chi _{\\gamma ,k}$. Applying Lemma \\re\n{lamda-equi} to estimate this term b\n\\begin{equation*}\nc\\Big\\|\\Big(\\sum\\limits_{k\\in \\mathbb{Z}^{n}}2^{js}\\lambda _{j,k}\\chi _{j,k\n\\Big)_{j>\\gamma }\\mid \\dot{K}_{q}^{\\alpha ,p}(\\ell _{\\beta })\\Big\\|+c\\Big\\\n\\sum\\limits_{k\\in \\mathbb{Z}^{n}}\\lambda _{0,k}\\chi _{\\gamma ,k}\\mid \\dot{K\n_{q}^{\\alpha ,p}\\Big\\|,\n\\end{equation*\nwich is bounded b\n\\begin{equation*}\nc\\Big\\|\\Big(2^{js}\\widetilde{\\varphi _{j-\\gamma }}\\ast f\\Big)_{j>\\gamma\n}\\mid \\dot{K}_{q}^{\\alpha ,p}(\\ell _{\\beta })\\Big\\|+c\\Big\\|\\widetilde{\\Phi \n\\ast f\\mid \\dot{K}_{q}^{\\alpha ,p}\\Big\\|.\n\\end{equation*\nBy Theorem \\ref{loc-mean-char}, we obtai\n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\\n\\leq c\\big\\|f\\mid \\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}\\big\\|^{\\prime }\\leq \n\\big\\|f\\mid \\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}\\big\\|,\n\\end{equation*\nwhere we use the caracterization of Herz-type Triebel-Lizorkin spaces by\nlocal means. Applying Lemma A.4 of \\cite{FJ90}, see also Lemma 8.3 of \\cit\n{Bow07}, we obtain \n\\begin{equation*}\n\\text{inf}_{\\gamma }(f)_{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta\n),\\gamma }^{\\ast }\\approx \\sup (f)_{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha \n,\\beta ),\\gamma }^{\\ast }.\n\\end{equation*\nHence for $\\gamma >0$ sufficiently large we obtain by applying Lemma \\re\n{lamda-equi}, \n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)_{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta\n),\\gamma }^{\\ast }\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|\\approx\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|\n\\end{equation*\nand \n\\begin{equation*}\n\\big\\|\\sup (f)_{\\min (q,\\frac{n}{\\frac{n}{q}+\\alpha },\\beta ),\\gamma }^{\\ast\n}\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|\\approx \\big\\|\\sup (f)\\mid \n\\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|\n\\end{equation*\nfor any $s\\in \\mathbb{R},0-n\/q$. Therefore, \n\\begin{equation*}\n\\big\\|\\text{inf}_{\\gamma }(f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\\n\\approx \\big\\|f\\mid \\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}\\big\\|\\approx \\big\\\n\\sup (f)\\mid \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}\\big\\|.\n\\end{equation*\nUse these estimates and repeating the proof of Theorem 2.2 in \\cite{FJ90} or\nTheorem 2.1 in \\cite{SiYY10} then complete the proof of Theorem \\re\n{phi-tran}. \\ \\ \\rule{3mm}{3mm}\n\n\\begin{remark}\n$\\mathrm{From}$ $\\mathrm{these}$ $\\mathrm{to}$ $\\mathrm{prove}$ $\\mathrm{the}\n$ $\\mathrm{embeddings}\n\\begin{equation*}\n\\dot{K}_{q}^{\\alpha _{2},r}F_{\\theta }^{s_{2}}\\hookrightarrow \\dot{K\n_{s}^{\\alpha _{1},p}F_{\\beta }^{s_{1}}\n\\end{equation*\n$\\mathrm{we}$ $\\mathrm{need}$ $\\mathrm{only}$ $\\mathrm{to}$ $\\mathrm{prove}\n\\begin{equation*}\n\\dot{K}_{q}^{\\alpha _{2},r}f_{\\theta }^{s_{2}}\\hookrightarrow \\dot{K\n_{s}^{\\alpha _{1},p}f_{\\beta }^{s_{1}},\n\\end{equation*\n$\\mathrm{under}$ $\\mathrm{the}$ $\\mathrm{same}$ $\\mathrm{restrictions}$ \n\\mathrm{on}$ $\\mathrm{parameters}$ $s_{1},s_{2},\\alpha _{1},\\alpha\n_{2},s,p,q,\\beta ,r,\\theta $.\n\\end{remark}\n\nWe end this section with one more lemma, which is basically a consequence of\nHardy's inequality in the sequence Lebesgue space $\\ell _{q}$.\n\n\\begin{lemma}\n\\label{lq-inequality}\\textit{Let }$00\\ $\\textit{depending only on }$a$\\textit{\\ and }$q$\nsuch tha\n\\begin{equation*}\n\\Big(\\sum\\limits_{k=0}^{\\infty }\\delta _{k}^{q}\\Big)^{1\/q}+\\Big\n\\sum\\limits_{k=0}^{\\infty }\\eta _{k}^{q}\\Big)^{1\/q}\\leq c\\text{ }\\Big\n\\sum\\limits_{k=0}^{\\infty }\\varepsilon _{k}^{q}\\Big)^{1\/q}.\n\\end{equation*}\n\\end{lemma}\n\n\\section{Sobolev embeddings for $\\dot{K}_{q}^{\\protect\\alpha ,p}F_{\\protec\n\\beta }^{s}$ spaces}\n\nIt is well-known tha\n\\begin{equation*}\nF_{q,\\infty }^{s_{2}}\\hookrightarrow F_{s,\\beta }^{s_{1}}\n\\end{equation*\nif $s_{1}-n\/s=s_{2}-n\/q$, where $0-n\/s\\ $\\textit{and }$\\alpha _{2}>-n\/q$. \\textit{We suppose that \n\\begin{equation}\ns_{1}-n\/s-\\alpha _{1}=s_{2}-n\/q-\\alpha _{2}. \\label{newexp1}\n\\end{equation}\n\n\\noindent \\textit{Let }$00$ does not depend on $N$. No\n\\begin{equation*}\n\\big\\|\\lambda ^{N}\\mid \\dot{K}_{q}^{\\alpha _{2},r}f_{\\theta }^{s_{2}}\\big\\\n^{r}=\\sum_{k=-\\infty }^{\\infty }2^{\\alpha _{2}kr}\\Big\\|\\Big\n\\sum_{v=0}^{\\infty }2^{vs_{2}\\theta }|\\lambda _{v,1}^{N}|^{\\theta }\\chi\n_{v,1}\\Big)^{1\/\\theta }\\chi _{k}\\Big\\|_{q}^{r}.\n\\end{equation*\nAgain we can rewrite the last statement as follows\n\\begin{eqnarray*}\n&&\\sum_{k=1-N}^{0}2^{\\alpha _{2}kr}\\Big\\|\\Big\n\\sum_{v=2}^{N-2}2^{(s_{2}-s_{1}+\\frac{1}{s}+\\alpha _{1})v\\theta }\\chi _{v,1\n\\Big)^{1\/\\theta }\\chi _{k}\\Big\\|_{q}^{r} \\\\\n&=&\\sum_{k=1-N}^{0}2^{\\alpha _{2}kr}\\Big\\|2^{(s_{2}-s_{1}+\\frac{1}{s}+\\alpha\n_{1})(1-k)}\\chi _{1-k,1}\\Big\\|_{q}^{r}=c\\text{ }N,\n\\end{eqnarray*\nwhere the constant $c>0$ does not depend on $N$. If the embeddings\n\\eqref{Sobolev-emb1} holds then for any $N\\in \\mathbb{N}$, $N^{\\frac{1}{p}\n\\frac{1}{r}}\\leq C$. Thus, we conclude that $00$. Therefore for any $x\\in C_{k}\n\\begin{eqnarray*}\n\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|^{t}\\chi _{v,m}(x) &\\leq\n&2^{nv}\\int_{B\\left( 0,2^{c_{n}-v+2}\\right) }\\sum_{m\\in \\mathbb{Z\n^{n}}|\\lambda _{v,m}|^{t}\\chi _{v,m}(y)dy \\\\\n&=&2^{nv}\\Big\\|\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}\\chi\n_{B\\left( 0,2^{c_{n}-v+2}\\right) }\\Big\\|_{t}^{t}.\n\\end{eqnarray*\nHenc\n\\begin{eqnarray*}\n&&2^{\\alpha _{1}k}\\Big\\|\\sum_{v=0}^{c_{n}-k+1}2^{vs_{1}}\\sum_{m\\in \\mathbb{Z\n^{n}}\\lambda _{v,m}\\chi _{v,m}\\chi _{k}\\Big\\|_{s} \\\\\n&\\leq &c\\text{ }2^{(\\alpha _{1}+\\frac{n}{s})k\n\\sum_{v=0}^{c_{n}-k+1}2^{v(s_{1}+\\frac{n}{t})}\\Big\\|\\sum_{m\\in \\mathbb{Z\n^{n}}|\\lambda _{v,m}|\\chi _{v,m}\\chi _{B\\left( 0,2^{c_{n}-v+2}\\right) }\\Big\\\n_{t}.\n\\end{eqnarray*\nWe may choose $t>0$ such that $\\frac{1}{t}>\\max (\\frac{1}{q},\\frac{1}{q}\n\\frac{\\alpha _{2}}{n})$. Using \\eqref{newexp1} and Lemma \\ref{lq-inequality}\nto estimate $I^{r}$ b\n\\begin{eqnarray*}\n&&c\\sum_{v=0}^{\\infty }2^{v(s_{2}-\\frac{n}{q}-\\alpha _{2}+\\frac{n}{t})r\n\\Big\\|\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}\\chi _{B\\left(\n0,2^{c_{n}-v+2}\\right) }\\Big\\|_{t}^{r} \\\\\n&\\leq &c\\sum_{v=0}^{\\infty }2^{v(s_{2}-\\frac{n}{q}-\\alpha _{2}+\\frac{n}{t})r\n\\Big(\\sum_{i\\leq -v}\\Big\\|\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi\n_{v,m}\\chi _{i+c_{n}+2}\\Big\\|_{t}^{\\sigma }\\Big)^{r\/\\sigma } \\\\\n&\\leq &c\\sum_{v=0}^{\\infty }2^{v\\frac{nr}{d}}\\Big(\\sum_{i\\leq -v}2^{i\\frac\nn\\sigma }{d}+\\alpha _{2}\\sigma i}\\Big\\|\\sup_{j}\\sum_{m\\in \\mathbb{Z\n^{n}}2^{s_{2}j}|\\lambda _{j,m}|\\chi _{j,m}\\chi _{i+c_{n}+2}\\Big\\\n_{q}^{\\sigma }\\Big)^{r\/\\sigma },\n\\end{eqnarray*\nby H\\\"{o}lder's inequality, with $\\sigma =\\min (1,t)$ and $\\frac{n}{d}=\\frac\nn}{t}-\\frac{n}{q}-\\alpha _{2}$. Again, we apply Lemma \\ref{lq-inequality} to\nobtai\n\\begin{equation*}\nI^{r}\\leq c\\sum_{i=0}^{\\infty }2^{-\\alpha _{2}ir}\\Big\\|\\sup_{j}\\sum_{m\\in \n\\mathbb{Z}^{n}}2^{s_{2}j}|\\lambda _{j,m}|\\chi _{j,m}\\chi _{-i+c_{n}+2}\\Big\\\n_{q}^{r}\\leq c\\big\\|\\lambda \\mid \\dot{K}_{q}^{\\alpha _{2},r}f_{\\infty\n}^{s_{2}}\\big\\|^{r}.\n\\end{equation*\n\\textit{Estimation of }$II$\\textit{.} We see that it suffices to show that\nfor any $k\\leq c_{n}+1\n\\begin{eqnarray*}\n2^{k\\alpha _{1}}\\Big\\|\\sum_{v=c_{n}-k+2}^{\\infty }2^{vs_{1}}\\sum_{m\\in \n\\mathbb{Z}^{n}}\\lambda _{v,m}\\chi _{v,m}\\chi _{k}\\Big\\|_{s} &\\leq &C^{s\/q\n\\text{ }2^{k\\alpha _{2}}\\Big\\|\\sup_{v\\geq c_{n}-k+2}\\sum_{m\\in \\mathbb{Z\n^{n}}2^{vs_{2}}\\lambda _{v,m}\\chi _{v,m}\\chi _{\\widetilde{C}_{k}}\\Big\\|_{q}\n\\\\\n&=&\\delta ,\n\\end{eqnarray*\nwhere $\\widetilde{C}_{k}=\\{x\\in \\mathbb{R}^{n}:2^{k-2}<\\left\\vert\nx\\right\\vert <2^{k+2}\\}$ and $C=2\\max (\\frac{2^{\\frac{1}{q}}}{1-2^{\\frac{n}{\n}-\\frac{n}{q}}},\\frac{2^{\\frac{n}{s}}}{2^{\\frac{n}{s}}-1})$. This claim can\nbe reformulated as showing tha\n\\begin{equation*}\n\\int_{C_{k}}2^{k\\alpha _{1}s}\\delta ^{-s}\\Big(\\sum_{v=c_{n}-k+2}^{\\infty\n}2^{vs_{1}}\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}(x)\\Big\n^{s}dx\\leq 1.\n\\end{equation*\nThe left-hand side can be rewritten us \n\\begin{eqnarray*}\n&&\\int_{C_{k}}\\delta ^{-s}\\Big(\\sum_{v=c_{n}-k+2}^{\\infty }2^{v(\\frac{n}{s}\n\\frac{n}{q})+(\\alpha _{1}-\\alpha _{2})(v+k)+vs_{2}+k\\alpha _{2}}\\sum_{m\\in \n\\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}(x)\\Big)^{s}dx \\\\\n&\\leq &\\int_{C_{k}}\\delta ^{-s}\\Big(\\sum_{v=c_{n}-k+2}^{\\infty }2^{v(\\frac{\n}{s}-\\frac{n}{q})+vs_{2}+k\\alpha _{2}}\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda\n_{v,m}|\\chi _{v,m}(x)\\Big)^{s}dx=T_{k},\n\\end{eqnarray*\nsince $\\alpha _{2}\\geq \\alpha _{1}$. Let us prove that $T_{k}\\leq 1$ for any \n$k\\leq c_{n}+1$. Our estimate use partially some decomposition techniques\nalready used in \\cite{Vybiral09}.\n\n\\noindent \\textit{Case 1.} $\\sup_{v\\geq c_{n}-k+2,m}2^{vs_{2}+k\\alpha\n_{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)\\leq \\delta $. In this case we obtai\n\\begin{equation*}\nT_{k}\\leq C^{s}\\int_{C_{k}}\\Big(\\delta ^{-1}\\sup_{v\\geq c_{n}-k+2}\\sum_{m\\in \n\\mathbb{Z}^{n}}2^{vs_{2}+k\\alpha _{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)\\Big\n^{q}dx\\leq 1.\n\\end{equation*}\n\n\\noindent \\textit{Case 2.} $\\sup_{v\\geq c_{n}-k+2,m}2^{vs_{2}+k\\alpha\n_{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)>\\delta $. We can distinguish two cases\nas follows:\n\n$\\bullet $ $\\delta ^{-1}\\sup_{v\\geq c_{n}-k+2,m}2^{vs_{2}+k\\alpha\n_{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)=\\infty $, then there is nothing to prove.\n\n$\\bullet $ $\\delta <\\sup_{v\\geq c_{n}-k+2,m}2^{vs_{2}+k\\alpha _{2}}|\\lambda\n_{v,m}|\\chi _{v,m}(x)<\\infty $. Let $N\\in \\mathbb{N}$ be such that \n\\begin{equation*}\n2^{\\frac{nN}{q}}<\\delta ^{-1}\\sup_{v\\geq c_{n}-k+2}\\sum_{m\\in \\mathbb{Z\n^{n}}2^{vs_{2}+k\\alpha _{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)<2^{\\frac{(N+1)n}{\n}}.\n\\end{equation*\n\\textit{Subcase 2.1.} $k\\geq c_{n}-N+2$. We split the sum over $v\\geq\nc_{n}-k+2$ into two parts, \n\\begin{equation*}\n\\sum_{v=c_{n}-k+2}^{\\infty }\\cdot \\cdot \\cdot =\\sum_{v=c_{n}-k+2}^{N}\\cdot\n\\cdot \\cdot +\\sum_{v=N+1}^{\\infty }\\cdot \\cdot \\cdot .\n\\end{equation*\nLet $x\\in C_{k}\\cap Q_{v,m}$ and $y\\in Q_{v,m}$. We have $|x-y|\\leq 2\\sqrt{n\n2^{-v}<2^{c_{n}-v}$ and from this it follows that \n2^{k-2}<|y|<2^{c_{n}-v}+2^{k}<2^{k+2}$, which implies that $y$ is located in \n$\\widetilde{C}_{k}$. The\n\\begin{equation*}\n|\\lambda _{v,m}|^{q}=2^{nv}\\int_{\\mathbb{R}^{n}}|\\lambda _{v,m}|^{q}\\chi\n_{v,m}(y)dy\\leq 2^{nv}\\int_{\\widetilde{C}_{k}}|\\lambda _{v,m}|^{q}\\chi\n_{v,m}(y)dy,\n\\end{equation*\nif $x\\in C_{k}\\cap Q_{v,m}$. But this immediately implies tha\n\\begin{eqnarray*}\n\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|^{q}\\chi _{v,m}(x) &\\leq\n&2^{nv}\\int_{\\widetilde{C}_{k}}\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda\n_{v,m}|^{q}\\chi _{v,m}(y)dy \\\\\n&=&2^{nv}\\big\\|\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}\\chi _\n\\widetilde{C}_{k}}\\big\\|_{q}^{q}\\leq 2^{(\\frac{n}{q}-s_{2})qv-\\alpha\n_{2}qk}\\delta ^{q}.\n\\end{eqnarray*\nTherefore for any $x\\in C_{k}\n\\begin{equation*}\n\\delta ^{-1}\\sum_{v=c_{n}-k+2}^{N}2^{v(\\frac{n}{s}-\\frac{n}{q\n)+vs_{2}+k\\alpha _{2}}\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi\n_{v,m}(x)\\leq \\sum_{v=c_{n}-k}^{N}2^{\\frac{n}{s}v}\\leq C\\text{ }2^{\\frac{n}{\n}N}\n\\end{equation*\nand \n\\begin{eqnarray*}\n&&\\delta ^{-1}\\sum_{v=N+1}^{\\infty }2^{v(\\frac{n}{s}-\\frac{n}{q\n)+vs_{2}+k\\alpha _{2}}\\sum_{m\\in \\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi\n_{v,m}(x) \\\\\n&=&2^{(\\frac{n}{s}-\\frac{n}{q})N}\\delta ^{-1}\\sum_{v=N+1}^{\\infty }2^{(v-N)\n\\frac{n}{s}-\\frac{n}{q})+vs_{2}+k\\alpha _{2}}\\sum_{m\\in \\mathbb{Z\n^{n}}|\\lambda _{v,m}|\\chi _{v,m}(x) \\\\\n&\\leq &c\\text{ }2^{(\\frac{n}{s}-\\frac{n}{q})N}\\delta ^{-1}\\sup_{v\\geq\nc_{n}-k+2}\\sum_{m\\in \\mathbb{Z}^{n}}2^{vs_{2}+k\\alpha _{2}}|\\lambda\n_{v,m}|\\chi _{v,m}(x) \\\\\n&\\leq &C\\text{ }2^{\\frac{n}{s}N}.\n\\end{eqnarray*\nHenc\n\\begin{equation*}\nT_{k}\\leq C^{s}\\int_{C_{k}}2^{nN}dx\\leq C^{s}\\int_{C_{k}}\\Big(\\delta\n^{-1}\\sup_{v\\geq c_{n}-k+2}\\sum_{m\\in \\mathbb{Z}^{n}}2^{vs_{2}+k\\alpha\n_{2}}|\\lambda _{v,m}|\\chi _{v,m}(x)\\Big)^{q}dx\\leq 1.\n\\end{equation*\n\\textit{Subcase 2.2.} $k\\alpha\n_{1}+n\/s$. We only need to estimate the part $T_{k}$. H\\\"{o}lder's\ninequality implies tha\n\\begin{eqnarray*}\nT_{k} &\\leq &\\Big\\|\\delta ^{-1}\\sum_{v=c_{n}-k+2}^{\\infty }2^{(\\frac{n}{s}\n\\frac{n}{q}+\\alpha _{1}-\\alpha _{2})(v+k)}2^{vs_{2}+k\\alpha _{2}}\\sum_{m\\in \n\\mathbb{Z}^{n}}|\\lambda _{v,m}|\\chi _{v,m}\\chi _{k}\\Big\\|_{q}^{s} \\\\\n&\\leq &\\Big\\|\\delta ^{-1}\\sup_{v\\geq c_{n}-k+2}\\sum_{m\\in \\mathbb{Z\n^{n}}2^{vs_{2}+k\\alpha _{2}}|\\lambda _{v,m}|\\chi _{v,m}\\chi _{k}\\Big\\\n_{q}^{s}=C^{-s\/q},\n\\end{eqnarray*\nwhere the last inequality follows by the fact that $\\alpha _{2}+n\/q>\\alpha\n_{1}+n\/s$. The remaining case can be easily solved. The proof is complete. \\\n\\ \\rule{3mm}{3mm\n\nAs a corollary of Theorems \\ref{phi-tran} and \\ref{embeddings3}, we have the\nfollowing Sobolev embedding for $\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$\nspaces.\n\n\\begin{theorem}\n\\label{embeddings3.1}\\textit{Let }$\\alpha _{1},\\alpha _{2},s_{1},s_{2}\\in \n\\mathbb{R},0-n\/s\\ $\\textit{and }$\\alpha _{2}>-n\/q$. \\textit{We suppose that \n\\begin{equation*}\ns_{1}-n\/s-\\alpha _{1}=s_{2}-n\/q-\\alpha _{2}.\n\\end{equation*\n\\noindent \\textit{Let }$0-n\/q$\\textit{. }Then there exists a linear isomorphism $T$\nwhich maps $\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$ onto $\\dot{K}_{q}^{\\alpha\n,p}f_{\\beta }^{s}$. Moreover, there is an unconditional basis in $\\dot{K\n_{q}^{\\alpha ,p}F_{\\beta }^{s}$.\n\\end{theorem}\n\nThe mapping $T$ is generated by an appropriate wavelet system. A proof of\nthis theorem can be found in Xu \\cite{Xu09} for the non-homogeneous\nHerz-type Triebel-Lizorkin spaces and $\\alpha >0$. This result is also true\nfor the spaces $\\dot{K}_{q}^{\\alpha ,p}F_{\\beta }^{s}$, with $\\alpha >-n\/q$.\nIndeed, the problem can be reduced to proof the $\\dot{K}_{q}^{\\alpha\n,p}f_{\\beta }^{s}$-version of Lemma 3.5 in Xu \\cite{Xu09}. Therefore we need\nto recall the definition of molecules.\n\n\\begin{definition}\n\\label{Atom-Def}Let $K,L\\in \\mathbb{N}_{0}$ and let $M>0$. A $K$-times\ncontinuously differentiable function $a\\in C^{K}(\\mathbb{R}^{n})$ is called \n[K,L,M]$-molecule concentrated in $Q_{v,m}$, if for some $v\\in \\mathbb{N\n_{0} $ and $m\\in \\mathbb{Z}^{n}$\n\n\\begin{equation}\n|D^{\\alpha }a(x)|\\leq 2^{v|\\alpha |}(1+2^{v}|x-2^{-v}m|)^{-M}\\text{,\\quad\nfor\\quad }0\\leq |\\alpha |\\leq K,x\\in \\mathbb{R}^{n} \\label{diff-cond}\n\\end{equation\nand i\n\\begin{equation}\n\\int\\limits_{\\mathbb{R}^{n}}x^{\\alpha }a(x)dx=0,\\text{\\quad for\\quad }0\\leq\n|\\alpha |-n\/q$\\textit{. }Furthermore, let $K,L\\in \\mathbb{N}_{0}$ and\nlet $M>0$ wit\n\\begin{equation*}\nL>n(\\frac{1}{\\min (1,q,\\beta )}-1)-1-s,\\text{\\quad }K\\text{ arbitrary and }\n\\text{ large enough}.\n\\end{equation*\nIf $a_{vm}$ are $\\left[ K,L,M\\right] $-molecules concentrated in $Q_{v,m}$\nand $\\lambda =\\{\\lambda _{v,m}\\in \\mathbb{C}:v\\in \\mathbb{N}_{0},m\\in \n\\mathbb{Z}^{n}\\}\\in \\dot{K}_{q}^{\\alpha ,p}f_{\\beta }^{s}$, then the su\n\\begin{equation}\n\\sum_{v=0}^{\\infty }\\sum_{m\\in \\mathbb{Z}^{n}}\\lambda _{v,m}a_{vm}\n\\label{new-rep}\n\\end{equation\nconverges in $\\mathcal{S}^{\\prime }(\\mathbb{R}^{n})$.\n\\end{lemma}\n\n\\textbf{Proof.} We use the arguments of \\cite{Triebel06}, see also \\cite{K10\n. Let $\\varphi \\in \\mathcal{S(}\\mathbb{R}^{n})$. We get from the moment\nconditions $\\mathrm{\\eqref{mom-cond}}$ for fixed $v\\in \\mathbb{N}_{0}\n\\begin{eqnarray*}\n&&\\int\\limits_{\\mathbb{R}^{n}}\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\lambda\n_{v,m}a_{vm}(y)\\varphi (y)dy \\\\\n&=&\\int\\limits_{\\mathbb{R}^{n}}\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\lambda\n_{v,m}2^{-v(L+1)}a_{vm}(y)\\Big(\\varphi (y)-\\sum\\limits_{\\left\\vert \\beta\n\\right\\vert 0$ is at our disposal. Let $0h>-1-L\n. Since $a_{vm}$ are $\\left[ K,L,M\\right] $-molecules, then \n2^{-v(L+1)}\\left\\vert a_{vm}(y)\\right\\vert \\leq 2^{hv}2^{-v(L+1+h)}\\left(\n1+2^{v}\\left\\vert y-2^{-v}m\\right\\vert \\right) ^{-M}$. Therefore, The sum \n\\sum\\limits_{i=-\\infty }^{0}\\cdot \\cdot \\cdot $ can be estimated b\n\\begin{equation}\nc\\text{ }2^{-v(L+1+h)}\\sum\\limits_{i=-\\infty\n}^{0}\\int\\limits_{C_{i}}\\sum\\limits_{m\\in \\mathbb{Z}^{n}}2^{hv}\\left\\vert\n\\lambda _{v,m}\\right\\vert \\left( 1+2^{v}\\left\\vert y-2^{-v}m\\right\\vert\n\\right) ^{L+S-M}(1+\\left\\vert y\\right\\vert )^{-S}dy. \\label{Max-estimate}\n\\end{equation\nSince $M$ can be taken large enough, by Lemma 4 in \\cite{K10} we obtai\n\\begin{equation*}\n\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\left\\vert \\lambda _{v,m}\\right\\vert \\left(\n1+2^{v}\\left\\vert y-2^{-v}m\\right\\vert \\right) ^{L+S-M}\\leq c\\mathcal{M}\\Big\n\\sum\\limits_{m\\in \\mathbb{Z}^{n}}\\left\\vert \\lambda _{v,m}\\right\\vert \\chi\n_{v,m}\\Big)(y)\n\\end{equation*\nfor any $y\\in C_{i}\\cap Q_{v,l}$ with $l\\in \\mathbb{Z}^{n}$. We split $S$\ninto $R+T$ with $R+\\alpha <0$ and $T$ large enough such that $T>\\max (-R\n\\frac{n(q-t)}{q})$. Then $\\mathrm{\\eqref{Max-estimate}}$ is bounded b\n\\begin{equation*}\nc\\ 2^{-v(L+1+h)}\\sum\\limits_{i=-\\infty }^{0}2^{-iR}\\int\\limits_{C_{i}\n\\mathcal{M}\\Big(\\sum\\limits_{m\\in \\mathbb{Z}^{n}}2^{vh}\\left\\vert \\lambda\n_{v,m}\\right\\vert \\chi _{v,m}\\Big)(y)(1+\\left\\vert y\\right\\vert )^{-T}dy.\n\\end{equation*\nSince we have in addition the factor $(1+\\left\\vert y\\right\\vert )^{-T}$, it\nfollows by H\\\"{o}lder's inequality that this expression is bounded b\n\\begin{eqnarray*}\n&&c\\ 2^{-v(L+1+h)}\\sum\\limits_{i=-\\infty }^{0}2^{-iR}\\Big\\|\\mathcal{M}\\Big\n\\sum\\limits_{m\\in \\mathbb{Z}^{n}}2^{hv}\\left\\vert \\lambda _{v,m}\\right\\vert\n\\chi _{v,m}\\Big)\\chi _{i}\\Big\\|_{q\/t} \\\\\n&\\leq &c\\text{ }2^{-v(L+1+h)}\\sum\\limits_{i=-\\infty }^{0}2^{-i(\\alpha +R)\n\\Big\\|\\sum\\limits_{m\\in \\mathbb{Z}^{n}}2^{hv}\\left\\vert \\lambda\n_{v,m}\\right\\vert \\chi _{v,m}|\\dot{K}_{q\/t}^{\\alpha ,\\infty }\\Big\\| \\\\\n&\\leq &c\\text{ }2^{-v(L+1+h)}\\Big\\|\\lambda |\\dot{K}_{q\/t}^{\\alpha\n,p}f_{\\infty }^{h}\\Big\\|,\n\\end{eqnarray*\nwhere the first inequality follows by the boundedness of the\nHardy-Littlewood maximal operator $\\mathcal{M}$ on $\\dot{K}_{q\/t}^{\\alpha\n,\\infty }$. Using a combination of the arguments used above, the sum \n\\sum\\limits_{i=1}^{\\infty }\\cdot \\cdot \\cdot $ can be estimated by $c$ \n2^{-v(L+1+h)}\\big\\|\\lambda |\\dot{K}_{q\/t}^{\\alpha ,p}f_{\\infty }^{h}\\big\\|$.\nSince $L+1+h>0$, the convergence of $\\mathrm{\\eqref{new-rep}}$ is now clear\nby the embeddings \n\\begin{equation*}\n\\dot{K}_{q}^{\\alpha ,p}f_{\\infty }^{s}\\hookrightarrow \\dot{K}_{q\/t}^{\\alpha\n,p}f_{\\infty }^{h},\n\\end{equation*\nsee Theorem \\ref{embeddings3}. The proof is completed. \\ \\ \\rule{3mm}{3mm}\n\nLet $w$ denote a positive, locally integrable function and $0-n$. Then in view of the\nfact that $L^{p}=\\dot{K}_{p}^{0,p}$, we hav\n\\begin{equation*}\n\\big\\|f\\mid F_{p,\\beta }^{s}(w_{\\gamma })\\big\\|\\approx \\big\\|f\\mid \\dot{K\n_{p}^{\\frac{\\gamma }{p},p}F_{\\beta }^{s}\\big\\|\\text{.}\n\\end{equation*}\n\nApplying Corollary \\ref{embeddings3.1} in some particular cases yields the\nfollowing embeddings, see for the case of .\n\n\\begin{corollary}\n\\textit{Let }$s_{1},s_{2}\\in \\mathbb{R}$, $0-n\\ $\\textit{and }$\\gamma\n_{2}>-n$. \\textit{We suppose that } \n\\begin{equation*}\ns_{1}-\\frac{n+\\gamma _{1}}{s}=s_{2}-\\frac{n+\\gamma _{2}}{q}\n\\end{equation*\nan\n\\begin{equation*}\n\\gamma _{2}\/q\\geq \\gamma _{1}\/s.\n\\end{equation*}\nThe\n\\begin{equation*}\nF_{q,\\infty }^{s_{2}}(w_{\\gamma _{2}})\\hookrightarrow F_{s,\\beta\n}^{s_{1}}(w_{\\gamma _{1}}).\n\\end{equation*}\n\\end{corollary}\n\n\\begin{remark}\n$\\mathrm{We}$ $\\mathrm{refer}$ $\\mathrm{the}$ $\\mathrm{reader}$ $\\mathrm{to}$\n$\\mathrm{the}$ $\\mathrm{recent}$ $\\mathrm{paper}$ $\\mathrm{\\cite{HarLkr08}}$ \n$\\mathrm{for}$ $\\mathrm{further}$ $\\mathrm{results}$ $\\mathrm{about}$ \n\\mathrm{Sobolev}$ $\\mathrm{embeddings}$ $\\mathrm{for}$ $\\mathrm{weighted}$ \n\\mathrm{spaces}$ $\\mathrm{of}$ $\\mathrm{Besov}$ $\\mathrm{type}$ $\\mathrm\nwhere}$ $\\mathrm{the}$ $\\mathrm{weight}$ $\\mathrm{belongs}$ $\\mathrm{to}$ \n\\mathrm{some}$ $\\mathrm{Muckenhoupt}$ $\\mathcal{A}_{\\varrho }$ $\\mathrm\nclass.}$ $\\mathrm{Notice}$ $\\mathrm{that}$ $\\mathrm{this}$ $\\mathrm{results}$\n$\\mathrm{are}$ $\\mathrm{given}$ $\\mathrm{in\\ \\cite[Theorem \\ 1.2]{MV12}}$ \n\\mathrm{but}$ $\\mathrm{under}$ $\\mathrm{the}$ $\\mathrm{restrictions}$ \n10$, or expander graphs, with $d, \\lambda_2$ being the largest and second largest eigenvalue of its adjacency matrix, satisfying\n$d \\ge \\lambda_2\/(\\delta \\sqrt{\\mu (1-\\mu +2\\delta\\mu)}).$\n\\begin{figure}[b]\n\\centering\n\\begin{tikzpicture}\n\\begin{scope}[every node\/.style={circle,thick,draw,minimum size=0.5cm}]\n \\node[fill=red] (X) at (-1,2) { };\n \\node[fill=red] (A) at (0,2) { };\n \\node[fill=red] (B) at (1,2) { };\n \\node[fill=red] (C) at (2,2) { };\n \\node[fill=red] (D) at (3,2) { };\n \\node[fill=red] (E) at (4,2) { };\n \\node[fill=blue] (F) at (5,2) { };\n \\node[fill=blue] (G) at (6,2) { };\n \\node[pattern=dots,pattern color=blue] (H) at (7,2) { };\n \\node[pattern=dots,pattern color=blue] (I) at (8,2) { };\n \\node[pattern=dots,pattern color=blue] (J) at (9,2) { };\n \\node[pattern=dots,pattern color=blue] (K) at (10,2) { };\n \\node[pattern=dots,pattern color=blue] (L) at (11,2) { };\n\\end{scope}\n\\begin{scope}[>={Stealth[black]},\n every edge\/.style={draw=black, very thick}]\n \\path (X) edge node {} (A);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\path (D) edge node {} (E);\n \\path (E) edge node {} (F);\n \\path (F) edge node {} (G);\n \\path (G) edge node {} (H);\n \\path (H) edge node {} (I);\n \\path (I) edge node {} (J);\n \\path (J) edge node {} (K);\n \\path (K) edge node {} (L);\n\\end{scope}\n\\end{tikzpicture}\n\\begin{minipage}{1\\textwidth}\n~\\\\~\\\\\n\\end{minipage}\n\n \n\\begin{tikzpicture}\n\\begin{scope}[every node\/.style={circle,thick,draw,minimum size=0.5cm}]\n \\node[fill=red] (X) at (-1,2) { };\n \\node[fill=red] (A) at (0,2) { };\n \\node[fill=red] (B) at (1,2) { };\n \\node[fill=red] (C) at (2,2) { };\n \\node[fill=red] (D) at (3,2) { };\n \\node[fill=red] (E) at (4,2) { };\n \\node[fill=blue] (F) at (5,2) { };\n \\node[pattern=dots,pattern color=blue] (G) at (6,2) { };\n \\node[pattern=dots,pattern color=blue] (H) at (7,2) { };\n \\node[fill=blue] (I) at (8,2) { };\n \\node[pattern=dots,pattern color=blue] (J) at (9,2) { };\n \\node[] (K) at (10,2) { };\n \\node[] (L) at (11,2) { };\n\\end{scope}\n\\begin{scope}[>={Stealth[black]},\n every edge\/.style={draw=black, very thick}]\n \\path (X) edge node {} (A);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\path (D) edge node {} (E);\n \\path (E) edge node {} (F);\n \\path (F) edge node {} (G);\n \\path (G) edge node {} (H);\n \\path (H) edge node {} (I);\n \\path (I) edge node {} (J);\n \\path (J) edge node {} (K);\n \\path (K) edge node {} (L);\n\\end{scope}\n\\end{tikzpicture}\n\\begin{minipage}{0.88\\textwidth}\n\\caption{\\small This figure shows an example from \\cite{Alon2015}. The colors red\/blue correspond to the experts labeled `1'\/'0'. The dotted vertices indicate their label after the dissemination process, the unmarked vertices are decided randomly. In the first line graph we consider the iterative strong adversary, where only the rightmost blue vertex determines the label of all remaining vertices. In the second line we consider the non-iterative setting, each blue expert can at most convince two non-experts. If $1\/2+\\delta>3(1\/2-\\delta)$ and $n$ is large, the adversary can not hope to convince more that half of all vertices. \n}\n\\label{fig:LitRef}\n\\end{minipage} \n\\end{figure}\n\\paragraph{Iterative Dissemination}\nIn \\cite{Alon2015} the authors also introduced an iterative version of the model with a more dynamic dissemination process. The \\emph{iterative model} also starts with labeled experts and all non-experts are labeled according to the majority of their neighbouring experts. Ties that involve at least one expert are broken uniformly at random. All non-experts without any expert neighbours, however, do not form their opinion right away, but remain unlabeled. This process is then iterated by considering all vertices with label `1' and all vertices with label `0', \nuntil all vertices are labeled. \n\nA natural question is whether iterativity helps or hinders the adversary.\nIntuitively, iterativity ought to be beneficial for the adversary.\nIf a graph is not robust against a non-iterative adversary, then there is a choice of expert sets such that after one round of dissemination there are more vertices that are labeled '0' than '1'.\nThe remaining vertices without expert neighbours are then either decided randomly (non-iterative) or there are subsequent rounds of dissemination (iterative). As there now are more '0' labeled vertices that '1' labeled vertices, deciding the label of the remaining vertices by dissemination should be beneficial for the adversary.\nIndeed, the authors of \\cite{Alon2015} provided examples where this is the case. For example, they showed that for suitable values of~$\\mu$ and~$\\delta$, a line graph is robust against the non-iterative strong\/weak adversaries, but not against their iterative versions, see Fig.~\\ref{fig:LitRef}. \n\nHowever, in \\cite{Alon2015} an additional example, where for the weak adversary the opposite is true, was constructed. Consider a graph that is a disjoint union of a star and a $d$-regular expander graph.\nPlace one expert in the center of the star and distribute the other experts as evenly as possible on the expander. In the non-iterative setting, each expert in the expander will spread its label to $d$ many non-experts. If the expert in the center of the star is labeled `0', all vertices in the star are labeled `0' as well, outweighing the difference between `1's and `0's in the expander.\nIn the iterative setting however, each expert does not only spread its label to $d$ many other vertices, but all vertices in the expander will be labeled at the end of the dissemination, roughly in the same ratio as that of the experts in the beginning.\nNow the difference in `1's and `0's is so large that even if all vertices in the star were labeled with `0' can sway the majority. \n\nGuided by the intuition described previously, it seems that no such construction can work for a strong adversary. In the previous example of the graph consisting of an expander and a star the adversary can place all '1'-labeled experts on the star and all others in the expander. Then, all vertices in the expander will be labeled '0' resulting in a clear majority. Consequently, in~\\cite{Alon2015} the following conjecture concerning the effect of iterativity in that case was made. \n\\begin{conjecture}[\\cite{Alon2015}]\\label{conjectureAlon}\nIn the case of a strong adversary an iterative propagation can never harm the adversary.\n\\end{conjecture}\nEquivalently, the conjecture states that there is no graph \nthat is robust against the iterative strong adversary and simulaneously not robust against the non-iterative version -- in this precise sense iterativity does not harm\/can only help the adversary.\n\n\n\\paragraph{Related Results}\nBesides of \\cite{Alon2015}, where this model for opinion formation was introduced, there is one more work that studies questions in this precise framework. In his doctoral thesis \\cite{Daknama2018}, Daknama studied resilience properties of random graphs. 'Local resilience' in this context refers to the largest number of edges, which are adjacent to any vertex, that can be removed so that the graph still is robust against the strong adversary. In~\\cite{Daknama2018} it was shown that one can delete up to a fraction of $2(1-\\mu+2\\delta \\mu)\\delta\/(1+2\\delta)$ of all edges at each vertex without affecting robustness. \n\nThere are also other directly related studies in opinion forming, which, however, do not use the exact model presented here. These papers include studies on word of mouth \\cite{Young2009}, group recommendation \\cite{Andersen2008, Grandi2016,lev2017group,Faliszewski2016} and informational cascades \\cite{Bikhchandani1992, Bikhchandani1998, Watts2002, Alon2012, Feldman2014}. For further references see also \\cite{Alon2015}.\n\n\\paragraph{Result} \nThe contribution of this paper is to refute Conjecture \\ref{conjectureAlon}. The idea is to consider a graph that has non-robustness against the non-iterative strong adversary in a very weak way. More concretely, the majority for '0' labeled vertices is only achieved if a majority of vertices without expert neighbours is labeled '0'. If we consider iterativity, then the adversary has no clear advantage in a subsequent round of the dissemination, as there are roughly equally many '1'- and '0'-labeled vertices. Additionally, we can construct the graph in a way such that the vertices without expert neighbours are connected to vertices that are labeled '1' in the first round of the dissemination, so that the adversary \\emph{gets harmed}.\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}\n\\begin{scope}[every node\/.style={circle,thick,draw,text width=0.5cm,align=center}]\n \\node[minimum size=1cm,, inner sep=1pt] (A) at (0,2) {{\\large I} \\raisebox{.5pt}{\\textcircled{\\raisebox{-.9pt} {1}}}};\n \\node[minimum size=1.5cm] (B) at (3,2) {{\\Large J} \\raisebox{.5pt}{\\textcircled{\\raisebox{-.9pt} {1}}}};\n \\node[minimum size=1.5cm] (C) at (8,2) {{\\Large P} \\raisebox{.5pt}{\\textcircled{\\raisebox{-.9pt} {1}}}};\n \\node[minimum size=1cm, inner sep=1pt] (D) at (11,2) {{\\large O} \\raisebox{.5pt}{\\textcircled{\\raisebox{-.9pt} {1}}}};\n \\node[inner sep=0pt] (E) at (2,-1) {{\\large D} \\raisebox{.5pt}{\\textcircled{\\raisebox{-.9pt} {0}}}};\n\\end{scope}\n\n\\begin{scope}[>={Stealth[black]},\n every node\/.style={fill=white,shape=rectangle},\n every edge\/.style={draw=black, very thick}]\n \\path (A) edge node {$p_{IJ}$} (B);\n \\path (B) edge node {$p_{JP}$} (C);\n \\path (A) edge[bend left=40] node {$p_{IP}$} (C);\n \\path (D) edge node {$1$} (C);\n \\path (D) edge[bend left=40] node {$1$} (B);\n \\path (A) edge[bend left=60] node {$1$} (D);\n \\path (B) edge node {$1\\textbf{ : }1$} (E); \n\\end{scope}\n\\end{tikzpicture}\n\\begin{minipage}{0.85\\textwidth}\n\\caption{\\small This figure shows the graph $G$. The numbers on the edges and in the vertices give the probability that an edge is present between\/in the components. For example, any edge with one vertex in $I$ and one in $J$ exists independently with probability $p_{IJ}$. Every vertex in $D$ has exactly one distinct neighbour in $J$ and no other neighbours, i.e., every vertex in $D$ has degree 1 and no two vertices in $D$ have a common neighbour. }\n\\label{fig:Result}\n\\end{minipage}\n\\end{figure}\nConsider the following graph that implements these ideas. Let $0<\\mu,\\delta<1\/2$ and $0< \\varepsilon_1<2\\delta\/(1\/2+\\delta)$, $0<\\varepsilon_2<(1\/2-\\delta)\/(1\/2+\\delta)$ as well as $00$ such that $G$ is whp robust against the iterative strong adversary, but not against the non-iterative strong adversary.\n\\end{theorem}\nNote that $\\delta>1\/6$ is a necessary constraint for our construction, but we are certain that there is an example for smaller $\\delta$ as well.\nPermissible values in Theorem \\ref{conjecture} are, e.g.~$\\mu=\\delta=1\/5,\\ \\varepsilon_1=10^{-2},\\ d=10^{-4}$ and $\\varepsilon_2=10^{-6}.$ \nThe remainder of this paper will consist of the proof of Theorem~\\ref{conjecture}. We first state and prove a well known description of the edge distribution of random graphs and then show the claimed (non-) robustness. \n\n\\section{Proof}\nFor a graph $G=(V,E)$ let $N(v)=\\{w\\in V\\mid (v,w)\\in E\\}$ be the set of neighbours of $v$. We begin with a statement about the distribution of edges in random graphs.\n\\begin{lemma}\\label{exp_lem_2}\\label{exp_lem}\nLet $\\varepsilon>0$. The Erd\\H{o}s-R\\'enyi random graph $G(n,p)$ with vertex set $V$ and $p\\ge \\varepsilon$ has whp the following property. For any set $S\\subseteq V$ of size $|S|\\ge \\varepsilon n$\nthere is a set $X_S\\subset V\\setminus S$ of size \nat most ${4\\varepsilon^{-3}(\\ln \\varepsilon^{-1} +2)}$ \nsuch that\n$$\\forall v\\in (V\\setminus S)\\setminus X_S: \\big ||N(v)\\cap S|-p|S|\\big| \\le \\varepsilon p|S|.$$\n\\end{lemma}\nSimilar versions of Lemma \\ref{exp_lem} with (somehow) different bounds exist in the literature, see for example \\cite[Lem.~IV.1 and IV.3]{Fountoulakis2010}. However, as we did not find the exact statement we will need in the literature we include a proof. We will utilize the following Chernoff bound.\n\\begin{theorem}[\\cite{Arora2009}, Cor 7.11]\\label{Chernoff}\nLet $X$ be a binomially distributed random variable. Then\n$${P}\\Big(|X-\\mathbb{E}[X]|>\\delta\\mathbb{E}[X]\\Big)\\le 2\\exp\\left(-\\min\\{\\delta^2,\\delta\\}\\mathbb{E}[X]\/4\\right ), \\qquad \\delta > 0.$$\n\\end{theorem}\\begin{proof}[Proof of Lemma \\ref{exp_lem_2}]\nLet $S\\subseteq V, |S|\\ge \\varepsilon n$ and let \n$$X_S=\\big\\{v\\in V\\setminus S\\bigm\\vert \\big ||N(v)\\cap S|-p|S|\\big| > \\varepsilon p|S|\\big\\}$$\nbe the set of vertices not satisfying the claim of the lemma. The number of neighbours of any vertex $v\\in V\\setminus S$ is a binomially distributed random variable, $|N(v)\\cap S|\\ =\\text{Bin}(|S|,p)$, and the expected number of neighbours of $v$ in $S$ is $p|S|$. Thus the probability of $v\\in X_S$ can be bounded with Theorem \\ref{Chernoff} by\n$${P}\\big (\\big||N(v)\\cap S|-p|S|\\big| > \\varepsilon p|S|\\big )\\le \\exp \\left (- \\varepsilon^2 p|S|\/4\\right ). $$\nLet furthermore $t\\in \\mathbb{N}$; the probability that $t$ distinct vertices are in $X_S$ is at most $\\exp(-\\varepsilon^2p|S|\/4\\cdot t)$ as the events of vertices being elements of $X_S$ are independent. There are $\\binom{n}{k}\\le \\left (\\frac{en}{k}\\right )^k$ possibilities to choose $S$, a set of size $k\\ge \\varepsilon n$. Hence the probability that for fixed $k\\ge \\varepsilon n$ there is a set $S, |S|= k $ such that $|X_S|=t$ is by union bound at most \\begin{align*}\n\\exp\\left (k\\ln (en\/k)-\\frac{\\varepsilon^2pk}{4}\\cdot t\\right )\\le \\exp\\left (k\\left (-\\ln \\varepsilon +1-\\frac{\\varepsilon^3}{4}\\cdot t\\right )\\right ),\n\\end{align*}\nwhere we used the assumption that $p\\ge \\varepsilon$. Thus, if $t\\ge{4{\\varepsilon^{-3}}(\\ln \\varepsilon^{-1} +2)}$ this expression is $\\le e^{-k}$ and summing over $k\\ge \\varepsilon n$ yields the claim.\n\\end{proof}\n\nThis concludes the preparations. Next we prove the main theorem, by proving the two claims separately. We show the robustness of $G$ against the iterative strong adversary first.\n\\begin{lemma}\\label{lemma_robust}\nFor all $0<\\mu<1\/2$ and $1\/6<\\delta<1\/2$ there are values $\\varepsilon_1,\\,\\varepsilon_2,\\, d>0$ such that $G$ is whp robust against the iterative strong adversary.\n\\end{lemma}\n\\begin{proof}\nLet $0<\\mu<1\/2,\\ 1\/6<\\delta<1\/2$ and $\\varepsilon_1, \\varepsilon_{2},\\ d>0$ such that \n\\begin{align}\\label{ass:e1}\n\\varepsilon_1<\\min\\left \\{\\frac{\\delta\\mu}2, \\frac{4\\delta}{1\/2+\\delta}-1\\right\\}\n\\end{align} and furthermore\n\\begin{align}\\label{ass:d}\nd<\\min\\left \\{\\frac{\\varepsilon_1\\delta}{1\/2+\\delta},\\frac{\\varepsilon_1\\delta\\mu}4 ,\\frac{1-\\mu -2\\delta\\mu}{3}\\right \\}\n\\end{align}\nas well as\n\\begin{align}\\label{ass:e2}\n\\varepsilon_2<\\min \\left\\{\\frac{d}{6}\\left (\\frac{4\\delta}{1\/2+\\delta}-1-\\varepsilon_1\\right ),\\frac{1\/2-\\delta}{1+2\\delta}\\right\\}.\n\\end{align}\nWe will show that for any choice of experts, at the end of the dissemination the majority will be labeled '1' thus proving robustness. Let therefore $\\E=\\E_1\\cup\\E_0$ be any set of experts as chosen by the iterative strong adversary and define\n$$i_1:=|I\\cap \\mathcal{E}_{1}|, \\quad j_{1}:=|J\\cap \\mathcal{E}_{1}|, \\quad o_{1}:=|O\\cap \\mathcal{E}_{1}|, \\quad p_1:=|P\\cap \\mathcal{E}_{1}|, \\quad d_1:=|D\\cap \\mathcal{E}_{1}|$$\nas well as\n$$i_0:=|I\\cap \\mathcal{E}_{0}|, \\quad j_{0}:=|J\\cap \\mathcal{E}_{0}|, \\quad o_{0}:=|O\\cap \\mathcal{E}_{0}|, \\quad p_0:=|P\\cap \\mathcal{E}_{0}|, \\quad d_0:=|D\\cap \\mathcal{E}_{0}|.$$\n By definition of the model we have that $|\\mathcal{E}_{1}|=\\left( {1}\/{2}+\\delta\\right )\\mu n$ as well as $|\\mathcal{E}_{0}|=\\left( {1}\/{2}-\\delta\\right )\\mu n$ and therefore\n\\begin{align}\\label{eq:1}\ni_1+j_1+o_1+p_1+d_1=\\left(\\frac{1}{2}+\\delta\\right )\\mu n\\quad\\text{and}\\quad i_0+j_0+o_0+p_0+d_0=\\left(\\frac{1}{2}-\\delta\\right )\\mu n,\n\\end{align}\nwhich readily implies that\n\\begin{align}\\label{eq:2}\n2\\delta\\mu n=\\frac{2\\delta}{1\/2+\\delta}(i_1+j_1+p_1+o_1+d_1).\n\\end{align}\nWe will see that the iterative dissemination will be finished after two rounds only. We start by determining the label of each vertex in the different components after the first round of dissemination.\nThis is decided by the difference in `0'\/`1' labeled expert neighbours. Consider the difference $$\n \\Delta(v):=|N(v)\\cap \\mathcal{E}_{1}|-|N(v)\\cap \\mathcal{E}_{0}|, \\quad v\\in V.\n$$ \nIn particular, $\\Delta(v)> 0$ means that $v\\in V\\setminus (\\mathcal{E}_1\\cup \\mathcal{E}_0)$ will be labeled `1' and $\\Delta(v)<0$ means it will be labeled `0'. Note that vertices $v$ with $\\Delta(v)=0$ could be either labeled randomly (if they have the same positive number of '0'\/'1' labeled neighbours) or not at all in this round. \n \nWe begin with a vertex $v\\in O$. Using the construction of $G$ and \\eqref{eq:1} we get\n\\begin{align*}\n|N(v)\\cap \\mathcal{E}_{1}|&=i_1+ j_1+ p_1+o_1=\\left (\\frac{1}{2}+\\delta\\right )\\mu n-d_1\n\\end{align*}\nand similarly \n\\begin{align*}\n|N(v)\\cap\\mathcal{E}_{0}|&=i_0+j_0+p_0+o_0=\\left (\\frac{1}{2}-\\delta\\right )\\mu n-d_0.\n\\end{align*}\nCombining these two equations, $d<\\varepsilon_1<\\delta\\mu\/2$ given by \\eqref{ass:d} and \\eqref{ass:e1} implies\n$$\\Delta(v)=2\\delta\\mu n-(d_1-d_0)>0\\qquad \\forall~v\\in O.$$\nWe continue with $v\\in J$. Using Lemma \\ref{exp_lem}, we get that for all $\\varepsilon>0$ whp there is $J_P\\subset J,~ |J_P|\\le 4\\varepsilon^{-3}(\\ln \\varepsilon^{-1} +2)$ such that \n$$\\big||N(v)\\cap P\\cap \\E_1|-p_{JP}\\cdot p_1\\big|\\le\\varepsilon\\cdot p_{JP}\\cdot p_1+ \\varepsilon n \\quad \\text{for all } v\\in J\\setminus J_P.$$\nAs $\\varepsilon>0$ is arbitrary we infer that \n$$|N(v)\\cap P\\cap \\E_1|=p_{JP}\\cdot p_1+o(n)\\quad \\text{for all } v\\in J\\setminus J_P.$$\nCompletely analogous calculations for $I$ and $\\E_0$ yield that whp there is $J'\\subset J,~ |J'|~=o(n)$ such that for all $v\\in J\\setminus J'$ \n\\begin{align*}\n|N(v)\\cap \\mathcal{E}_{1}|&=p_{IJ}\\cdot i_1+ j_1+ p_{JP}\\cdot p_1+o_1+o(n)\\\\&=\n\\left (\\frac{1}{2}+\\delta\\right )\\mu n-(1-p_{IJ})i_1-(1- p_{JP})p_1-d_1+o(n)\n\\end{align*}\nand\n\\begin{align*}\n|N(v)\\cap\\mathcal{E}_{0}|&=p_{IJ}\\cdot i_0+ j_0+ p_{JP}\\cdot p_0+o_0+o(n)\\\\&=\n\\left (\\frac{1}{2}-\\delta\\right )\\mu n-(1-p_{IJ})i_0-(1- p_{JP})p_0-d_0+o(n).\n\\end{align*}\nComputing the difference of the above expressions we get for all $v\\in J\\setminus J'$\n\\begin{align*}\n\\Delta(v)&=2\\delta\\mu n-(1-p_{IJ})(i_1-i_0)-(1- p_{JP})(p_1-p_0)-(d_1-d_0)+o(n)\\\\\n&=2\\delta\\mu n-\\left (\\frac{2\\delta}{1\/2+\\delta}-\\varepsilon_1\\right )\\Big ((i_1-i_0)+(p_1-p_0)\\Big )-(d_1-d_0)+o(n)\\\\\n&= 2\\delta\\mu n-\\frac{2\\delta}{1\/2+\\delta}\\Big ((i_1-i_0)+(p_1-p_0)\\Big )+\\varepsilon_1\\Big ((i_1-i_0)+(p_1-p_0)\\Big )-(d_1-d_0)+o(n).\n\\end{align*}\nApplying \\eqref{eq:2} and \\eqref{eq:1} we can obtain a lower bound for $\\Delta(v),v\\in J\\setminus J'$\n\\begin{align*}\n\\Delta(v)&\\ge \\frac{2\\delta}{1\/2+\\delta}(j_1+o_1+d_1+i_0+p_0)+\\varepsilon_1\\Big ((i_1-i_0)+(p_1-p_0)\\Big )-d_1+o(n)\\\\\n&\\ge \\varepsilon_1(i_1+j_1+p_1+o_1+d_1)+\\left ( \\frac{2\\delta}{1\/2+\\delta}-\\varepsilon_1\\right )(i_0+p_0)-d_1.\n\\end{align*}\nAccording to \\eqref{ass:e1} and \\eqref{ass:d} we have $\\varepsilon_1<2\\delta\/(1\/2+\\delta)$ as well as $d<\\varepsilon_1\\delta\\mu \/4$ and therefore\n\\begin{align*}\n\\Delta(v)\\ge \\varepsilon_1\\cdot 2\\delta\\mu n -dn>0\\qquad \\forall~ v\\in J\\setminus J'.\n\\end{align*}\nNext we look at $v\\in I.$ Using again Lemma \\ref{exp_lem} and~\\eqref{eq:1} we infer that whp there is $I'\\subset I,~ |I'|~=o(n)$ such that for all $v\\in I\\setminus I'$\n\\begin{align*}\n|N(v)\\cap \\mathcal{E}_{1}|&=i_1+p_{IJ}\\cdot j_1+p_{IP}\\cdot p_1+o_1+o(n)\\\\&=\\left (\\frac{1}{2}+\\delta\\right )\\mu n-\\left (1-p_{IJ}\\right )j_1-\\left (1-p_{IP}\\right )p_1-d_1+o(n)\n\\end{align*}\nand \n\\begin{align*}\n|N(v)\\cap\\mathcal{E}_{0}|&=i_0+p_{IJ}\\cdot j_0+p_{IP}\\cdot p_0+o_0+o(n)\\\\&=\\left (\\frac{1}{2}-\\delta\\right )\\mu n-\\left (1-p_{IJ}\\right )j_0-\\left (1-p_{IP}\\right )p_0-d_0+o(n).\n\\end{align*}\nBy combining those bounds we obtain for all $v\\in I\\setminus I'$\n\\begin{equation}\\label{eq:5}\n\\begin{aligned}\n \\Delta(v)\n &=\n 2\\delta \\mu n-\\left (1-p_{IJ}\\right )(j_1-j_0)-\\left (1-p_{IP}\\right )(p_1-p_0)-(d_1-d_0)+o(n) \\\\\n & \\hspace{-6mm}=\n 2\\delta\\mu n-\\frac{2\\delta}{1\/2+\\delta}\\Big ((j_1-j_0)+(p_1-p_0)\\Big )-(d_1-d_0)+\\varepsilon_{1}(j_1-j_0)-\\varepsilon_{2}(p_1-p_0)+o(n).\n\\end{aligned}\n\\end{equation}\nBefore we conclusively determine $\\Delta(v)$ for $ v\\in I\\setminus I'$ we look at vertices $v\\in P$. Using once more Lemma~\\ref{exp_lem} and~\\eqref{eq:1} we infer that whp there is $P'\\subset P,~ |P'|~=o(n)$ such that for all $v\\in P\\setminus P'$\n\\begin{align*}\n|N(v)\\cap \\mathcal{E}_{1}|&=p_{IP}\\cdot i_1+p_{JP}\\cdot j_1+ p_1+o_1+o(n)\\\\&=\\left (\\frac{1}{2}+\\delta\\right )\\mu n-\\left (1-p_{IP}\\right )i_1-\\left (1-p_{JP}\\right )j_1-d_1+o(n)\n\\end{align*}\nand \n\\begin{align*}\n|N(v)\\cap\\mathcal{E}_{0}|&=p_{IP}\\cdot i_0+p_{JP}\\cdot j_0+ p_0+o_0+o(n)\\\\&=\\left (\\frac{1}{2}-\\delta\\right )\\mu n-\\left (1-p_{IP}\\right )i_0-\\left (1-p_{JP}\\right )j_0-d_0+o(n).\n\\end{align*}\nTogether these two expressions yield for all $v\\in P\\setminus P'$\n\\begin{align}\\label{eq:4}\n\\Delta(v)&= 2\\delta\\mu n-\\frac{2\\delta}{1\/2+\\delta}\\Big((i_1-i_0)+(j_1-j_0)\\Big)-(d_1-d_0)+\\varepsilon_{1}(j_1-j_0)-\\varepsilon_{2}(i_1-i_0)+o(n).\n\\end{align}\nWe argue next, that either ``$\\Delta(v)<0 \\text{ for some }v\\in I\\setminus I'\\,$'' or ``$\\Delta(v)<0 \\text{ for some }v\\in P\\setminus P'\\,$'' but never both. To see this, observe that \n$$\\text{``}\\Delta(v)<0 \\text{ for some }v\\in I\\setminus I' \\quad\\text{and}\\quad \\Delta(v)<0 \\text{ for some }v\\in P\\setminus P \\, \\text{''}$$ implies that \n\\begin{align}\\label{eq:3}\n(i_1-i_0)+(j_1-j_0) \\quad \\text{and}\\quad (j_1-j_0)+(p_1-p_0) \\quad \\text{are both}\\quad \\ge((1\/2+\\delta)\\mu-\\varepsilon_1 )n.\n\\end{align}\nOtherwise~\\eqref{ass:d} and \\eqref{ass:e2} assert that $\\varepsilon_2< d\/6$ as well as $d< \\varepsilon_1\\delta\/(1\/2+\\delta)$ and therefore either by \\eqref{eq:5}\n\\begin{align*}\n\\Delta(v)&\\ge \\frac{2\\delta}{1\/2+\\delta}\\varepsilon_1 n-dn-\\varepsilon_{2}n>0,\\qquad\\text{for all } v\\in I\\setminus I'\n\\end{align*}\nor by \\eqref{eq:4}\n\\begin{align*}\n\\Delta(v)&\\ge \\frac{2\\delta}{1\/2+\\delta}\\varepsilon_1 n-dn-\\varepsilon_{2}n>0,\\qquad \\text{for all } v\\in P\\setminus P'.\n\\end{align*}\nHowever, as $(i_1-i_0)+(p_1-p_0)\\le (1\/2+\\delta)\\mu n$ we obtain from~\\eqref{eq:3} that $j_1-j_0\\ge (\\delta\\mu -\\varepsilon_1)n$. Then~\\eqref{ass:e1},\\eqref{ass:d} and \\eqref{ass:e2} imply that $\\varepsilon_1< \\delta\\mu\/2, \\, d<\\varepsilon_1\\delta\\mu\/$ and $\\varepsilon_2< d\/6$ and thus \\eqref{eq:5} yields\n\\begin{align*}\n\\Delta(v)&\\ge \\varepsilon_1\\cdot (\\delta\\mu-\\varepsilon_1 )n-dn- \\varepsilon_{2}n>0,\\qquad \\text{for all } v\\in I\\setminus I'.\n\\end{align*}\nSummarizing, we have shown that $\\Delta(v)>0$ for all $v\\in (J\\setminus J')\\cup O$ and either $\\Delta(v)>0$ for all $v\\in I\\setminus I'$ or $\\Delta(v)>0$ for all $v\\in P\\setminus P'$. \n\n\nIn the rest of the proof we consider the second round of the iterative dissemination process. We will distinguish two cases. Assume first that $j_0+ d_0< (d\/2-\\varepsilon_2)n$. As $\\Delta(v)>0$ for all $v\\in J\\setminus J'$ we infer that at most $(d\/2-\\varepsilon_2)n+o(n)$ vertices in $D$ will be labeled `0' after the second round of the dissemination process, all other vertices in $D$ will be labeled `1'. Thus counting the total number of vertices labeled `1' after the process, we get in this case for $n$ large enough\n\\begin{align*}\n\\#(\\text{vertices labeled `1'})&>|I\\setminus I'|\\ +\\ |J\\setminus J'|\\ +\\ |O|\\ +\\left (\\frac{d}{2}+\\varepsilon_2-o(1)\\right) n-|\\mathcal{E}_0|\\\\\n&=(1-d)\\frac{n}{2}+\\frac{dn}{2}+\\varepsilon_2n-o(n)>\\frac{n}{2}.\n\\end{align*}\nWe are left with the case $j_0+d_0\\ge (d\/2-\\varepsilon_2)n$. Observe that $d_1<(d\/2+\\varepsilon_2)n$ as otherwise the conclusion of the previous case applies. We revisit $\\Delta(v),\\ v\\in P\\setminus P'$ using \\eqref{eq:4} and \\eqref{eq:2} \n\\begin{align*}\n\\Delta(v)&=\\frac{2\\delta}{1\/2+\\delta}(p_1+o_1+d_1+i_0+j_0)-(d_1-d_0)+\\varepsilon_{1}(j_1-j_0)-\\varepsilon_{2}(i_1-i_0)+o(n)\\\\\n&\\ge \\left (\\frac{2\\delta}{1\/2+\\delta}-\\varepsilon_1\\right )j_0+d_0 - \\left (1-\\frac{2\\delta}{1\/2+\\delta}\\right ) d_1-\\varepsilon_{2}i_1+o(n).\n\\end{align*}\nUsing the assumptions $j_0+d_0\\ge (d\/2-\\varepsilon_2)n$ and $d_1<(d\/2+\\varepsilon_2)n$, this simplifies to \n\\begin{align*}\n\\Delta(v)&> \\left (\\frac{4\\delta}{1\/2+\\delta}-1-\\varepsilon_1\\right )dn\/2-3\\varepsilon_{2}n+o(n).\n\\end{align*}\nAssumption \\eqref{ass:e2} guarantees that $\\Delta(v)>0,\\ v\\in P\\setminus P'$ and thus in this case for $n$ large enough\n\\begin{align*}\n\\#( \\text{vertices labeled '1'})&>|I\\setminus I'|\\ +\\ |J\\setminus J'|\\ +\\ |O|\\ +\\ |P\\setminus P'|\\ -\\ |\\mathcal{E}_0|\\\\\n&=(1-d)n-\\left (\\frac{1}{2}-\\delta\\right )\\mu n-o(n)>\\frac{n}{2},\n\\end{align*}\nand the proof is completed.\n\\end{proof}\n\nThe next lemma together with Lemma \\ref{lemma_robust} implies Theorem \\ref{conjecture}.\n\\begin{lemma}\\label{lemma_non_robust}\nFor all $0<\\mu<1\/2$ and $1\/6<\\delta<1\/2$ there are values $\\varepsilon_1,\\varepsilon_2, d>0$ such that whp $G$ is not robust against the non-iterative strong adversary.\n\\end{lemma}\n\\begin{proof}\nLet $0<\\mu<1\/2$ and $1\/6<\\delta<1\/2$ and $\\varepsilon_1,\\, \\varepsilon_2,\\, d>0$ as given in \\eqref{ass:e1} to \\eqref{ass:e2}. \nWe show that $G$ is indeed not robust by giving a suitable choice of the expert set. Set $\\E=\\E_1\\cup\\E_0$ with $\\mathcal{E}_{1}=I$ and $\\mathcal{E}_{0}=O$. By definition, these sets have matching cardinalities. \nWe compute the quantity $$\\Delta(v)=|N(v)\\cap \\E_1|-|N(v)\\cap \\E_2|$$ for vertices $v\\in P$ to find their labels.\nUsing $\\varepsilon_2< (1\/2-\\delta)\/(1+2\\delta)$ by \\eqref{ass:e2} and Theorem \\ref{Chernoff} we readily obtain that whp for all $v\\in P$\n\\begin{align*}\n\\Delta(v)\\le (p_{IP}+ o(1))|\\E_1|-|\\E_0|=\\left (\\frac{1\/2-\\delta}{1\/2+\\delta}-\\varepsilon_{2}+ o(1)\\right )|\\E_1|-|\\E_0|=-(\\varepsilon_2+o(1))|\\E_1|<0.\n\\end{align*}\nTherefore the set of vertices labeled `0' contains $O\\cup P$ which has cardinality $(1-d)\\frac{n}{2}$. However, vertices in $D$ do not have any expert neighbours and as we are in the non-iterative setting, those vertices will be decided uniformly at random. Hence with probability $1\/2$ there will be at least $dn\/2+1$ vertices labeled `0' in $D$ and therefore $G$ is not robust against the non-iterative strong adversary.\n\\end{proof}\n\\phantomsection\n\\bibliographystyle{abbrv}\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkdsc b/data_all_eng_slimpj/shuffled/split2/finalzzkdsc new file mode 100644 index 0000000000000000000000000000000000000000..7075edaeb4c4415457b213b8f3d35ff939968101 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzkdsc @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\n\nDespite over forty years of theoretical and observational studies, we still do not fully understand the emission mechanism responsible for the observed pulsar radiation. \n\n\nIn an attempt to fully understand the high energy nonthermal emission from pulsars, both direct and indirect methods have been used. Direct models attempt to describe fully the source physics that subsequently generates the radiation we see. Conversely, inverse models use the observed radiation in an attempt to infer information about the source physics and emission geometry itself. In this paper, we present the methodology for a generic inverse method, whose aim is to restrict pulsar geometric parameters ($\\alpha, \\chi$, and emission location), using few assumptions and many observational constraints. Inverse models have had a long history within pulsar astrophysics. The models effectively utilise the link between lightcurve morphology and local source geometry, to place restrictions on various source parameters, i.e., to restrict the pulsar's magnetic inclination ($\\alpha$), the observer's viewing angle ($\\chi$), and\/or to determine the emission location within the pulsar magnetosphere. Perhaps the best known of these models is the Rotating Vector Model (RVM) (\\cite{RC1969}), which constrains pulsar inclination and viewing angle. It does so by assuming that the {\\it radio} linear polarisation is described by a polarisation vector fixed to local B field lines, (located close to the magnetic pole), which sweeps by an observer's line of sight as the pulsar rotates.\n\nInverse methods in general, have evolved over time. Early ideas were to associate the width of pulse profiles with the relativistically beamed opening angle of isotropic radiation, from point sources corotating within the pulsars magnetosphere (e.g., \\cite{S1970}, \\cite{Z1971}, \\cite{ZS1972}). \\cite{M1983} proposed a model for pulsed optical $\\rightarrow$ $\\gamma$-ray emission based on relativistic electron beams producing highly beamed emission in the far (measured radially) magnetosphere. Assuming emission from only the (purely dipolar) last open field lines of the {\\it equator} of an orthogonal rotator, the arrival phase of emitted photons (assumed tangential to the local B, subject to aberration) were mapped onto observer phase. Aligning this radius-to-phase mapping with the relative peak arrival phases for the Vela pulsar, Morini estimated that the radio, optical and $\\gamma$-ray emission for the Vela pulsar, originated at distances of $\\sim 0, 0.5$ and $0.7$ $R_{LC}$ from the neutron star surface. The advent of readily available computer power allowed the construction of radius-to-phase maps, as the assumption of tangential beaming means that the mapping is independent of the specific emission process. In the late 1980s, Smith and co-workers (\\cite{S1986}, \\cite{SJDP1988}) extended this idea somewhat, producing radius-to-phase maps for the whole equatorial {\\it plane} of an orthogonal {\\it retarded} dipole. RY95 (\\cite{RY1995}) succeeded in reproducing peak separations of the gamma ray pulsars, together with the radio-peak to gamma-ray peak relative phase offset, while simultaneously restricting pulsar inclination and viewing angle. \\cite{DRH2004} describe how light flight time delays can be responsible for asymmetries in pulse profiles. DR03 \\citep{DR2003} developed a `Two Pole Caustic' model (TPC), which can produce double peaked (and to a more limited extent, single peak) light curves. \nWhile certain restrictive assumptions are sometimes necessary and justifiable, we have designed a general method tailored specifically at restricting unconstrained pulsar parameters. We effectively carry out a large scale computational search through pulsar parameter space to constrain pulsar parameters. The aim was to create a methodology of testing and refining hypotheses of how pulsars work. In this paper, we apply the model to pulsed optical emission from isolated neutron stars.\n\n\nFollowing a successful two years of operation, the Fermi Gamma-ray observatory has identified over fifty gamma-ray pulsars (\\cite{A2010b} \\& \\cite{SP2010}), of which 21 are radio-quiet gamma-ray pulsars and surprisingly, 11 are millisecond pulsars (\\cite{A2009}, \\cite{R2010}). From these observations, the polar-cap model has been effectively ruled out (\\cite{A2009}, \\cite{VHG2009}), although the polar cap will play a role in the initial particle acceleration. This leaves the slot-gap, outer-gap (extended outer gap for millisecond pulsars) and variations of the striped pulsar wind (\\cite{AS1979}, \\cite{CHR1986a}, \\cite{P2009} \\& \\cite{KSG2002}), as the main contenders for a theory of high-energy pulsar emission.\n\n\nIt has long been recognised that the polarisation of pulsar radiation gives an indication of the geometry of the emission zone. \\cite{TG2010} showed recently that the core and conal components of the radio emission from PSRs B1839+09, B1916+14 and B2111+46 were at low altitudes $<$ 5\\% of the light cylinder radius. \\cite{WW2009} mapped the magnetopshere of PSR B1055-52, indicating that the emission height was around $\\sim$700 km above the neutron star's surface. In a more general way, \\cite{TGG2010} have looked the geometry for radio emission assuming curvature radiation. Our objective in this paper, is to investigate the use of optical polarisation data to map the regions within the magnetosphere at which the pulsar emission can originate. The advantage of optical radiation is that its emission mechanism, incoherent synchrotron, has a simple relationship between its polarisation profile and the underlying geometry. It is also the only region where we have decent high-energy polarisation data, and such data is highly sensitive to the local emission physics conditions.\n\n\n\\section[]{Description of the Model}\n\nHistorically, inverse models have focused on restricting various geometrical parameters, such as pulsar inclination angle, viewing angle and\/or the specific emission site responsible for the observed emission. These variables are assumed unknown initially and define a parameter search space, where the aim is to restrict them by comparison with observation. However, all such models use rather restrictive initial conditions to constrain these parameters, for example, tangential beaming from specific subsections of the magnetosphere using predefined emissivity functions. \nBy removing these restrictive initial conditions and using modern compute power, we believe that the domain of applicability of inverse models can be increased. For example, we can apply the inverse approach to constrain and test emission properties, as well as geometric properties of pulsar high energy emission. Effectively, one considers the various emission parameters as being undefined and adds these parameters to the overall search space to be constrained. Extending inverse modelling in this way may provide a new way of constraining pulsar variables, and of approaching the pulsar problem in general. We describe in this section the overall approach we have designed and the specific inverse method we have developed to investigate pulsed nonthermal optical emission from pulsars.\n\n\nOur approach is composed of a number of conceptually distinct steps, outlined in the flowchart in Figure~\\ref{flow.diagram}. The main steps are subdivided into physical (P), computational (C), statistical (S), and mapping (M) elements. Briefly, these steps constitute: (1) the application of a plausible physical model responsible for the emission, (2) the computational implementation of this model, resulting in the creation of phase resolved lightcurves ($\\xi(\\Phi,\\alpha,\\chi)$)\\footnote{Emission from a given pulsar is simulated and phase resolved ($\\Phi$) lightcurves produced for a range of viewing angles ($\\chi$). This process is repeated using different inclinations ($\\alpha$) for the magnetic axis (which is not in general constrained from observations). $\\xi$ can represent numerous observational properties, in this case the set of Stokes parameters I,Q,U,V.}, (3) the statistical comparison of the simulated lightcurves to observations, which results in a restriction of $(\\alpha,\\chi)$ parameter space enabling, (4) an `inverse mapping' into the magnetosphere, locating the regions of emission responsible the best fitting lightcurves in the first place. \\\\\n\\begin{figure}\n\\centering\n{\\includegraphics[scale=0.4]{flow_inverse.eps}}\n\\caption[Representation of `Inverse Mapping'\/`Search Algorithm' approach in flow diagram form]{\nRepresentation of `Inverse Mapping'\/`Search Algorithm' approach in flow diagram form. The approach\nis conceptually divided into physical, computational, statistical and mapping\/inverse mapping components.}\n\\label{flow.diagram} \n\\end{figure}\n\nPhase resolved Stokes parameters are produced for all possible values of $0\\deg \\le \\alpha \\le 180\\deg$ and $0\\deg \\le \\chi \\le 180\\deg$, in finite increments $\\Delta \\alpha$ and $\\Delta \\chi$, based on the assumption that a simple radiative emission process is occurring globally within the magnetosphere. The computational model divides the magnetosphere into a grid of points and the phase resolved angular distribution of photons from each point ($\\bm{R}$) is recorded as $\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$. The summation of $\\xi$ from all grid points creates a mapping from magnetospheric location to phase resolved observer profiles (as a function of $(\\alpha,\\chi)$). The statistical component selects the $\\xi(\\alpha,\\chi)$ which best fits the observations and $\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$ allows an `inverse' mapping of the photon paths back into the magnetosphere.\\\\\n\n\nThese steps constitute what we refer to as the `Inverse mapping' or search algorithm approach to isolating the regions within the magnetosphere which may be responsible for pulsed optical emission. The main steps in the process are as follows: \\\\\n\n\\newcounter{listno}\n\\begin{list}{\\bfseries\\upshape P--\\Roman{listno}}\n {\\usecounter{listno}\n \\setlength{\\rightmargin}{0.01\\columnwidth}\n \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item \nParticle motion (outside of accelerated regions) is dominated by the magnetic field, making the field an important element in any model $\\Rightarrow$ we propose a magnetic field structure in the form of a retarded dipole, as the standard Deutsch formalism, \\cite{ML1999}. \n\n\\item It is plausible that synchrotron radiation is responsible for pulsed optical emission, with the observed power law spectral form indicating that the underlying particle spectrum is also of this form. Using the premise of first order approximations, together with no a-priori assumptions regarding favourable emission locations, we assume that each point in the magnetosphere can emit synchrotron radiation from an underlying particle power law, where the total number density of particles at any one point is equivalent to the Goldreich-Julian density ($n_{GJ}\\propto \\Omega \\cdot B_z(r)$).\n\n\n\\item Each point $\\bm{R}$ is allowed to radiate synchrotron radiation, which is characterised by the properties of the particle population at that point - essentially the energy distribution of the particles (a power law from {\\bf P-II} above), and the pitch angle distribution (hereafter PAD) of the particles. The pitch angle distribution is perhaps the least well known parameter of pulsar emission models. Since the PAD is not well constrained, we assume various forms for the PAD and analyse the resultant lightcurves to choose which PADs may best represent the emission. For all PADs, we assume that the emitting particles are symmetrically distributed about the magnetic axis, with a cutoff occurring at a specific pitch angle, beyond which no emission occurs, {\\small PAD}$_{co}$.\n\\end{list}\n\n\\newcounter{listno1}\n\\begin{list}{\\bfseries\\upshape C--\\Roman{listno1}}\n{\\usecounter{listno1}\n \\setlength{\\rightmargin}{0.01\\columnwidth}\n \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Computationally, we represent the 3D volume of the magnetosphere as a grid. Each point on the grid is specified by local parameters such as $(\\bm{R},\\bm{B},n_{GJ},\\bm{v}_{co})$, where $\\bm{R}$, $\\bm{B}$, $n_{GJ}$, and $\\bm{v}_{co}$ are the spatial coordinate, the magnetic field, the local particle density and the corotational velocity respectively, at that point. The grid itself, is based on a spherical geometry, consisting of a series of fixed concentric spheres. The origin of the spheres is located at the centre of the neutron star, with points located such that $\\bm{R}=(R,\\theta,\\phi)$ in usual spherical coordinates.\n\n\\item At each grid point ($\\bm{R}$), a power law distribution of particles, having a specific PAD, emits synchrotron radiation into a certain section of sky of solid angle $d\\Omega$. Relativistic aberration, beaming and doppler shifts will alter the direction of emission, the irradiated solid angle and frequency\nof radiation, as seen by a sky based observer compared to an observer in the particle rest frame. These effects are considered and the emission from each point $\\bm{R}$, as seen by a sky based observer, is recorded in increments of $\\Delta \\chi = 1\\deg$, for the range of $0\\deg \\le \\chi \\le 180\\deg$.\n\n\\item Since one aim of this approach is to restrict both the inclination angle $\\alpha$ {\\it and} the viewing angle $\\chi$, step {\\bf C-II} is repeated for all inclination angles with $0\\deg \\le \\alpha \\le 180\\deg$, in discrete increments of $\\Delta \\alpha$.\n\\end{list}\n\n\\newcounter{listno2}\n\\begin{list}{\\bfseries\\upshape S--\\Roman{listno2}}\n {\\usecounter{listno2}\n \\setlength{\\rightmargin}{0.01\\columnwidth}\n \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Steps {\\bf C-II} and {\\bf C-III} create a phase space $(\\alpha,\\chi)$, where each point in this phase space contains phase resolved lightcurves of the form:\n\\begin{equation}\n\\xi(\\bm{R}, \\Phi) = \\xi(R,\\theta,\\phi,\\Phi),\n\\end{equation}\nwith $0\\deg \\le \\chi \\le 180\\deg$, $0\\deg \\le \\alpha \\le 180\\deg$, and where $\\xi$ represents each of the four Stokes parameters $(I,Q,U,V)$. A $\\chi$-squared goodness of fit test is performed between each of the simulated Stokes parameters and the observed Stokes parameters, at each point in $(\\alpha,\\chi)$ phase space, to produce the best fitting $(\\alpha,\\chi)$ combination.\n\\end{list}\n\n\\newcounter{listno3}\n\\begin{list}{\\bfseries\\upshape M--\\Roman{listno3}}\n {\\usecounter{listno3}\n \\setlength{\\rightmargin}{0.01\\columnwidth}\n \\setlength{\\leftmargin}{0.07\\columnwidth}}\n\n\\item Knowing the best fitting $(\\alpha,\\chi)$ combination, $\\xi(R,\\theta,\\phi,\\Phi,\\alpha,\\chi)$ is used to inverse map the constituent photons back into the magnetosphere, thereby locating the origin of the emission, which creates the best fitting lightcurves ($\\xi$). The visualisation of these regions will hopefully yield an indication of the originating locations.\n\\end{list}\n\nIn the following sections, each of these steps in illustrated in more detail, applying the search algorithm to a simulated pulsar with Crab like parameters: ($P=33$ ms, $\\dot{P}=4.209\\times10^{-13}\\rm{ss^{-1}}$ and $B_{\\rm{surf}}=3.8 \\times 10^{12}$ G).\n\n\\section[]{Description of the Physical Model}\n\nIt is assumed in this modelling that at any point in the magnetosphere, radiation arises through synchrotron radiation from a power law spectrum of particles: the particle index, p, is fixed via the observed photon spectral index s, through the relation, s=(p+1)\/2, where:\n\n\\begin{equation}\nF_\\nu \\propto \\nu^{-s} \\quad \\Rightarrow \\quad N(E) \\propto E^{-p}.\n\\end{equation}\n\nWhen simulating emission over a range of frequencies (e.\\,g., for U, B, V bands), contributions are summed only from those particles whose energies, are such that they contribute significantly to emission at the {\\it observed} frequency $\\nu$.\n\n\\subsection{Synchrotron Radiation}\n\nSynchrotron radiation from a single particle, spiralling around a local magnetic field ($\\bmath B$), at a pitch angle $\\alpha$ (where $\\hat{n} \\cdot \\hat{B} = \\cos \\alpha$ with $\\hat{n}$ the instantaneous velocity of the particle), produces a continuum spectrum of frequencies with a peak emissivity close to a critical frequency, $f_c$ where:\n\n\\begin{equation}\nf_c(E,B,\\theta) = \\frac{3}{2} f_B \\sin \\theta \\left( \\frac{E}{E_o} \\right)^2,\n\\label{fc}\n\\end{equation}\n\nwith $f_B=e B\/2\\pi m$, being the particle cyclotron frequency. To first order, the spectrum exhibits a quadratic rise and exponential drop as the frequency passes through the critical value. At lower frequencies, the spectrum consists of discrete harmonics of the fundamental (equation~\\ref{fundamental_freq}) frequency below which no emission is possible.\n\n\\begin{equation}\nf_f = \\frac{f_B}{\\sin^2 \\theta} \\frac{E_o}{E} \\label{fundamental_freq}.\n\\end{equation}\n\nThe emitted radiation is in general, elliptically polarised with the principle axes of the polarisation ellipse aligned parallel and perpendicular to the projection of $\\bmath{B}$ on the plane transverse to the emission direction, $\\hat{l}$ (see Figure~\\ref{particle_geometry}). The angular dependence of the polarised emissivity depends strongly on the angle between the line of sight ($\\hat{l}$) and $\\hat{n}$, where we write, $\\hat{n} \\cdot \\hat{l} = \\cos \\psi$ with $sgn(\\psi)=\\alpha-\\theta$. The case $\\psi = 0$, results in linear polarisation perpendicular to $\\bmath{B}$, $\\psi > 0$ denotes elliptical polarisation having the major axis perpendicular to the projection of $\\bmath{B}$, and $\\psi < 0$ indicates right handed polarisation and $\\psi > 0$ left handed.\n\n\\begin{figure}\n\\center\n{\\includegraphics[scale=0.4]{emission_source_geometry-1.eps}} \n\\caption{Important source and observer geometric parameters relating to synchrotron emission.}\n\\label{particle_geometry}\n\\end{figure}\n\nFor a population of particles, one must integrate emission over the velocity distribution, which amounts to an integration over the pitch angle distribution and over the range of particle energies present. Results for a power law spectrum of particles ($N(E) \\propto E^{-\\gamma}$) are well known; the emitted radiation spectrum is also a power law, with the photon spectral index related to the particle index, where $\\alpha=(\\gamma+1)\/2$ and $\\gamma < 1\/3$, in order to maintain a finite integral. This monotonic spectrum is of course a special case, which results from integrating over a population having infinitely wide energy bounds. So that practically, the formalism can be applied only at frequencies which are unaffected by any issues related to the nature of finite particle energy bounds. \n\n\\subsection{A Truncated Power Law Spectrum of Particles}\n\nIn any real situation, one is dealing with particles having finite upper and lower energy bounds, which alter the spectrum significantly from the classical monotonic scenario. For the purposes of this work, the angle dependant and integrated polarisation properties of synchrotron emission from a truncated power law spectrum of particles, with $N(E) \\propto E^{-\\gamma}$ is considered. In this spectrum, any given particle's energy is confined between lower ($E_1$) and upper ($E_2$) limits, such that $E_1 \\le E \\le E_2$ having a (potentially) isotropic, axially symmetric (about $\\bm{B}$) pitch angle distribution. Such a distribution is rather general and has been dealt with in detail by GLW74 \\citep{GLW1974}. They derive emissivity properties using the polarisation tensor for light, which represents the cross correlated quadratic components of the electromagnetic wave field as a rank 2 tensor, $\\rho_{\\alpha\\beta}$ (where $\\alpha$ and $\\beta$ represent the component directions in the transverse plane). The tensor $\\rho_{\\alpha\\beta}$, contains the complete polarisation content of the radiation and can therefore be related to the more commonly used Stokes parameters, as illustrated in equation~\\ref{polarisation_tensor}. For reference purposes, the underlying polarised emissivity properties are given here as equations~\\ref{stokes}, reproduced from GLW74.\n\n\\begin{equation}\n\\left.\n\\parbox{0.5\\textwidth}{\n\\begin{eqnarray*}\nI & = & \\frac{A\\mu e^2c}{2\\sqrt{2}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)({\\nu}_H \\rm{sin} \\theta)^{(\\gamma+1)\/2} \\\\\n \t & &\\cdot\\:\\: \\nu^{-(\\gamma-1)\/2} \\left[ \\mathcal{J}_{(\\gamma+1)\/2} \\right]_{x_2}^{x_1} \\nonumber \\\\ \nQ & = & \\frac{A\\mu e^2c}{2\\sqrt{2}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)\n ({\\nu}_H \\rm{sin}\\theta)^{(\\gamma+1)\/2} \\\\\n & & \\cdot\\:\\: \\nu^{-(\\gamma-1)\/2}\\left[ \\mathcal{L}_{(\\gamma+1)\/2} \\right]_{x_2}^{x_1} \\nonumber \\\\\nU & = & 0 \\nonumber \\\\\nV & = & \\frac{A\\mu e^2c}{\\sqrt{3}}(\\frac{3}{2})^{\\gamma\/2}\\phi(\\theta)\n \\rm{cot} \\theta ({\\nu}_H \\rm{sin}\\theta)^{\\gamma\/2+1}\n \\nu^{-(\\gamma\/2)} \\nonumber \\\\\n & & \\times \\left[\\mathcal{R}_{(\\gamma\/2+1)}+\n (1+g(\\theta))(\\mathcal{L}_{\\gamma\/2} -\\frac{1}{2}\\mathcal{J}_{\\gamma\/2}) \n \\right]_{x_2}^{x_1} \\label{stokes}\n\\end{eqnarray*}}\n\\right. \n\\end{equation}\nwhere:\n\\begin{equation}\n\\mathcal{J}_{n}(x)=\\int_o^x \\xi^{n-2} F(\\xi) d\\xi, \\hs \\hs \\mathcal{L}_{n}(x)=\\int_o^x \\xi^{n-2} F_p(\\xi) d\\xi\n\\end{equation}\n\\begin{equation}\n\\mathcal{R}_{n}(x)=\\int_o^x \\xi^{n-2} F_s(\\xi) d\\xi, \\hs \\hs F(\\xi)= \\int_\\xi^\\infty K_{5\/3}(y)dy\n\\end{equation}\n\\begin{equation}\nF_p(\\xi)= \\xi K_{2\/3}(\\xi), \\hs \\hs F_s(\\xi)= \\xi K_{1\/3}(\\xi)\n\\end{equation}\nso that:\n\\begin{equation}\n\\mathcal{J}_{n} \\equiv \\int_o^\\infty \\xi^{n-2} F(\\xi) d\\xi \\hs \\rm{etc.,}\n\\end{equation}\nwhere the $K_\\alpha$, are modified Bessel functions and the variable $x \\equiv f\/f_c$, where $f_c$ is the critical frequency given by \\ref{fc}:\n\nThe effect of the $truncated$ power law energy spectrum is seen in the fact that the functions modifying the observed Stokes parameters (i.e., $\\mathcal{J}_n(x), \\mathcal{L}_n(x)$ and $\\mathcal{R}_n(x)$), must be evaluated between upper ($x_1$) and lower ($x_2$) limits of the parameter `x', where $x=x(B,\\theta,E)$. Each polarised emissivity parameter is dependant on a functional combination of $\\phi(\\theta), f_B \\sin \\theta, f, \\gamma$, and this functionality is modulated by finite integrals over the particle population (having integrand `x').\n\n\nAs analysed in detail by GLW, having a finite particle energy range results in an emission spectrum with lower ($f_l$) and upper ($f_u$) frequency bounds. This spectrum possesses roughly two spectral breaks at frequencies $f_a$ and $f^{(i)}_b$, where $f_{lo} E_1$), and replacing the upper limit of integration with $x_1^{'}$, where $x_1^{'}=f\/f_c(E_1^{'})$. It is this replacement, which results in the augmented spectral index of $\\alpha=-\\gamma$ at frequencies below $f_a$.\n\nThe different energy cut-off effects delineate different regions in which different physical effects alter the emission characteristics. The effects are encoded in the integration x-factors, both in the form of $x_l$ and the extent of the range $(x_l,x_2)$, over which emission is allowed. Figure~\\ref{spectrum_with_corrxn_factors} shows the expected form of the emissivity for a truncated power law particle population emitting synchrotron radiation.\\footnote{We note that the photon indices quoted above are asymptotic values evaluated in the limit of `small', using power law expansion representations of the functions $\\mathcal{J}_{n}(x), \\mathcal{L}_{n}(x)$ and $\\mathcal{R}_{n}(x)$ derived by GLW74. The transition frequencies and photon indices are therefore asymptotic and indicative of the actual spectrum, which should transit smoothly through the transition frequencies and deviate smoothly from the asymptotic spectral indices.}\n\nIn our approach to modelling emission from a truncated power law particle population, we follow the GLW74 approach and describe the spectral variability in terms of correction factors, $C^i$, $i=1,2,4$ applied to the standard power law dependences (see Appendix \\ref{appendix_1}). We note that in limited circumstances, it is possible for $x_a>x^{(i)}_b$, so that the spectral emissivity has a lower and upper bound with a single internal transition frequency. This situation would arise when the upper and lower bounds are sufficiently close together. For completeness, we include the relevant correction factors in Appendix \\ref{appendix_1} (as this situation was not discussed in GLW74).\n\n\n\\subsection{Effect of source motion}\n\nThe description of the polarisation parameters above is valid when the magnetic field is stationary relative to an observer. In the case of emission from synchrotron radiating particles constrained to move along magnetic field lines in a pulsar magnetosphere, the underlying field structure is (relativistically) rotating. To describe the effect of the source motion on the observed polarised emission, we consider how a general Lorentz boost transforms the radiation fields. Also, to extract angular dependences and for frame transformations, it is convenient to use the polarisation tensor, $\\rho_{\\alpha\\beta}$, representation of light. \n\nIn our notation, a noninertial observer in the corotating frame $S$ (basis $O_{xyz}$), sees a stationary magnetic field $\\bm{B}$ and views this field along a direction $\\bm{\\hat{n}}$, such that $\\bm{B}_f \\cdot \\bm{\\hat{n}} \\propto \\cos \\psi$. The observer sees light from a particle, if $\\bm{\\hat{n}}$ is sufficiently close to $\\bm{\\tau}(t)$, where $\\bm{\\tau}$ describes the trajectory of the charged particle (see Figure~\\ref{particle_geometry}). The electromagnetic field of the light oscillates in a plane transverse to $\\bm{\\hat{n}}$ (the `observer plane' K), and may be expressed in component form along two mutually perpendicular axes $(\\bm{\\hat{i}}_1, \\bm{\\hat{i}}_2)$ in the plane K. Frame $S'$ observes the magnetic field $\\bm{B}_f$ to move at an arbitrary constant velocity $\\bm{\\beta}=\\bm{v}\/c$. An observer in $S'$ will therefore see light emitted at an aberrated direction $\\bm{\\hat{n}}'$ about the boosted field direction $\\bm{B}'$. The components of the electromagnetic wave will also be boosted ($\\bm{E}',\\bm{B}'$). A general Lorentz boost, where frame $S^{\\prime}$ moves at an arbitrary constant velocity $\\bm{\\beta}$ to frame $S$, is given as $\\Lambda^\\mu_\\nu$ (equation~\\ref{lorentzboost}) where a four-vector $\\vec{x}$ transforms as ${x'}^\\mu = {\\Lambda^{\\mu}}_\\nu x^\\nu$.\n\n\\begin{equation}\n\\label{lorentzboost}\n\\Lambda^\\mu_\\nu = \\left[ \\begin{array}{llll}\n \\gamma & -\\beta_x\\gamma & -\\beta_y\\gamma & -\\beta_z\\gamma \\\\\n -\\gamma\\beta_x & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_x}^2 & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_y & \\frac{\\gamma-1}{\\beta^2}\\beta_z\\beta_x \\\\\n -\\gamma\\beta_y & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_y & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_y}^2 & \\frac{\\gamma-1}{\\beta^2}\\beta_z\\beta_y \\\\\n -\\gamma\\beta_z & \\frac{\\gamma-1}{\\beta^2}\\beta_x\\beta_z & \\frac{\\gamma-1}{\\beta^2}\\beta_y\\beta_z & 1+\\frac{\\gamma-1}{\\beta^2}{\\beta_z}^2\\\\\n\\end{array} \\right] \n\\end{equation}\n\n\n\\noindent The electric and magnetic fields, transform as the components of the antisymmetric tensor $F_{\\mu\\nu}$, (equation~\\ref{antisym_eb}), so that ${F'}_{ik}={\\Lambda^{\\beta}}_k {\\Lambda^{\\alpha}}_iF^{\\alpha \\beta}$. The transformed wave four-vector ($\\bm{k'}$), electric ($\\bm{E'}$) and magnetic ($\\bm{B'}$) fields are given in equations~\\ref{E_trans} $-$ \\ref{k_trans}. \n\n\\begin{equation}\nF_{\\mu\\nu} = \\left[ \\begin{array}{cccc} \n 0 & -E_x & -E_y & -E_z \\\\\n E_x & 0 & B_z & -B_y \\\\\n E_y & -B_z & 0 & B_x \\\\\n E_z & B_y & B_x & 0 \n \\label{antisym_eb}\n \\end{array} \\right]\n\\end{equation} \n \n\\begin{eqnarray}\n\\bm{E}' & = & \\gamma(\\bm{E} + \\left[ \\bm{\\beta H} \\right]) \n - \\frac {\\gamma^2}{(\\gamma+1)}\\bm{\\beta} (\\bm{\\beta E}) \\label{E_trans}\\\\\n\\bm{B}' & = & \\gamma(\\bm{B} + \\left[ \\bm{\\beta B} \\right]) \n - \\frac{\\gamma^2}{(\\gamma+1)}\\bm{\\beta} (\\bm{\\beta B}) \\label{B_trans}\n\\end{eqnarray}\n\n\\begin{equation}\n{k'}_i = \\frac{\\omega}{c}\\gamma(1-\\bm{\\beta}\\bm{n})\n \\left\\{ \n 1,\\left( \\frac{\\bm{\\beta-\\hat{n}}}{1-(\\bm{\\beta}\\bm{n})}-\n \\frac{1}{(\\gamma+1)}\\frac{\\bm{\\beta}(\\bm{\\beta n})}{1-(\\bm{\\beta}\\bm{n})}\n \\right) \\right\\} \\nonumber\n\\end{equation}\n\n\\begin{equation}\n= \\left( \\frac{{\\omega}'}{c},\\frac{-{\\omega}'}{c}\\bm{\\hat{n}'} \\right) \\label{k_trans}\n\\end{equation}\n\n\nIt is now necessary to describe the observed intensity and polarisation in terms of quantities seen in the observers frame (S'). For this, the polarisation tensor representation ($\\rho_{\\alpha\\beta}$) of light is most useful, given by:\n\\begin{equation}\n\\label{polarisation_tensor1}\n\\rho_{\\alpha \\beta} = \\frac{cR^2}{4\\pi}\\fte_{\\,\\,\\alpha} {\\fte}^{\\,\\,\\,*}_{\\,\\,\\beta},\n\\end{equation}\n\\noindent where $\\fte_{\\,\\,\\alpha}$ and $\\fte_{\\,\\,\\beta}$ are two orthogonal components of the electromagnetic wave field. The polarisation tensor can also be expressed in terms of the more widely used Stokes parameters, as in equation~\\ref{polarisation_tensor}.\n\n\\begin{equation}\n\\label{polarisation_tensor}\n\\rho = \\left[ \\begin{array}{cc} \n \\rho_{11} & \\rho_{12} \\\\\n \\rho_{21} & \\rho_{22} \\\\\n \\end{array} \\right] \\propto \n \\left[ \\begin{array}{cc} \n \\frac{1}{2}(I+Q) & \\frac{1}{2}(U-iV) \\\\\n \\frac{1}{2}(U+iC) & \\frac{1}{2}(I-Q) \\\\\n \\end{array} \\right] \n\\end{equation}\n\n\n\\begin{figure*}\n\\center\n{\\includegraphics[scale=0.6]{nonrotating.eps}\n\\hfill\n\\includegraphics[scale=0.6]{rotating.eps}} \n\\caption{Effect of a relativistic source motion on the observed Stokes parameters, determined here for $B=1.9 \\times 10^{6}$ G, $\\beta=0.87$.}\n\\label{boosting_on_stokes}\n\\end{figure*}\n\nSince the fields of the wave remain transverse in any frame, it is apparent that the tensor $\\rho_{\\alpha\\beta}$ will remain two dimensional in any new frame. Also, because the transformation is real, the real and imaginary parts of the tensor transform independently. These properties allow us to boost and rotate the electric field components from plane K in frame S, to plane K' in frame S'. Before boosting $\\rho_{\\alpha\\beta}$, any reference to $\\bm{B}$ in the expression for $\\bm{E'}$ is removed, using the substitution $\\bm{B}=[\\uv{n}\\rm{E}]$. Also, a right handed orthonormal basis set (denoted $\\rm{O}_{j_1j_2j_3}$) is defined in the observers (S') plane (K'). The projection of the pulsars rotation axis ($\\bm{\\Omega}$) on the plane K' is chosen as a uniform reference direction $\\uv{j_1}$, so that the basis set designated as $\\rm{O}_{j_1j_2j_3}$, can be defined through equation~\\ref{j_basis}.\n\n\\begin{equation}\n\\bm{j}_1=\\frac{[\\bm{n}'\\uv{\\Omega}]}{\\mid[\\bm{n}'\\uv{\\Omega}]\\mid} ; \\quad \\bm{j}_2=[\\bm{n}'\\bm{j}_1]\n ; \\quad \\bm{j}_3=\\bm{n}' \\label{j_basis}\n\\end{equation}\n\n$\\uv{j_1}$ and $\\uv{j_2}$ are orthogonal and in the observers plane (K'), perpendicular to the boosted emission direction $\\uv{j_3}$. We now express $\\bm{E'} = \\bm{E_1}'\\uv{j_1} + \\bm{E_2}'\\uv{j_2}$, where $\\bm{E_1}' = (\\bm{E}' \\uv{j_1})$ and $\\bm{E_2}' = (\\bm{E}' \\uv{j_2})$. Since $\\bm{E}= iE_1 \\uv{i_1} + E_2 \\uv{i_2}$, it is possible to write $\\bm{E}'$ in the $\\rm{O}_{j_1j_2}$ basis explicitly in terms of $\\bm{E_1}$ and $\\bm{E_2}$ as follows:\n\n\\begin{eqnarray}\n({E}_1'\\bm{j}_1, \\, {E}_2'\\bm{j}_2) & = & (\\{iE_1x_{11}+E_2x_{21}\\}\\bm{j}_1, \\, \\nonumber \\\\\n & & \\{iE_1x_{12}+E_2x_{22}\\}\\bm{j}_2),\n\\label{transformed}\n\\end{eqnarray}\n\nwhere the x factors ($x_{11}$ etc.), are given by equation~\\ref{transformation_factors}, and are dot products between the $\\rm{O}_{j_1j_2}$ basis vectors and the $E_1$ and $E_2$ components of $\\bm{E}'$.\n\n\\begin{eqnarray}\n x_{\\alpha\\beta}=\\frac{1}{(1+(\\bm{\\beta}\\bm{n}'))}\\Bigg\\{ \\frac{\\bm{l}_\\alpha\\bm{j}_\\beta}{\\gamma}+ \\nonumber \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(\\bm{\\beta}\\bm{l}_\\alpha)\\left[ (\\bm{n}'\\bm{j}_\\beta) +\\gamma(\\bm{\\beta}\\bm{j}_\\beta)\\left( 1+\\frac{\\gamma}{(\\gamma+1)}(\\bm{\\beta}\\bm{n}') \\right) \\right] \\Bigg\\} \\label{transformation_factors}\n\\end{eqnarray}\n\nEquation~\\ref{transformed} expresses how the components of $\\bm{E}$ and $\\bm{B}$ in plane K, written in terms of the basis ($\\bm{\\hat{i}}_1, \\bm{\\hat{i}}2$), transform into components of $\\bm{E}'$ and $\\bm{B}'$ in plane $K'$, expressed along a basis $\\bm{\\hat{j}}_1, \\bm{\\hat{j}}_2$. Using equation~\\ref{polarisation_tensor}, the components of the emission-polarisation tensor ${\\rho'}_{\\alpha \\beta}$ in frame $S'$ can be determined as follows:\n\n\\begin{equation}\n \\begin{array}{lll} \n\\rho_{11} = \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,1}x_{11}x^{*}_{22} + \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,2}x_{21}x^*_{21}, \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\\\\n\\rho_{12} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x_{21}x^*_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{11}x^*_{12} \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: +i(\\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,2}x_{11}x^*_{22} - \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{21}x^*_{12}), \\\\\n\\\\\n\\rho_{21} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x^*_{21}x_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x^*_{11}x_{12} \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: +i(\\fte_{\\,\\,\\,1}^{\\,\\,\\,*}{\\fte}_{\\,\\,\\,2}x_{11}^*x_{22} - \\fte_{\\,\\,\\,2}^{\\,\\,\\,*}{\\fte}_{\\,\\,\\,1}x^*_{21}x_{12}), \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\\\\n\\\\\n\\rho_{22} = \\fte_{\\,\\,\\,2}{\\fte}^{\\,\\,\\,\\,*}_{\\,\\,\\,2}x_{22}x^*_{22} + \\fte_{\\,\\,\\,1}{\\fte}^{\\,\\,\\,*}_{\\,\\,\\,1}x_{12}x^*_{12}.\n \\end{array} \n\\end{equation}\n\n\nThe polarisation tensor can be related to the Stokes parameters in the standard way (see equation~\\ref{polarisation_tensor}), and using these relations, the Stokes parameters observed in frame $S'$, can be expressed in terms of the field components of ($\\bm{E}, \\bm{B}$), as calculated in frame $S$. In the same way, $(I',Q',U',V')$ in $S'$ can be related to $(I,Q,U,V)$ in frame $S$. Carrying out the algebra the results are as follows:\n\n\\noindent\n\\parbox{0.8\\columnwidth}{\n\\begin{eqnarray}\nI' & = & \\frac{1}{2} \\left\\{ (x_{11}^2+x_{12}^2)(I+Q)+(x_{21}^2+x_{22}^2)(I-Q) \\right\\} \\nonumber \\\\\nQ' & = & \\frac{1}{2} \\left\\{ (x_{11}^2-x_{12}^2)(I+Q)-(x_{22}^2-x_{21}^2)(I-Q) \\right\\} \\nonumber \\\\ \nU' & = & \\hspace{1em}\\frac{}{} (x_{12}x_{11})(I+Q)+(x_{22}x_{21})(I-Q) \\nonumber \\\\ \nV' & = & \\hspace{1.3em}(x_{22}x_{11}-x_{12}x_{21}) \\, V \\nonumber \n\\end{eqnarray}}\n\\hspace{0.3cm}\n\\parbox{0.05\\columnwidth}{$\\left.\n \\begin{array}{c} \n \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \n \\end{array}\n \\right\\} $} \n\\hfill \n\\parbox{0.05\\columnwidth}{\\begin{eqnarray}\\label{test-label} \\end{eqnarray}}\n\nA significant source motion will alter the observed polarisation, the intensity and magnitude of the Stokes parameters will change, and this will be accompanied by a Doppler shift in the observed frequency. The appearance of a non-zero Stokes U in the observer frame is not in itself significant, as the relative magnitude of Stokes Q and U, depend on the orientation of your reference axes in the observer plane. Figure~\\ref{boosting_on_stokes} shows the source and observer polarisation spectrum for a source magnetic field of $1.9\\times10^6$ G, with a source velocity of $\\beta=0.87$. \n\n\n\\section[4]{Description of the Computational Model}\n\\label{sec:4}\n\nThe above physical emission model is simulated within the pulsar's magnetospheric environment. The three dimensional magnetosphere is represented computationally in spherical polar coordinates, and it is assumed that emission arises from within the open volume of the magnetosphere, which is determined numerically. As such, emission is only simulated from those grid points lying in the open volume, where the grid itself is specified using coordinates $(r,\\theta,\\phi)$. The number of grid points on any given spherical surface is controlled by the parameters $N_\\theta$ and $N_\\phi$, (the number of $\\theta$ and $\\phi$ divisions). The parameter $N_r$ specifies the number of divisions in the radial direction, and the grid itself is devised in such a way that each grid point represents an equal volume of space, so that in essence, the discretisation of the magnetosphere is homogeneous with respect to volume. \n\nThe emission is calculated at all points in the open volume, which requires the determination of {\\it local} parameters. These include the magnetic field strength, $\\bm{B}$, the total number of particles at a specific point, $n_{GJ}$, the assumed pitch angle distribution, local co-rotational velocity and the range of particle gamma factors which contribute to observable emission at that particular point. All necessary dependencies are calculated at each grid point to create the local emission profile, which effectively determines how the polarised spectral emissivity varies across the extent of the local pitch angle distribution. Each profile is subject to the required frame transformations (described above), which determine the emissivity profile as seen by a stationary observer external to the rotating magnetosphere.\n\n\\begin{figure*}\n\\center\n{\\includegraphics[scale=0.4]{grid_mix_2.eps}} \n\\caption{Centre: The concentric spherical Cartesian grid imposed on the pulsar magnetosphere, showing points only in the open volume of an aligned rotator ($\\uv{\\Omega}\\cdot\\uv{\\mu}=1$). Right: At each individual location ($\\bm{R}$), we calculate the radiation emitted from the local particle distribution, taking the local physical conditions into account - e.g., $\\bm{B}, \\bm{v}_{co}$, the power law particle distribution, the specific pitch angle distribution etc. Left: Representation of how emission from a particle pitch angle distribution extends over a range of viewing angles ($\\chi$) and phase ($\\Phi$). Emission is recorded computationally into discrete bins of extent $\\Delta\\chi \\times \\Delta\\Phi$ (as described in section \\ref{sec:4}).}\n\\label{spherical_grid}\n\\end{figure*}\n\nThe pitch angle distribution at any open volume point, generates emission which extends over a finite solid angle. This solid angle of emission has a finite extent latitudinally and longitudinally with respect to an observer, which directly correspond to a range in observer viewing-angle, $\\chi$ and temporal phase, $\\Phi$, respectively. Discretising this observer space\\footnote{Representing $0\\deg \\le \\chi \\le 180\\deg$ and $0 \\le \\Phi \\le 1$ with step-sizes of $\\Delta \\chi$ and $\\Delta \\Phi$.} allows emission from any point to be recorded as a function of viewing angle and phase. For any given model pulsar (i.e., for a specific magnetic inclination, $\\alpha$), the code records the phase resolved polarimetry in the form of an array, denoted as $\\xi(R,\\theta,\\phi,\\Phi,\\alpha,\\chi)$ ($=\\xi(\\bm{R},\\Phi,\\alpha,\\chi)$). This means that for each grid point location, $\\bm{R}$, the intensity and phase resolved Stokes parameters for the range of viewing angles over which emission is seen, are recorded.\\footnote{The magnetosphere is of course rotating, so at any point in time, the correct phase of emission is determined taking light flight time, aberration and rotation into account.}\\\\\n\nThe indices in the $\\xi$ array are discrete and represents a record of the phase resolved Stokes parameters seen by observers from point $\\bm{R}$. Once all grid points have been sampled, the array $\\xi$ can be summed from all appropriate locations to form $\\xi(\\alpha, \\chi, \\Phi)$\\footnote{Represented in Figure~\\ref{flow.diagram} as the array $\\xi(\\alpha_A,\\chi_B,\\Phi)$.} - i.e., the phase resolved Stokes parameters observed from the entire magnetosphere for a pulsar inclination of $\\alpha$. \\\\\n\nA single run of this code will create files which contain $\\xi(\\alpha,\\chi,\\Phi)$ for a single, specific pulsar inclination $\\alpha$. The search algorithm approach attempts to restrict $\\alpha$ and $\\chi$ by simulating emission for the range of all {\\it possible} $\\alpha$ values\\footnote{By explicit assumption, there is no a priori restriction on $\\alpha$ (which is generally the case, as observations constrain neither $\\alpha$ nor $\\chi$).} ($0\\deg \\le \\alpha \\le 180\\deg$), and subsequently selecting those simulated lightcurves which best fit the observational data (which should correspond to a unique combination of $\\alpha$ and $\\chi$). To generate lightcurves for different possible inclinations of a given pulsar, $\\alpha$ is varied from $0\\deg \\le \\alpha \\le 180\\deg$ in discrete steps $\\Delta \\alpha$. This generates a large pool of phase resolved model lightcurves, each dependant on a unique combination of $\\alpha$ and $\\chi$ - for example, with $\\Delta \\alpha = 10\\deg$ and $\\Delta\\chi=1\\deg$, one generates $180\\times18= 3240$ distinct lightcurves ($\\times 4$ when polarimetry is considered). \n\n\n\n\n\\section{Results}\n\nThe search algorithm approach was partly motivated on the premise that any simulated Stokes parameters $(\\xi)$, would be a variable function of the phase space $(\\alpha,\\chi,$ {\\small PAD}, {\\small PAD}$_{co}$). The simulated Stokes parameters vary as a smooth function of all phase space variables $\\xi(\\alpha,\\chi,$ {\\small PAD}, {\\small PAD}$_{co}$\\footnote{PAD refers to the pitch angle distribution which can either be isotropic or have a Gaussian structure; {\\small PAD}$_{co}$ defines the maximum pitch angle.} ), thereby allowing for the selection of a best fitting parameter set.\nThis being the case, one can restrict the pulsar geometry ($(\\alpha,\\chi)$ combination), which best fits observations and subsequently carry out the inverse mapping step to locate the associated source region within the pulsar magnetosphere.\\\\\n\nSimulated Stokes parameters are by definition dependent on all phase space variables $(\\alpha,\\chi,$ {\\small PAD},{\\small PAD}$_{co}$). As described in section \\ref{sec:4}, this phase space is sampled discretely, thereby generating a large amount of model lightcurves. We find that the simulated Stokes parameters ($\\xi$), vary smoothly as a function of this parameter space, showing systematic behaviour as we vary the observed energy band, viewing angle $\\chi$, pulsar inclination $\\alpha$, and particle pitch angle distributions. We discuss initially, the form of the simulated Stokes parameters, how these vary as a function of parameter space and how this variation can be used to select a best fitting parameter combination (the `search algorithm'). We subsequently describe our restrictions on pulsar geometry ($(\\alpha,\\chi)$ combination), and discuss the inverse mapping step which locates the associated source region within the pulsar magnetosphere.\\\\\n\nWe note here, that our simulations resolve pulsar phase into 200 bins, and we subsequently quote the phase extent of a single pulsar cycle as going from 0 $\\rightarrow$ 1. As choice of a reference phase is arbitrary, we define phase 0 to be the phase at which the magnetic pole associated with the quoted ${\\alpha}$ value would be seen by an external observer. From symmetry, phase 0.5 is the phase at which the magnetic pole at ${\\alpha=180\\deg-\\alpha}$ would be seen. We refer to the magnetic pole at inclination $\\alpha$ as pole 1 and the magnetic pole at $180\\deg-\\alpha$ as pole 2.\n\n\\subsection{Form and variation of simulated Stokes parameters}\n\nThe search algorithm method, in sampling such a large parameter space, inherently creates a large amount of data. It is instructive initially to discuss the overall form of the simulated Stokes parameters and to describe how they vary as a function of the phase space parameters $\\alpha, \\chi,$ {\\small PAD}, and {\\small PAD}$_{co}$. Some useful systematic trends are evident which will be discussed.\n\n\\subsubsection{{\\small PAD}$\\:$and {\\small PAD}$_{co}$}\n\nIt is found that similar trends exist for the variations of $\\xi$ between the different {\\small PAD } combinations (i.e., each of the {\\small PAD } functions based on isotropic, circular, linear and Gaussian profiles all have similar $\\xi(\\alpha,\\chi)$). We find that varying the {\\small PAD}$_{co}$ has a larger effect on the polarisation than varying the {\\small PAD} itself. The general trend is for larger {\\small PAD}$_{co}$ values to generate broader peaks, but the peaks are located at the same phase locations so that the overall variation of $\\xi(\\alpha,\\chi)$ is similar for all {\\small PAD} s. Given this degenerate behaviour, we will focus our analysis on the isotropic pitch angle distribution, which should reflect the overall general trends in the variation of $\\xi(\\alpha,\\chi)$.\n\n\\subsubsection{Stokes I: Single and Double Peaks}\n\nWe have found that simulated lightcurves only have a single peak for any inclination $\\alpha \\lesssim 50\\deg$, but a double peak profile is seen to emerge for $\\alpha \\gtrsim 50\\deg$. The intensity of the secondary peak increases as $\\alpha \\rightarrow 90\\deg$, where the main:secondary peak ratio varies smoothly with viewing angle. This allows for an immediate restriction in terms of predicted inclination and viewing angle for single and double peaked pulsar profiles.\\\\\n\n\n\n \n\n\\begin{figure*}\n\\vspace{-10em}\n\\centering\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_70B.eps}\n\\end{minipage}\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_80B.eps}\n\\end{minipage}\n\\begin{minipage}{0.49\\linewidth}\n\\hspace{8em}\n\\includegraphics[viewport=110 300 500 690, width=1.0\\linewidth]{StokesI_90B.eps}\n\\end{minipage}\n\\vspace{14em}\n\\caption[Primary results 4]{Relative intensity (Stokes I) at all viewing angles (V. A.) for successive pulsar inclinations. Parameter sets are (clockwise from top left) $(\\alpha)= 70\\deg, 80\\deg, 90\\deg$ with ({\\small PAD}, {\\small PAD}$_{co}$)$=$(isotropic, $20\\deg$) in each case. The secondary peak increases its intensity and viewing angle extent, as $\\alpha \\rightarrow 90\\deg$. At $\\alpha=90\\deg$, the `secondary peak' is of equal intensity to the main peak. In terms of restricting parameter space, this evolution shows that the observed main:secondary peak ratio of $\\bm{ \\sim$ $\\bm 0.3}$ (for the Crab pulsar), can only be reproduced for a restricted range of viewing angles (for $\\bm{\\Delta \\chi \\sim 5\\deg}$ about a central $\\bm \\chi_o$) for $\\bm{ \\alpha \\gtrsim 60\\deg}$}.\n\\label{3d-peak2}\n\\end{figure*}\n\n\n\n\n\n\nUsing $\\xi$=I$(\\bm{R},\\alpha,\\chi,\\Phi)$, it is found that the main emission contribution to the secondary peak, comes from pole 2 if $ \\chi\\lesssim90\\deg$, and from pole 1 if $ \\chi\\gtrsim 90\\deg$, indicating that as $ \\alpha\\gtrsim 50\\deg$, the secondary peak is formed from the magnetic pole {\\it opposite} the pole contributing to the main peak. Figure~\\ref{10} shows some phase resolved plots of the I(pole 1):I(pole 2) ratios for an inclination of $\\alpha=80\\deg$, at different viewing angles, which illustrates the relative contributions of emission over each pole to the final integrated lightcurve.\\\\\n\n\\begin{figure}\n\\centering\n{\\includegraphics[scale=0.33,angle=-90,clip]{ratios_80_50.eps}}\n\\hfill\n{\\includegraphics[scale=0.33,angle=-90,clip]{ratios_80_130.eps}}\n\\caption[Primary results 6]{Intensity observed for $(\\alpha)=80\\deg$ showing relative relative flux contribution of emission from pole 1 and pole 2 for different viewing angles. Here pole 1 refers to the magnetospheric regions above pole 1 and not just to the polar cap regions. Results illustrate that, depending on whether $\\chi\\gtlt90\\deg$, different poles will contribute to main and interpulse emission.}\n\\label{10}\n\\end{figure}\n\n\\subsection{General Observational Restrictions}\n\nCertain features of Crab pulsar emission are clear and distinct and any successful modelling of this object should be able to recreate at least some of these main features:\n\n\\newcounter{listno5}\n\\begin{list}{ [\\alph{listno5}] }\n {\\usecounter{listno5}\n {\\setlength{\\rightmargin}{0.1\\columnwidth}}\n {\\setlength{\\leftmargin}{0.1\\columnwidth}}}\n\n\\item {\\bf Double peaks:} The Crab pulsar exhibits a double peaked structure at all wavelengths. Only simulated dipole inclinations with $\\alpha \\gtrsim 50\\deg$ are capable of producing double peak emission. This is our first restriction of phase space.\\\\\n\n\\item {\\bf Relative peak intensity and bridge emission:} The double peaked structure is accompanied by the requirements of explaining the relative intensity of the peaks. A secondary peak whose intensity and extent of visibility (range of $\\chi$ values over which it is visible) is a (monotonic) function of inclination $\\alpha$, appears within the simulations for $\\alpha \\gtrsim 50\\deg$. The ratio of main:secondary peak intensity of 1:0.3, is possible for a narrow range of $\\chi$ associated with any $\\alpha \\gtrsim 60\\deg$. This range of $\\chi$ extends no more than $\\sim 2.5\\deg$ about a central $\\chi=\\chi_o$. We find that $\\chi_o$ is a smooth function of pulsar inclination ($\\chi_o=\\chi_o(\\alpha)$), $\\chi_o$ decreasing steadily as $\\alpha$ increases.\\footnote{$\\chi_o \\gtrsim 80\\deg$ for $\\alpha = 60\\deg$, $\\chi_o \\sim 47\\deg$ for $\\alpha=80\\deg$, as can be seen in Figure~\\ref{3d-peak2}.} Our second restriction can be summarised as follows: only for $\\alpha \\gtrsim 60\\deg$ and for viewing angles $\\Delta \\chi \\sim 2.5\\deg$ about $\\chi_o=\\chi_o(\\alpha)$, are simulated lightcurves able to reproduce the main:secondary peak ratio of $\\sim$ 1:0.3. Bridge emission is also a function of $\\alpha$ where the large bridge emission seen at lower inclinations ($\\alpha$), tends to disappear for the more orthogonal rotators. \\\\\n\n\n\\item {\\bf Peak phase separation:} The Crab pulsar's peak separation is $\\sim 0.4$ in phase. Results indicate that the simulated model is unable to produce a double peak structure separated by less than $0.4$ in phase, where in fact most viewing angles have peak separations of 0.5. In hindsight, this is a result of the inherent symmetry of the physical model, as it simulates emission from the open volume of both poles of a symmetric dipolar magnetic field structure. The influence of relativistic beaming can lead to asymmetries in pulse structures, so one would sensibly infer that a global emission model viewed at $\\alpha\\ne 90\\deg, \\chi \\ne 90\\deg$ should subsequently exhibit non-symmetric behaviour. It seems therefore, that the explanation for the symmetry in these model results, most likely lies in the averaging of emission over such a large volume. The large width of the peaks is also most likely due to the large volume of emission. It is hoped that the inclusion of more general physical constraints into the model, should lead to a removal of the symmetry.\\\\\n\n\\end{list}\n\n\\subsection{Inverse Mapping - A two pole emitter?}\n\n\nOur results given that the model results are capable of more successfully reproducing lightcurve flux profiles than the associated polarisation \nprofiles, inverse mapping results are given for a selected ($\\alpha,\\chi$) combination, which compares most favourably to the observed \ndouble peak structure of the Crab pulsar and the observed QU relationship. A value of $(\\alpha,\\chi)=(70\\deg,45\\deg)$ is chosen \nfor ({\\small PAD}, {\\small PAD}$_{co}$)=(isotropic, $20\\deg$), these values were based on a best fit in the $\\chi^2$ sense between the simulated data and the \\cite{SJDP1988} \nobservations. In Figure \\ref{Phase_Plots} we show the intensity and degree of linear and circular polarisation as a function of phase. We note that the intensity distribution is wider than normal observed - see for example \\cite{2009MNRAS.397..103S} this can be explained by our initial approach of filling the open field region with radiating particles. \\cite{2009MNRAS.397..103S} also showed an increase in the degree of linear polarisation happening prior to the main pulse which we see although not at exactly the same phase relationship. From Figure \\ref{PA_Plots} we see the expected swing in polarisation angle around the two peaks and QU plots morphologically similar to \\cite{2009MNRAS.397..103S} and \\cite{S1986}. Our preferred values of $\\alpha$ and $\\chi$ are broadly consistent with \\cite{2004ApJ...601..479N}, which provided a robust estimate of the pulsar inclination angle.\n\nThe inverse mapping element of the search algorithm allows us to decompose light-curves, with the option of selecting different phase resolved regions (on our current resolution phase is divided into 200 bins), which we can subsequently retrace back into the \\mag, isolating the originating locations of the component photons. This will localise the magnetospheric regions which contribute to emission at a particular phase, one of the main motivations for the search algorithm.\\\\\n\nWe try and present the results of the inverse mapping in an interpretable sense here in 2 and 3 dimensions, although the task is not an easy one. The selected $(\\alpha,\\chi,${\\small PAD},{\\small PAD}$_{co}$) combination chosen to illustrate the inverse mapping is $(70\\deg, 52\\deg,\\:$isotropic, $20\\deg$), which has a main peak at phase $\\sim 0.1$ and an interpulse at phase $\\sim 0.6$. We choose specific phase regions ($\\Delta\\Phi=5$) centred on the arrival phase of the main peak and the inter-pulse, so as to obtain a view of the magnetospheric regions which contribute to `peak' intensity.We note however on the basis of Figure \\ref{10} that the fit is a smoothly varying function of $\\alpha,\\chi$,{\\small PAD} and {\\small PAD}$_{co}$. In Figure \\ref{POD1_Plot} we show the effect of changing the {\\small PAD}$_{co}$ showing intensity vs phase for {\\small PAD}$_{co}$ values of 1$\\deg$ and 5$\\deg$. \\\\ \n\n\\begin{figure*}\n\\centering\n{\\includegraphics[scale=0.65,angle=0,clip]{INT-both.eps}}\n\n\\hfill\n\n\\caption[Phase Plots]{Phase plots for 70$\\deg$ inclination and viewing angles 45$\\deg$ and 20$\\deg$. We show the total optical intensity, and degree of linear\/circular polarisation. }\n\\label{Phase_Plots}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\n\n{\\includegraphics[scale=0.53,angle=0]{PA-both.eps}}\n\n\\caption[PA Plots]{Illustrative polarisation angle vs phase and QU plot for 70$\\deg$ inclination and viewing angles 45$\\deg$ and 20$\\deg$. }\n\\label{PA_Plots}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{minipage}[a]{1.0\\linewidth}\n\\centering\n\\includegraphics[scale=0.6,angle=0]{POD_john.eps}\n\\caption[POD1 Plots]{Phase and intensity relationship for 70$\\deg$ inclination and viewing angle 45$\\deg$ for two ${\\small PAD}$ of 1 and 5 showing the ragged nature of the intensity for the lower pitch angles. }\n\\label{fig:figure1}\n\\end{minipage}\n\\hspace{0.5cm}\n\\label{POD1_Plot}\n\\end{figure*}\n\n\n\n\n\n\nCarrying out the inverse mapping step for $(\\alpha,\\chi)=(70\\deg,45\\deg)$, Figure~\\ref{main-peak-3d} presents a 3D colour coded intensity map of the originating location of emitted radiation seen by an observer at $\\chi=52\\deg$ relative to the model pulsar. Figure~\\ref{main-peak-3d} (left) shows the origin of emission which composes the main peak and Figure~\\ref{main-peak-3d} (right) shows the origin of emission composing the interpulse (as an aid to perspective, the circles define the light cylinder distance and the brownish field lines are the boundary of the open volume\/closed volume region within the pulsar magnetosphere). Immediate points of note, are that the main peak is primarily composed of emission from one pole (which we term pole 1), while the interpulse is composed of emission from the opposite pole (here defined as pole 2). Also, the most intense emission is located in tightly constrained volumes in the inner magnetosphere.\\\\ \n\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.31,clip]{FinalImageB.eps}\n\\caption[3D representation of inverse mapped regions for main peak emission]{Three dimensional representation of the regions contributing to {\\bf main peak} (left) and {\\bf secondary peak} (right) emission for a model (Crab) pulsar with parameters ($\\alpha,\\chi$,{\\small PAD},{\\small PAD}$_{co}$)=($80\\deg,25\\deg,\\:$isotropic, $20\\deg$). The intensity scale shows the emission contributions at the regions mapped. For reference, the circles define the light cylinder boundary and the light brown lines are the open volume\/closed volume boundary. Emission is seen to be concentrated in the lower magnetosphere from a single pole.}\n\\label{main-peak-3d}\n\\end{figure*}\n\nIt is immediately apparent that these results show localised emission regions and clearly indicate a two pole emission model. Figure~\\ref{main-peak-3d} illustrates rather dramatically, that the first order assumption of a global truncated power law synchrotron emission model (even for a relatively wide pitch angle distribution of $20\\deg$) can produce a rather limited spatial extent of maximum emission within the pulsar magnetosphere, while matching certain features observed in the Crab pulsar profile. The main location of emission is well away from the standard outer gap location and closer to the polar cap centre, but a closer analysis (described subsequently) yields that the emission is at distances of $\\sim 0.2R_{LC}$ from the polar cap, further from the polar cap surface than polar cap models generally estimate. \\\\\n \n\nIn an independent analysis, we attempt to localise relative emission contribution as a function of magnetospheric location. We subdivide the magnetospheric volume into concentric spherical shells of width $\\Delta\\rho$ ($\\rho$ being the spherical radius), and also into coaxial cylindrical shells of thickness $\\Delta r$ ($r$ in cylindrical coordinates). We integrate emission from all points within these shells to obtain a radial profile (spherically and cylindrically respectively) of absolute contributions to the total emission. The spherical profile is presented in the left hand panels of Figures~\\ref{13} and \\ref{14} with the cylindrical contributions shown on the right. Figure~\\ref{13} shows emission contributions to the main emission peak at phase $\\sim 0.1$, Figure.~\\ref{14} shows emission contributions to the secondary peak at phase $\\sim 0.6$. The top panels in these figures show the total emission from over both magnetospheric poles, the middle panels show emission regions associated with pole 1\\footnote{Pole 1 corresponds to the magnetospheric region above the \nmagnetic pole at an inclination $\\alpha$} only and the bottom panels show emission from pole 2\\footnote{Pole 2 refers to emission\nassociated with the pole at inclination $180\\deg + \\alpha$.} only.\\\\\n\nFrom the previous figures, certain trends are apparent. Pole 1 emission dominates for the main pulse and pole 2 emission for the secondary, each pole having distinct emissivity trends, both radially (cylindrically) and spherically. The poles dominating emission tend to have their maximum contributions located at very small spherical and cylindrical ($\\sim 0.2 R_{LC}$) radii, whereas the poles which contribute less to the total emission are located further from the star. The high peak in the emissivity, may be due to the influence of higher local charge densities lower down in the magnetosphere. Generally, the collective interpretation is that the main contribution to emission is very localised closer to the neutron star surface\\footnote{The poles contributing less to the emission of a given peak also seem to have their emission localised, but at a greater distance from the star.} than to the light cylinder. This tends to agree with models attempting to explain optical emission through a localised effect such as the slot gap. Indeed our results are consistent with the emission altitude predicted by \\cite{2010MNRAS.405..509D}. Furthermore we note that the slot-gap models primarily localise emission in the transverse (co-latitude) direction whereas we also restrict emission in altitude.\n\n \n\\subsection{Conclusion}\n\nOptical polarisation studies can be used to determine the local geometry of the emission region. In particular from this work we have\n\\begin{itemize}\n\\item A simple synchrotron model for the emission gives reasonable agreement with observations.\n\\item A prediction that the emission is low in the magnetosphere at an altitude in the region 30-40 stellar radii, that is away from both the polar cap and likely outer gap regions. \n\\item The linear polarisation peaks on the rising edge of the main pulse, consistent with the findings of \\cite{SJDP1988,2009MNRAS.397..103S}.\n\\item The radiation for the most part is circularly polarised with an interplay between linear and circular polarisation on the rising edge of the main pulse. On this basis observations of pulsed circular polarisation from optical pulsars would provide a significant geometrical restrict on pulsar parameters although we accept more detail work is needed of more realistic emission zones. \n\\item Pulse widths which are significantly greater than those observed although this probably stems from our requirement to fill to open field region with an emitting plasma.\n\\item Due the inherent symmetry of this model we do not see any significant bridge emission at the preferred orientation and viewing angle. \n\n\n\\end{itemize}\n\nOur future work will entail restricting emissions to localities around the last open field lines thereby removing the inherent symmetry existing within the model. In this way we hope to able to make firmer predictions in relation to the optical emission specifically and more generally comment on correlations between the optical and radio emission \\cite{2009MNRAS.397..103S}. As well as the emission location and pulsar orientation our approach can also restrict the {\\small PAD} and {\\small PAD}$_{co}$. We also intend to make more detailed comparison between these predictions and measurements of the pulsed circular polarisation. To-date there have been no measurements of optical circular polarisation from any pulsar. A new instrument, the Galway Astronomical Stokes Polarimeter (GASP), \\cite{2010EPJWC...505003K} has this capability and observations are planned to observe the Crab pulsar in late 2011.\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank the Higher Education Authority Programme for Research in Third Level Institutions (PRTLI) Cosmogrid project and National University of Ireland's Millenium Fund for financial support for POC for the duration of this work and the PRTLI 4 E-Inis project for financial support for JMcD for the duration of this work. Finally we would also like to thanks the anonymous referee whose comments made significant improvements to an earlier version of this paper.\n\n\n\\pagebreak\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe fix a nonnegative real number $T$ as well as two positive integers $k$ and $d$, and let $\\R^+:=[0,+\\infty)$. Let $\\mathbbm{1}_{A}$ represent the indicator function of a set $A$, and $\\langle x,y\\rangle$ the inner product of $x,y\\in\\R^k$. The Euclidean norms of a vector $y\\in\\R^k$ and a matrix $z\\in \\R^{k\\times d}$ are defined by $|y|$ and $|z|$, respectively.\n\nAssume that $(\\Omega,\\F,\\mathbbm{P})$ is a completed probability space carrying a standard $d$-dimensional Brownian motion $(B_t)_{t\\geq 0}$, and that $(\\F_t)_{t\\geq 0}$ is the natural $\\sigma$-algebra filtration generated by $(B_t)_{t\\geq 0}$ and $\\F=\\F_T$. For each $p>0$, denote by $\\Lp$ the set of all $\\R^k$-valued and $\\F_T$-measurable random vectors $\\xi$ such that $\\E[|\\xi|^p]<+\\infty$, by ${\\s}^p(0,T;\\R^k)$ (or $\\s^p$ simply) the set of $\\R^k$-valued, $(\\F_t)$-adapted and continuous processes $(Y_t)_{t\\in\\T}$ such that\n$$\\|Y\\|_{{\\s}^p}:=\\left( \\E\\left[\\sup_{t\\in\\T} |Y_t|^p\\right] \\right)^{1\\wedge 1\/p}<+\\infty,$$\nand by ${\\rm M}^p(0,T;\\R^{k\\times d})$ (or ${\\rm M}^p$ simply) the set of $(\\F_t)$-progressively measurable ${\\R}^{k\\times d}$-valued processes $(Z_t)_{ t\\in\\T}$ such that\n$$\\|Z\\|_{{\\rm M}^p}:=\\left\\{ \\E\\left[\\left(\\int_0^T |Z_t|^2\\ {\\rm d}t\\right)^{p\/2}\\right] \\right\\}^{1\\wedge 1\/p}<+\\infty.\n$$\nIt is well known that for each $p\\geq 1$, $\\s^p$ and ${\\rm M}^p$ are both Banach spaces respectively endowed with the norms $\\|\\cdot\\|_{{\\s}^p}$ and $\\|\\cdot\\|_{{\\rm M}^p}$. And, for each $p\\in (0,1)$, $\\s^p$ and ${\\rm M}^p$ are both complete metric spaces with the resulting distances $(Y,Y')\\mapsto \\|Y-Y'\\|_{{\\s}^p}$ and $(Z,Z')\\mapsto \\|Z-Z'\\|_{{\\rm M}^p}$ respectively.\n\nWe recall that a process $(Y_t)_{t\\in\\T}$ belongs to the class (D) if the family of variables $\\{|Y_\\tau|:\\tau\\ {\\rm is\\ an}$ $(\\F_t){\\rm - stopping\\ time\\ bounded\\ by}\\ n\\}$\nis uniformly integrable.\n\nIn this paper, we are interested in solving the following multidimensional backward stochastic differential equation (BSDE for short):\n\\begin{equation}\ny_t=\\xi+\\int_t^Tg(s,y_s,z_s){\\rm d}s-\\int_t^Tz_s {\\rm d}B_s,\\ \\ t\\in\\T,\n\\end{equation}\nwhere $\\xi\\in\\LT$ is called the terminal condition, $T$ is called the time horizon, and the random function $$g(\\omega,t,y,z):\\Omega\\tim \\T\\tim {\\R}^{k }\\tim\n{\\R}^{k\\times d}\\longmapsto {\\R}^k$$\nis $(\\F_t)$-progressively measurable for each $(y,z)$, called the generator of BSDE (1). Furthermore, the triple $(\\xi,T,g)$ is usually called the parameters of BSDE (1).\\vspace{0.1cm}\n\nThroughout this paper, we use the following definitions on solutions of (1).\\vspace{0.1cm}\n\n{\\bf Definition 1}\\ \\ A solution of BSDE (1) is a pair of $(\\F_t)$-progressively measurable processes $(y_t,z_t)_{t\\in\\T}$ with values in ${\\R}^k\\times {\\R}^{k\\times d}$ such that $\\ps$, $\\int_0^T|z_t|^2\\ {\\rm d}t<+\\infty$, $\\int_0^T|g(t,y_t,z_t)|\\ {\\rm d}t<+\\infty$, and (1) holds true for each $t\\in\\T$.\\vspace{0.2cm}\n\n{\\bf Definition 2}\\ \\ Assume that $(y_t,z_t)_{t\\in\\T}$ is a solution of (1). If $(y_t,z_t)_{t\\in\\T}\\in {\\s}^p(0,T;\\R^{k})\\times {\\rm M}^p(0,T;\\R^{k\\times d})$ for some $p>1$, then it is called an $L^p$ solution of BSDE (1); if $(y_t)_{t\\in\\T}$ belongs to the class (D) and\n$(y_t,z_t)_{t\\in\\T}\\in {\\s}^\\beta(0,T;\\R^{k})\\times {\\rm M}^\\beta(0,T;\\R^{k\\times d})$ for each $\\beta\\in (0,1)$, then it is called an $L^1$ solution of BSDE (1).\n\nIt is well known that nonlinear BSDEs were initially introduced in 1990 by \\citet{Par90}. They put forward and proved an existence and uniqueness result for $L^2$ solution of multidimensional BSDEs under the Lipschitz assumption of $g$ as well as the square integrability assumption of $\\xi$ and $g(t,0,0)$. From then on, the BSDE theory has attracted more and more interests, and due to the closely connections with many questions, it has gradually become a very powerful tool in many fields including stochastic control, financial mathematics, nonlinear mathematical expectation and partial differential equations, see \\cite{Bah10,Buc00,Chen00,Delb10,El97,Hu11,Hu08,Jia10,Kob00,\nMor09,Par99,Peng97,Tang98,Xing12} and so on.\n\nThere is no doubt that the existence and uniqueness of the solution is one of the most fundamental and kernel problems in the study on the theory and application of BSDEs. From the beginning, many researchers have attempted to improve the result of the $L^2$ solution of \\cite{Par90} by weakening the Lipschitz hypothesis on $g$, see, for example, \\cite{Bri07,Chen00,El97,Fan10,\nFan13,FJD13,Ham03,Jia08,Jia10,Lep97,Mao95,Xu15} for a survey.\nAt the same time, the existence and uniqueness of the $L^p\\ (p>1)$ solution for BSDEs has been extensively investigated by \\cite{Bri03,El97,Fan15,FJ14}, etc. Starting around 1998, the existence and uniqueness of the bounded solution and the solution whose exponential moments of certain order exist have also been becoming one of emphasis in the study on BSDE theory, one can see \\cite{Bri06,Bri08,Delb11,Hu15,Kob00,Lep98,Mor09,Rich12} for this topic, where the generator $g$ may have a quadratic or superquadratic growth in $z$.\n\nOn the other hand, in 1997, \\citet{Peng97} introduced the notion of $g$-martingales by solutions of BSDEs, which can be viewed as some kind of nonlinear martingales. Since the classical theory of martingales is carried in the integrable space, the question of solving a BSDE with only integrable parameters comes up naturally. In this spirit, some recent works including \\cite{Bri02,Bri03,Bri06,Fan12,FL10,Peng97,Xiao12,Xing12} investigated the existence and uniqueness of the $L^1$ solution of BSDEs. In particular, we would like to mention that \\citet{Bri03} established a general existence and uniqueness result of $L^1$ solution for a multidimensional BSDE with generator $g$ satisfying a monotonicity condition (see (H1) with $\\rho(x)=\\mu x$ in Section 3) as well as a general growth condition in $y$ (see (H2) in Section 3), and a Lipschitz condition together with a sublinear growth condition in $z$ (see (H3) in Section 3). Here, we also mention that multidimensional BSDEs are more difficult to handle than the one-dimensional case since for multidimensional BSDEs we usually can not establish or employ the comparison theorem of solutions. And, it is well known that the $L^1$ solution is more difficult to treat than the $L^p\\ (p>1)$ solution.\n\nThe present paper focus on the $L^1$ solution of multidimensional BSDEs. First of all, we will extend the existence and uniqueness result of the $L^1$ solution established in \\cite{Bri03} by weakening the monotonicity condition of $g$ in $y$ to a one-sided Osgood condition (see (H1) in Section 3). Under a Osgood condition of $g$ in $y$ and a Lipschitz condition of $g$ in $z$, \\citet{FJD13} first proved the existence and uniqueness of $L^2$ solution for multidimensional BSDEs. Recently, \\citet{Fan15} further extended this result and established the existence and uniqueness of $L^p\\ (p>1)$ solution for a multidimensional BSDE with generator $g$ satisfying a $p$-order weak monotonicity condition (see (H1a)$_p$ in Section 2) as well as a general growth condition in $y$, and a Lipschitz condition in $z$. We point out that in the case of $p=1$, the $p$-order weak monotonicity condition used in \\cite{Fan15} becomes the one-sided Osgood condition used in this paper. From this point of view, it is very natural to investigate the $L^1$ solution of multidimensional BSDEs under the one-sided Osgood condition of $g$ in $y$. However, when we use this condition instead of the usual monotonicity condition, some essential difficulty arise especially in the proof of existence of the $L^1$ solution. By virtue of Gronwall's inequality, Bihari's inequality and the relationship between convergence in $L^1$ and in probability together with two updated apriori estimates established in \\citet{Fan15}, we first consider the case when the generator $g$ is independent of $z$ (see Proposition 5 in Section 3). Then, making use of two estimates established in \\citet{Xu15} and \\citet{FJ14} together with Bihari's inequality, by a delicate argument involved in a Piciard's iterative procedure and a technique dividing the time interval $\\T$ we prove the existence of the $L^1$ solution for the general case (see Theorem 2 in Section 3). Here, we mention that it is interesting that in the case of the $\\alpha$ defined in (H3) values in $[1\/2,1)$, the one-sided Osgood condition need to be replaced with a $p$-order ($p>1$) one-sided Mao's condition (see (H1b)$_p$ in Section 2).\n\nThe second objective of this paper is to put forward and prove a stability theorem for the $L^1$ solutions of multidimensional BSDEs with generators of one-sided Osgood type. To the best of our knowledge, this is the first time for the $L^1$ solution of multidimensional BSDEs. It is not very hard to obtain a stability result of $L^p\\ (p>1)$ solutions for multidimensional BSDEs since by classical techniques one can establish and employ apriori estimates on the $L^p$ solution when $p>1$ (see, for example, Theorem 2 in \\citet{Fan15} for more details). However, it is well known that when $p=1$ the apriori estimates with respect to the first component of the $L^1$ solution are not valid any longer especially when the generator $g$ depends on $z$, which brings intrinsic difficulty when one tries to establish the stability of $L^1$ solutions. This may be the reason that by far there is still no reported work on the stability of $L^1$ solutions for multidimensional BSDEs even when $g$ only satisfies the monotonicity condition or the uniformly Lipschitz condition in $y$ other than the one-sided Osgood condition. In this paper, we will fill up the gap. More specifically, enlightened by the proof of the existence of the $L^1$ solution in this paper, we will first introduce some auxiliary BSDEs by virtue of a Picard's iterative procedure as a bridge and then use a very delicate argument to establish a stability theorem of the $L^1$ solutions for multidimensional BSDEs with generators of one-sided Osgood type (see Theorem 4 in Section 5), where expect for Gronwall's inequality, Bihari's inequality, the relationship between convergence in $L^1$ and in probability, and the technique dividing the time interval, the induction technique and the sharp apriori estimates established in \\cite{Fan15}, \\cite{Xu15} and \\cite{FJ14} all play important roles.\n\nIn addition, in this paper we also investigate, for the first time, the existence and uniqueness together with the stability of the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs (see Theorem 3 in Section 4 and Theorem 5 in Section 5).\n\nThe remainder of this paper is organized as follows. In Section 2 we gather several updated apriori estimates with respect to the solutions of multidimensional BSDEs and two technical lemmas. In section 3 we state and prove the existence and uniqueness result of $L^1$ solutions for the multidimensional BSDEs, and in Section 4 we are interested in solving the multidimensional BSDEs in $\\s^1\\times {\\rm M}^1$ and provide two examples to illustrate our theoretical results. Finally, in Section 5 we put forward and prove a stability theorem of $L^1$ solutions as well as solutions in $\\s^1\\times {\\rm M}^1$ for the multidimensional BSDEs.\n\n\n\\section{Preliminaries}\n\nIn this section, we first introduce several sharp apriori estimates with respect to solutions of multidimensional BSDEs, which will play very important roles in the proof of our main results. For this, let us introduce the following assumptions with respect to the generator $g$:\\vspace{0.3cm}\n\n{\\bf (A1)}\\ \\ $\\as,\\ \\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d},\\ \\ \\left\\langle y,\ng(\\omega,t,y,z)\\right\\rangle\\leq \\mu |y|^2+\\nu |y||z|+|y|f_t+\\varphi_t,\\ \\ $\\vspace{0.3cm}\\\\\nwhere $\\mu$ and $\\nu$ are two positive constants, $f_t$ and $\\varphi_t$ are two $(\\F_t)$-progressively measurable and nonnegative processes satisfying\n$$\n\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty\\ \\ \\ {\\rm and}\\ \\ \\ \\E\\left[\\left(\\int_0^T \\varphi_t\\ {\\rm d}t\\right)^{p\/2}\\right]<+\\infty.\\vspace{0.2cm}\n$$\n\n{\\bf (A2)}\\ \\ $\\as,\\ \\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d}$,\\vspace{0.25cm}\\\\\n\\hspace*{3cm}$|y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0},\ng(\\omega,t,y,z)\\right\\rangle\\leq \\psi(|y|^p)+\\nu |y|^{p-1}|z|+|y|^{p-1}f_t,$\\vspace{0.35cm}\\\\\nwhere $\\nu>0$ is a constant, $f_t$ is an $(\\F_t)$-progressively measurable and nonnegative process satisfying\n$$\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty,$$\nand $\\psi(\\cdot):\\R^+\\mapsto \\R^+$ is a nondecreasing and concave function with $\\psi(0)=0$.\\vspace{0.3cm}\n\n{\\bf (A3)}\\ \\ $\\as,\\RE\\ (y,z)\\in \\R^k\\times\\R^{k\\times d},\\ \\ \\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, g(\\omega,t,y,z)\\right\\rangle\\leq\n\\phi^{{1\\over p}}(|y|^p)+\\nu|z|+f_t,\\ \\ $\\vspace{0.3cm}\\\\\nwhere $\\nu>0$ is a constant, $f_t$ is an $(\\F_t)$-progressively measurable and nonnegative process satisfying\n$$\\E\\left[\\left(\\int_0^T f_t\\ {\\rm d}t\\right)^p\\right]<+\\infty,$$\nand $\\phi(\\cdot):\\R^+\\mapsto \\R^+$ is a nondecreasing and concave function with $\\phi(0)=0$.\\vspace{0.2cm}\n\nThe above assumptions (A2) and (A3) are respectively related to following assumptions (H1a)$_p$ and (H1b)$_p$, which are put forward and used in \\citet{Fan15} at the first time. Assumptions (H1a)$_p$ and (H1b)$_p$ will also be employed in this paper.\\vspace{0.2cm}\n\n{\\bf (H1a)$_p$} $g$ satisfies a $p$-order weak monotonicity condition in $y$, i.e., there exists a nondecreasing and concave function $\\kappa(\\cdot):\\R^+\\mapsto \\R^+$ with $\\kappa(0)=0$, $\\kappa(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}=+\\infty$ such that $\\as$, $\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d}$,\n$$\n|y_1-y_2|^{p-1}\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},g(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\rangle\\leq \\kappa(|y_1-y_2|^p),\n$$\nwhere and hereafter, $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}:=\\lim\\limits_{\\epsilon \\rightarrow 0}\\int_{0}^{\\epsilon} {{\\rm d}u\\over \\kappa(u)};\\vspace{0.2cm}$\n\n{\\bf (H1b)$_p$} $g$ satisfies a $p$-order one-sided Mao's condition in $y$, i.e., there exists a nondecreasing and concave function $\\varrho(\\cdot):\\R^+\\mapsto \\R^+$ with $\\varrho(0)=0$, $\\varrho(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\varrho(u)}=+\\infty$ such that $\\as$, $\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d}$,\n$$\n\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\rangle\\leq \\varrho^{{1\\over p}}(|y_1-y_2|^p).\\vspace{0.2cm}\n$$\n\nThe following Propositions 1-2 are respectively Propositions 2-3 in \\citet{Fan15}, and the following Proposition 3 comes from Proposition 1 in \\citet{Xu15}.\\vspace{0.1cm}\n\n{\\bf Proposition 1} Let $p>0$ and (A1) hold. Suppose that $(y_t,z_t)_{t\\in\\T}$ is a solution of BSDE (1) such that $y_\\cdot\\in {\\s}^p(0,T;\\R^{k})$. Then $z_\\cdot$ belongs to ${\\rm M}^p(0,T;\\R^{k\\times d})$, and $\\ps$, for each $0\\leq u\\leq t\\leq T$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left.\\left(\\int_t^T |z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right|\\F_u\\right]&\\leq & \\Dis\nC_{\\mu,\\nu,p,T}\\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^p\\right|\\F_u\\right]+C_p\\E\\left[\\left.\n\\left(\\int_t^T f_s\\ {\\rm d}s\\right)^p\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&& \\Dis+C_p\\E\\left[\\left.\\left(\\int_t^T \\varphi_s\\ {\\rm d}s\\right)^{p\/2}\\right|\\F_u\\right],\n\\end{array}\n$$\nwhere $C_{\\mu,\\nu,p,T}>0$ is a constant depending on $(\\mu,\\nu,p,T)$, and $C_p>0$ is a constant depending only on $p$.\\vspace{0.3cm}\n\n{\\bf Proposition 2} Let $p>1$ and (A2) hold. Suppose that $(y_t,z_t)_{t\\in\\T}$ is an $L^p$ solution of BSDE (1). Then, there exists a constant $C_{\\nu,p}>0$ depending only on $\\nu, p$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.2cm}\n$$\n\\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^{p}\\right|\\F_u\\right]\\leq\ne^{C_{\\nu,p}(T-t)}\\left\\{\\E[\\left.|\\xi|^p\\right|\\F_u]+\\int_t^T\n\\psi(\\E[\\left.|y_s|^p\\right|\\F_u]) \\ {\\rm\nd}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^{p}\\right|\\F_u\\right]\\right\\}.\\vspace{0.3cm}\n$$\n\n{\\bf Proposition 3} Let $g$ satisfy (A2) with $p=2$. Suppose that $(y_t,z_t)_{t\\in\\T}$ is an $L^2$ solution of BSDE (1). Then, there exists a constant $C_{\\nu}>0$ depending only on $\\nu$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.1cm}\n$$\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\left.\\sup\\limits_{r\\in\n[t,T]}|y_r|^2\\right|\\F_u\\right]+\\E\\left[\\left.\\int_t^T\n|z_s|^2\\ {\\rm d}s\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&\\leq & \\Dis\ne^{C_{\\nu}(T-t)}\\left\\{\\E\\left[\\left.|\\xi|^2\\right|\\F_u\\right]+\n\\int_t^T\n\\psi\\left(\\E\\left[\\left.|y_s|^2\\right|\\F_u\\right]\\right)\\ {\\rm d}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^2\\right|\\F_u\\right]\\right\\}.\n\\end{array}\\vspace{0.2cm}\n$$\n\nIn the same way as that in Lemmas 2-3 of \\citet{FJ14}, we can prove the following proposition. The proof is omitted here.\\vspace{0.1cm}\n\n{\\bf Propositions 4}\\ Let $p>1$ and (A3) hold. Suppose that $(y_t,z_t)_{t\\in\\T}$ is a solution of BSDE (1) such that $y_\\cdot\\in {\\s}^p(0,T;\\R^{k})$. Then, $z_\\cdot$ belongs to ${\\rm M}^p(0,T;\\R^{k\\times d})$ and there exists a positive constant $C_{\\nu,p}$ depending on $\\nu$ and $p$ such that $\\ps$, for each $0\\leq u\\leq t\\leq T$,\\vspace{0.1cm}\n$$\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\left.\\sup\\limits_{s\\in\n[t,T]}|y_s|^{p}\\right|\\F_u\\right]+\\E\\left[\\left.\n\\left(\\int_t^T|z_s|^2\\ {\\rm d}s\\right)^{p\/2}\\right|\\F_u\\right]\\vspace{0.1cm}\\\\\n&\\leq & \\Dis e^{C_{\\nu,p}(T-t)}\\left\\{\\E\\left[\\left.|\\xi|^p\\right|\n\\F_u\\right]+\\int_t^T\n\\phi\\left(\\E\\left[\\left.|y_s|^p\\right|\\F_u\\right]\\right) \\ {\\rm d}s+\\E\\left[\\left.\\left(\\int_t^T f_s\\ {\\rm\nd}s\\right)^p\\right|\\F_u\\right]\\right\\}.\n\\end{array}\\vspace{0.3cm}\n$$\n\nNow, let us introduce two technical lemmas, which will be used later. Firstly, the following Lemma 1 gives a sequence of upper bounds for linear growth functions, which comes from \\citet{Fan10}.\\vspace{0.2cm}\n\n{\\bf Lemma 1}\\ Suppose that $\\bar\\kappa(\\cdot):\\R^+\\mapsto \\R^+$ increases at most linearly, i.e., there exists a constant $A>0$ such that $$\\bar\\kappa(x)\\leq A(x+1),\\ \\ \\RE\\ x\\in \\R^+.$$\nThen for each $m\\geq 1$, we have\n$$\n\\bar\\kappa(x)\\leq (m+2A)x+\\bar\\kappa\\left({2A\\over m+2A}\\right),\\ \\ \\RE\\ x\\in \\R^+.\\vspace{0.2cm}\n$$\n\nThe following Lemma 2 can be regarded as a backward version of classical Bihari's inequality, which can be proved by classical methods. The proof is omitted.\\vspace{0.2cm}\n\n{\\bf Lemma 2}\\ (Bihari's inequality)\\ Let the nonnegative function $u(\\cdot):[0,T]\\mapsto \\R^+$ satisfy\n$$\nu(t)\\leq u_0 +\\int_t^T \\bar\\psi(u(s))\\ {\\rm d}s,\\ \\ t\\in \\T,\n$$\nwhere $u_0$ is a positive real number, $\\bar\\psi(\\cdot):\\R^+\\mapsto \\R^+$ is a continuous and nondecreasing function, $\\bar\\psi(0)=0$, $\\bar\\psi(u)>0$ for $u>0$ and $\\int_{0^+} {1\\over \\bar\\psi(u)}\\ {\\rm d}u=+\\infty$. Then, for each $t\\in \\T$, we have\n$$\nu(t)\\leq \\Psi^{-1}(\\Psi(u_0)+T-t),\n$$\nwhere\n$$\n\\Psi(x):=\\int_1^x {1\\over \\bar\\psi(u)}\\ {\\rm d}u, \\ \\ x>0\\vspace{0.2cm}\n$$\nis a strictly increasing function valued in $\\R$, and $\\Psi^{-1}$ is the inverse function of $\\Psi$. In particular, if $u_0=0$, then $u(t)=0$ for each $t\\in \\T$.\\vspace{0.2cm}\n\nTo the end of this section, we would like to especially mention that even though Propositions 1-4 mentioned above appear similar, there are some distinguish differences among both their conditions and conclusions. They will also play different roles in the proof of our main results.\n\n\n\n\\section{Existence and uniqueness of the $L^1$ solution}\n\nWe first introduce the following assumptions\non the generator $g$ used in \\citet{Fan15}, \\citet{Fan13} and \\citet{Bri03}:\\vspace{0.2cm}\n\n{\\bf (H1)} $g$ satisfies a one-sided Osgood condition in $y$, i.e., there exists a nondecreasing and concave function $\\rho(\\cdot):\\R^+\\mapsto \\R^+$ with $\\rho(0)=0$, $\\rho(u)>0$ for $u>0$ and $\\int_{0^+} {{\\rm d}u\\over \\rho(u)}=+\\infty$ such that $\\as$,\n$$\n\\RE\\ y_1,y_2\\in \\R^k,z\\in\\R^{k\\times d},\\ \\ \\left\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\right\\rangle\\leq \\rho(|y_1-y_2|).\\vspace{0.2cm}\n$$\n\n{\\bf (H2)} $g$ has a general growth with respect to $y$, i.e, \\vspace{0.2cm}\n$$\\RE\\ r>0,\\ \\ \\E\\left[\\int_0^T \\bar\\phi_r(t)\\ {\\rm d}t\\right]<+\\infty\\ \\ {\\rm with}\\ \\ \\bar\\phi_r(t):=\\sup\\limits_{|y|\\leq r}\n|g(\\omega,t,y,0)|;$$\nFurthermore, $\\as$, $\\RE\\ z\\in {\\R^{k\\times d}},\\ \\ \\ y\\longmapsto g(\\omega,t,y,z)$ is continuous.\\vspace{0.2cm}\n\n{\\bf (H3)}\\ $g$ is Lipschitz continuous in $z$, uniformly with respect to $(\\omega,t,y)$, i.e., there exists a constant $\\lam\\geq 0$ such that $\\as$,\n$$\\RE\\ y\\in \\R^k,z_1,z_2\\in\\R^{k\\times d},\\ \\ |g(\\omega,t,y,z_1)-g(\\omega,t,y,z_2)|\\leq \\lam |z_1-z_2|;$$\nFurthermore, $g$ has a sublinear growth in $z$, i.e., there exist two constants $\\gamma>0$ and $\\alpha\\in (0,1)$ as well as an $(\\F_t)$-progressively measurable and nonnegative process $(g_t)_{t\\in\\T}$ satisfying $\\E\\left[\\int_0^Tg_t\\ {\\rm d}t\\right]<+\\infty$ such that $\\as$, \\vspace{0.05cm}\n$$\n\\RE\\ y\\in \\R^k,z\\in\\R^{k\\times d},\\ \\ |g(\\omega,t,y,z)-g(\\omega,t,y,0)|\\leq \\gamma(g_t(\\omega)+|y|+|z|)^\\alpha.\\vspace{0.2cm}\n$$\n\n{\\bf Remark 1} For later use, it follows from Proposition 1 in \\citet{Fan15} that for each $p>1$,\n$${\\rm (H1b)}_p \\Longrightarrow {\\rm (H1)}\\Longrightarrow {\\rm (H1a)}_p,$$\nand when $p=1$, they are same. In addition, the functions $\\rho(\\cdot)$, $\\kappa(\\cdot)$ and $\\varrho(\\cdot)$ in (H1), (H1a) and (H1b) all increase at most linearly since they are all nondecreasing and concave function valued $0$ at $0$. Here and hereafter we will always denote by $A$ the linear-growth constants of them, i.e.,\n$$\\rho(x)\\leq A(x+1),\\ \\ \\kappa(x)\\leq A(x+1),\\ \\ \\varrho(x)\\leq A(x+1),\\ \\ \\RE\\ x\\in\\R^+.$$\nFinally, by Proposition 1 in \\citet{Fan15} we also point out that the concavity condition of $\\rho(\\cdot)$ and $\\varrho(\\cdot)$ defined respectively in assumptions (H1) and (H1b)${}_p$ can be replaced with the continuity condition.\\vspace{0.2cm}\n\nThe following Theorems 1-2 are the main results of this section.\\vspace{0.1cm}\n\n{\\bf Theorem 1} Assume that the generator $g$ satisfies assumptions (H1) and (H3). Then for each $\\xi\\in\\LT$, BSDE (1) admits at most one solution $(y_\\cdot,z_\\cdot)$ such that $y_\\cdot$ belongs to the class (D) and $z_\\cdot$ belongs to $\\bigcup_{\\beta>\\alpha}\\M^\\beta$,\nwhich leads to that it admits at most one $L^1$ solution.\\vspace{0.1cm}\n\n{\\bf Proof}\\ \\ Assume that (H1) and (H3) hold and that both $(y_t,z_t)_{t\\in\\T}$ and $(y'_t,z'_t)_{t\\in\\T}$ are solutions of BSDE (1) such that both $(y_t)_{t\\in\\T}$ and $(y'_t)_{t\\in\\T}$ belong to the class (D), and both $(z_t)_{t\\in\\T}$ and $(z'_t)_{t\\in\\T}$ belong to $\\M^\\beta$ for some $\\beta\\in(\\alpha, 1)$.\\vspace{0.1cm}\n\nWe first show that $(y_t-y'_t)_{t\\in\\T}\\in \\s^{\\beta\/\\alpha}$. In fact, let us fix $n\\geq 1$ and denote $\\tau_n$ the stopping time\n$$\n\\tau_n:=\\inf\\left\\{t\\in\\T:\\int^t_{0}(|z_s|^2+|z'_s|^2)\n\\,\\mathrm{d} s\\geq n\\right\\}\\wedge T.\n$$\nCorollary 2.3 in \\citet{Bri03} leads to the following inequality with setting $\\hat{y}_\\cdot:=y_\\cdot-y'_\\cdot$ and $\\hat{z}_\\cdot:=z_\\cdot-z'_\\cdot$, and $t\\in\\T$,\n\\begin{equation}\n\\Dis|\\hat{y}_{t\\wedge\\tau_n}|\\leq \\Dis |\\hat{y}_{\\tau_n}|+\\int^{\\tau_n}_{t\\wedge\\tau_n}\n\\left\\langle {\\hat{y}_s\\over |\\hat{y}_s| }\\mathbbm{1}_{|\\hat{y}_s|\\neq 0},g(s,y_s,z_s)-g(s,y'_s,z'_s)\\right\\rangle\\\n\\mathrm{d}s -\\int^{\\tau_n}_{t\\wedge\\tau_n}\\left\\langle {\\hat{y}_s\\over |\\hat{y}_s|}\\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\\hat{z}_s\\mathrm{d}B_s\\right\\rangle.\n\\end{equation}\nIt follows from assumptions (H1) and (H3) that $\\ass$,\n\\begin{equation}\n\\begin{array}{ll}\n&\\Dis \\left\\langle{\\hat{y}_s\\over |\\hat{y}_s|} \\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\ng(s,y_s,z_s)-g(s,y'_s,z'_s)\\right\\rangle\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\left\\langle{\\hat{y}_s\\over |\\hat{y}_s|} \\mathbbm{1}_{|\\hat{y}_s|\\neq 0},\ng(s,y_s,z_s)-g(s,y'_s,z_s)\\right\\rangle\n+|g(s,y'_s,z_s)-g(s,y'_s,z'_s)|\\vspace{0.1cm}\\\\\n\\leq &\\Dis\\rho(|\\hat{y}_s|)\n+2\\gamma\\left(g_s+|y'_s|+|z'_s|+|z_s|\\right)^\\alpha.\n\\end{array}\n\\end{equation}\nThen, combining (2) with (3) we can deduce that for each $n\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat{y}_{t\\wedge\\tau_n}|\\leq\\E\\left[|\\hat{y}_{\\tau_n}|\n+\\left.\\int^{\\tau_n}_{t\\wedge\\tau_n}\\rho(|\\hat{y}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right]+G(t),\n\\end{equation}\nwhere\n$$\nG(t):=2\\gamma\\E\\left[\\left.\\int^{T}_{0}\\left(g_s+\n|y'_s|+|z'_s|+|z_s|\\right)^\\alpha\\ \\mathrm{d}s\\right|\\F_t\\right].\n$$\nFurthermore, since $y'_\\cdot$ belongs to the class (D), both $z_\\cdot$ and $z'_\\cdot$ belong to ${\\rm M}^\\beta$ with $\\beta>\\alpha$, and $\\E\\left[\\int_0^Tg_t\\ {\\rm d}t\\right]<+\\infty$, we can use Doob's inequality, H\\\"{o}lder's inequality and Jensen's inequality to obtain that\n\\begin{equation}\n\\E\\left[\\sup_{t\\in\\T}|G(t)|^{\\beta\/\\alpha}\\right]<+\\infty.\n\\end{equation}\nThus, since $\\hat y_\\cdot$ belongs to the class (D) and $\\rho(\\cdot)$ increases at most linearly, we can send $n$ to $+\\infty$ in (4) and use Lebesgue's dominated convergence theorem, in view of $\\tau_n\\To T$ as $n\\To \\infty$, $\\hat{y}_T=0$ and Remark 1, to get that for each $t\\in \\T$,\n$$\n|\\hat{y}_t|\\leq G(t)+\\E\\left[\\left.\\int_t^T\n\\rho(|\\hat{y}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right]\\leq AT+G(t)+A\\int_t^T\\E\\left[\\left.\n|\\hat{y}_s|\\right|\\F_t\\right]\\mathrm{d}s,\n$$\nand then\n$$\n\\E\\left[\\left.|\\hat{y}_r|\\right|\\F_t\\right]\\leq AT+G(t)+A\\int_r^T\n\\E\\left[\\left.|\\hat{y}_s|\\right|\\F_t\\right]{\\rm d}s,\\quad r\\in [t,T].\\vspace{0.2cm}\n$$\nGronwall's inequality yields that $\\E\\left[\\left.|\\hat{y}_r|\\right|\\F_t\\right]\\leq\n\\left(AT+G(t)\\right)\\cdot e^{A(T-r)},\\ r\\in [t,T]$, form which, by letting $r=t$, we have\n\\begin{equation}\n|\\hat{y}_t|\\leq \\left(AT+G(t)\\right)\\cdot e^{AT}.\n\\end{equation}\nThis inequality together with (5) leads to\n\\begin{equation}\n\\hat y_\\cdot=(y_\\cdot-y'_\\cdot)\\in \\s^p\\ \\text{with}\\ p:=\\beta\/\\alpha>1.\n\\end{equation}\n\nIn the sequel, note that $(\\hat y_t, \\hat z_t)_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat y_t=\\int^T_t\\hat g(s,\\hat y_s,\\hat z_s)\\ \\mathrm{d} s-\\int^T_t\\hat z_s\\,\\mathrm{d} B_s,\\quad t\\in\\T,\n\\end{equation}\nwhere for each $(y,z)\\in\\R^k\\times\\R^{k\\times d}$,\n$\\hat g(t,y,z):=g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z'_t)$.\nIt follows from assumptions (H1) and (H3) on $g$ together with Remark 1 that $\\as$, for each $(y,z)\\in \\R^k\\times\\R^{k\\times d}$,\n\\begin{equation}\n\\begin{array}{lll}\n\\left\\langle y,\\hat g(t,y,z)\\right\\rangle&\\leq&\\Dis \\left\\langle y, g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z+z'_t)\\right\\rangle+|y||g(t,y'_t,z+z'_t)-g(t,y'_t,z'_t)|\\\\\n&\\leq&\\Dis \\bar\\kappa(|y|^2)+\\lam |y||z|\\leq \\Dis A|y|^2+\\lam |y||z|+A,\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis |y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0},\\hat g(t,y,z)\\right\\rangle\n&\\leq&\\Dis |y|^{p-1}\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, g(t,y+y'_t,z+z'_t)\n-g(t,y'_t,z+z'_t)\\right\\rangle\\\\\n&&\\Dis +|y|^{p-1}|g(t,y'_t,z+z'_t)-g(t,y'_t,z'_t)|\n\\vspace{0.1cm} \\\\\n&\\leq&\\Dis \\kappa(|y|^p)+\\lam |y|^{p-1}|z|,\n\\end{array}\n\\end{equation}\nwhere the functions $\\bar\\kappa(\\cdot)$ and $\\kappa(\\cdot)$ are respectively defined in (H1a)$_2$ and (H1a)$_p$. Thus, on one hand, inequality (9) means that the generator $\\hat g$ of BSDE (8) satisfies assumption (A1) with $\\mu=A,\\ \\nu=\\lam,\\ f_t\\equiv 0\\ \\text{and}\\ \\varphi_t\\equiv A$. It then follows from Proposition 1 together with (7) that $(\\hat y_t, \\hat z_t)_{t\\in\\T}$ is an $L^p$ solution of BSDE (8). On the other hand, inequality (10) means that the generator $\\hat g$ of BSDE (8) also satisfies assumption (A2) with $\\psi(\\cdot)=\\kappa(\\cdot),\\ \\nu=\\lam\\ \\text{and}\\ f_t\\equiv 0$. It then follows from Proposition 2 with $u=0$ that there exists a positive constant $C_{\\lam,p,T}$ depending only on $\\lam$, $p$ and $T$ such that for each $t\\in \\T$,\\vspace{0.1cm}\n\\begin{equation}\n\\E\\left[\\sup_{s\\in [t,T]}|\\hat y_s|^p\\right]\\leq C_{\\lam,p,T} \\int_t^T\\kappa\\left(\\E\\left[|\\hat y_s|^p\\right]\\right)\\ {\\rm d}s\\leq C_{\\lam,p,T} \\int_t^T\\kappa\\left(\\E\\left[\\sup_{u\\in [s,T]}|\\hat y_u|^p\\right]\\right)\\ {\\rm d}s.\\vspace{0.1cm}\n\\end{equation}\nThus, in view of the fact that $\\int_{0^+} {{\\rm d}u\\over \\kappa(u)}=+\\infty$, Bihari's inequality (Lemma 2) yields that\n\\begin{equation}\n\\E\\left[\\sup_{t\\in [0,T]}|y_t-y'_t|^p\\right]=\\E\\left[\\sup_{t\\in [0,T]}|\\hat y_t|^p\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nFinally, by (9), Remark 1 and Lemma 1 we can check that the generator $\\hat g$ of BSDE (8) satisfies assumption (A1) with\n$\\mu=m+2A,\\ \\nu=\\lam,\\ f_t\\equiv 0\\ {\\rm and}\\ \\ \\varphi_t=\\bar\\kappa({2A\\over m+2A})$\nfor each $m\\geq 1$. It then follows from Proposition 1 with $u=t=0$ that there exists a positive constant $C_{m,\\lam,p,T}$ depending on $m$, $\\lam$, $p$ and $T$, and a positive constant $C_{p}$ depending only on $p$ such that for each $m\\geq 1$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T |\\hat z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right]\\leq C_{m,\\lam,p,T}\\E\\left[\\sup\\limits_{t\\in [0,T]}|\\hat y_t|^p\\right]+C_{p}\\left(\\bar\\kappa({2A\\over m+2A})\\cdot T\\right)^{p\/2}.\n\\end{equation}\nThus, in view of (12) and the fact that $\\bar\\kappa(\\cdot)$ is a continuous function with $\\bar\\kappa(0)=0$, sending $m\\To\\infty$ in the previous inequality we deduce that\n$$\n\\E\\left[\\left(\\int_0^T |z_s-z'_s|^2\\\n{\\rm d}s\\right)^{p\/2}\\right]=\\E\\left[\\left(\\int_0^T |\\hat z_s|^2\\ {\\rm\nd}s\\right)^{p\/2}\\right]=0.\n$$\nThe proof of Theorem 1 is then complete. \\vspace{0.3cm}\\hfill $\\Box$\n\nWe now turn to the existence part of our study. We will prove the following result.\\vspace{0.1cm}\n\n{\\bf Theorem 2} Assume that the generator $g$ satisfies assumptions (H1)-(H3). In the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, we further assume that there exists a constant $\\bar p>1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies\n\\begin{equation}\n\\int_{0^+}{u^{\\bar p-1}\\over \\rho^{\\bar p}(u)}{\\rm d}u=+\\infty.\\vspace{0.2cm}\n\\end{equation}\nThen for each $\\xi\\in\\LT$, BSDE (1) admits an $L^1$ solution $(y_t,z_t)_{t\\in\\T}$.\\vspace{0.3cm}\n\n{\\bf Remark 2} It follows from Proposition 1 in \\citet{Fan15} that if $g$ satisfies (H1) with a $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$, then $g$ must satisfy (H1b)$_{\\bar p}$. And vice versa. This fact will be perfectly utilized later.\\vspace{0.2cm}\n\nThe following Proposition 5 is the first step to prove Theorem 2, which studies the case where the generator $g$ does not depend on the variable $z$.\\vspace{0.2cm}\n\n{\\bf Proposition 5} Let the generator $g$ be independent of $z$ and satisfy assumptions (H1)-(H2). Then for each $\\xi\\in\\LT$, BSDE (1) admits an $L^1$ solution $(y_t,z_t)_{t\\in\\T}$.\\vspace{0.2cm}\n\n{\\bf Proof}\\ \\ Assume that $\\xi\\in\\LT$, $g$ is independent of $z$ and assumptions (H1) and (H2) hold. For each $n\\geq 1$, we denote $q_n(x)=xn\/(n\\vee |x|)$ and set\n$$\n\\xi^n:=q_n(\\xi),\\quad g^n(t,y):=g(t,y)-g(t,0)+q_n(g(t,0)).\n$$\nNote that both $|\\xi^n|$ and $|g^n(t,0)|$ are bounded by $n$, and that the generator $g^n(t,y)$ satisfies (H1) and (H2) for each $n\\geq 1$. It then follows from\nCorollary 2 with $p=2$ in \\citet{Fan15} that the following BSDE\n\\begin{equation}\ny^n_t=\\xi^n+\\int^{T}_{t}g^n(s,y^n_s)\\,\\mathrm{d} s-\\int^{T}_{t}z_s^n\\,\\mathrm{d} B_s,\\quad t\\in\\T\n\\end{equation}\nadmits a unique $L^2$ solution $(y^n_t,z^n_t)_{t\\in\\T}$.\\vspace{0.2cm}\n\nFor each $n,i\\geq 1$, we set $\\hat{y}^{n,i}_\\cdot:=y^{n+i}_\\cdot-y^n_\\cdot$, $\\hat z^{n,i}_\\cdot:=z^{n+i}_\\cdot-z^n_\\cdot$ and $\\hat\\xi^{n,i}:=\\xi^{n+i}-\\xi^n$. It follows from Corollary 2.3 in \\cite{Bri03} that for each $n,i\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat{y}^{n,i}_t|\\leq \\Dis |\\hat{\\xi}^{n,i}|+\\int^{T}_{t}\n\\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^n(s,y^n_s)\\right\\rangle\n\\ \\mathrm{d} s\\Dis -\\int^{T}_{t}\\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0},\\hat z^{n,i}_s\\,\\mathrm{d} B_s\\right\\rangle.\n\\end{equation}\nIt follows from assumption (H1) and definition of $g^n(t,y)$ that $\\ass$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^n(s,y^n_s)\\right\\rangle\n&\\leq &\\Dis \\left\\langle {\\hat y^{n,i}_s\\over |\\hat y^{n,i}_s|}\\mathbbm{1}_{|\\hat y^{n,i}_s|\\neq 0}, g^{n+i}(s,y^{n+i}_s)-g^{n+i}(s,y^n_s)\\right\\rangle\n\\vspace{0.1cm}\\\\\n&&+|g^{n+i}(s,y^n_s)-g^n(s,y^n_s)|\\vspace{0.1cm} \\\\\n&\\leq &\\Dis \\rho(|\\hat y^{n,i}_s|)+|g(s,0)| \\mathbbm{1}_{|g(s,0)|>n}.\n\\end{array}\n\\end{equation}\nThen, combining (16) and (17), in view of Fubini's Theorem and Jensen's inequality, we can get that for each $n,i\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat y_t^{n,i}|\\leq H_n(t)+\\int^{T}_{t}\\rho\\left(\\E\\left[\\left.|\\hat y^{n,i}_s|\\right|\\F_t\\right]\\right)\\mathrm{d}s,\n\\end{equation}\nwhere\n$$H_n(t):=\\E\\left[|\\xi|\\mathbbm{1}_{|\\xi|>n} +\\left.\\int^{T}_{0}|g(s,0)|\\mathbbm{1}_{|g(s,0)|>n}\n\\mathrm{d}s\\right|\\F_t\\right].\\vspace{0.1cm}\n$$\n\nIn the sequel, by virtue of Fubini's theorem and Jensen's inequality, it follows from (18) that for each $n,i\\geq 1$ and $t\\in\\T$,\n$$\n\\E\\left[\\left.|\\hat y_r^{n,i}|\\right|\\F_t\\right]\\leq H_n(t)+\\int^{T}_{r}\\rho\\left(\\E\\left[\\left.|\\hat y^{n,i}_s|\\right|\\F_t\\right]\\right)\\mathrm{d}s,\\quad r\\in [t,T].\n$$\nThen in view of Remark 1, Gronwall's inequality yields that for each $n,i\\geq 1$ and $(\\omega,t)\\in \\Omega\\times\\T$,\n\\begin{equation}\n|\\hat{y}_t^{n,i}|=\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\n\\right|\\F_t\\right]\\leq \\left(H_n(t)+AT\\right)\\cdot e^{A(T-t)}.\n\\end{equation}\nThis inequality together with Lemma 6.1 in \\cite{Bri03} leads to that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis\\sup\\limits_{n\\geq 1}\\E\\left[\\sup\\limits_{i\\geq 1}\\sup\\limits_{t\\in\\T} |\\hat{y}_t^{n,i}|^\\beta\\right]&\\leq &\\Dis C\\left(1+\\sup\\limits_{n\\geq 1} \\E\\left[\\sup\\limits_{t\\in\\T} |H_n(t)|^\\beta\\right]\\right)\\vspace{0.1cm}\\\\\n&\\leq & \\Dis C\\left(1+{1\\over 1-\\beta}\\sup\\limits_{n\\geq 1}(\\E[H_n(T)])^\\beta\\right)<+\\infty,\n\\end{array}\n\\end{equation}\nwhere $C>0$ is a constant depending only on $A,T$ and $\\beta$. This means that the sequence\n$$\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_t^{n,i}|^{\\beta'}\\right\\}_{n=1}^{+\\infty}\n$$\nis uniformly integrable for each $\\beta'\\in (0,1)$.\\vspace{0.1cm}\n\nOn the other hand, by virtue of Fubini's theorem and Jensen's inequality, it follows from (18) that for each $n,i\\geq 1$ and $t\\in\\T$,\n$$\n\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\\right|\\F_u\\right]\\leq\nH_n(u)+\\int_t^T\\rho\\left(\\E\\left[\\left.|\\hat{y}_{s}^{n,i}|\n\\right|\\F_u \\right]\\right){\\rm d}s,\\ \\ u\\in[0,t],\n$$\nand then, in view of the fact that $\\rho(\\cdot)$ is a nondecreasing function,\n\\begin{equation}\n\\Dis h_n(t)\\leq \\sup_{u\\in\\T}H_n(u)+\\int_t^T\\rho\\left(h_n(s)\\right){\\rm d}s,\\ \\ t\\in\\T,\n\\end{equation}\nwhere\n$$\nh_n(t):=\\sup_{i\\geq 1}\\sup\\limits_{0\\leq u\\leq t}\n\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\n\\right|\\F_u\\right]\\vspace{0.2cm}\n$$\nare all nonnegative functions. Thus, by virtue of Lemma 2 we can deduce that for each $n\\geq 1$ and $\\omega\\in \\Omega$,\n\\begin{equation}\nh_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T),\\ \\ t\\in\\T,\n\\end{equation}\nwhere\n\\begin{equation}\n\\Theta(x):=\\int_1^x {1\\over \\rho(x)}{\\rm d}x,\\ x>0 \\vspace{0.2cm}\n\\end{equation}\nis a strictly increasing and continuous function valued in $\\R$, and $\\Theta^{-1}$ is the inverse function of $\\Theta$. Furthermore, by the maximum inequality with respect to sub-martingale and Lebesgue's dominated convergence theorem we have, for each $\\epsilon>0$, as $n\\To \\infty$,\n\\begin{equation}\n\\mathbbm{P}\\left(\\left\\{\\sup_{t\\in\\T} |H_n(t)|\\geq \\epsilon\\right\\}\\right)\\leq {1\\over \\epsilon}\\E[|H_n(T)|]\\longrightarrow 0.\n\\end{equation}\nThus, noticing by the definition of $h_n(t)$ together with (22) that\n$$\n\\sup_{t\\in\\T}\\sup_{i\\geq 1} \\E\\left[|\\hat{y}_{t}^{n,i}|\\right]\n\\leq \\sup_{t\\in\\T} h_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T)\n$$\nand\n$$\n\\sup_{t\\in\\T}\\sup_{i\\geq 1} |\\hat{y}_{t}^{n,i}|=\\sup_{t\\in\\T}\\sup_{i\\geq 1}\\E\\left[\\left.|\\hat{y}_{t}^{n,i}|\\right|\\F_t\\right]\n\\leq \\sup_{t\\in\\T} h_n(t)\\leq \\Theta^{-1}(\\Theta(\\sup_{t\\in\\T}H_n(t))+T),\n\\vspace{0.2cm}\n$$\nfrom (24) we can deduce that for each $\\epsilon>0$,\n$$\n0\\leq \\Lim \\mathbbm{P}\\left(\\left\\{\\sup_{i\\geq 1} \\sup_{t\\in\\T}\\E\\left[|\\hat{y}_{t}^{n,i}|\\right]\\geq \\epsilon\\right\\}\\right)\\leq\n\\Lim \\mathbbm{P}\\left(\\left\\{ \\sup_{t\\in\\T}|H_n(t)|\\geq \\Theta^{-1}(\\Theta(\\epsilon)-T)\\right\\}\\right)=0\n$$\nand\n\\begin{equation}\n0\\leq \\Lim \\mathbbm{P}\\left(\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|\\geq \\epsilon\\right\\}\\right)\\leq\n\\Lim \\mathbbm{P}\\left(\\left\\{ \\sup_{t\\in\\T}|H_n(t)|\\geq \\Theta^{-1}(\\Theta(\\epsilon)-T)\\right\\}\\right)=0.\n\\end{equation}\nThat is to say,\n\\begin{equation}\n\\Lim \\sup_{i\\geq 1} \\sup_{t\\in\\T} \\E\\left[|\\hat{y}_{t}^{n,i}|\\right]=0\n\\end{equation}\nand $\\left\\{\\sup_{i\\geq 1}\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|\\right\\}_{n=1}^{+\\infty}$\nconverges to $0$ in probability as $n\\To\\infty$.\\vspace{0.2cm}\n\nCombining (20) and (25) we get that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\Lim \\sup_{i\\geq 1}\\E\\left[\\sup_{t\\in\\T} |\\hat{y}_{t}^{n,i}|^\\beta\\right]=0.\n\\end{equation}\nIt then follows from (19), (26) and (27) that there exists a process $(y_t)_{t\\in \\T}$ which belongs to the class (D) and the space $\\bigcap_{\\beta\\in (0,1)}\\s^\\beta$ such that\n\\begin{equation}\n\\Lim \\sup_{t\\in\\T}\\E\\left[|y_t^{n}-y_t|\\right]=0\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in\\T} |y_t^{n}-y_t|^\\beta\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nFurthermore, note that $(\\hat y_t^{n,i},\\hat z_t^{n,i})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{n,i}=\\hat \\xi^{n,i}+\\int^{T}_{t}\\hat g^{n,i}(s,\\hat y_s^{n,i})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{n,i}\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $n,i\\geq 1$ and $y\\in\\R^k$,\n$\n\\hat g^{n,i}(t,y):=g^{n+i}(t,y+y^n_t)-g^n(t,y^n_t).\n$\nIt follows from assumption (H1) on $g$ together with Remark 1 that $\\as$, for each $n,i\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{n,i}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{n+i}(t,y+y^n_t)-g^{n+i}(t,y^n_t)\\rangle\n+|y||g^{n+i}(t,y^n_t)-g^n(t,y^n_t)|\\vspace{0.1cm}\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y||g(t,0)|\\mathbbm{1}_{|g(t,0)|>n},\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, by (31), Remark 1 and Lemma 1 we can check that the generator $\\hat g^{n,i}$ of BSDE (30) satisfies assumption (A1) with\n$\np=\\beta,\\ \\mu=m+2A,\\ \\nu=0,\\ f_t=|g(t,0)|\\mathbbm{1}_{|g(t,0)|>n}\\ {\\rm and}\\ \\ \\varphi_t=\\kappa({2A\\over m+2A})\n$\nfor each $m\\geq 1$ and $\\beta\\in (0,1)$. It then follows from Proposition 1 with $u=t=0$ and $p=\\beta$ that for each $m,n,i\\geq 1$, there exist a constant $C_{m,A,\\beta,T}>0$ depending only on $m, A,\\beta$ and $T$, and a constant $C>0$ such that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left(\\int_0^T |\\hat z_s^{n,i}|^2\\ {\\rm\nd}s\\right)^{\\beta\/2}\\right]&\\leq &\\Dis C_{m,A,\\beta,T}\\E\\left[\\sup\\limits_{t\\in [0,T]}|\\hat y_t^{n,i}|^\\beta\\right]+\nC\\left(\\kappa({2A\\over m+2A})\\cdot T\\right)^{\\beta\/2}\\vspace{0.1cm}\\\\\n&&\\Dis +C\\E\\left[\\left(\\int^{T}_{0}\n|g(t,0)|\\mathbbm{1}_{|g(t,0)|>n}\\mathrm{d}t\\right)^\\beta\n\\right].\n\\end{array}\n\\end{equation}\nThus, taking superemum with respect to $i$ and sending first $n\\To\\infty$ ($m$ being fixed) and then $m\\To\\infty$ in (32), by virtue of (27), (H2), Lebesgue's dominated convergence theorem and the fact that $\\kappa(\\cdot)$ is continuous function with $\\kappa(0)=0$, we can deduce that for each $\\beta\\in (0,1)$,\n$$\n\\Lim \\sup\\limits_{i\\geq 1}\\E\\left[\\left(\\int_0^T |z^{n+i}_s-z^{n}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=\\Lim \\sup\\limits_{i\\geq 1}\\E\\left[\\left(\\int_0^T |\\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right)^{\\beta\/2}\\right]=0,\n$$\nwhich means that there exists a process $(z_t)_{t\\in \\T}$ which belongs to $\\bigcap_{\\beta\\in (0,1)}\\s^\\beta$ such that\\vspace{0.1cm}\n\\begin{equation}\n\\RE\\ \\beta\\in (0,1),\\ \\ \\Lim \\E\\left[\\left(\\int_0^T |z^n_s-z_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\n\nFinally, since $\\int^{T}_{t}z^n_s\\ \\mathrm{d} B_s$ converges to $\\int^{T}_{t}z_s\\ \\mathrm{d} B_s$ under the uniform convergence in probability (ucp for short) by (33) and since $y\\mapsto g(t,y)$ is continuous and (29) holds true, we can easily check by taking limit in both sides of BSDE (15) under ucp that $(y_t,z_t)_{t\\in\\T}$ is an $L^1$ solution of BSDE (1). \\vspace{0.2cm}\\hfill $\\Box$\n\n{\\bf Remark 3}\\ Compared with the existing works, the argument between (18)-(29) involved in Gronwall's inequality, Bihari's ineqality and the relationship between convergence in $L^1$ and in probability seems to be new. \\vspace{0.2cm}\n\nWith Proposition 5 in the hand we can prove the main existence result.\\vspace{0.2cm}\n\n{\\bf Proof of Theorem 2}\\ \\ Assume that $\\xi\\in\\LT$ and the generator $g$ satisfies assumptions (H1)-(H3). Furthermore, in the case of $\\alpha\\in [1\/2,1)$ in (H3) we also assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. We will use some kind of Picard's iterative procedure. \\vspace{0.1cm}\n\nLet us set $(y^0_\\cdot,z^0_\\cdot):=(0,0)$. Note by (H3) that $g$ has a sublinear growth with respect to $z$. By virtue of H\\\"{o}lder's inequality and assumptions (H1) and (H2) of $g$, it is not hard to verify that the generator $g(t,y,z_t)$ satisfies (H1) and (H2) for each $(z_t)_{t\\in\\T}\\in\\M^1$. Thus, with the help of Proposition 5, we can define the process sequence $\\{(y^n_t,z^n_t)_{t\\in\\T}\\}_{n=1}^\\infty$ recursively,\n\\begin{equation}\ny^{n}_t=\\xi+\\int^{T}_{t}g(s,y^{n}_s,z^{n-1}_s)\\,\\mathrm{d} s-\\int^{T}_{t}z^{n}_s\\,\\mathrm{d} B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $n\\geq 1$, $(y^n_t,z^n_t)_{t\\in\\T}$ belongs to the space $\\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$, and $(y^n_t)_{t\\in\\T}$ belongs to the class (D).\\vspace{0.2cm}\n\nFor each $n,i\\geq 0$, set $\\hat{y}^{n,i}_\\cdot:=y^{n+i}_\\cdot-y^n_\\cdot$ and $\\hat z^{n,i}_\\cdot:=z^{n+i}_\\cdot-z^n_\\cdot$. Arguing as in the proof of Theorem 1, we can get, in view of (H1) and (H3), that for each $n,i\\geq 1$,\n$$|\\hat{y}^{n,i}_t|\\leq \\left(AT+G^{n,i}(t)\\right)\\cdot e^{AT},\\quad t\\in\\T,$$\nwhere\n\\begin{equation}\nG^{n,i}(t):=2\\gamma\\E\\left[\\left.\\int^{T}_{0}\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha\\ \\mathrm{d}s\\right|\\F_t\\right]\\in \\s^q\n\\end{equation}\nas soon as $\\alpha q<1$ with $q>1$. Hence, for each $n,i\\geq 1$, we know that $(\\hat y_t^{n,i})_{t\\in\\T}$ belongs to the space $\\s^q$ as soon as $\\alpha q<1$ with $q>1$. In the sequel, we will deal with two cases respectively:\n$$\n(i)\\ \\ \\alpha\\in (0,1\/2); \\quad \\quad (ii)\\ \\ \\alpha\\in [1\/2,1).\n$$\n\n{\\bf Case $(i)$}: In this case, we can pick $q=2$, then for each $n,i\\geq 1$,\n\\begin{equation}\n(\\hat y_t^{n,i})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nNote that for each $n,i\\geq 1$, $(\\hat y_t^{n,i},\\hat z_t^{n,i})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{n,i}=\\int_t^T \\bar{g}^{n,i}(s,\\hat{y}_s^{n,i})\\ {\\rm\nd}s-\\int_t^T \\hat{z}_s^{n,i}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$, $\\bar{g}^{n,i}(s,y):=g(s,y+y_s^n,z^{n+i-1}_s)-\ng(s,y_s^n,z^{n-1}_s)$. It follows from (H1), (H3) and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\bar{g}^{n,i}(t,y)\\rangle &\\leq &\\Dis \\kappa(|y|^2)+2\\gamma|y|\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha\\\\\n&\\leq & \\Dis A|y|^2+2\\gamma|y|\\left(g_s+\n|y^n_s|+|z^{n-1}_s|+|z^{n+i-1}_s|\\right)^\\alpha+A,\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\langle y,\\bar{g}^{n,i}(t,y)\\rangle \\leq \\kappa(|y|^2)+\\lam\n|y||z^{n+i-1}_t-z^{n-1}_t|=\\kappa(|y|^2)+ \\lam |y||\\hat\nz^{n-1,i}_t|,\\vspace{0.2cm}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, by (35) and (38) we know that the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) satisfies assumption (A1) with\n$p=2,\\ \\mu=A,\\ \\nu=0,\\ f_t=2\\gamma\\left(g_t+\n|y^n_t|+|z^{n-1}_t|+|z^{n+i-1}_t|\\right)^\\alpha\\ {\\rm and}\\ \\varphi_t\\equiv A$.\nThen, in view of (36), it follows from Proposition 1 with $p=2$ that $\\hat z^{n,i}_\\cdot\\in {\\rm M}^2$. Consequently, for each $n,i\\geq 1$, $(\\hat y^{n,i}_t,\\hat z^{n,i}_t)_{t\\in \\T}$ is an $L^2$ solution of BSDE (37).\\vspace{0.2cm}\n\nOn the other hand, it follows from (39) that for each $n\\geq 2$ and $i\\geq 1$, the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) also satisfies assumption (A2) with\n$\np=2,\\ \\psi(u)=\\kappa(u),\\ \\nu=0,\\ f_t=\\lam |\\hat z^{n-1,i}_t|.\n$\nThen, by Proposition 3 with $u=0$ and H\\\"{o}lder's inequality we can deduce that there exists a constant $C>0$ such that for each $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\delta T:=\\min\\left\\{{\\ln 2\\over C}, {1\\over 16\\lam^2}, {\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\\nT_j:=(T-j\\delta T)\\vee 0,\\ \\ \\RE\\ j=1,2,\\cdots\\vspace{0.2cm}\n$$\nThen for each $t\\in [T_1,T]$, we have\n\\begin{equation}\ne^{C(T-t)}\\leq 2, \\ \\ \\lam^2e^{C(T-t)}(T-t)\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (40) with (41) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [T_1,T]$,\\vspace{0.2cm}\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right].\\vspace{0.2cm}\n\\end{equation}\n\nFurthermore, note by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (42) and (41) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T | \\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nBy picking $n=2$ and $i=m-2$ in (43) we get that for each $t\\in [T_1,T]$ and $m\\geq 3$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T | z^{m-1}_s-z^1_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 2}\\E\\left[\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right]+{1\\over 2}\\E\\left[\\int_t^T | z^{m-1}_s-z^2_s|^2\\ {\\rm d}s\\right],\n\\end{array}\n$$\nfrom which we can obtain by induction that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{m\\geq 3} \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm\nd}s\\right]\\leq 8AT+\\E\\left[\\int_0^T |z^2_s-z^1_s|^2\\ {\\rm d}s\\right]<+\\infty.\n\\end{equation}\nIn addition, for each $n\\geq 2$, $i\\geq 1$ and $t\\in \\T$ we have\\vspace{0.1cm}\n\\begin{equation}\n\\Dis{1\\over 4}\\E\\left[\\int_t^T\n|\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right]\\leq \\Dis {1\\over 2}\\E\\left[\\int_t^T\n\\left(|z^{n-1+i}_s-z^2_s|^2+|z^{n-1}_s-z^2_s|^2\\right)\\ {\\rm d}s\\right]\\leq \\Dis \\sup_{m\\geq 1} \\E\\left[\\int_t^T | z^m_s-z^2_s|^2\\ {\\rm d}s\\right]\\vspace{-0.1cm}.\n\\end{equation}\nCombining (43), (45) and (44) yields that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\Dis\\sup_{n\\geq 2}\\sup_{i\\geq 1}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^2\\right]+\\E\\left[\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right]\\right)\\leq \\Dis 12AT+\\E\\left[\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right]<+\\infty.\\vspace{0.1cm}\n\\end{equation}\n\nNow, in view of (46), by first taking supremum with respect to $i$ and then taking $\\limsup$ with respect to $n$ in (42) and finally using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ together with Bihari's inequality, we can deduce the existence of processes $(Y_t,Z_t)_{t\\in [T_1,T]}\\in {\\s}^2(T_1,T;\\R^k)\\times {\\rm M}^2(T_1,T;\\R^{k\\times d})$ such that\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in [T_1,T]}|(y^n_t-y^1_t)-Y_t|^2+\\int_{T_1}^T\n|(z^n_t-z^1_t)-Z_t|^2\\ \\mathrm{d}t\\right]=0.\n\\end{equation}\nThus, note that $(y^1_t,z^1_t)_{t\\in\\T}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^1_t)_{t\\in\\T}$ belongs to the class (D). By passing to the limit in ucp for BSDE (34), in view of (47), (H2), (H3) and Lebesgue's dominated convergence theorem, we deduce that\n$\n(y_t,z_t)_{t\\in [T_1,T]}:=(Y_t+y^1_t,Z_t+z^1_t)_{t\\in [T_1,T]}\n$\nis an $L^1$ solution to the\nBSDE with parameters $(\\xi,T,g)$ on $[T_1,T]$.\\vspace{0.1cm}\n\nFinally, noticing that the $\\delta T>0$ depends only on $\\lam$ and $A$, we can find a minimal integer $N\\geq 1$ such that $T_N=0$. Thus, we can repeat, in finite steps, the above procedure to obtain an $L^1$ solution to BSDE (1) on $[T_2,T_1]$, $[T_3,T_2]$, $\\cdots$, $[0,T_{N-1}]$, and then we find an $L^1$ solution to BSDE (1) on $[0,T]$.\\vspace{0.2cm}\n\n{\\bf Case $(ii)$}: In this case, we can pick a $q\\in (1,\\bar p\\wedge {1\\over \\alpha})$, then for each $n,i\\geq 1$,\n\\begin{equation}\n(\\hat y_t^{n,i})_{t\\in\\T}\\in \\s^q.\n\\end{equation}\nNote that $q<\\bar p$ and that we also assume that equality (14) holds in this case. It follows from Proposition 1 in \\citet{Fan15} that $g$ also satisfies (H1b)$_{\\bar p}$ and then (H1b)$_q$. Note further that (37) and (38) is also true. In the same way as that in case $(i)$, it follows from Proposition 1 with $p=q$ that, in view of (48) and (35), $\\hat z^{n,i}_t\\in {\\rm M}^q$. Consequently, for each $n,i\\geq 1$, $(\\hat y^{n,i}_t,\\hat z^{n,i}_t)_{t\\in \\T}$ is an $L^q$ solution of BSDE (37).\\vspace{0.2cm}\n\nFurthermore, it follows from (H1b)$_q$ and (H3) of the generator $g$ together with Remark 1 that $\\as$, for each $y\\in \\R^k$,\\vspace{0.2cm}\n$$\n\\left\\langle {y\\over |y|}\\mathbbm{1}_{|y|\\neq 0}, \\bar{g}^{n,i}(t,y)\\right\\rangle \\leq \\varrho^{1\\over q}(|y|^q)+\\lam |\\hat z^{n-1,i}_t|,\n$$\nwhere $\\varrho(\\cdot)$ is defined in (H1b)$_q$. Then, for each $n\\geq 2$ and $i\\geq 1$, the generator $\\bar g^{n,i}(t,y)$ of BSDE (37) satisfies assumption (A3) with\n$\np=q,\\ \\phi(u)=\\varrho(u),\\ \\nu=0,\\ f_t=\\lam |\\hat z^{n-1,i}_t|.\n$\nThen, by Proposition 4 with $u=0$ and $p=q$ together with H\\\"{o}lder's inequality we can deduce that there exists a constant $C_q>0$ depending only on $q$ such that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&& \\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\\\\n&\\leq &\\Dis e^{C_q(T-t)}\\left\\{\\int_t^T\\varrho\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^q \\right]\\right){\\rm d}s+\\lam^q (T-t)^{q\/2}\\E\\left[\\left(\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\overline{\\delta T}:=\\min\\left\\{{\\ln 2\\over C_q}, \\left({1\\over 16\\lam^q}\\right)^{2\\over q},{\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\ \\bar T_j:=(T-j\\overline{\\delta T})\\vee 0,\\ \\ \\RE\\ j=1,2,\\cdots\\vspace{0.2cm}\n$$\nThen for each $t\\in [\\bar T_1,T]$, we have\n\\begin{equation}\ne^{C_q(T-t)}\\leq 2, \\ \\ \\lam^q e^{C_q(T-t)}(T-t)^{q\/2}\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (49) with (50) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\begin{array}{ll}\n& \\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\hat\ny^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\vspace{0.1cm}\\\\\n\\leq &\\Dis 2\\int_t^T\\varrho\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\hat y^{n,i}_r|^q \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\left(\\int_t^T |\\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right].\n\\end{array}\n\\end{equation}\n\nFurthermore, note by Remark 1 that $\\varrho(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (51) and (50) yields that for each $n\\geq 2$, $i\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T |\\hat z^{n,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\n&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\left(\\int_t^T | \\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]\\right)\\cdot e^{2A(T-t)}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\left(\\int_t^T | \\hat z^{n-1,i}_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right],\\vspace{-0.2cm}\n\\end{array}\n\\end{equation}\nfrom which, in view of $q\\in (1,2)$ and the basic inequality\n$$\\left(\\int_t^T(a_s+b_s)^2\\ \\mathrm{d}s\\right)^{q\/2}\\leq 2\\left[\\left(\\int_t^T a^2_s\\ \\mathrm{d}s\\right)^{q\/2}+\\left(\\int_t^T b^2_s\\ \\mathrm{d}s\\right)^{q\/2}\\right]$$\nfor each $a_s,b_s\\in L^2([t,T])$, by a similar argument to that in case $(i)$ we can deduce that for each $n\\geq 1$ and $t\\in [\\bar T_1,T]$,\n\\begin{equation}\n\\Dis\\sup_{n\\geq 2}\\sup_{i\\geq 1}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\hat y^{n,i}_r|^q\\right]+\\E\\left[\\left(\\int_t^T | \\hat z^{n,i}_s|^2\\ {\\rm\nd}s\\right)^{q\/2}\\right]\\right)\\\\\n\\leq \\Dis 12AT+\\E\\left[\\left(\\int_0^T |z^2_s- z^1_s|^2\\ {\\rm d}s\\right)^{q\/2}\\right]<+\\infty.\n\\end{equation}\n\nNow, in view of (53), by first taking supremum with respect to $i$ and then taking $\\limsup$ with respect to $n$ in (51) and finally using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ together with Bihari's inequality, we can deduce the existence of processes $(Y_t,Z_t)_{t\\in [\\bar T_1,T]}\\in {\\s}^q(\\bar T_1,T;\\R^k)\\times {\\rm M}^q(\\bar T_1,T;\\R^{k\\times d})$ such that\n\\begin{equation}\n\\Lim \\E\\left[\\sup_{t\\in [\\bar T_1,T]}|(y^n_t-y^1_t)-Y_t|^q+\\left(\\int_{\\bar T_1}^T\n|(z^n_t-z^1_t)-Z_t|^2\\mathrm{d}t\\right)^{q\/2}\\right]=0.\n\\end{equation}\nThus, note that $(y^1_t,z^1_t)_{t\\in\\T}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^1_t)_{t\\in\\T}$ belongs to the class (D). By passing to the limit in ucp for BSDE (34), in view of (54), (H2), (H3) and Lebesgue's dominated convergence theorem, we deduce that\n$\n(y_t,z_t)_{t\\in [\\bar T_1,T]}:=(Y_t+y^1_t,Z_t+z^1_t)_{t\\in [\\bar T_1,T]}\n$\nis an $L^1$ solution to the\nBSDE with parameters $(\\xi,T,g)$ on $[\\bar T_1,T]$.\\vspace{0.1cm}\n\nFinally, noticing that the positive real number $\\overline{\\delta T}$ depends only on $q$, $\\lam$ and $A$, we can find a minimal integer $\\bar N\\geq 1$ such that $T_{\\bar N}$=0. Thus, we can repeat, in finite steps, the above procedure to obtain an $L^1$ solution to BSDE (1) on $[\\bar T_2,\\bar T_1]$, $[\\bar T_3,\\bar T_2]$, $\\cdots$, $[0,\\bar T_{\\bar N-1}]$, and then we find an $L^1$ solution to BSDE (1) on $[0,T]$. The proof of Theorem 2 is finally completed.\\vspace{0.2cm} \\hfill $\\Box$\n\n{\\bf Remark 4}\\ \\ We would like to mention that it is interesting that in the case of $\\alpha$ in (H3) values in $[1\/2,1)$, the assumption (H1) in Theorem 2 needs to be replaced with the stronger assumption (H1)${}_{\\bar p}$. The main reason is to ensure obtaining the key inequality (49). This indicates the difference between Propositions 3 and 4. In addition, we point out that how to divide appropriately the time interval $\\T$ is also one of key problems in the proof of Theorem 2.\n\n\n\\section{Existence and Uniqueness of the solution in $\\s^1\\times{\\rm M^1}$ and examples}\n\nIn this section, by virtue of Theorems 1 and 2 we will establish an existence and uniqueness result of the solution in the space $\\s^1\\times {\\rm M}^1$ (a new type of $L^1$ solution) for multidimensional BSDEs with generators of one-sided Osgood type. This is the first time to the best of our knowledge. We will also provide two examples in this section to illustrate our theoretical results.\\vspace{0.2cm}\n\n{\\bf Theorem 3} Assume that the generator $g$ satisfies assumptions (H1)-(H3). In the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, we also assume that there exists a constant $\\bar p>1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies (14). If the following assumption (H4) holds true:\\\\\n\n{\\bf (H4)}\\ $\\Dis \\E\\left[\\sup\\limits_{t\\in\\T}\\left(\\E\\left[|\\xi|\n+\\left.\\int_0^T |g(s,0,0)|\\ {\\rm d}s\\right|\\F_t\\right]\\right)\\right]<+\\infty.\n\\vspace{0.4cm}\n$\n\n\\noindent then BSDE (1) admits a unique solution $(y_t,z_t)_{t\\in\\T}$ in $\\s^1(0,T;\\R^k)\\times{\\rm M}^1(0,T;\\R^{k\\times d})$.\\vspace{0.3cm}\n\n{\\bf Proof}\\ \\ It is clear that the uniqueness follows from Theorem 1 directly. Note that if (H4) holds true, then $\\xi\\in \\LT$. It follows from Theorems 1-2 that BSDE (1) admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$, i.e., $(y_t)_{t\\in\\T}$ belongs to the class (D) and $(y_t,z_t)_{t\\in\\T}\\in \\s^\\beta\\times {\\rm M}^\\beta$ for each $\\beta\\in (0,1)$. Hence, in order to complete the proof of Theorem 3, it remains to show, under (H1)-(H4),\n$$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times{\\rm M}^1(0,T;\\R^{k\\times d}).$$\nIn fact, let us fix $n\\geq 1$ and denote $\\tau_n$ the stopping time\n$$\n\\tau_n:=\\inf\\left\\{t\\in\\T:\\int^t_{0}|z_s|^2\\\n\\mathrm{d}s\\geq n\\right\\}\\wedge T.\n$$\nBy Corollary 2.3 in \\cite{Bri03} we know that for each $t\\in\\T$,\\vspace{0.1cm}\n\\begin{equation}\n|y_{t\\wedge \\tau_n}|\\leq |y_{\\tau_n}|+\\int_{t\\wedge \\tau_n}^{\\tau_n}\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0}, g(s,y_s,z_s)\\right\\rangle{\\rm d}s-\\int_{t\\wedge \\tau_n}^{\\tau_n}\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0},z_s {\\rm d}B_s\\right\\rangle.\\vspace{0.1cm}\n\\end{equation}\nAnd, it follows from (H1) and (H3) that $\\ass$,\n\\begin{equation}\n\\left\\langle {y_s\\over |y_s|}\\mathbbm{1}_{|y_s|\\neq 0}, g(s,y_s,z_s)\\right\\rangle\\leq \\rho(|y_s|)+|g(s,0,0)|+\\gamma(g_s+|z_s|)^{\\alpha}.\n\\end{equation}\nThus, in view of (55) and (56), using a similar argument to that in the proof of Theorem 1 we can get that for each $t\\in \\T$,\n\\begin{equation}\n|y_t|\\leq (AT+\\bar G(t))\\cdot e^{AT},\n\\end{equation}\nwhere\n$$\\bar G(t):=\\E\\left[|\\xi|+\\left.\\int_0^T|g(s,0,0)|\\ \\mathrm{d}s+\\gamma\\int_0^T(g_s+|z_s|)^{\\alpha}\\\n\\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T.\\vspace{0.2cm}\n$$\nFurthermore, it follows from (H4) and a similar argument to obtain (5) that $\\bar G(\\cdot)\\in \\s^1(0,T;\\R^k)$ and then, in view of (57), $(y_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)$. Finally, note by (H1) and (H3) together with Remark 1 that $\\as$, for each $(y,z)\\in \\R^k\\times\\R^{k\\times d}$,\n$$\n\\langle y, g(t,y,z) \\rangle\\leq \\bar\\kappa(|y|^2)+\\lambda |y||z|+|y||g(t,0,0)|\\leq A|y|^2+\\lambda |y||z|+|y||g(t,0,0)|+A,\n$$\nwhere the function $\\bar\\kappa(\\cdot)$ is defined in (H1a)$_2$. It follows from Proposition 1 with $p=1$ and $u=t=0$ that $(z_t)_{t\\in\\T}\\in {\\rm M}^1(0,T;\\R^{k\\times d})$. Theorem 3 is then proved.\\vspace{0.2cm} \\hfill $\\Box$\n\nBy Theorems 1-3 and Remarks 1 and 2 the following corollary is immediate.\\vspace{0.1cm}\n\n{\\bf Corollary 1} Assume that the generator $g$ satisfies assumption (H1b)$_p$ for some $p>1$, (H2) and (H3). Then, for each $\\xi\\in\\LT$, BSDE (1) admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. Furthermore, if (H4) holds true, then\n$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d})$.\\vspace{0.2cm}\n\n{\\bf Remark 5} Note that if the generator $g$ satisfies the monotonicity condition used in \\citet{Bri03}, then it must satisfy (H1b)$_p$ for all $p>1$. Theorems 1-3 of this paper generalize Theorems 6.2 and 6.3 in \\cite{Bri03}.\\vspace{0.2cm}\n\n{\\bf Example 1} Let $k=1$ and\n$$\ng(\\omega,t,y,z)=h(|y|)-e^{|B_t(\\omega)|\\cdot y}+(e^{-y}\\wedge 1)\\cdot \\sin |z|+{1\\over \\sqrt{t}}\\mathbbm{1}_{t>0},\n$$\nwhere\n$$\nh(x)=\\left\\{\n\\begin{array}{lll}\n-x|\\ln x|& ,&0 \\delta;\\\\\n0& ,&{\\rm other\\ cases}\n\\end{array}\\right.\\vspace{0.1cm}\n$$\nwith $\\delta>0$ small enough.\n\nIt is not hard to check that $g$ satisfies assumptions (H2) and (H3) with $\\lam =1$ and any $\\alpha\\in (0, 1\/2)$. Furthermore, note that $e^{-\\beta y}$ is decreasing in $y$ for each $\\beta\\geq 0$, $h(\\cdot)$ is concave and sub-additive and then the following inequality holds: $\\as$,\n$$\n\\RE\\ y_1,y_2,z,\\ \\\n\\left\\langle {y_1-y_2\\over |y_1-y_2|}\\mathbbm{1}_{|y_1-y_2|\\neq 0},\ng(\\omega,t,y_1,z)-g(\\omega,t,y_2,z)\\right\\rangle\\leq\nh(|y_1-y_2|)\n$$\nwith $\\int_{0^+} {{\\rm d}u\\over h(u)}=+\\infty$.\nIt follows that $g$ also satisfies assumption (H1). Then, by Theorems 1-2 we know that for each $\\xi\\in\\LT$, the BSDE with the parameters $(\\xi,T,g)$ admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. Moreover, by Theorem 3 we also know that if (H4) holds true for $\\xi$ and $g(t,0,0)$, then $$(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d}).$$\n\n{\\bf Example 2}\\ Let $y=(y_1,\\cdots,y_k)$ and\n$g(\\omega,t,y,z)=(g_1(\\omega,t,y,z),\\cdots,g_k(\\omega,t,y,z))$, where for each $i=1,\\cdots,k$,\n$$\ng_i(\\omega,t,y,z):=e^{-y_i}+\\bar h(|y|)+\\left(|z|^2\\wedge |z|^{2\/3}\\right)+|B_t(\\omega)|,\n$$\nwith\n$$\n\\bar h(x)=\\left\\{\n\\begin{array}{lll}\n-x|\\ln x|^{1\/p}& ,&0 \\delta;\\\\\n0& ,&{\\rm other\\ cases}\n\\end{array}\\right.\\vspace{0.2cm}\n$$\nwith $\\delta>0$ small enough and $p>1$.\n\nIn the same way as in Example 1, we can check that this generator $g$ satisfies assumptions (H1b)$_p$ with function $\\bar h(\\cdot)$, (H2) and (H3) with $\\lam =1$ and $\\alpha=2\/3$. It then follows from Corollary 1 that for each $\\xi\\in\\LT$, the BSDE with the parameters $(\\xi,T,g)$ admits a unique $L^1$ solution $(y_t,z_t)_{t\\in\\T}$. And, if (H4) holds true for $\\xi$ and $g(t,0,0)$, then $(y_t,z_t)_{t\\in\\T}\\in \\s^1(0,T;\\R^k)\\times {\\rm M}^1(0,T;\\R^{k\\times d}).$\n\n\\section{Stability of the $L^1$ solutions and the solutions in $\\s^1\\times {\\rm M}^1$}\n\nIn this section, enlightened by the proof of Proposition 5 and Theorems 2-3, we shall put forward and prove the stability theorems of the $L^1$ solutions and the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs with generators of one-sided Osgood type. To the best of our knowledge, this is the first time for the $L^1$ solution of multidimensional BSDEs.\\vspace{0.2cm}\n\nIn the sequel, for each $m\\in \\N$, let $\\xi^m\\in \\LT$ and let $(y_t^m, z_t^m)_{t\\in [0,T]}$ be\nan $L^1$ solution of\nthe following BSDEs depending on parameter $m$:\n\\begin{equation}\ny_t^m=\\xi^m+\\int_t^T g^m(s,y_s^m,z_s^m)\\ {\\rm d}s -\\int_t^T z_s^m{\\rm d}B_s,\\quad t\\in [0,T].\n\\end{equation}\nFurthermore, we introduce the following assumptions:\\vspace{0.2cm}\n\n{\\bf (B1)}\\ All $g^m$ satisfy assumptions (H1)-(H3)\n with the same parameters $\\rho(\\cdot)$, $\\lambda$, $\\gamma$, $g_t$ and $\\alpha$. Furthermore, in the case of $\\alpha\\in [1\/2,1)$ we assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. \\vspace{0.2cm}\n\n{\\bf (B2)}\\ There exists a nonnegative real number sequence\n$\\{a_m\\}_{m=1}^{+\\infty}$ satisfying $\\lim\\limits_{m\\To \\infty}a_m=0$ such that\n$\\as$, for each $m\\geq 1$,\n\\begin{equation}\n\\RE\\ (y,z)\\in \\R^k\\times \\R^{k\\times d},\\ \\ |g^m(\\omega,t,y,z)-g^0(\\omega,t,y,z)|\\leq a_m.\n\\end{equation}\nAnd,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[|\\xi^{m}-\\xi^{0}|\\right]=0.\n\\vspace{0.3cm}\n\\end{equation}\n\nThe following Theorem 4 is the stability theorem of $L^1$ solutions.\\vspace{0.1cm}\n\n{\\bf Theorem 4}\\ Under assumptions (B1) and (B2), we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^m-y_t^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|^\\beta+\\left(\\int_0^T |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\\vspace{0.1cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ For each $ m\\in \\N$, set $(y^{m,0}_\\cdot,z^{m,0}_\\cdot):=(0,0)$ and, similar to the beginning part of the proof of Theorem 2, define recursively the process sequence $\\{(y^{m,n}_\\cdot,z^{m,n}_\\cdot)\\}_{n=1}^{+\\infty}$ by the $L^1$ solutions of the following BSDEs\n\\begin{equation}\ny_t^{m,n}=\\xi^m+\\int_t^T g^{m,n}(s,y_s^{m,n},z_s^{m,n-1})\\ {\\rm d}s -\\int_t^T z_s^{m,n}{\\rm d}B_s,\\ \\ t\\in [0,T],\\vspace{0.2cm}\n\\end{equation}\nwhere $(y^{m,n}_t,z^{m,n}_t)_{t\\in [0,T]}\\in \\s^\\beta\\times \\M^\\beta$ for each $\\beta\\in (0,1)$ and $(y^{m,n}_t)_{t\\in\\T}$ belongs to the class (D) for each $m,n\\in \\N$.\\vspace{0.2cm}\n\nIn the sequel, note that $\\as$,\n\\begin{equation}\n|y_t^m -y_t^0| \\leq |y_t^m -y_t^{m,n}| + |y_t^{m,n} -y_t^{0,n}|+|y_t^{0,n} -y_t^0|\\leq \\Dis 2\\sup_{m\\geq 0}|y_t^{m,n}-y_t^m|+|y_t^{m,n} -y_t^{0,n}|\n\\end{equation}\nand\n\\begin{equation}\n|z_t^m -z_t^0|\\leq |z_t^m -z_t^{m,n}| + |z_t^{m,n} -z_t^{0,n}|+|z_t^{0,n} -z_t^0|\\leq \\Dis 2\\sup_{m\\geq 0}|z_t^{m,n}-z_t^m|+|z_t^{m,n} -z_t^{0,n}|.\n\\end{equation}\nWe will estimate, respectively, every term of the right hand in (64) and (65).\\vspace{0.2cm}\n\nFirstly, the following Proposition 6 gives the estimates with respect to the second term of the right hand in (64) and (65).\\vspace{0.1cm}\n\n{\\bf Proposition 6}\\ \\ For each $n\\geq 1$, we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^{m,n}-y_t^{0,n}|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^{m,n}-y_t^{0,n}|^\\beta+\\left(\\int_0^T |z_s^{m,n}-z_s^{0,n}|^2\\ {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\n\n{\\bf Proof}\\ \\ We first consider the case of $n=1$. Let us fix $k,m\\geq 1$ and denote $\\tau^m_k$ the stopping time\n$$\n\\tau^m_k:=\\inf\\left\\{t\\in\\T:\\int^t_{0}(|z^{m,1}_s|^2\n+|z^{0,1}_s|^2)\\ \\mathrm{d}s\\geq k\\right\\}\\wedge T.\n$$\nCorollary 2.3 in \\cite{Bri03} leads to the following inequality\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis|\\hat{y}^{m,1}_{t\\wedge\\tau^m_k}|&\\leq &\\Dis |\\hat{y}^{m,1}_{\\tau^m_k}|+\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\n\\left\\langle {\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s| }\\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^0(s,y^{0,1}_s,0)\\right\\rangle\\\n\\mathrm{d}s\\vspace{0.1cm}\\\\\n&&\\Dis -\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\\left\\langle {\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|}\\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},\\hat{z}^{m,1}_s\\ \\mathrm{d} B_s\\right\\rangle,\\ \\ t\\in\\T,\n\\end{array}\n\\end{equation}\nwhere and hereafter\n$$\\hat{y}^{m,1}_\\cdot:=y^{m,1}_\\cdot-y^{0,1}_\\cdot\\ \\ {\\rm and}\\ \\ \\hat{z}^{m,1}_\\cdot:=z^{m,1}_\\cdot-z^{0,1}_\\cdot.$$\nAnd, it follows from assumption (H1) of $g^m$ and (59) that $\\ass$,\n\\begin{equation}\n\\begin{array}{ll}\n&\\Dis \\left\\langle{\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|} \\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^0(s,y^{0,1}_s,0)\n\\right\\rangle\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\left\\langle{\\hat{y}^{m,1}_s\\over |\\hat{y}^{m,1}_s|} \\mathbbm{1}_{|\\hat{y}^{m,1}_s|\\neq 0},g^m(s,y^{m,1}_s,0)-g^m(s,y^{0,1}_s,0)\n\\right\\rangle+|g^m(s,y^{0,1}_s,0)-g^0(s,y^{0,1}_s,0)|\n\\vspace{0.1cm}\\\\\n\\leq &\\Dis \\rho(|\\hat{y}^{m,1}_s|)+a_m.\n\\end{array}\n\\end{equation}\nThen, combining (68) with (69) we can deduce that for each $k,m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,1}_{t\\wedge\\tau^m_k}|\\leq a_mT+\\E\\left[\n|\\hat{y}^{m,1}_{\\tau^m_k}|\n+\\left.\\int^{\\tau^m_k}_{t\\wedge\\tau^m_k}\n\\rho(|\\hat{y}^{m,1}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in\\T.\n\\end{equation}\nSince $\\hat y^{m,1}_\\cdot$ belongs to the class (D), and $\\rho(\\cdot)$ increases at most linearly, we can send $k$ to $+\\infty$ in (70) and use Lebesgue's dominated convergence theorem, in view of $\\tau^m_k\\To T$ as $k\\To \\infty$ and $\\hat y^{m,1}_T=\\xi^m-\\xi^0$, to get that for each $m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,1}_t| \\leq H_m(t) +\\E\\left[\\left.\\int_t^T\\rho(|\\hat{y}^{m,1}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T,\n\\end{equation}\nwhere\n$$H_m(t):=a_mT+\\E\\left[\\left.|\\xi^m-\\xi^0|\n\\right|\\F_t\\right].\\vspace{0.2cm}$$\nIn the sequel, note by Lemma 6.1 in \\cite{Bri03} and assumption (B2) that\n\\begin{equation}\n\\RE\\ \\beta\\in (0,1),\\ \\ \\sup_{m\\geq 1}\\E\\left[\\sup_{t\\in\\T}|H_m(t)|^\\beta\n\\right]\\leq {1\\over 1-\\beta}\\sup_{m\\geq 1} \\left(\\E\\left[H_m(T)\\right]\\right)^\\beta<+\\infty\n\\end{equation}\nand\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[|H_m(T)|\\right]=0.\\vspace{0.2cm}\n\\end{equation}\nArguing as that from (18) to (29) we can deduce that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\left(\\sup\\limits_{t\\in [0,T]}\\E\\left[|\\hat y_t^{m,1}|\\right]+\n\\E\\left[\\sup\\limits_{t\\in [0,T]} |\\hat y_t^{m,1}|^\\beta\\right]\\right)=0.\n\\end{equation}\nFurthermore, note that for each $m\\geq 1$, $(\\hat y_t^{m,1},\\hat z_t^{m,1})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{m,1}=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^{m,1}(s,\\hat y_s^{m,1})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{m,1}\\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\n\\hat g^{m,1}(t,y):=g^{m}(t,y+y^{0,1}_t,0)-g^0(t,y^{0,1}_t,0).\n$\nIt follows from assumption (H1) on $g^m$ together with (59) that $\\as$, for each $m\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{m,1}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{m}(t,y+y^{0,1}_t,0)-g^{m}(t,y^{0,1}_t,0)\\rangle +|y||g^{m}(t,y^{0,1}_t,0)-g^0(t,y^{0,1}_t,0)|\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y|a_m,\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$. Thus, in view of (74)-(76), Proposition 1, (B2) and the assumption of $\\kappa(\\cdot)$, a similar argument to that from (30) to (33) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T |\\hat z^{m,1}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\nFrom (74) and (77), we know that (66) and (67) hold true for $n=1$.\\vspace{0.2cm}\n\nNow, let us fix arbitrarily a $n\\geq 2$ and assume that (66) and (67) hold true for $n-1$. In the sequel, we will prove that they also hold for $n$. For each $m\\geq 1$, define\n$$\\hat{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^{0,n}_\\cdot\\ \\ {\\rm and}\\ \\ \\hat{z}^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^{0,n}_\\cdot.$$\nThen, $(\\hat y_t^{m,n},\\hat z_t^{m,n})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\hat{y}_t^{m,n}=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^{m,n}(s,\\hat y_s^{m,n})\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^{m,n}\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\n\\hat g^{m,n}(t,y):=g^{m}(t,y+y^{0,n}_t,z^{m,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t).\n$\nIt follows from assumption (H1) of $g^m$ together with Remark 1 and (59) that $\\as$, for each $m\\geq 1$ and $y\\in \\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat g^{m,n}(t,y)\\rangle&\\leq &\\Dis \\langle y,g^{m}(t,y+y^{0,n}_t,z^{m,n-1}_t)-\ng^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\\rangle\\vspace{0.1cm}\\\\\n&&\\Dis +|y||g^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\n-g^m(t,y^{0,n}_t,z^{0,n-1}_t)|\\vspace{0.1cm}\\\\\n&&\\Dis +|y||g^{m}(t,y^{0,n}_t,z^{0,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t)|\\vspace{0.1cm}\\\\\n&\\leq &\\Dis \\kappa(|y|^2)+|y|\\cdot(\\triangle_t^{m,n}+a_m),\n\\end{array}\n\\end{equation}\nwhere the function $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and, in view of (H3),\n\\begin{equation}\n\\triangle_t^{m,n}:=|g^{m}(t,y^{0,n}_t,z^{m,n-1}_t)\n-g^m(t,y^{0,n}_t,z^{0,n-1}_t)|\\leq \\lambda |z^{m,n-1}_t-z^{0,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\triangle_t^{m,n}\\leq 2\\gamma (g_t+|y^{0,n}_t|+|z^{m,n-1}_t|+|z^{0,n-1}_t|)^\\alpha.\n\\end{equation}\nSince (67) holds true for $n-1$, from (80) we know that the sequence of random variables $\\{\\int_0^T \\triangle_t^{m,n}\\ {\\rm d}t\\}_{m=1}^{+\\infty}$ converges in probability to $0$ as $m\\To \\infty$, and from (81) and H\\\"{o}lder's inequality that for each $q>1$ satisfying $\\alpha q<1$,\n\\begin{equation}\n\\sup_{m\\geq 1}\\E\\left[\\left(\\int_0^T \\triangle_t^{m,n}\\ {\\rm d}t\\right)^q\\right]\\leq K_n\\left(1+\\sup_{m\\geq 1}\\E\\left[\\left(\\int_0^T|z^{m,n-1}_t|^2 \\ {\\rm d}t\\right)^{\\alpha q\\over 2}\\right] \\right)<+\\infty,\n\\end{equation}\nwhere $K_n>0$ is a constant independent of $m$. Hence, for each $q'>1$ satisfying $\\alpha q'<1$, we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T\\triangle_t^{m,n} \\\n{\\rm d}t\\right)^{q'}\\right]=0.\\vspace{0.1cm}\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$ and (79) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\hat{y}^{m,n}_t\\over |\\hat{y}^{m,n}_t|} \\mathbbm{1}_{|\\hat{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^0(t,y^{0,n}_t,z^{0,n-1}_t)\n\\right\\rangle\\leq \\rho(|\\hat{y}^{m,n}_t|)+\\triangle_t^{m,n}+a_m.\n\\end{equation}\nArguing as that from (68) to (71), in view of (82), (83) and (84), we can obtain that for each $m\\geq 1$,\n\\begin{equation}\n|\\hat{y}^{m,n}_t| \\leq H^n_m(t) +\\E\\left[\\left.\\int_t^T\\rho(|\\hat{y}^{m,n}_s|)\\ \\mathrm{d}s\\right|\\F_t\\right],\\ \\ t\\in \\T,\n\\end{equation}\nwhere\n$$H^n_m(t):=a_mT+\\E\\left[\\left.|\\xi^m-\\xi^0|\n+\\int_0^T\\triangle_s^{m,n} \\\n{\\rm d}s\\right|\\F_t\\right].\\vspace{0.1cm}$$\nFurthermore, in view of (72), (73), (82), (83) and (85), a similar argument to that from (18) to (29) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\left(\\sup\\limits_{t\\in [0,T]}\\E\\left[|\\hat y_t^{m,n}|\\right]+\n\\E\\left[\\sup\\limits_{t\\in [0,T]} |\\hat y_t^{m,n}|^\\beta\\right]\\right)=0.\n\\end{equation}\nFinally, in view of (78), (79), (83), Proposition 1, (B2) and the assumption of $\\kappa(\\cdot)$, using a similar argument to that from (30) to (33) yields that for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty} \\E\\left[\\left(\\int_0^T |\\hat z^{m,n}_s|^2\\\n{\\rm d}s\\right)^{\\beta\/2}\\right]=0.\n\\end{equation}\nIn view of (86) and (87), we have proved that (66) and (67) hold also true for $n$. Thus, by induction the proof of Proposition 6 is completed.\\hfill $\\Box$\\vspace{0.2cm}\n\nNext, we turn to the estimates with respect to the first term of the right hand in (64) and (65).\\vspace{0.2cm}\n\n{\\bf Proposition 7}\\ \\ In the case when the $\\alpha$ defined in (H3) values in $(0,1\/2)$, there exists a positive real number $\\delta T>0$ depending only on $\\lam$ and $A$ such that for each $t\\in [T_1, T]$ with $T_1:=(T-\\delta T)\\vee 0$,\n\\begin{equation}\n\\Lim \\sup_{m\\geq 0}\\E\\left[\\sup_{r\\in [t,T]}|y_r^{m,n}-y_r^m|^2+\\int_t^T |z_s^{m,n}-z_s^m|^2\\ {\\rm d}s\\right]=0.\n\\end{equation}\nIn the case when the $\\alpha$ defined in (H3) values in $[1\/2,1)$, for each $q\\in (1,\\bar p\\wedge {1\\over \\alpha})$, there exists a positive real number $\\overline{\\delta T}>0$ depending only on $q$, $\\lam$ and $A$ such that for each $t\\in [\\bar T_1, T]$ with $\\bar T_1:=(T-\\overline{\\delta T})\\vee 0$,\n\\begin{equation}\n\\Lim \\sup_{m\\geq 0}\\E\\left[\\sup_{r\\in [t,T]}|y_r^{m,n}-y_r^m|^q+\\left(\\int_t^T |z_s^{m,n}-z_s^m|^2\\ {\\rm d}s\\right)^{q\/2}\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ We only prove the case when the $\\alpha$ in (H3) values in $(0,1\/2)$. In view of the proof of case $(ii)$ in Theorem 2, another case can be proved in the same way.\\vspace{0.2cm}\n\nNow, we assume that $\\alpha\\in (0,1\/2)$ and set $\\bar{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^m_\\cdot$ and $\\bar z^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^m_\\cdot$ for each $m,n\\geq 0$. Note that for each $m\\geq 0$ and $n\\geq 1$, $(\\bar y_t^{m,n},\\bar z_t^{m,n})_{t\\in\\T}$ is a solution of the following BSDE:\n\\begin{equation}\n\\bar{y}_t^{m,n}=\\int_t^T \\bar{g}^{m,n}(s,\\bar{y}_s^{m,n})\\ {\\rm\nd}s-\\int_t^T \\bar{z}_s^{m,n}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$,\n$\\bar{g}^{m,n}(s,y):=g^m(s,y+y_s^m,z^{m,n-1}_s)-\ng^m(s,y_s^m,z^m_s).$\nIt follows from (H1) of $g^m$ and Remark 1 that $\\as$, for each $y\\in\\R^k$,\\vspace{-0.2cm}\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\bar{g}^{m,n}(t,y)\\rangle &\\leq &\\langle y,g^m(t,y+y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^{m,n-1}_t)\n\\rangle\\\\\n&& \\Dis +|y||g^m(t,y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^m_t)|\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+|y|\\bar\\triangle^{m,n}_t\n\\end{array}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and in view of (H3) of $g^m$,\n\\begin{equation}\n\\bar\\triangle^{m,n}_t:=g^m(t,y_t^m,z^{m,n-1}_t)-g^m(t,y_t^m,z^m_t)\\leq \\lam |z^{m,n-1}_t-z^m_t|=\\lam |\\bar z^{m,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\bar\\triangle^{m,n}_t\\leq 2\\gamma \\left(g_t+\n|y^m_t|+|z^m_t|+|z^{m,n-1}_t|\\right)^\\alpha.\\vspace{0.2cm}\n\\end{equation}\nIn view of $\\alpha\\in (0,1\/2)$ and (93), using H\\\"{o}lder's inequality, Jensen's inequality and Doob's inequality yields that for each $m\\geq 0$ and $n\\geq 1$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\bar\\triangle^{m,n}_t\\ {\\rm d}t\\right)^2\\right]<+\\infty\\ \\ {\\rm and\\ then}\\ \\\n\\E\\left[\\left.\\int_0^T \\bar\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\in \\s^2.\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$ and (91) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\bar{y}^{m,n}_t\\over |\\bar{y}^{m,n}_t|} \\mathbbm{1}_{|\\bar{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^m(t,y^m_t,z^m_t)\n\\right\\rangle\\leq \\rho(|\\bar{y}^{m,n}_t|)+\\bar\\triangle_t^{m,n}.\n\\end{equation}\nThus, in view of (95) and (94), arguing as that from (2) to (7), by virtue of Gronwall's inequality we can obtain that for each $m\\geq 0$ and $n\\geq 1$,\n\\begin{equation}\n(\\bar y_t^{m,n})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nFurthermore, in view of (96), (91), (94) and Remark 1, it follows from Proposition 1 that $(\\bar z_t^{m,n})_{t\\in\\T}\\in {\\rm M}^2$. Thus, $(\\bar y_t^{m,n},\\bar z_t^{m,n})_{t\\in\\T}$ is an $L^2$ solution of BSDE (90) for each $m\\geq 0$ and $n\\geq 1$.\\vspace{0.1cm}\n\nIn the sequel, in view of (91) and (92), by Proposition 3 and H\\\"{o}lder's inequality we can deduce the existence of a constant $C>0$ such that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [0,T]$,\n\\begin{equation}\n\\hspace*{-0.8cm}\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\bar\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\bar y^{m,n}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right\\}.\n\\end{array}\n\\end{equation}\nNow, let\n$$\n\\delta T:=\\min\\left\\{{\\ln 2\\over C}, {1\\over 16\\lam^2}, {\\ln 2\\over 2A}\\right\\}\\ \\ {\\rm and}\\ \\\nT_1:=(T-\\delta T)\\vee 0.\\vspace{0.2cm}\n$$\nThen for each $t\\in [T_1,T]$, we have\n\\begin{equation}\ne^{C(T-t)}\\leq 2, \\ \\ \\lam^2e^{C(T-t)}(T-t)\\leq {1\\over 8},\\ \\ e^{2A(T-t)}\\leq 2.\n\\end{equation}\nCombining (97) with (98) yields that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\\vspace{0.2cm}\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\bar\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\bar y^{m,n}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{equation}\nFurthermore, note by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality with (99) and (98) yields that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm\nd}s\\right]&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\bar z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nBy the above inequality (100) together with the inequality\n$$|\\bar z^{m,n-1}_s|^2\\leq 2(|z^{m,n}_s-z_s^m|^2+|z^{m,n}_s-z_s^{m,n-1}|^2)\n=2(|\\bar z^{m,n}_s|^2+|\\tilde{z}^{m,n}_s|^2),$$\nwe can get that for each $m\\geq 0$, $n\\geq 2$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+{1\\over 2}\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\leq 4AT+{1\\over 2}\\E\\left[\\int_t^T\n|\\tilde{z}^{m,n}_s|^2\\ {\\rm d}s\\right],\n\\end{equation}\nwhere and hereafter, for each $m\\geq 0$ and $n\\geq 1$ we define\n$$\\tilde{y}^{m,n}_\\cdot:=y^{m,n}_\\cdot-y^{m,n-1}_\\cdot\\ \\ {\\rm and}\\ \\ \\tilde{z}^{m,n}_\\cdot:=z^{m,n}_\\cdot-z^{m,n-1}_\\cdot.\n\\vspace{-0.1cm}$$\n\nNext, as a key step in the proof of Proposition 7 we will show that\n\\begin{equation}\n\\RE\\ t\\in [T_1,T],\\ \\ \\sup_{m\\geq 0}\\sup_{n\\geq 2} \\E\\left[\\int_t^T | \\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]<+\\infty.\n\\end{equation}\nNote that for each $m\\geq 0$ and $n\\geq 2$, $(\\tilde y_t^{m,n},\\tilde z_t^{m,n})_{t\\in\\T}$ is an $L^1$ solution of the following BSDE:\n\\begin{equation}\n\\tilde{y}_t^{m,n}=\\int_t^T \\tilde{g}^{m,n}(s,\\tilde{y}_s^{m,n})\\ {\\rm\nd}s-\\int_t^T \\tilde{z}_s^{m,n}{\\rm d}B_s,\\ \\ \\ t\\in \\T,\n\\end{equation}\nwhere for each $y\\in \\R^k$,\n$\\tilde{g}^{m,n}(s,y):=g^m(s,y+y_s^{m,n-1},z^{m,n-1}_s)-\ng^m(s,y_s^{m,n-1},z_s^{m,n-2}).$\nIt follows from (H1) of $g^m$ and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\left\\langle y,\\tilde{g}^{m,n}(t,y)\\right\\rangle &\\leq &\\left\\langle y,g^m(t,y+y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n\\right\\rangle\\vspace{0.1cm}\\\\\n&& \\Dis +|y||g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-2}_t)|\\vspace{0.1cm}\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+|y|\\tilde\\triangle^{m,n}_t,\n\\end{array}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and in view of (H3) of $g^m$,\n\\begin{equation}\n\\Dis \\tilde\\triangle^{m,n}_t := \\Dis g^m(t,y_t^{m,n-1},z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z^{m,n-2}_t)\n\\leq \\Dis \\lam |z^{m,n-1}_t-z^{m,n-2}_t|=\\lam |\\tilde z^{m,n-1}_t|\n\\end{equation}\nas well as\n\\begin{equation}\n\\tilde\\triangle^{m,n}_t\\leq 2\\gamma \\left(g_t+\n|y_t^{m,n-1}|+|z^{m,n-1}_t|+|z^{m,n-2}_t|\\right)^\\alpha.\\vspace{0.2cm}\n\\end{equation}\nIn view of $\\alpha\\in (0,1\/2)$ and (106), using H\\\"{o}lder's inequality, Jensen's inequality and Doob's inequality yields that for each $m\\geq 0$ and $n\\geq 2$,\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\tilde\\triangle^{m,n}_t\\ {\\rm d}t\\right)^2\\right]<+\\infty\\ \\ {\\rm and\\ then}\\ \\\n\\E\\left[\\left.\\int_0^T \\tilde\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\in \\s^2.\n\\end{equation}\nOn the other hand, note from assumption (H1) of $g^m$ and (104) that $\\as$,\n\\begin{equation}\n\\left\\langle{\\tilde{y}^{m,n}_t\\over |\\tilde{y}^{m,n}_t|} \\mathbbm{1}_{|\\tilde{y}^{m,n}_t|\\neq 0},g^m(t,y^{m,n}_t,z^{m,n-1}_t)\n-g^m(t,y_t^{m,n-1},z_t^{m,n-2})\n\\right\\rangle\\leq \\rho(|\\tilde{y}^{m,n}_t|)+\\tilde\\triangle_t^{m,n}.\n\\end{equation}\nThus, in view of (108) and (107), arguing as that from (2) to (7), by virtue of Gronwall's inequality we can obtain that for each $m\\geq 0$, $n\\geq 2$ and $t\\in\\T$,\n\\begin{equation}\n|\\tilde y_t^{m,n}|\\leq \\left(AT+\\E\\left[\\left.\\int_0^T \\tilde\\triangle^{m,n}_s\\ {\\rm d}s\\right|\\F_t\\right]\\right)\\cdot e^{AT}\n\\end{equation}\nand then\n\\begin{equation}\n(\\tilde y_t^{m,n})_{t\\in\\T}\\in \\s^2.\n\\end{equation}\nIn view of (110), (104), (107) and Remark 1, it follows from Proposition 1 with $p=2$ and $u=0$ that there exists a constant $C_{A,T}>0$ depending only on $A,T$ such that for each $m\\geq 0$, $n\\geq 2$ and $t\\in \\T$,\n\\begin{equation}\n\\Dis\\E\\left[\\left(\\int_t^T |\\tilde z_s^{m,n}|^2\\ {\\rm d}s\\right)\\right]\\leq \\Dis C_{A,T}\\left\\{\\E\\left[\\sup_{r\\in [t,T]}|\\tilde y_r^{m,n}|^2\\right]+\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,n}\\ {\\rm d}s\\right)^2\\right]+(AT)^2\\right\\}<+\\infty.\n\\end{equation}\nThus, for each $m\\geq 0$ and $n\\geq 2$, $(\\tilde y_t^{m,n},\\tilde z_t^{m,n})_{t\\in\\T}$ is an $L^2$ solution of BSDE (103). Furthermore, in view of (104) and (105), by Proposition 3 and H\\\"{o}lder's inequality we can deduce that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [0,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n&&\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\tilde\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis e^{C(T-t)}\\left\\{\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\tilde y^{m,n}_r|^2 \\right]\\right){\\rm d}s+\\lam^2 (T-t)\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right\\},\n\\end{array}\n\\end{equation}\nwhere the constant $C>0$ is the same as in (97).\nThen, combining (112) with (98) yields that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\\tilde\ny^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T |\\tilde z^{m,n}_s|^2\\ {\\rm d}s\\right]\n\\leq \\Dis 2\\int_t^T\\kappa\\left(\\E\\left[\\sup_{r\\in\n[s,T]}|\\tilde y^{m,n}_r|^2 \\right]\\right){\\rm d}s+{1\\over 8}\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right].\n\\end{equation}\nNote by Remark 1 that $\\kappa(x)\\leq A(x+1)$ for each $x\\geq 0$. Gronwall's inequality together with (113) and (98) yields that for each $m\\geq 0$, $n\\geq 3$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\begin{array}{lll}\n\\Dis \\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\tilde y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\tilde z^{m,n}_s|^2\\ {\\rm\nd}s\\right]\n&\\leq &\\Dis \\left(2AT+{1\\over 8}\\E\\left[\\int_t^T |\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\right)\\cdot e^{2A(T-t)}\\vspace{0.1cm}\\\\\n&\\leq &\\Dis 4AT+{1\\over 4}\\E\\left[\\int_t^T\n|\\tilde z^{m,n-1}_s|^2\\ {\\rm d}s\\right]\\vspace{0.1cm}\\\\\n&\\leq &\\Dis {16AT\\over 3}+{1\\over 4^{n-2}}\\E\\left[\\int_t^T |\\tilde z^{m,2}_s|^2\\ {\\rm d}s\\right].\n\\end{array}\n\\end{equation}\nFurthermore, combining (109) and (111) with $n=2$, by virtue of Doob's inequality we obtain the existence of a constant $K_{A,T}>0$ depending only on $A$ and $T$ such that for each $m\\geq 0$ and $t\\in [T_1,T]$,\n\\begin{equation}\n\\E\\left[\\int_t^T|\\tilde z^{m,2}_s|^2\\ {\\rm d}s\\right]\\leq K_{A,T}\\left(1+\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]\\right).\n\\end{equation}\nAnd, in view of (106) with $n=2$ and the fact of $2\\alpha\\in (0,1)$, it follows from H\\\"{o}lder's inequality and Jensen's inequality that for each $m\\geq 0$ and $t\\in [T_1,T]$,\n$$\n\\begin{array}{lll}\n\\Dis \\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]\n&\\leq & \\Dis 16\\gamma^2T^{2-2\\alpha}\\left(\\E\\left[\\int_t^T g_s{\\rm d}s\\right]\\right)^{2\\alpha}+16\\gamma^2T^2\\E\\left[\\sup_{s\\in [t,T]}|y^{m,1}_s|^{2\\alpha}\\right]\\\\\n&&\\Dis +16\\gamma^2T^{2-\\alpha}\\E\\left[\\left(\\int_t^T |z^{m,1}_s|^2\\ {\\rm d}s\\right)^{\\alpha}\\right],\n\\end{array}\n$$\nfrom which as well as (67) with $n=1$, we can deduce that\n\\begin{equation}\n\\RE\\ t\\in [T_1,T],\\ \\ \\sup_{m\\geq 0}\\E\\left[\\left(\\int_t^T \\tilde \\triangle_s^{m,2}\\ {\\rm d}s\\right)^2\\right]<+\\infty.\n\\end{equation}\nThus, the inequality (102) follows from (114)-(116).\\vspace{0.1cm}\n\nFinally, combining (101) and (102) we can deduce that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{n\\geq 2}\\sup_{m\\geq 0}\\left(\\E\\left[\\sup\\limits_{r\\in [t,T]}|\n\\bar y^{m,n}_r|^2\\right]+\\E\\left[\\int_t^T | \\bar z^{m,n}_s|^2\\ {\\rm d}s\\right]\\right)<+\\infty.\n\\end{equation}\nThus, by first taking supremum with respect to $m$ and then taking $\\limsup$ with respect to $n$ in (99) as well as using Fatou's lemma, the monotonicity and continuity of the function $\\kappa(\\cdot)$ and Bihari's inequality, we can get (88). The proof of Proposition 7 is then completed.\\vspace{0.2cm} \\hfill $\\Box$\n\nNow, we come back to the proof of Theorem 4, and only consider the case when the $\\alpha$ in (H3) values in $(0,1\/2)$. Another case can be proved in the same way.\\vspace{0.2cm}\n\nFirstly, in view of (64) and (65) we have, for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\sup_{s\\in [t,T]}\\E\\left[|y_s^m -y_s^0|\\right] \\leq 2\\sup_{m\\geq 0}\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^m|\\right]+\\sup_{s\\in [t,T]}\\E\\left[|y_s^{m,n} -y_s^{0,n}|\\right]\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\E\\left[\\sup_{s\\in [t,T]}|y_s^m -y_s^0|^\\beta\\right] \\leq 2\\sup_{m\\geq 0}\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^m|^{\\beta}\\right]+\\E\\left[\\sup_{s\\in [t,T]}|y_s^{m,n}-y_s^{0,n}|^{\\beta}\\right]\n\\end{equation}\nas well as\n\\begin{equation}\n\\Dis \\E\\left[\\left(\\int_t^T|z_s^m -z_s^0|^2{\\rm d}s\\right)^\\beta\\right]\\leq \\Dis 2\\sup_{m\\geq 0}\\E\\left[\\left(\\int_t^T|z_s^{m,n}-z_s^m|^2{\\rm d}s\\right)^\\beta\\right]\n+\\E\\left[\\left(\\int_t^T|z_s^{m,n}-z_s^{0,n}|^2{\\rm d}s\\right)^\\beta\\right].\n\\end{equation}\nThen, letting first $m\\To\\infty$ ($n$ being fixed) and then $n\\To\\infty$ in above three inequalities (118)-(120), in view of (66) and (67) in Proposition 6 and (88) in Proposition 7 we obtain that for each $t\\in [T_1,T]$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{s\\in [t,T]}\\E\\left[|y_s^m-y_s^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{s\\in [t,T]} |y_s^m-y_s^0|^\\beta+\\left(\\int_t^T |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\nIn the sequel, set $T_j:=(T-j\\delta T)\\vee 0$ for each $j\\geq 2$.\nNoticing that the positive real number $\\delta T$ depends only on $\\lam$ and $A$, we can find a minimal integer $N\\geq 1$ such that $T_N=0$. In view of (121) and (122), if $N=1$, then (61) and (62) have been proved. Otherwise, we consider the BSDEs with parameters $(y^m_{T_1},T_1,g^m)$ and $(y_{T_1},T_1,g)$ defined on the time interval $[T_2,T_1]$. Note by (121) that\n$$\\lim\\limits_{m\\To\\infty} \\E\\left[|y^m_{T_1}-y_{T_1}|\\right]=0.$$\nRepeating the above arguments will yield that for each $t\\in [T_2,T_1]$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{s\\in [t,T_1]}\\E\\left[|y_s^m-y_s^0|\\right]=0,\n\\end{equation}\nand for each $\\beta\\in (0,1)$,\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{s\\in [t,T_1]} |y_s^m-y_s^0|^\\beta+\\left(\\int_t^{T_1} |z_s^m -z_s^0|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n\\end{equation}\nThus, in view of (121)-(124), if $N=2$, then (61) and (62) have also been proved. Otherwise, we can consider successively the BSDEs on $[T_3,T_2]$, $\\cdots$, $[0,T_{N-1}]$, and finally complete the proof of Theorem 4 by repeating the above procedure.\\hfill $\\Box$\\vspace{0.2cm}\n\n{\\bf Remark 6}\\ \\ The whole idea of the proof of Theorem 4 is involved in, by virtue of (63), introducing $(y_\\cdot^{m,n},z_\\cdot^{m,n})$ as a bridge between $(y_\\cdot^m,z_\\cdot^m)$ and $(y_\\cdot^0,z_\\cdot^0)$, and then, by virtue of (118)-(120), proving Propositions 6 and 7 respectively. This whole idea should be new. In addition, Propositions 6 and 7 are not easy to prove, especially a delicate argument has been done to obtain the inequality (117).\\vspace{0.3cm}\n\nIn the sequel, we will investigate the stability theorem of the solutions in the space $\\s^1\\times {\\rm M}^1$ for multidimensional BSDEs.\\vspace{0.2cm}\n\nNow, for each $m\\in \\N$, let $\\xi^m\\in \\LT$ and let $(y_t^m, z_t^m)_{t\\in [0,T]}$ be\na solution in $\\s^1\\times {\\rm M}^1$ for\nthe following BSDEs depending on parameter $m$:\n\\begin{equation}\ny_t^m=\\xi^m+\\int_t^T g^m(s,y_s^m,z_s^m)\\ {\\rm d}s -\\int_t^T z_s^m{\\rm d}B_s,\\quad t\\in [0,T].\n\\end{equation}\nFurthermore, we introduce the following assumptions:\\vspace{0.2cm}\n\n{\\bf (B3)}\\ All $g^m$ and $\\xi^m$ satisfy assumptions (H1)-(H4) with the same parameters $\\rho(\\cdot)$, $\\lambda$, $\\gamma$, $g_t$ and $\\alpha$. Furthermore, in the case of $\\alpha\\in [1\/2,1)$ we assume that $g$ satisfies (H1) with a function $\\rho(\\cdot)$ satisfying (14) for some $\\bar p>1$. \\vspace{0.2cm}\n\n{\\bf (B4)}\\ There exists a nonnegative real number sequence\n$\\{a_m\\}_{m=1}^{+\\infty}$ satisfying $\\lim\\limits_{m\\To \\infty}a_m=0$ such that $\\as$, for each $m\\geq 1$,\\vspace{-0.1cm}\n\\begin{equation}\n\\RE\\ (y,z)\\in \\R^k\\times \\R^{k\\times d},\\ \\ |g^m(\\omega,t,y,z)-g^0(\\omega,t,y,z)|\\leq a_m.\n\\end{equation}\nAnd\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi^{0}|\\right|\\F_t\\right]\\right]=0.\\vspace{0.2cm}\n\\end{equation}\n\nThe following Theorem 5 is the stability theorem of the solutions in $\\s^1\\times {\\rm M}^1$.\\vspace{0.1cm}\n\n{\\bf Theorem 5}\\ Under assumptions (B3) and (B4), we have\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|+\\left(\\int_0^T |z_s^m -z_s^0|^2\\ {\\rm d}s\\right)^{1\/2} \\right]=0.\\vspace{0.1cm}\n\\end{equation}\n\n{\\bf Proof}\\ \\ First of all, it follows from (62) of Theorem 4 that the sequence of variables $\\{\\sup_{t\\in [0,T]} |y_t^m-y_t^0|\\}_{m=1}^{+\\infty}$ converges in probability to $0$ as $m\\To\\infty$. In the sequel, we will first prove that it is also uniformly integrable, and then\n\\begin{equation}\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t^0|\\right]=0.\\vspace{0.2cm}\n\\end{equation}\nIn fact, set $\\hat y^m_\\cdot:=y^m_\\cdot-y^0_\\cdot$ and $\\hat z^m_\\cdot:=z^m_\\cdot-z^0_\\cdot$. Then, for each $m\\geq 1$, $(\\hat y^m_t,\\hat z^m_t)_{t\\in \\T}$ is a solution in the space $\\s^1\\times {\\rm M}^1$ for the following BSDE:\n\\begin{equation}\n\\hat{y}_t^m=\\xi^{m}-\\xi^0+\\int^{T}_{t}\\hat g^m(s,\\hat y_s^m,\\hat z_s^m)\\ \\mathrm{d} s-\\int^{T}_{t}\\hat{z}_s^m\\ \\mathrm{d}B_s, \\quad t\\in\\T,\n\\end{equation}\nwhere for each $y\\in\\R^k$,\n$\\hat g^m(t,y,z):=g^m(t,y+y^0_t,z+z^0_t)\n-g^0(t,y^0_t,z^0_t).$\nIt follows from assumptions (H1) and (H3) of $g^m$ together with (126) and Remark 1 that $\\as$, for each $y\\in\\R^k$,\n\\begin{equation}\n\\begin{array}{lll}\n\\langle y,\\hat{g}^m(t,y,z)\\rangle &\\leq &\\langle y,g^m(t,y+y_t^0,z+z^0_t)-g^m(t,y_t^0,z+z^0_t)\n\\rangle+|y||g^m(t,y_t^0,z+z^0_t)-g^m(t,y_t^0,z^0_t)|\\\\\n&& \\Dis +|y||g^m(t,y_t^0,z^0_t)-g^0(t,y_t^0,z^0_t)|\\\\\n&\\leq & \\Dis \\kappa(|y|^2)+\\lam |y||z|+|y|a_m\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\n\\left\\langle{\\hat{y}^m_t\\over |\\hat{y}^m_t|} \\mathbbm{1}_{|\\hat{y}^m_t|\\neq 0},g^m(t,y^m_t,z^m_t)\n-g^0(s,y_t^0,z_t^0)\\right\\rangle\\leq \\rho(|\\hat{y}^m_t|)+\\hat\\triangle_t^m+a_m,\\vspace{0.2cm}\n\\end{equation}\nwhere $\\kappa(\\cdot)$ is defined in (H1a)$_2$, and\n\\begin{equation}\n\\hat\\triangle^m_t:=|g^m(t,y_t^0,z^m_t)-g^m(t,y_t^0,z^0_t)| \\leq 2\\gamma \\left(g_t+\n|y_t^0|+|z^m_t|+|z^0_t|\\right)^\\alpha.\n\\end{equation}\nNext, in view of (132) and Remark 1, using Corollary 2.3 in \\cite{Bri03} and Gronwall's inequality we get that for each $m\\geq 1$ and $t\\in\\T$,\n\\begin{equation}\n|\\hat y_t^m|\\leq \\left(\\E\\left[\\left.|\\xi^m-\\xi^0|\\right|\\F_t\n\\right]+AT+\\E\\left[\\left.\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right|\\F_t\\right]+a_mT\\right)\\cdot e^{AT}.\n\\end{equation}\nOn one hand, it follows from (127) that $\\{\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi^{0}|\\right|\\F_t\\right]\\}_{m=1}^{+\\infty}$ is uniformly integrable. On the other hand, in view of (133), for each $q>1$ such that $\\alpha q<1$, H\\\"{o}lder's inequality yields the existence of a positive constant $C$ independent of $m$ such that\n\\begin{equation}\n\\E\\left[\\left(\\int_0^T \\hat\\triangle^m_s\\ {\\rm d}s\\right)^q\\right]1$ such that the function $\\rho(\\cdot)$ in (H1) satisfies (14). Suppose that for each $m\\geq 1$, $\\xi^m,\\xi\\in \\LT$ and $(y^m_t,z^m_t)_{t\\in\\T}$ and $(y_t,z_t)_{t\\in\\T}$ are respectively the unique $L^1$ solution of BSDE$(\\xi^m,T,g)$ and BSDE$(\\xi,T,g)$. If $\\lim\\limits_{m\\To\\infty}\\E\\left[|\\xi^m-\\xi|\\right]=0,$ then\n$$\n\\lim\\limits_{m\\To \\infty}\\sup\\limits_{t\\in [0,T]}\\E\\left[|y_t^m-y_t|\\right]=0,\n$$\nand for each $\\beta\\in (0,1)$,\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t|^\\beta+\\left(\\int_0^T |z_s^m -z_s|^2 {\\rm d}s\\right)^{\\beta\/2} \\right]=0.\n$$\nMoreover, if for each $m\\geq 1$, $\\xi^m$ and $g(t,0,0)$ satisfy assumptions (H4), and\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup_{t\\in\\T}\\E\\left[\\left.\n|\\xi^{m}-\\xi|\\right|\\F_t\\right]\\right]=0,\n$$\nthen\n$$\n\\lim\\limits_{m\\To \\infty}\\E\\left[\\sup\\limits_{t\\in [0,T]} |y_t^m-y_t|+\\left(\\int_0^T |z_s^m -z_s|^2\\ {\\rm d}s\\right)^{1\/2} \\right]=0.\\vspace{0.3cm}\n$$\n\n{\\bf Remark 8}\\ By Remarks 5 and 2, we know that Theorems 4-5 and Corollary 2 give the stability of the $L^1$ solutions of multidimensional BSDEs investigated in \\citet{Bri03}.\\vspace{0.1cm}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Gravito-electromagnetic analogy based on tidal tensors}\n\nThe topic of the gravito-electromagnetic analogies has a long story,\nwith different analogies being unveiled throughout the years. Some\nare purely formal analogies, like the splitting of the Weyl tensor\nin electric and magnetic parts, e.g. \\cite{Maartens}; but others\n(e.g \\cite{DSX,CostaHerdeiro2008,Jantzen,Natario,Ruggiero:2002})\nstem from certain physical similarities between the gravitational\nand electromagnetic interactions. The linearized Einstein equations\n(see e.g. \\linebreak \\cite{DSX,Ruggiero:2002,Gravitation and Inertia}), in the\nharmonic gauge $\\bar{h}_{\\alpha\\beta}^{\\,\\,\\,\\,\\,\\,,\\beta}=0$, take\nthe form $\\square\\bar{h}^{\\alpha\\beta}=-16\\pi T^{\\alpha\\beta}\/c^{4}$,\nsimilar to Maxwell equations in the Lorentz gauge: $\\square A^{\\beta}=-4\\pi j^{\\beta}\/c$.\nThat suggests an analogy between the trace reversed time components\nof the metric tensor $\\bar{h}_{0\\alpha}$ and the electromagnetic\n4-potential $A_{\\alpha}$. Defining the 3-vectors usually dubbed gravito-electromagnetic\nfields, the time components of these equations may be cast in a Maxwell-like\nform, e.g. eqs (16)-(22) of \\cite{Ruggiero:2002}. Furthermore (on\ncertain special conditions, see section \\ref{sec:Linearized-Gravity}) \\linebreak\ngeodesics, precession and forces on gyroscopes are described in terms\nof these fields in a \\linebreak form similar to their electromagnetic counterparts,\ne.g. \\cite{Ruggiero:2002}, \\linebreak \\cite{Gravitation and Inertia}. Such analogy may\nactually be cast in an exact form using the 3+1 splitting of spacetime\n(see \\cite{Jantzen,Natario}).\n\nThese are analogies comparing physical quantities (electromagnetic\nforces) from one theory with inertial gravitational forces (i.e. fictitious\nforces, that can be gauged away by moving to a freely falling frame,\ndue to the equivalence principle); it is clear that (non-spinning)\ntest particles in a gravitational field move with zero acceleration\n$DU^{\\alpha}\/d\\tau=0$; and that the spin 4-vector of a gyroscope\nundergoes Fermi-Walker transport $DS^{\\alpha}\/d\\tau=S_{\\sigma}U^{\\alpha}DU^{\\sigma}\/d\\tau$,\nwith no real torques applied on it. In this sense the gravito-electromagnetic\nfields are pure coordinate artifacts, attached to the observer's frame.\n\nHowever, these approaches describe also (not through the {}``gravito-electromagnetic''\nfields themselves, but through their derivatives; and, again, under\nvery special conditions) tidal effects, like the force applied on\na gyroscope. And these are covariant effects, implying physical gravitational\nforces. \n\nHerein we will discuss under which precise conditions a similarity\nbetween gravity and electromagnetism occurs (that is, under which\nconditions the physical analogy $\\bar{h}_{0\\mu}\\leftrightarrow A_{\\mu}$\nholds, and Eqs. like (16)-(22) of \\cite{Ruggiero:2002} have a physical\ncontent). For that we will make use of the tidal tensor formalism\nintroduced in \\cite{CostaHerdeiro2008}. The advantage of this formalism\nis that, by contrast with the approaches mentioned above, it is based\non quantities which can be covariantly defined in both theories ---\ntidal forces (the only physical forces present in gravity) --- which\nallows for a more transparent comparison between the electromagnetic\n(EM) and gravitational (GR) interactions.\n\n\\begin{table}[h]\n\\caption{\\label{analogy}The gravito-electromagnetic analogy based on tidal\ntensors. }\n\n\n\\begin{centering}\n\\setlength{\\arrayrulewidth}{0.8pt}\\begin{tabular}{>{\\centering}p{39.6ex}c>{\\centering}p{41.9ex}c}\n\\hline \n\\multicolumn{2}{c}{\\raisebox{3ex}{}\\raisebox{0.5ex}{Electromagnetism}} & \\multicolumn{2}{c}{\\raisebox{3ex}{}\\raisebox{0.5ex}{Gravity}}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Worldline deviation: & & \\raisebox{3ex}{}Geodesic deviation: & \\tabularnewline\n\\raisebox{6ex}{}\\raisebox{2ex}{${\\displaystyle \\frac{D^{2}\\delta x^{\\alpha}}{d\\tau^{2}}=\\frac{q}{m}E_{\\,\\,\\,\\beta}^{\\alpha}\\delta x^{\\beta}},\\,\\,\\, E_{\\,\\,\\,\\beta}^{\\alpha}\\equiv F_{\\ \\mu;\\beta}^{\\alpha}U^{\\mu}$} & \\raisebox{2ex}{(1a) \\,\\,} & \\raisebox{5.5ex}{}\\raisebox{2ex}{${\\displaystyle \\frac{D^{2}\\delta x^{\\alpha}}{d\\tau^{2}}=-\\mathbb{E}_{\\,\\,\\,\\beta}^{\\alpha}\\delta x^{\\beta}},\\,\\,\\,\\mathbb{E}_{\\,\\,\\,\\beta}^{\\alpha}\\equiv R_{\\,\\,\\,\\mu\\beta\\nu}^{\\alpha}U^{\\mu}U^{\\nu}$} & \\raisebox{2ex}{(1b)}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Force on magnetic dipole: & & \\raisebox{3ex}{}Force on gyroscope: & \\tabularnewline\n\\raisebox{6ex}{}\\raisebox{2ex}{${\\displaystyle F_{EM}^{\\beta}=\\frac{q}{2m}B_{\\alpha}^{\\,\\,\\,\\beta}S^{\\alpha}},\\,\\,\\, B_{\\,\\,\\,\\beta}^{\\alpha}\\equiv\\star F_{\\ \\mu;\\beta}^{\\alpha}U^{\\mu}$} & \\raisebox{2ex}{(2a)} & \\raisebox{6ex}{}\\raisebox{2ex}{~~${\\displaystyle F_{G}^{\\beta}=-\\mathbb{H}_{\\alpha}^{\\,\\,\\,\\beta}S^{\\alpha}},\\,\\,\\,\\mathbb{H}_{\\,\\,\\,\\beta}^{\\alpha}\\equiv\\star R_{\\,\\,\\,\\mu\\beta\\nu}^{\\alpha}U^{\\mu}U^{\\nu}$} & \\raisebox{2ex}{(2b)}\\tabularnewline\n\\hline \n\\raisebox{3ex}{}Maxwell Equations: & & \\raisebox{3ex}{}Eqs. Grav. Tidal Tensors: & \\tabularnewline\n\\raisebox{3ex}{}$E_{\\,\\,\\,\\alpha}^{\\alpha}=4\\pi\\rho_{c}$ & (3a) & \\raisebox{3ex}{}$\\mathbb{E}_{\\,\\,\\,\\alpha}^{\\alpha}=4\\pi\\left(2\\rho_{m}+T_{\\,\\,\\alpha}^{\\alpha}\\right)$ & (3b)\\tabularnewline\n\\raisebox{3.5ex}{}$E_{[\\alpha\\beta]}=\\frac{1}{2}F_{\\alpha\\beta;\\gamma}U^{\\gamma}$ & (4a) & \\raisebox{3.5ex}{}$\\mathbb{E}_{[\\alpha\\beta]}=0$ & (4b)\\tabularnewline\n\\raisebox{3.5ex}{}$B_{\\,\\,\\,\\alpha}^{\\alpha}=0$ & (5a) & \\raisebox{3.5ex}{}$\\mathbb{H}_{\\,\\,\\,\\alpha}^{\\alpha}=0$ & (5b)\\tabularnewline\n\\raisebox{5.5ex}{}\\raisebox{2ex}{$B_{[\\alpha\\beta]}=\\frac{1}{2}\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}-2\\pi\\epsilon_{\\alpha\\beta\\sigma\\gamma}j^{\\sigma}U^{\\gamma}$} & \\raisebox{2ex}{(6a)} & ~~\\raisebox{5.5ex}{}\\raisebox{2ex}{$\\mathbb{H}_{[\\alpha\\beta]}=-4\\pi\\epsilon_{\\alpha\\beta\\sigma\\gamma}J^{\\sigma}U^{\\gamma}$}~~ & \\raisebox{2ex}{(6b)}\\tabularnewline\n\\hline\n\\end{tabular}\\\\\n \n\\par\\end{centering}\n\n\\begin{raggedright}\n{\\scriptsize $\\rho_{c}=-j^{\\alpha}U_{\\alpha}$ and $j^{\\alpha}$ are,\nrespectively, the charge density and current 4-vector; $\\rho_{m}=T_{\\alpha\\beta}U^{\\alpha}U^{\\beta}$\nand $J^{\\alpha}=-T_{\\,\\beta}^{\\alpha}U^{\\beta}$ are the mass\/energy\ndensity and current (quantities measured by the observer of 4-velocity\n$U^{\\alpha}$); $T_{\\alpha\\beta}\\equiv$ energy-momentum tensor; $S^{\\alpha}\\equiv$\nspin 4-vector; $\\star\\equiv$ Hodge dual. We use $\\tilde{e}_{0123}=-1$.}\n\\par\\end{raggedright}\n\\end{table}\n\n\nThe tidal tensor formalism unveils a new gravito-electromagnetic analogy,\nsummarized in Table \\ref{analogy}, based on exact and covariant equations.\nThese equations make clear key differences, and under which conditions\na similarity between the two interactions may occur.\n\nEqs. (1) are the worldline deviation equations yielding the relative\nacceleration of two neighboring particles (connected by the infinitesimal\nvector $\\delta x^{\\alpha}$) with the \\textit{same} 4-velocity $U^{\\alpha}$\n(and the same $q\/m$ ratio, in the electromagnetic case). These equations\nmanifest the physical analogy between electric tidal tensors: $\\mathbb{E}_{\\alpha\\beta}\\leftrightarrow E_{\\alpha\\beta}$.\n\nEq. (2a) yields the electromagnetic force exerted on a magnetic dipole\nmoving with 4-velocity $U^{\\alpha}$, and is the covariant generalization\nof the usual 3-D expression $\\mathbf{F_{EM}}=\\nabla(\\mathbf{S}.\\mathbf{B})q\/2m$\n(valid only in the dipole's proper frame); Eq. (2b) is exactly the\nPapapetrou-Pirani equation for the gravitational force exerted on\na spinning test particle. In both (2a) and (2b), Pirani's supplementary\ncondition $S_{\\mu\\nu}U^{\\nu}=0$ is assumed (c.f. \\cite{CostaHerdeiro2009}).\nThese equations manifest the physical analogy between magnetic tidal\ntensors: $B_{\\alpha\\beta}\\leftrightarrow\\mathbb{H}_{\\alpha\\beta}$.\n\nTaking the traces and antisymmetric parts of the EM tidal tensors,\none obtains Eqs. (3a)-(6a), which are explicitly covariant forms for\neach of Maxwell equations. Eqs. (3a) and (6a) are, respectively, the\ntime and space projections of Maxwell equations $F_{\\ \\ \\ ;\\beta}^{\\alpha\\beta}=4\\pi j^{\\alpha}$;\ni.e., they are, respectively, covariant forms of $\\nabla\\cdot\\mathbf{E}=4\\pi\\rho_{c}$\nand $\\nabla\\times\\mathbf{B}=\\partial\\mathbf{E}\/\\partial t+4\\pi\\mathbf{j}$;\nEqs. (4a) and (5a) are the space and time projections of the electromagnetic\nBianchi identity $\\star F_{\\ \\ \\ ;\\beta}^{\\alpha\\beta}=0$; i.e.,\nthey are covariant forms for $\\nabla\\times\\mathbf{E}=-\\partial\\mathbf{B}\/\\partial t$\nand $\\nabla\\cdot\\mathbf{B}=0$. These equations involve only tidal\ntensors and sources, which can be seen substituting the following\ndecomposition (or its Hodge dual) in (4a) and (6a): \\begin{equation}\nF_{\\alpha\\beta;\\gamma}=2U_{[\\alpha}E_{\\beta]\\gamma}+\\epsilon_{\\alpha\\beta\\mu\\sigma}B_{\\,\\,\\,\\gamma}^{\\mu}U^{\\sigma}\\\n.\\label{Fdecomp}\\end{equation}\nIt is then straightforward to obtain the \\emph{physical} gravitational\nanalogues of Maxwell equations: one just has to apply the same procedure\nto the gravitational tidal tensors, i.e., write the equations for\ntheir traces and antisymmetric parts (that is more easily done decomposing\nthe Riemann tensor in terms of the Weyl tensor and source terms, see\n\\cite{CostaHerdeiro2007} sec. 2), which leads to Eqs. (3b) - (6b).\nUnderlining the analogy with the situation in electromagnetism, Eqs.\n(3b) and (6b) turn out to be the time-time and and time-space projections\nof Einstein equations $R_{\\mu\\nu}=8\\pi(T_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}T_{\\,\\,\\,\\alpha}^{\\alpha})$,\nand Eqs. (4b) and (5b) the time-space and time-time projections of\nthe algebraic Bianchi identities $\\star R_{\\ \\ \\ \\gamma\\beta}^{\\gamma\\alpha}=0$.\n\n\n\\subsection{Gravity vs Electromagnetism}\n\n\\emph{Charges} --- the gravitational analogue of $\\rho_{c}$ is $2\\rho_{m}+T_{\\,\\,\\alpha}^{\\alpha}$\n($\\rho_{m}+3p$ for a perfect fluid) $\\Rightarrow$ in gravity, pressure\nand all material stresses contribute as sources.\n\n\\emph{Ampere law} --- in stationary (in the observer's rest frame)\nsetups, $\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}$ vanishes and equations\n(6a) and (6b) match up to a factor of 2 $\\Rightarrow$ currents of\nmass\/energy source gravitomagnetism like currents of charge source\nmagnetism.\n\n\n\\emph{Symmetries of Tidal Tensors} --- The GR and EM tidal tensors\ndo not generically exhibit the same symmetries, signaling fundamental\ndifferences between the two interactions. In the general case of fields\nthat are time dependent in the observer's rest frame (that is the\ncase of an intrinsically non-stationary field, or an observer moving\nin a stationary field), the electric tidal tensor $E_{\\alpha\\beta}$\npossesses an antisymmetric part, which is the covariant derivative\nof the Maxwell tensor along the observer's worldline; there is also\nan antisymmetric contribution $\\star F_{\\alpha\\beta;\\gamma}U^{\\gamma}$\nto $B_{\\alpha\\beta}$. These terms consist of time projections of\nEM tidal tensors (cf. decomposition \\ref{Fdecomp}), and contain the\nlaws of electromagnetic induction. The gravitational tidal tensors,\nby contrast, are symmetric (in vacuum, in the magnetic case) and spatial,\nmanifesting the absence of analogous effects in gravity.\n\n\\emph{Gyroscope vs. magnetic dipole} --- According to Eqs. (2), both\nin the case of the magnetic dipole and in the case of the gyroscope,\nit is the magnetic tidal tensor, \\emph{as seen by the test particle}\n($U^{\\alpha}$ in Eqs. (2) is the gyroscope\/dipole 4-velocity), that\ndetermines the force exerted upon it. Hence, from Eqs. (6), we see\nthat the forces can be similar only if the fields are stationary (besides\nweak) in the gyroscope\/dipole frame, i.e., when it is at {}``rest''\nin a stationary field. Eqs. (2) also tell us that in gravity the angular\nmomentum $S$ plays the role of the magnetic moment $\\mu=S(q\/2m)$;\nthe relative minus sign manifests that masses\/charges of the same\nsign attract\/repel one another in gravity\/electromagnetism, as do\ncharge\/mass currents with parallel velocity. \n\n\n\\section{Linearized Gravity\\label{sec:Linearized-Gravity}}\n\nIf the fields are stationary in the observer's rest frame, the GR\nand EM tidal tensors have the same symmetries, which by itself does\nnot mean a close similarity between the two interactions (note that\ndespite the analogy in Table 1, EM tidal tensors are linear, whereas\nthe GR ones are not). But in two special cases a matching between\ntidal tensors occurs: ultrastationary spacetimes (where the gravito-magnetic\ntidal tensor is linear, see \\cite{CostaHerdeiro2008} Sec. IV) and\nlinearized gravitational perturbations, which is the case of interest\nfor astronomical applications. \n\nConsider an arbitrary electromagnetic field $A^{\\alpha}=(\\phi,\\mathbf{A})$\nand arbitrary perturbations around Minkowski spacetime in the form%\n\\footnote{In the previous sections we were putting $c=1$. In this section we\nre-introduce the speed of light in order to facilitate comparison\nwith relevant literature.%\n}\n\n\\begin{equation}\nds^{2}=-c^{2}\\left(1-2\\frac{\\Phi}{c^{2}}\\right)dt^{2}-\\frac{4}{c}\\mathcal{A}_{j}dtdx^{j}+\\left[\\delta_{ij}+2\\frac{\\Theta_{ij}}{c^{2}}\\right]dx^{i}dx^{j}\\ .\\label{Linear pert}\\end{equation}\n\n\n\\textbf{Tidal effects.} --- The GR and EM tidal tensors from these\nsetups will be in general very different, as is clear from equations\n(3-6), and as one may check from the explicit expressions in \\cite{CostaHerdeiro2008}.\n\nBut if one considers time independent fields, and a static observer\nof 4-velocity $U^{\\mu}=c\\delta_{0}^{\\mu}$, then the \\emph{linearized}\ngravitational tidal tensors match their electromagnetic counterparts\nidentifying $(\\phi,A^{i})\\leftrightarrow(\\Phi,\\mathcal{A}^{i})$ (in\nexpressions below colon represents partial derivatives; $\\epsilon_{ijk}\\equiv$\nLevi Civita symbol): \\begin{equation}\n\\mathbb{E}_{ij}\\simeq-\\Phi_{,ij}\\stackrel{\\Phi\\leftrightarrow\\phi}{=}E_{ij},\\quad\\mathbb{H}_{ij}\\simeq\\epsilon_{i}^{\\,\\,\\, lk}\\mathcal{A}_{k,lj}\\stackrel{\\mathcal{A}\\leftrightarrow A}{=}B_{ij}\\ .\\label{MatchingLinear}\\end{equation}\n This suggests the physical analogy $(\\phi,A^{i})\\leftrightarrow(\\Phi,\\mathcal{A}^{i})$,\nand defining the {}``gravito-electro-magnetic fields'' $\\mathbf{E_{G}}=-\\nabla\\Phi$\nand $\\mathbf{B_{G}}=\\nabla\\times\\bm{\\mathcal{A}}$, in analogy with\nthe electromagnetic fields $\\mathbf{E}=-\\nabla\\phi,$ $\\mathbf{B}=\\nabla\\times\\mathbf{A}$.\nIn terms of these fields we have $\\mathbb{E}_{ij}\\simeq(E_{G})_{i,j}$\nand $\\mathbb{H}_{ij}\\simeq(B_{G})_{i,j}$, in analogy with the electromagnetic\ntidal tensors $E_{ij}=E_{i,j}$ and $B_{ij}=B_{i,j}$. \n\nThe matching (\\ref{MatchingLinear}) means that a gyroscope at rest\n(relative to the static observer) will feel a force $F_{G}^{\\alpha}$\nsimilar to the electromagnetic force $F_{EM}^{\\alpha}$ on a magnetic\ndipole, which in this case take the very simple forms (time components\nare zero): \\begin{equation}\n\\mathbf{F_{EM}}=\\frac{q}{2mc}\\nabla(\\mathbf{B}.\\mathbf{S});\\ \\ \\ \\ \\ F_{G}^{j}=-\\frac{1}{c}\\mathbb{H}^{ij}S_{i}\\approx-\\frac{1}{c}(B_{G})^{i,j}S_{i}\\ \\Leftrightarrow\\ \\mathbf{F_{G}}=-\\frac{1}{c}\\nabla(\\mathbf{B_{G}}.\\mathbf{S})\\ .\\label{FG_Stationary}\\end{equation}\nHad we considered gyroscopes\/dipoles with different 4-velocities,\nnot only the expressions for the forces would be more complicated,\nbut also the gravitational force would significantly differ from the\nelectromagnetic one, as one may check comparing Eqs. (12) with (17)-(20)\nof \\cite{CostaHerdeiro2008}. This will be exemplified in section\n\\ref{Translational-vs.-Rotational}.\n\nThe matching (\\ref{MatchingLinear}) also means, by similar arguments,\nthat the relative acceleration between two neighboring masses $D^{2}\\delta x^{i}\/d\\tau^{2}=-\\mathbb{E}^{ij}\\delta x_{j}$\nis similar to the relative acceleration between two charges (with\nthe same $q\/m$): $D^{2}\\delta x^{i}\/d\\tau^{2}=E^{ij}\\delta x_{j}(q\/m)$,\n\\emph{at the instant} when the test particles have 4-velocity $U^{\\alpha}=c\\delta_{0}^{\\alpha}$\n(i.e., are \\emph{at rest} relative to the static observer $\\mathcal{O}$).\n\n\\textbf{Gyroscope precession.} --- The evolution of the spin vector\nof the gyroscope is given by the Fermi-Walker transport law, which,\nfor a gyroscope at rest reads $DS^{i}\/d\\tau=0$; hence, we have, in\nthe coordinate basis, Eq. (\\ref{PrecessGen}a). The last term of Eq.\n(\\ref{PrecessGen}a) vanishes if we express $\\mathbf{S}$ in the local\northonormal tetrad $e^{\\hat{\\alpha}}$: $S^{i}=S^{\\hat{i}}e_{\\,\\hat{i}}^{i}$,\nwhere to linear order $e_{\\,\\hat{i}}^{i}=\\delta_{\\ \\hat{i}}^{i}-\\Theta_{\\ \\ \\hat{i}}^{i}\/c^{2}$;\nin this fashion we obtain Eq. (\\ref{PrecessGen}b), which is similar\nto the precession of a magnetic dipole in a magnetic field $d\\mathbf{S}\/dt=q\\mathbf{S}\\times\\mathbf{B}\/2mc$:\n\\begin{equation}\n\\frac{dS^{i}}{dt}=-c\\Gamma_{0j}^{i}S^{j}=-\\frac{1}{c}\\left[(\\mathbf{S}\\times\\mathbf{B_{G}})^{i}+\\frac{1}{c}\\frac{\\partial\\Theta_{\\ }^{ij}}{\\partial t}S_{j}\\right]\\ \\ (a);\\ \\ \\ \\ \\ \\ \\frac{dS^{\\hat{i}}}{dt}=-\\frac{1}{c}(\\mathbf{S}\\times\\mathbf{B_{G}})^{\\hat{i}}\\ \\ (b).\\label{PrecessGen}\\end{equation}\n Thus, in the special case of gyroscope precession, the linear gravito-electromagnetic\nanalogy holds even if the fields vary with time.\n\n\\textbf{Geodesics.} --- The space part of the equation of geodesics\n$U_{\\ ,\\beta}^{\\alpha}U^{\\beta}=-\\Gamma_{\\beta\\gamma}^{\\alpha}U^{\\beta}U^{\\gamma}$\nis given, to first order in the perturbations and in test particle's\nvelocity, by ($a^{i}\\equiv d^{2}x^{i}\/dt^{2})$:\\begin{eqnarray}\n\\mathbf{a} & = & \\nabla\\Phi+\\frac{2}{c}\\frac{\\partial\\bm{\\mathcal{A}}}{\\partial t}-\\frac{2}{c}\\mathbf{v}\\times(\\nabla\\times\\bm{\\mathcal{A}})-\\frac{1}{c^{2}}\\left[\\frac{\\partial\\Phi}{\\partial t}\\mathbf{v}+2\\frac{\\partial\\Theta_{\\ j}^{i}}{\\partial t}v^{j}\\mathbf{e_{i}}\\right]\\ .\\label{geoGeneral}\\end{eqnarray}\nComparing with the electromagnetic Lorentz force:\\begin{equation}\n\\mathbf{a}=\\frac{q}{m}\\left[-\\nabla\\phi-\\frac{1}{c}\\frac{\\partial\\mathbf{A}}{\\partial t}+\\frac{\\mathbf{v}}{c}\\times(\\nabla\\times\\mathbf{A})\\right]=\\frac{q}{m}\\left[\\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}\\right]\\,,\\label{Lorentz}\\end{equation}\nthese equations do not manifest, in general, a close analogy. Note\nthat the last term of (\\ref{geoGeneral}), which has no electromagnetic\nanalogue, is, for the problem at hand (see next section), of the same\norder of magnitude as the second and third terms. But when one considers\nstationary fields, then (\\ref{geoGeneral}) takes the form $\\mathbf{a}=-\\mathbf{E_{G}}-2\\mathbf{v}\\times\\mathbf{B_{G}}\/c$\nanalogous to (\\ref{Lorentz}).\n\nNote the difference between this analogy and the one from the tidal\neffects considered above: in the case of the latter, the similarity\noccurs only when the \\emph{test particle} sees time independent \\emph{fields}\n(fields $\\equiv$ derivatives of potentials\/of metric perturbations);\nfor geodesics, it is when \\emph{the observer} (not the test particle!)\nsees a time independent \\emph{potential }($\\phi$)\\emph{\/metric perturbations}($\\Phi,\\Theta_{ij}$).\n\n\n\\subsection{\\label{Translational-vs.-Rotational}Translational vs. Rotational\nMass Currents}\n\nThe existence of a similarity between gravity and electromagnetism\nthus relies on the time dependence of the mass currents: if the currents\nare (nearly) stationary, for instance from a spinning celestial body,\nthe gravitational field generated is analogous to a magnetic field;\nan example is the gravitomagnetic field due to the rotation of the Earth,\ndetected on LAGEOS data by \\cite{Ciufolini Lageos} (and which is\nalso the subject of experimental scrutiny by the Gravity Probe B and\nthe upcoming LARES missions). But when the currents seen by the observer\nvary with time --- e.g. the ones resulting from translation of the\ncelestial body, considered in \\cite{SoffelKlioner} --- then the dynamics\ndiffer significantly.\n\n\\textbf{Rotational Currents.} --- We will start by the well known\nanalogy between the electromagnetic field of a spinning charge (charge\n$Q$, magnetic moment $\\mu$) and the gravitational field (in the\nfar region $r\\rightarrow\\infty$) of a rotating celestial body (mass\n$m$, angular momentum $J$), see Fig.\\,\\ref{fig1}\n\n\\begin{figure}[h]\n\n\n\\begin{centering}\n\\includegraphics[width=123.2mm]{figv3}\n\n\\par\\end{centering}\n\n\\caption{Spinning charge vs. spinning mass }\n\n\n\\begin{centering}\n\\label{fig1} \n\\par\\end{centering}\n\\end{figure}\n\n\nThe electromagnetic field of the spinning charge is described by the\n4-potential $A^{\\alpha}=(\\phi,\\mathbf{A})$, given by (\\ref{GravPot}a).\nThe spacetime around the spinning mass is asymptotically described\nby the linearized Kerr solution, obtained by putting in (\\ref{Linear pert})\nthe perturbations (\\ref{GravPot}b) :\\begin{equation}\n\\phi=\\frac{Q}{r}\\ ,\\ \\ \\ \\mathbf{A}=\\frac{1}{c}\\frac{\\bm{\\mathbf{\\mu}}\\times\\mathbf{r}}{r^{3}}\\ \\ \\ (a);\\ \\ \\ \\ \\ \\ \\Phi=\\frac{M}{r}\\ ,\\ \\ \\ \\bm{\\mathcal{A}}=\\frac{1}{c}\\frac{\\mathbf{J}\\times\\mathbf{r}}{r^{3}}\\ ,\\ \\ \\Theta_{ij}=\\Phi\\delta_{ij}\\ \\ \\ (b).\\ \\ \\ \\label{GravPot}\\end{equation}\nFor the observer at rest $\\mathcal{O}$ the gravitational tidal tensors\nasymptotically match the electromagnetic ones, identifying the appropriate\nparameters:{\\small \\[\n\\mathbb{E}_{ij}\\simeq\\frac{M}{r^{3}}\\delta_{ij}-\\frac{3Mr_{i}r_{j}}{r^{5}}\\stackrel{M\\leftrightarrow Q}{=}E_{ij};\\ \\ \\ \\ \\mathbb{H}_{ij}\\simeq\\frac{3}{c}\\left[\\frac{(\\mathbf{r}.\\mathbf{J})}{r^{5}}\\delta_{ij}+2\\frac{r_{(i}J_{j)}}{r^{5}}-5\\frac{(\\mathbf{r}.\\mathbf{J})r_{i}r_{j}}{r^{7}}\\right]\\stackrel{J\\leftrightarrow\\mu}{=}B_{ij}\\]\n}(all the time components are zero for this observer). This me{\\small ans\nthat} $\\mathcal{O}$ will find a similarity between \\emph{physical}\n(i.e., tidal) gravitational forces and their electromagnetic counterparts:\nthe gravitational force $F_{G}^{i}=-\\mathbb{H}^{ji}S_{j}\/c$ exerted\non a gyroscope carried by $\\mathcal{O}$ is similar to the force $F_{EM}^{i}=qB^{ji}S_{j}\/2mc$\non a magnetic dipole; and the worldline deviation $D^{2}\\delta x^{i}\/d\\tau^{2}=-\\mathbb{E}^{ij}\\delta x_{i}$\nof two masses dropped from rest is similar to the deviation between\ntwo charged particles with the same $q\/m$.\n\nMoreover, observer $\\mathcal{O}$ will see test particles moving on\ngeodesics described by equations analogous to the electromagnetic\nLorentz force (see Fig. \\ref{fig1}). \n\n\\textbf{Translational Currents.} --- For the observer $\\bar{\\mathcal{O}}$\nmoving with velocity $\\mathbf{w}$ relative to the mass\/charge of\nFig. \\ref{fig1}, however, the electromagnetic and gravitational interactions\nwill look significantly different. For simplicity we will specialize\nhere to the case where $\\mathbf{J}=\\bm{\\mu}=0$, so that the mass\/charge\ncurrents seen by $\\bar{\\mathcal{O}}$ arise solely from translation.\nTo obtain the electromagnetic 4-potential $A^{\\bar{\\alpha}}$ in the\nframe $\\bar{\\mathcal{O}}$, we apply the boost $A^{\\bar{\\alpha}}=\\Lambda_{\\ \\alpha}^{\\bar{\\alpha}}A^{\\alpha}=(\\bar{\\phi},\\bar{\\mathbf{A}})$,\nwhere $\\Lambda_{\\ \\alpha}^{\\bar{\\alpha}}\\equiv\\partial\\bar{x}^{\\bar{\\alpha}}\/\\partial x^{\\alpha}$,\nusing the expansion of Lorentz transformation (as done in e.g. \\cite{WillNordvedt1972}):\n\\begin{equation}\nt=\\bar{t}\\left(1+\\frac{w^{2}}{2c^{2}}+\\frac{3w^{4}}{8c^{4}}\\right)+\\left(1+\\frac{w^{2}}{2c^{2}}\\right)\\frac{\\mathbf{\\bar{x}}.\\mathbf{w}}{c^{2}};\\ \\ \\ \\ \\mathbf{x}=\\mathbf{\\bar{x}}+\\frac{1}{2c^{2}}(\\bar{\\mathbf{x}}.\\mathbf{w})\\mathbf{w}+\\left(1+\\frac{w^{2}}{2c^{2}}\\right)\\mathbf{w}\\bar{t}\\ ,\\label{Boost}\\end{equation}\nyielding, to order $c^{-2}$, $A^{\\bar{\\alpha}}=(\\bar{\\phi},\\mathbf{\\bar{A}})$,\nwith $\\bar{\\phi}=Q(1+w^{2}\/2c^{2})\/r$ and $\\mathbf{\\bar{A}}=-Q\\mathbf{w}\/rc$.\nTo obtain $A^{\\bar{\\alpha}}$ in the coordinates ($\\bar{x}^{i},\\bar{t}$)\nof $\\bar{\\mathcal{O}}$, we must also express $r$ (which denotes\nthe distance between the source and the point of observation, in the\nframe $\\mathcal{O}$) in terms of $R\\equiv|\\mathbf{\\bar{r}}+\\mathbf{w}\\bar{t}|$,\ni.e., the distance between the source and the point of observation\nin the frame $\\bar{\\mathcal{O}}$. Using transformation (\\ref{Boost}),\nwe obtain: $r^{-1}=R^{-1}[1-(\\mathbf{w}.\\mathbf{R})^{2}\/(2R^{2}c^{2})]$,\nand finally the electromagnetic potentials seen by $\\bar{\\mathcal{O}}$:\\begin{equation}\n\\bar{\\phi}=\\frac{Q}{R}\\left(1+\\frac{w^{2}}{2c^{2}}-\\frac{(\\mathbf{w}.\\mathbf{R})^{2}}{4R^{2}c^{2}}\\right);\\ \\ \\ \\ \\ \\ \\mathbf{\\bar{A}}=-\\frac{1}{c}\\frac{Q}{R}\\mathbf{w}\\ .\\label{EMF Obar}\\end{equation}\nThe metric of the spacetime around a point mass, in the coordinates\nof $\\bar{\\mathcal{O}}$, is also obtained using transformation (\\ref{Boost}),\nwhich is accurate to Post Newtonian order, by an analogous procedure.\nFirst we apply the Lorentz boost $g_{\\bar{\\alpha}\\bar{\\beta}}=\\Lambda_{\\ \\bar{\\alpha}}^{\\alpha}\\Lambda_{\\ \\bar{\\beta}}^{\\beta}g_{\\alpha\\beta}$\nto the metric (\\ref{GravPot}) (with $\\mathcal{A}=0$); then, expressing\n$r$ in terms of $R$, we finally obtain (note that, although we are\nnot putting the bars therein, indices $\\alpha=0,i$ in the following\nexpressions refer to the coordinates of $\\bar{\\mathcal{O}}$): \\begin{eqnarray}\ng_{00} & = & -1+2\\frac{M}{Rc^{2}}+\\frac{4Mw^{2}}{Rc^{4}}-\\frac{M(\\mathbf{w}.\\mathbf{R})^{2}}{c^{4}R^{3}}\\equiv-1+\\frac{2\\bar{\\Phi}}{c^{2}};\\ \\ \\nonumber \\\\\n\\ g_{0i} & = & \\frac{4Mw_{i}}{Rc^{3}}\\equiv-\\frac{2\\bar{\\mathcal{A}}_{i}}{c^{2}};\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ g_{ij}=\\left[1+2\\frac{M}{Rc^{2}}\\right]\\delta_{ij}\\equiv\\left[1+2\\frac{\\bar{\\Theta}}{c^{2}}\\right]\\delta_{ij}\\ ,\\label{GObar}\\end{eqnarray}\nwhere we retained terms up to $c^{-4}$ in $g_{00}$, up to $c^{-3}$\nin $g_{i0}$ and $c^{-2}$ in $g_{ij}$, as usual in Post-Newtonian\napproximation. This matches, to linear order in $M$, Eqs. (5) of\n\\cite{SoffelKlioner} for the case of one single source; or e.g. Eqs.\n(11) of \\cite{Nordvedt1988} (in the case of the latter, an additional\ngauge choice, Eq. (19) of \\cite{WillNordvedt1972}, was made). The\nmetric (\\ref{GObar}), like the electromagnetic potential (\\ref{EMF Obar}),\nis now time dependent, since $\\mathbf{R}(\\bar{t})=\\mathbf{\\bar{r}}+\\mathbf{w}\\bar{t}$.\n\nThe gravitational tidal tensors seen by $\\bar{\\mathcal{O}}$ are ({\\small $\\mathbb{E}_{\\alpha0}=\\mathbb{E}_{0\\alpha}=\\mathbb{H}_{\\alpha0}=\\mathbb{H}_{0\\alpha}=0$}):{\\small \\begin{eqnarray}\n\\mathbb{E}_{ij} & = & -\\bar{\\Phi}_{,ij}-\\frac{2}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{\\mathcal{A}}_{(i,j)}-\\frac{1}{c^{2}}\\frac{\\partial^{2}}{\\partial\\bar{t}^{2}}\\bar{\\Theta}\\delta_{ij}\\nonumber \\\\\n & = & \\frac{M\\delta_{ij}}{R^{3}}\\left[1+\\frac{3w^{2}}{c^{2}}-\\frac{9}{2}\\frac{(\\mathbf{R}.\\mathbf{w})^{2}}{c^{2}R^{2}}\\right]-\\frac{3MR_{i}R_{j}}{R^{5}}\\left[1+\\frac{2w^{2}}{c^{2}}-\\frac{5(\\mathbf{R}.\\mathbf{w})^{2}}{2c^{2}R^{2}}\\right]\\nonumber \\\\\n & & -\\frac{3Mw_{i}w_{j}}{c^{2}R^{3}}+\\frac{6Mw_{(i}R_{j)}(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{5}};\\label{Egij}\\\\\n\\mathbb{H}_{ij} & = & \\epsilon_{i}^{\\,\\,\\, lk}\\bar{\\mathcal{A}}_{k,lj}-\\frac{1}{c}\\epsilon_{ij}^{\\ \\ l}\\frac{\\partial\\bar{\\Theta}_{,l}}{\\partial\\bar{t}}\\ =\\frac{M}{cR^{3}}\\left[3\\epsilon_{ij}^{\\ \\ k}w_{k}-\\frac{3}{R^{2}}(\\mathbf{R}.\\mathbf{w})\\epsilon_{ij}^{\\ \\ k}R_{k}-\\frac{6}{R^{2}}(\\mathbf{R}\\times\\mathbf{w})_{i}R_{j}\\right],\\label{Hij}\\end{eqnarray}\n}which significantly differ from the electromagnetic ones ({\\small $E_{0\\alpha}=B_{0\\alpha}=0$}):{\\small \\begin{eqnarray}\nE_{ij} & = & -\\bar{\\phi}_{,ij}-\\frac{1}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{A}_{i;j}\\ =\\ E_{i,j}\\nonumber \\\\\n & = & \\frac{Q\\delta_{ij}}{R^{3}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{3}{4}\\frac{(\\mathbf{R}.\\mathbf{w})^{2}}{c^{2}R^{2}}\\right]-\\frac{3QR_{i}R_{j}}{R^{5}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{5(\\mathbf{R}.\\mathbf{w})^{2}}{4c^{2}R^{2}}\\right]\\nonumber \\\\\n & & -\\frac{Qw_{i}w_{j}}{2c^{2}R^{3}}+\\frac{3Qw_{[i}R_{j]}(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{5}};\\label{Eij}\\\\\nE_{i0} & = & -\\frac{1}{c}\\frac{\\partial}{\\partial\\bar{t}}\\bar{\\phi}_{;i}-\\frac{1}{c^{2}}\\frac{\\partial^{2}\\bar{A}_{i}}{\\partial\\bar{t}^{2}}\\ \\equiv\\frac{1}{c}\\ \\frac{\\partial E_{i}}{\\partial\\bar{t}}\\ =\\ \\frac{Q}{cR^{3}}\\left[w_{i}-\\frac{3(\\mathbf{R}.\\mathbf{w})R_{i}}{R^{2}}\\right];\\label{Ei0}\\\\\nB_{ij} & = & \\epsilon_{i}^{\\ lm}\\bar{A}_{m;lj}\\ \\equiv\\ B_{i,j}=\\frac{Q}{cR^{3}}\\left[\\epsilon_{ij}^{\\ \\ k}w_{k}-\\frac{3}{R^{2}}(\\mathbf{R}\\times\\mathbf{w})_{i}R_{j}\\right];\\label{Bij}\\\\\nB_{i0}\\ & = & \\frac{1}{c}\\frac{\\partial B_{i}}{\\partial\\bar{t}}=-\\ \\frac{3Q}{c^{2}R^{5}}(\\mathbf{R}.\\mathbf{w})(\\mathbf{R}\\times\\mathbf{w})_{i}\\ .\\label{Bi0}\\end{eqnarray}\n}Note in particular that, unlike their gravitational counterparts,\n$E_{\\alpha\\beta}$ and $B_{\\alpha\\beta}$ are not symmetric, and have\nnon-zero time components. The antisymmetric parts $E_{[ij]}=E_{[i,j]}$\nand $B_{[ij]}=B_{[i,j]}$ above are (vacuum) Maxwell equations $\\nabla\\times\\mathbf{E}=-(1\/c)\\partial\\mathbf{B}\/\\partial t$\nand $\\nabla\\times\\mathbf{B}=(1\/c)\\partial\\mathbf{E}\/\\partial t$,\nimplying that a time varying electric\/magnetic field endows the magnetic\/electric\ntidal tensor with an antisymmetric part. For instance, a time varying\nelectric field will always induce a force on a magnetic dipole. The\nfact that $\\mathbb{E}_{\\alpha\\beta}$ and $\\mathbb{H}_{\\alpha\\beta}$\nare symmetric reflects the absence of analogous gravitational effects.\nThe time component $B_{i0}$ means that the force on a magnetic dipole\n(magnetic moment $\\mu=q\/2m$) will have a time component $(F_{EM})_{0}=(1\/c)\\bm{\\mu}.\\partial\\mathbf{B}\/\\partial t$,\nwhich (see \\cite{CostaHerdeiro2009} sec. 1.2) is minus the power\ntransferred to the dipole by Faraday's law of induction (and is reflected\nin the variation of the dipole's proper mass $m=-P^{\\alpha}U_{\\alpha}\/c^{2}$).\nAgain, this is an effect which has no gravitational counterpart: $\\mathbb{H}_{\\alpha0}=\\mathbb{H}_{0\\alpha}=0$,\nthus $(F_{G})_{0}=0$, and the proper mass of the gyroscope is a constant\nof the motion. \n\nThe space part of the geodesic equation for a test particle of velocity\n$\\mathbf{v}$ is:{\\small \\begin{eqnarray}\n\\mathbf{a} & = & \\nabla\\bar{\\Phi}+\\frac{2}{c}\\frac{\\partial\\bm{\\bar{\\mathcal{A}}}}{\\partial\\bar{t}}-2\\mathbf{v}\\times(\\nabla\\times\\bm{\\bar{\\mathcal{A}}})-\\frac{3}{c^{2}}\\frac{\\partial}{\\partial\\bar{t}}\\left(\\frac{M}{R}\\right)\\mathbf{v}\\label{GeoTrans}\\\\\n & = & -\\frac{M}{R^{3}}\\left[1+\\frac{2w^{2}}{c^{2}}-\\frac{3(\\mathbf{R}.\\mathbf{w})^{2}}{2c^{2}R^{2}}\\right]\\mathbf{R}+\\frac{3M(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{3}}\\mathbf{w}-\\frac{4M}{c^{2}R^{3}}\\mathbf{v}\\times(\\mathbf{R}\\times\\mathbf{w})+\\frac{3}{c^{2}}\\frac{M}{R^{3}}(\\mathbf{R}.\\mathbf{w})\\mathbf{v}\\ ,\\nonumber \\end{eqnarray}\n}which matches equation (10) of \\cite{SoffelKlioner}, or (7) of \\cite{Nordvedt1973},\nagain, in the special case of only one source, and keeping therein\nonly linear terms in the perturbations and test particle's velocity\n$\\mathbf{v}$. \n\nComparing with its electromagnetic counterpart{\\small \\[\n\\left(\\frac{m}{q}\\right)\\mathbf{a}=\\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}=\\frac{Q}{R^{3}}\\left[1+\\frac{w^{2}}{2c^{2}}-\\frac{3(\\mathbf{R}.\\mathbf{w})^{2}}{4c^{2}R^{2}}\\right]\\mathbf{R}-\\frac{1}{2}\\frac{Q(\\mathbf{R}.\\mathbf{w})}{c^{2}R^{3}}\\mathbf{w}+\\frac{Q}{c^{2}R^{3}}\\mathbf{v}\\times(\\mathbf{R}\\times\\mathbf{w})\\]\n}we find them similar to a certain degree (up to some factors), except\nfor the last term of (\\ref{GeoTrans}). That term signals a difference\nbetween the two interactions, because it means that there is a velocity\ndependent acceleration which is parallel to the velocity; that is\nin contrast with the situation in electromagnetism, where the velocity\ndependent accelerations arise from magnetic forces, and are thus always\nperpendicular to $\\mathbf{v}$.\n\nAs expected from Eqs. (\\ref{PrecessGen}) (and by contrast with the\nother effects), the precession of a gyroscope carried by $\\bar{\\mathcal{O}}$,\nEq. (\\ref{PrecessTrans}b) takes a form analogous to the precession\nof a magnetic dipole, Eq. (\\ref{PrecessTrans}a), if we express $\\mathbf{S}$\nin the local orthonormal tetrad $e^{\\hat{i}}$, non rotating relative\nto the inertial observer at infinity, such that $S^{i}=(1-M\/R)S^{\\hat{i}}$:\n\\begin{equation}\n\\frac{d\\mathbf{S}}{d\\bar{t}}=\\frac{q}{2m}\\frac{Q}{c^{2}R^{3}}\\left[\\mathbf{S}\\times(\\mathbf{R}\\times\\mathbf{w})\\right]\\ \\ (a);\\ \\ \\ \\ \\ \\ \\frac{dS^{\\hat{i}}}{d\\bar{t}}=\\frac{2M}{c^{2}R^{3}}\\left[(\\mathbf{R}\\times\\mathbf{w})\\times\\mathbf{S}\\right]^{\\hat{i}}\\ \\ (b)\\ .\\label{PrecessTrans}\\end{equation}\nIf instead of the gyroscope comoving with observer $\\bar{\\mathcal{O}}$\n(with constant velocity $\\mathbf{w}$), we had considered a gyroscope\nmoving in a circular orbit, then an additional term would arise in\nanalogy with Thomas precession for the magnetic dipole; for a circular\ngeodesic that term amounts to $-1\/4$ of expression (\\ref{PrecessTrans}b),\nand we would obtain the well known equation for geodetic precession\n(e.g. \\cite{Gravitation and Inertia}).\n\n\n\\section{Conclusion}\n\nWe conclude our paper by discussing some of the implications of our\nconclusions in the approaches usually found in literature. In the\nframework of linearized theory, e.g. \\cite{Ruggiero:2002,Gravitation and Inertia},\nEinstein equations are often written in a Maxwell-like form; likewise,\ngeodesics, precession and gravitational force on a spinning test particle\nare cast (in terms of 3-vectors defined in analogy with the electromagnetic\nfields $\\mathbf{E}$ and $\\mathbf{B}$) in a form similar to, respectively,\nthe Lorentz force on a charged particle, the precession and the force\non a magnetic dipole. \n\nWe have concluded that the actual physical similarities between gravity\nand electromagnetism (on which the physical content of such approaches\nrelies) occur only on very special conditions. For tidal effects,\nlike the forces on a gyroscopes\/dipoles, the analogy manifest in Eqs.\n(\\ref{FG_Stationary}) holds only when the \\emph{test particle} sees\ntime independent \\emph{fields}. In the example of analogous systems\nconsidered in section \\ref{Translational-vs.-Rotational}, this means\nthat the center of mass of the gyroscope\/dipole must not move relative\nto the central body. In the case of the analogy between the equation\nof geodesics and the Lorentz force law (see Fig. \\ref{fig1}), as\nmanifest in equation (\\ref{geoGeneral}), it is in the \\emph{potentials\/metric\nperturbations}, as seen by \\emph{the observer} (not the test particle!),\nthat the time independence is required. The latter condition is not\nas restrictive as the one of the tidal effects: consider for instance\nobservers moving in circular orbits around a static mass\/charge; such\nobservers see an unchanging spacetime, and unchanging electromagnetic\npotentials, so, for them, the equation of geodesics and Lorentz force\ntake similar forms (such analogy may actually be cast in an exact\nform, see \\cite{Natario,Jantzen}). However, those observers see a\ntime-varying electric field $\\mathbf{E}$ (constant in magnitude,\nbut varying in direction), which, by means of equations (4) and (6),\nimplies that the tidal tensors are not similar to the gravitational\nones%\n\\footnote{The electromagnetic field $F^{\\alpha\\beta}$ is not constant along\nthe worldline of an observer moving in a circular orbit (radius $R$,\nangular velocity $\\bm{\\Omega}$, velocity $\\mathbf{w}=\\bm{\\Omega}\\times\\mathbf{R}$)\naround a point charge. Its variation endows the magnetic tidal tensor\nwith an antisymmetric part, and the electric tidal tensor with a time\ncomponent: $dF^{0i}\/d\\tau=Qw^{i}\/cR^{3}=-2E^{[i0]}=-\\epsilon^{ijk}B_{[jk]}$.\nThis means that they significantly differ from the GR tidal tensors\nseen by an observer in circular motion around a point mass.\n\nNote that both the GR and the EM tidal tensors for these analogous\nproblems can be obtained from, respectively, Eqs. (\\ref{Egij})-(\\ref{Hij})\nand (\\ref{Eij})-(\\ref{Bi0}), making therein $\\mathbf{R}.\\mathbf{w}=0$\n(corresponding to circular motion), despite the fact that these expressions\nwere originally derived for an observer with constant velocity. This\nis because, as can be seen from their definitions in Table \\ref{analogy},\nit is the 4-velocity $U^{\\alpha}$ (regardless of the way it varies),\nat the given point, that determines the tidal tensors. %\n}. \n\nFinally, as a consequence of this analysis, a distinction, from the\npoint of view of the analogy with electrodynamics, between effects\nrelated to (stationary) rotational mass currents, and those arising\nfrom translational mass currents, becomes clear: albeit in the literature\nboth are dubbed {}``gravitomagnetism'', one must note that, while\nthe former are clearly analogous to magnetism, in the case of the\nlatter the analogy is not so close.\n\n\\vskip 20pt\n\\leftline{{\\normalfont\\bfseries Acknowledgments}}\n\\vskip 4pt\nWe thank the anonymous referee for very useful comments and suggestions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{intro}\nDiluted magnetic semiconductors (DMS) combine the benefits of semiconducting and magnetic materials and hence are most promising candidates for future spintronics applications \\cite{Dietl2014, Dietl2010}. In particular, III-Mn-V DMS, as for instance the archetypal compound Ga$_{1-x}$Mn$_{x}$As, have been the subject of intensive research in the last two decades, motivated by both fundamental and technological interests \\cite{Ohno1998}. \nIn Ga$_{1-x}$Mn$_{x}$As, as well as in the related compound Ga$_{1-x}$Mn$_{x}$P, a few atomic percent of the nonmagnetic gallium host sublattice atoms are replaced with magnetic manganese atoms acting as acceptors. Both materials exhibit a long-range magnetic order of substitutional Mn ions mediated by holes \\cite{Jungwirth2006, Jungwirth2007, Scarpulla2005, Winkler2011}. Crystalline defects---in particular Mn interstitials (Mn\\textsubscript{I}), which act as double donors and thereby compensate the hole doping, as well as As antisites (As\\textsubscript{Ga})---play a substantial role for both the strongly interrelated electronic and magnetic properties of DMS. Therefore, the preparation of high-quality samples is a delicate procedure and constitutes an obstacle to further enhancement of the Curie temperature $T_\\mathrm{C}$ beyond $190\\,$K for Ga$_{1-x}$Mn$_{x}$As \\cite{Chen2009} and $65\\,$K for Ga$_{1-x}$Mn$_{x}$P \\cite{Farshchi2006}. \nMoreover, the effects of disorder \\cite{Berciu2001} and carrier-carrier interactions \\cite{Richardella2010} make the theoretical description difficult \\cite{Sato2010}, and there is still no consensus on the development of spontaneous magnetization in these materials. \nIn literature, a broad spectrum of sometimes opposing theoretical approaches exists \\cite{Samarth2012}, ranging from the assumption of free charge carriers \\cite{Koenig2003, Dietl2001, Dietl2000, Koenig2000} to the opposite case of strongly localized carriers \\cite{Dobrowolska2012, Bhatt2002, Berciu2001}. While the first picture assumes the Fermi energy to lie within a merged band consisting of the valence band and a strongly broadened impurity band, the second model proposes the existence of a separate impurity band.\nThe latter, so-called impurity-band model is expected to be valid for weakly-doped samples. In this context, Kaminski and Das Sarma \\cite{Kaminski2002, Kaminski2003} have developed an analytic polaron percolation theory for DMS ferromagnetism in the limit of strong charge carrier localization and for an inhomogeneous spatial distribution of magnetic impurities. In this model, the spins of localized charge carriers can polarize the surrounding magnetic impurities, leading to the emergence of bound magnetic polarons (BMP), which grow in size for decreasing temperatures and at low enough temperatures overlap until an infinite cluster is formed and spontaneous magnetization occurs \\cite{Kumar2013}.\nAnother conceivable scenario is that the p-d Zener model, which is equivalent to a weak-coupling RKKY picture and believed to be appropriate for more metallic systems with free charge carriers, can also be applied on the insulator side of the metal-insulator transition (MIT) of Ga$_{1-x}$Mn$_{x}$As. In this case, the hole localization length remains much greater than the average distance between the acceptors \\cite{Ohno2008, Dietl2006}. Large mesoscopic fluctuations in the local value of the density of states near the MIT are expected to lead to a nano-scale phase separation into ferromagnetic and paramagnetic regions below $T_\\mathrm{C}$, whereby the paramagnetic (hole-poor) regions can persist down to low temperatures, coexisting with ferromagnetic (hole-rich) bubbles. \\newline\nMotivated by the still ongoing debate about the particular mechanisms of ferromagnetism and the crucial role of point defects, the present study focuses on a systematic investigation of low-frequency electronic transport properties on a series of Ga$_{1-x}$Mn$_{x}$As samples with different Mn contents and growth parameters. We apply fluctuation spectroscopy, a method which is sensitive to electric inhomogeneities on the nano- and micrometer scale and has been proven to be a versatile tool for identifying electronic phase separation and magnetically driven percolation, as observed, e.g., for perovskite manganites \\cite{Podzorov2000} and more recently for the semimetallic ferromagnet EuB\\textsubscript{6} \\cite{Das2012}. Here, we aim to compare the noise behavior of metallic Ga$_{1-x}$Mn$_{x}$As films ($x>2$ \\%) with rather insulating ($x<2$ \\%) samples in order to gain further insight into possible percolative transitions and electronic phase separation. Moreover, we investigate another compound, Ga$_{1-x}$Mn$_{x}$P, an even better candidate for the observation of a percolative transition. Despite its chemical similarity with Ga$_{1-x}$Mn$_{x}$As, Ga$_{1-x}$Mn$_{x}$P has a Mn acceptor level lying four times deeper within the band gap, i.e.\\ about $0.4\\,$eV above the valence band edge \\cite{Clerjaud1985}. Therefore, it is obvious that the charge carriers are of a much more localized nature than in Ga$_{1-x}$Mn$_{x}$As. Nevertheless, hole-mediated ferromagnetism has also been demonstrated in Ga$_{1-x}$Mn$_{x}$P \\cite{Farshchi2006}. In addition, Ga$_{1-x}$Mn$_{x}$P is very similar to Ga$_{1-x}$Mn$_{x}$As concerning the magnetic anisotropy, spin-polarization and the scaling of $T_\\mathrm{C}$ as a function of the Mn concentration \\cite{Zhou2015}. A comparison of weakly doped Ga$_{1-x}$Mn$_{x}$As with a Ga$_{1-x}$Mn$_{x}$P sample ($x=3.5$ \\%) may provide valuable information about possible commonalities and differences with regard to their noise characteristics and possible electronic inhomogeneities. Apart from varying the Mn content of the above mentioned compounds, an alternative approach for the control of magnetic and electronic properties is the irradiation with He ions, leading to the introduction of deep traps into the system and thereby to an increasing disorder. In this work, we therefore also address the question of which consequences ion irradiation has for the low-frequency resistance noise characteristics and whether signatures of an electronic phase separation or introduced defects can be observed for irradiated samples. It should be noted that fluctuation spectroscopy in general is highly suitable for studying the energy landscape of defects in semiconducting thin films \\cite{Lonsky2016, Lonsky2015} and semiconductor heterostructures \\cite{Mueller2006, Mueller2006a, Muller2015}. \n\n\\section{Experimental Details}\n\\subsection{Sample Growth} \\label{growth}\nElectronic transport measurements have been performed on a total of seven Ga$_{1-x}$Mn$_{x}$As thin film samples and one Ga$_{1-x}$Mn$_{x}$P sample, see Table \\ref{tab:table1} for an overview. For all films, the corresponding Curie temperatures were determined by magnetization measurements \\cite{Zhou2016, Hamida2014, Yuan2017a, Zhou2015}.\nIn general, a crucial parameter is the Mn content $x$, but other factors as thermal annealing or induced disorder by ion irradiation also have a strong influence on the Curie temperature $T_\\mathrm{C}$ and the crystalline defect characteristics \\cite{Hayashi2001, Potashnik2001, Esch1997, Sinnecker2010}.\nThe samples in this study were prepared in two different ways. Metallic Ga$_{1-x}$Mn$_{x}$As samples with $x=4$ \\% were grown by low-temperature molecular beam epitaxy (LT-MBE) on semi-insulating GaAs(001) substrates in a Mod Gen II MBE system with the lowest possible As\\textsubscript{2}-partial pressure of about $2\\times 10^{-6}\\,$mbar at PTB in Braunschweig \\cite{Hamida2014}. After the growth of a $100\\,$nm high-temperature (HT) GaAs buffer layer at $T_\\mathrm{g}=560\\,^\\circ$C, the temperature was decreased to $270\\,^\\circ$C for the subsequent LT Ga$_{1-x}$Mn$_{x}$As growth. Post-growth annealing at $200\\,^\\circ$C (18 hours in ambient atmosphere) was performed for one of the samples in order to enhance $T_\\mathrm{C}$. The total Mn concentration $x$ was calculated from the molecular flux ratio of Mn and Ga measured in the MBE at the position of the wafer and compared with reflection high-energy electron diffraction (RHEED) and energy-dispersive X-ray spectroscopy (EDX) measurements. \nMoreover, three Ga$_{1-x}$Mn$_{x}$As samples with a nominal Mn concentration of $x=6$ \\% were grown by LT-MBE on semi-insulating GaAs(001) using a Veeco Mod III MBE system in Nottingham \\cite{Zhou2016, Wang2008}. In this case, thermal annealing was performed at $190\\,^\\circ$C for 48 hours in ambient atmosphere and the Mn content was determined from the Mn\/Ga flux ratio. Two of the films were irradiated with different doses of He ions after growth. This particular method allows to control the hole concentration and thus the electronic as well as the magnetic properties without changing the Mn content of a sample. The He-ion energy was chosen as $4\\,$keV. The fluences were $2.5\\times 10^{13}\\,$cm$^{-2}$ and $3.5\\times 10^{13}\\,$cm$^{-2}$ for the two irradiated samples. A better measure for the effect of irradiation on material properties than the fluence is the so-called displacement per atom (DPA), i.e.\\ the number of times that an atom in the target is displaced during irradiation. This allows for a comparison with data reported in the literature, in which other ion species and energies are used. For the two irradiated samples, the DPA was $1.6\\times 10^{-3}$ and $2.24\\times 10^{-3}$, respectively \\cite{Zhou2016}. During ion irradiation, the films were tilted by $7^{\\circ}$ to avoid channeling. The irradiation parameters result in defects distributed roughly uniformly in the whole Ga$_{1-x}$Mn$_{x}$As layer as confirmed by simulations using the SRIM (Stopping and Range of Ions in Matter) code \\cite{Ziegler1985}. No measurable increase of Mn interstitials was observed by Rutherford backscattering spectroscopy (RBS) \\cite{Zhou2016}. Previous studies show that also the sheet concentration of substitutional Mn atoms remains constant \\cite{Winkler2011}, which is why we conclude that the main effect of He-ion irradiation is to introduce deep traps and thereby compensate the holes. It is well established that these defects reside in the As sublattice and most of them are primary defects related to vacancies and interstitials \\cite{Sinnecker2010, Pons1985}. In our case, atomic force microscopy (AFM) and RBS measurements do not show any indications of irradiation induced surface reconstruction. Further details can be found in Ref.\\ \\cite{Zhou2016}. \nFinally, two Ga$_{1-x}$Mn$_{x}$As samples with low Mn contents of $1.8$ \\% and $1.2$ \\% as well as a Ga$_{1-x}$Mn$_{x}$P sample with $x=3.5$ \\% were fabricated by ion implantation combined with pulsed laser melting in Dresden \\cite{Yuan2017a, Zhou2015}. Ion implantation is a common materials engineering technique for introducing foreign ions into a host material. In this case, Mn ions are implanted into GaAs or GaP wafers. The subsequent laser pulse drives a rapid liquid-phase epitaxial growth. The implantation energy was set to $100\\,$keV for GaAs \\cite{Yuan2017a} and $50\\,$keV for GaP \\cite{Yuan2016, Zhou2015}. The wafer normal was tilted by $7^{\\circ}$ with respect to the ion beam to avoid a channeling effect. A coherent XeCl laser (with $308\\,$nm wavelength and $28\\,$ns duration) was employed to recrystallize the samples, and the energy densities were optimized to achieve high crystalline quality and the highest Curie temperature. The optimal laser energy density is $0.30\\,$J\/cm$^2$ for Ga$_{1-x}$Mn$_{x}$As and $0.45\\,$J\/cm$^2$ for Ga$_{1-x}$Mn$_{x}$P. The Mn concentration was determined by secondary ion mass spectroscopy. In contrast to films grown by LT-MBE, neither Mn interstitials nor As antisites are observed in samples prepared by ion implantation combined with pulsed laser melting \\cite{Yuan2017a, Cho2007}. Transmission electron microscopy (TEM) studies prove the high crystalline quality of the films and exclude the presence of any extended lattice defects, amorphous inclusions and precipitates of other crystalline phases \\cite{Yuan2017a}. \nFor some selected films, an array of $50\\times 50\\, \\mu$m$^2$ Hall bars was defined by photolithography followed by wet chemical etching. The quality of Hall effect measurements thereby improves significantly due to a well-defined contact geometry and, as explained in more detail below, the resistance noise magnitude as the desired measurement signal increases due to smaller sample volumes according to Hooge's law \\cite{Hooge1969, Hooge1976}. For all samples, electrical contacts were made by soldering In\/Sn on top of the films.\nCharge carrier concentrations obtained from Hall effect measurements are given in Table \\ref{tab:table1}. Due to the use of two different fabrication techniques and since the LT-MBE samples from Braunschweig and Nottingham were grown at different substrate temperatures and As-fluxes, care has to be taken in comparing the values for the hole concentration $p$ of films of different origin. Apart from that, as expected, the hole density increases after thermal annealing for the $x=4$ \\% samples and decreases with increasing ion irradiation dose for the $x=6$ \\% samples, cf.\\ Section \\ref{ssecmetallic} for more details. Furthermore, there is also a clear correlation between $T_\\mathrm{C}$ and $p$. Finally, we point out that extensive studies on all present samples, including magnetization measurements and standard thin film characterization techniques, have been published elsewhere, cf.\\ Ref.\\ \\cite{Hamida2014, Zhou2016, Yuan2017a, Zhou2015}. \n\\begin{table*}\n\\caption{\\label{tab:table1}Overview of the investigated thin film samples and related parameters, including: information about the manganese content $x$, whether samples were grown by low-temperature molecular beam epitaxy (LT-MBE) or ion implantation combined with pulsed laser melting (II+PLM), the institute where samples have been fabricated, which kind of post-treatment was given, the values of the film thickness, the Curie temperature $T_\\mathrm{C}$ as determined by magnetization measurements, and the hole density $p(T=300\\,\\mathrm{K})$ obtained from Hall effect measurements.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\nMn content&Fabrication&Source&Remarks&Thickness&$T_\\mathrm{C}$ & $p(T=300\\,\\mathrm{K})\\ [1\/\\mathrm{cm}^3]$\\\\\n\\hline\nGa$_{1-x}$Mn$_{x}$As\\\\\n4.0 \\% & LT-MBE & PTB & as-grown, micro-structured& $25\\,$nm & $70\\,$K & $8.0\\times 10^{19}$\\\\\n4.0 \\% & LT-MBE & PTB & annealed ($200\\,^\\circ$C, $18\\,$h), micro-structured& $25\\,$nm & $110\\,$K & $1.2\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed ($190\\,^\\circ$C, $48\\,$h)& $25\\,$nm & $125\\,$K & $9.6\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed, He-ion irradiated (low dose)& $25\\,$nm & $75\\,$K & $8.0\\times 10^{20}$\\\\\n6.0 \\% & LT-MBE & Nottingham & annealed, He-ion irradiated (high dose)& $25\\,$nm & $50\\,$K & $5.6\\times 10^{20}$\\\\\n1.8 \\% & II+PLM & HZDR & as-grown, micro-structured& $60\\,$nm & $60\\,$K & $2.8\\times 10^{20}$\\\\\n1.2 \\% & II+PLM & HZDR & as-grown, micro-structured& $60\\,$nm & $31\\,$K & $1.0\\times 10^{20}$\\\\\n\\hline\nGa$_{1-x}$Mn$_{x}$P\\\\\n3.5 \\% & II+PLM & HZDR & as-grown& $34\\,$nm & $45\\,$K & $3.1\\times 10^{20}$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\\subsection{Measurements}\nElectronic transport measurements have been performed using both AC and DC techniques. Experiments were carried out in a continuous-flow cryostat with variable temperature insert. Magnetic fields were applied perpendicular to the film plane.\nFor some of the lithographically patterned Ga$_{1-x}$Mn$_{x}$As samples, low-frequency noise spectroscopy was conducted in a five-terminal AC setup, where the sample is placed in a bridge-circuit in order to suppress the constant DC voltage offset and to minimize external perturbations \\cite{Scofield1987}. Other samples were measured in a four-terminal AC or DC setup. As a few samples showed a frequency-dependent resistivity at low temperatures, the excitation frequency was reduced from $227\\,$Hz down to $17\\,$Hz, or a DC noise measurement setup was utilized, which was verified to yield the same results as AC noise measurements. \nThe fluctuating voltage signal is preamplified and processed by a spectrum analyzer yielding the voltage noise power spectral density (PSD) $S_{V}(\\omega)$ defined by:\n\\begin{equation}\nS_{V}(\\omega)=2\\lim\\limits_{T \\to \\infty}\\frac{1}{T}\\left| \\int_{-T\/2}^{T\/2} {\\rm d}t \\, e^{-i\\omega t} \\, \\delta V(t) \\right|^2,\n\\end{equation} \nwhere $\\delta V(t)$ represents the fluctuating voltage drop across the sample and $\\omega = 2\\pi f$ the angular frequency. \nCare was taken that all spurious sources of noise were minimized or eliminated. \nAs required by Hooge's empirical law \\cite{Hooge1969, Hooge1976},\n\\begin{equation}\nS_{V}(f)=\\frac{\\gamma_{H}\\cdot V^2}{n \\Omega f^{\\alpha}},\n\\end{equation} \nthe magnitude of the voltage noise scales as $S_V \\propto V^2 \\propto I^2$, where $I$ represents the current flowing through the sample. \nHere, $n$ is the charge carrier density and $\\Omega$ the 'noisy' sample volume, i.e.\\ $n \\Omega = N_c$ gives the total number of charge carriers in the material causing the observed $1\/f$ noise. $\\alpha$ describes the frequency exponent which is commonly in the range $0.8\\leq \\alpha \\leq 1.4$ for $1\/f$-type fluctuations. \nThe Hooge parameter $\\gamma_H$ is widely used to compare the noise level of different systems and covers a range of $\\gamma_H=10^{-6}$--$10^{7}$ for different materials \\cite{Raquet2001}. For bulk semiconductors, $\\gamma_H$ usually is of order $10^{-2}$--$10^{-3}$. Moreover, it is useful to normalize the magnitude of the voltage fluctuations with respect to the applied current, $S_{R}=S_{V}\/I^2$, and to the resistance of the sample, resulting in $S_{R}\/R^2$.\nExemplary noise spectra for a Ga$_{1-x}$Mn$_{x}$As sample are presented in Fig.$\\,$\\ref{PTB_CURRENT_DEP} for three different currents in a log-log plot. The dashed line represents a $1\/f^{\\alpha}$ function with a slope of $\\alpha =1.2$. The inset clarifies the quadratic scaling of $S_{V}$ with the current $I$ as predicted by Hooge's law. \nFurther details about the fluctuation spectroscopy technique can be found in \\cite{Raquet2001,Mueller2011}. \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_1019_8_currentdep.eps}\n\\caption{Exemplary current-dependent noise spectra $S_{V}(f)$ acquired for the as-grown Ga$_{1-x}$Mn$_{x}$As sample with $x=4$ \\%. Inset: A quadratic dependence (red line: linear fit to the data) between $S_{V}$ and $I$ verifying Hooge's empirical law.} \n\\label{PTB_CURRENT_DEP}%\n\\end{figure}%\n\n\\section{Results and Discussion}\n\n\\subsection{Ga\\textsubscript{1-x}Mn\\textsubscript{x}As films with high Mn content} \\label{ssecmetallic}\n\nAt first, we will discuss measurements on metallic films with Mn contents $x>2$ \\%. Resistivity curves of as-grown and annealed Ga$_{1-x}$Mn$_{x}$As samples with $x=4$ \\% are depicted in Fig.$\\,$\\ref{PTB_PSD}(a). As expected, typical maxima \\cite{Matsukura1998} are observed in the vicinity of the samples' respective Curie temperature $T_\\mathrm{C}$. After post-growth annealing, the resistivity significantly decreases while $T_\\mathrm{C}$ increases compared to as-grown films. This effect can be explained by the out-diffusion of Mn interstitials to the film surface \\cite{Edmonds2004}. Commonly, these interstitials act as donors and compensate the hole-mediated ferromagnetism. \nFig.$\\,$\\ref{PTB_PSD}(b) shows the corresponding normalized magnitude of the resistance fluctuations $S_{R}\/R^2$ at $1\\,$Hz as a function of temperature in a semi-logarithmic representation. Remarkably, the PSD varies over several orders of magnitude for the two films and shows, in contrast to the resistivity, no significant features around $T_\\mathrm{C}$. More specifically, there are no major changes in the noise behavior throughout the entire temperature range. In addition, a constant external out-of-plane magnetic field up to $7\\,$T does not lead to any changes in the normalized resistance noise (not shown, cf.\\ Ref.\\ \\cite{Lonsky2017}). Possible contributions from mixed phases could be overshadowed by fluctuations related to thermally-activated impurity switching processes. However, more likely, due to the high concentration of Mn substitutional atoms, charge carriers are delocalized and the formation of magnetic polarons is not to be expected for these metallic samples. The observed $1\/f$-type noise is likely to be dominated by switching processes related to crystalline defects, which can also be seen in a strong variation of the Hooge parameter $\\gamma_H$ for the two different samples. At room temperature, we obtain $\\gamma_H=1\\times 10^{-2}$ for the as-grown and $\\gamma_H=3\\times 10^0$ for the annealed film. Apparently, thermal annealing has a strong influence on the noise magnitude, leading to the presumption that slow fluctuation processes related to Mn interstitials might play an important role. Since thermal annealing reduces the density of Mn interstitials, one might expect a lower PSD for annealed samples, but the opposite is the case. The same behavior is observed for two $x=7$ \\% samples with a thickness of $d=25\\,$nm (not shown). Strikingly, for two further $x=4$ \\% films with a higher thickness of $d=100\\,$nm and similar values of $T_{\\mathrm{C}}$, a slight decrease of the noise magnitude is observed after thermal annealing, indicating that the rearrangement of Mn interstitials due to diffusion processes towards the surface \\cite{Edmonds2004}, which become passivated due to oxidation or by binding with surplus As atoms, and concomitant surface effects may play an important role for the changes in the PSD after thermal annealing.\n \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_Noise_Res_4.eps}\n\\caption{(a) Temperature-dependent resistivity showing characteristic maxima around $T_\\mathrm{C}$ (marked by arrows) and (b) normalized resistance noise magnitude of as-grown and annealed Ga$_{1-x}$Mn$_{x}$As samples with 4 \\% Mn content. No features in the noise power are visible around $T_\\mathrm{C}$.} \n\\label{PTB_PSD}%\n\\end{figure}%\nIn order to deduce the characteristic energies of the switching processes contributing to the $1\/f$-type noise, we apply the phenomenological model of Dutta, Dimon and Horn (DDH) \\cite{Dutta1979}. In this model, a certain distribution of activation energies $D(E,T)$ determines the temperature dependence of both the noise magnitude and the frequency exponent. An essential requirement for the applicability of this model is to check whether $\\alpha(T)$ calculated after\n\\begin{equation} \\label{eq:DDH}\n\\alpha(T)=1-\\frac{1}{\\ln(2\\pi f\\tau _0)} \\left[\\frac{\\partial \\ln(\\frac{S_R}{R^2}(f,T))}{\\partial \\ln(T)}-g^{\\prime}(T)-1\\right]\n\\end{equation} \nagrees with the measured values. Here, $\\tau _0$ represents an attempt time, usually between $10^{-14}$ and $10^{-11}\\,$s, corresponding to typical inverse phonon frequencies. Moreover, it is $g^{\\prime}(T)=\\frac{\\partial \\ln(g(T))}{\\partial \\ln(T)} \\equiv b$, where $g(T)=a \\cdot T^{b}$ accounts for an explicit temperature dependence of the distribution of activation energies $D(E,T)$, which can be caused by a change of the number of thermally activated switching events (excitation of defect states) or of the coupling of fluctuations to the resistivity with temperature. As can be seen in Fig.$\\,$\\ref{PTB_DDH}(a) and (b), we find a good qualitative or even quantitative agreement between model and experiment for both $x=4$ \\% samples. This allows for calculating the distribution of activation energies,\n\\begin{equation}\nD(E) \\propto \\frac{2\\pi f}{k_B T}\\frac{1}{g(T)}\\frac{S_R}{R^2}(f,T), \n\\end{equation}\nof thermally activated fluctuators, which is shown in Fig.$\\,$\\ref{PTB_DDH}(c) and (d). Here, it is $E=-k_B T \\ln(2\\pi f \\tau_{0})$. Both samples show a similar behavior, namely an increase of $D(E)$ towards higher activation energies, which we interpret as a superposition of several thermally activated processes with different energies, that can be attributed to various kinds of defects. In both cases, four local maxima in $D(E)$ are observed and marked by black arrows in Fig.$\\,$\\ref{PTB_DDH}(c) and (d), whereby the values of the corresponding activation energies are remarkably similar for the two samples. It is plausible to assume that, due to the low growth temperatures utilized during LT-MBE, which are required in order to prevent phase separation within the material, a great variety of defects, such as Mn interstitials and As antisites, contribute to the distribution of activation energies. Although the energies of the local maxima are comparable to typical impurity binding energies in GaAs, care has to be taken when assigning the energies to specific defect states due to band gap renormalization in heavily doped semiconductors, which is accompanied by a shift of the respective binding energies. \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{PTB_samples_DDH_DE_4percent.eps}\n\\caption{(a) and (b) Application of the phenomenological DDH model (red curves) to the resistance noise data of two metallic Ga$_{1-x}$Mn$_{x}$As samples with $x=4$ \\%, both showing a reasonable agreement between calculated and measured values for the frequency exponent $\\alpha(T)$. The function $g(T)$ and the attempt time $\\tau_0$ are indicated in each case. (c) and (d) Calculated distribution of activation energies $D(E)$ for both samples, showing a similar trend. Local maxima are indicated by black arrows.}\n\\label{PTB_DDH}%\n\\end{figure\n\nSince no signatures of electronic phase separation can be identified for conventional metallic Ga$_{1-x}$Mn$_{x}$As samples with high Mn content ($x=4$ \\%), we next focus on samples irradiated by He ions, whereby disorder in the films is enhanced by the introduction of deep traps, i.e.\\ the Fermi level is shifted by means of carrier compensation in order to change the conductivity from metallic to insulating. \nFig.$\\,$\\ref{GAMNAS_RES}(a) shows resistivity data of three Ga$_{1-x}$Mn$_{x}$As samples with $x=6$ \\%. Point defects were introduced by the irradiation with an energetic He-ion beam using different doses as described in Section \\ref{growth} (fluences: $2.5\\times 10^{13}\\,$cm$^{-2}$ and $3.5\\times 10^{13}\\,$cm$^{-2}$, hereinafter referred to as \"low dose\" and \"high dose\", respectively). The corresponding Curie temperatures determined from magnetization measurements are marked by arrows. While the unirradiated sample shows the lowest resistivity and metallic behavior with a relatively high $T_\\mathrm{C}$ of $125\\,$K, the resistivity increases strongly as a function of He-ion irradiation dose. At the same time, $T_\\mathrm{C}$ decreases down to $75\\,$K for the sample irradiated with a low He-ion dose and even further to $50\\,$K for a high irradiation dose. The typical maximum in resistivity \\cite{Edmonds2002} becomes less pronounced for more insulating samples. These major changes in resistivity can be explained by an increase of the displacement per atom (DPA) for higher irradiation doses, which results in a decrease of the hole concentration. This is confirmed by Hall effect measurements at room temperature for the three samples (cf.\\ Table \\ref{tab:table1}).\nThe normalized temperature-dependent resistance noise PSD $S_{R}\/R^2$ at $1\\,$Hz for the $x=6$ \\% samples is shown in Fig.$\\,$\\ref{GAMNAS_RES}(b) in a semi-logarithmic plot. The unirradiated sample has the lowest noise level over the entire temperature range. At temperatures below $100\\,$K, the PSD is nearly independent of temperature, followed by a slight increase above $100\\,$K towards room temperature. Likewise, below $80\\,$K, the film irradiated with a low He-ion dose shows the same constant noise magnitude, while there is a much stronger increase of nearly two orders of magnitude towards higher temperatures. The sample irradiated with the high dose shows the highest noise level of all three films below $100\\,$K, also followed by a characteristic increase for $T$ approaching room temperature. The great variation in the PSD for the different samples can be explained by the introduction of deep traps into the As sublattice. An exchange of charge carriers between such traps and the rest of the conducting material, resulting in fluctuations of the hole concentration, can cause the observed $1\/f$-type noise. As shown in previous studies \\cite{Zhou2016}, the concentration of Mn interstitials should not change as a function of irradiation dose, which implies that these interstitials are not the cause for the variation in the PSD as a function of He-ion irradiation dose, although they might still contribute to the $1\/f$ noise. In addition, the noise data shown in Fig.$\\,$\\ref{GAMNAS_RES}(b) suggest a crossover between two temperature regimes: A temperature-independent region below about $100\\,$K and a characteristic increase of the noise magnitude, where the number of activated defects increases towards higher temperatures. The Hooge parameter $\\gamma_{H}$ at room temperature for these samples is of order $\\gamma_{H}=10^{3}$--$10^{5}$ and therefore several orders of magnitude larger than for 'clean' semiconductors. Furthermore, $\\gamma_{H}$ is also several orders of magnitude higher than for the $x=4$ \\% samples, presumably due to the higher Mn content, a higher concentration of traps in the case of the irradiated samples and different growth parameters (substrate temperature and annealing procedure). \\newline \nApart from that, no features around $T_\\mathrm{C}$ were observed except a slight increase below $T_\\mathrm{C}$ for the highest He-ion dose, which might be a hint for weak electronic phase separation or the increasing localization of charge carriers, and a pronounced peak at $80\\,$K. As shown in Ref.\\ \\cite{Lonsky2017}, the application of the phenomenological model by Dutta, Dimon and Horn \\cite{Dutta1979} allows to assign the enhanced noise magnitude at temperatures between 60 and 100\\,K to a distinct peak in the distribution of activation energies $D(E)$ at energies of about $180\\,$meV. This energy can very likely be attributed to traps introduced by the strong He-ion irradiation. No changes in the normalized noise power were found in external magnetic fields up to $7\\,$T (cf.\\ Ref.\\ \\cite{Lonsky2017}). Except the above-mentioned weak increase of the PSD below $T_\\mathrm{C}$ for the sample irradiated with the high dose, there are no indications for an electronic phase separation. As suggested by the large values of $\\gamma_{H}$, possible contributions attributed to a percolative magnetic phase transition may be overshadowed by disorder effects or impurity switching processes. A different approach in order to find indications for a possible electronic phase separation is to tune the Mn content of Ga$_{1-x}$Mn$_{x}$As, which will be discussed in the following section.\n\\begin{figure\n\\centering \t\n\\includegraphics[width=8.5 cm]{HZDR_Heirrad_PSD_AND_Res.eps}\n\\caption{(a) Temperature-dependent resistivity data of three Ga$_{1-x}$Mn$_{x}$As ($x=6$ \\%) samples with different He-ion irradiation doses. (b) Normalized noise PSD as a function of temperature for the three films in a logarithmic representation. No magnetic field dependence or significant features around $T_\\mathrm{C}$ were observed. Arrows indicate the corresponding Curie temperatures.} \n\\label{GAMNAS_RES}%\n\\end{figure}%\n\n\\subsection{Ga\\textsubscript{1-x}Mn\\textsubscript{x}As films with low Mn content and Ga\\textsubscript{1-x}Mn\\textsubscript{x}P}\n\nIn this section, we focus on weakly doped Ga$_{1-x}$Mn$_{x}$As samples with localized charge carriers and compare the results with the metallic films and another insulating Ga$_{1-x}$Mn$_{x}$P reference sample. Fig.\\ \\ref{HZDR_PSD_IN}(a) depicts the resistivity of two as-grown Ga$_{1-x}$Mn$_{x}$As samples with $x=1.8$ \\% and $x=1.2$ \\% and a Ga$_{1-x}$Mn$_{x}$P film with $x=3.5$ \\%. In contrast to the metallic samples, no or only weak features around $T_\\mathrm{C}$ are visible in the resistivity. However, for all three films, the $1\/f$ noise magnitude is significantly enhanced just below the respective Curie temperature, as can be seen in Fig.$\\,$\\ref{HZDR_PSD_IN}(b), where the normalized resistance noise at $1\\,$Hz is plotted versus temperature. For the $x=1.8$ \\% film, only a weak increase occurs around $T_\\mathrm{C}$, followed by a sharp decrease towards lower temperatures. Although this compound is still to be considered as metallic, in the phase diagram it is located very close to the metal-insulator transition (MIT) \\cite{Yuan2017a}, which is why weak signatures of an electronic phase separation are conceivable. The Ga$_{1-x}$Mn$_{x}$As film with $x=1.2$ \\% exhibits nearly the identical noise magnitude between $50$ and $170\\,$K, which is easily comprehensible since the same fabrication technique has been employed, the Mn content is very similar and hence the defect landscape contributing to the resistance noise is comparable. The calculated Hooge parameters at room temperature for both samples are also comparable, namely $\\gamma_{H}=2\\times 10^{0}$ and $\\gamma_{H}=5\\times 10^{1}$ for the $x=1.2$ \\% and the $x=1.8$ \\% films, respectively. However, the peak in the vicinity of $T_\\mathrm{C}$ is much more pronounced for the $x=1.2$ \\% sample, because this compound is situated right on the edge of the metal-insulator transition, cf.\\ studies of electrical and magnetic properties on the present samples in Ref.\\ \\cite{Yuan2017a}. In detail, this sample still exhibits a global ferromagnetic behavior below $T_\\mathrm{C}$, but electronic transport measurements indicate its insulating character. Within the framework of this study, no noise measurements on Ga$_{1-x}$Mn$_{x}$As samples situated on the insulating side of the MIT could be performed, because the maximum possible current $I$ was not sufficient to measure $1\/f$-type spectra reliably. Instead, the investigated Ga$_{1-x}$Mn$_{x}$P sample is suggested to provide a reference example for the signatures of an electronic phase separation in fluctuation spectroscopy measurements of a diluted magnetic semiconductor with localized charge carriers. For this sample, the Hooge parameter at room temperature amounts to $\\gamma_{H}=1\\times 10^{4}$. As can be seen in Fig.\\ \\ref{HZDR_PSD_IN}(b), the Ga$_{1-x}$Mn$_{x}$P film exhibits a pronounced peak just below $T_\\mathrm{C}$, where the noise level increases by more than one order of magnitude in a small temperature interval. \nIn analogy to previous studies on the semimetallic ferromagnet EuB\\textsubscript{6} \\cite{Das2012}, the diverging behavior of the resistance noise PSD for the Ga$_{1-x}$Mn$_{x}$P sample can be described by a Lorentz function with a peak at $35.5\\,$K and a width $\\Delta T=2.5\\,$K. In the case of EuB\\textsubscript{6}, Das et al.\\ attribute this sharp peak to a magnetic polaron percolation. As suggested by Kaminski and Das Sarma, such a behavior is also to be expected for Ga$_{1-x}$Mn$_{x}$As or Ga$_{1-x}$Mn$_{x}$P samples with strongly localized charge carriers \\cite{Kaminski2002, Kaminski2003}. Due to the high defect concentration in DMS, it is assumed that the charge carrier concentration is highly inhomogeneous and as ferromagnetism is mediated by charge carriers, upon decreasing the temperature, the ferromagnetic transition will first occur locally within the regions with higher carrier concentration. Upon lowering the temperature, these finite-size clusters will grow and merge until the entire sample becomes ferromagnetic via a percolation transition. \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{HZDR_insulsamples_PSD.eps}\n\\caption{(a) Resistivity curves of two Ga$_{1-x}$Mn$_{x}$As samples with low Mn content ($x=1.8$ \\% and $x=1.2$ \\%) and a Ga$_{1-x}$Mn$_{x}$P reference sample ($x=3.5$ \\%). (b) Temperature-dependent noise magnitude of the three samples, all showing an enhanced $1\/f$ noise just below $T_\\mathrm{C}$. Curie temperatures are marked by arrows.} \n\\label{HZDR_PSD_IN}%\n\\end{figure}%\nHowever, in contrast to semimetallic EuB\\textsubscript{6}, where the ferromagnetic transition is accompanied by a drastic reduction of $\\rho(T)$ and a colossal magnetoresistance effect, the temperature dependence of the resistivity of the investigated insulating DMS samples is monotonic. For samples located in the vicinity of the metal-insulator transition, only a small kink is observable in $\\rho(T)$ around $T_\\mathrm{C}$. In general, a ferromagnetic percolation transition is accompanied by an increase of the electrical conductivity, but, at the same time, this may be compensated by the increase of the resistivity with decreasing temperature due to the semiconducting nature of the material \\cite{Kaminski2003}. In the case of strongly localized charge carriers, the decrease in their hopping rate upon decreasing temperature overcomes the decrease in the hopping activation energy due to the ferromagnetic transition. Strikingly, although no features can be observed in the resistivity at $T_\\mathrm{C}$, resistance noise, which is very sensitive to the microscopic current distribution in the sample, shows a strong peak for the Ga$_{1-x}$Mn$_{x}$P sample. Due to the less localized nature of holes in the present weakly doped Ga$_{1-x}$Mn$_{x}$As samples, the possible percolation transition and thus the enhancement of the PSD are less pronounced. In addition, in the vicinity of the maximum in $S_{R}\/R^2$, we find a strong deviation between the calculated frequency exponents from Eq.\\ (\\ref{eq:DDH}) and the experimentally determined values, which is shown in Fig.\\ \\ref{HZDR_GAMNAP_ANALYS}(a) and (b) for the Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%) and the Ga$_{1-x}$Mn$_{x}$P samples, respectively. This non-applicability of the DDH model (marked by gray shaded areas) is another indication for a percolative transition \\cite{Kogan1996}, since the assumptions of this phenomenological approach are not compatible with the nonlinear electronic transport behavior around the percolation threshold. The deviations between calculated and experimentally determined frequency exponents and the divergence in the PSD in the vicinity of the percolation threshold $p_c$ can be understood within the frame of a random resistor network (RRN) model \\cite{Kogan1996}. The reduced number of effective current paths results in the suppression of cancellation of uncorrelated resistance fluctuations along different paths, which are abundant far away from $p_c$ \\cite{Das2012}. Around $p_c$ the current density is strongly inhomogeneous and the most significant contribution to the resistance noise comes from so-called bottlenecks which connect large parts of the infinite cluster. Here, the current density is higher than in other parts of the network. \nRammal et al.\\ have shown that near the percolation threshold $p_c$, the PSD diverges as $S_{R}\/R^2\\propto(p-p_c)^{-\\kappa}$, while the resistance $R$ behaves as $R\\propto (p-p_c)^{-t}$ \\cite{Rammal1985}. Here, $\\kappa$ and $t$ are critical percolation exponents derived from a RRN model, and $p$ is the fraction of unbroken bonds of a RRN. Due to the non-accessibility of these exponents in an experiment, it is common to link the PSD and the resistance via $S_{R}\/R^2 \\propto R^{w}$ with $w = \\kappa \/t$ \\cite{Yagil1992}. The corresponding analysis for the two relevant thin films is shown in Fig.$\\,$\\ref{HZDR_GAMNAP_ANALYS}(c) and (d), yielding a critical exponent $w=3.7\\pm 0.3$ for Ga$_{1-x}$Mn$_{x}$P and $w=7.1\\pm 0.3$ for Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%). While for Ga$_{1-x}$Mn$_{x}$P this is in fair agreement with typical values for the exponent $w$, e.g.\\ $w=2.9\\pm 0.5$ for perovskite manganites \\cite{Podzorov2000}, the value for the Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%) sample is exceptionally high. We note that no clear systematic changes of shape, position and height of the peak in the temperature-dependent PSD as a function of the applied out-of-plane magnetic field $B$ can be observed. It is assumed that possible changes as a function of the external field are too weak in order to be resolved. \n\\begin{figure\n\\centering\n\\includegraphics[width=8.5 cm]{HZDR_DDHanalysis_insulating.eps}\n\\caption{(a) and (b) Comparison between experimentally determined and calculated (DDH model) frequency exponents for the Ga$_{1-x}$Mn$_{x}$As sample with $x=1.2$ \\% and the Ga$_{1-x}$Mn$_{x}$P film. Strong deviations occur in the gray shaded areas around the percolative transition. (c) and (d) Log-log plot of PSD versus resistance for Ga$_{1-x}$Mn$_{x}$As ($x=1.2$ \\%) between $24$ and $32\\,$K and for Ga$_{1-x}$Mn$_{x}$P between $35$ and $45\\,$K. The solid black lines correspond to linear fits yielding $S_{R}\/R^2\\propto R^{w}$ with $w=7.1\\pm 0.3$ for the weakly doped Ga$_{1-x}$Mn$_{x}$As film and $w=3.7\\pm 0.3$ for the Ga$_{1-x}$Mn$_{x}$P sample.} \n\\label{HZDR_GAMNAP_ANALYS}%\n\\end{figure}%\nWe emphasize that the discussed picture of percolating bound magnetic polarons is expected to be only valid within the impurity-band model and is not compatible with the p-d Zener model, which assumes ferromagnetism being mediated by a Fermi sea of itinerant holes. The impurity-band model suggests that even for strong Mn doping the Fermi energy is located within the separate impurity band and only the degree of localization of the charge carriers will change. \nOur results support the view of holes being trapped in localized impurity band states for weakly doped Ga$_{1-x}$Mn$_{x}$As as well as for Ga$_{1-x}$Mn$_{x}$P, whereas for metallic Ga$_{1-x}$Mn$_{x}$As samples with higher $x$, where no signatures of a percolation mechanism were observed in the electronic noise, the widely-held view of delocalized holes within the valence band mediating ferromagnetism is more appropriate. We note that, for instance, DC transport and optical studies \\cite{Jungwirth2007} as well as first-principle calculations \\cite{Bae2016} corroborate the applicability of the two different models on Ga$_{1-x}$Mn$_{x}$As for the respective Mn concentration ranges. Moreover, because of a strong variation of the Mn energy level among different III-Mn-V combinations, it is unlikely that all materials can be treated within a single model \\cite{Khalid2014}. As a consequence of the higher degree of hole localization in Ga$_{1-x}$Mn$_{x}$P, the peak in $S_{R}\/R^2$ is more pronounced as compared to the insulating Ga$_{1-x}$Mn$_{x}$As sample. \nAn alternative interpretation of our findings for the Ga$_{1-x}$Mn$_{x}$As with small $x$ valid within the framework of the p-d Zener model could be an electronic phase separation in the vicinity of the metal-insulator transition \\cite{Ohno2008, Dietl2006}, cf.\\ Section \\ref{intro}. In this case, the nano-scale phase separation results in the coexistence of ferromagnetic bubbles (metallic, hole-rich regime) and a paramagnetic matrix (insulating, hole-poor regime). However, this kind of electronic phase separation should persist in a broad temperature range below $T_\\mathrm{C}$ for insulating Ga$_{1-x}$Mn$_{x}$As samples with low Mn doping, which is not expected to result in such a pronounced and sharp peak in the temperature-dependent noise power spectral density as it was observed in this work. It should also be noted that all investigated samples show global ferromagnetism below $T_\\mathrm{C}$, i.e.\\ there are no mixed phases consisting of ferromagnetic clusters and superparamagnetic grains. Yuan et al.\\ have shown that these mixed phases exist for Ga$_{1-x}$Mn$_{x}$As samples with $x\\leq 0.9$ \\%, but not for $x\\geq 1.2$ \\% \\cite{Yuan2017a}. It is desirable to study the resistance noise behavior of such mixed phases in future. \n\n\\section{Summary and Conclusion}\nIn this work, we investigated the resistance noise on a series of Ga$_{1-x}$Mn$_{x}$As films with different manganese and defect concentrations and an insulating Ga$_{1-x}$Mn$_{x}$P reference sample, all of which exhibit global ferromagnetism below $T_\\mathrm{C}$. By applying the phenomenological model by Dutta, Dimon and Horn, we calculated the distribution of activation energies $D(E)$ for several samples and discussed a superposition of different types of defects contributing to the measured $1\/f$-type noise. From the comparison of metallic and insulating samples we conclude that resistance noise in metallic Ga$_{1-x}$Mn$_{x}$As samples ($x>2$ \\%) is mainly dominated by impurity switching processes and no prominent features occur around $T_\\mathrm{C}$ even in the presence of an external out-of-plane magnetic field, while insulating samples, in particular Ga$_{1-x}$Mn$_{x}$P, show a sharp peak in the noise magnitude around $T_\\mathrm{C}$ which can be attributed to percolation processes in the material, for which we find a scaling behavior $S_{R}\/R^2 \\propto R^{w}$. Consequently, for Ga$_{1-x}$Mn$_{x}$P, we infer that the picture of percolating magnetic polarons within the impurity-band model is applicable, while for Ga$_{1-x}$Mn$_{x}$As, this picture seems to be valid only for low Mn doping $x$. These findings for samples with localized charge carriers are supported by clear deviations between the calculated (within the DDH model) and experimentally determined frequency exponents $\\alpha$ around the percolative transition. Besides varying the Mn content, another approach to tune Ga$_{1-x}$Mn$_{x}$As samples from the metallic side of the phase diagram towards the insulating regime is to irradiate the films with He ions. It was shown that fluctuation spectroscopy is sensitive to the changes in the defect landscape of irradiated samples, but no clear signs of electronic phase separation could be observed. \\newline\n We have shown that a deeper understanding of defect physics and electronic phase separation in DMS can be obtained from fluctuation spectroscopy measurements. We suggest similar studies on other magnetic semiconductors which are supposed to exhibit percolation transitions or an electronic phase separation, like In$_{1-x}$Mn$_{x}$As or Mn$_{x}$Ge$_{1-x}$ \\cite{Park2002}.\n\n\\section*{Acknowledgements}\nThe ion implantation was done at the Ion Beam Center (IBC) at HZDR. S.\\ Z.\\ acknowledges the financial support by the Helmholtz Association (VH-NG-713). We thank R.\\ P.\\ Campion from the University of Nottingham for supplying the Ga$_{1-x}$Mn$_{x}$As material for the He-ion irradiated samples.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{\\@startsection{section}{0}{\\z@}{5.5ex plus .5ex minus\n 1.5ex}{2.3ex plus .2ex}{\\large\\bf}}\n\\def\\subsection{\\@startsection{subsection}{1}{\\z@}{3.5ex plus .5ex minus\n 1.5ex}{1.3ex plus .2ex}{\\normalsize\\bf}}\n\\def\\subsubsection{\\@startsection{subsubsection}{2}{\\z@}{-3.5ex plus\n-1ex minus -.2ex}{2.3ex plus .2ex}{\\normalsize\\sl}}\n\n\\renewcommand{\\@makecaption}[2]{%\n \\vskip 10pt\n \\setbox\\@tempboxa\\hbox{\\small #1: #2}\n \\ifdim \\wd\\@tempboxa >\\hsize \n \\small #1: #2\\par \n \\else \n \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}\n \\fi}\n\n\\makeatother\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkhkn b/data_all_eng_slimpj/shuffled/split2/finalzzkhkn new file mode 100644 index 0000000000000000000000000000000000000000..e01a568bd0d0af712fb60c30f4599dfae42fe36b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzkhkn @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe quasi-one-dimensional (Q1D) charge order, \nwhich is well-known as the stripe order, has been \none of the significant issues in high-T$_{\\mathrm{c}}$ cuprates (HTC) for \nrecent years. The incommensurate peaks observed by the neutron \nscattering in La$_{\\mathrm{2-x}}$Ba$_{\\mathrm{x}}$CuO$_4$ \nare to be explained on the assumption that the static \nstripe order is present around 1\/8-filling.\\cite{Tranquada1995} The presence \nof the static stripe order in the ground state has been proved \ntheoretically by the density matrix renormalization group method on the basis \nof the Q1D {\\it t}-{\\it J} model.\\cite{White1998} \nOriginally, since the beginning of HTC \nstudies many experimentalists have pointed out the so-called 1\/8-problem in \nLa$_{\\mathrm{2-x}}$Ba$_{\\mathrm{x}}$CuO$_4$, i.e., the remarkable suppression \nof superconducting transition temperature at $x \\sim 1\/8$. \nMany theorists have \nthought that the strong on-site Coulomb interaction makes electrons \nalmost localized with the long-period spin and charge correlations \naround 1\/8-filling. They have studied these long-period correlations by the\n numerical or analytical methods on the basis of the \ntwo-dimensional (2D) Hubbard models and have clarified \nthat some long-period correlated \nstates can be the ground states around certain fillings.\n\\cite{Poilblanc1989,Zaanen1989,Kato1990,Salkola1996,Mizokawa1997,Machida1999,Ichioka1999,Kaneshita2001,Varlamov2002,Yanagisawa2002} These theoretically predicted long-period correlation \nstates are consistent with \nthe stripe state deduced from the neutron scattering experiments.\\cite{Tranquada1999} \n\nThe theoretical results on the basis of the 2D Hubbard models \nmean that the strong \non-site interaction is enough to stabilize the stripe state near 1\/8-filling \nas far as the ground state is considered. \nHowever, at finite temperature the strong \nfluctuations characteristic to the low dimensional strongly \ncorrelated electron systems exist. These strong fluctuations are composed of \nthe many elements, e.g., the anti-ferromagnetic (AF) fluctuation and \nthe charge ordering fluctuations with certain classes of super-lattice \nstructures. These strong fluctuations \naccompany with the instabilities of the spin and charge orderings \nand simultaneously tend to interrupt them. \nThe stripe with such strong fluctuations in metallic state \nis called a dynamical stripe.\\cite{Zaanen1996,Zaanen2000,Zaanen2003,Hasselmann2002} \nThe angle resolved photo-emission spectroscopy (ARPES) shows that \nthe stripes in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ has more dynamic \ncharacter than those in La$_{\\mathrm{2-x}}$Sr$_{\\mathrm{x}}$CuO$_4$. \nThe discrimination whether the stripe is static or dynamic is to be \njudged from the characteristic energy of the fluctuation mode. \nIf the fluctuation mode has low characteristic energy, \ni.e., the stripe fluctuates slowly, the mode \ncan easily couple with phonon and the static stripe long-range \norder will appear. On the other hand, \nif the fluctuation mode has high characteristic energy, \ni.e., the stripe fluctuates fast, the mode cannot couple with phonon and \nthe system will stay metallic together with the fluctuation. \nWhen we consider these fluctuations in the analysis of these models \nwithout the long-ranged Coulomb interaction, \nwe can expect that the metallic state should be recovered \nat higher temperature. In this state, however, \nthe electronic property is to be still affected by the stripe instability \nas far as its dynamical factor is investigated . \n\nIn this paper, we analyze the 2D three-band Hubbard model \non the basis of the unrestricted fluctuation exchange approximation (UFLEX). \nUFLEX can consider a strong correlation effect and a number of \nthe spatial inhomogeneities without any assumptions about the ground state. \nThese considered effects contain not only a static component \nbut a dynamic one. As a result, the electronic state \nreproduced on the basis of UFLEX is affected by \na number of spatially inhomogeneous fluctuations simultaneously. \nIn fact, when we calculate the one-particle spectral functions \nand the spin correlation functions for our fully self-consistent solutions, \nwe can find that the one-particle spectra of the electronic state \nreflect these exotic state accompanied with the anti-ferromagnetic \nor the further long-periodic instabilities. Especially, near $1\/8$-filling \nthe spin and charge correlation functions show the existence of the \nQ1D fluctuation at finite frequency. This fast Q1D fluctuation is originated \nfrom the striped spatial inhomogeneity. Thus, \nour results suggest that the spatial inhomogeneities should be concerned \nin the strongly correlated electron system when the electronic behavior \nat a finite frequency is discussed. \n\n\\section{2D three-band Hubbard model}\n\nOur 2D three-band Hubbard model Hamiltonian, $H$, is composed of {\\it d}-electrons at each Cu site and \n{\\it p}-electrons at O site. We consider only the on-site Coulomb repulsion $U$ between \n{\\it d}-electrons at each Cu site. Then, $H$ is divided \ninto the non-interacting part, $H_0$, and the interacting part, $H_1$, as \n\\begin{equation}\nH=H_0+H_1%\n-\\mu \\sum_{{\\mathbf k} \\sigma}\n(d_{{\\mathbf k} \\sigma}^{\\dagger}d_{{\\mathbf k} \\sigma}\n+p_{{\\mathbf k} \\sigma}^{x \\dagger}p_{{\\mathbf k} \\sigma}^x\n+p_{{\\mathbf k} \\sigma}^{y \\dagger}p_{{\\mathbf k} \\sigma}^y).\n\\end{equation} \nHere $d_{{\\mathbf k} \\sigma}(d_{{\\mathbf k} \\sigma}^{\\dagger})$ and \n$p_{{\\mathbf k} \\sigma}^{x(y)}(p_{{\\mathbf k} \\sigma}^{x(y) \\dagger})$ are \nthe annihilation (creation) operator for {\\it p}- and \n{\\it p}$^{x(y)}$-electron of momentum ${\\mathbf k}$ and spin $\\sigma$, \nrespectively. $\\mu$ is the chemical potential.\nThe non-interacting part $H_0$ is represented by \n\\begin{equation}\nH_0= {\\sum_{{\\mathbf k} \\sigma}}\n\\left(\nd_{{\\mathbf k} \\sigma}^\\dagger \\, \np_{{\\mathbf k} \\sigma}^{x \\dagger} \\, \np_{{\\mathbf k} \\sigma}^{y \\dagger}\n\\right) \\left(\n\\begin{array}{ccc} \\Delta_{dp} & \\zeta_{\\mathbf k}^x & \\zeta_{\\mathbf k}^y \\\\ \n-\\zeta_{\\mathbf k}^x & 0 & \\zeta_{\\mathbf k}^p \\\\\n-\\zeta_{\\mathbf k}^y & \\zeta_{\\mathbf k}^p & 0 \\\\\n\\end{array} \\right)\\!\n\\left( \\begin{array}{c} d_{{\\mathbf k} \\sigma} \\\\ p_{{\\mathbf k} \\sigma}^x\n\\\\ p_{{\\mathbf k} \\sigma}^y \\\\\n\\end{array} \\right),\n\\end{equation}\nwhere $\\Delta_{dp}$ is the hybridization gap energy\nbetween {\\it d}- and {\\it p}-orbitals. \nWe take the lattice constant of the square\nlattice formed of Cu sites as the unit of length, and we can represent\n$\\zeta_{\\mathbf k}^{x(y)}=2{\\rm i}\\,t_{dp} \\sin \\frac{k_{x(y)}}{2}$ and\n$\\zeta_{\\mathbf k}^p=-4t_{pp} \\sin \\frac{k_x}{2} \\sin \\frac{k_y}{2}$,\nwhere $t_{dp}$ is the transfer energy between a {\\it d}-orbital and a\nneighboring {\\it p}$^{x(y)}$-orbital and $t_{pp}$ is that between a\n{\\it p}$^x$-orbital and a {\\it p}$^y$-orbital. In this\nstudy, we take $t_{dp}$ as the unit of energy. The residual part,\n$H_1$, is described as\n\\begin{equation}\nH_1= \\frac{U}{N} \\sum_{{\\mathbf k} {\\mathbf k}^\\prime} \\sum_{\\mathbf q}\n d_{{\\mathbf k}+{\\mathbf q} \\uparrow}^{\\dagger} d_{{\\mathbf\n k}^\\prime- {\\mathbf q} \\downarrow}^{\\dagger} d_{{\\mathbf k}^\\prime\n \\downarrow} d_{{\\mathbf k} \\uparrow}.\n\\end{equation}\nwhere $U$ is the on-site Coulomb repulsion between {\\it d}-orbitals and\n$N$ is the number of ${\\mathbf k}$-space lattice points in the first\nBrillouin zone (FBZ). \n\n\\section{UFLEX}\n\nWe have developed UFLEX in order to analyze the spatially inhomogeneous \nsystem. In our UFLEX formulation we introduce 'cluster' momentum \nbesides 'usual' momentum. The notion of this cluster momentum is \nsuggested from the one in dynamical cluster approximation, \nin which the nonlocal correlations \nare considered beyond dynamical mean field theory.\n\\cite{Hettler1998,Maier2000,Imai2002}\n\nHowever, our cluster momentum is \nnot directly related with a coarse graining of the Brillouin zone. When \nthe most stable state of our system is spatially homogeneous, only a cluster \nmomentum, ${\\mathbf K}={\\mathbf 0}$, is enough to obtain \nthe {\\it true} solution for our problem. In that case, \nUFLEX is consistent to 'usual' FLEX.\\cite{Bickers1989} \nThe advantage of our introduction of cluster momenta is \nits flexibility in the correspondence to the problem. \nIf the problem have some important classes of the spatial inhomogeneities, \nwe will only need to provide the cluster momenta \ncorresponding to those classes. For example, when \nthe most important instability is the anti-ferromagnetic one, \ntwo cluster momenta, ${\\mathbf K}={\\mathbf 0}$ and ${\\mathbf K}\n=(\\frac{\\pi}{2},\\frac{\\pi}{2})$, are enough. Of course, the stripe \ninstabilities have their characteristic \ncluster momenta. Thus, when we anticipate to encounter \nthe stripe instabilities in our problem, we need to provide \nonly those cluster momenta.\n\nFirst, we define the unrestricted perturbed Green function, \n$G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$.\nHere we use the abbreviation of Fermion Matsubara frequencies, \n$\\epsilon_n=\\pi T(2n+1)$ with integer $n$, where $T$ is the temperature. $\\mathbf{K}$ indicates \nthe cluster momentum. The set, $\\{\\mathbf{K}\\}$, is chosen so that \nany sum, $\\mathbf{k}+\\mathbf{K}$, exists in the set, $\\{\\mathbf{k}\\}$. \n$G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$ is calculated by the Dyson equation :\n\\begin{equation}\n\\left[G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)\n\\right]^{-1}=\\delta_{\\mathbf{K}} \\cdot \n\\left\\{G_d^{\\,\\sigma\\,(0)}\n(\\mathbf{k},\\mathrm{i}\\epsilon_n) \\right\\}^{-1}\n-\\Sigma_{\\mathbf{K}}^\\sigma(\\mathbf{k},\\mathrm{i}\\epsilon_n), \\\\\n\\label{eq:1}\n\\end{equation}\nwhere $\\delta_{\\mathbf{K}}$ means Kronecker's delta, \ni.e. $\\delta_{\\mathbf{K}=\\mathbf{0}}=1$ \nand $\\delta_{\\mathbf{K} \\neq \\mathbf{0}}=1$.\n$\\left[\\cdots\\right]^{-1}$ indicates \nthe inverse operation defined so as to satisfy the identity : \n\\begin{equation}\n\\delta_{\\mathbf{K}}= \\sum_{\\mathbf{K}^\\prime}G_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}\n(\\mathbf{k}+\\mathbf{K}^\\prime-\\mathbf{K},\\mathrm{i}\\epsilon_n)\n\\left[G_{d\\,\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)\\right]^{-1}\n\\end{equation}\nfor all ${\\mathbf{k}}$ and $n$. $G_d^{\\,\\sigma\\,(0)}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$ \nin Eq.~(\\ref{eq:1}) is the unperturbed Green function derived by diagonalizing of $H-H_1$ as \n\\begin{align}\n& \\left\\{G_d^{\\,\\sigma\\,(0)}(\\mathbf{k},z-\\mu)\\right\\}^{-1}= \\nonumber \\\\\n& z-\\Delta_{dp}-\\frac{2z(2-\\cos k_x-\\cos k_y)+\n 8t_{pp}(1-\\cos k_x)(1-\\cos k_y)}\n{z^2-4t_{pp}^2(1-\\cos k_x)(1-\\cos k_y)}.\n\\end{align}\nIn order to estimate our unrestricted self-energy, $\\Sigma_{\\mathbf{K}}^\\sigma\n(\\mathbf{k},\\mathrm{i}\\epsilon_n)$, in Eq.~(\\ref{eq:1}), \nwe adopt the UFLEX as follows.\n\\begin{align}\n& \\Sigma_{\\mathbf{K}}^\\sigma\n(\\mathbf{k},\\mathrm{i}\\epsilon_n) = \\nonumber \\\\\n& \\frac{TU}{N}\\sum_{\\mathbf{q}\\,n^\\prime}\nG_{d\\,\\mathbf{K}}^{\\,-\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_{n^\\prime})\\left.e^{ \\mathrm{i}\\epsilon_{n^\\prime}\\eta}\n\\right|_{\\eta \\rightarrow +0} \\nonumber \\\\\n& +\\frac{T}{N}\n\\sum_{\\mathbf{q}\\,\\mathbf{K}^\\prime\\,m}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}\n(\\mathbf{q}-\\mathbf{k}+\\mathbf{K}^\\prime-\\mathbf{K},\n\\mathrm{i}\\omega_m-\\mathrm{i}\\epsilon_n)\nV_{\\mathbf{K}^\\prime}^{(\\mathrm{pp})}\n(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\nonumber \\\\\n& +\\frac{T}{N}\n\\sum_{\\mathbf{q}\\,\\mathbf{K}^\\prime\\,m}\\left[\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,-\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_n-\\mathrm{i}\\omega_m)\nV_{\\mathbf{K}^\\prime}^{\\sigma\\,(\\mathrm{ph1})}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\right. \\nonumber \\\\\n& \\hspace{5.5em}+\\left. G_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_n-\\mathrm{i}\\omega_m)\nV_{\\mathbf{K}^\\prime}^{\\sigma\\,(\\mathrm{ph2})}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\right],\n\\label{eq:2}\n\\end{align}\n\\begin{align}\n\\label{eq:4}\n& V_{\\mathbf{K}}^{(\\mathrm{pp})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\ \n& U^2\\phi_{\\mathbf{K}}(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\ \n& -U^2\\sum_{\\mathbf{K}^\\prime}\n\\phi_{\\mathbf{K}-\\mathbf{K}^\\prime}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\left[\\,\n\\delta_{\\mathbf{K}^\\prime}\n+U\\phi_{\\mathbf{K}^\\prime}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m)\\right]^{-1}, \\\\\n\\label{eq:5}\n& V_{\\mathbf{K}}^{\\sigma(\\mathrm{ph1})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\\n& -U^2\\chi_{\\mathbf{K}}^{\\sigma\\, -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\ \n&+U^2\\sum_{\\mathbf{K}^\\prime}\n\\chi_{\\mathbf{K}-\\mathbf{K}^\\prime}^{\\sigma\\, -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\left[\\delta_{\\mathbf{K}^\\prime}\n-U\\chi_{\\mathbf{K}^\\prime}^{\\sigma\\, -\\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m)\n\\right]^{-1}, \n\\end{align}\nand\n\\begin{align}\n\\label{eq:6}\n& V_{\\mathbf{K}}^{\\sigma \\,(\\mathrm{ph2})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\\n& U^2\\sum_{\\mathbf{K}^\\prime}\n\\chi_{\\mathbf{K}-\\mathbf{K}^\\prime}^{-\\sigma -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\\n& \\times \\left[\\,\\delta_{\\mathbf{K}^\\prime}\\right.\n-\\!U^2\\!\\sum_{\\mathbf{K}^{\\prime \\prime}}\n\\chi_{\\mathbf{K}^\\prime-\\mathbf{K}^{\\prime \\prime}}^{\\sigma \\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m) \\nonumber \\\\\n& \\hspace{7em}\n\\times \\left. \\chi_{\\mathbf{K}^{\\prime \\prime}}^{-\\sigma -\\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^{\\prime \\prime},\\mathrm{i}\\omega_m)\n\\right]^{-1},\n\\end{align}\nwhere $\\omega_m=2m\\,\\pi T$ with integer $m$ are Boson Matsubara frequencies, \n\\begin{align}\n\\label{eq:3}\n& \\phi_{\\mathbf{K}}(\\mathbf{q},\\mathrm{i}\\omega_m)= \\nonumber \\\\ \n& \\frac{T}{N}\\,\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\, n}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{q}-\\mathbf{k}, \n\\mathrm{i}\\omega_m-\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,\\mathbf{K}^\\prime}^{-\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n),\n\\end{align}\nand\n\\begin{align}\n\\label{eq:7}\n& \\chi_{\\mathbf{K}}^{\\sigma \\sigma^\\prime}(\\mathbf{q},\\mathrm{i}\\omega_m)= \\nonumber \\\\ \n& -\\frac{T}{N}\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\,n}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\sigma}\n(\\mathbf{q}+\\mathbf{k},\\mathrm{i}\\omega_m+\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,\\mathbf{K}^\\prime}^{\\sigma^\\prime}(\\mathbf{k}-\\mathbf{K}^\\prime\n,\\mathrm{i}\\epsilon_n) .\n\\end{align}\nIn order to obtain the solution satisfying the conserving law, \nwe need to solve \nEqs.~(\\ref{eq:1},\\ref{eq:2},\\ref{eq:4},\\ref{eq:5},\\ref{eq:6},\\ref{eq:3},\n\\ref{eq:7}) fully self-consistently. This self-consistent procedure is \nshown in Fig.~\\ref{figure:10} diagrammatically.\n\\begin{figure}\n\\includegraphics[width=6.1cm]{fig1.eps}\n\\caption{The self-consistent procedure in UFLEX. \n$i$ represents the iteration count, and $\\varepsilon$ is the small value.}\n\\label{figure:10}\n\\end{figure}\n\nIn numerical calculations we divide the FBZ into $16 \\times 16$ meshes. \nWe take $8\\times 8$ cluster momenta so that we can reproduce a \nhomogeneous state, inhomogeneous states \nwith 2-, 4-, or 8-lattice period charge density wave (CDW) \nor spin density wave (SDW) along \n$a(x)$-axis or $b(y)$-axis, or an AF state. We prepare $2^{11}=2048$ \nMatsubara frequencies for temperature $T=0.020 \\sim 180{\\mathrm{K}}$. \nThe other parameters :\n$t_{dp}=1.0 \\sim 0.80\\,{\\mathrm{eV}}$, \n$t_{pp}=0.60 \\sim 0.48\\,{\\mathrm{eV}}$, and \n$\\Delta_{dp}=0.70 \\sim 0.56\\,{\\mathrm{eV}}$, \n$U=4.0 \\sim 3.2\\,{\\mathrm{eV}}$, which are all common for our results.\n\n\\section{Fermi surface evolution with hole- or electron-doping}\n\\label{Fermi}\n\nWe have obtained the fully self-consistent solutions for \n$0.000 \\leq \\delta_{\\mathrm{h}} \\leq 0.057$, \n$0.112 \\leq \\delta_{\\mathrm{h}} \\leq 0.231$, \nand $0.000 \\leq \\delta_{\\mathrm{e}} \\leq 0.249$, \nwhere $\\delta_{\\mathrm{h}} \\equiv 1-n_{\\mathrm{d}}-n_{\\mathrm{p}}$ and \n$\\delta_{\\mathrm{e}} \\equiv n_{\\mathrm{d}}+n_{\\mathrm{p}}-1$. \n$n_{\\mathrm{d}}$ and $n_{\\mathrm{p}}$ represent the number of \n$d$- and $p$-electrons, respectively. \nWe could not obtain any convergent solution for \n$0.057 < \\delta_{\\mathrm{h}} < 0.112$. According to the elaborated neutron scattering experiments on La$_{\\mathrm{2-x-y}}$Ba$_{\\mathrm{x}}$Sr$_{\\mathrm{y}}$CuO$_4$,\\cite{Fujita2002PRB,Fujita2002PRL,Tranquada2004} \nin this region the inhomogeneous states with longer \nthan 8-lattice period SDW along diagonal, which is called diagonal \nstripe state, could be realized. Our present \ncalculation would be short of provided cluster momenta to reproduce \nthe inhomogeneous states with the diagonal fluctuations. Hereafter, we \nrestrict our discussion to the states with the vertical fluctuations. \n\nIn this section we investigate the Fermi \nsurface evolution with doping in detail. In order to determine the Fermi \nsurface, we define the one-particle spectral \nweight as \n\\begin{equation}\nA^{\\sigma}(\\mathbf{k},E) \\equiv -\\frac{1}{\\pi}\\mathrm{Im}\n\\sum_{\\zeta=d,\\,p^x,p^y}\\left.\nG_{\\zeta\\,\\mathbf{0}}^\\sigma(\\mathbf{k},\\mathrm{i}\\epsilon_n)\n\\right|_{\\mathrm{i}\\epsilon_n \\rightarrow E},\n\\label{eq:8}\n\\end{equation}\nwhere\n\\begin{align}\n& G_{p^{x,y}\\,\\mathbf{0}}^\\sigma(\\mathbf{k},z-\\mu) \\nonumber \\\\\n& = \\left\\{z[z-\\Delta_{dp}-\\Sigma_{\\mathbf{0}}^\\sigma(\\mathbf{k},z)]-2(1-\\cos k_{x,y})\\right\\} \\nonumber \\\\ \n& \\hspace{1em}\\times\\!\\left\\{[z^2-4t_{pp}^2(1-\\cos k_x)(1-\\cos k_y)][z-\\Delta_{dp}-\\Sigma_{\\mathbf{0}}^\\sigma(\\mathbf{k},z)]\\right. \\nonumber \\\\\n& \\hspace{2.5em} \n\\left.-2z(2-\\cos k_x-\\cos k_y)\\!-\\!8t_{pp}(1-\\cos k_x)(1-\\cos k_y)\\right\\}\n^{\\!-1}. \n\\label{eq:9}\n\\end{align}\nIn Eq.~(\\ref{eq:8}) we use Pad$\\acute{\\mathrm{e}}$ approximation \nfor the method of analytic continuation. In order to \ndetermine the Fermi surface on the basis of our calculated results, \nwe should calculate the one particle spectral weight at $E=0$, \n$A^{\\sigma}(\\mathbf{k},0)$, and find the $\\mathbf{k}$-points at which \n$A^{\\sigma}(\\mathbf{k},0)$ becomes large. \nBecause the thermal fluctuation with CDW, SDW, or anti-ferromagnetic \ninstabilities can generate the branched energy dispersion, we show \nour calculated results as follows. We choose the both $\\mathbf{k}$-points \non which the one-particle spectral weight at $E=0$ has \nthe largest and the second largest peaks for fixed $k_x$ \nas far as $0 < k_y \\leq k_x < \\pi$. And we repeat this operation for the other \nparts of FBZ. In the following figures, the points on which \nthe one-particle spectral weight at $E=0$ has\nthe largest and the second largest peaks are to be \nindicated with black and gray circles, respectively. \nOur results have been obtained at high temperature, and the \nthermal fluctuations make the Fermi level vague. \nHowever, if we have empty spaces surrounded by these points, \nwe can expect the branched energy dispersion near the Fermi level. The \nbranched energy dispersion is made by a spatial inhomogeneous instability. \nFurthermore, these points can suggest the Fermi surface connectivities at lower \ntemperature. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig2.eps}\n\\caption{Fermi surface evolutions near $1\/8$-filling. \nLeft sides : (a) for $\\delta_{\\mathrm{h}}=0.112$. \n(b) for $\\delta_{\\mathrm{h}}=0.127$. \nRight sides : The gray-scale images interpolated from raw data are attached \non account of the guide to the eye. The brighter spots have the larger spectral weights. (c) Illustration of nesting vectors. Dashed lines represent the planes connected with one another by the nesting vectors, $(\\pi,\\pi)$ or $(\\pi,-\\pi)$. Dotted lines represent the planes connected with one another by the nesting vectors, $(\\pi\/2,0)$ or $(0,\\pi\/2)$. \nThe illustration of nesting vectors \nis also applicable to the other figures representing \nFermi surface evolutions.}\n\\label{figure:1}\n\\end {figure}\nFirstly, we should mention our results for the hole-doped states \naround $1\/8$-filling, in which the dynamical stripe state might be realized. \nAs shown in Fig.~\\ref{figure:1}, \nnear $1\/8$-filling we have eight empty spots around the intersections \nof the planes indicated by dashed and dotted lines, which relate \nto the anti-ferromagnetic and the 4-lattice period \nCDW instabilities, respectively. \nIt means that the one-particle spectral weights are lost due \nto the fluctuations both of the anti-ferromagnetic instability and of \nthe 4-lattice period CDW instability. \nThese instabilities occur when some parts of the Fermi surface are \nconnected with one another by the nesting vectors. The parts of Fermi \nsurface connected with one another by the nesting vectors, $(\\pi,\\pi)$ \nor $(\\pi,-\\pi)$, are called 'hot spots'. On hot spots the quasi-particles \nplay important roles in the transport phenomena. \nTherefore, we expect some anomalous electronic behaviors near $1\/8$-filling, \nwhen the one-particle spectral weights are lost on hot spots. \nThis is the reason why so-called $1\/8$-problem occurs around $1\/8$-filling. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig3.eps}\n\\caption{Fermi surface evolutions for lightly-hole-doped states. (a) for $\\delta_{\\mathrm{h}}=0.039$. (b) for $\\delta_{\\mathrm{h}}=0.057$.}\n\\label{figure:2}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig4.eps}\n\\caption{Fermi surface evolutions for near half-filling. (a) for $\\delta_{\\mathrm{h}}=0.022$. (b) for half-filling. (b) for $\\delta_{\\mathrm{e}}=0.021$.}\n\\label{figure:4}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig5.eps}\n\\caption{Fermi surface evolutions for lightly-electron-doped states. (a) for $\\delta_{\\mathrm{e}}=0.045$. (b) for $\\delta_{\\mathrm{e}}=0.066$. (b) for $\\delta_{\\mathrm{e}}=0.086$.}\n\\label{figure:5}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig6.eps}\n\\caption{Fermi surface evolutions for heavily-electron-doped states. (a) for $\\delta_{\\mathrm{e}}=0.106$. (b) for $\\delta_{\\mathrm{e}}=0.144$. (b) for $\\delta_{\\mathrm{e}}=0.200$.}\n\\label{figure:6}\n\\end{figure}\nSecondly, we show our results for lightly-hole-doped states, \nin which the anti-ferromagnetic instability might exist. \nIn this case we have no more eight empty spots but four empty spots \ncentered on $(\\pm\\pi\/2,\\pm\\pi\/2)$ as shown in Fig.~\\ref{figure:2}. These \nempty spots are called 'Fermi arc's, which is believed to appear when \nthe strong anti-ferromagnetic fluctuation exists. In fact, these Fermi arcs \nhave been observed in La$_{1.97}$Sr$_{0.03}$CuO$_4$ \nby ARPES and interpreted as the evidence of the \nanti-ferromagnetic long-range order.\\cite{TYoshida2003} In our calculating \nresults, the spectral weights in these Fermi arcs are not completely zero, and \nthe anti-ferromagnetic long-range order does not exist. However, the existence \nof these Fermi arcs should be originated from \nthe strong anti-ferromagnetic fluctuation, evolving \ntoward the anti-ferromagnetic long-range order.\n \nWhen the doped holes become less, electrons should \nlocalize due to strong coulomb repulsion among them in real materials. \nHowever, in our three-band Hubbard model electrons cannot localize even \naround near half-filling and the Fermi surface are remained, as \nshown in Fig.~\\ref{figure:4}. If we could reproduce the Fermi surface \nevolution for the whole doping region, \nthe one electron spectral weight at the \nFermi level should be almost vanished near half-filling. \nThis result means that our formulation on \nthe basis of UFLEX is not efficient to describe the Mott-Hubbard \nmetal-insulator transition. Unfortunately this is the \nside-effect of UFLEX, in which we overestimate the thermal fluctuations \nand recover the itinerancy of electrons. It would be our future issue to \ndescribe the localized electrons but with strong fluctuations appropriately. \n\nOur results for electron-doped states will \nbe highlighted. The ARPES have already clarified the doping \ndependence of the Fermi surface in \nNd$_{\\rm 2-x}$Ce$_{\\rm x}$CuO$_{4 \\pm \\delta}$.\\cite{Armitage2002,Armitage2003,Shen2004} Except for completely vanished Fermi surface near half-filling, \ntheir observed evolution of Fermi surface with electron doping are consistent with our calculated result. As shown in Fig.~\\ref{figure:5}, \nin lightly-electron-doped region the eight pockets emerge \naround the cross-sectional point of Fermi surface with FBZ boundaries. \nThis reflects that the vertical charge fluctuation exists in the \nlightly-electron-doped region \nas well as near $1\/8$-hole-doped region. We can guess that \nthe vertical charge fluctuation gets to play an important role \nwhen the localized electron recovers its itinerancy \nas doped electron increases. \nWhen the doped-electron density density increases more, \neach two of eight pockets on the same \nFermi surface section come close to each other and unite into one \nas shown in Fig.~\\ref{figure:6}. Thus, \nin heavily-electron-doped region we have four pockets \naround the cross-sectional point of Fermi surface with $k_x=\\pm k_y$. \nThis means that the diagonal spin fluctuation, which is \nnot completely anti-ferromagnetic, \nis strengthened instead of the vertical fluctuation. \nThis diagonal spin fluctuation can cause pseudo-gap phenomena in electron-doped \nhigh-T$_{\\rm c}$ as observed at low temperature.\\cite{Onose2004} \nIt contrasts with the case of the hole-doped region, and the difference \nshould be caused by both a large $t_{pp}$. Such a $t_{pp}$ makes \nthe big difference of the energy dispersions between the hole-doped \nregion and the electron-doped region. If the electron were localized, \nthe difference of the energy dispersions should \nappear as the different charge distributions. \nIn our results the electron could not be localized, \nbut the charge fluctuations appear in the different way between the hole-doped \nregion and the electron-doped region instead. Hence, we can insist that \nthe inhomogeneity of the strongly correlated system should be considered. \n\n\\section{Spin and charge correlations}\n\nIn this section we show how the Fermi surface evolution \nwith doping is related to the development \nof the spin and charge correlations. The momentum dependences of \nspin and charge correlation functions \nreflect the spin and charge spatial inhomogeneities, respectively. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig7.eps}\n\\caption{The doping dependence of $S(\\mathbf{q})$ at its maximum momentum \n$\\mathbf{Q}_{\\mathrm{max}}$.}\n\\label{figure:7}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig8.eps}\n\\caption{The momentum dependences of $D(\\mathbf{q})$ at hole-doped regions.}\n\\label{figure:8}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig9.eps}\n\\caption{The momentum dependences of $D(\\mathbf{q})$ at electron-doped regions.}\n\\label{figure:9}\n\\end {figure}\nAt first we sum up our obtained results \nabout the static spin correlations as both the incommensurabilities and \nthe intensities of the peaks in the calculated elastic neutron scattering. \nThey are shown in Fig.~\\ref{figure:7}, where\n\\begin{equation}\nS(\\mathbf{q}) \\equiv\n\\sum_{\\mathbf{K}}\n\\chi_{-\\mathbf{K}}^{+-}\n(\\mathbf{q},0)\n\\left[\n\\delta_{\\mathbf{K}}\n-U\\chi_{\\mathbf{K}}^{+-}(\\mathbf{q}-\\mathbf{K},0)\n\\right]^{-1}.\n\\end{equation}\nIn the hole-doped region the incommensurabilities $\\delta_{\\rm inc}$, \ndefined by \n\\begin{equation}\n\\delta_{\\mathrm{inc}} \n\\equiv \\mathrm{Max}\\{[\\mathbf{Q}_{\\mathrm{max}}-\n(\\pi,\\pi)]_x,[\\mathbf{Q}_{\\mathrm{max}}-(\\pi,\\pi)]_y\\},\n\\end{equation}\nare almost proportional to $\\delta_{\\rm h}$. This has been already observed by \nthe neutron scattering experiment on La$_{\\mathrm{2-x-y}}$Ba$_{\\mathrm{x}}$Sr$_{\\mathrm{y}}$CuO$_4$.\\cite{Yamada1998,Fujita2002PRB,Fujita2002PRL} The ratio of \n$\\delta_{\\rm inc}$ to $\\delta_{\\rm h}$ in our calculating results are almost \nhalf of the one in these experiments. However, we believe that this deviation \nis originated from the difference between the real hole number \n$\\delta_{\\rm h}$ introduced in CuO$_2$ plane, which is correspond to the \none in our results, and the instoicheiometric number $\\delta$ used in \nthe analysis in Ref.~\\citen{Yamada1998}.\n\nOn the other hand, in the electron-doped region \nwe cannot recognize such a relationship between $\\delta_{\\mathrm{inc}}$ and \n$\\delta_{\\mathrm{e}}$. It suggests that our spin correlation functions do not \nsimply reflect the Fermi surface nesting but rather the existence of \ninhomogeneities as discussed in Section~\\ref{Fermi}. \n\nIn order to clear our insistence, we calculate the static \ncharge correlation functions at some regions. The momentum dependences \nof these correlation functions are shown in \nFigs.~\\ref{figure:8} and \\ref{figure:9}, where\n\\begin{equation}\nD(\\mathbf{q}) \\equiv \n\\frac{T}{N}\\,\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\, n}\nG_{d\\,\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{q}-\\mathbf{k}, \n-\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,-\\mathbf{K}^\\prime}^{-\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n).\n\\end{equation}\nFocusing on Fig.~\\ref{figure:8}(b) and Fig.~\\ref{figure:9}(b), \nwe can recognize that $D(\\mathbf{q})$ are slightly enhanced around the \nsymmetric lines, $q_x=0$ and $q_y=0$. These momentum dependences of \n$D(\\mathbf{q})$ suggest that the two collective modes \nwith the momenta $(\\pm 1,0)$ and $(0,\\pm 1)$ exist. These collective modes \ncan be translated as the instabilities toward \nvertical charge orderings along $x$-axis and $y$-axis, respectively. \nIn fact, when $\\delta_{\\mathrm{h}}=0.127$ and \n$\\delta_{\\mathrm{e}}=0.066$, the Fermi surfaces have eight pockets around \n$(\\pm\\pi,\\pm\\pi\/4)$ and $(\\pm\\pi\/4,\\pm\\pi)$ as shown in Fig.~\\ref{figure:1}(b) \nand Fig.~\\ref{figure:4}(b), respectively. As discussed in \\ref{Fermi}, \nthese anomalous Fermi surfaces are caused from the charge inhomogeneities, \nwhich appear as shown in Fig.~\\ref{figure:8}(b) and Fig.~\\ref{figure:9}(b). \n\n\\section{Conclusion}\n \nIn this paper, we have obtained fully self-consistent solutions \nfor the 2D three-band Hubbard model on the basis of UFLEX, in \nwhich the inhomogeneous distribution of d-electrons are allowed. \nBoth in hole-under-doped and electron-doped regions the one-particle \nspectral weights behave anomalously. Furthermore, \nthe spin and charge correlation functions \nreflect the existence of super-lattice structures around 1\/8-filling. \nOur numerical solutions show that the metallic state with spatially \ninhomogeneous spin or charge fast fluctuation can exist at high-temperature. \nThe electronic state in one of our microscopically derived \nsolutions corresponds to the dynamical stripe state mentioned in some \npioneering works.\\cite{Zaanen1996,Zaanen2000,Zaanen2003,Hasselmann2002} \nIn our 2D model, the electrons cannot have \nthe long-range order at finite temperature because of the strong \nfluctuation characteristic of low-dimensionality. However, \nin three-dimensional real materials, the spatially inhomogeneous fluctuation \nmay tend to have the long-range order and form the striped state, \nobserved in the neutron scattering experiments. This long-range order \nformation in three-dimensional real materials might not be reproduced \nby our simple model in this paper. The existences of the \nphonon or the long-range Coulomb potential might have an \nimportant role for these long-range order formation. \nThe theoretical researches concerning these complicated factors are \nto be expected as the future problem.\n\n\\section*{Acknowledgments}\nThe authors are grateful to Professor K. Yamaji, Professor Y. Aiura, Professor H. Eisaki, Dr. I. Nagai, Dr. M. Miyazaki, and Dr. S. Koike for their invaluable comments. The computation in this work has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The early development of the code for this computation has been achieved by using the IBM RS\/6000--SP at TACC and VT-Alpha servers at NeRI in AIST.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this talk we review recent developments on the interaction of vector mesons with baryons and nuclei. For this purpose we use effective field theory using \na combination of effective Lagrangians to account for hadron interactions, implementing exactly unitarity in coupled channels. This approach has turned out to be a very efficient tool to \n face many problems in Hadron Physics. Using this coupled channel unitary approach, usually referred to as chiral unitary approach, because the Lagrangians used account for chiral symmetry, the \ninteraction of the octet of\npseudoscalar mesons with the octet of stable baryons has been studied and \nleads to $J^P=1\/2^-$\nresonances which fit quite well the spectrum of the known low lying resonances\nwith these quantum numbers \n\\cite{Kaiser:1995cy,angels,ollerulf,carmenjuan,hyodo,ikeda}. \n New resonances are sometimes predicted, the most notable\nbeing the $\\Lambda(1405)$, where all the\nchiral approaches find two poles close by \n\\cite{Jido:2003cb,Borasoy:2005ie,Oller:2005ig,Oller:2006jw,Borasoy:2006sr,Hyodo:2008xr,Roca:2008kr}, rather than one, for which \nexperimental support is presented in \\cite{magas,sekihara}. Another step forward in this\n direction has been the interpretation\nof low lying $J^P=1\/2^+$ states as molecular systems of two pseudoscalar mesons and one baryon\n\\cite{alberto,alberto2,kanchan,Jido:2008zz,KanadaEn'yo:2008wm}. \n\nMuch work has been done using pseudoscalar\nmesons as building blocks, but more recently, vectors instead of\npseudoscalars are also being considered. In the baryon sector the\ninteraction of the $\\rho \\Delta$ interaction has been recently addressed in\n\\cite{vijande}, where three degenerate $N^*$ states and three degenerate\n$\\Delta$ states around 1900 MeV, with $J^P=1\/2^-, 3\/2^-, 5\/2^-$, are found. The extrapolation to SU(3) with the interaction of the vectors of the nonet with\nthe baryons of the decuplet has been done in \\cite{sourav}. The\nunderlying theory for this study is the hidden gauge formalism\n\\cite{hidden1,hidden2,hidden4}, which deals with the interaction of vector mesons and\npseudoscalars, respecting chiral dynamics, providing the interaction of\npseudoscalars among themselves, with vector mesons, and vector mesons among\nthemselves. It also offers a perspective on the chiral Lagrangians as limiting\ncases at low energies of vector exchange diagrams occurring in the theory. \n\n\n In the meson sector, the interaction of $\\rho \\rho$ within this formalism has\nbeen addressed in \\cite{raquel}, where it has been shown to lead to the\ndynamical generation of the $f_2(1270)$ and $f_0(1370)$ meson resonances. The extrapolation to SU(3) of the work \nof \\cite{raquel} has been done in \\cite{gengvec}, where many resonances are\nobtained, some of which can be associated to known meson states, while there are\npredictions for new ones.\n\n In this talk we present the results of the interaction of the nonet\n of vector mesons with the\n octet of baryons \\cite{angelsvec}, which have been obtained\n using the unitary approach in coupled\nchannels. The scattering amplitudes lead to poles in the\ncomplex plane which can be associated to some well known resonances. Under the\napproximation of neglecting the three momentum of the particles versus their\nmass, we obtain degenerate states of $J^P=1\/2^-,3\/2^-$ for the case of the\ninteraction of vectors with the octet of baryons. This degeneracy \nseems to be followed qualitatively by the experimental spectrum, although in\nsome cases the spin partners have not been identified. Improvements in the theory follow from the consideration of the decay of these states into a pseudoscalar meson and a baryon, and some results are presented here. \n\nMoreover we report on composite states of hidden charm emerging from the interaction of vector mesons and baryons with charm. \nWe shall also report on some states coming from the three body system \n $\\Delta \\rho \\pi$ which can shed some light on the status of some $\\Delta$ states of $J^P=5\/2^+$ in the vicinity of 2000 MeV. Finally, we devote some attention to new developments around vector mesons in a nuclear medium, specifically the $K^*$ and $J\/\\psi$ in the medium. \n\n\n\n \n\n\\section{Formalism for $VV$ interaction}\n\nWe follow the formalism of the hidden gauge interaction for vector mesons of \n\\cite{hidden1,hidden2} (see also \\cite{hidekoroca} for a practical set of Feynman rules). \nThe Lagrangian involving the interaction of \nvector mesons amongst themselves is given by\n\\begin{equation}\n{\\cal L}_{III}=-\\frac{1}{4}\\langle V_{\\mu \\nu}V^{\\mu\\nu}\\rangle \\ ,\n\\label{lVV}\n\\end{equation}\nwhere the symbol $\\langle \\rangle$ stands for the trace in the SU(3) space \nand $V_{\\mu\\nu}$ is given by \n\\begin{equation}\nV_{\\mu\\nu}=\\partial_{\\mu} V_\\nu -\\partial_\\nu V_\\mu -ig[V_\\mu,V_\\nu]\\ ,\n\\label{Vmunu}\n\\end{equation}\nwith $g$ given by $g=\\frac{M_V}{2f}$\nwhere $f=93\\,MeV$ is the pion decay constant. The magnitude $V_\\mu$ is the SU(3) \nmatrix of the vectors of the nonet of the $\\rho$\n\\begin{equation}\nV_\\mu=\\left(\n\\begin{array}{ccc}\n\\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}}&\\rho^+& K^{*+}\\\\\n\\rho^-& -\\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}}&K^{*0}\\\\\nK^{*-}& \\bar{K}^{*0}&\\phi\\\\\n\\end{array}\n\\right)_\\mu \\ .\n\\label{Vmu}\n\\end{equation}\n\nThe interaction of ${\\cal L}_{III}$ gives rise to a contact term coming from \n$[V_\\mu,V_\\nu][V_\\mu,V_\\nu]$\n\\begin{equation}\n{\\cal L}^{(c)}_{III}=\\frac{g^2}{2}\\langle V_\\mu V_\\nu V^\\mu V^\\nu-V_\\nu V_\\mu\nV^\\mu V^\\nu\\rangle\\ ,\n\\label{lcont}\n\\end{equation}\n and on the other hand it gives rise to a three \nvector vertex from \n\\begin{equation}\n{\\cal L}^{(3V)}_{III}=ig\\langle (\\partial_\\mu V_\\nu -\\partial_\\nu V_\\mu) V^\\mu V^\\nu\\rangle\n\\label{l3V}=ig\\langle (V^\\mu\\partial_\\nu V_\\mu -\\partial_\\nu V_\\mu\nV^\\mu) V^\\nu\\rangle\n\\label{l3Vsimp}\\ ,\n\\end{equation}\n\nIn this latter case one finds an analogy with the coupling of vectors to\n pseudoscalars given in the same theory by \n \n\\begin{equation}\n{\\cal L}_{VPP}= -ig ~tr\\left([\nP,\\partial_{\\mu}P]V^{\\mu}\\right),\n\\label{lagrVpp}\n\\end{equation}\nwhere $P$ is the SU(3) matrix of the pseudoscalar fields. \n\nIn a similar way, we have the Lagrangian for the coupling of vector mesons to\nthe baryon octet given by\n\\cite{Klingl:1997kf,Palomar:2002hk} \n\n\n\\begin{equation}\n{\\cal L}_{BBV} =\n\\frac{g}{2}\\left(tr(\\bar{B}\\gamma_{\\mu}[V^{\\mu},B])+tr(\\bar{B}\\gamma_{\\mu}B)tr(V^{\\mu})\n\\right),\n\\label{lagr82}\n\\end{equation}\nwhere $B$ is now the SU(3) matrix of the baryon octet \\cite{Eck95,Be95}. Similarly,\none has also a lagrangian for the coupling of the vector mesons to the baryons\nof the decuplet, which can be found in \\cite{manohar}.\n\n\nWith these ingredients we can construct the Feynman diagrams that lead to the $PB\n\\to PB$ and $VB \\to VB$ interaction, by exchanging a vector meson between the\npseudoscalar or the vector meson and the baryon, as depicted in Fig.\\ref{f1} .\n\n\\begin{figure}[tb]\n\\epsfig{file=f1a.eps, width=7cm} \\epsfig{file=f1b.eps, width=7cm}\n\\caption{Diagrams obtained in the effective chiral Lagrangians for interaction\nof pseudoscalar [a] or vector [b] mesons with the octet or decuplet of baryons.}%\n\\label{f1}%\n\\end{figure}\n\n As shown in \\cite{angelsvec}, in the limit of small three momenta of the vector mesons, which we consider, the vertices of Eq. (\\ref{l3Vsimp}) and Eq. (\\ref{lagrVpp}) give rise to the same expression. This makes the work technically easy allowing the use of many previous results. \n\n \n\n A small amendment is in order in the case of vector mesons, which\n is due to the mixing of $\\omega_8$ and the singlet of SU(3), $\\omega_1$, to give the\n physical states of the $\\omega$ and the $\\phi$.\n In this case, all one must do is to take the\n matrix elements known for the $PB$ interaction and, wherever $P$ is the\n $\\eta_8$, multiply the amplitude by the factor $1\/\\sqrt 3$ to get the\n corresponding $\\omega $ contribution and by $-\\sqrt {2\/3}$ to get the\n corresponding $\\phi$ contribution. Upon the approximation consistent with\n neglecting the three momentum versus the mass of the particles (in this\n case the baryon), we can just take the $\\gamma^0$ component of \n eq. (\\ref{lagr82}) and\n then the transition potential corresponding to the diagram of Fig. 1(b ) is\n given by\n \n \\begin{equation}\nV_{i j}= - C_{i j} \\, \\frac{1}{4 f^2} \\, (k^0 + k'^0)~ \\vec{\\epsilon}\\vec{\\epsilon\n} ',\n\\label{kernel}\n\\end{equation}\n where $k^0, k'^0$ are the energies of the incoming and outgoing vector mesons. \n The same occurs in the case of the decuplet. \n \n The $C_{ij}$ coefficients of Eq. (\\ref{kernel}) can be obtained directly from \n \\cite{angels,bennhold,inoue}\n with the simple rules given above for the $\\omega$ and the $\\phi$, and\n substituting $\\pi$ by $\\rho$ and $K$ by $K^*$ in the matrix elements. The\n coefficients are obtained both in the physical basis of states or in the\n isospin basis. Here we will show results in isospin\n basis. \n\n The next step to construct the scattering matrix is done by solving the\n coupled channels Bethe Salpeter equation in the on shell factorization approach of \n \\cite{angels,ollerulf}\n \\begin{equation}\nT = [1 - V \\, G]^{-1}\\, V,\n\\label{eq:Bethe}\n\\end{equation} \nwith $G$ the loop function of a vector meson and a baryon which we calculate in\ndimensional regularization using the formula of \\cite{ollerulf} and similar\nvalues for the subtraction constants. The $G$ function is convoluted with the \nspectral function for the vector mesons to take into account their width. \n\n \n\n\n\n The iteration of diagrams implicit in the Bethe Salpeter equation in the case\n of the vector mesons propagates the $\\vec{\\epsilon}\\vec{\\epsilon }'$ term \n of the interaction, thus,\nthe factor $\\vec{\\epsilon}\\vec{\\epsilon }'$ appearing in the potential $V$\nfactorizes also in the $T$ matrix for the external vector mesons. This has as a consequence that the interaction is spin independent and we find degenerate states in $J^P=1\/2^-$ and $J^P=3\/2^-$.\n\n\\section{Results} \n\n The resonances obtained are summarized in Table \\ref{tab:octet} .\n As one can see in Table \\ref{tab:octet} there are states which one can easily\nassociate to known resonances. There are ambiguities in other cases. One can also\nsee that in several cases the degeneracy in spin that the theory predicts is\nclearly visible in the experimental data, meaning that there are \nseveral states with about 50 MeV \nor less mass difference\nbetween them. In some cases, the theory predicts quantum numbers for \nresonances which have no spin and parity associated. It would be interesting to\npursue the experimental research to test the theoretical predictions. \n\n\n\n\n\n\n\\begin{table}[ht]\n \\renewcommand{\\arraystretch}{1.5}\n \\setlength{\\tabcolsep}{0.2cm}\n\\begin{tabular}{c|c|cc|ccccc}\\hline\\hline\n$S,\\,I$&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n \\vspace*{-0.3cm}\n & pole position & \\multicolumn{2}{c|}{real axis} & & & & & \\\\\n & {\\small (convolution)} &\\multicolumn{2}{c|}{{\\small\n (convolution)}} & \\\\\n & & mass & width &name & $J^P$ & status & mass & width \\\\\n \\hline\n$0,1\/2$ & --- & 1696 & 92 & $N(1650)$ & $1\/2^-$ & $\\star\\star\\star\\star$ & 1645-1670\n& 145-185\\\\\n & & & & $N(1700)$ & $3\/2^-$ & $\\star\\star\\star$ &\n\t1650-1750 & 50-150\\\\\n & $1977 + {\\rm i} 53$ & 1972 & 64 & $N(2080)$ & $3\/2^-$ & $\\star\\star$ & $\\approx 2080$\n& 180-450 \\\\\t\n & & & & $N(2090)$ & $1\/2^-$ & $\\star$ &\n $\\approx 2090$ & 100-400 \\\\\n \\hline\n$-1,0$ & $1784 + {\\rm i} 4$ & 1783 & 9 & $\\Lambda(1690)$ & $3\/2^-$ & $\\star\\star\\star\\star$ &\n1685-1695 & 50-70 \\\\\n & & & & $\\Lambda(1800)$ & $1\/2^-$ & $\\star\\star\\star$ &\n1720-1850 & 200-400 \\\\\n & $1907 + {\\rm i} 70$ & 1900 & 54 & $\\Lambda(2000)$ & $?^?$ & $\\star$ & $\\approx 2000$\n& 73-240\\\\\n & $2158 + {\\rm i} 13$ & 2158 & 23 & & & & & \\\\\n \\hline\n$-1,1$ & $ --- $ & 1830 & 42 & $\\Sigma(1750)$ & $1\/2^-$ & $\\star\\star\\star$ &\n1730-1800 & 60-160 \\\\\n & $ --- $ & 1987 & 240 & $\\Sigma(1940)$ & $3\/2^-$ & $\\star\\star\\star$ & 1900-1950\n& 150-300\\\\\n & & & & $\\Sigma(2000)$ & $1\/2^-$ & $\\star$ &\n$\\approx 2000$ & 100-450 \\\\\\hline\n$-2,1\/2$ & $2039 + {\\rm i} 67$ & 2039 & 64 & $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ &\n$1950\\pm15$ & $60\\pm 20$ \\\\\n & $2083 + {\\rm i} 31 $ & 2077 & 29 & $\\Xi(2120)$ & $?^?$ & $\\star$ &\n$\\approx 2120$ & 25 \\\\\n \\hline\\hline\n \\end{tabular}\n\\caption{The properties of the 9 dynamically generated resonances stemming from the vector-baryon octet interaction and their possible PDG\ncounterparts.}\n\\label{tab:octet}\n\\end{table}\n\n\n\n\n\n\n The predictions made here for resonances not observed should be a stimulus for\nfurther search of such states. In this\nsense it is worth noting the experimental program at Jefferson Lab \n\\cite{Price:2004xm} to investigate the $\\Xi$ resonances. We are\nconfident that the predictions shown here stand on solid grounds and anticipate much\nprogress in the area of baryon spectroscopy and on the understanding of the\nnature of the baryonic resonances. \n\n\\section{Incorporating the pseudoscalar meson-baryon channels}\n\nImprovements in the states tabulated in Table \\ref{tab:octet} have been done by incorporating intermediate states of a pseudoscalar meson and a baryon \\cite{garzon}. This is done by including the diagrams of Fig.~(\\ref{box}). However, arguments of gauge invariance \\cite{Rapp:1997fs,Peters:1997va,Rapp:1999ej,Urban:1999im,Cabrera:2000dx,Cabrera:2002hc,kanchan1,kanchan2} demand that the meson pole term be accompanied by the corresponding Kroll Ruderman contact term, see Fig. \\ref{fig:vbpb}.\n\n\n\n\\begin{center}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.5]{intbox.eps} \n\n\\end{center}\n\\caption{ Diagram for the $VB \\rightarrow VB$ interaction incorporating the intermediate pseudoscalar-baryon states.}\n\\label{box}\n\\end{figure}\n\\end{center} \n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[scale=0.5]{vbpb.eps}\n\\end{center}\n\\caption{Diagrams of the $VB\\rightarrow PB$. (a) meson pole term, (b) Kroll Ruderman contact term.}\n\\label{fig:vbpb}\n\\end{figure}\n\n\nIn the intermediate B states of Fig. \\ref{box} we include baryons of the octet and the decuplet. The results of the calculations are a small shift and a broadening of the resonances obtained with the base of vector-baryon alone.The results are shown in Fig. \\ref{res1} and Tables \\ref{tab:pdg12} and \\ref{tab:pdg32}. \n\nIn Fig.~\\ref{res1} we see two peaks for the state of $S=0$ and $I=1\/2$, one around 1700 MeV, in channels $\\rho N$ and $K^* \\Lambda$, and another peak near 1980 MeV, which appears in all the channels except for $\\rho N$. \nWe can see that the mixing of the PB channels affects differently the two spins, $J^P=1\/2^-$ and $3\/2^-$, as a consequence of the extra mechanisms contributing to the $J^P=1\/2^-$. The effect of the box diagram on the $J^P=3\/2^-$ sector is small, however the PB-VB mixing mechanism are more important in the $J^P=1\/2^-$ sector. Indeed, the Kroll Ruderman term only allows the $1\/2^-$ pseudoscalar baryon intermediate states in the box. The most important feature is a shift of the peak around 1700 MeV, which appears now around 1650 MeV. This breaking of the degeneracy is most welcome since this allows us to associate the $1\/2^-$ peak found at 1650 MeV with the $N^*(1650)(1\/2^-)$ while the peak for $3\/2^-$ at 1700 MeV can be naturally associated to the $N^*(1700)(3\/2^-)$. \n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[scale=1.5]{spin1.eps} \n\\end{center}\n\\caption{$|T|^2$ for the S=0, I=1\/2 states. Dashed lines correspond to tree level only and solid lines are calculated including the box diagram potential. Vertical dashed lines indicate the channel threshold.}\n\\label{res1}\n\\end{figure}\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|c|cc|ccccc}\n\\hline\\hline\n$S,\\,I$\t&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n \t& pole position\t& \\multicolumn{2}{c|}{real axis} & & & & & \\\\\n \t& $M_R+i\\Gamma \/2$\t& mass & width &name & $J^P$ & status & mass & width \\\\\n\\hline\n$0,1\/2$ & $1690+i24^{*}\t\t\t$\t& 1658 & 98 \n\t\t& $N(1650)$ & $1\/2^-$ & $\\star\\star\\star\\star$ \t& 1645-1670\t& 145-185\\\\\n \t\t\n \t& $1979+i67\t\t\t$\t& 1973 & 85 \n \t\t& $N(2090)$ & $1\/2^-$ & $\\star$ \t & $\\approx 2090$ & 100-400 \\\\\n\\hline\n$-1,0$\t& $1776+i39\t\t\t\t$\t& 1747 & 94 \n \t\t& $\\Lambda(1800)$ & $1\/2^-$ & $\\star\\star\\star$ \t\t& 1720-1850 & 200-400 \\\\\n \t\t\n & $1906+i34^{*}\t\t\t$ \t& 1890 & 93 \n & $\\Lambda(2000)$ & $?^?$ & $\\star$ \t\t\t\t\t& $\\approx 2000$ & 73-240\\\\\n\n & $2163+i37\t\t\t\t$ \t& 2149 & 61 & & & & & \\\\\n\\hline\n$-1,1$ & $ -\t\t\t$\t& 1829& 84 \n\t\t& $\\Sigma(1750)$ & $1\/2^-$ & $\\star\\star\\star$ & 1730-1800 & 60-160 \\\\\n \t\t& $ -\t\t\t $ \t& 2116 & 200-240 \n \t\t& $\\Sigma(2000)$ & $1\/2^-$ & $\\star$ \t\t\t& $\\approx 2000$ & 100-450 \\\\\n\\hline\n$-2,1\/2$& $2047+i19^{*}\t$\t& 2039 & 70 \n\t\t& $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ \t& $1950\\pm15$ & $60\\pm 20$ \\\\\n\n & $ -\t\t\t$\t& 2084 & 53 \n & $\\Xi(2120)$ & $?^?$ & $\\star$ \t\t\t& $\\approx 2120$ & 25 \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{The properties of the nine dynamically generated resonances and their possible PDG\ncounterparts for $J^P=1\/2^-$. The numbers with asterisk in the imaginary part of the pole position are obtained without the convolution for the vector mass distribution of the $\\rho$ and $K^*$.}\n\\label{tab:pdg12}\n\\end{center}\n\\end{table}\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|c|cc|ccccc}\n\\hline\\hline\n$S,\\,I$\t&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n \t& pole position\t& \\multicolumn{2}{c|}{real axis} & & & & & \\\\\n \t& $M_R+i\\Gamma \/2$\t& mass & width &name & $J^P$ & status & mass & width \\\\\n\\hline\n$0,1\/2$ & $1703+i4^{*}\t\t\t$\t& 1705 & 103 \n \t\t& $N(1700)$ & $3\/2^-$ & $\\star\\star\\star$ \t\t& 1650-1750 & 50-150\\\\\n \t\t\n \t& $1979+i56\t\t\t$\t& 1975 & 72 \n \t& $N(2080)$ & $3\/2^-$ & $\\star\\star$ & $\\approx 2080$ & 180-450 \\\\\t\n\\hline\n$-1,0$\t& $1786+i11\t\t\t\t$\t& 1785 & 19 \n\t\t& $\\Lambda(1690)$ & $3\/2^-$ & $\\star\\star\\star \\star$ \t& 1685-1695 & 50-70 \\\\\n \t\t\n & $1916+i13^{*}\t\t\t$ \t& 1914 & 59 \n & $\\Lambda(2000)$ & $?^?$ & $\\star$ \t\t\t\t\t& $\\approx 2000$ & 73-240\\\\\n\n & $2161+i17\t\t\t\t$ \t& 2158 & 29 & & & & & \\\\\n\\hline\n$-1,1$ & $ -\t\t\t$\t& 1839& 58 \n \t\t& $\\Sigma(1940)$ & $3\/2^-$ & $\\star\\star\\star$ & 1900-1950 & 150-300\\\\\n \t\t& $ -\t\t\t $ \t& 2081 & 270 & & & & & \\\\ \n\\hline\n$-2,1\/2$& $2044+i12^{*}\t$\t& 2040 & 53 \n\t\t& $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ \t& $1950\\pm15$ & $60\\pm 20$ \\\\\n\n & $2082+i5^{*} \t$\t& 2082 & 32 \n & $\\Xi(2120)$ & $?^?$ & $\\star$ \t\t\t& $\\approx 2120$ & 25 \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{The properties of the nine dynamically generated resonances and their possible PDG\ncounterparts for $J^P=3\/2^-$. The numbers with asterisk in the imaginary part of the pole position are obtained without the convolution for the vector mass distribution of the $\\rho$ and $K^*$.}\n\\label{tab:pdg32}\n\\end{center}\n\\end{table}\n\n\n\n\\section{The $\\Delta \\rho \\pi$ system and $\\Delta$ $J^P=5\/2^+$ states around 2000 MeV}\nIn Refs.~\\cite{vijande,sourav} it was shown that the $\\Delta \\rho$ interaction\ngave rise to $N^*$ and $\\Delta$ states with degenerate spin-parity\n$1\/2^-,3\/2^-,5\/2^-$. In a recent work~\\cite{Xie:2011uw}, one extra\n$\\pi$ was introduced in the system, and via the Fixed Center\nApproximation to the Faddeev Equations, the new system was studied\nand new states were found.\n\n\\begin{center}%\n\\begin{table}[ptbh]\n\\caption{Assignment of $I=3\/2$ predicted states to $J^{P}=1\/2^{+}%\n,3\/2^{+},5\/2^{+}$ resonances. Estimated PDG masses for these\nresonances as well as their extracted values from references\n\\cite{Man92} and \\cite{Vra00} (in brackets) are shown for\ncomparison. N. C. stands for a non cataloged resonance in the PDG\nreview}\n\\begin{tabular}\n[c]{c|ccccc} \\hline\nPredicted & \\multicolumn{5}{c}{PDG data}\\\\\n& & & & & \\\\\nMass (MeV) & Name & $J^{P}$ & Estimated Mass (MeV) & Extracted Mass (MeV) & Status\\\\\n1770 & $\\Delta(1740)$ & $5\/2^{+}$ & & $1752\\pm32$ & N.C.\\\\\n& & & & $(1724\\pm61)$ & \\\\\n& $\\Delta(1600)$ & $3\/2^{+}$ & $1550-1700$ & $1706\\pm10$ & *** \\\\\n& & & & $(1687\\pm44)$ & \\\\\n& $\\Delta(1750)$ & $1\/2^{+}$ & $\\approx1750$ & $1744\\pm36$ & * \\\\\n& & & & $(1721\\pm61)$ & \\\\\\hline\n$1875$ & $\\Delta(1905)$ & $5\/2^{+}$ & $1865-1915$ & $1881\\pm18$ & ****\\\\\n& & & & $(1873\\pm77)$ & \\\\\n& $\\Delta(1920)$ & $3\/2^{+}$ & $1900-1970$ & $2014\\pm16$ & ***\\\\\n& & & & $(1889\\pm100)$ & \\\\\n& $\\Delta(1910)$ & $1\/2^{+}$ &\n$1870-1920$ & $1882\\pm10$ & ****\\\\\n& & & & $(1995\\pm12)$ &\n\\\\\\hline\n\\end{tabular}\n\\label{threebody}\n\\end{table}\n\\end{center}\n\nWe show in Table \\ref{threebody} the two $\\Delta^*$ states obtained and\nthere is also a hint of another $\\Delta^*$ state around $2200$ MeV.\nExperimentally, only two resonances $\\Delta_{5\/2^+}(1905)(****)$ and\n$\\Delta_{5\/2^+}(2000)(**)$, are cataloged in the Particle Data Book\nReview~\\cite{pdg2010}. However, a careful look at\n$\\Delta_{5\/2^+}(2000)(**)$, shows that its nominal mass is in fact\nestimated from the mass $(1724\\pm61)$, $(1752\\pm32)$ and\n$(2200\\pm125)$ respectively, extracted from three independent\nanalyses of different character~\\cite{Man92,Vra00,Cut80}. Moreover a\nrecent new data analysis~\\cite{Suz10} has reported a\n$\\Delta_{5\/2^+}$ with a pole position at $1738$ MeV.\n\nOur results give quantitative theoretical support to the existence\nof two distinct $5\/2^+$ resonances, $\\Delta_{5\/2^+}(\\sim 1740)$ and\n$\\Delta_{5\/2^+}(\\sim 2200)$, apart from the one around 1905 MeV . We propose that these two resonances\nshould be cataloged instead of $\\Delta_{5\/2^+}(2000)$. This proposal\ngets further support from the possible assignment of the other\ncalculated baryon states in the $I=1\/2,3\/2$ and $J^P=1\/2^+,3\/2^+$\nsectors to known baryonic resonances. In particular the poorly\nestablished $\\Delta_{1\/2^+}(1750)(*)$ may be naturally interpreted\nas a $\\pi N_{1\/2^-}(1650)$ bound state.\n\n\\section{Hidden charm baryons from vector-baryon interaction}\nFollowing the idea of \\cite{angelsvec} in \\cite{wu} is was found that several baryon states emerged as hidden charm composite states of mesons and baryons with charm. In particular along the lines discussed here, we find a hidden charm baryon that couples to $J\/\\psi N$ and other channels, as shown in Tables \\ref{jpsicoupling},\\ref{jpsiwidth}. This will play a role later on when we discuss the $J\/\\psi$ suppression in nuclei. To do the calculations in the charm sector an extension to SU(4) of the hidden gauge Lagrangians is made, but the symmetry is explicitly broken when considering the exchange of heavy vector mesons, where the appropriate reduction in the Feynman diagrams is taken into account. \n\n \\begin{table}[ht]\n \\renewcommand{\\arraystretch}{1.1}\n \\setlength{\\tabcolsep}{0.4cm}\n\\begin{center}\n\\begin{tabular}{cccccc}\\hline\n$(I, S)$& $z_R$ & \\multicolumn{4}{c}{$g_a$}\\\\\n\\hline\n$(1\/2, 0) $ & & $\\bar{D}^{*} \\Sigma_{c}$ & $\\bar{D}^{*} \\Lambda^{+}_{c}$& $J\/\\psi N$ \\\\\n & $4415-9.5i$ & $2.83-0.19i $ &$-0.07+0.05i $ & $-0.85+0.02i$ \\\\\n & &$ 2.83 $ &$0.08 $ & $0.85$ \\\\\n\\hline\n$(0, -1) $ & & $\\bar{D}^{*}_{s} \\Lambda^{+}_{c}$ & $\\bar{D}^{*} \\Xi_{c}$ & $\\bar{D}^{*} \\Xi'_{c}$ & $J\/\\psi \\Lambda$\\\\\n & $4368-2.8i $ & $1.27-0.04i $ &$ 3.16-0.02i $ & $-0.10+0.13i $ & $0.47+0.04i $ \\\\\n & & $1.27 $ & $3.16 $ & $0.16 $ & $0.47 $ \\\\\n & $4547-6.4i $ & $0.01+0.004i$ & $0.05-0.02i$ & $2.61-0.13i $ & $-0.61-0.06i $ \\\\\n & & $0.01 $ & $0.05 $ & $2.61$ & $0.61 $ \\\\\n\\hline\\end{tabular} \\caption{Pole position ($z_R$) and coupling\nconstants ($g_a$) to various channels for the states from\n$PB\\rightarrow PB$ including the $J\/\\psi N$ and $J\/\\psi\\Lambda$\nchannels. }\n\\label{jpsicoupling}\n\\end{center}\n \\renewcommand{\\arraystretch}{1.1}\n \\setlength{\\tabcolsep}{0.4cm}\n\\begin{center}\n\\begin{tabular}{ccccc}\\hline\n$(I, S)$ & $z_R$ & \\multicolumn{2}{c}{Real axis} & $\\Gamma_i$ \\\\\n & & $M$ & $\\Gamma$ & \\\\\n\\hline\n$(1\/2, 0)$ & & & & $J\/\\psi N$\\\\\n & $4415-9.5i$ & $4412$ & $47.3$ & $19.2$ \\\\\n\\hline\n$(0, -1)$ & & & & $J\/\\psi\\Lambda$\\\\\n & $4368-2.8i$ & $4368 $& $28.0 $ & $5.4$ \\\\\n & $4547-6.4i $ & $4544 $& $36.6 $ & $13.8$ \\\\\n\\hline\\end{tabular} \\caption{Pole position ($z_R$), mass ($M$),\ntotal width ($\\Gamma$, including the contribution from the light\nmeson and baryon channel) and the decay widths for the $J\/\\psi N$\nand $J\/\\psi\\Lambda$ channels ($\\Gamma_i$). The unit are in MeV}\n\\label{jpsiwidth}\n\\end{center}\n\\end{table} \n\n\\section{The properties of $K^*$ in nuclei}\nMuch work about the vector mesons in nuclei has been done in the last decade\n\\cite{rapp,hayano,mosel,na60,wood,nucl-th\/0610067,nanova}, mostly for the $\\rho,\\phi,\\omega$, which can be studied looking for dileptons. Maybe this technical detail was what prevented any attention being directed to the renormalization of the $K^*$ in nuclei. However, recently this work has been addressed in \\cite{lauraraquel} with very interesting results. \n\n\n\n\nThe $K^{*-}$ width in vacuum is determined by the imaginary part of the free $\\bar K^*$ self-energy at rest, ${\\rm Im} \\Pi^0_{\\bar K^*}$, due to the decay of the $\\bar{K}^*$ meson into $\\bar{K}\\pi$ pairs: $\\Gamma_{K^{*-}}=-\\mathrm{Im}\\Pi_{\\bar{K}^*}^{0}\/m_{\\bar K^*}=42$ MeV \\cite{lauraraquel}. Note that this value is quite close to the experimental value $\\Gamma^{\\rm exp}_{K^{*-}}=50.8\\pm 0.9$ MeV.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5cm]{Fig2.eps}\n\\hfill\n\\caption{Self-energy diagrams from the decay of the $\\bar{K}^*$ meson in the medium.}\n\\label{fig:1}\n\\end{center}\n\\end{figure}\n\n\n\nThe $\\bar{K}^*$ self-energy in matter, on one hand, results from its decay into ${\\bar K}\\pi$, $\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (a)}}$, including both the self-energy of the antikaon \\cite{Tolos:2006ny} and the pion \\cite{Oset:1989ey,Ramos:1994xy} (see first diagram of Fig.~\\ref{fig:1} and some specific contributions in diagrams $(a1)$ and $(a2)$ of Fig.~\\ref{fig:3}). Moreover, vertex corrections required by gauge invariance are also incorporated, which are associated to the last three diagrams in Fig. \\ref{fig:1}.\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{new_diagrams.eps}\n\\caption{Contributions to the $\\bar K^*$ self-energy, depicting their different\ninelastic sources.}\n\\label{fig:3}\n\\end{center}\n\\end{figure}\n\n\n\n\nThe second contribution to the $\\bar K^*$ self-energy comes from its interaction\nwith the nucleons in the Fermi sea, as displayed in diagram (b) of \nFig.~\\ref{fig:3}. This accounts for the direct quasi-elastic process $\\bar K^* N \\to \\bar K^* N$ as well as other absorption channels $\\bar K^* N\\to \\rho Y, \\omega Y, \\phi Y, \\dots$ with $Y=\\Lambda,\\Sigma$. \nThis contribution is determined by\nintegrating the medium-modified $\\bar K^* N$ amplitudes, $T^{\\rho,I}_{\\bar\nK^*N}$, over the Fermi sea of nucleons,\n\\begin{eqnarray}\n\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}(q^0,\\vec{q}\\,)&=&\\int \\frac{d^3p}{(2\\pi)^3} \\, n(\\vec{p}\\,)\\,\n\\left [~{T^\\rho}^{(I=0)}_{\\bar K^*N}(P^0,\\vec{P})+3 {T^\\rho}^{(I=1)}_{\\bar K^*N}(P^0,\\vec{P})\\right ] \\ ,\n \\label{eq:pid}\n\\end{eqnarray}\nwhere $P^0=q^0+E_N(\\vec{p}\\,)$ and $\\vec{P}=\\vec{q}+\\vec{p}$ are the\ntotal energy and momentum of the $\\bar K^*N$ pair in the nuclear\nmatter rest frame, and the values $(q^0,\\vec{q}\\,)$ stand for the\nenergy and momentum of the $\\bar K^*$ meson also in this frame. The\nself-energy $\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}$ has to be determined self-consistently\nsince it is obtained from the in-medium amplitude\n${T}^\\rho_{\\bar K^*N}$ which contains the $\\bar K^*N$ loop function\n${G}^\\rho_{\\bar K^*N}$, and this last quantity itself is a function of the complete self-energy\n$\\Pi_{\\bar K^*}^{\\rho}=\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (a)}}\n+\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}$. \n\n\n\n\n\nWe note that the two contributions to the $\\bar K^*$ self-energy, coming from\nthe decay of\n$\\bar K \\pi$ pairs in the medium [Figs.~\\ref{fig:3}(a1) and \\ref{fig:3}(a2)] or\nfrom the $\\bar K^* N$ interaction [Fig.~\\ref{fig:3}(b)] provide different\nsources\nof inelastic $\\bar K^* N$ scattering, which add incoherently in the $\\bar K^*$\nwidth. \nAs seen in the upper two rows of Fig.~\\ref{fig:3}, the $\\bar K^* N$\namplitudes mediated by intermediate $\\bar K N$ or $\\pi\nY$ states are not unitarized. Ideally, one would like to treat the vector\nmeson-baryon ($VB$) and pseudoscalar meson-baryon ($PB$) states on the same\nfooting. However, at the energies of interest, transitions of the type $\\bar K^*\nN \\to \\bar K N$ mediated by pion exchange may place this pion on its mass shell,\nforcing one to keep track of the proper analytical cuts contributing to the\nimaginary part of the amplitude and making the iterative process more\ncomplicated. A technical solution can be found by calculating the box diagrams\nof Figs.~\\ref{fig:3}(a1) and \\ref{fig:3}(a2), taking all the cuts into account\nproperly, and adding the resulting $\\bar K^* N \\to \\bar K^* N$ terms to the $VB\n\\to V^\\prime B^\\prime$ potential coming from vector-meson exchange, in a\nsimilar way as done for the study of the vector-vector interaction in\nRefs.~\\cite{raquel,gengvec}. As we saw in the former sections, the generated resonances barely change their position for spin 3\/2 and only by a moderate amount in some cases for spin 1\/2. Their widths are somewhat enhanced due to the opening of the newly\nallowed $PB$ decay channels \\cite{garzon}.\n\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Fig5_colouronline.eps}\n\\hfill\n\\caption{ $\\bar K^*$ self-energy for\n $\\vec{q}=0 \\, {\\rm MeV\/c}$ and $\\rho_0$. }\n\\label{fig:auto-spec}\n\\end{center}\n\\end{figure}\n\n\n \nThe full $\\bar K^*$ self-energy as a function of the $\\bar K^*$ energy for zero\nmomentum at normal nuclear matter density is shown in \nFig.~\\ref{fig:auto-spec}. We explicitly indicate the contribution to the\nself-energy coming from the self-consistent calculation of the $\\bar K^* N$\neffective interaction (dashed lines) and the self-energy from the $\\bar K^*\n\\rightarrow \\bar K \\pi$ decay mechanism (dot-dashed lines), as well as the\ncombined result from both sources (solid lines).\n\nAround $q^0= 800-900$ MeV we observe an enhancement of the width as well as some structures in the real part of the $\\bar K^*$ self-energy. The origin of these structures can be traced back to the coupling of the $\\bar K^*$ to the in-medium $\\Lambda(1783) N^{-1}$ and $\\Sigma(1830) N^{-1}$ excitations, which dominate the $\\bar K^*$ self-energy in this energy region. However, at lower energies where the $\\bar K^* N\\to V B$ channels \nare closed, or at large energies beyond the resonance-hole excitations,\nthe width of the $\\bar K^*$ is governed by the $\\bar K \\pi$ decay mechanism in dense matter. \n\n\nAs we can see, the $\\bar K^*$ feels a moderately attractive optical potential and acquires a width of $260$ MeV, which is about five times its width in vacuum. In the next section we devise a method to measure this large width experimentally. \n \n\n\n\\section{Transparency ratio for $\\gamma A \\to K^+ K^{*-} A'$}\n\nThe width of the $\\bar K^*$ meson in nuclear matter can be analyzed experimentally by means of the nuclear transparency ratio. The aim is to compare the cross sections of the photoproduction reaction $\\gamma A \\to K^+ K^{*-} A'$ in different nuclei, and tracing the differences to the in medium $K^{*-}$ width.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig8.eps}\n\\caption{Nuclear transparency ratio for $\\gamma A \\to K^+ K^{*-} A'$ for\ndifferent nuclei}\n\\label{fig:ratio}\n\\end{center}\n\\end{figure}\n\n\nThe normalized nuclear transparency ratio is defined as\n\\begin{equation}\nT_{A} = \\frac{\\tilde{T}_{A}}{\\tilde{T}_{^{12}C}} \\hspace{1cm} ,{\\rm with} \\ \\tilde{T}_{A} = \\frac{\\sigma_{\\gamma A \\to K^+ ~K^{*-}~ A'}}{A \\,\\sigma_{\\gamma N \\to K^+ ~K^{*-}~N}} \\ .\n\\end{equation}\nThe quantity $\\tilde{T}_A$ is the ratio of the nuclear $K^{*-}$-photoproduction cross section\ndivided by $A$ times the same quantity on a free nucleon. This describes the loss of flux of $K^{*-}$ mesons in the nucleus and is related to the absorptive part of the $K^{*-}$-nucleus optical potential and, therefore, to the $K^{*-}$ width in the nuclear medium. In order to remove other nuclear effects not related to the absorption of the $K^{*-}$, we evaluate this ratio with respect to $^{12}$C, $T_A$.\n\nIn Fig.~\\ref{fig:ratio} we show the transparency ratio for different nuclei and for two energies in the center of mass reference system, $\\sqrt{s}=3$ GeV and $3.5$ GeV, or, equivalently, two energies of the photon in the lab frame of $4.3$ GeV and $6$ GeV respectively. There is a very strong attenuation of the $\\bar{K}^*$ survival probability coming from the decay or absorption channels $\\bar{K}^*\\to \\bar{K}\\pi$ and $\\bar{K}^*N\\to \\bar K^* N, \\rho Y, \\omega Y, \\phi Y, \\dots$, with increasing nuclear-mass number $A$. The reason is the larger path that the $\\bar{K}^*$ has to follow before it leaves the nucleus, having then more chances of decaying or being absorbed.\n\n\\section{$J\/\\psi$ suppression}\n$J\/\\psi$ suppression in nuclei has been a hot topic \\cite{Vogt:1999cu}, among others for its possible interpretation as a signature of the formation of quark gluon plasma in heavy ion reactions \\cite{Matsui:1986dk}, but many other interpretations have been offered \\cite{Vogt:2001ky,Kopeliovich:1991pu,Sibirtsev:2000aw}. In a recent paper \\cite{raquelxiao} a study has been done of different $J\/\\psi N$ reactions which lead to $J\/\\psi$ absorption in nuclei. The different reactions considered are \nthe transition of $J\/\\psi N$ to $VN$ with $V$ being a light vector, $\\rho, \\omega,\\phi$, together with the inelastic channels, \n$J\/\\psi N \\to \\bar D \\Lambda_c$ and $J\/\\psi N \\to \\bar D \\Sigma_c$. \nAnalogously, we consider the mechanisms where the exchanged $D$ collides with a nucleon and gives $\\pi \\Lambda_c$ or $\\pi \\Sigma_c$. The cross section has a peak around $\\sqrt s=4415$ MeV, where the $J\/\\psi N$ couples to the resonance described in Tables \\ref{jpsicoupling} and \\ref{jpsiwidth}. With the inelastic cross section obtained we study the transparency ratio for electron induced $J\/\\psi$ production in nuclei at about 10 GeV and find that 30 - 35 \\% of the $J\/\\psi$ produced in heavy nuclei are absorbed inside the nucleus. This ratio is in line with depletions of $J\/\\psi$ in matter observed in other reactions. This offers a novelty in the interpretation of the $J\/\\psi$ suppression in terms of hadronic reactions, which has also been advocated before \\cite{Sibirtsev:2000aw}. Apart from novelties in the details of the calculations and the reaction channels considered, we find that the presence of the resonance that couples to $J\/\\psi N$ produces a peak in the inelastic $J\/\\psi N$ cross section and a dip in the transparency ratio. A measure of this magnitude could lead to an indirect observation of the existence of resonances that couple to $J\/\\psi N$. \n\nIn Fig. \\ref{crosec} we can see the total, elastic and inelastic cross sections in $J\/\\psi N$ to $J\/\\psi N$ through intermediate vector-baryon states. We can clearly see the peak around 4415 MeV produced by the hidden charm resonance dynamically generated form the interaction of $J\/\\psi N$ with its coupled $VB$ channels. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{crossection3.eps}\n\\caption{The total, elastic and inelastic cross sections in $J\/\\psi N$ to $J\/\\psi N$ through intermediate vector-baryon states.}\\label{crosec}\n\\end{center}\n\\end{figure}\n\nIn Fig. \\ref{figcro} we can see the cross section for $J\/\\psi N \\to \\bar{D} \\Lambda_c$ and $J\/\\psi N\\to \\bar{D} \\Sigma_c$. We can see that numerically the first cross section is sizeable, bigger than the one from the $VB$ channels. \n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{sigtot_sum1aa.eps}\n\\includegraphics[width=0.5\\textwidth]{sigtot_sum2aa.eps}\n\\caption{The cross section for $J\/\\psi N \\to \\bar{D} \\Lambda_c$ (up) and $J\/\\psi N\\to \\bar{D} \\Sigma_c$ (down).}\\label{figcro}\n\\end{center}\n\\end{figure}\n\nThe cross sections for $J\/\\psi N$ to $\\bar D \\pi \\Lambda_c$ or $\\bar D \\pi \\Sigma_c$ are small in size in the region of interest and are not plotted here. \n\nIn Fig. \\ref{sigin} we plot the total $J\/\\psi N$ inelastic cross section, as the sum of all inelastic cross sections from the different sources discussed before.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{sigtot_sum5.eps}\n\\caption{The total inelastic cross section of $J\/\\psi N$.}\\label{sigin}\n\\end{center}\n\\end{figure}\n\nFinally, in Fig. \\ref{transg} we show the transparency ratio of $^{208}Pb$ versus the one of $^{12}C$ as a function of the energy. We find sizeable reductions in the rate of $J\/\\psi$ production in electron induced reactions, and particularly a dip in the ratio that could be searched experimentally. \nIt should be noted that the calculation of the transparency ratio done so far does not consider the shadowing of the photons and assumes they can reach every point without being absorbed. However, for $\\gamma$ energies of around 10 GeV, as suggested here, the photon shadowing cannot be ignored. Talking it into account is easy since one can multiply the ratio $T_A'$ by the ratio of $N_{eff}$ for the nucleus of mass $A$ and $^{12}C$. This ratio for $^{208}Pb$ to $^{12}C$ at $E_\\gamma =$10 GeV is of the order $0.8$ but with uncertainties \\cite{Bianchi:1995vb}. We should then multiply $T_A'(^{208}Pb)$ in Fig. \\ref{transg} by this extra factor for a proper comparison with experiment.\nAlthough a good resolution should be implemented, the prospect of finding new states through this indirect method should serve as a motivation to perform such experiments.\n\n\\section{Conclusions}\n\nWe have made a survey of recent developments along the interaction of vector mesons with baryons and the properties of some vector mesons in a nuclear medium. We showed that the interaction is strong enough to produce resonant states which can qualify as quasibound states of a vector meson and a baryon in coupled channels. This adds to the wealth of composite states already established from the interaction of pseudoscalar mesons with baryons. At the same time we offered the results of new pictures that mix the pseudoscalar-baryon states with the vector-baryon states and break the spin degeneracy that the original model had. The method of vector-baryon interaction extended to the charm sector also produced some hidden charm states which couple to the $J\/\\psi N$ channel and had some repercussion in the $J\/\\psi$ suppression in nuclei. We also showed results for the spectacular renormalization of the $K^*$ in nuclei, where the width becomes as large as 250 MeV at normal nuclear matter density and we made suggestions of experiments that could test this large change.\n \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{totTA_g32.eps}\n\\caption{The transparency ratio of $J\/\\psi$ photoproduction as a function of the energy in the CM of $J\/\\psi$ with nucleons of the nucleus. Solid line: represents the effects due to $J\/\\psi$ absorption. Dashed line: includes photon shadowing \\cite{Bianchi:1995vb}.}\\label{transg}\n\\end{center}\n\\end{figure}\n\n\\section*{Acknowledgments} \nThis work is partly supported by DGICYT contract number\nFIS2011-28853-C02-01, the Generalitat Valenciana in the program Prometeo, 2009\/090. L.T. acknowledges support from Ramon y Cajal Research Programme, and from FP7-PEOPLE-2011-CIG under contract PCIG09-GA-2011-291679. We acknowledge the support of the European Community-Research Infrastructure\nIntegrating Activity\nStudy of Strongly Interacting Matter (acronym HadronPhysics3, Grant Agreement\nn. 283286)\nunder the Seventh Framework Programme of EU.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\section{Introduction}\nJamming attacks pose a serious threat to the continuous operability of wireless communication systems \\cite{economist2021satellite, topgun}. \nEffective methods to mitigate such attacks are of paramount importance\nas wireless systems become increasingly critical to modern infrastructure~\\cite{popovski2014ultra, pirayesh2022jamming}.\nIn the uplink of massive multi-user multiple-input multiple-output (MU-MIMO) systems, effective jammer mitigation becomes possible by the asymmetry in the number of antennas between the basestation (BS), which has many antennas, and a mobile jamming device, which typically has one or few antennas.\nOne possibility, for instance, is to project the receive signals on the subspace \northogonal to the jammer's channel~\\cite{marti2021snips,yan2016jamming}. \nUnfortunately, such methods require accurate knowledge of the jammer's channel.\nIf a jammer transmits permanently and with a static signature (often called barrage jamming), the~BS~can estimate its channel, for instance during a dedicated period in which the user equipments (UEs) are not transmitting~\\cite{marti2021snips} or in which they transmit predefined symbols~\\cite{yan2016jamming}.\nIn contrast to barrage jamming, however, a smart jammer might jam the system only at specific time instants, such as when the UEs are transmitting data symbols,\nand thereby prevent the BS from estimating the jammer's channel using simple estimation algorithms.\n\n\n\\subsection{State of the Art}\n\nMulti-antenna wireless systems offer the unique potential to effectively mitigate jamming attacks.\nConsequently, a variety of multi-antenna methods have been proposed for the mitigation of jamming attacks in MIMO systems\n\\cite{pirayesh2022jamming, marti2021snips, shen14a, hoang2021suppression, yan2016jamming,zeng2017enabling, vinogradova16a, do18a, akhlaghpasand20a, akhlaghpasand20b, marti2021hybrid, wan2022robust, darsena2022anti}.\nCommon to all of them~is the assumption---in one way or other---that information about the jammer's transmit characteristics\n(e.g., the jammer's channel, or the covariance matrix between the UE transmit signals and the jammed receive signals)\ncan be estimated using some specific subset of the receive samples.\\footnote{The method of \\cite{vinogradova16a} is to some extent an exception as it estimates the~UEs' subspace and projects the receive signals thereon. This method, however, dist-inguishes the UEs' from the jammer's subspace based on the receive power, thereby presuming that the UEs and the jammer transmit with different power.}\n\\fref{fig:traditional}~illustrates the approach of such methods: \nThe data phase is preceded by an augmented \ntraining phase in which the jammer's transmit characteristics as well as the channel matrix are estimated.\nThis augmented training phase may (i) complement a traditional pilot phase with a dedicated period during which the UEs do not transmit in order to \nenable jammer estimation (e.g., \\cite{marti2021snips, shen14a, hoang2021suppression}) or (ii) consist of an extended pilot phase so that there exist pilot sequences that are unused by\nthe UEs and on whose span the receive signals can be projected to estimate the jammer subspace~\\mbox{(e.g., \\cite{do18a, akhlaghpasand20a, akhlaghpasand20b}).}\nThe estimated jammer characteristics are then used to perform jammer-mitigating data detection. \nSuch an approach succeeds in the case of barrage jammers, but is unreliable for estimating the propagation characteristics of smart jammers, see \\fref{sec:example}: \nA smart jammer can evade estimation and, thus, circumvent mitigation by not transmitting during the training phase, for instance because it is aware of the defense mechanism\nor simply because it jams in short bursts only. \nFor this reason, our proposed method does not estimate the jammer channel based on a dedicated training phase, \nbut instead utilizes the entire transmission period and unifies jammer estimation and mitigation, channel estimation and data detection; see \\fref{fig:maed}.\n\nMany studies have already shown how smart jammers can disrupt wireless communication systems by\ntargeting only specific parts\nof the wireless transmission process \\cite{miller2010subverting, miller2011vulnerabilities, clancy2011efficient, sodagari2012efficient, \nlichtman2013vulnerability, lichtman20185g, lichtman2016lte, girke2019towards,lapan2012jamming} \ninstead of using barrage jamming.\nJammers that target only the pilot phase have received considerable attention \n\\cite{miller2010subverting,miller2011vulnerabilities,clancy2011efficient,sodagari2012efficient,lichtman2013vulnerability}, \nas such attacks can be more energy-efficient than barrage jamming in disrupting communication systems that do not \ndefend themselves against jammers~\\cite{clancy2011efficient,sodagari2012efficient,lichtman2013vulnerability}.\nHowever, if a jammer is active during the pilot phase, then a BS that \\emph{does} defend itself against attacks\ncan estimate the jammer's channel by exploiting knowledge of the UE transmit symbols during the pilot phase, for instance with the aid of unused pilot sequences~\\cite{do18a, akhlaghpasand20a, akhlaghpasand20b}.\nTo disable such jammer-mitigating communication systems, a smart jammer might thus refrain from jamming the pilot phase and only target \nthe data phase, even if such jamming attacks have not received much attention so far \n\\cite{lichtman2016lte, girke2019towards}.\nOther threat models that have been analyzed include attacks on control \nchannels \\cite{lichtman2013vulnerability, lichtman2016lte, lichtman20185g}, the beam alignment procedure \\cite{darsena2022anti}, \nor the time synchronization phase~\\cite{lapan2012jamming}, but this paper will not consider such protocol or control channel attacking schemes. \n\n\\begin{figure}[tp]\n\\vspace{-1mm}\n\\centering\n~\n\\subfigure[Existing methods separate jammer estimation (JEST) and channel~estimation (CHEST) from the jammer-resilient\ndata detection (DET). They~are ineffective against jammers that jam the data phase but not the training~phase.]{\n\\includegraphics[width=0.95\\columnwidth]{figures\/sota.pdf}\n\\label{fig:traditional}\n}\n\\newline\n\\subfigure[Our method unifies jammer estimation and mitigation, channel estimation, and data detection \nto deal with jammers regardless of their activity~pattern.]{\n\\includegraphics[width=0.95\\columnwidth]{figures\/maed.pdf}\n\\label{fig:maed}\n}\n\\caption{\nThe approach to jammer mitigation taken by existing methods (a) compared to the proposed method (b).\nIn the figure, $\\bmy_1,\\dots,\\bmy_K$ are the receive signals, \nand $\\hat\\Hj, \\hat\\bH$, and $\\hat\\bS_D$ are the estimates of the jammer channel, the UE channel matrix, and the UE transmit symbols, respectively.\n}\n\\vspace{-2mm}\n\\label{fig:maed_vs_trad}\n\\end{figure}\n\n\n\\subsection{Contributions}\nTo mitigate smart jammers, we propose a novel approach that does not depend on the jammer being active during \n\\textit{any} specific period.\nLeveraging the fact that a jammer cannot change its subspace instantaneously,\nwe utilize a problem formulation which unifies jammer estimation and mitigation, \nchannel estimation, and data detection, instead of dealing with these tasks independently (cf.~\\fref{fig:maed}).\nWe support the soundness of the proposed optimization problem by proving that its global minimum is unique and recovers the transmitted data symbols, given that certain sensible conditions are satisfied. \nBy building on techniques for joint channel estimation and data detection\n\\cite{vikalo2006efficient, xu2008exact, kofidis2017joint, castaneda2018vlsi, yilmaz2019channel, he2020model, song2021soft},\nwe then develop two efficient iterative algorithms for approximately solving the optimization \nproblem. The first algorithm is called MAED (short for MitigAtion, Estimation, and Detection) and \nsolves the problem approximately using forward-backward splitting (FBS) \\cite{goldstein16a}. \nThe second algorithm is called SO-MAED (short for Soft-Output MAED) and extends MAED with a more informative \nprior on the data symbols to produce soft symbol estimates. SO-MAED also relies on deep unfolding to optimize its parameters {\\cite{song2021soft, hershey2014deep, balatsoukas2019deep, goutay2020deep, monga2021algorithm}.\nWe use simulations with different propagation models to demonstrate that MAED and SO-MAED effectively mitigate a wide variety of \nna\\\"ive and smart jamming attacks without requiring any knowledge about the attack type.\n\n\\subsection{Notation}\nMatrices and column vectors are represented by boldface uppercase and lowercase letters, respectively.\nFor a matrix~$\\bA$, the transpose is $\\tp{\\bA}$, the conjugate transpose is $\\herm{\\bA}$, \nthe Moore-Penrose pseudoinverse is $\\pinv{\\bA}$,\nthe entry in the $\\ell$th row and $k$th column is $[\\bA]_{\\ell,k}$, \nthe $k$th column is $\\bma_k$,\nthe submatrix consisting of the columns from $n$ through $m$ is $\\bA_{[n:m]}$,\nand the Frobenius norm is $\\| \\bA \\|_F$.\nThe $N\\!\\times\\!N$ identity matrix is $\\bI_N$.\nFor a vector~$\\bma$, the $\\ell_2$-norm is $\\|\\bma\\|_2$, the real part is $\\Re\\{\\bma\\}$, the imaginary part is $\\Im\\{\\bma\\}$, \nand the span is $\\textit{span}(\\bma)$.\nThe $k$th standard unit vector is denoted $\\bme_k$, where the dimension is implicit. \nExpectation with respect to a random vector~$\\bmx$ is denoted by \\Ex{\\bmx}{\\cdot}.\nWe define $i^2=-1$. \nThe complex $n$-hypersphere of radius $r$ is denoted by $\\mathbb{S}_r^n$,\nand~$[n:m]$ are the integers from $n$ through~$m$.\n\n\n\n\\section{System Setup}\\label{sec:setup}\nWe consider the uplink of a massive MU-MIMO system in which $U$ single-antenna UEs transmit data to \na $B$ antenna BS in the presence of a single-antenna jammer. \nThe channels are assumed to be frequency flat and block-fading with coherence time $K=T+D$.\nThe first $T$ time slots are used to \ntransmit pilot symbols; the remaining $D$ time slots are used to transmit data symbols.\nThe UE transmit matrix is $\\bS = [\\bS_T,\\bS_D]$, where $\\bS_T\\in \\opC^{U\\times T}$\nand $\\bS_D\\in\\setS^{U\\times D}$ contain the pilots~and the transmit symbols, respectively. The set $\\setS$ is the transmit constellation, which is normalized to have unit average symbol energy.\nWe assume that the jammer does not prevent the UEs and the BS from establishing synchronization,\nwhich allows us to use the following discrete-time input-output relation:\n\\begin{align}\n\t\\bY = \\bH\\bS + \\Hj\\tp{\\bsj} + \\bN. \\label{eq:io}\n\\end{align}\nHere, $\\bY\\in\\opC^{B\\times K}$ is the BS receive matrix that contains the \\mbox{$B$-dimensional} receive vectors over all $K$ time slots, \n\\mbox{$\\bH\\!\\in\\!\\opC^{B\\times U}$} models the MIMO uplink channel,\n$\\Hj\\in\\opC^B$ models the channel between the jammer and the BS, $\\bsj=\\tp{[\\tp{\\bsj_T},\\tp{\\bsj_D}]}\\in\\opC^K$ contains the jammer transmit symbols over all $K$ time slots, \nand $\\bN\\in\\opC^{B\\times K}$ models thermal noise consisting of independently and identically distributed (i.i.d.) circularly-symmetric complex Gaussian entries with variance~$N_0$.\nUnless stated otherwise, we assume that the jammer's transmit symbols $\\bsj$ are independent of $\\bS$. \nNo other assumptions about the distribution of $\\bsj$ are made;\nin particular, we do not assume that these entries are~i.i.d.\n\nIn what follows, we use plain symbols for the true channels and transmit signals, variables with a tilde for optimization variables, and quantities with a hat for (approximate) solutions to optimization problems, e.g., $\\hat\\bS_D$ is the the estimate of the UE transmit symbol matrix~$\\bS_D$ as determined by solving an optimization problem with respect to $\\tilde\\bS_D$.\n\n\\section{Motivating Example} \\label{sec:example}\nWe start by considering the motivating example of \\fref{fig:example},\nwhich shows uncoded bit error-rates (BERs) of different receivers for an i.i.d. Rayleigh \nfading MU-MIMO system with \\mbox{$B=128$} BS antennas and $U=32$ UEs that transmit 16-QAM symbols under~a jamming attack.\nIn \\fref{fig:example:success} the system is attacked by a barrage jammer that\ntransmits i.i.d. Gaussian symbols and whose \nreceive power exceeds that of the average UE by 30\\,dB. The ``LMMSE'' curve \nshows the performance of a non-mitigating receiver that estimates the UE channel matrix \nbased on orthogonal pilots with a least squares (LS) estimator followed by a linear minimum mean square error (LMMSE) detector. \nThe \\mbox{``JL-LMMSE''} curve shows the performance of an identical receiver but for \na jammerless system without a jamming attack. \nThe ``geniePOS'' receiver is furnished with ground-truth knowledge of the jammer channel~$\\Hj$. This baseline method \nnulls the jammer by orthogonally projecting the receive signals on the orthogonal \ncomplement of $\\textit{span}(\\Hj)$ using the matrix $\\bP_\\Hj = \\bI_B - \\Hj\\pinv{\\Hj}$, \nwhere $\\pinv{\\Hj}=\\herm{\\Hj}\/\\|\\Hj\\|_2^2$,~as\n\\begin{align}\n\t\\bP_\\Hj\\bY \n\t&= \\bP_\\Hj\\,\\bH\\bS + \\bP_\\Hj\\,\\Hj\\tp{\\bsj} + \\bP_{\\Hj}\\,\\bN \\label{eq:pos} \\\\\n\t&= \\bP_\\Hj\\,\\bH\\bS + \\bP_\\Hj\\,\\bN,\n\\end{align}\nsince $\\bP_\\Hj\\,\\Hj=\\mathbf{0}$.\nThe result is an effective jammerless system with receive signal $\\bY_{\\bP} = \\bP_\\Hj \\bY$, \neffective channel matrix \\mbox{$\\bH_{\\bP} = \\bP_\\Hj\\bH$}, and (colored) noise \n$\\bN_\\bP = \\bP_\\Hj \\bN \\sim \\setC\\setN(\\mathbf{0},\\No\\bP_\\Hj)$.\nFinally, geniePOS performs LS channel estimation and subsequent LMMSE data detection in this \nprojected system \\cite{marti2021snips}. \nThe ``POS'' receiver works analogously to geniePOS, except that it is not furnished with ground-truth \nknowledge of the jammer channel---instead, this method estimates the jammer subspace $\\Hj\/\\|\\Hj\\|_2$\nbased on ten receive samples in which the UEs do not transmit and only the jammer is active. \nIf the matrix received in that period is denoted by $\\bY_\\text{J}$, then the jammer subspace is estimated\nas the left-singular vector of the largest singular value of $\\bY_\\text{J}$.\n\\fref{fig:example:success} shows that geniePOS effectively mitigates the jammer, achieving a performance\nvirtually identical to that of the jammer-free \\mbox{JL-LMMSE} receiver. Indeed, geniePOS nulls the jammer \nperfectly, so that the only performance loss comes from the loss of one degree-of-freedom\nin the receive signal. POS is not as effective,~since it nulls the jammer only imperfectly due\nto its noisy estimate of the jammer subspace. However, this method still mitigates the jammer with\na loss of less than 2\\,dB in SNR (at $0.1\\%$ BER) compared to the jammer-free JL-LMMSE receiver.\n\\begin{remark}\nWe point out that reserving time slots for jammer estimation in which the UEs can not transmit\ndirectly reduces the achievable data~rates.\t\n\\end{remark}\n\nContrastingly, in \\fref{fig:example:fail} the attacking (smart) jammer is aware of the POS receiver's mitigation scheme\nand suspends transmission during the time slots that are used to estimate its subspace. \nThe POS receiver's subspace estimate is thus based entirely on noise and is completely independent\nof the jammer's true channel~$\\Hj$. Consequently, the mitigation mechanism fails spectacularly, yielding\na bit error-rate identical to the non-mitigating LMMSE receiver.\n\\begin{figure}[tp]\n\\centering\n\\!\\!\n\\subfigure[mitigation of a barrage jammer]{\n\\includegraphics[height=3.85cm]{figures\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_100Trials_success}\n\\label{fig:example:success}\n}\\!\\!\\!\n\\subfigure[failed mitigation of a smart jammer]{\n\\includegraphics[height=3.85cm]{figures\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_100Trials_fail}\n\\label{fig:example:fail}\n}\\!\\!\n\\caption{Example that illustrates how methods that estimate the jammer's channel based on \na subset of samples fail when facing a smart jammer.}\n\\label{fig:example}\n\\end{figure}\n\n\n\\section{Joint Jammer Estimation and Mitigation, Channel Estimation, and Data Detection}\nThe foregoing example has demonstrated the danger of estimating the jammer's subspace\n(or other characteristics of~the jammer, such as its spatial covariance) \nbased on a certain subset of receive samples when facing a smart jammer.\nWe therefore propose a method that does not depend on the jammer being active during any specific period.\nThis independence is achieved by considering the receive signal over an entire coherence interval \nat once and exploiting the fact that the jammer subspace stays fixed within that period, \nregardless of the jammer's activity pattern or transmit sequence. \nSpecifically, we first propose a novel optimization problem that combines a tripartite goal of \n(i) mitigating the jammer's interference by locating its subspace $\\textit{span}(\\Hj)$\nand projecting the receive matrix $\\bY$ onto the orthogonal complement of~that subspace, \n(ii) estimating the channel matrix~$\\bH$, and (iii) recovering the data matrix $\\bS_D$.\nWe then establish the soundness of the proposed optimization problem by proving that, \nunder certain sensible conditions, and assuming negligible thermal noise, \nits minimum is unique and corresponds to the desired solution; in particular, \nthe problem recovers the data matrix~$\\bS_D$. \nFinally, we develop efficient iterative algorithms that approximately solve the proposed optimization problem.\n\n\n\\subsection{The Optimization Problem}\n\nWe start our derivation by considering the maximum-likelihood problem for joint channel estimation and data detection in the absence of\njamming, which is \\cite{vikalo2006efficient}\n\\begin{align}\n\t \\big\\{\\hat\\bH, \\hat\\bS_D\\big\\}\n\t&= \\argmin_{\\substack{\\hspace{1.3mm}\\tilde\\bH\\in\\opC^{B\\times U}\\\\ \\tilde\\bS_D\\in\\setS^{U\\times D}}}\\!\n\t\\big\\|\\bY - \\tilde\\bH \\tilde\\bS \\big\\|^2_F, \\label{eq:ml_jed}\n\\end{align}\nwhere we define $\\tilde\\bS \\triangleq [\\bS_T,\\tilde\\bS_D]$ for brevity and leave the dependence on\n$\\tilde\\bS_D$ implicit. This objective already integrates the goals of estimating the channel matrix\nand detecting the data symbols: If the noise $\\bN$ is small enough to be negligible, \nthe problem is minimized by the true channel and data matrices,\n\\begin{align}\n\t\\|\\bY - \\bH \\bS \\|^2_F \\approx 0,\n\\end{align}\nwhere the pilot matrix $\\bS_T$ ensures uniqueness.\\footnote{If the noise $\\bN$ \nis not strictly equal to zero, then the channel estimate $\\hat\\bH$ for which \\eqref{eq:ml_jed} is minimized\ndoes not coincide \\emph{exactly} \nwith the true channel matrix~$\\bH$. But thanks to the discrete search space, the minimizing data \nestimate $\\hat\\bS_D$ still coincides exactly with the true data matrix $\\bS_D$ if $\\bN$ is small enough.}\nHowever, in case of a jamming attack, the jammer will cause a residual\n\\begin{align}\n\t\\|\\bY - \\bH\\bS\\|^2_F &= \\|\\Hj\\tp{\\bsj} + \\bN\\|^2_F \\approx \\|\\Hj\\tp{\\bsj}\\|^2_F \\gg 0 \\label{eq:jed_residual}\n\\end{align}\nwhen plugging the true channel and data matrices into \\fref{eq:ml_jed}, and there is no reason to assume that \nthere exists no tuple $\\{\\tilde\\bH,\\tilde\\bS_D\\}$ with $\\tilde\\bS_D\\neq\\bS_D$ such that \n$\\|\\bY - \\tilde\\bH \\tilde\\bS\\|^2_F < \\|\\bY - \\bH\\bS\\|^2_F$.\nNote, however, that the residual $\\Hj\\tp{\\bsj}$ in \\eqref{eq:jed_residual} is a rank-one matrix whose\ncolumns are all in $\\textit{span}(\\Hj)$, regardless of the jamming signal~$\\bsj$.\nConsider therefore what happens when we take the~matrix\\footnote{The dependence\nof $\\tilde\\bP$ on $\\tilde\\bmp$ is left implicit here and throughout the paper.}\n\\begin{align}\n\\tilde\\bP\\triangleq\\bI-\\tilde\\bmp\\herm{\\tilde\\bmp},~\\tilde\\bmp\\in \\mathbb{S}_1^B,\n\\end{align} \nwhich projects a signal \nonto the orthogonal complement of some arbitrary one-dimensional subspace $\\textit{span}(\\tilde\\bmp)$,\nand then apply that projection to the objective of \\eqref{eq:ml_jed} as follows: \n\\begin{align}\n\t\\|\\tilde\\bP(\\bY - \\tilde\\bH\\tilde\\bS)\\|^2_F. \\label{eq:ml_p_jed}\n\\end{align}\nIf we now plug the true channel and data matrices into \\fref{eq:ml_p_jed} (still assuming negligibility of the noise $\\bN$), then we obtain\n\\begin{align}\n\t\\|\\tilde\\bP(\\bY - \\bH\\bS)\\|^2_F \n\t&= \\|\\tilde\\bP\\Hj\\tp{\\bsj} + \\tilde\\bP\\bN\\|^2_F \\\\\n\t&\\approx \\|\\tilde\\bP\\Hj\\tp{\\bsj}\\|^2_F \\geq 0, \n\\end{align}\nwith equality if and only if $\\tilde\\bmp$ is collinear with $\\Hj$. \nIn other words, the unit vector $\\tilde\\bmp$ which in combination with the true channel and data matrices minimizes \\eqref{eq:ml_p_jed}\nis collinear with the jammer's channel, in which case $\\tilde\\bP$ is the POS matrix $\\bP_\\Hj$ from~\\eqref{eq:pos}.\n\nThus, if the noise $\\bN$ is negligible, and if\n(i)~$\\tilde\\bP$ is the projection onto the orthogonal complement of $\\textit{span}(\\Hj)$,\n(ii)~$\\tilde\\bH$ is the true channel matrix,\nand (iii) $\\tilde\\bS$ contains the true data matrix, \nthen the tuple $\\{\\tilde\\bmp,\\tilde\\bH,\\tilde\\bS\\}$ minimizes \\eqref{eq:ml_p_jed}.\nThese~are, of course, exactly the goals which we want to attain.\n %\nWe thus formulate our joint jammer estimation and mitigation, channel estimation, and data detection problem as follows:\n\\begin{align}\n\t \\big\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\big\\}\n\t&= \\argmin_{\\substack{\\tilde\\bmp\\in \\mathbb{S}_1^B\\hspace{1.4mm}\\\\ \\hspace{1.3mm}\\tilde\\bH_\\bP\\in\\opC^{B\\times U}\\\\ \\tilde\\bS_D\\in\\setS^{U\\times D}}}\\!\n\t\\big\\|\\tilde\\bP\\bY - \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F.\\!\n\t\\label{eq:obj1}\n\\end{align}\nNote that, compared to \\eqref{eq:ml_p_jed}, we have absorbed the projection matrix $\\tilde\\bP$ directly into \nthe unknown channel matrix $\\tilde\\bH_\\bP$, which replaces the product $\\tilde\\bP\\tilde\\bH$ in \\eqref{eq:ml_p_jed}. \nThis approach avoids the issue that the columns~of~$\\tilde\\bH$ would be indeterminate with respect to the length of their components in the direction of $\\tilde\\bmp\\approx\\Hj$, \nso that there would be no distinction between channel estimates $\\tilde\\bH + \\Hj\\tp{\\tilde{\\bsj}}$\nwith different jamming sequences~$\\tilde{\\bsj}$.\n\n\\subsection{Theory}\\label{sec:theory}\nWe have derived the optimization problem \\eqref{eq:obj1} based on intuitive but non-rigorous arguments.\nThus, we will now support the soundness of \\eqref{eq:obj1} by proving that, under certain sensible conditions, \nand assuming that the noise is negligible, its solution is unique \nand guaranteed to recover the true data matrix.\n\nWe make the following assumptions: \nThe channel matrix $\\bH$ has full column rank $U$, \n the jammer channel $\\Hj$ is not included in the columnspace of $\\bH$, \nand the pilot matrix $\\bS_T$ has full row rank $U$.\nIn addition, we define a concept which may seem cryptic at first, but which will be clarified later.\n\n\\begin{defi} \\label{def:eclipse}\nWe say that the jammer is \\emph{eclipsed} in a given coherence interval if there exists a matrix $\\tilde{\\bS}_D\\in\\setS^{U\\times D}$\nsuch that \\mbox{$\\text{rank}(\\bS_D - \\tilde{\\bS}_D)\\leq 1$} and $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$.\n\\end{defi}\nWe can now state our result; the proof is in \\fref{app:proof1}.\n\\begin{thm} \\label{thm:maed}\nIn the absence of noise, $\\bN=\\mathbf{0}$, and if the jammer is not eclipsed, then the problem in \\eqref{eq:obj1} has the unique solution $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$. \n(In fact, $\\hat{\\bmp}$ is unique only up to an immaterial complex rotation, $\\hat{\\bmp}=\\alpha\\bmp, |\\alpha|=1$.)\n\\end{thm}\n\n\nIn other words, as long as the jammer is not ecplised, the problem in \\eqref{eq:obj1} is uniquely minimized by the true jammer subspace, projected channel matrix, and data matrix. \nWe now shed light on the notion of eclipsedness, and we answer in the affirmative the important question of whether\none can expect the jammer to typically be un-eclipsed.\nIn essence, the jammer is eclipsed if its jamming signal $\\bmw$ is such that multiple possible ``explanations''\nof the receive signal $\\bY$ exist which are consistent with the pilot matrix $\\bS_T$ \nand under some of which the jammer is not recognized as the jammer; cf. the discussion of \\eqref{eq:eclipsing_equation}\nin \\fref{app:proof1}.\nThis is best explained by considering the two emblematic cases of an eclipsed jammer:\n\\subsubsection{An inactive jammer (or no jammer)} \nClearly, if $\\bsj=\\mathbf{0}$, then $\\tp{\\bsj_D}=\\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ for all $\\tilde{\\bS}_D$, \nincluding \\mbox{$\\tilde{\\bS}_D=\\bS_D$}.~In~this case, there is a mismatch between the jammerless actual wireless transmission and the jammed model in \\eqref{eq:io}. \nSince there is no jammer subspace to identify, the choice of the projection~$\\tilde\\bP$ is undetermined, so that \n\\fref{thm:maed} no longer~applies. \n\n\\subsubsection{The jammer transmits a valid pilot sequence}\nIf the jammer transmits the $k$th UE's pilot sequence in the training phase and constellation symbols\nin the data phase, then there are no formal grounds for the receiver to distinguish between the jammer and the $k$th UE. \nIt can readily be shown that, besides the desired solution \n$\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$, there exists then another solution to~\\eqref{eq:obj1}\nwhich identifies the $k$th UE as the jammer, nulls that UE by setting $\\hat\\bmp = \\bmh_k\/\\|\\bmh_k\\|$, \nand instead identifies the jammer as the $k$th UE by estimating\n\\begin{align}\n\t\\hat\\bH_\\bP &= \\hat\\bP [ \\bmh_1, \\dots, \\bmh_{k-1}, \\Hj, \\bmh_{k+1}, \\dots, \\bmh_U ], \\\\\n\t\\hat\\bS_D &= \\tp{[ \\tp{\\bms_{D,1}}, \\dots, \\tp{\\bms_{D,k-1}}, \\tp{\\bmw_D}, \\tp{\\bms_{D,k+1}}, \\dots, \\tp{\\bms_{D,U}} ]},\n\\end{align}\nwhere $\\bms_{D,u}$ is the $u$th row of $\\bS_D$.\n\nIn addition to these two emblematic cases, eclipsing can also happen in more accidental cases where the symbol\nerror matrix $\\bS_D - \\tilde\\bS_D$ has rank one for some $\\tilde\\bS_D$ such that by some (un)fortunate\ncoincidence $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ holds true. \n\nHowever, we will now show that if the jammer does not know the pilot sequences, e.g., because\nthey are drawn at random by the BS and secretly communicated to the UEs, then an active jammer (where $\\bsj\\neq \\mathbf{0}$)\nis typically not eclipsed. To show this, we consider a case in which the pilot matrix $\\bS_T$ is square; the proof is relegated to \\fref{app:proof2}.\n\n\\begin{thm} \\label{thm:maed2}\n\tIf the pilot matrix $\\bS_T$ is drawn uniformly over the set of $U\\times U$ unitary matrices and if $\\bsj\\neq\\mathbf{0}$ is\n\tindependent of $\\bS_T$,\tthen the probability that the jammer is eclipsed is zero. \n\\end{thm}\n\n\n\\begin{remark} \\label{rem:rare}\n\tIt is by no means necessary to use random pilots to avoid eclipsing. \n\tAnother sufficient condition for the jammer to be eclipsed only with zero probability \n\tis if $\\tp{\\bmw_D}$ is independent of $\\tp{\\bsj_T}\\pinv{\\bS_T}$ and if\n\tat least one of the marginals of $\\tp{\\bmw_D}$ or of $\\tp{\\bsj_T}\\pinv{\\bS_T}$ \n\thas no mass points. \n\tIn essence, unless the jammer choses its input sequence as some (partially randomized) \n\tfunction of the pilot matrix $\\bS_T$, eclipsing is the rare exception, not the norm. \n\tIn this regard, see also the simulation results in \\fref{sec:results}.\n\\end{remark}\n\n\\begin{remark} \nThe fact that error-free communication in the presence of jamming can be assured if \nthe BS and UEs~share~a common secret that enables them to use a randomized~communication scheme, but \nnot otherwise, is reminiscent of~information-theoretic results which prove a similar \ndichotomy on a more fundamental level. See \\cite[Sec. V]{lapidoth1998reliable} and references therein.\n\\end{remark}\n\n\n\\section{Forward-Backward Splitting with a Box Prior}\n\nWe now provide the first of two algorithms for approximately solving the\njoint jammer estimation and mitigation, channel estimation, and data detection problem in \\eqref{eq:obj1}. \nNote first of all that the objective is quadratic in $\\tilde\\bH_\\bP$, \nso we can derive the optimal value of $\\tilde\\bH_\\bP$ as a function of $\\tilde\\bP$ and $\\tilde\\bS$ as\n\\begin{align}\n\t\\hat\\bH_\\bP = \\tilde\\bP\\bY\\pinv{\\tilde\\bS}, \n\\end{align}\nwhere $\\pinv{\\tilde\\bS}=\\herm{\\tilde\\bS}\\inv{(\\tilde\\bS\\herm{\\tilde\\bS})}$. \nSubstituting $\\hat\\bH_\\bP$ back into \\eqref{eq:obj1} yields \nan optimization problem which only depends on $\\tilde\\bmp$ and $\\tilde\\bS_D$:\n\\begin{align}\n\t\\big\\{\\hat\\bmp, \\hat\\bS_D\\big\\} = \n\t\\argmin_{\\substack{\\tilde\\bmp\\in \\mathbb{S}_1^B\\hspace{1.4mm}\\\\ \\tilde\\bS_D\\in\\setS^{D\\times U}}}\n\t\\big\\|\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS)\\big\\|^2_F. \\label{eq:obj3}\n\\end{align}\nSolving \\eqref{eq:obj3} remains difficult due to its combinatorial nature, so we resort to solving it approximately. \nFirst, we relax the constraint set $\\setS$ to its convex hull $\\setC\\triangleq\\textit{conv}(\\setS)$ as in \\cite{castaneda2018vlsi}.\nThis can be viewed as replacing the probability mass function over the constellation\n$\\setS$, which represents the true symbol prior, with a box prior that is uniform \nover $\\setC$ and zero elsewhere \\cite{jeon2021mismatched}. \nWe then approximately solve this~relaxed problem formulation in an iterative fashion by alternating between \na forward-backward splitting descent step in $\\tilde\\bS$ and a minimization step in $\\tilde\\bP$.\n\n\\subsection{Forward-Backward Splitting Step in $\\tilde\\bS$} \\label{sec:fbs}\nForward-backward splitting (FBS) \\cite{goldstein16a}, also called proximal gradient descent,\nis an iterative method for solving convex optimization problems of the form\n\\begin{align}\n\t\\argmin_{\\tilde\\bms}\\, f(\\tilde\\bms) + g(\\tilde\\bms), \\label{eq:fbs1}\n\\end{align}\nwhere $f$ is convex and differentiable, and $g$ is convex but not necessarily\ndifferentiable, smooth, or bounded. Starting from an initialization vector $\\tilde\\bms^{(0)}$, \nFBS solves the problem in~\\eqref{eq:fbs1} iteratively by computing \n\\begin{align}\n\t\\tilde\\bms^{(t+1)} = \\proxg\\big(\\tilde\\bms^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bms^{(t)}); \\tau^{(t)}\\big). \\label{eq:fbs2}\n\\end{align}\nHere, $\\tau^{(t)}$ is the stepsize at iteration $t$, $\\nabla f(\\tilde\\bms)$ is the gradient~of $f(\\tilde\\bms)$, \nand $\\proxg$ is the proximal operator of $g$, defined as \\cite{parikh13a}\n\\begin{align}\n\t\\proxg(\\bmx, \\tau) = \\argmin_{\\tilde\\bmx} \\tau g(\\tilde\\bmx) + \\frac12 \\|\\bmx - \\tilde\\bmx\\|_2^2.\n\\end{align}\nFor a suitable sequence of stepsizes $\\{\\tau^{(t)}\\}$, FBS solves convex optimization problems exactly.\nFBS can also be used to approximately solve non-convex\nproblems, although there are typically no guarantees for optimality or even convergence~\\cite{goldstein16a}.\nFor the optimization problem in \\fref{eq:obj3}, we define $f$ and $g$ as \n\\begin{align}\n\tf(\\tilde\\bS) &= \\big\\|\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS)\\big\\|^2_F\n\\end{align}\nand\n\\begin{align}\n\tg(\\tilde\\bS) &= \\begin{cases}\n\t\t0 &\\text{if }\\,\\tilde\\bS_{[1:T]}=\\bS_T \\text{ and } \\tilde\\bS_{[T+1:K]}\\in\\setC^{U\\times D}\n\t\t\\!\\!\\!\\\\\n\t\t\\infty &\\text{else}.\n\t\\end{cases}\n\\end{align}\nThe gradient of $f$ in $\\tilde\\bS$ is given by \n\\begin{align}\n\t\\nabla f(\\tilde\\bS) = -\\herm{(\\bY\\pinv{\\tilde\\bS})}\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS), \\label{eq:gradient}\n\\end{align}\nand the proximal operator for $g$ is simply the orthogonal projection onto $\\setC$, which \nacts entrywise on $\\tilde\\bS$~as\n\\begin{align}\n\t[\\proxg(\\tilde\\bS)]_{u,k} = \\begin{cases}\n\t\t[\\bS_T]_{u,k} &\\text{ if } k\\in[1:T] \\\\\n\t\t\\text{proj}_\\setC([\\tilde\\bS_{u,k}]) &\\text{ else,}\n\t\\end{cases} \\label{eq:proxg} \n\\end{align}\nwhere the function $\\text{proj}_\\setC$ is given as \n\\begin{align}\n\t\\text{proj}_\\setC(x) =\\, & \\min\\{\\max\\{\\Re(x),-\\lambda\\},\\lambda\\} \\nonumber\\\\\n\t&+ i\\min\\{\\max\\{\\Im(x),-\\lambda\\},\\lambda\\},\n\\end{align}\nwith $\\lambda=\\sqrt{\\sfrac{1}{2}}$ for a QPSK constellation \nand $\\lambda=\\sqrt{\\sfrac{9}{10}}$ for a 16-QAM constellation, see \\fref{fig:constellations}. \nTo select the per-iteration stepsizes~$\\{\\tau^{(t)}\\}$, we use the Barzilai-Borwein method \n\\cite{barzilai1988two}. \n\n\\subsection{Minimization Step in $\\tilde\\bP$}\nAfter each FBS step in $\\tilde\\bS$, we minimize \\eqref{eq:obj3}\nwith respect to the vector~$\\tilde\\bmp$. Defining the residual matrix\n$\\tilde\\bE\\triangleq \\bY(\\bI_K - \\pinv{\\tilde{\\bS}}\\tilde{\\bS})$\nand performing standard algebraic manipulations yields\n\\begin{align}\n\t\\hat\\bmp &= \\argmin_{\\tilde\\bmp\\in \\mathbb{S}_1^B} \\big\\|\\tilde\\bP \\tilde\\bE \\big\\|^2_F\t\\\\\n\t&= \\argmax_{\\tilde\\bmp\\in \\mathbb{S}_1^B} \\, \\herm{\\tilde\\bmp} \\tilde\\bE \\herm{\\tilde\\bE} \\tilde\\bmp. \\label{eq:rayleigh}\n\\end{align}\nIt follows that the vector $\\hat\\bmp$ minimizing \\eqref{eq:obj3} for a fixed~$\\tilde\\bS$ is the unit vector that maximizes the \nRayleigh quotient of $\\tilde\\bE \\herm{\\tilde\\bE}$. \nThe~solution is the unit-length eigenvector \n$\\bmv_1(\\tilde\\bE \\herm{\\tilde\\bE})$ associated with the largest eigenvalue of $\\tilde\\bE \\herm{\\tilde\\bE}$~\\cite[Thm.\\,4.2.2]{horn2013matrix},\n\\begin{align}\n\t\\hat\\bmp=\\bmv_1(\\tilde\\bE \\herm{\\tilde\\bE}).\n\\end{align}\nCalculating this eigenvector in every iteration of our algorithm would be computationally expensive, \nso we approximate it using a single power iteration \\cite[Sec.\\,8.2.1]{GV96}, i.e., \nwe estimate \n\\begin{align}\n\t\\hat\\bmp^{(t+1)} = \\frac{\\tilde\\bE^{(t+1)} \\herm{(\\tilde\\bE^{(t+1)})}\\hat\\bmp^{(t)}}{\\|\\tilde\\bE^{(t+1)} \\herm{(\\tilde\\bE^{(t+1)})}\\hat\\bmp^{(t)}\\|_2},\n\\end{align}\nwhere the power method is initialized with the subspace estimate $\\hat\\bmp^{(t)}$ from the previous algorithm iteration.\n\n\\subsection{Preprocessing}\nIf the algorithm starts directly with a gradient descent step in the direction of \\eqref{eq:gradient}, \none runs the risk of advancing significantly into the wrong direction---especially if the jammer is extremely strong,\nsince a strong jammer will also lead to a large gradient amplitude. Empirically, we observe that such a large initial digression \ncan be problematic (if, e.g., the jammer is $\\geq\\!50$\\,dB stronger than the average UE).\nIt might therefore be tempting to start the algorithm directly with a projection step: \nIf one initializes $\\tilde\\bS^{(0)}=\\mathbf{0}_{U\\times D}$, then $\\tilde\\bE^{(0)}=\\bY$, so that the algorithm starts by \nnulling the dimension of $\\bY$ which contains the most energy. In the presence of a strong jammer, this is a sensible strategy\nsince this dimension then corresponds to the jammer subspace. However, if the received jamming energy is \nsmall compared to the energy received from the UEs (e.g., because the jammer does not transmit at all during a given coherence interval), \nthen such a projection would inadvertently null the strongest user. \nTo thread the needle between these two cases---largely removing a strong jammer before the first gradient step, \nbut not removing any legitimate UEs when a strong jammer is absent---we propose to start with a \\emph{regularized} projection step:\nThe algorithm starts by a projection onto the orthogonal complement of the eigenvector of the largest eigenvalue\nof \n\\begin{align}\n\t\\bY\\herm{\\bY} + \\mathbf{\\Gamma}, \\label{eq:regularizer}\n\\end{align}\nwhere $\\mathbf{\\Gamma}\\in\\opC^{B\\times B}$ is a constant regularization matrix. The basic idea is that this regularization matrix is\nstill overshadowed by very strong jammers, so that these are largely nulled within the preprocessing, \nwhile, in the presence of only a weak jammer (or no jammer), the regularization matrix has a sufficiently diverting impact \non the eigenvectors to prevent the nulling of a legitimate UE. \nThere are countless ways of choosing such a regularization matrix. (Note, however, that $\\mathbf{\\Gamma}$ should \nnot be a multiple of the identity matrix $\\bI_B$, which does not affect the eigenvectors of \\eqref{eq:regularizer}.) For simplicity, we set $\\mathbf{\\Gamma}$ to the all-zero matrix, except for \nthe top left entry, which is set to $0.1BUK$. \n\n\\subsection{Algorithm Complexity} \nWe now have all the ingredients for MAED, which is summarized in \\fref{alg:maed}. Its only input is the receive matrix~$\\bY$, \nas it does not even require an estimate of the thermal noise variance $\\No$.\nMAED is initialized with $\\tilde\\bS^{(0)}=[\\bS_T, \\mathbf{0}_{U\\times D}]$ and $\\tau^{(0)}=\\tau=0.1$, \nand runs for a fixed number of $t_{\\max}$~iterations.\n\nThe complexity of MAED is dominated by the eigenvector calculation in the preprocessing step\n(which could be reduced by approximating it using the power method) as well \nas the gradient computation in line 5 of \\fref{alg:maed}, \nwhich has a complexity of $O(3BUK+2U^2K+U^3)$.\nThe overall complexity of MAED is therefore \\mbox{$O(t_{\\max}(3BUK+2U^2K+U^3))$.}\nNote, however, that MAED detects $D$ data vectors at once.\nThus, the computational complexity per detected symbol is only \\mbox{$O(t_{\\max}(3BK+2KU+U^2)\/D)$.}\n\n\n\\section{Soft-Output Estimates with Deep Unfolding}\nMAED, wich corresponds to the algorithm proposed in~\\cite{marti2022smart} (newly adding the preprocessing step), \nalready attains the goal of mitigating smart jammers, see \\fref{sec:results}. \nHowever, its detection performance can be suboptimal, especially when higher-order transmit constellations \nsuch as 16-QAM are used. \nThe culprit is the box prior of MAED, which does not fully exploit the \ndiscrete nature of the transmit constellation. In particular, the box prior is uninformative about \nthe constellation symbols in the interior.\nTo improve detection performance, especially in the cases where MAED performs suboptimally, \nwe now provide a second algorithm for approximately solving the problem \nin \\eqref{eq:obj1}. This second algorithm builds on MAED but replaces the proximal operator in \\eqref{eq:proxg}, which enforces\nMAED's box prior, by an approximate posterior mean estimator (PME) based on the discrete symbol prior as in \\cite{song2021soft}. \nSince the PME also enables meaningful soft-output estimates of the bits that underlie the transmitted data symbols, we refer to \nthis second algorithm as soft-output MAED (SO-MAED).\n\n\\begin{algorithm}[tp]\n\\caption{MAED}\n\\label{alg:maed}\n\\begin{algorithmic}[1]\n\\setstretch{1.0}\n\\State {\\bfseries input:} $\\bY$\n\\State \\text{initialize} $\\tilde\\bS^{(0)} \\!=\\! \\left[\\bS_T, \\mathbf{0}_{U\\!\\times\\! D} \\right]\\!, \n\\tilde\\bmp^{(0)} \\!=\\! \\bmv_1(\\bY \\herm{\\bY} \\!+ \\mathbf{\\Gamma} ), \\tau^{(0)} \\!= \\tau$\n\\State $\\tilde\\bP^{(0)} = \\bI_B - \\tilde\\bmp^{(0)}\\tilde\\bmp^{(0)}{}^\\text{H}$\n\\For{$t=0$ {\\bfseries to} $t_{\\max}-1$}\n\t\\State $\\nabla f(\\tilde\\bS^{(t)}) = -\\herm{\\big(\\bY\\tilde\\bS^{(t)}{}^\\dagger\\big)} \\tilde\\bP^{(t)}\\bY(\\bI_K - \\tilde\\bS^{(t)}{}^\\dagger \\tilde\\bS^{(t)})$\n\t\\State $\\tilde\\bS^{(t+1)} = \\proxg\\big(\\tilde\\bS^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bS^{(t)})\\big)$ \n\t\\State $\\tilde\\bE^{(t+1)} = \\bY(\\bI_K - \\tilde\\bS^{(t+1)}{}^\\dagger \\tilde\\bS^{(t+1)})$\n\t\\State $\\tilde\\bmp^{(t+1)} = \\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\/\\|\\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\\|_2$\n\t\\State $\\tilde\\bP^{(t+1)} = \\bI_B - \\tilde\\bmp^{(t+1)}\\tilde\\bmp^{(t+1)}{}^\\text{H}$\n\t\\State compute $\\tau^{(t+1)}$ by following \\cite[Sec.\\,4.1]{goldstein16a}\n\t\\EndFor\n\t\\State \\textbf{output:} $\\tilde\\bS^{(t_{\\max})}_{[T+1:K]}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Approximate Posterior-Mean Estimation}\n\n\nTo replace the proximal operator following the gradient descent step in \\eqref{eq:fbs2}\nwith a more appropriate data symbol estimator which takes into account the discrete constellation~$\\setS$,\nwe model the per-iteration outputs of the gradient descent step\n\\begin{align}\n\t\\tilde\\bX^{(t)} = \\tilde\\bS^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bS^{(t)})\n\\end{align} \nas \n\\begin{align}\n\t\\tilde\\bX^{(t)} &= \\bS + \\bZ^{(t)} = \\big[ \\bS_T, \\bS_D \\big] + \\big[ \\bZ_T^{(t)}, \\bZ_D^{(t)} \\big], \\label{eq:symbol_noise}\n\\end{align}\ni.e., as the true transmit symbol matrix $\\bS$ corrupted by an additive error~$\\bZ^{(t)}$.\nIf the distribution of $\\bZ^{(t)}$ were known, \none could compute the posterior mean $\\mathbb{E}[\\bS\\,|\\,\\tilde\\bX^{(t)}]$\nand use it as a constellation-aware replacement of the proximal step \\eqref{eq:proxg}, \n\\begin{align}\n\t\\tilde\\bS^{(t+1)} = \\mathbb{E}\\big[\\bS\\,|\\,\\tilde\\bX^{(t)}\\big].\n\\end{align}\nUnfortunately, the distribution of $\\bZ^{(t)}$ is unknown in practice, but the~submatrix $\\bS_T$ of $\\bS$ (and hence its mean)\t\nis known at the receiver. To estimate the mean of the submatrix $\\bS_D$, we assume that\nthe entries of $\\bZ_D^{(t)}$ are distributed independently~of $\\bS$ as i.i.d.\\ circularly-symmetric \ncomplex Gaussians with variance~$\\nu^{(t)}$, \n\\begin{align}\n\t\\big[\\bZ^{(t)}\\big]_{u,k}\\sim\\setC\\setN(0,\\nu^{(t)}).\n\\end{align}\nThe variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$ are treated as algorithm parameters\nand will be optimized using deep unfolding as detailed below. \n\nBased on this idealized model, we utilize a three-step procedure as in \\cite{song2021soft} for computing \nsymbol estimates. First, we use~\\fref{eq:symbol_noise} to compute log-likelihood ratios (LLRs)\nfor every transmitted bit. We then convert these LLRs to the probabilities of the respective bits being $1$. \nThis step also provides the aforementioned soft-output estimates. \nFinally, we convert the bit probabilities back to symbol estimates by calculating the symbol mean. \n\\mbox{The LLRs can be computed following \\cite{collings2004low, jeon2021mismatched} as}\n\\begin{align}\n\tL_{i,u,k}^{(t)} = \\frac{\\ell\\big(\\tilde X_{u,k}^{(t)}\\big)\\!}{\\nu^{(t)}}, \n\t~ i\\!\\in\\![1\\!:\\!\\log_2\\!|\\setS|],~ u\\!\\in\\![1\\!:\\!U],~ k\\!\\in\\![T\\!+\\!1\\!:\\!K],\n\t\t \\label{eq:llrs}\n\\end{align}\nwhere $\\ell(\\cdot)$ is specified in Table~\\ref{tab:llr} (cf.~\\fref{fig:constellations}).\nThe LLR values are exact for QPSK and use the max-log \napproximation for 16-QAM~\\cite{jeon2021mismatched}.\nThe LLRs can then be converted to probabilities~via\n\\begin{align}\n\tp_{i,u,k}^{(t)} = \\frac12\\left( 1 + \\tanh{\\left(\\frac{L_{i,u,k}^{(t)}}{2}\\right)} \\right).\n\t\\label{eq:bit_probabs}\n\\end{align}\nFinally, the probabilities of \\eqref{eq:bit_probabs} can be used to compute symbol estimates according to Table~\\ref{tab:symbol_estimates}.\n\n\nTo summarize, SO-MAED replaces MAED's proximal operator in \\eqref{eq:proxg}\nwith the symbol estimator that consists of \\eqref{eq:llrs}, \\eqref{eq:bit_probabs}, and Table~\\ref{tab:symbol_estimates}.\nWe refer to this symbol estimation as posterior-mean approximation (PMA) and denote it as \n\\begin{align}\n\t\\tilde\\bS^{(t+1)} = \\pma(\\tilde\\bX^{(t)},\\nu^{(t)}),\t\t\n\\end{align}\nwhere the subscript $\\setS$ makes explicit the dependence of the PMA on the symbol constellation.\nSince the PMA involves only scalar computations, its complexity is negligible compared to the\nmatrix-vector and matrix-matrix operations of SO-MAED. The complexity order of SO-MAED is \ntherefore identical to that of MAED, namely \\mbox{$O(t_{\\max}(3BUK+2U^2K+U^3))$.}\n\n\n\\subsection{Deep Unfolding of SO-MAED}\nThe procedure outlined in the previous subsection requires the variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$ \nof the per-iteration estimation errors~$\\bZ^{(t)}$, which are generally unknown. We treat these variances \nas parameters of SO-MAED and optimize them using deep unfolding \\cite{song2021soft, hershey2014deep, balatsoukas2019deep, goutay2020deep, monga2021algorithm}.\nDeep unfolding is an emerging paradigm under which iterative algorithms are unfolded into artificial neural networks\nwith one layer per iteration, so that the algorithm parameters can be\nregarded as trainable weights of that network. These weights are then learned from training data with \nstandard deep learning tools~\\cite{Baydin2018automatic, tensorflow2015whitepaper}. \n\n\nTo improve stability of learning, we use the error precisions $\\{\\rho^{(t)}\\}_{t=0}^{t_{\\max}-1}$ \ninstead of the variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$\nas parameters of the unfolded network, with $\\rho^{(t)}=1\/\\nu^{(t)}$.\nIn addition, we also regard the gradient step sizes $\\{\\tau^{(t)}\\}_{t=0}^{t_{\\max}-1}$\nas trainable weights (instead of computing them according to the Barzilai-Borwein method). \nFurthermore, we add a momentum term with per-iteration weights $\\{\\lambda^{(t)}\\}_{t=0}^{t_{\\max}-1}$ to our gradient descent procedure.\nFinally, inspired by the Bussgang decomposition \\cite{bussgang52a, minkoff85a}, we add per-iteration scale factors \n$\\{\\alpha^{(t)}\\}_{t=0}^{t_{\\max}-1}$ to the output of \\eqref{eq:symbol_noise}, \nwith the goal of accommodating uncorrelatedness (if not independence) between $\\bZ_D^{(t)}$ and $\\bS_D$ in \\eqref{eq:symbol_noise}.\nThe final algorithm is summarized in \\fref{alg:so_maed}.\n\n\nWe implement the unfolded algorithm in TensorFlow \\cite{tensorflow2015whitepaper} and train it \nusing as cost function the empirical binary cross-entropy (BCE) between the transmitted bits \nand the estimated bit probabilities \\eqref{eq:bit_probabs} from the last iteration as the output of our network.\nThe loss function is given as\n\\begin{align}\n\t\\sum_{\\text{sample}\\in\\setD} \\beta_{\\text{sample}} \\!\\left( \\sum_{i=1}^{\\log_2|\\setS|} \\sum_{u=1}^U \\sum_{k=T+1}^K \\text{BCE}(b_{i,u,k}, p_{i,u,k}^{t_{\\max}}) \\right)\\!\\!,\\!\n\\end{align}\nwhere \n\\begin{align}\n\t\\text{BCE}(b,p(b)) = b \\log_2(p(b)) + (1-b)\\log_2(1-p(b)),\n\\end{align}\nand where $\\beta_{\\text{sample}}$ are the weights given to the different samples in the training set $\\setD$.\nWe only learn a single set of weights per system dimensions $\\{U,B,K\\}$, which is used for all \nsignal-to-noise ratios (SNRs) and, most importantly, \nall jamming attacks (since a receiver does not typically know in advance\nwhich type of a jamming attack it is facing). For this reason, we train using samples from different SNRs and \ndifferent jamming attacks. \nWe also have to avoid overfitting to a specific type of jamming attack.\nIf our evaluation in \\fref{sec:results} would feature only the exact same types of jammers \nthat were used for training, this would raise questions about the ability of SO-MAED to generalize to \njamming attacks which differ from those explicitly included in the training set.\nHowever, the principles underlying the SO-MAED algorithm are essentially invariant with respect to the type of a \njamming attack. For this reason, we only train on a single type of jammers, namely pilot jammers, cf. \\fref{sec:setup} (which we have empirically recognized to be the most difficult to mitigate), while evaluating the trained algorithm on many other jammer types besides pilot jammers, cf. \\fref{sec:results}.\n The attacks used for training also comprise different jammer receive strengths,\nnamely $\\{-\\infty\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$ \ncompared to the average UE. \n\\begin{figure}[tp]\n\\centering\n\n\\subfigure[QPSK\\hspace{1cm}]{\n\\includegraphics[height=3.4cm]{figures\/constellation_qpsk}\n\\label{fig:constellations:qpsk}\n}\n\\subfigure[16-QAM\\hspace{1cm}]{\n\\includegraphics[height=3.4cm]{figures\/constellation_16qam}\n\\label{fig:constellations:16qam}\n}\n\\caption{Transmit constellations $\\setS$ (including the used Gray mapping) and their convex hulls $\\setC$. \n$\\lambda=\\sqrt{\\sfrac{1}{2}}$ for QPSK and $\\lambda=\\sqrt{\\sfrac{9}{10}}$ for 16-QAM.\n}\n\\label{fig:constellations}\n\\end{figure}\n\\begin{table}[tp!]\n\\centering\n\\caption{LLR computation according to \\cite[Tbl. 1]{collings2004low}, \\cite{jeon2021mismatched}}\n\\vspace{-5mm}\n\\setstretch{1.1}\n\\begin{tabular}[t]{@{}lcl@{}}\n\\toprule\n&\\!\\!Bit $i$\\!\\!& $\\ell(x)$ \\\\\n\\midrule\n\\multirow{2}{*}{\\!QPSK\\!}& $1$ & $4\\lambda\\Re\\{x\\}$ \\\\ \n& $2$ & $4\\lambda\\Im\\{x\\}$ \\vspace{1mm} \\\\\n\\multirow{4}{*}{\\!16-QAM\\!}& $1$ & $\\frac{2\\lambda}{3} \\left( 4\\Re\\{x\\} + \\left|\\Re\\{x\\} \\!-\\! \\frac{2\\lambda}{3} \\right| - \\left|\\Re\\{x\\} \\!+\\! \\frac{2\\lambda}{3} \\right| \\right)\\!$ \\\\\n& $2$ & $\\frac{4\\lambda}{3} \\left( \\frac{2\\lambda}{3} - \\left|\\Re\\{x\\} \\right| \\right) $ \\\\\n& $3$ & $\\frac{2\\lambda}{3} \\left( 4\\Im\\{x\\} + \\left|\\Im\\{x\\} \\!-\\! \\frac{2\\lambda}{3} \\right| - \\left|\\Im\\{x\\} \\!+\\!\\frac{2\\lambda}{3} \\right| \\right)\\! $ \\\\\n& $4$ & $\\frac{4\\lambda}{3} \\left( \\frac{2\\lambda}{3} - \\left|\\Im\\{x\\} \\right| \\right) $ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{5mm}\n\\label{tab:llr}\n\\centering\n\\caption{Mapping the probabilities in \\eqref{eq:bit_probabs} to symbol estimates\n\\cite{jeon2021mismatched, tomasoni2006low}}\n\\vspace{-5mm}\n\\setstretch{1.1}\n\\begin{tabular}[t]{@{}lcc@{}}\n\\toprule\n& $\\Re\\{\\hat{s}\\}$\\!& $\\Im\\{\\hat{s}\\}$ \\\\\n\\midrule\nQPSK & $\\lambda(2p_{1}-1)$ & $\\lambda(2p_{2}-1)$ \\\\\n16-QAM& $\\lambda(2p_1-1)(3-2p_2)$ & $\\lambda(2p_3-1)(3-2p_4)$ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\label{tab:symbol_estimates} \n\\end{table\nAlso with regard to the jammer receive strength, the evaluation in \\fref{sec:results} will consider jammers with different strengths\nthan have been used for training.\nThe sample weights $\\beta_{\\text{sample}}$ are used to prevent certain training samples (e.g., those at low SNR with strong jammers)\nfrom dominating the learning process by drowning out the loss contribution from training samples with inherently lower BCE. \nFor this, we fix a ``baseline performance'' and select the weight of a training sample as the inverse of this sample's BCE loss \naccording to the baseline. The baseline is set by an untrained version of SO-MAED with reasonably initialized weights\n(its performance in general already exceeds that of~MAED).\nFor training, we use the Adam optimizer \\cite{kingma2014adam} from Keras with default values. The batch size \nstarts at one sample, but is increased first to five and then to ten samples whenever the training loss does not improve for two consecutive\nepochs.\n \n \n\\begin{algorithm}[tp]\n\\caption{SO-MAED}\n\\label{alg:so_maed}\n\\begin{algorithmic}[1]\n\\setstretch{1.0}\n\\State {\\bfseries input:} $\\bY, \\{\\tau^{(t)}, \\alpha^{(t)}, \\lambda^{(t)}, \\rho^{(t)} \\}_{t=0}^{t_{\\max} -1}$\n\\State \\text{init} $\\tilde\\bS^{(0)} \\!=\\! \\left[\\bS_T, \\mathbf{0}_{U\\!\\times\\! D} \\right], \n\\tilde\\bmp^{(0)} \\!=\\! \\bmv_1(\\bY \\herm{\\bY} \\!+ \\mathbf{\\Gamma} ), \\mathbf{\\Delta}^{(-1)} = \\mathbf{0}$\n\\State $\\tilde\\bP^{(0)} = \\bI_B - \\tilde\\bmp^{(0)}\\tilde\\bmp^{(0)}{}^\\text{H}$\n\\For{$t=0$ {\\bfseries to} $t_{\\max}-1$}\n\t\\State $\\nabla f(\\tilde\\bS^{(t)}) = -\\herm{\\big(\\bY\\tilde\\bS^{(t)}{}^\\dagger\\big)} \\tilde\\bP^{(t)}\\bY(\\bI_K - \\tilde\\bS^{(t)}{}^\\dagger \\tilde\\bS^{(t)})$\n\t\\State $\\mathbf{\\Delta}^{(t)} = - \\tau^{(t)} \\nabla f(\\tilde\\bS^{(t)}) + \\lambda^{(t)} \\mathbf{\\Delta}^{(t-1)}$\n\t\\State $\\tilde\\bX^{(t)} = \\tilde\\bS^{(t)} + \\mathbf{\\Delta}^{(t)} $\n\t\\State $\\tilde\\bS^{(t+1)} = \\pma\\big( \\alpha^{(t)}\\tilde\\bX^{(t)}, 1\/\\rho^{(t)} \\big)$ \n\t\\State $\\tilde\\bE^{(t+1)} = \\bY(\\bI_K - \\tilde\\bS^{(t+1)}{}^\\dagger \\tilde\\bS^{(t+1)})$\n\t\\State $\\tilde\\bmp^{(t+1)} = \\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\/\\|\\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\\|_2$\n\t\\State $\\tilde\\bP^{(t+1)} = \\bI_B - \\tilde\\bmp^{(t+1)}\\tilde\\bmp^{(t+1)}{}^\\text{H}$\n\t\\EndFor\n\t\\State \\textbf{output:} $\\tilde\\bS^{(t_{\\max})}_{[T+1:K]}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\section{Simulation Results} \\label{sec:results}\n\\subsection{Simulation Setup} \\label{sec:setup}\nWe simulate a massive MU-MIMO system with $B=128$~BS antennas, \n$U=32$ single-antenna UEs, and one single-antenna jammer. \nThe UEs transmit for $K=160$ time slots, where the first\n$T=32$ slots are used for orthogonal pilots $\\bS_T$ in the form of a\n $32\\times32$ Hadamard matrix with unit symbol energy.\nThe remaining $D=128$ slots are used to transmit QPSK or 16-QAM payload data.\nUnless noted otherwise, the channels are modelled as i.i.d. Rayleigh fading. \nWe define the average receive signal-to-noise ratio (SNR) as \n\\begin{align}\n\\textit{SNR} \\define \\frac{\\Ex{\\bS}{\\|\\bH\\bS\\|_F^2}}{\\Ex{\\bN}{\\|\\bN\\|_F^2}}.\n\\end{align}\nWe consider four different jammer types: \n(J1) barrage jammers that transmit i.i.d.\\ jamming symbols during the entire coherence interval,\n(J2) pilot jammers that transmit i.i.d.\\ jamming symbols during the pilot phase but do not jam the data phase, \n(J3) data jammers that transmit i.i.d.\\ jamming symbols during the data phase but do not jam the pilot phase, \nand (J4) sparse jammers that transmit i.i.d.\\ jamming symbols during some fraction $\\alpha$ of randomly selected bursts of unit length (i.e., one time slot).\nThe jamming symbols are either circularly symmetric complex Gaussian or drawn uniformly from the transmit constellation \n(i.e., QPSK or 16-QAM).\nThey are also independent of the UE transmit symbols $\\bS$, unless stated otherwise.\nWe quantify the strength of the jammer's interference relative to the strength of the average UE, either as the ratio \nbetween total receive~\\textit{energy}\n\\begin{align}\n\\rE \\define \\frac{\\Ex{\\bsj}{\\|\\Hj\\bsj\\|_2^2}}{\\frac1U\\Ex{\\bS}{\\|\\bH\\bS\\|_2^2}},\n\\end{align}\nor as the ratio between receive \\textit{power during those phases that the jammer is jamming}\n\\begin{align}\n\t\\rP \\triangleq \\frac{\\rE}{\\gamma},\n\\end{align}\nwhere $\\gamma$ is the jammer's duty cycle and equals $1,\\frac{T}{K},\\frac{D}{K}$, or~$\\alpha$\nfor barrage, pilot, data, or sparse jammers, respectively. This allows us to either precisely \ncontrol the jammer energy (for jammers which are assumed to be essentially energy-limited)\nor the transmit intensity (for jammers which may want to pass themselves off as a legitimate UE, for instance).\n\n\n\\subsection{Performance Baselines} \\label{sec:baseline}\nWe set the number of iterations for MAED and \\mbox{SO-MAED} to $t_{\\max}=20$ and \nemphasize again that we use only two different sets of weights for \\mbox{SO-MAED}: one for QPSK transmission and one\nfor 16-QAM transmission. Neither \\mbox{SO-MAED} nor MAED is adapted to the different jammer scenarios.\nWe compare our algorithms to the following baseline methods: \nThe first baseline is the ``LMMSE'' method from \\fref{sec:example}, which does not mitigate the jammer and \nseparately performs least-squares (LS) channel estimation and LMMSE data detection.\nThe second baseline is the ``geniePOS'' method from \\fref{sec:example}, which projects the receive\nsignals onto the orthogonal complement of the true jammer subspace \nand then separately performs LS channel estimation and LMMSE data detection in this projected space.\nThe last baseline, \\mbox{``JL-SIMO,''} serves as a lower bound for attainable error-rate performance. \nThis method operates in a jammerless but otherwise equivalent system and implements (with perfect channel knowledge) \nthe single-input multiple-output (SIMO) bound\ncorresponding to the idealized case in which no inter-user interference is present. \n\n\n\\begin{figure*}[tp]\n\\!\\!\\!\\!\\!\n\\subfigure[strong barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_barrage_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:static}\n}\\!\\!\n\\subfigure[strong pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_pilot_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:pilot}\n}\\!\\!\n\\subfigure[strong data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_data_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:data}\n}\\!\\!\n\\subfigure[strong sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_sparse_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{QPSK} transmission in the presence of a \\emph{strong} ($\\rE=30$\\,dB) jammer which\ntransmits Gaussian symbols (a) during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:qpsk_strong_jammers}\n\\end{figure*}\n\\begin{figure*}[tp]\n\\vspace{-1mm}\n\\!\\!\\!\\!\\!\n\\subfigure[strong barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:static}\n}\\!\\!\n\\subfigure[strong pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_pilot_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:pilot}\n}\\!\\!\n\\subfigure[strong data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_data_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:data}\n}\\!\\!\n\\subfigure[strong sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_sparse_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{16-QAM} transmission in the presence of a \\emph{strong} ($\\rE\\!=\\!30$\\,dB) jammer which transmits Gaussian symbols~(a)~during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:strong_jammers}\n\\end{figure*}\n\\begin{figure*}[h!]\n\\vspace{-1mm}\n\\!\\!\\!\\!\\!\n\\subfigure[weak barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_barrage_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:static}\n}\\!\\!\n\\subfigure[weak pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_pilot_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:pilot}\n}\\!\\!\n\\subfigure[weak data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_data_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:data}\n}\\!\\!\n\\subfigure[weak sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_sparse_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{16-QAM} transmission in the presence of a \\emph{weak} ($\\rP=0$\\,dB) jammer\nwhich transmits 16-QAM symbols (a) during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:weak_jammers}\n\\end{figure*}\n\n\\begin{figure*}[tp]\n\\!\\!\\!\\!\\!\n\\subfigure[barrage jammers (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_barrage_jammers\/128x32_16QAM_I1_D128_barrage_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:static}\n}\\!\\!\n\\subfigure[pilot jammers (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_pilot_jammers\/128x32_16QAM_I1_D128_pilot_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:pilot}\n}\\!\\!\n\\subfigure[data jammers (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_data_jammers\/128x32_16QAM_I1_D128_data_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:data}\n}\\!\\!\n\\subfigure[sparse jammers (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_sparse_jammers\/128x32_16QAM_I1_D128_sparse_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) performance curves of SO-MAED in the presence of jammers with different receive powers\ncompared to the average UE,\n$\\rP\\in\\{-20\\,\\text{dB}, -10\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$. \nThe subfigures correspond to the different jammer types (J1)\\,-\\,(J4) and show one curve \nper~jammer power (plotted with 25\\% opacity to depict the degree of overlap between curves). \nCurves that level off into an error floor are labeled with their jammer power, e.g.,\nin \\fref{fig:many:static}, the barrage jammer with receive power $\\rP=-20$\\,dB has an error floor while all other barrage jammers\nhave virtually identical BER curves.\n}\n\\label{fig:many_jammers}\n\\end{figure*}\n\n\\subsection{Mitigation of Strong Gaussian Jammers}\nWe first investigate the ability of MAED and SO-MAED to mitigate strong jamming attacks. \nFor this, we simulate Gaussian jammers with \\mbox{$\\rE=30$\\,dB} of all four types introduced in Section~\\ref{sec:setup} \nand evaluate the performance of our algorithms compared to the baselines of Section~\\ref{sec:baseline} for QPSK transmission (\\fref{fig:qpsk_strong_jammers}) \nas well as for 16-QAM transmission (\\fref{fig:strong_jammers}).\nWe note at this point that the performances of geniePOS and JL-SIMO are independent of the considered jammer type: geniePOS uses the genie-provided\njammer channel to null the jammer perfectly, regardless of its transmit sequence, and JL-SIMO operates on a jammer-free system.\nUnsurprisingly, the jammer-oblivious LMMSE baseline performs significantly worse than the jammer-robust geniePOS baseline under all attack scenarios,\nwith the data jamming attack turning out to be the most harmful and the pilot jamming attack the least harmful.\nBoth MAED and SO-MAED succeed in mitigating all four jamming attacks with highest effectiveness, even outperforming the genie-assisted geniePOS method by a\nconsiderable margin.\\footnote{The potential for MAED and SO-MAED to outperform geniePOS is a consequence of the superiority of joint channel estimation and data detection over \nseparating channel estimation from data detection.}\nTheir efficacy is further reflected in the fact that SO-MAED and MAED approach the performance of the jammerless and MU interference-free\nJL-SIMO lower bound to within less than $2$\\,dB and $3$\\,dB at $0.1$\\%~BER, respectively, in all considered~scenarios.\n\nThe behavior is largely similar when 16-QAM instead of QPSK is used as transmit constellation~(\\fref{fig:strong_jammers}). \nHowever, due to the decreased informativeness of the box prior for such higher-order constellations, MAED performs now closer \nto geniePOS, while SO-MAED still performs within $2$\\,dB (at $0.1$\\% BER) of the JL-SIMO lower bound.\nThe increased performance gap between them notwithstanding, both MAED and SO-MAED are able to effectively mitigate all four attack types.\n\n\n\n\n\\subsection{Mitigation of Weak Constellation Jammers} \\label{sec:results:weak}\nWe now turn to the analysis of more restrained jamming attacks in which the jammer transmits constellation symbols \nwith relative power $\\rP=0$\\,dB during its on-phase (to pass itself off as just another UE, for instance \\cite{vinogradova16a}).\nSimulation results for 16-QAM transmission under all four types of jamming attacks are shown in~\\fref{fig:weak_jammers}. \nBecause of the weaker jamming attacks, the jammer-oblivious LMMSE baseline now performs closer to the jammer-resistant geniePOS baseline \nthan it does in \\fref{fig:strong_jammers}.\nMAED again mitigates all attack types rather successfully, outperforming geniePOS in the low-SNR regime but slightly leveling off at high SNR.\nInterestingly, MAED shows worse performance under these weak jamming attacks than under the strong jamming attacks of \\fref{fig:strong_jammers}.\nThe reason is the following: MAED searches for the jamming subspace by looking for the dominant dimension \nof the iterative residual error $\\tilde\\bE^{(t)}$, see~\\fref{eq:rayleigh}. If the received jamming energy is small compared to the \nreceived signal energy, then it becomes hard to distinguish the residual errors caused by the jamming signal from those caused by \nerrors in estimating the channel and data matrices $\\tilde\\bH_\\bP^{(t)}$ and $\\tilde\\bS_D^{(t)}$.\nNote in contrast that, due to its superior signal prior, the equivalent performance loss of SO-MAED is only so small as to be \nvirtually unnoticeable. Thus, SO-MAED outperforms MAED by a large margin and still approaches the JL-SIMO bound by less than 2dB at a BER of 0.1\\%.\\footnote{We \nmention that MAED does not suffer such a performance loss under weak jamming attacks when the transmit constellation is QPSK, \nsince in that case the box signal prior of MAED is sufficiently accurate, see also \\cite{marti2022smart}.}\n\n\n\\subsection{How Versatile is SO-MAED Really?}\nIn the remainder of our evaluation, we focus mostly on \\mbox{SO-MAED}, since it is clearly the better of the two proposed algorithms. \nTo show that our approach indeed succeeds in mitigating arbitrary jamming attacks without need for fine-tuning of the algorithm or its parameters, \n\\fref{fig:many_jammers} depicts performance results for a series of jamming attacks spanning a dynamic range from $\\rP=-20$\\,dB to \n$\\rP=80$\\,dB. Specifically, \\fref{fig:many_jammers} shows results for all four jammer types, where every \nsubfigure plots BER curves for jamming attacks with \n$\\rP\\in\\{-20\\,\\text{dB}, -10\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$.\nThe purpose of these plots is to illustrate that, apart from jamming attacks where the jammer is significantly weaker than \nthe average UE\\footnote{A jammer that is much weaker than the average UE resembles\na non-transmitting, and thus eclipsed, jammer; see Sections \\ref{sec:theory} and \\ref{sec:results:eclipsed}.}, \nthe curves are virtually indistinguishable, meaning that the performance of SO-MAED is virtually independent of the specific\ntype of jamming attack that it is facing.\n\n\n\\subsection{Eclipsed Jammers} \\label{sec:results:eclipsed}\n\n\n\\begin{figure}[tp]\n\\centering\n\\!\\!\\!\\!\\!\n\\subfigure[no jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_all-zero_eclipsed_gaussian_rho0_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:no_jam}\n}\\!\\!\\!\n\\subfigure[weak UE-impersonating jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_pilot_eclipsed_gaussian_rho0_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:weak_impers}\n}\\!\\! \\\\\n\\!\\!\\!\\!\\!\n\\subfigure[strong UE-impersonating jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_pilot_eclipsed_gaussian_rho30_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:strong_impers}\n}\\!\\!\\!\n\\subfigure[data-dependent jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_row-difference_eclipsed_gaussian_rho30_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:row_eclipsed}\n}\\!\\!\n\\caption{Uncoded bit error-rate of SO-MAED for different types of eclipsed jammers: (a) no jammer, \n(b) $\\rP=0$\\,dB jammer impersonating the $j$th UE by transmitting its pilot sequence \n($\\text{UE}_j$ denotes the BER of the impersonated UE, and $\\overline{\\text{UE}}_j$ the BER among all other UEs),\n(c) $\\rP=30$\\,dB jammer impersonating the $j$th UE by transmitting its pilot sequence, \nand (d) jammer causes eclipsing by transmitting a jamming sequence that depends on the UE transmit matrix~$\\bS$.\nDashed lines represent the BER of the impersonated UE, transparent lines represent the BER among \nthe UEs that are not impersonated by the jammer.\n}\n\\label{fig:eclipsed}\n\\vspace{-2mm}\n\\end{figure}\n\nUp to this point, the jamming signal $\\bmw$ has always been generated independently from the UE transmit matrix $\\bS$.\nThe strong performance results of both MAED and SO-MAED have supported the claim in Remark~\\ref{rem:rare} that,\nin this case, eclipsing is the (rare) exception, not the norm. \nWe now turn to an empirical analysis of how SO-MAED behaves when eclipsing \\emph{does}\noccur (\\fref{fig:eclipsed}). To this end, we consider scenarios in which the jammer is eclipsed \nbecause there is no jamming activity (\\fref{fig:eclipsed:no_jam}), \nbecause the jammer transmits a UE's pilot sequence (\\fref{fig:eclipsed:weak_impers}, \\fref{fig:eclipsed:strong_impers}),\nor because the jamming sequence~$\\bmw$ depends on the transmit matrix $\\bS$ (which in reality would be unknown to the jammer) \nin a way that causes eclipsing (\\fref{fig:eclipsed:row_eclipsed}). \n\n\nIn the case of no jammer (\\fref{fig:eclipsed:no_jam}), or no jamming activity within a coherence interval,\nwe see that SO-MAED still reliably detects the transmit data. However, it now suffers from an error floor\n(albeit significantly below $0.1$\\% BER). The reason for this error floor is that, in the absence of \njamming energy to guide the choice of the nulled direction $\\tilde\\bmp$, there is the temptation to instead ``cover up''\ndetection errors (similar to the phenomenon discussed in \\fref{sec:results:weak}).\nHowever, the low level of the error floor shows that this potential pitfall does not cause a systematic breakdown \nof SO-MAED. \nWe emphasize also that SO-MAED does not simply null the strongest UE. Such (degenerate) behavior would only occur\nif one UE were \\emph{far} stronger than the others. With any reasonable power control scheme, UE nulling is not an \nissue.\\footnote{This is exemplified by our experiments with i.i.d. Rayleigh fading chanels, which also exhibit minor imbalances in receive power between different UEs. Cf. also our results in \\fref{sec:beyond}, \nwhere we use $\\pm1.5$\\,dB power control.}\nIn the case of a jammer that impersonates the $j$th UE by transmitting its pilot sequence in the training phase \nand constellation symbols in the data phase, with the same power ($\\rP=0$\\,dB) as the average~UE, \nSO-MAED indeed suffers a performance breakdown (\\fref{fig:eclipsed:weak_impers}). However, \ncloser analysis shows that this error floor is caused solely by errors in detecting the symbols of the \nimpersonated UE. \nThis is not surprising: The jammer is statistically indistinguishable from the $j$th UE,\nso that is impossible to reliably separate the UE transmit symbols from the fake jammer transmit symbols. \nIn this regard, we refer again to the information-theoretic discussion of \\mbox{\\cite[Sec. V]{lapidoth1998reliable}}.\nSuch impersonation attacks could be forestalled by using encrypted pilots~\\cite{basciftci2015securing}.\nIf the jammer transmits the $j$th UE's pilot sequence and constellation symbols, but with much more\npower ($\\rP=30$\\,dB), then the iterative detection procedure of \\mbox{SO-MAED} will separate the jammer subspace from the \n$j$th UE's subspace (\\fref{fig:eclipsed:strong_impers}), since, being so much stronger than any UE, \nthe jammer subspace will dominate the residual matrix $\\tilde\\bE^{(t)}$ in~\\fref{eq:rayleigh}.\nFinally, \\fref{fig:eclipsed:row_eclipsed} shows results for a case where the jammer knows $\\bS$\nand choses~$\\tilde\\bS_D$ to differ from $\\bS_D$ in a single row (with valid constellation symbols in the differing row), \nso that $\\textit{rank}(\\bS_D-\\tilde\\bS_D)=1$. It first draws $\\bmw_T\\sim\\setC\\setN(\\mathbf{0},\\bI_D)$ and then\nsets $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ to cause eclipsing. \nThe jammer strength is $\\rP=30$\\,dB. \nThe results show an error floor at roughly 0.2\\% BER caused by the presence of an alternative spurious solution.\nHowever, the results in Figs.~\\ref{fig:qpsk_strong_jammers}\\,--\\,\\ref{fig:many_jammers} show \nthat, when the jammer has to select~$\\bmw$ without knowing $\\bS_D$, such accidental eclipsing is \nextremely rare.\n\n\n\n\\subsection{Beyond i.i.d. Rayleigh Fading} \\label{sec:beyond}\nSo far, our experiments were based on i.i.d. Rayleigh fading channels, but our method does not depend\nin any way on this particular channel model. \nTo demonstrate that MAED and SO-MAED are also applicable in scenarios that deviate strongly from the i.i.d. Rayleigh model,\nwe now evaluate our algorithms on realistic mmWave channels generated \nwith the commercial Wireless InSite ray-tracer \\cite{Remcom}. The simulated scenario is depicted in \\fref{fig:remcom_scenario}.\nWe simulate a mmWave massive MU-MIMO system with a carrier frequency of $60$\\,GHz and a bandwidth \nof $100$\\,MHz. The BS is placed at a height of $10$\\,m and consists of a horizontal uniform linear array with $B=128$ omnidirectional antennas spaced at half \na wavelength. The omnidirectional single-antenna UEs and the jammer are located at a height of $1.65$\\,m and placed in a $150^\\circ$ sector \nspanning $180$\\,m$\\times$$90$\\,m in front of the BS; see \\fref{fig:remcom_scenario}. The~UEs~and the jammer are drawn at random from a grid with $5$\\,m pitch while ensuring\nthat the minimum angular separation between any two UEs, as well as between the jammer and any UE, is $2.5^\\circ$.\nWe assume $\\pm1.5$\\,dB power control, so that the ratio between the maximum and minimum per-UE receive power is~2.\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figures\/sector_marked_cropped}\n\\caption{Simulated scenario. The location of the BS his highlighted by the white circle while the red squares depict all possible UE locations.}\n\\label{fig:remcom_scenario}\n\\vspace{-2mm}\n\\end{figure}\n\\begin{figure}[tp]\n\\centering\n\\!\\!\\!\\!\\!\n\\subfigure[barrage jammer (J1)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_barrage_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:barrage}\n}\\!\\!\\!\n\\subfigure[pilot jammer (J2)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_pilot_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:pilot}\n}\\!\\! \\\\\n\\!\\!\\!\\!\\!\n\\subfigure[data jammer (J3)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_data_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:data}\n}\\!\\!\\!\n\\subfigure[sparse jammer (J4)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_sparse_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:sparse}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{QPSK} transmission over realistic mmWave channels in the presence of a \\emph{strong} ($\\rE=30$\\,dB) jammer.}\n\\label{fig:remcom_results}\n\\vspace{-2mm}\n\\end{figure}\nThe high correlation exhibited by these mmWave channels slows convergence of MAED and SO-MAED, so we increase their number of \niterations to $t_{\\max}=30$. We also retrain the parameters from SO-MAED on mmWave channels (while making a clear split between the training\nset and the~evaluation~set).\n\nThe results for QPSK transmission in the presence of $\\rE=30$\\,dB are shown in \\fref{fig:remcom_results}. \nThe performance hierarchy is identical as in the equivalent Rayleigh-fading setup of \\fref{fig:qpsk_strong_jammers}:\ngeniePOS is clearly outperformed by MAED, which is in turn outperformed by SO-MAED. \nHowever, the more challenging nature of mmWave channels amplifies performance differences: \nDue to its artificial immunity from the high inter-user interference of mmWave channels, JL-SIMO is now in a class of its own. \nHowever, MAED and SO-MAED gain almost $4$\\,dB and $6$\\,dB in SNR on geniePOS at $0.1\\%$ BER, respectively, \nregardless of the jammer type. This shows that MAED and SO-MAED are also well suited \nfor scenarios that deviate significantly from the i.i.d. Rayleigh model.\n\\vspace{-1mm}\n\n\n\n\\section{Conclusions}\n\nWe have proposed an approach for the mitigation of smart jamming attacks on the massive\nMU-MIMO uplink and supported its basic soundness with theoretical results. \nIn contrast to existing mitigation methods, our approach does not rely on jamming activity \nduring any particular time instant.\nInstead, our approach utilizes a newly proposed problem formulation which exploits the fact \nthat the jammer's subspace remains constant within a coherence interval.\nWe have developed two efficient iterative algorithms, MAED and SO-MAED, which approximately solve the \nproposed optimization problem. \nOur simulation results have shown that MAED and SO-MAED are able to effectively mitigate a wide range of jamming attacks.\nIn particular, they succeed in mitigating attack types like data jamming and sparse jamming,\nfor which---to the best of our knowledge---no mitigation methods have existed so far.\n\nFuture work could focus on jammer mitigation with iterative detection and decoding, \nfor which the soft-output estimates of SO-MAED are well suited. Other directions for \nfuture work are the extension to mutli-antenna jammers as well as to jammer mitigation\nin wideband systems.\n\n\\appendices\n\n\\vspace{-1mm}\n\n\\section{Proof of \\fref{thm:maed}} \\label{app:proof1}\nObviously, if $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$, then\n\\begin{align}\n\t\\hat\\bP\\bY - \\hat\\bH_\\bP \\hat\\bS \n\t&= \\bP\\bY - \\bP\\bH \\bS \\\\\n\t&= \\bP(\\bH\\bS + \\Hj\\tp{\\bsj}) - \\bP\\bH \\bS \\\\\n\t&= \\bP \\Hj\\tp{\\bsj} = \\mathbf{0}, \n\\end{align}\nand so the objective value of \\eqref{eq:obj1} is zero. Since the objective function in \\eqref{eq:obj1} is nonnegative, \nit follows that $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$ is a solution to \\eqref{eq:obj1}. \nIt remains to prove uniqueness. For this, we rewrite the objective in \\eqref{eq:obj1} as\n\\begin{align}\n\t\\big\\|\\tilde\\bP\\bY \\!- \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F \n\t= \\big\\|\\tilde\\bP\\bY_T \\!- \\tilde\\bH_\\bP \\bS_T \\big\\|^2_F + \\big\\|\\tilde\\bP\\bY_D \\!- \\tilde\\bH_\\bP \\tilde\\bS_D \\big\\|^2_F\n\t\\label{eq:decomposition}\n\\end{align}\nThe objective can only be zero if both terms on the right-hand-side (RHS) of \\eqref{eq:decomposition} are zero.\nThe first term is zero iff\n\\begin{align}\n\t& \\tilde\\bP\\bY_T - \\tilde\\bH_\\bP \\bS_T = \\mathbf{0},\n\\end{align}\nwhich implies\n\\begin{align}\n\t\\tilde\\bH_\\bP = \\tilde\\bP\\bY_T \\pinv{\\bS_T}, \\label{eq:optimal_h}\n\\end{align}\nsince $\\bS_T$ has full row rank.\nPlugging this back into the second term on the RHS of \\eqref{eq:decomposition} gives\n\\begin{align}\n\t& \\tilde\\bP\\bY_D - \\tilde\\bH_\\bP \\tilde\\bS_D \\\\\n\t&= \\tilde\\bP(\\bY_D - \\bY_T \\pinv{\\bS_T} \\tilde\\bS_D) \\\\\n\t&= \\tilde\\bP \\left(\\bH\\bS_D + \\Hj\\tp{\\bsj_D} - (\\bH\\bS_T + \\Hj\\tp{\\bsj_T}) \\pinv{\\bS_T} \\tilde\\bS_D \\right) \\\\\n\t&= \\tilde\\bP \\left(\\bH [\\bS_D- \\tilde\\bS_D] + \\Hj[\\tp{\\bsj_D} - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D] \\right). \\label{eq:term_mat}\n\\end{align}\nThe second term on the RHS of \\eqref{eq:decomposition} (and, hence, the objective) is zero if and only if\nthe matrix in \\eqref{eq:term_mat} is the zero matrix.\nThe projector $\\tilde{\\bP}$ can null a matrix of (at most) rank one. \nIt follows that the objective function in \\eqref{eq:decomposition} can be zero only if \n\\begin{align}\n\t\\bH [\\bS_D- \\tilde\\bS_D] + \\Hj[\\tp{\\bsj_D} - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D] \\label{eq:eclipsing_equation}\n\\end{align}\nis a matrix of (at most) rank one. Since $\\bH$ has full column rank and $\\Hj$ is not included in the columnspace of $\\bH$, \nthis requires that either $\\text{rank}(\\bS_D- \\tilde\\bS_D)=1$ and $\\tp{\\bsj_D} - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D=\\mathbf{0}$, \nor that $\\bS_D - \\tilde\\bS_D=\\mathbf{0}$. But the first case is precluded since it implies that the jammer is eclipsed,\nin contrast to our assumption. \n\nIn the second case, we have $\\tilde\\bS_D = \\bS_D$, so the estimated data matrix coincides with the true data matrix.\nIn that case,~\\eqref{eq:term_mat} is\n\\begin{align}\n\t\\tilde\\bP \\Hj[\\tp{\\bsj_D} - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D], \n\\end{align}\nwhich (again by the assumption that the jammer is not ecplised) is zero if and only if $\\tilde\\bmp$ is collinear with $\\bmj$, \nmeaning that $\\tilde\\bmp = \\alpha \\bmp, |\\alpha|=1$. This means that also the estimated jammer subspace coincides with the\ntrue jammer subspace.\nFinally, plugging this value of $\\tilde\\bmp$ back into \\eqref{eq:optimal_h} yields\n\\begin{align}\n\t\\tilde\\bH_\\bP &= \\tilde\\bP\\bY_T \\pinv{\\bS_T} \\\\\n\t&= \\tilde\\bP (\\bH\\bS_T + \\Hj\\tp{\\bsj_T}) \\pinv{\\bS_T} \\\\\n\t&= \\tilde\\bP\\bH\\bS_T \\pinv{\\bS_T} = \\bH_\\bP, \n\\end{align}\nshowing that also the estimated channel matrix coincides with the projection of the true channel matrix. \nWe have thereby shown that $\\big\\|\\tilde\\bP\\bY - \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F$ is zero if and only if\n$\\tilde\\bS_D=\\bS_D$, $\\tilde\\bmp = \\alpha\\bmp, |\\alpha|=1$, and $\\tilde\\bH_\\bP = \\bH_\\bP$.\n\\hfill $\\blacksquare$ \n\n\n\n\n\n\\section{Proof of \\fref{thm:maed2}} \\label{app:proof2}\n$\\bS_T$ is unitary, so the jammer eclipses if\n\\begin{align}\n\t\\tp{\\bsj_D} &= \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D \\\\\n\t&= \\tp{\\bsj_T}\\bS_T^H\\tilde{\\bS}_D = (\\underbrace{\\bS_T\\bsj_T^\\ast}_{\\triangleq \\bmx})^H \\tilde{\\bS}_D\n\\end{align}\nfor some $\\tilde{\\bS}_D\\in\\setS^{U\\times D}$ such that $\\text{rank}(\\bS_D - \\tilde{\\bS}_D)\\leq 1$.\nSince~$\\bS_T$ is Haar distributed\\footnote{The uniform distribution over unitary matrices is called Haar distribution.}, \nthe vector $\\bmx$ is distributed uniformly over the\ncomplex $U$-dimensional sphere of radius $\\|\\bmw\\|$ \\cite[p. 16]{meckes2019random}.\nIn particular, $\\bmx$ is independent of $\\bmw_D$.\nWe can now show that \n\\begin{align}\n\t\\tp{\\bmw_D} \\neq \\bmx^H\\tilde{\\bS}_D \n\\end{align}\nholds with probability one by showing that already for the first entry, we have\n\\begin{align}\n\tw_{D,1} \\neq \\bmx^H \\tilde\\bms_{D,1} \\label{eq:first_entry_criterion}\n\\end{align}\nwith probability one, where $ \\tilde\\bms_{D,1}$ is the leftmost column of~$\\tilde{\\bS}_D$:\nWe can interpret $\\bmx^H \\tilde\\bms_{D,1}=\\|\\tilde\\bms_{D,1}\\| \\langle \\tilde\\bms_{D,1}\/\\|\\tilde\\bms_{D,1}\\|, \\bmx \\rangle$ \nas the top left entry of a matrix product $\\|\\tilde\\bms_{D,1}\\|\\bX\\bZ$, \nwhere $\\bX$ is a Haar-distributed matrix whose first row is $\\bmx^H$, and where $\\bZ$ is an unitary matrix whose first \ncolumn is $\\tilde\\bms_{D,1}\/\\|\\tilde\\bms_{D,1}\\|$. \nIt then follows from \\cite[p. 7]{meckes2019random} that $\\bX\\bZ$ is Haar distributed.\nThus, the distribution of its top left entry and hence also of $\\bmx^H\\tilde\\bms_{D,1}$ has no mass points. \nSince $w_{D,1}$ is independent of $\\bmx$, \\eqref{eq:first_entry_criterion} must therefore hold with probability one. \n\\hfill $\\blacksquare$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\t\\label{S1}\n\\paragraph{The topos approach.} The topos approach to quantum theory was initiated by Isham in \\cite{Isham97} and Butterfield and Isham in \\cite{IB98,IB99,IB00,IB02}. It was developed and broadened into an approach to the formulation of physical theories in general by Isham and the author in \\cite{DI(1),DI(2),DI(3),DI(4)}. The long article \\cite{DI08} gives a more-or-less exhaustive\\footnote{and probably exhausting...} and coherent overview of the approach. More recent developments are the description of arbitrary states by probability measures \\cite{Doe08} and further developments \\cite{Doe09} concerning the new form of quantum logic that constitutes a central part of the topos approach. For background, motivation and the main ideas see also Isham's article in this volume \\cite{Ish10}.\n\nMost of the work so far has been done on standard non-relativistic quantum theory. A quantum system is described by its \\emph{algebra of physical quantities}. Often, this can be assumed to be $\\BH$, the algebra of all bounded operators on the (separable) Hilbert space $\\Hi$ of the system. More generally, one can use a suitable operator algebra. For conceptual and pragmatical reasons, we assume that the algebra of physical quantities is a \\emph{von Neumann algebra $\\N$} (see e.g. \\cite{KR83a,Tak79}). For our purposes, this poses no additional technical difficulty. Physically, it allows the description of quantum systems with symmetry and\/or superselection rules. The reader unfamiliar with von Neumann algebras can always assume that $\\N=\\BH$.\n\nQuantum theory usually is identified with the Hilbert space formalism together with some interpretation. This works fine in a vast number of applications. Moreover, the Hilbert space formalism is very rigid. One cannot just change some part of the structure since this typically brings down the whole edifice. Yet, there are serious conceptual problems with the usual instrumentalist interpretations of quantum theory which become even more severe when one tries to apply quantum theory to gravity and cosmology. For a discussion of some of these conceptual problems, see Isham's article in this volume.\n\nThe topos approach provides not merely another interpretation of the Hilbert space formalism, but a mathematical reformulation of quantum theory, based upon structural and conceptual considerations. The resulting formalism is \\emph{not} a Hilbert space formalism. The fact that such a reformulation is possible at all is somewhat surprising. (Needless to say, many open questions remain.)\n\n\\paragraph{Daseinisation.} In this article, we mainly focus on how the new topos formalism relates to the standard Hilbert space formalism. The main ingredient is the process that we coined \\emph{daseinisation}. It will be shown how daseinisation relates familiar structures in quantum theory to structures within the topos associated with a quantum system. Hence, daseinisation gives the `translation' from the ordinary Hilbert space formalism to the topos formalism.\n\nIn our presentation, we will keep the use of topos theory to an absolute minimum and do not assume any familiarity with category theory beyond the very basics. Some notions and results from functional analysis, e.g. the spectral theorem, will be used. As an illustration, we will show how all constructions look like concretely for the algebra $\\N=\\mc B (\\bbC^3)$ and the spin-$z$ operator, $\\hat S_z\\in\\mc B (\\bbC^3)$, of a spin-$1$ particle. \n\nIn fact, there are two processes called daseinisation. The first is \\emph{daseinisation of projections}, which maps projection operators to certain subobjects of the state object. (The state object and its subobjects will be defined below.) This provides the bridge from ordinary Birkhoff-von Neumann quantum logic to a new form of quantum logic that is based upon the internal logic of a topos associated with the quantum system. The second is \\emph{daseinisation of self-adjoint operators}, which maps each self-adjoint operator to an arrow in the topos. Mathematically, the two forms of daseinisation are related, but conceptually they are quite different.\n\nThe topos approach emphasises the r\\^ole of \\emph{classical perspectives} onto a quantum system. A classical perspective or \\emph{context} is nothing but a set of commuting physical quantities, or more precisely the abelian von Neumann algebra generated by such a set. One of the main ideas is that \\emph{all} classical perspectives should be taken into account simultaneously. But given e.g. a projection operator, which represents a proposition about the value of a physical quantity in standard quantum theory, one is immediately faced with the problem that the projection operator is contained in some contexts, but not in all. The idea is to approximate the projection in all those contexts that do not contain it. Likewise, self-adjoint operators, which represent physical quantities, must be approximated suitably to all contexts. Daseinisation is nothing but a process of systematic approximation to all classical contexts.\n\nThis article is organised as follows: in section \\ref{S2}, we discuss propositions and their representation in classical physics and standard quantum theory. Section \\ref{S3} presents some basic structures in the topos approach to quantum theory. In section \\ref{S4}, the representation of propositions in the topos approach is discussed -- this is via daseinisation of projections. Section \\ref{S5}, which is the longest and most technical, shows how physical quantities are represented in the topos approach; here we present daseinisation of self-adjoint operators is discussed. Section \\ref{S6} concludes.\n\n\\section{\\mbox{Propositions and their representation in}\\\\ classical physics and standard quantum\\\\ theory}\t\\label{S2}\n\\paragraph{Propositions.} Let $S$ be some physical system, and let $A$ denote a physical quantity pertaining to the system $S$ (e.g. position, momentum, energy, angular momentum, spin...). We are concerned with propositions of the form ``the physical quantity $A$ has a value in the set $\\De$ of real numbers'', for which we use the shorthand notation ``$\\Ain{\\De}$''. In all applications, $\\De$ will be a Borel subset of the real numbers $\\bbR$.\n\nArguably, physics is fundamentally about what we can say about the truth values of propositions of the form ``$\\Ain{\\De}$'' (where $A$ varies over the physical quantities of the system and $\\De$ varies over the Borel subsets of $\\bbR$) when the system is in a given state. Of course, the state may change in time and hence the truth values also will change in time.\\footnote{One could also think of propositions about \\emph{histories} of a physical system $S$ such that propositions refer to multiple times, and the assignment of truth values to such propositions (in a given, evolving state). For some very interesting recent results on a topos formulation of histories see Flori's recent article \\cite{Flo08}.}\n\nSpeaking about propositions like ``$\\Ain{\\De}$'' requires a certain amount of conceptualising. We accept that it is sensible to talk about physical systems as suitably separated entities, that each such physical system is characterised by its physical quantities, and that the range of values of a physical quantity $A$ is a subset of the real numbers. If we want to assign truth values to propositions, then we need the concept of a state of a physical system $S$, and the truth values of propositions will depend on the state. If we regard these concepts as natural and basic (and potentially as prerequisites for doing physics at all), then a proposition ``$\\Ain{\\De}$'' \\emph{refers to the world} in the most direct conceivable sense.\\footnote{Of course, as explained by Isham in his article \\cite{Ish10}, it is one of the central motivations for the whole topos programme to find a mathematical framework for physical theories that is not depending fundamentally on the real numbers. In particular, the premise that physical quantities have real values is doubtful in this light. How this seeming dilemma is solved in the topos approach to quantum theory will become clear later in section \\ref{Sec_RepOfPhysQuants}.} We will deviate from common practice, though, by insisting that also in quantum theory, a proposition is about `how things are', and is not to be understood as a counterfactual statement, i.e., it is not merely about what we would obtain as a measurement result if we were to perform a measurement.\n\n\\paragraph{Classical physics, state spaces and realism.} In classical physics, a state of the system $S$ is represented by an element of a set, namely by a point of the state space $\\mathcal{S}$ of the system.\\footnote{We avoid the usual synonym `phase space', which seems to be a historical misnomer. In any case, there are no phases in a phase space.} Each physical quantity $A$ is represented by a real-valued function $f_A:\\mathcal{S}\\rightarrow\\bbR$ on the state space. A proposition ``$\\Ain{\\De}$'' is represented by a certain subset of the state space, namely the set $f_A^{-1}(\\De)$. We assume that $f_A$ is (at least) measurable. Since $\\De$ is a Borel set, the subset $f_A^{-1}(\\De)$ of state space representing ``$\\Ain{\\De}$'' is a Borel set, too.\n\nIn any state, each physical quantity $A$ has a value, which is simply given by evaluation of the function $f_A$ representing $A$ at the point $s\\in\\mc{S}$ of state space representing the state, i.e., the value is $f_A(s)$. Every proposition ``$\\Ain{\\De}$'' has a Boolean truth value in each given state $s\\in\\mc{S}$:\n\\begin{equation}\n\t\t\tv(\\SAin{\\De};s)=\\left\\{\n\t\t\t\\begin{tabular}\n\t\t\t\t\t\t[c]{ll\n\t\t\t\t\t\t$true$ & if $s\\in f_ A^{-1}(\\De)$\\\\\n\t\t\t\t\t\t$false$ & if $s\\notin f_A^{-1}(\\De)$.\n\t\t\t\\end{tabular}\n\t\t\t\\ \\right.\n\\end{equation}\nLet ``$\\Ain{\\De}$'' and ``$B\\varepsilon\\Gamma$'' be two different propositions, represented by the Borel subsets $f_A^{-1}(\\De)$ and $f_B^{-1}(\\Gamma)$ of $\\mc{S}$, respectively. The union $f_A^{-1}(\\De)\\cup f_B^{-1}(\\Gamma)$ of the two Borel subsets is another Borel subset, and this subset represents the proposition ``$\\Ain{\\De}$ or $B\\varepsilon\\Gamma$'' (disjunction). Similarly, the intersection $f_A^{-1}(\\De)\\cap f_B^{-1}(\\Gamma)$ is a Borel subset of $\\mc{S}$, and this subset represents the proposition ``$\\Ain{\\De}$ and $B\\varepsilon\\Gamma$'' (conjunction). Both conjunction and disjunction can be extended to arbitrary countable families of Borel sets representing propositions. The set-theoretic operations of taking unions and intersections distribute over each other. Moreover, the negation of a proposition ``$\\Ain{\\De}$'' is represented by the complement $\\mc{S}\\backslash f_A^{-1}(\\De)$ of the Borel set $f_A^{-1}(\\De)$. Clearly, the empty subset $\\emptyset$ of $\\mc{S}$ represents the trivially false proposition, while the maximal Borel subset, $\\mc{S}$ itself, represents the trivially true proposition. The set $\\mc{B(S)}$ of Borel subsets of the state space $\\mc{S}$ of a classical system thus is a Boolean $\\sigma$-algebra, i.e., a $\\sigma$-complete distributive lattice with complement. Stone's theorem shows that every Boolean algebra is isomorphic to the algebra of subsets of a suitable space, so Boolean logic is closely tied to the use of sets.\n\nThe fact that in a given state $s$ each physical quantity has a value and each proposition has a truth value makes classical physics a \\emph{realist} theory.\\footnote{Here, we could enter into an interesting, but potentially never-ending discussion on physical reality, ontology and epistemology etc. We avoid that and posit that for the purpose of this paper, a realist theory is one that, in any given state, allows to assign truth values to all propositions of the form ``$\\Ain{\\De}$''. Moreover, we require that there is a suitable logical structure, in particular a deductive system, in which we can argue about (representatives of) propositions.}\n\n\\paragraph{Representation of propositions in standard quantum theory.} In contrast to that, in quantum physics it is not possible to assign real values to all physical quantities at once. This is the content of the Kochen-Specker theorem. In the standard Hilbert space formulation of quantum theory, each physical quantity $A$ is represented by a self-adjoint operator $\\A$ on some Hilbert space $\\Hi$. The range of possible real values that $A$ can take is given by the spectrum $\\spec{\\A}$ of the operator $\\A$. Of course, there is a notion of states in quantum theory: in the simplest version, they are given by unit vectors in the Hilbert space $\\Hi$. There is a particular mapping taking self-adjoint operators and unit vectors to real numbers, namely the evaluation\n\\begin{equation}\n\t\t\t(\\A,\\ket\\psi)\\longmapsto\\bra\\psi\\A\\ket\\psi.\n\\end{equation}\nIn general (unless $\\ket\\psi$ is an eigenstate of $\\A$), this real value is \\emph{not} the value of the physical quantity $A$ in the state described by $\\psi$. It rather is the expectation value, which is a statistical and instrumentalist notion. The physical interpretation of the mathematical formalism of quantum theory fundamentally depends on measurements and observers.\n\nAccording to the spectral theorem, propositions like ``$\\Ain{\\De}$'' are represented by projection operators $\\P=\\hat E[\\Ain{\\De}]$. Each projection $\\P$ corresponds to a unique closed subspace $U_\\P$ of the Hilbert space $\\Hi$ of the system and vice versa. The intersection of two closed subspaces is a closed subspace, which can be taken as the definition of a conjunction. The closure of the subspace generated by two closed subspaces is a candidate for the disjunction, and the function sending a projection $\\P$ to $\\hat 1-\\P$ is an orthocomplement. Birkhoff and von Neumann suggested in their seminal paper \\cite{BvN36} to interpret these mathematical operations as providing a logic for quantum systems.\n\nAt first sight, this looks similar to the classical case: the Hilbert space $\\Hi$ now takes the r\\^ole of a state space, while its closed subspaces represent propositions. But there is an immediate, severe problem: if the Hilbert space $\\Hi$ is at least two-dimensional, then the lattice $\\mc{L(H)}$ of closed subspaces is \\emph{non-distributive} (and so is the isomorphic lattice of projections). This makes it very hard to find a proper semantics of quantum logic.\\footnote{We cannot discuss the merits and shortcomings of quantum logic here. Suffice it to say that there are more conceptual and interpretational problems, and a large number of developments and abstractions from standard quantum logic, i.e., the lattice of closed subspaces of Hilbert space. An excellent review is \\cite{DCG02}.}\n\n\\section{Basic structures in the topos approach to quantum theory}\t\\label{S3}\n\\subsection{Contexts}\nAs argued by Isham in this volume, the topos formulation of quantum theory is based upon the idea that one takes the collection of \\emph{all} classical perspectives on a quantum system. No single classical perspective can deliver a complete picture of the quantum system, but the collection of all of them may well do. As we will see, it is also of great importance how the classical perspectives relate to each other.\n\nThese ideas are formalised in the following way: we consider a quantum system as being described by a \\emph{von Neumann algebra} $\\N$ (see e.g. \\cite{KR83a}). Such an algebra is always given as a subalgebra of $\\BH$, the algebra of all bounded operators on a suitable Hilbert space $\\Hi$. We can always assume that the identity in $\\N$ is the identity operator $\\hat 1$ on $\\Hi$. Since von Neumann algebras are much more general than just the algebras of the form $\\BH$, and since all our constructions work for arbitrary von Neumann algebras, we present the results in this general form. For a very good introduction to the use of operator algebras in quantum theory see \\cite{Emch84}. Von Neumann algebras can be used to describe quantum systems with symmetry and\/or superselection rules. Throughout, we will use the very simple example of $\\N=\\mc{B}(\\bbC^3)$ to illustrate the constructions.\n\nWe assume that the self-adjoint operators in the von Neumann algebra $\\N$ associated with our quantum system represent the \\emph{physical quantities} of the system. Since a quantum system has non-commuting physical quantities, the algebra $\\N$ is \\emph{non-abelian}. The $\\bbR$-vector space of self-adjoint operators in $\\N$ is denoted as $\\N_\\sa$.\n\nA classical perspective is given by a collection of commuting physical quantities. Such a collection determines an \\emph{abelian subalgebra}, typically denoted as $V$, of the non-abelian von Neumann algebra $\\N$. An abelian subalgebra is often called a \\emph{context}, and we will use the notions classical perspective, context and abelian subalgebra synonymously. We will only consider abelian subalgebras that\n\\begin{itemize}\n\t\\item[(a)] are von Neumann algebras, i.e., they are closed in the weak topology. The technical advantage is that the spectral theorem holds in both $\\N$ and its abelian von Neumann subalgebras, and that the lattices of projections in $\\N$ and its abelian von Neumann subalgebras are complete;\n\t\\item[(b)] contain the identity operator $\\hat 1$.\n\\end{itemize}\nGiven a von Neumann algebra $\\N$, let $\\VN$ be the set of all its abelian von Neumann subalgebras which contain $\\hat 1$. By convention, the trivial abelian subalgebra $V_0=\\bbC\\hat 1$ is not contained in $\\VN$. If some subalgebra $V'\\in\\VN$ is contained in a larger subalgebra $V\\in\\VN$, then we denote the inclusion as $i_{V'V}:V'\\rightarrow V$. Clearly, $\\VN$ is a partially ordered set under inclusion, and as such is a simple kind of \\emph{category} (see e.g. \\cite{McL98}). The objects in this category are the abelian von Neumann subalgebras, and the arrows are the inclusions. Clearly, there is at most one arrow between any two objects. We call $\\VN$ the \\emph{category of contexts} of the quantum system described by the von Neumann algebra $\\N$.\n\nConsidering the abelian parts of a non-abelian structure may seem trivial, yet in fact it is not, since the context category $\\VN$ keeps track of the relations between contexts: whenever two abelian subalgebras $V,\\tilde V$ have a non-trivial intersection, then there are inclusion arrows\n\\begin{equation}\n\t\t\tV\\longleftarrow V\\cap\\tilde V\\longrightarrow\\tilde V\n\\end{equation}\nin $\\VN$. Every self-adjoint operator $\\A\\in V\\cap\\tilde V$ can be written as $g(\\hat B)$ for some $\\hat B\\in V_{\\sa}$, where $g:\\bbR\\rightarrow\\bbR$ is a Borel function, and, at the same time, as another $h(\\hat C)$ for some $\\hat C\\in\\tilde V_\\sa$ and some other Borel function $h$.\\footnote{This follows from the fact that each abelian von Neumann algebra $V$ is generated by a single self-adjoint operator, see e.g. Prop. 1.21 in \\cite{Tak79}. A Borel function $g:\\bbR\\rightarrow\\bbC$ takes a self-adjoint operator $\\A$ in a von Neumann algebra to another operator $g(\\A)$ in the same algebra. If $g$ is real-valued, then $g(\\A)$ is self-adjoint. For details, see \\cite{KR83a,Tak79}.} The point is that while $\\A$ commutes with $\\hat B$ and $\\A$ commutes with $\\hat C$, the operators $\\hat B$ and $\\hat C$ do not necessarily commute. In that way, the context category $\\VN$ encodes a lot of information about the algebraic structure of $\\N$, not just between commuting operators, and about the relations between contexts.\n\nIf $V'\\subset V$, then the context $V'$ contains less self-adjoint operators and less projections than the context $V$, so one can describe less physics from the perspective of $V'$ than from the perspective of $V$. The step from $V$ to $V'$ hence involves a suitable kind of \\emph{coarse-graining}. We will see later how daseinisation of projections resp. self-adjoint operators implements this informal idea of coarse-graining.\n\n\\begin{example}\t\t\t\\label{Ex1}\nLet $\\Hi=\\bbC^3$, and let $\\N=\\mc{B}(\\bbC^3)$, the algebra of all bounded linear operators on $\\bbC^3$. $\\mc{B}(\\bbC^3)$ is the algebra $M_3(\\bbC)$ of $3\\times 3$-matrices with complex entries, acting as linear transformations on $\\bbC^3$. Let $(\\psi_1,\\psi_2,\\psi_3)$ be an orthonormal basis of $\\bbC^3$, and let $(\\P_1,\\P_2,\\P_3)$ be the three projections onto the one-dimensional subspaces $\\bbC\\psi_1,\\bbC\\psi_2$ and $\\bbC\\psi_3$, respectively. Clearly, the projections $\\P_1,\\P_2,\\P_3$ are pairwise orthogonal, i.e., $\\P_i\\P_j=\\delta_{ij}\\P_i$.\n\nThere is an abelian subalgebra $V$ of $\\mc{B}(\\bbC^3)$ generated by the three projections. One can use von Neumann's double commutant construction and define $V=\\{\\P_1,\\P_2,\\P_3\\}''$ (see \\cite{KR83a}). More concretely, $V=\\rm{lin}_{\\bbC}(\\P_1,\\P_2,\\P_3)$. Even more explicitly, one can pick the matrix representation of the projections $\\P_i$ such that\n\\begin{equation}\n\t\t\t\\P_1=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_2=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_3=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 1\n\t\t\t\\end{array}\\right).\t\t\t\n\\end{equation}\nThe abelian algebra $V$ generated by these projections then consists of all diagonal $3\\times 3$-matrices (with complex entries on the diagonal). Clearly, there is no larger abelian subalgebra of $\\mc{B}(\\bbC^3)$ that contains $V$, so $V$ is maximal abelian.\n\nEvery orthonormal basis $(\\tilde\\psi_1,\\tilde\\psi_2,\\tilde\\psi_3)$ determines a maximal abelian subalgebra $\\tilde V$ of $\\mc{B}(\\bbC^3)$. There is some redundancy, though: if two bases differ only by a permutation and\/or phase factors of the basis vectors, then they generate the same maximal abelian subalgebra. Evidently, there are uncountably many maximal abelian subalgebras of $\\mc{B}(\\bbC^3)$.\n\nThere also are non-maximal abelian subalgebras. Consider for example the algebra generated by $\\P_1$ and $\\P_2+\\P_3$. This algebra is denoted as $V_{\\P_1}$ and is given as $V_{\\P_1}=\\rm{lin}_\\bbC(\\P_1,\\P_2+\\P_3)=\\bbC\\P_1+\\bbC(\\hat 1-\\P_1)$. Here, we use that $\\P_2+\\P_3=\\hat 1-\\P_1$. There are uncountably many non-maximal subalgebras of $\\mc{B}(\\bbC^3)$.\n\nThe trivial projections $\\hat 0$ and $\\hat 1$ are contained in every abelian subalgebra. An algebra of the form $V_{\\P_1}$ contains two non-trivial projections: $\\P_1$ and $\\hat 1-\\P_1$. There is the trivial abelian subalgebra $V_0=\\bbC\\hat 1$ consisting of multiples of the identity operator only. By convention, we will not consider the trivial algebra (it is not included in the partially ordered set $\\VN$).\n\nThe set $\\VN=\\mc{V}(\\mc{B}(\\bbC^3))=:\\mc{V}(\\bbC^3)$ of abelian subalgebras of $\\mc{B}(\\bbC^3)$ hence can be divided into two subsets: the maximal abelian subalgebras, corresponding to orthonormal bases as described, and the ones of the form $V_{\\P_1}$ that are generated by a single projection.\\footnote{In other (larger) algebras than $\\mc{B}(\\bbC^3)$, the situation is more complicated, of course.} Clearly, given two different maximal abelian subalgebras $V$ and $\\tilde V$, neither one contains the other. Similarly, if we have two non-maximal algebras $V_{\\P_1}$ and $V_{\\P_2}$, neither one will contain the other, with one exception: if $\\P_2=\\hat 1-\\P_1$, then $V_{\\P_1}=V_{\\P_2}$. This comes from the more general fact that if a unital operator algebra contains an operator $\\A$, then it also contains the operator $\\hat 1-\\A$.\n\nOf course, non-maximal subalgebras can be contained in maximal ones. Let $V$ be the maximal abelian subalgebra generated by three pairwise orthogonal projections $\\P_1,\\P_2,\\P_3$. Then $V$ contains three non-maximal abelian subalgebras, namely $V_{\\P_1},V_{\\P_2}$ and $V_{\\P_3}$. Importantly, each non-maximal abelian subalgebra is contained in many different maximal ones: consider two projections $\\Q_2,\\Q_3$ onto one-dimensional subspaces in $\\bbC^3$ such that $\\P_1,\\Q_2,\\Q_3$ are pairwise orthogonal. We assume that $\\Q_2,\\Q_3$ are such that the maximal abelian subalgebra $\\tilde V$ generated by $\\P_1,\\Q_2,\\Q_3$ is different from the algebra $V$ generated by the three pairwise orthogonal projections $\\P_1,\\P_2,\\P_3$. Then both $V$ and $\\tilde V$ contain $V_{\\P_1}$ as a subalgebra. This argument makes clear that $V_{\\P_1}$ is contained in every maximal abelian subalgebra that contains the projection $\\P_1$.\n\nWe take inclusion of smaller into larger abelian subalgebras (here: of non-maximal into maximal ones) as a partial order on the set $\\mc{V}(\\bbC^3)$.\n\nSince $\\bbC^3$ is finite-dimensional, the algebra $\\mc{B}(\\bbC^3)$ is both a von Neumann and a $C^*$-algebra. The same holds for the abelian subalgebras. We do not have to worry about weak closedness. On infinite-dimensional Hilbert spaces, these questions become important. In general, an abelian von Neumann subalgebra $V$ is generated by the collection $\\PV$ of its projections (which all commute with each other, of course) via the double commutant construction.\n\\end{example}\n\n\\subsection{Gel'fand spectra and the spectral presheaf}\n\\paragraph{Introduction.} We will now make use of the fact that each abelian ($C^*$-) algebra $V$ of operators can be seen as an algebra of continuous functions on a suitable topological space, the \\emph{Gel'fand spectrum} of $V$. At least locally, for each context, we thus obtain a mathematical formulation of quantum theory that is similar to classical physics, with a state space and physical quantities as real-valued functions on this space. The `local state spaces' are then combined into one global structure, the \\emph{spectral presheaf}, which serves as a state space analogue for quantum theory.\n\n\\paragraph{Gel'fand spectra.} Let $V\\in\\VN$ be an abelian subalgebra of $\\N$, and let $\\Si_V$ denote its Gel'fand spectrum. $\\Si_V$ consists of the multiplicative states on $V$, i.e., the positive linear functionals $\\ld:V\\longrightarrow\\bbC$ of norm $1$ such that\n\\begin{equation}\n\t\t\t\\forall\\A,\\hat B\\in V:\\ld(\\A\\hat B)=\\ld(\\A)\\ld(\\hat B).\n\\end{equation}\nThe elements $\\ld$ of $\\Si_V$ also are pure states of $V$ and algebra homomorphisms from $V$ to $\\bbC$. (It is useful to have these different aspects in mind.) The set $\\Si_V$ is a compact Hausdorff space in the weak$^*$ topology \\cite{KR83a,Tak79}.\n\nLet $\\ld\\in\\Si_V$ be given. It can be shown that for any self-adjoint operator $\\A\\in V_\\sa$, it holds that\n\\begin{equation}\n\t\t\t\\ld(\\A)\\in\\spec{\\A},\n\\end{equation}\nand every element $s\\in\\spec{\\A}$ is given as $s=\\tilde\\ld(\\A)$ for some $\\tilde\\ld\\in\\Si_V$. Moreover, if $g:\\bbR\\rightarrow\\bbR$ is a Borel function, then\n\\begin{equation}\t\t\t\\label{FUNC}\n\t\t\t\\ld(g(\\A)=g(\\ld(\\A)).\n\\end{equation}\nThis implies that an element $\\ld$ of the Gel'fand spectrum $\\Si_V$ can be seen as a \\emph{valuation} on $V_\\sa$, i.e., a function sending each self-adjoint operator in $V$ to some element of its spectrum in such a way that equation \\eq{FUNC}, the FUNC principle, holds. Within each context $V$, all physical quantities in $V$ (and only those!) can be assigned values at once, and the assignment of values commutes with taking (Borel) functions of the operators. Every element $\\ld$ of the Gel'fand spectrum gives a different valuation on $V_\\sa$.\n\nThe well-known \\emph{Gel'fand representation theorem} shows that the abelian von Neumann algebra $V$ is isometrically $*$-isomorphic (i.e., isomorphic as a $C^*$-algebra) to the algebra of continuous, complex-valued functions on its Gel'fand spectrum $\\Si_V$.\\footnote{The Gel'fand representation theorem holds for abelian $C^*$-algebras. Since every von Neumann algebra is a $C^*$-algebra, the theorem applies to our situation.} The isomorphism is given by\n\\begin{eqnarray}\n\t\t\tV &\\longrightarrow& C(\\Si_V)\\\\\n\t\t\t\\A &\\longmapsto& \\overline{A},\n\\end{eqnarray}\nwhere $\\overline{A}(\\ld):=\\ld(\\A)$ for all $\\ld\\in\\Si_V$. The function $\\overline{A}$ is called the \\emph{Gel'fand transform} of the operator $\\A$. If $\\A$ is a self-adjoint operator, then $\\overline{A}$ is real-valued, and $\\overline{A}(\\Si_V)=\\spec{\\A}$.\n\nThe fact that the self-adjoint operators in an abelian (sub)algebra $V$ can be written as real-valued functions on the compact Hausdorff space $\\Si_V$ means that the Gel'fand spectrum $\\Si_V$ plays a r\\^ole exactly like the state space of a classical system. Of course, since $V$ is abelian, not all self-adjoint operators $\\A\\in\\N_{\\sa}$ representing physical quantities of the quantum system are contained in $V$. We will interpret the Gel'fand spectrum $\\Si_V$ as a \\emph{local state space} at $V$. Local here means `at the abelian part $V$ of the global non-abelian algebra $\\N$'.\n\nFor a projection $\\P\\in\\PV$, we have\n\\begin{equation}\n\t\t\t\\ld(\\P)=\\ld(\\P^{2})=\\ld(\\P)\\ld(\\P),\n\\end{equation}\nso $\\ld(\\P)\\in\\{0,1\\}$ for all projections. This implies that the Gel'fand transform $\\overline{P}$ of $\\P$ is the characteristic function of some subset of $\\Si_V$.\\footnote{The fact that a characteristic function can be continuous shows that the Gel'fand topology on $\\Si_V$ is pretty wild: the Gel'fand spectrum of an abelian von Neumann algebra is \\emph{extremely disconnected}, that is, the closure of each open set is open.} Let $A$ be a physical quantity that is represented by some self-adjoint operator in $V$, and let ``$\\Ain{\\De}$'' be a proposition about the value of $A$. Then we know from the spectral theorem that there is a projection $\\hat E[\\Ain{\\De}]$ in $V$ that represents the proposition. We saw that $\\ld(\\hat E[\\Ain{\\De}])\\in\\{0,1\\}$, and by interpreting $1$ as `true' and $0$ as `false', we see that a valuation $\\ld\\in\\Si_V$ assigns a Boolean truth-value to each proposition ``$\\Ain{\\De}$''.\n\nThe precise relation between projections in $V$ and subsets of the Gel'fand spectrum of $V$ is as follows: let $\\P\\in\\PV$, and define\n\\begin{equation}\n\t\t\t\tS_\\P:=\\{\\ld\\in\\Si_V\\mid\\ld(\\P)=1\\}.\n\\end{equation}\nOne can show that $S_\\P\\subseteq\\Si_V$ is \\emph{clopen}, i.e., closed and open in $\\Si_V$. Conversely, every clopen subset $S$ of $\\Si_V$ determines a unique projection $\\P_S\\in\\PV$, given as the inverse Gel'fand transform of the characteristic function of $S$. There is a lattice isomorphism\n\\begin{eqnarray}\t\t\t\\label{alpha}\n\t\t\t\\alpha:\\PV &\\longrightarrow& \\mc{C}l(\\Si_V)\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto& S_\\P\n\\end{eqnarray}\nbetween the lattice of projections in $V$ and the lattice of clopen subsets of $\\Si_V$. Thus, starting from a proposition ``$\\Ain{\\De}$'', we obtain a clopen subset $S_{\\hat E[\\Ain{\\De}]}$ of $\\Si_V$. This subset consists of all valuations, i.e., pure states $\\ld$ of $V$ such that $\\ld(\\hat E[\\Ain\\De])=1$, which means that the proposition ``$\\Ain\\De$'' is true in the state\/under the valuation $\\ld$. This strengthens the interpretation of $\\Si_V$ as a local state space.\n\nWe saw that each context $V$ provides us with a local state space, and one of the main ideas in the work by Isham and Butterfield \\cite{IB98,IB99,IB00,IB02} was to form a single global object from all these local state spaces.\n\nIn order to keep track of the inclusion relations between the abelian subalgebras, we must relate the Gel'fand spectra of larger and smaller subalgebras in a natural way. Let $V,V'\\in\\VN$ such that $V'\\subseteq V$. Then there is a mapping\n\\begin{eqnarray}\t\t\t\\label{Restriction}\n\t\t\tr:\\Si_V &\\longrightarrow& \\Si_{V'}\\\\\t\t\t \\nonumber\n\t\t\t\\ld &\\longmapsto& \\ld|_{V'},\n\\end{eqnarray}\nsending each element $\\ld$ of the Gel'fand spectrum of the larger algebra $V$ to its restriction $\\ld|_{V'}$ to the smaller algebra $V'$. It is well-known that the mapping $r:\\Si_V\\rightarrow\\Si_{V'}$ is continuous with respect to the Gel'fand topologies and surjective. Physically, this means that every valuation $\\ld'$ on the smaller algebra is given as the restriction of a valuation on the larger algebra.\n\n\\paragraph{The spectral presheaf.} We can now define the central object in the topos approach to quantum theory: the \\emph{spectral presheaf} $\\Sig$. A \\emph{presheaf} is a contravariant, $\\Set$-valued functor, see e.g. \\cite{McL98}.\\footnote{We will only make minimal use of category theory in this article. In particular, the following definition should be understandable in itself, without further knowledge of functors etc.} Our base category, the domain of this functor, is the context category $\\VN$. To each object in $\\VN$, i.e., to each context $V$, we assign its Gel'fand spectrum $\\Sig_V:=\\Si_V$, and to each arrow in $\\VN$, i.e., each inclusion $i_{V'V}$, we assign a function from the set $\\Sig_V$ to the set $\\Sig_{V'}$. This function is $\\Sig(i_{V'V}):=r$ (see equation \\eq{Restriction}). It implements the concept of coarse-graining on the level of the local state spaces (Gel'fand spectra).\n\nThe spectral presheaf $\\Sig$ associated with the von Neumann algebra $\\N$ of a quantum system is the analogue of the state space of a classical system. The spectral presheaf is not a set, and hence not a space in the usual sense. It rather is a particular, $\\Set$-valued functor, built from the Gel'fand spectra of the abelian subalgebras of $\\N$, and the canonical restriction functions between them.\n\nWe come back to our example, $\\N=\\mc{B}(\\bbC^3)$, and describe its spectral presheaf:\n\n\\begin{example}\t\t\t\\label{Ex_SigOfB(C3)}\nLet $\\N=\\mc{B}(\\bbC^3)$ as in Ex. \\ref{Ex1}, and let $V\\in\\mc{V}(\\bbC^3)$ be a maximal abelian subalgebra. As discussed above, $V$ is of the form $V=\\{\\P_1,\\P_2,\\P_3\\}''$ for three pairwise orthogonal rank-$1$ projections $\\P_1,\\P_2,\\P_3$. The Gel'fand spectrum $\\Si_V$ of $V$ has three elements (and is equipped with the discrete topology, of course). The spectral elements are given as\n\\begin{equation}\n\t\t\t\\ld_i(\\P_j)=\\delta_{ij}\\ \\ \\ \\ (i=1,2,3).\n\\end{equation}\nLet $\\A\\in V$ be an arbitrary operator. Then $\\A=\\sum_{i=1}^3 a_i\\P_i$ for some (unique) complex coefficients $a_i$. We have $\\ld_i(\\A)=a_i$. If $\\A$ is self-adjoint, then the $a_i$ are real and are the eigenvalues of $\\A$.\n\nThe non-maximal abelian subalgebra $V_{\\P_1}$ has two elements in its Gel'fand spectrum. Let us call them $\\ld'_1$ and $\\ld'_2$, where $\\ld'_1(\\P)=1$ and $\\ld'_1(\\hat 1-\\P)=0$, while $\\ld'_2(\\P)=0$ and $\\ld'_2(\\hat 1-\\P)=1$. The restriction mapping $r:\\Si_V\\rightarrow\\Si_{V'}$ sends $\\ld_1$ to $\\ld'_1$. Both $\\ld_2$ and $\\ld_3$ are mapped to $\\ld'_2$.\n\nAnalogous relations hold for the other non-maximal abelian subalgebras $V_{\\P_2}$ and $V_{\\P_3}$ of $V$. Since the spectral presheaf is given by assigning to each abelian subalgebra its Gel'fand spectrum, and to each inclusion the corresponding restriction function between the Gel'fand spectra, we have a complete, explicit description of the spectral presheaf $\\Sig$ of the von Neumann algebra $\\mc{B}(\\bbC^3)$.\n\nThe generalisation to higher dimensions is straightforward. In particular, if $\\rm{dim}(\\Hi)=n$ and $V$ is a maximal abelian subalgebra of $\\mc{B}(\\bbC^n)$, then then Gel'fand spectrum consists of $n$ elements. Topologically, the Gel'fand spectrum $\\Si_V$ is a discrete space. (For infinite-dimensional Hilbert spaces, where more complicated von Neumann algebras than $\\mc{B}(\\bbC^n)$ exist, this is not true in general.)\n\\end{example}\n\n\\section{Representation of propositions in the topos approach -- Daseinisation of projections}\t\\label{S4}\nWe now consider the representation of propositions like ``$\\Ain{\\De}$'' in our topos scheme. We already saw that the spectral presheaf $\\Sig$ is an analogue of the state space $\\mc{S}$ of a classical system. In analogy to classical physics, where propositions are represented by (Borel) subsets of state space, in the topos approach propositions will be represented by suitable subobjects of the spectral presheaf.\n\n\\paragraph{Coarse-graining of propositions.} Consider a proposition ``$\\Ain{\\De}$'', where $A$ is some physical quantity of the quantum system under consideration. We assume that there is a self-adjoint operator $\\A$ in the von Neumann algebra $\\N$ of the system that represents $A$. From the spectral theorem, we know that there is a projection operator $\\P:=\\hat E[\\Ain{\\De}]$ that represents the proposition ``$\\Ain{\\De}$''. As is well-known, the mapping from propositions to projections is many-to-one. One can form equivalence classes of propositions to obtain a bijection. In a slight abuse of language, we will usually refer to \\emph{the} proposition represented by a projection. Since $\\N$ is a von Neumann algebra, the spectral theorem holds. In particular, for all propositions ``$\\Ain\\De$'', we have $\\P=\\hat E[\\Ain\\De]\\in\\PN$. The projections representing propositions are all contained in the von Neumann algebra. If we had chosen a more general $C^*$-algebra instead, this would not have been the case.\n\nWe stipulate that in the topos formulation of quantum theory all classical perspectives\/contexts must be taken into account at the same time, so we have to adapt the proposition ``$\\Ain{\\De}$'' resp. its representing projection $\\P$ to all contexts. The idea is very simple: in every context $V$, we pick the strongest proposition implied by ``$\\Ain{\\De}$'' that can be made from the perspective of this context. On the level of projections, this amounts to taking the \\emph{smallest} projection in any context $V$ that is larger than or equal to $\\P$.\\footnote{Here, we use the interpretation that $\\P<\\Q$ for two projections $\\P,\\Q\\in\\PN$ means that the proposition represented by $\\P$ implies the proposition represented by $\\Q$. This is customary in quantum logic, and we use it to motivate our construction, though the partial order on projections will \\emph{not} be the implication relation for the form of quantum logic resulting from our scheme.} We write, for all $V\\in\\VN$,\n\\begin{equation}\t\t\t\\label{Eq_dastooVP}\n\t\t\t\\dastoo{V}{P}:=\\bigwedge\\{\\Q\\in\\PV\\mid\\Q\\geq\\P\\}\n\\end{equation}\nfor this approximation of $\\P$ to $V$. If $\\P\\in\\PV$, then clearly the approximation will give $\\P$ itself, i.e. $\\dastoo{V}{P}=\\P$. If $\\P\\notin\\PV$, then $\\dastoo{V}{P}>\\P$. For many contexts $V$, it holds that $\\dastoo{V}{P}=\\hat 1$.\n\nSince the projection $\\dastoo{V}{P}$ lies in $V$, it corresponds to a proposition about some physical quantity described by a self-adjoint operator \\emph{in} $V$. If $\\dastoo{V}{P}$ happens to be a spectral projection of the operator $\\A$, then the proposition corresponding to $\\dastoo{V}{P}$ is of the form ``$\\Ain\\Ga$'' for some Borel set $\\Ga$ of real numbers that is larger than the set $\\De$ in the original proposition ``$\\Ain\\De$''.\\footnote{We remark that in order to have $\\dastoo{V}{P}\\in\\PV$, it is sufficient, but \\emph{not} necessary that $\\A\\in V_\\sa$.} The fact that $\\Ga\\supset\\De$ shows that the mapping\n\\begin{eqnarray}\n\t\t\t\\delta^o_V:\\PN &\\longrightarrow& \\PV\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto& \\dastoo{V}{P}\n\\end{eqnarray}\nimplements the idea of coarse-graining on the level of projections (resp. the corresponding propositions).\n\nIf $\\dastoo{V}{P}$ is not a spectral projection of $\\A$, it still holds that it corresponds to a proposition ``$B\\varepsilon\\Ga$'' about the value of some physical quantity $B$ represented by a self-adjoint operator $\\hat B$ in $V$, and since $\\P<\\dastoo{V}{P}$, this proposition can still be seen as a coarse-graining of the original proposition ``$\\Ain\\De$'' corresponding to $\\P$ (even though $A\\neq B$).\n\nEvery projection is contained in some abelian subalgebra, so there is always some $V$ such that $\\dastoo{V}{P}=\\P$. From the perspective of this context, no coarse-graining takes place.\n\nWe call the original proposition ``$\\Ain\\De$'' that we want to represent the \\emph{global} proposition, while a proposition ``$B\\varepsilon\\Ga$'' corresponding to the projection $\\dastoo{V}{P}$ is called a \\emph{local} proposition. `Local' here again means `at the abelian part $V$' and has no spatio-temporal connotation.\n\nWe can collect all the local approximations into a mapping\n\\begin{equation}\n\t\t\t\\P\\mapsto(\\dastoo{V}{P})_{V\\in\\VN}\n\\end{equation}\nwhose component at $V$ is given by \\eq{Eq_dastooVP}. From every global proposition we thus obtain one coarse-grained local proposition for each context $V$. In the next step, we will consider the whole collection of coarse-grained local propositions, relate it to a suitable subobject of the spectral presheaf, and regard this object as the representative of the global proposition.\n\n\\paragraph{Daseinisation of projections.} Let $V$ be a context. Since $V$ is an abelian von Neumann algebra, the lattice $\\PV$ of projections in $V$ is a distributive lattice. Moreover, $\\PV$ is complete and orthocomplemented. We saw in \\eq{alpha} that there is a lattice isomorphism between $\\PV$ and $\\mc{C}l(\\Sig_V)$, the lattice of clopen subsets of $\\Sig_V$.\n\nWe already have constructed a family $(\\dastoo{V}{P})_{V\\in\\VN}$ of projections from a single projection $\\P\\in\\PN$ representing a global proposition. For each context $V$, we now consider the clopen subset $\\alpha(\\dastoo{V}{P})=S_{\\dastoo{V}{P}}\\subseteq\\Si_V$. We thus obtain a family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ of clopen subsets, one for each context. One can show that whenever $V'\\subset V$, then\n\\begin{equation}\n\t\t\tS_{\\dastoo{V}{P}}|_{V'}=\\{\\ld|_{V'} \\mid \\ld\\in S_{\\dastoo{V}{P}}\\}=S_{\\dastoo{V'}{P}},\n\\end{equation}\nthat is, the clopen subsets in the family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ `fit together' under the restriction mappings of the spectral presheaf $\\Sig$ (see Thm. 3.1 in \\cite{DI08}). This means that the family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ of clopen subsets, together with the restriction mappings between them, forms a \\emph{subobject} of the spectral presheaf. This particular subobject will be denoted as $\\ps{\\das{P}}$ and is called the \\emph{daseinisation of $\\P$}.\n\nA subobject of the spectral presheaf (or any other presheaf) is the analogue of a subset of a space. Concretely, for each $V\\in\\VN$, the set $S_{\\dastoo{V}{P}}$ is a subset of the Gel'fand spectrum of $V$. Moreover, the subsets for different $V$ are not arbitrary, but fit together under the restriction mappings of the spectral presheaf.\n\nThe collection of all subobjects of the spectral presheaf is a complete \\emph{Heyting algebra} and is denoted as $\\Sub{\\Sig}$. It is the analogue of the power set of the state space of a classical system.\n\nLet $\\ps S$ be a subobject of the spectral presheaf $\\Sig$ such that the component $S_V\\subseteq\\Sig_V$ is a clopen subset for all $V\\in\\VN$. We call such a subobject a \\emph{clopen} subobject. All subobjects obtained from daseinisation of projections are clopen. One can show that the clopen subobjects form a complete Heyting algebra $\\Subcl{\\Sig}$ (see Thm. 2.5 in \\cite{DI08}). This complete Heyting algebra can be seen as the analogue of the $\\sigma$-complete Boolean algebra of Borel subsets of the state space of a classical system. In our constructions, the Heyting algebra $\\Subcl{\\Sig}$ will be more important than the bigger Heyting algebra $\\Sub{\\Sig}$, just as in classical physics, where the Boolean algebra of measurable subsets of state space is technically more important than the algebra of all subsets.\n\nStarting from a global proposition ``$\\Ain\\De$'' with corresponding projection $\\P$, we have constructed the clopen subobject $\\ps{\\das{P}}$ of the spectral presheaf. This subobject, the analogue of the measurable subset $f_A^{-1}(\\De)$ of the state space of a classical system, is the representative of the global proposition ``$\\Ain\\De$''. The following mapping is called \\emph{daseinisation of projections}:\n\\begin{eqnarray}\n\t\t\\ps\\delta:\\PN &\\longrightarrow& \\Subcl{\\Sig}\\\\\t\t\t\\nonumber\n\t\t\\P &\\longmapsto& \\ps{\\das{P}}.\n\\end{eqnarray}\nIt has the following properties:\n\\begin{itemize}\n\t\\item[(1)] If $\\P<\\Q$, then $\\ps{\\delta(\\P)}<\\ps{\\delta(\\Q)}$, i.e., daseinisation is order-preserving;\n\t\\item[(2)] the mapping $\\ps{\\delta}:\\PN\\rightarrow\\Subcl{\\Sig}$ is injective, that is, two (inequivalent) propositions correspond to two different subobjects;\n\t\\item[(3)] $\\ps{\\delta(\\hat 0)}=\\ps 0$, the empty subobject, and $\\ps{\\delta(\\hat 1)}=\\Sig$. The trivially false proposition is represented by the empty subobject, the trivially true proposition is represented by the whole of $\\Sig$.\n\t\\item[(4)] for all $\\P,\\Q\\in\\PN$, it holds that $\\ps{\\delta(\\P\\vee\\Q)}=\\ps{\\delta(\\P)}\\vee\\ps{\\delta(\\Q)}$, that is, daseinisation preserves the disjunction (Or) of propositions;\n\t\\item[(5)] for all $\\P,\\Q\\in\\PN$, it holds that $\\ps{\\delta(\\P\\wedge\\Q)}\\leq\\ps{\\delta(\\P)}\\wedge\\ps{\\delta(\\Q)}$, that is, daseinisation does not preserve the conjunction (And) of propositions;\n\t\\item[(6)] in general, $\\ps{\\delta(\\P)}\\wedge\\ps{\\delta(\\Q)}$ is not of the form $\\ps{\\delta(\\hat R)}$ for a projection $\\hat R\\in\\PN$, and daseinisation is not surjective.\n\\end{itemize}\nThe domain of the mapping $\\ps\\delta$, the lattice $\\PN$ of projections in the von Neumann algebra $\\N$ of physical quantities, is the lattice that Birkhoff and von Neumann suggested as the algebraic representative of propositional quantum logic (\\cite{BvN36}; in fact, they considered the case $\\N=\\BH$.) Thus daseinisation of projections can be seen as a `translation' mapping between ordinary, Birkhoff-von Neumann quantum logic, which is based upon the non-distributive lattice of projections $\\PN$, and the topos form of propositional quantum logic, which is based upon the distributive lattice $\\Subcl{\\Sig}$. The latter more precisely is a Heyting algebra.\n\nThe quantum logic aspects of the topos formalism and the physical and conceptual interpretation of the relations above are discussed in some detail in \\cite{Doe09}, building upon \\cite{IB98,DI(2),DI08}. The resulting new form of quantum logic has many attractive features and avoids a number of the well-known conceptual problems of standard quantum logic \\cite{DCG02}.\n\n\\begin{example}\nLet $\\Hi=\\bbC^3$, and let $S_z$ be the physical quantity `spin in $z$-direction'. We consider the proposition ``$S_z\\varepsilon(-0.1,0.1)$'', i.e., ``the spin in $z$-direction has a value between $-0.1$ and $0.1$''. (Since the eigenvalues are $-\\frac{1}{\\sqrt 2},0$ and $\\frac{1}{\\sqrt 2}$, this amounts to saying that the spin in $z$-direction is $0$.) The self-adjoint operator representing $S_z$ is\n\\begin{equation}\n\t\t\t\\hat S_z=\\frac{1}{\\sqrt{2}}\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & -1\n\t\t\t\\end{array}\\right),\n\\end{equation}\nand the eigenvector corresponding to the eigenvalue $0$ is $\\ket\\psi=(0,1,0)$. The projection $\\P:=\\hat E[S_z\\varepsilon(-0.1,0.1)]=\\ket\\psi\\bra\\psi$ onto this eigenvector hence is\n\\begin{equation}\n\t\t\t\\P=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right).\n\\end{equation}\nThere is exactly one context $V\\in\\mc{V}(\\bbC^3)$ that contains the operator $\\hat S_z$, namely the (maximal) context $V_{\\hat S_z}=\\{\\P_1,\\P_2,\\P_3\\}''$ generated by the projections \n\\begin{equation}\n\t\t\t\\P_1=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_2=\\P=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_3=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 1\n\t\t\t\\end{array}\\right).\t\t\t\n\\end{equation}\nIn the following, we consider different kinds of contexts $V$ and the approximations $\\dasout{\\P}{V}$ of $\\P=\\P_2$ to these contexts.\n\n\\textbf{(1) $V_{\\hat S_z}$ and its subcontexts:} Of course, $\\dasout{\\P}{V_{\\hat S_z}}=\\P$. Let $V_{\\P_1}=\\{\\P_1,\\P+\\P_3\\}''$, then $\\dasout{\\P}{V_{\\P_1}}=\\P+\\P_3$. Analogously, $\\dasout{\\P}{V_{\\P_3}}=\\P+\\P_3$, but $\\dasout{\\P}{V_{\\P}}=\\P$, since $V_\\P$ contains the projection $\\P$.\n\n\\textbf{(2) Maximal contexts which share the projection $\\P$ with $V_{\\hat S_z}$, and their subcontexts:} Let $\\Q_1,\\Q_3$ be two orthogonal rank-$1$ projections that are both orthogonal to $\\P$ (e.g. projections obtained from $\\P_1,\\P_3$ by a rotation about the axis determined by $\\P$). Then $V=\\{\\Q_1,\\P,\\Q_3\\}''$ is a maximal algebra that contains $\\P$, so $\\dasout{\\P}{V}=\\P$. Clearly, there are uncountably many contexts $V$ of this form.\n\nFor projections $\\Q_1,\\P,\\Q_3$ as described above, we have $\\dasout{\\P}{V_{\\Q_1}}=\\P+\\Q_3$, $\\dasout{\\P}{V_{\\Q_3}}=\\Q_1+\\P$ (and $\\dasout{\\P}{V_\\P}=\\P$, as mentioned before).\n\n\\textbf{(3) Contexts which contain a rank-$2$ projection $\\Q$ such that $\\Q>\\P$:} We first note that there are uncountably many projections $\\Q$ of this form.\n\\begin{enumerate}\n\t\\item [(a)] The trivial cases of contexts containing $\\Q>\\P$ are (1) maximal contexts containing $\\P$; and (2) the context $V_{\\Q}=V_{\\hat 1-\\Q}$.\n\t\\item [(b)] There are also maximal contexts which do not contain $\\P$, but do contain $\\Q$: let $\\Q_1,\\Q_2$ be two orthogonal rank-$1$ projections such that $\\Q_1+\\Q_2=\\Q$, where $\\Q_1,\\Q_2\\neq\\P$. Let $\\Q_3:=\\hat 1-(\\Q_1+\\Q_2)$. For any given $\\Q$, there are uncountably many contexts of the form $V=\\{\\Q_1,\\Q_2,\\Q_3\\}$ (since one can rotate around the axis determined by $\\Q_3$).\n\\end{enumerate}\nFor all these contexts, $\\dasout{\\P}{V}=\\Q$, of course.\n\n\\textbf{(4) Other contexts:} All other contexts neither contain $\\P$ nor a projection $\\Q>\\P$ that is not the identity $\\hat 1$. This means that the smallest projection larger than $\\P$ contained in such a context is the identity, so $\\dasout{\\P}{V}=1$.\n\nWe have constructed $\\dasout{\\P}{V}$ for all contexts. Recalling that for each context $V$, there is a lattice isomorphism $\\alpha$ from the projections to the clopen subsets of the Gel'fand spectrum $\\Si_V$ (see \\eq{alpha}), we can easily write down the clopen subsets $S_{\\dasout{\\P}{V}}\\;(V\\in\\mc V (\\bbC^3))$ corresponding to the projections $\\dasout{\\P}{V}\\;(V\\in\\mc V (\\bbC^3))$. The daseinisation $\\ps{\\das{P}}$ of the projection $\\P$ is nothing but the collection $(S_{\\dasout{\\P}{V}})_{V\\in\\mc V (\\bbC^3)}$ (together with the canonical restriction mappings described in \\eq{Restriction} and Example \\ref{Ex_SigOfB(C3)}).\n\\end{example}\n\n\n\\section{Representation of physical quantities -- Daseinisation of self-adjoint operators}\t\\label{S5}\t\\label{Sec_RepOfPhysQuants}\n\\subsection{Physical quantities as arrows}\t\t\t\\label{SubS_PhysQuantitiesAsArrows}\nIn classical physics, a physical quantity $A$ is represented by a function $f_A$ from the state space $\\mc{S}$ of the system to the real numbers. Both the state space and the real numbers are sets (with extra structure), so they are objects in the topos $\\Set$ of sets and functions.\\footnote{We will not give the technical definition of a topos here. The idea is that a topos is a category that is structurally similar to $\\Set$, the category of small sets and functions. Of course, $\\Set$ itself is a topos.} The function $f_A$ representing a physical quantity is an arrow in the topos $\\Set$.\n\nThe topos reformulation of quantum physics uses structures in the topos $\\SetVNop$, the topos of presheaves over the context category $\\VN$. The objects in the topos, called presheaves, are the analogues of sets, and the arrows between them, called natural transformations, are the analogues of functions. We will now represent a physical quantity $A$ by a suitable arrow, denoted $\\dasB{\\A}$, from the spectral presheaf $\\Sig$ to some other presheaf related to the real numbers. Indeed, there is a real-number object $\\ps{\\bbR}$ in $\\SetVNop$ that is very much like the familiar real numbers: for every $V$, the component $\\ps{\\bbR}_V$ simply is the ordinary real numbers, and all restriction functions are just the identity. But the Kochen-Specker theorem tells us that it is impossible to assign (sharp) real values to all physical quantities, which suggests that the presheaf $\\ps{\\bbR}$ is \\emph{not} the right codomain for our arrows $\\dasB{\\A}$ representing physical quantities.\n\nInstead, we will be using a presheaf, denoted as $\\Rlr$, that takes into account that (a) one cannot assign sharp values to all physical quantities in any given state in quantum theory; and (b) we have coarse-graining. We first explain how coarse-graining will show up: if we want to assign a `value' to a physical quantity $A$, then we again have to find some global expression, involving all contexts $V\\in\\VN$. Let $\\A\\in\\N_\\sa$ be the self-adjoint operator representing $A$. For those contexts $V$ that contain $\\A$, there is no problem, but if $\\A\\notin V$, we will have to approximate $\\A$ by a self-adjoint operator \\emph{in} $V$. Actually, we will use two approximations: one self-adjoint operator in $V$ approximating $\\A$ from below (in a suitable order, specified below), and one operator approximating $\\A$ from above. Daseinisation of self-adjoint operators is nothing but the technical device achieving this approximation.\n\nWe will show that the presheaf $\\Rlr$ basically consists of real intervals, to be interpreted as `unsharp values'. Upon coarse-graining, the intervals can only get bigger. Assume we have given a physical quantity $A$, some state of the system and two contexts $V,V'$ such that $V'\\subset V$. We use the state to assign some interval $[a,b]$ as an `unsharp value' to $A$ at $V$. Then, at $V'$, we will have a bigger interval $[c,d]\\supseteq[a,b]$, being an even more unsharp value of $A$. It it important to note, though, that every self-adjoint operator is contained in some context $V$. If the state is an eigenstate of $\\A$ with eigenvalue $a$, then at $V$, we will assign the `interval' $[a,a]$ to $A$. In this sense, the eigenvector-eigenvalue link is preserved, regardless of the unsharpness in values introduced by coarse-graining. As always in the topos scheme, it is of central importance that the interpretationally relevant structures -- here, the `values' of physical quantities -- are global in nature. One has to consider all contexts at once, and not just argue locally at some context $V$, which will necessarily only give partial information.\n\n\\paragraph{Approximation in the spectral order.} Let $\\A\\in\\N_\\sa$ be a self-adjoint operator, and let $V\\in\\VN$ be a context. We assume that $\\A\\notin V_\\sa$, so we have to approximate $\\A$ in $V$. The idea is very simple: we take a pair of self-adjoint operators, consisting of the largest operator in $V_\\sa$ smaller than $\\A$ and the smallest operator in $V_\\sa$ larger than $\\A$. The only non-trivial point is the question which order on self-adjoint operators to use.\n\nThe most commonly used order is the \\emph{linear order}: $\\A\\leq\\hat B$ if and only if $\\bra\\psi\\A\\ket\\psi\\leq\\bra\\psi\\hat B\\ket\\psi$ for all vector states $\\bra\\psi\\_\\ket\\psi:\\N\\rightarrow\\bbC$. Yet, we will approximate in another order on the self-adjoint operators, the so-called \\emph{spectral order} \\cite{Ols71,deG04}. This has the advantage that the spectra of the approximated operators are subsets of the spectrum of the original operator, which is not the case for the linear order in general.\n\nThe spectral order is defined in the following way: let $\\A,\\hat B\\in\\N_\\sa$ be two self-adjoint operators in a von Neumann algebra $\\N$, and let $(\\hat E^\\A_r)_{r\\in\\bbR},(\\hat E^{\\hat B}_r)_{r\\in\\bbR}$ be their spectral families (see e.g. \\cite{KR83a}). Then\n\\begin{equation}\t\t\t\\label{Def_SpecOrder}\n\t\t\t\\A\\leq_s\\hat B\\text{\\ \\ \\ if and only if\\ \\ \\ }\\forall r\\in\\bbR:\\hat E^\\A_r\\geq\\hat E^{\\hat B}_r.\n\\end{equation}\nEquipped with the spectral order, $\\N_\\sa$ becomes a boundedly complete lattice. The spectral order is coarser than the linear order, i.e., $\\A\\leq_s\\hat B$ implies $\\A\\leq\\hat B$. The two orders coincide on projections and for commuting operators.\n\nConsider the set $\\{\\hat B\\in V_\\sa \\mid \\hat B\\leq_s\\A\\}$, i.e., those self-adjoint operators in $V$ that are spectrally smaller than $\\A$. Since $V$ is a von Neumann algebra, its self-adjoint operators form a boundedly complete lattice under the spectral order, so the above set has a well-defined maximum with respect to the spectral order. It is denoted as\n\\begin{equation}\t\t\\label{dastoiVA}\n\t\t\t\\dastoi{V}{A}:=\\bigvee\\{\\hat B\\in V_\\sa \\mid \\hat B\\leq_s\\A\\}.\n\\end{equation}\nSimilarly, we define\n\\begin{equation}\t\t\\label{dastooVA}\n\t\t\t\\dastoo{V}{A}:=\\bigwedge\\{\\hat B\\in V_\\sa \\mid \\hat B\\geq_s\\A\\},\n\\end{equation}\nwhich is the smallest self-adjoint operator in $V$ that is spectrally larger than $\\A$.\n\nUsing the definition \\eq{Def_SpecOrder} of the spectral order, it is easy to describe the spectral families of the operators $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$:\n\\begin{equation}\t\t\t\\label{Eq_SpecFamOutDas}\n\t\t\t \\forall r\\in\\bbR:\\hat E^{\\dastoo{V}{A}}_r=\\dasinn{\\hat E^\\A_r}{V}\n\\end{equation}\nand\n\\begin{equation}\n\t\t\t\\forall r\\in\\bbR:\\hat E^{\\dastoi{V}{A}}_r=\\dasout{\\hat E^\\A_r}{V}.\n\\end{equation}\nIt turns out (see \\cite{deG07,DI(2)}) that the spectral family defined by the former definition is right-continuous. The latter expression must be amended slightly if we want a right-continuous spectral family: we simply enforce right continuity by setting\n\\begin{equation}\t\t\t\\label{Eq_SpecFamInnDas}\n\t\t\t\\forall r\\in\\bbR:\\hat E^{\\dastoi{V}{A}}_r=\\bigwedge_{s>r}\\dasout{\\hat E^\\A_s}{V}.\n\\end{equation}\nThese equations show the close mathematical link between the approximations in the spectral order of projections and self-adjoint operators.\n\nThe self-adjoint operators $\\dastoi{V}{A},\\dastoo{V}{A}$ are the best approximations to $\\A$ in $V_\\sa$ with respect to the spectral order. $\\dastoi{V}{A}$ approximates $\\A$ from below, $\\dastoo{V}{A}$ from above. Since $V$ is an abelian $C^*$-algebra, we can consider the Gel'fand transforms of these operators to obtain a pair\n\\begin{equation}\n\t\t\t(\\fu{\\dastoi{V}{A}},\\fu{\\dastoo{V}{A}})\n\\end{equation}\nof continuous functions from the Gel'fand spectrum $\\Si_V$ of $V$ to the real numbers. One can show that \\cite{deG05b,DI(3)}\n\\begin{equation}\n\t\t\t\\spec{\\dastoi{V}{A}}\\subseteq\\spec{\\A},\\ \\ \\ \\ \\spec{\\dastoo{V}{A}}\\subseteq\\spec{\\A},\n\\end{equation}\nso the Gel'fand transforms actually map into the spectrum $\\spec{\\A}$ of the original operator $\\A$. Let $\\ld\\in\\Si_V$ be a pure state of the algebra $V$.\\footnote{We remark that in general, this cannot be identified with a state of the non-abelian algebra $\\N$, so we still have a local argument here.} Then we can evaluate $(\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld))$ to obtain a pair of real numbers in the spectrum of $\\A$. The first number is smaller than the second, since the first operator $\\dastoi{V}{A}$ approximates $\\A$ from below in the spectral order (and $\\dastoi{V}{A}\\leq_s\\A$ implies $\\dastoi{V}{A}\\leq\\A$), while the second operator $\\dastoo{V}{A}$ approximates $\\A$ from above. The pair $(\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld))$ of real numbers is identified with the interval $[\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld)]$ and interpreted as the component at $V$ of the unsharp value of $\\A$ in the (local) state $\\ld$.\n\n\\paragraph{The presheaf of `values'.} We now see in which way intervals show up as `values'. If we want to define a presheaf encoding this, we encounter a certain difficulty. It is no problem to assign to each context $V$ the collection $\\mathbb{IR}_V$ of real intervals, but if $V'\\subset V$, then we need a restriction mapping from $\\mathbb{IR}_V$ to $\\mathbb{IR}_{V'}$. Since both sets are the same, the naive guess is to take an interval in $\\mathbb{IR}_V$ and to map it to the same interval in $\\mathbb{IR}_{V'}$. But we already know that the `values' of our physical quantities come from approximations of a self-adjoint operator $\\A$ to all the contexts. If $V'\\subset V$, then in general we have $\\dastoi{V'}{A}<_s\\dastoi{V'}{A}$,\\footnote{This, of course, is nothing but coarse-graining on the level of self-adjoint operators.} which implies $\\fu{\\dastoi{V'}{A}}(\\ld|_{V'})\\leq\\fu{\\dastoi{V}{A}}(\\ld)$ for $\\ld\\in\\Si_V$. Similarly, $\\fu{\\dastoo{V'}{A}}(\\ld|_{V'})\\geq\\fu{\\dastoo{V}{A}}(\\ld)$, so the intervals that we obtain get bigger when we go from larger contexts to smaller ones, due to coarse-graining. Our restriction mapping from the intervals at $V$ to the intervals at $V'$ must take this into account.\n\nThe intervals at different contexts clearly also depend on the state $\\ld\\in\\Si_V$ that one picks. Moreover, we want a presheaf that is not tied to the operator $\\A$, but can provide `values' for all physical quantities and their corresponding self-adjoint operators.\n\nThe idea is to define a presheaf such that at each context $V$, we have all intervals (including those of the form $[a,a]$), plus all possible restrictions to the same or larger intervals at smaller contexts $V'\\subset V$. While this sounds daunting, there is a very simple way to encode this: let $\\downarrow\\!\\!V:=\\{V'\\in\\VN \\mid V'\\subseteq V\\}$. This is a partially ordered set. We now consider a function $\\mu:\\downarrow\\!\\!V\\rightarrow\\bbR$ that \\emph{preserves the order}, that is, $V'\\subset V$ implies $\\mu(V')\\leq\\mu(V)$. Analogously, let $\\nu:\\downarrow\\!\\!V\\rightarrow\\bbR$ denote an \\emph{order-reversing} function, that is, $V'\\subset V$ implies $\\mu(V')\\geq\\mu(V)$. Additionally, we assume that for all $V'\\subseteq V$, it holds that $\\mu(V')\\leq\\nu(V')$. The pair $(\\mu,\\nu)$ thus gives one interval $(\\mu(V'),\\nu(V'))$ for each context $V'\\subseteq V$, and `going down the line' from $V$ to smaller subalgebras $V',V'',...$, these intervals can only get bigger (or stay the same). Of course, each pair $(\\mu,\\nu)$ gives a specific sequence of intervals (for each given $\\ld$). In order to have all possible intervals and sequences built from them, we simply consider the collection of all pairs of order-preserving and -reversing functions.\n\nThis is formalised in the following way: let $\\Rlr$ be the presheaf given\n\\begin{itemize}\n\t\\item [(a)] on objects: for all $V\\in\\VN$, let\n\t\\begin{equation}\n\t\t\t\t\\Rlr_V:=\\{(\\mu,\\nu) \\mid \\mu,\\nu:\\downarrow\\!\\!V\\rightarrow\\bbR,\\mu\\text{ order-preserving, }\\nu\\text{ order-reversing},\\ \\mu\\leq\\nu\\};\n\t\\end{equation}\n\t\\item [(b)] on arrows: for all inclusions $i_{V'V}:V'\\rightarrow V$, let\n\t\\begin{eqnarray}\n\t\t\t\t\\Rlr(i_{V'V}):\\Rlr_V &\\longrightarrow& \\Rlr_{V'}\\\\\t\t\t\\nonumber\n\t\t\t\t(\\mu,\\nu) &\\longmapsto& (\\mu|_{V'},\\nu|_{V'}).\n\t\\end{eqnarray}\n\\end{itemize}\nThe presheaf $\\Rlr$ is where physical quantities take their `values' in the topos version of quantum theory. It is the analogue of the set of real numbers, where physical quantities in classical physics take their values.\n\n\\paragraph{Daseinisation of a self-adjoint operator.} We now start to assemble the approximations to the self-adjoint operator $\\A$ into an arrow from the spectral presheaf $\\Sig$ to the presheaf $\\Rlr$ of values. Such an arrow is a so-called natural transformation (see e.g. \\cite{McL98}), since these are the arrows in the topos $\\SetVNop$. For each context $V\\in\\VN$, we define a function, which we will denote as $\\dasB{\\A}_V$, from the Gel'fand spectrum $\\Sig_V$ of $V$ to the collection $\\Rlr_V$ of pairs of order-preserving and -reversing functions from $\\downarrow\\!\\!V$ to $\\bbR$. In a second step, we will see that these functions fit together in the appropriate sense to form a natural transformation.\n\nThe function $\\dasB{\\A}_V$ is constructed as follows: let $\\ld\\in\\Sig_V$. In each $V'\\in\\downarrow\\!\\!V$, we have one approximation $\\delta^i(\\A)_{V'}$ to $\\A$. If $V''\\subset V'$, then $\\delta^i(\\A)_{V''}\\leq_s\\delta^i(\\A)_{V'}$, which implies $\\delta^i(\\A)_{V''}\\leq\\delta^i(\\A)_{V'}$. One can evaluate $\\ld$ at $\\delta^i(\\A)_{V'}$ for each $V'$, giving an order-preserving function\n\\begin{align}\t\t\t\\label{mu_ld}\n\t\t\t\t\\mu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t \\nonumber\n\t\t\t\tV' &\\longmapsto \\ld|_{V'}(\\delta^i(\\A)_{V'})=\\ld(\\delta^i(\\A)_{V'}).\n\\end{align}\nSimilarly, using the approximations $\\delta^o(\\A)_{V'}$ to $\\A$ from above, we obtain an order-reversing function\n\\begin{align}\t\t\t\\label{nu_ld}\n\t\t\t\t\\nu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t\\nonumber\n\t\t\t\tV' &\\longmapsto \\ld|_{V'}(\\delta^o(\\A)_{V'})=\\ld(\\delta^o(\\A)_{V'}).\n\\end{align}\nThis allows us to define the desired function:\n\\begin{align}\t\t\t\\label{dasB(A)_V}\n\t\t\t\\dasB{\\A}_V:\\Sig_V &\\longrightarrow \\Rlr_V\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto (\\mu_\\ld,\\nu_\\ld).\n\\end{align}\n\nObviously, we obtain one function $\\dasB{\\A}_V$ for each context $V\\in\\VN$. The condition for these functions to form a natural transformation is the following: whenever $V'\\subset V$, one must have $\\dasB{\\A}_V(\\ld)|_{V'}=\\dasB{\\A}_{V'}(\\ld|_{V'})$. In fact, this equality follows trivially from the definition. We thus have arrived at an arrow\n\\begin{equation}\n\t\t\t\\dasB{\\A}:\\Sig\\longrightarrow\\Rlr\n\\end{equation}\nrepresenting a physical quantity $A$. The arrow $\\dasB{\\A}$ is called the \\emph{daseinisation of $\\A$}. This arrow is the analogue of the function $f_A:\\mc{S}\\rightarrow\\bbR$ from the state space to the real numbers representing the physical quantity $A$ in classical physics.\n\n\\paragraph{Pure states and assignment of `values' to physical quantities.} Up to now, we have always argued with elements $\\ld\\in\\Sig_V$. This is a local argument, and we clearly need a global representation of states. Let $\\psi$ be a unit vector in Hilbert space, identified with the pure state $\\bra\\psi\\_\\ket\\psi:\\N\\rightarrow\\bbC$ as usual. In the topos approach, such a pure state is represented by the subobject $\\ps{\\delta(\\P_\\psi)}$ of the spectral presheaf $\\Sig$, i.e., the daseinisation of the projection $\\P_\\psi$ onto the line $\\bbC\\psi$ in Hilbert space. Details can be found in \\cite{DI08}.\n\nThe subobject $\\ps\\wpsi:=\\ps{\\delta(\\P_\\psi)}$ is called the \\emph{pseudo-state} associated to $\\psi$. It is the analogue of a point $s\\in\\mc{S}$ of the state space of a classical system. Importantly, the pseudo-state is not a global element of the presheaf $\\Sig$. Such a global element would be the closest analogue of a point in a set, but as Isham and Butterfield observed \\cite{IB98}, the spectral presheaf $\\Sig$ has no global elements at all -- a fact that is equivalent to the Kochen-Specker theorem. Nonetheless, subobjects of $\\Sig$ of the form $\\ps\\wpsi=\\ps{\\delta(\\P_\\psi)}$ are minimal subobjects in an appropriate sense and as such are as close to points as possible (see \\cite{DI08,Doe08}).\n\nThe `value' of a physical quantity $A$, represented by an arrow $\\dasB{\\A}$, in a state described by $\\ps\\wpsi$ then is\n\\begin{equation}\n \\dasB{\\A}(\\ps\\wpsi).\n\\end{equation}\nThis of course is the analogue of the expression $f_A(s)$ in classical physics, where $f_A$ is the real-valued function on the state space $\\mc{S}$ of the system representing the physical quantity $A$, and $s\\in\\mc{S}$ is the state of the system. \n\nThe expression $\\dasB{\\A}(\\ps\\wpsi)$ turns out to describe a considerably more complicated object than the classical expression $f_A(s)$. One can easily show that $\\dasB{\\A}(\\ps\\wpsi)$ is a subobject of the presheaf $\\Rlr$ of `values', see section 8.5 in \\cite{DI08}. Concretely, it is given at each context $V\\in\\VN$ by\n\\begin{equation}\n (\\dasB{\\A}(\\ps\\wpsi))_V=\\dasB{\\A}_V(\\ps\\wpsi_V)=\\{\\dasB{\\A}_V(\\ld) \\mid \\ld\\in\\ps\\wpsi_V\\}.\n\\end{equation}\nAs we saw above, each $\\dasB{\\A}_V(\\ld)$ is a pair of functions $(\\mu_\\ld,\\nu_\\ld)$ from the set $\\downarrow\\!\\!V$ to the reals, and more specifically to the spectrum $\\spec{\\A}$ of the operator $\\A$. The function $\\mu_\\ld$ is order-preserving, so it takes smaller and smaller values as one goes down from $V$ to smaller subalgebras $V',V'',...$, while $\\nu_\\ld$ is order-reversing and takes larger and larger values. Together, they determine one real interval $[\\mu_\\ld(V'),\\nu_\\ld(V')]$ for every context $V'\\in\\downarrow\\!\\!V$. Every $\\ld\\in\\ps\\wpsi_V$ determines one such sequence of nested intervals. The `value' of the physical quantity $A$ in the given state is the collection of all these sequences of nested intervals for all the different contexts $V\\in\\VN$.\n\nWe conjecture that the usual expectation value $\\bra\\psi\\A\\ket\\psi$ of the physical quantity $A$, which of course is a single real number, can be calculated from the `value' $\\dasB{\\A}(\\ps\\wpsi)$. It is easy to see that each real interval showing up in (the local expressions of) $\\dasB{\\A}(\\ps\\wpsi)$ contains the expectation value $\\bra\\psi\\A\\ket\\psi$.\n\nIn \\cite{Doe08}, it was shown that arbitrary quantum states (not just pure ones) can be represented by probability measures on the spectral presheaf. It was also shown how the expectation values of physical quantities can be calculated using these measures.\n\nInstead of closing this section by showing how the relevant structures look like concretely for our standard example $\\N=\\mc{B}(\\bbC^3)$, we will introduce an efficient way of calculating the approximations $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$ first, which works uniformly for all contexts $V\\in\\VN$. This will simplify the calculation of the arrow $\\dasB{\\A}$ significantly.\n\n\\subsection{Antonymous and observable functions}\nOur aim is to define two functions $g_\\A,f_\\A$ from which we can calculate the approximations $\\dastoi{V}{A}$ resp. $\\dastoo{V}{A}$ to $\\A$ for all contexts $V\\in\\VN$. More precisely, we will describe the Gel'fand transforms of the operators $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$ (for all $V$). These are functions from the Gel'fand spectrum $\\Si_V$ to the reals. The main idea, due to de Groote, is to use the fact that each $\\ld\\in\\Si_V$ corresponds to a maximal \\emph{filter} in the projection lattice $\\PV$ of $V$. Hence, we will consider the Gel'fand transforms as functions on filters, and by `extending' each filter in $\\PV$ to a filter in $\\PN$ (non-maximal in general), we will arrive at functions on filters in $\\PN$, the projection lattice of the non-abelian von Neumann algebra $\\N$. First, we collect a few basic definitions and facts about filters:\n\n\\paragraph{Filters.} Let $\\mathbb{L}$ be a lattice with zero element $0$. A subset $F$ of elements of $\\mathbb{L}$ is a \\emph{(proper) filter} (or \\emph{(proper) dual ideal}) if (i) $0\\notin F$, (ii) $a,b\\in F$ implies $a\\wedge b\\in F$ and (iii) $a\\in F$ and $b\\geq a$ imply $b\\in F$. When we speak of filters, we always mean proper filters. We write $\\mc{F}(\\mathbb{L})$ for the set of filters in a lattice $\\mathbb{L}$. If $\\N$ is a von Neumann algebra, then we write $\\mc{F}(\\N)$ for $\\mc{F}(\\PN)$.\n\nA subset $B$ of elements of $\\mathbb{L}$ is a \\emph{filter base} if (i) $0\\neq B$ and (ii) for all $a,b\\in B$, there exists a $c\\in B$ such that $c\\leq a\\wedge b$.\n\nMaximal filters and maximal filter bases in a lattice $\\mathbb{L}$ with $0$ exist by Zorn's lemma, and each maximal filter is a maximal filter base and vice versa. The maximal filters in the projection lattice of a von Neumann algebra $\\N$ are denoted by $\\mc{Q}(\\N)$. Following de Groote \\cite{deG05c}, one can equip this set with a topology and consider its relations to well-known constructions (in particular, Gel'fand spectra of abelian von Neumann algebras). Here, we will not consider the topological aspects.\n\nIf $V$ is an abelian subalgebra of a von Neumann algebra $\\N$, then the projections in $V$ form a distributive lattice $\\PV$, that is, for all $\\P,\\Q,\\hat{R}\\in\\PV$ we have\n\\begin{eqnarray}\n\t\t\t\\P\\wedge(\\Q\\vee\\hat{R}) &=& (\\P\\wedge\\Q)\\vee(\\P\\wedge\\hat{R}),\\\\\n\t\t\t\\P\\vee(\\Q\\wedge\\hat{R}) &=& (\\P\\vee\\hat{Q})\\wedge(\\P\\vee\\hat{R}).\n\\end{eqnarray}\nThe projection lattice of a von Neumann algebra is distributive if and only if the algebra is abelian.\n\nA maximal filter in a complemented, distributive lattice $\\mathbb{L}$ is called an \\emph{ultrafilter}, hence $\\mc{Q}(\\mathbb{L})$ is the space of ultrafilters in $\\mathbb{L}$. We will not use the notion `ultrafilter' for maximal filters in \\emph{non}-distributive lattices like the projection lattices of non-abelian von Neumann algebras.\n\nThe characterising property of an ultrafilter $U$ in a distributive, complemented lattice $\\mathbb{L}$ is that for each element $a\\in\\mathbb{L}$, either $a\\in U$ or $a^{c}\\in U$, where $a^{c}$ denotes the complement of $a$. This can easily be seen: a complemented lattice has a maximal element $1$, and we have $a\\vee a^{c}=1$ by definition. Let us assume that $U$ is an ultrafilter, but $a\\notin U$ and $a^{c}\\notin U$. In particular, $a\\notin U$ means that there is some $b\\in U$ such that $b\\wedge a=0$. Using distributivity of the lattice $\\mathbb{L}$, we get\n\\begin{equation}\n\t\t\tb=b\\wedge(a\\vee a^{c})=(b\\wedge a)\\vee(b\\wedge a^{c})=b\\wedge a^{c},\n\\end{equation}\nso $b\\leq a^{c}$. Since $b\\in U$ and $U$ is a filter, this implies $a^{c}\\in U$, contradicting our assumption. If $\\mathbb{L}$ is the projection lattice $\\PV$ of an abelian von Neumann algebra $V$, then the maximal element is the identity operator $\\hat{1}$ and the complement of a projection is given as $\\P^{c}:=\\hat{1}-\\P$. Each ultrafilter $q\\in\\mc{Q}(V)$ hence contains either $\\P$ or $\\hat{1}-\\P$ for all $\\P\\in\\PV$.\n\n\\paragraph{Elements of the Gel'fand spectrum and filters.} We already saw that for all projections $\\P\\in\\PV$ and all elements $\\ld$ of the Gel'fand spectrum $\\Si_V$ of $V$, it holds that $\\ld(\\P)\\in\\{0,1\\}$. Given $\\ld\\in\\Si_V$, it is easy to construct a maximal filter $F_\\ld$ in the projection lattice $\\PV$ of $V$: let\n\\begin{equation}\n\t\t\tF_\\ld:=\\{\\P\\in\\PV\\mid\\ld(\\P)=1\\}.\n\\end{equation}\nIt is clear that $\\P\\in F_\\ld$ implies $\\Q\\in F_\\ld$ for all projections $\\Q\\geq\\P$. Let $\\P_{1},\\P_{2}\\in F_{\\ld}$. From the fact that $\\ld$ is a multiplicative state of $V$, we obtain $\\ld(\\P_{1}\\wedge\\P_{2})=\\ld(\\P_{1}\\P_{2})=\\ld(\\P_{1})\\ld(\\P_{2})=1$, so $\\P_{1}\\wedge\\P_{2}\\in F_{\\ld}$. Finally, we have $1=\\ld(\\hat 1)=\\ld(\\P+\\hat 1-\\P)=\\ld(\\P)+\\ld(\\hat 1-\\P)$ for all $\\P\\in\\PV$, so either $\\P\\in F_\\ld$ or $\\hat 1-\\P\\in F_\\ld$ for all $\\P$, which shows that $F_{\\ld}$ is a maximal filter in $\\PV$, indeed. It is straightforward to see that the mapping\n\\begin{align}\t\t\t\\label{Eq_MaxFilterFromld}\n\t\t\t\\beta:\\Si_V &\\longrightarrow \\mathcal{Q}(V)\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto F_\\ld\n\\end{align}\nfrom the Gel'fand spectrum of $V$ to the set $\\mathcal{Q}(V)$ of maximal filters in $\\mathcal{P}(V)$ is injective. This is the first step in the construction of a homeomorphism between these two spaces. De Groote has shown the existence of such a homeomorphism in Thm. 3.2 of \\cite{deG05c}.\n\nGiven a filter $F$ in the projection lattice $\\PV$ of some context $V$, one can `extend' it to a filter in the projection lattice $\\PN$ of the full non-abelian algebra $\\N$ of physical quantities. One simply defines the \\textit{$\\N$-cone over $F$} as\n\\begin{equation}\n\t\t\t\\mc C_{\\N}(F):=\\uparrow\\!\\!F=\\{\\Q\\in\\PN \\mid \\exists\\P\\in F:\\P\\leq\\Q\\}.\n\\end{equation}\nThis is the smallest filter in $\\PN$ containing $F$.\n\nWe need the following technical, but elementary lemma:\n\\begin{lemma}\t\t\t\\label{L_deltaT^-1(D)=Cone(D)}\nLet $\\N$ be a von Neumann algebra, and let $\\mc{T}$ be a von Neumann subalgebra such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$. Let \n\\begin{align}\n\t\t\t\\delta_{\\mc{T}}^{i}:\\PN &\\longrightarrow \\mc{P(T)}\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto \\dastoi{\\mc{T}}{P}:=\\bigvee\\{\\Q\\in\\mc{P(T)} \\mid \\Q\\leq\\P\\}.\n\\end{align}\t\t\t\nThen, for all filters $F\\in\\mc{F}(\\mc{T})$,\n\\begin{equation}\n\t\t\t(\\delta_{\\mc{T}}^{i})^{-1}(F)=\\mc{C}_{\\N}(F).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIf $\\Q\\in F\\subset\\mc{P(T)}$, then $(\\delta_{\\mc{T}}^{i})^{-1}(\\Q)=\\{\\P\\in\\PN\\mid\\delta^{i}(\\hat{P})_{\\mc{T}}=\\Q\\}$. Let $\\P\\in\\PN$ be such that there is a $\\Q\\in F$ with $\\Q\\leq\\P$, i.e., $\\P\\in\\mc{C}_{\\N}(F)$. Then $\\delta^{i}(\\hat{P})_{\\mc{T}}\\geq\\Q$, which implies $\\delta^{i}(\\hat{P})_{\\mc{T}}\\in F$, since $F$ is a filter in $\\mc{P(T)}$. This shows that $\\mc{C}_{\\N}(F)\\subseteq(\\delta_{\\mc{T}}^{i})^{-1}(F)$. Now let $\\P\\in\\PN$ be such that there is no $\\Q\\in F$ with $\\Q\\leq\\P$. Since $\\delta^{i}(\\P)_{\\mc{T}}\\leq\\P$, there also is no $\\hat{Q}\\in F$ with $\\Q\\leq\\delta^{i}(\\P)_{\\mc{T}}$, so $\\P\\notin(\\delta_{\\mc{T}}^{i})^{-1}(F)$. This shows that $(\\delta_{\\mc{T}}^{i})^{-1}(F)\\subseteq\\mc{C}_{\\N}(F)$.\n\\end{proof}\n\n\\paragraph{Definition of antonymous and observable functions.}\nLet $\\N$ be a von Neumann algebra, and let $\\A$ be a self-adjoint operator in $\\A$. As before, we denote the spectral family of $\\A$ as $(\\hat E^\\A_r)_{r\\in\\bbR}=\\hat E^\\A$.\n\\begin{definition}\nThe \\emph{antonymous function of $\\A$} is defined as\n\\begin{align}\n\t\t\tg_\\A:\\mc F (\\N) &\\longrightarrow \\spec{\\A}\\\\\n\t\t\tF &\\longmapsto \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^\\A_r\\in F\\}.\n\\end{align}\nThe \\emph{observable function of $\\A$} is\n\\begin{align}\n\t\t\tf_\\A:\\mc F (\\N) &\\longrightarrow \\spec{\\A}\\\\\n\t\t\tF &\\longmapsto \\on{inf}\\{r\\in\\bbR \\mid \\hat E^\\A_r\\in F\\}.\n\\end{align}\n\\end{definition}\nIt is straightforward to see that the range of these functions indeed is the spectrum $\\spec{\\A}$ of $\\A$.\n\nWe observe that the mapping described in \\eq{dastoiVA} can be generalised: when approximating a self-adjoint operator $\\A$ with respect to the spectral order in a von Neumann subalgebra of $\\N$, then one can use an arbitrary, not necessarily abelian subalgebra $\\mc T$ of the non-abelian algebra $\\N$. The only condition is that the unit elements in $\\N$ and $\\mc T$ coincide. We define\n\\begin{align}\n\t\t\t\\delta^i_{\\mc T}:\\N_\\sa &\\longrightarrow \\mc T_\\sa\\\\\t\t\\nonumber\n\t\t\t\\A &\\longmapsto \\dastoi{\\mc T}{A}=\\bigvee\\{\\hat B\\in\\mc T_\\sa \\mid \\hat B\\leq_s\\A\\}.\n\\end{align}\nAnalogously, generalising \\eq{dastooVA}\n\\begin{align}\n\t\t\t\\delta^o_{\\mc T}:\\N_\\sa &\\longrightarrow \\mc T_\\sa\\\\\t\t\\nonumber\n\t\t\t\\A &\\longmapsto \\dastoo{\\mc T}{A}=\\bigwedge\\{\\hat B\\in\\mc T_\\sa \\mid \\hat B\\geq_s\\A\\}.\n\\end{align}\nThe following proposition is central:\n\\begin{proposition}\t\t\t\\label{P_g_AEncodesAllg_delta^i(A)}\nLet $\\A\\in\\N_{sa}$. For all von Neumann subalgebras $\\mc{T}\\subseteq\\N$ such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$ and all filters $F\\in\\mc{F(T)}$, we have\n\\begin{equation}\n\t\t\tg_{\\dastoi{\\mc{T}}{A}}(F)=g_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWe hav\n\\begin{eqnarray*}\n\t\t\tg_{\\delta^{i}(\\A)_{\\mc{T}}}(F) &=& \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\dastoi{\\mc{T}}{A}}\\in F\\}\\\\\n\t\t\t&\\stackrel{\\eq{Eq_SpecFamInnDas}}{=}& \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\bigwedge_{\\mu>r}\\delta^{o}(\\hat{E}_{\\mu}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=& \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\delta^{o}(\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=& \\sup\\{r\\in\\bbR \\mid \\delta^{i}(\\hat{1}-\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=& \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\hat A}\\in(\\delta_{\\mc{T}}^{i})^{-1}(F)\\}\\\\\n\t\t\t&\\stackrel{\\text{Lemma \\ref{L_deltaT^-1(D)=Cone(D)}}}{=}& \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\hat A}\\in\\mc{C}_{\\N}(F)\\}\\\\\n\t\t\t&=& g_{\\A}(\\mc{C}_{\\N}(F))\n\\end{eqnarray*}\n\\end{proof}\n\nThis shows that the antonymous function $g_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ of $\\A$ encodes in a simple way \\emph{all} the antonymous functions $g_{\\delta^{i}(\\A)_{\\mc{T}}}:\\mc{F(T)}\\rightarrow\\spec{\\delta^{i}(\\A)}_{\\mc{T}}$ corresponding to the approximations $\\delta^{i}(\\A)_{\\mc{T}}$ (from below in the spectral order) to $\\A$ to von Neumann subalgebras $\\mc{T}$ of $\\N$. If, in particular, $\\mc{T}$\nis an abelian subalgebra, then the set $\\mc{Q(T)}$ of \\emph{maximal} filters in $\\mc{D(T)}$ can be identified with the Gel'fand spectrum of $\\mc{T}$ (see \\cite{deG05b}), and $g_{\\delta^{i}(\\A)_{\\mc{T}}}|_{\\mc{Q(T)}}$ can be identified with the Gel'fand transform of\n$\\delta^{i}(\\A)_{\\mc{T}}$. Thus we get\n\n\\begin{corollary}\nThe antonymous function $g_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ encodes all the Gel'fand transforms of the inner daseinised operators of the form $\\dastoi{V}{A}$, where $V\\in\\VN$ is an abelian von Neumann subalgebra of $\\N$ (such that $\\hat 1_V=\\hat 1_\\N$). Concretely, if $\\mc{Q}(V)$ is the space of maximal filters in $\\PV$ and $\\Si_V$ is the Gel'fand spectrum of $V$, then we have an identification $\\beta:\\Sig_V\\rightarrow\\mc{Q}(V)$, $\\ld\\mapsto F_\\ld$ (see \\eq{Eq_MaxFilterFromld}). Let $\\fu{\\dastoi{V}{A}}:\\Si_V\\rightarrow\\spec{\\dastoi{V}{A}}$ be the Gel'fand transform of $\\dastoi{V}{A}$. We can identify $\\fu{\\dastoi{V}{A}}$ with $g_{\\dastoi{V}{A}}|_{\\mc{Q}(V)}$, and hence, from Prop. \\ref{P_g_AEncodesAllg_delta^i(A)},\n\\begin{equation}\t\t\t\\label{EqInnDasFromAnton}\n\t\t\t\\fu{\\dastoi{V}{A}}(\\ld)=g_{\\dastoi{V}{A}}(F_\\ld)=g_{\\A}(\\mc{C}_{\\N}(F_\\ld)).\n\\end{equation}\n\\end{corollary}\n\nNot surprisingly, there is a similar result for observable functions and outer daseinisation. This result was first proved in a similar form by de Groote in \\cite{deG07}.\n\n\\begin{proposition}\t\t\t\\label{P_f_AEncodesAllf_delta^o(A)}\nLet $\\A\\in\\N_{sa}$. For all von Neumann subalgebras $\\mc{T}\\subseteq\\N$ such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$ and all\nfilters $F\\in\\mc{F(T)}$, we have\n\\begin{equation}\n\t\t\tf_{\\dastoo{\\mc{T}}{A}}(F)=f_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWe have\n\\begin{eqnarray}\n\t\t\tf_{\\dastoo{\\mc{T}}{A}}(F) &=& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\dastoo{\\mc{T}}{A}}\\in F\\}\\\\\n\t\t\t&\\stackrel{\\eq{Eq_SpecFamOutDas}}{=}& \\inf\\{r\\in\\bbR \\mid \\delta^{i}(\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\hat A}\\in(\\delta_{\\mc{T}}^{i})^{-1}(F)\\}\\\\\n\t\t\t&\\stackrel{\\text{Lemma \\ref{L_deltaT^-1(D)=Cone(D)}}}{=}& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\hat A}\\in\\mc{C}_{\\N}(F)\\}\\\\\n\t\t\t&=& f_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{eqnarray}\n\\end{proof}\n\nThis shows that the observable function $f_\\A:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ of $\\A$ encodes in a simple way \\emph{all} the observable functions $f_{\\dastoo{\\mc{T}}{A}}:\\mc{F}(\\N)\\rightarrow\\spec{\\dastoo{\\mc{T}}{A}}$ corresponding to approximations $\\dastoo{\\mc{T}}{A}$ (from above in the spectral order) to $\\A$ to von Neumann subalgebras $\\mc{T}$ of $\\N$. If, in particular, $\\mc{T}=V$ is an abelian subalgebra, then we get\n\\begin{corollary}\nThe observable function $f_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ encodes all the Gel'fand transforms of the outer daseinised operators of the form $\\dastoo{V}{A}$, where $V\\in\\VN$ is an abelian von Neumann subalgebra of $\\N$ (such that $\\hat 1_V=\\hat 1_\\N$). Using the identification $\\beta:\\Sig_V\\rightarrow\\mc{Q}(V)$, $\\ld\\mapsto F_{\\ld}$ between the Gel'fand and the space of maximal filters (see \\eq{Eq_MaxFilterFromld}), we can identify the Gel'fand transform $\\fu{\\dastoo{V}{A}:\\Sig_V\\rightarrow\n\\spec{\\dastoo{V}{A}}}$ of $\\dastoo{V}{A}$ with $f_{\\dastoo{V}{A}}|_{\\mc{Q(T)}}$, and hence, from Prop. \\ref{P_f_AEncodesAllf_delta^o(A)},\n\\begin{equation}\n\t\t\t\\fu{\\dastoo{V}{A}}(\\ld)=f_{\\dastoo{V}{A}}(F_{\\ld})=f_\\A(\\mc{C}_{\\N}(F_{\\ld})).\n\\end{equation}\n\\end{corollary}\n\n\\paragraph{Vector states from elements of Gel'fand spectra.} As emphasised in section \\ref{SubS_PhysQuantitiesAsArrows}, an element of $\\ld$ of the Gel'fand spectrum $\\Sig_V$ of some context $V$ is only a local state: it is a pure state of the abelian subalgebra $V$, but it is not a state of the full non-abelian algebra $\\N$ of physical quantities. We will now show how certain local states $\\ld$ can be extended to global states, i.e., states of $\\N$.\n\nLet $V$ be a context that contains at least one rank-$1$ projection.\\footnote{Depending on whether the von Neumann algebra $\\N$ contains rank-$1$ projections, there may or may not be such contexts. For the case $\\N=\\BH$, there of course are many rank-$1$ projections and hence contexts $V$ of the required form.} Let $\\P$ be such a projection, and let $\\psi\\in\\Hi$ be a unit vector such that $\\P$ is the projection onto the ray $\\bbC\\psi$. ($\\psi$ is fixed up to a phase.) We write $\\P=\\P_\\psi$. Then\n\\begin{equation}\t\t\t\\label{F_P_psi}\n\t\t\tF_{\\P_\\psi}:=\\{\\Q\\in\\PV \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nclearly is an ultrafilter in $\\PV$. It thus corresponds to some element $\\ld_\\psi$ of the Gel'fand spectrum $\\Sig_V$ of $V$. Being an element of the Gel'fand spectrum, $\\ld_\\psi$ is a pure state of $V$. The cone\n\\begin{equation}\n\t\t\t\\mc C_{\\N}(F_{\\P_\\psi})=\\{\\Q\\in\\PN \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nis a maximal filter in $\\PN$. It contains all projections in $\\N$ that represent propositions ``$\\Ain\\De$'' that are (totally) true in the vector state $\\psi$ on $\\N$.\\footnote{Here, we again use the usual identification between unit vectors and vector states, $\\psi\\mapsto\\bra\\psi\\_\\ket\\psi$.} This vector state is uniquely determined by the cone $\\mc C_{\\N}(F_{\\P_\\psi})$. The `local state' $\\ld_\\psi$ on the context $V\\subset\\N$ can thus be extended to a `global' vector state. On the level of filters, this extension corresponds to the cone construction.\n\nFor a finite-dimensional Hilbert space $\\Hi$, one necessarily has $\\N=\\BH$. Let $V$ be an arbitrary maximal context, and let $\\ld\\in\\Sig_V$. Then $\\ld$ is of the form $\\ld=\\ld_{\\P_\\psi}$ for some unit vector $\\psi\\in\\Hi$ and corresponding rank-$1$ projection $\\P_\\psi$. Hence, every such $\\ld_{\\P_\\psi}$ can be extended to a vector state $\\psi$ on the whole of $\\BH$.\n\n\\paragraph{The eigenstate-eigenvalue link.} We will now show that the arrow $\\dasB{\\A}$ constructed from a self-adjoint operator $\\A\\in\\N_{\\sa}$ preserves the eigenstate-eigenvalue link in a suitable sense. We employ the relation between ultrafilters in $\\PV$ of the form $F_{\\P_\\psi}$ (see equation \\eq{F_P_psi}) and maximal filters in $\\PN$ established in the previous paragraph, but `read it backwards'.\n\nLet $\\A$ be some self-adjoint operator, and let $\\psi$ be an eigenstate of $\\A$ with eigenvalue $a$. Let $V$ be a context that contains $\\A$, and let $\\P_\\psi$ be the projection determined by $\\psi$, i.e., the projection onto the one-dimensional subspace (ray) $\\bbC\\psi$ of Hilbert space. Then $\\P_\\psi\\in V$ from the spectral theorem. Consider the maximal filter\n\\begin{equation}\n\t\t\tF:=\\{\\Q\\in\\PN \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nin $\\PN$ determined by $\\P_\\psi$. Clearly, $F\\cap\\PV=\\{\\Q\\in\\PV \\mid \\Q\\geq\\P_\\psi\\}$ is an ultrafilter in $\\PV$, namely the ultrafilter $F_{\\P_\\psi}$ from equation \\eq{F_P_psi}, and $F=\\mc C_{\\N}(F_{\\P_\\psi})$. Let $\\ld_\\psi$ be the element of $\\Sig_V$ (i.e., local state of $V$) determined by $F_{\\P_\\psi}$. The state $\\ld_\\psi$ is nothing but the restriction of the vector state $\\bra\\psi\\_\\ket\\psi$ (on $\\N$) to the context $V$, and hence \n\\begin{equation}\n\t\t\t\\bra\\psi\\A\\ket\\psi=\\ld_\\psi(\\A)=a.\n\\end{equation}\nSince $\\A\\in V_{\\sa}$, we have $\\dastoi{V}{A}=\\dastoo{V}{A}=\\A$. This implies that \n\\begin{align}\t\t\t\\nonumber\n\t\t\t\\dasB{\\A}_V(\\ld_\\psi)(V) &= [g_{\\dastoi{V}{A}}(\\ld_\\psi),f_{\\dastoo{V}{A}}(\\ld_\\psi)]\\\\\t\t\t\\nonumber\n\t\t\t&= [g_\\A(\\ld_\\psi),f_\\A(\\ld_\\psi)]\\\\\n\t\t\t&= [\\fu{A}(\\ld_\\psi),\\fu{A}(\\ld_\\psi)]\\\\\t\t\\nonumber\n\t\t\t&= [\\ld_\\psi(\\A),\\ld_\\psi(\\A)]\\\\\t\t\\nonumber\n\t\t\t&= [a,a],\n\\end{align}\ni.e., the arrow $\\dasB{\\A}$ delivers the (interval only containing the) eigenvalue $a$ for an eigenstate $\\psi$ (resp. the corresponding $\\ld_\\psi$) at a context $V$ which actually contains $\\A$. In this sense, the eigenstate-eigenvalue link is preserved, and locally at $V$ the value of $\\A$ actually becomes a single real number as expected.\n\n\\begin{example}\t\t\t\\label{Ex_ValueOfS_z}\nWe return to our example, the spin-$1$ system. It is described by the algebra $\\mc{B}(\\bbC^3)$ of bounded operators on the three-dimensional Hilbert space $\\bbC^3$. In particular, the spin-$z$ operator is given by\n\\begin{equation}\n\t\t\t\\hat S_z=\\frac{1}{\\sqrt{2}}\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & -1\n\t\t\t\\end{array}\\right).\n\\end{equation}\nThis is the matrix expression for $\\hat S_z$ with respect to the basis $e_1,e_2,e_3$ of eigenvectors of $\\hat S_z$. Let $\\P_1,\\P_2,\\P_3$ be the corresponding rank-$1$ projections onto the eigenspaces. The algebra $V_{\\hat S_z}:=\\{\\P_1,\\P_2,P_3\\}''$ generated by these projections is a maximal abelian subalgebra of $\\mc{B}(\\bbC^3)$, and it is the only maximal abelian subalgebra containing $\\hat S_z$.\n\nLet us now consider how the approximations $\\dastoi{V}{S_z}$ and $\\dastoo{V}{S_z}$ of $\\hat S_z$ to other contexts $V$ look like. In order to do so, we first determine the antonymous and the observable function of $\\hat S_z$. The spectral family of $\\hat S_z$ is given by\n\\begin{equation}\t\t\t\\label{SpecFamOfSz}\n\t\t\t\\hat E^{\\hat S_z}_\\ld=\\left\\{\n\t\t\t\\begin{tabular}\n\t\t\t\t\t\t[c]{ll\n\t\t\t\t\t\t$\\hat 0$ & if $\\ld<-\\frac{1}{\\sqrt 2}$\\\\\n\t\t\t\t\t\t$\\P_3$ & if $-\\frac{1}{\\sqrt 2}\\leq\\ld<0$\\\\\n\t\t\t\t\t\t$\\P_3+\\P_2$ & if $0\\leq\\ld<\\frac{1}{\\sqrt 2}$\\\\\n\t\t\t\t\t\t$\\hat 1$ & if $\\ld\\geq \\frac{1}{\\sqrt 2}$.\n\t\t\t\\end{tabular}\n\t\t\t\\ \\right.\n\\end{equation}\nThe antonymous function $g_{\\hat S_z}$ of $\\hat S_z$ is\n\\begin{align}\n\t\t\tg_{\\hat S_z}:\\mc{F}(\\mc{B}(\\bbC^3)) &\\longrightarrow \\spec{\\hat S_z}\\\\ \\nonumber\n\t\t\tF &\\longmapsto \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\in F\\},\n\\end{align}\nand the observable function is\n\\begin{align}\n\t\t\tf_{\\hat S_z}:\\mc{F}(\\mc{B}(\\bbC^3)) &\\longrightarrow \\spec{\\hat S_z}\\\\ \\nonumber\n\t\t\tF &\\longmapsto \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\in F\\}.\n\\end{align}\nLet $\\Q_1,\\Q_2,\\Q_3$ be three pairwise orthogonal rank-$1$ projections, and let $V=\\{\\Q_1,\\Q_2,\\Q_3\\}''\\in\\mc V (\\mc B (\\bbC^3))$ be the maximal context determined by them. Then the Gel'fand spectrum $\\Sig_V$ has three elements; let us denote them $\\ld_1,\\ld_2$ and $\\ld_3$ (where $\\ld_i(\\Q_j)=\\delta_{ij}$). Clearly, $\\Q_i$ is the smallest projection in $V$ such that $\\ld_i(\\Q_i)=1\\;(i=1,2,3)$. Each $\\ld_i$ defines a maximal filter in the projection lattice $\\PV$ of $V$:\n\\begin{equation}\n\t\t\tF_{\\ld_i}=\\{\\Q\\in\\PV \\mid \\Q\\geq\\Q_i\\}.\n\\end{equation}\nThe Gel'fand transform $\\fu{\\dastoi{V}{S_z}}$ of $\\dastoi{V}{S_z}$ is a function from the Gel'fand spectrum $\\Sig_V$ of $V$ to the spectrum of $\\hat S_z$. Equation \\eq{EqInnDasFromAnton} shows how to calculate this function from the antonymous function $g_{\\hat S_z}$:\n\\begin{equation}\n\t\t\t\t\\fu{\\dastoi{V}{S_z}}(\\ld_i)=g_{\\hat S_z}(\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})),\n\\end{equation}\nwhere $\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})$ is the cone over the filter $F_{\\ld_i}$, i.e.,\n\\begin{align}\t\t\t\\nonumber\n\t\t\t\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i}) \n\t\t\t&= \\{\\hat R\\in\\mc P (\\mc B(\\bbC^3)) \\mid \\exists \\Q\\in F_{\\ld_i}:\\hat R\\geq\\Q\\}\\\\\n\t\t\t&= \\{\\hat R\\in\\mc P (\\mc B(\\bbC^3)) \\mid \\hat R\\geq\\Q_i\\}.\n\\end{align}\nNow it is easy to actually calculate $\\fu{\\dastoi{V}{S_z}}$: for all $\\ld_i\\in\\Sig_V$,\n\\begin{align}\t\t\t\\label{deltai(Sz)FromAnton}\t\\nonumber\n\t\t\t\\fu{\\dastoi{V}{S_z}}(\\ld_i)\n\t\t\t&= \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\in\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})\\}\\\\\n\t\t\t&= \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\geq\\Q_i\\}.\n\\end{align}\nAnalogously, we obtain $\\fu{\\dastoo{V}{S_z}}$: for all $\\ld_i$,\n\\begin{align}\t\t\t\\label{deltao(Sz)FromAnton}\t\\nonumber\n\t\t\t\\fu{\\dastoo{V}{S_z}}(\\ld_i)\n\t\t\t&= \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\in\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})\\}\\\\\n\t\t\t&= \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\geq\\Q_i\\}.\n\\end{align}\nUsing the expression \\eq{SpecFamOfSz} for the spectral family of $\\hat S_z$, we can see directly that the values $\\fu{\\dastoi{V}{S_z}}(\\ld_i),\\fu{\\dastoo{V}{S_z}}(\\ld_i)$ lie in the spectrum of $\\hat S_z$ as expected. A little less obviously, $\\fu{\\dastoi{V}{S_z}}(\\ld_i)\\leq\\fu{\\dastoo{V}{S_z}}(\\ld_i)$, so we can think of the pair of values as an interval $[\\fu{\\dastoi{V}{S_z}}(\\ld_i),\\fu{\\dastoo{V}{S_z}}(\\ld_i)]$.\n\nWe had assumed that $V=\\{\\Q_1,\\Q_2,\\Q_2\\}''$ is a \\emph{maximal} abelian subalgebra, but this is no actual restriction. A non-maximal subalgebra $V'$ has a Gel'fand spectrum $\\Sig_{V'}$ consisting of two elements. All arguments work analogously. The important point is that for each element $\\ld$ of the Gel'fand spectrum, there is a unique projection $\\Q$ in $V'$ corresponding to $\\ld$, given as the smallest projection in $V'$ such that $\\ld(\\Q)=1$.\n\nThis means that we can now write down explicitly the natural transformation $\\dasB{S_z}:\\Sig\\rightarrow\\Rlr$, the arrow in the presheaf topos representing the physical quantity `spin in $z$-direction', initially given by the self-adjoint operator $\\hat S_z$. For each context $V\\in\\mc V (\\mc B (\\bbC^3))$, we have a function\n\\begin{align}\t\t\t\\label{dasB(S_z)_V}\n\t\t\t\\dasB{S_z}_V:\\Sig_V &\\longrightarrow \\Rlr_V\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto (\\mu_{\\ld},\\nu_{\\ld}),\n\\end{align}\ncompare \\eq{dasB(A)_V}. According to \\eq{mu_ld}, $\\mu_\\ld:\\downarrow\\!\\!V\\rightarrow\\bbR$ is given as\n\\begin{align}\n\t\t\t\t\\mu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t \\nonumber\n\t\t\t\tV' &\\longmapsto \\ld(\\delta^i(\\hat S_z)_{V'})=\\fu{\\delta^i(\\hat S_z)_{V'}}(\\ld).\n\\end{align}\nLet $\\Q_{V'}\\in\\mc P (V')$ be the projection in $V'$ corresponding to $\\ld$, that is, the smallest projection in $V'$ such that $\\ld(\\Q_{V'})=1$. Note that this projection $\\Q_{V'}$ depends on $V'$. For each $V'\\in\\downarrow\\!\\!V$, the element $\\ld$ (originally an element of the Gel'fand spectrum $\\Sig_V$) is considered as an element of $\\Sig_{V'}$, given by the restriction $\\ld|_{V'}$ of the original $\\ld$ to the smaller algebra $V'$. Then \\eq{deltai(Sz)FromAnton} implies, for all $V'\\in\\downarrow\\!\\!V$,\n\\begin{equation}\n\t\t\t\t\\mu_\\ld(V')=\\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}\\geq\\Q_{V'}\\}.\n\\end{equation}\nFrom \\eq{nu_ld}, we obtain\n\\begin{align}\n\t\t\t\t\\nu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t\\nonumber\n\t\t\t\tV' &\\longmapsto \\ld(\\delta^o(\\hat S_z)_{V'})=\\fu{\\delta^o(\\hat S_z)_{V'}}(\\ld),\n\\end{align}\nand with \\eq{deltao(Sz)FromAnton}, we get for all $V'\\in\\downarrow\\!\\!V$,\n\\begin{equation}\n\t\t\t\t\\nu_\\ld(V')=\\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}\\geq\\Q_{V'}\\}.\n\\end{equation}\n\nWe finally want to calculate the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$ of the physical quantity `spin in $z$-direction' in the (pseudo-)state $\\ps\\wpsi$. As described in section \\ref{SubS_PhysQuantitiesAsArrows}, $\\psi$ is a unit vector in the Hilbert space (resp. a vector state) and $\\ps\\wpsi=\\ps{\\delta(\\P_\\psi)}$ is the corresponding pseudo-state, a subobject of $\\Sig$. The `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$ is given at $V\\in\\mc V (\\mc B (\\bbC^3))$ as\n\\begin{equation}\n (\\dasB{\\hat S_z}(\\ps\\wpsi))_V=\\dasB{\\hat S_z}_V(\\ps\\wpsi_V)\n =\\{\\dasB{\\hat S_z}_V(\\ld) \\mid \\ld\\in\\ps\\wpsi_V\\}.\n\\end{equation}\nHere, $\\ps\\wpsi_V=\\alpha(\\delta^o(\\P_\\psi)_V)=S_{\\delta^o(\\P_\\psi)_V}$ is a (clopen) subset of $\\Sig_V$ (compare \\eq{alpha}). This means that for each context $V$, the state determines a collection $\\ps\\wpsi_V$ of elements of the Gel'fand spectrum $\\Sig_V$ of $V$. We then evaluate the component $\\dasB{\\hat S_z}_V$ of the arrow\/natural transformation representing spin-$z$, given by \\eq{dasB(S_z)_V} (and subsequent equations), on all the $\\ld\\in\\ps\\wpsi_V$ to obtain the component at $V$ of the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$. Each $\\ld\\in\\ps\\wpsi$ gives a sequence of intervals; one interval for each $V'\\in\\downarrow\\!\\!V$ such that if $V''\\subset V'$, the interval at $V''$ contains the interval at $V'$.\n\\end{example}\n\nThe example can easily be generalised to other operators and higher dimensions. Other finite-dimensional Hilbert spaces present no further conceptual difficulty at all. Of course, infinite-dimensional Hilbert spaces bring a host of new technical challenges, but the main tools used in the calculation, the antonymous and observable functions ($g_\\A$ resp. $f_\\A$), are still available. Since they encode the approximations in the spectral order of an operator $\\A$ to \\emph{all} contexts $V\\in\\VN$, the natural transformation $\\dasB{\\A}$ corresponding to a self-adjoint operator $\\A$ can be written down efficiently, without the need to actually calculate the approximations to all contexts separately.\n\n\\section{Conclusion}\t\\label{S6}\nWe have shown how daseinisation relates central aspects of the standard Hilbert space formalism of quantum theory to the topos formalism. Daseinisation of projections gives subobjects of the spectral presheaf $\\Sig$. These subobjects form a Heyting algebra, and every pure state allows to assign truth-values to all propositions. The resulting new form of quantum logic is contextual, multi-valued and intuitionistic. For more details on the logical aspects, see \\cite{DI08,Doe07b,Doe09}.\n\nThe daseinisation of self-adjoint operators gives arrows from the state object $\\Sig$ to the presheaf $\\Rlr$ of `values'. Of course, the `value' $\\dasB{\\A}(\\ps\\wpsi)$ of a physical quantity in a pseudo-state $\\ps\\wpsi$ is considerably more complicated than the value of a physical quantity in classical physics, which is just a real number. The main point, though, is that in the topos approach all physical quantities \\emph{do} have (generalised) values in any given state -- something that clearly is not the case in ordinary quantum mechanics. Moreover, we have shown in Example \\ref{Ex_ValueOfS_z} how to calculate the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$.\n\nRecently, Heunen, Landsman and Spitters suggested a closely related scheme using topoi in quantum theory (see \\cite{HLS09} as well as their contribution to this volume and references therein). All the basic ingredients are the same: a quantum system is described by an algebra of physical quantities, in their case a Rickart $C^*$-algebra, associated with this algebra is a spectral object whose subobjects represent propositions, and physical quantities are represented by arrows in a topos associated with the quantum system. The choice of topos is very similar to ours: as the base category, one considers all abelian subalgebras (i.e., contexts) of the algebra of physical quantities and orders them partially under inclusion. In our scheme, we choose the topos to be contravariant, $\\Set$-valued functors (called \\emph{presheaves}) over the context category, while Heunen et al. choose \\emph{co}variant functors. The use of covariant functors allows the construction of the spectral object as the topos-internal Gel'fand spectrum of a topos-internal abelian $C^*$-algebra canonically defined from the external non-abelian algebra of physical quantities. \n\nDespite the similarities, there are some important conceptual and interpretational differences between the original contravariant approach and the covariant approach. These differences and their physical consequences will be discussed in a forthcoming article \\cite{Doe09b}.\n\nSumming up, the topos approach provides a reformulation of quantum theory in a way that seemed impossible up to now due to powerful no-go theorems like the Kochen-Specker theorem. In spite of the Kochen-Specker theorem, there is a suitable notion of state `space' for a quantum system in analogy to classical physics: the spectral presheaf. \n\n\\paragraph{Acknowledgements.} I would like to thank Chris Isham for the great collaboration and his constant support. 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Providence RI 02912, USA, \\email{braval@math.brown.edu} \\and\nIMU, IITP and National Research University\nHigher School of Economics\nDepartment of Mathematics, 20 Myasnitskaya st, Moscow 101000 Russia,\n\\email{fnklberg@gmail.com}}\n\\date{Received: \/ Revised version: }\n\\maketitle\n\\begin{abstract}\nLet $G$ be a semi-simple simply connected group over $\\CC$. Following \\cite{GLOI} we use the $q$-Toda integrable system obtained by quantum group version\nof the\nKostant-Whittaker reduction (cf. \\cite{Et} and \\cite{Sev}) to define the notion of $q$-Whittaker functions $\\Psi_{\\chlam}(q,z)$.\nThis is a family of invariant polynomials on the maximal torus $T\\subset G$ (here $z\\in T$) depending\non a dominant weight $\\check\\lam$ of $G$ whose coefficients are rational functions in a variable $q\\in \\CC^*$.\nFor a conjecturally the same (but a priori different) definition of the $q$-Toda system these functions were studied\nby B.~Ion in \\cite{Ion} and by I.~Cherednik in \\cite{C} (we shall denote the $q$-Whittaker functions from \\cite{C}\nby $\\Psi'_{\\chlam}(q,z)$). For $G=SL(N)$ these functions were extensively studied in \\cite{GLOI}-\\cite{GLOIII}.\n\nWe show that when $G$ is simply laced, the function $\\hatPsi_{\\check\\lam}(q,z)=\\Psi_{\\chlam}(q,z)\\cdot{\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$\n(here $I$ denotes the set of vertices of the Dynkin diagram\nof $G$) is equal to the character of a certain finite-dimensional $G[[\\st]]\\rtimes\\CC^*$-module $D(\\check\\lam)$\n(the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\\Psi_{\\chlam}$ replaced by $\\Psi'_{\\chlam}$\n(cf. \\cite{San} and \\cite{Ion}); however our proofs are algebro-geometric (and rely on our previous work \\cite{BF11}) and thus they are completely\ndifferent from \\cite{San} and\n\\cite{Ion} (in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\\chlam)$).\n\\subclass{19E08}\n\\end{abstract}\n\n\\sec{int}{Introduction}\n\\ssec{}{The $q$-Whittaker functions}Let $G$ be a semi-simple, simply connected group over $\\CC$ with Lie algebra\n$\\grg$; we choose a pair of opposite Borel subgroups $B, B_-$ of $G$ with unipotent radicals\n$U, U_-$; the intersection $B\\cap B_-$ is a maximal torus $T$ of $G$. It will be convenient for us to denote\nthe weight lattice of $T$ by $\\check \\Lam$ and the coweight lattice by $\\Lam$. In this paper\nwe study certain invariant polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ (the invariance is with respect\nto the Weyl group $W$ of $G$). Here $z\\in T$, $q\\in \\CC^*$ and $\\check\\lam:T\\to \\CC^*$ is a dominant weight of $G$. The function\n $\\Psi_{\\chlam}(q,z)$ is a polynomial function of $z$ with coefficients which are rational functions of $q$ (in fact, later\n were are going to work with a certain modification $\\hatPsi_{\\chlam}(q,z)$ of $\\Psi_{\\chlam}(q,z)$ which will be polynomial in\n$q$).\n\nThe definition of $\\Psi_{\\chlam}(q,z)$ is as follows. Let $\\check{G}$ denote the Langlands dual\ngroup of $G$ with its maximal torus $\\check{T}$. In \\cite{Et} and \\cite{Sev} the authors define (by adapting the so called\nKostant-Whittaker reduction to the case of quantum groups) a homomorphism\n$\\calM:\\CC[T]^W\\to \\End_{\\CC(q)}\\CC(q)[\\check T]$ called the quantum difference Toda integrable system associated with\n$\\check G$.\nFor each $f\\in \\CC[T]^W$ the operator $\\calM_f:=\\calM(f)$ is indeed a difference operator: it is a $\\CC(q)$-linear combination of\nshift operators $\\bfT_{\\check\\beta}$ where $\\check\\beta\\in\\check\\Lam$ and\n$$\n\\bfT_{\\check\\beta}(F(x))=F(q^{\\check\\beta}x).\n$$\n\n\\noindent\n{\\bf Remark.} In principle the constructions of \\cite{Et} and \\cite{Sev} depend on a choice of orientation of the Dynkin diagram of\n$\\check{G}$; however one can deduce from the main result of \\cite{ferm} that the resulting homomorphism is independent of this choice.\n\n\\medskip\n\\noindent\nIn particular, the above operators can be restricted to operators acting in the space of functions on the lattice\n$\\check\\Lam$ by means\nof the embedding $\\check\\Lam\\hookrightarrow \\check T$ sending every $\\chlam$ to $q^{\\chlam}$. For any $f\\in \\CC[T]^W$\nwe shall\ndenote the corresponding operator by $\\calM_f^{\\on{lat}}$. The following conjecture should probably be not very difficult; however, at the moment we don't know how to prove it:\n\\conj{whit-exist-co}\n\\begin{enumerate}\n\\item\nThere exists a unique collection of $\\CC(q)$-valued polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ satisfying the following properties:\n\na) $\\Psi_{\\chlam}(q,z)=0$ if $\\chlam$ is not dominant.\n\nb) $\\Psi_{0}(q,z)=1$.\n\nc) Let us consider all the functions $\\Psi_{\\chlam}(q,z)$ as one function $\\Psi(q,z):\\check \\Lam\\to \\CC(q)$ depending\non $z\\in T$. Then for every $f\\in \\CC[T]^W$ we have\n$$\n\\calM^{\\on{lat}}_f(\\Psi(q,z))=f(z)\\Psi(q,z).\n$$\n\\item The polynomials $\\Psi_{\\chlam}(q,z)$ are $W$-invariant.\n\\end{enumerate}\n\\end{conjec}\nOf course, the second statement follows from the ``uniqueness\" part of the first.\n\nSome remarks about the literature are necessary here. First of all, \\refco{whit-exist-co} is easy for $G=SL(N)$. In this case, the functions\n$\\Psi_{\\chlam}(q,z)$ are extensively studied in \\cite{GLOI}-\\cite{GLOIII}. Second, for general $G$ there exists another definition of the\n$q$-Toda system using double affine Hecke algebras, studied for example in \\cite{C}. Since it is not clear to us\nhow to prove that the definition of $q$-Toda from \\cite{C} and the definition of \\cite{Et} and \\cite{Sev} are the same,\nwe shall denote the operators from \\cite{C} by $\\calM_f'$. It is easy to see\nthat $\\calM_f=\\calM_f'$ for $G=SL(N)$.\\footnote{In fact, as we are going to explain later, the results of this\npaper together with the results of \\cite{Ion} imply that $\\calM_f=\\calM_f'$\nfor any $G$, but we would like to have a more direct proof of this fact.}\nSimilarly we shall denote by $(\\calM^{\\on{lat}}_f)'$ their ``lattice'' version. Then it is shown in \\cite{C} that the existence part of\n\\refco{whit-exist-co} holds for any $G$ if the operators $\\calM^{\\on{lat}}_f$ are replaced by $(\\calM^{\\on{lat}}_f)'$.\nWe shall denote the corresponding polynomials by $\\Psi_{\\chlam}'(q,z)$.\n\n\nThe main result of this paper will imply the following:\n\\th{whit-exist}\n\\begin{enumerate}\n\\item\nThere exists a collection of $W$-invariant polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ with coefficients in $\\CC(q)$ satisfying a), b) and c) above.\n\\item\nLet $\\hatPsi_{\\check\\lam}(q,z)=\\Psi_{\\chlam}(q,z)\\cdot{\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$. Then $\\hatPsi_{\\chlam}(q,z)$ is a polynomial\n function on $\\AA^1\\x T$.\n\\end{enumerate}\n\\eth\nWe are going to construct the above polynomials explicitly by algebro-geometric\nmeans. Thus we prove the existence part of~\\refco{whit-exist-co}.\n\nWe shall usually refer to the polynomials $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$ as $q$-Whittaker functions (following\n\\cite{GLOI}-\\cite{GLOIII}).\nIt is not difficult to see that\n$$\n\\lim\\limits_{q\\to 0}\\Psi_{\\chlam}=\\lim\\limits_{q\\to 0}\\hatPsi_{\\chlam}=\\chi(L(\\chlam))\n$$\nwhere $\\chi(L(\\lam))$ stands for the character of the irreducible representation $L(\\chlam)$ of $G$ with highest\nweight $\\chlam$.\n\nThe main purpose of this paper is to give several (algebro-geometric and representation-theoretic) interpretations of\nthe functions $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$; as a byproduct we shall show that $\\hatPsi_{\\chlam}(q,z)$ is {\\em positive},\ni.e. it is a linear combination of the functions $\\chi(L(\\chmu))$ with coefficients in $\\ZZ_{\\geq 0}[q]$ (this also implies that\n$\\Psi_{\\chlam}$ is a linear combination of the $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[[q]]$). All of our results are known\nfor the polynomials $\\Psi_{\\chlam}'$ (and thus, in particular, we can show that $\\Psi_{\\chlam}=\\Psi_{\\chlam}'$) due to\n\\cite{C} and~\\cite{San},~\\cite{Ion} but our proofs are totally different from {\\em loc. cit.}\n\\ssec{}{Weyl modules}\nRecall the notion of Weyl ${\\mathfrak g}[\\st]$-module $\\CW(\\check\\lambda)$\nfor dominant $\\check\\lambda\\in\\Lambda^\\vee_+$, see e.g.~\\cite{CFK}.\nIt is the maximal $G$-integrable ${\\mathfrak g}[\\st]$-quotient module of\n$\\on{Ind}_{\\fu[\\st]\\oplus\\ft}^{{\\mathfrak g}[\\st]}{\\mathbb C}_{\\check\\lambda}$\nwhere $\\fu\\subset {\\mathfrak g}$ is the nilpotent radical of a Borel subalgebra, containing $\\grt$. There is also a natural\nnotion of {\\em dual Weyl module} $\\calW(\\chlam)^{\\vee}$ (one has to replace the induction by coinduction and ``quotient module\"\nby ``submodule\"). Both $\\calW(\\chlam)$ and $\\calW(\\chlam)^{\\vee}$ are endowed with a natural action of\n$\\CC^*$ by ``loop rotation\". When restricted to $G\\x \\CC^*$ the module\n$\\calW(\\chlam)$ becomes a direct sum of finite-dimensional representations and the character $\\chi(\\calW(\\chlam))$ makes sense;\nmoreover it is a linear combination of $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[[q]]$. Also we have\n$\\chi(\\calW(\\chlam))=\\chi(\\calW(\\chlam)^{\\vee})$.\n\nLet $\\AA^{\\chlam}$ denote the space of all formal linear combinations $\\sum\\gam_i x_i$ where\n$x_i\\in \\AA^1$ and $\\gam_i$ are dominant weights of $G$ such that $\\sum \\gam_i=\\chlam$.\nThe character of $\\CC[\\AA^{\\chlam}]$ with respect to the natural action of\n$\\CC^*$ is equal to ${\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$. According to~\\cite{CFK} there exists\nan action of $\\CC[\\AA^{\\chlam}]$ on $\\calW(\\chlam)$ such that\n\n1) This action commutes with $G[\\st]\\rtimes \\CC^*$;\n\n2) $\\calW(\\chlam)$ is finitely generated and free\nover $\\CC[\\AA^{\\chlam}]$.\n\nLet $D(\\chlam)$ be the fiber of $\\calW(\\chlam)$ at $\\chlam\\cdot 0\\in \\AA^{\\chlam}$. This module is called\na Demazure module (for reasons explained in~\\cite{CL} and~\\cite{FL}).\nThis is a finite-dimensional $G[\\st]\\rtimes \\CC^*$-module\n(in fact, it is easy to see that the action of $G[\\st]$ on $D(\\chlam)$ extends to an action of $G[[\\st]]$).\nWe are going to prove the following\n\n\\th{demazure}Assume that $G$ is simply laced. Then\n\\begin{enumerate}\n\\item\n\\eq{dem-1}\n\\chi(\\calW(\\chlam))=\\Psi_{\\chlam}(q,z)\n\\end{equation}\n\\item\n\\eq{dem-2}\n\\chi(D(\\chlam))=\\hatPsi_{\\chlam}(q,z).\n\\end{equation}\n\\end{enumerate}\nIn particular, $\\hatPsi_{\\chlam}(q,z)$ is positive in the sense discussed above.\n\\eth\nWhen $G$ is not simply laced, the above result is still true, if one replaces $G[[\\st]]$ by some twisted (in the sense of Kac-Moody groups) version\nof it; we shall not give the details here (cf. \\refss{non-simp} for a discussion of the non-simply laced case).\n\n\\reft{demazure}(2) is proved in \\cite{Ion} for $\\hatPsi_{\\chlam}'$ instead of $\\hatPsi_{\\chlam}$.\n\\footnote{It is important to emphasize that the definition of Demazure modules used in this paper (as fibers of Weyl modules)\nis not obviously equivalent to the standard definition used in \\cite{Ion}; however, the equivalence of the two definitions\nis proved in~\\cite{CL} in type $A$, and in~\\cite{FL} in general.}\nThus \\reft{demazure} together with \\cite{Ion} imply the following\n\\cor{twotoda}\nAssume that $G$ is simply laced. Then we have $\\hatPsi_{\\chlam}'=\\hatPsi_{\\chlam}$. Hence\nfor any $f\\in \\CC[T]^W$ we have $\\calM_f=\\calM'_f$.\n\\end{corol}\nAs was mentioned earlier we would like to have a more direct proof of this result\n(independent of the results of \\cite{Ion} and this paper).\nWe would also like to emphasize that our proof of \\reft{demazure} is geometric (in fact it follows easily from the main\nresult of \\cite{BF11}) and thus it is quite different from the proof in \\cite{Ion}. Also, \\refc{twotoda} is\nwrong if $G$ is not simply laced, cf.~\\refss{non-simp}.\n\\ssec{}{Geometric interpretation and spaces of (quasi-)maps}\nTo prove \\reft{demazure} it is clearly enough to prove \\refe{dem-1}.\nThis will be done by interpreting both the LHS and the RHS in terms of algebraic geometry.\n\nLet us first do it for the LHS. The quotient $G[[\\st]]\/T\\cdot U_-[[\\st]]$\ncan naturally be regarded as a scheme\nover $\\CC$. Any weight $\\chlam$ defines a $G[[\\st]]\\rtimes \\CC^*$-equivariant line bundle on this scheme in the standard way.\nWe shall prove\n\\th{weyl}\nThere is a natural isomorphism $\\Gam(G[[\\st]]\/T\\cdot U_-[[\\st]],\\calO(\\chlam))\\simeq \\calW(\\chlam)^{\\vee}$. Similarly,\n$\\Gam(G[[\\st]]\/B_-[[\\st]],\\calO(\\chlam))\\simeq D(\\chlam)^{\\vee}$.\n\\eth\n{\\bf Remark.} \\reft{weyl} is not difficult; it can be thought of as an analog of Borel-Weil-Bott theorem for $G[[\\st]]$. Let us\nalso stress, that while the dual Weyl module $\\calW(\\chlam)^{\\vee}$ has a natural action of $G[[\\st]]$, the Weyl module\n$\\calW(\\chlam)$ itself only has an action of $G[\\st]$.\n\n\\medskip\n\\noindent\nOn the other hand, there is a well known connection between the quotient $G[[\\st]]\/T\\cdot U_-[[\\st]]$ and the space of based maps\n$\\PP^1\\to G\/B$. Moreover, in \\cite{BF11} we have given a construction of the universal eigen-function of the operators\n$\\calM_f$ via the geometry of the above spaces of maps. Using this construction, we can obtain \\refe{dem-1} from \\reft{weyl}\nby a (simple) sequence of formal manipulations. Technically, in order to perform this we shall need to consider a\ncompactification of the space of maps by the corresponding space of quasi-maps.\n\n\\ssec{non-simp}{The case of non-simply laced $G$}\nFormally, the above results do not hold when $G$ is not simply laced. However, it is\neasy to adjust all the results to the non-simply laced case following~Section~7 of~\\cite{BF11}; in particular, in the non-simply laced case the\nfunctions $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$ should be interpreted as\nthe characters of global (resp. local) Weyl modules for the distinguished\nmaximal parahoric subalgebra in a certain twisted affine algebra corresponding\nto ${\\mathfrak g}$ (cf.~Section~7 of~\\cite{BF11} for more detail). The relevant theory\nof Weyl modules and their relation to Demazure\nmodules in the twisted case is developed in \\cite{FoKu}.\nOn the other hand, the character of {\\em nontwisted} local Weyl modules are\nidentified with $\\hatPsi'_{\\chlam}$ in~\\cite{Len}.\n\\ssec{}{Plan of the paper} This paper is organized as follows.\n In \\refs{2} we discuss certain line\nbundles on the space of (quasi-)maps and relate those to sections of a line bundle on $G[[\\st]]\/T\\cdot U_-[[\\st]]$.\n\\refs{3} is devoted to the proof of certain cohomology vanishing on the space of quasi-maps.\nIn \\refs{4} we give an interpretation of $\\Psi_{\\chlam}$ via quasi-maps. Finally in \\refs{5} we give a proof of\n\\reft{demazure}.\n\n\n\n\n\\sec{2}{Quasimaps' scheme}\n\nWe follow the notations of~\\cite{BF11}, unless specified otherwise.\n\n\\ssec{ind}{Ind-scheme $\\fQ$}\nGiven $\\beta\\geq\\alpha\\in\\Lambda_+$ (the cone of positive integral combinations\nof the simple coroots) we consider the closed embedding\n$\\varphi_{\\alpha,\\beta}:\\ \\QM_{\\mathfrak g}^\\alpha\\hookrightarrow\\QM_{\\mathfrak g}^\\beta$ adding the\ndefect $(\\beta-\\alpha)\\cdot0$ at the point $0\\in\\bC$. We denote by\n$\\fQ$ the direct limit of this system.\n\nRecall that $V_{\\check\\omega_i},\\ i\\in I$, are the fundamental ${\\mathfrak g}$-modules,\nand $\\QM^\\alpha_{\\mathfrak g}$ is equipped with a closed embedding\n$\\psi_\\alpha:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\prod_{i\\in I}{\\mathbb P}\\Gamma(\\bC,V_{\\check\\omega_i}\\otimes\\CO(\\langle\\alpha,\n\\check\\omega_i\\rangle))$.\nGiven a ${\\mathfrak g}$-weight $\\check\\lambda=\n\\sum_{i\\in I}d_i\\check\\omega_i\\in\\Lambda^\\vee$ we define a line bundle\n$\\CO(\\check\\lambda)^\\alpha$ on $\\QM^\\alpha_{\\mathfrak g}$ as\n$\\psi_\\alpha^*\\bigotimes_{i\\in I}\\CO(d_i)$.\nNote that if $\\check\\lambda$ is dominant, i.e. $d_i\\geq0\\ \\forall i$,\nthen $\\CO(\\check\\lambda)^\\alpha$ is the inverse image of $\\CO(1)$ on\n${\\mathbb P}\\Gamma(\\bC,V_{\\check\\lambda}\\otimes\\CO(\\langle\\alpha,\n\\check\\lambda\\rangle))$ under the natural morphism\n$\\QM^\\alpha_{\\mathfrak g}\\to{\\mathbb P}\\Gamma(\\bC,V_{\\check\\lambda}\\otimes\\CO(\\langle\\alpha,\n\\check\\lambda\\rangle))$.\nClearly,\n$\\varphi_{\\alpha,\\beta}^*\\CO(\\check\\lambda)^\\beta\\simeq\\CO(\\check\\lambda)^\\alpha$.\nThe resulting line bundle on the ind-scheme $\\fQ$ is denoted\n$\\CO(\\check\\lambda)$.\n\n\\ssec{pro}{Infinite type scheme $\\bQ$}\nWe denote ${\\mathbb C}[[t^{-1}]]$ by $R$, and ${\\mathbb C}((t^{-1}))$ by $F$. Recall that\n$R_n=R\/(t^{-n})$. We denote the projection $R\\twoheadrightarrow R_n$ by $p_n$.\nThe ${\\mathbb C}$-points of the infinite type scheme $\\overline{G\/U_-}(R)$\nare the collections of\nvectors $v_{\\check\\lambda}\\in V_{\\check\\lambda}\\otimes R,\\ \\check\\lambda\\in\n\\Lambda^\\vee_+$ (dominant ${\\mathfrak g}$-weights), satisfying the Pl\\\"ucker equations.\nWe denote by $\\widehat\\bQ\\subset\\overline{G\/U_-}(R)$ the open subscheme\nformed by all the maps $\\on{Spec}R\\to\\overline{G\/U_-}$ whose restriction\nto the generic point of $\\on{Spec}R$ lands into\n$G\/U_-\\subset\\overline{G\/U_-}(R)$.\nIt is equipped with a free action\nof the Cartan torus $T:\\ h(v_{\\check\\lambda})=\\check\\lambda(h)v_{\\check\\lambda}$.\nThe quotient scheme $\\bQ=\\widehat\\bQ\/T$ is a closed subscheme in\n$\\prod_{i\\in I}{\\mathbb P}(V_{\\check\\omega_i}\\otimes R)$. Any weight $\\check\\lambda\\in\n\\Lambda^\\vee$ gives rise to a line bundle $\\CO(\\check\\lambda)$ on $\\bQ$.\n\n\\ssec{indpro}{The embedding $\\fQ\\hookrightarrow\\bQ$}\nWe fix a coordinate $t$ on $\\bC$ such that $t(0)=0,\\ t(\\infty)=\\infty$.\nFor $\\alpha\\in\\Lambda_+$ we define a $T$-torsor\n$\\widehat\\QM{}^\\alpha_{\\mathfrak g}\\stackrel{p}{\\to}\\QM^\\alpha_{\\mathfrak g}$ as follows.\nThe ${\\mathbb C}$-points of $\\widehat\\QM{}^\\alpha_{\\mathfrak g}$ are the collections\n$(v_{\\check\\lambda}\\in\\CL_{\\check\\lambda}\\subset V_{\\check\\lambda}\\otimes\\CO_\\bC),\\\n\\check\\lambda\\in\\Lambda^\\vee_+$, such that\n\n(a)\n$(\\CL_{\\check\\lambda}\\subset\nV_{\\check\\lambda}\\otimes\\CO_\\bC)_{\\check\\lambda\\in\\Lambda^\\vee_+}\\in\\QM^\\alpha_{\\mathfrak g}$;\n(b) $v_{\\check\\lambda}\\in\\Gamma(\\bC-0,\\CL_{\\check\\lambda})$ are the\nnonvanishing sections satisfying the Pl\\\"ucker equations.\n\nThe projection $p$ forgets the sections $v_{\\check\\lambda}$.\nThe action of $T$ on $\\widehat\\QM{}^\\alpha_{\\mathfrak g}$ is defined as follows:\n$h(v_{\\check\\lambda}\\in\\CL_{\\check\\lambda})=\n(\\check\\lambda(h)v_{\\check\\lambda}\\in\\CL_{\\check\\lambda})$.\n\nTaking a formal expansion of $v_{\\check\\lambda}$ at $\\infty\\in\\bC$ we obtain\na closed embedding $s_\\alpha:\\ \\widehat\\QM{}^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\widehat\\bQ$. Clearly, $s_\\alpha$ is $T$-equivariant, and gives rise to the\nsame named closed embedding $s_\\alpha:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\\bQ$.\nEvidently, for $\\beta\\geq\\alpha$ we have\n$s_\\alpha=s_\\beta\\circ\\varphi_{\\alpha,\\beta}$. Hence we obtain the closed\nembedding $s:\\ \\fQ\\hookrightarrow\\bQ$. The restriction of the line bundle\n$\\CO(\\check\\lambda)$ on $\\bQ$ to $\\fQ$ coincides with the line bundle\n$\\CO(\\check\\lambda)$ on $\\fQ$.\n\n\\ssec{open}{Open subschemes $\\fQ_\\infty\\subset\\fQ$ and $\\bQ_\\infty\\subset\\bQ$}\nWe define an open subscheme\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}\\subset\\QM^\\alpha_{\\mathfrak g}$ formed by all the\nquasimaps without defect at $\\infty\\in\\bC$. Clearly,\n$\\varphi_{\\alpha,\\beta}(\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g})\\subset\n\\overset{\\circ}\\QM{}^\\beta_{\\mathfrak g}$. The direct limit of this system is denoted by\n$\\fQ_\\infty$; it is an open sub ind-scheme of $\\fQ$.\n\nNote that $s(\\fQ_\\infty)\\subset G(R)\/T\\cdot U_-(R)\\subset\\bQ$. We are going to denote the\nopen subscheme $G(R)\/T\\cdot U_-(R)\\subset\\bQ$ by $\\bQ_\\infty$. For $n\\geq1$,\nwe have a natural projection $p_n:\\ \\bQ_\\infty\\to G\/U_-(R_n)\/T=:\\bQ_n$.\n\n\\begin{lemma}\n\\label{codim2}\nThe restriction\n$\\Gamma(\\fQ,\\CO(\\check\\lambda))\\to\\Gamma(\\fQ_\\infty,\\CO(\\check\\lambda))$\nis an isomorphism for any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to prove that the restriction\n$\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))\\to\n\\Gamma(\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\alpha\\in\\Lambda_+$. Since the complement of\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ in $\\QM^\\alpha_{\\mathfrak g}$ has codimension two, it\nsuffices to know that $\\QM^\\alpha_{\\mathfrak g}$ is normal. However, locally in the\n\\'etale topology, $\\QM^\\alpha_{\\mathfrak g}$ is isomorphic to the product of the\nZastava space $Z^\\alpha_{\\mathfrak g}$ and the flag variety $\\CB_{\\mathfrak g}$. Finally, the\nnormality of $Z^\\alpha_{\\mathfrak g}$ is proved in~\\cite[Corollary~2.10]{BF11}.\n\\end{proof}\n\nThe following conjecture is not needed in this paper, but it might be useful for future purposes.\n\\conj{codim two}\nThe restriction $\\Gamma(\\bQ,\\CO(\\check\\lambda))\\to\n\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\end{conjec}\n\nLet us make a few remarks about \\refco{codim two}.\nAs in the proof of~Lemma~\\ref{codim2}, it suffices to know that the scheme\n$\\bQ$ is normal. According to~\\cite{D},~\\cite{GK}, the formal completion of\n$\\bQ$ at a closed point $x\\in\\bQ$ is isomorphic to the product of the formal\ncompletion of a certain $\\QM^\\alpha_{\\mathfrak g}$ at a closed point\n$\\phi\\in\\QM^\\alpha_{\\mathfrak g}$, and countably many copies of the formal disc.\nSo the normality of the formal neighborhood of every closed point follows from the normality of $\\QM^\\alpha_{\\mathfrak g}$.\nUnfortunately, since $\\bQ$ is not noetherian it does not imply the normality of $\\bQ$ itself.\n\n\n\\medskip\n\\noindent\nThe group $\\BG_m$ acts on $\\fQ$ and $\\bQ$ by loop rotations, and the line\nbundles $\\CO(\\check\\lambda)$ are $\\BG_m$-equivariant. Hence $\\BG_m$ acts\non the global sections of these line bundles. We will denote by\n$\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))\\subset\\Gamma(\\fQ,\\CO(\\check\\lambda))$\nthe subspace of $\\BG_m$-finite sections.\n\n\n\\th{isoq}\nThe restriction $\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\to\n\\widetilde\\Gamma(\\fQ_\\infty,\\CO(\\check\\lambda))=\n\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\eth\n\n\\begin{proof}\nThe closed embedding $\\varphi_{\\alpha,\\beta}:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\QM^\\beta_{\\mathfrak g}$ lifts in an evident way to the same named closed embedding of\n$T$-torsors $\\widehat\\QM{}^\\alpha_{\\mathfrak g}\\hookrightarrow\\widehat\\QM{}^\\beta_{\\mathfrak g}$.\nWe denote the limit of this system by $\\widehat\\fQ$, a $T$-torsor over $\\fQ$.\nThe construction of~\\refss{indpro} defines a $T$-equivariant closed embedding\n$s:\\ \\widehat\\fQ\\hookrightarrow\\widehat\\bQ_\\infty:=G\/U_-(R)$. We have to prove\nthat the restriction ${\\mathbb C}[\\widehat\\bQ_\\infty]\\to\n\\widetilde{\\mathbb C}[\\widehat\\fQ_\\infty]=\\widetilde{\\mathbb C}[\\widehat\\fQ]$ is an isomorphism.\nHere $\\widetilde{\\mathbb C}[\\widehat\\fQ_\\infty]$ (resp. $\\widetilde{\\mathbb C}[\\widehat\\fQ]$)\nstands for the ring of $\\BG_m$-finite functions on $\\widehat\\fQ_\\infty$\n(resp. $\\widehat\\fQ$).\n\nTo this end we mimick the argument of~\\cite[Section~2]{BF11}.\nWe choose a {\\em regular} dominant $\\mu\\in\\Lambda^+$, and consider the\ncorresponding $T$-fixed point $t^\\mu\\in\\Gr_G$. Its stabilizer $\\on{St}_\\mu$\nin $G[t^{-1}]$ has the unipotent radical $\\on{RadSt}_\\mu$, and the quotient\n$\\on{St}_\\mu\/\\on{RadSt}_\\mu$ is canonically isomorphic to $T$.\nThe quotient $G[t^{-1}]\/\\on{St}_\\mu$ is the $G[t^{-1}]$-orbit\n$\\sW_{G,\\mu}\\subset\\Gr_G$ of $t^\\mu$ (see~\\cite[Section~2.4]{BF11}),\nand the quotient $G[t^{-1}]\/\\on{RadSt}_\\mu$ is a $T$-torsor\n$\\widehat\\sW_{G,\\mu}$.\n\n{\\em NB:} The group denoted $\\on{St}_\\mu$ in~\\cite[Section~2.6]{BF11} is\nthe intersection of our present $\\on{St}_\\mu$ with the first congruence\nsubgroup $G_1\\subset G[t^{-1}]$.\n\nIn modular terms, $\\sW_{G,\\mu}$ parametrizes the $G$-bundles on $\\bC$\nof isomorphism type $W\\mu$ equipped with a trivialization on $\\bC-0$\n(see~\\cite[Proof of~Theorem~2.8]{BF11}). Such a bundle ${\\mathcal F}_G$\npossesses a canonical Harder-Narasimhan flag $HN({\\mathcal F}_G)$. Note that this flag\nis complete, i.e. it is a reduction to the Borel, since $\\mu$ is regular.\nIn particular, the fiber ${\\mathcal F}_{G,\\infty}$ of ${\\mathcal F}_G$ at $\\infty\\in\\bC$ is\nequipped with a\ncanonical reduction to the Borel. Now $\\widehat\\sW_{G,\\mu}$ parametrizes\nthe data as above along with a further reduction of ${\\mathcal F}_{G,\\infty}$ to the\nunipotent radical of the Borel.\n\nIn complete similarity with~\\cite[Lemma~2.7]{BF11} we have\n\n\\begin{lemma}\n\\label{2.7}\n(1) Fix $n\\geq1$, and let $\\mu\\in\\Lambda^+_{\\on{reg}}$ satisfy the following\ncondition: $\\langle\\mu,\\check\\alpha\\rangle\\geq n$ for every positive root\n$\\check\\alpha$ of ${\\mathfrak g}$. Then the image of $\\on{RadSt}_\\mu$ in\n$G[t^{-1}]\/G_n=G(R_n)$ is equal to $U_-(R_n)$. In particular, we have a\nnatural map $\\pi_{\\mu,n}:\\ \\widehat\\sW_{G,\\mu}\\to G(R_n)\/U_-(R_n)$.\n\n(2) Under the assumption of (1), for every $ks^{\\lam}_{\\mu}>> \\widehat\\QM{}^\\alpha_{\\mathfrak g}\\\\\n@V\\pi_{\\mu,n}VV @VVp_n\\circ s_\\alpha V\\\\\n G(R_n)\/U_-(R_n)@>\\on{id}>> G(R_n)\/U_-(R_n)\n \\end{CD}\n\\end{equation}\n($s_\\alpha$ was constructed in~\\refss{indpro}).\n\n(2) The map $(s^\\lambda_\\mu)^*:\\ {\\mathbb C}[\\widehat\\QM{}^\\alpha_{\\mathfrak g}]\\to\n{\\mathbb C}[\\widehat\\sW{}^\\lambda_{G,\\mu}]$ induces an isomorphism on functions of\ndegree $0$ and $\\alpha\\in\\Lambda_+^{\\check\\lambda}$ we have\n$H^n(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=0$.\n\n(2) For $n>0$ and $\\check\\lambda\\in\\Lambda^\\vee$ we have\n$\\widetilde{H}{}^n(\\fQ,\\CO(\\check\\lambda))=0$.\n\n(3) For $\\check\\lambda\\not\\in\\Lambda^\\vee_+$ we have\n$\\widetilde{H}{}^0(\\fQ,\\CO(\\check\\lambda))=0$.\n\\eth\n\n\\begin{proof}\n(3) is clear, and (2) follows from (1). We prove (1).\n\nAccording to~\\cite[Proposition~5.1]{BF11}, $Z^\\alpha_{\\mathfrak g}$ is a Gorenstein\nvariety with rational singularities. Since $\\QM^\\alpha_{\\mathfrak g}$ is, locally in\n\\'etale topology, isomorphic to $Z^\\alpha_{\\mathfrak g}\\times\\CB_{\\mathfrak g}$, we conclude that\n$\\QM^\\alpha_{\\mathfrak g}$ is a Gorenstein variety with rational singularities as well.\n(It is here that we use the assumption that $G$ is simply laced.)\nLet us denote the dualizing sheaf of $\\QM^\\alpha_{\\mathfrak g}$ by $\\omega^\\alpha$.\n\n\\begin{lemma}\n\\label{givlee}\n$\\omega^\\alpha\\simeq\\CO(-\\alpha^*-2\\check\\rho)$.\n\\end{lemma}\n\n\\begin{proof}\nIn case $G=\\SL(N)$, the lemma is proved in~\\cite[Theorem~3]{GL}.\nFor arbitrary simply laced $G$ we first prove that $\\omega^\\alpha\\simeq\n\\CO(\\check\\lambda)$ for some $\\check\\lambda$. It is enough to check this\non the open subscheme $\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ since the complement\nis of codimension two. We have the morphism of evaluation at $\\infty\\in\\bC:\\\n\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}\\stackrel{ev_\\infty}{\\longrightarrow}\\CB_{\\mathfrak g}$.\nIt is a $G$-equivariant fibration with fibers isomorphic to $Z^\\alpha_{\\mathfrak g}$.\nSince the big cell $U\\cdot e_-\\subset\\CB_{\\mathfrak g}$ is a free orbit of $U$, we have\n$ev_\\infty^{-1}(U\\cdot e_-)\\simeq Z^\\alpha_{\\mathfrak g}\\times U$. The canonical class\nof $Z^\\alpha_{\\mathfrak g}$ is trivial (see~\\cite[Proof of~Proposition~5.1]{BF11}), hence\nthe canonical class of $ev_\\infty^{-1}(U\\cdot e_-)$ is trivial as well.\nThus $\\omega^\\alpha$ has a nowhere vanishing section $\\sigma$ on\n$ev_\\infty^{-1}(U\\cdot e_-)$. Hence the class of $\\omega^\\alpha$ on\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ is a linear combination of the pullbacks\nunder $ev_\\infty$ of the Schubert divisors on $\\CB_{\\mathfrak g}$. The pullback of an\nirreducible Schubert divisor being $\\CO(\\check\\omega_i)$ we conclude that\nthere exists $\\check\\lambda$ such that $\\omega^\\alpha\\simeq\\CO(\\check\\lambda)$.\n\nIt remains to check $\\check\\lambda=-\\alpha^*-2\\check\\rho$. We will do this on\nanother open subscheme $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}\\subset\\QM^\\alpha_{\\mathfrak g}$ with the\ncomplement of codimension two. Namely, $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$\nis the moduli space of quasimaps with defect at most a simple coroot (or no\ndefect at all). Note that $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$ is smooth,\nand the Kontsevich resolution is an isomorphism over it. Let us fix a\nquasimap without defect $\\phi\\in\\QM^{\\alpha-\\alpha_i}_{\\mathfrak g}$, and consider a curve\n$C^\\phi_i\\subset\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$ formed by all the\nquasimaps $\\phi(\\alpha_i\\cdot c),\\ c\\in\\bC$ (twisting $\\phi$ by an arbitrary point of $\\bC$).\nIt is easy to see that $\\deg(\\CO(\\check\\omega_j)|_{C^\\phi_i})=\\delta_{ij}=\n\\langle\\alpha_i,\\check\\omega_j\\rangle$. Hence it remains to check that\n$\\deg(\\omega^\\alpha|_{C^\\phi_i})=-\\langle\\alpha_i,\\alpha^*+2\\check\\rho\\rangle$.\nThis is done in~\\cite[Proposition~4.4]{FK}. Although {\\em loc. cit.} is\nformulated for $G=\\SL(N)$, its proof goes through word for word for arbitrary\nsimple $G$.\n\nThe lemma is proved. \\end{proof}\n\nWe are ready to finish the proof of the theorem. For\n$\\alpha\\in\\Lambda^{\\check\\lambda}_+$ the line bundle\n$\\CL=\\CO(\\check\\lambda)\\otimes(\\omega^\\alpha)^*$ on $\\QM^\\alpha_{\\mathfrak g}$ is\nvery ample. We have to prove that $H^n(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=\nH^n(\\QM^\\alpha_{\\mathfrak g},\\CL\\otimes\\omega^\\alpha)=0$ for $n>0$. According\nto~\\cite[Proposition~5.1]{BF11}, $\\QM^\\alpha_{\\mathfrak g}$ has rational singularities.\nLet $\\pi:\\ X\\to\\QM^\\alpha_{\\mathfrak g}$ be a resolution of singularities. Then for the\ncanonical line bundle $\\omega_X$ of $X$ we have $R\\pi_*\\omega_X=\\omega^\\alpha$.\nHence $H^n(\\QM^\\alpha_{\\mathfrak g},\\CL\\otimes\\omega^\\alpha)=\nH^n(X,\\pi^*\\CL\\otimes\\omega_X)=0$ (for $n>0$) by Kawamata-Viehweg vanishing\nsince $\\pi^*\\CL$ is nef and big.\n\nThis completes the proof of the theorem.\n\\end{proof}\n\n\n\\sec{4}{$q$-Whittaker functions}\n\n\\ssec{charac}{The character of $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$}\nRecall~\\cite[Introduction]{BF11} that $\\fJ_\\alpha(q,z)$\nis the character of $T\\times\\BG_m$-module ${\\mathbb C}[Z^\\alpha_{\\mathfrak g}]$, a rational\nfunction on $T\\times\\BG_m$. Let $x_i$ stand for the character of the dual\ntorus $\\check T$ corresponding to the simple coroot $\\alpha_i$. For\n$\\alpha\\in\\Lambda_+$ the corresponding character of $\\check T$ is denoted by\n$x^\\alpha$. We consider the formal generating functions\n$J_{\\mathfrak g}(q,z,x)=\\sum_{\\alpha\\in\\Lambda_+}x^\\alpha\\fJ_\\alpha$, and\n$\\fJ_{\\mathfrak g}(q,z,x)=\\prod_{i\\in I}x_i^{\\log(\\check\\omega_i)\/\\log q}J_{\\mathfrak g}(q,z,x)$,\ncf.~\\cite[Equation~(18)]{BF0}.\n\nAccording to~\\cite[Corollary~1.6]{BF11}, the function $\\fJ_{\\mathfrak g}(q,z,x)$\nis an eigen-function of the quantum difference Toda integrable system\nassociated with ${\\mathfrak g}$. For example, if $G=\\SL(N)$, the function\n$\\fJ_{\\mathfrak g}(q,z,x)$ is an eigenfunction of the operator\n$\\fG=T_1+T_2(1-x_1)+\\ldots+T_N(1-x_{N-1})$, cf.~\\cite[Equation~(16)]{BF0},\nwhere $T_k(F(q,z,x_1,\\ldots,x_{N-1}))=F(q,z,x_1,\\ldots,x_{k-2},q^{-1}x_{k-1},\nqx_k,x_{k+1},\\ldots,x_{N-1})$.\n\nNote that if we plug $x=q^{\\check\\lambda}$ into $J_{\\mathfrak g}(q^{-1},z,x)$ or into\n$\\fJ_{\\mathfrak g}(q^{-1},z,x)$, then for $\\check\\lambda\\in\\Lambda^\\vee_+$ these formal\nseries converge, and we have $\\fJ_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda}):=\\prod_{i\\in I}\n(q^{\\langle\\alpha_i,\\check\\lambda\\rangle})^{\\log(\\check\\omega_i)\/\\log q}\nJ_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda})=\nz^{\\check\\lambda}J_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda})$ (a formal Taylor series in\n$q$ with coefficients in Laurent polynomials in $z$).\n\nThe following lemma is a reformulation of~\\cite[Proposition~2]{GL}:\n\n\\begin{lemma}\n\\label{gile}\nThe class of $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ in\n$K_{T\\times\\BG_m}(pt)$ equals\n$$\\sum_{\\substack{\\gamma+\\beta=\\alpha\\\\ w\\in W}}\nz^{w\\check\\lambda}q^{\\langle\\gamma,\\check\\lambda\\rangle}\n\\fJ_\\gamma(q^{-1},wz)\\fJ_\\beta(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\pi:\\ \\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha))\\to\\QM^\\alpha_{\\mathfrak g}$\n(resp. $\\varpi:\\ M^\\alpha_{\\mathfrak g}\\to Z^\\alpha_{\\mathfrak g}$) be the Kontsevich resolution,\nsee e.g.~\\cite[Appendix]{FFKM} (resp.~\\cite[Proof of~Proposition~5.1]{BF11}).\nSince the singularities of $\\QM^\\alpha_{\\mathfrak g}$ (resp. $Z^\\alpha_{\\mathfrak g}$) are rational,\nwe have $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=\nR\\Gamma(\\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha)),\n\\pi^*\\CO(\\check\\lambda))$ (resp. ${\\mathbb C}[Z^\\alpha_{\\mathfrak g}]={\\mathbb C}[M^\\alpha_{\\mathfrak g}]$).\nHence we have to express the character of\n$R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ via the characters of\n${\\mathbb C}[M^\\beta_{\\mathfrak g}]$. This is done in~\\cite[Proof of~Proposition~2]{GL} via\nthe Atiyah-Bott-Lefschetz localization to $T\\times\\BG_m$-fixed points of\n$\\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha))$. As usually, we have to\nadd that {\\em loc. cit.} deals with $G=\\SL(N)$, however, the proof\ngoes through word for word for arbitrary semisimple $G$.\n\\end{proof}\n\n\\ssec{lim charac}{The character of $\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))$}\nBy the proof of~\\reft{isoq} and~Lemma~\\ref{2.8}(2), the character\n$\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$ is the limit of the characters\n$\\chi (R^0\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nas $\\alpha\\to\\infty$. By~\\reft{vanish}(1), as $\\alpha\\to\\infty$, the limit\nof the characters $\\chi (R^{>0}\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nvanishes. Thus, the character $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$\nis the limit of the characters $\\chi (R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nas $\\alpha\\to\\infty$. We define\n$\\fJ_\\infty(q,z):=\\displaystyle{\\lim_{\\alpha\\to\\infty}\\fJ_\\alpha(q,z)}$\n(it is easy to see that the latter limit exists).\n\n\\prop{brav}\n$$\n\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=\n\\sum_{w\\in W}\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.\n$$\n\\end{propos}\n\n\\begin{proof}\nAs $\\alpha$ goes to $\\infty$, the formula of~Lemma~\\ref{gile} goes to\n$$\\sum_{\\substack{\\gamma\\in\\Lambda_+\\\\ w\\in W}}\nz^{w\\check\\lambda}q^{\\langle\\gamma,\\check\\lambda\\rangle}\n\\fJ_\\gamma(q^{-1},wz)\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}=$$\n$$\\sum_{w\\in W}\nz^{w\\check\\lambda}J_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}=$$\n$$\\sum_{w\\in W}\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.$$\n\\end{proof}\n\n\\cor{gera}\nLet $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=\\Psi_{\\chlam}(q,z)$. Then the functions\n$\\Psi_{\\chlam}(q,z)$ satisfy all the conditions of \\refco{whit-exist-co}.\n\\end{corol}\n\n\\begin{proof}\nPart 2 of \\refco{whit-exist-co} is obvious by construction. Also \\refco{whit-exist-co}(1b) is obvious.\nAccording to~\\reft{vanish}(2), $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=0$\nif $\\check\\lambda\\not\\in\\Lambda^\\vee_+$, which proves \\refco{whit-exist-co}(1a).\n\nLet us prove \\refco{whit-exist-co}(1c). The function\n$\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})$ on the lattice $\\Lambda^\\vee$ is an\neigenfunction of the quantum difference Toda restricted to the lattice.\nAccording to~\\refp{brav}, $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$ is\na linear combination of the functions $\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})$\nwith coefficients independent of $\\check\\lambda$. Hence $\\Psi_{\\chlam}(q,z)$\nis an eigenfunction of the quantum difference Toda as well.\n\\end{proof}\n\n\n\n\n\n\\sec{5}{Weyl modules}\n\n\\ssec{nameless}{} Recall that $R={\\mathbb C}[[t^{-1}]]$. We introduce a new variable\n$\\st=t^{-1}$, so that $R={\\mathbb C}[[\\st]]$. We set $\\widetilde R:={\\mathbb C}[\\st]\\subset R$.\nThe proalgebraic group $G(R)$ acts naturally on the profinite dimensional\nvector space $\\Gamma(\\fQ,\\CO(\\check\\lambda))$. The continuous dual\n$\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$ coincides with the graded dual\n$\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$, and is equipped with a\nnatural action of $G(\\widetilde R):\\ g\\cdot v^*(v):=v^*(\\tau g\\cdot v)$.\nHere $g\\mapsto\\tau g$ is the Chevalley antiinvolution of $G$ identical on $T$.\nThe derivative of these actions gives rise to the actions of ${\\mathfrak g}(R)$\nand ${\\mathfrak g}(\\widetilde R)$. According to~\\reft{isoq}, the\n${\\mathfrak g}(\\widetilde R)$-module $\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$ coincides\nwith the graded dual $\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))^\\vee$.\n\nWe denote the preimage of the big cell $U\\cdot e_-\\subset\\CB_{\\mathfrak g}$ in\n$G\/U_-\\to\\CB_{\\mathfrak g}$ by $C\\subset G\/U_-$. We denote the open subscheme\n$C(R)\/T\\subset G(R)\/T\\cdot U_-(R)=\\bQ_\\infty$ by $\\overset{\\circ}{\\bQ}$.\nWe have the restriction morphism of ${\\mathfrak g}(R)$-modules\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\hookrightarrow\n\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$. Now $C(R)$ is a free\norbit of $B(R)\\subset G(R)$, and\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))=\n\\on{CoInd}_{\\fu(R)\\oplus\\ft}^{{\\mathfrak g}(R)}{\\mathbb C}_{\\check\\lambda}$. The graded dual\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))^\\vee=\n\\on{Ind}_{\\fu(\\widetilde R)\\oplus\\ft}^{{\\mathfrak g}(\\widetilde R)}{\\mathbb C}_{\\check\\lambda}$.\n\n\\begin{lemma}\n\\label{?}\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\subset\n\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$ is the maximal\n$G$-integrable ${\\mathfrak g}(R)$-submodule. Equivalently,\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))^\\vee$ is the maximal $G$-integrable\n${\\mathfrak g}(\\widetilde R)$-quotient module of\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))^\\vee$.\n\\end{lemma}\n\n\\begin{proof}\nNote that $\\bQ_\\infty$ is the $G$-saturation of $\\overset{\\circ}{\\bQ}$.\nLet $v\\in\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$ lie in a\nfinite-dimensional ${\\mathfrak g}$-submodule\n$V\\subset\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$. The action of\n${\\mathfrak g}$ on $V$ integrates to the action of $G$. Let us view $v$ as a\n$\\check\\lambda$-covariant function on $C(R)$. We have to check that\n$v$ is the restriction of a $\\check\\lambda$-covariant function $\\hat v$\non $G\/U_-(R)$ to $C(R)$. Given a point $y\\in G\/U_-(R)$ we can find $g\\in G$\nsuch that $g(y)\\in C(R)$. Then we define $\\hat{v}(y):=u(gy)$ where we view\n$u:=gv\\in V$ as a $\\check\\lambda$-covariant function on $C(R)$. Clearly, this\nis well defined, i.e. independent of a choice of $g$.\n\\end{proof}\n\nRecall the notion of Weyl ${\\mathfrak g}(\\widetilde R)$-module $\\CW(\\check\\lambda)$\nfor dominant $\\check\\lambda\\in\\Lambda^\\vee_+$, see e.g.~\\cite{CFK}.\nIt is the maximal $G$-integrable ${\\mathfrak g}(\\widetilde R)$-quotient module of\n$\\on{Ind}_{\\fu(\\widetilde R)\\oplus\\ft}^{{\\mathfrak g}(\\widetilde R)}{\\mathbb C}_{\\check\\lambda}$\n({\\em loc. cit.}). Thus~Lemma~\\ref{?} implies the first part of \\reft{demazure}.\n\nOn the other hand, taking into account \\reft{isoq} we also get\n\\prop{!}\nFor $\\check\\lambda\\in\\Lambda^\\vee_+$, we have a natural isomorphism of\n${\\mathfrak g}(\\widetilde R)$-modules $\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee\\simeq\n\\CW(\\check\\lambda)$. \\qed\n\\end{propos}\n\nCombining this with \\refc{gera} we get the following\n\\cor{weyl-whit}\n$\\chi(\\calW(\\chlam))=\\Psi_{\\chlam}(q,z)$.\n\\end{corol}\nThis is actually the statement of \\reft{demazure}(1).\nTo prove \\reft{demazure}(2) let us\nrecall that the Demazure module $D(\\check\\lambda)$ is a certain\n${\\mathfrak g}(\\widetilde R)$-submodule of an irreducible integrable level one\nrepresentation of\n${\\mathfrak g}_{\\on{aff}}$, see e.g.~\\cite[2.2]{FL}. In addition, according\nto~\\cite{CFK},~\\cite{FL} there exists\nan action of $\\CC[\\AA^{\\chlam}]$ on $\\calW(\\chlam)$ such that\n\n1) This action commutes with $G(R)\\rtimes \\CC^*$.\n\n2) $\\calW(\\chlam)$ is finitely generated and free\nover $\\CC[\\AA^{\\chlam}]$.\n\n3) The fiber of $\\calW(\\chlam)$ at $\\chlam\\cdot 0$ is isomorphic to $D(\\chlam)$.\n\n\\medskip\n\\noindent\nThus we get the following corollary, which is actually the statement of \\reft{demazure}(2)\n(as was mentioned in the introduction it was proved in~\\cite{GLOIII} for $G=\\SL(N)$):\n\\cor{dem}\nThe product $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))\\cdot\\displaystyle\n{\\prod_{i\\in I}\\prod_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}=\\hatPsi_{\\chlam}(q,z)$\nis equal to\nthe character of the (finite dimensional) Demazure module $D(\\check\\lambda)$. In particular, it is a finite\nlinear combination of $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[q]$.\n\\end{corol}\n\n\\ssec{}{Geometric interpretation of the $\\CC[\\AA^{\\chlam}]$-action}\nWe conclude the paper by giving an interpretation of the $\\CC[\\AA^{\\chlam}]$-action on $\\calW(\\chlam)$ in terms of\n\\reft{demazure}(1). This will enable us to prove the second assertion of \\reft{weyl}.\nIt would be nice to prove that this action is free directly by geometric means (without referring to\n\\cite{FL}).\n\nLet $T(R)_1$ denote the first congruence subgroup in $T(R)$ (i.e. the kernel of the natural map\n$T(R)\\to T$). Let $\\grt(R)_1$ denote its (abelian) Lie algebra (i.e. the kernel of the natural map\n$\\grt(R)\\to \\grt$). We denote by $\\grt(\\tilR)_1\\subset \\grt(R)_1$ the corresponding subspace (consisting of all\nmappings $\\AA^1\\to \\grt$ which are equal to $0$ at $0$). Then for every $\\chlam\\in\\Lam^{\\vee}_+$ there exists a natural epimorphism\n$\\pi_{\\chlam}:U(\\grt(R)_1)=\\Sym\\grt(R)_1)\\to \\CC[\\AA^{\\chlam}]$ defined by the following formula:\n$$\n\\pi_{\\chlam} (h\\st^n)(\\sum_i \\gam_i x_i)=\\sum_i \\langle h,\\gam_i)x_i^n.\n$$\nHere $h\\in \\grt$ and $\\sum_i \\gam_i x_i\\in \\AA^{\\chlam}$.\n\nClearly, the group $T(R)_1$ acts (on the right) on the scheme $\\bQ_{\\infty}=G(R)\/T\\cdot U_-(R)$. Hence we get a natural action\nof $\\Sym(\\grt(R)_1)$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))$ for every $\\chlam\\in \\Lam^{\\vee}$.\nThe following result is easy to prove; we leave\nthe details to the reader:\n\\prop{config}\n\\begin{enumerate}\n\\item The above action\nof $\\Sym(\\grt(\\tilR)_1)$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))$ factors through $\\pi_{\\chlam}$.\n\\item\nThe resulting action\nof $\\CC[\\AA^{\\chlam}]$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))^{\\vee}=\\calW(\\chlam)$\ncoincides with the action considered in~\\cite{CFK} and \\cite{FL}.\n\\end{enumerate}\n\\end{propos}\nFrom \\refp{config} we immediately get the following\n\\cor{}\nWe have\n$\\Gam(G(R)\/B_-(R),\\calO(\\chlam))\\simeq D(\\chlam)^{\\vee}$ (this is the second assertion of \\reft{weyl}).\n\\end{corol}\n\\begin{proof}\nIt follows from \\refp{config} and from the fact that $D(\\chlam)$ is the fiber of $\\calW(\\chlam)$ over\n$\\chlam\\cdot 0\\in \\CC[\\AA^{\\chlam}]$ that $D(\\chlam)^{\\vee}$ is isomorphic to the invariants of $\\grt(\\tilR)$ on $\\calW(\\chlam)^{\\vee}$.\nSince $\\grt(\\tilR)_1$ is dense in $\\grt(R)_1$, it follows that\n$$\n(\\calW(\\chlam)^{\\vee})^{\\grt(\\tilR)_1}=(\\calW(\\chlam)^{\\vee})^{\\grt(R)_1}.\n$$\nFrom \\refp{!} we get\n$$\n(\\calW(\\chlam)^{\\vee})^{\\grt(R)_1}=\\Gam(G(R)\/T\\cdot U_-(R),\\calO(\\chlam))^{\\grt(R)_1}=\\Gam(G(R)\/B_-(R),\\calO(\\chlam)).\n$$\n\n\n\\end{proof}\n\n\\begin{acknowledgement}\nThis paper emerged as a result of numerous conversations held between the first author and\nA.~Borodin in IHES in June 2011. In particular, A.~Borodin has introduced the first author to the notion of $q$-Whittaker\nfunctions and brought to his attention the papers \\cite{GLOI}-\\cite{GLOIII}. The first author wishes\nalso to thank IHES staff for exceptionally pleasant working conditions and hospitality.\nWe are grateful to B.~Feigin and S.~Loktev for\ntheir numerous patient explanations about Weyl modules. The observation that\n$\\Gamma(G[[\\st]]\/T\\cdot U_-[[\\st]],\\CO(\\check\\lambda))$ is a dual Weyl module is essentially due to them.\nWe would like to thank K.~Schwede and other mathoverflowers for their\nexplanations about Kodaira vanishing. The first author is also deeply indebted to Patrick\nClark for his patient explanations about Macdonald polynomials and to P.~Etingof and I.~Cherednik for explanations about the\n$q$-Toda system.\nM.F. was partially supported by the RFBR grants 12-01-00944, 12-01-33101,\n13-01-12401\/13,\nthe National Research University Higher School of Economics' Academic Fund\naward No.12-09-0062 and\nthe AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.\nThis study comprises research findings from the ``Representation Theory\nin Geometry and in Mathematical Physics\" carried out within The\nNational Research University Higher School of Economics' Academic Fund Program\nin 2012, grant No 12-05-0014.\n\\end{acknowledgement}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzllkc b/data_all_eng_slimpj/shuffled/split2/finalzzllkc new file mode 100644 index 0000000000000000000000000000000000000000..73e862dc0fe19f201c073f5a99d5d1319dbe89b5 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzllkc @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe numerical integration of functions in many dimensions has been a central topic in numerical\nanalysis for a long time. Current schemes such as adaptive cubature and adaptive Monte Carlo \nperform best for smooth integrand functions. However integrands with discontinuities can arise\nquite naturally in a variety of contexts, such as the calculation of the fermionic self-energy\nin condensed-matter physics or the study of multiphase flows in the context of computational fluid dynamics. \n\nIn one dimension discontinuities are easily accommodated within an adaptive framework simply\nby dividing the region of integration into sub-regions on which the integrand is smooth. \nIn this paper we show how this process can be extended to higher dimensionalities. Our method is applicable\nto integrals which are discontinuous on any number of hyperplanes that contain the origin, and in any number of \ndimensions. We limit our attention to integrals over the entire ${\\mathbb{R}^N}$ --- this\nis not a material limitation as integrals over a proper hyperrectangle can be straightforwardly mapped onto\n${\\mathbb{R}^N}$. We assume that the discontinuities in question arise from terms of the form sign$(C_{\\vec{x}})$ where\n$C_{\\vec{x}}$ is any linear combination of the coordinates. These is precisely the form of the discontinuities \nencountered in the Green's functions of fermionic systems. \n\nWe write the integral in question as \n\\begin{equation}\nI = \\int_{\\mathbb{R}^N} \\prod_{i=1}^{M} F_i(\\vec{x})d\\vec{x}, \n\\label{eq:integral}\n\\end{equation}\nwhere $F_i(\\vec{x})$ is discontinuous on the hyperplane with equation $\\vec{a_i}\\cdot \\vec{x} = 0$. \nWe construct the $M\\times N$ \\emph{discontinuity} matrix $\\mat{C}$ such that $C_{ij} = (\\vec{a_i})_j$. \nWe will here assume $M \\geq N$ --- we will comment on this at end the next section. \n\n\\section{Method}\n\nLet $S$ be the set of all $M\\times M$ diagonal matrices with diagonal components $\\pm 1$ ($|S| = 2^M$). \nTo determine the regions on which the integrand is continuous we thus have to solve the homogeneous system of simultaneous\ninequalities \n\\begin{equation}\n\\mat{C}\\mat{S}_i\\vec{x} \\geq 0,\n\\label{eq:system}\n\\end{equation}\nfor every $\\mat{S}_i \\in S$. Each inequality defines a closed-half space; the solution to the system is \nthe intersection of these half-spaces which can be interpreted geometrically as a convex polytope in \nits half-space representation (see \\cite[p. 31]{Grunbaum67}). In the case of a homogeneous system the resultant polytope is \nin fact a polyhedral (infinite) convex cone~\\cite{skeleton}. \n\nLet $P$ be the set of cones obtained by solving Eq.~\\eqref{eq:system}.\nA set of vectors $W_K = \\left\\{\\vec{w}_1, \\vec{w}_2, \\ldots ..\\vec{w}_p \\right\\}$ is a \\emph{skeleton} of a cone $K$\nif $\\vec{x} = \\sum_{i=1}^p \\lambda_i \\vec{w_i}$ belongs to $K$ for every $\\lambda_i \\geq 0$~\\cite{skeleton}. \nThe duality in the representation of a cone as either a system of linear inequalities or a conical combination \nof the skeleton is the essence of the well known Weyl-Minkowski theorem on cones. As we are only interested in \nsubspaces of ${\\mathbb{R}^N}$ with dimension $N$ --- lower-dimensional subspaces correspond to polyhedral facets which do \nnot contribute to the integral --- we can assume that $p \\geq N$. \nThe skeleton of an acute cone is unique up to scalar multiplication of the vectors~\\cite{skeleton}. \nOnce the normalization of the skeleton is fixed, each point in $K$ can be specified through its $\\lambda$ coefficients $(\\lambda_1, \\ldots, \\lambda_p)$.\n\nHaving achieved our goal of partitioning ${\\mathbb{R}^N}$ into regions where the integrand is \ncontinuous we now have to consider how to perform the integration over a cone $K\\in P$. When $p=N$ the polytope \nconstitutes an $N$-simplex which can be readily mapped onto the positive orthant by exploiting the bijection between\n$\\vec{x} \\in K$ and $\\vec{\\lambda}$. When $p>N$ the situation is more complex, for the skeleton is linearly dependent\nand there is no bijection to be exploited. To overcome this problem, each cone $K$ is decomposed into $N$-simplices\n$\\gamma^{K}_1, \\gamma^{K}_2., \\ldots$ which can then be individually mapped onto the positive orthant. The\nset of all simplices $\\digamma = \\left\\{ \\gamma^K_1, \\gamma^K_2, \\ldots | K \\in P \\right\\}$ evidently partitions ${\\mathbb{R}^N}$; the original \nintegration problem has thus been broken down into multiple, separate integrations, one over each \nsimplex in $\\digamma$. The method is inherently parallel --- barring error control considerations each region \nof integration can be processed independently of the others.\n\nTo control the precision of the calculation we use an unsophisticated two-pass scheme. The first pass consists of a \ncrude integration over every simplex $\\gamma \\in \\digamma$, with a relative precision of $10\\%$, yielding a\nresult $\\mu^{(1)}_\\gamma$ with an associated error $\\sigma^{(1)}_\\gamma$. From the $\\mu^{(1)}_\\gamma$ we determine the \nsimplex which contributes the most; let $\\mu_{\\textrm{max}} = \\textrm{max} \\left\\{|\\mu_\\gamma|, \\gamma \\in \\digamma \\right\\}$. \nTo achieve a requested relative precision $f$ on the entire integral $I$ we then repeat the integration, now evaluating \neach simplex to an \\emph{absolute} precision given by $\\epsilon_{\\textrm{abs}} = f\\mu_\\textrm{max}\/\\sqrt\\nu$, where $\\nu = |\\digamma|$\ndenotes the total number of simplices, ensuring obviously we do not re-evaluate the regions for which $\\sigma^{(1)}_\\gamma < \\epsilon_\\textrm{abs}$.\nThe end result $I=\\sum \\mu^{(2)}_\\gamma$ is then associated with an absolute error\n\\begin{equation}\n\\sigma_I = \\sqrt{\\sum_{\\gamma \\in \\digamma} (\\sigma^{(2)}_\\gamma)^2\/\\nu}.\n\\end{equation}\nIn practice small deviations of the resultant precision for the requested precision may occur when there are\nsignificant cancellations. This is not a particularly grave disadvantage as the actual error is always known.\n\nFinally, we return to the question of the number of constraints. We have been assuming that the number of rows $M$ \nof the constraint matrix $\\mat{C}$ is larger than the dimension of the integral, ignoring the case of an integrand which\nhas discontinuities on fewer than $N$ planes. This is dealt with by padding the rows of $\\mat{C}$ with arbitrary vectors \n(so long as they are not parallel to any other vectors) until $M\\geq N$. This trick has the disadvantage of causing \nunnecessary divisions of the region of integration but is necessary to guarantee the existence of cones. \n\n\\section{Implementation}\n\nThe process outlined above is implemented in \\verb|C++| with support for matrices provided by GSL~\\cite{gsl}. The\ninput is the integrand and the matrix $\\mat{C}$ construct as above specifying the discontinuities. The first step is \nthe decomposition of ${\\mathbb{R}^N}$ into the polyhedral cones $P$. To this end we use \\verb|skeleton|~\\cite{skeleton}\nwhich implements a modified version of the Motzkin-Burger algorithm. This package is called from our\ncode and returns the vectors comprising the skeleton of the polyhedral cones in $P$. \n\nTo cut the polyhedral cones in $K$ into $N$-simplices we first project the vectors in $W_K$ onto the\ncone's $(N-1)$-dimensional base. By `base' here we mean the subspace obtained by subtracting from\nall $\\vec{w} \\in W_k$ their components along the axis of the cone and then expressing them as linear\ncombinations of $(N-1)$ orthonormal vectors. We can then construct the desired decomposition of \n$K$ into $N$-simplices by triangulating the points in the $(N-1)$-dimensional base and then adding\nthe origin to these $(N-1)$-simplices. In general this triangulation is not unique. There are\nseveral algorithms to handle the triangulation of the base. We use the Quickhull algorithm\nimplemented in Qhull~\\cite{qhull}. \n\nEach point $\\vec{x}$ of the simplex can be written as a conical combination of the $(\\vec{\\lambda})$ \nand its skeleton vectors. To map the positive orthant onto the unit hypercube we use the rule\n$\\lambda_i = 1\/u_i - 1$. Depending on the integrand other rules may be more suitable but this\nwas chosen for its simplicity. \n\nThe final step is the integration itself. We use \\verb|HIntLib|~\\cite{hintlib1,hintlib2}, a sophisticated\n\\verb|C++| library that among other things implements adaptive cubature with a variety of rules and a range of Monte\nCarlo methods. It would be perhaps more efficient to use an adaptive code that can directly handle the \nsimplicial geometry, such as \\verb|CUBPACK|~\\cite{cubpack1,cubpack2} but for practical reasons this \napproach was not followed here. The integrations are performed in parallel using $\\verb|OpenMP|$ (\\verb|HIntLib|'s\nnative parallelization is not used).\n\n\n\\section{Results}\n\nWe test the method with a variety of integrands and for various dimensionalities. To do so we also have\nto prescribe the discontinuities. To streamline the discussion we express Eq.~\\eqref{eq:integral} as\n\\begin{equation}\nI = \\int_{\\mathbb{R}^N} \\prod_{i=1}^{M} F(g_i(\\vec{x}))d\\vec{x}, \n\\label{eq:integral2}\n\\end{equation}\nwhere $\\vec{g}(\\vec{x}) = \\mat{C}\\vec{x}$. A variety of test-matrices $\\mat{C}$ are considered --- they are listed in the Appendix. \nAll integrations are done using \\verb|HIntLib|'s adaptive routines and its implementation of the embedded degree-$7$ rule of Genz and Malik~\\cite{Genz80}. \nWe define the following integrand test-functions\n\\begin{align}\nF_1 (u) &= \\frac{1}{u - \\alpha + i \\beta \\textrm{sign}(u)} \\label{eq:f1} \\\\\nF_2 (u) &= \\frac{1}{u^2 - \\alpha + i \\beta \\textrm{sign}(u)} \\label{eq:f2}.\n\\end{align}\nWe note that $F_1$ is actually the non-interacting Green's function for the Anderson impurity model\nin the flat-band approximation~\\cite{Hewson97}. Our method was developed with this integrand\nin mind --- we also consider the integrand $F_{2}$ to illustrate the more general applicability of the \nmethod. As $F_{1}, F_{2}$ are complex-valued, we consider for brevity only the real part of Eq.~\\eqref{eq:integral2}. \n\nWe compare the speed-up afforded by our partitioning in terms of the number of integrand evaluations \nrequired to achieve a given relative error. In doing so, we seemingly ignore the computational effort \nrequired for the partitioning itself. In practice this turns out to be essentially insignificant, owing\nto the large computational cost of the integrations. Nevertheless the time spent in partitioning\ncan be reduced by noting that the solution of Eq.~\\eqref{eq:system} and subsequent triangulation can be\ncarried out in parallel for each $\\mat{S_i} \\in S$.\n\nUnless otherwise stated all integrations are carried out to $\\epsilon_{\\textrm{rel}} \\approx 10^{-4}$. Due to the\ntwo-pass technique for controlling the precision of the integration it may happen that the resultant error\nis (sometimes significantly) less than requested. This is to be expected when the $\\mu_\\gamma$ have mostly the\nsame sign, i.e.\\ $|I| \\gg \\mu_{\\textrm{max}}$; in such cases the target absolute error --- which is based on $\\mu_{\\textrm{max}}$ ---\nis smaller than necessary. The resultant error may be less than the requested in\nanother way: To obtain an estimate over each simplex $\\gamma$, \\verb|HIntLib| requires a minimum number $N_\\textrm{min}$\nof integrand evaluations. When this yields an estimate of the integral more precise than requested, nothing\ncan be done to reduce the number of evaluations.\n\n\\begin{table}\n\\caption{Results for $F_1$, $\\alpha = -0.2,\\beta=0.1$. \\label{table:f1}}{\n\\begin{tabular}{l | c | c | c | r}\n$N$ & $M$ & $N_{p}$ & $N_{H}$ & $N_{p}\/N$ \\\\\n\\hline\n2 & 3 & $5.3 \\times 10^4$ & $2.2 \\times 10^6$ & $41.4$ \\\\\n2 & 4 & $8.6 \\times 10^4$ & $8.3 \\times 10^5$ & $9.6$ \\\\\n2 & 5 & $1.3 \\times 10^5$ & $1.2 \\times 10^6$ & $9.0$ \\\\\n2 & 6 & $1.7 \\times 10^5$ & $1.0 \\times 10^6$ & $6.3$ \\\\\n3 & 5 & $3.3 \\times 10^6$ & $>3.0\\times 10^9$ & $>906.7$ \\\\\n3 & 6 & $7.2 \\times 10^6$ & $1.6 \\times 10^9$ & $224.9$ \\\\\n3 & 7 & $9.6 \\times 10^6$ & $2.6 \\times 10^9$ & $265.5$ \\\\\n3 & 8 & $1.4 \\times 10^7$ & $2.5 \\times 10^9$ & $172.9$ \\\\\n3 & 9 & $2.0 \\times 10^7$ & $2.7 \\times 10^9$ & $132.6$ \\\\ \n4 & 7 & $5.8 \\times 10^8$ & $>3.0\\times 10^9$ & $>5.2$ \\\\\n5 & 9 & $4.3 \\times 10^{10}$ & - & -\n\\end{tabular}}\n\\end{table}\n\\begin{table}\n\\caption{Results for $F_2$, $\\alpha = -0.2, \\beta = 0.1$.\\label{table:f2}}{\n\\begin{tabular}{l | c | c | c | c | c | c| r}\n$N$ & $M$ & $N_{p}$ & $\\epsilon^{*}_\\textrm{rel}$ & $N_{H}$ & $N^{*}_{H}$ & $N_{p}\/N$ & $N^{*}_{p}\/N$ \\\\\n\\hline\n2 & 3 & $3.6\\times 10^4$ & $1.8\\times10^{-7}$ & $1.3 \\times 10^4$ & $2.5 \\times 10^5$ & $0.35$ & $7.0$ \\\\\n2 & 4 & $4.8\\times 10^4$ & $4.6\\times10^{-6}$ & $4.0 \\times 10^5$ & $8.1 \\times 10^6$ & $8.3$ & $169$ \\\\\n2 & 5 & $6.0\\times 10^4$ & $4.6\\times10^{-6}$ & $6.5 \\times 10^5$ & $1.4 \\times 10^7$ & $10.8$ & $236$ \\\\\n2 & 6 & $7.2\\times 10^4$ & $1.6\\times10^{-6}$ & $1.8 \\times 10^6$ & $1.2 \\times 10^8$ & $24.9$ & $161$ \\\\\n3 & 5 & $1.7\\times 10^5$ & $8.8\\times10^{-5}$ & $9.8 \\times 10^8$ & $1.2 \\times 10^9$ & $5820$ & $7418$ \\\\\n3 & 6 & $2.5\\times 10^5$ & $7.7\\times10^{-5}$ & $1.7 \\times 10^8$ & $2.7 \\times 10^9$ & $6750$ & $10660$ \\\\\n3 & 7 & $3.8\\times 10^5$ & $6.1\\times10^{-5}$ & $>3.0 \\times 10^9$ & $>3.0 \\times 10^9$ & $7804$ & $>7804$ \\\\\n3 & 8 & $4.8\\times 10^5$ & $8.3\\times10^{-5}$ & $>3.0 \\times 10^9$ & $>3.0 \\times 10^9$ & $6244$ & $>6244$ \\\\\n3 & 9 & $5.8\\times 10^5$ & $7.0\\times10^{-5}$ & $>3.0 \\times 10^9$ & $>3.0 \\times 10^9$ & $>5172$ & $>5172$ \\\\\n4 & 7 & $2.7\\times 10^6$ & $9.8\\times10^{-5}$ & $>3.0 \\times 10^9$ & $>3.0 \\times 10^9$ & $>1111$ & $>5172$ \\\\\n5 & 9 & $4.8\\times 10^{7}$& $9.9\\times10^{-5}$ & $-$ & $-$ & $-$ & $-$\n\\end{tabular}}\n\\end{table}\n\nOur results are presented in Tables~\\ref{table:f1}, ~\\ref{table:f2}. In both tables $N$ denotes the dimension of integration, $M$ the number of hyperplane discontinuities, \n$N_p$ the number of function evaluations required using the partitioning technique and $N_H$ the number of function\nevaluations required to achieve the requested precision without utilising our partitioning scheme. To make the comparison\nmeaningful we obtain $N_{H}$ using the same adaptive cubature routines in \\verb|HIntLib| with the embedded degree-7 Genz-Malik rule that were employed to carry out the simplicial integrations, \nwith each point $\\vec{x} \\in {\\mathbb{R}^N}$ being mapped to $\\vec{t} \\in [-1, 1]^N$ through $x_i = t_i\/(1-t^2_i)$. We emphasise\nthat our proposed integration method can be used in conjunction with any integration algorithm and is not tied to this\nspecific Genz-Malik rule.\n\nWe note that for many of the integrations in Table~\\ref{table:f2} \nwe were unable to reduce the precision below a certain level. Thus for each integrand we report the relative precision $\\epsilon^{*}_\\textrm{rel}$ \\emph{actually} reached\nby our partitioning scheme. We feel it is not clear whether it would be fairer to judge the efficacy of our method method\nby comparing $N_p$ to the function evaluations required to achieve the target relative precision of $10^{-4}$ or the `accidental'\nprecision $\\epsilon^{*}_\\textrm{rel}$. We thus report both quantities, the latter denoted by $N^{*}_{H}$. From our\nresults it's obvious that even when calculating the integral to a precision much greater than required our method \ngreatly reduces the samples required of the integrand.\n\nFor the integrations attempted without partitioning ${\\mathbb{R}^N}$ we had to impose a maximum of $3\\times10^9$ integrand evaluations to prevent the integrator\nfrom exhausting the $8$~GB of RAM we had at our disposal. Apart from requiring fewer integrand samples, our partitioning\nmethod also drastically reduces the amount of memory required. This is because each the grid for each simplex can be discarded\nafter the integration is complete rather than having to concurrently store data for all previous grid refinements. We have\nhowever refrained from trying to quantify the improvement in the memory requirements as this is sensitive to the\ndetails of our implementation, our choice of integration routines and the benchmarking itself rather non-trivial, given the parallel nature\nof the program. Nevertheless in Table~\\ref{table:f2} it is evident that the integration becomes unmanageable without\nour method even in only four dimensions. \n\nAs the number of simplices into which ${\\mathbb{R}^N}$ is partitioned increases very rapidly with $N$, the success of the method depends on whether the advantages\nof a smooth integrand outweigh the cost of having to set up a new adaptive grid for each simplex, and the \n`unnecessary' function evaluations due to the crudeness of our error management. It is is evident that it does;\nin Table~\\ref{table:f1} we see that partitioning reduces the number of required integrand evaluations\nby $1-3$ orders of magnitude and in Table~\\ref{table:f2} by up to $4$ orders of magnitude. \n\n\n\\section{Conclusions}\n\nWe have described a method enabling the numerical integration of functions featuring hyperplane discontinuities\nwith existing adaptive cubature schemes. We showed how to construct a set $P$ of convex polyhedral cones\nthat partition ${\\mathbb{R}^N}$. Each cone $K \\in P$ is then partitioned into simplices $\\gamma^{K}_1,\\gamma^K_2, \\ldots$\nwhich comprise the set $\\digamma$ which partitions ${\\mathbb{R}^N}$. Each simplex $\\gamma \\in \\digamma$ can then\nbe mapped onto a hypercube and integrated using any existing multidimensional numerical integration algorithm. \nWe adopted a two-pass scheme to control the precision of our calculation. This allowed the essentially\ncomplete paralellization of the integrations. \n\nOur method can dramatically accelerate the evaluation of such multidimensional integrals. The reduction\nin the number of integrand samples required to obtain an estimate for the integral becomes more pronounced\nas the dimensionality $N$ increases, and is reduced by several orders of magnitude compared to a naive integration.\nMemory requirements are also greatly improved, allowing the evaluation of integrands in higher dimensions \nthan would be otherwise possible. \n\nThe method can be improved by coupling it directly to an integrator aware of the underlying simplicial \ngeometry, thereby eliminating the need for a simplex-hypercube mapping. Our precision control can also be potentially replaced with a more \nadvanced scheme in which no integrand evaluations are discarded and the threads dynamically synchronised. \nThis could improve the performance of our method but coordinating the threads will require some effort\nprogramming-wise.\n\nThe author would like to thank Alex C Hewson, Paul Carter and Ioannis Pesmazoglou for helpful discussions. This work was\ngenerously supported by the Engineering and Physical Sciences Research Council.\n\n\\section*{APPENDIX}\nWe list here the test matrices $\\mat{C}_{c \\times N}$ pertinent to Eq.~\\eqref{eq:integral2}.\n\\begin{equation*}\nC_{3\\times2}=\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n1 & 1 \n\\end{pmatrix} \\,\nC_{4\\times2}=\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n2 & 1 \\\\\n1 & -1 \n\\end{pmatrix} \\,\nC_{5\\times2}=\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n2 & 1 \\\\\n1 & -1 \\\\\n-1 & 2 \n\\end{pmatrix} \\,\nC_{6\\times2}=\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n2 & 1 \\\\\n1 & 1 \\\\\n1 & -1 \\\\\n-1 & 2 \n\\end{pmatrix}\n\\end{equation*}\n\n\\begin{equation*}\nC_{5\\times3} = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n1 & 1 & -1 \\\\\n-1 & 2 & 1 \n\\end{pmatrix} \\,\nC_{6\\times3} = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n1 & -1 & 1 \\\\\n1 & 1 & -1 \\\\\n-1 & 2 & 1 \n\\end{pmatrix} \\,\nC_{7\\times3} = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n2 & 1 & -1 \\\\\n1 & 1 & 1 \\\\\n1 & 1 & -1 \\\\\n-1 & \\frac{1}{2} & 2 \n\\end{pmatrix}\n\\end{equation*}\n\\begin{equation*}\nC_{8\\times3} = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n2 & 1 & -1 \\\\\n1 & 1 & 1 \\\\\n1 & 1 & -1 \\\\\n\\frac{1}{2} & \\frac{1}{2} & 1 \\\\\n-1 & \\frac{1}{2} & 2 \n\\end{pmatrix} \\,\nC_{9\\times3} = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n2 & 1 & -1 \\\\\n\\frac{1}{2}& 2 & 1 \\\\\n-1 & 1 & \\frac{1}{2} \\\\\n2 & -1 & 1 \\\\\n1 & 1 & -1 \\\\\n-1 & \\frac{1}{2} & 2 \n\\end{pmatrix}\n\\end{equation*}\n\\begin{equation*}\nC_{7\\times4} = \\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 1 \\\\ \n1 & 1 & 1 & 1 \\\\ \n1 & 2 & 1 & 2 \\\\ \n1 & -2 & 2 & 1 \n\\end{pmatrix}\\,\nC_{9\\times5} = \\begin{pmatrix}\n1 & 0 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 & 0 \\\\ \n0 & 0 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 0 & 1 \\\\ \n\n1 & 1 & 1 & 1 & 1 \\\\ \n\\frac{1}{2}& 1 & \\frac{1}{2} & 1 &\\frac{1}{2} \\\\ \n-1 & -1 & \\frac{1}{2} & 1 &2 \\\\ \n2 & 1 &-\\frac{1}{2} & 2 &-\\frac{1}{2}\n\\end{pmatrix}\n\\end{equation*}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nTransients are celestial objects whose brightness varies on timescales from seconds to years, for example due to explosions of their progenitors. These objects help astronomers to investigate physics under extreme conditions, and there are currently a large number of surveys being carried out to search for extra-galactic transients by regularly monitoring the night sky. \nDetected objects include classical transients, such as supernovae (SNe) that are luminous optical transients from either a thermonuclear explosion of a white dwarf, or the core-collapse (CC) of a massive star, as well as new and exotic transients. e.g. the kilonova AT2017gfo \\citep{GW170817MMA,sss17a,dlt17ck,vista,master,decam,lcogtkn, grawita, Smartt2017}. To correctly classify them, spectroscopy is required, however spectroscopic follow-up resources are scarce, and only a fraction of the currently discovered SNe can be classified in this way \\citep{Kulkarni2018}.\n\nInvestigating the multi-band light curves (LCs) of the transients can also provide information about the mechanism that drives the explosions, such as shock cooling, circumstellar matter (CSM) interaction, radioactive decay or a central engine such as a magnetar. The information is encoded in the LCs in different aspects, e.g. their timescales, shapes and absolute luminosities, both for individual objects and for curated samples of specific types.\nMuch progress has been made by linking multi-wavelength light curves to models, both numerical hydrodynamical models and simpler semi-analytical estimates. For instance, for both SNe Ia and stripped envelope CC SNe, whose emission during the photospheric phase is normally powered by the radioactive decay of synthesized \\Ni, \\citet{arnett1982} made a semi-analytic model that can reconstruct the bolometric LCs for physical parameters of the explosion and the progenitors, such as the amount of ejected nickel mass, the ejecta mass and the kinetic energy. Therefore, exploring a large sample of SNe provides the opportunity to unveil their underlying physical origins.\nSuch analysis has been performed in a number of studies, e.g. \\cite{barbarino, taddia2018csp, prentice2019, lyman16}, but the different studies often use different approaches and codes which makes the direct comparisons less straightforward.\nA generic code-package to handle light curve fitting for transients of different types, from different surveys, observed with varying cadences, is therefore useful to provide comparable results.\nFor this purpose, we present {\\tt HAFFET}\\xspace, a data-driven model fitter for transients. \nOften some spectroscopic information is also available for the SNe and needed to perform parts of the analysis, and {\\tt HAFFET}\\xspace also includes procedures for example for measuring ejecta velocities from spectra.\n\nThis paper aims to serve as a description of {\\tt HAFFET}\\xspace and its capabilities at the time of its first release.\nThis is complemented with a web-based\nusers guide, which the reader should always consult for more up-to-date information\\footnote{\\href{https:\/\/haffet.readthedocs.io\/en\/latest\/}{https:\/\/haffet.readthedocs.io\/en\/latest\/}}. This online documentation also contains more technical information and the code itself, whereas this paper aims to describe some of the possibilities in the code-package.\nFurthermore, as a scientific proof-of-concept of {\\tt HAFFET}\\xspace, we are in a companion paper exploring a large sample of Type Ib and Ic SNe, observed by the Zwicky Transient Facility \\citep[ZTF;][]{Bellm2019, Graham2019}, and spectroscopically classified as part of the Bright Transient Survey (BTS) \\citep{fremling2020,perley2020}. That paper \n(Yang et al. in preparation) contains the analysis of the objects that are used here to demonstrate the capabilities of {\\tt HAFFET}\\xspace, and is \nhereafter referred to as Paper I. \nWe have also estimated the luminosities for ZTF SNe Ic-BL using HAFFET in \\cite{Corsi2022} and in \nAnand et al. (in preparation). \n\nThis paper is organized as follows. \nIn Section~\\ref{sec:haffet} we present the code design of {\\tt HAFFET}\\xspace. In Section~\\ref{sec:data}, we describe how {\\tt HAFFET}\\xspace handle meta and observational data, and in Section~\\ref{sec:fitting} we discuss in detail the different available models designed for different types of observational data. In Section~\\ref{sec:application} we further explore the sample of ZTF SNe Ib and Ic with {\\tt HAFFET}\\xspace, complementary to Paper I. Finally, Section~\\ref{sec:conclusion} presents our conclusions and provides a short discussion where we put our results in context.\n\n\\section{HAFFET}\n\\label{sec:haffet}\n\nIn this section, we introduce the code structure as well as the key components of {\\tt HAFFET}\\xspace.\n\n\\subsection{Code Design}\n\\label{sec:cd}\n\nIn Figure~\\ref{fig:flowchart}, we illustrate the main steps as they were generally performed using {\\tt HAFFET}\\xspace for stripped envelope SNe in Paper I: \n\n\\begin{enumerate}\n \\item {\\tt HAFFET}\\xspace started a Python class named {\\tt snelist} with a meta table\\footnote{In {\\tt HAFFET}\\xspace, we use pandas dataframes for table construction: \\url{https:\/\/pandas.pydata.org\/pandas-docs\/stable\/reference\/api\/pandas.DataFrame.html}} that includes a list of objects. \n \n \\item For each object in the meta table, the meta information, e.g. coordinates, redshifts, etc, are standardized, serving as the basis for a new class named {\\tt snobject}.\n \n \\item For {\\tt snobject}, a number of built-in functions are available to parse both photometric and spectroscopic data from local or remote sources.\n \n \\item To anticipate the photometric features, e.g. the peak phase and magnitude, a continuous approach is provided as an optional choice, employing either the gaussian process (GP) regression or analytical SN LC model fittings.\n \n \\item Once the epochs from various bands have been matched (via binning or interpolations), the spectral energy distribution (SED) and the (pseudo)-bolometric LCs can be estimated. The host galaxy extinction can also be estimated via color comparison to templates.\n \n \\item Depending on the type of transient, the bolometric LCs can be fitted with a variety of analytical models: we have included the classical Arnett model, and also provide routines for users to import external models from, e.g. MOSFIT \\citep{mosfit} and redback\\footnote{\\href{https:\/\/redback.readthedocs.io\/en\/latest\/}{https:\/\/redback.readthedocs.io\/en\/latest\/}}.\n \n \\item Each spectrum was smoothed, normalized, and the continuum removed in order to show the emission and absorption features. \n For a certain selected line, the absorption components are automatically identified and are\n then fitted with various models for measuring spectral properties, in particular the minimum of the absorption features to estimate the ejecta velocities.\n \n \\item All of the photometric and spectral properties from {\\tt snobject} may be called by {\\tt snelist} and dispersed into parameter space for sample exploration.\n \n\\end{enumerate}\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{sdapy.pdf}\n \\caption{Flowchart of the main steps in the {\\tt HAFFET}\\xspace process, performed on the stripped envelop SNe in Paper I. As shown, after getting observational data for a single SN, {\\tt snobject} is called to fit on multi-band LCs, which could be used to estimate photometric properties, as well as provide interpolated magnitudes for estimating color and bolometric LC. A further step is to fit bolometric LC with, e.g. the Arnett model, and spectral lines can be also fitted for photospheric velocities, which are often used to help break the degeneracy of the Arnett model. Finally, all the fitted parameters from model fittings are collected by the {\\tt snelist} for a sample study.}\n \\label{fig:flowchart}\n\\end{figure*}\n\nThe functions and their usages of {\\tt snobject} and {\\tt snelist} can be found in the online documentation\\footnote{For example, \\url{https:\/\/haffet.readthedocs.io\/en\/latest\/snobject.html} for {\\tt snobject}.}. In this paper, we give a theoretical overview of {\\tt HAFFET}\\xspace, and users should consult the corresponding functions for each section on the documentation page for more information and application examples. \n\n\\subsection{Data Cache and fitting options}\n\\label{sec:cache}\n\n{\\tt HAFFET}\\xspace handles data and fits for various SNe on a case-by-case basis, and below we describe in detail how it operates.\nFirst, a global variable called {\\tt ZTFDATA}\\footnote{{\\tt HAFFET}\\xspace access to ZTF data via ztfquery \\citep{ztfquery}, that follows the IRSA data structure for data storage. In order to be consistent, {\\tt HAFFET}\\xspace adopted the same global variable and data structure not only for ZTF data but also other resources such as the ATLAS forced photometry.} has to be defined (often in the shell profiles; if not, it will use the current folder). This is the directory for {\\tt HAFFET}\\xspace to manage cached data, e.g. the photometric data downloaded from online resources.\nIn order to increase code efficiency, it is important to properly cache the samplings as model fitting may take a long time, especially when using the Monte Carlo method. \nWe store and load the cached fitting samplings in {\\tt hdf5} format with the joblib\\footnote{\\href{https:\/\/joblib.readthedocs.io\/en\/latest\/index.html}{https:\/\/joblib.readthedocs.io\/en\/latest\/index.html}} package, and all these cached files are stored following a dedicated data structure.\n\nThen, according to the intrinsic properties and data qualities, different SN data may be fitted to models with different options, e.g. whether there is a tail in the SN LCs, or what is the range of photospheric phases for the Arnett model fitting, or if the automatic identification of absorption features is correct.\n{\\tt HAFFET}\\xspace provides default settings in a file for general usage, as well as a possibility for the users to specify parameters for individual SNe\\footnote{The parameter files can be found in the defined {\\tt ZTFDATA} directory, with {\\tt default\\_par.txt} for the general options, and {\\tt individual\\_par.txt} for specific SNe.}.\nAn alternative option to adjust fitting options for particular SNe is to use {\\tt HAFFET}\\xspace with a Graphical User Interface (GUI), see Figure~\\ref{fig:gui}. All the data, fitting parameters, and fitting samplings can be stored and loaded, which is universal to different modes.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.8\\textwidth]{gui1.png}\n \\includegraphics[width=0.45\\textwidth]{gui2.png}\n \\includegraphics[width=0.45\\textwidth]{gui3.png}\n \\caption{A view of the {\\tt HAFFET}\\xspace GUI showing the model fittings on multi-band LCs, spectral lines and the constructed bolometric LC. In the upper panel we show the control window of the GUI.}\n \\label{fig:gui}\n\\end{figure*}\n\n\\section{Data Input}\n\\label{sec:data}\n\nTwo different kinds of data, i.e. meta and observational data, might be employed in the fitting procedure.\n\n\\subsection{Meta data}\n\\label{sec:meta}\n\nUsers may want to supply additional information for the SNe in addition to the observational data, for example spatial coordinates for forced photometry services or the redshifts to correct observing data to the rest frame.\nIn {\\tt HAFFET}\\xspace, both the Bright Transient Survey explorer\\footnote{\\url{https:\/\/sites.astro.caltech.edu\/ztf\/bts\/explorer.php}} (BTS) catalog and the Open Astronomy Catalog\\footnote{\\url{https:\/\/github.com\/astrocatalogs\/OACAPI}} (OAC) tables are available, providing meta data for the SNe. Users can also create their own tables specifically for their needs, with proper column names following this specific scheme\\footnote{More keywords of the scheme can be found at the {\\tt snelist} part of \\href{https:\/\/github.com\/saberyoung\/HAFFET\/blob\/master\/sdapy\/data\/default_par.txt}{https:\/\/github.com\/saberyoung\/HAFFET\/blob\/master\/sdapy\/data\/default\\_par.txt}.}:\n\n\\begin{verbatim}\n... ...\nidkey : objid # str # meta key as index\nidkey1 : alias # str # meta key as alias\nrakey : ra # str # meta key for ra\ndeckey : dec # str # meta key for dec\nzkey : z # str # meta key for redshift\ndistkey: dist # str # meta key for distance\n... ...\n\\end{verbatim}\n\nBelow is an illustration showing how only columns with certain names can be appropriately identified and processed \nto the {\\tt HAFFET}\\xspace meta table.\nHere, the object ID and coordinates would be read correctly, but the redshifts would be incorrect because of the erroneous column name (redshift instead of $z$).\n\n\\begin{verbatim}\nobjid,ra,dec,redshift\nZTF18aajpjdi,13:18:06.62,+27:52:24.2,0.0746\nZTF18aahtkze,11:14:19.24,+34:28:43.7,0.0693\nZTF18aapictz,11:59:37.00,+27:31:50.0,0.084\n... ...\n\\end{verbatim}\n\nThe Object ID is the only item \nrequired as the identifier of an object. \nIn most cases, an object would have multiple names, e.g. the survey name and the IAU name. \nIn order to compile all of these names that may be used to search for a certain object, an alias ID column is established as an option. \n\nCoordinates are essential mainly when using forced photometry services, and for getting Milky Way extinctions. There are built-in functions that can handle coordinates with different formats via astropy\\footnote{\\url{https:\/\/docs.astropy.org\/en\/stable\/index.html}}, as well as the healpix\\footnote{\\url{https:\/\/healpy.readthedocs.io\/en\/latest\/index.html}} indices for mollweide plots. \n\nThe redshift for each object should be well determined; otherwise all the \nanalysis will be done in the observer frame.\nUnless distances are specified, redshifts were converted to distances using a flat cosmology with H$_0=70$~km~s$^{-1}$~Mpc$^{-1}$ and $\\Omega_{\\rm{m}} = 0.3$. This was done using the redshifts in the CMB frame where the dipole parameters were obtained from \\cite{Fixsen}. We also included corrections for peculiar velocities using the formalism of \\cite{2015MNRAS.450..317C}.\nFor each individual SN we include an uncertainty on the peak magnitude corresponding to a 150~km~s$^{-1}$ uncertainty included in the peculiar velocity correction and a systematic $\\pm3$~km~s$^{-1}$~Mpc$^{-1}$ error on the adopted Hubble constant. \nFor the most nearby objects, with the largest peculiar motion corrections, the relative errors on the absolute magnitudes are the largest.\nComparing the estimates for peculiar velocity corrections for our SNe (Paper I) within $z < 0.015$ with values estimated using \\cite{2015ApJ...807...35M} and the corrections provided by NED for the Virgo, Great attractor and Shapley cluster \\citep{Mould2000ApJ...529..786M} we find a rms of 0.19 mag, so for these low-z SNe we add another 0.15 mag to the peak luminosity uncertainty in our error budget.\n\nThe transient type matters for some of the analysis, for example which spectral line should be used to determine the photospheric velocity, or which intrinsic colour template should be compared with to estimate the host galaxy reddening. \n{\\tt HAFFET}\\xspace is able to supply this from the BTS or OAC catalogs.\nThere is also a straightforward photometric categorization procedure within {\\tt HAFFET}\\xspace as implemented from \nastrorapid\\footnote{\\url{https:\/\/astrorapid.readthedocs.io\/en\/latest\/index.html}}. Moreover, since {\\tt HAFFET}\\xspace provides spectral line identification, this is aiding in a more secure classification of the transient types.\n\nWithin {\\tt HAFFET}\\xspace we correct all photometry for Galactic extinction, using the Milky Way (MW) color excess $E(B-V)_{\\mathrm{MW}}$ toward the position of the SNe.\nThese are all obtained from \\cite{2011ApJ...737..103S}\\footnote{By default, {\\tt HAFFET}\\xspace use Milky Way extinction in the meta table. If it is not available, and the coordinates are properly parsed, {\\tt HAFFET}\\xspace turn to get extinctions from \\cite{2011ApJ...737..103S} map via dustmaps \\citep{dustmaps}, i.e. \\url{https:\/\/dustmaps.readthedocs.io\/en\/latest\/}}. All reddening corrections are applied using the \\cite{1989ApJ...345..245C} extinction law with $R_V=3.1$. \n\nHost galaxy extinction is more uncertain. One method to estimate host-galaxy extinction is to make use of the fact that SNe often have similar intrinsic colors after peak \\citep[e.g.,][]{Drout2011}. This was developed by \\cite{stritzingercolors} and \\cite{taddia2018csp} for stripped envelope SNe and Type II SNe. \nFor super-luminous SNe (SLSNe), \\citet[][their table A4]{zhihao} investigated the $g-r$ colours at peak for a large sample of ZTF objects, which is used by {\\tt HAFFET}\\xspace as templates for SLSNe once the object has both $g$- and $r$-band data.\nTo estimate the values for the host-galaxy extinction, we use these intrinsic SN color-curve templates and attribute the difference between the observed and the intrinsic color to the host galaxy reddening.\n\n\\subsection{Observational data}\n\\label{sec:obsdata}\n\nOnce a list of objects is constructed using an appropriate meta table, each object's observational data must be assigned. \nBesides the image (see Fig. \\ref{fig:finder}),\n{\\tt HAFFET}\\xspace can be utilized to work with photometric and spectroscopic data, and in this section we describe how to input these data to {\\tt HAFFET}\\xspace.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{finder.png}\n \\caption{The image of the field of SN 2020bcq as a finder that was automatically made by {\\tt HAFFET}\\xspace. The images are queried with the PS1 Image Cutout Service\\tablenotemark{a}, while the SDSS field stars (the green open circles) are downloaded via astroquery\\tablenotemark{b}. The position of SN 2020bcq is marked as the red source in the middle of the inset.}\n \\label{fig:finder}\n\\tablenotetext{a}{\\url{https:\/\/outerspace.stsci.edu\/display\/PANSTARRS\/PS1+Image+Cutout+Service}}\n\\tablenotetext{b}{\\url{https:\/\/astroquery.readthedocs.io\/en\/latest\/}}\n\\end{figure*}\n\n\n\\subsubsection{Using private data and arbitrary input formats}\n\\label{sec:private}\n\nDifferent datasets may have significantly different data formats, depending on how the data was reduced and presented. \nFor example, certain codes favor the \\texttt{json} format, while others make use of the \\texttt{csv} format.\nMany sources may also use various keywords to describe the data, e.g. different notations of magnitudes or fluxes with distinct units.\n\nBy asking for particular previously determined keywords, {\\tt HAFFET}\\xspace uses a pandas dataframe to cope with pure text forms and streamline the procedure.\nIn the example shown below, the photometric data file should use the same keywords in order for them to be properly read\\footnote{More keywords of the scheme can be found at the {\\tt snobject} part of \\href{https:\/\/github.com\/saberyoung\/HAFFET\/blob\/master\/sdapy\/data\/default_par.txt}{https:\/\/github.com\/saberyoung\/HAFFET\/blob\/master\/sdapy\/data\/default\\_par.txt}.}:\n\n\\begin{verbatim}\n... ...\njdkey : jdobs # str # julian date\nmagkey : mag # str # AB magnitude\nemagkey : emag # str # magnitude error\nlimmagkey: limmag# str # limiting magnitude\nfluxkey : flux # str # flux (micro Jy)\nefluxkey : eflux # str # flux error\nfilterkey: filter# str # filters\n... ...\n\\end{verbatim}\n\nIt is important to note that \\texttt{jdkey} and \\texttt{filterkey} are strictly required, and either mag\/emag or flux\/eflux should be provided after that.\nThen, to convert between magnitudes and fluxes, built-in functions can be utilized. \n{\\tt HAFFET}\\xspace uses a default zeropoint of 23.9, which is appropriate for converting fluxes measured in micro Janskys.\n\nIt frequently happens that LCs from different sources do not match up. This can be due to a variety of factors, such as the methodology used to estimate the photometry, the way standard stars were chosen, or not correcting for the different filter functions. Therefore, {\\tt HAFFET}\\xspace makes a \\texttt{source} column whenever a LC is appropriately included in order to display the data origins.\n\nSimilar plain text files with either 2 or 3 columns are required for input of spectra, as well as an observation time that might be used later to estimate the line velocity evolution. \nThe three columns present the wavelengths, fluxes, and their errors. When making spectral fits, {\\tt HAFFET}\\xspace has built-in methods to mimic flux errors depending on the spectral quality if errors are not provided.\nThe flux units for spectra could be arbitrary, and {\\tt HAFFET}\\xspace is able to absolute calibrate them with interpolated photometric magnitudes.\n\n\\subsubsection{Using data from public online sources}\n\\label{sec:gm}\n\nSince open web servers offer data in a consistent data format, {\\tt HAFFET}\\xspace contains built-in functions for obtaining data from them, i.e. the ZTF and ATLAS Forced Photometry Services, the Open Astronomy Catalog API, and the Growth marshal\/fritz broker for the internal ZTF groups\\footnote{Users can easily add sources from their own API in the {\\tt HAFFET}\\xspace package following a consistent data format, and feel free to contact us for adding them into the GUI.}. A valid set of authorizations is required by {\\tt HAFFET}\\xspace to access these services if a user wants to utilize them.\n\nAs shown in Fig. \\ref{fig:ztfbase}, for the ZTF Forced Photometry Services the user can use {\\tt HAFFET}\\xspace to check if the object is covered by multiple fields, and if so, correct the baseline of different fields in the same filter. More details can be found in \\citet[][see their Fig. 4]{yuhan2019}.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{baseline.png}\n \\caption{Baseline correction for ZTF forced photometry LCs exemplified by SN 2020bcq. This target was observed in three filters (g, r and i band), and two fields (field 674 and 171). Observations associated with different fcqf ID (using Eq. 1 of \\citealp{yuhan2019}) and counts are shown in distinctive colors. The two red vertical dashed lines are set based on the user option for the selection of baseline range. The observations within the range are used to set the baseline flux levels, which are aligned by {\\tt HAFFET}\\xspace so that observations from different fields for the same filter are mutually consistent.}\n \\label{fig:ztfbase}\n\\end{figure*}\n\n\\section{Model Fittings}\n\\label{sec:fitting}\n\n{\\tt HAFFET}\\xspace is built to fit functions to a variety of datasets, including spectral absorption features and many kinds of LCs.\nDespite having very different data, these can be fitted using the same techniques.\nIn {\\tt HAFFET}\\xspace we therefore specify a number of \"models\" that provide the objectives for the data to approach as well as so-called \"engines\" that are used to prepare the dataset for certain fits.\n\nThere are currently nine engines available in {\\tt HAFFET}\\xspace, and these are shown in \nTable~\\ref{tab:engines}, along with a variety of built-in models for each of them, see Table~\\ref{tab:models}.\nIn this section, we describe all built-in engines and models, show how to tune their hyperparameters, and explain how the user can define their own engines and models.\n\n\\begin{table*}\n\\caption{Table of built-in engines defined in {\\tt HAFFET}\\xspace.}\n\\begin{center}\n\\begin{tabular}{cc} \\hline\\hline\nEngine name & Description \\\\ \n\\hline\nmultiband\\_early & multi-band LCs at early phases \\\\ \nmultiband\\_main & multi-band LCs at around the peak epoch \\\\ \nbol\\_early & bolometric LCs at early phases \\\\ \nbol\\_main & bolometric LCs at around the peak epoch \\\\ \nbol\\_tail & bolometric LCs at the tail phases \\\\ \nbol\\_full & bolometric LCs for the full range \\\\ \nsed & spectral energy distribution data, \\\\ \n& i.e. multi-band photometry or spectra \\\\ \nspecline & spectral line absorption feature \\\\ \nspecv\\_evolution & spectral\/photospheric line velocities \\\\\n\\hline\n\\end{tabular}\n\\label{tab:engines}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{Table of built-in models defined in {\\tt HAFFET}\\xspace. Here we list only the basic models, and in {\\tt HAFFET}\\xspace there are also models with multiple components, e.g. the Arnett model at the main peak plus the tail model, and a boundary of time between them is set as a free parameter as well. For such a case, we use a bol\\_full engine to prepare bolometric LCs for model fitting.}\n\\begin{center}\n\\begin{tabular}{cccc} \\hline\\hline\nModel Name & Engine & Description & Reference(s) \\\\ \n\\hline\nbazin & multiband\\_main & parametric model suited for SNe Ia LCs & \\cite{bazin} \\\\ \nvillar & multiband\\_main & parametric model for a broad range of LC & \\cite{villar} \\\\ \n & & morphologies, e.g. with bumps or plateau & \\\\ \nrisepl & multiband\\_early & power law model for the early rise of SNe & \\cite{Miller_2020} \\\\ \nsalt2 & multiband\\_main & empirical model for SNe Ia LCs & \\cite{salt2} \\\\\narnett & bol\\_main & radioactive model for bolometric LCs & \\cite{arnett1982} \\\\ \n & & of SNe Ia or CC at around the peak epochs & \\\\ \ntail & bol\\_tail & radioactive model for bolometric LCs & \\cite{Wygoda2019} \\\\ \n & & of SNe Ia or CC at the tail phases & \\\\ \nsbo & bol\\_early & shock cooling tail model for SNe IIb\/Ib & \\cite{piro2021} \\\\ \nmagnetar & bol\\_main & magnetar model for SLSNe & \\cite{Nicholl_2017} \\\\ \ngauss & specline & Gaussian model for spectral line profile & \\cite{gauss} \\\\ \nvoigt & specline & Voigt model for spectral line profile & \\cite{voigt} \\\\ \nexponential & specv\\_evolution & exponential model for velocity evolution & \\cite{Fremling2018} \\\\\n\\hline\n\\end{tabular}\n\\label{tab:models}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.6\\textwidth]{model_fitter.pdf}\n \\caption{Flowchart illustrating model fitting procedures of {\\tt HAFFET}\\xspace. As shown, {\\tt HAFFET}\\xspace first obtain model function and priors, as well as data to be fitted that were prepared by the engines. Then a minimizing process is performed to constrain model parameters. The output could be used as prior for MCMC routines. Finally the fitted samplings can be called by {\\tt HAFFET}\\xspace to, e.g. produce parameter contours, and reproduce observational data.}\n \\label{fig:modelfit}\n\\end{figure*}\n\n\\subsection{Defining models}\n\\label{sec:model}\n\nA \\texttt{python} file that contains analytical functions and a \\texttt{json} file that specifies the engine and parameters are used to define each model in {\\tt HAFFET}\\xspace.\nBelow is an example using the \\texttt{json} file of the power-law fits (which is further discussed in section~\\ref{sec:multiearly}):\n\n\\begin{verbatim}\nmodelmeta = {\n ...\n 'powerlaw_multiple':\n {\n 'engine': 'multiband_early',\n 'alias':\n [\n 'pl',\n 'powerlaw',\n 'Powerlaw',\n ],\n 'description': 'Power law fits',\n 'func': 'functions.powerlaw_full', \n 'parname' :\n [\n r'$t_\\mathrm{fl}$', \n r\"$A$\", \n r\"$\\alpha$\", \n r\"$C$\",\n ],\n 'par' :\n [\n 'texp', \n 'a', \n 'alpha', \n 'c',\n ],\n 'bestv':\n {\n 'texp' : -18,\n 'a' : 100,\n 'alpha' : 2, \n 'c' : 60,\n },\n 'bounds':\n {\n 'texp' : [-100, 0],\n 'a' : [0, 10000],\n 'alpha' : [0, 10],\n 'c' : [0, 10000],\n },\n 'same_parameter': ['texp'],\n 'fit_error' : True,\n },\n ...\n}\n\\end{verbatim}\n\nAs shown, the \\texttt{powerlaw\\_multiple} model would be called to fit data prepared by the \\texttt{multiband\\_early} engine with the \\texttt{powerlaw\\_full} functions having four free parameters. As discussed in Section~\\ref{sec:optimize}, there are two different likelihoods available in {\\tt HAFFET}\\xspace. By default, {\\tt HAFFET}\\xspace uses a Gaussian likelihood, while if we set \\texttt{fit\\_error} = True, it will include one more term to the likelihood as additional model uncertainties. \nThe \\texttt{same\\_parameter} item decides whether to fit multiple band LCs simultaneously or independently. If the \\texttt{same\\_parameter} is left blank, {\\tt HAFFET}\\xspace will fit the LCs in different bands one at a time. If some other parameters are specified, such as \\texttt{t\\_exp} for example, {\\tt HAFFET}\\xspace fit all bands at once with the same \\texttt{t\\_exp}, which is the epoch of first light for such a case.\n\n\\subsection{Built-in models}\n\\label{sec:builtin}\n\nIn the sections that follow we outline the fundamental concepts of the built-in models while highlighting the possibility of experimenting with various fitting choices, such as different free parameters, likelihoods, or fitting patterns. \n\n\\subsubsection{multiband\\_early models}\n\\label{sec:multiearly}\n\nPower-law fits are working with multiple-band photometric data prepared by the 'multiband\\_early' engine. \nThese fits can for example be used to estimate explosion epochs and rise times, but the precision by which we can constrain the first light is also of importance for many other investigations (for example for coincidence studies with neutrino signals or gravitational wave detections).\n\\cite{Miller_2020} developed a Bayesian framework to model the early rise of LCs as a power law in time in their Eq. 1:\n\\begin{equation}\n f_b(t) = C + H[t_\\mathrm{fl}] A_b (t - t_\\mathrm{fl})^{\\alpha_b},\n \\label{eqn:flux_model}\n\\end{equation}\nwhere C is the baseline flux present\nin the reference image prior to SN discovery, $t_\\mathrm{fl}$ is the time of first light, $H[t_\\mathrm{fl}]$ is the Heaviside function equal to 0 for t $< t_\\mathrm{fl}$ and 1 for t $\\geq t_\\mathrm{fl}$, $A_\\mathrm{b}$ is a constant of proportionality in filter b, and $\\alpha_\\mathrm{b}$ is the power-law index in filter b. \n\n\\cite{Miller_2020} applied this framework to a subset of normal SNe Ia from ZTF to simultaneoulsy model the evolution in both the ZTF $g$ and $r$ filters.\nThis assumes that the epoch of first light is \nthe same in the two ZTF bands, which is a fair assumption given the ZTF cadence and the similarity of the SN ejecta opacity at these wavelengths. \nIn {\\tt HAFFET}\\xspace we follow the same methodology to characterize the early emission in the \\texttt{powerlaw\\_multiple} model (see section \\ref{sec:model}), but also provide the possibility to fit the different band LCs independently by removing \\texttt{same\\_parameter} in the \\texttt{power\\_single} model. \n\nAs in \\cite{Olling2015}, \\cite{Miller_2020} only include observations up to 40\\% of the peak\namplitude of the SN LC, and note that this particular choice, instead of 30\\% or 50\\%, does slightly affect the final inference for the model parameters.\nFor comparison we adopt the same settings as default in the \\texttt{multiband\\_early} engine, where the peak amplitude in each band should be previously estimated via the gaussian process or other \\texttt{multiband\\_main} models. \n\nThe explosion epoch can also be fitted to bolometric models as a free parameter, or estimated as the middle of the last non-detection and the first detection epochs.\nIn {\\tt HAFFET}\\xspace, we provide users the option to choose their method of estimation.\n\n\\subsubsection{multiband\\_main models and Gaussian process}\n\\label{sec:multimain}\n\nThere are various spline techniques available to interpolate the multiple band photometric data around the main peak, i.e. Gaussian Process (GP) regression and the analytic model fittings. \n\nIn {\\tt HAFFET}\\xspace, we provide a non-parametric data-driven interpolation technique (i.e. GP) to serialize the LCs. In order to include color effects, we perform time-series forecasting on the joint multi-band (e.g. $g$ and $r$) fluxes convolved with the wavelengths. \nWe use a flat mean function\\footnote{Other mean functions are also available in {\\tt HAFFET}\\xspace, such as the Bazin or Villar models.} for the flux form, and establish two kernels, i.e. a constant kernel for the wavelengths, and a stationary Matern32 kernel for the fluxes, with the {\\tt GEORGE}\\footnote{\\href{https:\/\/george.readthedocs.io}{https:\/\/george.readthedocs.io}} package.\n\nThe GP interpolation becomes less informative with larger uncertainties and when the data are sparser. For estimating physical parameters for normal SNe, an alternative approach is then to fit the data using analytic functions, such as those used by \\cite{bazin, Karpenka, kessler1, kessler2, villar}.\nFor this purpose we included the \\cite{bazin} and \\cite{villar} models in {\\tt HAFFET}\\xspace. The 'Bazin' model is appropriate for SN LCs without fluctuations, whereas the 'Villar' model is also useful to fit plateaus and bumps.\nFor the Bazin model, the flux form is determined by 5 parameters:\n\n\\begin{equation}\n\\label{eqn:bazin}\nF(t) = \\mathrm{A} \\frac{e^{-(t-t_0)\/\\tau_\\mathrm{fall}}}{1+e^{-(t-t_0)\/\\tau_\\mathrm{rise}}} + \\mathrm{B}\n\\end{equation}\nwhere \\citet[][their fig.~8]{taddia2015sdss} illustrate the different parameters and how they contribute to the light curve properties.\nWe can obtain the date of maximum from the derivative of this function as $t_\\mathrm{max} = t_0 + \\tau_\\mathrm{rise} \\mathrm{log} \\frac{-\\tau_\\mathrm{rise}}{\\tau_\\mathrm{rise}+\\tau_\\mathrm{fall}}$ \\citep{bazin}.\n\nThe functional form of the Villar model is similar to the Bazin model, but incorporates a plateau component by adding two additional free parameters:\n\n\\begin{equation}\n\\label{eqn:villar}\nF(t) = \\begin{cases} \n \\frac{A+\\beta(t-t_0)}{1+e^{-(t-t_0)\/\\tau_\\mathrm{rise}}} & t 0$ and we set $\\bigtriangleup = 1$ in this paper.\n\n\\subsection{Problem Formulation}\nTo ensure that each requested FoV is rendered and transmitted within the VR interaction latency, we aim to optimize the total QoE under fixed VR interaction latency constraint via determining the optimal association between MEC and VR user group, and optimal rendering MEC for model migration. \n\nThe proposed MEC rendering schemes aim at maximizing the long-term total QoE under VR interaction latency constraint in the continuous time slots with respect to the policy $\\pi$ that maps the current state information $S_t$ to the probabilities of selecting possible actions in $A_t$. Therefore, based on the QoE of each VR user, an optimization problem (P1) is formulated as\n\\begin{align}\n (\\rm{P1})~\\max_{\\pi(A_t|S_t)}&\\sum_{t=0}^{\\infty}\\sum_{k = 1}^{K_{\\text{VR}}}\\gamma^{t}\\mathbb{E}_{\\pi}[\\rm{PSNR}_{k}]\\\\\n &T_{k} \\leq T_k^{\\text{th}},\n\\end{align}\nwhere $\\gamma\\in[0, 1)$ is the discount factor which can determine the weight of the future QoE, and $\\gamma = 0$ means that the agent just concerns the immediate reward. The state $S_t$ contains the index of the requested FoV, the location of each VR user, and the computation ability of each MEC. The action $A_{t}$ includes the optimal association between MEC and VR user group, and optimal rendering MEC for model migration.\n\nSince the dynamics of the wireless VR system is Markovian in continuous time slots, this is a Partially Observable Markov Decision Process (POMDP) problem which is generally intractable. Here, the parital observation refers to that the MECs can only know the previous FoV requests and the location of each VR user in the environment, while they are unable to know all the information of the communication environment, including, but not limited to, the channel conditions, and the FoV request in the current time slot. Furthermore, the traditional optimization methods may need the global information to achieve the optimal solution, which not only increase the overhead of signal transmission, but also increase the computation complexity. Approximate solutions will be discussed in Section III.\n\n\n\n\\begin{figure*}[!t]\n \\centering\n \\includegraphics[width=7 in]{prediction_learning_system.pdf}\n \\caption{Decoupled learning strategy for MEC rendering schemes in the wireless VR network.}\n \\label{basic_modules}\n\\end{figure*}\n\n\\section{Deep Reinforcement Learning-Based MEC Rendering Scheme}\nKnowing the deep neural networks as one of the most impressive non-linear approximation functions, DRL is an effective method to optimally solve POMDP problems \\cite{Jiang2}. In this section, to solve (P1), a decoupled learning strategy is proposed for FoV prediction and MEC rendering association, as shown in Fig. 5. Specifically, a RNN model based on GRU is used to predict FoV preference of each user over time. Then, four DRL algorithms, including centralized DQN, distributed DQN, centralized AC, and distributed AC, are proposed to select the FoV rendering MEC and the associated MEC for downlink transmission.\n\n\n\\subsection{FoV Prediction}\nBrownian Motion is used to simulate the eye movement of VR users over time, and assuming that the uplink received FoV preference of the $k$th VR user at the $t$th time slot is $\\widehat{{F}_{t}^{\\text{oV}}}^{k}\\in\\{1,2,...,N_{\\text{FoV}}\\}$. In order to detect dynamics in FoV preference of each VR user, the proposed learning scheme aims at utilizing not only the information presents in the most recent observation $O_{t}=\\{O_t^{1},O_t^{2},...,O_t^{\\text{K}_{\\text{VR}}}\\}$, where $O_t^{k} = \\{\\widehat{{F}_{t}^{\\text{oV}}}^{k}\\}$, but also the historical information in the previous observations ${H}_{t}=\\{O_{t-T_{0}+1},...,O_{t-2},O_{t-1}\\}$ given a memory window $T_{0}$. To recognize FoV preference over time, a RNN model with parameters $\\pmb{\\theta}_{\\text{RNN}}$, and specifically a GRU architecture, is leveraged. $\\pmb{\\theta}_{\\text{RNN}}$ is consisted of both the GRU internal parameters and the weights of the softmax layer. We choose RNN due to its ability in capturing time correlation of FoV preference over time, which can help learn the time-varying FoV preference for better prediction accuracy.\n\nAs shown in Fig. 5, the GRU layer includes multiple standard GRU units and historical observations $[O_{t-T_{0}+1},...,O_{t-1},]$ sequentially inputted into the RNN predictor. For the $k$th VR user, the GRU layer is connected to an output layer which is consisted of a softmax non-linearity with ${N}_{\\text{FoV}}$ output values, which represents the predicted probability $\\mathcal{P}\\{\\widehat{{F}_{t}^{\\text{oV}}}^{k} = \\hat{f}|[O_{t-T_{0}+1}^{k},...,O_{t-1}^{k}], \\pmb{\\theta}_{\\text{RNN}}\\}$ of the $\\hat{f}$th FoV ($\\hat{f} = 1,...,{N}_{\\text{FoV}}$) for the $t$th time slot given historical observations $[O_{t-T_{0}+1}^{k},...,O_{t-1}^{k}]$. \n\nTo adapt the model parameter $\\pmb{\\theta}_{\\text{RNN}}$, standard Stochastic Gradient Descent (SGD) via BackPropagation Through Time (BPTT) \\cite{BPTT} is deployed. At the $(t+1)$th time slot, the parameters $\\pmb{\\theta}_{\\text{RNN}}$ of the RNN predictor can be updated as\n\\begin{equation}\n \\pmb{\\theta}_{\\text{RNN}}^{t+1} = \\pmb{\\theta}_{\\text{RNN}}^{t} - \\lambda_{\\text{RNN}}\\nabla L_{\\text{RNN}}(\\pmb{\\theta}_{\\text{RNN}}^{t}),\n\\end{equation}\nwhere $\\lambda_{\\text{RNN}}\\in(0,1]$ is the learning rate, $\\nabla L_{\\text{RNN}}(\\pmb{\\theta}_{\\text{RNN}}^{t})$ is the gradient of the loss function $L_{\\text{RNN}}(\\pmb{\\theta}_{\\text{RNN}}^{t})$ to train the RNN predictor. $L_{\\text{RNN}}(\\pmb{\\theta}_{\\text{RNN}}^{t})$ can be obtained by averaging the cross-entropy loss as \n\\begin{equation}\n L_{\\text{RNN}}^{t}(\\pmb{\\theta}_{\\text{RNN}}) \\!=\\! -\\!\\!\\!\\!\\!\\!\\!\\!\\sum_{t^{'} = t-T_b+1}^{t}\\!\\!\\!\\!\\!\\log\\left(\\!\\mathcal{P}\\{\\widehat{{F}^{\\text{oV}}_{t^{'}}} = \\widetilde{{F}^{\\text{oV}}_{t^{'}}}|O_{t^{'}-T_{0}}^{t^{'}},\\pmb{\\theta}_{\\text{RNN}}\\}\\!\\right),\n\\end{equation}\nwhere\n\\begin{equation}\n O_{t^{'}-T_{0}}^{t^{'}} = [O_{t^{'}-T_{0}+1},...,O_{t^{'}-1},O_{t^{'}}],\n\\end{equation}\nand $T_b$ is the randomly selected mini-batch size.\n\nThrough FoV prediction, MECs are able to know the FoV preference of each VR user in advance. The VR users with the same predicted FoVs can be grouped together. After FoV rendering, the MECs will multicast or unicast the required FoVs to VR users selecting the same FoV, or a single VR user selecting unique FoV, respectively.\n\n\\subsection{Deep Reinforcement Learning}\nThe main purpose of Reinforcement Learning (RL) is to select proper MECs for MEC rendering schemes. Through a series of action strategies, MECs are able to interact with the environment, and obtain rewards due to their actions, which help to improve their action strategies. After plenty of iterations, MECs can learn the optimal policy that maximizes the long-term rewards. \n\nWe define $S\\in\\mathcal{S}$, $A\\in\\mathcal{A}$, and $R\\in\\mathcal{R}_{e}$ as any state, action and reward from their corresponding sets, respectively. According to the observed environmental state $S_t$ at the $t$th time slot, MECs choose specific actions $A_t$ from the set $\\mathcal{A}$ and receive rewards $R_t$, which are regarded as a metric to measure whether the selected actions are good. Thus, the purpose of RL algorithm is to find an optimal policy $\\pi$ which can maximize the long-term reward for $A = \\pi(S)$. The optimization function can be formulated as $$ and the detailed descriptions of the state, action and reward of problem (P1) are introduced as follows.\n\\begin{itemize}\n \\item State: At the $t$th time slot, the network state can be denoted as \n \\begin{align}\n S_{t} &= (\\widetilde{\\mathcal{F}_{t}^{\\text{oV}}}, \\mathcal{L}_{k,i}^{t}, \\mathcal{F}_{i}^{\\text{MEC}})\\in \\mathcal{S}, \\\\ \\nonumber\n \\text{with}~\\widetilde{\\mathcal{F}_{t}^{\\text{oV}}} &= \\{\\widetilde{{F}_{t}^{\\text{oV}}}^{1}, \\widetilde{{F}_{t}^{\\text{oV}}}^{2},...,\\widetilde{{F}_{t}^{\\text{oV}}}^{K_{\\text{VR}}}\\}, \\\\ \\nonumber\n \\mathcal{L}_{k,i}^{t} &= \\{ {l}_{k,1}^{t}, {l}_{k,2}^{t},..., {l}_{k,B}^{t}, \\},\\\\ \\nonumber\n \\mathcal{F}_{i}^{\\text{MEC}} &= \\{F_{1}^{\\text{MEC}}, F_{2}^{\\text{MEC}},..., F_{B}^{\\text{MEC}}\\},\n \\end{align}\n where $\\widetilde{{F}_{t}^{\\text{oV}}}^{k}$ is the index of the predicted FoV of the $k$th VR user at the $t$th time slot. $l_{k,i}^{t}$ is the distance between the $k$th VR user and the $i$th MEC at the $t$th time slot. $F_{i}^{\\text{MEC}}$ is the computation capability of the $i$th MEC.\n \n \\item Action: The action space can be written as\n \\begin{align}\n {A}_{t} &= \\{\\check{\\mathcal{A}}_{k,q}^{t}, \\acute{\\mathcal{A}}_{k,i}^{t}\\}\\in\\mathcal{A}, \\\\ \\nonumber\n \\text{with}~\\check{\\mathcal{A}}_{k,q}^{t} &= \\{\\check{A}_{k,1}, \\check{A}_{k,2},...,\\check{A}_{k,{N}_{\\text{FoV}}}\\},\\\\ \\nonumber\n \\acute{\\mathcal{A}}_{k,i}^{t} &= \\{\\acute{A}_{k,1}, \\acute{A}_{k,2},...,\\acute{A}_{k,K_{\\text{VR}}}\\}, \\nonumber\n \\end{align}\n where $\\check{A}_{k,q}^{t}\\in\\{0, 1\\}$ and $\\acute{A}_{k,i}^{t}\\in\\{0, 1\\}$ represent whether the $k$th MEC will render the $q$th FoV and serve the $i$th VR user at the $t$th time slot, respectively. For instance, if $\\check{A}_{k,q}^{t} = 1$ and $\\check{A}_{k,j}^{t} = 0$ , the $k$th MEC will render and migrate the $q$th FoV to the $j$th ($j\\neq k$) MEC choosing the same FoV. If $\\acute{A}_{k,i}^{t} = 1$, the $k$th MEC will support the downlink transmission of the $i$th VR user, otherwise, not.\n \n \\item Reward: The immediate reward $R_t$ is designed as\n \\begin{equation}\n {R}_{t}(S_t,A_t) = \\sum\\limits_{k=1}^{K_{\\text{VR}}} \\text{PSNR}_{k}^{t}.\n \\end{equation}\n\\end{itemize}\n\nThus, the discounted accumulation of the long-term reward can be denoted as\n\\begin{equation}\n {V}(S,\\pi) = \\sum_{t = 1}^{\\infty}(\\gamma)^{t - 1}{R}_{t}(S_t,A_t),\n\\end{equation}\nwhere $\\gamma\\in[0,1)$ is the discount factor.\n\nWhen the number of MECs and VR users are small, RL algorithm can efficiently obtain the optimal policy. However, when a large number of MECs and VR users exist, the state and action spaces will be scaled proportionally, which will inevitably result in massive computation latency and severely affect the performance of the RL algorithm. To address this issue, deep learning is introducted to RL, namely, deep reinforcement learning (DRL), through interaction with the environment, DRL can directly control the behavior of each agent, and solve complex decision-making problem. In DRL algorithm, two methods can be used to obtain optimal policy. One is called\nvalue-based optimization, such as DQN, which indirectly optimizes the policy by optimizing value function. While the other is policy-based optimization, such as AC, which can directly optimize the policy. In the following sections, four DRL algorithms are introduced in detail.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=3.0 in]{DQN.pdf}\n \\caption{The DQN diagram of the MEC rendering scheme.}\n \\label{basic_modules}\n\\end{figure}\n\n\\subsubsection{Centralized DQN}\n\nAs a value-based DRL algorithm, DQN combines a neural network with Q-learning and approximates the state-action value function via the deep neural network (DNN). Using DQN algorithm, a fraction of states are sampled and the neural network is applied to train a sufficiently accurate state-action value function, which is able to effectively solve the problem of high dimensionality in state space. Furthermore, the DQN algorithm uses the experience replay to train the learning process of RL. When updating the DQN algorithm, some experiences in the experience replay will be selected randomly to learn, so that the correlation among the training samples can be broke and the efficiency of the neural network can be improved. In addition, through averaging the selected samples, the distribution of training samples can be smoothed, which avoids the training divergence.\n\nAs shown in Fig. 6, the action-state value function $V_{\\text{DQN}}(S,A)$ in the DQN agent can be parameterized by using a function ${V}_{\\text{DQN}}(S,A;\\pmb{\\theta}_{\\text{DQN}})$, where $\\pmb{\\theta}_{\\text{DQN}}$ is the weight matrix of the DNN with multiple layers. Consider the conventional DNN, where the neurons between two adjacent layers are fully connected, which is so-called fully-connected layers. The input of the DNN is the variables in state $S_t$; the hidden layers are Rectifier Linear Units (ReLUs) through utilizing the function $f(x) = \\max(0,x)$; the output layer is consisted of linear units, which are all available actions in $A_t$. The exploitation is obtained by performing propagation of ${V}_{\\text{DQN}}(S,A;\\pmb{\\theta}_{\\text{DQN}})$ with respect to the observed state $S_t$. Moreover, the parameter $\\pmb{\\theta}_{\\text{DQN}}$ can be updated by using SGD as\n\\begin{equation}\n \\pmb{\\theta}_{\\text{DQN}}^{t+1} = \\pmb{\\theta}_{\\text{DQN}}^{t} - \\lambda_{\\text{DQN}}\\nabla L_{\\text{DQN}}(\\pmb{\\theta}_{\\text{DQN}}^{t}),\n\\end{equation}\nwhere $\\lambda_{\\text{DQN}}\\in(0,1]$ is the learning rate, $\\nabla L_{\\text{DQN}}(\\pmb{\\theta}_{\\text{DQN}}^{t})$ is the gradient of the loss function $L_{\\text{DQN}}(\\pmb{\\theta}_{\\text{DQN}}^{t})$ utilized to train the state-action value function. The loss function can be defined as\n\\begin{equation}\n L_{\\text{DQN}}(\\pmb{\\theta}_{\\text{DQN}}^{t}) =(\\hat{V}_{\\text{DQN}}-{V}_{\\text{DQN}}(S_i,A_i;\\pmb{\\theta}_{\\text{DQN}}^{t}))^{2},\n\\end{equation}\nwhere\n\\begin{equation}\n \\hat{V}_{\\text{DQN}} = R_{i+1}+\\gamma\\max_{A} {V}_{\\text{DQN}}(S_{i+1}, A; \\Bar{\\pmb{\\theta}}_{\\text{DQN}}^{t}).\n\\end{equation}\n$(S_i,A_i,S_{i+1},R_{i+1})$ are randomly selected previous samples for some $i\\in\\{t-M_r,...,t\\}$ with respect to a so-called minibatch. $M_r$ is the replay memory. $\\Bar{\\pmb{\\theta}}_{\\text{DQN}}^{t}$ is the so-called target Q-network which is utilized to estimate the future value of the Q-function in the update rule. Meanwhile, $\\Bar{\\pmb{\\theta}}_{\\text{DQN}}^{t}$ is periodically copied from the current value $\\pmb{\\theta}_{\\text{DQN}}^{t}$ and kept fixed for some episodes. The use of minibatch, rather than a single sample, to update the state-action value function ${V}_{\\text{DQN}}(S,A;\\pmb{\\theta}_{\\text{DQN}})$ is able to improve the convergent reliability of value function.\n\nThrough deriving the loss function in (29) and calculating the expectation of the selected previous samples in minibatch, ${V}_{\\text{DQN}}^{*}(S,A)$ can be obtained. The DQN algorithm is presented in Algorithm 1. \n\n\n\n\\begin{algorithm}[t]\n\\begin{algorithmic}[1]\n\\caption{DQN to dynamic decision-making and optimization of the MEC rendering scheme}\n\\STATE Initialize replay memory $D$ to capacity $\\hat{\\mathcal{N}}$, learning rate $\\lambda_{\\text{DQN}}\\in(0,1]$ and discount factor $\\gamma\\in[0,1)$.\n\\STATE Initialize state-action value function ${V}_{\\text{DQN}}(S,A;\\pmb{\\theta}_{\\text{DQN}})$, the parameters of primary Q-network $\\pmb{\\theta}_{\\text{DQN}}$ and target Q-network $\\Bar{\\pmb{\\theta}}_{\\text{DQN}}$.\n\\FOR{episode = 1,...,$M$}\n \\STATE Input the network state $S$ of the MEC rendering scheme.\n \\FOR{t = 1,...,T}\n \\STATE Use $\\epsilon$-greedy algorithm to select a random action $A_t$ from action space $\\mathcal{A}$.\n \\STATE Otherwise, select $A_t = \\max\\limits_{A\\in \\mathcal{A}}{V}(S_t,A;\\pmb{\\theta}_{\\text{DQN}})$.\n \\STATE The selected MECs render the predicted or uplink received FoVs and multicast\/unicast them to VR users according to the selected action $A_t$.\n \\STATE MECs observe reward $R_t$ and new state $S_{t+1}$.\n \\STATE Store transition $(S_{t},A_{t},R_{t},S_{t+1})$ in replay memory $D$.\n \\STATE Sample random minibatch of transitions $(S_j,A_j,R_j,S_{j+1})$ from replay memory $D$.\n \\IF{$j+1$ is terminal}\n \\STATE $y_j^{target} = R_j$.\n \\ELSE\n \\STATE $y_j^{target} = R_{j+1} + \\gamma\\max\\limits_{A}{V}_{\\text{DQN}}(S_{j+1},A;\\pmb{\\theta}_{\\text{DQN}})$.\n \\ENDIF\n \\STATE Perform a gradient descent step and update parameters $\\pmb{\\theta}_{\\text{DQN}}$ according to (28).\n \\STATE Update parameter $\\Bar{\\pmb{\\theta}}_{\\text{DQN}}$ of the target network every $\\Bar{K}$ steps.\n \\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Distributed DQN}\nIn the centralized DRL algorithm, it learns a single optimization policy centrally at the central controller, which requires the global observations, rewards, and actions of each MEC. When the number of MECs and VR users increase, the size of the proposed model and parameters can expand exponentially. In this case, the GPU memory in central controller not only needs to hold the model and batch of data, but also the intermediate outputs of the feedforward computation. With dense VR users, GPU memory can be easily overloaded in practice, especially for the GPUs with lower computation capability. Meanwhile, as the number of MECs and VR users scaling up, the centralized DRL can become inefficient due to the following issues. First, the training time is bound by the gradient computation time, and the frequency of parameter updating grows linearly with the number of MECs and VR users. Second, as the frequency of parameter updating grows, it could potentially slow down the optimization process and result in problems with convergence \\cite{Ddqn1}. \n\nUnlike the centralized DRL algorithm, the global objective in distributed DRL algorithm is the combination of each agent's local objective, and each agent needs to optimize its own objective. In the distributed DQN method, each agent learns independently from the other agents. When one of the agents selects an action based on the current state, the other agents can be approximated as part of the environment \\cite{DQ_OFDMA}.\n\nIn our model, the central controller stores a copy of the model parameter $\\pmb{\\theta}_{\\text{DDQN}}$. The $i$th MEC obtains the latest model parameter $\\pmb{\\theta}_{\\text{DDQN}}$ from the central controller with $\\widetilde{\\pmb{\\theta}}_{i} = \\pmb{\\theta}_{\\text{DDQN}}$. Based on the observed state $S_{t}^{i}$, it will select an action $A_{t}^{i}$ in all available actions in $\\mathcal{A}^i$. As a result, the environment will make a transition to the new state $S_{t+1}^{i}$ and a reward $R_{t}^{i}$ will be generated and fed back to the $i$th MEC. During training, the parameter $\\widetilde{\\pmb{\\theta}}_{i}$ of the $i$th MEC can be updated as \n\\begin{equation}\n \\widetilde{\\pmb{\\theta}}_{i}^{t+1} = \\widetilde{\\pmb{\\theta}}_{i}^{t} - \\lambda_{\\text{DDQN}}\\nabla L_{i}(\\widetilde{\\pmb{\\theta}}_{i}^{t}),\n\\end{equation}\nwhere $\\lambda_{\\text{DDQN}}\\in(0,1]$ is the learning rate, $L_{i}(\\widetilde{\\pmb{\\theta}}_{i}^{t})$ is the loss function of the $i$th MEC, which can be denoted as\n\\begin{equation}\n L_i(\\widetilde{\\pmb{\\theta}}_{i}^{t}) =(\\hat{V}_{\\text{DDQN}}-{V}_{\\text{DDQN}}(S_{j}^{i},A_{j}^{i};\\widetilde{\\pmb{\\theta}}_{i}^{t}))^{2},\n\\end{equation}\nwhere\n\\begin{equation}\n \\hat{V}_{\\text{DDQN}} = R_{j+1}^{i} + \\gamma\\max_{A^{i}}{V}_{\\text{DDQN}}(S_{j+1}^{i},A^{i};\\bar{\\widetilde{\\pmb{\\theta}}}_{i}^{t}).\n\\end{equation}\n$(S_{j}^{i},A_{j}^{i},S_{j+1}^{i},r_{j+1}^{i})$ are randomly selected previous samples for $j\\in\\{t-M_r,..,t\\}$ of the $i$th MEC. $\\bar{\\widetilde{\\pmb{\\theta}}}_{i}^{t}$ is the target Q-network which is used to estimate the future value of the state-action value function in the update rule. Furthermore, through deriving the loss function in (32) and computing the expectation of the selected samples, ${V}_{\\text{DDQN}}^{*}(S^{i},A^{i})$ can be obtained. In addition, the updated parameter $\\widetilde{\\pmb{\\theta}}_{i}$ of the $i$th MEC will be transmitted to the central controller and the model parameter $\\pmb{\\theta}_{\\text{DDQN}}$ can be updated as\n\\begin{equation}\n \\pmb{\\theta}_{\\text{DDQN}} = \\frac{1}{K_{\\text{DDQN}}^{\\text{MEC}}}\\sum\\limits_{i=1}^{K_{\\text{DDQN}}^{\\text{MEC}}}\\widetilde{\\pmb{\\theta}}_{i},\n\\end{equation}\nwhere $K_{\\text{DDQN}}^{\\text{MEC}}$ is the number of the MECs associated with the VR user groups.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=3.5 in]{actor_critic.pdf}\n \\caption{The Actor-Critic diagram of the MEC rendering scheme.}\n \\label{basic_modules}\n\\end{figure}\n\n\\subsubsection{Centralized AC }\nIn the DQN algorithm, the optimal policy of the MEC rendering scheme is indirectly obtained through optimizing the state-action value function. However, unlike the DQN algorithm, AC algorithm is able to directly optimize the policy of the MEC rendering scheme. \n\nThe core idea of the AC algorithm is to combine the advantages of Q-learning (value-based function) and the policy-gradient (policy-based function) algorithms. Consequently, the fast convergence of the value-based function and the directness of the policy-based function are all taken into consideration \\cite{AC1, AC2}. As shown in Fig. 7, the AC network is consisted of two independent networks, namely, an actor network and a critic network. Through learning the relationship between the environment and the rewards, the critic network is able to get the potential rewards of the current state. Then, the critic network will guide the actor network to select proper actions and update the actor network in each epoch. Therefore, the AC algorithm is usually developed as a two-time-scale algorithm, including critic updating step and actor updating step, which leads to slow learning efficiency. A parameterized policy $\\pi(A_t|S_t;\\pmb{\\theta}_{\\text{AC}})$ is learned to select actions according to the current environment state. Then, the critic network will obtain the reward feedback from the environment and use the state-value function $V_{\\text{AC}}(S_t;\\pmb{w}_{\\text{AC}})$ to evaluate the performed action. Meanwhile, a time-difference (TD) error is generated to reflect the performance of the performed action.\n\nIn particular, after performing action $A_t$ based on $S_t$ with policy $\\pi$, the critic network uses TD error to evaluate the action under the current state, which can be expressed as\n\\begin{equation}\n \\delta_t = R_t + \\gamma{V}_{\\text{AC}}(S_{t+1};\\pmb{w}_{\\text{AC}}^{t}) - V_{\\text{AC}}(S_t;\\pmb{w}_{\\text{AC}}^{t}).\n\\end{equation}\nThen, $\\pmb{w}_{\\text{AC}}^{t}$ can be updated as\n\\begin{equation}\n \\pmb{w}_{\\text{AC}}^{t+1} = \\pmb{w}_{\\text{AC}}^{t} + \\lambda_{\\text{critic}}\\delta_{t}\\nabla_{\\pmb{w}_{\\text{AC}}}V_{\\text{AC}}(S_t;\\pmb{w}_{\\text{AC}}^{t}).\n\\end{equation}\nwhere $\\lambda_{\\text{critic}}\\in(0,1]$ is the learning rate of the critic network.\n\nMeanwhile, in the actor network, the policy gradient method is usually adopted, which directly selects actions via parameterized policy. The parameter $\\pmb{\\theta}_{\\text{AC}}^{t}$ can be updated as\n\\begin{equation}\n \\pmb{\\theta}_{\\text{AC}}^{t+1} = \\pmb{\\theta}_{\\text{AC}}^{t} + \\lambda_{\\text{actor}}\\delta_{t}\\nabla_{\\pmb{\\theta}_{\\text{AC}}} \\log\\pi(A_t|S_t;\\pmb{\\theta}_{\\text{AC}}^{t}),\n\\end{equation}\nwhere $\\lambda_{\\text{actor}}\\in(0,1]$ is the learning rate of the actor network.\n\nCorrespondingly, the parameters in the actor and critic network will be iteratively updated to maximize the objective function. The detailed AC algorithm of MEC rendering scheme is proposed in Algorithm 2.\n\n\n\n\n\\begin{algorithm}[t]\n\\begin{algorithmic}[1]\n\\caption{Actor-Critic to dynamic decision-making and optimization of wireless VR system}\n\\STATE Initialize learning rate $\\lambda_{\\text{critic}}\\in(0,1]$, $\\lambda_{\\text{actor}}\\in(0,1]$ and discount factor $\\gamma\\in[0,1)$.\n\\STATE Initialize parameters $\\pmb{\\theta}_{\\text{AC}}$ and $\\pmb{w}_{\\text{AC}}$ for the actor and critic network, respectively.\n\\STATE Input the network state $S$ of the MEC rendering scheme.\n\\FOR{t = 1,...,T}\n \\STATE According to $\\pi(A|S_t;\\theta)$, select the action $A\\in\\mathcal{A}$.\n \\STATE The selected MECs render the required FoVs and multicast\/unicast them to VR users due to the selected action $A_t$.\n \\STATE MECs calculate the immediate reward $R_t$ and obtain the environment state $S_{t+1}$.\n \\STATE Store transition $(S_t,A_t,R_t,S_{t+1})$.\n \\STATE Calculate TD error $\\delta_t$ according to (35).\n \\STATE Update the parameters $\\pmb{w}_{\\text{AC}}$ of the critic network via (36).\n \\STATE Update the parameters $\\pmb{\\theta}_{\\text{AC}}$ of the actor network via (37).\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsubsection{Distributed AC}\nUnlike the centralized AC algorithm, the agent in the distributed AC algorithm performs action and obtains reward based on its own observed state. For the critic network in each agent, it shares its estimate of the value function with others through the central controller. While for the actor network in each agent, it performs individually without the need to infer the policies of others \\cite{Zhang_dis_AC}. \n\n\n\n\nIn our model, the $i$th MEC obtains the latest critic model parameter $\\pmb{w}_{\\text{DAC}}$ from the central controller, and let its own critic parameter $\\Bar{\\pmb{w}}_{t}^{i} = \\pmb{w}_{\\text{DAC}}$. At the $t$th time slot, according to the current environment state $S_{t}^{i}$ obtained by the $i$th MEC, a parameterized policy $\\pi_{i}(S_{t}^{i};\\Bar{\\pmb{\\theta}}_{t}^{i})$ is learned to select action $A_{t}^{i}$. Then, the critic network in the $i$th MEC will receive the reward feedback by the environment and evaluate the state-value function $V_{\\text{DAC}}(A_{t}^{i}|S_{t}^{i}; \\Bar{\\pmb{w}}_{t}^{i})$. Similarly, the TD error $\\delta_{t}^{i}$ of the $i$th MEC can be calculated to judge the performance of the performed action $A_{t}^{i}$, and the parameter $\\Bar{\\pmb{w}}_{t}^{i}$ of the critic network of the $i$th MEC can be updated as\n\\begin{equation}\n \\Bar{\\pmb{w}}_{t+1}^{i} = \\Bar{\\pmb{w}}_{t}^{i} + \\Bar{\\lambda}_{\\text{critic}}\\delta_{t}^{i}\\nabla_{\\Bar{\\pmb{w}}^{i}}V_{\\text{DAC}}(S_t^{i};\\Bar{\\pmb{w}}_{t}^{i}),\n\\end{equation}\nwhere $\\Bar{\\lambda}_{\\text{critic}}\\in(0,1]$ is the learning rate of the critic network, and\n\\begin{equation}\n \\delta_t^{i} = R_t^{i} + \\gamma{V}_{\\text{DAC}}(S_{t+1}^{i};\\Bar{\\pmb{w}}_{t}^{i}) - V_{\\text{DAC}}(S_t^{i};\\Bar{\\pmb{w}}_{t}^{i}).\n\\end{equation}\nFurthermore, for the parameter $\\Bar{\\pmb{\\theta}}_{t}^{i}$ of the actor network of the $i$th MEC, it can be updated via\n\\begin{equation}\n \\Bar{\\pmb{\\theta}}_{t+1}^{i} = \\Bar{\\pmb{\\theta}}_{t}^{i} +\\Bar{\\lambda}_{\\text{actor}}\\delta_{t}^{i}\\nabla_{\\Bar{\\pmb{\\theta}}^{i}}\\log\\pi(A_t^{i}|S_t^{i};\\Bar{\\pmb{\\theta}}_{t}^{i}),\n\\end{equation}\nwhere $\\Bar{\\lambda}_{\\text{actor}}\\in(0,1]$ is the learning rate of the actor network. In addition, the updated parameter $\\Bar{\\pmb{w}}_{t}^{i}$ in the critic network of the $i$th MEC will be sent to the central controller and the critic model parameter $\\pmb{w}_{\\text{DAC}}$ can be updated as\n\\begin{equation}\n \\pmb{w}_{\\text{DAC}} = \\frac{1}{K_{\\text{DAC}}^{\\text{MEC}}}\\sum\\limits_{i=1}^{K_{\\text{DAC}}^{\\text{MEC}}}\\Bar{\\pmb{w}}^{i},\n\\end{equation}\nwhere $K_{\\text{DAC}}^{\\text{MEC}}$ is the number of the MECs associated with the VR user groups. Correspondingly, the parameters in the actor and critic network will be iteratively updated to maximize the objective function.\n\n\n\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=3.0in]{prediction_reward.jpg}}\\label{fig_first_case}\n\t\\hfil\n\t\\subfloat[]{\\includegraphics[width=3.0in]{prediction_accuracy.jpg}}\\label{fig_second_case}\n\t\\caption{(a) Total reward of FoV prediction of each epoch via GRU. (b) FoV prediction accuracy of GRU for varying number of VR users.}\n\t\\label{basic_modules}\n\\end{figure}\n\n\n\\section{Simulation Results}\nIn this section, we examine the effectiveness of our proposed schemes with learning algorithms via simulation. For the learning algorithms, we set the learning algorithms use fully-connected neural network with two hidden layers and each layer has 128 ReLU units, we set the memory as 20, the minibatch size as 64, the learning rate for RNN as 0.005, the number of MECs as 8, the number of VR users as 8, $N_{\\text{FoV}} = 8$, $D(t) = 3$, $\\sigma^{2} = -110~\\text{dBm}$, $\\gamma = 0.9$, $\\alpha = \\beta = 3$, $\\lambda_{\\text{DQN}} = 0.05$, $\\lambda_{\\text{actor}} = 0.005$, $\\lambda_{\\text{critic}} = 0.05$, $T^{th} = 30 ~\\text{ms}$, $\\mathcal{R} = 1080\\text{p}$, $\\mathcal{C}_{\\mathcal{R}} = 200$, $F_{max}^{\\text{MEC}} = 5~\\text{GHz}$, $F_{min}^{\\text{MEC}} = 4~\\text{GHz}$, $F^{\\text{VR}} = 2~\\text{GHz}$, $f^{\\text{MEC}} = f^{\\text{VR}} = 1000~ \\text{Cycles\/bit}$, and $R^{\\text{fiber}} = 10~ \\text{Gb\/s}$. Consider a limited square area whose side length is 100 meters.\n\n\\subsection{FoV Prediction}\nIn the FoV prediction scheme, Brownian motion is deployed to simulate the eye movement of VR users. To obtain high accuracy in predicting FoV preference of each VR user in continuous time slots, a RNN model basd on GRU architecture is deployed. Fig. 8 (a) plots the total reward of FoV prediction of each epoch via RNN and Fig. 8 (b) plots the FoV prediction accuracy of RNN for varying number of VR users, respectively. It is observed that the prediction accuracy of the RNN remains $96\\%$ despite the increasing number of VR devices. This is because RNN utilizes a memory window with length 20 to store with the input observations, which can capture the FoV preference of VR users in the past time slots.\n\n\n\\subsection{MEC Rendering Scheme}\nFour DRL algorithms, including centralized DQN, distributed DQN, centralized AC, and distributed AC, are proposed to select proper MECs to render and transmit the required FoVs to VR users. For simplicity, we use ``w\/ Pred\", ``w\/o Pred\", ``w\/ Migra\", and ``w\/o Migra\" to represent ``with prediction\", ``without prediction\", ``with migration\", and ``without migration\" in the figures, respectively. To guarantee the fairness of each VR user, we use average QoE and VR interaction latency in the performance results.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=3.0 in]{qoe_final.jpg}\n \\caption{Total reward of the MEC rendering with prediction and migration scheme of each epoch via centralized\/distributed DQN\/AC learning algorithms.}\n \\label{basic_modules}\n\\end{figure}\n\n\n\nFig. 9 plots the total reward of the MEC rendering with prediction and migration scheme of each epoch via centralized\/distributed DQN\/AC learning algorithms. Each result is averaged over 100 training trails. It is observed that the total reward and the convergence speed of these four DRL learning algorithms follows: Centralized DQN $>$ Distributed DQN $>$ Centralized AC $>$ Distributed AC. This is due to the experience replay mechanism and randomly sampling in DQN, which use the training samples efficiently and smooth the training distribution over the previous behaviours. As the model parameters in AC algorithm are updated in two steps, including critic step and actor step, the convergence speed of the AC algorithm is lower. Apparently, the convergence speed of the centralized learning algorithms is faster than that of the distributed learning algorithms. This is because the distributed learning needs more time to learn from each agent with only local observation and reward, whereas centralized learning can learn from global observations and rewards.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=3.0 in]{uplink_dqn.jpg}\n \\caption{Average VR interaction latency of various MEC and VR rendering schemes via centralized DQN algorithm for varying uplink transmission latency.}\n \\label{basic_modules}\n\\end{figure}\n\nFig. 10 plots the average VR interaction latency of various MEC and VR rendering schemes via centralized DQN algorithm for varying uplink transmission latency. We observe that all the MEC rendering schemes outperform that of the VR rendering schemes, with around $40~\\text{ms}$ gain. This is because the processing ability of the MECs is much higher than that of the VR devices, and the data size of the FoV is smaller than that of the stitched 2D picture, which jointly decrease the rendering and downlink transmission latency. We also observe that the average VR interaction latency of the MEC rendering with prediction and migration scheme remains the same with increasing the uplink transmission latency, as the MECs do not need to wait for the uplink transmission of requested FoV from the VR devices before performing rendering.\n\nIn Fig. 10, we also compare our proposed learning-based schemes with those without learning. By comparing with the MEC\/VR rendering scheme with nearest association scheme plotted using dash lines, we see our proposed learning-based MEC\/VR rendering schemes achieve substantial gain in terms of VR interaction latency. This is due to that in the non-learning scheme, the VR user needs to transmit its requested FoV through uplink transmission and is always associated with the nearest MEC. Thus, it is possible that the MEC with low processing ability is selected to render the required FoV, which can increase the rendering latency.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=3.0 in]{different_learning_algorithms.jpg}\n \\caption{Average VR interaction latency of the MEC rendering with prediction and migration scheme and the VR rendering scheme via centralized\/distributed DQN\/AC learning algorithms for varying uplink transmission latency.}\n \\label{basic_modules}\n\\end{figure}\n\n\n\nFig. 11 plots the average VR interaction latency of the MEC rendering with prediction and migration scheme and the VR rendering scheme via centralized\/distributed DQN\/AC learning algorithms for varying uplink transmission latency. It is observed that for the MEC rendering scheme achieves much lower latency (about $40~\\text{ms}$) compared to VR rendering scheme. It is also seen that for the same rendering scheme, either the MEC or VR, the centralized DQN algorithm can achieve the minimum average VR interaction latency. This can be explained by the fact that the centralized learning algorithm learns a single policy common to the whole wireless VR system based on the global observations, while in the distributed learning algorithm, each agent only learns its own policy based on local observation.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=3.0in]{number_vr_user_qoe.jpg}}\\label{fig_first_case}\n\t\\hfil\n\t\\subfloat[]{\\includegraphics[width=3.0in]{number_vr_user.jpg}}\\label{fig_second_case}\n\t\\caption{Average QoE and VR interaction latency of MEC rendering with prediction with\/without migration schemes via centralized\/distributed DQN\/AC learning algorithms with increasing number of VR users.}\n\t\\label{basic_modules}\n\\end{figure}\n\nFig. 12 plots the average QoE and VR interaction latency of MEC rendering with prediction with\/without migration schemes via centralized\/distributed DQN\/AC learning algorithms with increasing number of VR users, respectively. With increasing the number of VR users, the average QoE of VR user first decreases then becomes nearly stable as shown in Fig. 12 (a), whereas the average VR interaction latency first increases, then becomes nearly stable as shown in Fig. 12 (b). This is because with increasing number of VR users, more MECs are activated to provide the downlink transmission for more requested different FoVs, which increase the interference among those transmission. As the number of VR users becomes too large, all the MECs become active to serve all VR users to render most of FoVs, the rendering latency and interference among VR users become stable. \n\n\n\nInterestingly, we notice that for both centralized DQN and AC algorithms, we can see the performance gain of the MEC rendering with migration scheme over that without migration scheme in Fig. 12 (a) and (b). This is because the MECs with higher computing ability will be selected to render the same required FoV for migration, which decreases the rendering latency. Importantly, all the learning-based MEC rendering with prediction schemes substantially outperform the conventional non-learning based MEC rendering with nearest association scheme.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=3.0in]{different_mec_number_qoe.jpg}}\\label{fig_first_case}\n\t\\hfil\n\t\\subfloat[]{\\includegraphics[width=3.0in]{different_mec_number.jpg}}\\label{fig_second_case}\n\t\\caption{Average QoE and VR interaction latency of MEC rendering with prediction with\/without migration schemes via centralized\/distributed DQN\/AC learning algorithms with increasing number of MECs.}\n\t\\label{basic_modules}\n\\end{figure}\n\nFig. 13 plots the average QoE and VR interaction latency of MEC rendering with prediction with\/without migration schemes via centralized\/distributed DQN\/AC learning algorithms with increasing number of MECs, respectively. With increasing the number of MECs, the average QoE of VR user first increases then becomes nearly stable as shown in Fig. 13 (a), whereas the average VR interaction latency first decreases, then becomes nearly stable as shown in Fig. 13 (b). This is because as the number of MECs increases, the VR users will have more MEC choices to be selected, thus, nearer MECs with higher execution ability can be utilized to render the required FoVs, which reduces the rendering and downlink transmission latency. However, as the number of MECs becomes too large, all MECs may be activated for rendering and downlink transmission, which leaves little gain for improvement.\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\nIn this paper, a decoupled learning strategy was developed to optimize real-time VR video streaming in wireless network, which considered FoV prediction and rendering MEC association. Specifically, based on GRU architecture, a RNN model was used to predict FoV preference of each VR user over time. Then, based on the correlation between the location and predicted FoV request of VR users, centralized and distributed DRL strategies were proposed to determine the optimal association between MEC and VR user group, and optimal rendering MEC for model migration, so as to maximize the long-term QoE of VR users. Simulation results shown that our proposed MEC rendering with prediction and migration scheme based on RNN and DRL algorithms substantially improved the long-term QoE of VR users and the VR interaction latency.\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn a seminal paper, Rieder, Lebowitz and Lieb investigated the properties of chains of $N$ \nharmonic oscillators, interacting at their ends with stochastic heat baths \\cite{Lebow}.\nThese authors proved that while energy flows from hot to cold baths, the kinetic \ntemperature profile decreases exponentially in the direction of the hotter bath, rather \nthan increasing, and in the bulk its slope vanishes as $N$ grows. Thus, in case the kinetic \ntemperature equals the thermodynamic temperature, heat flows against\nthe direction of energy, in the bulk of such 1D systems. Moreover, \nno steady state is reached, because at boundaries, heat flows in the opposite direction.\nTaken as a paradox without explanation in Ref.\\cite{Lebow}, this fact reveals that, in \nharmonic chains of oscillators, the kinetic temperature does not correspond to the \nthermodynamic temperature, or the energy flux does not represent a heat flux, or both.\nUnexpected phenomena that seem to contradict the hydrodynanmic laws of transport, {\\em e.g.}\\ \ncurrents going against the density gradient, \na phenomenon called ``uphill diffusion'', can be observed in several experimental settings;\nsee e.g.\\ Ref.\\cite{ColGiaGibVer} for further references and a nonequilibrium model \nwith phase transition exhibiting uphill diffusion. However, the thermodynamic relevance of \nsuch models is still under investigation.\n\nAs a matter of fact, temperature and heat pertain to macroscopic objects with microscopic states corresponding\nto Local Thermodynamic Equilibrium (LTE); \nthey cannot be directly identified with mechanical quantities such as kinetic energy and\nenergy flux, Ref.\\cite{LandauV} \\S 9, \\cite{Chibbaro} Chapters 3, 4 \nand 5. LTE is the essence of Thermodynamics: it can be viewed at once as the precondition \nfor the existence of the thermodynamic fields, such as temperature and heat, and as the \nnatural state of objects obeying the thermodynamic laws. The microscopic conditions under \nwhich LTE is expected to hold are extensively discussed in the literature, {\\em e.g.}\\ \n\\cite{Prigogine} Section 15.1, \\cite{Spohn} Section \n2.3, \\cite{Bellissard} Section 3.3, \\cite{Kreuzer} Chapter 1. In short, LTE requires\nthe existence of three well separated time and space scales, so that: {\\bf 1.}\\ a macroscopic \nobject can be subdivided in mesoscopic cells that look like a point to macroscopic \nobservers, while containing a large number of molecules; {\\bf 2.}\\ boundary effects\nare negligible compared to bulk effects, so that the contributions of neighboring cells \nto the mass and energy of a given cell are inappreciable within a cell; \n{\\bf 3.}\\ particle interactions allow the cells to thermalize (positions and velocities \nbecome respectively uniformly and Maxwell-Boltzmann distributed) within times that \nare mere instants on the macroscopic scale. \n\nThat macroscopic observables are not affected by microscopic fluctuations, despite the exceedingly \ndisordered and energetic microscopic motions, is essential for mesoscopic quantities to be sufficiently \nstable that thermodynamic laws apply, {\\em e.g.}\\ \\cite{LandauV}, \\S 1 and \\S 2. This is the case for \na quantity that is spatially weakly inhomogeneous, when the number $N$ of particles in a cell \nis large, and the molecular interactions randomize positions and momenta so that, for \ninstance, the fluctuations of a quantity $\\phi$ of size $O(N)$ are order $O(\\sqrt{N})$. \nThe bulk of the cell then dominates in- and out-fluxes, and variations of $\\phi$ \nare sufficiently slow on the mesoscopic scale. Quantitatively, the space and time scales \nfor which this description holds depend on the properties of the microscopic components \nof the systems of interest, \\cite{Prigogine,Kreuzer,Spohn,Bellissard,Pigo1,Pigo2}. \n\nUnder the LTE condition, matter can be considered a continuum, obeying hydrodynamic laws,\n{\\em i.e.}\\ balance equations for locally conserved quantities, such \nas mass, momentum and energy \\cite{DeGroot,CR98,RC02}. For small to moderate driving, \nthey take a linear form, in which the local material properties are expressed\nby the linear transport coefficients. Locality implies that such \ncoefficients do not depend on the conditions of the system far away from \nthe considered region. The thermal conductivity of an iron bar at a given temperature at a given \npoint in space does not depend on the conditions of the bar far from that region; cutting the bar \nin two, or joining it to another bar, without changing the local state, leaves unchanged its local \nproperties.\n\nFluctuations remain of course present in systems made of particles; they are larger for larger systems, \nthey may be observed \\cite{AURIGA1,AURIGA2}, and they play a major role \nin many circumstances, see {\\em e.g.}\\ Refs.\\cite{Mish,HickMish}. This\nmotivates a considerable fraction of research in statistical physics, {\\em e.g.}\\ \\cite{StochTherm,CasatiRev},\nconcerning scales much smaller than the macroscopic ones, or occurring in \nlow dimensional (1D and 2D) systems \\cite{Li1,Dhar,Li2,LepriEd,CasatiJ}. In these phenomena, the linear transport \ncoefficients do not always \nseem to exist \\cite{LepriEd}, the robustness and universality of the thermodynamic laws appear \nto be violated, and the behaviours appear to be strongly affected by boundary conditions and by all parameters \nthat characterize a given object \\cite{ExpAnomFourier,JeppsR,GibRon,Zhao1,Zhao2,onorato,Saito}. \nIt is also well known that chains of oscillators behave more like some kind of \n(non-standard) fluids than like solids, because of the loss of crystalline structure, caused by \ncumulative position fluctuations \\cite{Peierls}. Consequently, a fluid-like (possibly fluctuating) \ndescription has been adopted in a number of papers, cf.\\ Refs.\\cite{NaRam02,MaNa06}.\n\nIn driven systems, the situation is problematic also because equipartition may be violated \\cite{Breaking,Sengers,PSV17},\nthe statistic describing the state of the system is model dependent, and the ergodic properties of the particles dynamics are only partially understood \\cite{GGnoneq,EWSR2016}. Hence, \nthere is no universally accepted microscopic notion of nonequilibrium temperature \\cite{MR,Jou1,Jou2,JouBook,Criado,He,PSV17}. Further, a microscopic definition of heat flux requires a clear distinction between energy transport due to macroscopic motions (convection), and transport without macroscopic motions (conduction), cf.\\ Chapter 4 of Ref.\\cite{Zema}, and Section III.2 and Chapter XI of Ref.\\cite{DeGroot}. In 1D systems, this may not be always possible \\cite{inpreparat}.\n\nOne possible interpretation of these facts is that LTE is violated in some situations, hence that \nthermodynamic concepts may be inappropriate \\cite{JeppsR,GibRon}. Another interpretation is that \nthermodynamic notions should be modified to treat small and strongly nonequilibrium systems, see \n{\\em e.g.}\\ \\cite{MR,Jou1,Jou2,He}.\nIt is therefore interesting to investigate the validity and universality of the mechanical counterparts \nof thermodynamic quantities, in situations in which LTE is not expected to hold, and ``anomalous'' \nphenomena have been reported. \n\nWe address such questions considering chains of $N$ Lennard-Jones oscillators interacting with \ndeterministic baths at their ends, and without on-site potentials. Our central findings are that:\n\\begin{itemize}\n\\item\nthermostats at different temperatures induce $O(N)$ distortions of the \nequilibrium lattice, resulting in highly in-homogeneous chains;\n\\item\nthermostats induce collective order $O(N)$ fluctuations, \n{\\em i.e.}\\ ``macroscopic'' motions. Negligible incoherent $O(\\sqrt{N})$ vibrations typical of \n3D equilibrium systems are thus replaced by kind of convective motions, even in \nchains bounded by still walls.\n\\end{itemize}\n\nThese observations combined with the results of Ref.\\cite{Lebow} and further literature, \n{\\em e.g.}\\ Refs.\\cite{MR,Jou1,Jou2,JouBook,He,Breaking}, suggest \nthat microscopic definitions appropriate for 3-dimensional equilibrium thermodynamic quantities, need \nextra scrutiny in 1D. In particular, we illustrate how $O(N)$ fluctuations and lattice distortions \naffect the collective behaviour of 1D systems, considering the notion of heat flux, $J$ \nsay, given by Eq.(23) of Ref.\\cite{LLP-PhyRep}. This way, we confirm from a different standpoint the conclusions\nreached in previous studies on the inapplicability of standard hydrodynamics \\cite{Hurtado,Politi}. We find \nthat:\n\\begin{itemize}\n\\item\n$J$ is not spatially uniform in steady states. Variations of $J$ decrease if the baths temperature difference is reduced \nat constant $N$, but they do not if the mean temperature gradient is reduced increasing $N$ at constant baths \ntemperatures. \n\\item\nDividing $J$ by the local \nmass density partially balances the lattice inhomogeneity and yields an approximately uniform quantity.\n\\end{itemize}\nThese observations should be combined with those of Refs.\\cite{Politi,inpreparat}, according to which\ncollective and molecular motions are correlated, making hard to disentangle convection from conduction,\nbecause single particles push their neighbors, producing kind of convective cascades.\nThat difficulties do not ease when $N$ grows, indicates \nthat LTE, hence thermodynamic quantities, cannot be established in our 1D systems. We relate this fact \nwith the $O(N)$ growth of fluctuations and lattice distortions.\n\n\n\n\\section{Chains of Lennard-Jones oscillators} \\label{sectII}\nConsider a 1D chain of $N$ identical moving particles of equal mass $m$, \nand positions $x_{i}$, $i=1,...,N$. Add two particles with fixed positions, $x_{0}=0$ and $x_{N+1}=(N+1)a$, where \n$a$ is the lattice spacing. Let nearest neighbors interact via the Lennard-Jones potential (LJ):\n\\be\nV_{1}(r)=\\epsilon \\left [ \\left (\\frac{a}{r}\\right)^{12} - 2 \\left (\\frac{a}{r}\\right )^{6} \\right ],\n\\label{LJ}\n\\ee \nwhere $r$ is the distance between nearest neighbors: \\cla{$r=|x_{i}-x_{i-1}|$} and $\\epsilon>0$ is \nthe depth of the potential well. Thus, $x_{i}=ai$, with \n$\\, i=0,\\ldots, N+1$, is a configuration of stable mechanical equilibrium for the system. We also \nconsider interactions involving first and second nearest neighbors, with second \npotential given by \\cite{1st2ndpaper}:\n\\be\n V_{2}(s)=\\epsilon \\left [ \\left (\\frac{2a}{s}\\right)^{12} - 2 \\left (\\frac{2a}{s}\\right )^{6} \\right ],\n\\label{1st2nd}\n\\ee \nwhere \\cla{$s=|x_{i}-x_{i-2}|$}. Further, we add two particles with fixed positions $x_{-1}=-a$ and $x_{N+2}=(N+2)a$.\nWith potential $V = V_{1}+V_{2}$, the system has the usual stable mechanical equilibrium configuration $x_{i}=ai$, \n$\\, i=-1,\\ldots, N+2$.\nThe first and last moving particles are in contact with two Nos\\'e-Hoover thermostats, at kinetic temperatures \n$T_{L}$ (on the left) and $T_{R}$ (on the right) and with relaxation times $\\theta_L$ and $\\theta_R$. \nIntroducing the forces\n\\be\n\\label{F1F2}\nF_{1}(r)= \\frac{\\partial V_{1}}{\\partial r }(r) ,\\quad F_{2}(s)= \\frac{\\partial V_{2}}{\\partial s }(s) \\, ,\n\\ee \nthe equations of motion are given by:\n\\bea\n&&m \\ddot{x}_{1} = F_{1}(x_{1}) - F_{1}(x_{2}-x_{1})-\\xi_{1} \\dot{x}_{1}, \\\\\n&&m \\ddot{x}_{i} = F_{1}(x_{i}-x_{i-1}) - F_{1}(x_{i+1}-x_{i}), \\, i=2,...,N-1, \\;\\;\\;\\;\\; \\\\\n&&m \\ddot{x}_{N} = F_{1}(x_{N}-x_{N-1}) - F_{1}(x_{N+1}-x_{N})-\\xi_{N} \\dot{x}_{N},\n\\label{eqsmot}\n\\eea\nwith\n\\be\n\\dot{\\xi}_{1} = \\frac{1}{\\theta_{L}^{2}}\\left ( \\frac{m \\dot{x}_{1}^{2} }{T_{L}}-1\\right ),\\quad \\dot{\\xi}_{N} = \\frac{1}{\\theta_{R}^{2}}\\left ( \\frac{m \\dot{x}_{N}^{2} }{T_{R}}-1\\right ),\n\\ee\nin the case of nearest neighbors interaction. For first and second neighbors interactions, we have:\n\\begin{widetext}\n\\bea\n&& \\hskip -18pt m \\ddot{x}_{1} = F_{1}(x_{1}) - F_{1}(x_{2}-x_{1}) + F_{2}(x_{1}+a) - F_{2}(x_{3}-x_{1}) -\\xi_{1}\\dot{x}_{1}, \\nonumber \\\\\n&& \\hskip -18pt m \\ddot{x}_{2} = F_{1}(x_{2}-x_{1}) - F_{1}(x_{3}-x_{2}) + F_{2}(x_{2}) - F_{2}(x_{4}-x_{2})-\\xi_{2}\\dot{x}_{2},\\nonumber \\\\\n&& \\hskip -18pt m \\ddot{x}_{i} = F_{1}(x_{i}-x_{i-1}) - F_{1}(x_{i+1}-x_{i}) + F_{2}(x_{i}-x_{i-2}) - F_{2}(x_{i+2}-x_{i}), \\qquad i=3,\\ldots, N-2, \\\\\n&& \\hskip -18pt m \\ddot{x}_{N-1} = F_{1}(x_{N-1}-x_{N-2}) - F_{1}(x_{N}-x_{N-1}) + F_{2}(x_{N-1}-x_{N-3}) - F_{2}(x_{N+1}-x_{N-1})-\\xi_{N-1}\\dot{x}_{N-1},\\nonumber \\\\\n&& \\hskip -18pt m \\ddot{x}_{N} = F_{1}(x_{N}-x_{N-1}) - F_{1}(x_{N+1}-x_{N}) + F_{2}(x_{N}-x_{N-2}) - F_{2}(x_{N+2}-x_{N})-\\xi_{N}\\dot{x}_{N},\\nonumber \n\\label{eqsmot2}\n\\eea\n\\end{widetext}\nwith\n\\be\n\\begin{split}\n\\dot{\\xi}_{l} &= \\frac{1}{\\theta_{L}^{2}}\\left ( \\frac{m \\dot{x}_{l}^{2} }{T_{L}}-1\\right ),\\, \nl=1,2,\\\\\n \\dot{\\xi}_{l} &= \\frac{1}{\\theta_{R}^{2}}\\left ( \\frac{m \\dot{x}_{l}^{2} }{T_{R}}-1\\right ),\\, l=N-1,N.\n\\end{split}\n\\ee\nThe hard-core nature of the LJ potentials preserves the order of particles:\n$0< x_{1}400$. The label \nof the particle corresponding to the maximum lattice distortion is fitted by \n$i_{\\mbox{\\small mld}} = 0.6063 N - 6.804$ with $R^2=0.9997$.\n}\n\\end{figure}\nIn the lower panel of Fig.\\ref{page67June} and in Fig.\\ref{page89June}, square root fits and linear \nfits are compared for $N$ ranging from 64 to 6000. \nThe square root fits are appropriate for small $N$, while at large $N$ the linear \nfit takes over. The\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Fig2a.eps}\n\\includegraphics[width=0.5\\textwidth]{Fig2b.eps}\n\\caption{\\label{page67June} Upper panel: standard deviations of the particles vibrations \nabout their average position, in lattice vectors units, for the case of Fig.\\ref{pag2June}.\nLower panel: dependence on $N$ (ranging from 64 to 6000) of the maximum standard deviation together with a linear\nfit for $N>400$ (continuous blue line) and one square root fit for lattices with $N<2100$ (dashed red line). \nGrowing linearly with $N$, collective vibrations look like convective motions. \nThe label of the particle corresponding to the maximum fluctuation amplitude is fitted by \n$i_{\\mbox{\\small mfa}} = 0.7398 N - 6.75$ with $R^2=0.9993$.\n}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Fig3a.eps}\n\\includegraphics[width=0.5\\textwidth]{Fig3b.eps}\n\\caption{\\label{page89June} Upper panel: dependence on $N$ of the standard deviations of the\nvibrations of particles at 1\/3 of the chain. Lower panel: dependence on $N$ of the standard \ndeviations of the vibrations at 2\/3 of the chain. In both cases, a square root and a linear \nfit are drawn. The square root fit holds at small $N$. At large $N$ the linear fit takes over. \nIn both panels $N$ ranges from 64 to 6000.\nParticles motions look more like some kind of convection rather than like microscopic lattice\nvibrations.}\n\\end{figure}\nsize of these vibrations appears even more striking observing that displacing by a large amount \none of them, a whole collection of particles must be correspondingly displaced. \nIndeed, the \nrepulsive part of the LJ potential does not allow particles' order to be modified, as noted also in Ref.\\cite{Politi}. \nTherefore, the motion of particles about their average positions is not an irregular motion about fixed \npositions. In accord with the observations on \npersistent correlations, this motion looks like a kind of convection, \nalthough LTE and standard hydrodynamics do not hold \\cite{GibRon,NaRam02,MaNa06,Hurtado,Politi}.\nIt follows that, in these cases, energy transport cannot be directly related to ``heat'' flows. \n\\begin{figure}\n\\hskip -20pt\n\\centering\n\\includegraphics[width=0.5\\textwidth]{Fig4.eps} \n\\caption{\\label{MeanDispNoGrad} {Equilibrium simulations. Plot of the displacement\nof the mean position of particle $i$ from its mechanical equilibrium position, $\\langle x_i \\rangle - i a$, \nfor various values of $N$, for first and second neighbors interactions when $T_L=T_R=5$.\nThe deviations from the mechanical equilibrium are negligible.}}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Fig5a.eps}\n\\includegraphics[width=0.5\\textwidth]{Fig5b.eps}\n\\caption{\\label{equilibrio} Equilibrium simulations $(T_L=T_R=5)$ for $N$ ranging from 512 to \\cla{5000}. Upper panel: standard deviations of the particles vibrations \nabout their average position $(x_i- \\langle x_i \\rangle)$, in lattice vectors units.\nLower panel: dependence on $N$ of the maximum standard deviation, together with linear and square root fits. \nThis dependence on $N$ should not be confused with the $O(\\sqrt{i})$ dependence on $i$ of Ref.\\cite{Peierls}.}\n\\end{figure}\n\nThe situation is different for $T_L=T_R$. Figure \\ref{MeanDispNoGrad} shows \nthat the lattice deformations are much smaller than the lattice spacing $a$,\nand can be neglected. The computed values of \n$(\\langle x_i \\rangle -ia)$ practically vanish and do not depend on $N$. \nThe standard deviation of \nthe vibrations about the mean position is represented in upper panel of \nFig.\\ref{equilibrio} and it appears to be closer to $O(\\sqrt{N})$ than to $O(N)$ {as can be seen in lower panel of Fig.\\ref{equilibrio}.\nIn this case, in which there is no net energy transport, the system also behaves more like a fluid than\nlike a solid in sense closer to that of \\cite{Peierls}, although our results refers to a different situation.\n\n\n\\section{Heat flux}\n\n\\label{sec:heatflux}\nIn order to understand the effect of $O(N)$ fluctuations and lattice distortions on the\nbehaviour of usual microscopic quantities, let us consider the ``heat flux'' given\nby Eq.(23) of Ref.\\cite{LLP-PhyRep}. For the case of first and second nearest \nneighbors interactions, that expression must be modified as follows:\n\\be\n\\begin{split}\nJ_i=&\\frac 1 2 (x_{i+1}-x_i)F_1(x_{i+1}-x_i)(\\dot{x}_{i+1} + \\dot{x}_i) \\\\\n& + (x_{i+2}-x_i)F_2(x_{i+2}-x_i)(\\dot{x}_{i+2} + \\dot{x}_i)+ \\dot{x}_i h_i\\, ,\n\\label{exactJ}\n\\end{split}\n\\ee\nwhere $F_1$ and $F_2$ are defined by Eq.(\\ref{F1F2}) and $h_i$ is the energy of the $i$-th particle.\n\nThe quantity $J_i$ is only apparently ``local'' because it quantifies a flow through the position of particle $i$, \nand not through a fixed position in space. Moreover, it implicitly\nrequires small position fluctuations and small lattice deformations, because Eq.(\\ref{exactJ})\nis obtained through Fourier analysis for spatially homogeneous systems, in the limit of small \nwave vectors, \\cite{LLP-PhyRep,Dhar}. For instance, denoting by $k$ the wave-vector, Eq.(23) of \nRef.\\cite{LLP-PhyRep} follows from Eq.(21) only if $k(x_{n+1}-x_n)$ is small. \nOn the contrary, in our cases, this quantity strongly varies in space and time, and average lattice\ndistortions are of order $O(N)$, cf.\\ Section \\ref{fluid}. Therefore, one \nexpects $J_i$ to fail, and it is interesting to investigate how that \nis realized, varying the relevant model parameters.\n\nFor chains with nearest neighbors Lennard-Jones interactions ($F_2\\equiv 0$ \nin Eq.(\\ref{exactJ})),\nwe find that while the steady state heat flow should not depend on position, the time \naverage of $J_i$ substantially changes with $i$, cf.\\ Fig.\\ref{flussi}. \n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Fig6a.eps} \n\\includegraphics[width=0.5\\textwidth]{Fig6b.eps} \n\\caption{\\label{flussi} \nChains with nearest neighbors Lennard-Jones interactions.\nUpper Panel: flux $\\langle J_i \\rangle$ computed according to (\\ref{exactJ}), for $N=64$, $T_L=1$, $T_R=4$. \nLower Panel: $\\langle J_i \\rangle$ for $N=64$, $T_L=1$, $T_R=64$.}\n\\end{figure}\nTo quantify this phenomenon, we introduce the relative variation of $\\langle {J}_i \\rangle$, \n$$\n\\delta= \\left|{\\max_i \\langle {J}_{i}\\rangle-\\min_i \\langle {J}_{i}\\rangle \\over \\bar{J}} \\right| ~, ~~\\mbox{where } ~\n\\bar{J}={1 \\over N} \\sum_{i}\\langle {J}_{i}\\rangle ~ ,\n$$ \nIn Tables \\ref{tab2bis} and \\ref{tab4}, for average temperature gradients similar to those commonly found in the \nliterature \\cite{LLP,Kabu}, we observe that $\\delta$ tends to grow with the temperature gradient, \nat fixed $N$. In general, however, reducing the average gradient by increasing the system size, does not lead to smaller \n$\\delta$ \\cite{observ}.\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n $T_{R}$ & $\\delta_1$ & $\\delta_2$ \\\\\n\\hline\n 1.1 & 0.0240 & 0.0199 \\\\\n\\hline\n 1.5 & 0.0091 & 0.0077 \\\\\n\\hline\n 2 & 0.0142 & 0.0145 \\\\\n\\hline\n 4 & 0.0480 & 0.0481 \\\\\n\\hline\n 8 & 0.0831 & 0.0829 \\\\\n\\hline\n16 & 0.1060 & 0.1062 \\\\\n\\hline\n32 & 0.1199 & 0.1201 \\\\\n\\hline\n64 & 0.1229 & 0.1232\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab2bis} Relative variation $\\delta$ \nof the flux $J_i$ \\cla{for} $N=64$ particles with first and second nearest neighbors\ninteractions. $T_L=1$ while $T_R$ takes eight different values. $\\delta_1$ is \ncomputed averaging over $2\\cdot 10^9$ time steps, $\\delta_2$ over $4 \\cdot 10^9$ time steps. }\n\\end{table}\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n $T_{R}$ & $N=64$ & $N=128$ & $N=256$ \\\\\n\\hline\n 1.1 & 0.0240 & 0.0117 & 0.0110656 \\\\\n\\hline\n 1.5 & 0.0091 & 0.0297 & 0.0317283 \\\\\n\\hline\n 2 & 0.0142 & 0.0534 & 0.0555437 \\\\\n\\hline\n 4 & 0.0480 & 0.0817 & 0.104345 \\\\\n\\hline\n 8 & 0.0831 & 0.0659 & 0.0907829 \\\\\n\\hline\n16 & 0.1060 & 0.0683 & 0.0485491 \\\\\n\\hline\n32 & 0.1199 & 0.1560 & 0.0643797 \\\\\n\\hline\n64 & 0.1229 & 0.2306 & 0.195046 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab4} Relative variation $\\delta$ of the average fluxes $\\langle J_i\\rangle $ defined by Eq.(\\ref{exactJ}).\nChains with $N=64$, $N=128$ and with $N=256$ particles, with nearest neighbors interactions are considered. \nAverages are computed over $2\\cdot 10^9$ time steps. $T_L=1$, while $T_R$ takes eight different values.}\n\\end{table}\nWe conclude that under our conditions the quantity $J_i$ represents neither a heat nor an energy flow,\nand that this is not a consequence of the size of temperature gradients, but of the size of fluctuations.\nThese increase with growing $N$, thus preventing LTE and standard hydrodynamics in the large $N$ limit \\cite{Hurtado,GibRon,Politi}.\nIn the next section we propose a modification of $J_i$ that, taking into account \nthe deformations of the lattice, is more stable than $J_i$ along the chain.\n\n\n\\section{Concluding remarks}\nIn this work we have presented numerical results concerning several kinds of 1D\nsystems of nonlinear oscillators, in contact with two Nos\\'e-Hoover thermostats.\nScrutinizing the behaviour of mechanical quantities that are commonly considered \nin the specialized literature, we have investigated the fluctuations and lattice \ndistortions, which are expected to prevent the establishment of ``thermodynamic'' \nregimes \\cite{GibRon,JeppsR,JouBook,Politi}. \n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Fig7.eps}\n\\caption{\\label{normJ}\nNormalized energy flux $J^n$ (o) and the flux $J$ (+) defined by \\eqref{exactJ} for chains of different lengths \n(N=128, 512, 1000, 2000) and with $T_L=1$ and $T_R=10$. \nAlthough $J^n$ \nis not exactly constant, at large $N$ it enjoys small fluctuations about a given average value.}\n\\end{figure}\n\nThermodynamic properties emerge indeed from\nthe collective behavior of very large assemblies of interacting particles, if correlations \ndecay rapidly compared to observation time scales, and if boundary effects are negligible. \nWhile this is often the case of 3D mesoscopic cells containing large numbers of properly \ninteracting particles, it is not obvious in 1D systems.\n\nIn particular, we have observed that temperature differences at the boundaries produce \n$O(N)$ fluctuations and deformations of the lattice, that result in strongly inhomogeneous systems. \nThis should be taken into account when defining the heat conductivity. \nFurthermore, these $O(N)$ effects imply that larger $N$ is not going to make our systems closer \nto thermodynamic systems, when $N$ is increased, consequently standard hydrodynamics does not apply \\cite{Hurtado,GibRon,Politi}.\n\nIn the light of the above observations, the definition of energy transport (\\ref{exactJ}) may be \nmodified in order to take into account the lattice deformations. One possibility that may considered\nis to normalize the energy flow by the average distance between particles, introducing\n\\be\nJ^n_i = {J_i \\over \\langle x_{i+1} - x_i \\rangle } ~, \\qquad i = 2, ... , N-2\\ .\n\\ee\nThe result, shown in Fig.\\ref{normJ},\nindicates that this avenue deserves further investigation, and will be the subject of future works.\n\n\\centerline{\\bf ACKNOWLEDGMENTS } \nThe authors are grateful to Carlos Mejia-Monasterio for extensive discussions\nand enlightening remarks. The authors are grateful to Antonio Politi for very useful suggestions.\nAntonio Politi, in particular has suggested the normalization of the energy flux.\nThis work is partially supported by Gruppo Nazionale per la\nFisica Matematica (GNFM-INdAM). C.G. and C.V. acknowledge financial supports\nfrom ``Fondo di Ateneo per la Ricerca 2016\" and ``Fondo di Ateneo per la Ricerca 2017\"-\nUniversit\\`a\\ di Modena e Reggio Emilia. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\Ac{cs} is a signal processing technique aiming at recovering a high-dimensional sparse vector from a (noisy) system of linear equations \\cite{donoho2006compressed,CandesTao}. Joint sparsity refers to multiple vectors having the same support set\\footnote{The support set of a vector consists of the indices of the vector's nonzero entries.}, whose cardinality is typically much lower than the signal dimension. There are two prominent \\ac{cs} scenarios \\cite{cotter2005sparse,duarte2005joint} in the context of joint sparsity: (i) the \\ac{mmv} problem, where the measurement matrices are identical, and (ii) the \\ac{dcs} problem, where the measurement matrices are independent. Joint sparsity arises in a number of real-world scenarios, e.g., when multiple sensors or antennas observe the same signal corrupted by different channels and noise (e.g., \\cite{cotter2005sparse,duarte2005joint}). \nA prime example is radio frequency identification where the observed vectors are the received signals at different antennas (of the same receiver) \\cite{mayer2016exploiting}. Additionally, typical applications are magnetic resonance imaging \\cite{liang2009parallel}, distributed networks \\cite{wimalajeewa2014omp}, wireless communications \\cite{mayer2016exploiting}, and direction of arrival estimation \\cite{tzagkarakis2010multiple}. \n\nIn this work, we investigate an \\ac{amp} solution for joint sparse recovery when there is possible correlation between the signals (and the noise). We then evaluate this algorithm in the context of single-pixel color imaging \\cite{duarte2008sp}. In particular, we show the potential of joint recovery that exploits the correlation between the red, green, blue (RGB) color intensity channels. \n\n\\subsection{Related Work}\nSeveral methods for jointly sparse recovery have been proposed in the literature \n\\cite{cotter2005sparse,tropp2006algorithms1,tropp2006algorithms2,wipf2007empirical,davies2012rank,schniter2010turbo,wimalajeewa2014omp,ziniel2013efficient,mayer2015bayesian,zhao2015joint,lu2016independent,kim2011belief}. \\Ac{amp} was introduced in \\cite{donoho2009message,DonohoITWmessagepassingI,DonohoITWmessagepassingII} as a large system relaxation of loopy belief propagation to solve a random linear system with sparsity constraint. Scalar \\ac{bamp}, its Bayesian version \\cite{maleki2010approximate,montanari2012graphical}, uses the signal prior explicitly and is an efficient approximate \\ac{mmse} estimator. The turbo \\ac{bamp} methods in \\cite{schniter2010turbo,ziniel2013efficient,mayer2015bayesian}, and their generalization in \\cite{rangan2017} for clustered sparse signals, improve the recovery performance by exchanging extrinsic information about the current support estimate in each message passing iteration.\nIn \\cite{zhao2015joint,lu2016independent,zhu2016performance}, joint sparsity is directly enforced by an appropriate vector estimator (denoiser) function for the \\ac{bg} prior. \n\nThe \\ac{se} formalism developed in \\cite{DonohoITWmessagepassingI, DonohoITWmessagepassingII, bayati2011dynamics} analytically predicts the recovery performance of (B)\\acs{amp} algorithms. \\ac{se} was employed to analyze \\ac{bamp} for joint sparsity with a vector estimator and to point out the difference between the \\ac{dcs} and \\ac{mmv} scenarios in \\cite{lu2016independent}. Recent works rigorously prove the \\ac{se} for non-separable non-linearities \\cite{berthier2017} and a class of sliding-window denoisers \\cite{ma2017} with Gaussian i.i.d. measurement matrices. Furthermore, the \\ac{se} of the Vector AMP has been derived for a large class of right orthogonally invariant random sensing matrices \\cite{rangan2017vamp}. (We highlight that the acronym Vector AMP should not to be confused with the vector-prior version of BAMP, considered in this paper for the MMV\/DCS problems.) \n\nIn \\cite{zhu2016performance}, the replica method (a statistical physics tool for large disordered systems) is used to calculate the \\ac{mmse} of the \\ac{cs} measurement (note that \\cite{zhu2016performance} refers to \\ac{mmv} and \\ac{dcs} as MMV-2 and MMV-1, respectively).\nThe replica trick non-rigorously simplifies the high-dimensional integral for the \\ac{mmse} of the Bayesian estimator of the \\ac{cs} channel, thereby leading to the free energy as a function of the \\ac{mse}. The local maxima in the free energy function correspond to stable fixed points of \\ac{bp} and \\ac{bamp} and thus predict the expected \\ac{mse} of \\ac{bamp}. The replica analysis in \\cite{zhu2016performance} is performed for the \\ac{bg} signal prior with uncorrelated isotropic unitary signal and uncorrelated isotropic Gaussian noise distribution, i.e., with a single noise parameter.\n\n\\subsection{Contributions}\nWe consider the vector-prior \\ac{bamp} algorithm for the \\ac{dcs} and \\ac{mmv} problems, which uses an appropriate vector \\ac{mmse} estimator function and Onsager correction term to exploit joint sparsity structure, the signal distribution, and the noise covariance. We provide an analytical performance prediction for the \\ac{bamp} algorithm with a \\ac{bg} signal prior with arbitrary signal and noise correlation by (i) incorporating a linear joint decorrelation of the measurements, (ii) showing the equivariance of \\ac{Vbamp} w.r.t.\\ invertible linear transformations, (iii) extending the replica analysis from \\cite{zhu2016performance} to arbitrary diagonal noise covariance matrices.\n\nIn particular, the joint decorrelation yields a simpler equivalent measurement model with diagonal signal and noise covariance matrix (under mild conditions, one of the covariance matrices can be made the identity matrix). The simplified model naturally provides the measurement \\acp{snr} of each signal vector and substantially reduces the complexity of the \\ac{Vbamp} iterations. We further show that the \\ac{Vbamp} algorithm is equivariant to invertible linear transformations, thus, it preserves its properties across iterations in the transformed domain and delivers a result equivalent to that obtained with the original measurements {and covariance parameters}. For the widely used \\ac{bg} prior, we prove that the \\ac{Vbamp} iterations (and the corresponding \\ac{se}) preserve the diagonal structure of the (effective) noise covariance, thus implying that a $B$-dimensional state (instead of $B(B+1)\/2$ dimensions) is sufficient and that every \\ac{mmv} problem can be transformed into an equivalent \\ac{dcs} problem. Finally, we extend the replica analysis in \\cite{zhu2016performance} to the case of anisotropic noise (i.e., $B$ noise parameters instead of just $1$).\nThe replica analysis yields the $B$ measurement-wise \\acp{mse} of the \\ac{Vbamp} estimate in its {fixed} points.\n{We use both real-world and synthetic images to compare MMV-BAMP to state-of-the-art scalar recovery algorithms and to joint sparsity-aware algorithms in the context of single-pixel color imaging.}\n\n\\subsection{Outline}\nThe remainder of this paper is organized as follows. In Section \\ref{sec:vbamp}, we discuss the \\ac{Vbamp} algorithm, the estimator function for the multivariate \\ac{bg} signal prior, and the multivariate state evolution of \\ac{Vbamp}. In Section \\ref{sec:sec3}, the joint decorrelation of the signal and the noise vectors is investigated in the context of \\ac{Vbamp} and state evolution; the multivariate \\ac{bg} signal prior is studied as special case. In Section \\ref{sec:replicanalysis}, we present the multivariate free energy formula for arbitrary diagonal noise covariance matrices (the details of the replica analysis are relegated to the appendix). Section \\ref{sec:anisodyn} provides a qualitative discussion and open questions regarding the effects of signal correlation and the increasing number of jointly sparse vectors on the dynamics of \\ac{Vbamp}. Section \\ref{sec:sin_pix} evaluates the MMV-BAMP algorithm on a simplified single pixel imaging problem, highlighting the benefits of exploiting signal correlation across channels. We close with conclusions in Section \\ref{sec:concl}.\n\n\\subsection{Notation}\\label{subs:not}\nUppercase (lowercase) boldface letters denote matrices (vectors), and serif letters denote random quantities. For a matrix $\\mbf{A}$ (vector $\\mbf{a}$), $\\mbf{A}_{i}$ ($a_i$) denotes its $i$th row ($i$th entry) and $\\mbf{a}_{i}$ its $i$th column.\nThe all zero matrix and the identity matrix of dimension $M\\times N$ are denoted by $\\mbf{0}_{M\\times N}$ and $\\mathbf{I}_{M\\times N}$, respectively (we omit the subscript if the dimensions are clear from the context). The Dirac delta (generalized) function is $\\delta(\\mathbf{x})$. The normal distribution with mean $\\boldsymbol\\mu$ and covariance matrix $\\boldsymbol{\\Sigma}$ is denoted by $\\mcal{N}(\\boldsymbol\\mu,\\boldsymbol{\\Sigma})$ and $\\mcal{N}(\\mbf{x};\\boldsymbol\\mu,\\boldsymbol{\\Sigma})$ denotes the value of this normal \\ac{pdf} at $\\mbf{x}$. The outer product of a column vector $\\mathbf{x}$ with itself is denoted by $\\boldsymbol{\\langle} \\mathbf{x} \\boldsymbol{\\rangle} = \\mathbf{x} \\mathbf{x}^T$. For a vector $\\mathbf{x}=(x_1,\\ldots,x_B)^T$, $\\diag(\\mathbf{x})$ and $\\diag(x_1,\\ldots,x_B)$ denote the diagonal matrix whose $i$th diagonal element equals $x_i$. For a matrix $\\mathbf{X}$, $D(\\mathbf{X})$ is the diagonal matrix whose diagonal is identical to that of $\\mathbf{X}$, i.e., $D(\\cdot)$ is the orthogonal projection that zeros the off-diagonal elements. The Kronecker product of two matrices is denoted by $\\otimes$.\n\n\n\\section{BAMP with Vector Denoiser}\n\\label{sec:vbamp}\n\n\\subsection{Measurement Model}\n\nWe consider the measurement model\n\\begin{equation}\n \\mathbf{y}(b) = \\mbf{A}(b)\\mathbf{x}(b) + \\mathbf{w}(b)\\,, \\label{eq:matrixmeasurement}\n\\end{equation}\nwith $\\mathbf{y}(b)\\in\\mbb{R}^M$, $\\mathbf{x}(b)\\in\\mbb{R}^N$, $\\mathbf{w}(b)\\in\\mbb{R}^M$, and $\\mbf{A}(b)\\in\\mbb{R}^{M\\times N}$, for $b=1,\\ldots,B$.\nWe denote the measurement rate by $R = {M}\/{N}$.\nWe assume that the measurement matrices $\\mbf{A}(b)$ are realizations of Gaussian or Rademacher random matrices \\cite{foucart2013mathematical} with normalized columns.\nIf the measurement matrices $\\mbf{A}(b)$ are identical (i.e., $\\mbf{A}(b) = \\mbf{A}$, $b=1,\\dots,B$) we have an \\ac{mmv} scenario; if they are mutually independent then we have a \\ac{dcs} scenario.\nWe define the length-$B$ column vectors\n\\begin{align}\n \\vec{\\xv}_n &= (x_n(1),\\ldots,x_n(B))^T, \\nonumber\\\\\n \\vec{\\yv}_m &= (y_m(1),\\ldots,y_m(B))^T, \\\\\n \\vec{\\wv}_m &= (w_m(1),\\ldots,w_m(B))^T \\nonumber\n\\end{align}\n(similar notation will be used throughout the paper).\nJoint sparsity (cf.\\ JSM-2 in \\cite{duarte2005joint}) with sparsity (or nonzero probability) $\\epsilon$ requires that $\\vec{\\xv}_n=\\mathbf{0}$ with probability $1-\\epsilon$ and $\\vec{\\xv}_n\\ne\\mathbf{0}$\nwith probability $\\epsilon$.\nIn this work, we focus on signals with multivariate \\ac{bg} \\ac{pdf}, i.e., \n\\begin{equation}\nf_{\\vec{\\xvs}_n}(\\vec{\\xv}_n) = f_{\\vec{\\xvs}}(\\vec{\\xv}_n) =\n(1-\\epsilon) \\,\n\\delta(\\vec{\\xv}_n)\n+ \\epsilon\\, \\mcal{N}(\\vec{\\xv}_n;\\mathbf{0},\\boldsymbol{\\Sigma}_{\\xvecs}) , \\label{eq:signal_prior}\n\\end{equation}\n\\ac{iid} over $n$; here, $\\boldsymbol{\\Sigma}_{\\xvecs}$ is the covariance matrix of $\\vec{\\xv}_n$ given that it is non-zero vector.\nThe additive noise in \\eqref{eq:matrixmeasurement} is assumed to be \\ac{iid}\\ Gaussian over $m$ with zero mean and covariance $\\boldsymbol{\\Sigma}_{\\wvecs}$,\n\\begin{equation}\n\t\\vec{\\wvs}_m \\sim \\mcal{N}(\\mathbf{0}, \\boldsymbol{\\Sigma}_{\\wvecs}) \\,.\n\\end{equation}\n\n\n\\subsection{Vector-prior BAMP for MMV\/DCS} \\label{subsec:vbamp}\n\n\\begin{algorithm}[t]\n\t\\caption{\\acs{Vbamp} for MMV\/DCS}\n\t\\label{alg:vbamp}\n\t\\begin{algorithmic}[1]\n\t\t\\State {\\bf input:} $\\mathbf{y}(b)$, $\\mbf{A}(b)$, $\\boldsymbol{\\Sigma}_{\\xvecs}$, $\\varepsilon$, $t_{\\max}$, $\\varepsilon_{\\mathrm{tol}}$ \n\t\t\\State $t=0$, $\\hat{\\vec{\\xv}}_n^t=\\mathbf{0}_{B\\times 1}$, $\\vec{\\rv}_m^t=\\vec{\\yv}_m$, $\\forall m, n$\n\t\t\\Do\n\t\t\\State $t \\leftarrow t+1$ \n\t\t\\State $\\mathbf{u}^{t-1}(b)= \\hat{\\xv}^{t-1}(b) + \\mbf{A}(b)^T \\mathbf{r}^{t-1}(b)$, $\\forall b$ \n\t\t\\State $\\boldsymbol{\\Sigma}_{\\vvecs}^{t-1} = \\begin{cases} {\\boldsymbol{\\Sigma}_{\\vec{r}}^{t-1}}\n &\\mbox{ for MMV} \\\\ D\\big( {\\boldsymbol{\\Sigma}_{\\vec{r}}^{t-1}}\n \\big) &\\mbox{ for DCS} \\end{cases}$\n\t\t\\State $\\hat{\\vec{\\xv}}_n^t = F(\\vec{\\uv}_n^{t-1} ; \\boldsymbol{\\Sigma}_{\\vvecs}^{t-1})$, $\\forall n$ \n\t\t\\State $\\vec{\\rv}_m^{t} = \\vec{\\yv}_m - \\big(\\mbf{A}(1) \\hat{\\xv}^t(1),\\ldots,\\mbf{A}(B) \\hat{\\xv}^t(B)\\big)_m \\hspace*{15mm}$ \n\t\t\\hspace*{10mm} $\\phantom{=}+ \\frac{1}{M} \\sum_{n=1}^{N} F'(\\vec{\\uv}_n^{t-1} ; \\boldsymbol{\\Sigma}_{\\vvecs}^{t-1}) \\vec{\\rv}_m^t$, $\\forall m$\n\t\t\\doWhile{$\\sum_{b=1}^{B}\\left\\|\\hat{\\xv}^{t}(b)\\!-\\!\\hat{\\xv}^{t-1}(b) \\right\\|^2_2 \\!> \\! \\varepsilon_{\\mathrm{tol}} \\sum_{b=1}^{B}\\left\\|\\hat{\\xv}^{t-1}(b)\\right\\|^2_2$ {\\bf and} $t < t_{\\max}$} \n\t\t\\State\n\t\t\\Return $\\hat{\\xv}(b)=\\hat{\\xv}^t(b)$, \n\t\t$\\forall b$\n\t\\end{algorithmic}\n\\end{algorithm}\n\nThe \\ac{Vbamp} method for joint sparse recovery of $\\mathbf{x}(b)$, $b=1,\\dots,B$,\n\\cite{kim2011belief,zhao2015joint} is summarized in Algorithm \\ref{alg:vbamp}\n(superscript $t$ indicates the iteration index).\nNote that scalar \\ac{Vbamp} (i.e., when $B=1$) is a special case of Algorithm \\ref{alg:vbamp} where \\ac{mmv} and \\ac{dcs} are equivalent.\nThe vector-prior BAMP follows similar steps as ordinary scalar \\ac{bamp} \\cite{donoho2009message,maleki2010approximate,montanari2012graphical,DonohoITWmessagepassingI,DonohoITWmessagepassingII,bayati2011dynamics}. According to the decoupling principle \\cite{montanari2012graphical}, which holds in the asymptotic regime where $M,N\\rightarrow \\infty$ while $\\frac{M}{N} = R$, the \\ac{Vbamp} algorithm decouples the \\ac{cs} measurements \\equref{eq:matrixmeasurement} according to\n\\begin{equation}\n\\vec{\\uv}_n^t=\\vec{\\xv}_n + \\vec{\\vv}_n^t ,\n\\end{equation}\nwhere the effective noise vector is distributed as ${\\vec{\\vvs}_n^t \\sim \\mcal{N}(\\mathbf{0}, \\boldsymbol{\\Sigma}_{\\vvecs}^t)}$. \nThe effective noise covariance is estimated via the empirical covariance {$\\boldsymbol{\\Sigma}_{\\vec{r}}^{t}=\\cov\\!\\left\\{ \\vec{\\rv}_m^{t-1} \\right\\}$} from vectors $\\vec{\\rv}_m^{t-1}$ in line 6 of Algorithm \\ref{alg:vbamp}. It has been shown in \\cite{lu2016independent} that in the \\ac{dcs} scenario only the diagonal entries of the covariance matrix are retained due to the mixing effected by the $B$ {mutually} independent measurement matrices. In the following, we will simplify notation by\noccasionally dropping the indices $t$ and $n$.\n\nThe vector denoiser in \\ac{Vbamp} (line 7 of Algorithm \\ref{alg:vbamp}) amounts to a vector \\ac{mmse} estimator of $\\vec{\\xv}_n$ given the decoupled measurements $\\vec{\\uv}_n$.\nUsing Bayes' theorem, the denoiser can be written as:\n\\begin{equation}\n\\begin{aligned}\nF(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs}) &= \\Exp_{\\vec{\\xvs}} \\left\\{\\vec{\\xvs} \\mid \\vec{\\uvs} = \\vec{\\uv} ; \\boldsymbol{\\Sigma}_{\\vvecs} \\right\\}\\\\\n&= \\frac{\\int_{\\mathbb{R}^{B}} \\vec{\\zv} \\,\\mathcal{N}(\\vec{\\uv};\\vec{\\zv},\\boldsymbol{\\Sigma}_{\\vvecs}) f_{\\vec{\\xv}}(\\vec{\\zv}) \\,d \\vec{\\zv}}{\\int_{{\\mathbb{R}^{B}}} \\mathcal{N}(\\vec{\\uv};\\vec{\\zv},\\boldsymbol{\\Sigma}_{\\vvecs}) f_{\\vec{\\xv}}(\\vec{\\zv}) \\,d \\vec{\\zv}} \\,,\n\\end{aligned}\n\\label{eq:F}\n\\end{equation}\n{where the covariance of the effective noise is $\\boldsymbol{\\Sigma}_{\\vvecs} = \\boldsymbol{\\Sigma}_{\\vec{r}}$ (\\ac{mmv}) or $\\boldsymbol{\\Sigma}_{\\vvecs} = D(\\boldsymbol{\\Sigma}_{\\vec{r}})$ (\\ac{dcs})}.\nFor the multivariate \\ac{bg} prior \\equref{eq:signal_prior}, the vector denoiser becomes\n\\begin{align}\nF(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs}) = \\mathbf{W}\\vec{\\uv} \\qquad\\text{with }\\;\n\\mathbf{W} = \\frac{F_N(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})}{F_D(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})}\\boldsymbol{\\Sigma}_{\\xvecs} \\boldsymbol{\\Sigma}_{\\uvecs}^{-1} .\n\\label{eq:2_Fvec_bernoulli_gaussian}\n\\end{align}\nHere, $\\boldsymbol{\\Sigma}_{\\uvecs} = \\boldsymbol{\\Sigma}_{\\xvecs} + \\boldsymbol{\\Sigma}_{\\vvecs}$ and\n\\begin{align}\\label{eq:BGprior}\nF_N(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs}) & = \\epsilon\\, \\mcal{N}(\\vec{\\uv};\\mathbf{0},\\boldsymbol{\\Sigma}_{\\uvecs}),\\\\\nF_D(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs}) & = (1-\\epsilon)\\, \\mcal{N}(\\vec{\\uv};\\mathbf{0},\\boldsymbol{\\Sigma}_{\\vvecs}) + \\epsilon\\, \\mcal{N}(\\vec{\\uv};\\mathbf{0},\\boldsymbol{\\Sigma}_{\\uvecs})\\nonumber\n\\end{align}\nThe denoiser \\equref{eq:2_Fvec_bernoulli_gaussian} consists of a multivariate Gaussian Wiener estimator followed by a joint shrinkage operation.\n\nThe \\ac{Vbamp} residual is computed in line 8 of Algorithm \\ref{alg:vbamp}.\nAs in the original \\ac{amp} derivation \\cite{maleki2010approximate}, the Onsager correction term for the residual $\\vec{\\yv}_m - \\big(\\mbf{A}(1) \\hat{\\xv}(1),\\ldots,\\mbf{A}(B) \\hat{\\xv}(B)\\big)_m$ is computed via the derivative of the estimator. \nIn the asymptotic regime, the Onsager term \n\\begin{equation}\\label{eq:onsager}\n\t\\frac{1}{M} \\sum_{n=1}^{N} F'(\\vec{\\uv}_n ; \\boldsymbol{\\Sigma}_{\\vvecs}) \\,\\vec{\\rv}_m \n\\end{equation}\nrenders the decoupled measurement vectors $\\vec{\\uv}_n$ Gaussian with mean $\\vec{\\xv}_n$ and covariance $\\boldsymbol{\\Sigma}_{\\vvecs}$ \\cite{kim2011belief,zhu2016performance}.\nHere, the Jacobian matrix $F'(\\vec{\\uv} ; \\boldsymbol{\\Sigma}_{\\vvecs}) = {dF(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})}\/{d\\vec{\\uv}^T} $ of the estimator $F(\\vec{\\uv} ; \\boldsymbol{\\Sigma}_{\\vvecs})$ is given by\n\\begin{equation}\nF'(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})\n\t\t\t\t\\! =\\!\n\\mathbf{W} \\!-\\! \\left(1 \\!-\\! \\frac{F_N(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})}{F_D(\\vec{\\uv}; \\boldsymbol{\\Sigma}_{\\vvecs})}\\right) \\mathbf{W} \\vec{\\uv}_n \\vec{\\uv}_n^T (\\boldsymbol{\\Sigma}_{\\uvecs}^{-1}-\\boldsymbol{\\Sigma}_{\\vvecs}^{-1})\\,.\n\\end{equation}\n\nThe algorithm runs until the relative change in the estimated signal is below a certain threshold $\\varepsilon_{\\mathrm{tol}}$ or the maximum number of iterations $t_{\\max}$ is reached.\nCompared to scalar \\ac{bamp}, the vector \\ac{Vbamp} algorithm involves the following crucial modifications:\n\\begin{itemize}\n\\item a multivariate prior (possibly with joint sparsity structure and correlation);\n\\item the estimator acts on vectors rather than scalars \\equref{eq:F} and both correlated signal and correlated additive noise are taken into consideration (more precisely, the full signal and noise vector \\ac{pdf} is taken into account);\n\\item an\n\t Onsager term obtained as the sum of Jacobian matrices (cf.\\ \\equref{eq:onsager}).\n\\end{itemize}\n\n\\subsection{State Evolution}\n\n\\ac{se} was originally proposed in \\cite{donoho2009message} for scalar (B)AMP and extended to the \\ac{mmv} and \\ac{dcs} scenarios (e.g., in \\cite{lu2016independent}); it allows to characterize analytically the expected behavior of \\ac{Vbamp}\n(note that the Onsager term in \\cite{lu2016independent} is flawed \neven though the multivariate \\ac{se} is correct).\nIn particular, the \\ac{se} equation predicts the evolution of the effective noise covariance (the state)\nfor any signal prior {$f_{\\vec{\\xvs}}(\\vec{\\xv}_n)$} as\n\\begin{align}\n\\boldsymbol{\\Sigma}_{\\vvecs}^{t+1} \n= \\begin{cases}\n\\boldsymbol{\\Sigma}_{\\wvecs} + \\frac{1}{R} \\Exp_{\\vec{\\xvs},\\vec{\\vvs}} \\left\\{ \\boldsymbol{\\langle} \\mathbf{e}(\\vec{\\xvs},\\vec{\\vvs}) \\boldsymbol{\\rangle} \\right\\} & \\mbox{for MMV},\\\\\nD\\!\\left(\\boldsymbol{\\Sigma}_{\\wvecs} + \\frac{1}{R} \\Exp_{\\vec{\\xvs},\\vec{\\vvs}} \\left\\{ \\boldsymbol{\\langle} \\mathbf{e}(\\vec{\\xvs},\\vec{\\vvs})\\boldsymbol{\\rangle}\\right\\}\\right) & \\mbox{for DCS} ,\n\\end{cases} \\label{eq:stateevolution}\n\\end{align}\nwhere $\\mathbf{e}(\\vec{\\xvs},\\vec{\\vvs})=F(\\vec{\\xvs}+\\vec{\\vvs};\\boldsymbol{\\Sigma}_{\\vvecs}^{t}) - \\vec{\\xvs}$ is the error achieved by the\n\\ac{mmse} estimator $F(\\vec{\\uv};\\boldsymbol{\\Sigma}_{\\vvecs}^t)$ and $\\vec{\\vvs} \\sim \\mcal{N}(\\mathbf{0},\\boldsymbol{\\Sigma}_{\\vvecs}^t)$.\nThe state in the \\ac{mmv} scenario is in general $B(B+1)\/2$ dimensional (since the covariance matrix is symmetric).\nFrom \\equref{eq:stateevolution}, the \\ac{mse} prediction directly follows as\n\\begin{align*}\n\\cov \\{\\vec{\\uv}^t_n - \\vec{\\xv}_n\\} &= \\boldsymbol{\\Sigma}_{\\vvecs}^t , \\\\\n\\widehat{\\mathrm{MSE}}{}^t(b) &= R\\,(\\boldsymbol{\\Sigma}_{\\vvecs}^t - \\boldsymbol{\\Sigma}_{\\wvecs})_{b,b} ,\n\\end{align*}\nwith the \\ac{mse} per channel being defined as \n\\begin{equation*}\n\\mathrm{MSE}^t(b) = \\frac{1}{N}\\|\\hat{\\xv}^t(b) - \\mathbf{x}(b)\\|_2^2 .\n\\end{equation*}\n\n\n\\section{Diagonalized Vector-prior BAMP} \\label{sec:sec3}\n\n\\subsection{Joint Diagonalization for MMV} \\label{sec:gevd}\n\n\\begin{algorithm}[!t]\n\n\t\\caption{joint diagonalization transformation}\n\t\\label{alg:gevd}\n\t\\begin{algorithmic}[1]\n\t\t\\State Given $\\boldsymbol{\\Sigma}_{\\xvecs},\\boldsymbol{\\Sigma}_{\\wvecs}$\n\t\t\\State \\hspace*{5mm} find $\\mathbf{P}$ such that $\\mathbf{P}\\Pv^T=\\boldsymbol{\\Sigma}_{\\wvecs}$\n\t\t\\State \\hspace*{5mm} $\\mathbf{G} = \\mathbf{P}^{-1} \\boldsymbol{\\Sigma}_{\\xvecs} \\mathbf{P}^{-T}$\n\t\t\\State \\hspace*{5mm} find eigendecomposition $\\mathbf{Q} \\boldsymbol{\\Lambda} \\mathbf{Q}^T=\\mathbf{G}$\n\t\t\\State \\hspace*{5mm} $\\mathbf{T} = \\boldsymbol{\\Lambda}^{-1\/2}\\mathbf{Q}^T\\mathbf{P}^{-1}$\n\t\\end{algorithmic}\n\\end{algorithm}\n\nThe \\ac{Vbamp} algorithm in Section \\ref{subsec:vbamp} can deal with arbitrary signal and noise correlations $\\boldsymbol{\\Sigma}_{\\xvecs}$ and $\\boldsymbol{\\Sigma}_{\\wvecs}$, which in general results in a nondiagonal $\\boldsymbol{\\Sigma}_{\\vvecs}^t$ in the \\ac{mmv} scenario.\nIn the decoupled measurements $\\vec{\\uv}=\\vec{\\xv} + \\vec{\\vv}$, it means that there are $\\mcal{O}(B^2)$ \\ac{snr} relations and $B(B+1)\/2$ states: each $\\mathsf{x}_n(b)$ correlates with all $\\mathsf{x}_n(b')$, $b'\\in\\{1,\\ldots,B\\}\\setminus\\{b\\}$, and it is influenced simultaneously by all effective noise components $\\mathsf{v}(b')$, $b'\\in\\{1,\\ldots,B\\}$.\n\nUnder the assumption that the covariance matrices $\\boldsymbol{\\Sigma}_{\\xvecs}$ and $\\boldsymbol{\\Sigma}_{\\wvecs}$ are full rank and using the fact that covariance matrices are symmetric and positive definite and \\cite[Thm.~7.6.1.]{horn2012matrix}, there exists a nonsingular (but generally non-orthogonal) matrix $\\mathbf{T}$ that simultaneously diagonalizes the covariance matrices of the signal $\\vec{\\xv}$ and the noise $\\vec{\\wv}$.\nThe computation of $\\mathbf{T}$ is described in Algorithm \\ref{alg:gevd}. \nIn the transformed model\n\\begin{align}\n\\tilde{\\vec{\\yv}}_m = \\mathbf{T} \\vec{\\yv}_m , \\quad \\tilde{\\vec{\\xv}}_n &= \\mathbf{T} \\vec{\\xv}_n , \\quad \\tilde{\\vec{\\wv}}_m = \\mathbf{T} \\vec{\\wv}_m \\label{eq:transformedmodel2} ,\n\\end{align}\nwe thus have \n\\begin{align*}\n\\boldsymbol{\\Sigma}_{\\tilde{\\xvecs}} & = \\mathbf{T} \\boldsymbol{\\Sigma}_{\\xvecs} \\mathbf{T}^T = \\mathbf{I}_{B\\times B} , \\\\\n\\boldsymbol{\\Sigma}_{\\tilde{\\wvecs}} & = \\mathbf{T} \\boldsymbol{\\Sigma}_{\\wvecs} \\mathbf{T}^T = \\boldsymbol{\\Lambda}^{-1}=\\epsilon\\diag \\left(\\frac{1}{\\snr(1)}, \\ldots, \\frac{1}{\\snr(B)}\\right) .\n\\end{align*}\nHere, the per-channel SNRs are defined as\n\\begin{equation*}\n\t\\mathrm{SNR}(b) = \\frac{\\Exp_{\\mbsf{x}}\\! \\left\\{\\|\\mbf{A}(b)\\tilde{\\mbsf{x}}(b)\\|^2_2\\right\\}}{\\Exp_{\\mbsf{w}} \\!\\left\\{\\|\\tilde{\\mbsf{w}}(b)\\|^2_2\\right\\}} = \\epsilon\\,\\boldsymbol{\\Lambda}_{b,b}.\n\\end{equation*}\nNote that the decorrelation can be applied also in the \\ac{dcs} scenario, given that only the noise covariance $\\boldsymbol{\\Sigma}_{\\wvecs}$ is nondiagonal and the signal covariance $\\boldsymbol{\\Sigma}_{\\xvecs}$ is diagonal.\nWe emphasize that in case \\ac{Vbamp} operates on the transformed measurements, \nthe change in the prior distribution has to be accounted for in a nontrivial manner.\nThat is, the \\ac{mmse} estimator \\equref{eq:F} and its derivative will have a different form.\nConsider the \\ac{se} equation \\equref{eq:stateevolution} that describes the expected evolution of the effective noise covariance over the \\ac{Vbamp} iterations.\nIn the \\ac{mmv} scenario,\neven if $\\boldsymbol{\\Sigma}_{\\wvecs}$ and $\\boldsymbol{\\Sigma}_{\\vvecs}^t$ are diagonal, $\\boldsymbol{\\Sigma}_{\\vvecs}^{t+1}$ in general will not be diagonal because the estimator $F(\\vec{\\uv}^t_n,\\boldsymbol{\\Sigma}_{\\vvecs}^t)$ operates on the overall vector $\\vec{\\uv}^t_n$ ({nonetheless} the diagonalization described in Algorithm \\ref{alg:gevd} could be performed repeatedly in each iteration).\nHowever, we will see shortly that in the particular case of the \\ac{bg} prior this is no longer the case.\nA direct calculation reveals that\n\\begin{equation*}\n\\cov\\{ \\vec{\\yvs}_m \\} =\n\\begin{cases}\n\\boldsymbol{\\Sigma}_{\\wvecs} + \\frac{1}{R}\\cov\\{\\vec{\\xvs}_n\\} &\\mbox{for MMV}, \\\\\n\\boldsymbol{\\Sigma}_{\\wvecs} + \\frac{1}{R} D\\left( \\cov\\{\\vec{\\xvs}_n\\} \\right) &\\mbox{for DCS}.\n\\end{cases} \n\\end{equation*}\nThus, if we know either the noise or signal covariance then the other can be estimated directly through the measurement covariance $\\cov\\{\\vec{\\yvs}_m \\}$. Alternatively, when both covariances are unknown and the signal is drawn from a BG prior we can use the \\ac{em} AMP approach introduced in \\cite{SchniterEMAMP} to estimate both sets of parameters within the iterations.\n \n\\subsection{Equivariance of \\ac{Vbamp} for MMV}\nWe next establish the fact that for MMV both \\ac{Vbamp} and its \\ac{se} are equivariant w.r.t.\\ invertible linear transformations of the input. The proof of this result is provided in Appendix \\ref{app:vbamp_equivariance}.\n\n\\vspace*{2mm}\n\\begin{theorem}\\label{thm:equivar}\n\tAlgorithm \\ref{alg:vbamp} for \\ac{mmv} and its \\ac{se} are equivariant w.r.t.~invertible linear transformations. \n\n\tDenote one \\ac{Vbamp} iteration by $(\\hat{\\vec{\\xv}}_n^{t+1}, \\vec{\\rv}_m^{t+1}, \\boldsymbol{\\Sigma}_{\\vvecs}^{t+1})=\\text{V}(\\vec{\\yv}_m, \\hat{\\vec{\\xv}}_n^t, \\vec{\\rv}_m^t, \\boldsymbol{\\Sigma}_{\\vvecs}^t)$. For any nonsingular $\\mathbf{T}$, we have for all $m$ and $n$\n\\[\n\\text{V}(\\mathbf{T}\\vec{\\yv}_m, \\mathbf{T}\\hat{\\vec{\\xv}}_n^t, \\mathbf{T}\\vec{\\rv}_m^t, \\mathbf{T}\\boldsymbol{\\Sigma}_{\\vvecs}^t\\mathbf{T}^T) = (\\mathbf{T}\\hat{\\vec{\\xv}}_n^{t+1}, \\mathbf{T}\\vec{\\rv}_m^{t+1}, \\mathbf{T}\\boldsymbol{\\Sigma}_{\\vvecs}^{t+1}\\mathbf{T}^T).\n\\]\nFurthermore, the SE equation \\eqref{eq:stateevolution} translates to the transformed domain as\n\\begin{align}\n&\\mathbf{T}\\boldsymbol{\\Sigma}_{\\vvecs}^{t+1}\\mathbf{T}^T = \\mathbf{T}\\boldsymbol{\\Sigma}_{\\wvecs}\\mathbf{T}^T \\nonumber\\\\\n& \\hspace*{10mm}+ \\frac{1}{R} \\Exp_{\\vec{\\xvs},\\vec{\\vvs}} \\left\\{ \\boldsymbol{\\langle} F(\\mathbf{T}(\\vec{\\xvs}+\\vec{\\vvs});\\mathbf{T}{\\boldsymbol{\\Sigma}_{\\vvecs}}^{t}\\mathbf{T}^T) - \\mathbf{T}\\vec{\\xvs}\\boldsymbol{\\rangle} \\right\\} .\\label{eq:transformedSE1}\n\\end{align}\n\\end{theorem}\n\\vspace*{1mm}\n\nNote that \\equref{eq:transformedSE1} holds for any signal prior in the Bayesian setting, i.e., when the estimator is the \\ac{mmse} estimator.\nAssume that \\ac{Vbamp} converges to $\\hat{\\vec{\\xv}}_n$ with inputs $\\vec{\\yv}_n$, $\\boldsymbol{\\Sigma}_{\\xvecs}$, and $\\boldsymbol{\\Sigma}_{\\wvecs}$;\nthen, Theorem \\ref{thm:equivar} implies that \\ac{Vbamp} with inputs $\\mathbf{T}\\vec{\\yv}_n$, $\\mathbf{T}\\boldsymbol{\\Sigma}_{\\xvecs}\\mathbf{T}^T$, and $\\mathbf{T}\\boldsymbol{\\Sigma}_{\\wvecs}\\mathbf{T}^T$ \nconverges to the solution $\\mathbf{T}\\hat{\\vec{\\xv}}_n$. \n\n\\subsection{Bernoulli-Gauss Prior} \\label{sec:bg}\n\nFor the \\ac{bg} prior, after applying the transformation $\\mathbf{T}$, the equivalent measurement model becomes\n\\begin{equation}\n\\tilde{\\mathbf{y}}(b) = \\mbf{A}(b) \\tilde{\\mathbf{x}}(b) + \\tilde{\\mathbf{w}}(b) \\,, \\quad \\forall b \\label{eq:equivalentmodel}\n\\end{equation}\nwith signal and noise \\acp{pdf}\n\\begin{align}\nf_{\\tilde{\\vec{\\xvs}}}(\\tilde{\\vec{\\xv}}_n) &=\n(1-\\epsilon) \\,\\delta(\\tilde{\\vec{\\xv}}_n)\n+ \\epsilon\\, \\mcal{N}(\\tilde{\\vec{\\xv}}_n;\\mathbf{0},\\mathbf{I}) , \\label{eq:diagonalbgpdf} \\\\\nf_{\\tilde{\\vec{\\wvs}}}(\\tilde{\\vec{\\wv}}_m) &=\n\\mcal{N}(\\tilde{\\vec{\\wv}}_m;\\mathbf{0},\\boldsymbol{\\Lambda}^{-1}) \\label{eq:diagonalnoisepdf} .\n\\end{align}\nThat is, we retain a \\ac{bg} prior in the transformed domain, only with uncorrelated components. \nThis is a distinctive feature of the \\ac{bg} prior and in general doesn't hold for other types of distributions.\n\nIn Appendix \\ref{app:bg_diagonality} we demonstrate that \nfor the decorrelated model \\equref{eq:equivalentmodel} with \\ac{bg} prior \\equref{eq:diagonalbgpdf}--\\equref{eq:diagonalnoisepdf}, the \\ac{Vbamp} iterations under the MMV model preserve the diagonal structure of $\\boldsymbol{\\Sigma}_{\\tilde{\\vvecs}}^t$.\nIt follows that for \\ac{cs} measurements with multivariate \\ac{bg} signal prior, the decorrelation transformation has to be done only once before recovery;\ndetermining $\\mathbf{T}$ itself is of negligible computational effort unless $B$ is very large.\nThese observations have the following implications:\n\\begin{itemize}\n\t\\item The computation of \\equref{eq:2_Fvec_bernoulli_gaussian} and \\equref{eq:onsager} is significantly simplified, \n\tleading to complexity reductions by a factor of $B$.\n\\item The dimension of the \\ac{se} equations is $B$ instead of $B(B+1)\/2$.\n\tIn other words, $B(B+1)\/2$ effective noise covariance parameters in $\\boldsymbol{\\Sigma}_{\\vvecs}$ are reduced to $B$ effective noise variances, which\n\texplicitly characterize the \\ac{mse} for each signal vector estimate as\n\t\\begin{equation*}\n\t\\widehat{\\mse}{}^t(b) = R\\, (\\boldsymbol{\\Sigma}_{\\tilde{\\vvecs}}^{t} - \\boldsymbol{\\Sigma}_{\\tilde{\\wvecs}})_{b,b}.\n\t\\end{equation*}\n\t\\item Every \\ac{mmv} problem has an equivalent \\ac{dcs} problem with possibly rescaled \\acp{snr}.\n\tFurthermore, the analysis of \\ac{dcs} also covers that of \\ac{mmv}.\n\\end{itemize}\n\n\n\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=0.9\\linewidth]{.\/VBAMP-figure0.pdf}\n\\vspace*{2mm}\n\t\\caption{{Free energy function at different rates $R$ for $B=1$, $\\sigma_{\\ws}^2=-35\\dB$, and sparsity $\\epsilon=0.1$. Red squares and black triangles indicate local maxima and minima, respectively.}} \\label{fig:freeenergycurves_noisy}\n\\end{figure}\n\n\\section{Replica Analysis}\\label{sec:replicanalysis}\n\nIn \\cite{zhu2016performance}, the replica method was used to determine the \\ac{mse} performance of \\ac{Vbamp} for the measurement \\equref{eq:matrixmeasurement} and the \\ac{bg} prior \\equref{eq:signal_prior}, assuming $\\boldsymbol{\\Sigma}_{\\xvecs} = \\mathbf{I}$ and isotropic uncorrelated noise, i.e., $\\boldsymbol{\\Sigma}_{\\wvecs}=\\sigma_{\\mathsf{w}}^2 \\mathbf{I}$.\nIn this special case \\ac{mmv} and \\ac{dcs} (referred to as {MMV-2} and {MMV-1}, respectively, in \\cite{zhu2016performance}) are equivalent.\nThe analysis is quite sophisticated and the generalization to arbitrary signal and noise correlations seems infeasible.\nHowever, due to the joint diagonalization approach from Section \\ref{sec:sec3}, it suffices to extend the replica analysis to the case with $\\boldsymbol{\\Sigma}_{\\xvecs} = \\mathbf{I}$ and $\\boldsymbol{\\Sigma}_{\\wvecs}=\\diag( \\sigma_{\\ws}^2(1),\\ldots,\\sigma_{\\ws}^2(B) )$.\nIn particular, the replica method is capable of predicting the fixed points of \\ac{Vbamp} in the asymptotic regime ($N,M\\rightarrow\\infty$, $R=M\/N=\\mathrm{const.}$), as a function of the set of $B$ \\acp{mse}\\cite{krzakala2012probabilistic,krzakala2012statistical}.\nWe note that rigorous equivalence between the replica method and \\ac{se} is not always guaranteed and requires additional technicalities \\cite{reeves2016replica}.\nAssuming $\\boldsymbol{\\Sigma}_{\\xvecs} = \\mathbf{I}$ and $\\boldsymbol{\\Sigma}_{\\wvecs} = \\diag\\big(\\sigma_{\\ws}^2(1),\\ldots, \\sigma_{\\ws}^2(B)\\big)$, we compute in \nAppendix \\ref{app:replica}, following the derivation in \\cite{zhu2016performance}, the free energy $\\mcal{F}(\\vec{\\Ev})$\nas a function of the \\ac{mse} vector $\\vec{\\Ev} = (E(1),\\ldots,E(B))^T$ with $E(b)= \\mathrm{MSE}(b)$, resulting in\n\\begin{equation}\\label{eq:freeenergy}\n\\begin{split}\n& \\mcal{F}(\\vec{\\Ev}) = \n\t\t(1-\\epsilon)\\,\\zeta(\\gamma) + \\epsilon\\,\\zeta\\Big(\\frac{\\gamma}{1+\\gamma}\\Big) \\\\\n& {}-\\frac{R}{2} \\sum_{b=1}^{B}\\left(\\log \\frac{2\\pi R}{\\gamma(b)} \n\t\t+ \\gamma(b) \\sigma_{\\ws}^2(b) \n\t\t- \\frac{1-\\epsilon}{R} \\frac{\\gamma(b)}{1+\\gamma(b)} \n\t\t\\right) \\!.\n\\end{split}\n\\end{equation}\nIn this expression we used \n\\begin{align*}\n\\zeta(\\eta) \n& = \\int \\log \\left( \\epsilon \\prod_{b=1}^{B}(1+\\gamma(b))^{-\\frac12} \\right. \\\\\n&\\qquad\t+ \\left. (1-\\epsilon) \\exp\\!\\bigg(\\!-\\frac12 \\sum_{b=1}^{B} \\eta(b)\\,h^2(b)\\bigg) \\right)\\mcal{D}\\mathbf{h}\n\\end{align*}\nwith\n\\begin{equation*}\n\t\\gamma(b) = \\frac{R}{E(b)+R\\sigma_{\\ws}^2(b)};\n\\end{equation*}\nfurthermore, $\\mcal{D}\\mathbf{h} = \\mcal{N}(\\mathbf{h};\\mathbf{0},\\mathbf{I})\\,dh_1\\dots dh_B$ denotes the multivariate standard Gaussian measure.\n\n\nThe stationary points of $\\mcal{F}(\\vec{\\Ev})$ correspond to fixed points of belief propagation \\cite{heskes2003stable}, and hence to those of \\ac{Vbamp} in the asymptotic regime \\cite{zhu2016performance}. Thus, we can determine the component-wise \\ac{mse}s of \\ac{Vbamp} by evaluating \\equref{eq:freeenergy} and finding the largest components of $\\vec{\\Ev}$ that correspond to a local maximum of $\\mcal{F}(\\vec{\\Ev})$ \\cite{yedidia2005constructing,yedidia2001bethe}. Note that for isotropic noise ($\\sigma_{\\ws}^2(b) = \\sigma_{\\ws}^2$ $\\forall b$), the free energy in \\equref{eq:freeenergy} simplifies to the result obtained in \\cite{zhu2016performance} with one-dimensional argument $E=E(1)=\\ldots=E(B)$. Replica curves for the isotropic case with $B=1$ and $B=10$ are shown in Figure~\\ref{fig:freeenergycurves_noisy} and \\ref{fig:freeenergy_B10} respectively. It is important to point out here that all the plots are the result of numerical integrations (and not Monte Carlo simulations). In the free energy function, local maxima correspond to stable fixed points and local minima to unstable fixed points, whereas the global maximum of $\\mathcal{F}(E)$ corresponds to the \\ac{mmse}. \\ac{Vbamp} typically achieves the largest \\ac{mse} associated with a local maximum.\n\n\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=0.9\\linewidth]{.\/VBAMP-figure1.pdf}\n\\vspace*{2mm}\n\t\\caption{{One-dimensional free energy function for the isotropic case with $B=10$ jointly sparse \\ac{bg} vectors at rates around the phase transition rate ($\\boldsymbol{\\Sigma}_{\\wvecs} = -35\\dB \\mathbf{I}_B$).}}\\label{fig:freeenergy_B10}\n\\end{figure}\n\n\n\\subsection{MMSE Gap}\n\nIn the \\ac{cs} regime of small $\\epsilon$ and nonzero noise variance, the \\ac{mmse} estimate $\\hat{\\xv}$ for a single measurement features a first order phase transition (PT) characterized by an abrupt change of the \\ac{mse} at a certain rate $R_{\\PT}$: for rates less than $R_{\\PT}$, the \\ac{mse} tends to be large, whereas for rates larger than $R_{\\PT}$ the \\ac{mse} tends to be small and plateaus to fixed nonzero value. This phenomenon can be seen in Figure \\ref{fig:freeenergycurves_noisy}: for rates below $R\\approx 0.16$, where the free energy has a single maximum at an \\ac{mse} of about $-12\\dB$ whereas for rates larger than $R\\approx 0.17$ a second local maximum at \\acp{mse} less then about $-37\\dB$ appears. \n\nA similarly abrupt phase transition does not appear to occur when the number of measurements $B$ is sufficiently large. Figure \\ref{fig:SEcurves_B} shows the \\ac{se} curves for various $B$ with $\\epsilon=0.1$ and $\\boldsymbol{\\Sigma}_{\\xvecs}=\\mathbf{I}_B$. Observe that the ``bump'' in the \\ac{se} curve for small $B$ and large \\ac{mse}, which corresponds to the first \nfixed point, flattens out with increasing $B$. For large enough $B$ we observe that the \\ac{se} curve ceases to exhibit a first order PT, so that the \\ac{mse} changes smoothly with increasing rate $R$.\n\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=0.9\\linewidth]{.\/VBAMP-figure2.pdf}\n\\vspace*{2mm}\n\t\\caption{Noisy \\ac{se} curves for different number of jointly sparse \\ac{bg} signals ($\\epsilon=0.1$, $\\boldsymbol{\\Sigma}_{\\xvecs}=\\mathbf{I}_B$, $\\boldsymbol{\\Sigma}_{\\wvecs}=-35\\dB \\mathbf{I}_B$, $R=0.25$).} \\label{fig:SEcurves_B}\n\\end{figure}\n\nThe same conclusion can be obtained by investigating the behavior of the free energy functions. \\ac{Vbamp} typically achieves the largest \\ac{mse} which corresponds to a local maximum in the free energy, whereas the \\ac{mse} at the global maximum of the free energy is the \\ac{mmse}. As pointed out in \\cite{zhu2016performance}, whenever the free energy function has a second local maximum at a larger \\ac{mse} than the global maximum, \\ac{Vbamp} is not Bayesian-optimal (i.e., does not reach the \\ac{mmse}). For $B=1$, in Figure~\\ref{fig:freeenergycurves_noisy}, a second local (non-global) maximum appears and thus BAMP is not MMSE optimal in the rate region $0.19$\\SI{5}{\\minute} we observe Doniach-Sunjic inelastic background, indicating Eu metal inclusions in the film. Further deposition leads to an increase in background and indicates \\ce{Eu^0}. We conclude, that only a small amount of EuO is formed at this temperature for the initial growth. For extended growth the Eu metal inclusions turn the stoichiometry to Eu-rich.\n\nNext, we study a stepwise Eu deposition at $T_P=$\\SI{500}{\\celsius} (Fig.~\\ref{fig:fig1}(d)). \nFor $t=$\\SI{1}{\\minute}, we observe a spectrum with a mixture of Eu$^{2+}$ (85\\% )and Eu$^{3+}$ (15\\%) components, while contributions of Eu metal are absent. For continued growth the spectral weight from Eu$^{3+}$ decreases (1\\% at $t=$\\SI{20}{\\minute}). Already for $t>$\\SI{5}{\\minute}, we find that the XPS spectra of adsorbed Eu-metal are nearly indistinguishable from stoichiometric EuO reference data (compare Fig.\\ref{fig:fig1}(a)). This demonstrates that the Eu metal is oxidized into a Eu$^{2+}$ rich (Eu$^{2+}$,Eu$^{3+}$) mixture at the \\ce{SrTiO3} interface. For extended growth only stoichiometric EuO (Eu$^{2+}$) is observed.\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip,width=0.47\\textwidth]{fig\/Fig2_results_temp}\n\t\t\\includegraphics[width=0.47\\textwidth]{fig\/Fig2b_Temperature_Dependence_Boxes}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\tChemical quantification of the Eu $ 3d_{5\/2} $ core-level for metallic Eu$ ^{0} $ (a) and over-oxidized Eu$ ^{3+} $ (b) content as function of $ t $ and $ T_{P} $. (c) Exponential thickness dependence exemplary shown for a sample grown at $ T_P = \\SI{500}{\\degreeCelsius} $. (d) A reduction in $ d_0 $ and $ \\tau $ is observed at $ t \\geq \\SI{700}{\\degreeCelsius} $ at the same temperature where a \\ce{Eu^3+} rich growth is observed. (e) We summarize the findings in a block diagram depicting the overall trend of the resulting film thickness and stoichiometry as function of $ T_P $.\n\t\t}\n\t}\n\t\\label{fig:fig2}\n\\end{figure}\n\nIn the following, the stoichiometry of the grown film is quantified by fitting a linear combination of the reference spectra to the observed Eu \\textit{3d}$ _{5\/2} $ spectra. First, we discuss the Eu metal content of the grown films (Fig.~\\ref{fig:fig2}(a)). \nIn addition to the findings presented for $ T_{P} = \\SI{250}{\\degreeCelsius} $, we find for $ T_P\\geq\\SI{350}{\\celsius}$ no \\ce{Eu^0} metal in the spectra. We conclude that now Eu re-evaporation is a dominant process, which is in line with previous reports utilizing the adsorption limited growth mode on YSZ~\\cite{steeneken_new_2002}.\n\nWe observe an exponential decay of the \\ce{Eu^{3+}} fraction with $t$ for films prepared at {$T_P= \\,$\\SI{250}{\\celsius},} \\SI{350}{\\celsius}, \\SI{400}{\\celsius} and \\SI{500}{\\celsius} (Fig.~\\ref{fig:fig2}(b)). The Beer-Lambert law predicts an exponential decay of the intensity for the case of a buried layer with the growth over-layer thickness. Therefore, we argue that only the interface is responsible for the \\ce{Eu^{3+}} formation. It is also noted, that the \\ce{Eu^3+} content is higher at $ t = \\SI{1}{\\minute} $ for increasing $ T_{P} $. Yet, the total amount of \\ce{Eu^{3+}} is below one ML of \\ce{Eu2O3}, which is a negligible amount for extended film growth ($ T_{P} = \\SI{500}{\\degreeCelsius} $).\nFor $ T_P=\\SI{600}{\\celsius}$, the \\ce{Eu^{3+}} content increases with $t$ in the initial growth phase ($ t\\leq \\SI{5}{\\minute}$). For the extended growth ($ t \\geq \\SI{5}{\\minute} $) the content of \\ce{Eu^3+} decreases again and at $ t =\\SI{20}{\\minute} $ only \\ce{Eu^2+} is found in the spectrum. Increasing the temperature further to \\SI{700}{\\celsius}, and \\SI{800}{\\celsius}, the content of \\ce{Eu^{3+}} increases monotonically with time and no EuO growth phase is observed. Therefore, redox growth for stoichiometric EuO on \\ce{SrTiO3} is possible only in the range {$ T_P =\\, $\\SIrange{350}{600}{\\celsius}}.\n\n\\subsection{Thickness dependence}\nExemplary shown for a sample grown at $ T = \\SI{500}{\\degreeCelsius} ,$ we find the temporal dependence of the thickness, $ d(t) $ (see supplementary), to follow an exponential law of the form $ d(t) = d_0 ( 1 - \\exp( - t \/ \\tau ) ) $ (Fig.~\\ref{fig:fig2}(c)). Here $ d_0 $ is the final thickness and $ \\tau $ is the time constant. We apply this model to all samples (Fig~\\ref{fig:fig2}(d)) and find that both $ d_0 $ and $ \\tau $ increase monotonically for $ T \\leq \\SI{600}{\\degreeCelsius} $. Above this temperature a significant reduction is observed for both $ d_0 $ and $ \\tau $, simultaneous to the transition to \\ce{Eu^3+}-rich growth as observed in the chemical analysis.\n\nWe find that EuO rich films (i.e. $ \\SI{350}{\\degreeCelsius} \\leq T_{P} \\leq \\SI{600}{\\degreeCelsius}$) can be grown with up to $ d=\\SI{25}{nm} $ and $\\tau=$\\SI{25}{\\minute}. The fact that the time constants are in the range of many minutes allows a precise control of the film thickness by stopping the growth at a suitable time. The resulting chemical composition and thickness are compiled in a block diagram as a function of temperature (Fig.~\\ref{fig:fig2}(e)).\n\n\\subsection{Structural profile}\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig5_Structure_in_situ}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t\\textit{In situ} structure determination by LEED of the \\ce{SrTiO3}(001) substrate (a) and the redox-grown EuO film (b). We observe a \\ang{45} rotation of the EuO basis. (c-e) RHEED as a function of time shows a weak island growth mode and (f-h) structural transitions as a function of $ T_P $ from \\ce{Eu^0} to \\ce{Eu^2+} and \\ce{Eu^3+} respectively. The RHEED beam is parallel to \\ce{SrTiO3}(110) using \\SI{20}{keV} electrons.\n\t\t}\n\t}\n\t\\label{fig:fig3}\n\\end{figure}\n\n\nFor the \\textit{in situ} stuctural analysis we first present LEED (Fig.~\\ref{fig:fig3}(a)) for the \\ce{SrTiO3} substrate. We observe clear and sharp reflexes. The LEED pattern is four-fold symmetric, reflecting the symmetry of the perovskite lattice. Indicated by blue arrows is the basis of the reciprocal lattice. The EuO film ($ T_P = \\SI{600}{\\degreeCelsius} $, Fig.~\\ref{fig:fig3}(b)) is well-ordered and exhibits LEED reflexes, with a basis rotated by \\ang{45} to that of \\ce{SrTiO3}, the epitaxial relationship is {EuO(110)\/\\ce{SrTiO3}(100)}. The LEED reflexes are wider indicating a small degree of mosaicity.\n\nIn Fig.~\\ref{fig:fig3}(c-e) we show RHEED of the substrate, Eu deposition for $ t = \\SI{3}{\\minute} $ and $ t = \\SI{20}{\\minute} $ at $ T_{P} = \\SI{600}{\\degreeCelsius}$. The substrate shows RHEED streaks and two sharp spots on the Laue-circle, indicating a flat and well oriented substrate. At three minutes Eu deposition we observe RHEED streaks and a weak transmission pattern. For continued growth the transmission pattern dominates the RHEED reflexes, indicating a island type growth mode.\n\nThe RHEED reflexes as a function of $ T_{P} = $ \\SIlist{250;600;800}{\\degreeCelsius} are shown in Fig.~\\ref{fig:fig3}(f-h). For Eu-rich growth at \\SI{250}{\\degreeCelsius} we observe the hexagonal RHEED reflexes of the Eu-metal crystal. At elevated temperature the expected EuO related reflexes dominate the RHEED pattern, as described above. Finally at $ T_{P} = \\SI{800}{\\degreeCelsius} $ the structure shows the much bigger unit cell, and therefore shorter distances between RHEED reflexes, of the cubic \\ce{Eu2O3} crystal and a complex transmission pattern.\n\nIn conclusion, we observe clear structural transitions that are perfectly in line with the results from the chemical analysis by XPS. The EuO films grow by a redox-driven process epitaxially and are, except for a small surface roughness, flat. \n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig3_Structure}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t\\textit{Ex situ} X-ray analysis of a sample grown at \\SI{600}{\\degreeCelsius}. The reciprocal space map (a) shows the {\\ce{SrTiO3} (113)} and{ EuO (204)} peak in close proximity, a consequence of the epitaxial relation of {EuO(110)\/\\ce{SrTiO3}(100)}. The X-ray reflectometry shows that the film has a low roughness of \\SI{1.32}{nm} (b).\n\t\t}\n\t}\n\t\\label{fig:fig4}\n\\end{figure}\n\nIn Fig.~\\ref{fig:fig4}(a), we show the \\textit{ex situ} X-ray analysis, starting with a RSM of a sample grown at $ T_{P} = \\SI{600}{\\degreeCelsius} $. The Bragg peaks from the \\ce{SrTiO3}(113) and the EuO (204) peak are shown. The simultaneous observation of both reflexes is in line with the expected rotation of both lattices {EuO(110)$ \\parallel $\\ce{SrTiO3}(100)}, since the reflexes would otherwise be observed at an angle of \\SI{45}{\\degree}. The EuO(204) reflex has a rocking curve width of $ \\delta\\omega = \\SI{1.1}{\\degree} $ in line with the widened LEED reflexes. We calculate the lattice parameters for the in-plane and out-of-plane lattice constants and find that both \\ce{SrTiO3} and EuO ($ a_{\\ce{SrTiO3}} = \\SI{3.901}{\\angstrom} $ and $ a_{\\ce{EuO}} = \\SI{5.14}{\\angstrom} $) are close to the literature values ($ a_{\\ce{SrTiO3}} = \\SI{3.905}{\\angstrom} $ and $ a_{\\ce{EuO}} = \\SI{5.14}{\\angstrom} $ \\cite{karlsruhe_inorganic_2017}).\n\nIn Fig.~\\ref{fig:fig4} (b), we depict the corresponding XRR curve and a fit to the measured data points using the Parrat formalism. The grown film exhibits a total thickness of $ d = \\SI{16.7}{nm} $, while the roughness is $ a = \\SI{1.3}{nm} $. Hence we conclude, that the redox-growth process produces well oriented, i.e. \\SI{45}{\\degree} rotated, EuO films on \\ce{SrTiO3} with a small degree of roughness.\n\n\\subsection{Magnetic Properties}\nThe magnetic properties of samples grown at {$ T_{P} = $~\\SIrange{250}{800}{\\degreeCelsius}} are depicted in Fig~\\ref{fig:fig5}(a) as hysteresis loops and as function of temperature $M(T)$ in Fig.~\\ref{fig:fig5}(b). Since EuO is the only ferromagnetic component in the stack, it is expected, that the saturation magnetization $ M_S $ is proportional to the EuO thickness. Eu metal and \\ce{Eu2O3} exhibit small paramagnetic moments only. In the chemical analysis we showed that the EuO content and the thickness both depend on $ T_{P} $ in a complex manner. Consequently, the magnetic analysis cannot be expected to follow a simple linear temperature dependence\n\nAll $M(H)$ curves of $ T_{P} = $\\SIlist{350;400;500;600}{\\degreeCelsius} increase monotonically in $ M_{S} $, while the coercive field, $ H_{C} $, decreases simultaneously. Unordered and thin films are generally considered to cause an increase of the coercive field~\\cite{muller_thickness_2009}. This finding suggests, that the amount of ferromagnetic EuO increases as a function of the temperature and forms a more ordered lattice at higher temperatures.\n\nThe samples grown at $ T_{P} = $\\SIlist{700;800}{\\degreeCelsius} exhibit no hysteresis, which is in line with the chemical analysis that reveals a \\ce{Eu2O3}-rich (and therefore paramagnetic) composition.\n\nThe temperature dependence of the normalized magnetic moment is shown in Fig~\\ref{fig:fig5}(b). We find for the samples grown at $ T_{P} = $\\,\\SIrange{350}{600}{\\degreeCelsius} a Brillouin-like shape that closely follows a simulation with a Curie temperature of $ T_C = \\SI{69}{K} $, the literature value for EuO \\cite{matthias_ferromagnetic_1961}. Significant deviations are observed for samples grown at high temperature, where a paramagnetic behavior is measured and for $ T_{P} = \\SI{250}{\\degreeCelsius} $, where a pronounced metallic tail indicates the presence of Eu metal ions included in a EuO film~\\cite{suitsAnnealingStudyEuO1971, altendorf_oxygen_2011}.\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig4_Magnetism}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\n\t\t\tMagnetic properties of redox-grown \\ce{Eu}-oxide thin films on \\ce{SrTiO3} as (a) function of applied magnetic field and (b) temperature. EuO rich films, as found in the chemical analysis, exhibit a hysteresis, while high \\ce{Eu^{3+}} contents lead to paramagnetism. The inclusion of \\ce{Eu^{0}} as seen at $ T_{P} = \\SI{250}{\\degreeCelsius} $ causes a magnetic tail in the temperature dependence.\n\t\t\t}\n\t}\n\t\\label{fig:fig5}\n\\end{figure}\n\n\n\\section{Discussion}\n\nUnlike the classical EuO synthesis processes -- for which oxygen gas is supplied during a reactive MBE growth process -- here the oxide substrate itself acts as the supplier of oxygen: The \\ce{SrTiO3} substrate is reduced by the presence of Eu while the reactant oxidizes to EuO, \\ce{Eu3O4} or \\ce{Eu2O3}. These processes can be assessed with an Ellingham analysis (see supplementary). Indeed, we find that at equilibrium the most likely formed oxide is \\ce{Eu2O3}. However, we observe the formation of EuO in the intermediate temperature range. We therefore describe the complex redox growth of EuO\/\\ce{SrTiO3} as an interplay of three factors: (i) the kinetics of the oxygen anion reservoir from the substrate, (ii) the kinetics of the Eu metal on the surface of \\ce{SrTiO3} (and its concomitant re-evaporation) and (iii) the thermodynamics of the interface reactions.\n\n\nAs seen in Fig.~\\ref{fig:fig2}(c) the thickness of a redox-grown Eu oxide film is limited. This is surprising, as a normal diffusion type growth would have lead to a $ d \\propto \\sqrt{t} $ type behavior~\\cite{einsteinUberMolekularkinetischenTheorie1905}. We explain the thickness limit as a consequence of the insulating properties of the Eu oxides: The \\ce{O^{2-}} ions are charged and have to cross an insulating film of already grown Eu-oxides. This is well described in the context of a Mott-Cabrera type growth mode as was also found for the oxidation of Fe~\\cite{cabrera_theory_1949, kruger_room_1964}. In the Mott-Cabrera type growth, the potential that the \\ce{O^{2-}} ion is submitted to depends on the thickness of the film and its ion related resistivity. This explains the sudden reduction of $ d_0 $ and $ \\tau $ at the transition to \\ce{Eu^3+} rich growth in Fig.~\\ref{fig:fig2}(d), as the resistance is larger for the higher Eu oxides. Also, the re-evaporation of Eu from the surface can be expected to be much faster.\n\n\\section{Outlook}\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig6_other_oxides}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t (a) XPS analysis of the Eu$ 3d_{5\/2} $ core-level for redox-grown Eu oxides, (b) calculated thickness $ d $ and oxygen ion conductivity $ \\sigma $ for a selection of oxides grown at $ T_{P} = \\SI{500}{\\degreeCelsius} $ and $ t = \\SI{5}{\\minute} $. A qualitative correlation between $ d $ and $ \\sigma $ is observed.\n\t\t}\n\t}\n\t\\label{fig:fig6}\n\\end{figure}\n\nIn order to explore the redox-driven growth of EuO more generally, we study the initial growth ($ t = \\SI{5}{\\minute} $) at elevated temperatures ($ T_{P} = \\SI{500}{\\degreeCelsius} $) of Eu(O) on YSZ, \\ce{(LaAlO3){0.3}(Sr2TaAlO6){0.7}} (LSAT), \\SI{10}{nm} SrO grown on \\ce{SrTiO3}, \\ce{LaAlO3} (LAO), and MgO and compare it to the previous results of \\ce{SrTiO3}.\n\nAgain we study the Eu \\textit{3d}$ _{5\/2} $ core-level, as shown in Fig.~\\ref{fig:fig6}(a). In the case for MgO and LAO we observe \\ce{Eu^0} in the film even at this high $ T_P $. For SrO and LSAT, we observe only \\ce{Eu^2+}, whereas for MgO and YSZ we even observe some \\ce{Eu^3+}. This is surprising, as preparations on YSZ are often reported in adsorption limited growth conditions and the interfacial over-oxidation is not mentioned \\cite{sutarto_epitaxial_2009}.\n\n\nWe obtain the film thickness, $ d $, (see supplementary) for all redox-grown films and compare them to estimates for the ionic conductivities of the respective oxide substrate~\\cite{skinnerOxygenIonConductors2003,gunkelInfluenceChargeCompensation2012,rudolphUeberLeitungsmechanismusOxydischer1959,nguyenEffectOxygenVacancy2000,mitoffElectronicIonicConductivity1962,moosDefectChemistryDonorDoped2005}. We find, that $ d $ qualitatively scales with the ionic conductivities of the underlying substrate (Fig.~\\ref{fig:fig6}(b)). We attribute the high discrepancy between SrO conductivity and $ d $ of redox-grown EuO to the fact, that the SrO thin film grown on a \\ce{SrTiO3} will have a higher oxygen mobility than a bulk crystal, due to crystal defects.\n\nFrom this analysis we are able to differentiate substrates into active substrates, with a relevant redox process and passive substrates, where additional oxygen needs to be supplied. We find that \\ce{SrTiO3} is the most active substrate for redox-growth. LAO, SrO thin films, LSAT and YSZ might pose as suitable templates for thin films and are expected to lead to thinner EuO over-layers than would be expected for \\ce{SrTiO3}, while MgO and LAO act as passive substrates. \n\n\\section{Conclusions}\nIn summary, we report a novel route for the synthesis of stoichiometric and single-crystalline EuO films by supplying no gaseous oxygen. Instead, utilizing the oxidic substrate as source of oxygen is the key to a reliable and simplified preparation scheme. We have identified the parameter window in which\nEuO can be grown on \\ce{SrTiO3}(001) and reduce the complexity of the typically applied distillation growth mode. \n\nThe prepared films show the expected chemical, structural and magnetic properties of stoichiometric EuO in the temperature range $ T_{P} = $\\,\\SIrange{350}{600}{\\degreeCelsius}. By changing the growth temperature, the total thickness of the EuO film can be varied from \\SIrange[range-phrase=--]{9}{25}{nm} for $ t = \\SI{60}{\\minute} $, and by stopping the growth earlier the thickness can be varied freely. Thus the redox-driven EuO growth method allows to prepare thicker films compared to a topotractic synthesis mechanism. All redox-grown EuO films grown in the suitable parameter window exhibit bulk ferromagnetic properties with no metal inclusions. With regard to the structural properties, we have observed flat and well oriented films with the epitaxial relationship of {EuO(110)$ \\parallel $\\ce{SrTiO3}(100)}. \n\nFinally, we demonstrate that the redox-driven EuO growth scheme can be successfully applied not only to \\ce{SrTiO3}, but also to other oxide substrates. We believe, that the universality of a redox-controlled oxide thin film synthesis may also open up exciting perspectives for other topical oxide materials and their integration into complex oxide heterostructures.\n\n\n\\begin{acknowledgments}\nWe thank David~N.~M\u00fcller for fruitful discussions about solid state oxide chemistry. We thank O. Petracic and the J\u00fclich Centre for Neutron science for providing measurement time at the magnetometers.\nM.~M. acknowledges financial support from HGF under contract No. VH-NG-811.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{intro}\n\nSpin-exchange pumped NMR gyroscopes~\\cite{Thrasher2019,thrasher2019PRApp,Walker2016,Kornack2005,Jiang2018,Karwacki1980} use the precession of spin-polarized nuclei to measure~rotation. \nA vapor cell contains one or more isotopes of noble gas atoms and an~alkali-metal~vapor. The alkali-metal vapor is optically pumped using circularly polarized near-infrared laser light. Spin-exchange collisions between the polarized alkali-metal and noble gas atoms polarize the noble gas nuclei.~When subject to a magnetic (or ``bias'') field, the polarized species undergo Larmor precession. By~simultaneously measuring the Larmor precession of each entity, magnetic field correlations can be removed, thereby increasing sensitivity to non-magnetic spin-dependent phenomena, such as inertial~rotation\n\nThe precession of a polarized gas about a constant bias field constitutes an inertial reference frame, i.e., the gas has (ideally) no way of knowing whether or not its container is rotating. If the precession of the polarized gas is measured relative to a fixed point in the laboratory frame, such as with a pick-up~coil, then rotation of the pick-up coil about the bias field will change the measured rate of precession of the polarized gas\n\nThe application of polarized gases as inertial sensors is promising because of their low intrinsic~noise, miniaturize-ability, low power consumption, insensitivity to acceleration, and intrinsic scaling from experimental observable to rotation that is independent of experimental parameters~\\cite{Donley2013,Brinkmann62}. Interest in novel navigation systems which operate in global positioning system (GPS) denied environments has increased as of late. To date, the United States, Russia, China, and India have successfully demonstrated the capability to destroy their own satellites~\\cite{Chaudhury2019}. Such capability could potentially be used to remove GPS satellites. Other GPS denied environments include subterranean and submarine travel. One obstacle to harnessing geothermal energy resources is the limited ``in-hole'' navigation of horizontal drilling rigs. The future development of autonomous vehicles also relies on robust navigation, including in GPS denied environments. NMR gyroscopes will likely first be used to provide long-term corrections to another miniaturized gyroscope whose short term noise performance is superior but whose long term drift is inferior to the NMR gyroscope. A proof-of-concept experiment along these lines was recently performed using a classical and quantum accelerometer~\\cite{Cheiney2018}. \n\nImportant sources of systematic errors in spin-exchange pumped NMR gyroscopes are longitudinal (i.e., parallel to the bias field) spin-exchange fields produced by polarized alkali-metal atoms or noble gas nuclei~\\cite{Thrasher2019,Bulatowicz2013,Walker2016}. Such longitudinal spin-exchange fields are not identical for all noble gas species, and therefore cannot be removed in the same way as classical magnetic field correlations. Longitudinal spin-exchange fields are well suppressed by synchronous spin-exchange optical pumping~\\cite{Thrasher2019,Korver2015}, wherein the alkali-metal vapor is optically pumped transverse to a low duty cycle pulsed bias field. Each bias field pulse produces $2\\pi$ precession of the alkali-metal atoms such~that, despite having a gyromagnetic ratio in excess of $10^3$x that of the noble gas nuclei, the alkali-metal atoms can effectively co-precess with the noble gas nuclei. \n\nThis manuscript describes a $^{131}$Xe-$^{129}$Xe synchronous spin-exchange optically pumped NMR gyroscope which uses modulation of the alkali-metal ($^{85}$Rb) polarization to drive both Xe isotopes' NMR simultaneously. The Rb is also used as an embedded magnetometer to detect the Xe precession. Compared to our recent work~\\cite{Thrasher2019,thrasher2019PRApp} describing NMR excitation by modulating the bias field, polarization modulation (PM) further suppresses the influence of time-averaged longitudinal Rb spin-exchange fields~\\cite{Korver2015} by moving such fields from DC to AC. We study how modulation of the embedded Rb magnetometer causes signal mixing on our detection which leads to an effective change in scale factor~\\cite{ThrasherThesis}.\n\n\\subsection{Bloch Equation} \\label{BEsec}\nThe spin dynamics of the polarized noble gas transverse ($K_+ = K_x+iK_y$) and longitudinal ($K_z$) spins are described by the Bloch equations \n\\begin{subequations}\n\\begin{eqnarray}\n{dK_+\\over dt}=-(\\mp i\\Omega_z^K+\\Gamma_2)K_+ +\\Gamma_S^K S_+ + \\mp i\\Omega_+^K K_z,\\label{BE1}\n\\\\\n{dK_z\\over dt}=\\mp (\\Omega_y^K K_x-\\Omega_x^K K_y)-\\Gamma_1 K_z,\\label{BE2}\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\mathbf{\\Omega}^K$ is the Xe resonance frequency arising from both the Larmor precession $\\gamma^K \\mathbf{B}$ and from rotation $\\omega^R \\mathbf{\\hat{z}}$, $\\Gamma_1$ ($\\Gamma_2$) is the longitudinal (transverse) relaxation rate, $\\mathbf{S}$ is the Rb polarization, and~$\\Gamma_S^K$ is the spin-exchange rate constant. Let superscript $a$ represent $^{129}$Xe and superscript $b$ represent $^{131}$Xe. Since $\\gamma^a <0$ and $\\gamma^b > 0$ we find it useful to write the gyromagnetic ratio $\\gamma$ (and hence $\\Omega$) as a~positive~value. The sign is written explicitly out front (top sign is for $a$, bottom sign is for $b$). \n\nWe null the transverse fields experienced by the Xe, including the spin-exchange field $b_S^K S_+$ (where $b_S^K$ is the spin-exchange coefficient characterizing the influence of the Rb polarization on the Xe)~\\cite{Korver2015}. This suppresses $K_z$ such that $K_+$ is much more sensitive to $\\Omega_z^K$ than $\\Omega_+^K$. Assuming our transverse optical pumping produces negligible $S_z$, the Xe resonance frequencies are then $\\Omega_z^K=\\gamma^K(B_{z0}+B_p)\\mp\\omega^R$, where $B_{z0}$ is the stray field inside our magnetic shields and $B_p$ is the field from the bias pulses. Since the Xe precess on the order of only $2\\pi\/10^3$ radians per pulse, the pulsed field can be approximated as a continuous field with $B_p= \\omega_p\/\\gamma^S$ for pulsing frequency $\\omega_p$.\n\nThe transverse Rb polarization ($S_+$) is used to polarize the Xe via spin-exchange optical pumping, and the longitudinal Rb polarization ($S_z$) is used to detect the Xe precession. For small Rb precession ($\\Omega^S<< \\Gamma'$), the time-average solution to the Bloch equation for the Rb polarization can be expanded~as\n\\begin{equation} \\label{Seq}\n\\mathbf{S} = {\\mathbf{R}\\over \\Gamma'} +{\\mathbf{\\Omega}^S \\times \\mathbf{R} \\over\\Gamma'^2}+{\\mathbf{\\Omega}^S \\times \\mathbf{\\Omega}^S \\times \\mathbf{R}\\over\\Gamma'^3}+..., \n\\end{equation}\nwhere $\\mathbf{R}= \\int d\\nu \\Phi(\\nu) \\sigma(\\nu) \\mathbf{p} \/A$ is the pumping rate, $\\sigma(\\nu)$ is the cross section of the atomic transition, $\\int d\\nu \\Phi(\\nu)=P\/h\\nu$ is the total photon flux for a beam of power P incident on area A, $\\mathbf{p}$ is the photon spin, $\\Gamma'$ is the total relaxation rate (including pumping), and $\\mathbf{\\Omega}^S = \\gamma^S (\\mathbf{B}_0+b_a^S \\mathbf{K}^a+b_b^S \\mathbf{K}^b)$. The~magnetic field experienced by the Rb includes the stray fields $\\mathbf{B}_0$ and the spin-exchange fields ($b_K^S$ is the spin-exchange coefficient characterizing the influence of the Xe polarization on the Rb), but it does not include the bias~field. This is because the bias field is applied as a sequence of low duty cycle pulses, the~area of which correspond to $2\\pi$ precession of the Rb atom. These equations assume negligible back polarization from the Xe to the Rb~\\cite{Limes2018}, and that $\\mathbf{K}$ precesses slowly such that $S_z$ responds adiabatically. We optically pump along $\\hat{x}$ such that $\\mathbf{R}=R(t)\\hat{x}$, and the solution to Equation~(\\ref{Seq}) is\n\\begin{subequations} \n\\label{allequations} \n\\begin{eqnarray}\nS_+ = {R(t)\\over\\Gamma'}+i{R(t) \\Omega_z^S\\over \\Gamma'^2} = {R(t) \\over \\Gamma'}e^{i\\epsilon_z},\n\\\\\nS_z = -{R(t)\\over \\Gamma'^2}(\\Omega_y^S -{\\Omega_z^S\\over \\Gamma'} \\Omega_x^S)={-R(t)\\over \\Gamma'^2 }\\text{Im}[\\gamma^S b_K^S K_+e^{-i\\epsilon_z}],\n\\end{eqnarray}\n\\end{subequations}\nwhere \n\\begin{equation} \\label{ezdef}\n\\epsilon_z = \\tan^{-1}({ S_y \\over S_x}) \\equiv \\tan^{-1}({ B_{z0} \\over B_w})<<1\n\\end{equation}\n\nis the magnetometer phase shift. $B_w=\\Gamma'$\/$\\gamma^S$ is the magnetic width of the magnetometer. A stray $B_z$ effectively rotates the quantization axis of the Rb magnetometer thus causing a phase shift ($\\epsilon_z$) in the measurement of a rotating $B_\\perp$.\n\nThe Xe NMR can be driven by modulating the transverse Rb polarization near each isotope's resonance frequency while the bias pulsing frequency is kept fixed. Such an excitation scheme is desirable as it effectively AC couples $S_z$. Hence, we Fourier expand the Rb polarization as $S_+(t) = e^{i\\epsilon_z} \\sum_{p,q} s_{pq} e^{i(p\\omega_1+q\\omega_2) t}$ where $s_{pq}$ is the Fourier coefficient of $S_+$ at $p\\omega_1+q\\omega_2$. With the substitution $K_{+} = K_{\\perp}e^{\\pm i(\\omega_d t +\\delta)}$, where $\\delta$ is the phase shift of the Xe relative to the phase of $S_+$ and $\\omega_d$ is the drive~frequency, we find the real part of Equation ~(\\ref{BE1}) to be\n\\begin{linenomath}\n\\begin{equation}\n{d K_{\\perp}\\over dt} =-\\Gamma_2 K_{\\perp}+\\Gamma^K_S \\sum_{p,q} s_{pq} \\cos[(\\omega_d+p\\omega_1+q\\omega_2)t + \\delta \\pm \\epsilon_z].\n\\end{equation}\n\\end{linenomath}\n\nThe steady state solution of this equation is $K_{\\perp} = \\Gamma^K_S s_{pq} \/\\Gamma_2$ when $\\omega_d$ is chosen to satisfy the resonance condition $\\omega_d=-p_0\\omega_1-q_0\\omega_2$ for some $p_0$ and $q_0$ and when $\\delta,\\epsilon_z<<1$. Similarly, the~imaginary part of Equation~(\\ref{BE1}) (once again for $\\delta,\\epsilon_z<<1$) is\n\\begin{linenomath}\n\\begin{equation}\n{d \\delta \\over dt}= - \\Delta - \\Gamma_2 (\\delta \\mp \\epsilon_z),\n\\end{equation}\\label{fundgyro}\n\\end{linenomath}\nwhere $\\Delta =\\omega_d- \\gamma B_z\\pm \\omega^R$. The sign in front of $\\epsilon_z$ is isotope dependent because the Xe isotopes precess in opposite directions. We can solve this equation in the Fourier domain to find\n\\begin{linenomath}\n\\begin{equation}\n\\tilde{\\delta} = -{\\tilde{\\Delta} \\mp \\Gamma_2 \\tilde{\\epsilon}_z \\over \\Gamma_2+i\\omega},\\label{fundgyro1}\n\\end{equation}\n\\end{linenomath}\nwhere the notation $\\tilde{f} = f(\\omega)$ is used. \n\nBy monitoring $S_z$, we can obtain measurements of $\\delta$ for each Xe species. These measurements of $\\delta$ are used to extract the Xe resonance frequencies. We perform comagnetometry using these Xe resonance frequencies to remove magnetic field correlations and arrive at a measurement of inertial~rotation. In~Section~\\ref{apparatus} we detail the implementation of our Xe excitation scheme, and in Section~\\ref{result} we demonstrate the performance of our system.\n\n\\section{Materials and Methods} \\label{apparatus}\nAn 8 mm inner diameter cubic Pyrex cell filled with 68 Torr enriched Xe and 85 Torr N$_2$ with a~hydride coating~\\cite{Kwon1981} is mounted in a ceramic housing with holes for laser light to enter the cell. The~ceramic housing has four symmetric faces which fit together like a jigsaw puzzle (and so we call these ``jig'' heaters). On each face are printed conductive traces through which AC current at $\\sim$150 kHz is passed to heat the ceramic. The conductive traces are arranged to produce minimal stray magnetic fields including gradient magnetic fields. The $\\sim$1 mm gap between the vapor cell and ceramic heating jig is shimmed with a 1.5 mm thermally conductive and slightly compressible gap fill (model: TG~977, manufacturer: T-Global Technology, see Figure~\\ref{cellpics}). The ceramic is wrapped with aerogel (a high temperature insulating material) and secured with Kapton tape and then fitted into a 3D printed (high~temperature nylon) cartridge with holes to allow laser light to enter the cell. The~compressible nature of the aerogel produces a friction fit keeping the ceramic jig structure fixed within the cartridge. The cartridge itself is mounted in a 3D printed (ABS plastic) rig. This rig has support arms which extend out three of the magnetic shield portholes described below. These support arms are secured directly to an optical table on which the entire apparatus is mounted.\n\\vspace{-6pt}\n\\begin{figure}[h\n\\centering\n\\includegraphics[width =0.8\\linewidth]{cellpics.pdf}\n\\caption[Vapor cell heater and cartridge mount photographs.]{Vapor cell heater and cartridge mount photographs. (\\textbf{a}) Four ceramic jig heater sides with aerogel pillows attached to outer faces. AC power enters via MMCX connectors at the bottom of each jig heater. (\\textbf{b}) 1 cm$^3$ vapor cell with stem tucked into fiberglass insulation. A temperature sensor sits between the insulation and outer cell wall. (\\textbf{c}) Cartridge with cell installed. (\\textbf{d}) Gap fill shim on one jig heater face. (\\textbf{e}) View looking into the jig heater with the cell installed. Note the four pieces of gap fill between each outer cell wall and the jig heater inner faces.}\\label{cellpics}\n\\end{figure}\n\nThe three layer $\\mu$-magnetic shield we use is cylindrical with eight access ports (two along the axis of symmetry and six oriented tangentially). Since the tangential access ports are not located halfway between the two ends of the cylindrical shields, and since pump laser light must enter through the~ports, we must place our cell such that it is not equidistant from the end caps. The asymmetry in distance to end caps informs the design of our bias pulsing coil set. To minimize coupling to the shield end caps and maximize uniformity across the volume of the cell, the pulsing coil set consists of two pairs of square Helmholtz coils with differing side lengths wound in series with opposite~polarity. The purpose of the ancillary counter wound coils is to suppress the field produced by the coil set at the nearest end cap. See \\cite{KorverThesis} for specific design details.\n\nThe bias field requires short pulses ($<$5 $\\upmu$sec) of $\\sim$1 Ampere peak current. The circuit used to drive the pulsing coil was custom-made and is described in \\cite{KorverThesis}. The circuit used to drive the shim coils was also custom-made and is described in \\cite{WyllieThesis}.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width =1\\linewidth]{setupv4.pdf}\n\\caption[Experimental setup for PM excitation]{Experimental setup for PM excitation (not to scale). DM: dichroic mirror, Pol: polarizer, HWP: half wave plate, QWP: quarter wave plate, PBS: polarizing beam splitter, WP: Wollaston prism, PD:~photodiode, EOM: electro-optic modulator, TS: two-axis translation stage with lens. The three-axis magnetic shim and pulsing coils are not shown. The setup fits on a four foot square optical table.}\\label{polmodsetup}\n\\end{figure}\n\nTo perform optical pumping of the Rb, the outputs of two distributed feedback laser diodes tuned near the Rb D1 transition (one on either side of resonance) are overlapped (see Figure~\\ref{polmodsetup}). This is accomplished by polarizing pump A so that it is mostly transmitted by a polarizing beam splitter (PBS) and pump B so that it is mostly reflected by the same PBS. The combined beam is then sent through a quarter wave plate and then separated into two beams using a PBS once again. The orientation of the quarter wave plate is chosen such that both pump A and pump B have half their power in each output beam. A telescope is used to couple the beams into individual EOMs. Prior to each EOM is a polarizer and half wave plate. The polarizer ensures that the light incident to the EOM crystal is purely~linear. The half wave plate is used to align the light polarization relative to the EOM crystal~axis. The maximum and minimum voltages of the EOM drive waveform are chosen to produce $\\pm\\lambda\/2$ retardance. The quarter wave plate at the output of the EOM converts the EOM output at $V_{max}(V_{min})$ to be $\\sigma^+(\\sigma^-)$. The collimated output of each EOM is coupled into the vapor cell from opposing directions. The overall beam resizing is set to somewhat overfill the aperture of the ceramic~heater. Fine tuning of each pump beam's pointing is controlled using long focal length lenses mounted on two axis translation stages just outside the magnetic shield. The position of each steering lens is chosen to optimize the magnetometer gain. The power and detuning of each pump laser is chosen to approximately cancel the AC Stark effect. While we could also suppress the AC Stark effect by pumping with a single laser tuned on resonance, we would not get good spatial uniformity of the Rb polarization without significantly increasing our laser power due to optical thickness effects. To~detect $S_z$, approximately one mW of linearly polarized light from the output of a third distributed feedback laser diode, tuned near the Rb D2 line, is directed through the center of the cell and parallel to $\\hat{z}$ onto a balanced Faraday detector.\n\n\nWe choose to modulate the Rb polarization by switching between just two polarization states, $\\sigma^{\\pm}$. We modulate the x-component of $\\mathbf{R}$ according to\n\\begin{equation}\nR(t) = R_{0}\\;\\text{sign}[\\cos\\left({\\omega_d^a+\\omega_d^b\\over 2}t+2\\cos({\\omega_d^a-\\omega_d^b\\over 2}t)\\right)].\n\\end{equation}\n\nThis waveform is advantageous\nin that the smallest separation between reversals is larger than the finite response times of the EOMs and of the optical pumping of the Rb atoms. Modulating the polarization as a sum of two sine waves would also keep the modulations within the bandwidth of the EOMs and the Rb magnetometer, but we find that producing such a PM waveform in our optically thick vapor cell is very challenging. We use the same modulated cosine waveform to apply a~compensation field, the amplitude of which is set to cancel the spin-exchange field experienced by the Xe from the Rb. Doing so suppresses the production of $K_z$ and narrows the NMR linewidths, $\\Gamma_2$~\\cite{Korver2015}.\n\nFrom Equation~(\\ref{allequations}) we find that the detected longitudinal Rb polarization $S_z$ is \n\\begin{equation}\\label{Szdet1}\nS_z = -{R(t)\\over \\Gamma'^2}[b_a^S K^a_{\\perp} \\sin(\\delta^{a}+\\alpha^a - \\epsilon_z)+ b_b^S K^b_{\\perp} \\sin(\\delta^{b}+\\alpha^b + \\epsilon_z)],\n\\end{equation}\nwhere $\\delta=\\phi-\\alpha$ is the difference between the drive phase $\\alpha=\\int dt \\omega_d$ and the Xe precession phase $\\phi$ for each isotope, $\\epsilon_z$ is the magnetometer phase shift, and $R\\sim S_+$ is the optical pumping rate of the Rb. Although lock-in detection can be accomplished on $S_z$ as it appears in Eq.~\\ref{Szdet1}, the phase sensitivity is diminished due to the presence of $R(t)$ which effectively mixes some Xe signal to DC. A more effective approach is to \"rectify\" the $S_z$ signal such that $R(t)$ is removed. This is accomplished by multiplying $S_z$ by $R^{-1}(t)$. The resulting signal is then sent into two separate lock-in amplifiers, each referenced to a different isotope's drive frequency. From these demodulations, we arrive at the measured phases $\\delta^a-\\epsilon_z$ and $\\delta^b+\\epsilon_z$.\n\n\\section{Results} \\label{result}\n\\vspace{-6pt}\n\\subsection{NMR Excitation and Detection}\n\n\nFigure~\\ref{polmodsqsigs} shows the time series of the measured $S_z$ with and without rectification when we drive both isotopes near resonance simultaneously using $S_+$. We see that rectification reveals the sinusoidal precession of each isotope. The outlying data on the rectified signal, which occur when the polarization is reversed, are due to the finite response time of the Rb magnetometer. Although rectification collects the many Xe signal sidebands into the two Xe carrier frequencies (see power spectrum on right), it~also maps 1\/$f$ detection noise on $S_{z}$ to the carrier frequencies. The mapping of low-frequency $S_{z}$ noise to the carrier frequencies can be prevented by high-pass filtering $S_z$ with a 1 Hz corner prior to rectification. \n\n\\begin{figure}[t\n\\centering\n\\includegraphics[width=0.75\\linewidth]{myrect2.pdf}\n\\caption[Square wave PM signals]{Measured square wave PM signals. Top: $S_+(t)$. Middle: $S_z(t)$. Bottom: Rectified $S_z(t)$. The plot on the right shows the amplitude spectral density with (red) and without (blue) rectification.}\\label{polmodsqsigs}\n\n\\vspace*{\\floatsep}\n\n\\includegraphics[width=0.75\\linewidth]{polmodnmr5.pdf}\n\\caption[NMR of each species]{Measured NMR lineshapes of each species. Filled circles are $K_x$, open circles are $K_y$, and lines are Lorentzian fits.}\\label{polmodnmr}\n\\end{figure}\n\n\nFigure~\\ref{polmodnmr} shows the detected NMR signals for each isotope. These data were acquired by driving one isotope on resonance while varying the other isotope's drive frequency and recording its $K_x$ and $K_y$ derived using demodulation. We see that the lineshapes are nearly Lorentzian with linewidths (half-width at half-max) of $\\sim$15 mHz and amplitudes of approximately 60 $\\upmu$G. The on-resonance amplitudes are in agreement with estimates similar to those outlined in~\\cite{Walker2016}. The implied $T_2^a$, $T_2^b$ from the fits are in good agreement with independent measurements of each isotope's $T_1$. The $\\sim$15 mHz linewidths are only possible because of two features of our experiment; (i) the use of a Rb hydride cell coating (without which $T_2^b$ would be substantially shorter and $T_2^a$ much longer)~\\cite{Kwon1981}, and (ii) the application of a magnetic compensation field $B_x$ that cancels the Rb spin-exchange field experienced by the Xe. \n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.75\\linewidth]{polmodpn3.pdf}\n\\caption[Phase noise]{Measured phase noise of each species. Traces labeled ``Real'' are recorded when the Xe isotopes are excited on resonance. Traces labeled ``Artificial'' are recorded when the Xe isotopes are both driven off resonance (not excited) and an AC $B_y$ is applied to the magnetometer at the off-resonance drive frequencies.}\\label{polmodpn}\n\\end{figure}\n\nFigure~\\ref{polmodpn} demonstrates the amplitude spectral density of the phase noise measured for each isotope under simultaneous resonant excitation conditions. We see that for frequencies less than 1 Hz the spectra are dominated by $1\/f$ noise which is about $\\rho=\\gamma^a\/\\gamma^b$ greater for isotope $a$ (black traces) than for isotope $b$ (red traces) suggesting the dominant source of $1\/f$ noise is magnetic in~nature. Also shown is the phase noise measured when the Xe isotopes are driven off resonance (not excited) and a so-called ``artificial'' Xe signal is applied as an AC field, $B_y$. The amplitudes of this signal $A^a\\sin(\\omega_d^a t)+A^b\\sin(\\omega_d^b t)$ are chosen to produce the same size magnetometer signal as real Xe. We~note that this artificial Xe signal is planar, unlike the real Xe signal which rotates. The artificial signal allows us to measure the signal-to-noise ratio (SNR) of the Rb magnetometer. These signals do not show $1\/f$ dependence because, unlike the real Xe phase, the SNR of the magnetometer does not depend on the bias magnetic field to first order. The detection phase noise from each artificial Xe measurement is uncorrelated and limits the possible field suppression when performing comagnetometry. The SNR for each isotope is $\\sim$5000$\\sqrt{\\text{Hz}}$. While the detection of artificial Xe is insensitive to $1\/f$ bias magnetic field noise, $1\/f$ $S_z$ noise can still be mapped to the artificial Xe frequencies via rectification, and so the addition of a 1 Hz high-pass filter prior to rectification was essential for realizing such an SNR.\n\n\n\\subsection{Comagnetometry}\n\n\nWe perform comagnetometry by subtracting magnetic field correlations between the two isotope's precession frequencies. Since our device measures phase, we need to know or measure the transfer function from phase to frequency. In Section~\\ref{BEsec} we derived the transfer function (see Equation~(\\ref{fundgyro1})). We measured the transfer function of each isotope by recording the response of the measured phase $\\delta$ to sinusoidal modulation of $B_z$. Figure~\\ref{polmodtf} shows the measured transfer function for isotope $a$. We~use a~chirp waveform to modulate the bias field $B^{mod}_z(t)=B_0\\sin(2\\pi[e^{t\/T_2}-1 - t\/T_2])$ (where~$T_2 = 1\/2\\pi \\Gamma_2$), the time series of which is shown in the inset of Figure~\\ref{polmodtf}. This modulation waveform allows us to measure the transfer function from 0.002 to 0.1 Hz with good SNR in a single data acquisition. The~transfer function is the ratio $\\gamma^K\\tilde{B}_z^{mod}\/\\tilde{\\delta}$. We fit the data according to Equation~(\\ref{fundgyro1}) and find excellent agreement with the linewidth derived from the fits in Figure~\\ref{polmodnmr}.\n\n\\vspace{-6pt}\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.70\\linewidth]{transfunc4.pdf}\n\\caption[Transfer function of $^{129}$Xe]{Transfer function of $^{129}$Xe. Inset depicts the normalized chirp waveform used to modulate the bias field. Black crosses are measured data, while the blue line is a fit of the form $\\sqrt{(\\Gamma_2^{a})^2+f^2}$.}\\label{polmodtf}\n\\end{figure}\n\\unskip\n\nAlthough conversion from phase to frequency for the measured Xe phases is possible using a~measured transfer function, feedback is desirable because (in the high gain limit) the performance of the comagnetometry becomes insensitive to changes in the transfer function. We used the measured precession phase of isotope $a$ to stabilize the bias field and the measured transfer function of isotope $b$~to convert its measured phase noise to frequency noise. Under such conditions the frequency noise of isotope $b$ is proportional to rotation. We apply a feedback field, $B_f$, to hold the measured phase of isotope $a$, $(\\delta^a-\\epsilon_z)$, equal to zero. The field stabilization can then be written $\\tilde{B}_f = \\tilde{G}(\\tilde{\\delta}^a-\\tilde{\\epsilon}_z)$. In the high gain limit, this becomes\n\\begin{linenomath}\n\\begin{equation}\n\\lim_{G \\to \\infty}\\tilde{B}_{tot} = {1\\over \\gamma^a}(\\tilde{\\omega}^R+i\\omega\\tilde{\\epsilon}_z),\n\\end{equation}\n\\end{linenomath}\nwhere $B_{tot}=B_{z0}+B_p+B_f$, and where we made substitutions using Equation~(\\ref{fundgyro1}) and $\\tilde{\\Delta}^a =- \\gamma^a \\tilde{B}_z- \\tilde{\\omega}^R$ (since $\\omega_d^a$ is held constant). The measured phase of isotope $b$, $(\\delta^b+\\epsilon_z)$, converted to frequency is\n\\begin{linenomath}\n\\begin{equation}\n\\tilde{\\omega}^b \\equiv (\\tilde{\\delta}^b+\\tilde{\\epsilon}_z)(\\Gamma_2^b+i\\omega)= (\\rho^{-1}+1)(\\tilde{\\omega}^R+i\\omega\\tilde{\\epsilon}_z).\n\\end{equation}\n\\end{linenomath}\n\nWe see that in the high gain limit, when correcting the bias field to keep the measured phase of isotope $a$ equal to zero, the rotation is simply $\\tilde{\\omega}^R = \\rho\\;\\tilde{\\omega}^b\/(1+\\rho)$ assuming $\\tilde{\\epsilon}_z$ is negligible. \n\n\n\nThe best performance we observed with bias field feedback activated is shown in Figure~\\ref{polmodcomag}. The~feedback consisted of two analog inverted zero gain stages. The influence of bias field feedback is dramatic from 0.1 to 200 mHz. The servo suppresses $\\tilde{\\omega}^a$ to below 1 $\\upmu$Hz\/$\\sqrt{\\text{Hz}}$ at low frequency, which is nearly $10^4$x less than the open loop noise. Because magnetic noise dominates each isotope's precession, servoing the measured phase of isotope $a$ also greatly suppresses $\\tilde{\\omega}^b$. We observe at least 100$\\times$ improvement in $\\tilde{\\omega}^b$ due to feedback. The modified Allan deviation~\\cite{Allan1981} suggests a rotation ARW sensitivity of $\\sqrt{2}\\;15\\;\\upmu$Hz\/$\\sqrt{\\text{Hz}}{\\rho\\over 1+\\rho}\\sim 16\\; \\upmu$Hz\/$\\sqrt{\\text{Hz}}$ and a rotation bias instability of 1 $\\upmu$Hz ${\\rho\\over 1+\\rho}\\sim 800$~nHz. The size of ARW is within a factor of 3 of the ratio of the measured linewidths divided by the SNRs (shown in Figure~\\ref{polmodpn}). The peaks in the Allan deviation at 15 and 100 s of integration are due to low-frequency narrow-band large-amplitude noise peaks in $\\tilde{\\omega}^b$ which we attribute to the PM~waveform. The bias instability is limited by $\\tau^{1\/2}$ trending noise of unknown origin. We find that feedback causes the measured phase of isotope $b$ to trend linearly in time. The source of this frequency bias is uncertain. Although the trend is very stable over the course of a data run, the trend is not consistent between data runs. The bias instability demonstrated in Figure~\\ref{polmodcomag} was difficult to reproduce. Typically, the bias instability we measured was a few $\\upmu$Hz.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.9\\linewidth]{comag3.pdf}\n\\caption[Comagnetometry noise and stability]{Measured comagnetometry noise and stability. Left: the amplitude spectral density of frequency noise. The cross marks indicate open loop frequency noise. The solid lines are frequency noise when the measured phase noise of isotope $a$ is used to stabilize bias field. Right: modified Allan deviation of $\\omega^b$. Filled circles are measured data. Solid line shows the quadrature sum of $7\\times 10^{-5}\\tau^{-1}$, $15\\;\\upmu\\text{Hz}\/\\sqrt{\\text{Hz}}\\tau^{-1\/2}$, and $30\\;\\text{nHz}\\sqrt{\\text{Hz}}\\tau^{1\/2}$ trends.}\\label{polmodcomag}\n\\end{figure}\n\n\\subsection{Cross Talk}\nOnce we measured the stability of the PM comagnetometer, we desired to know the fidelity with which our detection separated signals from the two Xe isotopes. It is possible that the detection channel designed to measure isotope $a$'s phase was really measuring a linear combination of isotope $a$~and $b$'s phases. If such ``cross talk'' were present then the scale factor (or how we convert the measured precession frequencies to rotation) would change~\\cite{ThrasherThesis}. Suppose there exists cross talk in both channels such that $\\omega^a = \\gamma^a B_z + \\beta \\omega^b - \\omega^R$ and $\\omega^b = \\gamma^b B_z +\\beta' \\omega^a + \\omega^R$ where $\\beta$ and $\\beta'$ represent the cross talk between detection channels. Solving for $\\omega^R$ we find\n\\begin{linenomath}\n\\begin{equation}\n\\omega^R = {\\omega^b (\\rho+\\beta')- \\omega^a (\\rho \\beta +1) \\over 1+\\rho},\n\\end{equation} \\label{polmodcteq}\n\\end{linenomath}\nwhere if $\\beta=\\beta'=0$ we return to the expected expression for rotation without cross talk. Non-zero cross talk is undesirable because the accuracy with which it is known (or measured) limits the accuracy of conversion from measured precession frequencies to rotation (or any other non-magnetic spin-dependent interaction). A measurement of cross talk is vital since an important alleged feature of our comagnetometer is that it has a scale factor which is determined solely by $\\rho$.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.65\\linewidth]{pmct3.pdf}\n\\caption[Measurement of cross talk]{Measurement of cross talk for detection with rectification. The Xe drive frequencies are set such that the two isotopes are not driven on resonance at the same effective bias field magnitude. The~in-phase (filled symbols) and out-of-phase (un-filled symbols) components of each isotope's detection channel (black and red correspond to isotopes $a$ and $b$, respectively) are shown as the bias pulsing frequency is varied. We see that when isotope $a$ is resonant ($f_{2\\pi}\\sim 15.8$ kHz) the signal on isotope $b$'s detection channel is not flat despite isotope $b$ being driven many linewidths off resonance. Similarly, when isotope $b$ is resonant ($f_{2\\pi}\\sim 16$ kHz) the signal on isotope $a$'s detection channel is not flat despite isotope $a$ being driven many linewidths off resonance.}\\label{polmodmct}\n\\end{figure}\n\nWe characterize the cross talk present in our detection channels by looking for changes in the detected signal for one isotope when the drive of the other isotope is changed. We do this by detuning one isotope's drive frequency by $\\sim300$ mHz and then scanning the bias pulse repetition~rate. The~effective resonance frequency of each Xe isotope depends on the pulsing frequency ($\\omega_{2\\pi}$) as $\\omega^K = \\omega_{2\\pi}\\gamma^K\/\\gamma^S$. The 300 mHz detuning ensures that both isotopes are not simultaneously on resonance for a given pulsing frequency. When a Xe isotope is far from resonance, we expect the measured signals for that isotope's detection channel to be relatively flat if there is no cross talk. Cross~talk manifests if the signals from the off-resonanct isotope are not flat when the other isotope is scanned through resonance. Figure~\\ref{polmodmct} shows a phase sensitive measurement of cross talk. When~isotope $a$ is excited and isotope $b$ is not, isotope $b$'s detection channel exhibits non-zero signal, and vice versa. We estimate $\\beta = Q_{pp}^b\/Q_{pp}^a = 0.17$ (when isotope $a$ is on resonance) and $\\beta' =Q^a_{pp}\/Q^b_{pp}=0.07$ (when isotope $b$ is on resonance), where $Q^K_{pp}$ is the peak-to-peak quadrature signal of isotope $K$.\n \n\n\nWe observe cross talk on our detection channels even when Xe is not excited and artificial Xe signals are applied, suggesting that our observed cross talk is not due to physical interactions between Xe isotopes. We believe the measured cross talk stems from imperfect rectification of $S_z$ due to optical pumping transients (i.e., gain reversals) and unaccounted-for phase shifts from high-pass filtering prior to rectifying. The optical pumping transients stem from the few ms finite response time of the magnetometer. Indeed, by implementing a sample-and-hold algorithm to ignore data acquired during optical pumping transients, the cross talk is suppressed. The cross talk is further suppressed when the high-pass filter is removed. Doing so, however, reduces the detection SNR by an order of magnitude since rectification maps $1\/f$ $S_z$ noise to the Xe carrier frequencies. \n\n\\section{Discussion}\n\nWe demonstrated a novel spin-exchange pumped $^{131}$Xe-$^{129}$Xe NMR gyroscope. The~production of longitudinal spin-exchange fields and the systematic uncertainty inherent to them is greatly suppressed by synchronous spin-exchange optical pumping. The simultaneous precession of each transversely polarized noble gas is continuously monitored using the optically pumped Rb~atoms. The measured Larmor resonance frequencies are highly correlated with bias magnetic field fluctuations. \n\nComputing the stationary sensor's perceived inertial rotation when oriented East-West and assuming a scale factor determined solely by $\\rho$ allows for an ARW sensitivity of $\\sim$16\\; $\\upmu$Hz\/$\\sqrt{\\text{Hz}}$ and bias instability of $\\sim$800 nHz. The finite short term sensitivity appears limited by the detection~SNR. What limits the SNR is not known. The photon shot noise limited magnetometer performance should support an SNR more than 100x greater than we measure. It is possible that fluctuations in pump frequency and intensity could limit our SNR by producing noise in the AC Stark shift and the Rb polarization. We confirmed that the fidelity of modulation produced by the EOMs is sufficient to support 4x the measured SNR. The source of drift which limits the bias instability is also unknown. Cell~temperature, which was not stabilized during these measurements, could be an important contribution to long term stability. We see no direct indications of first order quadrupole from $^{131}$Xe interactions with electric field gradients at the cell walls. We showed that cross talk exists between phase sensitive detection channels for each noble gas. Such cross talk influences the device's effective scale factor. Future work will include detecting $S_z$ in a way such that gain modulation is suppressed as demonstrated in~\\cite{Thrasher2019,thrasher2019PRApp}. This device's scale factor can be verified by orienting its sensitive axis along North-South and measuring Earth's rate of rotation. \n\n\n\n\\begin{acknowledgments}\nWe would like to thank Michael Larsen for insightful discussions.\n\nThis research was funded by the National Science Foundation grant numbers PHY-1607439 and PHY-1912543 and Northrop Grumman Mission Systems' University Research Program.\n\nAll authors contributed equally in all respects to this manuscript. All authors have read and agreed to the published version of the manuscript.\n\nThe authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sect:intro}\n\nMaterials that are composed of collections of separate, macroscopic solid grains \nbelong to the general classification of {\\it granular materials}. Examples of such materials are common, including sand, gravel, medicinal pills, coins, and breakfast cereal.\nGranular media are important to numerous industries ranging \nfrom mining to pharmaceuticals. In geophysics, granular \nmaterials are a central problem in understanding\nthe physics of earthquakes and tectonic faulting. \nEarthquake fault zones produce granular wear material continuously as a\nfunction of shear and grinding between the fault surfaces. The wear \nmaterial, known as fault gouge, varies in thickness from 10's of cm to 1000 m\nand plays a critical role in determining the fault zone frictional\nstrength, the stability of fault slip, and the size of the rupture \nnucleation dimension. \n\n\nGranular media display a variety of complex \nstatic and dynamic properties that distinguish them from conventional\nsolids and liquids. \nThe complexity of granular media lies primarily in the collective \nproperties of a macroscopic number of grains and how they interact with \neach other. The conditions under which a granular medium is stable \nor flows and the nature of this flow depend critically on the \ndistributions of grain size and shape as well as the interactions \nbetween the grains. The practical importance of granular media combined\nwith the richness of their physical properties has led to a great deal\nof interest from both theoretical and experimental points of view\n\\cite{DEG, jaeger, kadanoff}.\n\n\nAn important class of granular materials consists of nearly rigid particles that possess the following property: if a moderate force is applied, the particles start to move, and only after a substantial increase of the force the particles deform significantly. In other words, for loads which are not very high, the deformations inside a particle\nare small compared with the displacement of the particle center of mass. \nConsequently, the particle shapes change very little, so that\neach particle can be associated with a region of space that is inaccessible\nto any other particle. This gives rise to constraints on the admissible positions of particles. These {\\it impenetrability constraints} are also known as\ngeometric, kinematic, and excluded volume constraints. \n\n\n\nA physical phenomenon related to appearance of constraints is jamming. A particle is jammed when its motion is completely obstructed by the neighbors, so the whole cluster of neighboring particles can only \nmove together as a rigid body. \nThe corresponding mathematical notion of rigidity (\\cite{W}) can be applied to various physical (sphere packings, frameworks (trusses)), as well as mathematical objects. In particular, an important mathematical object associated with any particle packing is a {\\it contact graph} defined as follows:\nvertices of this graph are particle centers of mass, while edges represent interparticle contacts. \n\n\n\n\n\n\nThe simplest physical model that exhibits jamming is a classical hard sphere packing. The particles\nin this model are represented by rigid spheres, and the only interparticle forces are reactions of constraints. Rigidity of hard sphere packings is studied in \\cite{Co}. This problem can be formulated as a problem of detecting rigidity of the {\\it cable framework} associated with the contact graph of the packing. The framework is obtained by replacing edges of the contact graph with the cables, and vertices with flexible hinges. The lengths of the cables can increase but not decrease, which models the impenetrability constraints. Recently, a linear programming algorithm for detecting rigidity in hard sphere packings (equivalently, cable frameworks) was proposed in \\cite{D}. \n\nIn this work, we also use {\\it bar frameworks}. A bar framework\nis obtained from a graph by replacing the edges with rigid bars, and vertices with hinges.\nA bar framework and the associated graph are called rigid if the only possible vertex motions correspond to rigid body motions of the whole framework. We note that both bar and cable frameworks can be associated\nto the same graph. To generate the bar framework, the edges of a graph are replaced with rigid bars that can only translate and rotate. In the case of the cable framework, one replaces edges with cables that can either move as rigid bodies or stretch. Thus every motion of a bar framework is also permitted by the cable framework, but the converse is not true in general. Therefore, it is possible that a bar framework associated to a graph is rigid, while the cable framework corresponding to the same graph is not. Both\ncable frameworks and bar frameworks are special case of the so-called tensegrity frameworks studied in \\cite{CoW}. In a tensegrity framework, properties of edges can vary, e.g. some edges may be bars, others may be cables or struts (that can shrink, but not stretch).\n\n\nIt appears that the currently available mathematical results \\cite{Co, CoW, D} on statics of discrete particle systems with geometric constraints deal only with hard particle packings. \nTo the best of authors' knowledge, there are\nno results on frictional packings, and even elastic frictionless packings have yet not been studied. \nThe present work differs from \\cite{Co, CoW, D} in several respects. First, all these studies\ndeal with rigid particles. We consider a somewhat more realistic situation of geometrically\nconstrained particles with elastic interactions defined by a quadratic potential energy.\nSecond, while \\cite{Co, CoW, D} focus on jamming, we are interested in generic contact patterns of the the energy minimizing configurations.\nThe packings that we study are not jammed. Their contact graphs are such that the associated bar framework is rigid, but the packing can still deform when external boundary conditions are applied.\n\nThe third difference is in the type of the boundary conditions. The conditions in\n\\cite{D} are periodic or hard wall conditions. The periodic conditions are commonly used to\nminimize influence of the boundaries in the problem. However, presence of walls\nis a major factor that determines bulk behavior of granular materials. Therefore it seems better to use boundary conditions corresponding to engineering and physical experiments, where the walls are rigid and may be moving. \n\n\n\n \nA frequently observed property of granular materials is concentration of the bulk\ndeformation in thin layers called shear bands. Within a band, the contact forces are\nweak, and the relative displacements can be on the order of particle size or larger.\nFor quasi-static flows driven by small shear rates, \nthe corresponding patterns are called micro-bands (\\cite{Kuhn1}). In that paper, shear\nband structures were studied by means of numerical simulations. The simulations\nin \\cite{Kuhn1} show that the typical size and number of bands in quasi-static shear depend on the imposed shear rate. For small shear rates, the bands have length and width comparable with the particle size. The distribution of these micro-bands within the material is rather uniform. As the shear rate increases, the band structure exhibits coarsening: the number of bands becomes smaller, and the length of each band increases. For sufficiently large shear rates, a single macroscopic shear band appears. \n\n\nHere, we are interested in the case of small external boundary conditions. In that case, a micro-band\ncan be formed by weakening of a single contact, or a small group of neighboring contacts. All weak\ncontacts form a subnetwork of the whole contact network. Such networks of weak contacts\n(corresponding to the micro-band patterns in \\cite{Kuhn1}) were studied numerically in \\cite{Radjai}. \nThe goal of this paper is to describe some generic geometric features of micro-band, or weak contact\nnetworks in dense packings of nearly rigid particles. In small deformation, pattern formation may be caused\nby the local jamming (which mathematically amounts to impenetrability constraints), and friction. \nWe are concerned with the role of constraints, while friction in neglected. \nThe notion of high density at this point is rather intuitive. It could mean, for instance, that each particle is\nin contact with at least three other particles, and the packing is jammed in the reference configuration, subject to zero boundary conditions. Below we make the notion of high density more precise (see the second paragraph on p. 4), using the relationship between the contact graph and the Delaunay graph (see e.g. \\cite{Ed}), generated by the set of particle centers. \n\n\nIn two dimensions, particles are represented by disks $D_i$ of radii $a_i$ with centers ${\\bf x}^i$, $i=1, 2,\\ldots, N$. The initial reference configuration is deformed by applying prescribed small displacements to the boundary particles. Assuming that the deformations inside of the individual particles are small, and neglecting rotational degrees of freedom, one can characterize the deformations of $D_i$ by the displacements ${\\bf u}^i$ of their centers. The elastic interaction forces are modeled as\nin classical mechanics of point particles: the force exerted by $D_j$ on $D_i$ is applied\nat ${\\bf x}^i$, its direction is along the line joining ${\\bf x}^i$ and ${\\bf x}^j$, and its magnitude\ndepends linearly on ${\\bf u}^j-{\\bf u}^i$.\n\nWe further assume that\nthe granular material is pre-stressed (or, equivalently, the material is under confining stress). This means that in the reference state, the particles are squashed into each other as a result of applied external pressure. \nFurther compression is supposed to be impossible (requires infinite energy), which\nintroduces impenetrability constraints into the problem.\nTo model impenetrability, one can, for instance, require that \n\\begin{equation}\n\\label{0.1}\n|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^j+{\\bf u}^j)|\\geq a_i+a_j,\n\\end{equation}\nfor each pair of particles. Since the the packing is dense, and the displacements are \nexpected to be small, it makes sense to require that a particle cannot escape a cage formed by its neighbors. Therefore, the contacts that exist in the reference configuration may be broken, but no new contacts are created after applying\nexternal boundary conditions to the reference configuration. \nAn important consequence of this assumption is as follows. If the displacements satisfy ({0.1}) for each pair of particles {\\it in contact}, then for any pair of particles ({0.1}) is automatically satisfied. Indeed, if two particles are not in contact\nin the reference configuration, they cannot come into contact in the deformed configuration, and the distance between\nthem must be larger than the sum of their radii. In the sequel, we use this assumption in the course of proving the main result of this paper (Theorem \\ref{main-theorem}). \n\n\nNext, we introduce a network model which describes a granular material under the above assumptions. The vertices of the network are the particle centers, and the edges represent\nparticle contacts. \nThe collection of vertices ${\\bf x}^i$, $i=1, 2, \\ldots, N$, and edges forms the contact network (graph) $\\Gamma$. We suppose that $\\Gamma$ is a triangulation of a connected, convex polygonal domain $\\Omega$. \nThis assumption is realistic, since, for example, a periodic 2D packing of disks is triangular. \nAnother natural triangulation generated by ${\\bf x}^i, i=1, 2,\\ldots, N$ is the Delaunay graph $G$. In principle, \n$\\Gamma$ and $G$ may be different, since some edges in $G$ may not correspond to contacts. In the present case, we suppose that $\\Gamma$ and $G$ coincide, which corresponds to \"maximally dense\" packings. \n\nFor small displacements ${\\bf u}^i$, the quadratic constraints (\\ref{0.1}) can be approximated by\ntheir linearizations near ${\\bf u}^i={\\bf u}^j=0$, which leads to the linearized impenetrability constraints\n\\begin{equation}\n\\label{i3-intro}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}\\geq 0,\n~~i=1,2,\\ldots, N\n\\end{equation}\nfor each pair of vertices $i,j$ connected by an edge of $\\Gamma$.\nIn (\\ref{i3-intro}), \n$\n{\\bf q}^{ij}=({\\bf x}^j-{\\bf x}^i)\/\\left|{\\bf x}^j-{\\bf x}^i\\right|\n$\nare unit vectors\nthat point from ${\\bf x}^i$ to ${\\bf x}^j$ along the line of centers.\nNote that if the position of $D_i$ is fixed (${\\bf u}^i=0$), then\n${\\bf u}^j$ satisfying (\\ref{i3-intro}) must lie in the half-plane ${\\bf u}\\cdot {\\bf q}^{ij}\\geq 0$, so that $D_j$ would be moving away from $D_i$. \n\nFor certain boundary conditions the deformed packing can become more loose than the reference packing. On the macroscale, this can be observed as swelling of the specimen caused by the increase in the volume of the\nvoid space between the particles. Such swelling is typical in shear deformation, where the overall volume increase, known as dilatation, is observed in experiments. To increase the void volume, some of the contacts\npresent in the reference configuration must be broken in the deformed configuration. Therefore, \namong all the contacts (satisfying (\\ref{i3-intro})), we further distinguish two types of contacts: broken and solid-like (see Fig. 1). We call a contact broken if\n\\begin{equation}\n\\label{broken}\n({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}>0,\n\\end{equation}\nand solid-like if\n\\begin{equation}\n\\label{solid-intro}\n({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}=0.\n\\end{equation}\n\nThe solid-like contacts correspond to two possible types of pair motions. The first type \nis a rigid motion of a pair, in which case the contact is called {\\it stuck}.\n\\begin{figure}[htbp]\n\\label{contacts}\n\\begin{center}\n\\input{hpackfig7-bis.pstex_t}\n\\caption{Left: a broken contact. Center: a solid-like sheared contact. Right: a solid like stuck contact\n(infinitesimal rigid rotation of a pair). The arrows indicate displacements.}\n\\end{center}\n\\end{figure}\n\nThe second type is a local shear motion. In the local coordinates of one particle, it is either the motion of the second particle in the direction perpendicular to the line of centers, or an infinitesimal rotation (rolling). The corresponding contacts are called {\\it sheared}. In our idealized model, friction in neglected, and any tangential force would lead to immediate separation of particles, because for disk-shaped particles, the contact surface is a point. We, however, still call these contacts solid-like, because in reality, these contacts are subject to friction forces, the contact surface has a positive area, and the particles in a sheared contact will not separate until the tangential force reaches the static friction threshold. In the case of rolling, the particles stay in contact and the pair is capable of bearing a compressive load.\n\n\nPhysically, vertices of the network can be realized as unit point masses and edges can be realized as elastic springs. Elastic force of the spring $(i,j)$ is determined by the pair potential $H(t_{ij})$, where $t_{ij}=({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}$. \nThe potential is an important ingredient of our model, and therefore\nwe discuss it in detail. To motivate the choice\nof $H$, we first \nrecall the classical hard sphere potential\n$H_{hs}$, which in our notation is defined by\n\\begin{equation}\n\\label{hs}\nH_{hs}(t_{ij})=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\n0 & \\mbox{if}~t_{ij}\\geq 0.\\\\\n\\end{array}\n\\right.\n\\end{equation}\n$H_{hs}$ models the following two options: \n(i) moving non-deformable (hard spheres) particles toward each other requires infinite energy\n(a vertical line at $t_{ij}=0$), \n(ii) moving particles apart requires no energy. Note that (\\ref{hs}) already incorporates the constraints\n(\\ref{i3-intro}) by requiring infinite potential energy to violate the constraints.\n\\begin{figure}[htbp]\n\\label{potentials}\n\\begin{center}\n\\input{hpackfig4.pstex_t}\n\\caption{\na) Hard sphere potential $H_{hs}$; b) The elastic potential $H(t_{ij})$ combines a vertical wall at $t_{ij}=0$ and a quadratic function with the vertex at $d$. }\n\\end{center}\n\\end{figure}\n\n\nElastic interaction between $D_i$ and $D_j$, together with constraints (\\ref{i3-intro})\ncan be modeled by the following potential\n\\begin{equation}\n\\label{actual}\nH(t_{ij}, d)=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\n\\frac 12 Cd^{-3}(t_{ij}-d)^2 & \\mbox{if}~t_{ij}\\geq 0.\\\\\n\\end{array}\n\\right.\n\\end{equation}\n\\noindent\nHere $d$ characterizes the cut-off distance of the potential, and $C$ determines\nthe magnitude of the pre-stress potential (the value of the potential when $t_{ij}=0$.\nThe potential (\\ref{actual}) is shown on Fig. 2, together with the hard-sphere potential.\nThe formula (\\ref{actual}) describes two options: (i) moving particles toward each other \nrequires infinite energy; (ii) movement of particles apart from each other is caused\nby finite, linear elastic force ${\\bf f}^{ij}=-\\partial H\/\\partial {\\bf u}^i$; This force is repulsive for\nsmall distances ($t_{ij}0$, then\n$\\lim_{d\\to\\infty} H(t_{ij}, d)=0$ uniformly\non $(0, d_0]$. In the paper, we do not pass to this limit. Instead we choose $d$ sufficiently large and fix it, so\nthat $H$ is close to $H_{hs}$. Also, for technical simplicity, we set $C=1$ in (\\ref{actual}), which corresponds\nto an appropriate rescaling. \n\n\nIn reality, once the distance between $D_i$ and\n$D_j$ is greater than the sum of their radii $a_i+a_j$, the pair interaction force\nis zero. In our model, we still have a small repulsive force for all $a_i+a_j\\leq t_{ij}\\leq d$. So, the particles in our model\nwould continue accelerating away from each other even after separating. This favors separation of particles, and could lead to\nincrease in the number of broken contacts. Since our goal is estimating the number of solid-like contacts from below, this increase is acceptable. In fact, our estimate holds \nfor all $d$ sufficiently large.\nWe also mention that elastic contact force predicted by the classical Hertz theory\nis a non-linear function of $t_{ij}$. Our model is chosen for simplicity, and can be viewed\nas an approximation of Hertz theory, valid for sufficiently small displacements.\n\n\nIn general, the cut-off parameter $d$ may be different for different pairs of particles\nin contact. Therefore, we define a pair interaction energy\n\\begin{equation}\n\\label{energy}\nh(t_{ij}, d_{ij})= \\frac 12d^{-3}(t_{ij}-d_{ij})^2,\n\\end{equation}\nand consider \n\\begin{equation}\n\\label{actual-gen}\nH(t_{ij}, d_{ij})=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\nh(t_{ij}, d_{ij}) & \\mbox{if}~t_{ij}\\geq 0,\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{reg}\nd_{ij}=d\\delta_{ij},~~~~\\delta_{ij}\\in [1\/2, 1].\n\\end{equation}\nThe formula (\\ref{actual-gen}) is more general than (\\ref{actual}). \nThe choice of $d_{ij}$ in (\\ref{reg}) ensures that for all pairs $(i,j)$, the points where $h(t_{ij}, d_{ij})=0$\nand located in the interval $[1\/2d, d]$. The number 1\/2 is of no particular significance. \nAny number in the interval $(0, 1)$ would work just as well. We only need all $d_{ij}$ to have the same sign and comparable magnitudes controlled by $d$.\nFinally, the total elastic interaction energy $Q$ of the network is obtained by summing up $h(t_{ij}, d_{ij})$\nover all pairs $(i,j)$ corresponding to the edges of the network. \n\n\nThe equilibrium state of a granular material corresponds to a minimum of $Q$ subject to the constraints\n(\\ref{i3-intro}) and the appropriate boundary conditions. Since the functional $Q$ is quadratic, and the\nconstraints and boundary conditions are linear, this is a quadratic programming problem, studied in optimization theory (e.g., \\cite{Hestenes}). In the language of optimization theory, the solid-like contacts (\\ref{solid-intro}) correspond to the so-called {\\it active} constraints, while\nthe constraints corresponding to the broken contacts (\\ref{broken}) are called {\\it inactive}. The question\naddressed in this paper concerns the total number and spatial distribution of each type of constraints in the energy-minimizing configuration of the network. It appears that no general results of this type are currently available in optimization theory. The present study makes use of the geometric features of the contact graph, in particular its rigidity properties (see above), to investigate the energy minimizer. We also note the connection between our constrained variational problem and continuum variational inequalities\n\\cite{Duvaut}, \\cite{Kikuchi}, \\cite{Kinderlehrer}, \\cite{Panagi}. Our problem can be viewed as a discrete\nvariational inequality. \n\n\n\n\nThe main result of the paper is Theorem \\ref{main-theorem} in Section \\ref{sect:optimality}.\nThere we consider a packing whose\ncontact graph in the reference configuration is a triangulation of a convex polygonal domain. The packing is deformed by imposing displacement boundary conditions at the packing boundary. The boundary conditions model the motion of rigid walls in engineering experiments. We prove that for generic contact graphs (precisely defined in Definition \\ref{reg-treg}) and generic pre-stresses (corresponding to the choice of $\\delta_{ij}$ in (\\ref{energy}), \nthe constrained energy minimizer for sufficiently large $d$ provides a packing with at least two solid-like contacts per each particle. There also some (non-generic) choices of $\\delta_{ij}$ for which theorem does not provide a definite conclusion.\n\nThe network of solid-like contacts is the load-bearing structure. The network of broken contacts can be associated with the micro-bands \\cite{Kuhn1}, \\cite{Kuhn2} that appear during small shear deformations. The result implies that no particle can lose contact with {\\it all} of its neighbors, which eliminates ``micro-avalanches\". Put another way, loss of structural integrity in dense packings is evolutionary rather than catastrophic, so that shearing with a small displacement will first lead to dilatation, during which the packing becomes more loose everywhere, and only then local avalanches may occur.\n\n\nAnother useful consequence of theorem \\ref{main-theorem} is as follows. It provides a lower bound on the order parameter, recently introduced in \\cite{AT, VTA} as one of the main ingredients of the new phenomenological theory of dense granular flows proposed by Aronson and Tsimring. The order parameter \n$\\rho$ characterizes the phase transition from solid to fluidized state. To define $\\rho$ at an arbitrary point ${\\bf x}$ of $\\Omega$, one begins by fixing a mesoscopic averaging volume $V$ of characteristic size $h$\n(e.g., a disk of radius $h$ centered at ${\\bf x}$). Then, all solid-like contacts within $V$ should be counted. Next, the obtained number $n_s$ of solid-like contacts is divided by the number $n$ of all contacts within $V$\nto obtain $\\rho$. So, $\\rho$ is a mesoscopic average, which in general depends on $h$.\nIn many systems, such as periodic elastic composites, the results of mesoscopic averaging is practically independent of $h$ for $h$ larger than a certain characteristic length. For disordered granular materials this\nis not necessarily true. Therefore, a rigorous mathematical theory may require study of a family of order parameters parametrized by $h$. In Section 6, we define such a family of order\nparameters using the notion of a $k$-neighborhood of a vertex of\n$\\Gamma$ as a discrete analogue of $V$. Specifying an integer $k$ in our definition corresponds to\nchoosing $h$ in the continuous case. \n\n\n\nThe paper is organized as follows. In Sect. 2 we formulate the main constrained minimization problem.\nThe problem contains two types of constraints: impenetrability constraints (\\ref{i3-intro}) and the\nboundary constraints (see (\\ref{b1}), (\\ref{b1-bis})), corresponding to the external boundary conditions. Elimination of these boundary constraints\nleads to a reduced minimization problem. In Sect. 3 we recall some facts concerning first-order rigidity of graphs. In Sect. 4 we show existence of a unique minimizer of the reduced\nproblem. Optimality conditions for the reduced problem are stated and analyzed in Sect. 5, where we also state and prove the main theorem \\ref{main-theorem}. In Sect. 6 we introduce a definition of the order parameter in the spirit of \\cite{VTA} and give a lower bound on the\norder parameter that follows from the main theorem. Finally, conclusions are provided in Sect.\n7.\n\n\n\n\n\n\n \n \n\n\n\n\\section{Formulation of the problem}\n\\subsection{Elastic interactions with impenetrability constraints}\n\\label{sect-constraints}\nIn 2D, consider a packing of spheres $D_i$ of radii\n$a_i$ with centers ${\\bf x}^i, i=1,2,\\ldots, N$. (All vectors in this\npaper are column vectors and we use superscript `{\\sf T}' to indicate\ntransposition). The packing fills a bounded region. After an infinitesimal motion, the position of the center of $D_i$ is \n${\\bf y}^i$. We write\n${\\bf y}^i={\\bf x}^i+{\\bf u}^i$ where ${\\bf u}^i$ are displacements.\nThe vertices ${\\bf x}^i, {\\bf x}^j$ are connected by an edge if and only if $D_i, D_j$ are in contact. In this case, we call ${\\bf x}^i$ and ${\\bf x}^j$ neighbors. We denote \nby \n$\n{\\mathcal N}_i\n$\nthe set of $j\\in \\{1, 2,\\ldots, N\\}$ such that ${\\bf x}^j$ is a neighbor of ${\\bf x}^i$.\nOrientation of contacts (equivalently, edges) is prescribed by the unit vectors\n\\begin{equation}\n\\label{orient}\n{\\bf q}^{ij}=\\frac{{\\bf x}^j-{\\bf x}^i}{|{\\bf x}^j-{\\bf x}^i|}.\n\\end{equation}\nThe vertices ${\\bf x}^i$ and edges $(i,j)$ define the contact graph\n$\\Gamma$. \nLet $E$ denote the number of edges of $\\Gamma$.\nThe edge set ${\\mathcal E}$ of $\\Gamma$ is given by\n$\\{(i,j): j\\in{\\mathcal N}_i, i =1,2, \\ldots, N\\}$. To each edge $(i,j)$\nwe can associate a pair potential energy $h(t_{ij}, d_{ij})$ defined in (\\ref{energy}).\nSumming up all these energies we obtain the total \nelastic interaction energy of the network. It is a quadratic form\n\\begin{equation}\n\\label{q-basic}\nQ({\\bf u}^1,{\\bf u}^2, \\ldots {\\bf u}^N)=\\sum_{i=1}^N\n\\sum_{j\\in {\\mathcal N}_i} h(t_{ij}, d_{ij})=\n\\frac 12 d^{-3}\\sum_{i=1}^N\n\\sum_{j\\in {\\mathcal N}_i}\n\\left (({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}-d_{ij}\\right)^2,\n\\end{equation}\non the displacements ${\\bf u}^i, i=1,2,\\ldots, N$.\nIn (\\ref{q-basic}), $d, d_{ij}$ are parameters specified by (\\ref{actual-gen}), (\\ref{reg}). \n\nOur objective is to determine the displacements ${\\bf u^i}$, $i=1,2, \\ldots,\nN$ so that the the energy functional $Q$ is minimized subject to two\ntypes of constraints. The first type of constraints consists of\n{\\em linearized impenetrability constraints}. These are obtained by formally\nlinearizing the condition\nthat the distance between two spheres in contact cannot decrease. Consider\ntwo spheres $D_i, D_j$ in contact. In the reference configuration,\n\\begin{equation}\n\\label{i1}\n|{\\bf x}^i-{\\bf x}^j|=a_i+a_j.\n\\end{equation}\nAssuming that $D_i, D_j$ cannot overlap, we have\n\\begin{equation}\n\\label{i2}\n|{\\bf y}^i-{\\bf y}^j|\\geq a_i+a_j.\n\\end{equation}\nThese are the impenetrability constraints.\nWe linearize (\\ref{i2}) by writing\n\\begin{eqnarray*}\n|{\\bf y}^i-{\\bf y}^j|^2 & = & |{\\bf x}^i-{\\bf x}^j+{\\bf u}^i-{\\bf u}^j|^2\\\\\n & = &|{\\bf x}^i -{\\bf x}^j|^2 +\n 2({\\bf x}^i - {\\bf x}^j) \\cdot ({\\bf u}^i - {\\bf u}^j) +\n |{\\bf u}^i - {\\bf u}^j|^2 \\\\\n & = & (a_i + a_j)^2 + \n 2({\\bf x}^i - {\\bf x}^j) \\cdot ({\\bf u}^i - {\\bf u}^j) +\n |{\\bf u}^i - {\\bf u}^j|^2. \n\\end{eqnarray*}\nNow for for ``small'' $|{\\bf u}^i - {\\bf u}^j|$ we can neglect quadratic term\n$|{\\bf u}^i - {\\bf u}^j|^2$, and (\\ref{i2}) yields $2({\\bf x}^i - {\\bf x}^j)\\cdot ({\\bf u}^i -{\\bf u}^j) \\geq\n0$, which in turn is equivalent to $({\\bf u}^j - {\\bf u}^i)\\cdot\n{\\bf q}^{ij} \\geq 0$ where $ {\\bf q}^{ij}$ is as defined in\n(\\ref{orient}).\nTherefore, the first set of constraints we impose on the displacements\n${\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N$ is\n\\begin{equation}\n\\label{i3}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}\\geq 0, ~~~j\\in {\\mathcal N}_i,\n~~i=1,2,\\ldots, N.\n\\end{equation}\nThe second type of constraints corresponds to the boundary conditions.\nParticles located at the packing boundary have {\\em prescribed} displacements.\nIn the sequel we refer to these particles as {\\it boundary particles}. The corresponding vertices of $\\Gamma$ are called {\\it boundary vertices}.\nOther particles are referred to as {\\it interior}, or sometimes, {\\it free}, \nand the corresponding vertices of $\\Gamma$ as {\\it interior vertices}. \n\nAll boundary particles are divided into several groups,\nnumbered $1,2,\\ldots,M$. Let $I_m$ denote the set of indices\nof the particles in group $m$ for $m=1,2, \\ldots, M$.\nEach sphere in a certain group is in contact with at least one\nother sphere from the same group. \nEach group moves as a single rigid body. We assume that the prescribed boundary displacements are of the form\n\\begin{equation}\n\\label{b1}\n{\\bf u}^i={\\bf R}^m({\\bf x}^i),~~~i\\in I_m, ~~m=1,2,\\ldots, M,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{b1-bis}\n{\\bf R}^m({\\bf x}^i)={\\bf c}^m+\\alpha^m K({\\bf x}^i-{\\bf x}^{\\star,m}),\n~~~i\\in I_m, ~~m=1,2, \\ldots, M,\n\\end{equation}\nand ${\\bf c}^m, {\\bf x}^{\\star,m}$ are given vectors, $\\alpha^m$ is a\ngiven scalar, and\n$K$ is the matrix denoting clockwise rotation by $\\pi\/2$.\nThe functions ${\\bf R}^m$ are called {\\em infinitesimal rigid\ndisplacements},\nparametrized by a scalar $\\alpha^m$, and\nvectors ${\\bf c}^m$ and ${\\bf x}^{\\star,m}$. We refer the\nreader to Sect. \\ref{sect:rigidity} for more details\non rigid displacements.\n\nOur description above leads to the\\\\\n\n\\vspace*{0.5\\baselineskip}\n\\noindent\n\\underline{\\bf Main problem:} \n\\begin{eqnarray}\n\\label{mpr1}\n\\mbox{minimize}&~~& Q({\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N)\\\\\n\\label{mpr2}\n\\mbox{subject to}&~~& \\mbox{linearized impenetrability\n constraints (\\ref{i3})}\\\\\n\\label{mpr3}\n &~~& \\mbox{and boundary conditions (\\ref{b1}).}\n\\end{eqnarray}\n\n\n\\subsection{Feasible region}\nLet us define the configuration space $U$. Points of this space\nare denoted by ${\\bf U}=(({\\bf u}^1)^{\\mbox{\\scriptsize\\sf T}}, ({\\bf u}^2)^{\\mbox{\\scriptsize\\sf T}}, \\ldots,\n({\\bf u}^N)^{\\mbox{\\scriptsize\\sf T}})^{\\mbox{\\scriptsize\\sf T}}$.\n\n\\noindent\n{\\bf Remark}. To avoid this heavy notation, we simply write\n$$\n{\\bf U}=({\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N),\n$$\nwhen no confusion can occur.\n\n\nDimension of $U$ is $2N$. \n{\\it Feasible region} ${\\mathcal F}$ is the subset of $U$\nin which all the constraints (\\ref{i3}) and (\\ref{b1})\nare satisfied. The points satisfying (\\ref{i3}) form a polyhedral\n(not necessarily bounded) region. The boundary of this region\nconsists of parts of the hyperplanes (subspaces of dimension $2N-1$) defined\nby \n\\begin{equation}\n\\label{i3-eq}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}=0,~~~~j\\in{\\mathcal N}_i,~i=1, 2, \\ldots, N.\n\\end{equation}\nBecause of the close relation to rigidity, we refer to (\\ref{i3-eq}) as\n$R$-{\\it equations}. \nEquations\n(\\ref{b1}) define\n$M$ planes $S_m, m=1, \\ldots, M$. Dimensions\nof $S_m$ depend on the number of the boundary particles in the $m$-th group. \n\nFor each point of ${\\bf U}\\in {\\mathcal F}$, \nsome of the constraints (\\ref{i3}) are satisfied as equations. These constraints\nare called {\\it active}. The corresponding edges of the contact graph $\\Gamma$ \nare called active as well. \nThe rest of (\\ref{i3}) are satisfied as strict inequalities. These are {\\it inactive}\nconstraints (respectively, edges).\n\n\\subsection{Elimination of constraints corresponding to boundary conditions}\n\\label{sect:elimination}\nThe quadratic form $Q$ in (\\ref{q-basic}) can be written in a convenient\nform in terms of a certain matrix $R^r$. To define $R^r$, we\nindex the edges of $\\Gamma$ by $l$,\n$l=1,2, \\ldots, E$. Let $(i_l, j_l) \\in {\\mathcal E}$ be the edge\nof $\\Gamma$ corresponding to $l$ for $l=1,2,\\ldots, E$. \nLet $R^r$ be the $E\\times 2N$ matrix whose $l$-th row is defined by\n\\begin{equation}\n\\label{row}\nR^r_{lm} = \\left\\lbrace\\begin{array}{ll}\n +({\\bf q}^{i_lj_l})_1 &\\mbox{if $m=2(j_l-1)+1$}\\\\\n +({\\bf q}^{i_lj_l})_2 &\\mbox{if $m=2(j_l-1)+2$}\\\\\n -({\\bf q}^{i_lj_l})_1 &\\mbox{if $m=2(i_l-1)+1$}\\\\\n -({\\bf q}^{i_lj_l})_2 &\\mbox{if $m=2(i_l-1)+2$}\\\\\n 0 &\\mbox{otherwise}\n \\end{array}\\right .\n\\end{equation}\nfor $l=1,2, \\ldots, E$. \n\n\n\\noindent\n{\\bf Remarks.} \n1. $R^r$ is the (first-order) rigidity matrix,\na well known object in geometric rigidity theory (see e.g. \\cite{CoW, W}).\n\n\\noindent\n2. Consider\nvertices ${\\bf x}^{i_l}, {\\bf x}^{j_l}$ and the edge $l$ connecting them.\nThe corresponding row ${\\bf r}^l$ of $R^r$ has $2N$ entries. We can view ${\\bf r}^l$\nas a string of $N$ pairs of numbers, the first pair corresponding to ${\\bf x}^1$,\nthe second to ${\\bf x}^2$ and so on. For simplicity, we shall call a pair\nof entries corresponding to a particular vertex ${\\bf x}^i$\na {\\it place corresponding to ${\\bf x}^i$}.\n\nThen we can interpret equation (\\ref{row}) as follows. A row ${\\bf r}^l$ has zeros at all places, except two. The non-zero entries are $-{\\bf q}^{i_l, j_l}$, written as a two-dimensional row vector\nat the place corresponding to ${\\bf x}^{i_l}$; and ${\\bf q}^{i_l, j_l}$,\nwritten as a two-dimensional row\nat the place corresponding to ${\\bf x}^{j_l}$.\n\n\n\\noindent\n3. A row of $R^r$ corresponds to an edge of $\\Gamma$. Therefore it is natural to call a row active (respectively, inactive)\nif a corresponding edge is active (respectively, inactive).\n\n\nNow define the vector ${\\bf d}\\in{\\mathbb R}^E$ by\n\\begin{equation}\n\\label{d}\n{\\bf d}=-(d_{i_1j_1},~d_{i_2j_2},~ \\ldots, ~d_{i_Ej_E}), \n\\end{equation}\nwhere $d_{i_l, j_l}$ are chosen according to (\\ref{reg}). \nWith these notations the quadratic form $Q$ in (\\ref{q-basic})\ncan be written as\n\\begin{equation}\n\\label{q}\nQ({\\bf U})=d^{-3}\\frac 12 (R^r{\\bf U}+{\\bf d})\\cdot( R^r{\\bf U}+{\\bf d}). \n\\end{equation}\n\n\nWe now eliminate the boundary conditions (\\ref{b1}) from the main\nproblem (\\ref{mpr1}, \\ref{mpr2}, \\ref{mpr3}).\nLet $N_b= \\sum_{m=1}^M \\mbox{card}(I_m)$.\nThen the equations (\\ref{b1}) simply state that the $2N_b$\ncomponents of ${\\bf U}$ corresponding to the $N_b$\nboundary vertices have prescribed\ndisplacements. Without loss of generality assume\nthat the {\\em last} $2N_b$ components of $U$ correspond\nto the boundary vertices.\nLet us partition ${\\bf U}$ as\n\\begin{equation}\n\\label{split1} \n{\\bf U}=[~{\\bf z}~|~{\\bf w}~],\n\\end{equation} \nwhere ${\\bf z}=({\\bf u}^1, {\\bf u}^2, \\ldots,\n{\\bf u}^{N-N_b})$ corresponds\nto displacement vectors of interior vertices, and \n${\\bf w}= ({\\bf u}^{N-N_b+1}, {\\bf u}^{N-N_b+2}, \\ldots,\n{\\bf u}^{N})$ corresponds to the displacements of the\nboundary vertices.\nThe equality constraint (\\ref{b1}) is now simply\n\\begin{equation}\n\\label{b3}\n{\\bf w}={\\bf g}\n\\end{equation} \nwhere $g\\in {\\mathbb R}^{2N_b}$ is the vector of displacements prescribed by\nthe right-hand-sides of (\\ref{b1}).\nThe matrix $R^r$ can be partitioned similarly to (\\ref{split1}):\n\\begin{equation}\n\\label{split2}\nR^r=[~R~|~R^b~],\n\\end{equation}\nwhere dimensions of $R$ and $R^b$ are $E\\times 2(N-N_b)$ and $E\\times\n2N_b$, respectively. \nDenote \n\\begin{equation}\n\\label{a}\n{\\bf a}=R^b{\\bf g}.\n\\end{equation}\nUsing (\\ref{split1})--(\\ref{a}) in (\\ref{q}) and in\n(\\ref{i3}) we can reduce the main problem\n(\\ref{mpr1}, \\ref{mpr2}, \\ref{mpr3})\nto\\\\\n\n\\vspace*{0.5\\baselineskip}\n\\noindent\n\\underline{{\\bf Reduced problem:}}\n\\begin{eqnarray}\n\\label{pr1}\n\\mbox{minimize}&~~&F({\\bf z})=\\frac 12 d^{-3}\\left(R{\\bf z}+{\\bf a}+{\\bf d}\\right)\n\\cdot\\left( R{\\bf z}+{\\bf a}+{\\bf d}\\right)\\\\\n\\label{pr2}\n\\mbox{subject to}&~~&R{\\bf z}+{\\bf a}\\geq 0.\n\\end{eqnarray}\n\\vspace*{0.5\\baselineskip}\nThe minimization in (\\ref{pr1}) is taken over all ${\\bf z}\\in\n{\\mathbb R}^{(N-N_b)}$.\n\n\\section{First-order rigidity}\n\\label{sect:rigidity}\nA rigid motion is a composition of a translation and rotation:\n\\begin{equation}\n\\label{r1}\n{\\bf y}({\\bf x})={\\bf c}+{\\bf x}^\\star+O({\\bf x}-{\\bf x}^\\star),\n\\end{equation}\nwhere $O$ is an orthogonal (rotation) matrix, ${\\bf c}$ is a translation vector,\n${\\bf x}^\\star$ is a center of rotation. If $O$ is close to identity $I$ (infinitesimally small rotation), then\n$$\nO\\approx I+A,\n$$\nwhere $A$ is a skew matrix ($a_{ij}=-a_{ji}$). \n\n\nSuppose that in a two-dimensional rigid motion, the rotation angle $\\alpha$ is close to zero. \nThen\n$$\nO=\n\\left(\n\\begin{array}{cc}\n\\cos \\alpha & \\sin \\alpha \\\\\n-\\sin \\alpha & \\cos \\alpha\\\\\n\\end{array}\n\\right)\\approx\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & 1\\\\\n\\end{array}\n\\right)\n+\n\\alpha\n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n-1 & 0\\\\\n\\end{array}\n\\right)=I+\\alpha K,\n$$\nwhere \n$$\nK=\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n-1 & 0\\\\\n\\end{array}\n\\right)\n$$ \nis a clockwise rotation by $\\pi\/2$. In that case, (\\ref{r1}) becomes\n\\begin{equation}\n\\label{r2}\n{\\bf y}({\\bf x})={\\bf c}+{\\bf x}+\\alpha K({\\bf x}-{\\bf x}^\\star)=\n\\left(\n\\begin{array}{c}\nc_1+x^\\star_1-\\alpha (x-x^\\star)_2\\\\\nc_2+x^\\star_2+\\alpha (x-x^\\star)_1\\\\\n\\end {array}\n\\right).\n\\end{equation}\n\n\nLet ${\\bf u}={\\bf y}({\\bf x})-{\\bf x}$ denote the displacement. We can write\n(\\ref{r2}) as\n\\begin{equation}\n\\label{r3}\n{\\bf u}({\\bf x})={\\bf c}+\\alpha K({\\bf x}-{\\bf x}^\\star).\n\\end{equation}\n\n\\begin{definition}\nWe call (\\ref{r3}) an infinitesimal rigid displacements in 2D.\n\\end{definition}\n\n\n\nNext, let $G$ be a graph. \nConsider all motions of vertices of $G$ that preserve the lengths of the edges.\nIf the only such motions are the rigid body motions of the whole graph, \nthen the graph is called {\\it rigid}.\nA graph $\\Gamma$ is {\\it first-order rigid} \\cite{W} if all solutions\nof the $R$-system (\\ref{i3-eq})\nare infinitesimally rigid displacements. \nIn addition, $\\Gamma$ is {\\it independent}\nif\nthe rows of the rigidity matrix $R^r$ are linearly independent. Graphs that are both\nfirst-order rigid and independent are called {\\it isostatic} (\\cite{W}). Intuitively, an isostatic graph is minimally rigid, that is removing any edge results in loss of rigidity. Another notion of rigidity\nis {\\it generic rigidity}, see \\cite{W}. According to Thm. 49.1.7 from \\cite{W},\ngeneric rigidity for a neighborhood in a configuration space is equivalent to the first-order rigidity for some specific configuration in that neighborhood. \n\nWith the definition of $R^r$ and ${\\bf U}$ in Sect. \\ref{sect:elimination}\nthe system (\\ref{i3-eq}) can be written as\n\\begin{equation}\n\\label{r-sys-a}\nR^r{\\bf U}=0.\n\\end{equation}\n\nNote the connection between $R$-system and constraints, \nas well as the functional of the main problem\n(\\ref{mpr1},{\\ref{mpr2},{\\ref{mpr3}).\n\nThe following definition (see \\cite{W} for a $d$-dimensional definition) is useful for verifying rigidity of graphs.\n\\begin{definition}\n\\label{henneberg}\nFor a graph $\\Gamma$, the Henneberg 2-construction in 2D is a sequence of graphs $G_1, G_2, \\ldots, G_n$ such that:\\newline\n\\noindent\n(i) $G_{k+1}$ is obtained from $G_k$ by either vertex addition (attaching a new vertex by 2 edges); or edge splitting (replacing and edge from $G_k$\nwith a new vertex joined to its ends and to 1 other vertex);\\newline\n\\noindent\n(ii) $G_k$ is a complete graph on $k$ vertices, and $G_n=\\Gamma$.\n\\end{definition}\nThe following result\nis stated in (\\cite{W}, thm. 49.1.13):\n\\begin{theorem}\n\\label{hen-thm}\nIf a graph $G\\subset {{\\mathbb R}}^2$ is obtained by a Henneberg $2$-construction, then $G$ is \ngenerically isostatic. \n\\end{theorem}\n\\noindent\n{\\bf Remark 1}.\nIn veiw of the definitions above, Theorem \\ref{hen-thm} implies that a graph obtained by Henneberg $2$-construction is first-order rigid and independent (minimally rigid).\n\n\nIn the present case, the rows of the rigidity matrix $R^r$ are not linearly independent, but the row rank is maximal. This means that we typically have more edges than needed to ensure rigidity of $\\Gamma$. In this situation, the following theorem (thm. 49.1.14 from \\cite{W}) is useful.\n\\begin{theorem}\n\\label{gluing}\nIf two graphs $G_1$ and $G_2$ are generically rigid planar graphs sharing at least $2$ vertices,\nthen the graph $G$ obtained by combining all vertices and edges of $G_1, G_2$ is generically rigid.\n\\end{theorem}\n\n\\noindent\n{\\bf Remark 2}. Because of the relation between generic rigidity and first-order rigidity, Theorem\n\\ref{gluing} implies that combining first-order rigid graphs $G_1$, $G_2$ as in this Theorem\nyields a first-order rigid graph $G$. We shall use Theorem \\ref{gluing} to obtain first-order rigidity of\ntriangulations. Indeed, one triangle $G_1$ is first-order rigid. Adding another triangle $G_2$ so that\n$G_1$ and $G_2$ share an edge, yields a first-order rigid graph. Then we can proceed sequentially.\nGiven a first-order rigid triangulation $G_k$, we construct $G_{k+1}$ by combining $G_k$ with a triangle. This new triangle either shares two vertices with $G_k$, or all three vertices. In the first case, $G_{k+1}$ would have\none more vertex and two more edges than $G_k$. In the second case, $G_{k+1}$ has the same number of vertices as $G_k$, and one more edge. By theorem \\ref{gluing}, any planar triangulation obtained by this sequential procedure is first-order rigid.\n\n\n\n\n\n\\section{Existence and uniqueness of minimizers of the reduced problem}\nLet $\\Omega$ be a bounded connected domain in ${\\mathbb R}^2$ with a polygonal boundary.\nFirst we show that, under certain assumptions on geometry of $\\Gamma$, the matrix $R$ has full column rank.\n\n\nWe shall say that $\\Gamma$ is a triangulation if edges of $\\Gamma$ partition $\\Omega$ into a disjoint\nunion of triangles. \n\nLet $X$ be a set of interior vertices of $\\Gamma$, containing at least two elements. Consider a graph\n$G_X\\subset \\Gamma$ defined as follows. Vertices of $G_X$ are all elements of $X$. Edges of \n$G_X$ are those edges of $\\Gamma$ that join two vertices from $X$. We also assume that $X$ is chosen\nso that $G_X$ is a connected graph.\n\\begin{definition}\n\\label{solid}\nThe contact graph $\\Gamma$ is {\\it cell-connected} if for each $G_X\\subset \\Gamma$ as above, there\nexist two vertices ${\\bf x}^1, {\\bf x}^2$ in $G_X$, and two interior vertices $\\hat{{\\bf x}}^1, \\hat{{\\bf x}}^2$ in $\\Gamma\\backslash G_X$, such that the quadrilateral with vertices \n${\\bf x}^1, {\\bf x}^2, \\hat{{\\bf x}}^1, \\hat{{\\bf x}}^2$ is a union of two {\\it adjacent} triangles of $\\Gamma$.\n\\end{definition}\nAn example illustrating the definition if shown in Fig. 3.\n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig9.pstex_t}\n\\caption{a) --A cell-connected graph. Solid dots indicate vertices of $G_X$;\nb) -- The cell connection with three edges and four vertices is shown separately; \nc)-- A triangulation that is not cell connected. Here the vertices of the small triangle inside\nare connected to other vertices by pairs of collinear edges.}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{definition}\n\\label{reg-treg}\nWe call $\\Gamma$ a {\\it regular triangulation} if \\newline\n(i) $\\Gamma$ can be obtained by sequential addition of triangles in the way described in the Remark 2;\\newline\n\\noindent\n(ii) every interior vertex, connected to a boundary vertex, is also connected to at least one other\nboundary vertex, and the corresponding edges are non-collinear;\\newline\n\\noindent\n(iii) $\\Gamma$ is cell-connected.\n\\end{definition}\n\n\n\n\n\\noindent\n{\\bf Remark. } \nNote that i) in Definition \\ref{reg-treg} implies that $\\Gamma$ is first-order rigid.\nThe property ii) states that every edge connecting a boundary vertex \nwith an interior vertex must be a part of the boundary of a triangle, containing two boundary vertices and one interior vertex. \n\nInformally, Definition \\ref{solid} (or (iii) in Definition \\ref{reg-treg} )\ncan be interpreted as a strong connectivity property. According to Definition \\ref{solid}, a generic connected subgraph\n$G_X$ is connected to the ``rest of $\\Gamma$\" not just by an edge, but by a \"more robust\" cell structure that consists of three edges, with one edge bracing the other two, (see Fig 3.). Note also that either ${\\bf x}^1$ or ${\\bf x}^2$ are connected to the vertices of $\\Gamma\\backslash G_X$ by a pair of non-collinear edges. This observation is important in the proof of Proposition \\ref{full-rank} below. It is not difficult to see\nthat a periodic triangular planar graph satisfies iii). However, there are triangulations with a mean coordination number four that do not satisfy iii). An example of such a graph is shown in Fig. 3 c).\n\n\n\n\\begin{proposition}\n\\label{full-rank}\nSuppose that $\\Gamma$ is a regular triangulation. Then ${\\mbox rank}~R=2(N-N_b)$.\n\\end{proposition}\n\n\\noindent\n{\\it Proof}. Consider a subgraph $\\Gamma_{max}\\subset \\Gamma$ constructed inductively as follows. \nBegin with $\\Gamma_1$ that consists of all boundary vertices. On the next step, add an interior vertices connected to $\\Gamma_1$ by two or more non-collinear edges. Also, add exactly two non-collinear edges that connect this vertex to $\\Gamma_1$. Call\nthe resulting graph $\\Gamma_2$. Generally, given $\\Gamma_k$, $k\\geq 2$, define\n$\\Gamma_{k+1}=\\Gamma_k\\cup S_k$, where $S_k$ consists of an interior vertex ${\\bf x}^k$, not contained in\n$\\Gamma_k$ but connected to $\\Gamma_k$ by at least two non-collinear edges, together with\na pair of non-collinear edges connecting ${\\bf x}^k$ to $\\Gamma_k$. Since the graph $\\Gamma$ has a finite number of vertices, the\nprocess terminates after a finite number of steps. The resulting graph is $\\Gamma_{max}$. The construction\nis illustrated in Fig. \n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig15.pstex_t}\n\\caption{The subgraphs $\\Gamma_1$ (left) and $\\Gamma_2$ (right). The boundary vertices are shown\nin gray. The interior vertices of $\\Gamma_1$ and $\\Gamma_2$ are shown in black. The edges of\n$\\Gamma_1$, $\\Gamma_2$ are shown by solid lines. The other vertices of $\\Gamma$ are represented by\nthe unfilled circles. The edges of $\\Gamma$ not included into $\\Gamma_1, \\Gamma_2$ are represented by the dotted lines.}\n\\end{center}\n\\end{figure}\n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig16.pstex_t}\n\\caption{Two different subgraphs $\\Gamma_{max}$ constructed inductively. Both subgraphs\nare constructed starting with $\\Gamma_2$ in Fig. 4.}\n\\end{center}\n\\end{figure}\nWe claim that $\\Gamma_{max}$ contains all vertices of $\\Gamma$. To obtain\na contradiction, suppose that there are vertices not included into $\\Gamma_{max}$.\nDenote the set of these vertices by $X$, and denote by $G_X$ the graph formed by vertices in\n$X$ and all edges of $\\Gamma$ that connect these vertices. Let $G_X^c$ be any connected component\nof $G_X$. If $G_X^c$ is a single point ${\\bf x}_g$, then, since $\\Gamma$ is a triangulation, there must be at least three edges incident at ${\\bf x}_g$, and at least two of these edges must be non-collinear. \nThus ${\\bf x}_g$ must be in $\\Gamma_{max}$, which gives a contradiction.\nNext suppose that $G_X^c$ contains two or more vertices. By Definition \\ref{solid}, (see also the Remark following that Definition), there must be a pair of vertices of ${\\bf x}^1, {\\bf x}^2$ in $G_X^c$ connected to two vertices in $\\Gamma\\backslash G_X$ by three edges, and at least one of ${\\bf x}^1, {\\bf x}^2$ must\nbe connected to $\\Gamma\\backslash G_X$ by two non-collinear edges. Denote this vertex by ${\\bf x}_g$. It must be included into $\\Gamma_{max}$ which gives a contradiction and proves the claim.\n\n\nNext, we claim that the number of edges in $\\Gamma_{max}$ is $2(N-N_b)$. Indeed, each interior\nvertex in $\\Gamma_2$ has exactly two non-collinear edges incident at it. Then, on each of the next steps, we add an interior vertex together with two non-collinear edges incident at it. Since\nthe number of free vertices in $\\Gamma_{max}$ is $N-N_b$, the claim is proved.\n\n\nFinally, we claim that the rows of $R$ corresponding to the edges of $\\Gamma_{max}$ are linearly independent. Let the matrix of these rows be denoted by\n$R_{max}$. This is a square $2(N-N_b)$ matrix. We claim that an appropriate\nrow-reduction reduces $R_{max}$ to a matrix $R^\\prime_{max}$ that has\nblock-diagonal form: for each\nvertex ${\\bf x}^i$ there are exactly two rows ${\\bf r}^{i,1}, {\\bf r}^{i,2}$ of $R^\\prime_{max}$,\nand two linearly independent unit vectors ${\\bf q}^{ij_1}$ and ${\\bf q}^{ij_2}$,\nsuch that ${\\bf r}^{i,1}$ (${\\bf r}^{i,2}$) contains\n${\\bf q}^{ij_1}$ (${\\bf q}^{ij_2}$) at a place\ncorresponding to ${\\bf x}^i$ while all other entries in these rows are zero.\n\n\nTo see this, consider first a ``basic unit\" of $\\Gamma_2$: an interior vertex ${\\bf x}^i$ and two non-collinear edges\nincident at it. Let the corresponding unit vectors be ${\\bf q}^{i1}$, ${\\bf q}^{i2}$. Recall that these edges connect ${\\bf x}^i$ to two boundary vertices. Consequently,\nthe rows ${\\bf r}^1, {\\bf r}^2$ corresponding to the above pair of edges have zeros\nat all places, except two places corresponding to ${\\bf x}^i$. \nThe non-zero entries of ${\\bf r}^1$ (${\\bf r}^2$) are two components of ${\\bf q}^{i1}$\n(${\\bf q}^{i2}$).\nSince ${\\bf q}^{i1}$, ${\\bf q}^{i2}$ are linearly independent, so are\n${\\bf r}^1, {\\bf r}^2$. Furthermore, linear combinations\nof ${\\bf r}^1, {\\bf r}^2$ can be used to eliminate non-zero entries in other rows.\nBy adding an appropriate linear combination of ${\\bf r}^1, {\\bf r}^2$ to a row with\nsome unit vector ${\\bf q}^{ij}$ at a place corresponding to ${\\bf x}^i$, we can obtain\nzeros at this place. Hence, by using ${\\bf r}^1, {\\bf r}^2$ as pivots in Gaussian elimination\nwe can obtain rows whose only non-zero entries are at a place corresponding\nto a vertex that was added to $\\Gamma_2$ on the next step of the iteration. These rows, in turn, can be used as pivots. Continuing\nwith row reduction, we can eventually reduce all rows of $R_{max}$ to this form\n(this follows from the fact that $\\Gamma_{max}$ contains all vertices of $\\Gamma$). \nThe proposition is proved.\n\n\\noindent\n{\\bf Remark}. It is interesting to compare $R$ and the rigidity matrix $R^r$. It is \nwell-known that for a first-order rigid graph, the null space of $R^r$ is non-trivial\nand consists of infinitesimal rigid displacements. The proposition above shows that\nthe null space of $R$ is trivial. The main difference in structure between these two \nmatrices is that $R$ contains special rows, that might be called {\\it broken}. These\nrows correspond to edges connecting an interior vertex to a boundary one. \nA typical row of $R^r$ has four non-zero entries, while each broken row\nhas only two. These entries occur at a place corresponding to an interior vertex.\nIf an interior vertex is connected to two boundary vertices, then the regular\ntriangulation property of $\\Gamma$ ensures that the corresponding broken rows are non-collinear, and can\nbe used in the sequential Gaussian elimination, as done in the proof. \n\n\n\\begin{proposition}\n\\label{existence}\nConsider the problem (\\ref{pr1},\\ref{pr2}). Suppose that $\\Gamma$ is a regular triangulation,\nthe feasible set of (\\ref{pr1},\\ref{pr2}) is non-empty,\nand that the unconstrained minimizer of $F({\\bf z})$ is not feasible. \nThen\nthe problem (\\ref{pr1},\\ref{pr2}) admits a unique minimizer that is a point\non the boundary of its feasible set.\n\\end{proposition}\n\n\\noindent\n{\\it Proof.} \\vspace{0.1cm}\nThe problem (\\ref{pr1},\\ref{pr2}) has a feasible\npoint $\\bar {\\bf z}$. Then the problem (\\ref{pr1},\\ref{pr2}) has a unique\nminimizer ${\\bf z}^*$ as we now demonstrate.\nLet the set ${\\mathcal L}(\\bar {\\bf z})$\nbe defined by\n\\begin{equation*}\n{\\mathcal L}(\\bar {\\bf z}) = \\{{\\bf z}\\in{\\mathbb R}^{2(N-N_b)}:\nF({\\bf z}) \\leq F(\\bar {\\bf z})\\}.\n\\end{equation*}\nBy Proposition \\ref{full-rank} the matrix $R$ has full column rank. Therefore, the set\n${\\mathcal L}(\\bar {\\bf z})$ is an ellipsoid, which is a closed,\nbounded, convex set.\nThe set\n${\\mathcal H}$ of ${\\bf z} \\in {\\mathbb R}^{2(N-N_b)}$ satisfying the constraint\n(\\ref{pr2})\nis a closed half space. Therefore, ${\\mathcal L}(\\bar {\\bf z}) \\cap {\\mathcal H}$\nis a nonempty, closed, bounded convex set. Indeed, the reduced problem\n(\\ref{pr1},\\ref{pr2}) is equivalent to the problem\n\\begin{eqnarray}\n\\label{preq1}\n\\mbox{minimize}&~~&F({\\bf z})\\\\\n\\label{preq2}\n\\mbox{subject to}&~~&{\\bf z} \\in {\\mathcal L}(\\bar {\\bf z})\\cap{\\mathcal H}.\n\\end{eqnarray}\nNow by the continuity of $F$ and the compactness of\n${\\mathcal L}(\\bar {\\bf z})\\cap{\\mathcal H}$ we see that the problem\n(\\ref{preq1},{\\ref{preq2}) and hence the reduced problem\n(\\ref{pr1},\\ref{pr2}) has a minimizer ${\\bf z}^*$. Since $R$ has full\ncolumn rank, $R^{\\mbox{\\scriptsize\\sf T}} R$ is positive definite. The positive definiteness\nof the matrix $R^{\\mbox{\\scriptsize\\sf T}} R$ implies that $F$ is strictly convex\non ${\\mathcal L}(\\bar {\\bf z})\\cap {\\mathcal H}$, from which we conclude that\n${\\bf z}^*$ must be unique. \n\n\nNow if $R{\\bf z}^\\star+{\\bf a}>0$, then ${\\bf z}^\\star$ must be the\nunconstrained minimizer of $F$. This contradicts the assumption that\nthe unconstrained\nminimizer is not feasible. Therefore, some components of\n$R{\\bf z}^\\star+{\\bf a}$ must be zero, and thus ${\\bf z}^\\star$ must be on the boundary\nof the feasible region.\n\n\n\n\n\n\n\n\\section{Optimality conditions for the reduced problem}\n\\label{sect:optimality}\nThe reduced problem (\\ref{pr1},\\ref{pr2}) has a convex objective\nfunction and linear constraints. For such problems \nit is possible to state optimality conditions \nthat are both necessary and sufficient \\cite{NW}.\nTo be specific define the Lagrangian $L$ for the problem\n({\\ref{pr1},\\ref{pr2}) by\n\\begin{equation}\n\\label{L}\nL({\\bf z}, \\boldsymbol \\lambda)=\\frac 12 d^{-3}(R{\\bf z}+{\\bf a}+{\\bf d})\\cdot(R{\\bf z}+{\\bf a}+{\\bf d})-\nd^{-3}\\boldsymbol \\lambda\\cdot (R{\\bf z}+{\\bf a}).\n\\end{equation}\n\nThen ${\\bf z}^*$ solves problem (\\ref{pr1},\\ref{pr2}) if and only if\nthere exists $\\boldsymbol\\lambda^*$ such that\n${\\bf z}={\\bf z}^*$ and $\\boldsymbol\\lambda =\\boldsymbol\\lambda^*$\nsatisfy the Karush, Kuhn, Tucker (KKT) conditions\n\\begin{eqnarray*}\n&& \\nabla_{z} L ({\\bf z},\\boldsymbol\\lambda) = 0\\\\\n&& R{\\bf z} +{\\bf a}\\geq 0\\\\\n&& \\boldsymbol\\lambda\\cdot (R{\\bf z}+{\\bf a})=0\\\\\n&& \\boldsymbol\\lambda\\geq 0\n\\end{eqnarray*}\nor equivalently\n\\begin{eqnarray}\n&& R^{^{\\mbox{\\scriptsize\\sf T}}}\\left(R {\\bf z}+{\\bf a}+{\\bf d}-\\boldsymbol \\lambda\\right)=0\\label{cp1}\\\\\n&& R{\\bf z}+{\\bf a}\\geq 0\\label{cp2}\\\\\n&& \\boldsymbol\\lambda\\cdot (R{\\bf z}+{\\bf a})=0\\label{cp3}\\\\\n&& \\boldsymbol\\lambda\\geq 0\\label{cp4}.\n\\end{eqnarray}\nSee \\cite[Chapter 12]{NW}.\n\n\n\n\nBefore stating and proving the main result (Theorem \\ref{main-theorem}), we list all \nthe assumptions, including both new and previously used.\n\n\\noindent\nA1. Consider the problem of minimizing (\\ref{q-basic}) subject only to the boundary conditions\n(\\ref{b1}, \\ref{b1-bis}), but not the constraints (\\ref{i3-intro}). We assume that the minimizer\nof that problem is not feasible, that is, this minimizer does not\nsatisfy the impenetrability constraints (\\ref{i3-intro}). Further we assume that the feasible region is not\nempty, which means that there is at lest one point ${\\bf z}$ satisfying all the inequality constraints\n(\\ref{cp2}).\n\n\\noindent\nA2. The network $\\Gamma$ is a regular triangulation (as defined in Definition \\ref{reg-treg}).\n\n\\noindent\nA3. The boundary conditions are prescribed so that \n\\begin{equation}\n\\label{upper}\n|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^i+{\\bf u}^j)|\\leq a_i+a_j+\\min_{k=1,2,\n\\ldots N}a_k,\n\\end{equation}\nfor each pair ${\\bf x}^i$, ${\\bf x}^j$ of {\\it boundary vertices in contact}. \n\n\nLet us provide some comments on the nature of assumptions A1--A3.\nAssumption A1 means that minimizing the energy of the spring network subject only to boundary conditions leads to a configuration in which at least one spring is compressed (and thus violates the impenetrability constraints). \n\n\nAssumption A2 concerns the contact geometry. The edges of the network split\nthe domain of the problem (a polygon) into elementary cells (triangles). Near the boundary, the cells must be compatible with the geometry of the boundary in the following sense. If an interior vertex is connected to a boundary vertex, then it is also connected with another\nboundary vertex, located next to the first boundary vertex. Hence, every triangular cell adjacent to the exterior boundary must contain one free vertex and two boundary vertices. \n\n\nAssumption A3 means that the boundary conditions (\\ref{b1}, \\ref{b1-bis}) are chosen to prevent particles from escaping through the gaps made by displacing the boundary particles. Clearly, if two boundary particles belong to the same group, then no gap can appear between them, and $|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^i+{\\bf u}^j)|=a_i+a_j$. Formation of gaps would be possible between two boundary\nparticles from different groups which are in contact in the reference configuration. If the parameters of rigid body motions in the boundary conditions (\\ref{b1}, \\ref{b1-bis}) are prescribed arbitrarily, then the two particles may move away from each other, and open a gap large enough for a third particle to slip through. Assumption A3 prohibits formation of such gaps. \n\n\n\n\n\\begin{theorem}\n\\label{main-theorem}\nSuppose that assumptions A1-A3 hold. Then there exist $d^\\star>0$ and a vector $\\boldsymbol\\delta=\n(\\delta_1, \\delta_2, \\ldots, \\delta_E)\\in {{\\mathbb R}}^E: \\delta_l\\in (-1, -1\/2), l=1, 2, \\ldots, E$, such that for each ${\\bf d}=d\\boldsymbol\\delta$ with $d>d^\\star$,\nthe unique minimizer of (\\ref{pr1},\\ref{pr2}) has the following property. Each\ninterior vertex ${\\bf x}^i$ of $\\Gamma$ has at least\ntwo active edges incident at it. The corresponding unit vectors\n${\\bf q}^{i,j_1}, {\\bf q}^{i,j_2}$ must be linearly independent.\n\\end{theorem}\n\n\\noindent\n{\\it Proof.}\\newline\n\\noindent\n{\\it Step 1}. We claim that A3 implies that there is $c_0>0$, which\ndepends on the boundary conditions, but is independent\nof the choice of $d_{ij}$ in (\\ref{q-basic}), such that each feasible displacement\n${\\bf u}^i, i=1,2, \\ldots, N$, satisfies\n\\begin{equation}\n\\label{up-u}\n|u^i_k|\\leq c_0,\n\\end{equation}\n$k=1,2$. Indeed, first we observe that if\n${\\bf u}^i, {\\bf u}^j$ satisfy the linearized constraint (\\ref{i3-intro}),\nthen they also satisfy the distance constraint (\\ref{0.1}) (the converse is not\ntrue in general). Then any feasible collection of displacements also satisfies\nthe distance constraints (\\ref{0.1}) for each pair of neighboring vertices. Now we recall the assumption made in the introduction to conclude that (\\ref{0.1}) must hold for all pairs of vertices. Fix $l\\in \\{1,2,\\ldots, N\\}$, corresponding to an interior vertex, and consider a smaller packing ${\\mathcal P}$ of particles, containing only $D_l$ and\nall boundary particles. In the reference configuration, $D_l$ is completely surrounded by boundary particles. Then the boundary conditions are prescribed according\nto A3, the boundary particles still completely confine $D_l$, so that\n${\\bf x}^l$ must displace to ${\\bf x}^l+{\\bf w}^l$ that lies inside\na certain bounded domain $\\Omega^\\prime$ that depends only on boundary conditions.\nSince ${\\bf x}^i_k$ are bounded, this implies that the claim is true for\nall displacements ${\\bf w}^l$ which are feasible for the smaller packing ${\\mathcal P}$.\nClearly the set of all such displacements is larger than the set of all ${\\bf u}^l$\nfeasible under all constraints (\\ref{0.1}), and the latter set is larger than the set of all ${\\bf u}^l$ feasible under the linearized constraints (\\ref{i3-intro}).\nThis proves the claim.\n\n\\noindent \n{\\it Step 2}.\nLet \n\\begin{equation}\n\\label{vi}\n{\\bf v}^i=\\sum_{j\\in{\\mathcal N}_i}{\\bf q}^{ij},~~~i=1,2,\\ldots, 2(N-N_b).\n\\end{equation}\nFirst, we prove the theorem under the additional assumption\n\\begin{equation}\n\\label{a4}\n\\mbox{For each}~i=1,2,\\ldots, 2(N-N_b),~~{\\bf v}^i\\ne s{\\bf q}^{ij},\n\\end{equation}\nwhere\n$j\\in {\\mathcal N}_i, s\\in {{\\mathbb R}}$. \n\n\nWe note that (\\ref{a4}) implies that \n\\begin{equation}\n\\label{ii}\n|{\\bf v}^i|\\geq v_0>0\n\\end{equation}\nwith $v_0$ independent of $i$. Indeed, $s$ can be zero, so validity of (\\ref{a4}) means in particular\nthat all ${\\bf v}^i$ are non-zero. Since there is finitely many ${\\bf v}^i$, (\\ref{ii}) holds.\n\n\n \nConsider solutions of the KKT system\n(\\ref{cp1},\\ref{cp2},\\ref{cp3},\\ref{cp4}). From (\\ref{cp3}), (\\ref{cp4})\nit follows that $\\lambda_j=0$ if the $j$-th constraint is inactive.\nLet $\\theta_j=(R{\\bf z}+{\\bf a})_j$.\nIf the $j$-th constraint is active then $\\theta_j=0$,\nwhile $\\lambda_j$ is arbitrary. Suppose that a feasible point\n${\\bf z}^\\star$ is given. Then\n$\\theta_j$ are given. To solve (\\ref{cp1}) we need to find\n$\\boldsymbol \\lambda$. Denote\nby ${\\bf r}^k, k=1,2,\\ldots, E$\nthe rows of $R$ (the columns of $R^{\\mbox{\\scriptsize\\sf T}}$), and suppose\nthat the rows ${\\bf r}^1, {\\bf r}^2\\ldots, {\\bf r}^S$ correspond to\nthe active constraints, and that the rows\n${\\bf r}^{S+1}, {\\bf r}^{S+2},\\ldots, {\\bf r}^{E}$ correspond to the\ninactive constraints. Choose ${\\bf d}=(-d, -d, \\ldots, -d)$. \nThen (\\ref{cp1}) can be written as\n\\begin{equation}\n\\label{cp1-1}\n-\\sum_{l=1}^S {\\bf r}^l \\lambda_l+\\sum_{l=S\n+1}^E{\\bf r}^l\\theta_l+d\\sum_{l=1}^E {\\bf r}^l=0.\n\\end{equation}\n\nPick a vertex ${\\bf x}^i$ of $\\Gamma$ and consider the restriction\nof each ${\\bf r}^l$ in (\\ref{cp1-1}) to the two components\ncorresponding to ${\\bf x}^i$.\nThen we have\n\\begin{equation}\n\\label{cp-local}\n-\\sum_{j\\in {\\mathcal N}_i}^\\prime\\lambda_{ij} {\\bf q}^{ij}+\\sum_{j\\in {\\mathcal N}_i}^{\\prime\\prime}\\theta_{ij} {\\bf q}^{ij}+\nd{\\bf v}^i=0,\n\\end{equation}\nwhere the first sum is taken over active edges incident at ${\\bf x}^i$, while the second sum is over the inactive edges\nincident at ${\\bf x}^i$. \n\nNext, we determine the minimal number of active edges needed for (\\ref{cp-local}) to hold. We can look at\n(\\ref{cp-local}) as a local problem in which ${\\bf u}^i$ may vary, while ${\\bf u}^j, j\\in {\\mathcal N}_i$ are fixed.\nDenote by ${\\mathcal F}_i\\subset {\\mathbb R}^2$ the feasible region of\nthis local problem.\nBy A3, ${\\mathcal F}_i$ is a polygon, each side\nof which corresponds to one or more constraints being active. \n\nIn the generic case, one constraint per side is active. In the non-generic case, two or more\nactive constraints correspond to the same side. Since our goal is estimating the number of\nactive constraints from below, it is sufficient to consider only the generic case,\ncorresponding to the ``worst case scenario\".\nIn the generic case there are only three possibilities.\\newline\n\\noindent\n{\\it Case 1}. ${\\bf u}^i$ is inside ${\\mathcal F}_i$. All edges incident at ${\\bf x}^i$ are inactive.\\newline\n\\noindent\n{\\it Case 2}. ${\\bf u}^i$ belongs to only one of the sides of $\\partial{\\mathcal F}_i$. One edge is active.\\newline\n\\noindent\n{\\it Case 3}. ${\\bf u}^i$ is a vertex of ${\\mathcal F}_i$. Two edges are active.\n\n \nConsider case 1.\nThen (\\ref{cp-local}) cannot hold for $d$ sufficiently large. Indeed, $|{\\bf v}^i|\\geq v_0>0$ by assumption, while\n$|\\sum_{j\\in {\\mathcal N}_i}\\theta_{ij} {\\bf q}^{ij}|$ is bounded from above independent of $d$ in view of (\\ref{up-u}).\n\nConsider case 2.\nLet us number the active edge by $(i,1)$. Then (\\ref{cp-local}) can be written as\n\\begin{equation}\n\\label{cp-loc-1}\n-\\lambda_{i1}{\\bf q}^{i1}+\\sum_{j\\in {\\mathcal N}_i, j>1} \\left(({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}\\right){\\bf q}^{ij}+\nd{\\bf v}^i=0.\n\\end{equation}\nEnlarging $d$, if necessary, we see that (\\ref{cp-loc-1}) can hold only if\n\\begin{equation}\n\\label{crit1} \n{\\bf v}^i=s{\\bf q}^{i1},\n\\end{equation}\n where\n$s<0$. Since\n(\\ref{crit1}) is not allowed by (\\ref{vi}), (\\ref{cp-local}) cannot hold for sufficiently large $d$.\n\nConsider case 3. Number the two active edges by $(i, 1)$, $(i,2)$. The equation (\\ref{cp-local}) is\n\\begin{equation}\n\\label{cp-loc-2}\n-\\lambda_{i1}{\\bf q}^{i1}-\\lambda_{i2}{\\bf q}^{i2}+\n\\sum_{j\\in {\\mathcal N}_i, j>2} \\left(({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}\\right){\\bf q}^{ij}+\nd{\\bf v}^i=0.\n\\end{equation}\nFor this to hold for large $d$, ${\\bf v}^i$ must be a non-positive linear combination\nof ${\\bf q}^{i1}, {\\bf q}^{i2}$. These two vectors are linearly independent, otherwise\ntheir intersection would not be a vertex of ${\\mathcal F}_i$. So, Case 3 is possible, provided\n${\\bf v}^i$ lies in the negative cone of two active edges.\n\n\\noindent\n{\\it Step 3}. Now we remove the assumption (\\ref{a4}). For each $i=1, 2, \\ldots, (N-N_b)$, and each $\\boldsymbol\\delta\\in {{\\mathbb R}}^E$, define a two-dimensional\nvector\n$\\tilde{\\bf v}^i$ to be the restriction of $R^{^{\\mbox{\\scriptsize\\sf T}}} \\boldsymbol\\delta\\equiv\\sum_{l=1}^E \\delta_l{\\bf r}^l$ to a place $i$. The theorem will be proved is we show that there is\na choice of $\\boldsymbol\\delta$ such that $\\delta_{l}\\in [1\/2, 1]$, and $\\tilde{\\bf v}^i$ has property (\\ref{a4}). \nIndeed, if such $\\boldsymbol\\delta$ is found, we could choose ${\\bf d}=d\\boldsymbol\\delta$, where $d>0$ is sufficiently large, and repeat the arguments made in the first step, using \n$\\tilde{\\bf v}^i$ instead of ${\\bf v}^i$. \n\n\nTo show existence of $\\boldsymbol\\delta$, consider the cube $C_E=\\{{\\bf y}\\in {{\\mathbb R}}^E: y_l\\in (1\/2, 1), l=1, 2, \\ldots, E\\}$. Pick any point ${\\bf y}^\\star\\in C_E$. Since $C_E$ is open, there is a Euclidean open ball\n$B({\\bf y}^\\star, \\rho)\\subset C_E$, with the radius $\\rho>0$. Consider the image of $B({\\bf y}^\\star, \\rho)$\nunder the mapping $R^{^{\\mbox{\\scriptsize\\sf T}}}$. Since $R$ has full rank, $R^{^{\\mbox{\\scriptsize\\sf T}}}$ is surjective, and is therefore an open mapping.\nThus, $R^{^{\\mbox{\\scriptsize\\sf T}}}(B({\\bf y}^\\star, \\rho))$ contains a Euclidean open ball $B(R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star, \\rho^\\star)$ of a positive radius $\\rho^\\star$ depending only on $R^{^{\\mbox{\\scriptsize\\sf T}}}$ and $\\rho$, but not on ${\\bf y}^\\star$. If $R^{^{\\mbox{\\scriptsize\\sf T}}} {\\bf y}^\\star$ has\nproperty (\\ref{a4}), we choose $\\boldsymbol\\delta={\\bf y}^\\star$ and we are done. Otherwise,\nnote that for each $i=1, 2, \\ldots, (N-N_b)$, the ball $B(R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star, \\rho^\\star)$ contains a non-empty two-dimensional\nEuclidean open ball $B_i$ centered at the restriction of $R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star$ to the place $i$. Since for each $i$\nthe set $\\{{\\bf v}\\in {{\\mathbb R}}^2: {\\bf v}=s{\\bf q}^{ij}, s\\in {{\\mathbb R}}, j\\in {\\mathcal N}^i\\}$ is a union of a finite number of lines, it cannot contain a two-dimensional ball. Therefore, for each $i=1, 2, \\ldots, (N-N_b)$ \nthere must be a vector $\\tilde{\\bf v}^i\\in B_i$ having property (\\ref{a4}). Now we can define \n$\\sum_{l=1}^E \\delta_l{\\bf r}^l$ via its restrictions $\\tilde{\\bf v}^i$. Next, by construction, we can find\na vector $\\boldsymbol\\delta\\in B({\\bf y}^\\star, \\rho)\\subset C_E$ such that $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta=\\sum_{l=1}^E \\delta_l{\\bf r}^l$. \n\nThe theorem is proved.\n\n\\noindent\n{\\bf Remark}. \nThe choice \nof $\\boldsymbol\\delta$ in the proof is based on the following criterion. Consider the vector $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta\\in {{\\mathbb R}}^{2(N-N_b)}$. The proof works if\n\\begin{equation}\n\\label{last}\nR^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta\\ne({\\bf v}_1, {\\bf v}_2, \\ldots, {\\bf v}_{N-N_b}),\n\\end{equation}\nwhere at least one of the two-dimensional vectors ${\\bf v}_i, i=1, 2, \\ldots, (N-N_b)$ is of the form\n${\\bf v}_i=s{\\bf q}^{ij}$, for some real $s$ and $j\\in {\\mathcal N}_i$. In other words, if ${\\bf v}_i$ \nis inadmissible, then it must lie on the line through the origin with direction vector ${\\bf q}^{ij}$. \nFor each fixed $i$, the inadmissible set $V_i=\\{{\\bf v}_i\\in {\\mathbb R}^2: {\\bf v}_i=s{\\bf q}^{ij}\\}$ has Hausdorff dimension one (a finite union of lines in the plane), while the admissible set is the two-dimensional complement of $V_i$ in ${\\mathbb R}^2$. Therefore, the set of inadmissible vectors in the right hand side of (\\ref{last}) has dimension $2(N-N_b)-1$ while the set of admissible vectors $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta$ is of dimension $2(N-N_b)$. Thus the admissible $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta$ are generic, and the theorem holds for a generic (in the sense of Definition \\ref{reg-treg}) packing under a generic pre-stress (the latter is determined by a generic choice of $\\boldsymbol\\delta$).\n\n\n\n\n\n\\section{Order parameter}\nRecently, a phenomenological theory of slow dense granular flows was proposed in\n\\cite{AT, VTA}. A key quantity in that theory is the order parameter, defined as the ratio\nof the number of solid-like contacts to the number of all contacts within a given control volume.\nIn \\cite{VTA}, a contact is considered solid-like if two particles are jammed together for longer\nthan a characteristic collision time. The relevant characteristic\ntime is $\\tau=a\/v_a$, where $a$ is particle radius and $v_a$ is the speed of sound in\na solid material of the particles. Our model corresponds to the instantaneous material\nresponse, when $\\tau$ is much smaller than other relaxation times in the system, such\nas the ratio of the sample size to a typical particle velocity.\n\nAn obvious type of pair motion leading to a solid-like contact is a rigid displacement\n(a pair of particles infinitesimally moves as a rigid body). We shall call this type of contacts {\\it stuck}. If a contact between $D_i$ and $D_j$ is stuck, then $({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}=0$, which is easy to check using the definition of rigid\ndisplacements. This means that the impenetrability constraint for the corresponding\nedge of $(i,j)$ of the network is satisfied as an equation (the edge is active). \nHowever, not every\nactive edge corresponds to a stuck contact. Another type of a local motion that produces\n$({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}=0$ is an infinitesimal shear motion when\n${\\bf u}^i-{\\bf u}^j$ is orthogonal to ${\\bf q}^{ij}$. The corresponding contact is\ncalled {\\it sheared}. Note also that infinitesimal shear\nis the same as infinitesimal rotation, so this type of motion includes infinitesimal rolling\nas well as shear sliding. \n\n\nWe consider both sheared and stuck contact as solid-like, because stuck contacts are stable, while sheared contacts in an actual granular material will be subject to friction. Friction can be viewed as partially stabilizing, at least when the shearing force\nis below the static friction threshold. Such non-sliding frictional contacts are considered as solid-like in the simulations performed in \\cite{VTA}. In addition, some heuristic arguments and numerical simulations \npresented in \\cite{GG1, GG2}, suggest that friction enhances elastic behavior of sufficiently large samples. Therefore, it makes sense to think of the network of solid-like contacts as the\nmain load-bearing structure and call this network {\\it strong}. \nIn contrast, a broken contact satisfying (\\ref{broken}) corresponds to\na local weakening in the material because in this case two particles separate completely.\nWe can think of the network of all broken contacts as {\\it weak}. \nMoreover, division of contacts into broken and solid-like corresponds to\nthe division of constraints into active and inactive, as done in optimization theory.\nTherefore, this division is natural mathematically, and also makes sense from the physics point of view. \n\n\nIn addition, the definition in \\cite{VTA} does not sufficiently clarify the nature of averaging. \nThe notion of an order parameter in static problems should not use time averaging. \nThe result of spatial averaging depends on the size of the sample that is being averaged. Thus, if the order\nparameter is obtained by, say, spatial averaging, then it must depend on both location and size of the \n``control volume\". In the discrete situation, the size of the averaging sample can be measured\nby the minimal number of edges connecting a pair of vertices within the sample.\n\n\nThis suggests a definition of the size-dependent order parameter. To state\nthis definition we first define the averaging sample. \n\\begin{definition}\n\\label{k-n}\nA vertex ${\\bf x}^j$ is in the k-th neighborhood of ${\\bf x}^i$\nif $\\Gamma$ contains a path connecting ${\\bf x}^i$ and ${\\bf x}^j$ with\nno more than $k$ edges.\n\\end{definition}\nNow, to each $k$-neighborhood we can associate a value of an order parameter. \n\\begin{definition}\n\\label{order}\nFor each ${\\bf x}^i$ and each non-negative integer $k\\leq N$, the size dependent\norder parameter $\\rho({\\bf x}^i, k)$ is defined by\n\\begin{equation}\n\\label{op}\n\\rho({\\bf x}^i, k)=\\frac{\\sum_{k} n_s}{\\sum_k n},\n\\end{equation}\nwhere the numerator is the number of active edges in $k$-neighborhood of ${\\bf x}^i$,\nand denominator is the number of all edges in that neighborhood.\n\\end{definition}\n\n\nTheorem \\ref{main-theorem} implies the lower bound \n\\begin{equation}\n\\label{simple}\n\\rho({\\bf x}^i,N)\\geq \\frac{N}{E}\n\\end{equation}\non the order parameter associated with the maximal, $N$-th neighborhood of each interior vertex ${\\bf x}^i$.\nIndeed, counting active edges (two per vertex) gives $2N$ edges, each counted at most twice.\nIn particular, (\\ref{simple}) means that the order parameter $\\rho({\\bf x}^i, N)$\nis bounded from below by the reciprocal of the mean coordination number of the network. \n \n\n\\section{Conclusions}\n\nWe have studied a network model of quasi-static deformation of dense pre-stressed\ngranular materials. \nIn our model, the packing was represented by a network of linear elastic springs.\nEach spring corresponds to a contact between two particles.\nGeometric impenetrability constraints within the packing were modeled by the\nlinearized impenetrability constraints on the displacements of the vertices of the\nnetwork. The constraints have the form of linear inequalities, that can be satisfied either\nas an equality (an active constraint), or as a strict inequality (inactive constraint).\nConstraints are in one-to one correspondence with the interpaticle contacts. An active constraint\ncorresponds to a relatively stable solid-like contact. Inactive constraints represent the relatively weak\nbroken contacts.\nThe question addressed in the paper is to estimate the total number and distribution of the solid-like\ncontacts in the energy-minimizing configuration. We showed that each interior vertex of the network has at least two \nsolid-like contacts corresponding to it. This result qualitatively\nreproduces the micro-band structure obtained in \\cite{Kuhn1, Kuhn2} by numerical simulations. We also discussed the connection between our result and a lower bound on the\norder parameter \\cite{AT, VTA}. In the paper, we proposed a definition of the\norder parameter that is similar to the one introduced in \\cite{VTA}, but differs from it in the interpretation of the so-called solid-like contacts. On the one hand, our definition appears to be in accord with a physical picture of granular statics, recently\nproposed in \\cite{GG1, GG2}. On the other hand, it is a naturally related to optimization theory. \n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Derivation of equation (2)}\nIn this appendix we show that eqs.\\ (20)-(22) in \\cite{BrAn98} are\nidentical with the string Bethe equations of Takahashi and that\n(\\ref{centre}) follows from these equations.\n\nLet us first cite the main results of \\cite{BrAn98}: The $n$-complex\nis parameterized by an $n$-string of the form\n\\begin{equation}\n \\phi_{a,j}^{(n)} = \\phi_a^{(n)} + (n + 1 - 2j) \\frac{\\text{i} u}{4}\n \\quad, \\quad j = 1, \\dots, n .\n\\end{equation}\n[Note the slight change of notation compared to \\cite{BrAn98}: We made\nthe replacement $\\phi_0^{(n)} \\rightarrow \\phi_a^{(n)}$ (and\n$\\phi_j^{(n)} \\rightarrow \\phi_{a,j}^{(n)}$) which allows us to include\nscattering of two different $n$ complexes of the same length. $a$\nenumerates the $n$-strings.] The S-matrix of an unbound particle\nwith an $n$-complex is\n\\begin{equation}\n S_{k \\phi_a^{(n)}}^{u (n)} = \\frac{\\sin k - \\phi_a^{(n)} - \n n \\textstyle{\\frac{\\text{i} u}{4}}}{\\sin k - \\phi_a^{(n)} +\n n \\textstyle{\\frac{\\text{i} u}{4}}},\n\\end{equation}\nand the S-matrix of an $m$-complex with an $n$-complex is\n\\begin{equation}\n S_{\\phi_b^{(m)} \\phi_a^{(n)}}^{(m) (n)}\n \\: = \\: \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t |n - m| \\textstyle{\\frac{\\text{i} u}{4}}}\n {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t |n - m| \\textstyle{\\frac{\\text{i} u}{4}}} \\:\n \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t (n + m) \\textstyle{\\frac{\\text{i} u}{4}}}\n {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t (n + m) \\textstyle{\\frac{\\text{i} u}{4}}} \\:\n\t \\prod_{l=1}^{\\min(m,n) - 1} \\left(\n \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t (|n - m| + 2l) \\textstyle{\\frac{\\text{i} u}{4}}}\n {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t (|n - m| + 2l) \\textstyle{\\frac{\\text{i} u}{4}}} \\right)^2.\n\\end{equation}\nDiagonalizing the transfer matrix leads to the following set of\nequations\n\\begin{eqnarray} \\label{bac1}\n e^{\\text{i} k_j L} & = & \\prod_{\\delta = 1}^{M^u}\n \\frac{\\lambda_\\delta - \\sin k_j - \\textstyle{\n\t \\frac{\\text{i} u}{4}}}{\\lambda_\\delta - \\sin k_j +\n\t \\textstyle{\\frac{\\text{i} u}{4}}}\n \\prod_{(n,a)} S_{\\phi_a^{(n)} k_j}^{(n) u}, \\\\ \\label{bac2}\n \\prod_{\\delta \\ne \\lambda}^{M^u}\n \\frac{\\lambda_\\gamma - \\lambda_\\delta -\n\t \\textstyle{\\frac{\\text{i} u}{2}}}{\\lambda_\\gamma -\n\t \\lambda_\\delta + \\textstyle{\\frac{\\text{i} u}{2}}}\n\t & = &\n \\prod_{j = 1}^{N^u}\n \\frac{\\lambda_\\gamma - \\sin k_j -\n\t \\textstyle{\\frac{\\text{i} u}{4}}}{\\lambda_\\gamma -\n\t \\sin k_j + \\textstyle{\\frac{\\text{i} u}{4}}},\n\t \\\\ \\label{bac3}\n e^{\\text{i} q^{(n)}(\\phi_a^{(n)})L} & = &\n \\prod_{(m,b) \\ne (n,a)} S_{\\phi_b^{(m)}\n\t \\phi_a^{(n)}}^{(m) (n)}\n \\prod_{j=1}^{N^u} S_{k_j \\phi_a^{(n)}}^{u (n)},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n q^{(n)} (\\phi) = - 2 \\text{Re} \\arcsin(\\phi + n \\text{i} u\/4) \n + (n + 1) \\pi =\n - (\\arcsin(\\phi + n \\text{i} u\/4)\n + \\arcsin(\\phi - n \\text{i} u\/4)) + (n + 1) \\pi\n\\end{equation}\nand\n\\begin{equation}\n S_{\\phi_a^{(n)} k_j}^{(n) u} =\n \\left( S_{k_j \\phi_a^{(n)}}^{u (n)} \\right)^{-1}.\n\\end{equation}\n\nEqs. (\\ref{bac1})-(\\ref{bac3}) are the ``new'' equations of Braak and\nAndrei ((20)-(22) or (BAC) in \\cite{BrAn98}). In order to show that\n(\\ref{bac1})-(\\ref{bac3}) agree with Takahashi's string Bethe equations\nlet us now adjust notations. Let\n\\begin{equation}\n U = u\/4 \\quad, \\quad {\\Lambda'}_a^m = \\phi_a^{(m)} \\quad,\n \\quad \\alpha = a \\quad, \\quad \\beta = b.\n\\end{equation}\nLet us further introduce the functions \\cite{Takahashi72}\n\\begin{eqnarray}\n e(x) & = & \\frac{x + \\text{i}}{x - \\text{i}} \\\\\n E_{nm} (x) & = & \\left\\{ \\begin{array}{l} \n {\\displaystyle\n e \\left( \\frac{x}{|n - m|} \\right)\n e^2 \\left( \\frac{x}{|n - m| + 2} \\right)\n\t\t \\cdots\n e^2 \\left( \\frac{x}{n + m - 2} \\right)\n e \\left( \\frac{x}{n + m} \\right)\n\t\t \\: \\text{for} \\quad n \\ne m,} \\\\[3ex]\n {\\displaystyle\n e^2 \\left( \\frac{x}{2} \\right)\n e^2 \\left( \\frac{x}{4} \\right)\n\t\t \\cdots\n e^2 \\left( \\frac{x}{2n - 2} \\right)\n e \\left( \\frac{x}{2n} \\right)\n\t\t \\: \\text{for} \\quad n = m.}\n\t\t \\end{array} \\right.\n\\end{eqnarray}\nThen eqs.\\ (\\ref{bac1})-(\\ref{bac3}) turn into\n\\begin{eqnarray} \\label{tbac1}\n e^{\\text{i} k_j L} & = & \\prod_{\\delta = 1}^{M^u}\n\t e \\left( \\frac{\\sin k_j - \\lambda_\\delta}{U} \\right)\n \\prod_{(n,\\alpha)}\n\t e \\left( \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t \\\\ \\label{tbac2}\n \\prod_{j = 1}^{N^u}\n e \\left( \\frac{\\lambda_\\gamma - \\sin k_j}{U} \\right)\n\t & = &\n - \\prod_{\\delta = 1}^{M^u}\n e \\left( \\frac{\\lambda_\\gamma - \\lambda_\\delta}{2U} \\right),\n\t \\\\ \\label{tbac3} \\nonumber\n \\exp \\{ - \\text{i} L\n [\\arcsin({\\Lambda'}_\\alpha^n + n \\text{i} U) & + &\n\t \\arcsin({\\Lambda'}_\\alpha^n - n \\text{i} U) +\n\t (n + 1) \\pi] \\} \\\\ & = &\n - \\prod_{j=1}^{N^u}\n\t e \\left( \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right)\n \\prod_{(m,\\beta)}\n\t E_{nm} \\left(\n\t \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U} \\right).\n\\end{eqnarray}\nThe spin rapidities $\\lambda_\\gamma$ in (\\ref{tbac1}) and (\\ref{tbac2})\nmay generally be complex. In order to obtain a set of equations which\ncontains only real unknowns and which transforms into a set of linear\nintegral equations in the thermodynamic limit we have to employ\nTakahashi's string hypothesis for $\\Lambda$ strings: As the number $N$\nof electrons becomes large the spin rapidities are driven to string\npositions characterized by their length $n$ and their real center\n$\\Lambda_\\alpha^n$. Following Takahashi \\cite{Takahashi72} we will\nuse the notation $\\Lambda_\\alpha^{n,j}$ instead of $\\lambda_\\gamma$.\n$\\Lambda_\\alpha^{n,j}$ is the $j$-th spin rapidity involved in an\n$n$-$\\Lambda$ string with center $\\Lambda_\\alpha^n$,\n\\begin{equation}\n \\Lambda_\\alpha^{n,j} = \\Lambda_\\alpha^n + (n + 1 - 2j) \\text{i}U.\n\\end{equation}\nFollowing again Takahashi let us assume that in the thermodynamic\nlimit all $\\lambda_\\gamma$ are grouped into strings with an accuracy\nof ${\\cal O}(\\exp( - \\delta N))$, where $\\delta$ is some positive\nnumber. Then eqs.\\ (\\ref{tbac1})-(\\ref{tbac3}) lead to\n\\begin{eqnarray} \\label{ttbac1}\n e^{\\text{i} k_j L} & = & \\prod_{(n,\\alpha)}\n\t e \\left( \\frac{\\sin k_j - \\Lambda_\\alpha^n}{nU} \\right)\n \\prod_{(n,\\alpha)}\n\t e \\left( \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t \\\\ \\label{ttac2}\n \\prod_{j = 1}^{N^u}\n e \\left( \\frac{\\Lambda_\\alpha^n - \\sin k_j}{nU} \\right)\n\t & = &\n - \\prod_{(m,\\beta)}\n E_{nm} \\left( \\frac{\\Lambda_\\alpha^n - \\Lambda_\\beta^m}{U}\n\t \\right), \\\\ \\label{ttbac3} \\nonumber\n \\exp \\{ - \\text{i} L\n [\\arcsin({\\Lambda'}_\\alpha^n + n \\text{i} U) & + &\n\t \\arcsin({\\Lambda'}_\\alpha^n - n \\text{i} U) +\n\t (n + 1) \\pi] \\} \\\\ & = &\n - \\prod_{j=1}^{N^u}\n\t e \\left( \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right)\n \\prod_{(m,\\beta)}\n\t E_{nm} \\left(\n\t \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U} \\right).\n\\end{eqnarray}\n\nTaking logarithms we arrive at the following form of the string\nBethe equations, which is suitable for considering the thermodynamic\nlimit,\n\\begin{eqnarray} \\label{t1}\n k_j L & = & 2 \\pi I_j - \\sum_{n=1}^\\infty \\sum_{\\alpha = 1}^{M_n}\n \\theta \\left(\n\t\t \\frac{\\sin k_j - \\Lambda_\\alpha^n}{nU} \\right)\n - \\sum_{n=1}^\\infty \\sum_{\\alpha = 1}^{M_n'}\n \\theta \\left(\n\t\t \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t\t \\\\ \\label{t2}\n \\sum_{j=1}^{N - 2M'} \\theta \\left(\n\t\t \\frac{\\Lambda_\\alpha^n - \\sin k_j}{nU} \\right) & = &\n\t\t 2 \\pi J_\\alpha^n +\n\t\t \\sum_{m=1}^\\infty \\sum_{\\beta = 1}^{M_m}\n\t\t \\Theta_{nm} \\left(\n\t\t \\frac{\\Lambda_\\alpha^n - \\Lambda_\\beta^m}{U} \\right),\n\t\t \\\\ \\label{t3}\n L [\\arcsin({\\Lambda'}_\\alpha^n + \\text{i}nU)\n + \\arcsin({\\Lambda'}_\\alpha^n - \\text{i}nU)] & = &\n\t 2 \\pi {J'}_\\alpha^n +\n\t\t \\sum_{j=1}^{N - 2M'} \\theta \\left(\n\t\t \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right) +\n\t\t \\sum_{m=1}^\\infty \\sum_{\\beta = 1}^{M_m'}\n\t\t \\Theta_{nm} \\left(\n\t\t \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U}\n\t\t \\right).\n\\end{eqnarray}\nHere we assumed $L = 2 \\times \\text{odd}$ to be even.\n$I_j$, $J_\\alpha^n$, and ${J'}_\\alpha^n$ are integer or half-odd\ninteger numbers, $N$ is the total number of electrons, $M' =\n\\sum_{n=1}^\\infty n M_n'$, $\\theta(x) = 2 \\arctan(x)$, and\n\\begin{equation} \\label{defthetas}\n \\Theta_{nm} (x) = \\left\\{ \\begin{array}{l}\n\t{\\displaystyle\n \\theta \\left( \\frac{x}{|n - m|} \\right) +\n 2 \\theta \\left( \\frac{x}{|n - m| + 2} \\right) + \\cdots +\n 2 \\theta \\left( \\frac{x}{n + m - 2} \\right) +\n \\theta \\left( \\frac{x}{n + m} \\right), \\: \\text{if} \\quad\n\tn \\ne m,} \\\\[3ex]\n\t{\\displaystyle\n 2 \\theta \\left( \\frac{x}{2} \\right) +\n 2 \\theta \\left( \\frac{x}{4} \\right) + \\cdots +\n 2 \\theta \\left( \\frac{x}{2n - 2} \\right) +\n \\theta \\left( \\frac{x}{2n} \\right), \\: \\text{if} \\quad n = m.}\n\t\\end{array} \\right.\n\\end{equation}\nThe branch of $\\arcsin(x)$ in (\\ref{t3}) is fixed as $- \\pi\/2 \\le\n\\text{Re} (\\arcsin (x)) \\le \\pi\/2$. $M_n$ and $M_n'$ are the numbers of\n$\\Lambda$ strings of length $n$, and $\\Lambda'$ strings of length $n$\nin a specific solution of the system (\\ref{t1})-(\\ref{t3}). The\ninteger (half-odd integer) numbers in (\\ref{t1})-(\\ref{t3}) have\nranges\n\\begin{eqnarray} \\label{r1}\n && - \\frac{L - 1}{2} \\le I_j \\le \\frac{L - 1}{2}, \\\\ \\label{r2}\n && |J_\\alpha^n| \\le \\frac{1}{2}\n \\left(N - 2M' - \\sum_{m=1}^\\infty t_{nm} M_m - 1 \\right), \\\\\n\t\\label{r3}\n && |{J'}_\\alpha^n| \\le \\frac{1}{2}\n \\left(L - N + 2M' - \\sum_{m=1}^\\infty t_{nm} M_m' - 1 \\right),\n\\end{eqnarray}\nwhere $t_{mn} = 2 \\min (m,n) - \\delta_{mn}$. Each set of numbers\n$\\{I_j\\}, \\{J_\\alpha^n\\}, \\{{J'}_\\alpha^n\\}$ is in one-to-one\ncorrespondence with a set of rapidities $\\{k_j\\}, \\{\\Lambda_\\alpha^n\\},\n\\{{\\Lambda'}_\\alpha^n\\}$, which in turn unambiguously specifies\none Bethe eigenstate of the Hubbard Hamiltonian. Thus the ground state\nand all excited states can be constructed by specifying a set of\nnumbers $\\{I_j\\}, \\{J_\\alpha^n\\}, \\{{J'}_\\alpha^n\\}$ and then taking the\nthermodynamic limit.\n\nIt is our aim to prove eq.\\ (\\ref{centre}) for the charge singlet\nexcitation over the half-filled ground state. The ground state is\ncharacterized by $N = L$, $M_1 = L\/2$. In this case the inequalities\n(\\ref{r1}) and (\\ref{r2}) lead to unique distributions of the quantum\nnumbers $I_j$ and $J_\\alpha^1$,\n\\begin{equation} \\label{ijg}\n I_j = - (L + 1)\/2 + j \\quad, \\quad j = 1, \\dots, L \\quad, \\quad\n J_\\alpha^1 = - (L + 2)\/4 + \\alpha \\quad, \\quad\n \\alpha = 1, \\dots, L\/2.\n\\end{equation}\nEqs.\\ (\\ref{t1}) and (\\ref{t2}) reduce to\n\\begin{eqnarray} \\label{g1}\n && L k_j = 2\\pi I_j - \\sum_{\\alpha = 1}^{L\/2}\n \\theta \\left( \\frac{\\sin k_j - \\Lambda_\\alpha}{U}\n\t\t \\right) \\quad, \\quad j = 1, \\dots, L,\\\\\n\t\t \\label{g2}\n && \\sum_{j=1}^L \\theta \\left( \\frac{\\Lambda_\\alpha - \\sin k_j}{U}\n\t\t \\right) = 2\\pi J_\\alpha^1 +\n \\sum_{\\beta = 1}^{L\/2} \\theta \\left(\n\t \\frac{\\Lambda_\\alpha - \\Lambda_\\beta}{2U} \\right)\n\t \\quad, \\quad \\alpha = 1, \\dots, L\/2.\n\\end{eqnarray}\nIn the thermodynamic limit eqs.\\ (\\ref{g1}) and (\\ref{g2}) turn\ninto the well known integral equations \\cite{LiWu68}\n\\begin{eqnarray} \\label{gi1}\n \\rho(k) & = & \\frac{1}{2\\pi} + \\frac{1}{\\pi} \\cos k\n \\int_{- \\infty}^\\infty d \\Lambda \\;\n\t\t \\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t \\sigma(\\Lambda), \\\\ \\label{gi2}\n \\sigma(\\Lambda) & = & \\frac{1}{2\\pi^2} \\int_{-\\pi}^\\pi dk \\;\n\t\t \\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t - \\frac{1}{\\pi} \\int_{-\\infty}^\\infty d \\Lambda' \\;\n\t\t \\frac{2U}{4U^2 + (\\Lambda - \\Lambda')^2}\n\t\t \\sigma(\\Lambda')\n\\end{eqnarray}\nfor the densities $\\rho(k_j) = 1\/(L(k_{j+1} - k_j))$ and\n$\\sigma(\\Lambda_\\alpha) = 1\/(L(\\Lambda_{\\alpha + 1} - \\Lambda_\\alpha))$.\n\nThe charge singlet is characterized by $N = L$, $M_1 = L\/2 - 1$ and\n$M_1' = 1$ \\cite{EsKo94a,EsKo94b}. Thus $M' = 1$, and the number of\nunbound electrons is $L - 2M' = L - 2$. We will denote quantities which\ndescribe the charge singlet by a tilde. Eqs.\\ (\\ref{r2}) and (\\ref{r3})\nuniquely determine the set $\\{\\tilde J_\\alpha^1\\}$ and the number\n$\\tilde {J'}^1$,\n\\begin{equation}\n \\tilde J_\\alpha^1 = - L\/4 + \\alpha \\quad\n \\alpha = 1, \\dots, L\/2 - 1 \\quad, \\quad\n \\tilde {J'}^1 = 0.\n\\end{equation}\nThe set $\\{\\tilde I_j\\}$, however, is not uniquely determined by\nthe inequality (\\ref{r3}). There are $L \\choose 2$ inequivalent such\nsets, which are parameterized by two vacancies $I_1^h$ and $I_2^h$ in the\ndistribution (\\ref{ijg}) of the numbers $I_j$, which characterize the\nground state. These two vacancies determine two charge rapidities\n$k_1^h$ and $k_2^h$ via eqs.\\ (\\ref{g1}) and (\\ref{g2}). In the\nthermodynamic limit the charge rapidities densely fill the interval\n$[-\\pi,\\pi]$, and $k_1^h$ and $k_2^h$ become the two free parameters of\nthe charge singlet excitation. We see already at this stage, that\nthere can not be a third free parameter, since $\\tilde{J'}^1 = 0$ is\nfixed. For the charge singlet excitation eqs.\\ (\\ref{t1})-(\\ref{t3})\nreduce to\n\\begin{eqnarray} \\label{e1}\n && L \\tilde k_j = 2\\pi \\tilde I_j - \\sum_{\\alpha = 1}^{L\/2 - 1}\n \\theta \\left(\n\t\t \\frac{\\sin \\tilde k_j - \\tilde \\Lambda_\\alpha}{U}\n\t\t \\right)\n - \\theta \\left( \\frac{\\sin \\tilde k_j - \\Lambda'}{U}\n\t\t \\right)\n\t\t \\quad, \\quad j = 1, \\dots, L - 2,\\\\\n\t\t \\label{e2}\n && \\sum_{j=1}^{L -2} \\theta \\left(\n \\frac{\\tilde \\Lambda_\\alpha - \\sin \\tilde k_j}{U}\n\t\t \\right) = 2\\pi \\tilde J_\\alpha^1 +\n \\sum_{\\beta = 1}^{L\/2 - 1} \\theta \\left(\n\t \\frac{\\tilde \\Lambda_\\alpha -\n\t \\tilde \\Lambda_\\beta}{2U} \\right)\n\t \\quad, \\quad \\alpha = 1, \\dots, L\/2 - 1, \\\\ \\label{e3}\n && L(\\arcsin(\\Lambda' + \\text{i} U) +\n \\arcsin(\\Lambda' - \\text{i} U)) =\n \\sum_{j=1}^{L-2} \\theta \\left(\n\t \\frac{\\Lambda' - \\sin \\tilde k_j}{U} \\right).\n\\end{eqnarray}\nLet us subtract (\\ref{g1}) from (\\ref{e1}) for $I_l = \\tilde I_j$,\n$j = 1, \\dots, L-2$, and (\\ref{g2}) from (\\ref{e2}) for $\\alpha = 1,\n\\dots, L\/2 - 1$. Then, taking the thermodynamic limit, we obtain\na pair of integral equations for the shift functions\n\\begin{equation}\n F^c (k_j) = \\frac{\\tilde k_j - k_j}{k_{j+1} - k_j} \\quad, \\quad\n F^s (\\Lambda_\\alpha) = \\frac{\\tilde \\Lambda_\\alpha -\n \\Lambda_\\alpha}{\\Lambda_{\\alpha + 1} - \\Lambda_\\alpha}.\n\\end{equation}\nThese integral equations are\n\\begin{eqnarray} \\label{f1}\n F_{CS}^c (k) & = & - \\frac{1}{2} - \\frac{1}{2\\pi}\n \\theta \\left( \\frac{\\sin k - \\Lambda'}{U}\n\t\t\t\\right)\n\t\t\t+ \\frac{1}{\\pi} \\int_{-\\infty}^\\infty\n\t\t\td \\Lambda \\;\n\t\t\t\\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t\tF_{CS}^s (\\Lambda), \\\\ \\label{f2}\n F_{CS}^s (\\Lambda) & = & 1 + \\frac{1}{2\\pi} \\sum_{l = 1}^2\n \\theta \\left(\n\t\t\t \\frac{\\Lambda - \\sin k_l^h}{U} \\right) -\n\t\t\t \\frac{1}{\\pi} \\int_{-\\infty}^\\infty\n\t\t\t d \\Lambda' \\;\n\t\t\t \\frac{2U}{4U^2 + (\\Lambda - \\Lambda')^2}\n\t\t\t F_{CS}^s (\\Lambda').\n\\end{eqnarray}\nHere we have supplied an index ``CS'' to emphasize that we are dealing\nwith the charge singlet excitation. The solution $F_{CS}^c (k)$ of\n(\\ref{f1}) can be found on page 526 of \\cite{EsKo94b}. It is a single\nvalued function of $\\sin k$. In the derivation of (\\ref{gi2}) and\n(\\ref{f2}) we made use of the following elementary lemma,\n\\begin{equation} \\label{lemma}\n \\int_{-\\pi}^\\pi dk \\; f(\\sin k) \\cos k = 0,\n\\end{equation}\nwhich holds for arbitrary single valued functions $f(x)$. This lemma\nfollows from the identities $\\sin (\\pi - k) = \\sin k$ and\n$\\cos (\\pi - k) = - \\cos k$.\n\nNote that there are still three free parameters, $k_1^h$, $k_2^h$\nand $\\Lambda'$, in (\\ref{f1}) and (\\ref{f2}). $\\Lambda'$ becomes\nfixed as a function of $k_1^h$ and $k_2^h$ by considering eq.\\\n(\\ref{e3}), which in the thermodynamic limit turns into\n\\begin{eqnarray} \\nonumber\n \\lefteqn{\n L(\\arcsin(\\Lambda' + \\text{i}U) + \\arcsin(\\Lambda' - \\text{i}U))}\n \\\\ & = &\n L \\int_{-\\pi}^\\pi dk \\; \\theta \\left( \\frac{\\Lambda' - \\sin k}{U}\n \\right) \\rho (k) - 2 \\int_{-\\pi}^\\pi dk \\;\n\t\\frac{U}{U^2 + (\\Lambda' - \\sin k)^2} F_{CS}^c (k) \\cos k\n\t- \\sum_{l=1}^2 \\theta \\left(\\frac{\\Lambda' - \\sin k_l^h}{U}\n\t\\right) + {\\cal O} \\left( \\frac{1}{L} \\right) \\nonumber\n \\\\ & = & \\label{rhs}\n \\frac{L}{2\\pi} \\int_{-\\pi}^\\pi dk \\;\n \\theta \\left( \\frac{\\Lambda' - \\sin k}{U} \\right)\n\t- \\sum_{l=1}^2 \\theta \\left(\\frac{\\Lambda' - \\sin k_l^h}{U}\n\t\\right) + {\\cal O} \\left( \\frac{1}{L} \\right).\n\\end{eqnarray}\nIn oder to obtain the second of the above equalities we have used\n(\\ref{gi1}), (\\ref{f1}) and the lemma (\\ref{lemma}). Let us calculate\nthe integral\n\\begin{equation}\n I (\\Lambda') = \\frac{1}{2\\pi} \\int_{-\\pi}^\\pi dk \\;\n \\theta \\left( \\frac{\\Lambda' - \\sin k}{U}\n\t\t\t \\right) .\n\\end{equation}\non the right hand side of (\\ref{rhs}). First note that $I(0) = 0$.\nThe derivative of $I (\\Lambda')$ can be represented as\n\\begin{equation} \\label{integral}\n I'(\\Lambda') = \\frac{1}{\\pi} \\int_{-\\pi}^\\pi dk \\;\n \\frac{U}{U^2 + (\\Lambda' - \\sin k)^2}\n = \\text{Re} \\left\\{ \\frac{1}{2\\pi \\text{i}}\n\t\t\t \\oint dz \\frac{4}{z^2 + 2z(U - \\text{i}\n\t\t\t \\Lambda') - 1} \\right\\} ,\n\\end{equation}\nwhere the contour of integration is the unit circle. Let\n\\begin{equation}\n p(z) = z^2 + 2z(U - \\text{i} \\Lambda') - 1 = (z - z_1)(z - z_2).\n\\end{equation}\nWe see that the poles of the integrand in (\\ref{integral}) are related\nas\n\\begin{equation} \\label{poles}\n z_1 = - 1\/z_2.\n\\end{equation}\nThus, only one of these poles, say $z_1$, lies inside the unit circle.\nUsing (\\ref{poles}) we obtain $I'(\\Lambda')$ as a function of\n$z_1$,\n\\begin{equation} \\label{phiprime}\n I'(\\Lambda') = 2 \\text{Re} \\left\\{\n \\frac{2}{z_1 + z_1^{-1}} \\right\\} .\n\\end{equation}\nLet us parameterize the pole at $z_1$ as $z_1 = e^{\\text{i} \\alpha}$.\nSince $z_1$ is located inside the unit circle, $\\text{Im} (\\alpha) > 0$.\nUsing $p(z_1) = 0$ we find that\n\\begin{equation}\n \\sin \\alpha = \\Lambda' + \\text{i} U \\quad,\n \\quad \\text{Im} (\\alpha) > 0.\n\\end{equation}\nThis equation fixes $\\alpha$ modulo $2\\pi$. Now $U > 0$ and $\\text{Im}\n(\\alpha) > 0$, and therefore\n\\begin{equation}\n \\cos \\text{Re} (\\alpha) = \\frac{U}{\\sinh \\text{Im} (\\alpha)} > 0.\n\\end{equation}\nWe conclude that $- \\pi\/2 < \\text{Re} (\\alpha) < \\pi\/2$. Thus (see the\ndefinition below (\\ref{defthetas})) $\\alpha = \\arcsin (\\Lambda' +\n\\text{i} U)$. Integrating (\\ref{phiprime}) with respect to $\\Lambda'$\nand using $I (0) = 0$ to fix the integration constant we arrive at\n\\begin{equation}\n I(\\Lambda') = 2 \\text{Re} (\\alpha)\n = 2 \\text{Re} (\\arcsin(\\Lambda' + \\text{i}U))\n\t\t = \\arcsin(\\Lambda' + \\text{i} U) +\n\t\t \\arcsin(\\Lambda' - \\text{i} U).\n\\end{equation}\nWe may now insert this result into eq.\\ (\\ref{rhs}). This yields\n\\begin{equation}\n \\theta \\left(\\frac{\\Lambda' - \\sin k_1^h}{U} \\right) =\n - \\theta \\left(\\frac{\\Lambda' - \\sin k_2^h}{U} \\right).\n\\end{equation}\nDividing by 2 and taking $\\tan$ gives\n\\begin{equation}\n 2 \\Lambda' = \\sin k_1^h + \\sin k_2^h.\n\\end{equation}\nSo we have accomplished our task to show that (2) follows from eqs.\\\n(20)-(22) of \\cite{BrAn98}.\n\\section{}\nLet us show that for one $k$-$\\Lambda$ string (and several real $k_j$\nand real $\\lambda_\\gamma$) eq.\\ (7) of \\cite{BrAn98} follows from\neqs.\\ (5), (6), (8) of \\cite{BrAn98}. In case of a single\n$k$-$\\Lambda$-two string the numbers $N^u$ and $M^u$ of \\cite{BrAn98}\nare $N^u = N - 2$, $M^u = M - 1$, where $N$ is the total number of\nelectrons and $M$ is the total number of down spins. There are $N - 2$\nreal charge rapidities $k_1, \\dots, k_{N-2}$, which correspond to\nunbound particles. The string is characterized by two charge rapidities\n$k^+$ and $k^-$ and by one spin rapidity $\\Lambda$. $k^\\pm$ and\n$\\Lambda$ satisfy\n\\begin{eqnarray}\n \\sin k^+ = \\Lambda - \\text{i} u\/4 + \\varepsilon^+ \\, \\\\\n \\sin k^- = \\Lambda + \\text{i} u\/4 - \\varepsilon^- \\,\n\\end{eqnarray}\nwhere $\\varepsilon^+$ and $\\varepsilon^-$ become small as the length\n$L$ of the lattice becomes large. Eqs.\\ (5)-(8) of \\cite{BrAn98} are\n\\begin{eqnarray} \\label{5}\n e^{\\text{i} k_j L} & = & \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n {\\Lambda - \\sin k_j + \\text{i} u\/4}\n \\prod_{\\delta = 1}^{M-1}\n \\frac{\\lambda_\\delta - \\sin k_j - \\text{i} u\/4}\n {\\lambda_\\delta - \\sin k_j + \\text{i} u\/4},\n \\quad j = 1, \\dots, N - 2 , \\\\ \\label{6}\n \\prod_{\\delta = 1 \\atop \\delta \\ne \\gamma}^{M-1}\n \\frac{\\lambda_\\gamma - \\lambda_\\delta - \\text{i} u\/2}\n {\\lambda_\\gamma - \\lambda_\\delta + \\text{i} u\/2}\n & = & \\prod_{j=1}^{N-2}\n \\frac{\\lambda_\\gamma - \\sin k_j - \\text{i} u\/4}\n {\\lambda_\\gamma - \\sin k_j + \\text{i} u\/4},\n \\quad \\gamma = 1, \\dots, M - 1 , \\\\ \\label{7}\n \\prod_{\\delta = 1}^{M-1}\n \\frac{\\Lambda - \\lambda_\\delta - \\text{i} u\/2}\n {\\Lambda - \\lambda_\\delta + \\text{i} u\/2}\n & = & \\frac{\\varepsilon^+}{\\varepsilon^-} \\prod_{j=1}^{N-2}\n \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n {\\Lambda - \\sin k_j + \\text{i} u\/4}, \\\\ \\label{8}\n e^{\\text{i}(k^+ + k^-)L}\n & = & \\frac{\\varepsilon^+}{\\varepsilon^-}\n\t \\prod_{\\delta = 1}^{M-1}\n \\frac{\\lambda_\\delta - \\Lambda - \\text{i} u\/2}\n {\\lambda_\\delta - \\Lambda + \\text{i} u\/2}.\n\\end{eqnarray}\nThe ratio\n\\begin{equation}\n \\frac{\\varepsilon^+}{\\varepsilon^-} = - \\;\n \\frac{\\Lambda - \\sin k^+ - \\text{i} u\/4}\n {\\Lambda - \\sin k^- + \\text{i} u\/4}\n\\end{equation}\nis denoted by $e^{\\text{i} \\varphi}$ in \\cite{BrAn98}. We want to show\nthat (\\ref{7}) algebraically follows from (\\ref{5}), (\\ref{6}) and\n(\\ref{8}). First note that momentum conservation implies that\n\\begin{equation} \\label{momentum}\n \\exp \\left({\\textstyle \\text{i} L \\left(\n \\sum_{j=1}^{N-2} k_j + k^+ + k^-\n \\right)} \\right) = 1.\n\\end{equation}\nMultiplying all eqs.\\ (\\ref{6}) we get\n\\begin{equation} \\label{prodisone}\n \\prod_{j=1}^{N-2} \\prod_{\\delta = 1}^{M-1}\n \\frac{\\lambda_\\delta - \\sin k_j - \\text{i} u\/4}\n {\\lambda_\\delta - \\sin k_j + \\text{i} u\/4} =\n \\prod_{\\delta = 1}^{M-1}\n \\prod_{\\gamma = 1 \\atop \\gamma \\ne \\delta}^{M-1}\n\t\\frac{\\lambda_\\delta - \\lambda_\\gamma - \\text{i} u\/2}\n\t {\\lambda_\\delta - \\lambda_\\gamma + \\text{i} u\/2} = 1.\n\\end{equation}\nNow let us multiply all eqs.\\ (\\ref{5}). Taking into account\n(\\ref{prodisone}) we get\n\\begin{equation}\n e^{\\text{i}L \\sum_{j=1}^{N-2} k_j} =\n \\prod_{j=1}^{N-2} \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n {\\Lambda - \\sin k_j + \\text{i} u\/4}.\n\\end{equation}\nLet us multiply this by (\\ref{8}) and use (\\ref{momentum}). Then\n\\begin{equation}\n 1 = \\exp \\left({\\textstyle \\text{i} L \\left(\n \\sum_{j=1}^{N-2} k_j + k^+ + k^-\n \\right)} \\right) =\n \\frac{\\varepsilon^+}{\\varepsilon^-} \\prod_{\\delta = 1}^{M-1}\n \\frac{\\lambda_\\delta - \\Lambda - \\text{i} u\/2}\n {\\lambda_\\delta - \\Lambda + \\text{i} u\/2}\n \\prod_{j=1}^{N-2}\n\t \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n\t {\\Lambda - \\sin k_j + \\text{i} u\/4},\n\\end{equation}\nwhich is equivalent to (\\ref{7}) (or (7) in \\cite{BrAn98}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum chromodynamics (QCD) has emerged as a phenomenologically accurate theory \nthat describes strong interactions \namong the six quark flavors that are bound into two families of hadrons, namely mesons and \nbaryons. From experiments, we understand that strong interactions \nenjoy C (charge conjugation), P (parity), T (time reversal) discrete symmetries of nature. Therefore, \nQCD must also obey such symmetries, both separately and any combinations formed thereof \\citep{KimCarosi2010}. \nHowever, CP symmetry is broken in QCD due to the presence of the following term in the \nQCD Lagrangian \\citep{Wilczek1978}\n\\begin{equation}\n\\mathfrak{L}_{\\text{int}}=\\left(\\frac{\\theta g^2}{32\\pi^2}\\right)\\text{tr}\\ \nG^{\\mu\\nu}_a\\tilde{G}_{a\\mu\\nu}\n\\end{equation}\nwhere $\\theta$ is a periodic parameter, $g$ is the QCD coupling constant, $G_{\\mu\\nu}$ is the \ncolor field strength tensor, and $\\tilde{G}_{\\mu\\nu}$ is its dual. The value of the $\\theta$-parameter \nis not set theoretically, but it can be measured from \nthe electric dipole moment of the neutron ($d_n$), for which many theoretical estimates exist but \nwe only quote one, $\\|d_n\\|\\sim 2.7\\times 10^{-16}\\bar{\\theta}e$ cm \\citep{Baluni1979}. Here $\\bar{\\theta}\n=\\theta+\\text{arg det }m_q$, where $m_q$ is the quark mass matrix. The latest experimental estimate of \n$\\|d_n\\| < 2.9\\times10^{-26}e$ cm \\citep{Bakeretal2006} constrains $\\|\\bar{\\theta}\\| \\lesssim 10^{-11}$ \n\\citep{KimCarosi2010}. This inexplicably small value of $\\bar{\\theta}$ gave rise to \nthe \\emph{strong CP problem}. One of the solutions, also the most favoured, to this problem \nwas envisioned by Peccei \\& Quinn \\citep{PecceiQuinn1977}, whereby the $\\bar{\\theta}$ parameter is driven precisely to \nzero under a global chiral symmetry, later named $U(1)_{\\text{PQ}}$. The pseudo-Nambu-Goldstone \nboson that results upon the spontaneous breakdown of this symmetry was dubbed the axion \n\\citep{Wilczek1978,Weinberg1978}. Not unlike the Higgs boson, the axion has proven to be extremely \ndifficult to observe as it couples only weakly to ordinary matter and radiation.\n\nDespite several attempts to experimentally observe the axion, it remains elusive to this day. \nNevertheless, the experimental efforts have not gone in vain, but have been able to place serious \nconstraints on the coupling strength of the axion to photons $g_{a\\gamma\n\\gamma} < 10^{-10}\\text{ GeV}^{-1}$. Stringent constraints have been placed on the mass of the axion \n$10^{-6} \\lesssim m_a \\lesssim 10^{-3}$ eV with the lower limit arising \nfrom cosmology \\citep{Preskilletal1983} and the upper limit\\footnote{Due to large uncertainties in the axion mass derived for the DFSZ \nmodel \\citep{DineFischlerSrednicki1981,Zhitnitskii1980} from SN 1987A observations \n($0.004 \\lesssim m_a \\lesssim 0.012$ eV), a more relaxed upper limit is \n$m_a \\lesssim 0.01$ eV \\citep{Raffelt2004}} from the neutrino flux recorded \nfor SN 1987A, which placed strong limits on the cooling flux through other channels namely, \nright-handed neutrinos or axions \\citep{Raffelt2004}. If $g_{a\\gamma\\gamma} > 10^{-10}\\text{ GeV}^{-1}$, \nthe production of axions through the Primakoff process will significantly alter the core He burning timescales of \npost main sequence stars, a possibility excluded by the ratio of horizontal branch stars in \nglobular clusters \\citep{Raffelt1996}. Several reviews on the properties of axions have been forthcoming in the past \ndecade, for example see \\citep{Raffelt1999,Asztalosetal2006,KimCarosi2010}, to which we point the \nreader for a more detailed and comprehensive exposition.\n\nStill, there is no denying the fact that none of the laboratory experiments conducted \nthus far have been able to secure a positive detection of this mysterious particle. \nThe \ndetection of very weakly coupled particles demands extremely sensitive laboratory experiments. \nSo far, only a handful of experiments, namely the Cern Axion Solar Telescope (CAST) \n\\citep{Andriamonjeetal2007}, Axion Dark Matter Experiment (ADMX) \\citep{Asztalosetal2004,Asztalosetal2010}, \nand Rochester-Brookhaven-Fermilab \n(RBF) collaboration \\citep{Panfilisetal1987,Wuenschetal1989} have \nbeen able to surpass the astrophysically derived limits on $g_{a\\gamma\\gamma}$ in the above \nquoted axion mass range. Yet, the sensitivity envelope needs to be pushed even further by a \nfew orders of magitude to be able to draw any definitive conclusions about the existence of \nthe axion. Plans are afoot to modifiy the existing experiments and devise new ones to \nimprove upon current limits on $g_{a\\gamma\\gamma}$ (see Section IV).\n\nUnlike the laboratory experiments, the odds are in favour for detecting axions \nin astrophysical systems. This optimism stems from the fact that the \naxion to photon conversion probability scales with large magnetic field strengths and \nlonger coherence lengths \\citep{Cheloucheetal2009}, such that $P_{a\\rightarrow\\gamma}\n\\propto g^2B^2L^2$, where $L$ is the length over which both the photon and axion fields \nare in phase. Thus, there is a very good chance of finding the axion in \nstrongly magnetized compact objects, namely magnetic white dwarfs (mWDs) and neutron \nstars (NSs). The possibility in the latter case has been expounded by many (see for \nexample \\citep{RaffeltStodolsky1988,LaiHeyl2006,Cheloucheetal2009,PshirkovPopov2009}; also see \n\\citep{Jimenezetal2011} where constraints on $g_{a\\gamma\\gamma}$ are derived \nfrom the dimming of radiation by photon-axion conversion in astrophysical sources), however, the \ncase of the mWDs has not been investigated in great detail and warrants further \nstudy. \n\\subsection{Magnetic white dwarfs}\nAfter the discovery of the first mWD by Kemp \\citep{Kempetal1970}, the number \nof white dwarfs with magnetic fields ranging from a few kG to $10^3$ MG has grown to about \n170. The size of this subpopulation is only 3\\% of the total population of known WDs \ncomprising of 5447 objects\\footnote{G.P. McCook and E.M. Sion, web version of the Villanova \nWhite Dwarf Catalog, http:\/\/www.astronomy.villanova.edu\/WDCatalog\/index.html}. The main channel for identifying magnetism \nin WDs is through Zeeman spectropolarimetry, which not only allows one to discern the \nstrength of the field but also the direction of the field lines, and also through \ncyclotron spectroscopy (see for e.g. \\citep{WickramasingheFerrario2000} for a review \non isolated and binary mWDs). Nevertheless, reconstruction of the field topology has \nproven to be very difficult, mainly due to its highly non-dipolar structure. Over the \nlast decade Zeeman tomography of mWDs has enjoyed some success in elucidating the \nunderlying field structure. This technique is based on calculating a database of \nmodel spectra, where different field geometries comprising of single\/multiple dipole, \nand higher multipoles, that may also be off-centered and misaligned with the rotational axis, are \nconsidered. Then a least-squares fit using the pre-calculated synthetic spectra is performed \nthrough a highly optimized algorithm on the phase-resolved Zeeman spectra to obtain the \ncomplex field structures \\citep{Euchneretal2002}. The \ngenerality of the models not only allows greater flexibility but also renders a closer \nfit to the actual field geometry of the source for a given rotational phase.\n\nThe presence of even a small degree of circular polarization in the spectrum of a WD is a \nstrong indicator of the object possessing a magnetic field upwards of $10^6$ G \n\\citep{Kemp1970}. The degree of circular polarization typically reaches up to $\\sim 5\\%$, \nand sometimes beyond that in a few selective objects, near absorption features and also \nin the continuum. Continuum circular polarization stems from the magnetic circular \ndichroism of the atmosphere, \nwhere the left and right circularly polarized waves propagating through a magnetized \nmedium encounter unequal opacities \\citep{Angel1977}. A relatively higher degree of \ncircular polarization also appears near the red and blue shifted wings of the Zeeman \nsplit absorption lines ($\\sigma_+$ and $\\sigma_-$ components, \\citep{WickramasingheFerrario2000}).\n\nOn the other hand, most observations of mWDs indicate that the linear polarization \ncomponent never exceeds that of the circular one, and the spectrum remains dominantly \ncircularly polarized until field strengths $\\geq 10^8$ G are reached \\citep{Angel1977}. \nIn a magneto-active plasma, the plane of linear polarization undergoes many Faraday \nrotations, an effect that arises due to the magnetic birefringence of the medium, so \nthat on average the degree of linear polarization of the emergent radiation is much reduced \n\\citep{SazonovChernomordik1975}.\n\nThe very fact that linear polarization is of the order of a few percent ($\\sim 5\\%$) in the \ncontinuum spectra of most mWDs can be exploited to draw meaningful conclusions on the extent of axion \ninteraction with photons traversing the magnetized plasma of mWDs. We explain how this can be \nimplemented in the next section.\n\\subsection{Plan of this study}\nThe purpose of this study is to conduct a survey of the $m_a-g_{a\\gamma\\gamma}$ parameter \nspace by modelling photon-axion oscillations in the magnetosphere of a mWD. To this end, \nwe model the field structure of a strongly magnetized WD PG 1015+014, for which high \nresolution optical spectropolarimetric observations are available \\citep{Euchneretal2006}. In the \nsame article, the authors also conduct a phase-resolved Zeeman tomographic analysis and derive \na best-fit model of the magnetic field topology. Despite fitting the spectrum with a range of \nfield geometries, they were only able to pin down the field geometry for a single rotational phase \nby fitting it with a superposition of three off-centered and non-aligned dipoles of unequal surface \nfield strengths (see Table \\ref{tab:pg1015} for model parameters). To model the effect of photon-axion oscillation \nin the magnetosphere on the emergent polarization, we propagate an unpolarized photon of a given \nenergy from the photosphere through the encompassing magnetosphere, that has been populated by \na diffuse, cold ionized H gas. The emergent intensity and polarization is then averaged over the \nwhole surface of the star. Finally, we compare the degree of polarization from our model simulation \nto what is observed in mWDs with field strengths in excess of a few $10^6$ G, \nfor example PG 1015+014, and draw conclusions on the strength of the coupling constant for \na given axion mass.\n\\begin{table}\n\\caption{\\textit{Top}: Magnetic field geometry adopted from the spectropolarimetric analysis by \\citep{Euchneretal2006} \nof the the mWD PG 1015+014. The model comprises of three off-centered and non-aligned dipoles \n$D_1,D_2,D_3$ with unequal surface field strengths $B_s$, polar ($\\theta_B$) and azimuthal ($\\phi_B$) angles \nof the magnetic field axes. The center positions of the dipoles relative to the center of the star are \ngiven by ($a_x,a_y,a_z$). \\textit{Bottom}: Collection of parameters used in the study.}\n\\begin{center}\n\\begin{tabular}{l r r r}\n\\multicolumn{4}{c}{\\textbf{Model Parameters}} \\\\\n\\hline\n\\hline\n& $D_1$ & $D_2$ & $D_3$ \\\\\n\\hline\n$B_s$ (MG) & -40 & 92 & -38 \\\\\n$\\theta_B({}^\\circ)$ & 44 & 63 & 63 \\\\\n$\\phi_B({}^\\circ)$ & 339 & 276 & 134 \\\\\n$a_x(R_\\star)$ & 0.04 & -0.012 & 0.27 \\\\\n$a_y(R_\\star)$ & 0.35 & -0.136 & 0.080 \\\\\n$a_z(R_\\star)$ & 0.33 & -0.28 & 0.21 \\\\\n\\hline\n$R_\\star$ & \\multicolumn{3}{r}{$7\\times10^8$ cm} \\\\\n$\\theta_k$ & \\multicolumn{3}{r}{$23^\\circ$} \\\\\n$Y_e$ & \\multicolumn{3}{r}{1} \\\\ \n$B_Q$ & \\multicolumn{3}{r}{$4.413\\times10^{13}$ G} \\\\\n$T$ & \\multicolumn{3}{r}{$10^4$ K} \\\\\n$g_\\star$ & \\multicolumn{3}{r}{$10^8\\text{ cm s}^{-2}$} \\\\\n$\\rho_0$ & \\multicolumn{3}{r}{$10^{-10}\\text{ g cm}^{-3}$} \\\\\n$\\rho_\\infty$ & \\multicolumn{3}{r}{$10^{-20}\\text{ g cm}^{-3}$} \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n\\label{tab:pg1015}\n\\end{table}\n\nIn the following Section, we formulate the key equations describing the interaction of the axion \nwith photons, geometry of the aggregate magnetic field, and structure of the plasma permeating the \nmagnetosphere. The lack of understanding of the density profile of the magnetospheric plasma \nintroduces some level of inaccuracy in any treatment of mWDs. We take the simplest approach and \ndescribe the plasma density by the barometric law for an isothermal atmosphere. In Section III we present \nthe main results of the study along with a comparison to some of the results obtained from \nlab experiments (see Fig. \\ref{fig:exclusion}). A discussion of the results is provided in \nSection IV.\n\\section{Model equations}\nThe interaction of the axion field with an external electromagnetic field is given by the \nfollowing Lagrangian density \\citep{RaffeltStodolsky1988}, in natural units where $\\hbar=c=1$,\n\\begin{eqnarray}\n\\mathfrak{L}=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\frac{1}{2}(\\partial_\\mu a\n\\partial^\\mu a - m_a^2 a^2)-\\frac{1}{4}g_{a\\gamma\\gamma}F_{\\mu\\nu}\n\\tilde{F}^{\\mu\\nu}a& \\nonumber \\\\\n& +\\frac{\\alpha^2}{90m_e^4}\\left[\\left(F_{\\mu\\nu}F^{\\mu\\nu}\n\\right)^2+\\frac{7}{4}\\left(F_{\\mu\\nu}\\tilde{F}^{\\mu\\nu}\\right)^2\\right]&\n\\label{eq:lagrangian}\n\\end{eqnarray}\nwhere the first term describes the external electromagnetic field, with $F_{\\mu\\nu}$ \nas the antisymmetric electromagnetic field strength tensor and $\\tilde{F}^{\\mu\\nu}=\n\\frac{1}{2}\\varepsilon^{\\mu\\nu\\rho\\sigma}F_{\\rho\\sigma}$ as its dual. The second term \nis simply the Klein-Gordon equation for the axion field $a$ where $m_a$ represents its \nmass. The next term is the interaction Lagrangian density, which upon simplification, \nusing the given definitions, yields $\\mathfrak{L}_{\\text{int}}=g_{a\\gamma\\gamma}\na \\mathbf{E}\\cdot\\mathbf{B}$, where $g_{a\\gamma\\gamma}$ is the photon-axion coupling \nstrength, $\\mathbf{E}$ is the polarization vector of the \nphoton field, and $\\mathbf{B}$ is the external magnetic field. Quantum corrections to \nthe classical electromagnetic field, due to the constant creation and annihilation of \nelectron-positron pairs in vacuum, are given in the last term of \\eqref{eq:lagrangian} by \nthe Euler-Heisenberg effective Lagrangian, causing the vacuum to be birefringent.\n\\subsection{Off-centered non-aligned three dipole model}\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig1}\n\\caption{This illustrates the coordinate system used to obtain the aggregate magnetic field. \nHere $\\Sigma_0$ represents the coordinate system centered on the star with the \n$\\mathbf{\\hat{z}}$-axis aligned with the rotational axis. $\\Sigma_i$ represents the coordinate system in which the \ndifferent dipole magnetic field equations are written. This system is misaligned with $\\Sigma_0$ \nby a polar angle $\\theta_{B,i}$ and an azimuthal angle $\\phi_{B,i}$, and then it is displaced from \nthe center by the vector $\\mathbf{a}_i$. For clarity we have chosen the vector $\\mathbf{a}_i$ to \nlie along the $\\mathbf{\\hat{z}}'$-axis (color online).}\n\\label{fig:coord}\n\\end{figure}\nWe start by writing down the aggregate magnetic field $\\mathbf{B}_0$ in the coordinate \nsystem $\\Sigma_0$ where the $\\mathbf{\\hat{z}}$-axis coincides with the rotational axis of the star \n\\begin{equation}\n\\mathbf{B}_0=\\sum_{i=1}^3R_i^T\\mathbf{B}_i'\n\\end{equation}\nThe three off-centered dipole fields $\\mathbf{B}_i'$ are first rotated before they are added \ntogether by operating on each of them with a rotation matrix $R^T=R_{z,i}^TR_{y,i}^T$, \nwhere the superscript $T$ indicates the transpose, and the rotation matrices are \ngiven as follows\n\\begin{eqnarray}\nR_{z,i}=&\\left(\\begin{array}{ccc}\n\\cos \\phi_{B,i} & \\sin \\phi_{B,i} & 0 \\\\\n-\\sin \\phi_{B,i} & \\cos \\phi_{B,i} & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{array}\\right)& \\\\\nR_{y,i}=&\\left(\\begin{array}{ccc}\n\\cos\\theta_{B,i} & 0 & -\\sin\\theta_{B,i} \\\\\n0 & 1 & 0 \\\\\n\\sin\\theta_{B,i} & 0 & \\cos\\theta_{B,i} \\\\\n\\end{array}\\right)&\n\\label{eq:Ry}\n\\end{eqnarray}\nHere the polar and azimuthal angles $\\theta_{B,i}$ and $\\phi_{B,i}$, respectively, are defined \nwith respect to the axis of rotation. In cartesian coordinates, the dipole fields are \nexpressed as\n\\begin{equation}\n\\mathbf{B}'_i=\\frac{B_{s,i}R_\\star^3}{2r'^5_i}(3x'_iz'_i\\mathbf{\\hat{x}'}\n+3y'_iz'_i\\mathbf{\\hat{y}'}+\\{3{z'_i}^2-{r'_i}^2\\}\\mathbf{\\hat{z}'})\n\\end{equation}\nwhere the fields are shifted from the coordinate center, such that $\\mathbf{r}_i'=\n\\mathbf{r}-\\mathbf{a}_i$. In the above equation, $\\mathbf{B}_{s,i}$ is the surface field \nstrength of the $i^{\\text{th}}$ dipole field component, $R_\\star\\simeq7\\times10^8$ cm is the radius of the WD, and $r_i'$ is the \nmagnitude of the radial vector in the coordinate system $\\Sigma_i$. The profile of the aggregate field $B_0$ as a function of distance \nis shown in Fig. \\ref{fig:los}.\n\\subsection{Fully ionized pure H atmosphere}\nThe presence of a magnetic field necessarily introduces anisotropy in the plasma dielectric tensor \n$\\matlabel{\\varepsilon}_p$. In the case of a nonuniform field, none of the dielectric tensor components \nvanish, as compared to the homogeneous case. Below we write all the dielectric components, \nwhich one can easily derive from Maxwell's equations, for completeness.\n\\begin{align}\n\\matlabel{\\varepsilon}_p & = \\left(\\begin{array}{ccc}\n\\varepsilon_{11} & \\varepsilon_{12} & \\varepsilon_{13} \\\\\n\\varepsilon_{21} & \\varepsilon_{22} & \\varepsilon_{23} \\\\\n\\varepsilon_{31} & \\varepsilon_{32} & \\varepsilon_{33} \\\\\n\\end{array}\\right) & \\\\\n\\varepsilon_{11} & = 1-\\sum_{s=e,p}\\hat{\\omega}_{p,s}^2\\left[\\frac{1-\\hat{\\omega}_{c,s}^2\n\\hat{B}_{0x}^2}{1-\\hat{\\omega}_{c,s}^2}\\right] \\nonumber & \\\\\n& \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}_{c,p}^2)\n\\hat{B}_{0x}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] & \\\\\n\\varepsilon_{12} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2\n)}(i\\hat{B}_{0z}+\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0y}) & \\\\\n\\varepsilon_{13} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0y}-\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0z}) & \\\\\n\\varepsilon_{21} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0z}-\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0y}) & \\\\\n\\varepsilon_{22} & \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}\n_{c,p}^2)\\hat{B}_{0y}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] & \\\\\n\\varepsilon_{23} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0x}+\\hat{\\omega}_{c,e}\\hat{B}_{0y}\\hat{B}_{0z}) & \\\\ \n\\varepsilon_{31} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0y}+\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0z}) & \\\\ \n\\varepsilon_{32} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0x}-\\hat{\\omega}_{c,e}\\hat{B}_{0y}\\hat{B}_{0z}) & \\\\\n\\varepsilon_{22} & \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}\n_{c,p}^2)\\hat{B}_{0z}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] &\n\\end{align}\nIn the above set of equations, $\\hat{\\omega}_{c,s}=q_sB_0\/\\omega m_s c$ is the normalized cyclotron \nfrequency for species $s=(e,p)$, where $e$ and $p$ signify electrons and protons; $\\hat{\\omega}_{p,s} \n= \\sqrt{4\\pi n_s\/m_s\\omega^2}$ is the normalized plasma frequency, where $n_p = n_e = Y_e\\rho\/m_p$ \nare the electron and proton number densities, $Y_e$ is the electron fraction, and $\\rho$ is the \nproton mass density of the plasma; the normalized magnetic field components are defined as \n$\\hat{B}_{0,i=x,y,z} = B_{0,i}\/B_0$.\n\\subsubsection{Vacuum corrections}\nDue to the polarizability of the vacuum in strong magnetic fields, the plasma dielectric \ntensor $\\matlabel{\\varepsilon}_p$, and the inverse permeability tensor $\\matlabel{\\mu}^{-1}$ \nare modified \\citep{MeszarosVentura1979,LaiHo2003b}, such that $\\matlabel{\\varepsilon}_{p+v} = \n\\matlabel{\\tilde{\\varepsilon}} = \\matlabel{\\varepsilon}_p + \\Delta\\matlabel{\\varepsilon}_v$ \nand $\\matlabel{\\mu}^{-1}_{p+v} = \\matlabel{\\tilde{\\mu}}^{-1} = \\matlabel{I} + \n\\Delta\\matlabel{\\mu}^{-1}_v$, where\n\\begin{eqnarray}\n& \\Delta\\matlabel{\\varepsilon}_v = (a_v-1)\\matlabel{I} + q_v\\hat{\\mathbf{B}}_0\\hat{\\mathbf{B}}_0 & \\\\\n& \\Delta\\matlabel{\\mu}^{-1}_v = (a_v-1)\\matlabel{I} + m_v\\hat{\\mathbf{B}}_0\\hat{\\mathbf{B}}_0 & \\\\\n& a_v = 1-2\\delta_v\\ \\ \\ q_v = 7\\delta_v\\ \\ \\ m_v=-4\\delta_v\\ \\ \\ \\delta_v = \n\\frac{\\alpha}{45\\pi}\\left(\\frac{B_0}{B_Q}\\right)^2 & \\nonumber\n\\end{eqnarray}\nand $B_Q = 4.413\\times10^{13}$ G is the quantum critical field for which the separation in energy \nbetween Landau levels of the electron exceeds its rest mass.\n\\subsubsection{Plasma density profile}\nThat many mWDs are surrounded by hot coronae has been suggested by many to explain the polarized \nflux of those WDs that show comparable degree of linear and circular polarization \n\\citep{SazonovChernomordik1975,InghamBrecherWasserman1976,ZheleznyakovSerber1994}. The thermal electrons in the hot \ntenuous plasma with temperature $T\\sim10^{6-8}$ K radiate at the cyclotron frequency that falls in the \noptical wavelength for field strengths of $B\\sim10^8$ G. This radiation appears \nto be polarized both linearly and circularly, depending on the orientation of the line of sight to the \nmagnetic field, and traverses the corona without any apsorption. Furthermore, slightly polarized \nradiation emanating from the photosphere, with very low degree of linear polarization due \nto Faraday rotation, gets added to that generated in the corona, as a result increasing the amount \nof flux that is polarized linearly. Several hot isolated WDs, with effective temperatures in \nexcess of $\\simeq25,000$ K, emitting X-rays were detected by ROSAT \\citep{Flemingetal1996}, however \nall cases were linked to subphotospheric thermal emission \\citep{Musielaketal2003}. Although the \nnon-detection of any coronal emission may indicate the absence of a hot tenuous corona, it is not \nat all unreasonable to suggest the presence of a tenuous cold plasma of fully ionized H. In this \nstudy, we envisage that the mWDs are encompassed by cold isothermal electron-proton coronae with \nthe following barometric density profile,\n\\begin{equation}\n\\rho(r) = \\rho_0\\exp\\left(-\\frac{r-R_\\star}{H_\\rho}\\right) + \\rho_\\infty\n\\end{equation}\nwhere $\\rho_0$ is the density near the surface of the star, $\\rho_\\infty$ is the density \nthat remains far away from the star as the strength of the magnetic field becomes significantly \nweaker than that at the surface, and $H_\\rho = 2k_BT\/m_pg_\\star \\simeq 1.65\\times10^4$ cm is the density \nscale height with an effective temperature $T\\simeq10^4$ K and surface gravity \n$\\log g_\\star(\\text{cm\/s}^2) = 8$. There is no clear agreement on the surface plasma \ndensity with $10^{-11} \\lesssim \\rho_0 \\lesssim 10^{-6}\\text{ g cm}^{-3}$. Here, we \nassume that the plasma is sufficiently tenuous with $\\rho_0 = 10^{-10}\\text{ g cm}^{-3}$ \nand $\\rho_\\infty = 10^{-20}\\text{ g cm}^{-3}$.\n\\subsection{Axion-photon mode evolution in an inhomogeneous magnetized plasma} \nWe are interested in knowing the evolution of the axion field and the polarization vector \nas the radiation propagates out from the surface of the star, traversing the region with an \ninhomogeneous plasma density and magnetic field. Here we follow the discussion given in \n\\citep{RaffeltStodolsky1988,LaiHeyl2006}, and derive the photon field mode evolution from the \nEM wave equation\n\\begin{equation}\n\\nabla\\times(\\matlabel{\\tilde{\\mu}}^{-1}\\cdot\\nabla\\times\\mathbf{E}) = \\frac{\\omega^2}{c^2}\n\\matlabel{\\tilde{\\varepsilon}}\\cdot\\mathbf{E}\n\\end{equation}\nNext, we assume the ansatz $\\mathbf{E}=\\tilde{\\mathbf{E}}\\exp(ikz)$ where the wave is propagating \nalong the rotational axis of the star, which in this case is also the line of sight direction, \nand the wavenumber $k=\\omega\/c$. Plugging this ansatz into \nthe wave equation, and ignoring second order derivatives, we find\n\\begin{equation}\n\\frac{d}{dz}\n\\left(\\begin{array}{c}\n\\tilde{E}_x \\\\\n\\tilde{E}_y\n\\end{array}\\right) = \\left(\\begin{array}{cc}\n\\chi_{11} & \\chi_{12} \\\\\n\\chi_{21} & \\chi_{22} \n\\end{array}\\right)\\left(\\begin{array}{c}\n\\tilde{E}_x \\\\\n\\tilde{E}_y\n\\end{array}\\right)\n\\label{eq:modeevol}\n\\end{equation}\nwhere the matrix elements are given below\n\\begin{eqnarray}\n\\chi_{11} & = & \\Upsilon^{-1}_3\\left[k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4-\n\\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \\Upsilon_1\n\\Upsilon_5^{-1} \\right. \\nonumber \\\\\n& & \\left. \\times\\left(k^2\\tilde{\\varepsilon}_{21}-\\Upsilon_2-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4\\}\n\\right)\\right] \\\\\n\\chi_{12} & = & \\Upsilon^{-1}_3\\left[k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2-\n\\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \\Upsilon_1\n\\Upsilon_5^{-1} \\right. \\nonumber \\\\\n& & \\left. \\times\\left(k^2\\tilde{\\varepsilon}_{22}-\\Upsilon_6-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2\\}\n\\right)\\right] \\\\\n\\chi_{21} & = & \\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \n\\Upsilon_5^{-1} \\nonumber \\\\ \n& & \\times \\left(k^2\\tilde{\\varepsilon}_{21}-\\Upsilon_2-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4\\}\n\\right) \\\\\n\\chi_{22} & = & \\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \n\\Upsilon_5^{-1} \\nonumber \\\\ \n& & \\times \\left(k^2\\tilde{\\varepsilon}_{22}-\\Upsilon_6-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2\\}\n\\right) \\\\\n\\Upsilon_1 & = & \\frac{d}{dz}(m_v\\hat{B}_x\\hat{B}_y)+i2km_v\\hat{B}_x\\hat{B}_y \\\\\n\\Upsilon_2 & = & ik\\frac{d}{dz}(m_v\\hat{B}_x\\hat{B}_y)-k^2m_v\\hat{B}_x\\hat{B}_y \\\\\n\\Upsilon_3 & = & -\\frac{d}{dz}(a_v+m_v\\hat{B}_y^2)+i2k(a_v+m_v\\hat{B}_y^2) \\\\\n\\Upsilon_4 & = & -ik\\frac{d}{dz}(a_v+m_v\\hat{B}_y^2)+k^2(a_v+m_v\\hat{B}_y^2) \\\\\n\\Upsilon_5 & = & -\\frac{d}{dz}(a_v+m_v\\hat{B}_x^2)+i2k(a_v+m_v\\hat{B}_x^2) \\\\\n\\Upsilon_6 & = & -ik\\frac{d}{dz}(a_v+m_v\\hat{B}_x^2)+k^2(a_v+m_v\\hat{B}_x^2)\n\\end{eqnarray}\n\\subsubsection{Line of sight geometry}\n\\begin{figure}[t!]\n\\includegraphics[width=0.5\\textwidth]{fig2}\n\\caption{This figure illustrates the coordinate system used to obtain the photon-axion \nmode evolution along a given LOS (top), and the decline of the magnetic field strength with distance \n$s$ from the surface (bottom). Here the LOS vector is represented by $s$ that is tilted \nat an angle $\\theta_k$ to the rotation axis, and is in parallel to $z'$ which is not to be \nconfused with $\\hat{z}'$ in Fig. \\ref{fig:coord}. Several different \npoints on the star's surface with spherical coordinates ($R_\\star,\\theta_R,\\phi_R$) are \nchosen and then averaged to determine the final polarization of the photon leaving the \nmagnetosphere (color online).}\n\\label{fig:los}\n\\end{figure}\n The Zeeman tomography analysis of mWD PG 1015+014 indicates that the line of sight (LOS) \nis inclined at an angle $\\theta_k=23^\\circ$ to the rotational axis of the star. \nFollowing \\citep{DupaysRoncadelli2006}, we modify the matrix Eq.(\\ref{eq:modeevol}) to \nobtain the mode evolution of the photon-axion system in a coordinate system oriented \nalong the LOS (see Fig. \\ref{fig:los}). Again, we assume the ansatz $a\\propto\\exp{(ik's-i\\omega t)}$\n\\begin{equation}\ni\\frac{d}{ds}\\left(\\begin{array}{c}\na \\\\\nE_{x'} \\\\\nE_{y'} \\\\\n\\end{array}\\right) = \\left(\\begin{array}{ccc}\n\\Delta_a-k' & \\Delta_{Mx'} & \\Delta_{My'} \\\\\n\\Delta_{Mx'} & i\\chi_{11}'-k' & i\\chi_{12}' \\\\\n\\Delta_{My'} & i\\chi_{21}' & i\\chi_{22}'-k'\\end{array}\\right)\n\\left(\\begin{array}{c}\na \\\\\nE_{x'} \\\\\nE_{y'} \\\\\n\\end{array}\\right) \n\\label{eq:newmodeevol}\n\\end{equation}\nwhere $\\Delta_a=m_a^2\/2\\omega$, $\\Delta_{Mx'}=-g_{a\\gamma\\gamma}B_x\/2$, \n$\\Delta_{My'}=-g_{a\\gamma\\gamma}B_y\/2$. \nNotice that Eq.(\\ref{eq:modeevol}) applies to a system for which the LOS vector coincides with the rotational \naxis of the star. For a different LOS vector, such as shown in Fig. \\ref{fig:los}, we perform a rotation of \nthe plasma dielectric tensor around the $\\mathbf{\\hat{y}}$-axis by an angle $\\theta_k$, \n$\\matlabel{\\tilde{\\varepsilon}}' = R_y^T(\\theta_k)\\matlabel{\\tilde{\\varepsilon}}R_y(\\theta_k)$ where $R_y$ is given in \nEq.(\\ref{eq:Ry}). \n\nThe total degree of polarization can be found by integrating Eq.(\\ref{eq:newmodeevol}) from a given point on the surface outwards \nto a distance beyond which the amplitude of photon-axion oscillations and plasma effects become \nnegligible, and then by averaging the Stokes parameters \\citep{RybickiLightman2004} over \nthe whole observable hemisphere. \n\\begin{eqnarray}\nI & = & \\|E_{x'}\\|^2 + \\|E_{y'}\\|^2 \\\\\nQ & = & \\|E_{x'}\\|^2 - \\|E_{y'}\\|^2 \\\\\nU & = & E_{x'} E_{y'}^\\ast + E_{y'} E_{x'}^\\ast \\\\\nV & = & -i (E_{x'} E_{y'}^\\ast - E_{y'} E_{x'}^\\ast)\n\\end{eqnarray}\nThe mode amplitudes\nare in general complex, and in the above set of equations $\\ast$ gives the\ncomplex conjugate. We sample the Stokes vector \nfrom different points on the surface, with coordinates ($R_\\star,\\theta_R,\\phi_R$), that are spread around the LOS vector \n$\\mathbf{\\hat{k}}$ with $\\Delta\\phi_R=30^\\circ$ and $\\Delta\\theta_R=10^\\circ$, where $10^\\circ \\leq \\theta_R \n\\leq 80^\\circ$ and $0^\\circ \\leq \\phi_R \\leq 330^\\circ$. Because the sampling in the azimuthal \nangle is sparse for larger polar angles, we take a weighted average, as shown below for one of the \nStokes parameters, to determine the average degree of polarization of the whole hemisphere\n\\begin{equation}\n\\langle I\\rangle = \\frac{\\sum_{\\theta_R,\\phi_R}I(\\theta_R,\\phi_R)\\sin\\theta_R}{\\sum_{\\theta_R}\\sin\\theta_R}\n\\end{equation}\n\\section{Results}\nIn the following, we look at how an unpolarized photon emitted from the photosphere \nof a mWD gets polarized as it traverses through the magnetosphere. Photon-axion \ninteraction and the intervening plasma make the medium birefringent, consequently, altering \nthe state of polarization of the unpolarized photon. We obtain the degree of polarization from \nthe averaged Stokes parameters\n\\begin{eqnarray}\nP_L & = & \\frac{\\sqrt{\\langle Q\\rangle^2 + \\langle U\\rangle^2}}{\\langle I\\rangle} \\\\\nP_C & = & \\frac{\\langle V\\rangle}{\\langle I\\rangle} \n\\end{eqnarray}\nwhere $P_L$ and $P_C$ represent linear and circular polarization. \n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig3}\n\\caption{Polarization evolution of an unpolarized photon along a given LOS as it starts at \nthe photosphere and propagates through the magnetosphere. In this case, $E_\\gamma = 3$ \neV, $m_a = 10^{-5}$ eV, $g_{a\\gamma\\gamma} = 10^{-9}\\mbox{ GeV}^{-1}$. The magnetic field \ngeometry assumed is that of mWD PG 1015+014 (color online).}\n\\label{fig:stokes}\n\\end{figure}\n\nIn Fig. \\ref{fig:stokes}, we present the evolution of the Stokes vector with distance $s$ \nfrom the surface of the star for the case of radiation with $E_\\gamma = 3$ eV, and axion parameters \n$m_a=10^{-5}\\text{ eV}, g_{a\\gamma\\gamma}=10^{-9}\\text{ GeV}^{-1}$. The oscillations in the solution arise due to the \nmixing of the axion and photon eigenstates, an effect analogous to neutrino oscillations due to the MSW \neffect \\citep{MikheevSmirnov1986,Wolfenstein1978}. However, notice that the interaction is non-resonant because a 50\\% drop in intensity would be \nobserved if the axion and photon modes were to achieve maximal mixing and undergo level crossing. Eventually, \nas the photons travel farther away from the surface, the decline in the magnetic field strength reduces the probability \nof conversion, hence the diminishing of intensity variation. We find that the change in \npolarization is primarily brought about by the axion interaction with the photon. In the event this \ninteraction is made negligible, no significant polarization or change in intensity of the emergent radiation \nis found. The origin of circular polarization in mWDs, as alluded \nto earlier, is understood in terms of the difference in opacities for the two modes of radiation, making \nthe plasma dichroic, in the presence of a magnetic field. Linear polarization, on the other hand, was \nexplained by the cyclotron radiation that emanates from the tenuous corona composed of an ionized plasma. \nIn this study, since the treatment of radiative transfer effects is very simplistic only an upper limit can be\nplaced on how strongly the axion couples to photons, as shown in the next section.\n\\subsection{Constraints on $g_{a\\gamma\\gamma}$}\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig4}\n\\caption{The final state of polarization of radiation after traversal from the WD's magnetosphere for different \n$m_a$ and $g_{a\\gamma\\gamma}$. Here $\\langle I \\rangle$ is the average Stokes intensity, and $P_L$ and $P_C$ are \nthe degrees of linear and circular polarizations. These results apply to the case of mWD PG 1015+014 (color online).}\n\\label{fig:composite}\n\\end{figure}\nAxion production in the mWD magnetosphere can enhance the degree of linear polarization of the \nobserved optical radiation. The goal here is to not determine \nthe precise value of the photon-axion coupling strength but only constrain it from above. To this \nend, we look at the amount of linear polarization that is produced for a given $m_a$ and $g_{a\\gamma\\gamma}$.\nThe underlying assumption here is that all of the observed linear polarization is generated due to \nphoton-axion interaction, and not by the plasma, which effectively yields the absolute upper limit on \n$g_{a\\gamma\\gamma}$. In Fig. \\ref{fig:composite}, we plot the emergent intensity and state of polarization \nfor different axion masses and for photons in the UV - optical waveband with energies between $2 - 5$ eV. \nThe $m_a$ and $g_{a\\gamma\\gamma}$ in Fig. \\ref{fig:composite} were chosen specifically so that $P_L \\gtrsim 0.05$ \nfor all photon energies. \n\nIn Fig. \\ref{fig:exclusion}, we use the same parameters to draw an exclusion region in the $m_a-g_{a\\gamma\\gamma}$ \nparameter space, along with regions excluded by lab experiments and astrophysical considerations. The shaded region \nin red excludes all $m_a$ - $g_{a\\gamma\\gamma}$ values for the case of mWD PG 1015+014, that is for a typical \nsurface field strength $B\\simeq10^8$ G and degree of linear polarization $P_L\\sim 5\\%$. We find that for \nthe range of masses that are of relevance, in particular, to the axion models, the constraints on \n$g_{a\\gamma\\gamma}$ from this study are superseded by that from horizontal branch (HB) stars. Still, the \nlimiting linear polarization criterion used in this study is able to probe smaller $g_{a\\gamma\\gamma}$ \nvalues in comparison to works that only look at radiation dimming (for e.g. see \\citep{Jimenezetal2011}).\n\nThe constraints can be further improved by looking at mWDs with higher magnetic field strengths. The highest \nfield strength that has ever been discovered in a mWD is $B\\simeq1000$ MG in two such objects namely, \nPG 1031+234 and SDSS J234605+385337 \\citep{Jordan2009}. Both objects show linear polarization as low as \n$\\sim1\\%$ for some rotational phases \\citep{PiirolaReiz1992,Vanlandinghametal2005}. Based on these two \nfacts and assuming that the magnetic field geometry of these two mWDs is at least as complex as that found \nin PG 1015+014, we produce two exclusion regions shown in Fig. \\ref{fig:exclusion} with colors blue \nand green. The former corresponds to a surface field strength $B=1000$ MG with the same level of linear polarization \nas before, and the latter studies the case with $P_L\\simeq1\\%$. For these two cases, we have only looked at \n$m_a \\leq 10^{-5}$ eV since higher mass values don't constrain $g_{a\\gamma\\gamma}$ better than limits derived \nfrom HB stars and CAST (Phase-I). On the other hand, we have extended our treatment to smaller particle masses \nwith $m_\\phi\\leq10^{-6}$ eV where $m_\\phi$ should be interpreted as the mass of any light pseudoscalar \nboson that is characteristically very much similar to the axion but isn't a CDM particle.\n\nIt is worth mentioning that the change in $g_{a\\gamma\\gamma}$ is not linear with the change in magnetic \nfield strength, as evident from the comparison between the red and blue regions in Fig. \\ref{fig:exclusion}.\nFor higher field strengths one observes higher degree of polarization of the emerging radiation. Naively, one \nwould expect the plane of polarization to rotate by an amount that is $\\mathcal{O}(g_{a\\gamma\\gamma}^2 \nB^2l^2)$, which is valid strictly in the absence of plasma when the photon and axion are treated as \nmassless particles \\citep{VanBibberetal1987,RaffeltStodolsky1988}. Therefore, for a fixed degree of \npolarization an increase in $B$ should also decrease $g_{a\\gamma\\gamma}$ by the same factor, when \n$l$, the length over which the magnetic field remains homogeneous, is kept constant. However, as \nshown by \\citep{Heyletal2003} in the case of NSs, an increase in magnetic field strength also \nincreases the level of polarization by effectively shifting the polarization-limiting radius, \nthe distance beyond which the two polarization modes couple and which depends weakly on \nthe magnetic field strength $R_{pl}\\propto B^{2\/5}$, farther away from the star. The farther the \npolarization-limiting radius, the more coherently the polarization states from different LOSs add, \nyielding a higher degree of polarization.\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig5}\n\\caption{Exclusion plot in the $m_a-g_{a\\gamma\\gamma}$ parameter space. The red region corresponds to \nthe case of mWD PG 1015+014 with $B\\simeq10^8$ G and limiting linear polarization $P_L\\sim5\\%$. The \nblue and green regions correspond to the case with $B\\simeq10^9$ G, but with $P_L\\sim5\\%$ and \n$P_L\\sim1\\%$ respectively. The mass of the \naxion is constrained to $10^{-6} \\lesssim m_a \\lesssim 10^{-2}$ eV from cosmology and SN 1987A measurments.\nThe photon-axion coupling constant is capped from above with $g_{a\\gamma\\gamma} < 10^{-10}\n\\text{ GeV}^{-1}$ by the number of horizontal branch stars in globular clusters. KSVZ \n\\citep{Kim1979,ShifmanVainshteinZakharov1980} and DFSZ \\citep{DineFischlerSrednicki1981,Zhitnitskii1980} \nare two different theoretical models that predict how $g_{a\\gamma\\gamma}$ scales with $m_a$. \nOther exclusion regions are from lab experiments by the CAST experiment \\citep{Andriamonjeetal2007}, \nADMX group \\citep{Asztalosetal2004,Asztalosetal2010} and by Rochester-Brookhaven-Fermilab collaboration \n\\citep{Panfilisetal1987,Wuenschetal1989} (color online).} \n\\label{fig:exclusion}\n\\end{figure}\n\\section{Discussion}\nThis study looks at how the production of axions in mWD magnetospheres can alter the \nstate of polarization of the observed radiation. We find that unpolarized photons of \nphotospheric origin become linearly polarized upon their traversal through the \ninhomogeneous magnetic field of a mWD. We have modeled the magnetospheric plasma, as fully \nionized pure H with a barometric profile. Since the majority of mWDs are strongly \ncircularly polarized and only show a relatively \nsmall degree of linear polarization, at most $5\\%$, we have used this observation to constrain the \ncoupling strength $g_{a\\gamma\\gamma}$ of axions to photons. We find that for the case where \nthe plasma component only contributes negligibly to the state of polarization, the \ncoupling strength $g_{a\\gamma\\gamma}$ increases with the mass of the axion $m_a$. The level \nof linear and circular polarization observed in mWDs is sensitive to the properties of the \nmagnetospheric plasma. The limits on $g_{a\\gamma\\gamma}$ can be improved by modelling all \nthe radiative transfer effects in WD atmospheres and fitting the model spectra to real observations.\n\nMagnetic fields stronger than that of mWDs exist in NSs. Going back to the argument of how \nastrophysical objects, compared to laboratory experiments, benefit from longer coherence \nlengths (see Section I), in comparing mWDs with NSs, one finds that the latter are $\\sim 10^4$ \ntimes more efficient in converting photons to axions and vice-versa. A number of studies have expounded \non the subject of propagation of polarized radiation through the NS magnetosphere, where \nthey have considered IR\/Optical radiation \\citep{ShannonHeyl2006}, and thermal X-rays \n\\citep{LaiHo2003a,LaiHo2003b} produced at the surface of the NS. Unfortunately, no X-ray \npolarimetry observations have been conducted partly due to the very low flux in X-rays \nfrom these objects, and also because none of the high energy telescopes are equipped \nwith a polarimeter. X-ray polarimetry has been neglected for the last 30 years but it \nis hoped that some of the future space missions \\citep{Mulerietal2010}, for example the Gravity and Extreme \nMagnetism Small Explorer (GEMS) \\citep{Jahoda2010}, will fill this \nvoid in X-ray astronomy. In any case, as discussed by \\citep{RaffeltStodolsky1988,\nLaiHeyl2006,Cheloucheetal2009}, NSs are excellent laboratories for the detection of \nany light, weakly coupled pseudoscalar particle. \n\\subsection{Outlook}\nThe ADMX\\footnote{http:\/\/www.phys.washington.edu\/groups\/admx\/experiment.html} \nproject, that employs a microwave cavity to search for cold dark matter axions, \nwill begin its phase II of testing in the year 2012. With the new \nupgrades the ADMX project will be able to exclude $g_{a\\gamma\\gamma}$ up to the DFSZ line in \nthe same mass range as before. Although outside of the range of axion masses probed \nin this study, the modified CAST experiment has been able to exclude axions with \n$g_{a\\gamma\\gamma} \\gtrsim 2.2\\times10^{-10}\\mbox{ GeV}^{-1}$ for $m_a \\lesssim 0.4$ \neV, becoming the first experiment to ever cross the KSVZ line \\citep{Ariketal2009,Cast2011}. \nThe currently running CAST experiment in its phase-II will \nbe able to exclude axions with $m_a\\lesssim 1.15$ eV with unprecedented sensitiviy in \nthis mass range. An improved version of the \\textit{light shining through wall} (LSW) experiment \n(\\citep{VanBibberetal1987}, also see \\citep{RedondoRingwald2011} for a recent review on \nsuch experiments) using Fabry-Perot optical cavities to resonantly enhance photon-axion conversion \nhas been proposed \\citep{HoogeveenZiegenhagen1991,Muelleretal2009,Sikivieetal2007}. The projected limit in sensitivity to \n$g_{a\\gamma\\gamma} \\gtrsim 2.0\\times 10^{-11}\\mbox{ GeV}^{-1}$ typically for \naxion masses $m_a \\lesssim 10^{-4}$ eV achieved using 12 Tevatron superconducting \ndipoles appears quite promising. Further improvements in experiment design and optimization techniques \nyielding increased sensitivity to even smaller coupling strengths have also been suggested by many workers \nin the field, for example the use of the dipole magnets, each providing a field strength of 5 T, \nfrom the Hadron Electron Ring Accelerator (HERA) at DESY in \nHamburg in a 20+20 configuration can potentially exclude \n$g_{a\\gamma\\gamma} \\gtrsim 10^{-11}\\text{ GeV}^{-1}$ for $m_a < 10^{-4}$ eV \\citep{Ringwald2003,Ariasetal2010}. \nAnother proposed line of investigation to search for axion like particles (ALPs) is the use of resonant microwave cavities which are \nmuch similar in design to the optical LSW experiments discussed above \\citep{Hoogeveen1992,Caspersetal2009,JaeckelRingwald2008}. This \nmethod has already been employed to search for hidden sector photons \\citep{Poveyetal2010} and can prove to be a powerful \ntool in the case of axions.\n\nFinally, the simplistic model assumed for the mWD atmosphere only yields an absolute upper bound \non $g_{a\\gamma\\gamma}$. A much tighter constraint can be obtained by adopting a more realistic \natmospheric model and solving the equations of radiative transfer with the photon-axion oscillations \nincluded. Such an analysis is outside the scope of this study, but it is hoped that the novel \nmethod discussed in this work will prove to be extremely useful in better constraining the \nproperties of any ALP.\n\\begin{acknowledgements}\n We would like to thank the referee for significantly improving the \n quality of this work. R.G. acknowledges support by the NSERC CGS-D3 scholarship. \n The Natural Sciences and Engineering Research Council of Canada,\n Canadian Foundation for Innovation and the British Columbia\n Knowledge Development Fund supported this work. Correspondence and\n requests for materials should be addressed to\n J.S.H. (heyl@phas.ubc.ca). This research has made use of NASA's\n Astrophysics Data System Bibliographic Services\n\\end{acknowledgements}\n\\bibliographystyle{prsty}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlnmd b/data_all_eng_slimpj/shuffled/split2/finalzzlnmd new file mode 100644 index 0000000000000000000000000000000000000000..755ddc6de47be82fdd1ce68536cf05985aacd0dc --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzlnmd @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $X$ be a curve, $\\widetilde{X} \\stackrel{\\nu}\\rightarrow X$ its normalization, $\\mathcal{O} = \\mathcal{O}_X$ and $\\widetilde\\mathcal{O} =\\nu_*(\\mathcal{O}_{\\widetilde{X}})$.\n Generalizing an original idea of K\\\"onig \\cite{koe}, we define a sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$ on $X$ called \\emph{K\\\"onig's order} such that the ringed space\n$\\mathbb{X} = (X, \\mathcal A} \\def\\kN{\\mathcal N)$ has the following properties.\n\n\\medskip\n\\noindent\n1.~The non--commutative curve $\\mathbb{X}$ is ``smooth'' in the sense that \\linebreak\n$\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) < \\infty$, where $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$ is the category of coherent $\\mathcal A} \\def\\kN{\\mathcal N$--modules on $X$. In fact, $\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\le 2n$, where $n$ is a certain (purely commutative) invariant of $X$ called \\emph{level}. If the original curve $X$ has only nodes and cusps as singularities, the sheaf $\\mathcal A} \\def\\kN{\\mathcal N$ coincides with \\emph{Auslander's order}\n $$\n \\left(\n \\begin{array}{cc}\n \\mathcal{O} & \\widetilde\\mathcal{O} \\\\\n \\mathcal I} \\def\\kV{\\mathcal V & \\widetilde\\mathcal{O}\n \\end{array}\n \\right)\n $$\n introduced in \\cite{bd}, where $\\mathcal I} \\def\\kV{\\mathcal V$ is the ideal sheaf of the singular locus of $X$.\n\n\\medskip\n\\noindent\n2.~The non--commutative curve $\\mathbb{X}$ is a \\emph{non--commutative} (or \\emph{categorical}) \\emph{resolution of singularities} of $X$, see \\cite{vdb,ku} for the definitions.\nThe category $\\mathop{\\mathsf{Coh}}\\nolimits(X)$ of coherent sheaves on $X$ is a Serre quotient of $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$. Moreover, the triangulated category $\\mathop{\\mathsf{Perf}}\\nolimits(X)$ of perfect complexes on $X$ admits an exact fully faithful embedding $\\mathop{\\mathsf{Perf}}\\nolimits(X) \\hookrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ such that its composition with the Verdier localization $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\rightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ is isomorphic to the canonical inclusion functor.\nIf the curve $X$ is Gorenstein, the constructed categorical resolution of singularities of $X$ turns out to be \\emph{weakly crepant} in the sense of Kuznetsov \\cite{ku}.\n\n\\medskip\n\\noindent\n3.~We show that the triangulated category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$\nis a recollement of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D^b(Q-\\mathrm{mod})$,\nwhere $Q$ is a certain \\emph{quasi-hereditary} artinian ring (in particular, of finite global dimension), determined ``locally'' by the singularity types of the singular points of $X$. In the case of simple curve singularities, we describe the corresponding algebras $Q$\nexplicitly in terms of quivers and relations.\n\n\n\n \\medskip\n\\noindent\n4.~Assume $X$ is projective over some field $\\mathbbm{k}$. According to Orlov \\cite{or}, the Rouquier dimension \\cite{rou} of the triangulated category\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ is equal to \\emph{one}. Let $\\widetilde\\mathcal F} \\def\\kS{\\mathcal S$ be a vector bundle on $\\tilde{X}$ such that\n$\\langle\\widetilde{\\mathcal F} \\def\\kS{\\mathcal S}\\rangle_2 = D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $\\mathcal F} \\def\\kS{\\mathcal S = \\nu_*(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S)$. We show that\n$\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\mathcal{O}_Z\\bigr\\rangle_{n+2}\n$\nwhere $\\mathcal{O}_Z$ is the structure sheaf of the singular locus of $X$ (with respect to the reduced scheme structure) and\n$n$ is the level of $X$.\n\n\n\n\\medskip\n\\noindent\n5.~If our original curve $X$ is moreover rational, then we show that $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ admits a\n\\emph{tilting object} $\\mathcal H} \\def\\kU{\\mathcal U$ such that the finite dimensional $\\mathbbm{k}$--algebra $\\Lambda = \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{D^b(\\mathbb{X})}(\\mathcal H} \\def\\kU{\\mathcal U)\\bigr)^{\\mathrm{op}}$ is quasi--hereditary. In particular, we get an exact fully faithful embedding\n $\\mathop{\\mathsf{Perf}}\\nolimits(X) \\hookrightarrow D^b(\\Lambda-\\mathrm{mod})$,\n giving an affirmative answer on a question posed to the first--named author by Valery Lunts.\n\n\\medskip\n\\noindent\n\\emph{Acknowledgement}. The work on this article has been started during the stay of the second--named author at the Max--Planck--Institut f\\\"ur Mathematik in Bonn. Its final version was prepared during the visit of the second-- and the third--named author to the Institute of Mathematics of the University of Cologne. The first--named author would like to thank Valery Lunts for the invitation and iluminative discussions during his visit to the Indiana University Bloomington. We are thankful to the referees for their helpful comments.\n\n\n\\section{Local description of K\\\"onig's order}\\label{Sec:LocalStory}\n\nLet $(O, \\mathfrak{m})$ be a reduced Noetherian local ring of Krull dimension one, $K$ be its total ring of fractions and\n$\\widetilde{O}$ be the normalization of $O$.\n\n\\begin{proposition}\\label{P:normalizelocal} Consider the ring $O^\\sharp = \\mathop{\\mathrm{End}}\\nolimits_{O}(\\mathfrak{m})$. Then the following properties hold.\n\\begin{itemize}\n\\item $O^\\sharp \\cong \\left\\{x \\in K \\big| x \\mathfrak{m} \\subset \\mathfrak{m} \\right\\}$. Moreover,\n$O \\subseteq O^\\sharp \\subseteq \\widetilde{O}$ and $O = O^\\sharp$ if and only if $O$ is regular.\n\\item Assume that $O$ is not regular. Then the canonical morphisms of $O$--modules\n$$\n\\mathfrak{m} \\stackrel{\\varphi}\\longrightarrow \\Hom_{O}(O^\\sharp, O) \\quad \\mbox{and} \\quad\nO^\\sharp \\stackrel{\\psi}\\longrightarrow \\Hom_{O}(\\mathfrak{m}, O)\n$$\nare isomorphisms.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nFor the first part, see for example \\cite[Proposition 4]{com} or \\cite[Theorem 1.5.13]{deJongPfister}. To show the second part, note that $\\varphi$ assigns\nto an element $a \\in \\mathfrak{m}$ a morphism $O^\\sharp \\stackrel{\\varphi_a}\\longrightarrow O$, where $\\varphi_a(x) = ax$. It is clear that $\\varphi$ is injective. Since $\\Hom_{O}(O^\\sharp, O)$ viewed as a subset of $K$ is a proper ideal in $O$, it is contained in $\\mathfrak{m}$.\nHence, $\\varphi$\n is also surjective, hence bijective.\n\nNext, the canonical morphism $\\Hom_O(\\mathfrak{m}, \\mathfrak{m}) \\stackrel{\\psi}\\longrightarrow \\Hom_O(\\mathfrak{m}, O)$ is injective. On the other hand, there are no surjective morphisms $\\mathfrak{m} \\rightarrow O$ (otherwise, $O$ would be a discrete valuation domain), hence the image of any morphism $\\mathfrak{m} \\rightarrow O$ belongs to $\\mathfrak{m}$ and $\\psi$ is surjective.\n\\end{proof}\n\nFrom now on, let $O$ be an \\emph{excellent} reduced Noetherian ring of Krull dimension one (see for example\n\\cite[Section 8.2]{liu} for the definition and main properties of excellent rings). As before, $K$ denotes its total ring of fractions and\n$\\widetilde{O}$ is the normalization of $O$. Let $X = \\Spec(O)$ and $Z$ be the singular locus of $X$ equipped\nwith the reduced scheme structure. In other words,\n$$Z =\n\\bigl\\{\\mathfrak{m}_1, \\dots, \\mathfrak{m}_t\\bigr\\}\n= \\bigl\\{\\mathfrak{m} \\in \\Spec(O) \\; \\big| \\; O_\\mathfrak{m} \\; \\mbox{is not regular}\\bigr\\}\n$$\n(the condition that $O$ is excellent implies that $Z$ is indeed a finite set).\n\n\\begin{proposition}\\label{P:DetailsKoenigResol} Let\n$\nI = I_Z = \\mathfrak{m}_1 \\cap \\dots \\cap \\mathfrak{m}_t\n$\nbe the vanishing ideal of $Z$ and $O^\\sharp = \\mathop{\\mathrm{End}}\\nolimits_{O}(I)$. Then the following properties are true.\n\\begin{itemize}\n\\item $O^\\sharp \\cong \\left\\{x \\in K \\big| x \\mathfrak{m} \\subset \\mathfrak{m} \\right\\}$. Moreover,\n$O \\subseteq O^\\sharp \\subseteq \\widetilde{O}$ and $O = O^\\sharp$ if and only if $O$ is regular.\n\\item Assume that $O$ is not regular. Then the canonical morphisms of $O$--modules\n$\n\\mathfrak{m} \\stackrel{\\varphi}\\longrightarrow \\Hom_{O}(O^\\sharp, O)$ and\n$O^\\sharp \\stackrel{\\psi}\\longrightarrow \\Hom_{O}(\\mathfrak{m}, O)$\nare isomorphisms.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nFor the first part, see again \\cite[Proposition 4]{com} or \\cite[Theorem 1.5.13]{deJongPfister}. To prove the second, observe that the maps $\\varphi$ and $\\psi$ are well--defined and compatible with localizations with respect to a maximal ideal.\nHence, Proposition \\ref{P:normalizelocal} implies the claim.\n\\end{proof}\n\nWe define a sequence of overrings $O_i$ of the initial ring $O$ by the following recursive procedure:\n\\begin{itemize}\n\\item $O_1 = O$.\n\\item $O_{i+1} = O_i^\\sharp$ for $i \\ge 1$.\n\\end{itemize}\nSince the ring $O$ is excellent, the normalization $\\widetilde{O}$ is finite over $O$, see for example \\cite[Theorem 6.5]{ddone} or \\cite[Section 8.2]{liu}. Hence,\nthere exists $n \\in \\mathbb{N}$ (called the \\emph{level} of $O$) such that we have a finite chain of overrings\n$$\nO_1 \\subset O_2 \\subset \\dots \\subset O_n \\subset O_{n+1}\n$$\nwith $O_1 = O$ and\n$O_{n+1} = \\widetilde{O}$.\n\n\\begin{definition}\\label{D:KoenigsOrder}\nThe ring $A := \\mathop{\\mathrm{End}}\\nolimits_{O}(O_1 \\oplus O_2 \\oplus \\dots \\oplus O_{n+1})^{\\mathrm{op}}$ is called the \\emph{K\\\"onig's order} of $O$.\n\\end{definition}\n\n\n\\begin{proposition}\\label{P:KoenigOrderMatrDescr}\nFor any $1 \\le i, j \\le n+1$ pose $A_{ij} := \\Hom_{O}(O_i, O_j)$. Then the following properties are true.\n\\begin{itemize}\n\\item For $i \\le j$ we have: $A_{ij} \\cong O_j$.\n\\item For $i > j$ we have: $A_{ij} \\cong I_{i,j} := \\Hom_{O_j}(O_i, O_j)$. In particular, $I_{n+1,1} \\cong C := \\Hom_{O}(\\widetilde{O}, O)$\nis the \\emph{conductor ideal}.\n\\item Next, $I_i:= I_{i+1,i}$ is the ideal of the singular locus of $\\Spec(O_i)$ and\nthe ring $\\bar{O}_i := O_i\/I_i$ is semi--simple.\n\\item Moreover, the ideal $I_{n+1, k}$ is projective over $\\mathcal{O}_{n+1}$ for any $1 \\le k \\le n$.\n\\item The ring $A$ admits the following ``matrix description'':\n\\begin{equation}\\label{E:KoenigOrder}\nA \\cong\n\\left(\n\\begin{array}{llccc}\nO_1 & O_2 & \\dots & O_n & O_{n+1} \\\\\nI_1 & O_2 & \\dots & O_n & O_{n+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\nI_{n,1} & I_{n, 2} & \\dots & O_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\end{array}\n\\right)\n\\end{equation}\nand $A \\otimes_O K \\cong \\mathsf{Mat}_{n+1, n+1}(K)$. In other words, $A$ is an \\emph{order} in the semi-simple algebra\n$\\mathsf{Mat}_{n+1, n+1}(K)$.\n\\item For any $2 \\le i \\le n+1$ and $1 \\le j \\le n$ we have inclusions\n\\begin{itemize}\n\\item $I_{i,1} \\subset I_{i,2} \\subset \\dots \\subset I_{i,i-1} \\subset O_i \\subset \\dots \\subset O_{n+1}$\n\\item $I_{n+1,j} \\subset I_{n,j} \\subset \\dots \\subset I_{j+1,j} \\subset O_j$\n\\end{itemize}\ndescribing the ``hierarchy'' between the entries in every row and every column in the matrix description\n(\\ref{E:KoenigOrder}) of the ring $A$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nWe have the following canonical isomorphisms of $O$--modules:\n$$\nO_j \\cong \\Hom_{O_i}(O_i, O_j) \\stackrel{\\cong}\\longrightarrow \\Hom_{O}(O_i, O_j)\n$$\nprovided $i \\le j$ as well as\n$$\nI_{i,j} := \\Hom_{O_j}(O_i, O_j) \\stackrel{\\cong}\\longrightarrow \\Hom_{O}(O_i, O_j)\n$$\nfor $i > j$. Proposition \\ref{P:DetailsKoenigResol} implies that the ideal $I_i = I_ {i+1, i}$ is indeed the ideal of the singular locus of $\\Spec(O_i)$, hence the quotient $\\bar{O}_i = O_i\/I_i$ is semi--simple.\nSince the ring $O_{n+1}$ is regular and the ideal $I_{n+1,k}$ is torsion free as $O_{n+1}$--module, it is projective over $O_{n+1}$.\n\nFinally, for any $1 \\le j \\le n$ and $1 \\le i \\le n+1$ the inclusion $O_j \\subset O_{j+1}$ induces embeddings of $O$--modules\n\\begin{align*}\n \\Hom_{O}(O_{j+1}, O_i)& \\hookrightarrow \\Hom_{O}(O_{j}, O_i)\\\\\n \\intertext{and}\n \\Hom_{O}(O_{i}, O_j)&\\hookrightarrow \\Hom_{O}(O_{i}, O_{j+1}).\n\\end{align*}\n\\vskip-1.5em\n\\end{proof}\n\n\\begin{remark}\nThe idea to study such a ring $A$ is due to K\\\"onig \\cite{koe}, who considered a similar but slightly different construction.\n\\end{remark}\n\n\\smallskip\n\\noindent\nFor any $1 \\le i \\le n+1$ let $e_i = e_{i,i}$ be the $i$-th standard idempotent of $A$ with respect to the presentation (\\ref{E:KoenigOrder}). For $1 \\le k \\le n$ we denote\n\\begin{itemize}\n\\item $\\varepsilon_{k} := \\sum\\limits_{i = k+1}^{n+1} e_i$, $J_k := A \\varepsilon_k A$ and\n$Q_k := A\/J_k$.\n\\item In what follows we write $e = e_{n+1}$, $J = A e A = J_{n+1}$ and $Q := A\/J = Q_n$.\n\\end{itemize}\n\n\\begin{theorem}\\label{T:GlobalDimensionKoenigOrder} The global dimension of $A$ is finite:\n $\\mathrm{gl.dim}(A) \\le 2n$. Moreover, the artinian ring $Q = Q_O$ is quasi--hereditary (hence, its global dimension is finite, too).\n\\end{theorem}\n\n\\begin{proof} A straightforward calculation shows that for every $2 \\le k \\le n+1$ the two--sided ideal $J_{k-1}$ has the following matrix description:\n$$\nJ_{k-1} =\n\\left(\n\\begin{array}{llclllcc}\nI_{k,1} & I_{k,2} & \\dots &I_{k,k-1} & O_k & O_{k+1} & \\dots & O_{n+1} \\\\\n \\vdots& \\vdots & \\vdots &\\vdots & \\vdots& \\vdots & \\vdots& \\vdots\\\\\nI_{k,1} & I_{k,2} & \\dots &I_{k,k-1} & O_k & O_{k+1} & \\dots & O_{n+1} \\\\\nI_{k+1,1} & I_{k+1,2} & \\dots &I_{k+1,k-1} & I_{k+1,k} & O_{k+1} & \\dots & O_{n+1} \\\\\n \\vdots& \\vdots & \\vdots &\\vdots & \\vdots& \\vdots & \\vdots& \\vdots\\\\\n I_{n+1,1} & I_{n+1,2} & \\dots &I_{n+1,k-1} & I_{n+1,k} & I_{n+1,k+1} & \\dots & O_{n+1} \\\\\n\\end{array}\n\\right).\n$$\nIn other words, the $i$-th row of $J_{k-1}$ is the same as for $A$ provided $k \\le i \\le n+1$ and\nin the case $1 \\le i \\le k-1$ the $i$-th and the $k$-th rows of $J_{k-1}$ are the same. In particular, the ideal\n$J = J_n$ has the shape\n\\begin{equation*}\nJ =\n\\left(\n\\begin{array}{llccc}\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\nI_{n+1,1} & I_{n+1, 2} & \\dots & I_n & O_{n+1} \\\\\n\\end{array}\n\\right).\n\\end{equation*}\nConsider the projective left $A$--module $P := Ae$. Then we have an adjoint pair\n$$\n\\xymatrix{A-\\mathsf{mod} \\ar@\/^2ex\/[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} & & \\widetilde{O}-\\mathsf{mod} \\ar@\/^2ex\/[ll]|{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}\n}\n$$\n where\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(P, \\,-\\,)$ and $\\tilde\\mathsf F} \\def\\sS{\\mathsf S = P \\otimes_{\\widetilde{O}} \\,-\\,$. The functor $\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ is exact and has the following explicit description: if $M$ is an $\\widetilde{O}$--module then\n$$\n\\tilde\\mathsf F} \\def\\sS{\\mathsf S(M) = M^{\\oplus{(n+1)}} =\n\\left(\n\\begin{array}{c}\nM \\\\\nM\\\\\n\\vdots \\\\\nM\n\\end{array}\n\\right)\n$$\nwhere the left $A$--action on $M^{\\oplus{(n+1)}}$ is given by the matrix multiplication. Since for every\n$1 \\le k \\le n$ the $O$--module $I_{n+1, k}$ is also a projective $\\widetilde{O}$--module, we see that the left $A$--module $J e_k$\nbelongs to the essential image of $\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and is projective over $A$. It is clear that all right $A$--modules\n$e_k J$ are projective, too. Since $P$ is free over $\\widetilde{O} = \\mathop{\\mathrm{End}}\\nolimits_{A}(P)$, \\cite[Lemma 4.9]{bdg} implies that\n$\n\\mathrm{gl.dim}(A) \\le \\mathrm{gl.dim}(Q) + 2.\n$\n\n\\noindent\nNext, observe that for every $1 \\le k \\le n$ the ring $Q_k$ has the following ``matrix description'':\n$$\nQ_k \\cong\n\\left(\n\\begin{array}{llcc}\n\\dfrac{O_1}{I_{k+1,1}} & \\dfrac{O_2}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\dfrac{I_{2,1}}{I_{k+1,1}} & \\dfrac{O_2}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\end{array}\n\\right),\n$$\nwhere $\\dfrac{{O}_k}{I_k} =: \\bar{O}_k$ is semi--simple.\nFor $1 \\le k \\le n$ let $\\bar{e}_k$ be the image\nof the idempotent $e_k \\in A$ in the ring $Q_k = A\/J_k$. Observe that for $2 \\le k \\le n$\n$$L_k := J_{k-1}\/J_k = Q_k \\bar{e}_k Q_k =\nQ_k \\cong\n\\left(\n\\begin{array}{llcc}\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\dfrac{I_{k,1}}{I_{k+1,1}} & \\dfrac{I_{k,2}}{I_{k+1, 2}} & \\dots & \\dfrac{{O}_k}{I_k} \\\\\n\\end{array}\n\\right)\n\\subset Q_k$$\nis projective viewed both as a left and as a right\n$Q_k$--module (via the same argument as for $J$ and $A$). Moreover, $Q_k\/L_k \\cong Q_{k-1}$ and\n$\\bar{e}_k Q_k \\bar{e}_k = \\bar{O}_k$ is semi--simple.\nTherefore, $J_1\/J \\subset J_2\/J \\subset \\dots \\subset J_n\/J$ is a \\emph{heredity chain} in $Q$ and the ring\n$Q$ is \\emph{quasi--hereditary}, see \\cite{CPS,dr} or the appendix of Dlab in \\cite{dk} for the definition and main properties of quasi--hereditary rings. It is well--known that\n$\n\\mathrm{gl.dim}(Q) \\le 2(n-1),\n$\nsee \\cite[Statement 9]{dr}, \\cite[Theorem A.3.4]{dk} (or \\cite[Lemma 4.9]{bdg} for a short proof). The theorem is proven.\n\\end{proof}\n\n\\begin{remark}\nThe bound on the global dimension of $A$ given in Theorem \\ref{T:GlobalDimensionKoenigOrder} is not optimal.\nFor example, if $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{m(n)})$ with $m(n) = 2n$ (respectively $2n+1$) is a simple singularity of type\n$A_{m(n)-1}$, then the level of $O$ is $n$. On the other hand, $O_1 \\oplus \\dots \\oplus O_{n+1}$ is the additive generator of the category of maximal Cohen--Macaulay modules, see \\cite[Section 7]{Bass}, \\cite[Section 5]{lw} or\n\\cite[Section 9]{Yoshino}. Hence, by a result of Auslander and Roggenkamp \\cite{ar}, the global dimension of $A$ is equal to two.\n\n\\smallskip\n\\noindent\nIn the particular cases $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{2})$ (simple node) and $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{3})$ (simple cusp) the K\\\"onig's order $A$ coincides with the Auslander's order\n$\n\\left(\n\\begin{array}{cc}\nO & \\widetilde{O} \\\\\nC & \\widetilde{O}\n\\end{array}\n\\right)\n$\nintroduced in the work \\cite{bd}.\n\\end{remark}\n\n\\begin{remark}\nBasic properties of excellent rings (see \\cite[Section 6]{ddone} or \\cite[Section 8.2]{liu}) imply that\n$$Q_O := Q \\cong Q_{\\widehat{O}_1} \\times \\dots \\times Q_{\\widehat{O}_t},$$ where $\\widehat{O}_i$ is the completion of the local ring\n$O_{\\mathfrak{m}_i}$ for each $\\mathfrak{m}_i \\in \\mathrm{Sing}(O)$. In other words, the quasi--hereditary ring $Q$ depends only on\nthe \\emph{local singularity types} of $\\Spec(O)$.\n\\end{remark}\n\n\\section{K\\\"onig's order as a categorical resolution of singularities}\nFor a (left) Noetherian ring $B$ we denote by $B-\\mathsf{mod}$ the category of all finitely generated left $B$--modules and by $B-\\mathsf{Mod}$ the category of all left $B$--modules.\nAs in the previous section, let $O$ be an excellent reduced Noetherian ring of Krull dimension one and level $n$, $\\widetilde{O}$ be its normalization, $A$ be the K\\\"onig's order of $O$ and $Q$ be the quasi--hereditary artinian algebra attached to $O$. Let $e = e_{n+1}$ and $f = e_1$ be two standard idempotents of $A$, $P = Ae$, $T = Af$ and\n$J = AeA$.\nIt is clear that $\\widetilde{O} \\cong \\mathop{\\mathrm{End}}\\nolimits_A(P)$ and $O \\cong \\mathop{\\mathrm{End}}\\nolimits_A(T)$.\nWe also denote $T^\\vee := \\Hom_A(T, A) \\cong f A$ and $P^\\vee := \\Hom_A(P, A) \\cong e A$.\nThen we have the following diagram of categories and functors:\n\\begin{equation}\n\\xymatrix{O-\\mathsf{mod} \\ar@\/^2ex\/[rr]^{\\,\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\mathsf H} \\def\\sU{\\mathsf U\\,} && A-\\mathsf{mod} \\ar[ll]|{\\,\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && \\widetilde{O}-\\mathsf{mod} \\ar@\/^2ex\/[ll]^{\\,\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nwhere $\\mathsf F} \\def\\sS{\\mathsf S = T \\otimes_O \\,-\\,$, $\\mathsf H} \\def\\sU{\\mathsf U = \\Hom_O(T^\\vee, \\,-\\,)$, $\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(T, \\,-\\,)$ and similarly,\n$\\tilde\\mathsf F} \\def\\sS{\\mathsf S = P \\otimes_{\\widetilde{O}} \\,-\\,$, $\\tilde\\mathsf H} \\def\\sU{\\mathsf U = \\Hom_{\\widetilde{O}}(P^\\vee, \\,-\\,)$,\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\Hom_A(P, \\,-\\,)$. There is the same diagram for the categories of all modules\n$O-\\mathsf{Mod}$, $\\widetilde{O}-\\mathsf{Mod}$ and $A-\\mathsf{Mod}$. The following results are standard, see for example \\cite[Theorem 4.3]{bdg} and references therein.\n\n\\begin{theorem}\nThe pairs of functors $(\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf H} \\def\\sU{\\mathsf U)$ (and respectively $(\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf H} \\def\\sU{\\mathsf U)$) are adjoint and the functors $\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf H} \\def\\sU{\\mathsf U, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful. Both categories\n$O-\\mathsf{mod}$ and $\\widetilde{O}-\\mathsf{mod}$ are Serre quotients of $A-\\mathsf{mod}$:\n$$\nO-\\mathsf{mod} \\cong A-\\mathsf{mod}\/\\mathsf{Ker}(\\mathsf G} \\def\\sT{\\mathsf T) \\quad \\mbox{and} \\quad\n\\widetilde{O}-\\mathsf{mod} \\cong A-\\mathsf{mod}\/\\mathsf{Ker}(\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$$\nMoreover,\n$\n\\mathsf{Ker}(\\tilde\\mathsf G} \\def\\sT{\\mathsf T) = Q-\\mathsf{mod}.\n$\n\\end{theorem}\nThe described picture becomes even better when we pass to (unbounded) derived categories. Observe that\nthe functors $\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are exact. Their derived functors will be denoted\nby $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ respectively, whereas $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ is the left derived functor of $\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ is the right derived functor of $\\mathsf H} \\def\\sU{\\mathsf U$.\n\n\n\\begin{theorem} We have a diagram of categories and functors\n\\begin{equation}\\label{E:descinglobale}\n\\xymatrix{D(O-\\mathsf{Mod}) \\ar@\/^2ex\/[rr]^{\\,\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\sR\\mathsf H} \\def\\sU{\\mathsf U\\,} && D(A-\\mathsf{Mod}) \\ar[ll]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && D(\\widetilde{O}-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nsatisfying the following properties.\n\\begin{itemize}\n\\item The following pairs of functors $(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\sR\\mathsf H} \\def\\sU{\\mathsf U)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)$ and\n$(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs.\n\\item The functors $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$, $\\sR\\mathsf H} \\def\\sU{\\mathsf U$, $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item Both derived categories $D(O-\\mathsf{Mod})$ and $D(\\widetilde{O}-\\mathsf{Mod})$ are Verdier localizations of\n$D(A-\\mathsf{Mod})$:\n\\begin{itemize}\n \\item $\nD(O-\\mathsf{Mod}) \\cong D(A-\\mathsf{Mod})\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$.\n\\item\n$D(\\widetilde{O}-\\mathsf{Mod}) \\cong D(A-\\mathsf{Mod})\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$\n\\end{itemize}\n\\item Moreover,\n$\n\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T) = D_Q(A-\\mathsf{Mod}) \\cong D(Q-\\mathsf{Mod}).\n$\n\\item\nThe derived category $D(A-\\mathsf{Mod})$ is a categorical resolution of singularities of $X = \\Spec(O)$ in the sense of Kuznetsov \\cite[Definition 3.2]{ku}.\n \\item If $O$ is Gorenstein, then the restrictions of $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ on $\\mathop{\\mathsf{Perf}}\\nolimits(O)$ are isomorphic. Hence, the constructed categorical resolution is even \\emph{weakly crepant} in the sense of\n \\cite[Definition 3.4]{ku}.\n\\end{itemize}\nWe have a \\emph{recollement diagram}\n\\begin{equation}\\label{E:recollement}\n\\xymatrix{D(Q-\\mathsf{Mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D(A-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D(\\widetilde{O}-\\mathsf{Mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nand all functors can be restricted on the bounded derived categories $D^b(Q-\\mathsf{mod})$, $D^b(A-\\mathsf{mod})$\nand $D^b(\\widetilde{O}-\\mathsf{mod})$. In particular, we have \\emph{two} semi--orthogonal decompositions\n$$\nD(A-\\mathsf{Mod}) = \\bigl\\langle \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\n\\bigl\\langle \\mathsf{Im}(\\sR\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)\\bigr\\rangle.\n$$\nThe same result is true when we pass to the bounded derived categories.\n\\end{theorem}\n\n\\noindent\n\\emph{Comment on the proof}. The study of various derived functors related with a pair $(B, \\epsilon)$, where $B$ is a ring and $\\epsilon \\in B$ an idempotent (in particular, the recollement diagram (\\ref{E:recollement})) are due to Cline, Parshall and Scott \\cite[Section 2]{CPS}. We also refer to \\cite[Section 4]{bdg} (and references therein) for an exposition focussed on non--commutative resolutions of singularities.\nThe weak crepancy of the categorical resolution $D(A-\\mathsf{Mod})$ of $\\Spec(O)$ follows from \\cite[Theorem 5.10]{bdg}. In particular, the constructed categorical resolution of singularities\nfits into the setting of non--commutative crepant resolutions initiated by van den Bergh in \\cite{vdb}. \\qed\n\n\\section{Survey on the derived stratification of an artinian quasi--hereditary ring}\nThe derived category $D^b(Q-\\mathsf{mod})$ of the quasi--hereditary ring $Q$ introduced in Theorem \\ref{T:GlobalDimensionKoenigOrder} can be further stratified in a usual way \\cite{CPS}, which we briefly describe now adapting the notation for further applications. All details can be found in \\cite{CPS}, \\cite[Appendix]{dr} and \\cite{bdg}.\n\n\n\\medskip\n\\noindent\n1.~Recall that we had started with a reduced excellent Noetherian ring $O$ of Krull dimension one, attaching to it\na certain order $A$. Then we have constructed a heredity chain $J_n \\subset J_{n-1} \\subset \\dots \\subset\nJ_1 \\subset A$ of two--sided ideals and posed $Q_k := A\/J_k$ for $1 \\le k \\le n$. In this notation,\n$Q = Q_n$ is an artinian quasi--hereditary ring we shall study in this section and $Q_1 = \\bar{O}$ is a semi-simple ring (supported on the singular locus of $\\Spec(O)$).\n\n\n\\medskip\n\\noindent\n2.~For any $1 \\le k \\le n$, let\n$\\bar{e}_k$ be the image of the standard idempotent $e_k \\in A$ in $Q_k = A\/J_k$. Then $Q_k\/(Q_k \\bar{e}_k Q_k) \\cong Q_{k-1}$ for all $2 \\le k \\le n$.\n\n\\smallskip\n\\noindent\nLet\n$P_k = Q_k \\bar{e}_k$ be the projective left $Q_k$--module and\n$P_k^\\vee = \\Hom_{Q_k}(P_k, Q_k) = e_k Q_k$ be the projective right $Q_k$--module, corresponding to\nthe idempotent $\\bar{e}_k$. Then we have: $\\mathop{\\mathrm{End}}\\nolimits_{Q_k}(P_k) \\cong \\bar{O}_k$.\nThe functor\n$$\n\\mathsf G} \\def\\sT{\\mathsf T_k = \\Hom_{Q_k}(P_k,\\,-\\,): Q_k-\\mathsf{mod} \\longrightarrow \\bar{O}_k-\\mathsf{mod}\n$$\nis a bilocalization functor: the functors $\\mathsf F} \\def\\sS{\\mathsf S_k = P_k \\otimes_{\\bar{O}_k} \\,-\\,$ and $\\mathsf H} \\def\\sU{\\mathsf U_k = \\Hom_{Q_k}(P_k^\\vee, \\,-\\,)$ are respectively the left and the right adjoints of $\\mathsf G} \\def\\sT{\\mathsf T_k$. Both $\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf H} \\def\\sU{\\mathsf U_k$ are fully faithful. Since\nthe ring $\\bar{O}_k$ is semi--simple, $\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf H} \\def\\sU{\\mathsf U_k$ are also exact. The kernel of $\\mathsf G} \\def\\sT{\\mathsf T_k$ is the category of $Q_{k-1}$--modules.\n\n\n\n\n\\medskip\n\\noindent\n3.~Most remarkably, for any $2 \\le k \\le n$ we have a recollement diagram\n\\begin{equation*}\n\\xymatrix{D^b(Q_{k-1}-\\mathsf{mod}) \\ar[rr]|-{\\,\\mathsf J} \\def\\sW{\\mathsf W_{k}\\,} && D^b(Q_k-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^-{\\,\\mathsf J} \\def\\sW{\\mathsf W_k^*\\,} \\ar@\/_2ex\/[ll]_-{\\,\\mathsf J} \\def\\sW{\\mathsf W_k^{!}\\,} \\ar[rr]|-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T_k\\,}\n && D^b(\\bar{O}_k-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k\\,} \\ar@\/_2ex\/[ll]_-{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k\\,}}\n\\end{equation*}\nThis claim in particular includes the following statements.\n\\begin{itemize}\n\\item The functor $\\mathsf J} \\def\\sW{\\mathsf W_k$ (induced by the ring homomorphism $Q_k \\longrightarrow Q_{k-1}$) is fully faithful.\nThe essential image of $\\mathsf J} \\def\\sW{\\mathsf W_k$ coincides with the kernel of $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T_k$ and\n$D^b(Q_k-\\mathsf{mod})\/\\mathsf{Im}(\\mathsf J} \\def\\sW{\\mathsf W_k) \\cong D^b(\\bar{O}_k-\\mathsf{mod})$.\n\\item The functors $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k$ are fully faithful.\n\\end{itemize}\n\n\n\\medskip\n\\noindent\n4.~For all $1 \\le k \\le n$ we have:\n\\begin{itemize}\n\\item $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_k(\\bar{O}_k) \\cong \\mathsf F} \\def\\sS{\\mathsf S_k(\\bar{O}_k) \\cong P_k$.\n\\item $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_k(\\bar{O}_k) \\cong \\mathsf H} \\def\\sU{\\mathsf U_k(\\bar{O}_k) = \\Hom_{\\bar{O}_k}(P_k^\\vee, \\bar{O}_k) := E_k$ is the injective left $Q_k$--module corresponding to the idempotent $\\bar{e}_k$.\n\\end{itemize}\nThe functor $\\mathsf I} \\def\\sV{\\mathsf V_k: D^b(Q_k-\\mathsf{mod}) \\longrightarrow D^b(Q-\\mathsf{mod})$ induced by the ring epimorphism\n$Q \\longrightarrow Q_k$ is fully faithful. In fact, it admits a factorization $\\mathsf I} \\def\\sV{\\mathsf V_k = \\mathsf J} \\def\\sW{\\mathsf W_n \\dots \\mathsf J} \\def\\sW{\\mathsf W_{k+1}$.\nThe $Q$--module $\\Delta_k := \\mathsf I} \\def\\sV{\\mathsf V_k(P_k)$ (respectively $\\nabla_k := \\mathsf I} \\def\\sV{\\mathsf V_k(E_k)$) is called $k$-th \\emph{standard} (respectively \\emph{costandard}) $Q$--module.\n\n\\medskip\n\\noindent\n5.~The standard and costandard modules have in particular the following properties:\n$$\n\\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\Delta_i, \\Delta_j) = 0 = \\mathop{\\mathrm{Ext}}\\nolimits_Q(\\nabla_j, \\nabla_i) \\; \\mbox{for all} \\; 1 \\le i < j \\le n \\; \\mbox{and} \\; p \\ge 0\n$$\nand\n$$\n\\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\Delta_k, \\Delta_k) = 0 = \\mathop{\\mathrm{Ext}}\\nolimits_Q^p(\\nabla_k, \\nabla_k) \\; \\mbox{for all} \\; 1 \\le k \\le n \\; \\mbox{and} \\; p \\ge 1.\n$$\nMoreover, $\\mathop{\\mathrm{End}}\\nolimits_Q(\\Delta_k) \\cong \\mathop{\\mathrm{End}}\\nolimits_Q(\\nabla_k) \\cong \\bar{O}_k$ is semi--simple. The derived category\n$D^b(Q-\\mathsf{mod})$ admits two canonical semi--orthogonal decompositions:\n$$\n\\langle D_1, \\dots, D_n\\rangle = D^b(Q-\\mathsf{mod}) = \\langle D'_n, \\dots, D'_1\\rangle,\n$$\nwhere $D_k$ (respectively $D'_k$) is the triangulated subcategory of $D^b(Q-\\mathsf{mod})$ generated by the object\n$\\Delta_k$ (respectively $\\nabla_k)$. Note that we have the following equivalences of categories:\n$\nD_k \\cong D^b(\\bar{O}_k-\\mathsf{mod}) \\cong D_k'.\n$\n\n\\medskip\n\\noindent\n6.~The stratification of $D^b(Q-\\mathsf{mod})$ by the derived categories $D^b(\\bar{O}_k-\\mathsf{mod})$ can be summarized by the following diagram of categories and functors:\n$$\n\\xymatrix{\nD^b(\\bar{O}_1-\\mathsf{mod}) \\ar[d]_= & D^b(\\bar{O}_2-\\mathsf{mod}) \\ar[d]_{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_2} & & D^b(\\bar{O}_n-\\mathsf{mod}) \\ar[d]^{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf F} \\def\\sS{\\mathsf S_n} \\\\\nD^b(Q_1-\\mathsf{mod}) \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_2} & D^b(Q_2-\\mathsf{mod}) \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_3} & \\dots \\ar[r]^-{\\mathsf J} \\def\\sW{\\mathsf W_{n}} & D^b(Q_n-\\mathsf{mod})\\\\\nD^b(\\bar{O}_1-\\mathsf{mod}) \\ar[u]^= & D^b(\\bar{O}_2-\\mathsf{mod}) \\ar[u]^{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_2} & & D^b(\\bar{O}_n-\\mathsf{mod}) \\ar[u]_{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf H} \\def\\sU{\\mathsf U_n}\n}\n$$\n\\section{Derived stratification and curve singularities}\nRecall that we also have the following recollement diagram\n\\begin{equation*}\n\\xymatrix{D^b(Q-\\mathsf{mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D^b(A-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D^b(\\widetilde{O}-\\mathsf{mod}) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation*}\nwhere $\\mathsf I} \\def\\sV{\\mathsf V$ is induced by the ring epimorphism $A \\rightarrow Q$. Abusing the notation, we shall write\n$\\Delta_k = \\mathsf I} \\def\\sV{\\mathsf V(\\Delta_k)$ for all $1 \\le k \\le n$. This implies the following result.\n\n\\begin{theorem}\nThe derived category $D^b(A-\\mathsf{mod})$ admits two semi--orthogonal decompositions\n$\n\\bigl\\langle\\mathsf{Im}(\\mathsf I} \\def\\sV{\\mathsf V), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigl\\rangle = D^b(A-\\mathsf{mod}) =\n\\bigl\\langle\\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Im}(\\mathsf I} \\def\\sV{\\mathsf V)\\bigl\\rangle.\n$\n\\end{theorem}\n\n\\noindent\nNext, recall that we have a bilocalization functor $$\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T: D^b(A-\\mathsf{mod}) \\longrightarrow D^b(O-\\mathsf{mod}).$$\n\\begin{lemma}\\label{L:calcullocale}\nFor any $1 \\le k \\le n$ we have: $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\bar{O}_k$. Moreover, $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(Q) \\cong\nO_1\/C_1 \\oplus \\dots \\oplus O_n\/C_n$, where $C_k := I_{n+1,k} = \\Hom_O\\bigl(O_{n+1}, O_k\\bigr)$.\n\\end{lemma}\n\n\\begin{proof} The first result follows from the following chain of isomorphisms:\n$$\n\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) = \\Hom_A(Af, \\Delta_k) \\cong f\\cdot \\Delta_k \\cong \\bar{O}_k.\n$$\nThe proof of the second statement is analogous.\n\\end{proof}\n\n\\section{K\\\"onig's resolution in the projective setting}\\label{S:ProjectiveSetting}\n\n\\smallskip\n\\noindent\nLet $X$ be a reduced projective curve over some base field $\\mathbbm{k}$. In this section we shall explain the construction\nof K\\\"onig's sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$, ``globalizing'' the arguments of Section \\ref{Sec:LocalStory}.\n\\begin{itemize}\n\\item Let $\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ be the normalization of $X$ and $Z$ be the singular locus of $X$\n(equipped with the reduced scheme structure).\n\\item In what follows, $\\mathcal{O} = \\mathcal{O}_X$ is the structure sheaf of $X$, $\\mathcal K} \\def\\kX{\\mathcal X$ is the sheaf of rational functions on $X$, $\\widetilde{\\mathcal{O}}: = \\nu_*(\\mathcal{O}_{\\widetilde{X}})$\nand $\\mathcal I} \\def\\kV{\\mathcal V$ is the ideal sheaf of the singular locus $Z$.\n\\item We consider the sheaf of rings $\\mathcal{O}^\\sharp := \\mbox{\\textit{End}}_X(\\mathcal I} \\def\\kV{\\mathcal V)$ on the curve $X$.\n\\end{itemize}\nThe next result follows from the corresponding affine version (Proposition \\ref{P:DetailsKoenigResol}).\n\\begin{proposition}\nWe have inclusions of sheaves $\\mathcal{O} \\subseteq \\mathcal{O}^\\sharp \\subseteq \\widetilde{O}$, and $\\mathcal{O} = \\mathcal{O}^\\sharp$ if and only if $X$ is smooth. Moreover, there are isomorphisms of $\\mathcal{O}$--modules\n$\n\\mathcal I} \\def\\kV{\\mathcal V \\cong \\mbox{\\textit{Hom}}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O})$ and $\\widetilde{\\mathcal{O}} \\cong \\mbox{\\textit{Hom}}_X(\\mathcal I} \\def\\kV{\\mathcal V, \\mathcal{O})$.\n\\end{proposition}\n\n\\noindent\nNow we define a sequence of sheaves of rings $\\mathcal{O} \\subset \\mathcal{O}_k \\subset \\widetilde{\\mathcal{O}}$ by the following recursive procedure.\n\\begin{itemize}\n\\item First we pose: $\\mathcal{O}_1 := \\mathcal{O}$.\n\\item Assume that the sheaf of rings $\\mathcal{O}_k$ has been constructed. Then it defines a projective curve $X_k$ together with a finite birational morphism $\\nu_k: X_k \\longrightarrow X$ (partial normalization of $X$) such that $\\mathcal{O}_k = \\bigl(\\nu_k\\bigr)_*(\\mathcal{O}_{X_k})$.\n\\item Let $Z_k$ be the singular locus of the curve $X_k$ (as usual, with respect to the reduced scheme structure).\nThen we write\n$$\n\\mathcal{O}_{k+1} := \\mathcal{O}_k^\\sharp \\cong (\\nu_k)_*\\bigl(\\mbox{\\textit{End}}_{X_k}(\\mathcal I} \\def\\kV{\\mathcal V_{Z_k})\\bigr).\n$$\n\\end{itemize}\nThen there exists a natural number $n$ (called the \\emph{level} of $X$) such that we have a finite chain of sheaves of rings\n$$\n\\mathcal{O} = \\mathcal{O}_1 \\subset \\mathcal{O}_2 \\subset \\dots \\subset \\mathcal{O}_n \\subset \\mathcal{O}_{n+1} = \\widetilde{\\mathcal{O}}.\n$$\nObviously, the level of $X$ is the maximum of the levels of local rings $\\widehat{\\mathcal{O}}_x$,\nwhere $x$ runs through the set of singular points of $X$.\n\n\\begin{definition}\nThe sheaf of rings\n$\\mathcal A} \\def\\kN{\\mathcal N := \\mbox{\\textit{End}}_X\\bigl(\\mathcal{O}_1 \\oplus \\dots \\oplus \\mathcal{O}_{n+1}\\bigr)$ is called the \\emph{K\\\"onig's sheaf of orders} on $X$.\n\\end{definition}\n\n\\noindent\nIn what follows, we study the ringed space $\\mathbb{X} = (X, \\mathcal A} \\def\\kN{\\mathcal N)$. We denote by $\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})$ (respectively\n$\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})$) the category of coherent (respectively quasi--coherent) sheaves of $\\mathcal A} \\def\\kN{\\mathcal N$--modules on the curve $X$.\n\n\\begin{theorem} The sheaf of orders $\\mathcal A} \\def\\kN{\\mathcal N$ admits the following description:\n\\begin{equation}\\label{E:KoenigOrderSheaf}\n\\mathcal A} \\def\\kN{\\mathcal N \\cong\n\\left(\n\\begin{array}{llccc}\n\\mathcal{O}_1 & \\mathcal{O}_2 & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_1 & \\mathcal{O}_2 & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_{n,1} & \\mathcal I} \\def\\kV{\\mathcal V_{n, 2} & \\dots & \\mathcal{O}_n & \\mathcal{O}_{n+1} \\\\\n\\mathcal I} \\def\\kV{\\mathcal V_{n+1,1} & \\mathcal I} \\def\\kV{\\mathcal V_{n+1, 2} & \\dots & \\mathcal I} \\def\\kV{\\mathcal V_n & \\mathcal{O}_{n+1} \\\\\n\\end{array}\n\\right) \\subset \\mbox{\\textit{Mat}}_{n+1, n+1}(\\mathcal K} \\def\\kX{\\mathcal X),\n\\end{equation}\nwhere $\\mathcal I} \\def\\kV{\\mathcal V_{i, j} := \\mbox{\\textit{Hom}}_{X}(\\mathcal{O}_i, \\mathcal{O}_j)$ for all $1 \\le j < i\\le n+1$ and $\\mathcal I} \\def\\kV{\\mathcal V_k = \\mathcal I} \\def\\kV{\\mathcal V_{k+1, k}$ for $1 \\le k \\le n$. Moreover, $\\mathcal A} \\def\\kN{\\mathcal N \\otimes_\\mathcal{O} \\mathcal K} \\def\\kX{\\mathcal X \\cong \\mbox{\\textit{Mat}}_{n+1, n+1}(\\mathcal K} \\def\\kX{\\mathcal X)$. Next, we have:\n$$\n\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) \\le 2n,\n$$\nwhere $n$ is the level of $X$.\n\\end{theorem}\n\n\\begin{proof}\nThe result follows from the corresponding local statements in Proposition \\ref{P:KoenigOrderMatrDescr} and\nTheorem \\ref{T:GlobalDimensionKoenigOrder} and the fact that\n$$\n\\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\mathrm{gl.dim}\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) = \\max\\bigl\\{\\mathrm{gl.dim}(\\widehat{\\mathcal A} \\def\\kN{\\mathcal N}_x) \\, | \\, x \\in X_{\\mathrm{cl}}\\bigr\\},\n$$\nsee for instance \\cite[Corollary 5.5]{bdg}.\n\\end{proof}\n\n\nFor any $1 \\le i \\le n+1$, let $e_i \\in \\Gamma(X, \\mathcal A} \\def\\kN{\\mathcal N)$ be the $i$-th standard idempotent with respect to the matrix\npresentation (\\ref{E:KoenigOrderSheaf}). As in the affine case, we use the following notation.\n\\begin{itemize}\n\\item We write $e = e_{n+1}$ and $f = e_1$. Let $\\kP := \\mathcal A} \\def\\kN{\\mathcal N e$ and $\\mathcal{T} := \\mathcal A} \\def\\kN{\\mathcal N f$ be the corresponding locally projective left $\\mathcal A} \\def\\kN{\\mathcal N$--modules. Then we have the following isomorphisms of sheaves of $\\mathcal{O}$--algebras:\n\\begin{equation}\\label{E:keysheafisom}\n\\mathcal{O} \\cong \\mbox{\\textit{End}}_{\\mathbb{X}}(\\mathcal{T}) := \\mbox{\\textit{End}}_{\\mathcal A} \\def\\kN{\\mathcal N}(\\mathcal{T}) \\ \\mbox{ and } \\ \\widetilde{\\mathcal{O}} \\cong \\mbox{\\textit{End}}_{\\mathbb{X}}(\\kP) :=\n \\mbox{\\textit{End}}_{\\mathcal A} \\def\\kN{\\mathcal N}(\\kP).\n\\end{equation}\nWe shall also use the notation\n$$\n\\kP^\\vee := \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kP, \\mathcal A} \\def\\kN{\\mathcal N) \\cong e\\mathcal A} \\def\\kN{\\mathcal N \\ \\mbox{ and } \\ \\mathcal{T}^\\vee := \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal{T}, \\mathcal A} \\def\\kN{\\mathcal N) \\cong f\\mathcal A} \\def\\kN{\\mathcal N.\n$$\n\\item For any $1 \\le k \\le n$ we set\n$$\\varepsilon_k := \\sum\\limits_{i = k+1}^{n+1} e_i \\in \\Gamma(X, \\mathcal A} \\def\\kN{\\mathcal N).\n$$\nThen $\\mathcal J} \\def\\kW{\\mathcal W_k:= \\mathcal A} \\def\\kN{\\mathcal N \\varepsilon_k\\mathcal A} \\def\\kN{\\mathcal N$ denotes the corresponding sheaf of two--sided ideals in $\\mathcal A} \\def\\kN{\\mathcal N$.\n\\item The sheaves of $\\mathcal{O}$--algebras $\\kQ_k := \\mathcal A} \\def\\kN{\\mathcal N\/\\mathcal J} \\def\\kW{\\mathcal W_k$ are supported on the finite set $Z$ for all $1 \\le k \\le n$. In what follows, we shall identify them with the corresponding finite dimensional\n $\\mathbbm{k}$--algebras of global sections\n $\n Q_k := \\Gamma(X, \\kQ_k),\n $\nwhich have been shown to be quasi--hereditary, see Theorem \\ref{T:GlobalDimensionKoenigOrder}. As before, we shall write $\\mathcal J} \\def\\kW{\\mathcal W = \\mathcal J} \\def\\kW{\\mathcal W_n$ and $Q = Q_n$.\n\\item In a similar way, the torsion sheaf $\\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k$ will be identified with the corresponding ring of global sections\n$\n\\bar{O}_k:= \\Gamma\\bigl(X, \\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k),\n$\nwhich is a semi--simple finite dimensional $\\mathbbm{k}$--algebra, isomorphic to the ring of functions of the singular locus $Z_k$ of the partial normalization $X_k$ of our original curve $X$.\n\\end{itemize}\n\n\\begin{proposition}\\label{P:recollementSheaves} Consider the following diagram of categories and functors\n\\begin{equation}\n\\xymatrix{\\mathop{\\mathsf{Coh}}\\nolimits(X) \\ar@\/^2ex\/[rr]^{\\,\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\mathsf H} \\def\\sU{\\mathsf U\\,} && \\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X}) \\ar[ll]|{\\,\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && \\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X}) \\ar@\/^2ex\/[ll]^{\\,\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nwhere $\\mathsf F} \\def\\sS{\\mathsf S = \\mathcal{T} \\otimes_O \\,-\\,$, $\\mathsf H} \\def\\sU{\\mathsf U = \\mbox{\\textit{Hom}}_X(\\mathcal{T}^\\vee, \\,-\\,)$, $\\mathsf G} \\def\\sT{\\mathsf T = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal{T}, \\,-\\,)$ and similarly,\n$\\tilde\\mathsf F} \\def\\sS{\\mathsf S = \\kP \\otimes_{\\widetilde{O}} \\,-\\,$, $\\tilde\\mathsf H} \\def\\sU{\\mathsf U = \\mbox{\\textit{Hom}}_{\\widetilde{X}}(P^\\vee, \\,-\\,)$,\n$\\tilde\\mathsf G} \\def\\sT{\\mathsf T = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kP, \\,-\\,)$. Here we identify (using the functor $\\nu_*$) the category $\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})$ with the category of coherent $\\widetilde{O}$--modules on the curve $X$. Then the following results are true.\n\\begin{itemize}\n\\item The pairs of functors $(\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T), (\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf H} \\def\\sU{\\mathsf U)$ and $(\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T), (\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs. The functors $\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf H} \\def\\sU{\\mathsf U, \\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item The functors $\\mathsf G} \\def\\sT{\\mathsf T$ and $\\tilde\\mathsf G} \\def\\sT{\\mathsf T$ are bilocalization functors. Moreover, $\\mathsf{Ker}(\\mathsf F} \\def\\sS{\\mathsf S) \\cong\nQ-\\mathsf{mod}$.\n\\item The pairs of functors $(\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ and $(\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf H} \\def\\sU{\\mathsf U)$ between\n$\\mathop{\\mathsf{Coh}}\\nolimits(X)$ and $\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})$ form adjoint pairs, too. Moreover, these functors admit the following ``purely commutative'' descriptions:\n$$\n\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\nu_*,\\ \\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf H} \\def\\sU{\\mathsf U \\simeq \\nu^!,\\ \\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\widetilde\\mathcal{O}} \\nu^*(\\,-\\,)\n\\ \\mbox{\\emph{ and }} \\ \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U \\simeq \\nu_*(\\mathcal C} \\def\\kP{\\mathcal P^\\vee \\otimes_{\\widetilde{\\mathcal{O}}} \\,-\\,),\n$$\nwhere $\\mathcal C} \\def\\kP{\\mathcal P := \\mbox{\\textit{Hom}}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O}) = \\mathcal I} \\def\\kV{\\mathcal V_{n+1,1}$ is the conductor ideal sheaf.\n\\end{itemize}\nThe same results are true when we replace each category of coherent sheaves by the corresponding category of quasi--coherent sheaves.\n\\end{proposition}\n\n\\begin{proof}\nThe proofs of the first two parts follow from standard local computations. Let $\\mathcal F} \\def\\kS{\\mathcal S$ be a coherent $\\mathcal{O}$--module and\n$\\mathcal G} \\def\\kT{\\mathcal T$ a coherent $\\widetilde{O}$--module (identified with the corresponding coherent sheaf on $\\widetilde{X}$).\nThen we have:\n$$\n\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf F} \\def\\sS{\\mathsf S(\\mathcal F} \\def\\kS{\\mathcal S) = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal A} \\def\\kN{\\mathcal N e, \\mathcal A} \\def\\kN{\\mathcal N f \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S) \\cong \\bigl(e \\mathcal A} \\def\\kN{\\mathcal N f\\bigr)\\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S \\cong\n\\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S \\cong \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\mathcal{O}} \\bigl(\\widetilde\\mathcal{O} \\otimes_{\\mathcal{O}} \\mathcal F} \\def\\kS{\\mathcal S\\bigr)\n$$\nand\n$$\n\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S(\\mathcal G} \\def\\kT{\\mathcal T) = \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\mathcal A} \\def\\kN{\\mathcal N f, \\mathcal A} \\def\\kN{\\mathcal N e \\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T) \\cong\n\\bigl(f \\mathcal A} \\def\\kN{\\mathcal N e\\bigr)\\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T \\cong \\widetilde\\mathcal{O} \\otimes_{\\widetilde\\mathcal{O}} \\mathcal G} \\def\\kT{\\mathcal T \\cong \\mathcal G} \\def\\kT{\\mathcal T.\n$$\nThis proves the isomorphisms of functors $\\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathcal C} \\def\\kP{\\mathcal P \\otimes_{\\widetilde\\mathcal{O}} \\nu^*(\\,-\\,)$ and\n$\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\nu_*$. Since $\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ and $\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\mathsf H} \\def\\sU{\\mathsf U$ are right adjoints of $\\tilde\\mathsf G} \\def\\sT{\\mathsf T \\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S$\n respectively, the remaining isomorphisms are true as well.\n\\end{proof}\n\n\\noindent\nThe next statement summarizes the main properties of the K\\\"onig's resolution in the projective framework.\n\\begin{theorem}\\label{T:main} We have a diagram of categories and functors\n\\begin{equation}\\label{E:descinglobale2}\n\\xymatrix{D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr) \\ar@\/^2ex\/[rr]^{\\,\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S\\,} \\ar@\/_2ex\/[rr]_{\\,\\sR\\mathsf H} \\def\\sU{\\mathsf U\\,} && D(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X}) \\ar[ll]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\,}\n \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,} && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nsatisfying the following properties.\n\\begin{itemize}\n\\item The pairs of functors $(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T, \\sR\\mathsf H} \\def\\sU{\\mathsf U)$, $(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)$ and\n$(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T, \\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ form adjoint pairs.\n\\item The functors $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$, $\\sR\\mathsf H} \\def\\sU{\\mathsf U$, $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S$ and $\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U$ are fully faithful.\n\\item Both derived categories $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr)$ and $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr)$ are Verdier localizations of\n$D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)$:\n\\begin{itemize}\n \\item $\nD\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(X)\\bigr) \\cong D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T)$.\n\\item\n$D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\cong D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)\/\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T).\n$\n\\end{itemize}\n\\item Moreover,\n$\n\\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T) \\cong D(Q-\\mathsf{Mod}).\n$\n\\item\nThe derived category $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr)$ is a categorical resolution of singularities of $X$ in the sense of Kuznetsov \\cite[Definition 3.2]{ku}.\n \\item If $X$ is Gorenstein, then the restrictions of $\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S$ and $\\sR\\mathsf H} \\def\\sU{\\mathsf U$ on $\\mathop{\\mathsf{Perf}}\\nolimits(X)$ are isomorphic. Hence, the constructed categorical resolution is even \\emph{weakly crepant} in the sense of\n \\cite[Definition 3.4]{ku}.\n\\end{itemize}\nWe have a \\emph{recollement diagram}\n\\begin{equation}\\label{E:recollement2}\n\\xymatrix{D(Q-\\mathsf{Mod}) \\ar[rr]|{\\,\\mathsf I} \\def\\sV{\\mathsf V\\,} && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf I} \\def\\sV{\\mathsf V^*\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf I} \\def\\sV{\\mathsf V^{!}\\,} \\ar[rr]|{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T\\,}\n && D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr) \\ar@\/^2ex\/[ll]^{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf H} \\def\\sU{\\mathsf U\\,} \\ar@\/_2ex\/[ll]_{\\,\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf F} \\def\\sS{\\mathsf S\\,}}\n\\end{equation}\nand all functors can be restricted on the bounded derived categories $D^b(Q-\\mathsf{mod})$, $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$\nand $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. In particular, we have \\emph{two} semi--orthogonal decompositions\n$$\nD\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\n\\bigl\\langle \\mathsf{Im}(\\sR\\mathsf H} \\def\\sU{\\mathsf U), \\mathsf{Ker}(\\mathsf D} \\def\\sQ{\\mathsf Q\\tilde\\mathsf G} \\def\\sT{\\mathsf T)\\bigr\\rangle.\n$$\nThe same result is true when we pass to the bounded derived categories.\n\\end{theorem}\n\n\\begin{corollary}\nFor each $1 \\le k \\le n$ let $D_k$ (respectively $D'_k$) be the full subcategory of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ generated\nby the $k$--th standard module $\\Delta_k$ (respectively, the $k$--th costandard module $\\nabla_k$). Then we have\nequivalences of categories\n$\nD_k \\cong D^b(\\bar{O}_k-\\mathsf{mod}) \\cong D'_k\n$\nand semi--orthogonal decompositions\n\\begin{equation}\\label{E:semiorthdec}\n\\bigl\\langle D_1, \\dots, D_n, \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle =\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U), D'_n, \\dots, D'_1\\bigr\\rangle.\n\\end{equation}\nBoth triangulated categories $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$ and $\\mathsf{Im}(\\sR\\tilde\\mathsf H} \\def\\sU{\\mathsf U)$ are equivalent\nto the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. Note that they are \\emph{different} viewed as subcategories\nof $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$.\n\\end{corollary}\n\n\\begin{remark}\nAs in the setting at the beginning of this section, let $X$ be a reduced excellent curve, $\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ its normalization and\n$\\mathcal C} \\def\\kP{\\mathcal P := \\mathit{Hom}_X(\\widetilde{\\mathcal{O}}, \\mathcal{O})$ the conductor ideal. Then $\\mathcal C} \\def\\kP{\\mathcal P$ is also a sheaf of ideals in $\\widetilde\\mathcal{O}$, hence the scheme $S = V(\\mathcal C} \\def\\kP{\\mathcal P) \\stackrel{\\eta}\\hookrightarrow X$ is a \\emph{non--rational locus of $X$ with respect to $\\nu$} in the sense of Kuznetsov and Lunts \\cite[Definition 6.1]{kl}. Starting with the Cartesian diagram\n$$\n\\xymatrix{\n\\widetilde{S} \\ar[r]^{\\tilde\\eta} \\ar[d]_{\\tilde{\\nu}} & \\widetilde{X} \\ar[d]^{\\nu} \\\\\nS \\ar[r]^\\eta & X\n}\n$$\none can construct a partial categorical resolution of singularities of $X$ obtained by the ``naive gluing'' of the\nderived categories $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(S)\\bigr)$, see \\cite[Section 6.1]{kl}.\nIt would be interesting to compare the obtained triangulated category with the derived category $D\\bigl(\\mathop{\\mathsf{Qcoh}}\\nolimits(\\widehat\\mathbb{X})\\bigr)$ of the non--commutative curve\n$\n\\widehat\\mathbb{X} = \\left(\nX,\n\\left(\n\\begin{array}{cc}\n\\mathcal{O} & \\widetilde\\mathcal{O} \\\\\n\\mathcal C} \\def\\kP{\\mathcal P & \\widetilde\\mathcal{O}\n\\end{array}\n\\right)\n\\right),\n$\nsee also \\cite[Section 8]{bd}. Next, \\cite[Theorem 6.8]{kl} provides a recipe to construct a categorical resolution of singularities of $X$, which however, involves some non--canonical choices.\nIt is an interesting question to compare these categorical resolutions with K\\\"onig's resolution $\\mathbb{X}$ constructed in our article. Another important problem is to give an ``intrinsic description'' of the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$, i.e.~to provide a list of properties describing it uniquely up to a triangle equivalence.\nWe follow here the analogy with non--commutative crepant resolutions, see \\cite[Conjecture 5.1]{bo} and \\cite[Conjecture 4.6]{vdb}. All such resolutions are known to be derived equivalent in certain cases, see for example \\cite[Theorem 6.6.3]{vdb}. Recall that K\\\"onig's resolution $\\mathbb{X}$ is weakly crepant in the case the curve $X$ is Gorenstein.\n\\end{remark}\n\n\\section{Purely commutative applications}\nResults of the previous sections allow to deduce a number of interesting ``purely commutative'' statements.\nLet $X$ be a reduced projective curve over some base field $\\mathbbm{k}$ and\n$\\widetilde{X} \\stackrel{\\nu}\\longrightarrow X$ be its normalization. According to Orlov \\cite{or}, the Rouquier dimension of the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ is equal to one. In fact, Orlov constructs an explicit vector bundle\n$\\widetilde\\mathcal F} \\def\\kS{\\mathcal S$ on $\\widetilde{X}$ such that $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr) = \\langle \\widetilde\\mathcal F} \\def\\kS{\\mathcal S\\rangle_2$ (here we follow the notation of Rouquier's seminal article \\cite{rou}).\n\n\\begin{theorem} Let $\\mathcal F} \\def\\kS{\\mathcal S := \\nu_*(\\widetilde{\\mathcal F} \\def\\kS{\\mathcal S})$ be the direct image of the Orlov's generator of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$. Then the following results are true.\n\\begin{itemize}\n\\item Let $Z$ be the singular locus of $X$ (with respect to the reduced scheme structure) and $\\mathcal{O}_Z$ be the corresponding structure sheaf. Then\n\\begin{equation}\\label{E:1stEstderDim}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\mathcal{O}_Z\\bigr\\rangle_{n+2},\n\\end{equation}\nwhere $n$ is the level of $X$.\n\\item Let $\\kS = \\mathcal{O}_1\/\\mathcal C} \\def\\kP{\\mathcal P_1 \\oplus \\dots \\oplus \\mathcal{O}_n\/\\mathcal C} \\def\\kP{\\mathcal P_n$, where $\\mathcal C} \\def\\kP{\\mathcal P_k := \\mbox{\\textit{Hom}}_X(\\mathcal{O}_{n+1}, \\mathcal{O}_k)$ is the conductor ideal sheaf of the $k$--th partial normalization of $X$ for $1 \\le k \\le n$. Then we have:\n\\begin{equation}\\label{E:2ndEstderDim}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\bigl\\langle \\mathcal F} \\def\\kS{\\mathcal S \\oplus \\kS\\bigr\\rangle_{d+3},\n\\end{equation}\nwhere $d$ is the global dimension of the quasi--hereditary algebra $Q$ associated with $X$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nAccording to Theorem \\ref{T:main}, the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ admits a semi--orthogonal decomposition\n$$\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle D^b(Q-\\mathsf{mod}), \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle.\n$$\nMoreover, the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ is the Verdier localization of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ via the functor $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T$. This implies that whenever we have an object $\\kX$ of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ with\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\langle \\kX\\rangle_m$ then $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr) = \\langle \\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\kX)\\rangle_m$.\nAccording to Proposition \\ref{P:recollementSheaves} we have:\n$$\n(\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T\\cdot\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) \\cong \\mathsf G} \\def\\sT{\\mathsf T\\tilde\\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) \\cong \\nu_*(\\widetilde\\mathcal F} \\def\\kS{\\mathcal S) =: \\mathcal F} \\def\\kS{\\mathcal S.\n$$\nNext, Lemma \\ref{L:calcullocale} implies that for all $1 \\le k \\le n$ we have:\n$$\n\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_k) \\cong \\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k.\n$$\nLet $\\nu_k: X_k \\longrightarrow X$ be the $k$--th partial normalization of $X$ and $Z_k = \\left\\{y_1, \\dots, y_p\\right\\}$ be the singular locus of $X_k$ (as usual, equipped with the reduced scheme structure). Then\n$$\\mathcal{O}_k\/\\mathcal I} \\def\\kV{\\mathcal V_k \\cong (\\nu_{k})_*\\bigl(\\mathcal{O}_{X_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{Z_k}\\bigr) \\cong\n(\\nu_{k})_*\\bigl(\\mathcal{O}_{Z_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{y_1} \\oplus \\dots \\oplus \\mathcal{O}_{Z_k}\/\\mathcal I} \\def\\kV{\\mathcal V_{y_p}\\bigr).\n$$\nObserve that if $y \\in Z_k$ and $x = \\nu_k(y)$ then $(\\nu_{k})_*(\\mathcal{O}_{X_k}\/\\mathcal I} \\def\\kV{\\mathcal V_y) \\cong (\\mathcal{O}\/\\mathcal I} \\def\\kV{\\mathcal V_x)^{\\oplus l}$,\nwhere $l = \\mathsf{deg}\\bigl[\\mathbbm{k}_y: \\mathbbm{k}_x\\bigr]$. Therefore,\n$$\n\\mathsf{add}\\bigl(\\mathsf G} \\def\\sT{\\mathsf T(\\Delta_1) \\oplus \\dots \\oplus \\mathsf G} \\def\\sT{\\mathsf T(\\Delta_n)\\bigr) = \\mathsf{add}(\\mathcal{O}_Z)\n$$\nand (\\ref{E:1stEstderDim}) is just a consequence of \\cite[Lemma 3.5]{rou}. The equality (\\ref{E:2ndEstderDim}) follows in a similar way from Lemma \\ref{L:calcullocale} and \\cite[Proposition 7.4]{rou}.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{C:boundRouquierDim}\nLet $X$ be a reduced quasi--projective curve over some base field $\\mathbbm{k}$. Then there is the following upper bound\non the Rouquier dimension of $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$:\n\\begin{equation}\\label{E:boundRouquierDim}\n\\mathsf{dim}\\Bigl(D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)\\Bigr) \\le \\mathsf{min}(n+1, d +2),\n\\end{equation}\nwhere $n$ is the level of $X$ and $d$ is the global dimension of the quasi--hereditary algebra $Q$ associated with\n$X$.\n\\end{corollary}\n\n\\begin{remark}\nIn the case $X$ is rational with only simple nodes or cusps as singularities, the bound (\\ref{E:boundRouquierDim}) has been obtained in \\cite[Theorem 10]{bd}. Note that $n = 1$ and $d=0$ is this case. We do not know whether\nthe estimates (\\ref{E:1stEstderDim}) and (\\ref{E:2ndEstderDim}) are strict.\n\\end{remark}\n\n\n\\noindent\nThe following result gives an affirmative answer on a question posed to the first--named author by Valery Lunts.\n\\begin{theorem}\\label{T:tilting}\nFor any reduced rational projective curve $X$ over some base field $\\mathbbm{k}$ there exists a finite dimensional\nquasi--hereditary $\\mathbbm{k}$--algebra $\\Lambda$ having the following properties.\n\\begin{itemize}\n\\item There exists a fully faithful exact functor\n$\n\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf I} \\def\\sV{\\mathsf V}\\longrightarrow D^b(\\Lambda-\\mathsf{mod})\n$ and a Verdier localization $D^b(\\Lambda-\\mathsf{mod})\n \\stackrel{\\sP}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr),\n$\nsuch that $\\sP \\mathsf I} \\def\\sV{\\mathsf V \\simeq \\mathsf{Id}_{\\mathop{\\mathsf{Perf}}\\nolimits(X)}$.\n\\item The triangulated category $D^b(\\Lambda-\\mathsf{mod})$ is a recollement of the triangulated categories $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$ and $D^b(Q-\\mathsf{mod})$, where $Q$ is the quasi--hereditary algebra associated with $X$.\n\\item We have: $\\mathrm{gl.dim}(\\Lambda) \\le d +2$, where $d = \\mathrm{gl.dim}(Q)$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof} According to Theorem \\ref{T:main}, there exists a fully faithful exact functor\n$\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ and a Verdier localization $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) \\stackrel{\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)$ such that $\\mathsf D} \\def\\sQ{\\mathsf Q\\mathsf G} \\def\\sT{\\mathsf T \\cdot \\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S \\simeq \\mathsf{Id}_{\\mathop{\\mathsf{Perf}}\\nolimits(X)}$.\nIt suffices to show that the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ has a tilting object. Recall that we have constructed a semi--orthogonal decomposition\n\\begin{equation}\\label{sorth}\nD^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr) = \\bigl\\langle \\langle\\kQ\\rangle,\\,\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)\\bigr\\rangle,\n\\end{equation}\nwhere $\\langle \\kQ\\rangle \\cong D^b(Q-\\mathsf{mod})$ is the triangulated subcategory generated by\n$\\kQ = \\mathcal A} \\def\\kN{\\mathcal N\/\\mathcal J} \\def\\kW{\\mathcal W$ and $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S) \\cong D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\widetilde{X})\\bigr)$.\n\nSince the curve $X$ is rational and projective, we have: $\\widetilde{X} = \\widetilde{X}_1 \\cup \\dots \\cup\n\\widetilde{X}_t$, where $\\widetilde{X}_k \\cong \\mathbbm{P}^1_{\\mathbbm{k}}$ for all $1 \\le k \\le t$.\nThen $$\\widetilde\\mathcal B} \\def\\kO{\\mathcal O := \\bigl(\\mathcal{O}_{\\widetilde{X}_1}(-1) \\oplus \\mathcal{O}_{\\widetilde{X}_1}\\bigr) \\oplus \\dots \\oplus\n\\bigl(\\mathcal{O}_{\\widetilde{X}_t}(-1) \\oplus \\mathcal{O}_{\\widetilde{X}_t}\\bigr)$$ is a tilting bundle on $\\widetilde{X}$ and the algebra $E:= \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{\\widetilde{X}}(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O)\\bigr)^{\\mathrm{op}}$ is isomorphic to the direct product of $t$ copies of the path algebra of the Kronecker quiver $\\kron{\\scriptscriptstyle\\bullet}\\bu$.\nThen $\\mathcal B} \\def\\kO{\\mathcal O:= \\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O) \\cong \\mathsf L} \\def\\sY{\\mathsf Y\\mathsf F} \\def\\sS{\\mathsf S(\\widetilde\\mathcal B} \\def\\kO{\\mathcal O)$ is a tilting object in\nthe triangulated category $\\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$.\n\nThe semi--orthogonal decomposition \\eqref{sorth} implies that $\\Hom_{D^b(\\mathbb{X})}(\\kY, \\kX) = 0$ for any $\\kX \\in \\langle \\kQ\\rangle$ and $\\kY \\in \\mathsf{Im}(\\mathsf L} \\def\\sY{\\mathsf Y\\tilde\\mathsf F} \\def\\sS{\\mathsf S)$.\n\nIt is clear that $\\mathop{\\mathrm{Ext}}\\nolimits_{\\mathbb{X}}^p(\\kQ, \\kQ) = 0$ for $p \\ge 1$ and $Q \\cong \\mathop{\\mathrm{End}}\\nolimits_{\\mathbb{X}}(\\kQ)^{\\mathrm{op}}$.\nSince the ideal $\\mathcal J} \\def\\kW{\\mathcal W$ is locally projective\nas a left $\\mathcal A} \\def\\kN{\\mathcal N$--module, we have: $\\mbox{\\textit{Ext}}_{\\mathbb{X}}^p(\\kQ,\\,-\\,) = 0$ for $p \\ge 2$. Moreover, since\n$\\mathcal B} \\def\\kO{\\mathcal O$ is locally projective and $\\kQ$ is torsion, we also have vanishing $\\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) = 0$.\nSince $\\Hom_{\\mathbb{X}}(\\kX_1, \\kX_2) \\cong \\Gamma\\bigl(X, \\mbox{\\textit{Hom}}_{\\mathbb{X}}(\\kX_1, \\kX_2)\\bigr)$ the local--to--global spectral sequence implies that\n$\n\\mathop{\\mathrm{Ext}}\\nolimits^p_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) = 0\n$\nunless $p = 1$ and \\begin{equation}\\label{E:local-to-global}\n\\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O) \\cong \\Gamma\\bigl(X, \\mbox{\\textit{Ext}}^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)\\bigr).\n\\end{equation}\nSumming up, the complex $\\mathcal H} \\def\\kU{\\mathcal U := \\kQ[-1] \\oplus \\mathcal B} \\def\\kO{\\mathcal O$ is \\emph{tilting} in the derived category $D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$. A result of Keller \\cite{kel} implies that the derived categories\n$D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(\\mathbb{X})\\bigr)$ and $D^b(\\Lambda-\\mathsf{mod})$ are equivalent, where $\\Lambda := \\bigl(\\mathop{\\mathrm{End}}\\nolimits_{D^b(\\mathbb{X})}(\\mathcal H} \\def\\kU{\\mathcal U)\\bigr)^{\\mathrm{op}}$. Finally, observe that\n$\n\\Lambda\n\\cong\n\\left(\n\\begin{array}{cc}\nQ & W \\\\\n0 & E\n\\end{array}\n\\right),\n$\nwhere $W := \\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)$ viewed as a ($Q$--$E$)--bimodule. Since the algebra $Q$ is quasi--hereditary and $E$ is directed, the algebra $\\Lambda$ is quasi--hereditary as well. According to \\cite[Corollary 4']{pr}, we have:\n$\\mathrm{gl.dim}(\\Lambda) \\le \\mathrm{gl.dim}(Q) + 2$.\n\\end{proof}\n\n\\begin{remark} In a recent work \\cite[Theorem 4.10]{Wei}, the following inversion of Theorem \\ref{T:tilting} was obtained. Assume $X$ is a projective curve over an algebraically closed field $\\mathbbm{k}$ and $\\Lambda$ a finite dimensional $\\mathbbm{k}$--algebra of finite global dimension such that there exist functors\n$$\\mathop{\\mathsf{Perf}}\\nolimits(X) \\stackrel{\\mathsf I} \\def\\sV{\\mathsf V}\\longrightarrow D^b(\\Lambda-\\mathsf{mod}) \\stackrel{\\sP}\\longrightarrow D^b\\bigl(\\mathop{\\mathsf{Coh}}\\nolimits(X)\\bigr)\n$$\n with $\\mathsf I} \\def\\sV{\\mathsf V$ fully faithful, $\\sP$ essentially surjective and $\\sP \\mathsf I} \\def\\sV{\\mathsf V \\simeq \\mathsf{Id}$. Then $X$ is rational. This result can be shown by examining the Grothendieck groups of the involved triangulated categories.\n\\end{remark}\n\n\\begin{remark}\nIn the case $X$ has only simple nodes or cusps as singularities, Theorem \\ref{T:tilting} has been obtained\nin \\cite[Theorem 9]{bd}. See also \\cite[Definition 3]{bd} for an explicit description of the algebra $\\Lambda$ is this case.\n\\end{remark}\n\n\n\n\\begin{remark}\\label{R:bimodule}\nNow we outline how the $Q$--$E$--bimodule $W = \\mathop{\\mathrm{Ext}}\\nolimits^1_{\\mathbb{X}}(\\kQ, \\mathcal B} \\def\\kO{\\mathcal O)$ from the proof of Theorem \\ref{T:tilting} can be explicitly determined. The isomorphism\n(\\ref{E:local-to-global}) implies that $W$ can be computed locally and we may assume that $X = \\Spec(O)$ and\n$O$ is a complete local ring. We follow the notation of Section \\ref{Sec:LocalStory}. For any $1 \\le k \\le n$ the\nleft $A$--module $R_k:= Q e_k$ has projective resolution\n$$\n0 \\longrightarrow Je_k \\longrightarrow A e_k \\longrightarrow R_k \\longrightarrow 0.\n$$\nThis yields the following isomorphisms of $\\widetilde{O}$--modules:\n\\begin{equation}\\label{E:localcomputation}\nW_k = \\mathop{\\mathrm{Ext}}\\nolimits^1_A(R_k, P) \\cong \\dfrac{\\Hom_A(J e_k, A e)}{\\Hom_A(A e_k, A e)} \\cong \\dfrac{\\Hom_O(C_k, \\widetilde{O})}{\\Hom_O(O_k, \\widetilde{O})} \\cong \\dfrac{C_k^\\vee}{\\widetilde{O}},\n\\end{equation}\nwhere $P = A e$ and $C_k = \\Hom_{O}(\\widetilde{O}, O_k) = \\Hom_{O_k}(\\widetilde{O}, O_k)$ is the conductor ideal of the\npartial normalization\n$O_k \\subset \\widetilde{O}$. Since $\\widetilde{O}$ is regular, we have a (non--canonical) isomorphism of\n$\\widetilde{O}$--modules $\\dfrac{C_k^\\vee}{\\widetilde{O}} \\cong \\dfrac{\\widetilde{O}}{C_k}$.\nSince $\\widetilde{O} \\cong \\mathop{\\mathrm{End}}\\nolimits_A(P)$, this leads to a description of the right $E$--action on $W$. To say more about the left action of $Q$ on $W$, we need an explicit description of the algebra $Q$.\n\\end{remark}\n\n\\section{Quasi--hereditary algebras associated with simple curve singularities}\nLet $\\mathbbm{k}$ be an algebraically closed field of characteristic zero.\nIn this section we compute the algebra $\\kQ$ for the\n \\emph{simple plane curve singularities} in the sense of Arnold \\cite{avg}. These singularities are in\n one--to--one correspondence with the simply laced Dynkin graphs.\n\n\\begin{proposition}\nThe algebra $Q$ associated with the simple singularity $O = \\mathbbm{k}\\llbracket u, v\\rrbracket\/(u^2 - v^{m+1})$\nof type $A_{m}$ is the path algebra of the following quiver\n\\begin{align*}\n &\\xymatrix{ 1 \\ar@\/_\/[r]_{\\beta} \\def\\eps{\\varepsilon_1}& 2\\ar@\/_\/[l]_{\\alpha} \\def\\de{\\delta_1} \\ar@\/_\/[r]_{\\beta} \\def\\eps{\\varepsilon_2} &\n \t3 \\ar@\/_\/[l]_{\\alpha} \\def\\de{\\delta_2} \\ar@{.}[r] & (n-1) & n \\ar@\/_\/[l]+<4ex,.9ex>*{}_-{\\alpha} \\def\\de{\\delta_{n-1}}\n \t\\ar@{<-}@\/^\/[l]+<4ex,-.9ex>*{}^-{\\beta} \\def\\eps{\\varepsilon_{n-1}}\t}\n \\end{align*}\n where $n=\\big[\\frac{m+1}2\\big]$ with the relations\n \\begin{align*}\n \t&\\ \\beta} \\def\\eps{\\varepsilon_k\\alpha} \\def\\de{\\delta_k=\\alpha} \\def\\de{\\delta_{k+1}\\beta} \\def\\eps{\\varepsilon_{k+1}\\ \\text{ if }\\ 1\\le k*{}_{\\alpha} \\def\\de{\\delta_{n-1}}\n \t\\ar@{<-}@\/^\/[l]+<4ex,-.9ex>*{}^{\\beta} \\def\\eps{\\varepsilon_{n-1}} }\n \\]\n with the relations\n \\begin{align*} \t&\\ \\beta} \\def\\eps{\\varepsilon_k\\alpha} \\def\\de{\\delta_k=\\alpha} \\def\\de{\\delta_{k+1}\\beta} \\def\\eps{\\varepsilon_{k+1}\\ \\text{ if }\\ 1\\le k0$. As one can see that if the thermodynamic quantities are evaluated in the time interval between $t=0$ and any time $t$, always the change in entropy, entropy production, and entropy extraction rates become greater than zero revealing such systems are inherently irreversible. Moreover, we show that when a distinct temperature difference is not retained between the hot and cold baths, in absence of load, ${\\dot e}_{p}={\\dot h}_{d}=0$ showing that the system is reversible. In the presence of load and when the viscous friction decreases in time, we show that the entropy $S(t)$ monotonously increases with time and saturates to a constant value as $t$ further steps up. The entropy production rate ${\\dot e}_{p}$ decreases in time and at steady state (in the presence of load), ${\\dot e}_{p}={\\dot h}_{d}>0$ which agrees with the results shown in the works \\cite{muuu177}. On the contrary, when the viscous friction increases in time, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. \n\n\n\n\nMost of the previous studies have also focused on exploring the thermodynamic feature of systems such as Brownian heat engines by assuming temperature invariance viscous friction. In reality, the viscous friction of a medium tends to decrease as the temperature of the medium increases. This is because as the intensity of the background temperature increases, the force of interaction between neighboring molecules decreases. In this paper, considering a spatially varying viscosity, the non-equilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. The potential is also coupled with a spatially varying temperature. In this case, the direction of the particle velocity is dictated by the magnitude of the external load of $f$. As one can note that the steady state velocity of the engine is positive when $f$ is smaller and the engine acts as a heat engine. In this regime ${\\dot e}_{p}={\\dot h}_{d}>0$. When $f$ steps up, the velocity of the particle steps down, and at stall force, we find that ${\\dot e}_{p}={\\dot h}_{d}=0$ revealing that the system is reversible at this particular choice of parameter. For large force, the current is negative and the engine acts as a refrigerator. In this region ${\\dot e}_{p}={\\dot h}_{d}>0$. In the absence of load, ${\\dot e}_{p}={\\dot h}_{d}>0$ as long as a distinct temperature difference is retained between the hot and cold baths. \n\n\n\nAt this point, we want also to stress that most of the previous works have focused on calculating\nthe thermodynamic features of different model systems\nby considering additive noise. On the contrary, most realistic\n systems such as neuron system can be also described by Langevin\nequations with multiplicative noise were in this case, the noise\namplitude varies spatially \\cite{mar12}. In this paper, we\nstudy how thermodynamic features of such systems behave.\n\n\nThe rest of the paper is organized as follows: in Section II, we derive the expression for various thermodynamic relations to a Brownian particle walking in overdamped and underdamped media. In Section III, the role of viscous friction is studied by considering viscous friction that varies spatially and temporally. In section IV, we explore the model system in the presence of multiplicative noise. Section V deals with a summary and conclusion.\n\n\n\n\n\n\n\n\n\n\\section{Free energy, entropy production and entropy extraction rates } \nRecently, the dependence for entropy production, free energy, and entropy extraction rates on the system parameters was explored \\cite{muuu177} by considering a Brownian particle that walks in a medium where its viscous friction is insensitive to time or position. However, in most realistic systems, the viscous friction of the medium varies spatially or temporally. To address this issue, let us consider a Brownian particle that moves in an underdamped medium along the potential $U(x)=U_{s}(x)+fx$ where $U_{s}(x)$ and $f$ are the periodic potential and the external force, respectively. Next, the relation for the entropy production and extraction rates will be derived considering a spatially varying viscous friction.\n\n\n\\subsection{Underdamped case}\n{\\it Derivation for entropy production and entropy extraction rates.\\textemdash} Let us consider a single Brownian particle that is arranged to undergo a random walk in an underdamped medium. Here the single-particle and\nits trajectory are considered to be the system as contrasted with the underdamped medium which provides friction and acts as a heat bath. The dynamics of the system is governed by \n\\begin{equation}\nm{dv\\over dt} = -\\gamma (x,t){dx\\over dt}+ {d U(x) \\over dx} + \\sqrt{2k_{B}\\gamma (x,t) T(x)}\\xi(t).\n\\end{equation}\nFor simplicity, Boltzmann constant $k_{B}$ is assumed to be unity. The random noise $\\xi(t)$ is assumed to be Gaussian white noise satisfying the relations \n$\\left\\langle \\xi(t) \\right\\rangle =0$ and $\\left\\langle \\xi(t) \\xi(t') \\right\\rangle=\\delta(t-t')$. The viscous friction $\\gamma (x,t)$ and $T(x)$ are assumed to vary spatially along the medium.\n\nFor underdamped case, the Fokker-Plank equation is given by\n\\begin{eqnarray}\n{\\partial P\\over \\partial t}&=&-{\\partial (vP) \\over \\partial x}-{1 \\over m}{\\partial(U'(x)P) \\over \\partial v}+ \\nonumber \\\\\n&&{\\gamma(x,t) \\over m}{\\partial (vP) \\over \\partial v}+{\\gamma(x,t) T(x) \\over m^2}{\\partial^2 P \\over \\partial v^2}\n\\end{eqnarray}\nwhere $P(x,v,t)$ is the probability of finding the particle at particular position $x$, velocity $v$ and time $t$.\n\n\n \n \nFor convenience, Eq. (2) can be rearranged as \n\\begin{eqnarray}\n{\\partial P\\over \\partial t}&=& -(k+{\\partial J' \\over \\partial v} )\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nk=v{\\partial P \\over \\partial x} ={\\partial J \\over \\partial x}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nJ'= -{\\gamma (x,t)\\over m}vP+{1\\over m}(U'P) -{\\gamma (x,t) T(x)\\over m^2}{\\partial P\\over \\partial v}.\n\\end{eqnarray}\nFrom Eqs. (4) and (5), one gets \n\\begin{eqnarray}\n{\\partial P \\over \\partial v}= -{m^{2}J'\\over \\gamma (x,t)T(x)}+{mU'P\\over \\gamma (x,t) T(x)}-{m vP\\over T(x)}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\partial P \\over \\partial x}= {k\\over v}.\n\\end{eqnarray}\nNext we derive the expressions for the entropy production by considering two cases. \n\n\n\n{\\it Case1.\\textemdash} Here we want to stress that the approach by Lee. $et$. $al$. \\cite{mar1} and Ge. $et$. $al$. \\cite{mar2} indicate that under time reversal operation, the total entropy production and extraction rate can be obtained. Particularly, the analysis by Ge. $et$. $al$. \\cite{mar2} shows that the entropy production rate is given by\n\\begin{eqnarray}\n{\\dot e}_{p}&=&-\\int {1\\over T(x) \\gamma(x,t)}\\left[F-{T(x) \\gamma(x,t)\\nabla_{v}\\ln(P)\\over m}\\right]^2P dx dv \\nonumber \\\\\n&=&-\\int {1\\over T(x) \\gamma(x,t)}\\left[F-{T(x) \\gamma(x,t){\\partial P\\over \\partial v}\\over m P}\\right]^2P dx dv\n\\end{eqnarray}\nwhere $F=-T(x) \\gamma(x,t)+U'$.\nSubstituting Eq. (6) into Eq. (8), one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv \n\\end{eqnarray}\nOn the other hand, the entropy extraction rate can be found via the method devloped by Ge. $et$. $al$. \\cite{mar2} as\n\\begin{eqnarray}\n{\\dot h}_{d}&=&-\\int {1\\over T(x) }\\left[T(x) \\gamma(x,t)+{T(x) \\gamma(x,t)\\nabla_{v}\\ln(P)\\over m}\\right]P v dxdv \\nonumber \\\\\n&=&-\\int {1\\over T(x) }\\left[T(x) \\gamma(x,t)+{T(x) \\gamma(x,t){\\partial P\\over \\partial v}\\over m P}\\right]P v dxdv.\n\\end{eqnarray}\nSubstituting Eq. (6) into Eq. (10) leads to\n \\begin{eqnarray}\n{\\dot h}_{d} &=& \\int {(U'J-vmJ') \\over T(x) } dx dv.\n\\end{eqnarray} Here ${\\dot e}_{p}={d e_{p}\\over dt} $ and ${\\dot h}_{d}={d h_{d}\\over dt}$ denote the entropy production and extraction rates.\nThe above expressions (for the entropy production and extraction rates) can be also derived at the ensemble level via the approach stated in the work \\cite{mu7}. \n\n{\\it Case2.\\textemdash} One can also rederive the expressions for the entropy production and extraction rates at the ensemble level by first analyzing the entropy of the system at the trajectory level as\n\\begin{eqnarray}\ns(t)=-ln P(x,v,t)\n\\end{eqnarray}\nwhere $x(t)$ denotes the stochastic trajectory. \nThe rate of entropy\nchange at trajectory level is then given by\n\\begin{eqnarray}\n{\\dot s}(t)=-{\\partial_{t} P(x,v,t) \\over P(x,v,t)}-{\\partial_{x} P(x,v,t) \\over P(x,v,t)}{\\dot x}-{\\partial_{v} P(x,v,t) \\over P(x,v,t)}{\\dot v}.\n\\end{eqnarray}\n\nSubstituting Eqs . (6) and (7) into Eq. (13), the entropy production and dissipation\nrates at trajectory level are given as\n\\begin{eqnarray}\n{\\dot e}_{p}^{*}=-{\\partial_{t} P(x,v,t) \\over P(x,v,t)}+{m^{2}J'\\over \\gamma (x,t)T(x)P(x,v,t)}{\\dot v}-{k \\over P(x,v,t)}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d}^{*}={mU'\\over \\gamma (x,t) T(x)}{\\dot v}-{m\\gamma(x,t) v\\over T(x)}{\\dot v}.\n\\end{eqnarray}\nBecause averaging overall trajectories yields $\\left\\langle {\\dot v}|x\\right\\rangle={J'\\over P(x,t,v)}$ and $\\int \\partial_{t} P(x,v,t ) =0$, after some algebra one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv -\\int P v dv\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=& \\int {m(U'-v\\gamma (x,t))J' \\over T(x)\\gamma (x,t) } dx dv,\n\\end{eqnarray} respectively.\nWhen a periodic boundary condtion is imposed, Eqs. (9) and (16) as well as Eqs. (11) and (17) converge to\n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {m^{2}J'^{2} \\over P T(x) \\gamma(x,t)} dx dv \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=& -\\int {vmJ' \\over T(x) } dx dv.\n\\end{eqnarray} \n\nThe heat dissipation rate ${\\dot H}_{d}$ can be calculated \\cite{am4,am5} as \n\\begin{eqnarray}\n{\\dot H}_{d} \n&=&-\\left\\langle \\left(-\\gamma(x,t){\\dot x}+ \\sqrt{2k_{B}\\gamma(x,t) T(x)}\\right).{\\dot x}\\right\\rangle \\nonumber \\\\\n&=&-\\left\\langle m{vdv\\over dt} +v U'(x) \\right\\rangle.\n\\end{eqnarray}\nOur previous analysis also suggests \\cite{mu17,muu17,muuu17} that the entropy extraction rate ${\\dot h}_{d}$ can be expressed as \n\\begin{eqnarray}\n{\\dot h}_{d} \n&=&-\\int \\left({m{vdv\\over dt} +v U'(x) \\over T(x)} \\right)P dxdv.\n\\end{eqnarray}\n One should note that Eq. (21) is exact and does not depend on any boundary condition. \nSince ${d S(t)\\over dt}$ and ${\\dot h}_{d} $ are computable, the entropy production rate can be readily obtained as\n\\begin{eqnarray}\n{\\dot e}_{p}&=&{d S(t)\\over dt}+{\\dot h}_{d}. \n\\end{eqnarray}\nIn the long time limit, ${d S(t)\\over dt}=0$ which implies ${\\dot e}_{p}={\\dot h}_{d}>0$ at steady state and ${\\dot e}_{p}={\\dot h}_{d}=0$ at stationary state.\n\n\n\n\nOnce the expressions for \n${\\dot S}(t)$, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ are computed as a function of time $t$, the analytic expressions for the change in entropy production, heat dissipation and total entropy can be found analytically via \n$\n\\Delta h_d(t)= \\int_{0}^{t}{\\dot h}_{d}(t)dt $,\n$\\Delta e_{p}(t)= \\int_{0}^{t} {\\dot e}_{p}(t) dt $ and \n$\\Delta S(t) =\\int_{0}^{t} {\\dot S}(t)dt $\nwhere $\\Delta S(t)=\\Delta e_p(t)-\\Delta h_d(t)$. \n\n \n{\\it Derivation for free energy.\\textemdash} Our next objective is to write the expression for the free energy in terms of ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$ where ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$ are the terms that are associated with ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$. As discussed before, the heat dissipation rate is either given by Eq. (20) (for any cases) or if a periodic boundary condition is imposed, ${\\dot H}_{d}(t)$ is given by \n\\begin{eqnarray}\n{\\dot H}_{d} &=&-\\int m(v)J' dx dv\n\\end{eqnarray}\nwhich is notably different from Eq. (19), due to the term $T(x)$. \nThe term associated to ${\\dot e}_{p}$ is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& -\\int {m^{2} J'^{2} \\over P \\gamma(x,t)} dx dv.\n\\end{eqnarray}\nThe entropy balance equation \n\\begin{eqnarray}\n{d S^T(t)\\over dt}&=&{\\dot E}_{p}-{\\dot H}_{d}\n\\end{eqnarray}\nis associated to Eqs. (11) or (17) except the term $T(x)$.\nOnce again, employing the expressions for \n${\\dot S}^T(t)$, ${\\dot E}_{p}(t)$ and ${\\dot H}_{d}(t)$, one can get \n$\\Delta H_d(t)= \\int_{0}^{t}{\\dot H}_{d}(t)dt$, \n$\\Delta E_{p}(t)= \\int_{0}^{t} {\\dot E}_{p}(t) dt $ and \n$\\Delta S(t)^T =\\int_{0}^{t} {\\dot S}(t)^{T}dt$\nwhere $\\Delta S(t)^T=\\Delta E_p(t)-\\Delta H_d(t)$. \n\n\nOn the other hand, the expression for the internal energy has a form \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int ({\\dot K}+ v U'_{s}(x))P(x,v,t)dvdx\n\\end{eqnarray}\nwhere ${\\dot K}=m{vdv\\over dt}$ and $U'_{s}$ denote the rate of kinetic and potential energy, respectively. The network work done by the system \n\\begin{eqnarray}\n{\\dot W}&=& \\int v f P(x,v,t)dvdx\n\\end{eqnarray}\nexplicitly depends on the velocity $V$ and the load $f$. \nIn terms of ${\\dot H}$ and ${\\dot W}$, the rate of the internal energy is given by \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}\n\\end{eqnarray}\nand after some algebra, the first law of thermodynamics can be written as \n\\begin{eqnarray}\n\\Delta E_{in}= -\\int_{0}^{t}( {\\dot H}_{d}(t)+{\\dot W}) dt.\n\\end {eqnarray}\n\nRearranging some terms, one gets \nthe rate of free energy as ${\\dot F}={\\dot E}-T{\\dot S}$ for isothermal case and ${\\dot F}={\\dot E}-{\\dot S}^T$ for nonisothermal case where ${\\dot S}^T={\\dot E}_{p}-{\\dot H}_{d}$. The rate of free energy dissipation \n\\begin{eqnarray}\n{\\dot F}&=&{\\dot E}_{in}- {\\dot S}^T \\nonumber \\\\\n&=&{\\dot E}_{in}-{\\dot E}_{p}+{\\dot H}_{d}\n\\end{eqnarray}\ncan be expressed as a definite integral as\n\\begin{eqnarray}\n\\Delta F(t)&=&-\\int_{0}^{t} \\left( {\\dot W}+ {\\dot E}_{p}(t) \\right)dt.\n\\end{eqnarray} \nFor isothermal case, at quasistatic limit where the velocity approaches zero $ v=0$, ${\\dot E}_{p}(t) =0$ and ${\\dot H}_{d}(t) =0$ and far from quasistatic limit \n$E_{p}={\\dot H}_{d}>0$ which is expected as the particle operates irreversibly. \n\n\\subsection{Overdamped case}\n\n{\\it Derivation for free energy, entropy production and entropy extraction rates.\\textemdash} For overdamped case,\n as discussed by Sancho. $et$ .$at$ \\cite{am3} and Jayannavar $et$ .$at$ \\cite{am33}, \nEq. (1) converges to \n\\begin{eqnarray}\n\\gamma(x,t){dx\\over dt}&=&{-\\partial U(x)\\over \\partial x} -{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)}+\\nonumber \\\\\n&&\\sqrt{2k_{B}\\gamma(x,t) T(x)}\\xi(t)\n\\end{eqnarray}\nwhich corresponds to the Stratonovich interpretation \\cite{am1,am2}.\n The corresponding Fokker Planck equation is given by \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({U'(x)\\over \\gamma(x,t)}+{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)^2}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T(x)\\over \\gamma(x,t)}{\\partial P(x,t)\\over \\partial x}\\right)\n\\end{eqnarray}\nwhich can be rewritten as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&-{\\partial J \\over \\partial x}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nJ&=&-\\left({U'(x)P(x,t)\\over \\gamma(x,t)}+{P(x,t)(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)^2}\\right)-\\nonumber \\\\\n&&\\left({T(x)\\over \\gamma(x,t)}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\n\nThe rate of entropy\nchange at trajectory level is given by\n\\begin{eqnarray}\n{\\dot s}(t)=-{\\partial_{t} P(x,t) \\over P(x,t)}-{\\partial_{x} P(x,t) \\over P(x,t)}{\\dot x}.\n\\end{eqnarray}\nOn the other hand, from Eq. (35) one gets \n\\begin{eqnarray}\n{\\partial P \\over \\partial x}= -{\\gamma (x,t) J\\over T(x)}-{U'(x)P(x,t)\\over T(x)}-{P(x,t)(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}.\n\\end{eqnarray}\n\n\nSubstituting Eq. (37) into Eq. (36), the entropy production and dissipation\nrates at trajectory level are given as\n\\begin{eqnarray}\n{\\dot e}_{p}^{*}=-{\\partial_{t} P(x,t) \\over P(x,t)}+{\\gamma (x,t)J\\over T(x)P(x,t)}{\\dot v}\n\\end{eqnarray}\nand \n \\begin{eqnarray}\n{\\dot h}_{d}^{*}={U'(x)\\over T(x)}+{(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}.\n\\end{eqnarray}\nBecause averaging overall trajectories yields $\\left\\langle {\\dot x}|x\\right\\rangle={J\\over P(x,t)}$ and $\\int \\partial_{t} P(x,t ) =0$, after some algebra one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {\\gamma(x,t) J^{2} \\over P T(x)} dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({JU'(x)\\over T(x)}+{J(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)T(x)}\\right) dx \n\\end{eqnarray} respectively. One should note that Eqs. (18) and (19) (underdamped case) as well as Eqs. (40) and (41) (overdamped case) approach \n\\begin{eqnarray}\n{\\dot h}_{d} ={\\dot e}_{p}\n&=&\\int \\left({J U'(x) \\over T(x)} \\right)dx\n\\end{eqnarray}\nat steady state ($v {dv\\over dt}=0$), \nas long as a \nperiodic boundary condition is imposed.\nFrom Eqs. (40) and (41), it is evident that when detailed balance conditions are satisfied the velocity of equivalently the current $J=0$ and as a result $ {\\dot e}_{p}=0$ and $ {\\dot h}_{d}=0$. Far from equilibrium, $J>0$, and in this case when the viscous friction decreases either spatially or temporally, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ approach to a constant value. When the system is driven out of equilibrium and when viscous friction increases spatially and temporally, $ {\\dot e}_{p}$ and $ {\\dot h}_{d}$ \nmonotonously step up.\nMoreover, from Eq. (41), the heat dissipation rate is derived as \n\\begin{eqnarray}\n{\\dot H}_{d} &=& \\int \\left(JU'(x)+{J(\\gamma'(x,t)T(x)+\\gamma(x,t)T'(x))\\over 2 \\gamma(x,t)} \\right) dx .\n\\end{eqnarray}\nOn the other hand, the term ${\\dot E}_{p}$ is related to ${\\dot e}_{p}$ and it is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& \\int {\\gamma(x,t) J^{2} \\over P } dx.\n\\end{eqnarray}\nThe new entropy balance equation has a simple form \n\\begin{eqnarray}\n{d S^T(t)\\over dt}&=&{\\dot E}_{p}-{\\dot H}_{d}.\n\\end{eqnarray}\nFurthermore, the internal energy \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int J U'_{s}(x) dx\n\\end{eqnarray}\nhas functional dependence on the current $J$ and the potential profile $U_s$. \nThe total work done is then given by \n\\begin{eqnarray}\n{\\dot W}&=& \\int \\left(Jf+{J(\\gamma'(x,t)T(x))\\over 2 \\gamma(x,t)} +{JT'(x)\\over 2 }\\right)dx.\n\\end{eqnarray}\nThe first law of thermodynamics can be written as \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}.\n\\end{eqnarray}\n The change in the internal energy reduces to \n$\n\\Delta E_{in}= -\\int_{0}^{t}( {\\dot H}_{d}(t)+{\\dot W}) dt.\n$\nOnce again \nthe rate of free energy dissipation can be written as \n${\\dot F}={\\dot E}_{in}- {\\dot S}^T \n={\\dot E}_{in}-{\\dot E}_{p}+{\\dot H}_{d}$. The change in the free energy is then given by \n$\\Delta F(t)=-\\int_{0}^{t} \\left( {\\dot W}+ {\\dot E}_{p}(t) \\right)dt.\n$\n\n \n \n\\section{ Time dependent viscous friction }\n\nSome viscous fluids show a change in viscosity when\ntime changes. This is because as the fluid shear stress\nchanges in time, so does the viscosity. Often, the dynamics\nof systems with self-organized criticality also can be explored\nby considering time dependent diffusion (viscous\nfriction) and drift terms \\cite{marr1,marr2, marr3}. Some studies have also\nfocused on calculating the mean first passage time by considering a time dependant diffusion term \\cite{muuuu17}. To explore the non-equilibrium thermodynamic features of a Brownian particle that hops in a medium where its\nviscosity depends on time, we consider a Brownian particle\nthat walks on a periodic isothermal medium (in the\npresence or absence of load) where its viscosity is given\nby \n\\begin{eqnarray}\n\\gamma(t)&=&{1\\over g(1+t^z)}.\n\\end{eqnarray}\nIn this case, the corresponding Fokker Planck equation in overdamped medium is given as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({f\\over \\gamma(t)}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T\\over \\gamma(t)}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nImposing a periodic boundary condition $P(0,t)=P(L,t)$ and let us choose a Fourier cosine series \n\\begin{eqnarray}\nP(x,t)=\\sum_{n=0}^\\infty b_{n}(t)cos\\left({n\\pi \\over L_0}(x+{f\\over \\gamma(t)})\\right)\n\\end{eqnarray}\nas a possible solution. \nAfter some algebra, we get \n the probability distribution as \n\\begin{eqnarray}\nP(x,t)&=&\\sum_{n=0}^\\infty \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right)\\right]\\zeta.\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\zeta&=&e^{-{(n\\pi )^2T\\left(gt+{gt^{z+1}\\over(z+1)}\\right) \\over L^2}}.\n\\end{eqnarray}\nHere $f$ is the external load and $T$ is the temperature of the medium.\nThe current is then given by \n\\begin{eqnarray}\nJ(x,t)&=&-\\left[{f P(x,t)\\over \\gamma(t)} + {T\\over \\gamma(t)}{\\partial P(x,t) \\over \\partial x}\\right].\n\\end{eqnarray}\nThe current $J(x,t)>0$, only when $f \\ne 0$ since $\\gamma(x)$ is not the necessary parameter to keep the system out of equilibrium. \n As stated before, $\n{\\dot e}_{p}= {\\dot h}_{d}+{d S(t)\\over dt}$ where \n$\n{d S(t)\\over dt}=-\\int {J\\over P(x,t)}{\\partial \\over \\partial x} P(x,t) dx$. \nAfter some algebra, we write \n\\begin{eqnarray}\n{d S(t)\\over dt}&=&-\\int J {\\sum_{n=0}^\\infty {{n\\pi\\over 2} \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right) \\right] \\zeta \\over \\sum_{n=0}^\\infty \\cos\\left[{n\\pi \\over L_0}\\left(x+f(gt+{gt^{z+1}\\over(z+1)})\\right)\\right]\\zeta}} dx.\n\\end{eqnarray}\n The entropy production and entropy extraction rates are given by the relations \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int {J^{2} \\over P(x,t) T g(1+t^z) } dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({Jf\\over T}\\right) dx.\n\\end{eqnarray} \nSubstituting Eqs. (52) and (54) into Eqs. (56) and (57), one can explore how ${\\dot e}_{p}$ and ${\\dot h}_{d}$ depend on time. For $z \\ge 0$, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ decrease and approach to a constant value at steady state. When $z<0$, ${\\dot e}_{p}$ and ${\\dot h}_{d}$ step up continuously. \nHereafter, for simplicity, the parameter $g$ is considered to be a constant. \n Furthermore, the heat dissipation rate is given by \n\\begin{eqnarray}\n{\\dot H}_{d} &=& \\int \\left(Jf \\right) dx \n\\end{eqnarray}\nwhile the term ${\\dot E}_{p}$ is given by \n\\begin{eqnarray}\n{\\dot E}_{p}&=& \\int {J^{2} \\over P(x,t)g(1+t^z) } dx.\n\\end{eqnarray}\nOn the other hand, the internal energy has a form \n\\begin{eqnarray}\n{\\dot E}_{in} = \\int J U'_{s}(x) dx.\n\\end{eqnarray}\nThe total work done is then given by \n\\begin{eqnarray}\n{\\dot W}&=& \\int \\left(Jf \\right)dx.\n\\end{eqnarray}\nThe first law of thermodynamics can be written as \n\\begin{eqnarray}\n{\\dot E}_{in} = -{\\dot H}_{d}(t)-{\\dot W}\n\\end{eqnarray}\nHereafter, whenever we plot any figures, we use the following dimensionless \nload ${\\bar f}=fL_{0}\/T_{c}$, ${\\bar U}=U\/T_{c}$, temperature ${\\bar \\tau}=T(x) \/T_{c}$ where $T_c$ is the reference temperature.\nWe also introduced dimensionless parameter ${\\bar x}=x\/L_{0}$, ${\\bar v}=vm\/ \\gamma L_{0}$ and ${\\bar t}=t \\gamma \/m$. Hereafter the bar will be dropped.\nFrom now on all the figures will be plotted in terms of the dimensionless parameters.\n\nThe expression for the rate of entropy production as well as entropy extraction rate can be readily calculated via Eqs. (56) and (57). In the absence of load $f=0$, ${\\dot e}_{p} = {\\dot h}_{d}=0$. This is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. When a distinct temperature difference is retained between the hot and cold baths, in absence of load, ${\\dot e}_{p}={\\dot h}_{d}=0$ showing that such a system is inherently reversible. In the presence of load and when the viscous friction increases in time (see Fig. 1), ${\\dot e}_{p}$ and ${\\dot h}_{d}$ decrease in time and as time further\nsteps up ${\\dot e}_{p}$ and ${\\dot h}_{d}$ monotonously increase in time as shown in Fig. 1. Figure 1 is plotted by fixing $\\tau=1.0$, $f=1.0$ and $z=-0.5$. On the contrary, in the presence of load and when the viscous friction decreases in time (see Fig. 2), ${\\dot e}_{p}$ and ${\\dot h}_{d}$ monotonously decrease with time and saturate to a constant value as $t$ further steps up. Fig. 2 is plotted by fixing $\\tau=1.0$, $f=1.0$ and $z=1.0$. \n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n9.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n10.eps}\n}\n\\caption{ (Color online) (a) The entropy extraction rate ${\\dot h}_{d}(t)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) The plot of entropy production rate ${\\dot e}_{p}(t)$ as a function of $t$.\n${\\dot e}_{p}(t)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The two figures exhibit that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ decrease in time and as time further steps up, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase. In the both figures, the parameters are fixed as $f=1.0$, $\\tau=1.0$ and $z=-0.5$. \n} \n\\label{fig:sub}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n7.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n8.eps}\n}\n\\caption{ (Color online) (a) ${\\dot h}_{d}(t)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) ${\\dot e}_{p}(t)$ as a function of $t$.\n${\\dot e}_{p}(t)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The figure exhibits that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ decrease in time and as time further steps up, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ approach a constant value. In the both figures, the parameters are fixed as $f=1.0$, $\\tau=1.0$ and $z=1.0$. \n} \n\\label{fig:sub}\n\\end{figure}\n\n\n\n\n\\section{ Spatially varying viscous friction }\n\n\nMost of the previous studies have focused on exploring the thermodynamic feature of systems such as Brownian heat engines by assuming temperature invariance viscous friction. However, various studies have indicated that the viscous friction of a medium tends to decrease as the temperature of the medium increases \\cite{mar15, mar16, aa15}. Particularly in the liquid medium, the viscosity decreases as the intensity of the background temperature steps up. This is because when the temperature of the medium increases, more molecules start vibrating, and as a result their speed increases. This speedy motion of the molecules creates a reduction in interaction time between neighboring molecules. At the macroscopic level, there will be a reduction in the intermolecular force, and hence reduced viscosity of the fluid. Consequently, when the temperature of the viscous medium decreases, the viscous friction in the medium decreases. \n\n\n\n\n\n\n\nIn this paper, considering a spatially varying viscosity, the non-equilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. The potential is also coupled with a spatially varying temperature. \n\n{\\it The model .\\textemdash} Let us consider a Brownian particle that walks in a piecewise linear potential with an external load $U(x)=U_{s}(x) + fx$, where the ratchet potential\n$U_{s}(x)$ is given by\n\\begin{equation}\n U_{s}(x)=\\left\\{\n \\begin{array}{ll}\n 2U_{0}\\left({x\\over L_{0}}\\right),& \\text{if}~~~ 0 \\le x \\le {L_{0}\\over 2};\\\\\n 2U_{0}\\left(1-{x\\over L_{0}}\\right),& \\text{if} ~~~{L_{0}\\over 2} \\le x \\le L_{0}.\n \\end{array}\n \\right.\n\\end{equation}\nHere $U_{0}$ and $L_{0}$ denote the barrier height and the width of the ratchet potential,\nrespectively. $f$ designates the external force. \nThe potential exhibits its maximum value\n$U_{0}$ at $x={L_{0}\\over 2}$ and its minima at $x=0$ and $x={L_{0}}$.\nThe \nspatially varying temperature is arranged as \n\\begin{equation}\nT(x)=\\left\\{\n\\begin{array}{ll}\nT_{h},& \\text{if} ~~~0 \\le x \\le {L_{0}\\over 2};\\\\\nT_{c},& \\text{if} ~~~ {L_{0}\\over 2} \\le x \\le L_{0}\n\\end{array}\n\\right.\n\\end{equation}\n as shown in Fig. 1. The potential $U_{s}(x)$ and \n$T(x)$ are assumed to be periodic with a period $L_0$, $U_{s}(x+L_{0})=U_{s}(x)$ and $T(x+L_{0})=T(x)$.\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=8cm]{am14.eps}\n}\n\\caption{(Color online) Schematic diagram for a particle that walks in a piecewise linear potential\nin the absence of an external load. }\n\\label{fig:ratchet}\n\\end{figure}\n\nFor a fluid such as blood, it is reasonable to\nassume that when the temperature of the blood sample increases by $1.0$ degree Celsius, its viscosity steps down by $2.0$ degree Celsius \\cite{aa15}. Thus let us consider viscous friction that varies as \n \\begin{equation}\n \\gamma(x)=\\gamma'-C(T(x)-T_c)\n\\end{equation}\nwhere $C$ is a constant which is less than one. In the next two sections, we will explore the model system in the overdamped and underdamped limits.\n\n\\subsection{ Underdamped case in the absence of ratchet potential }\nIn this section, we consider an important model system where a colloidal particle that undergoes a biased random walk in a spatially varying thermal arrangement in the presence of external load $f$ with no potential. \nSolving Eq. (3) at steady state, the general expression for the probability distribution is obtained as\n\\begin{equation} \nP(x,v)=\\frac{e^{-\\frac{m (f-(\\gamma+c T_c) v+c v T[x])^2}{2 T[x] (\\gamma+c T_c-c\nT[x])^2}} \\sqrt{\\frac{m}{T[x]}}}{\\sqrt{2 \\pi }}.\n \\end{equation}\n The average velocity is found to be \n\\begin{equation} \nv={f\\over \\gamma(x) }.\n\\end{equation}\n In the absence of force, the velocity of the particle approaches zero. \n\n \n\nVia Eqs. (9) and (11), the entropy production and extraction rates are calculated as \n \\begin{eqnarray} \n{\\dot h}_{d}(t)&=&{\\dot e}_{p}(t) \\nonumber \\\\\n&=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma T_c}+\\frac{1}{(\\gamma+c (T_c-T_h)) T_h}\\right)\n \\end{eqnarray}\nOne can see that in the limit where the load approaches the stall force, ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. \tExploiting Eq. (62), it is evident that ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase when the temperature difference between the hot and cold baths decreases. This is feasible since when the temperature difference between the heat baths steps up, the magnitude of the viscose friction decreases. \n\t\n\tThe rate of heat dissipation \nis calculated employing Eq. (20) and it converges to\n \\begin{eqnarray} \n{\\dot H}_{d}(t)&=&{\\dot E}_{p}(t)\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\n\tIn the limit where the load approach zero,\n\t${\\dot H}_{d}(t)={\\dot E}_{p}(t)=0$ showing that at quasistatic limit the system is reversible.\n\tOn the other hand, the rate of work done is given by \n\\begin{eqnarray} \n{\\dot W}(t)&=&{\\dot E}_{p}(t) \\nonumber \\\\ &=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\n\tFor isothermal case $T_{h}=T_{c}$, one gets $v=f\/\\gamma$, ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=f^2L_{0}\/\\gamma T_{c}$ and ${\\dot H}_{d}(t)={\\dot E}_{p}(t)=f^2L_{0}\/\\gamma$.\n\t\n\n\t\n\n \n\n\n\n\\subsection{ Overdamped case }\n\nIn the presence of ratchet potential, in the overdamped limit, the closed-form expression for the steady-state current can be given as\n\\begin{equation}\nJ= -{\\varsigma_{1}\\over \\varsigma_{2}\\varsigma_{3}+(\\varsigma_{4}+\\varsigma_{5})\\varsigma_{1}}\n\\end{equation}\nwhere the expressions for $\\varsigma_{1}$, $\\varsigma_{2}$, $\\varsigma_{3}$, and $\\varsigma_{4}$\nare given by\n\\begin{eqnarray}\n\\varsigma_{1}& = &-1+e^{{L_0(f-{2U_{0}\\over L_0})\\over 2T_{c}}+{L_0(f+{2U_{0}\\over L_0})\\over 2T_{h}}}\\\\ \\nonumber\n\\varsigma_{2}& = &{e^{-{{fL_{0}(T_{c}+T_{h})\\over T_{c}}+2U_{0}\\over 2T_{h}}}\\left(e^{fL_{0}\\over\n 2T_{c}}-e^{U_{0}\\over T_{c}}\\right)L_0\\over fL_{0}-2U_{0}}\\\\ \\nonumber\n &&-{\\left(e^{-{fL_{0}+2U_{0}\\over 2T_{h}}}-1\\right)L_{0}\\over fL_{0}+2U_{0}} \\\\ \\nonumber\n\\varsigma_{3} & = &{e^{-{U_{0}\\over T_{c}}+{fL_{0}+2U_{0}\\over 2T_{h}}}\n\\left(e^{fL_{0}\\over 2T_{c}}-e^{U_{0}\\over T_{c}}\\right)L_0T_{c} \\gamma\\over fL_{0}-2U_{0}}+\\\\ \\nonumber\n &&{\\left(e^{fL_{0}+2U_{0}\\over 2T_{h}}-1\\right)L_{0}T_{h}(\\gamma +c(T_c-T_h))\\over fL_{0}+2U_{0}}\\\\ \\nonumber\n\\varsigma_{4}& = &(\\gamma +c(T_c-T_h)){L_{0}^2\\left(fL_{0}+2((-1+e^{-{fL_{0}+2U_{0}\\over 2T_{h}}})T_{h}+U_{0})\\over\n 2(fL_{0}+2U_{0})^2\\right)}\n\\end{eqnarray}\nand $\\varsigma_{5}=L_{0}^2(t_1+t_2+t_3t_4)$. Here $t_1$, $t_2$, $t_3$ and $t_4$ are given by \n \\begin{eqnarray}\n t_1&=&{\\gamma \\over 2fL_{0}-4U_{0}}\\\\ \\nonumber\n t_2&=&{\\gamma T_c(-1+e^{-{{fL_{0}-2U_0}\\over 2T_{c}}})T_c\\over (fL_{0}-2U_{0})^2}\\\\ \\nonumber\n t_{3}&=&{(-1+e^{-{{fL_{0}-2U_0}\\over 2T_{h}}})T_h\\over (f^2L_{0}^2-4U_{0}^2)} \\\\ \\nonumber\n t_{4}&=&{{e^{-{fL_{0}(T_{c}+T_{h})\\over 2T_{c}T_{h}}-{U_{0}\\over T_{h}}}(e^{{fL_{0}\\over\n 2T_{c}}}-e^{{U_0\\over T_{c}}})(\\gamma +c(T_c-T_h))}\\over (f^2L_{0}^2-4U_{0}^2)}.\n \\end{eqnarray}\n\tThe steady state current converges to zero (at quasistatic limit) when \n\\begin{eqnarray}\nf'= {2U_{0}(T_{h}-T_{c})\\over (L_{0}(T_{h}+T_{c}))}.\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n1.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n2.eps}\n}\n\\caption{ (Color online) (a) The dependence of $J$ on $U_{0}$ for fixed $\\tau=2.0$, $\\gamma'=1$ and $f=0.5$. The parameter $C$ is also fixed as $0.4$ (solid line), $0.2$ (dashed line) and $0.04$ (dotted line). \n (b) The plot $J$ as a function of $f $ for parameter choice $U_{0}=2.0$ and $\\tau=2.0$. \n The parameter $C$ is fixed as $0.4$, (solid line), $0.2$ (dashed line) and $0.04$ (dotted line). }\n\\label{fig:sub}\n\\end{figure}\nNext, let us explore the dependence for the thermodynamic quantities on the model parameters. In Fig. 4a, the current as a function of potential height is plotted. The current exhibits a maximum value at a particular barrier height. As shown in Fig. 4b, the current monotonously decreases with the load. When $f0$ while when $f>f'$, $J<0$.\n\n\n\n\nOnce the expression for steady-state current is obtained, the values for ${\\dot h}_{d}$ and ${\\dot e}_{p}$ can be readily evaluated via Eq. (42). \nAt steady state ($v {dv\\over dt}=0$), to both underdamped and overdamped cases, one finds \n\\begin{eqnarray}\n{\\dot h}_{d} ={\\dot e}_{p}\n&=&\\int \\left({J U'(x) \\over T(x)} \\right)dx.\n\\end{eqnarray}\nThe rate of heat extraction is given by \n\\begin{eqnarray}\n{\\dot H}_{d}=&=&\\int \\left({J U'(x)} \\right)dx.\n\\end{eqnarray}\nAt quasistatic limit ( $f\\to f'$), ${\\dot h}_{d} ={\\dot e}_{p}=0$ as well as ${\\dot H}_{d}=0$ since at this limit $J=0$.\n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n3.eps}}\n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n4.eps}\n}\n\\caption{ (Color online) (a) The plot for ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $U_{0}$ for parameter choice of $\\tau=2.0$, $C=0.04$ and $f=0.5$. \n (b) The plot ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $f$ for parameter choice of $U_{0}=2.0$, $C=0.04$ and $\\tau=2.0$. } \n\\label{fig:sub}\n\\end{figure}\nLet us explore how the rate of entropy production ${\\dot e}_{p}(t)$ and the rate of entropy extraction ${\\dot h}_{d}(t)$ behave. The plot of ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $U_{0}$ is depicted in Fig. 5a. The entropy production and extraction rates take a zero value at the stall force (zero velocity), ${\\dot e}_{p}(t)={\\dot h}_{d}(t)=0$ which implies that at the stall force the system is reversible. The plot ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ as a function of $f$ is depicted in Fig. 5b. As depicted in the figure, the entropy production and extraction rates decrease as the load increases and attains a zero value at the stall force. As the load further increases, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ step up. \nThe entropy production and extraction rates increase as $C$ and the temperature difference between the two baths decreases. On the other hand, entropy production and extraction rates decrease as $\\tau$ increases and attain a zero value at a particular $\\tau$. As the temperature further increases, ${\\dot e}_{p}(t)$ and ${\\dot h}_{d}(t)$ increase\n\nIf one considers a periodic boundary condition at steady state in the absence of ratchet potential $U_{0}=0$, the results obtained quantitively agree with the underdamped case (Section IV ) and one gets \n\t\\begin{eqnarray} \n{\\dot h}_{d}(t)&=&{\\dot e}_{p}(t) \\nonumber \\\\\n&=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma T_c}+\\frac{1}{(\\gamma+c (T_c-T_h)) T_h}\\right) \\end{eqnarray}\n\tand \n\\begin{eqnarray} \n{\\dot H}_{d}(t)&=&{\\dot E}_{p}(t) \\nonumber \\\\ &=&\\frac{1}{2} f^2 L_{0} \\left(\\frac{1}{\\gamma}+\\frac{1}{\\gamma+c (T_c-T_h)}\\right).\n \\end{eqnarray}\t\n\t\t\t\t\n\nLet now explore the energetics of the model system. When the Brownian particle along the reaction coordinate, the heat that is taken from the hot heat bath $Q_{h}$ is given as \n\\begin{eqnarray}\nQ_{h} &=&U_{0}+{fL_{0}\\over 2}\n \\end{eqnarray}\n\twhile \n\tthe rate of heat flow into the cold heat bath $Q_{h}$ can be found as \n\t\\begin{eqnarray}\nQ_{c} &=&U_{0}-{fL_{0}\\over 2}\n \\end{eqnarray}\n\twhich implies the work done is given by \n\t\\begin{eqnarray}\nW &=& Q_{h}-Q_{c}= fL_{0}.\n \\end{eqnarray}\n\nLet us now explore how the efficiency $\\eta$ and the coefficient of performance of the refrigerator $P_{ref}$ behave. When the engine acts as a heat engine, the efficiency is given by\n\\begin{eqnarray}\n\\eta= {W\\over Q_{h}}={fL_{0}\\over U_{0}+fL_{0}\/2}.\n\\end{eqnarray}\nAt quasistatic limit, plugging Eq. (76) into Eq. (84), one gets \n\\begin{eqnarray}\n\\eta= 1-{T_{c}\\over T_{h}}\n\\end{eqnarray}\nwhich is the efficiency of the Carnot heat engine. When the engine performs as a refrigerator, the coefficient of performance of the refrigerator $P_{ref}$ is given by \n\\begin{eqnarray}\nP_{ref}= { Q_{c}\\over W}={U_{0}-fL_{0}\/2 \\over fL_{0}}\n\\end{eqnarray}\nand at quasistatic limit, plugging Eq. (76) into Eq. (86), $P_{ref}$ approaches Carnot refrigerator\n\\begin{eqnarray}\nP_{ref}= {T_{c}\\over T_{h}-T_{c}}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\n \n\n\\section{ Multiplicative noise }\nMost of the previous works have focused on calculating the thermodynamic features of different model systems by considering additive noise. Most realistic\n systems such as neuron system can be also described by Langevin\nequations with multiplicative noise were in this case, the noise\namplitude varies spatially \\cite{mar12}. Considering multiplicative noise, \nthe intrinsic noise-induced ordering phase transition has been also studied in the work \\cite{mar10}. \nIn this work, we study how entropy, entropy production, and extraction rate depend on the strength of the background noise by solving the model exactly.\n\nFor the case where the temperature is position-dependent $T(x)= \\sqrt{D}|x|^{{-z \\over 2}}$, in the absence any external potential, the corresponding Fokker Planck equation is given as \n\\begin{eqnarray}\n{\\partial P(x,t)\\over \\partial t}&=&{\\partial \\over \\partial x}\\left({T'(x))\\over 2 \\gamma}\\right)P(x,t)+\\nonumber \\\\\n&&{\\partial \\over \\partial x}\\left({T(x)\\over \\gamma}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nThe probability current is given as \n\\begin{eqnarray}\nJ&=&-\\left({P(x,t)T'(x))\\over 2 \\gamma}\\right)-\\nonumber \\\\\n&&\\left({T(x)\\over \\gamma}{\\partial P(x,t)\\over \\partial x}\\right).\n\\end{eqnarray}\nThe solution for the probability distribution is well known \\cite{mulu1} and it is given by \n\\begin{eqnarray}\nP(x,t) = {|x|^{z \\over 2} e^{-{|x|^{z+2}\\over D(z+2)^2 t}}\\over \\sqrt{4\\pi D t}}.\n\\end{eqnarray}\n \nFrom Eqs. (42) and (43), one gets \n\\begin{eqnarray}\n{\\dot e}_{p}&=& \\int { \\gamma J^{2} \\over P T(x) } dx \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n{\\dot h}_{d} &=&\\int\\left({J(T'(x))\\over 2 T(x)}\\right) dx. \n\\end{eqnarray}\n\n\n\nThe expression for the entropy production and extraction rates can be found by substituting Eqs. (87) and (88) into Eqs. (89) and (90).\n The plot ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ as a function of $t$ for parameter choice $\\tau=1$, $D=1.0$ and $z=-4.0$ is depicted in Figs. 6a and 6b. The figures depicts that ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ decrease as time increases and in long time limit, it approaches its stationary value ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. Only in the long time limit, $t\\to \\infty$, ${d S(t)\\over dt}=0$ since ${\\dot e}_{p}(t)= {\\dot h}_{d}(t)=0$. This can be intuitively comprehended on physical grounds. For isothermal case, in the long time limit, the system approaches stationary state and only at this particular state, $\\Delta h_d=0$, $\\Delta S=0$ or $\\Delta e_p=0$ (at stationary state). However when the particle operates at finite time, the system operates irreversibility and in this regime, the second law of thermodynamics states that $\\Delta S(t)>0$. As it can be seen from Fig. 6 that if the thermodynamic quantities are evaluated in the time interval between $t=0$ and any time $t$, always the inequality \n$\\Delta h_d(t)=h_d(t)-h_d(0)>0$, $\\Delta S(t)=S(t)-S(0)>0$ or $\\Delta e_p(t)=e_p(t)-e_p(0)>0$ holds true and as time progresses the change in this parameters increases. \nIn fact, in small $t$ regimes, ${\\dot e}_{p}(t)$ becomes much larger than ${\\dot h}_{d}(t)$ (see Figs. 6a and 6b) showing that the entropy production is higher (than entropy extraction) in the first few period of times. When time increases, more entropy will be extracted ${\\dot h}_{d}(t)>{\\dot e}_{p}(t)$. Over all, since the system produces enormous amount of entropy at initial time, in latter time or any time $t$, $\\Delta e_p(t)>\\Delta h _d(t)$ and hence $\\Delta S(t)>0$. \n\\begin{figure}[ht]\n\\centering\n{\n \n \\includegraphics[width=6cm]{n11.eps}} \n\\hspace{1cm}\n{\n \n \\includegraphics[width=6cm]{n12.eps}\n}\n\\caption{ (Color online) The plot ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ as a function of $t$ for parameter choice $\\tau=1$, $D=1.0$ and $z=-4.0$ is depicted in Figs. 6a and 6b, respectively. The figures depicts that ${\\dot h}_{d}(t)$ and ${\\dot e}_{p}(t)$ decrease as time increases and in long time limit, it approaches its stationary value ${\\dot h}_{d}(t)={\\dot e}_{p}(t)=0$. \n} \n\\label{fig:sub}\n\\end{figure}\n\n\n\n\\section{Summary and conclusion}\n\n\n\nThe influence of viscous friction on the thermodynamic properties of a Brownian particle that walks in overdamped and underdamped media is studied. The viscous friction is considered to vary either spatially or temporally. \nBy extending Seifert stochastic approach to underdamped and overdamped media, the general expressions for entropy production, free energy, and entropy extraction rates are derived. To explore \nthe non-equilibrium thermodynamic features of a Brownian particle that hops in medium where its viscosity depends on time, a Brownian particle that walks on a periodic isothermal medium (in the presence or absence of load) is considered. The analytical results depict that in the absence of load, the entropy production rate balances the entropy extraction rate which is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. It is shown that when a distinct temperature difference is not retained between the hot and cold baths, in absence of load, the entropy production still balances the entropy extraction rate revealing the system is reversible. When the external load is zero and \nwhen the viscous friction decreases in time, the entropy monotonously increases with time and saturates to a constant value as $t$ further steps up. The entropy production rate decreases in time and at steady state (in the presence of load), ${\\dot e}_{p}={\\dot h}_{d}>0$ which agrees with the results shown in the works \\cite{muuu177}. On the contrary, when the viscous friction increases in time, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. \n\n\n\nFor a system where the viscous friction of a medium tends to decrease as the temperature of the medium increases, the non-equilibrium thermodynamic features of the model system are explored. In this case, the load $f$ dictates the direction of the particle velocity. The steady-state velocity of the engine is positive when $f$ is smaller and the engine acts as a heat engine. In this regime the entropy production and extraction rates become nonzero. When $f$ steps up, the velocity of the particle steps down and at stall force, the entropy production rate balances the entropy extraction rate revealing the system is reversible at this particular choice of parameter. For large loads, the current is negative and the engine acts as a refrigerator. In this region the entropy production and extraction rates become nonzero. In the absence of load, the entropy production and extraction rates become larger than zero as long as a distinct temperature difference is retained between the hot and cold baths. We further explore the thermodynamic features of such systems by considering a multiplicative noise wherein\ncase the noise amplitude varies spatially. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn conclusion, in this work, we derive several thermodynamic relations to a Brownian particle moving in underdamped and overdamped media by considering viscous friction that varies temporally and spatially. We believe that the present theoretical work serves as a basic tool to understand the nonequilibrium thermodynamics.\n\n\n \n\\section*{Acknowledgment}\nI would like to thank Blaynesh Bezabih and Mulu Zebene for their\nconstant encouragement. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nTwo of the mainstreams of Noncommutative Geometry \nconcentrate around the notions of a\nprojective module \\cite{conbook,coq} and \nof a quantum group \\cite{frt,cmp}. Quite\nrecently (see \\cite{bmq,md1,mp}), the concept of a quantum principal \nbundle was systematically developed with quantum groups (Hopf algebras) in the\nrole of structure groups.\nHence, since both projective modules and quantum principal bundles \nserve as starting\npoints for quantum geometric considerations, \nthe conceptual framework provided by the notion of a \nquantum principal bundle has a good chance of unifying \nthose two branches of Noncommutative Geometry. \n\nIn the classical differential geometry,\nit is hard to overestimate the interplay between Lie groups and $K$-theory.\nTherefore, it is natural to expect that establishing a similar\ninteraction in the noncommutative case is necessary for better understanding of\nquantum geometry. \nIt is already known that \nthe classification of quantum principal bundles over manifolds depends only on\n the classical subgroups of quantum \nstructure groups \\cite{md2}. This leads to \nthe following questions: Is the classification of (general)\nquantum principal bundles over\nnoncommutative spaces richer then the classification of classical-group bundles\nover noncommutative spaces? Is there \na bimodule that can be obtained \nas the bimodule of intertwiners (noncommutative analogue of\nequivariant vector valued functions on a total space) only from a bundle with a \nnoncommutative structure Hopf algebra? More generally, when does a \ndeformation of a group into a quantum group entail essential consequences in the\ngeometry (e.g., in the classification of bundles or in the Yang--Mills theory)?\n\nSince this article is to a great extent a\nfollow-up of \\cite {bmq}, much of the mathematical\nand physical motivations listed there can also\nbe considered as motivations for this work, and so\nwill not be repeated here. Let us just emphasize\nthat our main purpose is to specify and analyze a class of\nconnections on quantum principal\nbundles (called strong connections) that enjoy\nsome additional properties making\nthem more like their classical counterparts, and\n(taking advantage of the notion of a strong connection) to discuss a link\nbetween the two approaches to noncommutative\ndifferential geometry based on quantum principal bundles and\nprojective modules. The study of the precise relationship between those two\napproaches is thought of as a move towards answering the questions \nmentioned above.\n\nWe begin this article by fixing the notation and recalling the fundamentals of\n quantum bundles and Yang--Mills theory on projective modules. \nIn the first section, in addition\nto this vocabulary review, we also study the definition\nof a quantum principal bundle using snake diagrams (see Remark~\\ref{snake} \nand Proposition~\\ref{galois}). Taking advantage of Remark~\\ref{snake}, we prove \n(Corollary~\\ref{dcor}) that the fundamental vector field compatibility condition\n(see Point~3 in Definition~\\ref{qpbdef}) implies its stronger version. \n(The latter version of\nthe fundamental vector field compatibility condition was assumed in \nExample~4.11\\bmq .)\n\nThe formalism \nused in this paper is a generalization of the\ncorresponding formalism used in classical\ndifferential geometry. The calculations showing this, \nthough often very instructive, are straightforward\n and we will not fully elaborate on\nthat fact later on. Differential geometry on\nquantum principal bundles is still in the process\nof being born --- the umbilical cord has hardly\nbeen cut yet --- and it seems premature at\nthis point to make precise categorical\nstatements establishing the relationship\nbetween classical and quantum differential\ngeometries. As Yu.I.Manin mentioned in a\nsimilar context (see p.86 in \\cite{ma}), ``Here,\none should not act too hastily since even in\nsupergeometry this program was started only\nrecently and revealed both rich content and\nsome puzzling new phenomena.\"\n\nIn Section~2, \nwe define and provide examples of strong connections.\nProposition~\\ref{scprop}\nallows one to interpret strong connections on trivial quantum bundles\nas those induced from the base space, and to produce examples of strong\nconnections in the case of trivial quantum bundles with the universal calculus. \nIn the Introduction to the preliminary version of \\bmq ,\none can read regarding inducing connection forms from the\nbase space that ``...in the general non-commutative or quantum case there\nwould appear to be slightly more possibilities...\" than in the classical case.\nExamples of connections that are not strong (and thus realize the\n just mentioned ``quantum possibility\") are supplied as well. More precisely,\nwe construct both strong and non-strong connections on a very simple (yet\n rich enough) example of a `discrete bundle' and on a quantum version of \nthe Hopf fibration $S\\sp2\\ra RP\\sp2$.\n(As a byproduct of our considerations we obtain a \\mbox{$q$-deformation}\n of~$RP\\sp2$.)\nIt might be worthwhile to note that the latter construction does not employ\nthe trivial-bundle or the quantum-group-quotient techniques. We end this section\nwith presenting an example of a non-strong connection in the set-up of a strict\nmonoidal category dual to the category of sets with Cartesian product.\n\nIn the subsequent section, we describe the action of (global) \ngauge transformations on the space of connections on a bundle with\nthe universal differential calculus, and show that this action \npreserves the strongness of a connection.\n\nIn Section~4, we use the notion of a strong connection to justify the\ndefinition of a global curvature form. (We need to assume that a connection \nis strong if we want to show that its curvature form has\ncertain properties that classical curvature\nforms possess automatically; e.g., the usual relation to the square of the exterior\ncovariant derivative.) \n\nIn Section~5, we present a link between gauge theory on a\nquantum principal bundle and Yang--Mills theory\non a projective module. First we show how, in the case of a free module,\nto incorporate the Yang--Mills action constructed in \\cite{conri,ri}\ninto the quantum bundle picture. Then, to obtain a\nHermitian metric compatible connection\non a free module from a strong $U\\sb q(2)$-connection on a\ntrivial quantum principal bundle, we mimic the\n classical geometry formula which permits one to\ndetermine the values of a connection form on a Hopf\nalgebra of smooth functions on a matrix Lie group\nby knowing its values on the matrix of\ngenerators. (Note that usually one thinks of a\nconnection form as a map sending smooth\n vector fields to elements of a Lie algebra, but\nwe can also view it as a map from the Hopf\nalgebra of smooth functions on a Lie group into\nthe space of smooth \\mbox{1-forms}.) It turns out that the connections\ncompatible with a particular $q$-dependent Hermitian structure can be \nidentified with the strong $U\\sb q(2)$-connections that satisfy certain condition.\nWe close this section by concluding that, in the setting under consideration,\nthe moduli space of critical points of $U\\sb q(2)$ and $U(2)$--Yang--Mills\ntheory coincide.\nThus, at least in this case, the $q$-deformation of the structure group alone\nhas no essential\nbearing on the Yang--Mills theory. This seems to bring us a step closer \nto answering the question posed at the end of the first paragraph: \nOne should expect geometrically interesting effects of the noncommutativity \nof a Hopf algebra in Yang--Mills theory\nonly for non-trivial bundles (comodule algebras that are not crossed product \nalgebras) or non-strong connections. \n\nFinally, in the Appendix, we examine the\nadvantages of adding a twist to the definition\nof a quantum associated bundle\nformulated in \\bmq\\ and point out the\npossibility of using the axiomatic definition\nof a frame bundle to try to define its\nnoncommutative analogue (cf.\\\nSection~5.1 in \\cite{bmq}).\n\n\\parindent0pt\n\\section{Preliminaries}\n\nThe notation used throughout this article is quite\nstandard and \nnot much different from that\nof \\bmq . Nevertheless, to eschew any possible misunderstanding or\nconfusion, we enclose a table of basic notations:\n\n\\bi\n\n\\item [{$[\\! ...]_{X}$}] an equivalence class\ndefined by $X$\n\n\\item [$\\delta_{m,n}$] equals 1 iff $m=n$, and 0\notherwise (Kronecker symbol)\n\n\\item [$\\frak g\\;\\;\\;\\:$] Lie algebra of a Lie group $G$\n\n\\item [$k\\;\\;\\;\\,$] field of characteristic zero \n(except for Proposition~\\ref{spera})\n\n\\item [$\\ot\\;\\;\\;$] tensor product over $k$\n\n\\item [$\\tau\\;\\;\\;\\,$] flip ($\\tau (u\\ot v):= v\\ot u$)\n\n\\item [$\\Omega\\sp{1}\\! A\\;$] first order universal\ndifferential calculus ($\\Omega\\sp{1}\\! A:=\\kr\\,\nm_{A}$ , $da:=1\\ot a-a\\ot 1 $)\n\n\\item [$\\Omega A\\;\\,$] differential envelope of $A$ \n\n\\item [$\\Omega (\\! A)$] differential algebra over $A$ (i.e.~a quotient of $\\hO A$\nby some differential ideal)\n\n\\item [$\\; m_{X}\\;$] multiplication on $X$, or in $X$\n(We will simply write $m$ for the multiplication in a Hopf algebra.)\n\n\\item [$\\Delta\\;\\;\\;$] comultiplication: $\\de (a)=a\\1\\ot a\\2$\n(Sweedler sigma notation with suppressed summation sign,\ncf.\\ Section~1.2 in~\\cite{swe})\n\n\\item [$\\de_{n}\\;$] comultiplication applied $n$\ntimes (Due to coassociativity we do not have\nto remember where \\de\\ is put in consecutive\ntensor products; $\\de\\sb na=a\\1\\ot\\cdots\\ot a\\sb{(n+1)}\\,$.)\n\n\\item [$\\eta_{Y}\\;\\;$] unit map of an algebra $Y$ (often suppressed)\n\n\\item [$\\epsilon\\;\\;\\;\\,$] counit (unless otherwise\nobvious from the context)\n\n\\item [$S\\;\\;\\,$ ] antipode, i.e.~$a\\1 S(a\\2)=\\epsilon (a)=S(a\\1)a\\2\\,$\n\n\\item [$A\\sp{op}\\;$] algebra identical with algebra $A$ as a vector space\nbut with the multiplication defined by $m\\sb{A\\sp{op}}=m\\sb A\\ci\\tau$\n\n\\item [$\\rho_{R}\\;\\;$] right coaction: $\\rho_{R}(x) = x\\0\\ot x\\1 $\n(Sweedler sigma notation for comodules with \nsuppressed summation sign, cf.\\ p.32--3 in~\\cite{swe})\n\\item [$\\dr\\;\\,$] right coaction on the\n`total space' of a quantum principal bundle\n\n\\item [$\\Delta_{\\cal R}\\;\\,$] right coaction on a right-covariant\ndifferential algebra of\nthe `total space' given by $\\mbox{$\\forall\\, m\\in{\\Bbb N}:\\;$}\n\\mbox{$\\dsr\n(p\\sb 0dp\\sb 1\\cdots dp\\sb m)$}=\\mbox{$(p\\sb 0)\\0d(p\\sb 1)\\0\\cdots\nd(p\\sb m)\\0\\ot (p\\sb 0)\\1\\cdots (p\\sb m)\\1$}$, where for\nany \\mbox{$n\\in\\{ 0,\\cdots ,m\\}$}, \\mbox{$(p\\sb n)\\0\\te (p\\sb n)\\1\n=\\dr p\\sb n$} (A differential algebra is right-covariant iff\n \\dsr\\ determined by\nthe above formula is well-defined; cf.\\ (21) and Section~4.2 in~\\bmq .)\n\n\\item [$ad_{R}\\;$] $:= (id\\ot m)\\ci (id\\ot S\\ot id)\\ci\n(\\tau\\ot id)\\ci\\de_{2}$ (right adjoint coaction;\n$ad\\sb Ra=a\\2\\te S(\\! a\\1\\! )a\\3\\,$)\n\n\\item [$P\\sp{co A}$] the space of right coinvariants ($P\\sp{co A}:=\\{p\\in P |\n\\dr p=p\\te 1\\}$)\n\n\\item [$*_{\\rho}\\;\\;\\,$] convolution: $\\forall\\, f\\!\\in\\!\n\\mbox{Hom}\\sb k(Q,X),\ng\\!\\in\\! \\mbox{Hom}\\sb k(A,Y): f*_{\\rho}g\n= \\m\\ci (f\\te g)\\ci\\rho_{R}$, where $(Q,\\rho\\sb R)$ is a right\n$A$-comodule and $\\m :X\\te Y\\ra Z$ is a multiplication map\n(If $\\rho_{R}$ equals $\\dsr$, \\dr\\ or \\de ,\nwe will use $*_{\\cal R}$, $*_{R}$ or $*$ respectively\nto denote the corresponding convolution.)\n\n\\item [$f\\sp{-1}\\;$] unless otherwise obvious from\nthe context, convolution inverse of $f$, i.e.~ \\\\\n $f\\sp{-1}(a\\1)f(a\\2)=(f\\sp{-1}\\! *\\! f)(a)\n=\\epsilon (a)=(f\\! *\\! f\\sp{-1})(a)=f(a\\1)f\\sp{-1}(a\\2)\\,$\\\\\n(In general, one has:\n $x\\sb{(0)}\\ot\\cdots\\ot x\\sb{(n)}\\ot\nf(x\\sb{(n+1)})g(x\\sb{(n+2)})\\ot\nx\\sb{(n+3)}\\ot\\cdots\\ot x\\sb{(m)}\\\\\n=x\\sb{(0)}\\ot\\cdots\\ot x\\sb{(n)}\\ot\n(f*g)(x\\sb{(n+1)})\\ot x\\sb{(n+2)}\\ot\\cdots\\ot\nx\\sb{(m-1)}\\;$ and\\\\\n${\\cal F}(x\\0)g(x\\1)\\ot x\\2\\ot\\cdots\\ot x\\sb{(m)}\n=({\\cal F}*\\sb\\rho g)(x\\0)\\ot x\\1\\ot\\cdots\\ot x\\sb{(m-1)}$.)\n\n\\ei\n\nAll algebras are assumed to be unital and associative.\nNow, let us recall the basic notions and\nconstructions of \\bmq\\ necessary to\nestablish the language used in this paper.\n\\bde [4.9 \\bmq ]\\label{qpbdef}\nLet $P$ be an algebra over a field $k$, $A$ a\nHopf algebra over the same field, $N_{P}\n\\inc\\Omega\\sp{1}\\! P$ a $P$-bimodule\ndefining the first order differential calculus\n\\op , $M_{A} \\inc\\ker\\,\\epsilon$ an $ad\\sb R$-invariant right\nideal defining the bicovariant differential calculus\n\\oa , and\n\\[\n\\dr : P\\lra P\\ot A\n\\]\nan algebra homomorphism making $P$ a right $A$-comodule\n algebra.\\\\ Then $(P,A,\\dr ,N_{P},M_{A})$ is called a quantum\nprincipal bundle iff:\n\n\\be\n\n\\item $\\can :P\\ot P\\ni t\\stackrel{\\mbox{\\em\\scriptsize def.}}{\\longmapsto}\n(m\\te id)\\cc (id\\te\\dr )\\, t\\in P\\ot A$ is a surjection (freeness condition),\n\n\\item $\\dsr (N_{P})\\inc N_{P}\\ot A$\n(right covariance of the differential\nstructure),\n\n\\item $\\can (N_{P})\\inc P\\ot M_{A}$\n (fundamental vector field compatibility condition),\n\n\\item $\\ker\\,\\mbox{\\em\\sf T} \\inc P\\hO\\sp1(P\\sp{co A}) P$ (exactness condition), \nwhere \n$\\hO\\sp1(P\\sp{co A}) := \\Omega\\sp 1 P\\sp{co A}\/(N\\sb\n P\\cap\\Omega\\sp 1 P\\sp{co A})$\n\\[\n\\mbox{and $\\;$ \\em\\sf T}: \\op\\ni{[\\alpha ]}\n_{N_{P}}\\stackrel{\\mbox{\\em\\scriptsize def.}}{\\longmapsto}\n\\llp(id\\te\\pi_{A})\\cc \\can\\lrp\\,\\alpha \\in P\\ot (\\ker\\,\\epsilon\\, \/\\, M_{A})\n\\]\n (the map $\\pi\\sb A :\n\\ker\\,\\epsilon\\ra\\ker\\,\\epsilon\\, \/\\, M_{A}$\nis the canonical projection, and\n$\\alpha\\!\\in\\!\\ker\\, m\\sb P$).\n \\ee\n\\ede\nFor simplicity, as well as to emphasize the\nanalogy with the classical situation, a quantum\nprincipal bundle is often denoted by\n$P(B,A)$, where $B:=P\\sp{co A}$ is the `base space' of the bundle. \nThe map {\\sf T} (denoted by\n$\\widetilde{~}\\sb{N\\sb P}$\nin \\bmq ) can\nbe more explicitly described by the formula\n\\[\n\\hsp{31mm}\\T (pdq)=pq\\0\\ot\n[q\\1 ]\\sb{M\\sb A} - pq\\ot 1 .\\hsp{44mm}\n\\mbox{(cf.\\ (24) in \\bmq )}\n\\]\n\n\\bre\\label{snake}{\\em\nLet $\\;\\T\\!\\sb U:\\mbox{Ker}\\, m\\sb P\\ra P\\ot\\mbox{Ker}\\,\\he\\;$ and\n$\\;\\T\\!\\sb{NM}:N\\sb P\\ra P\\ot M\\sb A\\;$ be the appropriate restrictions \nof \\cnl . It is straightforward to check that the following\ndiagrams are commutative diagrams (of left \\mbox{$P$-modules}) with exact rows\nand columns:\\\\\n\n\\beq\\label{d1}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\n\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb{U}&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb R\n&\\mapright{}&0&&&\\phantom{.}\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&\\mbox{Ker}\\, m\\sb P&\\mapright{}\n&P\\ot P&\\mapright{m\\sb P}&P&\\mapright{}&0\\cr\n&&\\mapdown{\\T\\!\\sb{U}}&&\\mapdown{\\T\\!\\sb R}&&\\mapdown{id}\\cr\n0&\\mapright{}&P\\ot\\mbox{Ker}\\,\\he\n&\\mapright{}&P\\ot A&\\mapright{m\\sb P\\ci (id\\ot\\he)}\n&P&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&\\mbox{Coker}\\,\\T\\!\\sb{U}&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb R\n&\\mapright{}&0&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\n\\beq\\label{d2}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb{NM}&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb U\n&\\mapright{}&\\mbox{Ker}\\,\\T\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&N\\sb P&\\mapright{}\n&\\Omega\\sp 1P&\\mapright{\\pi\\sb P}&\\op&\\mapright{}&0\\cr\n&&\\mapdown{\\T\\!\\sb{NM}}&&\\mapdown{\\T\\!\\sb U}&&\\mapdown{\\T}\\cr\n0&\\mapright{}&P\\ot M\\sb A&\\mapright{}&P\\ot\\mbox{Ker}\\,\\he&\\mapright{id\\ot\\pi\\sb A}\n&P\\ot(\\mbox{Ker}\\,\\he\\, \/\\, M\\sb A)&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&\\mbox{Coker}\\,\\T\\!\\sb{NM}&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb U\n&\\mapright{}&\\mbox{Coker}\\,\\T&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\nApplying the Snake Lemma (e.g., see Section~1.2 in \\cite{aki}) to both diagrams,\nwe obtain the following two exact sequences:\n\\beq\\label{s1}\n0\\lra\\mbox{Ker}\\,\\T\\!\\sb U\\lra\\mbox{Ker}\\,\\T\\!\\sb R\\lra 0\n\\lra\\mbox{Coker}\\,\\T\\!\\sb U\\lra\\mbox{Coker}\\,\\T\\!\\sb R\\lra 0\n\\phantom{vhvhvjhvhvsccsj......}\n\\eeq\n\\beq\\label{s2}\n0\\lra\\mbox{Ker}\\,\\T\\!\\sb{NM}\\lra\\mbox{Ker}\\,\\T\\!\\sb U\\lra\\mbox{Ker}\\,\\T \n\\lra\\mbox{Coker}\\,\\T\\!\\sb{NM}\\lra\\mbox{Coker}\\,\\T\\!\\sb U\\lra\\mbox{Coker}\\,\\T\\lra 0\n\\eeq\\ \\\\\nObserve that the freeness condition means exactly that $\\mbox{Coker}\\,\\T\\!\\sb R=0$,\nwhich, by (\\ref{s1}), is equivalent to $\\mbox{Coker}\\,\\T\\!\\sb U=0$ (see (33) in\n\\bmq ). Note also that $\\mbox{Ker}\\,\\T\\!\\sb{NM}=N\\sb P\\cap\\mbox{Ker}\\,\\T\\!\\sb U$.\n}\\hfill{$\\Diamond$}\\ere\n\n\\bco\\label{dcor}\n$\\!$Let $(P,A,\\dr ,N_{P},M_{A})$ be a quantum\nprincipal bundle. Then \\mbox{$\\can(\\! N\\sb P\\! )\\! =\\! P\\te M\\sb A$} \n(cf.~Example~4.11 in \\bmq\\ and the discussion below it).\n\\eco\n\n{\\it Proof.} From the exactness condition, we know that \n$\\mbox{Ker}\\,\\T=\\pi\\sb P(P\\Omega\\sp 1\\! B.P)$. On the other hand, since\n$P\\Omega\\sp 1\\! B.P\\inc\\mbox{Ker}\\,\\T\\!\\sb U$ and the map\n$\\pi\\sb U:\\mbox{Ker}\\,\\T\\!\\sb U\\ra\\mbox{Ker}\\,\\T$ in (\\ref{s2}) is a restriction\nof $\\pi\\sb P$ to $\\mbox{Ker}\\,\\T\\!\\sb U$, we can conclude that $\\pi\\sb U$ is\nsurjective. Consequently, by the freeness condition and the exactness of (\\ref{s2}),\n$\\mbox{Coker}\\,\\T\\!\\sb{NM}=0$, i.e.~\\cnl$(\\! N\\sb P\\! )=P\\te M\\sb A$.\n\\epf\\\\\n\nThe above corollary makes the following definition of a trivial quantum principal\nbundle equivalent to the definition proposed in Example~4.11 in \\bmq .\n\n\\bde\\label{trbu}\nA quantum principal bundle $(P,A,\\dr ,N\\sb P,M\\sb A)$ is called trivial iff there\nexists a convolution invertible map\n(trivialization) $\\Phi\\in \\mbox{\\em Hom}\\sb k(A,P)$ such that\n\\beq\\label{trcov}\n\\dr\\ci\\Phi = (\\Phi\\ot id)\\ci\\de\n\\eeq\n(i.e.~$\\Phi$ is right-covariant) and $\\Phi (1) = 1$. In such a case, $P$ is also\ncalled a crossed product or cleft extension (see p.273 in \\cite{sch2}).\n\\ede\n\\bde [1.1 \\cite{sch2}]\\label{galois}\nLet $P$ be a right $A$-comodule algebra and $B$ be the algebra of all right\ncoinvariants. The comodule $P$ is called an $A$-Galois extension iff the canonical\nleft \\mbox{$P$-algebra} and right $A$-coalgebra map\n\\[\n\\mbox{\\em\\sf T}\\!\\sb B:=(m\\sb P\\te id)\\ci(id\\te\\sb B\\dr):\nP\\ot\\sb B P\\ni p\\ot\\sb Bq\\longmapsto pq\\0\\ot q\\1\\in P\\ot A\n\\]\nis bijective. \n\\ede\n\\bpr\\label{diagram}{\\em\\footnote{\nThis proposition is implicitly proved in \\cite{tbt} (see Lemma~3.2 and the text\nabove it). The diagrammatic proof presented here was created during the author's\ndiscussion with Markus Pflaum.}}\nLet $P$, $A$ and $B$ be as above. A comodule algebra $P$ is an $A$-Galois extension\nif and only if $P(B,A)$ is a quantum principal bundle with the universal calculus.\n\\epr\n{\\it Proof.} Consider the following commutative diagram (of left $P$-modules)\nwith exact rows and columns:\\\\\n\n\\beq\\label{d3}\n\\def\\normalbaselines{\\baselineskip24pt\n\\lineskip3pt \\lineskiplimit3pt }\n\\def\\mapright#1{\\smash{\n\\mathop{\\!\\!\\!{-\\!\\!\\!}-\\!\\!\\!\\longrightarrow\\!\\!\\!}\n\\limits^{#1}}}\n\\def\\mapdown#1{\\Big\\downarrow\n\\rlap{$\\vcenter{\\hbox{$\\scriptstyle#1$}}$}}\n\\matrix{0&\\mapright{}&P\\hO\\sp1\\! B.P&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb R\n&\\mapright{}&\\mbox{Ker}\\,\\T\\!\\sb B&&&\\phantom{.}\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n0&\\mapright{}&P\\hO\\sp1\\! B.P&\\mapright{}\n&P\\ot P&\\mapright{}&P\\ot\\sb B P&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{\\T\\!\\sb R}&&\\mapdown{\\T\\!\\sb B}\\cr\n0&\\mapright{}&0\n&\\mapright{}&P\\ot A&\\mapright{id}\n&P\\ot A&\\mapright{}&0\\cr\n&&\\mapdown{}&&\\mapdown{}&&\\mapdown{}\\cr\n&&0&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb R\n&\\mapright{}&\\mbox{Coker}\\,\\T\\!\\sb B&\\mapright{}&0\\cr\n}\n\\eeq\\ \\\\\n\nAgain, we can apply the Snake Lemma to obtain an exact sequence\n\\beq\\label{s3}\n0\\lra P\\hO\\sp1\\! B.P\\lra\\mbox{Ker}\\,\\T\\!\\sb R\\lra\\mbox{Ker}\\,\\T\\!\\sb B\\lra 0\\lra\n\\mbox{Coker}\\,\\T\\!\\sb R\\lra\\mbox{Coker}\\,\\T\\!\\sb B\\lra 0\\; .\n\\eeq\nAssume first that $P$ is an $A$-Galois extension. Then \n$\\mbox{Ker}\\,\\T\\!\\sb B=0=\\mbox{Coker}\\,\\T\\!\\sb B$, and, from the exactness of \n(\\ref{s3}), we can infer that $\\mbox{Coker}\\,\\T\\!\\sb R=0$ (freeness condition)\nand $\\mbox{Ker}\\,\\T\\!\\sb R=P\\hO\\sp1\\! B.P\\,$. \nOn the other hand, by the exactness of (\\ref{s1}),\nwe have $\\mbox{Ker}\\,\\T\\!\\sb U=\\mbox{Ker}\\,\\T\\!\\sb R\\,$. Hence\nthe exactness condition follows, and we can conclude that $P(B,A)$ is a quantum\nprincipal bundle with the universal calculus.\n\nConversely, assume that $P(B,A)$ is a quantum\nprincipal bundle with the universal calculus. Then \n$\\mbox{Ker}\\,\\T\\!\\sb R=\\mbox{Ker}\\,\\T\\!\\sb U=P\\hO\\sp1\\! B.P$\nand $\\mbox{Coker}\\,\\T\\!\\sb R=0\\,$. Consequently, again due to the exactness of\n(\\ref{s3}), we have that $\\mbox{Ker}\\,\\T\\!\\sb B=0=\\mbox{Coker}\\,\\T\\!\\sb B\\,$,\ni.e.~$P$ is an $A$-Galois extension.\n\\epf\n\n\\bde [\\bmq ]\\label{condef}\nA left $P$-module projection $\\Pi$ on \\op\\ is called a connection on\\linebreak\n $P(B,A)$~iff\n\\be\n\\item $\\ker\\,\\Pi = \\Omega\\sp{1}_{hor}(P)$\n($\\,\\mbox{\\em Im}\\,\\Pi$ is called the space of vertical\n forms),\n\\item $\\dsr\\ci\\Pi = (\\Pi\\ot id)\\ci\\dsr$ (right covariance).\n\\ee\\ede \n\nDue to Proposition 4.10 in \\bmq , a connection\nform can be defined in the following way:\n\n\\bde \\label{confordef}\nA $k$-homomorphism $\\omega : A\\ra\\op$ is called a\nconnection form on $P(B,A)$ iff it satisfies the\nfollowing properties:\n\\be\n\\item $\\omega (k\\oplus M_{A}) = 0$ (compatibility with\nthe differential structure),\n\\item $\\mbox{\\em\\sf T}\n\\ci\\omega = (id\\ot\\pi_{A})\\ci \\llp 1\\ot\n(id - \\epsilon)\\lrp$ (fundamental vector field condition),\n\\item $\\dsr\\ci\\omega = (\\omega\\ot id)\\ci\nad_{R}$ (right adjoint covariance).\n\\ee\\ede\n\nFor every $P(B,A)$, there is a one-to-one\ncorrespondence between connections and\nconnection forms. In particular, the connection\n$\\Pi\\sp{\\omega}$ associated to\na connection form $\\omega$ is given by the\nformula: $\\Pi\\sp{\\omega}=m\\sb{\\Omega\\sp 1(P)}\\ci\n(id\\ot\\omega )\\ci\\T$ ((47) in \\bmq ).\nSince $\\Pi\\sp{\\omega}$\nis a left $P$-module homomorphism, to calculate\n$\\Pi\\sp{\\omega}$ it suffices to\nknow its values on exact forms, and on exact forms\n(47)\\bmq\\ simplifies to\n\\beq\\label{confor}\n\\Pi\\sp\\omega\\ci d = id*_{R}\\omega .\n\\eeq\n\nThe following four definitions are based on Appendix~A\\bmq .\n\n\\bde\nLet $\\Omega (P)$ be any differential algebra\nhaving \\op\\ as in Definition~\\ref{qpbdef}. For all $n\\in\\Bbb{N}$, the space \n$\\llp P\\ob P\\lrp\\sp{n}$ is called the space of horizontal\n$n$-forms and is denoted by $\\Omega\\sp{n}_{hor}(P)$.\nThe space of horizontal \\mbox{0-forms} is identified with~$P$.\n\\ede\n\n\\bde\nLet $\\Omega (B)$ be the differential algebra obtained by the restriction of\n$\\Omega (P)$. For all $n\\in\\Bbb{N}$, the space $\\Omega\\sp{n}\\!\n(\\! B)P$ is called the\nspace of strongly horizontal $n$-forms and is denoted\nby $\\Omega\\sp{n}_{shor}(P)$. The space of strongly horizontal \\mbox{0-forms}\nis identified with $P$.\n\\ede\n\nNote that in the classical case $\\Omega\\sp{*}_{hor}(P)$\nand $\\Omega\\sp{*}_{shor}(P)$ coincide.\n\n\\bde\\label{tfdef}\n Let $\\, (V,\\rho_{R})\\,$ be a right $A\\sp{op}$-comodule algebra \n(see Remark~\\ref{bialgebra}). Then\\linebreak \n\\mbox{$\\phi\\in \\mbox{\\em Hom}_{k}\\llp V,\\Omega (P)\\lrp$}\n is called a pseudotensorial form on $P$ iff\n\\[\n\\dsr\\ci\\phi = (\\phi\\ot id)\\ci\\rho_{R}\\, .\n\\]\nA pseudotensorial form taking values in $\\Omega\\sp{*}_{hor}(P)$ \n(in $\\Omega\\sp{*}_{shor}(P)$) is called a tensorial (strongly tensorial)\n form on $P$. The space of all pseudotensorial,\ntensorial and strongly tensorial \\mbox{$n$-forms} ($n\\geq 0$) will be denoted by\n $PT_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$,\n$T_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$ and\n $ST_{\\rho}\\llp V,\\Omega\\sp{n}\\! (\\! P)\\lrp$ respectively.\n\\ede\n\n\\bde [(68)\\bmq ]\\label{Ddef}\nLet $\\Pi$ be a connection on $P$. The $k$-homomorphism\n$D$ from $\\Omega\\sp{*}\\! P$ to $\\Omega_{hor}\\sp{*+1}\\! P$\ngiven by\n\\beq\\label{689}\nD : p\\sb{0}dp\\sb{1}\\cdots dp\\sb{n}\\longmapsto\n(id-\\Pi )(dp\\sb{0})\\cdots (id-\\Pi )(dp\\sb{n})\\, ,\n\\eeq\nwhere $n\\geq 0\\,$, is called the exterior covariant derivative associated to~$\\Pi$.\n\\ede \n\nTo complete this vocabulary review, we recall some basic definitions used\nin the Yang--Mills theory on projective modules. We choose here right rather\nthan left modules, but one should bear in mind that the formulation of this\nformalism for left modules is analogous.\n\n\\bpr[cf.~p.369 in \\cite{ms}]\\label{spera}\nLet $\\cal B$ be an associative unital algebra over a commutative ring $k$.\nLet $\\cal L\\sb k$ be a $k$-Lie subalgebra of the space of all $k$-derivations of\n$\\cal B$, and let $\\cal E$ be any right $\\cal B$-module admitting a connection. \nIf $\\hO(\\! \\cal B)$ is a differential graded subalgebra of \n\\mbox{$\\cal B\\oplus\\bigoplus\\sb{n\\geq 1}\\sp\\infty \n\\mbox{\\em Hom}\\sb k(\\mbox{\\large$\\wedge$}\\sp n\\cal L\\sb k,\\, \\cal B)$} \nwith the differential given by (see the first section in~\\cite{dkm}):\n\\bea\n(d\\ha)(X\\sb 0,X\\sb 1,\\cdots ,X\\sb n)\\!\\!\\!\n&=&\\!\\!\\!\\!\\sum\\sb{0\\leq i\\leq n}(-)\\sp iX\\sb i\\ha\n(X\\sb 0,\\cdots,X\\sb{i-1},X\\sb{i+1},\\cdots ,X\\sb n)\\\\\n&+&\\!\\!\\!\\!\\!\\!\\sum\\sb{0\\leq r\\,$}),\nlet $\\cal I\\sb0$ be a two-sided $*$-ideal of \\pf\\ generated by\n\\begin{eqnarray*}\n&&\\left\\{x\\sb m-x\\sb m\\sp*\\, ,\\;\\mbox{$\\sum\\sb j$}a\\sb jx\\sb j\\sp2-r\\sp2\\right\\},\\;\nm,j\\in\\{1,2,3\\}\\, ,\\; a\\sb j,r\\in\\Bbb R\\, ,\\; a\\sb j,r>0\\, ,\n\\end{eqnarray*}\nand let $\\cal I\\sb1$ be a two-sided $*$-ideal of \\pf\\ generated by\n\\begin{eqnarray*}\n&&\\mbox{\\large\\{}x\\sb m-x\\sb m\\sp*\\, ,\\; x\\sb1\\sp2+x\\sb2\\sp2+\n2\\mbox{$\\frac{1+q\\sp4}{(1+q\\sp2)\\sp2}$}x\\sb3\\sp2-1\\, ,\\; \nx\\sb1x\\sb2-x\\sb2x\\sb1-2i\\mbox{$\\frac{1-q\\sp4}{(1+q\\sp2)\\sp2}$}x\\sb3\\sp2\\, ,\\\\ \n&&\\ \\; x\\sb1x\\sb3-\\mbox{$\\frac{q\\sp{-2}+q\\sp2}{2}$}x\\sb3x\\sb1-\ni\\mbox{$\\frac{q\\sp{2}-q\\sp{-2}}{2}$}x\\sb3x\\sb2\\, ,\\;\nx\\sb2x\\sb3-\\mbox{$\\frac{q\\sp{-2}+q\\sp2}{2}$}x\\sb3x\\sb2+\ni\\mbox{$\\frac{q\\sp{2}-q\\sp{-2}}{2}$}x\\sb3x\\sb1\\mbox{\\large\\}}\\, ,\\\\\n&&\\ \\; m\\in\\{1,2,3\\}\\, ,\\; q\\in\\Bbb R\\, ,\\; 1\\geq |q|>0\\, .\n\\end{eqnarray*}\n(The second generator of $\\cal I\\sb1$ resembles the right-hand side of the formula\n(164) in \\cite{sm} that describes the metric on the $q$-Minkowski space discussed\nin Section~7.2 of~\\cite{sm}.)\nThe algebras $\\pf \/\\cal I\\sb0$ and $\\pf \/\\cal I\\sb1$ can be\nregarded as noncommutative two-spheres. Indeed, $\\pf \/\\cal I\\sb0$ is \n`the most noncommutative two-sphere', and $\\pf \/\\cal I\\sb1$ corresponds to\nthe equator sphere ($c=\\infty$) given by (7b) in \\cite{po} \n(see Remark~\\ref{equator}). Obviously, for $q=\\pm 1$, the algebra \n$\\pf \/\\cal I\\sb1$ corresponds to the usual $S\\sp2$. Since both $\\pf \/\\cal I\\sb0$\nand $\\pf \/\\cal I\\sb1$ can be used in the same way to construct connections\non a noncommutative Hopf fibration, we denote, for the sake of brevity,\n$\\pf \/\\cal I\\sb\\nu$ by \\pn , where $\\nu\\in\\{0,1\\}$. We also put\n$x\\sb{0j}=[x\\sb j]\\sb{\\cal I\\sb0}\\,$, $x\\sb{1j}=[x\\sb j]\\sb{\\cal I\\sb1}\\,$,\n\\mbox{$a\\sb{0j}=a\\sb j$}, $j\\in\\{1,2,3\\}\\, ,\\; \na\\sb{11}=1\\, ,\\, a\\sb{12}=1\\, ,\\, a\\sb{13}=2\\frac{1+q\\sp4}{(1+q\\sp2)\\sp2}\\, ,\\,\nr\\sb0=r\\, ,\\, r\\sb1=1\\,$, and thus define the coefficients \n$\\{a\\sb{\\nu j}\\, ,\\, r\\sb\\nu\\}\\sb{\\nu\\in\\{0,1\\},\\, j\\in\\{1,2,3\\}}\\,$.\nUnless stated otherwise, all the following statements of this example will be\nvalid for any of the two values of $\\nu$. The proposition below allows one\nto turn \\pn\\ into a right $A$-comodule algebra, where \n$A=\\mbox{Map}(\\Bbb Z\\sb2,\\Bbb C)$.\n\n\\bpr\\label{coaction}\nLet \\pf\\ and $A$ be as above. Also, let \\dr\\ be a coaction of $A$ on \\pf\\ \nmaking it a right $A$-comodule algebra. If \n$\\dr\\ci *=(*\\ot\\overline{\\phantom{a}})\\ci\\dr\\, $, where $\\overline{\\phantom{a}}$\ndenotes the complex conjugation, and\n\\beq\\label{co}\n\\dr :\\pf\\ni x\\sb j\\longmapsto x\\sb j\\ot (1-2\\hd)\\in\\pf\\te A\\, ,\n\\eeq\nwhere \\hd\\ is the map such that $\\hd (-1)=1$ and $\\hd (1)=0$, then \n$\\dr (\\cal I\\sb\\nu)\\inc\\cal I\\sb\\nu\\te A$.\n\\epr\n{\\it Proof.} Clearly, $\\dr (x\\sb j-x\\sb j\\sp*)\\in\\cal I\\sb\\nu\\te A\\,$ for any\n$j\\in\\{ 1,2,3\\}$. Also, for any $j,l\\in\\{1,2,3\\}$, we have \n\\[\n\\dr (x\\sb jx\\sb l)=\\dr (x\\sb j)\\dr (x\\sb l)=\nx\\sb jx\\sb l\\ot (1-2\\hd)\\sp2=x\\sb jx\\sb l\\te 1\\, .\n\\]\nHence $\\dr (\\cal I\\sb\\nu)\\inc\\cal I\\sb\\nu\\te A$, as claimed.\n\\epf\\\\\n\nIt follows now that a $*$-algebra homomorphism \n$\\hD\\sb\\nu :\\pn\\ra\\pn\\te A$ given by the formula \n$\\hD\\sb\\nu x\\sb{\\nu j}=x\\sb{\\nu j}\\ot (1-2\\hd)$ makes \\pn\\\na right $A$-comodule algebra.\n\n\\bpr\\label{bundle}\nLet \\pn\\ be a right $A$-comodule algebra as above. Then \n$(\\pn ,A,\\hD\\sb\\nu ,0,0)$ is a quantum principal bundle with the universal\ndifferential calculus.\n\\epr\n{\\it Proof.} By Proposition~\\ref{diagram}, it suffices to show that the\ncanonical map $\\mbox{\\sf T}\\!\\sb B:\\pn\\te\\sb{B\\sb\\nu}\\pn\\ra\\pn\\te A$, where\n$B\\sb\\nu:=\\pn\\sp{co A}$, is bijective. First, however, let us prove the\nfollowing\n\\ble\\label{bundlel}\nLet $B\\sb\\nu$ be as above. Then $B\\sb\\nu$ is the space spanned by monomials\nfrom \\pn\\ whose total degree is even, i.e.\n\\beq\\label{bn}\nB\\sb\\nu =\\left\\{\\sum\\sb{k\\geq 1}\\;\\sum\\sb{i\\sb1,...,i\\sb{2k}}\na\\sb{i\\sb1...i\\sb{2k}}x\\sb{\\nu i\\sb1}\\cdots x\\sb{\\nu i\\sb{2k}}\\in\\pn\\, \n\\mbox{\\Large$|$}\\; \na\\sb{i\\sb1...i\\sb{2k}}\\in\\Bbb C,\\, i\\sb1,...,i\\sb{2k}\\in\\{1,2,3\\}\\right\\}\\, .\n\\eeq\n\\ele\n{\\it Proof.} To simplify notation, let us denote the right hand side of \n(\\ref{bn}) by $\\widetilde{B\\sb\\nu}$. Thanks to (\\ref{co}), it is clear that\nevery element of $\\widetilde{B\\sb\\nu}$ is right coinvariant. It is also clear\nthat every element of \\pn\\ can be written as \n$b\\sb0+\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc\\widetilde{B\\sb\\nu}$. (Observe that any number\n$c\\in\\Bbb C$ can be expressed as \n$c=cr\\sb\\nu\\sp{-2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\\sp2\\in\n\\widetilde{B\\sb\\nu}$.) Furthermore, since\n\n\\[\n\\hD\\sb\\nu p-p\\te 1=b\\sb0\\te 1+\\sum\\sb{j=1}\\sp3\n(b\\sb jx\\sb{\\nu j}\\te 1-2b\\sb jx\\sb{\\nu j}\\te\\hd )-b\\sb0\\te 1-\\sum\\sb{j=1}\\sp3\nb\\sb jx\\sb{\\nu j}\\te 1=-2\\left(\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}\\right)\\te\\hd\\, ,\n\\]\n\nwe can conclude that $\\hD\\sb\\nu p=p\\te 1\\imp p=b\\sb0\\in\\widetilde{B\\sb\\nu}\\,$.\nHence $B\\sb\\nu=\\widetilde{B\\sb\\nu}$, as claimed.\n\\epf\\\\\n\nNow, consider a left \\pn -module map \n$\\widetilde{\\T }\\!\\sb B :\\pn\\te A\\ra\\pn\\te\\sb{B\\sb\\nu}\\pn $ given by the formula:\n\n\\[\n\\widetilde{\\T }\\!\\sb B (1\\te a)\\longmapsto\\left\\{\n\\begin{array}{ll}\n1\\te\\sb{B\\sb\\nu}1 & \\mbox{for $a=1$}\\\\\n\\mbox{$\\frac{1}{2}(1\\te\\sb{B\\sb\\nu}1-\nr\\sb\\nu\\sp{-2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\\ot\\sb{B\\sb\\nu}x\\sb{\\nu i})$} \n& \\mbox{for $a=\\hd $}\\, .\n\\end{array}\n\\right.\n\\]\n\nRecall that every element of \\pn\\ can be written as \n$b\\sb0+\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu$. Therefore, since\n$\\widetilde{\\T }\\!\\sb B\\ci\\T\\!\\sb B$ is a left \\pn -module map, it suffices to\ncheck that \n\\[\n(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)\\llp 1\\ot\\sb{B\\sb\\nu} \n(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp=\n1\\ot\\sb{B\\sb\\nu} (b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\n\\]\nfor arbitrary $\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu$. \nWith the help of Lemma~\\ref{bundlel}, we have\n\\bea\n&&\n(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)\\llp 1\\ot\\sb{B\\sb\\nu} \n(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp\n\\\\ &&\n=b\\sb0(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)(1\\ot\\sb{B\\sb\\nu}1)\n\\mbox{$+\\sum\\sb{j=1}\\sp3$}\nb\\sb j(\\widetilde{\\T }\\!\\sb B\\cc\\T\\!\\sb B)(1\\ot\\sb{B\\sb\\nu}x\\sb{\\nu j})\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb j\n\\widetilde{\\T }\\!\\sb B(x\\sb{\\nu j}\\te 1-2x\\sb{\\nu j}\\te\\hd )\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}\nb\\sb j\\llp x\\sb{\\nu j}\\ot\\sb{B\\sb\\nu} 1\n-2x\\sb{\\nu j}\\widetilde{\\T }\\!\\sb B(1\\te\\hd )\\lrp\n\\\\ &&\n=b\\sb0\\ot\\sb{B\\sb\\nu}1+\\mbox{$\\sum\\sb{j=1}\\sp3$}\n\\llp b\\sb jx\\sb{\\nu j}\\ot\\sb{B\\sb\\nu}1-b\\sb jx\\sb{\\nu j}\\ot\\sb{B\\sb\\nu}1\n+r\\sb\\nu\\sp{-2}\\mbox{$\\sum\\sb{i=1}\\sp3$}(a\\sb{\\nu i}b\\sb jx\\sb{\\nu j}x\\sb{\\nu i}\n\\ot\\sb{B\\sb\\nu}x\\sb{\\nu i})\\lrp\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}\\mbox{$\\sum\\sb{i=1}\\sp3$}\nr\\sb\\nu\\sp{-2}a\\sb{\\nu i}\\ot\\sb{B\\sb\\nu}b\\sb jx\\sb{\\nu j}x\\sb{\\nu i}\\sp2\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j}\n\\mbox{$\\sum\\sb{i=1}\\sp3$}r\\sb\\nu\\sp{-2}a\\sb{\\nu i}x\\sb{\\nu i}\\sp2)\n\\\\ &&\n=1\\ot\\sb{B\\sb\\nu}(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\, .\n\\eea\nThus we have shown that $\\widetilde{\\T }\\!\\sb B\\ci\\T\\!\\sb B=id$.\nFurthermore, it is straightforward to verify that\n$\\T\\!\\sb B\\ci\\widetilde{\\T }\\!\\sb B=id$. Hence $\\widetilde{\\T }\\!\\sb B$ is \nthe inverse of $\\T\\!\\sb B$ and consequently $\\T\\!\\sb B$ is bijective,\nas needed.\n\\hfill{$\\rule{7pt}{7pt}$}\n\n\\bre\\label{finite}\\em\nNote that, since $\\mbox{Map}(\\Bbb Z\\sb2,\\Bbb C)$ is finite dimensional,\nthe injectivity of $\\T\\!\\sb B$ follows immediately from its surjectivity\nand Theorem~1.3 in \\cite{sch2}.\n\\hfill{$\\Diamond$}\\ere\n\n\\bre\\label{equator}\\em\nRecall that a classical point of an algebra $B$ over a field $k$ is defined\nas an algebra homomorphism from $B$ to $k$. For $q\\neq\\pm 1$, the space of all\nclassical points of $P\\!\\sb1$ is parameterized by all pairs $(x,y)\\in\\Bbb R\\sp2$\nsubject to the relation $x\\sp2+y\\sp2=r\\sb1=1$. Any such pair yields an algebra\nhomomorphism $f:P\\!\\sb1\\ra\\Bbb C$ via the formulas \n$f(x\\sb{11})=x$, \\mbox{$f(x\\sb{12})=y$}, $f(x\\sb{13})=0\\,$. It is clear that the \nclassical `subspace' of $P\\!\\sb1$ is precisely its equator. Hence the name \n`equator sphere'. (To see the correspondence between $P\\!\\sb1$ and the \n$C\\sp*$-algebra defined by (7b)\\cite{po}, put $\\mu=q$ and \n\\beq\\label{corr}\nx\\sb{11}=\\frac{i}{2}(B\\sp*-B)\\, ,\\; x\\sb{12}=-\\frac{1}{2}(B\\sp*+B)\\, ,\\;\nx\\sb{13}=\\frac{-1-q\\sp2}{2}A\\, ;\n\\eeq\n cf.~the beginning of Section~7 in \\cite{po}.)\nNow, note that the quantum sphere employed in Section~5.2 of~\\bmq\\ \n($c=0$ in \\cite{po})\ncan be, in the same manner, regarded as a north pole sphere. On the other hand,\nthe quantum principal bundles considered in Proposition~\\ref{bundle} were \nconstructed to generalize the usual Hopf fibration $S\\sp2\\ra RP\\sp2$ \n(set $q=\\pm 1$ in the bundle $(P\\!\\sb1,A,\\hD\\sb1,0,0)$) where\n$\\Bbb Z\\sb2$ moves the points on the sphere to their antipodal counterparts\n(see p.69 in \\cite{tr}). \nOn the north pole sphere used in \\bmq , there is no other classical point \nto which the north pole could be moved under the free action of $\\Bbb Z\\sb2\\,$.\nThis is why, in order to deform the \nHopf fibration $S\\sp2\\ra RP\\sp2$, we used here the equator sphere instead. \n\\footnote{\nI am grateful to Stanis\\l aw Zakrzewski for explaining these things to me.}\n\\hfill{$\\Diamond$}\\ere\n\n\\bpr\\label{sconnection}\nLet $\\pn (B\\sb\\nu ,A)$ be a quantum principal bundle as in \nProposition~\\ref{bundle} and Lemma~\\ref{bundlel}. Also, let\n$\\ho\\in\\mbox{\\em Hom}\\sb{\\Bbb C}(A,\\hO\\sp1\\pn )$ be a homomorphism defined by\nthe formula\n\\[\n\\ho (a)=\\left\\{\n\\begin{array}{ll}\n0 & \\mbox{for $a=1$}\\\\\n\\mbox{$-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3\na\\sb{\\nu i}x\\sb{\\nu i}dx\\sb{\\nu i}$} \n& \\mbox{for $a=\\hd $}\\, .\n\\end{array}\n\\right.\n\\]\nThen \\ho\\ is a strong connection form on $\\pn (B\\sb\\nu ,A)$.\n\\epr\n{\\it Proof.} To prove that \\ho\\ is a connection form it suffices to verify that\n$(\\T\\cc\\ho )\\hd =1\\te\\hd\\ $. (Other conditions of Definition~\\ref{confordef} are\nimmediately satisfied.) We have\n\n\\bea\n(\\T\\cc\\ho )\\,\\hd\n&=&\n-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3\na\\sb{\\nu i}x\\sb{\\nu i}\\T\\!\\sb R(1\\te x\\sb{\\nu i}-x\\sb{\\nu i}\\te 1)\n\\\\ &=&\n-\\frac{1}{2r\\sb\\nu\\sp2}\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu i}\n(x\\sb{\\nu i}\\te 1-2x\\sb{\\nu i}\\te\\hd\\ -x\\sb{\\nu i}\\te 1)\n\\\\ &=&\n1\\ot\\hd\\, . \\phantom{+\\frac{1}{2}\\ot 1}\n\\eea\nHence \\ho\\ is indeed a connection form. Our next step is to show that\n\\ho\\ is strong (see Definition~\\ref{scdef}). With the help of formula\n(\\ref{confor}) and the Leibniz rule, for any \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu\\,$, we have\n\\bea\n(\\hP\\sp\\omega\\cc d)(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\n&=&\n-2\\sum\\sb{j=1}\\sp3b\\sb jx\\sb{\\nu j}\\ho (\\hd )\n\\\\ &=&\nr\\sb\\nu\\sp{-2}\\sum\\sb{j=1}\\sp3\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}b\\sb jx\\sb{\\nu j}\nx\\sb{\\nu i}dx\\sb{\\nu i}\n\\\\ &=&\nr\\sb\\nu\\sp{-2}\\sum\\sb{j=1}\\sp3\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}b\\sb j\n\\llp d(x\\sb{\\nu j}x\\sb{\\nu i}\\sp2)-d(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\lrp\n\\\\ &=&\n\\sum\\sb{j=1}\\sp3b\\sb jdx\\sb{\\nu j}-r\\sb\\nu\\sp{-2}\\!\\!\\!\\sum\\sb{i,j\\in\\{1,2,3\\}}\n\\!\\!\\! a\\sb{\\nu i}b\\sb jd(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\, .\n\\eea\nApplying the Leibniz rule again, we obtain\n\\[\n(id-\\hP\\sp\\omega)\\llp d(b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j})\\lrp\n=\ndb\\sb0+\\sum\\sb{j=1}\\sp3db\\sb j.x\\sb{\\nu j}\n+r\\sb\\nu\\sp{-2}\\!\\!\\!\\sum\\sb{i,j\\in\\{1,2,3\\}}\\!\\!\\!\na\\sb{\\nu i}b\\sb jd(x\\sb{\\nu j}x\\sb{\\nu i}).x\\sb{\\nu i}\\in\\hO\\sb{shor}\\sp1\\pn\\, .\n\\]\nTaking advantage of the fact that any $p\\in\\pn\\ $ can be expressed as\n$b\\sb0+\\mbox{$\\sum\\sb{j=1}\\sp3$}b\\sb jx\\sb{\\nu j}$ for some \n$\\{b\\sb l\\}\\sb{l\\in\\{0,...,3\\}}\\inc B\\sb\\nu\\,$, we can conclude that \\ho\\\nis strong.\n\\epf\\\\\n\n\\bpr\\label{wconnection}\nLet $\\pn (B\\sb\\nu ,A)$ and \\ho\\ be as in the proposition above. A homomorphism\n\\linebreak\\mbox{$\\widetilde{\\ho }\\in\\mbox{\\em Hom}\\sb{\\Bbb C}(A,\\hO\\sp1\\pn )$} \n defined by\nthe formula\n\\[\n\\widetilde{\\ho }(a)=\\left\\{\n\\begin{array}{ll}\n0 & \\mbox{for $a=1$}\\\\\n\\ho (\\hd )+dx\\sb{\\nu l}\\sp2 \n& \\mbox{for $a=\\hd $}\\, ,\n\\end{array}\n\\right.\n\\]\nwhere $l\\in\\{1,2,3\\}$, is a connection 1-form of a connection that is not strong.\n\\epr\n{\\it Proof.} Let $l$ be any fixed element of $\\{1,2,3\\}$. Since \n$\\dsr(dx\\sb{\\nu l}\\sp2)=dx\\sb{\\nu l}\\sp2\\ot 1$ and $\\T(dx\\sb{\\nu l}\\sp2)=0$,\nit is clear that $\\widetilde{\\ho }$ is a connection 1-form. To prove that\n$\\hP\\sp{\\tilde{\\omega}}$ is not a strong connection, we will demonstrate that\n$(id-\\hP\\sp{\\tilde{\\omega}})(dx\\sb{\\nu l})\n\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\, $.\nWith the help of formula (\\ref{confor}), we have:\n\\[\n(id-\\hP\\sp{\\tilde{\\omega}})(dx\\sb{\\nu l})=dx\\sb{\\nu l}+2x\\sb{\\nu l}\n\\widetilde{\\ho }(\\hd )=dx\\sb{\\nu l}+2x\\sb{\\nu l}\\ho (\\hd )\n+2x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2=(id-\\hP\\sp\\omega)(dx\\sb{\\nu l})\n+2x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\n\\]\nTherefore, as $\\hP\\sp\\omega$ is a strong connection, it is enough to show that\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\, $.\nTo this end, let us put \n$B\\sb\\nu\\sp c:=\\left\\{\\sum\\sb{j=1}\\sp3b\\sb j.x\\sb{\\nu j}\\in\\pn\\ |\\; \nb\\sb j\\in B\\sb\\nu\\, ,\\; j\\in\\{1,2,3\\}\\right\\}$. Clearly, \n$\\pn =B\\sb\\nu +B\\sb\\nu\\sp c\\,$. An argument similar to that used in the proof of\nLemma~\\ref{bundlel} shows that $B\\sb\\nu\\cap B\\sb\\nu\\sp c=0$. \nIt follows now that $\\pn =B\\sb\\nu\\oplus B\\sb\\nu\\sp c\\,$. As we consider here\nthe universal calculus on \\pn , we have an isomorphism\n$\n\\psi :\\hO\\sp1\\pn\\ni dp.u\\to [p]\\sb{\\Bbb C}\\ot u\\in\\pn \/\\Bbb C\\ot\\pn\\, .\n$\nFurthermore, we have\n\\begin{eqnarray}\\label{split}\n\\hO\\sp1\\pn \n&=&\n\\psi\\sp{-1}\\llp\\psi (\\hO\\sp1\\pn )\\lrp \n\\nonumber \\\\ &=&\\psi\\sp{-1}\n(B\\sb\\nu \/\\Bbb C\\ot\\pn )\\oplus\\psi\\sp{-1}\n\\llp (B\\sb\\nu\\sp c\\oplus\\Bbb C)\/\\Bbb C\\ot\\pn\\lrp\n\\nonumber \\\\ &=&\ndB\\sb\\nu .\\pn\\oplus d(B\\sb\\nu\\sp c\\oplus\\Bbb C).\\pn \n\\nonumber \\\\ &=&\n\\hO\\sp1B\\sb\\nu .\\pn\\oplus d(B\\sb\\nu\\sp c).\\pn\\, .\n\\end{eqnarray}\nOn the other hand, taking into account the isomorphism\n$\\hO\\sp1\\pn\\cong\\pn\\ot (\\pn \/\\Bbb C)$, one can show \n(with some help of the representation presented in Point~III.(a) of Proposition~4\nin \\cite{po} and formulas (\\ref{corr})) that\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\neq 0$. Therefore, since\n$x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2=dx\\sb{\\nu l}\\sp3-dx\\sb{\\nu l}.x\\sb{\\nu l}\\sp2\n\\in d(B\\sb\\nu\\sp c).\\pn\\,$, we can conclude, by virtue of (\\ref{split}),\nthat $x\\sb{\\nu l}dx\\sb{\\nu l}\\sp2\\;\\; \/\\!\\!\\!\\!\\!\\in\\hO\\sb{shor}\\sp1\\pn\\,$,\n as desired.\n\\epf\\\\\n\n\\bre\\label{vertical}\\em\nThe strong connection form \\ho\\ defined in Proposition~\\ref{sconnection} is\nnon-trivial. Indeed, taking (\\ref{confor}) into account, one can see that\n\\[\n\\hP\\sp{\\omega}(dx\\sb{\\nu l})=-2x\\sb{\\nu l}\\ho (\\hd )=r\\sb\\nu\\sp{-2}\n\\sum\\sb{i=1}\\sp3a\\sb{\\nu i}x\\sb{\\nu l}x\\sb{\\nu i}dx\\sb{\\nu i}\\, ,\n\\]\nwhere, as before, $l$ is any fixed element of $\\{1,2,3\\}$.\nOn the other hand, it can be checked that \n$\\left\\{[x\\sb{\\nu i}]\\sb{\\Bbb C}\\right\\}\\sb{i\\in\\{1,2,3\\}}$\nare linearly independent and that $x\\sb{\\nu l}\\sp2\\neq 0$. \n(Again, one can take advantage of the representation presented in Point~III.(a) \nof Proposition~4 in \\cite{po} and formulas (\\ref{corr}).) Hence, with the\nhelp of an isomorphism $\\hO\\sp1\\pn\\cong\\pn\\ot (\\pn \/\\Bbb C)$, it follows that\n$\\hP\\sp\\omega(dx\\sb{\\nu l})\\neq 0$. Consequently, the space of vertical forms\n(i.e.~Im $\\hP\\sp\\omega$) is non-zero.\n\\hfill{$\\Diamond\\;\\blacklozenge$}\\ere \n\\eex\n\nWe end our display of examples with a strict monoidal category\n(see Section~6.1 in~\\cite{shst}) example of a weak connection.\n\n\\bex\\label{formal}\\em\nSimilarly to Example~\\ref{z4}, \nthis example is concerned with a translation of the concept\nof the canonical connection on a homogeneous space to a different set-up.\nOnly this time, the groups employed are neither\nAbelian, nor finite. The former makes our bundle look more interesting,\nthe latter forces us to replace the algebraic tensor product by the\nappropriate dual of the Cartesian product. More precisely, let\n$\\frak M$ be the image of the category of sets under the contravariant\nfunctor $\\mbox{Map}(\\,\\cdot\\, ,k)$. The tensor product $\\ot\\sb{\\frak M}$\ndefined by\n\\[\n\\mbox{Map}(X,k)\\ot\\sb{\\frak M}\n\\mbox{Map}(Y,k)=\\mbox{Map}(X\\!\\times\\! Y,k)\n\\]\nmakes $\\frak M$ a strict monoidal category. Moreover, if $X$ is a group,\nthen $\\mbox{Map}(X,k)$ is an \\mbox{$\\frak M$-Hopf} algebra, where the\ndefinition of an \\mbox{$\\frak M$-Hopf} algebra is the same as\nthat of a Hopf algebra but with the tensor product taken to be\n$\\ot\\sb{\\frak M}\\,$. In what follows we define several gauge theoretic\nnotions in the setting of the category $\\frak M\\,$:\n\\be\n\\item $\\Omega\\sp 1\\sb{\\frak M}X:=\\{\nF\\!\\in\\!\\mbox{Map}(X\\!\\times\\! X,k)\\, |\\,\n\\forall\\, x\\!\\in\\! X:F(x,x)=0\\}$ (\\mbox{$\\mbox{Map}(X,k)$-bimodule} of\n\\mbox{$\\frak M$-differential} \\mbox{\\mbox{1-forms}})\n\n\\item $\\Delta\\sb{\\frak M-R}:=R\\sp *$ (right coaction), where\n$R:X\\times G\\ra X$ is a right free action of the group $G$ on $X$\n(e.g., see p.55 in~\\cite{tr}; $G$ will denote\na group and $R$ its right free action throughout the rest of this example)\n\n\\item $\\Delta\\sb{\\frak M-{\\cal R}}:{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}}\\ra\\{\n K\\!\\in\\!\\mbox{Map}(X\\!\\times\\! X\n\\!\\times\\! G,k)\\; |\\;\\forall\\; x\\!\\in\\! X,\\, g\\!\\in\\!\n G:K(x,x,g)=0\\}$ (right coaction\non \\mbox{$\\frak M$-differential} \\mbox{\\mbox{1-forms}}), where\n$\\mbox{\\large$\\forall$}\\; F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}}\\lc\\,\nx,y\\!\\in\\! X\\lc\\, g\\!\\in\\! G:\n(\\Delta\\sb{\\frak M-{\\cal R}}F)(x,y,g):=F\\llp R(x,g)\\bc R(y,g)\\lrp$\n\n\\item A triple $\\llp\\mbox{Map}(X,k)\\lc\\mbox{Map}(G,k)\\lc\n\\Delta\\sb{\\frak M-R}\\lrp$ is called an \\mbox{$\\frak M$-principal}\nbundle and, for simplicity, denoted by $(X,G,R)$.\n\n\\item ${\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}} :=\\mbox{\\large$\\{$} F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}} \\;\n|\\;\\mbox{\\large$\\forall$}\\; x\\!\\in\\!\n X\\lc\\, g\\!\\in\\! G: F\\llp R(x,g)\\lc\\, x\\lrp\n=0\\mbox{\\large$\\}$}$ (horizontal \\mbox{$\\frak M$-differential}\n\\mbox{\\mbox{1-forms}})\n\\item $B:=\\mbox{Map}(X\/G,k)$ (base space of $(X,G,R)$)\n\\item ${\\mbox{$\\Omega\\sp 1\\sb{\\frak M-shor}X$}} :=\\mbox{\\large$\\{$} F\\!\\in\\!{\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}}\\;\n|\\;\\mbox{\\large$\\forall$}\\; x,y\\!\\in\\!\nX\\lc\\, g\\!\\in\\! G:\nF\\llp R(x,g)\\lc\\, y\\lrp =F(x,y)\\mbox{\\large$\\}$}$ (strongly\nhorizontal \\mbox{$\\frak M$-differential}\n\\mbox{\\mbox{1-forms}})\n\\item Let $(X,G,R)$ be an \\mbox{$\\frak M$-principal} bundle and let\n\\[\n\\widetilde\\Pi :=\\widetilde\\Pi\\sb\n1\\times\\widetilde\\Pi\\sb 2\n\\in\\mbox{$\\mbox{Map}(X\\!\\times\\! X,X\\!\\times\\!\n X)$}\n\\]\n be an idempotent satisfying\nthe following conditions:\n\\bi\n\\item [a)] $\\mbox{\\large$\\forall$}\\; x,y\\in\n X:\\widetilde\\Pi\\sb 1(x,y)=x$,\n\\item [b)] $\\mbox{\\large$\\forall$}\\; F\\in{\\mbox{$\\Omega\\sp 1\\sb{\\frak M}X$}} :\\;\nF\\ci\\widetilde\\Pi =0\\iff F\\in{\\mbox{$\\Omega\\sp 1\\sb{\\frak M-hor}X$}}$,\n\\item [c)] $\\mbox{\\large$\\forall$}\\; x,y\\!\\in\\! X\\lc\\, g\\!\\in\\!\nG: \\widetilde\\Pi\\sb 2\\llp\nR(x,g)\\,\\lc\\, R(y,g)\\lrp =R\\llp\\widetilde\\Pi\n\\sb 2(x,y)\\,\\lc\\, g\\lrp$.\n\\ei\nThen $\\widetilde\\Pi\\sp *$ is called an\n\\mbox{$\\frak M$-connection}\nand is denoted by $\\Pi\\sb{\\frak M}$.\n\\item An \\mbox{$\\frak M$-connection} $\\Pi\n\\sb{\\frak M}$ is called a strong\n \\mbox{$\\frak M$-connection} iff\n\\[\n\\mbox{\\large\\boldmath$\\forall$}\\,\n x,y\\!\\in\\! X\\lc\\; g\\!\\in\\! G:\\;\n\\widetilde\\Pi\\sb 2\\llp R(x,g)\\,\\lc\\, y\\lrp\\\\\n =\n\\widetilde\\Pi\\sb 2(x,y)\\, .\n\\]\n\\ee\nThe above definitions were constructed so that the \\mbox{$\\frak M$-objects}\nthus defined become the corresponding\n`quantum objects' (the universal differential calculus assumed)\nwhen both $X$ and $G$ are finite and the \\mbox{$\\frak M$-tensor} product\nis the same as the algebraic tensor product. In particular,\none can rethink and equivalently\ndescribe Example~\\ref{z4} in \\mbox{$\\frak M$-terms}.\n Doing so blurs the view of general principles making that example work,\nbut it allows one to have a better insight into its concrete mathematical fabric.\n\nClearly, if $H$ is a subgroup of $G$ acting\non $G$ on the right by the group\nmultiplication (let us denote this action by\n$R\\sb G$), then\n$(G,H,R\\sb G)$ is an \\mbox{$\\frak M$-principal}\n bundle. Furthermore,\nany surjection \\mbox{$j:G\\ra H$} satisfying\n $j\\ci\\imath = id$, where\n\\mbox{$\\imath :H\\hookrightarrow G$} is the\n inclusion, and\n\\beq\\label{jad}\n\\forall\\, g\\in G,h\\in H:\\, j(h\\sp {-1}gh)=h\\sp\n {-1}j(g)h\n\\eeq\nyields an \\mbox{$\\frak M$-connection} on\n$(G,H,R\\sb G)$. Indeed, let\n$\\widetilde\\Pi\\sp j(g,r):=\\llp g,\\,\ngj(g\\sp{-1}r)\\lrp$, for any\n$g,r\\in G$. Then, for all $g,r\\in G\\bc\\,\n h\\in H$, we have:\n\\be\n\\item $(\\widetilde\\Pi\\sp j)\\sp 2(g,r)=\\lblp g,\\,\n gj\\llp g\\sp{-1}gj(g\\sp{-1}r)\\lrp\\lbrp\n=\\llp g,\\, gj(g\\sp{-1}r)\\lrp =\\widetilde\\Pi\\sp j(g,r)$,\nwhere the middle equality is implied by the formula $j\\ci\\imath =id$.\n\n\\item $\\widetilde\\Pi\\sp j\\sb 1(g,r)=g$ (obvious).\n\n\\item $F\\in\\Omega\\sp 1\\sb{\\frak M-hor} G\n\\Rightarrow F\\ci\\widetilde\\Pi\\sb j=0$ (obvious).\n\n\\item For any $F\\in\\mbox{Map}( G\\!\\times\\! G,k)$, the implication\n\\[\nF\\ci\\widetilde\\Pi\\sp j=0\\Rightarrow\nF\\in\\Omega\\sp 1\\sb{\\frak M-hor} G\n\\]\nis a consequence of the fact that\n$\\widetilde\\Pi\\sb j$ restricted to\n\\mbox{$\\bigcup\\sb{g\\in G}(g,gH )$}\ncoincides with the inclusion of\n\\mbox{$\\bigcup\\sb{g\\in G}(g,gH )$} in $ G\\times G$.\n\n\\item $\\widetilde\\Pi\\sp j\\sb 2(gh,rh)=\nghj(h\\sp{-1}g\\sp{-1}rh)=gj(g\\sp{-1}r)h\n=\\widetilde\\Pi\\sp j\\sb 2(g,r)h$, where\nthe middle step follows from\n(\\ref{jad}).\n\\ee\nHence, $(\\widetilde\\Pi\\sp j)\\sp *$ is an \\mbox{$\\frak M$-connection}.\nMoreover, much as in Example~\\ref{z4}, if\n$(\\widetilde\\Pi\\sp j)\\sp *$ is a strong\n\\mbox{$\\frak M$-connection}, then, for all $g\\in G , h\\in H$, $hj(g)=j(hg)$.\n\nNext, we proceed to consider a special case of $G$ and $H$.\nTo do so, first we need a definition of an algebraic formal group:\n\n\\bde [cf.\\ \\cite{ha} and Appendix~A in \\cite{mgtqg}]\\label{afg}\nLet $\\frak g$ be a finite dimensional Lie algebra ($\\dim \\frak g=n$),\nand let $\\{ E\\sp\\nu\\}\\sb{\\nu\\in\\{ 1,\\cdots ,n\\}}$\nbe a basis of $\\frak g$.\nAlso, let $F$ be the formal group law\n(see, e.g., Section~9.1 and Section~1.1 in~\\cite{ha}) given by the\nBaker--Campbell--Hausdorff formula determined by $\\frak g$\nand the basis $\\{ E\\sp\\nu\\}\\sb{\\nu\\in\\{ 1,\\cdots ,n\\}}$\n(see Appendix~A in~\\cite{mgtqg}), so that to every\npair of $n$-tuples of formal power series in a finite number of\nvariables with no free term we can assign another such $n$-tuple, i.e.\n\\bea\n&&\nF\\lblp\n\\llp p\\sb 1(t\\sb 1,\\cdots ,t\\sb l)\\lc\\cdots\\lc p\\sb n(t\\sb 1,\\cdots\n,t\\sb l)\\lrp\\lbc\n\\llp q\\sb 1(s\\sb 1,\\cdots ,s\\sb m)\\lc\\cdots\\lc q\\sb n(s\\sb 1,\\cdots\n,s\\sb m)\\lrp\\lbrp \\\\ &&\n=\\llp r\\sb 1(t\\sb 1,\\cdots ,t\\sb l,s\\sb 1,\\cdots ,s\\sb m)\\lc\\cdots\\lc\nr\\sb n(t\\sb 1,\\cdots ,t\\sb l,s\\sb 1,\\cdots ,s\\sb m)\\lrp\\, .\n\\eea\nSymbolically, we will write \\mbox{\n$\\llp\\, p\\sb 1(t\\sb 1\\, ,\\,\\cdots\\, ,\\, t\\sb l)\\,\\lc\\,\\cdots\\,\\lc\\,\n p\\sb n(t\\sb 1\\, ,\\,\\cdots\\, ,\\, t\\sb l)\\,\\lrp$}\nas $\\exp (p\\sb\\nu E\\sp\\nu )$ and \\linebreak\\mbox{$F\\llp\\exp\n(p\\sb\\nu E\\sp\\nu )\\lc\\exp (q\\sb\\mu E\\sp\\mu )\\lrp$}~as~\\mbox{\n$\\exp (p\\sb\\nu E\\sp\\nu )\\exp (p\\sb\\nu E\\sp\\nu )$}.\n (The Einstein convention of summation over repeating indices is\nassumed here and throughout the rest of this example.)\nThe group generated with the use of $F$ by the $n$-tuples\n$\\{\\exp (t\\sb mX)\\}\\sb{m\\in{\\Bbb N},X\\in\\frak g}\\,$, where\n$\\{ t\\sb m\\}\\sb{m\\in{\\Bbb N}}$\nare formal power series in one variable with all\nbut the linear coefficients vanishing, is called the algebraic formal group\nassociated to $\\frak g$ and is denoted by $E\\sb{\\frak g}$.\n\\ede\n\nNow, let $G={\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$, $H={\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$\nand let \\mbox{$j:{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}\\ra{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$} be the surjection defined by\n$j\\llp\\exp (p\\sb l E\\sp l +p\\sb\\mu E\\sp\\mu)\\lrp = \\exp (p\\sb l E\\sp l)$,\nwhere $\\{E\\sp l\\}\\sb{l\\in\\{ 1,2,3\\}}$ is a fixed basis of ${\\frak s\\frak u}(2)$, \nand $\\{ E\\sp\\mu\\}\\sb{\\mu\\in\\{ 1,2,3\\}}$ is a fixed basis of\n$i{\\frak s\\frak u}(2)$. It is clear that \n$({\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} ,{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}} ,R\\sb{E\\sb{\\frak s\\frak l(2,\\Bbb C)}})$ is an\n\\mbox{$\\frak M$-principal} bundle, and $j\\ci\\imath = id$.\nThus, to see that $j$ induces an\n\\mbox{$\\frak M$-connection}, it suffices to prove the following:\n\n\\ble\\label{lemex}\n$\\;\\mbox{\\large$\\forall$}\\, g\\in {\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} ,\n h\\in {\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\, : \\; j(h\\sp{-1}gh)=h\\sp{-1}j(g)h$\n\\ele\n{\\it Proof.} Since the formal power series\ndetermining elements of ${\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$ are generated by the\nBaker--Campbell--Hausdorff formula, we know that\n\\bea\n\\mbox{\\large$\\forall$}\\, g\\!\\in\\!{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}\\;\\,\\mbox{\\large$\\exists$}\n\\,\\epsilon\\sb g>0,n\\!\\in\\!{\\Bbb N}\\;\\,\\mbox{\\large$\\forall$}\\,\n\\overline{x}\\in\\{ (z\\sb 1,\\cdots ,z\\sb n)\\in\\Bbb R\\sp n\\, |\n\\; z\\sb 1\\sp 2+\\cdots +z\\sb n\\sp 2<\\epsilon\\sb g\\sp 2\\}:&&\n\\\\ \n V\\sb{\\overline{x}}(g):=\\exp\\llp p\\sb l\n(\\overline{x})E\\sp l + p\\sb\\mu (\\overline{x})E\\sp\n\\mu\\lrp\\in SL(2,{\\Bbb C})\\, .&&\n\\eea\nThis means that the formal power series\n$\\{ p_{l},p_{\\mu}\\}\\sb{l,\\mu\\in\\{ 1,2,3\\}}$\ndefining $g$ are convergent when evaluated at\n any $\\overline{x}$ in an $\\epsilon$-neighborhood\nof $0\\in {\\Bbb R}\\sp n$.\nNow, let $g$ be an arbitrary element of ${\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$\nand $h$ be any element of\n${\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$ of the form $h=\\exp (tX)$, where \n\\mbox{$X\\!\\in\\!{\\frak s\\frak u}(2)$} and $t$\nis understood as a formal power series in one variable. Also, let \n$\\epsilon :=\\min\\{\\epsilon\\sb{h\\sp{-1}j(g) h}\\; ,\\;\n\\epsilon\\sb{j(h\\sp{-1} gh)}\\} $,\n $ \\overline{y} :=(\\overline{t} ,\\overline{x})\\in\n{\\Bbb R}\\sp{n+1}$ and $\\widetilde{j}$ be\nthe map defined on all elements of $SL(2,{\\Bbb C})$ of the form\n $\\exp(t\\sb l E\\sp l + t\\sb\\mu E\\sp\\mu)$, \n$\\forall\\, l,\\mu :\\, t\\sb l,t\\sb\\mu\\in {\\Bbb R}, $ \nby the formula\n \\[\n\\widetilde{j} \\llp\\exp (t\\sb l\n E\\sp l+t\\sb\\mu E\\sp \\mu )\\lrp =\\exp\n(t\\sb l E\\sp l)\\, .\n\\]\nThen, using the fact that the splitting ${\\frak s\\frak l}(2,{\\Bbb C}) = \n{\\frak s\\frak u}(2)\\oplus i{\\frak s\\frak u}(2)$ is $ad$-invariant,\nfor every $\\overline{y}$ of length smaller than $\\epsilon$, we have\n\\bea\n&& (V\\sb{\\overline{y}} \\cc j)\\,\\llp\\exp( -tX)\\exp( p\\sb l\n E\\sp l+p\\sb\\mu E\\sp\\mu)\\exp(tX)\\lrp \\\\\n&& = (\\widetilde{j}\\cc V\\sb{\\overline{y}})\\,\n\\llp\\exp(-tX)\\exp(\np\\sb lE\\sp l +p\\sb\\mu E\\sp\\mu)\\exp(tX)\n\\lrp \\\\\n&& =\\widetilde{j}\\lblp\\exp( -\\overline{t}X)\\exp\n\\llp p\\sb l\n (\\overline{x})E\\sp l + p\\sb\\mu(\n\\overline{x})E\\sp\\mu\\lrp\\exp(\\overline{t}X)\\lbrp \\\\\n&& =\\widetilde{j}\\lblp\\exp\\llp p\\sb l\n(\\overline{x})\n\\exp(-\\overline{t}X) E\\sp l\\exp(\\overline{t}X) +\np\\sb\\mu(\\overline{x}) \\exp( -\\overline{t}X)\n E\\sp\\mu\n\\exp(\\overline{t}X)\\lrp\\lbrp \\\\\n&& = \\exp\\llp p\\sb l(\\overline{x})\\exp(-\n\\overline{t}X)\n E\\sp l\\exp(\\overline{t}X)\\lrp \\\\\n&& = \\exp(-\\overline{t}X)\\exp\\llp p\\sb\nl(\\overline{x})E\\sp l\\lrp\\exp(\\overline{t} X) \\\\\n&& = V\\sb{\\overline{y}} \\llp\\exp( -tX)\\exp( p\\sb\nl E\\sp l)\\exp(tX)\\lrp \\\\\n&& = V\\sb{\\overline{y}} \\lblp\\exp( -tX)\n j\\llp\\exp( p\\sb l\nE\\sp l+p\\sb\\mu E\\sp\\mu)\\lrp\\exp(tX)\n\\lbrp .\n\\eea\nThus, the formal power series defining\n$j(h\\sp{-1} gh)$ and $h\\sp{-1}j(g)h$\ncoincide when evaluated on an open\n neighborhood of $0\\in{\\Bbb R}\\sp {n+1}$,\n and hence are identical. To end the proof,\nwe need to note that $ad:\n{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\ra\\mbox{Aut}({\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}} )$ is a\nhomomorphism and, since every element of\n${\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}$ is generated by elements of the form\n$\\exp(tX)$, the formula\n$j(h\\sp{-1}gh) = h\\sp{-1}j(g)h$ is \nvalid for all $g\\in{\\mbox{$E\\sb{\\frak s\\frak l (2,{\\Bbb C})}$}}$ and $h\\in{\\mbox{$E\\sb{\\frak s\\frak u(2)}$}}\\,$, as claimed.\n\\hfill $\\Box$\n\\bigskip\n\nFinally, since $h=\\exp (tY)$, $g=\\exp(sZ)$,\n where $Y:=\\mbox{\\scriptsize $\\pmatrix{0&1\\cr -1&0}$}$, \n$Z:=\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&-1}$}$, do not satisfy\n$j(hg) = hj(g)$, the \\mbox{$j$-induced}\n\\mbox{$\\frak M$-connection} is non-strong, as\ndesired.\\footnote{\nI am very grateful to Philip Ryan \nfor pointing out that $Y,Z$\nprovide the desired counterexample to the formula $j(hg) = hj(g)$.}\n\nIt seems proper to mention at this point\n that it would be interesting to see to what extent gauge theory on\nquantum principal bundles can\nwork in some more interesting categories and whether \nthe monoidal reconstruction (see\n Section~8.2 in \\cite{shst}) can be\nextended to reconstruct bundles and\nconnections.\n\\hfill{$\\blacklozenge$}\\eex\n\n\\section{Gauge Transformations}\n\nThe next natural step is to determine how\nstrong connections behave under quantum gauge transformations. To do so,\nwe must first define gauge transformations of a quantum principal bundle.\nOne can define the group of quantum\ngauge transformations as the group of convolution-invertible elements\nof $\\mbox{Hom}\\sb k(A,P)$ which intertwine\n$\\Delta\\sb R$ with $ad\\sb R$, and satisfy $f(1)=1$. (The same definition is\nconsidered in Proposition~5.2 of~\\cite{tbt}.)\nThen one can define their action on connection forms in a way analogous\n to the action of their classical counterparts on the classical connection\n forms (see \\cite{booss}). Quantum gauge\ntransformations defined in this manner\ngeneralize locally defined quantum gauge transformations\ndiscussed in Section~3 of~\\bmq\\ (cf.\\ Section~3 in~\\cite{md4}).\n\n\\bde\\label{gtrans}\nLet $P(B,A)$ be a quantum principal bundle.\nA $k$-homomorphism \\mbox{$f:A\\ra P$} is called a gauge\ntransformation iff\\vspace*{3mm}\n\n\\hsp{5mm} 1. $f$ is convolution invertible,\n\n\\hsp{5mm} 2. $\\dr\\ci f = (f\\ot id)\\ci ad\\sb R,$\n\n\\hsp{5mm} 3. $f(1)=1$.\n\\ede\n\n\\bpr\\label{gtprop}\nThe set of all gauge transformations of a\nquantum principal bundle\nis a group with respect to convolution.\nWe denote this group by $GT(P)$.\n\\epr\n{\\it Proof.} A routine sigma notation calculation\nverifies the following lemma:\n\n\\ble\\label{adlem}\n$\\,$ Let $\\,\\{ (Q\\sb i,\\rho\\sb i)\\}\\sb{i\\in\\{ 1,2\\} }\\,$,\n$\\,V\\sb{12}\\,$ and $\\,m\\sb{12}\\,$ be as in Lemma~\\ref{r-inv}.\nThen, for all\\linebreak\n\\mbox{$\\mbox{\\em f}\\sb 1\\in\\mbox{\\em Hom}\\sb k(A,Q\\sb 1)$},\n\\mbox{$\\mbox{\\em f}\\sb 2\n\\in\\mbox{\\em Hom}\\sb k(A,Q\\sb 2)$},\n\\[\n\\llp (\\mbox{\\em f}\\sb 1\\te id)\\ci ad\\sb R\\lrp *\n\\llp (\\mbox{\\em f}\\sb 2\\te id)\\ci ad\\sb R\\lrp =\n\\llp (\\mbox{\\em f}\\sb 1\\! *\\!\\mbox{\\em f}\\sb 2)\\ot\nid\\lrp\\ci ad\\sb R\\, .\n\\]\n\\ele\n\nHence, since the map\n\\[\n(\\dr\\ci ):\\mbox{Hom}\\sb k(A,P)\\ni\\mbox{f}\\longmapsto\\dr\\ci\\mbox{f}\\in\n\\mbox{Hom}\\sb k(A,P\\te A)\n\\]\n is an algebra homomorphism (cf.\\ Lemma~4.0.2 in \\cite{swe}),\nthe set of all gauge\n transformations is closed\nunder the convolution. Furthermore,\nit follows from the same reason that\n\\mbox{$\\dr\\ci\\mbox{f}\\,\\sp{-1}=(\\dr\\ci\\mbox{f}\\, )\\sp{-1}$}. Therefore,\nas $f(1)=1$ implies $f\\sp{-1}(1)=1$,\nby putting $\\mbox{f}\\sb 1=f$ and $\\mbox{f}\\sb 2=f\\sp{-1}$\nin Lemma~\\ref{adlem}, we can also conclude the existence of the inverse.\n \\hfill $\\Box$\n\\bigskip\n\nWhen defined in this way,\nquantum gauge transformations are unwilling to preserve the\nproperty $\\omega(M\\sb A) = 0$ (see Definition~\\ref{confordef} and\nProposition~\\ref{gact}) defining a connection\nform $\\omega$ on a bundle with a general differential calculus. This is the case\neven if one assumes that the gauge transformations\nsatisfy an additional condition\n$\\llp(f\\ot f\\sp{-1})\\ci\\de\\lrp (M_{A})\\inc N_{P}\\,$.\n(A related discussion can be found around formula (48)\nin \\cite{bmq}; note that this condition\nis satisfied in the classical situation,\nin which $M_{A}=(\\mbox{Ker}\\,\\epsilon )\\sp{2}$ ---\nsee Example 1 on p.132 in \\cite{wor}.) Therefore,\n when dealing with quantum gauge transformations, we will assume the universal\ndifferential calculus.\n\n\\bpr\\label{gact}\nLet $f\\in GT(P)$ and $\\omega\\in\\cal C(P)$, where $\\cal C(P)$ denotes the space of\nall connection forms on a quantum principal bundle $(P,A,\\dr ,0,0)$. \nThen the formula\n\\[\nG\\!\\sb f\\omega =f*\\omega * f\\sp{-1} + f*(d\\ci f\\sp{-1})\n\\]\n(cf.~(20) in \\cite{wz}) defines an action $G:GT(P)\\times\\cal C(P)\\lra\\cal C(P)\\,$.\n\\epr\n{\\it Proof.} Let us verify first that $G\\!\\sb f\\omega$ is indeed a connection form.\n\\be\n\\item $G\\!\\sb f\\omega (k) = 0$. Obvious.\n\\item Taking into account that \\T\\ is a\nleft $P$-module morphism (see Point~4 in\nDefinition~\\ref{qpbdef}) and remembering that for the universal differential\ncalculus $N\\sb P\\! =\\! 0\\! =\\! M\\sb A$, one can see that\n\n\\bea\n&& (\\T\\ci G\\sb{\\! f}\\omega)(a) \\\\\n&& = f(a\\sb{(1)})(\\T\\cc\\omega)(a\\2)\n(\\Delta\\sb R\\cc f\\sp{-1})(a\\3 ) \\\\\n&& + \\llp(m\\ot id)\\ci (id\\ot\\dr)\\lrp\\llp\nf(a\\sb{(1)})\\ot\n f\\sp{-1}(a\\2) - \\epsilon(a)\\ot 1\\lrp \\\\\n&& = \\lblp f(a\\sb{(1)})\\ot\n\\llp a\\2-\\epsilon(a\\2)\n\\lrp\\lbrp\\llp f\\sp{-1}(a\\4 )\\ot\nS(a\\3 )a\\sb{(5)}\\lrp \\\\\n&& + f(a\\sb{(1)})f\\sp{-1}(a\\3 )\\ot\nS(a\\2)a\\4 - 1\\ot\\epsilon(a) \\\\\n&& = f(a\\sb{(1)}) f\\sp{-1}(a\\3 )\n \\ot\\llp\\epsilon(a\\sb {(2)})\na\\4 - S(a\\2)a\\4 \\lrp \\\\\n&& + f(a\\sb{(1)})f\\sp{-1}(a\\3 )\\ot\nS(a\\2)a\\sb\n{(4)} - 1\\ot\\epsilon(a) \\\\\n&& = 1\\ot\\llp a-\\epsilon(a)\\lrp .\n\\eea\n\\item $\\dsr\\ci G\\sb{\\! f}\\omega = (G\\sb{\\! f}\\omega\\ot id)\\ci ad\\sb R$\ncan be proved much as Proposition~\\ref{gtprop}.\n\\ee\nNow, to complete the proof, it suffices to note that\n$G\\sb{\\! f*g}\\omega = G\\sb{\\! f}G\\sb{\\! g}\\omega$.\n\\hsp{\\fill}$\\Box$\n\\bigskip\n\nThe action of the gauge group $GT(P)$ on the space of connections can be derived\nfrom its action on connection forms. It is explicitly described by\n\n\\bpr\\label{gtpi}\nLet $GT(P)$ be as in Proposition~\\ref{gact}.\nDenote by ${\\cal P}(P)$ the space of\nall connections on $P$, and by $\\Upsilon$ the bijection providing the\ncorrespondence between connections and connection forms, i.e., let\n\\[\n\\Upsilon : {\\cal P}(P)\\ni\\Pi\\longmapsto\\sigma\\sb\\Pi\\ci\\llp 1\\ot\n (id -\\epsilon )\\lrp\\in\\cal C(P)\\, ,\n\\]\nwhere $\\sigma\\sb\\Pi :\nP\\ot\\mbox{\\em Ker}\\,\\epsilon\\ra\\Omega\\sp 1\\! P$ is\nthe unique left \\mbox{$P$-module} homomorphism satisfying\n \\mbox{$\\mbox{\\em\\sf T}\\ci\\sigma\\sb\\Pi =id$} and\n\\mbox{$\\sigma\\sb\\Pi\\ci\\mbox{\\em\\sf T}=\\Pi$}.\n{\\em\\footnote{ \nSee Proposition~4.4 and the paragraph above it in \\bmq .}}\nThen the map\n${\\cal G} : GT(P)\\times{\\cal P}(P)\\lra{\\cal P}(P)$ given by\n\\beq\\label{gpi}\n{\\cal G}\\sb f\\Pi\n= m\\sb{\\Omega\\sp 1P}\\ci\\lblp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\lbrp +\n m\\sb{\\Omega\\sp 1P}\\ci\\llp id\\,\\ot\\, f\\! *\\! (d\\!\n\\ci\\! f\\sp{-1})\\lrp\\ci\\mbox{\\em\\sf T}\n\\eeq\nis the action of the gauge group $GT(P)$ on the space of connections\n${\\cal P}(P)$, and the following diagram commutes:\n\\bcd\nGT(P)\\times{\\cal P}(P) @>(id,\\Upsilon)>> GT(P)\\times\\cal C(P)\\\\\n@V{\\cal G}VV @VGVV\\\\\n{\\cal P}(P) @>\\Upsilon >>\\cal C(P)\\\\\n\\ecd\n\\epr\n{\\it Proof.} Both assertions of the proposition follow from the formula:\n\\beq\\label{gtpifor}\n\\mbox{\\large$\\forall$}\\,\nf\\in GT(P),\\,\\Pi\\in{\\cal P}(P):\\; {\\cal G}\\sb f\\Pi =\n(\\Upsilon\\sp{-1}\\ci G\\!\\sb f\\ci\\Upsilon) (\\Pi )\n\\eeq\nSince both sides of the above equation are left \\mbox{$P$-module}\nhomomorphisms, in order to prove~(\\ref{gtpifor}),\nit suffices to show that,\nfor any $f\\in GT(P)$ and $\\Pi\\in\\cal P(P)$, we have\n\\[\n(\\cal G\\sb f\\Pi )\\ci d =\n\\llp(\\Upsilon\\sp{-1}\\cc G\\!\\sb f\\cc\\Upsilon) (\\Pi )\\lrp\\ci d\\, ,\n\\]\nthat is, \n\\bea\\label{gtpifor2}\n\\hsp{2.25mm} &&\n m\\sb{\\Omega\\sp 1P}\\ci\\Lblp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\Lbrp\\ci\\mbox{\\sf T}\n\\ci d\\hsp{57mm}\n\\mbox{(\\theequation )}\\addtocounter{equation}{1} \\\\\n&&\n = m\\sb{\\Omega\\sp 1P}\\ci\\lblp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\lbrp\\ci d\n\\hsp{29.5mm} \\mbox{(pure gauge terms cancel).}\n\\eea\nTo do so, let us calculate the value of the left hand side of\n (\\ref{gtpifor2})\non arbitrary $p\\in P$:\n\\bea\n&&\n\\LAblp m\\sb{\\Omega\\sp 1P}\\ci\\LAlp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\LArp\\ci\\mbox{\\sf T}\n\\ci d\\LAbrp (p)\n\\\\ &&\n=\\LAblp m\\sb{\\Omega\\sp 1P}\\ci\\LAlp id\\,\\ot\\, f\\! *\\!\\Llp\\sigma\\sb\\Pi\\cc\n\\llp 1\\te (id - \\epsilon )\\lrp\\Lrp\\! *\\! f\\sp{-1}\\LArp\\LAbrp\n(p\\0\\ot p\\sb{(1)} - p\\ot 1)\n\\hsp{11mm} \\mbox{(see (24)\\bmq )}\n\\\\ &&\n=p\\0f(p\\sb{(1)})\\sigma\\sb\\Pi\\llp1\\ot p\\2-\\epsilon (p\\2)\n\\ot 1\\lrp f\\sp{-1}(p\\3 )\n\\\\ &&\n=\\sigma\\sb\\Pi\\llp p\\0f(p\\sb{(1)})\\ot p\\2-\np\\0f(p\\sb{(1)})\\epsilon (p\\2)\n\\ot 1\\lrp f\\sp{-1}(p\\3 )\n\\\\ &&\n=\\llp\\sigma\\sb\\Pi\\ci\\mbox{\\sf T}\\ci d\\ci\n (id\\! *\\sb R\\! f)\\lrp (p\\0)\\,\nf\\sp{-1}(p\\sb{(1)})\n\\hsp{41 mm} (\\mbox{$\\, id\\! *\\sb{\\! R}\\! f$ is right-covariant})\n\\\\ &&\n=\\lblp\\llp\\Pi\\cc d\\cc\n (id\\! *\\sb R\\! f)\\lrp*\\sb Rf\\sp{-1}\\lbrp\\, (p)\n\\\\ &&\n = \\Lblp m\\sb{\\Omega\\sp 1P}\\ci\\Llp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\Lrp\\Lbrp (1\\ot p - p\\ot 1)\n\\\\ &&\n = \\Lblp m\\sb{\\Omega\\sp 1P}\\ci\\Llp id\\,\\ot\\,\\llp\\Pi\\cc d\\cc\n(id\\! *\\sb R\\! f)\n\\lrp\\! *\\sb R\\! f\\sp{-1}\\Lrp\\ci d\\Lbrp (p)\n\\eea\n\nHence, we can conclude that (\\ref{gtpifor2}) is true,\nand the proposition\nfollows. \\hfill{$\\Box$}\n\n\\bre{\\em\nThe left \\mbox{$B$-module} isomorphism $(id*\\sb Rf):P\\lra P$\ncan be regarded as\na quantum version of a gauge transformation \n understood as an appropriate\ndiffeomorphism of the total space of a principal bundle\n(see p.339 in \\cite{booss} and Definition~5.1 in~\\cite{tbt}).\nIn fact, the map $f\\to id*\\sb Rf$ is a group isomorphism\n(see Corollary~5.3 in \\cite{tbt}).\n(Observe that $(id*\\sb Rf\\sp{-1})\\ci\n(id*\\sb Rf)=id=(id*\\sb Rf)\\ci (id*\\sb Rf\\sp{-1}).\\,$)\n\\hfill{$\\Diamond$}}\\ere\n\nHaving defined and described quantum gauge\ntransformations and their\naction on the space of connections and connection forms,\n we can now show that these\ntransformations preserve the strongness of\n a connection, i.e.\n(provided $M\\sb A= 0 =N\\sb P$) their action\nis well-defined on the space\n\\spp\\ of all strong connections on a\nquantum bundle $P$. (Obviously, their action\n is then also well-defined on the space $\\cal S\\cal C(P)$\nof all strong connection forms.)\n\n\\bpr\nLet $P(B,A)$ be a quantum principal\n bundle with the universal differential calculus. Then\n$\\mbox{\\large$\\forall$}\\, f\\in GT(P)$:\n $\\Pi\\in\\spp \\,\\Leftrightarrow\\, \\cal G\\sb f\\Pi \\in\\spp$.\n\\epr\n{\\it Proof.}\nNote first that, for any right $A$-comodule coaction\n\\mbox{$\\rho\\sb{\\cal C}:{\\cal C}\\lra{\\cal C}\\ot A$}, any \\linebreak\n\\mbox{$\\alpha\\sb 1\\in\\mbox{Hom}\\sb k\\llp {\\cal C},\\Omega\n\\sp m(X)\\lrp$}\nand \\mbox{$\\alpha\\sb 2\\in\\mbox{Hom}\\sb k\\llp{\\cal C},\n\\Omega\\sp n(X)\\lrp$} (where $m\\geq 0,\\, n\\geq 0$\nand $\\Omega (X)$ is any differential algebra), there is the familiar\ngraded Leibniz rule:\n\\[\nd\\ci (\\alpha\\sb 1\\! *\\sb{\\rho\\sb{\\cal C}}\\!\\alpha\\sb 2)=(d\\ci\\alpha\\sb\n 1)*\\sb{\\rho\\sb{\\cal C}}\\alpha\\sb 2\n+ (-)\\sp m\\alpha\\sb 1*\\sb{\\rho\\sb{\\cal C}}(d\\ci\\alpha\\sb 2)\n\\]\nNow, using similar calculations as in the proof of Proposition~\\ref{gtpi} \nand the Leibniz rule for the convolution, one can see that,\nfor any $f\\!\\in\\!\n GT(P),\\,\\Pi\\!\\in\\! {\\cal P}(P)$,\n\\bea\n\\hsp{20mm} && (id-{\\cal G}\\sb f\\Pi)\\ci d\n\\hsp{90mm} \\mbox{(see (\\ref{gpi}))}\n\\\\ &&\n= d - \\llp\\Pi\\ci d\\ci (id\\! *\\sb R\\! f)\\lrp *\\sb Rf\\sp{-1}\n- id*\\sb Rf*(d\\cc f\\sp{-1}) \\\\\n\\hsp{14mm} && = \\llp d\\ci (id\\! *\\sb R\\! f)\\lrp\\, *\\sb R\\, f\\sp{-1}\n- \\llp\\Pi\\ci d\\ci (id\\! *\\sb R\\! f)\\lrp *\\sb Rf\\sp{-1}\\\\\n\\hsp{14mm} && = \\llp (id-\\Pi )\\ci d\\ci\n(id\\!*\\sb R\\! f)\\lrp\\, *\\sb R\\, f\\sp{-1} \\, .\n\\eea\nHence, $\\mbox{\\large$\\forall$}\\,\nf\\!\\in\\! GT(P):\\,\\Pi\\!\\in\\!\\spp \\Rightarrow {\\cal G}\\sb\nf\\Pi\\!\\in\\!\\spp $, and since ${\\cal G}$ is a\ngroup action on ${\\cal P}(P)$, $\\mbox{\\large$\\forall$}\nf\\!\\in\\! GT(P):\\, {\\cal G}\\sb f\\Pi\\!\\in\\!\\spp\n\\Rightarrow \\Pi\\!\n\\in\\!\\spp $. \\hfill $\\Box$\n\\bigskip\n\n\\section{Curvature}\n\nWe are ready now to examine the properties of the exterior\n covariant derivative (see Definition~\\ref{Ddef})\nassociated with a strong connection.\n This will lead to the definition of a (global)\n curvature form on a quantum principal bundle. \nThe following proposition and corollaries describe\nthe composition of the covariant exterior derivative with strongly\ntensorial differential forms\n(see Definition~\\ref{tfdef}). They and Proposition~\\ref{curadj} are analogous\nto the corresponding local (i.e.~valid only for trivial quantum bundles)\nstatements made in \\bmq . \nAs we do not know how to characterize all\n differential algebras for which $D$ can be well-defined,\nwe will simply assume here, whenever needed, that $\\Omega(P)$ \nis a right-covariant differential algebra such that $D$ is well-defined. \n\n\\bpr [cf.\\ (17) and (76) in \\bmq ]\\label{Dformula}\nLet $(P,A,\\dr ,N\\sb P,M\\sb A)$ be a quantum principal bundle with a\nconnection form $\\omega$, and let $\\hO (P)$ be a differential algebra\nsuch that\\linebreak \n$\\op = \\hO\\sp1\\! P\/N\\sb P$ and the exterior covariant derivative\n$D\\sp\\omega$ associated to \\ho\\ is well-defined by formula~(\\ref{689}). \nThen, for all \n\\mbox{$\\phi\\in ST\\sb\\rho\\llp V,\\Omega\\sp n(\\! P)\\lrp ,\\, n\\in \\{0\\}\\cup{\\Bbb N}$},\n\\[\n D\\sp\\omega\\ci\\phi =d\\ci\\phi - (-)\\sp n \\phi *\\sb\\rho\\omega\\, .\n\\]\n\\epr\n{\\it Proof.}\nNote that, for any $d\\alpha\\in\\Omega\\sp n(B)$,\n$n\\in{\\Bbb N}$, $p\\in P$, we have\n\\bea\n\\hsp{6mm}&& (d-D\\sp\\omega)(d\\alpha .p)\n\\\\ &&\n=(-)\\sp nd\\alpha dp - (-)\\sp nd\\alpha .(id - \\Pi\\sp\\omega )(dp)\n\\hsp{70mm}\\mbox{(see (\\ref{confor}))}\n\\\\ &&\n=(-)\\sp nd\\alpha .p\\0\\omega(p\\sb{(1)})\n\\\\ &&\n=(-)\\sp n(id*\\sb{\\cal R}\\omega )(d\\alpha .p)\\, .\n\\eea\nOn the other hand, for all $v\\in V$, $\\,\\phi (v)$\ncan be written as a finite sum\n$\\sum\\sb id\\alpha\\sb i.p\\sb i$ for some closed differential forms\n$d\\alpha\\sb i\\in\\Omega\\sp n(B)$ and 0-forms\n$p\\sb i\\in P$. Hence, with the help of Lemma~\\ref{rcovlem}, we obtain\n\\[\n \\llp(d-D\\sp\\omega)\\cc\\phi\\lrp\\, (v)\n=\\llp(-)\\sp n(id*\\sb{\\cal R}\\omega )\\cc\\phi\\lrp\\, (v)\n=(-)\\sp n(\\phi *\\sb\\rho\\omega )(v)\\, ,\n\\]\nand the assertion follows.\n\\hfill{$\\Box$}\n\n\\bco [cf.\\ (7)\\cite{bmq}]\\label{Dcor}\nLet $\\hO (P)$ be as in the proposition above. \nIf $D$ is the exterior covariant derivative\nassociated to a strong connection, then, for every \\nin ,\nwe have $D\\ci\\lblp ST\\sb\\rho\\llp V, \\Omega\\sp n(\\! P)\\lrp\\lbrp\\inc ST\\sb\\rho\\llp\nV,\\Omega\\sp{n+1}(\\! P)\\lrp .$\n\\eco\n{\\it Proof.}\nIn the same way as in the case of the differential envelope, it can be directly\ncalculated that $D$ is always right-covariant (see Appendix~A in \\bmq ), and thus,\nfor all connections, $D$ composed with a pseudotensorial differential\n form is a tensorial differential form, i.e. \n\\[\n\\mbox{\\large\\boldmath$\\forall$}\\, n\\in\\{ 0\\}\\!\\cup\\!{\\Bbb N} :\\;\nD\\ci\\lblp PT\\sb\\rho\\llp V, \\Omega\\sp n(\\! P)\\lrp\\lbrp\\inc T\\sb\\rho\\llp\nV,\\Omega\\sp{n+1}(\\! P)\\lrp\\, .\n\\]\n Hence, to prove the assertion of this corollary,\nit suffices to note that, if $D$ is associated to a strong connection, then,\nas can be seen from the second line of the first calculation in\nProposition~\\ref{Dformula}, $D$ evaluated on a strongly horizontal differential\nform yields a strongly horizontal differential form.\n\\hfill{\\mbox{$\\Box$}}\n\n\\bre\\label{scdefrem}\\em\nWith the help of $D$ one can equivalently define a strong\nconnection as a connection\nwhose exterior covariant derivative maps $P$ into strongly horizontal forms,\ni.e.\\ $D(P)\\inc\\Omega\\sp 1\\sb{shor}(P)$. Obviously, since $D$ can be defined on $P$\nfor any differential calculus, such a definition works on any quantum\nprincipal bundle.\n\\hfill{$\\Diamond$}\\ere\n\n\\bco [cf.\\ Section~3 in \\cite{bmq}]\\label{D2cor}\nLet \\ho\\ be a strong connection form, and let $D\\sp\\omega$ and $\\hO (P)$ be as in \nProposition~\\ref{Dformula}. Then\n\\beq\\label{D2gen}\n\\mbox{\\large$\\forall$}\\,\\phi\\in ST\\sb\\rho\\llp\nV,\\Omega(P)\\lrp\\; :\\; (D\\sp\\omega)\\sp 2\\ci\\phi\n= -\\phi *\\sb\\rho(d\\ci\\omega +\\omega*\\omega).\n\\eeq\n\\eco\n\nFormula (\\ref{D2gen}) is, in particular, true for a classical principal bundle\n$\\widetilde{P}(M,G)$. (A classical principal bundle is a special case of a\nquantum principal bundle when we replace the algebraic tensor product by the\nappropriately completed tensor product.)\nIt is a generalization of the classical formula\n\\beq\\label{D2clas}\nD\\sp 2\\alpha =\n\\varrho\\sp{'}(F)\\wedge\\alpha\\, ,\n\\eeq\n where $\\alpha$ is a differential form on $\\widetilde{P}$\n with values in a finite dimensional vector space $W$, the homomorphism\n\\mbox{$\\varrho\\sp{'} : \\frak g\\ra\\frak g\\frak l(W)$} \nis the Lie algebra representation induced\nby a homomorphism \\mbox{$\\varrho :G \\ra GL(W)$},\nand $F$ is the curvature form of a connection defining $D$ (e.g., \nsee (19) in~\\cite{tr}). More precisely,\n taking $V$ to be the tensor algebra of the dual vector space\n$W\\sp*$, defining $\\rho\\sb R$~by\n\\[\n\\mbox{\\large$\\forall\\,$} v\\in W\\sp*\\, :\\,\n\\rho\\sb R(v)(w,g)=v\\llp\\varrho(g\\sp{-1})w\\lrp\\, ,\n\\]\nputting \n\n\\[\n\\phi\\sb\\alpha (v) : \n\\left\\{\\begin{array}{ll}\n{\\displaystyle\\bigwedge\\sp{\\deg\\alpha}T\\sb\n{\\tilde{p}}\\widetilde{P}\\ni X\\sb{{\\tilde{p}}}\n\\longmapsto\\left\\{\n\\begin{array}{ll}\n(v\\cc\\alpha) (X\\sb{{\\tilde{p}}})\n& \\mbox{for $v\\in W\\sp*$}\\\\\n0 & \\mbox{otherwise}\n\\end{array}\n\\right.} & \\mbox{for $\\deg\\alpha > 0$}\\\\ \n\\phantom{.} & \\phantom{.}\\\\\n{\\displaystyle\\widetilde{P}\\ni\\widetilde{p}\n\\longmapsto(\\check{v}\\cc\\alpha)\n(\\widetilde{p})\\in\\Bbb R}\n & \\mbox{for $\\deg\\alpha = 0\\,$,}\\\\\n\\end{array}\\right.\n\\]\\ \\\\\n\nwhere $\\check{v}$ is the polynomial function on $W$ corresponding to the tensor $v$,\nand remembering that every $\\frak g$-valued\ndifferential form $\\vartheta$ on $\\widetilde{P}$ can be viewed as an\n\\mbox{$\\epsilon_{C\\sp{\\!\\infty\\!}(\\! G\\!)}$-d}erivation\n\\mbox{$\\psi_{\\vartheta} : C\\sp{\\infty}(G)\\ra\\bigwedge\\sp*(\\widetilde{P})$} \ngiven by $\\psi_{\\vartheta}(a)X_{\\tilde{p}} = \\vartheta(X_{\\tilde{p}}) a$\n(or $\\psi_{\\vartheta}(a)\\widetilde{p} =\\vartheta (\\widetilde{p}) a$\n if $\\deg\\alpha = 0$), we can rewrite (\\ref{D2clas}) as\n\\[\nD\\sp 2\\cc\\phi\\sb\\alpha =\n-\\phi\\sb\\alpha\\! *\\sb\\rho\\! F\\, .\n\\]\nNote that, since every vector space\n$W$ is a Lie group and there is a canonical\nisomorphism between $W$ and the Lie algebra of $W$,\nwe could define $\\phi\\sb\\alpha$ in the\nsame way as we define~$\\psi\\sb\\vartheta$. But then we would not have\n$\\phi\\sb\\alpha (1) = \\delta\\sb{0,\\deg\\alpha}$, which we need in Appendix~A\n(see Proposition~\\ref{asssecprop}).\nIn the classical case, we can replace $F$ by $\\; d\\ci\\omega +\n\\omega*\\omega\\;$ or\n$\\; D\\ci\\omega\\;$ in the last formula, but we need to put\n$\\; F=d\\ci\\omega + \\omega *\\omega\\;$ to obtain (\\ref{D2gen}).\nAlso, formula (\\ref{D2clas}) can be considered as a motivating factor in defining\n the curvature of a connection on a projective module \nas the square of a covariant derivative\n(see p.554 in \\cite{conbook}; \ncovariant derivative $\\equiv$ connection on a projective\nmodule). Therefore, it is natural to define the (global)\ncurvature form of a connection $\\Pi\\sp\\omega$\nin the following way:\n\n\\bde \\label{curdef}\nLet $\\omega$ be a connection \\mbox{1-form} on a\nquantum principal bundle. The\ndifferential form \\mbox{$d\\cc\\omega +\n\\omega\\! *\\!\\omega$} is called the\ncurvature form of $\\omega$ and is denoted by\n$F\\sb\\omega\\,$.\n\\ede\n Clearly, if $\\omega$ is a strong connection\n form, then, at least in the case of a trivial\n bundle, for any differential algebra\n$\\Omega(P)$, even if $D$ is not well-defined,\n $F\\sb\\omega$ is horizontal\n(\\mbox{$F\\sb\\omega =\\Phi\\sp{-1}\\!\\! *\\! F\\sb\n \\beta\\! *\\!\\Phi$},\nwhere $\\Phi$ and $\\beta$ are as in\nProposition~\\ref{scprop} and $F\\sb\\beta :=\nd\\cc\\beta +\\beta\\! *\\!\\beta$\nis a local curvature\nform, cf.~(11)\\cite{bmq}). Moreover, unlike the\nexpression $D\\cc\\omega$,\nthe so-defined\ncurvature form has the desired (at least from\nthe point\n of view of Yang--Mills theory) transformation\nproperties, i.e.~we have:\n\\nopagebreak\n\\bpr [cf.\\ (13)\\cite{bmq}, (20)\\cite{wz}]\\label{curadj}\nLet $P(B,A)$ be a quantum principal bundle with\nthe universal differential calculus,\n$\\omega\\in\\cal C(P)$ and $f\\in GT(P)$. Then\n\\[\nF\\sb{G\\!\\sb f\\omega} =\nf*F\\sb\\omega*f\\sp{-1}\\, .\n\\]\n\\epr\n{\\it Proof.} Straightforward.\\hfill{$\\Box$}\n\n\\section{\\mbox{\\boldmath$U\\sb q(2)$}--Yang--Mills Theory on a Free Module}\n\nTo begin with, let us show that it is possible to define an action functional\non quantum bundle connections in such a way that, at least in the case of\na trivial quantum bundle, \nit agrees with the Yang--Mills\n action functional constructed in Section~1 of both \\cite{conri} and \\cite{ri}.\n(Clearly, we assume that the `base space' of a quantum bundle is an algebra over\n which modules are considered in~\\cite{conri,ri}.)\nConsiderations in \\cite{bmc} and Proposition~\\ref{curadj}\n suggest that, if $A$ is a matrix quantum group and $T$ its matrix of generators\n(fundamental representation, cf.\\ p.628 in~\\cite{cmp}),\nit is reasonable to define \nan action on a quantum principal bundle $P(B,A)$ to be\n(compare with the Lagrangian given by (6.65) in \\cite{md2} or (4.1) in \\cite{md4})\n\\beq\\label{ym}\nYM(\\omega) = - \\llp{\\cal T}\\ci Tr\\ci (F\\sb\\omega*F\\sb\\omega)\\lrp(T),\n\\eeq\nwhere \n$Tr$ is the usual matrix trace, and\n\\mbox{${\\cal T}:\\Omega P\\ra k$} is a linear map vanishing on\n$[P,\\Omega P]$. (To ensure that we have an ample supply\nof connections, throughout this section we will\nuse the universal differential calculus.)\nClearly, remembering the property of $T$ that,\nfor any $\\mbox{f}\\sb 1$ and $\\mbox{f}\\sb 2\\,$,\n$\n(\\mbox{f}\\sb 1\\! *\\! \\mbox{f}\\sb 2)(T\\sb{ab})=\\sum\\sb c\n\\mbox{f}\\sb 1(T\\sb{ac})\\, \\mbox{f}\\sb 2(T\\sb{cb})\\, ,\n$\none can see that\n\\[\n\\mbox{\\large$\\forall$}\\,\nf\\!\\in\\! GT(P) , \\omega\\!\\in\\!\\cal C(P) :\\,\nYM(G\\!\\sb f\\omega) = YM (\\omega)\\, .\n\\]\nSimilarly, if $P(B,A)$\nis a trivial bundle with a trivialization\n$\\Phi$ (which we will also assume for\nthe rest of this section), then\nfor any $\\omega\\,$,\n\\beq\\label{ymbeta}\nYM(\\omega) = -\\llp {\\cal T}\\ci\nTr\\ci(\\Phi*F\\sb\\omega*F\\sb\\omega\n*\\Phi\\sp{-1})\\lrp(T) =\n-\\llp {\\cal T}\\ci Tr\\ci\n(F\\sb\\beta*F\\sb\\beta)\\lrp(T) =:YM(\\beta )\\, ,\n\\eeq\n where $\\beta$ is given by formula (\\ref{betadef})\n(see Remark~\\ref{secrem}) and $F\\sb\\beta $\nis the curvature form associated to it (see the end of the previous section).\nAs to \\ct , observe that one should expect to have a lot of information\nvested in it: projection from the universal to a non-universal calculus and\nmetric (Hodge star). In what follows, we specify \\ct\\ in such a way as to\nincorporate the Yang--Mills functional presented in \\cite{conri,ri} into the quantum\nbundle framework. One ought to bear in mind that to obtain a $q$-deformed\nYang--Mills theory in the spirit of quantum groups, one should invent another\n\\ct\\ or change the formula (\\ref{ym}) altogether. Here, however, we investigate\nwhat effects can entail from the deformation of the structure group alone.\n\n\\bde\\label{tld}\nLet \\ctb\\ be a faithful invariant trace on $B$ (as in \\cite{conri,ri}),\n$\\cal L$ be a finite dimensional Lie subalgebra of $\\,\\mbox{\\em Der}(B)$, \n$\\{ X\\sb l\\}\\sb{l\\in\\{1,...,\\mbox{\\em\\scriptsize dim}\\cal L\\}}$ be its basis, and\n$\\widetilde{X}\\sb l=X\\sb l\\ot id$,\\linebreak\n $l\\in\\{1,...\\, ,\\mbox{\\em dim}\\,\\cal L\\}$. \nWe put\n\\[\n\\ct\\ (\\alpha )=\n\\left\\{\n\\begin{array}{ll}\n\\hsp{3mm}(\\ctb\\te\\epsilon)\\; (\\alpha) & \\mbox{if $\\deg\\alpha =0$}\\\\\n\\phantom{1} & \\phantom{1}\\\\\n\\hsp{3mm}\\displaystyle 0 & \\mbox{if $\\deg\\alpha =2m-1$ }\\\\\n\\phantom{1} & \\phantom{1}\\\\\n{\\displaystyle\\sum\\sb{l\\sb 1<\\cdots 0$.}}\n\\[\\Delta T =T\\buildrel .\\over\\ot T,\\;\\;\n\\epsilon (T) =I,\\;\\; S(T)=T\\sp\\dagger ,\\]\nwhere\n\\[\nT=\\pmatrix{a&b\\cr -q\\sp{-1}b\\sp*t\\sp* &\n a\\sp*t\\sp*}\\,\n \\]\nand $\\stackrel{.}{\\ot}$ is the matrix\nmultiplication with the product of its\nentries replaced by the tensor product $\\ot\\,$;\nand\n\\[\n\\Delta t =t\\ot t , \\;\\; \\epsilon(t) =1\n,\\;\\; S(t)=t\\sp*,\\;\\; S(t\\sp*) =t.\n\\]\n\\ede\n\nThe next step is to establish an equivalence between a certain\nclass of strong connections on a trivial $U\\sb q(2)$-bundle\nand the space of hermitian connections on the associated free module.\nSince any $\\beta\\in\n\\mbox{Hom}\\sb{{\\Bbb C}}(U\\sb q(2),\\Omega B)$ satisfying\n$\\beta({\\Bbb C})=0$ is\na connection form on the base space $B$ of a\n trivial $U\\sb q(2)$ bundle (see Remark~\\ref{secrem}\n and the paragraph below) and the\n$\\beta$-induced connection $\\nabla$ on $B\\sp 2$\n depends only on $\\beta(T)$, we will\nrestrict our attention to connection forms\nthat are, in some way, uniquely\ndetermined by $\\beta(T)$. In the classical situation, $\\beta$ is an\n\\mbox{$\\epsilon$-derivation} and thus is automatically\ndetermined by $\\beta (T)$.\nMimicking this classical differential geometry formula for $\\beta$,\nwe define an auxiliary map \n\\mbox{$\n\\widetilde\\hb :\\Bbb C\\langle a,b,t,a\\sp*, b\\sp*,t\\sp*,1\\rangle\n\\ra\\hO\\sp1\\! B\n$}\nby\\\\\n\n\\begin{eqnarray}\\label{betafor}\n&& \n\\hsp{-2mm}\\widetilde{\\beta}(\na\\sp{k\\sb1}a\\sp{*\\, l\\sb1}b\\sp{m\\sb1}b\\sp{*\\, n\\sb1}t\\sp{p\\sb1}t\\sp{*\\, r\\sb1}\n\\cdots \na\\sp{k\\sb s}a\\sp{*\\, l\\sb s}b\\sp{m\\sb s}b\\sp{*\\, n\\sb s}t\\sp{p\\sb s}t\\sp{*\\, r\\sb s})\n\\nonumber \\\\ && \\ \\nonumber \\\\ &&= \n\\left\\{\\begin{array}{ll}\n0 &\\mbox{for $\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i\\geq 2$ }\n\\\\ \\ & \\ \\\\\nq\\sp{\\sum\\sb{i=j+1}\\sp s(l\\sb i-k\\sb i)}\\;\\widetilde{b} \n& \\mbox{for $m\\sb j=1$, $j\\in\\{ 1,...,s\\}$, \n$\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0$\\phantom{........}}\n\\\\ \\ & \\ \\\\\nq\\sp{\\sum\\sb{i=j+1}\\sp s(l\\sb i-k\\sb i)}\\;\\widetilde{b}\\,\\sp* \n&\\mbox{for $n\\sb j=1$, $j\\in\\{ 1,...,s\\}$, \n$\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0$ }\n\\\\ \\ & \\ \\\\\n{\\displaystyle\\sum\\sb{i=1}\\sp sk\\sb i\\widetilde{a}\n+\\sum\\sb{i=1}\\sp sl\\sb i\\widetilde{a}\\,\\sp*\n+\\sum\\sb{i=1}\\sp sp\\sb i\\widetilde{t} \n+\\sum\\sb{i=1}\\sp sr\\sb i\\widetilde{t}\\,\\sp*}\n& \\mbox{for $\\displaystyle\\sum\\sb{i=1}\\sp sm\\sb i+n\\sb i=0\\, ,$}\n\\end{array} \\right .\n\\end{eqnarray}\\ \\\\\n\nwhere, a priori, $\\widetilde{a}\\, ,\\,\\widetilde{b}$ and $\\widetilde{t}$\nare any differential forms in $\\Omega\\sp 1\\! B$. \nOne can verify that\\linebreak\n \\mbox{$\\widetilde\\hb (\\cal J+\\cal J\\sp*)=0$} if\n$\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\n\\widetilde{t}+\\widetilde{t}\\,\\sp*$. Hence, since\n\\[\n\\widetilde\\beta \n(aa\\sp*+bb\\sp*-1)=0=\\widetilde\\beta (tt\\sp*-1)\\;\\;\\mbox{\\boldmath$\\Rightarrow$}\n\\;\\;\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\\widetilde{t}+\\widetilde{t}\\,\\sp*\\, ,\n\\]\nwe can conclude that (\\ref{betafor})\ndefines $\\hb\\in\\mbox{Hom}\\sb{\\Bbb C}(U\\sb q(2),\\hO\\sp1\\! B)$ such that\n$\\hb (1)=0$\n(i.e., a \\mbox{$U\\sb q(2)$-connection} form) if and only if\n$\\widetilde{a}+\\widetilde{a}\\,\\sp*=0=\\widetilde{t}+\\widetilde{t}\\,\\sp*$.\nMoreover, it is straightforward to check that\n\\beq\\label{dag}\n\\beta(T)\\sp\\dagger\\pmatrix{1&0\\cr 0&q}+\\pmatrix{1&0\\cr 0&q}\n\\beta(T) =0\\, ,\n\\eeq\n and conversely, that for every $M\\in M\\sb 2\\llp\\Omega\\sp1\\! B\\lrp$ satisfying \n(\\ref{dag}) we can find unique\n$\\widetilde{a}$, $\\widetilde{b}$, $\\widetilde{t}$ such that $\\beta (T) = M$\n(cf.~Proposition~1 in~\\cite{av}).\nThus we have proved the following:\n\n\\bpr\\label{hermitian}\nLet \n\\mbox{$\\cal C\\sb B=\\{\\hb\\!\\in\\!\\mbox{\\em Hom}\\sb{\\Bbb C}(U\\sb q(2),\\hO\\sp1\\! B)\n\\, |\\;\n\\hb\\ \\mbox{satisfies (\\ref{beta2}) for some}\\;\\widetilde{a},\n\\widetilde{b},\\widetilde{t}\\in\\hO\\sp1\\! B\\}$} and \n$\\cal C\\cal C(B\\sp2)$ denote the space of Hermitian connections on $B\\sp2$\n(see Point~1 and Point~3 of Definition~\\ref{pmdef}), where\n\\beq\\label{beta2}\n\\beta(a\\sp ka\\sp{*l}b\\sp m b\\sp{*n}t\\sp\npt\\sp{*r}):= \\left\\{\\begin{array}{ll}\n0 &\\mbox{for $m+n\\geq 2$ }\\\\\n\\widetilde{b} & \\mbox{for $m=1$, $n=0$}\\\\\n\\widetilde{b}\\,\\sp* &\\mbox{for $m=0$, $n=1$\n }\\\\\n(k-l)\\widetilde{a}+(p-r)\\widetilde{t}\n & \\mbox{for $m=n=0\\, ,$}\n\\end{array} \\right .\n\\eeq\nand $\\langle\\; ,\\;\\rangle$ is\ngiven by the formula $\\langle\\xi , \\zeta\\rangle =\\xi\\sp\\dagger \n\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}\\zeta$.\nThe map \n\\[\n\\psi :\\cal C\\sb B\\ni\\hb\\longmapsto d+\\hb (T)\\in\\cal C\\cal C(B\\sp2)\n\\]\nis a bijection.\n\\epr\n\n\\bco\\label{hercor}\nLet $P(B,U\\sb q(2))$ be a trivial quantum principal bundle with the\nuniversal differential calculus. There exists a one-to-one correspondence\nbetween the elements of $\\cal C\\cal C(B\\sp2)$, i.e.~the universal\ncalculus connections on $B\\sp2$ that are compatible with the Hermitian\nmetric given by the matrix $\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}$, \nand the strong connections\non $P(B,U\\sb q(2))$ whose `pullback' on the base space $B$ (given\nby (\\ref{betadef})) satisfies (\\ref{beta2}).\n\\eco\n\nThis way, we obtain the Yang--Mills theory of connections compatible with the\n\\mbox{$q$-d}ependent Hermitian structure on $B\\sp 2$. Now, in order to handle\nthe critical points of $YM$ the same way they were dealt with in \\cite{ri},\nlet us assume (as in \\cite{ri}, p.535--6) that $B$ is a smooth dense \n$*$-subalgebra of a $C\\sp*$-algebra, and that it is equipped with a faithful\ninvariant trace~\\ctb . As was argued in Section~1 of\n\\cite{conri}, the Yang--Mills functional\ndoes not depend on the choice of\na Hermitian metric --- we can gauge $q$ out of the picture. \nHence, the critical points of the \\mbox{$U\\sb q(2)$--Yang--Mills} action\nfunctional are simply the critical points\nof the Yang--Mills action (see p.536 in~\\cite{ri}), if there are any,\n`rotated' by the appropriate $q$-dependent gauge\ntransformation (see Section~1 in~\\cite{conri}). \n(More explicitly, as the Hermitian metric is given by\n$\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&q}$}$, \nthe corresponding gauge transformation is\n$\\nabla\\to\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&\\sqrt{q}}$}\\sp{-1}\n\\nabla\\mbox{\\scriptsize $\\pmatrix{1&0\\cr 0&\\sqrt{q}}$}$.)\nConsequently, we have:\n\n\\bco\\label{moduli}\nIn the above described setting, the $U\\sb q(2)$\nand $U(2)$--Yang--Mills theories have\n the same moduli spaces of critical points (cf.\\ Section~4 in\n\\cite{ri} and p.582 in~\\cite{conbook}). \n\\eco\n\nAnother way to remove $q$\nfrom the picture is to alter (\\ref{beta2}) by replacing $\\widetilde{b}\\,\\sp*$ by\n$q\\widetilde{b}\\,\\sp*$. Then, however, we would lose the\ngeometrical interpretation of the\n action of the $\\mbox{$q$-deformation}$ of $U(2)$\non the space of\ncompatible connections as the action of\n the gauge transformation. Also, the formula\n$\\beta(T\\sp*) = \\beta(T)\\sp* $ \nwould no longer be\ntrue. (Caution: at least in the general case, we cannot\nclaim that $\\beta$ is a \\mbox{$*$-morphism} even when it commutes\nwith the $*$ on the generators.) In any case, we can see\n that the Yang--Mills theory remains unchanged\nfor any $q\\in{\\Bbb R}\\sb +$. A~similar\n situation was discussed in the context of\nquantum group gauge theories on classical\n spaces in the last two paragraphs of\nSection~2 in \\cite{bmc}.\n\n\\bre\\label{sl}\\em\nThe reason for employing in the considerations\nabove $U\\sb q(2)$ rather than \n$SU\\sb q(2)$ is that, when using $SU\\sb q(2)$, formula\n(\\ref{beta2}) entails\n$Tr\\,\\beta (T) = 0$ for all $\\beta$, and although the tracelessness of\n $\\beta (T)$ is automatically\npreserved by the entire $GL(B\\sp2)$ in the\nclassical case, we cannot claim the\nsame in general (cf.~Introduction and Section~3 in~\\cite{arar}). \nNor can we claim that the\ntracelessness of $\\beta (T)$ is preserved by the \ngauge group $U(B\\sp2)$ of unitary automorphisms of $B\\sp 2$\n(see Remark~\\ref{ua}). Besides, $U\\sb q(2)$-connections\ngiven by (\\ref{beta2}) can be regarded as $\\langle \\; ,\\;\\rangle$-compatible\n connections, and vice versa, which makes a clear analogy with\nthe classical situation, where Hermitian metrics are $U(2)$-structures\n(cf.\\ p.13 in \\cite{tr}), and $U(2)$-connections are automatically compatible\n with the corresponding Hermitian structures (cf.~p.94--5 in~\\cite{tr}).\n\\hfill{$\\Diamond$}\\ere\n\n Finally, let us mention that\nan alternative approach would be to work with a non-universal differential\ncalculus instead of assuming (\\ref{betafor}), which is put in\nthe theory by hand. But that is yet another story.\n\\bigskip\n\\bigskip\n\n{\\parindent=0pt\n{\\bf\\Large Appendix}\\vspace*{-6pt}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n Physics of QCD jets is one of the most important testing grounds\nfor the theory of strong interactions. The perturbative evolution of a\nquark-gluon jet is at present well understood (see, e.g., \\cite{BPQ}). The known\ndetails include in particular the famous colour coherence phenomenon which\nleads to a probabilistic picture of timelike jet evolution where the daughter\npartons are emitted into a gradually shrinking cones. The nonperturbative\naspects of a jet evolution are much less understood. From the experience\nof a sum rule approach applied to the analysis of heavy resonance properties\n \\cite{SVZ} it is clear, that when the virtuality of a daughter parton is\n of order of\n1 {\\mbox {GeV}} one has to take into account the nonperturbative corrections\ndue to the nonperturbative vacuum fields giving rise to the QCD vacuum\ncondensates.\nAnother situation in which one deals with dissipative effects is a jet\npropagation in, e.g., nuclear medium. Although physically these situations\nare very different, we expect the corresponding theoretical formalism\nto be the same.\n\n In both cases the new interaction vertices lead to the appearance of new\ndimensionful parameters. This means in turn that a scaling description\nof jet evolution where the interaction vertices are determined only by\ndimensionless quantities (energy or virtuality ratios) is no longer valid.\nPhysically what happens is a beginning of a string formation taking energy\nfrom the perturbative component and converting it to nonperturbative degrees\nof freedom. The first model description of a jet evolution taking into account\nthe nonperturbative energy loss was proposed long ago by Dremin \\cite{Dr}.\n It was\nbased on the analogy with the physics oh high energy electromagnetic showers\nin the medium, where apart from a scale-invariant evolution due to a photon\nbremmstrahlung and pair creation there appears a scale-noninvariant\nenergy dissipation due to the ionization of the atoms in the medium \\cite{Be}.\nLater the corresponding modified evolution equations were analytically solved\nin \\cite{DL1}, where the expressions for the parton multiplicities in quark and\n gluon\njets and the energy loss by the perturbative component were computed.\nA discussion of this approach and also of the latest Monte-Carlo calculations,\n\\cite{GE} where the nonperturbative component was described by QCD effective\nlagrangeans can be found in the recent review \\cite{DL2}. A serious drawback\n of Dremin equations \\cite{Dr} is, however, a disregard of the kinematical\nand color interference effects in the jet evolution which are known to be\nof crucial importance for describing the jet characteristics. The main goal of\n the present paper is to introduce a formalism\nproviding a possibility of an analytical treatment of the jet evolution\ntaking into account both color interference and nonperturbative energy loss.\n\nBefore writing down the modified evolution equations let us recall, that color\ncoherence in QCD jets can be described using various approximations. The simplest\nis a leading logarithmic one (DLA, see \\cite{BPQ}). The calculations in this\napproximation are relatively simple, but the energy of the jet is not conserved.\nThis is due to the eikonal description used in this approximation, when the\nenergy loss of the projectile is considered to be negligible. A more refined\nanalysis is made within a modified leading logarithmic approximation (MLLA),\nwhere the total energy is conserved. The Dremin equations \\cite{Dr}\ncorrespond to a case, where the energy is perturbatively conserved. Thus to\ncalculate an absolute value of the dissipative energy loss one has to modify\nan MLLA formalism. The corresponding equations can be written (see Appendix A),\nbut unfortunately we were not able to find their solutions. Thus in what\nfollows we shall concentrate on a simpler DLA case, where the energy\nnonconservation is built in already at the level of perturbation theory.\nIn this case one can not calculate the absolute energy loss, but the relative\none is fairly well-defined.\n\n\n\\section{DLA Equation for the Generating Functional}\n\nLet us turn to a description of a formalism which generalizes the known\neffective methods of describing the QCD jet evolution including color\ncoherence effect \\cite{BPQ} by providing a possibility of considering\nthe new possible nonperturbative sources of energy dissipation.\n\nTechnically it is most convenient to use a generating functional.\n$G(\\{u\\})$ from which one can calculate both an exclusive cross section\n\\begin{equation}\nd\\sigma^{excl}_N=\\left(\\prod^N_{i=1}\\,d^3k_i\\,\\frac{\\delta}{\\delta u(k_i)}\n\\right)\\,G(\\{u\\})\\Big|_{u=0}d\\sigma_0.\n\\end{equation}\nand an inclusive one\n\\begin{equation}\nd\\sigma^{incl}_N=\\left(\\prod^N_{i=1}\\,d^3k_i\\,\\frac{\\delta}{\\delta u(k_i)}\n\\right)\\,G(\\{u\\})\\Big|_{u=1}d\\sigma_0.\n\\end{equation}\nIn the following we shall use the notations\n$dK=\\gamma_0^2\\frac{dk}{k}\\,\\frac{d^2k_\\perp}{2\\pi k^2_\\perp}$ for the\nusual DLA integration measure and $\\Gamma(p,\\theta)$ for the possible gluon\nemission phase space domain, where $p$ is an initial gluon energy and\n$\\theta$ is an opening angle of a jet. Let us remind, that in DLA\napproximation the generating functional $G(p,\\theta;\\{u\\})$ satisfies\nthe following master equation\n\n\\begin{flushleft}\n$G(p,\\theta;\\{u\\}) = u(p)e^{-w(p,\\theta)}+$\n\\end{flushleft}\n\\begin{equation}\\label{Master}\n\\int_{\\Gamma(p,\\theta)} {dk \\over k}\n{d^2 k_{\\perp} \\over {{k}_{\\perp}}^2} {2 C_V \\alpha_s \\over \\pi}\ne^{-w(p,\\theta)+w(p,\\theta_k)} G(p,\\theta_k;\\{u\\}) G(k,\\theta_k;\\{u\\}) ,\n\\end{equation}\nwhere\n$$\n w(p,\\theta)= \\int_{\\Gamma(p,\\theta)} dK\n$$\nis a total probability of an emission of a gluon having an energy $p$\ninto the cone having an opening angle $\\theta$, and $\\exp (-w(p,\\theta))$\nis a formfactor giving a probability of a non-emission of the same gluon.\nThen a sandard procedure (\\cite{BPQ}) leads to the following equation\non the generating functional:\n\\begin{equation}\\label{Gold}\nG(p,\\theta;\\{u\\})=u(p)\\,exp\\left(\\fspace{p}\\,[G(k,\\theta_k;\\{u\\})-1]\\right).\n\\end{equation}\n\nLet us now discuss the possible nonperturbative modifications of the equations\nfor the generating functional. The aim is to describe a partial convertion\nof the perturbative degrees of freedom (gluons) and their energy\ninto the nonperturbative\nones by the process presumably analogous to the ionization in the case of\nelectromagnetic showers discussed in the Introduction. Physically the\npicture for the jet is that the dissipation pumps some energy from the\ncone corresponding to the perturbative jet evolution reducing the gluon\nmultiplicity and energy inside it. Thus we have some additional nonperturbative\nvertex coupling the perturbative gluons with the nonperturbative vacuum or\nnuclear medium.\nThe presence of such a coupling can be related to reducing a probability of an\nemission of a perturbative gluon by taking away a certain portion of its energy\nbefore the perturbative emission. Techically it is convenient to introduce a\nfollowing modification of the gluon emission probability:\n\\begin{equation}\n w(p,\\theta;\\beta)= \\int_{\\Gamma(p-\\beta,\\theta)} dK,\n\\end{equation}\nwhere the constant $\\beta$ describes the additional damping of a perturbative\n gluon emission due to a dissipative interaction with the nonperturbative\nvacuum fluctuations or nuclear medium introduced by compressing a possible\n phase space domain\nfor the gluon emission from $\\Gamma(p,\\theta)$ to $\\Gamma(p-\\beta,\\theta)$.\n The corresponding modification of the master equation\n(\\ref{Master}) reads\n\\begin{flushleft}\n$G(p,\\theta;\\{u\\}) = u(p)e^{-w(p,\\theta;\\beta)}+$\n\\end{flushleft}\n\\begin{equation}\n\\int_{\\Gamma(p-\\beta,\\theta)} {dk \\over k}\n{d^2 k_{\\perp} \\over {{k}_{\\perp}}^2} {2 C_V \\alpha_s \\over \\pi}\ne^{-w(p,\\theta;\\beta)+w(p,\\theta_k;\\beta)}\nG(p,\\theta_k;\\{u\\}) G(k,\\theta_k;\\{u\\}) ,\n\\end{equation}\nThen instead of the Eq.~(\\ref{Gold}) we get\n\\begin{equation}\\label{Gnew}\nG(p,\\theta;\\{u\\})=u(p)\\,exp\\left(\\fspace{p-\\beta}\\,[G(k,\\theta_k;\\{u\\})-1]\n\\right),\n\\end{equation}\nwhere the initial conditions for this equation read $G(p,\\theta;\\{u\\})|_{u=1}=1$.\n\nWe suggest that $\\beta \\ll p$ for all physically meaningful values of energy\nand in the following we shall consider the terms of the first order in the\n small parameter $\\varepsilon=\\beta\/p$. In Appendix B we illustrate\nthe thus arising perturbation theory by deriving the differential equation on\n the\nleading correction to the generating functional.\n\n\\section{Mean multiplicity}\n\nLet us begin this section by looking at the simplest jet characteristic, a\nmean multiplicity of partons in a jet (in the following we are considering only\ngluons). Following the standard steps we obtain a following integral equation\non the mean multiplicity ${\\bar{n}}(p,\\theta)$:\n\\begin{equation}\n{\\bar{n}}(p,\\theta)=1+\\fspace{p-\\beta}\\,{\\bar{n}}(k,\\theta_k).\n\\end{equation}\\label{phas}\nIn the following it will be convenient to rewrite the integration over the phase\nspace as\n\\begin{equation}\n\\fspace{p-\\beta}=\\int_{\\Gamma(p-\\beta,\\theta)}\\frac{dk}{k}\\,\n\\frac{d^2k_\\perp}{2\\pi k^2_\\perp}\\gamma_0^2=\n\\int\\limits^{y-\\varepsilon}_0 d\\xi\\,\\int\\limits_0^\\xi dy'\\gamma_0^2,\n\\end{equation}\nwhere\n$$\ny=\\ln\\frac{p\\theta}{Q_0},\ny'=\\ln\\frac{k\\theta_k}{Q_0},\n\\xi=\\ln\\frac{k\\theta}{Q_0}.\n$$\nWorking in the first order in $\\varepsilon$ we can write\n$$\n{\\bar{n}} (p,\\theta)={\\bar{n}}_0 (y)+\\varepsilon {\\bar{n}}_1 (y).\n$$\nThe equations for the functions ${\\bar{n}}_0(y)$ and ${\\bar{n}}_1(y)$\n take the form\n\\begin{equation}\n{\\bar{n}}_0 (y)=1+\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_0 (y')\n\\end{equation}\nand\n\\begin{equation}\n {\\bar{n}}_1 (y)=\n\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_1 (y')\\,(e^{y-y'}-1)\\,-\\,\\int\\limits^y_0 dy'\\,\\gamma_0^2\\, {\\bar{n}}_0 (y')\n\\end{equation}\nThe first one has a well known solution\n$$\n{\\bar{n}}_0 (y)= \\mbox{ch}(\\gamma_0 y),\n$$\nand the second one can be rewritten in the differential form\n\\begin{equation}\\label{dif}\n{\\bar{n}}_1 ''(y)- {\\bar{n}}_1'(y)\n=\\gamma_0^2 {\\bar{n}}_1 (y)+\\gamma_0^2(\\mbox{ch}(\\gamma_0y)\n-\\gamma_0\\mbox{sh}(\\gamma_0 y))\n\\end{equation}\nwith the initial conditions\n$ {\\bar{n}}(0)=0 , {\\bar{n}}'(0)=-\\gamma_0^2.$\nThe solution reads\n\\begin{equation}\n\\bar n_1=\\gamma_0^2\\mbox{ch}(\\gamma_0y)-\\gamma_0\\mbox{sh}(\\gamma_0y)+\n+\\gamma_0\\frac{\\lambda_2e^{\\lambda_1y}-\\lambda_1e^{\\lambda_2y}}\n{\\sqrt{1+4\\gamma_0^2}}.\n\\end{equation}\nIn the limit of $(y\\gg1,\\gamma_0\\ll1)$ we have\n\\begin{equation}\\label{n1}\n{\\bar{n}}\\approx\\frac{1}{2}e^{\\gamma_0y}(1-\\varepsilon\\gamma_0-\n2\\varepsilon\\gamma_0^3e^y)=\n\\frac{1}{2}\\left({\\frac{Q}{Q_0}}\\right)^{\\gamma_0}\\left(1-\\gamma_0\n\\frac{\\beta}{p}-2\\gamma_0^3\\frac{\\beta\\theta}{Q_0}\\right)\n\\end{equation}\nThe above result is in accord with the expectation based on physical reasoning.\nNamely, the dissipation should lead to a decrease in perturbative multiplicity.\nLet us note, that from Eq. (\\ref{n1}) it is clear, that in order for\nperturbation theory to be applicable one should have\n$\\beta \\ll p\/\\gamma_0$ and $\\beta \\ll Q_0\/\\gamma_0^3$. This means that apart\nfrom the expected expansion parameter $\\varepsilon$ there appears a new one\n$\\varepsilon'= 2\\gamma_0^2\\frac{\\beta\\theta}{Q_0}$. An interesting feature\nof the above answer is a multiplicative combination of the dissipation scale\n$\\beta$ and a perturbative coupling constant hidden in $\\gamma_0$.\n\n\\section {Energy distrubution of particles in a jet}\n\nLet us now turn to a computation of a one-particle energy distribution function\n$\\bar D(k,\\theta)$. It can be obtained from the generating functional in the\nfollowing way:\n\\begin{equation}\n\\bar D(k,\\theta)=k\\frac{\\delta}{\\delta u(k)}G(p,\\theta;\\{u\\})\\Big|_{u=1},\n\\end{equation}\nMaking use of (\\ref{Gnew}) and (\\ref{phas}) we get\n\\begin{equation}\\label{en}\n\\bar D(l,y)=\\delta(l)+\\int\\limits^{l-\\varepsilon}_0 dl' \\int\\limits^y_0 dy'\n\\gamma_0^2 \\bar D(l',y'),\n\\end{equation}\nwhere $l=\\ln(p\/k)=\\ln(1\/x)$ and $l'=\\ln(k'\/k)$. The integration is performed\nover $k'$ and $\\theta_{k'}$. Let us stress that in the above equation (\\ref{en})\nthe distrubution function $\\bar D$ is also $\\beta$ - dependent. Expanding\nit in the small parameter $\\varepsilon$\n$$\n\\bar D(l,y)=\\bar D_0(l,y)+\\varepsilon \\bar D_1(l,y)\n$$\nwe rewrite (\\ref{en}) in the form\n\\begin{equation}\n\\bar D_0(l,y)+\\varepsilon\\bar D_1(l,y)=\\delta(l)+\n\\int\\limits^{l-\\varepsilon}_0 dl' \\int\\limits^y_0 dy'\\gamma_0^2\n(\\bar D_0(l,y)+\\varepsilon e^{l-l'}\\bar D_1(l,y)).\n\\end{equation}\nConsidering the zero and first order in $\\varepsilon$ we have\n\\begin{equation}\\label{D0eq}\n\\bar D_0(l,y)=\\delta(l)+\\int\\limits^{l}_0 dl' \\int\\limits^y_0 dy'\n\\gamma_0^2 \\bar D_0(l',y').\n\\end{equation}\n\\begin{equation}\\label{D1eq}\n\\bar D_1(l,y)=\\int\\limits^{l}_0 dl' \\int\\limits^y_0 dy'\\gamma_0^2 e^{l-l'}\n\\bar D_1(l',y')-\\int^y_0 dy'\\gamma_0^2 \\bar D_0(l,y').\n\\end{equation}\nThe solution of equation (\\ref{D0eq}) reads\n\\begin{equation}\n\\bar D_0(l,y)=\\delta(l)+\\gamma_0\\sqrt{\\frac{y}{l}}I_1(2\\gamma_0\\sqrt{yl})\n\\end{equation}\nand that for $\\bar D_1$ is (for details see Appendix C):\n\\begin{equation}\\label{ensol}\n\\bar D_1(l,y)=-y\\gamma_0^2\\delta(l)-\\gamma_0^2(e^l+1)\\frac{y}{l}\nI_2(2\\gamma_0\\sqrt{yl})+\\gamma_0^3\\left(\\frac{y}{l}\\right)^{3\/2}(e^l-1)\nI_3(2\\gamma_0\\sqrt{yl}).\n\\end{equation}\nThe plot of the resulting distribution for the jet energy $20 {\\mbox { GeV}}$\nis given at Fig.~1. We see that, as anticipated, the effect of dissipation\nshows itself through the noticeable reduction of distribution function.\nLet us also mention that here we also have a multiplicative dependence on both\nperturbative and nonperturbative factors.\n\n\n\\section{Conclusions}\n\nIn this paper we have proposed a phenomenological nonperturbative\nmodification of the equations describing the evolution of QCD jets\nand taking into account the crucial feature of the colour coherence\nof the QCD cascades\nby accounting for nonperturbative energy dissipation. The corresponding\nmodification of the MLLA and DLA formalism exploiting the generating\n functional was proposed. The calculation of simplest jet characteristics\nsuch as mean multiplicity and energy spectrum of particles in a jet in a\nDLA approximation has demonstrated an expected decrease in multiplicity\n and corresponding changes in the energy distribution.\n\n An interesting feature of the result is an unusual\nperturbative damping of the introduced nonperturbative energy dissipation\nappearing in the multiplicative dependance on some power of the QCD coupling\nconstant times the dissipative scale.\nIt is tempting to relate this feature with the successes of the soft\nblanshing hypothesis, where it is assumed that the nonperturbative\neffects are not crucially essential for the jet characteristics. It is\ninteresting to see, whether such damping is present within a more realistic\nMLLA description. Work in this direction is in progress.\n\n\n\\section{Acknowledgements}\nWe are grateful to I.M.~Dremin and I.V.~Andreev for the useful discussions.\nA.L. is grateful to P.V.~Ruuskanen for kind hospitality in the University\nof Helsinki, where part of this work was done. The research was supported by\nRussian Fund for Fundamental Research, Grant 93-02-3815.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nPolymer materials, while possessing some unique and attractive qualities, such as low weight, high strength, resistance to chemicals, ease of processing, are for the most part\ninsulators. If methods could be devised to turn common insulating polymers into conductors, that would open great prospects for using such materials in many more areas than they are\ncurrently used. These areas may include organic solar cells, printing electronic circuits, light-emitting diodes, actuators, supercapacitors, chemical sensors, and biosensors \\cite{Long_2011}.\n\n\nSince the reliable methods for carbon nanotubes (CNT) fabrication had been developed in the 1990s, growing attention has been paid to\nthe possibility of dispersing CNTs in polymers, where CNTs junctions may form a percolation network and turn insulating polymer into a good conductor when a percolation threshold is overcome.\nAn additional benefit of using such polymer\/CNTs nanocomposites instead of intrinsically conducting polymers, such as polyaniline \\cite{Polyaniline}\nfor example, is that dispersed CNTs, besides providing electrical conductivity, enhance\npolymer mechanical properties as well.\n\nCNTs enhanced polymer nanocomposites have been intensively investigated experimentally, including composites conductivity \\cite{Eletskii}. As for the theoretical research in this area, the results are more modest. \nIf one is concerned with nanocomposite conductivity, its value \ndepends on many factors, among which are the polymer type, CNTs density, nanocomposite preparation technique, CNTs and their junctions geometry, a possible presence of defects\nin CNTs and others. Taking all these factors into\naccount and obtaining quantitatively correct results in modeling is a very challenging task since the resulting conductivity is formed at different length scales: at the microscopic level it is influenced by the\nCNTs junctions contact resistance and at the mesoscopic level it is determined by percolation through a network of CNTs junctions. Thus a consistent multi-scale method for the modeling of conductivity,\nstarting from atomistic first-principles calculations of electron transport through CNTs junctions is necessary.\n\nDue to the complexity of this multi-scale task, the majority of investigations in the area are carried out in some simplified forms, this is especially true for the underlying part of the modeling: determination\nof CNTs junction contact resistance. For the contact resistance either experimental values as in \\cite{Soto_2015} or the results of phenomenological Simmons model as in \\cite{Xu_2013, \nYu_2010, Jang_2015, Pal_2016} are usually taken, or even an arbitrary value of contact resistance reasonable by an order of magnitude may be set \\cite{Wescott_2007}. In \\cite{Bao_2011, Grabowski_2017} the tunneling probability through a CNT junction is modeled using a rectangular potential barrier and the quasi-classical approximation.\n\nThe authors of \\cite{Castellino_2016} employed an oversimplified two-parameter expression for contact resistance, with these parameters fitted to the experimental data.\nThe best microscopic attempt, that we are aware of, is using the semi-phenomenological tight-binding approximation for the calculations of contact resistance \\cite{Penazzi_2013}. But in \\cite{Penazzi_2013} just\nthe microscopic part of the nanocomposites conductivity problem is addressed, and the conductivity of nanocomposite is not calculated. Moreover, in \\cite{Penazzi_2013} the coaxial CNTs configuration is only considered,\nwhich is hardly realistic for real polymers.\n\nThus, the majority of investigations are concentrated on the mesoscopic part of the task:\nrefining a percolation model or phenomenologically taking into account different geometry peculiarities of CNTs junctions. Moreover, comparison with experiments is missing in some publications on this topic.\nThus, a truly multi-scale research, capable of providing quantitative results comparable with experiments, combining fully first-principles calculations of contact resistance on the microscopic level with a percolation model on \nthe mesoscopic level seems to be missing. \n\nIn our previous research \\cite{comp_no_pol}, we proposed an efficient and precise method for fully first-principles calculations of CNTs contact resistance and combined it with a Monte-Carlo statistical percolation model to\ncalculate the conductivity of a simplified example network of CNTs junctions without polymer filling. \nIn the current paper, we are applying the developed approach to the modeling of conductivity of the CNTs enhanced polymer polyimide R-BAPB. \n\nR-BAPB (Fig. \\ref{fig_RBAPB-struct}) is a novel polyetherimide synthesized using 1,3-bis-(3$'$,4-dicarboxyphenoxy)-benzene (dianhidride R) and 4,4$'$-bis-(4$''$-aminophenoxy)diphenyl (diamine BAPB). It is thermostable polymer with extremely high thermomechanical properties (glass transition temperature $T_g= 453-463$~K, melting temperature $T_m= 588$~K, Young's modulus $E= 3.2$~GPa) \\cite{Yudin_JAPS}. This polyetherimide could be used as a binder to produce composite and nanocomposite materials demanded in shipbuilding, aerospace, and other fields of industry. The two main advantages of the R-BAPB among other thermostable polymers are thermoplasticity and crystallinity. R-BAPB-based composites could be produced and processed using convenient melt technologies.\n\nCrystallinity of R-BAPB in composites leads to improved mechanical properties of the materials, including bulk composites and nanocomposite fibers. It is well known that carbon nanofillers could act as nucleating agents for R-BAPB, increasing the degree of crystallinity of the polymer matrix in composites. As it was shown in experimental and theoretical studies\n\\cite{Yudin_MRM05, Larin_RSCADV14, Falkovich_RSCADV14,Yudin_CST07}, the degree of crystallinity of \ncarbon nanofiller enhanced R-BAPB may be comparable to that of bulk polymers.\n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_Larin1\/RBAPB.png}\n\\caption{The chemical structure of R-BAPB polyimide.} \n\\label{fig_RBAPB-struct}\n\\end{figure}\n\nOrdering of polymer chains relative to nanotube axes could certainly influence a conductance of the polymer filled nanoparticle junctions. However, it is expected that such influence will depend on many parameters, including the structure of a junction, position, and orientation of chain fragments on the nanotube surface close to a junction, and others. Taking into account all of these parameters is a rather complex task that requires high computational resources for atomistic modeling and ab-initio calculations, as well as complex analysis procedures. Thus, on the current stage of the study, we consider only systems where the polymer matrix was in an amorphous state, i.e. no sufficient polymer chains ordering relative to nanotubes were observed.\n\n\n\n\n\n\\section*{Description of the multiscale procedure} \n\nThe modeling of polymer nanocomposite electrical conductivity is based on a multi-scale approach, in which different simulation models are used at different scales. For the electron transport in polymer composites with a conducting filler, the lowest scale corresponds to the contact resistance between tubes. The contact resistance is determined at the atomistic scale by tunneling of electrons between the filler particles via a polymer matrix, and hence, analysis of contact resistance requires knowledge of the atomistic structure of a contact. Therefore, at the first step, we develop an atomistic model of the contact between carbon nanotubes in a polyimide matrix using the molecular dynamics (MD) method. This method gives us the structure of the intercalated polymer molecules between carbon nanotubes for different intersection angles between the nanotubes. One should mention, that since a polymer matrix is soft, the contact structure varies with time and, therefore, we use molecular dynamics to sample these structures. \n\nBased on the determined atomistic structures of the contacts between nanotubes in the polymer matrix we calculate electron transport through the junction using electronic structure calculations and the formalism of the Green's matrix. Since this analysis requires first-principles methods, one has to reduce the size of the atomistic structure of a contact to acceptable values for the first-principles methods, and we developed a special procedure for cutting the contact structure from MD results. First-principles calculations of contact resistance should be performed for all snapshots of an atomistic contact structure of MD simulations, and an average value and a standard deviation should be extracted. In this way, one can get the dependence of a contact resistance on the intersection angle and contact distance.\n\nUsing information about contact resistances we estimate the macroscopic conductivity of a composite with nanotube fillers. For this, we used a percolation model based on the Monte Carlo method to construct a nanotube network in a polymer matrix. In this model, we used distributions of contact resistances, obtained from the first-principles calculations for the given angle between nanotubes. Using this Monte Carlo percolation model one can investigate the influence of non-uniformities of a nanotube distribution on macroscopic electrical conductivity. \n\nIn the A section, we will describe the details of molecular dynamics modeling of the atomistic structure of contacts between nanotubes. In the B section, we present the details of first-principles calculations of electron transport for estimates of contact resistance. Finally, in the C section, we present the details of the Monte Carlo percolation model.\n\n\\subsection*{A. Preparation of the composite atomic configurations}\n\nInitially, two metallic CNTs with chirality (5,5) were constructed and separated by 6~\\AA. The CNTs consisted of 20 periods along the axis, and each one had the total length of 4.92~nm.\nThe broken bonds at the ends of the CNTs were saturated with Hydrogen atoms.\nThe distance 6 \\AA\\ was chosen, because starting with this distance polymer molecules are able to penetrate the space between CNTs. The three configurations of CNTs junctions were prepared: the first one with parallel CNTs axes (angle between nanotube axes $\\varphi=0^\\circ$),\nthe second one with the axes crossing at 45 degrees ($\\varphi=45^\\circ$), and the third one with perpendicular axes ($\\varphi=90^\\circ$).\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_em1.png}&\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_compress1.png}\\\\\n\\end{tabular}\n\\caption{The snapshots of the the nanocomposite system with the parallel orientation of carbon nanotubes at the initial state (left picture) and after the compression procedure (right picture). The black lines represent the periodic simulation cell.} \\label{fig_Larin-conf}\n\\end{figure}\n\nTo produce the polymer filled samples, we used \nthe procedure similar to that employed for the simulations of the thermoplastic polyimides and polyimide-based nanocomposites in the previous works\n\\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14, Nazarychev_MA16, Larin_RSCADV15}. First, partially coiled R-BAPB chains with the polymerization degree $N_p=8$, which corresponds to the polymer regime onset \\cite{Lyulin_MA13, Lyulin_SM14}, were added to the simulation box at random positions avoiding overlapping of polymer chains. This results in the initial configuration of samples with a rather low overall density ($\\rho\\sim 100~$ kg\/m$^3$) (Fig. \\ref{fig_Larin-conf}). Then the molecular dynamics simulations were performed to compress the systems generated, equilibrate them and perform production runs.\n\nThe molecular dynamics simulations were carried out using Gromacs simulation package \\cite{gromacs1, gromacs2}. The atomistic models used to represent both the R-BAPB polyimide and CNTs were parameterized using the Gromos53a6 forcefield \\cite{gromos}. Partial charges were calculated using the Hartree-Fock quantum-mechanical method with the 6-31G* basis set, and the Mulliken method was applied to estimate the values of the particle charges from an electron density distribution. As it was shown recently, this combination of the force field and particle charges parameterization method allows one to reproduce qualitatively and quantitatively the thermophysical properties of thermoplastic polyimides \\cite{Nazarychev_EI}. The model used in the present work was successfully utilized to study structural, thermophysical and mechanical properties of the R-BAPB polyimide and R-BAPB-based nanocomposites \\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14}.\n\nAll simulations were performed using the NpT ensemble at temperature $T=600$~K, which is higher than the glass transition temperature of R-BAPB. The temperature and pressure values were maintained using Berendsen thermostat and barostat \\cite{Berendses_1, Berendsen_2} with relaxation times $\\tau_T= 0.1$~ps and $\\tau_p=0.5$~ps respectively. The electrostatic interactions were taken into account using the particle-mesh Ewald summation (PME) method \\cite{PME1, PME2}. \n\nThe step-wise compression procedure allows one to obtain dense samples with an overall density close to the experimental polyimide density value ($\\rho \\approx 1250-1300$~kg\/m$^3$), as shown in Fig. \\ref{fig_Larin-conf}. The system pressure $p$ during compression was increased in a step-wise manner up to $p=1000$~bar and decreased then to $p=1$~bar. After compression and equilibration, the production runs were performed to obtain the set of polymer filled CNT junction configurations.\n\nAs the conductance of polymer filled CNT junctions is influenced by the density and structure of a polymer matrix in the nearest vicinity of a contact between CNTs, the relaxation of the overall system density was used as the system equilibration criterion. To estimate the equilibration time, the time dependence of the system density was calculated as well as the density autocorrelation function $C_\\rho(t)$:\n\\begin{equation}\nC_\\rho(t)=\\frac{\\langle \\rho(0) \\rho(t)\\rangle}{\\langle \\rho^2 \\rangle},\n\\end{equation}\nwhere $\\rho(t)$ is the density of the system at time $t$ and $\\langle \\rho^2 \\rangle$ is the average density of the system during the simulation.\n\n\nAs shown in Fig. \\ref{fig_Larin-density}a, the system density does not change sufficiently during simulation after the compression procedure. At the same time, the analysis of the density auto-correlation functions shows some difference in the relaxation processes in the systems studied (see Fig. \\ref{fig_Larin-density}b). In the case of the system where CNTs were placed parallel to each other ($\\varphi=0^\\circ$), $C_\\rho(t)$ could be approximated by the exponential decay function $C_\\rho(t)=\\exp(-t\/\\tau)$ with relaxation time $\\tau = 4$~ps. The density relaxation in the systems with crossed CNTs ($\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) was found to be slower. For these two systems density the auto-correlation functions could be approximated by a double exponential function $C_\\rho(t)=A\\exp(-t\/\\tau_1) + (1-A)\\exp(-t\/\\tau_2)$, and the relaxation times determined using this fitting were $\\tau_1=2.7$~ps and $\\tau_2=12.2$~ns (for $\\varphi=90^\\circ$), and $\\tau_1=9.5$~ps and $\\tau_2=24.6$~ns (in case of $\\varphi=45^\\circ$).\n\nNevertheless, the results obtained after the analysis of the system density relaxation allow us to choose the system equilibration time to be 100 ns, which is higher than the longest system density relaxation times determined by the density autocorrelation function analysis. \nThe same simulation time was used in our previous works to equilibrate the nanocomposite structure after switching on electrostatic interactions \\cite{Larin_RSCADV14, Nazarychev_EI, Larin_RSCADV15}. The equilibration was followed by the 150 ns long production run. To analyze the polymer filled CNT junction conductance, 31 configurations of each simulated system, separated by 5 ns intervals, were taken from the production run trajectory.\n\n\\begin{figure}[h]\n\\begin{tabular}{ccc}\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_left.pdf}\\\\\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_right.pdf}\\\\\n\\end{tabular}\n\\caption{The time dependence of the system density $\\rho$ (a) and the density auto-correlation functions $C_\\rho(t)$ (b) for the systems with various angles between nanotube axes $\\varphi$. The dots correspond to the calculated data. The solid lines correspond to the fitting of $C_\\rho(t)$ with the exponential (in case of $\\varphi=0^\\circ$) or double exponential (in case of of $\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) functions.} \n\\label{fig_Larin-density}\n\\end{figure}\n\nAfter the configurational relaxation is finished, we have to prepare polymer filled CNT junctions configuration for the first-principles calculations of contact resistance. \nThe method we used for the calculations of a contact resistance is based on the solution of the ballistic electronic transport problem, finding the Volt-Ampere characteristic $I(V)$ of a device and\nderiving the contact resistance from the linear part of $I(V)$ corresponding to the low voltages. For this purpose, we employed the Green's function method for \nsolving the scattering problem and the Landauer-Buttiker approach to find the current through a scattering region coupled to two semi-infinite leads, as described in \\cite{Datta}.\nSpecific details of how these techniques are applied in the case of crossed CNTs can be found in \\cite{comp_no_pol}.\n\n\\subsection*{B. The first-principles calculations of the contact resistance of CNTs junctions filled with polymer} \\label{sec_meth_fp}\n\nFor the preparation of a device for the electronic transport calculations, we first form that part of the device which consists of the atoms belonging to the CNTs used in\nthe CNTs+polymer relaxation. Regions with the same geometry as in \\cite{comp_no_pol}\nare cut from the initial 20-period long CNTs, and the rest of the atoms belonging to the CNTs are discarded. This is done to make possible a direct comparison of the results obtained\nfor the polymer filled CNTs junctions with the results for CNTs junctions without polymer reported in \\cite{comp_no_pol} for the same separation of CNTs equal to 6 \\AA. \n\nNote that\nthe CNTs parts of the scattering device contain atoms shifted from their positions in ideal CNTs due to the influence of the adjacent polymer molecules, and these shifts are time-dependent\nas a result of thermal fluctuations. \n\nThe cut regions contain two fragments of CNTs each 9 periods long, and in the case of the CNTs parallel configuration, the CNTs overlap by 7 periods.\nIn the nonparallel configurations, one of the CNTs is rotated around the axis perpendicular to the CNTs axes in the parallel configuration and passing through the\ngeometrical center of a device in the parallel configuration. \n\nAfter the construction of the CNTs part of the scattering region, we attach to it leads that consist of 5 period long fragments of an ideal CNT. The CNTs parts of the scattering regions with the attached leads for the three considered configurations are shown in Fig. \\ref{fig_CNT_conf}.\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig1\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The CNTs parts of the junctions. Left: the parallel configuration, center: CNTs axes are crossing at 45 degrees,\nright: the perpendicular configuration. The leads atoms are colored by green. } \\label{fig_CNT_conf}\n\\end{figure}\n\nAfter the preparation of the CNT parts of the junctions, we still have 17766 atoms in a device. A system with such a large number of atoms cannot be treated by fully first-principles\natomistic methods. On the other hand, keeping all those atoms for a precision calculation of the contact resistance of polymer filled CNTs junctions is not necessary,\nas only those polymer atoms which are close enough to a CNT will serve as tunneling bridges and give a contribution to the junctions conductivity. Thus,\nfor the calculations of the contact resistance, only those atoms were kept which are closer to the CNTs than a certain distance $d$. It has been established by numerical experiments\nthat if the value of $d$ is taken equal to the CNTs separation $d=6$ \\AA\\ this is quite sufficient, and taking into account more distant atoms does not change the contact resistance significantly.\n\nThe procedure of sorting the polymer atoms is as follows. In our molecular dynamics simulations we used 27 separate polymer molecules, consisting of 8 monomers each.\nIf at least one of the atoms of a polymer molecule was closer to the CNTs part of a junction than $d=6$ \\AA, the whole molecule was kept for a while, and discarded \notherwise. Having applied this first part of the procedure, we kept 4 polymer molecules for the parallel configuration, 8 molecules for the perpendicular configuration and\n11 molecules for the 45 degrees configuration. \n\nThen we looked at the monomers of each molecule that survived the first round of selection. The same procedure was applied\nto monomers as the one used earlier for molecules: if at least one of the monomer atoms was closer to the CNTs part of a junction than $d=6$ \\AA, \nthe whole monomer was kept for a while, and \ndiscarded otherwise. \n\nAfter the second round of selection with monomers was over, we dealt in the same manner with the individual residues comprising a monomer. The broken\nbonds that appeared in the second and third stages were saturated with hydrogen atoms. The described procedure resulted in the following numbers of atoms in the whole\ndevice, including the central scattering region and the leads: 881 for the parallel configuration, 1150 for the perpendicular configuration, and 1074 for the 45 degrees configuration.\nThe atomic configurations obtained using the described procedure for the first time steps in the corresponding series are presented in Fig. \\ref{fig_junction_conf}.\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig2\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The atomic configurations for the first-principles calculations of the polymer filled CNTs junctions contact resistance. The configurations are for the first time steps in the corresponding series.\nLeft: the parallel configuration, center: CNTs axes are crossed at 45 degrees,\nright: the perpendicular configuration. The Carbon atoms are gray, the Nitrogen atoms are blue, the Oxygen atoms are red, and the Hydrogen atoms are light gray. The leads atoms are colored by green. } \n\\label{fig_junction_conf}\n\\end{figure}\n\nA fully ab-initio method for electronic structure investigations utilizing a localized pseudo-atomic basis set, as described in \\cite{Ozaki_DFT} and implemented in \\cite{OpenMX}, \nwas used for the calculations of the electronic structures of the whole device and the leads.\nWe used basis set s2p2d1, the Pseudo Atomic Orbitals (PAO) cutoff radius equal to 6.0 a. u., and the cutoff energy of 150 Ry.\nThe pseudo-potentials generated according to Morrison, Bylander, and Kleinman scheme \\cite{pseudopot} were used. For the \ndensity functional calculations, the exchange-correlation functional was used in PBE96 form \\cite{PBE96}.\n\nUsing the electronic structures of the whole device and the leads we calculated energy the dependent transmission function\n through the device. Then the dependence $I(V)$ of the current $I$ on the voltage $V$ between the leads was determined\nwith the Green's function approach as described in detail in \\cite{Datta}. Finally, the Landauer-Buttiker approach was used to find the current through a polymer filled CNTs junction.\n\nSolving a scattering problem for a nano-device at arbitrary voltages is a computationally very complex task since it requires achieving self-consistency for both\nelectron density and induced electrostatic potential simultaneously. Fortunately, for contact resistance calculations one can take\nadvantage of the fact that the required voltages are very low. \n\nAccording to the experimental evidence, the size of a nanocomposite specimen used in conductivity experiments is about 10 mm \\cite{exp_CNT_length_1}, and the typical voltages applied across such specimen do not exceed 100 V \\cite{CNT_book}. Then for the size of a central scattering region about 1 nm, the voltage drop is about 10$^{-5}$ V, which is well within the range where the simplified approach is applicable.\n\nThe question of the modeling of quantum transport in the limit of low voltages was discussed in detail in \\cite{comp_no_pol}, where it was demonstrated that in the case of moderate voltages between leads,\nthe scattering probability $T(E)$ is not sensitive to the details of the electrostatic potential distribution $V({\\mathbf r})$ in the central scattering region, and\nsome physically reasonable approximation may be chosen for $V({\\mathbf r})$. \n\nThis is due to the fact that the difference of the Fermi functions\n$f(\\varepsilon - \\mu_L)$ and $f(\\varepsilon - \\mu_R)$ \nfor the left and right leads with corresponding the chemical potentials $\\mu_L$ and $\\mu_R$, present in the original Landauer-Buttiker formula:\n\\begin{equation}\nI=\\frac{2e}{h} \\int T(\\varepsilon) \\left (f(\\varepsilon - \\mu_L) - f(\\varepsilon - \\mu_R) \\right )d\\varepsilon ,\n\\label{LB_formula}\n\\end{equation}\nwhere $e$ is the elementary charge, $h$ is the Planck constant, $\\varepsilon$ is the electron energy and $T(\\varepsilon)$ \nis the energy dependent transmission probability,\n is reduced in this case to a very narrow and sharp peak centered at the Fermi level\nof the device. \n\nIn addition to the analysis performed in \\cite{comp_no_pol}, in this article, to verify the accuracy of our approach, we made contact resistance calculations for a simple test CNT junction in a coaxial configuration, using both the simplified method we suggest and the full NEGF method, where not only electron charge density but the electric potential was converged as well. The interlead voltage used in those test calculations was\nset to $10^{-4}$ V, and the gap between the CNTs tips was 0.94 \\AA. The atomic configurations for the test calculations are presented in Fig. \\ref{fig_NEGF_compare}. The consistent NEGF calculations produced 1.71~$\\cdot 10^{-5}$ S for the conductance of the junctions shown in Fig. \\ref{fig_NEGF_compare}, while modeling without searching for convergence of potential yielded 1.72 $\\cdot 10^{-5}$ S. \n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_conf_compare\/fig_compare.pdf}\n\\caption{The atomic configuration used in this work to validate the method of calculating quantum transport by comparing it againts the consistent NEGF approach.} \\label{fig_NEGF_compare}\n\\end{figure}\n\nThus, in our case, a very complex task of finding the $I(V)$ characteristic of a nano-device can be significantly simplified without the loss\nof precision. For the $I(V)$ calculations in the current paper we employed the abrupt potential model introduced in \\cite{comp_no_pol}: the potentials $V_L$ for the\nleft lead and $V_R$ for the right lead were set and were used for all atoms of the corresponding CNT to which that lead belonged. As for the polymer atoms, both $V_L$ and\n$V_R$ can be safely used for them, and at the considered voltages, adopting these two options, as we have checked by direct calculations, leads to the differences in\ncurrent not exceeding 0.1\\%.\n\n\\subsection*{C. The percolation model} \\label{sec_percolation}\nDetermination of the conductivity of a polymer-CNT system can be implemented in 2 stages. First, a percolation cluster is formed, and the second stage implies solving the matrix problem for a random resistor circuit (network). \n\nAt the first stage, the modeling area -- a cube of the linear size $L$ -- is filled with CNTs. For this task, permeable capsules (cylinders with hemispheres at the ends) with a fixed length and diameter were chosen as filling objects corresponding to CNTs. The cube is filled by the successive addition of CNTs until a fixed bulk density of CNTs \n\\begin{equation}\n\\eta = \\frac{((4\/3)\\pi R^3 + \\pi R^2 h)N}{L^3} \\label{cnt_vol_frac}\n\\end{equation}\nin the cube is reached, where $R$ is the radius of the cylinder and hemisphere, $h$ is the height of the cylinder, and $N$ is the number of CNTs in the cube. The percolation problem for permeable capsules was previously solved in \\cite{Xu_16}, and in \\cite{Schilling_15} capsules with a semipermeable shell were considered.\n\nIn the percolation problem, we use periodic boundary conditions as, for example, in \\cite{Bao_2011}. \nWe use the method of finding a percolation threshold based on the Newman and Ziff algorithm \\cite{Newman_2001}, where the identification of a percolation cluster is made at the stage of its formation.\n\n\nWhen a percolation cluster is formed, the obtained CNT configuration is transformed into a resistor circuit (2nd stage). \n\nThe contributions to a conductance matrix resulting from inner resistance of CNTs and the junctions tunneling resistance are usually discussed in connection with constructing conductivity percolation algorithms. Direct measurements of CNTs resistance per unit length are available. In \\cite{Chiodarelli_11}, the inner resistance of CNTs is estimated as $15 \\cdot 10^3$ $\\Omega\/\\mu$m. The results of \\cite{Ebbesen_96} give specific CNTs resistance in the range $(12-86) \\cdot 10^3$ $\\Omega\/\\mu$m. Taking into account that the characteristic CNTs lengths in nanocomposites are about several $\\mu$m \\cite{exp_CNT_length_1,CNT_book}, this results in the inner CNTs resistance approximately $10^4-10^5$ $\\Omega$, which is at least one order of magnitude less than the tunneling resistance obtained in this work. The specific results on tunneling resistance will be discussed below in section \\ref{RND}. Thus, in our percolation model, the inner resistance of CNTs is neglected, and only tunneling resistance of CNTs junctions is taken into account. This can significantly reduce the requirements for computational time. \n\nWhen contact resistance is determined only by tunneling, the principle of compiling the matrix for the percolation problem will be as follows. \nFirst, the matrix (N, N) is compiled from the bonds of percolation elements, where N is the number of CNTs participating in percolation. \nThen this matrix is transformed into the conductivity matrix according to the second Kirchhoff law \n\\begin{equation}\n\\sum_j G_{ij} (V_i-V_j) = 0 \\label {second_Kirchoff}\n\\end{equation}\n- the sum of currents for all internal elements of a percolation network is zero -- where $G_{ij}$ are the elements of the conductance matrix ${\\mathbf G}$ and $V_i$ is the component of the voltage vector ${\\mathbf V}$ corresponding to the $i$-th contact point in a percolation network \\cite{Kirkpatrick_73}. \nThe voltages on the left and right borders of a simulation volume are set to $V_L=1$ V and $V_R=0$ V, respectively. \n\nNow finding the conductivity of the system is reduced to the problem $\\mathbf{GV}=\\mathbf{I}$, where\n$\\mathbf{I}$ is the vector of the currents between the contact points. \n\nThe dimension of the matrix problem can be further reduced to \n(N-2, N-2) by excluding boundary elements. \n\nAfter solving the equation (\\ref{second_Kirchoff}), with the elements of the $\\mathbf{G}$ matrix obtained by the first-principles calculations, we obtain the voltage vector for all internal elements. \nThen, knowing this vector, we sum up all the currents on each of the boundaries. The currents on the left $I_L$ and right $I_R$ \nboundaries of a simulation volume are equal in magnitude and opposite in sign $I_L= - I_R$. Knowing these currents, we determine the conductance of the simulation system as \n$G=|I_L|\/(V_L-V_R) = |I_R|\/(V_L-V_R)$. \nAnd then the conductivity of the composite is calculated as $\\sigma=GL\/S$, where $L$ is the distance between the faces of a simulation volume where voltage is applied, and $S$ is the area of that kind of face. In our case, for the simulation volume of a cubic shape, $S=L^2$, and $\\sigma=G\/L$.\n\nTo calculate the conductivity, the following system parameters were selected: the length of a nanotube is $l=3$ $\\mu$m, the diameter of a CNT is $D=30$ nm, the aspect ratio $l\/D=100$, and the size of the system is 4 $\\mu$m. The same values were used in \\cite{Yu_2010}. We adopted those values to test our realization of the percolation algorithm against the previously obtained results \\cite{Yu_2010}.\nThen for the given parameters for each fixed tube density, the Monte Carlo method (100 implementations of various configurations of CNT networks) was used to calculate the system conductivity. \n\nThe quality of CNTs dispersion is one of the key factors that affect the properties of nanocomposites, and a lot of efforts is taken\nto achieve a homogeneous distribution of fillers \\cite{Mital_14}. In this work, we take into consideration the effect of inhomogeneity of a CNTs distribution on composite conductivity.\nThe spatial density of nanotubes $\\rho_{CNT}$, in this case, has one peak with a Gaussian distribution:\n\\begin{equation} \n\\rho_{CNT}=\\rho_0 \\cdot exp(-(\\mathbf {r-r_0})^2\/\\rho_\\sigma^2), \\label {agglo}\n\\end{equation}\nwhere ${\\mathbf r_0}$ coincides with the geometrical center of a simulation volume, and $\\rho_\\sigma = L\/12$. The value of the $\\rho_0$ parameter is chosen so that the CNTs volume fraction in the inhomogeneous case is the same as in the homogeneous distribution.\n\n\n\\section*{Results and discussion}\\label{RND}\nTo find the contact resistance of polymer filled CNTs junctions one first needs to find their Volt-Ampere characteristics $I(V)$ and to determine the voltage range where\n$I(V)$ is linear and is not sensitive to the specific distribution of an electrostatic potential in the scattering region. In Fig. \\ref{fig_I_V} the $I(V)$ plot\nfor the first time step in the atomic geometry series for the parallel configuration is shown.\n\\begin{figure}[b]\n\\begin{tabular}{lr}\n\\includegraphics[width=3.5cm]{figs\/I_V_test\/I_V.pdf} &\n\\includegraphics[width=3.5cm]{figs\/I_V_test_low_V\/I_V_low_V.pdf}\n\\end{tabular}\n\\caption{The Volt-Ampere characteristic for the polymer filled CNTs junction corresponding to the first time step in the series for the parallel configuration.\nLeft frame: maximum inter-lead voltage is $10^{-3}$ V, right frame: $10^{-4}$ V. The circles correspond to the results of calculations,\nthe lines are guides for an eye.} \n\\label{fig_I_V}\n\\end{figure}\n\nIt is clearly seen from Fig. \\ref{fig_I_V} that up to about $10^{-4}$ V the $I(V)$ characteristic is linear, and after that value, it starts to deviate from a\nsimple linear dependence. Thus, for the calculations of a contact resistance $R$ and its inverse, a junction conductivity $G$,\nwe used the electrical current values obtained for the inter-lead voltage equal to $10^{-4}$ V. Note, that according to our estimates in section \\ref{sec_meth_fp}, a characteristic voltage drop on the length of a CNTs tunneling junction is about $10^{-5}$ V which is well within the region where the linear $I(V)$ is observed.\n\nThe time dependences of the junctions conductances for the three considered configurations are presented in Fig. \\ref{fig_G_t}. One might expect that the \nshifts of both CNTs atoms and polymer atoms in the central scattering region due to thermal fluctuations would lead to fluctuations of junctions conductances $G$, but \nquantitative characteristics of this phenomenon such as minimum $G_{min}$, maximum $G_{max}$, mean values $\\langle G \\rangle$ and a standard deviation $G_\\sigma$ can only be captured by highly precise fully atomistic \nfirst-principles methods, like those employed in the current paper. The resulting fluctuations of conductance are very high. For the parallel CNTs configuration\nthe minimum value, $G_{min} = 2.4 \\cdot 10^{-8}$ S, and the maximum value, $G_{max} = 6.8 \\cdot 10^{-6}$ S, differ by more than two orders of magnitude, for the 45 degrees and perpendicular\nconfigurations the corresponding ratios are about 30. The same strong variations of conductance over time were reported in \\cite{Penazzi_2013} for the coaxial\nCNTs configuration, where the results were obtained using a semi-empirical tight-binding approximation. Thus, it is obvious that for the precise determination\nof a conductance of polymer filled CNTs junctions one needs to use fully atomistic approaches, and phenomenological methods taking atomic configurations into account \non the average are not reliable.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/G_t\/g_t.pdf}\n\\caption{The time dependence of the polymer filled junctions conductance $G$ in S. Red color correspond to the parallel configuration, the green lines -- to the perpendicular configuration\nand the blue lines -- the 45 degrees configuration. The results of the calculations are shown by circles, the saw-tooth lines serve as a guide for an eye.\nThe straight solid lines designate the mean values of conductance $\\langle G \\rangle$, and the dashed ones -- $\\langle G \\rangle \\pm G_\\sigma$.} \n\\label{fig_G_t}\n\\end{figure}\n\n\n\n\\begin{table*}[h]\n \\centering\n\\caption{The results of statistical analysis of the CNTs junctions conductances in S, for CNTs separation equal to 6 \\AA \\ and different values of the CNTs crossing angles $\\varphi$, \nwithout polymer from \\cite{comp_no_pol}, and with polyimide R-BAPB filling obtained in the current paper.\n}\n\\begin{tabular}{cccccc}\n\\textrm{$\\varphi$}&\n\\textrm{no polymer}&\n\\textrm{}&\n\\textrm{polymer present}&\n\\textrm{}&\n\\textrm{}\\\\\n\\textrm{}&\n\\textrm{the results of \\cite{comp_no_pol}}&\n\\textrm{$G_{min}$}&\n\\textrm{$G_{max}$}&\n\\textrm{$\\langle G \\rangle$}&\n\\textrm{$G_\\sigma$}\\\\\n\\hline\n& & & & &\\\\ \n0 & 3.6$\\cdot10^{-13}$ &2.4$\\cdot10^{-8}$ &6.8$\\cdot10^{-6}$ & 1.8$\\cdot10^{-6}$ & 1.6$\\cdot10^{-6}$\\\\\n& & & & & \\\\\n\\hline\n& & & & &\\\\ \n0.2$\\pi$ & 1.4$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.25$\\pi$ & --- &4.8$\\cdot10^{-9}$ &1.4$\\cdot10^{-7}$ &3.4$\\cdot10^{-8}$ &2.7$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.3$\\pi$ & 1.2$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.5$\\pi$ & 4.2$\\cdot10^{-14}$ &2.2$\\cdot10^{-9}$ &4.3$\\cdot10^{-8}$ & 1.4$\\cdot10^{-8}$ & 1.1$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\end{tabular}\n\\label{tab_cond}\n\\end{table*}\n\nTo assign the tunneling resistance to a polymer filled CNT junction the following algorithm was used. First, for each junction that had to be used in the percolation algorithm, a uniformly distributed random number $\\varphi$ in the range [0, $\\pi\/2$] was generated. The value of the intersection angle for that junction was assigned to the obtained random number. The mean values and standard deviations for CNTs tunneling resistances and conductances calculated for the different atomic configurations corresponding to the different time steps are known for $\\varphi = 0, \\pi\/4$, and $\\pi\/2$. Analyzing figure 4 of \\cite{comp_no_pol}, one can see that though an angle dependence of current and hence conductivity is a rather complex function, in the first approximation one can adopt a roughly piece-wise linear character for this function with the minimum located at $\\varphi=0.25\\pi$. Thus the logarithm of the mean value of conductance $\\mu_\\varphi$ for the generated $\\varphi$ was set by linear interpolation between the logarithms of the mean values of conductances for $\\varphi=0$ and $\\varphi=\\pi\/4$ or $\\varphi=\\pi\/4$ and $\\varphi=\\pi\/2$ presented in Table \\ref{tab_cond}. The same algorithm was applied to finding the standard deviation values $\\sigma_\\varphi$ for the generated $\\varphi$. After the statistical parameters for the generated $\\varphi$ are estimated, the conductivity of the junction is set to a random number generated using the normal distribution with the parameters $\\mu_\\varphi$ and $\\sigma_\\varphi$.\n\n\nIn \\cite{comp_no_pol}, the conductances were reported for the CNTs junctions with almost the same geometry as the CNTs parts of the devices considered in the current paper. The only difference \nbetween the configurations is that in this work the carbon atoms belonging to the CNTs part of the central scattering region are shifted somewhat from their equilibrium positions\ndue to the interaction with polymer. The maximum values of those shifts along the $x$, $y$, and $z$ coordinates lie in the range $0.2-0.5$ \\AA. This gives us the possibility to directly compare the current results to the data \nfrom \\cite{comp_no_pol} and,\nthus, elucidate the influence of polymer filling on the junctions conductance. The corresponding data and the results of a basic statistical\nanalysis for the case of the polymer filled junctions are provided in Tab. \\ref{tab_cond}.\n\nFirst, as was expected, filling CNTs junctions with polymer creates carrier tunneling paths and increases junctions conductance by 6-7 orders of magnitude.\nSecond, it is evident that the CNTs axes crossing angle is crucial for the junctions conductivity when polymer is present as it was the case without\npolymer \\cite{comp_no_pol}. At the same time, the sharp dependence of polymer filled junctions conductance on the CNTs crossing angle is somewhat different\nfrom the analogous dependence for junctions without polymers. While in the latter case this dependence is sharply non-monotonous, with a pronounced minimum\nat the angles around $0.25\\pi$, in the former case there is a significant difference between the conductance values for the parallel and nonparallel configurations, but\nthe configurations with the angle $\\varphi$ between CNTs angles equal to $0.25\\pi$ and $0.5\\pi$ have very close conductances, and their mean values averaged over\ntime $\\langle G \\rangle_{45}$ and $\\langle G \\rangle_{per}$ lie within the\nranges $\\langle G \\rangle \\pm G_\\sigma$ of each other. Moreover, in contrast to the geometries without polymer, for the polymer filled CNTs junctions $\\langle G \\rangle_{per}$ is lower than $\\langle G \\rangle_{45}$ by a factor of 2.4.\n\nNote also that for the parallel configuration, the polymer influence on the junction conductance is more pronounced than for the nonparallel ones.\nFor the parallel configuration, adding polymer to a junction of CNTs separated by 6.0 \\AA\\ with initial conductance of 3.6$\\cdot 10^{-13}$ S \nproduces conductance mean value equal to 1.8$\\cdot 10^{-6}$ S. This gives the factor 0.5$\\cdot 10^7$; the value of the analogous factor for the perpendicular configuration is 0.33$\\cdot 10^6$.\n\nThe probable reason for the more effective conductance increase, when polymer is added, for the configurations with smaller angles between CNTs axes, is\nthat the smaller is an intersection angle, the larger is the overlap area between CNTs where polymer can penetrate and, thus, create tunneling bridges.\nThe higher fluctuation of conductance over time for the parallel configuration can be explained by the same reason: a larger CNTs overlap area gives\nmore freedom for polymer atoms to adjust their positions.\n\nThe dependence of the calculated composite conductivity $\\sigma$ on CNTs volume fraction $\\eta$ $\\sigma(\\eta)$ is presented in Fig. \\ref{fig_cond_pol}. \nThe value of the percolation threshold $\\eta_{thresh}$ is estimated in this work as $\\eta_{thresh} = 0.007$. To test our realization of the percolation algorithm against the previous results of \\cite{Yu_2010} we calculated the composite conductivity using\nthe fixed CNTs junction conductance equal to 1 M$\\Omega$ for all junctions in a percolation network. Our results presented in\nFig. \\ref{fig_cond_pol} by the red circles coincide within graphical accuracy to the results of \\cite{Yu_2010} shown by the red squares.\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/fig9\/cnt_9.pdf}\n\\caption{ The conductivity of CNT enhanced nanocomposites above the percolation threshold obtained in this work. The symbols of different shapes and colors are used to designate the following results. The red circles: the fixed CNTs tunneling junctions resistance of $R=1$ M$\\Omega$ is used, the red squares: the conductivity results for the fixed 1 M$\\Omega$ tunneling resistance from \\cite{Yu_2010}, the black triangles: the same as the red circles but for $R=0.54$ M$\\Omega$ corresponding to the mean value of the tunneling junction resistance for the parallel configuration from Table \\ref{tab_cond}, the blue rhombi: the angle dependence of the CNTs junctions resistance is taken into account, the green pentagons: CNTs agglomeration is considered in addition to the angle dependence. The red line is a guide for an eye.}\n\\label{fig_cond_pol}\n\\end{figure}\nThe 1 M$\\Omega$, used in various sources, for example, \\cite{Yu_2010}, is not an arbitrary value, but rather a typical contact resistance of CNTs junctions filled with polymer for simple geometries. In this work, we obtained for the parallel configurations $1\/\\langle G \\rangle =0.54$ M$\\Omega$. The $\\sigma(\\eta)$ dependence for the fixed tunneling resistance of $0.54$ M$\\Omega$ is shown in Fig. \\ref{fig_cond_pol} by the black triangles.\n\nTaking into account the angle dependence of CNTs junctions conductances with the statistical parameters according to Table\n\\ref{tab_cond}, leads to the lowering of composite conductivity just above the percolation threshold by the factor of about 30. This number correlates with the ratio of the mean conductances for the parallel, $\\langle G \\rangle_{par}$, and 45$^\\circ$, $\\langle G \\rangle_{45}$, configurations: $f_G = \\langle G \\rangle_{par}\/ \\langle G \\rangle_{45} = 53$, but is higher than $f_G$ due to the presence of junctions with $\\varphi < \\pi\/4$. \n\nIf agglomeration of CNTs, modeled by the inhomogeneity of their distribution according to formula (\\ref{agglo}) and the parameter values discussed in section \\ref{sec_percolation}, is taken into account in addition to the angle dependence of conductance, the composite conductivity is further reduced above a percolation threshold by the factor of 2.5. Lowering of conductivity of composites with agglomerated CNTs above a percolation threshold was also mentioned in \\cite{Bao_2011}. The calculated results for the conductivity of a percolation network of agglomerated CNTs are shown in Fig. \\ref{fig_cond_pol} by the green pentagons.\n\nWe believe that in this work we have identified some of the key factors that influence nanocomposites electrical conductivity: the geometry of tunneling junctions and changes of atomic configurations due to thermal fluctuations. Among other causes that may affect conductivity, but are not considered in this work, are the presence of defects in CNTs, a distribution of CNTs over chiralities, lengths, aspect ratios, different separations between CNTs. \n\n\nUntil the specific experiments on conductivity for R-BAPB polyimide are not available, we can make a preliminary comparison of our modeling results to the available experimental results for different composites. The calculated conductivity of composite just above the percolation threshold at $\\eta=0.0075$ is equal to $3.6\\cdot10^{-3}$ S\/m.\nThis is a reasonable value that falls into the range of experimentally observed composites conductivities (for the comprehensive compilation of experimental results see Table 1 of \\cite{Eletskii}). To make a quantitative comparison of modeling results with experiments the full details of nanocomposites structure are necessary, including the CNTs parameters mentioned in the previous paragraph. All these factors can be easily incorporated into the approach proposed in this work if sufficient computational resources are available.\n\n\\section*{Conclusions}\n\nWe have proposed a physically consistent, computationally simple, and at the same time precise, multi-scale method for calculations of electrical conductivity of CNT enhanced\nnanocomposites. The method starts with the atomistic determination of the positions of polymer atoms intercalated between CNTs junctions, proceeds with the fully first-principles calculations of polymer-filled CNTs junctions conductance at the microscale and finally performs\nmodeling of percolation through an ensemble of CNTs junctions by the Monte-Carlo technique. \n\nThe developed approach has been applied to the modeling of electrical conductivity of polyimide R-BAPB + single wall (5,5) CNTs nanocomposite.\n\nOur major contributions to the field are the following. We have proposed a straightforward\nmethod to calculate a contact resistance and conductance for polymer-filled CNTs junctions with arbitrary atomic configurations without resorting to any simplifying assumptions. We have demonstrated that a consistent multiscale approach, based on solid microscopic physical methods can give reasonable results, lying within the experimental range, for the conductivity of composites and suggested a corresponding work-flow. \n \n\nIt is shown that a contact resistance\nand nanocomposite conductivity is highly sensitive to the geometry of junctions, including an angle between CNTs axes and subtle thermal shifts of polymer atoms in an inter-CNT's gap.\nThus, we argue that for the precision calculations of nanocomposites electrical properties rigorous atomistic quantum-mechanical approaches are indispensable.\n\nWe have to admit though, that we have not considered all possible factors that may influence CNT junctions conductivity on the micro-level. We concentrated on the CNTs crossing angle factor because it seems to be the\nmost influential. The additional factors may include, for example, defects in CNTs,\nCNTs overlap lengths and others. On the other hand, the proposed approach may be used to take all those factors into account, provided sufficient computational resources are available.\n\n\\section*{Acknowledgments}\n\nThis work was supported by the State Program \"Organization of\nScientific Research\" (project 1001140) and by the Research Center \"Kurchatov Institute\" (order No.~1878~of~ 08\/22\/2019).\n\nThis work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC \"Kurchatov Institute\", \\url{http:\/\/ckp.nrcki.ru\/}.\n\n\n\n\\section*{Introduction}\nPolymer materials, while possessing some unique and attractive qualities, such as low weight, high strength, resistance to chemicals, ease of processing, are for the most part\ninsulators. If methods could be devised to turn common insulating polymers into conductors, that would open great prospects for using such materials in many more areas than they are\ncurrently used. These areas may include organic solar cells, printing electronic circuits, light-emitting diodes, actuators, supercapacitors, chemical sensors, and biosensors \\cite{Long_2011}.\n\n\nSince the reliable methods for carbon nanotubes (CNT) fabrication had been developed in the 1990s, growing attention has been paid to\nthe possibility of dispersing CNTs in polymers, where CNTs junctions may form a percolation network and turn insulating polymer into a good conductor when a percolation threshold is overcome.\nAn additional benefit of using such polymer\/CNTs nanocomposites instead of intrinsically conducting polymers, such as polyaniline \\cite{Polyaniline}\nfor example, is that dispersed CNTs, besides providing electrical conductivity, enhance\npolymer mechanical properties as well.\n\nCNTs enhanced polymer nanocomposites have been intensively investigated experimentally, including composites conductivity \\cite{Eletskii}. As for the theoretical research in this area, the results are more modest. \nIf one is concerned with nanocomposite conductivity, its value \ndepends on many factors, among which are the polymer type, CNTs density, nanocomposite preparation technique, CNTs and their junctions geometry, a possible presence of defects\nin CNTs and others. Taking all these factors into\naccount and obtaining quantitatively correct results in modeling is a very challenging task since the resulting conductivity is formed at different length scales: at the microscopic level it is influenced by the\nCNTs junctions contact resistance and at the mesoscopic level it is determined by percolation through a network of CNTs junctions. Thus a consistent multi-scale method for the modeling of conductivity,\nstarting from atomistic first-principles calculations of electron transport through CNTs junctions is necessary.\n\nDue to the complexity of this multi-scale task, the majority of investigations in the area are carried out in some simplified forms, this is especially true for the underlying part of the modeling: determination\nof CNTs junction contact resistance. For the contact resistance either experimental values as in \\cite{Soto_2015} or the results of phenomenological Simmons model as in \\cite{Xu_2013, \nYu_2010, Jang_2015, Pal_2016} are usually taken, or even an arbitrary value of contact resistance reasonable by an order of magnitude may be set \\cite{Wescott_2007}. In \\cite{Bao_2011, Grabowski_2017} the tunneling probability through a CNT junction is modeled using a rectangular potential barrier and the quasi-classical approximation.\n\nThe authors of \\cite{Castellino_2016} employed an oversimplified two-parameter expression for contact resistance, with these parameters fitted to the experimental data.\nThe best microscopic attempt, that we are aware of, is using the semi-phenomenological tight-binding approximation for the calculations of contact resistance \\cite{Penazzi_2013}. But in \\cite{Penazzi_2013} just\nthe microscopic part of the nanocomposites conductivity problem is addressed, and the conductivity of nanocomposite is not calculated. Moreover, in \\cite{Penazzi_2013} the coaxial CNTs configuration is only considered,\nwhich is hardly realistic for real polymers.\n\nThus, the majority of investigations are concentrated on the mesoscopic part of the task:\nrefining a percolation model or phenomenologically taking into account different geometry peculiarities of CNTs junctions. Moreover, comparison with experiments is missing in some publications on this topic.\nThus, a truly multi-scale research, capable of providing quantitative results comparable with experiments, combining fully first-principles calculations of contact resistance on the microscopic level with a percolation model on \nthe mesoscopic level seems to be missing. \n\nIn our previous research \\cite{comp_no_pol}, we proposed an efficient and precise method for fully first-principles calculations of CNTs contact resistance and combined it with a Monte-Carlo statistical percolation model to\ncalculate the conductivity of a simplified example network of CNTs junctions without polymer filling. \nIn the current paper, we are applying the developed approach to the modeling of conductivity of the CNTs enhanced polymer polyimide R-BAPB. \n\nR-BAPB (Fig. \\ref{fig_RBAPB-struct}) is a novel polyetherimide synthesized using 1,3-bis-(3$'$,4-dicarboxyphenoxy)-benzene (dianhidride R) and 4,4$'$-bis-(4$''$-aminophenoxy)diphenyl (diamine BAPB). It is thermostable polymer with extremely high thermomechanical properties (glass transition temperature $T_g= 453-463$~K, melting temperature $T_m= 588$~K, Young's modulus $E= 3.2$~GPa) \\cite{Yudin_JAPS}. This polyetherimide could be used as a binder to produce composite and nanocomposite materials demanded in shipbuilding, aerospace, and other fields of industry. The two main advantages of the R-BAPB among other thermostable polymers are thermoplasticity and crystallinity. R-BAPB-based composites could be produced and processed using convenient melt technologies.\n\nCrystallinity of R-BAPB in composites leads to improved mechanical properties of the materials, including bulk composites and nanocomposite fibers. It is well known that carbon nanofillers could act as nucleating agents for R-BAPB, increasing the degree of crystallinity of the polymer matrix in composites. As it was shown in experimental and theoretical studies\n\\cite{Yudin_MRM05, Larin_RSCADV14, Falkovich_RSCADV14,Yudin_CST07}, the degree of crystallinity of \ncarbon nanofiller enhanced R-BAPB may be comparable to that of bulk polymers.\n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_Larin1\/RBAPB.png}\n\\caption{The chemical structure of R-BAPB polyimide.} \n\\label{fig_RBAPB-struct}\n\\end{figure}\n\nOrdering of polymer chains relative to nanotube axes could certainly influence a conductance of the polymer filled nanoparticle junctions. However, it is expected that such influence will depend on many parameters, including the structure of a junction, position, and orientation of chain fragments on the nanotube surface close to a junction, and others. Taking into account all of these parameters is a rather complex task that requires high computational resources for atomistic modeling and ab-initio calculations, as well as complex analysis procedures. Thus, on the current stage of the study, we consider only systems where the polymer matrix was in an amorphous state, i.e. no sufficient polymer chains ordering relative to nanotubes were observed.\n\n\n\n\n\n\\section*{Description of the multiscale procedure} \n\nThe modeling of polymer nanocomposite electrical conductivity is based on a multi-scale approach, in which different simulation models are used at different scales. For the electron transport in polymer composites with a conducting filler, the lowest scale corresponds to the contact resistance between tubes. The contact resistance is determined at the atomistic scale by tunneling of electrons between the filler particles via a polymer matrix, and hence, analysis of contact resistance requires knowledge of the atomistic structure of a contact. Therefore, at the first step, we develop an atomistic model of the contact between carbon nanotubes in a polyimide matrix using the molecular dynamics (MD) method. This method gives us the structure of the intercalated polymer molecules between carbon nanotubes for different intersection angles between the nanotubes. One should mention, that since a polymer matrix is soft, the contact structure varies with time and, therefore, we use molecular dynamics to sample these structures. \n\nBased on the determined atomistic structures of the contacts between nanotubes in the polymer matrix we calculate electron transport through the junction using electronic structure calculations and the formalism of the Green's matrix. Since this analysis requires first-principles methods, one has to reduce the size of the atomistic structure of a contact to acceptable values for the first-principles methods, and we developed a special procedure for cutting the contact structure from MD results. First-principles calculations of contact resistance should be performed for all snapshots of an atomistic contact structure of MD simulations, and an average value and a standard deviation should be extracted. In this way, one can get the dependence of a contact resistance on the intersection angle and contact distance.\n\nUsing information about contact resistances we estimate the macroscopic conductivity of a composite with nanotube fillers. For this, we used a percolation model based on the Monte Carlo method to construct a nanotube network in a polymer matrix. In this model, we used distributions of contact resistances, obtained from the first-principles calculations for the given angle between nanotubes. Using this Monte Carlo percolation model one can investigate the influence of non-uniformities of a nanotube distribution on macroscopic electrical conductivity. \n\nIn the A section, we will describe the details of molecular dynamics modeling of the atomistic structure of contacts between nanotubes. In the B section, we present the details of first-principles calculations of electron transport for estimates of contact resistance. Finally, in the C section, we present the details of the Monte Carlo percolation model.\n\n\\subsection*{A. Preparation of the composite atomic configurations}\n\nInitially, two metallic CNTs with chirality (5,5) were constructed and separated by 6~\\AA. The CNTs consisted of 20 periods along the axis, and each one had the total length of 4.92~nm.\nThe broken bonds at the ends of the CNTs were saturated with Hydrogen atoms.\nThe distance 6 \\AA\\ was chosen, because starting with this distance polymer molecules are able to penetrate the space between CNTs. The three configurations of CNTs junctions were prepared: the first one with parallel CNTs axes (angle between nanotube axes $\\varphi=0^\\circ$),\nthe second one with the axes crossing at 45 degrees ($\\varphi=45^\\circ$), and the third one with perpendicular axes ($\\varphi=90^\\circ$).\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_em1.png}&\n\\includegraphics[width=4cm]{figs\/fig_Larin_Config\/composite_compress1.png}\\\\\n\\end{tabular}\n\\caption{The snapshots of the the nanocomposite system with the parallel orientation of carbon nanotubes at the initial state (left picture) and after the compression procedure (right picture). The black lines represent the periodic simulation cell.} \\label{fig_Larin-conf}\n\\end{figure}\n\nTo produce the polymer filled samples, we used \nthe procedure similar to that employed for the simulations of the thermoplastic polyimides and polyimide-based nanocomposites in the previous works\n\\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14, Nazarychev_MA16, Larin_RSCADV15}. First, partially coiled R-BAPB chains with the polymerization degree $N_p=8$, which corresponds to the polymer regime onset \\cite{Lyulin_MA13, Lyulin_SM14}, were added to the simulation box at random positions avoiding overlapping of polymer chains. This results in the initial configuration of samples with a rather low overall density ($\\rho\\sim 100~$ kg\/m$^3$) (Fig. \\ref{fig_Larin-conf}). Then the molecular dynamics simulations were performed to compress the systems generated, equilibrate them and perform production runs.\n\nThe molecular dynamics simulations were carried out using Gromacs simulation package \\cite{gromacs1, gromacs2}. The atomistic models used to represent both the R-BAPB polyimide and CNTs were parameterized using the Gromos53a6 forcefield \\cite{gromos}. Partial charges were calculated using the Hartree-Fock quantum-mechanical method with the 6-31G* basis set, and the Mulliken method was applied to estimate the values of the particle charges from an electron density distribution. As it was shown recently, this combination of the force field and particle charges parameterization method allows one to reproduce qualitatively and quantitatively the thermophysical properties of thermoplastic polyimides \\cite{Nazarychev_EI}. The model used in the present work was successfully utilized to study structural, thermophysical and mechanical properties of the R-BAPB polyimide and R-BAPB-based nanocomposites \\cite{Larin_RSCADV14, Falkovich_RSCADV14, Nazarychev_EI, Lyulin_MA13, Lyulin_SM14}.\n\nAll simulations were performed using the NpT ensemble at temperature $T=600$~K, which is higher than the glass transition temperature of R-BAPB. The temperature and pressure values were maintained using Berendsen thermostat and barostat \\cite{Berendses_1, Berendsen_2} with relaxation times $\\tau_T= 0.1$~ps and $\\tau_p=0.5$~ps respectively. The electrostatic interactions were taken into account using the particle-mesh Ewald summation (PME) method \\cite{PME1, PME2}. \n\nThe step-wise compression procedure allows one to obtain dense samples with an overall density close to the experimental polyimide density value ($\\rho \\approx 1250-1300$~kg\/m$^3$), as shown in Fig. \\ref{fig_Larin-conf}. The system pressure $p$ during compression was increased in a step-wise manner up to $p=1000$~bar and decreased then to $p=1$~bar. After compression and equilibration, the production runs were performed to obtain the set of polymer filled CNT junction configurations.\n\nAs the conductance of polymer filled CNT junctions is influenced by the density and structure of a polymer matrix in the nearest vicinity of a contact between CNTs, the relaxation of the overall system density was used as the system equilibration criterion. To estimate the equilibration time, the time dependence of the system density was calculated as well as the density autocorrelation function $C_\\rho(t)$:\n\\begin{equation}\nC_\\rho(t)=\\frac{\\langle \\rho(0) \\rho(t)\\rangle}{\\langle \\rho^2 \\rangle},\n\\end{equation}\nwhere $\\rho(t)$ is the density of the system at time $t$ and $\\langle \\rho^2 \\rangle$ is the average density of the system during the simulation.\n\n\nAs shown in Fig. \\ref{fig_Larin-density}a, the system density does not change sufficiently during simulation after the compression procedure. At the same time, the analysis of the density auto-correlation functions shows some difference in the relaxation processes in the systems studied (see Fig. \\ref{fig_Larin-density}b). In the case of the system where CNTs were placed parallel to each other ($\\varphi=0^\\circ$), $C_\\rho(t)$ could be approximated by the exponential decay function $C_\\rho(t)=\\exp(-t\/\\tau)$ with relaxation time $\\tau = 4$~ps. The density relaxation in the systems with crossed CNTs ($\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) was found to be slower. For these two systems density the auto-correlation functions could be approximated by a double exponential function $C_\\rho(t)=A\\exp(-t\/\\tau_1) + (1-A)\\exp(-t\/\\tau_2)$, and the relaxation times determined using this fitting were $\\tau_1=2.7$~ps and $\\tau_2=12.2$~ns (for $\\varphi=90^\\circ$), and $\\tau_1=9.5$~ps and $\\tau_2=24.6$~ns (in case of $\\varphi=45^\\circ$).\n\nNevertheless, the results obtained after the analysis of the system density relaxation allow us to choose the system equilibration time to be 100 ns, which is higher than the longest system density relaxation times determined by the density autocorrelation function analysis. \nThe same simulation time was used in our previous works to equilibrate the nanocomposite structure after switching on electrostatic interactions \\cite{Larin_RSCADV14, Nazarychev_EI, Larin_RSCADV15}. The equilibration was followed by the 150 ns long production run. To analyze the polymer filled CNT junction conductance, 31 configurations of each simulated system, separated by 5 ns intervals, were taken from the production run trajectory.\n\n\\begin{figure}[h]\n\\begin{tabular}{ccc}\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_left.pdf}\\\\\n\\includegraphics[width=7.25cm]{figs\/fig_Larin_dens\/fig_right.pdf}\\\\\n\\end{tabular}\n\\caption{The time dependence of the system density $\\rho$ (a) and the density auto-correlation functions $C_\\rho(t)$ (b) for the systems with various angles between nanotube axes $\\varphi$. The dots correspond to the calculated data. The solid lines correspond to the fitting of $C_\\rho(t)$ with the exponential (in case of $\\varphi=0^\\circ$) or double exponential (in case of of $\\varphi=45^\\circ$ and $\\varphi=90^\\circ$) functions.} \n\\label{fig_Larin-density}\n\\end{figure}\n\nAfter the configurational relaxation is finished, we have to prepare polymer filled CNT junctions configuration for the first-principles calculations of contact resistance. \nThe method we used for the calculations of a contact resistance is based on the solution of the ballistic electronic transport problem, finding the Volt-Ampere characteristic $I(V)$ of a device and\nderiving the contact resistance from the linear part of $I(V)$ corresponding to the low voltages. For this purpose, we employed the Green's function method for \nsolving the scattering problem and the Landauer-Buttiker approach to find the current through a scattering region coupled to two semi-infinite leads, as described in \\cite{Datta}.\nSpecific details of how these techniques are applied in the case of crossed CNTs can be found in \\cite{comp_no_pol}.\n\n\\subsection*{B. The first-principles calculations of the contact resistance of CNTs junctions filled with polymer} \\label{sec_meth_fp}\n\nFor the preparation of a device for the electronic transport calculations, we first form that part of the device which consists of the atoms belonging to the CNTs used in\nthe CNTs+polymer relaxation. Regions with the same geometry as in \\cite{comp_no_pol}\nare cut from the initial 20-period long CNTs, and the rest of the atoms belonging to the CNTs are discarded. This is done to make possible a direct comparison of the results obtained\nfor the polymer filled CNTs junctions with the results for CNTs junctions without polymer reported in \\cite{comp_no_pol} for the same separation of CNTs equal to 6 \\AA. \n\nNote that\nthe CNTs parts of the scattering device contain atoms shifted from their positions in ideal CNTs due to the influence of the adjacent polymer molecules, and these shifts are time-dependent\nas a result of thermal fluctuations. \n\nThe cut regions contain two fragments of CNTs each 9 periods long, and in the case of the CNTs parallel configuration, the CNTs overlap by 7 periods.\nIn the nonparallel configurations, one of the CNTs is rotated around the axis perpendicular to the CNTs axes in the parallel configuration and passing through the\ngeometrical center of a device in the parallel configuration. \n\nAfter the construction of the CNTs part of the scattering region, we attach to it leads that consist of 5 period long fragments of an ideal CNT. The CNTs parts of the scattering regions with the attached leads for the three considered configurations are shown in Fig. \\ref{fig_CNT_conf}.\n\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig1\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig1\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The CNTs parts of the junctions. Left: the parallel configuration, center: CNTs axes are crossing at 45 degrees,\nright: the perpendicular configuration. The leads atoms are colored by green. } \\label{fig_CNT_conf}\n\\end{figure}\n\nAfter the preparation of the CNT parts of the junctions, we still have 17766 atoms in a device. A system with such a large number of atoms cannot be treated by fully first-principles\natomistic methods. On the other hand, keeping all those atoms for a precision calculation of the contact resistance of polymer filled CNTs junctions is not necessary,\nas only those polymer atoms which are close enough to a CNT will serve as tunneling bridges and give a contribution to the junctions conductivity. Thus,\nfor the calculations of the contact resistance, only those atoms were kept which are closer to the CNTs than a certain distance $d$. It has been established by numerical experiments\nthat if the value of $d$ is taken equal to the CNTs separation $d=6$ \\AA\\ this is quite sufficient, and taking into account more distant atoms does not change the contact resistance significantly.\n\nThe procedure of sorting the polymer atoms is as follows. In our molecular dynamics simulations we used 27 separate polymer molecules, consisting of 8 monomers each.\nIf at least one of the atoms of a polymer molecule was closer to the CNTs part of a junction than $d=6$ \\AA, the whole molecule was kept for a while, and discarded \notherwise. Having applied this first part of the procedure, we kept 4 polymer molecules for the parallel configuration, 8 molecules for the perpendicular configuration and\n11 molecules for the 45 degrees configuration. \n\nThen we looked at the monomers of each molecule that survived the first round of selection. The same procedure was applied\nto monomers as the one used earlier for molecules: if at least one of the monomer atoms was closer to the CNTs part of a junction than $d=6$ \\AA, \nthe whole monomer was kept for a while, and \ndiscarded otherwise. \n\nAfter the second round of selection with monomers was over, we dealt in the same manner with the individual residues comprising a monomer. The broken\nbonds that appeared in the second and third stages were saturated with hydrogen atoms. The described procedure resulted in the following numbers of atoms in the whole\ndevice, including the central scattering region and the leads: 881 for the parallel configuration, 1150 for the perpendicular configuration, and 1074 for the 45 degrees configuration.\nThe atomic configurations obtained using the described procedure for the first time steps in the corresponding series are presented in Fig. \\ref{fig_junction_conf}.\n\\begin{figure}[b]\n\\begin{tabular}{ccc}\n\\includegraphics[width=2.25cm]{figs\/fig2\/l_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/c_frame.pdf}&\n\\includegraphics[width=2.25cm]{figs\/fig2\/r_frame.pdf}\\\\\n\\end{tabular}\n\\caption{The atomic configurations for the first-principles calculations of the polymer filled CNTs junctions contact resistance. The configurations are for the first time steps in the corresponding series.\nLeft: the parallel configuration, center: CNTs axes are crossed at 45 degrees,\nright: the perpendicular configuration. The Carbon atoms are gray, the Nitrogen atoms are blue, the Oxygen atoms are red, and the Hydrogen atoms are light gray. The leads atoms are colored by green. } \n\\label{fig_junction_conf}\n\\end{figure}\n\nA fully ab-initio method for electronic structure investigations utilizing a localized pseudo-atomic basis set, as described in \\cite{Ozaki_DFT} and implemented in \\cite{OpenMX}, \nwas used for the calculations of the electronic structures of the whole device and the leads.\nWe used basis set s2p2d1, the Pseudo Atomic Orbitals (PAO) cutoff radius equal to 6.0 a. u., and the cutoff energy of 150 Ry.\nThe pseudo-potentials generated according to Morrison, Bylander, and Kleinman scheme \\cite{pseudopot} were used. For the \ndensity functional calculations, the exchange-correlation functional was used in PBE96 form \\cite{PBE96}.\n\nUsing the electronic structures of the whole device and the leads we calculated energy the dependent transmission function\n through the device. Then the dependence $I(V)$ of the current $I$ on the voltage $V$ between the leads was determined\nwith the Green's function approach as described in detail in \\cite{Datta}. Finally, the Landauer-Buttiker approach was used to find the current through a polymer filled CNTs junction.\n\nSolving a scattering problem for a nano-device at arbitrary voltages is a computationally very complex task since it requires achieving self-consistency for both\nelectron density and induced electrostatic potential simultaneously. Fortunately, for contact resistance calculations one can take\nadvantage of the fact that the required voltages are very low. \n\nAccording to the experimental evidence, the size of a nanocomposite specimen used in conductivity experiments is about 10 mm \\cite{exp_CNT_length_1}, and the typical voltages applied across such specimen do not exceed 100 V \\cite{CNT_book}. Then for the size of a central scattering region about 1 nm, the voltage drop is about 10$^{-5}$ V, which is well within the range where the simplified approach is applicable.\n\nThe question of the modeling of quantum transport in the limit of low voltages was discussed in detail in \\cite{comp_no_pol}, where it was demonstrated that in the case of moderate voltages between leads,\nthe scattering probability $T(E)$ is not sensitive to the details of the electrostatic potential distribution $V({\\mathbf r})$ in the central scattering region, and\nsome physically reasonable approximation may be chosen for $V({\\mathbf r})$. \n\nThis is due to the fact that the difference of the Fermi functions\n$f(\\varepsilon - \\mu_L)$ and $f(\\varepsilon - \\mu_R)$ \nfor the left and right leads with corresponding the chemical potentials $\\mu_L$ and $\\mu_R$, present in the original Landauer-Buttiker formula:\n\\begin{equation}\nI=\\frac{2e}{h} \\int T(\\varepsilon) \\left (f(\\varepsilon - \\mu_L) - f(\\varepsilon - \\mu_R) \\right )d\\varepsilon ,\n\\label{LB_formula}\n\\end{equation}\nwhere $e$ is the elementary charge, $h$ is the Planck constant, $\\varepsilon$ is the electron energy and $T(\\varepsilon)$ \nis the energy dependent transmission probability,\n is reduced in this case to a very narrow and sharp peak centered at the Fermi level\nof the device. \n\nIn addition to the analysis performed in \\cite{comp_no_pol}, in this article, to verify the accuracy of our approach, we made contact resistance calculations for a simple test CNT junction in a coaxial configuration, using both the simplified method we suggest and the full NEGF method, where not only electron charge density but the electric potential was converged as well. The interlead voltage used in those test calculations was\nset to $10^{-4}$ V, and the gap between the CNTs tips was 0.94 \\AA. The atomic configurations for the test calculations are presented in Fig. \\ref{fig_NEGF_compare}. The consistent NEGF calculations produced 1.71~$\\cdot 10^{-5}$ S for the conductance of the junctions shown in Fig. \\ref{fig_NEGF_compare}, while modeling without searching for convergence of potential yielded 1.72 $\\cdot 10^{-5}$ S. \n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{figs\/fig_conf_compare\/fig_compare.pdf}\n\\caption{The atomic configuration used in this work to validate the method of calculating quantum transport by comparing it againts the consistent NEGF approach.} \\label{fig_NEGF_compare}\n\\end{figure}\n\nThus, in our case, a very complex task of finding the $I(V)$ characteristic of a nano-device can be significantly simplified without the loss\nof precision. For the $I(V)$ calculations in the current paper we employed the abrupt potential model introduced in \\cite{comp_no_pol}: the potentials $V_L$ for the\nleft lead and $V_R$ for the right lead were set and were used for all atoms of the corresponding CNT to which that lead belonged. As for the polymer atoms, both $V_L$ and\n$V_R$ can be safely used for them, and at the considered voltages, adopting these two options, as we have checked by direct calculations, leads to the differences in\ncurrent not exceeding 0.1\\%.\n\n\\subsection*{C. The percolation model} \\label{sec_percolation}\nDetermination of the conductivity of a polymer-CNT system can be implemented in 2 stages. First, a percolation cluster is formed, and the second stage implies solving the matrix problem for a random resistor circuit (network). \n\nAt the first stage, the modeling area -- a cube of the linear size $L$ -- is filled with CNTs. For this task, permeable capsules (cylinders with hemispheres at the ends) with a fixed length and diameter were chosen as filling objects corresponding to CNTs. The cube is filled by the successive addition of CNTs until a fixed bulk density of CNTs \n\\begin{equation}\n\\eta = \\frac{((4\/3)\\pi R^3 + \\pi R^2 h)N}{L^3} \\label{cnt_vol_frac}\n\\end{equation}\nin the cube is reached, where $R$ is the radius of the cylinder and hemisphere, $h$ is the height of the cylinder, and $N$ is the number of CNTs in the cube. The percolation problem for permeable capsules was previously solved in \\cite{Xu_16}, and in \\cite{Schilling_15} capsules with a semipermeable shell were considered.\n\nIn the percolation problem, we use periodic boundary conditions as, for example, in \\cite{Bao_2011}. \nWe use the method of finding a percolation threshold based on the Newman and Ziff algorithm \\cite{Newman_2001}, where the identification of a percolation cluster is made at the stage of its formation.\n\n\nWhen a percolation cluster is formed, the obtained CNT configuration is transformed into a resistor circuit (2nd stage). \n\nThe contributions to a conductance matrix resulting from inner resistance of CNTs and the junctions tunneling resistance are usually discussed in connection with constructing conductivity percolation algorithms. Direct measurements of CNTs resistance per unit length are available. In \\cite{Chiodarelli_11}, the inner resistance of CNTs is estimated as $15 \\cdot 10^3$ $\\Omega\/\\mu$m. The results of \\cite{Ebbesen_96} give specific CNTs resistance in the range $(12-86) \\cdot 10^3$ $\\Omega\/\\mu$m. Taking into account that the characteristic CNTs lengths in nanocomposites are about several $\\mu$m \\cite{exp_CNT_length_1,CNT_book}, this results in the inner CNTs resistance approximately $10^4-10^5$ $\\Omega$, which is at least one order of magnitude less than the tunneling resistance obtained in this work. The specific results on tunneling resistance will be discussed below in section \\ref{RND}. Thus, in our percolation model, the inner resistance of CNTs is neglected, and only tunneling resistance of CNTs junctions is taken into account. This can significantly reduce the requirements for computational time. \n\nWhen contact resistance is determined only by tunneling, the principle of compiling the matrix for the percolation problem will be as follows. \nFirst, the matrix (N, N) is compiled from the bonds of percolation elements, where N is the number of CNTs participating in percolation. \nThen this matrix is transformed into the conductivity matrix according to the second Kirchhoff law \n\\begin{equation}\n\\sum_j G_{ij} (V_i-V_j) = 0 \\label {second_Kirchoff}\n\\end{equation}\n- the sum of currents for all internal elements of a percolation network is zero -- where $G_{ij}$ are the elements of the conductance matrix ${\\mathbf G}$ and $V_i$ is the component of the voltage vector ${\\mathbf V}$ corresponding to the $i$-th contact point in a percolation network \\cite{Kirkpatrick_73}. \nThe voltages on the left and right borders of a simulation volume are set to $V_L=1$ V and $V_R=0$ V, respectively. \n\nNow finding the conductivity of the system is reduced to the problem $\\mathbf{GV}=\\mathbf{I}$, where\n$\\mathbf{I}$ is the vector of the currents between the contact points. \n\nThe dimension of the matrix problem can be further reduced to \n(N-2, N-2) by excluding boundary elements. \n\nAfter solving the equation (\\ref{second_Kirchoff}), with the elements of the $\\mathbf{G}$ matrix obtained by the first-principles calculations, we obtain the voltage vector for all internal elements. \nThen, knowing this vector, we sum up all the currents on each of the boundaries. The currents on the left $I_L$ and right $I_R$ \nboundaries of a simulation volume are equal in magnitude and opposite in sign $I_L= - I_R$. Knowing these currents, we determine the conductance of the simulation system as \n$G=|I_L|\/(V_L-V_R) = |I_R|\/(V_L-V_R)$. \nAnd then the conductivity of the composite is calculated as $\\sigma=GL\/S$, where $L$ is the distance between the faces of a simulation volume where voltage is applied, and $S$ is the area of that kind of face. In our case, for the simulation volume of a cubic shape, $S=L^2$, and $\\sigma=G\/L$.\n\nTo calculate the conductivity, the following system parameters were selected: the length of a nanotube is $l=3$ $\\mu$m, the diameter of a CNT is $D=30$ nm, the aspect ratio $l\/D=100$, and the size of the system is 4 $\\mu$m. The same values were used in \\cite{Yu_2010}. We adopted those values to test our realization of the percolation algorithm against the previously obtained results \\cite{Yu_2010}.\nThen for the given parameters for each fixed tube density, the Monte Carlo method (100 implementations of various configurations of CNT networks) was used to calculate the system conductivity. \n\nThe quality of CNTs dispersion is one of the key factors that affect the properties of nanocomposites, and a lot of efforts is taken\nto achieve a homogeneous distribution of fillers \\cite{Mital_14}. In this work, we take into consideration the effect of inhomogeneity of a CNTs distribution on composite conductivity.\nThe spatial density of nanotubes $\\rho_{CNT}$, in this case, has one peak with a Gaussian distribution:\n\\begin{equation} \n\\rho_{CNT}=\\rho_0 \\cdot exp(-(\\mathbf {r-r_0})^2\/\\rho_\\sigma^2), \\label {agglo}\n\\end{equation}\nwhere ${\\mathbf r_0}$ coincides with the geometrical center of a simulation volume, and $\\rho_\\sigma = L\/12$. The value of the $\\rho_0$ parameter is chosen so that the CNTs volume fraction in the inhomogeneous case is the same as in the homogeneous distribution.\n\n\n\\section*{Results and discussion}\\label{RND}\nTo find the contact resistance of polymer filled CNTs junctions one first needs to find their Volt-Ampere characteristics $I(V)$ and to determine the voltage range where\n$I(V)$ is linear and is not sensitive to the specific distribution of an electrostatic potential in the scattering region. In Fig. \\ref{fig_I_V} the $I(V)$ plot\nfor the first time step in the atomic geometry series for the parallel configuration is shown.\n\\begin{figure}[b]\n\\begin{tabular}{lr}\n\\includegraphics[width=3.5cm]{figs\/I_V_test\/I_V.pdf} &\n\\includegraphics[width=3.5cm]{figs\/I_V_test_low_V\/I_V_low_V.pdf}\n\\end{tabular}\n\\caption{The Volt-Ampere characteristic for the polymer filled CNTs junction corresponding to the first time step in the series for the parallel configuration.\nLeft frame: maximum inter-lead voltage is $10^{-3}$ V, right frame: $10^{-4}$ V. The circles correspond to the results of calculations,\nthe lines are guides for an eye.} \n\\label{fig_I_V}\n\\end{figure}\n\nIt is clearly seen from Fig. \\ref{fig_I_V} that up to about $10^{-4}$ V the $I(V)$ characteristic is linear, and after that value, it starts to deviate from a\nsimple linear dependence. Thus, for the calculations of a contact resistance $R$ and its inverse, a junction conductivity $G$,\nwe used the electrical current values obtained for the inter-lead voltage equal to $10^{-4}$ V. Note, that according to our estimates in section \\ref{sec_meth_fp}, a characteristic voltage drop on the length of a CNTs tunneling junction is about $10^{-5}$ V which is well within the region where the linear $I(V)$ is observed.\n\nThe time dependences of the junctions conductances for the three considered configurations are presented in Fig. \\ref{fig_G_t}. One might expect that the \nshifts of both CNTs atoms and polymer atoms in the central scattering region due to thermal fluctuations would lead to fluctuations of junctions conductances $G$, but \nquantitative characteristics of this phenomenon such as minimum $G_{min}$, maximum $G_{max}$, mean values $\\langle G \\rangle$ and a standard deviation $G_\\sigma$ can only be captured by highly precise fully atomistic \nfirst-principles methods, like those employed in the current paper. The resulting fluctuations of conductance are very high. For the parallel CNTs configuration\nthe minimum value, $G_{min} = 2.4 \\cdot 10^{-8}$ S, and the maximum value, $G_{max} = 6.8 \\cdot 10^{-6}$ S, differ by more than two orders of magnitude, for the 45 degrees and perpendicular\nconfigurations the corresponding ratios are about 30. The same strong variations of conductance over time were reported in \\cite{Penazzi_2013} for the coaxial\nCNTs configuration, where the results were obtained using a semi-empirical tight-binding approximation. Thus, it is obvious that for the precise determination\nof a conductance of polymer filled CNTs junctions one needs to use fully atomistic approaches, and phenomenological methods taking atomic configurations into account \non the average are not reliable.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/G_t\/g_t.pdf}\n\\caption{The time dependence of the polymer filled junctions conductance $G$ in S. Red color correspond to the parallel configuration, the green lines -- to the perpendicular configuration\nand the blue lines -- the 45 degrees configuration. The results of the calculations are shown by circles, the saw-tooth lines serve as a guide for an eye.\nThe straight solid lines designate the mean values of conductance $\\langle G \\rangle$, and the dashed ones -- $\\langle G \\rangle \\pm G_\\sigma$.} \n\\label{fig_G_t}\n\\end{figure}\n\n\n\n\\begin{table*}[h]\n \\centering\n\\caption{The results of statistical analysis of the CNTs junctions conductances in S, for CNTs separation equal to 6 \\AA \\ and different values of the CNTs crossing angles $\\varphi$, \nwithout polymer from \\cite{comp_no_pol}, and with polyimide R-BAPB filling obtained in the current paper.\n}\n\\begin{tabular}{cccccc}\n\\textrm{$\\varphi$}&\n\\textrm{no polymer}&\n\\textrm{}&\n\\textrm{polymer present}&\n\\textrm{}&\n\\textrm{}\\\\\n\\textrm{}&\n\\textrm{the results of \\cite{comp_no_pol}}&\n\\textrm{$G_{min}$}&\n\\textrm{$G_{max}$}&\n\\textrm{$\\langle G \\rangle$}&\n\\textrm{$G_\\sigma$}\\\\\n\\hline\n& & & & &\\\\ \n0 & 3.6$\\cdot10^{-13}$ &2.4$\\cdot10^{-8}$ &6.8$\\cdot10^{-6}$ & 1.8$\\cdot10^{-6}$ & 1.6$\\cdot10^{-6}$\\\\\n& & & & & \\\\\n\\hline\n& & & & &\\\\ \n0.2$\\pi$ & 1.4$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.25$\\pi$ & --- &4.8$\\cdot10^{-9}$ &1.4$\\cdot10^{-7}$ &3.4$\\cdot10^{-8}$ &2.7$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.3$\\pi$ & 1.2$\\cdot10^{-14}$ & --- & --- & --- &\\\\\n& & & & &\\\\ \n\\hline\n& & & & &\\\\ \n0.5$\\pi$ & 4.2$\\cdot10^{-14}$ &2.2$\\cdot10^{-9}$ &4.3$\\cdot10^{-8}$ & 1.4$\\cdot10^{-8}$ & 1.1$\\cdot10^{-8}$\\\\\n& & & & &\\\\ \n\\end{tabular}\n\\label{tab_cond}\n\\end{table*}\n\nTo assign the tunneling resistance to a polymer filled CNT junction the following algorithm was used. First, for each junction that had to be used in the percolation algorithm, a uniformly distributed random number $\\varphi$ in the range [0, $\\pi\/2$] was generated. The value of the intersection angle for that junction was assigned to the obtained random number. The mean values and standard deviations for CNTs tunneling resistances and conductances calculated for the different atomic configurations corresponding to the different time steps are known for $\\varphi = 0, \\pi\/4$, and $\\pi\/2$. Analyzing figure 4 of \\cite{comp_no_pol}, one can see that though an angle dependence of current and hence conductivity is a rather complex function, in the first approximation one can adopt a roughly piece-wise linear character for this function with the minimum located at $\\varphi=0.25\\pi$. Thus the logarithm of the mean value of conductance $\\mu_\\varphi$ for the generated $\\varphi$ was set by linear interpolation between the logarithms of the mean values of conductances for $\\varphi=0$ and $\\varphi=\\pi\/4$ or $\\varphi=\\pi\/4$ and $\\varphi=\\pi\/2$ presented in Table \\ref{tab_cond}. The same algorithm was applied to finding the standard deviation values $\\sigma_\\varphi$ for the generated $\\varphi$. After the statistical parameters for the generated $\\varphi$ are estimated, the conductivity of the junction is set to a random number generated using the normal distribution with the parameters $\\mu_\\varphi$ and $\\sigma_\\varphi$.\n\n\nIn \\cite{comp_no_pol}, the conductances were reported for the CNTs junctions with almost the same geometry as the CNTs parts of the devices considered in the current paper. The only difference \nbetween the configurations is that in this work the carbon atoms belonging to the CNTs part of the central scattering region are shifted somewhat from their equilibrium positions\ndue to the interaction with polymer. The maximum values of those shifts along the $x$, $y$, and $z$ coordinates lie in the range $0.2-0.5$ \\AA. This gives us the possibility to directly compare the current results to the data \nfrom \\cite{comp_no_pol} and,\nthus, elucidate the influence of polymer filling on the junctions conductance. The corresponding data and the results of a basic statistical\nanalysis for the case of the polymer filled junctions are provided in Tab. \\ref{tab_cond}.\n\nFirst, as was expected, filling CNTs junctions with polymer creates carrier tunneling paths and increases junctions conductance by 6-7 orders of magnitude.\nSecond, it is evident that the CNTs axes crossing angle is crucial for the junctions conductivity when polymer is present as it was the case without\npolymer \\cite{comp_no_pol}. At the same time, the sharp dependence of polymer filled junctions conductance on the CNTs crossing angle is somewhat different\nfrom the analogous dependence for junctions without polymers. While in the latter case this dependence is sharply non-monotonous, with a pronounced minimum\nat the angles around $0.25\\pi$, in the former case there is a significant difference between the conductance values for the parallel and nonparallel configurations, but\nthe configurations with the angle $\\varphi$ between CNTs angles equal to $0.25\\pi$ and $0.5\\pi$ have very close conductances, and their mean values averaged over\ntime $\\langle G \\rangle_{45}$ and $\\langle G \\rangle_{per}$ lie within the\nranges $\\langle G \\rangle \\pm G_\\sigma$ of each other. Moreover, in contrast to the geometries without polymer, for the polymer filled CNTs junctions $\\langle G \\rangle_{per}$ is lower than $\\langle G \\rangle_{45}$ by a factor of 2.4.\n\nNote also that for the parallel configuration, the polymer influence on the junction conductance is more pronounced than for the nonparallel ones.\nFor the parallel configuration, adding polymer to a junction of CNTs separated by 6.0 \\AA\\ with initial conductance of 3.6$\\cdot 10^{-13}$ S \nproduces conductance mean value equal to 1.8$\\cdot 10^{-6}$ S. This gives the factor 0.5$\\cdot 10^7$; the value of the analogous factor for the perpendicular configuration is 0.33$\\cdot 10^6$.\n\nThe probable reason for the more effective conductance increase, when polymer is added, for the configurations with smaller angles between CNTs axes, is\nthat the smaller is an intersection angle, the larger is the overlap area between CNTs where polymer can penetrate and, thus, create tunneling bridges.\nThe higher fluctuation of conductance over time for the parallel configuration can be explained by the same reason: a larger CNTs overlap area gives\nmore freedom for polymer atoms to adjust their positions.\n\nThe dependence of the calculated composite conductivity $\\sigma$ on CNTs volume fraction $\\eta$ $\\sigma(\\eta)$ is presented in Fig. \\ref{fig_cond_pol}. \nThe value of the percolation threshold $\\eta_{thresh}$ is estimated in this work as $\\eta_{thresh} = 0.007$. To test our realization of the percolation algorithm against the previous results of \\cite{Yu_2010} we calculated the composite conductivity using\nthe fixed CNTs junction conductance equal to 1 M$\\Omega$ for all junctions in a percolation network. Our results presented in\nFig. \\ref{fig_cond_pol} by the red circles coincide within graphical accuracy to the results of \\cite{Yu_2010} shown by the red squares.\n\\begin{figure}\n\\includegraphics[width=8cm]{figs\/fig9\/cnt_9.pdf}\n\\caption{ The conductivity of CNT enhanced nanocomposites above the percolation threshold obtained in this work. The symbols of different shapes and colors are used to designate the following results. The red circles: the fixed CNTs tunneling junctions resistance of $R=1$ M$\\Omega$ is used, the red squares: the conductivity results for the fixed 1 M$\\Omega$ tunneling resistance from \\cite{Yu_2010}, the black triangles: the same as the red circles but for $R=0.54$ M$\\Omega$ corresponding to the mean value of the tunneling junction resistance for the parallel configuration from Table \\ref{tab_cond}, the blue rhombi: the angle dependence of the CNTs junctions resistance is taken into account, the green pentagons: CNTs agglomeration is considered in addition to the angle dependence. The red line is a guide for an eye.}\n\\label{fig_cond_pol}\n\\end{figure}\nThe 1 M$\\Omega$, used in various sources, for example, \\cite{Yu_2010}, is not an arbitrary value, but rather a typical contact resistance of CNTs junctions filled with polymer for simple geometries. In this work, we obtained for the parallel configurations $1\/\\langle G \\rangle =0.54$ M$\\Omega$. The $\\sigma(\\eta)$ dependence for the fixed tunneling resistance of $0.54$ M$\\Omega$ is shown in Fig. \\ref{fig_cond_pol} by the black triangles.\n\nTaking into account the angle dependence of CNTs junctions conductances with the statistical parameters according to Table\n\\ref{tab_cond}, leads to the lowering of composite conductivity just above the percolation threshold by the factor of about 30. This number correlates with the ratio of the mean conductances for the parallel, $\\langle G \\rangle_{par}$, and 45$^\\circ$, $\\langle G \\rangle_{45}$, configurations: $f_G = \\langle G \\rangle_{par}\/ \\langle G \\rangle_{45} = 53$, but is higher than $f_G$ due to the presence of junctions with $\\varphi < \\pi\/4$. \n\nIf agglomeration of CNTs, modeled by the inhomogeneity of their distribution according to formula (\\ref{agglo}) and the parameter values discussed in section \\ref{sec_percolation}, is taken into account in addition to the angle dependence of conductance, the composite conductivity is further reduced above a percolation threshold by the factor of 2.5. Lowering of conductivity of composites with agglomerated CNTs above a percolation threshold was also mentioned in \\cite{Bao_2011}. The calculated results for the conductivity of a percolation network of agglomerated CNTs are shown in Fig. \\ref{fig_cond_pol} by the green pentagons.\n\nWe believe that in this work we have identified some of the key factors that influence nanocomposites electrical conductivity: the geometry of tunneling junctions and changes of atomic configurations due to thermal fluctuations. Among other causes that may affect conductivity, but are not considered in this work, are the presence of defects in CNTs, a distribution of CNTs over chiralities, lengths, aspect ratios, different separations between CNTs. \n\n\nUntil the specific experiments on conductivity for R-BAPB polyimide are not available, we can make a preliminary comparison of our modeling results to the available experimental results for different composites. The calculated conductivity of composite just above the percolation threshold at $\\eta=0.0075$ is equal to $3.6\\cdot10^{-3}$ S\/m.\nThis is a reasonable value that falls into the range of experimentally observed composites conductivities (for the comprehensive compilation of experimental results see Table 1 of \\cite{Eletskii}). To make a quantitative comparison of modeling results with experiments the full details of nanocomposites structure are necessary, including the CNTs parameters mentioned in the previous paragraph. All these factors can be easily incorporated into the approach proposed in this work if sufficient computational resources are available.\n\n\\section*{Conclusions}\n\nWe have proposed a physically consistent, computationally simple, and at the same time precise, multi-scale method for calculations of electrical conductivity of CNT enhanced\nnanocomposites. The method starts with the atomistic determination of the positions of polymer atoms intercalated between CNTs junctions, proceeds with the fully first-principles calculations of polymer-filled CNTs junctions conductance at the microscale and finally performs\nmodeling of percolation through an ensemble of CNTs junctions by the Monte-Carlo technique. \n\nThe developed approach has been applied to the modeling of electrical conductivity of polyimide R-BAPB + single wall (5,5) CNTs nanocomposite.\n\nOur major contributions to the field are the following. We have proposed a straightforward\nmethod to calculate a contact resistance and conductance for polymer-filled CNTs junctions with arbitrary atomic configurations without resorting to any simplifying assumptions. We have demonstrated that a consistent multiscale approach, based on solid microscopic physical methods can give reasonable results, lying within the experimental range, for the conductivity of composites and suggested a corresponding work-flow. \n \n\nIt is shown that a contact resistance\nand nanocomposite conductivity is highly sensitive to the geometry of junctions, including an angle between CNTs axes and subtle thermal shifts of polymer atoms in an inter-CNT's gap.\nThus, we argue that for the precision calculations of nanocomposites electrical properties rigorous atomistic quantum-mechanical approaches are indispensable.\n\nWe have to admit though, that we have not considered all possible factors that may influence CNT junctions conductivity on the micro-level. We concentrated on the CNTs crossing angle factor because it seems to be the\nmost influential. The additional factors may include, for example, defects in CNTs,\nCNTs overlap lengths and others. On the other hand, the proposed approach may be used to take all those factors into account, provided sufficient computational resources are available.\n\n\\section*{Acknowledgments}\n\nThis work was supported by the State Program \"Organization of\nScientific Research\" (project 1001140) and by the Research Center \"Kurchatov Institute\" (order No.~1878~of~ 08\/22\/2019).\n\nThis work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC \"Kurchatov Institute\", \\url{http:\/\/ckp.nrcki.ru\/}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlytj b/data_all_eng_slimpj/shuffled/split2/finalzzlytj new file mode 100644 index 0000000000000000000000000000000000000000..b086688f574fbfe07a9b6ba463b5cd189681f0b7 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzlytj @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro} \nIt is well known that the so called Next Linear Collider (NLC) will provide\nopportunities for both discovery and precision measurements~\\cite{nlc}.\nWith the construction of the next generation of $e^+e^-$ linear \ncolliders, with a center of mass energy up to 1500 GeV and which will \nbe able to operate also in $\\gamma\\gamma$, $\\gamma e^-$ and $e^-e^-$ modes, new \nperspectives arise in detecting new physics beyond the electroweak standard \nmodel in processes having non-zero initial electric charge (and non-zero lepton \nnumber). For instance, new resonances of doubly charged scalar and vector \nbosons will enhance the cross section and in this case, it is possible that the \nmeasurement of the left-right asymmetry ($A_{RL}$) in electron-electron \nscattering can be a better parameter to signal out those resonances (if any) \nthan the total cross section itself. On the other hand, although the First Muon \nCollider (FMC) will be one of the type $\\mu^-\\mu^+$ (it is a circular \nmachine)~\\cite{gunion}, if this sort of technology is dominated perhaps a Linear \nMuon Collider (LMC) will be feasible.\nIn this work we will consider mainly $e^-e^-$ processes but we will also \ncomment the muon case.\n\n\\subsection{Lepton M\\o ller scattering at high energies}\n\\label{subsec:moller}\nThe appealing features for studying the parity violating asymmetries between\nthe scattering of left- and right-handed polarized electrons on a variety of \ntargets was pointed out some years ago by Derman and Marciano~\\cite{dm}. \nIn particular, in the context of the electroweak standard model (ESM) the \nmeasurement of the $A_{RL}$ asymmetry determined the relative \nsign between the weak and electromagnetic interactions. \n\nIt is well known that the left-right asymmetry in electron-electron M\\o ller \nscattering (fixed target) and unpolarized target is rather small \n$\\sim10^{-7}$ (see below)~\\cite{dm}. \nNotwithstanding, since the scattering has a large cross section, we \ncan expect that fixed target experiences like those at SLAC~\\cite{slac} will\nhave enough number of events to determine such a small number. The typical beam\nenergy of such experiences is 50 GeV and both the incident electron beam and \nthe target electron can be polarized. \nOn the other hand, collider experiences were stopped years ago at very low \nenergies, around 1.2 GeV~\\cite{co1}. \nAlthough the $A_{RL}$ asymmetry is larger in colliders \nthan in fixed target experiments, the former have a \nsmaller cross section. Hence, it seems at first sight that it will be \ndifficult a measurement of that asymmetry with large statistics in \ncolliders. \n\nIn general we have processes like \n\\begin{equation}\nl_1(p_1,\\lambda_1) + l_2(p_2,\\lambda_2)\\to l'_1(q_1,\\Lambda_1) + \nl'_2(p'_2,\\Lambda_2)\n\\label{ll}\n\\end{equation}\nwhen $l_{1,2}=e,\\mu$ and $l'_{1,2}=e,\\mu,\\tau,L$, with $L$ an exotic\nlepton. In Eq.~(\\ref{ll}) $p_i,q_i$ ($\\lambda_i,\\Lambda_i$) denote \nthe initial and final lepton momenta (helicities). In this work we \nwill consider only processes like\n\\begin{equation}\nl(p_1,\\lambda_1) + l(p_2,\\lambda_2)\\to l(q_1,\\Lambda_1) + \nl(q_2,\\Lambda_2)\n\\label{ll1}\n\\end{equation}\nNote that if we were considering quarks in the final state, there must be a \nmixing of ordinary quarks with exotic ones since with the known quarks we can \nnot build a doubly charged state. \n\nOur main goal is to compare the left-right asymmetry in the context of the\nelectroweak standard model in fixed target experiments with the respective one in colliders\nexperiments. Since we are not neglecting the masses of the fermions we\ncan use the same amplitudes for the case of $e^-e^-$ and $\\mu^-\\mu^-$\nscattering. We also study the behaviour of that quantity when a vector or\nscalar bilepton is considered in collider experiences.\n\n\\subsection{4-spinor helicity eigenstates and left-right asymmetry}\n\\label{subsec:sub1}\n\nIn this work we will not neglect the ESM Higgs-lepton \ninteractions because in collider \nexperiences with muons the Higgs contributions are not, in principle, \nnegligible. \nBeside, in the future we will consider processes in which the \nfinal lepton can be $\\tau$ or even an exotic \nlepton $L^-$ (or even exotic quarks $J$). \nHence, we will work with massive 4-spinor eigenvectors of the helicity \noperator, defined as\n\\begin{equation}\n\\Sigma=\n\\left(\n\\begin{array}{cc}\n\\vec{\\sigma}\\cdot\\vec{p}\/\\vert\\vec{p}\\vert & 0 \\\\\n0 &\\vec{\\sigma}\\cdot\\vec{p}\/\\vert\\vec{p}\\vert\n\\end{array}\n\\right),\n\\label{helicity}\n\\end{equation}\nwith the gamma matrices in the chiral representation, that are given by:\n \n\\begin{equation}\nu_{lR}=N_l\\,\\left(\n\\begin{array}{c}\n(1+\\eta_l)\\cos(\\theta_l\/2)\\,e^{-i\\phi_l\/2} \\\\\n(1+\\eta_l)\\sin(\\theta_l\/2)\\,e^{i\\phi_l\/2} \\\\\n(-1+\\eta_l)\\cos(\\theta_l\/2)\\,e^{-i\\phi_l\/2}\\\\\n(-1+\\eta_l)\\sin(\\theta_l\/2)\\,e^{i\\phi_l\/2}\n\\end{array}\n\\right),\\quad u_{lL}=N_l\\,\\left(\n\\begin{array}{c}\n(-1+\\eta_l)\\sin(\\theta_l\/2)\\,e^{-i\\phi_l\/2} \\\\\n-(-1+\\eta_l)\\cos(\\theta_l\/2)\\,e^{i\\phi_l\/2} \\\\\n(1+\\eta_l)\\sin(\\theta_l\/2)\\,e^{-i\\phi_l\/2}\\\\\n-(1+\\eta_l)\\cos(\\theta_l\/2)\\,e^{i\\phi_l\/2}\n\\end{array}\n\\right),\n\\label{e1a}\n\\end{equation}\n\nwhere we have defined \n\\begin{equation}\nN_l=\\sqrt{\\frac{E_l}{(1+\\eta_l^2)}}, \\quad \\eta_l^2=\\frac{E_l+m_l}{E_l-m_l},\n\\label{e2}\n\\end{equation}\nbeing $E_l$ and $m_l$ the total energy and rest mass for the lepton $l$. \n($\\hbar=1$ and $c=1$ throughout this work.) In Eq.~(\\ref{e1a}) \n$u_R,u_L$ denote positive energy spinors\nwith positive (right) and negative (left) helicity, respectively. \nSimilarly expressions can be obtained for the negative energy solutions but we \nwill not write them explicitly.\n\nThe left-right asymmetry is defined as\n\\begin{equation}\nA_{RL}(ll\\to ll)=\\frac{d\\sigma_R-d\\sigma_L}{d\\sigma_R+d\\sigma_L},\n\\label{asy1}\n\\end{equation}\nwhere $d\\sigma_{R(L)}$ is the differential cross section for one right\n(left)-handed lepton $l$ scattering on an unpolarized lepton $l$.\nThat is\n\\begin{equation}\nA_{RL}(ll\\to ll)=\\frac{(d\\sigma_{RR}+d\\sigma_{RL})-(d\\sigma_{LL}\n+d\\sigma_{LR})}{(d\\sigma_{RR}+d\\sigma_{RL})+(d\\sigma_{LL}\n+d\\sigma_{LR})},\n\\label{asy2}\n\\end{equation}\nwhere $d\\sigma_{ij}$ denotes the cross section for incoming leptons with\nhelicity $i$ and $j$, respectively, and they are given by \n\\begin{equation}\nd\\sigma_{ij}\n\\propto\\sum_{kl}\\vert M_{ij;kl}\\vert^2,\\quad i,j;k,l=L,R.\n\\label{dsigma}\n\\end{equation}\n\nWhen both leptons are \nidentical we can use the property $M_{RL;RL}=M_{LR;LR}$ (which implies \n$d\\sigma_{RL}=d\\sigma_{LR}$) assured by rotational invariance in the former \ncase. Hence, for calculating the asymmetry $A_{RL}$ in Eq.~(\\ref{asy2}) we\nuse\n\\begin{mathletters}\n\\label{sigmas}\n\\begin{equation}\nd\\sigma_{RR}\\propto\\vert M_{RR;RR}\\vert^2+\\vert M_{RR;LR}\\vert^2+\n\\vert M_{RR;RL}\\vert^2+\\vert M_{RR;LL}\\vert^2,\n\\label{s1}\n\\end{equation}\n\n\\begin{equation}\nd\\sigma_{RL}\\propto\\vert M_{RL;RR}\\vert^2+\n\\vert M_{RL;RL}\\vert^2+\\vert M_{RL;LR}\\vert^2+\n\\vert M_{RL;LL}\\vert^2,\n\\label{s2}\n\\end{equation}\n\n\\begin{equation}\nd\\sigma_{LR}\\propto\\vert M_{LR;RR}\\vert^2+\\vert M_{LR;RL}\\vert^2+\n\\vert M_{LR;LR}\\vert^2+\\vert M_{LR;LL}\\vert^2,\n\\label{s3}\n\\end{equation}\nand\n\\begin{equation}\nd\\sigma_{LL}\\propto\\vert M_{LL;RR}\\vert^2+\\vert M_{LL;RL}\\vert^2+\n\\vert M_{LL;LR}\\vert^2+\\vert M_{LL;LL}\\vert^2.\n\\label{s4}\n\\end{equation}\n\\end{mathletters}\n\nAnother interesting possibility is the case when both leptons are \npolarized. We can define an asymmetry $A_{R;RL}$ in which\none beam is always in the same polarization state, say right-handed, and \nthe other is either\nright- or left-handed polarized (similarly we can define $A_{L;LR}$):\n \\begin{equation}\nA_{R;RL}=\\frac{d\\sigma_{RR}-d\\sigma_{RL}}{d\\sigma_{RR}+d\\sigma_{RL}},\\qquad\nA_{L;RL}=\\frac{d\\sigma_{LR}-d\\sigma_{LL}}{d\\sigma_{LL}+d\\sigma_{LR}}.\\qquad\n\\label{ar}\n\\end{equation}\n\nWe can define also an asymmetry when one incident particle is right-\nhanded and the other is left-handed and the final states are right- and left or\nleft- and right-handed:\n\\begin{equation}\nA_{RL;RL,LR}=\\frac{d\\sigma_{RL;RL}-d\\sigma_{RL;LR}}{d\\sigma_{RL;RL} \n+ d\\sigma_{RL;LR}}\n\\label{ar3}\n\\end{equation}\nor similarly, $A_{LR;RL,LR}$. These asymmetries, in Eqs.(\\ref{ar}) and \n(\\ref{ar3}), are also dominated by QED \ncontributions. However, this will not be the case if a bilepton resonance\ndoes exist at typical energies of the NLC. To show this fact is the goal of \nthe next section.\nThese asymmetries can be calculated for both fixed target (FT) and \ncolliders (CO) experiments.\n\nThis work is organized as following. In Sec.~\\ref{sec:sm} we study the\nleft-right asymmetries for both fixed target and collider experiences\nin the context of the ESM and for $e^-e^-\\to e^-e^-$. We also briefly consider\nthe case of $\\mu^-\\mu^-$ colliders. The same asymmetries for colliders\nare shown in Sec.~\\ref{sec:331} but considering the contributions of \nvector and scalar bileptons. Our conclusions appear in the last section\nwhile we left the appendices \\ref{sec:aa}--\\ref{sec:higgs} for showing the \nseveral amplitudes and cross sections we have used in this work.\n\n\\section{Left-right asymmetries in the electroweak standard model}\n\\label{sec:sm}\nLet us consider the lepton-lepton (diagonal) process in the context of \nthe electroweak standard model at tree level. The relevant part of the \nLagrangian of this model is\n\\begin{equation}\n{\\cal L}_F=-\\sum_i\\,\\frac{g\\,m_i}{2M_W}\\,\\bar\\psi_i\\psi_i\\,H^0-\ne\\sum_iq_i\\bar\\psi_i\\gamma^\\mu\\psi_iA_\\mu-\\frac{g}{2\\cos\\theta_W}\\,\n\\psi_i\\gamma^\\mu(g^i_V-g^i_A\\gamma^5)\\psi_iZ_\\mu,\n\\label{su21}\n\\end{equation}\n$\\theta_W\\equiv\\tan^{-1}(g'\/g)$ is the weak mixing angle, $e=g\\sin\\theta_W$\nis the positron electric charge with $g$ such that\n\\begin{equation}\ng^2=\\frac{8G_F M^2_W}{\\sqrt2};\\quad {\\rm or}\\quad \ng^2\/\\alpha=4\\pi\\sin^2\\theta_W,\n\\label{gf}\n\\end{equation}\nwith $\\alpha\\approx 1\/128$; and the vector and axial neutral couplings are\n\\begin{equation}\ng^i_V\\equiv t_{3L}(i)-2q_i\\sin^2\\theta_W,\\quad\ng^i_A\\equiv t_{3L}(i),\n\\label{gva}\n\\end{equation}\nwhere $t_{3L}(i)$ is the weak isospin of the fermion $i$ and $q_i$ is the \ncharge of $\\psi_i$ in units of $e$. \n\n\\subsection{$e^-e^-$ Fixed target experiences}\n\\label{subsec:ftsmee}\n\nWith the spinors in Eqs.~(\\ref{e1a}) we have obtained all the helicity\namplitudes of Ref.~\\cite{dm} when $m_e\\to 0$; even with the actual \nvalue of $m_e$ the numerical value for the $A_{RL}(ee)$ asymmetry coincides \nwith that obtained in Ref.~\\cite{dm} ({\\it i.e.}, at tree level) as it \nshould be. In fact, neglecting the lepton masses we obtain the asymmetry for \na fixed target (FT) experience in the ESM context:\n\\begin{equation}\nA^{{\\rm FT,ESM}}_{RL}(ll\\to ll)\\approx-\\frac{G_F Q^2}{\\sqrt2\\pi\\alpha}\\,\n\\frac{1-y}{1+y^4+(1-y)^4}\\,(1-4\\sin^2\\theta_W),\n\\label{assi0}\n\\end{equation}\nwhere $Q^2=-(p'_1-p_1)^2$, $y=\\sin^2(\\theta\/2)$ and the others constants are the \nwell know $\\alpha$, $G_F$ and $\\sin^2\\theta_W$. \nFor FT experiences we have $Q^2=-q^2=y(p_1+p'_1)=y(2m^2_{l_1}+~\n2m_{l_1}E_{\\rm beam})$. In Eq.~(\\ref{assi0}) the approximation $m^2_e\\ll \nQ^2\\ll M^2_Z$ was used. \nAlso terms of the order $m_l\/E_{\\rm beam}$ were \nneglected. This approximation is valid even for $E_{\\rm beam}\\approx1$ TeV \n(typical energies of the NLC) since in this case $Q^2\\approx 0.5\\; \n({\\rm GeV})^2$. \n\nWe see that for a FT experiment, the $A_{RL}$ asymmetry is small,\nbeside of the factor $1-~4\\sin^2\\theta_W$ we have a small $Q^2$~\\cite{cm}.\nUsing the helicity amplitudes given in Appendices~\\ref{sec:aa} and \n\\ref{sec:ab}, with $E_{\\rm beam}=50$ GeV, $\\theta\\approx90^o$ ($y\n=1\/2$) and for 100\\% beam polarization, we obtain, \n\\begin{equation}\nA^{{\\rm FT,ESM}}_{RL}(ee\\to ee)\\approx-3\\times 10^{-7},\n\\label{aft1}\n\\end{equation}\n\nHence, for FT experiments we have confirmed the value obtained in Ref.~\\cite{dm} for \nthe case of $e^-e^-$ at tree level:\n\\begin{equation}\nA^{\\rm FT,ESM}_{RL}(ee\\to ee)\\approx -6\\times 10^{-9}\n\\left( E_{\\rm beam}\/1\\,{\\rm GeV}\\right).\n\\label{aft2}\n\\end{equation}\nRadiative corrections reduce the tree level value \nfor the $A^{\\rm FT,ESM}_{RL}$ asymmetry in $e^-e^-$ by $40\\pm 3\\%$~\\cite{cm}.\nAlthough the asymmetry is small in FT experiments, the cross section of the \nM\\o ller scattering is high: $\\sigma_T\\approx 10^3$ nb. In Eq.(\\ref{ac1}) \nwe show the differential cross section for the fixed target case. \n\nThe $A^{\\rm FT,ESM}_{R;RL}$ asymmetry defined in Eq.~(\\ref{ar}) \nis shown in Fig.~1 as a function of $\\sin^2(\\theta\/2)$. This \nasymmetry is dominated in the context of the ESM by purely QED \neffects~\\cite{bincer,ft1}. However, we will see that it sensible to\npurely electroweak effects. In fact, for massless electrons we have \n(see Table I)\n\\begin{equation}\nA^{\\rm FT,ESM}_{R;RL}(ee\\to ee)=\\frac{\\frac{1-y^4-(1-y)^4}{y^2(1-y)^2}+\ns\\,\\beta_W (g^2_V+g^2_A+4g_Vg_A)+O(\\beta_W^2)}\n{\\frac{1+y^4+(1-y)^4}{y^2(1-y)^2} +\ns\\,\\beta_W (3g^2_V+g^2_A+4g_Vg_A)+O(\\beta_W^2)},\n\\label{ar2}\n\\end{equation}\nwhere $\\beta_W=g^2\/4\\pi\\alpha M^2_Z\\approx5.2\\times10^{-4}$. \nNotice however that there are electroweak contributions which do not have\nthe suppression factor $1-4s^2_W$. Hence, the last effects are greater than\nin the case when the target is unpolarized {\\it i.e.,} for the case of \nthe asymmetry defined in Eq.~(\\ref{asy2}). This is due to the fact that the \ncontributions proportional to $g^2_A$ do not cancel out. When we\ncalculate the asymmetry with the target unpolarized the cancelation\noccurs and we obtain the expression given in Eq.~\\ref{assi0}.\nIn Fig.~1 it appears the $A_{R;RL}$ asymmetry when all amplitudes in\nAppendix~\\ref{subsec:foton} are considered.\nIn Fig.~2 we show the same asymmetry but only considering\nthe photon amplitudes which are given in Appendix~\\ref{subsec:foton}.\nWe note that, in fact, the electroweak contributions are important as can be \nseen by comparing Figs.~1 and 2.\n\nFor theoretical purely QED calculations in the Born approximation see Refs.~\\cite{bincer,rusos} and for radiative correction \nsee Ref.~\\cite{rusos}. For experimental data see Ref.~\\cite{ft1}. \nThe difference with the theoretical calculations~\\cite{bincer,rusos} is due to \nthe electroweak effects that we have considered. This indicates that this \nasymmetry is sensitive to massive vector boson exchanges.\nMeasurements of the QED contribution to the $A_{R;RL}$ asymmetry \n({\\it i.e.,} with both the incident beam and target longitudinally polarized) \nin a fixed target experience with a beam energy up to near 20 GeV, are given \nin Ref.~\\cite{ft1}.\n\n\\subsection{$e^-e^-$ Collider experiences}\n\\label{subsubsec:coeesm}\n\nNext, let us consider a collider (CO) experiment with the \ninteractions given in Eq.~(\\ref{su21}). In this case the transferred \nmomentum is\n\\begin{eqnarray}\nQ^2&=&-(p_1'-p_1)^2\\nonumber \\\\ &= &\n-m^2_{l'_1}-m^2_{l_1}+\\frac{s}{2}-2\\cos\\theta\\left[\\left(\\frac{s}{4}-m^2_{l_1}-\nm^2_{l'_{1}}\\right)\n\\left(\\frac{s}{4}-2m^2_{l_1}\\right)\\right]^{\\frac{1}{2}},\n\\nonumber \\\\ &\\to & s\\,\\sin^2\\frac{\\theta}{2},\\qquad {\\rm when}\\;\nm_{l_i}\\to0.\n\\label{cine}\n\\end{eqnarray}\nHence, now $Q^2$ is not a small factor. \n\nAs we said before, in the ESM the total cross section is $\\approx 1000$ nb in \nFT and $\\approx 10^{-3}$ nb in collider experiences at $\\sqrt{s}=300$ GeV. \n(See the Eqs.(\\ref{ac1}) and (\\ref{ac2}) \nfor the differential cross section in the context of the electroweak standard \nmodel for collider and fixed target experiment, respectively.) \nSo, the required luminosity, or the parameter to be measured, in colliders must \nbe larger than in fixed target experiments. \nAccording to Czarnecki and Marciano~\\cite{cm} the number of scattering events\nrequired for a $10^{-8}$ statistical accuracy in FT is of the order of \n$10^{16}$.\nHowever, since the cross section for M\\o ller scattering at FT is large\nat low $Q^2$ the above requirement does not intimidate.\nOn the other hand, in collider experiments the cross section is rather smaller\nthan that of FT at high energies. However, as in this case the asymmetry is \n$\\sim10^{5}$ times the corresponding value for FT it is \npossible that $10^5$ events\/year, which is the number of events \nexpected for a luminosity of ${\\cal L}\\approx10^{33}\\,\n{\\rm cm}^{-2}{\\rm s}^{-1}$, might be enough to measure the \n$A_{RL}$ asymmetry in collider experiments. \nIn addition, if new physics do really exist at \nthe hundred's GeV level the cross section in colliders can serve to detect new \nresonances. If this is indeed the fact, the cross section will have the\ntypical resonance enhancement(s).\nRecall also that the CO measurements \nof $A_{RL}$ in M\\o ller scattering unlike the case of FT experiments \nwill not have a background on $eN$ scattering. \n\nThus, it is worth to consider the contribution to the $A_{RL}$\nasymmetry at the energies of next linear colliders (electrons up \nto 1.5 TeV) and also in muon colliders (muons up to 4 TeV). Although large \npolarization implies sacrifice in luminosity at a muon collider~\\cite{gunion}, \nnew physics (if any) with clear signature could be distinguished even with low \nluminosity. Anyway, the asymmetry depends more on the polarization than on the \nluminosity.\n\nAt energies $\\sqrt{s}=300$ GeV and $\\theta\\approx90^o$ \n(in the center of mass frame) the asymmetry is \n\\begin{equation}\nA^{\\rm CO, ESM}_{RL}(ee\\to ee)\\approx -0.05,\n\\label{asec}\n\\end{equation}\nCompare with the value for the FT \nexperiment given in Eq.~(\\ref{aft1}). \nIn Fig.~3 we show the $A^{{\\rm CO,ESM}}_{RL}$ asymmetry as a \nfunction of $\\sqrt{s}$ and $\\theta$. Notice that this asymmetry is a smooth\nfunction of both variables.\n\n\nWe can integrate in the scattering angle and define the asymmetry \n$\\overline{A}_{RL}$ as\n\\begin{equation}\n\\overline{A}_{RL}(ll\\to ll)=\\frac{(\\int d\\sigma_{RR}+\\int d\\sigma_{RL})-(\\int \nd\\sigma_{LL}\n+\\int d\\sigma_{LR})}{(\\int d\\sigma_{RR}+\\int d\\sigma_{RL})+(\\int d\\sigma_{LL}\n+\\int d\\sigma_{LR})},\n\\label{aint}\n\\end{equation}\nwhere $\\int d\\sigma_{ij}\\equiv \\int^{175^o}_{5^o}d\\sigma_{ij}$. This integrated\nasymmetry for $ee\\to ee$, in the electroweak standard model \n($\\overline{A}^{\\rm CO,ESM}_{RL}(ll\\to ll)$), appears in Fig.~4 as function \nof $\\sqrt{s}$. We see that this integrated asymmetry varies slowly with\n$\\sqrt{s}$ and it is almost constant at the value $-0.02$ above\n$\\sqrt{s}=400$ GeV. \n\n\\subsection{$\\mu^-\\mu^-$ Experiences}\n\\label{subsubsec:mumusm}\n\nAn hypothetical fixed target $\\mu^-\\mu^-$ experience at the same \n$E_{\\rm beam}$ energy used in Sec.~\\ref{subsec:ftsmee} gives \n$A_{RL}(\\mu\\mu)\\approx5.4\\times~10^{-5}$ for $\\theta\\approx90^o$. In this case\n$Q^2_{\\rm muon}=(m_e\/m_\\mu)Q^2_{\\rm electron}\\approx4.2\\,{\\rm GeV}^2$. \nA more realistic case is the $\\mu^-\\mu^-$ collider where, for the same \nexperimental conditions than in the previous subsection, we obtain \n\\begin{equation}\nA^{\\rm CO,ESM}_{LR}(\\mu\\mu\\to\\mu\\mu)\\approx-0.1436.\n\\label{asmc}\n\\end{equation}\nNotice the large value for the asymmetry for the muons with respect to the \nelectron case given in Eq.~(\\ref{asec}).\n\nIn the electron case the Higgs contributions are negligible. For the muon case\nif we do not take into account the Higgs amplitudes given in Appendix~\n\\ref{subsec:h0} the asymmetry in Eq.~(\\ref{asmc}) has the value of $-0.1437$\nshowing a slight dependence on the Higgs contributions.\nThere are also effects of the lepton mass in the $\\gamma,Z$ contributions \nto the asymmetry that are the main responsible for the different values in \nEqs.~(\\ref{asec}) and (\\ref{asmc}).\nIndependently of the numerical value of the Higgs contributions, \nthe important issue is that all its main contributions (see the amplitudes\ngiven in given in Sec.~\\ref{subsec:h0} cancel out in the\n$A_{RL}$ asymmetry defined in Eqs.~(\\ref{asy2}). In fact, as \nwe will see in the next section, even in models with larger scalar-lepton\ncouplings the extra scalar contributions cancel out.\n\n\\section{Left-right asymmetries in bilepton models}\n\\label{sec:331}\n\nAn example of possible new physics are models with a doubly charged vector \nboson~\\cite{331}. Then, the charged current interactions in terms of the \nphysical basis is given by\n\\begin{equation}\n-\\frac{g}{\\sqrt2}\\left[\\bar\\nu_L E^{\\nu\\dagger}_LE^l_Ll_LW^+_\\mu+ \n\\bar l^c_L\\gamma^\\mu E^{lT}_RE^\\nu_L \\nu_L V^+_\\mu-\n\\bar l^c_L E^{lT}_RE^l_L l_L U^{++}_\\mu\\right]+H.c.,\n\\label{cc331}\n\\end{equation}\nwith $l'_L=E^l_Ll_L,\\quad l'_R=E^l_Rl_R,\\quad \\nu'_L=E^\\nu_L\\nu_L$,\nthe primed (unprimed) fields denoting symmetry (mass) eigenstates.\nWe see from Eq.~(\\ref{cc331}) that for massless neutrinos we have no mixing\nin the charged current coupled to $W^+_\\mu$ but we still have mixing \nin the charged currents coupled to $V^+_\\mu$ and $U^{++}_\\mu$. \nThat is, if neutrinos are massless we can always chose \n$E^{\\nu\\dagger}_LE^l_L=1$. However, \nthe charged currents coupled to $V^+_\\mu$ and $U^{++}_\\mu$ are not diagonal in \nflavor space and the mixing matrix ${\\cal K}=E^{lT}_RE^\\nu_L$ has\nthree angles and three phases. (An arbitrary $3\\times 3$ unitary matrix has \nthree angles and six phases. In the present case, however, the matrix \n${\\cal K}$ is determined entirely by the charged lepton sector, so we \ncan rotate only three phases~\\cite{liung}).\n\nIf the bilepton $U^{++}_\\mu$ is not too heavy, say, with a mass of the order of\na few hundred GeVs it is possible that it will appear as a resonance in the next \nlinear colliders (NLC). \nAssuming that the matrix ${\\cal K}$ is almost diagonal we can neglect the mixing \neffects in processes involving the same initial and final leptons as $l^-l^-\\to \nl'^-l'^-$. The helicity amplitudes for the $U$-exchange in the $s$-channel are \ngiven, at tree level, in the Appendix~\\ref{sec:ad}. \nWe do not known yet the total width $\\Gamma_U$ of the $U^{--}$ bilepton, \nhowever, we know that at least it has to decay into $l^-l'^-$:\nIf these were the only channels we have for the width\n\\begin{equation}\n\\Gamma(U^{--}\\to l^-_il^-_j)=\\frac{G_FM^3_U}{6\\sqrt2\\pi}\\,\\vert {\\cal K}_{ij}\n\\vert^2,\n\\label{gammaul}\n\\end{equation}\nand if ${\\cal K}_{ij}$ is almost diagonal the main decays will be $U\\to e^-e^-,\n\\mu^-\\mu^-,\\tau^-\\tau^-$ and with $M_U=300$ GeV, we have \n$\\Gamma(U\\to{\\rm leptons})\n\\approx 36$ GeV. However, in models with such a vector bilepton there are \nseveral Higgs fields and also exotic quarks. Hence, we should have to take into\naccount $U^{--}\\to {\\rm scalars}$ and also $U^{--}\\to d_3\\bar{J}^{-5\/3}, \n\\bar{u}_{1}(\\bar{u}_2)J^{-4\/3}$, with $d_3$ and $u_{1,2}$ denoting symmetry \neigenstates of the known quarks and $J^Q$ ($\\bar{J}^Q$) \ndenotes an exotic quark (anti-quark) of electric charge $Q$. \nThe left-handed mixing matrices, defined by the relation among the symmetry \neigenstates (primed fields) with the mass eigenstates (unprimed fields)\n{\\it i.e.}, $U'_L=V^u_LU_L$ and $D'_L=V^d_LD_L$, survive in the Lagrangian \ndensity. Thus, the partial width is of the form (neglecting the masses of the \nquarks $J$)\n\\begin{equation}\n\\Gamma(U^{--}\\to d_3\\bar{J}^{-5\/3})=\\frac{CG_FM^3_U}{6\\pi\\sqrt2}\\vert\n(V^d_L)_{3i}\\vert^2,\n\\label{gammauq}\n\\end{equation}\nand similarly for the other decays into $u$'s quarks. Notice that the mixing\nmatrix in Eq.~(\\ref{gammauq}) is not necessarily equal to the Cabibbo-\nKobayashi-Maskawa mixing matrix which is defined as $V_{\\rm CKM}=\nV^{u\\dagger}_LV^d_L$. The parameters in both\nEqs.~(\\ref{gammaul}) and (\\ref{gammauq}) are constrained for other processes\nbut here we will not consider this constraints. A realistic calculation will \nhave to take into account the masses of the exotic quarks $J$. At present, \nhowever, there is no experimental data bounds on the values of these masses.\nOf course, there are also trilinear vector interactions \n$U^{--}\\to W^-V^-$ that it will occur if $M_V+M_W45 $ GeV~\\cite{h2,h22} but we recall that the \nphenomenology of the $H^{++}$ derives from its couplings. The processes which\nhave to be considered are $\\gamma\/Z\\to H^{++}H^{--}$ and \n$H^{--}\\to l^-l'^-$~\\cite{gunion2}. In the model of \nRef.~\\cite{331} the coupling $W^-W^-$ with\ndoubly charged scalars does not exist (the vertex always includes single \ncharged vector bilepton $V^-$, {\\it i.e.}, the coupling which really exist is \nwith $V^-W^-$). \n\nAlthough the bilepton model we consider here are gauge\ninvariant theories which has its own scalar spectrum we will\nconsider in this work the bilepton contributions as an extra \ncontributions to the electroweak standard model ones due to the fact\nthat in any case the neutral Higgs contributions are smaller than that\nof the doubly charged vector bilepton. \nWe denote this \nby using the $\\{{\\rm ESM}\\}$ and $\\{{\\rm ESM+U}\\}$ labels in \ncross sections and asymmetries.\nHowever, as we will show, the doubly charged scalar bilepton contributions,\ndenoted by $\\{{\\rm ESM+H}\\}$, cancel out in the numerator of the $A_{RL}$\nasymmetry. Hence, in practice our calculations can be considered a truly\neffect of the bilepton model of Ref.~\\cite{331}.\n\n\\subsection{$e^-e^-$ High energy colliders}\n\\label{subsubsec:nlc}\n\nBy using the expression in Eqs.(\\ref{ac1}) and (\\ref{dcsu}) we can compute\nthe effect of such bilepton fields. In Fig.~5 we show the ratio\n$\\sigma^{\\rm CO,ESM+U}\/\\sigma^{\\rm CO,ESM}$ for the vector bilepton \ncase as a function of $\\sqrt{s}$ and \n$\\Gamma_U$. We observe that this ratio becomes appreciable for $\\sqrt{s}$\nnear the $U$-resonance as expected. In this case we have chosen and arbitrary\n(but reasonable) value of $M_U=300$ GeV to illustrate the cross section \nbehavior. We also note that this quantity is smoothly dependent on\nthe $\\Gamma_U$ values. \nSince the cross section could be large at the $U$-peak, if the mass of the $U$ \nis lower than the energies of future collider experiences, it is interesting to \nknow the value of the $A_{RL}$ asymmetry at these energies. \n\nHere, we will not show the analytic expression of $A^{\\rm CO}_{RL}$, but at \nthe same conditions than Eq.~(\\ref{asec}) it gives\n\\begin{equation}\nA^{\\rm CO,ESM+U}_{RL}(ee\\to ee)= -0.099,\n\\label{asec331}\n\\end{equation}\nand we see that it is larger than the respective value obtained in the \ncontext of the ESM (see Eq.~(\\ref{asec})). \nA more illustrative quantity is the angular-integrated \nasymmetry defined in Eq.~(\\ref{aint}) and\ndenoted here by $\\overline{A_{RL}}\\,^{\\rm CO,ESM+U}(ee\\to ee)$.\nWe show it in Fig.~6 as a function of \n$\\sqrt s$ and $\\Gamma_U$. \nWe can observe that the dependence of $\\Gamma_U$\ncancels out in $\\overline{A_{RL}}\\,^{\\rm CO,ESM+U}(ee\\to ee)$, so that even if \nwe do not use a realistic calculation for $\\Gamma_U$ based in all possible decay \nchannels predicted by the model, the result will be, in fact, independent of it. \nAs expect the asymmetry is larger in the $U$-pole. \nIn Fig.~7 we quote the quantity\n\\begin{equation}\n \\delta\\, \\overline{A}_{RL}(ee\\to ee)\\equiv \n(\\overline{A}\\,^{\\rm CO,ESM+U}_{RL}-\\overline{A}\\,^{\\rm CO,ESM}_{RL})\/\n\\overline{A}\\,^{\\rm CO,ESM}_{RL},\n\\label{deltadef}\n\\end{equation} \nwith the $\\overline{A}$'s defined in \nEq.~(\\ref{aint}), as a function of $\\sqrt{s}$ and \n$\\Gamma_U$. As observed above this quantity is independent of $\\Gamma_U$.\n$\\delta \\,\\overline{A_{RL}}$ is large at the $U$-resonance, however, we would \nlike to stress \nthat it remains appreciably large even far from the $U$-peak. That particular \nbehaviour suggest that this quantity could be the one to be considered in the \nsearch for new physics, like the bilepton $U^{--}$, in future colliders.\n\nNext let us consider the scalar contribution in the model. The $s$-channel\namplitudes appear in Appendix~\\ref{sec:higgs}. In this model the amplitudes\nin Appendix.~\\ref{subsec:h0} are used by making $v\\to v_S$,\nand, as we said before, in models with several scalar multiplets it is\npossible that one of the neutral scalars does not contribute significantly to \nthe $W^-$ and $Z$ masses. It means that the vacuum expectation value (VEV) \nof such a scalar can be of the order of a few GeV. \nIn this case there are some scalar contributions\nwhich are proportional to $m_l\/v_S$ with $m_l$ denotes the lepton mass and\n$v_S$ is a VEV which gives mass for the leptons. \n\nHere we will use $M^{++}_{H_i}=50,\\, 300$ GeV \n(for both $i=1,2$ although the masses of the triplet and singlet might be \ndifferent) and $v_S=10$ GeV only as an illustration. We also take for\ncolliders $\\sqrt{s}=300$ GeV.\nTaken into account the ESM plus the scalar contribution (but not the \nvector $U^{--}$ contributions) we obtain\n\\begin{equation}\nA^{\\rm CO,ESM+H}_{RL}(ee\\to ee)=-0.05, \n\\label{hee}\n\\end{equation}\nfor $y=1\/2$. We see that this value is the same\nof the pure ESM contribution given in Eq.~(\\ref{asec}). \nThis is not a surprise since \nthe purely scalar contributions cancel out in both the neutral Higgs\n(as in Appendix.~\\ref{subsubsec:mumusm}) and also in the doubly \ncharged Higgs contributions as can be seen from the amplitudes in \nAppendix~\\ref{sec:higgs} and the definition of the asymmetry in \nEq.~(\\ref{asy2}). However, in the muon-muon case the interference terms\namong the neutral Higgs and $\\gamma,Z$ contributions are relatively\nsignificant as we will see in the next subsection.\n\n\\subsection{$\\mu^-\\mu^-$ High energy colliders}\n\\label{subsec:mumuco} \nWe can calculate all the above asymmetries in the case of a possible\n$\\mu^-\\mu^-$ collider. For instance taking into account the contributions\nof the $U^{--}$ vector bilepton we obtain\n\\begin{equation}\nA^{\\rm CO,ESM+U}_{RL}(\\mu\\mu\\to\\mu\\mu)= -0.1801,\n\\label{asmcu}\n\\end{equation}\nwhile the contribution of the $H^{--}$ scalar bilepton gives\n\\begin{equation}\nA^{\\rm CO,ESM+H}_{RL}(\\mu\\mu\\to \\mu\\mu)=-0.1436,\n\\label{amm}\n\\end{equation}\nfor $M_H^{++}=50$ and $300$ GeV \n\nAs discussed above, in the model with bileptons the VEV of the neutral scalar\ncoupled mainly with leptons does not need to be equal to the VEV of the\nESM ($v\\sim246$ GeV) but it can have a lower value, say $v_S\\sim10$ GeV.\nFor the muon case we calculate the asymmetry in Eq.~(\\ref{asmcu}) with and\nwithout the ESM Higgs scalar contributions given in Appendix~\\ref{subsec:h0}. \nThe respective values are $-0.1436$ and $-0.1442$. \nIf the mass of the $U^{--}$ were 500 GeV the asymmetry in Eq.~(\\ref{asmcu})\nhas the value $-0.0133$.\n\n\\section{Conclusions}\n\\label{sec:con}\n\nWe have considered in this work \nthat collider ex\\-pe\\-ri\\-ments could be suitable for studying the left-right\nasymmetries in lepton-lepton scattering.\nIn fact, we have shown that in this case the asymmetries are larger than those\nwhich appear in fixed target experiments. \nIn particular we have shown that \nthere are significant difference between electron and muon asymmetries, being \nthe muon ones larger than the electron ones due to mass effects. However, more \ndetails were given here for the electron case.\nFor instance, the $A_{RL}$ asymmetry, when the contributions of the $U^{--}$ \nvector bilepton are considered, is larger than the ESM value, as can be seen \nfrom Fig.~7 for the electron-electron case. \nOn the other hand, the purely scalar bilepton contributions cancel out in the\nasymmetries as can be seen from the expression given in \nAppendices~\\ref{subsec:h0} and \\ref{sec:higgs}. Only the interference terms\nbetween the scalars and $\\gamma,Z$ amplitudes survive and these are not\nnegligible in the muon-muon case.\n\nConcerning the asymmetry $A_{R;RL}$ defined in \nEq.~(\\ref{ar}) {\\it i.e.,} when both beams are polarized, we see from Table II \nthat it is given by\n\\begin{equation}\nA^{\\rm CO,ESM+U}_{R;RL}(ee\\to ee)\\approx -1+O(\\beta_W),\n\\label{ar4}\n\\end{equation}\nand comparing with Eq.~(\\ref{ar2}) we see that the maximal value of\n$A^{\\rm CO,ESM+U}_{R;RL}(ee\\to ee)$ and its dependence on $y$ are different in \nmodels with vector bileptons and the electroweak standard model \n(almost purely QED) case.\nAt the $U$-resonance $d\\sigma_{RL}\\gg d\\sigma_{RR}$, hence $A^{{\\rm CO},\nESM+U}_{R;RL}(ee\\to ee)\\approx-1$. In the ESM by the contrary we have \n$d\\sigma_{RR}\\gg d\\sigma_{RL}$, hence $A_{R;RL}^{\\rm CO,ESM}(ee\\to ee)\n\\approx+1$. Similarly, $A^{\\rm CO,ESM+U}_{L;RL}(ee\\to ee)\\approx-1$ and \n$A^{\\rm CO,ESM}_{L;RL}(ee\\to ee)\\approx+1$.\nAlthough this asymmetry is large in both the ESM (see Fig.~1) and the bilepton \nmodel the sign of it is opposite and could be useful for discovering such a\nvector field. \n\nSummarizing, we have calculated in this work several left-right asymmetries in \nthe context of the ESM and in models with both vector and scalar bileptons.\n\nFinally a remark. In some models with vector bileptons like $U^{--}$ the decay \n$\\mu^-\\to e^-e^-e^+$ (and similar ones) does occur. The branching ratio for \nthis decay is $\\sim10^{-12}$~\\cite{pdg}.\nThis bound strongly limits the ${\\cal K}_{e\\mu}$ couplings. The branching \nfraction for $\\mu\\to 3e$ decay is\n\\begin{equation}\nB(\\mu\\to 3e)\\propto \\left(\\frac{{\\cal K}_{e\\mu}{\\cal K}_{ee}}{M^2_U}\\right)^2\\,\n\\frac{1}{G_F^2}.\n\\label{mu3e}\n\\end{equation}\nFor $M_U=300$ GeV and ${\\cal K}_{ee}\\approx1$ the experimental value for the\nabove branching ratio implies ${\\cal K}_{e\\mu}\\sim10^{-6}$. \nThe doubly charged scalar will also contribute to these rare decays.\n\n\\acknowledgments \nThis research was partially supported by Conselho Nacional de \nCi\\^encia e Tecnologia (J.C.M. and V.P.) and fully financed by Funda\\c c\\~ao \nde Amparo \\`a Pesquisa do Estado de S\\~ao Paulo (M. C. R.). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec-intro}\n\n\\subsection{The EKG system}\\label{sec-einstein-kg-eq}\nThe Einstein-Klein-Gordon (EKG) system takes the form of\n\\begin{subequations}\n\\begin{equation}\n\\breve R_{\\alpha \\beta} - \\frac{1}{2} \\breve R \\breve g_{\\alpha \\beta} = \\mathcal{T}_{\\alpha \\beta}(\\phi), \\label{eq-Einstein-source}\n\\end{equation}\n\\begin{equation}\n \\mathcal{T}_{\\alpha \\beta}(\\phi)= \\breve D_\\alpha \\phi \\breve D_\\beta \\phi - \\frac{1}{2} \\breve g_{\\alpha \\beta} \\left( \\breve D^\\mu \\phi \\breve D_\\mu \\phi + m^2 \\phi^2 \\right), \\label{def-energy-Mom-kg}\n \\end{equation}\n \\end{subequations}\nwhere $\\breve R_{\\alpha \\beta}$ and $\\breve R$ denote the Ricci and scalar curvature of an unknown Lorentzian\nmetric $\\breve g_{\\alpha \\beta}$ respectively, and $\\mathcal{T}_{\\alpha \\beta}(\\phi)$ is the energy momentum tensor for a massive scalar field $\\phi$. The Bianchi identities \nimply that the scalar field $\\phi$ satisfies the Klein-Gordon (KG) equation\n\\begin{equation} \\label{eq-kg}\n\\Box_{\\breve g} \\phi - m^2 \\phi = 0. \n\\end{equation}\n$\\breve D$ denotes the covariant derivative associated to $\\bar g_{\\alpha \\beta}$ and the geometric wave operator is $\\Box_{\\breve g} = \\breve D_\\alpha \\breve D^\\alpha$. We use $m$ to denote the mass of the KG field.\n\nOur first motivation for studying the EKG system comes form the problem of the nonlinear stability of the supersymmetric backgrounds posed by S.T. Yau. Consider the Kaluza-Klein spacetime of the form $\\mathcal{M} = \\mathbb{R}^{1+3} \\times S^1$. Then the five-dimensional metric depending on $S^1$ periodically can be Fourier-expanded. The projections of the five-dimensional Einstein equations onto the zero (massless) modes and the non-zero (massive) modes give rise to a wave-Klein-Gordon system. Subject to the zero mode perturbation, both of the work by Wyatt \\cite{Wyatt-17}, and Branding-Fajman-Kroencke \\cite{Branding-Fajman-Kroencke-18} have recently contributed on this aspect. \\cite{Wyatt-17} is related to the nonlinear stability of the Kluza-Klein spacetime with the base on the Minkowski space, while \\cite{Branding-Fajman-Kroencke-18} is concerning with the one base on the Milne model. There is also related result by Choquet-Bruhat and Moncrief \\cite{Bruhat-Moncrief-U1} who establish the stability of $U(1)$-symmetric spacetimes to $1+3$-dimensions.\n\nThe backgrounds we consider are the following family of cosmological vacuum spacetimes. Let $\\bar M$ be a $4$-manifold of the form $ I \\times \\Sigma,$ where $I \\subset \\mathbb{R}$ is an interval, $\\Sigma$ is a compact $3$-manifold admitting an Einstein metric $\\gamma$ with negative Einstein constant $\\lambda$. We choose $\\lambda =-2$ so that the Einstein metric $\\gamma$ is a hyperbolic metric with sectional curvature $-1$. Then $(\\bar M, \\bar\\gamma)$ with $\\bar\\gamma$ given by $$\\bar\\gamma = -dt^2 + t^2 \\gamma$$ is a solution to the vacuum Einstein equations and known as a $1+3$-dimensional \\emph{Milne model}. Such model undergoing accelerated expansion in the future direction is locally isometric to the $k=-1$ vacuum Friedmann-Lem\\^{i}atre-Robertson-Walker (FLRW) model. Andersson and Moncrief \\cite{A-M-04} first consider the nonlinear stability of this model (called hyperbolic cone spacetimes there), and show that for the constant mean curvature (CMC) Cauchy data for the vacuum Einstein equations close to the standard data for the hyperbolic cone spacetimes, the maximal future Cauchy development is globally foliated by the CMC Cauchy surfaces and causally geodesically complete to the future. This also motivates the current project. We refer to \\cite{Anderson-CMC, Andersson-CMC-02, Andersson-04-cmc, Gerhardt-CMC} and references therein for more backgrounds on the existence of CMC foliations.\n\nThere are several results concerning the decay of the KG field \\cite{Katayama-wave-kg, Klainerman-KG-85, Shatah-KG, Sideris-KG}. In contrast to the wave equation, the KG equation is not conformal invariant and hence does not commute well with the symmetry of scaling. Due to this fact, Klainerman \\cite{Klainerman-KG-85} employs the hyperbloid foliations which respect the Lorentz invariance of the KG operator and derives the asymptotic behavior of the KG field by the energy method alone. Furthermore, Katayama \\cite{Katayama-wave-kg} somehow overcomes the incompatibility between the wave and KG field. Other method including Fourier analysis for the wave-Klein-Gordon system is established by Ionescu-Pausader \\cite{Ionescu-Pausader-17}. Global solutions for the semilinear KG equations in the FLRW spacetimes are studied by Galstian-Yagdjian \\cite{Galstian-Yagdjian}.\n\n\nLet us review some nonlinear stability results of the Minkowski spacetime. Christodoulou and Klainerman \\cite{Christodoulou-K-93} initiate a covariant proof based on the Bel-Robinson energy. Their proof relies upon the geometric foliations of spacetime, including maximal $t = \\text{const}$ slices and a family outgoing null cones, and null condition hidden inside the Bianchi equations. Lindblad and Rodnianski \\cite{Lind-Rod-05, Lind-Rod-10} devise another proof based on the wave coordinate gauge, under which the Einstein equations satisfy the weak null condition. This method is extended to the case of massive Einstein-Vlasov system by Lindblad and Taylor \\cite{Lind-Taylor-17}. All of these works require the full symmetries of the Minkowski spacetime, including the conformal symmetries, suggesting that these methods can not apply to the EKG system straightforwardly. Recently, inspired by \\cite{Katayama-wave-kg} and \\cite{Lind-Rod-05, Lind-Rod-10}, Ma and LeFloch \\cite{Ma-Lefoch-16, Ma-Lefloch-EKG} make use of the hyperboloid method to address an energy argument for both of the wave equation and KG equation uniformly, and then prove the nonlinear stability for the EKG system. In addition to the $L^\\infty-L^\\infty$ estimate for the wave equations \\cite{Katayama-wave-kg, Lind-Rod-05, Lind-Rod-10}, Ma and LeFloch \\cite{Ma-Lefoch-16} also demonstrate an $L^\\infty-L^\\infty$ estimate for the KG equation, which is crucial in dealing with the nonlinear couplings \\cite{Ma-Lefloch-EKG}. Via this hyperboloid method, the massive Einstein-Vlasov system is alternatively understood by Fajman, Joudioux and Smulevici \\cite{F-J-S-17}. As the KG field and the massless Vlasov filed are lack of the conformal symmetries as well. Taylor \\cite{Taylor-17} gives a proof of the stability for the massless Einstein-Vlasov system with compact support assumptions. The proof in fact reduces to a semi global problem since the matter is shown to be supported in a strip going to null infinity.\n \nThe works of Andersson and Moncrief \\cite{A-M-04, A-M-11-cmc} also constitute nonlinear stability results for the vacuum Einstein equations. Compared to \\cite{A-M-04}, the backgrounds considered in \\cite{A-M-11-cmc} are extended to a family of $1+n$-dimensional Milne model. The proof of \\cite{A-M-04} is based on the Bel-Robinson energy and its higher-order generalization. Our proof possesses some similarities to that of \\cite{A-M-04} as far as we both treat the Bianchi equations as evolution equations to derive the energy estimates for the Weyl field. While in \\cite{A-M-11-cmc}, the authors turn to the Einstein evolution equations and use a wave equation type of energy introduced in \\cite{A-M-03-local} for the energy estimates. Related to this, Andersson and Fajman \\cite{Andersson-Fajman-17} show the future stability of the $1+3$-dimensional Milne model for the massive Einstein-Vlasov system. We remark that the work of \\cite{A-M-04, A-M-11-cmc, Andersson-Fajman-17} all base on the constant mean curvature, spatially harmonic (CMCSH) gauge, with which the local existence theorem is proved in \\cite{A-M-03-local}. \n\nIn this paper, we are concerning the nonlinear stability of the $1+3$-dim Milne model for the EKG system. We work within the CMC gauge with zero shift. The local existence theorem then follows by adjusting the argument in \\cite{Christodoulou-K-93} where the maximal gauge with zero shift is used. Our main difficulty lies in the fact that the KG field is not scale invariant and hence scaling does not qualify as a commutator vector field. We now recast the KG equation in a first order form and focus on the $1+3$ form of the Bianchi-Klein-Gordon equations. More precisely, we split the normal and tangential derivatives in the formulae of Bianchi-Klein-Gordon equations, treating them as evolution equations, and propose an energy argument in which the high order energies contain only high order of spatially covariant derivatives. Practically, we commute only the spatially covariant derivatives with the Bianchi-Klein-Gordon equations in $1+3$ from. As a result, the high order derivatives of the KG field along the normal direction possess no good estimates, and hence normal derivative is referred as a bad derivative. Due to this covariant proof, we avoid some unnecessarily nonlinear couplings that will arise when working in coordinates (see \\cite{Ma-Lefloch-EKG} for the Minkowski case). Meanwhile, we find that the ``genuine'' coupling structures admit some cancellations, such that in the Bianchi equations, the nonlinear couplings involve no the high order of bad (normal) derivatives of the KG field. On the other hand, the difficulty does not diminish because if we proceed to the top order energy estimates, some nonlinear couplings become borderline terms. Another difficulty comes from the resulted borderline terms when commuting the spatially covariant derivatives with the KG equation. Fortunately, these borderline terms exhibit linearizable feature. Motivated by \\cite{Christodoulou-08} (see also \\cite{Miao-Pei-Yu, Wang-Wei-17, Wang-Yu-13, Wang-Yu-16}), we resolve these difficulties by the method of linearization and the hierarchy of energy. In particular, the $L^\\infty-L^\\infty$ estimate for the KG equation proposed in \\cite{Ma-Lefoch-16} turns out to be helpful in the procedure of linearization. Besides, we reveal the subtle relation between the lapse and the KG field, with which we manage to improve the estimates step by step and finally reduce the nonlinear borderline terms into linear ones. \nAlternatively, if we use the CMCSH gauge, no new difficulty occurs and our method works as well. \n\n\\subsection{Main result}\\label{sec-main-result}\n\\begin{theorem}\\label{main-thm-into}\nLet $\\Sigma$ be a compact hyperbolic $3$-manifold without boundary. Assume that $(\\Sigma, \\tilde g_0, \\tilde k_0, \\phi_0, \\phi_1)$ is a CMC data set satisfying the constraint equations of the EKG system \\eqref{eq-Einstein-source}-\\eqref{def-energy-Mom-kg}, \\eqref{eq-kg}. \nLet $-3t_0^{-1}= \\tau_0= \\text{tr} \\tilde k_0 = \\text{constant} <0,$ with $t_0 > \\max \\{1, 9 m^{-1} \\}$. Then there is an $\\varepsilon > 0,$ so that if \n\\begin{equation}\\label{intro-initial-data}\n\\begin{split} \n&\\|\\tilde R_{0ij} + 2 t_0^{-2} \\tilde g_{0ij} \\|_{H^3(\\Sigma, \\tilde g_0)} + \\| \\tilde k_0 + t_0^{-1} \\tilde g_{0ij} \\|_{H^4(\\Sigma, \\tilde g_0)} \\leq \\varepsilon, \\\\\n& \\|\\phi_0\\|_{H^5(\\Sigma, \\tilde g_0)} + \\|\\phi_1\\|_{H^4(\\Sigma, \\tilde g_0)} \\leq \\varepsilon,\n\\end{split}\n\\end{equation}\nwhere $\\tilde R_{0ij}$ denotes the Ricci curvature of $\\tilde g_0$, the maximal development $(\\bar M, \\breve g)$ of the EKG data set $(\\Sigma, \\tilde g_0, \\tilde k_0, \\phi_0, \\phi_1)$ has a global CMC foliation in the expanding direction (to the future of $(\\Sigma, \\tilde g_0)$ in CMC time $\\tau = \\text{tr} \\tilde k$) and $(\\bar M, \\breve g)$ is future causally geodesically complete.\n\\end{theorem}\n\\begin{remark}[The existence of CMC data]\\label{rk-cmc}\nWe also refer to \\cite{Constrain-Bruhat-Isenberg} for results concerning the existence of initially CMC data in the current setting. We assume $|\\phi_0|_{L^\\infty(\\Sigma)}$ to be small enough, so that $\\left( \\frac{2}{3} \\tau_0^2 - 2m^2 \\phi_0^2\\right) > 0$ (that is, $\\mathcal{B}_{\\tau_0, \\phi_0} >0$ by the convention in \\cite{Constrain-Bruhat-Isenberg}). In our case, $\\Sigma$ has negative Yamabe invariant and $\\mathcal{B}_{\\tau_0, \\phi_0} >0$, then the existence CMC solutions of the constraint equations for the EKG system on such compact manifolds is covered by \\cite{Constrain-Bruhat-Isenberg}.\n\\end{remark}\n\n\\begin{remark}[Generalize to non-CMC data]\\label{rk-non-cmc}\nConsidering generalized data, which is not necessarily CMC, but close to the initially CMC data induced by the background solution, the existence of a maximal globally hyperbolic development of this initial data is assured in \\cite{Ringstrom-09}. Under the smallness condition, the existence of a CMC surface in such a development is shown in the vacuum case by Fajman-Kroencke \\cite{Fajman-Kroencke-15} (Theorem 2.9). Based on these results, we can now generalize our main theorem to the case of non-CMC data.\n\nThe rescaled version of data (non-CMC) \\eqref{intro-initial-data} is $\\|R_{0ij} + 2 g_{0}\\|_{H^3(\\Sigma, g_0)} + \\|k_0+g_0\\|_{H^{4}(\\Sigma, g_0)} \\lesssim \\varepsilon$ (c.f. \\eqref{rescale-metric-1}-\\eqref{def-rescale-mass} for the rescaling).\nTo match the condition in \\cite{Fajman-Kroencke-15}, one needs the smallness for $\\|g_0-\\gamma\\|_{H^s(\\Sigma, \\gamma)} + \\|k_0+\\gamma\\|_{H^{s-1}(\\Sigma, \\gamma)}$, $s>3\/2+1$, where $\\gamma$ is the hyperbolic metric. We take $s=5$ and choose the harmonic coordinates with respect to $g_0$ and $\\gamma$. Hence, in our setting, it involves controlling $\\|g_{0ij}-\\gamma_{ij}\\|_{H^5(\\Sigma, \\gamma)}$ in terms of $\\|R_{0ij} + 2 g_{0ij}\\|_{H^3(\\Sigma, g_0)}$. Denote by $\\mathcal{A} = 2{\\bf D}(R_{0ij} + 2g_{0ij})$, where ${\\bf D}$ is the Frechet derivative. Recall that from \\cite{A-M-04, A-M-11-cmc}, for a TT (transverse and traceless) tensor $h$, $$\\mathcal{A} h= -\\Delta_{\\gamma} h -2h.$$\nLet $\\lambda$ be the eigenvalues of $\\mathcal{A}$. It is known that, the hyperbolic metric $\\gamma$ ($n\\geq 3$) is strictly stable, i.e. $\\lambda >0$ (\\cite{Besse-Einstein} $\\S$ 12 {\\bf H}). The operator $\\mathcal{A}$ is an isomorphism.\nThus the smallness of $\\|R_{0ij} + 2 g_{0ij}\\|_{H^3(\\Sigma, g_0)}$ implies smallness for $\\|g_{0ij}-\\gamma_{ij}\\|_{H^5(\\Sigma, \\gamma)}$ and hence the rescaled version of \\eqref{intro-initial-data} entails that $\\|g_{0ij}-\\gamma_{ij}\\|_{H^5(\\Sigma, \\gamma)} + \\|k_{0ij}+\\gamma_{ij}\\|_{H^{4}(\\Sigma, \\gamma)} \\lesssim \\varepsilon$. Now we can prove along the lines of the corresponding argument presented in \\cite{Ringstrom-09, Fajman-Kroencke-15} to show the existence of a CMC surface in the non vacuum setting. Therefore, the data in Theorem \\ref{main-thm-into} can be generalized to be non-CMC.\n\\end{remark} \n \nBefore offering an overview for the proof, we introduce some notations.\nLet $(\\bar M, \\breve g)$ be the maximal development of the EKG data given above. $(\\bar M, \\breve g)$ is differmorphic to $\\mathbb{R} \\times \\Sigma,$ where $\\Sigma$ is a compact Cauchy surface and it is globally foliated by a CMC foliation $\\{\\Sigma_\\tau, \\tau \\in I\\}$. We rescale the CMC time function and define $t= -\\frac{3}{\\tau}.$ Then as $\\tau \\rightarrow 0^-, t \\rightarrow +\\infty$. Let $\\partial_t$ denote the coordinate vector field corresponding to $t$, then $\\breve g(\\partial_t, \\partial_t) = -N^2$, where $N$ is called lapse, and we set $\\hat N = N-1$. Let $\\tilde g_{ij}$ be the induced Riemannian metric on $\\Sigma_t$, $\\tilde\\nabla$ the corresponding covariant derivative. $T$ is the unit timelike vector field normal to $\\Sigma_t$, and we have $\\partial_t = NT$. The second fundamental form is denoted by $\\tilde k_{ij}$, while $\\hat{\\tilde k}_{ij}$ being the traceless part of $\\tilde k_{ij}$. In the CMC gauge, $\\text{tr} \\tilde k = \\tau = -\\frac{3}{t}.$ \n\nLet $g_{ij}=t^{-2} \\tilde g_{ij}$ be the spatially normalized metric, $\\nabla$ the corresponding connection. \nAs shown in \\cite{A-M-04} ((4.20a)-(4.20c)), the following variables are scale-free (the indices refer to a coordinate frame).\n\\begin{equation}\\label{rescale-metric-1}\n\\begin{split}\ng_{ij} &=t^{-2} \\tilde g_{ij} \\quad\\, g^{ij}=t^{2} \\tilde g^{ij} \\\\ \nk_{ij} &=t^{-1} \\tilde k_{ij} \\quad \\text{tr} k = t \\text{tr} \\tilde k\\\\\n\\bar T &= t T \\quad \\quad\\, \\mathrm{d}} %{\\text{d} \\mu_g =t^{-3} \\mathrm{d}} %{\\text{d}\\mu_{\\tilde g}.\n\\end{split}\n\\end{equation}\nIt is convenient to introduce the logarithmic time $\\bar \\tau = \\ln t$, which has the property that\n$\\partial_{\\bar \\tau} = t\\partial_t$ is scale free.\nWe also define the rescaled mass of the KG field.\n\\begin{equation}\\label{def-rescale-mass}\n\\bar m(t) = tm.\n\\end{equation}\nWe remark that such rescaling for the mass of the KG field is transitional, aiming to make the KG energy well-behaved with respect to the above scale-free variables.\nFinally, we let $\\phi$ be the KG field, and $E_{ij}, H_{ij}$ be the electric and magnetic parts of the Weyl field $W_{\\mu \\alpha \\nu \\beta}$. Note that, $E_{ij}, H_{ij}$ are scale-free as well.\n\n\n\\subsection{Strategy of the proof}\\label{sec-main-proof}\n{\\bf Energy scheme}. As is explained before, the scaling $\\partial_{\\bar \\tau}$ (and hence $\\bar T$) does not qualify as a commutator vector field for the KG equation. Therefore, we only commute the KG equation with the spatial covariant derivative $\\nabla$. Technically, we address an $1+3$ splitting for the KG equation, which takes the form of\n\\begin{align*}\n& \\lie_{\\partial_{\\bar \\tau}} \\bar T \\phi - \\left( \\text{tr} k N +1 \\right) \\bar T \\phi -N \\nabla^i \\nabla_i \\phi + \\bar m^2(t) N \\phi = \\nabla_i N \\nabla^i \\phi.\n\\end{align*}\n$\\lie$ means taking Lie derivative.\nThus in such $1+3$ forms, the high order of the KG equation \\eqref{eq-kg-N-l-1+3-general-simply} can be derived by calculating the commutator $[\\lie_{\\partial_{\\bar \\tau}}, \\nabla]$ (c.f. Section \\ref{sec-commuting-id}), which involves only $N$, $\\hat k$ and their spatial derivatives. The Bianchi equation are treated in an analogous fashion. We then establish the energy inequalities for these $1+3$ evolution equations (c.f. Section \\ref{sec-ee-kg-1+3} and \\ref{sec-ee-Bianchi-1+3}). The energies are defined as below: Fixing an integer $I$, for $t>t_0 > 3 m^{-1}$, the energy norms for $\\phi$ and $W$ ($E, H$) are\n\\begin{equation}\\label{intro-def-energy-l-homo-kg}\nE_l (\\phi, t) = \\|\\nabla^l \\bar T \\phi \\|^2_{L^2(\\Sigma_t)} + \\|\\nabla^l \\nabla \\phi \\|^2_{L^2(\\Sigma_t)} + \\|\\nabla^l (\\bar m(t) \\phi)\\|^2_{L^2(\\Sigma_t)}, \\, l \\leq I,\n\\end{equation}\n\\begin{equation}\\label{intro-def-energy-l-homo-Weyl}\nE_l (W, t) = \\|\\nabla^l E\\|^2_{L^2(\\Sigma_t)} + \\|\\nabla^l H \\|^2_{L^2(\\Sigma_t)}, \\quad l \\leq I-1.\n\\end{equation}\nThe notation $\\nabla^l$ means taking the derivative $\\nabla$ $l$ times. \n\nIn other words, we give up the standard method of the energy-momentum tensor. Alternatively, we recast the KG equation in a first order form and treat it as a transport equation rather than a wave equation. This turns out to be more compatible with the energy scheme which commuting merely with the spatial derivatives.\n\n{\\bf The coupling structures}. In the strategy of the $1+3$ splitting, the normal derivative $\\bar T$ is referred as a bad derivative, while the spatial derivative $\\nabla$ is a good derivative. This is implied by the following facts: The energies \\eqref{intro-def-energy-l-homo-kg} do not control $\\bar T^2\\phi$. However, making use of the KG equation which expresses $\\bar T^2 \\phi$ in terms of $\\bar T\\phi$ and $\\nabla^l \\phi, \\, 0 \\leq l \\leq 2$, in turn yields almost $t^{\\frac{1}{2}}$ growth for $\\bar T^2\\phi$. On the other hand, implied by Sobolev inequalities, terms involving the spatial derivatives $\\nabla^i \\bar T \\phi, \\nabla^j \\phi$, $i\\leq I-2, j \\leq l-1$ decay almost like $t^{-\\frac{1}{2}}$. Meanwhile, $\\nabla^l \\phi \\sim t^{-1} \\nabla^l (\\bar m (t) \\phi)$, $l\\leq I-2$ decay almost like $t^{-\\frac{3}{2}}$.\n\nOne of our key observations is that the nonlinear couplings avoid $\\bar T^2\\phi$, due to some cancellations (see \\eqref{stru-source-J-bianchi-0} and \\eqref{stru-source-AJ-bianchi-0-1}). Therefore, we get couplings such as $\\nabla \\bar T \\phi \\nabla \\phi E$ and $\\nabla^2\\phi \\bar T\\phi E$, which generally decay fast enough. Now, let us pay more attention to the estimates for $\\nabla^2 \\phi \\bar T\\phi E$ and its higher order versions $\\nabla^l (\\nabla^2 \\phi \\bar T \\phi E), l \\leq I-1$. Notice that, for the lower order cases: $l \\leq I-2$, $\\|\\nabla^l \\nabla^2 \\phi\\|_{L^2(\\Sigma_t)}$ is estimated as $t^{-1} \\|\\nabla^{l+2} (\\bar m(t) \\phi)\\|_{L^2(\\Sigma_t)}$. \nUnfortunately, such estimate is forbidden for the top order case: $l=I-1$, since it requires one more derivative than our purposes. That is to say, $\\|\\nabla^{I-1} \\nabla^2 \\phi\\|_{L^2(\\Sigma_t)}$ has no additional $t^{-1}$ factor. As a result, the top order coupling $\\nabla^{I-1} (\\nabla^2 \\phi \\bar T \\phi E)$ will provide a borderline term $\\nabla^{I-1}\\nabla^2 \\phi \\bar T\\phi E$ which requires more work.\n\n{\\bf Hierarchy of energy}.\nWe start with some weak bootstrap assumptions (with $\\delta (0<\\delta < \\frac{1}{6})$ loss in the decay rate of $t$), and finally close the bootstrap argument by sharpening the estimates (Section \\ref{sec-close-b-t}). In particular, the estimate for $\\|\\hat k\\|_{L^2(\\Sigma_t)}$ is retrieved by the transport equation for $\\hat k$ (Section \\ref{sec-closing-bt-k}).\n\nWhen commuting $\\nabla^l$ $(1 \\leq l \\leq I)$ with the KG equation, we encounter the borderline terms $BL_l$, taking the forms of $BL_l = \\sum_{a=1}^l BL_l^a$, with $1 \\leq a \\leq l$ and\n\\begin{equation}\\label{def-BLl-123l}\n\\begin{split}\nBL_l^a = & \\bar m (t) \\nabla^{l-a+1} N * \\nabla^l (\\bar m (t) \\phi) * \\nabla^{a-1} \\bar T \\phi \\\\\n&\\pm \\bar m (t) \\nabla^{l-a+1} N * \\nabla^l \\bar T \\phi * \\nabla^{a-1}( \\bar m (t) \\phi ).\n\\end{split}\n\\end{equation}\nWe confirm the inevitability of these borderline terms in Remark \\ref{rk-borderline-term}. Meanwhile, we should notice that $BL_l$ will vanish if the scalar field is massless.\n\nBecause of the weak bootstrap assumptions, $BL_l, 1 \\leq l \\leq I$ bear $\\delta$ loss in the decay rate. We propose an $L^\\infty - L^\\infty$ estimate for the KG field, which is due to the technical ODE estimate in \\cite{Ma-Lefoch-16}, to improve $\\|T\\phi\\|_{L^\\infty}$ and $\\|\\phi\\|_{L^\\infty}$ and then retrieve the $\\delta$ loss in $BL_l, 1 \\leq l \\leq I$. But this costs the regularity: the $L^\\infty - L^\\infty$ estimate requires the strongest decay for $\\nabla^2 \\phi$, which further forces us to perform two more orders of energy estimates. That is, in \\eqref{intro-def-energy-l-homo-kg}-\\eqref{intro-def-energy-l-homo-Weyl}, we take $I=4$ rather than $I=2$.\n\nWe observe the linear structure in \\eqref{def-BLl-123l}: $\\nabla^{a-1} \\phi$ is of lower order (less than $l^{\\text{th}}$ order) and also $\\nabla^{l-a+1} \\hat N$ depends mainly on lower order energies of the KG field. In more details, the refined estimates for the lapse (Lemma \\ref{lem-N-ineq-source}) tell that $\\|\\nabla^a \\hat N \\|_{L^2(\\Sigma_t)}, a \\leq 2$ relies crucially on the zeroth order energy $E_0(\\phi, t)$, and $\\|\\nabla^{a} \\hat N\\|_{L^2(\\Sigma_t)}, 2 < a \\leq l$ is primarily related to $E_{a-2}(\\phi, t)$, which is again of lower order. Enlightened by these features, we proceed by an argument of hierarchy of energy estimates, which are carried out in Section \\ref{sec-0-ee-kg} (for the KG field). The idea is that, the information already obtained for lower order energies will help to sharpen the estimates for $\\nabla^{l-a+1} \\hat N$, $\\nabla^{a-1} \\phi$ and reduce the nonlinear structure of \\eqref{def-BLl-123l} to a linear one. In addition, this idea of linearization also applies to the borderline terms (like $\\nabla^{I-1}\\nabla^2 \\phi \\bar T \\phi E$) that arise in the top order energy estimate for the Weyl field, since we can now take advantage of the improved estimates for $\\phi$, c.f. \\eqref{estimate-Sl11-2-1}.\n\nWe here sketch the linearization in the energy argument. Generically, the energy inequality for the KG field before linearization takes the form of\n\\begin{align*}\n \\partial_{\\bar\\tau} E_l(\\phi, t) + E_l(\\phi, t) \\lesssim t^{-1} E_l(\\phi, t) + t \\mathcal{E}_4(\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_4(\\phi, t) E^{\\frac{1}{2}}_l(\\phi, t),\n\\end{align*}\nwhere the second term on the right hand side is a rough bound for the borderline term \\eqref{def-BLl-123l}, and $\\mathcal{E}_4(\\phi, t)$ means the energies of the KG field up to the fourth order. Here the first quadratic term $ \\mathcal{E}_4(\\phi, t)$ is due to the fact that $\\nabla^i N$ depends quadratically on $\\phi$ (c.f. Lemma \\ref{lem-N-ineq-source} and Remark \\ref{rk-rough-lapse}). We aims to linearize the borderline terms via the hierarchy of energy estimate and the $L^\\infty - L^\\infty$ estimate for the KG field, so that the energy inequality is improved as (c.f. Section \\ref{sec-0-ee-kg})\n\\begin{align*}\n\\partial_{\\bar\\tau} E_l(\\phi, t) + E_l(\\phi, t) \\lesssim & t^{-1} E_l(\\phi, t) + t^{-\\frac{1}{2}} \\varepsilon M \\varepsilon I E^{\\frac{1}{2}}_l(\\phi, t),\\\\\n \\lesssim & \\left( t^{-1}+ \\varepsilon M \\right) E_l(\\phi, t) + \\varepsilon^2 I^2 t^{-1}.\n\\end{align*}\nHere $I$ is a constant depending on the initial data. In particular, the previously quadratic term is now replaced by a linear one. Changing to $t$, we have \n\\begin{equation*}\n\\begin{split}\n& \\partial_t ( t E_l(\\phi, t)) \\lesssim \\left( t^{-2} + \\varepsilon M t^{-1} \\right) (t E_l(\\phi, t)) + \\varepsilon^2 I^2 t^{-1}.\n\\end{split}\n\\end{equation*} \nThe Gr\\\"{o}nwall inequality then yields that $tE_l(\\phi, t) \\lesssim \\varepsilon^2 I^2 t^{C_M \\varepsilon}$, where $C_M$ depends linearly on $M$. Then the energy argument are closed.\n\n\n\n\nThe paper is organized as follows. In Section \\ref{preliminaries}, we describe some conventions and notations, and show the background materials. In Section \\ref{sec-energy-estimate-1+3}, we conduct the $1+3$ splitting of the EKG system and establish the energy scheme. In Section \\ref{sec-preliminary-estimate}, we introduce the bootstrap assumptions, and present some preliminary estimates that follows from the Sobolev inequalities. The energy estimates for the KG field and the Weyl field are carried out in Section \\ref{sec-energy-kg} and \\ref{sec-energy-Bianchi}. Section \\ref{sec-global-existence} is devoted to close the bootstrap argument and prove the global existence theorem, geodesic completeness. Finally, a number of useful definitions, identities and ODE estimates are collected in the appendix. \n\n \n \n \n \n{\\bf Acknowledgements}: The Einstein-Klein-Gordon project studied in this paper was suggested by L. Andersson. I am grateful to him\nfor many enlightening discussions. I thank V. Moncrief for many helpful discussions on this topic and suggesting the usage of CMC gauge in early 2014. I thank C. Liu and S. Ma for reading and useful comments as well.\nThe author was supported by a Humboldt Foundation post-doctoral fellowship at the Albert Einstein Institute during the period 2014-16, when part of this work was done. She is also supported by NSFC (Grant No. 11701482), NSF of Fujian Province (Grant No. 2018J05010) and Fundamental Research Funds for the Central Universities (Grant No. 20720170002).\n\n\n\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\\subsection{Notations and conventions}\\label{notations}\nIndices:\n\\begin{itemize}\n\\item $\\alpha, \\beta, \\cdots, \\mu, \\nu, \\cdots$: Greek indices (spacetime indices) run over $0,\\cdots,3.$\n\\item $i,j, \\cdots, p,q, \\cdots$: Latin indices (spatial indices) run over $1,\\cdots,3.$\n\\end{itemize}\nGeometry related:\n\\begin{itemize}\n\\item $(\\bar M, \\bar{g}_{\\mu\\nu}, D):$ the spacetime manifold with $D$ the connection.\n\\item $\\left( \\Sigma_t, g_{ij}, \\nabla \\right):$ the induced spatial manifold with $\\nabla$ the connection.\n\\item $\\Box_{\\bar g}=D^\\alpha D_{\\alpha}:$ the spacetime wave operator. \n\\item $\\Delta = \\nabla^i \\nabla_i$: the spatial Laplacian operator.\n\\item $\\mathrm{d}} %{\\text{d} \\mu_{\\bar g}, \\mathrm{d}} %{\\text{d} \\mu_g$: the volume form with respect to $\\bar g$ or $g$. \n\\item $\\bar R_{\\mu \\alpha \\nu \\beta}, \\bar R_{\\mu \\nu}, \\bar R:$ the components of the Riemannian, Ricci, scalar curvature tensor of $\\left( \\bar{M}, \\bar g_{\\mu \\nu} \\right).$\n\\item $R_{imjn}, R_{ij}, R:$ the components of the Riemannian, Ricci, scalar curvature tensor of $\\left( \\Sigma_t, g_{ij} \\right).$\n\\item $\\bar R_{imjn}, \\bar R_{ij}:$ projection of $\\bar R_{\\mu \\alpha \\nu \\beta}$, $\\bar R_{\\mu \\nu}$ to $(\\Sigma_t, g_{ij}).$\n\\end{itemize}\nFoliation related:\n\\begin{itemize}\n\\item $\\{T, e_i\\}:$ $T$ denotes the future directed, unit vector normal to $\\Sigma_t,$ and $\\{ e_i \\}$ an orthonormal frame tangent to $\\Sigma_t$. \n\\item $\\tau, t, \\bar \\tau:$ $\\tau$ is the CMC time $\\tau = \\text{tr} k$; $t=-\\frac{3}{\\tau}$; $\\bar \\tau = \\ln t$.\n\\item $\\partial_t, \\partial_{\\bar \\tau}:$ coordinate vector field corresponding to $t, \\bar \\tau$.\n\\end{itemize}\nSimplified conventions\n\\begin{itemize}\n\\item $\\nabla_{I_l} \\psi$: the $l^{\\text{th}}$ order of covariant derivative $\\nabla_{i_1} \\cdots \\nabla_{i_l}\\psi$ and the multi index $I_l=\\{ i_1 \\cdots i_l\\}$ is used.\n\\item $\\nabla_{i_1} \\cdots \\stackrel{j}{\\nabla}_p \\cdots \\nabla_{i_l} \\psi $: $\\nabla_{I_l} \\psi$ with the $j^{\\text{th}}$ $\\nabla_{i_j}$ being replaced by $\\nabla_p$.\n\\item $\\mathcal{D} \\phi$: any element in $\\{\\bar T \\phi, \\nabla \\phi, \\bar m (t) \\phi\\}$.\n\\item $f_1 \\lesssim f_2$: $f_1 \\leq C f_2$ with some {\\it universal} constant $C$.\n\\item $f_1 \\sim f_1$: $f_1$, $f_2$ are equivalent in the sense that $f_1 \\lesssim f_2$ and $f_2 \\lesssim f_1$.\n\\item $C_M$: a constant depending linearly on some controlling quantity $M$.\n\\item $A*B$: any finite sum of products of $A$ and $B$, with each product being a contraction (with respect to $g$) between two $\\Sigma_t$-tensors $A$ and $B$\\footnote{In the estimates, we will only employ the formula $\\|A * B\\| \\lesssim \\|A\\| \\|B\\| $ which allows ourselves to ignore the detailed product structure at this point. $\\|\\cdot \\|$ denotes the norm associated to $g.$}.\n\\end{itemize}\n\n \n \n \n \n \n\\subsection{The background materials}\\label{sec-sec-background}\n Let $\\Sigma$ be a spacelike hypersurface and let $\\bar{M}= I \\times \\Sigma$ be an $n+1$-dim manifold with Lorentz metric $\\breve{g}$ of signature $-\\, + \\cdots +$. We introduce local coordinates $(\\tau, x^i, i=1, \\cdots, n)$ on $\\bar{M}$ so that $\\tau$ is a time function and $x^i$ are coordinates on the level sets $\\Sigma_\\tau$ of $\\tau$. \n \n Let $\\partial_\\tau=\\partial \/\\partial \\tau$ be the coordinate vector fields corresponding to $\\tau$. The lapse function ${}_{\\tau}N$ and shift vector field $X$ of the foliation $\\{\\Sigma_\\tau\\}$ are defined by $\\partial_\\tau={}_{\\tau}NT+\\tilde X_\\tau,$ where $T$ is the unit timelike vector field normal to $\\Sigma_\\tau$. Assume $T$ is future directed so that ${}_{\\tau}N>0$. We make the gauge choice $\\tilde X_\\tau=0,$ so that\nthe spacetime metric $\\breve{g}$ with vanishing shift takes the form\n\\begin{equation}\\label{metric-form-tau}\n\\breve g_{\\mu\\nu} = -{}_{\\tau}N^2 \\mathrm{d}} %{\\text{d} \\tau^2 + \\tilde g_{ij} \\mathrm{d}} %{\\text{d} x^i \\mathrm{d}} %{\\text{d} x^j.\n\\end{equation}\nThe second fundamental form $\\tilde k_{ij}$ of $\\Sigma_\\tau$ is given by $\\tilde k_{ij} = - \\frac{1}{2}\\lie_{T} \\tilde g_{ij}.$\nThe {\\bf CMC} gauge is referred to as\n\\begin{equation}\\label{cmc-gauge-tau}\n\\text{tr} \\tilde k = \\tau.\n\\end{equation}\nWe call $\\tau$ the CMC time function.\nFor notational convenience, we define a rescaled time function $t:=-\\frac{3}{\\tau}$ so that the corresponding lapse ${}_{t} N$\n\\begin{equation}\\label{def-dt}\n{}_{t}N = \\frac{3}{t^2} {}_{\\tau} N, \\quad \\partial_t = {}_{t}N T.\n\\end{equation}\nWe will drop the subscript $t$, and denote the lapse for the $t$-foliation by $N$. We also let\n\\begin{equation}\\label{def-hat-N}\n\\hat N = N-1.\n\\end{equation}\nIn terms of the $t$-foliation,\n\\begin{equation}\\label{k-decomp}\n\\text{tr} \\tilde k = - \\frac{3}{t}, \\quad \\tilde k_{ij} = -\\frac{1}{t} \\tilde g_{ij} + \\hat{\\tilde k}_{ij},\n\\end{equation}\nwhere $\\hat{\\tilde k}_{ij}$ is the trace free part of $\\tilde k_{ij}$. \n\nPerforming the rescaling \\eqref{rescale-metric-1}, the rescaled curvature is related to the original one by\n\\begin{align*}\n R_{imjn} &= t^{-2} \\tilde R_{imjn}, \\quad R_{ij } = \\tilde R_{ij}, \\quad R = t^2 \\tilde R.\n\\end{align*}\nAnd we define the rescaled spacetime metric as well:\n\\begin{align*}\n\\bar g_{\\mu \\nu} &=t^{-2} \\breve g_{\\mu \\nu}, \\quad \\, \\bar g^{\\mu \\nu}=t^{2} \\breve g^{\\mu \\nu}, \\quad \\bar T = t T, \\\\ \n\\bar R_{\\mu \\alpha \\nu \\beta} &= t^{-2} \\breve R_{\\mu \\alpha \\nu \\beta}, \\quad \\bar R_{\\mu \\nu } = \\breve R_{\\mu \\nu}, \\quad \\bar R = t^2 \\breve R.\n\\end{align*}\n\n\n\n\nWe remark that, if $\\tau \\in [\\tau_0, 0)$ with $\\tau_0 <0,$ then $t \\in [ t_0, + \\infty),$ where $ t_0=-\\frac{3}{\\tau_0}.$ Letting $m$ be the mass of the KG field \\eqref{eq-kg}, we require that $|\\tau_0| < m$, i.e. $t_0 > 3 m^{-1}$.\n\n\n\n\\subsubsection{Lorenzian geometric equations}\\label{sec-einstein-eq}\nRecall the first and second variation equations for the rescaled metric\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{L}_{\\partial_{\\bar \\tau}} g_{ij} =2 \\hat N g_{ij} -2N \\hat k_{ij}, \\label{eq-evolution-1}\n\\end{equation}\n\\begin{equation}\n\\mathcal{L}_{\\partial_{\\bar \\tau}} \\hat k_{ij} + \\hat k_{ij} = \\hat N g_{ij} -\\nabla_i \\nabla_j N + N \\left( \\bar{R}_{i \\bar T j \\bar T} - \\hat k_{ip} \\hat k_{j}^p \\right). \\label{eq-evolution-2}\n\\end{equation}\n\\end{subequations}\nHere the indices $i, j$ refer to the frame $\\{\\partial_i, \\, i=1,2,3\\}.$\nAnd the Gauss-Codazzi equations are\n\\begin{equation}\\label{Gauss-Riem-hat-k}\n\\begin{split}\nR_{imjn} =&- ( g \\odot g )_{imjn}+ ( g \\odot \\hat k )_{imjn} - \\frac{1}{2} ( \\hat k \\odot \\hat k )_{imjn} + \\bar{R}_{imjn},\n\\end{split}\n\\end{equation}\nwhere the Kulkarni-Nomizu product $\\odot$ is defined by: Let $\\xi, \\eta$ be symmetric $(0, 2)$-tensors,\n\\begin{equation}\\label{def-odot}\n( \\xi \\odot \\eta )_{imjn} = \\xi_{ij} \\eta_{mn} - \\xi_{in} \\eta_{jm} + \\eta_{ij} \\xi_{mn} - \\eta_{in} \\xi_{jm}.\n\\end{equation}\n\\begin{equation}\n\\nabla_i \\hat k_{jm} -\\nabla_j \\hat k_{im}= \\bar R_{\\bar T mij}. \\label{Codazzi-curl-k}\n\\end{equation}\nThe trace of the Gauss-Codazzi equations \\eqref{Gauss-Riem-hat-k}-\\eqref{Codazzi-curl-k} are\n\\begin{subequations}\n\\begin{equation}\nR_{ij} = \\hat k_{il} \\hat k_{j}^l + \\hat k_{ij} - 2 g_{ij}+ \\bar{R}_{i \\bar T j \\bar T} + \\bar{R}_{ij},\\label{Gauss-Ricci-hat-k}\n\\end{equation}\n\\begin{equation}\n\\nabla^i \\hat k_{ij} - \\nabla_j \\text{tr} k= - \\bar R_{\\bar T j}.\\label{Codazzi-div-k}\n\\end{equation}\n\\end{subequations}\nThe double trace of the Gauss equations \\eqref{Gauss-Riem-hat-k} is\n\\begin{equation}\\label{Gauss-Ricci-trace-hat -k}\nR - |\\hat k|^2 + 6 = 2\\bar{R}_{\\bar T \\bar T} + \\bar{R},\n\\end{equation}\n\n\n\\subsubsection{The background spacetime}\nLet $(\\Sigma, \\gamma)$ be a compact manifold without boundary which if of hyperbolic type, with $\\gamma$ being the standard hyperbolic metric of sectional curvature $-1$. The hyperbolic cone spacetimes (or $1+3$-dimensional Milne model) $(\\bar M, \\bar \\gamma)$ is the Lorentzian cone over $(\\Sigma, \\gamma)$, i.e.\n\\begin{equation}\\label{def-bar-gamma}\n\\bar M = (0, \\infty) \\times \\Sigma, \\quad \\bar \\gamma = - \\mathrm{d}} %{\\text{d} \\rho^2 + \\rho^2 \\gamma.\n\\end{equation}\nThe family of hyperboloids $\\Sigma_\\rho$ given by $\\rho = \\text{constant}$ has normal $T= \\partial_{\\rho}.$ The vector field $\\rho \\partial_{\\rho}$ is a timelike homothetic killing field. A calculation shows that $\\tilde k_{ij} = - \\frac{1}{\\rho} \\tilde g_{ij},$ where $\\tilde g_{ij} = \\rho^2 \\gamma_{ij},$ and the mean curvature $\\text{tr} \\tilde k =- \\frac{3}{\\rho}.$ The $\\rho$-foliation has lapse $N=1$.\n\n\n\n\\subsection{Local existence}\\label{sec-local}\n\nIt is known that in the CMC gauge with zero shift, the Einstein equations are non-strictly hyperbolic. However, following the work of Christodoulou-Klainerman \\cite{Christodoulou-K-93}, where they work in maximal gauge with zero shift, we can prove the local existence theorem for EKG. The method is also analogous to the proof based on wavelike coordinates \\cite{Bruhat} or CMCSH gauge \\cite{A-M-03-local}, in the sense that we will show that to solve the local existence theorem for the original EKG system, it suffices to solve a 'reduced' system. Additionally, this approach is irrelevant to the matter field. We sketch the proof in Appendix \\ref{sec-local-cmc}.\n\n\n\\begin{theorem}\\label{thm-local-existence}\nAssume $\\Sigma$ be a compact Riemannian manifold of hyperbolic type. Let $(g_0, k_0, \\phi_0, \\phi_1)$ be the rescaled CMC data (letting $\\text{tr}_{g_0} k_0 = -3$) verifying the constraint equations of EKG system and the following conditions:\n\\begin{itemize}\n\\item [1] The Ricci curvature of $g_{0}$, $R_{0ij}$ satisfies $R_{0ij} + 2 g_{0ij} \\in H^3(\\Sigma, g_0),$\n\\item [2] $k_0$ is a symmetric $(0,2)$-tensor such that the traceless part verifying $\\hat k_0 \\in H^4(\\Sigma, g_0),$\n\\item [3] $(\\phi_0, \\phi_1)$ $\\in H^5(\\Sigma, g_0) \\times H^4(\\Sigma, g_0)$.\n\\end{itemize}\nThen there are $0 3 m^{-1}$), these two energy norm are indeed equivalent: $\\frac{1}{2} E_i (\\phi, t) < \\bar E_i (\\phi, t)< \\frac{3}{2} E_i (\\phi, t)$.\n\n\n\n\n\\subsubsection{The zero order case}\\label{sec-0-ee-kg-1+3}\n\\begin{theorem}\\label{Thm-energy-id-KG-0} \nLet $\\phi$ be a solution to the KG equation \\eqref{eq-kg}, then the $0^{\\text{th}}$-order energy admits \n\\begin{equation}\\label{energy-id-0-kg}\n\\begin{split}\n&\\partial_{\\bar \\tau} \\tilde E_0(\\phi, t) + \\tilde E_0(\\phi, t)+ \\int_{\\Sigma_{t}} 2 N |\\nabla \\phi|^2 + {}_0LK =0,\n\\end{split}\n\\end{equation}\nwhere the error term ${}_0LK$ is given by\n\\begin{equation}\\label{def-0-LK}\n{}_0LK = 3 N \\phi \\bar T \\phi - 4 \\hat N \\phi \\bar T \\phi - 2N \\hat k_{ij} \\nabla^i \\phi \\nabla^j \\phi -2 \\nabla_i N \\nabla^i \\phi \\bar T \\phi.\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\nSuch an energy inequality for the KG equation is also introduced by Galstian-Yagdjian \\cite{Galstian-Yagdjian} in the FLRW spacetime. However, they pursue global results with low regularity, and hence do not need the high order energy identities.\n\\end{remark}\n\n\n\n\\begin{proof}\nWe multiply $2\\bar T\\phi$ on \\eqref{eq-rescale-kg-1+3-0}, noticing that $$2N \\bar m^2(t) \\phi \\bar T \\phi =\\partial_{\\bar \\tau} (\\bar m^2(t) \\phi^2) - 2 \\bar m^2(t) \\phi^2,$$ and\n\\begin{align*}\n-2N \\nabla^i \\nabla_i \\phi \\bar T \\phi &= -\\nabla_i \\left(2N \\nabla^i \\phi \\bar T \\phi \\right) +2 \\nabla^i \\phi \\nabla_i \\left( N \\bar T \\phi \\right) \\\\\n&=-\\nabla_i \\left(2N \\nabla^i \\phi \\bar T \\phi \\right) + 2 \\nabla^i \\phi \\lie_{\\partial_{\\bar \\tau}} \\nabla_i \\phi,\n\\end{align*}\nwhere in the second equality above we have used the commuting identity \\eqref{id-commuting-nabla-T-l-simplify} with $l=1$. Additionally, making use of the identity $\\lie_{\\partial_{\\bar \\tau}} g^{ij} = -2\\hat N g^{ij} + 2N \\hat k^{ij}$, and hence $$2 \\nabla^i \\phi \\lie_{\\partial_{\\bar \\tau}} \\nabla_i \\phi = \\partial_{\\bar \\tau} |\\nabla \\phi|^2-2N \\hat k_{ij} \\nabla^i \\phi \\nabla^j \\phi + 2\\hat N |\\nabla \\phi|^2,$$ we arrive at \n\\begin{equation}\\label{energy-estimate-1+3-step1-1}\n\\begin{split}\n&\\partial_{\\bar \\tau} \\rho_0(\\phi) - 2 \\bar m^2(t) \\phi^2 + 4 N \\left(\\bar T \\phi\\right)^2+ 2 \\hat N (\\bar T \\phi)^2 + 2 \\hat N |\\nabla \\phi|^2 \\\\\n&-2N \\hat k_{ij} \\nabla^i \\phi \\nabla^j \\phi -2 \\nabla_i N \\nabla^i \\phi \\bar T \\phi -2\\nabla_i \\left(N \\nabla^i \\phi \\bar T \\phi \\right) =0.\n\\end{split}\n\\end{equation}\n\nNow let us focus on the quadratic term $4 N \\left(\\bar T \\phi\\right)^2 - 2 \\bar m^2(t) \\phi^2$ in \\eqref{energy-estimate-1+3-step1-1}. We will additionally address an integration by part (or rather extract a divergence form) and use the KG equation to make the sign right. Performing an integration by part,\n \\begin{equation*}\n\\begin{split}\n N \\left(\\bar T \\phi\\right)^2 =& \\partial_{\\bar \\tau} \\left( \\phi \\bar T \\phi \\right) - \\phi \\lie_{\\partial_{\\bar \\tau}} \\bar T \\phi.\n\\end{split}\n\\end{equation*}\nWe then substitute the KG equation \\eqref{eq-rescale-kg-1+3-0} into the following formula,\n\\begin{align*}\n4 N \\left(\\bar T \\phi \\right)^2 - 2 \\bar m^2(t) \\phi^2 =& N \\left( \\bar T \\phi \\right)^2 + 3 \\partial_{\\bar \\tau} ( \\phi \\bar T \\phi) - 3 \\phi \\lie_{\\partial_{\\bar \\tau}} \\bar T \\phi - 2 \\bar m^2(t) \\phi^2\\\\\n=& N \\left(\\bar T \\phi \\right)^2+ \\bar m^2(t) N \\phi^2 + \\partial_{\\bar \\tau} (3 \\phi \\bar T \\phi) + 2 \\bar m^2(t) \\hat N \\phi \\\\\n& + 3 \\phi \\left( 2N \\bar T \\phi + \\hat N \\bar T \\phi - N \\Delta \\phi - \\nabla_i N \\nabla^i \\phi \\right).\n\\end{align*}\nThe first two term now has a good sign, and the extra quadratic term $- 3N \\phi \\Delta \\phi$,\n\\begin{align*}\n- 3N \\phi \\Delta \\phi &= -3 \\nabla_i \\left( N \\phi \\nabla^i \\phi \\right) +3 N |\\nabla \\phi|^2 + 3 \\phi \\nabla^i N \\nabla_i \\phi.\n\\end{align*}\n We remark that the error terms arising in this procedure are all of lower order: they indeed have additional $t^{-1}$ factor, for instance, $\\phi \\cdot N \\bar T \\phi $ can be rewritten as $t^{-1} \\bar m (t) \\phi \\cdot N \\bar T \\phi.$\nPutting the two identities above together, we have\n\\begin{equation*}\n\\begin{split}\n&\\partial_{\\bar \\tau} \\left( \\rho_0(\\phi) + 3 \\phi \\bar T \\phi \\right) + N \\rho_0 (\\phi) +2 N |\\nabla \\phi|^2 + 2 \\hat N \\rho_0 (\\phi) \\\\\n& -2 \\nabla_i \\left( N \\phi \\nabla^i \\phi \\right) -2\\nabla_i \\left(N \\nabla^i \\phi \\bar T \\phi \\right) + 6N \\phi \\bar T \\phi \\\\\n& + 2 \\hat N \\phi \\bar T \\phi -2N \\hat k_{ij} \\nabla^i \\phi \\nabla^j \\phi -2 \\nabla_i N \\nabla^i \\phi \\bar T \\phi =0.\n\\end{split}\n\\end{equation*}\nIntegrating on $\\Sigma_{t}$ and making use of the identity \\eqref{id-div-T}, there is\n\\begin{equation}\\label{energy-estimate-1+3-step3-1}\n\\begin{split}\n& \\partial_{\\bar \\tau} \\int_{\\Sigma_{t}} \\left( \\rho_0(\\phi) + 3 \\phi \\bar T \\phi \\right) \\mathrm{d}} %{\\text{d} \\mu_g + \\int_{\\Sigma_{t}} 2 N |\\nabla \\phi|^2 \\mathrm{d}} %{\\text{d} \\mu_g \\\\\n&+ \\int_{\\Sigma_{t}} N \\left( \\rho_0(\\phi) +3 \\phi \\bar T \\phi \\right) + 3 N \\phi \\bar T \\phi - 7 \\hat N \\phi \\bar T \\phi \\\\\n&+ \\int_{\\Sigma_{t}} - \\hat N \\rho_0 (\\phi) -2N \\hat k_{ij} \\nabla^i \\phi \\nabla^j \\phi -2 \\nabla_i N \\nabla^i \\phi \\bar T \\phi =0.\n\\end{split}\n\\end{equation}\nWe then achieve \\eqref{energy-id-0-kg}. \n\\end{proof}\n\nAs a remark, \\eqref{energy-id-0-kg} can also be deduced by the method of energy-momentum tensor. However, if we proceed to the higher order case, it will be natural and simpler to commute the spatially covariant derivative $\\nabla$ with the KG equation in $1+3$ form \\eqref{eq-rescale-kg-1+3-0}. And this turns out to be more compatible with the energy with only high order of spatial energy. \n\n\\subsubsection{The higher order case}\\label{sec-high-ei-kg-1+3}\n\n\\begin{lemma}\\label{lemma-High-KG-1+3}\nLet $l \\geq 1,$ the $l^{\\text{th}}$-order KG equation $\\nabla_{I_l} \\left( N \\left( \\Box_{\\bar g} \\phi -\\bar m^2 (t) \\phi \\right) \\right) =0$ can be decomposed in the following $1+3$ form\n\\begin{equation}\\label{eq-kg-N-l-1+3-general-simply}\n\\begin{split}\n& \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} \\bar T \\phi - \\left( N \\text{tr} k +1 \\right) \\nabla_{I_l} \\bar T \\phi - N \\nabla_{I_l} \\Delta \\phi + \\bar m^2(t) N \\nabla_{I_l} \\phi \\\\\n &\\quad + \\bar m^2(t) \\mathcal{N}_{I_l}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) - \\mathcal{N}_{I_{l+1}} (\\nabla \\phi)=0,\n\\end{split}\n\\end{equation}\nwhere $\\mathcal{N}_{I_l} (\\phi), \\mathcal{N}_{I_{l+1}} (\\nabla \\phi)$ are defined as \\eqref{def-eq-kg-N-l-1+3-error}, and $\\mathcal{K}\\mathcal{N}^\\prime_{I_l} (\\bar T \\phi)$ is defined by \n\\begin{equation}\\label{def-KN-prime-l}\n\\begin{split}\n \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi)& = \\mathcal{K}\\mathcal{N}_{I_l}(\\bar T \\phi) + \\sum_{a+1+b=l } \\text{tr} k \\nabla_{I_a} \\nabla N * \\nabla_{I_{b}} \\bar T \\phi,\n\\end{split}\n\\end{equation}\nand $\\mathcal{K}\\mathcal{N}_{I_l}(\\bar T \\phi)$ is defined as in \\eqref{id-commuting-nabla-N-T-l-simplify}.\n\\end{lemma}\n\n\\begin{remark}[Inevitability of the borderline terms in the high order KG equation]\\label{rk-borderline-term}\nWe first remark that \n\\begin{equation*}\n\\bar m^2(t) \\mathcal{N}_{I_l} (\\phi) =\\bar m^2(t) \\left( \\nabla_{I_l} (N \\phi) - N\\nabla_{I_l} \\phi \\right),\n\\end{equation*}\nwill be one of the borderline terms in the energy estimate for KG equation.\n\nOne may argue that this borderline term is artificial, since if we commuting $\\nabla_{I_l}$ straightforwardly with the original KG equation, as\n\\begin{equation}\\label{eq-high-KG-original-1+3}\n\\begin{split}\n \\nabla_{I_l} \\left( \\lie_{\\bar T} \\bar T \\phi - \\left(N^{-1} + \\text{tr} k \\right) \\bar T \\phi - \\Delta \\phi + \\bar m^2(t) \\phi - N^{-1} \\nabla_i N \\nabla^i \\phi \\right) =0,\n\\end{split}\n\\end{equation}\nthen the aforementioned borderline term $\\bar m^2 (t) \\mathcal{N}_{I_l} (\\phi)$ does not exist. In fact, such borderline term will alternatively arise from the commuting term $ [\\lie_{\\bar T}, \\nabla_{I_l}] \\bar T \\phi = \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l} (\\bar T \\phi)$, c.f. \\eqref{def-commuting-K-l}. \nAfter commuting $\\lie_{\\bar T}$ with $\\nabla_{I_l}$ and multiplying by $N$ on \\eqref{eq-high-KG-original-1+3}, one has\n\\begin{equation*}\n\\begin{split}\n & \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} \\bar T \\phi -\\left( N \\text{tr} k +1 \\right) \\nabla_{I_l} \\bar T \\phi - N\\nabla_{I_l} (\\Delta \\phi) + \\bar m^2(t) N \\nabla_{I_l} \\phi \\\\\n =& N [\\lie_{\\bar T}, \\nabla_{I_l}] (\\bar T \\phi) +N \\nabla_{I_l} \\left( N^{-1} \\nabla_i N \\nabla^i \\phi \\right).\n\\end{split}\n\\end{equation*}\nThis almost takes the same form as \\eqref{eq-kg-N-l-1+3-general-simply} (ignoring the lower order terms), except that the previous $\\bar m^2(t) \\mathcal{N}_{I_l} (\\phi)$ is now replaced by $-N [\\lie_{\\bar T}, \\nabla_{I_l}] (\\bar T \\phi)$. However, taking a close look at $-N[\\lie_{\\bar T}, \\nabla_{I_l}] (\\bar T \\phi)$, we will find out that this is exactly the missing borderline term, c.f. \\eqref{def-commuting-K-l}, \n\\begin{equation*}\n\\begin{split}\n -N [\\lie_{\\bar T}, \\nabla_{I_l}] (\\bar T \\phi)\n =& -\\mathcal{N}_{I_l} (\\bar T^2 \\phi) - \\mathcal{K}\\mathcal{N}_{I_l} (\\bar T \\phi), \\quad l \\geq 1,\n\\end{split}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\begin{split}\n -\\mathcal{N}_{I_l} (\\bar T^2 \\phi) =& \\sum_{a +b+1 = l} \\nabla_{I_a} \\nabla N * \\nabla_{I_{b}} \\bar T^2 \\phi.\n\\end{split}\n\\end{equation*}\nWe further make use of the KG equation to substitute $\\nabla_{I_{b}} \\bar T^2 \\phi$,\n\\begin{equation*}\n\\begin{split}\n- \\mathcal{N}_{I_l} (\\bar T^2 \\phi) = & \\sum_{a +b+1 = l} \\bar m^2(t) \\nabla_{I_a} \\nabla N * \\nabla_{I_{b}} \\phi + \\text{l.o.t.}.\n\\end{split}\n\\end{equation*}\nThe first one on the right hand side above is exactly the borderline term $\\bar m^2 (t) \\mathcal{N}_{I_l} (\\phi)$. Note that, in the energy argument, this term will be multiplied with $2 \\nabla^{I_l} \\bar T \\phi$, and hence the whole term takes the form of $2 \\bar m^2(t) \\nabla^{I_l} \\bar T \\phi * \\mathcal{N}_{I_l} (\\phi)$.\n\nSecondly, let us remark that the other borderline term will also originate from something related to $ [\\lie_{\\bar T}, \\nabla_{I_l} ] \\phi$. As in the proof of Corollary \\ref{coro-energy-id-KG-l}, to establish the energy identity, we multiply $2 \\nabla^{I_l} \\bar T \\phi$ on the high order KG equation \\eqref{eq-kg-N-l-1+3-general-simply}. Let us focus on the massive term \\eqref{eq-1+3-KG-multiplying-3} for the moment,\n\\begin{equation*}\n\\begin{split}\n & 2 \\bar m^2(t) N \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\\\\n =& \\partial_{\\bar \\tau} \\left( \\bar m^2(t) |\\nabla_{I_l} \\phi|^2 \\right) - 2 \\bar m^2(t) N \\nabla^{I_{l}} \\phi [\\lie_{\\bar T}, \\nabla_{I_l} ] \\phi \\\\\n & - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2+ 2 \\bar m^2(t) \\hat N | \\nabla_{I_l} \\phi |^2 - 2 \\bar m^2(t) N \\hat k* \\nabla^{I_l} \\psi * \\nabla_{I_l} \\phi.\n \\end{split}\n\\end{equation*}\nIn the above identity, $\\partial_{\\bar \\tau} \\left( \\bar m^2 (t) | \\nabla_{I_l} \\phi|^2 \\right) $ will contribute to the energy norm, $- 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2$ will be absorbed (c.f. \\eqref{eq-step-2-0-kg}), and the other terms in the $3^{\\text{rd}}$ line are cubic lower order terms. While the final one $-2\\bar m^2 (t) \\nabla^{I_{l}} \\phi [\\lie_{\\bar T}, \\nabla_{I_l} ] \\phi$, as shown before, leads to the borderline term $-2\\bar m^2 (t) \\nabla^{I_{l}} \\phi * \\mathcal{N}_{I_l} (\\bar T \\phi),$ which can been seen as somehow ``dual'' to the previous one $2\\bar m^2 (t) \\nabla^{I_l} \\bar T \\phi * \\mathcal{N}_{I_l} (\\phi)$. \n\nThe terms discussed above are the only two borderline cases. More explicitly, they take the forms of\n\\begin{equation*}\n\\begin{split}\n &\\sum_{a +b+1 = l} 2\\bar m^2 (t) \\nabla_{I_a} \\nabla N * \\left( \\nabla_{I_{b}} \\phi * \\nabla^{I_l} \\bar T \\phi - \\nabla_{I_{b}} \\bar T \\phi * \\nabla^{I_{l}} \\phi \\right),\n \\end{split}\n\\end{equation*}\nwhich can not be cancelled.\n\\end{remark}\n\\begin{proof}\nTake the derivative $\\nabla_{i_1}$ on the KG equation in $1+3$ form \\eqref{energy-id-0-kg}, by the commuting Lemma \\ref{lemma-commuting-application} (with $l=1$ in \\eqref{id-commuting-nabla-N-T-l-simplify}),\n\\begin{equation*}\n\\begin{split}\n& 0= \\lie_{\\partial_{\\bar \\tau}} \\nabla_{i_1} \\bar T \\phi -\\nabla_{i_1} \\bar T \\phi - N \\text{tr} k \\nabla_{i_1} \\bar T \\phi + \\bar m^2(t) N \\nabla_{i_1} \\phi - N \\nabla_{i_1}\\nabla^p \\nabla_p \\phi \\\\\n & +\\bar m^2(t) \\nabla_{i_1} N \\phi - \\text{tr} k \\nabla_{i_1} N \\bar T \\phi - \\left(\\nabla_{i_1} N \\nabla^p \\nabla_p \\phi + \\nabla_{i_1} \\left(\\nabla^p N \\nabla_p \\phi \\right) \\right).\n\\end{split}\n\\end{equation*}\nThen \\eqref{eq-kg-N-l-1+3-general-simply}-\\eqref{def-eq-kg-N-l-1+3-error} with $l=1$ follows.\n\nSuppose the commuting Lemma \\ref{lemma-commuting-application} (with $l=1$ in \\eqref{id-commuting-nabla-N-T-l-simplify}) holds for $l=n-1,$ that is, the $(n-1)^{\\text{th}}$ order KG equation in $1+3$ form takes the form\n\\begin{equation}\\label{eq-kg-N-n-1-1+3-general}\n\\begin{split}\n& \\lie_{\\partial_{\\bar \\tau}} \\nabla_{i_2} \\cdots \\nabla_{i_n} \\bar T \\phi - \\left( \\text{tr} k N +1 \\right) \\nabla_{i_2} \\cdots \\nabla_{i_n} \\bar T \\phi - N\\nabla_{i_2} \\cdots \\nabla_{i_n} \\nabla^p \\nabla_p \\phi \\\\\n &+ \\bar m^2(t) N \\nabla_{i_2} \\cdots \\nabla_{i_n} \\phi + \\bar m^2(t) \\mathcal{N}_{I_{n-1}}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_{n-1}}(\\bar T \\phi) + \\mathcal{N}_{I_{n}} (\\nabla \\phi) =0,\n\\end{split}\n\\end{equation}\nNow taking the derivative $\\nabla_{i_1}$ on \\eqref{eq-kg-N-n-1-1+3-general}, and using the commuting Lemma \\ref{lem-commu-lie} with $l=n-1$, we have\n\\begin{equation*}\n\\begin{split}\n &\\nabla_{i_1} (\\lie_{\\partial_{\\bar \\tau}} \\nabla_{i_2} \\cdots \\nabla_{i_{n}} \\bar T \\phi) =\\lie_{\\partial_t} \\nabla_{I_n} \\bar T \\phi + \\sum_{m=2}^n \\lie_{\\partial_{\\bar \\tau}} \\Gamma^p_{i_1 i_m} \\nabla_{i_2} \\cdots \\stackrel{m}{\\nabla}_p \\cdots \\nabla_{i_n} \\phi.\n\\end{split}\n\\end{equation*}\nHence, we obtain\n\\begin{equation}\\label{eq-kg-N-l-1+3-general}\n\\begin{split}\n& \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_n} \\bar T \\phi - ( \\text{tr} k N +1 ) \\nabla_{I_n} \\bar T \\phi - N\\nabla_{I_n} \\nabla^p \\nabla_p \\phi + \\bar m^2(t) N \\nabla_{I_n} \\phi \\\\\n&+ \\sum_{m=2}^n \\lie_{\\partial_{\\bar \\tau}} \\Gamma^p_{i_1 i_m} \\nabla_{i_2} \\cdots \\stackrel{m}{\\nabla}_p \\cdots \\nabla_{i_n} \\phi - \\text{tr} k \\nabla_{i_1} N \\nabla_{i_2} \\cdots \\nabla_{i_n} \\bar T \\phi \\\\\n & - \\nabla_{i_1} N \\nabla_{i_2} \\cdots \\nabla_{i_n} \\nabla^p \\nabla_p \\phi + \\bar m^2(t) \\nabla_{i_1} N \\nabla_{i_2} \\cdots \\nabla_{i_n} \\phi\\\\\n &+ \\nabla_{i_1} \\left( \\bar m^2(t) \\mathcal{N}_{I_{n-1}}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_{n-1}}(\\bar T \\phi) + \\mathcal{N}_{I_{n}} (\\nabla \\phi) \\right) =0.\n\\end{split}\n\\end{equation}\nIn views of the definition for $\\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi), \\mathcal{N}_{l+1} (\\nabla \\phi), \\mathcal{N}_{l} (\\phi)$ and $\\mathcal{K}\\mathcal{N}_{I_l}(\\bar T \\phi)$, we rearrange the $2^{\\text{nd}}$-$4^{\\text{th}}$ lines in \\eqref{eq-kg-N-l-1+3-general}, thus \\eqref{eq-kg-N-l-1+3-general-simply}-\\eqref{def-eq-kg-N-l-1+3-error} holds for $l =n.$\n\\end{proof}\n\n\nThe high order KG equation in $1+3$ form \\eqref{eq-kg-N-l-1+3-general-simply}-\\eqref{def-eq-kg-N-l-1+3-error} leads to the partial energy identities of high order derivatives.\n\\begin{corollary}\\label{coro-energy-id-KG-l}\nLet $\\phi$ be a solution to the KG equation \\eqref{eq-kg}, and $l \\geq 1,$ then the $l^{\\text{th}}$-order tilde energy $\\tilde{E}_l(\\phi, t) $ (See definition in \\eqref{def-tilde-energy-norm-l-kg}) admits\n\\begin{equation}\\label{energy-id-l-kg-rescale}\n\\begin{split}\n &\\partial_{\\bar \\tau} \\tilde E_l (\\phi, t) + \\tilde E_l (\\phi, t) + \\int_{\\Sigma_t} 2 N |\\nabla_{I_{l+1}} \\phi|^2 \\mathrm{d}} %{\\text{d} \\mu_g \\\\\n = & \\int_{\\Sigma_t} \\left( BL_l + {}_lLK_1 + {}_lLK_2 + {}_lLK_3 \\right) \\mathrm{d}} %{\\text{d} \\mu_g,\n\\end{split}\n\\end{equation}\nwhere the lower order terms ${}_lLK_1, {}_lLK_2, {}_lLK_3$ are given below\n\\begin{equation}\\label{energy-estimate-kg-1-id-l-o-t-1}\n\\begin{split}\n {}_lLK_1 =& \\left( \\hat N + N\\hat k \\right) | \\nabla_{I_l} \\mathcal{D} \\phi|^2 + \\left( 3N-3\\hat N + N \\hat k \\right) \\nabla^{I_l} \\phi \\nabla_{I_l} \\bar T \\phi \\\\\n & - 2N \\nabla^{I_l} \\bar T \\phi \\mathcal{R}_{I_l} (\\phi) - N \\nabla^{I_l} \\phi \\mathcal{R}_{I_l}(\\phi),\n\\end{split}\n\\end{equation}\nwhere $\\mathcal{D} \\phi \\in \\{ \\bar T \\phi, \\nabla \\phi, \\bar m (t) \\phi\\}$, and\n\\begin{equation}\\label{energy-estimate-kg-1-id-l-o-t-2}\n\\begin{split}\n {}_lLK_2 = &\\nabla^{I_{l}} \\mathcal{D} \\phi \\left( \\mathcal{N}_{I_{l+1}}(\\mathcal{D} \\phi) + \\mathcal{N}_{I_{l}}(\\mathcal{D} \\phi) + \\mathcal{K}\\mathcal{N}_{I_l}(\\mathcal{D} \\phi) + \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) \\right),\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{energy-estimate-kg-1-id-l-o-t-3}\n\\begin{split}\n {}_lLK_3 = & \\nabla^{I_l} \\bar T \\phi \\mathcal{K}\\mathcal{N}_{I_l} (\\phi) + \\nabla^{I_l}\\phi \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi)\\\\\n & + \\nabla^{I_l}\\phi \\left( \\mathcal{N}_{I_{l}}(\\bar T \\phi) + \\mathcal{N}_{I_{l+1}} (\\nabla \\phi) \\right),\n\\end{split}\n\\end{equation}\nand the borderline terms $BL_l$ are given by\n\\begin{equation}\\label{def-energy-estimate-kg-id-bl} \nBL_l= 2\\bar m^2 (t) \\left( \\nabla^{I_l} \\bar T \\phi \\mathcal{N}_{I_l}(\\phi) - \\nabla^{I_l} \\phi \\mathcal{N}_{I_l}(\\bar T \\phi) \\right).\n\\end{equation}\nWe refer to \\eqref{def-commuting-KN-l}-\\eqref{def-commuting-K-l}, \\eqref{def-N-l}, \\eqref{def-R-l-commute-nabla-laplacian}, \\eqref{def-KN-prime-l} and \\eqref{def-eq-kg-N-l-1+3-error}-\\eqref{def-hat-k-N-A-Bianchi} for definitions of $\\mathcal{K}\\mathcal{N}_{I_l}(\\mathcal{D}\\phi) \\cdots$\n\\end{corollary}\n\n\\begin{remark}\\label{rk-CMCSH gauge-error-shift}\nIf working within the CMCSH gauge, $\\partial_t=NT+X_t, \\, \\partial_{\\bar \\tau} = N \\bar T +X$, where $X=t X_t$ denotes the rescaled shift. And extra terms as $-\\nabla_{I_l} \\left( \\lie_{X} \\bar T \\phi \\right) + \\nabla_{I_l} \\left(X^2 \\cdot \\bar T\\phi \\right)$ must be added in the KG equation \\eqref{eq-rescale-kg-1+3-0}. In the energy argument, these terms will yield the leading terms $\\nabla_i \\left( X^i |\\nabla_{I_l} \\bar T \\phi|^2 \\right) \\pm {}^X\\Pi * |\\nabla_{I_l} \\bar T \\phi|^2 \\pm \\nabla_{I_{l-2}} \\left( {}^X\\Pi * \\nabla \\bar T \\phi \\right) * \\nabla_{I_l} \\bar T \\phi$, where ${}^X\\Pi_{ij}=\\nabla_iX_j + \\nabla_jX_i$ denotes the deformation tensor of $X$. The first term vanishes after integrating on $\\Sigma_t$, while the other two terms can never be borderline terms, as we expect that $X$ shares the same (or better) estimates with $\\hat N$, c.f. Remark \\ref{rk-CMCSH gauge-shift}. \n\\end{remark}\n\n\n\n\\begin{proof}\nThe proof is in spirit of that of Theorem \\ref{Thm-energy-id-KG-0}.\nWe multiply $2 \\nabla_{I_l} \\bar T \\phi$ on the $l^{\\text{th}}$-order KG equation in $1+3$ form \\eqref{eq-kg-N-l-1+3-general-simply}-\\eqref{def-eq-kg-N-l-1+3-error}. At first, for $2 \\nabla^{I_l} \\bar T \\phi \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} \\bar T \\phi,$ \n\\begin{equation}\\label{eq-1+3-KG-multiplying-1}\n\\begin{split}\n &2 \\nabla^{I_l} \\bar T \\phi \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} \\bar T \\phi \\\\\n =& \\partial_{\\bar \\tau} \\left( | \\nabla_{I_l} \\bar T \\phi|^2 \\right) + 2\\hat N |\\nabla_{I_l} \\bar T \\phi|^2 - 2 N \\hat k * \\nabla_{I_l} \\bar T \\phi * \\nabla^{I_l} \\bar T \\phi.\n \\end{split}\n\\end{equation}\nSecondly, for $-2 \\nabla^{I_l} \\bar T \\phi N \\nabla_{I_l} \\nabla^p \\nabla_p \\phi$, by the commuting identity \\eqref{commuting-nabla-laplacian-l-simplify}, \n\\begin{equation}\\label{eq-1+3-KG-multiplying-2-0}\n\\begin{split}\n &-2 \\nabla^{I_l} \\bar T \\phi N \\nabla_{I_l} \\nabla^p \\nabla_p \\phi=-2 \\nabla^{I_l} \\bar T \\phi N \\left( \\Delta \\nabla_{I_l} \\phi + \\mathcal{R}_l(\\phi) \\right)\\\\\n =&- \\nabla_i \\left( 2 N \\nabla^i \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\right) + 2 N \\nabla^{I_{l+1}} \\phi \\nabla_{I_{l+1}} \\bar T \\phi \\\\\n & + \\nabla^{I_l} \\bar T \\phi * \\nabla N * \\nabla_{I_{l+1}} \\phi - N \\nabla^{I_l} \\bar T \\phi \\mathcal{R}_{I_l} (\\phi).\n \\end{split}\n\\end{equation}\nApplying the commuting identity \\eqref{id-commuting-nabla-T-l-simplify} to $\\nabla_{I_{l+1}} \\bar T \\phi$, $2 N \\nabla^{I_{l+1}} \\phi \\nabla_{I_{l+1}} \\bar T \\phi$ in \\eqref{eq-1+3-KG-multiplying-2-0} can be further reduced in an analogous way to \\eqref{eq-1+3-KG-multiplying-1}. Therefore,\n\\begin{equation}\\label{eq-1+3-KG-multiplying-2}\n\\begin{split}\n &-2 \\nabla^{I_l} \\bar T \\phi N \\nabla_{I_l} \\nabla^p \\nabla_p \\phi =- \\nabla_i \\left( 2 N \\nabla^i \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\right) \\\\\n &\\quad + \\partial_{\\bar \\tau} \\left( | \\nabla_{I_{l+1}} \\phi|^2 \\right) + 2 \\hat N | \\nabla^{l+1} \\phi |^2 - 2 N \\hat k * \\nabla^{I_{l+1}} \\phi * \\nabla_{I_{l+1}} \\phi \\\\\n& \\quad -2 N \\nabla^{I_{l+1}} \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_{l+1}}(\\phi)+ \\nabla^{I_l} \\bar T \\phi \\left( \\nabla N * \\nabla_{I_{l+1}} \\phi - N \\mathcal{R}_{I_l} (\\phi) \\right).\n \\end{split}\n\\end{equation}\nThirdly, $2 \\bar m^2(t) \\nabla^{I_l} \\bar T \\phi N \\nabla_{I_l} \\phi$ can be treated in the same way, thus\n\\begin{equation}\\label{eq-1+3-KG-multiplying-3}\n\\begin{split}\n &2\\bar m^2 (t) \\nabla^{I_l} \\bar T \\phi N \\nabla_{I_l} \\phi \\\\\n =& \\partial_{\\bar \\tau} \\left( \\bar m^2(t) | \\nabla_{I_l} \\phi|^2 \\right) -2 \\bar m^2(t) N \\nabla^{I_{l}} \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l}(\\phi)\\\\\n & - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2+2 \\bar m^2(t) \\hat N |\\nabla_{I_l} \\phi|^2 - 2 \\bar m^2(t) N \\hat k * \\nabla_{I_l} \\phi * \\nabla^{I_l} \\phi .\n \\end{split}\n\\end{equation}\nTherefore, putting \\eqref{eq-1+3-KG-multiplying-1}-\\eqref{eq-1+3-KG-multiplying-3} together,\n\\begin{align*}\n &\\partial_{\\bar \\tau} \\left( \\rho_{l}(\\phi) \\right) +4 N | \\nabla_{I_l} \\bar T \\phi|^2 - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2 \\\\\n & - \\nabla_i \\left( 2 N \\nabla^i \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\right) + \\text{error}=0,\n\\end{align*}\nwhere the error terms are given by\n\\begin{equation}\\label{def-error-kg-id-l}\n\\begin{split}\n \\text{error} =& 2 \\hat N |\\nabla_{I_l} \\mathcal{D} \\phi|^2 - 2 N \\hat k * |\\nabla^{I_l} \\mathcal{D} \\phi|^2 \\\\\n & + \\nabla^{I_l} \\bar T \\phi \\nabla N * \\nabla_{I_{l+1}} \\phi - N \\nabla^{I_l} \\bar T \\phi \\mathcal{R}_{I_l} (\\phi) \\\\\n & -2 N \\left( \\nabla^{I_{l+1}} \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_{l+1}}(\\phi) + \\bar m^2(t) \\nabla^{I_{l}} \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l}(\\phi) \\right)\\\\\n &+ 2 \\nabla^{I_l} \\bar T \\phi \\left( \\bar m^2(t) \\mathcal{N}_{I_l}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) - \\mathcal{N}_{I_{l+1}} (\\nabla \\phi) \\right).\n \\end{split}\n\\end{equation}\n$ \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l} (\\phi)$ (c.f. \\eqref{def-commuting-K-l}) will now split into two parts:\n\\begin{equation*}\n\\begin{split}\n \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l} (\\phi) &= N^{-1} \\mathcal{N}_{I_l} (\\bar T \\phi ) + N^{-1} \\mathcal{K}\\mathcal{N}_{I_l} (\\phi), \\,\\, l \\geq 1,\n\\end{split}\n\\end{equation*}\nwhere the definitions of $\\mathcal{N}_{I_l} (\\bar T \\phi ), \\mathcal{K}\\mathcal{N}_{I_l} (\\phi)$ are shown in \\eqref{def-eq-kg-N-l-1+3-error} and \\eqref{def-commuting-KN-l}.\nHence, the $3^{\\text{rd}}$ line of \\eqref{def-error-kg-id-l} contains $$-2 \\nabla^{I_{l+1}} \\phi * \\mathcal{N}_{l+1}(\\bar T\\phi) - 2 \\bar m^2(t) \\nabla^{I_{l}} \\phi * \\mathcal{N}_{l}(\\bar T \\phi). $$ We note that $\\bar m^2 (t) = m^2 t^2$. The extra $t^2$ makes $\\bar m^2(t) \\nabla^{I_l} \\phi * \\mathcal{N}_{l}(\\bar T \\phi)$ borderline term, while $\\nabla^{I_{l+1}} \\phi * \\mathcal{N}_{l+1}(\\bar T \\phi)$ is a lower order term. The other borderline term is $2\\bar m^2 (t) \\nabla^{I_l} \\bar T \\phi * \\mathcal{N}_{I_l}(\\phi)$ which is originated from $\\bar m^2(t) [\\nabla_{I_l}, N] \\phi$.\nWe further apply the identity \\eqref{id-div-T} to achieve\n\\begin{equation}\\label{energy-estimate-kg-1-step1}\n\\begin{split}\n &\\partial_{\\bar \\tau} \\rho_l(\\phi) + 4 N | \\nabla_{I_l} \\bar T \\phi|^2 - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2 \\\\\n & - \\nabla_i \\left( 2 N \\nabla^i \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\right) -BL_l + \\text{l.o.t.}_1=0.\n\\end{split}\n\\end{equation}\nwhere we had separated the error terms into two parts: the borderline terms \n\\begin{equation}\\label{def-b-l-K}\nBL_l = - 2 \\bar m^2(t) \\nabla^{I_l} \\phi * \\mathcal{N}_{I_l}(\\bar T \\phi) + 2 \\bar m^2(t) \\nabla^{I_l} \\bar T \\phi * \\mathcal{N}_{I_l}(\\phi) \n\\end{equation}\nand the lower order terms\n\\begin{equation}\\label{def-lot-1-K}\n\\begin{split}\n \\text{l.o.t.}_1 =& 2 \\hat N |\\nabla_{I_l} \\mathcal{D} \\phi|^2 - 2N \\hat k * |\\nabla_{I_l} \\mathcal{D} \\phi|^2 - 2 \\nabla^{I_{l+1}} \\phi * \\mathcal{N}_{I_{l+1}}(\\bar T \\phi) \\\\\n & - 2 \\left( \\nabla^{I_{l+1}} \\phi * \\mathcal{K}\\mathcal{N}_{I_{l+1}}(\\phi) +\\bar m^2(t) \\nabla^{I_{l}} \\phi * \\mathcal{K}\\mathcal{N}_{I_l}(\\phi)\\right)\\\\\n &- \\nabla^{I_l} \\bar T \\phi \\left( \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) + \\mathcal{N}_{I_{l+1}} (\\nabla \\phi) +N\\mathcal{R}_{I_l} (\\phi) \\right).\n\\end{split}\n\\end{equation}\nHere we note that $ \\nabla^{I_l} T\\phi \\nabla N * \\nabla_{I_{l+1}} \\phi$ in \\eqref{def-error-kg-id-l} has been covered by $\\nabla^{I_{l+1}} \\phi * \\mathcal{N}_{I_{l+1}}(\\bar T \\phi) $ in \\eqref{def-lot-1-K}. \n\n\nIn the second step, we will concentrate on the quadratic terms in \\eqref{energy-estimate-kg-1-step1},\n\\begin{equation}\\label{trace-part-energy-id-kg-l}\n 4 N | \\nabla_{I_l} \\bar T \\phi|^2 - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2.\n\\end{equation}\nWe again apply the commuting identity \\eqref{id-commuting-nabla-T-l-simplify} to $\\nabla_{I_{l}} \\bar T \\phi$,\n\\begin{equation*}\n\\begin{split}\n&N |\\nabla_{I_l} \\bar T \\phi|^2 = \\partial_{\\bar \\tau} \\left( \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) + \\bar m^2(t) N |\\nabla_{I_l} \\phi|^2+ 3 \\hat N \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\\\\n&- N \\hat k \\nabla^{I_l} \\bar T \\phi * \\nabla_{I_l} \\phi - N \\nabla^{I_l} \\bar T \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l} (\\phi) + 2 N \\nabla^{I_l}\\phi \\nabla_{I_l} \\bar T \\phi\\\\\n&+ \\nabla^{I_l}\\phi \\left( - N \\nabla_{I_l} \\Delta \\phi + \\bar m^2(t) \\mathcal{N}_{I_l}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) - \\mathcal{N}_{I_{l+1}} (\\nabla \\phi) \\right).\n\\end{split}\n\\end{equation*}\nUsing the high order KG equation \\eqref{eq-kg-N-l-1+3-general-simply}, we have\n\\begin{equation}\\label{eq-step-2-0-kg}\n\\begin{split}\n&4 N |\\nabla_{I_l} \\bar T \\phi|^2 - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2 = N |\\nabla_{I_l} \\bar T \\phi|^2 + \\partial_{\\bar \\tau} \\left( 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) \\\\\n& + \\bar m^2(t) N |\\nabla_{I_l} \\phi|^2 + 3N \\nabla^{I_l}\\phi \\nabla_{I_l} \\bar T \\phi - 3N \\nabla^{I_l}\\phi \\nabla_{I_l} \\Delta \\phi + \\text{l.o.t.}_2,\n\\end{split}\n\\end{equation}\nwhere the lower order terms are given by\n\\begin{equation}\\label{energy-estimate-kg-1-step2-1-lot-2}\n\\begin{split}\n& \\text{l.o.t.}_2= (3N+9\\hat N) \\nabla^{I_l}\\phi \\nabla_{I_l} \\bar T \\phi - N \\hat k \\nabla^{I_l} \\bar T \\phi * \\nabla_{I_l} \\phi \\\\\n& -N \\nabla^{I_l} \\bar T \\phi * \\mathcal{K}\\mathcal{N}\\mathcal{T}_{I_l}(\\phi) + 2 \\bar m^2(t) \\hat N |\\nabla_{I_l} \\phi|^2 \\\\\n&+ \\nabla^{I_l}\\phi \\left( \\bar m^2(t) \\mathcal{N}_{I_l}(\\phi) - \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) - \\mathcal{N}_{I_{l+1}} (\\nabla \\phi) \\right).\n\\end{split}\n\\end{equation}\nApplying the commuting identity between $\\Delta$ and $\\nabla_{I_l}$ \\eqref{commuting-nabla-laplacian-l-simplify} to $\\Delta \\nabla_{I_l} \\phi$ above,\nwe have\n\\begin{equation}\\label{energy-estimate-kg-1-step2-2}\n\\begin{split}\n & - 3 N\\nabla^{I_l}\\phi \\nabla_{I_l} \\Delta \\phi = - 3N \\nabla^{I_l}\\phi \\left( \\Delta \\nabla_{I_l} \\phi - \\mathcal{R}_{I_l}(\\phi) \\right) \\\\\n=& -3 \\nabla^{i} \\left( N \\nabla^{I_l}\\phi \\nabla_i \\nabla_{I_l}\\phi \\right) + 3 N \\nabla^{I_{l+1}}\\phi \\nabla_{I_{l+1}} \\phi + \\text{l.o.t.}_3,\n\\end{split}\n\\end{equation}\nwhere the lower order terms are given by\n\\begin{equation}\\label{lot-kg-1-step2-2}\n\\text{l.o.t.}_3 =3 \\nabla^{I_l}\\phi\\nabla^{i} N \\nabla_i \\nabla_{I_l}\\phi + 3\\nabla^{I_l}\\phi N \\mathcal{R}_{I_l}(\\phi).\n\\end{equation}\nPutting \\eqref{eq-step-2-0-kg}-\\eqref{energy-estimate-kg-1-step2-1-lot-2} and \\eqref{energy-estimate-kg-1-step2-2}-\\eqref{lot-kg-1-step2-2} together, we obtain\n\\begin{equation}\\label{energy-estimate-kg-1-step2-4}\n\\begin{split}\n&4 N |\\nabla_{I_l} \\bar T \\phi|^2 - 2 \\bar m^2(t) |\\nabla_{I_l} \\phi|^2 \\\\\n=& N |\\nabla_{I_l} \\bar T \\phi|^2 + \\bar m^2(t) N |\\nabla_{I_l} \\phi|^2 + 3 N |\\nabla_{I_{l+1}} \\phi|^2 + 3N \\nabla^{I_l}\\phi \\nabla_{I_l} \\bar T \\phi \\\\\n&+ \\partial_{\\bar \\tau} \\left( 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) -3 \\nabla^{i} \\left( N \\nabla^{I_l}\\phi \\nabla_i \\nabla_{I_l}\\phi \\right) + \\text{l.o.t.}_2 + \\text{l.o.t.}_3.\n\\end{split}\n\\end{equation}\n\nCombining \\eqref{energy-estimate-kg-1-step2-4} with \\eqref{energy-estimate-kg-1-step1}-\\eqref{def-lot-1-K}, we arrive at\n\\begin{equation}\\label{energy-estimate-kg-l-id}\n\\begin{split}\n &\\partial_{\\bar \\tau} \\left( \\rho_l (\\phi) + 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) + N \\left( \\rho_l (\\phi) + 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) \\\\\n & + 2 N |\\nabla_{I_{l+1}} \\phi|^2 - \\nabla^{i} \\left( 3N \\nabla^{I_l}\\phi \\nabla_i \\nabla_{I_l}\\phi + 2 N \\nabla^i \\nabla_{I_l} \\phi \\nabla^{I_l} \\bar T \\phi \\right) \\\\\n & -BL_l + \\text{l.o.t.}_1 + \\text{l.o.t.}_2 + \\text{l.o.t.}_3,\n\\end{split}\n\\end{equation}\nwhere $BL_l$, $\\text{l.o.t.}_1, \\cdots, \\text{l.o.t.}_3$ are defined in \\eqref{def-b-l-K}, \\eqref{def-lot-1-K}, \\eqref{energy-estimate-kg-1-step2-1-lot-2}, \\eqref{lot-kg-1-step2-2}.\nIntegrate \\eqref{energy-estimate-kg-l-id} on $\\Sigma_t$,\n\\begin{equation*}\n\\begin{split}\n &\\partial_{\\bar \\tau} \\int_{\\Sigma_t} \\left( \\rho_l (\\phi) + 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) + \\int_{\\Sigma_t} \\left( \\rho_l (\\phi, t) + 3 \\nabla^{I_l} \\bar T \\phi \\nabla_{I_l} \\phi \\right) \\\\\n & + \\int_{\\Sigma_t} 2 N |\\nabla_{I_{l+1}} \\phi|^2 -BL_l - {}_lLK_1 - {}_lLK_2- {}_lLK_3 =0,\n\\end{split}\n\\end{equation*}\nwhere ${}_lLK_i$ are defined as in \\eqref{energy-estimate-kg-1-id-l-o-t-1}-\\eqref{energy-estimate-kg-1-id-l-o-t-3}.\n And we denote $\\mathcal{D} \\phi \\in \\{ \\bar T \\phi, \\nabla \\phi, \\bar m \\phi\\}$\nAs we can see, the $2 N | \\nabla_{I_{l+1}} \\phi |^2$ above has a good sign and it will contribute a positive double integral in the energy estimate.\nWe then achieve \\eqref{energy-id-l-kg-rescale}-\\eqref{energy-estimate-kg-1-id-l-o-t-3}.\n\\end{proof}\n\n\n\n\\subsection{Energy identities for the $1+3$ Bianchi equations}\\label{sec-ee-Bianchi-1+3}\n \\subsubsection{The Weyl fields and Bianchi equations}\nThe Weyl tensor $W_{\\alpha \\beta \\gamma \\delta}$ being the traceless part of the curvature tensor is \n\\begin{equation}\\label{def-Weyl}\n\\begin{split}\nW_{\\alpha \\beta \\gamma \\delta} = &\\bar R_{\\alpha \\beta \\gamma \\delta} - \\frac{1}{2} ( \\bar g \\odot Ric(\\bar g) )_{\\alpha \\beta \\gamma \\delta} +\\frac{1}{12} \\bar R ( \\bar g \\odot \\bar g )_{\\alpha \\beta \\gamma \\delta},\n\\end{split}\n\\end{equation}\nwhere $Ric(\\bar g)$ is the Ricci tensor of $\\bar g$, and $\\odot$ is defined as \\eqref{def-odot}.\nWe define the left and right Hodge duals of $W$ by\n\\begin{equation}\\label{eq-Bianchi-J}\n{}^{*}W_{\\alpha \\beta \\gamma \\delta} = \\frac{1}{2} \\epsilon_{\\alpha \\beta \\mu \\nu} W^{\\mu \\nu}{}_{\\! \\gamma \\delta},\n\\quad W^*_{\\alpha \\beta \\gamma \\delta} = \\frac{1}{2} W_{\\alpha \\beta}{}^{\\! \\mu \\nu} \\epsilon_{\\mu \\nu \\gamma \\delta}.\n\\end{equation}\nHowever, by virtue of the algebraic properties of $W$ the two duals coincide: ${}^{*}W = W^*$.\nThe Bianchi identities entail the divergence equations\n\\begin{subequations}\n\\begin{equation}\nD^\\alpha W_{\\alpha \\beta \\gamma \\delta} = J_{\\beta \\gamma \\delta}, \\label{eq-Bianchi-J}\n\\end{equation}\n\\begin{equation}\nD^\\alpha {}^{*}W_{\\alpha \\beta \\gamma \\delta} = J^*_{\\beta \\gamma \\delta}, \\label{eq-Bianchi-J-*}\n\\end{equation}\n\\end{subequations}\n where\n \\begin{equation}\\label{def-J}\n J_{\\beta \\gamma \\delta} = \\frac{1}{2} \\left( D_\\gamma \\bar R_{\\delta \\beta} - D_{\\delta} \\bar R_{\\gamma \\beta}\\right) - \n\\frac{1}{12} \\left(\\bar g_{\\beta \\delta} D_\\gamma \\bar R - \\bar g_{\\beta \\gamma} D_\\delta \\bar R \\right),\n\\end{equation}\nand then\n\\begin{equation}\\label{def-J-*}\n J^*_{\\beta \\gamma \\delta} = \\frac{1}{2} J_\\beta{}^{\\! \\mu \\nu} \\epsilon_{\\mu \\nu \\gamma \\delta}.\n\\end{equation} \nThe source terms $J_{\\beta \\gamma \\delta}, J^*_{\\beta \\gamma \\delta}$ are related to the matter field, since the Ricci tensor $\\bar R_{\\alpha \\beta}$ is given by the KG field, see \\eqref{eq-ricci-kg-phi}.\n\nThe electric and magnetic parts $E(W), H(W)$ of the Weyl fields $W$, with respect to the foliation $\\Sigma_t$ are defined by\n\\begin{equation}\\label{def-electric-magnetic}\nE(W)_{\\alpha \\beta} = W_{\\alpha \\mu \\beta \\nu} \\bar T^\\mu \\bar T^\\nu, \\quad H(W)_{\\alpha \\beta} = {}^{*}W_{\\alpha \\mu \\beta \\nu} \\bar T^\\mu \\bar T^\\nu.\n\\end{equation} \nThe tensors $E$ and $H$ are $T$-tangent, i.e. $E_{\\alpha \\beta}T^\\alpha = H_{\\alpha \\beta}T^\\alpha=0$ and trace free $\\bar g^{\\alpha \\beta} E_{\\alpha \\beta} = \\bar g^{\\alpha \\beta} H_{\\alpha \\beta}=0$. It follows that $g^{ij} E_{ij} =g^{ij} H_{ij}=0$.\nThe following identities relate $W, {}^{*}W, E=E(W), H=H(W),$ (c.f. Page 144 in \\cite{Christodoulou-K-93}),\n\\begin{align*}\nW_{ijp \\bar T} &= -\\epsilon_{ij}{}^{\\! m} H_{mp}, \\quad \\quad \\quad {}^{*}W_{ijp \\bar T} = \\epsilon_{ij}{}^{\\! m} E_{mp},\\\\\nW_{ijpq} &= -\\epsilon_{ijm} \\epsilon_{pqn} E^{mn}, \\quad \\,\\, {}^{*}W_{ijpq} = \\epsilon_{ijm} \\epsilon_{pqn} H^{mn}.\n\\end{align*} \n\nWe note that, the spacetime hodge dual $\\ast$, and $J_{\\alpha \\beta \\gamma}$, $E_{ij}, H_{ij}$ are all scale-free.\nThe Gauss and Codazzi equations can be written in terms of $E$ and $H$ \n\\begin{subequations}\n\\begin{equation}\nR_{ij} + 2 g_{ij} - \\hat k_{im} \\hat k^m{}_{\\! j} - \\hat k_{ij} =E_{ij} +\\frac{1}{2} \\bar R_{ij} + \\frac{1}{6}(3\\bar R_{TT} + \\bar R) g_{ij}. \\label{eq-Gauss-E}\n\\end{equation}\n\\begin{equation}\n\\text{curl} k_{ij} = -H_{ij}. \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\label{eq-Codazzi-H}\n\\end{equation}\n\\end{subequations}\nFor the energy estimate scheme of the Weyl field, we will focus on the following (rescaled) $1+3$ splitting of Bianchi equations \\eqref{eq-Bianchi-J}-\\eqref{eq-Bianchi-J-*} (c.f. Proposition 7.2.1 in \\cite{Christodoulou-K-93} or Corollary 3.2 in \\cite{A-M-04})\n\\begin{subequations}\n\\begin{equation}\n\\begin{split}\n \\lie_{\\partial_{\\bar \\tau}} E_{ij} =&N\\text{curl} H_{ij}- \\left( \\nabla N \\wedge H \\right)_{ij} \\\\\n &- \\frac{5N}{2} \\left( E \\times k \\right)_{ij} - \\frac{2N}{3} \\left( E \\cdot k \\right) g_{ij}- \\frac{N}{2}\\text{tr} k E_{ij} - NJ_{i \\bar T j}, \\label{eq-1+3-bianchi-T-E}\n \\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n \\lie_{\\partial_{\\bar \\tau}} H_{ij} =&-N \\text{curl} E_{ij} +\\left( \\nabla N \\wedge E \\right)_{ij} \\\\\n &- \\frac{5N}{2} \\left( H \\times k \\right)_{ij} - \\frac{2N}{3} \\left( H \\cdot k \\right) g_{ij} - \\frac{N}{2}\\text{tr} k H_{ij} - NJ^*_{i \\bar T j}. \\label{eq-1+3-bianchi-T-H}\n \\end{split}\n\\end{equation}\n\\end{subequations}\nSee Appendix for the definition of $\\wedge, \\times,$ div and curl. \n\nWe shall allow ourselves to use $W$ to represent the electric part $E$ or the magnetic part $H$.\nDefine the $l^{\\text{th}}$-order energy norm for $W$\n\\begin{equation}\\label{def-energy-l-homo-Weyl}\nE_i (W, \\tau) = \\int_{\\Sigma_t} \\left( |\\nabla_{I_i} E|_g^2 + |\\nabla_{I_i} H |_g^2 \\right) \\mathrm{d}} %{\\text{d} \\mu_g.\n\\end{equation}\nLet $E^{\\tilde g}_i (W, t)$ be the energy norm associated to $\\tilde g$, i.e. replacing $g$ by $\\tilde g$ in \\eqref{def-energy-l-homo-Weyl}. Then, $t^2 E_i (W, t)= t^{3+2i} E^{\\tilde g}_i (W, t)$. And define\n\\begin{equation}\\label{def-energy-density-l-homo-Weyl}\n\\rho_i (W) = |\\nabla_{I_i} E|_g^2 + |\\nabla_{I_i} H |_g^2.\n\\end{equation}\n\n\\subsubsection{The zeroth order energy identity}\\label{sec-0-einstin-1+3}\n\n\n\\begin{theorem}\\label{lemma-energy-id-1-Bianchi}\nFor solutions to the Bianchi equation in $1+3$ form \\eqref{eq-1+3-bianchi-T-E}-\\eqref{eq-1+3-bianchi-T-H}, there is the energy identity\n\\begin{equation}\\label{energy-id-0-bianchi}\n\\begin{split}\n&\\partial_{\\bar \\tau} E_0 (W, \\bar \\tau) + 2E_0 (W, \\bar \\tau) = E_0(W, {\\bar \\tau}_0) + \\int_{\\Sigma_{t}} \\left( {}_0LW + S_0 \\right) \\mathrm{d}} %{\\text{d} \\mu_g,\n\\end{split}\n\\end{equation}\nwhere the error terms $LW_0$ and the source terms $S_0$ are given by\n\\begin{equation}\\label{def-e-s-1-bianchi}\n\\begin{split}\n{}_0LW =&3 \\hat N \\left( |E|^2+|H|^2 \\right) + \\nabla N * E*H \\\\\n& \\pm N \\hat k * \\left(E*E + H*H \\right), \\\\\nS_0 =& - 2 N\\left( J_{i \\bar T j} E^{ij} + J^*_{i \\bar T j} H^{ij} \\right).\n\\end{split}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe multiply $2 E_{ij}$ on \\eqref{eq-1+3-bianchi-T-E} and $2 H_{ij}$ on \\eqref{eq-1+3-bianchi-T-H} to derive \n\\begin{equation} \\label{eq-1+3-bianchi-T-EH-1}\n\\begin{split}\n& \\partial_{\\bar \\tau} \\left( |E|^2+|H|^2 \\right) -\\frac{2N}{3} \\text{tr} k \\left( |E|^2+|H|^2 \\right) \\\\\n=&\\text{div} \\left( 2N \\left( E \\wedge H \\right) \\right) + 6 \\nabla N \\cdot \\left( E \\wedge H \\right) \\\\\n &+ 4N \\hat k * \\left( E*E+ H*H\\right) -4\\hat N \\left( E^2 + H^2 \\right) \\\\\n&- 3N \\left( E \\times \\hat k \\right)\\cdot E -3N \\left( H \\times \\hat k \\right)\\cdot H \\\\\n&- 2N J_{i \\bar T j} E^{ij} - 2N J^*_{i \\bar T j} H^{ij}.\n\\end{split}\n\\end{equation}\nHere we have used the identity \\eqref{wedge-curl-div} relating curl, $\\cdot$, div, $\\wedge$ and the splitting,\n\\begin{equation}\\label{eq-E-time-K}\n\\left( E \\times k \\right)_{ij} =\\left( E \\times \\hat k \\right)_{ij} - \\frac{\\text{tr} k}{3}E_{ij}.\n\\end{equation} \nIntegrating on $\\Sigma_{t}$, we have\n\\begin{equation}\\label{eq-1+3-bianchi-T-EH-3}\n\\begin{split}\n& \\partial_{\\bar \\tau} \\int_{\\Sigma_{t}} \\rho_0(W) \\mathrm{d}} %{\\text{d} \\mu_g + \\int_{\\Sigma_{t}} 2N \\rho_0(W) -3\\hat N \\rho_0 (W) \\mathrm{d}} %{\\text{d} \\mu_g \\\\\n= & \\int_{\\Sigma_{t}} 4N \\hat k * \\left( E*E+ H*H\\right) -4\\hat N \\left( E^2 + H^2 \\right) \\\\\n& \\quad + 6 \\nabla N \\cdot \\left( E \\wedge H \\right) - 2N J_{i \\bar T j} E^{ij} - 2N J^*_{i \\bar T j} H^{ij}.\n\\end{split}\n\\end{equation}\nThis gives \\eqref{energy-id-0-bianchi}.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Higher order energy identities}\\label{sec-h-einstin-1+3}\nIn this section, we always require $l \\leq 3$.\nTo proceed to the high order case, we need a high order version of the identity \\eqref{wedge-curl-div}, which relates curl, $\\cdot$, div, $\\wedge$.\n\\begin{lemma}\\label{lemma-div-curl}\n Let $l \\geq 0$, and $H_{ij}, E_{ij}$ be any two $(0, 2)$ symmetric and trace free tensors on $\\Sigma_t$. Then there is the identity\n\\begin{equation}\\label{wedge-curl-div-general}\n\\begin{split}\n& \\nabla_{I_l} \\left( \\text{curl} H \\right)_{ij} \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\left( \\text{curl} E \\right)_{ij} \\nabla^{I_l} H^{ij} \\\\\n=& \\sum_{a \\leq l} \\nabla_q \\left( \\nabla_{I_a} H * \\nabla_{I_a} E \\right)^q \\pm \\left( \\hat{\\mathcal{R}}_{I_{l-1}}(H) * \\nabla_{I_l} E + \\hat{\\mathcal{R}}_{I_{l-1}}(E) * \\nabla_{I_l} H \\right),\n\\end{split}\n\\end{equation}\nwhere $\\hat{\\mathcal{R}}_{I_{l-1}}(A)$ is defined as \n\\begin{equation}\\label{Def-hat-R-div-curl}\n\\begin{split}\n \\hat{\\mathcal{R}}_{I_{l-1}}(A) =& \\sum_{a+b = l-1} \\nabla_{I_a} \\hat R_{imjn} * \\nabla_{I_b} A, \n\\end{split}\n\\end{equation}\nwith $\\hat R_{imjn} = R_{imjn} +( g \\odot g )_{imjn} $ being the error term in the Riemanian curvature. By virtue of \\eqref{Gauss-Riem-hat-k},\n\\begin{equation}\\label{def-hat-R}\n\\begin{split}\n\\hat R_{imjn} =& ( g \\odot \\hat k )_{imjn} - \\frac{1}{2} ( \\hat k \\odot \\hat k )_{imjn} + \\bar{R}_{imjn}.\n\\end{split}\n\\end{equation}\n\\end{lemma}\nThe proof for this Lemma is presented in the Appendix \\ref{sec-app-id}.\n\n\nTheorem \\ref{lemma-energy-id-1-Bianchi} can be generalized to the high order case as below.\n\\begin{corollary}\\label{coro-energy-id-l-Bianchi}\nFor solutions to the Bianchi equations in $1+3$ form \\eqref{eq-1+3-bianchi-T-E}-\\eqref{eq-1+3-bianchi-T-H} with $l \\geq 1$, there is the energy identity\n\\begin{equation}\\label{energy-id-l-bianchi}\n\\begin{split}\n & \\partial_{\\bar \\tau} E_l (W, \\bar \\tau) + 2 E_l(W, \\bar \\tau) = \\int_{\\Sigma_{t}} \\left( {}_lLW_1 + {}_lLW_2 + {}_lLW_3 + S_l \\right) \\mathrm{d}} %{\\text{d} \\mu_g,\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\\label{def-LB-1-2-S}\n\\begin{split}\n{}_lLW_1 = & \\left( \\mathcal{K}\\mathcal{N}_{I_l}(W) - \\hat{\\mathcal{K}}\\mathcal{N}_{I_l} (W) \\right) \\nabla^{I_l} W, \\\\\n{}_lLW_2 = & - \\hat N |\\nabla_{I_l} W|^2 \\pm \\sum_{a \\leq l} \\mathcal{N}_{I_{a}}(W) \\nabla^{I_a} W, \\\\\n{}_lLW_3 = & \\pm \\hat{\\mathcal{R}}_{I_{l-1}} (W) \\nabla^{I_l} W, \\\\\nS_l=& - \\nabla_{I_l} \\left( N J_{i \\bar T j}\\right) \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\left( N J^*_{i \\bar T j}\\right) \\nabla^{I_l} H^{ij}.\n\\end{split} \n\\end{equation}\nWe refer to \\eqref{def-commuting-K-R-T-l-tensor}, \\eqref{def-eq-kg-N-l-1+3-error}-\\eqref{def-hat-k-N-A-Bianchi} and \\eqref{Def-hat-R-div-curl} for the definition of $\\mathcal{K}\\mathcal{N}_{I_l}(E_{ij}),$ $ \\hat{\\mathcal{R}}_{I_{l-1}} (E_{ij}),$ $\\mathcal{N}_{I_l} (E_{ij}), \\hat{\\mathcal{K}}\\mathcal{N}_{I_l} (E_{ij})\\cdots$\n\\end{corollary}\n\n\\begin{proof}\nPerform $\\nabla_{I_l}$ on \\eqref{eq-1+3-bianchi-T-E}-\\eqref{eq-1+3-bianchi-T-H}, and apply the commuting identity \\eqref{id-commuting-nabla-N-T-l-simplify-tensor} to $\\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} E_{ij}$ and $\\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} H_{ij}$. Then,\n\\begin{align*}\n& \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} E_{ij} = \\nabla_{I_l} \\left( N \\text{curl} H_{ij} + \\frac{N}{3} \\text{tr} k E_{ij} \\right) + \\mathcal{K}\\mathcal{N}_{I_l}(E_{ij})\\\\\n& \\quad - \\nabla_{I_l} \\left( \\left( \\nabla N \\wedge H \\right)_{ij} + \\frac{5N}{2} \\left( E \\times \\hat k \\right)_{ij} + \\frac{2N}{3} \\left( E \\cdot \\hat k \\right) g_{ij}+ N J_{i \\bar T j} \\right).\n\\end{align*}\nHere the splitting \\eqref{eq-E-time-K} is used. We rearrange and collect similar terms\n\\begin{equation}\\label{eq-evolu-E-1}\n\\begin{split}\n \\lie_{\\partial_{\\bar \\tau}} \\nabla_{I_l} E_{ij} =& N \\nabla_{I_l} \\left( \\text{curl} H_{ij} \\right) + \\frac{N}{3} \\text{tr} k \\nabla_{I_l} E_{ij} - \\nabla_{I_l} \\left( N J_{i \\bar T j}\\right)\\\\\n&+ \\mathcal{K}\\mathcal{N}_{I_l}(E_{ij}) \\pm \\mathcal{N}_{I_{l}}(H_{ij}) - \\hat{\\mathcal{K}}\\mathcal{N}_{I_l} (E_{ij}).\n\\end{split} \n\\end{equation}\nMultiplying $2 \\nabla^{I_l} E^{ij}$ on \\eqref{eq-evolu-E-1}, we derive the transported equation for $|\\nabla_{I_l} E|^2,$\n\\begin{align*}\n&\\partial_{\\bar \\tau} \\left( |\\nabla_{I_l} E|^2 \\right) -\\frac{2N}{3} \\text{tr} k |\\nabla_{I_l} E|^2\\\\\n=& 2N \\nabla_{I_l} \\text{curl} H_{ij} \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\left( N J_{i \\bar T j}\\right) \\nabla^{I_l} E^{ij} -\\hat N \\nabla_{I_l} E * \\nabla^{I_l} E \\\\\n&+\\left( \\mathcal{K}\\mathcal{N}_{I_l}(E_{ij}) \\pm \\mathcal{N}_{I_{l}}(H_{ij}) - \\hat{\\mathcal{K}}\\mathcal{N}_{I_l} (E_{ij}) \\right) \\nabla^{I_l} E^{ij}.\n\\end{align*}\nWe had ignore irrelevant constant.\nSimilar argument also applies to \\eqref{eq-1+3-bianchi-T-H}.\n\nPutting the transported equations for $ |\\nabla_{I_l} E|^2$ and $ |\\nabla_{I_l} H|^2$ together, we have\n\\begin{equation*}\n\\begin{split}\n&\\partial_{\\bar \\tau} \\left( |\\nabla_{I_l} E|^2 + |\\nabla_{I_l} H|^2 \\right) - \\frac{2N}{3} \\text{tr} k \\left( |\\nabla_{I_l} E|^2 + |\\nabla_{I_l} H|^2 \\right) \\\\\n=& 2N \\left( \\nabla_{I_l} \\text{curl} H_{ij} \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\text{curl} E_{ij} \\nabla^{I_l} H^{ij} \\right) \\\\\n&+\\left( \\mathcal{K}\\mathcal{N}_{I_l}(W) \\pm \\mathcal{N}_{I_{l}}(W) - \\hat{\\mathcal{K}}\\mathcal{N}_{I_l}(W) \\right) \\nabla^{I_l} W - \\hat N \\nabla_{I_l} W * \\nabla^{I_l} W \\\\\n& - \\nabla_{I_l} \\left( N J_{i \\bar T j}\\right) \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\left( N J^*_{i \\bar T j}\\right) \\nabla^{I_l} H^{ij}.\n\\end{split} \n\\end{equation*}\nIn views of the identity \\eqref{wedge-curl-div-general} relating curl, $\\cdot$ and div, $\\wedge$, the second line above becomes\n\\begin{align*}\n& N \\left( \\nabla_{I_l} \\text{curl} H_{ij} \\nabla^{I_l} E^{ij} - \\nabla_{I_l} \\text{curl} E_{ij} \\nabla^{I_l} H^{ij} \\right)\\\\\n=& div \\pm \\hat{\\mathcal{R}}_{I_{l-1}} (W) * \\nabla^{I_l} W \\pm \\sum_{a \\leq l} \\nabla N * \\nabla_{I_a} E * \\nabla_{I_a} H.\n\\end{align*}\nHere $div$ denotes divergence forms. Integrating on $\\Sigma_{t}$, we have\n\\begin{equation}\\label{eq-1+3-bianchi-T-EH-3}\n\\begin{split}\n& \\partial_{\\bar \\tau} \\int_{\\Sigma_{t}} \\rho_l(W) \\mathrm{d}} %{\\text{d} \\mu_g + \\int_{\\Sigma_{t}} 2N \\rho_l(W) \\\\\n= & \\int_{\\Sigma_{t}} \\left( {}_lLW_1 + {}_lLW_2 + {}_lLW_3 + S_l \\right) \\mathrm{d}} %{\\text{d} \\mu_g.\n\\end{split}\n\\end{equation}\nwhere the lower order terms ${}_lLW_1, \\cdots, {}_lLW_3$ and source terms $S_l$ are defined in \\eqref{def-LB-1-2-S}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminary estimate}\\label{sec-preliminary-estimate}\n\nLet us remind the partial energy norm for the KG field $E_i(\\phi, t)$ \\eqref{def-energy-l-homo-kg-g} and for the Weyl tensor $E_i(W, t)$ \\eqref{def-energy-l-homo-Weyl}.\nWe also define the energy norm up to $l$ order as follows,\n\\begin{equation}\\label{def-energy-sum-inhomo}\n\\mathcal{E}_l (\\phi, t) = \\sum_{i\\leq l} E_i(\\phi, t), \\quad \\mathcal{E}_l (W, t) = \\sum_{i \\leq l} E_i(W, t).\n\\end{equation}\nBesides, the energies for $\\hat k$ are defined by\n\\begin{equation}\\label{def-energy-hat-k}\nE_i (\\hat k, t )= \\int_{\\Sigma_t} |\\nabla_{I_i} \\hat k|^2 \\mathrm{d}} %{\\text{d} \\mu_g, \\quad \\mathcal{E}_l(\\hat k, t )=\\sum_{i \\leq l} E_i(\\hat k, t ).\n\\end{equation}\n\n\\subsection{The continuity argument}\\label{sec-Boot-assum}\nFix a constant $0<\\delta < \\frac{1}{6}$ and let $M$ be a large constant to be determined.\nWe start with the following weak assumptions:\nThe bootstrap assumptions for the KG field $\\phi$,\n\\begin{equation}\\label{BT-KG}\n t E_i(\\phi, t) \\leq \\varepsilon^2M^2 t^{2\\delta}, \\quad i =1,\\cdots, 4;\n\\end{equation}\nThe bootstrap assumptions for the Weyl tensor $E(W), H(W)$, \n\\begin{equation}\\label{BT-E-H}\nt^2 E_i(W, t) \\leq \\varepsilon^2M^2 t^{2\\delta}, \\quad i=1, \\cdots, 3;\n\\end{equation}\nThe bootstrap assumptions for the $\\hat k$ and $\\hat N = N-1$,\n\\begin{equation}\\label{BT-k-N-L-infty}\nt^2 E_i (\\hat k, t ) \\leq \\varepsilon^2 M^2 t^{2\\delta}, \\, 0 \\leq i \\leq 4, \\quad t^2 \\|\\hat N\\|^2_{L^\\infty} \\leq \\varepsilon^2 M^2 t^{4\\delta}.\n\\end{equation}\n\nThe smallness condition for KG data \\eqref{intro-initial-data} given in Theorem \\ref{main-thm-into} entails that \\eqref{BT-KG} holds for $t=t_0$.\nConducting the elliptic estimates for $\\hat N$ (c.f. the second (elliptic) equation in \\eqref{eq-hat-lapse}) on the initial slice, we further show that the requirement for data \\eqref{intro-initial-data} implies that \\eqref{BT-k-N-L-infty} holds on $\\{ t = t_0\\}$. Moreover, in views of the Gauss equations \\eqref{Gauss-Riem-hat-k}-\\eqref{Gauss-Ricci-hat-k} and the Weyl tensor \\eqref{def-Weyl}, \\eqref{intro-initial-data} implies that \\eqref{BT-E-H} holds for $t=t_0.$ Let $[t_0, t']$ be the largest time interval on which \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty} still holds. We shall show that if $\\varepsilon$ is sufficiently small (depending on initial data and $\\delta$), then on $[t_0, t']$, \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty} implies the same inequality with the constant $\\varepsilon^2 M^2$ being replaced by $\\frac{\\varepsilon^2 M^2}{2}$, c.f. Section \\ref{sec-close-b-t}. It will then follow that the solution and the energy estimate \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty} can be extended to a larger time interval $[t_0,T^\\prime]$, thus contradicting the maximality of $t'$. This will imply that $t'=\\infty$ and the solution is global (see the formulation of Theorem \\eqref{main-thm-global}). We will in fact prove that for a sufficiently small $\\varepsilon$, the stronger estimate \\eqref{stronger-estimate-closing-bt} (Theorem \\eqref{main-thm-global}) holds true on the interval $[0,t']$. From now on, we always assume $t \\in [t_0, t']$.\n\nWe use the following convention for initial data for $\\psi$, \n\\begin{equation}\\label{def-initial-convention}\n\\varepsilon^2 I^2_l(\\psi): = t_0 \\mathcal{E}_l(\\psi, t_0).\n\\end{equation} \nIn application, $\\psi$ will be taken as $\\phi, \\hat k$ or $W$.\nAt the first stage, we choose $M$ large enough so that $I_4(\\phi)$, $I_3(W)$, $I_4(\\hat k) < 2M,$\nand $\\varepsilon$ is chosen small enough such that $\\varepsilon M <1$. \n\n\n\n\\subsection{Sobolev inequality}\\label{sec-Sobolev}\nRecall that $g_{ij}=t^{-2} \\tilde g_{ij}$ is the scale-free metric, with $\\nabla$ the corresponding connection. \nThroughout the paper, we simplify the notation $L^p(\\Sigma ,g)$ by $L^p(\\Sigma_t), \\, p>0$, and $\\int_{\\Sigma} \\mathrm{d}} %{\\text{d} \\mu_g$ by $\\int_{\\Sigma_t} $. \nBy the Sobolev embedding theorem on $(\\Sigma, g)$, we derive the following preliminary estimates: for $\\phi,$ (recalling that $\\mathcal{D} \\phi \\in \\{\\bar T \\phi, \\nabla \\phi, \\bar m(t) \\phi\\}$)\n\\begin{equation}\\label{BT-KG-L-infty-4-6}\n\\begin{split}\n \\sum_{j \\leq 2} \\|\\nabla^j \\mathcal{D} \\phi \\|_{L^\\infty} + \\sum_{i \\leq 3} \\|\\nabla^i \\mathcal{D} \\phi \\|_{L^4(\\Sigma_t)} + \\|\\nabla^i \\mathcal{D} \\phi \\|_{L^6(\\Sigma_t)} \\lesssim \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\nFor $W \\in \\{E, H\\}$ \\eqref{BT-E-H}, \n\\begin{equation}\\label{BT-Weyl-L-infty-4-6}\n\\begin{split}\n\\sum_{j \\leq 1} \\|\\nabla^j W\\|_{L^\\infty} + \\sum_{i \\leq 2} \\|\\nabla^i W\\|_{L^4(\\Sigma_t)} + \\|\\nabla^i W\\|_{L^6(\\Sigma_t)} & \\lesssim \\mathcal{E}_3 (W, t). \n\\end{split}\n\\end{equation}\nThe bootstrap assumption for $\\hat k$ \\eqref{BT-k-N-L-infty} leads to\n\\begin{equation}\\label{BT-k-L-infty-4-6}\n\\begin{split}\n\\sum_{j \\leq 2} \\|\\nabla^j \\hat k\\|_{L^\\infty} + \\sum_{i \\leq 3} \\|\\nabla^i \\hat k \\|_{L^4(\\Sigma_t)} + \\|\\nabla^i \\hat k \\|_{L^6(\\Sigma_t)} \\lesssim \\mathcal{E}_4 (\\hat k, t).\n\\end{split}\n\\end{equation}\n\n\\begin{remark}\nLet $g_0 = t^{-2}_0 \\tilde g_0$ be the initially normalized metric. \nUsing the estimate for $\\| \\hat k\\|_{L^\\infty},$ $\\| \\hat N\\|_{L^\\infty}$ \\eqref{BT-k-N-L-infty}, \\eqref{BT-k-L-infty-4-6} and the evolution equation for $g_{ij}$ \\eqref{eq-evolution-1}, we are able to prove that $ g$ is close to $ g_0$: $| g_{ij} - g_{0ij}| \\lesssim \\varepsilon M.$\nAnd the volumes of $g_{ij}$ and $g_{0ij}$ are equivalent: $\\exp(-\\varepsilon M) \\lesssim \\frac{\\mathrm{d}} %{\\text{d} \\mu_{g}}{ \\mathrm{d}} %{\\text{d} \\mu_{g_0}} \\lesssim \\exp(\\varepsilon M)$. Thus, there exists a uniform constant bounding all the Sobolev constants for $g_{ij}$.\n\\end{remark}\n\n\n\n\n\\subsection{The lapse}\\label{sec-N}\nIn this section, we present the elliptic estimates for the lapse. In particular, we highlight the subtle relation between the lapse and the KG field, which shall server as a key point in establishing the hierarchy of energy estimates for the KG field, c.f. Section \\ref{sec-0-ee-kg} .\n\nThe equation for lapse $N$ can be derived by taking the trace of the transport equation for $\\hat k_{ij}$ \\eqref{eq-evolution-2}. \n$\\hat N$ defined in \\eqref{def-hat-N} should be regarded as small quantity in the small data situation we are considering.\n\\begin{equation}\\label{eq-hat-lapse}\n\\Delta \\hat N - 3 \\hat N = N\\left( \\bar R_{\\bar T \\bar T} + | \\hat k |^2 \\right).\n\\end{equation}\n\nLet us turn to the elliptic estimates for $\\hat N$.\nDefine the energy for $\\hat N$:\n\\begin{equation}\\label{def-energy-hat-N}\nE_i (\\hat N, t )= \\int_{\\Sigma_t} |\\nabla^i \\hat N|_g^2 \\mathrm{d}} %{\\text{d} \\mu_g, \\quad \\mathcal{E}_l(\\hat N, t )=\\sum_{i \\leq l} E_i(\\hat N, t ).\n\\end{equation}\nAgain we have $E_i (\\hat N, t ) =t^{-3 + 2i} E^{\\tilde g}_i (\\hat N, t ),$ where $E^{\\tilde g}_i (\\hat N, t )$ is defined by replacing the $g$ in \\eqref{def-energy-hat-N} by $\\tilde g$.\n\n\\begin{lemma}\\label{lem-N-ineq-source}\nWith the bootstrap assumptions \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty}, we have \n \\begin{equation}\\label{N-inte-ineq-123-lemma}\n t^2 E_i(\\hat N, t ) \\lesssim \\varepsilon^2 M^2 t^{4\\delta}, \\quad 0<\\delta<\\frac{1}{6}, \\quad i \\leq 5.\n\\end{equation}\nMoreover, $\\hat N$ depends on the KG field in the exact manner as follows: letting \n \\begin{equation}\\label{lemma-N-inte-ineq-nabla3-source}\n\\|\\bar T \\phi\\|_{L^\\infty}^2 + \\|\\bar m (t) \\phi\\|_{L^\\infty}^2:= r^2(\\phi, t),\n\\end{equation}\nwe have\n\\begin{align}\n& \\mathcal{E}_2(\\hat N, t ) \\lesssim \\mathcal{E}^2_{1}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_0(\\phi, t), \\label{lemma-N-inte-ineq-2order-source} \\\\\n&E_3(\\hat N, t ) \\lesssim \\mathcal{E}^2_{2}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_1(\\phi, t), \\label{lemma-N-inte-ineq-nabla3-source}\\\\\n & E_4(\\hat N, t ) \\lesssim \\mathcal{E}^2_{3}(\\hat k,t) + \\mathcal{E}^2_2(\\phi, t) + r^2(\\phi, t) \\mathcal{E}_2(\\phi, t), \\label{lemma-N-inte-ineq-nabla4-source}\\\\\n&E_5(\\hat N, t ) \\lesssim \\mathcal{E}^2_{4}(\\hat k,t) + \\mathcal{E}_2(\\phi,t)\\mathcal{E}_3(\\phi,t) + r^2(\\phi, t)) \\mathcal{E}_3(\\phi,t).\\label{lemma-N-inte-ineq-nabla5-source}\n\\end{align}\n\\end{lemma}\n\\begin{remark}\\label{rk-rough-lapse}\nIf we do not take advantage of the special structure of the source term, the ellipticity of the equation for the lapse only gives \\eqref{lemma-N-inte-ineq-2order-source} and\n\\begin{equation*}\n\\begin{split}\n\\mathcal{E}_{l+2}(\\hat N, t ) \\lesssim &\\mathcal{E}^2_4(\\phi,t) + \\mathcal{E}^2_{l+1}(\\hat k,t) + \\mathcal{E}_{l+1}(\\hat N, t ), \\quad l \\leq 3.\n\\end{split}\n\\end{equation*}\nHence, we merely derive $\\mathcal{E}_{5}(\\hat N, t ) \\lesssim \\mathcal{E}^2_4(\\phi,t)$, ignoring lower order terms.\n\\end{remark}\n\\begin{remark}\\label{rk-CMCSH gauge-shift}\nIf one works within the CMCSH gauge, the elliptic equation for the shift $X$ interacts with the matter current $\\mathcal{T}_{\\bar T}^{i}(\\phi) = \\nabla^i \\phi \\bar T \\phi$:\n\\begin{equation}\\label{eq-shift}\n\\begin{split}\n\\Delta X^i + R^i_j X^j =& -2 \\nabla_j N \\hat k^{ij} - \\frac{\\text{tr} k}{3} \\nabla^i N + 2 N \\nabla^i \\phi \\bar T \\phi \\\\\n& - \\left( 2N \\hat k ^{pq}-\\nabla^p X^q \\right) \\left(\\Gamma_{pq}^i(g) - \\Gamma_{pq}^i(\\gamma) \\right).\n\\end{split}\n\\end{equation}\n Moreover, $\\nabla^i \\phi \\bar T \\phi \\sim t^{-1} \\nabla^i \\left(\\bar m (t) \\phi\\right) \\bar T \\phi.$ Therefore, the leading term on the right hand side of \\eqref{eq-shift} is $\\nabla^i N$, and we can expect the following estimate\n \\begin{equation}\\label{estimate-shift}\n\\begin{split}\n\\mathcal{E}_5\\left(X, t\\right) \\lesssim \\mathcal{E}_4(\\hat N, t) + t^{-2}\\mathcal{E}^2_4(\\phi, t) + l.o.t.,\n\\end{split}\n\\end{equation}\nwhere $\\mathcal{E}_5\\left(X, t\\right)$ is defined in the same way as \\eqref{def-energy-hat-N}.\n\\end{remark}\n\n\\begin{proof}\nWe multiply $\\hat N$ on \\eqref{eq-hat-lapse} to derive\n \\begin{equation}\\label{N-inte-1}\n\\begin{split}\n& \\int_{\\Sigma_t} |\\nabla \\hat N|^2 + 3 |\\hat N|^2 + |\\hat N|^2 |\\hat k|^2 \\\\\n=& - \\int_{\\Sigma_t} \\hat N \\left( |\\hat k|^2 + \\bar R_{\\bar T \\bar T} \\right) + |\\hat N|^2 \\bar R_{\\bar T \\bar T}.\n\\end{split}\n\\end{equation}\nMaking use of the $L^\\infty$ estimate for $\\phi$ \\eqref{BT-KG-L-infty-4-6}, the last term, $\\bar R_{\\bar T \\bar T} = (\\bar T \\phi)^2-\\frac{\\bar m^2(t)}{2} \\phi^2$, admits\n \\begin{equation*}\n \\int_{\\Sigma_t} |\\hat N|^2 |\\bar R_{\\bar T \\bar T}| \\lesssim \\int_{\\Sigma_t} |\\hat N|^2 \\left( |\\bar T \\phi|^2 + |\\bar m (t) \\phi|^2 \\right) \\lesssim \\int_{\\Sigma_t} \\varepsilon^2 M^2 t^{-1 + 2\\delta} |\\hat N|^2.\n\\end{equation*}\nAnd the Cauchy-Schwarz inequality shows that for some constant $c,$ \n \\begin{equation*}\n \\int_{\\Sigma_t} \\hat N \\left( |\\hat k|^2 + |\\bar R_{\\bar T \\bar T}| \\right) \n\\leq \\int_{\\Sigma_t} c |\\hat N|^2 + \\int_{\\Sigma_t} c^{-1} \\left( |\\hat k|^2 + |\\bar R_{\\bar T \\bar T}| \\right)^2.\n\\end{equation*}\nWe choose the constant $c<2$ so that $\\int_{\\Sigma_t} \\left( c + \\varepsilon^2 M^2 t^{-1 + 2\\delta} \\right) |\\hat N|^2$ is absorbed by the left hand side of \\eqref{N-inte-1}. \n \\begin{equation}\\label{estimate-nabla-N-1}\n \\begin{split}\n &\\int_{\\Sigma_t} |\\nabla \\hat N|^2 + |\\hat N|^2 \n\\lesssim \\int_{\\Sigma_t} \\left( |\\hat k|^2 + |\\bar R_{\\bar T \\bar T}| \\right)^2 \\\\\n& \\lesssim \\mathcal{E}^2_1(\\hat k, t) + \\int_{\\Sigma_t} \\left(|\\bar T \\phi|_{L^\\infty}^2 + |\\bar m (t) \\phi|_{L^\\infty}^2 \\right) \\left( |\\bar T \\phi|^2 + |\\bar m (t) \\phi|^2 \\right).\n\\end{split}\n\\end{equation}\nSubstituting the bootstrap assumption for $\\hat k$, and the assumed $L^2$ estimate for $\\bar T \\phi$ and $\\bar m (t) \\phi$, we have proved \\eqref{N-inte-ineq-123-lemma} up to 1st order.\n\n\nConcerning the higher order cases, we aim to prove by induction that\n\\begin{equation}\\label{estimate-nabla2-hat-N-i-induction}\n\\int_{\\Sigma_t} t^2 |\\nabla_{I_{i}} \\hat N|^2 \\lesssim \\varepsilon^2 M^2 t^{4\\delta}, \\quad 0\\leq i \\leq 5.\n\\end{equation}\nIt has been proved that \\eqref{estimate-nabla2-hat-N-i-induction} holds for $i=0, 1.$ Suppose \\eqref{estimate-nabla2-hat-N-i-induction} holds for $i\\leq l+1, \\, l \\leq 3$, we shall now prove that \\eqref{estimate-nabla2-hat-N-i-induction} holds for $i=l+2$ as well. \n \nApplying $\\nabla_{I_l} (0 \\leq l \\leq 3)$ to the equation for lapse \\eqref{eq-hat-lapse}, \n\\begin{equation}\\label{eq-lapse-nabla-l}\n\\begin{split}\n\\nabla_{I_l} \\Delta \\hat N =& 3 \\nabla_{I_l} N + \\sum_{a+b=l} \\nabla_{I_a} N * \\nabla_{I_b} \\left( \\bar R_{\\bar T \\bar T} + | \\hat k |^2 \\right).\n \\end{split}\n\\end{equation}\nNoting the commuting identity between $\\nabla_{I_l}$ and $\\Delta$ \\eqref{commuting-nabla-laplacian-l-simplify}-\\eqref{def-R-l-commute-nabla-laplacian},\nwe integrate by parts to derive that for $0 \\leq l \\leq 3,$\n\\begin{equation*}\n\\begin{split}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat N|^2 = & \\int_{\\Sigma_t} | \\nabla_{I_l} \\Delta \\hat{N}|^2 + \\mathcal{R}_{I_l} ( \\hat N ) \\nabla_{I_l} \\Delta \\hat N - \\mathcal{R}_{I_{l+1}} ( \\hat N) \\nabla_{I_{l+1}} \\hat N.\n\\end{split}\n\\end{equation*}\nSee \\eqref{def-R-l-commute-nabla-laplacian} for the definition of $\\mathcal{R}_{I_l} (\\hat N )$.\nThus, \n\\begin{equation*}\n\\begin{split}\n & \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat N|^2 \\lesssim \\int_{\\Sigma_t} |\\nabla_{I_l} \\Delta \\hat{N}|^2+| \\mathcal{R}_{I_l} (\\hat N ) |^2 +\\big| \\mathcal{R}_{I_{l+1}} (\\hat N ) * \\nabla_{I_{l+1}} \\hat N\\big|.\n\\end{split}\n\\end{equation*}\nBy \\eqref{eq-lapse-nabla-l} and Cauchy-Schwarz inequality,\n\\begin{equation*}\n\\begin{split}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat N|^2 \\lesssim& \\int_{\\Sigma_t} |\\nabla_{I_l} \\hat{N}|^2 + |\\mathcal{R}_{I_l} (\\hat N ) |^2 + | \\mathcal{R}_{I_{l+1}} (\\hat N ) * \\nabla^{I_{l+1}} \\hat N | \\\\\n &+ \\int_{\\Sigma_t} \\sum_{a+b =l} |\\nabla_{I_a} N * \\nabla_{I_b} \\bar R_{\\bar T \\bar T} |^2+ |\\nabla_{I_a} N * \\nabla_{I_b} (\\hat k)^2|^2.\n\\end{split}\n\\end{equation*}\nWe derive\n\\begin{equation}\\label{ineq-E-l+2-source}\n\\begin{split}\n\\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat N|^2 \\lesssim& \\int_{\\Sigma_t} |\\nabla_{I_{l}} \\hat{N}|^2 \\\\\n & + SN_1 +SN_2+SN_3 + RN_1 + RN_2,\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n RN_1 = & \\sum_{a+b =l \\atop a \\leq l-1} | \\nabla_{I_a} R_{imjn} * \\nabla_{I_b} \\hat N |^2, \\\\\nRN_2= & \\sum_{a+b =l+1 \\atop a \\leq l} |\\nabla_{I_a} R_{imjn} * \\nabla_{I_b} \\hat N * \\nabla_{I_{l+1}} \\hat N |, \\\\\n SN_1 = &\\sum_{a+b+c =l} \\big| \\nabla_{I_a} N * \\nabla_{I_{b}} \\bar T \\phi * \\nabla_{I_c} \\bar T\\phi \\big|^2, \\\\\nSN_2 = & \\sum_{a+b+c =l} \\big| \\nabla_{I_a} N * \\nabla_{I_{b}} (\\bar m (t) \\phi) * \\nabla_{I_c} (\\bar m (t) \\phi) \\big|^2, \\\\\nSN_3 = & \\sum_{a+b+c =l} \\big| \\nabla_{I_a} N * \\nabla_{I_{b}} \\hat k* \\nabla_{I_c}\\hat k \\big|^2.\n\\end{split}\n\\end{equation}\nViewing \\eqref{Gauss-Riem-hat-k}-\\eqref{Gauss-Ricci-hat-k} and \\eqref{def-Weyl}, the expansion for $R_{imjn}$ is rewritten as\n$$R_{imjn} = g*g \\pm \\hat k *g \\pm \\hat k* \\hat k +W_{imjn} \\pm \\bar R_{pq} *g \\pm \\bar R *g*g.$$\nIn the estimates, we only need the formula $\\|A*B\\| \\lesssim \\|A\\| \\|B\\|$, while the detailed product structure and constants are irrelevant. As a $(0, 4)$-tensor on $\\Sigma_t$, $R_{imjn}$ can be simplified as\n\\begin{equation}\\label{R-imjn-E}\nR_{imjn} \\sim 1 \\pm \\hat k \\pm |\\hat k|^2 + E \\pm |\\mathcal{D} \\phi|^2.\n\\end{equation}\n\nBy the assumption, \\eqref{estimate-nabla2-hat-N-i-induction} holds for $i\\leq l +1, l \\leq 3$,\nnamely,\n\\begin{equation}\\label{ineq-SN-0}\nt^2 \\mathcal{E}_{l+1} (\\hat N, t) \\lesssim \\varepsilon^2 M^2 t^{4\\delta}.\n \\end{equation}\n\nSince $SN_1$ and $SN_2$ can be treated in the same way, we only take $SN_2$ for example., \n\\begin{itemize}\n\\item {\\bf Case $SN_{21}$}: $a=0$. We always apply $L^4$ to all the four factors in $SN_2,$ for $l \\leq 3.$ That is, in this case, $SN_2$ becomes ($b+c = l$)\n\\begin{equation*}\n\\begin{split}\n & \\int_{\\Sigma_t} \\big|N \\nabla_{I_{b}} (\\bar m (t) \\phi) * \\nabla_{I_c} (\\bar m (t) \\phi) \\big|^2\n \\lesssim \\mathcal{E}^2_{l+1}(\\phi,t). \n\\end{split}\n\\end{equation*}\nThis estimate will be treated in a more subtle way later.\n\\item {\\bf Case $SN_{22}$}: $a \\geq 1$. Then $0 \\leq b,c \\leq l-1 \\leq 2$. We can apply $L^6$ to all the six factors in $SN_2$ ($a+b+c =l, a \\geq 1$)\n\\begin{equation*}\n\\begin{split}\n & \\int_{\\Sigma_t} \\big|( t\\nabla)_{I_a } \\hat N * \\nabla_{I_{b}} (\\bar m (t) \\phi) * \\nabla_{I_c} (\\bar m (t) \\phi) \\big|^2 \n \\lesssim \\mathcal{E}_{l+1}(\\hat N,t) \\mathcal{E}^2_{l}(\\phi,t). \n\\end{split}\n\\end{equation*}\n\\end{itemize}\nSimilar procedures apply to $SN_3$. Roughly speaking, $\\hat k$ enjoys better estimates than $\\phi$, and we derive \n\\begin{equation}\\label{esti-SN3}\n\\begin{split}\n\\int_{\\Sigma_t} SN_3 \\lesssim & \\mathcal{E}^2_{l+1}(\\hat k,t) + \\mathcal{E}_{l+1}(\\hat N,t) \\mathcal{E}^2_{l}(\\hat k,t).\n\\end{split}\n\\end{equation}\n\nWe summarize these estimates as\n\\begin{equation}\\label{esti-SN123}\n\\begin{split}\n\\int_{\\Sigma_t} \\sum_{i=1}^3 |SN_i| \\lesssim & \\mathcal{E}^2_{l+1}(\\phi,t) + \\mathcal{E}^2_{l+1}(\\hat k,t) \\\\\n& + \\mathcal{E}_{l+1}(\\hat N,t) \\left(\\mathcal{E}^2_{l}(\\hat k,t) + \\mathcal{E}^2_{l}(\\phi,t) \\right) \\lesssim \\varepsilon^2 M^2 t^{-2+ 4\\delta}.\n\\end{split}\n\\end{equation}\n\n\nFor the terms involving curvature $RN_1$, \n\\begin{itemize}\n\\item {\\bf Case $RN_{11}$}: $a=0, b=l$. In views of the expansion for $R_{imjn}$ \\eqref{R-imjn-E}, and the $L^\\infty$ estimates for $\\hat k, \\phi$, $RN_{11}$ is estimated as\n \\begin{equation*}\n \\begin{split}\n&\\int_{\\Sigma_t} \\left( 1 + |\\hat k|^2 + |\\hat k|^4 + |\\mathcal{D} \\phi|^4 + |E|^2 \\right) |\\nabla_{I_l} \\hat N|^2\\\\\n\\lesssim & \\mathcal{E}_{l}(\\hat N,t) \\left( \\mathcal{E}^2_{2}(\\hat k,t) + \\mathcal{E}_{2}(\\hat k,t) + \\mathcal{E}_{2}(W,t) + \\mathcal{E}^2_{2}(\\phi,t) +1 \\right). \n\\end{split}\n\\end{equation*}\n\\item {\\bf Case $RN_{12}$}: $1 \\leq a \\leq l-1.$ Then $1 \\leq b \\leq l-1,$ and $RN_{12}$ becomes ($a+b =l, 1 \\leq a ,b \\leq l-1, l \\leq 3$)\n \\begin{equation*}\n \\begin{split}\n & \\big| \\nabla_{I_a} \\left( |\\hat k| + |\\hat k|^2 + |\\mathcal{D} \\phi|^2 + E \\right)\\big|^2 \\big|\\nabla_{I_b} \\hat N |^2.\n\\end{split}\n\\end{equation*}\nWe can always apply $L^4$ to all the four factors or $L^6$ to all the six factors of each term in $RN_{12}$, so that\n \\begin{equation*}\n \\begin{split}\n \\int_{\\Sigma_t} |RN_{12}| \\lesssim & \\mathcal{E}_l(\\hat N,t) \\left( \\mathcal{E}^2_{l}(\\phi,t) +\\mathcal{E}^2_{l}(\\hat k,t) + \\mathcal{E}_{l}(W,t) + \\mathcal{E}_{l}(\\hat k,t) \\right).\n\\end{split}\n\\end{equation*}\n\\end{itemize}\n\n\nFor the term involving curvature $RN_2$, \n\\begin{itemize}\n\\item {\\bf Case $RN_{21}$}: $a=0, b=l+1$. Analogous to $RN_{11}$, $RN_{21}$ shares the estimate\n \\begin{equation*}\n \\begin{split}\n&\\int_{\\Sigma_t} |RN_{21}| \\lesssim \\mathcal{E}_{l+1}(\\hat N,t). \n\\end{split}\n\\end{equation*}\n\\item {\\bf Case $RN_{22}$}: $1 \\leq a \\leq l.$ Then $1 \\leq b \\leq l,$ and $RN_{22}$ is ($a+b =l+1, 1 \\leq a,b \\leq l, l \\leq 3$) \n \\begin{equation*}\n \\begin{split}\n& \\nabla_{I_a} \\left(|\\hat k| + |\\hat k|^2 + |\\mathcal{D} \\phi|^2 + |E| \\right) * \\nabla_{I_b} \\hat N *\\nabla_{I_{l+1}} \\hat N.\n\\end{split}\n\\end{equation*}\n\\begin{enumerate}\n\\item If $a \\leq 2$, then similar to $RN_{12}$, we apply $L^4, L^4, L^2$ to the three factors or $L^6, L^6, L^6, L^2$ to the four factors in each term of $RN_{22}$. \nso that\n \\begin{equation*}\n \\begin{split}\n \\int_{\\Sigma_t} |RN_{22}| \\lesssim & \\mathcal{E}_{l+1}(\\hat N,t) \\left( \\mathcal{E}_{a+1}(\\phi,t) +\\mathcal{E}_{a+1}(\\hat k,t) \\right) \\\\\n & + \\mathcal{E}_{l+1}(\\hat N,t) \\left( \\mathcal{E}^{\\frac{1}{2}}_{a+1}(W,t) + \\mathcal{E}^{\\frac{1}{2}}_{a+1}(\\hat k,t) \\right).\n\\end{split}\n\\end{equation*}\n\\item If $a=3$, then $l=3$ and $b=1$. We apply $L^2, L^\\infty, L^2$ to the three factors or $L^4, L^4, L^\\infty, L^2$ to the four factors in each term of $RN_{22}$ to derive (noting that here $l=3$)\n \\begin{equation*}\n \\begin{split}\n \\int_{\\Sigma_t} |RN_{22}| \\lesssim & \\mathcal{E}^{\\frac{1}{2}}_{l+1}(\\hat N,t) \\mathcal{E}^{\\frac{1}{2}}_{l}(\\hat N,t) \\left( \\mathcal{E}_{4}(\\phi,t) +\\mathcal{E}_{4}(\\hat k,t) \\right) \\\\\n & + \\mathcal{E}^{\\frac{1}{2}}_{l+1}(\\hat N,t) \\mathcal{E}^{\\frac{1}{2}}_{l}(\\hat N,t) \\left( \\mathcal{E}^{\\frac{1}{2}}_{3}(W,t) + \\mathcal{E}^{\\frac{1}{2}}_{3}(\\hat k,t) \\right).\n\\end{split}\n\\end{equation*}\n\\end{enumerate}\n\\end{itemize}\n In summary, $RN_1, RN_2$ enjoy the following estimate\n\\begin{equation}\\label{esti-RN1-RN2}\n\\begin{split}\n\\int_{\\Sigma_t} |RN_1| + |RN_2| \\lesssim & \\mathcal{E}_{l+1}(\\hat N,t) \\left( 1+ \\varepsilon M t^{-1+ 2\\delta} \\right) \\lesssim \\varepsilon^2 M^2 t^{-2+4\\delta}.\n\\end{split}\n\\end{equation}\n\n \n\n\nPutting all those estimates \\eqref{esti-SN123}-\\eqref{esti-RN1-RN2} together, we can close the inductive argument and complete the proof for \\eqref{N-inte-ineq-123-lemma}. In particular, we have proved that for $0 \\leq l \\leq 3,$\n\\begin{equation}\\label{estimate-lapse-nabla-l-3-4-5-1}\n\\begin{split}\nE_{l+2}(\\hat N,t) \\lesssim & \\int_{\\Sigma_t} \\sum_{b+c =l} \\big|N \\nabla_{I_{b}}\\bar T \\phi * \\nabla_{I_c} \\bar T \\phi \\big|^2\\\\\n &+\\int_{\\Sigma_t} \\sum_{b+c =l} \\big|N \\nabla_{I_{b}} (\\bar m (t) \\phi) * \\nabla_{I_c} (\\bar m (t) \\phi) \\big|^2\\\\\n & + \\mathcal{E}_{l+1}(\\hat N,t) \\left(1+ \\varepsilon M t^{-1+ 2\\delta}\\right) + \\mathcal{E}^2_{l+1}(\\hat k,t).\n \\end{split}\n\\end{equation}\n\n\nIn the above proof, we find out that when $l=0$, both $RN_1$ and $RN_{22}$ are absent. More precisely, we have, in views of \\eqref{estimate-nabla-N-1},\n \\begin{equation}\\label{ineq-hat-N-2order-source}\n\\begin{split}\n& E_2(\\hat N,t) \\lesssim \\mathcal{E}_{1}(\\hat N,t) + \\mathcal{E}^2_{1}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_0(\\phi, t) \\\\\n\\lesssim & \\mathcal{E}^2_{1}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_0(\\phi, t).\n\\end{split}\n\\end{equation}\nWhen $l=1,$ \\eqref{estimate-lapse-nabla-l-3-4-5-1} and the estimates for $\\mathcal{E}_2(\\hat N,t)$, \\eqref{estimate-nabla-N-1} and \\eqref{ineq-hat-N-2order-source}, show that\n \\begin{equation}\\label{N-inte-ineq-3order-source}\n\\begin{split}\n& E_3(\\hat N,t) \\lesssim \\mathcal{E}^2_{2}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_0(\\phi, t) \\\\\n& \\quad + \\int_{\\Sigma_t} r^2(\\phi, t) \\left( |\\nabla \\bar T \\phi|^2 + |\\nabla (\\bar m (t) \\phi)|^2 \\right),\n\\end{split}\n\\end{equation}\nwhich derives \\eqref{lemma-N-inte-ineq-nabla3-source}.\nWhen $l=2,$ we combine \\eqref{estimate-lapse-nabla-l-3-4-5-1} and the estimates for $\\hat N$ up to three derivatives \\eqref{estimate-nabla-N-1}, \\eqref{ineq-hat-N-2order-source}-\\eqref{N-inte-ineq-3order-source} to deduce that\n \\begin{equation}\\label{N-inte-ineq-4order-source}\n\\begin{split}\nE_4(\\hat N,t) \\lesssim & \\mathcal{E}^2_{3}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_1(\\phi, t)+ \\int_{\\Sigma_t} \\big| \\nabla \\bar T \\phi \\big|^4 + \\big| \\nabla ( \\bar m (t) \\phi) \\big|^4 \\\\\n& + \\int_{\\Sigma_t} r^2(\\phi, t) \\left( |\\nabla^2 \\bar T \\phi|^2 + |\\nabla^2 (\\bar m (t) \\phi)|^2 \\right).\n\\end{split}\n\\end{equation}\nBy the Sobolev inequality, we obtain \\eqref{lemma-N-inte-ineq-nabla4-source}.\nWhen $l=3,$ by \\eqref{estimate-lapse-nabla-l-3-4-5-1} and the estimates for $\\hat N$ up to four derivatives \\eqref{estimate-nabla-N-1}, \\eqref{ineq-hat-N-2order-source}-\\eqref{N-inte-ineq-4order-source}, there is\n \\begin{equation}\\label{N-inte-ineq-5order-source}\n\\begin{split}\n&E_5(\\hat N,t) \\lesssim \\mathcal{E}^2_{4}(\\hat k,t) + r^2(\\phi, t) \\mathcal{E}_2(\\phi, t)+\\mathcal{E}^2_2(\\phi, t) \\\\\n& + \\int_{\\Sigma_t} \\left( | \\nabla \\bar T \\phi * \\nabla^2 \\bar T \\phi | + | \\nabla ( \\bar m (t) \\phi) * \\nabla^2 ( \\bar m (t) \\phi) | \\right)^2 \\\\\n& \\quad + \\int_{\\Sigma_t} r^2(\\phi, t) \\left( |\\nabla^3 \\bar T \\phi|^2 + |\\nabla^3 (\\bar m (t) \\phi)|^2 \\right).\n\\end{split}\n\\end{equation}\nBy the Sobolev inequality, we prove \\eqref{lemma-N-inte-ineq-nabla5-source}. Note that, we only need the refined estimate up to four order derivatives for $\\hat N$ ($\\mathcal{E}_4(\\hat N,t)$) in the hierarchy of energy estimate for the KG field.\n\\end{proof}\n\n\\begin{remark}\\label{rk-TT-phi}\nIn views of the definition of partial energies \\eqref{def-energy-density-l-homo-kg-g}-\\eqref{def-energy-l-homo-kg-g}, there is no energy norm involving two normal derivatives, such as $\\|\\nabla^i \\bar T ^2\\phi\\|_{L^2(\\Sigma_t)}$. And hence we lose the control for $\\|\\bar T ^2\\phi\\|_{L^\\infty}$ as well. However, by the KG equation in $1+3$ form \\eqref{eq-rescale-kg-1+3-0}, there is\n\\begin{equation}\\label{TT-KG-L-infty}\n\\begin{split}\n\\|\\bar T ^2\\phi\\|_{L^\\infty} &\\lesssim \\|\\bar T\\phi\\|_{L^\\infty} + \\|\\bar m^2 (t) \\phi\\|_{L^\\infty} + \\|\\nabla^2\\phi\\|_{L^\\infty} \\\\\n&\\quad \\quad + \\|N^{-1} \\nabla \\hat N\\|_{L^\\infty} \\| \\nabla \\phi\\|_{L^\\infty}.\n \\end{split}\n\\end{equation}\nThus, by the bootstrap assumption \\eqref{BT-KG} and the $L^\\infty$ estimate \\eqref{BT-KG-L-infty-4-6} followed, we achieve \n\\begin{equation}\\label{TT-KG-L-2-infty-0}\n\\begin{split}\n\\|\\bar T^2\\phi\\|_{L^2(\\Sigma_t)} \\lesssim &\\bar m (t) \\mathcal{E}^{\\frac{1}{2}}_{1} (\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_{1} (\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_{2} (\\hat N, t) \\lesssim \\varepsilon M t^{\\frac{1}{2} + \\delta},\\\\\n\\|\\bar T^2\\phi\\|_{L^\\infty(\\Sigma_t)} \\lesssim & \\bar m (t) \\mathcal{E}^{\\frac{1}{2}}_{3} (\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_{2} (\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_{3} (\\hat N, t) \\lesssim \\varepsilon M t^{\\frac{1}{2} + \\delta}\n\\end{split}\n\\end{equation}\nThis will be used to derive the preliminary estimates for $N^\\prime$, see Lemma \\ref{lem-TN-ineq-source} below.\n\\end{remark}\n\nFrom now on, we let\n\\begin{equation}\\label{def-energy-mertic-rescale}\n\\mathcal{E}_{i} (g, t) = \\mathcal{E}_{i} (W, t) + \\mathcal{E}_{1+i} (\\hat k, t) + \\mathcal{E}_{2+i} (\\hat N, t).\n\\end{equation}\n\nThe hierarchy of energy estimates for the KG filed, namely the idea of linearization, also relies deeply on an $L^\\infty - L^\\infty$ estimate for the KG field, which indeed retrieves the $\\delta$ loss for $\\|\\phi\\|_{L^\\infty}, \\|T\\phi\\|_{L^\\infty}$. This will be explained in Section \\ref{sec-L-infty-L-infty-kg}. To continue these procedures, we first need $\\|N^\\prime\\|_{L^\\infty}$.\nRecall the definition\n \\begin{equation}\\label{def-N--prime}\nN^\\prime = \\partial_{\\bar \\tau} N,\n\\end{equation}\nand the elliptic equation for $N^\\prime$\n\\begin{equation*}\n\\begin{split}\n& \\Delta N^\\prime - 3 N^\\prime\n = N^\\prime \\left(\\bar R_{\\bar T \\bar T}(\\phi) + |\\hat k|^2 \\right) + N \\partial_{\\bar \\tau} \\left(\\bar R_{\\bar T \\bar T}(\\phi) + |\\hat k|^2 \\right) \\\\\n & - |\\nabla \\hat N |^2 -2 \\hat k^{ij} \\left( N\\nabla_i\\nabla_j \\hat N+ \\nabla_i \\hat N \\nabla_j \\hat N \\right) +2 \\hat N \\Delta \\hat N + 2N \\nabla^i \\hat N \\bar R_{\\bar T i} (\\phi).\n \\end{split}\n\\end{equation*}\n\n\\begin{lemma}\\label{lem-TN-ineq-source}\nFixing $0<\\delta<\\frac{1}{6},$ and assuming \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty}, we have for $N^\\prime$,\n \\begin{equation}\\label{N-prime-inte-ineq-123-lemma}\n \\int_{\\Sigma_t} |\\nabla N^\\prime|^2 + |N^\\prime|^2 + |\\nabla^2 N^\\prime|^2\n\\lesssim \\varepsilon^2 M^2 t^{4\\delta}.\n\\end{equation}\nAnd hence by Sobolev inequality,\n\\begin{equation}\\label{nabla-N-L-infty}\n\\|N^\\prime \\|_{L^\\infty} \\lesssim \\varepsilon M t^{ 2\\delta}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe multiply $N^\\prime$ on both hand sides of the elliptic equation for $N^\\prime$, and integrate by parts,\n \\begin{equation}\\label{N-prime-inte-1}\n\\begin{split}\n& \\int_{\\Sigma_t} |\\nabla N^\\prime|^2 + 3 |N^\\prime|^2= \\int_{\\Sigma_t} \\left( |\\nabla N |^2 - 2 \\hat N \\Delta N \\right) N^\\prime \\\\\n&\\quad \\quad - \\int_{\\Sigma_t} |N^\\prime|^2 \\left( |\\hat k|^2 + \\bar R_{\\bar T \\bar T} \\right) + N^\\prime N \\partial_{\\bar \\tau} \\left( |\\hat k|^2 + \\bar R_{\\bar T \\bar T} \\right)\\\\\n&\\quad \\quad + \\int_{\\Sigma_t} 2 N^\\prime \\hat k^{ij} \\left( N \\nabla_i\\nabla_j \\hat N+ \\nabla_i \\hat N \\nabla_j \\hat N \\right) -2 N^\\prime N \\nabla^i \\hat N \\bar R_{\\bar T i}.\n\\end{split}\n\\end{equation}\nThen by Cauchy-Schwarz inequality, and the $L^\\infty$ estimate for $\\hat k$, $\\mathcal{D} \\phi$,\n \\begin{equation*}\n \\begin{split}\n& \\int_{\\Sigma_t} |\\nabla N^\\prime|^2 + |N^\\prime|^2 \\lesssim\n\\int_{\\Sigma_t} \\left( \\partial_{\\bar \\tau} |\\hat k|^2 + \\partial_{\\bar \\tau} \\bar R_{\\bar T \\bar T} \\right)^2 + |\\nabla \\hat N |^2 |\\bar R_{\\bar T i}|^2 \\\\\n& \\quad \\quad \\quad \\quad + |\\nabla \\hat N|^4(1+|\\hat k|^2) + |\\nabla^2 \\hat{N}|^2 (|\\hat N|^2 + |\\hat k|^2).\n \\end{split}\n\\end{equation*}\nUsing the evolution equation for $\\hat k$ (c.f. \\eqref{eq-evolution-2}), and substituting the estimates for $\\hat N$ (c.f. Lemma \\ref{lem-N-ineq-source}) into the above formula, we arrive at \n \\begin{equation}\\label{N-prime-inte-ineq-1}\n \\begin{split}\n \\int_{\\Sigma_t} |\\nabla N^\\prime|^2 + |N^\\prime|^2 & \\lesssim \\mathcal{E}^2_1 (g,t) \\left(1+ \\mathcal{E}_1 (g,t) \\right) + \\mathcal{E}_1 (g,t)\\mathcal{E}^2_1 (\\phi,t) \\\\\n& + \\int_{\\Sigma_t} N |\\bar T \\phi|^2 |\\bar T^2\\phi|^2 +N \\bar m^2 (t) |\\bar m (t) \\phi|^2 |\\bar T \\phi|^2.\n \\end{split}\n\\end{equation}\nReferring to Remark \\ref{rk-TT-phi} for $\\|\\bar T^2 \\phi \\|_{L^2(\\Sigma_t)}$, there is\n \\begin{equation}\\label{N-prime-inte-ineq-2}\n \\int_{\\Sigma_t} |\\nabla N^\\prime|^2 + |N^\\prime|^2 \\lesssim t^2 \\mathcal{E}^2_2 (\\phi,t) + \\text{l.o.t.} \\lesssim \\varepsilon^2 M^2 t^{4\\delta}.\n\\end{equation}\n$\\text{l.o.t.}$ means lower order terms.\n\nTo proceed to the second order derivative, we make use of the B\\\"{o}chner identity, then\n \\begin{equation*}\n\\begin{split}\n& \\int_{\\Sigma_t} |\\nabla^2 N^\\prime|^2 = \\int_{\\Sigma_t} | \\Delta N^\\prime|^2 - R_{ij} \\nabla^i N^\\prime \\nabla^j N^\\prime.\n\\end{split}\n\\end{equation*}\nNoting the expansion for $R_{imjn}$ \\eqref{R-imjn-E}, $R_{ij}$ can be expanded in the same way.\nTherefore, with the help of Cauchy-Schwarz inequality, we obtain,\n \\begin{equation*}\n\\begin{split}\n \\int_{\\Sigma_t} |\\nabla^2 N^\\prime|^2 & \\lesssim \\int_{\\Sigma_t} \\left(\\partial_{\\bar \\tau} \\bar R_{\\bar T \\bar T} + \\partial_{\\bar \\tau} |\\hat k|^2 \\right)^2+ |N^\\prime|^2 \\left( |\\bar T \\phi|^4 + |\\bar m (t) \\phi|^4 + |\\hat k|^4 \\right) \\\\\n &+ |\\nabla \\hat N|^2 \\left( |\\nabla \\hat N|^2 \\left( 1+ |\\hat k|^2 \\right) +|\\bar T \\phi \\nabla \\phi|^2 \\right) + |\\nabla^2 N|^2 \\left( |\\hat N|^2 + |\\hat k|^2\\right) \\\\\n &+ |\\nabla N^\\prime|^2 \\left(1+ |\\hat k| + |\\hat k|^2 + |\\mathcal{D} \\phi|^2 + |E| \\right).\n\\end{split}\n\\end{equation*}\nSubstituting the estimates for $N$ (see Lemma \\ref{lem-N-ineq-source}) and the known results \\eqref{N-prime-inte-ineq-2}, we get\n \\begin{equation*}\n\\begin{split}\n \\int_{\\Sigma_t} |\\nabla^2 N^\\prime|^2 \\lesssim & t^2 \\mathcal{E}^2_2 (\\phi,t) + \\text{l.o.t.} \\lesssim \\varepsilon^2 M^2 t^{ 4\\delta}.\n\\end{split}\n\\end{equation*}\nFinally, the $L^\\infty$ estimate \\eqref{nabla-N-L-infty} is due to the Sobolev inequality.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Energy estimate for KG field}\\label{sec-energy-kg}\n\n\\subsection{The $L^\\infty - L^\\infty$ estimate for KG field}\\label{sec-L-infty-L-infty-kg}\nWe begin with the $L^\\infty$ estimates below which follows from the weak bootstrap assumption for the KG field:\n\\begin{align*}\n& \\sum_{i=0}^{2} \\|\\nabla^i \\mathcal{D} \\phi\\|_{L^\\infty} \\lesssim \\varepsilon M t^{- \\frac{1}{2} + \\delta}.\n\\end{align*}\nObviously, there is $\\delta$ loss in the above estimate.\nIn this section, we will apply the technical ODE estimate (c.f. Lemma 3.5 in \\cite{Ma-Lefoch-16}, of which a specific case for application is exhibited in Lemma \\ref{lemma-ode} of appendix) to retrieve the $\\delta$ loss.\n\\begin{corollary}[Improved $L^\\infty$ estimates of KG field]\\label{coro-Improved L-infty of Klein Gordon field}\nWith the bootstrap assumption \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty}, we have the improved estimate for $\\phi$,\n\\begin{equation}\\label{improved-L-infty-kg-1}\n|t^{\\frac{1}{2}} \\bar m (t) \\phi| + |t^{\\frac{1}{2}} \\bar T \\phi| \\lesssim \\varepsilon I_4 (\\phi) + \\varepsilon M.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nSet\n\\begin{equation}\\label{def-Phi}\n\\Phi = t^{\\frac{3}{2}} \\phi.\n\\end{equation}\nThen a direct calculation with respect to the KG equation in $1+3$ form \\eqref{eq-rescale-kg-1+3-0} leads to the second order ODE\n\\begin{equation}\\label{ode-kg}\n\\begin{split}\nT^2 \\Phi + m \\Phi =& t^{-\\frac{1}{2}} \\left( \\Delta \\phi +N^{-1} \\nabla_p N \\nabla^p \\phi \\right) - \\frac{3}{2} t^{-\\frac{1}{2}} N^{-3} N^\\prime t^{-1} \\bar m (t) \\phi \\\\\n&-3t^{-\\frac{1}{2}} N^{-1} \\hat N \\bar T \\phi +\\frac{3}{4} t^{-\\frac{1}{2}} N^{-2} t^{-1} \\bar m (t) \\phi.\n\\end{split}\n\\end{equation}\nWe use the $L^\\infty$ estimates for $\\Delta \\phi$,\n\\begin{align*}\n&\\|\\Delta \\phi\\|_{L^\\infty} \\lesssim t^{-1} \\|\\Delta (\\bar m (t) \\phi) \\|_{L^\\infty} \\lesssim t^{-1} \\mathcal{E}^{\\frac{1}{2}}_4(\\phi, t).\n\\end{align*}\nThen, the right hand side of \\eqref{ode-kg} can be bounded by \n\\begin{align*}\n|T^2 \\Phi + m \\Phi | \\lesssim& t^{-\\frac{3}{2}} \\left( \\mathcal{E}^{\\frac{1}{2}}_4(\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_2(N^\\prime, t) \\mathcal{E}^{\\frac{1}{2}}_2(\\phi, t) \\right) \\\\\n&+ t^{-\\frac{1}{2}} \\mathcal{E}^{\\frac{1}{2}}_3(\\hat N, t) \\mathcal{E}^{\\frac{1}{2}}_2(\\phi, t) \\lesssim \\varepsilon M t^{-2 + 3\\delta}.\n\\end{align*}\nWe integrate along the integral curves of $T$, which are the same as that of $\\partial_t$ but parametrized by arc\nlength $s$, i.e. $T=\\frac{\\partial}{\\partial s}$. Along this curve $\\mathrm{d}} %{\\text{d} s = N\\mathrm{d}} %{\\text{d} t$. It follows from the technical ODE estimate \\cite{Ma-Lefoch-16} (or Lemma \\ref{lemma-ode}) that\n\\begin{equation}\\label{ineq-ode-kg}\n|\\Phi(t)| + |T\\Phi(t)| \\lesssim \\varepsilon I_4(\\phi) + \\int_{t_0}^{t} \\varepsilon M t^{-2 + 3\\delta} N \\mathrm{d}} %{\\text{d} t.\n\\end{equation}\nThis further leads to the improved $L^\\infty$ estimates of $\\phi$ \\eqref{improved-L-infty-kg-1}.\n\\end{proof}\n\n\n\n\n\n\\subsection{The lower order terms}\\label{sec-e-lot-kg} \nIn this subsection, we will prove the following estimates for the lower order terms.\n\\begin{corollary}\\label{coro-LK-1-LK_3}\nWith the bootstrap assumption \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty}, the lower order terms ${}_0LK$ \\eqref{def-0-LK} and ${}_lLK_1-{}_lLK_3$ ($l \\leq 4$) \\eqref{energy-estimate-kg-1-id-l-o-t-1}-\\eqref{energy-estimate-kg-1-id-l-o-t-3} in the energy identities for KG field \n share the estimates\n\\begin{equation}\\label{estimate-0LK}\n \\int_{\\Sigma_t} |{}_0LK| \\lesssim t^{-1} E_0 (\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_1 (g, t) E_0 (\\phi, t),\n\\end{equation}\n\\begin{equation}\\label{estimate-L-1-2-3}\n\\begin{split}\n\\int_{\\Sigma_t} \\sum_{i=1}^3 |{}_lLK_i| \\lesssim & t^{-1}E_l (\\phi, t) + \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation}\n\\end{corollary}\n\n\\subsubsection{The estimate for ${}_0LK$}\\label{sec-e-lot-kg-LK0} \nThe formula for ${}_0LK$ is given in \\eqref{def-0-LK}. We should note that $\\phi =m^{-1} t^{-1} \\bar m (t) \\phi$, hence ${}_0LK$ can be rewritten as \n\\begin{equation*}\n\\begin{split}\n {}_0LK = & \\left( t^{-1} N + t^{-1} \\hat N + N \\hat k + \\nabla \\hat N \\right) |\\mathcal{D} \\phi|^2.\n\\end{split}\n\\end{equation*}\nTherefore, by $\\|\\hat N\\|_{L^\\infty}$ \\eqref{BT-k-N-L-infty} and $\\|\\hat k\\|_{L^\\infty}$ \\eqref{BT-k-L-infty-4-6}, we derive \\eqref{estimate-0LK}.\n\n\\subsubsection{The estimate for ${}_lLK_1$}\\label{sec-e-lot-kg-L1} \nSince $$\\nabla^{I_l} \\phi = m^{-1}t^{-1} \\nabla^{I_l} (\\bar m (t) \\phi),$$ the first line in ${}_lLK_1$ \\eqref{energy-estimate-kg-1-id-l-o-t-1} can be rewritten as \n\\begin{equation*}\n\\begin{split}\n {}_lLK^1_1 =& \\left( \\hat N + N\\hat k + t^{-1} N + t^{-1} \\hat N + t^{-1} N \\hat k \\right) | \\nabla_{I_l} \\mathcal{D} \\phi|^2,\n\\end{split}\n\\end{equation*}\nwhile second line in ${}_lLK_1$ \\eqref{energy-estimate-kg-1-id-l-o-t-1} is\n\\begin{equation*}\n\\begin{split}\n {}_lLK^2_1 = & - N \\nabla^{I_l} \\bar T \\phi \\mathcal{R}_{I_l} (\\phi) - t^{-1} N \\nabla^{I_l} (\\bar m (t) \\phi) \\mathcal{R}_{I_l}(\\phi).\n\\end{split}\n\\end{equation*}\n\nMaking use of $\\|\\hat N\\|_{L^\\infty}$ \\eqref{BT-k-N-L-infty} and $\\|\\hat k\\|_{L^\\infty}$ \\eqref{BT-k-L-infty-4-6}, we have \n\\begin{equation}\\label{estimate-LK-11}\n \\int_{\\Sigma_t} |{}_lLK^1_1| \\lesssim t^{-1} E_l (\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_1 (g, t) E_l (\\phi, t). \n\\end{equation}\n\nFor ${}_lLK^2_1$, comparing to the first term, the second term in ${}_lLK^2_1$ taking the form of $t^{-1} N \\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{R}_{I_l} (\\phi)$ is of lower order. We now consider the general form $N \\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{R}_{I_l} (\\phi)$, which can be further rewritten as\n\\begin{equation}\\label{def-L-23}\n\\begin{split}\nt^{-1} \\nabla^{I_l} \\mathcal{D} \\phi \\sum_{a +b= l, \\,\\, a \\leq l-1} N \\nabla_{I_a} R_{imjn} * \\nabla_{I_b} (\\bar m (t) \\phi).\n\\end{split}\n\\end{equation}\n\n\\begin{itemize}\n\\item {\\bf Case ${}_lLK^2_{1}$-I}: $a=0$. Then $b=l.$ In views of the expansion for $R_{imjn}$ \\eqref{R-imjn-E}, we have\n\\begin{equation}\\label{estimate-L231-1}\n\\begin{split}\n&\\int_{\\Sigma_t} t^{-1}\\big| \\nabla^{I_l} \\mathcal{D} \\phi * N R_{imjn} * \\nabla_{I_l} \\mathcal{D} \\phi \\big| \\\\\n \\lesssim & t^{-1} E_l(\\phi, t) +t^{-1} E_l(\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_2 (g, t) + \\mathcal{E}_2 (\\phi, t). \\right)\n\\end{split}\n\\end{equation}\n\\item {\\bf Case ${}_lLK^2_{1}$-II:} $a \\geq 1$. Then $1 \\leq a, b \\leq l-1 \\leq 3$.\n\\begin{enumerate}\n\\item If $a \\leq 2$, we can apply $L^4, L^4$ on $\\nabla_{I_a} R_{imjn} * \\nabla_{I_b} \\mathcal{D} \\phi$. To estimate $\\| \\nabla_{I_a} R_{imjn}\\|_{L^4(\\Sigma_t)}$,\nwe note that $\\nabla_{I_a} R_{imjn} = \\nabla_{I_a} ( \\hat k * \\hat k \\pm \\hat k \\pm E \\pm \\mathcal{D} \\phi * \\mathcal{D} \\phi ).$ Furthermore, to estimate \n$ \\|\\nabla_{I_a} \\left(\\mathcal{D} \\phi * \\mathcal{D} \\phi \\right)\\|_{L^4(\\Sigma_t)},$ or $ \\|\\nabla_{I_a} \\left(\\hat k* \\hat k \\right)\\|_{L^4(\\Sigma_t)}$, noticing that $[\\frac{a}{2}] + 2 \\leq [\\frac{l-1}{2}] + 2 \\leq 3 <4$ for $(l \\leq 4)$, we can apply $L^4, L^\\infty$ or $ L^\\infty, L^4$ to the two factors. Finally, we derive for $1 \\leq a \\leq 2$,\n\\begin{equation*}\n\\begin{split}\n\\| \\nabla_{I_a} R_{imjn} \\|_{L^4(\\Sigma_t)} \\| \\nabla_{I_b} \\mathcal{D} \\phi \\|_{L^4(\\Sigma_t)} \\lesssim & \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right) \\mathcal{E}^{\\frac{1}{2}}_l (\\phi, t).\n\\end{split}\n\\end{equation*}\n\n\\item If $a=3$, which occurs only when $l=4,$ and therefore $b=1$. In this case, we apply $L^2, L^\\infty$ on $\\nabla^3 R_{imjn} * \\nabla \\mathcal{D} \\phi$, and\n\\begin{equation*}\n\\begin{split}\n\\| \\nabla^3 R_{imjn} \\|_{L^2(\\Sigma_t)} \\|\\nabla \\mathcal{D} \\phi\\|_{L^\\infty} \\lesssim & \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_3 (\\phi, t) \\right) \\mathcal{E}^{\\frac{1}{2}}_3 (\\phi, t).\n\\end{split}\n\\end{equation*}\n\\end{enumerate}\nThus, it follows that for ${}_lLK^2_{1}$-II \n\\begin{equation}\\label{estimate-L231-2}\n\\begin{split}\n&\\int_{\\Sigma_t} t^{-1} \\big| \\nabla^{I_l} \\mathcal{D} \\phi * \\nabla_{I_a} R_{imjn} * \\nabla^{I_b} \\mathcal{D} \\phi \\big| \\\\\n\\lesssim & t^{-1} \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_3 (\\phi, t) \\right).\n\\end{split}\n\\end{equation}\n\\end{itemize}\nWe deduced for ${}_lLK^2_{1}$\n\\begin{equation*}\n\\begin{split}\n&\\int_{\\Sigma_t} | {}_lLK^2_{1} |\n\\lesssim t^{-1} E_l(\\phi, t) + t^{-1} \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation*}\nCombine with \\eqref{estimate-LK-11},\n\\begin{equation}\\label{estimate-LK1-2}\n\\begin{split}\n&\\int_{\\Sigma_t} | {}_lLK_{1} |\n\\lesssim t^{-1} E_l(\\phi, t) + t^{-1} \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation}\n\n\n\\subsubsection{The estimate for ${}_lLK_2$}\\label{sec-e-lot-kg-L2}\n${}_lLK_2$ \\eqref{energy-estimate-kg-1-id-l-o-t-2} can be further written as ${}_lLK_2 = {}_lLK_{21} + {}_lLK_{22}$, where (we have always ignored irrelevant constants) ${}_lLK_{2i}, \\, i=1,2$ are given as follows: ${}_lLK_{21} = {}_lLK^1_{21} + {}_lLK^2_{21}$\n\\begin{equation}\\label{def-LK-21}\n\\begin{split}\n{}_lLK^1_{21} := & \\nabla^{I_l} \\mathcal{D} \\phi \\sum_{a +b+1 = l+1} \\nabla_{I_{a}} \\nabla \\hat N * \\nabla_{I_{b}} \\mathcal{D} \\phi, \\\\\n {}_lLK^2_{21}:=& \\nabla^{I_l} \\mathcal{D} \\phi \\sum_{a +1+b = l} \\nabla_{I_{a}} \\nabla \\hat N * \\nabla_{I_b} \\mathcal{D} \\phi,\n \\end{split}\n\\end{equation}\n which comes from $\\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{N}_{l+1}(\\mathcal{D} \\phi),$ and $\\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{N}_{l}(\\mathcal{D} \\phi).$ And\n \\begin{equation}\\label{def-LK-22}\n\\begin{split}\n{}_lLK_{22} :=& \\nabla^{I_{l}} \\mathcal{D} \\phi \\sum_{a +b + 1 = l } \\nabla_{I_{a+1}} \\left( N \\hat k + \\hat N \\right) * \\nabla_{I_{b}} \\mathcal{D} \\phi,\n\\end{split}\n\\end{equation}\nwhich is $\\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{K}\\mathcal{N}_{I_l}(\\mathcal{D} \\phi)$.\nWe remark that $1 \\leq l \\leq 4$.\n \nFor ${}_lLK^1_{21}, {}_lLK^2_{21}$, $a+ b \\leq l,$ and $[\\frac{l}{2}] + 2 \\leq 4 (l \\leq 4),$ we can always apply $L^2, L^\\infty$ or $L^\\infty, L^2$ to the two factors in $ \\nabla_{I_{a}} \\nabla \\hat N * \\nabla_{I_b} \\mathcal{D} \\phi$ to derive \n\\begin{equation}\\label{estimate-LK211}\n\\begin{split}\n\\int_{\\Sigma_t} |{}_lLK_{21}| \n \\lesssim &E^\\frac{1}{2}_l(\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_5(\\hat N, t) \\mathcal{E}^{\\frac{1}{2}}_4(\\phi, t), \\lesssim \\varepsilon^3 M^3 t^{-2+4 \\delta}.\n\\end{split}\n\\end{equation}\n\n${}_lLK_{22}$ can be further split into the following two terms:\n\\begin{align*}\n {}_lLK^{1}_{22} &= \\nabla^{I_{l}} \\mathcal{D} \\phi \\sum_{p+1+q +b =l} \\nabla_{I_{p+1}} \\hat N *\\nabla_{I_{q}} \\hat k * \\nabla_{I_{b}} \\mathcal{D} \\phi, \\\\\n{}_lLK^{2}_{22} &= \\nabla^{I_{l}} \\mathcal{D} \\phi \\sum_{p+b=l, p\\geq 1} \\left( N \\nabla_{I_{p}} \\hat k + \\nabla_{I_{p}} \\hat N \\right) * \\nabla_{I_{b}} \\mathcal{D}\\phi.\n\\end{align*} \nFor ${}_lLK^{1}_{22}$, since $ b,q \\leq l-1$, and $p+1 \\leq l$ ($l \\leq 4$). Then we can apply $L^2, L^6, L^6, L^6$ to the four factors, so that\n\\begin{equation}\\label{estimate-LK221}\n\\begin{split}\n\\int_{\\Sigma_t} |{}_lLK^{1}_{22}| \n \\lesssim &\\mathcal{E}_l(\\phi, t) \\mathcal{E}_{l-1}(g, t) \\lesssim \\varepsilon^4 M^4 t^{-3+5 \\delta}.\n\\end{split}\n\\end{equation}\nAs ${}_lLK^{2}_{22}$, $[\\frac{l}{2}] + 2 \\leq 4 (l \\leq 4),$ we can always apply $L^2, L^2, L^\\infty$ or $L^2, L^\\infty, L^2$ to the three factors in ${}_lLK_{22}^{2}$\n\\begin{equation}\\label{estimate-LK222}\n\\begin{split}\n\\int_{\\Sigma_t} |{}_lLK^{2}_{22}| \n \\lesssim & E^\\frac{1}{2}_l(\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_3(g, t) \\mathcal{E}^{\\frac{1}{2}}_4(\\phi, t) \\lesssim \\varepsilon^3 M^3 t^{-2+4 \\delta}.\n\\end{split}\n\\end{equation}\n In summary, we have\n\\begin{equation}\\label{estimate-LK2-2}\n\\begin{split}\n& \\int_{\\Sigma_t} |{}_lLK_{22}| \\lesssim \\mathcal{E}_4(\\phi, t)\\mathcal{E}^{\\frac{1}{2}}_3(g, t)\n\\end{split}\n\\end{equation}\n\nCombining \\eqref{estimate-LK211} with \\eqref{estimate-LK2-2}, we conclude that for ${}_lLK_{2}$\n\\begin{equation}\\label{estimate-LK2}\n\\begin{split}\n&\\int_{\\Sigma_t}| {}_lLK_{2}|\n\\lesssim \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation}\n\n\n\n\\subsubsection{The estimate for ${}_lLK_3$}\\label{sec-e-lot-kg-L3}\nGenerally speaking, comparing to ${}_lLK_2$, ${}_lLK_3$ \\eqref{energy-estimate-kg-1-id-l-o-t-3} has additionally factor $t^{-1}$, for $\\nabla_{I_l} \\phi \\sim t^{-1} \\nabla_{I_l} \\left( \\bar m(t) \\phi \\right)$. To be more specific, the following terms in ${}_lLK_3$ admit the quantitatively equivalence \\footnote{Here $\\psi_1$ and $\\psi_2$ are quantitatively equivalent, denoted by $\\psi_1 \\dot\\sim \\psi_2$, in the sense that $\\psi_1$ and $\\psi_2$ share the same structures and they can be estimate in the same way.}:\n\\begin{subequations}\n\\begin{equation}\n \\nabla^{I_l} \\bar T \\phi \\mathcal{K}\\mathcal{N}_{I_l}(\\phi) + \\nabla^{I_l} \\phi \\mathcal{K}\\mathcal{N}^\\prime_{I_l}(\\bar T \\phi) \\dot\\sim t^{-1} \\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{K}\\mathcal{N}^\\prime_{I_l}( \\mathcal{D}\\phi) = t^{-1} {}_lLK_{22},\\label{eq-LK31-1}\n\\end{equation}\n\\begin{equation}\n \\nabla^{I_l} \\phi \\mathcal{N}_{I_l}(\\bar T \\phi) \\dot\\sim t^{-1} \\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{N}_l(\\mathcal{D} \\phi) = t^{-1} {}_lLK^2_{21},\\label{eq-LK31-2}\n\\end{equation}\n\\begin{equation}\n \\nabla^{I_l} \\phi \\mathcal{N}_{I_{l+1}}(\\nabla\\phi) \\dot\\sim t^{-1} \\nabla^{I_l} \\mathcal{D} \\phi \\mathcal{N}_{I_{l+1}}(\\mathcal{D} \\phi) = t^{-1} {}_lLK^1_{21}. \\label{eq-LK31-6}\n\\end{equation}\n\\end{subequations} \nDue to the estimates for ${}_lLK_{2}$ \\eqref{estimate-LK2}, ${}_lLK_{3}$ can be bounded as \n\\begin{equation}\\label{estimate-LK3}\n\\begin{split}\n&\\int_{\\Sigma_t} | {}_lLK_{3}|\n\\lesssim t^{-1} \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation}\n\n\n\n\n\n\n\n\\subsection{The borderline terms and hierarchy of energy}\\label{sec-0-ee-kg} \nIn this section, we focus on the borderline terms $BL_l$ \\eqref{def-energy-estimate-kg-id-bl}. We will take advantage of the relation between lapse and KG field (c.f. Lemma \\ref{lem-N-ineq-source}) and the enhanced $L^\\infty$ estimates \\eqref{improved-L-infty-kg-1} to conduct an hierarchy of energy estimate. Then each order of KG energy and lapse will be refined step by step, base on which the nonlinear terms $BL_l$ are reduced to linear ones. The idea of linearization and hierarchy of energy work together to close the energy argument for KG field. \n\n\\subsubsection{The zeroth order energy estimate}\\label{sec-0-ee-kg-1} \n\\begin{lemma}\\label{lemma-improved-0-energy-KG}\nWe have the improved estimate for the zeroth order energy \n\\begin{equation}\\label{eq-improved-energy-0-order}\nt E_0(\\phi,t) \\lesssim \\varepsilon^2I^2_0(\\phi).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nAccording to the zeroth order energy identity \\eqref{energy-id-0-kg} (see Theorem \\ref{Thm-energy-id-KG-0}) and the estimate for ${}_0LK$ \\eqref{estimate-0LK}, we obtain\n\\begin{equation*}\n\\partial_{\\bar \\tau} \\tilde E_0(\\phi, t) + \\tilde E_0(\\phi, t) \\lesssim t^{-1} E_0(\\phi, t) + E_0 (\\phi, t) \\mathcal{E}^{\\frac{1}{2}}_1 (g, t).\n\\end{equation*}\nWe then change to the $t$ time function to derive that\n\\begin{equation}\\label{ineq-energy-diff-0-order}\n\\partial_t ( t\\tilde E_0(\\phi, t)) \\lesssim t^{-2} (t E_0(\\phi, t)) + t^{-1} \\mathcal{E}^{\\frac{1}{2}}_1 (g, t) (tE_0 (\\phi, t)).\n\\end{equation}\nWe recall the definition for $\\mathcal{E}_1 (g, t)$.\nSubstitute the bound for $\\mathcal{E}_1 (g, t)$ \\eqref{BT-E-H}-\\eqref{BT-k-N-L-infty}, \\eqref{N-inte-ineq-123-lemma} and integrate along $t$ interval, noting that $ t\\tilde E_0(\\phi, t) \\sim t E_0(\\phi, t)$\n\\begin{equation}\\label{ineq-energy-0-order}\nt^\\prime E_0(\\phi, t^\\prime) \\lesssim \\varepsilon^2I^2_0(\\phi) + \\int^{t^\\prime}_{t_0} \\left( \\frac{1}{t^2} + \\frac{\\varepsilon M}{t^{2-2\\delta}} \\right) (t E_0(\\phi,t)) \\mathrm{d}} %{\\text{d} t, \\quad 0 < \\delta < \\frac{1}{6}.\n\\end{equation}\nAn application of the Gr\\\"{o}nwall inequality yields \\eqref{eq-improved-energy-0-order}.\n\\end{proof}\n\n\n\nWith the result of Lemma \\ref{lemma-improved-0-energy-KG}, we can further sharpen the estimates for lapse up to two derivatives.\n\\begin{corollary}\\label{lem-N-ineq-source-improved-2}\nThe estimate for $\\mathcal{E}_2(\\hat N,t)$ is improved as\n \\begin{equation}\\label{N-improved-estimate-source}\nt^2 \\mathcal{E}_2(\\hat N,t) \\lesssim \\varepsilon^2 I^2_4(\\phi).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nBy virtue of \\eqref{lemma-N-inte-ineq-2order-source} in Lemma \\ref{lem-N-ineq-source}, we substitute the enhanced $|\\bar m (t) \\phi |, |\\bar T \\phi|$ \\eqref{improved-L-infty-kg-1}, $E_0(\\phi,t)$ \\eqref{eq-improved-energy-0-order} and the bootstrap assumption for $\\mathcal{E}_1 (\\hat k, t)$ into \\eqref{lemma-N-inte-ineq-2order-source}, \n \\begin{align*}\n t^2 \\mathcal{E}_2(\\hat N,t) &\\lesssim \\varepsilon^2 I^2_0(\\phi) \\left( \\varepsilon^2 M^2 + \\varepsilon^2 I^2_4(\\phi) \\right) + \\varepsilon^4 M^4 t^{-2 + 4\\delta}\\lesssim \\varepsilon^2 I^2_4(\\phi).\n\\end{align*}\n\\end{proof}\n\n\n\n\\subsubsection{The high order energy estimates}\\label{sec-2-ee-kg}\nNow we make use of the previously improved $\\phi, \\hat N$ to further refine the estimate for $E_l(\\phi, t)$ step by step. \n\\begin{lemma}\\label{lemma-improved-l-energy-KG}\nThere is some constant $C_M$ depending linearly on $M$, such that\n\\begin{equation}\\label{improved-energy-l-order-1}\nt E_l(\\phi, t) \\lesssim \\left( \\varepsilon^2 I^2_4(\\phi) + \\varepsilon^3 M^3 \\right) t^{C_M \\varepsilon}, \\, 1 \\leq l \\leq 4.\n\\end{equation}\nMeanwhile, the lapse admits the enhanced estimates\n \\begin{equation}\\label{N-improved-estimate-source-to-4}\nt^2 \\mathcal{E}_{i+2}(\\hat N, t) \\lesssim \\left( \\varepsilon^2 I^2_{i}(\\phi) + \\varepsilon^3 M^3 \\right) t^{C_M \\varepsilon}, 1 \\leq i \\leq 2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nAfter substituting the estimates for the lower order terms ${}_lLK_1-{}_lLK_3$ (Corollary \\ref{coro-LK-1-LK_3}) into the energy identity for $\\phi$ \\eqref{energy-id-l-kg-rescale} (Corollary \\ref{coro-energy-id-KG-l}), we arrive at for $ l \\geq 1$ (noting that $\\tilde E_l(\\phi, t) \\sim E_l(\\phi, t)$)\n\\begin{equation}\\label{ineq-energy-diff-l-order-1}\n\\begin{split}\n\\partial_{\\bar \\tau} \\tilde E_l(\\phi, t) + \\tilde E_l(\\phi, t) \\lesssim & t^{-1} E_l (\\phi, t) + \\int_{\\Sigma_t} | BL_l |\\\\\n&+ \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right),\n\\end{split}\n\\end{equation}\nwhere the borderline terms $BL_l$ \\eqref{def-energy-estimate-kg-id-bl} take the form:\n$BL_l=BL_l^1 + \\cdots+BL_l^l$, and $BL_l^i, \\, i=1,\\cdots, l$ read\n\\begin{equation}\\label{def-BLl-1234}\n\\begin{split}\nBL_l^i =&\\bar m (t) \\nabla^{l-i+1} N *\\left( \\nabla^l (\\bar m (t) \\phi) * \\nabla^{i-1} \\bar T \\phi \\pm \\nabla^l \\bar T \\phi * \\nabla^{i-1}( \\bar m (t) \\phi )\\right).\n\\end{split}\n\\end{equation}\nThe special structure of $BL_l^i$ allows us to refine the estimates step by step.\n\n{\\bf Step I: the improvement for $E_1(\\phi, t)$}. \n With the help of Corollary \\ref{lem-N-ineq-source-improved-2} and the enhanced $\\|\\bar m (t) \\phi\\|_{L^\\infty}, \\|\\bar T \\phi\\|_{L^\\infty}$ \\eqref{improved-L-infty-kg-1}, the borderline term $BL_1= BL_1^1$, which reads $$\\bar m (t) \\nabla N * \\nabla (\\bar m (t) \\phi) * \\bar T \\phi - \\bar m (t) \\nabla N * \\nabla \\bar T \\phi * (\\bar m (t) \\phi)$$ is in fact a linear term.\n \nApplying $L^2, L^2, L^\\infty$ to the three factors in $BL_1,$ we have, by Cauchy-Schwarz inequality,\n\\begin{equation}\\label{ineq-energy-1-order-bl}\n\\begin{split}\n \\int_{\\Sigma_t} | BL_1| \\lesssim & t \\|\\bar T\\phi\\|_{L^\\infty} \\| \\nabla N \\|_{L^2(\\Sigma_t)} \\|\\nabla (\\bar m (t) \\phi) \\|_{L^2(\\Sigma_t)} \\\\\n &\\quad + t \\|\\bar m (t) \\phi\\|_{L^\\infty} \\| \\nabla N \\|_{L^2(\\Sigma_t)} \\| \\nabla \\bar T \\phi \\|_{L^2(\\Sigma_t)} \\\\\n \\lesssim & t^{\\frac{1}{2}} \\varepsilon M \\mathcal{E}^{\\frac{1}{2}}_1 (\\hat N, t) E_1^{\\frac{1}{2}}(\\phi,t) \n \\lesssim t^{-\\frac{1}{2}} \\varepsilon^2 M I_4(\\phi) E_1^{\\frac{1}{2}}(\\phi,t).\n \\end{split}\n\\end{equation}\nHence adding \\eqref{ineq-energy-1-order-bl} to \\eqref{ineq-energy-diff-l-order-1} with $l=1$,\n\\begin{equation*}\n\\begin{split}\n\\partial_{\\bar \\tau} \\tilde E_l(\\phi, t) + \\tilde E_l(\\phi, t) \\lesssim & \\left( t^{-1} + \\varepsilon^2 M^2 \\right) E_l (\\phi, t) + \\varepsilon^2 I_4^2 (\\phi) t^{-1} \\\\\n&+ \\mathcal{E}_4 (\\phi, t) \\left( \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_4 (\\phi, t) \\right).\n\\end{split}\n\\end{equation*}\nWe then change to $t$ function,\n\\begin{equation}\\label{ineq-energy-diff-1-order-1}\n\\begin{split}\n\\partial_t (t \\tilde E_l(\\phi, t)) \\lesssim & \\left( t^{-2} + \\varepsilon M t^{-1} \\right) (tE_l (\\phi, t) ) + \\varepsilon^2 I_4^2 (\\phi) t^{-1} \\\\\n&+ \\varepsilon^3 M^3 t^{-2+3\\delta},\n\\end{split}\n\\end{equation}\nIntegrate along $\\partial_t$, $\\tilde E_l(\\phi, t) \\sim E_l(\\phi, t)$,\n\\begin{equation}\\label{ineq-energy-1-order-2}\n\\begin{split}\nt^\\prime E_1(\\phi, t') \\lesssim& \\varepsilon^2 I^2_4(\\phi) t'^{\\varepsilon M} + \\varepsilon^3 M^3.\n \\end{split}\n\\end{equation}\nBy Gr\\\"{o}nwall inequality, there is some constant $C_M$ depending linearly on $M$, so that \\eqref{improved-energy-l-order-1} holds with $l=1$.\n\n\n{\\bf Step II: the improvement for $E_2(\\phi, t)$}. The borderline term $BL_2=BL_2^1 + BL_2^2$, where $BL_2^1, \\, BL_2^2$ read as below\n\\begin{equation*}\n\\begin{split}\nBL_2^1 =& \\bar m (t) \\nabla^2 (\\bar m (t) \\phi) * \\nabla^2 N * \\bar T \\phi + \\bar m (t) \\nabla^2 \\bar T \\phi * \\nabla^2 N * \\bar m (t) \\phi, \\\\\nBL_2^2 =& \\bar m (t) \\nabla^2 (\\bar m (t) \\phi) * \\nabla N * \\nabla \\bar T \\phi + \\bar m (t) \\nabla^2 \\bar T \\phi * \\nabla N * \\nabla (\\bar m(t) \\phi).\n\\end{split}\n\\end{equation*}\nNow, in additional to the enhanced $E_0(\\phi, t)$ \\eqref {ineq-energy-0-order}, $\\mathcal{E}_2(\\hat N, t)$ \\eqref{N-improved-estimate-source}, and $\\|\\bar m (t) \\phi\\|_{L^\\infty}$, $\\|\\bar T \\phi\\|_{L^\\infty}$ \\eqref{improved-L-infty-kg-1}, we should also utilize the newly improved $E_1(\\phi, t)$.\n\nFor $BL_2^1$, we follow the estimate for of $BL_1$ \\eqref{ineq-energy-1-order-bl},\n\\begin{equation}\\label{ineq-energy-2-order-bl-1}\n\\begin{split}\n\\int_{\\Sigma_t} | BL_2^1| \n \\lesssim & t^{-\\frac{1}{2}} \\varepsilon^2 M I_4(\\phi) E_2^{\\frac{1}{2}}(\\phi,t) \\lesssim \\varepsilon^2 M^2 E_2 (\\phi, t) + \\varepsilon^2 I_4^2 (\\phi) t^{-1}.\n \\end{split}\n\\end{equation}\n\nFor $BL_2^2$, we apply $L^2, L^4, L^4$ to the three factors. By Sobolev inequalities, $\\|\\nabla \\mathcal{D} \\phi \\|_{L^4(\\Sigma_t)} \\lesssim \\mathcal{E}_2^\\frac{1}{2} (\\phi, t)$. Note that $\\mathcal{E}^\\frac{1}{2}_2 (\\phi, t) \\lesssim \\mathcal{E}^\\frac{1}{2}_1(\\phi, t) + E^\\frac{1}{2}_2(\\phi, t)$, and $\\mathcal{E}^\\frac{1}{2}_1(\\phi, t)$ has been improved via Step I. Then, $BL_2^2$ is indeed a linear term,\n\\begin{equation}\\label{ineq-energy-2-order-bl-2}\n\\begin{split}\n\\int_{\\Sigma_t} | BL_2^2| \\lesssim & t \\| \\nabla N \\|_{L^4(\\Sigma_t)} \\|\\nabla \\bar T \\phi \\|_{L^4(\\Sigma_t)} \\|\\nabla^2 \\left( \\bar m (t) \\phi \\right) \\|_{L^2(\\Sigma_t)} \\\\\n&+ t \\| \\nabla N \\|_{L^4(\\Sigma_t)} \\| \\nabla \\left( \\bar m (t) \\phi \\right) \\|_{L^4(\\Sigma_t)} \\| \\nabla^2 \\bar T \\phi\\|_{L^2(\\Sigma_t)} \\\\\n &\\lesssim t \\mathcal{E}^{\\frac{1}{2}}_2 (\\hat N, t) \\left( \\mathcal{E}^\\frac{1}{2}_1(\\phi, t) + E^\\frac{1}{2}_2(\\phi, t) \\right) E_2^{\\frac{1}{2}}(\\phi,t) \\\\\n & \\lesssim \\varepsilon I_4(\\phi) \\left( \\left( \\varepsilon I_4(\\phi) +\\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) t^{-\\frac{1}{2} + C_M \\varepsilon} + E^\\frac{1}{2}_2(\\phi, t)\\right) E^\\frac{1}{2}_2 (\\phi,t) \\\\\n & \\lesssim \\varepsilon^2 I_4^2(\\phi) t^{-1+C_M \\varepsilon} + \\left( \\varepsilon^3 M^3 + \\varepsilon I_4(\\phi) \\right) E_2(\\phi,t).\n\\end{split}\n\\end{equation}\n\nWith \\eqref{ineq-energy-2-order-bl-1} and \\eqref{ineq-energy-2-order-bl-2}, the energy inequality \\eqref{ineq-energy-diff-l-order-1} with $l=2$ becomes\n\\begin{equation}\\label{ineq-energy-2-order-2}\n\\begin{split}\n\\partial_{\\bar \\tau} E_2(\\phi, t) + E_2(\\phi, t) \\lesssim & \\left( t^{-1} + \\varepsilon M \\right) E_2(\\phi,t) \\\\\n&+\\varepsilon^2 I_4^2(\\phi) t^{-1+C_M \\varepsilon} + \\varepsilon^3 M^3 t^{-2+3\\delta}.\n\\end{split}\n\\end{equation}\nChange to $t$,\n\\begin{equation*}\n\\begin{split}\n\\partial_t (t E_2(\\phi, t)) \\lesssim & \\left( t^{-2} + \\varepsilon Mt^{-1} \\right) (t E_2(\\phi,t) ) \\\\\n&+\\varepsilon^2 I_4^2(\\phi) t^{-1+C_M \\varepsilon} + \\varepsilon^3 M^3 t^{-2+3\\delta}.\n\\end{split}\n\\end{equation*}\nAnd the Gr\\\"{o}nwall inequality yields \\eqref{improved-energy-l-order-1} with $l=2.$\n\n\n{\\bf Setp III: the improvement for $\\hat N$}. In general, the refinement for $\\mathcal{E}_{l}(\\phi, t)$ leads to improvement for $\\mathcal{E}_{l+2}(\\hat N, t).$\nWe substitute the previously improved $\\mathcal{E}_2(\\phi, t)$ and $\\|\\bar m (t) \\phi\\|_{L^\\infty}$, $\\|\\bar T \\phi\\|_{L^\\infty}$ \\eqref{improved-L-infty-kg-1} into the corresponding inequality for $\\hat N$ \\eqref{lemma-N-inte-ineq-nabla3-source}-\\eqref{lemma-N-inte-ineq-nabla5-source} ($l=1$ corresponds to \\eqref{lemma-N-inte-ineq-nabla3-source}, $l=2$ corresponds to \\eqref{lemma-N-inte-ineq-nabla4-source}). Then,\n \\begin{equation*}\nt^2 E_{l+2}(\\hat N, t) \\lesssim \\varepsilon^3 M^3 t^{-2 + 4\\delta} + \\varepsilon^2 M^2 \\left( \\varepsilon^2 I^2_4(\\phi) + \\varepsilon^3 M^3 \\right) t^{C_M \\varepsilon}, \\, l \\leq 2,\n\\end{equation*}\nwhere $0<\\delta < \\frac{1}{6}.$ Hence \\eqref{N-improved-estimate-source-to-4} follows. \n\n\n{\\bf Step IV: the improvement for $E_3(\\phi, t)$}. We make use of the refined $E_l(\\phi, t), l \\leq 2$, $\\mathcal{E}_3(\\hat N, t)$ \\eqref{N-improved-estimate-source}, \\eqref{N-improved-estimate-source-to-4}, and $\\|\\bar m (t) \\phi\\|_{L^\\infty}$, $\\|\\bar T \\phi\\|_{L^\\infty}$ \\eqref{improved-L-infty-kg-1} to linearize the borderline term $BL_3$. Note that $BL_3=BL_3^1 + BL_3^2+BL_3^3$, where $BL_3^i,\\, i=1,2,3$ are shown in \\eqref{def-BLl-1234}.\n\nFor $BL_3^1$, we follow the argument for $BL_1$ \\eqref{ineq-energy-1-order-bl} and $BL_2^1$ \\eqref{ineq-energy-2-order-bl-1} to derive,\n\\begin{equation}\\label{ineq-energy-3-order-bl-1}\n\\begin{split}\n\\int_{\\Sigma_t} |BL_3^1| \n \\lesssim & t^{-\\frac{1}{2}} \\varepsilon M \\left( \\varepsilon I_4(\\phi) +\\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) t^{C_M \\varepsilon} E_3^{\\frac{1}{2}}(\\phi,t)\\\\ \\lesssim & \\varepsilon^2 M^2 E_3 (\\phi, t) + \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon}.\n \\end{split}\n\\end{equation}\nFor $BL_3^2$, we apply $L^2, L^4, L^4$ on the three factors, then\n\\begin{equation}\\label{ineq-energy-3-order-bl-2}\n\\begin{split}\n \\int_{\\Sigma_t} | BL_3^2| &\\lesssim t \\| \\nabla^2 N \\|_{L^4(\\Sigma_t)} \\| \\nabla \\bar T \\phi \\|_{L^4(\\Sigma_t)} \\| \\nabla^3 \\left( \\bar m (t) \\phi \\right)) \\|_{L^2(\\Sigma_t)} \\\\\n&+ t \\| \\nabla^2 N \\|_{L^4(\\Sigma_t)} \\|\\nabla \\left( \\bar m (t) \\phi \\right) \\|_{L^4(\\Sigma_t)} \\|\\nabla^3 \\bar T \\phi\\|_{L^2(\\Sigma_t)} \\\\\n &\\lesssim t \\mathcal{E}^{\\frac{1}{2}}_3 (\\hat N, t) \\mathcal{E}^\\frac{1}{2}_2(\\phi, t) E_3^{\\frac{1}{2}}(\\phi,t) \\\\\n & \\lesssim t^{-\\frac{1}{2} } \\varepsilon M \\left( \\varepsilon I_4(\\phi) +\\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right)^2 t^{C_M \\varepsilon} E^\\frac{1}{2}_3 (\\phi,t) \\\\\n & \\lesssim \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon} + \\left( \\varepsilon^3 M^3 + \\varepsilon^2 I^2_4(\\phi) \\right) E_3(\\phi,t).\n\\end{split}\n\\end{equation}\nFor $BL_3^3$, an analogous argument for $BL_2^2$ \\eqref{ineq-energy-2-order-bl-2} can be adapted here, noting that $\\mathcal{E}^\\frac{1}{2}_3 (\\phi, t) \\lesssim \\mathcal{E}^\\frac{1}{2}_2(\\phi, t) + E^\\frac{1}{2}_3(\\phi, t)$, and it leads to\n\\begin{equation}\\label{ineq-energy-3-order-bl-3}\n\\begin{split}\n \\int_{\\Sigma_t} |BL_3^3| \n\\lesssim & \\varepsilon^2 I_4^2(\\phi) t^{-1+C_M \\varepsilon} + \\left( \\varepsilon^3 M^3 + \\varepsilon^2 I^2_4(\\phi) \\right) E_3(\\phi,t).\n\\end{split}\n\\end{equation}\n\nCombining the above estimates for $BL_3$ and the energy inequality \\eqref{ineq-energy-diff-l-order-1} with $l=3$ yields\n\\begin{equation}\\label{ineq-energy-3-order}\n\\begin{split}\n \\partial_{\\bar \\tau} E_3(\\phi, t) + E_3(\\phi, t) \\lesssim& \\left( t^{-1} + \\varepsilon M \\right) E_3(\\phi,t) \\\\\n&+ \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon}.\n \\end{split}\n\\end{equation}\nThen \\eqref{improved-energy-l-order-1} with $l=3$ follows by Gr\\\"{o}nwall inequality.\n\n\n{\\bf Step V: the improvement for $E_4(\\phi, t)$}. Comparing to Step III, we now additionally take advantage of the previously improved $E_3(\\phi, t)$ and $E_4(\\hat N, t)$ to estimate the borderline term $BL_4$. It follows from \\eqref{def-BLl-1234} that $BL_4=BL_4^1 + BL_4^2+BL_4^3+BL_4^4$.\n\n$BL_4^1$ can be argued in the same way as $BL_1$ \\eqref{ineq-energy-1-order-bl}, $BL_2^1$ \\eqref{ineq-energy-2-order-bl-1}, and $BL_3^1$ \\eqref{ineq-energy-3-order-bl-1},\n\\begin{equation}\\label{ineq-energy-4-order-bl-1}\n\\begin{split}\n \\int_{\\Sigma_t} |BL_4^1|\n \\lesssim & \\varepsilon^2 M^2 E_4 (\\phi, t) + \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon}.\n \\end{split}\n\\end{equation}\nAnalogous to $BL_3^2$ \\eqref{ineq-energy-3-order-bl-2}, $BL_4^2$ and $BL_4^3$ share the following estimates:\n\\begin{equation}\\label{ineq-energy-4-order-bl-2-3}\n\\begin{split}\n \\int_{\\Sigma_t} |BL_4^2| + |BL_4^2| \n \\lesssim& \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon} \\\\\n & + \\left( \\varepsilon^3 M^3 + \\varepsilon^2 I^2_4(\\phi) \\right) E_4(\\phi,t).\n\\end{split}\n\\end{equation}\nFinally, following the argument for $BL_2^2$ \\eqref{ineq-energy-2-order-bl-2}, $BL_3^3$ \\eqref{ineq-energy-3-order-bl-3}, and noting that $\\mathcal{E}^\\frac{1}{2}_4 (\\phi, t) \\lesssim \\mathcal{E}^\\frac{1}{2}_3(\\phi, t) + E^\\frac{1}{2}_4(\\phi, t)$, we obtain for $BL_4^4$,\n\\begin{equation}\\label{ineq-energy-4-order-bl-4}\n\\begin{split}\n \\int_{\\Sigma_t} |BL_4^4| \n\\lesssim & \\varepsilon^2 I_4^2(\\phi) t^{-1+C_M \\varepsilon} + \\left( \\varepsilon^3 M^3 + \\varepsilon^2 I^2_4(\\phi) \\right) E_4(\\phi,t).\n\\end{split}\n\\end{equation}\nSummarizing the above estimates, we derive\n\\begin{equation}\\label{ineq-energy-diff-4-order}\n\\begin{split}\n \\partial_{\\bar \\tau} E_4(\\phi, t) + E_4(\\phi, t) \\lesssim& \\left( t^{-1} + \\varepsilon M \\right) E_4(\\phi,t) \\\\\n&+ \\left( \\varepsilon^2 I^2_4(\\phi) +\\varepsilon^3 M^3 \\right) t^{-1+C_M \\varepsilon}.\n \\end{split}\n\\end{equation} \nMaking use of the Gr\\\"{o}nwall inequality, we accomplish the proof for \\eqref{improved-energy-l-order-1} with $l=4.$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Energy estimate for Weyl field}\\label{sec-energy-Bianchi}\nWith improved energy for KG field \\eqref{eq-improved-energy-0-order}, \\eqref{improved-energy-l-order-1}, we are ready to improve energy estimates for Weyl tensor.\n\\begin{lemma}\\label{prop-energy-estimate-Bianchi}\nFor some constant $C_M$ depending linearly on $M$, there is,\n\\begin{equation}\\label{eq-energy-estimate-Bianchi-l-leq-2}\nt^2 E_l(W,t) \\lesssim \\varepsilon^2 I^2_{l}(W)+ \\varepsilon^3 M^3, \\quad 0 \\leq l \\leq 2.\n\\end{equation}\n\\begin{equation}\\label{eq-energy-estimate-Bianchi-l-3}\nt^2 E_3(W,t) \\lesssim \\left( \\varepsilon^2 I^2_{4}(\\phi) + \\varepsilon^2 I^2_{3}(W)+ \\varepsilon^3 M^3 \\right)t^{C_M \\varepsilon}.\n\\end{equation}\n\\end{lemma} \n\nWe start with the following Corollary which will be proved later in subsections \\ref{sec-estimate-error-bianchi} and subsection \\ref{sec-source-bianchi}.\n\\begin{corollary}\\label{coro-estimate-nonlinear-error-source-Bianchi}\nFor the nonlinear terms ${}_lLW_1, \\, {}_lLW_2$ and source term $S_l$ in the Bianchi identities \\eqref{energy-id-0-bianchi} (Lemma \\ref{lemma-energy-id-1-Bianchi}) and \\eqref{energy-id-l-bianchi} with $l \\leq 3$ (Corollary \\ref{coro-energy-id-l-Bianchi}), we have\n\\begin{equation}\\label{estimate-LW-1-2-Sl-leq-2}\n\\begin{split}\n& \\int_{\\Sigma_t} \\sum_{l=0}^3 | {}_lLW_1| + |{}_lLW_2| + \\sum_{l=0}^2 |S_l| \\\\\n\\lesssim & \\mathcal{E}^{\\frac{3}{2}}_3(g, t) +\\left( t^{-1} \\mathcal{E}^{\\frac{1}{2}}_3 (g, t) + \\mathcal{E}_3 (g, t)\\right) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\nAnd the top order source term $S_3$ admits\n\\begin{equation}\\label{estimate-S3}\n\\begin{split}\n&\\int_{\\Sigma_t} |S_{3}| \\lesssim \\varepsilon^2 M^2 E_3(W, t) + t^{-1} \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\\end{corollary}\n\nWith the help of Corollary \\ref{coro-estimate-nonlinear-error-source-Bianchi}, we now prove Lemma \\ref{prop-energy-estimate-Bianchi}.\n\\begin{proof}[Proof of Lemma \\ref{prop-energy-estimate-Bianchi}]\nSubstituting the results for $S_l, {}_lLW_1, {}_lLW_2 (0 \\leq l \\leq 2)$ \\eqref{estimate-LW-1-2-Sl-leq-2} and the bootstrap assumption \\eqref{BT-KG}-\\eqref{BT-k-N-L-infty}, and estimates for $\\mathcal{E}_i(\\hat N, t)$ Lemma \\ref{lem-N-ineq-source}, into the $l^{\\text{th}}$ order energy identities \\eqref{energy-id-l-bianchi}, we derive for $0 \\leq l \\leq 2,$\n\\begin{equation}\\label{eq-energy-inequality-Bianchi-leq-2}\n\\begin{split}\n \\partial_{\\bar \\tau} E_l(W,t) + 2 E_l(W, t) \\lesssim & \\varepsilon^3 M^3 t^{-3+4\\delta}, \\quad 0<\\delta < \\frac{1}{6}.\n \\end{split}\n\\end{equation}\nWe then change to the time function $t$ and multiplying $t$ on both hand side of \\eqref{eq-energy-inequality-Bianchi-leq-2} to obtain\n\\begin{equation*}\n\\begin{split}\n \\partial_t (t^2 E_l(W,t)) \\lesssim & \\varepsilon^3 M^3 t^{-2+4\\delta}, \\quad 0<\\delta < \\frac{1}{6}.\n \\end{split}\n\\end{equation*}\nThen \\eqref{eq-energy-estimate-Bianchi-l-leq-2} with $0 \\leq l \\leq 2$ follows from the Gr\\\"{o}nwall inequality.\nSubstituting the results for $S_3, {}_3LW_1, {}_3LW_2$ \\eqref{estimate-LW-1-2-Sl-leq-2}, \\eqref{estimate-S3} and the improved estimates for $\\mathcal{E}_4 (\\phi, t)$ (Lemma \\ref{lemma-improved-l-energy-KG}) into the energy identity \\eqref{energy-id-l-bianchi} with $l= 3$, we have\n\\begin{equation}\\label{eq-energy-inequality-Bianchi}\n\\begin{split}\n \\partial_{\\bar \\tau} E_3(W,t) + 2 E_3(W, t) \\lesssim & \\varepsilon M E_3(W, t) \\\\\n & + \\left( \\varepsilon^2 I^2_{4}(\\phi) + \\varepsilon^3 M^3 \\right)t^{-2+C_M \\varepsilon}.\n \\end{split}\n\\end{equation}\nChanging to $t$ and multiplying $t$ on both hand side of \\eqref{eq-energy-inequality-Bianchi}, we deduce \n\\begin{equation*}\n\\begin{split}\n \\partial_t (t^2 E_3(W,t)) \\lesssim & \\varepsilon M t^{-1} (t^2 E_3(W, t)) \\\\\n & + \\left( \\varepsilon^2 I^2_{4}(\\phi) + \\varepsilon^3 M^3 \\right)t^{-1+C_M \\varepsilon}.\n \\end{split}\n\\end{equation*}\nAn application of the Gr\\\"{o}nwall inequality then yields \\eqref{eq-energy-estimate-Bianchi-l-3}.\n\\end{proof}\n\nWe remark that the energy estimates for the Weyl field are carried out only up to three derivatives, then $l \\leq 3$ throughout this section.\n\\subsection{The nonlinear error terms}\\label{sec-estimate-error-bianchi}\nIn this section we present the estimates for the nonlinear error terms ${}_lLW_1, {}_lLW_2 (l \\leq 3)$.\n\\begin{corollary}\\label{coro-estimate-nonlinear-error-Bianchi}\nFor the nonlinear error terms ${}_lLW_1, {}_lLW_2$ ($l \\leq 3$), we have\n\\begin{equation}\\label{estimate-LW-1-2}\n\\int_{\\Sigma_t} |{}_0LW| + |{}_lLW_1| + |{}_lLW_2| \\lesssim \\mathcal{E}^{\\frac{3}{2}}_3(g, t) + \\mathcal{E}_3(g, t) \\mathcal{E}_3(\\phi, t). \n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nThe error terms ${}_lLW_1, {}_lLW_2$ can be rearranged as ${}_lLW_1=\\sum_{i=1}^3 {}_lLW_{1i},$\n\\begin{equation}\\label{def-LB-1-re}\n\\begin{split}\n{}_lLW_{11} &= \\sum_{a+b +c=l } \\nabla_{I_{a}} N *\\nabla_{I_{b}} \\hat k * \\nabla_{I_{c}} W * \\nabla^{I_{l}} W, \\\\\n{}_lLW_{12} & = \\sum_{a+1+b =l+1 } \\nabla_{I_{a+1}} \\hat N * \\nabla_{I_{b}} W * \\nabla^{I_{l}} W, \\\\\n{}_lLW_{13 } &= \\sum_{a+b=l } \\nabla_{I_{a}} \\hat N * \\nabla_{I_{b}} W * \\nabla^{I_{l}} W, \n\\end{split}\n\\end{equation}\n(note that, when $l=0$, the above ${}_lLW_1$ covers ${}_0LW$ \\eqref{def-e-s-1-bianchi}),\nand ${}_lLW_2$\n\\begin{equation}\\label{def-LB-2-re}\n\\begin{split}\n{}_lLW_{2} & = \\sum_{a+b =l-1} N \\nabla_{I_{a}} \\hat R_{imjn} * \\nabla_{I_{b}} W * \\nabla^{I_{l}} W.\n\\end{split}\n\\end{equation}\n\nFor ${}_lLW_{11}$, if $a=0$, noting that $[\\frac{l}{2}] + 2 \\leq 3 (l \\leq 3)$, then we can perform $L^\\infty, L^2, L^2$ or $L^2, L^\\infty, L^2$ on the three factors; If $a \\geq 1$, then $b,c \\leq l-1$ and $a \\leq l \\leq 3$, we can perform $L^6, L^6, L^6, L^2$ on the four factors. Putting these two cases together, we have\n\\begin{equation}\\label{estimate-LW11}\n\\int_{\\Sigma_t} |{}_lLW_{11}| \\lesssim \\mathcal{E}^{\\frac{3}{2}}_3(g, t) + \\mathcal{E}^{2}_3(g, t). \n\\end{equation}\n\nFor ${}_lLW_{12}$ and ${}_lLW_{13}$, noting that $[\\frac{l}{2}] + 2 \\leq 3 (l \\leq 3)$ and $[\\frac{l}{2}] +1+ 2 \\leq 4$ (to perform $L^\\infty$ on $\\nabla_{I_{a+1}} \\hat N$ in ${}_lLW_{12}$), we can always perform $L^2, L^\\infty, L^2$ or $L^2, L^\\infty, L^2$ on the three factors in each of ${}_lLW_{12}$, ${}_lLW_{13}$ so that\n\\begin{equation}\\label{estimate-LW1213}\n\\int_{\\Sigma_t} |{}_lLW_{12}| + |{}_lLW_{12}| \\lesssim \\mathcal{E}^{\\frac{3}{2}}_3(g, t). \n\\end{equation}\n\nFinally, for ${}_lLW_{2}$, we recall that $\\hat R_{imjn} = \\hat k \\pm |\\hat k|^2 \\pm E \\pm |\\mathcal{D} \\phi|^2$. Since, $a, b \\leq l-1$, we can always perform $L^4, L^4, L^2$ on the three factors (such as $\\nabla_{I_a} |\\hat k| * \\nabla_{I_b} W * \\nabla^{I_l} W$ ) or $L^6, L^6, L^6, L^2$ on the four factors ($\\nabla_{I_a} |\\hat k|^2 * \\nabla_{I_b} W * \\nabla^{I_l} W$) in ${}_lLW_{2}$. As a consequence, we derive \n\\begin{equation}\\label{estimate-LW1213}\n\\int_{\\Sigma_t} |{}_lLW_{2}| \\lesssim \\mathcal{E}^{\\frac{3}{2}}_3(g, t) + \\mathcal{E}_3(g, t) \\mathcal{E}_3(\\phi, t).\n\\end{equation}\n\nSummarize all these estimates, we obtain \\eqref{estimate-LW-1-2}\n\\end{proof}\n\n\n\n\n\n\n\\subsection{The nonlinear coupling terms}\\label{sec-source-bianchi}\nIn this section, we will prove the following estimate for the nonlinear couplings in the Bianchi equations.\n\\begin{corollary}\\label{coro-estimate-source-Bianchi}\nFor the coupling terms $S_l, l \\leq 3$, (see \\eqref{energy-id-0-bianchi} and \\eqref{energy-id-l-bianchi}), we have\n\\begin{equation}\\label{estimate-Sl-leq-2}\n \\int_{\\Sigma_t} |S_{l}| \\lesssim \\left( t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (g, t) + \\mathcal{E}_l (g, t)\\right) \\mathcal{E}_4 (\\phi, t), \\quad 0 \\leq l \\leq 2,\n\\end{equation}\nand the top order case $S_3$ admits \n\\begin{equation}\\label{estimate-Source-S3}\n \\int_{\\Sigma_t} |S_{3}| \\lesssim \\varepsilon^2 M^2 E_3(W, t) + t^{-1} \\mathcal{E}_4 (\\phi, t). \n\\end{equation}\n\\end{corollary} \n\nWe postpone the proof to subsection \\ref{sec-estimate-source-bianchi} and explore the structures for the couplings first.\n\\subsubsection{The structures of the couplings}\\label{sec-stru-source-bianchi}\nThe source terms $S_l, \\, l \\leq 3$ can be split into electric and magnetic parts: $S_l=S_{l1} + S_{l2}$, where\n\\begin{equation}\\label{def-S-re}\n\\begin{split}\nS_{l1} = & \\nabla_{I_l} \\left( N J_{i \\bar T j} \\right) \\nabla^{I_l} E^{ij}, \\\\\nS_{l2} = & \\nabla_{I_l} \\left( N J^*_{i \\bar T j} \\right) \\nabla^{I_l} H^{ij}.\n\\end{split}\n\\end{equation}\n Due to the fact that $E_{ij}$ is trace free and $T$-tangent, we have\n\\begin{equation}\\label{stru-source-J-bianchi-0}\n\\begin{split}\n J_{i \\bar T j} E^{ij} =& \\frac{1}{2} \\left( D_{\\bar T} \\bar R_{ij} - D_j \\bar R_{\\bar T i} \\right) E^{ij} \\\\\n =& \\frac{1}{2} \\left(D_{\\bar T} D_i \\phi D_j \\phi - D_i D_j \\phi \\bar T \\phi \\right) E^{ij} \\\\\n = & \\left( \\nabla \\bar T \\phi \\pm k_{i}^j * \\nabla \\phi \\right) * \\nabla \\phi * E \\pm \\nabla^2 \\phi * \\bar T \\phi * E \\pm (\\bar T \\phi)^2 \\hat k * E.\n\\end{split}\n\\end{equation}\nand \n\\begin{equation}\\label{stru-source-AJ-bianchi-0}\n\\begin{split}\n J^*_{i \\bar T j} H^{ij} \n =& \\frac{1}{4} \\left( D_\\mu \\bar R_{\\nu i} - D_\\nu \\bar R_{\\mu i} \\right) \\cdot \\epsilon^{\\mu \\nu} {}_{\\! \\bar T j} H^{ij} \\\\\n & \\quad \\quad - \\frac{1}{24} \\left( g_{i\\nu} D_\\mu \\bar R - g_{i\\mu} D_\\nu \\bar R \\right) \\cdot \\epsilon^{\\mu \\nu} {}_{\\! \\bar T j} H^{ij}\\\\\n =& \\frac{1}{4} \\left( D_p D_i \\phi D_q \\phi - D_q D_i \\phi D_p \\phi \\right) \\epsilon^{pq} {}_{\\! j} H^{ij} +Tr_1 +Tr_2, \n \\end{split}\n\\end{equation}\nwhere\n \\begin{equation*}\n\\begin{split}\nTr_1 &= \\frac{1}{4} \\bar m^2(t) \\phi D_p \\phi \\epsilon^{p} {}_{\\! ij} H^{ij} - \\frac{1}{4} \\bar m^2(t) \\phi D_q \\phi \\epsilon_i{}^{ q} {}_{\\! j} H^{ij}, \\\\\nTr_2 &= - \\frac{1}{24} D_p \\bar R \\epsilon^{p} {}_{\\! ij} H^{ij} + \\frac{1}{24} D_q \\bar R \\epsilon_i{}^{q} {}_{\\! j} H^{ij}.\n\\end{split}\n\\end{equation*}\nNotice that, $\\epsilon^{p} {}_{\\! ij}, \\epsilon_i{}^{\\! q} {}_{\\! j}$ are both antisymmetric in $i$ and $j$, while $H_{ij}$ is symmetric in $i,j.$ Hence, $Tr_1=Tr_2=0$. With the same reason, $ J^*_{i \\bar T j} H^{ij}$ can be further reduced as\n\\begin{equation}\\label{stru-source-AJ-bianchi-0-1}\n\\begin{split}\n J^*_{i \\bar T j} H^{ij} =& \\frac{1}{4} \\left( D_p D_i \\phi D_q \\phi - D_q D_i \\phi D_p \\phi \\right) \\epsilon^{pq} {}_{\\! j} H^{ij} \\\\\n= & \\left( \\nabla^2 \\phi \\pm \\hat k * \\bar T \\phi \\right) * \\nabla \\phi * H.\n\\end{split}\n\\end{equation}\n\nThe fact that $E_{ij}$ is trace free and $T$-tangent can be generalized to the case of high order derivatives. Obviously, $\\nabla_{I_l} E_{i j}$, which is projected to $\\Sigma_t$, is $T-$tangent.\nIn addition to this, we can derive that $ g^{ij}\\nabla_{I_l} E_{ij} =0$. Hence, the results of \\eqref{stru-source-J-bianchi-0} and \\eqref{stru-source-AJ-bianchi-0-1} can be generalized as\n\\begin{equation}\\label{stru-source-J-bianchi-1}\n\\begin{split}\n & \\nabla_{I_l} \\left( NJ_{i \\bar T j} \\right) \\nabla^{I_l} E^{ij} \\\\\n=&\\pm \\nabla_{I_l} \\left(N \\nabla \\bar T \\phi * \\nabla \\phi \\pm N \\nabla^2 \\phi * \\bar T \\phi \\right) * \\nabla^{I_l} E\\\\\n&\\pm \\nabla_{I_l} \\left( N\\left(k_i^j * \\nabla \\phi * \\nabla \\phi \\pm \\hat k* (\\bar T \\phi)^2 \\right) \\right) * \\nabla^{I_l} E,\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\\label{stru-source-AJ-bianchi-1}\n\\begin{split}\n & \\nabla_{I_l} \\left( N J^*_{i \\bar T j} \\right) \\nabla^{I_l} H^{ij} \\\\\n= & \\nabla_{I_l} \\left( N \\left( \\nabla^2 \\phi \\pm \\hat k* \\bar T \\phi \\right) * \\nabla \\phi \\right) * \\nabla^{I_l} H.\n\\end{split}\n\\end{equation}\n\n\\subsubsection{The estimate for the coupling terms}\\label{sec-estimate-source-bianchi}\nWe rewrite \\eqref{stru-source-J-bianchi-1} and \\eqref{stru-source-AJ-bianchi-1} as\n\\begin{equation}\\label{def-Sl1-re}\n\\begin{split}\nS_{l1} = \\sum_{a+b =l }& \\nabla_{I_a} N * \\nabla_{I_b} (\\nabla \\bar T \\phi * \\nabla \\phi) \\nabla^{I_l} E^{ij} \\\\\n&+ \\nabla_{I_a} N * \\nabla_{I_b} (\\nabla^2 \\phi * \\bar T \\phi) \\nabla^{I_l} E^{ij} \\\\\n&+ \\nabla_{I_a} N * \\nabla_{I_b} (\\nabla \\phi * \\nabla \\phi * k_i^j) \\nabla^{I_l} E^{ij} \\\\\n&+ \\nabla_{I_a} N * \\nabla_{I_b} ((\\bar T \\phi)^2 * \\hat k) \\nabla^{I_l} E^{ij},\n\\end{split}\n\\end{equation}\nand \n\\begin{equation}\\label{def-Sl2-re}\n\\begin{split}\nS_{l2} = \\sum_{a+b =l }& \\nabla_{I_a} N * \\nabla_{I_b} (\\nabla^2\\phi * \\nabla \\phi ) \\nabla^{I_l} H^{ij} \\\\\n&+ \\nabla_{I_a} N * \\nabla_{I_b} ( \\hat k* \\bar T \\phi * \\nabla \\phi ) \\nabla^{I_l} H^{ij}.\n\\end{split}\n\\end{equation}\nNotice that, then energy estimate argument for the Weyl tensor is done up to three order derivatives, hence in this section, we have $l \\leq 3.$\n\nFor the electric part $S_{l1},$ \n\\begin{itemize}\n\\item {\\bf Case $S_{l1}$-I}: $a=0.$ Then $b=l.$ $S_{l1}$-I consists\n\\begin{equation}\\label{def-S-l11-1234}\n\\begin{split}\n S_{l11}^1= &N \\nabla_{I_l} \\left( \\nabla \\bar T \\phi * \\nabla \\phi \\right) * \\nabla^{I_l} E,\\\\\n S^2_{l11} =& N \\nabla_{I_l} \\left( \\nabla^2 \\phi * \\bar T \\phi \\right) * \\nabla^{I_l} E,\\\\\nS^3_{l11} =& N \\nabla_{I_l} \\left( k_i^j * \\nabla \\phi * \\nabla \\phi \\right) * \\nabla^{I_l} E, \\\\\nS^4_{l11} =&N \\nabla_{I_l} \\left( (\\bar T \\phi)^2 \\hat k \\right) * \\nabla^{I_l} E.\n\\end{split}\n\\end{equation}\n\nFor $S_{l11}^1,$ we further rewrite it as \n\\begin{equation}\\label{def-S-l11-1}\nS_{l11}^1= \\sum_{p+q=l} (mt)^{-1} N \\nabla_{I_{p+1}} \\bar T \\phi * \\nabla_{I_{q+1}} (\\bar m (t) \\phi) * \\nabla^{I_l} E.\n\\end{equation}\n Since $[\\frac{l}{2}] + 1 + 2 \\leq 4 (l \\leq 3),$ we can apply $L^2, L^\\infty$ or $L^\\infty, L^2$ to the two factors in $\\nabla_{I_{p+1}} \\bar T \\phi *\\nabla_{I_{q+1}} \\phi$ and derive \n\\begin{equation}\\label{estimate-S11-1}\n\\begin{split}\n\\int_{\\Sigma_t} |S_{l11}^1| \\lesssim & t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (W, t) \\mathcal{E}_4 (\\phi, t) \\lesssim \\varepsilon^3 M^3 t^{-3+3\\delta}.\n\\end{split}\n\\end{equation}\n\nFor $S_{l11}^2,$ we further rewrite it as \n\\begin{align*}\nS_{l11}^2= \\sum_{p+q=l} N \\nabla_{I_{p+1}} \\nabla \\phi * \\nabla_{I_{q}} \\bar T \\phi * \\nabla^{I_l} E.\n\\end{align*}\n\\begin{enumerate}\n\\item In the case of $0 \\leq l \\leq 2$. We rearrange it as $$\\sum_{p+q=l \\leq 2} t^{-1} N \\nabla_{I_{p+2}} (\\bar m (t) \\phi) * \\nabla_{I_q} \\bar T \\phi * \\nabla^{I_l} E.$$\nWe apply $L^4, L^4, L^2$ (if $p \\leq 1$) or $L^2, L^\\infty, L^2$ (if $p=2$) on the three factors, since $p+q \\leq 2$. With the same argument for $S_{l11}^1$ \\eqref{estimate-S11-1}, we derive\n\\begin{equation}\\label{estimate-S011-leq-2}\n\\begin{split}\n\\int_{\\Sigma_t} |S_{l11}^2| \\lesssim & t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (W, t) \\mathcal{E}_4 (\\phi, t), \\, l \\leq 2.\n\\end{split}\n\\end{equation}\n\n\\item In the case of $l =3.$\n\n\\begin{itemize}\n \n\\item[i)] If $p \\leq l-1\\leq 2,$ then $\\nabla_{I_{p+2}}\\phi$ will never exceed the top ($4$) order derivative. We can perform the same argument as in the case of $0 \\leq l \\leq 2$ \\eqref{estimate-S011-leq-2}. The same bound $ t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (\\phi, t) \\mathcal{E}_4 (\\phi, t) $ can be obtained as well.\n\n\\item[ii)] If $p=l=3$, the argument for \\eqref{estimate-S011-leq-2} are not valid here. Because of the regularity, $\\nabla^5(\\bar m (t) \\phi)$ is not allowed, hence $\\nabla^4 \\nabla \\phi$ can not absorb an addition $t$ (or $\\bar m (t)$) as the previous case, which makes it the borderline case. However, as in the proof that leads to Lemma \\ref{lemma-improved-l-energy-KG}, the improved $\\|\\bar T\\phi\\|_{L^\\infty}$ and $E_{4}(\\phi,t)$ \\eqref{improved-energy-l-order-1} will help to linearize this borderline term, that is\n\\begin{equation}\\label{estimate-Sl11-2-1}\n\\begin{split}\n\\int_{\\Sigma_t} |S_{311}^2| =& \\int_{\\Sigma_t} \\big| N \\nabla^4 \\nabla \\phi * \\bar T \\phi * \\nabla^3 E \\big| \\\\\n\\lesssim & \\| \\nabla^4 \\nabla \\phi \\|_{L^2(\\Sigma_t)} \\|\\bar T \\phi\\|_{L^\\infty} \\|\\nabla^{3} E\\|_{L^2(\\Sigma_t)} \\\\\n \\lesssim & E^{\\frac{1}{2}}_4 (\\phi, t) \\cdot \\varepsilon M t^{-\\frac{1}{2}} \\cdot E^\\frac{1}{2}_3(W, t) \\\\\n \\lesssim & \\varepsilon^2 M^2 E_3(W, t) + t^{-1} E_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n \\end{itemize}\n\\end{enumerate}\n\n\nRelative to $S_{l11}^1$ and $S_{l11}^2$, $S_{l11}^3$ and $S_{l11}^4$ are in fact lower order terms. \n\n$S_{l11}^3$ can be rewritten as \n\\begin{align*}\nS_{l11}^3= \\sum_{p+q+h=l} N \\nabla_{I_h} k_i^j * \\nabla_{I_{p+1}} \\phi * \\nabla_{I_{q+1}} \\phi * \\nabla^{I_l} E.\n\\end{align*}\nIf $h=0,$ it becomes $\\sum_{p+q=l} N k_i^j * \\nabla_{I_{p+1}} \\phi * \\nabla_{I_{q+1}} \\phi * \\nabla^{I_l} E$, which is equivalent to (noting that $p+1, q+1 \\leq l+1 \\leq 4$)\n\\begin{equation}\\label{S-l-11-3-h=0}\n\\sum_{p+q=l} t^{-2} N \\nabla_{I_{p+1}} (\\bar m (t) \\phi) * \\nabla_{I_{q+1}}(\\bar m (t) \\phi) * \\nabla^{I_l} E,\n\\end{equation}\n Quantitatively, \\eqref{S-l-11-3-h=0} turns out to be equivalent to $t^{-1} S_{l11}^1$ \\eqref{def-S-l11-1}. Thus, similar to \\eqref{estimate-S11-1}, \\eqref{S-l-11-3-h=0} can be bounded by $t^{-2} \\mathcal{E}^{\\frac{1}{2}}_l (W, t) \\mathcal{E}_4 (\\phi, t)$.\nIf $h \\geq 1,$ we get\n\\begin{equation}\\label{S-l-11-3-h-geq-1}\n\\sum_{p+q+h=l \\atop h \\geq 1} N t^{-2} \\nabla_{I_h} \\hat k* \\nabla_{I_{p+1}}(\\bar m (t) \\phi ) * \\nabla_{I_{q+1}} (\\bar m (t) \\phi) * \\nabla^{I_l} E.\n\\end{equation}\nKnowing that $1 \\leq h \\leq l \\leq 3,$ and $0 \\leq p, q \\leq l-1 \\leq 2,$ we can apply $L^6, L^6, L^6, L^2$ \\eqref{S-l-11-3-h-geq-1} and derive the bound $t^{-2} \\mathcal{E}_l (g, t) \\mathcal{E}_4 (\\phi, t)$, which is of lower order than \\eqref{S-l-11-3-h=0}. As a summary, we have\n\\begin{equation}\\label{estimate-S11-3}\n\\begin{split}\n \\int_{\\Sigma_t} |S_{l11}^3|\n\\lesssim t^{-2} \\mathcal{E}^{\\frac{1}{2}}_l (g, t) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\n$S_{l11}^4$ can be further rewritten as \n\\begin{equation}\\label{def-S-l11-4}\nS_{l11}^4= \\sum_{p+q + h=l} N \\nabla_{I_p} \\bar T \\phi * \\nabla_{I_q} \\bar T \\phi *\\nabla_{I_h} \\hat k * \\nabla^{I_l} E.\n\\end{equation}\nNoting that $p,q,h \\leq l \\leq 3,$ we can apply $L^6, L^6, L^6, L^2$ to the four factors to derive \n\\begin{equation}\\label{estimate-S11-4}\n\\begin{split}\n \\int_{\\Sigma_t}|S_{l11}^4| \\lesssim \\mathcal{E}_l (g, t) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\n\n\n\\item {\\bf Case $S_{l1}$-II}: $a\\geq 1.$ Letting $a =c+1, c\\geq 0,$ $S_{l1}$-II composes of \n \\begin{equation}\\label{def-S-l12-1234}\n\\begin{split}\n S_{l12}^1= & \\sum_{c+1+b=l}\\nabla_{I_{c+1}} \\hat N * \\nabla_{I_b} \\left( \\nabla \\bar T \\phi * \\nabla \\phi \\right) * \\nabla^{I_l} E\\\\\n S^2_{l12} =& \\sum_{c+1+b=l} \\nabla_{I_{c+1}} \\hat N * \\nabla_{I_b} \\left( \\nabla^2 \\phi * \\bar T \\phi \\right) * \\nabla^{I_l} E\\\\\nS^3_{l12} =& \\sum_{c+1+b=l} \\nabla_{I_{c+1}} \\hat N * \\nabla_{I_b} \\left( k_i^j * \\nabla \\phi * \\nabla \\phi \\right) * \\nabla^{I_l} E \\\\\nS^4_{l12} =& \\sum_{c+1+b=l} \\nabla_{I_{c+1}} \\hat N* \\nabla_{I_b} \\left( (\\bar T \\phi)^2 \\hat k \\right) * \\nabla^{I_l} E.\n\\end{split}\n\\end{equation}\nIn fact, $S_{l1}$-II \\eqref{def-S-l12-1234} enjoys better estimate than $S_{l1}$-I \\eqref{def-S-l11-1234}. This can be roughly verified by comparing $N$ with $\\hat N$. \n$S^1_{l12}$ can be rearranged as (for $b=b_1 + b_2 \\leq l-1\\leq 2$, then $b_1 +1, b_2 +1 \\leq l \\leq 3$), \n\\begin{equation}\\label{Sl12-1}\n\\begin{split}\n& \\sum_{c+1+b_1 + b_2 =l } t^{-1}\\nabla_{I_{c+1}} \\hat N * \\nabla_{I_{b_1+1}} \\bar T \\phi * \\nabla_{I_{b_2+1}} (\\bar m (t) \\phi) * \\nabla^{I_l} E.\n\\end{split}\n\\end{equation}\nHere $c+1 \\leq 3$ and $b_1+1, b_2+1 \\leq 3$, we can apply $L^6, L^6, L^6, L^2$ to the four factors in \\eqref{Sl12-1} to derive\n\\begin{equation}\\label{estimate-Sl12-1}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l12}^1| \\lesssim t^{-1} \\mathcal{E}_3 (g, t) \\mathcal{E}_4(\\phi, t).\n\\end{split}\n\\end{equation}\n\n$S^2_{l12}$ can be rearranged as (for $b=b_1 + b_2 \\leq l-1\\leq 2$, then $b_1 +2 \\leq l+1 \\leq 4$), \n\\begin{equation}\\label{Sl12-1}\n\\begin{split}\n& \\sum_{c+1+b_1 + b_2 =l } t^{-1}\\nabla_{I_{c+1}} \\hat N * \\nabla_{I_{b_1+2}} (\\bar m (t) \\phi) * \\nabla_{I_{b_2}} \\bar T \\phi * \\nabla^{I_l} E.\n\\end{split}\n\\end{equation}\nHere $c+1 \\leq 3$ and $b_1, b_2 \\leq 2$. If $b_1 \\leq 1$, then $b_1 +2 \\leq 3,$ and we can apply $L^6, L^6, L^6, L^2$ to the four facts in in \\eqref{Sl12-1}; If $b_1 =2$, then $c=b_2=0$. We can use $L^\\infty, L^2, L^\\infty, L^2$ alternatively. Finally, $S_{l12}^2$ admits the estimate\n\\begin{equation}\\label{estimate-Sl12-2}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l12}^2| \\lesssim t^{-1} \\mathcal{E}_3 (g, t) \\mathcal{E}_4(\\phi, t).\n\\end{split}\n\\end{equation}\n\n$S^3_{l12}$ can be rearranged as ($c+1+b_1 + b_2 + b_3 =l $ and $b_2 +1, b_3 +1 \\leq l \\leq3$),\n \\begin{equation}\\label{Sl12-3}\n t^{-2} \\nabla_{I_{c+1}} \\hat N * \\nabla_{I_{b_1}} k_i^j * \\nabla_{I_{b_2+1}} (\\bar m (t) \\phi ) * \\nabla_{I_{b_3+1}} (\\bar m (t) \\phi) * \\nabla^{I_l} E.\n\\end{equation}\nHere $c+1, b_2+1, b_3+1 \\leq l \\leq 3$. If $b_1=0$, $k_i^j \\sim 1$, we apply $L^6, L^\\infty, L^6, L^6, L^2$ on the five factors in \\eqref{Sl12-3} and derive the bound $t^{-2} \\mathcal{E}_3 (g, t) \\mathcal{E}_4 (\\phi, t)$; If $1 \\leq b_1,$ noting that $b_1 \\leq l-1\\leq 2,$ we apply $L^6, L^\\infty, L^6, L^6, L^2$ on the five factors as well and the bound is replaced by $t^{-2} \\mathcal{E}^{\\frac{3}{2}}_3 (g, t) \\mathcal{E}_4 (\\phi, t)$ which is of lower order. Finally, we have\n\\begin{equation}\\label{estimate-Sl12-3}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l12}^3| \n \\lesssim t^{-2} \\mathcal{E}_3 (g, t) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\n$S^4_{l12}$ can be rearranged as ($c+1+b_1 + b_2 + b_3 =l $), \n\\begin{equation}\\label{Sl12-4}\n\\begin{split}\n& \\nabla_{I_{c+1}} \\hat N * \\nabla_{I_{b_1}} \\bar T \\phi * \\nabla_{I_{b_2}} \\bar T \\phi * \\nabla_{I_{b_3}} \\hat k * \\nabla^{I_l} E.\n\\end{split}\n\\end{equation}\nNow that, $c+b_1 + b_2 + b_3 =l -1 \\leq 2$, we can always apply $L^2, L^\\infty, L^\\infty, L^\\infty, L^2$ to the five factors of \\eqref{Sl12-4} and derive that \n\\begin{equation}\\label{estimate-Sl12-4}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l12}^4| \n \\lesssim \\mathcal{E}^{\\frac{3}{2}}_3 (g, t) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\\end{itemize}\n\nPutting {\\bf Case $S_{l1}$-I} and {\\bf Case $S_{l1}$-II} together, we have for $S_{l1},$\n\\begin{align}\n \\int_{\\Sigma_t} |S_{l1}| &\\lesssim \\left( t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (g, t) + \\mathcal{E}_l (g, t)\\right) \\mathcal{E}_4 (\\phi, t), \\quad 0 \\leq l \\leq 2, \\label{estimate-Sl1-l-leq-2}\\\\\n \\int_{\\Sigma_t} |S_{31}| &\\lesssim \\varepsilon^2 M^2 E_3(W, t) + t^{-1} \\mathcal{E}_4 (\\phi, t). \\label{estimate-Sl1-l=3}\n\\end{align}\n\n\nFor the magnetic part $S_{l2}$, \n\\begin{itemize}\n\\item {\\bf Case $S_{l2}$-I}: $a=0.$ Then $b=l.$ $S_{l2}$-I composes of \n\\begin{equation}\\label{def-S-l21-12}\n\\begin{split}\n S_{l21}^1= &N \\nabla_{I_{l}} \\left( \\nabla^2 \\phi * \\nabla \\phi \\right) * \\nabla^{I_{l}} H, \\\\\nS^2_{l21} =& N \\nabla_{I_{l}} \\left(\\hat k* \\nabla \\phi * \\bar T \\phi \\right) * \\nabla^{I_{l}} H.\n\\end{split}\n\\end{equation}\nNote that, $S^1_{l21}, S^2_{l21}$ possess the same structures as $S_{l11}^1, S_{l11}^4$ \\eqref{def-S-l11-1234} respectively if we replace $\\nabla^2 \\phi$ in $S^1_{l21}$ by $\\nabla \\bar T \\phi$, $\\nabla \\phi$ in $S^2_{l21}$ by $\\bar T \\phi$. In fact, the structures in $S^1_{l21}, S^2_{l21}$ is better, since they both have one more order of spatial derivative than $S_{l11}^1, S_{l11}^4$ respectively. Anyway, the analogous argument for $S_{l11}^1, S_{l11}^4$ can be applied to $S^1_{l21}, S^2_{l21}$ as well. Then \n\\begin{equation}\\label{estimate-Sl21-1-2}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l21}^1| + |S_{l21}^2|\n \\lesssim \\left( t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (g, t) + \\mathcal{E}_l (g, t)\\right) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\n\n\\item {\\bf Case $S_{l2}$-II}: $a\\geq 1.$ Letting $a =c+1, c \\geq 0,$ $S_{l2}$-II consists\n \\begin{equation}\\label{def-S-l22-12}\n\\begin{split}\n S_{l22}^1= & \\sum_{c+1+b=l}\\nabla_{I_{c+1}} \\hat N * \\nabla_{I_b} \\left( \\nabla^2 \\phi * \\nabla \\phi \\right) * \\nabla^{I_l} H, \\\\\nS^2_{l22} =& \\sum_{c+1+b=l} \\nabla_{I_{c+1}} \\hat N* \\nabla_{I_b} \\left( \\bar T \\phi * \\nabla \\phi * \\hat k \\right) * \\nabla^{I_l} H.\n\\end{split}\n\\end{equation}\nAgain, if replacing $\\nabla^2 \\phi$ in $S^1_{l22}$, $\\nabla \\phi$ in $S^2_{l22}$ by $\\nabla \\bar T \\phi, \\bar T \\phi$ respectively, one gets the same structures as $S_{l12}^1, S_{l12}^4$. Thus the estimate for $S_{l22}^1, S_{l22}^2$ will be just in an analogous to that for $S_{l12}^1, S_{l12}^4$. We obtain \n\\begin{equation}\\label{estimate-Sl22-12}\n\\begin{split}\n& \\int_{\\Sigma_t} |S_{l21}^1| + |S_{l22}^2| \\lesssim \\left( t^{-1} \\mathcal{E}_3 (g, t) + \\mathcal{E}^{\\frac{3}{2}}_3 (g, t)\\right) \\mathcal{E}_4 (\\phi, t).\n\\end{split}\n\\end{equation}\nThis is of course lower order than \\eqref{estimate-Sl21-1-2}.\n\\end{itemize} \nCombining the results for Case $S_{l2}$-I with Case $S_{l2}$-II, we have for $S_{l2},$\n\\begin{equation}\\label{estimate-S2}\n \\int_{\\Sigma_t} |S_{l2}| \\lesssim \\left( t^{-1} \\mathcal{E}^{\\frac{1}{2}}_l (g, t) + \\mathcal{E}_l (g, t)\\right) \\mathcal{E}_4 (\\phi, t).\n\\end{equation}\n\n\n\\section{The global existence}\\label{sec-global-existence}\n\\subsection{Closing the bootstrap argument}\\label{sec-close-b-t}\n\\subsubsection{For KG field and Weyl field}\nIn Lemma \\ref{lemma-improved-0-energy-KG} and Lemma \\ref{lemma-improved-l-energy-KG} and Lemma \\ref{prop-energy-estimate-Bianchi}, we have proven that, for some universal constant $C$, and some constant $C_M$ depending linearly on $M$,\n\\begin{equation*}\n\\begin{split}\nt E_0(\\phi,t) \\leq & C \\varepsilon^2 I^2_{0}(\\phi), \\\\\nt E_l(\\phi, t) \\leq&C \\left( \\varepsilon^2 I^2_4(\\phi) + \\varepsilon^3 M^3 \\right) t^{C_M \\varepsilon}, \\quad 1 \\leq l \\leq 4.\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\n\\begin{split}\nt^2 E_l(W,t) \\leq& C\\left( \\varepsilon^2 I^2_{l}(W)+ \\varepsilon^3 M^3 \\right), \\quad 0 \\leq l \\leq 2. \\\\\nt^2 E_3(W,t) \\leq& C\\left( \\varepsilon^2 I^2_{4}(\\phi) + \\varepsilon^2 I^2_{3}(W)+ \\varepsilon^3 M^3 \\right)t^{C_M \\varepsilon}.\n\\end{split}\n\\end{equation*}\nThen we choose $M$ satisfying $C( I^2_3(W) +I_4^2(\\phi) )\\leq \\frac{M^2}{4}$ and $\\varepsilon$ small enough such that $\\varepsilon CM \\leq \\frac{1}{4}$, $C_M \\varepsilon \\leq \\delta$. Hence, for $0\\leq i \\leq 4, 0 \\leq j \\leq 3,$\n\\begin{equation}\\label{eq-closing-b-t-KG-Weyl}\nt E_{i}(\\phi,t) \\leq \\frac{\\varepsilon^2 M^2}{2} t^{\\delta}, \\quad t^2 E_j(W,t) \\leq \\frac{\\varepsilon^2 M^2}{2} t^{\\delta},\n\\end{equation}\nwhich improve the bootstrap assumptions \\eqref{BT-KG}-\\eqref{BT-E-H}.\n\n\\subsubsection{For the second fundamental form}\\label{sec-closing-bt-k}\nWe will additionally take advantage of the transport equation for the second fundamental form to derive a bound for $\\|\\hat k\\|_{L^2(\\Sigma_t)}$. This helps to improve the weak bootstrap assumption for $\\hat k$ \\eqref{BT-k-N-L-infty}. \n\\begin{lemma}[Improvement for $\\hat k$]\\label{lem-hat-k-estimate-original-0}\nFixing $0<\\delta<\\frac{1}{6},$ we have for some universal constant $C$,\n\\begin{equation}\\label{eq-improved-hat-k-0}\n t^2 E_0(\\hat k, t) \\leq C \\varepsilon^2 I_0^2(\\hat k)+ C \\left( \\varepsilon I_0(\\phi) + \\varepsilon I_0(W) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) \\varepsilon M t^{\\delta}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nReminding the evolution equation for $\\hat k_{ij}$ \\eqref{eq-evolution-2},\nand noticing that\n\\begin{align*}\n\\bar R_{i\\bar T j \\bar T} \\hat k^{ij} \n=& \\left( E_{ij} - \\frac{1}{2}\\bar R_{ij} \\right) \\hat k^{ij} = \\left( E_{ij} - \\frac{1}{2} \\nabla_i \\phi \\nabla_j \\phi \\right) \\hat k^{ij},\n\\end{align*}\nwe multiply $\\hat k^{ij}$ and integrate on $\\Sigma_t,$ \n\\begin{equation}\\label{eq-transport-E0-hat-k-1} \n\\begin{split}\n\\partial_{\\bar \\tau} E_0(\\hat k, t) + 2 E_0(\\hat k, t) =&\\int_{\\Sigma_t} \\hat k^{ij} \\left( -2\\nabla_i \\nabla_j N + 2NE_{ij} \\right) \\\\\n& - \\hat N |\\hat k|^2 + 2N |\\hat k|^3-N \\nabla_i \\phi \\nabla_j \\phi \\hat k^{ij} .\n\\end{split}\n\\end{equation}\n\nTerms in the second line of \\eqref{eq-transport-E0-hat-k-1} are cubic, and hence easier.\nWe make use of $\\|\\hat k\\|_{L^\\infty}$ \\eqref{BT-k-L-infty-4-6} and $\\|\\hat N\\|_{L^\\infty}$ \\eqref{BT-k-N-L-infty}, then\n\\begin{align*}\n \\int_{\\Sigma_t} |\\hat N| |\\hat k|^2 + N |\\hat k|^3 &\\lesssim \\mathcal{E}^{\\frac{1}{2}}_1(g, t) E_0(\\hat k, t) \\lesssim \\varepsilon M t^{-1+2\\delta} E_0(\\hat k, t).\n\\end{align*}\nRewrite $\\nabla_i \\phi * \\nabla_j \\phi* \\hat k^{ij}$ as $(mt)^{-2} \\nabla_i (\\bar m (t) \\phi)* \\nabla_j (\\bar m (t) \\phi ) *\\hat k^{ij}$ and perform $L^4, L^4, L^2$ on the three factors\n\\begin{align*}\n\\int_{\\Sigma_t} | \\nabla_i \\phi \\nabla_j \\phi \\hat k^{ij}| \n \\lesssim & t^{-2} \\mathcal{E}_2(\\phi, t) E^{\\frac{1}{2}}_0(\\hat k, t) \\\\\n \\lesssim& \\varepsilon M t^{-1+\\delta} E_0(\\hat k, t) + \\varepsilon^3 M^3 t^{-5+\\delta}.\n\\end{align*}\n\n\nThe first line in \\eqref{eq-transport-E0-hat-k-1} involves quadratic terms. We will make use of the refined $\\mathcal{E}_2(\\hat N, t)$ \\eqref{N-improved-estimate-source}, and $E_0(W, t)$ \\eqref{eq-energy-estimate-Bianchi-l-leq-2} to linearize them:\n\\begin{align*}\n \\int_{\\Sigma_t} |\\nabla_i \\nabla_j N \\hat k^{ij}| \n &\\lesssim E^{\\frac{1}{2}}_2(\\hat N, t) E^{\\frac{1}{2}}_0(\\hat k, t) \\lesssim \\varepsilon I_0(\\phi) t^{-1} E^{\\frac{1}{2}}_0(\\hat k, t), \\\\\n\\int_{\\Sigma_t} |E_{ij} \\hat k^{ij}| \n&\\lesssim E^{\\frac{1}{2}}_0(W, t) E^{\\frac{1}{2}}_0(\\hat k, t) \\lesssim \\left( \\varepsilon I_0(W) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) t^{-1} E^{\\frac{1}{2}}_0(\\hat k, t).\n\\end{align*}\nNow, we substitute the bootstrap assumption for $E^\\frac{1}{2}_0(\\hat k, t)$ (see \\eqref{BT-k-N-L-infty}) to achieve\n\\begin{equation*}\n\\begin{split}\n\\partial_{\\bar \\tau} E_0(\\hat k, t) &+ 2 E_0(\\hat k, t) \\lesssim \\varepsilon M t^{-1+2\\delta} E_0(\\hat k, t) \\\\\n&+ \\left( \\varepsilon I_0(\\phi)+ \\varepsilon I_0(W) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right)\\varepsilon M t^{-2+\\delta}.\n\\end{split}\n\\end{equation*}\nChanging to $t$ and multiplying $t$ on the inequality, we achieve\n\\begin{equation}\\label{ode-K-1}\n\\begin{split}\n& \\partial_t (t^2 E_0(\\hat k, t)) \\lesssim \\varepsilon M t^{-2+2\\delta} (t^2 E_0(\\hat k, t)) \\\\\n&\\quad + \\left( \\varepsilon I_0(\\phi)+ \\varepsilon I_0(W) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right)\\varepsilon M t^{-1+\\delta}.\n\\end{split}\n\\end{equation}\nwhere $0<\\delta < \\frac{1}{6}$. By the Gr\\\"{o}nwall inequality, \n\\begin{equation*}\n\\begin{split}\n t^2 E_0(\\hat k, t) \\lesssim & \\exp(\\varepsilon M) \\left(\\varepsilon^2 I^2_0(\\hat k) + \\left( \\varepsilon I_0(\\phi) + \\varepsilon I_0(W) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) \\varepsilon M t^{\\delta} \\right).\n\\end{split}\n\\end{equation*}\n$\\varepsilon$ is chosen small enough such that $\\varepsilon M <1$, thus it implies \\eqref{eq-improved-hat-k-0}.\n\n\\end{proof}\n\n\n$\\hat k_{ij}$ being the trace free part of $k_{ij},$ satisfies the following elliptic system,\n\\begin{equation}\\label{div-curl-k-hat}\n\\begin{split}\n(\\text{div} \\hat k)_i &= - \\bar R_{\\bar T i},\\\\\n\\text{curl} \\hat k_{ij} &= - H_{ij}, \\\\\n\\text{tr} \\hat k &=0.\n\\end{split}\n\\end{equation}\nSee Appendix for the definition of div and curl. \n\n\\begin{lemma}[Improvement for derivative of $\\hat k$]\\label{lem-hat-k-estimate-original}\nFixing $0<\\delta<\\frac{1}{6},$ we have for $0 \\leq i \\leq 3$ and some universal constant $C$\n\\begin{equation}\\label{estimate-nabla-1-4-hat-k}\nt^2 E_{i+1}(\\hat k, t) \\leq C \\varepsilon^2 I_0^2(\\hat k)+ C\\left( \\varepsilon I_{3}(W) + \\varepsilon I_4(\\phi) + \\varepsilon^\\frac{3}{2} M^\\frac{3}{2} \\right) \\varepsilon M t^{\\delta}.\n\\end{equation} \n\\end{lemma}\n\n\\begin{proof}\nThe elliptic system \\eqref{div-curl-k-hat} gives the identity (c.f. Proposition 4.4.1 in \\cite{Christodoulou-K-93}), \n\\begin{equation}\\label{eq-div-curl-id-CK}\n\\int_{\\Sigma_t} |\\nabla \\hat k|^2 + 3 R_{mn} \\hat k^{im} \\hat k_{i}^{\\, n} - \\frac{R}{2} |\\hat k|^2 = \\int_{\\Sigma_t} |H_{ij}|^2 + \\frac{1}{2} |\\bar R_{\\bar T i}|^2.\n\\end{equation}\nNoticing the formula for $R$ \\eqref{Gauss-Ricci-trace-hat -k}, and the expansion \\eqref{R-imjn-E} for $R_{imjn}$, we also get\n\\begin{equation}\\label{R-ij-E}\nR_{ij} \\sim 1 \\pm \\hat k \\pm |\\hat k|^2 + E \\pm |\\mathcal{D} \\phi|^2, \\quad R \\sim 1 \\pm |\\hat k|^2 + E \\pm |\\mathcal{D} \\phi|^2.\n\\end{equation}\nUsing Cauchy-Schwarz inequality, we derive\n\\begin{equation}\\label{ineq-nabla-hat-k-source}\n\\begin{split}\n\\int_{\\Sigma_t} |\\nabla \\hat k|^2 \\lesssim & \\int_{\\Sigma_t} \\left(1+ |\\mathcal{D} \\phi|^2_{L^\\infty} + |\\hat k|_{L^\\infty} + |\\hat k|^2_{L^\\infty} +|E|_{L^\\infty} \\right) |\\hat k|^2 \\\\\n&+ \\int_{\\Sigma_t} |H|^2 + |\\bar T \\phi|^2_{L^\\infty} |\\nabla \\phi|^2.\n\\end{split}\n\\end{equation}\nThat is,\n\\begin{equation*}\n\\begin{split}\n\\int_{\\Sigma_t} |\\nabla \\hat k|^2 \\lesssim & \\left(1+ \\mathcal{E}_2(\\phi, t) + \\mathcal{E}^{\\frac{1}{2}}_2(g, t) + \\mathcal{E}_2(g, t) \\right) E_0(\\hat k, t) \\\\\n&+ E_0(W, t) +\\mathcal{E}_2(\\phi, t) \\mathcal{E}_0(\\phi, t).\n\\end{split}\n\\end{equation*}\nUsing the bootstrap assumption \\eqref{BT-KG}, the improved $E_0(W,t)$ \\eqref{eq-energy-estimate-Bianchi-l-leq-2}, we prove \\eqref{estimate-nabla-1-4-hat-k} with $i=0$, namely,\n\\begin{equation}\\label{ineq-nabla-hat-k-source-1}\n\\begin{split}\n\\int_{\\Sigma_t} t^2 |\\nabla \\hat k|^2 \\lesssim &t^2 E_0(\\hat k ,t)+ \\varepsilon^2 I^2_{0}(W) + \\varepsilon^3 M^3.\n\\end{split}\n\\end{equation}\n\nProceeding to the higher order cases, we wish to prove that for $0\\leq i \\leq 3$,\n\\begin{equation}\\label{estimate-nabla2-hat-k-i-induction}\n\\int_{\\Sigma_t} t^2 |\\nabla_{I_{i+1}} \\hat k|^2 \\lesssim t^2 E_0(\\hat k, t)+ \\left( \\varepsilon^2 I^2_{i}(W) + \\varepsilon^2 I^2_{4}(\\phi) + \\varepsilon^3 M^3 \\right) t^{C_M \\varepsilon}.\n\\end{equation}\nNow \\eqref{ineq-nabla-hat-k-source-1} shows that \\eqref{estimate-nabla2-hat-k-i-induction} holds for $i=0.$ Suppose \\eqref{estimate-nabla2-hat-k-i-induction} holds for $i\\leq l (l \\leq 2),$ we need to prove that \\eqref{estimate-nabla2-hat-k-i-induction} also holds for $i=l+1 (l \\leq 2).$ \n \nIn views of the commuting identity between $\\nabla_{I_l}$ and $\\Delta$ \\eqref{commuting-nabla-laplacian-l-Psi-i-ij}-\\eqref{Def-R-Psi-ij-commute-nabla-laplacian}, after an integration by parts, we calculate for $0 \\leq l,$ \n\\begin{align*}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat k|^2 & = \\int_{\\Sigma_t} |\\nabla_{I_l} \\Delta \\hat k|^2 \\pm \\mathcal{R}_{I_l}( \\hat k_{ij} ) * \\nabla_{I_l} \\Delta \\hat k \\pm \\mathcal{R}_{I_{l+1}} ( \\hat k_{ij} ) * \\nabla_{I_{l+1}} \\hat k.\n\\end{align*}\nApply the Cauchy-Schwarz inequality, then\n\\begin{align*}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat k|^2 & \\lesssim \\int_{\\Sigma_t} |\\nabla_{I_l} \\Delta \\hat k|^2 + \\big|\\mathcal{R}_{I_l}( \\hat k_{ij}) \\big|^2 + \\big|\\mathcal{R}_{I_{l+1}} ( \\hat k_{ij} )\\nabla_{I_{l+1}} \\hat k \\big|.\n\\end{align*}\nMeanwhile, \n\\begin{align*}\n\\nabla_{I_l} \\Delta \\hat k_{ij} &= \\nabla_{I_l} \\left( \\nabla^m \\bar R_{mi \\bar T j} - \\nabla_i \\bar R_{\\bar T j} +R_{i}{}^{\\!p} \\hat k_{pj} - R_i{}^{\\!p}{}_{\\!j}{}^{\\!q} \\hat k _{pq} \\right)\n\\end{align*}\nand $\\nabla^m \\bar R_{mi \\bar T j} = \\epsilon_{mi}{}^{\\!p} \\nabla^m H_{pj} - \\frac{1}{2} \\nabla_j \\bar R_{\\bar T i} +\\frac{1}{2} g_{ij} \\nabla^m \\bar R_{m \\bar T},$\ntherefore, \n\\begin{align*}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat k|^2 \\lesssim & \\int_{\\Sigma_t} |\\nabla_{I_{l+1}} H|^2 +|\\nabla_{I_{l+1}} \\left(\\bar R_{i \\bar T} \\right)|^2 \\\\\n & + \\big|\\mathcal{R}_{I_l} (\\hat k_{ij} ) \\big|^2 + \\big|\\mathcal{R}_{I_{l+1}} ( \\hat k_{ij}) \\nabla_{I_{l+1}} \\hat k \\big|.\n\\end{align*}\nNote that, $\\bar R_{\\bar T j} = \\nabla_j \\phi \\bar T \\phi + \\frac{\\bar m^2(t)}{2}\\phi^2 \\bar g_{\\bar T j},$ $ \\nabla_{I_l} (\\bar T^\\mu \\bar g_{\\mu i}) =0,$ then $\\nabla_{I_l} \\bar R_{\\bar T j} = \\nabla_{I_l} \\left( \\nabla_j \\phi \\bar T \\phi \\right).$ And in views of the expansion for $R_{imjn}$ \\eqref{R-imjn-E},\n\\begin{align*}\n \\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat k|^2 \\lesssim &\\int_{\\Sigma_t} | \\nabla_{I_{l+1}} H|^2 + \\sum_{a+b =l+1} |\\nabla_{I_{a+1}} \\phi \\nabla_{I_b} \\bar T \\phi|^2\\\\\n& \\quad \\quad \\quad \\quad + RK_1 + RK_2 + RK_3 + RK_4,\n\\end{align*}\nwhere $RK_1 - RK_4$ are defined as below,\n\\begin{align*}\nRK_1 =& \\left( 1+ |\\hat k|^2 + |\\hat k|^4 + |E|^2 + |\\mathcal{D}\\phi|^4 \\right) |\\nabla_{I_l} \\hat k|^2, \\\\\n RK_2 =& \\sum_{a+1+b=l} \\big|\\nabla_{I_{a+1}} \\left( |\\hat k| \\pm |\\hat k|^2 \\pm E \\pm |\\mathcal{D} \\phi |^2 \\right)\\big|^2 |\\nabla_{I_b} \\hat k|^2, \\\\\n RK_3 =& \\left( 1+ |\\hat k| + |\\hat k|^2 + |E| + |\\mathcal{D}\\phi|^2 \\right) |\\nabla_{I_{l+1}} \\hat k|^2, \\\\\n RK_4 =& \\sum_{a+1+b=l+1} \\big|( t\\nabla)_{I_{a+1}} \\left( |\\hat k| \\pm |\\hat k|^2 \\pm E \\pm |\\mathcal{D} \\phi|^2 \\right) \\big| | \\nabla_{I_b} \\hat k \\nabla_{I_{l+1}} \\hat k|.\n\\end{align*}\nNotice that $l \\leq 2$.\nFirst of all, \n\\begin{equation}\\label{esti-hat-k-l-H}\n\\int_{\\Sigma_t} | \\nabla_{I_{l+1}} H|^2 \\lesssim \\mathcal{E}_{l+1}(W, t), \\quad l \\leq 2. \n\\end{equation}\nViewing $[\\frac{l+1}{2}] +2 \\leq 3 (l \\leq 2)$, \n\\begin{equation}\\label{esti-hat-k-l-bar-R}\n\\begin{split}\n &\\int_{\\Sigma_t} \\sum_{a+b =l+1} |\\nabla_{I_{a+1}} \\phi * \\nabla_{I_b} \\bar T \\phi|^2\n \\lesssim \\mathcal{E}_3(\\phi, t) \\mathcal{E}_{l+1}(\\phi, t).\n\\end{split}\n\\end{equation}\n\n\nFor $RK_1$ and $RK_3,$ we apply $L^\\infty, L^2, L^2$ to derive\n\\begin{equation}\\label{esti-hat-k-l-RK13}\n \\int_{\\Sigma_t} RK_1 + RK_3\n \\lesssim \\mathcal{E}_l(\\hat k, t) + \\mathcal{E}_{l+1}(\\hat k, t).\n\\end{equation}\n\nFor $RK_2,$ since $1 \\leq a+1 \\leq l \\leq 2, b \\leq l-1 \\leq 1,$ we can always apply $L^4$ on each of the four factors (such as $ |\\nabla_{I_{a+1}} E|^2 | \\nabla_{I_b}\\hat k|^2 $) or $L^6$ on each of the six factors (such as $ |\\nabla_{I_{a+1}} (\\mathcal{D} \\phi)^2|^2 | \\nabla_{I_b}\\hat k|^2$). Then \n\\begin{equation}\\label{esti-hat-k-l-RK2}\n\\int_{\\Sigma_t} RK_2 \\lesssim \\left( \\mathcal{E}_{l+1}(g,t) + \\mathcal{E}^2_{l+1}(\\phi,t) \\right) \\mathcal{E}_{l}(\\hat k,t). \n\\end{equation}\n\nFor $RK_4$, the argument is similar to that for $RK_2$. \n\\begin{itemize}\n\\item If $a+1=l+1 \\leq 3,$ then $b=0,$ we get $$\\big|\\nabla_{I_{l+1}} \\left( |\\hat k| \\pm |\\hat k|^2 \\pm E \\pm |\\mathcal{D} \\phi|^2 \\right) \\big| * |\\hat k| |\\nabla_{I_{l+1}} \\hat k|.$$ For three factors (such as $\\nabla_{I_{l+1}} E * \\hat k * \\nabla_{I_{l+1}} \\hat k$), we apply $L^2, L^\\infty, L^2$; For four factors (such as $\\nabla_{I_{l+1}} \\hat k* \\hat k * \\nabla_{I_{l+1}} \\hat k$), we apply $L^2, L^\\infty, L^\\infty, L^2$ or $L^\\infty, L^2, L^\\infty, L^2$, since $[\\frac{l+1}{2}] +2 \\leq 3(l\\leq 2)$. Then we can deduce the bound\n\\begin{align*}\n \\lesssim &\\left( \\mathcal{E}^{\\frac{1}{2}}_{l+1}(W,t) + \\mathcal{E}^{\\frac{1}{2}}_{l+1}(\\phi,t) \\mathcal{E}^{\\frac{1}{2}}_{3}(\\phi,t) \\right) \\mathcal{E}^{\\frac{1}{2}}_{2}(\\hat k,t) E^{\\frac{1}{2}}_{l+1}(\\hat k,t) \\\\\n&+ \\left( \\mathcal{E}^{\\frac{1}{2}}_{2}(\\hat k,t) + \\mathcal{E}_{3}(\\hat k,t) \\right) \\mathcal{E}_{l+1}(\\hat k,t).\n\\end{align*}\n\n\\item If $1 \\leq a+1 \\leq l \\leq 2,$ then $1 \\leq b \\leq l \\leq 2,$ we apply $L^4, L^4, L^2$ or $L^6, L^6, L^6, L^2$ to derive the bound\n\\begin{align*}\n& \\left( \\mathcal{E}^\\frac{1}{2}_{l+1}(W,t)+ \\mathcal{E}^\\frac{1}{2}_{l+1}(\\hat k ,t) \\right) \\mathcal{E}_{l+1}(\\hat k,t) \\\\\n& + \\left( \\mathcal{E}_{l+1}(\\phi,t) + \\mathcal{E}_{l+1}(\\hat k,t) \\right) \\mathcal{E}_{l+1}(\\hat k,t).\n\\end{align*}\n\\end{itemize}\nIn summary, we have by Cauchy-Schwartz inequality,\n\\begin{equation}\\label{esti-hat-k-l-RK4}\n\\begin{split}\n\\int_{\\Sigma_t} RK_4 \\lesssim & \\left( \\mathcal{E}^\\frac{1}{2}_{l+1}(g,t) + \\mathcal{E}_{l+1}(\\phi,t) \\right) \\mathcal{E}_{l+1}(\\hat k,t) \\\\\n& + \\mathcal{E}_{l+1}(W,t) + \\mathcal{E}_{3}(\\phi,t)\\mathcal{E}_{l+2}(\\phi,t).\n \\end{split}\n\\end{equation}\n\nFrom \\eqref{esti-hat-k-l-H}-\\eqref{esti-hat-k-l-RK4}, we have for $ l \\leq 2,$\n\\begin{equation}\\label{eq-energy-hat-k-source}\n\\begin{split}\n\\int_{\\Sigma_t} |\\nabla_{I_{l+2}} \\hat k|^2 \\lesssim & \\mathcal{E}_{l+1}(W,t)+ \\mathcal{E}_{l+1}(\\hat k,t) + \\mathcal{E}^2_4(\\phi, t).\n \\end{split}\n\\end{equation}\nBy induction and combining the improvement for $W, \\phi$ with Lemma \\ref{lem-hat-k-estimate-original-0}, we proved \\eqref{estimate-nabla2-hat-k-i-induction} and hence \\eqref{estimate-nabla-1-4-hat-k}.\n\\end{proof}\n\n\nWe choose $M$ additionally satisfying $C\\left( I_0(\\hat k) + I_3(W) + I_4(\\phi) \\right) \\leq M\/4$ and $\\varepsilon$ sufficiently small such that $C \\varepsilon^\\frac{1}{2} M^\\frac{1}{2} \\leq 1\/4$, then \\eqref{eq-improved-hat-k-0} and \\eqref{estimate-nabla-1-4-hat-k} imply\n\\begin{equation}\\label{eq-energy-hat-k-improved}\nt^2 E_{i}(\\hat k,t ) \\leq \\frac{1}{2}\\varepsilon^2 M^2 t^{\\delta}, \\quad 0 \\leq i \\leq 4,\n\\end{equation}\nwhich improves the bootstrap assumption for $\\hat k$ in \\eqref{BT-k-N-L-infty}.\n\n\n\n\n\\subsubsection{Closing the bootstrap assumption for lapse}\nBy the Sobolev inequalities, the improved energy for $\\hat N$ \\eqref{N-improved-estimate-source} (see Corollary \\ref{lem-N-ineq-source-improved-2}) implies that \n \\begin{equation}\\label{BT-N}\n\\|\\hat N\\|_{L^\\infty} \\leq C \\varepsilon I_0(\\phi) t^{-1},\n\\end{equation}\nfor some universal constant $C$. We choose $C^2I^2_0(\\phi) \\leq M^2\/2$, then\n \\begin{equation}\\label{ineq-hat-k-improved-infty}\n \\|\\hat N\\|^2_{L^\\infty} \\leq \\frac{\\varepsilon^2 M^2}{2} t^{-2 + 4\\delta}, \\quad 0<\\delta < \\frac{1}{6},\n\\end{equation}\n which improves the bootstrap assumption for $\\|\\hat N\\|_{L^\\infty}$ in \\eqref{BT-k-N-L-infty}.\n \n \n \nWe are in a position to state the global existence theorem.\n\\begin{theorem}\\label{main-thm-global}\nLet $(\\Sigma, g_0, k_0, \\phi_0, \\phi_1)$ be CMC data for the EKG equations \\eqref{eq-Einstein-source}-\\eqref{def-energy-Mom-kg}, \\eqref{eq-kg} (also see the $1+3$ forms \\eqref{eq-1+3-bianchi-T-E}-\\eqref{eq-1+3-bianchi-T-H} and \\eqref{eq-rescale-kg-1+3-0}). Assume that $\\Sigma$ is a compact hyperbolic $3$-manifold without boundary and $\\text{tr}_{g_0} k_0 = -3$, with initial time $t_0 > \\max \\{9 m^{-1}, 1\\}$. Suppose that the initial data verifies the smallness condition: there is a constant $\\varepsilon_0 >0$ such that for all $\\varepsilon \\leq \\varepsilon_0$, \n\\begin{equation}\\label{initial-smallness}\nt_0 \\mathcal{E}_4(\\phi, t_0) + t_0^2 \\mathcal{E}_3(W, t_0) + t_0^2 \\mathcal{E}_4(\\hat k, t_0) \\leq \\varepsilon^2.\n\\end{equation}\nThen there exists a unique global solution $(W, \\phi)$ for $t \\geq t_0$ with the property that for some universal constant $C$ and $0<\\delta < \\frac{1}{6},$\n\\begin{equation}\\label{stronger-estimate-closing-bt}\nt \\mathcal{E}_4(\\phi, t) + t^2 \\mathcal{E}_3(W, t) \\lesssim \\varepsilon^2 t^{C \\varepsilon}, \\quad t^2 \\mathcal{E}_4(\\hat k, t) \\lesssim \\varepsilon^2 t^{\\delta}.\n\\end{equation}\n\\end{theorem}\nFinally, we can follow the argument presented in \\cite{A-M-04} (Theorem 6.2) to prove the geodesic completeness.\n \n \\begin{remark}\\label{re-asympototic-state}\nLet us the explore the asymptotic state of the rescaled metric $g$.\n As we can see from the Gauss equations \\eqref{Gauss-Riem-hat-k}, the decay estimates for $\\hat k, \\mathcal{D} \\phi, H$ yield that\n \\begin{align*}\n & |R_{imjn} + \\frac{1}{2}(g \\odot g) _{imjn}| \\lesssim \\varepsilon t^{-4+\\delta}.\n\\end{align*}\nThis shows that the sectional curvature $K$ of $g$ satisfies $K \\rightarrow -1.$\n\\end{remark}\n\\begin{remark}\\label{re-initial-data}\nBy the Gauss-Codazzi equation on the initial slice, the smallness condition for the data \\eqref{intro-initial-data} given in Theorem \\ref{main-thm-into} will enforce \\eqref{initial-smallness}.\n\\end{remark}\n \n \n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{section label}\nA system is called reciprocal when signal transmission between any two ports is independent with respect to propagation direction. Signal transmission between transmitter and receiver can be represented in terms of scattering ($S$) as $S^T = S$). In our daily life application such as antenna, passive electrical circuits, and components are reciprocal. Non-reciprocal systems, the opposite of reciprocal, exhibit different received-transmitted field ratios when the source and observation points interchange. Non-reciprocal components such as circulators, isolators, and gyrators play a pivotal role in many microwave wireless applications, including high-power transmitters, FDD system, emerging quantum computing and readout, defense and satellite communication, to name a few \\cite{Kord, caloz2018nonreciprocity}.\n\nSince 1950, there are significant amount of researches on non-reciprocity and over time different topologies have been developed to achieve nonreciprocity in the presence of permanent magnet, using nonreciprocal component, time modulation, parametric amplifier and so on \\cite{Kord, meta, Kodera1, alu_nonreciprocity, Alex}. Advances in solid state technology and significant research effort on artificially engineered material, known as metamaterial is to induce customized properties in a material that originally does not exist, have initiated a prime interest on achieving \"magnet-free\" non-reciprocity, where the time reversal symmetry can be broken in absence of static magnets. Over the past decade, research into metamaterials has been extended to a search for real-world applications, leading to the concept of meta-devices, defined as metamaterial-based devices that can operate in an active manner \\cite{Xiao_2020}. This article will mostly aim on the computational approach of designing nonreciprocal metamaterial leading to practical realization of co-simulation and machine learning based optimization. The proposed design in this article is the extension of the transistor based design first proposed by Toshiro et al. \\cite{Kodera2, ULMKodera} and later in our previous articles a nonreciprocal metasurface was investigated with a precise electronic tunability features \\cite{meta, Swadesh} as well as analytical modelling of polarization rotation in a nonreciprocal media. \\cite{swadeshmathpaper}. \n\nThe article is organized as follows. In Section II, an insight to the mechanism of floquet mode analysis and corresponding analytical approach will be provided, in Section III, the principle of ferromagnetic resonance, and in section IV, the design approaches and process flow of our nonreciprocal devices will be described. At last in section V our discussion will be mostly focused on simulated performance, benchmarking reciprocal and nonreciprocal design, highlighting major challenges, limitation, and future works. \n\n\\section{Floquet Mode Analysis and Mathematical Modeling}\n\\label{sec:formulation}\n\nIn this section, based on floquet mode analysis, a mathematical model is presented to characterize the unit cell of our designed metasurface. The Floquet port is used with planar-periodic structures. The analysis of the periodic structure is accomplished by analyzing a unit cell and master and slave boundaries are used to model a unit cell of a repeating\nstructure. The fields on the slave surface are constrained to be identical to those on the master surface, with phase shift. A coordinate system must be identified on the master and slave boundary to identify point-to-point correspondence and master and slave surfaces must be of identical shapes and sizes. A set of modes called floquet modes represent the fields on the port boundary. Fundamentally, Floquet modes can be assumed as plane waves where propagation direction is set by the frequency, and geometry of periodic structures. To simplify the discussion, a 3D structure has been developed and 1st 2 modes have been taken into consideration to propagate through the metamaterial unit cell. Floquet boundary at z direction has been calculated accordingly. \n\n\n\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.40\\textheight]{floquet_port.png}\n\\caption{Geometry of proposed metamaterial unit cell. Primary and secondary master and slave boundaries form the side walls of unit cell. In both ports floquet environment scattered and incident wave is shown with arrow direction}\n\\label{fig:floquet_port}\n\\end{figure}\n\nFigure \\ref{fig:floquet_port} shows the proposed metamaterial unit cell in the floquet port environment. $a_{mn}$ and $b_{mn}$ are the incident wave from the floquet port 1 and floquet port 2, respectively. On the other hand $c_{mn}$ and $d_{mn}$ are the scattered wave. In a normal incidence case where the 1st 2 modes, namely TE and TM is evaluated and can be expressed in S matrix. In Ansys HFSS simulation, floquet mode has been analyzed where transverse electric (TE) modes are y polarized and transverse magnetic (TM) modes are x polarized.\n\n\\begin{equation}\n\\left[ \n\\begin{array}{c}\n\\mathbf{c}_{mn}^{TE} \\\\ \n\\mathbf{c}_{mn}^{TM} \\\\ \n\\mathbf{d}_{mn}^{TE} \\\\ \n\\mathbf{d}_{mn}^{TM}%\n\\end{array}%\n\\right] =\\left[ \n\\begin{array}{cccc}\n\\mathbf{S}_{11}^{TE,TE} & \\mathbf{S}_{11}^{TE,TM} & \\mathbf{S}_{12}^{TE,TE}\n& \\mathbf{S}_{12}^{TE,TM} \\\\ \n\\mathbf{S}_{11}^{TM,TE} & \\mathbf{S}_{11}^{TM,TM} & \\mathbf{S}_{12}^{TM,TE}\n& \\mathbf{S}_{12}^{TM,TM} \\\\ \n\\mathbf{S}_{21}^{TE,TE} & \\mathbf{S}_{21}^{TE,TM} & \\mathbf{S}_{22}^{TE,TE}\n& \\mathbf{S}_{22}^{TE,TM} \\\\ \n\\mathbf{S}_{21}^{TM,TE} & \\mathbf{S}_{21}^{TM,TM} & \\mathbf{S}_{22}^{TM,TE}\n& \\mathbf{S}_{22}^{TM,TM}%\n\\end{array}%\n\\right] \\left[ \n\\begin{array}{c}\n\\mathbf{a}_{mn}^{TE} \\\\ \n\\mathbf{a}_{mn}^{TM} \\\\ \n\\mathbf{b}_{mn}^{TE} \\\\ \n\\mathbf{b}_{mn}^{TM}%\n\\end{array}%\n\\right],\n\\label{eq: matrix 1}\n\\end{equation}\nwhere, $\\mathbf{c}_{mn}^{TE}$ and $\\mathbf{a}_{mn}^{TE}$ defines the scattered and incident field respectively. Besides, $\\mathbf{S}_{ij}^{pol1,pol2}$, Where, i (pol1) and j (pol2) represents incident and received port (polarization) \nrespectively. For example, $\\mathbf{S}_{11}^{TE,TE}$ is the co polarized reflection case or in other words incident wave is TE(y) polarized from port 1 and received wave TE(y) polarized to port 1. Similarly, $\\mathbf{S}_{21}^{TM,TE}$ defines a cross polarization transmission case. The above equation \\ref{eq: matrix 1} can be written in a short form as below in terms of floquet modes. \n\n\\begin{equation}\n\\left[ \n\\begin{array}{c}\n\\mathbf{c}_{mn}^{TE} \\\\ \n\\mathbf{c}_{mn}^{TM} \\\\ \n\\mathbf{d}_{mn}^{TE} \\\\ \n\\mathbf{d}_{mn}^{TM}%\n\\end{array}%\n\\right] =[\\mathbf{S}]\\cdot \\left[ \n\\begin{array}{c}\n\\mathbf{a}_{mn}^{TE} \\\\ \n\\mathbf{a}_{mn}^{TM} \\\\ \n\\mathbf{b}_{mn}^{TE} \\\\ \n\\mathbf{b}_{mn}^{TM}%\n\\end{array}%\n\\right],\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{S}=\\left[ \n\\begin{array}{cccc}\nS(\\text{FP}1:1,\\text{FP}1:1) & S(\\text{FP}1:1,\\text{FP}1:2) & S(\\text{FP}1:1,%\n\\text{FP}2:1) & S(\\text{FP}1:1,\\text{FP}2:2) \\\\ \nS(\\text{FP}1:2,\\text{FP}1:1) & S(\\text{FP}1:2,\\text{FP}1:2) & S(\\text{FP}1:2,%\n\\text{FP}2:1) & S(\\text{FP}1:2,\\text{FP}2:2) \\\\ \nS(\\text{FP}2:1,\\text{FP}1:1) & S(\\text{FP}2:1,\\text{FP}1:2) & S(\\text{FP}2:1,%\n\\text{FP}2:1) & S(\\text{FP}2:1,\\text{FP}2:2) \\\\ \nS(\\text{FP}2:2,\\text{FP}1:1) & S(\\text{FP}2:2,\\text{FP}1:2) & S(\\text{FP}2:2,%\n\\text{FP}2:1) & S(\\text{FP}2:2,\\text{FP}2:2)%\n\\end{array}%\n\\right]\n\\label{eq: matrix2}\n\\end{equation}\nwhere FP is the short form of floquet port used here in eq. \\ref{eq: matrix2}. In summary, FP $p:u$, FP $q:u$ means $S_{pq}$ for $u$ polarized as incident and received wave define co polarization. Similarly, FP $p:v$, FP $q:u$ means $S_{pq}$ for incident mode as u and received wave as v polarized. In this explanation, $p$ and $q$ were used to mention port number and $v$ and $u$ to mention polarization type. \n\n\n\n\\begin{equation}\n\\left[ \n\\begin{array}{c}\n\\mathbf{c}_{mn}^{TE} \\\\ \n\\mathbf{c}_{mn}^{TM} \\\\ \n\\mathbf{d}_{mn}^{TE} \\\\ \n\\mathbf{d}_{mn}^{TM}%\n\\end{array}%\n\\right] =\\left[ \n\\begin{array}{cccc}\n\\mathbf{R}_{yy}^{(1)} & \\mathbf{R}_{yx}^{(1)} & \\mathbf{T}_{yy}^{(2)} & \\mathbf{T}%\n_{yx}^{(2)} \\\\ \n\\mathbf{R}_{xy}^{(1)} & \\mathbf{R}_{xx}^{(1)} & \\mathbf{T}_{xy}^{(2)} & \\mathbf{T}%\n_{xx}^{(2)} \\\\ \n\\mathbf{T}_{yy}^{(1)} & \\mathbf{T}_{yx}^{(1)} & \\mathbf{R}_{yy}^{(2)} & \\mathbf{R}%\n_{yx}^{(2)} \\\\ \n\\mathbf{T}_{xy}^{(1)} & \\mathbf{T}_{xx}^{(1)} & \\mathbf{R}_{xy}^{(2)} & \\mathbf{R}%\n_{xx}^{(2)}%\n\\end{array}%\n\\right] \\left[ \n\\begin{array}{c}\n\\mathbf{a}_{mn}^{TE} \\\\ \n\\mathbf{a}_{mn}^{TM} \\\\ \n\\mathbf{b}_{mn}^{TE} \\\\ \n\\mathbf{b}_{mn}^{TM}%\n\\end{array}%\n\\right]\n\\label{eq:matrix3}\n\\end{equation}\n\nEquation \\ref{eq:matrix3} has been represented in terms of reflection and transmission for the easier understanding of the floquet mode analysis. where for R\/T matrix, superscript describes the received port number and subscript\ndescribes the polarization. e.g. $T_{xx}^{1}$ equals the wave that is being\npropagated from port 2 and received at port 1, for generalized\nunderstanding, this can be mapped with $S_{12}$. In general the reflected and transmitted fields consist of both x- and y-\npolarized components. Transmission field $T_{xy}$ can be defined as the\nratio of the transmitted field with polarization $x$ to the incident field with polarization $y$ Based on same concept, $R_{xy}$ can be defined as the ratio of the reflected field with polarization $x$ to the incident field with polarization $y$. \n\nThe incident and transmitted fields are related through Jones transmission\nmatrix in cartesian basis as below \\cite{PhysRevA.82.053811, swadeshmathpaper}\n\n\\begin{equation}\n\\mathbf{T}=\\left[ \n\\begin{array}{cc}\n\\mathbf{T}_{xx} & \\mathbf{T}_{xy} \\\\ \n\\mathbf{T}_{yx} & \\mathbf{T}_{yy}%\n\\end{array}%\n\\right]\n\\end{equation}\n\nThe transmission matrix in linear basis can be mapped to transmission matrix\nin circular basis as below\n\n\\begin{equation}\n\\mathbf{T}_{CP}=\\left[ \n\\begin{array}{cc}\n\\mathbf{T}_{\\circlearrowright \\circlearrowright } & \\mathbf{T}%\n_{\\circlearrowright \\circlearrowleft } \\\\ \n\\mathbf{T}_{\\circlearrowleft \\circlearrowright } & \\mathbf{T}%\n_{\\circlearrowleft \\circlearrowleft }%\n\\end{array}%\n\\right] =\\frac{1}{2}\\left[ \\mathbf{L}\\right]\n\\end{equation}\n\n\\begin{equation}\n\\mathbf{T}_{CP}=\\left[ \n\\begin{array}{cc}\n\\mathbf{T}_{\\circlearrowright \\circlearrowright } & \\mathbf{T}%\n_{\\circlearrowright \\circlearrowleft } \\\\ \n\\mathbf{T}_{\\circlearrowleft \\circlearrowright } & \\mathbf{T}%\n_{\\circlearrowleft \\circlearrowleft }%\n\\end{array}%\n\\right] =\\frac{1}{2}\\left[ \\mathbf{L}\\right] =\\frac{1}{2}\\left[ \n\\begin{array}{cc}\nT_{xx}+T_{yy}+i(T_{xy}-T_{yx}) & T_{xx}-T_{yy}-i(T_{xy}+T_{yx}) \\\\ \nT_{xx}-T_{yy}+i(T_{xy}\\text{ }+\\text{ }T_{yx}) & \nT_{xx}+T_{yy}-i(T_{xy}-T_{yx})%\n\\end{array}%\n\\right]\n\\end{equation}\n\nBased on our above discussion, $\\mathbf{T}$ matrix in\ncircular basis can be generalized as below\n\n\\begin{eqnarray}\n\\mathbf{T}_{\\circlearrowright \\circlearrowright } &=&\\text{Incident RHCP to\nrecieved RHCP} \\\\\n\\mathbf{T}_{\\circlearrowright \\circlearrowleft } &=&\\text{Incident LHCP to\nrecieved RHCP} \\\\\n\\mathbf{T}_{\\circlearrowleft \\circlearrowright } &=&\\text{Incident RHCP to\nrecieved LHCP} \\\\\n\\mathbf{T}_{\\circlearrowleft \\circlearrowleft } &=&\\text{Incident LHCP to\nrecieved LHCP}\n\\end{eqnarray}\nwhere RHCP\/LHCP stands for right\/left handed circular polarization.\nOnce the transmission matrix in circular or linear basis is achieved, Faraday rotation and ellipticity can be calculated from there. Ellipticity, $\\delta _{F}$ is a measure of the polarization state of an\nelectromagnetic wave and it varies from $+1$ to $-1. $In terms of RHCP and\nLHCP, this can be written as%\n\\begin{equation}\n\\delta _{F}=\\frac{\\mid T_{\\circlearrowright \\circlearrowright }\\mid -\\mid\nT_{\\circlearrowleft \\circlearrowleft }\\mid }{\\mid T_{\\circlearrowright\n\\circlearrowright }\\mid +\\mid T_{\\circlearrowleft \\circlearrowleft }\\mid }\n\\end{equation}\nTherefore, from above equation, if $\\delta _{F}= +1$, wave is RHCP and $\\delta\n_{F}=-1$, the wave is LHCP, whereas, $\\delta _{F}= 0$ means Linear polarized wave wave.\n\nFaraday rotation can be calculated from the equation below \\cite{Lax, meta}\n\n\\begin{equation}\n\\theta _{F}=\\frac{1}{2}\\tan ^{-1}(\\frac{T_{\\circlearrowleft \\circlearrowleft\n}}{T_{\\circlearrowright \\circlearrowright }})\n\\end{equation}\n\nSimilarly, for ellipticity for a reflection based metasurface and Kerr rotation can be calculated similarly following the same steps as described above.\n\\begin{equation}\n\\delta _{K}=\\frac{\\mid R_{\\circlearrowright \\circlearrowright }\\mid -\\mid\nR_{\\circlearrowleft \\circlearrowleft }\\mid }{\\mid R_{\\circlearrowright\n\\circlearrowright }\\mid +\\mid R_{\\circlearrowleft \\circlearrowleft }\\mid }\n\\end{equation}\n\n\\begin{equation}\n\\theta _{K}=\\frac{1}{2}\\tan ^{-1}(\\frac{R_{\\circlearrowleft \\circlearrowleft\n}}{R_{\\circlearrowright \\circlearrowright }})\n\\end{equation}\n\n\\section{Ferromagnetic Resonance}\nThe concept related to the motion of the magnetic dipole in the presence of the magnetic field and a self-consistent RF magnetic field represents the principle idea of ferromagnetic resonance \\cite{Lax}. Microwave magnetism in a ferrite material is\nbased on the precession of the magnetic dipole moments which arise from unpaired electron\nspins along the axis of an externally applied static magnetic bias.\nFor resonance condition, it is necessary to apply the RF magnetic field (the microwave field) perpendicular to the dc field. The quantum mechanical phenomena can be described from Landau-Lifshitz-Gilbert equation as \\cite{Lax,Landau, Swadesh} \n\n\\begin{equation}\n\\frac{d\\mathbf{M}}{dt} = \\gamma (\\mathbf{M}\\times \\mathbf{H})-\\frac{\\alpha }{\\left | M \\right |}\\mathbf{M}\\times \\frac{d\\mathbf{M}}{dt},\n\\label{eq:gyro_5}\n\\end{equation}\nwhere $\\alpha$ is the Gilbert damping term, $\\gamma$ is the gyromagnetic ratio, $\\mathbf{M}$ = $\\textbf{i}_{z}$${M}_{0}$ + $\\textbf{m}$$e^{j\\omega t}$, and $\\mathbf{H}$ = $\\textbf{i}_{z}$${H}_{i}$ + $\\textbf{h}$$e^{j\\omega t}$, where $\\textbf{i}_{z}$ is the unit vector in the z direction, $\\textbf{m}$, and $\\textbf{h}$ are RF quantities, and ${H}_{i}$ is the total internal dc magnetic field as shown in Fig. \\ref{fig:E-spin}. The nearer the frequency of the microwave field to the natural precession frequency, the greater will be the energy absorbed by the spins \\cite{Lax}. \nThe longitudinal (z) component does not contribute to precession and therefore, to magnetism. Since, $\\textbf{H}_{0}$ $\\parallel$ $\\mathbf{\\widehat{z}}$, the z-component of $\\textbf{M}$ produced by $\\textbf{H}_{t}^{RF}$ would lead to $[M_{z}^{RF}(\\mu _{0}{H}+H^{RF})](\\widehat{\\mathbf{z}}\\times \\widehat{{\\mathbf{z}}})=0.$ Therefore, the only torque being produced by the transverse component at x-y plane can be stated as $(M_{t}^{RF}\\mu _{0}{H})(\\widehat{\\mathbf{t}}\\times \\widehat{\\mathbf{z}})\\neq 0.$\n\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.15\\textheight]{E-spin.png}\n\\caption{Magnetic dipole precession, arising from electron spinning in a ferrite material about the z axis of an externally applied static magnetic bias field, $H_0$, with effective unidirectional\ncurrent loops around z axis and magnetic dipole moment $m_\\rho$}.\n\\label{fig:E-spin}\n\\end{figure}\n\nThe magnetization vector described above has a singularity, when $\\omega$ = $\\omega_{0}$ = $\\gamma H_{0}$, which can be defined as the resonance condition. Resonance can be achieved by varying the operating frequency or the applied field until the precession frequency equals the microwave frequency.\nPermeability tensor of ferrite material can be represented as \\cite{Polder}\n\n\\begin{equation}\n\\overrightarrow{\\mathbf{\\mu}}=\\mu_0\n\\begin{bmatrix}\n\\mu & -jk & 0\\\\ \njk & \\mu & 0\\\\ \n0 & 0 & 1\n\\end{bmatrix},\n\\label{eq:gyro_8}\n\\end{equation}\nwhere $\\mu$ = $1$ +$\\frac{\\omega_0 \\omega_M}{\\omega_0^2-\\omega^2}$, $jk$ = $\\frac{j\\omega \\omega_M}{\\omega_0^2-\\omega^2}$, and $\\omega_M$ = $\\gamma 4\\pi M_0$.\n\n\\section{Nonreciprocal Device Design}\nActive devices have been used in the design of magnet-free non-reciprocal metamaterials. For example, a transistor amplifier can be viewed as an isolator that amplifies\nthe input signal in one direction and blocks it in the reverse direction and very popular in RF microwave devices, metamaterial and antenna design \\cite{swadeshring, Rabus2020}. In a series of publications between 2011 and 2019, Kodera et al. explored the unidirectional properties of transistors in the design of non-reciprocal metamaterials \\cite{Kodera1,Kodera2,Kodera3, Kodera4}. A ring resonator supports two counter propagating modes with the same resonance frequency, a result of reciprocity. However, when unidirectional component such as transistor is used in the ring gap, one of the modes is blocked and the ring loaded with transistor resonates only for one type of circularly polarized waves \\cite{Kord, meta}.\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.30\\textheight]{rotating_vector.png}\n\\caption{Non-reciprocal gyrotropy. Rotating dipole moment in (a) the proposed metamaterial structure consisting of slot-ring resonator loaded with active components (transistor). (b) travelling wave along the ring at four quarter-period spaced time instants. (c) Rotating electric field in the slot-ring and the corresponding electric dipole moment due to vectorial magnetic field pointed outward and inward. (d) Dimensions of the ring resonator (e) Unit super-cell of a 3 by 3 super-cell board consisting of 4 rings in a $90^0$-symmetric configuration (Top view). (f) Bias connection layout of the complete board (Bottom view) \\cite{meta, Swadesh}}\n\\label{fig:rotating_vector}\n\\end{figure}\nFigure. \\ref{fig:rotating_vector} shows travelling wave resonance of a slot ring resonator loaded with transistor and passive components mimicking the electron spin precession in a biased ferromagnetic material. The electrical length ($\\lambda$) of the ring resonator should be $2\\pi$ due to phase matching. The wave is traveling (as opposed to standing wave) due to the presence of the unilateral (transistor) element. Based on the applied electric bias to the active component, the electric dipole moment (red arrow) rotates along the circular ring at four quarter period spaced time instants. The direction of the electric field follows the direction of the dipole moment. Therefore, non-reciprocal gyrotropy is established in the slot-ring resonator. The circular ring resonates when the electrical length is equal to an integer multiple of $2\\pi$.\n\\begin{equation}\n\\beta l+\\alpha _{FET}=2m\\pi,\n\\label{eq:res_1}\n\\end{equation}\nwhere $\\beta$ is the propagation constant defined by $\\frac{2\\pi}{\\lambda_g}$, $\\alpha_{FET}$ is the physical distance of the ring gap as shown in Fig. \\ref{fig:rotating_vector} (d), and l is\nthe physical length of the slotted transmission line defined as $l$ = $(2\\pi - \\alpha_{FET})r_{mean}$. \nIn this research work, FR4 ($\\varepsilon_r$ = 4.4) substrate has been used with a thickness of 0.8mm. The major focus in this research work is the Faraday rotation, an important key performance indicator to quantify nonreciprocity, which is obtained based on the transmission properties. Ground plane have been designed on geometry of the same front side so the wave can propagate through the metasurface itself. Details design modelling has been discussed in our previous work \\cite{meta}.\n\nThe effective dielectric constant for a microstrip geometry is \n\\begin{equation}\n\\varepsilon _{e}=\\frac{\\varepsilon _{r}+1}{2}+\\frac{\\varepsilon _{r}-1}{2}\\left\\{ 1+12\\left ( \\frac{H}{W} \\right ) \\right\\}^{-\\frac{1}{2}},\n\\label{eq:res_2}\n\\end{equation}\nwhere Eq. \\ref{eq:res_2} follows the condition $\\frac{W}{H} \\geq 1$.\nand $Z_0$ can be calculated as\n\\begin{equation}\nZ_0 = \\frac{120\\pi}{\\sqrt{\\varepsilon _{e}}\\left [ \\frac{W}{H}+1.393+\\frac{2}{3}\\ln \\left ( \\frac{W}{H}+1.444 \\right ) \\right ]}.\n\\label{eq:res_3}\n\\end{equation}\nBased on the above analysis, initial parameters have been calculated considering a resonance frequency at 5.7 GHz and dielectric properties of the FR4 substrate. \n\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.50\\textheight]{reciprocal.png}\n\\caption{Simulated performance of ring resonator based metamaterial. (a) Simple circular ring resonator based unit cell on proposed metasurface design. (b) The design is reciprocal, hence, no nonreciprocal properties can be observed. (c) Both co pol and (d) cross pol components exhibit the reciprocal behaviour, resonance is observed at 5.69 GHz in this design. }\n\\label{fig:reciprocal}\n\\end{figure}\n\nFigure \\ref{fig:reciprocal} (a) shows the simulated performance of a ring resonator in a reciprocal condition i.e. without any active component connected. Based on our previous analysis, nonreciprocal properties can't be achieved as shown in \\ref{fig:reciprocal} (b), hence no Faraday or Kerr rotation. Propagating wave exhibits a linear polarization which can be shown from ellipticity. Figure \\ref{fig:reciprocal} (c) and (d) shows reflection and transmission behavior in terms of scattering parameter. One important observation to highlight for reciprocal device is that the co and cross pol components will be in line with respect to each other. For example, we observe no difference between $S_{11}^{xy}$ and $S_{11}^{yx}$ which is cross pol reflection parameter. Co and cross pol transmission parameter follows the similar pattern. \n\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.40\\textheight]{Dynamic_Link.png}\n\\caption{Electromagnetic co-simulation (a) Circuit configuration that was used to achieve nonreciprocity, nonreciprocal component such as transistor was used to bridge the ring resonator gap, other component are for impedance matching and biasing the transistor in various condition (b) 3D electromagnetic model of the metamaterial using floquet mode analysis. (c) S parameter file of the 3D model and circuit model has been connected using dynamic link and to evaluate co-simulation results.} \n\\label{fig:dynamink_link}\n\\end{figure}\nFigure \\ref{fig:dynamink_link} shows the setup of the co-simulation. The 3D electromagnetic structure of the metasurface have been solved in Ansys HFSS which is a passive solver. Besides, the circuit model has been solved independently in a specific bias condition. Later the S parameters are connected using the dynamic link to run the co-simulation and optimization.\n\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.25\\textheight]{flow_chart.png}\n\\caption{Process flow of the computational approach of the nonreciprocal device design}\n\\label{fig:processflow}\n\\end{figure}\nFigure \\ref{fig:processflow} shows the process-flow of design and optimization. At first step, the frequency of operation along with preferred dielectric material is selected. After that, the initial dimension has been calculated based on microwave equation discussed earlier in the section. All dimensions are used as variables for later optimization and sensitivity analysis. Lumped ports are used in the gap and floquet based S matrix have been solved for the metasurface. Besides, in a circuit solver S parameter from our designed circuit and solved S parameter of the 3D model are cascaded for the co-simulation. Once the initial design and result are obtained, based on the targeted goal and machine learning optimization algorithm design is optimized to achieve nonreciprocity at desired resonance. In this design, Quasi newton algorithm was used from HFSS. An in house script was developed and used during the optimization to calculate and optimize the performance at resonance on Key performance indicators such as Faraday, Kerr rotation, co-cross polarized components. \n\n\\section{Results and Discussion}\nIn this section the simulation results of the nonreciprocal metasurface will be discussed. From antenna 1 to antenna 2, the x-polarized wave is transmitted from region 1 and y-polarized probe (antenna) pick the signal at region 2 and if the time is reversed, from antenna 2 to antenna 1, the y-polarized wave is transmitted and x-polarized probe (antenna) pick the signal at region 1. For x-polarized wave and x-polarized probing, phase of the transmission coefficients are same when the time is reversed. Therefore, likewise magnetized ferrites, the co-polarized component does not show any change in their phase when the time is reversed, however, the cross-polarized components shows $180^\\circ$ phase difference at the resonance frequency so that the designed structure behaves like a magnetized ferrite where the permeability tensor has cross-components and they are out-of phase such that the non-reciprocal gyrotropic response is evident. In Fig. \\ref{fig:FR} (a) the nonreciprocal behaviour is evident where we see the magnitude of the cross polarized component is different and at resonance magnitude of co polarized component is negligible. Almost $90^0$ Faraday rotation at resonance has been achieved. Similarly, from fig \\ref{fig:FR} (b), at resonance $\\delta_F$ = 0, i.e. the wave is linear polarized. Figure \\ref{fig:KR} shows the Kerr rotation which is a reflection based measurements and can be explained similarly as we discussed for the Faraday rotation. The practical implementation and measurement procedures have been discussed in detail in our previous articles \\cite{meta, TIM, swadeshmathpaper}\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.25\\textheight]{FR.png}\n\\caption{(a) Faraday rotation and scattering parameter over frequency. co polarized components are same and at resonance those are almost 0. Cross polarized components are different and we observe maximum isolation at resonance. (b) Faraday rotation and ellipticity over frequency.}\n\\label{fig:FR}\n\\end{figure}\n\\vspace{0.0cm}\n\\begin{figure}[htbp]\n\\centering\n \n \\includegraphics[height=.25\\textheight]{KR.png}\n\\caption{(a) Kerr rotation and scattering parameter over frequency. co polarized components are same and at resonance those are almost 0. (b) Faraday rotation and ellipticity over frequency.}\n\\label{fig:KR}\n\\end{figure}\n\n\nDesign steps of a magnet-less nonreciprocal devices such as isolators, circulators, and leaky wave antennas are same however, design and optimization to achieve meaningful results are very challenging. For example, Impedance and phase matching between circular ring resonator and active\/passive component at microwave frequencies are the major challenges to create artificial non-reciprocity. The excited resonance is accompanied by loss in the presence of mismatch leading to nonzero standing wave ratio (SWR). For an infinite standing wave ratio, the device does not exhibit gyrotropy, hence, non-reciprocity. Therefore, full wave 3D simulation along with optimization cost functions were used in this work and highly recommended for any non-reciprocal device design. Lastly, the number of active\/passive components required to implement the artificial gyrotropy can be optimized and reduced. Characterize nonreciprocal devices are very susceptible where precise bias condition, setup test bench and careful extraction of data is very important to take into consideration. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nTo summarize, a detailed computational approach has been presented on a fully electronic non-reciprocal metamaterial. Necessary condition to achieve nonreciprocity, analytical modelling along with key steps required for the design has been proposed, discussed and implemented. With the precision level of engineering, magnet free non-reciprocal devices can be a probable substitute of conventional ferrite material biased by permanent magnet in industrial applications such as isolator, circulator, gyrator, leaky wave antenna, and 5G applications in future. We envision that the computational approaches in this work will be very helpful in the existing research area of magnet-free non-reciprocal metamaterials.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nCerebral palsy (CP) is the term for a group of lifelong neurological conditions which can cause difficulties with mobility, posture and coordination. CP can also cause problems with swallowing, speech articulation, vision, and has been associated with a diminished ability to learn new skills. There is significant variance in the severity of CP, with some individuals showing very minor symptoms whilst others may be severely disabled \\cite{nice}. CP is attributed to non-progressive damage to the brain in early infancy \\cite{bax_2005, Rosenbaum_CP2007} and is one of the most common physical disabilities in childhood. However, early diagnosis of CP can be difficult, with a confirmed diagnosis rarely made before 18 months of age \\cite{Marcroft2015}. The difficulty in providing an early diagnosis is problematic, as early intervention care is considered particularly important for those with emerging and diagnosed CP. \n \n\n \n Currently, the General Movements Assessment (GMA) is used to evaluate infant movement by manually observing spontaneous infant movements at a specific stage in development. In a typically developing infant ``Fidgety Movements\" (FMs) are detectable from 3 to 5 months post term \\cite{Prechtl1997} and consistently have a similar appearance. The absence of these movement characteristics consequently allows for abnormal FM patterns to be identified and classified \\cite{Einspieler2016}. However, the challenges associated with applying these assessments in practice depends upon the availability of appropriately trained clinicians.\nIn order to address the issues surrounding manual clinical assessment,\nseveral studies have been carried out which attempt to assess the viability of automating assessments to predict motor impairment based upon observed motion quality using computer vision-based approaches \\cite{Marcroft2015}. Examples such as \\cite{Adde2010} \\cite{Adde2013} \\cite{Adde2018} explore a per frame background subtraction method for analysis, whereas more recent methods \\cite{Stahl2012} \\cite{Orlandi2018}\n\\cite{Rahmati_2016}, propose the use of Optical Flow-based methods to track and assess infant movements. Whilst reasonable results are obtained, these methods typically struggle to deal with intra-class variation, as well as anomalies within the recorded video footage such as illumination changes, camera movement, subject-scaling, and resolution inconsistencies. This makes it difficult for these approaches to be adopted in a real-world clinical setting.\n\nOn the other hand, with the advancement of pose estimation techniques, high-quality skeletal poses can be extracted from video automatically. Recent work such as \\cite{mccay_pose, McCay_deepPose} proposed using histogram-based pose features to automate GMA by classifying infant movements into FM+ (normal) and FM- (abnormal). The pose-based features, namely Histograms of Joint Orientation 2D (HOJO2D) \\cite{mccay_pose} and Histograms of Joint Displacement 2D (HOJD2D) \\cite{mccay_pose} are computed from the orientation of the body segment and the displacement of the joints, respectively. Encouraging classification performance on traditional classifiers \\cite{mccay_pose} and deep learning frameworks \\cite{McCay_deepPose} were demonstrated. Wu et al. \\cite{Wu:MovCompIdx} proposed {\\it Movement Complexity Index} which determines the complexity of the body movements of the infant by computing the correlations between the movements of the joints using the Spearman Correlation Coefficient Matrix (SCCM). Although the method focuses on analyzing the features to predict the risk level of CP of the infant without the need of the training process as in machine learning based approaches, the features are computed from 3D skeletal data which requires specialized image sensing devices to capture those data. Furthermore, the method requires the user to specify a threshold level of the computed index to separate normal\/abnormal, and it is unclear if this can be generalized to other datasets.\n\nThe aforementioned studies suggest that an automated system could potentially help to reduce the time and cost associated with current manual clinical assessments, and also assist clinicians in making earlier and more confident diagnoses by providing additional information about the assessed infant movements. However, these methods are also not without their setbacks. One of the main issues with using machine-learning approaches in the medical domain is the problem of interpretable AI. Models are often seen as `black boxes' in which the underlying structures can be difficult to understand. There is an increasing requirement for the mechanisms behind why systems are making decisions to be transparent, understandable and explainable \\cite{holzinger2017need}. As such, we propose a new motion classification and visualization framework, which takes an RGB video as the input and analyzes the movement of individual body parts to determine if FMs are present (FM+) or absent (FM-), subsequently identifying normal or abnormal general movements from segments of the sequence. To make our proposed framework fully interpretable, an important aspect is the integration of an automatically generated visualization capable of relaying pertinent information to the assessor. The visualization highlights body-parts which are showing movement abnormalities, and are subsequently providing the most significant contribution towards the classification result. As such, our proposed contributions are summarized as:\n\\begin{itemize}\n \\item A new body-part based classification framework for the automated prediction of CP based upon body movement extracted from videos.\n \\item A visualization feature to highlight pertinent body-parts in the video to improve the model interpretability.\n\\end{itemize}\n\nExperimental results showed that out proposed fidgety movements prediction framework achieved 100\\% accuracy and outperforms the existing work on the benchmark MINI-RGBD \\cite{hesse2018computer} dataset. The details of our proposed classification and visualisation framework are discussed in Section \\ref{sec:Methodology}. Our evaluation is discussed in Section \\ref{sec:evaluation}. Our hope is that this contribution will aid in the adoption of such technologies in this domain, through accurate, quantifiable and explainable results. A demo video is available on \\url{https:\/\/youtu.be\/6CZZmWnT4mo}\n\n\\begin{comment}\n\\section{Related Works} \\label{sec:related}\n\n\nSeveral studies have been carried out which attempt to assess the viability of automating assessments to predict motor impairment based upon observed motion quality, such as the GMA, using computer vision-based approaches \\cite{Marcroft2015}. Examples such as \\cite{Adde2010} \\cite{Adde2013} \\cite{Adde2018} explore a per frame background subtraction method for analysis, whereas more recent methods \\cite{Stahl2012} \\cite{Orlandi2018} \\cite{Ihlen_DL_CP} \\cite{Rahmati_2016}, propose the use of Optical Flow-based methods to track and assess infant movements. Whilst reasonable results are obtained, these methods typically struggle to deal with intra-class variation, as well as anomalies within the recorded video footage such as illumination changes, camera movement, subject-scaling, and resolution inconsistencies. This makes it difficult for these approaches to be adopted in a real-world clinical setting. \n\n\n\nHowever, these machine learning approaches often suffer from a lack of interpretability and a large enough dataset to produce suitably accurate results. In order to address this we propose a pose-estimation based framework which takes advantage of\na traditional machine learning pipeline to retain the explainability of hand crafted features, meaning that our framework benefits from greater contextual interpretability.\n\n\nnew: Wu et al. \\cite{Wu:MovCompIdx} proposed Movement Complexity Index which XXYYZZ. Although the method focusing on analyzing the features to predict the risk level of CP of the infant without the need of the training process as in machine learning based approaches, the features are computed from 3D skeletal data which requires specialized image sensing devices to capture those data. Furthermore, the method require the user to specify a threshold level of the computed index to separate normal\/abnormal. Determining such a threshold automatically is not discussed in \\cite{Wu:MovCompIdx}. \n\\end{comment}\n\n\\section{Methodology} \\label{sec:Methodology}\nIn this section, we discuss the proposed classification and visualization framework illustrated in Figure \\ref{fig:overview}.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{Images\/overview.png}\n\\caption{The overview of the proposed prediction and visualization framework.}\n\\label{fig:overview}\n\\end{figure*}\n\n\\subsection{Pose-based motion features} \\label{sec:Pose-feature}\nThe first step of our proposed framework is extracting features from input video data. McCay et al. \\cite{mccay_pose} demonstrated the effectiveness of using histogram-based motion features, namely Histograms of\nJoint Orientation 2D (HOJO2D) and Histograms of Joint\nDisplacement 2D (HOJD2D), extracted from 2D skeletal poses in detecting FMs from videos. In this paper, an early fusion (i.e. concatenation) of the HOJO2D and HOJD2D is used as the input motion features, since better performance has been demonstrated \\cite{mccay_pose,McCay_deepPose}.\n\n\n\\subsection{Spatiotemporal Fidgety Movement Detection} \\label{sec:Spatiotemporal}\nIn order to detect the presence of FMs spatiotemporally, the motion features have to be extracted from 1) different body-parts and 2) different temporal segments individually. Inspired by this, we propose motion feature extraction from 5 different body-parts in the spatial domain, namely {\\it left arm, right arm, left leg, right leg, and head-torso}. For the temporal domain, we compute HOJO2D and HOJD2D features (8 bins) for the 5 body parts from every 100-frame segment. In doing so, each video is represented by multiple histogram-based motion features accordingly. For example, a 1000-frame video will be represented by 50 fused features of HOJO2D and HOJD2D.\n\nIn this work, we formulate the FMs detection problem as a binary classification. Since each video is annotated with FM+ or FM-, we label all the fused features extracted according to the holistic annotation of the video. When training the classifier all features are used, while the temporal location information is not used. In other words, no matter whether the features are extracted from the beginning or near the end of the video, they will be used to train a single classifier. This proposed approach provides distinct advantages over previous methods, i.e. 1) the classifier will be trained by more data samples rather than using only one histogram representation for the whole video as proposed in \\cite{mccay_pose, McCay_deepPose}, and 2) a focus on the presence\/absence of FMs while ignoring the temporal information when training the classifier.\n\nWe follow McCay et al. \\cite{mccay_pose} on using an ensemble classifier on MATLAB R2020a\nthat consists of a wide range of classifiers to boost the performance of the classification results. Given the multiple fused features extracted from a video, all the features will be classified as FM+ or FM-, this information is then used in visualizing the results (Section \\ref{sec:visualization}). As the features were extracted in sequential order in the temporal domain, the classification result on each histogram-based motion feature is essentially detecting FMs spatiotemporally.\n\n\\subsection{Late Fusion for Cerebral Palsy Prediction} \\label{sec:lateFusion}\nWhile the method presented in Section \\ref{sec:Spatiotemporal} provides precise information on the presence\/absence of FMs spatiotemporally, directly using all motion features as a cerebral palsy prediction for the whole video will result in sub-optimal performance since the temporal ordering is less important in the GMA than the presence\/absence of sustained FMs at any point in the sequence. To tackle this problem, we propose representing each of the 5 body parts using a single scalar score $s$, with this being the average score of the classification result (FM+ as $0$ and FM- as $1$) across all temporal segments for each body-part. Therefore, the range of $s$ will be between $0$ and $1$.\n\nHere, we propose to use a late fusion approach to train an ensemble classifier for cerebral palsy prediction. Specifically, each video is represented by using the 5 scores obtained from the body parts. The binary classifier will predict whether the motion in the video is considered {\\it normal} or {\\it abnormal}. \n\n\n\\subsection{Visualization} \\label{sec:visualization}\nWhile machine learning-based frameworks have obtained excellent performance in a wide range of visual understanding tasks, most of the existing frameworks can be considered black-box approaches since most of the classification frameworks only output the predicted label without specifying exactly what influences the classification decision. Whilst this is acceptable in typical computer vision tasks, it is less preferable in healthcare applications, since it is essential for the clinicians to verify the prediction as well. \n\nTo extract body part information from an input image, the CDCL \\cite{lin2019cross} pre-trained body segmentation model is used in this work. The body is segmented into 6 parts; head, torso, upper arms, lower arms, pelvis and upper legs, and lower legs. An example of the segmentation result is illustrated in Figure \\ref{fig:overview} (bottom left-hand corner). Specifically, given an input infant image, CDCL \\cite{lin2019cross} returns an image mask for segmentation. To align with those 5 body parts to be used in this work, we separate the segmentation masks for the arms and legs into the left and right masks. Here, k-means clustering is used to divide the pixels on each segment mask into two groups.\n\nIn order to make our proposed framework more interpretable, we include a visualization module that highlights the body-parts that are contributing to the classification decision. Our proposed method highlights the body-parts in {\\it red} to indicate the {\\it absence of fidgety movements} based on the scores computed in the body part abnormality detection explained in Section \\ref{sec:Spatiotemporal}, providing clinicians with an intuitive visualization such as the examples illustrated in Figure \\ref{fig:visualization_example}.\n\n\\begin{comment}\n\\begin{figure}\n \n \\centering\n \n \n \n \n \n \\includegraphics[width=0.9\\linewidth]{Images\/seg_syn_00002.jpg}\n \\caption{Segmentation result obtained using \\cite{lin2019cross}.}\n \\label{fig:CDCL_example}\n \n \n \n\\end{figure}\n\\end{comment}\n\n\n\n\\section{Evaluation} \\label{sec:evaluation}\nIn this section, we evaluate the effectiveness of our proposed method using the public dataset MINI-RGBD \\cite{hesse2018computer} with Fidgety movement annotation by an experienced GMs assessor in \\cite{mccay_pose}. We first compare the performance of our method on fidgety movement detection with baselines methods in Section \\ref{sec:res_classification}. Next, we present the visualization results as qualitative analysis in Section \\ref{sec:res_visaulization}. We follow the standard protocol as in \\cite{mccay_pose, McCay_deepPose, Wu:MovCompIdx} to conduct a leave-one-subject out cross-validation to ensure the results presented in this section are obtained base on {\\it unseen data} during the training process.\n\n\\subsection{Quantitative Evaluation on the Fidgety Movement Detection Results} \\label{sec:res_classification}\nTo demonstrate the overall performance of our proposed framework, we first evaluate the cerebral palsy prediction of the whole input video as explained in Section \\ref{sec:lateFusion}. We compared with the existing methods and the results are presented in Table \\ref{tabl:fused}. Using our framework, we achieved a perfect prediction with 100\\% accuracy. This highlights the effectiveness of our proposed framework over the previous work (\\cite{mccay_pose, McCay_deepPose,Wu:MovCompIdx}).\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ | c | c | c | c |}\n\\hline\nMethod & {Accuracy} & {Sensitivity} & {Specificity}\\\\\n\\hline\n\\textit{\\cite{mccay_pose} w\/ LDA} & 66.67\\% & 50.00\\% & 75.00\\%\\\\\n\\hline\n\n\\textit{\\cite{mccay_pose} w\/ SVM} & 83.33\\% & 50.00\\% & \\textbf{100.00\\%} \\\\\n\\hline\n\n\\textit{\\cite{mccay_pose} w\/ Decision Tree} & 75.00\\% & 50.00\\% & 87.50\\% \\\\\n\\hline\n\n\\textit{\\cite{mccay_pose} w\/ kNN (k=1)} & 75.00\\% & 25.00\\% & {\\bf 100.00\\%} \\\\\n\\hline\n\n\\textit{\\cite{mccay_pose} w\/ kNN (k=3)} & 50.00\\% & 00.00\\% & 75.00\\% \\\\\n\\hline\n\n\\textit{\\cite{mccay_pose} w\/ Ensemble} & 66.67\\% & 50.00\\% & 75.00\\%\\\\\n\\hline\n\n\\textit{FCNet \\cite{McCay_deepPose}} & 83.33\\% & 75\\% & 87.5\\% \\\\\n\\hline\n\\textit{Conv1D-1 \\cite{McCay_deepPose}} & 83.33\\% & 75\\% & 87.5\\% \\\\\n\\hline\n\\textit{Conv1D-2 \\cite{McCay_deepPose}}& 91.67\\% & 75\\% & {\\bf 100.00\\%} \\\\\n\\hline\n\\textit{Conv2D-1 \\cite{McCay_deepPose}} & 83.33\\% & 75\\% & 87.5\\% \\\\\n\\hline\n\\textit{Conv2D-2 \\cite{McCay_deepPose}} & 83.33\\% & 75\\% & 87.5\\% \\\\\n\\hline\n\\textit{Movement Complexity Index \\cite{Wu:MovCompIdx}} & 91.67\\% & {\\bf 100.00\\%} & 87.5\\% \\\\\n\\hline\n\\hline\n\\textbf{Our method} & \\textbf{100.00\\%} & \\textbf{100.00\\%} & \\textbf{100.00\\%} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\nClassification accuracy comparison between our proposed framework and baseline methods.}\n\\label{tabl:fused}\n\\end{table}\n\n\\subsection{Qualitative Evaluation on the Visualization Results} \\label{sec:res_visaulization}\nWe further provide qualitative results to demonstrate the effectiveness of our proposed framework. As presented in Section \\ref{sec:visualization}, we detect the absence (FM-) or presence (FM+) of fidgety movements of each body part in each temporal segment (see Section \\ref{sec:Spatiotemporal}). The body parts with a prediction of FM- will be highlighted in red. An example is illustrated in Figure \\ref{fig:visualization_example}. Readers are referred to \\url{https:\/\/youtu.be\/6CZZmWnT4mo}\nto evaluate the visual quality of the results. From the results, it can be seen that the highlighted body-parts generally show less complex or more repetitive movements in the videos annotated as FM-. As shown in Figure \\ref{fig:visualization_example} (a), the arms are showing a lack of movement and are subsequently predicted as FM- using our framework. On the other hand, the legs are predicted as FM+ as they are showing some movements in that temporal segment. Figure \\ref{fig:visualization_example} (c) show an example with monotonous arm and leg movements and our method highlights those body parts as FM- accordingly. For the videos annotated as FM+, such as the example shown in Figure \\ref{fig:visualization_example} (b), it can be seen that a much wider variety of movements can be observed. The visualization provides effective visual feedback to the user, as such clinicians can pay greater attention to the highlighted segments for further analysis. \n\n\\begin{figure}\n \\begin{subfigure}[b]{1.0\\linewidth}\n \\centering\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid4_1.png}\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid4_2.png}\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid4_3.png}\n \\caption{Positive (abnormal) example}\n \\end{subfigure} \\\\\n \\begin{subfigure}[b]{1.0\\linewidth}\n \\centering\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid6_1.png}\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid6_2.png}\n \\includegraphics[trim=3cm 2cm 6cm 2cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid6_3.png}\n \\caption{Negative (normal) example}\n \\end{subfigure}\\\\\n \\begin{subfigure}[b]{1.0\\linewidth}\n \\centering\n \\includegraphics[trim=3cm 3cm 6cm 1cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid11_1.png}\n \\includegraphics[trim=3cm 3cm 6cm 1cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid11_2.png}\n \\includegraphics[trim=3cm 3cm 6cm 1cm, clip, width=0.28\\linewidth]{Images\/visualization\/vid11_3.png}\n \\caption{Positive (abnormal) example}\n \\end{subfigure}\n \\caption{Examples of the video generated by our visualization module. Body-parts without fidgety movements are highlighted in red.}\n \\label{fig:visualization_example}\n\\end{figure}\n\n\\section{Conclusion} \\label{sec:Conclusion}\nIn this paper, we present a new framework for detecting fidgety movements of infants spatiotemporally using the pose-based features extracted from RGB videos. Experimental results demonstrated that the new method not only achieves perfect prediction with 100\\% accuracy, but also provides the user with visualization on how the machine-learning based framework made the overall prediction relating to the abnormality of the infant's movement. Whilst our system is able to provide rudimentary visual feedback to the user, additional visualization tools would be useful to exploit the extracted spatiotemporal information and provide additional predictive aid to clinicians for this complex diagnostic task. We intend to further implement our method on data gathered in a real-world clinical setting, as well as explore other relevant visualisation methods to provide meaningful and explainable feedback to the user.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intr}\nIn the percolation process sites or bonds of a regular lattice (or a graph) are independently open\nwith a probability, $p$ and one is interested in the statistics of clusters of connected\nsites. In the thermodynamic limit for dimensions $d \\ge 2$ there is a critical point at $p=p_c$,\nat which a second order phase transition takes place. Critical percolation is scale invariant and its scaling limit is also believed to be conformally invariant. This has been rigorously proved for some lattices\\cite{smirnov}. Conformal\ninvariance has been used to predict different relations about the order-parameter profiles,\ncorrelation functions, crossing probabilities, etc. of critical percolation. Some of the\nconformal results have been subsequently derived by rigorous mathematical methods, such as\nby Schramm-Loewner evolution\\cite{sle}.\n\nIn percolation the questions one usually asks concern the distribution of clusters, or the\nproperties of the largest ones, but comparatively less attention is paid to the total number\nof clusters, which is expected to scale with the volume of the system under consideration.\nTo the leading volume term there are corrections due to boundary effects, such as surfaces,\nedges and corners. The total number of clusters and its correction terms are generally\nnon-universal, thus they depend on the type of percolation (bond or site) and the details\nof the lattice. The only exception could be the corner contribution, which we are\ngoing to study in this paper.\n\nTo be specific we consider bond percolation in a $2d$ square lattice and denote by $\\Gamma$ a\nsubset of bonds. Initially we take this to be a simple closed curve consisting of straight edges\nand corners and will generalize later. We are interested in the number of clusters, $N_{\\Gamma}$,\nwhich intersect $\\Gamma$, in the scaling limit when $\\Gamma$ is large but still much smaller than the total size of the system. This problem is closely related to the\nentanglement entropy of the bond-diluted quantum Ising model which is defined\nin the same lattice and $\\Gamma$ represents the interface, which separates a subsystem from the\nrest of the system.\n\nIn the following we calculate $N_{\\Gamma}$ by different methods, both analytically and numerically.\nIn the scaling limit we consider\nthe random cluster representation of the $Q$-state Potts model and in the $Q \\to 1$ limit\nwe calculate $N_{\\Gamma}$. In particular we obtain a\nlogarithmically divergent contribution due to corners and its prefactor is calculated through conformal invariance.\nUsing different geometries of $\\Gamma$ we have performed large scale\nMonte Carlo simulations, the results of which are confronted with the conformal prediction.\nWe have also considered anisotropic percolation, in which the probabilities $(p_x,p_y)$ that horizontal and vertical bonds are open are different.\n\n\\section{Potts model representation}\n\\label{sec:Potts}\nBond percolation is well known to be related to the $Q \\to 1$ limit of the $Q$-state Potts model, \ndefined on a lattice with sites $i=1,2,\\dots,n$ and $m$ nearest neighbor bonds.\nThe partition function is given by\n\\be\nZ(Q)=\\sum_{s} \\prod_{\\left\\langle ij\\right\\rangle } \\exp \\left( K \\delta_{s_i,s_j} \\right)\\;,\n\\label{Potts}\n\\ee\nwhere the sum is over all spin configurations, $s=(s_1,s_2,\\dots,s_n)$, $s_i=1,2,\\dots, Q$,\nthe product runs over all nearest neighbor bonds, $\\delta_{s_i,s_j}$\nis the Kronecker symbol and $K$ is the reduced coupling, which is the ratio of the pair interaction and the temperature.\nUsing the identity: $\\exp\\left( K \\delta_{s_i,s_j} \\right)=1+\\frac{p}{1-p} \\delta_{s_i,s_j}$\nwith $p=1-e^{-K}$, the sum of products in $Z$ is written in terms of the so called Fortuin-Kasteleyn\nclusters\\cite{Fortuin-Kasteleyn}, denoted by $F$. In $F$ the edge of the lattice $i,j$ is occupied, if\na factor $\\frac{p}{1-p} \\delta_{s_i,s_j}$ is present and\nin any connected cluster the spins are in the same orientation. \nFor a given element of $F$ there are $N_{tot}(F) \\le n$ connected components and $M(F) \\le m$\noccupied bonds so that the partition function reads:\n\\beqn\n&Z(Q)=\\sum_{F} Q^{N_{tot}(F)}\\left( \\dfrac{p}{1-p}\\right)^{M(F)} \\cr\n&\\sim \\sum_{F} Q^{N_{tot}(F)} p^{M(F)} {(1-p)}^{m-M(F)}\n=\\left\\langle Q^{N_{tot}}\\right\\rangle\\;.\n\\label{Z}\n\\eeqn\nIn this random cluster representation $Q$ is a real parameter and percolation is recovered\nin the $Q \\to 1$ limit, when the critical point in the square lattice is given by $p_c=1\/2$.\nThe mean total number of clusters in percolation is:\n\\be\n\\left\\langle N_{tot}\\right\\rangle=\\left. \\dfrac{\\partial \\ln Z(Q)}{\\partial Q}\\right|_{Q=1}\\;.\n\\label{N_tot}\n\\ee\nNow impose a boundary condition that all the Potts spins are fixed (say, in the state $1$) on $\\Gamma$.\nThe partition function is now:\n\\be\nZ_{\\Gamma}(Q) \\sim \\left\\langle Q^{N_{tot}-N_{\\Gamma}}\\right\\rangle\\;,\n\\label{Z_Gamma}\n\\ee\nwhere $N_{\\Gamma}$ is the number of clusters which intersect $\\Gamma$. Hence\n\\be\n\\left\\langle N_{tot} - N_{\\Gamma} \\right\\rangle=\\left. \\dfrac{\\partial \\ln Z_{\\Gamma}(Q)}{\\partial Q}\\right|_{Q=1}\\;.\n\\label{N_tot-N_Gamma}\n\\ee\nOn the other hand at the critical point $p=p_c$ we can write:\n\\beqn\n\\ln Z(Q)&\\sim&A f_b(Q) \\cr\n\\ln Z_{\\Gamma}(Q)&\\sim&A f_b(Q)+L_{\\Gamma} f_s(Q) + C_{\\Gamma}(Q) \\ln L_{\\Gamma}\\;,\n\\label{log_Z}\n\\eeqn\nwhere $A \\propto n$ is the total area, $L_{\\Gamma}$ is the length of $\\Gamma$, and $f_b$ and\n$f_s$ are the bulk and surface free-energy densities, respectively, which are non-universal.\nThe last term represents the corner contribution.\nHence we obtain:\n\\be\n\\left\\langle N_{\\Gamma} \\right\\rangle=-f'_s(1)L_{\\Gamma}-C'_{\\Gamma}(1)\\ln L_{\\Gamma} \\;.\n\\label{N_Gamma}\n\\ee\nCardy and Peschel\\cite{cardypeschel} considered these corner contributions to the free energy of a general conformally invariant system in domains with a boundary. Their results apply equally to an exterior boundary, with corners with interior angle $\\gamma_k$, or an interior boundary, with $\\gamma_k$ replaced by $2\\pi-\\gamma_k$. An important property of percolation is its locality, that is the partition function $Z_\\Gamma(Q)$ is exactly the product of the partition function $Z^{\\rm int}_\\Gamma(Q)$ for the interior of $\\Gamma$, and $Z^{\\rm ext}_\\Gamma(Q)$ for the exterior. Thus we may apply the Cardy-Peschel result to each of these. Note that this would not be correct if we had given the clusters intersecting $\\Gamma$ a weight $\\not=1$. \nWe therefore find from Ref.~\\onlinecite{cardypeschel} that the prefactor of the logarithm in Eq.(\\ref{log_Z}) is\ngiven by:\n\\beqn\nC_{\\Gamma}(Q)&=&\\dfrac{c(Q)}{24} \\sum_k \\left[ \\left( \\dfrac{\\pi}{\\gamma_k}\\right)- \\left( \\dfrac{\\gamma_k}{\\pi}\\right)\\right. \\cr\n&+&\\left. \\left(\\dfrac{\\pi}{2 \\pi-\\gamma_k}\\right)-\\left(\\dfrac{2 \\pi-\\gamma_k}{\\pi}\\right)\\right]\\;,\n\\label{cardy_peschel}\n\\eeqn\nwhere $\\gamma_k$ is the interior angle at each corner, and $c(Q)$ is the central charge of the $Q$-state Potts model, and the two sets of terms come from the interior and exterior contribution\\cite{note}.\n\nUsing\\cite{CG} $c=(6-\\kappa)(3 \\kappa -8)\/2 \\kappa$ and $\\sqrt{Q}=-2 \\cos(4 \\pi\/\\kappa)$ we have:\n\\be\nc'(1)=\\dfrac{5 \\sqrt{3}}{4 \\pi}\\;\n\\label{c'}\n\\ee\nThe corner contribution to $\\left\\langle N_{\\Gamma} \\right\\rangle$ which is derived here for bond\npercolation is expected to be universal, thus to be valid for site percolation, too.\nWe are going to compare the conformal results in Eqs.(\\ref{N_Gamma},\\ref{cardy_peschel})\nand (\\ref{c'}) with those of numerical calculations for different forms of $\\Gamma$ in Sec.\\ref{sec:numerical}.\n\n\\section{Entanglement entropy of the diluted quantum Ising model}\n\\label{sec:entropy}\n\nThe problem studied in Sec.\\ref{sec:Potts} for percolation is closely related\nto the entanglement entropy of a bond-diluted quantum Ising model and here we follow Refs.~\\onlinecite{lin07,yu07} .\nThe model is defined by the Hamiltonian:\n\\be\n{\\cal H}=-\\sum_{\\left\\langle ij\\right\\rangle } J_{ij} \\sigma_i^x \\sigma_j^x -\\sum_i h \\sigma_i^z\\;,\n\\label{Ising}\n\\ee\nin terms of the $\\sigma_i^{x,z}$ Pauli matrices at site $i$. The first sum in Eq.(\\ref{Ising})\nruns over nearest neighbors and the $J_{ij}$ coupling equals $J>0$ with probability $p$ and\nequals $J=0$ with probability $1-p$. At the percolation transition point, $p_c$, for small\ntransverse field, $h$, there is a line of phase transition the critical properties of\nwhich are controlled by the percolation fixed point\\cite{senthil_sachdev}. The ground state of ${\\cal H}$\nis given by a set of ordered clusters, which are in the same form as for percolation. Now consider\na subsystem, $A$ the boundary of which is represented by $\\Gamma$ and calculate the entanglement\nentropy between the subsystem and the environment, which is given by\n${\\cal S}_{\\Gamma}=-{\\rm Tr}(\\rho_A \\log_2 \\rho_A)$ in terms of the reduced density matrix $\\rho_A$.\nHere we note, that in each ordered cluster the spins are in a maximally entangled GHZ\nstate, thus\nfor a given realization of disorder in the small $h$ limit all those clusters give a unit (1) contribution\nto the entanglement entropy, which intersect $\\Gamma$ and contain also at least one point of the environment.\nWe shall call them crossing clusters, thus ${\\cal S}_{\\Gamma}$ is given by the number of crossing\nclusters and we are interested in their average value: $\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle$.\nNote that ${\\cal S}_{\\Gamma} \\le N_{\\Gamma}$, and thus $\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle \\le \\left\\langle N_{\\Gamma} \\right\\rangle$. The singular corner contributions to $\\left\\langle N_{\\Gamma} \\right\\rangle$ in Eq.(\\ref{N_Gamma}), however, are\ndue to large clusters, which are present typically among the crossing clusters, too. Consequently\nwe expect the asymptotic form of the average entanglement entropy to be\n\\be\n\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle =a L_{\\Gamma}+b\\ln L_{\\Gamma} \\;,\n\\label{S_Gamma}\n\\ee\nwhere $a \\le -f'_s(1)$ and $b=-C'_{\\Gamma}(1)$. The leading term in Eq.(\\ref{S_Gamma}) represents the so called\narea law, to which there is a logarithmic correction, which is expected to be universal and given\nin Eqs. (\\ref{N_Gamma},\\ref{cardy_peschel}) and (\\ref{c'}).\n\n\\section{Numerical results}\n\\label{sec:numerical}\n\nWe have performed large scale numerical calculations for site and bond percolations at the critical\npoint on the square lattice\\cite{sedgewick}. The finite systems we have used have $L \\times L$ sites with $L$ up to\n8192 and with periodic boundary conditions. The different shapes\nof the subsystems used in the numerical calculations are illustrated in Fig.\\ref{fig_1}.\nIn all cases the linear size of the subsystem is of ${\\cal O}(L)$, so that $L_{\\Gamma} \\sim L$ and\nin the following we shall use $L$ instead of $L_{\\Gamma}$.\nFor a given shape of the subsystem and thus for its\nboundary $\\Gamma$ we have calculated the number of crossing clusters, ${\\cal S}_{\\Gamma}$, which is\nthen averaged:\ni) for a given percolation sample over the positions of $\\Gamma$ (typically $10^3$ positions), and\nii) over different samples. Typically we have used $10^5$ samples for each size, $L$, except the\nlargest ones, where we had at least $10^4$ samples. From the numerical data on $\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle$\nwe have deduced estimates for the prefactor $b$ by the so called \\textit{difference approach}. Here we use the relation:\n\\be\n\\Delta {\\cal S}_{\\Gamma}(L)=2 \\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle(L)-\n\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle(2 L)=b \\ln L + \\text{const}\\;.\n\\label{delta_S}\n\\ee\nIf $\\Gamma$ has a special shape, such as a square or a sheared square, (see the three shapes in the\nfirst row of Fig.\\ref{fig_1}) we can calculate the\ncorner contribution to $\\left\\langle {\\cal S}_{\\Gamma}\\right\\rangle$ directly, by comparing\nresults obtained at different geometries. This type of \\textit{geometrical approach}, which has been\nused in Ref.~\\onlinecite{kovacs_igloi12}, will be explained in more detail in the following subsection.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_1}\n\\end{center}\n\\caption{\n\\label{fig_1} (Color online) Shapes of the subsystems used in the numerical calculation: square, sheared square, anisotropic square; equilateral triangle, circle and section of a circle. In a lattice we consider the nearest neighbor\nbonds which are intersected by the contour of the subsystem and for each bond one of its sites\nbelongs to $\\Gamma$ (depending on the local orientation of the surface).}\n\\end{figure}\n\n\\subsection{Square subsystem}\n\\label{sec:square}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_2}\n\\end{center}\n\\caption{\n\\label{fig_2} (Color online) Critical site percolation on a $L=128$ lattice. The subsystem is a square of size\n$\\ell=L\/2=64$ and in this example $N_{\\square}=26$ clusters intersect $\\Gamma=\\square$ and the number of crossing clusters is\n$S_{\\square}=22$.}\n\\end{figure}\n\nThe first geometry we consider for $\\Gamma$ is a square of linear size $\\ell=L\/2$.\nThe cluster structure of critical site percolation is illustrated in Fig.\\ref{fig_2}, in which one can\nidentify the clusters which intersect $\\Gamma$ and also the crossing clusters. Here we use the\ngeometrical approach, in which the complete square is divided either to four neighboring squares\nwith $\\ell=L\/2$ or to orthogonal stripes of size $L\/2 \\times L$: altogether four stripes\ndue to the two different orientations. (This is illustrated in Ref.\\onlinecite{kovacs_igloi12} in the\nright panel of Fig.1.)\nThe two subsystems of different shapes have the same total boundary, however for stripes - due to periodic\nboundary conditions - there is no corner contribution. This is then obtained from the difference:\n\\be\n{\\cal S}_{\\square}^{cr}(L)={\\cal S}_{\\square}(L)-{\\cal S}_{str}(L)\\;.\n\\label{S_corner}\n\\ee\nNote that by this geometrical method the corner contribution is calculated for each sample,\ntherefore the average values have considerably less noise, than by the difference method.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_3}\n\\end{center}\n\\caption{\n\\label{fig_3} (Color online) Estimates for the prefactor of the corner contribution\n$\\left\\langle {\\cal S}_{\\square}^{cr}\\right\\rangle$ of the square subsystem obtained from two-point fits. The asymptotic values as $L\\to\\infty$ are the same for site and bond percolation\nand agree with the conformal prediction. Inset: Large scale topology of the clusters.\nThe square is divided into four quadrants and the different\ntopologies of clusters in the quadrants are illustrated: empty box $\\to$ no site; filled box $\\to$ at least one occupied\nsite.}\n\\end{figure}\n\nMaking use of the fact that asymptotically $\\left\\langle {\\cal S}_{\\square}^{cr}(L)\\right\\rangle\n=b \\ln L + \\text{const}$, we have calculated \neffective, size-dependent estimates for the prefactor by\ntwo-point fits, by comparing the average corner contributions in finite systems of size $L$ and $2L$. The\nresults are presented in Fig.\\ref{fig_3} both for site and bond percolations. With increasing $L$ the\neffective prefactors approach a common, universal limiting value of $b=-0.077(1)$. This is to be\ncompared with the conformal result in Eqs. (\\ref{N_Gamma},\\ref{cardy_peschel}) and (\\ref{c'}) :\n$-C'(1)=-5 \\sqrt{3}\/(36 \\pi)=-0.07657$, thus the agreement is\nsatisfactory. We note that a previous numerical estimate in Ref.~\\onlinecite{yu07} for smaller systems has obtained: $b=-0.06(1)$.\nThe corner contribution is related to the large scale topology of the clusters, as illustrated\nin the inset of Fig.\\ref{fig_3} and discussed in Sec.\\ref{sec:corner_pr}.\n\nWe have also studied the $p$-dependence of the corner contribution to $\\left\\langle {\\cal S}_{\\square} \\right\\rangle$\noutside the critical point by the geometrical approach. As can be seen in Fig.\\ref{fig_4} ${\\cal S}_{\\square}^{cr}$\nhas a peak around $p=p_c$ and close to $p_c$ the extrapolated curve can be well described by the scaling result:\n\\be\n{\\cal S}_{\\square}^{cr}(p) \\simeq b' \\ln(p_c-p)+const\\;,\n\\label{S_corner_p}\n\\ee\nwhere $b'=b \\nu$, with $\\nu=4\/3$ being the correlation length critical exponent for percolation. Indeed, assuming\nthe form in Eq.(\\ref{S_corner_p}) we have calculated effective, $p$-dependent prefactors by two-point fits,\nwhich are shown in the inset of Fig.\\ref{fig_4}. The extrapolated value for $p \\to p_c$ ($0.106(10)$)\nis consistent with the scaling prediction ($0.102$).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_4}\n\\end{center}\n\\caption{\n\\label{fig_4} (Color online) $\\Delta p=p-p_c$-dependence of the corner contribution to ${\\cal S}_{\\square}$\nfor different sizes, $L=16,32,\\dots,2048$ from below for bond percolation. Close to $p_c$ the extrapolated curve has a\nlogarithmic singularity, see Eq.(\\ref{S_corner_p}). In the inset the effective prefactors, $b'$, are shown as a function\nof $|\\Delta p|$. }\n\\end{figure}\n\n\\subsection{Sheared square subsystem}\n\\label{sec:sheared_square}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_5}\n\\end{center}\n\\caption{\n\\label{fig_5} (Color online) Estimates for the prefactor $-b$ for sheared squares as a function of\n$\\ln(\\gamma)$. The numerical results are for $L=128,~256,~\\dots,~4096$ from below, the full line\nrepresents the conformal result in Eq.(\\ref{sheared}). In the inset the ratio of the extrapolated numerical data \nand the conformal result is given.}\n\\end{figure}\n\nThe square subsystem used in the previous subsection is sheared now to a parallelogram, having\nthe same surface: $\\ell^2=L^2\/4$ and its (smaller) angle is $\\gamma$. The geometrical approach\nto calculate the corner contribution to ${\\cal S}_{\\Diamond}^{cr}$ can be extended in this case\nfor specific values of $\\gamma$ given by the condition: $\\tan(\\gamma)=1\/n$, with $n=0,1,\\dots,\\infty$\nbeing an integer. As shown in Fig.\\ref{fig_5} with increasing $L$ up to $L=4096$ the numerical results approach\nthe conformal prediction:\n\\be\nC'(1)=-\\frac{c'(1)}{12}\\left[4-\\pi\\left(\\frac{1}{\\gamma}+\\frac{1}{\\pi-\\gamma}+\\frac{1}{\\pi+\\gamma}+\\frac{1}{2\\pi-\\gamma}\\right)\\right]\\;.\n\\label{sheared}\n\\ee\nPerforming an extrapolation with an $\\ln L\/L$ correction term we have obtained an excellent agreement, as\nshown in the inset of Fig.\\ref{fig_5}. For other values of $\\gamma$, which do not fit to the geometrical approach we\nhave made calculations by the difference method. Also in these cases the numerical results are found to\nagree with the conformal prediction.\n\n\\subsection{Anisotropic percolation}\n\\label{sec:anisotrop}\nHere we consider anisotropic bond percolation, in which the probabilities are $p_x$ (horizontally) and $p_y$ (vertically)\nand the critical point is given by the condition: $p_x+p_y=1$. The system is a diagonally placed\nsquare with $2L^2$ sites, and the subsystem is also a diagonally oriented square, having\n$L^2\/2$ sites and its boundary contains $4 \\times L\/2$ sites, see the third figure in the first row of Fig.\\ref{fig_1}.\nThis anisotropic system with symmetric angles, $\\pi\/2$, is conjectured to be \nequivalent in the scaling limit to an isotropic system with asymmetric angles, $\\gamma$ and $\\pi-\\gamma$, such that\n\\be\n\\dfrac{p_y}{p_x}=\\dfrac{\\sin((\\pi-\\gamma)\/3)}{\\sin(\\gamma\/3)}\\;.\n\\label{p_y\/p_x}\n\\ee\nThis follows from the requirement that there exists a discretely holomorphic observable\\cite{ikhlef_cardy}. More recently Grimmett and Manolescu\\cite{grimmett} have proved that many properties of the scaling limit are the same if a more general inhomogenously anisotropic lattice is embedded in the plane according to this prescription.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_6}\n\\end{center}\n\\caption{\n\\label{fig_6} (Color online) Estimates for the prefactor $-b$ for anisotropic percolation\nfor finite systems with $L=32,~64,~\\dots,~1024$ from below. The opening angle of the\nequivalent sheared square, $\\gamma$ is given by Eq.(\\ref{p_y\/p_x}) and the full line\nrepresents the conformal result. In the inset the ratio of the finite-size results and\nthe conformal results are given.}\n\\end{figure}\n\nIn the anisotropic system we have chosen the probabilities in such a way, that the corresponding angle, $\\gamma$,\nhas satisfied the condition: $\\tan(\\gamma)=1\/n$, with $n=0,1,\\dots,\\infty$. In this way we could directly\ncompare the results for anisotropic systems with those obtained for sheared squares in Sec. \\ref{sec:sheared_square}.\nThe numerical results for the prefactor obtained on finite systems up to $L=1024$ are are shown in Fig.\\ref{fig_6}.\nFor the three largest systems there is no systematic size-dependence of our data, the error is purely statistical\nand the numerical results agree well with the conformal prediction.\n\n\\subsection{Corner probability}\n\\label{sec:corner_pr}\nFor the (sheared) square subsystem we used the geometrical method, in which\nthe $L \\times L$ system is divided for four $L\/2 \\times L\/2$ squares and for four $L\/2 \\times L$\nstripes and the difference in the number of crossing clusters in Eq.(\\ref{S_corner}) is just the corner contribution.\nHaving a general (connected) cluster it could occupy different parts of the four quadrants and its topology\ncould be of four different types as illustrated in the inset of Fig.\\ref{fig_3}. Among these the I, II and IV type\nof topology gives identical contribution both for stripes and squares. Clusters\nwith topology III, however, are crossing clusters for all the four possible stripes, but these are crossing clusters only\nfor three out of four squares. (In the left-bottom square there is no crossing.) Let us denote the occurrence\nprobability of type III cluster as $P_3(L)$, which is given by the ratio of such clusters in a $L \\times L$ square,\nwhich have points in three quadrants, but have no point in the fourth one. As argued above $P_3(L)$ is proportional\nto the corner contribution of crossing squares, more precisely:\n\\be\nP_3(L)=\\dfrac{1}{4} {\\cal S}_{\\square}^{cr}(L)=-\\dfrac{1}{4}C'(1)\\ln L +\\text{const.}\\;.\n\\label{P_3}\n\\ee\nThis results is valid for sheared squares with angle $\\gamma$, as well as for anisotropic percolation\nwith a square subsystem. In these cases the appropriate results of ${\\cal S}_{\\square}^{cr}(L)$ have to\nbe used.\nInterestingly this corner probability has a logarithmic $L$-dependence and its prefactor is known exactly.\n\n\\subsection{Other subsystem geometries}\n\\label{sec:Other}\nWe have also studied subsystems with different geometries: equilateral triangle, circle and section of a circle,\nthese are illustrated in the second row of Fig.\\ref{fig_1}.\nIn these cases the largest linear scale of the subsystem is fixed to $L\/2$, while an angle was varied. In the\ncalculations the difference approach in Eq.(\\ref{delta_S}) has been used.\n\n\\subsubsection{Equilateral triangle}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_7}\n\\end{center}\n\\caption{\n\\label{fig_7} (Color online) Prefactor of the logarithm for an equilateral triangle as a function\nof the base angle $\\gamma$ compared with the conformal result in Eq.(\\ref{triangle}).\nThe numerical results are extrapolated up to $L=4096$ for site percolation.\nThe statistical error is larger, than the difference between the conformal and the numerical results.}\n\\end{figure}\n\nFor an equilateral triangle with a base angle $\\gamma$ the extrapolated results for $-b$\nare shown in Fig.\\ref{fig_7} , which is compared with the conformal prediction:\n\\be\nC'(1)=-\\frac{c'(1)}{24}\\left[6-\\pi\\left(\\frac{2}\n{\\gamma}+\\frac{1}{\\pi-2\\gamma}+\\frac{1}{\\pi+2\\gamma}+\\frac{2}{2\\pi-\\gamma}\\right)\\right]\\;.\n\\label{triangle}\n\\ee\nAs seen in Fig. \\ref{fig_7} there is a satisfactory agreement, although the statistical error of the numerical\nresults is comparatively large, in particular for small and large angles.\n\n\\subsubsection{Circle and section of a circle}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_8}\n\\end{center}\n\\caption{\n\\label{fig_8} (Color online) Prefactor of the logarithm $-b$ for a section of a circle as a function of the\nangle $\\gamma$. The full line represents the conformal result in Eq.(\\ref{section}). The differences\nbetween the conformal and numerical results are of the order of the statistical error. In the inset the\ncorner contribution is shown for a circle: there is no systematic radius dependence.\nThe numerical results are extrapolated up to $L=4096$ (in the main figure) and\nup to $L=8192$ (in the inset) for site percolation.}\n\\end{figure}\n\nAccording to the conformal results, there is no logarithmic correction to $S_{\\bigcirc}$ for a circle shaped subsystem.\nIndeed the numerical results in the inset of Fig.\\ref{fig_8} are in agreement with this statement: $\\Delta S_{\\bigcirc}$\napproaches a finite limiting value of $0.65(5)$.\n\nIn contrast, for a section of a circle with an angle $0 < \\gamma < \\pi$ there is a logarithmic correction, the\nprefactor of which is given by conformal invariance:\n\\be\nC'(1)=-\\frac{c'(1)}{12}\\left[2-\\pi\\left(\\frac{1}{\\gamma}+\\frac{1}{2\\pi-\\gamma}\\right)\\right]\\;\n\\label{section}\n\\ee\nThe numerical results in Fig.\\ref{fig_8} are in agreement with the conformal prediction, although the statistical\nerror of the numerical results is comparatively large.\n\n\\subsection{Line segment}\n\\label{sec:line}\nAccording to the derivation in Sec.\\ref{sec:Potts} $\\Gamma$ does not have to be a closed curve. For example\nit could be a straight line of length $\\ell$. In that case we have only a corner contribution from two\nexterior angles, each $\\gamma=2 \\pi$, so that $C_{\\Gamma}(Q)=-c(Q)\/8$.\n\nTo study this problem we can use the geometrical approach, when the line is oriented parallel with one of the axes of the\nsquare lattice. Then the corner contribution is related to the difference\nbetween the cluster numbers obtained for a\nperiodic line of length $L$ and that of two segments of lengths $\\ell=L\/2$. In\nthis case the corner contribution is simply half of the number of common clusters\nbetween the two line segments.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.2in,angle=0]{Fig_9}\n\\end{center}\n\\caption{\n\\label{fig_9} (Color online) $\\Delta S$ for the line segment obtained by the difference approach for different\nlengths, $L$. In the inset the $p$-dependence of $S^{cr}$ is shown close to the critical point\nfor finite systems with $L=64, 128, 256$ and $512$ from below.}\n\\end{figure}\n\nIn the numerical calculation we have used the difference approach for a line segment of length $\\ell=L\/2$, which is\nplaced in random positions and random orientations with respect to the axes of the periodic square lattice of size $L \\times L$.\nThe average of $\\Delta S_{\\arrowvert}(L)$ is shown in Fig. \\ref{fig_9} as a function of $\\ln L$. The logarithmic dependence is clearly visible with a prefactor estimated as $b=0.086(1)$, which agrees fairly well with the conformal prediction: $-C'_{\\Gamma}(1)=\\frac{5\\sqrt{3}}{32\\pi}\\approx 0.08615$. We have also checked the $p$-dependence of\n$S^{cr}_{\\arrowvert}(p)$, which is shown in the inset of Fig. \\ref{fig_9}.\nA singularity is developed at the critical point as $L \\to \\infty$.\n\nWe note that if the line segment is put to the open boundary of the system, then the prefactor\nis different, being $b=\\frac{\\sqrt{3}}{4\\pi}\\approx 0.1378$ as obtained by conformal invariance\\cite{cardy01,yu07} and\nnumerically $b=0.15(2)$ \\cite{yu07}.\n\n\\section{Discussion}\n\\label{sec:disc}\nIn this paper we considered the logarithmic terms in the mean number $N_\\Gamma$ of clusters in critical percolation which intersect a curve $\\Gamma$ in the cases when it has sharp corners or end points. We have shown that the Cardy-Peschel result\\cite{cardypeschel} can be simply applied to compute the universal coefficients of these logarithmic terms, and that accurate numerical estimates agree very well with this, for a variety of shapes for $\\Gamma$. We also considered anisotropic bond percolation on the square lattice and showed that if Eq.~(\\ref{p_y\/p_x}) is used to deform the lattice then the predictions again agree with numerics if the correct effective corner angle is used. We have also pointed out a relation between the corner\ncontribution to $N_{\\Gamma}$ and the statistics of cluster shapes.\n\nOur study is related to recent investigations of shape dependent terms of different thermodynamic\nquantities, mainly the free energy, of $2d$ critical systems. For critical percolation, however, there is no\nlogarithmic corner contribution to the free energy, since the central charge is $c(Q=1)=0$. In the latter problem\nthe corner contribution to the cluster numbers and its higher moments are of interest, which are related\nto derivatives of $c(Q)$ at $Q=1$.\n\nThe excellent agreement of the numerical data with the theory serves as confirmation that critical percolation is indeed conformally invariant, of the Coulomb gas predictions for $c'(Q=1)$, and of the formula (\\ref{p_y\/p_x}). Extensions are obviously possible: for example by taking further derivatives with respect to $Q$ we can find results for the corner contributions to the higher moments of the distribution of the cluster numbers $N_\\Gamma$. We note that recently\\cite{Vasseur} an explicit example has been found of a correlation function in percolation which contains a multiplicative logarithm. As with our result, this may be understood\\cite{cardylogs} from the necessity of having to take a suitable derivative with respect to $Q$ at $Q=1$ in the Potts model.\n\n\\begin{acknowledgments}\nThis work has been supported by the Hungarian National Research Fund under Grants\nNo. OTKA K75324 and K77629. This work has been partly\ndone when two of the authors (J. C. and F. I.) were guests of the Galileo Galilei Institute in Florence\nwhose hospitality is kindly acknowledged.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzznevb b/data_all_eng_slimpj/shuffled/split2/finalzznevb new file mode 100644 index 0000000000000000000000000000000000000000..ee90fdb0507d6033485d5bade7155a17d385fd08 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzznevb @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nSmall-scale hydropower plants are in many cases of ``run-of-river\" type (ROR) meaning that any dam or barrage is small,\nusually just a weir, and generally little or no water can be stored. ROR hydropower plants preserve the natural flow of the river (besides of course at the location of the power plant) and are therefore among the most environmentally benign existing energy technologies. \nDue to a low installation cost, ROR hydropower plants are often cost-efficient and, as such, \nuseful both for rural electrification in developing countries and for new hydropower developments in industrialized countries \\cite{P02}.\nROR hydropower plants are common in smaller rivers but also exist in larger sizes such as the Niagara Falls hydroelectric plants (Canada\/USA), the Chief Joseph dam on the Columbia River (Washington, USA) or the Saint Marys Falls hydropower plant in Sault Sainte Marie (Michigan, USA).\n\nThe optimal sizing of ROR hydropower plants has been considered by several authors, see e.g. \\cite{AP07, BMM11, F83, GZV09}\nand the references therein. In the estimation of the performance of a given power plant, all these authors omit the cost of switching between different production modes. Doing so, the optimal management strategy can be found trivially by starting the machine when flow is sufficient and stopping it when flow is insufficient. However, in rivers with large and rapid flow fluctuations, which is typically the case in smaller unregulated rivers, such strategies can lead to a large number of switches, the cost of which can not be neglected. For example, starting and stopping the turbines induces wear and tear on the machines and may also require intervention from personnel. Moreover, each start and stop involves a risk which can be considered a cost. To give an example, the major breakdown in the Akkats hydropower plant (Lule river, Sweden) 2002 was caused by a turbine being stopped too quickly, resulting in rushing water destroying the foundation of both the turbine and the generator \\cite{AKKATS1, AKKATS2}.\n\nIn this paper we create fully automatic production schemes for ROR hydropower plants, with stochastic flow of water and with non-zero cost of switching between different states of production. Our method is based on optimal switching and provides, to the best of our knowledge, a novel way to handle hydropower production planning using stochastic control theory. Along the way of deriving the production strategies, we also create a model for the random flow of water based on stochastic differential equations (SDEs) and fit this model to historical data. The stochastic flow model can incorporate a given flow forecast in order to catch up short term fluctuations.\n\nOur main focus lies on the management of a single power plant without storage capacity or means to control the flow of water, i.e.,~a ROR hydropower plant. Although this is rather restrictive from a practitioner's point of view, we stress that the mathematical framework presented here can be used also for more intricate optimization hydropower production problems, e.g.,~those including dams or pumped-storage hydropower. We have chosen to stay within a simple setup here to highlight the specific features of optimal switching and intend to treat more involved hydropower plants in a series of forthcoming papers. \n\nThe rest of this paper is outlined as follows. In Section \\ref{sec:litreview} we give a literature survey and explain the contribution of this paper. Section \\ref{sec:optimalsw} contains an introduction to optimal switching problems and, in particular, an outline of the theory in the context of hydropower planning. Sections \\ref{sec:flow} and \\ref{sec:powerplants} contain models for the flow of water and the power plants under consideration, respectively. Section \\ref{sec:numericsPDE} contains a very brief outline of the numerical approach taken to solve the variational inequalities appearing. We thereafter present the result of our parameter estimation and the performance of the constructed strategy in Section \\ref{sec:results}. We end with Section \\ref{sec:discussion} in which we discuss our approach in general, and our results in particular, and make some concluding remarks.\n\n\n\\setcounter{equation}{0} \\setcounter{theorem}{0}\n\n\\subsection{Literature survey and our contribution} \\label{sec:litreview}\n\n\nOptimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) \\cite{BGSS20, BJK10,DHP10, AF12, AH09, HM12, HT07, K14, LNO14, LNO14b, LOO17, M16_1, M16_2, P18, P20, P20_2}. However, a literature survey shows that,\nalthough the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area. A possible explanation for this discrepancy could be the difficulty of formulating real problems in mathematical terms and, conversely, to interpret the theoretical results in practical terms. \n\nMost commonly, applications are found in the context of real options, see \\cite{BO94, BS85, CL08, DZ01, GKL17}. In \\cite{ACLP12} the authors provide a probabilistic numerical scheme for optimal switching problems and test their scheme on a fictitious problem of investment in electricity generation. In \\cite{CL10} valuation of energy storage is studied with the help of optimal switching and in \\cite{P20_3, PE17} the authors study how the framework can be used to track electricity demand. \n\n\nThis paper extends the use of optimal switching by applying the general framework described in Section \\ref{sec:optimalsw} to two canonical examples of ROR hydropower plants, consisting of one and two adjustable units, respectively, more thoroughly described in Section \\ref{sec:powerplants}.\nIn particular, we construct a management strategy by solving a variational inequality related to the optimal switching problem (see \\eqref{pde}). However, for this solution to be practically meaningful, we first adapt and calibrate the optimal switching problem to the case of ROR power plants.\n\nA main feature of optimal switching-based production planning is that it allows for random factors influencing the production strategy (for details see Section \\ref{sec:optimalsw} below). This randomness is in general given by a Markovian stochastic process and can incorporate any number of different variables. However, to reduce computational strain we will in this paper focus on how the flow of water impacts the production. \n\nPopular streamflow models include linear time series models, such as ARMA models with Gaussian or GARCH noise \\cite{MO13, MNK90, WGVM05}, and non-linear time series models, such as SETAR models \\cite{FFODN19}. Modern approaches also include neural networks and machine learning techniques, see, e.g.,~\\cite{MBFZ17, MS17} and the references therein. Another suitable approach, and the approach that we have adopted here due to its natural relation to the optimal switching framework, is SDEs driven by Gaussian white noise and\/or compound Poissonian impulses \\cite{BTU87}. \nIn particular, we develop a new stand-alone SDE-based model for the flow of water $Q$, based on historical data and driven by Gaussian white noise, which mimics the long term seasonal variations of the flow while still allowing for short term fluctuations and flow forecasts. \n\n\nWhen trying to maximize monetary profit, the electricity spot price $P$ at which the electricity produced is sold is of course also of interest when planning the production. In general, $P$ is an exogenous stochastic process which, in principle, can be incorporated into our model similarly to the water flow by constructing an SDE for $P$ and applying optimal switching theory. However, modelling electricity prices is a non-trivial task and prices are usually not It\\^o diffusions, but rather discontinuous jump-processes, see, e.g., \\cite{HL12, WBT04, WSW04}, making the operator in the variational inequality to be solved non-local (see, e.g., \\cite{LNO14, LNO14b}). For simplicity, we therefore let $P$ be a continuous \\textit{deterministic} process in the current paper. \nWe stress, however, that our approach readily can be extended to random electricity prices (and streamflow models with compound Poisson impulses) at the cost of computational complexity. \n\nWe have chosen not to let deeper knowledge of different types of generators and how they operate be a prerequisite for the understanding of this paper and we hence avoid going into any such technical details. A comprehensive survey of different turbines and generators and their distinct characteristics can be found in, e.g., \\cite{B15, B01, W84}.\n\n\n\\section{Optimal switching in the context of hydropower} \\label{sec:optimalsw}\n\n\nTailored to the setting of hydropower plants, the optimal switching problem can be described as follows. We consider a manager of a hydropower plant with several units, each unit being a sub-power plant, i.e.,~a turbine and a generator. Each unit can be started and stopped separately in order to adjust the production of the power plant to the supply of water or to the production demand. This implies that the manager has the option to run the plant in $m \\geq 2$ production modes, corresponding to running different combinations of units. Starting and stopping units induces wear and tear on the units and therefore the manager finds herself in a trade-off, weighing the benefits of changing production state against the costs induced by making these changes.\n\nLet $X = \\{X_t\\}_{t\\geq 0}$ denote a Markovian stochastic process representing the features which influence the production. For small hydropower plants, $X_t$ may represent the flow of the river, but it may also be interpreted as, e.g.,~production demand for a frequency regulating plant, the spot price of electricity, or a varying cost of production such as for an oil driven power plant. The process $X$ may be multi-dimensional and hence incorporate all of the above and more, but we will in this paper consider $X =(Q,P)$ where $Q$ and $P$ are one-dimensional processes representing flow of water and spot price of electricity, respectively. Moreover, we let $f_i(X_t,t)$ denote the instantaneous payoff generated in production mode $i$ at time $t$, when the state of the underlying process is $X_t$. Depending on the interpretation of $X$ and the choice of $f$, $f_i(X_t,t)$ can be interpreted as, e.g., the power delivered by a power plant or as the instantaneous monetary profit per unit time. Finally, we associate to each start and stop of a unit a cost $c_{ij}$ for switching from production mode $i$ to production mode $j$, where $i,j \\in \\{1,\\dots m\\}$.\n\nThe manager of the power plant controls the production by choosing a \\textit{management strategy}, i.e.,~a combination of a non-decreasing sequence of stopping times $\\{\\tau_k\\}_{k\\geq 0}$, where, at time $\\tau_k$, the manager decides to switch the production from its current mode to another, and a sequence of indicators $\\{\\xi_k\\}_{k\\geq 0}$, taking values in $\\{1,\\dots,m\\}$, indicating the mode to which the production is switched. More precisely, at $\\tau_k$ the production is switched from mode $\\xi_{k-1}$ to $\\xi_k$ and when starting in mode $i$ at time $t$, we have $\\tau_0 = t$ and $\\xi_0 = i$. We stress that $\\tau_i$ is required to be a \\textit{stopping time} and as such it is adapted to the filtration $\\mathcal F^X$ generated by the underlying process $X$. In less mathematical terms, this simply means that the decision to switch at time $t$ must be based solely on the information made available up to time $t$, i.e., the manager cannot``peek into the future'' when making her decision\\footnote{Although this is obviously impossible from a practical point of view, it must be stated as an explicit restriction in the mathematical formulation.}.\n\nA strategy $(\\{\\tau_k\\}_{k\\geq 0},\\{\\xi_k\\}_{k\\geq 0})$ can be represented by the random function $\\mu : [0, T] \\to \\{1,...,m\\}$ defined as\n\\begin{align*}\n\\mu_s \\equiv \\mu(s) = \\sum_{k\\geq 0} \\mathbb{I}_{[\\tau_{k}, \\tau_{k+1})}(s)\\xi_{k},\n\\end{align*}\nwhich indicates the mode of production at time $s$. Here, $\\mathbb{I}_{[\\tau_{k}, \\tau_{k+1})}(s)$ denotes the indicator function, i.e.,~$\\mathbb{I}_{[\\tau_{k}, \\tau_{k+1})}(s) = 1$ if $s\\in [\\tau_{k}, \\tau_{k+1})$ and 0 otherwise. When the production is run using a strategy $\\mu$, defined by $(\\{\\tau_k\\}_{k\\geq 0},\\{\\xi_k\\}_{k\\geq 0})$, over a finite horizon $[0, T]$, the total expected payoff is\n\\begin{align*}\nE\\biggl [\\int \\limits_0^T f_{\\mu_s}(X_s,s) ds - \\sum_{k\\geq 1,\\tau_k < T} c_{\\xi_{k-1},\\xi_k} \\biggr ].\n\\end{align*}\nSimilarly, given that the stochastic process $X$ starts from $x$ at time $t$, the profit made using strategy $\\mu$ starting in $\\xi_0=i$, over the time horizon $[t, T]$, is\n\\begin{align} \\label{eq:OSP}\nJ_i(x,t,\\mu) = E\\biggl [\\int \\limits_t^T f_{\\mu_s}(X_s,s) ds - \\sum_{k\\geq 1,\\tau_k < T} c_{\\xi_{k-1},\\xi_k}\\bigg| X_t = x \\biggr ].\n\\end{align}\n\nThe task of the manager is now to maximize the expected payoff, i.e.~to find the most profitable trade-off between switching to more efficient states and minimizing the total cost of switches. In general, this problem, often referred to as an optimal switching problem, consists in finding the value function\n\\begin{align*}\nu_i(x, t) = \\sup_{\\mu\\in\\mathcal A_i} J_i(x, t, \\mu),\n\\end{align*}\nwhere $\\mathcal A_i$ is a set of strategies starting from $\\xi_0=i$, and the optimal management strategy $\\mu^* \\in \\mathcal A_i$, defined by $(\\{\\tau^*_k\\}_{k\\geq 0},\\{\\xi^*_k\\}_{k\\geq 0})$,\nsuch that\n\\begin{align*}\nJ_i(x, t, \\mu^*) \\geq J_i(x, t, \\mu)\n\\end{align*}\nfor any other strategy $\\mu \\in \\mathcal A_i$. We note that the value function and optimal strategy of an optimal switching problem depends on the set of available modes $\\{1,\\dots,m\\}$, the (finite) time horizon $T$, the running payoff functions $f_i$, $i\\in \\{1,\\dots,m\\}$, the switching costs $c_{ij}$, $i,j\\in \\{1,\\dots,m\\}$, as well as the dynamics of the underlying stochastic process $X$.\n\nFrom the dynamic programming principle, see, e.g.,~\\cite{TY93}, one can derive that the value function $u(x, t) = (u_1(x, t), \\dots, u_m(x, t))$ satisfies the following system of variational inequalities\n\\begin{align}\\label{pde}\n&\\min \\left\\{ -\\partial _{t}u_{i}\\left( x,t\\right) - \\mathcal{L} u_{i}\\left( x,t\\right) -f_i\\left( x,t\\right),\n\\;u_{i}\\left( x,t\\right) -\\max\\limits_{j\\neq i}\\left\\{ u_{j}\\left(\nx,t\\right) -c_{ij}\\left( x,t\\right) \\right\\} \\right\\} = 0, \\\\\n&u_{i}(x,T) = g_{i}\\left( x\\right) , \\notag\n\\end{align}\nfor $x\\in \\mathbb{R}$, $t\\in [0,T]$, $i\\in \\{1,...,m\\}$ and where \n$$\\mathcal{L} u(x,t) : = \\sum_{i=1}^{d_X} b_i(x,t) \\frac{\\partial }{\\partial x_i} u(x,t)+ \\sum_{i,j=1}^{d_X} \\frac{1}{2} (\\sigma \\sigma^\\ast)_{i,j}(x,t) \\frac{\\partial^2}{\\partial x_i \\partial x_j} u(x,t),\n$$ \nis the infinitesimal generator of the underlying $d_X$-dimensional stochastic process $X$ with dynamics\n$$\ndX_t = b (X_t,t) dt + \\sigma(X_t,t) dW_t.\n$$\nThe operator $\\mathcal{L}$ is a linear operator whenever the process $X$ is driven by Gaussian white noise (which we assume in this work). \n\nMoreover, the optimal strategy from state $i$ is given iteratively by the solution to \\eqref{pde} by\n\\begin{align*}\n&\\tau^\\ast_0=0 \\qquad \\xi^\\ast_0=i \\\\\n&\\tau^\\ast_k = \\inf\\left \\{t > \\tau^\\ast_{k-1}: u_{\\xi^{\\ast}_{k-1}}(X_t,t) \\leq \\max_{j \\neq \\xi^\\ast_{k-1}}\\{ u_j(X_t,t) - c_{\\xi^\\ast_{k-1}j} \\} \\right \\}\n\\\\\n& \\xi^\\ast_k = \\underset{j}{\\text{argmax}} \\left \\{ u_{j}(X_{\\tau^\\ast_{k}},\\tau^\\ast_{k}) - c_{\\xi^\\ast_{k-1}j} \\right \\}.\n\\end{align*}\n\nLast, we remark that system \\eqref{pde} consists of $m$ PDEs with interconnected obstacles and that a unique Lipschitz continuous solution to this system exists under the assumptions of this paper, see, e.g., \\cite{ADPS09, AH09, LNO14b}. \n\n\\section{Modeling river flow with an SDE} \\label{sec:flow}\nIn this section we construct an SDE whose solution resembles the actual flow of water in the river under investigation.\nThis resemblance must of course hold in the long-term (seasonal) sense, but must also allow for short-term fluctuations due to inter-yearly variations. More specifically, we use historical data to find functions $b_s$ and $\\sigma_s$, such that the solution to the stochastic differential equation\n\\begin{align*\ndQ_t =& b_s(Q_t,t)\\, dt + \\sigma_s(Q_t,t)\\, d\\tilde{W}_t,\\quad \\text{if }t\\in [s,T], \\notag \\\\\nQ_s=&q\n\\end{align*}\nwhere $\\tilde{W}_t$ is a standard Brownian motion, is similar (in some appropriate sense) to the actual flow of water.\n\nResults indicate that log-transformation of river flow data may increase prediction accuracy of flow models, see \\cite{AYKA17}, and we therefore work with the logarithm of the flow rather than the flow directly. We treat the seasonal and short-term resemblance separately, as outlined below.\n\n\\subsection{Seasonal variations}\n\nStarting with the long term seasonal variations, we define $R_t = \\log Q_t$. We let $r_t$ be a function describing the expected value of the logarithm of the flow at time $t$, independent of current observations. Defined as such, $r_t$ reflects the expected seasonal variation in flow due to spring flood, autumn rains etc., but without any consideration taken to observations from the current year\\footnote{As an example, one would expect that heavy snowfall from January to March would increase the probability of a long and intensive spring flood. Such considerations are \\textbf{not} included in $r_t$.}, and we may thus estimate the deterministic function $r_t$ from historical flow. More precisely, we will construct $r_t$ as a one-week moving average of the logarithm of the flow, see Section \\ref{sec:parametervalues}. The choice of a one-week moving average here is a trade-off between capturing seasonal variations, such as the spring flood, without letting the mean flow depend too much on the flows of particular years.\n\n\\subsection{Short term fluctuations}\n\nNext, we consider the fluctuations around the expected yearly mean, $S_t=R_t-r_t$, and assume that these fluctuations are given by an Ornstein-Uhlenbeck process reverting towards $0$, i.e.\n\\begin{equation*\ndS_t = -\\kappa S_t \\, dt + \\sigma\\, dW_t,\n\\end{equation*}\nwhere $\\kappa>0$ and $\\sigma>0$ are constants to be determined and $W_t$ is a standard Brownian motion. By standard It\\^o calculus, the flow $Q_t = \\exp\\left(r_t+S_t\\right)$, then satisfies the following stochastic differential equation \n\\begin{equation} \\label{eq:sde}\ndQ_t = \\left(r'_t+\\frac{1}{2}\\sigma^2-\\kappa\\left(\\log Q_t - r_t\\right)\\right)Q_t\\, dt + \\sigma Q_t\\,dW_t.\n\\end{equation}\nNote that the particular form of \\eqref{eq:sde} ensures that $Q_t$ stays positive.\n\nTo estimate the parameters $\\kappa,\\sigma$ we consider the asymptotic variance and asymptotic autocorrelation for lag $\\tau$ of an Ornstein-Uhlenbeck process, which are given by\n\\begin{equation}\n\\text{Variance}=\\frac{\\sigma^2}{2\\kappa}\\qquad\\text{and}\\qquad\\text{Autocorrelation}=e^{-\\kappa\\tau}.\\label{eq:varacf}\n\\end{equation}\nFor a given set of historical flow data, we first calculate the logarithm of the data and subtract the running-mean $r_t$ from above to obtain an empirical time series for $S_t$. We calculate the sample autocorrelation function of the time series and estimate the value of $\\kappa$ by a linear regression with the logarithm of the sample autocorrelation function as the dependent variable and the lag as the covariate. Finally, we calculate the sample variance of the time series and deduce an estimate of $\\sigma$ from the first equation in \\eqref{eq:varacf} and the estimated value of $\\kappa$.\n\n\\subsection{Forecasts}\nThe SDE constructed so far respects seasonal variations whilst still allowing for inter-yearly forecasts. However, in order to perform optimally, our model must also be able to treat short-term fluctuations based on, e.g.,~weather forecasts or upstream measurements of the flow. Such forecasts will be included in the model by altering the drift in the dynamics of $Q$, i.e.,~by changing \\eqref{eq:sde} on a short time span close to the present time $s$. \n\nMore precisely, we let for $t \\geq s$\n\\begin{equation} \\label{eq:sde!}\nd Q_t = b_s(Q_t,t) \\, dt + \\sigma_s(Q_t, t) \\, dW_t,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:sigma}\n\\sigma_s(Q, t) = \\sigma Q\n\\end{equation}\nas before and\n\\begin{align} \\label{eq:b}\nb_s(Q, t) &=\n\\begin{cases}\nb_s^{f} (t)\t\t\t\t\t\t\t &\\textrm{if}\\quad s \\leq t \\leq s+l +\\ell,\\\\\n \\left(r'_{t}+\\frac{1}{2}\\sigma^2-\\kappa\\left(\\log Q- r_{t}\\right)\\right) Q\t& \\textrm{if}\\quad s+l+\\ell < t,\n \\end{cases}\n\\end{align}\nwhere $b^f_s$ is a function constructed from the long term (log-)mean $r$, the flow $Q_s$ at time $s$ and the forecast of the flow during $(s, s+l)$ as follows. In the above, $l$ and $\\ell$ are parameters representing the length of the forecast and the (estimated) time it takes for the forecasted flow to return to the long term (log-)mean $r$.\nStarting at time $s$ with current flow $Q_s$ and given a forecast $\\{F_r\\}_{s< r \\leq\ts+l}$ of the future flow at times $t \\in (s,s+l)$ we set\n$$\ng_s(t) = \\begin{cases}\n\\log(Q_s) & \\mbox{for $t=s$} \\\\\n\\log (F_t) & \\mbox{for $t\\in (s,s+l]$} \\\\\n\\log(F_{s+l}) +(t-(s+l)) \\frac{ r_{s+l+\\ell}- \\log(F_{s+l}) }{\\ell} & \\mbox{for $t \\in (s+l, s+l+\\ell] $}\n\\end{cases}\n$$\nand let\n$$\nb_s^{f} (Q,t)\t = \\left(g_s'({t})+\\frac{1}{2}\\sigma^2-\\kappa\\left(\\log Q - g_s({t})\\right)\\right)Q.\n$$\n\nMore explicitly, we calculate the drift $b^f_s$ as in \\eqref{eq:sde} but with the (log-) mean $r$ replaced by $g_s$, where $g_s(t)$ is given directly by the forecast for $t \\in (s, s+l]$, coincides with $r_t$ for $t > s+l+\\ell$ and is linearly interpolated between $s+l$ and $s+l+\\ell$. The impact of such forecasts are illustrated in Figure \\ref{fig:forecasts}.\n\nWe stress that as time evolves, the forecast will be updated and the function $b_s^f$ needs to be updated accordingly each time $s$ that a new forecast becomes available. As the starting time $s$ will be clear from context we will drop the subscript $s$ and simply write $b^f(Q,t)$ in the following, although this function varies with $s$ as parameter.\n\n\n\\section{Modeling the payoff structure of power plants} \\label{sec:powerplants}\nAs mentioned above, we will consider two canonical examples of ROR hydropower plants and outline the payoff structure of these in the current section. We will measure the performance in monetary units (m.u.), but one can easily modify the below to have, e.g.,~total electricity produced as the trait for optimization.\n\n\\subsection{Power plant I: One adjustable unit}\\label{sec:model1}\nWe consider first the simple case of a hydropower plant having a single unit. The unit is designed for the flow $Q_d$, but can be run over a wide flow range $[Q_{min}, Q_{max}]$ with lower efficiency. We assume that the unit, automatically and at negligible cost, adjusts to the available flow, and the task of the manager is thus to find out when to start and stop the unit. In the setting of optimal switching, we model the above power plant as follows. The power plant can be run in two states, `1' and `0', representing `on' and `off', respectively, and for each switch from $0$ to $1$ or from $1$ to $0$ the manager must pay a cost $c_{01}$ or $c_{10}$, respectively. We assume that the electricity output (in Watts) of the unit when in state $1$ is given by\n\\begin{equation*}\nW_1(Q) =\n\\begin{cases}\n0\t\t\t\t\t\t\t& \\textrm{if}\\quad Q < Q_{min},\\\\\nc\\, \\eta (Q) \\, Q\t\t\t\t\t\t& \\textrm{if}\\quad Q_{min} \\leq Q < Q_{max},\\\\\nc\\, \\eta (Q_{max}) Q_{max}\t\t\t\t\t& \\textrm{if}\\quad Q_{max} \\leq Q,\n\\end{cases}\n\\end{equation*}\nwhere the constant $c$ is simply given by $c = \\rho\\, g\\, h$, where $\\rho = 10^3$ kg\/m$^3$ is the (approximate) density of water, $g = 9.82$ m\/s$^2$ is the (approximate) gravitational constant, and $h$ is the water head in meters, and $\\eta(Q) \\in (0,1)$ is the flow-dependent efficiency. In practice, this latter function is given by the specific characteristics of the turbine and generator, but in general it is a concave down function with a maximum at the design flow $Q_d$, see, e.g., \\cite[Figure 9]{IPCC}. To mimic such behaviour, we let\n\\begin{align}\\label{eq:efficiency-curve-assumption}\n\\eta(Q) = \\alpha - \\beta \\left( \\frac{Q}{Q_d} - 1\\right)^2,\n\\end{align}\nwhere $\\alpha$ is the efficiency at design flow and $\\beta > 0$ quantifies the concavity. Multiplying $W_1$ with the spot electricity price and integrating over time gives the income when running the plant in state $1$.\n\nTo be able to determine an optimal strategy in terms of monetary profit, we need to consider not only the produced electricity, but also running-costs of the plant as well as electricity prices. \nWe assume that the running cost of the unit is $c_{run}$ per unit time, the electricity price at time $t$ is $P_t$, and that a large additional cost $c_{low}$ must be paid for each unit of time that the production unit is run with insufficient flow $Q $ totaltolerance}\n{\n\t\n\t\n\n\t\t$u_{old} = u$;\n\t\t\n\ttotalerror = 0;\n\t\n\t\\For{time from $t_{N-1}$ to $t_k$ }{\n\t\t\n\t\t\\While{spatialerror $>$ spatialtolerance}{\n\t\t\t\n\t\t\tspatialerror = 0;\n\t\t\t\n\t\t\t\\For{all spatial points $q_j$}{\n\t\t\t\n \t\t\t\t\\For{$i \\in \\{1,\\dots, m\\}$}{\n \t\t\t\t\n \t\t\t\t\tDummy variable $y$ = the Crank-Nicolson scheme for $u_i(q_j,t)$;\n \t\t\t\t\t\n \t\t\t\t\t$obstacle_i = \\max_{j \\neq i} \\{u_j - c_{ij}\\}$;\n \t\t\t\t\t\n \t\t\t \t\t$y = \\max \\{y,\\text{$obstacle_i$} \\}$;\n \t\t\t \t\t\n \t\t\t \t\tspatialerror = spatialerror + $|u_i(q_j,t)-y| $;\n \t\t\t \t\t\n \t\t\t \t\t$u_i(q_j,t)=y$;\n\t\t\t\t}\n \t\t\t}\n \t\t\tInterpolate linearly at the edges of the spatial grid\n\t\t}\n\t}\n\ttotalerror = $|u_{old} - u|$;\n }\n\\caption{A numerical algorithm for estimating solutions to \\eqref{pde}.}\n\\label{alg:pseudo}\n\\end{algorithm}\n\n\n\\section{Results} \\label{sec:results}\n\nThis section contains results of our parameter estimation and shows the performance of our PDE-based strategy. \n\n\\subsection{Parameter values} \\label{sec:parametervalues}\nWe test our model using flow data from the Swedish river S\u00e4var\u00e5n\\footnote{The choice of S\u00e4var\u00e5n as our data source is made for no other reason than that the river lies close to Ume\u00e5 University where most of the work on this paper was done.} during the years $1980-2018$, of which we use the first 35 years ($1980-2014$) for model calibration and the last four ($2015-2018$) for benchmarking. We also use data from $1980-2014$ for a long term evaluation of the performance of our scheme. Leap days are excluded in favour of a coherent presentation of the results.\n\nWe study optimization over a one year horizon with possibility to change the state of production once every day, i.e.~$T=365$ days and $\\Delta t= 1$ day. Our main reason for sticking with this coarse time-discretization is that the flow data available to us is given with one data point per day.\n\n\\begin{remark}\nFor hydropower plants with several units, i.e.,~plant II, the total flow of S\u00e4var\u00e5n is somewhat low and a non-regulated river with higher flow would be more suitable for our model. For simplicity of the presentation however, we have chosen to limit ourselves to a single river. Specific parameters, such as running costs, deterioration costs, etc., vary over time and between different hydropower plants and the parameters used here should, as with the choice of flow data, be seen merely as an example.\n\\end{remark}\n\n\\subsubsection*{Flow parameters}\nFrom the historical flow data\\footnote{Flow data were gathered from SMHI (http:\/\/vattenwebb.smhi.se\/station) on June 15, 2020.} of S\u00e4var\u00e5n, the estimated values of the parameters in \\eqref{eq:varacf} are $\\kappa = 0.0208$ and $\\sigma=0.1018$. Figure \\ref{fig:flowmodel} shows $e^{r_t}$ and the flow during $2015-2018$ and Figure \\ref{fig:simulatedflows} shows independent random realisations of \\eqref{eq:sde} with these parameters.\n\n\\begin{figure}[h!]\n\\begin{subfigure}{.50\\textwidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{flow.jpg}\n\\caption{The seasonal model $e^{r_t}$ together with actual flows $2015-2018$.}\n\\label{fig:flowmodel}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{simulated_flows.jpg}\n\\caption{The seasonal model $e^{r_t}$ together with 5 random realisations of \\eqref{eq:sde}.}\n\\label{fig:simulatedflows}\n\\end{subfigure}\n\\caption{A visual comparison of the seasonal model $e^{r_t}$, solutions to \\eqref{eq:sde} and actual flow during $2015-2018$.}\n\\end{figure}\n\n\\subsubsection*{Power plant parameters}\n\n\nThe parameters used in modeling our power plants are summarized and presented in Table \\ref{tab:constants}.\n\nStarting with the efficiency curve of our power plant units, i.e., the coefficients of \\eqref{eq:efficiency-curve-assumption}, Figure \\ref{fig:johej_eta} shows measured efficiency of two Swedish Kaplan units together with the assumed efficiency curve in \\eqref{eq:efficiency-curve-assumption} with least squared fitted coefficients $\\alpha$ and $\\beta$. Based on this data and the flow in S\u00e4var\u00e5n, we found it reasonable to choose the unit parameters as in Table \\ref{tab:constants}.\n\nThe running cost is estimated from \\cite{S12} to be approximately $1\/5$ of the electricity price. The power of our unit is approximately $500$ $kW$, and with $P_t=P_0\\equiv1$ it is thus reasonable to set $c_{run} = 100$~$m.u.\/h$. It is difficult to estimate the cost $c_{low}$ reflecting the running loss when the machine is run on too little water. The rotational speed of the turbine may drop so that the frequency of the produced electricity falls, resulting in a non-sellable production. We choose a value simply by multiplying $c_{run}$ with 10 and note that such a choice forces our algorithm to shut down the plant efficiently when the water supply fails.\n\n\n\nIt is not a trivial task to estimate a reasonable value of the switching costs. It may heavily depend on machine parameters related to the intake and specific properties of the turbine, tubes and the generator. Cost of personnel and environmental parameters such as local fish habitat may also be included, as well as the type of contract to which the electricity is sold. Due to these difficulties, we handle the switching costs as a parameter. In particular, we assume the switching costs to be constant and study the impact of varying this constant in Section \\ref{sec:results}.\n\nMoreover, when considering power plant 2, we assume that the cost of switching directly from state $0$ to state $2$ is cheaper than going through the intermediate state $1$, and vice versa. In particular, this implies that at any fixed time $t$, at most one switch is made. Exactly how much cheaper a direct switch from mode $0 \\to 2$ (or vice versa) should be compared to the alternative $0 \\to 1 \\to 2$ depends on the actual power plant under consideration. Here, we simply assume that the alternative route is $50 \\%$ more expensive. We summarize the switching costs in Table \\ref{tab:swcosts}. Note that we have assumed all switching costs to be positive. When applicable, negative switching costs representing a gain rather than a cost when, e.g.,~reducing production capacity or moving to a more environmentally friendly production mode, can be used as well.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale = 0.8]{efficiency.eps}\n\\caption{Efficiency values of two Swedish Kaplan units with design flow of approximately 100 $m^3\/s$ (red squares) and 170 $m^3\/s$ (black diamonds).\nThe curves are $\\eta(Q)$ defined in \\eqref{eq:efficiency-curve-assumption} with least square fitted coefficients.\nFor the smaller unit $\\alpha = 0.917$, $\\beta = 0.430$ (red dotted) and for the larger unit $\\alpha = 0.935$, $\\beta = 0.464$ (black solid).}\n\\label{fig:johej_eta}\n\\end{figure}\n\n\\begin{table\n\\begin{center}\n \\begin{tabular}{| l | l | l | l | }\n \\hline\n$P_0$ & 1 $[m.u. \/kWh]$ &$c_{low}$ & $ 1000 $ $[m.u.\/h]$ \\\\ \\hline\nh & 5 $[m]$ & $c_{run}$ & $ 100 $ $[m.u.\/h]$ \\\\ \\hline\nT & 365 [days] & $Q_{min}$ & 5 $[m^3\/s]$ \\\\ \\hline\n$\\alpha$ & 0.92 &$Q_{max}$ &13 $[m^3\/s]$ \\\\ \\hline\n$\\beta$ & 0.45 & $Q_{d}$ &10 $[m^3\/s]$ \\\\ \\hlin\n\\end{tabular}\n\\end{center}\n\\caption{Parameter values used in our numerical investigation.}\n\\label{tab:constants}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{| l | c | c | c |}\n \\hline\n$c_{ij} $ & $j= 0$ &$j= 1$ & $ j= 2 $ \\\\ \\hline\n$i=0 $ &$0$ & $C $ & $1.5C$ \\\\ \\hline\n$i=1$ &$ C$ & $0$& $C $\\\\ \\hline\n$i=2$ & $1.5C$ & $C$& $0$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Relative switching costs. $C$ is a fixed constant determined in Section \\ref{sec:results}.}\n\\label{tab:swcosts}\n\\end{table}\n\\subsubsection*{Forecasts}\nAs already stated in the introduction, our main purpose is to highlight the use of optimal switching theory in production planning for ROR hydropower plants. The ambition is \\textbf{not} to provide methods for forecasting river flow, and to avoid such discussions, we will simply use the true flow as forecast. However, we keep the stochastic component in the dynamics \\eqref{eq:sde!} unchanged so that, even with forecast applied, our model does not know the future flow with certainty. Instead, this ``forecast'' only provides our model with a more accurate estimate of the average flow during the validity period of the forecast. We depict the impact of forecasts in Figure \\ref{fig:forecasts}.\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{forecast1.jpg}\n \\caption{Plot of $e^{r(t)}$, the true flow for year 2015, \\\\and $e^{g_k(t)}$ for $k=120$, $l=5$ and $\\ell=10$.}\n \\label{fig:sub-first}\n\\end{subfigure} \\hfill\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{forecast1zoom.jpg}\n \\caption{Zoom of figure (a)}\n \\label{fig:sub-second}\n\\end{subfigure}\n\n\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{forecast2.jpg}\n \\caption{Plot of $e^{r(t)}$, the true flow for year 2015, \\\\and $e^{g_k(t)}$ for $k=120$, $l=10$ and $\\ell=20$.}\n \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{forecast2zoom.jpg}\n \\caption{Zoom of figure (c)}\n \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{The impact of forecasts in relation to the long-term expected value $r_t$. Note that a stochastic term is also present in the flow model \\eqref{eq:sde!} and hence the plotted figure above only corresponds to an estimate of the expected value of the future flow.}\n\\label{fig:forecasts}\n\\end{figure}\n\n\n\\subsection{Comparison of different strategies}\nWe will benchmark the performance of our strategy to the \\textit{a fortiori} optimum, i.e.,~the optimal strategy in hindsight, with all information available. From a practical point of view, this value can only be achieved with certainty by ``looking into the future'', using the future flow of the river when making decisions. As this is of course impossible in practice, our comparison is slightly skew to the disadvantage of the model presented here. However, this value, the theoretical maximum output of the plant, is indeed achievable and should therefore be considered as the ultimate goal in any attempt to construct a production strategy. We emphasize and stress that, although the benchmark strategy can be found only in hindsight, our PDE-based strategy uses no other information when making a decision at time $t_k$ than historical information up to that point and the forecast starting at $t_k$. \n\nAs a comparison, we also show the result of using a \\textit{na\\\"{\\i}ve strategy} in which the manager always switches to the production mode which momentarily has the highest payoff. To ease the presentation, we assume in all cases that the starting state is $0$, i.e.,~that the plant is ``off'' at the beginning of the planning period.\n\nTo get comparable results, we first calculate a fixed constant value $D$, depending on the power plant under consideration, but not on the switching costs or the flow of the river. More precisely, $D$ is calculated as the profit generated by the plant if it works at maximum capacity for a full year without interruptions (and starting in the most beneficial state), i.e.,\n$$\nD= f_i(Q_{max}, P_0, t) * 365,\n$$ \nwhere $i =1$ or $i=2$ depending on the plant under consideration and $t \\in [0,365]$ is arbitrary (since \\eqref{eq:payoff} and \\eqref{eq:f3} are independent of $t$). After determining $D$, the switching cost constant $C$ (cf. Table \\ref{tab:swcosts}) is taken as a fixed percentage of $D$. \n\nLastly, for comparison of strategies, we use the quotient $\\gamma(\\mu)$, defined as the total payoff from the strategy $\\mu$ divided by $D$,\n$$\n\\gamma(\\mu) = \\frac{J_0(q_0,0,\\mu)}{D},\n$$\nwhere $q_0$ is the flow at the starting time $t_0=0$ and $J_0$ is as defined in \\eqref{eq:OSP}. \n\n\nResults are provided for $T=365$ days with forecast lengths $l \\in \\{0,5,10\\}$ days and with linear return to the long-term mean $e^{r_t}$ over $\\ell=20$ days. Here, $l=0$ means no forecast. We give detailed descriptions of the suggested strategies for 2015 and show summarized results for $2016-2018$. Moreover, we provide results on the long term performance of our strategy by comparing results from the years $1980-2014$ for a fixed parameter set\n\\footnote{It should be noted that, for convenience, this data set is the same as that used for calibrating parameters. Thus, for each year in the long term evaluation, the data tested is part of the data used for calibration. However, a single year out of the 35 used has minimal impact on the end calibration and the long term results are therefore still valid.}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{savarp1resulty1}\n\n\\caption{Strategies for plant I during 2015 with $C\/D=0.01$. The black line represents water flow and the yellow and red lines represent running payoff for states 1 and 0, respectively. Yellow circles (asterisks) represent the action of moving to state 1 and red circles (asterisks) the action of moving to state 0 under the PDE strategy with $l=10$ (optimal strategy).}\n\\label{fig:p1y1results}\n\\end{figure}\n\n\\begin{table}\n \\centering\n\\begin{tabular}{l | l}\n\\textbf{Strategy} & \\textbf{(Day of action, Move to state)} \\\\\n\\hline\nOptimal & (102,1) \\quad (225,0) \\\\ \\hline\nPDE, $l=0$ & (103,1) \\quad (226,0) \\\\ \\hline\nPDE, $l=5$ & (103,1) \\quad (225,0) \\\\ \\hline\nPDE, $l=10$ & (103,1) \\quad (225,0) \\\\ \\hline\nNa\\\"ive & (102,1) \\quad (225,0) \\quad (262,1) \\quad (265,0) \\quad (324,1) \\quad (327,0) \\\\ \\hline\n\\end{tabular}\n\\caption{Strategies for plant I during 2015 with $C\/D=0.01$.}\n\\label{tab:strategyp1}\n\\end{table}\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{savarp1relativesy1}\n \\caption{2015}\n \\label{fig:sub-first}\n\\end{subfigure} \\hfill\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{savarp1relativesy2}\n \\caption{2016}\n \\label{fig:sub-second}\n\\end{subfigure}\n\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{savarp1relativesy3}\n \\caption{2017}\n \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\linewidth]{savarp1relativesy4}\n \\caption{2018}\n \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Relative payoff for plant I during $2015-2018$ as a function of $C\/D$.}\n\\label{fig:relativesp1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[scale=0.6]{averageresultsp1}\n \\caption{The PDE strategy consistently gives payoff close to the optimal. Here, $C\/D=0.01$ and $l=10$. The average quotient $\\gamma$ for plant I over the years $1980-2014$ is $0.2674 $, $0.2628$, and $0.2466$ for the optimal strategy, the PDE strategy and the na\\\"{\\i}ve strategy, respectively. (Lines are present for visual aid only.)} \n \\label{fig:averageresultsp1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{savarp2resulty1}\n\n\\caption{Strategies for plant II during 2015 with $C\/D=0.01$. The black line represents the water flow and the green, yellow and red lines represent running payoff for states 2, 1, and 0, respectively. Green circles (asterisks) represent the action of moving to state 2, yellow circles (asterisks) the action of moving to state 1, and red circles the action of moving to state 0 under the PDE strategy with $l=10$ (optimal strategy).}\n\\label{fig:p2y1results}\n\\end{figure}\n\n\\begin{table}\n \\centering\n\\begin{tabular}{l | l}\n\\textbf{Strategy} & \\textbf{(Day of action, Move to state)} \\\\\n\\hline\nOptimal & (102,1) \\quad (111,2) \\quad (159,1) \\quad (225,0) \\\\ \\hline\nPDE, $l=0$ & (103,1) \\quad (112,2) \\quad (162,1) \\quad (226,0) \\\\ \\hline\nPDE, $l=5$ & (103,1) \\quad (112,2) \\quad (162,1) \\quad (225,0) \\\\ \\hline\nPDE, $l=10$ & (103,1) \\quad (112,2) \\quad (162,1) \\quad (225,0) \\\\ \\hline\nNa\\\"ive & (102,1) \\quad (111,2) \\quad (159,1) \\quad (225,0) \\quad (262,1) \\quad (265,0) \\quad (324,1) \\quad (327,0) \\\\ \\hline\n\\end{tabular}\n\\caption{Strategies for plant II during 2015 with $C\/D=0.01$.}\n\\label{tab:strategyp2}\n\\end{table}\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{savarp2relativesy1}\n \\caption{2015}\n \\label{fig:sub-first}\n\\end{subfigure} \\hfill\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{savarp2relativesy2}\n \\caption{2016}\n \\label{fig:sub-second}\n\\end{subfigure}\n\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{savarp2relativesy3}\n \\caption{2017}\n \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{savarp2relativesy4}\n \\caption{2018}\n \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Relative payoff for plant II during $2015-2018$ as a function of $C\/D$.}\n\\label{fig:relativesp2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\includegraphics[scale=0.6]{averageresultsp2\n \\caption{The PDE strategy consistently gives payoff close to the optimal. Here, $C\/D=0.01$ and $l=10$. The average quotient $\\gamma$ for plant II over the years $1980-2014$ is $0.1708 $, $0.1624$, and $0.1244$ for the optimal strategy, the PDE strategy and the na\\\"{\\i}ve strategy, respectively. (Lines are present for visual aid only.)}\n \\label{fig:averageresultsp2}\n\\end{figure}\n\nThe different production schemes (optimal, PDE-based, na\\\"{\\i}ve) for plant I (plant II) during 2015, $l \\in \\{0,5,10\\}$ and $C\/D=0.01$ are presented in Figure \\ref{fig:p1y1results} (Figure \\ref{fig:p2y1results}) and Table \\ref{tab:strategyp1} (Table \\ref{tab:strategyp2}). The relative payoffs as a function of $C\/D$ for $l \\in \\{0,5,10\\}$ are given for all years in Figure \\ref{fig:relativesp1} (Figure \\ref{fig:relativesp2}). The long term performance of the PDE-based strategy for plant I (II) with $C\/D=0.01$ is presented in Figure \\ref{fig:averageresultsp1} (Figure \\ref{fig:averageresultsp2}).\n\nThe PDE-based strategy in most cases performs very close to the optimal strategy, with a difference of less than 2 \\% (5 \\%) from the theoretical maximum in the long term tests for plant I (plant II) with $C\/D=0.01$ and 10 days forecast. The most common mistake of the PDE-based strategy compared to the optimal is delaying the decision to change production mode. In all but a few cases, longer forecast also means better results. \n\n\n\\section{Discussion} \\label{sec:discussion}\n\n\nThe optimal switching theory is designed for maximizing the average payoff over a long period of time, but as our results show, it performs well also on single years. When comparing our strategy to the optimal one, we see that the differences between the strategies arise as our strategy occasionally recommends switching mode too late due to uncertainty regarding the future flow, see, e.g., Figure \\ref{fig:p2y1results}. Most often, the decision is only late by one day and with a finer time discretization, these differences would most likely disappear or, at least, the discrepancy would be much smaller. We also see that by introducing forecasts, we are able to remove this gap entirely in many cases, see Figures \\ref{fig:relativesp1} and \\ref{fig:relativesp2}. \n\nOur (artificial) forecast includes uncertainty from the first forecasted day and reduction in this uncertainty, which is reasonable and possible by, e.g., upstream measurements, would also help removing any delay in the decision making. Indeed, in our SDE model, we assumed the uncertainty to be the same regardless of if a forecast was introduced or not. If the uncertainty of the forecast is known, one could introduce a new parameter $\\sigma^f_k$ in a similar fashion as for $b^f_k$ and let the forecast influence also the stochastic volatility of the flow, having lower volatility close to the current time $t_k$ and increasing volatility further in the future. We have chosen not to alter the volatility $\\sigma$ during the forecast period, partly to keep our model as simple as possible, and partly to avoid the need to construct forecasts (which is outside the scope of the current paper). \n\nAlready without forecast, our PDE-based strategy outperforms the na\\\"{\\i}ve strategy in most cases, even for small values of $C\/D$, and in many cases also finds the truly optimal strategy or something very close to optimal. In the rare event that the na\\\"{\\i}ve strategy performs as good or better than the PDE-based strategy, it is because the na\\\"{\\i}ve strategy happens to be optimal. In these cases, the difference between the optimal and the PDE-based strategies is small. \n\nOur results show, as should be expected, that longer forecasts means better results. However, on a few occasions, this is not the case, see, e.g., Figure \\ref{fig:relativesp1} (c), where, for $C\/D \\geq 0.016$, we perform worse with a 10 day forecast than with a 5 day forecast. The reason for this (rather unsatisfactory) result is that, when large fluctuations up and down in the flow happens in a just a few days time, a longer forecast may capture both directions of the movement whereas a shorter only captures one direction, triggering a decision to open\/close if the current flow is at or close to a level at which different payoff functions intersect, whereas with longer forecast, the uncertainty introduced by forecasting also a rapid downward movement delays the decision slightly when the cost of switching is high. When run repeatedly over a large number of years, the decision made from the longer forecast would perform better \\textit{on average}, but it may come up short in a single year. Luckily, as in the comparison with optimal na\\\"{\\i}ve strategies, the deviation in the final payoff is small on these occasions.\n\nOur model is calibrated to a constant electricity spot price $P_0$ for convenience when interpreting results and we repeat that a time-varying deterministic electricity price causes no problems other than parsing the results. However, our model could also be calibrated to a random price process $P_t$ as well. There is no (theoretical) restriction in the number of underlying Markovian It\\^o processes our model can handle, so that allowing for such calibration is merely a question of computational power. In fact, not even the Markov property is a restriction as any discretized random process can be made Markovian at the cost of increasing its dimension. However, the cost of increasing the number of random sources is that the underlying optimization problem, which here is solved by PDE-methods, increases in dimensionality at the same rate. In practice, as long as the random sources are few, say 2 or 3, our approach based on numerical solutions of PDE can be used to find a solution. When the dimensionality increases even further, the PDE-methods become computationally heavy and other ways of attacking the resulting optimal switching problem may be preferable, e.g.~Monte Carlo-methods as in \\cite{ACLP12,BGSS20}. In the current setting, the algorithm for obtaining our strategy is run in only a few minutes on a standard laptop computer and is thus more than sufficient for the purpose of the current paper.\n\n\\subsection{Concluding remarks and future research} \\label{sec:concludingremarks}\nIn this paper we have, to the authors knowledge for the first time, used the mathematical optimal switching theory to create hydropower production plans which can incorporate random water flow and non-negligible costs of switching between different operational modes. Although our setup is somewhat simplified to keep the analysis of the results tractable, the results are satisfying, showing that automatic optimal switching schemes can perform close to the theoretical maximum already with small computational effort. Moreover, in our study the difference between our model and a na\\\"{\\i}ve approach increased with the number of available production modes, indicating that the decision support provided by optimal switching theory becomes increasingly valuable as the complexity of the underlying problem increases.\n\nAn interesting theoretical continuation of the work initiated in this paper would be to study hydropower plants with a dam or hydropower plants of pumped-storage type. At the practitioners' level, a natural next step would be to adapt the current scheme to a real hydropower plant and to benchmark its performance to the production strategy currently in use. \n\nIt is our firm believe that continued work in this direction, utilizing advanced stochastic control theory for production planning, will play an important role in making today's energy system more effective. Indeed, the theory outlined in this paper has potential to become as useful in the energy sector as the closely related theory of optimal stopping and singular- and non-singular control has become in financial mathematics.\n\n\n\\section*{Acknowledgements}\nThe work of Niklas L. P. Lundstr\\\"om was partially supported by the Swedish research council grant 2018-03743.\nMarcus Olofsson gratefully acknowledges the support from the Centre for Interdisciplinary Mathematics at Uppsala University.\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMilitary confrontation is one of the key factors that have shaped human history. While instances of large-scale interstate warfare have decreased in the post-World War era, internal wars and low-intensity conflicts carried out by non-state armed groups (NAGs) against nation-states have become increasingly common \\cite{San-Akca:2016,Maoz:2012,Horowitz:2014,Phillips:2018,LaFree:2009,Phillips:2015,Freilicha:2015,Pinker:2012,Gleditsch:2013,Gleditsch:2016,Byman:2001,Kalyvas:2010}. NAGs include rebel, insurgent, guerrilla, or terrorist groups that engage in violent activities targeting the government, citizens, or institutions of nation-states. Many of these NAGs are cultivated by a complex network of supporting host states (HSs) --- states external to the locus of the internal \nconflict between the NAG and the target government. Examples include U.S. support to the Contras in Nicaragua in the 1980s, Israeli and Iranian support for Kurdish rebels in Iraq during the 1960s and 1970s, and NATO support for the Libyan rebels in 2011 \\cite{San-Akca:2016}. The\nHSs support NAGs by providing them with military and economic aid, sanctuary to leaders or members of NAGs, logistics, or training.\nThe bipartite relation between NAGs and host states has become a common mode of foreign policy conduct \\cite{San-Akca:2016,Maoz:2012,Gleditsch:2016,Byman:2001,Kalyvas:2010}, yet we lack reliable real-time data on NAG-HS relations as most of these interactions are covert; public information usually emerges \\emph{ex-post facto.} The few existing studies have focused on specific NAG-HS relations \\cite{LaFree:2009,Freilicha:2015,Popovic:2017,Salehyan:2010,Gleditsch:2008}. \nThe entire ecosystem of NAG and HS interactions and its evolution over time remains an uncharted territory.\nRevealing the underlying patterns and mechanisms behind the NAG-HS support network is important for analytical insights and policy implications.\n\nWe construct the global network of bipartite support between NAGs and HSs involving all military conflicts over the period of 1945-2010 and measure its global topological characteristics. Two types of links are present in the network: A HS can provide a NAG with either intentional (``active'') or de facto (``passive\") support. Active support entails a deliberate decision by a HS to form an alliance with a NAG in order to further common interests, such as the Israeli and Iranian support for the Kurds in the 1960s and 1970s \\cite{maoz2006}. Passive support refers to situations where a NAG operates or extracts resources from a HS, while the HS does not engage in a deliberate action or create channels to provide any support.\nExamples of passive support include the IRA operations in the Irish community in the US during the 1980s and 1990s \\cite{guelke1996}, or the fundraising for Palestinian insurgents among Islamic communities in the US prior to September 11, 2001 \\cite{levitt2008}.\nLikewise, a NAG may operate in a HS whose institutions are weak and incapable of resisting the NAG's activities (e.g., the PLO operation in Lebanon over the 1970-1982 period) \\cite{San-Akca:2016, San-Akca:2014}. \n\n\nBy tracking the changes in the players and supporting relations, we find that the network evolves via a mechanism of generalized preferential attachment and detachment: highly connected players are more likely to both gain and lose connections relative to peripheral players. Such evolution leads to characteristic patterns typically associated with ecological networks, such as plant-pollinator networks, seed-frugivore networks \\cite{Bascompte:2003,Olesen:2007} and \nhost-parasite networks \\cite{Poulin:2010,Graham:2009}, which similarly involve tradeoffs through cooperative or exploitative relations. \nIn particular, we find a robust nested and modular architecture over a broad time span. Nestedness entails a nonrandom pattern of generalist-specialist structure, which is a crucial factor determining the structural stability of a network \\cite{Okuyama:2008,Bastolla:2009,Rohr:2014}. Modularity indicates the extent to which the network is organized into clearly delimited clusters. Such patterns are also frequently observed in socio-economic systems, from the network of bankers \\cite{May:2008} to the designer-manufacturer network of New York garment industry \\cite{Saavedra:2009}. \n\nUsing a temporal community detection algorithm we find that there are nine major modules that persist over time and many small, short-lived modules. The persistent modules reveal both regional and trans-regional composition and historical connections. These major modules display significant variation in their membership, whereas the transitory ones show high consistency. Furthermore, we identify the roles of nodes and find that if a NAG or HS is involved in both the active and passive support subnetworks they play the similar role of hub or peripehral in both subnetworks. The discussion section addresses some of the implications of these findings. We suggest that our understanding of robustness and interventions in ecological bipartite networks may be useful for developing efficient disruptions of malign relations in the NAG-HS network, where we can treat the support as mutualistic and parasitic relationships. \n\n\n\n\\section{Results}\nWe focus on the relations between NAGs and external state supporters, extracted from the Dangerous Companions Database covering the time period from 1945 to 2010 \\cite{San-Akca:2016}. This considers only violent NAGs (that is NAGs whose operations resulted in 25 or more deaths per year of operation). A HS is a country that provides any type of material support to a given NAG, including safe haven to NAG members, training, or military or financial aid. A temporal network is constructed from these bipartite relations, including both active and passive support. During the time period of study, the network experiences substantial growth with an almost monotonic increase peaking in the early 1990s, corresponding to first years in the post-Cold War era, followed by a decrease in both numbers of nodes and links (Fig. \\ref{NAG_CDF_H_N_INSET}a and Fig. S1 in SI), whereas the connectance (link density) shows an opposite trend (Fig. \\ref{NAG_CDF_H_N_INSET}b).\n\n\nFirst we track the assembly and disassembly of the NAG-HS network, focusing on the relative probabilities of acquisition and disassociation of actors (NAGs and HSs) and links (NAG-HS relations). Then, we show that the network evolves to have\na particular architecture exhibiting both nested and modular patterns.\nFinally, based on this characteristic architecture, we determine the roles of individual NAGs and HSs.\\\\\n\n\n\n\\noindent\\textbf{Fitness of actors.} \nWe consider two indicators of a NAG's or HS's fitness. The first indicator is based on duration over time, specifically the lifespan of a NAG or the hosting time of a HS (how long the state participates in the supporting activities in any capacity). Generally, we find that these follow an exponential-like distribution, as shown in Fig. \\ref{NAG_CDF_H_N_INSET}c and \\ref{NAG_CDF_H_N_INSET}d. The second indicator is based on a snapshot of time, specifically the extent of connectivity of the NAG or HS measured by its degree, $k_{NH}$ or $k_{HN}$ respectively. The insets of Fig. \\ref{NAG_CDF_H_N_INSET} show the Pearson coefficients $\\rho$ between the lifespan\/hosting time and node degree ($p<0.0001$ using a two-sided t-test); a NAG tends to endure longer with more supporting HSs, while the hosting time of a HS is more likely to increase the more NAGs rely on it.\n\nThese results suggest that ``successful\" NAGs---with longer survival time---depend on their ability to extract external support. Likewise, ``active\" HSs---states with longer record of support for NAGs---tend to be generalists in that they support more NAGs, compared to the activity duration of ``specialist\" HSs.\\\\\n\n\n\\begin{figure}[t]\n\\hskip 1.8cm\n\\includegraphics[width=0.75\\textwidth]{NAG_Fig1.png}\n\\caption{\\label{NAG_CDF_H_N_INSET} Evolving NAG-HS support network. \\textbf{a}, Changing size of the bipartite network over time, in terms of the number of all actors and the components. \\textbf{b}, Network connectance, defined as the ratio of the actual number of links against the maximum possible number of links ($n_{NAG}*n_{HS}$). Inset: fractions of links that correspond to active, passive and both types of support, respectively. \\textbf{c, d}, Cumulative probability distributions of life spans of NAGs and hosting time of host states (exponential-like distribution). Insets: Correlation of life span or hosting time and the number of attached actors (degree). }\n\\end{figure}\n\n\n\\noindent\\textbf{Network assembly and disassembly.} To understand how the actors (nodes) and the associated relations (links) attach to and detach from the network, we examine the assembling and disassembling processes that unfold on it. Specifically, we observe how actors acquire (``$+$\") and remove (``$-$\") counterpart actors by comparing the adjacency matrices in two consecutive years, as illustrated in Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}a. From the perspective of HSs, the histograms show that both attachment and detachment of NAGs occur more frequently at HSs with a lower degree (Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}b and d), but this is biased since the degree distribution of the incumbent HSs is already skewed before those events. It is thus particularly revealing to calculate the relative probability of a HS gaining or losing a link from a NAG, which is equal to the absolute probability divided by the probability from random selection \\cite{Newman:2001,Saavedra:2008} (with that any unevenness in the degree distribution is \noffset), over the entire time span of 66 years. A nonrandom distribution of relative probability is observable if there is an intrinsic bias in the attaching or detaching process. \n\nIndeed, both attachment and detachment of NAGs are found to occur preferentially at incumbent HSs of higher fitness as measured by their degrees. First, we consider the attachment of NAGs, that is, the relative probability for an incumbent HS to acquire links from newly arriving NAGs (NAGs that start to receive support from at least one HS) as a function of the degree of the HS, denoted $T^+(k_{HN})$, and find that a newcoming NAG is more likely to attach to a generalist (high-degree) HS. Explicitly, the relative probability $T^+(k_{HN}) = P^+(k_{HN})\/P_0^+(k_{HN})$ is defined as the ratio of the absolute probability $P^+(k_{HN})$ that a NAG joining the network in year $t$ connects to an incumbent HS that has already been supporting $k_{HN}$ NAGs divided by the probability $P_0^+(k_{HN})$ that a HS with degree $k_{HN}$ is uniformly selected at random (see Methods). $T^+(k_{HN})$ is shown \nin Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}c, which is approximately fit by $T^+(k_{HN})\\sim k^{0.86}_{HN}$, indicating a \n(slightly sublinear) preferential attachment (PA) process \\cite{Yule:1925,Barabasi:1999}.\nTurning to detachment, we find the same tendency holds. Specifically, \na NAG departing the supporting network (a NAG has no support from any HS or has been dismissed) is more likely to detach from a counterpart HS supporting more NAGs, as shown by the linearly increasing relative probability $T^-(k_{HN})\\sim k^{1.1}_{HN}$ with the degree of the latter (see Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}f). This (slightly superlinear) preferential detachment (PD) is somewhat counter-intuitive as it suggests that the network tends to disassemble at higher-fitness nodes. \n\n\\begin{figure}[t]\n\\hskip 1cm\n\\includegraphics[width=0.84\\textwidth]{fig2_new2_newfit.png}\n\\caption{\\label{NAG_REL_PROB_TK_RK_ALL2}Network assembly and disassembly. \\textbf{a}, Link addition (``$+$\") and removal (``$-$\") as the adjacency matrix evolves from $M_{t-1}$ to $M_t$ in consecutive years. $M_t$ is divided into four blocks ($B_1$ to $B_4$) by separating incumbent and newly joining actors. \\textbf{b, e}, Histograms for the frequencies of acquiring and losing links from NAGs by HSs with degree $k_{HN}$ (zero-degree for new HSs). Inset of panel b: frequencies of adding links in the four blocks. \\textbf{c}, Relative probability $T^+(k_{HN})$ that an incumbent HS with $k_{HN}$ previous connections acquires links from newly joining NAGs. \\textbf{d}, Relative probability $R^+(k_{HN})$ that an incumbent HS acquires new links from an incumbent NAG. \\textbf{f}, Relative probability $T^-(k_{HN})$ that an incumbent HS loses links from NAGs that depart the network. \\textbf{g}, Relative probability $R^-(k_{HN})$ that an incumbent HS loses links from an incumbent NAG.} \n\\end{figure}\n\nSecond,\nbesides the contribution from newcoming NAGs, an incumbent NAG (a NAG that receives support from at least one HS) of the supporting network may also add or delete links to another incumbent HS over time. To include such cases, \nwe measure the relative probability $R^+(k_{HN})$ (slight superlinear) that an incumbent HS that has $k_{HN}$ links in the year prior to $t$ will acquire links from incumbent NAGs in year $t$, excluding those from newcoming NAGs (Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}d). Similarly, Fig. \\ref{NAG_REL_PROB_TK_RK_ALL2}g shows the case for incumbent HSs losing links from incumbent NAGs (remaining in the network), as indicated by $R^-(k_{HN})$ (linear). All these attachment and detachment show that NAGs interact preferentially with HSs of greater fitness. We can also observe the tendencies from the perspective of NAGs, in terms of acquiring and losing links from HSs. Yet, the dependencies are more obscure (though the overall positive dependencies are still observable), due to the paucity of data points at some degrees (see Fig. S5 in SI). \n\nThe above analyses thus show that the assembly and disassembly of the supporting network, including joining and departure of NAGs, and rewiring between incumbent actors, proceed with propensities that are essentially dependent on nodal degree, which seems to reflect the fitness of the HS supporters. \nHowever, these analyses also show two conflicting processes:\nwhile the attachment is driven in attempt to maximize individual fitness, having higher fitness can also lead to higher possibility of losing counterpart actors, which has not been observed in other bipartite network disassembly \\cite{Saavedra:2008}. \nAs discussed later, this could reflect the fact that an actor with many alliances may benefit from having fewer interactions of higher quality.\n\\\\\n\n\\noindent\\textbf{Nested structures.} Despite significant variations in the network size (numbers of actors and links), the network exhibits a robust nested and modular architecture over a long time span. The measure of nestedness simply reflects to what extent actors share the same set of partners. It is calculated with the NODF metric (``nestedness metric based on overlap and decreasing fill\") \\cite{AlmeidNeto:2008}. To show how it evolves, we calculate the nestedness of the aggregated network in a 5-year sliding window, as shown in Fig. \\ref{NG_NODF_MOD_LL_CORR}a (this is to avoid statistical fluctuation due to the sparsity of entries in single yearly network slices). Between 1945 (end of World War II) and 1975 (end of Vietnam War), the nestedness keeps at a level that is comparable to that of a random network. Starting from 1975, the empirical nestedness score exceeds significantly the random scores, peaking in 1987 (towards the end of the Cold War). The increase parallels the cumulative broadening of degree distribution due to the preferential attachment and detachment processes (see the comparisons in Fig. S4 in SI). This is then followed by a dip and then upswing back to a similarly high level in 2010. For comparison, random networks are used as the null model, where the probability of each entry of the adjacency matrix being occupied is the average of the occupation probabilities of the row and column \\cite{Bascompte:2003}. The significantly high nestedness implies that actors (say NAGs) are more likely to share common partners (say HSs) which tends to make these common partners generalists. This renders a highly unbalanced structure of relationships: specialists (say NAGs only focusing on few HSs) are more likely attached to generalists (HSs supporting many NAGs) rather than other specialists (HSs supporting only few NAGs). \n\n\n \n\n\\begin{figure}[tb]\n\\hskip 0.3cm\n\\includegraphics[width=0.95\\textwidth]{NG_Fig3_new.png}\n\\caption{\\label{NG_NODF_MOD_LL_CORR}Evolution of structural measures of the temporal NAG-HS network: (a) nestedness NODF and (b) modularity Q. The nestedness is measured for the 5-year aggregated networks in a sliding window; the modularity is obtained by applying a temporal detection algorithm (note that the optimization algorithm can result in slightly different partitions of the same network). The results for the empirical network are compared with corresponding null models: I. average occupation probability (labeled ``null occu.\") and II. random slice permutation (labeled ``null perm.\"). Confidence interval of one standard deviation is used. (c) Negative correlation of modularity and nestedness in the NAG-HS supporting network, in comparison to the random relation between the two measures for the null (occu.) model in the inset. (d) Temporal \ncommunity detection algorithm. The example temporal bipartite network (circles for NAGs and squares for HSs) consists of four time slices that are linked by connecting nodes in one slice to themselves in the adjacent slices (demonstrated for nodes $i$ and $j$). The partitioning is based on the entire temporal network and the yearly modularity is then obtained according to the partition in each slice (each color representing one module). }\n\\end{figure}\n\nA heuristic explanation can be suggested for the formation of the nested architecture, following the principle of fitness maximization in ecology \\cite{Suweis:2013,San-Akca:2009}: For those NAGs and HSs maintaining mutualistic relations, the successful military action taken by a NAG may improve the status of the counterpart HS and thus increase the state's propensity and willingness to support more NAGs. \nAnd the successful NAG may also attract support from more HSs intending to confront their rivals in this way. Similarly, for parasitic NAG-HS relations, the feasibility of one NAG gaining resources from a HS may encourage more NAGs to attach to that state. \nThis mechanism is weakened by the detaching events, which however occur much less than the attaching events up to the middle of 1990s (see the number of links in Fig. S1a). This has led to ``net\" preferential attachment. \n\nA special note should be made to the indirect benefit for the NAGs in the nested configuration \\cite{Bastolla:2009,Maoz:2012,Phillips:2018}. The NAGs that engage in mutualistic support from the same HSs may avoid or reduce possible competition between these NAGs for territory or resources. Since successful military actions by a given NAG may improve the willingness of HS to support additional NAGs, NAGs can be thought to indirectly contribute to each other's support levels. This complies with the general ecological principle of mutualism \\cite{Bastolla:2009}: a more nested architecture is able to pack more actors in the community by minimizing competition among like-type actors. This higher capacity marks a more structurally stable architecture \\cite{Rohr:2014}. \\\\\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.98\\textwidth]{area_H_N_JJ_new.png}\n\\caption{\\label{NG_MOD_YR_HN}Temporal modular structure in the (a) HS and (b) NAG guilds. The height of the curve represents the number of actors in the submodule. The modules are numbered (vertical axis), among which 9 modules contain more than 3 different actors in each while the other 24 are small groups or individual actors. Inset: Positive correlation of the Jaccard indices of NAG- and HS-submodules of two of the major modules shows coherent fluctuation in their membership (Pearson coefficient $\\rho = 0.71$ with $p = 6.1 \\times 10^{-9}$ (two-sided t-test) for mod \\#19; $\\rho = 0.56$ with $p = 9.9 \\times 10^{-7}$ for mod \\#3).}\n\\end{figure}\n\n\n\n\n\n\\noindent\\textbf{Modular structures.} Concurrently, the network exhibits a nontrivial modular structure, in which actors are more densely connected to partners within the same module than across modules. The modularity measure is usually calculated by seeking a partition of an individual network that maximizes a modularity quality function \\cite{Newman:2006}. However, for this temporal network, partitioning the yearly slices of the temporal network individually would lose the inter-slice relation, making the modules in successive years independent of each other. Here we use a temporal community detection algorithm based on Markov stability, proposed by Mucha, et al \\cite{Mucha:2010}. Instead of partitioning separate time slices, this algorithm relies on the systematic optimization of a single quality function for all network slices in the 66 years. It has the merit of allowing the identification of the same module through all years and thus tracking the evolution of modules \\cite{Fortunato:2010}. \n\nThe modularity of the network each year as measured via the metric $Q(t)$ is significantly higher compared to that of corresponding null models, as shown in Fig. \\ref{NG_NODF_MOD_LL_CORR}b (note that the partition is not unique). There are two appropriate null models: 1. randomly permuted yearly slices (see Fig. S11a) and 2. randomly shuffled within-slice links that maintain the same degree sequence (see Fig. S11b). The comparisons demonstrate that both intra-slice linkage and the temporal order of network slices contribute to the nontrivial modular structure \\cite{Bassett:2011}. \nThe modularity value reaches a minimal value around 1990, a time point marking the end of the Cold War. \n\n\n\\begin{figure}[t]\n\\hskip -0.2cm\n\\includegraphics[width=1.02\\textwidth]{Y1970-2010_shapes.png}\n\\caption{\\label{Y1970-2000F} Partitioned NAG-HS networks for representative years (a) 1970 and (b) 2000. Nodes are coloured differently according to their bipartite module-membership, but the colour of the same module does not alter for different years. NAGs are represented by circles and HSs by squares.\n}\n\\end{figure}\n\n \n\n\nWe then seek a representative partition of the network. This is realized by using a stabilizing technique for robust module detection \\cite{Bassett:2013} and maximizing the Adjusted Mutual Information (AMI) (see Methods and Sec. 2 in SI). Our more detailed analyses are then based on this unique representative partition. The supporting network over the 66 years is partitioned into 33 modules, in which 9 major modules contain more than 3 different actors in each and the other 24 modules are separate nodes or tiny groups. In Fig. \\ref{NG_MOD_YR_HN}, we show the sizes (number of actors) of modules in all years in the HS- and NAG-guilds (the NAG- and HS-submodules of each bipartite module), respectively (see also the modular structures in representative years in Fig. S9 in SI). The contrast is sharp: while the major modules can persist for significantly long time spans, the others are merely temporary. \n\nWhile the 24 temporary modules are almost constant in their membership, all the major 9 modules have experienced significant turnovers (see module membership in 1970 and 2000 shown in Tab. S1 and S2 in SI). This is consistent with the general tendency found in the unipartite coauthorship and the phone-call networks \\cite{Palla:2007}: small modules survive when they are stationary in the membership, while large modules survive if they experience sufficient changes. For the 9 major modules, we further observe that the fluctuations in the NAG- and HS-submodules are coherently correlated. Quantitatively, we calculate the temporal Jaccard index to show the stationarity of the membership of the submodule, which compares the contents of in the successive submodules \n\\begin{equation}\n\\label{eq:jac}\nJ_{\\alpha}(t,t-1)=\\frac {|A_{\\alpha}(t)\\cap A_{\\alpha}(t-1)|}{|A_{\\alpha}(t)\\cup A_{\\alpha}(t-1)|}\n\\end{equation}\nwhere $A_{\\alpha}$ denotes the members of a specific submodule in focus \\cite{Palla:2007,Bassett:2013}. Thus, the Jaccard index describes the similarity of successive modules (auto-correlation), ranging from 0 (no common member) to 1 (identical). For all submodules of the 9 major modules, the Jaccard indices averaged over the active years turn out to be high (see values in Fig. S6 in the SI), indicating high cross-time stability of these modules. In individual modules, for 7 of the 9 major modules, the temporal Jaccard indices of the corresponding NAG- and HS-submodules, $J_{NAG}(t)$ and $J_{HS}(t)$, are positively correlated ($p<0.1$; two-sided t-test), as illustrated for the two modules with the highest correlation coefficients in the inset of Fig. \\ref{NG_MOD_YR_HN} (see Fig. S7 in the SI for all major modules). This implies that the constituents of most major bipartite modules are co-changing: NAGs and their attached states tend to join or leave a module together.\n\nViewing the nestedness and modularity measures jointly, we find a clear negative correlation between them through the 66 years, as shown in Fig. \\ref{NG_NODF_MOD_LL_CORR}c. This is strikingly different from that observed for the null model. Heuristically, on one hand, NAGs may favor a nested structure by attaching to common HSs so that the HS may increase the willingness to provide more support. On the other hand, the benefits of support could be diluted if the difference between the traits (including geographic, political and religious factors) of a pair of NAG and HS is too significant, which may lead them to fall into different modules. This is consistent with the ecological principle that at any given time, the network tends to find an optimal structure balancing nestedness and modularity, which trades off benefits against disadvantages \\cite{Bastolla:2009,Suweis:2013,Cai:2020}. \\\\\n\n\n\n\\noindent\\textbf{Roles of actors.} Based on the representative partition, we may further identify the roles of actors in the network setting. We calculate two characteristics of each actor, the standardized within-module degree $\\{d_i\\}$ and the participation coefficient $\\{c_i\\}$ \\cite{Guimera:2005}. The c-score is 0 if all the node's links are within its own module and approaches 1 if they are distributed evenly among modules. The two scores characterize the contribution of a node to the degree distribution (thus to the nestedness) and modular structure. Then the roles can be classified into four categories: peripherals, module hubs, connectors and network hubs. Figure \\ref{NAG_ROLES_ALL2}a shows the two scores of all actors in the networks in two typical years 1970 (I) and 2000 (II), respectively. It is notable that although the number of actors, including both NAGs and HSs, has increased from 98 to 160, the distribution of roles of all actors has remained very similar (mean value: $\\langle d^I \\rangle = \\langle d^{II} \\rangle = 0$, $\\langle c^I \\rangle = 0.1086$ vs $\\langle c^{II} \\rangle = 0.1402$, standard deviation: $\\sigma(d^I) = 0.9250$ vs $\\sigma(d^{II}) = 0.9582$, $\\sigma(c^I) = 0.2133$ vs $\\sigma(c^{II}) = 0.2348$). For each actor, we measure the fluctuation of its yearly role scores by the standard deviation throughout its lifespan (for NAG) or hosting time (for HS). As shown in Fig. \\ref{NAG_ROLES_ALL2}b, most actors turn out to be consistent in their d-scores, with $82\\%$ of all actors experiencing fluctuations less than 0.5. For the c-score, $67\\%$ of the actors remain the same or have a fluctuation less than 0.1. Most actors have thus kept their within-module degree (reflecting its fitness) unchanged and meanwhile stayed similar in the role of connecting different modules. In this sense, the roles of the actors (being hubs or peripherals) tend to be maintained in the network evolution regardless of addition and deletion of actors. \n\n\\begin{figure}[t]\n\\hskip 0.2cm\n\\includegraphics[width=0.95\\textwidth]{NAG_ROLES_ALL3.png}\n\\caption{\\label{NAG_ROLES_ALL2} Roles of actors. \\textbf{a}, Standardized within-module degree $d$ and participation coefficient $c$ for the network in year 1970 and 2000, respectively. According to the two scores, any node falls into one of the four predefined categories. \\textbf{b}, Histograms of fluctuation in d- (left panel) and c-scores (right panel) measured by the standard deviation. \\textbf{c}, Correlation of temporal d-scores (upper panel) and c-scores (lower panel) in the active and passive support subnetworks, for the actors that are involved in both relations. Insets: p-values of the temporal correlations. }\n\\end{figure}\n\nAnalogous to ecological mutualistic and parasitic relations, the NAG-HS network consists of active and passive support relations, which comprise two subnetworks, each containing only one type of support. Now we analyze the roles of the actors that are involved in both relations. We project the partition of the whole network onto the subnetworks (so the actor's module-membership in the subnetwork is determined by the partition of the whole network).\nWe calculate the d- and c-scores of actors in each year in each of the subnetworks, respectively, and then the Pearson coefficients between the yearly d-\/c-scores of all actors for the two subnetworks, as shown in Fig. \\ref{NAG_ROLES_ALL2}c. Except a few years, the correlation of the d-scores is significantly high in the majority of the 66 years $(p<0.01$; two-sided t-test). Similarly, the correlation of c-scores shows an increasing tendency over time from 1975 on and becomes significantly high from 1992 on $(p<0.01)$. \n\nThe correlation indicates that an actor involved in both active and passive relations generally plays a consistent role in both subnetworks. Concretely, if the actor plays a role as a peripheral\/connector\/module-hub\/network-hub in one subnetwork, it tends to play a similar role in the other subnetwork. In terms of ecology, this means that if a NAG relies heavily on mutually beneficial relations with supporting states, it is very likely to develop substantial parasitic relations as well, such as extracting funds, transporting weapons or finding safe heavens without the approval or notice of the HS \\cite{San-Akca:2009,Phillips:2015}. It is notable here that a HS supporting a set of NAGs actively may support another set of NAGs passively. For example, the US provided active support to the Mujahedin in Afghanistan in the 1980s, and it was exploited otherwise by the IRA, PLO, and Hamas during the same period.\\cite{guelke1996, levitt2008, abu1993}\n\n\n\n\n\\section*{Discussion}\n\nAs for any ecological system, identifying the resources that cultivate the actors is crucial for understanding the actors' life cycles and their collective behaviours. Although NAGs are not persistently owned or controlled by any state, they acquire resources from selected states in a nonrandom manner. The support network is self-organized through a process where every actor aims to maximize their individual fitness \\cite{Suweis:2013,Cai:2020}, which is known as ``cumulative advantages\" \\cite{deSollaPrice1976,Perc2014}. This is reflected by the preferential selection of partnerships and results in characteristic network patterns. This means that NAGs select HSs that are most likely to provide support for longer periods of time. For a similar reason, \nlong-lasting NAGs tend to attract HSs, which tend to cluster around the same NAGs. \n\n\nThe tendency of attaching to higher degree nodes is consistent with other network formation processes, such as in the unipartite scientific collaborations, the bipartite plant-pollinator relations \\cite{Bascompte:2003}, or the designer-contractor partnerships \\cite{Saavedra:2008}. Yet the generalized preferential detachment in the disassembly of the NAG-HS network is unique. It means that an actor with more associated partners (if mutualistic) or exploiters (if parasitic) may be more likely to break a link, which is most probably sub-optimal.\nThis may be due to the fact that an actor with more links tends to be more selective over time to minimize its dependence on others. Since supporting a NAG is an act of animosity and might attract retaliation, limiting interactions is the smart strategy to pursue by HSs. By the same token, if we consider that HSs might also constrain the operations of NAGs, the latter would try to keep their links limited. For example, Arafat was particularly careful about getting support from Syria and Egypt in the 1970s since both of these countries had strong leaders who prevented Palestinian infiltration to Israel from their own territory, for fear of Israeli retaliation. Instead he opted to form strong ties with Jordan (in the late 1960s) and with Lebanon in the 1970s and early 1980, because he believed that these states had weak governments that could be exploited \\cite{Sayigh:1997,Rubin:2003, maoz2006}.\n\n\nDespite continuous attachment and detachment, the self-organized nontrivial architecture persists over a wide time span. Moreover, the independent measures of nestedness and modularity turn out to be negatively correlated in the NAG-HS supporting network, but no correlation appears in the null model. While nestedness can result from the compound effect of PA and PD, modularity seems to be the consequence of matching of actors' traits, which are determined by geographical, political and religious factors. A similar negative correlation is also observed between the dyadic measures of a large ensemble of ecological mutualistic networks, across a wide range of geographical factors and constituents \\cite{Cai:2020}. The emergence of a nested, modular structure and their negative correlation suggests that this network of military support and mutualistic ecological networks may share a common ecological principle, that is, optimizing fitness in light of tradeoffs. \n\nThe temporal community detection algorithm confirms the existence of an almost unique partition of the support network (see Sec. 2 in SI). Most importantly, this approach is able to track the temporal changes of modules. Unlike the modules in the US congress roll call network \\cite{Mucha:2010}, partitioned by using the same algorithm, where a range of duration periods of modules appear, the modules in the NAG-HS supporting network are either long-lasting ones containing a variety of members or short-lived ones consisting of less than three pairs of interacting actors. This suggests that a module has to contain sufficiently many actors and internal interactions to persist; it cannot survive on only few isolated interactions.\n\nGeographic similarity is shared by most actors in each of the nine established modules (see Tab. S1 and S2 in SI for contents in two representative years), as most prominently\ndemonstrated by the HSs \nand would thus give credits to regional studies that focus essentially on internal dense interactions \\cite{San-Akca:2016,San-Akca:2009,Phillips:2015}. However, it is notable that remote actors beyond the region are also often present in the same module. Most of them are major powers with global force and financial projection capacity, connecting several modules, including US, UK, Russia (Soviet Union) and China. These were related to the Africa- or Mideast-centered regions (see Fig. \\ref{Y1970-2000F} for the partitions in 1970 and 2000). In addition, a few states may have transregional influences due to specific historical reasons, for instance, the leaders of GAM, a separatist group in Indonesia, are known to have lived in and operated from Sweden for most of the 80s and 90s before its dissolution in 2005 \\cite{Schulze:2004}. This suggests that the temporal partition of modules may reveal deeper relations of intimately connected actors beyond the simple methodology of regional division and thus justifies the modules as proper study objects.\n\n\nRevealing the nontrivial structure enhances our understanding of the roles of the actors. It allows us to capture the manner in which certain NAGs act as connectors, and in which some states form large connecting hubs for multiple NAGs. This may open the door for studying policies to cut off key actors that may cause malfunction of actors only within a module, or lead to cascades that would have extendable influence on multiple modules. These research avenues can benefit from the large body of recent studies on the percolation and cascading failure in modular and\/or nested networks \\cite{Saavedra:2011,Memmott:2004,Burgos:2007,KaiserBunbury:2010,Gleeson:2008,Snyder2020}, as well as on their structural stability and capacity \\cite{Rohr:2014}. \n\n\nIn all, quantifying the ecosystem of NAG-HS interactions provides a unique and important perspective towards the comprehension of the origin, evolution and termination of NAGs, the states supporting them, and their collective behaviours. Ecological organizing principles, originally discovered in natural systems, seem to be shared by this military network, as well as a variety of previously studied socio-economic networks \\cite{May:2008,Saavedra:2009}. We believe further investigations of the NAG-HS network will be fruitful for the domain of warfare and conflicts as well as developing theories to explain the origin of the ubiquitous network patterns beyond the domain.\n\n\n\n\n\n\n\n\n\n\\section*{Methods}\n\n\\textbf{Dataset of NAGs.} Our study objects are the bipartite adjacency matrices between NAGs and their attached states of support, constructed from the Dangerous Companions Database \\cite{San-Akca:2016}. The dataset covers the period between 1945 and 2010, which introduces a profile for each NAG with information on the foundation year, objectives and ideational characteristics. All NAGs are engaged in a violent conflict against one or more governments and only conflicts that caused more than 25 battle-related deaths (BRD) are collected, namely the major ones. \nA HS can provide multiple types of support to a NAG\neach of which is classified as active if it is intentional or passive (de facto) if it is inadvertent. \\\\ \n\n\n\\noindent\\textbf{Relative probability.} We calculate the relative probability $T^+(k_{HN})$ that a NAG added in year $t$ connects to an existing state which is connected to $k_{HN}$ NAGs in year $t-1$. The relative probability $T^+(k_{HN}) = P^+(k_{HN},t)\/P_0^+(k_{HN},t)$ is defined as the ratio of the absolute probability $P^+(k_{HN},t)$ that a NAG connects to a HS with degree $k_{HN}$ against the the probability $P_0^+(k_{HN},t)$ that a HS with degree $k_{HN}$ is uniformly selected at random, where $P_0^+(k_{HN},t) = n_s(k_{HN},t)\/N_s(t)$ with $n_s(k_{HN},t)$ being the number of HSs with degree $k_{HN}$ prior to the addition of this NAG and $N_s(t)$ being the number of all HSs. Then $T^+(k_{HN})$ can be estimated by making a histogram of the degrees $k_{HN}$ of the HSs to which each NAG is added, in which each sample is weighted by a factor of $N_s(t)\/n_s(k_{HN},t)$ \\cite{Newman:2001}. In the random case, $T^+(k_{HN})$ should equal 1 for all $k_{HN}$. If it is a nonconstant function of $k_{HN}$, some bias is involved in the attachment process. In particular, a linearly increasing function corresponds to the standard preferential attachment (PA); while a general monotonically increasing function corresponds to generalized PA. Similarly, we calculate the relative probability $T^-(k_{HN})$ that a NAG detaches from a HS with degree $k_{HN}$. Furthermore, the relative probability of a HS acquires or loses a connection from an incumbent NAG, namely, $R^+(k_{HN})$ and $R^-(k_{HN})$. From the perspective of NAGs, we can also calculate the relative probability that a NAG acquires or loses a connection from a HS (see Fig. S5 in SI). \\\\\n\n\n\n\\noindent\\textbf{Temporal detection of modules.} We use a temporal community detection algorithm, GenLouvain, based on the Markov stability, proposed by Mucha, et al \\cite{Mucha:2010}. The merit of the temporal algorithm is its ability to find a systematic optimal partition of the network by optimizing a single quality function throughout all years. Note that by using this optimization algorithm, there can still be arbitrariness due to different partitions $\\{P_s\\}$ of the same network that give comparable modularity values close to the optimum. A specific $s$th partition is denoted by $P_s(t)=\\cup_i M_{s,i}(t)$, where the subscript $i$ denotes the $i$th module and $t$th year. \n\n\nWe then seek a representative partition if the network has a truly nontrivial modular structure (for details see Sec. S2 in SI). This is realized by using a further stabilizing technique for robust module detection \\cite{Bassett:2013}. We construct a nodal association matrix $T$, each entry of which is the frequency of a pair of actors being partitioned in the same module according to the partitions $\\{P_s\\}$ (only statistically significant contributions are retained). We then partition this matrix $T$, which reveals the network modular structure from all partitions $\\{P_s\\}$. It turns out that all partitions $\\{\\tilde{P}_s\\}$ of $T$, after applying this technique, are almost identical to each other, indicating the existence of a highly robust modular structure of the studied network. Finally, to remove the remaining slight uncertainty, we select a representative partition $\\tilde{P}^*$, which has a maximum average Adjusted Mutual Information (AMI) with all other partitions $\\{\\tilde{P}_s\\}$ \\cite{Vinh:2010}.\\\\ \n\n\\noindent\\textbf{Roles of actors.} We calculate two characteristics of each actor, the standardized within-module degree and the participation coefficient \\cite{Guimera:2005}\n\\begin{eqnarray}\n\\label{eq:zscs}\nd_i&=&(k_{is}-\\langle \\{k_{is}\\} \\rangle)\/\\sigma(\\{k_{is}\\}) \\\\\nc_i&=&1-\\sum_{t=1}^{N_m}k_{it}\/k_i\n\\end{eqnarray}\nwhere $k_i$ is the degree of nodes $i$, $k_{is}$ is the within-module degree (number of links of node $i$ to other nodes in its own module $s$), $\\langle \\{k_{is}\\} \\rangle$ and $\\sigma(\\{k_{is}\\})$ are the average and standard deviation of within-module degree of all nodes in $s$, and $k_{it}$ is the number of links from node $i$ to nodes in module $t$.\n\n\n\\section*{Data Availability}\nThe ``Dangerous Companions\" Dataset used in this study is available at http:\/\/armedgroups.net.\n\n\\begin{sloppypar}\n\n\\section*{Code Availability}\nCodes for analyzing the network are available upon request. The code for measuring nestedness from the library ``BiMat\" is available at https:\/\/bimat.github.io. The code for temporal community detection from the library ``GenLouvain\" is available at https:\/\/github.com\/GenLouvain\/GenLouvain. \n\n\\end{sloppypar}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section*{Introduction}\nThe research presented in this paper is part of an ongoing investigation into the suitability of classical logic in the context of programming languages with control.\nRather than looking at how to encode known control features into calculi like the $`l$-calculus \\cite{Church'36,Barendregt'84}, Parigot's $\\lmu$-calculus \\cite{Parigot'92}, or $\\Lmu$\\Paper{\\footnote{The name $\\Lmu$ was first introduced in \\cite{Saurin'05}, that also introduced a different notation for terms, in placing names \\emph{behind} terms, rather than in front, as done by Parigot and de Groote; we use their notation here.}} \\cite{deGroote'94}, as has been done in great detail by others, we focus on trying to understand what is exactly the notion of computation that is embedded in calculi like $\\lmu$; we approach that problem here by presenting a fully abstract interpretation for that calculus into the (perhaps better understood) $`p$-calculus \\cite{Milner'92}.\n\n\\CLAC{\n\nIn the past, many \nresearchers \\Paper{to }investigate\\CLAC{d}\ninterpretations into the $`p$-calculus of various calculi that have their foundation in classical logic\\Paper{, as done in, for example, \\cite{Honda-Yoshida-Berger'04,vBCV-CLaC'08,CiminiCS'10,Beffara-Mogbil'12}}.\nFrom these papers it might seem that the interpretation of such `classical' calculi comes at a great expense; for example, to encode \\emph{typed} $\\lmu$, \\cite{Honda-Yoshida-Berger'04} defines an extension of Milner's encoding and considers a \\Paper{version of the $`p$-calculus that is strongly typed}\\CLAC{strongly typed $`p$-calculus}; \\cite{vBCV-CLaC'08} shows preservation of reduction in $\\X$ \\cite{Bakel-Lescanne-MSCS'08} only with respect to $\\Ctg$, the contextual ordering (so not with respect to $\\equivC$, contextual equivalence, nor with respect to weak bisimilarity); \\cite{CiminiCS'10} defines a non-compositional interpretation of $\\lmmt$ \\cite{Curien-Herbelin'00} that strongly depends on recursion, and does not regard the logical aspect\\Paper{ at all}.\n\n\\CLAC{In \\cite{Bakel-Vigliotti-IFIPTCS'12} we started our investigations by presenting an interpretation for de Groote's variant $\\Lmu$ into the $`p$-calculus \\cite{Milner'92} and proved a soundness result; here we show that this interpretation is fully abstract, but have to limit the interpretation to $\\lmu$ terms. \nWe study an output-based encoding of $\\lmu$ into the $`p$-calculus that is an extension of the one we defined for the $`l$-calculus \\cite{Bakel-Vigliotti-CONCUR'09} and is a natural variant of that for $\\Lmu$ in \\cite{Bakel-Vigliotti-IFIPTCS'12}.\nIn those papers, we have shown that our encoding respects \\emph{single-step explicit head reduction} (which only ever replaces the head variable of a term) modulo $\\equivC$\\Paper{; here we restate those properties with respect to $\\wbisimilar$, weak bisimilation}.\n\\CLAC{\n\nWe will here address the natural question that arises next: are two terms that are equal under the interpretation also operational equivalent, \\emph{i.e.}: is the interpretation \\emph{fully abstract}?\nWe answer that question positively, using a new approach to showing full abstraction, for our interpretation of $\\lmu$-terms (rather than $\\Lmu$ as used in \\cite{Bakel-Vigliotti-IFIPTCS'12}) and thereby also for the standard $`l$-calculus. \n\\Paper{Using a new technique, we will show that our interpretation is fully abstract, \\emph{i.e.}~two terms that are equal under the interpretation are also operational equivalent; our approach will be to first show that our interpretation respects single-step explicit head reduction $\\redxh$ modulo \\emph{weak bi-simularity} $\\wbisim$ and to \n\\CLAC{Following the approach of \\cite{Bakel-Vigliotti-IFIPTCS'12} we can show that our interpretation respects single-step explicit head reduction $\\redxh$ modulo \\emph{weak bisimularity} $\\wbisim$ (rather than $\\equivC$ as used in \\cite{Bakel-Vigliotti-IFIPTCS'12}; we omit the details here).\nWe \nextend this result to $\\equivwxh$, the equivalence relation generated by $\\redxh$ that equates also terms without (weak) normal form with respect to $\\redxh$.\nThe main proof of the full abstraction result is then achieved through showing that $\\equivwxh$ equates to $\\equivwbmu$, the equivalence relation generated by standard reduction that also equates terms without weak head normal form\\Paper{: this latter result is stated entirely within $\\lmu$}.\n\nThis technique is considerably different from the one used by Sangiorgi, who has shown a full abstraction result \\cite{Sangiorgi'94,Sangiorgi-Walker-Book'01} for Milner's encoding $\\MilSem[M] a $ of the lazy $`l$-calculus \\cite{Milner'92}.\n\\Paper{The characterisation of $\\MilSem[M] a \\wbisim \\MilSem[N] a $, left as open problem in \\cite{Milner'92}, was achieved through showing that the equivalence classes under weak-bisimilarity of Milner's encoding form a model for the lazy $`l$-calculus in the sense that it provides a denotational semantics, similar to Cor.~\\ref{semantics} below. \nTo achieve full abstraction, Sangiorgi proves that $\\MilSem[M] a \\wbisim \\MilSem[N] a $ if and only if $M \\AppBis N$, where $\\AppBis$ is the \\emph{applicative bisimularity} on $`l$-terms \\cite{Abramsky-Ong'93b}\\CLAC{. }\\Paper{, an operational notion of equivalence on terms of the lazy $`l$-calculus as defined by Abramsky and Ong, rather than $`b$-equality.\n\n\nHowever, this result comes at a price, since applicative bisimulation equates \\Paper{the }terms \\Paper{$ x (x \\Theta `D`D) \\Theta $ and $ x (`l y . x \\Theta `D`D y ) \\Theta $ (with $`D = `lx.xx$, and $\\Theta$ is such that, for every $N$, $\\Theta N$ is reducible to an abstraction) whereas these terms}\\CLAC{that} are not weakly bisimilar under \\Paper{the interpretation }$\\MilSem[`.] `. $: in order to achieve full abstraction, Sangiorgi had to extend Milner's encoding to $`L_c$, a $`l$-calculus enriched with constants and by exploiting a more abstract encoding into the \\emph{Higher Order} $`p$-calculus, a variant of the $`p$-calculus with higher-order communications.\nSangiorgi's result then essentially states that the interpretations of closed $`L_c$-terms $M$ and $N$ are contextually equivalent if and only if $M$ and $N$ are applicatively bisimilar; in \\cite{Sangiorgi'94} he shows that the interpretation of terms in $`L_c$ in the standard $`p$-calculus is weakly bisimilar if and only if they have the same L\\'evy-Longo tree\\Paper{ \\cite{Levy'76,Longo'83} (a lazy variant of B\\\"ohm trees \\cite{Barendregt'84})}.\n\n\nWe would like to stress that in order to achieve full abstraction for our interpretation, \\emph{we did not need to extend the interpreted calculus, and use a first order $`p$-calculus}.\nIn fact, \\CLAC{the main contribution of this paper and }novelty of our proof is the \\emph{structure of the proof} of the fact that our interpretation gives a fully abstract semantics.\nTo wit, we define a choice of operational equivalences for the $\\lmu$-calculus, both with and without explicit substitution.\nWe define the \\emph{weak explicit head equivalence} $\\equivwxh$ and show that this is exactly the relation that is naturally representable in the $`p$-calculus; we define \\emph{weak head equivalence} $\\equivxh$ and show that for $\\lmu$-terms without explicit substitution, $\\equivwxh$ corresponds to $\\equivxh$.\nThe relation $\\equivwxh$ essentially equates terms that have the same L\\'evy-Longo tree, but of course defined for $\\lmu$, which gets shown through a notion of weak approximation.\nWe then show that the relation $\\equivwA$, which expresses that terms have the same set of weak approximants, $\\equivwxh$, and $\\equivwbmu$ all correspond.\n\n\\CLAC{The combined results of \\cite{Bakel-Vigliotti-CONCUR'09,Bakel-Vigliotti-IFIPTCS'12} and the full abstraction}\\Paper{The} results we present here stress that the $`p$-calculus constitutes a very powerful abstract machine indeed: although the notion of structural reduction in $\\lmu$ is very different from normal $`b$-reduction, no special measures had to be taken in order to be able to express it through our interpretation.\nIn fact, the distributive character of application in $\\lmu$, and of both term and context substitution is dealt with \\Paper{entirely }by congruence in $`p$\\Paper{ (see also Example~\\ref{redex example})}, and both naming and $`m$-binding are dealt with entirely statically by the interpretation.\n\n\n\n\\Comment{\nThe results of \\cite{Bakel-Vigliotti-IFIPTCS'12} concentrated on the relation between $\\Lmux$ and the $`p$-calculus, and in particular on Soundness and Completeness (Theorem~\\ref{rtc soundness} here); in this paper, we will focus on full abstraction for $\\lmu$.\nWe will define a notion of equivalence $\\equivwxh$ between terms of $\\lmux$ that equates also terms that have no weak head-normal form, and show that terms are equivalent with respect to $\\equivwxh$ if and only if their images under $ \\PilmuTerm[`.] `. $ are contextually equivalent.\nWe will then generalise this to equivalences generated by head reduction, standard reduction, and weak-approximation, respectively, that all equate terms that have no weak head-normal form, and show that all coincide: this will lead to our main result: $M \\equivwbmu N \\Iff \\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $.\n\n\n\n \\vspace{2mm}\n\\noindent\n\\textbf{Organisation of this paper:}\nWe start with revisiting the $\\lmu$-calculus in Section~\\ref{lambda mu calculus} and define a notion of \\emph{head-reduction} $\\redh$.\nIn Section~\\ref{pi with pairing} we revisit the $`p$-calculus, enriched with \\emph{pairing}.\nIn Section~\\ref{lmux section} we define $\\lmux$, a version of $\\lmu$ with \\emph{explicit substitution}, as well as a notion of \\emph{explicit head reduction} and in Section~\\ref{lambda interpretation} define our \\emph{logical interpretation} of $\\lmux$ in to $`p$.\n\nWorking towards our full abstraction result, in Section~\\ref{weak reduction} we will define notions of weak reduction, in particular \\emph{weak head reduction} and \\emph{weak explicit head reduction}.\nWe then define the two notions of equivalence these induce, also equating terms without weak head-normal form and show that these notions coincide on pure $\\lmu$ terms (\\emph{i.e.}~without explicit substitutions).\nWe also define the equivalence $\\equivwbmu$ induced by $\\redbmu$ on pure $\\lmu$ terms, that also equates terms without weak head-normal form.\nIn Section~\\ref{approximation section}, we define a notion of \\emph{weak approximation} for $\\lmu$, and show the semantics this induces, $\\equivwA$, is fully abstract with respect to both $\\equivwh$ and $\\equivwbmu$.\nWe show that our logical interpretation is fully abstract with respect to weak bisimilarity $\\wbisim$ on processes and $\\equivwxh$, $\\equivwh$, $\\equivwA$, and $\\equivwbmu$ on pure $`l`m$-terms.\n\n\n\n\\vspace*{2mm}\n\\noindent\n\\textbf{Notation:}\nWe will use a vector notation $\\Vect{`.}$ as abbreviation for any sequence: for example, $\\Vect{x_i}$ stands for $x_1, \\ldots, x_n$, for some $n$, or for $\\Set{x_1, \\ldots, x_n}$, \\CLAC{and\n}$\\langle\\Vect{ `a_i := N_i `. `b_i }\\rangle$ for $ \\excontsub`a_1 := N_1 . `b_1 \\,$ $\\dots\\, \\excontsub `a_n := N_n . `b_n $,\\Paper{ $\\Vect{M_i \\eqh N_i}$ for $\\Forall 1 \\seq i \\seq n \\Pred [ M_i \\eqh N_i ] $,} etc.\nWhen possible, we will drop the indices.\n\n\n \\section{The $\\lmu$~calculus\\CLAC{ and explicit substitution}} \\label{lambda mu calculus}\n\nIn this section, we will briefly discuss Parigot's $\\lmu$-calculus~\\cite{Parigot'92}; we assume the reader to be familiar with the $`l$-calculus and its notion of reduction $\\bred$ and equality $=_{`b}$.\n\n$\\lmu$ is a proof-term syntax for classical logic, expressed in Natural Deduction, defined as an extension of the Curry type assignment system for the {\\LC} by adding the concept of \\emph{named} terms, and adding the functionality of a \\emph{context switch}, allowing arguments to be fed to subterms.\n\n\\Paper{In the next section we will define explicit head reduction for $\\lmux$, a variant of $\\lmu$ with explicit substitution \\emph{\\`a la} $\\Lx$ \\cite{Bloo-Rose'95}, and will show full abstraction results for $\\lmux$; since $\\lmux$ implements $\\lmu$-reduction, this implies that, automatically, our main results are also shown for standard reduction (with implicit substitution).\n\n \\begin{definition}[Syntax of $\\lmu$] \\label{lm-terms}\n\\Paper{The $\\lmu$-\\emph{terms} we consider are defined over the set of \\emph{variables} represented by Roman characters, and \\emph{names}, or \\emph{context} variables, represented by Greek characters, through the grammar:}%\n\\CLAC{The $\\lmu$-\\emph{terms} we consider are defined over the set of \\emph{variables} (Roman characters) and \\emph{names}, or \\emph{context} variables (Greek characters), through:}\n\\Paper{\n \\[ \\begin{array}{@{}rrl@{\\quad}l}\nM,N &::=& x & \\textit{variable} \\\\\n& \\mid & `l x.M & \\textit{abstraction} \\\\\n& \\mid & MN & \\textit{application} \\\\\n& \\mid & \\muterm`a.[`b]M & \\textit{context switch}\n \\end{array} \\]\n\n\\CLAC{\n \\[ \\begin{array}{rcl}\nM,N &::=& x \\mid `l x.M \\mid MN \\mid \\muterm`a.[`b]M\n \\end{array} \\]\n\nWe will occasionally write $\\Cmd$ for the pseudo-term $[`a] M$.\n \\end{definition}\n\\Paper{In fact, the main difference between $\\Lmu$ and $\\lmu$ is that in the former, $[`a] M$ is considered to be a term.}\n\nAs usual, $`l x.M$ binds $x$ in $M$, and $\\muterm `a .\\Cmd$ binds $`a $ in $\\Cmd$, and the notions of free variables $\\fv(M)$ and names $\\fn(M)$ are defined accordingly; the notion of $`a $-conversion extends naturally to bound names and we assume Barendregt's convention in that we assume that free and bound variables and names are always distinct, using $`a$-conversion when necessary.\n\\Paper\nAs usual, we call a term \\emph{closed} if it has no free variables.\n\nSimple type assignment for $\\lmu$ is defined as follows:\n \\begin {definition}[Types, Contexts, and Typing] \\label{types} \\label {typing for lmu}\n \\begin {enumerate}\n\n \\item\nTypes are defined by:\n \\[ \\begin{array}{rcl}\nA,B & ::= & \\tvar \\mid A \\arrow B\n \\end{array} \\]\nwhere $\\tvar$ is a basic type of which there are infinitely many.\n\n \\item\nA \\emph{context of inputs} $`G$ is a mapping from term variables to types, denoted as a finite set of \\emph{statements} $\\stat{x}{A}$, such that the \\emph{subject} of the statements ($x$) are distinct.\nWe write $\\G_1,\\G_2$ for the \\emph{compatible} union of $\\G_1$ and $\\G_2$ (if $\\stat{x}{A_1} \\ele \\G_1$ and $\\stat{x}{A_2} \\ele \\G_2$, then $A_1 = A_2$), and write $`G, \\stat{x}{A}$ for $`G, \\{\\stat{x}{A}\\}$.\n\n \\item\nContexts of \\emph{outputs} $`D$ as mappings from names to types, and the notions $`D_1,`D_2$ and $`a{:}A,`D$ are defined similarly.\n\n \\item\nType assignment for $\\lmu$ is defined by the following natural deduction system.\n \\[ \\kern-5mm\n \\begin{array}{@{}rl@{\\quad}rl}\n(\\Ax) : &\n\\Inf\t{ \\derLmu `G,x{:}A |- x : A | `D }\n&\n (`m) : &\n \\Inf\t[`a \\notele `D]\n\t{ \\derLmu `G |- M : B | `a{:}A,`b{:}B,`D\n\t}{ \\derLmu `G |- \\muterm`a.[`b]M : A | `b{:}B,`D }\n\\quad\n \\Inf\t[`a \\notele `D]\n\t{ \\derLmu `G |- M : A | `a{:}A,`D\n\t}{ \\derLmu `G |- \\muterm`a.[`a]M : A | `D }\n\\end{array} \\] \\[ \\begin{array}{@{}rl@{\\dquad}rl}\n(\\arrI) : &\n\\Inf\t[x \\notele `G]\n\t{ \\derLmu `G,x{:}A |- M : B | `D\n\t}{ \\derLmu `G |- `l x.M : A\\arrow B | `D }\n&\n(\\arrE) : &\n\\Inf\t{ \\derLmu `G |- M : A\\arrow B | `D\n\t \\quad\n\t \\derLmu `G |- N : A | `D\n\t}{ \\derLmu `G |- MN : B | `D }\n \\end{array} \\]\n\n \\end {enumerate}\n \\end{definition}\nSo, for the context $`G, \\stat{x}{A}$, we have either $\\stat{x}{A} \\in `G$, or $`G$ is not defined on $x$.\n\nIn $\\lmu$, reduction of terms is expressed via implicit substitution; a%\n\n\\CLAC{A}s usual, $M[N \\For x]$ stands for the substitution of all occurrences of $x$ in $M$ by $N$, and $M [ N{`.}`g \\For `a ]$, the \\emph{structural substitution}, for the term obtained from $M$ when every (pseudo) sub-term of the form $[`a]M'$ is replaced by $[`g]M' N$.\\Paper{\\footnote{This notion is often defined as $M [ N \\For `a ]$, where every (pseudo) sub-term of the form $[`a]M'$ is replaced by $[`a]M' N$; in our opinion, this creates confusion, since $`a$ gets `reintroduced'; it is in fact a new name, which is illustrated by the fact that, in the typed version $`a$ then changes type.}}\n\\CLAC{(We omit the formal definition here; see Def.~\\ref{definition lmux} for the variant with \\emph{explicit} structural substitution.)}\n\\Paper\nFor reasons of clarity, and because below we will present a version of $\\lmu$ that makes the substitution explicit, we define the $`m$-substitution formally.\n\n \\begin{definition} [Structural substitution] \\CLAC{\\hfill}\nWe define $M [ N{`.}`g \\For `a ]$ \\Paper{(where every sub-term $[`a]L$ of $M$ is replaced by $[`g]LN$) }by induction over the structure of (pseudo-)terms by:\n \\[ \\begin{array}{@{}r@{\\,}lcl@{\\quad}l}\n([`a]M) & [ N{`.}`g \\For `a ] & \\ByDef & [`g] ( M \\, [ N{`.}`g \\For `a ] ) N\n\t\\\\\n([`b]M)&[ N{`.}`g \\For `a ] & \\ByDef & [`b] ( M \\, [ N{`.}`g \\For `a ] ) & (`a\\not=`b)\n\t\\\\\n(\\muterm`b.\\Cmd) & [ N{`.}`g \\For `a ] & \\ByDef & \\muterm`b. ( \\Cmd \\, [ N{`.}`g \\For `a ] )\n\t\\\\\nx & [ N{`.}`g \\For `a ] & \\ByDef & x\n\t\\\\\n(`lx.M) & [ N{`.}`g \\For `a ] & \\ByDef & `lx . (M \\, [ N{`.}`g \\For `a ] )\n\t\\\\\n(M_1M_2) & [ N{`.}`g \\For `a ] & \\ByDef & M_1 \\, [ N{`.}`g \\For `a ]~M_2 \\, [ N{`.}`g \\For `a ]\n \\end{array}\\]\n \\end{definition}\n\nWe have the following rules of computation in $\\lmu$:\n\n\n \\begin{definition}[$\\lmu$ reduction] \\label{lmu reduction}\n\\CLAC{Reduction on $\\lmu$-terms is defined as the contextual closure of the rules:}\n\\Paper{$\\lmu$ has a number of reduction rules: two \\emph{computational rules}: }\n \\[ \\begin{array}{@{}rrcll}\n\\textit{logical } (`b): & (`l x . M ) N & \\red & M [ N \\For x ]\n\t\\\\\n\\textit{structural } (\\mu): & (\\muterm `a . \\Cmd ) N & \\red & \\muterm`g. ( \\Cmd [N{`.}`g \\For `a] )\n\n\\Paper{ \\end{array} \\]\nas well as the \\emph{simplification rules}:\n \\[ \\begin{array}{@{}rrcll}\n}\\CLAC{\\\\ }\n\\textit{renaming} : & \\muterm `d . [`b](\\muterm`g.[`a]M) & \\red & \\muterm `d . [`a] M[`b \\For `g]\n\t\\\\\n\\textit{erasing} : & \\muterm `a . [`a] M & \\red & M & (`a \\notele \\fn(M))\n \\end{array} \\]\n\\Paper{which are added mainly to simplify the presentation of results.\nWe use the contextual rules:\\footnote{Normally the contextual rules are not mentioned but are left implicit; here we need to state them, since we will below consider notions of reduction that do not permit all contextual rules.}\n \\[ \\begin {array}[t]{@{}r@{~~}c@{~~}l}\nM \\red N &\\Then&\n \\begin{cases}\nML &\\red& NL \\\\\nLM &\\red& LN \\\\\n`lx.M &\\red& `lx.N \\\\\n\\muterm`a.[`b]M & \\red & \\muterm`a.[`b]N\n \\end{cases}\n \\end {array} \\]}%\nWe use ${}\\rtcredbmu{}$ for the pre-congruence\\Paper{\\footnote{A \\emph{pre-congruence} is a reflexive and transitive relation that is preserved in all contexts; a \\emph{congruence} is symmetric pre-congruence.}} based on these rules, ${}\\eqbmu{}$ for the congruence, write $M \\rtcredbmu[\\nf] N$ if $M \\rtcredbmu N$ and $N$ is in normal form, $M \\rtcredbmu[\\hnf] N$ if $M \\rtcredbmu N$ and $N$ is in head-normal form, $M \\Shows $ if there exists a finite reduction path starting from $M$,\\Paper{\\footnote{Note that this does not imply that \\emph{all} paths are finite.}} and $M \\Diverges$ if this is not the case; we will use these notations for other notions of reduction as well.\n\n\n \\end{definition}\n\nThat this notion of reduction is confluent was shown in \\cite{Py-PhD'98}; so we have:\n\n \\begin{proposition} \\label{confluence eq}\nIf $M \\eqbmu N $ and $M \\rtcredbmu P $, then there exists $Q$ such that $ P \\rtcredbmu Q $ and $N \\rtcredbmu Q $.\n \\end{proposition}\n\n\n\n\\Paper\nFor convenience, Parigot also considers $[`a]M$ and $\\muterm`a.M$ as \\emph{pseudo}-terms, that represent \\emph{deactivation} and \\emph{activation}.\\footnote{In fact, \\cite{Parigot'92} formulates the renaming rule as $ [`b](\\muterm`g.M) \\red M[`b \\For `g] $.}\n\nThe intuition behind the structural rule is given by \\cite{deGroote'94}: ``\\emph{in a $\\lmu$-term $\\muterm`a.M$ of type $A \\arr B$, only the subterms named by $`a$ are \\emph{really} of type $A \\arr B$ (\\ldots); hence, when such a $`m$-abstraction is applied to an argument, this argument must be passed over to the sub-terms named by $`a$.}''\nWe can think of $[`a]M$ as storing the type of $M$ amongst the alternative conclusions by giving it the name $`a $.\n\n\\cite{Parigot'93} has shown that typeable terms are strongly normalisable.\nIt also defines the extensional rules:\n\\[ \\begin{array}{@{}r@{\\quad}rcl@{\\quad}l}\n(\\eta): & `lx.Mx & \\red & M & (x\\not\\in \\fv(M))\n\t\\\\\n(`h`m) : & \\muterm`a . [`b] M & \\red & `lx\\muterm`g . [`b] M[x{`.}`g \\For `a]\n \\end{array} \\]\nHere we do not consider these rules: the model we present through our interpretation is not extensional, and we can therefore not show that those rules are preserved by the interpretation (see Rem.~\\ref{not extensional}).\n\n \\begin{example} \\label{lmu double negation}\nAs an example illustrating the fact that this system is more powerful than the system for the \\LC, here is a proof of it is possible to inhabit Peirce's Law (due to \\cite{Ong-Stewart'97}):\n \\[ \\begin{array}{@{}c}\n \\Inf\t[\\arrI]\n\t{\\Inf\t[`m]\n\t\t{\\Inf\t[\\arrE]\n\t\t\t{\\Inf\t[\\Ax]\n\t\t\t\t{ \\derLmu x{:}(A\\arrow B)\\arrow A |- x : (A\\arrow B)\\arrow A | `a{:}A }\n\t\t\t \\Inf\t[\\arrI]\n\t\t\t\t{\\Inf\t[`m]\n\t\t\t\t\t{\\Inf\t[\\Ax]\n\t\t\t\t\t\t{ \\derLmu x{:}(A\\arrow B)\\arrow A,y{:}A |- y : A | `b{:}B }\n\t\t\t\t\t}{ \\derLmu x{:}(A\\arrow B)\\arrow A,y{:}A |- \\muterm`b.[`a]y : B | `a{:}A }\n\t\t\t\t}{ \\derLmu x{:}(A\\arrow B)\\arrow A |- `ly.\\muterm`b.[`a]y : A\\arrow B | `a{:}A }\n\t\t\t}{ \\derLmu x{:}(A\\arrow B)\\arrow A |- x(`ly.\\muterm`b.[`a]y) : A | `a{:}A }\n\t\t}{ \\derLmu x{:}(A\\arrow B)\\arrow A |- \\muterm`a.[`a](x(`ly.\\muterm`b.[`a]y)) : A | {} }\n\t}{ \\derLmu { } |- `lx.\\muterm`a.[`a](x(`ly.\\muterm`b.[`a]y)) : ((A\\arrow B)\\arrow A)\\arrow A | {} }\n \\end{array} \\]\nThe underlying logic of the system of Def.~\\ref{typing for lmu} corresponds to \\emph{minimal classical logic} \\cite{Ariola-Herbelin'03}.\n \\end{example}\n\nWe also consider the notion of head reduction; it is defined in \\cite{Wadsworth'76} for the $`l$-calculus by first defining the head-redex of a term as the subterm $(`lx.M)N$ in a term of the form\n \\[ `lx_1x_2\\dots x_n.((`lx.M)N)L_1L_2\\dots L_m \\quad (n \\geq 0, m \\geq 0) \\]\nHead reduction is then that notion in which each step is determined by contraction of the head redex (when it exists), and head-normal forms (the normal forms with respect to head reduction) are of the generic shape\n \\[ `lx_1x_2\\dots x_n.yL_1L_2\\dots L_m \\quad (n \\geq 0, m \\geq 0) \\]\nand $y$ in this term is called the head variable.\nIn $\\lmu$, given the naming and $`m$-binding features, the notion of head redex is not this easily defined; rather here we define head reduction by not allowing reductions to take place in the right-hand side of applications (in the context of the $`l$-calculus, this gives the original notion); we also define a notion of head-normal form for $\\lmu$.\n\n\n \\begin{definition} [Head reduction for $\\lmu$ (cf.~\\cite{Lassen'06})] \\label{head reduction definition}\n \\begin{enumerate}\n\n \\item\nWe define \\emph{head reduction} $\\redh$ as the restriction of $\\redbmu$ by removing the contextual rule:\n \\CLAC{$}\\Paper{\\[} \\begin{array}{rcl@{}}\nM \\red N &\\Then& LM \\red LN\n \\end{array} \\CLAC{$}\\Paper{\\]}\n\n \\item\nThe $\\lmu$ \\emph{head-normal forms} (\\HNF) are defined through the grammar:\n \\[ \\begin {array}{@{}rrl@{\\quad}l}\n\\lmuHNF & ::= &\n`lx. \\lmuHNF\n\t\\\\ &\\mid &\nxM_1\\dots M_n & (n \\geq 0)\n\t\\\\ &\\mid &\n\\muterm`a.[`b] \\lmuHNF\n\t& (`b \\not= `a \\textrm{ or } `a \\in \\lmuHNF, \\textrm{ and }\n\t\\lmuHNF \\not= \\muterm`g.[`d] \\lmuHNF' )\n \\end {array} \\]\n\n \\end{enumerate}\n \\end{definition}\n\\Paper{Notice that the $\\redbmu$ \\HNF s are $\\redh$-normal forms.}\n\nThe following is straightforward:\n \\begin{proposition} [$\\redh$ implements $\\lmu$'s head reduction]\n\\label{head reduction}\nIf $ M \\rtcredbmu N $ with $N$ in {\\HNF} (so $ M \\rtcredbmu[\\hnf] N $), then there exists $\\lmuHNF$ such that $ M \\rtcredh[\\nf] \\lmuHNF $ (so $\\lmuHNF$ is in $\\redh$-normal form) and $ \\lmuHNF \\rtcredbmu N $ without using $\\redh$.\n\n \\end{proposition}\n\n\\Paper\nNotice that\n $ \\begin{array}[t]{@{}lcl}\n`lf.(`lx.f(xx))(`lx.f(xx))\n\t& \\redh &\n`lf.f((`lx.f(xx))(`lx.f(xx)))\n \\end{array} $\nand this last term is in \\HNF, and in $\\redh$-normal form.\n\n\n\n\n\n \\section{The synchronous \\texorpdfstring{$`p$}{}-calculus with pairing} \\label{pi with pairing}\n\nThe notion of $`p$-calculus that we consider in this paper was already considered in \\cite{Bakel-Vigliotti-CONCUR'09} and is different from other systems studied in the literature \\cite{Honda-Tokoro'91} in that it adds \\emph{pairing} and uses a {\\proc{let}}-construct to deal with inputs of pairs of names that get distributed, similar to that defined in \\cite{Abadi-Gordon'97}; in contrast to \\cite{vBCV-CLaC'08,Bakel-Vigliotti-CONCUR'09}, we do not consider the asynchronous version of this calculus.\n\n\n \\begin{definition}[Processes] \\label{pi calculus}\n\\emph{Channel names} and \\emph{data} are defined by:\n \\[ \\begin{array}{rcl@{\\quad}l}\n& & a,b,c,\\dchan,x,y,z &\\textsl{names}\n \\end{array} \\hspace*{1cm} \\begin{array}{rcl@{\\quad}l}\n\\proc{p} & ::= & a\\mid \\PiPair(a,b) ~ & \\textsl{data}\n \\end{array} \\]\n\\Paper{(the pairing in data is \\emph{not} recursive.)}\nProcesses are defined by:\n\\Paper{\n \\[ \\begin{array}{@{}rrl@{\\quad}l}\n\\proc{P}, \\proc{Q} & ::= & \\Zero & \\textsl{nil}\n\t\\\\\n&\\mid& \\proc{P} \\Par \\proc{Q} & \\textsl{composition}\n\t\\\\\n&\\mid& \\Bang\\proc{P} & \\textsl{replication}\n\t\\\\ %\n&\\mid& \\New{a} \\proc{P} & \\textsl{restriction}\n\t\\\\\n&\\mid& \\In a(x) . \\proc{P} & \\textsl{input}\n\t\\\\\n&\\mid& \\Out a <\\procp> . \\proc{P} & \\textsl{output}\n\t\\\\\n&\\mid& \\Let = \\proc{p} in \\proc{P}\t& \\textsl{let construct}\n \\end{array} \\]\n}\\CLAC{\\[ \\begin{array}{@{}rrl@{\\quad}l}\n\\proc{P}, \\proc{Q} & ::= & \\Zero \\mid \\proc{P} {\\Par} \\proc{Q} \\mid \\Bang\\proc{P} \\mid \\New{a} \\proc{P} \\mid\n\\In a(x) . \\proc{P} \\mid \\Out a <\\procp> . \\proc{P} \\mid \\Let = \\proc{p} in \\proc{P}\n \\end{array} \\]\n}\nWe see, as usual, $`n$ as a binder, and call the name $n$ \\emph{bound} in $\\New n \\proc{P} $, $x$ bound in $\\In a(x) . \\proc{P} $ and $x,y$ bound in $\\Let = \\proc{p} in \\proc{P} $; we write $\\bn(\\proc{P})$ for the set of bound names in \\proc{P}; $n$ is \\emph{free} in $\\proc{P}$ if it occurs in $\\proc{P}$ but is not bound, and we write $\\fn(\\proc{P})$ for the set of free names in $\\proc{P}$.\n\\Paper{We call $a$ in $\\New a \\proc{P}$ a \\emph{hidden} channel.\nA {\\it context} $ \\Cont[ \\cdot]$ is a process with a hole $[ \\,]$; we call $\\In a (x) $ and $\\Out a <\\procp> $ \\emph{guards}, and call \\proc{P} in $\\In a (x) . \\proc{P} $ and $\\Out a <\\procp> . \\proc{P} $ a process \\emph{under guard}.}\n\n \\end{definition}\n\nNotice that data occurs only in two cases: $\\Out a <{\\proc{p}}> $ and $\\Let = \\proc{p} in \\proc{P} $, and that then $\\proc{p}$ is either a single name, or a pair of names; we therefore do not allow $\\In a ({ \\PiPair}) . \\proc{P} $, nor $\\Out a <{ \\PiPair<{ \\PiPair},d>}> . \\proc{P} $, nor $\\Out {\\PiPair} <{\\proc{p}}> . \\proc{P} $, nor $\\New \\PiPair \\proc{P} $, nor $\\Let <\\mbox{$\\PiPair$},y> = \\proc{p} in \\proc{P} $, etc.\n\\Paper{So substitution $\\proc{P}[p\\For x]$ is a partial operation, which depends on the places in $\\proc{P}$ where $x$ occurs; whenever we use $\\proc{P}[p\\For x]$, we will assume it is well defined.\nIt is worthwhile to point out that using pairing is not the same as working with the polyadic (or even dyadic) $`p$-calculus, because there each channel has a fixed arity, whereas we allow data to be sent, which is either a name or a pair of names.}\n\nWe abbreviate $\\In {a} (x) . \\Let = x in \\proc{P} $ by $\\In {a} (y,z) . \\proc{P} $, \\Paper{as well as }$\\New m \\New n \\proc{P} $ by $\\New mn \\proc{P} $, and write $\\Out a <\\procp> $ for $\\Out a <\\procp> . \\Zero $.\nAs in \\cite{Sangiorgi-Walker-Book'01}, we write $\\Eq a=b $ for the \\emph{forwarder} $ \\In a (x) . \\Out b $ \\Paper{and $\\BOut x (w) . \\proc{P} $ for $ \\New w ( \\Out x . \\proc{P} ) $}.\n\n \\begin{definition}[Structural Congruence]\nThe \\emph{structural congruence} is the smallest congruence generated by the rules:\n\\noindent\n \\[ \\begin{array}{c@{\\quad}c} \\begin{array}{rcll}\n\\proc{P} \\Par \\Zero &\\StrCon& \\proc{P}\n\t\\\\\n\\proc{P} \\Par \\proc{Q} &\\StrCon & \\proc{Q} \\Par \\proc{P}\n\t\\\\\n\\Bang \\proc{P} &\\StrCon& \\proc{P} \\Par \\Bang \\proc{P}\n\t\\\\\n\\New n \\Zero &\\StrCon& \\Zero\n\\CLAC{\t\\end{array} & \\begin{array}{rcll} }\n\\Paper{\\\\ }\n(\\proc{P} \\Par \\proc{Q} ) \\Par \\proc{R} &\\StrCon& \\proc{P} \\Par (\\proc{Q} \\Par \\proc{R})\n\t\\\\\n\\New m \\New n \\proc{P} &\\StrCon& \\New n \\New m \\proc{P}\n\t\\\\\n\\New n (\\proc{P} \\Par \\proc{Q}) &\\StrCon& \\proc{P} \\Par \\New n \\proc{Q} & (n \\notin \\fn(\\proc{P}))\n\t\\\\\n\\Let = \\PiPair(a,b) in \\proc{P} &\\StrCon& \\proc{P} [a \\For x,b \\For y]\n \\end{array} \\end{array}\n\\]\n\n \\end{definition}\n\nAs usual, we will consider processes modulo congruence and $`a$-conversion: this implies that we will not deal explicitly with the process $ \\Let = \\PiPair(a,b) in \\proc{P} $, but rather with $ \\proc{P} [a \\For x,b \\For y] $.\nBecause of rule $(\\proc{P} \\Par \\proc{Q} ) \\Par \\proc{R} \\StrCon \\proc{P} \\Par (\\proc{Q} \\Par \\proc{R}) $, we will \\Paper{normally }not write brackets in a parallel composition of more than two processes.\n\nComputation in the $`p$-calculus with pairing is expressed via the exchange of \\textsl{data}.\n \\begin{definition}[Reduction] \\label{pi reduction}\n\\Paper{ \\begin{enumerate}\n\n \\item}\nThe \\emph{reduction relation} over the processes of the $`p$-calculus is defined by the following (elementary) rules:\n \\[ \\begin{array}{rcl@{~}l}\n\\Out a <\\procp> . \\proc{P} \\Par \\In a (x) . \\proc{Q} &\\redPi& \\proc{P} \\Par \\proc{Q} [\\proc{p} \\For x]\n\t\\Paper{ & (\\textsl{synchronisation}) }\n\t\\\\\n\\proc{P} \\redPi \\proc{P'} &\\Then& \\New n \\proc{P} \\redPi \\New n \\proc{P}' \\quad\n\t\\Paper{ & (\\textsl{hiding}) }\n\t\\\\\n\\proc{P} \\redPi \\proc{P}' &\\Then& \\proc{P} \\Par \\proc{Q} \\redPi \\proc{P}' \\Par \\proc{Q}\n\t\\Paper{ & (\\textsl{composition}) }\n\t\\\\\n \\proc{P} \\StrCon \\proc{Q} \\And \\proc{Q} \\redPi \\proc{Q}' \\And \\proc{Q}' \\StrCon \\proc{P}'\n\t& \\Then& \\proc{P} \\redPi \\proc{P}'\n\t\\Paper{ & (\\textsl{congruence}) }\n \\end{array} \\]\n\\Paper{We write $ \\proc{P} \\redPi(c) \\proc{Q} $ if $\\proc{P}$ reduces to $\\proc{Q}$ in a single step via a synchronisation over channel $c$, and write $\\redPi(=_{`a})$ if we want to point out that $`a$-conversion has taken place during the synchronisation.\nWe say that $ \\proc{P} \\redPi(c) \\proc{Q} $ takes place \\emph{over a hidden channel} if $c$ is hidden in $\\proc{P}$.}\n\n\\Paper{ \\item\nWe say that a $\\proc{P}$ is \\emph{irreducible} (is in \\emph{normal form}) if it does not contain a possible synchronisation.\n\n \\end{enumerate}}\n \\end{definition}\nNotice that the first rule is only allowed if $\\proc{Q} [\\proc{p} \\For x]$ is a well-defined process.\n\\Comment{As usual, we write $\\rtcredPi[+]$ for the transitive closure of $\\redPi$, and $\\rtcredPi$ for its reflexive and transitive closure; we write $\\redPi(a)$ if we want to point out that a synchronisation took place over channel $a$}\n\\Paper\nAlso,\n \\[ \\begin{array}[t]{@{}lclclcl}\n\\Out a \\Par \\In a(x,y) . \\proc{Q}\n\t& \\ByDef &\n\\Out a <{\\PiPair(b,c)}> \\Par \\In a(z) . \\Let = z in \\proc{Q} \\\\\n\t& \\redPi &\n\\Let = \\PiPair(b,c) in \\proc{Q} \\\\\n\t& \\StrCon &\n\\proc{Q} [b \\For x,c \\For y]\n \\end{array} \\]\n\n\n\nThere are several notions of equivalence defined for the $`p$-calculus: the one we consider here, and will show is related to our encoding, is that of weak-bisimilarity.\n\n \\begin{definition} [Weak-bisimilarity] \\label{bisimilarity} \\label{other reduction}\n \\begin{enumerate}\n\n \\item\nWe write $\\proc{P} \\outson n $ and say that $\\proc{P}$ \\emph{outputs on} $n$ (or $\\proc{P}$ exhibits an output barb on $n$) if $\\proc{P} \\StrCon \\New \\Vect{b} ( \\Out n <\\procp> . \\proc{Q} \\Par \\proc{R} ) $, where $ n \\notele \\Vect{b}$, and $\\proc{P} \\inson n $ ($\\proc{P}$ \\emph{inputs on} $n$) if $\\proc{P} \\StrCon \\New \\Vect{b} ( \\In n (x) . \\proc{Q} \\Par \\proc{R} ) $, where $ n\\notele \\Vect{b}$.\n\n \\item\nWe write $\\proc{P} \\Outson n $ ($\\proc{P}$ \\emph{will output on} $n$) if there exists $\\proc{Q}$ such that $\\proc{P} \\rtcredPi \\proc{Q}$ and $\\proc{Q} \\outson n $, and $\\proc{P} \\Inson n $ ($\\proc{P}$ \\emph{will input on} $n$) if there exists $\\proc{Q}$ such that $\\proc{P} \\rtcredPi \\proc{Q}$ and $\\proc{Q} \\inson n $.\n\n \\item A \\emph{barbed bisimilarity} $\\bbisimilar$ is the largest symmetric relation such that $\\proc{P} \\bbisimilar \\proc{Q}$ satisfies:\n\n \\begin{itemize}\n\n \\item for every name $n$: if $\\proc{P} \\outson n $ then $\\proc{Q} \\Outson n $, and if $\\proc{P} \\inson n $ then $\\proc{Q} \\Inson n $;\n\n \\item for all $\\proc{P}'$, if $\\proc{P} \\rtcredPi \\proc{P}'$, then there exists $\\proc{Q}'$ such that $\\proc{Q} \\rtcredPi \\proc{Q}'$ and $\\proc{P}' \\bbisimilar \\proc{Q}'$;\n\n\n \\end{itemize}\n\n \\item \\emph{Weak-bisimilarity} is the largest relation $\\wbisimilar$ defined by: $ \\proc{P} \\wbisimilar \\proc{Q}$ if and only if $\\Cont[ \\proc{P}] \\bbisimilar \\Cont[\\proc{Q}]$ for any context $\\Cont[`.]$.\n\n \\end{enumerate}\n \\end{definition}\n\n\n\\Paper\nThe following is easy to show.\\footnote{This property was stated in \\cite{Bakel-Vigliotti-IFIPTCS'12} using contextual equivalence rather than weak bisimilarity.}\n \\begin{proposition} \\label{reduction hidden}\nLet $\\proc{P}, \\proc{Q}$ not contain $a$ and $a \\not= b$, then\n \\[ \\begin{array}{rcl}\n\\New a ( \\Out a <{\\proc{p}}> . \\proc{P} \\Par \\In a (x) . \\proc{Q} )\n\t& \\wbisim &\n\\proc{P} \\Par \\proc{Q}[\\proc{p} \\For x]\n\t\\\\ [1mm]\n\\New a ( \\Bang \\Out a <{\\proc{p}}> . \\proc{P} \\Par \\In a (x) . \\proc{Q} )\n\t& \\wbisim &\n\\proc{P} \\Par \\proc{Q}[\\proc{p} \\For x]\n \\end{array} \\]\n \\end{proposition}\nThis expresses that synchronisation over hidden (internal) channels is unobservable.\n\nThe following property is needed in the proof of Theorem~\\ref{soundness}.\n\n \\begin{lemma} [\\cite{Bakel-Vigliotti-JLC'14}] \\label {replication lemma}\nLet $x$ only be used as input channel in $\\proc{P}$ and $\\proc{Q}$, and not appear in \\proc{R}, then:\n\n \\[ \\begin{array}{@{}rcl@{}}\n\\New x (\\proc{P} \\Par \\proc{Q} \\Par \\Bang \\Out x . \\proc{R})\n\t&\\wbisimilar&\n\\New x (\\proc{P} \\Par \\Bang \\Out x . \\proc{R}) \\Par \\New x (\\proc{Q} \\Par \\Bang \\Out x . \\proc{R})\n\t\\\\\n \\New x ( \\Bang \\proc{P} \\Par \\Bang \\Out x . \\proc{R})\n\t&\\wbisimilar&\n\\Bang \\New x ( \\proc{P} \\Par \\Bang \\Out x . \\proc{R})\n\t\\\\\n\\New x ({ \\In c (y) . \\proc{P} \\Par \\Bang \\Out x . \\proc{R} })\n\t&\\wbisimilar&\n\\In c (y) . ({ \\New x ({ \\proc{P} \\Par \\Bang \\Out x . \\proc{R} }) })\n\t\\\\\n\\New x ({ \\Out c . \\proc{P} \\Par \\Bang \\Out x . \\proc{R} })\n\t&\\wbisimilar&\n \\Out c . ({ \\New x ({ \\proc{P} \\Par \\Bang \\Out x . \\proc{R} }) })\n \\end{array} \\]\n \\end{lemma}\n\n\n \\begin{lemma} \\label {replication lemma}\n \\begin{enumerate}\n\n \\item \\label{out distributes}\nAssume that $a$ does not occur in $\\proc{p}$, \\proc{P}, and is only used for input in \\proc{R} and \\proc{Q}. Then:\n \\[ \\begin{array}{lcl}\n\\New a (\\Bang \\Out a <\\procp> . \\proc{P} \\Par \\proc{Q} \\Par \\proc{R} )\n\t& \\wbisim &\n{ \\New a (\\Bang \\Out a <\\procp> . \\proc{P} \\Par \\proc{Q} ) \\Par \\New a (\\Bang \\Out a <\\procp> . \\proc{P} \\Par \\proc{R} ) }\n \\end{array} \\]\n\n\n\n\n \\item \\label{out splits}\nAssume \\proc{P}, \\proc{Q} only output on $a$ and $a$ does not appear in \\proc{R} or \\proc{p}. Then:\n \\[ \\begin{array}{lcl}\n\\New a ( \\Out a <\\procp> . \\proc{P} \\Par \\proc{Q} \\Par \\Bang \\In a (x) . \\proc{R} )\n\t& \\wbisim &\n{ \\New a ( \\New b ({ \\Out b <\\procp> . \\proc{P} \\Par \\proc{Q} \\Par \\Bang \\In b (x) . \\proc{R} }) \\Par \\Bang \\In a (x) . \\proc{R} ) }\n \\end{array} \\]\n\n \\item \\label{in distributes}\nAssume that $a$ does not appear in \\proc{P} and is only used for output in \\proc{R} and \\proc{Q}. Then:\n \\[ \\begin{array}{lcl}\n\\New a (\\Bang \\In a (x) . \\proc{P} \\Par \\proc{Q} \\Par \\proc{R} )\n\t& \\wbisim &\n{ \\New a (\\Bang \\In a (x) . \\proc{P} \\Par \\proc{Q} ) \\Par \\New a (\\Bang \\In a (x) . \\proc{P} \\Par \\proc{R} ) }\n \\end{array} \\]\n\n \\end{enumerate}\n\n \\end{lemma}\n\n \\begin{Proof} Straightforward. \\quad\\usebox{\\proofbox}\n \\end{Proof}\n\nThe following properties follow directly from the definition of $\\wbisim$ and Proposition~\\ref{reduction hidden}:\n\n \\begin{proposition} \\label{wbisim lemma}\n \\begin{enumerate}\n\n \\item If for all $\\proc{P}'$, $\\proc{Q}'$ such that $ \\proc{P} \\rtcredPi \\proc{P}' $, $ \\proc{Q} \\rtcredPi \\proc{Q}' $ over hidden channels, we have $ \\proc{P}' \\wbisim \\proc{Q}' $, then $ \\proc{P} \\wbisim \\proc{Q} $.\n\n \\item Let \\proc{P}, \\proc{Q} be such that no interaction is possible between them, then: $ \\proc{P} \\wbisim \\proc{P}' $ and $ \\proc{Q} \\wbisim \\proc{Q}'$ if and only if $ \\proc{P} \\Par \\proc{Q} \\wbisim \\proc{P}' \\Par \\proc{Q}' $.\n\n\n \\end{enumerate}\n \\end{proposition}\n\n\nThe $`p$-calculus is equipped with a rich type theory \\cite{Sangiorgi-Walker-Book'01}, from the basic type system for counting the arity of channels \\cite{Pierce-Sangiorgi'96} to session types \\cite{Honda'93} and sophisticated linear types in \\cite{Honda-Yoshida-Berger'04}.\nThe notion of type assignment we use here is the one first defined in \\cite{vBCV-CLaC'08} and differs from systems presented in the past in that types do not contain channel information, and in that it expresses \\emph{implication}, \\emph{i.e.}~has functional types and describes the `\\emph{input-output interface}' of a process.\n\n \\begin {definition}[Context assignment for $`p$ \\cite{vBCV-CLaC'08}] \\label{type assignment for pi}\nFunctional type assignment for the {\\PiC} is defined by the following sequent system:\\footnote{Type assignment is classical in nature (\\emph{i.e.}~not intuitionistic), since we can have more than one conclusion.}\n{\\def\\Turn{\\Turn}\n \\[ \\begin{array}{rlrlrl}\n(\\Zero): &\n \\Inf\t{ \\Pider {\\Zero} : `G |- `D }\n &\n(\\InRule) : &\n \\Inf\t{ \\Pider \\proc{P} : `G, x{:}A |- x{:}A,`D }\n\t{ \\Pider \\In a(x) . \\proc{P} : `G,a{:}A |- `D }\n &\n~ (\\OutRule) : &\n \\Inf\t[a \\not= b]\n\t{ \\Pider \\proc{P} : `G,b{:}A |- b{:}A,`D }\n\t{ \\Pider \\Out a . \\proc{P} : `G,b{:}A |- a{:}A,b{:}A,`D }\n \\\\ [5mm]\n(!): &\n \\Inf\t{ \\Pider \\proc{P} : `G |- `D }\n\t{ \\Pider \\Bang \\proc{P} : `G |- `D }\n &\n(`n): &\n \\Inf\t{ \\Pider \\proc{P} : `G,a{:}A |- a{:}A,`D }\n\t{ \\Pider \\New a \\proc{P} : `G |- `D }\n &\n(\\Par): &\n \\Inf\t{ \\Pider \\proc{P} : `G |- `D \\quad \\Pider \\proc{Q} : `G |- `D }\n\t{ \\Pider \\proc{P} \\Par \\proc{Q} : `G |- `D }\n\\\\ [5mm]\n&&(\\Let) : &\n \\multicolumn{3}{l}{\n \\Inf\t[y,z \\notele `D; x \\notele `G]\n\t{ \\Pider \\proc{P} : `G,y{:}B |- x{:}A,`D }\n\t{ \\Pider \\Let = z in \\proc{P} : `G,z{:}A\\arr B |- `D }\n}%\n\\\\ [5mm]\n& \\multicolumn{2}{r}{(\\PairOut)} : &\n \\multicolumn{3}{l}{\n\\Inf\t[a \\notele `D; a,c \\notele `G]\n\t{ \\Pider \\proc{P} : `G,b{:}A |- c{:}B,`D }\n\t{ \\Pider \\Out a . \\proc{P} : `G,b{:}A |- a{:}A\\arr B,c{:}B,`D }\n}%\n \\end{array} \\] }\n\nWe write $\\Pider \\proc{P} : `G |- `D $ if there exists a derivation using these rules that has this expression in the conclusion.\n \\end {definition}\nWe should perhaps stress that it is not known if this system has a relation with logic, other than the one established in \\cite{vBCV-CLaC'08}.\n\nNotice that the `\\emph{input-output interface of a $`p$-process}' property is nicely preserved by all the rules; handling of arrow types is restricted by the type system to the rules $(\\Let)$ and $(\\PairOut)$.\n\n\n \\begin {example}\nThe inference rules\n \\[ \\begin{array}{@{~}rlrl}\n(\\textit{Weak}): &\n\\Inf\t[`G' \\supseteq `G,`D' \\supseteq `D]\n\t{ \\Pider \\proc{P} : `G |- `D }\n\t{ \\Pider \\proc{P} : `G' |- `D' }\n &\n(\\PairIn) : &\n \\Inf\t[y,a \\not\\in `D, \\, x \\not\\in `G]\n\t{ \\Pider \\proc{P} : `G,y{:}B |- x{:}A,`D\n\t}{ \\Pider \\In a (x,y) . \\proc{P} : `G,a{:}A\\arr B |- `D }\n \\\\ [5mm]\n(\\OutRule\\,') : &\n\\Inf\t{ \\Pider \\Out a : `G,b{:}A |- a{:}A,b{:}A,`D }\n &\n(\\PairOut\\,') : &\n\\Inf\t{ \\Pider \\Out a : `G,b{:}A |- a{:}A\\arr B,c{:}B,`D }\n \\\\ [5mm]\n({\\forwarder}) : &\n\\Inf\t{ \\Pider \\BEq u=a : u{:}A |- a{:}A }\n &\n(\\PiOverline{`n}) : &\n\\Inf\t{ \\Pider \\PilmuTerm [Q] w : `G |- w{:}B,`D\n\t}{ \\Pider { \\BOut b (w) . { \\PilmuTerm [Q] w } } : `G |- b{:}B,`D }\n \\end{array} \\]\nare admissible.\nThat weakening is admissible follows by a straightforward reasoning over the structure of derivations; for the other rules, consider:\n \\[ \\begin{array}{ccc}\n\\Inf\t[\\InRule]\n\t{\\Inf\t[\\Let]\n\t\t{\\InfBox{ \\Pider \\proc{P} : `G,y{:}B |- x{:}A,`D } }\n\t\t{ \\Pider \\Let = z in \\proc{P} : `G,z{:}A\\arr B |- `D }\n\t}\n\t{ \\Pider \\In a (z) . \\Let = z in \\proc{P} : `G,a{:}A\\arr B |- `D }\n \\end{array} \\]\n \\[ \\begin{array}{ccc}\n \\Inf\t[\\OutRule]\n\t{\\Inf\t[\\Zero]\n\t\t{\\Pider {\\Zero} : `G,b{:}A |- b{:}A,`D }\n\t}{ \\Pider \\Out a . \\proc{P} : `G,b{:}A |- a{:}A,b{:}A,`D }\n &&\n\\Inf\t[\\PairOut]\n\t{\\Inf\t[\\Zero]\n\t\t{ \\Pider {\\Zero} : `G,b{:}A |- c{:}B,`D }\n\t}{ \\Pider \\Out a . {\\Zero} : `G,b{:}A |- a{:}A\\arr B,c{:}B,`D }\n \\end{array} \\]\n \\[ \\begin{array}{ccc}\n\\Inf\t[\\InRule]\n\t{\\Inf\t[\\OutRule\\,']\n\t\t{ \\Pider \\Out a : `G,w{:}A |- a{:}A,w{:}A }\n\t}{ \\Pider \\In u (w) . \\Out a : `G,u{:}A |- a{:}A }\n &&\n\\Inf\t[`n]\n\t{\\Inf\t[\\OutRule]\n\t\t{\\InfBox{ \\Pider \\PilmuTerm [Q] w : `G |- w{:}B,`D }\n\t\t}{ \\Pider \\Out b . \\PilmuTerm [Q] w : `G |- b{:}B,w{:}B,`D }\n\t}{ \\Pider \\New w ({ \\Out b . \\PilmuTerm [Q] w }) : `G |- b{:}B,`D }\n \\end{array} \\]\n \\end {example}\n\n\nSince weakening is included, we allow ourselves to be a little less precise when we construct derivations, and freely switch to multiplicative style where rules join contexts whenever convenient, by using, for example, the rule\n \\[ \\begin{array}{@{}rl}\n(\\mid): &\n \\Inf\t{ \\Pider \\proc{P}_1 : \\G_1 |- `D_1 \\quad \\dots \\quad \\Pider \\proc{P}_n : \\G_n |- `D_n }\n\t{ \\Pider \\proc{P}_1 \\Par\\dots \\Par \\proc{P}_n : \\G_1, \\ldots, \\G_n |- `D_1, \\ldots, `D_n }\n \\\\ [4mm]\n \\end{array} \\]\n\n\\Comment\nThe main soundness result for our notion of type assignment for $`p$ is stated as:\n \\begin {theorem}[Witness reduction \\cite{vBCV-CLaC'08}] \\label{Witness reduction}\nIf $\\Pider \\proc{P} : `G |- `D $ and $\\proc{P} \\redPi \\proc{Q}$, then $\\Pider \\proc{Q} : `G |- `D $.\n \\end {theorem}\n\n\n\n \\section{Context and background of this paper} \\label{context and background}\n\nIn the past, there have been several investigations of interpretation from the $`l$-calculus into the $`p$-calculus.\nResearch in this direction started by Milner's interpretation $\\MilTerm [`.] `. $ of $`l$-terms \\cite{Milner'92}; Milner's interpretation is input based and the interpretation of closed $`l$-terms respects large-step \\emph{lazy} reduction $\\redlazy$ \\cite{Abramsky'90} to normal form up to substitution; this was later generalised to $`b$-equality, but using weak bisimilarity \\cite{Sangiorgi-Walker-Book'01}.\n\nFor many years, it seemed that the first and final word on the interpretation of the $`l$-calculus has been said by Milner; in fact, input-based interpretations of the $`l$-calculus into the $`p$-calculus\nhave become the \\emph{de facto} standard, and most published systems are based on Milner's interpretation.\nThe various interpretations studied in \\cite{Sangiorgi-Walker-Book'01} constitute examples, also in the context of the higher-order $`p$-calculus; \\cite{Honda-Yoshida-Berger'04} used Milner's approach with a typed version of the $`p$-calculus; \\cite{Thielecke'97} used it in the context of continuation-passing style languages.\n\nMilner's interpretation of the $`l$-calculus into the (synchronous, monadic) $`p$-calculus is defined by:\n\n \\begin {definition}[Milner's interpretation \\cite{Milner'92}] \\label{Milner's interpretation}\nThe \\emph{input-based interpretation} of $`l$-terms into the $`p$-calculus\nIt is defined by:\n \\[ \\begin{array}{rcll}\n\\MilSem [v x] a & \\ByDef & \\Milvar x a & (x \\not= a)\n \\\\\n\\MilSem [l x . M] a & \\ByDef & \\MilTerm [l x . M] a & (b \\textit{ fresh})\n\\\\\n\\MilSem [a M N] a & \\ByDef & \\MilTerm [a M N] a & (c,z \\textit{ fresh})\n\\\\\n\\MilSem [x := M] & \\ByDef & \\MilSubX [M] x & (w \\textit{ fresh})\n \\end{array} \\]\nMilner calls $\\MilSem [x := M] $ an ``\\emph{environment entry}''; it could be omitted from the definition above, but is of use separately.\n \\end {definition}\n\nNotice that, in $\\MilSem [a M N] a $, the interpretation of the operand $N$ is placed under output, and thereby blocked from running; this comes at a price: now $`b$-reductions that occur in the operand can no longer be mimicked.\nCombined with using input to model abstraction, this makes that a redex can only be contracted if it occurs on the outside of a term (is the \\emph{top} redex): the modelled reduction is \\emph{lazy}.\n\nMilner states an Operational Soundness result:\n \\begin{theorem}[\\cite{Milner'92}] \\label{Milner's result}\nFor closed\n$M$, either $M \\diverges$ ($M$ diverges) and $\\MilSem[M] u \\diverges$, or $M \\redlazy `l y . R \\Vect{[N \\For x ]} $, and\n \\[ \\begin{array}{rcl}\n\\MilSem[M] u & \\rtcredPi & (\\Vect{`nx}) \\, ( \\MilSem[`ly.R] u \\Par \\Vect{\\MilSem[x:=N ]} ).\n \\end{array} \\]\n \\end{theorem}\n\n\n\nAlthough obviously the intention is that the substitutions $\\Vect{[N \\For x ]}$ are generated by the lazy reduction, the way the result is stated this need not be the case.\nThis `glitch' was fixed in \\cite{Bakel-Vigliotti-CONCUR'09} by reformulating Milner's result using \\emph{explicit} substitution.\nThat paper also\npresented a \\emph{logical}, \\emph{output-based} interpretation $\\PiLSem [`.] `. $ that interprets abstraction $`lx.M$ not using \\emph{input}, but via an asynchronous \\emph{output} which leaves the interpretation of the body $M$ free to reduce.\nThat interpretation is defined as:\n\n \\begin{definition} [Spine interpretation \\cite{Bakel-Vigliotti-CONCUR'09}] \\label{head interpretation} ~\n\\def\\PiSTerm{\\PiSTerm}\n \\[ \\begin {array}{rcll}\n\\PiLSem [v x] a & \\ByDef & \\PiHTerm [v x] a\n \\\\\n\\PiLSem [l x . M] a & \\ByDef & \\PiHTerm [l x . M] a\n \\\\\n\\PiLSem [a M N] a & \\ByDef & \\PiHTerm [a M N] a\n \\\\\n\\PiLSem [s M x := N] a & \\ByDef & \\PiHTerm [s M x := N] a\n \\end {array} \\]\n \\end{definition}\nFor this interpretation, \\cite{Bakel-Vigliotti-CONCUR'09} showed Operational Soundness and Type Preservation, but with respect to the notion of \\emph{explicit head reduction} $\\redxh$, similar to the notion defined in Def.~\\ref{explicit head reduction}, and the notion of type assignment defined in Def.~\\ref{type assignment for pi}.\nThe main results shown are (using $\\Diverges$ to denote non-termination):\n \\begin{theorem} [\\cite{Bakel-Vigliotti-CONCUR'09}] \\label{head reduction preservation}\n \\begin{enumerate}\n\n \\item If $M \\Diverges $ then $\\PiLSem[M] a \\Diverges$, and if $M \\redxh N $ then $ \\PiLSem[M] a \\rtcredPi\\equivC \\PiHTerm[N] a $.\n\n \\item If $\\der `G |- M : A $ then $ \\derPi \\PiLSem[M] a : `G |- a{:}A $.\n\n \\end{enumerate}\n \\end{theorem}\n\n\nAs argued in \\cite{Bakel-Vigliotti-CONCUR'09}, to show the above result, which formulates a direct \\emph{step-by-step} relation between $`b$-reduction and the synchronisation in the $`p$-calculus, it is necessary to make the substitution explicit.\nThis is a direct result of the fact that, in the $`p$-calculus, $`l$'s implicit substitution gets `implemented' \\emph{one variable at the time}, rather than all in one fell swoop.\nSince we aim to show a similar result for $\\lmu$, we will therefore define a notion of explicit substitution.\n\nAlthough termination is not studied in that paper, it is easily achieved through restricting the notion of reduction in the $`p$-calculus by not allowing reduction to take place inside processes whose output cannot be received, or by placing a guard on the replication as we do in this paper. \\\\\n\n\\noindent\nA natural extension of this line of research is to see if the $`p$-calculus can be used to interpret more complex calculi as well, as for example calculi that relate not to intuitionistic logic, but to classical logic, as $\\lmu$, $\\lmmt$, or $\\X$.\nThere are, to date, a number of papers on this topic.\n\nIn \\cite{Honda-Yoshida-Berger'04} an interpretation of Call-by-Value $\\lmu$ is defined that is based on Milner's.\nContrary to what is claimed in that paper, this interpretation itself is not based on Milner's \\CBV-interpretation, but rather is a variant of $\\MilTerm [`.] `. $; this implies that, \\emph{a priori}, a process like $\\sem{(`lx.x)((`lx.x)(`lx.x))} \\, a $ cannot be reduced: the top redex should not be contractable since the interpretation should respect the \\CBV-reduction, and the contraction of the right-hand redex cannot be modelled, since that term is placed under a guard.\nThe authors address this problem by considering \\emph{typed processes only}, and using a much more liberal notion of reduction on processes. The syntax of processes there considered is\n \\[ \\begin{array}{rcl}\nP &::=& \\Bang \\In x (\\Vect{y}) . \\proc{P} ~\\mid~ \\New \\Vect{y} ( \\Out x <{\\Vect{y}}> \\Par \\proc{P} ) ~\\mid~ \\proc{P} \\Par \\proc{Q} ~\\mid~ \\New x \\proc{P} ~\\mid~ \\Zero\n \\end{array} \\]\nand the notion of reduction on processes is extended to that of $\\searrow$, defined as the least compatible relation over typed processes (\\emph{i.e.}~closed under typed contexts), taken modulo $\\congruent$, that includes:\n \\[ \\begin{array}{rcl}\n\\Bang \\In x (\\Vect{y}) . \\proc{P} \\Par \\New \\Vect{a} ( \\Out x <{\\Vect{a}}> \\Par \\proc{Q} ) & \\red & \\Bang \\In x (\\Vect{y}) . \\proc{P} \\Par \\New \\Vect{a} ( \\proc{P}[\\Vect{a \\For y}] \\Par \\proc{Q} )\n \\end{array} \\]\nas the basic synchronisation rule, as well as\n\\[ \\begin{array}{rcl}\n\\Cont[{ \\New \\Vect{a} ( \\Out x <{\\Vect{a}}> \\Par \\proc{P} ) }] \\Par \\Bang \\In x (\\Vect{y}) . \\proc{Q} &\\searrow_r&\n\\Cont[{ \\New \\Vect{a} ( \\proc{P}[\\Vect{a \\For y}] \\Par \\proc{Q} ) }] \\Par \\Bang \\In x (\\Vect{y}) . \\proc{Q}\n\t\\\\\n\\New x ( \\Bang \\In x (\\Vect{y}) . \\proc{Q} ) & \\searrow_g & \\Zero \\qquad\n \\end{array} \\]\nwhere $\\Cont[`.]$ is an arbitrary (typed) context; note that $\\searrow$ synchronises with any occurrence of $\\Out x <{\\Vect{a}}> $, no matter what guards they may be placed under.\nThe resulting calculus is thereby very different from the original $`p$-calculus.\nTypes for processes prescribe usage of names, and name passing is restricted to \\emph{bound (private) name passing}.\\footnote{This is a feature of all related interpretations into the $`p$-calculus.}\n\nOn the relation between Girard's linear logic \\cite{Girard'87} and the $`p$-calculus,\n\\cite{Bellin-Scott'94} give a treatment of information flow in proof-nets; only a small fragment of Linear Logic was considered, and the translation between proofs and $`p$-calculus was left rather implicit as also noted by \\cite{Caires-Pfenning'10}.\n\nTo illustrate this, notice that Bellin and Scott use the standard syntax for the polyadic $`p$-calculus\n \\[ \\begin{array}{rrl@{\\quad}l}\n\\proc{P},\\proc{Q} & ::= &\n\\Zero\n\\mid \\proc{P} \\Par \\proc{Q}\n\\mid \\Bang \\proc{P}\n\\mid \\New a \\proc{P}\n\\mid \\In a ({ \\Vect{x} }) . \\proc{P}\n\\mid \\Out a < {\\Vect{p} }> . \\proc{P}\n \\end{array} \\]\nsimilar to the one we use here (see Def.~\\ref{pi calculus}) but for the fact that for us output is not synchronous, and there the $\\Let$-construct is not used.\nHowever, the encoding of a `cut' in linear logic\n\\[ \\def \\Turn} \\begin{array}[t]{@{}c@{\\quad}c@{\\quad}c{\\Turn}\n\\Inf\t{\\derLK {} |- x{:}A\\otimes B,y{:}{(A\\otimes B)^\\perp}\n\t \\quad\n\t \\Inf\t{\\derLK {} |- n{:}A,m{:}{A^\\perp}\n\t\t \\quad\n\t\t \\derLK {} |- z{:}B,w{:}{B^\\perp}\n\t\t}{\\derLK {} |- m{:}A^\\perp,w{:}B^\\perp,v{:}A\\otimes B }\n\t}{ \\derLK {} |- x{:}A\\otimes B,m{:}A^\\perp,w{:}{B^\\perp} }%\n\\]\n\\emph{i.e.}~the `term' $ x{:}A\\otimes B,m{:}A^\\perp, w{:}B^\\perp$, gets translated into a `language of proofs' which looks like:\n \\[\nCut^k(I,\\bigotimes^{n,z}_ v(I,I)m w z )x,(m,w) = (\\nu k) \\big( I[k\/y ] \\mid \\bigotimes^{n,z}_ v(I,I)m w z[k\/v ] \\big)\n \\]\nwhere the terms $Cut$ and $I$ are (rather loosely) defined in.\nNotice the use of arbitrary application of processes to channel names, and the operation of pairing; the authors do not specify how to relate this notation to the above syntax of processes they consider.\n\nHowever, even if this relationship is made explicit, also then a different $`p$-calculus is needed to make the encoding work.\nTo clarify this point, consider the translation in the $`p$-calculus of the term above, which according to the definition given in \\cite{Bellin-Scott'94} becomes:\n \\[\n(\\nu k) \\big ( x(a).\\underbrace{k(a)} \\mid (\\nu nz) (\\underbrace{\\Vect k(n,z).}\\big( n(b).m(b) \\mid z(b).w(b) \\big) ) ) .\n \\]\nAlthough intended, no communication is possible in this term.\nWe have `underbraced' the desired communication which is impossible, as the arity of the channel $k$ does not match.\nTo overcome this kind of problem, Bellin and Scott would need the \\Let-construct with use of pairs of names as we have introduced in this paper in Def.~\\ref{pi calculus}.\nMoreover, there is no relation between the interpreted terms and proofs stated in \\cite{Bellin-Scott'94} in terms of logic, types, or provable statements; here, we make a clear link between interpreted proofs and the logic through our notion of type assignment for the $`p$-calculus.\n\nIn \\cite{vBCV-CLaC'08} an interpretation into $`p$ of the sequent calculus $\\X$ is defined that enjoys the Curry-Howard isomorphism for Gentzen's {\\LK} \\cite{Gentzen'35}, which is shown to respect reduction.\nHowever, this result is only partial, as it is formulated as ``\\emph{if $ P \\redX Q $, then $\\PiSem[P] \\gtC \\PiSem[Q] $}'', allowing $\\PiSem[P] $ to have more observable behaviour than $\\PiSem[Q] $; the main reason for this is that reduction in $\\X$ is non-confluent.\nAlthough in \\cite{vBCV-CLaC'08} it is reasoned that this is natural in the context of non-confluent, symmetric sequent calculi, and is shown that the interpretation preserves types, it is a weaker result than could perhaps be expected.\n\nAn interpretation of $\\lmmt$ is studied in \\cite{CiminiCS'10}; the interpretation defined there strongly depends on recursion, is not compositional, and preserves only outermost reduction; no relation with types is shown.\n\n\n\n\n\\section{$\\lmux$: $\\lmu$ with explicit substitution}\n\\label{lmux section}\n\nOne of the main achievements of \\cite{Bakel-Vigliotti-CONCUR'09} is that it establishes a strong link between reduction in the $`p$-calculus and step-by-step \\emph{explicit substitution} \\cite{Bloo-Rose'95} for the $`l$-calculus, by formulating a result\nwith respect to explicit head reduction and the spine interpretation defined there.\n\nIn view of this, for the purpose of our interpretation it was natural to study a variant of $\\Lmu$ in \\cite{Bakel-Vigliotti-IFIPTCS'12} with explicit substitution as well; since here we work with $\\lmu$, we present here $\\lmux$, as a variant of $\\Lmux$ as presented in that paper.\nExplicit substitution treats substitution as a first-class operator, both for the logical and the structural substitution, and describes all the necessary steps to effectuate both.\n \\begin {definition}[$\\lmux$] \\label{definition lmux}\n\n \\begin {enumerate}\n\n \\item\nThe syntax of the \\emph{explicit $\\lmu$ calculus}, $\\lmux$, is defined by:\n \\[ \\begin {array}{@{}rrl}\nM,N & ::= & x \\mid `lx.M \\mid MN \\mid \\Sub M x := N \\mid\n\\muterm`a.[`b]M \\mid \\ContSub M `a := N . `g\n \\end {array} \\]\nWe consider the occurrences of $x$ in $M$ bound in $ \\Sub M x := N $, and those of $`a$ in $M$ in $ \\Sub M `a := N{`.}`g $; by Barendregt's convention, $x$ and $`a$ do not appear outside $M$.\n\n\n \\item\nThe reduction relation $\\redx$ on \\Paper{terms in }$\\lmux$ is defined \\CLAC{as the contextual closure of}\\Paper{through} the following rules\\Paper{ (for the sake of completeness, we list all)}:\n\n \\begin{enumerate} \\itemsep 0pt\n\n \\item Main reduction rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n (`l x.M) N &\\red & \\Sub M x := N \\\\\n(\\muterm `a . {\\Cmd} ) N & \\red & \\muterm`g . \\Sub {\\Cmd} `a := N{`.}`g & (`g \\textit{ fresh}) \\\\\n\\muterm`b.[`b]M &\\red& M & (`b \\not\\in \\fn(M)) \\\\ {}\n[`b]\\muterm`g.\\Cmd &\\red& \\Cmd[`b \\For `g]\n \\end {array} \\]\n\n \\item Term substitution rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\Sub x x := N &\\red & N \\\\\n\\Sub M x := N &\\red & M & (x \\not\\in \\fv(M)) \\\\\n\\Sub (`ly.M) x := N &\\red & `l y.(\\Sub M x := N ) \\\\\n\\Sub (PQ) x := N &\\red & (\\Sub P x := N )(\\Sub Q x := N ) \\\\\n\\Sub (\\muterm`a.[`b]M) x := N & \\red & \\muterm`a.[`b](\\Sub M x := N)\n \\end {array} \\]\n\n \\item Structural rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\Sub (\\muterm`d.\\Cmd) `a := N{`.}`g & \\red & \\muterm`d.(\\Sub {\\Cmd} `a := N{`.}`g ) \\\\\n\\Sub ([`a]M) `a := N{`.}`g & \\red & [`g](\\Sub M `a := N{`.}`g ) N \\\\\n\\Sub ([`b]M) `a := N{`.}`g & \\red & [`b](\\Sub M `a := N{`.}`g ) & (`a \\not= `b) \\\\\n\\Sub M `a := N{`.}`g & \\red & M & (`a \\not\\in \\fn(M)) \\\\\n\\Sub (`lx.M) `a := N{`.}`g & \\red & `lx.\\Sub M `a := N{`.}`g \\\\\n\\Sub (PQ) `a := N{`.}`g & \\red & (\\Sub P `a := N{`.}`g )( \\Sub Q `a := N{`.}`g )\n \\end {array} \\]\n\n\\Paper\n \\item Contextual rules\\CLAC{ ~ \\vspace*{-10pt} }\n \\[ \\begin {array}[t]{rcl}\nM \\red N &~\\Then ~~ &\n \\begin{cases}\n`lx.M &\\red& `lx.N \\\\\nML &\\red& NL \\\\\nLM &\\red& LN \\\\\n\\muterm`a.[`b]M & \\red & \\muterm`a.[`b]N \\\\\n\\Sub M x := L &\\red& \\Sub N x := L \\\\\n\\Sub L x := M &\\red& \\Sub L x := N \\\\\n\\Sub M `a := L{`.}`g &\\red& \\Sub N `a := L{`.}`g \\\\\n\\Sub L `a := M{`.}`g &\\red& \\Sub L `a := N{`.}`g \\\\ [1mm]\n \\end{cases}\n \\end {array} \\]\n\n\n \\end {enumerate}\n\n \\item \\label{redxsub definition}\nWe use $\\redxsub$ for the notion of reduction where only term substitution and structural rules are used (so not the main reduction rules)\\Paper{, and $\\eqx$ for the congruence generated by $\\redx$}.\n\n \\end {enumerate}\n\n \\end{definition}\n\n\\Paper\nThis is a system different from that of \\cite{Audebaud'94}, where a version with explicit substitution is defined for a variant of $\\lmu$ that uses de Bruijn indices \\cite{deBruijn'72}.\n\nNotice that since reduction in $\\lmux$ \\Paper{actually }is formulated via term rewriting rules \\cite{Klop'92}, reduction is allowed to take place also inside the substitution term.\n\\Paper{\n\nExplicit substitution describes explicitly the process of executing a $`b`m$-reduction, \\emph{i.e.}~expresses syntactically the details of the computation as a succession of atomic steps (like in a first-order rewriting system), where the implicit substitution of each $`b`m$-reduction step is split up into reduction steps.\nThereby t}\\CLAC{T}he following is straightforward:\n\n \\begin{proposition} [$\\lmux$ implements $\\lmu$-reduction]\n\\label{lmu vs lmux}\n \\begin{enumerate}\n \\item $ M \\redbmu N \\Implies M \\rtcredx N $.\n \\item $M \\ele \\lmu \\And M \\redx N \\Implies \\Exists L \\ele \\lmu \\Pred [ N \\rtcredxsub L ] $.\n \\end{enumerate}\n \\end{proposition}\n\n\\Paper\nThe notion of type assignment on $\\lmux$ is a natural extension of the system for the $\\lmu$-calculus of Def.~\\ref{typing for lmu} by adding rules $(\\TCut)$ and $(\\CCut)$.\n \\begin {definition}\nUsing the notion of types in Def.~\\ref{types}, type assignment for $\\lmux$ is defined by:\n \\[ \\begin {array}{@{}r@{~~}l@{\\dquad}r@{~~}l}\n(\\Ax) : &\n\\Inf\t{ \\derLmu `G,x{:}A |- x : A | `D }\n&\n (`m) : &\n \\Inf\t{ \\derLmu `G |- M : B | `a{:}A,`D\n\t}{ \\derLmu `G |- \\muterm`a.[`b]M : A | `b{:}B,`D }\n\\quad\n \\Inf\t{ \\derLmu `G |- M : A | `a{:}A,`D\n\t}{ \\derLmu `G |- \\muterm`a.[`a]M : A | `D }\n\\\\ [5mm]\n(\\arrI) : &\n\\Inf\t{ \\derLmu `G,x{:}A |- M : B | `D\n\t}{ \\derLmu `G |- `l x.M : A\\arrow B | `D }\n&\n(\\arrE) : &\n\\Inf\t{ \\derLmu `G |- M : A\\arrow B | `D\n\t \\quad\n\t \\derLmu `G |- N : A | `D\n\t}{ \\derLmu `G |- MN : B | `D }\n \\end{array} \\]\n \\[ \\begin {array}{@{}r@{~~}l@{\\dquad}r@{~~}l}\n(\\TCut): &\n\\Inf\t{ \\derLmu `G,x{:}A |- M : B | `D \\quad \\derLmu `G |- N : A | `D }\n\t{ \\derLmu `G |- {\\Sub M x := N } : B | `D }\t\t\n&\n(\\CCut): &\n\\Inf\t{ \\derLmu `G |- M : C | `a{:}A\\arr B,`D \\quad \\derLmu `G |- N : A | `D }\n\t{ \\derLmu `G |- {\\Sub M `a := N{`.}`g } : C | `g{:}B,`D }\n \\end {array} \\]\nWe write $\\derLmux `G |- M : A $ for judgements derivable in this system.\n\n \\end {definition}\n\n\n\\Comment\n \\begin{lemma}\nIf $M \\redx N$, and $\\derLmu `G |- M : A $ then $\\derLmu `G |- N : A $.\n \\end{lemma}\n\n \\begin{Proof}\n{ To do.}\n\n\n\nIn the context of head reduction and explicit substitution, we can economise further on how substitution is executed, and perform only those that are essential for the continuation of reduction.\nWe will therefore limit substitution to allow it to \\emph{only replace} the head variable of a term.\n(This principle is also found in Krivine's machine \\cite{Krivine'07}.)\\,\nThe results of \\cite{Bakel-Vigliotti-CONCUR'09} show that this is exactly the kind of reduction that the $`p$-calculus naturally encodes.\n\n \\begin{definition}[Explicit head reduction]\n \\label{explicit head reduction}\n\nThe\n\\emph{head variable} of $M$, $\\hv(M)$, is defined as expected, adding $ hv(\\Sub M x := N ) = \\hv(M) $ if $ \\hv(M) \\not= x $, and the \\emph{head name} $\\hn(M)$ is defined by $ \\hn(\\muterm`a.[`b] \\lmuHNF) = `b $, $ \\hn(\\Sub M x := N ) = \\hn(M) $, and $ \\hn(\\ContSub M `a := N . `g ) = \\hn(M) $ if $ \\hn(M) \\not= `a $.\n\nWe define \\emph{explicit head reduction} $\\redxh$ on $\\lmux$ as $\\redx$, but change and add a few rule\n{ (we only give the changes)}:\n\n\\Comment{\n \\begin{enumerate}\n \\item \\Paper{Main reduction rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n (`l x.M) N &\\red & \\Sub M x := N & \\\\\n(\\muterm `a . M ) N & \\red & \\muterm`g . \\Sub M `a := N{`.}`g & (`g \\textit{ fresh}) \\\\\n\\muterm`a.[`a]M &\\red& M & (`a \\not\\in \\fn(M)) \\\\\n\\muterm`a.[`b]\\muterm`g.\\Cmd &\\red& \\muterm`a.\\Cmd[`b \\For `g]\n \\end {array} \\]\n\n \\item \\label{term rules} \n\\label{changed rules}\nTerm substitution rules:\n \\Paper{ \\[ }\\CLAC{$} \\begin {array}{rcl@{\\quad}l}\n\\Paper{\\Sub x x := N &\\red & N \\\\\n\\Sub M x := N &\\red & M & (x \\not\\in \\fv(M)) \\\\\n\\Sub (`ly.M) x := N &\\red & `l y.(\\Sub M x := N ) & (x = \\hv(M)) \\\\ \n\\Sub (PQ) x := N &\\red & \\Sub {( \\Sub P x := N \\,Q )} x := N & (x = \\hv(P)) \\Paper{ \\\\\n\\Sub (\\muterm`a.[`b]M) x := N & \\red & \\muterm`a.[`b](\\Sub M x := N) & (x = \\hv(M)) \\\\ \n \\end {array} \\Paper{ \\] }\\CLAC{$}\n\n \\item\\label{structural rules}\nThere are only two structural rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\Sub (\\muterm`b.[`a]M) `a := N{`.}`g & \\red & \\muterm`b.[`g](\\ContSub M `a := N . `g ) N & (`a = \\hn(\\muterm`b.[`a]M)) \\\\\n\\Sub M `a := N{`.}`g & \\red & M & (`a \\not\\in \\fn(M)) \\\\\n \\end {array} \\]\n\n \\item\nWe remove the following contextual rules:\n \\[ \\begin {array}[t]{rcl}\nM \\red N &\\Then&\n \\begin{cases}\n LM &\\red& LN \\\\\n \\Sub L x := M &\\red& \\Sub L x := N \\\\\n \\Sub L `a := M{`.}`g &\\red& \\Sub L `a := N{`.}`g \\\\\n \\end{cases}\n \\end {array} \\]\nso no longer allow reduction inside the substitution or inside the right-hand side of an application.\n\n \\item \\label{substitution rules} \\label{new rules}\nWe add two substitution rules:\n \\[ \\kern-3mm \\begin {array}{@{}rcll}\n\\Sub {\\Sub M x := N } y := P &\\red&\n\t{ \\Sub { \\Sub { \\Sub M y := P } x := N } y := P }\n\t& (y = \\hv(M))\n\t\\\\\n\\Sub {\\Sub M `a := N{`.}`g } `b := L{`.}`d &\\red&\n\t{\n\t\\Sub { \\Sub { \\Sub M `b := L{`.}`d } `a := N{`.}`g } `b := L{`.}`d }\n\t& (`b = \\hn(M))\n \\end {array} \\]\n\n \\end {enumerate}\n\n\n \\begin{enumerate}\n \\item \\label{changed rules} \nWe replace the term substitution rule for application and add side-conditions:%\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\Sub (`ly.M) x := N &\\red & `l y.(\\Sub M x := N ) & (x = \\hv(`ly.M)) \\\\\n\\Sub (PQ) x := N &\\red & \\Sub {( \\Sub P x := N \\,Q )} x := N & (x = \\hv(PQ)) \\\\\n\\Sub (\\muterm`a.[`b]M) x := N & \\red & \\muterm`a.[`b](\\Sub M x := N) & (x = \\hv(\\muterm`a.[`b]M))\n \\end {array} \\]\n\n \\item\\label{structural rules}\nThere are only two structural rules:\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\Sub (\\muterm`b.[`a]M) `a := N{`.}`g & \\red & \\muterm`b.[`g](\\ContSub M `a := N . `g ) N\n\\\\\n\\Sub M `a := N{`.}`g & \\red & M & (`a \\not\\in \\fn(M)) \\\\\n \\end {array} \\]\n\n \\item\nWe remove the following contextual rules:\n \\[ \\begin {array}[t]{rcl}\nM \\red N &\\Then&\n \\begin{cases}\n LM &\\red& LN \\\\\n \\Sub L x := M &\\red& \\Sub L x := N \\\\\n \\Sub L `a := M{`.}`g &\\red& \\Sub L `a := N{`.}`g \\\\\n \\end{cases}\n \\end {array} \\]\n\n \\item \\label{substitution rules} \\label{new rules}\nWe add four substitution rules:\n \\[ \\kern-3mm \\begin {array}{@{}rcll}\n\\Sub {\\Sub M x := N } y := L &\\red& \\Sub { \\Sub { \\Sub M y := L } x := N } y := L\n\t& (y = \\hv(M)) \\\\\n\\Sub {\\ContSub M `a := N . `b } y := L &\\red& \\Sub { \\ContSub { \\Sub M y := L } `a := N . `b } y := L\n\t& (y = \\hv(M)) \\\\\n\\ContSub {\\ContSub M `a := N . `g } `b := L . `d &\\red& \\ContSub { \\ContSub { \\ContSub M `b := L . `d } `a := N . `g } `b := L . `d\n\t& (`b = \\hn(M)) \\\\\n\\ContSub {\\Sub M x := N } `b := L . `d &\\red& \\ContSub { \\Sub { \\ContSub M `b := L . `d } x := N } `b := L . `d\n\t& (`b = \\hn(M))\n \\end {array} \\]\n\n \\end {enumerate}\n \\end{definition}\nNotice that, for example, in case\\CLAC{~}\\ref{changed rules}, \\Comment{the fourth of }the clause\\Comment{s} postpones the substitution $ \\exsub x := N $ on $Q$ until such time that an occurrence of the variable $x$ in $Q$ becomes the head-variable of the full term, and that we no longer allow reduction inside the substitution or inside the right-hand side of an application.\n\n\\Paper\nNotice that we do not add rules like\n $ \\Sub { \\Sub M x := N } y := L \n\t\\red\n\\Sub { \\Sub M y := L } x := {\\Sub N y := L }\n$;\nas in \\cite{Bloo-Rose'95}, this would introduce undesired non-termination.\n\nRemark that we need to add the rules of case~\\ref{new rules}: if we take the term $\\Sub { \\Sub P x := Q } y := R $, under `normal' reduction $\\redx$, the innermost substitution has to complete first before the outermost can run.\nWhen moving towards explicit head reduction, this would mean that we cannot reduce, for example, $\\Sub { \\Sub yx x := Q } y := R $; the innermost substitution cannot advance, since $x$ is not the head variable, and the outermost cannot advance since it is not defined for the substitution term it has to work on, $\\Sub yx x := Q $ in this case.\nTo allow outermost substitution to `jump' the innermost, we need to add the extra rules.\nThis, potentially, introduces non-termination, but by demanding that the variable concerned is actually the head-variable of the term, we avoid this.\n\nNormal forms of explicit head reduction are naturally defined as follows:\n \\begin{definition} [cf.~\\cite{Lassen'06}]\nThe normal forms with respect to $\\redxh$ are defined through the grammar:\n \\[ \\begin {array}{@{}rrll}\n\\lmuNF & ::= &\n`lx. \\lmuNF\n\t\\\\ &\\mid &\nxM_1\\dots M_n & (n \\geq 0)\n\t\\\\ &\\mid &\n\\muterm `a . [`b] \\lmuNF & (`a \\not= `b \\Or `a \\in \\lmuNF', \\lmuNF \\not= \\muterm`g.[`d] \\lmuNF' )\n\t\\\\ &\\mid &\n\\Sub {\\lmuNF} x := M & (\\hv(\\lmuNF) \\not= x)\n\t\\\\ &\\mid &\n\\ContSub {\\lmuNF} `a := M . `g & (\\hn(\\lmuNF) \\not= `a)\n \\end {array} \\]\n \\end{definition}\n\n\n\nThe following proposition states the relation between explicit head reduction, head reduction, and explicit reduction.\n \\begin{proposition} \\label{explicit versus head}\n \\begin{enumerate}\n\n \\item\nIf $ M \\rtcredh N $, then there exists $L \\ele \\lmux$ such that $ M \\rtcredxh L $ and $ L \\rtcredxsub N $.\n\n\n \\item\nIf $ M \\rtcredxh[\\nf] N $ with $M \\ele \\lmu$, then there exists $ L \\ele \\lmu$ such that $ N \\rtcredxsub[\\nf] L $, and $ M \\rtcredh[\\nf] L $.\n\n \\item\n$M \\rtcredbmu[\\nf] N $ if and only if there exists $L \\ele \\lmux$ such that $ M \\rtcredxh[\\nf] L $ and $ L \\rtcredx[\\nf] N $.\n\n \\end{enumerate}\n \\end{proposition}\nThis result gives that we can show our main results for $\\lmux$ for reductions that reduce to \\HNF.\n\n\n\\Paper\nWe will give some examples that illustrate $\\lmux$ and $\\redxh$.\n \\begin{example} \\label{redxh examples} \\label{reductions example}\n \\begin{enumerate} \\itemsep1mm\n\n \\item\n$\n\\begin{array}[t]{lclcl}\n(`lx.xx)(`ly.y)\n\t& \\redxh &\n\\Sub xx x := `ly.y\n\t& \\redxh & \\\\\n\\Sub {(\\Sub x x := `ly.y x)} x := `ly.y\n\t& \\redxh &\n\\Sub {}(`ly.y)x x := `ly.y\n\t& \\redxh & \\\\\n\\Sub {\\Sub y y := x } x := `ly.y\n\t & \\redxh &\n\\Sub x x := `ly.y\n\t& \\redxh & \\\\\n\\Sub `ly.y x := `ly.y\n\t& \\redxh &\n`ly.y\n \\end{array} $\n\n \\item\nReduction in $\\redxh$ is not deterministic; notice that the term $\\LTerm$ can reduce in two ways:\n \\[ \\begin{array}{rcccl}\n\\LTerm\n\t&\\redxh&\n \\left \\{\n\\begin{array}{l}\n\\LTerm\n\t\\\\\n\\LTerm\n \\end{array}\n\\right \\}\n\t& \\redxh &\n\\LTerm\n \\end{array} \\]\n(the last step is not the only one possible).\n\n \\item \\label{mu reduction}\n$ \\begin{array}[t]{@{}lclcl}\n(\\muterm`a.[`b]\\muterm`d.[`a](`ly.y))(`lz.z)\n\t& \\redxh &\n\\Sub \\muterm`g.[`b]\\muterm`d.[`a](`ly.y) `a := `lz.z{`.}`g\n\t& \\redxh & \\\\\n\\Sub \\muterm`g.[`a](`ly.y)[`b\\For `d] `a := `lz.z{`.}`g\n\t& = &\n\\Sub \\muterm`g.[`a](`ly.y) `a := `lz.z{`.}`g\n\t& \\redxh & \\\\\n\\muterm`g.[`g](`ly.y)(`lz.z)\n\t& \\redxh &\n\\muterm`g.[`g]\\Sub y y := `lz.z\n\t& \\redxh & \\\\\n\\muterm`g.[`g]`lz.z\n\t& \\redxh &\n`lz.z\n \\end{array} $\n\n\n \\item\nSome reductions leave substitutions in place:\n\n\\[ \\begin{array}[t]{@{}lcl}\n`lf.(`lx.f(xx))(`lx.f(xx))\n\t& \\redxh &\n`lf.(\\Sub f(xx) x := `lx.f(xx) )\n \\end{array} \\]\n\nand the last term is in $\\redxh$-normal form.\n\n\\Comment\n \\item\n $ \\begin{array}[t]{@{}lclcl}\n\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y)\n\t& \\redxh &\n\\muterm`a.[`a]\\Sub q q := \\muterm`b.[`a]`ly.y\n\t& \\redxh & \\quad \\\\\n\\muterm`a.[`a]\\muterm`b.[`a]`ly.y\n\t& \\redxh &\n\\LTerm\n \\end{array} $\n\n\n \\item \\label{non-terminating reduction}\nOf course in $\\redxh$ we can have non-terminating reductions.\nWe know that in $\\redbmu$ and $\\redh$, $ (`lx.xx) (`lx.xx) $ reduces to itself; this is not the case for $\\redxh$, as is illustrated by:\n \\[ \\begin{array}[t]{@{}lclcl}\n`D`D ~=~ (`lx.xx) (`lx.xx)\n\t& \\redxh &\n\\Sub xx x := `D\n\t& \\rtcredxh & \\\\\n\\Sub {{}(\\Sub x x := `D x)} x := `D\n\t& \\redxh\n\t&\n\\Sub {{}(`ly.yy) x} x := `D\n\t& \\redxh & \\\\\n\\Sub {\\Sub yy y := x } x := `D\n\t& \\rtcredxh &\n\\Sub {\\Sub {(\\Sub y y := x y)} y := x } x := `D\n\t& \\redxh & \\\\\n\\Sub {\\Sub xy y := x } x := `D\n\t& \\redxh &\n\\Sub {\\Sub {\\Sub xy x := `D } y := x } x := `D\n\t& \\redxh & \\\\\n\\Sub {\\Sub {\\Sub {(\\Sub x x := `D y)} x := `D } y := x } x := `D\n\t& \\rtcredxh\n\t&\n\\Sub {\\Sub {{}(`lz.zz)y} y := x } x := `D\n\t& \\rtcredxh & \\\\\n\\Sub {\\Sub {\\Sub {{}(`lw.ww)z} z := y } y := x } x := `D\n\t& \\rtcredxh &\n\t\t\t\\dots\n \\end{array} \\]\n(notice the $`a$-conversions, needed to adhere to Barendregt's convention).\nThis reduction is deterministic and clearly loops.\nNotice that $`D`D$ does not run to itself; however,\n \\[ \\begin{array}[t]{@{}lclcl}\n\\Sub {\\Sub xy y := x } x := `D\n\t& \\rtcredxsub &\n\\Sub xx x := `D\n\t& \\rtcredxsub & `D`D\n \\end{array} \\]\nso, as stated by Proposition~\\ref{explicit versus head}, the standard reduction result can be achieved by reduction in $\\redxsub$ (we will use $`D$ again below).\n\n \\end{enumerate}\n \\end{example}\n\n \\Comment\n \\begin{example} \\label{reductions example}\nTo illustrate the\nnotions of reduction, take $ M = (`lf.(`lx.f(xx))(`lx.f(xx)))(`lab.a) $, and $N = `lxb.xx $; then\n \\begin{itemize}\n\n \\item $\\begin{array}[t]{@{}lclclclclc}\nM &\\rtcredbmu& NN\n\t&=& (`lxb.xx)N\n\t&\\rtcredbmu& `lb.NN\n\t&\\rtcredbmu& `lbb'.NN & \\dots\n \\end{array} $\n\n \\item $\\begin{array}[t]{@{}lclclclclc}\nM &\\rtcredh& NN\n\t&=& (`lxb.xx)N\n\t&\\rtcredh& `lb.NN\n\t&\\rtcredh& `lbb'.NN & \\dots\n \\end{array} $\n\n\n \\item $\\begin{array}[t]{@{}lclc}\nM &\\rtcredxh& \\Sub (`lx.f(xx))((`lx.f(xx))) f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub f(yy) y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub (`lab.a)(yy) y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub {\\Sub `lb.a a := yy } y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub `lb.yy y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub `lb.(`lz.f(zz))y y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub { \\Sub `lb.f(zz) z := y } y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub { \\Sub `lb.(`lab'.a)(zz) z := y } y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub { \\Sub {\\Sub `lbb'.a a := zz } z := y } y := `lx.f(xx) } f := `lab.a \\\\\n\t&\\rtcredxh& \\Sub {\\Sub { \\Sub `lbb'.zz z := y } y := `lx.f(xx) } f := `lab.a\n\t& \\dots\n \\end{array} $\n\n \\end{itemize}\n \\end{example}\n \n\n\n\n\n \\section{A logical interpretation of \\texorpdfstring{$\\lmux$}{}-terms to \\texorpdfstring{$`p$}{}-processes}\n\\label{lambda interpretation}\n\nWe will now define our logical,\\footnote{It is called \\emph{logical} because it has its foundation in the relation between natural deduction and Gentzen's sequent calculus\\CLAC{.}\\Paper{ \\LK; in particular, the case for application is based on the representation of \\emph{modus ponens}\n \\[ \\def \\Turn} \\begin{array}[t]{@{}c@{\\quad}c@{\\quad}c {\\Turn} \\begin{array}[t]{@{}c@{\\quad}c@{\\quad}c}\n\\Inf{ \\derLK `G |- A\\Arr B \\quad \\derLK `G |- A }{ \\derLK `G |- B }\n& \\textrm{ by } &\n \\Inf{ \\derLK `G |- A\\Arr B,`D \\quad \\Inf { \\derLK `G |- A,`D \\quad \\derLK `G,B |- `D }{ \\derLK `G,A\\Arr B |- `D }\n}{ \\derLK `G |- B,`D }\n \\end{array} \\]}}\noutput-based interpretation $\\PilmuTerm[M] a $ of the $\\lmux$-calculus into the $`p$-calculus\\Paper{. }\\CLAC{ (where $M$ is a $\\lmu$-term, and $a$ is the name given to its (anonymous) output), which is essentially the one presented in \\cite{Bakel-Vigliotti-IFIPTCS'12}, but no longer considers $[`a]M$ to be a term.\nThe reason for this change is the following: using the interpretation of \\cite{Bakel-Vigliotti-IFIPTCS'12},\n \\[ \\begin{array}{rcl}\n\\SPilmuTerm[\\muterm`a.`lx.x] a\n\t&=&\n\\PilmuTerm[U `a . {l x . {v x}}] a\n \\end{array} \\]\nis in normal form, and all inputs and outputs are restricted; thereby, it is weakly bisimilar to $\\Zero$ and to $\\SPilmuTerm[a {(l x . xx)} {(l x . xx)}] a $.\nSo using that interpretation, we cannot distinguish between \\emph{blocked} and \\emph{looping} computations, which clearly affects any full-abstraction result.\nWhen restricting our interpretation to $\\lmu$, this problem disappears: since naming has to follow $`m$-abstraction, $\\muterm`a.`lx.x$ is not a term in $\\lmu$.\nSince $\\lmu$ is a subcalculus of $\\Lmu$, this change clearly does not affect the results shown in \\cite{Bakel-Vigliotti-IFIPTCS'12} that all hold for the interpretation we consider here as well.\n\n}\nThe main idea behind the interpretation, as in \\cite{Bakel-Vigliotti-CONCUR'09}, is to give a name to the anonymous output of terms; it combines this with the inherent naming mechanism of $\\lmu$.\n\\Paper{As we will show in Thm.~\\ref{soundness},}\\CLAC{As shown in \\cite{Bakel-Vigliotti-IFIPTCS'12},} this encoding naturally represents explicit head reduction; we will need to consider weak reduction later for the full abstraction result, but not for soundness, completeness, or termination.\n\n \\begin{definition}[Logical interpretation\\Paper{ of $\\lmux$ terms}\\CLAC{ \\cite{Bakel-Vigliotti-IFIPTCS'12}}] \\label{lmu interpretation}\nThe interpretation of $\\lmux$ terms into the $`p$-calculus is defined by:\n \\[ \\begin {array}{rcl@{\\quad}l}\n\\SPilmuTerm [x] a & \\ByDef & \\PilmuTerm [v x] a & (u \\textit{ fresh})\n\t\\\\\n\\SPilmuTerm [`lx.M] a & \\ByDef & \\PilmuTerm [l x . M] a & (b \\textit{ fresh})\n\t\\\\\n\\SPilmuTerm [MN] a & \\ByDef & \\PilmuTerm [a M N] a & (c,v,d \\textit{ fresh})\n\t\\\\\n\\SPilmuTerm [\\Sub M x := N ] a & \\ByDef & \\PilmuTerm [S M x := N] a\n\t\\\\\n\\SPilmuTerm [x := N] {\\textcolor{white}{a}} & \\ByDef & \\PiExsub x := N & (w \\textit{ fresh})\n\t\\\\\n\\SPilmuTerm [\\muterm`g.\\Cmd] a & \\ByDef & \\PilmuTerm [m `g . \\Cmd] a & (s \\textit{ fresh})\n\t\\\\\n\\SPilmuTerm [{[`b]M}] a & \\ByDef & \\PilmuTerm [n `b M] a\n\t\\\\\n\\SPilmuTerm [\\Sub M `b := N{`.}`g ] a & \\ByDef & \\PilmuTerm [C M `b := N . `g] a\n\t\\\\\n\\SPilmuTerm [`a := M{`.}`g] {\\textcolor{white}{a}} & \\ByDef & \\PiExContsub `a := N . `g & (v,d \\textit{ fresh})\n \\end {array} \\]\n \\end{definition}\n\n \\Comment\nAs we will show below, this interpretation improves on the results of \\cite{Bakel-Vigliotti-CONCUR'09} in that we encode not only explicit head reduction for the $`l$-calculus, but also explicit head reduction for the $\\lmu$-calculus.\n\nIn \\cite{Bakel-Vigliotti-IFIPTCS'12}, this interpretation was defined for $\\Lmu$, where naming and $`m$-abstraction are separated; in fact, there the definition above contained:\n \\[ \\begin {array}{rcll}\n\\SPilmuTerm [\\muterm`g.\\Cmd] a & \\ByDef & \\PilmuTerm [m `g . \\Cmd] a , & s \\textit{ fresh}\n\t\\\\\n\\SPilmuTerm [[`b]M] a & \\ByDef & \\PilmuTerm [n `b M] a\n \\end {array} \\]\n\\Paper\nNote that the this encoding very elegantly expresses that the main computation in $\\muterm`g.\\Cmd$ is blocked: the name $s$ is fresh and bound, so the main output of $\\PilmuTerm [m `g . \\Cmd] a $ cannot be received.\nMoreover, n}%\n\\CLAC{\\noindent N}otice that\n \\CLAC{$ \\begin{array}{rclclcl}\n\\SPilmuTerm [\\muterm`g.[`b]M] a\n\t&\\ByDef&\n\\PilmuTerm [m `g . {{}[`b]M}] a\n\t&\\ByDef&\n\\PilmuTerm [m `g . n `b M] a\n\t& \\congruent &\n\\PilmuTerm [u `g . `b M] a\n \\end{array} $}%\n\\Paper{ \\[ \\begin{array}{rclclcl}\n\\SPilmuTerm [\\muterm`g.[`b]M] a\n\t&\\ByDef&\n\\PilmuTerm [m `g . {{}[`b]M}] a\n\t&\\ByDef&\n\\PilmuTerm [m `g . n `b M] a\n\t& \\congruent &\n\\PilmuTerm [u `g . `b M] a\n \\end{array} \\]\nwhich implies that we can add $ \\SPilmuTerm [\\muterm`g.[`b]M] a\n\t\\ByDef\n\\PilmuTerm [u `g . `b M] a $ to our encoding.\n\n\\Paper\nNote that we could have avoided the implicit renaming in the case for $`m$-abstraction by defining:\n $ \\SPilmuTerm [\\muterm`g.\\Cmd] a \\ByDef \\New {s} ({ \\PilmuTerm[\\Cmd] {s} \\Par \\BEq `g=a }) $\nwhich is operationally the same as $ \\PilmuTerm [m `g . \\Cmd] a $ (they are, in fact, weakly bisimilar) but then we could not show that terms in {\\HNF} are translated to processes in normal form (Lem.~\\ref{head pi normal form}), a property that is needed in the proof of termination.\n\nThere is a strong relation between this interpretation and the abstract machine defined in \\cite{Crolard'99}, but for the fact that that only represents lazy reduction.\n\nAs in \\cite{Bakel-Vigliotti-JLC'14}, we can make the following observations:\n\n \\begin{remark} \\label{natural observations}\n \\begin{itemize}\n\n \\item\nThe synchronisations generated by the encoding only involve processes of the shape:%\n \\[ \\begin{array}{c@{\\dquad}c@{\\dquad}c}\n\\Picaps & \\Out `b & \\In z (`b,y) . (\\proc{P} \\Par \\proc{Q})\n \\end{array} \\]\nso in particular, substitution is always well defined.\nThese synchronisations are of the shape:\n \\[ \\begin{array}{rcl}\n\\New c ({\\New y`b ({\\proc{P} \\Par \\Out c }) \\Par \\In c (v,d) . ({ \\proc{R} \\Par \\In d (w) . \\Out a }) })\n\t&\\redPi&\n\\New y`b ({\\proc{P} \\Par \\proc{R}[y\\For v] \\Par \\In b (w) . \\Out a })\n \\end{array} \\]\nand after the synchronisation over $c$, $\\proc{P}$ can receive over $y$ from $\\proc{R}[y\\For `a]$ and send over $b$ to $\\In b (w) . \\Out a $; or of the shape\n \\[ \\begin{array}[t]{@{}lcl@{}}\n\\New c ({\\New yb ({\\proc{P} \\Par \\Out c }) \\Par \\Picaps })\n\t&\\redPi&\n\\New yb ({\\proc{P} \\Par \\Out a })\n \\end{array} \\]\n\n \\item\nAll synchronisation takes place \\emph{only} over channels whose names are bound connectors in the terms that are interpreted.\n\n \\item\nTo underline the significance of our results, notice that the encoding is not trivial, since%\n \\[ \\begin{array}{rcl}\n\\SPilmuTerm [`lyz.y] a & = & \\PilmuTerm [l y . {l z . {v y}}] a\n\\\\\n\\SPilmuTerm [`lx.x] a &=& \\PilmuTerm [l x . {v x}] a\n \\end{array} \\]\nprocesses that differ under $\\wbisimilar$.\n\n \\end{itemize}\n \\end{remark}\n\n\nNotice that the context switches do not really influence the structure of the process that is created by the interpretation since they have no representation in $`p$, but are statically encoded through renaming.\nThis strengthens our view that, as far as our interpretation is concerned, $`m$-reduction is not a separate computational step, but is essentially static administration, a reorganisation of the applicative structure of a term, which is dealt with by our interpretation statically rather than by synchronisation between processes.\nIn other words, the point of view from the $`p$-calculus is that modelling $`b$-reduction involves a computational step, but context switches are dealt with by congruence; this is only possible, of course, because the interpretation of the operand in application uses replication.\n\nIn \\cite{Bakel-Vigliotti-CONCUR'09} the case for application in the interpretation for $`l$-terms was defined as:\n \\[ \\begin {array}{rcll}\n\\PiLSem [a M N] a & \\ByDef & \\PiHTerm [a M N] a\n \\end {array} \\]\nIn particular, there the input on name $c$ is \\emph{not replicated}: this corresponds to the fact that for $`l$-terms, in $\\PiHTerm[M] c $, the output $c$ is used \\emph{exactly once}, which is not the case for the interpretation of $\\lmu$-terms: for example, $`a$ might appear many times in $M$, and since $\\SPilmuTerm[u `a . `a M] a = \\PilmuTerm[u `a . `a M] a = \\PilmuTerm[{M[a\/`a]}] a $, then the name $a$ appears many times in the latter.\n\nWe would like to stress that, although inspired by logic, our interpretation does not depend on types \\emph{at all}; in fact, we can treat untypeable terms as well, and can show that $ \\SPilmuTerm [{}(`lx.xx)(`lx.xx)] a $ (perhaps the prototype of a non-typeable term) runs forever without generating output (see Example~\\ref{DD is a zero process\n; this already holds for the interpretation of \\cite{Bakel-Vigliotti-CONCUR'09}).\n\nNotice that, as is the case for Milner's interpretation and in contrast to the interpretation of \\cite{Bakel-Vigliotti-CONCUR'09}, a guard is placed on the replicated terms.\nThis is not only done with an eye on proving preservation of termination, but more importantly, to make sure that $ \\New x ( \\PiExSub x := N ) \\wbisim \\Zero $, a property we need for our full abstraction result: since a term can have named sub-terms, the interpretation will generate output not only for the term itself, but also for those named terms, so $ \\New x ( \\PilmuTerm[N] x ) $ -- as used in \\cite{Bakel-Vigliotti-CONCUR'09} -- \\emph{can have} observable behaviour, in contrast to here, where $ \\New x ( \\PiExSub x := N ) = \\New x ( \\PiExsub x := N ) $ \\emph{is} weakly bisimilar to $\\Zero$.\n\n\nObserve the similarity between\n \\[ \\begin{array}{rcll}\n\\Paper{\\hspace*{13mm} }\\SPilmuTerm[a M N] a &\\ByDef& \\PilmuTerm[a M N] a & \\textrm{and} \\\\\n\\SPilmuTerm [\\Sub M c := N{`.}`g ] a &\\ByDef&\n\\PilmuTerm [C M c := N . `g ] a \\\\ &\\ByDef&\n\\PilmuTerm [c M c := N . `g ] a\n \\end{array} \\]\nThe first communicates $N$ via the output channel $c$ of $M$\\CLAC{ (which might occur more than once inside $\\PilmuTerm[M] c $, so replication is needed)}, whereas the second communicates with all the sub-terms that have $c$ as output name, and changes the output name of the process to $`g$.\\Paper{\\footnote{A similar observation can be made for the interpretation of $\\lmu$ in $\\X$ \\cite{Bakel-Lescanne-MSCS'08}.}}\nIn other words, application is just a special case of explicit structural substitution;\nthis allows us to write $ \\PilmuTerm[A M N] a\n$ for $\\SPilmuTerm[a M N] a $.\n\\Paper{This very elegantly expresses exactly what the structural substitution does: it `connects' arguments with the correct position in a term.}\nThis stresses that the $`p$-calculus constitutes a very powerful abstract machine indeed: although the notion of structural reduction in $\\lmu$ is very different from normal $`b$-reduction, no special measures had to be taken in order to be able to express it; the component of our interpretation that deals with pure $`l$-terms is almost exactly that of \\cite{Bakel-Vigliotti-CONCUR'09} (ignoring for the moment that substitution is modelled using a guard, which affects also the interpretation of variables), but for the use of replication in the case for application.\n\n\n\\Paper\nThe operation of \\emph{renaming} we will use below is defined and justified via the following lemma, which states that we can safely rename the output of an interpreted $\\lmu$-term.\nFirst we need to show:\n\n \\begin{proposition} [\\cite{Bakel-Vigliotti-JLC'14}] \\label{aux-renaming lemma}\n \\[ \\begin{array}{rcl}\n\\New xb ({ \\In c(v,d) . ( \\proc{P} \\Par \\BEq d=e ) }) &\\wbisim& \\CLAC{ \\\\ }\n\\New a ({ \\BEq a=e \\Par \\New xb ({ \\In c(v,d) . ( \\proc{P} \\Par \\BEq d=a ) }) })\n \\end{array} \\]\n\n \\end{proposition}\n\nWe use this result to show the following:\n\n \\begin{lemma} [Renaming lemma] \\label{renaming lemma}\n \\begin{enumerate}\n\n \\item \\label{renaming outside}\nLet $ a \\not\\in \\fv(M)$, then\n $ \\begin{array}[t]{@{}lcl@{}}\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [M] a ) & \\wbisim & \\SPilmuTerm [M] e .\n \\end{array} $\n\n \\item \\label{renaming inside}\n $ \\begin{array}[t]{@{}lcl@{\\quad}l@{}}\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [M] b ) & \\wbisim & \\SPilmuTerm [{M[e \\For a]}] b & (b \\not= a)\n \\end{array} $.\n\n \\end{enumerate}\n \\end{lemma}\n\n\n \\begin{Proof}\nBy induction on the structure of $\\lmux$-terms.\n\n \\begin{description} \\itemsep 4pt\n\n \\item [$ M = x $]\n$ \\begin{array}[t]{@{}lclclcl}\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [v x] a )\n\t& \\ByDef &\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [v x] a )\n\t& \\wbisim &\n\\PilmuTerm [v x] e\n\t& \\ByDef &\n\\SPilmuTerm [v x] e\n \\end{array} $\n\n\n \\myitem [$ M = `lx.N $]\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [l x . N] a )\n\t& \\ByDef &\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [l x . N] a )\n\t& \\redPi (a) \\\\\n\\New axb ( \\BEq a=e \\Par \\PilmuTerm [N] b \\Par \\Out e )\n\t& \\wbisim & (a \\not= x \\Then a \\notele \\fv(N)) \\\\\n\\New a ( \\BEq a=e ) \\Par \\PilmuTerm [l x . N] e\n\t& \\wbisim &\n\\SPilmuTerm [l x . N] e\n \\end{array} $\n\n\n \\myitem [$ M = PQ $]\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [a P Q] a )\n\t& \\ByDef \\\\\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [X P Q] a )\n\t& \\wbisim & (\\Ref{aux-renaming lemma}, a \\not=c, a \\notele \\fv(P,Q)) \\\\\n\\PilmuTerm [X P Q] e\n\t& \\ByDef &\n\\SPilmuTerm [PQ] e\n \\end{array} $\n\n\n \\myitem [$ M = \\Sub P x := Q $]\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [s P x := Q] a )\n\t& \\ByDef & \\\\\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [s P x := Q] a )\n\t& \\wbisim & (\\IH\\Ref{renaming outside}, x \\not= a, a \\notele \\fv(P,Q)) & \\\\\n\\PilmuTerm [s P x := Q] e\n\t& \\ByDef &\n\\SPilmuTerm [s P x := Q] e\n \\end{array} $\n\n\n \\myitem [{$ M = \\muterm`b. [`g] N $}]\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [{\\muterm`b. [`g] N}] a )\n\t\\kern-2cm &&\n\t& \\ByDef & \\\\\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [u `b . `g N] a )\n\t& \\wbisim & \\multicolumn{3}{l}{ (\\IH\\Ref{renaming inside}, a \\not=`g, a \\notele \\fv(N)) } \\\\\n\\PilmuTerm [u `b . `g N] a \\, [e \\For a]\n\t& \\wbisim &\n\\PilmuTerm [u `b . `g N] e\n\t& \\ByDef &\n\\SPilmuTerm [u `b . `g N] e\n \\end{array} $\n\n\n \\myitem [$ M = \\Sub P `b := Q{`.}`g $]\n\\New a ( \\BEq a=e \\Par \\SPilmuTerm [\\Sub P `b := Q{`.}`g ] a )\n\t& \\ByDef & \\\\\n\\New a ( \\BEq a=e \\Par \\PilmuTerm [C P `b := Q . `g] a )\n\t& \\wbisim & (\\IH\\Ref{renaming outside}, a\\not= `b, a \\notele \\fv(P,Q)) \\\\\n\\PilmuTerm [C {P[e \\For a]} `b := {Q[e \\For a]} . `g] e\n\t& \\ByDef & \\\\\n\\PilmuTerm [C P `b := Q . `g] e\n\t& \\ByDef &\n\\SPilmuTerm [C P `b := Q . `g] e \\quad\\usebox{\\proofbox}\n \\end{array} $\n\n \\end{description}\n \\end{Proof}\n\nFor reasons of clarity, we use some auxiliary notions of equivalence, that are used in Thm~\\ref{soundness}.\n\n \\begin{definition} \\label{sbired definition}\n \\begin{enumerate}\n\n \\item\nWe define a \\emph{garbage collection} bisimilarity by: $ \\proc{P} \\redGC \\proc{Q} $\nif and only if there exists $\\proc{R}$ such that $ \\proc{P} = \\proc{Q} \\Par \\proc{R} $ and $ \\proc{R} \\wbisimilar \\Zero $.\n\nWe call a process that is weakly bisimilar to $\\Zero$ \\emph{garbage}.\n\n \\item\nWe define $\\redR$ as the largest, symmetric equivalence such that:\n\n \\begin{enumerate}\n\n \\item for all $\\proc{P}$ such that the name $a$ is the only free output of $\\proc{P}$ and is only used to output and only once: $ \\New a ( \\proc{P} \\Par \\Eq a=e ) \\redR \\proc{P}[e \\For a]$,\n\n \\item for all contexts $\\Cont$, if $ \\proc{P} \\redR \\proc{Q} $ then $ \\Cont [\\proc{P}] \\redR \\Cont [\\proc{Q}] $.\n\n \\end{enumerate}\n\nWe will use $\\redR$ if we want to emphasise that two processes are equivalent just using renaming, \\emph{i.e.}~define: $ \\New a ( \\proc{P} \\Par \\Eq a=e ) \\redR \\proc{P}[e \\For a]$.\n \\item\nWe define $ \\conredPi $ as $ \\lpired $.\n\n \\end{enumerate}\n \\end{definition}\nNotice that ${\\redGC} \\subset {\\wbisimilar}$ and ${\\redR} \\subset {\\wbisimilar}$.\n\n\n\nUsing this lemma, we can show the following:\n \\begin{example} \\label{redex example}\nThe interpretation of the $`b$-redex $(`lx.P)Q$ reduces as follows:\n \\[ \\def1.1} \\begin{array}{rcl{1.1} \\begin{array}[t]{@{}l\\CLAC{@{~}}c@{~}l}\n\\SPilmuTerm [a {(l x . P )} Q] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [a {l x . P} Q] a\n\t& \\redPi & (c) \\\\\n\\New c ( \\New bx ({ \\PilmuTerm[P] b \\Par \\BEq b=a \\Par \\PilmuTerm[x := Q] {} }) \\Par\n\\PiExContsub c := Q . a )\n\t& \\congruent & (c \\notele \\fn(P,Q)) \\\\\n\\New bx ({ \\PilmuTerm[P] b \\Par \\BEq b=a \\Par \\PiExSub x := Q }) \\Par \\New c ( \\PiExContSub c := Q . a )\n\t& \\wbisimG & (\\ast) \\\\\n\\New bx ({ \\PilmuTerm[P] b \\Par \\BEq b=a \\Par \\PiExSub x := Q })\n\t& \\wbisimR & (\\Ref{renaming lemma}) \\\\\n\\PilmuTerm[S P x := Q] a\n\t& \\ByDef &\n\\SPilmuTerm[s P x := Q] a\n \\end{array} \\]\n\nThis shows that $`b$-reduction is implemented in $`p$ by at least one synchronisation.\nNotice that, in step $(\\ast)$, the process\n $ \\New c ( \\PiExContSub c := Q . a ) \\ByDef \\New c ( \\PiExContsub c := Q . a ) $\nis weakly bisimilar to $\\Zero$.\n\n\\Comment\nRenaming is not always needed; for a concrete example, take\n \\[ \\begin{array}[t]{@{}l@{~}c@{~}l}\n\\SPilmuTerm [a {(l x . x )} (`ly.y)] a\n\t& \\ByDef & \\\\\n \\PilmuTerm [a {l x . x} `ly.y] a\n\t& \\redPi & (c) \\\\\n\\New bx ({ \\PilmuTerm[x] b \\Par \\BEq b=a \\Par \\PilmuTerm[x := `ly.y ] {} }) \\Par {}\n\\New c ( \\PiExContsub c := `ly.y . a )\n\t& \\wbisimG & \\\\\n\\New bx ( \\PilmuTerm[x] b \\Par \\BEq b=a \\Par \\PilmuTerm[x := `ly.y] {} )\n\t& \\ByDef & \\\\\n\\New bx ( \\PilmuTerm[v x] b \\Par \\BEq b=a \\Par \\PiExsub x := `ly.y )\n\t& \\redPi & (x) \\\\\n\\New bw ({ \\BEq w=b \\Par \\BEq b=a \\Par \\PilmuTerm[`ly.y ] w \\Par \\New x ( \\PiExsub x := `ly.y ) })\n\t& \\wbisimG & \\\\\n\\New bw ( \\BEq w=b \\Par \\BEq b=a \\Par \\PilmuTerm[`ly.y ] w )\n\t& \\ByDef & \\\\\n\\New bw ( \\BEq w=b \\Par \\BEq b=a \\Par \\PilmuTerm[l y . y] w )\n\t& \\redPi & (w) \\\\\n\\New bw ( \\BEq w=b \\Par \\BEq b=a ) \\Par \\PilmuTerm[l y . y] a\n\t& \\wbisimG &\n\\SPilmuTerm[l y . y] a\n \\end{array} \\]\nwhich shows that renaming is not needed here; however, we need it for cases like\n \\[ \\begin{array}[t]{@{}l@{~}c@{~}l}\n\\SPilmuTerm [a {(l x . x )} y] a\n\t& \\ByDef & \\\\\n \\PilmuTerm [a {l x . x} y] a\n\t& \\redPi & (c) \\\\\n\\New bx ({ \\PilmuTerm[x] b \\Par \\BEq b=a \\Par \\PilmuTerm[x := y ] {} }) \\Par\n\\New c ( \\PiExContsub c := y . a )\n\t& \\wbisimG & \\\\\n\\New bx ( \\PilmuTerm[x] b \\Par \\BEq b=a \\Par \\PilmuTerm[x := y] {} )\n\t& \\ByDef & \\\\\n\\New bx ( \\PilmuTerm[v x] b \\Par \\BEq b=a \\Par \\PiExsub x := y )\n\t& \\redPi & (x) \\\\\n\\New bw ({ \\BEq w=b \\Par \\BEq b=a \\Par \\PilmuTerm[v y] w \\Par \\New x ( \\PiExsub x := y ) })\n\t& \\wbisimG & \\\\\n\\New bw ( \\BEq w=b \\Par \\BEq b=a \\Par \\PilmuTerm[v y] w )\n\t& \\wbisimR & (\\Ref{renaming lemma}) \\\\\n\\PilmuTerm[v y] a\n\t& \\ByDef &\n\\SPilmuTerm[y] a\n \\end{array} \\]\n\nOn the other hand, as mentioned above, $`m$-reduction consists of a reorganisation of the structure of a term by changing its applicative structure.\nSince application is modelled through parallel composition, this implies that the interpretation of a $`m$-redex is essentially dealt with by congruence and renaming.\nFor example,\n \\[ \\begin{array}[t]{@{}lll}\n\\SPilmuTerm [a {(u `b . `b P)} Q] a\n\t& \\,\\ByDef & \\\\\n\\PilmuTerm [a {(u `b . `b P)} Q] a\n\t& =_{`a} & \\\\\n\\New `b ({ \\PilmuTerm [P] `b \\Par \\PiExContsub `b := Q . a })\n \\end{array} \\]\nWe can show, using Lem.~\\ref{replication lemma}, this last process is weakly bisimilar to\n \\[ \\begin{array}[t]{@{}lllll}\n\\New `g ( \\New `b ({ \\PilmuTerm [P] `g \\Par \\PiExContSub `b := Q . a }) \\Par\n\\PiExContSub `g := Q . a )\n\t& \\ByDef &\n\\SPilmuTerm [a {c P `b := Q . a} Q] a\n \\end{array} \\]\n(notice that\nwe have separated out the outside name of the term $P$, being $`b$, which we renamed to $`g$; this leaves two context substitutions, one dealing with the occurrences of $`b$ inside $P$, and one with $`g$.\\footnote{This corresponds to the behaviour of rule $(\\Riii)$ in $\\X$.})\nThis illustrates that structural substitution is dealt with by structural congruence and $`a$-conversion.\n\n\n\n \\end{example}\nThis example stresses that all synchronisations in the image of $\\PilmuTerm[`.] `. $ are over hidden channels, so by Proposition~\\ref{reduction hidden} are in $\\wbisim$.\n\nWe can also show that typeability is preserved: first, we need to prove a substitution lemma.\n\n \\begin{proposition}[Substitution] \\label{substitution lemma}\nIf $\\Pider \\proc{P} : `G,x{:}A |- x{:}A,`D $, then $\\Pider \\proc{P}[b\/x] : `G,b{:}A |- b{:}A,`D $ for $b$ fresh or $b{:}A \\ele `G \\union `D$.\n \\end{proposition}\n\n \\begin{Proof} Easy.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\n \\begin {theorem} [{$\\SPilmuTerm [`.] `. $} preserves $\\lmux$ types]\n \\label{typeability is preserved}\nIf $\\derLmux `G |- M : A | `D $, then $\\Pider \\SPilmuTerm [M] a : `G |- a{:}A,`D $.\n \\end {theorem}\n\n \\begin{Proof}\nBy induction on the structure of derivations in $\\TurnLmux$:\n\n \\begin {description} \\itemsep3mm\n\n \\item [$(\\Ax)$]\nThen $M= x$, and $ `G = `G ', x{:}A$.\nNotice that $\\PiSTerm[v x] a = \\PilmuTerm [v x] a $, and that\n \\[ \\begin {array}{@{}c}\n \\Inf\t[\\InRule]\n\t{\\Inf\t[\\Bang]\n\t\t{\\Comment{%\n\t\t\\Inf\t[\\InRule]\n\t\t\t{\\Inf\t[\\OutRule]\n\t\t\t\t{\\Inf\t[\\Zero]\n\t\t\t\t\t{\\Pider {\\Zero} : `G',w{:}A |- w{:}A }\n\t\t\t\t}{ \\Pider \\Out a : `G',w{:}A |- a{:}A,w{:}A }\n\t\t\t}{ \\Pider \\PiHTerm[v u] a : `G',u{:}A |- a{:}A }\n\t\t \\Inf\t[\\forwarder]\n\t\t\t{ \\Pider \\Eq u=a : `G',u{:}A |- a{:}A }\t\n\t\t}{ \\Pider \\Bang \\PiHTerm[v u] a : `G',u{:}A |- a{:}A }\n\t}{ \\Pider \\PilmuTerm[v x] a : `G',x{:}A |- a{:}A }\n \\end {array} \\]\n\n\n \\item [$(\\arrI)$]\nThen $M = `lx.N$, $A = C \\arr D$, and $\\derLmux `G,x{:}C |- N : D | `D $; by definition, $\\SPilmuTerm[l x . N ] a = \\PilmuTerm [l x . N ] a $.\nThen, by induction, $ \\D \\dcol \\Pider \\SPilmuTerm [N] b : `G,x{:}C |- b{:}D,`D $ exists, and we can construct:\n \\[ \\def\\Turn{\\Turn}\n \\Inf\t[`n]\n\t{\\Inf\t[`n]\n\t\t{\\Inf\t[\\mid]\n\t\t\t{ \\InfBox{\\D}{ \\Pider \\SPilmuTerm [N] b : `G,x{:}C |- b{:}D,`D }\n\t\t\t \\quad\n\t\t\t \\Inf\t[\\PairOut\\,']\n\t\t\t\t{ \\Pider \\Out a : x{:}C |- a{:}C\\arr D,b{:}D }\n\t\t\t}{ \\Pider \\SPilmuTerm [N] b \\Par \\Out a : x{:}C |- a{:}C\\arr D,b{:}D,`D }\n\t\t}{ \\Pider {\\New b ( \\SPilmuTerm [N] b \\Par \\Out a )} : `G,x{:}C |- a{:}C\\arr D,`D }\n\t}{ \\Pider {\\PilmuTerm[l x . N ] a } : `G |- a{:}C\\arr D,`D }\n \\]\nNotice that $\\PilmuTerm[l x . N ] a = \\SPilmuTerm [`lx . N ] a $.\n\n\n \\item [$(`m)$]\nThen $M = \\muterm `a.[`b]N$, and either:\n\n \\begin{description}\n\n \\item[$`a \\not= `b $]\nThen $\\derLmux `G |- N : A | `a{:}A,`b{:}B,`D $.\nBy induction, there exist a derivation for $ \\Pider \\SPilmuTerm [N] `b : `G |- `a{:}A,`b{:}B,`D $; then by Lem.~\\ref{substitution lemma}, also $ \\Pider \\PilmuTerm [u `a . `b N] a : `G |- a{:}A,`b{:}B,`D $ as well, and $\\PilmuTerm [u `a . `b N] a = \\SPilmuTerm [u `a . `b N] a $.\n\n \\item[$`a = `b $]\nThen $\\derLmux `G |- N : A | `a{:}A,`D $.\nBy induction, there exist a derivation for $ \\Pider \\SPilmuTerm [N] a : `G |- a{:}A,`a{:}A,`D $; then by Lem.~\\ref{substitution lemma}, also $ \\Pider \\PilmuTerm [u `a . `a N] a : `G |- a{:}A,`D $ as well, and $\\PilmuTerm [u `a . `a N] a = \\SPilmuTerm [u `a . `a N] a $.\n\n \\end{description}\n\n\n \\item [$(\\CCut)$]\nThen $M = \\ContSub P `a := Q . `g $ and we have $\\derLmux `G |- P : C | `a{:}A\\arr B,`D $ and $\\derLmux `G |- Q : A | `g{:}B,`D $ for some $B$.\nBy induction, there exist derivations $ \\D_1 \\dcol \\Pider { \\SPilmuTerm [P] a } : `G |- a{:}C,`a{:}A\\arr B,`D $ and, since $a$ is fresh,\n$ \\D_2 \\dcol \\Pider \\SPilmuTerm [Q] w : `G |- w{:}B,`D $, and we can construct\nthe derivation\n \\[ \\def\\Turn{\\Turn}\n\\kern-1mm \\begin{array}{@{}c}\n\\Inf\t[`n]\n\t{\\Inf\t[\\Par]\n\t\t{\\InfBox{\\D_1}{ \\Pider \\PilmuTerm [P] a : `G |- a{:}B\\arr A,`D }\n\t\t \\Inf\t[!]\n\t\t\t{\\Inf\t[\\PairIn]\n\t\t\t\t{\\Inf\t[\\mid]\n\t\t\t\t\t{\\Inf\t[!]\n\t\t\t\t\t\t{\\Inf\t[\\PiOverline{`n}]\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\\InfBox{\\D_2}{ \\Pider \\PilmuTerm [Q] w : `G |- w{:}B,`D }\n\t\t\t\t\t\t\t\t}{\n\t\t\t\t\t\t\t\t\\Pider \\BOut b (w) . { \\PilmuTerm [Q] w } : `G |- b{:}B,w{:}B,`D }\n\t\t\t\t\t\n\t\t\t\t\t\t}{ \\Pider \\Bang {\n\t\t\t\t\t\t\t\\BOut b (w) . { \\PilmuTerm [Q] w } } : `G |- b{:}B,`D }\n\t\t\t\t\t \\Inf\t[!]\n\t\t\t\t\t\t{\\Comment\n\t\t\t\t\t\t \\Inf\t[\\InRule]\n\t\t\t\t\t\t\t{\\Inf\t[\\OutRule]\n\t\t\t\t\t\t\t\t{\\Inf\t[\\Zero]\n\t\t\t\t\t\t\t\t\t{\\Pider {\\Zero} : w{:}A |- w{:}A }\n\t\t\t\t\t\t\t\t}{ \\Pider \\Out `g : w{:}A |- `g{:}A,w{:}A }\n\t\t\t\t\t\t\t}{ \\Pider \\Eq d=`g : d{:}A |- `g{:}A }\n\t\t\t\t\t\t \\Inf\t[\\forwarder]\n\t\t\t\t\t\t\t{ \\Pider \\Eq d=`g : d{:}A |- `g{:}A }\n\t\t\t\t\t\t}{ \\Pider \\BEq d=`g : d{:}A |- `g{:}A }\n\t\t\t\t\t}{ \\Pider \\PiExSub b := Q \\Par \\BEq d=`g : `G,d{:}A |- `g{:}A,b{:}B,`D }\n\t\t\t\t}{ \\Pider \\In `a (b,d) . ( \\PiExSub b := Q \\Par \\BEq d=`g ) : `G,`a{:}B\\arr A |- `g{:}A,`D }\n\t\t\t}{ \\Pider \\Bang \\In `a (b,d) . ( \\PiExSub b := Q \\Par \\BEq d=`g ) : `G,`a{:}B\\arr A |- `g{:}A,`D }\n\t\t}{ \\Pider \\PilmuTerm [P] a \\Par \\Bang \\In `a (b,d) . ( \\PiExSub b := Q \\Par \\BEq d=`g ) : `G,`a{:}B\\arr A |- `g{:}A,`D }\n\t}{ \\Pider \\PilmuTerm [c P `a := Q . `g] a : `G |- `g{:}A,`D }\n \\end{array} \\]\nand $ \\PilmuTerm [c P `a := Q . `g] a = \\SPilmuTerm[c P `a := Q . `g] a = \\PilmuTerm [C P `a := Q . `g] a $.\n\n \\item [\\CLAC{$\\arrE)$, $(\\TCut$}\\Paper{$(\\arrE)$, $(\\TCut)$}]\nNotice that since $ \\SPilmuTerm[a P Q] a = \\PilmuTerm[A P Q] a $ and $\\SPilmuTerm[\\Sub M x := N ] a = \\PilmuTerm[S M x := N ] a $, these cases are very similar to that for $(\\CCut)$.\n\\quad\\usebox{\\proofbox}\n \\end {description}\n \\end{Proof}\n\n\nWe can show a witness reduction result for our encoding, for which we need the property:\n\n \\begin{lemma}[Contraction \\cite{Bakel-Vigliotti-JLC'14}] \\label{Pi witness reduction}\n \\begin{enumerate}\n\n \\item If $ \\Pider \\New bc ( \\proc{P} \\Par \\Out a ) \\Par \\In a (x) . \\Out e : `G,a{:}C |- a{:}C,`D $, $a$ does not occur in $\\proc{P}$, and $a \\not= e$, then $ \\Pider \\New bc ( \\proc{P} \\Par \\Out e ) : `G |- `D $.\n\n \\item If $a$ does not occur in $\\proc{P}$ and $\\proc{Q}$ and $ \\Pider \\New bc ( \\proc{P} \\Par \\Out a ) \\Par \\In a (x,y) . \\proc{Q} : `G,a{:}C |- a{:}C,`D $, then \\\\ $ \\Pider \\New bc ( \\proc{P} \\Par \\proc{Q} [b \\For x,c \\For y] ) : `G |- `D $.\n\n \\end{enumerate}\n \\end{lemma}\n\nUsing this result, we can also show a witness reduction result:\n\n \\begin{theorem}\nIf $ \\Pider \\PilmuTerm[P] a : `G |- `D $, and $ \\PilmuTerm[P] a \\rtcredPi \\proc{Q} $, then $ \\Pider \\proc{Q} : `G |- `D $.\n \\end{theorem}\n\n \\begin{Proof}\nBy Remark~\\ref{natural observations}, Theorem~\\ref{typeability is preserved}, and Lemma~\\ref{Pi witness reduction}.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\n\nNotice that, as for Milner's and Sangiorgi's interpretations, ours is not extensional, since\n$ \\PilmuTerm[`D`D] a \\wbisim \\Zero $, but not $ \\PilmuTerm[`lx.`D`Dx] a \\wbisim \\Zero $ (see Lem.~\\ref{zero lemma} and \\ref{bmu reduction characterisation}).\n\n \\begin{remark} \\label{not extensional}\nWe could not have represented the extensional rules: note that\n \\[ \\begin{array}{@{}lcl}\n\\PilmuTerm [`lx.yx] a\n\t& \\ByDef &\n\\PilmuTerm [l x . {A y x}] a\n \\end{array} \\]\ndoes not reduce to $\\PilmuTerm[y] a $, and neither does\n \\[ \\begin{array}{@{}lclcl}\n\\PilmuTerm [{`m`a . [`b] y}] a\n\t& \\ByDef &\n\\PilmuTerm [u `a . `b y] a\n\t& = &\n\\PilmuTerm [y] `b\n \\end{array} \\]\nreduce to:\n \\[ \\begin{array}{@{}lclclcl}\n\\New xb ( \\PilmuTerm[y] `b \\Par \\Out a )\n\t& = &\n\\PilmuTerm [l x . {u `g . `b y}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [l x . {u `g . `b y}] a\n\t& \\ByDef &\n\\PilmuTerm [l x . {`m`g . {[`b]} y }] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [{`lx.`m`g . [`b] y }] a\n\t& = &\n\\PilmuTerm [{`lx.`m`g . [`b] y [x{`.}`g \\For `a]}] a\n \\end{array} \\]\n \\end{remark}\n\n\\Comment{ \\Double }\n\n\nTo illustrate the expressiveness and elegance of our interpretation, we give some examples:\n \\begin{example} \\label{our example}\n \\begin{enumerate} \\itemsep 1mm\n\n \\item\nThe interpretation of the fixed-point combinator runs as follows:\n\n\\[ \\begin{array}[t]{lcl}\n\\PilmuTerm [`lf.(`lx.f(xx))(`ly.f(yy))] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [l f . (`lx.f(xx))(`ly.f(yy))] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [l f . {a `lx.f(xx) `ly.f(yy)}] a \\quad\n\t& \\ByDef & \\\\\n\\PilmuTerm [l f . {d {l x . {{}f(xx)}} `ly.f(yy)}] a\n\t& \\redPi(c),\\wbisimG & \\\\\n\\PilmuTerm [l f . {S {}f(xx) x := `ly.f(yy)}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [l f . {t\n\t\t{a {v f} {(xx)}} x := `ly.f(yy)}] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [l {}f . {t {a {}f {(xx)}} x := `ly.f(yy)}] a\n \\end{array} \\]\n\n\\noindent\nThis last process is in normal form, \\emph{i.e.}~cannot be reduced further: since the only `reachable inputs' are $c$ and $f$, and the only `reachable output' is $x$, which is different from both $c$ and $f$, no synchronisation is possible.\nMoreover, this is not a lazy reduction.\n\n\\Comment\n \\item\nIn Fig.~\\ref{figure double} we run the interpretation of the term $(`lx.x)(\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y))$\nfrom Example~\\ref{redxh examples}\n$\\PilmuTerm[{}(`lx.x)(\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y))] a $,\n \\[ \\begin{array}{@{}lcl}\n(`lx.x)(\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y)) & \\red \\\\\n\\sub x x := \\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y) & \\red \\\\\n\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y) & \\red \\\\\n\\muterm`a.[`a](\\sub q q := \\muterm`b.[`a]`ly.y ) & \\red \\\\\n\\muterm`a.[`a]\\muterm`b.[`a]`ly.y & \\red \\\\\n\\muterm`a.[`a]`ly.y\n \\end{array} \\]\nas an example of a term that generates two outputs over $`a$, and highlights the need for the repeated use of replication; notice that the individual reduction steps return (translated) in that figure.\n\nNotice that\n \\[ \\begin{array}{@{}lcl}\n(`lx.x)(\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y))\n\t& \\redxh & \\\\\n\\Sub x x := \\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y)\n\t& \\redxh & \\\\\n\\muterm`a.[`a](`lq.q)(\\muterm`b.[`a]`ly.y)\n\t& \\redxh & \\\\\n\\muterm`a.[`a]\\Sub q q := \\muterm`b.[`a]`ly.y\n\t& \\redxh & \\quad \\\\\n \\multicolumn{3}{l}{\n\\muterm`a.[`a]\\muterm`b.[`a]`ly.y\n\t\\hfill \\redxh\n\\LTerm\n }\n \\end{array} \\]\n\n \\item\n $ \\begin{array}[t]{@{}lclcl}\n\\PilmuTerm[\\Sub M x := N L ] a & = \\\\\n\\PilmuTerm[A {S M x := N } L ] a & \\congruent \\\\\n\\PilmuTerm[S {a M L} x := N ] a & = \\\\\n\\PilmuTerm[\\Sub ML x := N ] a\n \\end{array} $\n\n\n \\item\nWhen running the reduction of Example \\ref{redxh examples}\\ref{mu reduction}, the naming features are taken care of by the interpretation:\n \\[ \\begin{array}{lcl}\n\\PilmuTerm[{}(\\muterm`a.[`b]\\muterm`d.[`a](`ly.y))(`lz.z)] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a {u `a . `b \\muterm`d.[`a](`ly.y)} `lz.z] a\n\t& = & \\\\\n\\PilmuTerm[a {\\muterm`d.[c](`ly.y)} `lz.z] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a {u `d . c `ly.y} `lz.z] a\n\t& = & \\\\\n\\PilmuTerm[a `ly.y `lz.z] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm[a (`ly.y) (`lz.z)] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a {l y . y} `lz.z] a\n\t& \\redPi (c) & \\\\\n\\New yb ({ \\PilmuTerm[y] b \\Par \\PiExSub y := `lz.z \\Par \\BEq b=a })\n\t& \\wbisimilar & (\\Ref{renaming lemma}) \\\\\n\\PilmuTerm[S y y := `lz.z] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[f {v y} y := {l z . z}] a\n\t& \\red (y) \\\\\n\\New w ( \\BEq w=a \\Par \\PilmuTerm[l z . z] w ) \\Par \\New y ( \\PiExSub y := `lz.z )\n\t& \\red (w) \\\\\n\\PilmuTerm[l z . z] w \\Par \\New w ( \\BEq w=a ) \\Par \\New y ( \\PiExSub y := `lz.z )\n\t& \\wbisim &\n\\PilmuTerm[`lz.z] a\n \\end{array} \\]\n\n\\Comment\n \\[ \\begin{array}{@{}lcl}\n\\SPilmuTerm[(\\muterm`a.[`b]\\muterm`d.[`a](`ly.y))(`lz.z)] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[A (\\muterm`a.[`b]\\muterm`d.[`a](`ly.y)) {`lz.z}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a {u `a . `b {\\muterm`d.[`a](`ly.y)}} {`lz.z}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[A {n `b {\\muterm`d.[c](`ly.y)}} {`lz.z}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[A {n `b {u `d . c (`ly.y)}} {`lz.z}] a\n\t& = & \\\\\n\\PilmuTerm[A {`ly.y} {`lz.z}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a {l y . y} {`lz.z}] a\n\t\\quad & \\redPi & (c) \\\\\n\\New yb ( \\PilmuTerm[y] b \\Par \\PiExSub y := `lz.z \\Par \\BEq b=a ) \\Par\n\\New c ( \\PiExContSub c := {`lz.z} . a )\n\t& \\wbisimG & \\\\\n\\New yb ( \\PilmuTerm[y] b \\Par \\PiExSub y := `lz.z \\Par \\BEq b=a )\n\t& \\ByDef & \\\\\n\\New yb ( \\PilmuTerm[v y] b \\Par \\PiExsub y := `lz.z \\Par \\BEq b=a )\n\t& \\redPi & (y) \\\\\n\\New bw ( \\BEq w=b \\Par \\PilmuTerm[`lz.z] w \\Par \\BEq b=a ) \\Par \\New y ( \\PiExSub y := `lz.z )\n\t& \\wbisimG & \\\\\n\\setcounter{indb}{1}\t\n \\New bw ( \\BEq w=b \\Par {} \\PilmuTerm[l z . z] w \\Par \\BEq b=a )\n\t& \\redPi & (w,b) \\\\\n\\PilmuTerm[l z . z] a \\Par {} \\New bw ( \\BEq w=b \\Par \\BEq b=a )\n\t& \\wbisimG &\n\\SPilmuTerm[l z . z] a\n \\end{array} \\]\n\n\n\n \\item\n$ \\begin{array}[t]{@{}lcl}\n\\SPilmuTerm [a {a P Q} R] a\n\t& \\ByDef,\\congruent &\n\\NewA cc' ( \\PilmuTerm [P] c' \\Par \\PiExContsub c' := Q . c \\Par\n\\PiExContsub c := R . a )\n \\end{array} $\n\n\\noindent\nso components of applications are placed in parallel under the interpretation.\nSimilarly, we have\n $\n\\SPilmuTerm [C {C M `a := Q . `b } `g := L . `d ] a\n=\n\\New `g`a ( \\PilmuTerm [M] a \\Par \\PiExContSub `a := Q . `b \\Par \\PiExContSub `g := L . `d )\n $, so repeated structural substitutions are also placed in parallel under the interpretation and can be applied independently.\n\n \\end{enumerate}\n \\end{example}\n\n \\section{Soundness, completeness, and termination}\n \\label{Soundness, completeness, and termination}\n\n\n\nWe can now show a reduction-preservation result for explicit head reduction for $\\lmux$, by showing that $\\PilmuTerm [`.] {`.} $ preserves $\\redxh$ up to weak bisimularity, stated using $\\equivC$ in \\cite{Bakel-Vigliotti-IFIPTCS'12}.\n\\Paper\nNotice that we prove the result for $\\lmux$ terms, do not require the terms to be closed, and that the result is shown for single step reduction.\n\n\n\n \\begin{theorem} [Soundness] \\label{soundness} \\label{head reduction simulation}\nIf $ M \\redxh N $, then $\\SPilmuTerm [M] a \\conredPi\n\\SPilmuTerm [N] a $.\n \\end {theorem}\n\n \\begin{Proof}\nBy induction on the definition of explicit head reduction\\Comment{; for convenience, we separate naming and $`m$-binding, so actually use the encoding of \\cite}.\n\n \\begin{description} \\itemsep5pt\n\n \\item [Main reduction rules]\n\n \\myitem[$ (`l x.M) N \\red \\Sub M x := N $]\n\\SPilmuTerm [a {(l x . M )} N ] a\n\t& \\ByDef & \\\\\n\\multicolumn{3}{@{}l}{ \\PilmuTerm [a {l x . M} N] a\n\t\\quad \\redPi(c),\\wbisimG } \\\\\n\\New b ({ \\PilmuTerm[M] b \\Par {\\PiExSub x := N } \\Par \\BEq b=a })\n\t& \\wbisimR (\\Ref{renaming lemma}) &\n\\PilmuTerm[S M x := N] a\n \\end{array} $\n\n\n \\myitem[{$(\\muterm `b . [`a]M ) N \\red \\muterm`g . \\Sub {[`a]M} `b := N{`.}`g , ~ `g \\textit{ fresh} $}]\n\\kern-20mm && \\kern10mm\n\\SPilmuTerm [a {(u `b . `a M)} N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [A {(\\muterm `b . [`a]M)} N] a\n\t& \\ByDef &\n\\PilmuTerm [A {(u `b . `a M)} N] a\n\t& =_{`a} & \\\\\n\\New `b ( \\PilmuTerm[M] `a \\Par \\PiExContSub `b := N . a )\n\t& = &\n\\PilmuTerm[u `g . `a {C M `b := N . `g }] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[u `g . `a {\\Sub M `b := N{`.}`g }] a\n\t& \\ByDef &\n\\SPilmuTerm[u `g . `a {s M `b := N{`.}`g }] a\n \\end{array} $\n\n\n \\myitem[{$ \\muterm`b.[`b]M \\red M \\hfill \\textit{ if } `b \\not\\in \\fn(M)$}]\n\\SPilmuTerm [\\muterm`b.[`b]M] a\n\t& \\ByDef &\n\\PilmuTerm [u `b . `b M] a\n\t& \\ByDef (`b \\not\\in \\fn(M) ) &\n\\PilmuTerm [M] a\n \\end{array} $\n\n\n \\myitem[{$\\muterm`a.[`b]\\muterm`g.[`d]M \\red \\muterm`a.[`d]M[`b \\For `g], `g \\not= `d$}]\n\\kern-35mm &&&& \\kern-10mm\n\\SPilmuTerm [\\muterm`a.[`b]\\muterm`g.[`d]M] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [u `a . `b {\\muterm`g.[`d]M}] a\n\t& \\ByDef &\n\\PilmuTerm [u `a . `b {u `g . `d M}] a\n\t& =&\n\\PilmuTerm [u `a . `d {M[`b \\For `g]}] a\n\t& \\ByDef& \\\\\n\\SPilmuTerm [u `a . `d {M[`b \\For `g]}] a\n \\end{array} $\n\n\n \\myitem[{$\\muterm`a.[`b]\\muterm`g.[`g]M \\red \\muterm`a.[`b]M[`b \\For `g]$}]\n\\kern-35mm &&&& \\kern-25mm\n\\SPilmuTerm [\\muterm`a.[`b]\\muterm`g.[`g]M] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [u `a . `b {\\muterm`g.[`g]M}] a\n\t& \\ByDef &\n\\PilmuTerm [u `a . `b {u `g . `g M}] a\n\t& =&\n\\PilmuTerm [u `a . `b {M[`b \\For `g]}] a\n\t& \\ByDef& \\\\\n\\SPilmuTerm [u `a . `b {M[`b \\For `g]}] a\n \\end{array} $\n\n\n \\item [Term substitution rules]\n\n \\myitem[$ \\Sub x x := N \\red N $]\n\\SPilmuTerm [S x x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [s {v x} x := N] a\n\t& \\redPi (x) & \\\\\n\\New w ( \\BEq w=a\n\\Par \\PilmuTerm [N] w ) \\Par \\New x ( \\PiExSub x := N )\n\t& \\wbisimR (\\Ref{renaming lemma}) & \\\\\n\\PilmuTerm [N] a \\Par \\New x ( {\\PiExSub x := N } )\n\t& \\wbisimG &\n\\SPilmuTerm [N] a\n \\end{array} $\n\n\n \\myitem[$ \\Sub M x := N \\red M, ~ x \\not\\in \\fv(M) $]\n\\SPilmuTerm [S M x := N] a\n\t& \\ByDef &\n\\PilmuTerm [s M x := N] a\n\t& \\congruent & \\\\\n\\PilmuTerm[M] a \\Par \\New x ( \\PiExsub x := N )\n\t& \\wbisimG &\n\\SPilmuTerm [M] a\n \\end{array} $\n\n\n\\myitem [$\\Sub (PQ) x := N \\red \\Sub {(\\Sub P x := N Q)} x := N , ~ x = \\hv(P) $]\n\\SPilmuTerm [S (PQ) x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {A P Q} x := N] a\n\t& \\wbisim (\\Ref{replication lemma}) & \\\\\n\\PilmuTerm [S {A {S P x := N} Q } x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {A {\\Sub P x := N } Q} x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {\\Sub P x := N Q } x := N] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [S {(a {\\Sub P x := N } Q)} x := N] a\n \\end{array} $\n\n\n \\myitem[$ \\Sub (`ly.M) x := N \\red `l y.(\\Sub M x := N ) , ~ x = \\hv(M) $]\n\\kern-16mm && \\kern10mm \\SPilmuTerm [S {(l y . M)} x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {l y . M} x := N] a\n\t& \\congruent & \\\\\n\\PilmuTerm [l y . {S M x := N}] a\n\t& \\ByDef &\n\\SPilmuTerm [l y . {S M x := N}] a\n \\end{array} $\n\n\n \\myitem[{$ \\Sub (\\muterm`a.[`b]M) x := N \\red \\muterm`a.[`b](\\Sub M x := N) , ~ x = \\hv(M)$}]\n\\kern-35mm && \\kern10mm\n\\SPilmuTerm [S {(u `a . `b M)} x := N] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {(u `a . `b M)} x := N] a\n\t& \\congruent (`a \\notele \\fn(N) ) &\n\\PilmuTerm [u `a . `b {S M x := N}] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [u `a . `b {S M x := N}] a\n \\end{array} $\n\n\n\n \\item [Structural rules]\n\n\\myitem[{$\\Sub (\\muterm`d.\\Cmd) `a := N{`.}`g \\red \\muterm`d.(\\Sub {\\Cmd} `a := N{`.}`g ) N$}] \\kern-10mm &&\n\\SPilmuTerm [S {(m `d . \\Cmd)} `a := N{`.}`g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {m `d . \\Cmd} `a := N . `g ] a\n\t& =_{`a} & (`d \\not\\in \\fn(\\exsub `a := {N{`.}`g} ) ) \\\\\n\\PilmuTerm [m `d . {C {\\Cmd} `a := N . `g }] a\n\t& \\ByDef &\n\\PilmuTerm [m `d . { {\\Cmd} \\excontsub `a := N . `g }] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [\\muterm`d.(\\Sub {\\Cmd} `a := N{`.}`g ) ] a\n \\end{array} $\n\n\n\n\\myitem[{$\\Sub ([`a]M) `a := N{`.}`g \\red [`g](\\Sub M `a := N{`.}`g ) N$}]\n\\kern-10mm && \\kern-10mm\n\\SPilmuTerm [S {(n `a M)} `a := N{`.}`g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {n `a M} `a := N . `g ] a\n\t& \\wbisim (\\Ref{replication lemma}) &\n\t\\\\\n\\PilmuTerm [n `g {A {C M `a := N . `g } N}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [n `g {A {M \\excontsub `a := N . `g } N}] a\n\t& \\ByDef &\n\\SPilmuTerm [A {C M `a := N . `g } N] `g\n\t& \\ByDef & \\\\\n\\SPilmuTerm [n `g {A {c M `a := N . `g } N}] a\n \\end{array} $\n\n\n\\myitem[{$\\Sub ([`b]M) `a := N{`.}`g \\red [`b]\\Sub M `a := N{`.}`g , ~ `b \\not= `a$}]\n\\kern-30mm && \\kern20mm\n\\SPilmuTerm [C {(n `b M)} `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {[`b]M} `a := N . `g ] a\n\t& \\ByDef &\n\\PilmuTerm [C {n `b M} `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [n `b {M \\excontsub `a := N . `g }] a\n\t& \\ByDef &\n\\SPilmuTerm [n `b {c M `a := N . `g }] a\n \\end{array} $\n\n \\myitem[$ \\Sub M `a := N{`.}`g \\red M , ~ `a \\not\\in \\fn(M) $]\n\\kern-7mm &&\n\\SPilmuTerm [C {M} `a := N . `g ] a\n\t& \\ByDef &\n\\PilmuTerm [C {M} `a := N . `g ] a\n\t& \\congruent & \\\\\n\\SPilmuTerm [M] a \\Par \\New `a ( \\PiExContSub `a := N . `g )\n\t& \\wbisimG &\n\\SPilmuTerm [M] a\n \\end{array} $\n\n\n \\myitem[$ \\Sub (`lx.M) `a := N{`.}`g \\red `lx.\\Sub M `a := N{`.}`g $]\n\\kern-7mm && \\kern-10mm\n\\SPilmuTerm [C {(l x . M)} `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {l x . M} `a := N . `g ] a\n\t& \\congruent & \\\\\n\\PilmuTerm [l x . {C M `a := N . `g }] a\n\t& \\ByDef &\n\\SPilmuTerm [l x . {C M `a := N . `g }] a\n \\end{array} $\n\n\n \\myitem[$ \\Sub (PQ) `a := N{`.}`g \\red \\Sub {(\\Sub P `a := N{`.}`g Q )} `a := N{`.}`g $]\n\\SPilmuTerm [C (PQ) `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {A P Q} `a := N . `g ] a\n\t& \\wbisim (\\Ref{replication lemma}) &\n\t\\\\\n\\PilmuTerm [C {A {C P `a := N . `g } Q } `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {C P `a := N . `g Q } `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [C {(a {C P `a := N . `g } Q)} `a := N . `g ] a\n \\end{array} $\n\n\n \\item [Substitution rules]\n\n \\myitem[$ \\Sub {\\Sub M x := N } y := P \\red \\Sub { \\Sub { \\Sub M y := P } x := N } y := P $]\n&& \\kern-15mm\n\\SPilmuTerm [S {S M x := N } y := P ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S {S M x := N } y := P ] a\n\t& \\wbisim (\\Ref{replication lemma}) &\n\t\\\\\n\\PilmuTerm [S {S {S M y := P } x := N } y := P] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [S {S {S M y := P } x := N } y := P] a\n \\end{array} $\n\n\n\\item[$ \\Sub {\\Sub M `a := N{`.}`g } `b := L{`.}`d \\red \\Sub { \\Sub { \\Sub M `b := L{`.}`d } `a := N{`.}`g } `b := L{`.}`d $] ~\n\n\n$ \\begin{array}{lclcl}\n\\SPilmuTerm [C {C M `a := N . `g } `b := L . `d ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {C M `a := N . `g } `b := L . `d ] a\n\t& \\wbisim (\\Ref{replication lemma}) &\n\t\\\\\n\\Paper{\\PilmuTerm [C {C {C M `b := L . `d } `a := N . `g } `b := L . `d ] a }\n\\CLAC{\\PilmuTerm [C {C {B M `b := L . `d } `a := N . `g } `b := L . `d ] a }\n\t& \\ByDef & \\\\\n\\SPilmuTerm [C {C {C M `b := L . `d } `a := N . `g } `b := L . `d ] a\n \\end{array} $\n\n\n \\item [Contextual rules] By induction. \\quad\\usebox{\\proofbox}\n\n\\Comment\n \\myitem[$ M \\red N \\Then ML \\red NL $]\n\\SPilmuTerm [a M L] a\n\t& \\ByDef &\n\\PilmuTerm [A M L] a\n\t& \\wbisim (\\IH) & \\\\\n\\PilmuTerm [A N L] a\n\t& \\ByDef &\n\\SPilmuTerm [a N L] a\n \\end{array} $\n\n\n \\myitem[$ M \\red N \\Then `lx.M \\red `lx.N $]\n\\SPilmuTerm [l x . M] a\n\t& \\ByDef &\n\\PilmuTerm [l x . M] a\n\t& \\wbisim (\\IH) & \\\\\n\\PilmuTerm [l x . N] a\n\t& \\ByDef &\n\\SPilmuTerm [l x . N] a\n \\end{array} $\n\n\n \\myitem[{$ M \\red N \\Then \\muterm`a.[`b]M \\red \\muterm`a.[`b]N$}]\n \\kern-10mm &&\n\\SPilmuTerm [u `a . `b M] a\n\t& \\ByDef &\n\\PilmuTerm [u `a . `b M] a\n\t& \\wbisim (\\IH) &\n\t\\\\\n\\PilmuTerm [u `a . `b N] a\n\t& \\ByDef &\n\\SPilmuTerm [u `a . `b N] a\n \\end{array} $\n\n\n \\myitem[$ M \\red N \\Then \\Sub M x := L \\red \\Sub N x := L $]\n\\kern-15mm &&\n\\SPilmuTerm [S M x := L] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [S M x := L] a\n\t& \\wbisim (\\IH) &\n\\PilmuTerm [S N x := L] a\n\t& \\ByDef &\n\\SPilmuTerm [S N x := L] a\n \\end{array} $\n\n\n \\myitem[$ M \\red N \\Then \\Sub M `a := L{`.}`g \\red \\Sub N `a := L{`.}`g $]\n\\kern-15mm &&\n\\SPilmuTerm [C M `a := L . `g] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C M `a := L . `g] a\n\t& \\wbisim (\\IH) &\n\\PilmuTerm [C N `a := L . `g] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [C N `a := L . `g] a \\quad\\usebox{\\proofbox}\n \\end{array} $\n\n\n \\end {description}\n \\end{Proof}\n\nRemark that, by Prop.~\\ref{reduction hidden}, all proper reductions in this proof are in $\\wbisim$.\nAlso, the proof \\CLAC{in \\cite{Bakel-Vigliotti-IFIPTCS'12} }shows that $`b$-reduction is implemented in $`p$ by at least one synchronisation.\n\nWe can now easily show:\n\n\n \\begin{theorem} [Operational Soundness\\Paper{ for $\\redwxh$}\\CLAC{ \\cite{Bakel-Vigliotti-IFIPTCS'12}}] \\label{Operational Soundness redxh} \\label{rtc soundness}\n\n \\begin{enumerate}\n \\item $M \\rtcredxh N \\Then \\SPilmuTerm [M] a \\wbisim \\SPilmuTerm [N] a $.\n\n \\item If $M \\Divergesxh $ then $\\PilmuTerm[M] a \\Diverges $.\n\n \\end{enumerate}\n \\end {theorem}\n\\Paper\n\n \\begin{Proof}\nThe first is shown by induction using Theorem~\\ref{head reduction simulation}, using Proposition~\\ref{reduction hidden}; the second follows from Example~\\ref{redex example}, and the fact that $`m$-reduction and substitution do not loop \\cite{Py-PhD'98} (\\emph{i.e.}~non-termination is caused only by $`b$-reduction).%\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\n\\CLAC{The proof in \\cite{Bakel-Vigliotti-IFIPTCS'12} shows that $`b$-reduction is implemented in $`p$ by at least one synchronisation.}\n\n\\Comment\nThis result states that our interpretation gives, in fact, a semantics for the explicit head reduction for $\\lmu$.\n By Lem.~\\ref{explicit versus head} and Prop.~\\ref{lmu vs lmux} we can show the same for $\\redh$ and $\\redbmu$.\n\nBy Lem.~\\ref{explicit versus head}, we can also show:\n\n \\begin{theorem} [Operational Soundness for $\\redh$] \\label{Operational Soundness redh} \\CLAC{~}\n\n \\begin{enumerate}\n \\item If $M \\rtcredh N $, then $ \\SPilmuTerm [M] a \\wbisimilar \\SPilmuTerm [N] a $.\n \\item $M \\divergesh \\Implies \\SPilmuTerm [M] a \\divergesPi $.\n \\end{enumerate}\n \\end {theorem}\n\n \\begin{Proof}\n \\begin{enumerate}\n \\item\n\nIf $M \\rtcredh N $, then, by Lem.~\\ref{explicit versus head}, there exists $L$ such that $ M \\rtcredxh L $ and $ L \\rtcredxsub N $; \\Paper{then, }by Theorem~\\ref{Operational Soundness redxh}, $ \\SPilmuTerm [M] a \\wbisimilar \\SPilmuTerm [L] a $.\n\n \\item\nSince every $`b$-reduction step in $\\redh$ invokes at least one step in $\\redxh$.\n\\quad\\usebox{\\proofbox}\n\n \\end{enumerate}\n \\end{Proof}\nand similarly, by Prop.~\\ref{lmu vs lmux}, we can show:\n\n \\begin{theorem} [Operational Soundness for $\\redbmu$] \\label{Operational Soundness}\n\n \\begin{enumerate}\n \\item\n$M \\rtcredbmu N \\Then \\SPilmuTerm [M] a \\wbisimilar \\SPilmuTerm [N] a $.\n \\item\n$M \\divergess \\Implies \\SPilmuTerm [M] a \\divergesPi $.\n \\end{enumerate}\n \\end {theorem}\n\n \\begin{Proof}\nIf $M \\rtcredbmu N $, then, by Prop.~\\ref{lmu vs lmux}, there exists $L$ such that $ M \\rtcredx L $ and $ L \\rtcredxsub N $; then, by Theorem~\\ref{Operational Soundness}, $ \\SPilmuTerm [M] a \\wbisimilar \\SPilmuTerm [L] a $,\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\\def \\Lmux\t{\\Lmux}\n\nWe can also show operational completeness for $\\redxh$.\n \\begin{theorem} [Operational Completeness for $\\redxh$] \\label{Operational Completeness for Lmux}\nIf $ \\PilmuTerm [M] a \\redPi \\proc{P} $, then there exists $N$ such that $ \\proc{P} \\wbisim \\PilmuTerm [N] a $ and $ M \\rtcredxh N $.\n \\end {theorem}\n\n \\begin{Proof}\nBy easy induction on the structure of terms, using the fact that all reductions that are possible in $\\PilmuTerm [M] a $ are generated by the interpretation, and correspond to $`b$-contractions or are the result of reduction over $y$ in\nthe process\n$ \\New y ( \\PilmuTerm [l x . M] y \\Par \\Eq y=a ) $, and is then the result of executing a substitution $\\Sub y y := `lx.M $.%\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\nThis in turn can be used to show:\n \\begin{lemma} \\label{Completeness}\n \\begin{enumerate}\n\n \\item\nLet $M$ be a term in $\\lmux$.\nIf $ \\PilmuTerm[M] a \\rtcredPi \\PilmuTerm[N] a $ then $ M \\rtcredxh N $.\n\n \\item\nLet $M \\ele \\lmu$, \\emph{i.e.}~a (pure) $\\lmu$-term.\nIf $ \\PilmuTerm[M] a \\redPi \\proc{P} $ then there exists $N \\ele \\lmux$ and $L \\ele \\lmu$ such that $ \\proc{P} \\wbisimilar \\PilmuTerm[N] a $, and $ M \\rtcredxh N $ and $ N \\rtcredxsub L $.\n\n \\end{enumerate}\n \\end {lemma}\n\n \\begin{Proof}\nThe first is an obvious consequence of Theorem~\\ref{Operational Completeness for Lmux}, the second follows from Lem.~\\ref{head reduction}, Lem.~\\ref{lmu vs lmux}, and Theorem~\\ref{Operational Completeness for Lmux}.\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\nSince renaming can be executed for abstractions, we can even generalise this result for lazy reduction $\\redlazy$ \\cite{Abramsky'90} and explicit lazy reduction $\\redxl$ \\cite{Bakel-Vigliotti-CONCUR'09} on closed $`l$-terms:\n \\begin{theorem}\nLet $M$ be closed. Then\n \\begin{enumerate}\n \\item $M \\redxl `ly.M' \\, \\Vect{\\exsub x := N } \\Implies \\CLAC{$ \\\\ $} \\SPilmuTerm [M] a \\rtcredPi\n y := P {(`nx)} \\, ( \\SPilmuTerm [{`ly.M}'] a \\Par \\Vect{ \\Bang \\SPilmuTerm [x := N] {} } ) $.\n\n \\item $M \\redlazy `ly.M' \\, \\Vect{[N \\For x]} \\Implies \\CLAC{$ \\\\ $} \\SPilmuTerm [M] a \\rtcredPi\n y := P {(`nx)} \\, ( \\SPilmuTerm [`ly.M'] a \\Par \\Vect{ \\SPilmuTerm [N] x } ) $.\n\n \\end{enumerate}\n \\end{theorem}\n\n\n\n\nWe can also show that equality with explicit substitution, $\\eqx$, is preserved under our encoding by weak bisimulation.\n\n \\begin{theorem} \\label{lambda mu x model}\nIf $M \\eqx N$, then $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $.\n \\end{theorem}\n\n \\begin{Proof}\nBy induction on the definition of $\\eqx$\\Paper\n; we only show the cases that are different from the proof of Theorem~\\ref{soundness}.\n\n \\begin{description} \\itemsep 4pt\n\n \\myitem [$ \\Sub (PQ) `a := N{`.}`g \\red (\\Sub P `a := N{`.}`g ) \\, (\\Sub Q `a := N{`.}`g) $]\n\\PilmuTerm[\\sub (PQ) x := N ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[S {A P Q} x := N ] a\n\t& \\wbisimilar (\\Ref{replication lemma}) \\\\\n\\New c ({ \\New x ({ \\PilmuTerm[P] c \\Par \\PiLExSub x := N }) \\Par\n\\quad {} \\\\ \\hfill\n\t\\New x ({ \\In c (v,d) . ({ \\PiExsub v := Q \\Par \\Eq d=a }) \\Par \\PiLExSub x := N })\n})\n\t& \\wbisimilar (\\Ref{replication lemma}) \\\\\n\\New c ({ \\New x ({ \\PilmuTerm[P] c \\Par \\PiLExSub x := N }) \\Par\n\t\\In c (v,d) . ({ \\New x ({ \\PiExsub v := Q \\Par \\PiExSub x := N }) \\Par \\Eq d=a })\n})\n\t& \\wbisimilar (\\Ref{replication lemma}) \\\\\n\\New c ({ \\New x ({ \\PilmuTerm[P] c \\Par \\PiExSub x := N }) \\Par\n\t\\In c (v,d) . ({ \\PiExsub v := {S Q x := N } \\Par \\Eq d=a }) })\n\t& \\ByDef & \\\\\n\\New c ({ \\New x ({ \\PilmuTerm[M] c \\Par \\PiExSub x := L }) \\Par\n\t\\In c (v,d) . ({ \\PiExsub v := {\\sub N x := L } \\Par \\Eq d=a }) })\n\t& \\ByDef & \\\\\n\\PilmuTerm[A {\\sub P x := N } {\\sub Q x := N }] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[{}(\\sub P x := N ) (\\sub Q x := N )] a\n \\end{array} $\n\n\n \\myitem [$ \\Sub (PQ) `a := N{`.}`g \\red (\\Sub P `a := N{`.}`g ) \\, (\\Sub Q `a := N{`.}`g) $]\n\\SPilmuTerm [C (PQ) `a := N . `g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [C {X P Q} `a := N . `g ] a\n\t& \\wbisim (\\Ref{replication lemma}) & \\\\\n\\PilmuTerm [Y {(C P `a := N . `g)} {C Q `a := N . `g}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [a {(C P `a := N . `g)} {C Q `a := N . `g}] a\n\t& \\ByDef & \\\\\n\\PilmuTerm [A {C P `a := N . `g } {(\\ContSub Q `a := N . `g )}] a\n\t& \\ByDef & \\\\\n\\SPilmuTerm [a {(C P `a := N . `g)} {(C Q `a := N . `g )}] a\n \\end{array} $\n\n \\item [If $ M \\red N $ then $ LM \\red LN $, $\\sub L x := M \\red \\sub L x := N $, and $ \\Sub L `a := M{`.}`g \\red \\Sub L `a := N{`.}`g $] ~\n\nBy induction.\n\\quad\\usebox{\\proofbox}\n\n\\Comment\n \\myitem [$ M \\red N \\Then LM \\red LN $]\n\\PilmuTerm[LM] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[X L M] a\n\t& \\wbisimilar & (\\IH) \\\\\n\\PilmuTerm[X L N] a\n\t& \\ByDef &\n\\PilmuTerm[LN] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then \\sub L x := M \\red \\sub L x := N $]\n\\PilmuTerm[\\sub L x := M ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[s L x := M] a\n\t& \\wbisimilar & (\\IH) \\\\\n\\PilmuTerm[s L x := N] a\n\t& \\ByDef &\n\\PilmuTerm[\\sub L x := N ] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then \\Sub L `a := M{`.}`g \\red \\Sub L `a := N{`.}`g $]\n\\PilmuTerm[\\Sub L `a := M{`.}`g ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[F L `a := M . `g ] a\n\t& \\wbisimilar & (\\IH) & \\\\\n\\PilmuTerm[F L `a := N . `g ] a\n\t& \\ByDef &\n\\PilmuTerm[\\Sub L `a := N{`.}`g ] a\n \\end{array} $\n\n\n \\myitem [$ \\sub x x := L \\red L $]\n\\PilmuTerm[\\, \\sub x x := L ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[s {v x} x := L] a\n\t& \\wbisimilar (\\Ref{reduction hidden}) \\\\\n\\New w ({ \\Eq w=a \\Par \\PilmuTerm[L] w }) \\Par \\New x ({ \\PiLExSub x := N })\n\t& \\wbisimilar (\\Ref{other reduction},\\Ref{renaming lemma}) &\n\\PilmuTerm[L] a\n \\end{array} $\n\n \\myitem [$ \\sub M x := L \\red M \\hfill (x \\notele \\fv(M) ) $]\n\\SPiLTerm [s M x := N] a\n\t& \\ByDef &\n\\PilmuTerm [S M x := N] a\n\t& \\wbisimilar & \\\\\n\\PilmuTerm[M] a \\Par \\New x ( \\PiLExSub x := N )\n\t~ \\wbisimilar (\\Ref{other reduction}) ~\n\\SPiLTerm[M] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then ML \\red NL $]\n\\PilmuTerm[ML] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[A M L] a\n\t& \\wbisimilar (\\IH) \\\\\n\\PilmuTerm[A N L] a\n\t& \\ByDef &\n\\PilmuTerm[NL] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then LM \\red LN $]\n\\PilmuTerm[LM] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[a L M] a\n\t& \\wbisimilar (\\IH) \\\\\n\\PilmuTerm[a L N] a\n\t& \\ByDef &\n\\PilmuTerm[LN] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then `lx.M \\red `lx.N $]\n\\PilmuTerm[`lx.ML] a\n\t& \\ByDef &\n\\PilmuTerm[l x . M] a\n\t& \\wbisimilar (\\IH) & \\\\\n\\PilmuTerm[l x . N] a\n\t& \\ByDef &\n\\PilmuTerm[`lx.N] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then \\sub M x := L \\red \\sub N x := L $]\n\\kern -11mm &&\n\\PilmuTerm[\\sub M x := L ] a\n\t& \\ByDef &\n\\PilmuTerm[S M x := L] a\n\t& \\wbisimilar (\\IH) \\\\\n\\PilmuTerm[S N x := L] a\n\t& \\ByDef &\n\\PilmuTerm[\\sub N x := L ] a\n \\end{array} $\n\n \\myitem [$ M \\red N \\Then \\sub L x := M \\red \\sub L x := N $]\n\\PilmuTerm[\\sub L x := M ] a\n\t& \\ByDef & \\\\\n\\PilmuTerm[s L x := M] a\n\t& \\wbisimilar (\\IH) \\\\\n\\PilmuTerm[s L x := N] a\n\t& \\ByDef &\n\\PilmuTerm[\\sub L x := N ] a\n \\end{array} $\n\n \\item [$ M \\red N \\Then M \\rtcred N $]\nImmediate.\n\n \\item [$ M \\rtcred M $]\nImmediate.\n\n \\item [$ M \\rtcred N \\And N \\rtcred L \\Then M \\rtcred L $]\nSince $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $ and $ \\PilmuTerm[N] a \\wbisimilar \\PilmuTerm[L] a $ implies $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[L] a $.\n\n \\item [$ M \\rtcred N \\Then M \\eqb N $]\nImmediate.\n\n \\item [$ M \\eqb N \\Then M \\eqb N $]\nSince $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $ implies $\\PilmuTerm[N] a \\wbisimilar \\PilmuTerm[M] a $.\n\n \\item [$ M \\eqb N \\And N \\eqb L \\Then M \\eqb L $]\nSince $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $ and $ \\PilmuTerm[N] a \\wbisimilar \\PilmuTerm[L] a $ implies $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[L] a $.\\QED\n\n\n \\end{description}\nThe steps to a reflexive, transitive closure and equivalence relation follow directly from $\\wbisimilar\n\n.\\quad\\usebox{\\proofbox}\n\n \\end{Proof}\n\nNow the following is an immediate consequence:\n\n \\begin{theorem}[Semantics] \\label{lambda model}\nIf $M \\eqbmu N$, then $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $.\n \\end{theorem}\n \\begin{Proof}\nBy induction on the definition of $\\eqbmu$.\nThe case $M \\rtcredbmu N$ follows from the fact that then, by Proposition~\\ref{lmu vs lmux}, also $M \\rtcredx N$, so by Theorem~\\ref{lambda mu x model}, we have $\\PilmuTerm[M] a \\wbisimilar \\PilmuTerm[N] a $.\nThe steps to an equivalence relation follow directly from $\\wbisimilar$.\\QED\n \\end{Proof}\n\nNotice that it is clear that we cannot prove the exact reversal of this result, since terms without head-normal form are all interpreted by $\\Zero$ (see also Lem.~\\ref{zero lemma}), but are not all related through $\\eqbmu$.\nUsing \\Paper{our notion of }weak equivalence, we can deal with the reverse part, and will do so in the last sections of this paper.\n\n\\Paper\n\nWe can show that interpretation of terms in $\\redxh$-normal form are in normal form as well, which is a property that we need here.\n \\begin{lemma} \\label{head pi normal form}\n$\\lmuNF$ is a $\\redxh$-nf implies $ \\PilmuTerm[\\lmuNF] a $ is irreducible.\n \\end{lemma}\n\n \\begin{Proof}\nBy induction on the structure of terms in $\\redxh$-normal form.\n\n \\begin{description}\n\n \\myitem[$\\lmuNF = xM_1\\dots M_n ~ (n \\geq 0) $]\n\\textrm{Then } \\PilmuTerm [xM_1\\dots M_n] a\n\t& \\ByDef & \\\\\n\\New c_n (\\PilmuTerm [xM_1\\dots M_{n-1}] c_n \\Par \\PiExContSub c_n := M_{n-1} . a )\n\t& \\ByDef & \\\\\n\\NewA c_n ({ \\NewA c_{n-1} (\\PilmuTerm [xM_1\\dots M_{n-2}] c_{n-1} \\Par \\PiExContSub c_{n-1} := M_n . c_n }) \\Par \\dots \\Par \\PiExContSub c_n := M_n . a )\n\t& \\congruent & \\\\\n\\NewA c_nc_{n-1} (\\PilmuTerm [xM_1\\dots M_{n-2}] c_{n-1} \\Par \\PiExContSub c_{n-1} := M_n . c_n \\Par \\dots \\Par \\PiExContSub c_n := M_n . a )\n\t& \\ByDef & \\\\\n \\New c_n\\dots c_1 (\\PilmuTerm [x] c_1 \\Par \\PiExContSub c_1 := M_1 . c_2 \\Par \\dots \\Par\n\\PiExContSub c_n := M_n . a ) \\\\ [1mm]\n \\end{array} $\n\n\\noindent\nSince\n \\[ \\begin{array}[t]{rcl}\n \\SPilmuTerm [x] c_1 &=& \\PilmuTerm [v x] c_1 \\\\\n \\PiExContSub c_i := M_i . c_{i+1} &=& \\PiExContFsub c_i := M_i . c_{i+1}\n \\end{array} \\]\nall $\\PilmuTerm [M_i] w $ appear under input on $c_i$, so no synchronisation inside one of the $\\PilmuTerm [M_i] w $ is possible; since all $c_i$ are fresh, all are different from $x$ and no synchronisation is possible over one of the $c_i$.\nSo this process is in normal form.\n\n \\item[$\\lmuNF = `lx. \\lmuNF' $]\nThen $ \\SPilmuTerm [l x . \\lmuNF'] a = \\PilmuTerm [l x . \\lmuNF'] a $, and, by induction, $\\PilmuTerm [\\lmuNF'] b $ is in normal form; since $a$ is fresh, that process does not input over $a$, so $ \\SPilmuTerm [l x . \\lmuNF'] a $ is normal form.\n\n \\item[{$\\lmuNF = \\muterm `a . [`b] \\lmuNF' ~ (`a \\not= `b \\Or `a \\in \\lmuNF', \\lmuNF' \\not= \\muterm`g.[`d] \\lmuNF'' )$}]\nThen $ \\SPilmuTerm [u `a . `b \\lmuNF'] a = \\PilmuTerm [u `a . `b \\lmuNF'] a $, this follows immediately by induction.\n\n \\item[$\\lmuNF = \\Sub \\lmuNF' x := M ~ (\\hv(\\lmuNF') \\not= x ) $]\nThen $ \\SPilmuTerm [s \\lmuNF' x := M ] a = \\PilmuTerm [s \\lmuNF' x := M ] a $.\nBy induction, $\\PilmuTerm [\\lmuNF'] a $ is in normal form; since $x$ is not the head-variable of $\\lmuNF$, the process $\\PilmuTerm [\\lmuNF'] a $ has no reachable input over $x$, so no synchronisation is possible over $x$; also, no synchronisation is possible inside $\\PilmuTerm [M] w $, as above.\n\n \\item[$\\lmuNF = \\Sub \\lmuNF' `a := M{`.}`g ~ (\\hn(\\lmuNF') \\not= `a ) $]\nThen $ \\SPilmuTerm [c {\\lmuNF'} `a := M . `g ] a = \\PilmuTerm [C {\\lmuNF'} `a := M . `g ] a $.\nBy induction, $\\PilmuTerm [\\lmuNF'] a $ is in normal form.\nNote that $`a$ is only a reachable output in $\\PilmuTerm [\\lmuNF'] a $ if $\\lmuNF'$ is an abstraction and $a = `a$; this is impossible, since we choose $a$ fresh.\nAs above, no synchronisation is possible inside $\\PilmuTerm [M] w $.\n\\quad\\usebox{\\proofbox}\n\n \\end{description}\n \\end{Proof}\n\nNotice that $ \\SPilmuTerm [u `a . `b {u `g . `d \\lmuNF}] a = \\PilmuTerm [u `a . `b {u `g . `d \\lmuNF}] a $, which is in normal form, so some reducible terms in $\\lmux$ are mapped to processes in normal form; this does not contradict the above result, of course.\n\nWe can now show the following termination results:\n \\begin{theorem} [Termination] \\label{lambda head termination}\n \\begin{enumerate}\n \\item If $M \\rtcredxh[\\nf] N $, then $ \\PilmuTerm [M] a \\convergesPi $.\n \\item If $M \\rtcredbmu[\\hnf] N $, then $ \\PilmuTerm [M] a \\convergesPi $.\n\n \\end{enumerate}\n \\end{theorem}\n\n \\begin{Proof}\n \\begin{enumerate}\n\n \\item\nBy Lem.~\\ref{head pi normal form}, if $N$ is in explicit head-normal from, then $ \\PilmuTerm [N] a $ is in normal form, and by Thm~\\ref{head reduction simulation}, $ \\PilmuTerm [M] a \\rtcredPi \\proc{P} $ with $\\proc{P} \\wbisim \\PilmuTerm [N] a $.\nSince in the proof of Thm~\\ref{head reduction simulation}, $\\wbisimG$ only removes processes in normal form, this implies that \\proc{P} is in normal form.\n \\item\nBy Lem.~\\ref{head reduction}, there exists $L$ in {\\HNF} such that $ M \\rtcredh[\\nf] L $;\nby Prop.~\\ref{explicit versus head}, there exists $\\lmuNF$ such that $ M \\rtcredxh[\\nf] \\lmuNF $; by the previous part, $ \\PilmuTerm [M] a \\convergesPi $.\n\\quad\\usebox{\\proofbox}\n\n \\end{enumerate}\n \\end{Proof}\n\nNotice also that this result is stronger than the formulation of the termination result for Milner's interpretation in \\cite{Sangiorgi-Walker-Book'01}, since it models reduction to head-normal form, not just lazy normal form.\nSince terms that have a normal form have a head-normal form as well, Theorem~\\ref{lambda head termination} immediately leads to:\n \\begin{corollary} \\label{lambda termination}\nIf $M \\convergesbmu $, then $ \\PiLmuSem [M] a \\convergesPi $.\n \\end{corollary}\n\n\n\n\n\n\\Paper{ \\section{Weak reduction for $\\lmu$ and $\\lmux$} \n \\label{weak reduction} }\n\\CLAC{ \\section{Weak \\Paper{reduction and }equivalences for $\\lmu$ and $\\lmux$}\n\\label{first full abstraction section} \n \\label{weak reduction}\n}\n\n\\label{weak equivalences}\n\n\n\\Paper{It seems widely accepted that bisimilarity-like equivalences have become the standard when studying interpretations of $`l$-calculi into the $`p$-calculus.\nThis creates a point of concern with respect to full abstraction.}\nSince $`D`D$ and $`W`W$ (where \\CLAC{$`D = `lx.xx$ and }$`W = `ly.yyy$\\Paper{; we will use $`W$ again below}) are closed terms that do not interact with any context, they are contextually equivalent; any well-defined interpretation of these terms into the $`p$-calculus, be it input based or output based, will therefore map those to processes that are weakly bisimilar to $\\Zero$, and therefore to weakly bisimilar processes.\nAbstraction, on the other hand, enables interaction with a context, and therefore the interpretation of $`lz.`D`D$ will \\emph{not} be weakly bisimilar to $\\Zero$.\nWe therefore cannot hope to model standard $`b`m$-equality in the $`p$-calculus in a fully-abstract way; rather, we need to consider a notion of reduction that considers \\emph{all} abstractions meaningful; therefore, the only kind of reduction on $`l$-calculi that can naturally be encoded into the $`p$-calculus is \\emph{weak} reduction.\n\n\\Paper\n\n\\DDinpi\n\n \\begin{example} \\label{DD is a zero process}\nConsider the reduction of $`D`D$ that was given in Example~\\ref{reductions example}; by Theorem~\\ref{head reduction simulation}, we have for example the reduction of $\\SPilmuTerm [`D `D] a \\wbisim \\SPilmuTerm [\\Sub {\\Sub xy y := x } x := `ly.yy ] a $ as shown in Figure~\\ref{DDinpi},\nwhich shows that the interpretation of $`D`D$ reduces without creating output over $a$ -- it always occurs inside a sub-process of the shape\n \\[ \\begin{array}{rcl}\n\\PiExContsub c := y . a\n \\end{array} \\]\nand does not input, since the head-variable is always bound, so $\\SPilmuTerm [`D `D] a $ is weakly bisimilar to $\\Zero$ (see also Lem.~\\ref{zero lemma}).\nTherefore,\n \\[ \\begin{array}{lclcl}\n\\PilmuTerm[`lz.`D`D] a\n\t&\\ByDef&\n\\PilmuTerm[l z . {`D`D}] a\n\t&\\wbisim& \\\\ &&\n\\New zb ( \\Zero \\Par \\Out a )\n\t&\\wbisim& \\\\ &&\n\\PilmuTerm[l z . {`W`W}] a\n\t&\\ByDef&\n\\PilmuTerm[`lz.`W`W] a\n \\end{array} \\]\nSo, for full abstraction, we are forced to consider $`lz.`D`D$ and $`lz.`W`W$ equivalent, and therefore, we need to consider \\emph{weak} equivalences on terms.\n \\end{example}\n\n\n\n\n\nGenerally, the concept of \\emph{weak} reduction refers to the variant of calculi that eliminate the contextual rule\n \\[ \\begin{array}{rcl}\nM \\red N &\\Then& `lx.M \\red `lx.N\n \\end{array} \\]\nFor the $`l$-calculus, then a closed normal form is an abstraction.\nNote that, in the context of $\\lmu$, this is no longer the case, since also context switches might occur inside the term.\n\nWe will now introduce the correct notions in $\\lmu$.\n\n\n \\begin{definition}\nWe define the notion $\\redwbmu$ of \\emph{weak $`b`m$-reduction} as in Def.~\\ref{lmu reduction}, the notion $\\redwh$ of \\emph{weak head reduction}\\footnote{This notion is also known as \\emph{lazy} reduction; for the sake of keeping our terminology consistent, we prefer to call it weak head reduction.} on $\\lmu$ as in Def.~\\ref{head reduction definition}, and the notion $\\redwxh$ of \\emph{weak explicit head reduction} on $\\lmux$ as in Def.~\\ref{explicit head reduction}, by (also) eliminating the rules:\n \\[ \\begin{array}{rcl}\n\\Sub (`ly.M) x := N &\\red & `l y.(\\Sub M x := N ) \\\\\n\\Sub (`lx.M) `a := N{`.}`g & \\red & `lx.(\\Sub M `a := N{`.}`g ) \\\\\nM \\red N &\\Then&\n`lx.M \\red `lx.N\n \\end{array} \\]\n \\end{definition}\n\nWe define the notion of weak head-normal forms, the normal forms with respect to weak head-reduction:\n\n \\begin{definition}[Weak head-normal forms for $\\lmu$] \\label{weak head reduction definition}\n \\begin{enumerate}\n\n \\item\nThe $\\lmu$ \\emph{weak head-normal forms} (\\WHNF) are defined through the grammar:\n \\[ \\begin {array}{rrl@{\\quad}l}\n\\lmuwHNF & ::= &\n`lx. M\n\t\\\\ &\\mid &\nxM_1\\dots M_n & (n \\geq 0)\n\t\\\\ &\\mid &\n\\muterm`a. [`b] \\lmuwHNF\n\t& (`a \\not= `b \\textrm{ or } `a \\in \\fn(\\lmuwHNF),\n\t\t\\Paper{\\textrm{ and }}\\lmuwHNF \\not= \\muterm `g.[`d] \\lmuwHNF' )\n \\end {array} \\]\n\n \\item\nWe say that $M$ \\emph{has a \\WHNF} if there exists $\\lmuwHNF$ such that $M \\rtcredwh \\lmuwHNF$.\n\n\\Comment{\n \\item\nWe define \\emph{values} as those \\WHNF s defined through:\n \\[ \\begin {array}{r\\CLAC{@{~}}r\\CLAC{@{~}}ll}\n\\lmuV & ::= & `lx. M\n\t\\\\ &\\mid &\n\\muterm`a. [`b] \\lmuV & (`a \\not= `b \\textrm{ or } `a \\not\\in \\lmuV, \\\\ &&& ~\n\t\\textrm{ and }\\lmuV \\not= \\muterm `g.[`d] \\lmuV' )\n \\end {array} \\]\n}\n\n \\end{enumerate}\n \\end{definition}\n\\Paper{As before, the normal forms of weak head reduction are the \\WHNF s.}\n\nThe main difference between \\HNF s and \\WHNF s is in the case of abstraction: where the definition of {\\HNF} only allows for the abstraction over a \\HNF, for \\WHNF s any term can be the body.\nMoreover, notice that both $\\LTerm$ and $\\LTerm$ are in \\WHNF.\n\nSince ${\\redwxh} \\subseteq {\\redxh}$, we can show the equivalent of Lem~\\ref{head reduction} and Thm.~\\ref{rtc soundness} also for \\emph{weak explicit head reduction}:\n\n \\Paper\n \\begin{proposition} \\label{weak head reduction}\nIf $ M \\rtcredbmu N $ with $N$ in {\\WHNF}, then there exists $\\lmuwxHNF$ such that $ M \\rtcredwh[\\nf] \\lmuwxHNF $ and $ \\lmuwxHNF \\rtcredbmu N $ without using $\\redwh$.\n \\end{proposition}\n\n \\begin{corollary} \\label{weak soundness}\n\n\\CLAC\n \\begin{theorem} [cf.~\\cite{Bakel-Vigliotti-IFIPTCS'12}] \\label{weak soundness}\n\n \\begin{enumerate}\n \\item\nIf $ M \\rtcredwxh N $, then $\\SPilmuTerm [M] a \\wbisim \\SPilmuTerm [N] a $.\n \\item\nIf $ M \\rtcredbmu N $ with $N$ in {\\WHNF}, then there exists $\\lmuwxHNF$ such that $ M \\rtcredwxh[\\nf] \\lmuwxHNF $ and $ \\lmuwxHNF \\rtcredx N $ without using $\\redwxh$.\n \\end{enumerate}\n\\CLAC\n \\end{theorem}\n\n \\Paper\n \\end{corollary}\n\n\n\nWe also define weak explicit head-normal forms.\n\n \\begin{definition}[Weak explicit head-normal forms\\Paper{ for $\\lmu$}] \\label{weak explicit head reduction definition}\n\n \\begin{enumerate}\n\n \\item\nThe $\\lmux$ \\emph{weak explicit head-normal forms \\CLAC{\\\\ }(\\WEHNF)} are defined through\\Paper{ the grammar}:\n \\[ \\def1.1} \\begin{array}{rcl{1.1} \\begin {array}{rrl@{\\qquad}l}\n\\lmuwxHNF & ::= &\n`lx. M \\, \\Vect{\\exsub y := N } \\, {\\Vexcontsub `s := Q . `t }\n\t\\\\ &\\mid &\nxM_1\\dots M_n \\, \\Vect{\\exsub y := N } \\, {\\Vexcontsub `s := Q . `t } & (n \\geq 0, ~ x \\notele \\Vect{y})\n\t\\\\ &\\mid &\n\\muterm `a . [`b] \\lmuwxHNF \\, \\Vect{\\exsub y := N } \\, {\\Vexcontsub `s := Q . `t }\n&\t\\\\ \\multicolumn{4}{r}\n\t{(`b \\notele \\Vect{`s}, ~ `a \\not= `b \\textrm{ or } `a \\in \\fn(\\lmuwxHNF),\n\t\\textrm{ and } \\lmuwxHNF \\not= `m`g.[`d] \\lmuwxHNF' ) }\n \\end {array} \\]\n\n \\item\nWe say that $M \\ele \\lmux$ \\emph{has an \\WEHNF} if there exists $\\lmuwxHNF$ such that $M \\rtcredwxh \\lmuwxHNF$.\n\n\\Comment{\n \\item\nWe define \\emph{values} as those weak head-normals defined through:\n \\[ \\begin {array}{rrll}\n\\lmuV & ::= & `lx. M\n\t\\\\ &\\mid &\n\\muterm`a. [`b] \\lmuV & (`a \\not= `b \\textrm{ or } `a \\not\\in \\lmuV, \\\\ &&& ~\n\t\\textrm{ and }\\lmuV \\not= \\muterm `g.[`d] \\lmuV' )\n \\end {array} \\]\n}\n\n \\end{enumerate}\n \\end{definition}\n\n \\begin{remark} \\label{substitution remark}\nIn the context of reduction (normal and weak), when starting from pure terms, the substitution operation can be left inside terms in normal form, as in\n \\[ \\begin{array}{rcl}\n(`lx.yM)NL &\\redxh& \\Sub yM x := N \\, L .\n \\end{array} \\]\nHowever, since, by Barendregt's convention, $x$ does not appear free in $L$, the latter term is operationally equivalent to $\\Sub yML x := N $; in fact, these two are equivalent under $\\equivwh$ (see Def.~\\ref{equivwh definition}), and also congruent when interpreted as processes.\n\\Paper\n \\[ \\begin{array}{lclcl}\n\\SPilmuTerm[a {S yM x := N } L] a\n\t& \\ByDef &\n\\PilmuTerm[A {S yM x := N } L] a\n\t& \\congruent & \\\\ &&\n\\PilmuTerm[S {A yM L} x := N] a\n\t& \\ByDef & \\\\ &&\n\\SPilmuTerm[S {a yM L} x := N] a\n \\end{array} \\]\n\nSince in weak reduction the reduction $\\Sub (`lx.M) y := N $ for $`lx . (\\Sub M y := N ) $ is not allowed, also this substitution can be considered to stay at the outside.\nTherefore, without loss of generality, for readability and ease of definition we will use a notation for terms that places all explicit substitutions on the outside.\\footnote{This is exactly the approach of Krivine's machine, where explicit substitutions are called \\emph{closures} that form an environment in which a term is evaluated.}\nSo actual terms can have substitutions inside, but they are written as if they appear outside.\nTo ease notation, we will use $\\Ssub$ for a set of substitutions of the shape $ \\exsub x := N $ or $ \\excontsub `a := N . `g $ when the exact contents of the substitutions is not relevant; we write $x \\ele \\Ssub $ if $\\exsub x := N \\ele \\Ssub $ and similarly for $`a \\ele \\Ssub $.\n\n \\end{remark}\n\nWe can show that the interpretation of a term without {\\WHNF} \\Paper{gives a process that }is weakly bisimilar to $\\Zero$.\n\n \\begin{lemma} \\label{zero lemma}\nIf $M$ has no {\\WEHNF} (so $M$ also has no \\WHNF), then $\\PilmuTerm[M] a \\wbisim \\Zero $.\n \\end{lemma}\n\n \\begin{Proof}\nIf $M$ has no {\\WEHNF}, then $M$ has no leading abstractions and all terms generated by reduction have a weak explicit head redex.\nIf $M = \\muterm `a . [`b] N $, then $\\PilmuTerm[M] a \\ByDef \\PilmuTerm[u `a . `b N] a \\wbisim \\Zero $, so also $ \\PilmuTerm[N] `b \\wbisim \\Zero $; therefore we can assume $M$ itself does not start with a context switch.\n\nWe reason by coinduction on the explicit weak head reduction sequence from $M$ and analyse the cases of weak explicit head reduction.\nFor example,\n \\[ \\def1.1} \\begin{array}{rcl{1.2} \\begin{array}{lcl}\n\\PilmuTerm [\\VSub {(`lx.P_1)P_2 \\dots P_n} y := Q \\, {\\Vexcontsub `a := R . `b }] a ~ \\ByDef\n\t\\\\ \\dquad\n\\NewA \\Vect{c} (\\PilmuTerm [l x . P_1] c_1 \\Par \\Comment{ {} \\\\ \\dquad \\dquad } {\\VPiExContSub c_{i-1} := P_i . c_i } \\Par {\\VPiExSub y := Q } \\Par \\, {\\VPiExContSub `a := R . `b })\n \\end{array} \\]\nwhere $c_{n-1} = a$\\CLAC{. }\\Paper{ and\n \\[ \\def1.1} \\begin{array}{rcl{1.1} \\begin{array}{rcl}\n\\PiExContSub c_{i-1} := P_i . c_i &=& \\PiExContFsub c_{i-1} := P_i . c_i \\\\\n\\PiExSub y_j := Q_j &=& \\PiExsub y_j := Q_j \\\\\n\\PiExContSub `a_k := R_k . `b_k &=& \\PiExContFsub `a_k := R_k . `b_k\n \\end{array} \\]}\nSince a synchronisation over $c_1$ is possible, the process is not in normal form.\nObserve that all outputs are over bound names or under guard, and since the result of the reduction has no head variable, no input is exposed.\nSo \\Paper{reduction of $\\PilmuTerm[M] a $ cannot exhibit an input or an output, so }$\\PilmuTerm[M] a \\wbisim \\Zero $.%\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\\Paper{\\begin{figure*\n\nThe reduction of $\\SPilmuTerm [a `D `D] a $ is given in Fig.~\\ref{DDnogarb}, which shows that the interpretation of $`D`D$ reduces without creating output over $a$; notice that the individual steps of the above reduction in $\\redxh$ in Example~\\ref{redxh examples} are respected in Fig.~\\ref{DDnogarb}.\n\nAs a direct consequence of this result,\\Paper{ as for Milner's and Sangiorgi's interpretations,} our interpretation is not extensional, since\n$ \\PilmuTerm[`D`D] a \\wbisim \\Zero $, whereas $ \\PilmuTerm[`lx.`D`Dx] a \\ByDef \\PilmuTerm[l x . {`D`Dx}] a \\not\\wbisim \\Zero $.\n\n\n\nWe can show the following property.\n\n \\begin{lemma} \\label{redwh to redwxh lemma} \\label{redbmu to redwxh lemma}\n\nLet $M$ and $N$ be pure $\\lmu$-terms; then\n$M \\rtcredwh[\\nf] N$ if and only if there exists $N'$, $\\Ssub$ such that $M \\rtcredwxh[\\nf] N' \\, \\Ssub $, and $ N' \\, \\Ssub\\rtcredxsub[\\nf] N $.\n\n\n \\end{lemma}\n\n\n\n\nWe will now define equivalences $\\equivwbmu$ and $\\equivwh$ between terms of $\\lmu$, and $\\equivwxh$ between terms of $\\lmux$ (the last two are defined coinductively as bisimulations), that are based on weak reduction, and show that the last two equate the same pure $\\lmu$-terms.\nThese notions all consider terms without {\\WHNF} equivalent.\nThis is also the case for the approximation semantics we present in the next section.\n\nFirst we define a weak equivalence generated by the reduction relation $\\redwbmu$.\n\n \\begin{definition} \\label{wbmu equivalence}\nWe define $\\equivwbmu$ as the smallest congruence that contains:\n \\[ \\begin{array}{rcl@{\\quad}l}\nM, N \\emph{ have no \\WHNF} &\\Then& M \\equivwbmu N \\\\\n(`l x . M ) N & \\equivwbmu & M [ N \\For x ] \\\\\n(\\muterm `a . \\Cmd ) N & \\equivwbmu & \\muterm`g. \\Cmd [N{`.}`g \\For `a] & (`g \\textit{ fresh}) \\\\\n\\quad \\muterm`a.[`b]\\muterm`g.[`d]M & \\equivwbmu & \\muterm`a.([`d]M[`b \\For `g]) \\\\\n\\muterm `a . [`a] M & \\equivwbmu & M & (`a \\notele M)\n\\Comment\nM \\equivwbmu N \\And N \\equivwbmu L &\\Then& N \\equivwbmu L \\\\\n \\\\\nM &\\equivwbmu& M \\\\\nM \\equivwbmu N & \\Then &\n\\multicolumn{2}{l}{ \\begin{cases}\nN \\equivwbmu M \\\\\n`lx.M \\equivwbmu `lx.N \\\\\nLM \\equivwbmu LN \\\\\nML \\equivwbmu NL \\\\\n\\muterm `a . [`b] M \\equivwbmu \\muterm `a . [`b] N \\\\\n \\end{cases} }\n\n \\end{array} \\]\n \\end{definition}\n\n\\Paper\nNotice that $`D`D \\equivwbmu `W`W$ and $`lz.`D`D \\equivwbmu `lz.`W`W$, but $`D`D \\not \\eqbmu `W`W$; moreover, $\\equivwbmu$ is closed under reduction.\nIn Section~\\ref{approximation section} we will show that two terms are equivalent in $\\equivwbmu$ if and only if they have the same set of weak approximants.\n\n\nSince reduction is confluent, the following is immediate.\n\n \\begin{proposition} \\label{confluence equiv}\nIf $M \\equivwbmu N$ and $M \\rtcredwbmu \\lmuwHNF$, then there exists $\\lmuwHNF'$ such that $\\lmuwHNF \\equivwbmu \\lmuwHNF'$ and $N \\rtcredwbmu \\lmuwHNF'$.\n \\end{proposition}\n\\Paper{Notice that Prop.~\\ref{confluence eq} is formulated with respect to $\\eqbmu$, not $\\equivwbmu$.}\n\nThe other two equivalences we consider are generated by \\emph{weak head reduction} and \\emph{weak explicit head reduction}.\nWe will show in Theorem~\\ref{equivwh is equivwxh} that these coincide for pure, substitution-free terms.\n\n \\begin{definition} [Weak head equivalence] \\label{equivwh definition}\nThe relation $\\equivwh$ is defined co-inductively as the largest symmetric relation such that: $M \\equivwh N$ if and only if either $M$ and $N$ have both no \\WHNF, or both $M \\rtcredwh[\\nf] M'$ and $ N \\rtcredwh[\\nf] N'$, and either:\n \\begin {itemize}\n\n \\item if $M' = xM_1\\dots M_n$ $(n \\geq 0)$, then $ N' = xN_1\\dots N_n$ and $ M_i \\equivwh N_i $ for all $1 \\seq i \\seq n$; or\n\n \\item if $M' =`lx.M''$, then $N' = `lx.N''$ and $M'' \\equivwh N''$; or\n\n \\item if $M' = \\muterm`a.[`b]M''$, then $N' = \\muterm`a.[`b]N''$ (so $`a \\not= `b$ or $`a \\ele \\fn(M'')$, $M'' \\not= \\muterm`g.[`d] R$, and similarly for $N''$), and $M'' \\equivwh N''$.\n\n \\end {itemize}\n \\end{definition}\nNotice that $`lz.`D`D \\equivwh `lz.`W`W$ because $`D`D \\equivwh `W`W$, since neither has a \\WHNF.\n\n\\Paper\nWe perhaps need to clarify the details of this definition.\nThe notion of weak head equivalence captures the fact that, once weak head reduction has finished, there are sub-terms that can be reduced further.\nThis process will generate (in principle) infinite terms and the equivalence expresses when this produces equal (infinite) terms.\nHowever, it also equates terms that have no \\WHNF.\nAs can be seen from Def.~\\ref{weak head reduction definition}, a context switch $\\muterm`a.[`b]N$ is in {\\WHNF} only if $N$ is; so when we state in the third case that $M \\rtcredwh[\\nf] \\muterm`a.[`b]M''$, by the fact that this reduction has terminated, we know that $M''$ \\emph{is} in \\WHNF.\n\n\nWe will now define a notion of weak explicit head equivalence, that, in approach, corresponds the weak head equivalence but for the fact that now explicit substitutions are part of terms.\n\n \\begin{definition} [Weak explicit head-equivalence] \\label{equivwxh definition}\nThe relation $\\equivwxh$ is defined co-in\\-ductively as the largest symmetric relation such that: $M \\equivwxh N$ if and only if either $M$ and $N$ have both no $\\redwxh$-normal form, or both $M \\rtcredwxh[\\nf] M' \\, \\Ssub $ and $ N \\rtcredwxh[\\nf] N' \\, \\Ssub' $, and either:\n \\begin {itemize}\n\n \\item if $M' = xM_1\\dots M_n $ $(n \\geq 0)$, then $ N' = xN_1\\dots N_n $ (so $x \\notele \\Ssub$, $x \\notele \\Ssub'$) and $M_i \\, \\Ssub \\equivwxh N_i \\, \\Ssub ' $ for all $1 \\seq i \\seq n$; or\n\n \\item if $M' =`lx.M''$, then $N' = `lx.N''$ and $M'' \\, \\Ssub \\equivwxh N'' \\, \\Ssub' $; or\n\n \\item if $M' = \\muterm`a.[`b]M''$, then $N' = \\muterm`a.[`b]N''$ (so $`a \\not= `b$ or $`a \\ele \\fn(M'')$, $M'' \\not= \\muterm`g.[`d] R$, so $`b \\notele \\Ssub$, $`b \\notele \\Ssub'$, and similarly for $N''$) and $M'' \\, \\Ssub \\equivwxh N'' \\, \\Ssub' $.\n\n \\end {itemize}\n \\end{definition}\nNotice that $\\muterm`a.[`b]`D`D \\equivwxh `D`D$.\n\n\nThe following results formulate the strong relation between $\\equivwh$ and $\\equivwxh$, and therefore between $\\redwh$ and $\\redwxh$.\nWe first show that pure terms that are equivalent under $\\equivwxh$ are also so under $\\equivwh$.\n\n\n \\begin{lemma}\nLet $M$ and $N$ be pure $\\lmu$-terms; then $M \\equivwh N $ if and only if there are $M'$, $N'$ such that $M' \\rtcredxsub[\\nf] M$ and $N' \\rtcredxsub[\\nf] N$, and $M' \\equivwxh N' $.\n\n \\begin{Proof}\n \\begin{description}\n\n \\item[only if]\nBy co-induction on the definition of $\\equivwh$.\nIf $M \\equivwh N $, then either:\n \\begin {itemize}\n\n \\item $M \\rtcredwh[\\nf] xM_1\\dots M_n$ and $N \\rtcredwh[\\nf] xN_1\\dots N_n$ and $ M_i \\equivwh N_i$, for all $1 \\seq i \\seq n$.\nThen, by Lem.~\\ref{redwh to redwxh lemma}, there are $\\Vect{M'_i}$ such that \\Paper{both}\n \\[ \\begin{array}{ccccc}\nM &\\rtcredwxh[\\nf]& xM'_1\\dots M'_n ~ \\Ssub &\\rtcredxsub& xM_1\\dots M_n\n\\\\\nN &\\rtcredwxh[\\nf]& xN'_1\\dots N'_n ~ \\Ssub' &\\rtcredxsub& xN_1\\dots N_n\n \\end{array} \\]\nBut then $M'_i \\, \\Ssub \\rtcredxsub[\\nf] M_i$ and $N'_i \\, \\Ssub' \\rtcredxsub[\\nf] N_i$, for all $1 \\seq i \\seq n$; then by induction, $M'_i \\, \\Ssub \\equivwxh N'_i \\, \\Ssub' $ for all $1 \\seq i \\seq n$.\nBut then $M \\equivwxh N$.\n\n\n\n\n \\end {itemize}\nThe other cases are similar.\n\n\n \\item[if]\nBy co-induction on the definition of $\\equivwxh$.\nIf there are $M'$, $N'$ such that $M' \\rtcredxsub[\\nf] M$ and $N' \\rtcredxsub[\\nf] N$, and $M' \\equivwxh N' $, then\neither:\n\n \\begin {itemize}\n\n \\item $M' \\rtcredwxh[\\nf] xM'_1\\dots M'_n ~ \\Ssub $, $N' \\rtcredwxh[\\nf] xN'_1\\dots N'_n ~ \\Ssub' $ and $ \\Vect{M'_i \\, \\Ssub \\equivwxh N'_i \\, \\Ssub' } $.\nLet, for all $1 \\seq i \\seq n$, $M'_i \\, \\Ssub \\rtcredxsub[\\nf] M_i$ and $N'_i \\, \\Ssub \\rtcredxsub[\\nf] N_i$ then by induction, $\\Vect{M_i \\equivwh N_i}$.\nNotice that we have $M' \\rtcredwxh[\\nf] xM'_1\\dots M'_n ~ \\Ssub \\rtcredxsub[\\nf]$ $xM_1\\dots M_n$.\nLet $M' = M'' \\, \\Ssub'' $, so $ M'' \\, \\Ssub'' \\rtcredwxh[\\nf] xM'_1\\dots M'_n ~ \\Ssub'\\,\\Ssub'' $, where $\\Ssub = \\Ssub'\\,\\Ssub'' $.\nLet $M'' \\, \\Ssub'' \\rtcredxsub[\\nf] M $, then by Lem.~\\ref{redwh to redwxh lemma}, we also have $M \\rtcredwxh[\\nf] xM''_1\\dots M''_n ~ \\Ssub' \\rtcredwh[\\nf] xM_1\\dots M_n $.\nThen, again by Lem.~\\ref{redwh to redwxh lemma}, $M \\rtcredwh[\\nf] xM_1\\dots M_n$; likewise, we have $N \\rtcredwh[\\nf] xN_1\\dots N_n$.\nBut then $M \\equivwh N$.\n\n\n\n\n \\end {itemize}\nThe other cases are similar.\\quad\\usebox{\\proofbox}\n\n \\end{description}\n \\end{Proof}\n\n \\end{lemma}\n\nNotice that this lemma in fact shows:\n\n \\begin{theorem} \\label{equivwh is equivwxh}\nLet $M,N \\ele \\lmu$, then $M \\equivwxh N \\Iff M \\equivwh N$.\n \\end{theorem}\n\n\\Comment\n\n \\begin{lemma} \\label{equivwxh implies equivwh}\nFor $M,N \\ele \\lmu$: $M \\equivwxh N \\Then M \\equivwh N$.\n \\end{lemma}\n\n \\begin{Proof}\nWe define $ {\\equivR} \\subseteq \\lmu^2 $ by: $ M \\equivR N $ if and only if $ M,N$ have no {\\WHNF} or there exist $M',N'\\ele \\lmux $ such that $ M' \\equivwxh N' $, $ M' \\rtcredxsub M $, and $ N' \\rtcredxsub N $.\nWe show that, for $M,N \\ele \\lmu$: $M \\equivwxh N \\Then M \\equivR N \\Then M \\equivwh N$.\n\\CLAC{ \\leftmargini 0pt}\n \\begin{description}\n\n \\item [$M \\equivR N \\Then M \\equivwh N$]\nLet $M',N'\\ele \\lmux$ be such that $M' \\equivwxh N'$, $M' \\rtcredxsub M$, and $N' \\rtcredxsub N $.\nWe reason by co-induction on the definition of $\\equivwxh$.\n\n \\begin {itemize} \\itemsep 3pt\n\n\n \\item $M$ and $N$ have both no \\WHNF; then also $M \\equivwh N$.\n\n\n \\item Assume $M' \\rtcredwxh[\\nf] xM'_1\\dots M'_n ~ \\Ssub $ and $N' \\rtcredwxh[\\nf] xN'_1\\dots N'_n ~ \\Ssub' $ such that $ M'_i \\, \\Ssub \\equivwxh N'_i \\, \\Ssub' $ for all $1 \\seq i \\seq n$.\n\nLet $M'_i \\, \\Ssub \\rtcredxsub[\\nf] M_i$ and $N'_i \\, \\Ssub \\rtcredxsub[\\nf] N_i$ (mark that $\\rtcredxsub[\\nf]$ just removes the explicit substitutions), then by definition of $\\equivR$ we have $M_i \\equivR N_i$, so by induction, $M_i \\equivwh N_i$, for all $1 \\seq i \\seq n$.\nThen also $xM'_1\\dots M'_n ~ \\Ssub \\rtcredxsub[\\nf] xM_1\\dots M_n$.\n\nAssume that $M'$ has some substitutions, so $M' = M'' \\, \\Ssub'' $, then also $ M'' \\, \\Ssub'' \\rtcredwxh[\\nf] xM'_1\\dots M'_n ~ \\Ssub'\\,\\Ssub'' $, where $\\Ssub = \\Ssub'\\,\\Ssub'' $.\n\nLet $M$ be the result of removing all substitutions in $M''$, so such that $M'' \\, \\Ssub'' \\rtcredxsub[\\nf] M $, then by Lem.~\\ref{redwh to redwxh lemma}, $M \\rtcredwh[\\nf] xM_1\\dots M_n$; likewise, we can show that $N \\rtcredwh[\\nf] xN_1\\dots N_n$.\nSo we have shown that $M_i \\equivwh N_i$ for all $1 \\seq i \\seq n$, and we obtain $M \\equivwh N$.\n\n\\Paper\n \\item Assume {$M' \\rtcredwxh[\\nf] `lx.M'_0 \\, \\Ssub $ and $N' \\rtcredwxh[\\nf] `lx.N'_0 \\, \\Ssub' $ such that $M'_0 \\, \\Ssub \\equivwxh N'_0 \\, \\Ssub' $}.\nLet $M'_0 \\, \\Ssub \\rtcredxsub[\\nf] M_0$ and $N'_0 \\, \\Ssub \\rtcredxsub[\\nf] N_0$,\nthen $M_0 \\equivR N_0$ by definition, so by induction, $M_0 \\equivwh N_0$.\nThen also $`lx.M'_0 \\, \\Ssub \\rtcredxsub[\\nf] `lx.M_0$.\n\nLet $M' = M'' \\, \\Ssub'' $, then also $ M'' \\, \\Ssub'' \\rtcredwxh[\\nf] `lx.M'_0 ~ \\Ssub'\\,\\Ssub'' $, where $\\Ssub = \\Ssub'\\,\\Ssub'' $.\nTake $M$ such that $M'' \\, \\Ssub'' \\rtcredxsub[\\nf] M $, then by Lem.~\\ref{redwh to redwxh lemma}, $M \\rtcredwh[\\nf] `lx.M_0$; likewise, we can show that $N \\rtcredwh[\\nf] `lx.N_0$.\nBut then $M \\equivwh N$.\n\n\n\n \\item Assume {$M' \\rtcredwxh[\\nf] \\muterm`a.[`b]M_0$ and $N' \\rtcredwxh[\\nf] \\muterm`a.[`b]N_0$, such that $M_0 \\equivh N_0 $ and both have a \\WHNF.}\nLet $M_0 \\, \\Ssub \\rtcredxsub[\\nf] M'_0$ and $N_0 \\, \\Ssub \\rtcredxsub[\\nf] N'_0$,\nthen $M'_0 \\equivR N'_0$, so by induction, $M_0 \\equivwh N_0$.\nThen also $\\muterm`a.[`b]M_0' \\, \\Ssub \\rtcredxsub[\\nf] \\muterm`a.[`b]M_0$.\nLet $M' = M'' \\, \\Ssub'' $, then also $ M'' \\, \\Ssub'' \\rtcredwxh[\\nf] \\muterm`a.[`b]M_0 ~ \\Ssub'\\,\\Ssub'' $, where $\\Ssub = \\Ssub'\\,\\Ssub'' $.\nTake $M$ such that $M'' \\, \\Ssub'' \\rtcredxsub[\\nf] M $, then by Lem.~\\ref{redwh to redwxh lemma}, $M \\rtcredwh[\\nf] \\muterm`a.[`b]M_0$; likewise, we can show that $N \\rtcredwh[\\nf] \\muterm`a.[`b]N_0$.\nBut then $M \\equivwh N$.\n\n\n\n \\item $M' \\rtcredwxh[\\nf] xM'_1\\dots M'_n $ or $M' \\rtcredwxh[\\nf] \\muterm`a.[`b]M_0$: similar\n\n\n \\end {itemize}\n\nSo $M \\equivR N$ implies $M \\equivwh N$.\n\n \\item [$M,N \\ele \\lmu \\Then ( M \\equivwxh N \\Then M \\equivR N )$]\nTake $M' = M$, and $N' = N$.\n\\quad\\usebox{\\proofbox}\n \\end{description}\n \\end{Proof}\n\n\nWe can also show the converse of the previous result, which is:\n\n \\begin{lemma} \\label{equivwh implies equivwxh}\nFor $M,N \\ele \\lmu$: $M \\equivwh N \\Then M \\equivwxh N$.\n \\end{lemma}\n\n \\begin{Proof}\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\nWe can now easily show:\n\n \\begin{theorem} \\label{equivwh is equivwxh}\nLet $M,N \\ele \\lmu$, then $M \\equivwxh N$ if and only if $M \\equivwh N$.\n \\end{theorem}\n\n \\Paper{\\begin{Proof}\nBy Lem.~\\ref{equivwxh implies equivwh} and~\\ref{equivwh implies equivwxh}.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}}\n\n\n\n\n\n\\CLAC{ \\section{Full abstraction for the logical interpretation} \\label{Full abstraction} \\label{approximation section}\n\nIn this section we will show our main result, that the logical encoding is fully abstract with respect to weak equivalence between pure $\\lmu$-terms.\nTo achieve this, we show in Thm.~\\ref{FA equivxh equivC} that $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $ \\emph{iff} $M \\equivwxh N$.\nWe are thus left with the obligation to show that $M \\equivwxh N$ \\emph{iff} $M \\equivwbmu N$.\nIn Thm.~\\ref{equivwh is equivwxh} we have shown that $M \\equivwxh N$ \\emph{iff} $M \\equivwh N$, for pure terms; to achieve $M \\equivwh N$ \\emph{iff} $M \\equivwbmu N$, we go through a notion of \\emph{weak approximation}; based on Wadsworth's approach \\cite{Wadsworth'76}, we define $\\equivwA$ that expresses that terms have the same weak approximants and show that $ M \\equivwh N $ \\emph{iff} $ M \\equivwA N $ \\emph{iff} $ M \\equivwbmu N $.\n\n\n}\n\nWe can \\Paper{also }show \\Paper{the following results that state }that if the interpretation of $M$ produces an output, then $M$ reduces by head reduction to an abstraction; similarly, if the interpretation of $M$ produces an input, then $M$ reduces by head reduction to a term with a head variable.\n\n\n \\begin{lemma} \\label{pi reduction characterisation}\n \\begin{enumerate} \\itemsep 0pt\n\n \\item\nIf $\\PilmuTerm [M] a \\Outson a $, then there exist $x, N$ and $\\Ssub$ such that $ \\PilmuTerm [M] a \\wbisim \\PilmuTerm [`lx.N\\, \\Ssub] a $, and $ M \\rtcredwxh[\\nf] `lx.N \\, \\Ssub $.\n\n \\item\nIf $\\PilmuTerm [M] a \\Outson c $, with $a \\not= c$, then there exist $`a, c, x, N$ and $\\Ssub$ such that $ \\PilmuTerm [M] a \\wbisim \\PilmuTerm [\\muterm`a.[c]`lx.N\\, \\Ssub] a $, and $ M \\rtcredwxh[\\nf] \\muterm`a.[c]`lx.N \\, \\Ssub $.\n\n\n \\item\nIf $\\PilmuTerm [M] a \\Inson x $, then there exist $\\Vect{z_j}, x, \\Vect{N_i}, c$ and $\\Ssub$ with $x \\notele \\Vect{z_j}$, $m \\geq 0$, and $n \\geq 0$ such that\n \\begin{itemize}\n \\item $\\PilmuTerm [M] a \\wbisim \\PilmuTerm [`lz_1\\dots z_m.xN_1\\dots N_n \\, \\Ssub] c $;\n \\item $ M \\rtcredwxh[\\nf] `lz_1\\dots z_m.xN_1\\dots N_n \\, \\Ssub $ if $a = c$;\n \\item $ M \\rtcredwxh[\\nf] \\muterm`a.[c]`lz_1\\dots z_m.xN_1\\dots N_n[a \\For `a] \\, \\Ssub $, if $a \\not= c$.\n \\end{itemize}\n\n \\end{enumerate}\n \\end{lemma}\n\n \\begin{Proof}\nStraightforward.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\n\n\\Comment\n \\[ \\begin{array}{lcl}\n\\SPilmuTerm [(`lx.x)(`ly.z)] a\n\t&=& \\\\\n\\PilmuTerm [a {l x . x} `ly.z ] a\n\t&=& \\\\\n\\PilmuTerm [s {v x} x := `ly.z ] a\n\t&=& \\\\\n\\New w ( \\BEq w=a \\Par \\PilmuTerm [l y . z ] w )\n\t&=& \\\\\n\\PilmuTerm [l y . {v z} ] a\n \\end{array} \\]\n\n\nAs to the reverse, we can show:\n\n \\begin{lemma} \\label{bmu reduction characterisation}\n \\begin{enumerate} \\itemsep 0pt\n\n \\item\nIf $ M \\rtcredwxh[\\nf] `lx.N \\, \\Ssub $, then $ \\PilmuTerm [M] a \\Outson a $.\n\n \\item\nIf $ M \\rtcredwxh[\\nf] \\muterm`a.[`b]`lx.N \\, \\Ssub $, then $ \\PilmuTerm [M] a \\Outson `b $.\n\n \\item\n$\\PilmuTerm [M] a \\Inson x $ if $ M \\rtcredwxh[\\nf] xN_1\\dots N_n \\, \\Ssub $ or $ M \\rtcredwxh[\\nf] \\muterm`a.[`b]xN_1\\dots N_n \\, \\Ssub $.\n\n \\end{enumerate}\n \\end{lemma}\n\n \\begin{Proof}\nStraightforward.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n\n\n}\n\n\\Paper{ \\section{Weak approximation for \\texorpdfstring{$\\lmu$}{}}\n\\label{approximation section}\n\nThe notions of \\emph{approximant} and \\emph{approximation} were first introduced by Wadsworth for the {\\LC}~\\cite{Wadsworth'76}, where they are used in order to better express the relation between equivalence of meaning in Scott's models and the usual notions of conversion and reduction.\nWadsworth defines approximation of terms through the replacement of any parts of a term remaining to be evaluated (\\emph{i.e.}~$`b$-redexes) by $\\bot$.\nRepeatedly applying this process over a reduction sequence starting with $M$ gives a set of approximants, each giving some - in general incomplete - information about the result of reducing $M$.\nOnce this reduction produces $ `lx.yN_1\\dots N_n $, all remaining redexes occur in $N_1, \\ldots, N_n$, which then in turn will be approximated.\n\nFollowing this approach, Wadsworth \\cite{Wadsworth'76} defines $\\SetAppr(M)$ (similar to Def.~\\ref{approximation} below) as the set of approximants of the $`l$-term $M$, which forms a meet semi-lattice.\nIn \\cite{Wadsworth'78}, the connection is established between approximation and semantics, by showing\n \\[ \\begin{array}{rcl}\n\\Sem{M}_{\\Dinfty} \\, p &=& \\bigsqcup \\, \\Set { \\Sem{A}_{\\Dinfty} \\, p \\mid A \\ele \\SetAppr(M) }.\n \\end{array} \\]\nSo, essentially, approximants are partially evaluated expressions in which the locations of incomplete evaluation (\\emph{i.e.}~where reduction \\emph{may} still take place) are explicitly marked by the element $\\bottom$; thus, they \\emph{approximate} the result of computations. Intuitively, an approximant can be seen as a `snapshot' of a computation, where we focus on that part of the resulting program which will no longer change, which corresponds to the (observable) \\emph{output}.\n\n\n\\CLAC{Essentially following \\cite{Wadsworth'76}, w}\\Paper{W}e now define a \\emph{weak approximation semantics} for $\\lmu$.\nApproximation for $\\lmu$ has been studied by others as well \\cite{Saurin'10,deLiguoro'13}; however, seen that we are mainly interested in \\emph{weak} reduction here, we will define \\emph{weak} approximants, which are normally not considered.\n\n \\begin{definition}[Weak approximation for $\\lmu$] \\label{weak approximation lmu} \\label{approximation}\n\n \\begin{enumerate} \\itemsep 0pt\n\n \\item\nThe set of $\\lmu$'s \\emph{weak approximants} $\\SetwApprlmu$ \\Paper{with respect to $\\redbmu$ }is defined through the grammar:\\Paper{\\footnote{For `normal' approximants, the case $`lx.\\ftAppr$ normally demands that $\\ftAppr \\not= \\bot$, as motived by the relation with $\\Dinfty$.}}\n\\Paper{ \\[ \\begin{array}{rrl@{\\quad}l}\n\\wAppr & ::= & \\bot \\\\\n\t&\\mid & x\\wAppr^1\\dots \\wAppr^n & (n \\geq 0) \\\\\n\t&\\mid & `lx.\\wAppr \\\\\n\t&\\mid & \\muterm`a.[`b]\\wAppr & (`a \\not= `b \\textrm{ or } `a \\in \\wAppr,\n\t\\wAppr \\not= \\muterm `g.[`d] \\wAppr', ~ \\wAppr \\not= \\bot )\n \\end{array} \\]\n}\\CLAC{\n \\[ \\begin{array}{rrl@{\\quad}l}\n\\wAppr & ::= &\n\\bot \\mid `lx.\\wAppr \\mid x\\wAppr^1\\dots \\wAppr^n & (n \\geq 0)\n\\\\\n\t&\\mid & \\muterm`a.[`b]\\wAppr & (`a \\not= `b \\textrm{ or } `a \\in \\wAppr, ~\n\t\\wAppr \\not=\t\\muterm `g.[`d] \\wAppr', ~ \\wAppr \\not= \\bot )\n \\end{array} \\]\n}\n\n \\item\nThe relation ${\\dirapp} \\subseteq \\SetwApprlmu \\times \\lmu $ is \\Paper{defined as }the smallest preorder that is the compatible extension of $\\bot \\dirapp M$.\\Paper{\\footnote{Notice that if $\\ftAppr_1 \\dirapp M_1$, and $\\ftAppr_2 \\dirapp M_2$, then $\\ftAppr_1\\ftAppr_2$ need not be an approximant; it is one if $\\ftAppr_1 = x\\ftAppr_1^1\\dots \\ftAppr_1^n$, perhaps prefixed with a number of context switches of the shape $\\muterm`a.[`b]$.}\n \\[ \\begin {array}{rcl}\n \\bottom &\\dirapp& M \\\\\nM \\dirapp M' & \\Then & `l x . M \\dirapp `l x . M' \\\\\nM \\dirapp M' & \\Then & \\muterm `g.[`d] M \\dirapp \\muterm `g.[`d] M' \\\\\nM_1 \\dirapp M'_1 \\And M_2 \\dirapp M'_2 & \\Then & M_1M_2 \\dirapp M_1'M_2'\n \\end {array} \\] \n\n\n \\item \\label{appr def}\n\\Paper{ The set of \\emph{weak approximants} of $M$, $\\SetwApprlmu(M)$, is defined as:\\footnote{Notice that we use $\\redbmu$ here, not $\\redwbmu$; the approximants are weak, not the reduction.}%\n \\[ \\begin{array}{rcl}\n\\Set{ \\wAppr \\ele \\SetwApprlmu \\mid \\Exists N \\ele \\lmu \\Pred[ M \\rtcredbmu N \\And \\wAppr \\dirapp N] } .\n \\end{array} \\]}\n\\CLAC{ $ \\begin{array}{@{}rcl}\n\\SetwApprlmu(M) & \\ByDef & \\Set{ \\wAppr \\ele \\SetwApprlmu \\mid \\Exists N \\ele \\lmu \\Pred[ M \\rtcredbmu N \\And \\wAppr \\dirapp N] } .\n \\end{array} $}\n\n \\item\n\\emph{Weak approximation equivalence} is defined through:\n $ \\begin{array}{rcl@{}}\nM \\equivwA N &\\ByDef& \\SetwApprlmu(M) = \\SetwApprlmu(N)\n \\end{array} $.\n\n \\end{enumerate}\n \\end{definition}\nNotice that, in part~\\ref{appr def}, the approximants are weak, not the reduction.\n\n\\Comment{ Notice that\n$ \\begin{array}[t]{ccccc}\n\\SetwApprlmu(`lz.`D`D)\n\t&=&\n\\Set{\\bot, `lz.\\bot}\n\t&=&\n\\SetwApprlmu(`lz.`W`W)\n\t\\\\\n\\SetwApprlmu(\\muterm `a . [`b] `D`D)\n\t&=&\n\\Set{\\bot}\n\t&=&\n\\SetwApprlmu(`D`D)\n \\end{array} $ \\\\ }\n\nThe relationship between the approximation relation and reduction is characterised by\\Paper{ the following result}:\n \\begin{lemma} \\label{approximation lemma redbmu}\n \\begin{enumerate}\n\n \\item If $\\wAppr \\dirapp M$ and $M \\rtcredbmu N$, then $\\wAppr \\dirapp N$.\n\n \\item If $ \\wAppr \\ele \\SetwApprlmu(N) $ and $ M \\rtcredbmu N $, then also $ \\wAppr \\ele \\SetwApprlmu(M) $.\n\n \\item If $\\wAppr \\ele \\SetwApprlmu(M)$ and $M \\redbmu N$, then there exists $L$ such that $N \\rtcredbmu L$ and $\\wAppr \\dirapp L$.\n\n \\item $M$ is a {\\WHNF} if and only if there exists $\\wAppr \\not= \\bot $ such that $ \\wAppr \\dirapp M $.\n\n \\end{enumerate}\n \\end{lemma}\n\n\\Paper{ \\begin{Proof}\nEasy. \\quad\\usebox{\\proofbox}\n \\end{Proof}}\n\n\n\\Paper\nWe could also have defined the set of approximants of a term coinductively:\n\n \\begin{definition} \\label{alternative approximation}\nWe define $\\SetwApprlmuAlt(M)$ coinductively by:\n\n \\begin {itemize}\n\n \\item If $ \\wAppr \\dirapp M $, then $ \\wAppr \\ele \\SetwApprlmuAlt(M)$.\n\n \\item\nif $M \\rtcredwh xM_1\\dots M_n$ $(n \\geq 0)$, then\n$ \\SetwApprlmuAlt(M) = \\Set { x\\wAppr^1\\dots \\wAppr^n \\mid \\wAppr^i \\ele \\SetwApprlmuAlt(M_i) , 1 \\seq i \\seq n } $.\n\n \\item if $M \\rtcredwh `lx.N$, then\n$ \\SetwApprlmuAlt(M) = \\Set { `lx.\\wAppr \\mid \\wAppr \\ele \\SetwApprlmuAlt(N) } $.\n\n \\item if $M \\rtcredwh \\muterm`a.[`b]N$, with $`a \\not= `b$ or $`a \\ele \\fn(N)$, $N \\not= \\muterm`g.[`d] L$, then\n$ \\SetwApprlmuAlt(M) = \\Set { \\muterm`a.[`b]\\wAppr \\mid \\wAppr \\ele \\SetwApprlmuAlt(N) } $.\n\n \\end {itemize}\n \\end{definition}\n\nWe can show that these definitions coincide:\n\n \\begin{lemma} \\label{coincide lemma}\n$\\SetwApprlmuAlt(M) = \\SetwApprlmu(M)$.\n \\end{lemma}\n\n \\begin{Proof}\n \\begin{description}\n \\item[$\\subseteq$]\nIf $\\wAppr \\ele \\SetwApprlmuAlt(M)$, then by Def.~\\ref{alternative approximation} either:\n\n \\begin{description}\n\n \\item [$ \\wAppr \\dirapp M $] Immediate.\n\n \\item [$ \\wAppr = x\\wAppr^1\\dots \\wAppr^n $]\nThen $ M \\rtcredwh xM_1\\dots M_n $, with $ \\wAppr^i \\ele \\SetwApprlmuAlt(M_i) $.\nBy coinduction, also $ \\wAppr^i \\ele \\SetwApprlmu(M_i) $; then, by Def.~\\ref{weak approximation lmu}, there exist $M'_i$ such that $ M_i \\rtcredbmu M'_i $ and $ \\wAppr^i \\dirapp M'_i $.\nThen, since ${\\rtcredwh} \\subseteq {\\rtcredbmu}$, in particular $ M \\rtcredbmu xM'_1\\dots M'_n$ and also $ \\wAppr \\dirapp xM'_1\\dots M'_n $, so $ \\wAppr \\ele \\SetwApprlmu(M) $.\n\n\n\n \\end{description}\nThe other cases are similar.\n\n \\item[$\\supseteq$]\nIf $ \\wAppr \\ele \\SetwApprlmu(M) $, then by Def.~\\ref{weak approximation lmu}, there exists $N$ such that $ M \\rtcredbmu N $ and $ \\wAppr^i \\dirapp N $.\nNow either:\n\n \\begin{description}\n\n \\item [$ \\wAppr \\dirapp M $] Trivial.\n\n \\item [$ \\wAppr = x\\wAppr^1\\dots \\wAppr^n $]\nSince $ x\\wAppr^1\\dots \\wAppr^n \\dirapp N $, $ N = xN_1\\dots N_n $, and $ \\wAppr^i \\dirapp N_i $.\nThen by Def.~\\ref{alternative approximation} $ \\wAppr^i \\ele \\SetwApprlmuAlt(N_i) $, and by induction, $ \\wAppr^i \\ele \\SetwApprlmu(N_i) $.\nBy Lem.~\\ref{weak head reduction}, there exist $M_i$ $(1 \\seq i \\seq n)$ such that $M \\rtcredwh xM_1 \\dots M_n \\rtcredbmu xN_1\\dots N_n $; since ${\\rtcredwh} \\subseteq {\\rtcredbmu}$, this implies that $M_i \\rtcredbmu N_i$; then by Lem.~\\ref{approximation lemma redbmu}, \n$ \\wAppr^i \\ele \\SetwApprlmu(M_i) $.\nThen, by Def.~\\ref{alternative approximation}, $ \\wAppr \\ele \\SetwApprlmuAlt(M) $.\n\n\n\n \\end{description}\nThe other cases are similar.\n \\end{description}\n \\end{Proof}\nAs a result, below we will use whichever definition is convenient.\n\nNotice that the first and latter cases in Def.~\\ref{alternative approximation} overlap for terms that are already in head-normal form. In fact, we can show:\n\n \\begin{lemma}\nIf $\\wAppr \\ele \\SetwApprlmu(M) $ and $ \\wAppr' \\dirapp \\wAppr $, then $ \\wAppr' \\ele \\SetwApprlmu(M) $.\n \\end{lemma}\n \\begin{Proof}\nBy Def.~\\ref{weak approximation lmu} and transitivity of $\\dirapp$.\\quad\\usebox{\\proofbox}\n \\end{Proof}\nBy Lem.~\\ref{coincide lemma}, this result also holds for $ \\SetwApprlmuAlt(`.) $.\n\n\n\n\nAs is standard in other settings, interpreting a $\\lmu$-term $M$ through its set of weak approximants $\\SetwApprlmu(M)$ gives a semantics.\n\n\n \\begin{theorem} [Weak approximation semantics] \\label{approx seman lmux}\nIf $M \\eqbmu N$, then $M \\equivwA N$%\n\\Comment{$\\SetApprlmu(M) = \\SetwApprlmu(N) $}.\n \\end{theorem}\n\n \\begin{Proof}\n\\CLAC{Using Prop.~\\ref{confluence eq} and Lem.~\\ref{approximation lemma redbmu}.\\quad\\usebox{\\proofbox}}\n\\Paper{\n\\setbox100=\\hbox{\\hspace*{\\leftmargini}\\textit{Proof. }}\n$\n\\begin {array}[t]{@{}lclcl}\nM \\eqbmu N \\And \\wAppr \\ele \\SetwApprlmu(M)\n\t& \\Then & \\\\\nM \\eqbmu N \\And \\Exists L \\Pred[ M \\rtcredbmu L \\And \\wAppr \\dirapp L ]\n\t& \\Then & (\\textit{\\ref{confluence eq}}) \\\\\n\\Exists L,K \\Pred[ L \\rtcredbmu K \\And N \\rtcredbmu K \\And \\wAppr \\dirapp L ]\n\t& \\Then & (\\textit{\\ref{approximation lemma redbmu}}) \\\\\n\\Exists K \\Pred[ N \\rtcredbmu K \\And \\wAppr \\dirapp K ]\n\t& \\Then &\n\\wAppr \\ele \\SetwApprlmu(N) \\quad\\usebox{\\proofbox}\n \\end{array} $\n}\n\\Comment{\n$ \\kern-6mm \\begin {array}[t]{@{}lllll} \\kern6mm\nM \\eqbmu N \\And \\wAppr \\ele \\SetwApprlmu(M)\n\t& \\Then &\n\\kern -10mm\nM \\eqbmu N \\And \\Exists L \\Pred[ M \\rtcredbmu L \\And \\wAppr \\dirapp L ]\n\t& \\Then (\\textit{\\ref{confluence eq}}) \\\\\n\\Exists L,K \\Pred[ L \\rtcredbmu K \\And N \\rtcredbmu K \\And \\wAppr \\dirapp L ]\n\t& \\Then (\\textit{\\ref{approximation lemma redbmu}}) &\n\\Exists K \\Pred[ N \\rtcredbmu K \\And \\wAppr \\dirapp L ]\n\t& \\Then & \\\\\n\\wAppr \\ele \\SetwApprlmu(N) \\quad\\usebox{\\proofbox}\n \\end{array} $\n}\n \\end{Proof}\n\nThe reverse implication of this result does not hold, since terms without {\\WHNF} (which have only $\\bot$ as approximant) are not all related by reduction.\nBut we can show the following full abstraction result:\n\n \\begin{theorem} [Full abstraction of $\\equivwbmu$ versus $\\equivwA$]\n\\label{full abstr approx seman Lmux}\n$M \\equivwbmu N$ if and only if $M \\equivwA N$.\n \\end{theorem}\n\n \\begin{Proof}\n\n\n \\begin {description}\n\n \\item[if]\nBy co-induction on the definition of the set of weak approximants.\n\\CLAC{ \\leftmarginii 0pt}\n\\Paper{%\nIf $\\SetwApprlmu(M) = \\Set{\\bottom} = \\SetwApprlmu(N)$, then both $M$ and $N$ have no {\\WHNF}, so $M \\equivwbmu N$.\nOtherwise, either:\n\n \\begin{description} \\itemindent -1em\n\n \\item[$x\\wAppr^1\\dots \\wAppr^n \\ele \\SetwApprlmu(M) \\And x\\wAppr^1\\dots \\wAppr^n \\ele \\SetwApprlmu(N)$] \\CLAC{ ~ }\n\nThen by Def.~\\ref{alternative approximation}\n$M \\rtcredwh xM_1\\dots M_n$ for some $M_i$ ($1 \\seq i \\seq n$) and $\\wAppr^i \\ele \\SetwApprlmu(M_i)$.\nLikewise, there exist $N_i$ ($1 \\seq i \\seq n$) such that $N \\rtcredwh xN_1\\dots N_n$ and $\\wAppr^i \\ele \\SetwApprlmu(N_i)$.\nSo $\\SetwApprlmu(M_i) = \\SetwApprlmu(N_i)$ and by induction $M_i \\equivwbmu N_i$, for $1 \\seq i \\seq n$.\nSince $\\equivwbmu$ is a congruence, also $xM_1\\dots M_n \\equivwbmu xN_1\\dots N_n$; since $\\equivwbmu$ is closed under reduction $\\redwbmu$, it is also under $\\redwh$, and we have $M \\equivwbmu N$.\n\n\n\\Comment{%\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n\n \\item[$x\\wAppr^1\\dots \\wAppr^n \\ele \\SetwApprlmu(M) \\And x\\wAppr^1\\dots \\wAppr^n \\ele \\SetwApprlmu(N)$] \\CLAC{ ~ }\n\nIf $x\\wAppr^1\\dots \\wAppr^n \\ele \\SetwApprlmu(M)$, then there exist $L$ such that $M \\rtcredbmu L$ and $x\\wAppr^1\\dots \\wAppr^n \\dirapp L$, so $L = xL_1\\dots L_n$ with $\\wAppr^i \\dirapp L_i$.\n\nThen, by Lem.~\\ref{redbmu to redwxh lemma}, $M \\rtcredwh xM_1\\dots M_n$ with, for $1 \\seq i \\seq n$, $M_i \\rtcredwbmu L_i$; then $\\wAppr^i \\ele \\SetwApprlmu(M_i)$.\n\nLikewise, there exist $L'$ such that $N \\rtcredbmu L'$ and $x\\wAppr^1\\dots \\wAppr^n \\dirapp L'$, so $L' = xL'_1\\dots L'_n$ with $\\wAppr^i \\dirapp L'_i$, and $N \\rtcredwh xN_1\\dots N_n$ with, for $1 \\seq i \\seq n$, $N_i \\rtcredwbmu L'_i$; then $\\wAppr^i \\ele \\SetwApprlmu(N_i)$.\n\nSo $\\SetwApprlmu(M_i) = \\SetwApprlmu(N_i)$, so by induction $M_i \\equivwbmu N_i$, for $1 \\seq i \\seq n$.\nBut then $xM_1\\dots M_n \\equivwbmu xN_1\\dots N_n$; since $\\equivwbmu$ is closed under reduction, also $M \\equivwbmu N$.\n\n\n\n\n \\end {description}\nThe other cases are similar.\n\n\n \\item[only if]\nAs the proof of Theorem~\\ref{approx seman lmux}, but using Proposition~\\ref{confluence equiv} rather than~\\ref{confluence eq}.\\quad\\usebox{\\proofbox}\n\n \\end {description}\n \\end{Proof}\n\nWe can also show that weak head equivalence and weak approximation equivalence coincide:\n \\begin{theorem} \\label{eqA is eqh}\n$M \\equivwh N $ if and only if $M \\equivwA N$.\n \\end{theorem}\n\n \\begin{Proof}\nStraightforward, by coinduction.\\quad\\usebox{\\proofbox}\n\n\\Comment\n \\begin{description} \\itemsep 4pt\n\n \\item[if : $ \\SetwApprlmu(M) = \\SetwApprlmu(N) \\Then M \\equivwh N $]\nBy co-induction on the definition of the set of weak approximants.\n\n \\begin{description}\n\n \\item [$ \\SetwApprlmu(M) = \\Set{\\bottom} = \\SetwApprlmu(N) $]\nThen both $M$ and $N$ have no \\WHNF, so $ M \\equivwh N $.\n\n \\item [$ \\wAppr = x\\wAppr^1\\dots \\wAppr^n $]\nThen $ M \\rtcredwh xM_1\\dots M_n $, and $ \\wAppr^i \\ele \\SetwApprlmu(M_i) $, for $1 \\seq i \\seq n$.\nSince $ \\SetwApprlmu(M) = \\SetwApprlmu(N) $, also $ N \\rtcredwh xN_1\\dots N_n $, with $ \\wAppr^i \\ele \\SetwApprlmu(N_i) $, so $ \\SetwApprlmu(M_i) = \\SetwApprlmu(N_i) $.\nThen, by coinduction, $ M_i \\equivwh N_i $, so $ M \\equivwh N $.\n\n\n]\n \\end{description}\nThe other cases are similar.\n\n \\item[only if : $M \\equivwh N \\Then M \\equivwA N $]\nBy co-induction on the definition of $\\equivwh$.\n \\begin{description}\n\n \\item [$M$ and $N$ have no {\\WHNF}]\nThen $ \\SetwApprlmu(M) = \\Set{\\bottom} = \\SetwApprlmu(N) $.\n\n \\item[$ M \\rtcredwh xM_1\\dots M_n $]\nThen also $N \\rtcredwh xN_1 \\dots N_n$, and $ M_i \\equivwh N_i $ for $1 \\seq i \\seq n$, and by coinduction, $ M \\equivwA N $, so $ \\SetwApprlmu(M_i) = \\SetwApprlmu(N_i)$.\nThen, by Def.~\\ref{alternative approximation}, we have $ \\SetwApprlmu(M) = \\SetwApprlmu(N)$.\n\n\n\n \\end{description}\nThe other cases are similar.\n\n \\end{description}\n\n \\end{Proof}\n\nWe can \\Paper{also }define $\\Sem{M}_{\\SetwApprlmu} = \\sqcup \\, \\Set{ \\wAppr \\mid \\wAppr \\ele \\SetwApprlmu(M) } $, with $\\sqcup$ the least-upper bound with respect to $\\dirapp$; then $\\Sem{`.}_{\\SetwApprlmu}$ corresponds to the ($\\lmu$ variant of) L\\'evy-Longo trees.\nCombined with the results shown in the previous section, we now also have the following result that states that all equivalences coincide:\n \\begin{corollary} \\label{all lmu weak equivalences}\nLet $M,N \\ele \\lmu$, then $M \\equivwxh N \\Iff M \\equivwh N \\Iff M \\equivwA N \\Iff M \\equivwbmu N $.\n \\end{corollary}\n\n\n\n\\Paper{ \\section{Full abstraction for the logical interpretation} \\label{Full abstraction} }\n\nWe now come to the main result of this paper, where we show a full abstraction result for our logical interpretation.\nFirst we show the relation between weak explicit head equivalence and weak bisimilarity.\n\n \\begin{theorem} [Full abstraction of $\\wbisim$ versus $\\equivwxh$]\n\\label{FA equivxh equivC}\nFor any $M,N \\ele \\lmux$:\n$\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $ if and only if $M \\equivwxh N$.\n \\end{theorem}\n\n{\n \\begin{Proof}\n\n\\CLAC{ \\leftmargini 0pt\n}\n\n \\begin{description} \\itemsep 4pt\n\n \\item[if]\nBy co-induction on the definition of $\\equivwxh$.\nLet $M \\equivwxh N$, then either $M$ and $N$ have both no $\\redwxh$-normal form, so, by Lem.~\\ref{zero lemma}, their interpretations are both weakly bisimilar to the process $\\Zero$; or\nboth $M \\rtcredwxh[\\nf] M' \\, \\Ssub $ and $ N \\rtcredwxh[\\nf] N' \\, \\Ssub' $ (let $\\Ssub = \\Vexsub y := P \\, \\Vexcontsub `a := Q . `b $, and $\\Ssub' = \\Vexsub y := P' \\, \\Vexcontsub `a := Q' . `b $), and either:\n\n \\begin{description}\n \\CLAC{ \\itemindent -5mm}\n\n \\item [$M' = xM_1\\dots M_n $ $(n \\geq 0)$, $ N = xN_1\\dots N_n $ and $M_i \\, \\Ssub \\equivwxh N_i \\, \\Ssub ' $, for all $1 \\seq i \\seq n$] ~\n\nWe have $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[xM_1\\dots M_n \\, \\Ssub] a $ and $\\PilmuTerm[N] a \\wbisim \\PilmuTerm[xN_1\\dots N_n \\, \\Ssub'] a $ by Corollary~\\ref{weak soundness}.\nNotice that\n \\[ \\begin{array}{lcl}\n\\PilmuTerm [xM_1\\dots M_n \\, \\Ssub] a &=&\n\\New \\Vect{cy`a} (\\PilmuTerm [v x] c_1 \\Par {\\VPiExContSub c_i := M_i . c_{i+1} } \\Par {} \\PiLmuSem[\\Ssub] )\n \\end{array} \\]\nwhere $c_n = a$ and\n \\[ \\begin{array}{rcl}\n\\PiLmuSem[\\Ssub] &=& {\\VPiExSub y := P } \\Par {\\VPiExContSub `a := Q . `b } \\\\\n\\PiExContSub c_i := M_i . c_{i+1} &=&\n{ \\PiExContFsub c_i := M_i . c_{i+1} } \\\\\n\\PiExSub y_j := P_j &=& \\PiExsub y_j := P_j \\\\\n\\PiExContSub `a_k := Q_k . `b_k &=&\n {\\PiExContFsub `a_k := Q_k . `b_k }\n \\end{array} \\]\nand similar for $\\PilmuTerm [xN_1\\dots N_n \\, \\Ssub'] a $.\nBy induction,\n \\[ \\begin{array}{rclccll}\n\\New \\Vect{y`a} (\\PilmuTerm[M_i] w \\Par {} \\PiLmuSem[\\Ssub]) & \\ByDef &\n\\PilmuTerm[M_i \\, \\Ssub] w & \\wbisim &\n\\PilmuTerm[N_i \\, \\Ssub'] w & \\ByDef &\n\\New \\Vect{y`a} (\\PilmuTerm[N_i] w \\Par {} \\PiLmuSem[\\Ssub'])\n \\end{array} \\]\nSince $\\wbisim$ is a congruence, also\n \\[ \\begin{array}{ccl}\n\\PiExContFsub c_i := M_i . c_{i+1} \\Par \\PiLmuSem[\\Ssub]\n\t& \\wbisim &\n\\PiExContFsub c_i := N_i . c_{i+1} \\Par \\PiLmuSem[\\Ssub']\n \\end{array} \\]\nfor all $1 \\seq i \\seq n$, so also\n $ \\PilmuTerm [xM_1\\dots M_n \\, \\Ssub] a \\wbisim \\PilmuTerm [xN_1\\dots N_n \\, \\Ssub'] a\n$ but then also $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $.\n\n\\Paper\n \\item [ $M' = `lx.M''$, $N' = `lx.N''$, and $M'' \\, \\Ssub \\equivwxh N'' \\, \\Ssub' $]\nBy Corollary~\\ref{weak soundness}, we have $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[`lx.M'' \\, \\Ssub ] a $ and $\\PilmuTerm[N] a \\wbisim \\PilmuTerm[`lx.N'' \\, \\Ssub' ] a $.\nNotice that\n \\[ \\begin{array}{rcl}\n\\PilmuTerm [`lx.M'' \\, \\Ssub ] a &\\ByDef&\n\\New \\Vect{y`a} ( \\PilmuTerm [l x . M''] a \\Par {} \\PiLmuSem[\\Ssub] )\n\t\\\\ [1mm]\n\\PilmuTerm [`lx.N'' \\, \\Ssub' ] a &\\ByDef&\n\\New \\Vect{y`a} ( \\PilmuTerm [l x . N''] a \\Par {} \\PiLmuSem[\\Ssub'] )\n \\end{array} \\]\nwith $\\Ssub$ and $\\Ssub'$ as in the previous part and $a$ not in $\\Ssub$ or $\\Ssub'$.\nBy induction,\n \\[ \\begin{array}{rclclcl}\n\\New \\Vect{y`a} ( \\PilmuTerm [M''] b \\Par {} \\PiLmuSem[\\Ssub] )\n\t& \\ByDef &\n\\PilmuTerm[M'' \\, \\Ssub] b\n\t& \\wbisim & \\CLAC{ \\\\ }\n\\PilmuTerm[N''\\, \\Ssub'] b\n\t& \\ByDef &\n\\New \\Vect{y`a} ( \\PilmuTerm [N''] b \\Par {} \\PiLmuSem[\\Ssub] )\n \\end{array} \\]\nSince $\\wbisim$ is a congruence, also\n$\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $.\n\n\n \\item [{$M' = \\muterm`g.[`d]M''$, $N' = \\muterm`g.[`d]N''$}]\nThen $M''$ and $N''$ themselves are in normal form and $M'' \\, \\Ssub \\equivwxh N'' \\, \\Ssub' $, with $\\Ssub$, $\\Ssub'$ as above.\nBy Corollary~\\ref{weak soundness}, $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[\\muterm`g.[`d]M'' \\, \\Ssub ] a $ and $\\PilmuTerm[N] a \\wbisim \\PilmuTerm[\\muterm`g.[`d]N'' \\, \\Ssub' ] a $.\nNotice that\n \\[ \\begin{array}{\\CLAC{l@{~}c@{~}l}\\Paper{rcl}cl}\n\\PilmuTerm [{\\muterm`g.[`d].M''} \\, \\Ssub ] a\n\t&\\ByDef& \\CLAC{ \\\\ }\n\\New \\Vect{y`a} ( \\PilmuTerm [{ M''[a \\For `g]}] `d \\Par {} \\PiLmuSem[\\Ssub] )\n\t&\\ByDef&\n\\PilmuTerm[{M''[a \\For `g]} \\, \\Ssub] `d\n\t\\\\ [2mm]\n\\PilmuTerm [{\\muterm`g.[`d].N''} \\, \\Ssub'] a\n\t&\\ByDef& \\CLAC{ \\\\ }\n\\New \\Vect{y`a} ( \\PilmuTerm [{ N''[a \\For `g]}] `d \\Par {} \\PiLmuSem[\\Ssub'] )\n\t&\\ByDef&\n\\PilmuTerm[{N''[a \\For `g]} \\, \\Ssub'] `d\n \\end{array} \\]\nBy induction, $\\PilmuTerm[{M''[a \\For `g]} \\, \\Ssub] `d \\wbisim \\PilmuTerm[{N''[a \\For `g]} \\, \\Ssub'] `d $;\nsince $\\wbisim$ is a congruence, also\n$\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $.\n\n\n\n\\CLAC{\\item [{$M' = `lx.M''$ or $M' = \\muterm`g.[`d]M''$}] Similar.}\n\n \\end{description}\n\n\n \\item[only if]\nWe distinguish the following cases.\n\n \\begin{enumerate}\n\n \\item $ \\PilmuTerm[M] a $ can never input nor output; then $ \\PilmuTerm[M] a \\wbisim \\Zero \\wbisim \\PilmuTerm[N] a $.\nAssume $M$ has a weak head-normal form, then by Lem.~\\ref{bmu reduction characterisation}, $ \\PilmuTerm[M] a $ is not weakly bisimilar to $\\Zero$; therefore, $M$ and $N$ both have no weak head-normal form.\n\n \\item $\\PilmuTerm[M] a \\Outson c $, then by Lem.~\\ref{pi reduction characterisation}, $\\PilmuTerm[M] a \\wbisim \\New xb ( \\PilmuTerm[M'] b \\Par \\Out c \\Par \\PiLmuSem[\\Ssub] ) $, and $ M \\rtcredwxh `lx.M' \\, \\Ssub $.\nSince $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $,\nalso $\\PilmuTerm[N] a \\Outson c $, so $ \\PilmuTerm[N] a \\wbisim \\New xb ( \\PilmuTerm[N'] b \\Par \\Out c \\Par \\PiLmuSem[\\Ssub'] ) $ and $ N \\rtcredwxh `lx.N' \\, \\Ssub' $.\nThen also $ \\PilmuTerm[M'] b \\Par \\PiLmuSem[\\Ssub] \\wbisim \\PilmuTerm[N'] b \\Par \\PiLmuSem[\\Ssub'] $, so $\\PilmuTerm[M' \\, \\Ssub] a \\wbisim \\PilmuTerm[N' \\, \\Ssub'] a $\nand by induction, $M' \\, \\Ssub \\equivwxh N' \\, \\Ssub'$; so also $M \\equivwxh N$ by definition.\n\n \\item\nIf $\\PilmuTerm [M] a \\not\\!\\Outson c $, but $\\PilmuTerm [M] a \\Inson x $, then by Lem.~\\ref{pi reduction characterisation}, $ \\PilmuTerm[M] a \\wbisim \\PilmuTerm [xM_1\\dots M_n \\, \\Ssub] a' $ and $ M \\rtcredwxh xM_1\\dots M_n \\, \\Ssub$.\nWe have\n \\[ \\begin{array}{lcl}\n\\PilmuTerm [xM_1\\dots M_n \\, \\Ssub] a' &=&\n\\New \\Vect{cy`a} (\\PilmuTerm [v x] c_1 \\Par {\\VPiExContSub c_i := M_i . c_{i+1} } \\Par \\PiLmuSem[\\Ssub])\n \\end{array} \\]\n\\Paper{where $c_n = a'$ and\n \\[ \\begin{array}{rcl}\n\\PiLmuSem[\\Ssub] &=& {\\VPiExSub y := P } \\Par {\\VPiExContSub `a := Q . `b } \\\\\n\\PiExContSub c_i := M_i . c_{i+1} &=& { \\PiExContFsub c_i := M_i . c_{i+1} } \\\\\n\\PiExSub y_j := P_j &=& \\PiExsub y_j := P_j \\\\\n\\PiExContSub `a_k := Q_k . `b_k &=& { \\PiExContFsub `a_k := Q_k . `b_k }\n \\end{array} \\]}\n\\CLAC{with $\\PiLmuSem[\\Ssub] $,\n$ \\PiExContSub c_i := M_i . c_{i+1} $,\n$ \\PiExSub y_j := P_j $, and\n$ \\PiExContSub `a_k := Q_k . `b_k $ are defined as above.}\n\nSince $\\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $, again by Lem.~\\ref{pi reduction characterisation}, $\\PilmuTerm[N] a \\wbisim \\PilmuTerm [xN_1\\dots N_n \\, \\Ssub'] a'' $ and $ N \\rtcredwxh xN_1\\dots N_n \\, \\Ssub' $.\nNotice that\n \\[ \\begin{array}{lcl}\n\\PilmuTerm [xN_1\\dots N_n \\, \\Ssub'] a'' & = &\n\\New \\Vect{cy`a} (\\PilmuTerm [v x] c_1 \\Par {\\VPiExContSub c_i := N_i . c_{i+1} } \\Par {} \\PiLmuSem[\\Ssub'])\n \\end{array} \\]\n\\Paper\nwhere $c_n = a''$ and\n \\[ \\begin{array}{rcl}\n\\PiLmuSem[\\Ssub'] &=& {\\VPiExSub y := P' } \\Par {\\VPiExContSub `a := Q' . `b } \\\\\n\\PiExContSub c_i := N_i . c_{i+1} &=& \\PiExContFsub c_i := N_i . c_{i+1} \\\\\n\\PiExSub y_j := P'_j &=& \\PiExsub y_j := P'_j \\\\\n\\PiExContSub `a_k := Q'_k . `b_k &=& { \\PiExContFsub `a_k := Q'_k . `b_k }\n \\end{array} \\]\n\n\\CLAC{with $\\PiLmuSem[\\Ssub'] $, $\\PiExContSub c_i := N_i . c_{i+1} $, $ \\PiExSub y_j := P'_j $, and $ \\PiExContSub `a_k := Q'_k . `b_k $ similar to above.}\nThen we have\n \\[ \\begin{array}{rcl}\n\\PilmuTerm [xM_1\\dots M_n \\, \\Ssub] a' &\\wbisim& \\PilmuTerm [xN_1\\dots N_n \\, \\Ssub'] a'' ,\n\\end{array} \\]\nso $a' = a''$ and $\\PilmuTerm [M_i' \\, \\Ssub ] w \\wbisim \\PilmuTerm [N_i' \\, \\Ssub'] w $; then by induction, $M_i' \\, \\Ssub \\equivwxh N_i' \\, \\Ssub' $, and \\Paper{then also }$M\\equivwxh N $.\n\\quad\\usebox{\\proofbox}\n\n\n \\end{enumerate}\n \\end{description}\n \\end{Proof}\n\n\nWe now obtain our main result:\n\n \\begin{theorem} [Full abstraction] \\label{main result}\nLet $M,N \\ele \\lmu$, then $ \\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $ if and only if \n M \\equivwbmu N $.\n\n\\Paper{\\begin{Proof}\nBy Corollary~\\ref{all lmu weak equivalences} and Theorem~\\ref{FA equivxh equivC}.\n\\quad\\usebox{\\proofbox}\n \\end{Proof}\n}\n\n \\end{theorem}\n\n\\Paper\nSince ${\\eqbmu} \\subseteq {\\equivwbmu}$, the following is immediate.\n\n \\begin{corollary} \\label{semantics}\nIf $M \\eqbmu N$, then $ \\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $.\n \\end{corollary}\nwhich states that our interpretation gives a semantics for $\\lmu$.\n\n\n \\subsection*{Conclusions and Future Work}\nWe have found a new, simple and intuitive interpretation of $\\lmu$-terms in $`p$ that respects head reduction with explicit substitution.\nFor this interpretation, we have shown that termination is preserved, and that it is sound and complete.\nWe have shown that, for our context assignment system that uses the type constructor $\\arrow$ for $`p$ and is based on classical logic, typeable $\\lmu$-terms are interpreted by our interpretation as typeable $`p$-processes, preserving the types.\n\n\n\n\n\\CLAC{ \\section*{Conclusions and future work}\nWe have studied the output based, logic-inspired interpretation of untyped $\\lmu$ with explicit substitution into the $`p$-calculus and shown that this interpretation is fully abstract with respect to weak equivalence between terms and weak bisimilarity between processes.\n\nWe have defined the\nweak equivalences $\\equivwbmu$, $\\equivwh$, $\\equivwxh$, and $\\equivwA$ on $\\lmu$ terms, and shown that these all coincide.\nWe then proved that $M \\equivwxh N \\Iff \\PilmuTerm[M] a \\wbisim \\PilmuTerm[N] a $, which, combined with our other results, essentially shows that $\\PilmuTerm[`.] `. $ respects equality between L\\'evy-Longo trees for $\\lmu$.}\n\nWe will investigate the relation between our interpretation and the {\\sc cps}-translation of Lafont, Reus, and Streicher \\cite{Lafont-Reus-Streicher'93}.\n\n\\Paper{ \n\\bibliography {references} \n}\n \\CLAC{ \\input {biblio} }\n\n\n\n\n\n\n \\end {document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA staged program of neutrino oscillation experiments is forseen using a new beam line facility, the ``Neutrinos at the Main Injector'' (NuMI) \\cite{numitdr}, at the Fermi National Accelerator Laboratory. The first experiment, MINOS \\cite{minos}, will perform definitive spectrum measurements which demonstrate the effect of $\\nu_\\mu$ oscillations. The experiment has run since early 2005 and published a first result \\cite{minos-prl} based upon $1.3\\times10^{20}$~protons-on-target, approximately 1\/10$^{th}$ the eventual exposure of the experiment. The second, NO$\\nu$A \\cite{nova}, has been approved to explore the phenomenon of CP violation in the transition $\\nu_\\mu\\rightarrow\\nu_e$ and possibly resolve the mass heirarchy of neutrinos. A third, MINER$\\nu$A \\cite{minerva}, is an experiment 1 km from the NuMI target to perform neutrino cross section measurements essential for long-baseline neutrino oscillation searches. \n \n\n\n\n\\section{NuMI Intensity and Upgrades}\n\nNuMI is a tertiary beam resulting from the decays of pion and kaon secondaries produced in the NuMI target. Protons of 120 GeV are fast-extracted (spill duration $\\sim10\\mu$sec) from the Main Injector (MI) accelerator and bent downward by 58~mrad toward Soudan, MN (see Figure~\\ref{fig:numi}). The beam line is designed to accept $4\\times10^{13}$~protons-per-pulse~(ppp). The beam line can be flexibly configured to achieve a variety of neutrino energies \\cite{flexybeam}. \nTo date, the average (peak) beam power has been 230~kW (310~kW), and a three-phase plan has been approved to upgrade the accelerator complex and NuMI line to increase this intensity to 430~kW by 2009, then 700~kW by 2012, then 1200~kW by 2013. \n\n\\begin{figure*}[htb]\n\\vspace{9pt}\n \\includegraphics[width=5.8in]{numi-beam-figure.eps}\n\\caption{Plan and elevation view of the NuMI beam at FNAL. }\n\\label{fig:numi}\n\\end{figure*}\n\nThe MI is fed multiple batches from the 8~GeV Booster accelerator. The Booster can deliver $(4-5)\\times10^{12}~p$\/batch. During normal operations, the MI beam is shared: 2 of 7 accelerated Booster batches are extracted to make antiprotons for the Tevatron collider, while the remaining 5 are extracted to the NuMI target. \n\nThe first phase of the intensity upgrades takes advantage of the fast (67~ms) cycle time to the Booster to inject as many as 11 batches of protons into the MI.\n Pairs of batches will be coalesced by a process of ``slip-stacking'' \\cite{kiyomi} into 6 double-batches around the MI circumference, and this beam accelerated to 120~GeV. Already, MINOS has received a 20\\% increase in beam power by slip-stacking of 2 batches and accelerating these along with 5 other single batches, and tests of 11-batch stacking have achieved as much as 3$\\times10^{13}$~ppp, which will improve as the coalescing and capture process is better understood. This phase should be fully-implemented in 2007 and achieve 430~kW. \n\nThe second phase will use the 8~GeV Recycler ring in the MI tunnel as a pre-injector. The Recycler, which has the same circumference as the MI, will be used to pre-load and slip-stack 11 Booster batches, then perform a fast transfer to the MI for acceleration. This reduces the MI cycle time from 2.2~sec. to 1.3~sec., increasing the beam power to NuMI to 700~kW (see Table~\\ref{table:power}). This phase is to be implemented at the end-of-Tevatron shutdown in 2009, and has been approved by the Laboratory.\n\nThe third phase, which is still in its conceptual stages, calls for use of the Accumulator, currently utilized for antiproton beam accumulation, as a proton beam accumulator which will permit 3 Booster batches to be coalesced in one Booster-equivalent circumference. Six such Accumulator batches can be injected into the Recycler, then the Main Injector. The plan calls for new injection and extraction lines to be built for the Accumulator (see Figure~\\ref{fig:accum}). As shown in Table~\\ref{table:power}, the anticipated beam power to NuMI is then 1.2~MW.\n\n\\begin{figure}[htb]\n\\vspace{9pt}\n \\includegraphics[width=3in]{accumulator.eps}\n\\vskip -1.cm\n\\caption{Plan view of the FNAL Accumulator ring, with new AP4 injection line from the Booster and AP5 extraction line to the MI.}\n\\label{fig:accum}\n\\vskip -1.cm\n\\end{figure}\n\nPhases II and III require upgrades to the NuMI line. These include the substitution of a graphite target with alternative water cooling schemes capable of withstanding the higher beam power and also additional cooling in the target station, decay tunnel, and beam stop. The new target design is facilitated by the fact that in the future the NuMI beam will operate at higher on-axis neutrino energy (which will obtain an ideal 1.8~GeV off-axis for the NO$\\nu$A experiment, see below) and the target will no longer be cantilevered inside the first focusing horn.\n\n\\begin{table*}[htb]\n\\caption{Present and projected NuMI beam power following 3 upgrades.}\n\\label{table:power}\n\\newcommand{\\hphantom{$-$}}{\\hphantom{$-$}}\n\\newcommand{\\cc}[1]{\\multicolumn{1}{c}{#1}}\n\\renewcommand{\\tabcolsep}{2pc}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{@{}llll}\n\\hline\n & Protons\/pulse & Cycle time & Power \\\\\n\\hline\n{Current Complex} & & & \\\\\n$\\cdot$ Shared Beam & 25$\\times10^{12}$ & 2.4 s & 200 kW \\\\\n$\\cdot$ NuMI Alone & 33$\\times10^{12}$ & 2.0 s & 310 kW \\\\\n{Phase I (2009)$^{(a)}$} & & & \\\\\n$\\cdot$ Shared Beam & 37$\\times10^{12}$ & 2.2 s & 320 kW \\\\\n$\\cdot$ NuMI Alone & 49$\\times10^{12}$ & 2.2 s & 430 kW \\\\\n{Phase II$-$Recycler (2012)$^{(a)}$} & 49$\\times10^{12}$ & 1.3 s & 700 kW \\\\\n{Phase III$-$Accumulator (2013)$^{(b)}$} & 83$\\times10^{12}$ & 1.3 s & 1200 kW \\\\\n\\hline\n & $^{(a)}$Approved &\\multicolumn{2}{l}{$^{(b)}$Still at conceptual stage.}\n\\end{tabular}\\\\[2pt]\n\\end{table*}\n\n\n\\section{MINOS}\n\nThe Main Injector Neutrino Oscillation Search (MINOS) \\cite{minos} began data-taking in 2005 and has accumulated $1.8\\times10^{20}$~POT. The first results \\cite{minos-prl} based on 1.2$\\times10^{20}$~POT were covered by B. Rebel at this workshop. MINOS is a two-detector experiment, with a 980~ton detector located 1~km from the NuMI target on site at FNAL (see Figure~\\ref{fig:numi}) to measure the neutrio flux directly, and a second, 5400~ton detector located in the Soudan mine in Minnesota, at a distance 735~km. The spectrum of neutrinos arriving in the far detector is studied to investigate whether neutrinos oscillate, decay, or otherwise disappear in flight. \n\n\\begin{figure}[htb]\n \\includegraphics[width=3.1in]{minos-result.eps}\n\\vskip -1.cm\n\\caption{First result from the MINOS experiment \\cite{minos-prl}. A $>6\\sigma$ deficit of neutrinos is observed in the far detector in Minnesota. }\n\\label{fig:minos-result}\n\\end{figure}\n\nMINOS has investigated the rate of charged-current $\\nu_\\mu$ interactions in the near and far detectors. Oscillations would result in an energy-dependent depletion of these interactions. The results of the first 1.2$\\times10^{20}$~POT are shown in Figure~\\ref{fig:minos-result}. Following commissioning of the beam line, the experiment acquired several data samples at different neutrino energies to study the beam spectrum and neutrino interactions in the near detector. Having thus tested the calculated spectrum of the neutrino flux, the Monte Carlo could be better trusted to describe any differences in the neutrino energy spectra in the near and far detectors. \nThe energy spectrum observed in the near detector was multiplied by an extrapolation factor \\cite{para} which corrects the near spectrum to the expected spectrum (in the absence of oscillations) at the far detector. A deficit of $\\nu_\\mu$'s with $E<6$~GeV of $>6\\sigma$ is observed. If interpreted as neutrino oscillations, the data indicate\n\n\\begin{equation}\n\\Delta m^2_{32}=(2.74^{+0.44}_{-0.26})\\times10^{-3}~\\mbox{eV}^2 \n\\end{equation}\nwith the mixing angle consistent with maximal mixing: $\\sin^2(2\\theta_{23})>0.87$. The expected resolution is $\\delta(\\Delta m_{32}^2)<10^{-4}~\\mbox{eV}^2$ at $16\\times10^{20}$~POT.\n\nWith further statistics, the experiment hopes to observe or place stringent limits on the transition $\\nu_\\mu\\rightarrow\\nu_e$. As shown in Figure~\\ref{fig:minos-nue}, the potential for 3$\\sigma$ discovery depends upon the values of the CP-violating phase $\\delta$ and the sign of $\\delta m_{32}^2$. \n\n\\begin{figure}[tb]\n \\includegraphics[width=2.9in]{minos-nue.eps}\n\\vskip -1.cm\n\\caption{Expected sensitivity of the MINOS experiment to the appearance of $\\nu_e$'s in its $\\nu_\\mu$ beam. }\n\\label{fig:minos-nue}\n\\end{figure}\n\n\n\n\\section{MINER$\\nu$A}\n\nThe MINER$\\nu$A experiment is a fine-grained scintillator detector like the KEK SciBar detector, but located in the NuMI detector hall 1~km from the NuMI target. It features 196 alternating planes of $U$ and $V$ strips, with each strip consisting of $1.7\\times3.3~\\mbox{cm}^2$ triangular solid extruded scintillator read out into a multi-anode PMT via scintillating fiber. The finer pitch, overlapping triangular strips gives excellent position resolution to reconstruct final states in neutrino-nucleus interactions. A coarse outer calorimeter is used for shower containment, and muon momentum analysis is performed in the downstream MINOS near detector. The ability of the detector to perform detailed analyses of neutrino final states may be seen in Figure~\\ref{fig:minerva-events}.\n\n\\begin{figure}[tb]\n\\vspace{9pt}\n \\includegraphics[width=3.in]{minerva-events.eps}\n\\vskip -0.5cm\n\\caption{Display of two simulated neutrino interactions in the MINER$\\nu$A detector: (left) quasielastic scattering, (right) resonant production, with the $\\Delta^+\\rightarrow p\\pi^0$ visible in the detector. }\n\\label{fig:minerva-events}\n\\end{figure}\n\nUncertainties in neutrino cross sections are substantial at low energy. Of particular note are the large uncertainties in the MINOS oscillation result due to such cross section uncertainties \\cite{minos-prl}: pion reabsorption by the struck nucleus can alter the apparent final state multiplicity and visible energy. Such nuclear effects will be studied using the 18 nuclear target sheets in MINER$\\nu$A.\n\n\n\\section{NO$\\nu$A}\n\nThe NuMI Off-Axis $\\nu_e$ Appearance (NO$\\nu$A) experiment has been approved by the Laboratory to search for and use the transition $\\nu_\\mu\\rightarrow\\nu_e$ to study CP violation in the neutrino sector. \n\n\\begin{figure}[t]\n\\vspace{9pt}\n\\vskip -0.5cm\n \\includegraphics[width=2.5in]{nue-matter.eps}\n\\vskip -1.cm\n\\caption{Probability for the transition $\\nu_\\mu\\rightarrow\\nu_e$ and $\\overline{\\nu}_\\mu\\rightarrow\\overline{\\nu}_e$ for 735~km baseline and $E_\\nu$=1.5~GeV as a function of the CP violating phase $\\delta$ under two assumptions for the mass ordering. Taken from \\cite{nova}.}\n\\label{fig:nue-matter}\n\\end{figure}\n\n\\begin{figure}[h]\n\\vspace{9pt}\n\\vskip -0.5cm\n \\includegraphics[width=2.6in]{nova-spectrum.eps}\n\\vskip -1.cm\n\\caption{The charged-current $\\nu_\\mu$ spectrum expected in NO$\\nu$A, along with the expected $\\nu_e$ signal from oscillations with $\\sin^2 2\\theta_{13}=0.1$ and $\\nu_e$ contamination from the beam. Also shown is the visible energy from neutral current $\\nu_\\mu$ interactions before any rejection criteria are applied. }\n\\label{fig:nova-spectrum}\n\\end{figure}\n\nThe NuMI long baseline and higher neutrino energy results in a splitting between the transition probability for $\\nu_\\mu\\rightarrow\\nu_e$ and $\\overline{\\nu}_\\mu\\rightarrow\\overline{\\nu}_e$, as shown in Figure~\\ref{fig:nue-matter}. This arises due to the matter effect, and results in an asymmetry even in the absence of CP violation. This fact makes the NO$\\nu$A experiment quite complementary to the JPARC T2K experiment, which will operate at lower $E_\\nu$ and shorter baseline: JPARC will measure the transition probability for $\\nu_\\mu\\rightarrow\\nu_e$, which measures the mixing angle $\\theta_{13}$ in the neutrino mixing matrix, and the NO$\\nu$A experiment could study both the $\\nu$ and $\\overline{\\nu}$ process to understand the mass heirarchy of the neutrino states, and possibly also measure the CP phase $\\delta$.\n\n\\begin{figure}[tb]\n\\vspace{9pt}\n \\includegraphics[width=2.9in]{nova-sensitivity-1.eps}\n \\includegraphics[width=2.9in]{nova-sensitivity-2.eps}\n\\vskip -1.cm\n\\caption{Curves showing the $3\\sigma$ discovery potential of the NO$\\nu$A experiment to observe the transition $\\nu_\\mu\\rightarrow\\nu_e$ after an exposure of $60\\times10^{20}$~POT. The upper plot is for an all-neutrino exposure, the lower for an equal admixture of $\\nu$ and $\\overline{\\nu}$ running.}\n\\label{fig:nova-sensitivity}\n\\end{figure}\n\n\nNO$\\nu$A must look for the appearance of charged-current $\\nu_e$ interactions in a $\\nu_\\mu$ beam, a challenge because because of the small transition probability and the prevalence of showers from neutral current $\\nu_\\mu$ interactions which can mimic the signal. As shown in Figure~\\ref{fig:nova-spectrum}, the $\\nu_\\mu$ off-axis spectrum can be centered at the oscillation maximum for $\\nu_\\mu\\rightarrow\\nu_e$, and the putative signal for $\\sin^2 2\\theta_{13}=0.1$ is significant over the intrinsic $\\nu_e$ in the beam which arises primarily from muon decays in flight. The challenge for the experiment will be the reduction of neutral current $\\nu_\\mu$ interactions, which can mimic the showers of charged-current $\\nu_e$ events. \n\nTo reduce neutral current backgrounds, the NO$\\nu$A detector is a fine-grained totally-active segmented calorimeter, consisting of alternating horizontally- and vertically-oriented planes of scintillator strips, 15.7~m in length and 6~cm in depth by 3.9~cm transverse width. The scintillator strips are made using extruded PVC shells filled with liquid scintillator. The two ends of a looped scintillating fiber which runs the 15.7~m length of the strip brings the light out to an APD. The light output for a minimum ionizing particle is 20~photo-electrons, to be compared with 2~photo-electrons noise from the APD. The detector will have 1654 planes of 384 strips, for a total mass of 25~kton. \n\n\nFigure~\\ref{fig:nova-sensitivity} shows the capability of the NO$\\nu$A experiment to observe the $\\nu_\\mu\\rightarrow\\nu_e$ transition with 3$\\sigma$ significance after an exposure of $60\\times10^{20}$~POT (equivalent to 7~years of Phase-III running). The curves are calculated using values derived from a full simulation of neutrino interactions in the NO$\\nu$A detector, namely 23\\% efficiency for identifying a $\\nu_e$ charged current interaction, as well as a rejection factor of 7$\\times10^{-4}$ (1.3$\\times10^{-3}$) for charged current (neutral current) $\\nu_\\mu$'s. \nRunning the NuMI beam in both $\\nu$ and $\\overline{\\nu}$ mode makes the discovery potential of the experiment less dependent upon the CP phase and mass heirarchy.\n\n\\section{Future Initiatives}\n\nTwo longer-term design studies are underway in the U.S. The first would consider the feasibility of building a high-resolution liquid Argon detector of order 50~kton to study $\\nu_\\mu\\rightarrow\\nu_e$ transitions in the NuMI beam \\cite{flare}. The design would build upon previous LAr experience, housing the detector in a massive cryogenic system as is used for liquid natural gas.\n\nThe second is a BNL-FNAL study \\cite{diwan} to send a neutrino beam to DUSEL. If the beam were initiated at FNAL, the Main Injector complex could serve as its source, but a new neutrino line would have to be built which points in the direction of DUSEL. The proposal is for an on-axis wide-band neutrino beam. At the FNAL site, such a beam line could be at most 400~m in length.\n\n\\section{Summary}\n\nNeutrino physics in the U.S. has enjoyed a renaissance with the commissioning of the NuMI line at FNAL. The MINOS experiment has already published a competitive measurement of $\\Delta m^2_{32}$, and still looks forward to collecting 10 times more data. In the future, we may look forward to oscillation and scattering results\n\nThe NO$\\nu$A experiment is an ambitious effort to use $\\nu_\\mu\\rightarrow\\nu_e$ to study the $\\nu$ mass heirarchy and search for CP violation in the lepton sector. It complements the shorter-baseline experiment at J-PARC or a reactor disappearance experiment. \n\nThe NuMI line is commissioned and performing well. As the Tevatron collider ramps down, Fermilab will be a dedicated neutrino facility, and the accelerator complex will be re-commissioned toward higher beam power. \n\nI wish to acknowledge my colleagues on the NuMI facility and MINOS, NO$\\nu$A, and MINER$\\nu$A experiments. R. Zwaska, A. Marchionni, M. Martens, and N. Grossman kindly provided information on the proton source upgrades.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDeep learning has been making significant progress in many fields in recent years especially in computer vision \\cite{krizhevsky2012imagenet} and speech recognition \\cite{hinton2012deep}. However, it remains a black box as of today. The hyper-parameter tuning is a hard and tedious process, which makes it more of an artistry than hard-core science. Moreover, its lack of interpretability poses a major holdback to its adoption in many practical scenarios. Some researchers try to combine deep learning with traditional machine learning methods such as Random Forest (RF). For example, RF is used to enhance interpretability and performance with deep learning models \\cite{frosst2017distilling, kontschieder2015deep}. There's also advancement in representation learning \\cite{feng2017autoencoder} and trying to make RF deep \\cite{zhou2017deep}.\n\nRF is the paradigm of Bagging algorithm in ensemble learning. In 2001, Breiman combined decision trees \\cite{breiman2017classification} into a forest \\cite{breiman2001random}. Features (columns) and data (rows) are randomly sampled to generate multiple decision trees and their predictions are aggregated. RF improves the prediction accuracy without significant increase in computation, and it is insensitive to multicollinearity. Its performance is robust against missing and imbalanced data, and it can well measure the roles of thousands of variables \\cite{breiman2001statistical}. Compared with models in deep learning, it has the advantage of small computation, outstanding generalization ability and strong interpretability, which makes it ever appealing to researchers and practitioners in spite of its long existence. \n\nHowever, there are also limitations of RF. First of all, the RF model can only be extended horizontally (more decision trees) but not vertically since the decision trees exist in parallel and cannot be stacked in layers in the same fashion as neurons in neural networks. Secondly, these decision trees have the same weight in voting for the final prediction despite that some of these trees may perform poorly. Lastly, all points in training data have the same weight and are treated equally in the sampling and training process, despite that some of the data are easy to classify while others are hard.\n\nIn this paper, we propose a dynamic boosted ensemble learning method based on random forest (DBRF) to overcome these limitations. It is a novel ensemble algorithm where such notion of hard example mining as in Boosting is incorporated into RF. Specifically, we propose to score the quality of each leaf node of every decision tree in the random forest and then vote to determine hard examples. By iteratively training and then removing easy examples from training data to train again, we evolve the model to focus on hard examples dynamically so as to learn decision boundaries better. After training, the test data can be cascaded through these learned random forests of each iteration in sequence to generate predictions, thus achieving more depth with RF. In addition, we propose to use evolution mechanism and smart iteration mechanism to enhance the performance of DBRF and conducted ablation test. We evaluate DBRF on public datasets and it outperforms RF and achieves state-of-the-art results compared to other deep models. Last but not least, we also perform visualizations to validate DBRF and analyze its effectiveness from a sampling point of view.\n\nThe contributions of this paper are listed as follows:\n\\begin{enumerate}\n\\item We propose a novel ensemble algorithm called DBRF to extend RF vertically by iteratively mining and training on hard examples, and thus incorporate boosting into RF training process. It outperforms RF.\n\\item We propose a criterion to measure the quality of a leaf node in a decision tree and a voting mechanism to determine hard examples.\n\\item We propose an evolution mechanism which filters out poor decision trees in a random forest, together with iteration mechanism to enhance the performance of DBRF and achieve state-of-the-art results on three UCI datasets.\n\\item The hard example mining process proposed with DBRF can be seen as a novel way of sampling. We demonstrate the effectiveness of using DBRF to deal with imbalanced data.\n\\end{enumerate}\n\nThe rest of the paper is organized as follows. Section 2 explains our motives. Section 3 presents our proposed model DBRF and the enhancement mechanisms. Section 4 reports the results of multiple evaluation experiments. Finally, Section 5 contains related work and Section 6 concludes the paper.\n\n\\section{Motivation}\nIn typical classification tasks, classifiers are trained to find the decision boundaries of labelled data using a set of features. The distribution of data near the decision boundaries largely affect the performance of the classifier and is generally where the overfitting and under-fitting trade-off is made. Those data whose predicted labels are not agreed upon across multiple classifiers, probably located near the decision boundaries, can be seen as hard examples. On the contrary, those data whose predicted labels are consistent and correct across all classifiers can be seen as easy examples. Naturally, the performance on hard examples determines how good a classifier is compared with others and it's ideal to focus on these hard examples during training.\n\nHard example mining is a common notion in many machine learning algorithms. A typical example is AdaBoost, in which wrongly classified examples are deemed as hard examples. The general idea is to assign more weight to those hard examples and train the model iteratively until convergence. This hard example mining notion has also been adopted in deep learning to successfully improve the performance of the model, especially in object recognition tasks, such as OHEM \\cite{shrivastava2016training}, S-OHEM \\cite{li2017s} and A-Fast-RCNN \\cite{wang2017fast}.\n\nThus, we are motivated to incorporate this hard example mining notion, or boosting, into RF. It can be integrated smoothly within the RF training process since a random forest is an ensemble of decision trees trained on randomly sampled data and features, which are weak classifiers. If all decision trees make right predictions on a part of the training data, these data can be considered as easy examples. Rather than assign more weight to hard examples, we can simply remove these data from the training data to achieve the same effect. When training the RF model again on the new set of data, we essentially make the model focus on hard examples and learn the decision boundaries better without making the classifier more complex and sacrificing generalization abilities. The idea is illustrated in Figure \\ref{f1}. \n\n\\begin{figure}[htb]\n\\centering\n\\caption{Illustration of removing easy examples to improve learning. When we train a classifier as shown on the left, one of the decision regions is not ideal, but when we remove the easy examples from the good decision region and then train a classifier as shown in the right figure. At this point, the ensemble of the two classifiers allows the training data to be perfectly divided.}\\label{f1}\n\\includegraphics[width=3in]{figure1}\n\\end{figure}\n\nOne point to clarify is that the aforementioned criterion to determine easy examples, i.e. data that are classified right across multiple classifiers, is not strict enough for removal because in extreme cases where data are imbalanced, we may be left with data of only one label. We will explain our proposed criterion in detail in Section 3. But basically, with RF, we are seeking to find good rules and define easy examples as ones that fit the good rules of all classifiers. In a decision tree, each leaf node represents a rule, thus we need to come up with a criterion to determine whether a leaf node is good or not.\n\n\n\\section{The Proposed Approach: DBRF}\nIn this section, we present out proposed approach DBRF based on RF, which drives the evolution of the model by iteratively updating the training data. Compared to RF, DBRF can greatly improve the performance. We will first present the general framework and the basic algorithm of our model and then propose two mechanisms to enhance the basic model. \n\nIn the following, we consider a typical classification task where $\\mathcal{X}$ and $\\mathcal{Y}$ denote input and output space respectively. For a decision tree $T$, $N$ denotes a decision node and $L$ a leaf node. A RF $F$ is a set of $T$ trained on data $D = \\{X, Y\\}$ where $X$ is points set in $\\mathcal{X}$ and $Y$ is corresponding labels set in $\\mathcal{Y}$.\n\n\\subsection{The General Framework}\n\n\\begin{figure}[htb]\n\\centering\n\\caption{Illustration of DBRF with three iterations. During training (above the dotted line), firstly, a random forest is trained on the training set, and then the leaf nodes are evaluated through the HEM process (Section \\ref{s3.2}). The hard examples (with bold borders) can be identified accordingly (by removing easy examples) as a new training set for the next iteration. Similarly, at each iteration during test (below the dotted line), easy examples are identified (except the last iteration) and their predictions are outputted.}\\label{f2}\n\\includegraphics[width=5in]{figure7}\n\\end{figure}\n\nFigure \\ref{f2} illustrates the basic structure of DBRF. We first split a dataset into training set and test set and then train a random forest of decision trees on the training set $D^{d}=\\{X, Y\\}$. Next, we use a criterion to measure the quality of each leaf node of each decision tree in the forest. In Figure \\ref{f2}, the leaf nodes circled in bold line are good ones which represent possible easy examples. By using the proposed hard example mining (HEM) method to be elaborated in Section \\ref{s3.2}, we can divide $D^{d}$ into two parts $\\{D^{d}_{e}, D^{d}_{h}\\}$, where $D^{d}_{e}$ denotes easy examples and $D^{d}_{h}$ hard examples. Only $D^{d}_{h}$ are preserved for the next iteration's training. This process keeps on iterating until a predetermined $n^{th}$ iteration is done.\n\nAt iteration i, we need to preserve the random forest model $F_{i}$ trained in current iteration and the evaluation scores of all leaf nodes of all decision trees in $F_{i}$, denoted as $\\Pi_{i}$. For predicting test set $D^{t}$, we first use $F_{1}$ to predict and then divide $D^{t}$ into $\\{D^{t}_{e}, D^{t}_{h}\\}$, according to $F_{1}$. For $D^{t}_{e}$ the easy data, the predictions made by $F_{1}$ are outputted as the final prediction result, while for the hard data $D^{t}_{h}$, we will feed them into $F_{2}$ for the next iteration's prediction. This process goes on until no longer contains data, or until the last iteration. In the last iteration n, the output of $F_{n}$ will be the final predicted labels of the corresponding data.\n\nThis training and test process can be further written in pseudo code shown in Algorithm \\ref{a1}.\n\n\\begin{algorithm}[htb]\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\t\\caption{Dynamic Boosted Random Forest}\\label{a1}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE Training set: $D^{d}$, Test set: $D^{t}$, Iterations $n$\n\t\t\\ENSURE Prediction of test set: $O$\n\t\t\n\t\t\\STATE \/\/ Training Procedure\n\t\t\\STATE $D \\gets D^{d}$\n\t\t\\FOR {$i = 1 \\to n$}\n\t\t\\STATE Train RF as $F_{i}$ on dataset $D$\n\t\t\\STATE Get scores of leaf nodes $\\Pi_{i}$ by HEM (Section \\ref{s3.2})\n\t\t\\STATE Split $D$ into easy data $D_{e}$ and hard data $D_{h}$ according to $F_{i}$ and $\\Pi_{i}$\n\t\t\\STATE $D \\gets D_{h}$\n\t\t\\STATE add $F_{i}$ to $F$-list \n\t\t\\STATE add $\\pi_{i}$ to $\\Pi$-list \n\t\t\\ENDFOR\n\t\t\\STATE\n\n\t\t\\STATE \/\/ Test Procedure\n\t\t\\STATE $D \\gets D^{t}$\n\t\t\\STATE $O \\gets \\varnothing$\n\t\t\\FOR {$i = 1 \\to n$}\n\t\t\\STATE $F_{i} \\leftarrow F$-list at i\n\t\t\\STATE $\\Pi_{i} \\leftarrow \\Pi$-list at i\n\t\t\\STATE Split $D$ into easy data $D_{e}$ and hard data $D_{h}$ according to $F_{i}$ and $\\Pi_{i}$\n\t\t\\STATE Predict $D_{e}$ as $O'$ by $F_{i}$\n\t\t\\STATE $D \\gets D_{h}$\n\t\t\\STATE $O = O \\cup O'$\n\t\t\\ENDFOR\n\t\t\\STATE Predict $D_{h}$ as $O'$ by $F_{n}$\n\t\t\\STATE $O = O \\cup O'$\n\t\t\n\t\t\\STATE \\textbf{return} $O$\n\t\\end{algorithmic}\n\\end{algorithm}\n\nWe propose to use two mechanisms to enhance the basic model of DBRF. To prevent the negative influence of poor decision trees in a random forest on the HEM process, we propose to use an evolution mechanism to eliminate them from the random forest, which will be further elaborated in Section 3.3. Furthermore, we propose a smart iteration mechanism to better guide the HEM process, as elaborated in Section 3.3. The model of DBRF with two mechanisms is illustrated in Figure \\ref{f3}.\n\n\\begin{figure}[htb]\n\\centering\n\\caption{Illustration of the mechanisms of DBRF. The evolution mechanism is used to eliminate poor decision trees from the random forest and the smart iteration mechanism is used to better guide the HEM process.}\\label{f3}\n\\includegraphics[width=5in]{figure3}\n\\end{figure}\n\n\\subsection{Hard Example Mining (HEM)}\\label{s3.2}\nA random forest consists of multiple decision trees. We use $D^{e}_{F}$ and $D^{h}_{F}$ to denote easy examples and hard examples of a random forest, and $D^{e}_{T}$ and $D^{h}_{T}$ to denote easy examples and hard examples of each decision tree. We propose that $D^{e}_{F}$ and $D^{h}_{F}$ can be generated as\n$$D^{e}_{F}= \\bigcap_{T\\in F} D^{e}_{T},\\ D^{h}_{F}=D - D^{e}_{F}$$\nmeaning that the intersection of easy examples of all the decision trees are considered as easy examples.\n\nA leaf node $L$ in a decision tree corresponds to a rule $R_{L}$. Suppose the path to reach the leaf node $L$ from the root in a decision tree $F$ is $W_{L}=\\{n_{1},n_{2},n_{3},...,n_{i}\\}$, where $n_{1}$ to $n_{i}$ denote the decision nodes along the path. The probability distribution in $\\mathcal{Y}$ in node $L$ is $\\pi_{L}$, $c$ is any label in $\\mathcal{Y}$, then we can define the rule $R_{L}$ corresponding to the leaf node $L$ as\n\n$$R_{L}: x\\ |\\ x \\in \\bigcap_{n_{i} \\in W_{L}} r(n_{i}) \\Rightarrow y = \\underset{c}{\\mathrm{\\textit{arg\\ max}}}\\ \\pi_{L}(c)$$\nwhere $r(\\cdot)$ is a function that represents the data that satisfy the rule of a decision node $n$. Then the easy examples of a decision tree can be defined as\n\n$$D^{e}_{T} = \\{x\\ |\\ x \\in \\bigcap_{L\\in T, T \\in F} (score(L) > \\sigma) \\}$$\nwhere $score(\\cdot)$ is a leaf node evaluation metric and $\\sigma$ is the threshold, which is implemented as the average score of leaf nodes in all decision trees in our model.\n\nNext, we propose several leaf node evaluation metrics $score(\\cdot)$. We may use support and confidence, as were used to evaluate association rules \\cite{agrawal1993mining} to score the leaf node. For the rule $R_{L}$ of a leaf node $L$, all data that satisfy the preconditions of the rule, i.e. $x \\in_{n_{i} \\in W_{L}} r(n_{i})$, compose candidate set $C$. Then\n$$score_{supp}(L)=\\frac{|C|}{|X|}$$\n$$score_{conf}(L)=\\frac{|\\{y = c,\\ x \\in C\\}|}{|C|}$$\nSince both support and confidence derive from association rules, we can merge them to be f1 score. \n$$score_{f1}(L)=2 \\cdot \\frac{score_{supp}(L) \\cdot score_{conf}(L)}{score_{supp}(L) + score_{conf}(L)}$$\n\nAlso, since Gini impurity (gini) and information gain (entropy) are the partitioning criteria used in decision tree, we can define gini score and entropy score of a leaf node as\n\n$$score_{gini}(L)=-Gini(L)=\\sum^{c}_{j=1} p_{j}^{2}-1$$\n$$score_{entropy}(L)=-Entropy(L)=\\sum^{c}_{j=1} p_{j}^{2}\\ log\\ p_{j}$$\nwhere $c_{j}$ is $j^{th}$ label in $\\mathcal{Y}$ and $p_{j}$ is the probability of $c_{j}$ .\n\nAll in all, we propose three leaf node evaluation metrics, namely \\textit{score-gini}, \\textit{score-entropy}, and \\textit{score-f1}. Their effects are explored in experiments, as discussed in Section 4.1.\n\n\\subsection{Mechanisms}\n\\subsubsection{Evolution Mechanism}\nIn RF, data and features are randomly sampled to generate multiple decision trees. Chances are that some of them are of poor quality, which have negative effect on the voting process in our proposed HEM method. To overcome this problem, we draw on the idea of genetic programming \\cite{banzhaf1998genetic} and propose to use a fitness formula to eliminate those decision trees with lower scores before determining easy and hard examples from the training data. In our implementation, we use the average evaluation metric score across all leaf nodes as the fitness score of each decision tree. The specific procedure is as follows.\n\\begin{enumerate}\n\\item At each iteration, calculate the fitness score of each decision tree in a random forest after training.\n\\item Set an elimination ratio (20\\% by default) and calculate the threshold, and then eliminate those decision trees whose score is lower than the threshold.\n\\item Determine the easy and hard examples by voting among the rest of the decision trees and remove them to generate a new training set for the next iteration.\n\\end{enumerate}\n\n\\subsubsection{Smart Iteration Mechanism}\nAt each iteration during training, the division of $D^{d}$ as $\\{D^{d}_{e}, D^{d}_{h}\\}$, might not be ideal in terms of validation accuracy. We propose to use the prediction accuracy on the training data $D^{d}$ as the validation accuracy (which can be generated using k-fold cross-validation as mentioned in Stacking \\cite{wolpert1992stacked, breiman1996stacked}), to judge the quality of the division and annul the division or terminate training if rules are triggered. We propose two smart iteration rules as follows.\n\\begin{enumerate}\n\\item If the validation accuracy decreases for \\textit{N} consecutive times (five by default), apply early termination.\n\\item If the validation accuracy of easy examples $D^{e}$ is lower than that of the whole training data $D^{d}$, render this division invalid and continue to train the model on $D^{d}$ in the next iteration.\t\n\\end{enumerate}\n\n\n\\section{Experiments}\nTo evaluate DBRF, we compared its performance on public datasets against several popular or related methods, and performed ablation test to determine the effects of the three proposed mechanisms for enhancement. We further applied visualization to demonstrate the effectiveness of DBRF. \n\nIn these experiments, we used the default parameters defined in our model and did not fine-tune them. The default number of decision trees in a random forest is 200; the default splitting criterion in a decision tree is Gini impurity; the default number of iterations is 10; the default quality evaluation criterion for leaf nodes is \\textit{score-f1}; the other default parameters have already been mentioned in the Section 3.\n\n\\subsection{Comparison with Other Approaches}\nWe used three datasets from the UCI Machine Learning Repository\\footnote{\\url{http:\/\/archive.ics.uci.edu\/ml\/index.php}} in the comparison experiments, namely Adult, Letter and Yeast. We evaluated three versions of DBRF, namely DBRF-g, DBRF-e and DBRF-f, which uses \\textit{score-gini}, \\textit{score-entropy}, and \\textit{score-f1} as the quality evaluation criterion of leaf nodes respectively. For comparison, we chose two ensemble algorithms, GBDT \\cite{friedman2001greedy} and RF \\cite{breiman2001random}, two related state-of-the-art approaches, gcForest \\cite{zhou2017deep} and sNDF \\cite{kontschieder2015deep}, and multiplayer perceptron (MLP).\n\nFor fairer comparison, we compiled the open-source code of gcForest\\footnote{\\url{https:\/\/github.com\/kingfengji\/gcForest}} published by Professor Zhou's team and used exactly the same method to split a dataset into training set and test set. The evaluation metric of the experiments is accuracy, as shown in Table 1. Numbers in italics are directly cited from either the gcForest paper \\cite{zhou2017deep} or the sNDF paper \\cite{kontschieder2015deep}. We used PyTorch to reproduce sNDF and recorded the results on both Adult and Yeast datasets. We used scikit-learn\\footnote{\\url{http:\/\/scikit-learn.org\/stable\/}} library to evaluate GBDT.\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Comparison of test accuracy on UCI datasets.}\\label{t1}\n\\begin{tabular}{lccc}\n \\toprule\n & Adult & Letter & Yeast \\\\\n \\midrule\n sNDF & 85.58\\% & \\textit{97.08\\%} & 60.31\\% \\\\\n gcForest & \\textit{86.40\\%} & 97.12\\% & \\textit{63.45\\%}\\\\\n MLP & \\textit{85.25\\%} & \\textit{95.70\\%} & \\textit{55.60\\%} \\\\\n RF & \\textit{85.49\\%} & \\textit{96.50\\%} & \\textit{61.66\\%} \\\\\n GBDT & 86.34\\% & 96.32\\% & 60.98\\% \\\\\n \\midrule\n DBRF-g & 86.57\\% & 97.02\\% & 63.68\\% \\\\\n DBRF-e & 86.56\\% & 97.18\\% & \\textbf{64.13\\%} \\\\\n DBRF-f & \\textbf{86.62\\%} & \\textbf{97.25\\%} & 63.90\\% \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nAs Table \\ref{t1} shows, DBRF performs better than RF in all three datasets, which indicates that DBRF is indeed superior to RF. All DBRF versions achieve state-of-the-art results, proving evidence to the effectiveness of DBRF. Last but not least, the results in the last three rows indicate that \\textit{score-f1} may be the best fit for the quality evaluation criterion of leaf nodes, though not by a large margin over the other two.\n\n\\subsection{Decision Regions Visualization}\nWe visualized the decision boundaries learned by DBRF to verify that our proposed hard example mining method indeed achieves our intended purpose. \n\n\\textbf{To facilitate visualization, we selected the second and third column of the UCI Iris dataset}, and randomly splitted the dataset into 67\\% for training and 33\\% for testing. We conducted a comparison experiment between RF and DBRF using default parameters (for RF we used scikit-learn). RF achieves 98\\% accuracy on the training set but only 94\\% accuracy on the test set, which indicates overfitting. In contrast, though DBRF achieves 94\\% accuracy on the training set, it achieves a surprising 98\\% accuracy on the test set, suggesting that it has learned better decision boundaries. The decision boundaries learned by RF and DBRF are shown in Figure \\ref{f4}, which corroborates the above findings.\n\n\\begin{figure}[htb]\n\\centering\n\\caption{Visualization of decision boundaries. The two pictures on the left represent the RF's decision boundaries of the training set and test set, while the two pictures on the right represent the DBRF's decision boundary of the training set and test set, respectively.}\\label{f4}\n\\includegraphics[width=5in]{figure4}\n\\end{figure}\n\n\\subsection{Imbalanced Data}\nWe can better understand how our proposed hard example mining method makes DBRF more effective from a sampling point of view. Suppose we have a large dataset and the decision boundaries are determined by only a small proportion of the dataset. These are important data that ideally need to be preserved for training. However, traditional sampling methods such as random sampling used in RF have no clue whether a data point is important or not. Since they account for only a small amount, chances are that after sampling there are simply not enough important data left for a classifier to learn from. This is how a model may fail to learn the boundaries in spite of seemingly abundant data and features. Removing easy examples makes the proportion of important data higher, and thus makes it more likely that there will be sufficient amount of important data left after sampling. To the best of our knowledge, our proposed hard example mining method is a new way of sampling.\n\nThis is extremely useful when dealing with imbalanced datasets. We applied DBRF to Kaggle's classic imbalanced datasets, Credit Card Fraud Dataset\\footnote{\\url{https:\/\/www.kaggle.com\/mlg-ulb\/creditcardfraud}} to demonstrate this point. \n\nWe randomly selected 65\\% of the dataset as training data and 35\\% as test data. Figure \\ref{f5} shows how the proportion of positive and negative examples in the training data shifted after each iteration. We can see that in the initial training set (on the left of the x-axis), the data are extremely imbalanced with a negative-to-positive ratio of 565.38:1. As the iteration goes on, more data are removed from the training set and at the end (on the right of the x-axis), data are more balanced in the remaining hard examples for training with a negative-to-positive ratio of 3.62:1. \n\n\\begin{figure}[htb]\n\\centering\n\\caption{Stacked training data distribution in each iteration. The blue area (neg-pass) and the yellow area (pos-pass) at the bottom represent the proportions of negative and positive examples that were removed after each iteration. The red area (pos) and green area (neg) at the top represent the proportions of positive and negative examples that were kept as hard examples after each iteration.}\\label{f5}\n\\includegraphics[width=3in]{figure5}\n\\end{figure}\n\nThe whole process achieves the purpose of hard example mining and sampling at the same time. Essentially, it uses data's importance rather than its label as a guideline for sampling. Without much fine-tuning, DBRF achieves an AUC 97.55\\% on the test set, a very competitive score.\n\n\n\\subsection{Running time}\nThe three UCI datasets we used for evaluation differ in data size (1.5k, 20k, 50k), ranging from small to medium. To further consolidate our claim that the proposed model DBRF is superior to RF and can achieve state-of-the-art result while still being computational efficient, we conducted a series of more experiments on Kaggle's Credit Card Fraud dataset mentioned in section 4.3. It has a considerably larger data size of 280k, and is extremely imbalanced since only about 400 data have positive labels. Thus, we have to use AUC instead of accuracy as the evaluation metric. We also reported standard deviation and computation time where necessary.\n\nWe compared our DBRF model against several closely related methods in terms of AUC and training time. As the original data are given in time order, we performed shuffling before randomly splitting the data into training set and test set with a ratio of 2:1. To be more credible, we ran the experiments on three different data splits, and for each split we ran each method five times and recorded means and standard deviations etc.\n\nFor RF-based models (gcForest, RF and DBRF), we grew 200 trees in a random forest per iteration, considering there are 30 features in the dataset. For iterative models (sNDF, gcForest, DBRF and GBDT), we set iteration times as 10 except for GBDT, for we found that the validation AUC of the first three models didn't change much after roughly 10 iterations. And since after 10 iterations, GBDT still performed not well enough so we increased it to 200, when the validation AUC stabilized. All the training time reported were recorded at the end of maximum iterations. For MLP or deep neural network configurations, we used ReLU for activation function, cross-entropy for loss function, Adam for optimization. Via three-fold cross-validation, we set the MLP with 3 hidden layers, 200 neurons in the first layer, 100 in the second and 10 in the third. The other hyper-parameters were set as their default value.\n\n\\begin{table}[htb]\n\\centering\n\\caption{Comparison of AUC and running time on Credit Card Fraud dataset on three data splits.}\\label{t3}\n\\begin{tabular}{lcccc}\n \\toprule\n & \\multicolumn{2}{c}{AUC(Test)} & \\multicolumn{2}{c}{Time(s)} \\\\\n & Avg. & Std.($\\times 10^{-2}$) & Avg. & Std. \\\\\n \\midrule\nsNDF & 0.9348\/0.9290\/0.9264 & 1.2849\/1.3254\/1.1094 & 712.4\/717\/724.4 & 20.9\/34.0\/24.5 \\\\\ngcForest & 0.9580\/0.9702\/0.9548 & 0.2208\/0.3660\/0.1673 & 161\/152.2\/148.4 & 8.06\/8.11\/4.50 \\\\\nMLP & 0.9172\/0.8423\/0.9112 & 1.9070\/4.4489\/1.9399 & 51.7\/53.8\/62.2 & 9.98\/8.73\/8.89 \\\\\nRF & 0.9508\/0.9598\/0.9478 & 0.3050\/0.5299\/0.2218 & 61.2\/61.6\/61.54 & 1.42\/1.84\/1.43 \\\\\nGBDT & 0.9434\/0.8829\/0.7259 & 0.0009\/0.0032\/0.0063 & 147.4\/141.6\/135.2 & 8.57\/6.71\/3.06 \\\\\nDBRF & \\textbf{0.9762}\/\\textbf{0.9848}\/\\textbf{0.9730} & 0.0293\/0.0242\/0.0117 & 97.4\/81\/77.80 & 5.73\/2.83\/0.75 \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nFrom Table \\ref{t3} one can see that DBRF performs consistently better than all other methods in terms of AUC on test set on all three data splits. Besides, the training time of DBRF is quite competitive. In particular, it is more computationally efficient than sNDF and gcForest.\n\nWe also want to know how the computation time and the size of training data change over each iteration. We recorded computation time and the proportion of easy examples that are removed at each iteration during both training and testing process on the first data split. The results are shown in Table \\ref{t4}. For comparison, RF trains in 58.2s and costs 0.33s in testing on average.\n\n\\begin{table}[htb]\n\\centering\n\\caption{Computation time and the proportion of easy examples of the first three iteration of DBRF on the first data split.}\\label{t4}\n\\begin{tabular}{llllll}\n \\toprule\n & & Iter.1 & Iter.2 & Iter.3 & All\\\\\n \\midrule\n \\multirow{2}{2cm}{Time} & Train & 59.18s & 1.05s & 0.91s & 61.14s \\\\\n & Test & 2.86s & 0.16s & 0.16s & 3.18s \\\\\n \\multirow{2}{2cm}{Pass rate} & Train & 98.08\\% &0.16\\% & 0.12\\% & 98.36\\% \\\\\n & Test & 98.05\\% & 0.15\\% & 0.11\\% & 98.31\\% \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable \\ref{t5} shows that after the first iteration of DBRF, if we train other methods on the remaining training data, their overall test AUC all increased, which indicates the effectiveness of DBRF. Table \\ref{t6} records the test AUC of RF and DBRF after three iterations when the number of trees in a random forest is fixed. It shows that just like neural nets, depth is also beneficial to RF.\n\n\\begin{table}[htb]\n\\centering\n\\caption{Comparison of AUC of other methods trained alone and on top of the first iteration of DBRF.}\\label{t5}\n\\begin{tabular}{llllll}\n \\toprule\n & MLP & RF & GDBT & sNDF & gcForest\\\\\n \\midrule\n alone & 91.72\\% & 95.08\\% & 94.34\\% & 93.48\\% & 95.80\\% \\\\\n +DBRF & 95.98\\% & 97.40\\% & 97.38\\% & 96.69\\% & 97.37\\% \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[htb]\n\\centering\n\\caption{Comparison of AUC between RF and DBRF after three iterations when number of trees per forest is fixed.}\\label{t6}\n\\begin{tabular}{lllllll}\n \\toprule\n & 17 & 50 & 150 & 450 & 1350 & 4050\\\\\n \\midrule\n RF & - & 93.86\\% & 94.58\\% & 96.59\\% & 96.60\\% & 96.67\\% \\\\\n DBRF & 97.21\\% & 97.25\\% & 97.57\\% & 97.62\\% & 97.83\\% & - \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Related Work}\nEnsemble learning \\cite{zhou2012ensemble} is a powerful machine learning technique in which multiple learners are trained to solve the same problem. RF \\cite{breiman2001random}, GBDT \\cite{friedman2001greedy}, XGBoost \\cite{chen2016xgboost} are paradigms of ensemble learning, which are all tree-based learning algorithms. DBRF is a novel ensemble algorithm in that it incorporates boosting into the training process of RF. AdaBoost \\cite{freund1997decision} iteratively adds weight to the wrongly classified examples so as to focus training on these hard examples. In contrast, DBRF removes easy examples in each iteration to train on hard examples, which can effectively avoid the effects of unconcerned data while achieving similar boosting effect.\n\nDeep learning research community is also resorting to the strength of tree models. Kontschieder et al. proposed deep neural decision forests \\cite{kontschieder2015deep}, which is a novel approach that unifies decision trees with the representation learning known from deep convolutional networks, by training them in an end-to-end manner. In contrary, we attempt to make RF deep without training through back propagation and hyper-parameter tuning.\n\nDBRF uses RF as the base learner. The capacity of RF is extended vertically by iteratively mining and training on hard examples. Zhou proposed gcForest \\cite{zhou2017deep, feng2017autoencoder} has a cascade procedure similar to DBRF, but the specific approach is different. GcForest cascades the base learner by passing the output of one level of learners as input to the next level, which is similar to Stacking \\cite{wolpert1992stacked, breiman1996stacked}. DBRF cascades the base learner by using the quality of leaf nodes in one level as a criterion to dynamically evolve the training data to train the next level (model of the next iteration). \n\n\\section{Conclusion}\nIn this paper, to overcome some of the limitations of Random Forest (RF), we incorporate boosting into the training process of RF and propose a novel dynamic boosted ensemble learning method based on random forest (DBRF). Specifically, we propose a criterion to measure the quality of a leaf node in a decision tree and then vote to remove easy examples. By iteratively mining and training on hard examples, we evolve the model to learn decision boundaries better and thus extend RF vertically. We also propose evolution mechanism and smart iteration mechanism to enhance DBRF. Experiments show that DBRF outperforms RF and achieves state-of-the-art results. We also provide an explanation of its effectiveness from a sampling point of view.\n\nWe believe that DBRF is a very practical approach and is particularly useful in learning from imbalanced data. In future work, we plan to extend DBRF to learn from raw feature data such as images and sequences. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzofub b/data_all_eng_slimpj/shuffled/split2/finalzzofub new file mode 100644 index 0000000000000000000000000000000000000000..46b24ebd443b8fa03ecde413a8781aeba8ebd813 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzofub @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nA three-layer actively constrained layer (ACL) beams is an elastic beam consisting of a stiff elastic beam, a piezoelectric beam, and a viscoelastic beam which creates passive damping in the structure. The piezoelectric beam itself an elastic beam with electrodes at its top and bottom surfaces, insulated at the edges (to prevent fringing effects), and connected to an external electric circuit (see Fig. \\ref{ACL}). Piezoelectric structures are widely used in in civil, aeronautic and space space structures due to their small size and high power density. They convert mechanical energy to electric energy, and vice versa. ACL composites involve a piezoelectric layer and therefore utilize the benefits of it. Modeling these composites requires better understanding of the modeling of piezoelectric layer since the they are generally actuated through the piezoelectric layer. \n\nThere are mainly three ways to electrically actuate piezoelectric materials: voltage, current or charge. Piezoelectric materials have been traditionally activated by a voltage source \\cite{Baz,Cao-Chen,Rogacheva,Smith,Stanway}, and the references therein. However, it has been observed that the type of actuation changes the controllability characteristics of the host structure \\cite{O-M3}.\nCharge or current controlled piezoelectric beams show \\%85 less hysteresis (electrical nonlinearity) than the voltage actuated ones, i.e. see \\cite{Cao-Chen,Review,Main1,Main2,M-F}, and the references therein. In the case of voltage and charge actuation, the underlying control operator is unbounded in the energy space whereas in the case of current actuation (including magnetic effects) the control operator is bounded \\cite{O-M3}.\n\n\nAccurately modeling an ACL beam also requires understanding of how sandwich structures are modeled and how the interaction of the elastic layers are established. A three-layer sandwich beam consists of stiff outer layers and a viscoelastic core layer. The core layer is supposed to deform only in transverse shear. The bending is uniform for the whole composite. Many sandwich beam models have been proposed in the literature, i.e., see \\cite{Ditaranto,Mead,Rao,Sun,Trindade,Yan}, and the references therein. These models mostly differ by the assumptions on the viscoelastic layer. For example, the Mead-Marcus type models disregard the effects of longitudinal and rotational inertias \\cite{Mead}, and the Rao-Nakra type models preserve these effects since it is noted that these inertia effects are expected to have considerable importance especially at the high frequency modes for sandwich beams \\cite{Rao}.\n\n\n The models for the ACL beams proposed in the literature mostly use the sandwich beam assumptions for the interactions of the layers, i.e. \\cite{Stanway,Trindade}, and references therein. The massive majority of these models are actuated by a voltage source \\cite{Baz,F}. Moreover, the longitudinal vibrations were not taken into account. Only the bending of the composite is studied.\n\n In this paper, we use the Rao-Nakra thin-complaint layer approximation \\cite{Hansen3} for which the longitudinal and rotational inertia terms are kept. We obtain two different models, one for the charge actuated ACL beam, another one for the current actuated ACL beam. We assume the electrostatic approach together with the quadratic-through thickness electric potential so that the induced potential effect is taken into account.\n We show that the proposed model with charge actuation can be written in semigroup formulation (state-space formulation), and it is well-posed in the natural energy space. The biggest advantage of our models is being able to study controllability and stabilizability problems for ACL beams in the infinite dimensional setting. We show that the charge actuated model can be uniformly exponentially stabilized by choosing appropriate feedback controllers which are all mechanical and collocated in this paper due to the electrostatic assumption. This type of feedback controllers avoids the well-known \\emph{spillover} effect. Our stability result relies on the compact perturbation argument and a unique continuation result for the spectral problem by the multipliers technique. A similar argument was used earlier in \\cite{O-Hansen3,O-Hansen4} for different boundary conditions.\n \n\n \\begin{figure}\n\\begin{center}\n\\includegraphics[height=5.5cm]{ACL-curr-char.png} \n\\caption{For a current or charge-actuated ACL beam, when $i_s(t)$ or $\\sigma_s(t)$ is supplied to the electrodes of the piezoelectric layer, an electric field is created between the electrodes, and therefore the piezoelectric beam either shrinks or extends, and this causes the whole composite stretch and bend.} \n\\label{ACL} \n\\end{center} \n\\end{figure}\n\n\n\\section{Modeling Active-Constrained Layer\n (ACL) beams} \\label{modeling}\n\\noindent \\textbf{I- Charge or current-actuation:}\n\n\nThe ACL beam is a composite consisting of three layers that occupy the\nregion $\\Omega=\\Omega_{xy}\\times (0, h)=[0,L]\\times [-b,b] \\times (0,h)$ at equilibrium where $\\Omega_{xy}$ is a smooth bounded domain in the plane.\nThe total thickness $h$ is assumed to be small in comparison to the dimensions of $\\Omega_{xy}$.\nThe beam consists of a stiff\u009d layer, a compliant\u009d layer, and a piezoelectric layer, see Figure \\ref{ACL}.\n\nLet $0=z_00.$\n\nThe well-posedness of the model (\\ref{d4-non})-(\\ref{d-son-non}) with (\\ref{feedback}) is shown in \\cite{Ozkan3}. For that reason, we only analyze the well-posedness of (\\ref{reduced-nog})-(\\ref{reduced-BC-nog}) with (\\ref{feedback}).\n\n\nDefine the complex linear spaces\n\\begin{eqnarray}\n\\nonumber \\mathrm V&=&\\left(H^1_L(0,L)\\right)^2 \\times H^2_L(0,L),\\quad \\mathrm H= {\\mathbb X}^2 \\times H^1_L(0,L),\\quad \\mathcal{H} = \\mathrm V \\times \\mathrm H.\n\\end{eqnarray}\n\nThe natural energy associated with (\\ref{d4-non})-(\\ref{d-son-non}) is\n\\begin{eqnarray}\n\\nonumber &&\\mathrm{E}(t) =\\frac{1}{2}\\int_0^L \\left\\{\\rho_1 h_1 |\\dot v^1|^2 + \\rho_3 h_3 |\\dot v^3|^2 + m |\\dot w|^2 + \\alpha^1 h_1 |v^1_x|^2 + \\alpha^3 h_3 |v^3_x|^2 \\right. \\\\\n\\label{Energy-nat-non} &&\\left. + \\frac{{\\gamma^2 h_3}}{{{\\varepsilon_{3}}}}~(P_{\\xi} v^3_x) (\\bar v^3)_x + K_1 |\\dot w_x|^2 + K_2 |w_{xx}|^2 +G_2 h_2 |\\phi^2|^2 \\right\\}~ dx.\n\\end{eqnarray}\n This motivates definition of the inner product on $\\mathcal H$\n{ \\small{\\begin{eqnarray}\n\\nonumber && \\left<\\left[ \\begin{array}{l}\n u_1 \\\\\n u_2 \\\\\n u_3\\\\\n u_4 \\\\\n u_5 \\\\\n u_6\n \\end{array} \\right], \\left[ \\begin{array}{l}\n v_1 \\\\\n v_2 \\\\\n v_3\\\\\n v_4\\\\\n v_5\\\\\n v_6\n \\end{array} \\right]\\right>_{\\mathcal H}= \\left<\\left[ \\begin{array}{l}\n u_4\\\\\n u_5\\\\\n u_6\n \\end{array} \\right], \\left[ \\begin{array}{l}\n v_4\\\\\n v_5\\\\\n v_6\n \\end{array} \\right]\\right>_{\\mathrm H}+ \\left<\\left[ \\begin{array}{l}\n u_1 \\\\\n u_2 \\\\\n u_3\n \\end{array} \\right], \\left[ \\begin{array}{l}\n v_1 \\\\\n v_2\\\\\n v_3\n \\end{array} \\right]\\right>_{\\mathrm V}\\\\\n\\nonumber && =\\int_0^L \\left\\{\\rho_1 h_1 u_5 {\\dot {\\bar v}}_5 + \\rho_3 h_3 u_6 {\\dot {\\bar v}}_6+ \\mu h_3 \\dot u_7 {\\dot {\\bar v}_7} + m \\dot u_8{\\dot {\\bar v}_8}+ K_1 (u_8)_x(\\bar v_8)_x+ \\alpha^1 h_1 (u_1)_{x} (\\bar v_1)_x \\right. \\\\\n\\nonumber &&\\left. + \\alpha^3 h_3 (u_3)_{x} (\\bar v_3)_x+ \\frac{{\\gamma^2 h_3}}{{{\\varepsilon_{3}}}}~(P_{\\xi} (u_3)_x) (\\bar u_3)_x + K_2 (u_4)_{xx} (\\bar v_4)_{xx} +\\frac{G_2}{h_2} (-u_1+u_2 + H(u_4)_x)(-\\bar v_1+\\bar v_2 + H(\\bar v_4)_x)\\right\\}~dx.\n \\end{eqnarray}}}\n Since $P_\\xi$ is a positive operator and the term $\\|-u_1+u_2 + H(u_4)_x\\|_{L^2(0,L)}$ is coercive, see \\cite{Hansen3} for the details., $\\langle \\, , \\, \\rangle_{\\mathcal H} $ does indeed define an inner product.\n\n Let $\\vec y=(v^1, v^3, w)$ be the smooth solution of the system of (\\ref{d4-non})-(\\ref{d-son-non}) with (\\ref{feedback}). Multiplying the equations in (\\ref{d4-non}) by $\\tilde y_1, \\tilde y_2 \\in H^1_L(0,L)$ and $ \\tilde y_3 \\in H^2_L(0,L),$ respectively, and integrating by parts yields\n\\begin{subequations}\n\\small\n \\begin{empheq}[left={\\phantomword[r]{0}{} }]{align}\n\\nonumber &\\int_0^L \\left(\\rho_1h_1\\ddot v^1 \\tilde y_1 + \\alpha^1 h_1 v^1_{x} (\\tilde y_1)_x - G_2 \\phi^2 \\tilde y_1\\right)dx + s_1\\dot v_1(L) y_1(L)= 0, & \\\\\n \\nonumber &\\int_0^L \\left(\\rho_3h_3\\ddot v^3 \\tilde y_2 +\\alpha^3 h_3 v^3_{x} (\\tilde y_2)_x + \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi v^3_x) (\\tilde y_2)_x+G_2 \\phi^2 (\\tilde y_2)_x \\right)dx + \\frac{\\gamma s_3}{{\\varepsilon_{3}}} \\dot v_3(L) y_2(L) = 0, & \\\\\n \\nonumber & \\int_0^L \\left( m \\ddot w \\tilde y_3 + K_1 \\ddot{w}_{x} (\\tilde y_3)_x + K_2 w_{xx} (\\tilde y_3)_{xx}- G_2 H \\phi^2_x (\\tilde y_3)_x \\right)dx + k_1 \\dot w_x(L) (y_3)_x(L)+ k_2\\dot w(L) y_3(L) =0.&\n\\end{empheq}\n\\end{subequations}\nNow define the linear operators\n\\begin{eqnarray}\n\\nonumber \\left_{\\mathrm V'\\times \\mathrm V}=(y,\\psi)_{\\mathrm V \\times \\mathrm V}, \\quad \\forall y,\\psi\\in \\mathrm V\\\\\n\\nonumber \\left_{\\mathrm H'\\times \\mathrm H}= \\left[ \\begin{array}{c}\n 0_{2\\times 1}\\\\\n k_2y_3(L) \\psi_3(L)\n \\end{array} \\right], \\quad \\forall \\vec y,\\vec \\psi\\in \\mathrm H\\\\\n\\label{ops}\\left_{\\mathrm H'\\times \\mathrm H}= \\left[ \\begin{array}{c}\n s_1y_1(L) \\psi_1(L) \\\\\n \\frac{\\gamma s_3}{{\\varepsilon_{3}}} y_2(L) \\psi_2(L) \\\\\n k_1(y_3)_x(L) (\\psi_3)_x(L)\n \\end{array} \\right], \\quad \\forall \\vec y,\\vec \\psi\\in \\mathrm V.\n\\end{eqnarray}\nLet $\\mathcal M : H^1_L (0,L) \\to (H^1_L(0,L))'$ be a linear operator defined by\n\\begin{eqnarray}\\label{def-M}\\left< \\mathcal M \\psi, \\tilde \\psi \\right>_{(H^1_L(0,L))', H^1_L(0,L)}= \\int_0^L (m \\psi \\tilde {\\bar \\psi} + K_1 \\psi_x \\tilde {\\bar\\psi}_x) dx.\n\\end{eqnarray}\nFrom the Lax-Milgram theorem $\\mathcal M$ and $A$ are canonical isomorphisms from $H^1_L(0,L)$ onto $(H^1_L(0,L))'$ and from $V$ onto $V',$ respectively.\nAssume that $A y \\in V',$ then we can formulate the variational equation above into the following form\n\\begin{eqnarray} M\\ddot y + A y + D_0\\dot y + B_0\\dot y=0\n\\end{eqnarray}\nwhere $M=\\left[\\rho_1 h_1 I ~~\\rho_3 h_3 I~~ \\mathcal M\\right]$ is an isomorphism from $\\mathrm H$ onto $\\mathrm H'.$\nNext we introduce the linear unbounded operator by\n\\begin{equation}\n\\label{defA}\\mathcal A: {\\text{Dom}}(A)\\times V\\subset \\mathcal H \\to \\mathcal H\n\\end{equation}\nwhere $\\mathcal A= \\left[ {\\begin{array}{*{20}c}\n O_{3\\times 3} & -I_{3\\times 3} \\\\\n M^{-1}A & M^{-1} D_0 \\\\\n\\end{array}} \\right]$\nwith $\\nonumber{\\rm {Dom}}(\\mathcal A) = \\{(\\vec z, \\vec {\\tilde z}) \\in V\\times V, A\\vec z \\in \\mathrm V' \\}$\nand if ${\\rm Dom}(\\mathcal A)'$ is the dual of ${\\rm Dom}(\\mathcal A)$ pivoted with respect to $\\mathcal H,$ we define the control operators $B$ and $D$\n \\begin{eqnarray}\n\\label{defb_0} \\quad B \\in \\mathcal{L}(\\mathbb{C} , {\\rm Dom}(\\mathcal A)'), ~ \\text{with} ~ B= \\left[ \\begin{array}{c} 0_{3\\times 1} \\\\ M^{-1}B_0. \\end{array} \\right]\n \\end{eqnarray}\n\nWriting $\\varphi=[v^1, v^3, w, \\dot v^1, \\dot v^3, \\dot w]^{\\rm T},$ the control system (\\ref{d4-non})-(\\ref{d-son-non}) with the feedback controllers (\\ref{feedback}) can be put into the state-space form\n\\begin{eqnarray}\n\\label{Semigroup}\n\\dot \\varphi + \\mathcal A \\varphi + B\\varphi =0, \\quad\\varphi(x,0) = \\varphi ^0.\n\\end{eqnarray}\n\n \\begin{lem} \\label{skew}The operator $\\mathcal{A}$ defined by (\\ref{defA}) is maximal monotone in the energy space $\\mathcal H,$\n and ${\\rm Range}(I+\\mathcal A)=\\mathcal H.$\n\\end{lem}\n\n\\textbf{Proof:} Let $\\vec z \\in {\\rm Dom}(\\mathcal A).$ A simple calculation using integration by parts and the boundary conditions yields\n{\\small{\\begin{eqnarray}\n\\nonumber &&\\left<\\mathcal A \\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right], \\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right]\\right>_{\\mathcal H\\times \\mathcal H} = \\left< \\left[ \\begin{array}{c}\n-\\vec z_2 \\\\\nM^{-1} A \\vec z_1 \\end{array} \\right], \\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right]\\right>_{\\mathcal H}= \\left<-\\vec z_2, \\vec z_1\\right>_{V\\times V} + \\left_{H\\times H}\\\\\n\\nonumber &&\\quad\\quad = -\\overline{\\left_{V'\\times V}} + \\left_{H'\\times H}.\n\\end{eqnarray}\n}}\nSince $\\vec z=\\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right] \\in {\\rm Dom}(\\mathcal A) $, then $A\\vec z _1+ D_0z_2\\in V'$ and $\\vec z_2 \\in V$ so that\n\\begin{eqnarray*}&&\\left_{H'\\times H}=\\left_{V'\\times V} =\\left_{V'\\times V}+\\left< D_0z_2, \\vec z_2\\right>_{V'\\times V}.\n\\end{eqnarray*}\nHence\n${\\rm Re}\\left<\\mathcal A \\vec z, \\vec z\\right>_{\\mathcal H\\times \\mathcal H}=\\left< D_0z_2, \\vec z_2\\right>_{V'\\times V}\\ge 0.$\nWe next verify the range condition. Let $\\vec z=\\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right]\\in \\mathcal H.$ We prove that there exists a $\\vec y=\\left[ \\begin{array}{c}\n\\vec y_1 \\\\\n\\vec y_2 \\end{array} \\right]\\in \\mathcal {\\rm Dom} (\\mathcal A )$ such that $(I+\\mathcal A) \\vec y=\\vec z.$\nA simple computation shows that proving this is equivalent to proving ${\\rm Range} (M+A + D_0)=H',$ i.e., for every $\\vec f \\in H'$ there exists a unique solution $\\vec z \\in H$ such that $(M+A+D_0)\\vec z=\\vec f.$\nThis obviously follows from the Lax Milgram's theorem. $\\square$\n\n\\begin{prop} The operator $B$ is a monotone compact operator on $\\mathrm H.$\n\\end{prop}\n\n\\textbf{Proof:} Let $\\left[ \\begin{array}{c}\n\\vec y\\\\\n\\vec z \\end{array} \\right]\\in \\mathrm H.$ Then $\\left_{\\mathcal H}= k_2 |z_3(L)|^2.$\nThe compactness follows from the fact that $M^{-1}$ is a canonical isomorphism from $\\mathrm H$ to $\\mathrm H',$ and the fact that $B$ is a rank-one operator, hence compact from $\\mathrm H$ to $\\mathrm H'.$ $\\square$\n\n\\subsection{Description of ${\\rm Dom}(\\mathcal A)$}\n\n\n\\begin{prop}\\label{prop-dom}Let $\\vec u=(\\vec y, \\vec z)^{\\rm T}\\in \\mathcal H.$ Then $\\vec u \\in {\\rm Dom}(\\mathcal A)$ if and only if the following conditions hold:\n\\begin{eqnarray}\n\\nonumber && \\vec y\\in (H^2(0,L)\\cap H^1_L(0,L))^2 \\times (H^3(0,L) \\cap H^2_L(0,L)),\\quad \\vec z\\in V ~{\\rm such ~ that}~(y_1)_{x}=(y_2)_{x}=(y_4)_{xx}\\left. \\right|_{x=L}=0.\n\\end{eqnarray}\nMoreover, the resolvent of $\\mathcal A$ is compact in the energy space $\\mathcal H.$\n\\end{prop}\n\\vspace{0.1in}\n\n\\textbf{Proof:} Let $\\vec {\\tilde u}= \\left( \\begin{array}{c}\n\\vec {\\tilde y} \\\\\n\\vec {\\tilde z}\\\\\n \\end{array} \\right)\\in \\mathcal H$ and $\\vec {u}= \\left( \\begin{array}{c}\n\\vec {y} \\\\\n\\vec {z}\\\\\n \\end{array} \\right)\\in {\\rm Dom}(\\mathcal A)$ such that $\\mathcal A \\vec u=\\vec {\\tilde u}.$ Then we have\n $$ -\\vec z= \\vec {\\tilde y} \\in V, \\quad A\\vec y + D_0 z=M\\vec{\\tilde z},$$ and therefore,\n\\begin{eqnarray}\\label{sal}\\left< \\vec y, \\vec \\varphi\\right>_{V}= \\left< \\vec {\\tilde z}, \\vec \\varphi \\right>_{H} ~~{\\rm for ~~all ~~} \\vec \\varphi \\in V.\n\\end{eqnarray}\nLet $\\vec \\psi=[\\psi_1,\\psi_2,\\psi_3]^{\\rm T} \\in (C_0^\\infty(0,L))^4.$ We define $\\varphi_i=\\psi_i$ for $i=1,2,$ and $\\varphi_3=\\int_0^x \\psi_3 (s)ds.$ Since $\\vec \\varphi\\in V,$ inserting $\\vec \\varphi$ into the above equation yields\n\\begin{eqnarray} \\nonumber &&\\int_0^L \\left\\{ -\\alpha^1 h_1 (y_1)_{xx} \\bar \\psi_1 - \\left(\\alpha^3 h_3 (y_2)_{xx}+\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi (y_2)_x)_x\\right) \\bar \\psi_2- K_2 (y_3)_{xxx} \\bar \\psi_3 \\right\\}~dx\\\\\n\\nonumber && + \\frac{G_2}{h_2} (-y_1+y_2 + H(y_3)_x)(-\\bar \\psi_1+\\bar \\psi_2 + H(\\bar \\psi_3)_x)+s_1 (z_1)_x(L)(\\psi_1)_x(L) + \\frac{\\gamma s_3}{{\\varepsilon_{3}}} z_2(L)\\psi_2(L)+ k_1(z_3)_{x}(L) \\psi_3(L) \\\\\n \\nonumber &&= \\int_0^L \\left\\{ \\left(\\int_1^x m \\tilde z_4 ds + K_1 (\\tilde z_4)_x\\right) \\bar \\psi_4 +\\rho_1 h_1 \\tilde z_1 \\bar \\psi_1+ \\rho_3 h_3 \\tilde z_2 \\bar \\psi_2 + \\mu h_3 \\tilde z_3 \\bar \\psi_3 \\right\\}~dx\\\\\n \n \\end{eqnarray}\nfor all $\\vec \\psi\\in (C_0^\\infty(0,L))^3.$ Therefore it follows that $\\vec y\\in (H^2(0,L)\\cap H^1_L(0,L))^2 \\times (H^3(0,L) \\cap H^2_L(0,L)).$\n\nNext let $\\vec \\psi \\in \\mathrm H.$ We define\n\\begin{eqnarray} \\label{dumber}\\varphi_i=\\int_0^x \\psi_i(s)ds,\\quad i=1,\\ldots,3.\n\\end{eqnarray}\nObviously $\\vec \\varphi \\in \\rm V. $ Then plugging (\\ref{dumber}) into (\\ref{sal}) yields\n\\begin{eqnarray}&& \\nonumber 0=(\\alpha^1 h_1 (y_1)_{x}(L) + s_1 z_1(L)) \\bar \\psi_1(L)+\\alpha^3 h_3 (y_2)_{x}(L)+ \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} P_\\xi (y_2)_x(L) \\\\\n\\nonumber &&+ \\frac{\\gamma s_3}{{\\varepsilon_{3}}} z_2(L) \\bar \\psi_2(L)+ (k_1 (y_3)_{xx}(L) + k_1 (z_3)_x(L)) (\\bar \\psi_3)_x(L)\n\\end{eqnarray}\nfor all $\\psi\\in \\mathrm H.$ Hence,\n\\begin{eqnarray}\\nonumber &&\\alpha^1 h_1 (y_1)_{x}(L) + s_1 z_1(L)=\\alpha^3 h_3 (y_2)_{x}(L) + \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} P_\\xi (y_2)_x(L)+ \\frac{\\gamma s_3}{{\\varepsilon_{3}}} z_2(L)= k_1 (y_3)_{xx}(L) + k_1 (z_3)_x(L)=0.\n\\end{eqnarray}\n\nNow let $\\vec y=\\left[ \\begin{array}{c}\n\\vec y_1 \\\\\n\\vec y_2 \\end{array} \\right]\\in \\mathcal {\\rm Dom} (\\mathcal A )$ and $\\vec z=\\left[ \\begin{array}{c}\n\\vec z_1 \\\\\n\\vec z_2 \\end{array} \\right] $such that $(I+\\mathcal A) \\vec y=\\vec z.$ By Proposition \\ref{prop-dom} and Lemma \\ref{skew}, the compactness of the resolvent follows. $\\square$\n\n\n\n\\begin{lem} \\label{xyz} The eigenvalue problem\n\\begin{eqnarray}\n \\label{dbas-eig} &&\\left\\{\n \\begin{array}{ll}\n \\alpha^1 h_1 z^1_{xx} - G_2 \\phi^2 = \\lambda^2 \\rho_1 h_1 z^1,& \\\\\n \\alpha^3 h_3 z^3_{xx} + \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi(z_3)_x)_x + G_2 \\phi^2 = \\lambda^2 \\rho_3 h_3 z^3 & \\\\\n - K_2 u_{xxxx} + G_2 H \\phi^2_x=\\lambda^2 (m u - K_1 u_{xx}),&\n\n \\end{array} \\right.\n\\end{eqnarray}\nwith the overdetermined boundary conditions\n\\begin{eqnarray}\n \\nonumber && u(0)=u_x(0)=z^1(0)=z^3(0)=z^1(L)=z^1_x (L)=z^3(L)=z^3_x(L)=0,\\\\\n \\label{d-son-eig} && u(L)=u_x(L)=u_{xx}(L)=u_{xxx}(L)=0\n\\end{eqnarray}\nhas only the trivial solution.\n\\end{lem}\n\\vspace{0.1in}\n\n\\textbf{Proof:} Now multiply the equations in (\\ref{dbas-eig}) by $-x\\bar u_x+3\\bar u,$ $ -x\\bar z^3_x +2\\bar z^3,$ and $-x \\bar z^1_x+2\\bar z^1,$ respectively, integrate by parts on $(0,L),$\n\n\\begin{eqnarray}\n\\nonumber && 0=\\int_0^L\\left\\{ -\\alpha^1 h_1|z^1_x|^2 +(x\\bar z^3_{xx} -\\bar z^3_x) \\left(\\alpha^3 h_3 z^3_x +\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} P_\\xi z^3_x\\right) -3\\rho_1 h_1 \\lambda^2|z^1|^2 -3\\rho_3 h_3 \\lambda^2 |z^3|^2 -4m \\lambda^2 |u|^2 \\right. \\\\\n \\nonumber && - 2 K_1 \\lambda^2|u_x|^2 - G_2 h_2 \\phi^2 (x \\bar \\phi^2_x)-3G_2 h_2|\\phi^2|^2-K_2 \\bar u_{xxxx} (x u_x) + \\alpha^1_1 h_1 \\bar z^1_{xx} (x z^1_x) -\\rho_1h_1 \\lambda^2\\bar z^1 (xz^1_x)\\\\\n\\label{CH3-mult1-20} && \\left. -\\rho_3h_3 \\lambda^2 \\bar z^3 (x z^3_x) -\\lambda^2 (m\\bar u-K_1 u_{xx})(x u_x)\\right)~dx\\quad\\quad\n \\end{eqnarray}\nNow consider the conjugate eigenvalue problem corresponding to (\\ref{dbas-eig})-(\\ref{d-son-eig}). Now multiply the equations in the conjugate problem by $-x u_x-2 u,$ $-x z^1_x-3 z^1,$ and $ -x z^3_x-3 z^3,$ respectively, integrate by parts on $(0,L),$ and add them up:\n\\begin{eqnarray}\n\\nonumber && 0=\\int_0^L\\left\\{ 3\\alpha^1 h_1 |z^1_x|^2 -x z^3_{x} \\left(\\alpha^3 h_3 \\bar z^3_{xx} +\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi \\bar z^3_x)_x\\right) +3 z^3_{x} \\left(\\alpha^3 h_3 \\bar z^3_{x} +\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} P_\\xi \\bar z^3_x\\right) \\right. \\\\\n \\nonumber && +3\\bar\\lambda^2\\rho_1 h_1 |z^1|^2 +3\\rho_3 h_3 \\bar \\lambda^2|z^3|^2 +2 m \\bar \\lambda^2 |u|^2 + 2 K_1 \\bar \\lambda^2|u_x|^2 +2K_2 |u_{xx}|^2 + G_2 h_2 \\bar \\phi^2 (x \\phi^2_x)+3G_2 h_2|\\phi^2|^2\\\\\n\\label{CH3-mult1-21} && \\left.+K_2 \\bar u_{xxxx} (x u_x) - \\alpha^1_1 h_1 \\bar z^1_{xx} (x z^1_x) +\\rho_1h_1 \\bar \\lambda^2\\bar z^1 (xz^1_x)+\\rho_3h_3 \\bar \\lambda^2 \\bar z^3 (x z^3_x) +\\bar \\lambda^2 (m\\bar u-K_1 u_{xx})(x u_x)\\right)~dx\\quad\\quad\n \\end{eqnarray}\n\nAdding (\\ref{CH3-mult1-20}) and (\\ref{CH3-mult1-21}) yields,\n\\begin{eqnarray}\n\\nonumber && 0=\\int_0^L\\left\\{ 2\\alpha^1 h_1 |z^1_x|^2 +2\\alpha^3 h_3 |z^3_x|^2 + \\frac{2\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi z^3_x\\cdot \\bar z^3_x) +\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} x\\left( -z^3_{x} (P_\\xi \\bar z^3_x)_x +\\bar z^3_{xx} P_\\xi \\bar z^3_x\\right) \\right. \\\\\n \\nonumber && +3\\bar\\rho_1 h_1 (-\\lambda^2 + \\bar \\lambda^2) |z^1|^2 +3\\rho_3 h_3 \\bar (-\\lambda^2 + \\bar \\lambda^2)|z^3|^2 + (-4m \\lambda^2 + 2 m \\bar \\lambda^2) |u|^2 + 2K_1(- \\lambda^2 + \\bar \\lambda^2)|u_x|^2 \\\\\n\\label{CH3-mult1-22} && \\left.+2K_2 |u_{xx}|^2 + G_2 h_2 \\left(- \\phi^2 (x \\bar \\phi^2_x) + \\bar \\phi^2 (x \\phi^2_x)\\right)+ (-\\lambda^2+\\bar \\lambda^2) (m\\bar u-K_1 u_{xx})(x u_x)\\right)~dx\\quad\\quad\n \\end{eqnarray}\n\nFinally, adding (\\ref{CH3-mult1-20}) and (\\ref{CH3-mult1-21}),considering only the real part of the expression above and all eigenvalues are\nlocated on the imaginary axis, i.e. $\\lambda=\\mp \\imath \\nu,$ yields\n\\begin{eqnarray}\n\\nonumber && \\int_0^L\\left( K_2 |u_{xx}|^2+ m \\nu^2|u|^2+ \\alpha^1 h_1 |z^1_x|^2 +\\left(\\alpha^3 h_3 z^3_x + \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi z^3_x)\\right)\\cdot \\bar z^3_x \\right.\\\\\n\\label{damn1}&& \\quad \\left.+\\frac{\\gamma^2 h_3}{2{\\varepsilon_{3}}} x\\left( -z^3_{x} (P_\\xi \\bar z^3_x)_x +\\bar z^3_{xx} P_\\xi \\bar z^3_x\\right) \\right)~dx=0\n \\end{eqnarray}\n\nNow we should get rid of the last term which survived through the calculations. Let $P_\\xi z^3_x=\\eta.$ Then $\\eta-\\xi \\eta_{xx}=z^3_x,$ and\n\\begin{eqnarray}\n\\nonumber &&\\frac{\\gamma^2 h_3}{2{\\varepsilon_{3}}} \\int_0^L x\\left[ -z^3_{x} (P_\\xi \\bar z^3_x)_x +\\bar z^3_{xx} P_\\xi \\bar z^3_x\\right]~dx = \\frac{\\gamma^2 h_3}{2{\\varepsilon_{3}}} \\int_0^L x\\left[ (xz^3_{xx}+z^3_x)(P_\\xi \\bar z^3_x) +\\bar z^3_{xx} P_\\xi \\bar z^3_x\\right]~dx\\\\\n\\nonumber && \\quad = \\frac{\\gamma^2 h_3}{2{\\varepsilon_{3}}} \\int_0^L \\left[ (x(\\eta_x-\\xi \\eta_{xxx})+(\\eta-\\xi\\eta_{xx}))\\bar \\eta +x(\\bar\\eta_x-\\xi\\bar \\eta_{xxx}) \\eta\\right]~dx\\\\\n\\nonumber && \\quad = \\frac{\\gamma^2 h_3}{2{\\varepsilon_{3}}} \\int_0^L \\left[ x (\\eta\\cdot \\bar\\eta)_x + |\\eta|^2 + \\xi |\\eta_x|^2 + \\xi \\left(\\eta_{xx} (\\bar\\eta+x\\bar\\eta_x) + \\bar\\eta_{xx}(\\eta+\\eta_{xx})\\right)\\right]~dx\\\\\n\\label{damn2} && \\quad = -\\frac{\\gamma^2 h_3\\xi}{2{\\varepsilon_{3}}}\\int_0^L |\\eta_x|^2 dx,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{damn3} \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} \\int_0^L (P_\\xi z^3_x) \\cdot \\bar z^3_x ~dx&=& \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} \\int_0^L \\eta (\\bar \\eta -\\xi \\bar \\eta_{xx}) ~dx=\\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} \\int_0^L (|\\eta|^2 + \\xi |\\eta_x|^2) ~dx.\n \\end{eqnarray}\nPlugging (\\ref{damn2}) and (\\ref{damn3}) in (\\ref{damn1}) gives\n\\begin{eqnarray}\n\\nonumber && \\int_0^L\\left( K_2 |u_{xx}|^2+ m \\nu^2|u|^2+ \\alpha^1 h_1 |z^1_x|^2 +\\left(\\alpha^3 h_3 z^3_x + \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (P_\\xi z^3_x)\\right)\\cdot \\bar z^3_x \\right.\\\\\n\\label{damn4} && \\quad \\left.+ \\frac{\\gamma^2 h_3}{{\\varepsilon_{3}}} (|P_\\xi z^3_x|^2 + \\frac{\\xi}{2} |(P_\\xi z^3_x)_x|^2) \\right)~dx=0.\n \\end{eqnarray}\nThis together with (\\ref{d-son-eig}) imply that $u=z^1=z^3\\equiv 0$ by (\\ref{d-son-eig}). This also covers the case $\\lambda\\equiv 0. \\square$\n\n\n\n\nNow we consider the decomposition $\\mathcal{A}+B=(\\mathcal{A}_d+B)+ \\mathcal {A}_{\\phi}$ of the semigroup generator of the original problem (\\ref{defA}) where $\\mathcal{A}_d+B$ is the semigroup generator of the decoupled system, i.e. $\\phi^2\\equiv 0$ in (\\ref{d4-non})-(\\ref{d-son-non}),\n\\begin{eqnarray}\n \\label{dbas-dcpld} &&\\left\\{\n \\begin{array}{ll}\n m \\ddot w - K_1 \\ddot{w}_{xx}+K_2w_{xxxx} =0,&\\\\\n \\rho_1h_1 \\ddot v^1 -\\alpha^1 h_1 v^1_{xx} =0, & \\\\\n \\rho_3h_3 \\ddot v^3 -\\alpha^3 h_3 v^3_{xx} = 0, &\n \\end{array} \\right.\n\\end{eqnarray}\nwith the boundary and initial conditions\n \\begin{eqnarray}\n&& \\begin{array}{ll}\n\\nonumber \\left\\{v^1, v^3, w, w_x \\right\\}_{x=0}=0, ~ \\left\\{\\alpha^1 h_1v^1_x\\right\\}_{x=L}=-s_1 \\dot v^1(L),~~ \\left\\{ \\alpha^3 h_3 v^3_x \\right\\}_{x=L}=-s_3 \\dot v^3(L,t),&\\\\\n \\end{array} \\\\\n && \\begin{array}{ll}\n\\nonumber \\left\\{K_2 w_{xx} \\right\\}_{x=L} =-k_1\\dot w_x(L,t) , ~ \\left\\{K_1 \\ddot w_x -K_2 w_{xxx} \\right\\}_{x=L} =k_2 \\dot w(L,t),& t\\in \\mathbb{R}^+ \\\\\n \\end{array} \\\\\n&& \\begin{array}{ll}\n\\label{d-son-dcpld} (v^1,v^3, w, \\dot v^1, \\dot v^3, \\dot w)(x,0) =(v^1_0, v^3_0, w_0, v^1_1, v^3_1, w_1),& x \\in [0,L].\n \\end{array}\n\\end{eqnarray}\nThe operator $\\mathcal{A}_{\\phi}:\\mathcal{H}\\to\\mathcal{H}$ is the coupling between the layers defined as the following\n\\begin{eqnarray}\\label{opB}\\mathcal {A}_{\\phi}{\\bf y}=\n \\left( \\begin{array}{c}\n 0_{3\\times 1} \\\\\n \\mathcal M^{-1} \\left(H G_2 ~\\phi^2_x\\right) \\\\\n\\frac{G_2}{h_1 \\rho_1}\\phi^2\\\\\n\\frac{{\\gamma^2 h_3}}{{{\\varepsilon_{3}}}}~(P_{\\xi} v^3_x)_x-\\frac{G_2}{h_3 \\rho_3}\\phi^2\\\\\n \\end{array} \\right).\n \\end{eqnarray}\n where ${\\bf y}=(w,u^1, u^3, \\tilde w, \\tilde v^1, \\tilde v^3)$ and $\\phi^2=\\frac{1}{h_2}\\left(-u^1+u^3 + H u_x\\right).$\n Let $E_d(t)$ be natural energy corresponding to the system (\\ref{dbas-dcpld})-(\\ref{d-son-dcpld}), i.e. (\\ref{Energy-nat-non}) without the $P_\\xi$ and $\\phi^2$ terms.\n\n\n \\begin{thm} Let ${\\mathcal A}_d+B$ be the infinitesimal generator of the semigroup corresponding to the solutions of (\\ref{dbas-dcpld})-(\\ref{d-son-dcpld}). Then the semigroup $\\{e^{({\\mathcal A}_d+B) t}\\}_{t\\ge 0}$ is exponentially\nstable in $\\mathcal H$.\n \\end{thm}\n\n \\textbf{Proof:} Note that the equations in (\\ref{dbas-dcpld}) are completely decoupled. The exponential stability of the semigroup $e^{({\\mathcal A}_d+B)t}$ follows from the exponential stability of wave equations \\cite{O-Hansen4} and the Rayleigh beam equation \\cite{B-Rao}.\n\n\n\\begin{lem} \\label{compact} The operator $A_{\\phi}:\\mathcal{H}\\to \\mathcal{H}$ defined in (\\ref{opB}) is compact.\n\\end{lem}\n\n\\vspace{0.1in}\n{\\bf Proof:} When $(w,v^1, v^3, \\tilde w, \\tilde u^1, \\tilde u^3)\\in \\mathrm H,$ we have $w\\in H^2_L(0,L)$ and $v^1, v^3 \\in (H^1_L(0,L))^{2},$ and therefore $\\phi^2 \\in H^1_L(0,L).$ Since $\\mathcal M: H^2_L(0,L)\\to \\mathrm L^2(0,L)$ remains an isomorphism, the terms in (\\ref{opB}) satisfy\n\\begin{eqnarray}\n\\label{dumb5} && \\mathcal M^{-1} \\left(\\phi^2_x\\right) \\in H^2_L(0,L), ~~\\phi^2 \\in H^1_L(0,L),~~ (P_\\xi v^3_x)_x \\in H^1_0(0,L)\n \\end{eqnarray}\nand $H^2_L(0,L)\\times H^1_L(0,L)$ is compactly embedded in $H^1_L(0,L)\\times (\\mathrm L^2(0,L))^{2},$ and $H^1_0(0,L)$ is compactly embedded in $L^2(0,L).$ Hence the operator $A_{\\phi}$ is compact in $\\mathcal H$.\n\n \\begin{thm} \\label{main-thm} Then the semigroup $\\{e^{({\\mathcal A}+B) t}\\}_{t\\ge 0}$ is exponentially\nstable in $\\mathcal H.$\n \\end{thm}\n\n \\textbf{Proof:} The semigroup $\\mathcal A+B={\\mathcal A}_d +B + {\\mathcal A}_{\\phi}$ is strongly stable on $\\mathcal H$ by Lemma \\ref{xyz}, and the operator ${\\mathcal A}_{\\phi}$ is a compact in $\\mathcal H$ by\nLemma \\ref{compact}. Therefore, since the semigroup generated by $({\\mathcal A}_d +B + {\\mathcal A}_{\\phi})-{\\mathcal A}_{\\phi}$ is uniformly\nexponentially stable in $\\mathcal H,$ the semigroup $\\mathcal A=({\\mathcal A}_d + B+ {\\mathcal A}_{\\phi})$ is uniformly\nexponentially stable in ${\\mathcal H}$ by e.g., the perturbation theorem of \\cite{Trigg}.\n\nThe main difference between the voltage and charge controlled models is the existence of the $P_\\xi$ term in the charge controlled model. By using the same argument above (by taking $P_\\xi\\equiv 0$), the voltage controlled model can be easily shown to be exponentially stable with the same feedback controller (\\ref{feedback}).\n\n\\begin{thm} Then the semigroup $\\{e^{({\\mathcal A}+B) t}\\}_{t\\ge 0}$ corresponding to (\\ref{d4-non}) ($P_\\xi\\equiv 0$) with (\\ref{feedback}) is exponentially\nstable in $\\mathcal H.$\n \\end{thm}\n\n Several remarks are in order:\n\n\\begin{rmk} (i) The number of controllers for the bending motion may be reduced to one by taking $k_2=0.$ However, it is worthwhile to note that the multiplier used in \\cite{Rao} does not work in the proof of Lemma \\ref{xyz}. For that reason, the classical four boundary conditions for $u$ at $x=L$ in (\\ref{d-son-eig}) are still needed due to the existence of the coupling $\\phi^2$ in (\\ref{dbas-eig}). Removing $u(L)=0,$ and proving the same result in Lemma \\ref{xyz} is an open question.\n\n\\noindent (ii) If the moment of inertia term in (\\ref{d4-non})-(\\ref{d-son-non}) is set to zero, $K_1\\equiv 0,$ the bending motion is described by the Euler-Bernoulli-type equation. The exponential stabilizability (\\ref{d4-non})-(\\ref{d-son-non}) with $s_2\\equiv 0,$ i.e. one controller for each equation, is still an open problem.\n\\end{rmk}\n\\section{Generalization to the Multilayer ACL beam case}\nBy using the same methodology in Section \\ref{modeling}, we can obtain the model for the multilayer generalization of the ACL sandwich beams as in Figure \\ref{ACL-mult}.\nThe equations of motion are found to be\n\\begin{equation}\\left\\{ \\begin{array}{l}\n {\\bf{h}}_{\\mathcal O} {\\bf{p}}_{\\mathcal O} {\\ddot y}_{\\mathcal O} -{\\bf{h}}_{\\mathcal O} {\\bf{\\alpha}}_{\\mathcal O} ({y}_{\\mathcal O})_{xx} -{\\bf{\\gamma}}_{\\mathcal O}^2{\\bf{\\varepsilon}}_{\\mathcal O}^{-1} {\\bf{h}}_{\\mathcal O} ~(P_{\\xi_{\\mathcal O}} ({y}_{\\mathcal O})_{x})_x+ {\\bf{B}}^{\\rm T} {\\bf{ G}}_E \\psi_E = \\delta(x-L){\\bf \\gamma}_{\\mathcal O}{\\bf \\varepsilon}_{\\mathcal O}^{-1}\\sigma_{\\mathcal O}(t), \\\\\nm\\ddot w -K_1 \\ddot w_{xx} + K_2 w_{xxxx} - N^{\\rm T} {\\bf{h}}_E {\\bf{ G}}_E \\psi_E = 0 ~~~ {\\rm{in}}~~ (0, L)\\times \\mathbb{R}^+ \\\\\n {\\text{where}}~~ {\\bf{B}} { y}_{\\mathcal O}={\\bf{h}}_E \\phi_E-{\\bf{h}}_E N w',\n \\end{array} \\right.\n\\label{maincont}\n\\end{equation}\nwith clamped-free boundary conditions and initial conditions\n\\begin{eqnarray}\\label{bdryback10}\n & \\left\\{ \\begin{array}{l}\n \\left\\{{y}_{\\mathcal O}, w, w_x \\right\\}_{x=0}=0 ~~ {\\rm{on}}~ \\mathbb{R}^+,\\label{bdrycont3}\\\\\n \\left\\{\\alpha_{\\mathcal O} h_{\\mathcal O}y_{\\mathcal O} + {\\bf{\\gamma}}_{\\mathcal O}^2{\\bf{\\varepsilon}}_{\\mathcal O}^{-1} {\\bf{h}}_{\\mathcal O} ~P_{\\xi_{\\mathcal O}} ({y}_{\\mathcal O})_{x}\\right\\}_{x=L}=0,~~~ {\\rm{on}}~~ \\mathbb{R}^+,\\label{bdrycont4}\\\\\n \\left\\{K_2 w_{xx} \\right\\}_{x=L} =-M(t) , ~ \\left\\{K_1 \\ddot w_x -K_2 w_{xxx} \\right\\}_{x=L} = g(t), ~~ {\\rm{on}}~ \\mathbb{R}^+ ,\\label{bdrycont5}\\\\\nw(x,0)=w^0(x), ~\\dot w(x,0)=w^1(x), ~{y}_{\\mathcal O}(x,0)= {y}^0_{\\mathcal O}, ~ {\\dot y}_{\\mathcal O}(x,0)= {y}^1_{\\mathcal O} ~ {\\rm{on}}~~ (0,L). \\label{initialcont}\n \\end{array} \\right. &\n\\end{eqnarray}\n\n \\begin{figure}\n\\begin{center}\n\\includegraphics[height=5.5cm]{multilayer.png} \n\\caption{The composite consists of joint ACL layers. Odd layers are all piezoelectric and are actuated by a charge source.} \n\\label{ACL-mult} \n\\end{center} \n\\end{figure}\n\nThe model (\\ref {maincont}) consists of $2m+1$ alternating stiff and complaint (core) layers, with piezoelectric layers on outside. The piezoelectric layers have odd indices $1,3,\\ldots 2m+1$ and the even layers have even indices $2,4,\\ldots 2m$.\n\n In the above, $m,K_1, K_2$ are \\emph{positive} physical constants, $w$ represents the transverse displacement, $\\phi^i$ denotes the shear angle in the $i^{\\rm{th}}$ layer, $\\phi_E=[\\phi^2,\\phi^4,\\ldots,\\phi^{2m} ]^{\\rm T},$ $y^i$ denote the longitudinal displacement\nalong the center of the $i^{\\rm{th}}$ layer, and $y_{\\mathcal O}=[y^1,y^3, \\ldots, y^{2m+1}]^{\\rm T}.$ Define the following $n\\times n$ diagonal matrices\n\\begin{eqnarray}\n\\nonumber &{\\bf{p}}_{\\mathcal O}={\\rm{diag}}~ (\\rho_1, \\ldots,\\rho_{2m+1}), ~~{\\bf{h}}_{\\mathcal O}={\\rm{diag}}~(h_1, \\ldots, h_{2m+1}), ~~{\\bf{h}}_E={\\rm{diag}}~(h_2,\\ldots, h_{2m}),&\\\\\n\\nonumber & {\\bf{\\varepsilon}}_{\\mathcal O}={\\rm{diag}}~(\\varepsilon_3^1, \\ldots, \\varepsilon_3^{2m+1}), ~~{\\bf{\\xi}}_O={\\rm{diag}}~(\\xi_1,\\ldots, \\xi_{2m+1})~~{\\bf{\\alpha}}_{\\mathcal O}={\\rm{diag}}~(\\alpha_1, \\ldots, \\alpha_{2m+1}),&\\\\\n\\nonumber & {\\bf{ G}}_E={\\rm{diag}}~( G_2, \\ldots, G_{2m}), ~~{\\bf{\\sigma}}_{\\mathcal O}={\\rm{diag}}~(\\sigma_1, \\ldots, \\sigma_{2m+1})&\n\\end{eqnarray}\nwhere $h_i, \\rho_i, E_i,$ are positive and denote the thickness, density, and Young's modulus, respectively.\nAlso $G_i\\ge 0$ denotes shear modulus of the $i^{\\rm{th}}$ layer, and $ {\\tilde{ G}}_i\\ge 0$ denotes coefficient for damping in the corresponding compliant layer.\n\nThe vector $ N$ is defined as\n$N={\\bf{h}}_E^{-1}{\\bf{A }}{\\bf{h}}_{\\mathcal O} \\vec 1_{\\mathcal O} + \\vec 1_E$\nwhere ${\\bf{A}}=(a_{ij})$ and ${\\bf{B}}=(b_{ij})$ are the $m\\times(m+1)$ matrices\n$$a_{ij} = \\left\\{ \\begin{array}{l}\n1\/2,~~{\\rm{ if }}~~j = i~~{\\rm{ or }}~~j = i + 1 \\\\\n~~0,\\quad{\\rm{ otherwise}} \\\\\n\\end{array} \\right., ~~b_{ij}=\\left\\{ \\begin{array}{l}\n(-1)^{i+j+1},~~{\\rm{ if }}~~j = i~~{\\rm{ or }}~~j = i + 1 \\\\\n~~0, \\quad\\quad\\quad\\quad {\\rm{ otherwise}} \\\\\n\\end{array} \\right. $$\nand $ \\vec 1_{\\mathcal O}$ and $\\vec 1_E$ denote the vectors with all entries $1$ in $\\mathbb{R}^{m+1}$ and $\\mathbb{R}^{m},$ respectively.\n\nIn the above, the operator $P_{\\xi_{\\mathcal O}}$ is defined by\n\\begin{eqnarray}\\label{Lgamma-mat}&&P_{\\xi_{\\mathcal O}}:={\\rm{diag}}~(P_{\\xi_1}, \\ldots, P_{\\xi_{2m+1}})\\end{eqnarray}\nwhere $$P_{\\xi_i}=\\left(-\\xi_i D_x^2+I\\right)^{-1}, \\quad \\xi_i:=\\frac{{\\varepsilon_{1}}^i h_i^2}{12{\\varepsilon_{3}}^i}, ~~~i=1,3,\\ldots, 2m+1.$$\n\nLet ${\\bf{s}}_O={\\rm{diag}}~(s_1,\\ldots, s_{2m+1}),~$ $s_i,k_1, k_2>0,$ and\n \\begin{eqnarray}\\label{feedback-mult} {\\bf \\sigma}_{\\mathcal O}^i=-{\\bf s}_{\\mathcal O}\\dot {y}_{\\mathcal O}(L,t),~~M(t)= k_1\\dot w_x(L,t),~~ g(t)=k_2 \\dot w(L).\n\\end{eqnarray}\nThe well-posedness of the model (\\ref{maincont})-(\\ref{initialcont}) with (\\ref{feedback-mult}) can be established in a similar fashion as in Section \\ref{Section-stab}. The following result follows:\n\\begin{thm} The system (\\ref{maincont})-(\\ref{initialcont}) with (\\ref{feedback-mult}) is exponentially stable.\n\\end{thm}\\\\\n\\textbf{Proof:} The proof is analogous to the proof of Theorem \\ref{main-thm}.\n\n\\begin{rmk} The equations of motion for a voltage controlled multilayer ACL beam can be obtained similarly. One can also consider a hybrid type \\emph{hybrid} multilayer ACL beam model for which some layers are actuated by voltage sources, and the rest are actuated by charge sources. The equations of motion can be also obtained in a similar fashion.\n\\end{rmk}\n \\section{Conclusion and Future Research}\nIn this paper, it is shown that in the case of electrostatic assumption, charge, current, or voltage actuated ACL beams can be uniformly exponentially stabilized by using mechanical feedback controllers. This is not the case once we consider the quasistatic or fully dynamic approaches where the magnetic effects are accounted for. For example, the voltage-actuated ACL beam model obtained by the fully dynamic or quasistatic approaches is shown to be not uniformly exponentially stabilizable \\cite{Ozkan3}. The polynomial stability result obtained in \\cite{Ozkan1} for certain combinations of material parameters on a more regular space than the natural energy space is applicable to this problem yet it is still an open problem. Moreover, the fully dynamic model for current or charge actuation is obtained in \\cite{Ozkan4}, yet the stabilization problem is currently open and under investigation.\n\n\nEven though the quadratic-through-thickness assumption for the electric potential in the case of charge or current actuation makes the beam stiffer than voltage actuated ACL beam, this effect is not observed for higher order eigenvalues due to the compactness of the operator $P_\\xi$. \n\n\n\n\n\n\n\\bibliographystyle{spiebib}\n\n\n\\bibliographystyle{plain} \n \n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe observation of cosmic ray particles with energies higher than $10^{11}~GeV$\n\\cite{EHE} gives a serious challenge to the known mechanisms of acceleration. \nThe shock acceleration in different \nastrophysical objects typically gives maximal energy of accelerated protons\nless than $(1-3)\\cdot 10^{10}~GeV$ \\cite{NMA}. The unipolar induction can \nprovide the maximal energy $1\\cdot 10^{11}~GeV$ only for the extreme values \nof the parameters \\cite{BBDGP}. Much attention has recently been given to \nacceleration by ultrarelativistic shocks \\cite{Vie},\\cite{Wax}. The\nparticles here can gain a tremendous increase in energy,\nequal to $\\Gamma^2$, at a single reflection, \nwhere $\\Gamma$ is the Lorentz factor of the shock.\nHowever, it is \n known (see e.g. the simulation \nfor pulsar relativistic wind in \\cite{Hosh}) that particles entering \nthe shock region are captured there or at least have a small probability \nto escape. \n\n{\\em Topological defects, TD,} (for a review see \\cite{Book}) can naturally \nproduce particles of ultrahigh energies (UHE). The pioneering observation \nof this possibility was made by Hill, Schramm and Walker \\cite{HS} (for \na general analysis of TD as UHE CR sources see \\cite {BHSS} and for a \nreview \\cite{Sigl}).\n\nIn many cases TD become unstable and decompose to constituent fields, \nsuperheavy gauge and Higgs bosons (X-particles), which then decay \nproducing UHE CR. It could happen, for example, when two segments of \nordinary string, or monopole and antimonopole touch each other, when \nelectrical current in superconducting string reaches the critical value\nand in some other cases.\n\nIn most cases the problem with UHE CR from TD \nis not the maximal energy, but the fluxes. One very general reason \nfor the low fluxes consists in the large distance between TD. A dimension\nscale for this distance is the Hubble distance $H_0^{-1}$. However, in some \nrather exceptional cases this dimensional scale is multiplied to a small \ndimensionless value $r$. If a distance between TD is larger than \n UHE proton attenuation length (due to the GZK effect \\cite{GZK}), then \nthe flux at UHE is typically exponential suppressed. \n\n\n{\\em Ordinary cosmic strings} can produce particles when a loop annihilate \ninto double line \\cite{BR}. The produced UHE CR flux is strongly reduced due \nto the fact that a loop oscillates, and in the process of a collapse the \ntwo incoming parts of a loop touch each other in one point producing thus \nthe smaller loops, instead of two-line annihilation. However, this idea was \nrecently revived due to recent work \\cite{Vincent}. It is argued there that \nthe energy loss of the long strings is dominated by production of very \nsmall loops with the size down to the width of a string, which immediately \nannihilate into superheavy particles. A problem with this scenario is \ntoo large distance between strings (of order of the Hubble distance). \nFor a distance between an observer and a string being the same, the \nobserved spectrum of UHE CR has an exponential cutoff at energy \n$E \\sim 3\\cdot 10^{19}~eV$.\n\nSuperheavy particles can be also produced when two segments of\nstring come \ninto close contact, as in {\\it cusp} events \\cite{Bran}. \nThis process\nwas studied later by Gill and Kibble \\cite{GK}, and they concluded that\nthe resulting cosmic ray flux is far too small. \nAn interesting possibility suggested by Brandenberger \\cite{Bran} is the\n{\\em cusp} ``evaporation'' on cosmic strings. \nWhen the distance between two segments of the cusp\nbecomes of the order of the string width, the cusp may``annihilate\" \nturning into high energy particles, \nwhich are boosted by a very large Lorentz\nfactor of the cusp \\cite{Bran}. \nHowever, the resulting UHE CR flux is considerably smaller than one \nobserved \\cite{BBM}. \n\n{\\em Superconducting strings} \\cite{Witten} appear to\nbe much better suited for particle production.\nMoving through cosmic magnetic fields, such strings develop\nelectric currents and copiously produce charged heavy particles when the\ncurrent reaches certain critical value. \nThe CR flux produced by\nsuperconducting strings is affected by some model-dependent string\nparameters and by the history and spatial distribution of cosmic\nmagnetic fields. \nModels considered so far failed to account\nfor the observed flux \\cite{SJSB}.\n\n \n{\\em Monopole-antimonopole pairs } ($M{\\bar M}$) \ncan form bound states and eventually\nannihilate into UHE particles \\cite{Hill}, \\cite{BS}. \nFor an appropriate choice of the\nmonopole density $n_M$, this model is consistent with observations;\nhowever, the required (low) value of $n_M$ implies fine-tuning. \nIn the first phase transition $G \\to H \\times U(1)$ in the early \nUniverse the monopoles are produced with too high density. It must then be \ndiluted by inflation to very low density, precisely tuned to \nthe observed UHE CR flux. \n\n{\\em Monopole-string network} can be formed in the early Universe in the \nsequence of symmetry breaking \n\\begin{equation}\nG \\to H \\times U(1) \\to H \\times Z_N.\n\\label{eq:Z-N}\n\\end{equation}\nFor $N \\geq 3$ an infinite network of monopoles connected by strings is \nformed. The magnetic fluxes of monopoles in the network are channeled into \ninto the strings that connect them. The monopoles typically have additional \nunconfined magnetic and chromo-magnetic charges. When strings shrink the \nmonopoles are pulled by them and are accelerated. The \naccelerated monopoles produce extremely high energy gluons, which then \nfragment into UHE hadrons \\cite{BMV}. The produced flux is too small \nto explain UHE CR observation \\cite{BBV}.\n\n\n{\\em Cosmic necklaces} are TD which are formed \nin a sequence of symmetry breaking given by Eq.(\\ref{eq:Z-N}) when $N=2$. \n The first phase transition\nproduces monopoles, and at the second phase transition each monopole\ngets attached to two strings, with its magnetic flux channeled along the \nstrings. The resulting necklaces resemble ``ordinary'' cosmic strings\nwith monopoles playing the role of beads. Necklaces can evolve in such way\nthat a distance between monopoles diminishes and in the end all monopoles \nannihilate with the neighboring antimonopoles \\cite{BV}. The produced \nUHE CR flux \\cite{BV} is close to the observed one and the shape of the \nspectrum \nresembles one observed. The distance between necklaces can be much smaller \nthan attenuation length of UHE protons.\n\n{\\em Superheavy relic particles} can be sources of UHE CR \\cite{KR,BKV}.\nIn this scenario Cold Dark Matter (CDM) have a small admixture \nof long-lived superheavy particles. These particles must be heavy,\n$m_X > 10^{12}~GeV$, long-lived $\\tau_X > t_0$, where $t_0$ is the age \nof the Universe, and weakly interacting. The required life-time \ncan be \nprovided if this particle has (almost) conserved quantum number broken \nvery weakly due to warmhole \\cite{BKV} or instanton \\cite{KR} effects.\nSeveral mechanisms for production of such particles in the early Universe \nwere identified. Like other forms of non-dissipative CDM , X-particles \nmust accumulate in the halo of our Galaxy \\cite{BKV} and thus they produce \nUHE CR without GZK cutoff and without appreciable anisotropy. \n\n{\\em The UHE carriers} produced at the decay of superheavy \nrelic particles or from TD, can be be nucleons,\nphotons and neutrinos or neutralinos \\cite{BK}. Production of neutralinos\noccurs in particle cascade, which originates at the decay of superheavy \nX-particle in close analogy to QCD cascade \\cite{BK}. Though flux of \nUHE neutralino is of the same order as neutrino flux, its detection is more \nproblematic because of smaller cross-section. The other particles \ndiscussed as carrier of UHE signal are gluino \\cite{Farr,MN,BK}, in case it is \nLightest Supersymmetric Particle (LSP), and heavy monopole \\cite{Weil,MN}. \n\nIn this paper I will present the results obtained in our joint works with\nM.Kachelriess and A.Vilenkin about necklaces and superheavy relic \nparticles as possible sources of UHE CR. Neutralino and gluino as the \ncarriers of UHE signal will be also shortly discussed.\n\n\\section{Necklaces} \nNecklaces produced in a sequence of symmetry breaking \n$G \\to H \\times U(1) \\to H \\times Z_2$ form the infinite necklaces having \nthe shape or random walks and a distribution of closed loops. Each monopole \nin a necklace is attached to two strings. \n\nThe monopole mass $m$ and the string tension $\\mu$ are determined by\nthe corresponding symmetry breaking scales, $\\eta_s$ and $\\eta_m$ \n($\\eta_m>\\eta_s$): \n$m\\sim 4\\pi\\eta_m\/e,\\;\\;\\; \\mu\\sim 2\\pi\\eta_s^2$.\nHere, $e$ is the gauge coupling. The mass per unit length of string\nis equal to its tension, $\\mu$. Each string attached to a monopole\npulls it with a force $F \\sim \\mu$ in the direction of the string. \nThe monopole radius $\\delta_m$ and the string thickness $\\delta_s$ are\ntypically of the order $\\delta_m\\sim(e \\eta_m)^{-1}$, $~~\\delta_s\\sim\n(e \\eta_s)^{-1}$. \n\nAn important quantity for the necklace evolution is the dimensionless ratio\n\\begin{equation}\nr=m\/\\mu d,\n\\label{r}\n\\end{equation}\n\nWe expect the necklaces to evolve in a \nscaling regime. If $\\xi$ is the characteristic length scale of the network, \nequal to the typical separation of long strings and to their characteristic\ncurvature radius, then the force per unit length of string is \n$f \\sim \\mu\/\\xi$, and the acceleration is $a \\sim \n(r+1)^{-1}\\xi^{-1}$.\nWe assume that $\\xi$ changes on a Hubble time scale $\\sim t$. Then the \ntypical distance travelled by long strings in time $t$ should be \n$\\sim \\xi$, so that the strings have enough time to intercommute in a \nHubble time. This gives $at^2 \\sim\\xi$, or \n\\begin{equation}\n\\xi \\sim (r+1)^{-1\/2}t.\n\\label{xi}\n\\end{equation}\nThe typical string velocity is $v \\sim (r+1)^{-1\/2}$.\n\nIt is argued in Ref.(\\cite{BV}) that $r(t)$ is driven towards large value\n$r \\gg1$. However, for $r \\geq 10^6$ the characteristic velocity of the \nnetwork falls down below the virial velocity, and the necklaces will be \ntrapped by gravitational clustering of the matter. This may change \ndramatically the evolution of network. One possible interesting effect\nfor UHE CR can be enhancement of necklace space density within \nLocal Supercluster -- a desirable effect as far \nabsence of the GZK cutoff is concerned. However, we restrict our \nconsideration by the case $r < 10^6$. The distance between \nnecklaces is still small enough, $\\xi \\gtrsim 3~Mpc$, to assume their uniform \ndistribution, when calculating the UHE CR flux.\n\nSelf-intersections of long necklaces result in copious production of\nclosed loops. For $r\\gtrsim 1$ the motion of loops is not periodic,\nso loop self-intersections should be frequent and their fragmentation\ninto smaller loops very efficient.\nA loop of size $\\ell$ typically disintegrates on a timescale \n$\\tau\\sim r^{-1\/2}\\ell$. All monopoles trapped in the loop must, of course, \nannihilate in the end. \n\n\nAnnihilating $M{\\bar M}$ pairs decay into\nHiggs and gauge bosons, which we shall refer to collectively as\n$X$-particles. The rate of $X$-particle production is easy to\nestimate if we note that infinite necklaces lose a substantial\nfraction of their length to closed loops in a Hubble time. \nThe string length per unit volume is $\\sim \\xi^{-2}$, and the monopole\nrest energy released per unit volume per unit time is\n$r\\mu\/\\xi^2 t$. Hence, we can write\n\\begin{equation}\n\\dot{n}_X \\sim r^2 \\mu\/(t^3 m_X),\n\\label{xrate}\n\\end{equation}\nwhere $m_X\\sim e\\eta_m$ is the $X$-particle mass.\n\nX-particles emitted by annihilating monopoles decay into hadrons, photons \nand neutrinos, the latter two components are produced through decays of \npions.\n\nThe diffuse flux of ultra-high energy protons\ncan be evaluated as\n\\begin{equation}\nI_p(E)=\\frac{c\\dot{n}_X}{4 \\pi m_X} \\int_0^{t_0} dt\\,\\, W_N(m_X,x_g)\n\\frac{dE_g(E,t)}{dE}\n\\label{pflux}\n\\end{equation}\nwhere $dn_X\/dt$ is given by Eq.(\\ref{xrate}),\n$E$ is an energy of proton at observation and $E_g(E,t)$ is its energy at \ngeneration at cosmological epoch $t$, $x_g=E_g\/E$ and\n$W_N(m_X,x)$ is the fragmentation function of X-particle into nucleons\nof energy $E=xm_X$. The value of $dE_g\/dE$ can be calculated from the \nenergy losses of a proton on microwave background radiation (e.g. see \n\\cite{BBDGP}). In Eq.(\\ref{pflux}) the recoil protons are taken into \naccount, while in Ref.(\\cite{BV}) their contribution was neglected.\n\n\n\nThe fragmentation function $W_N(m_X,x)$ is calculated using the decay of \nX-particle into QCD \npartons (quark, gluons and their supersymmetric partners) with the \nconsequent development of the parton cascade. The cascade in this case is\nidentical to one initiated by $e^+e^-$ -annihilation.\nWe have used the fragmentation function in the gaussian form as \nobtained in MLLA approximation in \\cite{DHMT} and\n\\cite{ESW}. \n\n\nIn our calculations \nthe UHE proton flux is fully determined by only two parameters,\n$r^2\\mu$ and $m_X$. \nThe former is restricted by low energy diffuse gamma-radiation.\nIt results from e-m cascades initiated \nby high energy photons and electrons produced in \nthe decays of X-particles.\n\nThe cascade energy density predicted in our model is \n\\begin{equation}\n\\omega_{cas}=\\frac{1}{2}f_{\\pi}r^2\\mu \\int_0 ^{t_0}\\frac{dt}{t^3}\n\\frac{1}{(1+z)^4}=\\frac{3}{4}f_{\\pi}r^2\\frac{\\mu}{t_0^2},\n\\label{cas}\n\\end{equation}\nwhere $t_0$ is the age of the Universe (here and below we use\n$h=0.75$), $z$ is the redshift and $f_{\\pi} \\sim 1$ is the fraction of\nenergy \ntransferred to pions. In Eq.(\\ref{cas}) we took into account that half \nof the energy of pions is transferred to photons and electrons. \nThe observational bound on the cascade density, for the kind of\nsources we are considering here, is \\cite{B92} $\\omega_{cas} \\lesssim\n10^{-5}~eV\/cm^3$. This gives a bound on the parameter $r^2\\mu$.\n\n\nIn numerical calculations we used \n$r^2\\mu= 1\\times 10^{28}~GeV^2$, which results in \n$\\omega_{cas}=5.6 \\cdot 10^{-6}~eV\/cm^3$, somewhat below the observational \nlimit. Now we are left with one free parameter, $m_X$, which we fix at\n$1\\cdot 10^{14}~GeV$. The maximum energy\nof protons is then $E_{max} \\sim 10^{13}~GeV$. \nThe calculated proton \nflux is presented in Fig.1, together with a summary \nof observational data taken \nfrom ref.\\cite{akeno}. \n\nLet us now turn to the calculations of UHE gamma-ray flux from the decays \nof X-particles. The dominant channel is given by the decays of neutral \npions. The flux can be readily calculated as\n\\begin{equation}\nI_{\\gamma}(E)=\\frac{1}{4\\pi}\\dot{n}_X\\lambda_{\\gamma}(E)N_{\\gamma}(E)\n,\n\\label{gflux}\n\\end{equation}\nwhere $\\dot{n}_X$ is given by Eq.(\\ref{xrate}), \n$\\lambda_{\\gamma}(E)$ is the absorption length of a photon with energy \n$E$ due to $e^+e^-$ pair production on background radiation and \n$N_{\\gamma}(E)$ is the number of photons with energy E produced per one decay \nof X-particle. The latter is given by \n\\begin{equation}\nN_{\\gamma}(E)=\\frac{2}{m_X}\\int_{E\/m_X}^1 \\frac{dx}{x} W_{\\pi^0}(m_X,x)\n\\label{gnumber}\n\\end{equation}\nwhere $W_{\\pi^0}(m_X,x)$ is the fragmentation function of X-particles into\n$\\pi^0$ pions.\n\nAt energy $E>1\\cdot 10^{10}~GeV$ the \ndominant\ncontribution to the gamma-ray absorption comes from the radio background. The \nsignificance of this process was first noticed in \\cite{B70}(see also \nbook \\cite{BBDGP}). New calculations for this absorption \nwere recently\ndone \\cite{PB}. We have used the \nabsorption lengths from this work. \n\nWhen evaluating the flux \\ref{gflux} at $E > 1 \\cdot 10^{10}~GeV$ we neglected \ncascading of a primary photon, because pair production and inverse \ncompton scattering occur at these energies on radio background, and thus \nat each collision the energy of a cascade particle is halved.\nMoreover, assuming an intergalactic magnetic field \n$H \\geq 1\\cdot 10^{-9}$, the secondary \nelectrons and positrons loose their energy mainly due to synchrotron \nradiation and the emitted photons escape from the considered energy \ninterval \\cite{BGP}. \n\nThe calculated flux of gamma radiation is presented in Fig.1 by the curve\nlabelled $\\gamma$. One can see that at $E \\sim 1\\cdot 10^{11}~GeV$ the \ngamma ray flux is considerably lower than that of protons. This is \nmainly due \nto the difference in the attenuation lengths for protons ($110~Mpc$) and \nphotons ($2.6~Mpc$ \\cite{PB} and $2.2~Mpc$ \\cite{B70}). At higher energy \nthe attenuation length for protons dramatically decreases ($13.4~Mpc$ at \n$E=1 \\cdot 10^{12}~GeV$) and the fluxes of protons and photons become \ncomparable. \n\n\nA requirement for the models explaining the observed UHE events is that \nthe distance between sources must be smaller than the attenuation\nlength. Otherwise the flux at the \ncorresponding energy would be \nexponentially suppressed. This imposes a severe constraint on the\npossible sources. For example, in the case of protons with energy \n$E \\sim (2- 3)\\cdot 10^{11}~GeV$ \nthe proton \nattenuation length is $19~Mpc$ . If protons \npropagate rectilinearly, there should be several sources inside this\nradius;\notherwise all particles would arrive from the same direction.\nIf particles are strongly deflected in extragalactic magnetic fields, \nthe distance to the source should be even smaller. Therefore, the \nsources of \nthe observed events at the highest energy must be at a distance \n$R\\lesssim 15~Mpc$ in the case or protons. \n\nIn our model the distance between \nsources, given by Eq.(\\ref{xi}), satisfies this condition for \n$r>3\\cdot 10^{4}$.\nThis is in contrast to other potential sources,\nincluding supeconducting cosmic strings and powerful \nastronomical sources such as AGN, for which this condition imposes\nsevere restrictions. \n\nThe difficulty is even more pronounced in the case of UHE photons. \nThese particles \npropagate rectilinearly and their absorption length is shorter: \n$2 - 4~Mpc$ at $E \\sim 3\\cdot 10^{11}~GeV$. It is rather unrealistic to expect\nseveral powerful astronomical sources at such short distances. This \ncondition is very restrictive for topological defects as well. The\nnecklace model is rather exceptional regarding this aspect.\n\n\\section{UHE CR FROM RELIC QUASISTABLE PARTICLES}\n\nThis possibility was recognized recently in Refs(\\cite{KR,BKV}).\n\nOur main assumption is that Cold Dark Matter \n(CDM) has a small admixture of long-lived supermassive $X$-particles.\nSince, apart from very small scales, fluctuations grow identically\nin all components of CDM, the fraction of $X$-particles, $\\xi_X$, is \nexpected to be the same in all structures. In particular, $\\xi_X$ is\nthe same in the halo of our Galaxy and in the\nextragalactic space. Thus the halo density of $X$-particles is enhanced in \ncomparison with the extragalactic density.\nThe decays \nof these particles produce UHE CR, whose flux is dominated by the \nhalo component, and therefore has no GZK cutoff. Moreover,\nthe potentially dangerous e-m cascade radiation is suppressed.\n\nFirst, we address the elementary-particle and cosmological aspects\nof a superheavy long-living particle. Can the relic density \nof such particles be as high as required by observations of UHE CR?\nAnd can they have a lifetime comparable or larger than the age\nof the Universe?\n \nLet us assume that $X$-particle is a neutral fermion which belongs \nto a representation of the $SU(2)\\times U(1)$ group. We assume also\nthat the stability of $X$-particles is protected by a discrete\nsymmetry \nwhich is the remnant of a gauge symmetry and is\nrespected by all interactions except quantum\ngravity through wormhole effects. In other words, our particle is very\nsimilar to a very heavy neutralino with a conserved quantum number, $R'$, \nbeing the direct analogue of $R$-parity (see \\cite{BJV} and the\nreferences therein).\nThus, one can assume that the decay of $X$-particle occurs due to dimension 5\noperators, inversely proportional to the Planck mass $m_{\\rm Pl}$ and \nadditionally suppressed by a factor $\\exp(-S)$, where $S$ is the \naction of a wormhole which absorbs $R'$-charge. \nAs an example one can consider a term\n\\begin{equation}\n{\\cal L} \\sim \\frac{1}{m_{Pl}} \\bar{\\Psi}\\nu \\phi\\phi \\exp(-S),\n\\label{eq:d5}\n\\end{equation}\nwhere $\\Psi$ describes X-particle, and $\\phi$ is a $SU(2)$ scalar with vacuum \nexpectation value $v_{EW}=250$~GeV.\nAfter spontaneous symmetry breaking the term (\\ref{eq:d5}) results in \nthe mixing of $X$-particle and neutrino, and \nthe lifetime due to $X \\to \\nu +q + \\bar{q}$ , {\\it e.g.}, is given by\n\\begin{equation}\n\\tau_X \\sim \\frac{192(2\\pi)^3}{(G_Fv_{EW}^2)^2}\\frac{m_{\\rm Pl}^2}{m_X^3}\ne^{2S},\n\\label{eq:ltime}\n\\end{equation}\nwhere $G_F$ is the Fermi constant. The lifetime $\\tau_X >t_0$ for \n$X$-particle with $m_X \\geq 10^{13}$~GeV needs $S>44$. \nThis value is within the \nrange of the allowed values as discussed in Ref. \\cite{KLLS}.\n\nLet us now turn to the cosmological production of $X$-particles with \n$m_X \\geq 10^{13}$~GeV. Several mechanisms were identified in \\cite{BKV}, \nincluding \nthermal production at the reheating stage, production through the decay of \ninflaton field at the end of the \"pre-heating\"\nperiod following inflation, and through the decay of hybrid topological \ndefects, such as monopoles connected by strings or walls bounded by\nstrings. \n\nFor the thermal production, temperatures comparable to $m_X$ are needed. \nIn the case of a heavy decaying gravitino,\nthe reheating temperature $T_R$ (which is the highest temperature \nrelevant for our problem) \nis severely limited to value below $10^8- 10^{10}$~GeV, depending \non the gravitino mass (see Ref. \\cite{ellis} and references therein). \nOn the other hand, \nin models with dynamically broken supersymmetry, the lightest \nsupersymmetric particle is the gravitino. Gravitinos with mass \n$m_{3\/2} \\leq 1$~keV interact relatively strongly with the thermal bath,\nthus decoupling relatively late, and can be the CDM particle \\cite{grav}. \nIn this scenario all phenomenological\nconstraints on $T_R$ (including the decay of the second \nlightest supersymmetric particle) disappear and one can assume\n$T_R \\sim 10^{11} - 10^{12}$~GeV. In this \nrange of temperatures, $X$-particles are not in thermal equilibrium.\nIf $T_R < m_X$, the density $n_X$ of $X$-particles produced during the \nreheating phase at time $t_R$ due to $a+\\bar{a} \\to X+\\bar{X}$ is easily \nestimated as\n\\begin{equation}\nn_X(t_R) \\sim N_a n_a^2 \\sigma_X t_R \\exp(-2m_X\/T_R),\n\\label{eq:dens}\n\\end{equation} \nwhere $N_a$ is the number of flavors which participate in the production of \nX-particles, $n_a$ is the density of $a$-particles and $\\sigma_X$ is \nthe production cross-section. The density of $X$-particles at the\npresent epoch can be found by the standard procedure of calculating\nthe ratio $n_X\/s$, where \n$s$ is the entropy density. Then for $m_X = 1\\cdot 10^{13}$~GeV\nand $\\xi_X$ in the wide range of values $10^{-8} - 10^{-4}$, the required\nreheating temperature is $T_R \\sim 3\\cdot 10^{11}$~GeV.\n\nIn the second scenario mentioned above, non-equilibrium inflaton decay,\n$X$-particles are usually overproduced and a second period of \ninflation is needed \nto suppress their density.\n\nFinally, $X$-particles could be produced by TD such as strings or textures. \nParticle production occurs at string intersections or in collapsing texture \nknots. The evolution of defects is scale invariant, and roughly a constant \nnumber of particles $\\nu$ is produced per horizon volume $t^3$ per Hubble \ntime $t$. ($\\nu \\sim 1$ for textures and $\\nu \\gg 1$ for strings.) The main \ncontribution to to the X-particle density is given by the earliest epoch,\nsoon after defect formation, and we find \n$\\xi_X \\sim 10^{-6} \\nu (m_X\/10^{13}~GeV)(T_f\/10^{10}~GeV)^3$, where \n$T_f$ is the defect formation temperature. Defects of energy scale\n$\\eta \\gtrsim m_X$ could be formed at a phase transition at or slightly \nbefore the end of inflation. In the former case, $T_f \\sim T_R$ , while in \nthe latter case defects should be considered as \"formed\" when their typical \nseparation becomes smaller than $t$ (hence $T_f < T_R$). It should be noted \nthat early evolution of defects may be affected by friction; our estimate \nof $\\xi_X$ will then have to be modified. X particles can also be produced \nby hybrid topological defects: monopoles connected by strings or walls \nbound by strings. The required values of $n_X\/s$ can be obtained for a wide \nrange of defect parameters.\n\nThe decays of $X$-particles result in the production of nucleons with a \nspectrum $W_N(m_X,x)$, where $m_X$ is the mass of the X-particle \nand $x=E\/m_X$. The flux of nucleons $(p,\\bar{p},n,\\bar{n})$ from the halo and \nextragalactic space can be calculated as\n\\begin{equation}\nI_N^{i}(E)={1\\over{4\\pi}}{n_X^i\\over{\\tau_X}}R_i{1\\over{m_X}}W_N(m_X,x),\n\\label{eq:nhalo}\n\\end{equation}\nwhere index $i$ runs through $h$ (halo) and $ex$ (extragalactic), \n$R_i$ is the size of the halo $R_h$, or the attenuation length of \nUHE protons due to their collisions with microwave photons, \n$\\lambda_p(E)$, for the halo case and extragalactic case, respectively. We \nshall assume $m_Xn_X^h=\\xi_X\\rho_{\\rm CDM}^h$ and \n$m_Xn_X^{\\rm ex}=\\xi_X\\Omega_{\\rm CDM}\\rho_{\\rm cr}$, \nwhere $\\xi_X$ describes the fraction \nof $X$-particles in CDM, $\\Omega_{\\rm CDM}$ is the CDM \ndensity in units of the critical density $\\rho_{\\rm cr}$, and \n$\\rho_{CDM}^h \\approx 0.3~GeV\/cv^3$ is the CDM density in the halo.\nWe shall use the following values for these parameters: a large \nDM halo with $R_h=100$~kpc (a smaller halo with $R_h=50$~kpc is possible, \ntoo), $\\Omega_{CDM}h^2=0.2$, \nthe mass of $X$-particle in the range \n$10^{13}~{\\rm GeV}1\\cdot 10^{10}$~GeV,\non the radio background. The neutrino flux calculation is similar.\n\nBefore discussing the obtained results, we consider \nthe astrophysical constraints.\n\nThe most stringent constraint comes from electromagnetic \ncascade radiation, discussed in the previous section.\nIn the present case \nthis constraint is weaker, because the low-energy extragalactic\nnucleon flux \nis $\\sim 4$ times \nsmaller than that one from the Galactic halo (see Fig.~2). Thus \nthe cascade radiation is suppressed by the same factor. \n\nThe cascade energy density calculated by integration over cosmological epochs\n(with the dominant contribution given by the present epoch $z=0$) yields\nin our case\n\\begin{equation}\n\\omega_{\\rm cas}=\\frac{1}{5}r_X\\frac{\\Omega_{CDM}\\rho_{cr}}{H_0t_0}=\n6.3\\cdot10^2 r_X f_{\\pi}~{\\rm eV\/cm}^3.\n\\end{equation}\n\nTo fit the UHE CR observational data by nucleons from halo, \nwe need $r_X=5\\cdot 10^{-11}$. Thus the cascade energy density is \n$\\omega_{\\rm cas}=3.2\\cdot 10^{-8} f_{\\pi}$~eV\/cm$^3$, well below the\nobservational bound. \n\n\nLet us now discuss the obtained results.\nThe fluxes shown in Fig.~2 are obtained for $R_h=100$~kpc, \n$m_X=1\\cdot 10^{13}$~GeV and\n$r_X=\\xi_X t_0\/\\tau_X=5\\cdot10^{-11}$. This ratio $r_X$ allows very small \n$\\xi_X$ and $\\tau_X > t_0$. The fluxes \nnear the maximum energy $E_{\\rm max}=5\\cdot 10^{12}$~GeV were only roughly\nestimated (dotted lines on the graph). \n\nIt is easy to verify that the extragalactic nucleon flux at $E \\leq \n3\\cdot 10^{9}$~GeV is suppressed by a factor $\\sim 4$ and by a much larger \nfactor at higher energies due to energy losses. The flux of \nextragalactic photons is suppressed even stronger, because the attenuation \nlength for photons (due to absorption on radio-radiation) is much smaller\nthan for nucleons (see Ref. \\cite{PB}). This flux is not shown in the graph. \nThe flux of high energy gamma-radiation from the halo is by a factor $7$ \nhigher than that of nucleons and the neutrino flux, given in the Fig.2 as \nthe sum of the dominant halo component and subdominant extragalactic one,\nis twice higher than the gamma-ray flux.\n\nThe spectrum of the observed EAS is formed due to fluxes of gamma-rays and \nnucleons. The gamma-ray contribution to this spectrum is rather complicated.\nIn contrast to low energies, the photon-induced showers at \n$E>10^9$~GeV have the low-energy muon component as abundant as that \nfor nucleon-induced showers \\cite{AK}. However, the \nshower production by the photons is suppressed by the \nLPM effect \\cite{LPM}\nand by absorption in geomagnetic field (for recent calculations and \ndiscussion see \\cite{ps,Kasa} and references therein). \n\nWe wish to note that the excess of the gamma-ray flux over the nucleon\nflux from the halo is an unavoidable feature of this model. It follows\nfrom the more effective production of pions \nthan nucleons in the QCD cascades from the decay of $X$-particle. \n\nThe signature of our model might be the signal from the Virgo \ncluster. The virial mass of the Virgo cluster is\n$M_{\\rm Virgo} \\sim 1\\cdot 10^{15} M_{\\odot}$ and the distance to it \n$R= 20$~Mpc. If UHE protons (and antiprotons) propagate rectilinearly from \nthis source\n(which could be the case for $E_p \\sim 10^{11} - 10^{12}$~GeV), their \nflux is given by\n\\begin{equation}\nF_{p,\\bar{p}}^{\\rm Virgo}= r_X \\frac{M_{\\rm Virgo}}{t_0 R^2 m_X^2}W_N(m_X,x).\n\\end{equation} \nThe ratio of this flux to the diffuse flux from the half hemisphere is\n$6.4\\cdot 10^{-3}$. This signature becomes less pronounced at smaller \nenergies, when protons can be strongly deflected by intergalactic magnetic \nfields.\n\n\\section{LSP IS UHE CARRIER}\n\nLSP is the Lightest Supersymmetric Particle. It can be stable if R-parity \nis strictly conserved or unstable if R-parity is violated. To be able to \nreach the Earth from most remote regions in the Universe, the LSP must have \nlifetime longer than $\\tau_{LSP} \\gtrsim t_0\/\\Gamma$, where $t_0$ is the \nage of the Universe and $\\Gamma=E\/m_{LSP}$ is the Lorentz-factor of the LSP.\nIn case $m_{LSP} \\sim 100~GeV$, $\\tau_{LSP} > 1~yr$.\n\nTheoretically the best motivated candidates for LSP are the neutralino and \ngravitino. We shall not consider the latter, because it is practically \nundetectable as UHE particle. \n\n\nIn all elaborated SUSY models the gluino is not the LSP. Only, if\nthe dimension-three SUSY breaking terms are set\nto zero by hand, gluino with mass $m_{\\tilde g}={\\mathcal O}(1~GeV)$ can be the \nLSP \\cite{fa96}. There is some controversy if\nthe low-mass window $1~GeV \\lesssim m_{\\tilde g} \\lesssim 4~GeV$ for\nthe gluino is still allowed \\cite{pdg,aleph}. Nevertheless, we shall study\nthe production of high-energy gluinos and their interaction with matter\nbeing inspired by the recent suggestion \\cite{Farr} (see also \\cite{MN}), \nthat the atmospheric showers observed at the highest energies can be\nproduced by colorless hadrons containing gluinos. \nWe shall refer to any of such hadron as\n$\\tilde{g}$-hadron. Light gluinos as \nUHE particles with energy $E \\gtrsim 10^{16}$~eV were considered in \nsome detail in the literature in connection with Cyg X-3 \\cite{aur,BI}.\nAdditionally, we consider heavy gluinos with $m_{\\tilde g} \\gtrsim\n150$~GeV \\cite{MN}.\n\n\nUHE LSP are most naturally produced at the decays of unstable superheavy \nparticles, either from TD or as the relic ones \\cite{BK}. \n\nThe QCD parton cascade is not a unique cascade process. A cascade \nmultiplication of partons at the decay of superheavy particle appears \nwhenever a probability of production of extra parton has the terms \n$\\alpha \\ln Q^2$ or $\\alpha \\ln^2 Q^2$, where $Q$ is a maximum of parton \ntransverse momentum, i.e. $m_X$ in our case. Regardless of smallness of \n$\\alpha$, the cascade develops as long as\n $\\alpha \\ln Q^2 \\gtrsim 1$. Therefore, for \nextremely large $Q^2$ we are interested in, a cascade develops due to\nparton multiplication through $SU(2)\\times U(1)$ interactions as well. Like in \nQCD, the account of diagrams with $\\alpha \\ln Q^2$ gives the \nLeading Logarithm Approximation to the cascade fragmentation function.\n\nFor each next generation of cascade particles the virtuality of partons \n$q^2$ diminishes. When $q^2 \\gg m_{SUSY}^2$ , where $m_{SUSY}$ is a typical \nmass of supersymmetric particles, the number of supersymmetric partons \nin the cascade is the same as their ordinary partners. At $q^2 < \nm_{SUSY}^2$ the supersymmetric particles are not produced any more and \nthe remaining particles decay producing the LSP. In Ref.(\\cite{BK}) \na simple Monte Carlo simulation for SUSU cascading was performed \nand the spectrum of emitted LSP was calculated. LSP take away a \nconsiderable fraction of the total energy ($\\sim 40\\%$). \n\nThe fluxes of UHE LSP are shown in Fig.~3 for the case of their production \nin cosmic necklaces (see section II). When the LSP is neutralino, the \nflux is somewhat lower than neutrino flux. The neutralino-nucleon cross-\nsection, $\\sigma_{\\chi N}$, is also smaller than that for neutrino. For the \ntheoretically favorable masses of supersymmetric particles, \n$\\sigma_{\\chi N} \\sim 10^{-34}~cm^2$ at extremely high energies. If the the \nmasses of squarks are near their experimental bound, $M_{L,R} \\sim 180~GeV$, \nthe cross-section is 60 times higher.\n\n{\\em Gluino as the LSP} is another phenomenological option. Let us discuss \nshortly the status of the gluino as LSP.\n\nIn all elaborated SUSY models the gluino is not LSP, and this possibility \nis considered on purely phenomenological basis.\n Accelerator experiments give the lower limit on the gluino mass as \n$m_{\\tilde{g}} \\gtrsim 150$~GeV \\cite{pdg}. The upper limit of the \ngluino mass is given by cosmological and astrophysical constraints, as \nwas recently discussed in \\cite{MN}. In this work it \nwas shown that if the gluino provides the dark matter observed in our galaxy,\nthe signal from gluino annihilation and the abundance of anomalous heavy \nnuclei is too high. Since we are not interested in the case when gluino \nis DM particle, we can use these arguments to obtain an upper limit for \nthe gluino mass. Calculating the relic density of \ngluinos (similar as in \\cite{MN}) and using\nthe condition $\\Omega_{\\tilde{g}} \\ll \\Omega_{\\rm CDM}$, we obtained \n$m_{\\tilde{g}} \\ll 9$~TeV.\n\nNow we come to very interesting argument against existence \nof a light stable or quasistable gluino \\cite{VO}.\nIt is plausible that the {\\em glueballino} ($\\tilde{g}g$) is the lightest \nhadronic \nstate of gluino \\cite{aur,BI}. However, {\\em gluebarino\\\/}, i.e. the bound \nstate of gluino and three quarks, is almost stable because baryon number \nis extremely weakly violated. In Ref.~\\cite{VO} it is argued that the \nlightest gluebarino is the neutral state ($\\tilde{g}uud$). \n These charged gluebarinos are produced by cosmic\nrays in the earth atmosphere \\cite{VO}, and light gluino as LSP is \nexcluded by the search for heavy hydrogen or by proton decay \nexperiments (in case of quasistable gluino). In the case that the lightest \ngluebarino is neutral, see \\cite{fa96}, the arguments of \\cite{VO}\nstill work if a neutral gluebarino forms a bound state with the nuclei. \nThus, a light gluino is disfavored.\n\n\nThe situation is different if the gluino is heavy, \n$m_{\\tilde{g}}\\gtrsim 150~GeV$. This gluino can be unstable\ndue to weak R-parity violation \\cite{BJV} and have a lifetime \n$\\tau_{\\tilde{g}} \\gtrsim 1$~yr, {\\it i.e.\\\/}\nlong enough to be UHE carrier (see beginning of this section). \nThen the calculated relic density at the time \nof decay is not in conflict with the cascade nucleosynthesis and all \ncosmologically produced $\\tilde{g}$-hadrons decayed up to the present time.\nMoreover, the production of these gluinos by cosmic rays in the\natmosphere is ineffective because of their large mass.\n\nGlueballino, or more generally $\\tilde{g}$-hadron, looses its energy while \npropagating from a source to the Earth. The dominant energy loss of \nthe $\\tilde{g}$-hadron is due to pion production in collisions with microwave \nphotons. Pion production effectively starts at the same Lorentz-factor as \nin the case of the proton. This implies that the energy of the GZK \ncutoff is a factor $m_{\\tilde{g}}\/m_p$ higher than in case of the proton. \nThe attenuation length also increases because the fraction of energy lost \nnear the threshold is small, $\\mu\/m_{\\tilde{g}}$, where $\\mu$ is a pion \nmass. Therefore, even for light $\\tilde{g}$-hadrons, $m_{\\tilde{g}} \\gtrsim \n2~GeV$, the steepening of the spectrum is less pronounced than for \nprotons. \n\nThe spectrum of $\\tilde{g}$-hadrons from the cosmic necklaces accounted for \nabsorption in intergalactic space, is shown in Fig.~3.\n\nA very light UHE $\\tilde{g}$-hadron interacts with the nucleons in the \natmosphere similarly to UHE proton. The cross-section is reduced \nonly due to the radius of $\\tilde{g}$-hadron and is of order of \n$\\sim 1~mb$ \\cite{BI}. In case of very heavy $\\tilde{g}$-hadron the total \ncross-section can be of the same order of magnitude, but the \ncross-section with the large energy transfer, relevant for the \ndetection in the atmosphere, is very small \\cite{BK}. This is due to the \nfact that interaction of gluino in case of large energy transfer is \ncharacterized by large $Q^2$ and thus interaction is a deep inelastic \nQCD scattering. \n\nThus, only UHE gluino from low-mass window \n$1~GeV \\leq m_{\\tilde{g}}\\leq 4~GeV$ could be a candidate for observed \nUHE particles, but it is disfavored by the arguments given above. \n\n\\section{CONCLUSIONS}\n\nTopological Defects naturally produce particles with extremely high \nenergies, much in excess of what is presently observed. However, the fluxes \nfrom most known TD are too small. So far only necklaces \\cite{BV} and \nmonopole-antimonopole pairs \\cite{BS} can provide the observed flux of UHE CR. \n\nAnother promising sources of UHE CR are relic superheavy particles \n\\cite{KR,BKV}. These particles should be clustering in the halo of \nour Galaxy \\cite{BKV}, and thus UHE CR produced at their decays do not \nhave the GZK cutoff. The signatures of this model are dominance of \nphotons in the primary flux and Virgo cluster as a possible discrete source.\n\nApart from protons, photons and neutrinos the UHE carriers can be \nneutralinos \\cite{BK}, gluino \\cite{Farr,MN,BK} and monopoles \\cite{Weil,MN}.\nWhile neutralino is a natural candidate for the Lightest Supersymmetric \nParticle (LSP) in SUSY models, gluino can be considered as LSP only \nphenomenologically. LSP are naturally produced in the parton cascade at \nthe decay of superheavy X-particles. In case of neutralino both fluxes and \ncross-sections for interaction is somewhat lower than for neutrino. In case \nof gluinos the fluxes are comparable with that of neutralinos, but \ncross-sections for the production of observed extensive air showers are \nlarge enough only for light gluinos. These are disfavored, especially if \nthe charged gluebarino is lighter than the neutral one \\cite{VO}. \\\\*[1mm]\n\\begin{center}\nACKNOWLEDGEMENTS\n\\end{center}\n\\vspace{1mm}\nThis report is based on my recent works with Michael Kachelriess and \nAlex Vilenkin \\cite{BV,BKV,BK}. I am grateful to my co-authors for \npleasant and useful cooperation and for many discussions. \n\nMany thanks are to the organizers of the workshop for the most efficient \nwork. I am especially grateful to Jonathan Ormes for all his efforts as the \nChairman of the Organizing Committee and for inviting me to this most \ninteresting meeting.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe epitaxy of magnetically doped semiconductors constitutes a versatile mean of fabricating in a self-organized way semiconductor\/ferromagnet nanocomposites\\cite{Bonanni:2007_SST,Katayama:2007_pssa,Dietl:2008_JAP} with still widely unexplored but remarkable functionalities relevant to spintronics, nanoelectronics, photonics, and plasmonics. In these nanocomposite materials the presence of robust ferromagnetism correlates with the existence of nanoscale volumes containing a large density of magnetic cations, that is with the formation of condensed magnetic semiconductors (CMSs) buried in the host matrix and characterized by a high spin ordering temperature.\\cite{Bonanni:2010_CSR} The aggregation of CMSs and, therefore, the ferromagnetism of the resulting composite system shows a dramatic dependence on the growth conditions and co-doping with shallow impurities.\n\nIn particular, the understanding and control of CMSs in tetrahedrally coordinated semiconductor films containing transition metals (TMs) -- typical examples being (Ga,Mn)As,\\cite{Tanaka:2008_B} (Ga,Mn)N,\\cite{Martinez-Criado:2005_APL} (Ge,Mn),\\cite{Jamet:2006_NM} (Zn,Cr)Te,\\cite{Kuroda:2007_NM} and (Ga,Fe)N \\cite{Bonanni:2007_PRB,Bonanni:2008_PRL,Rovezzi:2009_PRB} -- have been lately carefully considered with the necessary aid of nanoscale characterization techniques. Indeed, the control over the CMSs formation as a function of the fabrication parameters, and the possibility to reliably produce on demand CMSs with a predefined size, structure, and distribution in the semiconductor host, are fundamental requisites for the exploitation of these nanostructures in functional devices. At the same time these studies draw us nearer to understand the origin of the ferromagnetic-like features -- persisting up to above room temperature (RT) -- found in a number of semiconductors and oxides. \\cite{Liu:2005_JMSME,Coey:2008_JPD}\n\nDilute zincblende (Ga,Mn)As grown by molecular beam epitaxy (MBE) is known to decompose upon annealing with the formation of embedded MnAs nanocrystals coherent with the GaAs matrix,\\cite{De_Boeck:1996_APL} and a striking spin-battery effect produced by these CMSs has been already proven.\\cite{Nam-Hai:2009_N}\nAs an example of the critical role played by growth parameters, MBE Ge$_{1-x}$Mn$_x$ grown below 130$^{\\circ}$C is seen to promote the self-assembling\nof coherent magnetic Mn-rich nanocolumns whilst a higher growth temperature leads to the formation of hexagonal Ge$_5$Mn$_3$ nanocrystals buried in the Ge host.\\cite{Devillers:2007_PRB}\n\nFollowing theoretical suggestions,\\cite{Dietl:2006_NM} it has recently been demonstrated experimentally that it is possible to change the charge state of TM ions in a semiconducting matrix and, therefore, the aggregation energy by co-doping with shallow donors or acceptors.\\cite{Kuroda:2007_NM,Bonanni:2008_PRL} In particular, it has been proven that in the model case of wurtzite (wz) (Ga,Fe)N fabricated by metalorganic vapor phase epitaxy (MOVPE) the Fermi-level tuning by co-doping with Mg (acceptor in GaN) or Si (donor in GaN) is instrumental in controlling the magnetic ions aggregation.\\cite{Bonanni:2008_PRL} \n\nThe same system has been thoroughly analyzed at the nanoscale by means of advanced electron microscopy as well as by synchrotron-based diffraction and absorption techniques. The structural characteristics have then been related -- together with the growth parameters -- to the magnetic properties of the material as evidenced by superconducting quantum interference device (SQUID) magnetometry.\\cite{Bonanni:2007_PRB,Bonanni:2008_PRL,Rovezzi:2009_PRB} It has been concluded that for a concentration of Fe below its optimized solubility limit ($\\sim0.4$\\% of the magnetic ions) the dilute system is predominantly paramagnetic (PM). For higher concentrations of the magnetic ions (Ga,Fe)N shows either chemical- (intermediate state) or crystallographic-phase separation.~\\cite{Bonanni:2007_PRB,Bonanni:2008_PRL,Rovezzi:2009_PRB} In the phase-separated layers a ferromagnetic (FM) behavior persisting far above RT is observed, and has been related to the presence of either Fe-rich regions coherent with the host GaN (in the intermediate state) or of Fe$_{x}$N nanocrystals in the GaN matrix. These investigations appear to elucidate the microscopic origin of the magnetic behavior of (Ga,Fe)N reported by other groups.~\\cite{Kuwabara:2001_JJAP,Kane:2006_pssb}\n\nAlong the above mentioned lines, in this work we consider further the MOVPE (Ga,Fe)N material system and we reconstruct the phase diagram of the Fe$_{x}$N nanocrystals buried in GaN as a function of the growth temperature. Synchrotron radiation x-ray diffraction (SXRD), extended fine structure x-ray absorption (EXAFS) and x-ray absorption near-edge fine structure (XANES), combined with high-resolution transmission electron microscopy (HRTEM) and SQUID magnetometry allow us to detect and to identify particular Fe$_{x}$N phases in samples fabricated at different growth temperatures $T_{\\mathrm{g}}$ as well as to establish a correlation between the existence of the specific phases and the magnetic response of the system. Our results imply, in particular, that self assembled nanocrystals with a high concentration of the magnetic constituent account for ferromagnetic-like features persisting up to above RT. These findings for (Ga,Fe)N do not support, therefore, the recent suggestions that high temperature ferromagnetism of -- the closely related -- oxides is brought about by spin polarization of defects, whereas the role of magnetic impurities is to bring the Fermi energy to an appropriate position in the band gap.~\\cite{Coey:2008_JPD} \n\nWe find that already a 5\\% variation in the growth temperature is critical for the onset of new Fe$_{x}$N species and we can confirm that an increase in the growth temperature promotes the aggregation of the magnetic ions, resulting in an enhanced density of Fe-rich nanocrystals in the matrix and in a consequent increase of the ferromagnetic response of the system. Moreover, we observe that while in the low-range of growth temperatures the Fe-rich nanoobjects tend to segregate close to the sample surface, at higher $T_{\\mathrm{g}}$ two-dimensional assemblies of nanocrystals form in a reproducible way at different depths in the layer, an arrangement expected to have a potential as template for the self-aggregation of metallic nanocolumns.\\cite{Fukushima:2006_JJAP} The non-uniform distribution of magnetic aggregates over the film volume here revealed, implies also that the CMS detection may be challenging and, in general, requires a careful examination of the whole layer, including the surface and interfacial regions.\n\nThe paper is organized as follows: in the next Section we give a concise summary of the MOVPE process employed to fabricate the (Ga,Fe)N phase-separated samples together with a brief description of the characterization techniques. A table with the relevant samples and relative parameters completes this part.\nThe central results of this work are reported in Section III and are presented in two sub-sections discussing respectively: i) the detection, identification and structural properties $vs.$ $T_{\\mathrm{g}}$ of the different Fe$_{x}$N nanocrystals in phase-separated (Ga,Fe)N, with the distribution of the nanocrystals in the sample volume, and ii) the magnetic properties of the specific families of Fe$_{x}$N phases. In Section IV we sum up the main conclusions and the prospects of this work.\n\n\\section{Experimental Procedure}\n\\subsection{Growth of (Ga,Fe)N}\nWe summarize here our study by considering a series of wurtzite (Ga,Fe)N samples fabricated by MOVPE in an AIXTRON 200 RF horizontal reactor. All structures have been deposited on $c$-plane sapphire substrates with TMGa (trimethylgallium), NH$_3$, and FeCp$_2$ (ferrocene) as precursors for, respectively, Ga, N and Fe, and with H$_2$ as carrier gas. \n\nThe growth process has been carried out according to a well established procedure,\\cite{Simbrunner:2007_APL} namely: substrate nitridation, low temperature (540$^{\\circ}$C) deposition of a GaN nucleation layer (NL), annealing of the NL under NH$_3$ until recrystallization and the growth of a $\\approx$~1~$\\mu$m thick device-quality GaN buffer at 1030$^{\\circ}$C. On the GaN buffer, Fe-doped GaN overlayers ($\\approx$~700~nm thick) have been deposited at different $T_{\\mathrm{g}}$ ranging from 800$^{\\circ}$C to 950$^{\\circ}$C, with a V\/III ratio of 300 [NH$_3$ and TMGa source flow of 1500 standard cubic centimeters per minute (sccm) and 5~sccm, respectively], with an average growth-rate of 0.21 nm\/s, and the flow-rate of the Fe-precursor set at 300~sccm. During the whole growth process the samples have been continuously rotated in order to promote the deposition homogeneity, while \\textit{in situ} and on line ellipsometry is employed for the real time control over the entire fabrication process. \n\nThe considered samples main parameters, including the Fe concentration, are displayed in Table~\\ref{Tab:table1}.\n\n\\begin{table} [h]\n\\begin{ruledtabular}\n\\caption{\\label{Tab:table1} Considered (Ga,Fe)N samples with the corresponding growth temperature, Fe concentration as evaluated by secondary ions mass spectroscopy (SIMS) as well as concentration of the dilute paramagnetic Fe$^{3+}$ ions $x_{\\mathrm{Fe}^{3+}}$ and a lower limit of the concentration of Fe ions $x_{\\mathrm{Fe}_N}$ contributing to the Fe-rich nanocrystals, as obtained from magnetization data.}\n\\label{Tab:table1}\n\\begin{tabular}{|c|cccc|}\nSample & $T_{\\mathrm{g}}$ & Fe concentration & $x_{\\mathrm{Fe}^{3+}}$ & $x_{\\mathrm{Fe}_N}$ \\\\\n & $^{\\circ}$C & [$10^{20}$ cm$^{-3}$] & [$10^{19}$ cm$^{-3}$] & [$10^{19}$ cm$^{-3}$] \\\\\n\\hline\nS690 & 800 & 1 & $3.2$ & $0.1$\\\\\nS687 & 850 & 2 & $2.9$ & $1.7$ \\\\\nS680 & 850 & 2 & $2.7$ & $1.5$\\\\\nS987 & 900 & 4 & $2.4$ & $1.6$ \\\\\nS691 & 950 & 4 & $2.9$ & $3.2$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Synchrotron x-ray diffraction -- experimental}\n Coplanar SXRD measurements have been carried out at the Rossendorf Beamline BM20 of the European Synchrotron Radiation Facility (ESRF) in Grenoble -- France, using a photon energy of 10.005~keV. The x-ray data correspond to the diffracted intensities in reciprocal space along the sample surface normals.\nThe beamline is equipped with a double-crystal Si(111) monochromator with two collimating\/focusing mirrors (Si and Pt-coating) for rejection of higher harmonics, allowing measurements in an energy range of 6 to 33~keV. The symmetric $\\omega=2\\theta$ scans are acquired using a heavy-duty 6-circle Huber diffractometer and the most intense peaks are found for 2$\\theta$ up to 40$^{\\circ}$.\n\n\\subsection{XAFS -- experimental and method}\\label{sec:xafs_exp}\n X-ray absorption fine structure (XAFS) measurements at the Fe K edge (7112~eV) are carried out at the ``GILDA'' Italian collaborating research group beam-line~\\cite{d'Acapito:1998_EN} (BM08) of ESRF under the same experimental conditions reported in Ref.~\\onlinecite{Rovezzi:2009_PRB}, collecting both the XANES and EXAFS spectra, and employing the following method for the analysis. \n \nA set of model compounds is established: Fe substitutional of Ga in GaN (Fe$_{\\rm Ga}$),~\\cite{Rovezzi:2009_PRB} $\\zeta$-Fe$_2$N,~\\cite{Rechenbach:1996_JAC} $\\varepsilon$-Fe$_3$N,~\\cite{Jacobs:1995_JAC} $\\gamma$-Fe$_4$N,~\\cite{Jacobs:1995_JAC} $\\alpha$-Fe~\\cite{Swanson1955} and $\\gamma$-Fe.~\\cite{Gorton1965} For these input structures the XANES absorption spectra are calculated using the {\\sc fdmnes} code~\\cite{Joly:2009_JPC} while the EXAFS scattering expansion signals are computed with the {\\sc feff8.4} code~\\cite{Ankudinov:1998_PRB} in order to de-correlate the structural results to a specific software choice. In both cases muffin-tin potentials and the Hedin-Lunqvist approximation for their energy-dependent part is used, with self-consistent potential calculation for enhancing the accuracy in the determination of the Fermi energy ($E_{\\rm F}$). \n\nX-ray polarization is taken into account for Fe$_{\\rm Ga}$ while unpolarized simulations are conducted for the other phases assuming a random orientation of the nanocrystals in the sample. In addition, for XANES the convergence of the results is tested against the increasing in the input cluster size ($>$150 atoms) and the method is validated by experimental values from Fe$_{\\rm Ga}$ and $\\alpha$-Fe. The resulting simulated spectra are then convoluted $\\textit{via}$ an energy-dependent function as implemented in {\\sc fdmnes}~\\cite{Joly:2009_JPC} plus a Gaussian experimental broadening of 1~eV and fitted to the normalized XANES experimental data in the energy range from -20 to 80~eV relative to $E_{\\rm F}$ with a linear combination analysis using the {\\sc Athena} graphical interface~\\cite{Ravel:2005_JSR} to {\\sc i\\-feffit}.~\\cite{Newville:2001_JSR} \n\nAll the possible combinations with a maximum of three spectra per fit (maximum of six fit parameters: amplitude and energy shift) are tested and the best fit is chosen on the basis of the $\\chi^2$ statistics, discarding unphysical results. Finally, the XANES results are independently checked through the quantitative analysis of the EXAFS data where the background-subtracted (via the {\\sc viper} program\\cite{Klementev:2001_JPDAP}) $k^2$-weighted fine-structure oscillations, $\\chi(k)$, are fitted in the Fourier-transformed space.\n\n\\subsection{High resolution transmission electron microscopy -- experimental}\n The HRTEM studies are performed on cross-sectional samples prepared by standard mechanical polishing followed by Ar$^{+}$ ion milling at 4~kV for about 1~h. Conventional diffraction contrast images in bright-field imaging mode and high-resolution phase contrast pictures were obtained from a JEOL 2011 Fast TEM microscope operating at 200~kV and capable of an ultimate point-to-point resolution of 0.19~nm and allowing to image lattice fringes with a 0.14~nm resolution. \n \nAdditionally, energy dispersive x-ray (EDS) analysis has been performed \\textit{via} an Oxford Inca EDS equipped with a silicon detector to obtain information on the local composition. Selected area electron diffraction (SAED) and fast Fourier transform (FFT) procedures are employed to study scattering orders and $d$-spacing for respectively the larger and the smaller nanocrystals. \n\n\\subsection{SQUID magnetometry -- experimental}\n The magnetic properties have been investigated in a\nQuantum Design MPMS XL 5 SQUID magnetometer between 1.85 and 400~K\nand up to 50~kOe following the methodology described\npreviously.\\cite{Stefanowicz:2010_PRB} \n\nThe difference between the\nmagnetization values measured up to 50~kOe at 1.8~K and 5~K is\nemployed to determine the concentration $x_{\\text{Fe$^{3+}$}}$ of\nparamagnetic Fe$^{3+}$ ions in the layers.\\cite{Pacuski:2008_PRL}\nThe lower limit of the concentration of Fe ions contributing to\nthe Fe-rich nanocrystals, as well as an assessment of their Curie\ntemperature is inferred from magnetization curves at higher\ntemperatures. \n\nFinally, measurements of field cooled (FC) and zero field\ncooled (ZFC) magnetization hint to the influence of the growth\ntemperature on the size distribution of the nanocrystals.\n\n\n\\section{Results}\n\\subsection{Fe$_{x}$N phases \\textit{vs.} $T_{\\mathrm{g}}$ in crystallographically separated (Ga,Fe)N}\n\n Before entering into the detailed discussion of our studies, we would like to point out that the reproducibility of the \ndata has been accurately tested and: i) different samples grown under the same conditions have been characterized, ii) all measurements (SXRD, HRTEM, etc...) have been repeated in different runs on the same samples and we can conclude that both the (Ga,Fe)N structures are stable over time and the formation of different phases is reproduced when the growth conditions are fidely replicated. \n\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{FigSXRD.eps} [h]\n \\caption{(Color online) SXRD spectra for (Ga,Fe)N layers deposited at different growth temperatures. Inset: peak at 35.3$^{\\circ}$ deconvoluted into two components assigned to diffraction maxima (111) of $\\varepsilon$-Fe$_3$N and (110) of $\\alpha$-Fe [experiment (dotted line) and fit (smooth line)].}\n \\label{fig:SXRD}\n\\end{figure}\n\nIn Fig.~\\ref{fig:SXRD} we report SXRD diffraction spectra for the (Ga,Fe)N samples grown at different temperatures, as listed in Table I. For the layer S690 fabricated at 800$^{\\circ}$C we have no evidence of secondary phases and only diffraction peaks originating from the sapphire substrate and from the GaN matrix are revealed, in agreement with HRTEM measurements showing no phase separation. Moreover, in order to test the stability of the dilute phase, we have annealed the samples up to $T_{\\mathrm{a}}$ = 900$^{\\circ}$C and $\\textit{in situ}$ SXRD measurements upon annealing do not reveal the onset of any secondary phases, as reported in Fig.~\\ref{fig:SXRD_ann}, in accord with the behavior of dilute Mn in GaN \\cite{Stefanowicz:2010_PRB} and in contrast with (Ga,Mn)As where post-growth annealing is found to promote the precipitation of MnAs nanocrystals. \\cite{Tanaka:2008_B}\n\n\\begin{figure} \n \\includegraphics[width=0.98\\columnwidth]{Fig_anneal2.eps} [h]\n \\caption{SXRD spectra for a dilute (Ga,Fe)N sample (S690) as grown and upon $\\textit{in situ}$ annealing at $T_{\\mathrm{a}}$ = 900$^{\\circ}$C for 1~h, indicating that post-growth annealing does not induce - in the SXRD sensitivity range - crystallographic decomposition.}\n \\label{fig:SXRD_ann}\n\\end{figure}\n\nMoving to a $T_{\\mathrm{g}}$ of 850$^{\\circ}$C (S687) different diffraction peaks belonging to secondary phases become evident. We have previously reported\\cite{Bonanni:2007_PRB} that when growing (Ga,Fe)N at this temperature, one dominant Fe-rich phase is formed, namely wurzite $\\varepsilon$-Fe$_3$N, for which we identify two main peaks corresponding to the (002) and the (111) reflexes, respectively. A closer inspection of the (111)-related feature and a fit with two gaussian curves centered at 35.2$^{\\circ}$ and 35.4$^{\\circ}$, gives evidence of the presence of the (110) reflex from cubic metallic $\\alpha$-Fe. Moreover, the broad feature appearing around 38$^{\\circ}$ is associated to the (200) reflex of face centered cubic ($fcc$) $\\gamma$'-Fe$_4$N, that crystallizes in an inverse perovskite structure.\\cite{Jack:1952_AC} From the position of the peak, we can estimate that these nanocrystals are strained.\n\n\\begin{figure} [htb]\n \\includegraphics[width=0.93\\columnwidth]{FigSXRD_950.eps} [h]\n \\caption{SXRD for a sample (S874) grown at 950$^{\\circ}$C evidencing the aggregation of (200) $\\gamma$-Fe in the (Ga,Fe)N layer.}\n \\label{fig:SXRD_950}\n\\end{figure}\n\nAs the growth temperature is increased to 900$^{\\circ}$C (S987) there is no contribution left from the (110) $\\alpha$-Fe phase, and the signal from the (111) of $\\varepsilon$-Fe$_3$N is significantly quenched, indicating the reduction in either size or density of the specific phase. Furthermore, an intense peak is seen at 34$^{\\circ}$, corresponding to the (121) contribution from orthorhombic $\\zeta$-Fe$_2$N. This phase crystallizes in the $\\alpha$-PbO$_2$-like structure, where the Fe atoms show a slightly distorted hexagonal close packing (\\textit{hcp}), also found for $\\varepsilon$-Fe$_3$N.\\cite{Jacobs:1995_JAC} \n\nThe structural resemblance of $\\varepsilon$-Fe$_3$N and the $\\zeta$-Fe$_2$N is remarkable, as the \\textit{hcp} arrangement in $\\varepsilon$-Fe$_3$N is nearly retained in $\\zeta$-Fe$_2$N.\\cite{Rechenbach:1996_JAC} This gives a hint of the likely direct conversion of phase from $\\varepsilon$-Fe$_3$N into $\\zeta$-Fe$_2$N. The diffraction peak from (200) $\\gamma$'-Fe$_4$N is still present at this temperature, but its position is slightly shifted to its bulk value. A similar behavior is observed for the diffraction from (200) $\\varepsilon$-Fe$_3$N (002), shifted from 32.78$^{\\circ}$ to 32.9$^{\\circ}$.\n\nAt a growth temperature of 950$^{\\circ}$C (S691) the diffraction peak of (200) $\\gamma$'-Fe$_4$N recedes, indicating the decomposition of this \\textit{fcc} phase at temperatures above 900$^{\\circ}$C, in agreement with the phase diagram for free standing Fe$_x$N,\\cite{Jacobs:1995_JAC} reporting cubic $\\gamma$'-Fe$_4$N as stable at low temperatures. Only the (002) $\\varepsilon$-Fe$_3$N- and the (121) $\\zeta$-Fe$_2$N-related diffraction peaks are preserved with a constant intensity and position with increasing temperature, suggesting that at high $T_{\\mathrm{g}}$ these two phases and their corresponding orientations, are noticeably stable. Furthermore, in samples grown at this $T_{\\mathrm{g}}$ the peak from (200) $\\gamma$-Fe is detected around 41.12$^{\\circ}$, as reported in Fig.~\\ref{fig:SXRD_950}, in agreement with the XAFS data discussed later in this Section.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=1.1\\columnwidth]{nanosize2.eps}\n \\caption{(Color online) Average size $vs.$ $T_{\\mathrm{g}}$ of nanocrystals in the different Fe$_x$N phases, as determined from SXRD.}\n \\label{fig:nanosize}\n\\end{figure}\n\nFollowing the procedure employed previously,\\cite{Lechner:2009_APL} and based on the Williamson-Hall formula method,\\cite{Williamson:1953_AcMetal} we obtain the approximate average nanocrystals size from the full-width at half maximum (FWHM) of the diffraction peaks in the radial ($\\omega\/2\\theta$) scans. The FWHM of the (002) $\\varepsilon$-Fe$_3$N, of the (200) $\\gamma$'-Fe$_4$N, and of the (121) $\\zeta$-Fe$_2$N diffraction peaks are comparable for samples grown at different temperatures, indicating that the average size of the corresponding nanocrystals is also constant, as summarized in Fig.~\\ref{fig:nanosize}. \n\nThe (111) $\\varepsilon$-Fe$_3$N signal intensity is seen to change abruptly when comparing the results for the sample grown at 850$^{\\circ}$C to those from the layers fabricated at higher temperatures. From the FWHM for this particular orientation we can estimate that the nanocrystal average size adjusts between 16.5 and 12.0~nm in the considered temperature range. At high temperatures, the size then remains constant up to 950$^{\\circ}$C. The size of the $\\alpha$-Fe nanocrystals can only be estimated for the sample grown at 850$^{\\circ}$C, where the corresponding diffraction peak can easily be resolved and suggests an average size of these objects larger than that of the other identified phases, as confirmed by the HRTEM images reported in Fig.~\\ref{fig:TEM}.\n\n\\begin{figure}[h]\n \\includegraphics[width=0.5\\textwidth]{lcf_fig-v2a}\n \\caption{(Color online) Normalized XANES spectra (main plot) and the amplitude of the Fourier-transforms (inset) of the $k^2$-weighted EXAFS in the range from 2.5 to 10.0 ~\\AA$^{-1}$ for three samples (points) grown at different temperatures, with their relative fits (lines) summarized in Table~\\ref{tab:xafs}.}\n \\label{fig:xafs}\n\\end{figure}\n\nThe XAFS study on the (Ga,Fe)N samples fabricated at different $T_{\\mathrm{g}}$ permits to have a structural description of the atomic environment around the absorbing species from a local point of view, complementary to SXRD. The experimental data are reported in Fig.~\\ref{fig:xafs} with the relative fits obtained by following the method described in Sec.~\\ref{sec:xafs_exp}. Qualitatively an evolution with $T_{\\mathrm{g}}$ is visible and it is quantitatively confirmed by the results summarized in Table~\\ref{tab:xafs}. \n\nIn particular, from the XANES analysis -- sensitive to the occupation site symmetry and multiple scattering effects -- it is possible to infer how the composition of the different phases evolves with increasing $T_{\\mathrm{g}}$: Fe$_{\\rm Ga}$ reduces, while $\\varepsilon$-Fe$_3$N increases up to 950$^\\circ$C when the precipitation is in favor of $\\zeta$-Fe$_2$N and $\\gamma$-Fe. This behavior is confirmed also by the EXAFS spectra given in the inset to Fig.~\\ref{fig:xafs}, where the first three main peaks in the fit range from $R_{\\rm min}$ to $R_{\\rm max}$ represent, respectively, the average Fe-N, Fe-Fe and Fe-Ga coordination. \n\nIn addition, the signal present at longer distances confirms the high crystallinity and permits to include important multiple scattering paths in the fits for a better identification of the correct phase. In fact, from the Fe-Fe distances it is possible to distinguish Fe$_x$N ($\\approx$~2.75~\\AA) from pure Fe phases ($\\approx$~2.57~\\AA), while the distinction between $\\alpha$-Fe and $\\gamma$-Fe is possible with the different multiple scattering paths generated from the body centered cubic ($bcc$) and from the $fcc$ structure, respectively.\n\n\\begin{table*}[htbp]\n\\begin{ruledtabular}\n \\caption{Quantitative results of the XAFS analysis (best fits). XANES: composition ($x$) and energy shift relative to $E_{\\rm F}$ ($\\Delta$$E$) for each structure; EXAFS: average distance ($R$) and Debye-Waller factor ($\\sigma^2$) for the first three coordination shells around the absorber. For each phase the coordination numbers are kept to the crystallographic ones and rescaled by the relative fractions found by XANES and a global amplitude reduction factor, $S_0^2$, of 0.93(5) as found for Fe$_{\\rm Ga}$. Error bars on the last digit are reported in parentheses.}\n \\label{tab:xafs}\n \n \\begin{tabular}{|c|cccccccc|cccccc|}\n \n \n Fit & \\multicolumn{8}{c|}{XANES} & \\multicolumn{6}{c|}{EXAFS} \\\\\n & \\multicolumn{2}{c}{Fe$_{\\rm Ga}$} & \\multicolumn{2}{c}{$\\zeta$-Fe$_2$N} & \\multicolumn{2}{c}{$\\varepsilon$-Fe$_3$N} & \\multicolumn{2}{c|}{$\\gamma$-Fe} & \\multicolumn{2}{c}{Fe-N} & \\multicolumn{2}{c}{Fe-Fe} & \\multicolumn{2}{c|}{Fe-Ga}\\\\\n & $x$ & $\\Delta$$E$ & $x$ & $\\Delta$$E$ & $x$ & $\\Delta$$E$ & $x$ & $\\Delta$$E$ & $R$ & $\\sigma^2$ & $R$ & $\\sigma^2$ & $R$ & $\\sigma^2$\\\\\n & & (eV) & & (eV) & & (eV) & & (eV) & (\\AA) & (10$^{-3}$\\AA$^2$) & (\\AA) & (10$^{-3}$\\AA$^2$) & (\\AA) & (10$^{-3}$\\AA$^2$)\\\\\n \\hline\n 1 & 0.9(1) & 1.3(5) & - & - & 0.1(1) & 2.9(9) & - & - & 1.99(1) & 5(2) & 2.75(5) & 13(5) & 3.20(1) & 7(1)\\\\ \n 2 & 0.6(1) & 1.1(5) & - & - & 0.4(1) & 1.8(5) & - & - & 2.00(2) & 4(1) & 2.76(2) & 9(4) & 3.20(1) & 8(1)\\\\ \n 3 & 0.2(1) & 1.0(5) & 0.4(1) & 4.5(5) & - & - & 0.4(1) & -0.3(5) & 1.95(4) & 10(9) & 2.60(5) & 15(9) & 3.18(2) & 4(2)\\\\ \n \n \n \\end{tabular}\n \\end{ruledtabular}\n \n\\end{table*}\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{Fig_TEM.eps}\n \\caption{Transmission electron micrographs of different Fe$_x$N phases: a) HRTEM image of a $\\varepsilon$-Fe$_{3}$N nanocrystal; (b) the corresponding FFT image, revealing that the \\textit{d}-spacing along the growth direction is about 0.216~nm. (c) HRTEM image on $\\alpha$-Fe nanocrystal in sample S687; (d) SAED pattern on the enclosing area. (e) HRTEM image of a $\\zeta$-Fe$_{2}$N nanocrystal; (f) the corresponding FFT image, revealing that the \\textit{d}-spacing along the growth direction is about 0.211~nm.}\n \\label{fig:TEM}\n\\end{figure}\n\nThe presence of the different Fe$_x$N phases detected with SXRD has been confirmed also by HRTEM measurements on the considered samples, as reported in Fig.~\\ref{fig:TEM}. All the HRTEM images presented here have been taken along the $[10\\overline{1}0]$ zone axis. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{distribution.eps} \n \\caption{(Color online) TEM images: distribution of the Fe-rich nanocrystals with increasing growth temperature. (a) $T_{\\mathrm{g}}$ = 800$^{\\circ}$C (S690) -- dilute (Ga,Fe)N; (b) $T_{\\mathrm{g}}$ = 850$^{\\circ}$C (S687), Fe-rich nanocrystals concentrated in proximity of the samples interface solely; $T_{\\mathrm{g}}$ = 950$^{\\circ}$C (S691) -- Fe-rich nanocrystals segregating in proximity of the sample surface (c) and of the interface between the GaN buffer and the Fe-doped layer (d); (e),(f) $T_{\\mathrm{g}}$ = 950$^{\\circ}$C (S876) -- to be compared with the TEM images of S691 in (c) and (d): reproducibility in the distribution of the Fe-rich nanocrystals for different samples grown at the same $T_{\\mathrm{g}}$.}\n \\label{fig:distribution}\n\\end{figure}\n\nBy using the SAED technique for the larger nanocrystals and a FFT combined with a subsequent reconstruction for the smaller objects, we have studied the foreign scattering orders and the \\textit{d}-spacings along the growth direction. As shown in Fig.~\\ref{fig:TEM}(a), the transitional Moir\\'e fringes indicate that there is a set of planes parallel to the GaN (002) ones with a similar \\textit{d}-spacing inside the nanocrystal. The corresponding FFT image shown in Fig.~\\ref{fig:TEM}(b) gives an additional diffraction spot close to GaN (002), corresponding to a \\textit{d}-spacing of 0.217~nm, matching the $\\textit{d}_{002}$ of $\\varepsilon$-Fe$_{3}$N. The $\\varepsilon$-Fe$_{3}$N phase is found in all the considered samples, with the exception of the one grown at 800$^{\\circ}$C (S690, dilute). The phase $\\varepsilon$-Fe$_{3}$N has the closest structure to wurtzite GaN and we can assume that the formation of $\\varepsilon$-Fe$_{3}$N is, thus, energetically favored.\\cite{Li:2008_JCG}\n\nThe micrograph displayed in Fig.~\\ref{fig:TEM}(c) has been obtained from the layer grown at 850$^{\\circ}$C and refers to a nanocrystal located in the proximity of the sample surface. The corresponding SAED pattern in Fig.~\\ref{fig:TEM}(d) reveals that the \\textit{d}-spacing of the lattice planes overlapping the GaN matrix has a value of 0.203~nm, matching the $\\textit{d}_{110}$ of $\\alpha$-Fe. For values of $T_{\\mathrm{g}}$ between 900 and 950$^{\\circ}$C, nanocrystals like the one represented in Fig.~\\ref{fig:TEM}(e) are found. The FFT image shown in Fig.~\\ref{fig:TEM}(f), reveals that the additional \\textit{d}-spacing is 0.211~nm, corresponding to the $\\textit{d}_{121}$ of $\\zeta$-Fe$_{2}$N.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{phasediagram.eps}\n \\caption{A phase diagram of (Ga,Fe)N as a function of the growth temperature.}\n \\label{fig:phasediagram}\n\\end{figure}\n\nIt should be underlined here that the size of the nanocrystals in the HRTEM images is smaller than the average value obtained from SXRD. This discrepancy originates from the fact that a cross-sectional TEM specimen must be rendered considerably thin, in order to achieve a sufficient transparency to the electron beam. Therefore, a nanocrystal is usually only partly enclosed in the investigated area. At the same time, in low magnification micrographs on thicker volumes the size of the objects becomes comparable to the average value determined by the SXRD studies.\n\nCross-sectional low-magnification TEM measurements permit to observe that while at the lower growth temperatures the Fe-rich nanoobjects tend to segregate close to the sample surface as seen in Fig.~\\ref{fig:distribution}(b), at higher $T_{\\mathrm{g}}$ two-dimensional assemblies of nanocrystals form in a reproducible way -- as proven by comparing Fig.~\\ref{fig:distribution}(c),(d) and Fig.~\\ref{fig:distribution}(e),(f) -- and this arrangement is expected to be instrumental as template for the self-aggregation of metallic nanocolumns.\\cite{Fukushima:2006_JJAP} \n\nSummarizing the SXRD, XAFS and HRTEM findings, a phase diagram of the Fe-rich phases formed in (Ga,Fe)N as a function of the growth temperature is constructed and reported in Fig.~\\ref{fig:phasediagram}, showing the dominant phases for each temperature interval.\n\nAccording to Ref.~\\onlinecite{Jack:1952_AC} when the concentration of the interstitial-atoms in the $\\varepsilon$ phase is increased by only 0.05 atoms\/100 Fe, a phase transition from $\\varepsilon$ to $\\zeta$ occurs. In this process the Fe atoms retain their relative positions but there is a slight anisotropic distortion of the $\\varepsilon$ lattice that reduces the symmetry of the (nano)crystal to $\\zeta$-orthorhombic. The hexagonal unit cell parameter a$_{hex}$ of $\\varepsilon$-Fe$_3$N splits into the parameters $b_{\\mathrm{orth}}$ and $c_{\\mathrm{orth}}$ in $\\zeta$-Fe$_2$N. Moreover, according to the Fe \\textit{vs.} N phase diagram the orthorhombic phase contains a higher percentage of nitrogen\\cite{Jack:1952_AC} compared to the hexagonal one, and this guides us to conjecture that the higher the growth temperature, the more nitrogen is introduced into the system.\n\nRemarkable is the fact that by increasing the growth temperature the (002) $\\varepsilon$-Fe$_3$N is preserved, while the (111) oriented nanocrystals are not detected. A focused study would be necessary to clarify the kinetic processes taking place between 850$^{\\circ}$C and 900$^{\\circ}$C. Moreover, it is still to be clarified whether the fact that the $\\varepsilon$-Fe$_3$N nanocrystals oriented along the growth direction are stable, while the ones lying out of the growth plane are not, may be related to differences in surface energy.\n\nThe Fe$_x$N phases found in our (Ga,Fe)N samples are listed in Table~\\ref{tab:table2}, together with their crystallographic structure, lattice parameters, $d$-spacing for the diffracted peaks, and magnetic properties.\n\n\\begin{table*} \n\\begin{ruledtabular}\n\\caption{\\label{tab:table2}Structural and magnetic parameters of\nthe Fe-rich phases found in the considered (Ga,Fe)N samples.}\n\\label{tab:table2}\n\\begin{tabular}{|c|cccc|cccc|}\n\n&&\\multicolumn{3}{c|}{Lattice parameter\\cite{Eck:1999_JCM}}&\\multicolumn{3}{c}{$d$-spacing}&\\\\\n&&\\multicolumn{3}{c|}{-------------------------------}&\\multicolumn{3}{c}{-------------------------------}&\\\\\n& Structure & $a$(nm)& $b$(nm)& $c$(nm)& literature value\\cite{icdd:2009}& SXRD& HRTEM& $\\mu_{B}$\\\\\n\\hline\n$\\gamma'$-Fe$_4$N & $\\textit{fcc}$ & 0.382 & -- & -- & 0.189 & 0.188-0.189 & 0.188 & 2.21~\\cite{Eck:1999_JCM}\\\\\n$\\varepsilon$-Fe$_3$N & wz & 0.469 & -- & 0.438 & 0.2189$_{(002)}$ & 0.2188$_{(002)}$ & 0.2178$_{(002)}$ & 2.0~\\cite{Leineweber:1999_JAC} \\\\\n& & & & & 0.208$_{(111)}$ & 0.206$_{(111)}$ & -- &\\\\\n$\\zeta$-Fe$_2$N & ortho & 0.443 & 0.554 & 0.484 & 0.2113 & 0.2114 & 0.211 & 1.5~\\cite{Eck:1999_JCM}\\\\\n$\\alpha$-Fe& $\\textit{bcc}$ & 0.286 & -- & -- & 0.202 & 0.204 & 0.203 & 2.2~\\cite{Keavney:1995_PRL} \\\\\n$\\gamma$-Fe& $\\textit{fcc}$ & 0.361 & -- & -- & 0.180$_{(200)}$ & 0.176$_{(200)}$ & -- & 0.3--1.6~\\cite{Shi:1996_PRB} \\\\ \n& & & & & 0.210$_{(111)}$ & -- & -- &\\\\\n\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nFurther focused studies are required in order to clarify the kinetic mechanisms of segregation and possibly the range of parameters that could allow the selectivity of the species in different two-dimensional regions of the doped layers. \n\n\\subsection{Magnetic properties of Fe$_x$N phases}\n\nAs reported in Table~\\ref{tab:table2}, the different Fe$_x$N phases we identify in the considered samples are expected to show specific magnetic responses. The $\\varepsilon$-Fe$_3$N phase, predominant in the samples grown at 850$^{\\circ}$C, is ferromagnetic with a Curie temperature $T_{\\mathrm{C}}$ of 575~K.\\cite{Leineweber:1999_JAC} \n\nThe $\\gamma$'-Fe$_4$N phase, also present though in lesser amount in these layers, is FM too, with a $T_{\\mathrm{C}}$ of 750~K.\\cite{Jack:1952_AC} For the samples deposited at temperatures above 850$^{\\circ}$C, the dominant and stable phase becomes $\\zeta$-Fe$_2$N.\n\nThe magnetic response of these (Ga,Fe)N layers is quite typical for semiconductors containing TM ions at concentration above or close to the solubility limits. Regardless of the prevailing diamagnetic component from the sapphire substrate -- that we compensate with the procedure detailed elsewhere \\cite{Stefanowicz:2010_PRB} -- the field dependency of magnetization $M(H)$ is characterized primarily by a dominant paramagnetic contribution at low temperatures from diluted substitutional Fe$^{3+}$ ions and by a superparamagnetic-like component saturating (relatively) fast and originating from various magnetically ordered nanocrystals with high Fe content. Among them, the FM hexagonal $\\varepsilon$-Fe$_{3-x}$N have the highest density, according to the SXRD and HRTEM studies discussed above. Despite the richness of different phases, it is relatively straightforward to separate these major components and to treat them -- to a large extent -- qualitatively.\n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[width=1.98\\columnwidth]{figANDhistANDfmt.eps}\n \\end{center}\n\\caption{(Color online) (a) Magnetic field dependence of the\nnanocrystals magnetization $M_{\\mathrm{N}}$ at selected temperatures for\nsample S687 ($T_{\\mathrm{g}}$ = 850$^{\\circ}$C). Each $M_{\\mathrm{N}}(H)$\ncurve has been measured from the maximum positive to the maximum negative\nfield, only and the dotted lines obtained by numerical invertion are guides\nfor the eye. The dashed lines represent the saturation level of\nmagnetization at each temperature.\n(b) Bullets - temperature dependence of the saturation\nmagnetization $M_{\\mathrm{N}}^{\\mathrm{Sat}}$ obtained from panel (a).\nDashed line - the Brillouin function for the magnetic moment of\n2$\\mu_{\\mathrm{B}}$ per Fe atom.\n(c) Three major contributions to the total magnetic signal (thick\nbrown solid) for sample S691 ($T_{\\mathrm{g}}$ = 950$^{\\circ}$C): i)\nparamagnetic from Fe$^{3+}$ (thick red dashed), ii) high-$T_{\\mathrm{C}}$\nsuperparamagnetic-like (thin green solid) from the nanocrystals and iii) slowly\nsaturating component (blue short dashed). Also here only a half of the\nfull hysteresis loop was measured and the dotted lines obtained by\nnumerical reflection are guides for the eye. (d) Bullets - temperature\ndependence of $M_{\\mathrm{N}}^{\\mathrm{Sat}}$ for sample S691. Dashed line\n- the Brillouin function for the magnetic moment of 2$\\mu_{\\mathrm{B}}$ per\natom Fe. The blue dotted line follows the excess of\n$M_{\\mathrm{N}}^{\\mathrm{Sat}}$ over the contribution from high\n$T_{\\mathrm{C}}$ ferromagnetic nanocrystals.}\n\\label{Fig:figANDhistANDfmt}\n\\end{figure*}\n\nWe begin by noting that the superparamagnetic-like component originates primarily from nanocrystals characterized by a relatively high magnitude of the spin ordering temperature, so that their magnetization $M_{\\mathrm{N}}(T,H)$ can be regarded as temperature independent at very low temperatures. This means that a temperature dependence of the magnetization in this range comes from dilute Fe$^{3+}$ ions, whose properties in GaN have been extensively investigated previously.\\cite{Pacuski:2008_PRL,Malguth:2008_pssb} \n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.97\\columnwidth]{figFMconc2.eps}\n \\end{center}\n \\caption{(Color online) Estimated lower limit for the Fe concentration that precipitates in the form of various Fe-rich nanocrystals $x_{\\mathrm{Fe}_N}$ as a function of the growth temperature.}\n \\label{fig:figFMconc2}\n\\end{figure}\n\nAccordingly, the concentration of these ions $x_{\\mathrm{Fe}^{3+}}$ can be obtained by fitting $g\\mu_BS$$x_{\\mathrm{Fe}^{3+}}N_0\\Delta B_{S}(\\Delta T,H)$ to the difference between the experimental values of the magnetization measured at 1.85 and 5.0~K, where $\\Delta B_{S}(\\Delta T,H)$ is the difference of the corresponding paramagnetic Brillouin functions\n$\\Delta B_{S}(\\Delta T,H)=B_{S}(1.85\\, \\mathrm{K},H)-B_{S}(5\\, \\mathrm{K},H)$. We consider the spin $S = 5\/2$, the corresponding Land\\'e factor $g=2.0$, and treat $x_{\\mathrm{Fe}^{3+}}$ as the only fitting parameter.\n\nThe values established in this way are listed in Table~\\ref{Tab:table1} for the studied samples and they are then employed to calculate the paramagnetic contribution at any temperature according to $M=g\\mu_{\\mathrm{B}}Sx_{\\mathrm{Fe}^{3+}}B_{S}(T,H)$, which is then subtracted from the experimental data to obtain the magnitude of the magnetization $M_{\\mathrm{N}}(T,H)$ coming from nanocrystals.\n\nFor the layers grown at $T_{\\mathrm{g}} <$ 900$^{\\circ}$C, $M_{\\mathrm{N}}(T,H)$ saturates at all investigated temperatures for a magnetic field above $\\sim10$~kOe, as evidenced in Fig.~\\ref{Fig:figANDhistANDfmt}(a), pointing to a predominantly ferromagnetic order within the nanocrystals. The values of saturation magnetization $M_{\\mathrm{N}}^{\\mathrm{Sat}}$ obtained in this way when plotted \\textit{vs.} temperature as in Fig.~\\ref{Fig:figANDhistANDfmt}(b) allow us to assess the corresponding $T_{\\mathrm{C}}$ from a fitting of the classical Brillouin function to the experimental points. Furthermore, assuming a value of the magnetic moment\n2$\\mu_{\\mathrm{B}}$ per Fe, as in $\\varepsilon$-Fe$_3$N,\\cite{Bouchard:1974_JAP} we determine the concentration\nof Fe ions $x_{\\mathrm{Fe}_N}$ contributing to the Fe-rich nanocrystals, as shown in Table~I.\n\nHowever, for the samples deposited at $T_{\\mathrm{g}} \\geq$ 900$^{\\circ}$C the magnitude of $M_{\\mathrm{N}}(H)$ saturates only at relatively high temperatures, namely around $T\\gtrsim 150$~K, whereas at low temperatures it shows the sizable contribution of a slowly saturating component, as shown in Fig.~\\ref{Fig:figANDhistANDfmt}(c) where magnetization data acquired at $1.85$~K for the layer S691 are reported.\n\nThis new contribution must arise from magnetically coupled objects with a spin arrangement other than ferromagnetic. According to the SXRD measurements previously discussed and summarized in Fig.~\\ref{fig:SXRD} the most likely candidate is orthorombic $\\zeta$-Fe$_2$N, antiferromagnetic below 9~K,\\cite{Hinomura:1996_INC} or slowly saturating weakly ferromagnetic below 30~K.\\cite{Nagamura:2004_STAM} In this case, in order to establish $M_{\\mathrm{N}}^{\\mathrm{Sat}}$ we employ the Arrot plot method. The value of $M_{\\mathrm{N}}^{\\mathrm{Sat}}(T)$ determined in this way is reported in Fig.~\\ref{Fig:figANDhistANDfmt}(d), and is seen to differ considerably from that of layers grown at lower $T_{\\mathrm{g}}$. \n\nWe are able to approximate the experimental values of $M_{\\mathrm{N}}^{\\mathrm{Sat}}(T)$ with a single Brillouin function only for $T\\gtrsim150$~K (dashed line in Fig.~\\ref{Fig:figANDhistANDfmt}(d)). This points to a lower value of $T_{\\mathrm{C}}\\cong 430$~K, indicating a shift of the chemical composition of $\\varepsilon$-Fe$_{3-x}$N from Fe$_3$N ($x\\cong 0$) for $T_{\\mathrm{g}} <$ 900$^{\\circ}$C to at most Fe$_{2.6}$N ($x\\cong 0.4$) for $T_{\\mathrm{g}}\\geq$ 900$^{\\circ}$C, as $T_{\\mathrm{C}}$ of $\\varepsilon$-Fe$_{3-x}$N decreases with increasing nitrogen content.\\cite{Bouchard:1974_JAP} \n\nMoreover, the gradually increasing values of $M_{\\mathrm{N}}^{\\mathrm{Sat}}(T)$ below $T\\lesssim 150$~K, marked as the hatched area in Fig.~\\ref{Fig:figANDhistANDfmt}(d), indicate the presence of even more diluted $\\varepsilon$-Fe$_{3-x}$N nanocrystals with $x$ ranging from 0.5 to 1 and with a wide spectrum of $T_{\\mathrm{C}}$. Importantly, since $\\varepsilon$-Fe$_{3-x}$N preserves its crystallographic structure and the changes of the lattice parameters are minor in the whole range $0\\leq x \\leq 1$, all various $\\varepsilon$-Fe$_{3-x}$N nanocrystals contribute to the same diffraction peak in the SXRD spectrum, and are detected there as a single compound.\n\nWe note that the presence of either $\\varepsilon$-Fe$_3$N or $\\zeta$-Fe$_2$N, characterized by a low spin ordering temperature, does not hinder the determination of the $x_{\\mathrm{Fe}^{3+}}$ values, as both compounds have a rather low magnetic moment of $0.1\\mu_B$ per Fe atom. Accordingly, the resulting variation of their magnetization is small comparing to the changes of the Fe$^{3+}$ paramagnetic signal at low temperatures.\n\nThe procedure exemplified above allows us to establish the lower limit for the Fe concentration that precipitates in the form of various Fe-rich nanocrystals ($x_{\\mathrm{Fe}_N}$), and that is determined by the magnitude of $M_{\\mathrm{N}}^{\\mathrm{Sat}}$ at low temperatures. We again assume $2\\mu_{\\mathrm{B}}$ per Fe atom, as in the dominant $\\varepsilon$-Fe$_3$N. These values are collected in Table~\\ref{Tab:table1} and plotted as the function of $T_{\\mathrm{g}}$ in Fig.~\\ref{fig:figFMconc2}. We see, that $x_{\\mathrm{Fe}_N}$ consistently increases with $T_{\\mathrm{g}}$, and that the growth temperature plays a more crucial role than the Fe-precursor flow rate~\\cite{Bonanni:2007_PRB} in establishing the total value of $x_{\\mathrm{Fe}_N}$.\n\nFinally, measurements of FC and ZFC magnetization confirm the superparamagnetic-like behavior of $M_{\\mathrm{N}}(T,H)$, as reported in Fig.~\\ref{fig:figFCZFC}. As seen in Fig.~\\ref{fig:figFCZFC}(a), the layer grown at the lowest temperature (S690) shows a minimal spread in $T_{\\mathrm{B}}$, having its maximum at $T_{\\mathrm{B}} < 100$~K, and accordingly a non-zero coercivity is evident only at low temperatures. \n\nIn contrast, for most of the studied layers a broad maximum on the ZFC curve, exemplified in Fig.~\\ref{fig:figFCZFC}(b), indicates a wide spread of blocking temperatures ($T_{\\mathrm{B}}$) -- reaching RT -- and consequently a broad distribution in the volume of the nanocrystals. These high values of $T_{\\mathrm{B}}$ are responsible for the existence of the open hysteresis in the $M(H)$ curves seen in Figs.~\\ref{Fig:figANDhistANDfmt}(a),(c) and thus of a non-zero coercivity. This observation again points to the growth temperature as to the key factor in the determination of the crystallographic structure, size and chemical composition of the Fe-rich nanocrystals.\n\n\\begin{figure} [h]\n \\begin{center}\n \\includegraphics[width=0.97\\columnwidth]{figFCZFC.eps}\n \\end{center}\n \\caption{(Color online) ZFC and FC curves measured in applied magnetic field of $200$~Oe for samples grown at (a) 800$^{\\circ}$C and (b) 950$^{\\circ}$C.}\n \\label{fig:figFCZFC}\n\\end{figure}\n\n\\section{Conclusions}\nThe previous\\cite{Bonanni:2007_PRB,Bonanni:2008_PRL,Rovezzi:2009_PRB} and\npresent studies allow us to draw a number of conclusions concerning the\nincorporation of Fe into GaN and about the resulting magnetic properties,\nexpected to be generic for a broad class of magnetically doped\nsemiconductors and oxides. These materials show magnetization consisting\ntypically of two components: i) a paramagnetic contribution appearing at\nlow temperatures and with characteristics typical for dilute magnetic\nsemiconductors containing weakly interacting randomly distributed magnetic\nmoments; ii) a puzzling ferromagnetic-like component persisting up to\nabove RT but with a value of remanence much smaller\nthan the magnitude of saturation magnetization.\n\nAccording to SQUID and electron paramagnetic resonance~\\cite{Bonanni:2007_PRB} measurements on\n(Ga,Fe)N, the concentration of Ga substituting the randomly distributed\nFe$^{3+}$ ions increases with the iron precursor flow rate\nreaching typically the value of 0.1\\%. Our results imply that the\nmagnitude of the paramagnetic response and, hence, the density of dilute\nFe cations, does not virtually depend on the growth temperature. However,\nthe incorporation of Fe can be enlarged by co-doping with Si donors,\nshifting the solubility limit to higher Fe concentrations.\\cite{Bonanni:2008_PRL}\n\nThe presence of ferromagnetic-like features can be consistently\ninterpreted in terms of crystallographic and\/or chemical phase separations\ninto nanoscale regions containing a large density of the magnetic\nconstituent. Our extensive SQUID, SXRD, TEM, EXAFS, and XANES measurements\nof MOVPE-grown (Ga,Fe)N indicate that at the lowest growth temperature\n$T_{\\mathrm{g}} = 800^\\circ$C, a large majority of the Fe ions occupy random\nGa-substitutional positions. However, in films grown at higher\ntemperatures, $850 \\leq T_{\\mathrm{g}} \\leq 950^\\circ$C, a considerable\nvariety of Fe-rich nanocrystals is formed, differing in the Fe to N ratio.\nIn samples deposited at the low end of the $T_{\\mathrm{g}}$ range,\nwe observe mostly $\\varepsilon-$Fe$_3$N precipitates but also inclusions of\nelemental $\\alpha$- and $\\gamma$-Fe as well as of $\\gamma'$-Fe$_4$N. In\nall these materials $T_{\\mathrm{C}}$ is well above RT,\nso that the presence of the corresponding nanocrystals explains the\nrobust superparamagnetic behavior of (Ga,Fe)N grown at $T_{\\mathrm{g}}\n\\geq 850^\\circ$C.\n\nWith the increase of the growth temperature nanocrystals of\n$\\zeta$-Fe$_2$N form and owing to antiferromagnetic interactions specific\nto this compound, the magnetization acquires a component linear in the\nmagnetic field. This magnetic response \nhas been previously observed and assigned to the Van Vleck paramagnetism\nof isolated Fe$^{2+}$ ions. In view of the present findings, however, its\ninterpretation in terms of antiferromagnetically coupled spins in nitrogen\nrich Fe$_x$N ($x \\le 2$) nanocrystals seems more grounded.\n\nThe total amount of Fe ions contributing to the formation of the Fe-rich\nnanocrystals is found to increase with the lowering of the growth rate\nand\/or with the raising of the growth temperature. At the same time,\nhowever, the size of individual nanocrystals appears not to vary with\nthe growth parameters. Furthermore, annealing of (Ga,Fe)N containing only\ndiluted Fe cations does not result in a crystallographic phase separation.\nAltogether, our findings indicate that the aggregation of Fe ions occurs\nby nucleation at the growth front and is kinetically limited.\nMoreover, according to the TEM results presented here, the spatial distribution of\nnanocrystals is highly non-random. They tend to reside in two-dimensional\nplanes, particularly at the film surface and at the interface between the GaN buffer and the nominally Fe-doped layer. \n\nAs a whole, these findings constitute a significant step\non the way to control the chemistry and local structure of\nsemiconductor\/ferromagnetic metal nanocomposites.\\\\\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nThe work was supported by the European Research Council through the FunDMS Advanced Grant within the \"Ideas\" 7th Framework Programme of the EC, and by the Austrian Fonds zur {F\\\"{o}rderung} der wissenschaftlichen Forschung -- FWF (P18942, P20065 and N107-NAN). We acknowledge the technical staff at the Rossendorfer Beamline (BM20) of the ESRF, and in particular C. B\\\"{a}htz and N. Jeutter for their valuable assistance. We also thank R. Jakie{\\l}a for performing SIMS measurements.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Sun's outer atmosphere, the corona, is a hot ($\\sim 10^6$ Kelvin), highly magnetized, dynamic plasma, with spatial scales ranging from the electron gyroradius of less than 1 cm to the solar radius of over $10^{13}$ cm. As such, it is one of the most fruitful objects for the study of astrophysical plasmas. One of the most fascinating puzzles of the Corona is how it is heated to temperatures $\\sim 1000$ times those of the underlying photosphere. It is believed that this process is magnetic in nature, but the details (e.g., the spatial distribution of heating and whether it is continuous or impulsive) are not known. One important tool for understanding these details is the reconstruction of solar temperature distributions called `Differential Emission Measures' (DEMs) from spectral images of the Sun.\n\nThe DEM, ${\\cal E} (T)$ characterizes coronal emission at a given temperature\\footnote{For most spectral lines; lines corresponding to forbidden transitions have more complicated density sensitivity, but that is beyond the scope of this paper. Interested readers may consult \\citet{delzannamasonasr2005}.}, and is defined in terms of the densities, $n(l)$, along the line-of-sight so that $\\int{\\cal E} (T) dT \\equiv \\int n^2(l)dl$ (column emission convention). Observed intensities, ${I}_n$,\nare given by integrating the instrument response, $R_n(T)$, against the DEM being observed:\n\\begin{equation}\\label{eq:demconvolution}\n\t{I}_n = \\int R_n(T){\\cal E} (T)dT\n\\end{equation}\n\nAlgorithms which reconstruct these temperature distributions must be very fast if they are to process the volume of data produced by modern solar observatories, or resolve the dynamics and fine spatial scales believed to be involved in coronal heating. For example, matching the real-time data rate of AIA \\citep{lemenetal_aia_soph_2012}, requires computation of over $10^5$ DEMs per second. \nThere are a number of existing algorithms for reconstructing DEMs from solar image data, but they are far too slow to meet this requirement. \nOne widely used method is the PINTofALE code \\citep{kashyap_drake_apj_1998}, which employs a Markov Chain Monte Carlo (MCMC) search taking seconds or minutes to compute a DEM. Another method, which is considered relatively fast and was recently applied to AIA data by \\citet{hannahkontaraap2012}, computes $\\sim 4$ DEMs per second (an updated version with substantially improved performance has recently been made available at \\url{http:\/\/www.astro.gla.ac.uk\/~iain\/demreg\/map\/}; we use this version in our comparisons).\n\nWe present a fast, iterative, regularized method for inferring DEMs using data from solar imagers such AIA and EIS. With one thread on a 3.2GHz processor, it is able to compute well over 1000 DEMs per second for an example solar active region observed by AIA, and over $100$ DEMs per second in our most difficult test cases. Moreover, we anticipate that with straightforward optimizations (e.g., conversion of computationally intensive portions of the code to C and parallelization), the performance of the code will be increased to $\\sim 10^5$ DEMs per second on a single workstation, sufficient to match AIA's real-time observing rate. \n\nThis paper describes the method, analyzes its fidelity, compares its performance and results with other DEM methods, and applies it to example solar data. We also touch on the limitations of the solar data to constrain the DEM being observed. It must be noted that the ability to recover the details of the original solar input DEM is limited, both because of the number of channels are limited and because the temperature response functions are broad, due to the width of the temperature-dependent emissivities of the spectral lines from which they are constructed. This is discussed in detail, for instance, in \\citet{craigbrownaap1976,judgeetalapj1997}.\n\n\n\\section{DEM Algorithm}\n\\subsection{First Pass}\\label{sub:firstpass}\nWe wish to infer the DEM, a continuous function, from a small number of observed intensities which are the result of the convolution of the DEM with the instrument response functions. This problem is inherently ill-posed due to the limited number of response functions and the loss of information resulting from the convolution (i.e., equation \\ref{eq:demconvolution}) . To resolve the ambiguity, we impose additional constraints on the DEM solution. The first is that the DEM can be expressed as a linear combination of some set of basis elements (e.g., a set of narrow temperature bins), $B_j$, with coefficients $e_j$ (i.e., ${\\cal E} (T) = \\sum_j e_j B_j$). The integral equation (\\ref{eq:demconvolution}) then becomes a matrix equation:\n\\begin{equation}\\label{eq:demmatrix}\n\t{I}_n = \\int R_n(T) {\\cal E} (T)dT = \\sum_k\\Big[ {e}_k \\int R_n(T) B_k(T)dT\\Big] \\equiv \\sum_k {e}_k \\gamma_n A_{nk},\n\\end{equation}\nWhere $\\gamma_n$ are a set of normalization constants for the response functions, which we choose to be their squared integral, square root. \n\nIf the number of basis elements is greater than number of instrument channels, the underconstrained inversion can be resolved by a Singular Value Decomposition \\citep[SVD;][]{nr}, which picks the coefficient vector with the smallest magnitude (i.e., least squared emission measure). This is a sort of smoothness constraint, since a solution which has emission concentrated into narrow peaks will have greater total squared EM than a solution which spreads the emission as broadly as the data will allow. Such a solution will also tend to reduce non-physical negative emission, since negative EM is likely to require excess positive emission measure elsewhere in order to produce the observed intensities, resulting in relatively high total squared EM.\n\nIf the number of basis elements is equal to the number of instrument channels, it is straightforward to invert equation \\ref{eq:demmatrix} to find the DEM coefficients, ${e}_k$. In that case, however, the quality of the inversion is highly sensitive to choice of basis. In particular, many choices of basis lead to an inversion matrix (i.e., $A_{jk}^{-1}$) that has large negative entries, causing large negative values in the resulting DEMs. \n\nThe tendency of the inverse matrix to have large negative values is reduced, however, if $A_{jk}$ is diagonally dominant (i.e., each diagonal entry of the matrix is the largest one in it's respective row\/column). In particular, if we choose the basis functions to be the instrument response functions themselves, square-normalized, we obtain a symmetric $A_{jk}$ matrix whose diagonal entries are unity, with all off-diagonal entries less than one.\nRemarkably, we find that this basis gives results identical to using a large number of narrow basis elements and an SVD\\footnote{Appendix \\ref{app:instresp_demderivation} demonstrates why this is the case}. It therefore satisfies the SVD's minimum squared emission measure constraint, and we use it to compute a first pass DEM as a starting points for our DEM inversion. It reduces the number of operations required for each pixel to only $\\sim N^2$ operations with $N$ instrument channels, increasing the speed of the inversion. \n\nEven when using the response functions as a basis, however, we continue to compute $A_{ij}^{-1}$ using an SVD. This allows us to enforce a minimum condition ($\\sim 10^{-12}$, typically\\footnote{This means the smallest singular value used to form the inverse matrix must be at least $10^{-12}$ times the largest singular value.}) on the inversion, ensuring that round-off error (from $A_{ij}$ being nearly singular) does not cause ringing. In effect, this combines highly similar channels (or linear combinations of channels, to be more precise) into a single channel rather than trying to fit them individually. \n\n\\subsection{Regularization}\\label{sub:reg}\nAnother constraint which DEM solutions must satisfy is a physically required positivity constraint. However, the first pass DEM inversion described in Section \\ref{sub:firstpass} can produce negative coefficients and therefore negative EM. This is particularly true when the errors in the observed intensities are significant, as the first-pass solutions exactly reproduce the input data and there is no guarantee that a set of noisy data intensities can be exactly reproduced by a purely positive DEM. To mitigate this issue, we once again seek to minimize the total squared emission measure, but we now allow the DEM to deviate from the input data at a specified $\\chi^2$ level. This is implemented by seeking a new DEM which minimizes the sum\n\\begin{equation}\n\t\\chi^2+\\lambda\\int [{\\cal E} (T)]^2dT = \\sum_j\\frac{\\Delta {I}_j}{\\sigma_j^2} + \\lambda\\sum_{jk}{e}_j {e}_k A_{jk},\n\\end{equation}\nwhere ${\\cal E} (T)$ is the DEM described above, and $\\lambda$ is a regularization parameter chosen to enforce the desired $\\chi^2$ threshold, $\\chi^2_0$. Using our basic DEM solution, we replace ${e}_i$ with a set of corresponding regularization corrections to the data values, $\\Delta {I}_i$, so that ${e}_j = \\sum_k A^{-1}_{jk} ({I}_k+\\Delta {I}_k)\/\\gamma_k$:\n\\begin{equation}\n\t\\chi^2+\\lambda\\int [{\\cal E} (T)]^2dT = \\sum_j\\frac{\\Delta {I}_j^2}{\\sigma_j^2} + \\lambda\\sum_{jk}({I}_j+\\Delta {I}_j)\\frac{A_{jk}^{-1}}{\\gamma_j\\gamma_k}({I}_k+\\Delta {I}_k).\n\\end{equation}\nThis nonlinear system of equations may be solved for the data corrections, $\\Delta{I}_j^2$, and regularization parameter $\\lambda$, satisfying the desired $\\chi^2$ threshold $\\chi_0^2$, in a variety of ways; we have found that a standard bisection search \\citep[see][for example]{nr} gives acceptable performance. The new regularized DEM is simply the first-pass inversion applied to the new corrected data values.\n\nRegularized solution of the DEM problem has recently been discussed by \\citet{hannahkontaraap2012}, but our regularization is simpler and faster owing to the basis set used for the DEM. Since DEMs constructed from the instrument response functions have only a small number of basis elements, however, they remain liable to producing negative emission in cases with sharp features. The solution to this problem is discussed next.\n\n\\subsection{Enforcing Non-Negativity}\nWe remove the remaining negative emission via the following iterative process: \n\\begin{enumerate}\n\t\\item Zero the negative EM in the current DEM, ${\\cal E}^{(n)}$, to create a new DEM, ${\\cal E}_+^{(n)}$. At the zeroth iteration, this is the regularized DEM from Section \\ref{sub:reg} ${\\cal E}^{(0)}=\\sum_j {e}_j B_j(T)=\\sum_{jk} B_j(T) A^{-1}_{jk} ({I}_k+\\Delta {I}_k)\/\\gamma_k$.\\label{enum:initialstep}\n\t\\item Compute the data intensities, ${I}_j^+=\\int {\\cal E}_+^{(n)}(T)R_j(T)dT$, corresponding to ${\\cal E}_+^{(n)}$.\\label{enum:iterationposintensities}\n\t\\item Take the difference between ${I}_j^+$ and the original ${I}_j$, $\\Delta {I}_j^+ = {I}_j^+-{I}_j$.\n\t\\item Compute correction DEM coefficients, $\\Delta {e}_j = \\sum_k A_{jk}^{-1}\\Delta {I}_k^+\/\\gamma_k$. $A_{jk}^{-1}$ is computed using an SVD at this step, and we enforce a relatively strong minimum condition number ($\\sim 0.1$) to reduce ringing in the correction.\\label{enum:iterationcorrectionstep}\n\t\\item Subtract the corresponding DEM corrections, $\\Delta {\\cal E}^{(n)}$ from ${\\cal E}_+^{(n)}$. By construction, this restores ${\\cal E}^{(n+1)} \\equiv {\\cal E}_+^{(n)}-\\Delta {\\cal E}^{(n)}$ to agreement with \n\tthe data, but reintroduces some negative emission.\n\t\\item Repeat from step \\ref{enum:initialstep} until ${I}_i^+$ matches ${I}_i$ to within the desired $\\chi^2$.\n\\end{enumerate}\n\nAfter we zero the negative EM in the initial DEM (i.e., at step \\ref{enum:initialstep} of the zeroth iteration), the DEM is no longer represented by a basis of instrument response functions, but rather by a continuous function (see step \\ref{enum:iterationposintensities} of the iteration). In practice, we choose to express it using an intermediate basis of closely spaced, narrow functions - usually triangle functions. \n\nTo further speed convergence of the iteration, we attempt a linear extrapolation at each step of the iteration. The extrapolation steps move the DEM vector further along the direction of the current iteration step, and are only accepted when they improve $\\chi^2$. \n\nWe find that the optimal regularization strength, and minimum condition number strength in step \\ref{enum:iterationcorrectionstep} of the iteration, vary with input DEM and data quality, so we try multiple values of these parameters, beginning with a light regularization and small minimum condition number threshold and moving to stronger regularization and larger minimum condition number threshold. Typical values of these parameters are $[0.9,0.5,0.01]$\\footnote{We define the regularization parameters to be p-values of the $\\chi^2$ distribution \\citep[see, for instance, Chapter 15 of][where they are referred to as $Q$, and $P$ refers to their complement]{nr}. With a regularization parameter of 0.9, for instance, the $\\chi^2$ of the regularized data with respect to the original data will correspond to a p-value of $0.9$.} and $[0.01,0.05,0.1]$, for the regularization strengths and minimum condition numbers, respectively. In broad terms, the operation of the algorithm can then be described as follows:\n\\begin{itemize}\n\\item Apply the light (e.g., 0.9) regularization to the data and compute the corresponding first-pass DEM.\n\\item Zero the negative emission in the first-pass DEM and compute the $\\chi^2$ of the resulting data with respect to the initial data. If $\\chi^2$ exceeds the desired threshold, attempt to iterate away the negative emission, using the smallest minimum condition threshold (e.g., 0.01), until the $\\chi^2$ threshold is reached.\n\\item If the iteration takes too long to reach the desired $\\chi^2$ threshold, retry the first two steps with the next strongest regularization strengths and minimum conditions (e.g., 0.5 and 0.05, respectively). Repeat until the $\\chi^2$ threshold is reached, or each pair of regularization strengths and minimum conditions have been tried.\n\\end{itemize}\n\nReaders who would like more implementation details are encouraged to examine our code, which we have made publicly available at \\url{http:\/\/solar.physics.montana.edu\/plowman\/firdems.tgz}. We also intend to submit the code to SolarSoft in the near future.\n\n\\section{Fidelity \\& Performance}\nWe have tested our DEM using a variety of example cases. The results are given in figures on the following pages, which show recovered DEMs, $\\chi^2$ agreement with the data, and computational time. Please note the following:\n\\begin{itemize}\n\\item All reported $\\chi^2$ have been reduced by dividing by the number of instrument channels being fit. \n\\item Unless otherwise noted, we use a one second exposure time for AIA, and 30 seconds for EIS. EIS pixels were assumed to be sampled to 2 square arcseconds.\n\\item AIA errors are assumed to be from read and shot noise alone, while EIS errors assume an\nadditional 10\\% error from other sources.\n\\item We use the 94, 131, 171, 193, 211, and 335 \\AA\\ AIA channels throughout. \n\\item EIS emissivities were computed assuming a density of $10^9 \\ \\mathrm{cm}^{-3}$, unless otherwise noted.\n\\item AIA response functions were computed by calling \\texttt{aia\\_get\\_response(\/temp, \/dn, \/evenorm)}. \n\\item Computation times are for a single thread running on a 3.2GHz Intel Xeon processor, in IDL. \n\\end{itemize}\n\nWherever possible, we use $\\log_{10}(T)$ as a temperature variable rather than $T$ (in Kelvin) itself. We believe this is the more natural parameter for the temperature, since we are interested in a temperature range spanning multiple orders of magnitude ($\\sim 10^5 - 10^7$ Kelvin). Similarly, our DEMs are scaled per unit $\\log_{10}(T)$, rather than per unit $T$. We believe this is the natural scaling when the DEMs are represented as functions of $\\log_{10}(T)$, because the area under the DEM curves is then the emission measure. Rescaling to unit $T$ may be accomplished by dividing by $T\\ln{(10)}$.\n\nWith few exceptions, we find that we are able to recover the test cases with good $\\chi^2$ (Reduced $\\chi^2$ of one or two), that the recovered DEMs are a reasonable qualitative match to the input DEMs. Typical times for AIA DEMs are approximately one millisecond, with some cases taking under 0.1 millisecond. The test cases are as follows:\n\\begin{itemize}\n\t\\item AIA inversion of Log-normal DEMs with widths of 0.2 at selected temperatures (Fig. \\ref{fig:aia_lognormaltest01_5e28}).\n\t\\item EIS inversion of Log-normal DEMs with widths of 0.2 at selected temperatures (Fig. \\ref{fig:eis_lognormaltest01_5e28}).\n\t\\item AIA inversion of Log-normal DEMs at temperatures ranging from $10^{5.5}$ to $10^{7.0}$ Kelvin, for widths of 0.2 (Fig. \\ref{fig:aia_gresp_02_5e28}) and 0.3 (Fig. \\ref{fig:aia_gresp_03_5e28}).\n\t\\item EIS inversion (using the 24 lines chosen by Warren et al.\\cite{warrenetal_apj_2011}) of Log-normal DEMs at temperatures ranging from $10^{5.5}$ to $10^{7.0}$ Kelvin, for widths of 0.2 (Fig. \\ref{fig:eis_gresp_02_5e28}) and 0.3 (Fig. \\ref{fig:eis_gresp_03_5e28}).\n\t\\item AIA inversion of DEMs produced by summing five Log-normal DEMs with randomly chosen centers, widths, and amplitudes (Fig. \\ref{fig:aia_multimod}).\n\t\\item EIS Active region DEM from \\cite{warrenetal_apj_2011} (Fig. \\ref{fig:warrendem_comparison}). A density of $10^{9.5} \\ \\mathrm{cm}^{-5}$ is assumed, to match their emissivities.\n\\end{itemize}\n\n\nWe find that narrow (compared with the temperature response functions in question) DEMs are the most difficult for our method to reconstruct, in terms of obtaining good $\\chi^2$ (i.e., $\\chi^2_R \\sim 1$) without negative emission. This can be seen by comparing Figures \\ref{fig:aia_gresp_02_5e28} and \\ref{fig:aia_gresp_03_5e28} (for AIA), or Figures \\ref{fig:eis_gresp_02_5e28} and \\ref{fig:eis_gresp_03_5e28} (for EIS). For AIA, there is somewhat more difficulty in recovering narrow DEMs at temperatures above $10^{6.5}$ Kelvin, but even in that case, acceptable $\\chi^2$ were achieved for the majority of noise realizations. Despite their relatively poor $\\chi^2$, these reconstructed DEMs are well-behaved and localized at the center of the injected DEM; the median $\\log_{10}(T)$ is recovered to within $0.1$ between $\\log_{10}(T) \\approx 5.8$ and $7.0$. The difficulty in achieving good $\\chi^2$ is absent in cases where there the emission is not concentrated near a single temperature, as can be seen in figure \\ref{fig:aia_multimod}. In all cases considered, the average time to compute a DEM was under 10 milliseconds, and less than 0.1 millisecond for some cases.\n\nIn the case of EIS, temperatures greater than $10^{6.8}$ Kelvin, above the highest temperature peak of the spectral lines used, are the most challenging. The median temperatures are accurately (i.e., $\\delta\\log_{10}(T)\\lesssim 0.1$) recovered over a range of $\\log_{10}(T)$, $5.6\\dots 6.8$. Due the larger number of channels, the computational time is somewhat longer, at $\\sim 10$ milliseconds for a narrow log-normal distribution. \n\n\\begin{figure}[!ht]\n\t\\caption{DEM inversions of Log-normal simulated DEMs of width 0.2 using the six AIA EUV channels. The input DEM is shown by the solid line, while each dotted line is a DEM inversion with randomly chosen read and shot noise.}\\label{fig:aia_lognormaltest01_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_dem_iterative_test_plots.eps}\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\caption{Same as figure \\ref{fig:aia_lognormaltest01_5e28}, but for the 24 EIS lines used in \\citet{warrenetal_apj_2011}}\\label{fig:eis_lognormaltest01_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_dem_iterative_test_plots.eps}\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\caption{AIA Response to Log-normal DEMs of width 0.2 and total EM $5.0 \\times 10^{28} \\textrm{cm}^{-5}$ at temperatures from 5.5 to 7.0 dex. Left: the recovered DEMs - each vertical slice of this plot is a DEM like one of the output curves of Fig. \\ref{fig:aia_lognormaltest01_5e28}, at the temperature of the corresponding x axis value. The solid lines on left show emission measure weighted median temperature (EMWMT). Center: $\\chi^2$ percentiles resulting from repeated MC trials of the read and shot noise, at each temperature. Right: The average time per DEM at each temperature.}\\label{fig:aia_gresp_02_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_gresp_width02_5e+28EM.eps}\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\caption{Same as Fig. \\ref{fig:aia_gresp_02_5e28}, but for Log-normal DEMs of width 0.3.}\\label{fig:aia_gresp_03_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_gresp_width03_5e+28EM.eps}\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[!ht]\n\t\\caption{EIS Response to Log-normal DEMs of width 0.2 and total EM $5.0 \\times 10^{28}\\textrm{cm}^{-5}$ at temperatures from 5.5 to 7.0 dex. Solid lines on left shows emission measure weighted median temperature (EMWMT). The 24 spectral lines from \\citet{warrenetal_apj_2011} were used.}\\label{fig:eis_gresp_02_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_gresp_width02_5e+28EM.eps}\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\caption{EIS Response to Log-normal DEMs of width 0.3 and total EM $5.0 \\times 10^{28} \\textrm{cm}^{-5}$ at temperatures from 5.5 to 7.0 dex. Solid lines on left shows emission measure weighted median temperature (EMWMT). The 24 spectral lines from \\citet{warrenetal_apj_2011} were used.}\\label{fig:eis_gresp_03_5e28}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_gresp_width03_5e+28EM.eps}\\end{center}\n\\end{figure}\n\n\nFigure \\ref{fig:aia_multimod} compares our DEM inversions (top set) with those of the optimized \\citet{hannahkontaraap2012} (bottom set) `demmap' code. We ran the their code without its positivity constraint option, finding better $\\chi^2$ were achieved by simply zeroing the negative emission. The tests DEMs are randomly generated multimodal distributions, as observed by AIA. Both codes were run in a single thread on the 3.2 GHz processor mentioned above. The results are qualitatively quite similar, although the Hannah \\& Kontar method appears to produce some spurious high-temperature emission. For these test cases, our DEM method is faster at $\\sim 10^{-4}$ seconds for most DEMs compared with $3.7\\times 10^{-3}$ seconds (without positivity constraint) for the Hannah \\& Kontar DEM. Our method also appears to produce reasonable $\\chi^2$ with zero negative emission more consistently than the Hannah \\& Kontar DEM inversions.\n\n\\begin{figure}[!ht]\n\t\\caption{AIA inversion of DEMs produced by summing five Log-normal DEMs with randomly chosen centers, widths, and amplitudes. Center locations are uniformly distributed between $10^{5.75}$ and $10^7$, widths between $0.1$ and $0.3$ and total EM between $5\\times 10^{27} \\textrm{cm}^{-5}$ and $5\\times 10^{28} \\textrm{cm}^{-5}$. Top: Results from our fast DEM Method. Bottom: Same for \\citet{hannahkontaraap2012} regularized DEM method, without positivity constraint.}\\label{fig:aia_multimod}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_dem_multimod_test_plots.eps}\\end{center}\n\n\t\\hrule\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_dem_multimod_test_plots_hannah.eps}\\end{center}\n\\end{figure}\n\n\n\nFigure \\ref{fig:warrendem_comparison} compares our DEM results with the PINTofALE MCMC results reported in Figure 4 of \\citet{warrenetal_apj_2011}. These results include all 24 EIS lines used by Warren et al, with their factor of 1.7 tweak to the Mg intensities, along with the XRT Open\/Al-thick filter. Our DEM curve is considerably different than that of \\citet{warrenetal_apj_2011}. In particular, we do not find a peak in the DEM at $\\log_{10}(T)\\approx 6.6$. Despite this difference, we obtain a very reasonable $\\chi^2$ of 1.16, which is fully consistent with the data. This suggests that the \\citet{warrenetal_apj_2011} reconstruction may not represent the full range of DEMs consistent with their data.\n\n\n\\begin{figure}[!ht]\n\t\\caption{Comparison of our DEM results (top) with \\citet{warrenetal_apj_2011} (Figure 4) MCMC DEM results (bottom). We include their factor of 1.7 tweak to the coronal Mg abundance. The DEMs matched the data with $\\chi^2$ of order unity.}\\label{fig:warrendem_comparison}\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_dem_warren_comparison.eps}\\end{center}\n\t\\begin{center}\\includegraphics[width=\\figwidth]{warren_dem.eps}\\end{center}\n\\end{figure}\n\n\n\\subsection{Example Solar DEM Analysis}\n\nFinally, we show example DEM analysis of solar data. Figure \\ref{fig:example_ardem} shows emission weighted median temperature (EMWMT) and total emission measure for an active region observed by EIS and AIA. The data cover the area of the large raster taken by EIS at around 01:30 on April 19, 2011. They are centered $\\sim 350$ arcseconds above disk center, with an approximately $230\\times510$ arcsecond field of view. The EIS data used fits to a set of five iron lines: Fe IX 188.497, Fe X 184.537, Fe XII 195.119, Fe XV 284.163, and Fe XVI 262.976. \n\nThe average time per AIA DEM was approximately $0.13$ milliseconds, and we achieve reasonable $\\chi^2$ for over 95\\% of the AIA DEMs (the 95th percentile $\\chi^2$ for EIS is 4.7). The active region plots in Figure \\ref{fig:example_ardem} are qualitatively quite similar, suggesting that the results give useful insight into the underlying solar temperature distribution. \n\nWe also show DEMs along the loop segment indicated by the dashed line in Figure \\ref{fig:example_ardem}. Figure \\ref{fig:example_loopdem} shows DEMs along the length of the loop segment (solid lines in Figure \\ref{fig:aia_extracted_loop}), as well as equivalent sets of DEMs offset to either side of the loop (dotted lines in Figure \\ref{fig:aia_extracted_loop}). Once again, the DEMs found by AIA and by EIS are reasonably consistent, and both EIS and AIA inversions achieved good $\\chi^2$ for over 95\\% of the points along the loop. \n\nThe DEMs shown along the loop axis are not made from background subtracted data, and they show little difference from the offset DEMs, which indicates that the loop emission has been swamped out be the bright background. In an upcoming work, we compute background subtracted DEMs for this loop and compare its temperature and density profile to a set of analytic strand heating models.\n\n\\begin{figure}[!ht]\n\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_aia_ardem.eps}\\end{center}\n\t\\caption{Left: EMWMT (hue) and Total EM (intensity) from AIA DEM inversion of Active region (covered by EIS fov on April 19, 2011). Middle: Same for EIS. Right: color scale for middle and left plots.}\\label{fig:example_ardem}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\begin{center}\\includegraphics[width=\\figwidth]{eis_aia_loopdem.eps}\\end{center}\n\t\\caption{Top: Loop DEM from AIA inversion of Active region (Center), offset 5 arcseconds to the left (left), 5 arcseconds to the right (right). Loop area indicated by dashed line in Figure \\ref{fig:example_ardem}, or in more detail in Figure \\ref{fig:aia_extracted_loop}. Bottom: Same for EIS.}\\label{fig:example_loopdem}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\begin{center}\\includegraphics[width=\\figwidth]{aia_extracted_loop.eps}\\end{center}\n\t\\caption{Detailed, straightened AIA images of loop outlined in Figure \\ref{fig:example_ardem}, for comparison with Figure \\ref{fig:example_loopdem}. Loop trace indicated by solid line, while $\\pm 5$ arcsecond offset background traces are indicated by dotted lines.}\\label{fig:aia_extracted_loop}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe have demonstrated a method for fast reconstruction of DEM distributions using coronal data from instruments such as EIS and AIA. This DEM method achieves reduced chi-squared of order unity with no negative emission in all but a few test cases. The most difficult test cases are narrow DEMs at high ($>10^{6.5}$ Kelvin) temperatures for AIA, and temperature regions with little spectral coverage for EIS. Even for the high temperature AIA cases, we achieve reasonable $\\chi^2$ for the majority ($\\sim 75\\%$) of noise realizations. Qualitatively, the reconstructed DEMs match the input DEMs well, although the ability to recover finer details of the input DEMs is inherently limited. The data, particularly for AIA, do not constrain the fine details of the DEMs. When interpreting DEMs, great care must be exercised to determine whether or not the features of interest are genuine aspects of the coronal temperature distribution.\n\nOur DEM method is fast enough to study the dynamics of the coronal temperature distributions in real time. For AIA data, the worst cases considered take under $10$ milliseconds, and some test cases execute in less than $0.1$ milliseconds. For the whole disk, at full AIA resolution using all six coronal EUV channels, we expect less than $\\sim 1$ hour with the code as currently implemented. We plan to convert the computationally intensive parts of the code to C and rewrite them to take advantage of multithreading, which should offer a factor of $\\sim 100$ performance gain on an eight core workstation. This would reduce the time for a set of full-resolution, full-disk AIA DEMs to under $\\sim 1$ minute, and achieve the objective of matching the AIA observing rate in real time. The software is available online at \\url{http:\/\/solar.physics.montana.edu\/plowman\/firdems.tgz}, and will be submitted to SolarSoft in the near future.\n\nWe also applied our DEM reconstruction to solar data, an active region observed by EIS and AIA on April 19, 2011. We found no difficulty achieving reduced $\\chi^2$ of order unity with no negative emission, and the average time per DEM was approximately $0.13$ milliseconds, or $44$ seconds for the entire active region at full AIA resolution. We find a relatively hot active region core with median temperature of around three million Kelvin, surrounded by cooler emission with median temperature of around two million Kelvin. We also plotted DEMs as a function of length along a coronal loop segment visible in the active region. These DEMs are dominated by a relatively bright background, making it unclear what emission is associated with the loop; background subtraction of the data will be necessary to isolate the loop emission from its surroundings.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\lettrine{L}{arge-scale} image retrieval techniques have been developing and improving greatly for more than a decade. Many of the current state-of-the-art approaches~\\cite{Nister-CVPR06,Chum-ICCV07,Jegou-IJCV10} are based on the bag-of-words (BOW) approach originally proposed by Sivic and Zisserman~\\cite{Sivic-ICCV03}. Another popular image representation arises from aggregating local descriptors like Fisher kernel~\\cite{Perronnin-CVPR10} and Vector of Locally Aggregated Descriptors (VLAD)~\\cite{Jegou-CVPR10}.\n\nThe BOW vectors are high dimensional (up to 64 million dimensions in \\cite{Mikulik-IJCV12}), so, due to the high memory and computational requirements, search is limited to a several million images on a single machine. There are more scalable approaches that tackle this problem by generating compact image representations~\\cite{Torralba-CVPR08,Perronnin-CVPR10,Jegou-CVPR10}, where the image is described by a short vector that can be additionally compressed into compact codes using binarization~\\cite{Torralba-CVPR08,Weiss-NIPS09}, product quantization~\\cite{Jegou-PAMI11}, or recently proposed additive quantization techniques~\\cite{Babenko-CVPR14}.\nIn this paper we propose and experimentally evaluate simple techniques that additionally boost retrieval performance, but at the same time preserve low memory and computational costs.\n\nShort vector image representations are often generated using the principal component analysis (PCA)~\\cite{Bishop06} technique to perform the dimensionality reduction over high-dimensional vectors. Jegou and Chum~\\cite{Jegou-ECCV12} study the effects of PCA on BOW representations. They show that both steps of PCA procedure, i.e., centering and selection of de-correlated (orthogonal) basis minimizing the dimensionality reduction error, improve retieval performance. Centering (mean subtraction) of BOW vectors provides a boost in performance by adding a higher value to the negative evidence: given two BOW vectors, a visual word jointly missing in both vectors provides useful information for the similarity measure~\\cite{Jegou-ECCV12}.\nAdditionnaly, they advocate the joint dimensionality reduction with multiple vocabularies to reduce the quantization artifacts underlying BOW and VLAD.\nThese vocabularies are created by using different initializations for the k-means algorithm, which may produce relatively highly correlated vocabularies.\n\nIn this paper, we propose to reduce the redundancy of the joint vocabulary representation (before the joint dimensionality reduction) by varying parameters of the local feature descriptors prior to the k-means quantization.\nIn particular, we propose: (i) different sizes of measurement regions for local description, (ii) different power-law normalizations of local feature descriptors, and (iii) different linear projections (PCA learned) to reduce the dimensionality of local descriptors. In this way, created vocabularies will be more complementary and joint dimensionality reduction of concatenated BOW vectors originating from several vocabularies will carry more information. Even though the proposed approaches are simple, we show that they provide significant boosts to retrieval performance with no memory or computational overhead at the query time.\n\n\n\\paragraph{Related work.}%\nThis paper can be seen as an extension of~\\cite{Jegou-ECCV12}, details of which are given later in Section~\\ref{sec:baseline}. A number of papers report results with short descriptors obtained by PCA dimensionality reduction. In~\\cite{Jegou-PAMI12} and~\\cite{Perronnin-CVPR10}, aggregated descriptors (VLAD and Fisher vector respectively) are used followed by PCA to produce low dimensional image descriptors. In a paper about VLAD~\\cite{Arandjelovic-CVPR13}, authors propose a method for adaptation of the vocabulary built on an independent dataset (adapt) and intra-normalization (innorm) method that $L_2$~normalizes all VLAD components independently, which suppresses the burstiness effect~\\cite{Jegou-CVPR09}. In~\\cite{Jegou-CVPR14}, a `democratic' weighted aggregation method for burstiness supression is introduced.\nIn this paper, we compare results of all the aforementioned methods using low dimensional descriptors $D'=128$.\\\\\n\nThe rest of the paper is organized as follows: Section~\\ref{sec:background_and_baseline} gives a brief overview of several methods: bag-of-words\\linebreak (BOW), efficient PCA dimensionality reduction of high dimensional vectors, and baseline retrieval with multiple vocabularies. Used datasets and evaluation protocols are established in Section~\\ref{sec:datasets}. Section~\\ref{sec:our_approach} introduces novel methods for joint dimensionality reduction of multiple vocabularies and presents extensive experimental evaluations. Main conclusions are given in Section~\\ref{sec:conclusion}.\n\n\\section{Background and baseline}\n\\label{sec:background_and_baseline}\n\nThis section gives a short overview of the background of bag-of-words based image retrieval and the method used in~\\cite{Jegou-ECCV12}. Key steps and ideas are discussed in higher detail to help understanding of the paper.\n\n\\subsection{Bag-of-words (BOW) image representation}\n\\label{sec:bow}\n\nFirst efficient image retrieval based on BOW image representation was proposed by Sivic and Zisserman~\\cite{Sivic-ICCV03}. They use local descriptors extracted in an image in order to construct a high-dimensional global descriptor. This procedure follows four basic steps:\n\n\\begin{enumerate}\n\n\\item For each image in the dataset, regions of interest are detected~\\cite{Mikolajczyk-IJCV04,Matas-BMVC02} and described by an invariant descriptor which is $d$-dimensional. In this work we use the multi-scale Hessian-Affine~\\cite{Perdoch-CVPR09} and MSER~\\cite{Matas-BMVC02} detectors, followed by SIFT~\\cite{Lowe-IJCV04} or RootSIFT~\\cite{Arandjelovic-CVPR12} descriptors. The rotation of the descriptor is either determined by the detected dominant orientation~\\cite{Lowe-IJCV04}, or by the gravity vector assumption~\\cite{Perdoch-CVPR09}.\nThe descriptors are extracted from different sizes of measurement regions~\\cite{Matas-BMVC02}, as described in detail in Section \\ref{sec:our_approach}. \n\n\\item Descriptors extracted from the training (independent) dataset (see Section \\ref{sec:datasets}) are clustered into $k$ clusters using the k-means algorithm, which creates a visual vocabulary. \n\n\\item For each image in the dataset, a histogram of occurrences of visual words is computed. Different weighting schemes can be used, the most popular is inverse document frequency (\\textit{idf}), which generates a $D$ dimensional BOW vector ($D = k$).\n\n\\item All resulting vectors are $L_2$~normalized, as suggested in~\\cite{Sivic-ICCV03}, producing final global image representations used for searching.\n\n\\end{enumerate}\n\n\n\\subsection{Efficient PCA of high dimensional vectors}\n\\label{sec:pca}\n\nIn most of the cases BOW image representations have very high number of dimensions ($D$ can take values up to 64 million~\\cite{Mikulik-IJCV12}). In these cases the standard PCA method (reducing $D$ to $D'$) computing the full covariance matrix is not efficient. The dual gram method (see Paragraph 12.1.4 in~\\cite{Bishop06}) can be used to learn the first $D'$ eigenvectors and eigenvalues. Instead of computing the $D\\times D$ covariance matrix $\\boldsymbol{C}$, the dual gram method computes the $n\\times n$ matrix $\\boldsymbol{Y}^{T}\\boldsymbol{Y}$, where $\\boldsymbol{Y}$ is a set of vectors used for learning, and $n$ is the number of vectors in the set $\\boldsymbol{Y}$. Eigenvalue decomposition is performed using the Arnoldi algorithm, which iteratively computes the $D'$ desired eigenvectors corresponding to the largest eigenvalues. This method is more efficient than the standard covariance matrix method if the number of vectors $n$ of the training set is smaller than the number of vector dimensions $D$, which is usually the case in the BOW approach.\n\nJegou and Chum~\\cite{Jegou-ECCV12} analyze the effects of PCA dimensionality reduction on the BOW and VLAD vectors. They show that even though PCA successfully deals with the problem of negative evidence (higher importance of jointly missing visual words in compared BOW vectors), it ignores the problem of co-occurrences (co-occurences lead to over-count some visual patterns when comparing two image vector representation, see~\\cite{Chum-CVPR10}). In order to tackle the aforementioned problem, they propose performing a whitening operation, similar to the one done in independent component analysis~\\cite{Comon94} (implicitly performed by the Mahalanobis distance), jointly with the PCA. In our experiments we will use dimensionality reduction from $D$ to $D'$ components, as done in~\\cite{Jegou-ECCV12}:\n\n\\begin{enumerate}\n\n\\item Every image vector $v = (v_1,\\ldots, v_D)$ is post-processed using power-law normalization~\\cite{Perronnin-CVPR10}: $v_i := |v_i|^{\\beta}\\times sign(v_i)$, with $0\\leq \\beta < 1$ as a fixed constant. Vector $v$ is $L_2$~normalized after processing. It has been shown~\\cite{Jegou-PAMI12} that this simple procedure reduces the impact of multiple matches and visual bursts~\\cite{Jegou-CVPR09}. In all our experiments $\\beta = 0.5$, denoted as signed square rooting (SSR).\n\n\\item First $D'$ eigenvectors of matrix $\\boldsymbol{C}$ are learned using power-law normalized training vectors $\\boldsymbol{Y} = [Y_1|\\ldots |Y_n]$, corresponding to the largest $D'$ eigenvalues $\\lambda_{1},\\ldots,\\lambda_{D'}$.\n\n\\item Every power-law normalized image descriptor used for searching $X$ is PCA-projected and truncated, and at the same time whitened and re-normalized to a new vector $\\hat{X}$ that is the final short vector representation with dimensionality $D'$:\n\\begin{equation}\n\\label{eq:pca_white}\n\\hat{X} = \\frac\n{\\text{ diag}(\\lambda_{1}^{-\\frac{1}{2}},\\ldots,\\lambda_{D'}^{-\\frac{1}{2}})\\boldsymbol{P}^TX}\n{\\left \\| \\text{ diag}(\\lambda_{1}^{-\\frac{1}{2}},\\ldots,\\lambda_{D'}^{-\\frac{1}{2}})\\boldsymbol{P}^TX \\right \\|},\n\\end{equation}\nwhere the $D\\times D'$ matrix $\\boldsymbol{P}$ is formed by the largest eigenvectors calculated in the previous step. Comparing two vectors after this dimensionality reduction with the Euclidian distance is now similar to using a Mahalanobis distance. It has been argued that the re-normalization step is critical for a better comparison metric, see~\\cite{Jegou-ECCV12}.\n\n\\end{enumerate}\n\nIn order to compare results in a fair manner, we will use $D'=128$ dimensions for all our experiments following the trend of previous research in short image representations.\n\n\\subsection{The baseline method}\n\\label{sec:baseline}\n\n\\begin{figure*}\\centering\n\\includegraphics[trim = 20mm 0mm 20mm 0mm, width=\\linewidth]{fig_related.pdf}\n\n\\caption{\\textbf{Baseline methods:} Left plots show mAP performance on Oxford5k (upper plot) and Holidays (lower plot) after straightforward concatenation of BOW vectors (no PCA dimensionality reduction performed) generated using multiple vocabularies. Note that dimensionality of BOW grows linearly with every new concatenation. Right plots present mAP performance on Oxford5k and Holidays after joint PCA dimensionality reduction of concatenated BOW representations to a $D'=128$ dimensional vector.}\n\n\\label{fig:related}\n\\end{figure*}\n\nThis paper builds upon the work~\\cite{Jegou-ECCV12}, which is briefly reviewed in this section.\nIn~\\cite{Jegou-ECCV12}, a joint dimensionality reduction of multiple vocabularies is proposed. Image representation vectors are separately SSR normalized for each vocabulary, concatenated and then jointly PCA-reduced and whitened as explained in the Section~\\ref{sec:pca}. The \\textit{idf} term is ignored, and it is noted that the influence is limited when used with multiple vocabularies. Results of this method are shown in Figure~\\ref{fig:related} (right plots). Comparing to the straightforward concatenation (Figure~\\ref{fig:related}, left plots) where the results do not noticeably improve after adding multiple vocabularies, it can be noticed that an improvement in performance is achieved even when keeping low memory requirements by using PCA dimensionality reduction. However, for some vocabularies (i.e. $k=2$k), performance is dropping after only few vocabularies used.\n\n\n\n\n\n\\section{Datasets and evaluation}\n\\label{sec:datasets}\n\nResults of our methods are evaluated on the datasets~\\cite{Philbin-CVPR07,Philbin-CVPR08,Jegou-ECCV08} that are widely used in the image retrieval area. Also, we compare our results with other approaches evaluated on the same datasets.\n\n\\paragraph{Oxford5k~\\cite{Philbin-CVPR07} and Paris6k~\\cite{Philbin-CVPR08}:}%\nBoth datasets contain a set of images (5062 for Oxford and 6300 for Paris) having 11 different landmarks together with distractors, downloaded from Flickr by searching for tags of popular landmarks. For each of the 11 landmarks there are 5 different query regions defined by a bounding box, meaning that there are 55 different query regions per dataset. The performance is reported as mean average precision (mAP), see~\\cite{Philbin-CVPR07} for more details. In our experiments we use Paris6k as a training dataset in order to learn the visual vocabulary and projections of PCA dimensionality reduction. When evaluating our methods on Oxford5k, we always use the data learned on Paris6k.\n\n\\paragraph{Oxford105k~\\cite{Philbin-CVPR07}:\nThis dataset is the combination of Oxford5k dataset and 99782 negative images crawled from Flickr using 145 most popular tags. This dataset is used to evaluate the search performance (reported as mAP) on a large scale. Paris6k is used as a training dataset for Oxford105k.\n\n\\paragraph{Holidays~\\cite{Jegou-ECCV08}:\nThis dataset is a selection of personal holidays photos (1491 images) from INRIA, including a large variety of scene types (natural, man-made, water and fire effects, etc.). A sample of 500 images from the whole dataset is selected for query purposes~\\cite{Jegou-ECCV08}. The performance is reported as mAP, like for Oxford5k and Oxford105k, after excluding the query image from the results. As a training dataset for vocabulary construction and image representation level PCA learning we use Paris6k dataset in all experiments. \n\n\n\n\n\\section{Sources of multiple codebooks}\n\\label{sec:our_approach}\n\nWe propose combining multiple vocabularies that are differing not just in random initialization of clustering procedure, but also in the data used for clustering. The feature data are alternated in the process of local features description.\nThis process is not trying to synthesize appearance deformations, but rather varying certain design choices in the pipeline of feature description, such as the relative size of the measurement region. \nVocabularies created in this manner will contain less redundancy. This is combined with joint PCA dimensionality reduction (as described in Sections~\\ref{sec:pca} and \\ref{sec:baseline}) in order to produce short-vector image representations that are used for searching the most similar images in the dataset.\n\nQuantization complexity for all vocabularies used in experiments is given in Table~\\ref{tab:complexity}. As stated in~\\cite{Jegou-ECCV12}, time necessary to quantize 2000 local descriptors of a query image, for four $k=8$k vocabularies, on 12 cores is 0.45s, using a multi-threaded exhaustive search implementation. Timings are proportional to the vocabulary size, i.e., to the number in the right column of Table~\\ref{tab:complexity}. \n\n\\setlength{\\tabcolsep}{10pt}\n\\begin{table}\n\\centering\n\n\\caption{\\textbf{Complexity of vocabularies used throughout the experiments:} Complexity is given as a number of vector comparisons per local descriptor during the construction of the final BOW image representation.}\n\n\\begin{tabular}{lr}\n\\\\\n\\hline\n\\textbf{Vocabulary} & \\textbf{Complexity}\\\\\n\\hline\n\\hline\n8k & 8192\\\\\n4k & 4096\\\\\n2k & 2048\\\\\n1k & 1024\\\\\n\\hline\n4k+2k+\\ldots+128 & 8064\\\\\n2k+1k+\\ldots+128 & 3968\\\\\n1k+512+256+128 & 1920\\\\\n512+256+128 & 896\\\\\n\\hline\n\\end{tabular\n\n\\label{tab:complexity}\n\n\\end{table}\n\n\\paragraph{Multiple measurement regions.}%\n\nAn affine invariant descriptor of an affine covariant region can be extracted from any affine covariant constructed measurement region~\\cite{Matas-BMVC02}. As an example of a measurement region that is, in general, of a different shape than the detected region, is an ellipse fitted to the regions, as proposed by~\\cite{Tuytelaars-BMVC00} and also used for MSERs~\\cite{Matas-BMVC02}. An important parameter is the relative scale of the measurement region with respect to the scale of the detected region. Since the output of the detector is designed to be repeatable, it is usually not discriminative. To increase the disriminability of the descriptor, it is commonly extracted from area larger than the detected region. In case of~\\cite{Perdoch-CVPR09}, the relative change in the radius is $r=3\\sqrt{3}$. The larger the region, the higher discriminability of the descriptor, as long as the measurement region covers a close-to-planar surface. On the other hand, larger image patches have higher chance of hitting depth discontinuities and thus being corrupted. An example of multiple measurement regions is shown in Figure~\\ref{fig:multiMR}. To take the best of this trade off, we propose to construct multiple vocabularies over descriptors extracted at multiple relative scales of the measurement regions. Including lower scales leverages the disadvantages of large measurement regions, while joint dimensionality reduction eliminates the dependencies between the representations. \n\n\\begin{figure*} \\centering\n\n\\begin{minipage}[c]{0.25\\linewidth} \\centering\n\\includegraphics[height=3cm]{fig_mes_reg_graff\/graffAmr.png}\n\\end{minipage}\n\\begin{minipage}[c]{0.74\\linewidth} \\centering\n\\setlength{\\tabcolsep}{2pt}\n\\begin{tabular}{ccccc}\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffA_0_50.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffA_0_75.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffA_1_00.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffA_1_25.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffA_1_50.png}\n\\\\\n$0.5{\\times}r$ &\n$0.75{\\times}r$ &\n$1{\\times}r$ &\n$1.25{\\times}r$ &\n$1.5{\\times}r$\n\\end{tabular}\n\\end{minipage}\n\n\\vspace{5pt}\n\n\\begin{minipage}[c]{0.25\\linewidth} \\centering\n\\includegraphics[height=3cm]{fig_mes_reg_graff\/graffBmr.png}\n\\end{minipage}\n\\begin{minipage}[c]{0.74\\linewidth} \\centering\n\\setlength{\\tabcolsep}{2pt}\n\\begin{tabular}{ccccc}\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffB_0_50.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffB_0_75.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffB_1_00.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffB_1_25.png} &\n\\includegraphics[height=2.2cm]{fig_mes_reg_graff\/graffB_1_50.png}\n\\\\\n$0.5{\\times}r$ &\n$0.75{\\times}r$ &\n$1{\\times}r$ &\n$1.25{\\times}r$ &\n$1.5{\\times}r$\n\\end{tabular}\n\\end{minipage}\n\n\\vspace{2mm}\n\\caption{\\textbf{Multiple measurement regions (mMeasReg):} A corresponding feature is detected in two images (left). Multiple measurement regions for a single detected feature are shown in each row. The normalized patches (right) show different image content described by the respective descriptor.} \\label{fig:multiMR}\n\n\\end{figure*}\n\nWe consider using different sizes of measurement regions: $0.5\\times r,\\: 0.75\\times r,\\: 1\\times r,\\: 1.25\\times r,\\: 1.5\\times r$; creating slightly different SIFT descriptors used to learn every vocabulary. Implementation is very simple and during online stage the computation has to be done only for the features from query image region. Though simple, this method provides significant improvement even when concatenating vocabularies of small sizes (i.e. $k=2\\text{k}$ and $k=1\\text{k}$), see Figure~\\ref{fig:mes_reg} (left plot). We also explore the use of vocabularies with different sizes. All BOW vectors in this case are weighted proportionally to the logarithm of their vocabulary size~\\cite{Jegou-ECCV12}. In each step we concatenate a new bundle of vocabularies with multiple sizes, calculated with a different measurement region. We notice improvement when using multiple vocabulary sizes as well, see Figure~\\ref{fig:mes_reg} (right plot). For presentation of results on both plots in Figure~\\ref{fig:mes_reg}, in every step we are adding a different vocabulary created on SIFT vectors with measurement regions in predefined order: $0.5\\times r,\\: 0.75\\times r,\\: 1\\times r,\\: 1.25\\times r,\\: 1.5\\times r$. This approach is denoted as mMeasReg.\n\n\\begin{figure*}\\centering\n\\includegraphics[trim = 20mm 0mm 20mm 0mm, width=\\linewidth]{fig_mes_reg.pdf}\n\n\\caption{\\textbf{Multiple measurement regions (mMeasReg):} mAP performance improvement on Oxford5k after PCA reduction to $D'=128$ of concatenated BOW vectors produced on vocabularies created using SIFT descriptors with different measurement regions: $0.5{\\times}r,\\: 0.75{\\times}r,\\: 1{\\times}r,\\: 1.25{\\times}r,\\: 1.5{\\times}r$.}\n\n\\label{fig:mes_reg}\n\\end{figure*}\n\n\\paragraph{Multiple power-law normalized SIFT descriptors.\nSIFT descriptors~\\cite{Lowe-IJCV04} were the popular choice in most of the image retrieval systems for a long time. Arandjelovic \\etal~\\cite{Arandjelovic-CVPR12} show that using a Hellinger kernel instead of standard Euclidian distance to measure the similarity between SIFT descriptors leads to a noticeable performance boost in retrieval system. The kernel is implemented by simply square rooting every component of SIFT descriptor. Using Euclidian distance on these new RootSIFT descriptors will give the same result as using Hellinger kernel on the original SIFT descriptors. \nIn general, a power-law normalization~\\cite{Perronnin-CVPR10} with any power $0 \\leq \\beta \\leq 1$ can be applied to the descriptors ($\\beta = 0.5$ resulting in RootSIFT~\\cite{Arandjelovic-CVPR12}). Voronoi cells constructed in power-law normalized descriptor spaces can be seen as non-linear hyper-surfaces separating the features in the original (SIFT) descriptor space. Concatenation of such feature space partitionings reduces the redundant information. \n\nThere is no additional memory required and the change can be done on-the-fly with virtually no additional computational cost using simple power operation. We consider building four different vocabularies using: SIFT and SIFT with every component to the power of 0.4, 0.5, 0.6 (denoted as $\\text{SIFT}^{0.4}$, $\\text{SIFT}^{0.5}$, $\\text{SIFT}^{0.6}$ respectively). Concatenation is done on single vocabularies (Figure~\\ref{fig:SIFTs}, left plot) and on a bundle of vocabularies with different sizes (Figure~\\ref{fig:SIFTs}, right plot). Adding all SIFT modifications to the process of vocabulary creation achieves noticeable improvement of retrieval performance in the case of all vocabulary sizes. We denote this method as mRootSIFT.\n\nCombining vocabularies of different SIFT exponents improves over combining different vocabularies of a single SIFT exponent. For example, for 4~$\\times$ 2k vocabularies, the mAP on Oxford5k is $46.5$ for 4~$\\times$ $\\text{SIFT}^{0.5}$, and $47.7$ (Figure~\\ref{fig:SIFTs} left) for exponent combination.\n\n\\begin{figure*}[t!]\\centering\n\\includegraphics[trim = 20mm 0mm 20mm 0mm, width=\\linewidth]{fig_SIFTs.pdf}\n\n\\caption{\\textbf{Multiple power-law normalized SIFT descriptors (mRootSIFT):} mAP performance improvement on Oxford5k after PCA reduction to $D'=128$ of concatenated BOW vectors produced on vocabularies created using multiple local feature descriptors: SIFT, $\\text{SIFT}^{0.4}$, $\\text{SIFT}^{0.5}$, $\\text{SIFT}^{0.6}$.}\n\n\\label{fig:SIFTs}\n\\end{figure*}\n\n\n\\paragraph{Multiple linear projections of SIFT descriptors.}%\nIn locality sensitive hashing (random) linear projections are commonly used to reduce the dimensionality of the space while preserving locality. The idea pursued in this part of the paper is to use linear projections on the feature descriptors (SIFTs) before the vocabulary construction via k-means. However, random projections do not reflect the structure of the descriptors, resulting in noisy descriptor space partitionings. We propose to use PCA learned linear projections of SIFTs, learned on different training sets or subsets. The projections learned this way account for the statistics given by the training sets and hence produce meaningful distances, while inserting different biases into the vocabulary construction.\n\n\\begin{figure*}[t!]\\centering\n\\includegraphics[trim = 20mm 0mm 20mm 0mm, width=\\linewidth]{fig_PCA.pdf}\n\n\\caption{\\textbf{Multiple linear projections of SIFT descriptors (mPCA-SIFT):} mAP performance improvement on Oxford5k after PCA reduction to $D'=128$ of concatenated BOW vectors produced on vocabularies created using different PCA-reduced SIFT descriptors. For more details about all three presented methods see Section~\\ref{sec:our_approach}.}\n\n\\label{fig:PCA}\n\\end{figure*}\n\n\nThe improvement is twofold: (i) increased performance measured by mAP, and (ii) shorter quantization time during query due to shorter local descriptors after dimensionality reduction. On the other side there is a small amount of storage required to save learned projection matrices for every vocabulary, which we reuse at query. We consider and evaluate three different approaches for learning the eigenvectors used to project SIFT vectors from $D$ to $D'$ dimensions:\n\\begin{enumerate}[nolistsep]\n\n\\item We learn eigenvectors on Paris6k dataset and reduce the dimension of SIFT descriptors to $D'=80,64,48,32$ in the respective order for every newly created vocabulary (m$\\text{PCA}_1$-SIFT). Results of this experiment are shown in Figure~\\ref{fig:PCA}, \\nth{1} row.\n\n\\item We learn eigenvectors on different datasets: Paris6k, Holidays, University of Kentucky benchmark (UKB), PASCAL VOC'07 training in the respective order for every newly created vocabulary (m$\\text{PCA}_2$-SIFT). Dimension of SIFT descriptors is reduced to $D'=80$ in all cases. For the mAP performance on Oxford5k, see Figure~\\ref{fig:PCA}, \\nth{2} row.\n\n\\item We learn eigenvectors on different datasets: Paris5k, Holidays, UKB, PASCAL VOC'07 training and reduce the dimension of SIFT descriptors differently for each dataset ($D'=80,64,48,32$ respectively) creating different vocabularies (m$\\text{PCA}_3$-SIFT). Performance is presented in Figure~\\ref{fig:PCA}, \\nth{3} row.\n\n\\end{enumerate}\nNote that first vocabulary in all three different approaches is produced using standard SIFT descriptors without PCA reduction. A new vocabulary is added in every step of the experiment having joint dimensionality reduction of 5 concatenated BOW vectors in the end.\n\n\\paragraph{Multiple feature detectors.\nIn the Video Google approach~\\cite{Sivic-ICCV03} the authors combine vocabularies created from two different feature types.\nIn this paper we attempt to combine Hessian-Affine~\\cite{Perdoch-CVPR09} and MSER~\\cite{Matas-BMVC02} detectors. Even though straightforward concatenation of BOW vectors created on $k=8$k vocabularies ($48.7$ mAP on Oxford5k) gives improvement over using single BOW representations with Hessian-Affine ($44.7$) and MSER ($40.1$) features, after joint PCA reduction there is a decrease of performance when combining features ($37.0$ mAP on Oxford5k) compared to only doing PCA reduction on a single Hessian-Affine vocabulary ($38.6$), and an increase in performance when compared to PCA-reduced BOW vectors built on a single MSER vocabulary ($24.4$). Similar conclusions are made when combining smaller vocabulary sizes, i.e., there is always a drop in performance when comparing PCA reduction on a single vocabulary with Hessian-Affine features and PCA on combined vocabularies with Hessian-Affine and MSER features; mAP drop: from $39.8$ to $39.1$, from $40.7$ to $38.7$, from $36.8$ to $35.1$ for $k=4$k, $2$k, $1$k respectively. We also experimented with combining Harris-Affine~\\cite{Mikolajczyk-IJCV05} with Hessian-Affine features in the same manner as with MSER, but the improvement is not significant. PCA reduction of a single $k=8$k vocabulary on Hessian-Affine yields $38.6$ mAP on Oxford5k while joint PCA after adding a vocabulary of the same size built on Harris-Affine improves mAP to $39.0$, which is smaller improvement than using two vocabularies built on Hessian-Affine features with different randomization ($40.0$ mAP).\n\n\\paragraph{Discussion.}%\nIn order to better understand the impact of using multiple vocabularies we count the number of unique assignments in the product vocabulary. It corresponds to the number of non-empty cells of the descriptor space generated by all vocabularies simultaneously. The maximum possible number of unique assignments is equal to the product of number of clusters (cells) of all joint vocabularies. The number is related to the precision of reconstruction of each feature descriptor from its visual word assignments. For combination of vocabularies with different SIFT exponents (mRootSIFT) the number of unique assignments for Oxford5k dataset is shown in Figure~\\ref{fig:unq}. The plots are similar for all vocabulary combinations.\n\n\\setlength{\\tabcolsep}{8pt}\n\\begin{table*}\n\\centering\n\n\\caption{\\textbf{Comparison with the state-of-the-art on short vector image representation ($D'=128$):} Results in the first section of the table are mostly obtained from the paper~\\cite{Jegou-PAMI12}, except for the recent method on triangulation embedding and democratic aggregation with rotation and normalization ($\\phi_\\Delta$+$\\psi_\\text{d}$+$\\text{RN}$) proposed in~\\cite{Jegou-CVPR14}. In the second section we present results from methods that are using joint PCA and whitening of high dimensional vectors as we do. Results marked with * are obtained after our reimplementation of the methods using feature detector and descriptor as described in Section~\\ref{sec:bow} and Paris6k as a learning dataset. In the last section of the table we present results of our methods. All methods are described in detail in Section~\\ref{sec:our_approach}.}\n\n\\begin{tabular}{llccc}\n\\\\\n\\hline\n\\textbf{Method} & \\textbf{Vocabulary} & \\textbf{Oxford5k} & \\textbf{Oxford105k} & \\textbf{Holidays}\\\\\n\\hline\n\\hline\nGIST~\\cite{Oliva-IJCV01} & N\/A & $-$ & $-$ & $36.5$\\\\\nBOW~\\cite{Sivic-ICCV03} & $k{=}20\\text{k}$ & $19.4$ & $-$ & $45.2$\\\\\nImproved Fisher~\\cite{Perronnin-CVPR10} & $k{=}64$ & $30.1$ & $-$ & $56.5$\\\\\nVLAD~\\cite{Jegou-CVPR10} & $k{=}64$ & $-$ & $-$ & $51.0$\\\\\nVLAD+SSR~\\cite{Jegou-PAMI12} & $k{=}64$ & $28.7$ & $-$ & $55.7$\\\\\n$\\phi_\\Delta$+$\\psi_\\text{d}$+RN~\\cite{Jegou-CVPR14} & $k{=}16$ & $43.3$ & $35.3$ & $61.7$\\\\\n\\hline\nmVocab\/BOW~\\cite{Jegou-ECCV12} & $k{=}4{\\times}8\\text{k}$ & $41.3{\/}41.4^{*}$ & $-{\/}33.2^{*}$ & $56.7{\/}63.0^{*}$\\\\\nmVocab\/BOW~\\cite{Jegou-ECCV12} & $k{=}2{\\times}(32\\text{k}{+}\\ldots{+}128)$ & $-{\/}42.9^{*}$ & $-{\/}35.1^{*}$ & $60.0\/64.5^{*}$\\\\\nmVocab\/VLAD~\\cite{Jegou-ECCV12} & $k{=}4{\\times}256$ & $-$ & $-$ & $61.4$\\\\\nmVocab\/VLAD+adapt+innorm~\\cite{Arandjelovic-CVPR13} & $k{=}4{\\times}256$ & $44.8$ & $37.4$ & $62.5$\\\\\n\\hline\nmMeasReg\/mVocab\/BOW & $k{=}5{\\times}2\\text{k}$ & $46.9$ & $38.9$ & $66.9$\\\\\nmMeasReg\/mVocab\/BOW & $k{=}4{\\times}(4\\text{k}{+}\\ldots{+}128)$ & $47.7$ & $39.2$ & $67.3$\\\\\nmRootSIFT\/mVocab\/BOW & $k{=}4{\\times}2\\text{k}$ & $47.7$ & $39.8$ & $64.3$\\\\\nmRootSIFT\/mVocab\/BOW & $k{=}4{\\times}(2\\text{k}{+}\\ldots{+}128)$ & $48.8$ & $41.4$ & $65.6$\\\\\nm$\\text{PCA}_3$-SIFT\/mVocab\/BOW & $k{=}5{\\times}2\\text{k}$ & $45.8$ & $38.1$ & $63.2$\\\\\nm$\\text{PCA}_1$-SIFT\/mVocab\/BOW & $k{=}5{\\times}(4\\text{k}{+}\\ldots{+}128)$ & $45.5$ & $37.8$ & $64.6$\\\\\n\\hline\n\\end{tabular\n\\label{tab:mAP}\n\\end{table*}\n\n\\subsection{Comparison with the state-of-the-art}\n\nComparison with the current methods dealing with short vector image representation is given in Table~\\ref{tab:mAP}. Authors of the baseline approach on multiple vocabularies (mVocab) did not provide results for Oxford5k and Oxford105k datasets using all of their proposed methods, so we reimplemented and presented the corresponding results. Compared to their best method on Oxford5k that achieves $42.9$ mAP, our best method ($48.8$ mAP) obtains significant relative improvement of $13.8\\%$. In fact, all our methods outperform mVocab baseline methods on Oxford5k by a noticeable margin, with an improvement of $6.1\\%$ in the case of our worst performing method. When evaluating large-scale retrieval on Oxford105k dataset our methods again outperform the baseline method, relative improvement is $17.9\\%$ for our best performing method, and $7.7\\%$ for the worst performing one. In order to make a fair comparison when evaluating on Holidays dataset we again reimplemented the baseline approach, using Paris6k for learning the vocabularies and PCA projections (as we did in all our methods). In this case, the relative improvement is $4.3\\%$ with our best method (from $64.5$ mAP to $67.3$ mAP). We also compare our methods to two recent state-of-the-art approaches on short representations~\\cite{Arandjelovic-CVPR13,Jegou-CVPR14}. On Oxford5k and Oxford105k we improve as much as $8.9\\%$ and $10.7\\%$, respectively, compared to VLAD based approach~\\cite{Arandjelovic-CVPR13}, and $12.7\\%$ and $17.3\\%$, respectively, compared to T-embedding based approach~\\cite{Jegou-CVPR14}. On Holidays dataset relative improvement is $7.7\\%$ compared to the former and $9.1\\%$ compared to the latter. Note that the dataset used for learning of the meta-data for Holidays is different: we use Paris6k, while both \\cite{Arandjelovic-CVPR13} and \\cite{Jegou-CVPR14} are using an independent dataset comprising of 60k images downloaded from Flickr.\n\n\\begin{figure}[b!]\\centering\n\\includegraphics[width=.9\\columnwidth]{fig_unq.pdf}\n\\caption{\\textbf{Number of unique assignments (vocabulary cells) for Oxford5k dataset when combining vocabularies built on multiple power-law normalized SIFT descriptors (mRootSIFT):} SIFT, $\\text{SIFT}^{0.4}$, $\\text{SIFT}^{0.5}$, $\\text{SIFT}^{0.6}$.}\n\\label{fig:unq}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nMethods for multiple vocabulary construction were studied and evaluated in this paper. Following~\\cite{Jegou-ECCV12}, the concatenated BOW image representations from multiple vocabularies were subject to joint dimensionality reduction to 128D descriptors. We have experimentally shown that generating diverse multiple vocabularies has crucial impact on search performance. Each of the multiple vocabularies was learned on local feature descriptors obtained with varying parameter settings. That includes\nfeature descriptors extracted from measurement regions of different scales, different power-law normalizations of the SIFT descriptors, and applying different linear projections to feature descriptors prior to k-means quantization. \nThe proposed vocabulary constructions improve performance over the baseline method~\\cite{Jegou-ECCV12}, where only different initializations were used to produce multiple vocabularies. More importantly, the {\\em all} proposed methods exceed the state-of-the-art results~\\cite{Arandjelovic-CVPR13,Jegou-CVPR14} by a large margin. The choice of the optimal combination of vocabularies to combine still remains an open problem.\n\n\\clearpage\n\n\\vspace{2mm} \\noindent {\\bf Acknowledgements.} The authors were supported by MSMT LL1303 ERC-CZ and ERC VIAMASS no. 336054 grants.\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzopvh b/data_all_eng_slimpj/shuffled/split2/finalzzopvh new file mode 100644 index 0000000000000000000000000000000000000000..f339ec9fdafe286ea58e03859d13f1ee1cf1270b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzopvh @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec1}\n\nThe discovery of Giant Magnetoteresistance in 1998 by the groups\nof Fert and Gr\\\"unberg led to new reading heads for hard disks\n\\cite{Fert,Grunberg}. Moreover for the first time, a device based\non magnetic phenomena replaced a conventional electronics device\nbased on the movement of the electrons charge and thus opened the\nway to the field of spintronics or magnetoelectronics. The aim is\nto replace conventional electronics with new devices where\nmagnetism plays a central role leading to smaller energy\nconsumption. Several architectures have been proposed\n\\cite{Zutic,Zabel} but only in 2009 Dash and collaborators managed\nto inject spin-polarized current from a metallic electrode into\nSi, which is a key issue in current research in this field.\nshowing that spintronic devices can be incorporated into\nconventional electronics \\cite{Dash}.\n\nIn order to maximize the efficiency of spintronic devices, the\ninjected current should have as high spin-polarization as possible\n\\cite{Zutic,Zabel}. To this respect half-metallic compounds have\nattracted a lot of interest (for a review see reference\n\\cite{Katsnelson}). These alloys are ferromagnets where the\nmajority spin channel is metallic while the minority-spin band\nstructure is that of a semiconductor leading to 100\\%\\\nspin-polarization of the electrons at the Fermi level and thus to\npossibly 100\\%\\ spin-polarized current into a semiconductor when\nhalf metals are employed as the metallic electrode. The term\nhalf-metal was initially used by de Groot et al in the case of the\nNiMnSb Heusler alloy \\cite{Groot}.\n\nAb-initio (also known as first-principles) calculations have been\nwidely used to explain the properties of these alloys and to\npredict new half-metallic compounds. An interesting case is the\ntransition-metal pnictides like CrAs and MnAs. Akinaga and\ncollaborators found in 2000 that when a CrAs thin film is grown on\ntop of a zinc-blende semiconductor like GaAs, the metallic film\nadopts the lattice of the substrate and it crystallizes in a\nmeta-stable half-metallic zinc-blende phase \\cite{Akinaga}\nstructure. Later CrAs was successfully synthesized in the\nzinc-blence structure in the form of multilayers with GaAs\n\\cite{Mizuguchi} and other successful experiments include the\ngrowth of zinc-blende MnAs in the form of dots \\cite{Ono} and CrSb\nin the form of films \\cite{Zhao,Li}.\n\nExperiments agree with predictions of ab-initio calculations\nperformed by several groups\n\\cite{MavropoulosZB,GalaZB,calculations,Shirai}. In the case of\nthe half-metallic ferromagnets like CrAs or CrSe, the gap in the\nminority-spin band arises from the hybridization between the\n$p$-states of the $sp$ atom and the triple-degenerated $t_{2g}$\nstates of the transition-metal and as a result the total\nspin-moment, $M_t$, follows the Slater-Pauling (SP) behavior being\nequal in $\\mu_B$ to $Z_t-8$ where $Z_t$ the total number of\nvalence electrons in the unit cell \\cite{MavropoulosZB}. Recently\ntheoretical works have appeared attacking also some crucial\naspects of these alloys like the exchange bias in\nferro-\/antiferromagnetic interfaces \\cite{Nakamura2006}, the\nstability of the zinc-blende structure \\cite{Xie2003}, the\ndynamical correlations \\cite{Chioncel2006}, the interfaces with\nsemiconductors \\cite{InterGala,Interfaces}, the exchange\ninteraction \\cite{Sasioglu-Gala}, the emergence of half-metallic\nferrimagnetism \\cite{Rapid} and the temperature effects\n\\cite{MavropoulosTemp}. An extended overview on the properties of\nthese alloys can be found in reference \\cite{Review}.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{lekk-fig1.eps}\n\\end{center} \\caption{(Color online) Schematic representation of the structure of the\nmultilayers under study. We have assumed that the growth direction\nof the zinc-blende structure is the (001) and the period consists\nof four CrAs and four InAs(CdSe) layers. Each plane contains atoms\nof a single chemical type plus a void site. We denote as Void1 the\nvacant site in the same layer with the Cr atoms at the interfaces,\nVoid2 in the interfacial As(Se) layer and Void3 in the interfacial\nIn(Cd) layer. The distance between two consecutive plane is\n$\\frac{1}{4}$ of the lattice constant. Note that in the case of\nthe CrAs\/CdSe we have two non-equivalent interfaces: (i) when the\nsequence of the atoms is ...-Cr-As-Cd-... denoted as CrAs\/CdSe-1\nand (ii) when the sequence is ...-Cr-Se-Cd-... denoted as\nCrAs\/CdSe-2. Finally we should note that we have assumed the\nlattice constant of the two semiconductors (0.606 nm).\n\\label{fig1}}\n\\end{figure}\n\n\\begin{table*}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the perfect CrAs and CrSe alloys and for the three perfect\nmultilayers under study. For the position of each atom in the\nmultilayer please refer to figure \\ref{fig1}.}\n \\begin{tabular}{l|ccccc} \\hline \\hline\nPerfect Systems& CrAs(Bulk) & CrSe(Bulk)& CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nCr & 3.267 & 3.825 & 3.248& 3.117 & 3.462 \\\\\nVoid1 & -0.029& 0.049& -0.008& -0.025& -0.002 \\\\\nAs(Se) [CrAs layer] & -0.382& -0.103(Se)& -0.383& -0.388 & \\\\\nAs(Se) [Interface] &&& -0.209& -0.365 & -0.093(Se)\\\\\n As(Se) [SemiConducting layer]& && -0.066& & -0.048(Se) \\\\\nVoid2 & 0.080& 0.162 & 0.043& 0.027& 0.056\\\\\n In(Cd)&& &-0.008& -0.045& 0.003 \\\\\n Void3&&& -0.022 &-0.033 & -0.001\n \\\\\\hline \\hline\n\\end{tabular}\n\\label{table1}\n\\end{table*}\n\nRecently we have published a work on the role of defects in the\ncase of half-metallic CrAs, CrSb and CrSe alloys crystallizing in\nthe zinc-blende structure and adopting the lattice constant of\nInAs binary semiconductor of 0.606 nm \\cite{Pouliasis}. All\ndefects under study in this reference, with the exception of void\nimpurities at Cr and $sp$ sites and Cr impurities at $sp$ sites,\nwere found to induce new states within the gap and the Fermi level\ncan be pinned within these new minority states destroying the\nhalf-metallic character of the perfect bulk compounds. These\nimpurity states are localized in space around the impurity atoms\nand very fast the bulk behavior is regained. But in realistic\ndevices interfaces with semiconductors will occur and the\ninteraction of these bulk-impurity states with interface states\ncan destroy the spin-polarization of the injected current.\nAb-initio calculations show that interfaces are in principle\nhalf-metallic for pnictides containing Cr or V atoms even when the\nsemiconductor is a II-VI like CdSe and not a III-V one like InAs,\nand no interface states occur \\cite{InterGala}. But impurities at\nthe interfaces can induce new impurity-interfaces states within\nthe minority spin-gap which can couple to the bulk impurity states\nand lead to loss of the half-metallic character.\n\nIn this communication we expand our previous study to cover also\nthe case of impurities at interfaces. We have decided to consider\nCrAs as the half-metallic spacer since it is the most-widely\nstudied transition-metal zinc-blende pnictide and both InAs and\nCdSe as the zinc-blende semiconducting spacer to cover both the\ncase of III-V and II-VI semiconductors. In and Cd atoms, as well\nas As and Se atoms, are in the same row of the periodic table and\nthey have one valence electron difference and thus both alloys\nhave the same lattice constant of 0.606 nm and for this value of\nthe lattice constant bulk CrAs shows a Fermi level exactly in the\nmiddle of the minority-spin gap making clear the appearance of\nimpurity states. For our calculation we employ the\n the Korringa-Kohn-Rostoker method (KKR) method \\cite{Pap02} as in the case of\n the perfect bulk \\cite{MavropoulosZB} and interfaces\n \\cite{InterGala} and we have treated the impurities as in\n references \\cite{Pouliasis,imp}.\n\n\nTo model the interface we have considered a multilayer with a\nperiod of 4 CrAs and 4 InAs(CdSe) monolayers (MLs) which is in\naccordance with the experimental data in reference\n\\cite{Mizuguchi} and a schematic illustration is shown in figure\n\\ref{fig1}. As growth direction we have chosen the (001) and thus\nits layer is made up of atoms of a single chemical type. Moreover\nat each layer we have considered a void to describe better the\nspace in our calculation. In the case of CrAs\/InAs multilayers all\ninterfaces are equivalent while in CrAs\/CdSe case we have two\nnon-equivalent interfaces; the interface where As is between the\nCr and Cd layers which we denote as Interface-1 and the interface\nwhere Se separates the Cr and Cd layers (Interface-2). Moreover we\nshould note that the distance between two successive layers is 1\/4\nof the lattice constant and thus atoms in successive layers are\nnearest atoms while within the same atomic layer the nearest atoms\nare second neighbors. We have also denoted as Void1 the vacant\nsite at the Cr interface layer, Void2 in the As(Se) interface\nlayer and Void3 in the In(Cd) interface layer. We have considered\nall possible impurities and in section \\ref{sec2} we present the\ncase of perfect bulk and mulitilayers while in section \\ref{sec3}\nwe present the properties of the Cr impurities at various sites at\nthe interface, in section \\ref{sec4} the case of As(Se)\nimpurities, in section \\ref{sec5} the case of voids and in section\n\\ref{sec6} the case of In(Cd) impurities. Finally in section\n\\ref{sec7} we summarize and conclude. Throughout the discussion we\ncompare our results with the case of impurities in perfect bulk\nCrAs presented in reference \\cite{Pouliasis}. In the case of the\nCrAs\/CdSe-2 interface the Cr atoms have As atoms from one side and\nSe atoms from the other side but as it was shown in\n\\cite{Pouliasis} both CrAs and CrSe show similar behavior with\nrespect to the defects-induced states.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig2.eps}\n\\caption{(Color online) Atom-resolved density of states (DOS) for\nthe bulk CrAs and the perfect multilayers under study presented in\nfigure \\ref{fig1}. Panels (a), (b), (c) and (d) refer to the\nCrAs\/InAs multilayer and (e),(f), (g) and (h) to the CrAs\/CdSe\nmultilayer. In panel (a) and (b) we have also included the DOS of\nCr and As atoms in perfect bulk CrAs for comparison. Note that in\nthe case of the CrAs\/CdSe we have two interfaces one containing As\none containing Se atoms (see figure \\ref{fig1} for explanation).\nThe Fermi energy has been set as the zero of the energy axis.\nPositive DOS values correspond to\n the spin-up (majority-spin) electrons and negative values to the spin-down\n (minority-spin) electrons.\\label{fig2}}\n\\end{figure}\n\n\n\n\\section{Perfect systems}\\label{sec2}\n\n\nWe will start our discussion from the perfect systems. In table\n\\ref{table1} we have gathered the atom-resolved spin magnetic\nmoment in $\\mu_B$ for the three interfaces and for the bulk CrAs\nand CrSe alloys. Notice that for the bulk systems, where we have\ntwo vacant sites, we denote as Void1 the vacant site with the\nsame symmetry as the Void1 site at the interface and the same\nstands for the Void2 site. Bulk CrAs has 11 valence electrons per\nunit cell while CrSe has 12 valence electrons and thus\nhalf-metallic CrAs has a total spin moment of 3 $\\mu_B$ and CrSe\nof 4 $\\mu_B$. As it was shown in \\cite{MavropoulosZB} Cr atoms in\nCrSe accommodate half of the extra electron in the majority-spin\nband leading to a larger Cr-spin moment which now reaches about\n3.8 $\\mu_B$ instead of $\\sim$3.3 $\\mu_B$ in CrAs. The other half\nelectron is accommodated in the majority-spin band of the Se and\nVoid2 atoms as it easily deduced from the spin moments presented\nin table \\ref{table1} (Void2 is not a real atom but orbitals from\nthe nearest Cr atoms penetrate in the Void2 site).\n\nIn the case of the CrAs\/InAs multilayer, both Cr atoms have\nexactly the same nearest environment as in bulk CrAs and almost an\nidentical spin magnetic moment. The same is true also for the As\natom within the CrAs layer (As layer between the two Cr layer as\nshown in figure \\ref{fig1}) which has four Cr and four Void1 sites\nas nearest neighbors as in the bulk CrAs and exactly the same spin\nmagnetic moment of -0.38 $\\mu_B$. As at the interface layer has\none Cr layer from one side an In layer from the other side. In\natoms have two valence electrons which mainly are of $s$ character\nand have no contribution to the In spin magnetic moment. The spin\nmoment at the In sites is induced by the As atoms since\n$p$-orbitals of As penetrate in the In sites. Thus As atoms at the\ninterface show a smaller absolute value of the spin magnetic\nmoment since now they have only two instead of four Cr nearest\nneighbors (the spin-moment at As is induced by the Cr atoms\nthrough the hybridization of the As-$p$ and Cr-$t_{2g}$ states)\nand In atoms at the interface show a negligible spin moment\ninduced by the As interface atoms which in reality is the\nreflection of the Cr spin moment. As atoms within the\nsemiconducting layer have four In atoms as nearest neighbors and a\nvery small spin magnetic moment.\n\nIn the case of the CrAs\/CdSe multilayer we have two inequivalent\ninterfaces and the two Cr layer are no-more equivalent. The Cr\natoms at the As interface behave similarly to bulk CrAs although\nthe spin moment is somewhat smaller while in the case of the Se\ninterface Cr atoms have two As and two Se atoms as nearest\nneighbors exhibiting a spin magnetic moment of about 3.5$\\mu_B$ a\nmean value between the bulk CrAs and CrSe atoms. Within the CrAs\nlayer As shows a spin magnetic moment similar to the CrAs\/InAs\nmultilayer. At the 1$^\\mathrm{st}$ interface As atoms show similar\nbehavior to the CrAs\/InAs case although their spin moment is\nlarger in absolute value since Cd atoms have one valence electron\nless that In ones and the leakage of charge from the As interface\natoms to the Cd ones is smaller than to the In ones. In the\n2$^\\mathrm{nd}$ interface Se atoms have one electron more than As\none and thus the Se-$p$ orbitals are less polarized from the\nCr-$t_{2g}$ ones and show a smaller spin magnetic moment by a\nfactor of four. Within the CdSe semiconducting layer the Se atoms\nbehave like the As atoms in the InAs layer in the first interface\nand show a small value of the spin moment. Finally Cd atoms show a\nnegligible spin-magnetic moment as the In ones.\n\nIn figure \\ref{fig2} we have gathered the density of states (DOS)\nin states\/eV for both multilayers under study and the bulk CrAs.\nIn the left column we present the case of the CrAs\/InAs\nmultilayer. Cr atoms at the interface show a similar DOS to the\nbulk case with a very small weight of occupied minority-spin\nstates while in the majority band both $e_g$ and a fraction of the\nthree $t_{2g}$ states are occupied. The gap is smaller in the case\nof the multilayer since the occupied minority-spin states are\nbroader in energy and this is reflected to all other atoms. As\natoms within the CrAs layer show an almost identical DOS to the\nbulk CrAs case as was also the case for the spin magnetic moments.\nAs we mote to the interface and then to the InAs layer the shape\nof the As states changes due to the different local environment\nand the bands move towards the Fermi level. The states shown are\nthe $p$ states since the $s$-states of all atoms are located at\nabout -9 to -12 eV below the Fermi level and are not shown in the\nfigures. Note that for the In atoms we have used a different scale\nin the vertical axis. In the case of the CrAs\/CdSe multilayer the\nhalf-metallicity is again present as for the CrAs\/InAs layer but\ndue to the character of the II-VI semiconductor the gap in the\nminority-spin band is smaller and the Fermi level is near the\nleft-edge of the gap as it is clearly seen in the case of the Cr\nDOS. The reduction of the gap-width is larger for the As than the\nSe interface and this agrees with the fact that the bulk CrSe\npresents a larger gap than the bulk CrAs due to the larger\nelectronegativity of the Se atom as it was shown in reference\n\\cite{MavropoulosZB}. As a consequence the states of the As atoms\nat the first interface have moved closer to the Fermi level in a\nrigid way with respect to the As atoms in the CrAs layer. Cd atoms\nat both interfaces show similar DOS to the In atoms in the\nCrAs\/InAs interface. Se atoms within the CdSe layer show a similar\nshape to the As atoms in the InAs layer while Se atoms at the\nsecond interface show states deeper in energy than the As atom in\nthe first interface in agreement with the larger gap-width shown\nby the Cr atoms in the second interface with respect to the first\nCrAs\/CdSe interface.\n\\begin{table*}\n\\centering \\caption{Atom-resolved spin magnetic moments for the\ncase of Cr impurity atoms at interfacial As(Se) and Void2 sites in\nthe case of bulk CrAs and the three interfaces under study. \"imp\"\nstands for impurity and \"1st\" stands for nearest neighbor atoms\n(similar for 2nd and 3rd). We present results for both cases of\ncoupling of the Cr impurity spin moment with respect to the spin\nmoments of the Cr nearest-neighbors (ferromagnetic-FM and\nantiferromagnetic-AFM cases). Note that As(Se) atoms can be found\nin the CrAs layer [CrAs], the interfaces [Inter] and the\nsemiconductiong layer [SC]. }\n \\begin{tabular}{l|c|c|c|c|c|c|c|c|c} \\hline \\hline\n\\underline{\\bf Cr at As(Se) site} & \\multicolumn{2}{c|}{BULK CrAs}\n& \\multicolumn{2}{c|}{CrAs\/InAs} &\n\\multicolumn{2}{c|}{CrAs\/CdSe-1}&\n\\multicolumn{2}{c|}{CrAs\/CdSe-2}\\\\\n& \\underline{FM} & \\underline{AFM} & \\underline{FM} &\n\\underline{AFM} & \\underline{FM} &\n\\underline{AFM} & \\underline{FM} & \\underline{AFM} \\\\\n Cr-imp & 4.388 & -3.690 & 4.334 & -3.802 & 4.248& -4.058 & 4.335 & -3.988 \\\\\n Cr-1st & 3.602 & 3.364 & 3.704 & 3.425 & 3.543 & 3.343 & 3.735 & 3.485 \\\\\n In(Cd)-1st & & & 0.043 & -0.067 & 0.044 & -0.108 & 0.063 & -0.104 \\\\\n As [CrAs]-3rd & -0.405 & -0.405 & -0.412 & -0.413 & -0.418 & -0.418 &\n -0.025 & -0.074\\\\\nAs(Se) [Inter]-3rd & & & -0202 &-0.237 & -0.305 & -0.352 &-0.105(Se) & -0.110(Se) \\\\\nAs(Se) [SC]-3rd & & & -0.025 & -0.067 & -0.032(Se) & -0.059(Se) &-0.417(Se) & -0.418(Se) \\\\\n\\hline \\hline \\underline{\\bf Cr at Void2 site} &\n\\multicolumn{2}{c|}{BULK CrAs} & \\multicolumn{2}{c|}{CrAs\/InAs} &\n\\multicolumn{2}{c|}{CrAs\/CdSe-1}&\n\\multicolumn{2}{c|}{CrAs\/CdSe-2}\\\\\n& \\underline{FM} & \\underline{AFM} &\\underline{FM} &\n\\underline{AFM} & \\underline{FM} &\n \\underline{AFM} & \\underline{FM} & \\underline{AFM} \\\\\nCr-imp & 3.901 & -2.908 & 3.821 & -3.262& 3.729& -3.263& 4.171 & -3.551 \\\\\n Cr-1st & 3.419 & 3.267& 3.412 &3.225 & 3.300 &3.103 & 3.467 &3.335 \\\\\n In(Cd)-1st & & & 0.014 & -0.176 & -0.118 &-0.030 & 0.023 & -0.038\\\\\n As [CrAs]-2nd & -0.341 &-0.285 & -0.364 &-0.323 & -0.371 & -0.330 & -0.470 &-0.371 \\\\\n As(Se) [Inter]-2nd & & & -0.167 &-0.141 & -0.308 &-0.214 & -0.102(Se) & -0.104(Se) \\\\\n As(Se) [SC]-2nd & & & -0.009 &-0.039& -0.013(Se) &-0.033(Se) & -0.033(Se) & -0.026(Se)\\\\\n\\hline \\hline\n\\end{tabular}\n\\label{table2}\n\\end{table*}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig3.eps}\n\\caption{(Color online) Cr-resolved DOS for the case of Cr\nimpurity atoms at interfacial As(Se) sites at in the case of the\nferromagnetic (FM) [left panel] and antiferromagnetic coupling\n(AFM) [right panel) between the Cr impurity atom and its Cr\nnearest-neighbors (see text for details). Details as in figure\n\\ref{fig2}. \\label{fig3}}\n\\end{figure}\n\n\n\\section{Cr impurities}\\label{sec3}\n\nWe will start our discussion from the case of Cr impurities. The\nfirst case under study is when Cr impurity atoms are located at\nAs(Se) or Void2 sites at the interface. With respect to the\nperfect Cr sites where the first neighbors were four sp atoms and\nfour voids, the Cr impurity atoms have now as nearest neighbors\ntwo Cr atoms and two In(Cd) atoms together with two Void1 and two\nVoid3 sites. Thus the local environment of the Cr impurities is\nfundamentally different. In the case of Cr impurities at As and\nVoid2 sites in the bulk, we have shown that the spin magnetic\nmoment could be either ferromagnetically (FM) coupled to the spin\nmoment of the Cr atoms at the perfect sites or\nantiferromagnetically (AFM) coupled with the latter one being the\nenergetically favorable case \\cite{Pouliasis}. The AFM was\noccurring due to the short distance between the Cr impurity atoms\nand its four nearest Cr neighbors (Cr as well as Mn atoms are know\nto have antiparallel spin moments under a critical distance). In\nthe case of the interfaces under study the Cr impurity atoms have\nnow two instead of four nearest Cr neighbors as in the bulk. But\nit seems that this is enough to have again two stable solutions, a\nFM and an AFM one, depending on the starting potential, for both\nCr at interfacial As(Se) and Void2 sites. The AFM solution seems\nenergetically more favorable although we have not converged the\nenergetics of the defects (for the reason see discussion in\nreference \\cite{Pouliasis}).\n\nIn figure \\ref{fig3} we have gathered the atom-resolved DOS for\nthe Cr impurity atoms and its nearest Cr neighbors for all three\ninterfaces under study and bulk CrAs for the case of Cr impurity\nat As(Se) site and for both the FM and AFM cases. Results are\nsimilar when the Cr impurity is located at the Void2 site and thus\nare not shown here. Also results are in all cases similar to the\nbulk one with minor changes at the position of the picks due to\nthe variation in the environment surrounding the impurities. In\nthe FM case the impurities show almost zero occupied minority-spin\nstates while the Cr nearest neighbors show a very small weight of\nminority-spin states below the Fermi level. In all cases the\nhalf-metallic character of the interface is not affected. In the\nAFM case the Cr impurities exhibit a large spin-splitting of the\nbands and almost all the spin-down states are occupied while\nalmost all spin-up states are empty. This leads to a large gap in\nthe spin-down band between the spin-down occupied bands of the\nimpurity atom and the unoccupied spin-down bands of the Cr nearest\nneighbors and the half-metallic character of the interface is not\naffected. In table \\ref{table2} we have gathered the atomic spin\nmagnetic moments for all cases mentioned above. We can see that\nspin moments behave similar to the case of the Cr impurities at As\nor Void2 sites in the perfect bulk. In the FM case the Cr impurity\natoms have a larger spin moment with respect to the Cr atoms in\nthe perfect systems since the weight of the minority occupied\nstates has vanished as a result of the hybridization with its\nfirst neighbors. The Cr atoms which are first neighbors of the\nimpurity atoms present spin moments almost identical to the\nperfect compounds in table \\ref{table1}. In the AFM case although\nit seems from the DOS that all five spin-down $d$-states of the\nimpurity atom are occupied the spin moment is less than 5 $\\mu_B$\nsince some spin-up states also appear below the Fermi level\nreflecting the band-structure of the Cr nearest neighbors. We\nshould note that spin moments are slightly smaller when the\nimpurity is located at a Void2 instead of a As(Se) site because\nalthough the nearest-neighbors are the same, the next-nearest and\nfurther neighbors are different. We should finally note here that\nthe AFM case is the most interesting case for applications since\nthe occurrence of half-metallic ferrimagnetism leads to smaller\nexternal fields and exhibiting smaller energy losses with respect\nto ferromagnets.\n\n\\begin{table}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of Cr impurity atoms at interfacial Void1, In(Cd) and\nVoid3 sites in the case of the three interfaces under study.\nNotation as in table \\ref{table2}. }\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf Cr at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nCr-imp & 3.522& 3.363 & 3.686 \\\\\n As(Se) [CrAs]-1st & -0.300 & -0.299 & -0.296 \\\\\n As(Se) [Inter]-1st & -0.151& -0.275& -0.042 \\\\\n Cr [Inter]-2nd & 3.448& 3.352 & 3.622\\\\ \\hline\n\\underline{\\bf Cr at In(Cd) site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ Cr-imp & 3.185 & 3.311& 4.015 \\\\\n As(Se) [SC]-1st & -0.157 & -0.095& -0.147 \\\\\n As(Se) ([Inter]-1st &-0.324 & -0.406& -0.254 \\\\\n Cr [Inter]-3rd & 3.328& 3.252 & 4.219 \\\\ \\hline\n\\underline{\\bf Cr at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nCr-imp & 3.508& 3.501& 3.833\\\\\n As(Se) [SC]-1st &-0.024 &-0.004 & 0.009 \\\\\n As(Se) [Inter]-1st & -0.135 & -0.275 & -0.025 \\\\\n Cr [Inter]-2nd & 3.478 & 3.384 & 3.635 \\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table3}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig4.eps}\n\\caption{(Color online) Density of states of the Cr impurity atom\nat Void3(upper panel) and In(Cd) (lower panel) interfacial sites\nfor all the interfaces under study. Details as in figure\n\\ref{fig2}. \\label{fig4}}\n\\end{figure}\n\nThe next case under study is when the Cr impurity is located at a\nVoid1 site. Although the Void1 site is located in the same layer\nas the Cr atoms, they are second neighbors having the same nearest\nneighbors (four sp atoms and four voids). Thus we do not expect Cr\nimpurities at Void1 sites to alter the electronic and magnetic\nproperties of the interface similarly to the situation of Cr\nimpurities at Void1 sites in the bulk CrAs \\cite{Pouliasis}. The\nDOS is similar to the Cr atoms in the perfect systems and thus we\ndo not present them since the gap is almost unaltered and the\nhalf-metallicity is not affected. The spin magnetic moment of the\nCr impurity atoms which are shown in table \\ref{table3} are\nslightly larger than the Cr atoms in the perfect system by about\n0.2 $\\mu_B$. The Cr nearest-neighbors have spin moment almost\nidentical to the Cr impurities and thus part of the charge of the\nCr impurity atoms is used to increase the spin moment of the Cr\nsecond neighbors with respect to the perfect interfaces.\n\nThe final case under study in this section is when the Cr impurity\noccurs at the In(Cd) sites at the interface or the Void3 sites\nwhich are located in the same layer with the In(Cd) ones. Although\none could think that these impurities should destroy the\nhalf-metallic character of the interface, this is not true. Cr\nimpurities at these sites have as nearest neighbors two As(Se)\natoms in the semiconducting space and two As(Se) site at the\ninterface. Thus their local environment is quite close to the one\nof the Cr atoms at the perfect interface sites. As a result the\nhalf-metallic character is not destroyed as shown by the DOS in\nfigure \\ref{fig4}. When the impurity is located at a Void3 site,\nthere is almost no-occupied spin-down states while when the\nimpurity is located at the In(Cd) site the weight of the occupied\nspin-down states is very small and the width of the gap is\ncomparable to the perfect compounds in figure \\ref{fig2}. In the\nlatter case we see that for the CrAs\/CdSe-2 interface the bands\nare slightly shifted to higher energy as in a rigid-band model\nwith respect to the other interfaces but the gap has even larger\nwidth. This is reflected also to the spin-magnetic moments\npresented in table \\ref{table3} where for this case the Cr\nimpurity spin moment exceeds the 4 $\\mu_B$ while for the other\ninterfaces it is slightly larger than the perfect systems.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig5.eps}\n\\caption{(Color online) In the case of As(Se) impurities at Void2\nsites we present the atom-resolved DOS of the nearest Cr and\nIn(Cd) neighbors for all three interfaces. Details as in figure\n\\ref{fig2}. \\label{fig5}}\n\\end{figure}\n\n\\section{As(Se) impuritites}\\label{sec4}\n\nNext we will present our results on the As impurities in the case\nof the CrAs\/InAs and CrAs\/CdSe-1 interfaces and on the Se\nimpurities in the case of the CrAs\/CdSe-2 interface. We have\nstudied all possible cases and in table \\ref{table4} we have\nconcentrated the atom-resolved spin magnetic moments for the\nimpurities and their nearest neighbors. The DOS of the impurities\nin general present no large variation with respect to the As(Se)\natom at the interface or the case of impurities at the perfect\nbulk and thus we present only the case of As(Se) atom at Void2\nsites, where the As(Se) impurity atom has the same\nnearest-neighbors as at the perfect interfacial As(Se) sites. In\nall other cases the impurity atom has four As(Se) atoms as nearest\nneighbors (two at the interface and two at the CrAs, when the\nimpurity is located at the Cr or Void1 sites, and two at the\nInAs(CdSe) layer, when it is located at the In(Cd) or Void3\nsites).\n\nWe will first discuss the case of As(Se) impurities at the Cr and\nVoid1 sites. At the same energy window with the $d$ electrons of\nthe Cr atoms are located the $p$ electrons of the As(Se) atoms.\nThe DOS of the impurity atoms is similar to the case of As(Se)\nimpurities at Cr and Void1 sites in bulk CrAs(CrSe) presented in\n\\cite{Pouliasis} and thus we do not present them. For the\ninterfacial As(Se) atoms in the perfect interfaces the minority\n$p$ states of As(Se) are completely occupied leading to small\nnegative spin moments (see table \\ref{table1}). The impurity\nAs(Se) atoms at Cr or Void1 sites, on the other hand, have four\nother As(Se) atoms as nearest neighbors and thus the $p$ states of\nthe As(Se) impurity atoms have to hybridize with the $p$ states of\nthe neighboring As(Se) atoms which are almost completely occupied\ninstead of the Cr $t_2g$ states for which only the majority states\nare occupied leading to occupancy of the antibonding $p$ spin-up\nstates and to positive spin magnetic spin moments as shown in\ntable \\ref{table4}. Due to the reorganization of the charge\ninduced by the impurity As(Se) states, the number of the occupied\nspin-up $p$-states of the neighboring As(Se) atoms increases and,\nalthough the spin magnetic moments remain negative, their absolute\nvalue decreases. In all cases discussed just above, the impurity\nstates are not located in the gap similar to the what happened for\nthe same impurities in the bulk systems (see \\cite{Pouliasis}) and\nthe half-metallicity is preserved.\n\n\n\\begin{table}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of As(Se) impurity atoms at various interfacial sites\nin the case of the three interfaces under study. Notation as in\ntable \\ref{table2}. }\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf As(Se) at Cr site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.108 & 0.118& 0.127 \\\\\n As(Se) [CrAs]-1st & -0.243 & -0.255 &-0.271 \\\\\n As(Se) [Inter)]-1st & -0.061 & -0.177 & -0.093 \\\\ \\hline\n\n\\underline{\\bf As(Se) at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.143 & 0.084 & 0.070 \\\\\nAs(Se) [CrAs]-1st & -0.222 &-0.267 & -0.123 \\\\\nAs(Se) [Inter]-1st & -0.114 & -0.208& -0.087 \\\\ \\hline\n\n\\underline{\\bf As(Se) at Void2 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.269& 0.207& -0.046\\\\\n Cr-1st& 3.361&3.212& 3.367\\\\\n In(Cd)-1st & 0.012& -0.007& -0.010\\\\\nAs(Se) [CrAs]-2nd& -0.318& -0.314& -0.494\\\\\n As(Se) [Inter]-2nd& -0.104& -0.229& -0.114\\\\\n As(Se) [SC]-2nd & 0.056&0.050& -0.074\\\\ \\hline\n\n\\underline{\\bf As(Se) at In(Cd) site} & CrAs\/InAs & CrAs\/CdSe-1\n& CrAs\/CdSe-2\n\\\\ As(Se)-imp & 0.038& 0.269& 0.048\\\\\n As(Se) [Inter]-1st&-0.128& -0.013& -0.584\\\\\n As(Se) [SC]-1st& -0.008&0.287& -0.249\\\\ \\hline\n\n\\underline{\\bf As(Se) at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\\\\ As(Se)-imp & 0.093&\n0.068& 0.197\\\\\n As(Se) [Inter]-1st& -0.069& -0.142&-0.003\\\\\n As(Se) [SC]-1st& -0.009 & -0.013& -0.008\\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table4}\n\\end{table}\n\nWhen the As(Se) atoms migrate to Void2 sites, their nearest\nneighbors remain two Cr and two In(Cd) atoms as for the perfect\nAs(Se), but the further neighbors change and the interfacial As\natoms are now next-nearest neighbors and the hybridization between\ntheir $p$ states is intense leading to occupation of the\nantibonding spin-up $p$ states and positive values of the spin\nmagnetic moments of the impurity atoms as shown in table\n\\ref{table4} for the case of the As impurities in CrAs\/InAs and\nCrAs\/CdSe-1 interfaces or to a large variation of the negative Se\nspin moment in the case the CrAs\/CdSe-2 interface (the spin moment\nof the impurity is half with respect to the Se atom at the perfect\nsites). The Cr atoms at the interface have now 5 instead of 4\nAs(Se) atoms as nearest neighbors but the hybridization of the\n$t_{2g}$ $d$-states is little affected and the nearest-neighboring\nCr spin moments are almost identical to the perfect interfaces. A\nsimilar behavior is exhibited by the spin moments of the As(Se)\nsecond neighbors. The behavior of the spin moments is reflected\nalso in the DOS presented in figure \\ref{fig5}. The DOS of the Cr\nnearest neighbors presents for all interfaces under study a very\nwide minority-spin gap similar to the perfect compounds with a\nlarge spin-splitting between the occupied majority- and the\nunoccupied minority-spin bands. The shape of the Cr DOS is similar\nto the Cr atoms at the perfect interfaces shown in figure\n\\ref{fig2} revealing the small influence of the hybridization with\nthe impurity As(Se) $p$-states. The same is also valid for the DOS\nof the nearest In(Cd) atoms shown also in the same figure although\nthe gap is smaller for these atoms similarly to what occurred in\nthe perfect interfaces.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig6.eps}\n\\caption{((Color online) In the case of Void impurity at an\ninterfacial Cr site we present the atom-resolved DOS of the\nnearest As(Se) neighbors in the CrAs layer and at the interface\nfor all three interfaces and for the bulk CrAs. Details as in\nfigure \\ref{fig2}. \\label{fig6}}\n\\end{figure}\n\nThe case of As(Se) impurities at In(Cd) and Void3 sites is more\ndifficult to be understood and although the half-metallicity is\npreserved the spin moments of the As(Se) impurity atoms and their\nnearest As(Se) atoms show large variations. In the case of the\nCrAs\/InAs interface the As impurity atom has 4 As atoms as nearest\nneighbors, in the case of the CrAs\/CdSe-1 atom the As impurity\natom has two As at the interface and two Se atoms within the CdSe\nlayer as nearest neighbors while in the case of the CrAs\/CdSe-2\ninterface the Se impurity atom has four Se atoms as nearest\nneighbors. Se atoms have one electron more than the As ones and\nthus in general a larger number of spin-up states is occupied as\nrevealed also from the spin magnetic moments in table\n\\ref{table1}. Thus especially when the impurity is located at the\nIn(Cd) site no trend can be found in table \\ref{table4} regarding\nthe spin magnetic moments and the situation is\ninterface-dependent.\n\n\n\n\n\\section{Void impurities}\\label{sec5}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig7.eps}\n\\caption{(Color online) In the case of Void impurity at an\ninterfacial In(Cd) site we present the atom-resolved DOS of the\nnearest As(Se) neighbors in the semiconducting layer and at the\ninterface for two of the studied interfaces. Details as in figure\n\\ref{fig2}. \\label{fig7}}\n\\end{figure}\n\nUp to now we have studied the case of the Cr and As(Se) impurities\nand in all cases the half-metallicity was preserved. In this\nsection we will study the effect of Voids. Voids are Schottky-type\ndefects with large formation energies. A Void impurity in reality\nmeans that an atom is missing leading to a reorganization of the\ncharge of the neighboring atoms. We will start our discussion from\nthe case of a Void at a Cr site and in figure \\ref{fig6} we\npresent the DOS of the nearest As(Se) neighbors for all three\ninterfaces and for the bulk CrAs. In the case of the bulk the\nmissing Cr atom means that As atoms have now three instead of four\nCr nearest neighbors and the $p$-bands of As move closer to the\nFermi level without crossing it. In the case of interfaces we\nshould distinguish between the As atoms within the CrAs layer\nwhich show a behavior identical to the bulk case and the As(Se)\natoms at the interface layer. The latter ones in the perfect\nmultilayers have two Cr atoms and two In(Cd) as nearest neighbors\nand when a Void occurs at a Cr site they lose one of the two Cr\nneighbors resulting to a large shift of the $p$-bands which almost\ncross the Fermi level. Although one could argue that\nhalf-metallicity is present, a small change of the lattice\nconstant would lead to loss of the half-metallic character of the\nperfect interfaces. As the bands move closer to the Fermi level\nthe weight of the spin-up states decreases while the bonding\nspin-down states are completely occupied leading to larger\nabsolute values of the negative spin moments of the nearest As(Se)\nneighbors with respect to the perfect case as shown in table\n\\ref{table5}.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig8.eps}\n\\caption{(Color online) In the case of Void impurity at an\ninterfacial As(Se) site we present the atom-resolved DOS of the\nnearest Cr and In(Cd) neighbors in the CrAs layer for all three\ninterfaces and for the bulk CrAs. Note that the In(Cd) DOS has\nbeen multiplied by a factor of 10. Details as in figure\n\\ref{fig2}.\\label{fig8}}\n\\end{figure}\n\nThe same phenomenon occurs also when the Void is located at an\nIn(Cd) site as shown in figure \\ref{fig7}. The missing In(Cd)\nneighbors lead to a shift of the $p$-bands of the\nnearest-neighboring As(Se) atoms towards higher energies and now\nclearly the spin-down band crosses the Fermi level destroying the\nhalf-metallicity. Especially in the case of the CrAs\/InAs\ninterface a spin-down pick is pinned exactly at the Fermi level.\nIn the case of the CrAs\/CdSe-1 interface the bands are more\nnarrow; the spin-down pick is located below the Fermi level but a\nsmall contraction of the lattice constant could lead to the\npinning of the Fermi level at this pick. As discussed for the case\nof Void impurities at Cr sites, also here the absolute values of\nthe negative spin moments of the neighboring As(Se) atoms increase\n(see table \\ref{table5}).\n\n\\begin{table} \\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of Void impurities at interfacial Cr, As(Se) and\nIn(Cd) sites in the case of the three interfaces under study.\nNotation as in table \\ref{table2}.}\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf Void at Cr site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\n Void-imp &\n-0.097& -0.129& -0.061\\\\ As [CrAs]-1st& -0.592&\n-0.606& -0.613\\\\ As(Se) [Inter]-1st& -0.350& -0.644& -0.159\\\\\n\\hline\n \\underline{\\bf Void at Ae(Se) site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nVoid-imp & 0.201& 0.154& 0.190\\\\\n Cr-1st & 3.748& 3.691& 3.812\\\\\n In(Cd)-1st &0.033& -0.008& 0.017 \\\\ \\hline\n\\underline{\\bf Void at In(Cd) site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\ Void-imp & -0.094& -0.073& 0.066\\\\\n As(Se) [Inter]-1st &\n-0.468& -0.535& -0.210\\\\\n As(Se) [SC]-1st & -0.141&\n-0.115& -0.127 \\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table5}\n\\end{table}\n\n\nThe last possible case which can occur is when the Void is located\nat the interface As(Se) site. The loss of the As(Se) atom only\nmarginally affects the DOS of the Cr atoms at the interface which\nnow have three instead of four neighboring atoms. As shown in\nfigure \\ref{fig8} Cr nearest-neighbors at the interface show\nsimilar behavior as in the bulk and the weight of the occupied\nminority-spin states is vanishing and the half-metallicity remains\nrobust. The In(Cd) nearest-neighbors exhibit also a large gap as\nshown in the same figure where the DOS of the In(Cd) has been\nmultiplied by 10 to be visible with respect to the Cr nearest\nneighbors. The loss of its As(Se) neighbor leads to an increase of\nthe local charge of the Cr nearest-neighbors (in a way the regain\nthe charge which was participating at the bonds with the missing\nAs(Se) atom). This extra local charge occupies spin-up states\nleading to and increase of the Cr spin moment with respect to the\nperfect interfaces by about 0.4-0.5 $\\mu_B$.\n\n\n\n\\begin{table} \\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of In(Cd) impurities at various interfacial sites in\nthe case of the three interfaces under study. Notation as in table\n\\ref{table2}.}\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf In(Cd) at Cr site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nIn(Cd)-imp & <0.001& -0.070& -0.025\\\\\n As(Se) [CrAs]-1st&\n-0.357& -0.459& -0.434\\\\ As(Se) [Inter]-1st& -0.146& -0.376&\n-0.074\\\\ Cr [CrAs]-3rd& 3.299& 3.456& 3.173\\\\\nCr [Inter]-3rd& 3.321& 3.176& 3.464\\\\ In(Cd) [Inter]-3rd\n& -0.014& -0.048& -0.006 \\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ In(Cd)-imp & 0.068 & 0.004 & 0.033\\\\ As(Se)\n[CrAs]-1st & -0.238 & -0.318 & -0.283\\\\ As(Se) [Inter]-1st &\n-0.130 & -0.269 & -0.059\\\\ Cr [CrAs]-2nd & 3.429 & 3.571 &\n3.590\\\\ Cr [Inter]-2nd & 3.427 & 3.272 & 3.262 \\\\In(Cd)\n[Inter]-2nd & -0.002 & -0.032& 0.004\\\\ \\hline\n\n\\underline{\\bf In(Cd) at As(Se) site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\n\\\\ In(Cd)-imp & 0.100& 0.148& 0.188\\\\ Cr-1st& 3.590&\n3.590& 3.746\\\\ In(Cd)-1st& -0.024& -0.001& 0.019\\\\ As(Se)\n[CrAs]-3rd -0.414& -0.421& -0.416\\\\ As(Se) [Inter]-3rd\n&-0.209& -0.384& -0.096\\\\ As(Se) [SC]-3rd& -0.040& -0.032&\n-0.023\\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void2 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ In(Cd)-imp & -0.041& 0.088& 0.063\\\\ Cr-1st &\n3.185& 3.256& 3.446\\\\ In(Cd)-1st& -0.006& -0.020&\n-0.007\\\\ As(Se) [CrAs]-2nd& -0.398& -0.362& -0.475\\\\ As(Se)\n[Inter]-2nd & -0.176& -0.288& -0.102\\\\ As(Se) [SC]-2nd&\n-0.044& -0.039& -0.067\\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\n\\\\ In(Cd)-imp & 0.058& -0.008& 0.045\\\\ As(Se) [Inter]-1st&\n-0.073& -0.230& -0.016\\\\ As(Se) [SC]-1st &-0.011&\n-0.028& -0.011 \\\\Cr [Inter]-2nd& 3.473& 3.286& 3.638\\\\\nIn(Cd) [Inter]-2nd& 0.001& -0.031& 0.004\\\\ In(Cd) [SC]-2nd &\n-0.003& 0.002& -0.038\\\\\n\n \\hline \\hline\n\\end{tabular}\n\\label{table6}\n\\end{table}\n\n\\section{In(Cd) impurities}\\label{sec6}\n\nFinally, in the last section we will present our results\nconcerning the case of In, for the CrAs\/InAs interface, and Cd,\nfor both CrAs\/CdSe interfaces, impurities at various sites. All\nthree interfaces show similar behavior and thus in figure\n\\ref{fig9} we present the DOS for all possible In impurities for\nthe CrAs\/InAs multilayer. We should note that with respect to the\nconservation of the half-metallicity this is the most interesting\ncase since for the other two CrAs\/CdSe interfaces the\nhalf-metallic character is conserved for all cases under study. In\ntable \\ref{table6} we have gathered the atom-resolved spin moments\nfor all cases under study and as it can be easily deduced from the\ntable the variation of the spin moments for the same position of\nthe In(Cd) impurity is similar for all three interfaces and thus\nwe will restrict our discussion to the CrAs\/InAs case.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig9.eps}\n\\caption{(Color online) In the case of In(Cd) impurities at\nvarious interfacial sites, we present the atom-resolved DOS of the\nimpurity atom and its neighbors. Note that the In(Cd) DOS has been\nmultiplied by a factor of 5 or 10 in all cases. Details as in\nfigure \\ref{fig2}. \\label{fig9}}\n\\end{figure}\n\nWe expect that the most frequent case to occur would be the In\nimpurity at the Cr site since such an impurity does not disrupt\nthe zinc-blende structure. In atoms have only two valence\nelectrons occupying the deep-energy-lying $s$-states and thus for\nthe energy window which we examine the $p$-states, which we\nobserve, have their origin at the nearest As neighbors whose\n$p$-states penetrate in the In sites (Cd has only one valence\n$s$-electron). Thus the In impurity acts similarly to a Void,\nalthough it does not lead to such large reorganization of the\ncharge of the neighboring atoms, leading to slightly larger spin\nmoment of the neighboring atoms with respect to the perfect\ninterfaces as shown in table \\ref{table6}. Due to the small weight\nof the In $p$-states we have multiplied the corresponding DOS with\na factor 5 or 10 in figure \\ref{fig9} to make it visible. With\nrespect to the case of Void impurity at the Cr site, here the\nshift of the bands of the nearest-neighboring As atoms is smaller\nkeeping the half-metallic character of the interface although the\ngap is considerably shrinking.\n\nWhen the In impurity is located at the Void1 site, the disturbance\nof the lattice is smaller with respect to the case just presented,\nalthough both Cr and Void1 sites have the same nearest-neighbors\nand as shown in figure \\ref{fig9} the width of the gap remains\nunchanged. Due to the negligible weight of the In $p$ states also\nthe occurrence of In impurities at Void2 and Void3 sites leads to\na slight variation of the spin moments but the half-metallicity is\npreserved and the gap retains a large width. To conclude we should\ndiscuss also the case of In impurities at As sites. As atoms at\nthe interface have two Cr atoms as nearest neighbors and the\nhybridization between the As $p$- and Cr $t_{2g}$-orbitals is\nstrong. The substitution of an As atom by an In one leads to\nreduced hybridization for the Cr orbitals and Cr atoms at the\ninterface regain the charge participating at the bonds with the\nmissing As atom. This extra charge is accommodated at the Cr\nspin-up states leading to larger spin magnetic moments of the Cr\natoms at the interface which now are about 3.59 $\\mu_B$ instead of\n3.25 $\\mu_B$ in the case of the perfect CrAs\/InAs interface\npresented in table \\ref{table1}. The In impurity atom and its\nnearest-neighboring In atoms have states within the gap, as shown\nin figure \\ref{fig9} but if we take into account that we have\nmultiplied the In DOS by 10 their real weight at the Fermi level\nis negligible with respect to the Cr majority-spin DOS. Thus we\ncan safely consider that the half-metallicity is conserved\nalthough as shown by the Cr DOS, the gap in the minority-spin band\nseriously shrinks and the Fermi level is near the right edge of\nthe gap.\n\n\n\\section{Conclusion}\\label{sec7}\n\nWe have studied using the Korringa-Kohn-Rostoker method the\nappearance of single impurities at interfaces between the\nhalf-metallic ferromagnet CrAs and the binary InAs and CdSe\nsemiconductors. In the case of bulk CrAs studied in reference\n\\cite{Pouliasis} we had shown that most impurities affect the\nhalf-metallic character of CrAs inducing states either at the\nedges of the gap or in the middle of the gap. But multilayers are\nvery thin as experiments show \\cite{Mizuguchi}\n and thus we cannot use the impurity calculations for the bulk to\n discuss the case of interfaces. We have studied al possible\n single impurities at the interfaces. Contrary to the bulk systems\n almost all defects were not affecting the half-metallic character\n of the perfect interfaces. The only exception were Void\n impurities at Cr or In(Cd) sites. The missing Cr or In(Cd) atom\n leads to a reorganization of the charge of the surrounding atoms\n and as a result the $p$ bands of the nearest neighboring As(Se)\n atom shift to higher energies crossing the Fermi level and\n leading to loss of the half-metallicity. This is the opposite\n behavior that the one exhibited by the Void impurities at Cr site in the\n bulk CrAs alloy where the DOS remained almost unchanged\n \\cite{Pouliasis}. The different behavior of the Void impurities\n should be attributed to the lower dimensionality of the\n interfaces with respect to the bulk. But Void impurities are\n Schottky-type and we expect them to exhibit high formation\n energies and thus their number should be small in the\n experimental systems. Thus contrary to the bulk CrAs and\n eventually thick films showing a bulk-like behavior,\n impurities are expected to affect only marginally the\n half-metallic character of the interfaces in the case of thin multilayers\nand the latter ones are promising for spintronic devices.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and known results}\n\n\nThe study of random field $\\xi_x$ from a lattice graph $\\mathbb L$ (usually $\\mathbb Z^d$\nor a Cayley tree $\\Gamma^k$) to a measure space $(E, \\mathcal E)$ is a\ncentral component of ergodic theory and statistical physics.\n\nIn many classical models\nfrom physics (e.g., the Ising model, the Potts model), $E$ is a finite set (i.e., with a finite underlying measure $\\lambda$), and $\\xi_x$\nhas a physical interpretation as the spin of a particle\nat location $x$ in a crystal lattice.\n\nFollowing \\cite{BiKo}, \\cite{BEvE}, \\cite{FS97}, \\cite{Ge}, \\cite{HKLR}, \\cite{HKa}, \\cite{HKb}, \\cite{KS}, \\cite{Sh}, \\cite{Z1}, let us\ngive basic definitions and some known facts related to (gradient) Gibbs measures.\n\n{\\bf $\\sigma$-algebra, Hamiltonian.}\nIn general, $(E, \\mathcal E)$ is a space with an infinite underlying measure $\\lambda$ (i.e. $\\mathbb L$ with counting measure),\nwhere $\\mathcal E$ is the Borel $\\sigma$-algebra of $E$ and $\\xi_x$ usually\nhas a physical interpretation as the spatial position of a particle at location $x$\nin a lattice. In \\cite{FS97} first such models were considered.\n\nThe prime examples of unbounded spin systems are harmonic\noscillators. Another example is the Ginzburg-Landau\ninterface model; which is obtained from the harmonic oscillators \\cite{Ge}, \\cite{Sh}.\n\nDenote by $\\Omega$ the set of functions from $\\mathbb L$ to $E$, such a function also is called a configuration.\n\nAssume random field $(\\xi_x)_{x\\in \\mathbb L}$ on $\\Omega$ given as the projection onto the coordinate $x\\in \\mathbb L$:\n$$\\xi_x(\\omega)=\\omega(x)=\\omega_x, \\ \\ \\omega \\in \\Omega.$$\n\nFor $\\Lambda\\subset \\mathbb L$, denote by $\\mathcal F_\\Lambda$\nthe smallest $\\sigma$-algebra with respect to which $\\xi_x$ is measurable for all $x\\in \\Lambda$. Write $\\mathcal T_\\Lambda=\\mathcal F_{\\mathbb L\\setminus\\Lambda}$.\n\nA subset of $\\Omega$,\nis called a cylinder set if it belongs to $\\mathcal F_\\Lambda$ for some finite set $\\Lambda \\subset \\mathbb L$.\n\nLet $\\mathcal F$ be the smallest\n$\\sigma$-algebra on $\\Omega$ containing the cylinder sets.\n\nWrite $\\mathcal T$ for the tail-$\\sigma$-algebra, i.e., intersection of $\\mathcal T_\\Lambda$\nover all finite subsets $\\Lambda$ of $\\mathbb L$ the sets in $\\mathcal T$ are called tail-measurable sets.\n\nAssume that we are given a family of measurable potential\nfunctions $\\Phi_\\Lambda:\\Omega\\to \\mathbb R\\cup \\{\\infty\\}$ (one for each finite subset $\\Lambda$ of $\\mathbb L$)\n each $\\Phi_\\Lambda$ is $\\mathcal F_\\Lambda$\nmeasurable.\n\nFor each finite subset $\\Lambda$ of $\\mathbb L$ define a Hamiltonian:\n$$ H_\\Lambda(\\sigma)=\\sum_{S\\subset \\mathbb L:\\atop S\\cap \\Lambda\\ne \\emptyset}\\Phi_{S}(\\sigma),$$\n where the sum is taken over finite subsets $S$.\n\n{\\bf Gibbs Measures.}\nTo define Gibbs measures and gradient Gibbs measures, we will need some additional\nnotation \\cite{Ge}, \\cite{Sh}.\n\nLet $(X, \\mathcal X)$ and $(Y, \\mathcal Y)$ be general measure spaces.\n\nA function $\\pi : \\mathcal X\\times Y\\to [0, \\infty]$ is called a probability kernel from $(Y, \\mathcal Y)$ to $(X, \\mathcal X)$ if\n\\begin{itemize}\n\\item[1.] $\\pi(\\cdot | y)$ is a probability measure on $(X, \\mathcal X)$ for each fixed $y\\in Y$, and\n\\item[2.] $\\pi(A | \\cdot)$ is $\\mathcal Y$-measurable for each fixed $A\\in \\mathcal X$.\n\\end{itemize}\n\nSuch a kernel maps each measure $\\mu$, on $(Y, \\mathcal Y)$ to a measure $\\mu\\pi$ on $(X, \\mathcal X)$ by\n$$\\mu\\pi(A)=\\int \\pi(A | \\cdot)d\\mu$$\n\nThe following is a probability kernel from $(\\Omega, \\mathcal T_\\Lambda)$ to $(\\Omega, \\mathcal F)$:\n$$\\gamma_\\Lambda(A,\\omega)=Z_\\Lambda(\\omega)^{-1}\\int\\exp(-H_\\Lambda(\\sigma_\\Lambda\\omega_{\\Lambda^c}))\n\\mathbf{1}_A(\\sigma_\\Lambda\\omega_{\\Lambda^c})\\nu^{\\otimes\\Lambda}(d\\sigma_\\Lambda),$$\nwhere $\\nu=\\{\\nu(i)>0, i\\in E\\}$ is a counting measure.\n\nA configuration $\\sigma$ has finite energy if $ \\Phi_\\Lambda(\\sigma) < \\infty$ for all finite $\\Lambda$. Moreover, $\\sigma$ is $\\Phi$-admissible\nif each $Z_\\Lambda(\\sigma)$ is finite and non-zero.\n\nGiven a measure $\\mu$ on $(\\Omega, \\mathcal F)$, define a new\nmeasure $\\mu\\gamma_\\Lambda$ by\n$$\\mu\\gamma_\\Lambda(A)=\\int\\gamma_\\Lambda(A, \\cdot)d\\mu$$\n\n\\begin{defn} A probability measure $\\mu$ on $(\\Omega, \\mathcal F)$ is called a Gibbs measure if $\\mu$ is supported on the set of $\\Phi$-admissible configurations in $\\Omega$ and for all finite subset $\\Lambda$ we have\n$$\\mu \\gamma_\\Lambda=\\mu.$$\n\\end{defn}\n{\\bf Gradient Gibbs measure.} For any configuration $\\omega = (\\omega(x))_{x \\in \\mathbb L} \\in E^\\mathbb L$ and edge $b = \\langle x,y \\rangle$ of $\\mathbb L$\nthe \\textit{difference} along the edge $b$ is given by $\\nabla \\omega_b = \\omega_y - \\omega_x$ and $\\nabla \\omega$ is called the \\textit{gradient field} of $\\omega$.\n\nThe gradient spin variables are now defined by $\\eta_{\\langle x,y \\rangle} = \\omega_y - \\omega_x$ for each $\\langle x,y \\rangle$.\n\nThe space of \\textit{gradient configurations} denoted by $\\O^\\nabla$. The measurable structure on the space $\\Omega^{\\nabla}$ is given by $\\sigma$-algebra $$\\mathcal{F}^\\nabla:=\\sigma(\\{ \\eta_b \\, \\vert \\, b \\in \\mathbb L \\}).$$\nNote that $\\mathcal F^\\nabla$ is the subset of\n$\\mathcal F$ containing those sets that are invariant under translations $\\omega\\to \\omega +c$ for $c\\in E$.\n\nSimilarly, we define\n$$\\mathcal T^\\nabla_\\Lambda=\\mathcal T_\\Lambda \\cap \\mathcal{F}^\\nabla, \\ \\ \\mathcal F^\\nabla_\\Lambda=\\mathcal F_\\Lambda \\cap \\mathcal{F}^\\nabla.$$\n\nLet $\\Phi$ be a translation invariant\ngradient potential. Since, given any $A\\in \\mathcal F^\\nabla$, the kernels $\\gamma^\\Phi_\\Lambda(A, \\omega)$ are $\\mathcal F^\\nabla$-measurable\nfunctions of $\\omega$, it follows that the kernel sends a given measure $\\mu$ on $(\\Omega, \\mathcal F^\\nabla)$ to another\nmeasure $\\mu\\gamma_\\Lambda^\\Phi$ on $(\\Omega, \\mathcal F^\\nabla)$.\n\n\\begin{defn} A measure $\\mu$ on $(\\Omega, \\mathcal F^\\nabla)$ is called a gradient Gibbs measure\nif for all finite subset $\\Lambda$ we have\n$$\\mu \\gamma^\\Phi_\\Lambda=\\mu.$$\n\\end{defn}\nNote that, if $\\mu$ is a Gibbs measure on $(\\Omega, \\mathcal F)$, then its restriction to $\\mathcal F^\\nabla$ is a gradient\nGibbs measure.\n\nA gradient Gibbs measure is said to be localized or smooth if it arises\nas the restriction of a Gibbs measure in this way. Otherwise, it is non-localized or\nrough.\n\nIt is known\\cite{Ge}, \\cite[Theorem 8.19.]{FV} that\nmany natural Gibbs measures on $\\mathbb Z^d$ are rough when $d\\in \\{1, 2\\}$.\n\n{\\bf Construction of gradient Gibbs measure on Cayley trees}. Following \\cite{KS}\nwe consider models where spin-configuration $\\omega$ is a function from the\nvertices of the Cayley tree $\\Gamma^k=(V, \\vec L)$ to the set $E= \\Z$, where\n$V$ is the set of vertices and $\\vec L$ is the set of oriented edges (bonds) of the tree\n(see Chapter 1 of \\cite{Ro} for properties of the Cayley tree).\n\nFor nearest-neighboring (n.n.) interaction potential $\\Phi=(\\Phi_b)_b$, where\n$b=\\langle x,y \\rangle$ is an edge, define symmetric transfer matrices $Q_b$ by\n\\begin{equation}\\label{Qd}\nQ_b(\\omega_b) = e^{- \\big(\\Phi_b(\\omega_b) + | \\partial x|^{-1} \\Phi_{\\{x\\}}(\\omega_x) + |\\partial y |^{-1} \\Phi_{\\{y\\}} (\\omega_y) \\big)},\n\\end{equation}\nwhere $\\partial x$ is the set of all nearest-neighbors of $x$ and $|S|$ denotes the number of elements of the set $S$.\n\nDefine the Markov (Gibbsian) specification as\n$$\n\\gamma_\\Lambda^\\Phi(\\sigma_\\Lambda = \\omega_\\Lambda | \\omega) = (Z_\\Lambda^\\Phi)(\\omega)^{-1} \\prod_{b \\cap \\Lambda \\neq \\emptyset} Q_b(\\omega_b).\n$$\n\nIf for any bond $b=\\langle x,y \\rangle$ the transfer operator $Q_b(\\omega_b)$ is\na function of gradient spin variable $\\zeta_b=\\omega_y-\\omega_x$ then the underlying potential $\\Phi$ is called\na \\textit{gradient interaction potential}.\n\n \\emph{Boundary laws} (see \\cite{Z1}) which allow to describe the set $\\mathcal{G}(\\gamma)$ of all Gibbs measures (that are Markov chains on trees).\n\n\\begin{defn}\n\tA family of vectors $\\{ l_{xy} \\}_{\\langle x,y \\rangle \\in \\vec L}$ with $l_{xy}=\\left(l_{xy}(i): i\\in \\Z\\right) \\in (0, \\infty)^\\Z$ is called a {\\em boundary law for the transfer operators $\\{ Q_b\\}_{b \\in \\vec L}$} if for each $\\langle x,y \\rangle \\in \\vec L$ there exists a constant $c_{xy}>0$ such that the consistency equation\n\t\\begin{equation}\\label{eq:bl}\n\tl_{xy}(i) = c_{xy} \\prod_{z \\in \\partial x \\setminus \\{y \\}} \\sum_{j \\in \\Z} Q_{zx}(i,j) l_{zx}(j)\n\t\\end{equation}\n\tholds for every $i \\in \\Z$.\n\nA boundary law is called {\\em $q$-periodic} if $l_{xy} (i + q) = l_{xy}(i)$ for every oriented edge $\\langle x,y \\rangle \\in \\vec L$ and each $i \\in \\Z$.\n\t\n\\end{defn}\n\nIt is known that there is a one-to-one correspondence between boundary laws\nand tree-indexed Markov chains if the boundary laws are {\\em normalisable} in the sense of Zachary \\cite{Z1}:\n\n\\begin{defn} A boundary law $l$ is said to be {\\em normalisable} if and only if\n\\begin{equation}\\label{Norm}\n\\sum_{i \\in \\Z} \\Big( \\prod_{z \\in \\partial x} \\sum_{j \\in \\Z} Q_{zx}(i,j) l_{zx}(j) \\Big) < \\infty\n\\end{equation} at any $x \\in V$.\n\\end{defn}\nFor any $\\Lambda \\subset V$ we define its outer boundary as\n\\begin{equation*}\n\\partial \\Lambda := \\{ x \\notin \\Lambda : \\langle x,y\\rangle \\, \\mbox{ for some } \\, y \\in \\Lambda\\}.\n\\end{equation*}\nThe correspondence now reads the following:\n\n\\begin{thm} \\cite{Z1}\nFor any Markov specification $\\gamma$ with associated family of transfer matrices $(Q_b)_{b \\in L}$ we have\n\\begin{enumerate}\n\\item Each {\\it normalisable} boundary law $(l_{xy})_{x,y}$ for $(Q_b)_{b \\in L}$ defines a unique tree-indexed Markov chain $\\mu \\in \\mathcal{G}(\\gamma)$ via the equation given for any connected set $\\Lambda \\subset V$\n\\begin{equation}\\label{BoundMC}\n\\mu(\\sigma_{\\Lambda \\cup \\partial \\Lambda}=\\omega_{\\Lambda \\cup \\partial \\Lambda}) = (Z_\\Lambda)^{-1} \\prod_{y \\in \\partial \\Lambda} l_{y y_\\Lambda}(\\omega_y) \\prod_{b \\cap \\Lambda \\neq \\emptyset} Q_b(\\omega_b),\n\\end{equation}\nwhere for any $y \\in \\partial \\Lambda$, $y_\\Lambda$ denotes the unique $n.n.$ of $y$ in $\\Lambda$.\n\\item Conversely, every tree-indexed Markov chain $\\mu \\in \\mathcal{G}(\\gamma)$ admits a representation of the form (\\ref{BoundMC}) in terms of a {\\it normalisable} boundary law (unique up to a constant positive factor).\n\\end{enumerate}\n\\end{thm}\n\n\nThe Markov chain $\\mu$ defined in \\eqref{BoundMC} has the transition probabilities\n\\begin{equation}\\label{last}\nP_{xy}(i,j)=\\mu(\\s_y = j \\mid \\s_x =i)\n= \\frac{l_{yx}(j) Q_{yx}(j, i)}{\\sum_s l_{yx}(s) Q_{yx}(s, i)}.\n\\end{equation}\nThe expressions \\eqref{last} may exist even in situations where the underlying boundary\nlaw $(l_{xy})_{x,y}$ is not normalisable. However, the Markov chain given by (\\ref{last}), in general, does not have an invariant probability measure.\nTherefore in \\cite{HKLR}, \\cite{HKa}, \\cite{HKb}, \\cite{KS}\nsome non-normalisable boundary laws are used to give gradient Gibbs measures.\n\n Now we give some results of above mentioned papers. Consider a model on Cayley tree $\\Gamma^k=(V, \\vec L)$, where the spin takes values in\nthe set of all integer numbers $\\mathbb Z$. The set of all configurations is $\\Omega:=\\mathbb Z^V$.\n\nFor $\\Lambda\\subset V$, fix a site $w \\in \\Lambda$. If the boundary law $l$ is assumed to be $q$-periodic, then take $s \\in \\mathbb{Z}_q=\\{0,1,\\dots,q-1\\}$ and define probability measure $\\nu_{w,s}$ on $\\mathbb{Z}^{\\{b \\in \\vec L \\mid b \\subset \\Lambda\\}}$ by\n$$\n\\nu_{w,s}(\\eta_{\\Lambda \\cup \\partial \\Lambda}=\\zeta_{\\Lambda \\cup \\partial \\Lambda})=\n$$\n$$Z^\\Lambda_{w,s}\\prod_{y \\in \\partial \\Lambda} l_{yy_\\L}\\Bigl (T_q(\ns+\\sum_{b\\in \\Gamma(w,y)}\\zeta_b)\n\\Bigr) \\prod_{b \\cap \\Lambda \\neq \\emptyset}Q_b(\\zeta_b),\n$$\nwhere $Z^\\Lambda_{w,s}$ is a normalization constant, $\\Gamma(w,y)$ is the unique path from $w$ to $y$\nand $T_q: \\mathbb{Z} \\rightarrow \\mathbb{Z}_q$ denotes the coset projection.\n\n\\begin{thm} \\cite{KS}\n\tLet $(l_{})_{ \\in \\vec L}$ be any $q$\n-periodic boundary law to some gradient interaction potential.\nFix any site $w \\in V$ and any class label $s \\in \\mathbb{Z}_q$. Then\n$$\t\\nu_{w,s}(\\eta_{\\Lambda \\cup \\partial \\Lambda}=\\zeta_{\\L\\cup\\partial\\L})\n\t=$$\n\\begin{equation}\nZ^\\Lambda_{w,s} \\prod_{y \\in \\partial \\Lambda} l_{yy_\\L}\\Bigl (T_q(\n\ts+\\sum_{b\\in \\Gamma(w,y)}\\zeta_b)\n\t\\Bigr) \\prod_{b \\cap \\Lambda \\neq \\emptyset}\n\tQ_b(\\zeta_b),\n\t\\end{equation}\ngives a consistent family of probability measures on the gradient space $\\Omega^\\nabla$.\nHere $\\Lambda$ with $w \\in \\L \\subset V$ is any finite connected set,\n$\\zeta_{\\L\\cup\\partial\\L} \\in \\Z^{\\{b \\in \\vec L \\mid b \\subset (\\L \\cup \\partial\\L)\\}}$ and $Z^\\Lambda_{w,s}$ is a normalization constant.\\\\\n\\end{thm}\n\tThe measures $\\nu_{w,s}$ will be called pinned gradient measures.\n\n\nIf $q$-periodic boundary law and the underlying potential are translation invariant then it is possible to obtain\nprobability measure $\\nu$ on the gradient space by mixing the pinned gradient measures:\n\n\\begin{thm}\\cite{KS}\t\nLet a $q$-periodic boundary law $l$ and its gradient interaction potential are translation invariant.\nLet $\\Lambda \\subset V$ be any finite connected set and let $w\\in \\Lambda$ be any vertex. Then the measure $\\nu $ with marginals given by\n\\begin{equation}\n\\nu (\\eta_{\\L\\cup\\partial \\L} = \\zeta_{\\L\\cup\\partial\\L}) = Z_\\L \\ \\left(\\sum_{s\\in\\Z_q} \\prod_{y \\in \\partial\\L} l \\big(s + \\sum_{b \\in \\Gamma(w,y)} \\zeta_{b}\\big) \\right)\\prod_{b \\cap \\L \\neq \\emptyset} Q(\\zeta_b),\n\\end{equation}\nwhere $Z_\\L$ is a normalisation constant, defines a translation invariant gradient Gibbs measure on $\\Omega^\\nabla$.\n\\end{thm}\n\n{\\bf SOS model.} The (formal) Hamiltonian of the SOS model is\n\\begin{equation}\\label{nu1}\n H(\\omega)=-J\\sum_{\\langle x,y\\rangle \\in L}\n|\\omega_x-\\omega_y|, \\ \\ \\omega \\in\\Omega,\n\\end{equation}\nwhere $J \\in \\mathbb R_+$ is a constant.\n\nIn \\cite{HKLR}, using Theorem 3 some gradient Gibbs measures are found.\n\n Let $\\beta>0$ be inverse temperature and $\\theta:= \\exp(-J\\beta)<1$.\n The transfer operator $Q$ then reads $Q(i-j)=\\theta^{\\vert i-j \\vert }$ for any $i,j \\in \\mathbb{Z}$,\n and a translation invariant boundary law, denoted by $\\mathbf z$, is any positive function on $\\mathbb{Z}$\n solving the consistency equation, whose values we will denote by $z_i$ instead of $z(i)$.\n By definition of the boundary law it is only unique up to multiplication with any positive prefactor.\n Hence we may choose this constant in a way such that we have $z_0=1$.\n\n Set $\\mathbb{Z}_0:= \\mathbb{Z} \\setminus \\{0\\}$. Then the boundary law equation (for translation-invariant case, i.e.\n $l_b\\equiv l$, for all $b\\in L$) reads\n\\begin{equation}\\label{nu11}\nz_i=\\left({\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}z_j\n\\over\n1+\\sum_{j\\in \\mathbb Z_0}\\theta^{|j|}z_j}\\right)^k, \\ \\ i\\in\\mathbb Z_0.\n \\end{equation}\n\nLet $\\mathbf z(\\theta)=(z_i=z_i(\\theta), i\\in \\mathbb Z_0)$ be a solution to (\\ref{nu11}).\n\nDenote $u_i=\\sqrt[k]{z_i}$ and assume $u_0=1$.\n\n\\begin{pro} \\cite{HKLR} If $z_0=1$ (i.e. $u_0=1$) then the equation (\\ref{nu11}) is equivalent to the following\n\\begin{equation}\\label{V}\nu_i^k={u_{i-1}+u_{i+1}-\\tau u_i\\over u_{-1}+u_{1}-\\tau}, \\ \\ i\\in \\mathbb Z,\n\\end{equation}\nwhere $\\tau=\\theta^{-1}+\\theta$.\n\\end{pro}\n\nIn general, solutions of (\\ref{V}) are not known. But in class of periodic solutions, some results are obtained.\nThe following theorem is proved for $k=2$ and $4$-periodic boundary laws:\n \t\n\\begin{thm} \\cite{HKLR} For the SOS model (\\ref{nu1}) on the binary tree (i.e. $k=2$) with parameter $\\tau=\\theta+\\theta^{-1}$ the following assertions hold\n\t\\begin{itemize}\n\t\t\\item[1.] If $\\tau \\leq 4$ then there is precisely one GGM associated to a 4-periodic boundary law.\n\t\t\\item[2.] If $4< \\tau \\leq 6$ then there are precisely two GGMs.\n\t\t\\item[3.] If $6<\\tau < 2+2\\sqrt{5}$ then there are precisely three GGMs.\n\t\t\\item[4.] If $\\tau \\geq 2+2\\sqrt{5}$ then there are precisely four such measures.\n\t\\end{itemize}\n\\end{thm}\n\nThe following theorem is proved for any $k\\geq 2$ and $3$-periodic boundary laws.\n\nDenote\n$$\\tau_0:={2k+1\\over k-1}.$$\n\n\\begin{thm} \\cite{HKLR} For the SOS-model on the $k$-regular tree, $k \\geq 2$, with parameter $\\tau$ there is $\\tau_c$ such that\n$0<\\tau_c<\\tau_0$ and the following holds:\n\\begin{itemize}\n\t\\item[1.] If $\\tau<\\tau_c$ then there is no any GGM corresponding to\n\ta nontrivial 3-periodic boundary.\n\t\\item[2.] At $\\tau=\\tau_c$ there is a unique GGM corresponding to a\n\tnontrivial $3$-periodic boundary law.\n\t\\item[3.] For $\\tau>\\tau_c$, $\\tau\\ne \\tau_0$ (resp. $\\tau=\\tau_0$) there are exactly\n\ttwo such (resp. one) GGMs.\n\\end{itemize}\t\n\n The GGMs described above are all different from the GGMs mentioned in Theorem 4.\n\\end{thm}\n\n{\\bf General case.}\n Assume that the transfer operator $\\{Q_b\\}_{b\\in \\mathbb L}$, defined in (\\ref{Qd}), is summable, i.e.\n$$\\sum_{i\\in \\mathbb Z}Q_b(i) <\\infty \\ \\ \\mbox{for all} \\ \\ b\\in \\mathbb L.$$\n\nThe following is the main result of \\cite{HKa}:\n\n\\begin{thm} For any summable $Q$ and any degree $k\\geq 2$ there is a finite period\n$q_0(k)$ such that for all $q\\geq q_0(k)$ there are GGMs of\nperiod $q$ which are not translation invariant.\n\\end{thm}\n\nMoreover, in \\cite{HKb} the authors provided general conditions in terms of the\nrelevant $p$-norms of the associated transfer operator $Q$ which ensure the existence of\na countable family of proper Gibbs measures. The existence of delocalized GGMs is proved, under natural conditions on $Q$.\nThis implies coexistence of both types of measures for large classes of models including the SOS-model,\nand heavy-tailed models arising for instance for potentials of logarithmic growth.\n\n\\section{4-periodic boundary laws for $k\\geq 2$}\n\nIn this section our goal is to find solutions of (\\ref{V}) which have the form\n\\begin{equation}\\label{up}\nu_n=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ n=2m,\\\\[2mm]\na, \\ \\ \\mbox{if} \\ \\ n=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nb, \\ \\ \\mbox{if} \\ \\ n=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $a$ and $b$ some positive numbers.\n\nThen from (\\ref{V}) for $a$ and $b$ we get the following system of equations\n\\begin{equation}\\label{ab}\n\\begin{array}{ll}\n(a+b-\\tau)b^k+\\tau b-2=0\\\\[2mm]\n(a+b-\\tau)a^k+\\tau a-2=0.\n\\end{array}\n\\end{equation}\n\nThe case $k=2$ is fully analyzed in \\cite{HKLR} and the following is proved\n\\begin{pro}\\label{pps} For $k=2$ the periodic solutions of the form (\\ref{up}) (i.e. solutions of the system (\\ref{ab}))\ndepend on the parameter $\\tau=2\\cosh(\\beta)$ in the following way.\n\t\\begin{enumerate}\n\t\t\\item If $\\tau \\leq 4$ then there is a unique solution.\n\t\t\\item If $4<\\tau \\leq 6$ then there are exactly two solutions.\n\t\t\\item If $6<\\tau < 2+2\\sqrt{5}$ then there are exactly four solutions.\n\t\t\\item If $\\tau \\geq 2+2\\sqrt{5}$ then there are exactly five solutions.\n\t\\end{enumerate}\n\twhere explicit formula of each solution is found.\n\\end{pro}\n\nNow we reduce the system (\\ref{ab}) to a polynomial equation with one unknown $a$. To do this\nfrom the first (resp. second) equation of (\\ref{ab}) find $a$ (resp. $b$):\n\n\\begin{equation}\\label{ab1}\n\\begin{array}{ll}\na=f(b):=\\tau-b+(2-\\tau b)b^{-k}\\\\[2mm]\nb=f(a).\n\\end{array}\n\\end{equation}\nThus the system (\\ref{ab}) is reduced to\n\\begin{equation}\\label{ab2}\na=f(f(a)).\n\\end{equation}\nNote that solutions of $a=f(a)$ are solutions to (\\ref{ab2}) too. It is easy to see that $a=f(a)$ is equivalent to\n\\begin{equation}\\label{ab3}\nQ(a):=2a^{k+1}-\\tau a^k+\\tau a-2=0\n\\end{equation}\n\n\nThe equation (\\ref{ab3}) has the solution $a=1$ independently of the parameters $(\\tau, k)$. Dividing both sides of (\\ref{ab3}) by $a-1$\nwe get\n\\begin{equation}\\label{uy22}\n2a^k+(2-\\tau)(a^{k-1}+a^{k-2}+\\dots+a)+2=0.\n\\end{equation}\nThe following lemma gives the number of solutions to equation (\\ref{uy22}) (compare with Lemma 4.7 in \\cite{HKLR}):\n\\begin{lemma}\\label{l6} For each $k\\geq 2$,\n\tthere is exactly one critical value of $\\tau$, i.e., $\\tau_c=\\tau_c(k):=2\\cdot {k+1\\over k-1}$, such that\n\t\\begin{enumerate}\n\t\t\\item if $\\tau<\\tau_c$ then (\\ref{uy22}) has no positive solution;\n\t\t\\item if $\\tau=\\tau_c$ then the equation has a unique solution $a=1$;\n\t\t\\item if $\\tau>\\tau_c$,\tthen it has exactly two solutions (both different from 1);\n\t\t\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFrom (\\ref{uy22}) we get\n\t\\[\\tau=\\psi_k(a):=2+\\frac{2(a^k+1)}{a^{k-1}+a^{k-2}+\\dots+a}. \\]\n\tWe have $\\psi_k(a)>2$, $a>0$ and $\\psi_k'(a)=0$ is equivalent to\n\t\\begin{equation}\\label{ho}\n\t\\sum_{j=1}^{k-1}(k-j)a^{k+j-1}-\\sum_{j=1}^{k-1}ja^{j-1}=0.\n\t\\end{equation}\n\tThe last polynomial equation has exactly one positive solution,\n\tbecause signs of its coefficients changed only one time,\n\tand at $a=0$ it is negative, i.e. -1 and at $a=+\\infty$ it is positive.\n\tMoreover, this unique solution is $a=1$, because putting $a=1$ in (\\ref{ho}) we get\n $$\\sum_{j=1}^{k-1}(k-j)-\\sum_{j=1}^{k-1}j=\\sum_{j=1}^{k-1}k-2\\sum_{j=1}^{k-1}j = k(k-1)-2\\cdot {k(k-1)\\over 2}=0.$$\n\n Thus $\\psi_k(a)$ has unique minimum\n\tat $a=1$, and $\\lim_{a\\to 0}\\psi_k(a)=\\lim_{a\\to +\\infty}\\psi_k(a)=+\\infty$.\nConsequently,\n\t\\[\\tau_c=\\tau_c(k)=\\min_{a>0}\\psi_k(a)=\\psi_k(1)=2\\cdot {k+1\\over k-1}.\\]\n\tThese properties of $\\psi_k(a)$ completes the proof.\n\\end{proof}\n\\begin{rk} The equation (\\ref{uy22}) was also considered in \\cite{BH}.\nLemma \\ref{l6} improves their result (see Theorem 5.2 of \\cite{BH}), because we found explicit\nformula for the critical value $\\tau_c$.\n\n\\end{rk}\nNow we want to find solutions of (\\ref{ab2}) which are different from solutions of $a=f(a)$ (i.e., $Q(a)=0$).\n\nBy simple calculations the equation (\\ref{ab2}) can be rewritten as\n\\begin{equation}\\label{ab4}\nP(a):=(2-\\tau a)[2-\\tau a+\\tau a^k-a^{k+1}]^k-a^{k^2}[\\tau a^{k+1}+(2-\\tau^2)a^k+\\tau^2a-2\\tau]=0.\n\\end{equation}\nRecall that $Q(a)$ divides $P(a)$. Now we shall find ${P(a)\\over Q(a)} $.\nIt is easy to see that $P(a)$ can be written as\n$$P(a)=(2-\\tau a)[a^{k+1}-Q(a)]^k-a^{k^2}[2a^k-\\tau a^{k+1}+\\tau Q(a)]$$\n$$=(2-\\tau a)\\sum_{j=0}^{k}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)-a^{k^2+k}(2-\\tau a)-\\tau a^{k^2}Q(a)$$\n$$=(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)+(2-\\tau a)a^{k^2+k}-a^{k^2+k}(2-\\tau a)-\\tau a^{k^2}Q(a)$$\n$$=(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)-\\tau a^{k^2}Q(a)$$\n$$=Q(a)\\left((2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j-1}(a)-\\tau a^{k^2}\\right).$$\n\nConsequently, the equation (\\ref{ab2}) in case $Q(a)\\ne 0$ is reduced to\n\\begin{equation}\\label{kt}\nU(a):=\\tau a^{k^2}-(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j-1}(a)=0.\n\\end{equation}\nBut $U(a)$ may have some roots coinciding with roots of $Q(a)$. This is result of the following lemma.\n\n\\begin{lemma} Let $a=\\hat a$ be a root of $Q(a)$. Then $\\hat a$ is a root of $U(a)$ iff\n$\\hat a={2k\\over \\tau (k-1)}$ and $\\tau$ satisfies\n\\begin{equation}\\label{ha}\n(k-1)^k\\tau^{k+1}-(k-1)2^{k-1}k^k\\tau^2+(2k)^{k+1}=0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} For the root $\\hat a$ we have $Q(\\hat a)=0$. Therefore from (\\ref{kt}), i.e. $U(\\hat a)=0$, we get\n$$\\tau \\hat a^{k^2}+(2-\\tau \\hat a)k \\hat a^{k^2-1}=0 \\ \\ \\Leftrightarrow \\ \\ \\tau \\hat a+(2-\\tau \\hat a)k=0 \\ \\ \\Leftrightarrow \\ \\ \\hat a={2k\\over \\tau (k-1)}.$$\n\nThen $Q(\\hat a)=Q\\left({2k\\over \\tau (k-1)}\\right)=0$ gives (\\ref{ha}).\n\n\\end{proof}\n\n\\begin{lemma}\\label{s3} For each fixed $k\\geq 2$ the equation (\\ref{ha}) has exactly two\nsolutions $\\tau_1(k)={2k\\over k-1}$ and $\\tau_2(k)>{2(k+1)\\over k-1}$.\n\\end{lemma}\n\\begin{proof} It is easy to see that $\\tau={2k\\over k-1}$ is a solution to (\\ref{ha}),\nfor any $k\\geq 2$. Moreover, the corresponding\n$\\hat a$ is $1$.\n\nDenote $L=(k-1)\\tau-2k$\n then from (\\ref{ha}) we get\n$$0=(L+2k)^{k+1}-2^{k-1}k^k(L+2k)^2+(k-1)(2k)^{k+1}$$\n$$=\\sum_{j=0}^{k+1}{k+1\\choose j}L^{k+1-j}(2k)^j-2^{k-1}k^kL^2-(2k)^{k+1}L-2^{k+1}k^{k+2}+(k-1)(2k)^{k+1}\n$$ $$=\\sum_{j=0}^{k}{k+1\\choose j}L^{k+1-j}(2k)^j+(2k)^{k+1}-2^{k-1}k^kL^2-(2k)^{k+1}L-2^{k+1}k^{k+2}+(k-1)(2k)^{k+1}$$\n$$=\\sum_{j=0}^{k}{k+1\\choose j}L^{k+1-j}(2k)^j-2^{k-1}k^kL^2-(2k)^{k+1}L$$\n$$=L\\left(\\sum_{j=0}^{k}{k+1\\choose j}L^{k-j}(2k)^j-2^{k-1}k^kL-(2k)^{k+1}\\right)$$\n$$=L\\left(\\sum_{j=0}^{k-2}{k+1\\choose j}L^{k-j}(2k)^j+{k(k-1)\\over 2}L-(k-1)(2k)^{k}\\right).$$\nThus $L=0$ (i.e. $\\tau=\\tau_1(k)$) or\n\\begin{equation}\\label{L}\\sum_{j=0}^{k-2}{k+1\\choose j}L^{k-j}(2k)^j+{k(k-1)\\over 2}L-(k-1)(2k)^{k}=0.\n\\end{equation}\nBy Lemma \\ref{l6} we know that $Q(a)=0$ has a solution different\nfrom 1 iff $\\tau>\\tau_c={2(k+1)\\over k-1}$. Therefore $L=(k-1)\\tau-2k>(k-1)\\cdot {2(k+1)\\over k-1}-2k=2>0$.\nThe polynomial equation (\\ref{L}) has exactly one positive solution, denoted by $L^*$,\n\tbecause signs of its coefficients changed only one time. Moreover, we have $L^*>2$, because at $L=2$ the LHS of (\\ref{L}) is negative:\n$$2^k[(k+1)^{k+1}-(3k+1)k^k]<0$$ and at $L=+\\infty$ it is positive. Thus $\\tau_2(k)={L^*+2k\\over k-1}$.\n\\end{proof}\n\\begin{ex} For $k=2,3,4$ we have\n$$\\tau_1(2)=4, \\ \\ \\tau_2(2)=2+2\\sqrt{5}.$$\n$$\\tau_1(3)=3, \\ \\ \\tau_2(3)=3\\sqrt{2}.$$\n$$\\tau_1(4)=8\/3, \\ \\ \\tau_2(4)\\approx 3.497.$$\n\n\\end{ex}\n\n\\begin{rk} It seems impossible to solve equation (\\ref{kt}) for each $k\\geq 2$. But one can use numerical\nmethods to give some its solutions for concrete values of parameters.\nTherefore, one can try to solve it for small values of $k$. In \\cite{HKLR} the case $k=2$ is fully analyzed.\nBelow we shall consider the case $k=3$. Cases $k\\geq 4$ remains open.\n\\end{rk}\n\n{\\bf Case $k=3$}. In this case the equation (\\ref{kt}) has the form\n\n$$g(a,\\tau):=(\\tau^2+2)a^8-(\\tau^2+2)\\tau a^7+\\tau^2 a^6+2(\\tau^2+2)\\tau a^5$$\n\\begin{equation}\\label{g}-4(2\\tau^2+1)a^4-(\\tau^2-8)\\tau a^3+6\\tau^2a^2-12\\tau a+8=0.\n\\end{equation}\nIt is well known (see \\cite{Pra}, p.28) that the number of positive\nroots of the polynomial (\\ref{g}) does not exceed the number of sign\nchanges of its coefficients.\nSince $\\tau>2$, the number of positive roots of\nthe polynomial (\\ref{g}) is at most $6$. Numerical analysis shows that\nfor some values of $\\tau$ there are exactly 6 solutions (see Fig.\\ref{k3}).\nIndeed, rewrite (\\ref{g}) as\n$$a=Y(a,\\tau):={(\\tau^2+2)\\tau a^7+4(2\\tau^2+1)a^4+(\\tau^2-8)\\tau a^3\n+13\\tau a-8\\over(\\tau^2+2)a^7+\\tau^2 a^5+2(\\tau^2+2)\\tau a^4+6\\tau^2 a+\\tau}.$$\nNote that, for fixed $\\tau>2$, the function $Y(a, \\tau)$ is continuous with respect\nto both arguments $a>0$, $\\tau>2$ and is a bounded function.\nMoreover, $Y(0,\\tau)=-{8\\over \\tau}$ and $Y(+\\infty, \\tau)=\\tau$.\n\n\\begin{figure}[h!]\n\\vspace{-.5pc} \\centering\n\\includegraphics[width=7.2cm]{k3tau6.eps}\n\\includegraphics[width=7.2cm]{k3tau6-davomi.eps}\n\\caption{Graph of function $Y(x,6)-x$, for $x\\in (0.3,0.9)$ and $x\\in (0.9,10)$. Hence, $x=Y(x,6)$ has 6 (the maximal number) positive solutions.}\\label{k3}\n\\end{figure}\n\n\n\\begin{rk} \\label{rk} For $k=3$ one can explicitly find all positive solutions of (\\ref{ab3}), i.e.\n$$a_1=1, \\ \\ 01.$$\nMoreover, $a_2a_3=1$.\n\\end{rk}\n\nIn case $a\\ne b$, we do not know explicit solutions of (\\ref{g}).\nSince $b=f(a)$ may be negative for some positive solutions $a$,\nwe have to check positivity of $b$ for each positive $a$. To avoid this difficulty, in case $k=3$ and $a\\ne b$, we solve (\\ref{ab}) as\nfollows.\n\nWe rewrite the system of equations (\\ref{ab}) for the case $k=3$.\n\\begin{equation}\\label{eq1.1}\\left\\{\n \\begin{array}{ll}\n (a+b-\\tau)a^3+\\tau a-2=0; \\\\\n (a+b-\\tau)b^3+\\tau b-2=0.\n \\end{array}\n\\right.\n\\end{equation}\n\\begin{lemma}\\label{new} For the system (\\ref{eq1.1}) there are critical\nvalues $\\tau^{(1)}_{cr}\\approx 3.13039$ and $\\tau^{(2)}_{cr}\\approx 4,01009$ of $\\tau$ such that the following assertions hold\n\\begin{enumerate}\n\\item If $\\tau\\in [2, \\tau^{(1)}_{cr}]$ then there is no any solution.\n\\item If $\\tau\\in (\\tau^{(1)}_{cr}, \\tau^{(2)}_{cr}]$ then there is precisely one solution.\n\\item If $\\tau\\in(\\tau^{(2)}_{cr},\\infty)$ then there are precisely two solution to (\\ref{eq1.1}).\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe add the first and second equations of (\\ref{eq1.1}), i.e.,\n$$(a+b-\\tau)(a^3+b^3)+\\tau(a+b)-4=0.$$\nIf $a+b-\\tau=0$ then there is one solution to (\\ref{eq1.1}), i.e., $a=b=\\frac{2}{\\tau}$. We have to find $(a,b), a\\neq b$ solutions, so we can suppose $a+b-\\tau\\neq 0$. Consequently,\n$$a^3+b^3=\\frac{4-\\tau(a+b)}{a+b-\\tau}.$$\nFrom the last equality, one gets\n\\begin{equation}\\label{eq1.2} 3ab=(a+b)^2+\\frac{\\tau(a+b)-4}{(a+b)(a+b-\\tau)}.\n\\end{equation}\nNow, we subtract the second equation of (\\ref{eq1.1}) from the first one. Then\n$$(a+b-\\tau)(a^3-b^3)+\\tau(a-b)=0.$$\nSince $a\\neq b$, both sides can be divided by $a-b$ and we obtain the following\n\\begin{equation}\\label{eq1.3} ab=(a+b)^2+\\frac{\\tau}{a+b-\\tau}.\n\\end{equation}\nLet $a+b=x$. Then by (\\ref{eq1.2}) and (\\ref{eq1.3}), we have\n$$2x^2+\\frac{3\\tau}{x-\\tau}=\\frac{\\tau x-4}{x(x-\\tau)}.$$\nThe last equation can be written as\n\\begin{equation}\\label{eq1.4} x^4-\\tau x^3+\\tau x+2=0.\\end{equation}\nBy Ferrari's method for solving a quartic equation, the equation (\\ref{eq1.4}) can be written as\n\\begin{equation}\\label{eq1.5} \\left(x^2-\\frac{\\tau}{2}x+\\frac{c(\\tau)}{2}\\right)^2-\n\\left[\\left(\\frac{\\tau^2}{4}+c(\\tau)\\right)x^2-\\left(\\frac{\\tau c(\\tau)}{2}+\\tau\\right)x+\\left(\\frac{c^2(\\tau)}{4}-2\\right)\\right]=0,\\end{equation}\nwhere $c(\\tau)$ is a real root of the following polynomial\n$$P(z):=z^3-(\\tau^2+8)z-3\\tau^2=0.$$\nDenote\n$$F(\\tau) = \\dfrac {-(\\tau^2+8)} {3}, \\quad R(\\tau)= \\dfrac {3\\tau^2} {2}.$$\nTo find a certain view of $c(\\tau)$, we shall find a real root of $P(z)$. By Cardano's formula, roots of $P(z)$ are:\n$$z_1(\\tau) =S(\\tau)+T(\\tau),$$\n$$z_2(\\tau)= - \\dfrac {S(\\tau) + T(\\tau)} 2 + \\dfrac {i \\sqrt 3} 2 (S(\\tau)- T(\\tau)),$$\n$$z_3(\\tau)= - \\dfrac {S(\\tau) + T(\\tau)} 2 - \\dfrac {i \\sqrt 3} 2 (S(\\tau)- T(\\tau)),$$\nwhere\n$$S(\\tau)= \\sqrt [3] {R(\\tau)+ \\sqrt {F^3(\\tau)+R^2(\\tau)} }, \\quad T(\\tau)= \\sqrt [3] {R(\\tau)- \\sqrt {F^3(\\tau)+ R^2(\\tau)} }.$$\nIt's known that the expression $D(\\tau)=F^3(\\tau)+R^2(\\tau)$ is called the discriminant of the equation.\n\n \\begin{itemize}\n\\item If $D(\\tau)>0$, then one root is real and two are complex conjugates.\n\\item If $D(\\tau)=0$, then all roots are real, and at least two are equal.\n\\item If $D(\\tau)<0$, then all roots are real and unequal.\n\\end{itemize}\nWe have\n$$D(\\tau)=F^3(\\tau)+R^2(\\tau)=-\\frac{1}{27}\\tau^6+\\frac{49}{36}\\tau^4-\\frac{64}{9}\\tau^2-\\frac{512}{27}.$$\nNote that there are two positive solutions of $D(\\tau)$ on $[2,+\\infty)$ i.e., $\\tau_1\\approx 2.994$ and $\\tau_2\\approx 5.45$ and $D(\\tau)<0$ for $\\tau\\in[2, \\tau_1)\\cup (\\tau_2, +\\infty)$ and $D(\\tau)>0$ for $\\tau\\in(\\tau_1, \\tau_2).$\nFor all cases, $S(\\tau)+T(\\tau)$ is a real solution and we can choose $c(\\tau)$ as $S(\\tau)+T(\\tau)$. In addition, $c(\\tau)=S(\\tau)+T(\\tau)>0$.\nThe equation (\\ref{eq1.5}) can be written as\n$$\\left(x^2+\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}-\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}-\\sqrt{\\frac{c^2(\\tau)}{4}-2}\\right)\\times$$\n$$\\times\\left(x^2-\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}+\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}+\\sqrt{\\frac{c^2(\\tau)}{4}-2}\\right)=0.$$\nIt's easy to check that\n$$x^2+\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}-\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}-\\sqrt{\\frac{c^2(\\tau)}{4}-2}>0, \\ x\\in \\mathbb{R}.$$\nHence, from the last equality we obtain\n$$x_{1,2}(\\tau)=\\frac{1}{2}\\left[\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\pm \\sqrt{\\left(\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\right)^2-2\\left(c(\\tau)+\\sqrt{c^2(\\tau)-8}\\right)}\\right].$$\n$$\\left(\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\right)^2-2\\left(c(\\tau)+\\sqrt{c^2(\\tau)-8}\\right)\\geq 0\\ \\Leftrightarrow \\ \\tau\\geq\\tau_1\\approx 2.994.$$\nFrom the equation (\\ref{eq1.3})\n$$ab=x^2_{i}(\\tau)+\\frac{\\tau}{x^2_{i}(\\tau)-\\tau}, \\ i\\in \\{1,2\\}.$$\nNamely,\n$$a(x_{i}(\\tau)-a)=x^2_{i}(\\tau)+\\frac{\\tau}{x^2_{i}(\\tau)-\\tau}.$$\nConsequently,\n$$a_{1,2}^{(i)}(\\tau)=\\frac{x_{i}(\\tau)\\pm\\sqrt{x^2_{i}(\\tau)-4\\left(x^2_i(\\tau)+\\frac{\\tau}{x_{i}(\\tau)-\\tau}\\right)}}{2}.$$\nFrom numerical analysis, $$x^2_{1}(\\tau)-4\\left(x^2_1(\\tau)+\\frac{\\tau}{x_{1}(\\tau)-\\tau}\\right)< 0, \\ \\tau\\in[\\tau_1,+\\infty).$$\nAlso,\n$$x^2_{2}(\\tau)-4\\left(x^2_2(\\tau)+\\frac{\\tau}{x_{2}(\\tau)-\\tau}\\right)\\geq 0, \\ \\tau\\in[\\tau^{(1)}_{cr},+\\infty), \\ \\tau^{(1)}_{cr}\\approx 3.13039.$$\nHence, we have only two cases $a\\in \\{a_1^{(2)}(\\tau), a_2^{(2)}(\\tau)\\}$. When $a_1^{(2)}(\\tau)$ (resp $a_2^{(2)}(\\tau)$) belongs to the interval $(0, x_{2}(\\tau))$ then $b_1^{(2)}(\\tau)$ (resp. $b_2^{(2)}(\\tau)$) is also positive. Again we use numerical analysis and obtain the following results: if $\\tau\\in[2,\\tau^{(1)}_{cr}]$ then $a_2^{(2)}(\\tau)\\geq x_2(\\tau)$, i.e., the equation $(\\ref{eq1.1})$ has no any positive solution such that $a\\neq b$. Also, for all $\\tau\\in(\\tau^{(1)}_{cr}, \\tau^{(2)}_{cr}]$ we have $a_2^{(2)}(\\tau)0, i\\in \\mathbb Z_0)$ be a solution to (\\ref{di1}). Denote\n\\begin{equation}\\label{lr}\nl_i\\equiv l_i(\\theta)=\\sum_{j=-\\infty}^{-1}\\theta^{|i-j|}z_j, \\ \\\nr_i\\equiv r_i(\\theta)=\\sum_{j=1}^{\\infty}\\theta^{|i-j|}z_j, \\ \\ i\\in\\mathbb Z_0.\n\\end{equation}\nIt is clear that each $l_i$ and $r_i$ can be a finite positive number or $+\\infty$.\n\n\\begin{lemma}\\label{l1} \\cite{HKLR} For each $i\\in \\mathbb Z_0$ we have\n\\begin{itemize}\n\\item $l_i<+\\infty$ if and only if $l_0<+\\infty$;\n\n\\item $r_i<+\\infty$ if and only if $r_0<+\\infty$.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{pro} \\label{pps: 1} Assume $h(0)=1$.\n\tA vector $\\mathbf z=(z_i,i\\in \\mathbb Z)$, with $z_0=1$, is a solution to (\\ref{di1})\nif and only if for $s_i=\\sqrt[k]{{z_i\\over h(i)}}$ the following holds\n\t\\begin{equation}\\label{Va}\n\th(i)s_i^k=\\frac{{s_{i-1}+s_{i+1}-\\tau s_i}}{s_{-1}+s_{1}-\\tau}, \\ \\ i\\in \\mathbb Z,\n\t\\end{equation}\n\twhere $\\tau=\\theta^{-1}+\\theta=2\\cosh(\\beta)$.\n\\end{pro}\n\\begin{proof} (cf. with the proof of Proposition 4.3 of \\cite{HKLR}).\nTake some $C>0$ and denote\n$$v_i=C\\cdot h^{1\/k}(i)\\left(\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}z_j\\right), \\ \\ i\\in \\mathbb Z.$$\n\nThen from (\\ref{di1}) we get $z_i=\\left({v_i\\over v_0}\\right)^k$ and consequently,\n\\begin{equation}\\label{di2}\n\\left({v_i\\over v_0}\\right)^k=\\frac{h(i)}{h(0)}\\left({\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}\\left({v_j\\over v_0}\\right)^k\n\\over\n1+\\sum_{j\\in \\mathbb Z_0}\\theta^{|j|}\\left({v_j\\over v_0}\\right)^k}\\right)^k, \\ \\ i\\in\\mathbb Z_0.\n \\end{equation}\nFrom (\\ref{di2}) we obtain\n$$v_i=C\\cdot \\sqrt[k]{h(i)}\\left(\\sum_{j=1}^{+\\infty}\\theta^jv_{i-j}^k+v_i^k+ \\sum_{j=1}^{+\\infty}\\theta^jv_{i+j}^k\\right), \\ \\ i \\in \\mathbb{Z}.\n$$\nBy the last equality we get\n$${v_{i-1}\\over \\sqrt[k]{h(i-1)}}+{v_{i+1}\\over \\sqrt[k]{h(i+1)}}-\\tau\\cdot {v_{i}\\over \\sqrt[k]{h(i)}}=C\\cdot (\\theta-{1\\over \\theta}) v^k_i.$$\nThis equality by the notation $s_i=\\sqrt[k]{{z_i\\over h(i)}}$ gives (\\ref{Va}). Conversely, from (\\ref{Va}) one gets (\\ref{di1}).\n\\end{proof}\n\\subsection{A case of 4-periodic external field for $k=2$.}\nHere we shall find solutions of (\\ref{Va}) for external field\n\\begin{equation}\\label{uh}\nh(i)=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ i=2m,\\\\[2mm]\nh_1, \\ \\ \\mbox{if} \\ \\ i=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nh_2, \\ \\ \\mbox{if} \\ \\ i=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $h_1$ and $h_2$ are some positive numbers and\na solution which has the form\n\n\\begin{equation}\\label{upa}\nu_n=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ n=2m,\\\\[2mm]\na, \\ \\ \\mbox{if} \\ \\ n=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nb, \\ \\ \\mbox{if} \\ \\ n=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $a$ and $b$ some positive numbers.\n\nThen from (\\ref{Va}) for $a$ and $b$ we get the following system of equations\n\\begin{equation}\\label{abd}\n\\begin{array}{ll}\n(a+b-\\tau)h_2b^k+\\tau b-2=0\\\\[2mm]\n(a+b-\\tau)h_1a^k+\\tau a-2=0.\n\\end{array}\n\\end{equation}\n\nFor simplicity we consider the case $k=2$ and $h_1=h_2=h$.\nIn this case, subtracting from the first equation of the system (\\ref{abd}) the second one we get\n\\[(b-a)[h(a+b)^2-h\\tau(a+b)+\\tau]=0.\\]\nThis gives three possible cases:\n\\begin{equation}\n\\label{b}\na=b, \\ \\ \\mbox{and} \\ \\ a+b=\\frac{1}{2h}(h\\tau\\pm \\sqrt{h\\tau(h\\tau-4)}) \\ \\ \\mbox{for} \\ \\ h\\tau\\geq 4.\n\\end{equation}\n\n \\textbf{Case $a=b$}. In this case from the first equation of (\\ref{abd}) we get\n\t\\begin{equation}\n\t\\label{a3}\n\t2ha^3-h\\tau a^2+\\tau a-2=0.\n\t\\end{equation}\n\\begin{lemma}\\label{lp} Any real solution of (\\ref{a3}) is positive.\n\\end{lemma}\n\\begin{proof} Since $h>0$, $\\tau>2$, if $a\\leq 0$ then LHS of (\\ref{a3}) is strictly negative.\n\\end{proof}\nRewrite the cubic equation (\\ref{a3}) as\n$$\na^{3}+\\a a^{2}+\\b a+\\c=0\n$$\nwhere $\\a=-\\tau\/2$, $\\b=\\tau\/2h$, $\\c=-1\/h$.\n\nLet\n$$\np=\\b-\\frac{\\a^{2}}{3} \\quad \\text { and } \\quad q=\\frac{2 \\a^{3}}{27}-\\frac{\\a \\b}{3}+\\c\n$$\nThen the discriminant $\\Delta$ of the cubic equation is\n$$\n\\Delta=\\frac{q^{2}}{4}+\\frac{p^{3}}{27}.\n$$\n\n\n\\begin{lemma} The following assertions hold\n\\begin{itemize}\n\\item[Case: $\\Delta>0 .$] In this case there is only one \\textbf{positive} real solution. It is\n$$\na_1=\\left(-\\frac{q}{2}+\\sqrt{\\Delta}\\right)^{\\frac{1}{3}}+\\left(-\\frac{q}{2}-\\sqrt{\\Delta}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6}\n$$\n\n\\item[Case: $\\Delta=0 .$] In this case there are two \\textbf{positive} real solutions. These roots are\n$$\na_{1}=-2\\left(\\frac{q}{2}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6} \\quad \\text { and } \\quad a_{2}=a_{3}=\\left(\\frac{q}{2}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6}\n$$\n\n\\item[Case: $\\Delta<0 .$] In this case $-p>0$ and there are three \\textbf{positive} real solutions:\n$$\n\\begin{array}{l}\na_{1}=\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\sin \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)\\right)+\\frac{\\tau}{6} \\\\\na_{2}=-\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\sin \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)+\\frac{\\pi}{3}\\right)+\\frac{\\tau}{6} \\\\\na_{3}=\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\cos \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)+\\frac{\\pi}{6}\\right)+\\frac{\\tau}{6}\n\\end{array}\n$$\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} The conditions of existence of real solutions are well-known\\footnote{https:\/\/en.wikipedia.org\/wiki\/Cubic$_-$equation}. The positivity of each solution follows from Lemma \\ref{lp}.\n\\end{proof}\n\n\n\\textbf{Case $a+b=\\frac{1}{2h}(h\\tau+ \\sqrt{h\\tau(h\\tau-4)})$}.\n\tIn this case from the second equation of (\\ref{abd}) we get\n\t\\[(h\\tau-\\sqrt{h\\tau(h\\tau-4)})a^2-2\\tau a +4=0.\\]\n\tNote that this equation has only positive real solutions.\nMoreover, one can see that if\n$$(\\tau, h)\\in A:=\\left\\{(\\tau, h)\\in \\mathbb R^2_+ \\,: \\,h\\geq \\left[\\begin{array}{ll}\n{\\tau^3\\over 8(\\tau^2-8)}, \\ \\ \\mbox{if} \\ \\ 2\\sqrt{2}<\\tau<4\\\\[2mm]\n{4\\over \\tau}, \\ \\ \\mbox{if} \\ \\ \\tau\\geq 4\n\\end{array}\\right.\\right\\}$$\nthen the quadratic equation has the following positive solutions\n\t\\[a_4=\\frac{\\tau-\\sqrt{\\tau^2-4h\\tau+4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau-\\sqrt{h\\tau(h\\tau-4)}}, \\ \\ a_5=\\frac{\\tau+\\sqrt{\\tau^2-4h\\tau+4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau-\\sqrt{h\\tau(h\\tau-4)}}. \\]\n\n\tUsing (\\ref{b}) we get $b_4=a_5$ and $b_5=a_4$.\n\n\\textbf{Case $a+b=\\frac{1}{2h}(h\\tau- \\sqrt{h\\tau(h\\tau-4)})$}.\n\tIn this case we obtain\n\t\\begin{equation*}\n\t(h\\tau+ \\sqrt{h\\tau(h\\tau-4)})a^2-2\\tau a+4=0\n\t\\end{equation*}\n\twhich\n\tfor\n$$(\\tau, h)\\in B:=\\left\\{(\\tau, h)\\in \\mathbb R^2_+ \\,: \\, \\tau\\geq 4, \\, {4\\over \\tau}\\leq h\\leq {\\tau^3\\over 8(\\tau^2-8)}\\right\\}$$ has the following solutions\n\t\\[a_6=\\frac{\\tau-\\sqrt{\\tau^2-4h\\tau-4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau+\\sqrt{h\\tau(h\\tau-4)}}, \\ \\ a_7=\\frac{\\tau+\\sqrt{\\tau^2-4h\\tau-4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau+\\sqrt{h\\tau(h\\tau-4)}}. \\]\n\tUsing (\\ref{b}) we get $b_6=a_7$ and $b_7=a_6$. Clearly all of these solutions are positive.\n\nDenote by $\\mu_i$ the gradient Gibbs measure corresponding to solution $(a_i, b_i)$, $i=1,2,\\dots,7$.\n\nThus depending on the values $(\\tau, h)$ related to $\\Delta$ and sets $A$, $B$ we have the following result.\n\n\\begin{thm} For the SOS model with 4-periodic external field there are up to seven 4-periodic gradient Gibbs measures $\\mu_i$, $i=1,\\dots,7$.\n\n\\end{thm}\n\n\n\\section*{ Acknowledgements}\n\n The author thanks C. K\\\"ulske for helpful discussions.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA classical kind of problem in algebraic geometry is to consider vanishing conditions on a linear system $\\mathcal L$, \nand to ask if the dimension of the resulting linear system is what one would expect based on the dimension of $\\mathcal L$ \nand the specific conditions imposed. That is, one asks if the desired vanishing imposes the {\\em expected number} of \nconditions on $\\mathcal L$. For example, if $\\mathcal L$ is the complete linear system of conics in $ \\ensuremath{\\mathbb{P}}^2$ \n(which is 5-dimensional) and $P$ is a point, then vanishing to multiplicity 2 at $P$ imposes three conditions on $\\mathcal L$; \nthat is, there is a 2-dimensional linear system of conics double at $P$ as expected. However, if we impose vanishing to \nmultiplicity 2 at each of two points $P_1$ and $P_2$, we expect $3+3 = 6$ conditions, i.e. we expect there to be no such \nconic, while in fact the double line passing through $P_1$ and $P_2$ is such a conic. Continuing in this direction leads \nto the well-known Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) Conjecture \\cite{segre,Harb,Gi,Hi}. \nA conjecture of Laface and Ugaglia addresses the corresponding situation in $ \\ensuremath{\\mathbb{P}}^3$ \\cite{LU}, while results of \nAlexander and Hirschowitz \\cite{AH} partially address the situation in $ \\ensuremath{\\mathbb{P}}^n$ for all $n\\geq2$, but much remains unknown.\n\nIt is not only vanishing conditions imposed by points that is of interest.\nGiven a general set of $r$ lines in $ \\ensuremath{\\mathbb{P}}^n$, one could ask if vanishing on all of these lines with multiplicity 1 \nimposes the expected number of conditions on the complete linear system of hypersurfaces of given degree $d$, \nand an affirmative answer was given by Hartshorne and Hirschowitz \\cite{HaHi}. Their paper led to much other \nwork, such as research in which lines of higher multiplicity are allowed (for just two recent examples, see \\cite{DHRST} and \\cite{BDSSS}). The work of Hartshorne and Hirschowitz also led to the paper \\cite{CCG}, which can be viewed as a direct precursor to the study of unexpected hypersurfaces.\n\n\n\nThus in recent years, a flurry of activity has emerged on this kind of problem. Some of it grew out of a striking \nexample in \\cite{DIV}, later formalized in \\cite{CHMN} and since then branching off in many different directions \n(for some examples, see \\cite{CM, DMO, DHRST, FGST, HaHa, HMNT, HMT, S1, S2, Tr}). The path that led to \nthis paper began \nby our looking for conditions on a variety $X$ that either automatically force the existence of unexpected hypersurfaces, \nor else force the conclusion that no unexpected hypersurfaces exist. The results we describe below give examples of \nsuch conditions. But more specifically we can ask: are there conditions on the Hilbert function that force either of \nthese outcomes? It turns out that if the Hilbert function forces $X$ to be degenerate in $ \\ensuremath{\\mathbb{P}}^n$ then $X$ \ndoes not admit any unexpected hypersurfaces of any kind in $ \\ensuremath{\\mathbb{P}}^n$ (see Corollary \\ref{degenerate}), \nregardless of whether it does in the smallest linear space containing it. Beyond that, additional conditions involving the \ngeometry of $X$ seem to be involved. Indeed, with a \nminor assumption on the geometry of $X$, there are such Hilbert functions (see Theorem \\ref{force unexp}),\nbut in the setting of non-degenerate, finite sets of points, we conjecture that there \nare no Hilbert functions that force any kind of unexpectedness (see Conjecture \\ref{conj about existence}).\n\nWe now describe our results in more detail.\nGiven a subscheme $X$ of $ \\ensuremath{\\mathbb{P}}^n_K$, \nits defining saturated homogeneous ideal $I_X\\subseteq R=K[ \\ensuremath{\\mathbb{P}}^n_K]=K[x_0,\\ldots,x_n]$ (where $K$ is a field)\nand integers $t\\ge m\\ge 1$, we define three numbers associated \nto $(X,t,m)$ (see Notation \\ref{dim not}). The {\\em actual dimension}, ${\\ensuremath{\\mathrm{adim}}}(X,t,m)$, is the dimension \nof the vector space of the forms in $I_X$ of degree $t$ vanishing at a general point $P$ with multiplicity $m$. That is,\n\\[\n{\\ensuremath{\\mathrm{adim}}}(X,t,m) = \\dim [I_X \\cap I_P^m]_t.\n\\]\nNext, the {\\em virtual dimension}, ${\\ensuremath{\\mathrm{vdim}}}(X,t,m)$, is the dimension of the linear system of the forms of \ndegree $t$ in $I_X$ minus the expected number of conditions imposed by taking $P$ with multiplicity $m$. That is,\n\\[\n{\\ensuremath{\\mathrm{vdim}}}(X,t,m) = \\dim [I_X]_t - \\binom{m-1+n}{n}.\n\\] \nFinally, the {\\em expected dimension} ${\\ensuremath{\\mathrm{edim}}}(X,t,m)$ is the maximum of ${\\ensuremath{\\mathrm{vdim}}}(X,t,m)$ and 0.\n\nOf course, ${\\ensuremath{\\mathrm{adim}}}(X,t,m) \\ge {\\ensuremath{\\mathrm{edim}}}(X,t,m) \\geq {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$. We say \nthat $X$ {\\em admits an unexpected hypersurface of degree $t$ vanishing at a general point $P$ with multiplicity $m$} when \n${\\ensuremath{\\mathrm{adim}}}(X,t,m)>{\\ensuremath{\\mathrm{edim}}}(X,t,m)$, i.e., when\n${\\ensuremath{\\mathrm{adim}}}(X,t,m)>0$ and ${\\ensuremath{\\mathrm{adim}}}(X,t,m)>{\\ensuremath{\\mathrm{vdim}}}(X,t,m)$. \n\nThe purpose of this paper is to get a better feel for when unexpected hypersurfaces are forced to occur \n(hence ``expecting the unexpected\"). We relate unexpectedness to algebraic and geometric properties (see for example \nProposition \\ref{p.lex segments are bad for unexpectedness} for the former and \nCorollary \\ref{degenerate} and Theorem \\ref{uct} for the latter)\nand we bring to bear methods not previously applied to unexpectedness (such as the use of generic \ninitial ideals and partial elimination ideals) to clarify when unexpectedness can and when it cannot be \nexpected. When it can, we also study to what extent\nit can, which we do by introducing AV sequences \nmeasuring the gap between the actual dimension and the virtual dimension (see Definition \\ref{def of AV}),\nwhich also allows us to frame our work in terms of persistence (i.e., how long does the gap remain\npositive?).\n\nIn Section \\ref{s.Background} we define these AV sequences. \nSpecifically, for a given subscheme $X$ of $ \\ensuremath{\\mathbb{P}}^n$ and a non-negative integer $j\\ge 0$, we define \n$AV_{X,j}: \\ensuremath{\\mathbb{Z}}_{>0}\\to \\ensuremath{\\mathbb{Z}}_{\\geq0}$ where\n$AV_{X,j}(m):={\\ensuremath{\\mathrm{adim}}}(X,m+j,m)-{\\ensuremath{\\mathrm{vdim}}}(X,m+j,m)$. Studying the difference ${\\ensuremath{\\mathrm{adim}}} - {\\ensuremath{\\mathrm{vdim}}}$ is \nnot new (see for instance \\cite{HMT}), but considering it as a sequence is novel. This sequence has interesting \nproperties and leads to compact formulas. \nSometimes we will consider the general case, but even the cases $j=0$ and $j=1$ are interesting \n(see Proposition \\ref{AV_X,0(alpha)} and Sections \\ref{s.irreducible ACM curve in P3} -- \\ref{s.Unmixed curves and unions with finite sets in P3}).\n\nIn Section \\ref{s. gin and unexp}, for any subscheme $X$ and integer $j\\ge 0$, we prove (Theorem \\ref{t. AV is an O-sequence}) \nthat the sequence $AV_{X,j}$ is actually an $O$-sequence, up to a shift. Indeed, it is the \nHilbert function of the $K$-algebra $R\/(\\ensuremath{\\mathrm{gin}}(I_X):x^{j+1})$, where $\\ensuremath{\\mathrm{gin}}(I_X)$ denotes the \ngeneric initial ideal with respect to the lexicographic order.\nWe use this fact to obtain results which ensure the non-existence of unexpected hypersurfaces. \nIn particular, if $X$ lies on a hyperplane or if $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal then $X$ does not admit {\\em any} unexpected hypersurfaces of any type (Corollary \\ref{degenerate} and Proposition \\ref{p.lex segments are bad for unexpectedness}), and hence \n${\\ensuremath{\\mathrm{adim}}}(X,t,m)$ {\\it always} has the expected value.\n\nThe fact that the AV sequence is actually an $O$-sequence raises many other related questions. \nIn Section \\ref{s.irreducible ACM curve in P3} we use a cohomological interpretation of the AV sequence, \ndescribed in Remark \\ref{cohom interp}, to prove that if $X$ is an irreducible, arithmetically Cohen-Macaulay \n(ACM) curve in $ \\ensuremath{\\mathbb{P}}^3$, then $AV_{X,1}(m)$ is unimodal and the first part is differentiable \n(see Theorem \\ref{part of conj}). Moreover, we have a great deal of experimental evidence which\nsuggests that if $X$ is smooth then this sequence is also finite and symmetric (see Conjecture \\ref{SI conj}).\n\nIn Section \\ref{ci section} we apply a new method to study the question of unexpected hypersurfaces, \nnamely the theory of {\\em partial elimination ideals} introduced by Green \\cite{Gr}. We consider the \ncase of a general codimension 2 complete intersection $C\\subset \\ensuremath{\\mathbb{P}}^n$ and we show the \nnon-vanishing of $[I_C \\cap I_P^m]_t$ for prescribed values of $t$ and $m$ (Proposition \\ref{p. codimension 2 complete intersection}). \nWe apply this to the case of $n=3$ to show for $t = (a-1)(b-1) + 1$ and $m = (a-1)(b-1)$, that a \ngeneral complete intersection curve $C$ of type $(a,b)$ with \n$2 k_{d-1} > \\dots > k_j \\geq j \\geq 1$ are integers.\n\\end{lemma}\n\n\\noindent Given the above $d$-Macaulay representation of $a$, we define\n\\[\na^{\\langle d \\rangle} = \\binom{k_d+1}{d+1} + \\binom{k_{d-1}+1}{d} + \\dots + \\binom{k_j+1}{j+1}\n\\] \nand set $0^{\\langle d \\rangle} = 0$.\n\n\\begin{theorem}[\\cite{BH} Theorem 4.2.10] \\label{macaulay thm}\nLet $K$ be a field and let $h : \\ensuremath{\\mathbb{Z}}_{\\geq0} \\rightarrow \\ensuremath{\\mathbb{Z}}_{\\geq0}$ be a numerical function. \nThen $h$ is the Hilbert function of some standard graded $K$-algebra if and only if\n\\[\nh(0) = 1 \\ \\ \\ \\hbox{ and } \\ \\ \\ h(d+1) \\leq h(d)^{\\langle d \\rangle}\n\\]\nfor all $d \\geq 1$.\n\\end{theorem}\n\n\\noindent An infinite sequence $a_0,a_1,a_2,\\ldots$ of non-negative integers with $a_i=h(i)$ for an $h$ \nsatisfying the conditions of \nTheorem~\\ref{macaulay thm} is called an {\\em $O$-sequence}. \nWe will regard a finite sequence $a_0,a_1,a_2,\\ldots, a_r$ as an infinite sequence\nby setting $a_i=0$ for $i>r$.\n\nHereafter, let $R = K[x_0,\\dots, x_n]$ be a polynomial ring over a field $K$.\nOur default assumption will be that $K$ is algebraically closed,\nbut sometimes we will need the characteristic to be zero, \nand sometimes we will need only that $K$ is infinite;\nin these cases we will say so explicitly.\n \n\\begin{notation}\nFor any subvariety (or subscheme) $V \\subseteq \\ensuremath{\\mathbb{P}}^n$ we write $I_V \\subseteq R$ for the saturated ideal of $V$\nand $\\mathcal{I}_V$ for the sheaf on $ \\ensuremath{\\mathbb{P}}^n$ corresponding to $I_V$. For a standard graded algebra $R\/I$ \nwe write $h_{R\/I}(t)$ for the {\\em Hilbert function} of $R\/I$, i.e. $h_{R\/I}(t) = \\dim_K [R\/I]_t$. When $I = I_V$ for \nsome subscheme $V$, we sometimes write $h_V(t)$ for $h_{R\/I_V}(t)$. We say that $V$ is {\\em arithmetically \nCohen-Macaulay (ACM)} if $R\/I_V$ is a Cohen-Macaulay ring.\n\nFor any integer function $h : \\ensuremath{\\mathbb{Z}}_{\\geq 0} \\to \\ensuremath{\\mathbb{Z}}$, the first difference $\\Delta h$ is the backward difference,\ndefined by setting $\\Delta h(0) = h(0)$ and $\\Delta h(t) = h(t)-h(t-1)$ for $t>0$. \nWhen $X$ is a finite set of points, it is well-known and easy to see that\n$h_X$ is strictly increasing until it becomes constant, hence there is a $j$ such that\n$1=h_X(0)<\\cdots {\\ensuremath{\\mathrm{edim}}}(X,t,m)$, we say that $X$ {\\em admits an unexpected \nhypersurface of degree $t$ for multiplicity $m$}. (In this case, note that ${\\ensuremath{\\mathrm{adim}}} (X,t,m)>0$,\nhence $t\\geq m$.) If $X \\subset \\ensuremath{\\mathbb{P}}^2$ is a finite set of \npoints which admits an unexpected hypersurface of degree $t$ for multiplicity $m=t-1$,\nthen following \\cite{CHMN} we say simply that {\\em $X$ admits an unexpected curve of degree $t$}.\n\\end{definition}\n\n\\begin{remark} \\label{equiv to unexp}\nAn equivalent condition for $X$ to admit an unexpected hypersurface of degree $t$ for \nmultiplicity $m$ is ${\\ensuremath{\\mathrm{adim}}}(X,t,m) > 0$ and ${\\ensuremath{\\mathrm{adim}}}(X,t,m) > {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$.\n\\end{remark}\n\n\\begin{remark}\nAny hypersurface of degree $t$ with an isolated singularity of multiplicity $t$ must be a \ncone (by Bezout's theorem). Thus ${\\ensuremath{\\mathrm{adim}}}(X,t,t)$ is the dimension of the vector space \nof cones over $X$ of degree $t$ with vertex at~$P$. If ${\\ensuremath{\\mathrm{adim}}} (X,t,t) > {\\ensuremath{\\mathrm{edim}}}(X,t,t)$, \nwe say that $X$ {\\em admits an unexpected cone of degree $t$}. See \\cite{HMNT, CM, HMT} for \nmore on unexpected cones. In particular, if $X$ has codimension two and is reduced, \nequidimensional and non-degenerate then the cone $S_P$ over $X$ with vertex $P$ is \nan unexpected cone of degree $t=\\deg X$ (\\cite{HMNT} Proposition 2.4).\n\\end{remark}\n\n\n\\begin{definition} \\label{def of AV}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a closed subscheme. Fixing a non-negative integer $j$, \nwe define the sequence $AV_{X,j}$ as follows:\n\\[\nAV_{X,j}(m) = \n\\begin{array}{cc}\n{\\ensuremath{\\mathrm{adim}}}(X,m+j,m) - {\\ensuremath{\\mathrm{vdim}}}(X,m+j,m), & m\\ge 1.\n\\end{array}\n\\] \n\\end{definition}\n\n\\begin{remark}\nRephrasing Remark \\ref{equiv to unexp}, if ${\\ensuremath{\\mathrm{adim}}} (X,t,m) > 0$ then $X$ admits \nan unexpected hypersurface of degree $t$ for multiplicity $m$ if and only if $AV_{X,j}(m) > 0$ for $j = t-m$.\n\n\\end{remark}\n\n\\begin{notation} \\label{Pm vs mP}\nLet $P \\in \\ensuremath{\\mathbb{P}}^n$ be a general point, with defining ideal $I_P$. We will \ndenote the scheme defined by $I_P^m$ in $ \\ensuremath{\\mathbb{P}}^n$ by $P^m$. We will \nsometimes consider the hyperplane section of $P^m$ by a hyperplane $H$ \ncontaining $P$, and we will denote the corresponding subscheme of $H$ by $mP$, thus $mP=P^m\\cap H$.\n\\end{notation}\n\n\nWe now give an interpretation of the sequence $AV_{X,j}$. Notice that, in the \nfollowing lemma, the ideal in the dimension of the quotient on the right changes with $m$. \n\n\\begin{lemma}\\label{l.AV seq 1} \nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a subscheme. Then\n\t\\[\n\tAV_{X,j}(m) = \\dim\\left[ \\faktor{R}{(I_X + I_P^m)}\\right]_{m+j} .\n\t\\] \n\\end{lemma}\n\\begin{proof}\n\tSet $t:=m+j$. From the short exact sequence\n\t\\[\n\t0\\to R\/(I_X \\cap I_P^m) \\to R\/I_X \\oplus R\/ I_P^m \\to R\/(I_X + I_P^m)\\to 0\n\t\\]\n\twe get the relation between the dimension of the modules in degree $t$\n\t\\[\n\t\\dim[R]_t- {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[R]_t+\\dim[I_X]_t - \\dim[R]_t+ \\dim [I_P^m]_t +\\dim[ R\/(I_X + I_P^m)]_t= 0.\n\t\\]\n\tTherefore\n\t\\[\n\t\\begin{array}{rcl}\n\t\\dim[ R\/(I_X + I_P^m)]_t&=& {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[I_X]_t + \\dim[R]_t-\\dim [I_P^m]_t =\\\\\n\t&=& {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[I_X]_t + h_{P^m}(t) .\n\t\\end{array}\n\t\\]\n\tSince $t\\ge m$ the Hilbert function of the fat point $P^m$ in degree $t$ is \n\t$h_{P^m}(t)={m+n-1\\choose n}$. (It reaches the degree $\\deg(P^m)$ in degree $m-1$.) So we get\n\t\\[\\dim[ R\/(I_X + I_P^m)]_t = {\\ensuremath{\\mathrm{adim}}} (X,t,m) - {\\ensuremath{\\mathrm{vdim}}} (X,t,m) \\] \n\tas desired.\n\\end{proof}\n\n\n\\begin{remark} \\label{cohom interp}\n\tWe can also give a cohomological interpretation for the sequence $AV_{X,j}(m)$. \n\t Let $X$ be an ACM subscheme in $ \\ensuremath{\\mathbb{P}}^n$ of dimension $\\geq 1$. Assume $m \\geq 0$ and $j \\geq 0$. Consider the exact sequence of sheaves\n\t\\[\n\t0 \\rightarrow \\mathcal I_{X \\cup P^m} \\rightarrow \\mathcal I_X \\rightarrow \\mathcal O_{P^m} \\rightarrow 0.\n\t\\]\n\tTwisting by $m+j$ and taking cohomology gives the exact sequence\n\t\\[\n\t0 \\rightarrow [I_{X \\cup P^m}]_{m+j} \\rightarrow [I_X]_{m+j} \\rightarrow H^0(\\mathcal O_{P^m}(m+j)) \\rightarrow H^1(\\mathcal I_{X \\cup P^m}(m+j)) \\rightarrow 0.\n\t\\]\n\t(Exactness on the right is because $X$ is ACM of dimension $\\geq 1$; see \\cite[pp. 9-11]{migbook}.) This gives\n\t\\[\n\th^1(\\mathcal I_{X \\cup P^m}(m+j)) = \\binom{(m-1)+n}{n} - \\dim [I_X]_{m+j} + \\dim [I_{X \\cup P^m}]_{m+j} = AV_{X,j}(m).\n\t\\]\n\\end{remark}\n\n\\begin{remark}\nGiven a subscheme $X \\subset \\ensuremath{\\mathbb{P}}^n$, it is natural to ask about the persistence \nof the unexpectedness imposed by $X$. For example, in \\cite[Corollary 2.12]{HMNT}, it is \nshown that a nondegenerate curve $C \\subset \\ensuremath{\\mathbb{P}}^3$ of degree $d=\\deg C$ admits \nan unexpected hypersurface of degree $t$ for multiplicity $t$ at a general point for all $t \\geq d$. \nThus fixing $0 = j = t-m$, and fixing $C$, we have the persistence of unexpectedness as long as $t \\geq d$.\n\nMany of the results in this paper give formulas for the sequences $AV_{X,j} (m)$. Leaving \naside the issue of whether ${\\ensuremath{\\mathrm{adim}}} (X,t,m) > 0$, this sequence can be interpreted both \nas a measure of unexpectedness (how much bigger is the actual dimension than what one \nwould expect?) and as a measure of persistence (how long is $AV_{X,j}(m)$ positive?). \nThe fact that these sequences are represented by simple formulas, as we will see in the coming sections, is a pleasant bonus.\n\\end{remark}\n\n\n\n\n\\section{Generic initial ideals and unexpectedness}\\label{s. gin and unexp}\nIn this section we relate the study of unexpected hypersurfaces of a subscheme $X\\subseteq \\ensuremath{\\mathbb{P}}^n$ \nto the generic initial ideal of $I_X$ with respect to the lexicographic order. Then, we prove that the $AV_{X,j}$ \nsequence, up to a shift, is an $O$-sequence. As a consequence of this result we are able to ensure the \nnon-existence of unexpected hypersurfaces in several cases.\n \nLet $R = K[x_0,x_1,\\ldots,x_n]$ be a standard graded polynomial ring. In this section we only \nrequire $K$ to be infinite. We assume the monomials of $R$ are ordered by $>_{lex}$, the \nlexicographic monomial order which satisfies $x_0> x_1 > \\cdots > x_n$.\t\nWe recall that a set $M\\subseteq R$ of monomials is a {\\em lex-segment} \nif the monomials have the same degree and they satisfy the condition that whenever $u,v$ \nare monomials with $u\\geq v$ and $v\\in M$, then $u\\in M$ \\cite{V}.\nIt is convenient to also refer to a vector subspace $W\\subseteq R$\nas a lex-segment if $W$ is spanned by a lex-segment in the previous sense.\nWe also recall that a homogeneous ideal $I\\subseteq R$ is a {\\em lex-segment ideal} if \nfor each degree $d$ the the homogeneous component $I_d$ of $I$ of degree $d$\nis a lex segment \\cite{V}; see also \\cite{Hu}.\n\nFor a graded ideal $I\\subseteq R$, we will denote by $\\ensuremath{\\mathrm{gin}}(I)$ the generic initial ideal of $I$ with respect\nto the monomial order $>_{lex}$. For an introduction to generic initial ideals, see for instance \\cite{Gr} and Section 15.9 in \\cite{E}.\nThe next lemma relates the actual and virtual dimensions of a scheme in terms of the generic initial ideal of its ideal. \n\n\n\\begin{lemma}\\label{l. adim vdim and Gin} Let $X\\subseteq \\ensuremath{\\mathbb{P}}^n$ be a subscheme. For any non-negative integers $t$ and $m$, we have \n\t\\begin{itemize}\n\t\t\\item[\\em (i)] ${\\ensuremath{\\mathrm{adim}}} (X,t,m)=\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^m]_t$, where $Q = (1,0,\\dots,0)$.\n\t\t\\item[\\em (ii)] ${\\ensuremath{\\mathrm{vdim}}} (X,t,m)={\\ensuremath{\\mathrm{vdim}}} (\\ensuremath{\\mathrm{gin}}(I_X),t,m)$.\n\t\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\n(i) Let $X'$ be the image of $X$ under a general linear change of variables, so $\\hbox{gin}(I_{X})=\\hbox{in}(I_{X'})$\nand so the point $Q$ is general for $X'$.\t\nMoreover, if $\\mu_1,\\mu_2$ are monomials of the same degree with $\\mu_1\\in[I_Q^m]_t$ and $\\mu_1>_{lex}\\mu_2$,\nthen $\\mu_2\\in I_Q^m$, hence $\\hbox{in} (I_{X'}\\cap I_Q^m)=\\hbox{in} (I_{X'})\\cap I_Q^m$. \n(To see this note that both sides of the equality are monomial ideals, and \nthat $\\hbox{in} (I_{X'}\\cap I_Q^m)\\subseteq \\hbox{in} (I_{X'})\\cap I_Q^m$ is clear.\nSo suppose that $w\\in \\hbox{in} (I_{X'})\\cap I_Q^m$ is a monomial. \nThen there is a form $W\\in I_{X'}$ with $w=\\hbox{in}(W)$. \nIt follows that the other terms of $W$ have lex order less than $w$, so\neach is in $I_Q^m$, hence $W\\in I_Q^m$, and we have $W\\in I_{X'}\\cap I_Q^m$\nso $w\\in \\hbox{in} (I_{X'}\\cap I_Q^m)$ giving $\\hbox{in} (I_{X'}\\cap I_Q^m)\\supseteq \\hbox{in} (I_{X'})\\cap I_Q^m$.)\nThus we have\n${\\ensuremath{\\mathrm{adim}}}(X,t,m)=\\dim [I_X\\cap I_P^m]_t \n\t\t\t = \\dim [I_{X'}\\cap I_Q^m]_t \n\t\t\t = \\dim [\\hbox{in}(I_{X'}\\cap I_Q^m)]_t \n\t\t\t = \\dim [\\hbox{in}(I_{X'})\\cap I_Q^m]_t \n\t\t\t = \\dim [\\hbox{gin}(I_{X})\\cap I_Q^m]_t$.\n\n(ii) This is a consequence of $h_X=h_{R\/(\\ensuremath{\\mathrm{gin}}(I_X))}$.\n\\end{proof}\n\n\n\\begin{remark} \\label{r: partial elimination ideal} \n\tLet $I=\\oplus_k I_k$ be a homogeneous ideal in $R$. \nThen we make the following definition (see \\cite[Definition 6.1]{Gr}):\n\t\t\\[\n\t\t\\widetilde K_d(I) =\\bigoplus_k[I\\cap I_Q^{k-d}]_k,\n\t\t\\]\nwhere $Q = (1,0,\\dots,0)$, so\n\t $\\widetilde K_d(I)$ is a graded module over $K[x_1,\\ldots,x_n]$. \n\t In particular, $\\widetilde K_0(I)$ is an ideal: it is obtained from $I$ by \\textit{eliminating} the variable $x_0$. \n\t Geometrically, it corresponds to the linear space of the cones having vertex at $Q$. Then Lemma \\ref{l. adim vdim and Gin} (i) can be rephrased as follows: \t\n\t \\[\n\t \\dim (I_X \\cap I_P^m)_t=\\dim \\left[\\widetilde K_{t-m}\\left(\\ensuremath{\\mathrm{gin}}(I_X) \\right)\\right]_t.\n\t \\]\n\\end{remark}\n\n\\begin{example}\\label{Fig1Fig2Example}\nAssume $\\hbox{\\rm char}(K)=0$. \nHere, given only the Hilbert function $h_X$ of a set of points $X\\subset \\ensuremath{\\mathbb{P}}^2$,\nwe show how information about $\\ensuremath{\\mathrm{gin}}(I_X)$ relates to whether or not $X$ has\nan unexpected curve.\nSo let $X\\subseteq \\ensuremath{\\mathbb{P}}^2$ be a set of 13 points with \n$h_X=(1, 3, 6, 10, 12, 13, 13, \\ldots)$,\nso $X$ lies on three independent quartics but no cubics (such examples exist, as we see later in this example). Then \n$\\dim [I_X]_6=15$, so ${\\ensuremath{\\mathrm{vdim}}} (X,6,5)=0$, hence $X$ admits an unexpected curve \nof degree~$6$ if and only if ${\\ensuremath{\\mathrm{adim}}}(X,6,5)>0$. \nFrom Lemma \\ref{l. adim vdim and Gin}(i), \n${\\ensuremath{\\mathrm{adim}}}(X,6,5)>0$ if and only if $\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^5]_6>0.$ In this case, the ideal defining the point $Q$ is $I_Q=(y,z)\\subseteq K[x,y,z]$, so the monomials of degree 6 contained in $[I_Q^5]_6$ are exactly those\nlexicographically less than or equal to $xy^5$.\nBut (in characteristic 0) generic initial ideals are (by \\cite{CaS}) strongly stable \n(see \\cite{AL} for the definition and properties of strong stability),\nhence if any monomial of degree 6 less than or equal to $xy^5$ is in $\\ensuremath{\\mathrm{gin}}(I_X)$,\nthen $xy^5$ is in $\\ensuremath{\\mathrm{gin}}(I_X)$ too. \nThus $X$ admits an unexpected curve of degree 6 if and only if $xy^5\\in \\ensuremath{\\mathrm{gin}}(I_X)$.\n\nNow we determine all the monomials in $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$, assuming that $X$ admits no unexpected curves in degrees strictly lower than 6. With this assumption and using the fact that unexpected curves have degree at least 4 \n\\cite{A, FGST}, from Lemma \\ref{l. adim vdim and Gin}(i) we have \n\\begin{itemize}\n\t\\item $[\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^3]_4=[\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^4]_5=(0)$, so if $x^ay^bz^c \\in [\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$ then $a\\ge 2$, and\n\\smallskip\n\t\\item ${\\ensuremath{\\mathrm{edim}}}(X,4,2)={\\ensuremath{\\mathrm{adim}}}(X,4,2)=0$ (by Bertini's Theorem), hence $[ \\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^2]_4=(0)$, so if $x^ay^bz^c \\in [\\ensuremath{\\mathrm{gin}}(I_X)]_{4}$ then $a\\ge 3.$\n\\end{itemize}\nCollecting this information, we get only one strongly stable ideal through degree 5 (with the given Hilbert function), that is \n$(x^4, x^3y, x^3z, \nx^2y^3, x^2y^2z)$. So, $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 5}$ is a lex segment.\n\nTaking generators up to degree 6 of the lex-segment ideal with Hilbert function $h_X$ we get\n\\[ L = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z, x^2yz^3, x^2z^4), \\]\nthus $X$ has an unexpected curve of degree 6 with a general multiple point of multiplicity 5 if and only if the component $[\\ensuremath{\\mathrm{gin}}(I_X)]_6$ \nfails to be equal to the degree 6 component of the lex-segment ideal with the same Hilbert function.\n\nAs promised, we now show that sets $X$ do arise.\nWe first give an example of a set $X$ of 13 points with no unexpected curves of degree 6 or less.\nWe get 12 points of the points of $X$\nas the complete intersection of a general cubic and quartic; add to this a general point $Q$\nto obtain $X$. It is easy to see that $X$ has the Hilbert function as specified above,\nand by direct computation with Macaulay2 \\cite{GS}, we find that\n$X$ has no unexpected curves of degree 6 or less. In this case, we find from Macaulay2 that\n$\\ensuremath{\\mathrm{gin}}(I_X)=(x^4, x^3y, x^3z,\nx^2y^3, x^2y^2z,\nx^2yz^3, x^2z^4,\nxy^6, xy^5z,\nxy^4z^3,\nxy^3z^5,\nxy^2z^7,\nxyz^9,\nxz^{11},\ny^{13})$, hence $xy^5$, as claimed, does not occur.\n\n\\begin{comment}\nMacaulay2 code for running the previous example\n\nR=QQ[x,y,z];\nC=random(3,R)\nQ=random(4,R)\nI=intersect(ideal(C,Q),ideal(x,y));\nfor i from 0 to 6 do print {i,hilbertFunction(i,I)}\nP=ideal(random(1,R),random(1,R));\nJ3=intersect(I,P^3);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J3)\n J4=intersect(I,P^4);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J4)}\nJ5=intersect(I,P^5);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J5)}\nloadPackage \"GenericInitialIdeal\"\nlexgin(I)\n\nNote 1: To check the claim about the HF of G_2 use:\nK=ideal(x^4, x^3*y, x^3*z, x^2*y^3, x^2*y^2*z, x*y^5, x*y^4*z);\n\\end{comment}\n\nWe now give two examples of an $X$ with the specified Hilbert function which do have unexpected sextics.\nReturning to the specfied Hilbert function, we see that two monomials of degree 6 are needed in the minimal set of generators of $\\ensuremath{\\mathrm{gin}}(I_X)$. If $X$ admits an unexpected curve of degree 6 with a general multiple point of multiplicity 5, we have already noticed that $xy^5\\in \\ensuremath{\\mathrm{gin}} I_X$.\nThus, the strongly stable property forces $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$ to be either\n\\[\n\\mathcal G_1 = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z,x^2yz^3, xy^5)\n\\]\nor \n\\[\n\\mathcal G_2 = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z, xy^5, xy^4z).\n\\]\nBy direct computation we see that a set of points $X$ with $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_2$ would have ${\\ensuremath{\\mathrm{adim}}}(X,6,5)=2$,\nbut by \\cite[Corollary 5.5]{CHMN} this would mean $X$ has an unexpected quintic, contrary to our assumption\nthat $X$ admits no unexpected curves in degrees strictly lower than 6.\nThus $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_2$ cannot occur if $X$ is a reduced set of 13 points.\n\nHowever, $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_1$ can occur; we give two examples.\nSpecifically, the following two sets of points $X_1$ and $X_2$ have $h_{X_1}=h_{X_2}=h_X$ \nand $[\\ensuremath{\\mathrm{gin}}(I_{X_i})]_{\\le 6}= \\mathcal G_1$. The lines (see Figure \\ref{f.line config X }) dual to the points $X_1$\ngive what \\cite{DMO} refers to as a $(1,1)$ tic-tac-toe arrangement:\n \n\\[\\begin{array}{ccc}\nX_1&:=&\\{(1,0,0), (0,1,0), (0,0,1), (1,1,0),(0,1,1),(1,0,1), (-1,1,0),(0,-1,1),(-1,0,1),\\\\\n&&(1,1,1), (-1,1,1),(-1,1,-1),(1,1,-1)\\}.\\\\\n\\end{array}\n\t\\]\nThe ideal defining $X_1$ is\n\\[\nI_{X_1}=(y^3z - yz^3, x^3z - xz^3, x^3y - xy^3).\n\\]\n\n \\begin{figure}[!ht]\n \t\\centering\n \t\\begin{tikzpicture}[scale=0.7]\n \\draw (-4,-1) -- (4,-1);\n \\draw (-4,0) -- (4,0);\t\n \\draw (-4,1) -- (4,1);\n \n \\draw (1,-4) -- (1,4);\n \\draw (0,-4) -- (0,4);\t\n \\draw (-1,-4) -- (-1,4); \t\n\n \\draw (-4,-3) -- (3,4);\n \\draw (-4,-4) -- (4,4);\n \\draw (-3,-4) -- (4,3);\n\n \\draw (-4,3) -- (3,-4);\n \\draw (-4,4) -- (4,-4);\n \\draw (-3,4) -- (4,-3);\n \n \t\\end{tikzpicture}\n \t\\caption{A sketch of the line configuration dual to the points of $X_1$ from Example \\ref{Fig1Fig2Example}. \n\t(The line at infinity, corresponding to the point $(0,0,1)$, is not shown).}\n \t\\label{f.line config X } \n \\end{figure}\n\nThe second set of points is\n\\[\\begin{array}{ccc}\nX_2&:=&\\{(1,0,0), (0,1,0), (0,0,1), (1,1,0),(0,1,1),(1,0,1), (-1,1,0),(0,-1,1),(-1,0,1),\\\\\n&&(2,1,1), (-2,1,1),(-2,1,-1),(2,1,-1)\\};\\\\\n\\end{array}\n\\]\nit is defined by the ideal\n\\[I_{X_2}=(y^3z - yz^3, x^3z - xz^3 - 3xy^2z, x^3y - xy^3 - 3xyz^2).\\]\n(The lines dual to $X_2$ are shown in Figure \\ref{f.line config X'}.)\n \n \\begin{figure}[!ht]\n \t\\centering\n \t\\begin{tikzpicture}[scale=0.7]\n \\draw (-4,-1) -- (4,-1);\n \\draw (-4,0) -- (4,0);\t\n \\draw (-4,1) -- (4,1);\n \n \\draw (1,-4) -- (1,4);\n \\draw (0,-4) -- (0,4);\t\n \\draw (-1,-4) -- (-1,4); \t\n \t\n \t\\draw (-2.5,-4) -- (1.5,4);\n \t\\draw (-4,-4) -- (4,4);\n \t\\draw (-1.5,-4) -- (2.5,4);\n \t\n \t\\draw (-2.5,4) -- (1.5,-4);\n \t\\draw (-4,4) -- (4,-4);\n \t\\draw (-1.5,4) -- (2.5,-4);\n \t\n \t\\end{tikzpicture}\n \t\\caption{A sketch of the line configuration dual to the points of $X_2$ from Example \\ref{Fig1Fig2Example}. \n\t(The line at infinity, corresponding to the point $(0,0,1)$, is not shown).}\n \t\\label{f.line config X'} \n \\end{figure}\n\\end{example}\n\n\nWe now show that the sequence $AV_{X,j}$ is the Hilbert \nfunction of a fixed standard graded algebra; since $AV_{X,j}(m)$ is defined for $m\\ge 1$ we need to shift it by 1.\n\n\\begin{theorem}\\label{t. AV is an O-sequence}\nFor any non-negative integer $j$, the sequence $AV_{X,j}$ \nshifted to the left by $1$ is an $O$-sequence. In particular, setting $J:=\\ensuremath{\\mathrm{gin}}(I_X)\\ :\\ x_0^{j+1},$ \nthe sequence $AV_{X,j}$ shifted to the left by $1$ coincides with the Hilbert function of $R\/J$, i.e.,\n\t\\[\n\tAV_{X,j}(d+1)=h_{R\/J}(d), \\ d\\ge 0.\n\t\\]\n\n\\end{theorem}\n\\begin{proof}Set $\\mathfrak q= (x_1,\\cdots, x_n)$ the ideal defining the point $Q=(1,0,\\ldots,0)$.\nFor any $m\\ge 1$ and for any non-negative integer $j$, from Lemma \\ref{l.AV seq 1}, reasoning as in the proof of Lemma \\ref{l. adim vdim and Gin}, we have\n$$AV_{X,j}(m)=\\dim\\left[ \\faktor{R}{(I_X + I_P^m)}\\right]_{m+j}=\\dim\\left[ \\faktor{R}{( \\ensuremath{\\mathrm{gin}}(I_X) + \\mathfrak q^m)}\\right]_{m+j} .$$\n\nFor a monomial ideal $T$, it is easy to show that we can write $\\left[T\\right]_{m+j}$ as the following direct sum (and so the summands have only 0 in common):\n\\[ \\left[T\\right]_{m+j}= \\left[x_0^{j+1}\\cdot (T : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[ T\\cap \\mathfrak q^m\\right]_{m+j}.\\]\nIn particular, \n$$\\left[R\\right]_{m+j}= \\left[x_0^{j+1}\\cdot (R : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[ R\\cap \\mathfrak q^m\\right]_{m+j}=\\left[x_0^{j+1}R\\right]_{m+j}\\bigoplus\\left[\\mathfrak q^m\\right]_{m+j},$$ \nso $\\dim \\left[{R\/\\mathfrak q^m}\\right]_{m+j}=\\dim \\left[x_0^{j+1}R\\right]_{m+j}$.\nSimilarly, \n\\begin{align*}\n\\left[\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m\\right]_{m+j}&=\\left[x_0^{j+1}\\cdot ((\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m) : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\\cap \\mathfrak q^m\\right]_{m+j}\\\\\n&=\\left[x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[\\mathfrak q^m\\right]_{m+j},\n\\end{align*}\nso $\\dim \\left[ {(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}=\\dim \\left[x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}) \\right]_{m+j}$.\nThus\n\t\\[\\begin{array}{rcl}\n\t\\dim\\left[\\dfrac{R}{\\ensuremath{\\mathrm{gin}}(I_{X}) +\\mathfrak q^m}\\right]_{m+j}&=& \\dim \\left[\\dfrac{R\/\\mathfrak q^m}{(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[{R\/\\mathfrak q^m}\\right]_{m+j} -\\dim \\left[ {(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[x_0^{j+1}R \\right]_{m+j} -\\dim \\left[ x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1})\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[R \\right]_{m-1} -\\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}\\right]_{m-1}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[R \/ (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1})\\right]_{m-1}.\\\\\n\t\\end{array}\n\t\\]\t \n\\end{proof}\n\nAs a consequence of Theorem \\ref{t. AV is an O-sequence} we get a new criterion\n(in this case, geometric) for \nthe non-existence of unexpected hypersurfaces. \n\n\\begin{corollary} \\label{degenerate}\n\tIf $X$ is a reduced subscheme of $ \\ensuremath{\\mathbb{P}}^n$ contained in a hypersurface of degree $d+1\\geq1$, \n\tthen for any $t\\geq d+m$ and $m\\ge 1$ we have \n\\[\n{\\ensuremath{\\mathrm{adim}}} (X,t,m) = {\\ensuremath{\\mathrm{vdim}}} (X,t,m).\n\\]\nIn particular, if $X$ is degenerate (meaning, $X$ is contained in a hyperplane, or, equivalently, $d=0$), then\n$X$ admits no unexpected hypersurfaces for multiplicity $m$ in degrees $t\\geq m$,\nand hence no unexpected hypersurfaces of any kind.\n\\end{corollary}\n\n\\begin{proof}\n\tSince $I_X$ has an element of degree $d+1$, $x_0^{d+1}\\in \\ensuremath{\\mathrm{gin}}(I_X)$. This implies \n\t$1\\in (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{t-m+1})$ when $t\\geq d+m, m\\geq 0$. \n\tFrom Theorem \\ref{t. AV is an O-sequence} and Definition \\ref{def of AV},\n\tfor $t=m+j, j\\geq d, m\\geq 1$, we have $0=AV_{X,j}(m)={\\ensuremath{\\mathrm{adim}}}(X,t,m) - {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$.\n\tThe last part, about unexpected hypersurfaces, follows from Definition \\ref{unexpDef}.\n\\end{proof}\n\nThe next corollary compares the sequences $AV_{X,j} (m) $ and $ AV_{X,j+1}(m)$ for a subscheme $X$.\n\n\\begin{corollary} \\label{ineq for AV}\n\t\tLet $X$ be a subscheme of $ \\ensuremath{\\mathbb{P}}^n$. Then\n\t\t\\[AV_{X,j} (m) \\ge AV_{X,j+1}(m).\\] \n\\end{corollary}\n\n\\begin{proof}From Theorem \\ref{t. AV is an O-sequence} we have, for any $i$,\n\\[\nAV_{X,i} (m) = \\dim \\left[R \/ (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{i+1})\\right]_{m-1}.\n\\]\nSo the statement is equivalent to proving that\n\\[\n\\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1} \\right]_{m-1}\\le \\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+2} \\right]_{m-1},\n\\]\nand this is trivial since we always have $\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1} \\subseteq \\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+2}.$ \n\\end{proof}\n\n\\begin{lemma} \\label{subscheme}\nLet $P$ be a general point in $ \\ensuremath{\\mathbb{P}}^n$. \nLet $Y_1$ be the zero-dimensional subscheme of $ \\ensuremath{\\mathbb{P}}^n$ defined by $I_P^m$ and let $Y_2$ be the subscheme of $Y_1$ defined by $I_P^{m-1}$. Fix a positive integer~$t \\geq m-1$ and consider the component $ [I_X]_t$, where $X$ is some subvariety of $ \\ensuremath{\\mathbb{P}}^n$. If $Y_1$ imposes $\\binom{m-1+n}{n}$ independent conditions on $[I_X]_t$ then $Y_2$ imposes $\\binom{m-2+n}{n}$ independent conditions on $[I_X]_t$.\n\\end{lemma}\n\n\\begin{proof}\nWe know that $\\deg(Y_1) = \\binom{m-1+n}{n}$ and $\\deg(Y_2) = \\binom{m-2+n}{n}$, and \n that $Y_1$ and $Y_2$ impose independent conditions on $[R]_t$ since $t \\geq m-1$ (by a regularity argument). In particular we know that \n\\[\n\\dim [I_{Y_1}]_t = \\dim [R]_t - \\binom{m-1+n}{n} \\ \\ \\hbox{ and } \\ \\ \\dim [I_{Y_2}]_t = \\dim [R]_t - \\binom{m-2+n}{n}.\n\\]\nConsider first the exact sequence\n\\[\n0 \\rightarrow I_{Y_1} \\rightarrow I_{Y_2} \\rightarrow A \\rightarrow 0,\n\\]\nwhere $A$ is the quotient, and is supported on $P$. We have\n\\[\n\\dim[A]_t = \\binom{m-1+n}{n} - \\binom{m-2+n}{n} = \\binom{m-2+n}{n-1}.\n\\]\nLet $\\mathcal A$ be the sheafification of $A$. From the above exact sequence and the fact that $t \\geq m-1$ (so $h^1(\\mathcal I_P^m(t)) = h^1(\\mathcal I_{Y_1}(t)) = 0$) and the fact that $I_{Y_1}$ and $I_{Y_2}$ are saturated, we get $h^0(\\mathcal A(t)) = \\dim [A]_t$. \n\nRemembering that $X$ and $P$ are disjoint, consider the exact sequence of sheaves\n\\[\n0 \\rightarrow \\mathcal I_{X \\cup Y_1}(t) \\rightarrow \\mathcal I_{X \\cup Y_2}(t) \\rightarrow \\mathcal A(t) \\rightarrow 0.\n\\]\nTaking cohomology we obtain\n\\[\n0 \\rightarrow [I_X \\cap I_P^m]_t \\rightarrow [I_X \\cap I_P^{m-1}]_t \\rightarrow [A]_t \\rightarrow \\cdots\n\\]\nHence\n\\[\n\\begin{array}{rcl}\n\\dim [I_X \\cap I_{Y_2}]_t & = & \\dim [I_X \\cap I_P^{m-1}]_t \\\\ \\\\\n& \\leq & \\dim [I_X \\cap I_P^m]_t + \\dim [A]_t \\\\ \\\\\n& = & \\dim [I_X]_t - \\binom{m-1+n}{n} + \\binom{m-2+n}{n-1} \\\\ \\\\\n& = & \\dim [I_X]_t - \\binom{m-2+n}{n}.\n\\end{array}\n\\]\nBut $Y_2$ cannot impose more conditions in degree $t$ than its degree, so we must have equality.\n\\end{proof}\n\nLike Corollary \\ref{degenerate}, the next result gives a criterion (again basically geometric), \nbased on one single piece of information,\nfor a subvariety $X \\subset \\ensuremath{\\mathbb{P}}^n$ to admit no unexpected hypersurfaces of any degree or multiplicity at a general point.\n\n\\begin{proposition} \\label{AV_X,0(alpha)}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a subscheme. Let $\\alpha:= \\alpha(I_X)$ be the least degree $t$ such that $[I_X]_t\\neq 0$. If $AV_{X,0} (\\alpha)=0$, then $X$ does not admit unexpected hypersurfaces, for any degree and multiplicity at a general point. \n\\end{proposition}\n\n\\begin{proof}\nBy Remark \\ref{equiv to unexp}, $X$ fails to admit an unexpected hypersurface \nof degree $t$ for multiplicity $m$ if and only if $T_{t,m}=\\min({\\ensuremath{\\mathrm{adim}}}(X,t,m), AV_{X,t-m}(m))=0$.\nConsider the table $T$ of values of $T_{t,m}$ for $t,m\\geq1$. It suffices to show we always have\n$T_{t,m}=0$. We will divide the table into three regions as follows:\n\\[\n\\begin{array}{cc|ccccc|ccccccccccccccccc}\n& t & 1 & 2 & 3 & \\dots & \\alpha-1 & \\alpha & \\alpha +1 & \\alpha+2 & \\alpha+3 & \\dots\\\\ \nm & &&&&&&&&& \\\\ \\hline\n1 & &&&&&&&&& \\\\ \n2 & &&&&&&&&& \\\\\n\\vdots & &&&&&&& & {\\rm\\Large III}& \\\\\n\\alpha-1 & &&&&&&&&& \\\\ \\cline{8-12}\n\\alpha & && && {\\rm\\Large I}&&&&& \\\\\n\\alpha +1 & &&&&&&&&& \\\\\n\\vdots & &&&&&&&& {\\rm\\Large II} & \\\\\n\n\\end{array}\n\\]\nAs always, $P$ denotes a general point.\n\nFor $t < \\alpha$ it is clear that $T_{t,m}={\\ensuremath{\\mathrm{adim}}}(X,t,m)=0$. \nThis takes care of region I.\n\nWe now show that all entries in region II are 0. The condition $AV_{X,0}(\\alpha) = 0$ gives us that $T_{\\alpha, \\alpha}=0$;\nthis the top left point of region II.\nIt also gives us that $Y_1$, defined by $I_P^\\alpha$, imposes $\\binom{\\alpha-1+n}{n}$ independent conditions on $[I_X]_\\alpha$.\nBut then by Corollary \\ref{ineq for AV} we have $T_{j+\\alpha,\\alpha}=AV_{X,j}(\\alpha) = 0$ for all $j \\geq 0$ \n(this gives the top row of region~II). \n\nIn the portion below the main diagonal of region II we have $m > t$, so $T_{t,m}={\\ensuremath{\\mathrm{adim}}}(X,t,m)$ is 0,\nsince there there can be no hypersurfaces with degree $t$ and multiplicity $m$ (unexpected or not). \nFor the portion on and above the main diagonal of region II, \nwe apply Theorem \\ref{t. AV is an O-sequence}, which says that $AV_{X,j}(m)$ is an $O$-sequence. \nRecall that the top row of region II gives us $AV_{X,j}(\\alpha) = 0$ for $j \\geq 0$.\nThen it follows for $i>0$ that $T_{\\alpha+i+j,\\alpha+i}=AV_{X,j}(\\alpha+i) = 0$ as well, since a Hilbert function that attains a value 0 cannot subsequently become non-zero. Thus starting from each entry on the top row of region II (all of which are 0), the entries \ndescending diagonally and to the right are all 0. Thus all entries in region II above the main diagonal are also 0.\n\nWe now show all entries in region III are 0.\nBecause $AV_{X,j}(\\alpha) = 0$, the scheme $Y_1$ defined by the (saturated) ideal $I_P^\\alpha$ \nimposes independent conditions on $[I_X]_t$ for all $t\\geq\\alpha$. Hence, applying Lemma \\ref{subscheme}\niteratively, so does the scheme $Y_2$ defined by $I_P^k$ for $\\alpha-1 \\geq k \\geq 1$.\nThus $T_{t,k}=AV_{X,t-k}(k) = 0$ for $t\\geq\\alpha$. Thus all entries in region III are 0, and we are done.\n\\end{proof}\n\nThe next proposition shows that if a subscheme $X$ admits an unexpected hypersurface of \ndegree $t$ vanishing with multiplicity $m$ at a general point, then $[\\ensuremath{\\mathrm{gin}}(I_X)]_t$ is not a lex-segment. \nThis gives a criterion (this time algebraic), again based on a single piece of information \n(admittedly more difficult to verify), for the non-existence of any unexpected hypersurfaces.\n\n\\begin{proposition}\\label{p.lex segments are bad for unexpectedness}\n\tLet $X$ be a subscheme of $ \\ensuremath{\\mathbb{P}}^n$. Assume there exists $j\\ge 0$ such that \n\t$[\\ensuremath{\\mathrm{gin}}(I_X)]_{m+j}$ is a lex-segment and $\\hbox{adim(X,m+j,m)}>0.$ Then \n\t $AV_{X,j}(m)=0.$ In particular, if $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal, then $X$ does not admit any unexpected hypersurfaces.\n\\end{proposition}\n\n\\begin{proof}\n Let $j\\ge 0$ be an integer such that $X$ admits a hypersurface of degree $m+j$ \n vanishing with multiplicity at least $m$ at a general point $P$. Then by Lemma \\ref{l. adim vdim and Gin}(i) \n we have ${\\ensuremath{\\mathrm{adim}}} (X,t,m)=\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^m]_t>0$, so\n there is a monomial $x_0^sM\\in \\ensuremath{\\mathrm{gin}}(I_X)$ where $M]\\in I_P^m$ so has degree $\\deg M=m+j-s\\geq m$, \n but $j\\geq s\\geq0$.\n Now, since $[\\ensuremath{\\mathrm{gin}}(I_X)]_{m+j}$ is a lex-segment, we have $x_0^sM \\in \\ensuremath{\\mathrm{gin}}(I_X)$. \n In order to prove the statement, it is enough by Theorem \\ref{t. AV is an O-sequence} to show that\n $\\dim [R\/(\\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1})]_{m-1}=0$. So, take any monomial \n $F\\in R_{m-1}$. Then $x_0^{j+1}F\\in[R]_{m+j}$ and $x_0^{j+1}F>_{lex} x_0^sM$. Thus, $x_0^{j+1}F\\in \\ensuremath{\\mathrm{gin}}(I_X)$,\n so $F\\in \\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1}$, hence $[R\/(\\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1})]_{m-1}=0$.\n\\end{proof}\n\n It is natural to ask if the converse of Proposition \\ref{p.lex segments are bad for \n\tunexpectedness} is true, namely if it is true for a finite set of points $X$ that if \n$\\ensuremath{\\mathrm{gin}}(I_X)$ is {\\em not} a lex-segment ideal then $X$ must admit some sort of \nunexpected hypersurface. Recalling that one must take with a grain of salt the \ngeneric initial ideal produced by a computer algebra program (is the change of \nvariables ``general enough?\"), a counterexample is given in Example \\ref{ex.root A_n}. \nIt would be interesting to have a theoretical procedure to determine if the generic initial \nideal of a finite set of points is a lex-segment ideal or not. \n\n\\begin{example}\\label{ex.root A_n} \nAssume the characteristic of the field $K$ is $0$. Let $X_n\\subseteq \\ensuremath{\\mathbb{P}}^n$ be the set of $\\binom{n+2}{2}$ points obtained from the root system $A_{n+1}$ as described in \\cite{HMNT}, section 3.1. Specifically, $X_n$ consists of the $\\binom{n+1}{2}$ points having one entry equal to 1, one equal to $-1$ and the rest 0, together with the $n+1$ coordinate points.\n\t \t\n\t\tThe initial degree of $I_{X_n}\\subseteq K[x_0, x_1, \\ldots, x_n]$ is $\\alpha(I_{X_n})=3$ for $n\\ge 2$, because the product of any 3 indeterminates vanishes at $X_n$ and $[I_{X_n}]_2= (0)$. To see that $X_n$ does not lie on a quadric we take $F:=\\sum c_{ab}x_ax_b\\in I_{X_n}$ and we show that $F=0$. Indeed, since $F$ vanishes at the coordinate points, $c_{ab}=0$ if $a= b$; also $c_{ab}=0$ for $a\\neq b$ because $F$ vanishes at the point having no zero entries at the positions $a$ and $b$. \n\t\t\n\t\tA computer calculation, by {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \\cite{cocoa}, showed that $AV_{X_n,0}(3)=0$ for $2\\le n\\le 12$. \n\t\tTherefore, by Proposition \\ref{AV_X,0(alpha)}, $X_n$ does not admit any unexpected hypersurface of any sort, for $2\\le n\\le 12$.\n\t\t\n\t\tThis result is consistent with \\cite{HMNT}, where a computer search did not turn up any unexpected hypersurfaces\n\t\tfor $X_n$ in the cases $2\\leq n\\leq 6$, $2 \\leq d \\leq 6$, $2 \\leq m \\leq d$.\n\t\t\n\t\tFurthermore, we checked with {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ that, for $4\\le n\\le 12$, the ideal defining the set of points $X_n\\subseteq \\ensuremath{\\mathbb{P}}^n$ has a generic initial ideal that is not a lex-segment ideal. In particular, in the cases $5 \\le n\\le 12$ we noticed that $\\left[\\ensuremath{\\mathrm{gin}}(I_{X_n})\\right]_3$ fails to be a lex-segment because $x_1x_n^2$ fails to belong to $\\ensuremath{\\mathrm{gin}}(I_{X_n})$.\n\t\t\n\t\tInterestingly, we have checked for $3\\le n\\le 12$ that the set of points $Y_n\\subseteq \\ensuremath{\\mathbb{P}}^n$, constructed from $X_n$ by replacing in the coordinates of its points all the ``-1\" with ``+1,\" admits an unexpected cone of degree 3, and $x_0x_n^2$ fails to belong to $\\ensuremath{\\mathrm{gin}}(I_{Y_n})$.\n\\end{example}\n\n\n\\section{On the sequence $AV_{X,1}(m)$ when $X$ is an irreducible ACM curve in $ \\ensuremath{\\mathbb{P}}^3$}\\label{s.irreducible ACM curve in P3}\n\nIn this section we will assume that $K$ has characteristic zero. We have already seen in \nTheorem \\ref{t. AV is an O-sequence} that for a subscheme $X \\subset \\ensuremath{\\mathbb{P}}^n$, the \nsequence $AV_{X,j}$ shifted to the left by 1 is an $O$-sequence. In this section we will abuse \nterminology and just say that the sequence is an $O$-sequence, often (but not always) suppressing \nthe shift. Furthermore, if the positive part of the sequence $AV_{X,j}$ is finite, we will ignore the terms \nthat are zero and just say that $AV_{X,j}$ is finite. In this section our focus is on the case $j=1$.\n\nRecall that an SI-sequence is a finite, non-zero, symmetric $O$-sequence such that the first half is a \ndifferentiable $O$-sequence (i.e., also the first difference of the first half is an $O$-sequence). \nThe significance of SI-sequences is that they characterize the $h$-vectors of arithmetically \nGorenstein subschemes of projective space whose artinian reductions have the Weak Lefschetz Property. \nIn codimension three they characterize the $h$-vectors of all arithmetically Gorenstein subschemes. \nNote that SI-sequences are automatically unimodal. See \\cite{Hrm} for properties of SI-sequences.\n\nWe have produced a great deal of experimental evidence for the following conjecture\nconcerning SI-sequences. We will shortly prove part of it.\nWe recall that an ACM scheme is always connected \\cite[Theorem 18.12]{E}, hence a smooth\nACM scheme is irreducible.\n\n\\begin{conjecture}\\label{SI conj}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a smooth ACM curve not lying on a quadric surface. \nThen the sequence $AV_{X,1}$ is an SI-sequence (shifted by 1). The last non-zero term in this sequence \nis $AV_{X,1}(\\deg X - 5)$, so the SI-sequence ends in degree $\\deg X - 6$.\n\\end{conjecture}\n\n \\begin{remark}\n We point out two things. First, \nwe have not yet seen a direct connection to Gorenstein algebras; we have only this numerical conjecture. It would be very interesting to tie Gorenstein algebras to the study of unexpected hypersurfaces in some way. Second, \n sometimes the SI-sequences that we obtain have codimension three and sometimes codimension four.\n \\end{remark}\n \n The case where $X$ {\\em does} lie on a quadric surface is contained in Theorem \\ref{part of conj}.\nWe first give examples to show that all of the other assumptions are needed in this conjecture. \n\n\\begin{example} \\label{hyp not hold}\nThe symmetry of the sequence $AV_{X,1}(m)$ requires all of the given assumptions. \nThe following examples, which were run in either {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \\cite{cocoa} or Macaulay2 \\cite{GS}, show that dropping a hypothesis can\nresult in the AV sequence either not being finite, or, if finite, not being symmetric.\n\n\\begin{itemize}\n\n\\item[(a)] ({\\em $X$ satisfies all assumptions.}) Assume that $X_1$ is a line in $ \\ensuremath{\\mathbb{P}}^3$ \nand $X$ is obtained as the residual to $X_1$ by two general cubic surfaces containing $X_1$. \nThen $X$ is a smooth ACM curve of degree 8 and genus 7, and the positive part of the sequence \n$AV_{X,1}$ is $(1,2,1)$. This satisfies Conjecture \\ref{SI conj}, and in particular it is symmetric\nwith its last value being in degree $8-6=2$.\n\n\\item[(b)] ({\\em $X$ is not ACM}.) Assume that $X_1$ is the disjoint union of two lines in \n$ \\ensuremath{\\mathbb{P}}^3$ and $X$ is linked to $X_1$ by a general choice of two cubic surfaces. \nThen $X$ is a smooth, non-ACM curve of degree 7 and genus 4, and the positive part of \nthe sequence $AV_{X,1}$ is $(1,2)$, which is not symmetric.\n\n\\item[(c)] ({\\em $X$ is not in $ \\ensuremath{\\mathbb{P}}^3$}.) Assume that $X$ is a smooth surface of \ndegree 8 in $ \\ensuremath{\\mathbb{P}}^4$ obtained from a plane by linking using two general hypersurfaces of \ndegree 3. (The example in (a) is a hyperplane section of this one.) Then the positive part of \nthe sequence $AV_{X,1}$ is $(1,3,4,4,\\dots)$, which is not finite. (Note that its first difference is the sequence in (a).)\n\n\\item[(d)] ({\\em $X$ is not equidimensional, but the curve part is ACM}.) Assume that $X$ is the \nresidual in $ \\ensuremath{\\mathbb{P}}^3$ of a line inside the complete intersection of two cubics (as in \n(a)). Let $Y$ be the union of $X$ with a general point. Then the positive part of $AV_{Y,1}$ is the sequence $(1,3,2)$,\nwhich is not symmetric.\n\n\\item[(e)] ({\\em $j = 0$ instead of $j=1$}.) Assume that $X$ is the curve in (a) but take $j=0$. \nThen the positive part of $AV_{X,0}$ is the sequence $(1, 4, 8, 11, 13, 14, 14, 14, \\dots)$,\nwhich is not finite.\n\n\\item[(f)] ({\\em $j = 2$ instead of $j = 1$}.) Assume that $X$ is the curve in (a) but take $j=2$. \nThen $AV_{X,2}$ is the sequence $(0,0,\\dots)$, and so is finite and vacuously symmetric, but does not end in the conjectured degree. However, linking the line \n$X_1$ from (a) using two surfaces of degree 4 gives a curve $X$ of degree 15 and genus 28, with the positive part of $AV_{X,2}$ being the sequence $(1,2,2)$,\nwhich is not symmetric.\n\n\\item[(g)] ({\\em $X$ is zero-dimensional}.) Assume that $X$ is the complete intersection in $ \\ensuremath{\\mathbb{P}}^3$ \nof three general quartic surfaces. Then the positive part of the sequence $AV_{X,1}$ is $(1, 4, 7, 8, 5)$\nwhich is not symmetric.\n\n\\item[(h)] ({\\em $X$ is ACM but not irreducible, 1}.) Let $Z \\subset \\ensuremath{\\mathbb{P}}^2$ be the set of 9 points \ncoming from the $B_3$ root system (see \\cite[Figure 2]{DIV} and \\cite{HMNT}). Let $X$ be the cone over $Z$ with vertex \nat a general point $Q$. Then $X$ is an ACM union of lines (whose general hyperplane section admits an \nunexpected quartic curve), and the positive part of the sequence $AV_{X,1}$ is $(1,3,4,4,4,\\dots)$, which is not finite.\n\n\\item[(i)] ({\\em $X$ is ACM but not irreducible, 2}). Let $X$ be as in (h), except now let $Z \\subset \\ensuremath{\\mathbb{P}}^2$ be a general \nset of 9 points. The Hilbert function of $Z$ (and hence $X$) is the same as it was in (h), but now one computes that \nthe positive part of the sequence $AV_{X,1}$ is $(1,3,3,3,\\dots)$, which is not finite. (Thus the AV sequence depends on\ngeometry beyond the Hilbert function.)\n\n\\item[(j)] ({\\em $X$ is ACM and irreducible but not smooth}). Let $Q$ be a general point and \nlet $X$ be the complete intersection of two general quartic surfaces in $I_Q^3$. Then $X$ is \nACM and we have verified that $X$ is irreducible (using Macaulay2), but the positive part of the \nsequence $AV_{X,1}$ is $(1, 4, 8, 12, 15, 16, 15, 12, 8, 4, 2, 2, 2, \\dots)$, which is not finite. (Thus we need \nsmoothness and not simply irreducibility in the statement of the conjecture.)\n\n\\end{itemize}\n\\end{example}\n\nWhile we are not able to prove the full conjecture, we at least show unimodality and the \ndifferentiability of the increasing part, using only irreducibility of $X$ and not necessarily \nsmoothness. What is missing is the finiteness, the symmetry and the degree of the last \npositive term in the sequence when $X$ is smooth. Notice that nothing about our \nargument fails for (i) or (j) in Example \\ref{hyp not hold}, so something more will be \nneeded to prove the rest of the conjecture.\n\n\\begin{theorem} \\label{part of conj} Let $X \\subset \\ensuremath{\\mathbb{P}}^3$ be an irreducible ACM curve. \n\n\\begin{itemize}\n\n\\item[(a)] If $X$ lies on a quadric surface, then for each $j\\geq1$ the sequence $AV_{X,j}(m)$ is zero.\n\n\\item[(b)] If $X$ does not lie on a quadric surface then the sequence $AV_{X,1}(m)$ is \nnon-zero and unimodal. Furthermore, the increasing part is a differentiable $O$-sequence.\n\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\t\n(a) This follows by Corollary \\ref{degenerate}.\n\n(b) We assume that \n\t\n\t\\begin{itemize}\n\t\t\\item $P \\in \\ensuremath{\\mathbb{P}}^3$ is a general point with defining ideal $I_P$, \n\t\t\n\t\t\\item $H$ is a general plane in $ \\ensuremath{\\mathbb{P}}^3$ containing $P$,\n\t\t\n\t\t\\item $L {\\color{blue} \\in I_P}$ is a linear form defining $H$,\n\t\t\n\t\t\\item $Z = X \\cap H$.\n\t\t\n\t\\end{itemize}\n\t\n\t\\noindent \n\tNote that $P$ may be taken to be a general point in $H$ with respect to the set $Z$. \n\tIn addition to the notation introduced in Remark~\\ref{cohom interp}, recall from Notation \\ref{Pm vs mP} that\n\t\\begin{quotation}\n\t\t{\\em we will denote the scheme defined by $I_{P|H}^m$ by $mP$, a fat point in the \n\t\tplane, to distinguish it from the fat point scheme $P^m$ in $ \\ensuremath{\\mathbb{P}}^3$ defined by $I_{P}^m$.} \n\t\\end{quotation}\n\tNotice that \n\t\\[\n\tI_{X \\cup P^{m+1}} : L = I_{X \\cup P^m} \\ \\ \\ \\hbox{ and } \\ \\ \\ (I_{X\\cup P^m} + (L))^{sat} = I_{Z \\cup mP}.\n\t\\]\n\tSince $Z = X \\cap H$ is a general hyperplane section of $X$, and $X$ is irreducible, $Z$ is \n\ta set of points in linearly general position in $H$ (in this case meaning no three points of $Z$ \n\tare collinear). Hence by \\cite{CHMN} Corollary 6.8, $Z$ does not admit any unexpected \n\tcurves in the plane. Considering the exact sequence of sheaves \n\t\\[\n\t0 \\rightarrow \\mathcal I_{Z \\cup (m+1)P} (m+2) \\rightarrow \\mathcal I_Z (m+2) \\stackrel{r_{m+2}}{\\longrightarrow} \\mathcal O_{(m+1)P} (m+2) \\rightarrow 0\n\t\\]\n\tfrom Remark \\ref{cohom interp} (where the ideal sheaves are on $H = \\ensuremath{\\mathbb{P}}^2$), the fact that $Z$ \n\tdoes not admit any unexpected curves means that $r_{m+2}$ has maximal rank on \n\tglobal sections. (For some values of $m$ it will be injective, and eventually it will be surjective.)\n\t\n\tNow consider the commutative diagram of sheaves (the rows are exact since $P\\not\\in X$ and the columns are exact since $X$ is irreducible):\n\t\\[\n\t\\begin{array}{cccccccccccccccc}\n\t&& 0 && 0 && 0 \\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t0 & \\rightarrow & \\mathcal I_{X \\cup P^m}(m+1) & \\rightarrow & \\mathcal I_X (m+1) & \\rightarrow & \\mathcal O_{P^m}(m+1) & \\rightarrow & 0 \\\\\n\t&& \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} && \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} && \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} \\\\\n\t0 & \\rightarrow & \\mathcal I_{X \\cup P^{m+1}}(m+2) & \\rightarrow & \\mathcal I_X (m+2) & \\rightarrow & \\mathcal O_{P^{m+1}}(m+2) & \\rightarrow & 0 \\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t0 & \\rightarrow & \\mathcal I_{Z \\cup (m+1)P}(m+2) & \\rightarrow & \\mathcal I_Z(m+2) & \\rightarrow & \\mathcal O_{(m+1)P}(m+2) & \\rightarrow & \\phantom{.} 0 .\\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t&& 0 && 0 && 0 \n\t\\end{array}\n\t\\]\n\tIn cohomology we obtain\n\t{\\footnotesize\n\t\t\\[\n\t\t\\begin{array}{cccccccccccccccc}\n\t\t&& 0 && 0 && 0 \\\\\n\t\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t\t0 & \\rightarrow & [ I_{X \\cup P^m}]_{m+1} & \\rightarrow & [ I_X ]_{m+1}& \\rightarrow & H^0(\\mathcal O_{P^m}(m+1)) & \\rightarrow & H^1(\\mathcal I_{X \\cup P^m}(m+1)) & \\rightarrow & 0 \\\\\n\t\t&& {\\big \\downarrow} && {\\big \\downarrow} && {\\big \\downarrow} && \\phantom{\\alpha_{m+1}} {\\big \\downarrow} \\alpha_{m+1} \\\\\n\t\t0 & \\rightarrow & [ I_{X \\cup P^{m+1}}]_{m+2} & \\rightarrow & [ I_X ]_{m+2}& \\rightarrow & H^0(\\mathcal O_{P^{m+1}}(m+2)) & \\rightarrow & H^1(\\mathcal I_{X \\cup P^{m+1}}(m+2)) & \\rightarrow & 0 \\\\\n\t\t&& \\downarrow && \\downarrow* && \\downarrow && \\downarrow \\\\\n\t\t0 & \\rightarrow & [ I_{Z \\cup (m+1)P}]_{m+2} & \\rightarrow & [ I_Z ]_{m+2}& \\stackrel{r_{m+2}}{\\longrightarrow} & H^0(\\mathcal O_{(m+1)P}(m+2)) & \\rightarrow & \n\t\t\\coker (r_{m+2}) & \\rightarrow & \\phantom{.} 0 \\\\\n\t\t&&&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t\t&&&& 0 && 0 && 0\n\t\t\\end{array} \n\t\t\\] }\nusing the fact that $X$ is ACM for the first vertical surjection (marked by an asterisk).\n\t\t\nTo begin, we focus only on the first line of the above commutative diagram. For $m=0$ we obtain\n\\[\nh^1(\\mathcal I_{X\\cup P^0} (1)) = h^1(\\mathcal I_{X} (1)) = 0\n\\]\nsince $X$ is ACM. For $m=1$ we obtain\n\\[\nh^1(\\mathcal I_{X \\cup P}(2)) = \n\\left \\{\n\\begin{array}{ll}\n0 & \\hbox{if $\\dim [I_X]_2 \\geq 1$} \\\\\n1 & \\hbox{if $\\dim [I_X]_2 = 0$.}\n\\end{array}\n\\right.\n\\]\nSince, by Remark \\ref{cohom interp} we have $h^1(\\mathcal I_{X \\cup P^m}(m+1)) = AV_{X,1}(m)$, \nand we know the latter is an $O$-sequence shifted by one thanks to Theorem \\ref{t. AV is an O-sequence}, \nthis gives that the start of the shifted $O$-sequence (the value 1) corresponds to $h^1(\\mathcal I_{X \\cup P}(2))$. \nThus the sequence is non-zero since $X$ does not lie on a quadric surface, so we have the first part of (b).\n(If we had merely assumed that $X$ is not degenerate, this also shows that the sequence is non-zero if and only if $X$ \ndoes not lie on a quadric surface.)\n\nSo for the rest of the proof we assume that $X$ does not lie on a quadric surface.\nApplying the Snake Lemma to the above commutative diagram, the fact that $r_{m+2}$ has \nmaximal rank means that also $\\alpha_{m+1}$ has maximal rank (in the right-hand column of \nthe diagram). Then applying Remark \\ref{cohom interp} as $m$ varies, since $r_{m+2}$ is initially \ninjective and then surjective, the same is true for the map $\\alpha_{m+1}$. This shows that the \nsequence $AV_{X,1}(m)$ is unimodal.\n\nFinally, we argue that the increasing part of the sequence $\\{h^1(\\mathcal I_{X \\cup P^m}(m+1))\\}$ is a \ndifferentiable $O$-sequence (shifted by 2).\n\nThanks to Theorem \\ref{t. AV is an O-sequence} and Remark \\ref{cohom interp}, we know \nthat the sequence $\\{ h^1(\\mathcal I_{X \\cup P^m}(m+1)) \\}$ is an $O$-sequence, and we have \njust seen that the map $\\alpha_{m+1}$ is injective as long as $\\coker (r_{m+2})$ is non-zero. So \nwe have only to show that $\\dim ( \\coker (r_{m+2}))$ is an $O$-sequence (shifted by~2). But \nbecause $r_{m+2}$ has maximal rank, when $\\coker (r_{m+2})$ is non-zero we have from the \nexactness of the bottom row of the above diagram that\n\\[\n\\begin{array}{rcl} \n\\displaystyle \\dim [ \\coker (r_{m+2}) ]_{m+2} & = & \\displaystyle \\binom{(m+1)-1+2}{2} - \\dim [I_Z]_{m+2} \\\\ \\\\\n& = & h_{R\/I_Z}(m+2) - [ 2(m+2) + 1 ] \n\\end{array}\n\\]\nas long as this number is positive. \n\nSo for $t \\geq 2$ {\\em we want to show that \n\\[\nk_{t-2} := h_{R\/I_Z}(t) - (2t+1) \n\\]\nis an $O$-sequence as long as it is positive}. Let $s = t-2$, so we want to show that \n$k_s$ is an $O$-sequence for $s \\geq 0$.\nSince $X$ does not lie on a quadric surface, $Z$ does not lie on a conic (since $X$ is ACM). \nThus $k_0 = \\dim [ \\coker (r_2) ]_2 = 6-5 = 1$. \n\nNow assume that $t \\geq 3$, so $s \\geq 1$. As long as $h_{R\/I_Z}(t) = \\binom{t+2}{2}$ \n(i.e., before $I_Z$ begins) this difference is $k_s = \\binom{s+2}{2}$, which is an $O$-sequence. So \nassume $h_{R\/I_Z}(t) < \\binom{t+2}{2}$. Consider the $t$-binomial expansion of $h_{R\/I_Z}(t)$. Since\n\\[\n2t+1 < h_{R\/I_Z}(t) < \\binom{t+2}{2},\n\\]\nwe have \n\\[\nh_{R\/I_Z}(t) = \\binom{t+1}{t} + \\binom{t}{t-1} + (\\hbox{terms in degrees $\\leq t-2$})\n\\]\nand \n\\[\n(2t+1) = \\binom{t+1}{t} + \\binom{t}{t-1}.\n\\]\nHence $k_s = h_{R\/I_Z}(t) - (2t+1)$ has an $s$-binomial expansion coming directly from the \n$t$-binomial expansion of $h_{R\/I_Z}(t)$, by removing the first two binomial coefficients. \nBut $h_{R\/I_Z}$ is an $O$-sequence, so it obeys Macaulay's bound (Theorem \\ref{macaulay thm}). \nThus $k_s$ does as well, and so is an $O$-sequence.\n\\end{proof}\n\n\n\\section{Codimension 2 complete intersections in $ \\ensuremath{\\mathbb{P}}^n$} \\label{ci section}\n\nIn this section we apply results of Section \\ref{s. gin and unexp} to the case of complete \nintersections of codimension 2 in $ \\ensuremath{\\mathbb{P}}^n$. In particular, as introduced in \nRemark \\ref{r: partial elimination ideal}, we get partial information on the actual dimension \nfor a complete intersection of codimension 2 in a certain degree by using the theory of \\textit{partial elimination ideals}. \n(See \\cite{Gr} for background on partial elimination ideals and for Sylvester matrices, which appear\nin the proof of Proposition \\ref{p. codimension 2 complete intersection}.)\nComplete intersections of codimension 2 have been investigated in several papers. Indeed, \nthe following result can be deduced from Proposition 6.8 in \\cite{Gr} and Corollary 3.9 in \\cite{CoS}. \nWe include here a proof for completeness of the exposition and to show explicitly how the \npartial elimination theory affects the existence of unexpected hypersurfaces. \n \nRecall that $R = K[x_0,x_1,\\ldots,x_n]$ denotes a standard graded polynomial ring and the monomials of $R$ are ordered by $>_{lex}$, the \nlexicographic monomial order which satisfies $x_0> x_1 > \\cdots > x_n$.\t\n\n\n\\begin{proposition}\\label{p. codimension 2 complete intersection}\n\tLet $C\\subseteq \\ensuremath{\\mathbb{P}}^n$ be a codimension 2 complete intersection. \n\tAssume $C$ is defined by two, sufficiently general, forms of degree $a,b$ respectively. Say $a\\le b$.\n\tSet, for any integer $0\\le j< a$,\n\\[\n\\begin{array}{rcl}\nt & := & (a-j)(b-j)+j \\hbox{ and } \\\\ \nm& := & (a-j)(b-j).\n\\end{array}\n\\]\nThen\n\t\\[\n\t{\\ensuremath{\\mathrm{adim}}} (X,t,m)> 0.\n\t\\]\n\\end{proposition}\n\\begin{proof}\n\tAfter a general change of coordinates, say\n\t\\[\n\t\\begin{array}{ccc}\n\tF &=& f_a+f_{a-1}x_0+f_{a-2}x_0^2+\\cdots +f_{1}x_0^{a-1}+ f_0 x_0^a\\\\\n\tG &=& g_b+g_{b-1}x_0+g_{b-1}x_0^2+\\cdots+ g_1x_0^{b-1}+ g_0 x_0^b\\\\\n\t\\end{array}\n\t\\]\n\tand define $J:=(F,G)$ to be the ideal generated by $F$ and $G$.\n\tLet $M$ be the following matrix of size $(a+b-2j)\\times (a+b-2j)$: \n\t{\\tiny\\[M:=\\left(\\begin{array}{ccccccccccccc}\n\t\tf_{a-j} & f_{a-j-1} & \\cdots & f_1 & f_{0} & 0 & \\cdots & 0 \\\\ \n\t\tf_{a-j+1} & f_{a-j} & \\cdots & f_{2} & f_{1} & f_0 & \\cdots & 0\\\\ \n\t\t\\vdots & \\vdots& \\ddots \\\\\n\t\tf_{a} & f_{a-1} & \\cdots & & & & \\\\\n\t\t0 & f_a& \\\\\n\t\t\\vdots& \\vdots \\\\\n\t\t0 & 0 & \\cdots &&&&\\cdots & f_0\\\\\n\t\tg_{b-j} & g_{b-j-1} & \\cdots & & & & \\cdots & 0 \\\\ \n\t\tg_{b-j+1} & g_{b-j} & \\cdots & & & & \\cdots & 0\\\\ \n\t\t\\vdots & \\vdots & \\ddots\\\\\n\t\tg_{b } & g_{b-1} & \\cdots & & & & \\\\\n\t\t0 & g_b& \\cdots \\\\\n\t\t\\vdots & \\vdots\\\\\n\t\t0 & 0 & \\cdots &&&& \\cdots & g_0\\\\\\\n\t\t\\end{array}\\right).\n\t\t\\]} \n(Note that the determinant of the matrix $M$ is one of the minors of the Sylvester matrix $Syl(F,G,x_0)$ of \n$F$ and $G$, and these minors belong to the partial elimination ideal $K_{j}(J)$. See in \nparticular Corollary 3.9 in \\cite{CoS} and Proposition 6.9 and the following Remark in \\cite{Gr}.) \nOne can check that $\\det(M)$ is homogeneous and it has degree $m$.\nLet $M_i$ denote the cofactor of $M$ corresponding to $(-1)^{i+1}$ times the determinant of the matrix that results from deleting the $i$-th row and the first column of $M$.\n\tConsider the following $a+b-2j$ polynomials\n\t\\[\n\t\\begin{array}{cccccccrrcccccc}\n\tF & =& f_a &+& f_{a-1}x_0 &+& \\cdots +& f_{a-j}x_0^j &+& f_{a-j-1}x_0^{j+1} &+& \\cdots\\\\\n\tx_0F &=& & & f_ax_0 & +&\\cdots +& f_{a-j+1}x_0^j & +& f_{a-j}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{j-1}F &=& & & & & & f_{a}x_0^j & +& f_{a-1}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{b-j-1}F &=& \\cdots & & & & & \\\\\n\tG &=& g_b &+& g_{b-1}x_0 &+& \\cdots +& g_{b-j}x_0^j & +& g_{b-j-1}x_0^{j+1} &+& \\cdots\\\\\n\tx_0G &=& & & g_bx_0 & +&\\cdots +& g_{b-j+1}x_0^j & +& g_{b-j}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{a-j-1}G &=& \\cdots & & & & & \\\\\n\t\\end{array}\n\t\\] \t\n\tMultiplying the above $a+b-2j$ polynomials respectively by $M_1, M_2, \\ldots, M_{a+b-2j}$ and taking the sum of them, we get\n\t\\[\\begin{array}{ccl}\n\tT:=\\left(\\sum\\limits_{i=1}^{b-j} M_ix_0^{i-1} \\right) F&+& \\left(\\sum\\limits_{i=1}^{a-j} M_{b-j+i}x_0^{i-1} \\right) G=\\\\ &=& f_aM_1+g_bM_{b-j+1} +x_0 \\left(\\cdots \\right) +\\cdots+ x_0^{j}(\\cdots) +x_0^{j+1}(\\cdots)+\\cdots\\\\\n\t\\end{array}\n\t\\] \t\n\tNote that in the form $T\\in K[x_1, \\ldots x_n][x_0]$ the coefficient of $x_0^j$ is $\\det(M)$. Also, note that all the coefficients \n\tof the powers of $x_0$ greater than $j$ are zero; for instance the coefficient of $x_0^{j+1}$ \n\tis the sum of the entries of the second column in $M$ multiplied by the cofactors of the entries \n\tin the first column. Of course $T$ belongs to the ideal generated by $F$ and $G$, therefore, since we performed a general change of variables, its leading term belongs to $\\ensuremath{\\mathrm{in}}\\hspace{1pt}(J)=\\ensuremath{\\mathrm{gin}} (I_C)$. As noted above, the form $T$ can be written as $T=T_{m+j}+x_0 \\left(T_{m+j-1} \\right) +\\cdots+ x_0^{j}(T_m)$ where $T_i\\in K[x_1, \\ldots x_n]_i$ and $T_m=\\det(M),$ thus\n\t$\\ensuremath{\\mathrm{in}}\\hspace{1pt}(T)=x_0^j\\cdot \\ensuremath{\\mathrm{in}}\\hspace{1pt}(\\det(M)) \\in \\ensuremath{\\mathrm{gin}}(I_C)$. \n\tSince $\\ensuremath{\\mathrm{in}}\\hspace{1pt}(T)$ vanishes with multiplicity $m$ at the point $Q:=(1,0,\\ldots, 0)$, by Lemma \\ref{l. adim vdim and Gin}, we are done.\n\\end{proof}\n\n\\begin{remark}\\label{r. compute edim CI} As a consequence of Proposition \n\\ref{p. codimension 2 complete intersection}, a general codimension 2 complete intesection \n$C\\subseteq \\ensuremath{\\mathbb{P}}^n$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$\n\twhenever $\\mbox{edim}\\left(C,t,m \\right)=0$, i.e. $\\mbox{vdim}\\left(C,t,m \\right)\\le 0$.\n\tIn order to compute $\\mbox{edim}\\left(C,t,m \\right)$, where $C$ is a complete \n\tintersection of type $(a,b)$ in $ \\ensuremath{\\mathbb{P}}^n$, take the short exact sequence\n\t\\[\n\t0\\to R(-a-b)\\to R(-a)\\oplus R(-b) \\to I_{C} \\to 0. \n\t\\]\n\tSo, computing the dimension of the graded pieces of degree $t$, we get\n\t\\[\\dim [I_C]_t= {t-a +n\\choose n}+{t-b +n\\choose n}- {t-a-b +n\\choose n}. \\]\n\tTherefore, we have\n\t\\[\n\t\\mbox{edim}\\left(C,t,m \\right)=\\max\\left\\{\\dim [I_C]_t-{m-1+n \\choose n},0\\right\\}.\n\t\\]\n\\end{remark}\n\n\n\\begin{remark}The case $j=0$ is covered by \\cite[Proposition 2.4]{HMNT}. The cone \nwith vertex $P$ over a codimension 2 complete intesection $C\\subseteq \\ensuremath{\\mathbb{P}}^n$ \nof type $(a,b)$ is an unexpected hypersurface for $C$ of degree $ab$ and multiplicity $ab$ at $P$. \n\\end{remark}\n\nIf $j\\ge 1$ it is enough to show that ${\\ensuremath{\\mathrm{vdim}}}\\big(C, (a-j)(b-j)+j, (a-j)(b-j)\\big)\\le 0$ to \nensure the existence of an unexpected hypersurface.\nIn the next proposition we deal with the case $j=1$ in $ \\ensuremath{\\mathbb{P}}^3$. Theorem \\ref{part of conj} \nshows that a general complete intersection of type $(2,b)$ never admits an unexpected hypersurface \nwith $j=1$. The following shows that when $a>2$ we do obtain unexpected hypersurfaces.\n\n\\begin{proposition} \\label{unexpected CI}\tLet $C\\subseteq \\ensuremath{\\mathbb{P}}^3$ be a general \ncodimension 2 complete intersection defined by two forms of degree $a,b$ respectively. \nSay $a\\le b$. \tSet\\ $t:=(a-1)(b-1)+1$ and $m:=(a-1)(b-1)$.\n\tIf $a >2$ then $C$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$. \n\\end{proposition}\n\\begin{proof}\n\nFrom Remark \\ref{r. compute edim CI} we have\n\t\\[\\mbox{vdim}\\left(C,t,m \\right)={t-a +3\\choose 3}+{t-b +3\\choose 3}- {t-a-b +3\\choose 3}-{t-2+3 \\choose 3}.\n\t\\]\n\tIf $a=b=3$ then $t=5$ and\n\t\\[\\mbox{vdim}\\left(C,t,m \\right)={5\\choose 3}+{5\\choose 3}-{6 \\choose 3}\\le 0.\n\t\\]\n\tA similar computation follows if $a=3$ and $b=4$. Note that, in this case $t+3-a-b=3$. \n\tSince the integer $t+3-a-b$ increases with $a$ and $b$, in particular if $(a,b)>(3,4)$ \n\twe have $t+3-a-b > 3$, so the binomial coefficient ${t +3-a-b\\choose 3}$ is not zero. Then, \n\tassuming $(a,b)>(3,4)$, after a standard computation we get\n\t\\[\n\t\\begin{array}{rl}\n\t\\mbox{vdim}\\left(C,t,m \\right)& = -\\dfrac{1}{2}a^2b-\\dfrac{1}{2}ab^2+a^2+b^2+4ab-6a-6b+9=\\\\\n\t&\\\\\n\t&=-\\dfrac{1}{2}(a-2)(b-2)(a+b-4)+1\\le 0.\\\\\n\t\\end{array}\n\t\\]\n\tThus $\\mbox{vdim}\\left(C,t,m \\right)\\le 0$.\n\\end{proof}\n\n\\begin{remark} \\label{ans question}\n\nQuestion 2.11 of \\cite{HMNT} asks the following: Let $Z$ be a non-degenerate set of points in linear general position in $ \\ensuremath{\\mathbb{P}}^n$, $n \\geq 3$. Is it true that there does not exist an unexpected hypersurface of any degree $t$ and multiplicity $m = t - 1$ at a general point? (All other possible combinations of $(n,t,m)$ are settled.)\n\nThe work in this section allows us to give a negative answer to this question. Indeed, let $X$ be a general complete intersection of type $(3,3)$ in $ \\ensuremath{\\mathbb{P}}^n$ ($n \\geq 3$). Then let $t = (3-1)(3-1)+1 = 5$ and $m = t-1 = 4$. From Remark \\ref{r. compute edim CI} we have \n\\[\n\\hbox{vdim } (X,t,m) = \\binom{n+2}{n} + \\binom{n+2}{n} - \\binom{n-1}{n} - \\binom{n+3}{n} \\leq 0\n\\]\nfor all $n \\geq 3$. On the other hand, Proposition \\ref{p. codimension 2 complete intersection} gives (with $a=b=3$ and $j=1$) that $X$ does lie on a hypersurface of degree 5 and multiplicity 4 at a general point, so this hypersurface is unexpected. Now take a set $Z$ consisting of sufficiently many general points on $X$, so that $[I_X]_5 = [I_Z]_5$. Then $Z$ is in linear general position since $X$ is irreducible, and so $Z$ admits an unexpected hypersurface and gives a negative answer to the question.\n\nWe believe that the natural extension of Proposition \\ref{unexpected CI} to $ \\ensuremath{\\mathbb{P}}^n$ is also true, but we do not have a proof.\n\\end{remark}\n\n\\begin{remark}\nOne could make the following objection to Remark \\ref{ans question} as an answer to Question 2.11 of \\cite{HMNT}. That is that the degree 5 component of $I_Z$ is the same as the degree 5 component of $I_X$, so the base locus of $[I_Z]_5$ is $X$ and the geometry is really only about $X$ and not about $Z$.\n\nTo respond to this, we make the following tweak. Take $n = 4$. We choose the same $X$ as above (now a surface in $ \\ensuremath{\\mathbb{P}}^4$), and we still take $t=5$, $m=4$. One checks that the Hilbert function of $R\/I_X$ is\n\\[\n1, 5, 15, 33, 60, 96, 141 ...\n\\]\nand we still have \n\\[\n\\hbox{vdim }(X,t,m) = \\binom{4+2}{4} + \\binom{4+2}{4} - \\binom{4-1}{4} - \\binom{4+3}{4} = -5.\n\\]\nBut $X$ contains a set $Z$ of 225 points giving a $(3,3,5,5)$ complete intersection, hence\n\\[\n\\dim [I_{Z}]_5 = 32 = \\dim [I_X]_5 + 2,\n\\]\nso $\\hbox{vdim}(Z,t,m) = -3$, hence all quintic hypersurfaces containing $Z$ with a general point of multiplicity 4\nare unexpected, and there is certainly at least one (this being the one that contains $X$).\nThere still remains an objection: computations show in this case that $X$ and $Z$ \nboth have a unique unexpected quintic\nfor multiplicity 4, so the one for $Z$ is the same one that we already got for $X$.\nNevertheless, what we have gained is that the component of the ideal for $Z$ in degree 5 is no longer the same as that of $X$, and indeed the base locus now is finite while the base locus in the original example was $X$ itself.\n\\end{remark}\n\n\\begin{comment}\nBut $X$ contains a set $Z''$ of 225 points giving a $(3,3,5,5)$ complete intersection, hence\n$Z''$ imposes 94 independent conditions on quintics, so $Z''$ contains a subset \n$Z'$ of 91 points imposing 91 conditions on quintics. We then have\n\\[\n\\dim [I_{Z'}]_5 = \\dim [I_X]_5 + 5 = 35\n\\]\nLet $Z$ be the base locus of these 35 quintics; thus $Z'\\subseteq Z\\subset Z''\\subset X$.\nBut now $\\hbox{vdim}(Z,t,m) = 0$, so the hypersurface is still unexpected.\n\nR=QQ[x,y,z,w,v];\nF=random(3,R);\nG=random(3,R);\nI=ideal(F,G);\nP=ideal(x,y,z,w);\nA=random(5,R);\nB=random(5,R);\nJ=ideal(F,G,A,B);\nJ4P=intersect(J,P^4);\nfor i from 0 to (numgens J4P)-1 do print {i, degree J4P_i}\nfor i from 0 to 10 do print {i,hilbaertFunction(i,J)}\nI4P=intersect(I,P^4);\n\\end{comment}\n\n\n\n\\section{Cones, unmixed curves and unions with finite sets of points in \n$ \\ensuremath{\\mathbb{P}}^3$} \\label{s.Unmixed curves and unions with finite sets in P3}\n\nIn this section, for the most part we restrict our attention to the case of subvarieties of $ \\ensuremath{\\mathbb{P}}^3$. \nWe recall that\n\\[\nAV_{X,0} (t) = {\\ensuremath{\\mathrm{adim}}} (X,t,t) - {\\ensuremath{\\mathrm{vdim}}} (X,t,t).\n\\]\nWe will determine the $AV_{Z,0}$ sequence for the cases where $Z=C$ is an \nequidimensional curve in $ \\ensuremath{\\mathbb{P}}^3$ and where $Z=C\\cup X$ is the union of a \ncurve $C$ and a finite set of points $X$. In particular, since $j=0$, we are focusing on the case of unexpected {\\em cones}.\nIn both cases the geometric information on $C$ provides a description of the {\\em persistence} of \nunexpected hypersurfaces. We recall the following result, which reflects the persistence of \nunexpectedness for cones for a non-degenerate curve in $ \\ensuremath{\\mathbb{P}}^3$.\n\n\\begin{theorem}[\\cite{HMNT} Corollary 2.12] \\label{HMNT cone}\nLet $C \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, equidimensional, non-degenerate curve \nof degree $d \\geq 2$ ($C$ may be reducible, singular, and\/or disconnected). Let \n$P \\in \\ensuremath{\\mathbb{P}}^3$ be a general point. Let $k \\geq d$ be a positive integer. Then \n$C$ admits an unexpected hypersurface of degree $k$ with multiplicity $k$ at $P$. When $k = d$, this hypersurface is unique.\n\\end{theorem}\n\nOur next result not only reproduces the persistence \ngiven by the result of Theorem~\\ref{HMNT cone} but also gives a measure of the unexpectedness in each degree. \n\n\\begin{theorem} \\label{uct}\nLet $C \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, equidimensional curve of degree $e$ and arithmetic genus $g$. Then\n\\[\nAV_{C,0}(t) = \\binom{e-1}{2} - g\n\\]\nfor all $t \\geq e$. Moreover, $AV_{C,0}(e) = 0$ if and only if $AV_{C,0}(t)=0$ for $t\\geq e$ if and only if $C$ lies in a plane.\n\\end{theorem}\n\n\\begin{proof}\nLet $P$ be a general point. We are considering hypersurfaces of degree $t$ with multiplicity $t$ at \n$P$, which are cones with vertex $P$ that contain $C$. Since the projection of $C$ from \n$P$ is contained in the hyperplane section of such a cone, we must have $t \\geq e$.\nWe first compute the virtual dimension of $I_{C \\cup P^t}$:\n\\[\n\\begin{array}{rcl}\n\\dim [I_C]_t - \\binom{t+2}{3} & = & \\binom{t+3}{3} - h_C(t) - \\binom{t+2}{3} =\\\\\n& = & \\binom{t+2}{2} - [te - g + 1].\n\\end{array}\n\\] \n(The fact that for $t \\geq e$ we have $h_C(t) = te - g + 1$ follows from the main \nresult of \\cite{GLP}.)\nOn the other hand, if $S$ is such a cone of degree $t$ and if $Q$ is a point of $C$, \nthen the line joining $P$ and $Q$ meets $S$ with multiplicity at least $t+1$, so it \nmust lie on $S$. That is, $S$ contains as a component the cone over $C$ with vertex \n$P$. Then the actual dimension of $I_{C \\cup P^t}$ is the (vector space) dimension of the \nlinear system of plane curves of degree $t-e$, i.e. it is $\\binom{t-e+2}{2}$. Then we obtain\n\\[\nAV_{C,0}(t) = \\binom{e-1}{2} - g\n\\]\nafter a simple calculation. The last part follows since $g = \\binom{e-1}{2}$ if and only if $C$ is a plane curve, \\cite[Proposition 2.1, Claim 1]{HMNT}.\n\\end{proof}\n\n\nEven if your interest is for the case when $Z$ is a finite set of points (which we consider in the next section),\nsituations involving curves (such as we are looking at in this section) \nsometimes force themselves into the picture in subtle ways,\nas the next example shows.\n\n\\begin{example}\nLet $X_1 , X_2 \\subset \\ensuremath{\\mathbb{P}}^3 $ be finite sets of points with $h$-vectors, respectively,\n\\[\n(1,3,6,5,3,3,2) \\ \\ \\ \\hbox{ and } \\ \\ \\ (1,3,6,6,3,3,2).\n\\]\nIn both cases, the two 3's constitute maximal growth, viewing the $h$-vector as a Hilbert function,\nand force ``many\" of the points to lie on a curve of degree 3 \\cite{BGM}. For the sake of this \nexample, let us assume that in both cases this curve is a twisted cubic, that it contains 18 points \nof each of $X_1$ and $X_2$ (with $h$-vector $(1,3,3,3,3,3,2)$ in both cases), and that the \nremaining 5 points of $X_1$ and the remaining 6 points of $X_2$ are chosen generically. \nThen in the first case there is a unique unexpected cone of degree 5 with vertex at a general \npoint, while in the second case there is no unexpected cone of degree 5. \nWe omit details here since we will study this \nkind of situation in the next section (see especially Example \\ref{1366332}).\n\\end{example}\n\nOur next result can again be viewed as a measure of unexpectedness for cones in \neach degree, and a statement about the persistence of unexpected cones.\n\n\n\\begin{theorem}\\label{t.C U X}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a finite set of points. Let $C$ be a reduced, \nequidimensional curve of degree $e$ and arithmetic genus $g$. Assume that $X$ \nis disjoint from $C$. \nLet $t$ be the smallest integer such that \n\\begin{enumerate}\n\\item[(i)] $|X| < \\binom{t+2}{2}$, and \n\\item[(ii)] $X$ imposes independent conditions on forms of degree $t$.\n\\end{enumerate}\nThen \n\\[\nAV_{X \\cup C,0}(t+e) = AV_{X,0}(t) + \\left [ \\binom{e-1}{2} - g \\right ].\n\\]\n\\end{theorem}\n\n\\begin{proof}\n\nLet $P$ be a general point. Let $S_P$ be the cone over $C$ with vertex $P$; we \nknow $\\deg S_P = e$, $S_P$ is reduced, and $S_P$ has multiplicity $e$ at $P$. Let $T_P$ \nbe a surface (unmixed) of degree $t+e$ containing $X \\cup C$ and having multiplicity $t+e$ at \n$P$. ($T_P$ is a cone over a suitable plane curve.) Note that $S_P$ is a component of $T_P$, \nand note that since $P$ is general, no point of $X$ lies on $S_P$. \nWrite $U_P$ for the residual to $S_P$ in $T_P$. Note that $\\deg U_P = t$, $U_P$ has \nmultiplicity $t$ at $P$ and $U_P$ contains $X$. We also know that ${\\ensuremath{\\mathrm{adim}}} (C,e,e) = 1$ \n(Theorem \\ref{HMNT cone}). These observations imply\n\\[\n\\dim [I_X \\cap I_P^t]_t = \\dim [I_{X \\cup C} \\cap I_P^{t+e}]_{t+e}.\n\\]\nWe also remark that since $X$ imposes independent conditions on forms of degree $t$ and \n$C$ imposes independent conditions on forms of degree $e$ (by \\cite{GLP}), we can conclude \n\\begin{equation} \\label{indep cond}\n\\hbox{\\em $X$ also imposes independent conditions on $[I_{C}]_{t+e}$.}\n\\end{equation}\n\nBy definition we have\n\\[\nAV_{X,0}(t) = \\dim [I_X \\cap I_P^t]_t - \\left [ \\dim [I_X]_t - \\binom{t+2}{3} \\right ].\n\\]\nNow we compute (using the observation about independent conditions)\n\\small{\n\\[\n\\begin{array}{lcl}\nAV_{X \\cup C,0}(t+e) & = \\displaystyle & \\displaystyle \\dim [I_{X \\cup C} \\cap I_P^{t+e}]_{t+e} - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle \\dim [I_X \\cap I_P^t]_t - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\left [ \\dim [I_X]_t - \\binom{t+2}{3} \\right ] - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle\nAV_{X,0}(t) + \\left [ \\binom{t+3}{3} - |X| - \\binom{t+2}{3} \\right ] - (\\dim [ I_{C} ]_{t+e} - |X|) + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{t+2}{2} - \\dim [I_C]_{t+e} + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{t+2}{2} - \\left [ \\binom{t+e+3}{3} - (e(t+e) - g + 1) \\right ] + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{e-1}{2} - g\n\\end{array}\n\\] }\n(the fourth line uses (\\ref{indep cond}) and the last line comes after a routine calculation).\n\\end{proof}\n\n\n\n\\begin{corollary} \\label{X U plane curve}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a finite set of points. Let $C$ be a reduced plane curve in \n$ \\ensuremath{\\mathbb{P}}^3$ of degree $d$ disjoint from $X$. Then $X$ has an unexpected cone of degree \n$t$ if and only if $X \\cup C$ has an unexpected cone of degree $t+d$. Furthermore, \n$AV_{X,0} (t) = \nAV_{X\\cup C,0}(t+d)$ for all $t$.\n\\end{corollary}\n\n\n\\begin{proof}\nThe arithmetic genus of a plane curve of degree $e$ is $\\binom{e-1}{2}$.\n\\end{proof}\n\n\n\n\\section{Finite sets of points in $ \\ensuremath{\\mathbb{P}}^3$} \\label{s.Finite sets of points in P3}\n\nIn this section we assume that $K$ has characteristic zero. \nOne of the original motivations for this paper was to determine if there are any Hilbert \nfunctions for non-degenerate sets of points that {\\em force} the existence of unexpected hypersurfaces of some sort. \n(A consequence of \nCorollary \\ref{degenerate} is that given a finite $O$-sequence $(1,a_1,\\dots,a_r)$, \none can trivially find a set of points in some projective space, with this $h$-vector, that does \nnot admit any unexpected hypersurfaces. One simply produces a set $X$ in $ \\ensuremath{\\mathbb{P}}^n$ for $n > a_1$ \nhaving this $h$-vector. Then $X$ is degenerate, hence admits no unexpected hypersurfaces. \nThus it is more interesting to consider non-degenerate sets.)\n\nIt now seems plausible that in a strict sense there are no such Hilbert functions. Indeed,\nwe make the following conjecture (but see Theorem \\ref{force unexp} and Corollary \\ref{LGP}, which show that such Hilbert functions\ndo arise when combined with a little geometric information).\n\n\\begin{conjecture} \\label{conj about existence}\nFor every possible $h$-vector $(1,n,a_2,\\dots,a_r)$ for a non-degenerate, finite set of \npoints in $ \\ensuremath{\\mathbb{P}}^n$, there is a set of points $X$ with that $h$-vector such that $X$ \ndoes not admit any unexpected hypersurfaces of any degree and multiplicity.\n\\end{conjecture}\n\nIn trying to prove Conjecture \\ref{conj about existence} we made the following observations. \nRecall that a {\\em distraction} is a construction that converts, in particular, an artinian monomial \nideal in $K[x_0,\\dots,x_{n-1}]$ to the ideal of a reduced set of points in $ \\ensuremath{\\mathbb{P}}^n$. It was \nintroduced in \\cite{hartshorne}. See also \\cite{MN2} for related constructions and results.\n\nOne way of constructing a reduced set of points with a given $h$-vector is \nto start with the artinian lex-segment ideal with Hilbert function $h$ and perform a distraction to \nproduce a set of points $X$. Experimentally, it seems that very often $\\hbox{gin}(I_X)$ is a \nlex-segment ideal. If this were always the case, we would be done by Corollary \\ref{recall gin}:\n\n\\begin{corollary} \\label{recall gin}\nLet $h= (1,n, a_2,\\dots,a_r)$ be a finite $O$-sequence. Let $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a \nset of points with this $h$-vector. If $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal in \n$R = K[x_0,x_1,\\dots,x_n]$, then $X$ does not admit any unexpected hypersurfaces, \nfor any degree and multiplicity.\n\\end{corollary}\n\n\\begin{proof}\nThis follows immediately from Proposition \\ref{p.lex segments are bad for unexpectedness}.\n\\end{proof}\n\nUnfortunately, we have verified that the $h$-vector {\\tt (1,3,6,10,5,5,2)} results in a set of \npoints for which $\\ensuremath{\\mathrm{gin}}(I_X)$ is not a lex-segment ideal, and in fact one can check on {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \nthat it admits an unexpected hypersurface of degree 4 with multiplicity 3. However, we were able to \nconfirm that the union in $ \\ensuremath{\\mathbb{P}}^3$ of 22 general points on a plane curve of degree 5 and 10 \ngeneral points in $ \\ensuremath{\\mathbb{P}}^3$ results in a set of points with the desired $h$-vector, which \ndoes not admit any unexpected hypersurfaces, so the conjecture is still true for this $h$-vector \neven though the distraction does not produce the desired set of points. This should be \ncontrasted with the end of Example \\ref{1366332}; in this case we verified that the \ndistraction does produce a set of points whose gin is a lex-segment. Thus the conjecture remains open.\n\nWe recall a result from \\cite{BGM}, modified to fit our context. For a set of points $X$ we \ndenote by $\\langle [I_X]_{\\leq d} \\rangle$ the ideal generated by the polynomials in $I_X$ of \ndegree $\\leq d$. Recall also that for a subscheme $Y$ of $ \\ensuremath{\\mathbb{P}}^n$ we denote by $h_Y (t)$ its Hilbert function.\n\n\\begin{proposition}[\\cite{BGM} Theorem 3.6] \\label{BGM thm}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, finite set of points with $h$-vector\n\\[\n(1,3,a_2,a_3,\\dots,a_k,d,d, a_{k+3},\\dots,a_r)\n\\]\nwhere $d \\leq k+1$. Then\n\\begin{itemize}\n\n\\item[\\rm (a)] $\\langle [I_X]_{\\leq k+1} \\rangle$ is the saturated ideal of a reduced curve, $V$, of \ndegree $d$ (not necessarily unmixed). Also, $I_X$ has no minimal generators in degree $k+2$, \nso $\\langle [I_X]_{\\leq k+1} \\rangle = \\langle [I_X]_{\\leq k+2} \\rangle$.\n\n\\item[\\rm (b)] Let $C$ be the unmixed, one-dimensional part of $V$. Let $X_1$ be the subset \nof $X$ on $C$ and let $X_2$ be the subset of $X$ not on $C$; note $X = X_1 \\cup X_2$. \nThen $\\langle [I_{X_1}]_{\\leq k+1} \\rangle = I_C$, and $V = C \\cup X_2$.\n\n\\item[\\rm (c)] $h_{X_1} (t) = h_X(t) - |X_2|$ for all $t \\geq k$.\n\n\\item[\\rm (d)] \n\\[\n\\Delta h_{X_1} (t) = \n\\left \\{\n\\begin{array}{ll}\n\\Delta h_C (t) & \\hbox{for } t \\leq k+2; \\\\\n\\Delta h_X (t) & \\hbox{ for } t \\geq k+1.\n\\end{array}\n\\right.\n\\]\n\n\\end{itemize}\n\n\n\\end{proposition}\n\n\nFrom now on we focus on the special case $j=0$, i.e. when the degree of the unexpected \nhypersurface is equal to the multiplicity at the general point. The following example shows \nthat knowing the $h$-vector of a finite set of points, and taking into account the fact that the \nbase locus of some component of $I_X$ contains a curve, is not enough\nto ensure that $X$ has an unexpected surface. \nWe generally have to know something more about the curve.\n\n\\begin{example} \\label{1366332}\nConsider the $h$-vector\n\\[\n(1,3,6,6,3,3,2).\n\\]\nThe values in degrees 4 and 5 force the existence of a cubic curve of some sort in the \nbase locus of $[I_X]_4$ and of $[I_X]_5$ for any finite set $X$ with this $h$-vector. We \nwill look at a few different kinds of cubic curves to see how the differences in the geometry \nof the curves gives different behavior with respect to unexpected hypersurfaces (specifically cones). \nOur goal is not to give an exhaustive list of possible sets of points \nwith this $h$-vector, but rather to highlight a few to see how they differ.\nSo consider the following sets of points sharing this $h$-vector. \n\n\\begin{itemize}\n\\item Let $X$ consist of 18 points on a twisted cubic $C$ (note that the \n$h$-vector of these 18 points is (1,3,3,3,3,3,2)) plus 6 general points. \nThen we claim that $X$ admits no unexpected cone of any degree. \n\nNotice that \n\\[\n\\dim [I_X ]_t = \\left \\{\n\\begin{array}{ll}\n4 & \\hbox{if } t = 3; \\\\\n16 & \\hbox{if } t = 4; \\\\\n34 & \\hbox{if } t = 5; \\\\\n\\binom{t+3}{3} - 24 & \\hbox{if } t \\geq 6 .\n\\end{array}\n\\right.\n\\]\nWe first consider the case $t \\leq 5$, and we notice ${\\ensuremath{\\mathrm{edim}}} (X,t,t) = 0$ in this case.\nSince any cone of degree $\\leq 5$ containing $X$ must also contain $C$, it also contains \nthe cone over $C$ (which is a surface of degree 3) as a component. But the projection \nfrom $P$ of the 6 general points gives 6 general points in the plane, and there is no conic \nthrough 6 general points. Thus $\\dim [I_{X \\cup P^t}]_t = 0$ for $t \\leq 5$ so ${\\ensuremath{\\mathrm{adim}}}(X,t,t) = {\\ensuremath{\\mathrm{edim}}}(X,t,t)=0$.\n\nNow let $t \\geq 6$. We have\n\\[\n{\\ensuremath{\\mathrm{vdim}}} (X,t,t) = \\binom{t+3}{3} - 24 - \\binom{t+2}{3} = \\binom{t+2}{2} - 24 > 0.\n\\]\nNow, the projection of the points on $C$ gives a set of points with \n$h$-vector \\linebreak $(1,2,3,3,3,3,3)$ so adding six general points gives a set with \n$h$-vector $(1,2,3,4,5,6,3)$, hence the general projection imposes independent \nconditions on plane curves of degree $t$. Thus the vector space dimension of this \nlinear system (hence the vector space dimension of the family of cones of degree $t$ with vertex $P$) is the expected one.\n\nNotice that if the $h$-vector had been $(1,3,6,6,3,3,3)$ then the projection would \nhave $h$-vector $(1,2,3,3,3,3,3,1)$ and the above argument would not work for $t=6$. \nIndeed, Theorem \\ref{force unexp} gives an unexpected sextic cone.\n\n\\medskip\n\n\\item Let $C$ be a set of three disjoint lines. Let $X$ consist of 6 points on one of the \nlines and 7 points on each of the remaining lines, chosen generally, together with 4 \ngeneral points in $ \\ensuremath{\\mathbb{P}}^3$. (The $h$-vector of the points on $C$ is (1,3,5,3,3,3,2), \nand the $h$-vector of $X$ is again $(1,3,6,6,3,3,2)$.) Then the expected dimension in \ndegree 5 is 0 as before, but since there is a pencil of conics through four general points in \nthe plane we obtain ${\\ensuremath{\\mathrm{adim}}}(X,5,5) = 2$, i.e. there is a pencil of unexpected cones of degree 5. \n\n\\medskip\n\n\\item Let $C$ be a smooth plane cubic curve and let $X$ consist of 17 points on $C$ \n(with $h$-vector (1,2,3,3,3,3,2)) plus a set $X_1$ of 7 general points in $ \\ensuremath{\\mathbb{P}}^3$. \nOne can check that $1 = AV_{X,0}(5) = AV_{C \\cup X_1,0} (5) = AV_{X_1,0}(2)$ in \naccordance with Corollary \\ref{X U plane curve} but that there is no unexpected hypersurface because ${\\ensuremath{\\mathrm{adim}}}(X_1,2,2) = 0$.\n\n\\medskip\n\n\\item Let $C$ be a smooth plane cubic curve in $ \\ensuremath{\\mathbb{P}}^3$, and let $\\lambda_1$ \nand $\\lambda_2$ be general lines in $ \\ensuremath{\\mathbb{P}}^3$. Let $X$ consist of 17 general \npoints on $C$, plus a subset $X_1$ of four general points on $\\lambda_1$ and three \ngeneral points on $\\lambda_2$. One can check that $X$ also has the $h$-vector \n$(1,3,6,6,3,3,2)$ so we expect no surface of degree 5 with a point of multiplicity 5 \nat a general point $P$. However, the cone over $C \\cup \\lambda_1 \\cup \\lambda_2$ \nis such a surface. But notice that the one-dimensional \ncomponent of the base locus of $[I_X]_5$ is only the plane cubic.\n\n\\end{itemize}\n\n\\noindent Thus the $h$-vector $(1,3,6,6,3,3,2)$ may or may not force an unexpected \ncone, depending mostly, but not entirely, on the cubic curve that is forced by the $h$-vector. \n\nIt is worth noting that the $h$-vector $(1,3,6,5,3,3,2)$ (analyzed as above) admits \nan unexpected cone even when the cubic curve is a twisted cubic, and $(1,3,6,6,3,3,2)$ \nadmits an unexpected cone even when the cubic is a plane cubic.\n\n\\end{example}\n\nAs mentioned at the beginning of this section, we do not believe that any finite \n$O$-sequence forces the existence of unexpected hypersurfaces for non-degenerate \nsets of points. However, some sequences force the existence of a curve in the base \nlocus of at least some components of the ideal, and if this curve is not a plane curve \nthen we {\\em can} find finite $O$-sequences that force unexpected hypersurfaces. \nThis idea is elementary, but a bit technical. Thus we will first look at an example, to \nmake the proof of Theorem \\ref{force unexp} clearer. \nWe will refer to the notation of Theorem \\ref{force unexp} in this example.\n\n\\begin{example}\nConsider sets of points $X$ with the $h$-vector\n\\[\n(1,3,6,9,8,7,\\underbrace{6,6,\\dots,6}_\\ell)\n\\]\nwhere $\\ell \\geq 6$. In the notation of Theorem \\ref{force unexp} we have $k = 5$, $d = 6$, $N = (6-4) + (9-5) + (8-6) + (7-6) = 9$ and $m = 3$. Because of the values of this $h$-vector, Proposition \\ref{BGM thm} applies. We get that for $6 \\leq t \\leq \\ell+5$, $(I_X)_{\\leq t}$ has a 1-dimensional base locus, $C$, of degree 6, together with a finite set $X_2$, which imposes independent conditions on hypersurfaces of degree $\\geq 5$. In fact\n\\[\n(I_X)_{\\leq t} = (I_{C \\cup X_2})_{\\leq t}\n\\]\nfor $6 \\leq t \\leq \\ell +5$.\nFurthermore, using Lemma \\ref{bound X2} as in the proof of Theorem \\ref{force unexp}, we see that $|X_2| \\leq N = 9$. {\\em Assume that $C$ is not a plane curve.}\n\nNow we look in degree 11, which is in the range $6 \\leq t \\leq \\ell+5$.\nApplying Theorem \\ref{t.C U X} we obtain\n\\[\nAV_{X,0}(11) = AV_{X_2 \\cup C,0}(11) = AV_{X_2,0} (5) + \\left [ \\binom{6-1}{2} - g \\right ] > 0, \n\\]\nwhere $g$ is the arithmetic genus of $C$. The fact that this is positive follows since $C$ is not a plane curve. \n\nWe remark that this works because the degree 11 is such that the value of the $h$-vector is still $6$ in that degree. Beyond degree $k+\\ell$ there is no longer a curve, and Theorem \\ref{t.C U X} no longer applies.\n\nNow let $P$ be a general point in $ \\ensuremath{\\mathbb{P}}^3$ and consider the projection from $P$ to a general $ \\ensuremath{\\mathbb{P}}^2$. The image of $X_2$ is thus a set of $\\leq 9$ points in the plane, and as such it lies on a plane curve of degree $m=3$ and hence also a plane curve of degree $k = 5$. The cone over this curve is a surface of degree 5 containing $X_2$ with multiplicity 5 at $P$. Together with the cone over $C$ (which has degree 6), we have a surface of degree 11 having multiplicity 11 at $P$. This means $\\hbox{adim} (X,11,11) > 0$. Since also $AV_{X,0}(11) >0$, $X$ admits an unexpected cone of degree 1.\n\\end{example}\n\n\\begin{lemma}[\\cite{BGM} Lemma 3.1] \\label{BGM lemma} \nLet $I \\subset R$ be an ideal satisfying $h_{R\/I}(t) = \\binom{t+m}{t}$ and $h_{R\/I}(t+1) = \\binom{t+1+m}{t+1}$. \nThen $[I]_t$ is the degree $t$ component of the saturated ideal of an $m$-dimensional linear space in $ \\ensuremath{\\mathbb{P}}^n$ (and similarly for $[I]_{t+1}$).\n\\end{lemma}\n\n\\begin{lemma} \\label{bound X2}\nLet $C$ be a reduced, unmixed, non-degenerate curve in $ \\ensuremath{\\mathbb{P}}^3$ of degree $d$. \nLet $h_C(t)$ be its Hilbert function. Then $\\Delta h_C(t)$ \nhas a sharp lower bound as follows: If $d=2$ or 3 then the lower bound is (respectively) \n\\[\n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & 5 & \\dots \\\\ \\hline\n& 1 & 3 & 2 & 2 & 2 & 2 & \\dots \n\\end{array}\n\\ \\ \\ \\hbox{ and } \\ \\ \\ \n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & 5 & \\dots \\\\ \\hline\n& 1 & 3 & 3 & 3 & 3 & 3 & \\dots \n\\end{array}\n\\]\nIf $d \\geq 4$ then the lower bound is\n\\[\n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & \\dots & d-3 & d-2 & d-1 & d & d & \\dots \\\\ \\hline\n& 1 & 3 & 4 & 5 & 6 & \\dots & d-1 & d & d & d & d & \\dots \n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{proof}\nIn all cases, since $C$ is non-degenerate we must have $\\Delta h_C(1) = 3$.\nIf $d=2$ and $C$ is non-degenerate then $C$ must be a pair of disjoint lines, and the first \ngiven $\\Delta h_C$ is its Hilbert function. (So this is precisely the Hilbert function and not a lower bound.) If $d=3$ and $\\Delta h_C(t) \\leq 2$ for any $t \\geq 2$ then by Macaulay it can \nnever grow to 3, which it must do since $\\deg C = 3$. Thus the first two cases are done.\n\nNotice that the given sequence is $\\Delta h_C$ for the curve $C$ consisting of the union of a \nplane curve of degree $d-1$ and a line, meeting at one point. Thus this sequence occurs.\n\nLet $I_C$ be the saturated ideal of $C$. Since $R\/I_C$ has depth $\\geq 1$, if $L$ is a \ngeneral linear form then the first difference of $h_{C}(t)$ is the Hilbert function of $R\/(I_C,L)$ \nand so is an $O$-sequence. We know $\\Delta h_{C}(t) = d$ for $t \\gg 0$. If $\\Delta h_{C}(2) \\leq 2$ then by \nMacaulay's theorem it can never grow to $d$, so we must have $\\Delta h_{C}(2) \\geq 3$. \n\nSuppose $\\Delta h_C(2) = 3$. In order to eventually reach $d$, by Macaulay's theorem \nwe must have $\\Delta h_C(t) = t+1$ for $2 \\leq t \\leq d-1$. Then by Lemma \\ref{BGM lemma} \n(taking $m=1$), $[(I_C + (L))\/(L)]_t$ is the degree $t$ component of a line in $K[x,y,z]$. \nThus since $I_C$ is saturated, $[I_C]_t$ is the degree $t$ component of a plane, i.e. $C$ \nis a plane curve of degree $d$. This is impossible since $C$ is non-degenerate.\n\nThe same argument applies for all degrees $3 \\leq t \\leq d-2$: we must have $\\Delta h_C(t) \\geq t+1$ \nin order to reach $d$, and if we have equality then $C$ must be a plane curve. Thus the stated Hilbert function is the smallest possible.\n\\end{proof}\n\n If Conjecture \\ref{conj about existence} is true, the following kind of result is the best that one can \n hope for, in terms of finding a Hilbert function that forces unexpectedness (but we do \n not claim that this result is optimal in any way). It says that for a certain class of Hilbert \n functions (which we define specifically via some numerical conditions) for which a curve \n is forced in some component of the ideal because of maximal growth, if you {\\em assume} \n that this curve is not a plane curve, then {\\em any} set of points with this Hilbert function \n must admit unexpected cones. Conjecture \\ref{conj about existence} thus implies that if, however, you allow the curve \n to be a plane curve then a set of points can be found for which there is no unexpected cone. \n\n\\begin{theorem}\n \\label{force unexp}\nLet $X$ be a set of points in $ \\ensuremath{\\mathbb{P}}^3$ with $h$-vector\n\\[\n(1,3,a_2, a_3, \\dots, a_k, \\underbrace{d, d , \\dots, d}_\\ell),\n\\]\nwhere $k \\geq 2$ and $a_k > d$. Assume\n\\[\n2 \\leq d \\leq \\min \\{ k+1, \\ell \\}.\n\\]\nIn case $d \\geq 4$, let \n\\[\nb_i = \n\\left \\{\n\\begin{array}{lll}\na_i - (i+2) & \\hbox{for $2 \\leq i \\leq d-2$} \\\\\na_i - d & \\hbox{for $d-1 \\leq i \\leq k$}. \\\\\n\\end{array}\n\\right.\n\\]\nIf $d = 2$ or $d=3$, we replace $i+2$ by the bounds given in the first two parts of Lemma \\ref{bound X2}.\n\nSet \n\\[\nN = \\sum_{i=2}^k b_i \n\\]\nand\n\\[\nm = \\min \\left \\{ i \\ | \\ \\binom{i+2}{2} >~N \\right \\}. \n\\]\nWe also assume $m \\leq k$.\n\nLet $C$ be the equidimensional curve of degree $d$ guaranteed by Proposition~\\ref{BGM thm}. \nIf $C$ is not a plane curve then $X$ admits an unexpected cone of degree $d+k$.\n\n\\end{theorem}\n\n\\begin{proof}\nSince $d \\leq k+1$, Proposition \\ref{BGM thm} applies in degree $k$.\nIn particular, we get from Proposition \\ref{BGM thm} (c) that $X_2$ \nimposes independent conditions on $[I_X]_s$ for any $s \\geq k$, hence it also imposes \nindependent conditions on the complete linear system of forms of degree $s$; we will use \nthe case $s=k$. \n\nNow we look in degree $d+k$. By Theorem \\ref{t.C U X} we have \n\\[\nAV_{C \\cup X_2} (d+k) = AV_{X_2,0}(k) + \\left [ \\binom{d-1}{2} - g \\right ] > 0 \n\\]\nsince $C$ is not a plane curve.\n\n\nNote that $N$ is an upper bound for $|X_2|$, thanks to Lemma \\ref{bound X2}. If we \ndenote by $\\pi_P$ the projection from a general point $P$ to a general plane, the assumption $m \\leq k$\nguarantees that $\\pi_P(X_2)$ lies on a curve of degree $k$. This means that $X_2$ lies on a \ncone of degree $k$ with vertex at $P$. If $S_P$ is the cone over $C$ with vertex $P$, the \nunion of these cones is a surface of degree $d+k$ with multiplicity $d+k$ at $P$. \nThus ${\\ensuremath{\\mathrm{adim}}} (C \\cup X_2,d+k,d+k) > 0$, so we have an unexpected cone of degree $d+k$ for \n$C \\cup X_2$. \nBut $k+1 \\leq d+k \\leq k+\\ell$ so $[I_X]_{d+k} = [I_{C \\cup X_2}]_{d+k}$, \nso also $X$ admits an unexpected cone of degree $d+k$.\n\\end{proof}\n\n\\begin{example}\nAs mentioned above, the preceding result is not meant to be optimal. \nConsider for instance the $h$-vector $(1,3,6,5,3,3,3)$. We have $d=3$, $\\ell = 3$, $N = 3 + 2 = 5$, $m = 2$, $k = 3$. \nThe cubic curve $C$ guaranteed in the base locus of $[I_X]_t$ for $t = 4,5,6$ is either a twisted cubic, \nthe union of a line and a conic (meeting in 0 or 1 points), or the union of three lines (meeting in a total of $<3$ points). \nConsidering possible Hilbert functions of such curves, the given $h$-vector forces a set $X_2$ of at most $N = 5$ points off the curve (as a result of Lemma~\\ref{bound X2}).\n\nThe theorem guarantees an unexpected cone of degree 6. Indeed, \n\\[\n\\dim [I_X]_6 - \\binom{6-1+3}{3} = 60 - 56 = 4\n\\]\nand since the projection of $\\leq 5$ points to $ \\ensuremath{\\mathbb{P}}^2$ lies on at least a 5-dimensional vector space of plane cubics, the cones over these cubics (with vertex at the general point $P$) together with the cone over $C$ confirm the conclusion that there is an unexpected sextic. \n\nHowever, these projected points also lie on at least one conic, so there is a quadric cone containing $X_2$, and together with the cone over $C$ we get an unexpected quintic cone (since $\\dim [I_X]_5 - \\binom{5-1+3}{3} = 0$), which is not covered by the theorem. \n\nIf we had allowed $C$ to be a plane cubic curve, Lemma \\ref{bound X2} would no longer hold: the lower bound in this case would be given by the sequence $(1,2,3,3,3 ,\\dots)$ so $|X_2|$ could also be 6.\n\\end{example}\n\nThe following result gives a geometric property for a set of points that is enough to find $h$-vectors that force unexpected cones.\n\n\\begin{corollary}\n\\label{LGP}\nLet $X$ be a set of points in $ \\ensuremath{\\mathbb{P}}^3$ in linear general position, and assume \nthat $X$ has $h$-vector given by the numerical conditions in Theorem \\ref{force unexp}. Then $X$ admits an unexpected cone.\n\\end{corollary} \n\n\\begin{proof}\nThe assumption of linear general position forces $C$ to be non-degenerate.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn \\cite{wein}, Weinstein introduced the notion of coisotropic submanifold of a Poisson manifold generalizing the notion of Lagrangian submanifold\nof symplectic manifold. A submanifold $C$ of a Poisson manifold $(P, \\pi)$ is called coisotropic, if $\\pi^{\\sharp}(TC)^0 \\subset TC,$ or, equivalently\n$\\pi (\\alpha, \\beta) = 0$, for all $ \\alpha, \\beta \\in (TC)^0$, where $(TC)^0$ is the conormal bundle of $C$. Moreover, Weinstein proved the following results.\n\\begin{enumerate}\n\\item A map $\\phi : P_1 \\rightarrow P_2$ between Poisson manifolds is a Poisson map if and only if its graph is a coisotropic submanifold of\n$P_1 \\times P_2^{-},$ where $P_2^{-}$ stands for the manifold $P_2$ with opposite Poisson structure.\n\\item If $\\phi : P \\rightarrow Q$ is a surjective submersion from a Poisson manifold $P$ to some manifold $Q$, then $Q$\nhas a Poisson structure for which $\\phi$ is a Poisson map if and only if\n$$\\{(x,y)| \\phi(x) = \\phi(y)\\} \\subset P \\times P$$ is a coisotropic submanifold of $P \\times P^{-}.$\n\\end{enumerate}\n\nTo define the coisotropic submanifold of a Poisson manifold, one does not require the Poisson tensor to be closed, that is, $[\\pi, \\pi] = 0,$\nwhere $[~,~]$ denotes the Schouten bracket on multivector fields. Therefore, the notion of coisotropic submanifolds make sense for any bivector field, or more generally, for any multivector field.\nExplicitly, if $M$ is a smooth manifold and $\\Pi \\in \\mathcal{X}^n(M) = \\Gamma \\bigwedge^n TM$ be an $n$-vector field on $M$, then a submanifold $C \\hookrightarrow M$ is called coisotropic with respect to $\\Pi$\nif \n$$\\Pi^{\\sharp} ({\\bigwedge}^{n-1}(TC)^0) \\subset TC ~~\\Leftrightarrow ~~\\Pi (\\alpha_1, \\ldots, \\alpha_n) = 0,~~ \\mbox{for all}~~ \\alpha_1, \\ldots, \\alpha_n \\in (TC)^0,$$\nwhere $\\Pi^\\sharp : \\bigwedge^{n-1}T^*M \\rightarrow TM$ is the bundle map induced by $\\Pi$.\n\nNambu-Poisson manifolds are generalization of Poisson manifolds. Recall that a Nambu-Poisson manifold of order $n$ is a manifold $M$ equipped with an $n$-vector field $\\Pi$ such that the induced bracket on functions satisfies the Fundamental identity (Definition \\ref{nambu-poisson}). The $n$-vector field $\\Pi$ of a Nambu-Poisson manifold is referred to as the associated Nambu tensor. Coisotropic submanifolds of a Nambu-Poisson manifold $M$ are those submanifolds which are coisotropic with respect to the Nambu tensor $\\Pi$. \n \nIn the present paper, we study some basic properties of coisotropic submanifolds of a manifold with respect to a given multivector field\nand generalize the results of Weinstein to the case of multivector field. More precisely, we prove the following results (Propositions \\ref{nambu map-coiso} and \\ref{coinduced-coiso}).\n\\begin{enumerate}\n\\item Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields and $\\phi: M \\rightarrow N$ be a smooth map. Then $\\phi_* \\Pi_M = \\Pi_N$ if and only if its graph\n$$\\text{Gr}(\\phi) := \\{(m, \\phi(m))| m \\in M\\}$$\nis a coisotropic submanifold of $M \\times N$ with resoect to $\\Pi_M \\oplus (-1)^{n-1} \\Pi_N$.\n\\item Let $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\\Pi_N$ such that $\\phi_* \\Pi_M = \\Pi_N$ if and only if \n$$R(\\phi) := \\{(x,y) \\in M \\times M|\\phi(x)=\\phi(y)\\}$$\nis a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$.\n\\end{enumerate}\n\nPoisson Lie group is a Lie group equipped with a Poisson structure such that the group multiplication map is a Poisson map. Equivalently, a Lie group equipped with a Poisson structure is a Poisson Lie group if the Poisson bivector field is multiplicative \\cite{lu-wein}. These definitions have no natural extension when one wants to define Poisson groupoid. Nevertheless, the notion of coisotropic submanifolds of Poisson manifolds was used by Weinstein \\cite{wein} to introduce the notion of Poisson groupoid. Recall that a Poisson groupoid is a Lie groupoid $G \\rightrightarrows M$ with a Poisson structure on $G$ such that the graph of the groupoid (partial) multiplication map is a coisotropic submanifold of $G \\times G \\times G^{-}.$ \n\nIn \\cite{xu}, P. Xu gave an equivalent formulation of Poisson groupoid which generalizes the multiplicativity condition for Poisson Lie group. More generally, In \\cite{pont-geng-xu}, the authors introduced the notion of multiplicative multivector fields on a Lie groupoid. Given a Lie groupoid $G \\rightrightarrows M$, an $n$-vector field $\\Pi \\in \\mathcal{X}^{n}(G)$ is called multiplicative, if the graph of the groupoid multiplication\nis a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$. In this terminology, a Poisson groupoid is a Lie groupoid equipped with a\nmultiplicative Poisson tensor. \n\nIn the present paper, we extend this approach to the case of Lie groupoid with a Nambu structure. We introduce the notion of a Nambu-Lie groupoid as a \nLie groupoid with a Nambu structure $\\Pi$ such that the Nambu tensor $\\Pi$ is multiplicative (Definition \\ref{nambu-lie groupoid}).\nWhen $G$ is a Lie group, this definition coincides with the definition of Nambu-Lie group given by Vaisman \\cite{vais}. Using results proved in \\cite{pont-geng-xu} for mutiplicative multivector fields on Lie groupoid, we deduce the following facts which are parallel to the case of Poisson groupoid. \nSuppose $(G \\rightrightarrows M, \\Pi)$ is a Nambu-Lie groupoid, then \n\\begin{enumerate}\n\\item $M \\hookrightarrow G$ is a coisotropic submanifold of $G$;\n\\item the groupoid inversion map $i: G \\rightarrow G$ is an anti Nambu-Poisson map; \n\\item there is a unique Nambu-Poisson structure $\\Pi_M$ on $M$ for which the source map is a Nambu-Poisson map (Proposition \\ref{inverse-basenambu}).\n\\end{enumerate}\n\nIt is well known that for a Nambu-Poisson manifold $M$ of order $n$, the space of $1$-forms admits a\nskew-symmetric $n$-bracket which satisfies the Fundamental identity modulo some restriction (\\cite{vais , gra-mar , bas-bas-das-muk}). Moreover, the bracket on forms and the de-Rham differential of the manifold satisfy a compatibility condition similar to that of a Lie bialgebroid. This motivates the authors \\cite{bas-bas-das-muk} to introduce a notion of weak Lie-Filippov bialgebroid of order $n$. If $M$ is a Nambu-Poisson manifold of order $n$, then $(TM, T^*M)$ provides such an example. Roughly speaking, a weak Lie-Filippov bialgebroid of order $n$ ($n > 2$) over $M$ is a Lie algebroid $A \\rightarrow M$ together with a skew-symmetric $n$-ary bracket\non the space of sections of the dual bundle $A^* \\rightarrow M$ and a bundle map $\\rho : \\bigwedge^{n-1}A^* \\rightarrow TM$ satisfying some conditions (cf. Definition \\ref{lie-fill-defn}). Moreover it is proved in \\cite{bas-bas-das-muk} that, if $(A, A^*)$ is a weak Lie-Filippov bialgebroid of order $n$ over $M$, then\nthere is an induced Nambu-Poisson structure of order $n$ on the base manifold $M$.\n\nIn the present paper, we prove that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids. \nExplicitly, if $C$ is a coisotropic submanifold of a Nambu-Poisson manifold $(M, \\Pi)$, then we show that the $n$-ary bracket on the space of $1$-forms\non $M$ restricts to the sections of the conormal bundle $(TC)^0 \\rightarrow C$ and the induced bundle map $\\Pi^{\\sharp} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ \nmaps $\\bigwedge^{n-1}(TC)^0$ to $TC$ (Proposition \\ref{coiso-n-bracket}). Therefore, if $G \\rightrightarrows M$ is a Nambu-Lie groupoid of order $n$ whose Lie algebroid is \n$AG \\rightarrow M,$ then as $M$ is a coisotropic submanifold of $G$, the space of sections of the dual bundle $A^*G \\cong (TM)^0 \\rightarrow M$ is equipped with a skew-symmetric $n$-ary bracket. \nMoreover, there is a bundle map $\\bigwedge^{n-1} A^*G \\rightarrow TM$ so that the pair $(AG, A^\\ast G),$ with the above data, satisfies the conditions of a weak Lie-Filippov bialgebroid. Thus we prove the following (cf. Theorem \\ref{nambu-grpd-bialgbd}). \n\\begin{thm}\n Let $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.\n\\end{thm}\n\nFinally, we compare the Nambu-Poisson structures on the base manifold $M$ induced from the Nambu-Lie groupoid $G \\rightrightarrows M$ and the weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \\ref{compare-two-NP-structure}).\n\nNext, we introduce the notion of a coisotropic subgroupoid $H \\rightrightarrows N$ of a Nambu-Lie groupoid $(G \\rightrightarrows M, \\Pi).$\nTo study the infinitesimal form of a coisotropic subgroupoid, we introduce the notion of coisotropic subalgebroid of a weak Lie-Filippov bialgebroid.\nThen we show that the Lie algebroid of a coisotropic subgroupoid $H \\rightrightarrows N$ is a coisotropic subalgebroid of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \\ref{coiso-subgrpd-coiso-subalgbd}). \n\n{\\bf Organization.} The paper is organized as follows. In section 2, we recall basic definitions and conventions. In section 3, we study some properties of coisotropic\nsubmanifolds of a manifold with respect to a given multivector field and in section 4 we introduce Nambu-Lie groupoids and study some of its basic properties.\nIn section 5, we show that the infinitesimal object corresponding to Nambu-Lie groupoid is weak Lie-Filippov bialgebroid and in section 6 we introduce coisotropic\nsubgroupoids of a Nambu-Lie groupoid and study their infinitesimal.\\\\\n{\\bf Acknowledgements.} The author would like to thank his Ph.D supervisor Professor Goutam Mukherjee for his guidance and carefully reading the manuscript.\n\\section{Preliminaries}\nIn this section, we recall some basic preliminaries from \\cite{duf-zung , mac-book , xu} and fix the notations that will be used throughout the paper .\n\nNambu-Poisson manifolds are $n$-ary generalizations of Poisson manifolds introduced by Takhtajan \\cite{takhtajan}.\n\\begin{defn}\\label{nambu-poisson}\nLet $M$ be a smooth manifold. A {\\it Nambu-Poisson structure} of order $n$ on $M$ is a skew-symmetric $n$-multilinear bracket\n\\begin{align*}\n \\{~, \\ldots, ~\\} : C^\\infty (M) \\times \\stackrel{(n)}{\\ldots} \\times C^\\infty (M) \\rightarrow C^\\infty (M)\n\\end{align*}\nsatisfying the following conditions:\n\\begin{enumerate}\n \\item {\\it Leibniz rule}:\\\\\n $\\{f_1, \\ldots, f_{n-1}, fg\\} = f \\{f_1, \\ldots, f_{n-1}, g \\} + \\{f_1, \\ldots, f_{n-1}, f \\} g; $\n \\item {\\it Fundamental identity}:\\\\\n $\\{f_1, \\ldots ,f_{n-1}, \\{g_1,\\ldots, g_n\\}\\} = \\sum_{k=1}^n \\{g_1,\\ldots,g_{k-1},\\{f_1,\\ldots,f_{n-1},g_k\\},\\ldots, g_n\\}; $\n\\end{enumerate}\nfor all $f_i, g_j, f, g \\in C^\\infty (M).$ A manifold together with a Nambu-Poisson structure of order $n$ is called a {\\it Nambu-Poisson manifold of order $n$}. Thus the space of smooth functions with this bracket forms a Nambu-Poisson algebra. A Nambu-Poisson manifold of order $2$ is nothing but a Poisson manifold \\cite{vais-book}. Since the bracket above is skew-symmetric and satisfies Leibniz rule, there exists an $n$-vector field $\\Pi \\in \\mathcal{X}^{n}(M)$, such that\n$$ \\{f_1, \\ldots, f_n\\} = \\Pi (df_1, \\ldots, df_n),$$\nfor all $ f_1, \\ldots, f_n \\in C^\\infty (M)$. Given any $(n-1)$ functions $ f_1, \\ldots, f_{n-1} \\in C^\\infty (M)$, the {\\it Hamiltonian vector field}\nassociated to these functions is denoted by $X_{f_1\\ldots f_{n-1}}$ and is defined by\n$$ X_{f_1\\ldots f_{n-1}} (g) = \\{f_1, \\ldots ,f_{n-1}, g\\}.$$\nNote that the Fundamental identity, in terms of Hamiltonian vector fields, is equivalent to the condition\n$$ [X_{f_1\\ldots f_{n-1}}, X_{g_1\\ldots g_{n-1}}] = \\sum_{k=1}^{n-1} X_{g_1\\ldots \\{f_1, \\ldots, f_{n-1},g_k\\}\\ldots g_{n-1}}.$$\nfor all $f_1, \\ldots, f_{n-1}, g_1, \\ldots, g_{n-1} \\in C^\\infty(M).$ The Fundamental identity can also be rephrased as\n$$ \\mathcal{L}_{X_{f_1\\ldots f_{n-1}}} \\Pi = 0,$$\nfor all $f_1, \\ldots, f_{n-1} \\in C^\\infty(M),$ which shows that every Hamiltonian vector field preserves the Nambu tensor. A Nambu-Poisson manifold is often denoted by $(M, \\{~, \\ldots, ~\\})$ or simply by $(M, \\Pi).$\n\\end{defn}\n\n\\begin{exam}\n\\begin{enumerate}\n\\item Let $M$ be an orientable manifold of dimension $n$ and $\\nu$ be a volume form on $M$. Define an $n$-bracket $\\{~, \\ldots, ~\\}$ on\n$C^\\infty(M)$ by the following identity\n$$ df_1 \\wedge \\cdots \\wedge df_n = \\{f_1, \\ldots, f_n\\} \\nu.$$\nThen $\\{~, \\ldots, ~\\}$ defines a Nambu-Poisson structure of order $n$ on $M$. Let $\\Pi_\\nu \\in \\Gamma{\\bigwedge^n TM}$ denotes the associated Nambu-Poisson\ntensor. If $\\Pi \\in \\Gamma{\\bigwedge^n TM}$ is any Nambu-Poisson structure of order $n$ such that $\\Pi \\neq 0$ at every point, then there\nexists a volume form $\\nu'$ on $M$ such that $\\Pi = \\Pi_{\\nu'}$. If $M = \\mathbb{R}^n$ and $\\nu = dx_1 \\wedge \\cdots \\wedge dx_n$ is the standard volume form, then one recovers the Nambu structure on $\\mathbb{R}^n$ originally discussed by Y. Nambu \\cite{nambu}.\n\\item Let $\\Pi$ be any $n$-vector field on an oriented manifold $M$ of dimension $n$. Then $\\Pi$ defines a Nambu structure of order $n$ on $M$ (see \\cite{ibanez1}).\n\\item Let $M$ be a manifold of dimension $m$ and $X_1, \\ldots, X_n$ be linearly independent vector fields such that $[X_i, X_j] = 0$ for all\n$i, j = 1, \\ldots, n$. Then the $n$-vector field $\\Pi = X_1 \\wedge \\cdots \\wedge X_n$ defines a Nambu structure of order $n$.\n\\item Let $(M, \\{~, \\ldots, ~\\})$ be a Nambu-Poisson manifold of order $n$. Suppose $k \\leqslant n-2$ and $F_1, \\ldots, F_k \\in C^\\infty(M)$ be any fixed functions on $M$. Define a $(n-k)$-bracket\n$\\{~, \\ldots, ~\\}'$ on $C^\\infty(M)$ by\n\\begin{align*}\n \\{f_1, \\ldots, f_{n-k}\\}' = \\{F_1, \\ldots, F_k, f_1, \\ldots, f_{n-k}\\}\n\\end{align*}\nfor $f_1, \\ldots, f_{n-k} \\in C^\\infty(M)$. Then $\\{~, \\ldots, ~\\}'$ defines a Nambu-structure of order $(n-k)$ on $M$. This Nambu structure\nis called the {\\it subordinate Nambu structure} of $(M, \\{~, \\ldots, ~\\})$ with subordinate function $F_1, \\ldots, F_k$.\n \\item If $\\Pi_i$ is a Nambu structure of order $n_i$ on a manifold $M_i$ $(i=1,2)$, then $\\Pi = \\Pi_1 \\wedge \\Pi_2$ is a Nambu structure of order $n_1 + n_2$ on $M_1 \\times M_2$ \\cite{duf-zung}.\n\\end{enumerate}\nMore examples of Nambu structures can be found in \\cite{ibanez2, vais}.\n\\end{exam}\n\n\nLet $(M, \\Pi)$ be a Nambu-Poisson manifold of order $n$. For each $m \\in M$, let $\\mathcal{D}_mM \\subset T_mM$ be the subspace of the tangent space at $m$ generated by all Hamiltonian vector fields at $m$. Since the Lie bracket of two Hamiltonians is again a Hamiltonian, therefore $\\mathcal{D}$ defines a {\\it (singular) integrable distribution} whose leaves are either $n$-dimensional submanifolds endowed with a volume form or just singletons \\cite{duf-zung}.\n\n\n\\begin{defn}\nLet $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields. A smooth map $\\phi: M \\rightarrow N$ is called {\\it $(\\Pi_M, \\Pi_N)$-map} if the induced brackets on functions satisfies:\n\\begin{align*}\n \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = \\phi^* \\{f_1, \\ldots, f_n\\}_N\n\\end{align*}\nfor all\n$f_1, \\ldots, f_n \\in C^\\infty(N),$ or equivalently, $\\phi_* \\Pi_M = \\Pi_N.$ The map $\\phi$ is called an {\\it anti $(\\Pi_M, \\Pi_N)$-map} if \n\\begin{align*}\n \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = (-1)^{n-1} \\phi^* \\{f_1, \\ldots, f_n\\}_N\n\\end{align*}\nfor all $f_1, \\ldots, f_n \\in C^\\infty(N)$. A $(\\Pi_M, \\Pi_N)$-map $\\phi : (M, \\Pi_M) \\longrightarrow (N, \\Pi_N)$ between Nambu-Poisson manifolds of the same order $n$ is called a {\\it Nambu-Poisson map} or a $N$-$P$-map.\n\\end{defn}\n\\begin{remark}\nThe condition for a $(\\Pi_M, \\Pi_N)$-map can also be expressed in terms of the induced bundle maps as\n\\begin{align*}\n\\Pi^{\\sharp}_{N, {\\phi(m)}} = T_m \\phi \\circ \\Pi^{\\sharp}_{M, {m}} \\circ T^*_m \\phi ~ ~~\\mbox{~~ for each}~~ m \\in M,\n\\end{align*}\nwhere $\\Pi^{\\sharp}_M :\\bigwedge^{n-1}T^*M \\rightarrow TM$ is the induced bundle map and is given by\n\\begin{align*}\n \\langle \\beta, \\Pi^{\\sharp}_M (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) \\rangle = \\Pi_M (\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta)\n\\end{align*}\nfor all $\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta \\in T_x^* M,$ $x \\in M$.\n\\end{remark}\n\n\\begin{defn}\n A {\\it Lie groupoid} over a smooth manifold $M$ is a smooth manifold $G$ together with the following structure maps:\n\\begin{enumerate}\n \\item two surjective submersions $\\alpha, \\beta : G \\rightarrow M $, called the {\\it source} map and the {\\it target} map respectively;\n \\item a smooth {\\it partial multiplication} map\n$$ G_{(2)} = \\{(g,h) \\in G \\times G | \\beta(g) = \\alpha (h) \\} \\rightarrow G , ~~ (g,h)\\mapsto gh ;$$\n \\item a smooth {\\it unit} map $\\epsilon : M \\rightarrow G$, $x \\mapsto \\epsilon_x ;$\n \\item and a smooth {\\it inverse} map $i : G \\rightarrow G$, $g \\mapsto g^{-1}$ with $\\alpha (g^{-1}) = \\beta (g)$ and $\\beta (g^{-1}) = \\alpha (g)$\n\\end{enumerate}\nsuch that, the following conditions are satisfied\\\\\n$\\hspace*{1.5cm}$(i) $\\alpha (gh) = \\alpha (g)$ and $\\beta(gh) = \\beta (h)$;\\\\\n$\\hspace*{1.5cm}$(ii) $(gh)k = g(hk),$ whenever the multiplications make sense;\\\\\n$\\hspace*{1.5cm}$(iii) $\\alpha(\\epsilon_x) = \\beta (\\epsilon_x) = x$, $\\forall x \\in M$;\\\\\n$\\hspace*{1.5cm}$(iv) $\\epsilon_{\\alpha(g)} g = g$ and $ g \\epsilon_{\\beta(g)} = g$, $\\forall g \\in G$;\\\\\n$\\hspace*{1.5cm}$(v) $g g^{-1} = \\epsilon_{\\alpha(g)}$ and $g^{-1} g = \\epsilon_{\\beta(g)}$, $\\forall g \\in G$.\n\\end{defn}\nA Lie groupoid $G$ over $M$ is denoted by $G \\rightrightarrows M$ when all the structure maps are understood.\n\n\\begin{remark}\nNote that the smooth structure on $G_{(2)}$ comes from the fact that \n$$G_{(2)} = (\\beta \\times \\alpha)^{-1}(\\Delta_M),$$ where $\\beta \\times \\alpha : G \\times G \\rightarrow M \\times M,~~(g,h) \\mapsto (\\beta(g), \\alpha (h))$\nand $ \\Delta_M = \\{(m,m)| m \\in M\\} \\subset M \\times M$ is the diagonal submanifold of $M \\times M.$ Then these conditions imply that the inverse\nmap $i : G \\rightarrow G,$ $g \\mapsto g^{-1}$ is also smooth \\cite{mac-book}. Moreover, $\\alpha$-fibers and $\\beta$-fibers are submanifolds of $G$ as both $\\alpha$ and $\\beta$ are surjective submersions.\n\\end{remark}\n\n\\begin{defn}\n Given a Lie groupoid $G \\rightrightarrows M$, define an equivalence relation $'\\sim'$ on $M$ by the following: two points $x, y \\in M$ are said to be\nequivalent, written as $x \\sim y$, if there exists an element $g \\in G$ such that $\\alpha(g) = x$, $\\beta(g) = y$. The quotient $M\/\\sim$ is called the {\\it orbit set} of $G$.\n\\end{defn}\n\n\\begin{defn}\n Given two Lie groupoid $G_1 \\rightrightarrows M_1$ and $G_2 \\rightrightarrows M_2$, a {\\it morphism} between Lie groupoids is a pair $(F, f)$ of smooth maps\n$F : G_1 \\rightarrow G_2$ and $f: M_1 \\rightarrow M_2$ which commute with all the structure maps of $G_1$ and $G_2$. In other words,\n$$ \\alpha_2 \\circ F = f \\circ \\alpha_1, ~ ~ \\beta_2 \\circ F = f \\circ \\beta_1, ~ ~ \\text{and} ~ ~ F(g_1 h_1) = F(g_1)F(h_1)$$\nfor all $(g_1,h_1) \\in (G_1)_{(2)}.$\n\\end{defn}\n\n\\begin{defn}\n Let $G \\rightrightarrows M$ be a Lie groupoid. A {\\it Lie subgroupoid} of it is a Lie groupoid $ H \\rightrightarrows N$ together with injective immersions $i : H \\rightarrow G$ and\n$i_0 : N \\rightarrow M$ such that $(i, i_0)$ is a Lie groupoid morphism.\n\\end{defn}\n\n\n\\begin{defn}\n Let $G \\rightrightarrows M$ be a Lie groupoid. A submanifold $\\mathcal{K}$ of $G$ is called a {\\it bisection} of the Lie groupoid, if $\\alpha|_{\\mathcal{K}} : \\mathcal{K} \\rightarrow M$ and\n$\\beta|_{\\mathcal{K}} : \\mathcal{K} \\rightarrow M$ are diffeomorphisms.\n\\end{defn}\n The existence of local bisections through any point $g \\in G$ is always guaranted. The space of bisections $\\mathcal{B}(G)$ form an infinite dimensional (Fr\\'{e}chet) Lie group under the multiplication of subsets induced from the partial multiplication of $G$. Note that the left (right) multiplication is defined only on $\\alpha$-fibers ($\\beta$-fibers), therefore, we can not define a diffeomorphism of $G$ using left (right) multiplication by an element, like a Lie group. However we can do so by using bisection instead of an element.\nGiven a bisection $\\mathcal{K} \\in \\mathcal{B}(G)$, let\n$l_{\\mathcal{K}}$ and $r_{\\mathcal{K}}$ be the diffeomorphisms on $G$ defined by\n$$ l_{\\mathcal{K}} (h) = gh, \\hspace{0.15cm} \\text{where} \\hspace{0.15cm} g \\in \\mathcal{K} \\hspace{0.15cm} \\text{is the unique element such that} \\hspace{0.15cm} \\beta(g) = \\alpha(h)$$\nand\n$$ r_{\\mathcal{K}} (h) = hg', \\hspace{0.15cm} \\text{where} \\hspace{0.15cm} g' \\in \\mathcal{K} \\hspace{0.15cm} \\text{is the unique element such that} \\hspace{0.15cm} \\alpha(g') = \\beta(h).$$\n\n\\begin{remark}\n Suppose $\\mathcal{K}$ is any (local) bisection of $G$ through $g \\in G$. Then the restriction of the map\n$l_{\\mathcal{K}}$ to $\\alpha^{-1} (\\beta(g))$ is the left translation $l_g$ by $g$:\n$$ l_g : \\alpha^{-1}(\\beta(g)) \\rightarrow \\alpha^{-1}(\\alpha(g)), ~ h \\mapsto gh .$$\n\\end{remark}\n\nThen we have the following result \\cite{xu}.\n\n\\begin{prop}\\label{left-invariant}\n Let $G \\rightrightarrows M$ be a Lie groupoid and $P$ be an $n$-vector field on $G$. Suppose for any $g \\in G$ with $\\beta(g) = u$, $P$ satisfies $P(g) = (l_{\\mathcal{G}})_* P(\\epsilon_u)$, where $\\mathcal{G} \\in \\mathcal{B}(G)$ is any arbitrary bisection through the point $g$. Then $P$ is left invariant.\n\\end{prop}\n\n\n\n\\begin{defn}\n A {\\it Lie algebroid} $(A, [~, ~], a)$ over a smooth manifold M is a smooth vector bundle $A$ over\nM together with a Lie algebra structure $[~, ~]$ on the space $\\Gamma{A}$ of the smooth sections of $A$ and a bundle map $a : A \\rightarrow T M $ , called the {\\it anchor}, such that\n\\begin{enumerate}\n \\item the induced map $ a : \\Gamma{A} \\rightarrow \\mathcal{X}^1(M) $ is a Lie algebra homomorphism, where $\\mathcal{X}^1(M)$ is the usual Lie algebra of vector fields on $M$.\n \\item For any $ X, Y \\in \\Gamma{A} $ and $f \\in C^\\infty (M)$, we have\n$$ [X, f Y ] = f [X, Y ] + (a(X)f )Y. $$\n\\end{enumerate}\n\\end{defn}\nWe may denote a Lie algebroid simply by $A$, when all the structures are understood. Any Lie algebra is a Lie algebroid over a point with zero anchor. The tangent bundle of any smooth manifold is a Lie algebroid with usual Lie bracket of vector fields and identity as anchor.\\\\\n{\\bf Lie algebroid of a Lie groupoid.}\nGiven a Lie groupoid $G \\rightrightarrows M$, its Lie algebroid consists of the vector bundle $AG \\rightarrow M$ whose fiber at $x \\in M$ coincides with the tangent space\nat the unit element $\\epsilon_x$ of the $\\alpha$-fiber at $x$. Then the space of sections of $AG$ can be identified with the left invariant vector fields\n\\begin{align*}\n \\mathcal{X}^1_{\\text{inv}}(G) = \\{ X \\in \\Gamma (T^\\alpha G) = \\Gamma (\\text{ker} (d\\alpha))| X_{gh} = (l_g)_* X_h, \\forall (g, h) \\in G_{(2)} \\}\n\\end{align*}\non $G$. Since the space of left invariant vector fields on $G$\nis closed under the Lie bracket, therefore it defines a Lie bracket on $\\Gamma AG$. The anchor $a$ of $AG$ is defined to be the differential of the target map $\\beta$ restricted to $AG$. \n\n\n\n\n\n\n\n\n\nLet $AG $ be the Lie algebroid of the Lie groupoid $G \\rightrightarrows M$. Given any $X \\in \\Gamma AG$, let $\\overleftarrow{X}$ be the corresponding left invariant vector field on $G$. Then there exists an $\\epsilon > 0$ and a $1$-parameter family of transformations $\\phi_t$ $(|t| < \\epsilon)$, generated by $\\overleftarrow{X}$ (\\cite{mac-book}). Suppose each\n$\\phi_t$ is defined on all of $M$, where $M$ is identified with a closed embedded submanifold of $G$ via the unit map. We denote the image of $M$ via $\\phi_t$\nby exp $tX$. Then exp $tX$ is a bisection of the groupoid (for all $|t| < \\epsilon$) and satisfies $1$-parameter group like conditions, namely\n\\begin{align*}\n \\text{exp} (t+s) X = \\text{exp} \\hspace*{0.1cm} tX \\cdot \\text{exp} \\hspace*{0.1cm}sX, \\hspace*{1cm} \\text {whenever} \\hspace*{0.5cm} |t| , |s| , |t+s| < \\epsilon,\n\\end{align*}\nwhere on the right hand side, we used the multiplication of bisections.\n\n\n\\section{Coisotropic submanifolds}\nLet $M$ be a manifold and $\\Pi \\in \\mathcal{X}^n(M)$ be a $n$-vector field on $M$. Let\n\\begin{align*}\n \\Pi^{\\sharp} : {\\bigwedge}^{n-1}T^*M \\rightarrow TM\n\\end{align*}\nbe the induced bundle map given by\n\\begin{align*}\n \\langle \\beta, \\Pi^{\\sharp} (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) \\rangle = \\Pi (\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta)\n\\end{align*}\nfor all $\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta \\in T_x^* M,$ $x \\in M$.\n\nWe recall the following definition from \\cite{pont-geng-xu}.\n\\begin{defn}\nA submanifold $C \\hookrightarrow M$ is said to be {\\it coisotropic} with respect to $\\Pi$,\nif\n\\begin{align*}\n \\Pi^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0) \\subset TC\n\\end{align*}\nwhere $$(TC)^0_x = \\{ \\alpha \\in T_x^* M | \\hspace*{0.1cm} \\alpha (v) = 0, \\forall v \\in T_xC\\},~~x \\in C,$$\nor equivalently,\n\\begin{align*}\n \\Pi_x (\\alpha_1, \\ldots, \\alpha_n) = 0, \\forall \\alpha_i \\in (TC)^0_x, x \\in C.\n\\end{align*}\n\\end{defn}\n\nWe have the following easy observation for coisotropic submanifolds of a Nambu-Poisson manifold.\n\\begin{prop}\n Let $(M, \\Pi)$ be a Nambu-Poisson manifold of order $n$ and $C$ be a closed embedded submanifold of $M$. Let $\\mathcal{I} (C) = \\{ f \\in C^\\infty(M)\\big| f|_C \\equiv 0 \\}$ denote the vanishing ideal of $C$. Then the followings are equivalent:\n\\begin{enumerate}\n\\item $C$ is a coisotropic submanifold;\n\\item $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra;\n\\item for every $f_1, \\ldots, f_{n-1} \\in \\mathcal{I} (C)$, the Hamiltonian vector field $X_{f_1...f_{n-1}}$ is tangent to $C$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n (1) $\\Rightarrow$ (2) Let $f_1, \\ldots, f_n \\in \\mathcal{I} (C) $.\nThen for any $x \\in C$, $d_xf_i \\in (TC)^0_x$, for all $i=1, \\ldots, n.$ Now since $C$ is a coisotropic submanifold, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}(x) = \\Pi_x (d_xf_1, \\ldots, d_xf_n) = 0,~ \\forall x \\in C.\n\\end{align*}\nHence $\\{f_1, \\ldots, f_n\\} \\in \\mathcal{I} (C)$. Therefore $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra.\n\n(2) $\\Rightarrow$ (3) Let $f_1, \\ldots, f_{n-1} \\in \\mathcal{I} (C)$ and $x \\in C$.\nLet $\\alpha \\in (TC)^0_x.$ Then there exists a function $g$ vanishing on $C$ such that $d_xg = \\alpha$. Since $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_{n-1}, g \\}(x) = 0.\n\\end{align*}\nThus,\n\\begin{align*}\n X_{f_1\\ldots f_{n-1}}\\big|_x (\\alpha) = X_{f_1\\ldots f_{n-1}}\\big|_x (d_xg) = \\{f_1, \\ldots, f_{n-1}, g \\}(x) = 0\n\\end{align*}\nand consequently, $X_{f_1\\ldots f_{n-1}}$ is tangent to $C$.\n\n(3) $\\Rightarrow$ (1) Let $x \\in C$ and $\\alpha_1, \\ldots, \\alpha_n \\in (TC)^0_x.$ Then there exist functions $f_1, \\ldots, f_n \\in \\mathcal{I}(C)$ such that\n$d_xf_i = \\alpha_i,$ $\\forall i = 1, \\ldots, n.$ Therefore,\n\\begin{align*}\n \\Pi_x (\\alpha_1, \\ldots, \\alpha_n) = \\Pi_x (d_xf_1, \\ldots, d_xf_n) = X_{f_1...f_{n-1}} \\big|_x (d_xf_n) = 0.\n\\end{align*}\nHence $C$ is a coisotropic submanifold of $M$.\n\\end{proof}\n\n\n\n\\begin{prop}\n Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two Nambu-Poisson manifolds of same order $n$ and $C \\hookrightarrow N$ be a coisotropic submanifold of $N$ with respect to $\\Pi_N$. If $\\phi: M \\rightarrow N$\nis a Nambu-Poisson map transverse to $C$, then $\\phi^{-1}(C)$ is a coisotropic submanifold of $M$ with respect to $\\Pi_M$ (the result holds true for manifolds with $n$-vector fields such that\n$\\phi_* \\Pi_M = \\Pi_N$).\n\\end{prop}\n\n\n\\begin{proof}\nSince $\\phi$ is transverse to $C$, therefore $\\phi^{-1}(C)$ is a submanifold of $M$. Moreover\n\\begin{align*}\n T(\\phi^{-1}(C)) = (T \\phi)^{-1} TC.\n\\end{align*}\nTherefore $T(\\phi^{-1}(C))^0 = (T \\phi)^{*} (TC)^0$. Observe that\n\\begin{align*}\n T\\phi (\\Pi_M^{\\sharp} \\big( {\\bigwedge}^{n-1} T(\\phi^{-1}(C))^0 \\big)) =& T\\phi ( \\Pi_M^{\\sharp} \\big((T \\phi)^{*} {\\bigwedge}^{n-1} (TC)^0 \\big))\\\\\n=& \\Pi_N^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0)\\\\\n\\subseteq & TC.\n\\end{align*}\nThus,\n\\begin{align*}\n \\Pi_M^{\\sharp} \\big( {\\bigwedge}^{n-1} T(\\phi^{-1}(C))^0 \\big) \\subseteq (T\\phi)^{-1} TC = T(\\phi^{-1}(C))\n\\end{align*}\nand hence $\\phi^{-1}(C)$ is coisotropic with respect to $\\Pi_M.$\n\\end{proof}\n\n\\begin{prop}\\label{image-coiso}\n Let $\\phi: (M, \\Pi_M) \\rightarrow (N, \\Pi_N)$ be a Nambu Poisson map between two Nambu-Poisson manifolds $(M, \\Pi_M)$ and $(N, \\Pi_N)$ and $C\\hookrightarrow M$ be a coisotropic submanifold\nof $M$. Assume that $\\phi(C)$ is a submanifold of $N$. Then $\\phi(C)$ is a coisotropic submanifold of $N$ (the result holds true for manifolds with $n$-vector fields such that $\\phi_{*}\\Pi_M = \\Pi_N$).\n\\end{prop}\n\n\\begin{proof}\nWe have $T(\\phi(C)) \\supseteq T\\phi (TC)$ and $(T\\phi)^* (T(\\phi(C)))^0 \\subseteq (TC)^0$. Therefore,\n\\begin{align*}\n \\Pi_N^{\\sharp} ({\\bigwedge}^{n-1} T(\\phi(C))^0) =& T\\phi (\\Pi_M^{\\sharp} ((T\\phi)^* {\\bigwedge}^{n-1} T(\\phi(C))^0)) \\hspace*{1cm}(\\text{since ~} \\phi \\text{~ is a N-P map})\\\\\n\\subseteq & T\\phi (\\Pi_M^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0))\\\\\n\\subseteq & T\\phi (TC) \\hspace*{1cm}(\\text{since~} C \\hookrightarrow M \\text{~ is coisotropic})\\\\\n\\subseteq & T(\\phi(C))\n\\end{align*}\nwhich shows that $\\phi(C)$ is a coisotropic submanifold of $N$.\n\\end{proof}\n\nUsing the terminology of coisotropic submanifold with respect to any multivector field allows us to extend the results of Weinstein \\cite{wein} from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field.\n\n\\begin{prop}\\label{nambu map-coiso}\n Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields and $\\phi: M \\rightarrow N$ be a smooth map. Then $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map, that is $\\phi_* \\Pi_M = \\Pi_N$ if and only if its graph\n\\begin{align*}\n \\text{Gr}(\\phi) = \\{(m, \\phi(m))| m \\in M \\}\n\\end{align*}\nis a coisotropic submanifold of $M \\times N$ with respect to $\\Pi_M \\oplus (-1)^{n-1} {\\Pi}_N$.\n\\end{prop}\n\n\\begin{proof}\n Let $C = \\text{Gr}(\\phi) \\subset M \\times N$. Then $C$ is a closed embedded submanifold of $M \\times N.$\nNote that, a tangent vector to the graph consist of a pair $(v_m, (T\\phi)(v_m))$, where $m \\in M$, $v_m \\in T_mM$. Therefore,\n$(TC)^0$ consists of a pair of covectors $(-(T\\phi)^*\\psi, \\psi)$, where $\\psi \\in T_{\\phi(m)}^*N.$\nTherefore, Gr($\\phi)$ is a coisotropic submanifold of $M \\times N$ with respect to $\\Pi_M \\oplus (-1)^{n-1} {\\Pi}_N$ if and only if\n$(\\Pi_M^{\\sharp} \\times (-1)^{n-1} \\Pi_N^{\\sharp})$ maps $(-(T\\phi)^*\\psi_1, \\psi_1) \\wedge \\cdots \\wedge (-(T\\phi)^*\\psi_{n-1}, \\psi_{n-1}) $ into $TC$,\nfor all $\\psi_1, \\ldots, \\psi_{n-1} \\in T^*_{\\phi(m)}N$ and $m \\in M$. In other words,\n\\begin{align*}\n(T\\phi) \\bigg(\\Pi_M^{\\sharp} (- (T\\phi)^*\\psi_1, \\ldots, - (T\\phi)^*\\psi_{n-1})\\bigg) = (-1)^{n-1} {\\Pi}_N^{\\sharp} (\\psi_1, \\ldots, \\psi_{n-1})\n\\end{align*}\nthat is,\n\\begin{align*}\n(T\\phi) \\bigg(\\Pi_M^{\\sharp} ( (T\\phi)^*\\psi_1, \\ldots, (T\\phi)^*\\psi_{n-1})\\bigg) = {\\Pi}_N^{\\sharp} (\\psi_1, \\ldots, \\psi_{n-1}).\n\\end{align*}\nThis is equivalent to the condition that $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map.\n\\end{proof}\n\n\\begin{defn}\n Let $(M, \\Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\\phi: M \\rightarrow N$ be a smooth surjective map. If there exist a Nambu-Poisson structure $\\Pi_N$ (of order $n$) on $N$\nwhich makes $\\phi$ into a Nambu-Poisson map, then $\\Pi_N$ is called the Nambu-Poisson structure {\\it coinduced} by the mapping $\\phi$.\n\\end{defn}\n\nThe following is a characterization of coinduced Nambu-Poisson structure.\n \n\\begin{prop}\\label{coinduced}\n Let $(M, \\Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\\phi: M \\rightarrow N$ be a smooth surjective map from $M$ to some manifold $N$. Then $N$ has a Nambu-Poisson structure coinduced by $\\phi$ if and only if for all $f_1, \\ldots, f_{n} \\in C^\\infty(N)$, the function\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the fibers of $\\phi$.\n\\end{prop}\n\n\n\\begin{proof}\n Let $f_1, \\ldots, f_n \\in C^\\infty(N)$. If the function $\\{ \\phi^*f_1, \\ldots, \\phi^*f_n \\}_M$ is constant along the $\\phi$-fibers, then there exists a function on $N$, which we denote by $\\{f_1, \\ldots, f_n\\}_N$ such that\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = \\phi^* \\{f_1, \\ldots, f_n\\}_N $. Clearly this bracket defines a coinduced Nambu-Poisson structure on $N$.\n\nConversely, suppose that there is a Nambu-Poisson bracket $\\{~, \\ldots, ~\\}_N$ on $N$ coinduced by $\\phi$. Then for any $y \\in N,$\n\\begin{align*}\n\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M(\\phi^{-1}\\{y\\}) =& (\\phi^* \\{f_1, \\ldots, f_n\\}_N )(\\phi^{-1}y)\\\\\n=& \\{f_1, \\ldots, f_n\\}_N (y)\n\\end{align*}\nproving $\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the $\\phi$-fibers.\n\\end{proof}\n\n\\begin{remark}\\label{rem-coinduced}\n Let $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a smooth map. Then there exists an $n$-vector field\n$\\Pi_N$ on $N$ such that $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map if and only if for all $f_1, \\ldots, f_n \\in C^\\infty(N)$, the function\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the fibers of $\\phi$.\n\\end{remark}\n\n\n\\begin{prop}\nLet $(M, \\Pi_M)$ be a Nambu-Poisson manifold and $\\phi : M \\rightarrow N$ be a surjective submersion with connected fibers. Let $\\text{ker} \\hspace*{0.1cm} \\phi_*(m)$ is spanned by local Hamiltonian vector fields (that is, $\\text{ker} \\hspace*{0.1cm} \\phi_*(m) \\subset \\mathcal{D}_mM $), for all $m \\in M$. Then \n$N$ has a Nambu-Poisson structure coinduced by $\\phi.$\n\\end{prop}\n\n\n\\begin{proof}\n Since $\\phi$ is a submersion, the fibers of $\\phi$ are submanifolds of $M$. Then for $y \\in N,$ $\\phi^{-1}(\\{y\\}) = C$ is a submanifold of $M$. Let $g_1, \\ldots, g_{n-1}$ be locally defined functions on $M$ such that $X_{g_1...g_{n-1}} \\in \\text{ker} \\hspace*{0.1cm} \\phi_{*}$. Let $f_1, \\ldots, f_n \\in C^\\infty(N).$ To prove that $\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}$\nis constant on the fibers, it is enough to prove that\n\\begin{align*}\n X_{g_1...g_{n-1}} \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\} = 0.\n\\end{align*}\nNote that\n\\begin{align*}\n X_{g_1...g_{n-1}} \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\} = \\sum_{k=1}^n \\{\\phi^*f_1, \\ldots, X_{g_1...g_{n-1}} (\\phi^*f_k), \\ldots, \\phi^*f_n\\}\n\\end{align*}\nand the functions $\\phi^*f_i$ are constant along the fibers. Hence by the Proposition \\ref{coinduced}, there exists a coinduced Nambu-Poisson structure on $N.$\n\\end{proof}\n\n\\begin{prop}\\label{coinduced-coiso}\nLet $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\\Pi_N$ such that\n$\\phi$ is a $(\\Pi_M, \\Pi_N)$-map if and only if\n $R(\\phi) = \\{(x,y) \\in M \\times M|\\phi(x)=\\phi(y)\\}$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$.\n\\end{prop}\n\n\\begin{proof}\nNote that $R(\\phi) = (\\phi \\times \\phi)^{-1} (\\Delta_N)$, where\n$\\Delta_N$ is the diagonal of $N \\times N$. Since $\\phi$ is surjective submersion $R(\\phi)$ is a submanifold of $M \\times M$.\nMoreover, for $(x,y) \\in R(\\phi)$\n\\begin{align*}\n T_{(x,y)} (R(\\phi)) = \\{(X, Y)\\in T_xM \\times T_yM | (T\\phi)_x (X) = (T\\phi)_y (Y)\\}.\n\\end{align*}\nTherefore, $T(R(\\phi))^0$ consists of covectors $(-(T\\phi)_x^*\\psi, (T\\phi)_y^*\\psi)$, where $\\psi \\in T_{\\phi(x)}^*N$.\n\nThus, $R(\\phi)$ be a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$\nif and only if for all $\\psi_1, \\ldots, \\psi_{n-1} \\in T_{\\phi(x)}^*N$ and $(x, y) \\in R(\\phi),$ $\\Pi_M^{\\sharp} \\oplus (-1)^{n-1}{\\Pi}_M^{\\sharp}$ maps \n$$(-(T\\phi)_x^*\\psi_1, (T\\phi)_y^*\\psi_1) \\wedge \\cdots \\wedge (-(T\\phi)_x^*\\psi_{n-1}, (T\\phi)_y^*\\psi_{n-1})$$ into $T(R(\\phi)).$ \nThat is\n$$(T\\phi)_x \\Pi_M^{\\sharp} (-(T\\phi)_x^*\\psi_1, \\ldots, -(T\\phi)_x^*\\psi_{n-1} )= (-1)^{n-1} (T\\phi)_y \\Pi_M^{\\sharp} ((T\\phi)_y^*\\psi_1, \\ldots, (T\\phi)_y^*\\psi_{n-1}),$$\nor equivalently,\n\\begin{align} \\label{firsteqn}\n (T\\phi)_x \\Pi_M^{\\sharp} ((T\\phi)_x^*\\psi_1, \\ldots, (T\\phi)_x^*\\psi_{n-1} ) = (T\\phi)_y \\Pi_M^{\\sharp} ((T\\phi)_y^*\\psi_1, \\ldots, (T\\phi)_y^*\\psi_{n-1})\n\\end{align}\nholds. Let $f_1, \\ldots ,f_n \\in C^\\infty(N)$ and $x \\in M.$ Then\n\\begin{align*}\n\\{\\phi^*f_1, \\ldots , \\phi^*f_n\\}_M (x) =& \\langle \\Pi_M^{\\sharp} (d_x (\\phi^*f_1) \\wedge \\cdots \\wedge d_x (\\phi^*f_{n-1})), d_x (\\phi^*f_n) \\rangle\\\\\n =& \\langle \\Pi_M^{\\sharp} \\big((T\\phi)_x^* \\psi_1 \\wedge \\cdots \\wedge (T\\phi)_x^* \\psi_{n-1}\\big), (T\\phi)_x^* \\psi_n \\rangle\\\\\n =& \\langle (T\\phi)_x \\Pi_M^{\\sharp} \\big((T\\phi)_x^* \\psi_1 \\wedge \\cdots \\wedge (T\\phi)_x^* \\psi_{n-1}\\big), \\psi_n \\rangle\n\\end{align*}\nwhere $\\psi_i = d_{\\phi(x)}f_i = d_{\\phi(y)}f_i\\in T_{\\phi(x)}^*N$, for all $1 \\leqslant i \\leqslant n.$ It follows from the Equation (\\ref{firsteqn}) that the function $\\{\\phi^*f_1, \\ldots , \\phi^*f_n\\}_M$ is constant along the $\\phi$-fibers if and only if $R(\\phi)$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M^{\\sharp} \\oplus (-1)^{n-1}{\\Pi}_M^{\\sharp}$. Hence the result follows by the Remark \\ref{rem-coinduced}. The uniqueness follows from the surjectivity of $\\phi.$\n\\end{proof}\n\n\\section{Nambu-Lie groupoids}\nIn this section, we recall the definition of multiplicative multivector fields on Lie groupoid (\\cite{pont-geng-xu}) and define Nambu-Lie groupoid (of order $n$) as a Lie groupoid with a multiplicative\n$n$-vector field which is also a Nambu-Poisson tensor.\n\n\\begin{defn}\nLet $G \\rightrightarrows M$ be a Lie groupoid and $\\Pi \\in \\mathcal{X}^n(G)$ be an $n$-vector field on $G$. Then $\\Pi$ is called {\\it multiplicative} if the graph of the groupoid multiplication\n\\begin{align*}\n \\{ (g, h, gh) \\in G \\times G \\times G | ~ \\beta(g) = \\alpha(h)\\}\n\\end{align*}\nis a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$.\n\\end{defn}\n \nThen we have the following characterization of multiplicative multivector fields \\cite{pont-geng-xu}:\n\\begin{thm} \\label{multiplicative}\n Let $G \\rightrightarrows M$ be a Lie groupoid and $\\Pi \\in \\mathcal{X}^n (G)$ be an $n$-vector field on $G$. Then $\\Pi$ is multiplicative if and only if the following conditions are satisfied.\n\\begin{enumerate}\n \\item $\\Pi$ is an affine tensor. In other words\n\\begin{align*}\n \\Pi(gh) = (r_{\\mathcal{H}})_* \\Pi(g) + (l_{\\mathcal{G}})_* \\Pi(h) - (r_{\\mathcal{H}})_* (l_{\\mathcal{G}})_* \\Pi(u)\n\\end{align*}\nwhere $u = \\beta(g) = \\alpha(h)$ and $\\mathcal{G}, \\mathcal{H}$ are (local) bisections through the points $g, h$ respectively.\n \\item $M$ is a coisotropic submanifold of $G$ with respect to $\\Pi$.\n \\item For all $g \\in G$, $\\alpha_* \\Pi(g)$ and $\\beta_* \\Pi(g)$ depend only on the base points $\\alpha(g)$ and $\\beta(g)$ respectively.\n \\item For all $f, f' \\in C^\\infty(M)$, the $(n-2)$-vector field $\\iota_{d(\\alpha^*f) \\wedge d(\\beta^*f')} \\Pi$ is zero. In other words,\n\\begin{align*}\n \\{~, \\ldots, \\alpha^*f, \\beta^*f' \\} = 0.\n\\end{align*}\n \\item For all $f_1, \\ldots, f_k \\in C^\\infty(M)$, $\\iota_{d(\\beta^*f_1) \\wedge \\cdots \\wedge d(\\beta^*f_k)} \\Pi$ is a left invariant $(n-k)$-vector field on $G$, $1 \\leqslant k < n.$\n\\end{enumerate}\n\\end{thm}\n\n\\begin{remark}\\label{lie-gp-multiplicative}\n Suppose $G$ be a Lie group considered as a Lie groupoid over a point. Then the conditions {\\it (3)} - {\\it (5)} of the Theorem \\ref{multiplicative} are satisfied automatically. The condition {\\it (2)} implies that $\\Pi (e) = 0$ (where $e$ is the identity element of the group), which together with condition {\\it (1)} implies\nthat $\\Pi$ satisfies the usual multiplicativity condition\n$$ \\Pi (gh) = (r_h)_* \\Pi(g) + (l_g)_* \\Pi(h) .$$\n\\end{remark}\n\n\n\n\\begin{defn}\\label{nambu-lie groupoid} \n A {\\it Nambu-Lie groupoid of order $n$} is a Lie groupoid $G \\rightrightarrows M$ with a multiplicative Nambu tensor $\\Pi \\in \\mathcal{X}^n(G)$ of order $n.$ \n\\end{defn}\nA Nambu Lie groupoid (of order $n$) will be denoted\nby $(G \\rightrightarrows M, \\Pi) $.\n\n\\begin{exam}\\label{exam-nlg}\n\\begin{enumerate}\n\\item Poisson groupoids \\cite{wein} are examples of Nambu-Lie groupoids with $n=2.$\n\\item Any Lie groupoid with zero Nambu structure is a Nambu-Lie groupoid.\n\\item Let $(G, \\Pi)$ be a Nambu-Lie group (of order $n$) \\cite{vais}. Thus $G$ is a Lie group equipped with a Nambu structure $\\Pi$ of order $n$ on $G$\nsuch that\n\\begin{align*}\n \\Pi(gh) = (r_h)_* \\Pi(g) + (l_g)_* \\Pi(h)\n\\end{align*}\nfor all $g, h \\in G$. Note that the right hand side of the above equality is equal to $m_* (\\Pi(g), \\Pi(h)),$ where $m_* : \\bigwedge^n T_{(g,h)} (G \\times G) \\rightarrow \\bigwedge^n T_{gh}G$\nis the map induced by the multiplication map $m : G \\times G \\rightarrow G$. Therefore,\n\\begin{align*}\n \\Pi(gh) = m_* (\\Pi(g), \\Pi(h)).\n\\end{align*}\nThus, the group multiplication map $m : G \\times G \\rightarrow G$ is a $(\\Pi \\oplus \\Pi, \\Pi)$-map. Therefore, by the Proposition \\ref{nambu map-coiso}, the graph of the group multiplication map is a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$. Hence $(G, \\Pi)$ is a Nambu-Lie groupoid over a point.\nConversely, if $(G, \\Pi)$ is a Nambu-Lie groupoid over a point, then the group multiplication map $m : G \\times G \\rightarrow G$ is a \n$(\\Pi \\oplus \\Pi, \\Pi)$-map. Hence $(G, \\Pi)$ is a Nambu-Lie group in the sense of \\cite{vais}. One can also see the equivalence between Nambu-Lie groupoid\nover a point and Nambu-Lie group by using Remark \\ref{lie-gp-multiplicative}.\n\\end{enumerate}\n\\end{exam}\n\nFor a Poisson groupoid the following facts are well known \\cite{wein}.\n\\begin{itemize}\n\\item The groupoid inversion map is a anti-Poisson map. \n\\item The Poisson structure on the total space induces a Poisson structure on the base such that the source map is a Poisson map and the target map is a anti-Poisson map. \n\\end{itemize}\n\nIn the next proposition we generalize the above facts to the Nambu-Poisson setting.\n\n\\begin{prop}\\label{inverse-basenambu}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid. Then\n\\begin{enumerate} \n\\item The inverse map $i : G \\rightarrow G$, $g \\mapsto g^{-1}$ is an anti-Nambu Poisson map.\n\\item There is a unique Nambu-Poisson structure on $M$ which we denote by $\\Pi_M$ for which $\\alpha$ is a Nambu-Poisson map and $\\beta$ is an anti Nambu-Poisson map.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(1) It is proved in \\cite{pont-geng-xu} that given a Lie groupoid $G \\rightrightarrows M$ with multiplicative $n$-vector field $\\Pi \\in \\mathcal{X}^n(G)$,\nthe groupoid inversion map $i : G \\rightarrow G$ satisfies\n\\begin{align*} \n i_* \\Pi = (-1)^{n-1} \\Pi.\n\\end{align*}\nHence the result follows as $\\Pi$ is a Nambu tensor.\n\n(2) Let $f_1, \\ldots, f_n \\in C^\\infty(M)$ be any functions on $M$. Then for any $g \\in G$, we have\n\\begin{align*}\n\\{\\alpha^* f_1, \\ldots, \\alpha^* f_n \\} (g) =& \\Pi (g) (d_g (\\alpha^* f_1), \\ldots, d_g (\\alpha^* f_n))\\\\\n =& \\Pi(g) (\\alpha^* (d_{\\alpha(g)} f_1), \\ldots, \\alpha^* (d_{\\alpha(g)} f_n))\\\\\n =& \\alpha_* \\Pi (g) (d_{\\alpha(g)} f_1, \\ldots, d_{\\alpha(g)} f_n).\n\\end{align*}\nSince $\\alpha_* \\Pi(g)$ depends only on the value of $\\alpha(g)$, it follows that the function $\\{ \\alpha^* f_1, \\ldots, \\alpha^* f_n \\}$ is constant on the $\\alpha$-fibers. Therefore, by the Proposition \\ref{coinduced}, there exists a Nambu-Poisson structure $\\Pi_M$ with the induced bracket denoted by $\\{ ~, \\ldots, ~\\}_M,$ on $M$\nfor which $\\alpha$ is a Nambu-Poisson map. Since $\\beta = \\alpha \\circ i$ and $i$ is anti Nambu-Poisson, therefore $\\beta$\nis an anti Nambu-Poisson map.\n\\end{proof}\n\n\\begin{remark}\nConsider the map $(\\alpha,\\beta): G \\rightarrow M \\times M.$\nSince we have $\\alpha_{*} \\Pi = \\Pi_M$ and $\\beta_{*}\\Pi = (-1)^{n-1} \\Pi_M$, using property {\\it (4)} of the Theorem \\ref{multiplicative} we obtain \n\\begin{align*}\n(\\alpha,\\beta)_{*} \\Pi = \\alpha_* \\Pi \\oplus \\beta_* \\Pi = \\Pi_M \\oplus (-1)^{n-1} \\Pi_M.\n\\end{align*}\n\\end{remark}\n\n\n\\begin{prop}\n Let $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid. If the orbit space $M\/\\sim$ is a smooth manifold, then $M\/\\sim$ carries a Nambu-Poisson structure such that the projection $q: M \\rightarrow M\/\\sim $ is a Nambu-Poisson map.\n\\end{prop}\n\n\\begin{proof}\nLet $\\Pi_M$ be the induced Nambu structure on the base $M$. For the projection map $q: M \\rightarrow M\/\\sim $, we have\n\\begin{align*}\n R(q) =& \\{(x,y) \\in M \\times M| q(x) = q(y)\\}\\\\\n=& \\{(\\alpha(g), \\beta(g))| g \\in G\\}\\\\\n=& (\\alpha,\\beta)(G).\n\\end{align*}\nConsider $G$ as a coisotropic submanifold of $G$ with respect to $\\Pi$ and also consider the map $(\\alpha, \\beta) : G \\rightarrow M \\times M$. By the above Remark we have $(\\alpha, \\beta)_* \\Pi = \\Pi_M \\oplus (-1)^{n-1} \\Pi_M.$ Therefore, by the Proposition \\ref{image-coiso}, $R(q) = (\\alpha,\\beta)(G)$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}\\Pi_M.$ Hence the result follows from the Proposition \\ref{coinduced-coiso}.\n\\end{proof}\n\n\n\n\\section{Infinitesimal form of Nambu-Lie groupoid}\n\nThe aim of this section, is to study the infinitesimal form of a\nNambu-Lie groupoid. We show that if $(G \\rightrightarrows M, \\Pi)$ is a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$, then\n$(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ introduced in \\cite{bas-bas-das-muk}. Before proceeding further, let us briefly recall from \\cite{bas-bas-das-muk} the notion of a weak Lie-Filippov bialgebroid.\n\nLie bialgebroids are generalization of both Poisson manifolds and Lie bialgebras. Recall that a Lie bialgebroid, introduced by Mackenzie and Xu \\cite{mac-xu} is also the infinitesimal form of a Poisson groupoid. It is defined as a pair $(A, A^\\ast)$ of Lie algebroids in duality, where the Lie bracket of $A$ satisfies the following compatibility condition expressed in terms of the differential $d_\\ast$ on $\\Gamma (\\bigwedge^\\bullet A)$ \n$$d_\\ast[X, Y] = [d_\\ast X, Y] + [X , d_\\ast Y],$$ \nfor all $X,~ Y \\in \\Gamma A.$ \n\nWe note that if $M$ is a Poisson manifold then the Lie algebroid structures on $TM$ and $T^\\ast M$ form a Lie bialgebroid. On the other hand, it is well known \\cite{kosmann,mac-xu} that if $(A, A^\\ast)$ is a Lie bialgebroid over a smooth manifold $M$ then there is a canonical Poisson structure on the base manifold $M$.\n\nThus it is natural to ask the following question which was posed in \\cite{bas-bas-das-muk}:\\\\ \n{\\it Does there exist some notion of bialgebroid associated to a Nambu-Poisson manifold of order $n > 2$ }? \n\nTo answer this question, the authors \\cite{bas-bas-das-muk} introduced the notion of weak Lie-Filippov bialgebroid.\n\nIt is well known \\cite{gra-mar, vais} that for a Nambu-Poisson manifold $M$ of order $n \\geqslant 2$, the space $\\Omega^1(M)$ of $1$-forms admits an $n$-ary bracket, called {\\it Nambu-form bracket}, such that the bracket satisfies almost all the properties of an $n$-Lie algebra (also known as Filippov algebra of order $n$) bracket \\cite{fil} except that the Fundamental identity is satisfied only in a restricted sense as described below.\n\nLet $(M, \\{~, \\ldots, ~\\})$ be a Nambu-Poisson manifold of order $n$ with associated Nambu-Poisson tensor $\\Pi$. Then one can define the Nambu form-bracket on the space of $1$-forms\n\\begin{align*}\n [~, \\ldots, ~]: \\Omega^1(M) \\times \\cdots \\times \\Omega^1(M) \\rightarrow \\Omega^1(M)\n\\end{align*}\n by the following\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_n] = \\sum_{k=1}^n (-1)^{n-k} \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge \\alpha_n)} \\alpha_k - (n-1) d (\\Pi(\\alpha_1, \\ldots, \\alpha_n))\n\\end{align*}\n\\begin{align}\\label{nambu-bracket}\n\\hspace*{1cm} = d (\\Pi(\\alpha_1, \\ldots, \\alpha_n)) + \\sum_{k=1}^n (-1)^{n-k} \\iota_{\\Pi^{\\sharp}(\\alpha_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge \\alpha_n)} d\\alpha_k\n\\end{align}\nfor $\\alpha_i \\in \\Omega^1(M), i=1,\\ldots ,n.$ Here $\\hat{\\alpha}_k$ in a monomial $\\alpha_1 \\wedge \\cdots \\wedge \\hat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n$ means that the symbol $\\alpha_k$ is missing in the monomial. The above bracket satisfies the following properties (\\cite{vais}).\n\\begin{enumerate} \n\\item The bracket is skew-symmetric.\n\\item $[df_1, \\ldots, df_n] = d \\{f_1, \\ldots, f_n \\}$.\n\\item $[\\alpha_1,\\ldots, \\alpha_{n-1}, f \\alpha_n] = f [\\alpha_1,\\ldots, \\alpha_{n-1}, \\alpha_n] + \\Pi ^{\\sharp} (\\alpha_1 \\wedge\\cdots \\wedge \\alpha_{n-1})(f) \\alpha_n$.\n\\item The bracket satisfies the Fundamental identity\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_{n-1}, [\\beta_1, \\ldots, \\beta_n]] = \\sum_{k=1}^n [\\beta_1, \\ldots , \\beta_{k-1},[\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta_k], \\ldots, \\beta_n]\n\\end{align*}\nwhenever the $1$-forms $\\alpha_i \\in \\Omega^1(M)$ are closed, $1 \\leqslant i \\leqslant n-1$ and for any $\\beta_j$.\n\\item $[\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}), \\Pi^{\\sharp}(\\beta_1 \\wedge \\cdots \\wedge \\beta_{n-1})]\\\\ \n= \\sum_{k=1}^{n-1} \\Pi^{\\sharp} (\\beta_1 \\wedge\\cdots \\wedge [\\alpha_1,\\ldots , \\alpha_{n-1}, \\beta_k] \\wedge \\cdots \\wedge \\beta_{n-1})$\\\\\nfor closed $1$-forms $\\alpha_i \\in \\Omega^1 (M)$ and for any $1$-forms $\\beta_j$. \n\\end{enumerate}\n\n\nThe Nambu-form bracket on $\\Omega^1(M)$, together with the usual Lie algebroid structure on $TM$ yields an example of a notion called a {\\it weak Lie-Filippov algebroid pair of order $n$}, $n>2,$ on a smooth vector bundle (cf. Definition $5.5$, \\cite{bas-bas-das-muk}). \n\nIn order to classify such structures, the authors formulate a notion of {\\it Nambu-Gerstenhaber algebra of order $n$}. It turns out, weak-Lie-Filippov algebroid pair structures of order $n$, $n>2$, on a smooth vector bundle $A$ over $M$, are in bijective correspondence with Nambu-Gerstenhaber brackets of order $n$ on the graded commutative, associative algebra $\\Gamma \\bigwedge^\\bullet A^*$ of multisections of $A^*$, where $A^*$ is the dual bundle (cf. Definition $5.7$, Theorem $5.8$, \\cite{bas-bas-das-muk}).\n\nMoreover, for a Nambu-Poisson manifold $M$ of order $n > 2$, the Nambu-Gerstenhaber bracket on $\\Omega^\\bullet(M)$, extending the Nambu-form bracket on $\\Omega^1(M)$ satisfies certain suitable compatibility condition similar to the compatibility condition of a Lie bialgebroid. This motivates the authors to introduce the notion of a {\\it weak Lie-Filippov bialgebroid structure} of order $n$ on a smooth vector bundle. \n\n\n\\begin{defn}\\label{lie-fill-defn}\nA {\\it weak Lie-Filippov bialgebroid of order $n>2$} over a smooth manifold $M$ consists of a pair $(A, A^*)$, where $A$ is a smooth vector bundle over $M$ with dual bundle $A^*$ satisfying the following properties:\n\\begin{enumerate}\n\\item $A$ is a Lie algebroid with $d_A$ being the differential of the Lie algebroid cohomology of $A$ with trivial representation;\n\\item the space of smooth sections $\\Gamma A^*$ admits a skew-symmetric $n$-ary bracket\n$$[~, \\ldots ,~]: {\\Gamma A^* \\times \\cdots \\times \\Gamma A^*} \\longrightarrow \\Gamma A^*$$\nsatisfying \n$$[\\alpha_1, \\ldots , \\alpha_{n-1}, [\\beta_1, \\ldots , \\beta_n]] = \\sum_{k=1}^n [\\beta_1, \\ldots , \\beta_{k-1}, [\\alpha_1, \\ldots , \\alpha_{n-1}, \\beta_k], \\ldots ,\\beta_n]$$\nfor all $d_A$-closed sections $\\alpha_i \\in \\Gamma A^*,~ 1 \\leqslant i \\leqslant n-1$ and for any sections $\\beta_j \\in \\Gamma A^*,~ 1\\leqslant j\\leqslant n;$\n\\item there exists a vector bundle map $\\rho : \\bigwedge^{n-1}A^* \\longrightarrow TM$, called the {\\it anchor} of the pair $(A, A^*)$, such that the identity\n$$ [\\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}), \\rho (\\beta_1 \\wedge \\cdots \\wedge \\beta_{n-1})] = \\sum_{k=1}^{n-1}\\rho (\\beta_1 \\wedge \\cdots \\wedge [\\alpha_1, \\ldots , \\alpha_{n-1}, \\beta_k] \\wedge \\cdots \\wedge \\beta_{n-1})$$\nholds for all $d_A$-closed sections $\\alpha_i \\in \\Gamma A^*,~ 1\\leqslant i \\leqslant n-1$ and for any sections $\\beta_j \\in \\Gamma A^*,~ 1\\leqslant j\\leqslant n-1;$\n\\item for all sections $\\alpha_i \\in \\Gamma A^*,~ 1\\leqslant i \\leqslant n$ and any $f \\in C^\\infty(M)$,\n$$[\\alpha_1, \\ldots , \\alpha_{n-1}, f\\alpha_n] = f [\\alpha_1, \\ldots , \\alpha_{n-1}, \\alpha_n] + \\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})(f)\\alpha_n$$ holds;\n\\item the following compatibility condition holds:\n$$ d_A[\\alpha_1, \\ldots , \\alpha_n] = \\sum_{k=1}^n [\\alpha_1, \\ldots , d_A\\alpha_k, \\ldots , \\alpha_n],$$ for any $\\alpha_i \\in \\Gamma A^\\ast,$ $1 \\leqslant i \\leqslant n$, where the bracket $[~, \\ldots , ~]$ on the right hand side is the graded extension of the bracket on $\\Gamma A^\\ast$.\n\\end{enumerate}\n\\end{defn}\n\nA weak Lie-Filippov bialgebroid (of order $n$) over $M$ is denoted by $(A, A^*)$ when all the structures are understood. A Lie bialgebroid is a Lie-Filippov\nbialgebroid of order $2$ such that the conditions (2) and (3) of the above definition has no restriction on $\\alpha$.\n\nIn \\cite{bas-bas-das-muk}, the authors have shown that for a Nambu-Poisson manifold $M$ of order $n > 2$, the pair $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid of order $n$ (cf. Corollary $6.3$, \\cite{bas-bas-das-muk}). It is also proved that if $(G, \\Pi)$ is a Nambu-Lie group \\cite{vais} of order $n$ with its Lie algebra $\\mathfrak{g},$ then $(\\mathfrak{g}, \\mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebroid of order $n$ over a Point.\n\nIt is known that the base of a Lie bialgebroid carries a natural Poisson structure. In \\cite{bas-bas-das-muk} it has been extended to the Nambu-Poisson set up.\n\n\\begin{prop} \\label{wlfb-base-nambu}(\\cite{bas-bas-das-muk})\nLet $(A, A^*)$ be a weak Lie-Filippov bialgebroid (of order n) over $M$. Then the bracket\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}_{(A, A^*)} := \\rho (d_A f_1 \\wedge \\cdots \\wedge d_A f_{n-1}) f_n\n\\end{align*}\ndefines a Nambu-Poisson structure of order n on $M$.\n\\end{prop}\n\n\nIt is known that, given a coisotropic submanifold $C$ of a Poisson manifold $M$, the conormal bundle $(TC)^0 \\rightarrow C$ is a Lie subalgebroid\nof the cotangent Lie algebroid $T^*M$ \\cite{wein}. If $M$ is a Nambu-Poisson manifold of order $n$ ($n \\geqslant 3$), the cotangent bundle $T^*M$ is not a Filippov\nalgebroid. However we have the following useful result.\n\n\\begin{prop}\\label{coiso-n-bracket}\n Let $C$ be a closed embedded coisotropic submanifold of a Nambu-Poisson manifold $(M, \\Pi)$ of order $n$. Then\n \\begin{enumerate}\n\\item the bundle map $\\Pi^{\\sharp} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ maps $\\bigwedge^{n-1} (TC)^0$ to $TC$;\n\\item the Nambu-form bracket on the space of $1$-forms $\\Omega^1(M)$ can be restricted to the sections of the conormal bundle $ (TC)^0 \\rightarrow C$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThe assertion $(1)$ follows from the definition of coisotropic submanifold. To prove $(2)$, let $\\alpha_1, \\ldots ,\\alpha_n \\in \\Gamma (TC)^0.$ We extend them to $1$-forms on $M$, which we denote by the same notation. Let $X \\in \\mathcal{X}^1(M)$ be such that $X\\big|_C$ is tangent to $C$. From the definition\nof Nambu-form bracket on $1$-forms, we have\n\\begin{align*}\n\\langle [\\alpha_1, \\ldots, \\alpha_n], X \\rangle = \\sum_{k=1}^n (-1)^{n-k} \\langle \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n)} \\alpha_k, X \\rangle\n- (n-1) \\langle d (\\Pi(\\alpha_1,\\ldots, \\alpha_n)), X \\rangle. \n\\end{align*}\nObserve that\n\\begin{align*}\n \\langle \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n)} \\alpha_k, X \\rangle \n=& \\mathcal{L}_{ \\Pi^\\sharp (\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n )} \\langle \\alpha_k, X \\rangle \\\\\n&- \\langle \\alpha_k, [\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n), X]\\rangle .\n\\end{align*}\nThis is zero on $C$, because, \n\\begin{itemize}\n\\item $\\langle \\alpha_k, X\\rangle$ is zero on $C$; \n\\item $\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha_k} \\wedge \\cdots \\wedge \\alpha_n)$ and $X$ are both tangent to $C$ and hence their Lie bracket is also tangent to C. Thus its pairing with $\\alpha_k$ vanish on $C$.\n\\end{itemize}\nNote that $\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})\\big|_C$ is tangent to $C$ and $\\alpha_n \\big|_C \\in (TC)^0.$ As a consequence, the function\n\\begin{align*}\n\\Pi (\\alpha_1,\\ldots, \\alpha_n) = \\langle \\alpha_n, \\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})\\rangle\n\\end{align*}\nis zero on $C.$ Therefore, the differential\n$d (\\Pi(\\alpha_1, \\ldots, \\alpha_n))$ restricted to $C$ is in $(TC)^0$, which in turn implies that the second term of the right hand side also vanish on $C$. Hence\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_n]\\big|_C \\in (TC)^0.\n\\end{align*}\nOne can check that the restriction to $C$ does not depend on the chosen extension. Hence it defines a bracket on the sections of the conormal bundle $(TC)^0 \\rightarrow C$.\n\\end{proof}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item Let $m_0 \\in M$ such that $\\Pi (m_0) = 0$. Then $\\{m_0\\}$ is a coisotropic submanifold of $M$. In this case, the conormal structure becomes $T_{m_0}^*M$, which is a Filippov algebra.\n\\item The Nambu structure of a Nambu-Lie group G vanishes at the identity element and therefore the dual $\\mathfrak{g}^*$ of the Lie algebra $\\mathfrak{g}$ of G has a Filippov algebra structure \\cite{vais}.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{remark}\\label{nlg-bracket-anchor}\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. By the Proposition \\ref{coiso-n-bracket}, we see that the space of sections of the conormal bundle $ A^*G = (TM)^0 \\rightarrow M$ admits a skew-symmetric $n$-bracket $[~,\\ldots,~]$ and there exists a bundle map\n$$\\rho := \\Pi^{\\sharp}\\big|_{{\\bigwedge}^{n-1} (TM)^0} : {\\bigwedge}^{n-1} A^*G = {\\bigwedge}^{n-1} (TM)^0 \\rightarrow TM ,$$ as $M$ is a coisotropic submanifold of $G$.\n\\end{remark}\n\n\n\n\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$.\n Let $f \\in C^\\infty(M).$ Then by part ($5$) of the Theorem \\ref{multiplicative}, $\\iota_{d(\\beta^*f)} \\Pi$ is a left invariant $(n-1)$-vector field\non $G$. Therefore, there exists an $(n-1)$-multisection $\\delta^0_\\Pi (f)\\in \\Gamma{\\bigwedge^{n-1}AG}$ of the Lie algebroid $AG$ such that\n\\begin{align*}\n \\iota_{d(\\beta^*f)} \\Pi = \\overleftarrow{\\delta^0_\\Pi (f)} .\n\\end{align*}\nThen we have the following result.\n\n\\begin{prop}\n Let $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ and $AG \\rightarrow M$ be its Lie algebroid. Then for any $ X \\in \\Gamma AG$,\n\\begin{align*}\n \\mathcal{L}_{\\overleftarrow{X}} \\Pi := [\\overleftarrow{X}, \\Pi]\n\\end{align*}\nis a left invariant $n$ vector field on $G$, where $\\overleftarrow{X}$ is the left invariant vector field on $G$ corresponding to $X$. Moreover $\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ corresponds to the $n$-multisection $- \\delta^1_\\Pi (X) \\in \\Gamma{\\bigwedge^n AG}$,\nthat is,\n\\begin{align*}\n \\mathcal{L}_{\\overleftarrow{X}} \\Pi = -\\overleftarrow{\\delta^1_\\Pi (X)}\n\\end{align*}\nwhere $\\delta^1_\\Pi (X) \\in \\Gamma{\\bigwedge^n AG}$ is given by\n\\begin{align*}\n\\delta^1_\\Pi (X) (\\alpha_1, \\ldots, \\alpha_n) = \\sum_{k=1}^{n} (-1)^{n-k} \\Pi^{\\sharp} (\\alpha_1 \\wedge \\cdots \\wedge \\hat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n) (X(\\alpha_k))\n- X([\\alpha_1,\\ldots, \\alpha_n])\n\\end{align*}\nfor $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma A^*G = \\Gamma (TM)^0.$\n\\end{prop}\n\n\\begin{proof}\nLet $\\mathcal{X}_t = \\text{exp} \\hspace*{0.05cm} tX$ be the one-parameter family of bisections generated by $X \\in \\Gamma AG.$ Let $g \\in G$ with $\\beta(g) = u$. Let\n$u_t = (\\text{exp} \\hspace*{0.05cm} tX) (u)$ be the integral curve of $\\overleftarrow{X}$ starting from $u$. If $\\mathcal{G}$ is any (local) bisection through $g$, then from the multiplicativity condition of $\\Pi$ (cf. Theorem \\ref{multiplicative}), we have\n\\begin{align*}\n \\Pi ( g u_t) = (r_{\\mathcal{X}_t})_* \\Pi(g) + (l_{\\mathcal{G}})_* \\Pi(u_t)- (r_{\\mathcal{X}_t})_* (l_{\\mathcal{G}})_* \\Pi(u).\n\\end{align*}\nTherefore,\n\\begin{align*}\n(r_{\\mathcal{X}^{-1}_t})_* \\Pi ( g u_t) - \\Pi(g) = (r_{\\mathcal{X}^{-1}_t})_* (l_{ \\mathcal{G}})_* \\Pi(u_t) - (l_{\\mathcal{G}})_* \\Pi (u).\n\\end{align*}\nTaking derivative at $t = 0$, one obtains\n\\begin{align*}\n (\\mathcal{L}_{\\overleftarrow{X}} \\Pi)(g) = (l_{\\mathcal{G}})_* ((\\mathcal{L}_{\\overleftarrow{X}} \\Pi)(u)).\n\\end{align*}\nTherefore, $\\mathcal{L}_{\\overleftarrow{X}} \\Pi $ is left invariant by the Proposition \\ref{left-invariant} and hence it corresponds to some $n$-multisection of $AG$. To show that\n$\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ corresponds to $- \\delta^1_\\Pi (X) \\in \\Gamma {\\bigwedge^n AG}$, we have to check that $\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ and $- \\overleftarrow{\\delta^1_\\Pi (X)}$ coincide on the unit space $M$ (both being left invariant). Since both of them are tangent to $\\alpha$-fibers, it is enough to show that they coincide on the conormal bundle $(TM)^0$.\nLet $\\alpha_1, \\ldots, \\alpha_n$ be any sections of $(TM)^0$ and $\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_n$ be their respective extensions to one forms on $G$. Observe that\n\\begin{align*}\n& (\\mathcal{L}_{\\overleftarrow{X}} \\Pi)\\big|_M (\\alpha_1, \\ldots, \\alpha_n)\\\\\n =& \\big[\\langle \\overleftarrow{X}, d (\\Pi(\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n)) \\rangle - \\sum_{k=1}^{n} \\Pi(\\tilde{\\alpha}_1 ,\\ldots, \\mathcal{L}_{\\overleftarrow{X}} \\tilde{\\alpha}_k,\\ldots, \\tilde{\\alpha}_n)\\big]\\big|_M\\\\\n =& \\big[\\langle \\overleftarrow{X}, [\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\langle \\overleftarrow{X}, \\iota_{\\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n)} d\\tilde{\\alpha}_k \\rangle\\\\\n & - \\sum_{k=1}^{n} (-1)^{n-k} \\langle \\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n), \\mathcal{L}_{\\overleftarrow{X}} \\tilde{\\alpha}_k \\rangle \\big]\\big|_M\\\\\n & \\hspace*{5cm} (\\text{from the Equation}~ (\\ref{nambu-bracket}))\\\\\n =& \\big[\\langle \\overleftarrow{X}, [\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\langle \\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n), d \\iota_{\\overleftarrow{X}} \\tilde {\\alpha}_k \\rangle \\big]\\big|_M\\\\\n & \\hspace*{5cm} (\\text{using Cartan formula})\\\\\n =& \\langle X, [\\alpha_1,\\ldots,\\alpha_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\Pi^{\\sharp} ({\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge {\\alpha_n}) (X (\\alpha_k))\\\\\n = & -\\delta^1_\\Pi (X) (\\alpha_1,\\ldots, \\alpha_n).\n\\end{align*}\n\\end{proof}\n\nTo make our notation simple, let us denote $\\delta^0_\\Pi, \\delta^1_\\Pi$ by the same symbol $\\delta_\\Pi$. We extend $\\delta_\\Pi$ to the graded algebra $\\Gamma{\\bigwedge^{\\bullet}A}$ of multisections of $AG$ by the following rule\n\\begin{align*}\n \\delta_\\Pi (P \\wedge Q) = \\delta_\\Pi (P) \\wedge Q + (-1)^{|P| (n-1)} P \\wedge \\delta_\\Pi (Q)\n\\end{align*}\nfor $P \\in \\Gamma{\\bigwedge^{|P|}A}, Q \\in \\Gamma{\\bigwedge^{|Q|}A}$. Then the operator\n\\begin{align*}\n \\delta_\\Pi : \\Gamma{{\\bigwedge}^kAG} \\rightarrow \\Gamma{{\\bigwedge}^{k+n -1} AG}\n\\end{align*}\nsatisfies\n\\begin{align*}\n \\delta_\\Pi ([P,Q]) = [ \\delta_\\Pi (P), Q] + (-1)^{(|P|-1)(n-1)} [P, \\delta_\\Pi (Q)].\n\\end{align*}\nNote that the operator $\\delta_\\Pi$ need not satisfy condition $\\delta_\\Pi \\circ \\delta_\\Pi = 0 .$\n\n\n\n\n\\medskip\n\nWe known that, Lie bialgebroids are infinitesimal form of Poisson groupoids. More precisely, given a Poisson groupoid $G \\rightrightarrows M$ with Lie algebroid $AG$, it is known that its dual bundle $A^*G$\nalso carries a Lie algebroid structure and $(AG, A^*G)$ forms a Lie bialgebroid. In the next theorem we show that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids.\n\n\n\n\n\n\n\\begin{thm}\\label{nambu-grpd-bialgbd}\nLet $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.\n\\end{thm}\n\n\n\\begin{proof}\nFrom the Remark \\ref{nlg-bracket-anchor}, we have the space of sections of the bundle $A^*G = (TM)^0 \\rightarrow M$ admits a skew-symmetric $n$-bracket\n$[~, \\ldots, ~]$ and there exists a bundle map $$\\rho : {\\bigwedge}^{n-1}A^*G \\rightarrow TM.$$\n\nLet $\\alpha_1, \\ldots, \\alpha_{n-1} \\in \\Gamma (TM)^0=\\Gamma{(A^*G)} $ with $d_A \\alpha_i = 0$, for all $i= 1, \\ldots ,n-1.$ Let $\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}$ be, respectively, their extensions to left invariant $1$-forms on $G$ such that $d \\tilde{\\alpha}_i = 0$, for all\n$i= 1, \\ldots ,n-1.$ Then the conditions $(2)$ and $(3)$ of the Definition \\ref{lie-fill-defn} of a weak Lie-Filippov algebroid pair follows from the weak Lie-Filippov bialgebroid structure $(TG, T^*G)$ (Note that $G$ is a Nambu-Poisson manifold). \n\nLet $f \\in C^\\infty(M).$ Then observe that\n\\begin{align*}\n [\\tilde {\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}, (\\beta^*f) \\tilde{\\alpha}_n ] = (\\beta^*f) [\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}, \\tilde{\\alpha}_n ] + \\Pi^{\\sharp}(\\tilde{\\alpha}_1 \\wedge \\cdots \\wedge \\tilde{\\alpha}_{n-1})(\\beta^*f)\\tilde{\\alpha}_n.\n\\end{align*}\nSince $(\\beta^*f) \\tilde{\\alpha}_n = \\widetilde{f \\alpha}_n$, by assertion $(2)$ of the Propostion \\ref{coiso-n-bracket}, we get\n\\begin{align*}\n[\\alpha_1,\\ldots, \\alpha_{n-1}, f \\alpha_n] = f [\\alpha_1, \\ldots, \\alpha_n] + \\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) (f) \\alpha_n\n\\end{align*}\nproving condition (4) of the Definition \\ref{lie-fill-defn}. Moreover the compatibility condition of the weak Lie-Filippov bialgebroid (condition $(5)$ of the Definition \\ref{lie-fill-defn}) follows from the observation that for any $\\alpha \\in \\Gamma{(A^*G)} = \\Gamma (TM)^0$ and any left invariant extension $\\tilde{\\alpha} \\in \\Omega^1(G)$, we have\n\\begin{align*}\n d_A \\alpha = (d \\tilde \\alpha)|_M.\n\\end{align*}\nThus, $(AG, A^*G)$ is a weak Lie-Filippov bialgebroid of order $n$.\n\\end{proof}\n\n\n\\begin{remark}If $(G, \\Pi)$ is a Nambu-Lie group with Lie algebra $\\mathfrak{g}$, the dual vector space $\\mathfrak{g}^*$\ncarries a Filippov algebra structure \\cite{vais}. Moreover the pair $(\\mathfrak{g} , \\mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebra (\\cite{bas-bas-das-muk, vais}).\nThe Lie-Filippov bialgebra $(\\mathfrak{g}, \\mathfrak{g}^*)$ is the infinitesimal form of the Nambu-Lie group $(G, \\Pi).$ \nA Lie-Filippov bialgebra $(\\mathfrak{g}, \\mathfrak{g}^*)$ can also be seen a Lie \nalgebra $\\mathfrak{g}$ together with a Filippov algebra structure on the dual vector space $\\mathfrak{g}^*$ such that the map $\\delta : \\mathfrak{g} \\rightarrow \\bigwedge^n \\mathfrak{g}$ dual\nto the Filippov bracket on $\\mathfrak{g}^*$, defines a $1$-cocycle of $\\mathfrak{g}$ with respect to the adjoint representation on $\\bigwedge^n \\mathfrak{g}$.\n\\end{remark}\n\nWe have seen that given a Nambu-Lie groupoid of order $n$, there is an induced Nambu-Poisson structure on the base manifold (cf. Proposition \\ref{inverse-basenambu}). On the other hand, \ngiven a weak Lie-Filippov bialgebroid, there is an induced Nambu-Poisson structure on the base (cf. Theorem \\ref{wlfb-base-nambu}).\nThe next proposition compares these Nambu-Poisson structures on the base induced from the Nambu Lie groupoid and its infinitesimal. \n\n\n\\begin{prop}\\label{compare-two-NP-structure}\nLet $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid (of order $n$) with associated weak Lie-Filippov bialgebroid $(AG, A^*G).$ Then the induced Nambu structures on $M$\ncoming from the Nambu-Lie groupoid and the weak Lie-Filippov bialgebroid are related by\n\\begin{align*}\n \\{ ~, \\ldots, ~\\}_M = (-1)^{n-1}\\{ ~, \\ldots, ~\\}_{(AG, A^*G)} .\n\\end{align*}\n\\end{prop}\n\n\n\\begin{proof}\nFor any functions $f_1, \\ldots, f_n \\in C^\\infty(M)$, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}_{(AG, A^*G)} =& \\Pi^{\\sharp}\\big|_M (d_A f_1 \\wedge \\cdots \\wedge d_A f_{n-1}) f_n\\\\\n=& \\Pi^{\\sharp} ( d (\\beta^* f_1) \\wedge \\cdots \\wedge d (\\beta^* f_{n-1}))\\big|_M f_n\\\\\n=& \\Pi (\\beta^* f_1 , \\ldots, \\beta^* f_{n-1}, \\beta^* f_n)\\big|_M\\\\\n=& (-1)^{n-1} \\big( \\beta^*\\{f_1, \\ldots, f_n\\} \\big) \\big|_M\\\\\n=& (-1)^{n-1} \\{f_1, \\ldots, f_n\\}_M.\n\\end{align*}\n\\end{proof}\n\\begin{remark}\n It is known that under some connectedness and simply connectedness assumption, any Lie bialgebra integrates to a Poisson-Lie group \\cite{lu-wein}, and any Lie bialgebroid\nintegrates to a Poisson groupoid \\cite{mac-xu2}. These results does not hold in the context of Nambu structures of order $\\geqslant 3$. Let $G$ be a connected and simply-connected Lie group with Lie algebra $\\mathfrak{g}.$ Given a Lie-Filippov bialgebra structure $(\\mathfrak{g}, \\mathfrak{g}^*)$ on $\\mathfrak{g}$, the $1$-cocycle $\\delta : \\mathfrak{g} \\rightarrow \\bigwedge^n \\mathfrak{g}$\ndual to the Filippov algebra bracket on $\\mathfrak{g}^*$ integrates a multiplicative $n$-vector field $\\Pi$ on the Lie group. However this $n$-vector field (for $n \\geqslant 3$) need not be a Nambu tensor \\cite{vais}, that is, need not be locally decomposable. Thus (weak) Lie-Filippov bialgebra\ndoes not integrate to a Nambu-Lie group in general. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Coisotropic subgroupoids of a Nambu-Lie groupoid}\n\nIn this final section, we introduce the notion of coisotropic subgroupoid of a Nambu-Lie groupoid \n and study the infinitesimal object corresponding to it.\n\n\\begin{defn}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid of order $n$. Then a subgroupoid $H \\rightrightarrows N$ is called a {\\it coisotropic subgroupoid} if $H$ is a coisotropic submanifold of $G$ with respect to $\\Pi$.\n\\end{defn}\n\n\\begin{exam}\n\\begin{enumerate}\n\\item For $n=2$, that is, when $G \\rightrightarrows M$ is a Poisson groupoid, this notion is same as the coisotropic subgroupoid of a Poisson groupoid introduced in \\cite{xu}.\n\\item Let $(G, \\Pi)$ be a Nambu-Lie group. Then a subgroup of $G$ is called coisotropic if it is also a coisotropic submanifold of $G$. Any coisotropic subgroup\nof $G$ is a coisotropic subgroupoid over a point.\n\\item Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid. Then by the Proposition \\ref{inverse-basenambu}, there exist an induced Nambu-structure\non $M$ for which the source map $\\alpha$ is a Nambu-Poisson map. Let $N \\hookrightarrow M$ be a coisotropic submanifold of $M$ with respect to this induced Nambu\nstructure. Consider the restriction $G\\big|_N := \\alpha^{-1}(N) \\cap \\beta^{-1}(N)$, then $G\\big|_N \\rightrightarrows N$ is a coisotropic subgroupoid.\n\\item Let $G \\rightrightarrows M$ be a Nambu-Lie groupoid. If the set of all elements of $G$ which has same source and target, is a submanifold of $G$, then it is a coisotropic subgroupoid.\n\\end{enumerate}\n\\end{exam}\n\nNote that, the infinitesimal object corresponding to a Nambu-Lie groupoid $(G\\rightrightarrows M , \\Pi)$ is the weak Lie-Filippov bialgebroid\n$(AG, A^*G)$. Therefore it is natural to ask how the Lie algebroid of a coisotropic subgroupoid $H \\rightrightarrows N$ is related to the weak Lie-Filippov bialgebroid $(AG, A^*G)$. To answer this question, we introduce a notion of {\\it coisotropic subalgebroid} of a weak Lie-Filippov bialgebroid and show that infinitesimal forms of coisotropic subgroupoids of a Nambu-Lie groupoid $(G \\rightrightarrows M, \\Pi)$ appear as coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$.\n\n\\begin{defn}\\label{coiso-subalg}\n Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid of order $n$ over $M$. Then a Lie subalgebroid $B \\rightarrow N$ of $A\\rightarrow M$ is\ncalled a {\\it coisotropic subalgebroid} if the anchor $\\rho: \\bigwedge^{n-1}A^* \\rightarrow TM$ and the $n$-bracket $[~, \\ldots, ~]$ on $\\Gamma{A^*}$\nsatisfy the following properties.\n\\begin{enumerate}\n\\item The anchor $\\rho$ maps $\\bigwedge^{n-1}B^{0} \\rightarrow TN$.\n\\item If $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*}$ with ${\\alpha_i}\\big|_N \\in B^{0}$ for all $i$, then $[\\alpha_1, \\ldots, \\alpha_n]\\big|_N \\in B^{0}$.\n\\item If $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*}$ with ${\\alpha_i}\\big|_N \\in B^{0}$ for all $i$ and $\\alpha_n\\big|_N = 0$, then $[\\alpha_1, \\ldots, \\alpha_n]\\big|_N = 0,$\nwhere $B_x^{0} = \\{\\gamma \\in A_x^* | \\gamma (v) = 0, \\forall v \\in B_x\\}$, is the annihilator of $B_x$, $x \\in N$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{exam}\n Let $M$ be a Nambu-Poisson manifold, then $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid over $M$. Let $N \\hookrightarrow M$ be a coisotropic submanifold. Then from the Proposition \\ref{coiso-n-bracket}, it follows that the tangent bundle $TN \\rightarrow N$ is a coisotropic subalgebroid. \n\\end{exam}\n\nIt is known that (Proposition \\ref{wlfb-base-nambu}, see also \\cite{bas-bas-das-muk}), the base of a weak Lie-Filippov bialgebroid carries a Nambu structure.\nThe next Proposition shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced Nambu structure.\n\n\\begin{prop}\n Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid over $M$ and $B \\rightarrow N$ be a coisotropic subalgebroid. Then $N$ is a coisotropic submanifold of $M$.\n\\end{prop}\n\n\\begin{proof}\nLet $a : A \\rightarrow TM$ denote the anchor of the Lie algebroid $A$ and $\\rho : \\bigwedge^{n-1}A^* \\rightarrow TM$ be the anchor of pair $(A, A^*).$\nWe first show that, $a^* (TN)^0 \\subseteq B^{0}$. This is true because, $ \\langle a^*\\xi_x, v \\rangle = \\langle \\xi_x, a(v)\\rangle = 0$\nfor $\\xi_x \\in (TN)_x^0$ and $v \\in B_x$.\n\nLet $\\Pi_{(A, A^*)}$ be the induced Nambu structure on $M$ coming from the weak Lie-Filippov bialgebroid $(A, A^*).$ Then the induced map\n$\\Pi^{\\sharp}_{(A, A^*)} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ is given by $\\Pi^{\\sharp}_{(A, A^*)} = \\rho \\circ \\bigwedge^{n-1}a^*$. Therefore, for any\n$\\xi_1, \\ldots, \\xi_{n-1} \\in (TN)^0$, we have\n$$ \\Pi^{\\sharp}_{(A, A^*)} (\\xi_1, \\ldots, \\xi_{n-1}) = \\rho (a^*\\xi_1, \\ldots, a^*\\xi_{n-1}) \\in TN $$\nas $a^*\\xi_i \\in B^{0}$ and $B$ is a coisotropic subalgebroid. Therefore $N$ is a coisotropic submanifold of $M$.\n\\end{proof}\n\n\nThe next proposition shows that the infinitesimal object corresponding to coisotropic subgroupoids are coisotropic subalgebroids.\n\n\\begin{prop}\\label{coiso-subgrpd-coiso-subalgbd}\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid with weak Lie-Filippov bialgebroid $(AG, A^*G)$. Let $H \\rightrightarrows N$\nbe a coisotropic subgroupoid of $G \\rightrightarrows M$ with Lie algebroid $AH \\rightarrow N$. Then $AH \\rightarrow N$ is a coisotropic subalgebroid.\n\\end{prop}\n\n\\begin{proof}\n Since $H \\rightrightarrows N$ is a Lie subgroupoid of $G \\rightrightarrows M$, therefore $AH \\rightarrow N$ is a Lie subalgebroid of\n$AG \\rightarrow M$. We claim that the anchor $\\rho = \\Pi^{\\sharp}\\big|_{\\bigwedge^{n-1}(TM)^0} = \\Pi^{\\sharp}\\big|_{\\bigwedge^{n-1} (A^*G)}$ of the weak\nLie-Filippov bialgebroid $(AG, A^*G)$ maps $\\bigwedge^{n-1}(AH)^0$ to $TN$. First observe that, for any $x \\in N$, $(AH)_x^0 = (TM)_x^0 \\cap (TH)_x^0$\nand $T_xN = T_xM \\cap T_xH$. Therefore, \n$\\rho$ maps $\\bigwedge^{n-1} (AH)^0$ to \n$$\\Pi^{\\sharp}({\\bigwedge}^{n-1} (TM)^0) \\cap \\Pi^{\\sharp}({\\bigwedge}^{n-1} (TH)^0) \\subseteq TM \\cap TH \\cong TN,$$ \nhere we have used the fact that $M$ and $H$ are both coisotropic submanifolds of $G$.\n\nLet $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*G} = \\Gamma{(TM)^0}$ such that $\\alpha_i\\big|_N \\in (AH)^0$, for all $i=1, \\ldots, n$. Let\n$\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_n$ be one forms on $G$ extending $\\alpha_1, \\ldots, \\alpha_n$ and are conormal to $H$. Then\nby the Proposition \\ref{coiso-n-bracket}, the $1$-form $[\\tilde{\\alpha_1}, \\ldots, \\tilde{\\alpha_n}]$ is conormal to both $M$ and $H$, as $M$\nand $H$ are both coisotropic submanifolds of $G$. Therefore, \n$$[\\tilde{\\alpha_1}, \\ldots, \\tilde{\\alpha_n}]\\big|_N \\in (TM)^0 \\cap (TH)^0 \\cong (AH)^0.$$ \nVerification of the last condition of the Definition \\ref{coiso-subalg} is similar. Hence $AH \\rightarrow N$ is a coisotropic subalgebroid of $(AG, A^*G).$\n\\end{proof}\n\n\\begin{corollary}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid and $H \\rightrightarrows N$ be a coisotropic subgroupoid. Then $N$ is a coisotropic submanifold of $M$.\n\\end{corollary}\n\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1} \\markboth{#1}{#1}\n \\leavevmode\\par}\n\\def\\paragraph#1{{\\bf #1\\ }}\n\n\\newtheorem{lemma}{Lemma}[section]\n\n\\newtheorem{theorem}[lemma]{Theorem}\n\n\\newtheorem{corollary}[lemma]{Corollary}\n\n\\newtheorem{definition}[lemma]{Definition}\n\n\\newtheorem{proposition}[lemma]{Proposition}\n\n\\newtheorem{remark}{Remark}[section]\n\n\\newtheorem{example}{Example}[section]\n\n\\newtheorem{conject}{Conjecture}[section]\n\n\\newtheorem{hypothesis}{Hypothesis}[section]\n\n\n\n\n\\def\\varepsilon{\\varepsilon}\n\\def\\<{\\langle}\n\\def\\>{\\rangle}\n\\def\\subset\\subset{\\subset\\subset}\n\\def {\\mathbb R} {{\\mathbb R}}\n\\def {\\mathbb Q} {{\\mathbb Q}}\n\\def {\\mathbb Z} {{\\mathbb Z}}\n\\def {\\mathbb C} {{\\mathbb C}}\n\\def {\\mathbb N} {{\\mathbb N}}\n\n\n\\def {\\mathbb S} {{\\mathbb S}}\n\\def {\\frac{1}{2}} {{\\frac{1}{2}}}\n\\def \\varepsilon {\\varepsilon}\n\\def \\sigma {\\sigma}\n\\def \\alpha {\\alpha}\n\\def \\beta {\\beta}\n\\def \\omega {\\omega}\n\\def \\Omega{\\Omega}\n\\def \\Theta {\\Theta}\n\\def \\delta {\\delta}\n\\def \\rho {\\rho}\n\\def \\phi {\\phi}\n\n\n\n\n\n\\title{Macroscopic models of collective motion with repulsion}\n\\author{Pierre Degond$^{1}$, Giacomo Dimarco$^{2}$, Thi Bich Ngoc Mac$^{3,4}$, Nan Wang $^{5} $ }\n\\date{}\n\\begin{document}\n\n\\maketitle\n\n\\begin{center}\n1- Department of Mathematics, Imperial college London, \\\\\nLondon SW7 2AZ, United Kingdom.\\\\\nemail:pdegond@imperial.ac.uk\\\\\n$\\mbox{}$ \\\\\n2-Department of Mathematics, University of Ferrara, 44100 Ferrara, Italy\\\\\nemail: giacomo.dimarco@unife.it \\\\\n$\\mbox{}$ \\\\\n3-Universit\\'e de Toulouse; UPS, INSA, UT1, UTM ;\\\\\nInstitut de Math\\'ematiques de Toulouse ; \\\\\nF-31062 Toulouse, France. \\\\\n$\\mbox{}$ \\\\\n4-CNRS; Institut de Math\\'ematiques de Toulouse UMR 5219 ;\\\\\nF-31062 Toulouse, France.\\\\\nemail: thi-bich-ngoc.mac@math.univ-toulouse.fr \\\\\n$\\mbox{}$ \\\\\n5-National University of Singapore, Department of Mathematics,\\\\\nLower Kent Ridge Road, Singapore 119076.\\\\\nemail: g0901252@nus.edu.sg \\\\\n\\end{center}\n\n\n\n\\vspace{0.5 cm}\n\\begin{abstract}\n\nWe study a system of self-propelled particles which interact with their neighbors via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors, up to some noise. A continuum model is derived starting from a mean-field kinetic description of the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a numerical validation of the continuum model by comparison with the particle model. We also provide comparisons with other self-propelled particle models with alignment and repulsion. \n\\end{abstract}\n\n\\medskip\n\\noindent\n{\\bf Acknowledgements:} PD is on leave from CNRS, Institut de Math\\'ematiques de Toulouse, France, and GD is on leave from Universit\\'e Paul Sabatier, Institut de Math\\'ematiques de Toulouse, France, where part of this research was conducted. TBNM wishes to thank the hospitality of the Department of Mathematics, Imperial College London for its hospitality. NW wishes to thank the Universit\\'e Paul Sabatier, Institut de Math\\'ematiques de Toulouse, France, for its hospitality. This work has been supported by the French 'Agence Nationale pour la Recherche (ANR)' in the frame of the contract 'MOTIMO' (ANR-11-MONU-009-01). \n\n\n\\medskip\n\\noindent\n{\\bf Key words: } Fokker-Planck equation, macroscopic limit, Von Mises-Fisher distribution, Generalized Collision Invariants, Non-conservative equations, Self-Organized Hydrodynamics, self-propelled particles, alignment, repulsion.\n\n\\medskip\n\\noindent\n{\\bf AMS Subject classification: } 35Q80, 35L60, 82C22, 82C70, 92D50\n\\vskip 0.4cm\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Introduction}\n\\label{sec:intro}\n\nThe study of collective motion in systems consisting of a large number of agents, such as bird flocks, fish schools, suspensions of active swimmers (bacteria, sperm cells ), etc has triggered an intense literature in the recent years. We refer to \\cite{Vicsek_Zafeiris_PhysRep12, Koch_FluidMech2011} for recent reviews on the subject. Many of such studies rely on a particle model or Individual Based Model (IBM) that describes the motion of each individual separately (see e.g in \\cite{Aoki_BullJapSocSciFish92, Chate_etal_PRE08, Chuang_etal_PhysicaD07, Couzin_etal_JTB02, Cucker_Smale_IEEETransAutCont07, Henkes_etal_PRE11, Mogilner_etal_JMB03, Motsch_Tadmor_JSP11, Szabo_PRL06}).\n\n\nIn this work, we aim to describe dense suspensions of elongated self-propelled particles in a fluid, such as sperm. In such dense suspensions, steric repulsion is an essential ingredient of the dynamics. A large part of the literature is concerned with dilute suspensions \\cite{Hernandez-Ortiz_etal_JPhysCondMat09, Koch_FluidMech2011, Pedley_etal_JFM88, Saintillan_Fluids08, Woodhouse_Goldstein_PRL12}. In these approaches, the Stokes equation for the fluid is coupled to the orientational distribution function of the self-propelled particles. However, these approaches are of ``mean-field type'' i.e. assume that particle interactions are mediated by the fluid through some kinds of averages. These approaches do not deal easily with short-range interactions such as steric repulsion or interactions mediated by lubrication forces. Additionally, these models assume a rather simple geometry of the swimmers, which are reduced to a force dipole, while the true geometry and motion of an actual swimmer, like a sperm cell, is considerably more complex. \n\nIn a recent work \\cite{Peruani_etal_PRE06}, Peruani et al showed that, for dense systems of elongated self-propelled particles, steric interaction results in alignment. Relying on this work, and owing to the fact that the description of swimmer interactions from first physical principles is by far too complex, we choose to replace the fluid-mediated interaction by a simple alignment interaction of Vicsek type \\cite{Vicsek_etal_PRL95}. In the Vicsek model, the agents move with constant speed and attempt to align with their neighbors up to some noise. Many aspects of the Vicsek model have been studied, such as phase transitions \\cite{Aldana_etal_PRL07, Chate_etal_PRE08, Degond_etal_JNonlinearSci13, Degond_etal_preprint13, Gretoire_Chate_PRL04, Vicsek_etal_PRL95}, numerical simulations \\cite{Motsch_Navoret_MMS11}, derivation of macroscopic models \\cite{Bertin_etal_JPA09, Degond_Motsch_M3AS08}.\n\nThe alignment interaction acting alone may trigger the formation of high particle concentrations. However, in dense suspensions, volume exclusion prevent such high densities to occur. When distances between particles become too small, repulsive forces are generated by the fluid or by the direct reaction of the bodies one to each other. These forces contribute to repel the particles and to prevent further contacts. To model this behavior, we must add a repulsive force to the Vicsek alignment model. Inspired by \\cite{Baskaran_Marchetti_PRL10, Henkes_etal_PRE11, Szabo_PRL06} we consider the possibility that the particle orientations (i.e the directions of the self-propulsion force) and the particle velocities may be different. Indeed, steric interaction may push the particles in a direction different from that of their self-propulsion force. \n\nWe consider an overdamped regime in which the velocity is proportional to the force through a mobility coefficient. The overdamped limit is justified by the fact that the background fluid is viscous and thus the forces due to friction are very large compared to those due to motion. Indeed, for micro size particles, the Reynolds number is very small ($\\sim 10^{-4}$) and thus the effect of inertia can be neglected. Finally, differently from \\cite{Baskaran_Marchetti_PRL10, Henkes_etal_PRE11, Szabo_PRL06} we consider an additional term describing the relaxation of the particle orientation towards the direction of the particle velocity. We also take into account a Brownian noise in the orientation dynamics of the particles. This noise may take into account the fluid turbulence for instance. Therefore, the particle dynamics \nresults from an interplay between relaxation towards the mean orientation of the surrounding particles, relaxation towards the direction of the velocity vector and Brownian noise. From now on we refer to the above described model as the Vicsek model with repulsion.\n\nStarting from the above described microscopic dynamical system we successively derive mean-field equations and hydrodynamic equations. Mean field equations are valid when the number of particles is large and describe the evolution of the one-particle distribution, i.e. the probability for a particle to have a given orientation and position at a given instant of time. Expressing that the spatio-temporal scales of interest are large compared to the agents' scales leads to a singular perturbation problem in the kinetic equation. Taking the hydrodynamic limit, (i.e. the limit of the singular perturbation parameter to zero) leads to the hydrodynamic model. Hydrodynamic models are particularly well-suited to systems consisting of a large number of agents and to the observation of the system's large scale structures. Indeed, the computational cost of IBM increases dramatically with the number of agents, while that of hydrodynamic models is independent of it. With IBM, it is also sometimes quite cumbersome to access observables such as order parameters, while these quantities are usually directly encoded into the hydrodynamic equations. \n\nThe derivation of hydrodynamic models has been intensely studied by many authors. Many of these models are based on phenomenological considerations \\cite{Toner_etal_AnnPhys05} or derived from moment approaches and ad-hoc closure relations \\cite{Baskaran_Marchetti_PRL10, Bertin_etal_JPA09, Ratushnaya_etal_PhysicaA07}. The first mathematical derivation of a hydrodynamic system for the Vicsek model has been proposed in \\cite{Degond_Motsch_M3AS08}. We refer to this model to as the Self-Organized Hydrodynamic (SOH) model. One of the main contributions of \\cite{Degond_Motsch_M3AS08} is the concept of ``Generalized Collision Invariants'' (GCI) which permits the derivation of macroscopic equations for a particle system in spite of its lack of momentum conservation. The SOH model has been further refined in \\cite{Degond_etal_MAA13, Frouvelle_M3AS12}. \n\nPerforming the hydrodynamic limit in the kinetic equations associated to the Vicsek model with repulsion leads to the so-called ``Self-Organized Hydrodynamics with Repulsion'' (SOHR) system. The SOHR model consists of a continuity equation for the density $\\rho$ and an evolution equation for the average orientation $\\Omega \\in {\\mathbb S}^{n-1}$ where $n$ indicates the spatial dimension. More precisely, the model reads\n\\begin{eqnarray}\n& \\partial_t\\rho + \\nabla_x \\cdot(\\rho U ) = 0, \\label{macro1_intro} \\\\\n& \\rho \\partial_t \\Omega + \\rho (V \\cdot \\nabla_x)\\Omega + P_{\\Omega^\\perp}\\nabla_x p(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \\label{macro2_intro}\t\\\\\n& \\vert \\Omega \\vert = 1,\t\\label{macro3_intro}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n& U = c_1v_0 \\Omega - \\mu \\Phi_0 \\nabla_x \\rho , \\quad V = c_2 v_0 \\Omega - \\mu \\Phi_0 \\nabla_x \\rho , \\label{macro4_intro}\\\\\n& p(\\rho ) = v_0 d \\rho + \\alpha \\mu \\Phi_0 \\big( (n-1)d + c_2 \\big) \\dfrac{\\rho^2}{2}, \\quad \\gamma = k_0 \\big( (n-1)d + c_2\\big). \\label{macro5_intro}\n\\end{eqnarray}\nThe coefficients $c_1$, $c_2$, $v_0$, $\\mu$, $\\Phi_0$, $d$, $\\alpha$, $k_0$ are associated to the microscopic dynamics and will be defined later on. The symbol $P_{\\Omega^\\bot}$ stands for the projection matrix $ P_{\\Omega^\\bot} = \\mbox{Id} - \\Omega \\otimes \\Omega$ of ${\\mathbb R}^n$ on the hyperplane $\\Omega^\\bot$. The SOHR model is similar to the SOH model obtained in \\cite{Degond_etal_MAA13}, but with several additional terms which are consequences of the repulsive force at the particle level. The repulsive force intensity is characterized by the parameter $\\mu \\Phi_0$. In the case $\\mu \\Phi_0 = 0$, the SOHR system is reduced to the SOH one. \n\nWe first briefly describe the original SOH model. Inserting (\\ref{macro4_intro}), (\\ref{macro5_intro}) with $\\mu \\Phi_0 = 0$ into (\\ref{macro1_intro}), (\\ref{macro2_intro}) leads to \n\\begin{eqnarray}\n& \\partial_t\\rho + c_1 v_0 \\, \\nabla_x \\cdot(\\rho \\Omega ) = 0, \\label{macro6_intro} \\\\\n& \\rho \\partial_t \\Omega + c_2 v_0 \\rho (\\Omega \\cdot \\nabla_x)\\Omega + v_0 d \\, P_{\\Omega^\\perp}\\nabla_x \\rho = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \\label{macro7_intro}\n\\end{eqnarray}\ntogether with (\\ref{macro3_intro}). This model shares similarities with the isothermal compressible Navier-Stokes (NS) equations. Both models consist of a non linear hyperbolic part supplemented by a diffusion term. Eq. (\\ref{macro6_intro}) expresses conservation of mass, while Eq. (\\ref{macro7_intro}) is an equation for the mean orientation of the particles. It is not conservative, contrary to the corresponding momentum conservation equation in NS. The two equations are supplemented by the geometric constraint (\\ref{macro3_intro}). This constraint is satisfied at all times, as soon as it is satisfied initially. The preservation of this property is guaranteed by the projection operator $P_{\\Omega^\\perp}$. A second important difference between the SOH model and NS equations is that the convection velocities for the density and the orientation, $v_0 c_1$ and $v_0 c_2$ respectively are different while for NS they are equal. That $c_1 \\not = c_2$ is a consequence of the lack of Galilean invariance of the model (there is a preferred frame, which is that of the fluid). The main consequence is that the propagation of sound waves is anisotropic for this type of fluids \\cite{Toner_etal_AnnPhys05}. \n\nThe first main difference between the SOH and the SOHR system is the presence of the terms $\\mu \\Phi_0 \\nabla_x \\rho$ in the expressions of the velocities $U$ and $V$. Inserting this term in the density Eq. (\\ref{macro1_intro}) results in a diffusion-like term $- \\mu \\Phi_0 \\, \\nabla_x \\cdot(\\rho ( \\nabla_x \\rho))$ which avoids the formation of high particle concentrations. This term shows similarities with the non-linear diffusion term in porous media models. Similarly, inserting the term $\\mu \\Phi_0 \\nabla_x \\rho$ in the orientation Eq. (\\ref{macro2_intro}) results in a convection term in the direction of the gradient of the density. Its effect is to force particles to change direction and move towards regions of lower concentration. The second main difference is the replacement of the linear (with respect to $\\rho$) pressure term $v_0 d \\, P_{\\Omega^\\perp}\\nabla_x \\rho$ by a nonlinear pressure $p(\\rho)$ in the orientation Eq. (\\ref{macro2_intro}). The nonlinear part of the pressure enhances the effects of the repulsion forces when concentrations become high. \n\nTo further establish the validity of the SOHR model (\\ref{macro1_intro})-(\\ref{macro5_intro}), we perform numerical simulations and compare them to those of the underlying IBM. To numerically solve the SOHR model, we adapt the relaxation method of \\cite{Motsch_Navoret_MMS11}. In this method, the unit norm constraint (\\ref{macro3_intro}) is abandonned and replaced by a fully conservative hyperbolic model in which $\\Omega$ is supposed to be in ${\\mathbb R}^n$. However, at the end of each time step of this conservative model, the vector $\\Omega$ is normalized. Motsch and Navoret showed that the relaxation method provides numerical solutions of the SOH model which are consistent with those of the particle model. The resolution of the conservative model can take advantage of the huge literature on the numerical resolution of hyperbolic conservation laws (here specifically, we use \\cite{Degond_Peyrard_99}). We adapt the technique of \\cite{Motsch_Navoret_MMS11} to include the diffusion fluxes. Using these approximations, we numerically demonstrate the good convergence of the scheme for smooth initial data and the consistency of the solutions with those of the particle Vicsek model with repulsion. \n\nThe outline of the paper is as follows. In section 2, we introduce the particle model, its mean field limit, the scaling and the hydrodynamic limit. In Section 3 we present the numerical discretization of the SOHR model, while in Section 4 we present several numerical tests for the macroscopic model and a comparison between the microscopic and microscopic models. Section 5 is devoted to draw a conclusion. Some technical proofs will be given in the Appendices.\n\n\n\n\\setcounter{equation}{0}\n\\section{ Model hierarchy and main results }\n\\label{sec:IBM}\n\n\\subsection{The individual based model and the mean field limit}\n\nWe consider a system of $N$-particles each of which is described by its position $ X_k(t) \\in {\\mathbb R}^n$, its velocity $v_k(t) \\in {\\mathbb R}^n,$ and its direction $ \\omega_k(t) \\in {\\mathbb S}^{n-1} ,$ where $ k \\in \\{ 1, \\cdots , N \\}$, $n$ is the spatial dimension and ${\\mathbb S}^{n-1}$ denotes the unit sphere. The particle ensemble satisfies the following stochastic differential equations \n\\begin{eqnarray} \n\\dfrac{dX_k}{dt} & = & v_k,\n\\label{overdamped1}\\\\\nv_k\t& = & v_0 \\, \\omega_k - \\mu \\, \\nabla_x \\Phi(X_k(t),t),\n\\label{overdamped2}\\\\\nd\\omega_k\t & = & P_{\\omega_k^\\perp}\\circ (\\nu \\, \\bar{\\omega}(X_k(t),t) \\, dt + \\alpha \\, v_k \\, dt + \\sqrt{2D} \\, dB^k_t ).\\label{overdamped3}\n\\end{eqnarray}\t\nEq. (\\ref{overdamped1}) simply expresses the spatial motion of a particle of velocity $v_k$. Eq (\\ref{overdamped2}) shows that the velocity $v_k$ is composed of two components: a self-propulsion velocity of constant magnitude $v_0$ in direction $\\omega_k$ and a repulsive force proportional to the gradient of a potential $\\Phi(x,t)$ with mobility coefficient $\\mu$. Equation (\\ref{overdamped3}) describes the time evolution of the orientation. The first term models the relaxation of the particle orientation towards the average orientation $\\bar{\\omega}(X_k(t),t)$ of its neighbors with rate $\\nu$. The second term models the relaxation of the particle orientation towards the direction of the particle velocity $v_k$ with rate $\\alpha$. Finally, the last term describes standard independent white noises $dB^k_t$ of intensity $\\sqrt{2D}$. The symbol $\\circ$ reminds that the equation has to be understood in the Stratonovich sense. Under this condition and thanks to the presence of $P_{\\omega^\\perp}$, the orthogonal projection onto the plane orthogonal to $\\omega$ (i.e $P_{\\omega^\\perp} = (\\mbox{Id} - \\omega \\otimes \\omega),$ where $\\otimes $ denotes the tensor product of two vectors and $\\mbox{Id}$ is the identity matrix), the orientation $\\omega_k$ remains on the unit sphere. We assume that $v_0$, $\\mu$, $\\nu$ $\\alpha$, $D$ are strictly positive constants. \n\nThe repulsive potential $\\Phi(x,t)$ is the resultant of binary interactions mediated by the binary interaction potential $ \\phi $. It is given by: \n\\begin{equation}\n\\Phi(x,t) = \\dfrac{1}{N} \\sum_{i=1}^{N} \\nabla \\phi \\Big( \\frac{|x - X_i|}{r} \\Big)\n\\label{eq:pot}\n\\end{equation}\nwhere the binary repulsion potential $\\phi(|x|)$ only depends on the distance. We suppose that $x \\mapsto \\phi(|x|)$ is smooth (in particular implying that $\\phi'(0) =0$ where the prime denotes the derivative with respect to $|x|$). We also suppose that \n$$ \\phi \\geq 0, \\qquad \\int_{{\\mathbb R}^n} \\phi(|x|) \\, dx < \\infty, $$\nin particular implying that $\\phi(|x|) \\to 0$ as $ |x| \\rightarrow \\infty$. The quantity $r$ denotes the typical range of $\\phi$ and the fact that $\\phi \\geq 0$ ensures that $\\phi$ is repulsive. In the numerical test Section, we will propose precise expressions for this potential force. \n\nThe mean orientation $\\bar{\\omega}(x,t)$ is defined by\n\\begin{equation}\n\\bar{\\omega}(x,t) = \\dfrac{{\\mathcal J}(x,t)}{\\vert {\\mathcal J}(x,t) \\vert}, \\qquad {\\mathcal J} (x,t) = \\frac{1}{N} \\sum\\limits_{i=1}^{N} K\\Big(\\frac{|x-X_i|}{R}\\Big) \\omega_i.\n\\label{eq:meanorient}\n\\end{equation}\t\nIt is constructed as the normalization of the vector $ {\\mathcal J} (x,t)$ which sums up all orientation vectors $\\omega_i$ of all the particles which belong to the range of the ``influence kernel'' $K(|x|)$. The quantity $R>0$ is the typical range of the influence kernel $K (|x|\/R)$, which is supposed to depend only on the distance. It measures how the mean orientation at the origin is influenced by particles at position $x$. Here, we assume that $x \\to K(|x|)$ is smooth at the origin and compactly supported. For instance, if $K$ is the indicator function of the ball of radius $1$, the quantity $\\bar \\omega (x,t)$ computes the mean direction of the particles which lie in the sphere of radius $R$ centered at $x$ at time $t$. \n\n\n\n\n\\begin{remark}\n(i) In the absence of repulsive force (i.e. $\\mu = 0$), the system reduces to the time continuous version of the Vicsek model proposed in \\cite{Degond_Motsch_M3AS08}. \\\\\n(ii) The model presented is the so called overdamped limit of the model consisting of (\\ref{overdamped1}) and (\\ref{overdamped3}) and where (\\ref{overdamped2}) is replaced by: \n\\begin{eqnarray} \\label{nonoverdamped4}\n\\epsilon \\dfrac{dv_k}{dt}\t & = & \\lambda_1 (v_0 \\omega_k - v_k) - \\lambda_2 \\nabla_x \\Phi(X_k(t),t).\n\\end{eqnarray}\t\nwith $\\mu = \\lambda_2\/\\lambda_1.$ Taking the limit $ \\epsilon \\rightarrow 0 $ in (\\ref{nonoverdamped4}), we obtain (\\ref{overdamped2}). As already mentioned in the Introduction, for microscopic swimmers, this limit is justified by the very small Reynolds number and the very small inertia of the particles. \n\\end{remark}\n\nWe now introduce the mean field kinetic equation which describes the time evolution of the particle system in the large $N$ limit. The unknown here is the one particle distribution function $f(x, \\omega, t)$ which depends on the position $x \\in {\\mathbb R}^n$, orientation $\\omega\\in {\\mathbb S}^{n-1}$ and time $t$. The evolution of $f$ is governed by the following system\n\\begin{eqnarray}\n& & \\hspace{-1cm} \\partial_t f + \\nabla_x \\cdot (v_f f) + \\nu \\, \\nabla_\\omega \\cdot( P_{\\omega^\\perp} \\bar{\\omega}_f f) + \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v_f f ) - D \\Delta_\\omega f = 0, \n\\label{eqmeanfield1}\\\\\n& & \\hspace{-1cm} v_f(x, t) = \tv_0 \\omega - \\mu \\nabla_x \\Phi_f(x,t), \n\\label{eqmeanfield2}\n\\end{eqnarray}\nwhere the repulsive potential and the average orientation are given by \t\n\\begin{eqnarray}\n& & \\Phi_f(x,t) = \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} \\phi\\left(\\dfrac{|x-y|}{r}\\right) \\,f(y, w, t) \\, dw \\, dy, \n\\label{mfeq1} \\\\\n & &\\bar{\\omega}_f (x,\\omega, t) = \\dfrac{\\mathcal J_f(x,t)}{|\\mathcal J_f(x,t)|}, \n\\label{mfeq2}\\\\\t \t\n& & \\mathcal J_f (x,t) = \t\\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} K\\left(\\dfrac{|x-y|}{R}\\right) \\, f(y, w, t) \\, w \\, dw \\, dy \n\\label{mfeq3}.\t\t \n\\end{eqnarray}\t\nEquation (\\ref{eqmeanfield1}) is a Fokker-Planck type equation. The second term at the left-hand side of (\\ref{eqmeanfield1}) describes particle transport in physical space with velocity $ v_f$ and is the kinetic counterpart of Eq. (\\ref{overdamped1}). The third, fourth and fifth terms describe transport in orientation space and are the kinetic counterpart of Eq. (\\ref{overdamped3}). The alignment interaction is expressed by the third term, while the relaxation force towards the velocity $v_f$ is expressed by the fourth term. The fifth term represents the diffusion due to the Brownian noise in orientation space. The projection $P_{\\omega^\\perp}$ insures that the force terms are normal to $\\omega$. The symbols $\\nabla_\\omega \\cdot$ and $\\Delta_\\omega$ respectively stand for the divergence of tangent vector fields to ${\\mathbb S}^{n-1}$ and the Laplace-Beltrami operator on ${\\mathbb S}^{n-1}$. Eq. (\\ref{eqmeanfield2}) is the direct counterpart of (\\ref{overdamped2}). \n\nEq. (\\ref{mfeq1}) is the continuous counterpart of Eq. (\\ref{eq:pot}). Indeed, letting $f$ be the empirical measure\n$$f = \\frac{1}{N} \\sum_{i=1}^N \\delta_{(x_i(t), \\omega_i(t))} (x,\\omega), $$\nin (\\ref{mfeq1}) (where $\\delta_{(x_i(t), \\omega_i(t))} (x,\\omega)$ is the Dirac delta at $(x_i(t), \\omega_i(t))$) leads to (\\ref{eq:pot}). Similarly, Eqs. (\\ref{mfeq2}), (\\ref{mfeq3}) are the continuous counterparts of (\\ref{eq:meanorient}) (by the same kind of argument). The rigorous convergence of the particle system to the above Fokker-Planck equation (\\ref{eqmeanfield1}) is an open problem. We recall however that, the derivation of the kinetic equation for the Vicsek model without repulsion has been done in \\cite{Bolley_etal_AML11} in a slightly modified context.\n\n\n\\subsection{Scaling}\n\t\nIn order to highlight the role of the various terms, we first write the system in dimensionless form. We chose $t_0$ as unit of time and choose \n$$ x_0 = v_0 t_0, \\qquad f_0 = \\frac{1}{x_0^n}, \\qquad \\phi_0 = \\frac{v_0^2 \\, t_0}{\\mu}, $$\nas units of space, distribution function and potential. We introduce the dimensionless variables:\n$$\n\\tilde{x} = \\dfrac{x}{x_0}, \\qquad \\tilde{t} = \\dfrac{t}{t_0}, \\qquad \\tilde{f} = \\dfrac{f}{f_0}, \\qquad \\tilde{\\phi} = \\dfrac{\\phi}{\\phi_0}, \n$$\nand the dimensionless parameters\n$$\n\\breve R = \\dfrac{R}{x_0}, \\qquad \\breve r = \\dfrac{r}{x_0}, \\qquad \\breve D = t_0 D, \\qquad \\breve \\nu = t_0 \\nu, \\qquad \\breve \\alpha = \\alpha x_0.\n$$\nIn the new set of variables $(\\tilde x,\\tilde t)$, Eq. (\\ref{eqmeanfield2}) becomes\t(dropping the tildes and the $\\breve{}$ for simplicity): \n$$\nv_f =\t \\omega - \\nabla_x \\Phi_f(x,t), \n$$\n\\noindent\t\nwhile $f$, $\\Phi_f$, $\\bar \\omega_f$, ${\\mathcal J}_f$are still given by (\\ref{eqmeanfield2}), (\\ref{mfeq1}), (\\ref{mfeq2}), (\\ref{mfeq3}) (now written in the new variables). \n\nWe now define the regime we are interested in. We assume that the ranges $R$ and $r$ of the interaction kernels $K$ and $\\phi $ are both small but with $R$ much larger than $r$. More specifically, we assume the existence of a small parameter $\\varepsilon \\ll 1$ such that:\n$$ R = \\sqrt{\\varepsilon} \\hat{R}, \\qquad r = \\varepsilon \\hat{r} \\qquad \\mbox{ with } \\qquad \\hat{R}, \\, \\hat{r} = \\mathcal O(1).$$ \nWe also assume that the diffusion coefficient $D$ and the relaxation rate to the mean orientation $\\nu$ are large and of the same orders of magnitude (i.e. $d = D\/\\nu = {\\mathcal O}(1)$), while the relaxation to the velocity $\\alpha$ stays of order $1$, i.e.\n$$ \\nu = \\dfrac{1}{\\varepsilon}, \\qquad d = \\dfrac{D}{\\nu} = {\\mathcal O}(1), \\qquad \\alpha = {\\mathcal O}(1). $$\nWith these new notations, dropping all 'hats', the distribution function $f^\\varepsilon(x,\\omega,t)$ (where the superscript $\\varepsilon$ now higlights the dependence of $f$ upon the small parameter $\\varepsilon$) satisfies the following Fokker-Plank equation \t\n\\begin{eqnarray}\n&&\\hspace{-1.2cm} \\varepsilon \\Big( \\partial_t f^\\varepsilon + \\nabla_x \\cdot (v_{f^\\varepsilon}^\\varepsilon f^\\varepsilon) \\Big)+ \\nabla_\\omega \\cdot( P_{\\omega^\\perp} \\bar \\omega^\\varepsilon_{f^\\varepsilon} f^\\varepsilon) + \\varepsilon \\alpha \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v^\\varepsilon_{f^\\varepsilon} f^\\varepsilon) - d \\Delta_\\omega f^\\varepsilon = 0, \n\\label{eqmeanfield5} \\\\\n&&\\hspace{-1cm} v^\\varepsilon_f =\t \\omega - \\nabla_x \\Phi_f^\\varepsilon(x,t),\t\n\\label{eqmeanfield6}\n\\end{eqnarray}\nwhere the repulsive potential and the average orientation are now given by\n\\begin{eqnarray*}\n& & \\hspace{-1cm} \\Phi_f^\\varepsilon(x,t) = \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} \\phi\\Big(\\dfrac{|x-y|}{\\varepsilon r } \\Big) \\, f^\\varepsilon(y, w, t) \\, dw \\, dy, \n\\\\\n& & \\hspace{-1cm} \t\\bar{\\omega}^\\varepsilon_f = \\dfrac{\\mathcal J^\\varepsilon_{f}(x,t)}{|\\mathcal J^\\varepsilon_f(x,t)|} , \\quad \\mathcal J^\\varepsilon_f (x,t) = \t \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} K\\Big(\\dfrac{|x-y|}{\\sqrt{\\varepsilon}R } \\Big) \\, f^\\varepsilon (y, w, t) \\, w \\, dw \\, dy \n\\end{eqnarray*}\t\n\n\nNow, by Taylor expansion and the fact that the kernels $K$, $\\phi$ only depend on $|x|$, we obtain (provided that $K$ is normalized to $1$ i.e. $\\int_{\\mathbb R} K(|x|) \\, dx = 1$) :\n\\begin{eqnarray}\n& & \\hspace{-1cm} v_f^\\varepsilon(x,t) =\\omega - \\Phi_0 \\nabla_x \\rho_f^\\varepsilon + \\mathcal O(\\varepsilon^2), \\label{expansion1}\\\\\n& & \\hspace{-1cm} \\bar{\\omega}^\\varepsilon_f (x,t) = G_f^0(x,t) + \\varepsilon G^1_f(x,t) + \\mathcal O(\\varepsilon^2), \\label{expansion2}\\\\\n& & \\hspace{-1cm} G_f^0(x,t) = \\Omega_f(x,t) , \\quad G^1_f(x,t) = \\dfrac{k_0}{ | J_f|} P_{\\Omega_f^\\perp} \\Delta_x J_f, \\nonumber\n\\end{eqnarray}\nwhere the coefficients $k_0, \\Phi_0$ are given by\n\\begin{equation}\n\\label{k0}\nk_0 = \\dfrac{R^2}{2n} \\int_{x\\in {\\mathbb R}^n} K (|x|)|x|^2 dx , \\quad \\Phi_0 = \\int_{x\\in {\\mathbb R}^n} \\phi(x) dx .\n\\end{equation}\nFor example, if $K$ is the indicator function of the ball of radius $1,$ then $ k_0 = \\vert {\\mathbb S}^{n-1} \\vert \/ 2n(n+2) ,$ where $\\vert {\\mathbb S}^{n-1} \\vert $ is the volume of the sphere ${\\mathbb S}^{n-1}$. In the cases $d = 2$ and $d = 3$, we respectively get $ k _0= \\pi\/8 $ and $ k_0 =2\\pi\/15 .$ The local density $\\rho_f,$ the local current density $J_f$ and local average orientation $\\Omega_f$ are defined by\n\\begin{eqnarray}\n& & \\hspace{-1cm} \\rho_f(x,t) = \\int_{{\\mathbb S}^{n-1} } f(x, w, t) \\, dw, \\quad\n\\label{f1}\\\\\t\n& & \\hspace{-1cm} \nJ_f(x,t) = \\int_{\\omega \\in {\\mathbb S}^{n-1}} f(x, w,t) \\, w \\, d w, \\quad\n\\Omega_f(x,t) = \\dfrac{J_f(x,t)}{|J_f(x,t)|}. \n\\label{f2} \n\\end{eqnarray}\nMore details about this Taylor expansion are given in Appendix \\ref{sec:taylorexpasion} .\tLet us observe that this scaling, first proposed in \\cite{Degond_etal_MAA13} is different from the one used in \\cite{Degond_Motsch_M3AS08} and results in the appearance of the viscosity term at the right-hand side of Eq. (\\ref{macro2_intro}). \n\nFinally, if we neglect the terms of order $\\varepsilon^2$ and we define the so-called collision operator $Q(f)$ by\n$$\n Q(f) = - \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\Omega_f f) + d \\Delta_\\omega f,\n$$\nthe rescaled system (\\ref{eqmeanfield5}), (\\ref{eqmeanfield6}) can be rewritten as follows\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\varepsilon \\Big ( \\partial_t f^\\varepsilon + \\nabla_x \\cdot (v^\\varepsilon_f f^\\varepsilon) + \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v^\\varepsilon_f f^\\varepsilon) + \\nabla_\\omega \\cdot (P_{\\omega^\\perp} G^1_{f^\\varepsilon} f^\\varepsilon) \\Big ) = Q(f^\\varepsilon), \n\\label{lastscaling1}\\\\\n&&\\hspace{-1cm} v_{f^\\varepsilon}(x, \\omega,t) = \\omega - \\Phi_0 \\nabla_x \\rho_{f^\\varepsilon} , \\quad G^1_{f^\\varepsilon}(x,t) = \\dfrac{k_0}{ | J_{f^\\varepsilon}|} P_{\\Omega_f^\\perp} \\Delta_x J_{f^\\varepsilon} \n\\label{lastscaling2}\n\t\\end{eqnarray}\t\n\n\n\\subsection{Hydrodynamic limit}\n\nThe aim is now to derive a hydrodynamic model by taking the limit $\\varepsilon \\rightarrow 0$ of system (\\ref{lastscaling1}), (\\ref{lastscaling2}) where the local density $\\rho_f,$ the local current $J_f$ and the local average orientation $\\Omega_f$ are defined by (\\ref{f1}), (\\ref{f2}). \n\nWe first introduce the von Mises-Fisher (VMF) probability distribution $M_{\\Omega}(\\omega)$ of orientation $\\Omega \\in {\\mathbb S}^{n-1}$ defined for $\\omega \\in {\\mathbb S}^{n-1}$ by: \n$$\nM_\\Omega(\\omega)=Z^{-1} \\, \\exp\\left(\\dfrac{\\omega \\cdot \\Omega}{d}\\right), \\qquad Z = \\int_{\\omega \\in {\\mathbb S}^{n-1}} \\exp\\left(\\dfrac{\\omega \\cdot \\Omega}{d}\\right) \\, d\\omega\n$$\nAn important parameter will be the flux of the VMF distribution, i.e. $\\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\omega d\\omega$. By obvious symmetry consideration, we have \n$$\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\, \\omega \\, d\\omega = c_1 \\Omega, \n$$\nwhere the quantity $c_1 = c_1(d)$ does not depend on $\\Omega$, is such that $0 \\leq c_1(d) \\leq 1$ and is given by \n\\begin{equation}\n\\label{c1}\nc_1(d) = \\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\, (\\omega \\cdot \\Omega) \\, d\\omega.\n\\end{equation}\nWhen $d$ is small, $M_\\Omega$ is close to a Dirac delta $\\delta_\\Omega$ and represents a distribution of perfectly aligned particles in the direction of $\\Omega$. When $d$ is large, $M_\\Omega$ is close to a uniform distribution on the sphere and represents a distribution of almost totally disordered orientations. \nThe function $d \\in {\\mathbb R}_+ \\mapsto c_1(d) \\in [0,1]$ is strictly decreasing with $\\lim_{d \\to 0} c_1(d) = 1$, $\\lim_{d \\to \\infty} c_1(d) = 0$. Therefore, $c_1(d)$ represents an order parameter, which corresponds to perfect disorder when it is close to $0$ and perfect alignment order when it is close to $1$. \n\n\\medskip\n\\noindent\nWe have following theorem:\n\n\\begin{theorem}\n\\label{thm:sm}\nLet $f^\\varepsilon$ be the solution of (\\ref{lastscaling1}), (\\ref{lastscaling2}). Assume that there exists $f$ such that \n\\begin{equation}\nf^\\varepsilon \\rightarrow f \\quad \\mbox{ as } \\quad \\varepsilon \\rightarrow 0, \n\\label{eq:converg}\n\\end{equation} \npointwise as well as all its derivatives. Then, there exist $ \\rho(x,t)$ and $\\Omega(x,t)$ such that\n\t\\begin{equation}\\label{f0}\n\t\tf(x,\\omega,t) = \\rho(x,t) M_{\\Omega(x,t)}(\\omega),\n\t\\end{equation}\t\t\t\nMoreover, the functions $\\rho(x,t), \\Omega(x,t)$ satisfy the following equations\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t\\rho + \\nabla_x \\cdot(\\rho U ) = 0, \n\\label{macro1} \\\\\n&&\\hspace{-1cm} \\rho \\big(\\partial_t \\Omega + (V \\cdot \\nabla_x)\\Omega \\big) + P_{\\Omega^\\perp}\\nabla_x p(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \n\\label{macro2}\t\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&\\hspace{-1cm} U = c_1 \\Omega - \\Phi_0 \\nabla_x \\rho , \\quad V = c_2 \\Omega - \\Phi_0 \\nabla_x \\rho ,\n\\label{macro21}\\\\\n&&\\hspace{-1cm} p(\\rho ) = d \\rho + \\alpha \\Phi_0 \\big( (n-1)d + c_2 \\big) \\dfrac{\\rho^2}{2}, \\quad \\gamma = k_0 \\big( (n-1)d + c_2 \\big).\n\\label{macro22}\n\\end{eqnarray}\nand the coefficients $c_1, c_2$ will be defined in formulas (\\ref{c1}), (\\ref{c2}) below.\t\t\n\\end{theorem}\n\n\n\\noindent\nGoing back to unscaled variables, we find the model (\\ref{macro1_intro})-(\\ref{macro5_intro}) presented in the Introduction. \n\n\n\\noindent\n{\\bf{Proof:} }\n The proof of this theorem is divided into three steps: (i) determination of the equilibrium states ; (ii) determination of the Generalized Collision Invariants ; (iii) hydrodynamic limit. We give a sketch of the proof for each step.\n\n\\medskip\n\\noindent\n{\\bf Step (i): determination of the equilibrium states} We define the equilibria as the elements of the null space of $Q$, considered as an operator acting on functions of $\\omega$ only. \n\n\\begin{definition} \\label{def:small_equi}\nThe set ${\\mathcal E}$ of equilibria of $Q$ is defined by\n$$\n{\\mathcal E} = \\big\\{ f \\in H^1({\\mathbb S}^{n-1}) \\, \\, | \\, \\, f \\geq 0 \\mbox{ and } Q(f) = 0 \\big\\} .\n$$\n\\end{definition}\n\\noindent\n\nWe have the following:\n\n\\begin{lemma}\nThe set ${\\mathcal E}$ is given by\n$$\n\\mathcal E = \\Big\\{ \\rho M_\\Omega(\\omega) \\, \\, | \\, \\, \\rho \\in {\\mathbb R}_+, \\, \\, \\Omega \\in {\\mathbb S}^{n-1} \\Big\\}\n$$\n\\label{lem:smallkin_equi}\n\\end{lemma}\n\n\\noindent\n\\vspace{-0.2cm}\nFor a proof of this lemma, see \\cite{Degond_Motsch_M3AS08}. The proof relies on writing the collision operator as\n$$\n Q(f) = \\nabla_\\omega \\cdot \\Big( M_{\\Omega_f} \\nabla_\\omega \\big( \\frac{f}{M_{\\Omega_f}} \\big) \\Big). \n$$\n\n\n\\medskip\n\\noindent\n{\\bf Step (ii): Generalized Collision Invariants (GCI).} We begin with the definition of a collision invariant. \n\n\\begin{definition}\nA collision invariant (CI) is a function $\\psi(\\omega)$ such that for all functions $f(\\omega)$ with sufficient regularity we have\n$$\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} Q (f) \\, \\psi \\, d\\omega = 0.\n$$\nWe denote by ${\\mathcal C}$ the set of CI. The set ${\\mathcal C}$ is a vector space.\n\\label{def:CI}\n\\end{definition}\n\n\n\\noindent\nAs seen in \\cite{Degond_Motsch_M3AS08}, the space of CI is one dimensional and spanned by the constants. Physically, this corresponds to conservation of mass during particle interactions. Since energy and momentum are not conserved, we cannot hope for more physical conservations. Thus the set of CI is not large enough to allow us to derive the evolution of the macroscopic quantities $\\rho$ and $\\Omega.$ To overcome this difficulty, a weaker concept of collision invariant, the so-called ``Generalized collisional invariant'' (GCI) has been introduced in \\cite{Degond_Motsch_M3AS08}. To introduce this concept, we first define the operator ${\\mathcal Q}(\\Omega, f)$, which, for a given $\\Omega \\in {\\mathbb S}^{n-1}$, is given by\n$$\n {\\mathcal Q}(\\Omega, f) = \\nabla_\\omega \\cdot \\Big( M_\\Omega \\nabla_\\omega \\big( \\frac{f}{M_\\Omega} \\big) \\Big). \n$$\nWe note that \n\\begin{equation}\nQ(f) = {\\mathcal Q}(\\Omega_f, f), \n\\label{eq:calQ_def}\n\\end{equation}\nand that for a given $\\Omega \\in {\\mathbb S}^{n-1}$, the operator $f \\mapsto \n {\\mathcal Q}(\\Omega, f)$ is a linear operator. Then we have the\n\n\n\\begin{definition}\nLet $\\Omega \\in {\\mathbb S}^{n-1}$ be given. A Generalized Collision Invariant (GCI) associated to $\\Omega$ is a function $\\psi \\in H^1({\\mathbb S}^{n-1})$ which satisfies:\n\\begin{equation}\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} {\\mathcal Q} (\\Omega, f) \\, \\psi(\\omega) \\, d\\omega = 0, \\quad \\forall f \\in H^1({\\mathbb S}^{n-1}) \\quad \\mbox{ such that } \\quad P_{\\Omega^\\bot} \\Omega_f = 0. \n\\label{eq:sm_def_GCI}\n\\end{equation}\nWe denote by ${\\mathcal G}_\\Omega$ the set of GCI associated to $\\Omega$.\n\\label{def:GCI}\n\\end{definition}\n\n\\noindent\nThe following Lemma characterizes the set of generalized collision invariants.\n\n\\begin{lemma}\nThere exists a positive function $h$: $[-1,1] \\to {\\mathbb R}$ such that \n\\begin{equation*}\n{\\mathcal G}_\\Omega = \\{ C + h(\\omega \\cdot \\Omega) \\beta \\cdot \\omega \\, \\, \\mbox{with arbitrary} \\, \\, C\\in {\\mathbb R} \\mbox{ and } \\beta \\in {\\mathbb R}^n \\mbox{ such that } \\beta \\cdot \\Omega = 0 \\}.\n \\end{equation*}\nThe function $h$ is such that $h(\\cos \\theta) = \\frac{g(\\theta)}{ \\sin \\theta}$ and $ g(\\theta)$ is the unique solution in the space $V$ defined by\n\\begin{eqnarray*}\n&&\\hspace{-1cm}\nV = \\{ g \\, \\, \\vert \\, \\, (n-2)(\\sin \\theta)^{\\frac{n}{2} - 2} g \\in L^2(0, \\pi), \\quad (\\sin \\theta)^{\\frac{n}{2} - 1} g \\in H^1_0(0, \\pi) \\},\n \\end{eqnarray*}\n(denoting by $H^1_0(0, \\pi) $ the Sobolev space of functions which are square integrable as well as their derivative and vanish at the boundary) of the problem\n\\begin{eqnarray*}\n&&\\hspace{-1cm}\n-\\sin^{2-n} \\theta \\, e^{ -\\frac{\\cos\\theta}{d}} \\, \\frac{d}{d \\theta} \\big( \\sin^{n-2} \\theta \\, e^{ \\frac{\\cos\\theta}{d}} \\, \\,\\frac{dg}{d \\theta}(\\theta) \\big) + \\dfrac{n-2}{\\sin^2 \\theta } \\, g(\\theta)= \\sin \\theta.\n \\end{eqnarray*}\nThe set ${\\mathcal G}_\\Omega$ is a $n$-dimensional vector space.\n\\end{lemma}\n\\noindent\nFor a proof we refer to \\cite{Degond_Motsch_M3AS08} for $n=3$ and to \\cite{Frouvelle_M3AS12} for general $n \\geq 2$. We denote by $\\psi_\\Omega$ the vector GCI \n\\begin{equation}\n\\psi_\\Omega = h(\\omega \\cdot \\Omega) \\, P_{\\Omega^\\bot} \\omega, \n\\label{eq:psi}\n\\end{equation}\n\n\\noindent\nWe note that, thanks to (\\ref{eq:calQ_def}) and (\\ref{eq:sm_def_GCI}), we have\n\\begin{equation}\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} Q(f) \\, \\psi_{\\Omega_f}(\\omega) \\, d\\omega = 0, \\quad \\forall f \\in H^1({\\mathbb S}^{n-1}). \n\\label{eq:intGCI}\n\\end{equation}\n\n\\medskip\n\\noindent\n{\\bf Step (iii): Hydrodynamic limit $\\varepsilon \\rightarrow 0.$} In the limit $\\varepsilon \\rightarrow 0$, we assume that (\\ref{eq:converg}) holds. Then , thanks to (\\ref{lastscaling1}), we have $Q(f) = 0 $. In view of Lemma \\ref{lem:smallkin_equi}, this implies that $f$ has the form (\\ref{f0}). We now need to determine the equations satisfied by $\\rho$ and $\\Omega$. \n\nFor this purpose, we divide Eq. (\\ref{lastscaling1}) by $\\varepsilon$ and integrate it with respect to $\\omega$. Writing (\\ref{lastscaling1}) as \n\\begin{equation}\n({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3)f^{\\varepsilon}=\\dfrac{1}{\\varepsilon} \\, Q(f^\\varepsilon), \n\\label{eq:kin_reduced}\n\\end{equation}\nwhere \n\\begin{equation}\n{\\mathcal T}_1 f =\\partial_t f + \\nabla_x \\cdot (v_f f), \\quad {\\mathcal T}_2 f = \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\, v_f \\, f), \\quad {\\mathcal T}_3 f = \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\, G^1_{f} \\, f), \n\\label{eq:T123}\n\\end{equation}\nwe observe that the integral of ${\\mathcal T}_2 f^\\varepsilon$ and ${\\mathcal T}_3 f^\\varepsilon$over $\\omega$ is zero since it is in divergence form and the integral of the right-hand side of (\\ref{eq:kin_reduced}) is zero since $1$ is a CI. The integral of ${\\mathcal T}_1 f^\\varepsilon$ gives: \n$$ \\partial_t \\rho_{f^\\varepsilon} + \\nabla_x \\cdot \\Big( \\int_{{\\mathbb S}^{n-1}} f^\\varepsilon(x,\\omega,t) \\, v_{f^\\varepsilon} (x,\\omega,t) \\, d\\omega \\Big) = 0. $$\nWe take the limit $\\varepsilon \\rightarrow 0$ and use (\\ref{eq:converg}) to get Eq. (\\ref{macro1}) with \n$$ U = \\int_{{\\mathbb S}^{n-1}} \\rho(x,t) \\, M_{\\Omega(x,t)} (\\omega) \\, v_{\\rho M_\\Omega} (x,\\omega,t) \\, d\\omega. $$\nUsing (\\ref{lastscaling2}), we get $v_{\\rho M_\\Omega} (x,\\omega,t) = \\omega - \\Phi_0 \\nabla_x \\rho(x,t)$. With (\\ref{c1}), this leads to the first equation (\\ref{macro21}). \n\n\nMultiplying (\\ref{eq:kin_reduced}) by $\\psi_{\\Omega_{f^\\varepsilon}}$, integrating with respect to $\\omega$ and using (\\ref{eq:intGCI}), we get\n$$ \\int_{{\\mathbb S}^{n-1}} ({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3)f^{\\varepsilon}(x,\\omega,t) \\, \\psi_{\\Omega_{f^\\varepsilon}} (x,\\omega,t) \\, d \\omega = 0. \n$$\n and taking the limit $\\varepsilon \\rightarrow 0,$ we get\n\\begin{equation}\n\\label{vitesse}\n\\int_{{\\mathbb S}^{n-1}} ( ({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3) (\\rho M_{\\Omega}))(x,\\omega,t) \\, \\psi_{\\Omega(x,t)} (\\omega) \\, d \\omega = 0. \n\\end{equation}\nThis equation describes the evolution of the mean direction $\\Omega.$ The computations which lead to (\\ref{macro2}) are proved in the Appendix \\ref{subsec:small_equi}.\nThe coefficient $c_2$ in (\\ref{macro2}) is defined by\n\\begin{eqnarray}\n\\label{c2}\n& & c_2(d) = \\dfrac{\\langle \\sin^2 \\theta \\cos \\theta \\, h \\rangle_{M_\\Omega} }{ \\langle \\sin^2 \\theta \\, h \\rangle_{M_\\Omega}}= \\dfrac{\\int_{0}^{\\pi} \\sin^n \\theta \\cos \\theta \\, M_\\Omega \\,h \\,d\\theta}{\\int_{0}^{\\pi} \\sin^n \\theta \\,M_\\Omega \\,h \\,d\\theta},\n\\end{eqnarray}\nwhere for any function $g(\\cos \\theta),$ we denote $\\langle g \\rangle$ by\n$$\n\\langle g \\rangle_{M_\\Omega} = \\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega)\\,g(\\omega \\cdot \\Omega) \\,d\\omega = \\dfrac{\\int_{0}^{\\pi} g(\\cos \\theta) \\,e^{\\frac{\\cos \\theta}{d}}\\, \\sin^{n-2} \\theta \\,d\\theta}{\\int_{0}^{\\pi} e^{\\frac{\\cos \\theta}{d}} \\,\\sin^{n-2} \\,\\theta \\,d\\theta}.\n$$\n\n\\medskip\n\\begin{remark}\nThe SOHR model (\\ref{macro1}), (\\ref{macro2}) can be rewritten as follows\n\\begin{eqnarray*}\n&&\\hspace{-1cm} \\partial_t \\rho + c_1 \\nabla_x \\cdot(\\rho \\Omega) = \\Phi_0 \\Delta_x \\left(\\dfrac{\\rho^2}{2}\\right), \n\\\\\n&&\\hspace{-1cm} \\partial_t \\Omega + (\\bar V \\cdot \\nabla_x)\\Omega + P_{\\Omega^\\perp}\\nabla_x h(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x \\Omega, \n\\end{eqnarray*}\nwhere the vectors $ \\bar V$ and the function $h(\\rho)$ are defined by\n\\begin{eqnarray*}\n\\bar V = c_2 \\Omega - (\\Phi_0 + 2 \\gamma) \\nabla_x \\rho, \\quad h'(\\rho) = \\frac{1}{\\rho} \\, p'(\\rho), \n\\end{eqnarray*}\nand where the primes denote derivatives with respect to $\\rho$. This writing displays this system in the form of coupled nonlinear advection-diffusion equations. \n \\end{remark}\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Numerical discretization of the SOHR model}\n\\label{sec:num_solution}\nIn this section, we develop the numerical approximation of the system (\\ref{macro1})-(\\ref{macro22}) in the two dimensional case. As mentioned above, this system is not conservative because of the geometric constraint $|\\Omega| = 1.$ Weak solutions of non-conservative systems are not unique because jump relations across discontinuities are not uniquely defined. This indeterminacy cannot be waived by means of an entropy inequality, by contrast to the case of conservative systems. In \\cite{Motsch_Navoret_MMS11} the authors address this problem for the SOH model. They show that the model is a zero-relaxation limit of a conservative system where the velocity $\\Omega$ is non-constrained (i.e. belongs to ${\\mathbb R}^n$). Additionally, they show that the numerical solutions build from the relaxation system are consistent with those of the underlying particle model, while other numerical solutions built directly from the SOH model are not. Here we extend this idea to the SOHR model. More precisely, we introduce the following relaxation model (in dimension $n=2$):\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t \\rho^\\eta + \\nabla_x \\cdot (\\rho^\\eta U^\\eta) = 0, \\label{relax1}\\\\\n&&\\hspace{-1cm} \\partial_t(\\rho^\\eta \\Omega^\\eta) + \\nabla_x \\cdot ( \\rho^\\eta V^\\eta \\otimes \\Omega^\\eta) + \\nabla_x p( \\rho^\\eta ) - \\gamma \\Delta_x(\\rho^\\eta \\Omega^\\eta) = \\dfrac{\\rho^\\eta}{\\eta} (1 - |\\Omega^\\eta|^2) \\Omega^\\eta, \\label{relax2} \\\\\n&&\\hspace{-1cm} U^\\eta = c_1 \\Omega^\\eta - \\Phi_0 \\nabla_x \\rho^\\eta , \\quad V^\\eta = c_2 \\Omega^\\eta - \\Phi_0 \\nabla_x \\rho^\\eta,\\label{relax22}\\\\\n&&\\hspace{-1cm} p(\\rho^\\eta) = d \\rho^\\eta + \\alpha \\Phi_0 \\big( d + c_2 \\big) \\dfrac{(\\rho^\\eta)^2}{2}, \\quad \\gamma = k_0 \\big( d + c_2 \\big).\n\\label{relax23}\n\\end{eqnarray}\nThe left-hand sides form a conservative system. We get the following proposition:\n\n\\begin{proposition}\n\\label{prop1}\nThe relaxation model (\\ref{relax1})-(\\ref{relax23}) converges to the SOHR model (\\ref{macro1})-(\\ref{macro22}) as $\\eta $ goes to zero.\n\\end{proposition}\n\n\\noindent\nThe proof of proposition \\ref{prop1} is given in Appendix \\ref{numerical:convergence}. This allows us to use well-established numerical techniques for solving the conservative system (i.e. the left-hand side of (\\ref{relax1}), (\\ref{relax2})). The scheme we propose relies on a time splitting of step $\\Delta t$ between the conservative part\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t \\rho^{\\eta} + \\nabla_x \\cdot (\\rho^{\\eta} U^{\\eta}) = 0, \n\\label{split1}\\\\\n&&\\hspace{-1cm} \\partial_t(\\rho^{\\eta} \\Omega^{\\eta}) + \\nabla_x \\cdot ( \\rho^{\\eta} V^{\\eta} \\otimes \\Omega) + \\nabla_x p(\\rho^{\\eta}) - \\gamma \\Delta_x (\\rho^{\\eta} \\Omega^{\\eta}) = 0, \n\\label{split2}\n\\end{eqnarray}\nand the relaxation part\n\\begin{eqnarray}\n&&\\hspace{-1cm}\\partial_t \\rho^\\eta = 0, \n\\label{relax3} \\\\\n&&\\hspace{-1cm}\\partial_t(\\rho^\\eta \\Omega^\\eta) = \\dfrac{\\rho^\\eta}{\\eta} (1 - |\\Omega^\\eta|^2) \\Omega^\\eta. \n\\label{relax4}\n\\end{eqnarray}\n\n\\medskip\n\\noindent\nSystem (\\ref{split1}-\\ref{split2}) can be rewritten in the following form (we omit the superscript $\\eta$ for simplicity)\n$$\n\tQ_t + (F(Q, Q_x))_x + (G(Q, Q_y))_y = 0,\n$$\nwhere\n$$\n\tQ = \\begin{pmatrix}\n\t\t\t\t\\rho \\\\ \\rho \\Omega_1 \\\\ \\rho \\Omega_2\n\t\t\\end{pmatrix}, \\quad\n\tF(Q, Q_x) = \\begin{pmatrix}\n\t\t\t\t \\rho U_1 \\\\\n\t\t\t\t\\rho \\Omega_1 V_1 + p(\\rho) - \\gamma \\partial_x(\\rho \\Omega_1), \\\\\n\t\t\t\t\\rho \\Omega_1 V_2 - \\gamma \\partial_x(\\rho \\Omega_2)\n\t\t\\end{pmatrix}, \n$$\n$$\n\tG(Q, Q_y) = \\begin{pmatrix}\n\t\t\t\t \\rho U_2 \\\\\n\t\t\t\t \\rho \\Omega_2 V_1 - \\gamma \\partial_y(\\rho \\Omega_1) \\\\\n\t\t\t\t\\rho \\Omega_2 V_2 + p(\\rho) - \\gamma \\partial_y(\\rho \\Omega_2)\n\t\t\t\t\\end{pmatrix}.\n$$\n\n\n\\medskip\n\\noindent\nWe consider now the following numerical scheme where we denoted $Q^*_{i,j}$ the approximation of $Q$ at time $t^{n+1} = (n+1) \\Delta t $ and position $x_i = i \\Delta x, y_j = j \\Delta y$:\n\\begin{eqnarray*}\n Q^{*}_{i,j} & = & Q^{n}_{i, j} - \\dfrac{\\Delta t}{\\Delta x} \\{F^n_{i +1\/2,j} - F^n_{i-1\/2,j}\\} \n- \\dfrac{\\Delta t}{\\Delta y} \\{ G^n_{i,j+1\/2} - G^n_{i,j-1\/2} \\}, \n\\end{eqnarray*}\nwhere the numerical flux $F^n_{i +1\/2,j}$ is given by\n\\begin{equation*}\n\t F^n_{i +1\/2,j} = \\dfrac{F^n(Q^n_{i,j}, Q^n_{xi, j}) + F^n(Q^n_{i+1,j}, Q^n_{x(i+1),j})}{2} - P^{i+\\frac{1}{2}}_2 \\Big( \\dfrac{\\partial F}{\\partial Q}(\\bar Q^n_{i,j}, \\bar{Q}^n_{xi,j} ) \\Big) (Q^n_{i+1,j} - Q^n_{i,j}),\n\\end{equation*}\nwith\n\\begin{equation*}\nQ^n_{xi,j} = \\frac{(Q^n_{i+1,j} - Q^n_{i,j} )}{\\Delta x} , \\quad \\bar{Q}^n_{i,j} = \\dfrac{Q^n_{i,j} + Q^n_{i+1, j}}{2},\t \\quad \\bar{Q}^n_{xi,j} = \\dfrac{Q^n_{xi,j} + Q^n_{x(i+1), j}}{2},\n\\end{equation*}\nand the analogous discretization holds for $G^n_{i ,j+ \\frac{1}{2}}$.\n\n\\noindent\nIn the above formula, $P^{i+\\frac{1}{2}}_2$ is a polynomial of matrices of degree $2$ calculated with the eigenvalues of the Jacobian matrices $\\dfrac{\\partial F}{\\partial Q}$ at an intermediate state depending on $(Q^n_{i,j}, Q^n_{xi, j})$ and $(Q^n_{i+1,j}, Q^n_{x(i+1), j})$ as detailed in \\cite{Degond_Peyrard_99}.\nTo ensure stability of the scheme, the time step $\\Delta t$ satisfies a Courant-Friedrichs-Lewy (CFL) condition computed as the minimum of the CFL conditions required for the hyperbolic and diffusive parts of the system. \n\n\nOnce the approximate solution of the conservative system is computed, equations (\\ref{relax3}) and (\\ref{relax4}) can be solved explicitly. In the limit $\\eta\\rightarrow 0$ they give\n$$\n \\rho^{n+1}=\\rho^{*}, \\qquad \\Omega^{n+1}=\\frac{\\Omega^{*}}{|\\Omega^{*}|}\n$$\nwhere $(\\rho^*,\\Omega^{*})$ is the numerical solution of system (\\ref{split1}-\\ref{split2}). This ends one step of the numerical scheme for the system (\\ref{relax1}-\\ref{relax2}).\n\n\n\\section{Numerical tests}\nThe goal of this section is to present some numerical solutions of the system (\\ref{macro1})-(\\ref{macro22}) which validate the numerical scheme proposed in the previous section. We will first perform a convergence test. We then successively compare the solutions obtained with the SOHR model with those computed by numerically solving the individual based model (\\ref{overdamped1}) in regimes in which the two models should provide similar results. We will finally perform some comparisons between the SOH and the SOHR system to highlight the difference between the two models. We will compare the SOHR model with another way to incorporate repulsion in the SOH Model, the so-called DLMP model of \\cite{Degond_etal_MAA13}.\n\nFor all the tests, we use the model in uscaled variables as described in the Introduction (see (\\ref{macro1_intro})-(\\ref{macro5_intro}). The potential kernel $\\phi$ is chosen as\n\\begin{equation}\n\\label{phi}\n\\phi(x) = \\begin{cases}\n (|x| -1)^2 \\quad \\mbox{if} \\quad |x| \\leq1,\\\\\n0 \\quad \\mbox{if} \\quad |x| > 1,\n\\end{cases}\n\\end{equation}\nwhich gives $\\Phi_0 = \\dfrac{\\pi}{6}$, while for $K$, by assumption normalized to $1,$ we choose the following form\n$$\nK(\\vert z \\vert ) = \\begin{cases}\n \\dfrac{1}{\\pi} \\quad \\mbox{if} \\quad |z| \\leq 1,\\\\\n0 \\quad \\mbox{if} \\quad |z| > 1.\n\\end{cases}\n$$\nThis leads to $k_0 = \\dfrac{1}{8}.$ The other parameters, which are fixed for all simulations if not differently stated, are :\n$$ v_0 = 1, \\,\\, \\mu = \\dfrac{1}{2}, \\,\\, \\alpha = 1, \\,\\, d = 0.1, \\,\\, L_x = 10, \\,\\, L_y = 10, $$\nwhich, in dimension $n=2$, lead to (after numerically computing the GCI and the associated integrals):\n$$ c_1 = 0.9486, \\,\\, c_2 =0.8486. $$\nIn the visualization of the results, we will use the angle $\\theta$ of the vector $\\Omega$ relative to the $x$-axis, i.e. $\\Omega = (\\cos \\theta, \\sin \\theta)$. \n\n\\subsection{Convergence test}\nThe first test is targeted at the validation of the proposed numerical scheme. For this purpose, we investigate the convergence when the space step $(\\Delta x, \\Delta y )$ tends to $(0,0)$, refining the grid and checking how the error behaves asymptotically. The initial mesh size is $\\Delta x = \\Delta y = 0.25$ while the time step is $\\Delta t = 0.001.$ We repeat the computation for $(\\dfrac{\\Delta x}{2}, \\dfrac{\\Delta y}{2}),$ $(\\dfrac{\\Delta x}{4}, \\dfrac{\\Delta y}{4}),$ $(\\dfrac{\\Delta x}{8}, \\dfrac{\\Delta y}{8})$. The convergence rate is estimated through the measure of the $L^1$ norm of the error for the vectors $(\\rho, \\cos\\theta)$ by using for each grid the next finer grid as reference solution. The initial data is\n\\begin{equation}\n\\label{Vortex}\n\\rho_0 = 1, \\quad \\theta_0(x,y) = \\begin{cases}\n\t\t\t\t\t\t\t\t\t\t\\arctan(\\dfrac{y_1}{x_1} ) + \\dfrac{\\pi}{2} sign(x_1) \\,\\, \\mbox{if} \\,\\, x_1 \\neq 0, \\\\\n\t\t\t\t\t\t\t\t\t\t\\pi \\,\\,\\mbox{if} \\,\\, x_1 = 0 \\,\\, \\mbox{and} \\,\\, y_1 > 0, \\\\\n 0 \\,\\, \\mbox{if} \\,\\, x_1 = 0 \\,\\, \\mbox{and} \\,\\, y_1 < 0,\n \\end{cases}\n\\end{equation}\nwhere\n$$ x_1 = x - \\dfrac{L_x}{2}, \\quad y_1 = y - \\dfrac{L_y}{2}.$$\nThe boundary conditions are fixed in time on the four sides of the square : $( \\rho^n, \\theta^n ) = (\\rho_0, \\theta_0 )$.\nThe error curves for the density and for $ \\cos \\theta $ are plotted in figure \\ref{fig:scheme} as a function of the space step in log-log scale at time $T = 1s.$ The slope of the error curves are compared to a straight line of slope 1.\nFrom the figure, we observe the convergence of the scheme with accuracy close to $1.$\n\n\\noindent\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[height=8cm,width=11cm]{errorMacro3.png}\n\\caption{$L^1$-error for the density $\\rho$ and the flux direction $ \\cos\\theta $ as a function of $\\Delta x$ in log-log scale. A straight line of slope $1$ is plotted for reference. This figure shows that the scheme is numerically of order $1$. \n}\n\\label{fig:scheme}\n\\end{figure}\n\n\n\\subsection{Comparison between the SOHR and the Vicsek model with repulsion}\n\nIn this subsection, we validate the SOHR model by comparing it to the Vicsek model with repulsion on two different test cases. We investigate the convergence of the microscopic IBM to the macroscopic SOHR model when the scaling parameter $\\varepsilon$ tend to zero. The scaled IBM is written: \n\\begin{eqnarray*} \n&&\\hspace{-1cm} \\dfrac{dX_k}{dt} = v_k, \\qquad \nv_k\t= \\omega_k - \\, \\nabla_x \\Phi(X_k(t),t),\\\\\n&&\\hspace{-1cm} d\\omega_k\t= P_{\\omega_k^\\perp} \\circ \\Big( \\frac{1}{\\varepsilon} \\, \\bar{\\omega}(X_k(t),t) \\, dt + \\alpha \\, v_k \\, dt + \\sqrt{\\frac{2d}{\\varepsilon}} \\, dB^k_t \\Big).\\\\\n&&\\hspace{-1cm} \\Phi(x,t) = \\dfrac{1}{\\varepsilon^2 \\, N} \\sum_{i=1}^{N} \\nabla \\phi \\Big( \\frac{|x - X_i|}{\\varepsilon \\, r} \\Big), \\\\\n&&\\hspace{-1cm} \\bar{\\omega}(x,t) = \\dfrac{{\\mathcal J}(x,t)}{\\vert {\\mathcal J}(x,t) \\vert}, \\qquad {\\mathcal J} (x,t) = \\frac{1}{N} \\sum\\limits_{i=1}^{N} K\\Big(\\frac{|x-X_i|}{\\sqrt{\\varepsilon} \\, R}\\Big) \\omega_i. \n\\end{eqnarray*}\t\nThe solution of the individual based model (\\ref{overdamped1}-\\ref{overdamped3}) is computed by averaging different realizations in order to reduce the statistical errors. The coefficient of the IBM are fixed to $r = 0.0625$ for the repulsive range, $R = 0.25$ for the alignment interaction range, while $N = 10^5$ particles are used for each simulation. The details of the particles simulation can be found in \\cite{Fehske_2007, Hockney_1988} for classical particle approaches or in \\cite{Motsch_Navoret_MMS11} for a direct application to the SOH model.\n\n\\medskip\n\\noindent\n\\paragraph{Riemann problem:}\nThe convergence of the two models is measured on a Riemann problem with the following initial data\n\\begin{equation}\n\\label{RM}\n(\\rho_l, \\theta_l ) = (0.0067, 0.7 ) ,\\quad (\\rho_r, \\theta_r ) = (0.0133, 2.3).\n\\end{equation}\nand with periodic boundary condition in $x$ and $y$. The parameters of the SOHR model are: $\\Delta t = 0.01, \\,\\, \\Delta x = \\Delta y = 0.25$. In figure \\ref{fig:macro_micro} we report the relative $L^1$ norm of the error for the macroscopic quantities $(\\rho, \\theta)$ between the SOHR model and the particle model with respect to the number of averages for different values of $\\varepsilon$ : $ \\varepsilon = 1$ (x-mark), $\\varepsilon = 0.5$ (plus), $\\varepsilon = 0.1$ (circle), $\\varepsilon = 0.05$ (square) at time $T = 1s.$ This figure shows, as expected, that the distance between the two solutions goes to zero when $\\varepsilon$ goes to zero.\nIn figure \\ref{fig:solution_RM} we report the density $\\rho$ and the flux direction $\\theta$ for the same Riemann problem along the $x$-axis for $\\varepsilon = 0.05$ at time $T = 1s$, the solution being constant in the $y$-direction. Again we clearly observe that the two models provide very close solutions, the small differences being due to the different numerical schemes employed for their discretizations.\n\n\n\\begin{figure}[!ht]\n\\subfloat[For density $\\rho$ \\label{dd}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_macro_micro_RM.png}\n }\n\\hfill\n\\subfloat[For $\\theta$ \\label{velo}]{%\n\\includegraphics[width=0.51\\textwidth]{theta_macro_micro_RM.png}\n }\n\\caption{Relative error between the macroscopic and the microscopic model for density (a) and $ \\theta $ (b) as a function of the number of averages for different values of $\\varepsilon$. The error decreases with both decreasing $\\varepsilon$ and increasing number of averages, showing that the SOHR model provides a valid approximation of the IBM for $\\rho$ and $\\theta$. }\n\\label{fig:macro_micro}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ \\label{density}]{%\n\\includegraphics[width=0.5\\textwidth]{density_RM.png}\n }\n\\hfill\n\\subfloat[ $\\theta$ \\label{velocity}]{%\n\\includegraphics[width=0.5\\textwidth]{theta_RM.png}\n }\n\\caption{Solution of the Riemann problem (\\ref{RM}) along the $x$-axis for the SOHR model (blue line) and for the IBM with $\\varepsilon = 0.05$ (red line) at $T = 1s.$ The agreement between the two models is excellent. For the SOHR model, the mesh size is $\\Delta x = \\Delta y = 0.0625$. }\n\\label{fig:solution_RM}\n\\end{figure}\n\n\\medskip\n\\noindent\n\\paragraph{Taylor-Green vortex problem:}\nIn this third test case, we compare the numerical solutions provided by the two models in a more complex case. The initial data are\n\\begin{equation}\n\\label{Taylor_Green}\n\\rho_0 = 0.01, \\quad \\Omega_0(x,y) = \\dfrac{\\tilde{\\Omega}_0(x,y)}{\\vert \\tilde{\\Omega}_0(x,y) \\vert},\n\\end{equation}\n where the vector $ \\tilde{\\Omega}_0 = (\\tilde{\\Omega}_{01}, \\tilde{\\Omega}_{02}) $ is given by\n\\begin{eqnarray*}\t\t\n\t&& \\tilde{\\Omega}_{01}(x,y) = \\dfrac{1}{3}\\sin( \\dfrac{\\pi}{5} x ) \\cos( \\dfrac{\\pi}{5} y ) + \\dfrac{1}{3} \\sin( \\dfrac{3\\pi}{10} x) \\cos( \\dfrac{3\\pi}{10} y ) + \\dfrac{1}{3} \\sin( \\dfrac{\\pi}{2} x) \\cos( \\dfrac{\\pi}{2} y),\\\\\n\t && \\tilde{\\Omega}_{02}(x,y) = - \\dfrac{1}{3} \\cos( \\dfrac{\\pi}{5} x) \\sin( \\dfrac{\\pi}{5} y )- \\dfrac{1}{3} \\cos( \\dfrac{3\\pi}{10} x ) \\sin( \\dfrac{3 \\pi}{10} y) - \\dfrac{1}{3} \\cos( \\dfrac{\\pi}{2} x) \\sin( \\dfrac{\\pi}{2} y ).\t\n\\end{eqnarray*}\nwith periodic boundary conditions in both directions. The numerical parameters for the SORH model are : $\\Delta x = \\Delta y = 0.2,$ $\\Delta t = 0.01$, while for the particle simulations we choose : $N = 10^5$ particles, $\\varepsilon = 0.05,$ $r = 0.04,$ $R = 0.2.$ \nIn figure \\ref{fig:dens_macro_micro_t06_GT} and \\ref{fig:velo_macro_micro_t06_GT}, we report the density $\\rho$ and the flux direction $\\Omega$ at time $t = 0.6$s. In both figures, the left picture is for the IBM and the right one for the SOHR model. Again, we find a very good agreement between the two models in spite of the quite complex structure of the solution.\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[ density $\\rho$ for the IBM \\label{density_micro_t06}]{%\n\\includegraphics[width=0.5\\textwidth]{density_micro_t06_TG.png}\n }\n\\hfill\n\\subfloat[density $\\rho$ for the SOHR model \\label{density_macro_t06}]{%\n\\includegraphics[width=0.5\\textwidth]{density_macro_t06_TG.png}\n }\n\\caption{Density $\\rho$ for Taylor-Green vortex problem \\ref{Taylor_Green} at time $t = 0.6s$. Left: IBM. Right: SOHR model. }\n\\label{fig:dens_macro_micro_t06_GT}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[ $\\Omega$ for the IBM \\label{velo_micro_t06}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_micro_t06_TG.png}\n }\n\\hfill\n\\subfloat[$\\Omega$ for the SOHR model \\label{velo_macro_t06}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_macro_t06_TG.png}\n }\n\\caption{Mean direction $\\Omega$ for Taylor-Green vortex problem \\ref{Taylor_Green} at time $t = 0.6s$. Left: IBM. Right: SOHR model. }\n\\label{fig:velo_macro_micro_t06_GT}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Comparison between the SOH and the SOHR model}\nIn this part, we show the difference between the SOH system (\\ref{macro6_intro}), (\\ref{macro7_intro}) and the SOHR one for different values of the repulsive force $\\Phi_0.$ We recall that the SOHR model reduces to the SOH one in the case in which the repulsive force is set equal to zero. To this aim, we rescale the repulsive force $\\Phi_0$ by\n$$ \\Phi_0 = F_0 \\int_{x \\in {\\mathbb R}^2} \\phi(x) dx $$\nand then we let $F_0$ vary. The repulsive potential $\\phi$ is still given by (\\ref{phi}), so that $\\Phi_0 = F_0 \\pi\/6$.\nThe other numerical parameters are chosen as follows:\n$ d = 0.05,$ $\\alpha = 0,$ $ k_0 = 1\/8, $ $\\mu = 1,$ $ L_x = 10, L_y = 10,$ $ \\Delta x = \\Delta y = 0.15,$ $ \\Delta t = 0.001.$\nThe initial data are those of the vortex problem (\\ref{Vortex}) except that we start with four vortices instead of only one. Periodic boundary conditions in both directions are used.\n\nFigure (\\ref{fig:solution_F5}) displays the solutions for the SOHR system for the density (left) and for the flux direction\n(right) at $T = 1.5s$ with $F_0 = 5.$ Figure (\\ref{fig:solution_F005}) displays the solutions for $F_0 = 0.05.$ The results are almost undistinguishable to those of the SOH model ($F_0=0$) and are not shown for this reason. \nThese figures show that when the repulsive force is large enough, the SOHR model can prevent the formation of high concentrations. By contrast, when this force is small, the SOHR model becomes closer to the SOH one and high concentrations become possible.\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ for $F_0 = 5$ \\label{F5_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F501500.png}\n }\n\\hfill\n\\subfloat[$\\Omega$ for $F_0 = 5$ \\label{F5_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F501500.png}\n }\n\\caption{Solution of the SOHR model for $F_0 = 5.$ Density $\\rho$ (fig \\ref{F5_density} ), flux direction $\\Omega$ (fig.\\ref{F5_velo} ) at $t= 1.5s$.}\n\\label{fig:solution_F5}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\subfloat[ density $\\rho$ for $F_0 = 0.05$ \\label{F005_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F00501500.png}\n }\n\\hfill\n\\subfloat[ $\\Omega$ for $F_0 = 0.05$ \\label{F005_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F00501500.png}\n}\n\\caption{Solution of the SOHR model for $F_0 = 0.05.$ Density $\\rho$ (fig \\ref{F5_density} ), flux direction $\\Omega$ (fig.\\ref{F5_velo} ) at $t = 1.5s$.}\n\\label{fig:solution_F005}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Comparison between the SOHR and the DLMP model}\n\nIn this final part, we want to compare the SOHR system to the hydrodynamic model proposed by Degond, Liu, Motsch and Panferov in \\cite{Degond_etal_MAA13} (referred to as DLMP model). This model is derived, in a similar fashion as the SOHR model, starting from a system of self-propelled particles which obey to alignment and repulsion. The main difference is that in the DLMP model, the particle velocity is exactly equal to the self propulsion velocity but the particles adjust their orientation to respond to repulsion as well as alignment. The resulting model is of SOH type and is therefore written (\\ref{macro6_intro}), (\\ref{macro7_intro}), but with an increased coefficient in front of the pressure term $P_{\\Omega^\\bot} \\nabla_x \\rho$, this coefficient being equal to $v_0 \\, d (1 + \\frac{d+c_2}{c_1}F_0)$. The initial conditions and numerical parameters are the same as in previous test\n\nIn figure \\ref{fig:solution_DLMP_F5}, we report the density $\\rho$ (left) and the flux direction $\\Omega$ (right) for $F_0 = 5$ for the DLMP model. Comparing figures (\\ref{fig:solution_F5}) with figure \\ref{fig:solution_DLMP_F5}, we observe that the solutions of the SOHR and of the DLMP model are different. The homogenization of the density seems more efficient with the SOHR model than with the DLMP model. This can be attributed to the effect of the nonlinear diffusion terms that are included in the SOHR model but not in the DLMP model. Therefore, the way repulsion is included in the models may significantly affect the qualitative behavior of the solution. In practical situations, when the exact nature of the interactions is unknown, some care must be taken to choose the right repulsion mechanism. \n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ \\label{F5DLMP_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F5DLMP01500}\n }\n\\hfill\n\\subfloat[$\\Omega$ \\label{F5DLMP_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F5DLMP01500}\n }\n\\caption{Solution of the DLMP model for $F_0 = 5.$ Density $\\rho$ (left) and flux direction $\\Omega$ (right) at $t = 1.5s$.}\n\\label{fig:solution_DLMP_F5}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Conclusion}\n\\label{sec:conclu}\nIn this paper, we have derived a hydrodynamic model for a system of self-propelled particles which interact through both alignment and repulsion. In the underlying particle model, the actual particle velocity may be different from the self-propulsion velocity as a result of repulsion interactions with the neighbors. Particles update the orientation of their self-propulsion seeking to locally align with their neighbors as in Vicsek alignment dynamics. The corresponding hydrodynamic model is similar to the Self-Organized Hydrodynamic (SOH) system derived from the Vicsek particle model but it contains several additional terms arising from repulsion. These new terms consist principally of gradients of linear or nonlinear functions of the density including a non-linear diffusion similar to porous medium diffusion. This new Self-Organized Hydrodynamic system with Repulsion (SOHR) has been numerically validated by comparisons with the particle model. It appears more efficient to prevent high density concentrations than other approaches based on simply enhancing the pressure force in the SOH model. In future work, this model will be used to explore self-organized motion in collective dynamics. To this effect, it will be calibrated on data based on biological experiments, such as recordings of collective sperm-cell motion. \n\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzozws b/data_all_eng_slimpj/shuffled/split2/finalzzozws new file mode 100644 index 0000000000000000000000000000000000000000..5af34a01c533ab0acabdfdd79fe0220ad5b193d9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzozws @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction.}\n\\label{_Intro_Section_}\n\n\n\\subsection{An overview}\n\nThis is the third article studying closed complex analytic subvarieties\nof compact holomorphically symplectic\\footnote{Holomorphically\nsymplectic means ``equipped with a holomorphic symplectic form''.\nSee \\ref{_holomorphi_symple_Definition_} for details.}\n K\\\"ahler manifolds. In the first article in series \n(\\cite{Verbitsky:Symplectic_I_}), we proved that, when a holomorphically\nsymplectic manifold $M$ is generic in its deformation class, all\nsubvarieties $X\\subset M$ are also holomorphically symplectic, \ni. e. restriction of holomorphic symplectic form to nonsingular\nstrata of $X$ is non-degenerate. In the second article\n\\cite{Verbitsky:Symplectic_II_}, we obtained a more precise result\nabout the structure of such $X$, which is related to the hyperk\\\"ahler\nstructure of $M$.\n\nBy Yau's proof of Calabi conjecture, $M$ admits a natural hyperk\\\"ahler\nmetric (\\ref{_symplectic_=>_hyperkahler_Proposition_};\nfor a definition of hyperk\\\"ahler manifold,\nsee \\ref{_hyperkahler_manifold_Definition_}). \nA hyperk\\\"ahler structure (which is\nessentially a quaternion action in the tangent bundle to $M$) gives\na rise to a whole family of complex structure on $M$, parametrized\nby ${\\Bbb C} P^1$. These complex structures are called {\\bf complex\nstructures induced by the hyperk\\\"ahler structure}\n(\\ref{_indu_comple_str_Definition_}).\n Denote the set of all induced complex structures\nby $\\c R_M$. A closed subset $X\\subset M$ is called \n{\\bf trianalytic} if $X$ is complex analytic with respect\nto all induced complex structures $L \\in \\c R_M$\n(\\ref{_trianalytic_Definition_}). For an induced\ncomplex structure $L$, we denote by $(M, L)$ the $M$ considered\nas a complex manifold, with complex structure $L$. In\n\\cite{Verbitsky:Symplectic_II_}, we proved that for all $L\\in \\c R_M$,\nwith exception of may be a countable set, all complex analytic subsets\nof $(M, L)$ are trianalytic (\\ref{_hyperkae_embeddings_Corollary_}). \n\n\nUnlike the second article \\cite{Verbitsky:Symplectic_II_},\nwhich supersedes results of the first \\cite{Verbitsky:Symplectic_I_},\nthe present one (the third) elaborates on the results of the second.\nWe start where \\cite{Verbitsky:Symplectic_II_} left. Consider\na complex subvariety $X \\subset (M, L)$ which happens to be\ntrianalytic. Such subvarieties are\ncalled {\\bf complex analytic subvarieties of trianalytic type}. We \ndescribe the deformations of $X$ in $(M, L)$ and \nthe Douady space of such deformations.\\footnote{The Douady space\n\\cite{_Douady_} is the same as Chow scheme, but in complex analytic\nsituation.}\n\nIn \\cite{Verbitsky:Symplectic_II_}, we gave a simple cohomological\ncriterion of trianaliticity, for arbitrary complex analytic\nsubvariety $X \\subset (M, L)$ \n(see \\ref{_G_M_invariant_implies_trianalytic_Theorem_} \nof this article). \\ref{_G_M_invariant_implies_trianalytic_Theorem_} \nimmediately implies that a complex analytic deformation of \na trianalytic subvariety is again trianalytic.\nThe main result of this article is the following theorem.\n\n\\hfill\n\n\\theorem \\label{_iso_intro:Theorem_}\nLet $M$ be a hyperk\\\"ahler manifold, $L$ an induced complex structure,\nand $X, X'\\subset (M, L)$ be closed complex analytic subvarieties in\nthe same deformation class. Assume that $X$ is\ntrianalytic.\\footnote{Then the subvariety $X'$ is also trianalytic,\nas implied by \\ref{_G_M_invariant_implies_trianalytic_Theorem_}.} \nConsider $X, X'$ as K\\\"ahler subvarieties of $(M, L)$,\nwith K\\\"ahler metric induced from $M$. Then \n\n\\begin{description}\n\\item[(i)] $X, X'$ are naturally\nisomorphic as complex varieties, and this isomorphism is compatible\nwith the K\\\"ahler metric.\n\n\\item[(ii)] The embedding $X\\stackrel i \\hookrightarrow M$ \nis compatible with metrics given by geodesics. \n\n\\item[(iii)] Also, every map \n$X\\stackrel \\phi{\\:\\longrightarrow\\:} (M, L)$\nis complex analytic, provided that $\\phi$ is a \ndeformation of $i$ in the space of isometries\n$X{\\:\\longrightarrow\\:} M$.\n\\end{description}\n\n\\hfill\n\n{\\bf Caution} The varieties $X, X'$ need not to be non-singular.\n\n\\hfill\n\n\\ref{_iso_intro:Theorem_} (i) follows from\n\\ref{_triana_subse_isome_Theorem_} and\n\\ref{_triana_subse_comple_ana_Theorem_}.\n\\ref{_iso_intro:Theorem_} \n(ii) follows from \\ref{_hype_embe_comple_geode:Corollary_},\nand (iii) from \\ref{_isome_embe_Proposition_}.\n\n\\hfill\n\nAs a corollary, we obtain the following interesting result\n(\\ref{_Doua_hyperka_Theorem_}).\n\n\\hfill\n\n\\theorem \\label{_defo_intro:Theorem_} \nLet $M$ be a compact holomorphically symplectic K\\\"ahler \nmanifold, and $X\\subset M$ a closed complex analytic subvariety.\nConsider the Douady space $D_M(X)$ of deformations of $X$ inside\n$M$. Then $D_M(X)$ is compact and is equipped with a natural\nhyperk\\\"ahler structure, in the sense of \n\\cite{_Verbitsky:Hyperholo_bundles_},\\footnote{If $D_M(X)$\nis non-singular, this means exactly that $D_M(X)$ is hyperk\\\"ahler.\nIn singular case, there is no satisfactory definition of a hyperk\\\"ahler\nstructure. The definition of \\cite{_Verbitsky:Hyperholo_bundles_}\nis a palliative, which is probably much stronger that the correct\ndefinition. See \\cite{_Simpson:hyperka-defi_}, \\cite{_Deligne:defi_}\nfor alternative definitions.} (see \\ref{_singu_hype_Definition_}).\n\n\\hfill\n\n\\remark This gives a new set of examples of hyperk\\\"ahler\nvarieties, in addition to those produced by \n\\cite{_Verbitsky:Hyperholo_bundles_}. The varieties obtained\nthrough \\cite{_Verbitsky:Hyperholo_bundles_} are usually not\ncompact; on contrary, our new examples are compact.\n\n\\hfill\n\nThe proof of \\ref{_iso_intro:Theorem_} is based on the\nfollowing argument. First, we show that every trianalytic\nsubmanifold $X\\subset M$ (not necessarily closed) is \n{\\bf completely geodesic} in $M$ (\\ref{_hype_embe_comple_geode:Corollary_}).\n\\footnote{{\\bf Completely geodesic} means \nthat geodesics in $X$ are also geodesics in $M$.}\nA simple geometric argument shows that a family $\\c X$ of completely\ngeodesic submanifolds admits a natural {\\bf connection}\n(\\ref{_conne_Definition_}, \n\\ref{_conne_in_fam_of_comple_geode_Proposition_}).\nThis connection is compatible with the holomorphic\nstructure if the family $\\c X$ is holomorphic\n(\\ref{_conne_in_fam_of_comple_geode_Proposition_}).\n\n\n\\hfill\n\nFor $M$ a compact hyperk\\\"ahler variety,\n$I$ an induced complex structure and $X$ a trianalytic subvariety of $M$,\nwe study the Douady space $D(X,I)$ which classifies complex analytic\ndeformations of $X$ in $(M,I)$, where $(M, I)$ is $M$\nconsidered as a complex manifold. We prove that\nthe real analytic subvariety $D(X)$ underlying\n$D(X,I)$ does not depend from the choice of induced\ncomplex structure. This endows $D(X)$ with a \n2-dimensional sphere of complex structures, which\ninduce quaternionic action in the Zariski tangent space. \nA real analytic variety with such a system of complex\nstructures is called {\\bf hypercomplex} \n(\\ref{_hypercomplex_Definition_}).\n\nWe return to the study of the families of trianalytic subvarieties.\nAs we have shown, the base of the universal family is \nhypercomplex. The fibers are trianalytic, and therefore,\nhypercomplex as well. The\nnatural connection in such a family is compatible with\nthe quaternionic action, because this connection is \n{\\bf holomorphic} with respect to each of \ninduced complex structures. Thus, a curvature of this\nconnection is ${\\Bbb H}^*$-invariant (\\ref{_curva_SU(2)_inva_Lemma_}). \nOn the other hand, the curvature lies in the tensor product\nof three representations of ${\\Bbb H}^*$ of weight one.\nA trivial linear algebra argument shows that such a\ntensor product does not contain non-trivial ${\\Bbb H}^*$-invariant\nvectors. This proves that the natural connection\nin the family of trianalytic subvarieties is {\\bf flat}\n(\\ref{_conne_in_triana_flat_Theorem_}).\n\nLet $\\c X\\stackrel \\pi {\\:\\longrightarrow\\:} S$ be a family of closed trianalytic\nsubvarieties in a compact hyperk\\\"ahler manifold $M$. Let\n$X_s^{ns}$ be the set of non-singular points in\n$X_s:= \\pi^{-1}(s)$, $s\\in S$. The submanifold $X_s^{ns}$\nis completely geodesic. Thus, the family $\\c X$ is equipped\nwith a natural connection. For every two points $s_1, s_2 \\in S$,\nwe show that this connection might be integrated to an\nisomorphism of hyperk\\\"ahler manifolds\n$\\Psi^{s_2}_{s_1}:\\; X^{ns}_{s_1} {\\:\\longrightarrow\\:} X^{ns}_{s_2}$\n(\\ref{_triana_subse_isome_Theorem_}). Since $X_s^{ns}$ is\ncompletely geodesic in $M$, its completion as a metric space\nis naturally isomorphic to a closure $X_s$ of $X_s^{ns}$ in $M$.\nSince the map $\\Psi^{s_2}_{s_1}:\\; X^{ns}_{s_1} {\\:\\longrightarrow\\:} X^{ns}_{s_2}$\nis an isometry, it is naturally extended to an isomorphism\nof metric spaces $\\bar \\Psi^{s_2}_{s_1}:\\; X^{ns}_{s_1} {\\:\\longrightarrow\\:}\nX^{ns}_{s_2}$ (\\ref{_triana_subse_isome_Theorem_}). From\nconstruction it follows that $\\Psi^{s_2}_{s_1}$ is compatible\nwith each of induced holomorphic structures. We extend\nthis assertion to $\\bar \\Psi^{s_2}_{s_1}$. This is done\nin two steps. A general type argument shows that\na homeomorphism of complex varieties is bimeromorphic\nif it is holomorphic in a dense open set. The leap from\nbimeromorphic to holomorphic is done via a convoluted\nalgebro-geometric argument involving normalization\nand finite unramified maps.\n\nWe finish this paper with the definition of singular hyperk\\\"ahler\nvarieties. We cite the previously known examples of singular hyperk\\\"ahler\nvarieties and then show that constructed above maps \n$\\bar \\Psi^{s_2}_{s_1}:\\; X^{ns}_{s_1} {\\:\\longrightarrow\\:} X^{ns}_{s_2}$\nare isomorphisms of hyperk\\\"ahler varieties. The Douady space\nof trianalytic subvarieties of a compact hyperk\\\"ahler manifold\nis proven to be hyperk\\\"ahler.\n\n\n\n\n\n\\subsection{Contents}\n\n\\begin{itemize}\n\n\\item Section \\ref{_Intro_Section_} is an introduction,\nindependent from the rest of this paper.\n\n\\item Section \\ref{_basics_Section_} is a compendium of results\npertaining to hyperk\\\"ahler geometry and Yang--Mills theory.\nWe define hyperk\\\"ahler manifolds, trianalytic subvarieties, \nhyperholomorphic bundles, and cite their most basic properties.\nA reader acquainted with a literature may skip this section.\n\n\\item Section \\ref{_nonsingu_preli_Section_} illustrates our results\nwith a simplicistic example of deformations of a smooth trianalytic\nsubvariety. This section is also independent from the rest,\nand its results are further superseded by\nmore general statements. It is safe to skip this section too.\n\n\\item Section \\ref{_comple_geode_defo_Section_} contains a study\nof completely geodesic submanifolds and their deformations.\nUsing an easy geometric argument,\nwe show that a family of completely geodesic submanifolds\nis equipped with a natural connection\n(\\ref{_conne_in_fam_of_comple_geode_Proposition_}). We study this\nconnection and show that, under certain additional assumptions,\nthis connection induces isometry (metric equivalence)\nof the fibers of this family. This section does not use\nresults of hyperk\\\"ahler geometry.\n\n\\item In Section \\ref{_comple_geode_hyperho_Section_}, \nwe prove that hyperk\\\"ahler\nsubmanifolds are always completely geodesic. We use arguments from \nYang--Mills theory and the theory of hyperholomorphic\nbundles (\\cite{_Verbitsky:Hyperholo_bundles_}).\n\n\\item Section \\ref{_triholo_Section_} shows that the deformational results\nof Section \\ref{_comple_geode_defo_Section_} can be applied to the\ndeformations of hyperk\\\"ahler submanifolds, not necessary closed. \nAgain, we use arguments from the theory of hyperholomorphic\nbundles. In Appendix to this section, we prove equivalence of real\nanalytic structures induced on a hyperk\\\"ahler manifold by different\ninduced complex structures. The proof is based on twistor\ngeometry (\\cite{_HKLR_}, \\cite{_NHYM_}).\n\n\\item Section \\ref{_Douady_Section_} studies the Douady deformation\nspace $D(X)$ of trianalytic subvariety $X$ of a compact hyperk\\\"ahler \nmanifold $M$.\nA general argument of K\\\"ahler geometry shows that the Douady\nspace is compact (\\cite{_Lieberman_}). Using this, we prove that\nthe underlying real analytic variety does not depend on the\ncomplex structure. As an application, we obtain that\nthe Douady space is {\\bf hypercomplex} (\\ref{_hypercomplex_Definition_}),\ni. e., has an integrable quaternionic action in the tangent space.\nIn the Appendix to this section, we give an independent\nproof of the compactness of the Douady space. We describe\nthe Douady space in terms of the space of isometric embeddings\nfrom $X$ to $M$. The Appendix is based on Wirtinger's\ninequality (\\ref{_Wirti_for_Kahle_Theorem_}). \nThe body of this section depends only on the result\nof the Appendix to Section \\ref{_triholo_Section_} \n(the equivalence of induced real analytic structures\non a given hyperk\\\"ahler manifold).\n\n\\item In Section \\ref{_Conne_in_fami_Section_},\nwe define a curvature of a connection in a family of\nmanifolds. We consider the natural connection on a family\nof trianalytic subvarieties of a compact hyperk\\\"ahler manifolds.\nUsing the results of Section \\ref{_Douady_Section_}\n(hypercomplex structure on the Douady space), we show that this connection\nis {\\bf flat} i. e. its curvature is zero. This result is\nnot used anywhere outside of Section \\ref{_singu_hype_Section_}\n(and even there, we don't really need it). However,\nthe flatness of the connection shows that the\nnatural isomorphism $\\Psi^{s_1}_{s_2}$ of the fibers\nof a family $\\c X {\\:\\longrightarrow\\:} S$ of trianalytic subvarieties\nis independent from the choice of a path $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} S$\nin its homotopy class.\n\n\\item Section \\ref{_isome=>holo_Section_} deals with the map \n$\\Psi^{s_1}_{s_2}:\\; X_{s_1} {\\:\\longrightarrow\\:} X_{s_2}$ \nof fibers of a family of trianalytic subvarieties of $M$,\nobtained by integrating the natural connection. This map is by\nconstruction a homeomorphism and is holomorphic outside of singularities,\nwith respect to each of induced complex structures. Using general argument,\nwe show that $\\Psi^{s_1}_{s_2}$ is bimeromorphic and induces an isomorphism\nof normalizations of $X_{s_1}$, $X_{s_2}$. Then, we apply the knowledge\nthat, outside of singularities,\n$X_{s_i}$ are completely geodesic in $M$. This allows us to \nshow that $\\Psi^{s_1}_{s_2}$ induces an isomorphism on the\nZariski tangent spaces. An algebro-geometric argument shows\nthat these properties are sufficient to prove that\n$\\Psi^{s_1}_{s_2}:\\; X_{s_1} {\\:\\longrightarrow\\:} X_{s_2}$ is a complex analytic\nisomorphism, with respect to each of induced complex structures.\n\nIn Section \\ref{_isome=>holo_Section_}, we use the results of \nSection \\ref{_comple_geode_defo_Section_} (structure of deformations\nof completely geodesic submanifolds) and Section \n\\ref{_triholo_Section_} (that these results may be applied to\nthe deformations of trianalytic submanifolds). \n\n\\item In Section \\ref{_singu_hype_Section_}, we give a definition\nof singular hyperk\\\"ahler varieties. After a short discussion,\nwe give examples of hyperk\\\"ahler varieties, stemming from the\ntheory of hyperholomorphic bundles (\\cite{_Verbitsky:Hyperholo_bundles_}).\nAll trianalytic subvarieties of hyperk\\\"ahler manifolds are\nhyperk\\\"ahler varieties, which is clear from the definition.\nAlso, the map $\\Psi^{s_1}_{s_2}:\\; X_{s_1} {\\:\\longrightarrow\\:} X_{s_2}$\nconstructed in Section \\ref{_isome=>holo_Section_}, is an\nisomorphism of hyperk\\\"ahler varieties. We show that\nthe Douady space of deformations of a trianalytic subvariety\nis also a hyperk\\\"ahler variety.\n\n\\end{itemize}\n\n\n\\section{Basic definitions and results.}\n\\label{_basics_Section_}\n\n\nThis section used mainly for reference, and contains a \ncompilation of results and definitions from literature. \nAn impatient reader is advised to skip it and proceed to the next \nsection.\n\n\\subsection{Hyperk\\\"ahler manifolds}\n\\label{_basics_hyperka_Section_}\n\n\n\nThis subsection contains a compression of \nthe basic and most known results \nand definitions from hyperk\\\"ahler geometry, found, for instance, in\n\\cite{_Besse:Einst_Manifo_} or in \\cite{_Beauville_}.\n\n\\hfill\n\n\\definition \\label{_hyperkahler_manifold_Definition_}\n(\\cite{_Besse:Einst_Manifo_}) A {\\bf hyperk\\\"ahler manifold} is a\nRiemannian manifold $M$ endowed with three complex structures $I$, $J$\nand $K$, such that the following holds.\n \n\\begin{description}\n\\item[(i)] the metric on $M$ is K\\\"ahler with respect to these complex \nstructures and\n \n\\item[(ii)] $I$, $J$ and $K$, considered as endomorphisms\nof a real tangent bundle, satisfy the relation \n$I\\circ J=-J\\circ I = K$.\n\\end{description}\n\n\\hfill \n\nThe notion of a hyperk\\\"ahler manifold was \nintroduced by E. Calabi (\\cite{_Calabi_}).\n\n\\hfill\n \nClearly, hyperk\\\"ahler manifold has the natural action of\nquaternion algebra ${\\Bbb H}$ in its real tangent bundle $TM$. \nTherefore its complex dimension is even.\nFor each quaternion $L\\in \\Bbb H$, $L^2=-1$,\nthe corresponding automorphism of $TM$ is an almost complex\nstructure. It is easy to check that this almost \ncomplex structure is integrable (\\cite{_Besse:Einst_Manifo_}).\n\n\\hfill\n\n\\definition \\label{_indu_comple_str_Definition_} \nLet $M$ be a hyperk\\\"ahler manifold, $L$ a quaternion satisfying\n$L^2=-1$. The corresponding complex structure on $M$ is called\n{\\bf an induced complex structure}. The $M$ considered as a complex\nmanifold is denoted by $(M, L)$.\n\n\\hfill\n\n\\definition \\label{_holomorphi_symple_Definition_}\nLet $M$ be a complex manifold and $\\Omega$ a closed \nholomorphic 2-form over $M$ such that \n$\\Omega^n=\\Omega\\wedge\\Omega\\wedge...$, is\na nowhere degenerate section of a canonical class of $M$\n($2n=dim_{\\Bbb C}(M)$).\nThen $M$ is called {\\bf holomorphically symplectic}.\n\n\\hfill\n\n\nLet $M$ be a hyperk\\\"ahler manifold; denote the\nRiemannian form on $M$ by $<\\cdot,\\cdot>$.\nLet the form $\\omega_I := $ be the usual K\\\"ahler\nform which is closed and parallel\n(with respect to the Levi-Civitta connection). Analogously defined \nforms $\\omega_J$ and $\\omega_K$ are\nalso closed and parallel. \n \nA simple linear algebraic\nconsideration (\\cite{_Besse:Einst_Manifo_}) shows that the form\n$\\Omega:=\\omega_J+\\sqrt{-1}\\omega_K$ is of\ntype $(2,0)$ and, being closed, this form is also holomorphic.\nAlso, the form $\\Omega$ is nowhere degenerate, as another linear \nalgebraic argument shows.\nIt is called {\\bf the canonical holomorphic symplectic form\nof a manifold M}. Thus, for each hyperk\\\"ahler manifold $M$,\nand induced complex structure $L$, the underlying complex manifold\n$(M,L)$ is holomorphically symplectic. The converse assertion\nis also true:\n\n\\hfill\n\n\\proposition \\label{_symplectic_=>_hyperkahler_Proposition_\n(\\cite{_Beauville_}, \\cite{_Besse:Einst_Manifo_})\nLet $M$ be a compact holomorphically\nsymplectic K\\\"ahler manifold with the holomorphic symplectic form\n$\\Omega$, a K\\\"ahler class \n$[\\omega]\\in H^{1,1}(M)$ and a complex structure $I$. \nThere is a unique hyperk\\\"ahler structure $(I,J,K,(\\cdot,\\cdot))$\nover $M$ such that the cohomology class of the symplectic form\n$\\omega_I=(\\cdot,I\\cdot)$ is equal to $[\\omega]$ and the\ncanonical symplectic form $\\omega_J+\\sqrt{-1}\\:\\omega_K$ is\nequal to $\\Omega$.\n\n\\hfill\n\n\\ref{_symplectic_=>_hyperkahler_Proposition_} immediately\nfollows from the conjecture of Calabi, pro\\-ven by\nYau (\\cite{_Yau:Calabi-Yau_}). \n\\blacksquare\n\n\n\\subsection{Hyperholomorphic bundles}\n\\label{_hyperholo_Subsection_}\n\n\nThis subsection contains several versions of a\ndefinition of hyperholomorphic connection in a complex\nvector bundle over a hyperk\\\"ahler manifold.\nWe follow \\cite{_Verbitsky:Hyperholo_bundles_}.\n\n\\hfill\n\nLet $M$ be a hyperk\\\"ahler manifold. We identify the group $SU(2)$\nwith the group of unitary quaternions. This gives a canonical \naction of $SU(2)$ on the tangent bundle, and all its tensor\npowers. In particular, we obtain a natural action of $SU(2)$\non the bundle of differential forms. Of a special interest\nto us are those forms which are $SU(2)$-invariant.\n\n\\hfill\n\n\\lemma \\label{_SU(2)_inva_type_p,p_Lemma_} \nLet $\\omega$ be a differential form over\na hyperk\\\"ahler manifold $M$. The form $\\omega$ is $SU(2)$-invariant\nif and only if it is of Hodge type $(p,p)$ with respect to all \ninduced complex structures on $M$.\n\n{\\bf Proof:} This is \\cite{_Verbitsky:Hyperholo_bundles_}, \nProposition 1.2. \\blacksquare\n\n\\hfill\n\nFurther in this article, we use the following statement.\n\n\n\\lemma \\label{_SU(2)_commu_Laplace_Lemma_}\nThe action of $SU(2)$ on differential forms commutes\nwith the Laplacian.\n\n{\\bf Proof:} This is Proposition 1.1\nof \\cite{_Verbitsky:Hyperholo_bundles_}. \\blacksquare\n\nThus, for compact $M$, we may speak of the natural action of\n$SU(2)$ in cohomology.\n\n\n\\hfill\n\n Let $B$ be a holomorphic vector bundle over a complex\nmanifold $M$, $\\nabla$ a connection \nin $B$ and $\\Theta\\in\\Lambda^2\\otimes End(B)$ be its curvature. \nThis connection\nis called {\\bf compatible with a holomorphic structure} if\n$\\nabla_X(\\zeta)=0$ for any holomorphic section $\\zeta$ and\nany antiholomorphic tangent vector $X$. If there exist\na holomorphic structure compatible with the given\nHermitian connection then this connection is called\n{\\bf integrable}.\n \n\\hfill\n \nOne can define a {\\bf Hodge decomposition} in the space of differential\nforms with coefficients in any complex bundle, in particular,\n$End(B)$.\n\n\\hfill\n \n\n\n\\theorem \\label{_Newle_Nie_for_bu_Theorem_}\nLet $\\nabla$ be a Hermitian connection in a complex vector\nbundle $B$ over a complex manifold. Then $\\nabla$ is integrable\nif and only if $\\Theta\\in\\Lambda^{1,1}(M, \\operatorname{End}(B))$, where\n$\\Lambda^{1,1}(M, \\operatorname{End}(B))$ denotes the forms of Hodge\ntype (1,1). Also,\nthe holomorphic structure compatible with $\\nabla$ is unique.\n\n{\\bf Proof:} This is Proposition 4.17 of \\cite{_Kobayashi_}, \nChapter I.\n$\\hbox{\\vrule width 4pt height 4pt depth 0pt}$\n\n\\hfill\n\n\\definition \\label{_hyperho_conne_Definition_}\nLet $B$ be a Hermitian vector bundle with\na connection $\\nabla$ over a hyperk\\\"ahler manifold\n$M$. Then $\\nabla$ is called {\\bf hyperholomorphic} if \n$\\nabla$ is\nintegrable with respect to each of the complex structures induced\nby the hyperk\\\"ahler structure. \n \nAs follows from \n\\ref{_Newle_Nie_for_bu_Theorem_}, $\\nabla$ is hyperholomorphic\nif and only if its curvature $\\Theta$ is of Hodge type (1,1) with\nrespect to any of complex structures induced by a hyperk\\\"ahler \nstructure.\n\nAs follows from \\ref{_SU(2)_inva_type_p,p_Lemma_}, \n$\\nabla$ is hyperholomorphic\nif and only if $\\Theta$ is a $SU(2)$-invariant differential form.\n\n\\hfill\n\n\\example \\label{_tangent_hyperholo_Example_}\n(Examples of hyperholomorphic bundles)\n\n\\begin{description}\n\n\\item[(i)]\nLet $M$ be a hyperk\\\"ahler manifold, $TM$ its tangent bundle\nequipped with Levi--Civita connection $\\nabla$. Then $\\nabla$\nis integrable with respect to each induced complex structure,\nand hence, Yang--Mills.\n\n\\item[(ii)] For $B$ a hyperholomorphic bundle, all its tensor powers\nare also hyperholomorphic.\n\n\\item[(iii)] Thus, the bundles of differential forms on a hyperk\\\"ahler\nmanifold are also hyperholomorphic.\n\n\\end{description}\n\n\n\\subsection{Stable bundles and Yang--Mills connections.}\n\n\nThis subsection is a compendium of the most\nbasic results and definitions from the Yang--Mills theory\nover K\\\"ahler manifolds, concluding in the fundamental\ntheorem of Uhlenbeck--Yau \\cite{_Uhle_Yau_}.\n\n\\hfill\n\n\\definition\\label{_degree,slope_destabilising_Definition_}\nLet $F$ be a coherent sheaf over\nan $n$-dimensional compact K\\\"ahler manifold $M$. We define\n$\\deg(F)$ as\n \n\\[ \n \\deg(F)=\\int_M\\frac{ c_1(F)\\wedge\\omega^{n-1}}{vol(M)}\n\\] \nand $\\text{slope}(F)$ as\n\\[ \n \\text{slope}(F)=\\frac{1}{\\text{rank}(F)}\\cdot \\deg(F). \n\\]\nThe number $\\text{slope}(F)$ depends only on a\ncohomology class of $c_1(F)$. \n\nLet $F$ be a coherent sheaf on $M$\nand $F'\\subset F$ its proper subsheaf. Then $F'$ is \ncalled {\\bf destabilizing subsheaf} \nif $\\text{slope}(F') \\geq \\text{slope}(F)$\n\nA holomorphic vector bundle $B$ is called {\\bf stable}\nif it has no destabilizing subsheaves.\n \n\\hfill\n\nLater on, we usually consider the bundles $B$ with $deg(B)=0$.\n\n\\hfill\n\n\nLet $M$ be a K\\\"ahler manifold with a K\\\"ahler form $\\omega$.\nFor differential forms with coefficients in any vector bundle\nthere is a Hodge operator $L: \\eta{\\:\\longrightarrow\\:}\\omega\\wedge\\eta$.\nThere is also a fiberwise-adjoint Hodge operator $\\Lambda$\n(see \\cite{_Griffi_Harri_}).\n \n\\hfill\n\n\\definition \\label{Yang-Mills_Definition_}\nLet $B$ be a holomorphic bundle over a K\\\"ahler manifold $M$\nwith a holomorphic Hermitian connection $\\nabla$ and a \ncurvature $\\Theta\\in\\Lambda^{1,1}\\otimes End(B)$.\nThe Hermitian metric on $B$ and the connection $\\nabla$\ndefined by this metric are called {\\bf Yang-Mills} if \n\n\\[\n \\Lambda(\\Theta)=constant\\cdot \\operatorname{Id}\\restrict{B},\n\\]\nwhere $\\Lambda$ is a Hodge operator and $\\operatorname{Id}\\restrict{B}$ is \nthe identity endomorphism which is a section of $End(B)$.\n\nFurther on, we consider only these Yang--Mills connections\nfor which this constant is zero.\n\n\\hfill\n\n \nA holomorphic bundle is called {\\bf indecomposable} \nif it cannot be decomposed onto a direct sum\nof two or more holomorphic bundles.\n\n\\hfill\n\nThe following fundamental \ntheorem provides examples of Yang-\\--Mills \\linebreak bundles.\n\n\\theorem \\label{_UY_Theorem_} \n(Uhlenbeck-Yau)\nLet B be an indecomposable\nholomorphic bundle over a compact K\\\"ahler manifold. Then $B$ admits\na Hermitian Yang-Mills connection if and only if it is stable, and\nthis connection is unique.\n \n{\\bf Proof:} \\cite{_Uhle_Yau_}. \\blacksquare\n\n\\hfill\n\n\\proposition \\label{_hyperholo_Yang--Mills_Proposition_}\nLet $M$ be a hyperk\\\"ahler manifold, $L$\nan induced complex structure and $B$ be a complex vector\nbundle over $(M,L)$. \nThen every hyperholomorphic connection $\\nabla$ in $B$\nis Yang-Mills and satisfies $\\Lambda(\\Theta)=0$\nwhere $\\Theta$ is a curvature of $\\nabla$.\n \n\\hfill\n\n{\\bf Proof:} We use the definition of a hyperholomorphic \nconnection as one with $SU(2)$-invariant curvature. \nThen \\ref{_hyperholo_Yang--Mills_Proposition_}\nfollows from the\n\n\\hfill\n\n\\lemma \\label{_Lambda_of_inva_forms_zero_Lemma_}\nLet $\\Theta\\in \\Lambda^2(M)$ be a $SU(2)$-invariant \ndifferential 2-form on $M$. Then\n$\\Lambda_L(\\Theta)=0$ for each induced complex structure\n$L$.\\footnote{By $\\Lambda_L$ we understand the Hodge operator \n$\\Lambda$ associated with the K\\\"ahler complex structure $L$.}\n \n \n{\\bf Proof:} This is Lemma 2.1 of \\cite{_Verbitsky:Hyperholo_bundles_}.\n\\blacksquare\n \n\\hfill\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $I$ an induced \ncomplex structure. \nFor any stable holomorphic bundle on $(M, I)$ there exists a unique\nHermitian Yang-Mills connection which, for some bundles,\nturns out to be hyperholomorphic. It is possible to tell when\nthis happens (though in the present paper we never use\nthis knowledge).\n\n\\hfill\n\n\\theorem\nLet $B$ be a stable holomorphic bundle over\n$(M,I)$, where $M$ is a hyperk\\\"ahler manifold and $I$\nis an induced complex structure over $M$. Then \n$B$ admits a compatible hyperholomorphic connection if and only\nif the first two Chern classes $c_1(B)$ and $c_2(B)$ are \n$SU(2)$-invariant.\\footnote{We use \\ref{_SU(2)_commu_Laplace_Lemma_}\nto speak of action of $SU(2)$ in cohomology of $M$.}\n\n{\\bf Proof:} This is Theorem 2.5 of\n \\cite{_Verbitsky:Hyperholo_bundles_}. \\blacksquare\n\n\n\\subsection{Generic holomorphically symplectic manifolds}\n\n\nIn this section, we define generic holomorphically symplectic manifolds.\nSuch manifolds, as seen later (\\ref{_hyperkae_embeddings_Corollary_}),\nadmit a hyperk\\\"ahler structure $\\c H$\nsuch that every closed complex analytic subvariety is trianalytic\nwith respect to $\\c H$\n(for the definition of trianalytic subvarieities,\nsee \\ref{_trianalytic_Definition_}). \nWe follow \\cite{Verbitsky:Symplectic_I_} \n(see also \\cite{Verbitsky:Symplectic_II_}).\n\n\\hfill\n\nLet $M$ be a compact holomorphically symplectic K\\\"ahler manifold. \nBy \\ref{_symplectic_=>_hyperkahler_Proposition_}, \n$M$ has a unique hyperk\\\"ahler structure with\nthe same K\\\"ahler class and holomorphic symplectic form. \nTherefore one can without ambiguity speak\nabout the action of $SU(2)$ on $H^*(M,{\\Bbb R})$ (see \n\\ref{_SU(2)_commu_Laplace_Lemma_}).\nOf course, this action essentially depends on the choice\nof K\\\"ahler class.\n\n\\hfill\n\n\\definition \\label{_generic_manifolds_Definition_}\nLet $\\omega\\in H^{1,1}(M)$ be the K\\\"ahler class \nof a holomorphically symplectic\nmanifold $M$. We say that $\\omega$ {\\bf induces the $SU(2)$-action\nof general type} when all elements of the group\n\\[ \\bigoplus\\limits_p H^{p,p}(M)\\cap H^{2p}(M,{\\Bbb Z})\\subset H^*(M)\\] \nare $SU(2)$-invariant. \nA holomorphically symplectic manifold $M$ is \ncalled {\\bf of general type} if there\nexists a K\\\"ahler class on $M$ which induces \nan $SU(2)$-action of general type.\n\n\\hfill\n\n\nAs \\ref{_gene_type_co_div_by2_Remark_} below\nimplies, holomorphically symplectic manifolds\nof general type have no Weil divisors. \nIn particular, such manifolds are never algebraic.\n\n\\hfill\n\n\\proposition \\label{_generic_are_dense_Proposition_}\nLet $M$ be a hyperk\\\"ahler manifold and $S$\nbe the set of induced complex structures over $M$. Denote by \n$S_0\\subset S$ the set of $L\\in S$ such that the natural \nK\\\"ahler metric on $(M,L)$ induces the $SU(2)$ action of \ngeneral type. Then $S_0$ is dense in $S$.\n\n{\\bf Proof:} This is Proposition 2.2 from\n\\cite{Verbitsky:Symplectic_II_}\n\\blacksquare\n\n\\hfill\n\nOne can easily deduce from the results in \n\\cite{_Todorov:Moduli_I_II_} and\n\\ref{_generic_are_dense_Proposition_} that the set of points \nassociated with holomorphically symplectic \nmanifolds of general type is dense in the classifying space\nof holomorphically symplectic manifolds of K\\\"ahler type.\n\n\n\n\\subsection{Trianalytic subvarieties in compact hyperk\\\"ahler\nmanifolds.}\n\n\nIn this subsection, we give a definition and a few basic properties\nof trianalytic subvarieties of hyperk\\\"ahler manifolds. \nWe follow \\cite{Verbitsky:Symplectic_II_}.\n\n\\hfill\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $\\dim_{\\Bbb R} M =2m$.\n\n\\hfill\n\n\\definition\\label{_trianalytic_Definition_}\nLet $N\\subset M$ be a closed subset of $M$. Then $N$ is\ncalled {\\bf trianalytic} if $N$ is a complex analytic subset \nof $(M,L)$ for any induced complex structure $L$.\n\n\\hfill\n\nThroughout this paper, we implicitly assume that our\ntrianalytic subvarieties are connected. Most results are trivially\ngeneralized to the general case.\n\n\\hfill\n\nLet $I$ be an induced complex structure on $M$,\nand $N\\subset(M,I)$ be\na closed analytic subvariety of $(M,I)$, $dim_{\\Bbb C} N= n$.\nDenote by $[N]\\in H_{2n}(M)$ the homology class \nrepresented by $N$. Let $\\inangles N\\in H^{2m-2n}(M)$ denote \nthe Poincare dual cohomology class. Recall that\nthe hyperk\\\"ahler structure induces the action of \nthe group $SU(2)$ on the space $H^{2m-2n}(M)$.\n\n\\hfill\n\n\\theorem\\label{_G_M_invariant_implies_trianalytic_Theorem_}\nAssume that $\\inangles N\\in H^{2m-2n}(M)$ is invariant with respect\nto the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic.\n\n{\\bf Proof:} This is Theorem 4.1 of \n\\cite{Verbitsky:Symplectic_II_} (see Subsection\n\\ref{_SU(2)-inv=>triana_Subsection_} for a sketch of a proof). \n\\blacksquare\n\n\\remark \\label{_triana_dim_div_4_Remark_}\nTrianalytic subvarieties have an action of quaternion algebra in\nthe tangent bundle. In particular,\nthe real dimension of such subvarieties is divisible by 4.\n\n\\hfill\n\n\n\\ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following\nimmediate corollary:\n\n\n\\corollary \\label{_hyperkae_embeddings_Corollary_}\nLet $M$ be a compact holomorphically symplectic \nmanifold of general type, $S\\subset M$ be its complex analytic\nsubvariety, and $\\omega$ be a K\\\"ahler class\nwhich induces an $SU(2)$-action of general type.\nConsider the hyperk\\\"ahler structure associated with \n$\\omega$ by \\ref{_symplectic_=>_hyperkahler_Proposition_}.\nThen $S$ is trianalytic with respect to $\\c H$.\n\n\\blacksquare\n\n\\remark \\label{_gene_type_co_div_by2_Remark_}\n{}From \\ref{_hyperkae_embeddings_Corollary_} and\n\\ref{_triana_dim_div_4_Remark_}, it follows that\na holomorphically symplectic manifold of general type\nhas no closed complex analytic subvarieties of odd dimension;\nin particular, such a manifold has no divisors.\n\n\n\\subsection{Wirtinger's inequality and its use in K\\\"ahler geometry}\n\n\nLet $M$ be a compact K\\\"ahler manifold, $X\\subset M$ a \nclosed real analytic subvariety.\nIn this section, we give criteria for $X$ to be \ncomplex analytic, in terms of certain integrals.\nWe follow \\cite{Verbitsky:Symplectic_II_} and \n\\cite{_Stolzenberg_}.\n\n\\hfill\n\n\\definition \\label{_volume_Riema_Definition_}\nLet $H$ be an ${\\Bbb R}$-linear space equipped with a\npositively defined scalar product, $\\dim H =h$. The exterrior\nform $\\operatorname{Vol}\\in \\Lambda^h(H)$ is called {\\bf a volume form}\nif the the standard hypercube with the side 1\nhas the volume 1 in the measure defined by $\\operatorname{Vol}$.\n\n\\hfill\n\nClearly, the volume form is defined up to a sign.\nThis sign is determined by the choice of orientation on $H$.\nIn the same manner \nwe define the top degree differential form $\\operatorname{Vol}$ called\n{\\bf a volume form} on any oriented \nRiemannian manifold. \n\n\\hfill\n\nLet $V$ be a Hermitian linear space, $W\\subset V$ be a \n${\\Bbb R}$-linear subspace, $\\dim_{\\Bbb R} W = 2n$. Consider \nspace $\\Lambda^{2n}(W)$ of volume forms on $M$. Let\n$\\omega$ be the imaginary part of the Hermitian form \non $V$. Consider the vectors $\\omega^n$, \n$\\operatorname{Vol} \\in \\Lambda^{2n}(W)$. Since $\\operatorname{Vol}$ is non-zero,\nand $\\Lambda^{2n}(W)$ is one-dimensional, we can speak\nof a fraction $\\frac{\\omega^n}{\\operatorname{Vol}}$, which is a real number,\ndefined up to a sign (the form $\\operatorname{Vol}$ is defined up to a sign).\nDenote by $\\Xi_W$ the number \n$\\left|\\frac{\\omega^n}{\\operatorname{Vol}}\\right|$.\n\n\\hfill\n\n\n\\lemma\\label{_Wirtinger_Lemma_}\n(Wirtinger's inequality) \nIn these assumptions, $\\Xi_W\\leq 2^n.$\nMoreover, if $\\Xi_W=2^n$, then $W$ is a \ncomplex subspace of $V$.\n\n{\\bf Proof:} \\cite{_Stolzenberg_} page 7. \\blacksquare\n\n\\hfill\n\nLet $M$ be a K\\\"ahler manifold, $N\\subset (M,I)$ be a closed\nreal analytic subvariety of even dimension, \n$N_{ns}\\subset N$ the set of non-singular \npoints of $N$. For each\n$x\\in N_{ns}$, consider $T_xN$ as a subspace of $T_xM$.\nDenote the corresponding number\n$\\Xi_{T_xN}$ by $\\Xi_L (x)$. The following statement is a direct\nconsequence of \\ref{_Wirtinger_Lemma_}:\n\n\\hfill\n\n\\proposition\\label{_N_is_analytic_if_eta_is_constant_Proposition_}\nLet $J$ be an induced complex structure.\nThe subset $N\\subset M$ is complex analytic \nif and only if \n\n\\[ \\forall x\\in N_{ns} \\;\\;\\;\\; \\Xi_L(x)=2^n. \\]\n\n\\blacksquare\n\n\\hfill\n\nLet $\\operatorname{Vol} N_{ns}$ be the volume form of $N_{ns}$, taken with respect to the\nRiemannian form (see \\ref{_volume_Riema_Definition_}).\n\n\n\\hfill\n\n\n\\theorem \\label{_Wirti_for_Kahle_Theorem_}\nLet $M$ be a K\\\"ahler manifold, $N\\subset M$, $\\dim N =2n$ a \nclosed real analytic subvariety, $N_{ns}\\subset M$ the set\nof its non-singular points. Assume that the improper integrals\n$\\int_{N_{ns}} \\operatorname{Vol} N_{ns}$, $\\int_{N_{ns}} \\omega^n$ exist.\nThen\n\\[ \n 2^n\\int_{N_{ns}} \\operatorname{Vol} N_{ns} \\geq \\int_{N_{ns}} \\omega^n \n\\]\nand the equality is reached if and only if $N$ is complex analytic\nin $M$.\n\n{\\bf Proof:} This is a direct consequence of \n\\ref{_N_is_analytic_if_eta_is_constant_Proposition_} \n\\blacksquare\n\n\\hfill\n\nWe use the term ``symplectic volume'' for the number\n$\\frac{1}{2^n}\\int_{N_{ns}} \\omega^n $ and ``Riemannian volume''\nfor $\\int_{N_ns} \\operatorname{Vol} N_{ns}$. Then, \n\\ref{_Wirti_for_Kahle_Theorem_} might be rephrased in the form\n``a real analytic cycle is complex analytic if and only if\nits symplectic volume is equal to its Riemannian volume''.\n\n\n\\subsection{$SU(2)$-invariant cycles in cohomology \nand trianalytic subvarieties}\n\\label{_SU(2)-inv=>triana_Subsection_}\n\n\nThis subsection consists of a sketch of a proof of \n\\ref{_G_M_invariant_implies_trianalytic_Theorem_}.\nWe follow \\cite{Verbitsky:Symplectic_II_}.\n\n\\hfill\n\nFor a K\\\"ahler manifold $M$, $m=dim_{\\Bbb C} M$ \nand a form $\\alpha\\in H^{2p}(M,{\\Bbb C})$, define \n\\[\n \\deg(\\alpha):=\\int_M \\omega^{m-p}\\wedge\\alpha,\n\\]\nwhere $\\omega$ is the K\\\"ahler form. \n\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $\\alpha$ a\ndifferential form.\nWe denote by $\\deg_L\\alpha$ the degree associated with\nan induced complex structure $L$.\n\n\\hfill\n\n\\proposition \\label{_G_M_invariant_cycles_over_Proposition_} \nLet $M$ be a compact hyperk\\\"ahler manifold and\n$\\alpha$ be an $SU(2)$-invariant form of non-zero degree.\nThen the dimension of $\\alpha$ is divisible by 4. Moreover, \n\\[ \n \\deg_{I}\\alpha = \\deg_{I'}\\alpha,\n\\]\nfor every pair of induced complex structures $I$, $I'$.\n\n{\\bf Proof:} This is Proposition 4.5 of \n\\cite{Verbitsky:Symplectic_II_}.\n\\blacksquare\n\n\\hfill\n\n\\ref{_G_M_invariant_implies_trianalytic_Theorem_} follows\nimmediately from \\ref{_G_M_invariant_cycles_over_Proposition_} \nand \\ref{_Wirti_for_Kahle_Theorem_}. By\n\\ref{_Wirti_for_Kahle_Theorem_}, a real analytic subvariety\n$N \\subset M$ is trianalytic if and only if its symplectic volume,\ntaken with respect to any of induced complex structures, is \ndefined and equal to its Riemannian volume.\nIn notation of \\ref{_G_M_invariant_implies_trianalytic_Theorem_},\nthe symplectic volume of $N$ taken with respect to $L$\nis equal to $\\frac {1}{2^n} \\deg_L\\inangles N$.\n{}From \\ref{_G_M_invariant_cycles_over_Proposition_},\nwe obtain that symplectic volume of $N$ is the same for all\ninduced complex structures. Since $N$ is complex analytic\nwith respect to $I$, the symplectic volume of $N$ is equal\nto the Riemannian volume of $N$, again by \n\\ref{_Wirti_for_Kahle_Theorem_}. This proves\n\\ref{_G_M_invariant_implies_trianalytic_Theorem_}.\n\n\n\n\\section[Deformations of non-singular trianalytic subvarieties.]\n{Deformations of non-singular trianalytic \\\\subvarieties.}\n\\label{_nonsingu_preli_Section_}\n\n\nLet $M$ be a holomorphically symplectic K\\\"ahler manifold,\n$X\\subset M$ a closed complex analytic submanifold \nof trianalytic type. We study\nthe deformations of $X$ in $M$, in order to prove \n\\ref{_iso_intro:Theorem_}. It turns out that\n\\ref{_iso_intro:Theorem_} is almost trivial in the case\n$X$ non-singular. In this section, we give the proof\nof \\ref{_iso_intro:Theorem_} (i) for non-singular $X$; the general\ncase is proven independently. We hope that a simple argument\nwill be insightful, even if we need to produce a separate proof\nfor the general case. This section is perfectly safe to skip.\n\n\\hfill\n\n\\remark For a smooth trianalytic submanifold $X\\subset M$,\n$X$ is obviously hyperk\\\"ahler, in a natural way. The quaternion\naction comes from quaternion action in $TM$, and the metric is\ninduced from $M$ too.\n\n\\hfill\n\nWe start from the following simple, but important, lemma.\n\n\\hfill\n\n\\lemma \\label{_TM-restrict_X_hyperholo:Lemma_}\nLet $M$ be a complex manifold equipped with a hyperk\\\"ahler metric,\n$X\\subset M$ a closed complex analytic submanifld of trianalytic\ntype. Consider the\nrestriction $TM\\restrict{X}$ of the tangent bundle to $M$ on $X$.\nWe equip $TM_\\restrict{X}$ with a connection $\\nabla$ coming\nfrom Levi-Civita connection in $TM$. Then $(TM\\restrict{X}, \\nabla)$\nis {\\bf hyperholomorphic}.\\footnote{See \nSubsection \\ref{_hyperholo_Subsection_} \nfor the definition of {\\bf hyperholomorphic}.\nBy \\ref{_hyperholo_Yang--Mills_Proposition_},\na hyperholomorphic connection is Yang-Mills.}\n\n\\hfill\n\n{\\bf Proof:} Consider the natural action of $SU(2)$ on the \nspace \n\\[ \n \\Lambda^2\\left(X, \\operatorname{End}\\left(TM\\restrict{X}\\right)\\right).\n\\] \nWe need to show that\nthe curvature $\\Theta$ of $\\nabla$ is $SU(2)$-invariant. Let\n$\\nabla_{LC}$ be the Levi--Civita connection on $TM$, and \n$(\\nabla_{LC})^2\\in \\Lambda^2(M, \\operatorname{End}(TM))$ be its curvature. Clearly,\n$\\Theta$ is a pull-back of $(\\nabla_{LC})^2$ to $X$. Therefore,\nit suffices to show that $(\\nabla_{LC})^2$ is $SU(2)$-invariant.\nThis is \\ref{_tangent_hyperholo_Example_}. \n\\blacksquare\n\n\n\\hfill\n\nAs a corollary, we obtain the following proposition.\n\n\\hfill\n\n\\proposition \\label{_TM-restrict-X_decompo_for_compa_Proposition_}\nIn assumptions of \\ref{_TM-restrict_X_hyperholo:Lemma_},\nlet $M$, $X$ be compact. Then the following statements are true.\n\n\\begin{description}\n\\item[(i)] The bundle $TM\\restrict{X}$ is naturally \nisomorphic to the direct sum $TM\\restrict X \\cong TX \\oplus NX$,\nwhere $N$ is the normal bundle to $X$ inside $M$. This\nisomorphism is compatible with the natural connections and Hermitian\nmetrics on $TM\\restrict X$, $TX$, $NX$. \n\n\\item[(ii)] $NX$ is hyperholomorphic.\n\n\\item[(iii)] For each section $\\gamma$ of $NX$, $\\gamma$ is nowhere\ndegenerate, and there is a natural decomposition\n\\begin{equation} \\label{_NX=O-gamma+rest:Equation_}\nNX = {\\cal O}_\\gamma \\oplus N_\\gamma X,\n\\end{equation}\nwhere ${\\cal O}_\\gamma$ is a trivial sub-bundle of $NX$ generated by\n$\\gamma$, and $N_\\gamma X$ its orthogonal complement. The \ndecomposition \\eqref{_NX=O-gamma+rest:Equation_} is compatible \nwith connection. \n\n\\item[(iv)] In assumptions of (iii),\nconsider the connection $\\nabla_\\gamma$ induced from\n$NX$ to ${\\cal O}_\\gamma X$. Then is $\\nabla_\\gamma$ flat.\n\\end{description}\n\n\\hfill\n\n\n\\remark In fact, \n\\ref{_TM-restrict-X_decompo_for_compa_Proposition_} (i)-(ii) holds true \nin assumptions of \\ref{_TM-restrict_X_hyperholo:Lemma_} for\ngeneral, not necessarily compact, $X$ and $M$ \n(see \\ref{_NX_splits_for_hype_Proposition_}, \n\\ref{_NX_hyperholo_Proposition_}).\nHowever, the proof is easier in compact case.\n\n\\hfill\n\n{\\bf Proof of \\ref{_TM-restrict-X_decompo_for_compa_Proposition_}.}\nConsider the embedding \n\\[ TX \\hookrightarrow TM\\restrict X \\]\nof bundles with connection. By \\ref{_tangent_hyperholo_Example_},\n$TX$ is hyperholomorphic, and hence Yang--Mills.\nBy \\ref{_TM-restrict_X_hyperholo:Lemma_}, $TM\\restrict X$ is\nhyperholomorphic as well. We obtain that\n$TX$ is a destabilizing subsheaf in $TX$.\nBy \\ref{_UY_Theorem_}, a Yang--Mills bundle \nis a direct sum of stable bundles.\\footnote{Bundles which are direct\nsum of stable are called {\\bf polystable}.}\nThus, $TM$ is a direct sum of $TX$ and its orthogonal\ncomplement $NX$ (see \\ref{_YM_exact_split_Proposition_} \nfor details). This proves\n\\ref{_TM-restrict-X_decompo_for_compa_Proposition_} (i).\nSince the curvature\nof $TM\\restrict X$ is decomposed onto a \ndirect sum of the curvature of $NX$ and the curvature of\n$TX$, the curvature of $NX$ is $SU(2)$-invariant.\nThis proves \\ref{_TM-restrict-X_decompo_for_compa_Proposition_} (ii).\n\n\\hfill\n\nThe condition (iii) follows from (ii) and \\ref{_UY_Theorem_}, \nsince every holomorphic section $\\gamma$ of a \nYang--Mills bundle $B$ of zero degree (such as hyperholomorphic bundles)\nspans a destabilizing subsheaf ${\\cal O}_\\gamma$. \nSince $B$ is Yang--Mills, it is polystable; thus, \n${\\cal O}_\\gamma$ is its direct summand.\nThis proves \\ref{_TM-restrict-X_decompo_for_compa_Proposition_} \n(iii). To prove (iv), we notice that ${\\cal O}_\\gamma$\nis a trivial holomorphic bundle with Yang--Mills connection.\nBy \\ref{_UY_Theorem_}, Yang--Mills connection is unique,\nand therefore, ${\\cal O}_\\gamma$ is flat. \n\\ref{_TM-restrict-X_decompo_for_compa_Proposition_} \nis proven. \n\\blacksquare\n\n\\hfill\n\nWe recall the following general results of the\ntheory of deformations of complex subvarieties.\nLet $X\\subset M$ be a closed complex analytic subvariety\nof a compact complex manifold. The {\\bf Douady space}\nof deformations of $X$ inside of $M$ is defined\n(\\cite{_Douady_}). We denote the Douady space by $D_M X$,\nor sometimes by $D(X)$. By definition, $D_M X$ is a\nfinite-dimensional complex analytic variety.\nThe points of $D_MX$ are identified with the \nsubvarieties $X'\\subset M$, where $X'$ is a deformation\nof $X$. Let $\\gamma:\\; {\\Bbb R} {\\:\\longrightarrow\\:} D_m X$ be a real analytic map.\nAssume that for $t=t_0$, $\\gamma(t)$ is a smooth subvariety of\n$M$. In deformation theory, the differential\n$\\frac{d\\gamma}{dt}(t_0)$ is interpreted as a \nholomorphic section of the normal bundle $N \\gamma (t_0)$.\nThus, the Zariski tangent space $T_{X} D_M (X)$\nis naturally embedded to the space of holomorphic\nsections of $N X$. The following proposition is an\neasy application of the Kodaira--Spencer theory.\n\n\\hfill\n\n\\proposition \\label{_compa_subva_iso_holo:Proposition_}\nLet $X \\subset M$ be a closed complex analytic submanifold of\na compact complex manifold $M$, and $D_M X$ the corresponding\nDouady space. Let $\\gamma:\\; {\\Bbb R} {\\:\\longrightarrow\\:} D_M X$ be a real analytic\nmap. Assume that for all $t_0\\in {\\Bbb R}$, the subvariety\n$\\gamma(t_0)$ is smooth. Assume also that the section \n\\[ \n \\nu_{t_0} = \n \\frac{d \\gamma}{dt}(t_0) \\in \\Gamma_{\\gamma(t_0)} (N \\gamma(t_0))\n\\]\nis nowhere degenerate. Finally, assume that $\\nu_{t_0}$\nsplits from $N \\gamma(t_0)$, i. e, there exists a decomposition\nof a holomorphic vector bundle\n\\[\n N \\gamma (t_0) = {\\cal O}_{\\nu_{t_0}} \\oplus N'\n\\]\nwhere ${\\cal O}_{\\nu_{t_0}}$ is the trivial subbundle of \n$N \\gamma(t_0)$ generated by $\\nu_{t_0}$.\nThen the subvarieties $\\gamma(t)\\subset M$ are naturally\nisomorphic for all $t$.\n\n{\\bf Proof:} Well known; see, for instance,\n\\cite{_Kodaira_Spencer_} \\blacksquare\n\n\\hfill\n\n\\proposition \\label{_compa_subva_iso_metric:Proposition_}\nIn assumptions of \\ref{_compa_subva_iso_holo:Proposition_},\nlet $M$ be K\\\"ahler. Consider the induced \nHermitian structure on $N\\gamma(t)$, for all $t\\in {\\Bbb R}$.\nAssume that for all $t\\in {\\Bbb R}$, the section\n\\[ \n \\nu_{t_0} = \n \\frac{d \\gamma}{dt}(t_0)\n\\]\nhas constant length. Assume also that the \northogonal decomposition \\[ TM\\restrict X = NX \\oplus NX^\\bot \\]\nis compatible with the Levi-Civita connection in \n$TM\\restrict X$. Consider the isomorphisms \n$\\psi_{t_1,t_2}:\\; \\gamma(t_1){\\:\\longrightarrow\\:} \\gamma(t_2)$ constructed in\n\\ref{_compa_subva_iso_holo:Proposition_}. Then \nthe maps $\\psi_{t_1,t_2}$ are compatible with the K\\\"ahler\nmetric induced from $M$.\n\n{\\bf Proof:} The proof follows from Kodaira--Spencer construction;\nfor a complete argument, see \n\\ref{_conne_in_fami_of_comple_geo_Proposition_} (ii). \n\\blacksquare\n\n\\hfill\n\nThe following theorem is the main result of this section.\n\n\\hfill\n\n\\theorem \\label{_iso_for_smooth_subva:Theorem_}\nLet $M$ be a compact holomorphically symplectic \nK\\\"ahler manifold,\nand $X\\subset M$ a complex submanifold\nof trianalytic type. Let $X'$ be a deformation of $X$ \nin $M$. Then there exists a complex analytic\nisomorphism $\\psi:\\; X {\\:\\longrightarrow\\:} X'$. Moreover, if\nthe K\\\"ahler metric on $M$ is hyperk\\\"ahler,\nthen $\\psi:\\; X {\\:\\longrightarrow\\:} X'$\nis compatible with the K\\\"ahler structure induced from $M$.\n\n\\hfill\n\n{\\bf Proof:} The deformations of $X$ are infinitesimally classified\nby the sections of $NX$. Let $\\gamma$ be such a section.\nApplying \\ref{_TM-restrict-X_decompo_for_compa_Proposition_},\nwe obtain that assumptions of\n\\ref{_compa_subva_iso_holo:Proposition_} and\n\\ref{_compa_subva_iso_metric:Proposition_} hold for the deformations\nof $X$ inside $M$. Now, \\ref{_iso_for_smooth_subva:Theorem_}\nis directly implied by \\ref{_compa_subva_iso_holo:Proposition_} and\n\\ref{_compa_subva_iso_metric:Proposition_}. \\blacksquare\n\n\\hfill\n\n\\remark \n\\ref{_iso_for_smooth_subva:Theorem_} \ngives a proof of \\ref{_iso_intro:Theorem_} \n(i), for the subvarieties $X\\subset M$ which are\nsmooth. We prove \\ref{_iso_for_smooth_subva:Theorem_} for\ngeneral subvarieties in \\ref{_triana_subse_comple_ana_Theorem_}.\n\n\n\\section[Completely geodesic embeddings of Riemannian manifolds.]\n{Completely geodesic embeddings \\\\of Riemannian manifolds.}\n\\label{_comple_geode_defo_Section_}\n\n\nBefore we proceed with the proof of \\ref{_iso_intro:Theorem_},\nwe have to prove a serie of preliminary \nresults from deformation theory. The arguments of\ndeformation theory are greatly simplified in the\nhyperk\\\"ahler case, because hyperk\\\"ahler embeddings are\ncompletely geodesic\\footnote{For a definition of completely geodesical\nembeddings, see \\ref{_comple_geode:Definition_}.}\n (\\ref{_hype_embe_comple_geode:Corollary_}).\nWe prove a number of simple statements from the deformation\ntheory of completely geodesical embeddings of K\\\"ahler manifolds.\n\n\\subsection{Completely geodesic submanifolds}\n\n\\nopagebreak\n\\hspace{5mm}\n\\proposition \\label{_comple_geodesi_basi_Proposition_\nLet $X \\stackrel \\phi\\hookrightarrow M$ be an embedding of Riemannian \nmanifolds (not necessarily compact) compatible with the Riemannian\nstructure.\\footnote{Such embeddings are called \n{\\bf Riemannian embeddings}.}\n Then the following conditions are equivalent.\n\n\\begin{description}\n\\item[(i)] For every point $x\\in X$, there exist a neighbourhod $U \\ni x$\nof $x$ in $X$ such that for all \n$x' \\in U$ there is a geodesic in $M$\ngoing from $\\phi(x)$ to $\\phi(x')$ which lies \nin $\\phi(X)\\subset M$.\n\n\\item[(ii)] Consider the Levi-Civita connection $\\nabla$ on $TM$,\nand restriction of $\\nabla$ to $TM \\restrict{X}$. Consider the\northogonal decomposition \n\\begin{equation} \\label{TM_decompo_Equation_} \n TM\\restrict{X} = TX \\oplus TX^\\bot. \n\\end{equation}\nThen, this decomposition is preserved by the connection $\\nabla$.\n\\end{description}\n\n{\\bf Proof:} Well known; see, for instance, \n\\cite{_Besse:Einst_Manifo_}.\n\\hbox{\\vrule width 4pt height 4pt depth 0pt}\n\n\\hfill\n\n\\definition\\label{_comple_geode:Definition_}\nLet $X \\stackrel i \\hookrightarrow M$ be a Riemannian embedding\nsatisfying either of the conditions of \n\\ref{_comple_geodesi_basi_Proposition_}. Then $i$ is called \n{\\bf a completely geodesic embedding}, and the image $i(X)\\subset M$\nis called {\\bf a completely geodesic submanifold}.\n\n\\hfill\n\n\n\\lemma \\label{_comple_geo_compa_holo_Lemma_}\nLet $M$ be a K\\\"ahler manifold, $X\\subset M$ a complex submanifold.\nIf $X$ is completely geodesic, then the decomposition\n\\eqref{TM_decompo_Equation_} \nis compatible with the holomorphic structure\non $TX$, $TM\\restrict X$.\n\n{\\bf Proof:} Clear. \\blacksquare\n\n\n\n\n\\subsection{Deformations of submanifolds}\n\\label{_defo_subva_conven_Subsection_}\n\n\nLet $\\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S$ be a family of complex\nmanifolds equipped with a map $\\c X \\stackrel \\phi {\\:\\longrightarrow\\:} M$.\nDenote the pre-image $\\pi^{-1}(s)\\subset \\c X$ by $X_s$.\nAssume that for all $s\\in S$\nthe restriction $\\phi_s:\\; X_s{\\:\\longrightarrow\\:} M$ of \n$\\c X \\stackrel \\phi {\\:\\longrightarrow\\:} M$\nto $X_s \\subset \\c X$ is a smooth embedding.\nAssume that for $s_0\\in S$, the image $\\phi_{s_0}(X_{s_0})\\subset M$\ncoinsides with $X$.\n\n\\hfill\n\n\\definition \nThe collection of data \n\\[ \\left( \\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S, \n \\phi, \\phi_{s_0}:\\; X_{s_0} \\oldtilde {\\:\\longrightarrow\\:} X\\right )\n\\]\nis called {\\bf a family of submanifolds of $M$},\nand {\\bf a family of deformations of $X$}.\nThe same definition can be formulated for $M$, $X$, $S$, $\\c X$\nreal analytic; in this case, we speak of {\\bf family of real\nanalytic submanifolds of $M$}. Also, the submanifolds might be \nreplaced by subvarieties in order to obtain the definition\nof a family of subvarieties.\n\n\n\\subsection{Section of a normal bundle arising from a deformation}\n\\label{_norma_vecto_Subsection_}\n\n\nLet $M$ be a complex or real analytic manifold and\n\\[ \\left( \\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S, \n \\phi:\\; \\c X {\\:\\longrightarrow\\:} M \\right)\n\\]\nbe a system of submanifolds.\nFor each tangent vector $t \\in T_{s_0} S$, deformation theory \ngives a canonical section $\\eta$ of the normal bundle $N X_{s_0}$.\nThis section is holomorphic if we work in complex situation.\nWe recall how this section is obtained.\n\nFor a sufficiently small neighbourhood of $x\\in \\c X$,\nwe can always find coordinates in $X_s$ which analytically\ndepend on $s$. For each point $x\\in X_{s_0}$, \ndenote by $x(s)$ the point of $X_s$ with the same\ncoordinates as $x$. Consider the vector\n$\\tilde \\eta_x :=\\frac{dx(s)}{ds}(t)\\in T_xM$, which is a derivative\nof $x(s)$ along $t$. The vector $\\tilde \\eta_x$ obviously depends\non the choice of \ncoordinates in $X_s$. Let $\\eta_x\\in NX_{s_0}\\restrict x$ be \nthe image of $\\tilde\\eta_x$ under the natural map \n$T_x M {\\:\\longrightarrow\\:} N X_{s_0}\\restrict x$. Clearly, $\\eta_x$ is\nindependent from the choice of coordinates. Gluing\n$\\eta_x$ together, we obtain the canonical \nsection $\\eta\\in NX_{s_0}$.\n\nWe need the following technical lemma in Section \n\\ref{_triholo_Section_}.\n\n\\hfill\n\n\\lemma \\label{_norma_sec_holom_Lemma_}\nLet $M$ be a complex analytic manifold and\n\\[ \\left( \\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S, \n \\phi:\\; \\c X {\\:\\longrightarrow\\:} M \\right)\n\\]\nbe a real analytic system of submanifolds. Assume that for \nall $s\\in S$, the submanifold $\\pi^{-1}(s) := X_s\\subset M$ \nis complex analytic.\nThen, for all $t\\in T_{s_0} S$, the corresponding section\n$\\eta \\in N X_{s_0}$ is holomorphic.\n\n\\hfill\n\n{\\bf Proof:} Shrinking $\\c X$ if necessary, we \ncan find a system of holomorphic\ncoordinates in $X_s$, $s\\in S$ \nwhich depends smoothly upon $s$. Consider the section\n$\\tilde \\eta\\in TM\\restrict{X_{s_0}}$ obtained by\nderiving $x(s)$ along $t$ as above. Since the map\n$x{\\:\\longrightarrow\\:} x(s)$ is by construction holomorphic,\nthe section $\\tilde\\eta$ is also holomorphic. Then,\n$\\eta$ is holomorphic by construction. \\blacksquare\n\n\n\\subsection{Connections in families of manifolds}\n\\label{_conne_Subsection_}\n\n\n\n\\hfill\n\n\\definition \\label{_conne_Definition_}\nLet $\\pi:\\; \\c X {\\:\\longrightarrow\\:} S$ be a family\nof real analytic manifolds parametrized by $S$. \nConsider the bundles $T\\c X$, $N_\\pi X$,\nwhere $N_\\pi X$ is a fiberwise normal bundle \nto the fibers of $\\pi$. There is a natural \nprojection $p:\\; T\\c X {\\:\\longrightarrow\\:} N_\\pi X$.\nThe {\\bf connection} in $\\c S$ is \na section of $p$, i. e. such an embedding\n$\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T\\c X$ that \n$p \\circ s = \\operatorname{Id} \\restrict {N_\\pi X}$. \n\nThis definition\nis naturally adopted to the case of complex analytic family\nof manifolds. If the \nsection $\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T\\c X$ is holomorphic,\nthe connection is called {\\bf a holomorphic connection}. \n\nThis definition makes sense only when the base variety $S$ is smooth.\nHowever, the connection in a family gives a connection on a pullback\nof this family, under all maps $s:\\; S' {\\:\\longrightarrow\\:} S$. Connections on the \npullback families are naturally compatible. We shall sometimes speak \nof {\\bf connection} in a family where the base $S$ is not smooth.\nThis means that for all maps $s:\\; S' {\\:\\longrightarrow\\:} S$, where $S'$ is smooth,\nthe pullback family is equipped with a connection, and these\nconnections are compatible. All the definitions and results\n(which we state and prove in the case of a smooth base) are\nnaturally adopted to the case when the base $S$ is singular.\n\n\n\\hfill\n\n\n\\proposition \\label{_conne_in_fam_of_comple_geode_Proposition_}\nLet \n\\[ \\left( \\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S, \n \\phi:\\; \\c X {\\:\\longrightarrow\\:} M \\right)\n\\]\nbe a family of submanifolds of a Riemannian manifold $M$.\nAssume that for all $s\\in S$, the submanifold \n$X_s:= \\psi(\\pi^{-1}(s)) \\subset M$ is completely geodesic. Then\nthe family $\\c X$ is equipped with a natural connection.\nMoreover, if $M$ is K\\\"ahler and the family $\\c X$ is complex\nanalytic, then the connection $\\nabla$ is holomorphic.\n\n\\hfill\n\n{\\bf Proof:} Let $T_\\pi X\\subset T\\c X$ be the bundle of vectors\ntangent to the fibers of $\\pi$.\nTo split the exact sequence\n\\[ \n 0 {\\:\\longrightarrow\\:} T_\\pi X \\stackrel i{\\:\\longrightarrow\\:} T\\c X {\\:\\longrightarrow\\:} T_\\pi X {\\:\\longrightarrow\\:} 0,\n\\]\nwe have to construct a surjection \n\\begin{equation} \\label{_surje_secti_for_conne_Equation_}\n T{\\c X} \\stackrel p {\\:\\longrightarrow\\:} T_\\pi X\n\\end{equation}\nsatisfying $i\\circ p = \\operatorname{Id}_{T_\\pi X}$.\nConsider the orthogonal decomposition \n\\begin{equation} \\label{_ortho_compo_for_norma_in_fami_Equation_}\n\\phi^*TM=\\phi^*N_M X_t \\oplus TX_t,\n\\end{equation}\nwhere $N_M X_t$ is the normal bundle to $\\phi(X_t)$ in $M$.\nThis gives a natural epimorphism \n\\[ \\phi^* TM \\stackrel e{\\:\\longrightarrow\\:} T_\\pi X.\\] Taking a composition\nof $e$ with \\[ d\\phi:\\; T\\c X {\\:\\longrightarrow\\:} \\phi^* TM,\\]\nwe obtain a surjection $p$ of \n\\ref{_surje_secti_for_conne_Equation_}. \n\nIn K\\\"ahler case, the decomposition\n\\eqref{_ortho_compo_for_norma_in_fami_Equation_}\nis holomorphic by \\ref{_comple_geo_compa_holo_Lemma_}. Thus, \nwe constructed a connection which is holomorphic.\n\\blacksquare\n\n\\hfill\n\nFor each real analytic path $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} S$,\n$t, t'\\in [0,1]$, we may {\\bf integrate} the connection\nalong $\\gamma$. This notion is intuitively\nclear (and well known). We recall briefly\nthe definition of the integral maps associated to connection,\nin order to fix the notation.\n\nRestricting the family $\\c X$ to the image of $\\gamma$\nin $S$ does not change the result of integration.\nTo simplify the exposition, we assume that $S$ coinsides with\nthe image $D \\cong [0,1]$ of $\\gamma$. In this case, the bundle $N_\\pi X$ \nis naturally isomorphic to a pullback $\\pi^* T D$. Denote by\n$\\nu$ a canonical unit section of $\\pi^* TD$ corresponding to\nthe unit tangent field to $[0,1]$. Then, $\\nabla \\nu$ is a vector\nfield in $T\\c X$. Integrating $\\nu$ to a \ndiffeomorphism, we obtain a map $\\exp(t \\nu)$ defined in an open\nsubset of $\\c X$, for $t\\in {\\Bbb R}$. Clearly, $\\exp(t\\nu)$ maps the points\nof $X_{t_1}$ to the points of $X_{t_1+t}$. The resulting\ndiffeomorphism we denote by \n\\[ \n \\Psi^{t_1}_{t_1+t}:\\; U_{t_1} {\\:\\longrightarrow\\:} U_{t+t_1}, \n\\]\nwhere $U_{t_1}$ is an intersection of $X_{t_1}$ with the domain \nof $\\exp(t\\nu)$. Clearly, the same definition applies to the\narbitrary families of manifolds with connection. When $\\c X$\nis a complex analytic family with a holomorphic connection,\nthe diffeomorphisms $\\Psi^t_{t'}$ are complex analytic.\n\n\n\n\n\n\\subsection{Deformations of completely\ngeodesic submanifolds (the main statement)}\n\n\nLet \n\\[ \\left( \\c X \\stackrel \\pi {\\:\\longrightarrow\\:} S, \n \\phi, \\phi_{s_0}:\\; X_{s_0} \\oldtilde {\\:\\longrightarrow\\:} X\\right )\n\\]\nbe a real analytic deformation of $X\\subset M$,\nand $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} S$ be a real analytic map.\nSlightly abusing the notation, we \nshall consider the fibers $X_s$ of $\\pi$ as subvarieties in $M$.\nAssume that $X_{\\gamma(t)}$ is a completely geodesic complex\nanalytic submanifold of $M$, for all\n$t$. Let $\\eta_t \\in \\Gamma \\left(N X_{\\gamma(t)}\\right)$ be a section of\nthe normal bundle to $X_{\\gamma(t)}$ corresponding to the vector \n$\\frac {d \\gamma}{dt} \\in T_{\\gamma(t)} S$. \nUsing the natural \nholomorphic embedding \n\\begin{equation} \\label{_N_embe_to_TM_Equation_} \n N X_{\\gamma(t)} \\hookrightarrow TM\\restrict{X_{\\gamma(t)}}\n\\end{equation}\nassociated with a decomposition \n$TM\\restrict{X_{\\gamma(t)}} = N X_{\\gamma(t)}\\oplus TX_{\\gamma(t)}$,\nwe may consider $\\eta_t$ as a section of\n$TM\\restrict{X_{\\gamma(t)}}$.\n\n\\hfill\n\n\\proposition \\label{_conne_in_fami_of_comple_geo_Proposition_}\nLet $t$, $t'\\in [0,1]$. \nLet $\\Psi^t_{t'}:\\; U_t {\\:\\longrightarrow\\:} U_{t'}$ be the map of\nSubsection \\ref{_conne_Subsection_} \nintegrating the connection $\\nabla$ of \n\\ref{_conne_in_fam_of_comple_geode_Proposition_},\nwhere $U_t$, $U_{t'}$ are open subsets \nof $X_{\\gamma(t)}$, $X_{\\gamma(t')}$ defined in \nSubsection \\ref{_conne_Subsection_}.\nAssume that for all $t$ the section \n$\\eta_t\\in TM\\restrict{X_{\\gamma(t)}}$ is \nparallel with respect to the natural\nconnection on $TM\\restrict{X_{\\gamma(t)}}$\nobtained from Levi--Civita on $TM$. Assume also that the Riemannian form\non $M$ is real analytic.\\footnote{This assumption is extraneous;\nwe use it to simplify the exposition. For the case we are interested\nin ($M$ a hyperk\\\"ahler manifold) the Riemannian form\nis real analytic by \\ref{_Riema_on_hype_real_ana_Corollary_}.}\nThen $\\Psi^t_{t'}$ satisfies the following conditions.\n\n\\begin{description}\n\n\\item[(i)] $\\Psi^t_{t'}$ is compatible\nwith a Riemannian structure on $U_t$, $U_{t'}$. \n\n\\item[(ii)] \nAssume also that $M$ admits a smooth Riemannian\ncompactification $\\bar M$ such that the family $\\c X$ \nadmits a compactification in $\\bar M$ (such is the case\nwhen $M$ and $\\c X$ are quasiprojective, and $\\phi$ is\nalgebraic). Then the maps $\\Psi^t_{t'}:\\; U_t {\\:\\longrightarrow\\:} U_{t'}$ of (i) \ncan be found in such a way that $U_t$ is the set \nof non-singular points of $X_{\\gamma(t)}$. \n\n\\item[(iii)] In assumptions of (ii), we can extend $\\Psi^t_{t'}$\nto an isomorphism of metric spaces \n\\[ \n \\bar \\Psi^t_{t'}:\\; \\bar X_{\\gamma(t)} {\\:\\longrightarrow\\:} \\bar X_{\\gamma(t')}, \n\\]\nwhere $\\bar X_{\\gamma(t)}$, $\\bar X_{\\gamma'(t)}$ are\nclosures of $X_{\\gamma(t)}$, $X_{\\gamma'(t)}$ in $\\bar M$, with induced metrics.\n\n\\end{description}\n\nThe next subsection is taken by the proof of \n\\ref{_conne_in_fami_of_comple_geo_Proposition_}.\n\n\n\\subsection{Deformations of completely\ngeodesic submanifolds (the proofs)}\n\n\nThe statement (i) is sufficient to prove\nin a small neighbourhood, by analytic continuation. Thus, we\nmay pick an open subset $U\\in {\\c X}$ in such a way that the restriction of\n$\\phi:\\; \\c X {\\:\\longrightarrow\\:} M$ to \n\\[ \n \\c X':= \\pi^{-1}(\\gamma([0,1])\\cap U\n\\]\nis an embedding. Shrinking $\\c X$ to $\\c X'$ and pulling\nfrom $M$ the Riemannian metric, we see that\n\\ref{_conne_in_fami_of_comple_geo_Proposition_} (i) is \nimplied by the following lemma.\n\n\\hfill\n\n\\lemma \\label{_integra_field_Killing_Lemma_}\nLet $\\c X \\stackrel \\pi{\\:\\longrightarrow\\:} [0,1]$ \nbe a real analytic family of manifolds, equipped with a Riemannian\nmetrics. Let $T _\\pi X\\subset T\\c X$ \nbe the relative tangent bundle consisiting\nof all vectors tangent to the fibers of $\\pi$, and \n$N _\\pi X= T \\c X \/T _\\pi X$ be the normal bundle \nto the fibration $\\pi$. Assume that the fibers \n$\\pi^{-1} (t) \\subset \\c X$ are completely geodesic \nin $\\c X$. Consider the connection\n$\\nabla:\\; N _\\pi X {\\:\\longrightarrow\\:} T\\c X$ of \n\\ref{_conne_in_fam_of_comple_geode_Proposition_}.\nLet $\\Psi^t_{t'}:\\; U_t {\\:\\longrightarrow\\:} U_{t'}$ be the\nmaps obtained by integrating the connection as in \nSubsection \\ref{_conne_Subsection_}\nand $\\eta_t \\in N _\\pi X$ be the normal fields arising\nfrom the deformation theory. Assume that $\\eta_t$'s\nare parallel with respect to the natural connection on $N _\\pi X$.\nThen the maps $\\Psi^t_{t'}$ are compatible with Riemannian metrics.\n\n\\hfill\n\n{\\bf Proof:} Let $\\eta\\in T \\c X$ be the tangent vector field\nobtained by gluing all $\\eta_t$ together. Then $\\Psi_t$ \ncan be considered as integral map of this vector field. Thus,\nto prove that $\\Psi_t$ is compatible with the Riemannian\nstructure, we have to show that \n$\\eta$ is {\\bf a Killing vector field}.\\footnote{A Killing field \nis a vector field which integrates to a diffeomorphism which is\ncompatible with a Riemannian metric.} Denote by\n$\\nabla_x:\\; T\\c X {\\:\\longrightarrow\\:} T\\c X$ the \naction of covariant derivative along the vector field $x$. \nBy \\cite{_Besse:Einst_Manifo_},\nTheorem 1.81, to prove that $\\eta$ is Killing it \nsuffices to prove that for all fields $a, b\\in T\\c X$,\nwe have\n\\begin{equation}\\label{_Killing_Equation_} \n (\\nabla_a \\eta, b) + (\\nabla_b \\eta,a) =0,\n\\end{equation}\nwhere $(\\cdot,\\cdot)$ is the Riemannian form.\nTake coordinates $(x_0, x_1,...x_n)$ \non $\\c X$ in such a way that $x_0$ comes from a\nprojection $\\pi:\\; \\c X {\\:\\longrightarrow\\:} [0,1]$ and \n$x_1, ... x_n$ are coordinates along the fibers.\nLet $\\frac{d}{d x_i}$ be the corresponding vector fields.\nTo prove that $\\eta$ is Killing it suffices to check\n\\eqref{_Killing_Equation_} \nfor $a$, $b$ coordinate vector\nfields. Since $\\eta$ is parallel along $X_t$, we have\n$\\nabla_{\\frac{d}{d x_i}} \\eta=0$ for $i= 1,..., n$.\nThus, it suffices to prove \\eqref{_Killing_Equation_} \nin case $a = \\frac{d}{dx_0}$. For appropriate choice\nof coordinates, $\\frac{d}{d x_0}=\\eta$; thus, \n\\eqref{_Killing_Equation_} is implied by\n\\begin{equation}\\label{_nabla_eta_coo_Equation_}\n \\left(\\nabla_\\eta \\eta,\\frac{d}{d x_i}\\right) =0, \n \\;\\; i= 0,\\dots n.\n\\end{equation}\nSince Levi-Civita connection is compatible with the metrics,\nwe have \n\\begin{equation}\\label{_Levi_Civi_compa_Equation_}\n D_\\eta \\left(\\eta, \\frac{d}{d x_i}\\right) =\n \\left(\\nabla_\\eta \\eta,\\frac{d}{d x_i}\\right) + \n \\left(\\eta,\\nabla_\\eta\\frac{d}{d x_i}\\right),\n\\end{equation}\nwhere $D_\\eta$ is the usual (directional) derivative along $\\eta$.\nFor $i=0$, this gives \n\\[ D_\\eta (\\eta, \\eta) =\n (\\nabla_\\eta \\eta,\\eta) + \n (\\eta,\\nabla_\\eta\\eta)=0\n\\]\n($\\eta$ is parallel, and hence has constant length).\nThis proves \\eqref{_nabla_eta_coo_Equation_} for $i=0$.\nSince $\\frac{d}{d x_i}$ are coordinate vector fields\nthey commute, and therefore, $\\nabla_\\eta\\frac{d}{d x_i}=\n-\\nabla_{\\frac{d}{d x_i}}\\eta=0$ for $i>0$. Thus,\nthe equation \\eqref{_Levi_Civi_compa_Equation_} for $i>0$ is \nreduced to \n\\[\n D_\\eta \\left(\\eta, \\frac{d}{d x_i}\\right) =\n \\left(\\nabla_\\eta \\eta,\\frac{d}{d x_i}\\right). \n\\]\nSince the vector fields\n$\\frac{d}{d x_i}$, $i>0$ are tangent to the fibration $\\pi$, the function\n$(\\eta, \\frac{d}{d x_i})$ is identically zero. This proves \n\\eqref{_nabla_eta_coo_Equation_} for $i>0$. \n\\ref{_integra_field_Killing_Lemma_} and consequently\n\\ref{_conne_in_fami_of_comple_geo_Proposition_}\n(i) is proven. \\blacksquare\n\n\\hfill\n\nTo prove\n\\ref{_conne_in_fami_of_comple_geo_Proposition_}\n(ii) we have to show that the connection of \n\\ref{_conne_in_fam_of_comple_geode_Proposition_} \ncan be integrated for all smooth points of $\\c X$.\nWe give a sketch of a simple geometric argument.\nBy (i), the maps $\\Psi_t$ are isometries. Thus, the \ndistance from the given point \nto the singular set of $X_t$ is invariant under \nthe maps $\\Psi_t$. To integrate the connection,\nwe write a tangent vector field which we\nsubsequently integrate. Since the normal field $\\eta_t$\nis parallel, this tangent field is uniformly bounded.\nOn a certain distance from the singular set $Sing(X_t)$, depending\non this uniform bound, the connection might be always \nintegrated. In more precise terms,\nfor all $\\epsilon>0$, all $x\\in X_t$, with the distance\nbetween $x$ and $Sing(X_t)$ no less than $\\epsilon$, there exist\n$\\delta>0$ and a map \n$\\Psi^t_{t+\\delta}:\\;X_t {\\:\\longrightarrow\\:} X_{t+\\delta}$ defined\nin a neighbourhood of $x$ which integrates the connection.\n{}From this statement and \\ref{_conne_in_fami_of_comple_geo_Proposition_} \n(i), \\ref{_conne_in_fami_of_comple_geo_Proposition_} (ii)\nfollows directly.\n\nFinally, to prove \\ref{_conne_in_fami_of_comple_geo_Proposition_} (iii)\nwe notice that $X_t$ is completely geodesic in $M$ for all $t$. Thus,\nthe completion of $X_t$ as a metric space coinsides with the\nclosure of $X_t$ in $\\bar M$. Every isometry of metric spaces \nextends to a completion, and thus, $\\Psi^t_{t'}$ extends to a closure of\n$X_t$, $X_{t'}$ in $\\bar M$.\n\\blacksquare\n\n\n\\section[Hyperholomorphic bundles and completely geodesical \nembeddings.]{Hyperholomorphic bundles \\\\and completely geodesical\nembeddings.} \n\\label{_comple_geode_hyperho_Section_}\n\n\n\n\\hfill\n\nIn this section, we prove that hyperk\\\"ahler embeddings are\ncompletely geodesic. \n\n\\subsection{Hyperholomorphic structure on the normal bundle}\n\nLet $M$ be a hyperk\\\"ahler manifold, not necessary compact,\nand $B$ a vector bundle.\nRecall that in Section \\ref{_basics_Section_}, we defined \n{\\bf hyperholomorphic connections} in $B$ \n(\\ref{_hyperho_conne_Definition_}). \nThese are connections $\\nabla$ such that the curvature \n$\\Theta:=\\nabla^2\\in \\Lambda^2(M,End(B))$ is an \n$SU(2)$-invariant 2-form with respect to the\nnatural $SU(2)$-action in $\\Lambda^2(M)$. \nThe hyperholomorphic connections are always Yang-Mills \n(\\ref{_hyperholo_Yang--Mills_Proposition_}). The Levi--Civita connection\nin the tangent bundle $TM$ is a prime example of a hyperholomorphic\nconnection (\\ref{_tangent_hyperholo_Example_}). \n\n\\hfill\n\n\\proposition \\label{_NX_hyperholo_Proposition_}\nLet $M$ be a hyperk\\\"ahler manifold, not necessary compact, and \n$X\\subset M$ a trianalytic submanifold. Consider the normal\nbundle $NX$, equipped with a connection $\\nabla$ \ninduced from the Levi--Civita\nconnection on $M$. Then $NX$ is hyperholomorphic.\n\n\\hfill\n\n{\\bf Proof:} Let $L$ be an induced complex structure $M$.\nConsider the manifold $(X, L)$ as a complex submanifold of $(M, L)$. The\nnormal bundle $NX = N(X,L)$ has a natural holomorphic structure,\nwhich is compatible with the connection $\\nabla$. Therefore,\nthe curvature $\\Theta\\in \\Lambda^2(X, End(NX))$ is of type\n$(1,1)$ with respect to the Hodge decomposition\ndefined by $L$. Thus, $\\Theta$ is of type $(1,1)$ with\nrespect to any of the induced complex structures.\nBy definition, this means that $\\nabla$ is hyperholomorphic.\n\\blacksquare\n\n\\subsection{Hyperk\\\"ahler embeddings are completely geodesic}\n\nIn assumptions of \\ref{_NX_hyperholo_Proposition_}, \nconsider the bundle $TM\\restrict X$ with metrics and connection\ninduced from $TM$. There is a natural exact sequence of \nholomorphic vector bundles over $(X, L)$:\n\\[\n 0{\\:\\longrightarrow\\:} TX {\\:\\longrightarrow\\:} TM\\restrict X {\\:\\longrightarrow\\:} NX {\\:\\longrightarrow\\:} 0 .\n\\]\nThe natural connection in each of these bundles is hyperholomorphic\n(\\ref{_NX_hyperholo_Proposition_}).\n\n\n\\hfill\n\n\\proposition \\label{_NX_splits_for_hype_Proposition_}\nLet $X$ be a hyperk\\\"ahler manifold, not necessary compact, $L$\ninduced complex structure, and \n\\begin{equation}\\label{_holo_seque_Equation_}\n 0{\\:\\longrightarrow\\:} E_1{\\:\\longrightarrow\\:} E_2 {\\:\\longrightarrow\\:} E_3{\\:\\longrightarrow\\:} 0\n\\end{equation}\n be an exact sequence\nof holomorphic vector bundles. Let $g_2$ be a Hermitian structure \non $E_2$. Consider the metrics $g_1$, $g_2$, $g_3$ and the connections\n$\\nabla_1$, $\\nabla_2$, $\\nabla_3$ on $E_1$, $E_2$, $E_3$ \ninduced by $g_2$. Assume that $\\nabla_1$, $\\nabla_2$, $\\nabla_3$\nare hyperholomorphic. Then the exact sequence\n\\eqref{_holo_seque_Equation_} splits, and moreover,\nthe orthogonal decomposition $E_2 = E_1 \\oplus E_1^\\bot$ is preserved by\n the connection $\\nabla_2$.\n\n\\hfill\n\n{\\bf Proof:} The same statement is well known \nfor the Yang--Mills connections: every exact sequence of \nholomorphic bundles with compatible Yang--Mills metrics\nsplits (see Appendix to this section). \nBy \\ref{_hyperholo_Yang--Mills_Proposition_}, \nhyperholomorphic connections\nare always Yang--Mills. \\blacksquare\n\n\\hfill\n\n\\definition\nLet $N \\stackrel i \\hookrightarrow M$ be an embedding of\nhyperk\\\"ahler manifolds. We say that $i$ is a {\\bf hyperk\\\"ahler\nembedding} if $i$ is compatible with the quaternionic structure\nand Riemannian metric.\n\n\\hfill\n\n\\corollary\\label{_hype_embe_comple_geode:Corollary_}\nLet $N \\stackrel i \\hookrightarrow M$ be a hyperk\\\"ahler embedding.\nThen $i$ is completely geodesic.\n\n{\\bf Proof:} Follows directly from \n\\ref{_comple_geodesi_basi_Proposition_} and \n\\ref{_NX_splits_for_hype_Proposition_} \\blacksquare\n\n\\hfill\n\n\\subsection{Appendix: every exact \nsequence of Yang--Mills bundles splits.}\n\n\n\n\\hspace{6mm}\n\\proposition \\label{_YM_exact_split_Proposition_}\nLet $X$ be a K\\\"ahler manifold, not necessary compact, and\n\\begin{equation}\\label{_holo_seque2_Equation_}\n 0{\\:\\longrightarrow\\:} E_1{\\:\\longrightarrow\\:} E_2 {\\:\\longrightarrow\\:} E_3{\\:\\longrightarrow\\:} 0\n\\end{equation}\n be an exact sequence\nof holomorphic vector bundles. Let $g_2$ be a Hermitian \nstructure on $E_2$.\nConsider the induced metrics $g_1$, $g_2$, $g_3$. \nAssume that either $g_1$ or $g_3$ is Yang--Mills. \nThen the exact sequence\n\\eqref{_holo_seque_Equation_} splits, and moreover,\nthe orthogonal complement $E_1^\\bot\\subset E_2$ is preserved by\n the connection $\\nabla_2$.\n\n\\hfill\n\n{\\bf Proof:} Consider the second fundamental form \n\\[ A\\in \\Lambda^{0,1}(X, \\operatorname{Hom}(E_1,E_3)) \\]\nof the exact sequence \\eqref{_holo_seque2_Equation_}.\nThe curvatures $\\Theta_i$ of $E_i$ are expressed through\n$A$ as follows (\\cite{_Griffi_Harri_}): \n\\begin{equation}\\label{_Theta_1_through_seco_Equation_}\n \\Theta_1 = \\Theta_2 \\restrict{E_1} + {}^t\\bar A\\wedge A,\n\\end{equation}\n\\begin{equation}\\label{_Theta_3_through_seco_Equation_}\n \\Theta_3 = \\Theta_2 \\restrict{E_3} - A\\wedge{}^t\\bar A.\n\\end{equation}\nThe 2-forms ${}^t\\bar A\\wedge A$, $A\\wedge{}^t\\bar A$ are {\\bf positive}\nunless $A=0$. Thus, the endomorphisms\n$\\Lambda \\left(A\\wedge{}^t\\bar A\\right)\\in End(E_1)$, \n$\\Lambda \\left({}^t\\bar A\\wedge A\\right)\\in End(E_3)$ have positive trace\n(again, unless $A=0$). By our assumption,\n$\\Lambda (\\Theta_2)=0$ and either $\\Lambda(\\Theta_1)=0$\nor $\\Lambda(\\Theta_3)=0$. Applying $\\Lambda$ to both sides\nof \\eqref{_Theta_1_through_seco_Equation_} and\n\\eqref{_Theta_3_through_seco_Equation_}, we obtain that either\n$\\Lambda \\left(A\\wedge{}^t\\bar A\\right)=0$ or\n$\\Lambda \\left({}^t\\bar A\\wedge A\\right)=0$. \nTherefore, $A=0$ and the exact sequence\n\\eqref{_holo_seque2_Equation_} splits.\n\\blacksquare\n\n\n\\section[Triholomorphic sections of hyperholomorphic bundles\nand deformations of trianalytic submanifolds.]\n{Triholomorphic sections \\\\of hyperholomorphic bundles\\\\\nand deformations of trianalytic submanifolds.}\n\\label{_triholo_Section_}\n\n\nIn the previous section, we proved that hyperk\\\"ahler embeddings are\ncompletely geodesic. In this section, we show that, furthermore,\nthe deformational results of Section\n\\ref{_comple_geode_defo_Section_} are fully applicable\nto the deformations of trianalytic submanifolds.\n\n\\subsection{Triholomorphic sections of normal bundle}\n\n\\hfill\n\n\\definition \nLet $M$ be a hyperk\\\"ahler manifold, $B$ a vector bundle\nequipped with a hyperholomorphic connection, and $\\alpha$ a section\nof $B$. Then $\\alpha$ is called {\\bf triholomorphic} \nif for each induced complex structure $L$ on $M$, $\\alpha$ \nis a holomorphic section of $B$ considered as a holomorphic\nbundle over $(M, L)$. \n\n\\hfill\n\nLet $M$ be a hyperk\\\"ahler manifold, not necessarily compact, and\n$X\\subset M$ a trianalytic submanifold, not necessarily closed.\nFix an induced complex structure $I$ on $M$. Let \n\\[ \n \\left( \\pi:\\; \\c X {\\:\\longrightarrow\\:} S, \n \\phi:\\; \\c X {\\:\\longrightarrow\\:} M, \\phi(\\pi^{-1}(s_0)) = X \\right)\n \\]\nbe a family of deformations of $(X,I) \\subset (M, I)$\n(see Section \\ref{_comple_geode_defo_Section_} for details). \nLet $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} S$ be a real analytic path in $S$, such\nthat $\\gamma(0) = s_0$. Assume that for all $t\\in [0,1]$, the\nsubmanifold $X_t = \\phi(\\gamma^{-1}(t))\\subset M$ is trianalytic.\nConsider the normal bundle $NX$ to $X$, with the metric and\nconnection induced from $M$. By \\ref{_NX_hyperholo_Proposition_}, \n$NX$ is hyperholomorphic. Let $\\eta$ be the section of \n$NX$ corresponding to $\\frac{d\\gamma}{dt}$ as in \nSubsection \\ref{_norma_vecto_Subsection_}.\n\n\\hfill\n\n\\proposition \\label{_norma_triholo_Proposition_}\nIn the above assumptions, $\\eta$ is a triholomorphic section of $NX$.\n\n\\hfill\n\n{\\bf Proof:} Let $L$ be an induced complex structure, and $(M, L)$,\n$(X, L)$ the manifolds $M$ and $X$ considered as complex \nmanifolds with the complex structure $L$. The normal bundle\n$N(X,L)$ is naturally identified with $NX$ as a real vector \nbundle. Therefore $\\eta$ can be considered as a section\nof $N(X,L)$. By \\ref{_real_ana_indu_on_hype_equiva_Proposition_}\n(see Appendix to this section), $\\c X$ can be considered\nas a real analytic deformation of $(X, L)$.\n{}From \\ref{_norma_sec_holom_Lemma_} \nit is clear that $\\eta$ is holomorphic\nas a section of $N(X,L)$. This proves \n\\ref{_norma_triholo_Proposition_}. \\blacksquare\n\n\\subsection{Triholomorphic sections are parallel}\n\n\\hfill\n\n\\proposition\\label{_triholo_parallel_Proposition_}\nLet $M$ be a hyperk\\\"ahler manifold, not necessary compact,\nand $B$ a vector bundle with a hyperholomorphic connection $\\nabla$.\nLet $\\nu$ be a trianalytic section of $B$. Then, $\\nu$ is parallel:\n\\begin{equation}\\label{_nu_para_Equation_}\n\\nabla\\nu=0\n\\end{equation}\n\n{\\bf Proof:} Let $L$ be an induced complex structure. Since \n$\\nu$ is triholomorphic, $\\bar \\partial_L \\nu =0$, where\n$\\bar \\partial_L:\\; B {\\:\\longrightarrow\\:} B\\times \\Lambda^{0,1}_L(M)$ is the\n$(0,1)$-part of the connection, taken with respect to $L$.\nTaking $L=I, -I$, we obtain \n\\begin{equation} \\label{_bar6_I+bar6_-I_Equation_}\n\\bar \\partial_I + \\bar\\partial_{-I}(\\nu)=0.\n\\end{equation}\nOn the other hand, $\\bar\\partial_{-I}= \\partial_I$, where $\\partial_I$ is the \n$(1,0)$-part of $\\nabla$ taken with respect to $I$. Thus,\n\\[ \\bar \\partial_I + \\bar\\partial_{-I} =\\bar \\partial_I + \\partial_{I} =\\nabla. \\]\n{}From \\eqref{_bar6_I+bar6_-I_Equation_} \nwe obtain that $\\nabla(\\nu)=0$. \\blacksquare\n\n\\hfill\n\n{}From \\ref{_norma_triholo_Proposition_} \nand \\ref{_triholo_parallel_Proposition_}, \nwe obtain that the section $\\eta \\in NX$ \nis parallel with respect to the connection. Then, \n\\ref{_conne_in_fami_of_comple_geo_Proposition_} (i)\ncan be applied to the following effect.\n\n\\hfill\n\n\\corollary \\label{_norma_sec_para_for_hype_from_it_Corollary_}\nLet $M$ be a hyperk\\\"ahler manifold, not necessarily compact, and\n$X\\subset M$ a trianalytic submanifold, not necessarily closed,\nand\n\\[ \n \\left( \\pi:\\; \\c X {\\:\\longrightarrow\\:} S, \n \\phi:\\; \\c X {\\:\\longrightarrow\\:} M, \\phi(\\pi^{-1}(s_0)) = X \\right)\n \\]\nbe a real analytic family of deformations of $X\\subset M$\n(see Section \\ref{_comple_geode_defo_Section_} for details). \nLet $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} S$ be a real analytic path in $S$, such\nthat $\\gamma(0) = s_0$. Assume that for all $t\\in [0,1]$, the\nsubmanifolds $X_t = \\phi(\\gamma^{-1}(t))\\subset M$ are trianalytic.\nFix an induced complex structure $I$ on $M$.\nLet $U_t\\subset (X_t, I)$, $t\\in [0,1]$ be the subsets constructed \nin Subsection \\ref{_conne_Subsection_}\\footnote{By \n\\ref{_hype_embe_comple_geode:Corollary_},\ntrianalytic submanifolds are completely geodesic, and thus,\n\\ref{_conne_in_fam_of_comple_geode_Proposition_} can be applied.}\nwith the corresponding holomorphic isomorphisms \n$\\Psi_t:\\; U_0{\\:\\longrightarrow\\:} U_t$. Then, assumptions of\n\\ref{_conne_in_fami_of_comple_geo_Proposition_} (i) hold.\nThus,\nfor all $t, t'\\in [0,1]$,\nthe maps $\\Psi^t_{t'}$ are isometries. Moreover,\n$\\Psi^t_{t'}$ are compatible with the hyperk\\\"ahler structure.\n\n\\hfill\n\n{\\bf Proof:} \nAssumptions of \\ref{_conne_in_fami_of_comple_geo_Proposition_} (i)\nhold by \\ref{_norma_triholo_Proposition_} \nand \\ref{_triholo_parallel_Proposition_}.\nThe maps $\\Psi^t_{t'}$ are isometries by \n\\ref{_conne_in_fami_of_comple_geo_Proposition_} (i).\nThe maps $\\Psi^t_{t'}$ are compatible with the \nhyperk\\\"ahler structure because the they are obtained by integrating\na certain connection in the family $\\c X$. This connection\nis constructed from the Hermitian metric, and thus, does\nnot depend from the choice of induced complex structure. Taking\ndifferent induced complex structures,\nwe obtain the same maps $\\Psi^t_{t'}$. \nThus, $\\Psi^t_{t'}$ is holomorphic\nwith respect to each of induced complex structures\n(see also \\ref{_norma_sec_holom_Lemma_}).\n\\blacksquare\n\n\\subsection{Appendix: real analytic structures on hyperk\\\"ahler\nmanifolds}\n\nConsider the real analytic structures on a given hyperk\\\"ahler\nmanifold arising from the different induced complex structures.\nWe prove that these real analytic structures are equivalent.\n\n\\hfill\n\n\\proposition \\label{_real_ana_indu_on_hype_equiva_Proposition_}\nLet $M$ be a hyperk\\\"ahler manifold, $I_1$, $I_2$ induced complex\nstructures. Let $(M, I_1)$ and $(M, I_2)$ be the corresponding\ncomplex manifolds, and $(M, I_1)_{\\Bbb R}$, $(M, I_2)_{\\Bbb R}$ be the real\nanalytic manifolds underlying $(M, I_1)$, $(M, I_2)$. Consider the\ntautological map $(M, I_1)_{\\Bbb R}\\stackrel \\phi {\\:\\longrightarrow\\:} (M, I_2)_{\\Bbb R}$.\nThen $\\phi$ is compatible with the real analytic structure.\n\n\\hfill\n\n{\\bf Proof:} Consider the {\\bf twistor space} for $M$ (see\n\\cite{_Besse:Einst_Manifo_}), $\\operatorname{Tw}(M)$, with the natural\nholomorphic map $\\pi:\\; \\operatorname{Tw}(M) {\\:\\longrightarrow\\:} {\\Bbb C} P^1$. Let $Sec(M)$\nbe the space of holomorphic sections of the map $\\pi$.\nThen $Sec(M)$ is identified naturally with an open subspace\nof a Douady space for $\\operatorname{Tw}(M)$, and thus, has a natural \ncomplex structure. According to \\cite{_HKLR_} (see \\cite{_NHYM_} \nfor details), $Sec(M)$ is a complexification of $(M,I)$, in the\nsense of Grauert. In other words, the complex valued real analytic functions\non $(M,I)$ are naturally identified with the germs of complex analytic\nfunctions on $Sec(M)$. Since $Sec(M)$ is defined independently\nfrom the choice of an induced complex structure, the \ntautological map $\\phi$ is an equivalence.\nThis proves \\ref{_real_ana_indu_on_hype_equiva_Proposition_}.\n \\blacksquare\n\n\\hfill\n\n\\ref{_real_ana_indu_on_hype_equiva_Proposition_} implies that we\nmay speak of a real analytic manifold underlying a given hyperk\\\"ahler\nmanifold. \n\n\\hfill\n\n\\corollary \\label{_Riema_on_hype_real_ana_Corollary_}\nLet $M$ be a hyperk\\\"ahler manifold. Consider the Riemannian\nform $g$ on $M$ as a section of the real analytic bundle of \nsymmetric 2-forms. Then $g$ is real analytic.\n\n\\hfill\n\n{\\bf Proof:} Let $I, J, K$ be the induced complex structures\nwhich form the standard basis in quaternions, and \n$\\omega_I$, $\\omega_J$, $\\omega_K$ be the corresponding \nK\\\"ahler forms. Then $\\Omega:=\\omega_J + \\sqrt{-1}\\:\\omega_K$ is the natural\nholomorphically symplectic form on $(M,I)$, and as such,\n$\\Omega$ is real analytic. Then, its real part $\\omega_J$ is\nalso real analytic. Since the complex structure operator $J$\nis real analytic, we obtain that the form\n$g(\\cdot,\\cdot):= - \\omega(\\cdot, J\\cdot)$ is also real analytic.\n\\blacksquare\n\n\n\n\n\n\\section{Douady spaces for trianalytic cycles and real analytic structure}\n\\label{_Douady_Section_}\n\n\n\\subsection{Real analytic structure on the Douady space.} \n\nIn this section, we consider the Douady space $D_M(X)$\nfor a subvariety $X$\nof a compact complex manifold $M$ equipped with\n a hyperk\\\"ahler structure.\nWe prove that, when $X$ is trianalytic, $D_M(X)$ is {\\bf\nhypercomplex} (\\ref{_hypercomplex_Definition_}). \nWe also show that the real analytic\nvariety underlying $D_M(X)$ does not change if we replace\na complex structure on $M$ by another induced complex structure.\nThese results are technical and we use them mainly to \nsimplify the exposition. \n\n\\hfill\n\nLet $M$ be a compact hyperk\\\"ahler manifold, and $X\\subset M$ a \nclosed trianalytic subvariety. For each induced complex\nstructure $L$, consider $(X, L)$ as a complex subvariety of $(M, L)$.\nLet $D_L(X)$ be the Douady deformation space of $(X, L)$ in\n$(M, L)$. The points $[X']$ of $D_L(X)$ correspond to the\nsubvarieties $X'\\subset (M, L)$ which are deformation of $X$.\nBy \\ref{_G_M_invariant_implies_trianalytic_Theorem_} (see also\n\\cite{Verbitsky:Symplectic_II_}), every such $X'$ is trianalytic. \n\nLet $I_1$, $I_2$ be induced complex structures on $M$.\nThe following proposition shows that a trianalytic subvariety\n$X'$ can be obtained as a deformation of $(X, I_1)$ in $(M, I_1)$\nif and only if $X'$ can be obtained\nas a deformation of $(X, I_2)$ in $(M, I_2)$.\n\n\\hfill\n\n\\proposition \\label{_X_in_Douady_indep_from_I_1_Proposition_} \nLet $I_1$, $I_2$ be induced complex\nstructures on a compact hyperk\\\"ahler manifold\n$M$, and $X\\subset M$ a closed trianalytic subvariety. \nConsider the corresponding Douady spaces $D_{I_1}(X)$,\n$D_{I_2}(X)$. Then, for a trianalytic subvariety $X'\\subset M$,\n$[X'] \\in D_{I_1}(X)$ if and only if $[X']\\in D_{I_2}(X)$.\n\n\\hfill\n\n{\\bf Proof:} Let us recall the notion of {\\bf degree} of\na trianalytic subvariety.\nLet $M_1$ be a K\\\"ahler manifold.\nBy {\\bf degree} of a subvariety $X \\subset M_1$ we understand a number\n\\[ \\deg X := \\int_X \\omega^{\\dim_{\\Bbb C} X}, \\]\nwhere $\\omega$ is the K\\\"ahler form. If $X\\subset M$ is \na trianalytic subvariety of a hyperk\\\"ahler manifold, we can\nassociate a number $\\deg X$ to each of induced complex structures.\nIn \\cite{Verbitsky:Symplectic_I_}, we prove that $\\deg X$\nis in fact independent from the choice of induced complex structure.\nThis enables us to speak of {\\bf degree} of a closed trianalytic\nsubvariety of a compact complex manifold.\n\nWe return to the proof of\n\\ref{_X_in_Douady_indep_from_I_1_Proposition_}.\nLet $D_1$ be the union of all components of the \nDouady space for $(M,I_1)$. By \\cite{_Lieberman_},\n\\cite{_Fukjiki_Kahler_}, $D_1$ is compact;\nin particular, $D_1$ has a finite number of connected components.\nLet $D_1 = \\cup D^i_1$, $i\\in \\Upsilon$ \nbe the decomposition of $D$ unto a union of its\nconnected components. \n\nLet $X'\\subset M$ be a closed trianalytic subvariety of $M$\nsuch that $[X'] \\in D_{I_2}(X)$. There exists a real analytic\npath $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} D_{I_2}(X)$ joining $[X]$ and $[X']$.\nFor all $t\\in [0,1]$, the point $\\gamma(t)$ lies in $D_1^i$ for some\n$i\\in \\Upsilon$. Let $S^i\\subset [0,1]$ be the set of all\n$t\\in [0,1]$ such that $\\gamma(t)\\in D_1^i$. We are going to\nshow that $S^i$ are compact for all $i$. This will clearly imply\nthat for all $i$ except one, $S^i$ is empty, thus proving that\n$[X]$ and $[X']$ lie in the same component of $D_1$.\n\nLet $D_{\\Bbb R}$ be the set of all real analytic deformations of $X$ in $M$\nwith natural topology. The forgetful map $\\psi:\\; D_1 {\\:\\longrightarrow\\:} D_{\\Bbb R}$\nis continous and injective. Since $D_1$ is compact, $\\psi$\nis a closed embedding. The composition \n$\\gamma\\circ \\psi:\\; [0,1] {\\:\\longrightarrow\\:} D_{\\Bbb R}$ is obviously continous,\nand thus, $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} D_1$ is also continous. \nThis implies that all $S_i$, $i\\in \\Upsilon$ are \nclosed, thus proving that all $S_i$ except one are empty.\n\\ref{_X_in_Douady_indep_from_I_1_Proposition_} is proven.\n\\blacksquare\n\n\\hfill\n\nLet $D_{I_1}(X)_{\\Bbb R}$, $D_{I_2}(X)_{\\Bbb R}$ be the real analytic \nvarieties underlying $D_{I_1}(X)$, $D_{I_2}(X)$. \n\\ref{_X_in_Douady_indep_from_I_1_Proposition_} gives a \ntautological bijection \\[ \\psi:\\; D_{I_1}(X)_{\\Bbb R}{\\:\\longrightarrow\\:} D_{I_2}(X)_{\\Bbb R}.\\]\n\n\\hfill\n\n\\proposition\\label{_Douady_real_ana_inde_from_indu_Proposition_}\nThe map $\\psi:\\; D_{I_1}(X)_{\\Bbb R}{\\:\\longrightarrow\\:} D_{I_2}(X)_{\\Bbb R}$\nis an isomorphism of real analytic varieties.\n\n\\hfill\n\n{\\bf Proof:} Let $I$ be an induced complex structure on $M$.\nFor a real analytic function $f$ on $M$, consider the function\n$\\hat f:\\; D_I(X) {\\:\\longrightarrow\\:} {\\Bbb R}$,\n\\[ [X'] \\stackrel{\\hat f}{\\:\\longrightarrow\\:} \\int_{X'} f \\cdot \\operatorname{Vol} X', \\]\nwith $\\operatorname{Vol} X'$ the volume form on $X'$. For\nall $f$, the function $\\hat f$ is real analytic. From the definition of\nDouady spaces it might be seen that converging \npower series of different $\\hat f$ generate the sheaf\n$\\c A_I$ of real analytic functions on $D_I(X)$. For an induced\ncomplex structure $I'$, the same set $\\{ \\hat f\\}$\ngenerates the sheaf $\\c A_{I'}$ \nof real analytic functions on $D_{I'}(X)$\n(\\ref{_real_ana_indu_on_hype_equiva_Proposition_}). Thus,\nthe sheaves $\\c A_I$ and $\\c A_{I'}$ coinside.\n\\blacksquare\n\n\\hfill\n\n\\ref{_Douady_real_ana_inde_from_indu_Proposition_}\nshows that we may speak of the real analytic variety underlying\n$D_M(X)$ without specifying an induced complex structure $I$.\n\n\\hfill\n\n\\definition \\label{_hypercomplex_Definition_}\nLet $Y$ be a real analytic variety equipped with\nthree complex structures $I$, $J$ and $K$. Assume that\nfor every point $y\\in Y$, the action of $I$, $J$, $K$ on the\nZariski tangent space $T_yY$ satisfies $I\\circ J = - J\\circ I =K$.\nThen $Y$ is called {\\bf a hypercomplex variety}.\n\n\\hfill\n\n\\remark \\label{_Douady_hyperc_Remark_}\n\\ref{_Douady_real_ana_inde_from_indu_Proposition_}\nimmediately implies that the induced complex structures on $M$ \nequip the Douady space $D_M(X)$ with a hypercomplex structure.\n\n\n\n\\subsection{Appendix: isometric embeddings are hyperk\\\"ahler.}\n\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$\na closed trianalytic subvariety.\nThere is an alternative way to describe the Douady space\n$D(X)$. The \\ref{_isome_embe_Proposition_} \nbelow describes $D(X)$ in terms of of the space of isometric embeddings\n$\\nu:\\; X {\\:\\longrightarrow\\:} M$. \n\n\\hfill\n\nLet $M$ be a compact K\\\"ahler manifold, and $X\\subset M$\nbe a closed analytic subvariety. One can make sense of the\n{\\bf fundamental class} of $X$, $[X] \\in H^{\\dim_{\\Bbb R}(X)}(M)$,\nlying in homology of $M$.\nConsider $X$ with an induced structure of a metric space,\nand let $X \\stackrel \\nu {\\:\\longrightarrow\\:} M'$ be an isometric embedding,\nwhere $M'$ is a compact Riemannian manifold. The same argument which\nallows us to define the fundamental class of $X$, allows\nus to define the fundamental class of $\\nu(X)$. \n\n\\hfill\n\n\\proposition \\label{_isome_embe_Proposition_}\nLet $M$ be a closed analytic subvariety, $X\\subset M$ a closed \ntrianalytic subvariety, and $\\nu:\\; X {\\:\\longrightarrow\\:} M$ an isometric embedding.\nAssume that the fundamental class of $\\nu(X)$ is equal to the fundamental\nclass of $X$. Then $\\nu(X)$ is a trianalytic subvariety of $M$.\n\n\\hfill\n\n{\\bf Proof:} We use notation \nintroduced in Subsection \\ref{_SU(2)-inv=>triana_Subsection_}.\nFor each induced complex structure $I$,\nwe have \n\\begin{equation}\\label{_deg_same_appl_nu_Equation_}\n \\deg_I (X) - \\deg_I (\\nu(X)),\n\\end{equation} \nbecause $\\nu(X)$ has the same fundamental class as $X$.\nSince $X$ is trianalytic,\napplying Wirtinger's inequality (\\ref{_Wirti_for_Kahle_Theorem_}), \nwe obtain that\n\\begin{equation} \n 2^n \\int_{X_{ns}} \\operatorname{Vol} X_{ns} = \\int_{X_{ns}} \\omega^n,\n\\end{equation}\nwhere $\\omega$ is the K\\\"ahler class of $(M,I)$, $n=\\frac{1}{2}\\dim_{\\Bbb R}(X)$\nand $X_{ns}$ is the nonsingular part of $(X,I)$.\nBy definition, we have\n\\begin{equation}\n \\int_{X_{ns}} \\omega^n = \\deg_I (X), \\ \\ \n \\int_{\\nu(X_{ns})} \\omega^n = \\deg_I (\\nu(X)).\n\\end{equation}\nSince $\\nu$\nis an isometry, and $\\operatorname{Vol}(X)$ is an invariant of a metric, \nwe have \n\\begin{equation} \\label{_Vol_same_appl_nu_Equation_}\n \\int_{X_{ns}} \\operatorname{Vol} X_{ns} = \n \\int_{\\nu(X_{ns})} \\operatorname{Vol} \\left(\\nu(X_{ns}) \\right).\n\\end{equation}\nTogether the equations \n\\eqref{_deg_same_appl_nu_Equation_}--\\eqref{_Vol_same_appl_nu_Equation_}\ngive \n\\[ \n 2^n\\int_{X_{ns}} \\operatorname{Vol} X_{ns} = \\int_{\\nu(X_{ns})} \\omega^n. \n\\]\nApplying Wirtinger's inequality (\\ref{_Wirti_for_Kahle_Theorem_}) once\nagain, we obtain that $\\nu(X)$ is complex analytic with respect to $I$.\nThis proves \\ref{_isome_embe_Proposition_}. \\blacksquare\n\n\\hfill\n\nWe just proved \\ref{_iso_intro:Theorem_} (iii). \nAs another application of \\ref{_isome_embe_Proposition_},\nwe give a direct proof that $D(X)$ is compact.\n\n\\hfill \n\n\\corollary\nLet $M$ be a compact hyperk\\\"ahler manifold, \n$X\\stackrel {\\bar i}\\hookrightarrow M$ be a closed trianalytic subvariety\nand $D(X)$ its Douady space. Then $D(X)$ is compact.\n\n\\hfill\n\n{\\bf Proof:} Let $X_{ns}$ be the non-singular part of $X$,\nand $i:\\; X_{ns} \\hookrightarrow M$ the natural embedding.\nThen $i$ is an isometry. Consider a deformation $\\nu$ of $i$ in class of\nisometries. The arument proving \\ref{_isome_embe_Proposition_}\nshows that the closure of $\\nu(X_{ns})$ is trianalytic in $M$.\nConsider the space $\\underline{Is(X_{ns})}$ of isometries \n$X_{ns}\\stackrel \\nu\\hookrightarrow M$. Let $Is(X_{ns})$ be\na connected component of this space, containing $i$. There\nis a natural continuous surjection \n$p:\\; Is(X_{ns}) {\\:\\longrightarrow\\:} D(X)$, which maps $\\nu \\in Is(X_{ns})$\nto a closure of $\\nu(X_{ns})$ in $M$. To prove that $D(X)$ is\ncompact it suffices to show that $Is(X_{ns})$ is compact.\nThis is an implication of the following general statement, which is clear.\n\n\\hfill\n\n\\claim \nLet $X$, $M$ be Riemannian manifolds, $M$ compact. Let\n$Is(X, M)$ be the space of isometries (maps which preserve the \ngeodesic distance) from $X$ to $M$. Then $Is(X, M)$ is compact.\n\n\\blacksquare\n\n\n\n\\section[Connections in the families of trianalytic subvarieties]\n{Connections in the families of trianalytic \\\\subvarieties}\n\\label{_Conne_in_fami_Section_}\n\n\n\\subsection{Introduction}\n\\label{_intro_conne_Subsection_}\n\nLet $M$ be a compact hyperk\\\"ahler manifold, and\n\\begin{equation} \\label{_fami_Equation_}\n \\left( \\pi:\\; {\\c X} {\\:\\longrightarrow\\:} S, \\phi:\\; {\\c X} {\\:\\longrightarrow\\:} M \n \\right )\n\\end{equation}\na real analytic family of subvarieties, not necessarily closed. \nAssume that for all $s\\in S$, the fiber $X_s = \\phi(\\pi^{-1} (s))$\nis trianalytic in $M$.\nBy \\ref{_conne_in_fam_of_comple_geode_Proposition_} \nand \\ref{_hype_embe_comple_geode:Corollary_}, the family \n${\\c X}$ is then equipped with a connection\n$\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T {\\c X}$. It is natural \nto ask whether this connection is {\\bf flat} \n(see \\ref{_flat_conne_in_fami_Definition_}). \nThe answer is affirmative, \nunder certain additional assumptions.\n\n\\hfill\n\n\\definition \\label{_admits_co_Definition_}\nWe say that the family \\eqref{_fami_Equation_} {\\bf admits \na compactification} if the following conditions hold.\n\n\\begin{description}\n\\item[(i)]\nFor each $s \\in S$, the closure $\\bar X_s$ of the fiber \n$X_s$ in $M$ is a trianalytic subvariety of $M$.\n\n\\item [(ii)] For all $s\\in S$, $\\bar X_s$ lie in the same \ncomponent of the Douady space.\n\n\\end{description}\n\n\\hfill\n\nWe show that in assumptions of \n\\ref{_admits_co_Definition_}, the connection\n\\[ \\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T {\\c X}\\] is indeed flat\n(\\ref{_conne_in_triana_flat_Theorem_}). \nHowever, we don't know whether it is flat\nfor general families of trianalytic submanifolds.\n\n\\hfill\n\n\n\\subsection{Flat connections and curvature.}\n\n\n\\hfill\n\n\\definition\n(curvature of a connection) \nLet $\\pi: {\\c X} {\\:\\longrightarrow\\:} S$ be a family of manifolds,\n$T_\\pi X \\subset T{\\c X}$ be the fiberwise tangent bundle and \n$N_\\pi X = T{\\c X} \/ T_\\pi X$ be a fiberwise normal bundle. \nLet $\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T{\\c X}$ be a connection in ${\\c X}$.\nThen the {\\bf curvature} of $\\nabla$ is the following\ntensor \n$\\Theta\\in \\operatorname{Hom}_{\\Bbb R}\\left ( \\Lambda^2_{\\Bbb R} \\left( N_\\pi X\\right), \nT_\\pi{\\c X}\\right)$.\nFor two sections $a,b \\in N_\\pi X$, consider the corresponding\nvector fields $\\nabla a$, $\\nabla b\\in T{\\c X}$. Consider the projection\n$T {\\c X} \\stackrel {pr}{{\\:\\longrightarrow\\:}} T{\\c X} \/ \\nabla N_\\pi X = T_\\pi X$.\nLet $\\Theta(a,b):= pr([\\nabla a, \\nabla b])$.\nClearly, $\\Theta$ is a tensor. \n\n\\hfill\n\n\\proposition\nLet $\\left( \\pi:\\; {\\c X} {\\:\\longrightarrow\\:} S \\right)$ \nbe a family of manifolds equipped with a connection $\\nabla$.\nThen the following conditions are equivalent.\n\n\\begin{description}\n\n\\item[(i)] The connection $\\nabla$ is flat.\n\n\\item[(ii)] Let $s_1, s_2\\in S$ be a pair of points,\nand $\\gamma, \\gamma':\\; [0,1] {\\:\\longrightarrow\\:} S$ real analytic paths.\nLet $U_0 \\subset X_{s_0}$, $U_1 \\subset X_{s_1}$ be the open\nsubsets such that the connection $\\nabla$ might be integrated to a map\n$\\Psi:\\; U_{s_0} {\\:\\longrightarrow\\:} U_{s_1}$ along $\\gamma$, and\n$U_0' \\subset X_{s_0}$, $U_1' \\subset X_{s_1}$ be \nsuch that $\\nabla$ might be integrated to a map\n$\\Psi':\\; U_{s_0}' {\\:\\longrightarrow\\:} U_{s_1}'$ along $\\gamma'$. Then\n$\\Psi$ coinsides with $\\Psi'$ in the intersection\n$U_{s_0} \\cap U'_{s_0}$.\n\n\\end{description}\n\n{\\bf Proof:} Well known. \\blacksquare\n\n\\hfill\n\n\n\\definition \\label{_flat_conne_in_fami_Definition_} \n Let ${\\c X} {\\:\\longrightarrow\\:} S$ be a family of manifolds, equipped with a\nconnection $\\nabla$. Then $\\nabla$ is called {\\bf flat}\nif its curvature is zero.\n\n\n\n\\subsection{Curvature of holomorphic connections}\n\n\nLet $\\pi:\\; {\\c X} {\\:\\longrightarrow\\:} S$ be a complex analytic family\nof manifolds. Then the vector bundles $T{\\c X}$ and\n$N_\\pi X$ are equipped with a natural holomorphic structure.\nThe connection $\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T {\\c X}$ is called\n{\\bf holomorphic} if the map $\\nabla:\\; N_\\pi X {\\:\\longrightarrow\\:} T{\\c X}$\nis holomorphic. \n\n\\hfill\n\n\\claim \\label{_holo_conne_curva_C-line_Claim_}\nLet $\\pi:\\; {\\c X} {\\:\\longrightarrow\\:} S$ be a family of \nmanifolds equipped with with a holomorphic connection\n$\\nabla$. Consider the curvature \n\\[ \\Theta\\in \\operatorname{Hom}_{\\Bbb R}\\left \n ( \\Lambda^2_{\\Bbb R} \\left(N_\\pi X\\right), T_\\pi{\\c X}\\right).\n\\]\nThen $\\Theta$ is ${\\Bbb C}$-linear; in other words, $\\Theta$ belongs\nto the subspace\n\n\\[ \\operatorname{Hom}_{\\Bbb C}\\left ( \\Lambda^2_{\\Bbb C} \\left(N_\\pi X\\right), T_\\pi{\\c X}\\right)\n \\subset \\operatorname{Hom}_{\\Bbb R}\\left \n ( \\Lambda^2_{\\Bbb R} \\left(N_\\pi X\\right), T_\\pi{\\c X}\\right).\n\\]\n\n{\\bf Proof:} Clear. \\blacksquare\n\n\n\\subsection{Flat connections in a family of trianalytic varieties. }\n\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$ \na closed trianalytic subvariety, $D(X)$ its Douady space\nand \n\\[ \\left( \\underline \\pi: \\; \\underline{{\\c X}} {\\:\\longrightarrow\\:} D(X), \n \\underline \\phi:\\; \\underline{{\\c X}} {\\:\\longrightarrow\\:} M \\right )\n\\]\na universal family of subvarieties corresponding to $D(X)$.\nLet ${\\c X}$ be the union of all smooth points in all fibers\nof $\\underline \\pi$, and $\\pi$, $\\phi$ be the restrictions of\n$\\underline \\pi$ and $\\underline \\phi$ to ${\\c X}$.\nConsider the corresponding family of manifolds\n\\[ \\left( \\pi: {{\\c X}} {\\:\\longrightarrow\\:} D(X), \n \\phi {{\\c X}} {\\:\\longrightarrow\\:} M \\right ).\n\\]\nWe are in the same situation as described in \nSubsection \\ref{_intro_conne_Subsection_}. \nThus, the family ${\\c X}$ is equipped with a natural \nconnection $\\nabla$.\n\n\\hfill\n\n\\theorem \\label{_conne_in_triana_flat_Theorem_}\nIn these assumptions, $\\nabla$ is flat.\n\n\\hfill\n\n{\\bf Proof:} The proof of \\ref{_conne_in_triana_flat_Theorem_}\ntakes the rest of this section. \n\n\\hfill\n\nClearly, $N_\\pi X = \\pi^* TS$. By \\ref{_Douady_hyperc_Remark_}, \n$TS$ has a natural quaternionic action, and thus, an action of\n$SU(2) \\subset {\\Bbb H}^*$. The bundle $T_\\pi X$ is also equipped \nwith a natural action of quaternions. This endows the bundle\n$\\operatorname{Hom}_{\\Bbb R}\\left ( \\Lambda^2_{\\Bbb R} \\right(N_\\pi X\\left), T_\\pi{\\c X}\\right)$\nwith an action of $SU(2)$.\n\n\\hfill\n\n\\lemma \\label{_curva_SU(2)_inva_Lemma_}\nLet $\\Theta$ be a curvature of $\\nabla$, \n$\\Theta \\in\\operatorname{Hom}_{\\Bbb R}\n\\left ( \\Lambda^2_{\\Bbb R} \\left( N_\\pi X\\right), T_\\pi{\\c X}\\right)$.\nThen $\\Theta$ is $SU(2)$-invariant.\n\n\\hfill\n\n{\\bf Proof:} Let $U(1) \\stackrel {\\tilde i}\\hookrightarrow SU(2)$ be \nan embedding corresponding to an algebra embedding \n${\\Bbb C} \\stackrel i\\hookrightarrow {\\Bbb H}$. Let $I$ be an\ninduced complex structure associated with $i$. The\nconnection $\\nabla$ is holomorphic with respect to $I$.\nTherefore $\\Theta$ is ${\\Bbb C}$-linear\nwith respect to the action of ${\\Bbb C}$ given by \n${\\Bbb C} \\stackrel i \\hookrightarrow {\\Bbb H}$\n(\\ref{_holo_conne_curva_C-line_Claim_}). \nThus, $\\Theta$ is $U(1)$-invariant with respect\nto the action of $U(1)$ given by \n$\\tilde i:\\; U(1)\\hookrightarrow SU(2)$.\nThe group $SU(2)$ is generated by the images of $\\tilde i$ for all\nalgebra embeddings $i:\\; {\\Bbb C} \\hookrightarrow {\\Bbb H}$. Thus,\n$\\Theta$ is $SU(2)$-invariant. \\blacksquare\n\n\\hfill\n\nConsider $N_\\pi X$, $T_\\pi X$ as representation of $SU(2)$. These representations\nare of weight 1. The bundle \n\\[ \n \\operatorname{Hom}_{\\Bbb R}\\left ( \\Lambda^2_{\\Bbb R} \\left(N_\\pi X\\right), T_\\pi{\\c X}\\right)\n\\] \nis \nagain a representation of $SU(2)$. Since \n$N_\\pi X$, $T_\\pi X$ are of weight one, \n$\\operatorname{Hom}_{\\Bbb R}\\left ( \\Lambda^2_{\\Bbb R} \\left(N_\\pi X\\right), T_\\pi{\\c X}\\right)$ is \na direct sum of representations of weight 3 and 1. Thus,\nthis bundle cannot have non-zero $SU(2)$-invariant sections.\nTherefore, ($\\Theta$ is $SU(2)$-invariant) \nimplies ($\\Theta =0$).\n\\ref{_conne_in_triana_flat_Theorem_} is proven. \\blacksquare\n\n\n\n\n\n\\section[Isometries of trianalytic subvarieties \nof a compact hy\\-per\\-k\\\"ah\\-ler manifold]{Isometries \nof trianalytic subvarieties \\\\ \nof a compact hyperk\\\"ahler manifold}\n\\label{_isome=>holo_Section_}\n\n\n\\subsection{Premises}\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$ a \nclosed trianalytic subvariety.\n{}From \\ref{_conne_in_fami_of_comple_geo_Proposition_},\n\\ref{_conne_in_triana_flat_Theorem_} and \n\\ref{_norma_sec_para_for_hype_from_it_Corollary_},\nwe immediately obtain the following theorem.\n\n\\hfill\n\n\\theorem \\label{_triana_subse_isome_Theorem_}\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$\na closed trianalytic subvariety, $D(X)$ the corresponding \nDouady space (\\ref{_Douady_hyperc_Remark_}). \nLet $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} D(X)$ \nbe a real analytic path. Denote by $X_t$ \nthe subvarieties corresponding to points $\\gamma(t)\\in D(X)$.\nConsider $X_t$ as metric spaces with the metric induced\nfrom $M$.\nThen, for each $t_1$, $t_2\\in [0,1]$,\nthere exist a natural isometry\n$\\Psi^{t_1}_{t_2}:\\; X_{t_1} {\\:\\longrightarrow\\:} X_{t_2}$, mapping\nnon-singular points to non-singular points, and\nacting compatible with the hyperk\\\"ahler structure \non $X_{t_1}^{ns}$.\\footnote{The non-singular part $X_t^{ns}$ \nis naturally equipped with a structure of a hyperk\\\"ahler manifold.}\nThis isometry depends only on the homotopy class of $\\gamma$.\n\n\\blacksquare\n\n\\hfill\n\nThis almost finishes the proof \\ref{_iso_intro:Theorem_} (i).\nIt remains to prove the following\ntheorem.\n\n\\hfill\n\n\\theorem \\label{_triana_subse_comple_ana_Theorem_}\nIn assumptions of \\ref{_triana_subse_isome_Theorem_},\nlet $I$ be an induced complex structure on $M$. Then\nthe isometry $\\Psi^{t_1}_{t_2}:\\; X_{t_1} {\\:\\longrightarrow\\:} X_{t_2}$\nis compatible with the complex analytic structure induced\nby $I$.\n\n\n\\hfill\n\nThis section is taken fully with the proof of \n\\ref{_triana_subse_comple_ana_Theorem_}\n\n\n\\subsection{Homeomorphisms of complex varieties and normalization.}\n\n\n\\hfill\n\n\\claim \\label{_conti_fu_extend_to_holo_Claim_}\n(\\cite{_Grauert_}) Let $X$ be a normal complex analytic variety, $U$\na dense open subset in $X$ and $f:\\; U {\\:\\longrightarrow\\:} {\\Bbb C}$ a bounded \nholomorphic function. Then $f$ can be extended to a holomorphic function\non $X$.\n\n\\blacksquare\n\n\n\\hfill\n\nThis statement has an immediate corollary.\n\n\\hfill\n\n\\corollary \\label{_conti_map_norma_holo_Corollary_}\nLet $\\phi:\\; X {\\:\\longrightarrow\\:} Y$ be a continous map of complex analytic\nvarieties. Assume that $\\phi$ is holomorphic in an open\ndense subset $U\\subset X$. Then, if $X$ is normal, $\\phi$\nis holomorphic.\n\n\\hfill\n\n{\\bf Proof:} Let $x\\in X$ be an aritrary point, $V$ its neighbourhood,\nsufficiently small. Taking \ncoordinates in a neighbourhood $W$ of $\\phi(V)$ and applying\n\\ref{_conti_fu_extend_to_holo_Claim_}, we find that\n$\\phi\\restrict {U\\cap V}$ extends continously to a \nholomorphic map $\\tilde \\phi$ from $U\\cap V$ to $W$. \nSince $U\\cap V$ is dense in $V$, $\\tilde \\phi$\ncoinsides with $\\phi\\restrict V$. Thus, $\\phi$ is\nholomorphic. \\blacksquare\n\n\\hfill\n\nThus, in the situation of \\ref{_triana_subse_comple_ana_Theorem_},\nwere $X_t$ normal, $\\Psi^{t}_{t'}$ would have been holomorphic\nand \\ref{_triana_subse_comple_ana_Theorem_} would have been proven.\nUnfortunately, we have no means to show that $X_t$ is normal.\nHowever, from \\ref{_conti_map_norma_holo_Corollary_}\nwe obtain some information about maps of arbitrary \nvarieties too.\n\n\\hfill\n\n\\corollary \\label{_conti_map_mero_Corollary_} \nLet $\\phi:\\; X {\\:\\longrightarrow\\:} Y$ be a continous map of complex analytic\nvarieties, which is holomorphic on an open dense subset \n$U\\subset X$. Then $\\phi$ is meromorphic.\n\n\\hfill\n\n{\\bf Proof:} Take a normalization $\\tilde X \\stackrel n {\\:\\longrightarrow\\:} X$.\nApplying \\ref{_conti_map_norma_holo_Corollary_} to the composition\n$n\\circ \\phi:\\; \\tilde X {\\:\\longrightarrow\\:} Y$, we obtain that\n$n\\circ \\phi$ is holomorphic. Thus, $\\phi$ is meromorphic\nas a composition of holomorphic\n$n\\circ \\phi$ and a meromorphic map $n^{-1}$. \\blacksquare\n\n\\hfill\n\nApplying \\ref{_conti_map_mero_Corollary_} to the situation\nof \\ref{_triana_subse_comple_ana_Theorem_}, we obtain the\nfollowing corollary.\n\n\\hfill\n\n\\corollary \\label{_Psi_bimero_Corollary_}\nIn the situation of \\ref{_triana_subse_isome_Theorem_},\nthe map \\[ \\Psi^{t_1}_{t_2}:\\; X_{t_1} {\\:\\longrightarrow\\:} X_{t_2} \\]\nis bimeromorphic for each induced complex structures.\n\n\\blacksquare\n\n\\hfill\n\nIt remains to make a leap from ``bimeromorphic'' to ``holomorphic''.\nThis is done in two steps. We prove a number of\nalgebro-geometric statements about the behaviour of $\\Psi^{t_1}_{t_2}$,\nconcluding with \\ref{_pi_i_properties_Proposition_}.\nIn \\ref{_alge_geo_suffi_pi_i_iso_Proposition_}, \nwe show that these statements are strong enough to show that\nthe $\\Psi^{t_1}_{t_2}$ is holomorphic. The premise\nof \\ref{_alge_geo_suffi_pi_i_iso_Proposition_} \nis purely algebro-geometric and its proof is independent\nfrom the rest of this section.\n\nWe finish this Subsection with the following statement, which \nwe use in \\ref{_pi_i_properties_Proposition_}.\n\n\\hfill\n\n\\proposition \\label{_Psi_iso_on_norma_Proposition_}\nThe map $\\Psi^{t_1}_{t_2}:\\; X_{t_1} {\\:\\longrightarrow\\:} X_{t_2}$ induces\nan isomorphism of normalizations\n\\[ \\tilde \\Psi^{t_1}_{t_2}:\\;\\tilde X_{t_1} {\\:\\longrightarrow\\:} \\tilde X_{t_2}. \\]\n\n{\\bf Proof:} From the definition of normalization \\cite{_Grauert_}, the\nfollowing lemma is evident.\n\n\\hfill\n\n\\lemma\nLet $\\Psi:\\; X {\\:\\longrightarrow\\:} Y$ be a homeomorphism of complex\nvarieties. Assume that in an open dense subset $U\\subset X$,\n$\\Psi$ is holomorphic. Then $\\Psi$\ninduces an isomorphism $\\tilde \\Psi:\\; \\tilde X {\\:\\longrightarrow\\:} \\tilde Y$ \nof normalizations, if the following statement holds.\n\n\\begin{description}\n\\item[(*)] There exist a Stein covering $\\{ U_i\\}$ of $X$ such that\n$\\{ \\Psi(U_i)\\}$ is a Stein covering for $Y$.\n\\end{description}\n\\blacksquare\n\nTo prove \\ref{_Psi_iso_on_norma_Proposition_}, it remains to show that\nthe property {\\bf (*)} holds for the map \n$\\Psi^{t_1}_{t_2}:\\; X_{t_1} {\\:\\longrightarrow\\:} X_{t_2}$. \nFor each point $x \\in X_{t_1}$, it suffices to \nconstruct a Stein neighbourhood $U$ of $X_{t_1}$, such that \n$\\Psi^{t_1}_{t_2}(U)$ is Stein. For a K\\\"ahler variety,\nconsider an open ball $B$ of radius $r$, taken with respect to the \nmetric defined by geodesics. Then, for $r$ sufficiently small,\n$B$ is Stein (\\cite{_Greene_}). Since $\\Psi^{t_1}_{t_2}$ is an \nisometry (\\ref{_triana_subse_isome_Theorem_}), \nan image of an open ball of radius $r$ is again \nan open ball of radius $r$. This gives a \nsystem of Stein neighbourhoods satisfying\n{\\bf (*)}. \\ref{_Psi_iso_on_norma_Proposition_}\nis proven. \\blacksquare\n\n\n\n\\subsection{Homeomorphisms of completely geodesic subvarieties \ninduce isomorphisms of Zariski tangent spaces.}\n\n\n\\hfill\n\n\\proposition\\label{_homeo_indu_iso_Zariski_Proposition_}\nLet $M_1$, $M_2$ be a K\\\"ahler manifolds,\n$X_1^{ns}\\subset M_1$, $X_2^{ns}\\subset M_2$ \nbe completely geodesic complex submanifolds, not necessarily\nclosed, and $X_1$, $X_2$ be the closures of $X_1^{ns}$, $X_2^{ns}$ in\n$M_1$, $M_2$. Assume that $X_1$, $X_2$ are complex analytic \nsubvarieties of $M_1$, $M_2$. Let $\\phi:\\; X_1{\\:\\longrightarrow\\:} X_2$ be a morphism\nof complex varieties, such that $\\phi\\restrict{X_1^{ns}}$ is an isometry.\nThen $\\phi$ induces an isomorphism of Zariski tangent spaces. \n\n\\hfill\n\n{\\bf Proof:}\nTo prove \\ref{_homeo_indu_iso_Zariski_Proposition_}, we interpret\nthe Zariski tangent space \n$T_x X_1$ in terms of the metric structure on $X_i^{ns}$, where $x$ \nis a point of $X_1$. Let $\\gamma:\\; [0,1] {\\:\\longrightarrow\\:} X_1$ be a path satisfying \n$\\gamma\\left([0,1] \\backslash\\{0.5\\}\\right) \\subset X_1^{ns}$,\n$\\gamma(0.5) =x$. \nThere is a natural topology on the total space $Tot(T X_1^{ns})$ ,\nwhich comes from the embedding \n$Tot(TX_1) \\stackrel i\\hookrightarrow Tot(TM_1)$.\nSince $X_1$ is completely geodesic, $i$ is an isometry.\nThis topology is compatible with the map \n$d\\phi:\\; TX_1 {\\:\\longrightarrow\\:} TX_2$, because $d\\phi$ is also an isometry.\nAssume that $\\gamma$ is differentiable \noutside of $\\{0.5\\}$ and \n\\[ \\lim\\limits_{t\\to +0.5} \\frac{d\\gamma}{dt} = \n \\lim\\limits_{t\\to -0.5} \\frac{d\\gamma}{dt}.\n\\]\n(the limits are taken in the metric completion of $TX_1$,\nwhich might be considered as a subset of $Tot(TM_1)$).\nThe Zariski tangent space $T_x X_1$ can be identified with\nequivalence classes of such paths as follows. Two \npaths $\\gamma, \\gamma'$ are equivalent if\n\\[ \n \\lim\\limits_{t\\to 0.5} \n \\frac{\\rho(\\gamma_1(t), \\gamma_2(t))}{(t-0.5)^2} =0,\n\\]\nwhere $\\rho$ is the distance function in $M_1$. Since the\ndistance in $M_1$ coinsides with the distance in $X_1$, and\n$\\phi$ is isometry, this equivalence relation is compatible\nwith $\\phi$. \\blacksquare\n\n\n\\subsection{Algebro-geometric properties of the map $\\Psi^{s_1}_{s_2}$.}\n\n\n\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$ a \nclosed trianalytic subvariety. Fix a choice of induced complex structure.\nLet $D(x)$ be the Douady space of $X$, and $s_1, s_2$ points\non $D(X)$, and $\\gamma$ a real analytic path in $D(x)$\nconnecting $s_1$ and $s_2$. Let $X_1 = X_{s_1}, X_2 = X_{s_2}$\nbe the subvarieties of $M$ corresponding to $s_1$, $s_2$,\nand $\\Psi:\\; X_1 {\\:\\longrightarrow\\:} X_2$ be the bimeromorphic map\nof \\ref{_Psi_bimero_Corollary_}. Let $X_1^{ns}$, $X_2^{ns}$ be the\nnon-singular part of $X_1$, and $\\Psi^{ns}$ be the restriction of $\\Psi$\nto $X_1^{ns}$. Let $\\Gamma^{ns}\\subset M\\times M$ be a graph of\n$\\Phi^{ns}$, and $\\Gamma$ its closure. Clearly, $\\Gamma$\nis a complex analytic subvariety of $M\\times M$.\nLet $\\pi_i:\\; \\Gamma {\\:\\longrightarrow\\:} X_i$ be the projections of\n$\\Gamma$ to $X_i$, which are obviously morphisms\nof complex varieties. \n\n\\hfill\n\n\\proposition\\label{_pi_i_properties_Proposition_} \nThe maps $\\pi_i$, $i= 1,2$, have the following properties.\n\n\\begin{description}\n\\item[(i)] $\\pi_i$ is finite.\n\\item[(ii)] $\\pi_i$ is dominant and induces isomorphism\nof normalizations\n\\item[(iii)] For every point $x\\in \\Gamma$, the differential\n$d \\pi_i :\\; T_x \\Gamma {\\:\\longrightarrow\\:} T_{\\pi_i(x)} X_i$ is an\nisomorphism.\n\\end{description}\n\n{\\bf Proof:} The statement \n(ii) follows from \\ref{_Psi_iso_on_norma_Proposition_}\nand (iii) from \\ref{_homeo_indu_iso_Zariski_Proposition_}. \nTo prove \\ref{_pi_i_properties_Proposition_} (i), consider the\nnormalization map $n:\\; \\tilde \\Gamma {\\:\\longrightarrow\\:} \\Gamma$. On the\nlevel of rings of functions, we have an embedding \n\\[ \n {\\cal O}(X_i) \\hookrightarrow {\\cal O}(\\Gamma) \\hookrightarrow\n {\\cal O}(\\tilde \\Gamma).\n\\]\nSince $\\tilde \\Gamma$ is a normalization of $X_i$ by (ii),\nthe ring ${\\cal O}(\\tilde \\Gamma)$ is finitely generated\nas a ${\\cal O}(X_i)$-module. Since ${\\cal O}(X_i)$ is Noetherian\n(\\cite{_Grauert_}), this implies that ${\\cal O}(\\Gamma)$\nis also finitely generated as a ${\\cal O}(X_i)$-module.\nThis proves \\ref{_pi_i_properties_Proposition_} (i).\n\\blacksquare\n\n\\hfill\n\nTo prove that the map $\\Psi:\\; X_1 {\\:\\longrightarrow\\:} X_2$, a.k.a.\n$\\Psi^{s_1}_{s_2}:\\; X_{s_1} {\\:\\longrightarrow\\:} X_{s_2}$\nis holomorphic, it suffices to show that $\\pi_i:\\; \\Gamma {\\:\\longrightarrow\\:} X_i$\nis an isomorphism. In the following Subsection, we show that\nconditions (i) -- (iii) of \\ref{_pi_i_properties_Proposition_} \nare {\\it a priori} sufficient to establish that $\\pi$ is \nan isomorphism. This will finish the proof of \n\\ref{_triana_subse_comple_ana_Theorem_}.\n\n\n\n\n\n\\subsection{Finite dominant unramified morphisms of complex varieties.}\n\n\n\\hfill\n\n\\proposition \\label{_alge_geo_suffi_pi_i_iso_Proposition_}\nLet $\\phi:\\; X {\\:\\longrightarrow\\:} Y$ be a map of complex varieties satisfying\nconditions (i)--(iii) of \\ref{_pi_i_properties_Proposition_}.\nThen $\\phi$ is an isomorphism.\n\n\\hfill\n\n{\\bf Proof:} The map $\\phi$ is one-to-one in general point (by (ii)). \nThus, to prove that $\\phi$ is an isomorphism it suffices to show\nthat $\\phi$ is etale. On the other hand, $\\phi$ is unramified by\n(iii). By definition of etale morphisms,\nto prove that $\\phi$ is etale\nit remains to show that $\\phi$ is flat. To conclude the\nproof of \\ref{_alge_geo_suffi_pi_i_iso_Proposition_}, we\nuse the following lemma.\n\n\\hfill\n\n\\lemma \\label{_unrami_domi_flat_Lemma_}\nLet $\\phi:\\; X {\\:\\longrightarrow\\:} Y$ be a dominant morphism of complex varieties.\nAssume that for every point $x\\in X$, the map $\\phi$ induces an\nisomorphism $d\\phi:\\; T_x X {\\:\\longrightarrow\\:} T_{\\phi(x)} Y$ of Zariski\ntangent spaces. Then $\\phi$ is flat.\n\n\\hfill\n\n{\\bf Proof:} Let $y=\\phi (x)$. Conside the associated morphism\nof local rings ${\\cal O}_y Y \\stackrel {\\phi_x} \\hookrightarrow \n{\\cal O}_x X$. To prove\nthat $\\phi$ is flat, it suffices to show that $\\phi$ is an isomorphism.\nLet ${\\mathfrak m}_x$, ${\\mathfrak m}_y$ be the maximal ideals\nin ${\\cal O}_x X$, ${\\cal O}_y Y$. Then\n${\\mathfrak m}_x\/{\\mathfrak m}_x^2$ is generated by $\\phi_x({\\mathfrak m}_y)$.\nThus, by Nakayama, $\\phi_x({\\mathfrak m}_y)$ generate ${\\mathfrak m}_x$,\nand we obtain ${\\mathfrak m}_x= \\phi_x({\\mathfrak m}_y)\\otimes {\\cal O}_x X$.\nConsider ${\\cal O}_x X$ as ${\\cal O}_y Y$-module. Then \n${\\cal O}_x X\/{\\mathfrak m}_y {\\cal O}_x X$ is one-dimensional.\nApplying Nakayama once more, we obtain that ${\\cal O}_x X$\nis an ${\\cal O}_y Y$-module generated by $1\\in{\\cal O}_x X$.\nThus, the map $\\phi_x$ is surjective. Since $\\phi$ is \ndominant, $\\phi_x$ is also injective. Thus, ${\\cal O}_x X$\nis a free ${\\cal O}_{\\phi(x)} Y$-module for every $x\\in X$.\nBy the local criterion of flatness, this implies that\n$\\phi$ is flat. \\ref{_unrami_domi_flat_Lemma_} is proven.\nThis finishes the proof of \\ref{_alge_geo_suffi_pi_i_iso_Proposition_}\nand \\ref{_triana_subse_comple_ana_Theorem_} \\blacksquare\n\n\n\n\\section{Singular hyperk\\\"ahler varieties.}\n\\label{_singu_hype_Section_}\n\n\nIt is an intriguing question, what is the \n``correct''\\footnote{``Correct'' in Platonic sense: some mathematicians\npresume that the unique ``correct'' definition of each and every\nsignificant mathematical object exists in itself and independently\nof human perception.}\n definition of a singular hyperk\\\"ahler variety. We don't pretend to\nanswer this question. Instead, we give an {\\it ad hoc} set of axioms\nwhich describe some known examples (deformation spaces of stable bundles\nand trianalytic subvarieties). It is likely that this {\\it ad hoc}\ndefinition is stronger than the ``correct'' one. A more elegant approach\nwas suggested by Deligne and Simpson \n(\\cite{_Deligne:defi_}, \\cite{_Simpson:hyperka-defi_}).\n\n\\hfill\n\n\\definition\\label{_singu_hype_Definition_}\n(\\cite{_Verbitsky:Hyperholo_bundles_}, Definition 6.5)\nLet $M$ be a hypercomplex variety (\\ref{_hypercomplex_Definition_}).\nThe following data define a structure of {\\bf hyperk\\\"ahler variety}\non $M$.\n\n\\begin{description}\n\n\\item[(i)] For every $x\\in M$, we have an ${\\Bbb R}$-linear \nsymmetric positively defined\nbilinear form $s_x:\\; T_x M \\times T_x M {\\:\\longrightarrow\\:} {\\Bbb R}$\non the corresponding real Zariski tangent space.\n\n\\item[(ii)] For each triple of induced complex structures\n$I$, $J$, $K$, such that $I\\circ J = K$, we have a \nholomorphic differential 2-form $\\Omega\\in \\Omega^2(M, I)$.\n\n\\item[(iii)] \nFix a triple of induced complex structure \n$I$, $J$, $K$, such that $I\\circ J = K$. Consider the\ncorresponding differential 2-form $\\Omega$ of (ii).\nLet $J:\\; T_x M {\\:\\longrightarrow\\:} T_x M$ be an endomorphism of \nthe real Zariski tangent spaces defined by $J$, and $Re\\Omega\\restrict x$\nthe real part of $\\Omega$, considered as a bilinear form on $T_x M$.\nLet $r_x$ be a bilinear form $r_x:\\; T_x M \\times T_x M {\\:\\longrightarrow\\:} {\\Bbb R}$ \ndefined by $r_x(a,b) = - Re\\Omega\\restrict x (a, J(b))$.\nThen $r_x$ is equal to the form $s_x$ of (i). In particular,\n$r_x$ is independent from the choice of $I$, $J$, $K$.\n\n\\end{description}\n\n\n\\noindent \\remark \\nopagebreak\n\\begin{description}\n\\item[(a)] It is clear how to define a morphism of hyperk\\\"ahler varieties.\n\n\\item[(b)]\nFor $M$ non-singular, \\ref{_singu_hype_Definition_} is\n equivalent to the usual\none (\\ref{_hyperkahler_manifold_Definition_}). \nIf $M$ is non-singular,\nthe form $s_x$ becomes the usual Riemann form, and \n$\\Omega$ becomes the standard holomorphically symplectic form.\n\n\\item[(c)] It is easy to check the following.\nLet $X$ be a hypercomplex subvariety of a hyperk\\\"ahler\nvariety $M$. Then, restricting the forms $s_x$ and $\\Omega$\nto $X$, we obtain a hyperk\\\"ahler structure on $X$. In particular,\ntrianalytic subvarieties of hyperk\\\"ahler manifolds are always\nhyperk\\\"ahler, in the sense of \\ref{_singu_hype_Definition_}.\n\n\\end{description}\n\n\\hfill\n\n{\\bf Caution:} Not everything which is seemingly hyperk\\\"ahler\nsatisfies the conditions of \\ref{_singu_hype_Definition_}.\nTake a quotient $M\/G$ os a hyperk\\\"ahler manifold by an action \nof finite group $G$, acting in accordance with hyperk\\\"ahler\nstructure. Then $M\/G$ is, generally speaking, {\\it not} hyperk\\\"ahler\n(in fact, $M\/G$ is {\\it never} hyperk\\\"ahler). For instance,\ntake a quotient of a 2-dimensional torus $T$ by $G=\\{\\pm 1\\}$\nacting as an involution $t{\\:\\longrightarrow\\:} -t$. This is a beautiful and well\nknown example of a hyperk\\\"ahler automorphism; the quotient space\nhas 16 isolated singular points, which, if blown up, give\na K3 surface. For $x$ a singular point of $T\/\\{\\pm 1\\}$,\nits Zariski tangent space has real dimension 6. \nOn the other hand, for a hypercomplex variety,\nthere is a quaternion action in every Zariski tangent\nspace, and thus, the dimension real dimension\nof Zariski tangent space must be divisible by 4. We obtain that\nthe space $T\/\\{\\pm 1\\}$ is not even hypercomplex. How this happens?\nWe take a twistor space $\\operatorname{Tw}(T)$ of $T$ and take a quotient of \n$\\operatorname{Tw}(T)$ by the natural action of $G=\\{\\pm 1\\}$, which is \nholomorphic. The quotient is a complex variety fibered over ${\\Bbb C} P^1$\nFor $\\operatorname{Tw}(T)\/\\{\\pm 1\\}$ to be hypercomplex, this fibration must be trivial,\nin a real analytic category. But the functor of forgetting the complex\nstructure does not commute with taking finite quotients! Thus, even\nif $Tw(T)$ is (as a real analytic space) trivially fibered over\n${\\Bbb C} P^1$, there is no way to push down this trivialization\nto $\\operatorname{Tw}(T)\/\\{\\pm 1\\}$. \n\n\\hfill\n\nThe following theorem, proven in \n\\cite{_Verbitsky:Hyperholo_bundles_} (Theorem 6.3), \ngives a convenient way to construct\nexamples of hyperk\\\"ahler varieties.\n\n\\hfill\n\n\\theorem \\label{_hyperho_defo_hyperka_Theorem_}\nLet $M$ be a compact hyperk\\\"ahler manifold, $I$ an induced\ncomplex structure and $B$ a stable holomorphic bundle over $(M, I)$.\nLet $D(B)$ be a deformation space of stable holomorphic structures on $B$.\nAssume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect\nto the standard action of $SU(2)$ on $H^*(M)$. Then $D(B)$ has a\nnatural structure of a hyperk\\\"ahler variety. \n\n\\nopagebreak\n\\blacksquare\n\n\\hfill\n\n\nThe following theorem is implicit in \n\\cite{_Verbitsky:Hyperholo_bundles_}.\n\n\\hfill\n\n\\theorem \\label{_hyperholo_functo_Theorem_}\nLet $M$ be a compact hyperk\\\"ahler manifold, $I$ an induced\ncomplex structure and $B_1, B_2, ..., B_n$ \nstable holomorphic bundles over $(M, I)$.\nLet $D(B_i)$ be a deformation space of stable \nholomorphic structures on $B_i$.\nAssume that $c_1(B_i)$, $c_2(B_i)$, $i=1, 2, ..., n$ \nare $SU(2)$-invariant, with respect\nto the standard action of $SU(2)$ on $H^*(M)$. \nLet $\\Pi$ be a natural tensor operation on the vector bundles,\nsuch that, e. g.,\n\n\\[ B_1, ... B_n {\\:\\longrightarrow\\:} B_1 \\otimes B_2 \\otimes \\Lambda^2 B_3 \\otimes S^7\n B_4 \\otimes ... \\otimes B_n ^*.\n\\]\nAssume that $\\Pi(B_1, ... B_n)$ cannot be decomposed to a direct sum\nof holomorphic bundles. Then $\\Pi(B_1, ... B_n)$ is stable, \nand the associated map\n\\[ D(B_1) \\times D(B_2) \\times ..., \\times D(B_n){\\:\\longrightarrow\\:} \n D\\left(\\Pi(B_1, ..., B_n)\\right)\n\\]\n(defined in a certain neighbourhood of \n$[B_1]\\times [B_2]\\times ..., \\times [B_n] \\in \nD(B_1) \\times D(B_2) \\times ..., \\times D(B_n)$\nis a morphism of hyperk\\\"ahler varieties.\n\n\\blacksquare\n\n\\hfill\n\n\\ref{_hyperholo_functo_Theorem_} gives a natural way to \nconstruct trianalytic subvarieites of hyperk\\\"ahler varieties. \n\n\\hfill\n\nThe following theorem is almost trivial; the reader is advised \nto invent his or her own proof instead of reading ours\n(which is by necessity sketchy).\n\n\\hfill\n\n\\theorem \\label{_Doua_hyperka_Theorem_}\nLet $M$ be a compact hyperk\\\"ahler manifold, $X\\subset M$\na trianalytic subvariety and $D(X)$ its Douady \nspace. Then $D(X)$ is a compact hyperk\\\"ahler variety.\n\n\\hfill\n\n{\\bf Proof:} Clearly, $D(X)$ is hypercomplex, in a natural way\n(\\ref{_Douady_hyperc_Remark_}). It remains to construct\nthe forms $s_x$ and $\\Omega$. Let $Y$ be a trianalytic \nsubvariety of $M$ which is a deformation of $X$,\nand $U$ be a sufficiently small neighbourhood of $[Y]\\in D(X)$. \nLet \n$\\left(\\pi:\\; {\\c X}_U {\\:\\longrightarrow\\:} U,\\phi:\\; {\\c X}_U {\\:\\longrightarrow\\:} M\\right)$ \nbe the universal family of subvarieties\nof $M$, attached to $U\\subset D(X)$. The space ${\\c X}_U$\nis hypercomplex and the map $\\pi$\nis compatible with the hypercomplex structure. \nMoreover, the map $\\phi:\\; {\\c X}_U {\\:\\longrightarrow\\:} M$ is an\nimmersion, so ${\\c X_U}$ is hyperk\\\"ahler (hyperk\\\"ahler structure\nis obtained as a pullback from $M$).\n\\ref{_conne_in_triana_flat_Theorem_}\nprovides a natural trivialization of \n${\\c X}_U {\\:\\longrightarrow\\:} U$. Thus, for each $y\\in Y$, \nthere exists a natural section $\\sigma_y:\\; U\\hookrightarrow {\\c X}_U$.\nBy \\ref{_triana_subse_comple_ana_Theorem_}, \nthis section is compatible with the hypercomplex structure.\nRestricting the hyperk\\\"ahler structure\nfrom ${\\c X_U}$ to $\\sigma_y(U)$, we obtain a hyperk\\\"ahler\nstructure on $U$. It is easy to check that this hyperk\\\"ahler\nstructure is independent from the choice of section\n$\\sigma_y$. Gluing the hyperk\\\"ahler structures from\ndifferent $U$, we obtain the proof of\n\\ref{_Doua_hyperka_Theorem_}. \\blacksquare\n\n\n\n\n\\hfill\n\n{\\bf Acknowledgements:} I am indebted to D. Kaledin for enlightening\ndiscussions; he also suggested some of the proofs. \nI am thankful to Tony Pantev for the frutiful discussions.\nMohan Ramachandran communicated me the reference\nto \\cite{_Greene_}. I am grateful to A. Beilinson and J. Bernstein \nfor the help with algebraic geometry. My gratitude to\nM. Finkelberg, D. Kazhdan and S.-T. Yau for their \ninterest.\n\n\n\\hfill\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Preliminaries}\n\n\nAs noticed in \\cite[Section 1.1]{as}, the probabilistic method is a\npowerful tool in tackling many problems in Discrete Mathematics\n(Combinatorics, Graph Theory, Number Theory and Combinatorial Geometry).\nMore recently, it has been applied in the development of \nefficient algorithmic techniques and in the study of various computational\nproblems. \n\n\nIn this paper we present three combinatorial\n identities whose proofs are based on a simple probability \ntechnique consisting on calculations of $k$-th moments ($k=1,2,3$) of \nsome discrete random variables. Our proofs consist of \nshowing that these identities essentially compute the moments \nof order $k$ $(k=1,2,3)$ of the discrete random variable \ndefined in \\cite{m1}. Notice that this random variable\nis a generalization of the complex-valued discrete random variable\ndefined in \\cite{ssa} by providing a statistical analysis for efficient \ndetection of signal components when missing data samples are present\n(cf. \\cite{sso}). \nOn the other hand, the author of this paper continued the \nresearch on the mentioned complex-valued discrete random variables\n\\cite{m8}. \n\n\nNotice that combinatorial identities and combinatorial problems \nappear in many areas of mathematics, notably in \nNumber Theory, Probability Theory, Topology, Geometry, \nMathematical Optimization, Computer Science, Ergodic Theory and Statistical \nPhysics. \n \n\nAs usually, throughout our considerations we use the \nterm ``multiset'' (often written as ``set'') to mean ``a totality having possible \nmultiplicities''; so that two (multi)sets will be counted as equal if \nand only if they have the same elements with identical multiplicities.\nLet $\\Bbb C$ and $\\Bbb R$ denote the fields of complex and real \nnumbers, respectively. For a given positive integer $N$, let ${\\mathcal M}_N$ \ndenote the collections of all multisets of the form \n $$\n\\Phi_N=\\{z_1,z_2,\\ldots ,z_N:\\, z_1,z_2,\\ldots ,z_N\\in\\Bbb C\\}.\\leqno (1)\n $$\nFurthermore, denote by ${\\mathcal M}$ the set consisting \nof all multisets of the form (1), i.e., \n $$\n{\\mathcal M}=\\bigcup_{N=1}^{\\infty}{\\mathcal M}_N. \n $$\n\n\nFollowing Definition 1.2 from \\cite{m8} \n(also see Definition 1.1 in \\cite{m7}), the random variable $X(m, \\Phi_N)$\nwas generalized in \\cite{m1} as follows.\n \n \\begin{definition} (\\cite[Definition 1.1]{m1})\nLet $N$ and $m$ be arbitrary nonnegative integers \nsuch that $1\\le m\\le N$.\nFor given not necessarily distinct complex numbers\n$z_1,z_2,\\ldots ,z_N$, let \n$\\Phi_N \\in {\\mathcal M}_N$ be a multiset defined by (1).\n Define the discrete complex-valued random variable $X(m,\\Phi_N)$ as\n \\begin{eqnarray*}\n && \\mathrm{Prob}\\left(X(m,\\Phi_N)\n = \\sum_{i=1}^m z_{n_i}\\right)\\\\\n(2)\\qquad\\qquad &= &\\frac{1}{{N\\choose m}}\\cdot \\big|\\{\\{t_1,t_2,\\ldots,t_m\\}\n\\subset\\{1,2,\\ldots,N\\}: \n\\sum_{i=1}^m z_{t_i}=\\sum_{i=1}^mz_{n_i}\\} \\big|\n\\quad\\qquad\\\\\n& =&:\\frac{q(n_1,n_2,\\ldots, n_m)}{{N\\choose m}},\n \\end{eqnarray*}\n where $\\{n_1,n_2,\\ldots, n_m\\}$ is an arbitrary fixed\n subset of $\\{1,2,\\ldots,N\\}$ such that $1\\le n_1a$, then\nthe conditions $k_1\\le n_1,k_2\\le n_2,\\ldots,k_s\\le n_s$ \nwhich appear under the first sum $\\sum\\cdot$ of (8) can be omitted.\n \\end{remark}\n\nA particular case of Identity 2.1 is Identity 2.5 given below\nwhich is a well known ``multinomial'' generalization of the \n{Vandermonde identity} (often called \n{\\it Vandermonde convolution formula} or {\\it Chu-Vandermonde convolution}) \n(see, e.g., \\cite{ri}). \n \n\n\\begin{identity} Let $n_1,n_2,\\ldots ,n_s$ $(s\\ge 2)$ be arbitrary positive \nintegers. If $m$ is a positive integer $m$ such that \n$m\\le \\sum_{i=1}^sn_i$, then\n $$\n\\sum_{\\sum_{i=1}^sk_i=m\\atop k_1\\le n_1,k_2\\le n_2,\\ldots, \nk_s\\le n_s}{n_1\\choose k_1}{n_2\\choose k_2}\\cdots \n{n_s\\choose k_s}={\\sum_{i=1}^s n_i\\choose m},\\leqno(12)\n $$ \nwhere the summation ranges over all nonnegative integers \n$k_i$ $(i=1,2,\\ldots , s)$ such that $k_1\\le n_1,k_2\\le n_2,\\ldots, \nk_s\\le n_s$ and $\\sum_{i=1}^sk_i=m$. \n \\end{identity}\n \n\\begin{proof}\nTaking $z_1=z_2=\\cdots =z_s=1$ into equality (8), we immediately obtain (12).\n\\end{proof}\n \n\\begin{remark} There are well known algebraic and combinatorial proofs\n of the identity (12) (see, e.g., \\cite{wi}). Notice also that \n for $s=2$ the identity (12) with $k_1=k$, $n_1=a$\nand $n_2=b$ simplifies to Chu-Vandermonde identity given by \n(see, e.g., \\cite[p. 67]{aa})\n $$\n\\sum_{k=0}^m{a\\choose k}{b\\choose m-k}={a+b\\choose m},\\leqno(13)\n $$\nwhich also holds for any complex numbers $a$ and $b$. \nNotice that\nthe identity (13) is named after A.T. Vandermonde (1772), although it was already \nknown in 1303 by the Chinese mathematician Zhu Shijie (Chu Shih-Chieh)\n(see \\cite[pp. 59--60]{ask} for the history).\nThis identity plays an important role in Combinatorics,\nCombinatorial Number Theory and Probability Theory (\\cite{gkp}, \n\\cite{gs} and \\cite{ri}). As indicated in \\cite[p. 8]{ri},\n Vandermonde convolution formula \nis perhaps the most widely used combinatorial identity. \nIn the literature there are many proofs of this identity and its \nseveral generalizations. A proof given in \\cite{vm} was established by giving \nprobabibilstic interpretations to the summands. \n \\end{remark}\n\nTaking $s=2$ and $z_1\/z_2=z$ ($z_2\\not= 0$) into (8), it \nsimplifies to the following Vandermonde-type convolution formula.\n\n\\begin{identity}\nLet $n_1$ and $n_2$ be arbitrary positive \nintegers and let $z$ be any complex numbers. \nIf $m$ is a positive integer such that \n$m\\le n_1+n_2$, then\n $$\n \\sum_{k=0}^m{n_1\\choose k}{n_2\\choose m-k}(kz+(m-k))\n={n_1+n_2\\choose m}\\frac{2m(n_1z+n_2)}{n_1+n_2}.\\leqno(14)\n $$\n\\end{identity}\n\n\\begin{remark} Observe that taking $z=1$, $n_1=a$ and \n$n_2=b$ into (14), it immediately reduces to \nChu-Vandermonde identity given by (13). \n \\end{remark}\n\nSubstituting $s=3$ $z_1\/z_3=z$ and $z_2\/z_3=w$ $(z_3\\not= 0)$ into (8), it \nreduces to the following Vandermonde-type convolution formula.\n\n\\begin{identity}\nLet $n_1$, $n_2$ and $n_3$ be arbitrary positive \nintegers and let $z$ and $w$ be arbitrary complex numbers. \nIf $m$ is a positive integer such that \n$m\\le n_1+n_2+n_3$, then\n \\begin{eqnarray*}\n && \\sum_{0\\le k_1+k_2\\le m\\atop k_1\\ge 0,k_2\\ge 0}{n_1\\choose k_1}\n{n_2\\choose k_2} {n_3\\choose m-k_1-k_2}(k_1z+k_2w+(m-k_1-k_2))\\qquad\\\\\n(15)\\qquad &=&\n{n_1+n_2+n_3\\choose m}\\frac{2m(n_1z+n_2w+n_3)}{n_1+n_2+n_3}.\\qquad\\qquad\n \\end{eqnarray*}\n\\end{identity}\n\nTaking $z=w=1\/2$ into (15), we obtain the following identity.\n\n\\begin{identity}\nLet $n_1$, $n_2$ and $n_3$ be arbitrary positive \nintegers and let $z_1$ and $z_2$ be arbitrary complex numbers. \nIf $m$ is a positive integer such that \n$m\\le n_1+n_2+n_3$, then\n \\begin{eqnarray*}\n && \\sum_{0\\le k_1+k_2\\le m\\atop k_1\\ge 0,k_2\\ge 0}(2m-k_1-k_2){n_1\\choose k_1}\n{n_2\\choose k_2} {n_3\\choose m-k_1-k_2}\\\\\n(16)\\qquad &=&\n\\frac{2m(n_1+n_2+2n_3)}{n_1+n_2+n_3}{n_1+n_2+n_3\\choose m}.\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n \\end{eqnarray*}\n\\end{identity}\n\nAnother special case of Identity 2.1 is given as follows.\n\n \n\\begin{identity} \nLet $s$ and $l$ be arbitrary positive integers, and let \n$m$ be a positive integer such that $m\\le sl$. Then\n $$\n\\sum_{\\sum_{i=1}^sk_i=m \\atop k_1\\le l,k_2\\le l,\\ldots, \nk_s\\le l}{l\\choose k_1}\n{l\\choose k_2}\\cdots {l\\choose k_s}(k_1+2k_2+\\cdots+sk_s)=\\frac{m(s+1)}{2}\n {sl\\choose m},\\leqno(17)\n $$\nwhere the summation ranges over all nonnegative integers \n$k_i$ $(i=1,2,\\ldots , s)$ such that $\\sum_{i=1}^sk_i=m$ and \n$k_1\\le l,k_2\\le l,\\ldots,k_s\\le l$.\n \\end{identity}\n\n \\begin{proof}\nSubstituting $z_i=i$ for all $i=1,2,\\ldots ,s$ and $n_1=n_2=\\cdots =n_s=l$\ninto (8), it immediately reduces to the equality (17). \n \\end{proof}\n\nAs a consequence of Identity 2.11, we obtain the following \n``supercongruence'' closely related to the remarkable \n{\\it Wolstenhlme's theorem} which asserts that\n $$\n{2p-1\\choose p-1}\\equiv 1\\pmod{p^3}\n $$ \nfor all primes $p\\ge 3$ (\\cite{wo}; also see \\cite[p. 3]{m9} and \\cite{m2}).\n\n\\begin{congruence} \nLet $p\\ge 5$ be a prime. Then for each positive integer $s$ there holds \n $$\n\\sum_{\\sum_{i=1}^sk_i=p\\atop k_1,k_2,\\ldots, k_s\\ge 0}{p\\choose k_1}\n{p\\choose k_2}\\cdots {p\\choose k_s}(k_1+2k_2+\\cdots+sk_s)\n\\equiv \\frac{s(s+1)p}{2}\\pmod{p^4}.\\leqno(18)\n $$\nIn particular,\n $$\n\\sum_{\\sum_{i=1}^sk_i=p\\atop k_1,k_2,\\ldots, k_s\\ge 0}{p\\choose k_1}\n{p\\choose k_2}\\cdots {p\\choose k_s}(k_1+2k_2+\\cdots+sk_s)\\equiv 0 \\pmod{p}.\n\\leqno(19)\n $$\n \\end{congruence}\n\\begin{proof} \nIf we substitute $l=m=p$ into equality (17), then its right hand\nside is equal to $\\frac{(s+1)p}{2}{sp\\choose p}$. Since by \nGlaisher's congruence \\cite[p. 21]{gl} \n(or more general, Ljunggren's congruence (\\cite{lj}; also see \n\\cite[the congruences (15), p. 7 and (35) and (36), p. 11]{m9}; \ncf. \\cite[Section 3.3]{m3} and \\cite{m10}), for any prime $p\\ge 5$ and \na positive integer $s$, we have \n $$\n{sp\\choose p}\\equiv s\\pmod{p^3}, \n $$ \nand hence, \n $$\n\\frac{(s+1)p}{2}{sp\\choose p}\\equiv \\frac{s(s+1)p}{2}\\pmod{p^4}.\n $$\nSubstituting the above congruence into (17) with $l=m=p$,\nwe immediately obtain the congruence (18). Finally, reducing the modulus \nin (18) to $(\\bmod{\\, p})$, implies (19). \n \\end{proof}\n\nLet us recall that a prime $p$ is said to be a {\\it Wolstenholme prime} \n(see, e.g., \\cite{mc}, \\cite[Section 7]{m9} and \\cite{m11}; this is \nSloane's sequence A088164 from \\cite{sl}) if it satisfies the congruence \n $$\n{2p-1\\choose p-1}\\equiv 1\\pmod{p^4}.\n $$ \nIt is well known (see \\cite[p. 21]{gl}, \\cite[p. 323]{gl2} and \n\\cite[p. 14]{m9}) that $p$ is a Wolstenholme prime if and only if $p$ divides\nthe numerator of the {\\it Bernoulli number} $B_{p-3}$.\nMoreover, these primes together with the primes $p$ such that \nthe Euler number $E_{p-3}$ is divisible by $p$, are closely\nrelated to the first case of {\\it Fermat Last Theorem} \n(see \\cite{va} and \\cite{m3}). It can be shown that for any Wolstenholme \nprime, the congruence (18) holds modulo $p^5$, i.e., we have the following\nassertion. \n \n\\begin{congruence} \nLet $p$ be a Wolstenholme prime. Then for each positive integer $s$ \nthere holds \n $$\n\\sum_{\\sum_{i=1}^sk_i=p\\atop k_1,k_2,\\ldots, k_s\\ge 0}{p\\choose k_1}\n{p\\choose k_2}\\cdots {p\\choose k_s}(k_1+2k_2+\\cdots+sk_s)\\equiv \\frac{s(s+1)p}{2}\n\\pmod{p^5}.\\leqno(20)\n $$\n \\end{congruence}\n\\begin{proof} Notice that by a result of Glaisher (\\cite[p. 21]{gl}, \n\\cite[p. 323]{gl2}; also see \\cite[the conguence (15), p. 7]{m9}\nand the {\\it Jacobsthal's congruence} \\cite{lj}),\nfor any Wolstenholme prime $p$,\n $$\n{sp\\choose p}\\equiv s\\pmod{p^4}. \n $$\nThen the rest of the proof is quite similar to that of the previous \nCongruence 2.12, and hence may be omitted.\n \\end{proof}\n\nAnother consequence of Identity 2.1 is given as follows.\n \n \\begin{identity}\nLet $n\\ge 2$ and $s$ be fixed positive integers and let \n$k=k_1+k_2n+\\cdots + k_sn^{s-1}$ be the base $n$ \nrepresentation of a positive integer $k 1$ \nsuggest $\\rho_{tc} > 0.5$, with phase $\\phi_{tc}$ near maximal,\ncould be a backup to $\\rho_{tt}$ for EWBG.\nIn making this plot, the simplifying assumption of\n$m_H = m_A = m_{H^+} = 500$ GeV is taken.\nMuch higher values would either run into issues of perturbativity,\nor damping by decoupling.\n\nFig.~1[left] scanned through realistic Yukawa matrices,\nbut a simplified texture can help elucidate the driving effect.\nSuppose\n $(Y_1)_{tc} \\neq 0,\\, (Y_2)_{tc} \\neq 0$ and\n $(Y_1)_{tt}= (Y_2)_{tt} \\neq 0$, \nwhile all other extra Yukawas vanish, i.e. altogether 3 complex parameters.\nIf one assumes $\\sqrt2 Y^{\\rm SM}$ is the linear sum of $Y_1$ and $Y_2$,\none can solve for $V^u_R$, while there is no need for $V^u_L$.\nOne can then arrive at the combination of $Y_1$ and $Y_2$ that is\northogonal to $Y^{\\rm SM}$. In this way, one finds~\\cite{fhs}\n\\begin{equation}\n {\\rm Im}\\big[ (Y_1)_{tc}(Y_2)_{tc}^* \\big]\n = -\\lambda_t\\, {\\rm Im}\\,\\rho_{tt}, \\quad \\rho_{ct}=0,\n\\label{eq:simpCPV}\n\\end{equation}\nwith $\\rho_{tc}$ remaining basically a free parameter.\nWe see from Eq.~(\\ref{eq:simpCPV}) that {\\it both} doublets participate\nin the CPV source for EWBG in 2HDM, which is \nreminiscent to the Jarlskog invariant for SM.\nWe can also see how 2HDM with extra Yukawas overcomes \nthe suppression factors in the Jarlskog invariant, given that\n$\\lambda_t$, $|{\\rm Im}\\,\\rho_{tt}|$ are both ${\\cal O}(1)$.\n\n\n\n\\section{\\boldmath Bonus 2: : Alignment from ${\\cal O}(1)$ Higgs Quartics}\n\n{\\it It is remarkable that the extra Yukawa coupling \n$\\rho_{tt}$ could account for BAU!} \n\nNote that such mechanism does not exist in\n2HDM-I or 2HDM-II, the 2HDMs that satisfy NFC, since NFC\nmeans there are essentially no new Yukawa couplings, \ndespite having a second Higgs doublet.\nWe now show~\\cite{hk2} that the prerequisite for 1st order EWPT, that\nextra Higgs quartic couplings are ${\\cal O}(1)$, could be behind\nthe observed approximate alignment.\n\nThe general $CP$-conserving Higgs potential~\\cite{haber} of 2HDM is, \n\\begin{eqnarray}\n V(\\Phi,\\, \\Phi')\n&= \\mu_{11}^2 |\\Phi|^2 +\\mu_{22}^2 |\\Phi'|^2\n - \\left(\\mu_{12}^2 \\Phi^\\dagger \\Phi' + {\\rm h.c.}\\right) \n \\ +\\frac{1}{2}\\eta_1^{}|\\Phi|^4 + \\frac{1}{2}\\eta_2^{}|\\Phi'|^4 + \\eta_3^{}|\\Phi|^2|\\Phi'|^2 \\nonumber \\\\\n & + \\eta_4^{}|\\Phi^\\dagger \\Phi'|^2\n + \\left\\{\\frac{1}{2}\\eta_5^{}(\\Phi^\\dagger \\Phi')^2\n + \\left[ \\eta_6^{}|\\Phi|^2 + \\eta_7^{}|\\Phi'|^2\\right] \\Phi^\\dagger \\Phi' + {\\rm h.c.}\\right\\},\n\n \\label{eq:pote2}\n\\end{eqnarray}\nwhere we take Higgs basis, i.e. $\\mu_{11}^2 < 0$ but $\\mu_{22}^2 >0$.\nWith the two minimization conditions\n\\begin{equation}\n \\mu_{11}^2 = -\\frac{1}{2}\\eta_1^{}v^2, \\quad\n \\mu_{12}^2 = \\frac{1}{2}\\eta_6^{}v^2,\n \\label{eq:minim}\n\\end{equation}\n$\\mu_{11}^2 < 0$ is exchanged for $v$, the usual\n``soft breaking parameter'' $\\mu_{12}^2$ is removed, and \nthe quartic coupling $\\eta_6$ is solely responsible for $\\Phi$--$\\Phi'$ mixing.\nThe $CP$-even Higgs mass matrix \n\\begin{equation}\n M_\\textrm{even}^2 =\n \\left[\\begin{array}{cc}\n \\eta_1^{}v^2 & \\eta_6^{} v^2 \\\\\n \\eta_6^{} v^2 & \\mu_{22}^2 + \\frac{1}{2} (\\eta_3^{} + \\eta_4^{} + \\eta_5^{})v^2\\\\\n \\end{array}\\right], \\quad\\quad\n R_\\gamma = \\left[\\begin{array}{cc}\n c_\\gamma & - s_\\gamma \\\\\n s_\\gamma & c_\\gamma \\\\\n \\end{array}\\right],\n\\end{equation}\nis diagonalized by $R_\\gamma$, \ni.e. $R^T_\\gamma M_\\textrm{even}^2 R_\\gamma$ is diagonal \nwith elements $m_H^2$, $m_h^2$.\nIn Eq.~(7), our $c_\\gamma \\equiv \\cos\\gamma$ corresponds to \n$\\cos(\\beta - \\alpha)$ in the 2HDM-II notation, and is the relative angle (mod. $\\pi\/2$) \nbetween the Higgs basis and the neutral Higgs mass basis.\n\nRather than give the formula for $m_H^2$, we note the mixing angle \n$c_\\gamma$ satisfies two relations,\n\\begin{equation}\n c_\\gamma^2 = \\frac{\\eta_1^{}{v^2} - m_h^2}{m_H^2 - m_h^2}, \\quad \\ \\ \n \\sin2\\gamma^{} = \\frac{2\\eta_6^{}v^2}{m_H^2 - m_h^2}.\n \\label{eq:npara}\n\\end{equation}\nIn alignment \\emph{limit} of $c_\\gamma \\to 0$, $s_\\gamma \\to -1$,\none has $\\eta_1 \\to m_h^2\/v^2 \\simeq 0.26$ in numerator of first term, \nwhere $m_h \\simeq 125$ GeV is used.\nFor $c_\\gamma$ small but nonvanishing, \n$m_H^2 - m_h^2 >$ several $v^2$ can weigh down \n$|\\eta_1 v^2 - m_h^2| < v^2$.\nSince $s_\\gamma \\to -1$ holds better than $c_\\gamma \\to 0$, \nthe second relation gives\n\\begin{equation}\n c_\\gamma^{} \\simeq \\frac{-\\eta_6^{}v^2}{m_H^2 - m_h^2}.\n \\quad\\quad \\ \\ {\\rm (near\\; alignment)}\n \\label{eq:Haber}\n\\end{equation}\nAlthough the result exists~\\cite{bechtle} in the literature, \n$|\\eta_6| \\ll 1$ is generally assumed, \nas it arises through loop effects in MSSM.\nBut we see that $c_\\gamma$ can be small for\n\\begin{equation}\n |\\eta_6| \\sim {\\cal O}(1)\\; ({\\rm or\\ smaller}), \\quad m_H^2 - m_h^2 > {\\it several}\\; v^2. \n \\label{eq:alignment}\n\\end{equation}\nNote that a low $m_h^2\/v^2 \\simeq 0.26$ is \\emph{not} required,\ni.e. $c_\\gamma$ can be small even if $m_h \\sim 300$ GeV.\n\nWhat drives alignment in 2HDM?\nFor $\\eta_{1,3,4,5}$, $\\mu_{22}^2\/v^2 \\sim {\\cal O}(1)$,\n$[M_{\\rm even}^2]_{22}$ has four ${\\cal O}(v^2)$ terms\nwhile $[M_{\\rm even}^2]_{11}$ has only one,\nhence $m_H^2 - m_h^2 > {\\rm several}\\; v^2$ is likely.\nHowever, $\\mu_{22}^2\/v^2 > 1$ would damp the \n1st order EWPT, hence sub-TeV exotic Higgs masses are preferred.\nSecond, $\\eta_6 \\sim {\\cal O}(1)$\nincreases $m_H^2 - m_h^2$ by level repulsion,\npushing $m_h^2\/v^2$ down from $\\eta_1 \\sim {\\cal O}(1)$.\nFinally, tuning $\\eta_6 < 1\/4 \\sim m_h^2\/v^2$ would give \n\\emph{extreme} alignment ($c_\\gamma \\to 0$) hence $\\eta_1 \\to 0.26$.\nThese observations are illustrated in Fig.~1[right] for \nallowed $\\eta_1$ vs $\\eta_6$ range,\nwhere custodial SU(2) is assumed to evade $\\Delta T$ constraint, \ni.e. $m_A^2 = m_{H^+}^2 = \\mu_{22}^2 + \\eta_3 v^2\/2$.\nWe vary $\\eta_4 = \\eta_5 \\in (0.5,\\, 2)$, \nso $m_H$ could be up to 100 GeV higher.\nHigh values of $\\eta_1$ are cut off by $\\Delta T$\n(via scalar--vector loop),\nand the two dashed lines mark $-c_\\gamma = 0.1$ and $0.2$,\nwhich are quite close to alignment; even $-c_\\gamma = 0.3$, \nclose to the bound from $\\Delta T$, is still allowed by \nobserved approximate alignment at LHC. \n\n{\\it ${\\cal O}(1)$ Higgs quartics could be behind \napproximate alignment}, or small $c_\\gamma$, regardless of\nwhether a $Z_2$ symmetry is used to enforce NFC or not,\nas our discussion is general.\nBut we have advocated that $\\rho_{tt} \\sim {\\cal O}(1)$ could explain BAU.\nIt is then intriguing to comment that sizable $\\rho_{tt}$\ncould possibly~\\cite{hk} help ``protect'' alignment:\nwith ${\\cal O}(1)$ Higgs quartics, bosonic loops\nwould reduce $\\Gamma_{h \\to ZZ^*}$, but\nthe top loop can bring $\\Gamma_{h \\to ZZ^*}$ back to SM value \nfor $\\rho_{tt}\\, c_\\gamma > 0$, consistent with what is observed.\nThis was our original motivation to understand the mechanism of alignment.\n\n\n\n\\section{\\boldmath Same-sign Top and Triple-top Signatures: $pp \\to tH\/A \\to tt\\bar c$, $tt\\bar t$}\n\nThe process $cg \\to tA$ was suggested long time ago as a direct probe~\\cite{hlmy}\nof the $ctA$ FCNH coupling, restricting to $m_A < 2m_t$ such that\n$A \\to t\\bar c$ (and $\\bar t c$) is at 100\\%.\nWe recently studied~\\cite{kmh} the $cg \\to tH\/A$ associated production \nthrough the $\\rho_{tc}$ coupling, followed by subsequent decay \n$H\/A \\to t\\bar c$, $\\bar tc$ and $t\\bar t$ final states involving \n$\\rho_{tc}$ and $\\rho_{tt}$ couplings, advocating \nthe signatures of same-sign top, $tt\\bar c$, and triple-top, $tt\\bar t$.\nThe same-sign top signature involves same-sign dileptons, together with\ntwo $b$-jets, missing energy, and additional jets. We find that,\nfor $\\rho_{tc} \\sim 1$, the second case for EWBG can\nbe probed with 300 fb$^{-1}$, but signature does not improve\nfor higher luminosity, unless background can be further controlled.\nGiven that $\\rho_{tt}$ is the favored driver for EWBG, \ntriple-top search at HL-LHC may be more interesting, and\npossesses more exquisite signatures:\nthree leptons, three $b$-jets, missing energy.\nThe backdrop of SM cross section at only fb level makes the case strong,\nwhere full HL-LHC data can cover up to 700 GeV mass range \nfor $\\rho_{tt} \\sim 1$, but $\\rho_{tc}$ needs to be \nnot much smaller than 0.5 for signal cross section.\n\n\n\n\\section{Conclusion: $H^0$, $A^0$, $H^\\pm$ in Our Time }\n\nWith ${\\cal O}(1)$ Higgs quartics for 1st order EWPT, \nthe extra Yukawa $\\rho_{tt}$ (or $\\rho_{tc}) \\sim {\\cal O}(1)$\n in general 2HDM is remarkably efficient for EWBG.\nThe ${\\cal O}(1)$ Higgs quartics support approximate alignment, \nand together with quark mass-mixing hierarchy \ncontrol low energy FCNH effect, without need for NFC.\nHaving $H$, $A$ and $H^\\pm$ sub-TeV in mass would be a boon to LHC search,\nthe discovery of which in $tt\\bar c$, $tt\\bar t$ final states would\ntouch upon Matter Asymmetry of the Universe.\n\n\\section*{Acknowledgments}\n\nWe thank our collaborators on three consecutive papers presented here\nfor an enjoyable 2017, and we are grateful to illuminating discussions \nwith Howie Haber at Moriond QCD 2018.\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn statistics we like to analyze data combined with some additional information in order to make predictions on the underlying population and\/or future expectations. Consider the following example.\nWhen comparing an assumed probability function $p(x|\\tau)$, where $\\tau$ is a continuous parameter, to experimental data it is common practice in classical statistics\nto define a certain confidence interval (how certain does one wants to be) after which interval ranges \nfor\nthe parameter $\\tau$ can be ruled out. One remains with possible values for $\\tau$ (see for example Ref.~\\cite{prosper}). Although it is certainly nice that we can rule out\ncertain ranges for $\\tau$, this by itself is not sufficient to give an expectation value for some observable, for example, the mean of the population.\nFor an introduction into statistics and probability theory the reader is referred to Ref.~\\cite{larsen}.\n\nIn order to come with an estimate for the mean one can use another common\ntechnique which is making a least-square fit. Although least-square fits enables one to easily derive some value for $\\tau$, care should be taken with its interpretation.\nLeast-square fits \nhappen in some cases to correspond to the most likely scenario for the \nprobability function (see for example Ref.~\\cite{least_square}) and although most-likely certainly contains valuable information, it is often used to make further \npredictions which can easily fail as shown by the following example.\n\nConsider for example 10 vases where the first vase has 91 red balls and 9 blue balls, \nthe second has 92 red balles and 8 red balles, etc.,\nand the last vase has 100 red balls. Blindfolded a ball is taken from a vase which turns out to be red. The most likely\nvase from which this red ball was taken is the vase having the 100 red balls. \nIf we use this most-likely scenario\/fit for making further \npredictions we could easily be making a huge mistake. Suppose we would estimate the probability for\ndrawing a blue ball from the same vase from which we drew the red ball we would estimate this \nprobability to be zero. However, although the most-likely probability after having drawn the red ball is indeed\nzero, the expecation value is certainly nonzero as there is a nonzero chance that we could have drawn\nthe balls from one of the other vases.\n\n\nBayesian statistics is one attractive option\nfor addressing the above problem. One makes an inventory of \npossible hypotheses and uses the data to calculate\nthe probability that a certain hypothesis is true. \nUsing all those probabilities one can derive a kind of \nexpected probability function. The basic Equations are the following (see e.g. Ref~\\cite{prosper}):\n\\begin{equation}\np(x_2 | x) = \\int \\rm{d}\\tau\\ p(x_2| \\tau)\\ \\mathcal{P}(\\tau | x), \n\\label{invariance}\n\\end{equation}\nwhere\n\\begin{equation}\nP(\\tau | x) \\propto \\int {\\rm d}x\\ p(x| \\tau)\\ \\mathcal{P}(\\tau).\n\\end{equation}\n$\\mathcal{P}(\\tau)$ which describes the initial probability for a certain \nhypothesis before any data was taken and is known as the prior. \n\nOne desired property is the so-called invariance under reparametrization. Consider for example\nthe two Equations above in the case of a normal distribution with a known mean and unknown standard\ndeviation $\\sigma$. One experimentator could make the choice for considering $p(x| \\sigma)$ and integrate in \nEquation~\\ref{invariance}\nover $\\sigma$ while another experimentator could make another choice such as $p(x| s=\\sigma^2)$ and\nintegrate over $s$. It is desired that both experimenters obtain the same result which can be \nimposed through choosing the prior with certain properties.\nThere are several popular choices such as \nJeffreys prior~\\cite{jeffrey1, jeffrey2}. Although they may fulfill some necessary requirements such as invariance under reparametrization,\nthey do lack an ab initio derivation. \n\n\n\n \nThere seems to be no fundamental method for estimating an expectation probability function from a set of \nexperimental data. When the experimentator assumes a probability function $p(x| \\tau)$ both classical as well as Bayesian statistics cannot ab initio provide an expected probability function in case\nno other information is known. It is the goal of this article to obtain some estimates for expectation values without making additional assumptions.\n\n\n\n\n\n\\section{Confidence level expectation estimated probabilities}\n\nAlthough there are statistical approaches with different view points there seems to be a common view \nwhen\nsomething is probable or not. Suppose that we have a data point at x=0 and we know that it is drawn from a\nnormal distribution with $\\sigma=1$ but with unknown $\\mu$. At different confidence levels we can deduct an\ninterval for the parameter $\\mu$ as shown in Table~\\ref{Table_confidence}. The Table illustrates that at full confidence one\ncannot limit $\\mu$ to a finite interval but also that the most likely value for $\\mu=0$.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{rrr}\nconfidence level & confidence level interval & interval difference wrt previous\\\\\n\\hline\n10\\% & [-0.13, 0.13] & [-0.13,0] \\& [0, 0.13]\\\\\n20\\% & [-0.25, 0.25] & [-0.25, -0.13] \\& [0.13, 0.25]\\\\\n30\\% & [-0.39, 0.39] &[-0.39, -0.25] \\& [0.25, 0.39]\\\\\n40\\% & [-0.52, 0.52] &[-0.52, -0.39] \\& [0.39, 0.52]\\\\\n50\\% & [-0.67, 0.67] &[-0.67, -0.52] \\& [0.52, 0.67]\\\\\n60\\% & [-0.84, 0.84] &[-0.84, -0.67] \\& [0.67, 0.84]\\\\\n70\\% & [-1.04, 1.04] &[-1.04, -0.84] \\& [0.84, 1.04]\\\\\n80\\% & [-1.28, 1.28] &[-1.28, -1.04] \\& [1.04, 1.28]\\\\\n90\\% & [-1.64, 1.64] &[-1.64, -1.28] \\& [1.28, 1.64]\\\\\n100\\%& [-$\\infty$, $\\infty$] &[-$\\infty$, -1.64] \\& [1.64, $\\infty$]\n\\end{tabular}\n\\end{center}\n\\caption{Confidence interval for a normal distribution with $\\sigma=1$ and with unknown mean given\na single data point at $x=0$.\\label{Table_confidence}}\n\\end{table}\n\nWhen we look at Table~\\ref{Table_confidence} from a different point of view then we notice \nthat each row makes an equal \ncontribution to our confidence (first column shows that each row contributes 10\\% with respect to previous row). \nAs confidence levels\/intervals is at the heart of any statistical method it\nmakes sense to use this for defining an expectation probability function. Expectation values\ncould be obtained by considering the intervals in the last column of each row in Table~\\ref{Table_confidence} as equally important. It is proposed that intervals contributing equally to confidence shall be\ntaken as equally contributing to the expectation value.\n\nBefore defining the expectation value, the following confidence level limit is defined as\n\\begin{equation}\n\\alpha(\\vec{x},\\tau) \\equiv \\int_{p(\\vec{y}|\\tau)>p(\\vec{x}|\\tau)} \\prod_{i=1}^n \\rm{d}y_i\\ p(\\vec{y}| \\tau),\n\\label{definition}\n\\end{equation}\nwhere $p(\\vec{x}|\\tau)=\\prod_i^n\\ p(x_i| \\tau)$, $x_i$ are the known measurements, and $\\tau$\nis a model parameter. Please\nnote that this definition also nicely deals with probability distributions that have multiple maxima\nor that have a maximum at its border.\n\nFor each value $\\alpha$ one can find either no, one, or more values for $\\tau$. \nThe number of solutions we indicate with \n$N(\\vec{x}, \\alpha)$. The expected probability distribution based on equal contributing confidence is now defined as\n\\begin{equation}\n\\left< \\mathcal{O} \\right>_c \\equiv \\frac{1}{K} \\int_{N(\\vec{x}, \\alpha) \\neq 0} \\rm{d}\\alpha \\ \\frac{1}{N(\\vec{x}, \\alpha)} \n\\sum_{i=1}^{N(\\alpha, \\vec{x})} \\mathcal{O}(\\tau_i(\\vec{x}, \\alpha)),\n\\end{equation}\nwhere\n\\begin{equation}\nK = \\int_{N(\\vec{x}, \\alpha) \\neq 0} \\rm{d}\\alpha.\n\\end{equation}\nThe amount to the expection to which regions of $\\tau$ contribute is determined by $\\alpha$. For this\nreason the expectation probability function is by definition invariant under reparametrization.\n\n\n\n\n\n\\section{Examples involving normal distribution}\n\n\\subsection{Normal distribution with unknown $\\mu$}\n\nIn order to derive the expectation probability function it turns out to be most\nconvenient for first compute $\\alpha$ after which one can use its derivative with respect to $\\mu$ to \nrewrite integral over $\\alpha$ into an integral over $\\mu$. After translating the integration variables\n$y_i$ one can write $\\alpha$ as\n\\begin{equation}\n\\alpha(\\vec{x},\\mu) = \\int_{p(\\vec{y}|0) > p(\\vec{x}|\\mu)} \\prod_{i=1}^n \\rm{d}y_i\\ p(\\vec{y}| 0).\n\\end{equation}\nUsing the\nJacobian for spherical coordiantes in $n$ dimensions, $2\\pi^{n\/2}\/\\Gamma(n\/2)\\ r^{n-1}$, see also Ref.~\\cite{jacobian1}\n, one finds\n\\begin{equation}\n\\alpha = \\frac{1}{\\Gamma (n\/2)} \\left[ \\Gamma(n\/2, 0) - \\Gamma (n\/2, - \\ln \\prod_k e^{-(x_k-\\mu)^2\/(2\\sigma^2)})\n\\right],\n\\end{equation}\nwhere $\\Gamma$ is the (upper) incomplete Gamma function. When taking the derivative with respect to $\\mu$ one \nobtains\n\\begin{equation}\n\\partial_\\mu \\alpha = \\frac{-(\\sqrt{2\\pi}\\sigma)^n}{\\sqrt{2}\\sigma\\ \\Gamma(n\/2)}\\ \n\\left[ \\sum_i^n \\frac{(x_i - \\mu)^2}{2\\sigma^2} \\right]^{n\/2 - 1\/2}\\ p(\\vec{x}| \\mu).\n\\end{equation}\n\n\nFor the expected probability function, one obtains\n\\begin{equation}\n\\left< p(x_2 | \\vec{x} ) \\right>_c = \\frac{1}{K} \\int \\rm{d}\\mu\\ p(x_2| \\mu)\\ P(\\mu, \\vec{x})\\ p(\\vec{x}| \\mu),\n\\end{equation}\nwhere\n\\begin{equation}\nP(\\mu, \\vec{x}) = \\frac{(\\sqrt{2\\pi}\\sigma)^n}{2\\sqrt{2}\\sigma\\ \\Gamma(n\/2)}\\ \n\\left[ \\sum_i^n \\frac{(x_i - \\mu)^2}{2\\sigma^2} \\right]^{n\/2 - 1\/2}.\n\\end{equation}\nIn general the factor $K$ is not equal to $1$ as for multiple different data points it is not feasible\nto find $\\alpha=0$. However, the factor $K$ can easily be computed by using the property that\n$\\int {\\rm d} x_2 \\ \\left< p(x_2 | \\vec{x} ) \\right>_c = 1$.\n\nIn the case of a single data point ($n=1$) one finds\n\\begin{equation}\n\\left< p(x_2 | x_1 ) \\right>_c = \\int \\rm{d}\\mu\\ p(x_2| \\mu)\\ P(\\mu, x_1)\\ p(x_1| \\mu),\n\\end{equation}\nwhere\n\\begin{equation}\nP( \\mu | x_1) = 1.\n\\end{equation}\nThe result above is similar to what one could obtain in Bayesian statistics but please do note that\nin case of multiple data points that $P$ does contain a dependence on the initial data $x_1$.\n\n\n\\subsection{Normal distribution with unknown $\\sigma$}\n\n\nThe treatment of this case is quite similar to the previous case. One finds\n\\begin{equation}\np(x_2 | \\vec{x} ) = \\frac{1}{K} \\int \\rm{d}\\sigma\\ p(x_2| \\sigma)\\ P(\\sigma , \\vec{x})\\ p(\\vec{x}| \\sigma),\n\\end{equation}\nwhere\n\\begin{equation}\nP(\\sigma,\\vec{x}) = \\frac{(\\sqrt{2\\pi}\\sigma)^n}{\\Gamma(n\/2)}\n\\left[ \\sum_i^n \\frac{(x_i-\\mu)^2}{2 \\sigma^2}\\right]^{n\/2-1}\\ \n\\left| \\partial_\\sigma p(\\vec{x}| \\sigma) \\right|.\n\\end{equation}\n\nIn case of a single data point, $n=1$, one finds\n\\begin{equation}\np(x_2 | x_1 ) = \\int \\rm{d}\\sigma\\ p(x_2| \\sigma)\\ P(\\sigma, x_1)\\ p(x_1| \\sigma),\n\\end{equation}\nwhere\n\\begin{equation}\nP(\\sigma, x_1) = 2 \\frac{|x_1-\\mu|}{\\sigma}.\n\\end{equation}\nWhen comparing these results with the Bayesian approach one concludes that for $n=1$\nit is not possible to identify a prior that does not depend on the\ninitial measurements. \n\n\n\n\\section{Binomial distribution}\n\n\nIn a production process one often encounters products that are produced with a certain spread. This\nspread can be binned in certain intervals. Some bins are considered to be accepTable while other\nbins are seen as failures. A question that often occurs is whether the production process is good enough. This is usually investigated by considering a subset of the production (batch) on which\na statistical analysis is carried out with the purpose of making a claim on the total production (population).\n\nAn often used approach is to argue that (1) the products deviations follow a certain distribution,\ne.g. normal distribution, (2) a fit is being made with the batch, and (3) using the fit an estimate\nis made for the failures in the population.\nIn this section we try to estimate the expection for a certain bin while not making any\nassumptions on underlying the probability distribution. As the integrals were too difficult to\ncompute analytically the results below were obtained via computing the integrals numerically.\n\nThe binomial distribution is defined as\n\\begin{equation}\np(k|n,p) = {n\\choose k}\\ p^k\\ (1-p)^{n-k}.\n\\end{equation}\nEquation~\\ref{definition} in this case becomes\n\\begin{equation}\n\\alpha = \\int_{p(l|n,p)> p(k|n,p)} {\\rm d} l\\ p(l|n,p),\n\\end{equation}\nand for the expected probability function we get\n\\begin{equation}\n\\left< p(k_2 | k_1) \\right> = \\frac{1}{K} \\int_{N \\neq0} {\\rm d}p'\\ \\frac{{\\rm d}\\alpha}{{\\rm d} p'}\np(k_2|n,p').\n\\end{equation}\n\n\\begin{table}\n\\begin{tabular}{r|rrrr}\n\\# counts & \\# counts \n& st. dev.& \\# counts &\nst. dev. \\\\\nfound in bin& most likely \n& most likely & expected&\nexpected\\\\\n\\hline\n0 &0&0&0.89&0.49\\\\\n1 &1&.42&1.68&0.59\\\\\n2 &2 & 0.44 & 2.46 & 0.62\\\\\n3 &3 & 0.47 & 3.24 & 0.64\n\\end{tabular}\n\\caption{Most likely and expected bin values in case in total 10 samples were measured with a number of counts observed in a \ncertain bin (column 1).\\label{Table10}}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{r|rrrr}\n\\# counts & \\# counts \n& st. dev.& \\# counts &\nst. dev. \\\\\nfound in bin& most likely \n& most likely & expected&\nexpected\\\\\n\\hline\n0 &0&0 \t&1.12\t&0.57\\\\\n5 &5&\t.57\t&6.5&0.83\\\\\n10 &10 \t& 0.66 \t& 12.14 & 0.90\\\\\n15 &15 \t& 0.71 \t& 17.58 & 0.94\n\\end{tabular}\n\\caption{Most likely and expected bin values in case in total 100 samples were measured with a number of counts observed in a \ncertain bin (column 1).\\label{Table100}}\n\\end{table}\n\n\nThe integral above can be calculated numerically and results for $n=10$ and $n=100$ are given in \nTable~\\ref{Table10} and Table~\\ref{Table100}. From a batch of $n$ samples the most left column indicates the number\nthat was observed (e.g. number of products with a failure), and the other columns indicate what one\ncan expect for the population based on most-likely or weighted by confidence.\n\nFrom the Tables one can see that in case a low number is observed from a large batch ($k< = \\int_{-10}^{10} {\\rm d}a \\int_{-10}^{10}{\\rm d}b\\ \\mathcal{P}(a,b | \\vec{x} )\\ V(b).\n\\end{equation}\n\nWhen carrying out the calculations one finds in this case that the expection value for $V$ becomes\n$\\left< V \\right>=5.77$ with $\\sigma = 5.1$. This value is substantially different from the most-likely value which is what we kind of\nexpected as all deviations would result in a shift to higher values. One problem of this approach\nis the assumption of having a constant prior over a finite interval. What is we would have parametrized\nour linear relation as $y=a^3 x + b$ or what if we would have chosen the interval $[-20,20]$ instead of the interval $[-10,10]$.\nHow would our answer change and what would be the correct one?\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm,angle=0]{plot_article.png}\n\\caption{A contour plot of $\\alpha$ \\label{contour}.} \n\\end{center}\n\\end{figure}\n\nFollowing the suggested approach in this article, weighting with confidence, we can keep the definition for $\\alpha$\nfor this problem (Equation~\\ref{definition}). In this case we write $\\alpha$ as ($n=3$)\n\\begin{equation}\n\\alpha = \\frac{1}{\\Gamma (n\/2)} \\left[ \\Gamma(n\/2, 0) - \\Gamma (n\/2, - \\ln \\prod_k e^{-(ax_k+b-y_k)^2\/(2)})\n\\right],\n\\end{equation}\nwhere $x_k$ and $y_k$ are the measured data points.\n\nIn contrast to $\\alpha$, the definition for the expectation value\nneeds to be adapted as the two model parameters (\\emph{multi-variate}), $a$ and $b$, can lead to a continuum of solutions\nat a certain $\\alpha$. This means that $N$ has to be replaced. It seems natural to replace $N$ by \na line integral in $ab$ space over the contours of equal $\\alpha$. We write\n\\begin{equation}\n\\left< V \\right> = \\frac{1}{K} \\int {\\rm d}\\alpha\\ \\frac{1}{\\int_{\\alpha} {\\rm d}l_{ab}} \\ \n\\int_{\\alpha} {\\rm d}l_{ab} \\ V(b).\n\\end{equation}\nPlease note that the line integral is yet again invariant under reparametrization as\nthe contour does not change under reparametrization (defined by $\\alpha$) and (e.g.)\n\\begin{eqnarray}\n\\int_{\\alpha} {\\rm d}l_{ab} \\ f(a,b) &=& \\int_{0}^{1} {\\rm d}t \\sqrt{ (a'(t))^2 + (b'(t))^2 }\\ f(a(t),b(t)) \\\\\n&=& \\int_0^1 {\\rm d}t \\sqrt{ ({a}'(h(t)))^2 + ({b}'(h(t)))^2 }\\ f(a(h(t)),b(h(t)))\\\\\n&=&\\int_{\\alpha} {\\rm d}l_{a'b'} \\ f(a',b').\n\\end{eqnarray}\n\n \nThe integral above can be computed numerically in the following manner. \nFirst one determines $\\alpha(a,b)$ on a grid.\nFrom this grid one can use scipy.skimage.measure to find the contours at certain levels of\n$\\alpha$ on the grid (see Figure~\\ref{contour}). After having obtained the contours one can carry out the integration, e.g.\nusing the trapezium rule. Following this method we found $\\left\\approx 5.7$\nwith $\\sigma = 5.1$. This result is very similar to what was achieved with the Bayesian approach.\n\n\n\n\n\n\n\n\\section{Conclusions}\n\n\nIn this article the difference between most-likely and expectation has been discussed. Standard\nleast mean square fits lead in some cases \nto a most likely estimate for the fitted parameters. Although\nmost-likely is certainly of interest, often we are also interested in what we can expect.\n\nIn this article it is proposed to use a kind of summation over hypotheses (or integrating\/summing over \nparameters) and taking as weights the amount to which they contribute to the confidence. The whole\nidea is that model parameter intervals making an equal contribution to confidence should also\nmake an equal contribution to the expection value.\nIt was shown\nthat the resulting estimate for expectation values does not require a choice for priors. Moreover, the\nresult is also invariant under reparametrization of the model parameters.\n\nIn the case of a normal distribution with a single data point and model parameter the mean, it was found that the result is similar\nto Bayesian statistics with a prior equal to $1$. However, in general it was found that the\ncorresponding prior would depend on the initial data which cannot occur in a Bayesian approach.\n\nThe binomial distribution was also studied. This distribution can be of interest\nwhen considering production processes. While not making any assumptions on the underlying\nprobability distributions, it is nevertheless possible to make some predictions on counts in bins which\ndiffer from most-likely estimates from classical statistics.\n\nIn the last section we tried to fit some data with a linear model, $y=ax+b$, having the two parameters.\nFor this multi-variate case we extended our definition for confidence weighted expectation value. By \nusing numerical techniques an expectation value could be obtained which turned out to be close\nto a choice that was made for a prior in combination with a Bayesian approach. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\n\nMeasurements at low energies may provide interesting indirect constraints on the masses of particles that are too heavy to be produced directly.\nThis is particularly true for Flavor Changing Neutral Currents (FCNC) processes which are highly suppressed in the Standard Model (SM) and can only occur through higher order diagrams.\n \nThe SM prediction for the branching fractions ($\\mathcal{B}$) of the FCNC decays \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace and \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace~\\footnote{In these proceedings the inclusion of charge-conjugate states is implicit.} have been computed~\\cite{Buras2010} to be \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace = $(3.2 \\pm 0.2) \\times 10^{-9}$ and \\BRof \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace = $(0.10 \\pm 0.01) \\times 10^{-9}$.\nHowever New Physics (NP) contributions can significantly enhance these values.\nFor instance, within Minimal Supersymmetric extensions of the SM (MSSM), in the large $\\tan \\beta$ approximation~\\cite{MSSM}, \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace is found to be proportional to $\\sim \\tan^6\\beta$, where $\\tan\\beta$ is the ratio of the vacuum expectation values of the two neutral CP-even Higgs fields.\nTherefore it could be strongly enhanced for large values of $\\tan \\beta$.\n\n\nThe best published limits from the Tevatron at $95\\%$C.L.\\ are obtained using 6.1 fb$^{-1}$ by the D0 collaboration~\\cite{d0_PLB}, \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace $<5.1 \\times 10^{-8}$ and using 2 fb$^{-1}$ by the CDF collaboration~\\cite{cdf_PRL}, \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace $<5.8 \\times 10^{-8}$ and \\BRof \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace $<1.8 \\times 10^{-8}$.\nThe CDF collaboration has also presented a preliminary result~\\cite{cdf_bsmumu_preprint} with 6.9 fb$^{-1}$ that lowers the limit to \\BRof \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace$<6.0 \\times 10^{-9}$ at 95\\% C.L.\\ . \nIn the same dataset an excess of \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace candidates is also observed compatible with a \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace = $(1.8^{+1.1}_{-0.9}) \\times 10^{-8}$ and with an upper limit \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace $< 4.0 \\times 10^{-8}$ at 95\\,\\% C.L.\nThe CMS collaboration has recently presented a preliminary result~\\cite{cms_preprint} with 1.14\\,fb$^{-1}$ and set the limits \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace $< 1.9 \\times 10^{-8}$ and \\BRof \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace $< 4.6 \\times 10^{-9}$ at 95\\,\\% C.L.\n\nThe LHCb collaboration has obtained limits \\cite{LHCb_paper} comparable with the Tevatron ones, \\BRof \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace $<5.4 \\times 10^{-8}$ and \\BRof \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace$<1.5 \\times 10^{-8}$ at 95$\\%$ C.L.\\ based on 37 pb$^{-1}$ of luminosity collected in the 2010 run.\nThis document presents an analysis of the LHCb data taken with an integrated luminosity of 300 pb$^{-1}$ in 2011. A combination of this data with the data used for the analysis published on the 2010 data based on 37 pb$^{-1}$ is also performed.\n\n\n\n\\section{The LHCb detector}\n\\label{sec:detector}\n\nThe LHCb detector~\\cite{LHCbdetector} is a single-arm forward spectrometer with an acceptance for charged tracks with pseudorapidity $\\eta$ of $2 < \\eta < 5$.\nIt consists of a vertex locator, a warm dipole magnet, a tracking system, two RICH detectors, a calorimeter system and a muon system. \n\nTrack momenta are measured to a precision of $\\delta p \/ p$ ranging $0.35$--$0.5\\,\\%$. \nThe RICH system provides charged hadron identification in a momentum range 2--100\\ensuremath{\\,{\\rm GeV}\\!\/\\!c}\\xspace.\nTypically, kaon identification efficiencies of over 90\\% can be attained for a $\\pi \\to K$ fake rate below 10\\%.\nThe calorimeter system identifies high transverse energy ($E_{\\rm T}$) hadron, electron and photon candidates and provides information for the trigger.\nThe muon system provides information for the trigger and muon identification with an efficiency of $\\sim 95$\\,\\% for a misidentification rate of about 1--2\\,\\% for momenta above 10\\ensuremath{\\,{\\rm GeV}\\!\/\\!c}\\xspace.\n\n\\section{Analysis Strategy}\n\nThe analysis presented in this document uses about 300 pb$^{-1}$ of data collected by LHCb between March and June 2011 at $\\sqrt{s}$ = 7\\,TeV. \nAssuming the SM branching fractions and $\\sigma_{b \\overline{b}} = 75\\pm14\\,\\mu$b as measured within the LHCb acceptance~\\cite{bbxsection}, approximately $3.4~(0.32)$\n$B^0_{s}\\to \\mu^+\\mu^-$ ($B^0 \\to \\mu^+\\mu^-$) events are expected to be reconstructed and selected in this sample.\nThe general structure of the analysis is unchanged with respect to Ref.~\\cite{LHCb_paper}, but has been improved in two main points: the optimization of the multi-variate analysis discriminant using geometrical and kinematic variables and the use of the RICH particle identification (PID) in the analysis.\n\nThe selection procedure treats signal and control\/normalization channels in the same way in order to minimize systematic uncertainties.\nAfter the selection, each event is given a probability to be signal or background in a two-dimensional space defined by the invariant mass and a Boosted Decision Tree (BDT) from the TMVA package~\\cite{tmva}.\nThe combination of variables entering the BDT is optimized using Monte Carlo (MC) simulation. \nHowever the probability for a signal or background event to have a given value of the BDT is obtained from data using trigger unbiased \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace events for signal and sideband \\ensuremath{B^0_{(s)}\\to \\mu^+\\mu^-}\\xspace candidates for background\n\nThe parameters describing the invariant mass line shape of the signal are extracted from data using control samples.\nThe average mass values are obtained from \\ensuremath{\\Bd\\to K^+\\pi^-}\\xspace and \\ensuremath{\\Bs\\to K^+K^-}\\xspace exclusive samples. \nThe \\ensuremath{B^0_s}\\xspace and \\ensuremath{B^0}\\xspace mass resolutions are estimated by interpolating the ones obtained with the dimuon resonances ($J\/\\psi, \\psi(2S)$ and $\\Upsilon(1S),\\Upsilon(2S)$ and $\\Upsilon(3S)$) and cross-checked via a fit to the invariant mass distribution of the \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace inclusive decays and of the \\ensuremath{\\Bd\\to K^+\\pi^-}\\xspace exclusive decay. \n\nThe number of expected signal events is obtained by normalizing to channels of known branching fractions, \\decay{\\Bu}{\\jpsi\\ensuremath{K^+}\\xspace}\\xspace, \\decay{\\Bs}{\\jpsi\\phi}\\xspace and \\decay{\\Bd}{\\ensuremath{K^+}\\xspace\\pi^-}\\xspace that are selected in a way as similar as possible to the signals in order to minimize the systematic uncertainty due to the different phase space.\nThe probability for a background event to have a given BDT and invariant mass value is obtained by a fit of the mass distribution of events in the mass sidebands, in bins of BDT.\nThe two-dimensional space formed by the invariant mass and BDT is binned, and for each bin we compute how many events are observed in data, how many signal events are expected for a given BF hypothesis and luminosity, and how many background events are expected for a given luminosity.\nThe compatibility of the observed distribution of events in all bins with the expected one for a given BF hypothesis is computed using the modified frequentist method method~\\cite{Read_02}, which allows to exclude a given hypothesis at a given confidence level.\nIn order to avoid unconscious biases, the data in the mass region defined by $M_{\\ensuremath{B^0}\\xspace} - 60 \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace $ and $ M_{\\ensuremath{B^0_s}\\xspace} + 60 \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace$ have been blinded until the completion of the analysis.\n\n\n\\section{Event selection}\n\nThe selection aims at reducing the data size to a manageable level while keeping the efficiency on the signal as high as possible.\nThe selection consists of loose requirements on track separation from the interaction point, decay vertex quality, compatibility of the reconstructed origin of the $B$ meson with the interaction point and muon identification. \nEvents passing the selection are considered $B^0_{(s)} \\to \\mu^+ \\mu^-$ candidates if their invariant mass lies within 60\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace of the nominal $B^0_{(s)}$ mass.\n\nA similar selection is applied to the normalization channels, in order to minimize systematic errors in the ratio of efficiencies.\nThe \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace inclusive sample is selected exactly in the same way as the \\ensuremath{B^0_{(s)}\\to \\mu^+\\mu^-}\\xspace signals (apart from the muon identification requirement) in order to be used as the main control sample in the calibration of the BDT and the invariant mass. \nThe $J\/\\psi \\to \\mu \\mu$ decay in the \\ensuremath{B^+\\to J\/\\psi K^+}\\xspace and \\ensuremath{B^0_s\\to J\/\\psi \\phi}\\xspace normalization channels is selected \nexactly in the same way as \\ensuremath{B^0_{(s)}\\to \\mu^+\\mu^-}\\xspace signals.\n$K^{\\pm}$ candidates are required to pass some track quality and impact parameter cuts.\n\nFiducial cuts are applied to reject fake signal candidates. \nAll tracks from selected candidates are required to have a momentum less than 1\\,TeV\/$c$. \nFurthermore, only $B$ candidates with lifetimes less than $5 \\times \\tau_{B_{(s)}}$ are accepted for further analysis.\nDimuon candidates coming from elastic di-photon production are effectively removed by requiring a minimal transverse momentum of the $B$ candidate of 500\\ensuremath{\\,{\\rm MeV}\\!\/\\!c}\\xspace.\n\n\\section{Signal and background likelihoods}\n\\label{sec:likelihoods}\n\n\nThe discrimination of the signal from the background is achieved through the combination of two independent variables: the BDT and the invariant mass. \nThe invariant mass in the search regions ($\\pm 60\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace$ around the $B^0_{(s)}$ masses) is divided into six equal-width bins, and the BDT into four equal-width bins distributed between zero and one. \n\nThe BDT combines information related with the topology and kinematics of the event: the $B^0_{(s)}$ lifetime, the minimum impact parameter of the two muons, the distance of closest approach of the two tracks, the $B^0_{(s)}$ impact parameter and $p_T$ and the isolation of the muons with respect to the other tracks of the event, the cosine of the polarization angle, the $B$ isolation, the minimum $p_{\\rm T}$ of the two muons.\nThose variables have been chosen on the basis of the performance on MC samples.\nThe BDT output is found to be fully independent of the invariant mass both for signal and for background.\nThe BDT is trained using Monte Carlo simulations (\\ensuremath{B^0_{(s)}\\to \\mu^+\\mu^-}\\xspace for signals and \\ensuremath{b\\bar{b}\\to \\mu \\mu X}\\xspace for background) and calibrated with data using $B^0_{(s)}\\to h^+h^{'-}$ selected as the signal events and triggered independently on the signal in order to avoid trigger bias. \n\n\n\\subsection{Calibration of the invariant mass likelihood for signal}\n\nThe invariant mass shape for the signal is parametrized as a Crystal Ball function. \nThe mass average value and the mass resolution are calibrated from data and the transition point of the radiative tail is taken from Monte Carlo simulation. \n\n\nThe average mass values are extracted from \\ensuremath{\\Bd\\to K^+\\pi^-}\\xspace and \\ensuremath{\\Bs\\to K^+K^-}\\xspace exclusive candidates\nand the results are the following:\n\\begin{eqnarray}\n m(\\ensuremath{B^0_s}\\xspace) & = & (5358.0 \\pm 1.0) \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace\\nonumber \\\\ \n m(\\ensuremath{B^0}\\xspace) & = & (5272.0 \\pm 1.0) \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace. \\nonumber\n\\end{eqnarray}\nThe mass resolutions are extracted from data by a linear interpolation between the measured resolution of charmonium and bottomonium resonances decaying to two muons (\\Jpsi, $\\psi$(2S) and $\\Upsilon(1S)$, $\\Upsilon(2S)$, $\\Upsilon(3S)$) as shown in Fig.~\\ref{fig:dimuon_reso}.\nThis result has been cross-checked with the mass resolution obtained via full fit to the \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace inclusive decay and the \\ensuremath{\\Bd\\to K^+\\pi^-}\\xspace exclusive decay. \nThe interpolation yields $\\sigma(\\ensuremath{B^0_s}\\xspace) = (24.6 \\pm 0.2_{\\rm stat} \\pm 1.0_{\\rm syst}) \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace$.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.48\\textwidth]{Charmonia.pdf\n\\includegraphics[width=.48\\textwidth]{Bottomia.pdf\n\\vspace{-4mm}\n\\caption{Fits to the dimuon invariant mass spectrum in the $[2.9, 3.9]\\ensuremath{\\,{\\rm GeV}\\!\/\\!c^2}\\xspace$ range (left) and in $[9,11]\\ensuremath{\\,{\\rm GeV}\\!\/\\!c^2}\\xspace$ (right).}\n\\label{fig:dimuon_reso}\n\\end{figure}\n\n\n\\subsection{Calibration of the BDT likelihood for signal}\n\nThe BDT shape for signal events is calibrated using a \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace inclusive sample composed only of events {\\it Triggered Independently of the Signal} (TIS)\\footnote{Here we consider events that are TIS with respect to the two first trigger levels (L0 and HLT1) but accept all HLT2 triggers. This is due to the lack of statistics for full (L0 $\\times$ HLT1 $\\times$ HLT2) TIS events. Corrections are computed from Monte Carlo simulations and cross-checked on control channels.} to avoid biases introduced by the hadronic trigger.\nThe number of \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace signal events in each BDT bin is extracted by fitting the $hh$ invariant mass distribution in the $\\pi\\pi$ mass hypothesis~\\cite{Bhhnote}.\nFigure~\\ref{fig:b2hh_fitinbins} depicts fits to the mass distribution of the full sample and for the three highest BDT bins for \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace TIS events is shown.\nThose yields are used to compute fractional yield of events as a function of the BDT bin as shown on Fig.~\\ref{fig:BDT_all}.\nCross-checks consist of one-dimensional fits to the inclusive sample and of fits to $B^0_{(s)} \\to KK,K\\pi, \\pi\\pi$ exclusive samples selected using the $K - \\pi$ separation capability of the RICH system.\nThe maximum spread among the different methods has been used as a systematic error in the fractional yield determination.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.42\\textwidth]{b2hh_allbins_300pb_new.pdf}\n\\includegraphics[width=.42\\textwidth]{b2hh_bin2_300pb_new.pdf}\n\\includegraphics[width=.42\\textwidth]{b2hh_bin3_300pb_new.pdf}\n\\includegraphics[width=.42\\textwidth]{b2hh_bin4_300pb_new.pdf}\n\\vspace{-4mm}\n\\caption{Full fit to the invariant mass distributions of \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace candidates \nin the $\\pi\\pi$ mass hypothesis for the whole sample (top left) and for the samples\nin bins 2,3,4 of the BDT (top right, bottom left, bottom right).}\n\\label{fig:b2hh_fitinbins}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.45\\textwidth]{BDT_sig_bkg_distribution.pdf\n\\vspace{-4mm}\n \\caption{Probability of signal events in bins of BDT obtained from the inclusive sample of TIS \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace events (solid squares). \nThe background probability (open circles) is obtained from the events in the sidebands.} \n\\label{fig:BDT_all}\n\\end{figure}\n\n\\subsection{Calibration of the combinatorial background Likelihood}\n\n\nThe BDT and invariant mass shapes for the combinatorial background inside the signal regions are calibrated with data by interpolating the number of expected events using fit to the invariant mass sidebands in each bin of BDT.\nThe lower sideband is potentially polluted by two contributions other than combinatorial background that can affect the fit: cascading $b \\to c \\mu \\to \\mu \\mu X$ decays below 4900\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace and \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace events with the two hadrons misidentified as muons above 5000\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace. \nThus the model used to extract the expected number of background events is chosen to be a single exponential fitted to the events in the region [4900, 5000]\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace and in the upper sideband [5418, 6000]\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace.\n\nIn Fig.~\\ref{fig:BDT_bkg_bins} the invariant mass distribution for events in each BDT bin is shown and different fit models used for estimating the expected number of background events in the signal regions are superimposed. \nThe fraction of combinatorial background expected per BDT bins is depicts on Fig.~\\ref{fig:BDT_all}.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.42\\textwidth]{BDTflat_Root_1_600_20110901.pdf}\n\\includegraphics[width=.42\\textwidth]{BDTflat_Root_2_600_20110901.pdf}\n\\includegraphics[width=.42\\textwidth]{BDTflat_Root_3_600_20110901.pdf}\n\\includegraphics[width=.42\\textwidth]{BDTflat_Root_4_600_20110901.pdf}\n\\vspace{-4mm}\n\\caption{\nDistribution of the $\\mu\\mu$ invariant mass for events in each BDT bin.\nThe curves show the models used to fit the sidebands and extract the expected number \nof background events in the signal regions.\nRed: linear fit in [4672,5212] \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace and [5418,6000]\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace sidebands.\nBlue: single exponential fit in the same ranges as the red curve.\nGreen: single exponential fit in the region [4900,5000] \\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace and [5418,6000]\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace.\nMagenta: single exponential fit using only the upper sideband ([5418,6000]\\ensuremath{\\,{\\rm MeV}\\!\/\\!c^2}\\xspace).\nThe green curve defines the model and ranges used in the analysis.\nThe \\ensuremath{B^0_s}\\xspace and \\ensuremath{B^0}\\xspace search windows, delimited by the vertical green and orange lines, \nhave not been considered in the fitting procedure.}\n\\label{fig:BDT_bkg_bins}\n\\end{figure}\n\n\n\\subsection{Peaking background Likelihood}\n\nThe peaking background coming from \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace events for which the two hadrons in the final states are misidentified as muons has been evaluated from data to be $N_{\\ensuremath{B^0_s}\\xspace} = (0.5\\pm 0.4)$ events in the \\ensuremath{B^0_s}\\xspace mass window and $N_{\\ensuremath{B^0}\\xspace} = (2.5 \\pm 2.0)$ events in the \\ensuremath{B^0}\\xspace one in the whole BDT range.\nThe mass lineshape of the peaking background is obtained from a MC sample of double misidentified \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace events and normalized to the number of events expected in the two search windows from data.\n\n\n\n\\section{Normalization}\n\\label{sec:norm_factors}\n\n\n\\begin{figure}[t] \n\\begin{center}\n \\includegraphics*[width=0.44\\textwidth]{300pb_BuNorm_Bpt.pdf}\n \\includegraphics*[width=0.44\\textwidth]{300pb_BsNorm_Bpt.pdf}\n\\vspace{-4mm}\n \\caption\n {Invariant mass distribution of the \\ensuremath{B^+\\to J\/\\psi K^+}\\xspace (left) and \\ensuremath{B^0_s\\to J\/\\psi \\phi}\\xspace (right) candidates used in the normalization procedure.} \n \\label{fig:num_Bu}\n\\end{center}\n\\end{figure}\n\n\nThe number of expected signal events is evaluated by normalizing with three complementary channels of known BF, $B^+\\to \\ensuremath{J\\!\/\\!\\psi}\\xspace K^+$, $B^0_s\\to \\ensuremath{J\\!\/\\!\\psi}\\xspace \\phi$ and $B^0 \\to K^+\\pi^-$.\nThe first two decays have similar trigger and muon identification efficiency to the signal but a different number of final-state particles, while the third channel has the same two-body topology but cannot be efficiently selected with the muon triggers. \n\n\nThe number of observed events $N_{\\ensuremath{\\ensuremath{B^0_{q}}\\to\\mu^+\\mu^-}\\xspace}$ is translated into a BF via the relation:\n\\begin{eqnarray}\n\\mathcal{B} &=& {\\mathcal{B}_{\\rm norm}}\\times\\frac{\\rm\n\\epsilon_{norm}^{REC}\n\\epsilon_{norm}^{SEL|REC}\n\\epsilon_{norm}^{TRIG|SEL}\n}{\\rm\n\\epsilon_{sig}^{REC}\n\\epsilon_{sig}^{SEL|REC}\n\\epsilon_{sig}^{TRIG|SEL}\n}\\times\\frac{f_{\\rm norm}}{f_{\\ensuremath{B^0_q}\\xspace}}\n\\times\\frac{N_{\\ensuremath{\\ensuremath{B^0_{q}}\\to\\mu^+\\mu^-}\\xspace}}{N_{\\rm norm}} = \\alpha_{\\rm norm} \\times N_{\\ensuremath{\\ensuremath{B^0_{q}}\\to\\mu^+\\mu^-}\\xspace}\\,, \\nonumber\n\\label{eq:normAlpha}\n\\end{eqnarray}\nwhere $f_{\\ensuremath{B^0_q}\\xspace}$ and $f_{\\rm norm}$ are the probabilities that a $b$-quark fragments into a $\\ensuremath{B^0_q}\\xspace$ and into the $b$-hadron relevant for the chosen calibration mode. \nWe use LHCb measurement for $f_d\/f_s = 3.745 \\pm 0.295$~\\cite{fdfs}.\n${\\cal B}_{\\rm norm}$ is the branching fraction and $N_{\\rm norm}$ the number of selected events of the normalization channel. \nThe efficiency is separated into three factors: $\\epsilon^{\\rm REC}$ is the efficiency to reconstruct all the tracks of the decay including the geometrical acceptance of the\ndetector; $\\epsilon^{\\rm SEL|REC}$ is the efficiency to select the events which have been reconstructed; $\\epsilon^{\\rm TRIG|SEL}$ is the efficiency of the trigger on reconstructed and selected events. \nThe ratios of reconstruction and selection efficiencies are estimated from simulations and checked with data, while the ratios of trigger efficiencies on selected events are determined from data~\\cite{TisTos}.\nFinally, $\\alpha_{\\rm norm}$ is the normalization factor.\n\n\nEach $N_{\\rm norm}$ is obtained with a fit to the invariant mass distribution of the normalization channel.\nThe invariant mass distributions for \\ensuremath{B^+\\to J\/\\psi K^+}\\xspace and \\ensuremath{B^0_s\\to J\/\\psi \\phi}\\xspace are shown in Fig.~\\ref{fig:num_Bu}, while the \\decay{\\Bd}{\\ensuremath{K^+}\\xspace\\pi^-}\\xspace yield is obtained via the \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace full fit as pictured in Fig.~\\ref{fig:b2hh_fitinbins} top left.\nThe normalization factors calculated using the three channels give compatible results; the final normalization factors are weighted averages and read $\\alpha_{B^0_{(s)} \\to \\mu\\mu}= (9.84 \\pm 0.91) \\times 10^{-10}$ and $\\alpha_{B^0 \\to \\mu\\mu}= (2.59 \\pm 0.15) \\times 10^{-10}$.\n\n\\section{Results}\n\\label{sec:results}\n\n\nThe expected number of combinatorial background events,\nthe number of peaking background events as well as the number of expected signal events assuming a given BF hypothesis are computed for each 24 bins.\nThe distribution of the invariant mass in the four BDT bins is shown in Fig.~\\ref{fig:fondo_bs} and~\\ref{fig:fondo_bd} for \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace candidates and \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace candidates, respectively. The observed events are in good agreement with background expectations.\nFor \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace the observed events are also in good agreement with the background expectations and the presence of \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace events according to SM predictions.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics*[width=0.81\\textwidth]{PlotBs.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{\\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace: distribution of selected dimuon events in the invariant mass plane for the four BDT bins. \nThe black dots are data, the light blue histogram shows the contribution of the combinatorial background, the green histogram shows the contribution of the \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace background and the red filled histogram the contribution of \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace signal events according to the SM rate.\nThe uncertainty on the expectations is shown by the hatched area. \nThe uncertainty on data in the first BDT bin is smaller than the size of the dots.}\n\\label{fig:fondo_bs}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics*[width=0.81\\textwidth]{PlotBd.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{\\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace: distribution of selected dimuon events in the invariant mass plane for the four BDT bins. \nThe black dots are data, the light blue histogram shows the contribution of the combinatorial background and the green histogram shows the contribution of the \\ensuremath{B^0_{q}\\to h^+h^{'-}}\\xspace background.\nThe uncertainty on the expectations is shown by the hatched area. \nThe uncertainty on data in the first BDT bin is smaller than the size of the dots.}\n\\label{fig:fondo_bd}\n\\end{figure}\n\n\nThe compatibility of the distribution of events in the invariant mass {\\it vs} BDT plane with a given branching fraction hypothesis is evaluated using the \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace method~\\cite{Read_02}. \nThis provides two estimators: \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace is a measure of the compatibility of the observed distribution with the signal hypothesis, while \\ensuremath{\\textrm{CL}_{\\textrm{b}}}\\xspace is a measure of the compatibility with the background-only hypothesis. \n\nFor the \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace decay, the distributions of expected \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace values are shown as dashed (black) lines in Fig.~\\ref{fig:CLsBsmm_bkg} (left) under the hypothesis to observe background plus SM events.\nThe green shaded areas cover the region of $\\pm 1 \\sigma$ of compatible observations.\nThe observed \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace as a function of the assumed branching fraction is shown as dotted (blue) line.\nFor the \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace decay, the distribution of possible values of \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace is shown as dashed (black) line in Fig.~\\ref{fig:CLsBsmm_bkg} (right) under the hypothesis to observe background events only.\nThe observed \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace as a function of the assumed branching fraction is shown as a dotted (blue) line.\n\n\nThe upper limits read:\n\\begin{align}\n\\BRof\\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace &< 1.3~(1.6)\\times10^{-8}~{\\rm at}~90\\,\\% ~(95\\,\\%)~{\\rm C.L.,} \\nonumber \\\\\n\\BRof\\ensuremath{B^0\\to\\mu^+\\mu^-}\\xspace &< 4.2~(5.2)\\times10^{-9}~{\\rm at}~90\\,\\% ~(95\\,\\%)~{\\rm C.L.,} \\nonumber\n\\end{align} \nwhile the expected values of the limits are $\\BRof\\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace < 1.2~(1.5)\\times10^{-8}$ under the hypothesis to observe background plus SM events and $\\BRof\\ensuremath{B^0\\to\\mu^+\\mu^-}\\xspace < 2.4~(3.1) \\times 10^{-9}~{\\rm at}~90\\,\\% ~(95\\,\\%)$ C.L.\\ under the hypothesis to observe background events only.\n\n\\FloatBarrier\n\nThe comparison of the observed distribution of events with the expected background distribution results in a p-value (1-\\ensuremath{\\textrm{CL}_{\\textrm{b}}}\\xspace) of 20\\,\\% (21\\,\\%) for the \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace (\\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace) decays.\nIn the case of \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace, the slightly low p-value is due to an excess of the observed \nevents in the most sensitive BDT bin with respect to the background expectations. \nA larger data sample will allow to clarify the situation. \nIn the case of \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace, when a signal is included at the level expected in the Standard Model the p-value increases to 50\\,\\%.\n\n\nA limit on \\BRof\\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace has been extracted after combining the analysis on 300\\ensuremath{\\,{\\rm pb}^{-1}}\\xspace of 2011 data with the published analysis~\\cite{LHCb_paper} on 37\\ensuremath{\\,{\\rm pb}^{-1}}\\xspace of 2010 data; the upper limits improve to:\n\\begin{eqnarray}\n\\BRof{\\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace} (2010+2011) &<& 1.2 (1.5) \\times 10^{-8}~{\\rm at}~90\\,\\% (95\\,\\%)~{\\rm C.L.} \\nonumber\n\\end{eqnarray} \nThe above 95\\% C.L.\\ upper limits are still about 4.7 times above the SM predictions for the \\ensuremath{B^0_s}\\xspace and 51 times for the \\ensuremath{B^0}\\xspace.\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.495\\textwidth]{BsSysCLsSmData2.pdf}\n\\includegraphics[width=0.495\\textwidth]{BdSystCLsBkgData_final.pdf}\n\\vspace{-4mm}\n\\caption\n{Expected distributions of possible values of \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace (dashed black line) under the hypothesis to observe a combination of background and signal events according to the SM rate in the case of \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace (left) and under the hypothesis to observe background only in the case of \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace (right).\nOn both graphs, the green areas cover the region of $\\pm 1 \\sigma$ of compatible observations.\nThe observed distributions of \\ensuremath{\\textrm{CL}_{\\textrm{s}}}\\xspace as a function of the assumed branching fraction are shown as dotted blue lines for \\ensuremath{\\Bs\\to\\mu^+\\mu^-}\\xspace (left) and \\ensuremath{\\Bd\\to\\mu^+\\mu^-}\\xspace (right).\nThe measured upper limits at 90\\% and 95\\% C.L.\\ are also shown.}\n\\label{fig:CLsBsmm_bkg}\n\\end{figure} \n\n\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpcfz b/data_all_eng_slimpj/shuffled/split2/finalzzpcfz new file mode 100644 index 0000000000000000000000000000000000000000..d08bb0e6e8e0e570320828034ab4d4db52ef4d12 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzpcfz @@ -0,0 +1,5 @@ +{"text":"\\chapter*{Introduction}\n\n\nA classical method to study algebraic surfaces of general type is to consider\nthe behaviour of pluricanonical maps (see e.g. the important paper \\cite{Bo}).\nIf $\\S$ is a (smooth, projective) minimal surface of general type (over an\nalgebraically closed field $\\K$), denoting by $K_{\\S}$ the canonical divisor on\n$\\S$ and by $\\ds_{\\S}=\\so_{\\S}(K_{\\S})$ the canonical sheaf, the $m^{th}$\ncanonical map is the rational map $\\varphi_{\\linsys{mK_{\\S}}}:\\S\\rat\\P^N$\ndetermined by the linear system $\\linsys{mK_{\\S}}$. More generally, one can\nconsider a map into a weighted projective space, induced by sections of\ndifferent multiples of $K_{\\S}$. More precisely, given positive numbers\n$\\w=(\\w_0,\\dots,\\w_n)$, we denote by $\\P(\\w)$ (or simply by $\\P$) the weighted\nprojective space of weights $\\w$, which is defined as $\\proj\\p$, where\n$\\p=\\p(\\w)$ is the polynomial ring $\\K[x_0,\\dots, x_n]$ graded by\n$\\deg(x_i)=\\w_i$. Then the choice of sections $\\sigma_i\\in\nH^0(\\S,\\ds_{\\S}^{\\w_i})$ determines a rational map $\\ph:\\S\\rat\\P$, called a\n{\\em weighted canonical projection}. $\\ph$ is called {\\em good birational} if\nit is a morphism which is birational onto its image $\\Y\\subset\\P$. Such a map\nalways factors through the canonical model $\\X$ of $\\S$ (which is defined as\n$\\proj\\R$, where $\\R:=\\bigoplus_{d\\ge0}H^0(\\S,\\ds_{\\S}^d)$ is the canonical\nring of $\\S$):\n\\[\\xymatrix{\\S \\ar[rr]^{\\ph}\\ar[dr]_{\\pr} & & \\Y\\subset\\P \\\\\n& \\X \\ar[ur]_{\\ps}}\\]\n$\\X$ is a surface with only rational double points as singularities (the\nbirational morphism $\\pr$ contracts $(-2)$--curves of $\\S$), which essentially\nbehaves like $\\S$. The advantage of considering $\\ps$ instead of $\\ph$ is that\nit is a finite morphism, induced by the morphism of graded rings\n\\[\\begin{split}\n\\rh:\\p(\\w) & \\to \\R \\\\\nx_i & \\mapsto \\sigma_i\n\\end{split}\\]\nIt follows from known results on pluricanonical maps that every surface of\ngeneral type admits a {good birational weighted canonical projection} to a $3$--dimensional weighted projective space\nwith weights $\\w_0,\\w_1\\le2$, $\\w_2\\le3$ and $\\w_3\\le5$.\n\nSo let's assume that $\\ph:\\S\\to\\P$ is a {good birational weighted canonical projection} with $\\dim\\P=3$. If $\\S$ is\nregular (i.e. $q(\\S):=h^1(\\S,\\so_{\\S})=0$) good results can be obtained by\nlooking for (graded) free resolutions of the canonical ring $\\R$, regarded as a\ngraded $\\p$--module via the morphism $\\rh$ defined before. In \\cite{C1} it is\nproved that $\\R$ always admits a length $1$ resolution of the form\n\\[0\\to\\bigoplus_{j=1}^h\\p(l_j-\\sw{\\w}-1)\\mor{\\alpha}\\bigoplus_{i=1}^h\\p(-l_i)\n\\to\\R\\to0\\]\n(with $\\sw{\\w}:=\\sum_{i=0}^3\\w_i$) where $\\alpha$ is a minimal symmetric matrix\nof homogeneous polynomials $\\alpha_{i,j}\\in\\p_{\\sw{\\w}+1-l_j-l_i}$ for some\nintegers $0=l_1\\sw{\\w}$. Moreover, $\\alpha$ satisfies the\nfollowing properties:\n\\begin{enumerate}\n\n\\item $\\f:=\\det(\\alpha)$ is an irreducible homogeneous polynomial defining\n$\\Y=\\proj\\p\/(\\f)\\subset\\P$;\n\n\\item (rank condition) denoting by $\\beta$ the matrix of cofactors of $\\alpha$,\n\\[(\\beta_{1,i}\\st i=1,\\dots,h)=(\\beta_{i,j}\\st i,j=1,\\dots,h)\\]\nas ideals of $\\p$.\n\\end{enumerate}\n\nThis last condition is seen to be equivalent to the fact that $\\cok\\alpha$\ncarries a natural structure of $\\p\/(\\f)$--algebra (``rank condition =\nring condition''). Then the important fact is that, conversely, if $\\alpha$ is\na minimal symmetric matrix satisfying the two conditions above (so that\n$\\cok\\alpha$ is a ring) and if moreover $\\proj(\\cok\\alpha)$ is a surface\nwith only rational double points as singularities, then $\\cok\\alpha$ is the\ncanonical ring of a surface of general type.\n\nUsing this result Catanese could prove interesting\napplications to moduli problems: for instance, he showed that the surfaces with\nnumerical invariants $K^2=7$, $p_g=4$ (and $q=0$, of course) such that the\ncanonical map is base point free, have a unirational irreducible moduli space.\n\nThe reason why this method works well only for regular surfaces is that only\nin this case is $\\R$ a Gorenstein ring (hence it has a resolution of length $1$\nas graded $\\p$--module). On the other hand, a closely related\napproach is possible also in the case of irregular surfaces: the idea is to\nlook for locally free resolutions of the coherent sheaf $\\R\\sh{}\\cong\n{\\ph}_*\\so_{\\S}$ on $\\P$. In particular, for projections into $\\P^3$ the\nfollowing theorem is proved in \\cite{CS} (see also \\cite{C2}).\n\n \\begin{CatSch}\nLet $\\S$ be a minimal surface of general type with $p_g(\\S)=p_g$, $q(\\S)=q$ and\n$K_{\\S}^2=K^2$, let $\\ph:\\S\\to\\Y\\subset\\P^3$ be a good birational canonical\nprojection, and consider the vector bundle\n\\[\\s{E}:=\\so(-2)^{p_g-q+K^2-9}\\oplus\\sd^1(-1)^q\\oplus(\\sd^2)^{p_g-4}\\]\non $\\P^3$. Then there exists an exact sequence of coherent sheaves on $\\P^3$\n\\[0\\to(\\so\\oplus\\s{E})\\dual(-5)\\mor{\\alpha=\\begin{pmatrix}\n\\alpha^{(1)} \\\\\n\\alpha'\n\\end{pmatrix}}\n\\so\\oplus\\s{E}\\to{\\ph}_*\\so_{\\S}\\to0,\\]\nsuch that $\\alpha$ is a minimal morphism which satisfies the following\nproperties:\n\\begin{enumerate}\n\n\\item $\\alpha$ is symmetric, i.e. $\\alpha=\\alpha\\dual(-5)$;\n\n\\item $\\f=\\det(\\alpha)$ is an irreducible polynomial (defining $\\Y=\n\\proj\\p\/(\\f)$);\n\n\\item $\\s{I}_r(\\alpha)=\\s{I}_r(\\alpha')$ (where $r:=\\rk{\\s{E}}$ and, if $\\beta:\n\\s{E}_1\\to\\s{E}_2$ is a morphism of vector bundles, $\\s{I}_k(\\beta)$ denotes\nthe sheaf of ideals defined by the image of the natural map\n$\\Lambda^k(\\s{E}_1)\\otimes\\Lambda^k(\\s{E}_2)\\dual\\to\\so_{\\P^3}$ induced by\n$\\Lambda^k(\\beta)$);\n\n\\item $\\X:=\\Spec(\\cok{\\alpha})$ is a surface with only rational double points\nas singularities.\n\\end{enumerate}\n\nConversely, if $\\alpha:(\\so\\oplus\\s{E})\\dual(-5)\\to\\so\\oplus\\s{E}$ is minimal\nand satisfies properties $1$, $2$, $3$, then $\\cok\\alpha$ is in a natural way a\ncoherent sheaf of commutative $\\so_{\\Y}$--algebras, and if $4$ also holds,\nthen, denoting by $\\pr:\\S\\to\\X$ a minimal resolution of singularities of $\\X$,\n$\\S$ is a minimal surface of general type with $p_g(\\S)=p_g$, $q(\\S)=q$ and\n$K_{\\S}^2=K^2$, $\\X$ is its canonical model and $\\ph:\\S\\mor{\\pr}\\X\\to\\Y$ is a\ngood birational canonical projection.\n \\end{CatSch}\n\nThe main purpose of this work is to extend the above result to the weighted\ncase. Before we can say how this is done, we have to discuss briefly the main\nelements in the proof of the above theorem. The essential ingredient is a\nfamous theorem by Beilinson (\\cite{B1}), which (in its more concrete version)\ngives, for every coherent sheaf $\\s{F}$ on $\\P^n$, a two--sided resolution\n(i.e., a complex $\\cp{\\brd}=\\cp{\\brd}(\\s{F})$ whose cohomology is $\\s{F}$ in\nposition $0$ and $0$ elsewhere) in terms of the vector bundles of twisted\ndifferential forms $\\sd_{\\P^n}^j(j)$. One can always take a minimal such\nresolution, and then it is explicitly given by\n\\[\\brd^i=\\bigoplus_{0\\le j\\le n}\\sd^j(j)\\otimes_{\\K}\nH^{i+j}(\\P^n,\\s{F}(-j)).\\] In our case Beilinson's theorem is applied to the\ncoherent sheaf $\\s{F}:= {\\ph}_*\\so_{\\S}(2)$ on $\\P^3$: the complex\n$\\cp{\\brd}(\\s{F})(-2)$ is then a resolution of ${\\ph}_*\\so_{\\S}$, which is\neasily seen to be quasi--isomorphic to one of the form $0\\to\n(\\so\\oplus\\s{E})\\dual(-5)\\mor{\\alpha}\\so\\oplus\\s{E}\\to0$ (with $\\alpha$ not\nnecessarily symmetric). The fact that $\\alpha$ can be actually chosen to be\nsymmetric is a consequence of the birationality of $\\ph$. Then the rest of the\nproof is rather straightforward, using a result which says that (under quite\ngeneral assumptions) the condition $\\s{I}_r(\\alpha)= \\s{I}_r(\\alpha')$ (which\nis just the sheafified version of the rank condition already discussed for\nmodules) is equivalent to the fact that $\\cok\\alpha$ has a natural structure of\na sheaf of commutative $\\so_{\\Y}$--algebras .\n\nThe principal difficulties in trying to extend both theorems to the weighted\ncase reside in the fact that weighted projective spaces are singular varieties,\npresenting several pathologies: in particular, the sheaves of the form $\\so(i)$\nor $\\sd^j(i)$ are not locally free in general. The technique used to overcome\nsuch problems is to consider, instead of the usual category of sheaves of\nmodules on $\\P$, a category of graded sheaves, which we are now going to\ndescribe, and which turns out to be much better behaved (a similar method is\nused in \\cite{B} and \\cite{GL}, where however a different graduation is used).\n\nLet $R=\\bigoplus_{d\\ge0}R_d$ be a noetherian positively graded ring ($R=\\p$ and\n$R=\\R$ are the examples we are interested in): one can consider a ``graded\nscheme'' $\\gproj R$ (we will write $\\gP$ for $\\gproj\\p$), which is equal to\n$\\proj R$ as a topological space, but which has a $\\Z$--graded structure sheaf\n\\[\\so_{\\gproj R}:=\\bigoplus_{d\\in\\Z}\\so_{\\proj R}(d).\\]\nIn other words, $\\gproj R$ is defined as $\\proj R$, except that in the\ndefinition of the structure sheaf one takes as stalk at a point $\\I{p}$\n($\\I{p}$ is a homogeneous, non maximal prime ideal of $R$) the whole local\n$\\Z$--graded ring $R_{\\I{p}}:=S^{-1}R$ (where $S$ denotes the set of\nhomogeneous elements of $R\\setminus\\I{p}$), instead of its degree $0$ part\n$R_{(\\I{p})}$. It is clear that there is a natural notion of graded sheaf of\nmodules on $\\gproj R$; in particular, if $M$ is a graded $R$--module, one\ncan consider the associated graded sheaf $M\\gsh{}:=\\bigoplus_{d\\ge0}M(d)\\sh{}$\n(as before, the stalk at $\\I{p}$ of $M\\gsh{}$ is $M_{\\I{p}}$ instead of\n$M_{(\\I{p})}$). Denoting $\\gproj R$ by $Z$ and $\\proj R$ by $Z_0$, we will\nuse the (abelian) category $\\gsmo{Z}$ of graded sheaves of $\\so_Z$--modules,\ninstead of the usual category $\\smo{Z_0}$ of sheaves of $\\so_{Z_0}$--modules\n(similar considerations can be made for the subcategories of quasi--coherent\nand coherent sheaves, of which natural definitions can be given also in the\ngraded case). The two categories are in any case closely related: sending a\ngraded sheaf $\\s{F}$ to $\\s{F}_0$ clearly defines an exact functor\n\\[\\usmf{Z}:\\gsmo{Z}\\to\\smo{Z_0}.\\]\nMoreover, it is easy to see that there exists (not unique) a functor\n\\[F:\\smo{Z_0}\\to\\gsmo{Z}\\]\nsuch that $\\usmf{Z}\\comp F\\cong{\\rm id}$. This is important for us, because if\n$\\s{G}$ is a sheaf on $Z_0$ and $\\cp{\\s{C}}$ is a resolution of $F(\\s{G})$,\nthen $\\cp{\\s{C}}_0$ will be a resolution of $\\s{G}$. One can also prove that\n$\\usmf{Z}$ is an equivalence if and only if $\\forall\\I{p}\\in\\gproj R$ the graded\nring $R_{\\I{p}}$ contains an invertible element of degree $1$. This is always\nthe case if $R=\\R$ (the canonical ring of a surface of general type), whereas\nif $R=\\p(\\w)$ this condition is satisfied if and only if $\\w=(1,\\dots,1)$.\n\nAs we mentioned before, the category $\\gsmo{Z}$ is usually nicer than\n$\\smo{Z_0}$, mainly because in the graded case there is a natural way to define\ntwist functors: if $\\s{F}$ is a graded sheaf and $d\\in\\Z$, one sets, as usual,\n$\\s{F}(d)_i:=\\s{F}_{d+i}$. These functors satisfy all the good properties one\nexpects: they are exact, $(d)\\comp(d')=(d+d')$, $(d)$ is isomorphic to\n$-\\otimes_{\\so_Z}\\so_Z(d)$, if $M$ is a graded $R$--module then $M(d)\\gsh{}\\cong\nM\\gsh{}(d)$, and so on. On the contrary, there is not a good notion of twist\nfunctors on $Z_0$ in general (for instance when $Z_0=\\P(\\w)$ is not isomorphic\nto a $\\P^n$): one can define $(d)$ as $-\\otimes_{\\so_{Z_0}}\\so_{Z_0}(d)$ or as\n$\\sHom_{\\so_{Z_0}}(\\so_{Z_0}(-d),-)$ (which is usually better), but in either\ncase they are not well behaved. Moreover, the sheaves $\\so_Z(j)$ are always\nlocally free by definition.\n\nIt was pointed out by the referee that (at least when $R_0$ is a\nfield) our category of graded (quasi)coherent sheaves on $Z$ is\nequivalent to the category considered in \\cite{AKO} (which appeared on\nthe web after the submission of the present paper) of (quasi)coherent\nsheaves on the quotient algebraic stack of $\\spec R\\setminus\\{R_+\\}$ by the action\nof the multiplicative group $\\mult$ induced by the graduation of $R$.\n\nThe use of graded sheaves allows us to prove a version of Beilinson's theorem\non a (graded) weighted projective space $\\gP=\\gP(\\w)$, which is much better\nthan the one of \\cite{Ca1}. As in the case of $\\P^n$, we prove it first in an\nabstract form (as an equivalence between the bounded derived category of graded\ncoherent sheaves and a suitable homotopy category of graded modules, see\n\\ref{wbeilthm}; similar results are also proved in \\cite{B} and\n\\cite{AKO}) and then give the explicit form of the minimal resolution of\nevery (graded coherent) sheaf, or, more generally, of every bounded complex of\nsheaves (\\ref{wbeilfor}). Such a resolution involves, besides the sheaves\n$\\sd_{\\gP}^j(j)$ (which can be defined in a natural way, and which are locally\nfree), also the sheaves $\\so_{\\gP}(j)$ for $n-\\sw{\\w}0$).\n \\end{defi}\n\n \\begin{lemm}\\label{qtrivcrit}\nA graded ring $A$ is {quasi--trivial} if and only if $A_{\\ne0}\\subset\\I{p}$ $\\forall\\I{p}\n\\in\\gspec A$.\n \\end{lemm}\n\n \\begin{proof}\nIt is an immediate consequence of the following algebraic fact: if $\\I{a}$ is\na homogeneous ideal of a graded ring $A$, then $\\sqrt{\\I{a}}=\\bigcap_{\\I{p}\\in\n\\V{\\I{a}}}\\I{p}$.\n \\end{proof}\n\n \\begin{lemm}\\label{stdcrit}\nA graded ring $A$ is {standard} if and only if it is isomorphic to\n$A_0[t,t^{-1}]$, where $\\deg(t)=1$.\n \\end{lemm}\n\n \\begin{proof}\nOne implication is obvious, since $t$ is invertible and of degree $1$ in\n$A_0[t,t^{-1}]$. Conversely, if $A$ is a graded ring and $a\\in A_1$ is\ninvertible, then the morphism of graded rings $\\varphi:A_0[t,t^{-1}]\\to A$\ndefined by $\\varphi(t)=a$ is easily seen to be an isomorphism.\n \\end{proof}\n\n \\begin{lemm}\\label{qstdgood}\nFor a graded ring $A$ the following implications are true:\n\\begin{gather*}\nA\\text{ standard}\\implies A\\text{ quasi--standard}\\implies A\\text{ good,}\\\\\nA\\text{ trivial}\\implies A\\text{ quasi--trivial}\\implies A\\text{ good.}\n\\end{gather*}\n \\end{lemm}\n\n \\begin{proof}\nThe implications on the left are obvious.\n\nIf $A$ is quasi--standard, by \\ref{goodcrit} we have to prove that if $\\I{p}\\in\\gspec A$\nis such that $\\I{p}_d=A_d$ for some $d\\in\\Z$, then also $\\I{p}_{-d}=\nA_{-d}$. Now, if $s\\in A_n$ ($n>0$) is invertible and $a\\in A_{-d}$, then\n$s^da^{n-1}\\in A_d\\subset\\I{p}$, whence $n>1$ and $a\\in\\I{p}$.\n\nIf $A$ is quasi--trivial, then $A_{\\ne0}\\subset\\I{p}$ $\\forall\\I{p}\\in\\gspec A$ by\n\\ref{qtrivcrit}, whence $A$ is {good} by \\ref{goodcrit}.\n \\end{proof}\n\nIf $A$ is a graded ring, $S\\subset A\\hom$ will be called a {\\em homogeneous\nmultiplicative system} if $0\\notin S$ and $a,b\\in S\\implies ab\\in S$. As usual,\none can define the localized graded ring $S^{-1}A$ and, for every graded\n$A$--module $M$, the graded $S^{-1}A$--module $S^{-1}M$. If $S=\\{a^n\\st n\\in\n\\N\\}$ for some $a\\in A\\hom$ not nilpotent and $T=A\\hom\\setminus\\I{p}$ for\nsome $\\I{p}\\in\\gspec A$, we will write $A_a$, $M_a$ and $A_{\\I{p}}$,\n$M_{\\I{p}}$ instead of $S^{-1}A$, $S^{-1}M$ and $T^{-1}A$,\n$T^{-1}M$.\n\n \\begin{exem}\\label{Rrqstd}\nIf $R$ is a positively graded ring (i.e. $R_d=0$ if $d<0$) and $r\\in R_n$\n($n>0$) is not nilpotent, then clearly the localization $R_r$ is a {quasi--standard} (and\nalso {standard} if $n=1$) graded ring.\n\nOn the other hand, $R$ itself is never quasi--standard, and it is {good} if and only\nif it is {quasi--trivial} (this follows easily from \\ref{goodcrit} and\n\\ref{qtrivcrit}).\n \\end{exem}\n\n \\begin{lemm}\\label{0loc=loc0}\nLet $A$ be a graded ring, $M$ a graded $A$--module and $S\\subset A$ a\nhomogeneous multiplicative system such that $\\forall d\\in\\Z$ $S_d=\n\\emptyset\\iff S_{-d}=\\emptyset$. Then the natural morphism of\n$S_0^{-1}A_0$--modules $f:S_0^{-1}M_0\\to(S^{-1}M)_0$ is an isomorphism.\n \\end{lemm}\n\n \\begin{proof}\n$f$ is injective: if $m\\in M_0$ and $s\\in S_0$ are such that $m\/s=0$ in\n$(S^{-1}M)_0$, there exist $d\\in\\Z$ and $t\\in S_d$ such that $tm=0$. By\nhypothesis there exist $t'\\in S_{-d}$, and so $t'tm=0$, which implies that\n$m\/s=0$ in $S_0^{-1}M_0$ because $t't\\in S_0$.\n\n$f$ is surjective: given $m\/s\\in(S^{-1}M)_0$ (with $m\\in M_d$ and $s\\in S_d$\nfor some $d\\in\\Z$) by hypothesis there exists $s'\\in S_{-d}$, whence $m\/s=\nf((s'm)\/(s's))$.\n \\end{proof}\n\n \\begin{coro}\\label{0p=p0}\nIf $A$ is a {good} graded ring and $M$ a graded $A$--module, then $\\forall\n\\I{p}\\in\\gspec A$ the natural morphism of $(A_0)_{\\I{p}_0}$--modules\n$(M_0)_{\\I{p}_0}\\to(M_{\\I{p}})_0$ is an isomorphism.\n \\end{coro}\n\n \\begin{proof}\nBy \\ref{goodcrit} the multiplicative system $A\\hom\\setminus\\I{p}$ satisfies the\ncondition of \\ref{0loc=loc0}.\n \\end{proof}\n\n \\begin{prop}\\label{goodlocgood}\nLet $A$ be a graded ring and $S\\subset A$ a homogeneous multiplicative system.\nIf $A$ is {good} or {quasi--standard} or {standard} or {quasi--trivial} or {trivial}, then so is\n$S^{-1}A$.\n \\end{prop}\n\n \\begin{proof}\nObvious if $A$ is {quasi--standard} or {standard} or trivial; if $A$ is quasi--trivial, the statement\nfollows immediately from the fact that $S\\subset A_0$.\n\nIf $A$ is good, given $\\I{q},\\I{q}'\\in\\gspec(S^{-1}A)$ such that $\\I{q}\\ne\n\\I{q}'$, we have to show that $\\I{q}_0\\ne\\I{q}'_0$. Since there is a one to one\ncorrespondence between $\\{\\I{p}\\in\\gspec A\\st\\I{p}\\cap S=\\emptyset\\}$ and\n$\\gspec(S^{-1}A)$ (given by $\\I{p}\\mapsto S^{-1}\\I{p}$), there exist (unique)\n$\\I{p},\\I{p}'\\in\\gspec A$ such that $\\I{p}\\cap S=\\I{p}'\\cap S=\\emptyset$,\n$\\I{q}=S^{-1}\\I{p}$, $\\I{q}'=S^{-1}\\I{p}'$ and $\\I{p}\\ne\\I{p}'$. This implies\n$\\I{p}_0\\ne\\I{p}'_0$ because $A$ is good, and we can assume that $\\I{p}_0\n\\subsetneq\\I{p}'_0$. So let $a\\in\\I{p}_0\\setminus\\I{p}'_0$: then clearly $a\\in\n\\I{q}_0$ and $a\\notin\\I{q}'_0$.\n \\end{proof}\n\n \\begin{lemm}\\label{Apqstd}\nGiven a graded ring $A$ and $\\I{p}\\in\\gspec A$, consider the ideal of $\\Z$\n\\[I(\\I{p}):=(i\\in\\Z\\st\\I{p}_i\\ne A_i).\\]\nThen $A_{\\I{p}}$ is {standard} (respectively quasi--standard) if and only if $I(\\I{p})=(1)$\n(respectively $I(\\I{p})\\ne(0)$, i.e. $A_{\\ne0}\\nsubseteq\\I{p}$).\n \\end{lemm}\n\n \\begin{proof}\nIf $A_{\\I{p}}$ is {standard} (respectively quasi--standard), then there exist $x\\in\n(A_{\\I{p}})_d$ invertible with $d=1$ (respectively $d>0$). $x$ can be written\nin the form $x=a\/s$ with $a,s\\in A\\hom\\setminus\\I{p}$ and $\\deg(a)=\\deg(s)+d$.\nSince $\\deg(a),\\deg(s)\\in I(\\I{p})$, also $d\\in I(\\I{p})$.\n\nConversely, if $I(\\I{p})=(d)$ with $d=1$ (respectively $d>0$), then there exist\nelements $a_i\\in A_{d_i}\\setminus\\I{p}$ ($i=1,\\dots,n$) such that $(d_1,\\dots,d_n)\n=(d)$. Therefore there exist $m_1,\\dots,m_n\\in\\Z$ such that $d=\\sum_{i=1}^n\nm_id_i$, whence $x:=\\prod_{i=1}^na_i^{m_i}\\in A_{\\I{p}}$ is an invertible\nelement of degree $d$, i.e. $A_{\\I{p}}$ is {standard} (respectively quasi--standard).\n \\end{proof}\n\n \\begin{prop}\\label{locgoodgood}\nIf $A$ is a graded ring such that $A_{\\I{p}}$ is {good} $\\forall \\I{p}\\in\n\\gspec A$, then $A$ is good, too. The same statement holds with {quasi--trivial} or\n{trivial} in place of good.\n \\end{prop}\n\n \\begin{proof}\nBy \\ref{goodcrit} given $\\I{p}\\in\\gspec A$ we have to prove that $\\I{p}=\n\\g{\\I{p}_0}$, or, equivalently, that $\\I{p}A_{\\g{\\I{p}_0}}=\\g{\\I{p}_0}\nA_{\\g{\\I{p}_0}}$. Since $A_{\\g{\\I{p}_0}}$ is good, this is true if and only if\n$(\\I{p}A_{\\g{\\I{p}_0}})_0=(\\g{\\I{p}_0}A_{\\g{\\I{p}_0}})_0$. Now, given $a\/s\\in\n(\\g{\\I{p}_0}A_{\\g{\\I{p}_0}})_0$ (where $a\\in(\\g{\\I{p}_0})_d$ and $s\\in\n(A\\setminus\\g{\\I{p}_0})_d$), by \\ref{specsurj} there exists $t\\in\n(A\\setminus\\g{\\I{p}_0})_{-d}$, whence $a\/s=(at)\/(st)\\in(\\I{p}A_{\\g{\\I{p}_0}})_0$\nbecause $at\\in(\\g{\\I{p}_0})_0=\\I{p}_0$.\n\nIf $\\I{p}\\in\\gspec A$ is such that $A_{\\I{p}}$ is quasi--trivial, then $A_{\\ne0}\n\\subset\\I{p}$ by \\ref{Apqstd}. Therefore if $A_{\\I{p}}$\nis {quasi--trivial} $\\forall \\I{p}\\in\\gspec A$, then $A$ is {quasi--trivial} by\n\\ref{qtrivcrit}.\n\nIf $A_{\\I{p}}$ is {trivial} $\\forall \\I{p}\\in\\gspec A$, let $a\\in A_{\\ne0}$:\n$\\forall \\I{p}\\in\\gspec A$ $\\exists s\\in A\\setminus\\I{p}=A_0\\setminus\\I{p}_0$ such\nthat $sa=0$, which implies that $a=0$, whence $A$ is trivial.\n \\end{proof}\n\nOn the other hand, if $A_{\\I{p}}$ is {quasi--standard} (respectively standard) $\\forall \\I{p}\\in\n\\gspec A$, then $A$ need not be {quasi--standard} (respectively standard), too, as the\nfollowing example shows.\n\n \\begin{exem}\\label{lstdnotqstd}\nThere exists a graded ring $A$ which is not quasi--standard, but such that $A_{\\I{p}}$ is\n{standard} $\\forall \\I{p}\\in\\gspec A$. To see this, let $B$ be a noetherian ring and\n$M$ a $B$--module such that $M_{\\I{q}}$ is a free $B_{\\I{q}}$--module of rank\none $\\forall\\I{q}\\in\\spec B$ and $\\forall n>0$ $M^{\\otimes n}$ is not a free\n$B$--module (this is equivalent to require that $M\\sh{}$ is an invertible sheaf\non $\\spec B$ whose equivalence class in $\\Pic(\\spec B)$ is not a torsion\nelement). Setting $M^{\\otimes n}:=\\Hom_B(M^{\\otimes(-n)},B)$ for $n<0$, it is\nthen clear that $A:=\\bigoplus_{d\\in\\Z}M^{\\otimes d}$ is a graded ring (with\n$A_0=B$) in a natural way. By construction each $A_{\\I{p}}$ is standard, but $A$ is\nnot quasi--standard: assume on the contrary that there is $n>0$ and $a\\in A_n$, $b\\in\nA_{-n}$ such that $ab=1$. Regarding $b$ as a morphism $b:M^{\\otimes n}\\to B$ of\n$B$--modules, $b$ is surjective (since $b(a)=1$) and hence it is an isomorphism\n(because $M^{\\otimes n}$ and $B$ are both projective $B$--modules of rank one),\nwhich contradicts the hypothesis on $M$.\n \\end{exem}\n\n \\begin{lemm}\\label{lqstdlocqstd}\nLet $A$ be a graded ring. If $\\I{p}\\in\\gspec A$ is such that $A_{\\I{p}}$ is\n{quasi--standard} (respectively standard), then there exists $t\\in A\\hom\\setminus\\I{p}$ such\nthat $A_t$ is also {quasi--standard} (respectively standard).\nIf moreover $A$ is good, then $t$ can be taken in $A_0\\setminus\\I{p}_0$.\n \\end{lemm}\n\n \\begin{proof}\nIf $a\/s\\in(A_{\\I{p}})_n$ (with $s\\in A_d\\setminus\\I{p}$, $a\\in A_{d+n}$ and $n>0$)\nis an invertible element, it is enough to prove that $(A_t)_n$ contains an\ninvertible element for some $t\\in A\\hom\\setminus\\I{p}$. By hypothesis there exist\n$s'\\in A_{d'}\\setminus\\I{p}$, $a'\\in A_{d'-n}$ and $s''\\in A_{d''}\\setminus\\I{p}$\nsuch that $aa's''=ss's''$. Then, setting $t:=ss's''\\in A_{d+d'+d''}\\setminus\n\\I{p}$, $as's''\/t\\in(A_t)_n$ is an invertible element (its inverse is\n$a'ss''\/t$).\n\nIf moreover $A$ is good, then (by \\ref{goodcrit}) $\\forall u\\in A\\hom\\setminus\n\\I{p}$ there exists $u'\\in A\\hom\\setminus\\I{p}$ such that $uu'\\in A_0$, and this\nimplies that in the above argument we can assume $d=d'=d''=0$, whence $t\\in A_0\n\\setminus\\I{p}_0$.\n \\end{proof}\n\n \\begin{defi}\n$A$ is a {\\em local graded ring}\\index{local graded ring} if $A$ is a graded\nring with a unique maximal homogeneous ideal.\n \\end{defi}\n\n \\begin{prop}\\label{gspecAgood}\nLet $\\varphi:A\\to B$ be a morphism of graded rings.\n\\begin{enumerate}\n\n\\item If $A$ is {quasi--standard} (respectively standard), then $B$ is {quasi--standard}\n(respectively standard), too.\n\n\\item If $A$ is local graded and $\\varphi$ is surjective (whence $B$ is also\nlocal graded), then $A$ is {quasi--standard} (respectively standard) if and only if $B$ is\n{quasi--standard} (respectively standard).\n\n\\end{enumerate}\n \\end{prop}\n\n \\begin{proof}\n\\begin{enumerate}\n\n\\item Clear, since $\\varphi$ preserves degrees and $\\varphi(a)$ is\ninvertible if $a$ is invertible.\n\n\\item Just observe that, denoting by $\\I{m}$ and $\\I{n}$ the maximal\nhomogeneous ideals of $A$ and $B$, $\\forall d\\in\\Z$ we have $\\I{m}_d=A_d$ if and\nonly if $\\I{n}_d=B_d$.\n\\end{enumerate}\n \\end{proof}\n\n \\begin{prop}\\label{Anoeth}\nFor a graded ring $A$ the following conditions are equivalent:\n\\begin{enumerate}\n\n\\item $A_0$ is a noetherian ring and $A$ is a finitely generated\n$A_0$--algebra;\n\n\\item $A$ is a noetherian ring;\n\n\\item $A$ satisfies the ascending chain condition for homogeneous ideals, or,\nequivalently, every homogeneous ideal of $A$ is finitely generated.\n\n\\end{enumerate}\nIf the above conditions are satisfied and $M$ is a finitely generated graded\n$A$--module, then $M_d$ is a finitely generated $A_0$--module $\\forall d\\in\\Z$.\n \\end{prop}\n\n \\begin{proof}\n(A proof of $2\\iff3$ can also be found in \\cite{NO}). First we show that $3$\nimplies the last statement. Clearly $M$ satisfies the ascending chain condition\nfor graded $A$--submodules. Since moreover for every $A_0$--submodule $N$ of\n$M_d$ we have $(AN)_d=N$, it follows that $M_d$ is a noetherian (hence finitely\ngenerated) $A_0$--module. Notice that, in particular, $A_0$ is a noetherian\nring.\n\nThe implication $1\\implies2$ follows from Hilbert's base theorem, whereas\n$2\\implies3$ is obvious. So let's prove that $3\\implies 1$. We have already\nproved that $A_0$ is noetherian. Let $A_+$ and $A_-$ be the ideals of $A$\ngenerated by $A_{>0}$ and $A_{<0}$ respectively: the hypothesis clearly implies\nthat there exist $a_1,\\dots,a_n\\in A\\hom_{>0}$ and $b_1,\\dots,b_m\\in\nA\\hom_{<0}$ such that $A_+=(a_1,\\dots,a_n)$ and $A_-=(b_1,\\dots,b_m)$. Let\n$r,s\\in\\Z$ be such that $r>\\deg(a_i)$ for $i=1,\\dots,n$ and $s<\\deg(b_i)$ for\n$i=1,\\dots,m$. Since each $A_d$ is a finitely generated $A_0$--module, there\nexist homogeneous elements $c_1,\\dots,c_l$ which generate\n$\\bigoplus_{sl$. Hence for $\\abs{d}>l$ we have $B_d=\n\\I{n}_0B_d$, so that $B_d=(0)$ by Nakayama's lemma ($B_d$ is a finitely\ngenerated $B_0$--module again by \\ref{Anoeth}). As $B$ is a domain, this\nactually implies that $B_{\\ne0}=(0)$, whence $A_{\\ne0}\\subset\\I{p}$, a\ncontradiction.\n \\end{proof}\n\n \\begin{exem}\nIf $A$ is a {good} noetherian graded ring, then it is not true in general that\n$A$ is {quasi--trivial} or $A_{\\I{p}}$ is {quasi--standard} $\\forall \\I{p}\\in\\gspec A$. Take for\ninstance\n\\begin{align*}\n& A=\\Z\/6\\Z[x,y]\/(2x,2y,xy-3) & \\deg(x)=-\\deg(y)=1.\n\\end{align*}\nIt is easy to see that $A_0=\\Z\/6\\Z$ and $\\gspec A=\\{(2),(x,y)\\}$\n(so that $A$ is good), but $A_{(2)}$ is {standard} and $A_{(x,y)}$ is trivial.\n \\end{exem}\n\n \\begin{prop}\\label{clopen}\nLet $A$ be a {good} noetherian graded ring. Then\n\\begin{align*}\n& \\QT{A}:=\\{\\I{p}\\in\\gspec A\\st A_{\\I{p}}\\text{ quasi--trivial}\\}, \\\\\n&\\QS{A}:=\\{\\I{p}\\in\\gspec A\\st A_{\\I{p}}\\text{ quasi--standard}\\}\n\\end{align*}\nare closed and open in $\\gspec A$ and $\\gspec A=\\QT{A}\\djun\\QS{A}$.\n \\end{prop}\n\n \\begin{proof}\nNotice first that the last statement follows from \\ref{locgqtqs}. Then, since\n$\\QT{A}=\\gspec A\\setminus\\QS{A}=\\V{(A_{\\ne0})}$ by \\ref{Apqstd}, it is enough to\nprove that $\\QT{A}$ is open. Now, if $\\I{p}\\in\\QT{A}$, then $\\forall a\\in\nA_{\\ne0}$ there exist $n\\in\\N$ and $t\\in A\\setminus\\I{p}\\subset A_0$ such that\n$ta^n=0$. Since $A$ is a finitely generated $A_0$--algebra by \\ref{Anoeth},\nthis implies that there exists $s\\in A\\setminus\\I{p}$ such that $\\forall a\\in\nA_{\\ne0}$ $\\exists n\\in\\N$ with $sa^n=0$. Therefore, $\\I{q}\\in\\D{s}$ and $a\\in\nA_{\\ne0}$ imply $a\\in\\I{q}$, so that $\\I{q}\\in\\QT{A}$. This proves that $\\D{s}\n\\subset\\QT{A}$, and so $\\QT{A}$ is open.\n \\end{proof}\n\nWith a proof completely analogous to that of the not graded case (see\n\\cite[prop. II.3.2]{H1}), one can prove the following result.\n\n \\begin{lemm}\\label{locnoethnoeth}\nLet $A$ be a graded ring and $M$ a graded $A$--module. If there exist $a_1,\n\\dots,a_n\\in A\\hom$ such that $(a_1,\\dots,a_n)=(1)$ and each $M_{a_i}$ is a\nnoetherian $A_{a_i}$--module, then $M$ is a noetherian $A$--module.\n \\end{lemm}\n\n\n \\section{Modules versus graded modules}\n\nIf $A$ is a graded ring, we will denote by $\\gmo{A}$\\index{Mod(A) @$\\gmo{A}$}\nthe (abelian) category of graded $A$--modules (with morphisms preserving\ndegrees) and by $\\gfmo{A}$\\index{mod(A) @$\\gfmo{A}$} its full subcategory of\nfinitely generated graded $A$--modules. $\\mo{A}$\\index{Mod(A)@$\\mo{A}$} and\n$\\fmo{A}$\\index{mod(A)@$\\fmo{A}$} will denote instead the categories of\n$A$--modules and of finitely generated $A$--modules respectively. Notice that\nif $A=A_0$ then $\\mo{A}$ can be identified with a full subcategory of $\\gmo{A}$\n(by regarding a module as concentrated in degree $0$).\n\n$\\forall n,m\\in\\Z$ the natural functors $(n):\\gmo{A}\\to\\gmo{A}$ (defined\nby $M(n)_d=M_{n+d}$) are exact autoequivalences and $(n)\\comp(m)=(n+m)$.\n\n$\\gmo{A}$ has both enough projectives ($\\forall n\\in\\Z$ the free graded\n$A$--module $A(n)$ is obviously projective) and enough injectives (this can be\neasily seen slightly adapting the standard proof that $\\mo{A}$ has enough\ninjectives).\n\nIf $M$ and $N$ are (finitely generated) graded $A$--modules, so is $M\\otimes_A\nN$. As usual $M\\otimes_A-:\\gmo{A}\\to\\gmo{A}$ is right exact and $\\forall n\\in\n\\Z$ the functors $(n)$ and $A(n)\\otimes_A-$ are isomorphic.\n\nWe will write $\\Hom_A(M,N)$\\index{Hom_A@$\\Hom_A$} instead of\n$\\Hom_{\\gmo{A}}(M,N)$, and we will denote by\n$\\gHom_A(M,N)$\\index{Hom_A@$\\gHom_A$} the graded $A$-module defined by\n\\begin{align*}\n& \\gHom_A(M,N)_d:=\\Hom_A(M,N(d))=\\Hom_A(M(-d),N) & \\forall d\\in\\Z.\n\\end{align*}\nClearly $\\gHom_A(M,N)_0=\\Hom_A(M,N)$ and the functor $\\gHom_A(A(-n),-)$ is\nisomorphic to $(n)$. The functors\n\\begin{gather*}\n\\Hom_A(-,-):\\gmo{A}\\opp\\times\\gmo{A}\\to\\mo{A_0},\\\\\n\\gHom_A(-,-):\\gmo{A}\\opp\\times\\gmo{A}\\to\\gmo{A}\n\\end{gather*}\nare left exact in both arguments. Notice that $\\gHom_A(M,N)=\\Hom_{\\mo{A}}(M,N)$\n(as $A$--modules) if $M$ is finitely generated, and that $\\gHom_A(M,N)\\in\n\\gfmo{A}$ and $\\Hom_A(M,N)\\in\\fmo{A_0}$ if $M,N\\in\\gfmo{A}$ and $A$ is\nnoetherian.\n\nA morphism of graded rings $\\varphi:A\\to B$ induces functors\n\\begin{align*}\n& B\\otimes_A-:\\gmo{A}\\to\\gmo{B}, & \\gHom_A(B,-):\\gmo{A}\\to\\gmo{B}.\n\\end{align*}\nIf $M\\in\\gfmo{A}$ then $B\\otimes_A M\\in\\gfmo{B}$ and $\\gHom_A(B,M)\\in\\gfmo{B}$\nif $A$ is noetherian and $\\varphi$ is finite. As in the not graded case (see\n\\cite[p. 693]{E}) it is straightforward to prove the following result.\n\n \\begin{lemm}\\label{adj}\n$B\\otimes_A-$ and $\\gHom_A(B,-)$ are respectively left and right adjoint of\nthe natural forgetful functor $\\gmo{B}\\to\\gmo{A}$.\n \\end{lemm}\n\nNow we are going to study the relation between $\\gmo{A}$ and $\\mo{A_0}$, where\n$A$ is a graded ring. Clearly taking the degree $0$ component defines an exact\nfunctor\n\\[\\umf{A}:\\gmo{A}\\to\\mo{A_0},\\]\nwhich sends $\\gfmo{A}$ into $\\fmo{A_0}$ if $A$ is noetherian (by \\ref{Anoeth}).\n\n \\begin{prop}\\label{adj0}\nThe functors\n\\begin{align*}\n& \\omf{A}:\\mo{A_0}\\to\\gmo{A}, & \\hmf{A}:\\mo{A_0}\\to\\gmo{A}\n\\end{align*}\nare respectively left and right adjoint of $\\umf{A}$, and they are both right\ninverse of $\\umf{A}$, i.e.\n\\[\\umf{A}\\comp\\omf{A}\\cong\\umf{A}\\comp\\hmf{A}\\cong{\\rm id}.\\]\n$\\omf{A}$ sends $\\fmo{A_0}$ into $\\gfmo{A}$, and the same is true for\n$\\hmf{A}$ if $A$ is {good} and noetherian.\n \\end{prop}\n\n \\begin{proof}\nTaking into account that given $M\\in\\gmo{A}$ and $N\\in\\mo{A_0}\\subset\\gmo{A_0}$\n\\begin{align*}\n& \\Hom_{A_0}(M_0,N)=\\Hom_{A_0}(M,N) & &\\text{and} &\n\\Hom_{A_0}(N,M_0)=\\Hom_{A_0}(N,M),\n\\end{align*}\nby \\ref{adj} everything is clear except the last statement.\n\nIf $A$ is {good} and noetherian, then there exist $a_1,\\dots,a_n\\in A_0$ such\nthat each $A_{a_i}$ is either {quasi--standard} or {quasi--trivial} and with\n$(a_1,\\dots,a_n)=(1)$: taking into account that $\\gspec A$ is quasi--compact\nbecause $A$ is noetherian, this follows immediately from \\ref{clopen},\n\\ref{lqstdlocqstd} and \\ref{locgoodgood} (just notice that if $a\\in A\\hom$ is\nsuch that $A_a$ is quasi--trivial, then $a\\in A_0$). By \\ref{locnoethnoeth} it is\nenough to prove that $\\gHom_{A_0}(A,N)_{a_i}$ is a finitely generated\n$A_{a_i}$--module $\\forall N\\in\\fmo{A_0}$ and $\\forall i=1,\\dots,n$. Since\n$\\gHom_{A_0}(A,N)_{a_i}\\cong\\gHom_{(A_0)_{a_i}}(A_{a_i},N_{a_i})$ (by\n\\cite[prop. 2.10]{E}) and since $(A_0)_{a_i}\\cong(A_{a_i})_0$ (by\n\\ref{0loc=loc0}), this means that we can assume that $A$ is either {quasi--standard} or\nquasi--trivial. If $M:=\\gHom_{A_0}(A,N)$ for some $N\\in\\fmo{A_0}$, $\\forall d\\in \\Z$ we\nhave $M_d=\\Hom_{A_0}(A_{-d},N)\\in\\fmo{A_0}$ (because $A_{-d}\\in \\fmo{A_0}$ by\n\\ref{Anoeth}), whence it is enough to show that $\\exists m>0$ such that $M$ is\ngenerated by $\\bigoplus_{-m\\le d\\le m}M_d$ as an $A$--module. Now, if $A$ is\n{quasi--standard} we can take $m$ such that there exists $s\\in A_{2m}$ invertible\n(because then $\\forall d\\in\\Z$ multiplication by $s$ is bijective between $M_d$\nand $M_{d+2m}$), whereas if $A$ is {quasi--trivial} we can take $m$ such that $A_d=(0)$\nfor $\\abs{d}>m$ (such an $m$ exists because $A$ is a finitely generated\n$A_0$--algebra by \\ref{Anoeth}).\n \\end{proof}\n\n \\begin{prop}\\label{modequivcrit}\n$\\umf{A}:\\gmo{A}\\to\\mo{A_0}$ is an equivalence of categories if and only if\n$A_{\\I{p}}$ is {standard} $\\forall \\I{p}\\in\\gspec A$.\n\nIf this is the case, then $\\omf{A}\\cong\\hmf{A}$ and they are quasi--inverse of\n$\\umf{A}$ and exact. Moreover, the restrictions of these functors give an\nequivalence of categories between $\\gfmo{A}$ and $\\fmo{A_0}$ if $A$ is\nnoetherian.\n \\end{prop}\n\n \\begin{proof}\nBy \\ref{adj0} it is clear that $\\umf{A}$ is an equivalence of categories if and\nonly if $\\omf{A}$ is quasi--inverse of $\\umf{A}$ (and in this case $\\omf{A}\\cong\n\\hmf{A}$ and they are exact since $\\umf{A}$ is). Therefore $\\umf{A}$ is an\nequivalence if and only if $\\forall M\\in\\gmo{A}$ the natural morphism (in\n$\\gmo{A}$) $\\alpha(M):A\\otimes_{A_0}M_0\\to M$ is an isomorphism,\ni.e. if and only if the morphism of $A_0$--modules\n\\[\\alpha(M)_d:A_d\\otimes_{A_0}M_0\\to M_d\\]\nis an isomorphism $\\forall d\\in\\Z$. First we notice that if $\\umf{A}$ is\nan equivalence, then $A$ is good: $\\forall d\\in\\Z$ we have\n$(A(d)\/(A_d)A(d))_0=0$, whence $(A_d)=A$, which implies that $\\I{p}_d\\ne A_d$\n$\\forall \\I{p}\\in\\gspec A$, so that $A$ is {good} by \\ref{goodcrit}.\nTherefore we can assume that $A$ is good, and so (taking into account\n\\ref{0p=p0}) $\\alpha(M)_d$ is an isomorphism if and only if\n\\[\\alpha(M_{\\I{p}})_d:(A_{\\I{p}})_d\\otimes_{(A_{\\I{p}})_0}(M_{\\I{p}})_0\\to\n(M_{\\I{p}})_d\\]\nis an isomorphism $\\forall \\I{p}\\in\\gspec A$. Now, if $A_{\\I{p}}$ is standard, then\nclearly $\\alpha(M_{\\I{p}})_d$ is an isomorphism, since if $t\\in(A_{\\I{p}})_1$\nis invertible, then multiplication by $t^d$ induces an isomorphism between\n$(M_{\\I{p}})_0$ and $(M_{\\I{p}})_d$ (and between $(A_{\\I{p}})_0$ and\n$(A_{\\I{p}})_d$). Conversely, if $\\umf{A}$ is an equivalence, then, in\nparticular, the natural multiplication map\n\\[\\alpha(A_{\\I{p}}(1))_{-1}:(A_{\\I{p}})_{-1}\\otimes_{(A_{\\I{p}})_0}\n(A_{\\I{p}})_1\\to(A_{\\I{p}})_0\\]\nis an isomorphism $\\forall \\I{p}\\in\\gspec A$. So there exist\n$a\\in(A_{\\I{p}})_1$ and $b\\in(A_{\\I{p}})_{-1}$ such that $ab\\notin\n\\I{p}A_{\\I{p}}$ (otherwise $\\im\\alpha(A_{\\I{p}}(1))_{-1}\\subseteq\\I{p}\nA_{\\I{p}}$), whence $a$ is invertible. Therefore $A_{\\I{p}}$ is {standard} $\\forall\n\\I{p}\\in\\gspec A$.\n\nThe last statement follows from the fact that if $A$ is noetherian, then\n$\\umf{A}$ sends $\\gfmo{A}$ into $\\fmo{A_0}$ and $\\omf{A}$ and $\\hmf{A}$ send\n$\\fmo{A_0}$ into $\\gfmo{A}$.\n \\end{proof}\n\n\n \\section{Graded schemes}\n\nIf $X$ is a topological space and $\\s{F}$ a sheaf of graded abelian groups on\n$X$, we will denote by $\\gsec(U,\\s{F})$\\index{Gamma(U,F)@$\\gsec(U,\\s{F})$} the\ngraded abelian group of sections of $\\s{F}$ over an open subset $U$ of $X$.\nNotice that if $\\s{F}_d$ ($d\\in\\Z$) are sheaves of abelian groups on $X$, then\nthe presheaf $\\s{F}:= \\bigoplus_{d\\in\\Z}\\s{F}_d$ is not a sheaf in general.\nHowever, $\\s{F}$ is certainly a sheaf if every open subset of $X$ is\nquasi--compact (which is true if and only if $X$ is noetherian).\n\n \\begin{defi}\nA {\\em graded ringed space}\\index{graded!ringed space} $X=(X,\\so_X)$ is a\ntopological space $X$ with a sheaf of graded rings\n$\\so_X=\\bigoplus_{d\\in\\Z}(\\so_X)_d$ on $X$.\n\nA morphism of graded ringed spaces\\index{morphism of!graded ringed spaces}\n$f=(f,f\\mrs):X\\to Y$ is a continuous map $f:X\\to Y$ together with a morphism\n(of degree $0$) $f\\mrs:\\so_Y\\to f_*\\so_X$ of sheaves of graded rings on $Y$.\n \\end{defi}\n\nIf $X$ is a graded ringed space, we will denote by $\\gsmo{\\so_X}$, or simply by\n$\\gsmo{X}$,\\index{Mod(X)@$\\gsmo{X}$} the (abelian) category of sheaves of\ngraded $\\so_X$--modules on $X$. With a proof completely analogous to that of\nthe not graded case, one can show that $\\gsmo{X}$ has enough injectives.\n\n$\\forall n\\in\\Z$ the natural {\\em twist functors} $(n):\\gsmo{X}\n\\to\\gsmo{X}$ (defined by $\\s{F}(n)_d:=\\s{F}_{n+d}$) are exact autoequivalences\nand $(n)\\comp(m)=(n+m)$.\n\n \\begin{defi}\nA sheaf of graded $\\so_X$--modules $\\s{F}$ is {\\em locally free} \\index{locally\nfree graded sheaf} if it is locally isomorphic to a sheaf of the form\n$\\bigoplus_{i\\in I}\\so_X(n_i)$ (where $n_i\\in\\Z$). $\\card{I}$ (if finite and\nconstant) is the {\\em rank} of $\\s{F}$.\n \\end{defi}\n\nAs in the case of graded modules over a graded ring, one can define the\nfollowing functors (the first right exact and the other two left exact in both\narguments):\\index{Hom_X@$\\Hom_X$}\\index{Hom_X @$\\gHom_X$}\n\\begin{gather*}\n-\\otimes_X-:=-\\otimes_{\\so_X}-:\\gsmo{X}\\times\\gsmo{X}\\to\\gsmo{X},\\\\\n\\Hom_X(-,-):=\\Hom_{\\gsmo{X}}(-,-):\\gsmo{X}\\opp\\times\\gsmo{X}\\to\n\\mo{\\gsec(X,\\so_X)_0},\\\\\n\\gHom_X(-,-):=\\bigoplus_{d\\in\\Z}\\Hom_X(-,-(d)):\\gsmo{X}\\opp\\times\n\\gsmo{X}\\to\\gmo{\\gsec(X,\\so_X)}.\n\\end{gather*}\nOne can consider moreover the functors\\index{Hom_X@$\\sHom_X$}\\index{Hom_X\n@$\\gsHom_X$}\n\\begin{gather*}\n\\sHom_X(-,-):\\gsmo{X}\\opp\\times\\gsmo{X}\\to\\smo{(\\so_X)_0},\\\\\n\\gsHom_X(-,-)=\\bigoplus_{d\\in\\Z}\\sHom_X(-,-(d)):\\gsmo{X}\\opp\\times\n\\gsmo{X}\\to\\gmo{X},\n\\end{gather*}\nwhere, for every open subset $U$ of $X$, $\\sHom_X(-,-)(U):=\n\\Hom_U(-\\rest{U},-\\rest{U})$ and $\\gsHom_X(-,-)(U):=\n\\gHom_U(-\\rest{U},-\\rest{U})$. By definition we have\n$\\gHom_X(-,-)_0=\\Hom_X(-,-)$ and $\\gsHom_X(-,-)_0=\\sHom_X(-,-)$, and clearly\n$\\forall n\\in\\Z$ the functors $(n)$, $\\so_X(n)\\otimes_X-$ and\n$\\gsHom_X(\\so_X(-n),-)$ are isomorphic. If $\\s{L}$ is a locally free graded\nsheaf of finite rank, we will write $\\s{L}\\dual$\\index{$\\dual$} for\n$\\gsHom_X(\\s{L},\\so_X)$. More generally, if $\\cp{\\s{L}}$ is a complex of\nlocally free graded sheaves of finite rank, $(\\cp{\\s{L}})\\dual$ will denote the\ncomplex $\\gsHom_X(\\cp{\\s{L}},\\so_X)$ (which in position $i$ has\n$(\\s{L}^{-i})\\dual$).\n\nGiven $\\s{F}\\in\\gsmo{X}$ and $i\\in\\N$, the $i^{\\rm th}$ right derived functor\nof $\\Hom_X(\\s{F},-)$, $\\gHom_X(\\s{F},-)$, $\\sHom_X(\\s{F},-)$ and\n$\\gsHom_X(\\s{F},-)$ will be denoted by\n$\\Ext^i_X(\\s{F},-)$,\\index{Ext^i_X@$\\Ext^i_X$}\n$\\gExt^i_X(\\s{F},-)$,\\index{Ext^i_X @$\\gExt^i_X$}\n$\\sExt^i_X(\\s{F},-)$\\index{Ext^i_X@$\\sExt^i_X$} and\n$\\gsExt^i_X(\\s{F},-)$\\index{Ext^i_X @$\\gsExt^i_X$}, respectively. Notice that\n$\\gExt^i_X(\\s{F},-)\\cong \\bigoplus_{d\\in\\Z}\\Ext^i_X(\\s{F},-(d))$ and that $\\forall\nn\\in\\Z$ there are natural isomorphisms\n\\[\\gExt^i_X(\\s{F},-)(n)\\cong\\gExt^i_X(\\s{F},-(n))\\cong\n\\gExt^i_X(\\s{F}(-n),-)\\]\n(whence, in particular, also $\\Ext^i_X(\\s{F},-(n))\\cong\n\\Ext^i_X(\\s{F}(-n),-)$). Analogous results hold, of course, also for\n$\\gsExt^i_X(\\s{F},-)$ and $\\sExt^i_X(\\s{F},-)$. It is also clear that\n$\\gsExt^i_X(\\so_X(n),-)=0$ if $i>0$. Moreover, if $\\s{L}$ is a locally free\ngraded sheaf of finite rank, $\\forall i\\in\\N$ there are natural isomorphisms\n\\begin{gather*}\n\\gExt^i_X(\\s{F}\\otimes\\s{L},-)\\cong\\gExt^i_X(\\s{F},-\\otimes\\s{L}\\dual),\\\\\n\\gsExt^i_X(\\s{F}\\otimes\\s{L},-)\\cong\\gsExt^i_X(\\s{F},-\\otimes\\s{L}\\dual)\\cong\n\\gsExt^i_X(\\s{F},-)\\otimes\\s{L}\\dual.\n\\end{gather*}\n\n$H^i(X,-)$\\index{H^i(X,-)@$H^i(X,-)$} ($i\\in\\N$) will denote the $i^{\\rm th}$\nright derived functor of the degree zero global section functor\n$\\gsec(X,-)_0:\\gsmo{X}\\to\\mo{\\gsec(X,\\so_X)_0}$. Since clearly\n$\\Hom_X(\\so_X(-n),-)\\cong\\gsec(X,-)_n$, there are natural isomorphisms\n$H^i(X,-(n))\\cong\\Ext^i_X(\\so_X(-n),-)$ $\\forall n\\in\\Z$. We define more generally\n$\\forall i\\in\\Z$\n\\[H^i(X,-):=R^i\\gsec(X,-)_0:D^+(\\gsmo{X})\\to\\mo{\\gsec(X,\\so_X)_0}.\\]\nAs usual, a morphism $f:X\\to Y$ of graded ringed spaces induces a left exact\nfunctor $f_*:\\gsmo{X}\\to\\gsmo{Y}$ and a right exact functor $f^*:\\gsmo{Y}\\to\n\\gsmo{X}$, which is left adjoint of $f_*$.\n\n \\begin{defi}\nA {\\em locally graded ringed space}\\index{locally graded ringed space} is a\ngraded ringed space $X$ such that $\\forall x\\in X$ the stalk $\\so_{X,x}$ is a\nlocal graded ring.\n\nA morphism of locally graded ringed spaces\\index{morphism of!locally graded\nringed spaces} $f:X\\to Y$ is a morphism of graded ringed spaces such that $\\forall\nx\\in X$ the induced map on stalks $f\\mrs_x: \\so_{Y,f(x)}\\to\\so_{X,x}$ is a\nlocal graded homomorphism of local graded rings.\n \\end{defi}\n\nGiven a noetherian graded ring $A$ one can consider the graded ringed space $X$\nsuch that $X=\\gspec A$ as a topological space and with structure sheaf $\\so_X$\nthat we are going to define. In general, for every graded $A$--module $M$ one\ncan consider the graded sheaf $M\\gsh{A}$ (usually denoted simply by\n$M\\gsh{}$)\\index{$\\gsh{A}$} such that if $U\\subseteq X$ is open, $M\\gsh{}(U)$\nis the set of functions $s:U \\to\\djun_{\\I{p}\\in U}M_{\\I{p}}$ satisfying the\nfollowing conditions $\\forall\\I{p} \\in U$:\n\\begin{enumerate}\n\n\\item $s(\\I{p})\\in M_{\\I{p}}$;\n\n\\item there exist $V\\subseteq U$ neighbourhood of $\\I{p}$ and $a\\in A\\hom$,\n$m\\in M$ such that $\\forall \\I{q}\\in V$, $a\\notin\\I{q}$ and $s(\\I{q})=m\/a$ in\n$M_{\\I{q}}$.\n\n\\end{enumerate}\nNotice that since $A$ is noetherian, $U$ is quasi--compact, and this implies\nthat $M\\gsh{}(U)=\\bigoplus_{d\\in\\Z}M\\gsh{}(U)_d$, where $M\\gsh{}(U)_d$ is the\nsubset of $M\\gsh{}(U)$ consisting of those sections which are locally of the\nform $m\/a$ with $m\\in M_{d+\\deg(a)}$.\n\nThen $\\so_X:=A\\gsh{}$ is a sheaf of graded rings, so that $X$ is a graded\nringed space, and $M\\gsh{}$ is a sheaf of graded $\\so_X$--modules for every\ngraded $A$--module $M$. Clearly $A(n)\\gsh{}\\cong\\so_X(n)$ $\\forall n\\in\\Z$.\n\nAs in the not graded case, it is not difficult to prove the following facts.\n\\begin{enumerate}\n\n\\item $\\forall \\I{p}\\in X$ the stalk $\\so_{X,\\I{p}}$ is isomorphic to $A_{\\I{p}}$,\nwhence $X$ is a locally graded ringed space; more generally, $(M\\gsh{})_{\\I{p}}\n\\cong M_{\\I{p}}$ if $M$ is a graded $A$--module.\n\n\\item Denoting (for $a\\in A\\hom$) by $\\D{a}$ the open subset $X\\setminus\\V{(a)}$\nof $X$, $(\\D{a},\\so_X\\rest{\\D{a}})$ is isomorphic (as locally graded ringed\nspace) to $\\gspec A_a$, and under this identification $(M\\gsh{A})\\rest{\\D{a}}\n\\cong(M_a)\\gsh{A_a}$.\n\n\\item A morphism $\\varphi:A\\to B$ of noetherian graded rings induces a morphism\n$f:\\gspec B\\to\\gspec A$ of locally graded ringed spaces.\n\n\\end{enumerate}\n\n \\begin{defi}\nAn {\\em affine graded scheme}\\index{affine!graded scheme} is a locally graded\nringed space which is isomorphic to $\\gspec A$ for some noetherian graded ring\n$A$.\n\nA {\\em graded scheme}\\index{graded!scheme} is a locally graded ringed space $X$\nwhich is noetherian as a topological space and in which every point has an open\nneighbourhood $U$ such that $(U,\\so_X\\rest{U})$ is an affine graded scheme.\n\nA morphism of graded schemes\\index{morphism of!graded schemes} is just a\nmorphism of locally graded ringed spaces.\n \\end{defi}\n\n \\begin{rema}\nOf course the category $\\gsch$\\index{Sch@$\\gsch$} of graded schemes includes as\na full subcategory the category of noetherian schemes (whose structure sheaves\nare considered concentrated in degree $0$).\n \\end{rema}\n\n \\begin{rema}\nIf $X$ is a graded scheme, one can also define in the obvious way the category\n$\\gsch_X$ of graded schemes over $X$. If $A$ is a noetherian graded ring,\n$\\gsch_{\\gspec A}$ will be usually denoted by $\\gsch_A$.\\index{Sch_A@$\\gsch_A$}\n \\end{rema}\n\nMuch of the theory of (noetherian) schemes extends naturally (with some obvious\nmodifications) to graded schemes. Here we just give the definitions and state\nthe results that we will need. Unless otherwise stated, the proofs are\ncompletely analogous to those in the not graded case.\n\n \\begin{defi}\nLet $X$ be a graded scheme. A sheaf of graded $\\so_X$--modules $\\s{F}$ is {\\em\nquasi--coherent}\\index{quasi--coherent graded sheaf} (respectively {\\em\ncoherent})\\index{coherent graded sheaf} if for some (and hence any) open cover\nof $X$ by affine graded schemes $U_i\\cong\\gspec A^i$, there exist graded\n$A^i$--modules (respectively finitely generated graded $A^i$--modules) $M^i$\nsuch that $\\s{F}\\rest{U_i}\\cong(M^i)\\gsh{}$.\n \\end{defi}\n\nIf $X$ is a graded scheme, $\\gqco{X}$\\index{Qcoh(X)@$\\gqco{X}$} (respectively\n$\\gcoh{X}$)\\index{Coh(X)@$\\gcoh{X}$} will be the full subcategory of $\\gsmo{X}$\nwhose objects are quasi--coherent (respectively coherent) graded sheaves. As\nusual, kernels, cokernels, images and extensions of quasi--coherent\n(respectively coherent) graded sheaves are also quasi--coherent (respectively\ncoherent), so that $\\gqco{X}$ and $\\gcoh{X}$ are abelian categories. $\\gqco{X}$\nhas enough injectives.\n\nThe restrictions of the twist functors, of $-\\otimes_X-$ and of $\\gsHom_X(-,-)$\nto $\\gqco{X}$ (respectively $\\gcoh{X}$) have image contained in $\\gqco{X}$\n(respectively $\\gcoh{X}$).\n\nIf $f:X\\to Y$ is a morphism of graded schemes, then $f^*(\\gqco{Y})\\subseteq\n\\gqco{X}$, $f^*(\\gcoh{Y})\\subseteq\\gcoh{X}$ and $f_*(\\gqco{X})\\subseteq\n\\gqco{Y}$ (but, of course, it is false in general that $f_*(\\gcoh{X})\\subseteq\n\\gcoh{Y}$).\n\n \\begin{rema}\nA locally free graded sheaf is quasi--coherent, and also coherent if it is of\nfinite rank. In the latter case it will be often called {\\em vector\nbundle}.\\index{vector bundle}\n\nIt is easy to prove that a sheaf of graded $\\so_X$--modules $\\s{F}$ is\nquasi--coherent (respectively coherent) if and only if it is locally isomorphic\nto the cokernel of a morphism of (locally) free sheaves (respectively of vector\nbundles).\n \\end{rema}\n\n \\begin{rema}\nA sheaf of graded $\\so_X$--modules $\\s{F}$ is locally free if and only if\n$\\forall x\\in X$ the stalk $\\s{F}_x$ is a free graded $\\so_{X,x}$--module.\n \\end{rema}\n\n \\begin{prop}\\label{gspecequiv}\nThe contravariant functor $\\gspec$ induces an antiequivalence between the\ncategory of noetherian graded rings and the category of affine graded schemes.\nIts quasi--inverse is defined on objects by $X\\mapsto \\gsec(X,\\so_X)$.\nMoreover, if $X$ is a graded scheme and $A$ is a noetherian graded ring, the\nnatural map\n\\[\\begin{split}\n\\Hom_{\\gsch}(X,\\gspec A) & \\to \\Hom_{\\g{\\cat{Ring}}}(A,\\gsec(X,\\so_X)) \\\\\nf & \\mapsto f\\mrs(\\gspec A)\n\\end{split}\\]\n(where $\\g{\\cat{Ring}}$ denotes the category of graded rings) is bijective.\n \\end{prop}\n\n \\begin{prop}\nIf $A$ is a noetherian graded ring, the associated graded sheaf functor\n$\\gsh{A}:\\gmo{A}\\to\\gsmo{\\gspec A}$ is exact and its right adjoint is the\nglobal sections functor $\\gsec(\\gspec A,-):\\gsmo{\\gspec A}\\to\\gmo{A}$.\nThese two functors induce quasi--inverse equivalences of categories between\n$\\gmo{A}$ (respectively $\\gfmo{A}$) and $\\gqco{\\gspec A}$ (respectively\n$\\gcoh{\\gspec A}$).\n \\end{prop}\n\n \\begin{prop}\\label{modaff}\nLet $\\varphi:A\\to B$ be a morphism of noetherian graded rings and let $f:Y:=\n\\gspec B\\to X:=\\gspec A$ be the induced morphism of graded schemes.\n\\begin{enumerate}\n\n\\item If $M,N\\in\\gmo{A}$, then $(M\\otimes_A N)\\gsh{}\\cong M\\gsh{}\\otimes_X\nN\\gsh{}$.\n\n\\item If $M\\in\\gmo{A}$ and $N\\in\\gmo{B}$, then $f^*(M\\gsh{A})\\cong\n(M\\otimes_AB)\\gsh{B}$ and $f_*(N\\gsh{B})\\cong N\\gsh{A}$.\n\\end{enumerate}\n \\end{prop}\n\n \\begin{lemm}\\label{secext}\nLet $X$ be a graded scheme, $f\\in\\gsec(X,\\so_X)_d$ ($d\\in\\Z$) and let $X_f$ be\nthe open subset of $X$ defined by $X_f:=\\{x\\in X\\st f_x\\notin\\I{m}_x\\}$ (where\n$\\I{m}_x\\subset\\so_{X,x}$ is the maximal homogeneous ideal). Then $\\forall\n\\s{F}\\in\\gqco{X}$ we have:\n\\begin{enumerate}\n\n\\item $\\forall s\\in\\gsec(X,\\s{F})$ such that $s\\rest{X_f}=0$, $\\exists n>0$ such that\n$f^ns=0$;\n\n\\item $\\forall t\\in\\gsec(X_f,\\s{F})$ $\\exists n>0$ and $\\exists s\\in\\gsec(X,\\s{F})$ such\nthat $s\\rest{X_f}=f^nt$.\n\\end{enumerate}\n \\end{lemm}\n\n \\begin{proof}\nThe proof is similar (but easier, since the graduation is already in the\nstructure sheaf, and there is no need of an extra line bundle) to that of\n\\cite[II, lemma 5.14]{H1}.\\footnote{Actually this result can be viewed as a\ngeneralization of the cited lemma, which just deals with the particular case of\n{standard} graded schemes (see \\ref{stdchar}).}\n \\end{proof}\n\n \\begin{lemm}\\label{injflasque}\nLet $X$ be a graded scheme and $\\s{I}\\in\\gsmo{X}$ an injective object. Then\n$\\s{I}$ is flasque (i.e., the restriction map $\\gsec(U,\\s{I})\\to\\gsec(V,\\s{I})$\nis surjective if $V\\subseteq U$ are open subsets of $X$).\n \\end{lemm}\n\n \\begin{defi}\nA morphism $f:X\\to Y$ of graded schemes is {\\em affine}\\index{affine!morphism}\nif for some (and then for any) open affine cover $\\{U_i\\st i\\in I\\}$ of $Y$,\n$f^{-1}(U_i)$ is affine $\\forall i\\in I$. $f$ is {\\em finite}\\index{finite\nmorphism} if moreover the induced maps of graded rings\n$\\so_Y(U_i)\\to\\so_X(f^{-1}(U_i))$ are finite. $f$ is a {\\em closed\nimmersion}\\index{closed immersion} if, as a map of topological spaces, it gives\na homeomorphism between $X$ and a closed subset of $Y$, and if $f\\mrs$ is\nsurjective.\n \\end{defi}\n\nGiven a graded scheme $Y$ and a quasi--coherent sheaf of graded\n$\\so_Y$--algebras $\\s{A}$, there exist unique (up to isomorphism) a graded\nscheme $X$ and a morphism $f:X\\to Y$ such that $f^{-1}(U)\\cong\\gspec\\s{A}(U)$\nand $f\\rest{f^{-1}(U)}$ is the morphism corresponding to the natural map\n$\\so_Y(U)\\to\\s{A}(U)$ for every open affine subset $U$ of $Y$, and such that\nthe inclusion morphism $f^{-1}(V)\\mono f^{-1}(U)$ corresponds to the\nrestriction map $\\s{A}(U)\\to\\s{A}(V)$ if $V\\subseteq U$ are open affine subsets\nof $Y$. The graded scheme $X$ will be denoted by $\\gSpec\\s{A}$.\\index{Spec\n@$\\gSpec$}\n\n \\begin{prop}\\label{gSpecA}\n\\begin{enumerate}\n\n\\item If $\\s{A}$ is a quasi--coherent (respectively coherent) sheaf of graded\n$\\so_Y$--algebras, then the natural morphism of graded schemes $f:X:=\n\\gSpec\\s{A}\\to Y$ is affine (respectively finite) and $\\s{A}\\cong f_*\\so_X$.\nConversely, if $f:X\\to Y$ is an affine (respectively finite) morphism of graded\nschemes, then $\\s{A}:=f_*\\so_X$ is a quasi--coherent (respectively coherent)\nsheaf of graded $\\so_Y$--algebras and $X\\cong\\gSpec\\s{A}$.\n\n\\item Let $f:X\\to Y$ be an affine morphism of graded schemes and let $\\s{A}:=\nf_*\\so_X$. Then\n\\[f_*:\\gqco{X}\\to\\gqco{Y}\\cap\\gsmo{\\s{A}}\\]\nis an exact equivalence of abelian categories. If moreover $f$ is finite, then\n$f_*$ restricts to an equivalence $f_*:\\gcoh{X}\\to\\gcoh{Y}\\cap\\gsmo{\\s{A}}$.\n\n\\end{enumerate}\n \\end{prop}\n\n \\begin{lemm}\\label{gcgqgc}\nLet $X$ be a graded scheme and let $\\phi:\\s{G}\\to\\s{F}$ be a surjective\nmorphism in $\\gqco{X}$ with $\\s{F}\\in\\gcoh{X}$. Then there exists a morphism\n$\\psi:\\s{H} \\to\\s{G}$ in $\\gqco{X}$ such that $\\s{H}\\in\\gcoh{X}$ and\n$\\phi\\comp\\psi$ is surjective.\n \\end{lemm}\n\n \\begin{lemm}\\label{gsExtgcoh}\nLet $X$ be a graded scheme and let $\\s{F},\\s{G}\\in\\gcoh{X}$. Then\n$\\gsExt^i_X(\\s{F},\\s{G})\\in\\gcoh{X}$ $\\forall i\\in\\N$.\n \\end{lemm}\n\n\n \\section{Good graded schemes}\n\n \\begin{defi}\\label{defgoodsch}\nLet $X$ be a graded scheme. A point $x\\in X$ is called\n{good}\\index{good!point} or {quasi--standard}\\index{quasi--standard!point} or\n{standard}\\index{standard!point} or {quasi--trivial}\\index{quasi--trivial!point} or\n{trivial}\\index{trivial!point} if the noetherian local graded ring $\\so_{X,x}$\nhas the same property.\n\n$X$ is called {good}\\index{good!graded scheme} or\n{quasi--standard}\\index{quasi--standard!graded scheme} or {standard}\\index{standard!graded\nscheme} or {quasi--trivial}\\index{quasi--trivial!graded scheme} or\n{trivial}\\index{trivial!graded scheme} if every point of $X$ has the same\nproperty.\n \\end{defi}\n\n \\begin{rema}\\label{locgsch}\nIt follows from \\ref{goodlocgood}, \\ref{locgoodgood} and \\ref{lqstdlocqstd}\nthat a graded scheme $X$ has one of the properties of \\ref{defgoodsch} if and\nonly if $\\forall x\\in X$ there is an open neighbourhood $U$ of $x$ which is\nisomorphic to $\\gspec A$ for some noetherian graded ring $A$ with the same\nproperty. Moreover, if $X$ is {good} or {quasi--trivial} or trivial, and if $U$ is any\nopen subset of $X$ such that $U\\cong\\gspec A$ for some noetherian graded ring\n$A$, then $A$ has the same property.\n \\end{rema}\n\nFrom now on we will consider only {good} graded schemes.\n\n \\begin{rema}\nSince a {trivial} graded scheme is just a noetherian scheme, we see that (usual)\nnoetherian schemes are {good} graded schemes.\n \\end{rema}\n\n \\begin{prop}\nIf $X$ is a {good} graded scheme, then $X_0:=(X,(\\so_X)_0)$ is a noetherian\nscheme. If $f:X\\to Y$ is a morphism of {good} graded schemes, then\n$f_0:=(f,f\\mrs_0): X_0\\to Y_0$ is a morphism of noetherian schemes.\n \\end{prop}\n\n \\begin{proof}\nIf $U\\subseteq X$ is an open subset such that $U\\cong\\gspec A$ for some {good}\nnoetherian graded ring $A$, then $U_0\\cong\\spec A_0$: this follows immediately\nfrom \\ref{0p=p0}. The last statement is obvious.\n \\end{proof}\n\n \\begin{rema}\nIf $A$ is a {good} noetherian graded ring, then $(\\gspec A)_0\\cong\\spec A_0$.\n \\end{rema}\n\n \\begin{prop}\\label{YgoodXgood}\nLet $f:X\\to Y$ be a morphism of graded schemes.\n\\begin{enumerate}\n\n\\item If $x\\in X$ is such that $f(x)$ is {quasi--standard} (respectively standard), then $x$\nis also {quasi--standard} (respectively standard). In particular, if $Y$ is {quasi--standard} or standard,\nthen so is $X$.\n\n\\item If $f$ is a closed immersion, then $x\\in X$ is {quasi--standard} (respectively\nstandard) if and only if $f(x)$ is {quasi--standard} (respectively standard).\n\\end{enumerate}\n \\end{prop}\n\n \\begin{proof}\nIt follows immediately from \\ref{gspecAgood}.\n \\end{proof}\n\n \\begin{prop}\nLet $X$ be a {good} graded scheme. Then\n\\begin{align*}\n& \\QT{X}:=\\{x\\in X\\st x\\text{ quasi--trivial}\\}, & \\QS{X}:=\\{x\\in X\\st x\\text{ quasi--standard}\\}\n\\end{align*}\nare closed and open in $X$ and $X=\\QT{X}\\djun\\QS{X}$. In particular, a\nconnected {good} graded scheme is either {quasi--standard} or quasi--trivial.\n \\end{prop}\n\n \\begin{proof}\nIt follows immediately from \\ref{clopen}, taking into account \\ref{locgsch}.\n \\end{proof}\n\n \\begin{rema}\nWe have already noticed that a {trivial} graded scheme is just a usual noetherian\nscheme, and it is very easy to see that a graded scheme $X$ is {quasi--trivial} if and\nonly if $X_0$ is a noetherian scheme and $(\\so_X)_d=0$ for $\\abs{d}>>0$.\n \\end{rema}\n\nAlso {standard} graded schemes are easy to characterize, as the following result\nshows.\n\n \\begin{prop}\\label{stdchar}\nIf $X$ is a noetherian scheme and $\\s{L}$ an invertible sheaf on $X$, then\n$(X,\\bigoplus_{d\\in\\Z}\\s{L}^{\\otimes d})$ is a {standard} graded scheme, which will\nbe denoted by $\\stdsch{X}{\\s{L}}$.\\index{X(L)@$\\stdsch{X}{\\s{L}}$}\n\nConversely, if $Z$ is a {standard} graded scheme, then there is an invertible sheaf\n$\\s{L}$ on $Z_0$ such that $Z\\cong\\stdsch{Z_0}{\\s{L}}$.\n\nIf $\\stdsch{X}{\\s{L}}$ and $\\stdsch{Y}{\\s{M}}$ are {standard} graded schemes,\nthen to give a morphism of graded schemes from $\\stdsch{X}{\\s{L}}$ to\n$\\stdsch{Y}{\\s{M}}$ is equivalent to giving a morphism of schemes $f:X\\to Y$\ntogether with an isomorphism $\\varphi:f^*\\s{M}\\isomor\\s{L}$ of invertible\nsheaves on $X$.\n \\end{prop}\n\n \\begin{proof}\nIf $X$ is a noetherian scheme and $\\s{L}$ is an invertible sheaf on $X$, then\n$\\forall x\\in X$ there is an affine open neighbourhood $U\\cong\\spec A$ of $x$\nwhere $\\s{L}$ is trivial. Then $\\stdsch{X}{\\s{L}}\\rest{U}\\cong\n\\gspec A[t,t^{-1}]$ (with $\\deg(t)=1$), whence $\\stdsch{X}{\\s{L}}$ is standard.\n\nConversely, if $Z$ is standard, then $\\s{L}:=(\\so_Z)_1$ is an invertible sheaf on\n$Z_0$ and the natural morphism\n$(\\so_Z)_{-1}\\to\\s{L}^{\\otimes-1}=\\sHom_{Z_0}(\\s{L},\\so_{Z_0})$ is an\nisomorphism (because it is an isomorphism on each stalk). It follows that there\nis a natural morphism of sheaves of graded rings\n$\\bigoplus_{d\\in\\Z}\\s{L}^{\\otimes d}\\to\\so_Z$, which is an isomorphism (again,\nbecause it is an isomorphism on each stalk), so that $Z\\cong\n\\stdsch{Z_0}{\\s{L}}$ as graded schemes.\n\nIf $g:\\stdsch{X}{\\s{L}}\\to\\stdsch{Y}{\\s{M}}$ is a morphism of {standard} graded\nschemes, then $f:=(g,g\\mrs_0):X\\to Y$ is a morphism of schemes and the morphism\nof $\\so_Y$--modules $g\\mrs_1:\\s{M}\\to f_*\\s{L}$ corresponds by adjunction to a\nmorphism $\\varphi:f^*\\s{M}\\to\\s{L}$ of $\\so_X$--modules, which is easily seen\nto be an isomorphism. Conversely, given $f:X\\to Y$ and an isomorphism\n$\\varphi:f^*\\s{M}\\isomor\\s{L}$, we can naturally extend $\\varphi$ to an\nisomorphism $\\tilde{\\varphi}:\\bigoplus_{d\\in\\Z}f^*(\\s{M}^{\\otimes d})\\cong\nf^*(\\bigoplus_{d\\in\\Z}\\s{M}^{\\otimes d})\\isomor\n\\bigoplus_{d\\in\\Z}\\s{L}^{\\otimes d}$ of sheaves of graded rings on $X$, which,\nby adjunction, corresponds to a morphism\n$\\alpha:\\bigoplus_{d\\in\\Z}\\s{M}^{\\otimes d}\\to\nf_*\\bigoplus_{d\\in\\Z}\\s{L}^{\\otimes d}$ of sheaves of graded rings on $Y$, thus\nyielding a morphism of graded schemes\n$g=(f,\\alpha):\\stdsch{X}{\\s{L}}\\to\\stdsch{Y}{\\s{M}}$.\n \\end{proof}\n\nIn the following we will be mainly interested in {quasi--standard} graded schemes. Even\nwhen we have an ordinary noetherian scheme, we will often regard it as a {standard}\ngraded scheme by choosing a line bundle on it (in the problems we are going to\nconsider, there will be a natural way of making such a choice).\n\nFor the rest of this section $X$ will be a {good} graded scheme: we are going\nto study the relation between graded sheaves on $X$ and sheaves on $X_0$. We\nwill denote by $\\smo{X_0}$\\index{Mod(X_0)@$\\smo{X_0}$} the category of sheaves\nof $\\so_{X_0}$--modules and by $\\qco{X_0}$\\index{Qcoh(X_0)@$\\qco{X_0}$}\n(respectively $\\coh{X_0}$)\\index{Coh(X_0)@$\\coh{X_0}$} its full subcategory of\nquasi--coherent (respectively coherent) sheaves.\n\nSending a graded sheaf of $\\so_X$--modules $\\s{F}$ to $\\s{F}_0$ defines an\nexact functor\n\\[\\usmf{X}:\\gsmo{X}\\to\\smo{X_0}.\\]\nIn analogy with the case of graded modules over a graded ring, we can prove the\nfollowing result.\n\n \\begin{prop}\\label{adjsheaf}\nThe functors\n\\begin{align*}\n& \\osmf{X}:\\smo{X_0}\\to\\gsmo{X}, & \\hsmf{X}:\\smo{X_0}\\to\\gsmo{X}\n\\end{align*}\nare respectively left and right adjoint of $\\usmf{X}$, and they are both right\ninverses of $\\usmf{X}$.\n\nMoreover, $\\usmf{X}$ sends $\\gqco{X}$ (respectively $\\gcoh{X}$) into\n$\\qco{X_0}$ (respectively $\\coh{X_0}$). $\\osmf{X}$ and $\\hsmf{X}$ send\n$\\qco{X_0}$ (respectively $\\coh{X_0}$) into $\\gqco{X}$ (respectively\n$\\gcoh{X}$).\n \\end{prop}\n\n \\begin{proof}\nIt follows easily from \\ref{adj0}.\n \\end{proof}\n\n \\begin{rema}\nBy \\ref{locsubcat} the above result implies that $\\smo{X_0}$ is equivalent to\n$\\gsmo{X}\/\\cat{C}$, where $\\cat{C}$ is the full subcategory of $\\gsmo{X}$ whose\nobjects are the graded sheaves $\\s{F}$ such that $\\s{F}_0=0$. Notice that\n$\\cat{C}$ is both left and right localizing. The same result holds, of course,\nalso for quasi--coherent and coherent sheaves.\n \\end{rema}\n\n \\begin{coro}\n$\\osmf{X}$ and $\\hsmf{X}$ identify $\\smo{X_0}$ (respectively $\\qco{X_0}$,\nrespectively $\\coh{X_0}$) with an additive (but not abelian in general)\nsubcategory of $\\gsmo{X}$ (respectively $\\gqco{X}$, respectively $\\gcoh{X}$).\n \\end{coro}\n\n \\begin{coro}\\label{esssurj}\n$\\usmf{X}:\\gsmo{X}\\to\\smo{X_0}$ is essentially surjective. The same is true for\nits restriction to quasi--coherent and coherent sheaves.\n \\end{coro}\n\n \\begin{prop}\\label{shequivcrit}\n$\\usmf{X}:\\gsmo{X}\\to\\smo{X_0}$ is an equivalence of categories if and only if\n$X$ is standard.\n\nIf this is the case, then the restriction of $\\usmf{X}$ also gives an\nequivalence between $\\gqco{X}$ (respectively $\\gcoh{X}$) and $\\qco{X_0}$\n(respectively $\\coh{X_0}$). Moreover, $\\osmf{X}$ and $\\hsmf{X}$ are isomorphic,\nexact and quasi--inverse of $\\usmf{X}$.\n \\end{prop}\n\n \\begin{proof}\nBy \\ref{adjsheaf} $\\usmf{X}$ is an equivalence if and only if $\\forall \\s{F}\n\\in\\gsmo{X}$ the natural morphism of $\\so_X$--modules $\\s{F}_0\n\\otimes_{\\so_{X_0}}\\so_X\\to\\s{F}$\nis an isomorphism. Then everything follows easily from \\ref{modequivcrit}.\n \\end{proof}\n\n \\begin{prop}\\label{goodcohom}\n$\\forall\\cp{\\s{F}}\\in D^+(\\gsmo{X})$ and $\\forall i\\in\\Z$ there is a natural\nisomorphism $H^i(X,\\cp{\\s{F}})\\cong H^i(X_0,\\cp{\\s{F}}_0)$.\n \\end{prop}\n\n \\begin{proof}\nLet $\\cp{\\s{I}}\\in D^+(\\gsmo{X})$ be an injective resolution of $\\cp{\\s{F}}$,\nso that (by definition)\n\\[H^i(X,\\cp{\\s{F}})\\cong H^i(\\gsec(X,\\cp{\\s{I}})_0)\\cong\nH^i(\\Gamma(X_0,\\cp{\\s{I}_0})).\\]\nOn the other hand, since each $\\s{I}^j$ is flasque by \\ref{injflasque} and\nsince $\\usmf{X}$ is an exact functor which sends flasque sheaves into flasque\nsheaves, $\\cp{\\s{I}_0}$ is a flasque resolution of $\\cp{\\s{F}}_0$ on $X_0$,\nwhich can be used to compute cohomology. Therefore\n$H^i(\\Gamma(X_0,\\cp{\\s{I}_0}))\\cong H^i(X_0,\\s{F}_0)$.\n \\end{proof}\n\n\n \\section{$\\gproj$ of a noetherian positively graded ring}\n\nWe will denote by $\\prng$\\index{PRing@$\\prng$} the full subcategory of the\ncategory of graded rings whose objects are noetherian positively graded rings\n(i.e., noetherian graded rings $R$ such that $R_d=0$ for $d<0$).\n\n \\begin{rema}\nIt follows from \\ref{Anoeth} that $R\\in\\prng$ is isomorphic to a quotient of a\npolynomial ring $R_0[x_0,\\dots,x_n]$, where $\\deg(x_i)>0$ and $R_0$ is a\nnoetherian ring.\n \\end{rema}\n\nIf $A$ is a noetherian ring and $\\w=(\\w_0,\\dots,\\w_n)\\in\\N_+^{n+1}$, the\nnoetherian positively graded ring $A[x_0,\\dots,x_n]$ where $\\deg(x_i)=w_i$ will\nbe denoted by $\\p_A(\\w)$ (or simply by $\\p(\\w)$\\index{P(w)@$\\p$, $\\p(\\w)$} if\nthere can be no doubt about $A$).\n\n \\begin{defi}\nGiven $R\\in\\prng$, $\\gproj R$\\index{Proj@$\\gproj$} is the open graded subscheme\n$\\gspec R\\setminus\\V{R_+}$ of $\\gspec R$.\n \\end{defi}\n\n \\begin{rema}\nSince $\\gproj R=\\bigcup_{r\\in R_+\\hom}\\D{r}$ and (if $r$ is not nilpotent)\n$\\D{r}\\cong\\gspec R_r$, we see that $\\gproj R$ is a {quasi--standard} graded scheme\n(actually it is just the open graded subscheme of $\\gspec R$ consisting of\n{quasi--standard} points). It is also clear that $(\\gproj R)_0=\\proj R$ and $\\so_{\\gproj\nR}=\\bigoplus_{d\\in\\Z}\\so_{\\proj R}(d)$.\n \\end{rema}\n\n \\begin{exem}\nIf $R=\\p_A(\\w)$ for some noetherian ring $A$ and some $\\w=(\\w_0,\\dots,\\w_n)\\in\n\\N_+^{n+1}$, $\\gproj R$ will be denoted by $\\gP_A(\\w)$ and $\\proj R$ by\n$\\P_A(\\w)$ (again, we will simply write $\\gP(\\w)$\\index{P(w) @$\\gP$, $\\gP(\\w)$}\nand $\\P(\\w)$\\index{P(w)@$\\P$, $\\P(\\w)$} if $A$ is clear from the context). It\nwill be called (graded) weighted projective space (over $A$) of\nweights $\\w$. Of course, if $\\w=(1,\\dots,1)$, it will be denoted by\n$\\gP^n$ or $\\P^n$.\n \\end{exem}\n\n \\begin{rema}\nClearly $\\gproj R$ is a {standard} graded scheme if $R$ is generated by $R_1$ as an\n$R_0$--algebra. The converse is false in general (examples are given by the\ncanonical rings of surfaces of general type, see \\ref{canmodstd}), but it is\ntrue if $R$ is of the form $\\p(\\w)$, as the following result shows.\n \\end{rema}\n\n \\begin{prop}\\label{stdmori}\nLet $A$ be a noetherian ring and $\\w=(\\w_0,\\dots,\\w_n)\\in\\N_+^{n+1}$. Then\n$\\I{p}\\in\\gP_A(\\w)$ is {standard} if and only if $\\gcd(\\w_i\\st x_i\\notin\\I{p})=1$.\nIn particular, $\\gP_A(\\w)$ is {standard} if and only if $\\w=(1,\\dots,1)$.\n \\end{prop}\n\n \\begin{proof}\nBy \\ref{Apqstd} it is enough to prove that $I(\\I{p})=(\\w_i\\st x_i\\notin\n\\I{p})$. Obviously $\\w_i\\in I(\\I{p})$ if $x_i\\notin\\I{p}$. Conversely, if $d>0$\nis such that $\\I{p}_d\\ne\\p_A(\\w)_d$, then there exists a monomial $x_0^{m_0}\n\\cdots x_n^{m_n}\\in\\p_A(\\w)_d\\setminus\\I{p}$. It follows that\n$d=\\sum_{i=0}^{n}m_i\\w_i\\in I(\\I{p})$, since $x_i\\notin\\I{p}$ if $m_i>0$.\n\nThe last statement follows at once, taking into account that $\\forall i=0,\\dots,n$\nthere exists $\\I{p}\\in\\gP_A(\\w)$ such that $(x_0,\\dots,\\hat{x_i},\\dots,x_n)\n\\subseteq\\I{p}$.\n \\end{proof}\n\n \\begin{rema}\nThe above result says that the (open) subset of {standard} points in a weighted\nprojective space coincides with the regular locus which was first considered in\n\\cite{M}: it is the locus where ``everything behaves well''(see also\n\\cite{BR}), and it is contained in the smooth locus (in general strictly, e.g.\nit is empty if $\\gcd(\\w_0,\\dots,\\w_n)\\ne1$).\n \\end{rema}\n\n \\begin{prop}\\label{projmor}\nLet $A$ be a noetherian ring, let $X\\in\\gsch_A$ and, for $i=0,\\dots,n$, let\n$s_i\\in\\gsec(X,\\so_X)_{\\w_i}$ for some $\\w=(\\w_0,\\dots,\\w_n)\\in\\N_+^{n+1}$.\nThen the $s_i$ induce a rational map $f:X\\rat\\gP_A(\\w)$ in $\\gsch_A$, defined\non the open subset\n\\[U:=\\{x\\in X\\st ((s_0)_x,\\dots,(s_n)_x)=\\so_{X,x}\\}\\subseteq X,\\]\nsuch that $f^*(x_i)=s_i\\rest{U}$ for $i=0,\\dots,n$. In particular, $f$ is a\nmorphism (i.e. $U=X$) if and only if $\\{s_0,\\dots,s_n\\}$ generate $\\so_X$ (and\nconversely, of course, given a morphism $g:X\\to\\gP_A(\\w)$ in $\\gsch_A$, the\nsections $\\{g^*(x_0),\\dots,g^*(x_n)\\}$ generate $\\so_X$).\n \\end{prop}\n\n \\begin{proof}\nThe $s_i$ clearly determine a morphism of graded $A$-algebras\n$\\varphi:\\p(\\w)=\\p_A(\\w) \\to\\gsec(X,\\so_X)$ (defined by $\\varphi(x_i):=s_i$),\nwhich (by \\ref{gspecequiv}) corresponds to a morphism $f':X\\to\\gspec\\p(\\w)$ in\n$\\gsch_A$. Since $f'$, as a map of topological spaces, is defined by\n$f'(x):=\\{a\\in\\p(\\w)\\st \\varphi(a)_x\\in\\I{m}_x\\}$, it is easy to see that\n${f'}^{-1}(\\gP(\\w))=U$, and then everything follows.\n \\end{proof}\n\n \\begin{rema}\n$U$ is {quasi--standard} (even {standard} if $\\w=(1,\\dots,1)$) by \\ref{YgoodXgood}, and in\nthis case the sections $s^i$ also determine a rational map $f_0:X_0\\rat\n\\P_A(\\w)$ (over $A$), defined on $U_0$. Notice that if $X$ is {standard} (say, by\n\\ref{stdchar}, $X\\cong\\stdsch{Y}{\\s{L}}$ for some noetherian scheme $Y$ and some\ninvertible sheaf $\\s{L}$ on $Y$), then $s_i\\in\\Gamma(Y,\\s{L}^{\\w_i})$.\nTherefore \\ref{projmor} implies in particular the well known result\nthat to give a morphism of (noetherian) schemes over $A$ from $Y$ to\n$\\P^n_A$ is equivalent to giving an invertible sheaf on $Y$ and $n+1$\nglobal sections which generate it.\n \\end{rema}\n\nIf $R\\in\\prng$ and $M\\in\\gmo{R}$, by abuse of notation we will denote\n$M\\gsh{R}\\rest{\\gproj R}$ again by $M\\gsh{R}$ (or simply by\n$M\\gsh{}$,\\index{$\\gsh{R}$} if there is no doubt about the ring). Also, if\n$\\s{F}\\in\\gsmo{\\gproj R}$, we will usually write\n$\\gsec(\\s{F})$\\index{Gammma(F)@$\\gsec(\\s{F})$} instead of $\\gsec(\\gproj\nR,\\s{F})$.\n\nA morphism $\\varphi:R\\to S$ in $\\prng$ induces a rational map of graded schemes\n$f:\\gproj S\\rat\\gproj R$, defined on the open subset $U:= \\{\\I{p}\\in\\gproj\nS\\st\\varphi(R_+)\\nsubseteq\\I{p}\\}$ of $\\gproj S$ ($f$ is just the restriction\nof the induced morphism $f':\\gspec S\\to\\gspec R$). Notice that $f$ is affine,\nsince $f'$ is. Observe also that $f$ is a morphism (i.e. $U=\\gproj S$) if and\nonly if $\\sqrt{(\\varphi(R_+))}\\supseteq S_+$. The following result is an\nimmediate consequence of \\ref{modaff}.\n\n \\begin{prop}\\label{affproj}\nLet $\\varphi:R\\to S$ be a morphism in $\\prng$ and let $f:U\\subseteq\\gproj S\\to\nX:=\\gproj R$ be the induced morphism of graded schemes.\n\\begin{enumerate}\n\n\\item If $M,N\\in\\gmo{R}$, then $(M\\otimes_R N)\\gsh{}\\cong M\\gsh{}\\otimes_X\nN\\gsh{}$.\n\n\\item If $M\\in\\gmo{R}$ and $N\\in\\gmo{S}$, then $f^*(M\\gsh{R})\\cong\n(M\\otimes_RS)\\gsh{S}\\rest{U}$ and $f_*(N\\gsh{S}\\rest{U})\\cong N\\gsh{R}$.\n\\end{enumerate}\n \\end{prop}\n\n \\begin{rema}\nThe above result (except that $f_*(N\\sh{S}\\rest{U})\\cong N\\sh{R}$ still holds)\nis not true in general with $\\proj$ instead of $\\gproj$.\n \\end{rema}\n\n \\begin{prop}\\label{prngmor}\nLet $\\varphi:R\\to S$ be a morphism in $\\prng$ and let $f:\\gproj S\\rat\\gproj R$\nbe the induced rational map. Then the following conditions are equivalent:\n\\begin{enumerate}\n\n\\item $\\varphi:R\\to S$ is finite;\n\n\\item $f$ is a morphism and $\\varphi_0:R_0\\to S_0$ is finite.\n\n\\end{enumerate}\nIf these conditions are satisfied, then $f:\\gproj S\\to\\gproj R$ (and also\n$f_0:\\proj S\\to\\proj R$) is finite.\n \\end{prop}\n\n \\begin{proof}\n\\begin{description}\n\n\\item [$1\\implies2$] By hypothesis there exist $s_1,\\dots,s_n\\in S\\hom$ which\ngenerate $S$ as an $R$--module. Now $\\forall s\\in S\\hom_+$ we can choose $m>0$\nsuch that $m\\deg(s)>\\deg(s_i)$ for $i=1,\\dots,n$. Then there exist $r_1,\n\\dots,r_n\\in R\\hom_+$ such that $s^m=\\sum_{i=1}^nr_is_i$, whence $s\\in\n\\sqrt{(\\varphi(R_+))}$, which proves that $f$ is a morphism. Moreover, the\n$s_i$ of degree $0$ generate $S_0$ as an $R_0$--module.\n\n\\item [$2\\implies1$] Since $S$ is a finitely generated $S_0$--algebra and\n$\\sqrt{(\\varphi(R_+))}\\supseteq S_+$, it is easy to see that there exists $m>0$\nsuch that $(\\varphi(R_+))_{>m}=S_{>m}$. By \\ref{Anoeth} and since $\\varphi_0$\nis finite, there exist $s_1,\\dots,s_n\\in S\\hom$ which generate\n$\\bigoplus_{0\\le d\\le m}S_d$ as an $R_0$--module, and then it is not difficult\nto prove that $s_1,\\dots,s_n$ generate $S$ as an $R$--module.\n\n\\end{description}\nAs for the last statement, it is enough to observe that $\\forall r\\in R\\hom_+$ non\nnilpotent\n\\[f\\rest{f^{-1}(\\D{r})}:f^{-1}(\\D{r})=\\D{\\varphi(r)}\\cong\n\\gspec S_{\\varphi(r)}\\to\\D{r}\\cong\\gspec R_r\\] is the morphism of affine graded\nschemes induced by the finite morphism of graded rings $\\varphi_r:R_r\\to\nS_{\\varphi(r)}$.\n \\end{proof}\n\nFor the rest of this section $R$ will be a noetherian positively graded ring,\n$\\gproj R$ will be denoted by $X$ and $R_0$ by $A$. We are going to see that a\ngeneralized version of Serre's correspondence holds between (finitely\ngenerated) graded $R$--modules and (quasi)coherent sheaves on $X$ (see also\n\\cite{B} and \\cite{AKO}).\n\n \\begin{prop}\\label{gmogqco}\n\\begin{enumerate}\n\n\\item $\\gsec(-):\\gsmo{X}\\to\\gmo{R}$ is right adjoint of the exact functor\n$\\gsh{}:\\gmo{R}\\to\\gsmo{X}$.\n\n\\item If $M\\in\\gmo{R}$ (respectively $\\gfmo{R}$), then $M\\gsh{}$ is\nquasi--coherent (respectively coherent).\n\n\\item If $\\s{F}\\in\\gqco{X}$, then the natural map $\\gsec(\\s{F})\\gsh{}\\to\\s{F}$\nis an isomorphism.\n\n\\end{enumerate}\n \\end{prop}\n\n \\begin{proof}\nAs for $1$, denoting by $j:X\\mono\\gspec R:=Y$ the inclusion, $\\forall M\\in\\gmo{R}$\nand $\\forall\\s{F}\\in\\gsmo{X}$ there are natural isomorphisms\n\\[\\Hom_R(M,\\gsec(\\s{F}))\\cong\\Hom_R(M,\\gsec(Y,j_*\\s{F}))\\cong\n\\Hom_Y(M\\gsh{},j_*\\s{F})\\cong\\Hom_X(M\\gsh{},\\s{F})\\] (using the fact that\n$\\gsec(Y,-)$ is right adjoint of $\\gsh{}$ and $j^*$ is left adjoint of $j_*$).\n$2$ is clear, since the same is true for $M\\gsh{}$ on $Y$, and the proof of $3$\nis similar to that of \\cite[II, prop. 5.15]{H1} (using \\ref{secext} instead of\n\\cite[II, lemma 5.14]{H1}).\n \\end{proof}\n\nIf $\\s{F}\\in\\gcoh{X}$, it is not true in general that $\\gsec(\\s{F})\\in\\gfmo{R}$\n(it can happen that $\\gsec(\\s{F})_d\\ne0$ $\\forall d\\in\\Z$, for instance if $\\s{F}$\nis supported at a point), but, as we are going to see, the truncations\n$\\gsec(\\s{F})_{>d}$ are finitely generated.\n\n \\begin{lemm}\\label{lbcohom}\nGiven $\\w\\in\\N_+^{n+1}$, let $\\p(\\w)=\\p_A(\\w)$ and $\\gP(\\w)=\\gP_A(\\w)$.\n\\begin{enumerate}\n\n\\item The natural map $\\p(\\w)\\to\\gsec(\\so_{\\gP(\\w)})$ is an isomorphism in\n$\\gmo{\\p(\\w)}$; in particular, $H^0(\\gP(\\w),\\so_{\\gP(\\w)}(m))\\cong\\p(\\w)_m$\n$\\forall m\\in\\Z$.\n\n\\item $H^i(\\gP(\\w),\\so_{\\gP(\\w)}(m))=0$ for $00$.\n\\end{enumerate}\n \\end{prop}\n\n \\begin{coro}\\label{gcohfg}\nIf $\\s{F}\\in\\gcoh{X}$, then $\\gsec(\\s{F})_{>d}\\in\\gfmo{R}$ $\\forall d\\in\\Z$.\n \\end{coro}\n\n \\begin{proof}\n$R=\\p(\\w)\/I$ for some $n\\in\\N$, some $\\w\\in\\N_+^{n+1}$ and some ideal $I\\subset\n\\p(\\w):=\\p_A(\\w)$, and (by \\ref{prngmor}) the projection $R\\to\\p(\\w)$ induces a\nfinite morphism $f:X\\to\\gP(\\w)$ (which is actually a closed immersion), whence\n$f_*\\s{F}\\in\\gcoh{\\gP(\\w)}$. Moreover, since $\\gsec(X,\\s{F})\\cong\\gsec(\\gP(\\w),\nf_*\\s{F})$ and since an $R$--module is finitely generated if and only if it is\nfinitely generated as a $\\p(\\w)$--module, we can assume $R=\\p(\\w)$ and $X=\n\\gP(\\w)$.\n\nBy part 1 of \\ref{cohcohom} there is an exact sequence of the form\n\\[0\\to\\s{R}\\to\\s{E}\\to\\s{F}\\to0\\]\nwith $\\s{E}=\\bigoplus_{j=1}^r\\so_{\\gP(\\w)}(-m_j)$ for some $r\\in\\N$ and some\n$m_j\\in\\Z$. Since $\\s{R}$ is also coherent, by part 3 of \\ref{cohcohom} $\\exists\nm_0\\in\\Z$ such that $H^1(\\gP(\\w),\\s{R}(m))=0$ for $m\\ge m_0$, so that the map\n$H^0(\\gP(\\w),\\s{E}(m))\\to H^0(\\gP(\\w),\\s{F}(m))$ is surjective. From this and\nfrom part 1 of \\ref{lbcohom} it follows that $M:=\\gsec(\\s{F})$ is generated by\n$M_{m_j$ for $j=0\\dots,r$. Now, $\\forall d\\in\\Z$ take $d'\\in\\Z$\nsuch that $d'\\ge l$ and $d'>d+\\w_j$ for $j=0\\dots,n$. Then it is clear that\n$M_{>d}$ is generated by $\\bigoplus_{dm.\n\\end{align*}\n \\end{lemm}\n\nWe are now ready to prove the generalized version of Serre's correspondence;\nsee \\ref{abelquot} for the notion of localizing subcategory and of quotient\nof an abelian category by a localizing subcategory.\n\n \\begin{prop}\\label{Serre}\nLet $\\cat{Z}(R)$ (respectively $\\cat{z}(R)$) be the full right localizing\nsubcategory of $\\gmo{R}$ (respectively $\\gfmo{R}$) whose objects are the\nmodules $M$ such that $M\\gsh{}=0$, and let $T:\\gmo{R}\\to\\gmo{R}\/\\cat{Z}(R)$\n(respectively $T':\\gfmo{R}\\to\\gfmo{R}\/\\cat{z}(R)$) be the natural functor. Then\nthe functor $\\gsh{}:\\gmo{R}\\to\\gsmo{X}$ induces an exact equivalence of\ncategories $F:\\gmo{R}\/\\cat{Z}(R)\\to\\gqco{X}$ (respectively $F':\n\\gfmo{R}\/\\cat{z}(R)\\to\\gcoh{X}$), whose quasi--inverse is the functor $T\\comp\n\\gsec(-)$ (respectively $T'\\comp\\gsec(-)_{>0}$).\n \\end{prop}\n\n \\begin{proof}\nThe statement about $\\gmo{R}$ and $\\gqco{X}$ is an immediate consequence of\n\\ref{locsubcat} and \\ref{gmogqco}. The other statement follows from\n\\ref{gcohfg}, if one observes that (by definition of quotient category)\n$\\gfmo{R}\/\\cat{z}(R)$ can be identified with a full subcategory of\n$\\gmo{R}\/\\cat{Z}(R)$ and that $\\forall M\\in\\gmo{R}$ the inclusion $M_{>0}\\mono M$\nis an isomorphism in $\\gmo{R}\/\\cat{Z}(R)$.\n \\end{proof}\n\n \\begin{rema}\nIt is not difficult to prove (using the fact that $R$ is a noetherian ring)\nthat $\\forall M,N\\in\\gmo{R}$\n\\[\\Hom_{\\gmo{R}\/\\cat{Z}(R)}(M,N)\\cong\\lim_{M'\\in I(M)}\\Hom_{\\gmo{R}}(M',N),\\]\nwhere $I(M):=\\{M'\\subseteq M\\st M\/M'\\in\\cat{Z}(R)\\}$. It follows (since if $L\n\\in\\gfmo{R}$, $L\\in\\cat{z}(R)$ if and only if $L_d=0$ for $d>>0$) that if\n$M\\in\\gfmo{R}$ then\n\\[\\Hom_{\\gmo{R}\/\\cat{Z}(R)}(M,N)\\cong\\lim_{n\\in\\N}\\Hom_{\\gmo{R}}(M_{>n},N).\\]\nTherefore, in the particular case $X=\\gP^n$ (and more generally when $X$ is\nstandard), considering the equivalence (given by \\ref{shequivcrit}) between\n$\\gcoh{X}$ and $\\coh{X_0}$, we obtain the usual description of $\\coh{X_0}$, as\nproved in \\cite{S}.\n \\end{rema}\n\nNow we will give also a graded version of Serre duality. Still denoting\n$\\gproj R$ by $X$ (and $\\proj R$ by $X_0$), we assume now that $R_0$ is a\nfield $\\K$ and that $\\dim X_0=n$. We recall that on $X_0$ there exists unique\n(up to isomorphism) a dualizing sheaf $\\ds_{X_0}$ (see\n\\cite[III, prop.7.5]{H1}).\n\n \\begin{defi}\nA dualizing sheaf on $X$ is a coherent graded sheaf\n$\\ds_X$\\index{omega_X@$\\ds_X$} together with a map $t:H^n(X,\\ds_X)\\to\\K$ such\nthat $\\forall\\s{F}\\in\\gcoh{X}$ the bilinear map\n\\[\\Hom_X(\\s{F},\\ds_X)\\times H^n(X,\\s{F})\\to H^n(X,\\ds_X)\\mor{t}\\K\\]\nis a perfect pairing of $\\K$--vector spaces (i.e., the induced map\n$\\Hom_X(\\s{F},\\ds_X)\\to H^n(X,\\s{F})\\dual$ is an isomorphism).\n \\end{defi}\n\n \\begin{prop}\\label{Serredual}\nA dualizing sheaf on $X$ exists unique (up to isomorphism), and it is given by\n$\\ds_X=\\gsHom_{X_0}(\\so_X,\\ds_{X_0})$. If moreover $X_0$ is Cohen--Macaulay and\nequidimensional and $\\sExt^i_{X_0}(\\so_{X_0}(m),\\ds_{X_0})=0$ for $i>0$ and\n$\\forall m\\in\\Z$, then $\\forall i\\ge0$ there are natural isomorphisms\n$\\Ext^i_X(\\s{F},\\ds_X)\\cong H^{n-i}(X,\\s{F})\\dual$.\n \\end{prop}\n\n \\begin{proof}\nUniqueness follows as usual from the uniqueness of an object representing a\nfunctor. Since, given $\\s{F}\\in\\gcoh{X}$, we have $H^n(X,\\s{F})\\cong\nH^n(X_0,\\s{F}_0)$ by \\ref{goodcohom} and $\\Hom_X(\\s{F},\\ds_X)\\cong\n\\Hom_{X_0}(\\s{F}_0,\\ds_{X_0})$ by \\ref{adjsheaf}, it is clear that $\\ds_X$ is a\ndualizing sheaf.\n\nAs for the last statement, remember that (by \\cite[III, thm. 7.6]{H1}) $X_0$ is\nCohen--Macaulay and equidimensional if and only if $\\forall\\s{G}\\in\\coh{X_0}$\nthere are natural isomorphisms $\\Ext^i_{X_0}(\\s{G},\\ds_{X_0})\\cong\nH^{n-i}(X_0,\\s{G})\\dual$. As before, if $\\s{F}\\in\\gcoh{X}$ then\n$H^{n-i}(X,\\s{F})\\cong H^{n-i}(X_0,\\s{F}_0)$, so that it is enough to prove that\n$\\Ext^i_X(\\s{F},\\ds_X)\\cong\\Ext^i_{X_0}(\\s{F}_0,\\ds_{X_0})$. Let $\\cp{\\s{I}}$ be\nan injective resolution of $\\ds_{X_0}$, and observe that then $\\cp{\\s{J}}:=\n\\gsHom_{X_0}(\\so_X,\\cp{\\s{I}})$ is an injective resolution of $\\ds_X$ (it is\ninjective because $\\hsmf{X}$ preserves injective objects, as it is easy to\ncheck, and it is a resolution because $\\gsExt^i_{X_0}(\\so_X,\\ds_{X_0})\\cong\n\\bigoplus_{m\\in\\Z}\\sExt^i_{X_0}(\\so_{X_0}(m),\\ds_{X_0})=0$ for $i>0$). Then,\nagain by \\ref{adjsheaf}, we have\n\\[\\Ext^i_X(\\s{F},\\ds_X)\\cong H^i(\\Hom_X(\\s{F},\\cp{\\s{J}}))\\cong\nH^i(\\Hom_{X_0}(\\s{F}_0,\\cp{\\s{I}}))\\cong\\Ext^i_{X_0}(\\s{F}_0,\\ds_{X_0}).\\]\n \\end{proof}\n\n\n \\section{Graded schemes and algebraic stacks}\\label{stacks}\n\nWe assume the reader is familiar with algebraic stacks (see e.g.\n\\cite{LM}); here we just recall that, if a (smooth) algebraic group\n$G$ acts on a scheme $Z$, the quotient stack $[Z\/G]$ is algebraic (the\nprojection $Z\\to[Z\/G]$ is a smooth presentation) and $\\coh{[Z\/G]}$\n(respectively $\\qco{[Z\/G]}$) is equivalent to $G$--$\\coh{Z}$\n(respectively $G$--$\\qco{Z}$), the category of (quasi)coherent\n$G$--equivariant sheaves on $Z$.\n\nIf $R\\in\\prng$ and $R_0=\\K$ is a field, \\ref{Serre} and \\cite[prop. 2.3]{AKO}\nimply that the category $\\gcoh{\\gproj R}$ (respectively $\\gqco{\\gproj R}$) is\nequivalent to $\\coh{[(\\spec R\\setminus\\{R_+\\})\/\\mult]}$ (respectively\n$\\qco{[(\\spec R\\setminus\\{R_+\\})\/\\mult]}$), where the action of\n$\\mult:=\\spec\\K[x,x^{-1}]$ is the natural one induced by the graduation of $R$\n(such that the quotient scheme is the usual $\\proj R$). We are going to see\nbriefly how this result can be generalized; a more accurate study of the\nrelation between graded schemes (considered in a more general sense, namely\nallowing the grading group to be different from $\\Z$) and algebraic stacks will\nappear in a future paper.\n\nLet $X$ be a graded scheme, which we assume for simplicity of finite type over\na field $\\K$: we are going to define a scheme $\\ung{X}$ together with an action\nof $\\mult$ on it. Assume first that $X$ is affine, say $X=\\gspec A$ for some\ngraded finitely generated $\\K$--algebra $A$. To avoid confusion, we denote by\n$\\ung{A}$ the ring obtained from $A$ by forgetting the graduation (i.e.,\n$\\ung{A}=\\ung{A}_0=A$). The graduation of $A$ yields an action\n$\\mult\\times\\spec\\ung{A}\\cong\\spec\\ung{A}[x,x^{-1}]\\to\\spec\\ung{A}$ of $\\mult$\non $\\ung{X}:=\\spec\\ung{A}$ (induced by the morphism of rings\n$\\ung{A}\\to\\ung{A}[x,x^{-1}]$ defined by $a\\mapsto ax^{\\deg(a)}$ for $a\\in\nA\\hom$). It is then easy to see that for general $X$ we can take an open cover\nof $X$ by affine graded schemes $U_i$ and glue the schemes $\\ung{U_i}$ to a\nscheme $\\ung{X}$ with an action $\\rho_0:\\mult\\times\\ung{X}\\to\\ung{X}$: this is\npossible (and the result does not depend, up to isomorphism, on the chosen\ncover) because the above construction is compatible with localization, in the\nsense that, if $a\\in A\\hom$ (so that $\\gspec A_a$ can be identified with an\nopen graded subscheme of $\\gspec A$), then $\\spec\\ung{A_a}$ can be identified\nwith an open subscheme of $\\spec\\ung{A}$ and the action of $\\mult$ on\n$\\spec\\ung{A_a}$ is the restriction of the action on $\\spec\\ung{A}$. It is also\nclear that, if $X=\\gproj R$, then $\\ung{X}=\\spec\\ung{R}\\setminus\\V{R_+}$ and\n$\\rho_0$ is the restriction of the action on $\\spec\\ung{R}$.\n\n \\begin{prop}\nThere is an equivalence of categories between $\\gcoh{X}$ (respectively\n$\\gqco{X}$) and $\\coh{[\\ung{X}\/\\mult]}$ (respectively $\\qco{[\\ung{X}\/\\mult]}$).\n \\end{prop}\n\n \\begin{proof}\nWe claim that $\\rho_0:\\mult\\times\\ung{X}\\to\\ung{X}$ can be extended to\nan action $\\rho:\\mult\\times\\tilde{X}\\to\\tilde{X}$ of $\\mult$ on the\n{standard} graded scheme $\\tilde{X}:=\\stdsch{\\ung{X}}{\\so_{\\ung{X}}}$\nand that there is a morphism of graded schemes $p:\\tilde{X}\\to X$ such\nthat the diagram\n\\[\\begin{CD}\n\\mult\\times\\tilde{X} @>{\\rho}>> \\tilde{X} \\\\\n@V{\\pi}VV @VVpV \\\\\n\\tilde{X} @>>p> X\n\\end{CD}\\]\n(where $\\pi$ is the projection) is cartesian in $\\gsch_{\\K}$ (and also\ncocartesian, so that $X$ is the quotient $\\tilde{X}\/\\mult$ as graded scheme).\nIf $X=\\gspec A$ is affine, $\\tilde{X}\\cong\\gspec\\ung{A}[t,t^{-1}]$,\n$\\mult\\times\\tilde{X}\\cong\\gspec\\ung{A}[t,t^{-1},x,x^{-1}]$ (with $\\deg(t)=1$\nand $\\deg(x)=0$) and $\\pi$ and $\\rho$ are induced by the morphisms of graded\nrings $\\iota,\\varphi:\\ung{A}[t,t^{-1}]\\to \\ung{A}[t,t^{-1},x,x^{-1}]$ defined\nby $\\iota(at^i):=at^i$ and $\\varphi(at^i):=at^ix^{\\deg(a)-i}$ for $a\\in A\\hom$\nand $i\\in\\Z$. We take as $p$ the morphism induced by the morphism of graded\nrings $\\alpha:A\\to\\ung{A}[t,t^{-1}]$ defined by $a\\mapsto at^{\\deg(a)}$ for\n$a\\in A\\hom$. Indeed, it is straightforward to check that the diagram of graded\nrings\n\\[\\begin{CD}\n\\ung{A}[t,t^{-1},x,x^{-1}] @<{\\varphi}<< \\ung{A}[t,t^{-1}] \\\\\n@A{\\iota}AA @AA{\\alpha}A \\\\\n\\ung{A}[t,t^{-1}] @<<{\\alpha}< A\n\\end{CD}\\]\nis cocartesian (and also cartesian), and the general claim follows easily from\nthis.\n\nThen, extending classical results of faithfully flat descent theory to the\ngraded setting, the fact that $p$ is faithfully flat (and even smooth) because\nthe same is true for $\\alpha$ (since $\\ung{A}[t,t^{-1}]$, as a graded\n$A$--module via $\\alpha$, is isomorphic to $\\bigoplus_{i\\in\\Z}A(i)$) implies\nthat $p^*$ gives an equivalence of categories between $\\gcoh{X}$ (respectively\n$\\gqco{X}$) and $\\mult$--$\\gcoh{\\tilde{X}}$ (respectively\n$\\mult$--$\\gqco{\\tilde{X}}$). As $\\tilde{X}$ is standard, the natural\ngeneralization of \\ref{shequivcrit} to the case of $\\mult$--equivariant sheaves\nallows to conclude that $\\mult$--$\\gcoh{\\tilde{X}}$ (respectively\n$\\mult$--$\\gqco{\\tilde{X}}$) is equivalent to $\\mult$--$\\coh{\\ung{X}}$\n(respectively $\\mult$--$\\qco{\\ung{X}}$), hence to $\\coh{[\\ung{X}\/\\mult]}$\n(respectively $\\qco{[\\ung{X}\/\\mult]}$).\n \\end{proof}\n\n \\begin{rema}\nIf $X$ is {quasi--standard} (in particular, if $X$ is of the form $\\gproj R$) then\n$[\\ung{X}\/\\mult]$ is a Deligne--Mumford stack (and even a scheme if $X$ is\nstandard), at least if $\\K$ is algebraically closed of characteristic $0$. In fact,\nwe can reduce to the case $X=\\gspec A$ with $A$ quasi--standard, say $s\\in A_d$ is\ninvertible for some $d>0$. Denoting by\n$\\roots{d}:=\\spec\\K[x]\/(x^d-1)\\cong\\Z\/d\\Z$ the subgroup of $\\mult$ of $d^{\\rm\nth}$ roots of unit, the above described action of $\\mult$ induces an action of\n$\\roots{d}$ on $\\spec(\\ung{A}\/(s-1))$ (which can be again extended to an action\nof $\\roots{d}$ on $\\gspec(\\ung{A}\/(s-1)[t,t^{-1}])$), and it can be proved that\n$[\\spec\\ung{A}\/\\mult]$ is isomorphic to $[\\spec(\\ung{A}\/(s-1))\/\\roots{d}]$ (for\nwhich the projection $\\spec(\\ung{A}\/(s-1))\\to[\\spec(\\ung{A}\/(s-1))\/\\roots{d}]$\nis an \\'etale presentation, and obviously an isomorphism if $d=1$). Observe\nthat the morphism of graded rings (induced by $\\alpha$)\n$A\\to\\ung{A}\/(s-1)[t,t^{-1}]\\cong A[t]\/(t^d-s)$ is \\'etale because $s$ is\ninvertible.\n\nIn particular, in the case of weighted projective space $\\gP(\\w)$ one obtains a\nsmooth Deligne--Mumford stack locally isomorphic to\n$[\\mathbb{A}^n\/\\roots{\\w_i}]$ (this is the point of view of \\cite{K}, where\nimportant results about bounded derived categories of coherent sheaves are\nextended from smooth varieties to smooth Deligne--Mumford stacks).\n \\end{rema}\n\n\n\n \\chapter{Beilinson's theorem on $\\gP(\\w)$}\n\n\nThroughout this chapter we will consider (graded) weighted projective spaces\nover a fixed field $\\K$.\\footnote{Probably much of what follows is still true\nif $\\K$ is replaced by an arbitrary noetherian ring.} Therefore, given $n\\in\\N$\nand $\\w\\in\\N_+^{n+1}$, we will write $\\p(\\w)$,\\index{P(w)@$\\p$, $\\p(\\w)$}\n$\\P(\\w)$\\index{P(w)@$\\P$, $\\P(\\w)$} and $\\gP(\\w)$\\index{P(w) @$\\gP$, $\\gP(\\w)$}\n(or simply $\\p$, $\\P$ and $\\gP$ if no confusion is possible) instead of\n$\\p_{\\K}(\\w)=\\K[x_0,\\dots,x_n]$ (with $\\deg(x_i)=\\w_i$),\n$\\P_{\\K}(\\w)=\\proj\\p(\\w)$ and $\\gP_{\\K}(\\w)=\\gproj\\p(\\w)$. $\\forall m\\in\\Z$ we set\nmoreover $\\pd_m=\\pd_m(\\w):=\\dim_{\\K}\\p(\\w)_m$.\\index{p_m(w)@$\\pd_m$,\n$\\pd_m(\\w)$} $\\w$ is called {\\em normalized}\\index{normalized} if\n$\\gcd(\\w_0,\\dots,\\hat{\\w}_i,\\dots,\\w_n)=1$ for $i=0,\\dots,n$. We recall that\n$\\P(\\w)$ is always isomorphic to $\\P(\\w')$ for some $\\w'$ normalized (of\ncourse, this is not true for $\\gP(\\w)$). If $I \\subseteq\\{0,\\dots,n\\}$, we set\n$\\sw{\\w_I}:=\\sum_{i\\in I}\\w_i$;\\index{w_I@$\\sw{\\w_I}$}\n$\\sw{\\w_{\\{0,\\dots,n\\}}}$ will be usually denoted by\n$\\sw{\\w}$.\\index{w@$\\sw{\\w}$} We will write $\\compl{I}$\\index{I^c@$\\compl{I}$}\nfor $\\{0,\\dots,n\\}\\setminus I$.\n\n\nWe will extend to weighted projective spaces a classical theorem by Beilinson\n(see \\cite{B1}), which gives equivalences between $D^b(\\coh{\\P^n})$ (the\nbounded derived category of coherent sheaves on $\\P^n$) and two homotopy\ncategories of modules. More precisely, let $\\p:=\\p(1,\\dots,1)$ and let\n$\\Lambda=\\bigoplus_{i\\in\\Z}\\Lambda_i$\\index{Lambda@$\\Lambda$} be the graded\n(skew--commutative) ring defined by $\\Lambda_i:=\\Lambda^i(\\p_1\\dual)$.\nMoreover, if $A$ is a graded ring, we will denote by $\\mco{[-n,0]}(A)$ the full\nsubcategory of $\\gfmo{A}$ whose objects are finite direct sums of graded\n$A$--modules of the form $A(j)$ for $-n\\le j\\le0$. Then the theorem can be\nstated in the following way.\n\n \\begin{Beil}\nThe natural additive functors $\\mco{[-n,0]}(\\p)\\to\\coh{\\P^n}$ and\n$\\mco{[-n,0]}(\\Lambda)\\to\\coh{\\P^n}$ (defined on objects by $\\p(j)\\mapsto\n\\so(j)$ and $\\Lambda(j)\\mapsto\\sd^{-j}(-j)$ for $-n\\le j\\le0$) extend to exact\nfunctors between triangulated categories\n\\begin{align*}\n& \\fo:K^b(\\mco{[-n,0]}(\\p))\\to D^b(\\coh{\\P^n}), &\n\\fde:K^b(\\mco{[-n,0]}(\\Lambda))\\to D^b(\\coh{\\P^n}).\n\\end{align*}\n$\\fo$ and $\\fde$ are equivalences of categories.\n \\end{Beil}\n\nIn particular, $D^b(\\coh{\\P^n})$ is generated (as a triangulated category) by\neach of the following two sets of vector bundles:\n\\begin{align*}\n& \\oset:=\\{\\so_{\\P^n}(j)\\st-n\\le j\\le0\\},\n& \\dset:=\\{\\sd_{\\P^n}^j(j)\\st0\\le j\\le n\\}.\n\\end{align*}\nThis means that $\\forall\\cp{\\s{F}}\\in D^b(\\coh{\\P^n})$ there exist bounded\ncomplexes $\\cp{\\bro}=\\cp{\\bro}(\\cp{\\s{F}})$ and $\\cp{\\brd}=\n\\cp{\\brd}(\\cp{\\s{F}})$ isomorphic to $\\cp{\\s{F}}$ (in $D^b(\\coh{\\P^n})$) and\nwith each $\\bro^i$ (respectively $\\brd^i$) isomorphic to a finite direct sum of\nsheaves of $\\oset$ (respectively $\\dset$). $\\cp{\\bro}$ and $\\cp{\\brd}$ are also\nunique up to isomorphism in $C^b(\\coh{\\P^n})$ if we require them to be minimal,\nand in this case their terms are given explicitly by\n\\begin{align*}\n\\bro^i & =\\bigoplus_{0\\le j\\le n}\\so(-j)\\otimes_{\\K}\nH^{i+j}(\\P^n,\\cp{\\s{F}}\\otimes\\sd^j(j)),\\\\\n\\brd^i & =\\bigoplus_{0\\le j\\le n}\\sd^j(j)\\otimes_{\\K}\nH^{i+j}(\\P^n,\\cp{\\s{F}}(-j)).\n\\end{align*}\n\nAs we said in the introduction, it turns out that, in order to get a good\nversion of this theorem on weighted projective spaces, it is better to consider\n$\\gcoh{\\gP(\\w)}$ instead of $\\coh{\\P(\\w)}$ (remember that, by \\ref{shequivcrit}\nand \\ref{stdmori}, these two categories are equivalent if and only if\n$\\w=(1,\\dots,1)$). The reason of this is that working with graded sheaves on\n$\\gP$ one avoids all the pathologies of $\\P$ (basically because the sheaves of\nthe form $\\so_{\\gP}(j)$ are always invertible, whereas the $\\so_{\\P}(j)$ are\nnot, in general).\n\nIn the weighted case the two sets of vector bundles defined above will be\nreplaced by the following two sets\\index{O@$\\goset$}\\index{D@$\\gdset$}\n\\begin{align*}\n\\goset & :=\\{\\so_{\\gP}(j)\\st-\\sw{\\w}0$, $\\forall\\s{A},\\s{B}\\in\\goset$ and\n$\\forall\\s{A},\\s{B}\\in\\gdset$ (here it is essential to use $\\gcoh{\\gP(\\w)}$\ninstead of $\\coh{\\P(\\w)}$), whereas essential surjectivity corresponds to the\nfact that $\\goset$ and $\\gdset$ generate $D^b(\\gcoh{\\gP})$ as a triangulated\ncategory. Now, this last fact follows easily from Hilbert's syzygy theorem, but\nit must be said that Beilinson proved it using a resolution of the diagonal in\n$\\P^n\\times\\P^n$, and we don't know if a similar resolution exists also in the\nweighted case. The drawback of this fact is that we cannot apply the technique\nused in \\cite{AO} to obtain the explicit form of the minimal resolutions. On\nthe other hand, once one knows that they exist unique, they can be determined\nin another way (in the case of $\\P^n$, this was already observed by Beilinson\nin \\cite{B2}). For instance, the fact that the coefficient of $\\so(-j)$ ($0\\le\nj\\le n$) in $\\bro^i(\\cp{\\s{F}})$ is given by\n$H^{i+j}(\\P^n,\\cp{\\s{F}}\\otimes\\sd^j(j))$ is a formal consequence of the fact\nthat $\\forall\\s{A}\\in O$ and $\\forall k\\in\\Z$ we have $H^k(\\P^n,\\s{A}\\otimes\\sd^j(j))\n=0$, except for $\\s{A}=\\so(-j)$ and $k=j$, when it is $\\K$. Therefore, also in\nthe weighted case it is enough to find objects of $D^b(\\gcoh{\\gP})$ with\nsimilar properties of cohomology vanishing as $\\sd^j(j)$ in the example above.\nIt turns out that such objects are given by suitable complexes obtained from\n$\\cp{\\ko}$, except for those which determine the coefficients of the\n$\\sd^j(j)$, which are again $\\so(-j)$. The presence of these complexes makes it\nless easy to compute the minimal resolution in general. As for the\nproblem of determining the differentials of the minimal resolutions,\nwe can prove very little: in the case of $\\P^n$, a method to construct\nthem has been found recently (see \\cite{EFS}), but it doesn't seem easy to\nextend it to the weighted case.\n\nOur approach is close to those of \\cite{B} (see also \\cite{GL}) and of\n\\cite{AKO}, where $\\gP(\\w)$ is replaced, respectively, by a different\ngraded variety (it is obtained grading $\\p(\\w)$ not over $\\Z$, but\nover an abelian group depending on $\\w$), and by its associated\nalgebraic stack (having an equivalent category of coherent sheaves, as\nwe have seen in \\ref{stacks}). In both papers, however, only the case\nof $\\goset$ (and not of $\\gdset$) is treated, and the explicit form of\nthe minimal resolution is not given. In \\cite{AKO} (where\nnoncommutative deformations of weighted projective spaces are also\nconsidered) another generating set for $D^b(\\gcoh{\\gP})$ (``dual'' to\n$\\goset$) is found, and we prove that it coincides (up to twists\nand shifts) with the above mentioned set of complexes which appear in\nthe explicit form of the minimal resolution given by $\\goset$.\n\nOur result also allows to obtain (minimal) resolutions of every\ncoherent sheaf on $\\P$ (although they are no more unique in general),\nand it is applied to prove a generalization of Horrock's splitting\ncriterion for vector bundles on $\\P^n$.\n\nMuch of the contents of this chapter is summarized (without proofs) in\n\\cite{Ca2}.\n\n\n \\section{Koszul complex and sheaves of differentials}\n\nLet $\\md_{\\p}:=\\md_{\\p\/\\K}$ be the $\\p=\\p(\\w)$--module of\n$\\K$--differentials of $\\p$ and $\\forall j\\in\\Z$ let $\\md_{\\p}^j:=\n\\Lambda^{j}(\\md_{\\p})$ (of course, $\\md_{\\p}^j=0$ if $j<0$ or $j>n+1$);\ndenoting by $d:\\p\\to\\md_{\\p}$ the universal derivation, $\\md_{\\p}^j$ is a free\n$\\p$--module with basis\n\\[\\{dx_I\\st I\\subseteq\\{0,\\dots,n\\};\\card{I}=j\\}\\]\n(where $dx_I:=dx_{i_1}\\wedge\\cdots\\wedge dx_{i_j}$ if $I=\\{i_1,\\dots,i_j\\}$\nwith $0\\le i_1<\\cdotsn$). It\nis also clear from the definition that $\\sd^0_{\\gP}\\cong\\so_{\\gP}$ and\n$\\sd^n_{\\gP}\\cong\\so_{\\gP}(-\\sw{\\w})$. The last statement follows immediately\nfrom \\ref{koselfdual}.\n \\end{proof}\n\n \\begin{prop}\\label{gPdualsh}\n$\\ds_{\\gP}:=\\sd^n_{\\gP}\\cong\\so_{\\gP}(-\\sw{\\w})$ is the dualizing sheaf on\n$\\gP$. Moreover, $\\forall\\s{F}\\in\\gcoh{\\gP}$ and $\\forall i>0$ there are natural\nisomorphisms $\\Ext_{\\gP}^i(\\s{F},\\ds_{\\gP})\\cong H^{n-i}(\\gP,\\s{F})\\dual$.\n \\end{prop}\n\n \\begin{proof}\nOne could easily adapt the standard proof for $\\P^n$ (see\n\\cite[III, thm. 7.1]{H1}). Alternatively, observe that $\\so_{\\P}(-\\sw{\\w})$ is\nthe dualizing sheaf on $\\P$ (by \\cite[cor. 6B.8]{BR}) and that\n$\\so_{\\gP}(-\\sw{\\w}))\\cong\\gsHom_{\\P}(\\so_{\\gP},\\so_{\\P}(-\\sw{\\w}))$: this is a\nconsequence of the fact that $\\so_{\\P}(-m-\\sw{\\w}))\\cong\n\\sHom_{\\P}(\\so(m),\\so(-\\sw{\\w}))$ $\\forall m\\in\\Z$ (we can clearly assume\n$\\gcd(\\w_0,\\dots,\\w_n)=1$, and then this is true by \\cite[lemma 4.1]{D} if $\\w$\nis normalized, whereas in general, if $\\P(\\w)\\cong\\P(\\w')$ with $\\w'$\nnormalized, it is easy to check, using \\cite[prop. 3C.7]{BR}, that $\\exists m'\\in\n\\Z$ such that $\\so_{\\P(\\w)}(m)\\cong\\so_{\\P(\\w')}(m')$, $\\so_{\\P(\\w)}(-\\sw{\\w})\n\\cong\\so_{\\P(\\w')}(-\\sw{\\w'})$ and $\\so_{\\P(\\w)}(-m-\\sw{\\w})\\cong\n\\so_{\\P(\\w')}(-m'-\\sw{\\w'})$). Since moreover\n$\\sExt^i_{\\P}(\\so(m),\\so(-\\sw{\\w}))=0$ for $i>0$ and $\\forall m\\in\\Z$ (as usual,\none can reduce to the case $\\w$ normalized, and then this is proved in\n\\cite[prop. 5.4]{D}), everything follows from \\ref{Serredual}.\n \\end{proof}\n\n \\begin{lemm}\\label{mondimfor}\n$\\forall\\w\\in\\N_+^{n+1}$ and $\\forall l\\in\\Z$ the following formula holds:\n\\[\\sum_{I\\subseteq\\{0,\\dots,n\\}}(-1)^{\\card{I}}\\pd_{l-\\sw{\\w_I}}(\\w)=\n\\delta_{l,0}.\\]\n \\end{lemm}\n\n \\begin{proof}\nThe statement being trivial for $l\\le0$, we can assume $l>0$. In this case\n$\\forall I\\subseteq\\{0,\\dots,n\\}$ by \\ref{lbcohom} we have $\\pd_{l-\\sw{\\w_I}}=\nh^0(\\gP,\\so(l-\\sw{\\w_I}))=\\chi(\\gP,\\so(l-\\sw{\\w_I}))$. Therefore, as\n$\\cp{\\sko}(l)$ is an exact complex,\n\\begin{multline*}\n\\sum_I(-1)^{\\card{I}}\\pd_{l-\\sw{\\w_I}}=\n\\sum_{j=0}^{n+1}(-1)^j\\sum_{\\card{I}=j}\\chi(\\gP,\\so(l-\\sw{\\w_I}))\\\\\n=\\sum_{j=0}^{n+1}(-1)^j\\chi(\\gP,\\sko^{-j}(l))=\\chi(\\gP,\\cp{\\sko}(l))=0.\n\\end{multline*}\n \\end{proof}\n\n \\begin{lemm}\\label{omegacohom}\n$\\forall j,l\\in\\Z$ the cohomology of $\\sd_{\\gP}^j(l)$ is given by the following\nformulas:\n\\begin{align*}\nh^i(\\gP,\\sd^j(l)) & =\\delta_{i,j}\\delta_{l,0} & \\text{if $0j}(-1)^{j+1-\\card{I}}\n\\pd_{l-\\sw{\\w_I}}+(-1)^j\\delta_{l,0}\\delta_{j,0};\\\\\nh^n(\\gP,\\sd^j(l)) & =h^0(\\gP,\\sd^{n-j}(-l)).\n\\end{align*}\n \\end{lemm}\n\n \\begin{proof}\nIt can be easily proved using the exact sequences $\\esd{j}$ and \\ref{lbcohom}.\n\nAlternatively, since $H^i(\\gP,\\sd^j(l))\\cong H^i(\\P,\\sd^j(l))$ by\n\\ref{goodcohom}, the result follows from \\cite[thm. 2.3.2]{Do}. Notice that\nthe last equality in the second equation follows from \\ref{mondimfor} and the\nthird equation is a consequence of Serre duality and of the fact that\n$(\\sd_{\\gP}^j)\\dual\\cong\\sd_{\\gP}^{n-j}(\\sw{\\w})$.\n \\end{proof}\n\n \\begin{coro}\\label{omcohvan}\n$\\forall j\\in\\Z$ we have $h^0(\\gP,\\sd^j(l))=0$ if $lj-n$. If moreover $j\\ne0,n$, then also $h^0(\\gP,\\sd^j(j))=\nh^n(\\gP,\\sd^j(j-n))=0$.\n \\end{coro}\n\n\n \\section{The theorem as equivalence of categories}\n\n \\begin{lemm}\\label{extvan}\n$\\forall j,j',l,l'\\in\\Z$ and $\\forall i>0$ we have the following results:\n\\begin{enumerate}\n\n\\item $\\Ext_{\\gP}^i(\\so(l),\\so(l'))=0$ if $l'>l-\\sw{\\w}$;\n\n\\item $\\Ext_{\\gP}^i(\\so(l),\\sd^j(j))=0$ if $ln-\\sw{\\w}-i$;\n\n\\item $\\Ext_{\\gP}^i(\\sd^j(l),\\sd^{j'}(j'))=0$ if $l0$, applying $\\Hom_{\\gP}(-,\\sd^{j'}(j'))$ to\n$\\esd{j}(l)$ yields the exact sequence\n\\[\\Ext_{\\gP}^i(\\bigoplus_{\\card{I}=j}\\so_{\\gP}(l-\\sw{\\w_I}),\\sd^{j'}(j'))\\to\n\\Ext_{\\gP}^i(\\sd^j(l),\\sd^{j'}(j'))\\to\n\\Ext_{\\gP}^{i+1}(\\sd^{j-1}(l),\\sd^{j'}(j')).\\]\nWe have\n$\\Ext_{\\gP}^i(\\bigoplus_{\\card{I}=j}\\so_{\\gP}(l-\\sw{\\w_I}),\\sd^{j'}(j'))=0$ by\nwhat we have just proved ($\\sw{\\w_I}\\ge j$ if $\\card{I}=j$) and\n$\\Ext_{\\gP}^{i+1}(\\sd^{j-1}(l),\\sd^{j'}(j'))=0$ by the inductive hypothesis,\nwhence $\\Ext_{\\gP}^i(\\sd^j(l),\\sd^{j'}(j'))=0$.\n \\end{proof}\n\n \\begin{coro}\\label{Beilextvan}\nIf $\\s{A},\\s{B}\\in\\goset$ or $\\s{A},\\s{B}\\in\\gdset$, then\n$\\Ext_{\\gP}^i(\\s{A},\\s{B})=0$ for $i>0$.\n \\end{coro}\n\n \\begin{lemm}\\label{Homsmd=Homsd}\n$\\forall j,j',l,l'\\in\\Z$ the functor $\\gsh{}$ induces an isomorphism\n\\[\\Hom_{\\p}(\\smd^j(l),\\smd^{j'}(l'))\\isomor\\Hom_{\\gP}(\\sd^j(l),\\sd^{j'}(l')).\\]\n \\end{lemm}\n\n \\begin{proof}\nFirst we prove the case $j=0$ (and $l=0$, which we can always assume), namely\nthat $\\smd^{j'}(l')_0\\cong\\Hom_{\\p}(\\p,\\smd^{j'}(l'))\\isomor\n\\Hom_{\\gP}(\\so,\\sd^{j'}(l'))\\cong H^0(\\gP,\\sd^{j'}(l'))$. The statement being\ntrivial for $j'<0$, we can assume $j'\\ge0$ and we proceed by induction on $j'$.\nSince $\\smd^0\\cong\\p$, the case $j'=0$ follows from \\ref{lbcohom}. If $j'>0$,\nfrom the exact sequence\n\\[0\\to\\smd^{j'}(l')\\to\\ko^{-j'}(l')\\cong\\bigoplus_{\\card{I}=j'}\\p(l'-\\sw{\\w_I})\n\\to\\smd^{j'-1}(l')\\]\nwe obtain a commutative diagram with exact rows\n\\[\\begin{CD}\n0 @>>> H_{\\p}(\\smd^{j'}(l')) @>>> H_{\\p}(\\ko^{-j'}(l')) @>>>\nH_{\\p}(\\smd^{j'-1}(l'))\\\\\n@. @VVV @VVV @VVV \\\\\n0 @>>> H_{\\gP}(\\sd^{j'}(l')) @>>> H_{\\gP}(\\sko^{-j'}(l'))\n@>>> H_{\\gP}(\\sd^{j'-1}(l'))\n\\end{CD}\\]\n(where $H_{\\p}(-):=\\Hom_{\\p}(\\p,-)$ and $H_{\\gP}(-):=\\Hom_{\\gP}(\\so,-)$). Since\nthe vertical arrows on the right and in the middle are isomorphisms by\ninduction, the one on the left is also an isomorphism by the five--lemma.\n\nNow we pass to the general case. As before, we can assume $j\\le n$ and we\nproceed by descending induction on $j$, the case $j=n$ following from what we\nhave just proved, since $\\smd^n(l)\\cong\\p(l-\\sw{\\w})$. If $j>> H'_{\\p}(\\smd^j(l)) @>>> H'_{\\p}(\\ko^{-j-1}(l)) @>>>\nH'_{\\p}(\\smd^{j+1}(l))\\\\\n@. @VVV @VVV @VVV \\\\\n0 @>>> H'_{\\gP}(\\sd^j(l)) @>>> H'_{\\gP}(\\sko^{-j-1}(l)) @>>>\nH'_{\\gP}(\\sd^{j+1}(l))\n\\end{CD}\\]\n(where $H'_{\\p}(-):=\\Hom_{\\p}(-,\\smd^{j'}(l'))$ and\n$H'_{\\gP}(-):=\\Hom_{\\gP}(-,\\sd^{j'}(l'))$). The result then follows, as before,\nfrom the inductive hypothesis using the five--lemma.\n \\end{proof}\n\nGiven $S\\subseteq\\Z$, we will denote by\n$\\mco{S}=\\mco{S}(\\p)$\\index{M_S@$\\mco{S}$} the full subcategory of $\\gfmo{\\p}$\nwhose objects are of the form $\\bigoplus_{l\\in S} \\p(l)^{a_l}$ ($a_l\\in\\N$) and\nby $\\mcom{S}$\\index{M_S @$\\mcom{S}$} the full subcategory of $\\gfmo{\\p}$ whose\nobjects are of the form $\\bigoplus_{0\\le j\\le n}\\smd^j(j)^{b_j}\\oplus X$ with\n$b_j\\in\\N$ and $X\\in\\mco{S}$ (notice that $\\mcom{S}=\n\\mcom{S\\cup\\{n-\\sw{w},0\\}}$). Similarly, $\\co{S}$\\index{O_S@$\\co{S}$}\n(respectively $\\com{S}$)\\index{O_S @$\\com{S}$} will denote the full subcategory\nof $\\gcoh{\\gP}$ whose objects are of the form $\\bigoplus_{l\\in S}\\so(l)^{a_l}$\n(respectively $\\bigoplus_{0\\le j\\le n}\\sd^j(j)^{b_j}\\oplus X$ with\n$X\\in\\co{S}$); we will write $\\co{}$\\index{O@$\\co{}$} and $\\com{}$\\index{O\n@$\\com{}$} instead of $\\co{\\Z}$ and $\\com{\\Z}$. By \\ref{Homsmd=Homsd} it is\nclear that $\\gsh{\\p}$ induces an equivalence of categories between $\\mco{S}$\n(respectively $\\mcom{S}$) and $\\co{S}$ (respectively $\\com{S}$). If $a,b\\in\\Z$,\nwe set $]a,b[:=\\{m\\in\\Z\\st a{\\alpha}>>\n\\Hom_{D^b(\\gcoh{\\gP})}(\\cp{\\s{F}},\\cp{\\s{G}}) \\\\\n@VV{\\beta}V @VV{\\beta'}V \\\\\n\\Hom_{K^b(\\gqco{\\gP})}(\\cp{\\s{F}},\\cp{\\s{G}}) @>{\\alpha'}>>\n\\Hom_{D^b(\\gqco{\\gP})}(\\cp{\\s{F}},\\cp{\\s{G}})\n\\end{CD}\\]\nwhere $\\beta$ is obviously an isomorphism. By \\ref{gcgqgc} the abelian\ncategories $\\gqco{\\gP}$ and $\\gcoh{\\gP}$ satisfy the hypotheses of\n\\ref{HomDA=HomDB}, whence $\\beta'$ is an isomorphism. Moreover, it follows from\n\\ref{Beilextvan} that $\\Ext_{\\gP}^i(\\s{F}^l,\\s{G}^m)=0$ for $i>0$ and $\\forall\nl,m\\in\\Z$, so that $\\alpha'$ is an isomorphism by \\ref{HomKA=HomDA}. Thus\n$\\alpha$ is an isomorphism, too.\n\nIt remains to prove that $\\gfo$ and $\\gfd$ are essentially surjective. As\n$D^b(\\gcoh{\\gP})$ is generated (as a triangulated category) by the complexes\nconcentrated in position $0$ and since every object of $\\gcoh{\\gP}$ has a\nfinite resolution with sheaves of the form $\\so(j)$ (by Hilbert's syzygy\ntheorem), it is enough to show that the $\\so(j)$ ($j\\in\\Z$) are in the\nessential images $\\im\\gfo$ and $\\im\\gfd$. First we prove by descending\ninduction on $j$ that $\\so(j)\\in\\im\\gfd$ for $-\\sw{\\w}0$\n(in a similar way one proceeds for $j\\le-\\sw{\\w}$).\n \\end{proof}\n\n \\begin{rema}\nIf $\\w=(1,\\dots,1)$, this result coincides with Beilinson's theorem (taking\ninto account that $\\gcoh{\\gP^n}$ is equivalent to $\\coh{\\P^n}$ by\n\\ref{shequivcrit} and \\ref{stdmori}), except that, in order to identify $\\gfd$\nwith $\\fde$, it is also necessary to consider the equivalence between\n$\\mcom{\\intd}=\\mcom{\\emptyset}$ and $\\mco{[-n,0]}(\\Lambda)$ (defined on objects\nby $\\smd^j(j)\\mapsto\\Lambda(-j)$). That this is an equivalence follows from the\nfact that $\\Hom_{\\P^n}(\\sd^j(j),\\sd^{j'}(j'))\\cong\\Lambda^{j-j'}(\\p_1\\dual)$.\nSince this is not true in the weighted case (although it still holds in a\nweaker form, see \\ref{omegahom}), and also because of the presence of $\\p(l)$\nfor $n-\\sw{\\w}l$ and $\\Hom_{\\gP}(\\so(j),\\so(j))\\cong\\K$, the sequence\n$(\\so(1-\\sw{\\w}),\\dots,\\so)$ (hence also $(\\so(k),\\dots,\\so(k+\\sw{\\w}-1))$\n$\\forall k\\in\\Z$) of $D^b(\\gcoh{\\gP})$ is strong exceptional (see \\ref{exc}), and\nalso full, since $\\goset$ generates $D^b(\\gcoh{\\gP})$, as we proved before. On\nthe other hand, while $(\\sd^n(n),\\dots,\\sd^0(0)=\\so)$ is a full and strong\nexceptional sequence if $\\w=(1,\\dots,1)$, the objects of $\\gdset$ usually do\nnot form an exceptional sequence, because in general\n$\\dim_{\\K}\\Hom_{\\gP}(\\sd^j(j),\\sd^j(j))>1$ (so that $\\sd^j(j)$ is not\nexceptional), as it is easy to see. In \\cite{AKO} also the full (but not\nstrong) exceptional sequence which is left dual (see \\ref{dualmut}) of\n$(\\so,\\dots,\\so(\\sw{\\w}-1))$ plays an important role; later we will study it\n(see \\ref{Odual} and \\ref{notstrong}).\n \\end{rema}\n\n \\begin{rema}\\label{Dmod}\nBoth in \\cite{B} and in \\cite{AKO} a different, although closely related,\nresult is proved, namely (translating to our setting) that there is an\nequivalence of categories between $D^b(\\gcoh{\\gP})$ and $D^b(\\fmo{B})$, where\n$B$ is the (noncommutative) endomorphism algebra of $\\bigoplus_{0\\le\nj<\\sw{\\w}}\\so(j)$.\n \\end{rema}\n\n\n \\section{Morphisms between sheaves of differentials}\n\nGiven $\\s{F},\\s{G}\\in\\gcoh{\\gP}$ and $S\\subseteq\\Z$, let\n$\\ohom{S}(\\s{F},\\s{G})$\\index{Hom^O_S@$\\ohom{S}$} be the subset of\n$\\Hom_{\\gP}(\\s{F},\\s{G})$ consisting of those morphisms which factor through an\nobject of $\\co{S}$. Since the composition of two morphisms of $\\gcoh{\\gP}$, one\nof which is in $\\ohom{S}$, is again in $\\ohom{S}$, we can consider the quotient\ncategory $\\coq{S}= \\gcoh{\\gP}\/\\co{S}$\\index{Q_S@$\\coq{S}$} with the same\nobjects as $\\gcoh{\\gP}$ and with\n$\\Hom_{\\coq{S}}(\\s{F},\\s{G}):=\\Hom_{\\gP}(\\s{F},\\s{G})\/\\ohom{S}(\\s{F},\\s{G})$\n(as usual, $\\coq{\\Z}$ will be simply denoted by $\\coq{}$).\\index{Q@$\\coq{}$}\n\n \\begin{rema}\\label{coqadd}\nIt is easy to see that $\\forall\\s{F},\\s{G}\\in\\gcoh{\\gP}$ the set\n$\\ohom{S}(\\s{F},\\s{G})$ is a subgroup of $\\Hom_{\\gP}(\\s{F},\\s{G})$, and that\ntherefore $\\coq{S}$ is an additive category.\n \\end{rema}\n\n \\begin{defi}\\label{mindef}\nA morphism in $\\gcoh{\\gP}$\n\\[f:\\bigoplus_{0\\le jl'$), i.e. $g\\comp f=0$.\n \\end{proof}\n\nDenoting by $\\pi^j:\\sko^{-j}(j)\\cong\\bigoplus_{\\card{I}=j}\\so_{\\gP}(j-\\sw{\\w_I})\n\\to\\bigoplus_{\\card{I}=j,\\sw{\\w_I}>j}\\so_{\\gP}(j-\\sw{\\w_I})$ the natural\nprojection, and observing that $n-\\sw{\\w}j}\\so_{\\gP}(j-\\sw{\\w_I}).\\]\n\n \\begin{lemm}\\label{omhomfact}\n$\\phi\\in\\Hom_{\\gP}(\\sd^j(j),\\sd^{j'}(j'))$ is in\n$\\ohom{\\intd}(\\sd^j(j),\\sd^{j'}(j'))$ if and only if it factors through\n$\\pi^j\\comp\\omd{j}(j)$. Moreover, if $j'>0$, this is the case if and only if\n$\\phi$ factors through $\\omd{j}(j)$.\n \\end{lemm}\n\n \\begin{proof}\nAssume that $\\phi\\in\\ohom{\\intd}(\\sd^j(j),\\sd^{j'}(j'))$ (the other implication\nbeing obvious), say $\\phi=\\phi'\\comp\\psi$ with $\\sd_{\\gP}^j(j)\\mor{\\psi}\\s{F}:=\n\\bigoplus_{n-\\sw{\\w}n-\\sw{\\w}$, $\\Ext_{\\gP}^1(\\sd^{j-1}(j),\\so(l))=0$ by\n\\ref{extvan}, we see that $\\psi=\\rho\\comp\\omd{j}(j)$ for some $\\rho\\in\n\\Hom_{\\gP}(\\bigoplus_{\\card{I}=j}\\so(j-\\sw{\\w_I}),\\s{F})$. On the other hand,\nsince $\\Hom_{\\gP}(\\so,\\so(l))=0$ for $l<0$, it is clear that $\\rho$ factors\nthrough $\\pi^j$, say $\\rho=\\rho'\\comp\\pi^j$. Therefore $\\phi=\\phi'\\comp\\rho'\n\\comp\\pi^j\\comp\\omd{j}(j)$. The last statement follows immediately from the\nfact that, if $j'>0$, then $\\Hom_{\\gP}(\\so,\\sd^{j'}(j'))=0$ (by\n\\ref{omcohvan}).\n \\end{proof}\n\nUnlike the case of $\\gP^n$, it is not true in general that\n$\\Hom_{\\gP}(\\sd^j(j),\\sd^{j'}(j'))$ (for $0\\le j,j'\\le n$) only depends on\n$j-j'$ (for instance, it can happen that $\\Hom_{\\gP}(\\sd^j(j),\\sd^{j'}(j'))\\ne\n0$ for some $j1$ and\nbecause $\\phi=0$ if $j\\le1$), we obtain\n\\[\\Hom_{\\coq{\\intd}}(\\sd^j(j),\\so)\\cong\\Hom_{\\gP}(\\hat{\\sko}^{-j}(j),\\so)\\cong\n\\Hom_{\\gP}(\\so\\otimes_{\\K}\\Lambda^j(\\p_1),\\so)\\cong\\Lambda^j(\\p_1\\dual).\\]\nBy duality this also proves the case $j=n$: indeed, we have more generally\n\\[\\Hom_{\\coq{\\intd}}(\\s{F},\\s{G})\\cong\n\\Hom_{\\coq{\\intd}}(\\s{G}\\dual(n-\\sw{\\w}),\\s{F}\\dual(n-\\sw{\\w}))\\]\n$\\forall\\s{F},\\s{G}\\in\\gcoh{\\gP}$ (because the same is true for $\\Hom_{\\gP}$ and\n$\\ohom{\\intd}$). Taking into account that $\\sd^j(j)\\dual(n-\\sw{\\w})\\cong\n\\sd^{n-j}(n-j)$, this implies\n\\[\\Hom_{\\coq{\\intd}}(\\sd^j(j),\\sd^{j'}(j'))\\cong\n\\Hom_{\\coq{\\intd}}(\\sd^{n-j'}(n-j'),\\sd^{n-j}(n-j)),\\]\nwhence $\\Hom_{\\coq{\\intd}}(\\sd^n(n),\\sd^{j'}(j'))\\cong\\Lambda^{n-j'}(\\p_1\\dual)$\nby the case already proved.\n\nTherefore we can assume $j'>0$ and $jl'$.\n \\end{proof}\n\n\n \\section{Uniqueness of the minimal resolution}\n\n \\begin{lemm}\\label{minnil}\nIf $f:\\s{L}\\to \\s{L}$ is a minimal morphism in $\\com{}$, then $f$ is nilpotent.\n \\end{lemm}\n\n \\begin{proof}\nLet $\\s{L}=\\bigoplus_{l\\in\\Z}\\so(l)^{a_l}\\bigoplus_{0>> \\tilde{\\s{L}}^{i-1}\n@>\\cdiff{\\tilde{\\s{L}}}^{i-1}>> \\tilde{\\s{L}}^i @>\\cdiff{\\tilde{\\s{L}}}^i>>\n\\tilde{\\s{L}}^{i+1} @>\\cdiff{\\tilde{\\s{L}}}^{i+1}>> \\tilde{\\s{L}}^{i+2} @>>>\n\\cdots \\\\\n@VV\\cp{f}V @. @VV{\\rm id} V\n@VV\\left(\\begin{smallmatrix}\n{\\rm id} \\\\\n-s^{-1}\\comp b^i\n\\end{smallmatrix}\\right)V\n@VV\\left(\\begin{smallmatrix}\n\\scriptstyle {\\rm id} \\\\\n\\scriptstyle 0\n\\end{smallmatrix}\\right)V\n@VV{\\rm id} V @. \\\\\n\\cp{\\s{L}}= @. \\cdots @>>> \\s{L}^{i-1} @>\\cdiff{\\s{L}}^{i-1}>>\n\\tilde{\\s{L}}^i\\oplus\\s{A}\n@>\\cdiff{\\s{L}}^i>> \\tilde{\\s{L}}^{i+1}\\oplus\\s{A}\n@>\\cdiff{\\tilde{\\s{L}}}^{i+1}>>\\tilde{\\s{L}}^{i+2} @>>> \\cdots \\\\\n@VV\\cp{g}V @. @VV{\\rm id} V\n@VV{\\left(\\begin{smallmatrix}\n{\\rm id} & 0\n\\end{smallmatrix}\\right)}V\n@VV{\\left(\\begin{smallmatrix}\n{\\rm id} & -c^i\\comp s^{-1}\n\\end{smallmatrix}\\right)}V\n@VV{\\rm id} V @. \\\\\n\\cp{\\tilde{\\s{L}}}= @. \\cdots @>>> \\tilde{\\s{L}}^{i-1}\n@>\\cdiff{\\tilde{\\s{L}}}^{i-1}>> \\tilde{\\s{L}}^i @>\\cdiff{\\tilde{\\s{L}}}^i>>\n\\tilde{\\s{L}}^{i+1} @>\\cdiff{\\tilde{\\s{L}}}^{i+1}>> \\tilde{\\s{L}}^{i+2} @>>>\n\\cdots\n\\end{CD}\\]\ndefine morphisms of complexes, which are inverse isomorphisms in $K^b(\\com{})$.\nIndeed, $\\cp{g}\\comp\\cp{f}={\\rm id}_{\\cp{\\tilde{\\s{L}}}}$ and $\\forall j\\in\\Z$\n${\\rm id}_{\\s{L}^j}-f^j\\comp g^j=h^{j+1}\\comp\\cdiff{\\s{L}}^j+\\cdiff{\\s{L}}^{j-1}\n\\comp h^j$, where $h^j:\\s{L}^j\\to \\s{L}^{j-1}$ is defined by\n\\begin{align*}\n& h^{i+1}:=\\begin{pmatrix}\n0 & 0\\\\\n0 & s^{-1}\n\\end{pmatrix}, & h^j=0\\text{ if }j\\ne i+1.\n\\end{align*}\nSince $\\rk{\\cp{\\tilde{\\s{L}}}}<\\rk{\\cp{\\s{L}}}$, by induction\n$\\cp{\\tilde{\\s{L}}}$ (and then also $\\cp{\\s{L}}$) is isomorphic to a minimal\ncomplex.\n\nIt remains to prove the last statement (which clearly implies the uniqueness of\nthe minimal complex in $C^b(\\com{})$), namely that if $\\cp{f}:\\cp{\\s{L}}\\to\n\\cp{\\s{M}}$ is a morphism in $C^b(\\com{})$ inducing an isomorphism in\n$K^b(\\com{})$ with $\\cp{\\s{L}}$ and $\\cp{\\s{M}}$ minimal, then $\\cp{f}$ is an\nisomorphism in $C^b(\\com{})$. Let $\\cp{g}:\\cp{\\s{M}}\\to\\cp{\\s{L}}$ in\n$C^b(\\com{})$ be such that $\\cp{f}$ and $\\cp{g}$ are inverse isomorphisms in\n$K^b(\\com{})$. $\\cp{g}\\comp\\cp{f}$ being homotopic to ${\\rm id}_{\\cp{\\s{L}}}$ means\nthat $\\forall i\\in\\Z$ there exist maps $h^i:\\s{L}^i\\to\\s{L}^{i-1}$ such that,\nsetting $a^i:=\\cdiff{\\s{L}}^{i-1}\\comp h^i+h^{i+1}\\comp\\cdiff{\\s{L}}^i$, we\nhave $g^i\\comp f^i={\\rm id}_{\\s{L}^i}-a^i$. By \\ref{compminmin} each $a^i$ is\nminimal, and then, by \\ref{minnil}, $\\exists m\\in\\N$ such that $(a^i)^m=0$. Thus,\nsetting $s^i:=\\sum_{j=0}^{m-1}(a^i)^j$, we obtain\n\\[{\\rm id}_{\\s{L}^i}=({\\rm id}_{\\s{L}^i})^m-(a^i)^m=({\\rm id}_{\\s{L}^i}-a^i)\\comp s^i=s^i\\comp\n({\\rm id}_{\\s{L}^i}-a^i)=(g^i\\comp f^i)\\comp s^i=s^i\\comp(g^i\\comp f^i),\\]\nso that $g^i\\comp f^i$ is an isomorphism. In the same way, the fact that\n$\\cp{f}\\comp\\cp{g}$ is homotopic to ${\\rm id}_{\\cp{\\s{M}}}$ implies that also $f^i\n\\comp g^i$ is an isomorphism, whence $f^i$ and $g^i$ are isomorphisms $\\forall i\n\\in\\Z$.\n \\end{proof}\n\n \\begin{rema}\\label{minunit}\nWith obvious modifications, the same result holds, of course, also for a\ncomplex a twist of which is in $K^b(\\com{})$.\n \\end{rema}\n\n \\begin{rema}\\label{minuniwrt}\nIt is clear from the proof of \\ref{minuni} that if $\\cp{\\s{L}},\\cp{\\s{M}}\\in\nK^b(\\com{})$ are isomorphic, $\\cp{\\s{M}}$ is minimal and $\\cp{\\s{L}}$ is\nminimal with respect to some $\\s{A}$ (with $\\s{A}=\\sd^j(j)$ for some $0{\\cp{\\tbo{\\s{U}^{\\le m}}{m}}}>> \\s{U}^m[-m]\n@>{\\cp{i}}>> \\s{U}^{\\le m} @>{\\cp{p}}>> \\s{U}^{{\\cp{\\tbo{\\s{V}^{\\le m}}{m}}}>> \\s{V}^m[-m] @>{\\cp{j}}>>\n\\s{V}^{\\le m} @>{\\cp{q}}>> \\s{V}^{>> R^{i-m}F(\\s{U}^m) @>R^iF(\\cp{i})>> R^iF(\\s{U}^{\\le m}) @>R^iF(\\cp{p})>>\nR^iF(\\s{U}^{>> 0 @. \\\\\n@. @VVR^{i-m}F(\\phi^m)V @VVR^iF(\\phi^{\\le m})V @VVR^iF(\\phi^{>> R^{i-m}F(\\s{V}^m) @>R^iF(\\cp{j})>> R^iF(\\s{V}^{\\le m}) @>R^iF(\\cp{q})>>\nR^iF(\\s{V}^{>> 0 @. .\n\\end{CD}\\]\nFor $i\\ne m$ we have $R^{i-m}F(\\s{U}^m)=R^{i-m}F(\\s{V}^m)=0$ by hypothesis, so\nthat $R^iF(\\cp{p})$ and $R^iF(\\cp{q})$ are isomorphisms (hence $R^iF(\\phi^{\\le\nm})$ can be identified with $R^iF(\\phi^{m$}\n\\end{cases}\\]\n(and similarly for $\\cp{\\s{V}}$) and that $R^iF(\\phi^{\\le m})$ can be\nidentified with $R^0F(\\phi^i)$ if $i\\le m$. It remains to prove that\n$R^iF(\\cp{\\tbo{\\s{U}^{\\le m+1}}{m+1}})=0$. Now, for $i \\ne m+1$ this is true\nbecause the target of the map is $R^{i-m-1}F(\\s{U}^{m+1}) =0$ by hypothesis,\nwhereas for $i=m+1$ observe that $\\cdiff{\\s{U}}^m= \\cp{\\tbo{\\s{U}^{\\le\nm+1}}{m+1}}[m+1]\\comp\\cp{i}[m]:\\s{U}^m\\to\\s{U}^{m+1}$, so that\n\\[R^0F(\\cdiff{\\s{U}}^m)=R^{m+1}F(\\cp{\\tbo{\\s{U}^{\\le m+1}}{m+1}})\\comp\nR^mF(\\cp{i}).\\]\nAs $R^0F(\\cdiff{\\s{U}}^m)=0$ because $\\cdiff{\\s{U}}^m$ is\nminimal (remember that $R^0F(\\s{E}\\ind{h})\\cong\\K$ and that each component of\n$\\cdiff{\\s{U}}^m$ from $\\s{E}\\ind{h}$ to $\\s{E}\\ind{h}$ is nilpotent by\n\\ref{minnil}) and as $R^mF(\\cp{i})$ is an isomorphism, it follows that also\n$R^{m+1}F(\\cp{\\tbo{\\s{U}^{\\le m+1}}{m+1}})=0$.\n\n$\\forall i\\in\\Z$, taking $m>i$ such that $\\s{U}^{-l}\\so(-l-\\sw{\\w_I}), &\n\\abd{l}^j\\cong\\bigoplus_{\\card{I}=1-j,\\sw{\\w_I}>-l-j}\\so(-l-\\sw{\\w_I}).\n\\end{align*}\n \\end{rema}\n\n \\begin{rema}\\label{bobdvan}\n$\\bo{l}^j=\\abo{l}^j=\\bd{l}^j=\\abd{l}^j=0$ if $j<-n$ or $j>0$. It is also\nimmediate to see that each $\\bo{l}^j$ (respectively $\\abo{l}^j$) contains\nonly terms of the form $\\so(i)$ with $0\\le i\\le-l$ (respectively\n$-l-\\sw{\\w}\\le i<0$). Similarly, each $\\bd{l}^j$ (respectively\n$\\abd{l}^j$) contains only terms of the form $\\so(i)$ with $j-l}\\so(1-l-\\sw{\\w_I})\\dual\\\\\n\\cong\\bigoplus_{\\card{I}=n+j+1,\\sw{\\w_I}>-l}\\so(l+\\sw{\\w_I}-1)\\cong\n\\bigoplus_{\\card{\\compl{I}}=-j,\\sw{\\w_{\\compl{I}}}<\\sw{\\w}+l}\n\\so(l+\\sw{\\w}-1-\\sw{\\w_{\\compl{I}}})\\\\\n\\cong\\bigoplus_{\\card{J}=-j,\\sw{\\w_J}\\le\\sw{\\w}+l-1}\\so(\\sw{\\w}+l-1-\\sw{\\w_J})\n\\cong\\bo{1-\\sw{\\w}-l}^j\n\\end{multline*}\nimply that $(\\cp{\\abo{l}})\\dual(-1)[n]\\cong\\cp{\\bo{1-\\sw{\\w}-l}}$ in\n$C^b(\\gcoh{\\gP})$. Therefore, in view of \\ref{altiso}, there is an isomorphism\n$(\\cp{\\bo{l}})\\dual\\cong(\\cp{\\abo{l}})\\dual\\cong\\cp{\\bo{1-\\sw{\\w}-l}}(1)[-n]$ in\n$D^b(\\gcoh{\\gP})$.\n \\end{proof}\n\n \\begin{lemm}\\label{vancrit}\nLet $\\cp{\\s{L}},\\cp{\\s{F}}\\in D^b(\\gcoh{\\gP})$ and $a\\in\\Z$. If\n$H^i(\\gP,\\s{L}^m\\lotimes\\cp{\\s{F}})=0$ for $i>a-m$ (respectively\n$ia$\n(respectively $i$ (the other one is similar). $\\cp{\\s{L}}$\nbeing bounded, it is enough to prove that\n$H^i(\\gP,\\s{L}^{a$ and $\\forall m\\in\\Z$. Since\nthis is obviously true for $m<<0$, we can proceed by induction on $m$. Applying\nthe functor $H^0(\\gP,-\\lotimes\\cp{\\s{F}})$ to the distinguished triangle\n$\\s{L}^{a$, $H^i(\\gP,\\s{L}^{a-m$), whence\nalso $H^i(\\gP,\\s{L}^{\\le m}\\lotimes\\cp{\\s{F}})=0$.\n \\end{proof}\n\n \\begin{theo}\\label{wbeilfor}\nGiven $\\cp{\\s{F}}\\in D^b(\\gcoh{\\gP})$ let $\\cp{\\gbro}=\\cp{\\gbro}(\\cp{\\s{F}})\\in\nC^b(\\co{\\into})$\\index{X@$\\cp{\\gbro}$} and\n$\\cp{\\gbrd}=\\cp{\\gbrd}(\\cp{\\s{F}})\\in C^b(\\com{\\intd})$\\index{Y@$\\cp{\\gbrd}$}\nbe the minimal complexes (which exist unique up to isomorphisms by\n\\ref{wbeiluni}) such that $\\cp{\\s{F}}\\cong\\cp{\\gbro}\\cong\\cp{\\gbrd}$ in\n$D^b(\\gcoh{\\gP})$. Then $\\forall i\\in\\Z$ we have\n\\begin{align*}\n\\gbro^i & =\\bigoplus_{-\\sw{\\w}-m$ and $\\forall m\\in\\Z$ (by \\ref{lbcohom}\nand \\ref{bobdvan}), we have $h^i(\\gP,\\cp{\\bo{l}}(j))=0$ if $i>0$ by\n\\ref{vancrit}. Moreover, also $h^i(\\gP,\\abo{l}^m(j))=0$ for $i<-m$ and $\\forall\nm\\in\\Z$, and so $h^i(\\gP,\\cp{\\bo{l}}(j))=h^i(\\gP,\\cp{\\abo{l}}(j))=0$ if $i<0$,\nagain by \\ref{vancrit}. In a similar way (using \\ref{omegacohom} instead of\n\\ref{lbcohom} in the case $0\\le j\\le n$) one can prove that\n$h^i(\\gP,\\cp{\\bd{l}}\\otimes\\s{E}\\ind{j})=0$ if $i\\ne0$ for $n-\\sw{\\w}-\\sw{\\w}$ if\n$\\card{I}\\le j$ and $\\sw{\\w_J}<\\card{J}-l$). Hence we can write\n$\\chi(\\gP,\\cp{\\bd{l}}\\otimes\\sd^j(j))=x-y$ with\n\\begin{align*}\nx & :=\\sum_{\\card{I}\\le j}\\sum_J(-1)^{\\card{I}-j+\\card{J}}\n\\pd_{j-l-\\sw{\\w_I}-\\sw{\\w_J}},\\\\\ny & :=\\sum_{\\card{I}\\le j}\\sum_{\\sw{\\w_J}\\ge\\card{J}-l}\n(-1)^{\\card{I}-j+\\card{J}}\\pd_{j-l-\\sw{\\w_I}-\\sw{\\w_J}}.\n\\end{align*}\nUsing \\ref{mondimfor} we obtain\n\\begin{multline*}\nx=\\sum_{\\card{I}\\le j}(-1)^{\\card{I}-j}\\sum_J(-1)^{\\card{J}}\n\\pd_{j-l-\\sw{\\w_I}-\\sw{\\w_J}}\\\\\n=\\sum_{\\card{I}\\le j}(-1)^{\\card{I}-j}\\delta_{\\sw{\\w_I},j-l}=\n(-1)^j\\sum_{\\card{I}\\le j,\\sw{\\w_I}=j-l}(-1)^{\\card{I}}.\n\\end{multline*}\nOn the other hand, observing that $j-l-\\sw{\\w_I}-\\sw{\\w_J}<0$ if $\\card{I}>j$\nand $\\sw{\\w_J}\\ge\\card{J}-l$, again by \\ref{mondimfor} we have\n\\begin{multline*}\ny=\\sum_{\\sw{\\w_J}\\ge\\card{J}-l}(-1)^{j+\\card{J}}\\sum_I(-1)^{\\card{I}}\n\\pd_{j-l-\\sw{\\w_I}-\\sw{\\w_J}}=\n\\sum_{\\sw{\\w_J}\\ge\\card{J}-l}(-1)^{j+\\card{J}}\\delta_{\\sw{\\w_J},j-l}\\\\\n=(-1)^j\\sum_{\\sw{\\w_J}\\ge\\card{J}-l,\\sw{\\w_J}=j-l}(-1)^{\\card{J}}=\n(-1)^j\\sum_{\\card{J}\\le j,\\sw{\\w_J}=j-l}(-1)^{\\card{J}}.\n\\end{multline*}\nThus $x=y$ and $\\chi(\\gP,\\cp{\\bd{l}}\\otimes\\sd^j(j))=0$.\n \\end{proof}\n\n \\begin{rema}\nOf course, in the particular case $\\w=(1,\\dots,1)$, the above resolutions\ncoincide with the usual Beilinson's resolutions on $\\P^n$, taking into account\nthat $\\gcoh{\\gP^n}$ is equivalent to $\\coh{\\P^n}$ by \\ref{shequivcrit} and\n\\ref{stdmori}. Indeed, in this case $n-\\sw{\\w}=-1$, and so\n$\\gbrd^i(\\cp{\\s{F}})=\\bigoplus_{0\\le j\\le n}\n\\sd^j(j)\\otimes_{\\K}H^{i+j}(\\gP^n,\\cp{\\s{F}}(-j))$. On the other hand,\n$\\cp{\\bo{j}}=\\sko^{\\ge j}(-j)$, which is isomorphic in $D^b(\\gcoh{\\gP^n})$ to\n$\\sd^{-j}(-j)[-j]$ for $-n=1-\\sw{\\w}\\le j\\le0$, whence $\\gbro^i(\\cp{\\s{F}})=\n\\bigoplus_{0\\le j\\le n}\\so(-j)\\otimes_{\\K}\nH^{i+j}(\\gP^n,\\cp{\\s{F}}\\otimes\\sd^j(j))$.\n \\end{rema}\n\nThe complexes $\\cp{\\bo{j}}$ and $\\cp{\\bd{j}}$ (including $\\so(-j)[j]$ for $0\\le\nj\\le n$) which appear in the resolutions of \\ref{wbeilfor} are uniquely\ndetermined as objects of $D^b(\\gcoh{\\gP})$. More precisely, we have the\nfollowing result.\n\n \\begin{prop}\nIf $\\cp{\\s{P}}\\in D^b(\\gcoh{\\gP})$ is such that for some $-\\sw{\\w}0$ and $\\forall l,m\\in\\Z$,\nit follows from \\ref{HomKA=HomDA} that\n$\\Hom_{K^b(\\gcoh{\\gP})}(\\cp{\\tilde{\\s{P}}},\\cp{\\s{P}})\\isomor\n\\Hom_{D^b(\\gcoh{\\gP})}(\\cp{\\tilde{\\s{P}}},\\cp{\\s{P}})$. Choosing $\\alpha:\n\\cp{\\tilde{\\s{P}}}\\to\\cp{\\s{P}}$ whose image in $D^b(\\gcoh{\\gP})$ is an\nisomorphism, it is clear that $\\cp{\\s{L}}:=\\mc{\\alpha}$ satisfies the following\nproperties: $\\cp{\\s{L}}\\iso0$ in $D^b(\\gcoh{\\gP})$ (i.e., $\\cp{\\s{L}}$ is an\nexact complex), $\\cp{\\s{L}}$ is minimal and $\\s{L}^i=\\sko^i(-a)$ $\\forall i\\in\\Z$.\nIt is easy to see that this implies that $\\cp{\\s{L}}\\cong\\cp{\\sko}(-a)$ in\n$C^b(\\gcoh{\\gP})$ ($\\s{L}^{\\ge-n}$ and $\\sko^{\\ge-n}(-a)$ are isomorphic as\ncomplexes because they are both $\\cp{\\gbro}$ resolutions of\n$\\so(-a-\\sw{\\w})[n+1]$, and it is immediate to check that an isomorphism\n$\\s{L}^{\\ge-n}\\to\\sko^{\\ge-n}(-a)$ extends to an isomorphism\n$\\cp{\\s{L}}\\to\\cp{\\sko}(-a)$). Therefore $\\cp{\\s{P}}$, which is a subcomplex of\n$\\cp{\\s{L}}\\cong\\cp{\\sko}(-a)$ by definition of mapping cone, can be identified\nwith $\\cp{\\bo{a}}$.\n\nIn a similar way (always using the $\\cp{\\gbro}$ resolution) one can prove that\n$\\cp{\\s{Q}}\\cong\\cp{\\bd{b}}$ or $\\cp{\\s{Q}}\\cong\\so(-b)[b]$.\n \\end{proof}\n\n \\begin{prop}\\label{Odual}\nThe left dual of $(\\so,\\dots\\so(\\sw{\\w}-1))$ (which is a full and strong\nexceptional sequence of $D^b(\\gcoh{\\gP})$ by \\ref{Oexc}) is the (full\nexceptional by \\ref{dualmut}) sequence\n$(\\cp{\\bo{1-\\sw{\\w}}}[1-\\sw{\\w}],\\dots,\\cp{\\bo{l}}[l],\\dots\\cp{\\bo{0}})$.\n \\end{prop}\n\n \\begin{proof}\nWe will prove more generally that $L^{(j)}\\so(i)\\cong\\cp{\\bo{-j}}(i-j)[-j]$ for\n$0\\le j\\le i<\\sw{\\w}$. We proceed by induction on $j$, the case $j=0$ being\nclear, since $\\cp{\\bo{0}}\\cong\\so$. For $j>0$, by definition, there is a\ndistinguished triangle in $D^b(\\gcoh{\\gP})$\n\\[L^{(j)}\\so(i)\\to\\cp{\\s{H}}\\mor{u}L^{(j-1)}\\so(i)\\to(L^{(j)}\\so(i))[1],\\]\nwhere $\\cp{\\s{H}}:=\\bigoplus_{k\\in\\Z}\n(\\Hom_{D^b(\\gcoh{\\gP})}(\\so(i-j)[k],L^{(j-1)}\\so(i))\\otimes_{\\K}\\so(i-j)[k])$\nand $u$ is the natural morphism. We can assume\n$L^{(j-1)}\\so(i)=\\cp{\\bo{1-j}}(i+1-j)[1-j]$ by induction, whence we have\n\\begin{multline*}\n\\s{H}^p=\n\\Hom_{D^b(\\gcoh{\\gP})}(\\so(i-j)[-p],L^{(j-1)}\\so(i))\\otimes_{\\K}\\so(i-j)\\\\\n\\cong H^p(\\gP,(L^{(j-1)}\\so(i))(j-i))\\otimes_{\\K}\\so(i-j)\\cong\nH^{p+1-j}(\\gP,\\cp{\\bo{1-j}}(1))\\otimes_{\\K}\\so(i-j)\n\\end{multline*}\n$\\forall p\\in\\Z$. Now, $h^{p+1-j}(\\gP,\\cp{\\bo{1-j}}(1))$ is the coefficient of\n$\\so(1-j)$ in $\\gbro^{p+1-j}(\\so(1))$ by \\ref{wbeilfor}, and, as\n$\\cp{\\gbro}(\\so(1))$ is clearly given by $\\sko^{<0}(1)[-1]$, we obtain\n\\[h^{p+1-j}(\\gP,\\cp{\\bo{1-j}}(1))=\\card{\\{I\\st\\card{I}=j-p,\\sw{\\w_I}=j\\}}.\\]\nUsing \\ref{HomKA=HomDA} it is immediate to see that $u$ is actually a morphism\nof $K^b(\\gcoh{\\gP})$; then we can assume that $L^{(j)}\\so(i)=\\MC{u}[-1]$,\nso that we have\n\\begin{multline*}\n(L^{(j)}\\so(i))^p=\\s{H}^p\\oplus(L^{(j-1)}\\so(i))^{p-1}=\n\\s{H}^p\\oplus\\bo{1-j}^{p-j}(i+1-j)\\\\\n\\cong\\so(i-j)^{\\card{\\{I\\st\\card{I}=j-p,\\sw{\\w_I}=j\\}}}\n\\bigoplus_{\\card{I}=j-p,\\sw{\\w_I}\\le j-1}\\so(i-\\sw{\\w_I})\\cong\n\\bigoplus_{\\card{I}=j-p,\\sw{\\w_I}\\le j}\\so(i-\\sw{\\w_I})\\\\\n=\\bo{-j}^{p-j}(i-j)=\\cp{\\bo{-j}}(i-j)[-j]^p.\n\\end{multline*}\nIt is also easy to check that the differential of $L^{(j)}\\so(i)$ can\nbe identified with that of $\\cp{\\bo{-j}}(i-j)[-j]$.\n \\end{proof}\n\n \\begin{rema}\\label{notstrong}\nOne can also prove directly that\n$(\\cp{\\bo{1-\\sw{\\w}}},\\dots,\\cp{\\bo{0}})$ is a (full) exceptional\nsequence of $D^b(\\gcoh{\\gP})$ (see \\ref{shiftexc}). Indeed, for\n$-\\sw{\\w}j$ and $\\delta_{k,0}$ if $l=j$, we see\nthat $(\\cp{\\bo{1-\\sw{\\w}}},\\dots,\\cp{\\bo{0}})$ is really an exceptional\nsequence (it is also full, since it clearly generates the same\ntriangulated subcategory as $\\{\\so,\\dots,\\so(\\sw{\\w}-1)\\}$). Moreover,\nthe above formula shows that, for general $\\w$, there do not exist\nintegers $k_i$ (for $-\\sw{\\w}1$, as we are going to\nsee.\n\n \\begin{lemm}\\label{Omegadecomp}\nLet $\\w=(\\w_0,\\dots,\\w_n)$ be such that $\\w_{m+1}=\\cdots=\\w_n=1$ for some $m\\le\nn$ and let $\\w':=(\\w_0,\\dots,\\w_m)$. Then, denoting by $\\iota:\\gP(\\w')\\to\n\\gP(\\w)$ the closed immersion of graded schemes induced by the natural\nepimorphism of graded rings $\\p(\\w)\\to\\p(\\w')$, $\\forall j\\in\\Z$ there is an\nisomorphism of graded sheaves on $\\gP(\\w')$\n\\[\\iota^*\\sd_{\\gP(\\w)}^j(j)\\cong\n\\bigoplus_{0\\le k\\le j}\\sd_{\\gP(\\w')}^k(k)^{\\binom{n-m}{j-k}}.\\]\n \\end{lemm}\n\n \\begin{proof}\nApplying $\\iota^*$ to the exact sequence\n\\[\\sko_{(\\w)}^{-j-2}\\cong\\bigoplus_{\\card{I}=j+2}\\so_{\\gP(\\w)}(-\\sw{\\w_I})\n\\mor{\\diff{\\cp{\\sko}_{(\\w)}}^{-j-2}}\\sko_{(\\w)}^{-j-1}\\cong\n\\bigoplus_{\\card{I}=j+1}\\so_{\\gP(\\w)}(-\\sw{\\w_I})\\to\\sd_{\\gP(\\w)}^j\\to0\\]\nand twisting by $j$ gives the exact sequence\n\\[\\iota^*\\sko_{(\\w)}^{-j-2}(j)\n\\mor{\\varphi:=\\iota^*\\diff{\\cp{\\sko}_{(\\w)}(j)}^{-j-2}}\n\\iota^*\\sko_{(\\w)}^{-j-1}(j)\\to\\iota^*\\sd_{\\gP(\\w)}^j(j)\\to0.\\]\nSince $\\iota^*\\so_{\\gP(\\w)}(l)\\cong\\so_{\\gP(\\w')}(l)$ $\\forall l\\in\\Z$, we have\n$\\iota^*\\sko_{(\\w)}^{-j-h}(j)\\cong\\bigoplus_{J\\subseteq\\{m+1,\\dots,n\\}}\n\\s{A}^h(J)$, where\n\\[\\s{A}^h(J):=\\bigoplus_{\\card{I}=j+h-\\card{J}}\n\\so_{\\gP(\\w')}(j-\\card{J}-\\sw{\\w'_I})\\cong\n\\sko_{(\\w')}^{\\card{J}-j-h}(j-\\card{J}).\\]\nAs $\\iota^*(x_l)=0$ for $l>m$, it is clear that the component of $\\varphi$ from\n$\\s{A}^2(J)$ to $\\s{A}^1(J')$ is $0$ if $J\\ne J'$ and can be identified with\n$\\diff{\\cp{\\sko}_{(\\w')}(j-\\card{J})}^{\\card{J}-j-2}$ if $J=J'$.\nTherefore\n\\begin{multline*}\n\\iota^*\\sd_{\\gP(\\w)}^j(j)\\cong\\bigoplus_{J\\subseteq\\{m+1,\\dots,n\\}}\n\\cok\\diff{\\cp{\\sko}_{(\\w')}(j-\\card{J})}^{\\card{J}-j-2}\\\\\n\\cong\\bigoplus_{J\\subseteq\\{m+1,\\dots,n\\}}\n\\sd_{\\gP(\\w')}^{j-\\card{J}}(j-\\card{J})\n\\cong\\bigoplus_{0\\le k\\le j}\\sd_{\\gP(\\w')}^k(k)^{\\binom{n-m}{j-k}}.\n\\end{multline*}\n \\end{proof}\n\n \\begin{coro}\\label{weightred}\nWith the same notation used in \\ref{Omegadecomp}, we have\n\\[h^i(\\gP(\\w),\\cp{\\s{F}}\\otimes\\cp{\\bd{j}})=\nh^i(\\gP(\\w'),L\\iota^*(\\cp{\\s{F}})\\otimes\\cp{\\bd{j}})\\]\n$\\forall i\\in\\Z$, $\\forall\\cp{\\s{F}}\\in D^b(\\gcoh{\\gP(\\w)})$ and for\n$n-\\sw{\\w}=m-\\sw{\\w'}f^i_j(\\so(j)[-i-j])>>\n\\sd^{j-1}(j-1)\\otimes_{\\K}H^0(\\gP,\\so(1)) \\\\\n@VV{{\\rm id}\\otimes\\varphi}V @VV{{\\rm id}\\otimes\\varphi}V \\\\\n\\sd^j(j)\\otimes_{\\K}H^{i+j}(\\gP,\\cp{\\s{F}}(-j)) @>f^i_j(\\cp{\\s{F}})>>\n\\sd^{j-1}(j-1)\\otimes_{\\K}H^{i+j}(\\gP,\\cp{\\s{F}}(1-j)).\n\\end{CD}\\]\nIt is then clear that it is enough to prove that $f^i_j(\\so(j)[-i-j])=\nf^{-j}_j(\\so(j))=\\upsilon$.\n\nLet's assume for simplicity that $\\w_i>1$ if and only if $0\\le i \\le m$ for\nsome $0\\le m\\le n$, so that $H^0(\\gP,\\so(1))\\cong\\p_1=\\langle x_{m+1},\\dots,x_n\n\\rangle$. It is immediate to check that the maps\n\\[\\begin{split}\ng^l:\\ko^{l-1}(j)=\\bigoplus_{\\card{I}=1-l}\\p(j)dx_I & \\to\n\\ko^l(j-1)\\otimes_{\\K}\\p_1=(\\bigoplus_{\\card{I}=-l}\\p(j-1)dx_I)\\otimes_{\\K}\\p_1\n\\\\\ndx_I & \\mapsto\n\\sum_{i\\in I,i>m}(-1)^{\\card{\\{k\\in I\\st k0$,\n$\\s{W}^{-j}=\\sd^j(j)$, $\\s{W}^{1-j}=\\sd^{j-1}(j-1)\\otimes_{\\K}\\p_1\\oplus\\s{L}$\nwith $\\s{L}\\in\\co{[n-\\sw{\\w},0]}$, that the component of $\\cdiff{\\s{W}}^{-j}$\nfrom $\\sd^j(j)$ to $\\sd^{j-1}(j-1)\\otimes_{\\K}\\p_1$ is given by $\\psi^{-j}$ and\nthat $\\s{W}^{\\ge2-j}\\in K^b(\\co{]n-\\sw{\\w},j-2]})$ (notice that $\\psi^0$ is\nsurjective, so that no term of the form $\\so(j-1)$ survives). Applying\n$\\cp{\\gbrd}$ to the distinguished triangle $\\s{W}^{<2-j}[-1]\\to\\s{W}^{\\ge2-j}\n\\to\\cp{\\s{W}}\\to\\s{W}^{<2-j}$ yields a distinguished triangle in\n$K^b(\\com{\\intd})$\n\\[\\cp{\\gbrd}(\\s{W}^{<2-j})[-1]\\to\\cp{\\gbrd}(\\s{W}^{\\ge2-j})\\to\n\\cp{\\gbrd}(\\cp{\\s{W}})\\to\\cp{\\gbrd}(\\s{W}^{<2-j}).\\] For $j-1\\le l\\le n$ and\n$\\forall i\\in\\Z$, as $H^i(\\gP,\\s{W}^m(-l))=0$ for $m\\ge 2-j$, by \\ref{vancrit} we\nhave $H^i(\\gP,\\s{W}^{\\ge2-j}(-l))=0$, whence $\\cp{\\gbrd}(\\s{W}^{\\ge2-j})$\ncontains no term of the form $\\sd^l(l)$ for $j-1 \\le l\\le n$. Since moreover\n$\\s{W}^{<2-j}$ can be identified with $\\cp{\\gbrd}(\\s{W}^{<2-j})$ and\n$\\cp{\\gbrd}(\\so(j))$ with $\\cp{\\gbrd}(\\cp{\\s{W}})$, it follows that\n$f^{-j}_j(\\so(j))$ is given by $\\psi^{-j}$. It is not difficult to check that\nits image in $\\Hom_{\\coq{\\intd}}$ is $\\upsilon$.\n \\end{proof}\n\n\n \\section{The theorem on $\\P(\\w)$}\n\nFrom \\ref{wbeilfor} we can obtain also a weaker version of the theorem for\n$\\coh{\\P}$.\n\n \\begin{prop}\nEach of the two sets of coherent sheaves on $\\P$\n\\begin{align*}\n& \\{\\so(j)\\st-\\sw{\\w}0$ if and only if $j+1-\n\\card{I(\\chi)}\\le\\card{J}\\le j$, and that in this case $n-\\sw{\\w}0$.\n\n \\begin{lemm}\\label{torcohom}\nLet $0\\ne\\s{F}\\in\\gcoh{\\gP}$ be a torsion--free sheaf. Then $\\exists k\\in\\Z$ such\nthat $h^0(\\gP,\\s{F}(k))\\ne0$ and $h^0(\\gP,\\s{F}(j))=0$ for $jc$ if\n$l'\\ge l$, and then $h^0(\\gP,\\s{F}(j))=0$ if $j\\le-dl$.\n \\end{proof}\n\n \\begin{prop}\\label{splitcrit}\nLet $\\s{F}\\in\\gcoh{\\gP}$ be a torsion--free sheaf. Then $\\s{F}$ decomposes as a\ndirect sum of line bundles $\\so(l)$ if and only if $H^i(\\gP,\\s{F}(j))=0$ for\n$00$, and $r^d=r't^j$ for some $0\\ne r'\\in R_0$.\nSince $f(rR_{i-j})=0$, we have also $f(r^dR_{i-dj})=f(r't^jR_{i-dj})=0$, whence\n$f(r'R_i)=0$ (because $t^j:R_{i-dj}\\isomor R_i$). In particular, $0=f(r's)=\nr'f(s)$, which implies $r'=0$ (whence $r=0$) or $f(s)=0$, a contradiction.\n \\end{proof}\n\n \\begin{coro}\nLet $\\s{G}\\in\\coh{\\P}$ be a torsion--free sheaf. Then $\\s{G}$ decomposes as a\ndirect sum of sheaves of the form $\\so(l)$ ($l\\in\\Z$) if and only if\n$H^i(\\P,\\sHom_{\\P}(\\so(j),\\s{G}))=0$ for $01$)\nare given by $\\chi(\\so_{\\S})+\\frac{m(m-1)}{2}K^2_{\\S}$ (where $\\chi(\\so_{\\S}):=\n\\chi(\\S,\\so_{\\S})=1+p_g(\\S)-q(\\S)$).\\index{chi(S_0,O_S_0)@$\\chi(\\S,\\so_{\\S})$}\nThus all the cohomology groups of all the powers of $\\ds_{\\S}$ are completely\ndetermined by $p_g(\\S)$, $q(\\S)$ and $K^2_{\\S}$: indeed, we have also\n$H^0(\\S,\\ds_{\\S}^m)=0$ if $m<0$, $H^1(\\S,\\ds_{\\S}^m)=0$ if $m\\ne0,1$ and, by\nSerre duality, $H^2(\\S,\\ds_{\\S}^m) \\cong H^0(\\S,\\ds_{\\S}^{1-m})\\dual$ $\\forall\nm\\in\\Z$. Notice that, in particular,\n$\\chi(\\S,\\ds_{\\S}^m)=\\chi(\\so_{\\S})+\\frac{m(m-1)}{2}K^2_{\\S}$ $\\forall m\\in\\Z$.\n\nAfter extending in a natural way the notion of weighted canonical\nprojection to the graded setting, we will show that the datum of a\n{good birational weighted canonical projection} $\\gph:\\gS\\to\\gY\\subset\\gP$ (where $\\gS$ is the {standard} graded\nscheme $\\stdsch{\\S}{\\ds_{\\S}}$ and $\\gP=\\gP(\\w)$ is a $3$--dimensional\nweighted projective space) is equivalent to the datum of a symmetric\nminimal morphism\n$\\alpha:(\\so\\oplus\\s{E})\\dual(-1-\\sw{\\w})\\to\\so\\oplus\\s{E}$ of vector bundles\non $\\gP$, satisfying some conditions, which are just the graded\nweighted versions of those appearing in the theorem by Catanese and\nSchreyer described in the introduction.\n\nTo this purpose, we first prove that for every good weighted canonical\nprojection $\\gph:\\gS\\to\\gY\\subset\\gP$ there exists a short exact\nsequence in $\\gcoh{\\gP}$\n\\[0\\to(\\so\\oplus\\s{E})\\dual(-1-\\sw{\\w})\\mor{\\alpha=\\begin{pmatrix}\n\\alpha^{(1)} \\\\\n\\alpha'\n\\end{pmatrix}}\\so\\oplus\\s{E}\\to\n\\gph_*\\so_{\\gS}\\to0,\\]\nwhere $\\s{E}=\\s{E}(\\gph)$ is a direct sum of vector bundles of the form\n$\\sd^j(j-2)$ for $j=0,1,2$ and $\\so(j-2)$ for $3-\\sw{\\w}2$).\nAnyway, eventually we manage (again, under the mild assumption\n$\\deg(\\Y)>2$) to determine the explicit form of the vector bundle\n$\\s{E}$, in which the coefficients of the $\\sd^j(j-2)$ depend only on\n$p_g(\\S)$, $q(\\S)$, $K^2_{\\S}$ and $\\w$ (the examples of chapter $4$\nwill show that instead the complicated expressions we get for the\ncoefficients of the $\\so(j-2)$ really depend on $\\gph$).\n\nWe also prove that an exact sequence as above is unique up to\nisomorphism, and this fact implies that $\\alpha$ can be chosen to be\nsymmetric, provided $\\gph$ is birational. Then, as in the case of\nprojections to $\\P^3$, the main result needed to complete the proof of\nthe theorem is that the rank condition\n$\\s{I}_{\\rk{\\s{E}}}(\\alpha)=\\s{I}_{\\rk{\\s{E}}}(\\alpha')$ is\nsatisfied if and only if $\\cok\\alpha$ is a graded sheaf of commutative\n$\\so_{\\gY}$--algebras.\n\n\n\n \\section{Weighted canonical projections}\n\nLet $\\S$ be a minimal surface of general type. The canonical ring of $\\S$ is\nthe positively graded ring\n\\[\\R=\\R(\\S,\\ds_{\\S}):=\\bigoplus_{d\\ge0}H^{0}(\\S,\\ds_{\\S}^d).\\]\\index{R@$\\R$}\nIt is well known that $\\R$ is a noetherian domain, so that $\\R\\in\\prng$. The\ncanonical model of $\\S$ is $\\X:=\\proj\\R$.\\index{X_0@$\\X$} $\\X$ is a surface\nwith only rational double points as singularities and there is a natural\nbirational morphism $\\pr:\\S\\to\\X$,\\index{pi_0@$\\pr$} which contracts\n$(-2)$--curves (i.e. curves $C\\cong\\P^1$ with $C^2=-2$, $CK_{\\S}=0$).\n$\\ds_{\\X}\\cong\\so_{\\X}(1)$ is an ample invertible sheaf ($\\X$ is Gorenstein),\nand there are isomorphisms $\\forall m\\in\\Z$:\n\\begin{align*}\n& \\pr^*\\ds_{\\X}^m\\cong\\ds_{\\S}^m, & {\\pr}_*\\ds_{\\S}^m\\cong\\ds_{\\X}^m.\n\\end{align*}\nThe singularities of $\\X$ being rational means that $R^1{\\pr}_*(\\so_{\\S})=0$,\nand this implies that $H^i(\\S,\\ds_{\\S}^m)\\cong H^i(\\X,\\ds_{\\X}^m)$\n$\\forall m\\in\\Z$.\n\nLet $\\gS$\\index{S@$\\gS$} be the {standard} graded scheme $\\stdsch{\\S}{\\ds_{\\S}}$\n(i.e. $\\gS$ has the same topological space as $\\S$ and $\\so_{\\gS}=\n\\bigoplus_{d\\in\\Z}\\ds_{\\S}^d$) and let $\\gX:=\\gproj\\R$.\\index{X@$\\gX$} Although\n$\\R$ is usually not generated by $\\R_1$ as an $\\R_0=\\K$--algebra, we have the\nfollowing result.\n\n \\begin{lemm}\\label{canmodstd}\n$\\gX$ is a {standard} graded scheme.\n \\end{lemm}\n\n \\begin{proof}\nSince the linear system $\\linsys{mK_{\\X}}$ is base point free for $m>>0$ ($m\\ge\n4$), $\\forall\\I{p}\\in\\gX=\\gproj\\R$ there exist $s\\in\\R_m\\setminus\\I{p}$ and $t\\in\n\\R_{m+1}\\setminus\\I{p}$ for some $m$. Then $t\/s\\in(\\R_{\\I{p}})_1$ is an invertible\nelement, i.e. $\\R_{\\I{p}}$ is standard.\n \\end{proof}\n\nBy \\ref{stdchar} the natural isomorphism $\\pr^*\\ds_{\\X}\\cong\\ds_{\\S}$ induces an\nextension of $\\pr$ to a morphism of graded schemes\n$\\gpr:\\gS\\to\\gX$\\index{pi@$\\gpr$} such that $\\gpr\\mrs$ is an isomorphism.\nNotice moreover that, since $\\ds_{\\X}\\cong \\so_{\\X}(1)$, \\ref{canmodstd} and\n\\ref{Serredual} imply that also $\\ds_{\\gX} \\cong\\so_{\\gX}(1)$; by definition of\n$\\gS$ it is also clear that $\\ds_{\\gS}\\cong \\so_{\\gS}(1)$.\n\nLet $\\w\\in\\N_+^{n+1}$ for some $n>0$. By \\ref{projmor} the choice of sections\n\\begin{align*}\n& \\sigma_i\\in H^0(\\gS,\\so_{\\gS}(\\w_i))\\cong H^0(\\S,\\ds_{\\S}^{\\w_i}) &\ni=0,\\dots,n\n\\end{align*}\ndetermines a rational map $\\gph:\\gS\\rat\\gP(\\w)$\\index{phi@$\\gph$} (and also\n$\\ph:\\S\\rat\\P(\\w)$),\\index{phi_0@$\\ph$} called a {\\em weighted canonical\nprojection}.\\index{weighted canonical projection} $\\gph$ always factors through\n$\\gpr:\\gS\\to\\gX$, i.e. $\\gph=\\gps\\comp\\gpr$\\index{psi@$\\gps$} (and\n$\\ph=\\ps\\comp\\pr$),\\index{psi_0@$\\ps$} where the rational map $\\gps:\\gX\n\\to\\gP(\\w)$ is the one induced by the morphism of graded rings\\index{rho@$\\rh$}\n\\[\\begin{split}\n\\rh:\\p(\\w) & \\to\\R\\\\\nx_i & \\mapsto\\sigma_i\n\\end{split}\\]\ndetermined by the chosen sections.\n\n \\begin{defi}\nA weighted canonical projection $\\gph$ (or $\\ph$) is called {\\em\ngood}\\index{good!weighted canonical projection} if it is a morphism; it is\ncalled {\\em birational}\\index{birational weighted canonical projection} if it\nis birational onto the image.\n \\end{defi}\n\n \\begin{rema}\n$\\ph$ is good if and only if $\\gph$ is good if and only if $\\gps$ (and $\\ps$)\nis a morphism. By \\ref{prngmor} this happens if and only if $\\rh$ is a finite\nmorphism, and in this case $\\gps$ and $\\ps$ are also finite.\n \\end{rema}\n\n \\begin{rema}\\label{agalg}\n$\\gph$ (respectively $\\ph$) is good birational if and only if $\\gps$\n(respectively $\\ps$) is a morphism which is birational onto the image. This\nhappens if and only if $\\rh$ is finite and $\\exists p\\in\\p_+\\hom$ such that\n$p\\notin\\ker\\rh$ and $\\rh_p: \\p_p\\to\\R_p$ (respectively\n$\\rh_{(p)}:\\p_{(p)}\\to\\R_{(p)}$) is surjective. It is thus clear that if $\\gph$\nis good birational, then $\\ph$ is good birational, too, but the converse is not\ntrue in general (see \\ref{gphagphag}).\n \\end{rema}\n\n \\begin{rema}\\label{psnorm}\nIf $\\ph$ is good birational, then $\\ps$ is the normalization morphism (because\nit is finite and birational and $\\X$ is normal).\n \\end{rema}\n\nKeeping the notation introduced so far, from now on we will assume that $\\gph:\n\\gS\\to\\gP=\\gP(\\w)$ is a good weighted canonical projection with $\\w\\in\\N_+^4$\n(i.e. $n=\\dim\\P=3$). We will denote by $\\gY$\\index{Y@$\\gY$} (respectively\n$\\Y$)\\index{Y_0@$\\Y$} the image of $\\gph$ in $\\gP$ (respectively of $\\ph$ in\n$\\P$). As $\\gY$ is a hypersurface in $\\gP$, we have $\\gY=\\gproj\\p\/(\\f)$ (and\n$\\Y=\\proj\\p\/(\\f)$), where $\\f\\in\\p=\\p(\\w)$\\index{f@$\\f$} is a homogeneous\nirreducible polynomial which generates $\\ker\\rh$.\\footnote{It is easy to see\nthat $\\deg(\\f)=(K^2_{\\S}\\prod_{i=0}^3\\w_i)\/\\deg(\\ps)$.}\n\nNow we are going to prove a result on relative duality for the\nmorphism $\\gps:\\gX\\to\\gP$, which will be needed to obtain a symmetric\nresolution of $\\gps_*\\so_{\\gX}\\cong\\gph_*\\so_{\\gS}$. For later use, we\nwill prove it under more general hypotheses. So, let $\\gamma:Z\\to\\gP$\nbe a finite morphism, where $Z$ is a (necessarily {quasi--standard} by \\ref{YgoodXgood})\ngraded scheme with $Z_0$ of dimension $2$.\\footnote{Actually, with obvious\nmodifications, everything holds for arbitrary dimensions of $Z$ and $\\gP$.\nIndeed, the proofs of the following results are essentially a rewriting of\nthose of \\cite[III, lemma 7.3, lemma 7.4, prop. 7.5]{H1}, where it is proved\nthat every closed subscheme of $\\P^n$ has a dualizing sheaf.} $\\forall i\\in\\N$ let\n$\\s{E}^i:=\\gsExt^i_{\\gP}(\\gamma_*\\so_Z,\\ds_{\\gP})$, and observe that\n$\\s{E}^i\\in\\gcoh{\\gP}$ by \\ref{gsExtgcoh} ($\\gamma_*\\so_Z$ is coherent because\n$\\gamma$ is a finite morphism).\n\n \\begin{lemm}\\label{gsExtvan}\nIf $H^{3-i}(Z,\\so_Z(j))=0$ for $j<<0$, then $\\s{E}^i=0$. In particular,\n$\\s{E}^0=0$.\n \\end{lemm}\n\n \\begin{proof}\nAs $\\s{E}^i\\cong\\gsec(\\s{E}^i)\\gsh{}$ by \\ref{gmogqco}, in order to prove that\n$\\s{E}^i=0$ it is enough to show that $H^0(\\gP,\\s{E}^i(d))=0$ for $d>>0$. By\n\\ref{secgsExt} $\\forall i\\in\\Z$ $\\exists m\\in\\N$ such that\n\\[H^0(\\gP,\\s{E}^i(d))\\cong\\Ext^i_{\\gP}(\\gamma_*\\so_Z,\\ds_{\\gP}(d))\\cong\n\\Ext^i_{\\gP}(\\gamma_*\\so_Z(-d),\\ds_{\\gP})\\]\n$\\forall d>m$. By Serre duality and since $\\gamma$ is an affine morphism we have\n\\[\\Ext^i_{\\gP}(\\gamma_*\\so_Z(-d),\\ds_{\\gP})\\cong\nH^{3-i}(\\gP,\\gamma_*\\so_Z(-d))\\dual\\cong\nH^{3-i}(Z,\\so_Z(-d))\\dual,\\]\nand everything follows.\n \\end{proof}\n\nAs clearly $\\s{E}^1\\in\\gcoh{\\gP}\\cap\\gsmo{\\gamma_*\\so_Z}$, by \\ref{gSpecA}\nthere exists unique (up to isomorphism) $\\s{D}\\in\\gcoh{Z}$ such that $\\s{E}^1\n\\cong\\gamma_*\\s{D}$.\n\n \\begin{lemm}\\label{HomExt}\n$\\forall\\s{F}\\in\\gcoh{Z}$ there is a functorial isomorphism\n\\[\\Hom_Z(\\s{F},\\s{D})\\cong\\Ext^1_{\\gP}(\\gamma_*\\s{F},\\ds_{\\gP}).\\]\n \\end{lemm}\n\n \\begin{proof}\n$\\forall\\s{G}\\in\\gqco{\\gP}$ let $\\gamma^!\\s{G}\\in\\gqco{Z}$ be the graded sheaf\n(which exists unique up to isomorphism by \\ref{gSpecA}) such that\n$\\gamma_*\\gamma^!\\s{G}\\cong\\gsHom_{\\gP}(\\gamma_*\\so_Z,\\s{G})$. It is easy to see\nthat $\\gamma^!:\\gqco{\\gP}\\to\\gqco{Z}$ is right adjoint of\n$\\gamma_*:\\gqco{Z}\\to\\gqco{\\gP}$.\n\nTaking an injective resolution $\\cp{\\s{I}}$ of $\\ds_{\\gP}$ in $\\gqco{\\gP}$, we\nhave natural isomorphisms\n\\[\\gamma_*H^i(\\gamma^!\\cp{\\s{I}})\\cong H^i(\\gamma_*\\gamma^!\\cp{\\s{I}})\\cong\nH^i(\\gsHom_{\\gP}(\\gamma_*\\so_Z,\\cp{\\s{I}}))\\cong\n\\gsExt^i_{\\gP}(\\gamma_*\\so_{Z},\\ds_{\\gP})=\\s{E}^i\\]\n$\\forall i\\in\\N$. In particular, it follows from \\ref{gsExtvan} that\n$H^0(\\gamma^!\\cp{\\s{I}})=0$. Since moreover $\\gamma^!\\s{I}^j$ is injective in\n$\\gqco{Z}$ ($\\Hom_{\\gqco{Z}}(-,\\gamma^!\\s{I}^j)\\cong\n\\Hom_{\\gqco{\\gP}}(\\gamma_*-,\\s{I}^j)$ is exact because composition of two exact\nfunctors), $\\gamma^!\\cp{\\s{I}}$ splits as $\\cp{\\s{J}}\\oplus\\cp{\\s{L}}$ with\n$\\cp{\\s{L}}$ exact and $\\s{J}^0=0$. Therefore\n\\[\\gamma_*\\s{D}\\cong\\s{E}^1\\cong\\gamma_*H^1(\\gamma^!\\cp{\\s{I}})\\cong\n\\gamma_*H^1(\\cp{\\s{J}}),\\]\nwhence $\\s{D}\\cong H^1(\\cp{\\s{J}})=\\ker\\cdiff{\\s{J}}^1$ by \\ref{gSpecA}. We\nhave then natural isomorphisms\n\\begin{multline*}\n\\Hom_{Z}(\\s{F},\\s{D})\\cong\\ker\\Hom_{Z}(\\s{F},\\cdiff{\\s{J}}^1)\\cong\nH^1(\\Hom_{Z}(\\s{F},\\cp{\\s{J}}))\\cong\\\\\n\\cong H^1(\\Hom_{Z}(\\s{F},\\gamma^!\\cp{\\s{I}}))\n\\cong H^1(\\Hom_{\\gP}(\\gamma_*\\s{F},\\cp{\\s{I}}))\\cong\n\\Ext^1_{\\gP}(\\gamma_*\\s{F},\\ds_{\\gP}).\n\\end{multline*}\n \\end{proof}\n\n \\begin{prop}\\label{gsExtdual}\nLet $\\gamma:Z\\to\\gP$ be a finite morphism of graded schemes with $Z_0$ of\ndimension $2$. Then $Z$ has a dualizing sheaf $\\ds_Z$ and\n$\\gamma_*\\ds_{Z}\\cong\\gsExt^1_{\\gP}(\\gamma_*\\so_{Z},\\ds_{\\gP})$.\n \\end{prop}\n\n \\begin{proof}\nUsing \\ref{HomExt}, Serre duality on $\\gP$ and the fact that $\\gamma$ is a\nfinite morphism, we see that $\\forall\\s{F}\\in\\gcoh{Z}$ there are natural\nisomorphisms\n\\[\\Hom_{Z}(\\s{F},\\s{D})\\cong\\Ext^1_{\\gP}(\\gamma_*\\s{F},\\ds_{\\gP})\\cong\nH^2(\\gP,\\gamma_*\\s{F})\\dual\\cong H^2(Z,\\s{F})\\dual.\\]\nThen $\\s{D}\\cong\\ds_Z$ by definition of dualizing sheaf, whence\n$\\gamma_*\\ds_{Z}\\cong\\gsExt^1_{\\gP}(\\gamma_*\\so_{Z},\\ds_{\\gP})$.\n \\end{proof}\n\n\n \\section{Construction of the resolution}\n\nKeeping the notation introduced so far, in this section we set\n$\\s{F}:=(\\gps_*\\so_{\\gX})(2)\\cong\\gps_*(\\so_{\\gX}(2))\\in\\gcoh{\\gP}$\n(the twist is necessary in order to have a symmetric Beilinson resolution).\n\n \\begin{lemm}\\label{Fsym}\n$R\\gsHom_{\\gP}(\\s{F},\\so(3-\\sw{\\w}))\\cong\\s{F}[-1]$ in $D^b(\\gcoh{\\gP})$. More\nexplicitly,\n\\[\\gsExt^i_{\\gP}(\\s{F},\\so_{\\gP}(3-\\sw{\\w}))\\cong\n\\begin{cases}\n\\s{F} & \\text{if $i=1$} \\\\\n0 & \\text{if $i\\ne1$}\n\\end{cases}.\\]\n \\end{lemm}\n\n \\begin{proof}\nAs $H^{3-i}(\\gX,\\so_{\\gX}(j))\\cong H^{3-i}(\\X,\\ds_{\\X}^j)=0$ for $j<0$ if\n$i\\ne1$, by \\ref{gPdualsh} and \\ref{gsExtvan}\n$\\gsExt^i_{\\gP}(\\s{F},\\so_{\\gP}(3-\\sw{\\w}))\\cong\n\\gsExt^i_{\\gP}(\\gps_*\\so_{\\gX},\\ds_{\\gP})(1)=0$ if $i\\ne1$, and by\n\\ref{gsExtdual}\n\\[\\gsExt^1_{\\gP}(\\s{F},\\so_{\\gP}(3-\\sw{\\w}))\\cong\n\\gsExt^1_{\\gP}(\\gps_*\\so_{\\gX},\\ds_{\\gP}(1))\\cong\\gps_*\\ds_{\\gX}(1)\\cong\\s{F}.\\]\n \\end{proof}\n\nLet $\\cp{\\gbrd}:=\\cp{\\gbrd}(\\s{F})$ and, for $3-\\sw{\\w}0$ or $i<-1$ and\n$\\cbrd^{-1}_{3-\\sw{\\w}-j}=\\cbrd^0_j:=\\cbrd_j$.\\index{y_j@$\\cbrd_j$}\n \\end{lemm}\n\n \\begin{proof}\nBy \\ref{Fsym} and \\ref{symbeilres} $(\\cp{\\gbrd})\\dual(3-\\sw{\\w})\\cong\n\\cp{\\gbrd}[-1]$ in $C^b(\\gcoh{\\gP})$. Therefore $\\cbrd^i_j=\n\\cbrd^{-i-1}_{3-\\sw{\\w}-j}$ for $3-\\sw{\\w}0$. Taking into account that\n$H^i(\\gP,\\s{F}(l))\\cong H^i(\\X,\\ds_{\\X}^{l+2})$, it is immediate to check that\n$H^i(\\gP,\\s{F}\\otimes\\bd{j}^m)=0$ for $3-\\sw{\\w}-m$. The\nresult then follows from \\ref{vancrit}.\n \\end{proof}\n\nBy \\ref{wbeilfor} $\\cp{\\gbrd}$ has therefore the following form:\n\\[0\\to\\sd^2(2)\\to\\begin{matrix}\n\\so(3-\\sw{\\w})^{\\chi(\\so_{\\S})+K^2_{\\S}}\\\\\n\\oplus\\sd^2(2)^{q(\\S)}\\oplus\\sd^1(1)^{p_g(\\S)}\\\\\n\\bigoplus_{3-\\sw{\\w}1,\\w_i+\\w_{i'}=2-j\\}}$ and\n$z^0_j=\\card{\\{i\\st\\w_i=2-j\\}}$.\\index{z^i_j@$z^i_j$}\n \\end{lemm}\n\n \\begin{proof}\nBy \\ref{wbeilfor} the only non trivial part is to show that\n$h^i(\\gP(\\w),\\cp{\\bd{j}}(2))=z^i_j$ for $n-\\sw{\\w}m$. Then, setting $\\w':=(\\w_0,\\dots,\\w_m)$, by\n\\ref{weightred} we have\n$h^i(\\gP(\\w),\\cp{\\bd{j}}(2))=h^i(\\gP(\\w'),\\cp{\\bd{j}}(2))$ (since\n$L\\iota^*\\so_{\\gP(\\w)}(2)\\cong\\iota^*\\so_{\\gP(\\w)}(2)\\cong\\so_{\\gP(\\w')}(2)$).\nThe complex on $\\gP(\\w')$ (obtained truncating $\\cp{\\sko}_{(\\w')}(2)[-1]$)\n\\[0\\to\\sd^2(2)\\to\\sko^{-2}(2)\\cong\\bigoplus_{i>> \\s{Z}^{-2}=\\sd^2(2) @>d^{-2}>> \\s{Z}^{-1} @>d^{-1}>> \\s{Z}^{0} @>d^0>>\n\\so(2)=\\s{Z}^1 @>>> 0\\\\\n@. @VV{\\alpha^{-2}}V @VV{\\alpha^{-1}}V @VV{\\alpha^0}V @VV{\\alpha^1}V @. \\\\\n0 @>>> \\tilde{\\s{Z}}^{-1}=\\so(1-\\sw{\\w}) @>\\tilde{d}^{-2}>> \\tilde{\\s{Z}}^0\n@>\\tilde{d}^{-1}>> \\tilde{\\s{Z}}^1 @>\\tilde{d}^0>> \\sd^1(1)=\\tilde{\\s{Z}}^2\n@>>> 0\n\\end{CD}\\]\nis a morphism of complexes. Since $\\Hom_{\\gP}(\\so(2),\\sd^1(1))=0$, we have\n$\\alpha^1=0$, and, by duality, also $\\alpha^{-2}=0$. Therefore $\\alpha^{-1}\n\\comp d^{-2}=0$, and so $\\alpha^{-1}$ factors through $\\im d^{-1}=\\ker d^0$. As\n$\\Ext^1_{\\gP}(\\so(2),\\tilde{\\s{Z}}^0)\\cong H^1(\\gP,\\tilde{\\s{Z}}^0(-2))=0$ by\n\\ref{lbcohom}, it follows that $\\alpha^{-1}$ actually factors through $d^{-1}$,\nsay $\\alpha^{-1}=\\beta\\comp d^{-1}$ for some $\\beta:\\s{Z}^0\\to\\tilde{\\s{Z}}^0$.\nSetting $\\gamma:=\\alpha^0-\\tilde{d}^{-1} \\comp\\beta$, we have $\\gamma\\comp\nd^{-1}=0$, which implies that $\\gamma=\\beta' \\comp d^0$ for some\n$\\beta':\\so(2)\\to\\tilde{\\s{Z}}^1$. As $\\Hom_{\\gP}(\\so(2),\\tilde{\\s{Z}}^1)\\cong\nH^0(\\gP,\\tilde{\\s{Z}}^1(-2))=0$ by \\ref{omcohvan}, we see that\n$\\alpha^0=\\tilde{d}^{-1}\\comp\\beta$, and this proves that $\\cp{\\alpha}$ is\nhomotopic to zero.\n \\end{proof}\n\nConsider the distinguished triangle of $K^b(\\gcoh{\\gP})$\n\\[\\cp{\\gbrd}[-1]\\mor{\\cp{\\tilde{\\varphi}}}\\cp{\\tilde{\\s{Z}}}\n\\mor{\\inc{\\tilde{\\varphi}}}\\mc{\\tilde{\\varphi}}\\mor{\\pro{\\tilde{\\varphi}}}\n\\cp{\\gbrd}\\mor{\\cp{\\tilde{\\varphi}}[1]}\\cp{\\tilde{\\s{Z}}}[1].\\]\nAs $\\cp{\\tilde{\\s{Z}}}\\iso0$ in $D^b(\\gcoh{\\gP})$, by \\ref{trcatprop}\n$[\\pro{\\tilde{\\varphi}}]$ is an isomorphism in $D^b(\\gcoh{\\gP})$. Moreover,\n$\\cp{\\tilde{\\varphi}}[1]\\comp\\cp{\\varphi}=0$ by \\ref{HomKb=0}, so that, again\nby \\ref{trcatprop}, $\\cp{\\varphi}=\\pro{\\tilde{\\varphi}}\\comp\\cp{\\eta}$ for some\n$\\cp{\\eta}:\\cp{\\s{Z}}\\to\\mc{\\tilde{\\varphi}}$ in $K^b(\\gcoh{\\gP})$. Again\nbecause $\\cp{\\s{Z}}\\iso0$ in $D^b(\\gcoh{\\gP})$, we see that $[\\inc{\\eta}]:\n\\mc{\\tilde{\\varphi}}\\to\\mc{\\eta}$ is an isomorphism, i.e. there are\nisomorphisms in $D^b(\\gcoh{\\gP})$\n\\[\\mc{\\eta}\\cong\\mc{\\tilde{\\varphi}}\\cong\\cp{\\gbrd}\\cong\\s{F}.\\]\n\n \\begin{prop}\\label{Fres}\nIn the notation introduced so far, let $\\cp{\\s{W}}\\in K^b(\\com{})$ be the\nminimal complex isomorphic to $\\mc{\\eta}$ ($\\cp{\\s{W}}$ exists unique up to\nisomorphism in $C^b(\\com{})$ by \\ref{minuni}). Then $\\cp{\\s{W}}\\cong\\s{F}$ in\n$D^b(\\gcoh{\\gP})$ and, assuming that $\\deg(\\f)>2$, we have $\\s{W}^i=0$ if\n$i\\ne-1,0$, $\\s{W}^{-1}\\cong(\\s{W}^0)\\dual(3-\\sw{\\w})$ and\n\\[\\s{W}^0=\\so(2)\\oplus\\so^{\\chi(\\so_{\\S})+K_{\\S}^2-\\pd_2}\\oplus\\sd^1(1)^{q(\\S)}\n\\oplus\\sd^2(2)^{p_g(\\S)-\\pd_1}\\bigoplus_{3-\\sw{\\w}2$, this last map is\ninjective, and so $\\so$ only survives in $\\s{W}^0$ with coefficient\n$h^0(\\gP,\\s{F})-h^0(\\gP,\\so(2))=\\chi(\\so_{\\S})+K_{\\S}^2-\\pd_2$. Now we consider\n$\\sd^1(1)$: we have\n\\begin{gather*}\n\\sd^1(1)\\otimes_{\\K}H^0(\\gP,\\so(1))\\subset\\s{Z}^{-1}\\subset\\s{V}^{-2},\\\\\n\\sd^1(1)\\otimes_{\\K}H^0(\\gP,\\s{F}(-1))\\subset\\gbrd^{-1}\\subset\\s{V}^{-1},\\\\\n\\sd^1(1)\\otimes_{\\K}H^1(\\gP,\\s{F}(-1))\\subset\\gbrd^0\\subset\\s{V}^0,\\\\\n\\sd^1(1)\\otimes_{\\K}H^2(\\gP,\\s{F}(-1))\\cong\n\\sd^1(1)\\otimes_{\\K}H^0(\\gP,\\gps_*\\so_{\\gX})\\dual\\subset\\gbrd^1\\subset\\s{V}^1,\n\\\\\n\\sd^1(1)\\otimes_{\\K}H^3(\\gP,\\so(-\\sw{\\w}))\\cong\n\\sd^1(1)\\otimes_{\\K}H^0(\\gP,\\so)\\dual\\subset\\tilde{\\s{Z}}^2=\\s{V}^2.\n\\end{gather*}\nThe only non trivial maps between them in $\\Hom_{\\coq{}}$ are the restriction\nof $\\eta^{-1}$ (which is the same as the restriction of $\\varphi^{-1}$) between\nthe first two terms and the restriction of $\\tilde{\\varphi}^2=\n(\\varphi^{-2})\\dual(3-\\sw{\\w})$ between the last two terms. The former is\n${\\rm id}_{\\sd^1(1)}\\otimes H^0(\\gP,\\gps\\mrs(1))$ (which is injective) and the\nlatter is ${\\rm id}_{\\sd^1(1)}\\otimes H^0(\\gP,\\gps\\mrs)\\dual$ (which is an\nisomorphism). It follows that $\\sd^1(1)$ only survives in $\\s{W}^0$ with\ncoefficient $h^1(\\gP,\\s{F}(-1))=q(\\S)$ and in $\\s{W}^{-1}$ with coefficient\n$h^0(\\gP,\\s{F}(-1))-h^0(\\gP,\\so(1))=p_g(\\S)-\\pd_1$ (notice that\n$(\\sd^1(1)^{p_g(\\S)-\\pd_1})\\dual(3-\\sw{\\w})\\cong\\sd^2(2)^{p_g(\\S)-\\pd_1}$, the\nexpected term in $\\s{W}^0$).\nIn a completely similar way (or by symmetry, see \\ref{unisym}) one can proceed\nin the case of $\\sd^2(2)$ and $\\so(3-\\sw{\\w})$.\n\nIt remains to consider the case of $\\so(j)$ for $3-\\sw{\\w}1$. Setting\n\\[h^i_j:=H^i(\\gP,\\gps\\mrs(2)\\otimes\\cp{\\bd{j}}):\nH^i(\\gP,\\cp{\\bd{j}}(2))\\to H^i(\\gP,\\s{F}\\otimes\\cp{\\bd{j}})\\]\nand remembering that $H^i(\\gP,\\s{F}\\otimes\\cp{\\bd{j}})\\cong\nH^{-1-i}(\\gP,\\s{F}\\otimes\\cp{\\bd{3-\\sw{\\w}-j}})\\dual$, it is clear that the\nmaps from $\\so(j)\\otimes_{\\K}V^i_j$ to $\\so(j)\\otimes_{\\K}V^{i+1}_j$ are given\nby ${\\rm id}_{\\so(j)}\\otimes d^i_j$, where\n\\begin{align*}\n& d^{-2}_j=\\begin{pmatrix}\nh^{-1}_j \\\\\n0\n\\end{pmatrix}, &\n& d^{-1}_j=\\begin{pmatrix}\n0 & h^0_j \\\\\n(h^0_{3-\\sw{\\w}-j})\\dual & g\n\\end{pmatrix}, &\nd^0_j=\\begin{pmatrix}\n(h^{-1}_{3-\\sw{\\w}-j})\\dual & 0\n\\end{pmatrix}\n\\end{align*}\nfor some $g:H^0(\\gP,\\cp{\\bd{j}}(2))\\to H^0(\\gP,\\cp{\\bd{3-\\sw{\\w}-j}}(2))\\dual$\n(observe that $d^{i+1}_j\\comp d^i_j=0$ because $\\cp{\\s{V}}$ is a complex and\nevery morphism $\\so(j)\\to\\s{E}\\to\\so(j)$ is $0$ by \\ref{indec} if $\\s{E}=\n\\sd^{j'}(j')$ or $\\s{E}=\\so(l)$ for some $l\\ne j$). Let's denote by $\\cre^i_j$\nthe coefficient of $\\so(j)$ in $\\s{W}^i$. As $h^{-1}_j$ is injective by\n\\ref{coeffinj} (whence also $(h^{-1}_{3-\\sw{\\w}-j})\\dual$ is surjective), we\nsee that $\\cre^i_j=0$ if $i<-1$ or $i>0$. Then by symmetry (see \\ref{unisym})\nit is enough to show that $\\cre^0_j=\\cre_j(\\gph)$. Let's assume first that\n$g=0$: in this case we have\n\\begin{multline*}\n\\cre^0_j=\\dim_{\\K}\\ker d^0_j\/\\im d^{-1}_j=\\dim V^0_j-\\dim V^1_j-\n\\dim\\im h^0_j-\\dim\\im h^0_{3-\\sw{\\w}-j}=\\\\\n=\\cbrd_j+z^0_{3-\\sw{\\w}-j}-z^{-1}_{3-\\sw{\\w}-j}-(z^0_j-k_j)-\n(z^0_{3-\\sw{\\w}-j}-k_{3-\\sw{\\w}-j})=\\cre_j(\\gph).\n\\end{multline*}\nSuppose now that $g\\ne0$: we must have, in particular, $h^0(\\gP,\\cp{\\bd{j}}(2))\n=z^0_j\\ne0$ and $h^0(\\gP,\\cp{\\bd{3-\\sw{\\w}-j}}(2))=z^0_{3-\\sw{\\w}-j}\\ne0$.\nAssuming for simplicity $\\w_0\\le \\w_1\\le \\w_2\\le \\w_3$, by \\ref{O2res} this\nhappens if and only if $\\sw{\\w}=2\\w_3-1$ (i.e. $\\w_3=\\w_0+\\w_1+\\w_2+1$) and\n$j=2-\\w_3$ (notice that then $j=3-\\sw{\\w}-j$), in which case $z^0_{2-\\w_3}=1$\n(whence $k_{2-\\w_3}$ can only be $0$ or $1$). If $k_{2-\\w_3}=0$ (so that\n$h^0_{2-\\w_3}$ is injective and $(h^0_{2-\\w_3})\\dual$ is surjective), it is\nclear that $\\im d^{-1}_j$ (and of course also $\\ker d^0_j$) doesn't depend on\n$g$; therefore also in this case $\\cre^0_j=\\cre_j(\\gph)$. On the other hand, if\n$k_{2-\\w_3}=1$ (i.e. $h^0_{2-\\w_3}=0$), $\\dim_{\\K}\\im d^{-1}_j$ increases by\n$1$ (with respect to the case $g=0$).\n\nThus, to conclude we have only to show that (under the hypothesis $\\sw{\\w}=\n2\\w_3-1$ and $h^0_{2-\\w_3}=0$) $g=0$ if and only if $\\cp{\\tilde{\\varphi}}[1]\n\\comp\\cp{\\varphi}=0\\in C^b(\\com{})$. Now, again by \\ref{O2res} it is clear that\nthe restriction $\\xi$ of $\\cdiff{\\s{V}}^{-1}$ from $\\s{Z}^0$ to\n$\\tilde{\\s{Z}}^0$ is just ${\\rm id}_{\\so(2-\\w_3)}\\otimes g$, whence $g=0$ if and\nonly if $\\xi=0$. Denoting by $\\cp{d}$ the differential of $\\cp{\\s{Z}}$, the\nfact that $\\Hom_{\\gP}(\\s{Z}^1,\\tilde{\\s{Z}}^0)=0$ implies that $\\xi=0$ if and\nonly if $\\xi\\comp d^{-1}=0$. Moreover, by definition of mapping cone $\\xi$ is\nalso the component of $\\eta^0$ from $\\s{Z}^0$ to $\\tilde{\\s{Z}}^0$, and the\ncondition that $\\eta$ is a morphism of complexes easily implies (since\n$\\Hom_{\\gP}(\\s{Z}^{-1},\\tilde{\\s{Z}}^{-1})=0$) that $\\xi\\comp d^{-1}=\n\\tilde{\\varphi}^0\\comp\\varphi^{-1}$. Then it is enough to observe that if this\nlast term is $0$, also $\\tilde{\\varphi}^1\\comp\\varphi^0=(\\tilde{\\varphi}^0\\comp\n\\varphi^{-1})\\dual(3-\\sw{\\w})=0$, so that $\\cp{\\tilde{\\varphi}}[1]\\comp\n\\cp{\\varphi}=0$ (in any case $\\tilde{\\varphi}^{i+1}\\comp\\varphi^i=0$ if $i\\ne\n-1,0$).\n \\end{proof}\n\n \\begin{rema}\\label{unisym}\nA minimal complex $\\cp{\\s{L}}\\in C^b(\\com{\\{1-\\sw{\\w},2\\}\\cup]3-\\sw{\\w},0[})$\nwith $\\s{L}^i=0$ for $i\\ne-1,0$ and quasi--isomorphic to $\\s{F}$ is unique up\nto isomorphism in $C^b(\\gcoh{\\gP})$ (hence it must coincide with the complex\n$\\cp{\\s{W}}$ defined in \\ref{Fres}). Indeed, if $\\cp{\\s{M}}$ is another complex\nwith the same properties, since $\\Ext^1_{\\gP}(\\s{L}^0,\\s{M}^{-1})=0$ (as it is\neasy to check), by \\ref{AfullDA} and \\ref{HomKA=HomDA} there are natural\nisomorphisms\n\\[\\Hom_{\\gP}(\\s{F},\\s{F})\\cong\\Hom_{D^b(\\gcoh{\\gP})}(\\cp{\\s{L}},\\cp{\\s{M}})\\cong\n\\Hom_{K^b(\\gcoh{\\gP})}(\\cp{\\s{L}},\\cp{\\s{M}}).\\]\nIt follows that $\\cp{\\s{L}}\\cong\\cp{\\s{M}}$ in $K^b(\\gcoh{\\gP})$, whence also in\n$C^b(\\gcoh{\\gP})$ by \\ref{minuni}.\n\nMoreover, it is clear a priori that $\\s{L}$ must satisfy $\\s{L}^{-1}\\cong\n\\s{L}^0(3-\\sw{\\w})$: applying $\\gsHom_{\\gP}(-,\\so(3-\\sw{\\w}))$ to the exact\nsequence $0\\to\\s{L}^{-1}\\to\\s{L}^0\\to\\s{F}\\to0$, by \\ref{Fsym} we obtain the\nexact sequence\n\\[0\\to(\\s{L}^0)\\dual(3-\\sw{\\w})\\to(\\s{L}^{-1})\\dual(3-\\sw{\\w})\\to\n\\gsExt^1_{\\gP}(\\s{F},\\so(3-\\sw{\\w}))\\cong\\s{F}\\to0.\\]\nThen the result follows from the uniqueness proved above.\n \\end{rema}\n\n\n \\section{Symmetric resolution}\n\n \\begin{defi}\\label{Edef}\nIf $\\gph:\\gS\\to\\gP$ is a good weighted canonical projection, let $\\s{E}(\\gph)$\nbe the vector bundle on $\\gP$ defined by\n\\[\\s{E}(\\gph):=\\so(-2)^{\\chi(\\so_{\\S})+K_{\\S}^2-\\pd_2}\\oplus\n\\sd^1(-1)^{q(\\S)}\\oplus(\\sd^2)^{p_g(\\S)-\\pd_1}\n\\bigoplus_{3-\\sw{\\w}2$,\nit follows from \\ref{Fres} that there is an exact sequence on $\\gP$\n\\begin{equation}\\label{symres}\n0\\to(\\so\\oplus\\s{E}(\\gph))\\dual(-1-\\sw{\\w})\\mor{\\alpha}\\so\\oplus\\s{E}(\\gph)\\to\n\\gph_*\\so_{\\gS}\\cong\\gps_*\\so_{\\gX}\\to0,\n\\end{equation}\nwhere $\\alpha$ is a minimal morphism.\n\nFrom now on we will assume that $\\gph$ is good birational (this implies that\n$\\deg(\\f)>2$, as it is easy to see). We are going to prove that, with this\nhypothesis, also $\\alpha$ in \\eqref{symres} can be taken to be symmetric (i.e.,\n$\\alpha=\\alpha\\dual(-1-\\sw{\\w})$).\n\n \\begin{lemm}\\label{gphagphag}\n$\\gph$ is good birational if and only if $\\gcd(\\w_0,\\w_1,\\w_2,\\w_3)=1$ and\n$\\ph$ is good birational.\n \\end{lemm}\n\n \\begin{proof}\nFirst observe that, since $\\gS$ is standard, given $s\\in\\gS$,\n$\\gph\\mrs_s:\\so_{\\gY,\\gph(s)}\\to\\so_{\\gS,s}$ is an isomorphism if and only if\n$(\\ph\\mrs)_s=(\\gph\\mrs_s)_0$ is an isomorphism and $\\gph(s)$ is standard. Therefore\nit is enough to prove that the (open) set $V$ of {standard} points of $\\gY$ is\nempty if and only if $d:=\\gcd(\\w_0,\\w_1,\\w_2,\\w_3)\\ne1$. By \\ref{YgoodXgood}\n$V=U\\cap\\gY$, where $U$ denotes the set of {standard} points of $\\gP$. Moreover, by\n\\ref{stdmori}, $U=\\emptyset$ if $d\\ne1$, whereas $\\gP\\setminus\nU\\subseteq\\V{(x_0x_1x_2x_3)}$ if $d=1$. To conclude, just notice that, if $\\ph$\nis birational, then $\\gY\\nsubseteq\\V{(x_0x_1x_2x_3)}$ (otherwise $\\Y$ would be\nisomorphic to a $2$--dimensional weighted projective space, which is a rational\nsurface).\n \\end{proof}\n\n \\begin{lemm}\\label{det=Y}\n$\\det(\\alpha)=\\lambda\\f$ for some $\\lambda\\in\\K^*$.\n \\end{lemm}\n\n \\begin{proof}\nSimilar to that of \\cite[lemma 2.11]{C1}.\n \\end{proof}\n\n \\begin{lemm}\\label{birend}\n$\\Hom_{\\gP}(\\gps_*\\so_{\\gX},\\gps_*\\so_{\\gX})\\cong\\K$.\n \\end{lemm}\n\n \\begin{proof}\nSince $\\gps_*\\so_{\\gX}\\cong\\R\\gsh{\\p}$,\n$\\Hom_{\\gP}(\\R\\gsh{\\p},\\R\\gsh{\\p})\\cong\\Hom_{\\p}(\\R,\\gsec(\\R\\gsh{\\p}))$ (by\n\\ref{gmogqco}) and $\\gsec(\\R\\gsh{\\p})\\cong\\R$ ($\\forall m\\in\\Z$ the natural map\n\\[\\R_m\\to\\gsec(\\R\\gsh{\\p})_m=H^0(\\gP,\\R\\gsh{\\p}(m))\\cong H^0(\\gX,\\so_{\\gX}(m))\n\\cong H^0(\\X,\\ds_{\\X}^m)\\]\nis an isomorphism), it is enough to prove that $\\Hom_{\\p}(\\R,\\R)\\cong\\K$.\n\nGiven $\\varphi\\in\\Hom_{\\p}(\\R,\\R)$, let $\\lambda:=\\varphi(1)\\in\\R_0\\cong\\K$. As\n$\\gps$ is birational, by \\ref{agalg} $\\exists p\\in\\p_+\\hom$ such that $p\\notin\n\\ker\\rh$ and $\\rh_p:\\p_p\\to\\R_p$ is surjective. This means that $\\forall r\\in\\R$\nwe can find $m\\in\\N$ and $p'\\in\\p$ such that $p^mr=\\rh(p')$. Therefore\n\\[p^m\\varphi(r)=\\varphi(p^mr)=\\varphi(\\rh(p'))=\\lambda\\rh(p')=p^m\\lambda r.\\]\nAs $\\R$ is a domain and $\\rh(p)\\ne0$, this implies that $\\varphi(r)=\\lambda r$,\nwhence $\\varphi=\\lambda{\\rm id}_{\\R}$.\n \\end{proof}\n\n \\begin{prop}\\label{symmres}\nIf $\\gph$ is a good birational weighted canonical projection, then there exists an exact sequence on $\\gP$\n\\[0\\to(\\so\\oplus\\s{E}(\\gph))\\dual(-1-\\sw{\\w})\\mor{\\alpha}\\so\\oplus\\s{E}(\\gph)\n\\to\\gph_*\\so_{\\gS}\\cong\\gps_*\\so_{\\gX}\\to0,\\]\nwhere $\\alpha$ is a minimal morphism and $\\alpha=\\alpha\\dual(-1-\\sw{\\w})$.\n \\end{prop}\n\n \\begin{proof}\nDenoting $\\so\\oplus\\s{E}(\\gph)$ by $\\s{A}$, $\\dual(-1-\\sw{\\w})$ by\n$\\tdual$\\index{$\\tdual$} and $\\gsExt^1_{\\gP}(-,\\so(-1-\\sw{\\w}))$ by $E$, we\nalready know that there is an exact sequence\n\\[0\\to\\s{A}\\tdual\\mor{\\beta}\\s{A}\\mor{\\pi}\\gph_*\\so_{\\gS}\\to0\\]\nwith $\\beta$ minimal. Applying $\\gsHom_{\\gP}(-,\\so(-1-\\sw{\\w}))$ we obtain the\nexact sequence\n\\[0\\to\\s{A}\\tdual\\mor{\\beta\\tdual}\\s{A}\\mor{\\pi'}E(\\gph_*\\so_{\\gS})\\to0\\]\nand applying it once more we have also the exact sequence\n\\[0\\to\\s{A}\\tdual\\mor{\\beta}\\s{A}\\mor{\\pi''}E(E(\\gph_*\\so_{\\gS}))\\to0\\]\n(since $\\dtdual$ is naturally isomorphic to ${\\rm id}$).\nNotice that there exists a unique isomorphism $\\iota:\\gph_*\\so_{\\gS}\\to\nE(E(\\gph_*\\so_{\\gS}))$ such that $\\pi''=\\iota\\comp\\pi$. Choosing an isomorphism\n$\\epsilon:\\gph_*\\so_{\\gS}\\isomor E(\\gph_*\\so_{\\gS})$, we know from \\ref{unisym}\nthat there exists (unique up to homotopy) an isomorphism of complexes\n\\[\\begin{CD}\n0 @>>> \\s{A}\\tdual @>{\\beta}>> \\s{A} @>{\\pi}>> \\gph_*\\so_{\\gS} @>>> 0 \\\\\n@. @VV{\\eta}V @VV{\\xi}V @VV{\\epsilon}V @. \\\\\n0 @>>> \\s{A}\\tdual @>{\\beta\\tdual}>> \\s{A} @>{\\pi'}>> E(\\gph_*\\so_{\\gS}) @>>> 0\n.\n\\end{CD}\\]\nApplying $\\gsHom_{\\gP}(-,\\so(-1-\\sw{\\w}))$ to the above diagram yields the\ncommutative diagram\n\\[\\begin{CD}\n0 @>>> \\s{A}\\tdual @>{\\beta}>> \\s{A} @>{\\pi}>> \\gph_*\\so_{\\gS} @>>> 0 \\\\\n@. @VV{\\xi\\tdual}V @VV{\\eta\\tdual}V @VV{E(\\epsilon)\\comp\\iota}V @. \\\\\n0 @>>> \\s{A}\\tdual @>{\\beta\\tdual}>> \\s{A} @>{\\pi'}>> E(\\gph_*\\so_{\\gS}) @>>> 0\n,\n\\end{CD}\\]\nand applying it once more we obtain the commutative diagram\n\\[\\begin{CD}\n0 @>>> \\s{A}\\tdual @>{\\beta}>> \\s{A} @>{\\pi}>> \\gph_*\\so_{\\gS} @>>> 0 \\\\\n@. @VV{\\eta}V @VV{\\xi}V @VV{E(E(\\epsilon)\\comp\\iota)\\comp\\iota}V @. \\\\\n0 @>>> \\s{A}\\tdual @>{\\beta\\tdual}>> \\s{A} @>{\\pi'}>> E(\\gph_*\\so_{\\gS}) @>>> 0\n.\n\\end{CD}\\]\nIt follows that $E(E(\\epsilon)\\comp\\iota)\\comp\\iota=\\epsilon$. On the other\nhand, by \\ref{birend} $E(\\epsilon)\\comp\\iota=\\lambda\\epsilon$ for some $\\lambda\n\\in\\K$. Therefore, as $E$ is $\\K$--linear,\n\\[\\epsilon=E(E(\\epsilon)\\comp\\iota)\\comp\\iota=E(\\lambda\\epsilon)\\comp\\iota=\n\\lambda E(\\epsilon)\\comp\\iota=\\lambda^2\\epsilon,\\]\nwhence $\\lambda=\\pm1$. Moreover, since $(\\eta,\\xi,\\epsilon)$ and $(\\xi\\tdual,\n\\eta\\tdual,\\lambda\\epsilon)$ are morphisms between the same complexes, so is\nalso\n\\[(\\eta+\\lambda\\xi\\tdual,\\xi+\\lambda\\eta\\tdual,\n\\epsilon+\\lambda^2\\epsilon=2\\epsilon).\\]\nAs $2\\epsilon$ is an isomorphism (here we are using the fact that the\ncharacteristic of $\\K$ is not $2$), this morphism is actually an isomorphism by\n\\ref{unisym}. This implies that $\\alpha:=(\\xi+\\lambda\\eta\\tdual)\\comp\\beta$ is\ninjective and satisfies $\\cok\\alpha\\cong\\cok\\beta$, so that there is an exact\nsequence\n\\[0\\to\\s{A}\\tdual\\mor{\\alpha}\\s{A}\\to\\gph_*\\so_{\\gS}\\to0.\\]\nBy \\ref{compminmin} it is also clear that $\\alpha$ is minimal. Moreover (using\nthe above commutative diagrams) we have\n\\[\\alpha\\tdual=[(\\xi+\\lambda\\eta\\tdual)\\comp\\beta]\\tdual=\\beta\\tdual\\comp\n(\\xi+\\lambda\\eta\\tdual)\\tdual=\\beta\\tdual\\comp\\xi\\tdual+\n\\lambda\\beta\\tdual\\comp\\eta=\\eta\\tdual\\comp\\beta+\\lambda\\xi\\comp\\beta=\n\\lambda\\alpha.\\]\nBy \\ref{det=Y} it cannot happen that $\\lambda=-1$ (otherwise $\\det(\\alpha)$\nwould be a square), i.e. $\\alpha\\tdual=\\alpha$.\n \\end{proof}\n\nThe following result will be used only in chapter $4$.\n\n \\begin{coro}\\label{resdet}\nLet $\\gph:\\gS\\to\\gP$ be a {good birational weighted canonical projection} and assume that $\\gamma:\\s{A}:=\n\\so\\oplus\\s{E}(\\gph)\\to\\gph_*\\so_{\\gS}$ is a morphism such that\n$H^1(\\gP,\\gamma(1))$ and $H^0(\\gP,\\gamma(i))$ $\\forall i\\in\\Z$ are surjective\n(this implies that $p_g(\\S)=\\pd_1$). Then\n\\begin{enumerate}\n\n\\item $\\gamma$ is surjective and $\\ker\\gamma\\cong\\s{A}\\dual(-1-\\sw{\\w})$;\n\n\\item if $\\beta:\\s{A}\\dual(-1-\\sw{\\w})\\to\\s{A}$ is a morphism such that $\\gamma\n\\comp\\beta=0$ and $H^0(\\gP,\\beta(i))$ is injective $\\forall i\\in\\Z$, then the\nsequence $0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\beta}\\s{A}\n\\mor{\\gamma}\\gph_*\\so_{\\gS}\\to0$ is exact.\n\n\\item if $0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\beta}\\s{A}\\mor{\\gamma}\\gph_*\\so_{\\gS}\n\\to0$ is an exact sequence, then there exists an isomorphism $\\delta:\n\\s{A}\\dual(-1-\\sw{\\w})\\to\\s{A}\\dual(-1-\\sw{\\w})$ such that $\\alpha:=\\beta\\comp\n\\delta$ is symmetric ($\\alpha=\\alpha\\dual(-1-\\sw{\\w})$) and the sequence\n\\[0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\alpha}\\s{A}\\mor{\\gamma}\\gph_*\\so_{\\gS}\\to0\\]\nis also exact.\n\\end{enumerate}\n \\end{coro}\n\n \\begin{proof}\n$\\gamma$ is surjective because $\\gamma=\\gsec(\\gamma)\\gsh{}$ by \\ref{gmogqco}\nand $\\gsec(\\gamma)$ is surjective by hypothesis. In a similar way we can prove\n$2$ (the hypothesis implies, by dimension reasons, that\n\\[0\\to H^0(\\gP,\\s{A}\\dual(i-1-\\sw{\\w}))\\mor{H^0(\\gP,\\beta(i))}\nH^0(\\gP,\\s{A}(i))\\mor{H^0(\\gP,\\gamma(i))}H^0(\\gP,\\gph_*\\so_{\\gS}(i))\\to0\\]\nis exact $\\forall i\\in\\Z$). Setting $\\s{J}:=\\ker\\gamma$, we claim that\n$\\Ext^1_{\\gP}(\\s{A},\\s{J})=0$. Indeed, $\\forall j\\in\\Z$ there is an exact sequence\n\\[\\Hom_{\\gP}(\\so(j),\\s{A})\\mor{\\Hom_{\\gP}(\\so(j),\\gamma)}\n\\Hom_{\\gP}(\\so(j),\\gph_*\\so_{\\gS})\\to\n\\Ext^1_{\\gP}(\\so(j),\\s{J})\\to\\Ext^1_{\\gP}(\\so(j),\\s{A}).\\] Since\n$\\Hom_{\\gP}(\\so(j),\\gamma)=H^0(\\gP,\\gamma(-j))$ is surjective and\n$\\Ext^1_{\\gP}(\\so(j),\\s{A})=0$ for $j\\ne-1$, we see that\n$\\Ext^1_{\\gP}(\\so(j),\\s{J})=0$ for $j\\ne-1$, whence it remains to show that\n$\\Ext^1_{\\gP}(\\sd^1(-1),\\s{J})=0$ (recall that $\\s{A}$ does not contain\n$\\sd^2$, as $p_g(\\S)=\\pd_1$). Applying $\\Hom_{\\gP}(-,\\s{J})$ to $\\esd{1}(-1)$\nwe obtain the exact sequence\n\\[\\Ext^1_{\\gP}(\\sko^{-1}(-1),\\s{J})\\to\\Ext^1_{\\gP}(\\sd^1(-1),\\s{J})\n\\to\\Ext^2_{\\gP}(\\so(-1),\\s{J})\\cong H^2(\\gP,\\s{J}(1)).\\]\nBy what we have already seen $\\Ext^1_{\\gP}(\\sko^{-1}(-1),\\s{J})=0$ and the\nexact sequence\n\\[H^1(\\gP,\\s{A}(1))\\mor{H^1(\\gP,\\gamma(1))}\nH^1(\\gP,\\gph_*\\so_{\\gS}(1))\\to H^2(\\gP,\\s{J}(1))\\to\nH^2(\\gP,\\s{A}(1))\\]\nshows that $H^2(\\gP,\\s{J}(1))=0$, since $H^2(\\gP,\\s{A}(1))=0$ and\n$H^1(\\gP,\\gamma(1))$ is surjective by hypothesis. Thus\n$\\Ext^1_{\\gP}(\\sd^1(-1),\\s{J})=0$ and then also $\\Ext^1_{\\gP}(\\s{A},\\s{J})=0$.\n\nBy \\ref{symmres} we know that there is an exact sequence of the form\n\\[0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\tilde{\\alpha}}\\s{A}\\mor{\\tilde{\\gamma}}\n\\gph_*\\so_{\\gS}\\to0\\]\nwith $\\tilde{\\alpha}$ symmetric (and minimal). Denoting by $\\cp{\\s{U}}$ the\ncomplex $0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\tilde{\\alpha}}\\s{A}\\to0$ and by\n$\\cp{\\s{V}}$ the complex $0\\to\\s{J}\\to\\s{A}\\to0$, the same argument used in\n\\ref{unisym} shows that $\\cp{\\s{U}}$ and $\\cp{\\s{V}}$ are isomorphic in\n$K^b(\\gcoh{\\gP})$. Then, in particular, there are morphisms of complexes\n$\\cp{g}:\\cp{\\s{U}}\\to\\cp{\\s{V}}$ and $\\cp{h}:\\cp{\\s{V}}\\to\\cp{\\s{U}}$ such that\n$\\cp{h}\\comp\\cp{g}$ is homotopic to ${\\rm id}_{\\cp{\\s{U}}}$. By \\ref{minuni} this\nimplies that $\\cp{h}\\comp\\cp{g}$ is an isomorphism in $C^b(\\gcoh{\\gP})$, whence\neach $h^i$ is surjective. Being a surjective endomorphism of a vector bundle,\n$h^0:\\s{A}\\to\\s{A}$ is actually an isomorphism, and then it is clear that also\n$h^{-1}:\\s{J}\\to\\s{A}\\dual(-1-\\sw{\\w})$ is an isomorphism. This proves $1$.\n\nAs for $3$, if $0\\to\\s{A}\\dual(-1-\\sw{\\w})\\mor{\\beta}\\s{A}\\mor{\\gamma}\n\\gph_*\\so_{\\gS}\\to0$ is an exact sequence, then by \\ref{unisym} there is an\nisomorphism of complexes\n\\[\\begin{CD}\n0 @>>> \\s{A}\\dual(-1-\\sw{\\w}) @>{\\tilde{\\alpha}}>> \\s{A} @>{\\tilde{\\gamma}}>>\n\\gph_*\\so_{\\gS} @>>> 0 \\\\\n@. @VV{b}V @VV{a}V @VV{{\\rm id}}V @. \\\\\n0 @>>> \\s{A}\\dual(-1-\\sw{\\w}) @>{\\beta}>> \\s{A} @>{\\gamma}>> \\gph_*\\so_{\\gS}\n@>>> 0.\n\\end{CD}\\]\nIt follows that the sequence $0\\to\\s{A}\\dual(-1-\\sw{\\w})\n\\mor{\\alpha:=a\\comp\\tilde{\\alpha}\\comp a\\dual(-1-\\sw{\\w})}\\s{A}\\mor{\\gamma}\n\\gph_*\\so_{\\gS}\\to0$ is exact and $\\alpha\\dual(-1-\\sw{\\w})=\\alpha$ because\n$\\tilde{\\alpha}\\dual(-1-\\sw{\\w})=\\tilde{\\alpha}$. Moreover, $\\alpha=\\beta\\comp\nb\\comp a\\dual(-1-\\sw{\\w})$ and $b\\comp a\\dual(-1-\\sw{\\w})$ is an isomorphism.\n \\end{proof}\n\n\n \\section{Rank condition}\n\nWe are going to prove a version of the result ``rank condition = ring\ncondition'' for a morphism of vector bundles on a graded scheme. As in\n\\cite{CS} (see also \\cite{JS}), we prove it under weaker hypotheses than we\nwill need: in particular, the morphism is not assumed to be symmetric.\n\nLet $Z$ be a graded scheme. If $\\s{E}$ is a vector bundle of rank $r$ on $Z$,\nthe line bundle $\\Lambda^r(\\s{E})$ will be denoted by $\\det(\\s{E})$. As usual,\nfor $0\\le k\\le r$ the natural bilinear map $\\Lambda^k(\\s{E})\\times\n\\Lambda^{r-k}(\\s{E})\\to\\det(\\s{E})$ is a perfect pairing.\n\nGiven a morphism of vector bundles $\\alpha:\\s{G}\\to\\s{F}$ on $Z$, $\\forall k\\in\\N$\nwe will denote by $\\s{I}_k(\\alpha)$\\index{I_k(alpha)@$\\s{I}_k(\\alpha)$} the\ngraded ideal sheaf of $k\\times k$ minors of $\\alpha$. More precisely,\n$\\s{I}_k(\\alpha)$ is defined as the image of the natural map\n$\\Lambda^k(\\s{G})\\otimes\\Lambda^k(\\s{F})\\dual\\to\\so_Z$ induced by\n$\\Lambda^k(\\alpha)$. $\\s{I}_k(\\alpha)$ is also called the $(\\rk{\\s{F}}-k)^{\\rm\nth}$ Fitting ideal of $\\cok\\alpha$.\n\nIn the following we will assume that $\\rk{\\s{G}}=\\rk{\\s{F}}:=r+1$. The morphism\n\\[\\det(\\alpha):=\\Lambda^{r+1}(\\alpha):\\det(\\s{G})\\to\\det(\\s{F})\\]\nwill be identified with the corresponding map $\\s{L}:=\\det(\\s{G})\\otimes\n\\det(\\s{F})\\dual\\to\\so_Z$ (whose image is $\\s{I}_{r+1}(\\alpha)$) and also with\nthe corresponding element of $H^0(Z,\\s{L}\\dual)$. The morphism\n\\[\\Lambda^r(\\alpha):\\Lambda^r(\\s{G})\\cong\\s{G}\\dual\\otimes\\det(\\s{G})\\to\n\\Lambda^r(\\s{F})\\cong\\s{F}\\dual\\otimes\\det(\\s{F})\\]\ncorresponds (as\n$\\Hom_Z(\\s{G}\\dual\\otimes\\det(\\s{G}),\\s{F}\\dual\\otimes\\det(\\s{F}))\\cong\n\\Hom_Z(\\s{F}\\otimes\\s{L},\\s{G})$) to a map $\\beta:\\s{F}\\otimes\\s{L}\\to\\s{G}$,\nwhich satisfies the identities\n\\begin{align*}\n& \\alpha\\comp\\beta={\\rm id}_{\\s{F}}\\otimes\\det(\\alpha):\\s{F}\\otimes\\s{L}\\to\\s{F}, &\n\\beta\\comp(\\alpha\\otimes\\s{L})={\\rm id}_{\\s{G}}\\otimes\\det(\\alpha):\n\\s{G}\\otimes\\s{L}\\to\\s{G}.\n\\end{align*}\n\n \\begin{lemm}\\label{Qref}\nIn the notation introduced so far, let $Y\\subseteq Z$ be the closed graded\nsubscheme defined by $\\s{I}_{r+1}(\\alpha)$ (i.e., $\\so_Y:=\n\\so_Z\/\\s{I}_{r+1}(\\alpha)$). Let moreover $\\s{Q}:=\\cok\\alpha$, which is in a\nnatural way a graded $\\so_Y$--module. If $\\det(\\alpha)$ is a {non--zerodivisor} (meaning\nthat $\\det(\\alpha)_z$ is a {non--zerodivisor} $\\forall z\\in Z$), then $\\s{C}:=\n\\gsHom_Y(\\s{Q},\\so_Y)$ satisfies $\\s{Q}\\cong\\gsHom_Y(\\s{C},\\so_Y)$ (as graded\n$\\so_Y$--modules).\n \\end{lemm}\n\n \\begin{proof}\nApplying the functor $-_Y:=-\\otimes_{\\so_Z}\\so_Y$ to the exact sequence $\\s{G}\n\\mor{\\alpha}\\s{F}\\to\\s{Q}\\to0$, and taking into account that\n$\\s{I}_{r+1}(\\alpha)\\s{Q}=0$ ($\\alpha\\comp\\beta={\\rm id}_{\\s{F}}\\otimes\\det(\\alpha)$\nimplies that $\\s{I}_{r+1}(\\alpha)\\s{F}\\subseteq\\im\\alpha$), we obtain the\nexact sequence in $\\gcoh{Y}$\n\\[\\s{G}_Y\\mor{\\alpha_Y}\\s{F}_Y\\to\\s{Q}_Y\\cong\\s{Q}\\to0,\\]\nand also, applying $\\gsHom_Y(-,\\so_Y)$, the exact sequence\n\\[0\\to\\s{C}\\to\\s{F}_Y\\dual\\mor{\\alpha_Y\\dual}\\s{G}_Y\\dual.\\]\nWe claim moreover that there is a long exact sequence in $\\gcoh{Y}$\n\\[\\cdots\\to\\s{F}_Y\\otimes\\s{L}_Y^{n+1}\\mor{\\beta_Y\\otimes\\s{L}_Y^n}\n\\s{G}_Y\\otimes\\s{L}_Y^n\\mor{\\alpha_Y\\otimes\\s{L}_Y^n}\\s{F}_Y\\otimes\\s{L}_Y^n\n\\mor{\\beta_Y\\otimes\\s{L}_Y^{n-1}}\\s{G}_Y\\otimes\\s{L}_Y^{n-1}\\to\\cdots\\] Indeed,\n$(\\alpha_Y\\otimes\\s{L}_Y^n)\\comp(\\beta_Y\\otimes\\s{L}_Y^n)=\n({\\rm id}_{\\s{F}\\otimes\\s{L}^n}\\otimes\\det(\\alpha))_Y=0$ because $\\det(\\alpha)_Y=0$,\nand if (for some $y\\in Y$) $\\sigma\\in(\\s{G}_Y\\otimes\\s{L}_Y^n)_y$ is such that\n$(\\alpha_Y\\otimes\\s{L}_Y^n)_y(\\sigma)=0$, then (choosing\n$\\tilde{\\sigma}\\in(\\s{G}\\otimes\\s{L}^n)_y$ lift of $\\sigma$) $\\exists\\tau\\in\n(\\s{F}\\otimes\\s{L}^{n+1})_y$ such that$(\\alpha\\otimes\\s{L}^n)_y(\\tilde{\\sigma})\n=({\\rm id}_{\\s{F}\\otimes\\s{L}^n}\\otimes\\det(\\alpha))_y(\\tau)$, which implies (since\n$\\alpha\\otimes\\s{L}^n$ is injective, because $(\\alpha\\otimes\\s{L}^n)\\comp\n(\\beta\\otimes\\s{L}^n)={\\rm id}_{\\s{F}\\otimes\\s{L}^n}\\otimes\\det(\\alpha)$ is\ninjective by hypothesis) that $\\tilde{\\sigma}=(\\beta\\otimes\\s{L}^n)_y(\\tau)$,\nwhence $\\sigma\\in\\im(\\beta_Y\\otimes\\s{L}_Y^n)_y$. So the sequence is exact at\n$\\s{G}_Y\\otimes\\s{L}_Y^n$, and in a completely similar way one can check that\nit is exact at $\\s{F}_Y\\otimes\\s{L}_Y^n$. Besides, observing that\n$\\s{I}_{r+1}(\\alpha)=\\s{I}_{r+1}(\\alpha\\dual)$, from $\\alpha\\dual$ we obtain in\nthe same way the long exact sequence\n\\[\\cdots\\to\\s{G}_Y\\dual\\otimes\\s{L}_Y^{1-n}\n\\mor{\\beta_Y\\dual\\otimes\\s{L}_Y^{1-n}}\\s{F}_Y\\dual\\otimes\\s{L}_Y^{-n}\n\\mor{\\alpha_Y\\dual\\otimes\\s{L}_Y^{-n}}\\s{G}_Y\\dual\\otimes\\s{L}_Y^{-n}\n\\mor{\\beta_Y\\dual\\otimes\\s{L}_Y^{-n}}\\s{F}_Y\\dual\\otimes\\s{L}_Y^{-n-1}\\to\\cdots\n\\]\nIn particular, it follows that $\\s{C}\\cong\\ker\\alpha_Y\\dual\\cong\n\\im(\\beta_Y\\dual\\otimes\\s{L}_Y)$, whence there is an exact sequence\n\\[\\s{F}_Y\\dual\\otimes\\s{L}_Y\\mor{\\alpha_Y\\dual\\otimes\\s{L}_Y}\n\\s{G}_Y\\dual\\otimes\\s{L}_Y\\to\\s{C}\\to0,\\]\nand applying $\\gsHom_Y(-,\\so_Y)$ to it we obtain the exact sequence\n\\[0\\to\\gsHom_Y(\\s{C},\\so_Y)\\to\\s{G}_Y\\otimes\\s{L}_Y^{-1}\n\\mor{\\alpha_Y\\otimes\\s{L}_Y^{-1}}\\s{F}_Y\\otimes\\s{L}_Y^{-1}.\\]\nTherefore $\\gsHom_Y(\\s{C},\\so_Y)\\cong\\ker(\\alpha_Y\\otimes\\s{L}_Y^{-1})\\cong\n\\im(\\beta_Y\\otimes\\s{L}_Y^{-1})\\cong\\cok\\alpha_Y\\cong\\s{Q}$.\n \\end{proof}\n\n \\begin{prop}\\label{RC}\nIn the notation of \\ref{Qref}, assume moreover that $\\s{F}=\\so_Z\\oplus\\s{E}$\n(for some vector bundle $\\s{E}$ of rank $r$) and that\n\\[\\alpha=\\begin{pmatrix}\n\\alpha^{(1)} \\\\\n\\alpha'\n\\end{pmatrix}:\\s{G}\\to\\s{F}=\\so_Z\\oplus\\s{E}.\\]\nIf $\\det(\\alpha)$ is a {non--zerodivisor} and ${\\rm depth}(\\s{I}_r(\\alpha'))\\ge2$ (i.e.,\n${\\rm depth}(\\s{I}_r(\\alpha')_z)\\ge2$ $\\forall z\\in Z$), then the following conditions\nare equivalent:\n\\begin{enumerate}\n\n\\item $\\s{Q}=\\cok\\alpha$ carries the structure of a graded sheaf of commutative\n$\\so_Y$--algebras, with unit given by the image of $1\\in\\gsec(Z,\\so_Z)\\subset\n\\gsec(Z,\\s{F})$ in $\\s{Q}$;\n\n\\item $\\s{I}_r(\\alpha')=\\s{I}_r(\\alpha)$.\n\\end{enumerate}\n \\end{prop}\n\n \\begin{proof}\nAs we can always reduce to make local computations, we will adopt the following\nnotation at a point $z\\in Z$: $R:=\\so_{Z,z}$, $F:=\\s{F}_z$ and $G:=\\s{G}_z$\n(since $\\s{F}$ and $\\s{G}$ are locally free, we have $F\\cong\n\\bigoplus_{1\\le i\\le r+1}R(m_i)$ and $G\\cong\\bigoplus_{1\\le i\\le r+1}R(n_i)$ for\nsome $m_i,n_i\\in\\Z$ with $m_1=0$, so that $\\s{L}_z\\cong R(n-m)$, where $m:=\n\\sum m_i$ and $n:=\\sum n_i$). Then $\\alpha_z$ and $\\beta_z$ can be identified\nwith maps $a:G\\to F$ and $b:F(n-m)\\to G$, which can also be considered as\n$(r+1)\\times(r+1)$ matrices (with entries $a_{i,j}\\in R_{m_i-n_j}$ and $b_{i,j}\n\\in R_{m-n+n_i-m_j}$). For simplicity $\\forall l\\in\\Z$ we will write $a$ and $b$\ninstead of $a(l)$ and $b(l)$. Thus, setting $d:=\\det(a)\\in R_{m-n}$, we have\n$a\\comp b=d{\\rm id}_F$ and $b\\comp a=d{\\rm id}_G$. $\\s{I}_r(\\alpha')_z$ and\n$\\s{I}_r(\\alpha)_z$ can be identified with the ideals of $R$\n\\[I':=(b_{i,1}\\st i=1,\\dots,r+1)\\subseteq I:=(b_{i,j}\\st i,j=1,\\dots,r+1).\\]\nAs $d\\in I'$ is a {non--zerodivisor} and ${\\rm depth}(I')\\ge2$, there exists $d'\\in I'$ such\nthat $(d,d')$ is a regular sequence. We set moreover $\\bar{R}:=R\/(d)$ and we\nwill denote more generally by $\\bar{-}$ the functor $-\\otimes_R\\bar{R}$.\nClearly $Q:=\\s{Q}_z=\\cok a$ and $C:=\\s{C}_z=\\gHom_{\\bar{R}}(Q,\\bar{R})$ satisfy\n$Q\\cong\\bar{Q}$ and $C\\cong\\bar{C}$. Finally, $\\iota:R=R(m_1)\\mono F$ and\n$p:F\\epi Q$ will denote, respectively, the natural inclusion and projection.\n\nFirst we prove that the natural map $\\so_Z\\to\\s{Q}$ induces an injection $\\so_Y\n\\mono\\s{Q}$, which is the same as proving that $\\ker(p\\comp\\iota)=(d)$. Now,\n\\[p\\comp\\iota(d)=p(d(1,0,\\dots,0))=p(a\\comp b(1,0,\\dots,0))=0\\]\nbecause $p\\comp a=0$. On the other hand, if $x\\in R$ is such that $p\\comp\n\\iota(x)=0$, then $\\exists g=(g_1,\\dots,g_{r+1})\\in G$ such that $\\iota(x)=a(g)$.\nTherefore we have\n\\[dg=b\\comp a(g)=b(\\iota(x))=x(b_{1,1},\\dots,b_{r+1,1}),\\]\nwhich shows that $xI'\\subseteq(d)$. Then, in particular, $xd'\\in(d)$, which\nimplies that $x\\in(d)$, because $(d,d')$ is a regular sequence.\n\nNext we show that the quotient $\\s{Q}\/\\so_Y$ is annihilated by\n$\\s{I}_r(\\alpha')$, i.e that $I'Q\/\\bar{R}=0$. As $Q\/\\bar{R}= F\/(\\im\na+\\im\\iota)$, we have to prove that $b_{i,1}F\\subseteq\\im a+\\im\\iota$ for\n$i=1,\\dots,r+1$. Let $a_{(i)}$ be the matrix equal to $a$, except that the\nfirst row is substituted by $(0,\\dots,1,\\dots,0)$ (with $1$ in position $i$):\nas $\\det(a_{(i)})=\\pm b_{i,1}$, $b_{(i)}$ (the matrix of cofactors of\n$a_{(i)}$) satisfies $a_{(i)}\\comp b_{(i)}=\\pm b_{i,1}{\\rm id}$. Thus $\\forall f\\in F$\nwe have\n\\[\\pm b_{i,1}f=a_{(i)}\\comp b_{(i)}(f)=a\\comp b_{(i)}(f)+(a-a_{(i)})\\comp\nb_{(i)}(f).\\]\nSince $\\im(a-a_{(i)})\\subseteq\\im\\iota$, we see that $b_{i,1}f\\in\\im a+\n\\im\\iota$.\n\nNow we can prove that $\\gsHom_Y(\\s{Q}\/\\so_Y,\\so_Y)=0$, i.e. that\n$\\gHom_{\\bar{R}}(Q\/\\bar{R},\\bar{R})=0$. Indeed, an element of\n$\\gHom_{\\bar{R}}(Q\/\\bar{R},\\bar{R})$ can be identified with an element $\\varphi\n\\in\\gHom_{\\bar{R}}(Q,\\bar{R})$ such that $\\varphi\\rest{\\bar{R}}=0$. Then\n(remembering that $I'Q\\subseteq\\bar{R}$) $\\forall q\\in Q$ we have $0=\\varphi(d'q)=\nd'\\varphi(q)$, from which it follows that $\\varphi(q)=0\\in\\bar{R}$ (again,\nbecause $(d,d')$ is a regular sequence).\n\nThus the natural map $\\s{C}=\\gsHom_Y(\\s{Q},\\so_Y)\\to\\gsHom_Y(\\so_Y,\\so_Y)\\cong\n\\so_Y$ is injective, so that $\\s{C}$ can be identified with its image, which we\nclaim to be the graded sheaf of ideals $\\s{I}_r(\\alpha')\/\\s{I}_{r+1}(\\alpha)$\nof $\\so_Y=\\so_Z\/\\s{I}_{r+1}(\\alpha)$. To see this, it is enough to check that\nif $c\\in C=\\gHom_{\\bar{R}}(Q,\\bar{R})\\subseteq\\bar{R}$, then $c(1)\\in\\bar{I'}$,\nand that every element of $I'$ is of the form $c(1)$ for some $c\\in C$. As $C$\nis naturally isomorphic to $\\im(\\bar{b}\\dual:\\bar{G}(m-n)\\dual\\to\n\\bar{F}\\dual)$, every $c\\in C$ corresponds to an element of the form\n$\\bar{b}\\dual(\\varphi)=\\varphi\\comp\\bar{b}$ for some $\\varphi\\in\n\\gHom_{\\bar{R}}(\\bar{G}(m-n),\\bar{R})$ ($\\varphi\\comp\\bar{b}:\\bar{F}\\to\\bar{R}$\nis null on $\\im\\bar{a}$, hence it induces $c:Q=\\bar{F}\/\\im\\bar{a}\\to\n\\bar{R}$). Then we have by definition\n\\[c(1)=\\varphi\\comp\\bar{b}(1,0,\\dots,0)=\n\\varphi(\\bar{b}_{1,1},\\dots,\\bar{b}_{r+1,1})=\\sum_{i=1}^{r+1}\\bar{b}_{i,1}x_i\\]\n(where $x_i:=\\varphi(0,\\dots,1,\\dots,0)$, with $1$ in position $i$). Since the\n$x_i\\in\\bar{R}$ can be arbitrary, the claim follows.\n\nWe are now ready to prove that the condition $\\s{I}_r(\\alpha')=\\s{I}_r(\\alpha)$\nis satisfied if and only if the natural inclusion $\\gsHom_Y(\\s{C},\\s{C})\n\\subseteq\\gsHom_Y(\\s{C},\\so_Y)$ is an equality, i.e. that $I'=I$ if and only if\n$\\gHom_{\\bar{R}}(C,C)=\\gHom_{\\bar{R}}(C,\\bar{R})$. Given $q\\in Q\\cong\n\\gHom_{\\bar{R}}(C,\\bar{R})$ and $c\\in C$, by \\ref{Qref} we have $q(c)=c(q)$.\nSince $Q\\cong\\im(\\bar{b}:\\bar{F}\\to\\bar{G}(m-n))$ and $C\\cong\\im\\bar{b}\\dual$, we\ncan find $f\\in\\bar{F}$ and $\\varphi\\in\\gHom_{\\bar{R}}(\\bar{G}(m-n),\\bar{R})$\nsuch that $q$ and $c$ correspond to $\\bar{b}(f)$ and $\\bar{b}\\dual(\\varphi)=\n\\varphi\\comp\\bar{b}$. This implies that $q(c)=c(q)=\\varphi\\comp\\bar{b}(f)$.\nTherefore the condition $\\gHom_{\\bar{R}}(C,C)=\\gHom_{\\bar{R}}(C,\\bar{R})$ is\nequivalent to $\\varphi\\comp\\bar{b}(f)\\in C=\\bar{I'}\\subseteq\\bar{R}$ $\\forall f\\in\n\\bar{F}$ and $\\forall\\varphi\\in\\bar{G}(m-n)\\dual$, which is clearly satisfied if\nand only if every entry of $\\bar{b}$ is in $\\bar{I'}$, i.e. if and only if $I'=\nI$.\n\\begin{description}\n\n\\item[$1\\implies2$] By what we have just seen, we have to prove that\n$\\gsHom_Y(\\s{C},\\s{C})=\\gsHom_Y(\\s{C},\\so_Y)$, i.e. that given $q\\in Q\\cong\n\\gHom_{\\bar{R}}(C,\\bar{R})$ and $c\\in C$, $q(c)=cq\\in C$. Now, in any case $cq\n\\in\\bar{R}$, and (by definition of $C$) $cq\\in C$ if and only if $(cq)q'\\in\n\\bar{R}$ $\\forall q'\\in Q$. As $Q$ is an $\\bar{R}$--algebra, we have $(cq)q'=cqq'=\n(qq')(c)\\in\\bar{R}$.\n\n\\item[$2\\implies1$] As $\\s{Q}\\cong\\gsHom_Y(\\s{C},\\so_Y)=\\gsHom_Y(\\s{C},\\s{C})$,\n$\\s{Q}$ has a natural structure of $\\so_Y$--algebra, with multiplication given\nby composition in $\\gsHom_Y(\\s{C},\\s{C})$. Given $q,q'\\in Q$, we have\n\\[{\\bar{d'}}^2qq'=\\bar{d'}q\\bar{d'}q'=\\bar{d'}q'\\bar{d'}q={\\bar{d'}}^2q'q\\]\n(since $\\bar{R}$ is obviously in the centre of $Q$, and $\\bar{d'}q,\\bar{d'}q'\n\\in\\bar{R}$ because $\\bar{d'}\\in\\bar{I'}=C$). As $\\bar{d'}$ is a {non--zerodivisor} in\n$Q$ (if $p\\in Q\\cong\\gHom_{\\bar{R}}(C,C)$ is such that $\\bar{d'}p=0$, then $\\forall\nc\\in C$ we have $0=(\\bar{d'}p)(c)=\\bar{d'}p(c)$, whence $p(c)=0$ because\n$\\bar{d'}$ is a {non--zerodivisor} in $\\bar{R}$), this implies that $qq'=q'q$, i.e.\nmultiplication is commutative. The statement about unit is immediate to check.\n\\end{description}\n \\end{proof}\n\n \\begin{rema}\\label{RCrem}\nIt follows from the proof that $1\\implies2$ is true also without the assumption\n${\\rm depth}(\\s{I}_r(\\alpha'))\\ge2$, provided one knows that $\\so_Y\\mono\\s{Q}$ and\n$\\gsHom_Y(\\s{Q}\/\\so_Y,\\so_Y)=0$.\n \\end{rema}\n\n \\begin{rema}\\label{birpts}\nIf the hypotheses of \\ref{RC} are satisfied, it is clear from the proof that\nthe natural map $\\so_{\\gY}\\mono\\s{Q}$ is an isomorphism precisely at those\npoints $y\\in\\gY$ where $\\s{I}_r(\\alpha')_y=\\so_{Z,y}$.\n \\end{rema}\n\n\n \\section{Main theorem}\n\n \\begin{theo}\\label{mthm}\nLet $\\S$ be a minimal surface of general type and $\\gS$ the {standard} graded\nscheme $\\stdsch{\\S}{\\ds_{\\S}}$ (see \\ref{stdchar}), let\n$\\gph:\\gS\\to\\gY\\subset\\gP=\\gP(\\w)$ (with $\\dim\\P=3$) be a good birational weighted canonical projection,\nand let $\\s{E}(\\gph)$ be the vector bundle on $\\gP$ defined in\n\\ref{Edef}. Then there exists an exact sequence on $\\gP$\n\\[0\\to(\\so\\oplus\\s{E}(\\gph))\\dual(-1-\\sw{\\w})\\mor{\\alpha=\\begin{pmatrix}\n\\alpha^{(1)} \\\\\n\\alpha'\n\\end{pmatrix}}\n\\so\\oplus\\s{E}(\\gph)\\to\\gph_*\\so_{\\gS}\\to0,\\]\nsuch that $\\alpha$ is a minimal morphism which satisfies the following\nproperties:\n\\begin{enumerate}\n\\item $\\alpha$ is symmetric, i.e. $\\alpha=\\alpha\\dual(-1-\\sw{\\w})$;\n\\item $\\f=\\det(\\alpha)$ is an irreducible polynomial (defining $\\gY=\n\\gproj\\p\/(\\f)$);\n\\item $\\s{I}_r(\\alpha)=\\s{I}_r(\\alpha')$ (where $r:=\\rk{\\s{E}(\\gph)}$);\n\\item $\\gX:=\\gSpec(\\cok{\\alpha})$ is a {standard} graded scheme and $\\X$ is\na surface with only rational double points as singularities.\n\\end{enumerate}\n\nConversely, given a vector bundle $\\s{E}$ on $\\gP$ with $\\s{E}(2)\\in\n\\com{]3-\\sw{\\w},0[}$ and a minimal morphism\n$\\alpha:(\\so\\oplus\\s{E})\\dual(-1-\\sw{\\w})\\to\\so\\oplus \\s{E}$ satisfying\nproperties $1$, $2$, $3$, then $\\cok\\alpha$ is in a natural way a graded\ncoherent sheaf of commutative $\\so_{\\gY}$--algebras, and if $4$ also holds,\nthen, denoting by $\\pr:\\S\\to\\X$ a minimal resolution of singularities of $\\X$,\n$\\S$ is a minimal surface of general type, $\\X$ is its canonical model, $\\pr$\nextends to a morphism of {standard} graded schemes\n$\\gpr:\\gS:=\\stdsch{\\S}{\\ds_{\\S}}\\to\\gX$, $\\gph:\\gS\\mor{\\gpr} \\gX\\to\\gY$ is a\n{good birational weighted canonical projection} and $\\s{E}=\\s{E}(\\gph)$.\n \\end{theo}\n\n \\begin{proof}\nThe first implication is now very easy to prove: by \\ref{symmres} there exists\n$\\alpha$ minimal satisfying $1$; moreover, $2$ follows from \\ref{det=Y}, $3$\nfrom \\ref{RC} and \\ref{RCrem} (clearly $\\so_{\\gY}\\mono\\gph_*\\so_{\\gS}$ and\n$\\gsHom_{\\gY}((\\gph_*\\so_{\\gS})\/\\so_{\\gY},\\so_{\\gY})=0$, because the support of\n$(\\gph_*\\so_{\\gS})\/\\so_{\\gY}$ is a proper closed subset of $\\gY$, as $\\gph$ is\nbirational), and $4$ from \\ref{gSpecA} and \\ref{canmodstd}.\n\nConversely, denoting $\\cok\\alpha$ by $\\s{Q}$, $\\dual(-1-\\sw{\\w})$ by $\\tdual$\nand $\\gsExt^1_{\\gP}(-,\\so(-1-\\sw{\\w}))$ by $E$, since $\\alpha$ is injective\n(because $\\det(\\alpha)\\ne0$), there is an exact sequence\n\\[0\\to(\\so\\oplus\\s{E})\\tdual\\mor{\\alpha}\\so\\oplus\\s{E}\\to\\s{Q}\\to0.\\]\nApplying $\\gsHom_{\\gP}(-,\\so(-1-\\sw{\\w}))$ we obtain the exact sequence\n\\[0\\to(\\so\\oplus\\s{E})\\tdual\\mor{\\alpha\\tdual}\\so\\oplus\\s{E}\\to E(\\s{Q})\\to0\\]\n($\\gsHom_{\\gP}(\\s{Q},\\so(-1-\\sw{\\w}))=0$ because $\\s{Q}\\in\\gsmo{\\gY}$). As\n$\\alpha\\tdual=\\alpha$, there is then a natural isomorphism $E(\\s{Q})\\cong\\s{Q}$.\nNotice that ${\\rm depth}(\\s{I}_r(\\alpha'))\\ge2$: clearly it is enough to show that\nif $y\\in\\gY$, then ${\\rm depth}(\\s{I}_r(\\alpha')_y)\\ge2$. As $\\f\\in\n\\s{I}_r(\\alpha')_y$ is irreducible, this is true if and only if $(\\f)=\n\\s{I}_{r+1}(\\alpha)_y\\subsetneq\\s{I}_r(\\alpha')_y=\\s{I}_r(\\alpha)_y$. If on the\ncontrary $(\\f)=\\s{I}_r(\\alpha)_y$, then $\\f$ would divide every entry of\n$\\beta_y$ (the matrix of cofactors of $\\alpha_y$), whence $\\f^{r+1}$ would\ndivide $\\det(\\beta_y)$, but $\\det(\\beta_y)=\\f^r$ (this follows from the fact\nthat $\\alpha_y\\beta_y=\\beta_y\\alpha_y=\\f{\\rm id}$). Therefore $\\s{Q}$ is a graded\ncoherent sheaf of commutative $\\so_{\\gY}$--algebras by \\ref{RC}, and, denoting\nby $\\gps:\\gX:=\\gSpec\\s{Q}\\to\\gY\\subset\\gP$ the natural map, by \\ref{gSpecA}\n$\\s{Q}\\cong\\gps_*\\so_{\\gX}$ and $\\gps$ is finite. Moreover, $\\gps$ is birational\nonto $\\gY$ because there exists an open subset $\\emptyset\\ne U\\subseteq\\gY$\nsuch that $(\\s{I}_r(\\alpha)\/\\s{I}_{r+1}(\\alpha))\\rest{U}=\\so_U$ (this follows\nfrom the fact that $(\\f)=\\s{I}_{r+1}(\\alpha)\\subsetneq\\s{I}_r(\\alpha)$ and $\\f$\nis irreducible), whence $\\s{Q}\\rest{U}=\\so_U$ by \\ref{birpts}. By\n\\ref{gsExtdual} there are natural isomorphisms in\n$\\gcoh{\\gP}\\cap\\gsmo{\\gps_*\\so_{\\gX}}$\n\\[\\gps_*\\ds_{\\gX}\\cong\\gsExt^1_{\\gP}(\\s{Q},\\so(-\\sw{\\w}))\\cong E(\\s{Q})(1)\\cong\n\\s{Q}(1).\\] Since also\n$\\gps_*(\\gps^*\\so_{\\gP}(1))\\cong\\so_{\\gP}(1)\\otimes\\gps_*\\so_{\\gX} \\cong\\s{Q}(1)$\nby projection formula, it follows from \\ref{gSpecA} that\n$\\ds_{\\gX}\\cong\\gps^*\\so_{\\gP}(1)\\cong\\so_{\\gX}(1)$. Therefore $\\ds_{\\X}\\cong\n(\\so_{\\gX})_1$ is an invertible sheaf (because $\\gX$ is standard). As $\\X$ has only\nrational double points as singularities, everything will follow if we prove\nthat $\\ds_{\\X}$ is ample: indeed, in this case $\\X$ is the canonical model of a\nminimal surface of general type $\\S$, the map $\\pr:\\S\\to\\X$ extends naturally\nto a morphism $\\gpr:\\gS\\to\\gX$ (where $\\gS:=\\stdsch{\\S}{\\ds_{\\S}}$), $\\gph:=\n\\gps\\comp\\gpr$ is a {good birational weighted canonical projection} and $\\s{E}=\\s{E}(\\gph)$ by the part already proved\nand by the uniqueness of the minimal resolution (see \\ref{unisym}).\n\nSo it remains to prove that $\\ds_{\\X}$ is ample, and it is clearly enough to\nshow that $\\ds_{\\X}^m$ is ample, where $m$ is the least common multiple of\n$\\{\\w_0,\\w_1,\\w_2,\\w_3\\}$. By definition this is true if and only if, given\n$\\s{G}\\in\\coh{\\X}$, $\\exists l\\in\\N$ such that the natural map\n\\[g_n:\\so_{\\X}\\otimes_{\\K}H^0(\\X,\\s{G}\\otimes\\ds_{\\X}^{nm})\\to\n\\s{G}\\otimes\\ds_{\\X}^{nm}\\]\nis surjective for $n\\ge l$. As $\\so_{\\P}(m)$ is an ample invertible sheaf (by\n\\cite[prop. 2.3]{D}), $\\exists l\\in\\N$ such that the natural map\n\\[h_n:\\so_{\\P}\\otimes_{\\K}H^0(\\P,{\\ps}_*\\s{G}\\otimes\\so_{\\P}(nm))\\to\n{\\ps}_*\\s{G}\\otimes\\so_{\\P}(nm)\\]\nis surjective for $n\\ge l$. Since $\\ps^*\\so_{\\P}(nm)\\cong\\ds_{\\X}^{nm}$ (this is\nan easy consequence of the fact that every stalk of $\\so_{\\gP}$ contains an\ninvertible element of degree $m$), by projection formula\n${\\ps}_*(\\s{G}\\otimes\\ds_{\\X}^{nm})\\cong{\\ps}_*\\s{G}\\otimes\\so_{\\P}(nm)$.\nMoreover, there is a natural isomorphism $H^0(\\X,\\s{G}\\otimes\\ds_{\\X}^{nm})\\cong\nH^0(\\P,{\\ps}_*(\\s{G}\\otimes\\ds_{\\X}^{nm}))$ (because $\\ps$ is finite), and then\nit is clear that $h_n$ factors through ${\\ps}_*g_n$, which is therefore\nsurjective if $n\\ge l$. By \\ref{gSpecA}, then, also $g_n$ is surjective for\n$n\\ge l$.\n \\end{proof}\n\n \\begin{rema}\\label{coeffdiff}\nAs we will see, the coefficients $\\cre_j(\\gph)$ of the sheaves $\\so(j-2)$ (for\n$3-\\sw{\\w}2$, so that the most natural choice (which\nwe make) is $\\w=(1,1,2,3)$.\n\nWe consider only the (more interesting) case of an irreducible polarization.\nIt turns out that even the problem of determining exactly the vector bundle\n$\\s{E}(\\gph)$ (i.e., of finding the coefficients $\\cre_j(\\gph)$ of the sheaves\n$\\so(j-2)$) requires a deep knowledge of the canonical ring $\\R$ of the\nsurface $\\S$. So we first study it: the hard part is to determine the ring of\ntheta functions on $A$ $\\R(A,\\Theta):=\\bigoplus_{n\\ge0}H^0(A,\\so_A(n\\Theta))$\n(because $\\R$ is simply an integral extension of it, generated by a single\nelement of degree $1$). We manage to do it, finding explicit generators and\nrelations, in a purely algebraic way. Once we have the canonical ring, we can\ncompute the $\\cre_j(\\gph)$ using ${\\rm \\check{C}}$ech cohomology (and we see\nthat they actually depend on the choice of $\\gph$). Then we choose $\\gph$ as\nsimple as possible and briefly describe the computations (which require the use\nof a computer) which allow us to determine a symmetric resolution $\\alpha$ of\n$\\gph_*\\so_{\\gS}$ as in \\ref{mthm}.\n\nWe will work over the field $\\C$ of complex numbers.\n\n\n \\section{Double covers of an abelian surface}\n\nLet $A$\\index{A@$A$} be an abelian surface and let\n$\\Theta$\\index{Theta@$\\Theta$} be a symmetric principal polarization on $A$.\nGiven a smooth divisor $B\\in\\linsys{2\\Theta}$,\\index{B@$B$} let $\\S$ be the\nsurface obtained as double cover of $A$ branched along $B$, and denote by\n$p:\\S\\to A$ the projection. Then $\\S$ is a minimal surface of general type and\n$p_g(\\S)=q(\\S)=2$, $K_{\\S}^2=4$. In fact, as $\\ds_{\\S}\\cong p^*\\so_A(\\Theta)$\nand $p_*\\so_{\\S}\\cong\\so_A\\oplus\\so_A(-\\Theta)$, we have\n\\[p_*\\ds_{\\S}^n\\cong\\so_A(n\\Theta)\\oplus\\so_A((n-1)\\Theta)\\]\n$\\forall n\\in\\Z$ ($\\so_A(n\\Theta)$ is the invariant part and $\\so_A((n-1)\\Theta)$\nthe antiinvariant if $n$ is even, and viceversa if $n$ is odd). $p$ being\nfinite it follows that\n\\[H^i(\\S,\\ds_{\\S}^n)\\cong H^i(A,p_*\\ds_{\\S}^n)\\cong H^i(A,\\so_A(n\\Theta))\\oplus\nH^i(A,\\so_A((n-1)\\Theta))\\]\n$\\forall i\\in\\N$. Therefore, since $h^0(A,\\so_A(n\\Theta))=n^2$ for $n>0$,\n$h^i(A,\\so_A(n\\Theta))=0$ for $n\\ne0$ and $h^1(A,\\so_A)=2$, we see that\n$p_g(\\S)=h^0(A,\\so_A(\\Theta))+h^0(A,\\so_A)=2$, $q(\\S)=h^1(A,\\so_A)+\nh^1(A,\\so_A(-\\Theta))=2$ (so that $\\chi(\\so_{\\S})=1$) and (for $n>1$)\n\\[h^0(\\S,\\ds_{\\S}^n)=h^0(A,\\so_A(n\\Theta))+h^0(A,\\so_A((n-1)\\Theta))=\nn^2+(n-1)^2=1+2n(n-1),\\] whence $\\S$ is of general type and $K_{\\S}^2=4$\n(because $h^0(\\S,\\ds_{\\S}^n)= \\chi(\\so_{\\S})+\\frac{n(n-1)}{2}K_{\\S}^2$).\nMoreover, $\\S$ contains no rational curve (because the same is true for $A$),\nwhich implies that $\\S$ is minimal and coincides with its canonical model, i.e.\n$\\S\\cong\\proj\\R$, where $\\R:= \\R(\\S,\\ds_{\\S})$ is the canonical ring of $\\S$.\nDenoting by $\\R'$ the (positively graded) ring\\index{R'@$\\R'$}\n\\[\\R'=\\R(A,\\Theta):=\\bigoplus_{n\\ge0}H^0(A,\\so_A(n\\Theta))\\]\nand by $\\xi$\\index{xi@$\\xi$} a generator of $H^0(A,\\so_A)\\subset\nH^0(\\S,\\ds_{\\S})=\\R_1$, $\\forall n\\in\\N$ we can write\n$\\R_n=\\R'_n\\oplus\\xi\\R'_{n-1}$, so that $\\R=\\R'\\oplus\\xi\\R'$. Then\nmultiplication in $\\R$ is given by\n\\[(x+\\xi y)(x'+\\xi y')=xx'+syy'+(xy'+x'y)\\xi\\]\n$\\forall x,x',y,y'\\in\\R'$, where $s\\in H^0(A,\\so_A(2\\Theta))=\\R'_2$\\index{s@$s$}\nis a section whose zero divisor is $B$.\n\n\n \\section{The ring of theta functions of an abelian surface}\n\nWe will assume that the polarization $\\Theta$ is irreducible, so that $A$ is\nisomorphic to the Jacobian of a (smooth projective) curve $C$\\index{C@$C$} of\ngenus $2$. Being hyperelliptic, $C$ is a double cover of $\\P^1$ branched on six\n(distinct) points. Denoting by $[y_0,y_1]$ the homogeneous coordinates on\n$\\P^1$, we can write $C=\\proj R$, where $R$\\index{R@$R$} is the graded ring\n\\begin{align*}\n& R:=\\C[y_0,y_1,z]\/(z^2-F) & \\deg(y_0)=\\deg(y_1)=1,\\deg(z)=3\n\\end{align*}\nand $F$\\index{F@$F$} is a homogeneous polynomial of degree $6$ vanishing\nexactly at the branch points (notice that $R$ can be identified with the\ncanonical ring $\\R(C,\\ds_C)$ of $C$). Up to composing with an automorphism of\n$\\P^1$, we can assume that three of the branch points are $0$, $1$, $\\infty$,\nwhence we can write $F$ as\n\\[F=y_0y_1 (y_1^4+\\lambda y_0y_1^3+\\mu y_0^2y_1^2+\\nu y_0^3y_1+\\epsilon y_0^4).\\]\nfor some $\\lambda,\\mu,\\nu,\\epsilon\\in\\C$\\index{lambda@$\\lambda$}\\index{mu@$\\mu$}\n\\index{nu@$\\nu$}\\index{epsilon@$\\epsilon$} with $1+\\lambda+\\mu+\\nu+\\epsilon=0$.\n\nWe will denote by $h:C\\to\\P^1$ the double cover and by $\\iota:C\\to C$ the\nhyperelliptic involution (given, in weighted homogeneous coordinates\n$[y_0,y_1,z]$, by $[a,b,c]\\mapsto[a,b,-c]$). We will identify the abelian\nsurface $A=\\Pic^0(C)$\\index{A@$A$} with $\\Pic^2(C)$ using the isomorphism given\nby $\\s{L}\\mapsto\\s{L}\\otimes \\so_C(2c_{\\infty})$, where $c_{\\infty}:=[0,1,0]\\in\nC$. We recall that $\\Pic^2(C)$ is naturally isomorphic to a blow--down of\n$C^{(2)}$ (the twofold symmetric product of $C$, whose closed points are in\ncanonical bijection with the set of effective divisors of degree $2$ on $C$):\nthe natural map $q:C^{(2)} \\to\\Pic^2(C)$ is an isomorphism everywhere, except\nthat it contracts to a point the exceptional divisor $E$, corresponding to the\nlinear system $g^1_2$ of $C$. Letting $r:C\\times C\\to C^{(2)}$ be the quotient\nmap and setting $D_1:= \\{c_{\\infty}\\}\\times C$ and\n$D_2:=C\\times\\{c_{\\infty}\\}$, we can take as theta divisor $\\Theta:=q\\comp\nr(D_1)=q\\comp r(D_2)\\subset A$.\\index{Theta@$\\Theta$}\n\nDenoting by $\\theta$\\index{theta@$\\theta$} a generator of\n$H^0(A,\\so_A(\\Theta))$, the exact sequences\n\\[0\\to\\so_A((n-1)\\Theta)\\mor{\\cdot\\theta}\\so_A(n\\Theta)\\to\\so_{\\Theta}(n\\Theta)\n\\to0\\]\ninduce exact sequences for $n>1$ (as $H^1(A,\\so_A(m\\Theta))=0$ for $m>0$)\n\\[0\\to H^0(A,\\so_A((n-1)\\Theta))\\mor{\\cdot\\theta}H^0(A,\\so_A(n\\Theta))\\to\nH^0(\\Theta,\\so_{\\Theta}(n\\Theta))\\to0,\\] whereas for $n=1$ we have\n$\\theta:H^0(A,\\so_A)\\isomor H^0(A,\\so_A(\\Theta))$ and\n$H^0(\\Theta,\\so_{\\Theta}(\\Theta))\\isomor H^1(A,\\so_A)\\cong\\C^2$. As $\\Theta \\cong\nC$ and (by adjunction) $\\so_{\\Theta}(\\Theta)\\cong\\ds_{\\Theta}$,\n$H^0(\\Theta,\\so_{\\Theta}(n\\Theta))\\cong H^0(C,\\ds_C^n)\\cong R_n$. Letting $R'$ be\nthe (graded) subring of $R$ given by $R':=\\bigoplus_{d\\ne1}R_d$\\index{R'@$R'$}\nwe have therefore that $R'\\cong\\R'\/(\\theta)$. Since $\\theta$ is a {non--zerodivisor} in\n$\\R'$, the ``hyperplane section principle'' (see \\cite[p. 218]{R}) implies that\n$\\R'$ has the same generators as $R'$ plus $\\theta$, and that the relations of\n$\\R'$ are given by lifts of those of $R'$ (in particular, the two rings have\nthe same number of relations and in the same degrees).\n\nSo, as a first step to compute $\\R'$, we will find generators and\nrelations of $R'$. Setting $r_i:=y_0^{2-i}y_1^i\\in R'_2$ ($i=0,1,2$), $s_i:=\ny_0^{3-i}y_1^i\\in R'_3$ ($i=0,1,2,3$) and $t_i:=y_iz\\in R'_4$ ($i=0,1$), it is\nclear that $R'$ is minimally generated (as a graded $\\C$--algebra), by the\n$r_i$, $s_i$, $t_i$ and $z$. As for relations, we have the following result.\n\n \\begin{lemm}\\label{curvering}\nThe kernel $J$ of the natural epimorphism of graded rings\n\\[P:=\\C[r_0,r_1,r_2,s_0,s_1,s_2,s_3,z,t_0,t_1]\\epi R'\\]\nis minimally generated by the following $37$ homogeneous polynomials (of which\n$1$ has degree $4$, $6$ have degree $5$, $17$ have degree $6$, $10$ have degree\n$7$ and $3$ have degree $8$):\n\\begin{align*}\n& r_1^2-r_0r_2 & \\text{(of degree $4$);}\\\\\n& r_is_j-r_{i+1}s_{j-1} \\text{ for $i=0,1$ and $j=1,2,3$} &\n\\text{(of degree $5$);}\\\\\n& r_it_1-r_{i+1}t_0 \\text{ for $i=0,1$} & \\text{(of degree $6$);}\\\\\n& s_is_j-r_0^{3-\\lfloor\\frac{i+j+1}{2}\\rfloor}\nr_1^{i+j-2\\lfloor\\frac{i+j}{2}\\rfloor}r_2^{\\lfloor\\frac{i+j}{2}\\rfloor}\n\\text{ for $0\\le i\\le j\\le3$} & \\text{(of degree $6$);}\\\\\n& s_iz-r_it_0 \\text{ for $i=0,1,2$ and } s_3z-r_2t_1 &\n\\text{(of degree $6$);}\\\\\n& z^2-r_1(r_2^2+\\lambda r_1r_2+\\mu r_0r_2+\\nu r_0r_1+\\epsilon r_0^2) & \\text{(of\ndegree $6$);}\\\\\n& s_it_j-r_0^{2-\\lfloor\\frac{i+j+1}{2}\\rfloor}\nr_1^{i+j-2\\lfloor\\frac{i+j}{2}\\rfloor}r_2^{\\lfloor\\frac{i+j}{2}\\rfloor}z\n\\text{ for $0\\le i\\le3$ and $j=0,1$} & \\text{(of degree $7$);}\\\\\n& zt_i-s_{i+1}(r_2^2+\\lambda r_1r_2+\\mu r_0r_2+\\nu r_0r_1+\\epsilon r_0^2)\n\\text{ for $i=0,1$} & \\text{(of degree $7$);}\\\\\n& t_it_j-r_1r_{i+j}(r_2^2+\\lambda r_1r_2+\\mu r_0r_2+\\nu r_0r_1+\\epsilon r_0^2)\n\\text{ for $0\\le i\\le j\\le1$} & \\text{(of degree $8$).}\n\\end{align*}\n \\end{lemm}\n\n \\begin{proof}\nDenoting by $J'$ the (homogeneous) ideal generated by the polynomials in the\nabove list, it is clear that $J'\\subseteq J$. $\\forall n\\in\\N$ let $\\tilde{R}_n$\nbe the subspace of $P_n$ defined in the following way: $\\tilde{R}_n:=P_n$ if\n$n\\le3$, whereas for $m\\ge2$\n\\begin{gather*}\n\\tilde{R}_{2m}:=\\langle r_0^{m-i-j-2k}r_1^jr_2^it_0^k\\st\n0\\le j,k\\le 1;0\\le i\\le m-j-2k\\rangle\\oplus\\langle r_2^{m-2}t_1\\rangle,\\\\\n\\tilde{R}_{2m+1}:=\\langle r_0^{m-1-i-j}r_1^jr_2^is_0^{1-k}z^k\\st\n0\\le j,k\\le 1;0\\le i1$, $\\dim_{\\C}\\tilde{R}_n=2n-1$, so that\n$\\dim_{\\C}\\tilde{R}_n=\\dim_{\\C}R'_n$ $\\forall n\\in\\N$. Moreover, it follows easily\nfrom the definition that $\\tilde{R}_n+J'_n=P_n$ $\\forall n\\in\\N$. Therefore\n\\[\\dim_{\\C}J'_n\\ge\\dim_{\\C}P_n-\\dim_{\\C}\\tilde{R}_n=\\dim_{\\C}P_n-\\dim_{\\C}R'_n=\n\\dim_{\\C}J_n,\\]\nwhich implies (since $J'_n\\subseteq J_n$) that $J'_n=J_n$ $\\forall n\\in\\N$, i.e.\nthat $J'=J$. The given list of polynomials is also a minimal set of generators\nof $J$: indeed, it is not difficult to check that, if $J''$ denotes the ideal\ngenerated by the whole list except one polynomial (say of degree $n$), then\n$J''_n+\\tilde{R}_n\\subsetneq P_n$.\n \\end{proof}\n\nBy what we said before, it follows that $\\R'$ has $11$ generators ($\\theta$ of\ndegree $1$, $3$ of degree $2$, $5$ of degree $3$ and $2$ of degree $4$) and\n$37$ relations (in the same degrees as those of $R'$) and that, for instance,\ndenoting by $\\rho_i$ lifts of $r_i$, since the only relation in degree $4$ in\n$R'$ is $r_1^2=r_0r_2$, $\\R'$ has also a single relation in degree $4$, and it\nis of the form $\\rho_1^2=\\rho_0\\rho_2+\\theta\\varphi$ for some $\\varphi\\in\n\\R'_3$. In order to determine this $\\varphi$ and the analogous terms in the\nother relations, we need to study the ring $\\R'$ more directly.\n\nConsider the divisor $E':=r^*(E)$ of $C\\times C$ and observe that $E'=\n\\{(c,\\iota(c))\\st c\\in C\\}$.\n\n \\begin{lemm}\\label{Thetacohom}\nLet $D:=D_1+D_2+E'$. Then $\\forall n\\in\\Z$ there are natural isomorphisms\n\\[H^0(A,\\so_A(n\\Theta))\\cong H^0(C\\times C,\\so_{C\\times C}(nD))^+,\\]\nwhere the superscript $^+$ denotes the subspace of invariants under the\nnatural action of $\\Z\/(2)$ induced by the map which exchanges the two factors\nof $C\\times C$.\n \\end{lemm}\n\n \\begin{proof}\nSince $q_*\\so_{C^{(2)}}\\cong\\so_A$, by projection formula $\\so_A(n\\Theta)\\cong\nq_*q^*\\so_A(n\\Theta)$, whence $H^0(A,\\so_A(n\\Theta))\\cong\nH^0(C^{(2)},q^*\\so_A(n\\Theta))$. Observe moreover that\n\\[H^0(C\\times C,r^*q^*\\so_A(n\\Theta))\\cong\nH^0(C^{(2)},q^*\\so_A(n\\Theta)\\otimes r_*\\so_{C\\times C}).\\]\nAs the invariant part of $r_*\\so_{C\\times C}$ is $\\so_{C^{(2)}}$, it follows\nthat\n\\[H^0(A,\\so_A(n\\Theta))\\cong H^0(C^{(2)},q^*\\so_A(n\\Theta))\\cong\nH^0(C\\times C,r^*q^*\\so_A(n\\Theta))^+.\\]\nTo conclude, just notice that $q^*\\so_A(\\Theta)\\cong\\so_{C^{(2)}}(\\Theta'+E)$\n(where $\\Theta':=r(D_1)=r(D_2)$), $r^*\\so_{C^{(2)}}(\\Theta')\\cong\n\\so_{C\\times C}(D_1+D_2)$ and $r^*\\so_{C^{(2)}}(E)\\cong\\so_{C\\times C}(E')$,\nso that $r^*q^*\\so_A(\\Theta)\\cong\\so_{C\\times C}(D)$.\n \\end{proof}\n\nWe can therefore identify the elements of $H^0(A,\\so_A(n\\Theta))$ with\nsymmetric rational functions on $C\\times C$ having as poles at most $nD$. We\nwill denote by $[y'_0,y'_1,z']$ the coordinates on the second factor of\n$C\\times C\\cong\\proj(R\\times_{\\C}R)$ (where $(R\\times_{\\C}R)_n:=\nR_n\\otimes_{\\C}R_n$). Our aim is to find functions corresponding to generators\nof $\\R'$ (then it will be not difficult to determine the relations, because in\nthis way multiplication in the ring is simply given by multiplication of\nfunctions). Now, some of them are easy to find: it is clear that $\\theta:=1$ is\na generator of $H^0(A,\\so_A(\\Theta))$. As for $H^0(A,\\so_A(2\\Theta))$, besides\n$1=\\theta^2$ we have for instance $\\frac{y_1\\otimes y'_1}{y_0\\otimes y'_0}$ and\n$\\frac{y_1\\otimes y'_0+y_0\\otimes y'_1}{y_0\\otimes y'_0}$ (which have as poles\n$2D_1+2D_2$). On the other hand, as we will see, a fourth generator of\n$H^0(A,\\so_A(2\\Theta))$ is not so simple (it has poles also at $2E'$), and the\nsame is true for two generators of degree $3$ and one of degree $4$. Therefore\nwe need a systematic procedure which allows us to determine all sections of $n\n\\Theta$ (at least for $n\\le 4$). Before we describe it, we need to know\nexplicitly the morphism $\\R'\\epi R'$: it is not difficult to prove the\nfollowing result.\n\n \\begin{lemm}\nThe map $\\R'_n=H^0(A,\\so_A(n\\Theta))\\epi R'_n$ can be identified with\n\\[\\gamma\\mapsto y_0^n((\\frac{y'_0}{z'})^n\\gamma\\rest{y'_0=z'=0,y'_1=1})\\]\n(notice that $\\frac{y'_0}{{z'}^2}\\rest{y'_0=z'=0,y'_1=1}=1$).\n \\end{lemm}\n\nFor instance, it follows that (as expected) $\\theta$ and $\\theta^2$ are mapped\nto $0$, whereas $\\frac{y_1\\otimes y'_1}{y_0\\otimes y'_0}$ goes to $y_0y_1=r_1$\nand $\\frac{y_1\\otimes y'_0+y_0\\otimes y'_1}{y_0\\otimes y'_0}$ to $y_0^2=r_0$\n(so the still missing generator of degree $2$ will be a lift of $r_2$).\n\nIn order to shorten notation in the following we will write $a\\odot\nb$\\index{$\\odot$} for $a(y_0,y_1,z)\\otimes b(y'_0,y'_1,z')+b(y_0,y_1,z)\\otimes\na(y'_0,y'_1,z')$ and $a\\wedge b$ for $a(y_0,y_1,z)\\otimes\nb(y'_0,y'_1,z')-b(y_0,y_1,z)\\otimes a(y'_0,y'_1,z')$. Consider the\n$\\Z\/(2)\\times\\Z\/(2)$ Galois cover $H:=h\\times h: C\\times C\\to\\P^1\\times\\P^1$:\nsetting $D'_1:=\\{[0,1]\\}\\times\\P^1$ and $D'_2:= \\P^1\\times\\{[0,1]\\}$, we have\n$H^*(D'_i)=2D_i$ and $H^*(\\Delta_{\\P^1})=\\Delta_C +E'$ ($\\Delta_X$ denotes the\ndiagonal in $X\\times X$), whence\n\\[nD=H^*(\\tilde{D}_n)-n\\Delta_C-(n-2\\lfloor\\frac{n}{2}\\rfloor)(D_1+D_2),\\]\nwhere $\\tilde{D}_n:=n\\Delta_{\\P^1}+\\lfloor\\frac{n+1}{2}\\rfloor(D'_1+D'_2)$.\nThus we see that $H^0(C\\times C,\\so_{C\\times C}(nD))$ can be identified with a\nsubspace of $H^0(C\\times C,\\so_{C\\times C}(H^*(\\tilde{D}_n)))$. More precisely,\nevery global section $\\gamma$ of $\\so_{C\\times C}(H^*(\\tilde{D}_n))$ can be\nwritten in the form\n\\[\\gamma=\\frac{G}\n{(y_0\\otimes y'_0)^{\\lfloor\\frac{n+1}{2}\\rfloor}(y_1\\wedge y'_0)^n}\\]\nwith $G\\in(R\\times_{\\C}R)_{n+\\lfloor\\frac{n+1}{2}\\rfloor}$, and then the\ncondition $\\gamma\\in H^0(C\\times C,\\so_{C\\times C}(nD))^+$ is equivalent to the\nfollowing: $G$ vanishes on $\\Delta_C$ of order $n$, $G$ vanishes on $D_1+D_2$\nif $n$ is odd, and $G$ is symmetric or antisymmetric according to whether $n$\nis even or odd. The problem of determining the $G$ satisfying the above\ncondition eventually translates into a problem of linear algebra: we are going\nto see how only for $n=2$ (the cases $n=3$ and $n=4$ are similar, only slightly\nmore complicated). So we are looking for symmetric $G\\in R_3\\otimes_{\\C}R_3$\nvanishing on $\\Delta_C$ of order $2$. Passing to affine coordinates\n$(y=y_1\/y_0,\\tilde{z}=z\/y_0)$, $(y'=y'_1\/y'_0,\\tilde{z}'=z'\/y'_0)$ and dropping\ntensor products, we want the $g$ vanishing on $\\Delta_C$ of order $2$ and of\nthe form\n\\[g=u(y,y')+v(y)\\tilde{z}'+\\tilde{z}v(y')+2k\\tilde{z}\\tilde{z}'\\]\nwith $u(y,y')=u(y',y)$ of degree $\\le3$ in each variable, $\\deg(v)\\le3$ and $k\n\\in\\C$. Now, the condition $g\\rest{\\Delta}=0$ immediately implies $v=0$, so\nthat $g=u(y,y')+2k\\tilde{z}\\tilde{z}'$. Setting $f(y):=F(1,y)$, we have\n$(\\tilde{z}-\\tilde{z}')^2=f(y)+f(y')-2\\tilde{z}\\tilde{z}'$, so that $g$\nvanishes on $\\Delta_C$ of order $2$ if and only if the same is true for\n\\[g':=g+k(\\tilde{z}-\\tilde{z}')^2=u(y,y')+k[f(y)+f(y')].\\]\nThis last condition is obviously satisfied if and only if $g'(y,y')$ is\ndivisible by $(y-y')^2$. Actually, as $g'(y,y')=g(y',y)$, it is enough to\nrequire that $g'$ is divisible by $(y-y')$, i.e. that $g'(y,y)=u(y,y)+2kf(y)=\n0$. Now it is clear that this gives a linear system of $7$ equations (since\n$\\deg(g'(y,y))=6$) with $11$ unknowns ($k$ and the coefficients of $u$, which\nare $10$ because $u$ is symmetric). It turns out that the space of solutions\nhas the expected dimension $4=h^0(A,\\so_A(2\\Theta))$, and a base is given by\nthe following values of $k$ and $u(y,y')$ (the first three solutions correspond\nto the already known functions $1$, $\\frac{y_1\\odot y'_0}{y_0\\otimes y'_0}$ and\n$\\frac{y_1\\otimes y'_1}{y_0\\otimes y'_0}$):\n\n$k=0$, $u=(y-y')^2$;\n\n$k=0$, $u=(y+y')(y-y')^2$;\n\n$k=0$, $u=yy'(y-y')^2$;\n\n$k=-1$, $u=y^2{y'}^2(y+y')+2\\lambda y^2{y'}^2+\\mu yy'(y+y')+2\\nu yy'+\n\\epsilon(y+y')$.\n\nCarrying out a similar analysis also for $n=3$ and $n=4$, one eventually\nobtains the following result.\n\n \\begin{prop}\nIdentifying, as above, global sections of $\\so_A(n\\Theta)$ with rational\nfunctions on $C\\times C$, $\\R'$ is generated (as a graded $\\C$--algebra)\nby the following (homogeneous) elements:\n\\[\\theta:=1\\text{ of degree $1$;}\\]\n\\begin{align*}\n& \\rho_0:=\\frac{y_1\\odot y'_0}{y_0\\otimes y'_0}, &\n\\rho_1:=\\frac{y_1\\otimes y'_1}{y_0\\otimes y'_0},\n\\end{align*}\n\\begin{multline*}\n\\rho_2:=(y_1^3\\odot y'_0{y'_1}^2+\\lambda y_0y_1^2\\odot y'_0{y'_1}^2+\n\\mu y_0y_1^2\\odot {y'_0}^2y'_1+\\nu y_0^2y_1\\odot{y'_0}^2y'_1\\\\\n+\\epsilon y_0^2y_1\\odot{y'_0}^3-z\\odot z')\/[(y_0\\otimes y'_0)(y_1\\wedge y'_0)^2]\n\\end{multline*}\nof degree $2$;\\index{rho_i@$\\rho_i$}\n\\begin{align*}\n& \\sigma_0:=\\frac{z\\wedge{y'_0}^3}{(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)}, &\n& \\sigma_1:=\\frac{z\\wedge{y'_0}^2y'_1}{(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)},\n&\n\\sigma_2:=\\frac{z\\wedge y'_0{y'_1}^2}{(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)},\n\\end{align*}\n\\begin{multline*}\n\\sigma_3:=[y_1^2z\\wedge{y'_0}^2{y'_1}^3+y_0y_1z\\wedge(3y'_0{y'_1}^4+\n4\\lambda{y'_0}^2{y'_1}^3+3\\mu{y'_0}^3{y'_1}^2+2\\nu{y'_0}^4y'_1+\\epsilon{y'_0}^5)\\\\\n+y_0^2z\\wedge(\\mu{y'_0}^2{y'_1}^3+2\\nu{y'_0}^3{y'_1}^2+3\\epsilon{y'_0}^4y'_1)]\/\n[(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)^3],\n\\end{multline*}\n\\begin{multline*}\n\\zeta:=(y_1z\\odot y'_0z'-y_0y_1^3\\odot y'_0{y'_1}^3-\\lambda y_0y_1^3\\odot\n{y'_0}^2{y'_1}^2-\\mu y_0y_1^3\\odot{y'_0}^3y'_1-\\nu y_0^2y_1^2\\odot{y'_0}^3y'_1\n\\\\\n-\\epsilon y_0^3y_1\\odot{y'_0}^3y'_1)\/[(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)^2]\n\\end{multline*}\nof degree $3$;\\index{sigma_i@$\\sigma_i$}\\index{zeta@$\\zeta$}\n\\[\\tau_0:=\\frac{y_1^3\\wedge z'}{(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)},\\]\n\\begin{multline*}\n\\tau_1:=[\\epsilon y_0^3y_1^2\\wedge{y'_0}^2z'+\n(y_1^5+2\\lambda y_0y_1^4+3\\mu y_0^2y_1^3+4\\nu y_0^3y_1^2+3\\epsilon y_0^4y_1)\\wedge\ny'_0y'_1z'\\\\\n+(3y_0y_1^4+2\\lambda y_0^2y_1^3+\\mu y_0^3y_1^2)\\wedge{y'_1}^2z']\/\n[(y_0^2\\otimes{y'_0}^2)(y_1\\wedge y'_0)^3]\n\\end{multline*}\nof degree $4$.\\index{tau_i@$\\tau_i$}\n\nMoreover, the images in $R'\\cong\\R'\/(\\theta)$ of $\\rho_i$, $\\sigma_i$, $\\zeta$\nand $\\tau_i$ are, respectively, $r_i$, $s_i$, $z$ and $t_i$.\n \\end{prop}\n\n \\begin{rema}\\label{invar}\n$\\theta$, $\\rho_i$ and $\\zeta$ are invariant with respect to the involution of\n$A$ given by multiplication by $-1$, whereas $\\sigma_i$ and $\\tau_i$ are\nantiinvariant.\n \\end{rema}\n\n \\begin{rema}\nThe above functions are not uniquely determined by the condition of being lifts\nof the corresponding generators of $R'$: for instance, the $\\rho_i$ could be\nchanged by adding multiples of $\\theta^2=1$. On the other hand, the $\\sigma_i$\nare unique if we require them to be antiinvariant, since the elements of\n$\\R'_{<3}$ are all invariant.\n \\end{rema}\n\nNow we have to determine the relations of $\\R'$, and it is clear that this\nbecomes a problem of linear algebra (which can be easily solved by a computer).\nFor instance, the unique relation of degree $4$ must be of the form $\\rho_1^2=\n\\rho_0\\rho_2+\\theta\\varphi$ with (since $\\varphi$ is invariant)\n\\[\\varphi=a_1\\theta^3+a_2\\theta\\rho_0+a_3\\theta\\rho_1+a_4\\theta\\rho_2+a_5\\zeta\n\\]\nfor some $a_i\\in\\C$. The condition to impose on the $a_i$ is then simply the\nvanishing of $\\rho_1^2=\\rho_0\\rho_2+\\theta\\varphi$, i.e. of its numerator,\nwhich is a polynomial in $y_0$, $y_1$, $y'_0$, $y'_1$, $z$, $z'$ (not in this\ncase, but in higher degrees one has of course to use the fact that $z^2=\nF(y_0,y_1)$ and ${z'}^2=F(y'_0,y'_1)$), whose coefficients depend linearly on\nthe $a_i$. The result of these computations is the following.\n\n \\begin{prop}\\label{thetaring}\n$\\R':=\\R(A,\\Theta)\\cong Q'\/I'$, where $Q'$ is the polynomial algebra\n\\[Q':=\\C[\\theta,\\rho_0,\\rho_1,\\rho_2,\\sigma_0,\\sigma_1,\\sigma_2,\\sigma_3,\\zeta,\n\\tau_0,\\tau_1]\\] with $\\deg(\\theta)=1$, $\\deg(\\rho_i)=2$,\n$\\deg(\\sigma_i)=\\deg(\\zeta)=3$, $\\deg(\\tau_i)=4$, and $I'$ is the homogeneous\nideal which is minimally generated by the following polynomials (such that\n$\\deg(\\rel{1})=4$, $\\deg(\\rel{i})=5$ for $2\\le i\\le7$, $\\deg(\\rel{i})=6$ for\n$8\\le i\\le24$, $\\deg(\\rel{i})=7$ for $25\\le i\\le34$ and $\\deg(\\rel{i})=8$ for\n$35\\le i\\le37$):\\index{rel'_i@$\\rel{i}$|(}\n\\begin{itemize}\n\n\\item [] $\\rel{1}:=\\rho_1^2-\\rho_0\\rho_2+\\theta(-2\\zeta-\\mu\\theta\\rho_1+\n\\epsilon\\theta^3)$;\n\n\\item [] $\\rel{2}:=\\rho_0\\sigma_1-\\rho_1\\sigma_0-\\theta^2\\sigma_2$;\n\n\\item [] $\\rel{3}:=\\rho_0\\sigma_2-\\rho_1\\sigma_1+\\theta\\tau_0$;\n\n\\item [] $\\rel{4}:=\\rho_0\\sigma_3-\\rho_1\\sigma_2+\n\\theta(2\\tau_1-\\epsilon\\theta\\sigma_0-2\\nu\\theta\\sigma_1-\\mu\\theta\\sigma_2)$;\n\n\\item [] $\\rel{5}:=\\rho_1\\sigma_1-\\rho_2\\sigma_0+\n\\theta(-2\\tau_0+\\mu\\theta\\sigma_1+2\\lambda\\theta\\sigma_2+\\theta\\sigma_3)$;\n\n\\item [] $\\rel{6}:=\\rho_1\\sigma_2-\\rho_2\\sigma_1-\\theta\\tau_1$;\n\n\\item [] $\\rel{7}:=\\rho_1\\sigma_3-\\rho_2\\sigma_2+\\epsilon\\theta^2\\sigma_1$;\n\n\\item [] $\\rel{8}:=\\rho_0\\tau_1-\\rho_1\\tau_0+\\theta(\\mu\\rho_1\\sigma_1+\n2\\lambda\\rho_1\\sigma_2+2\\rho_1\\sigma_3+\\epsilon\\theta^2\\sigma_1)$;\n\n\\item [] $\\rel{9}:=\\rho_1\\tau_1-\\rho_2\\tau_0+\\theta(-2\\epsilon\\rho_0\\sigma_1-\n2\\nu\\rho_1\\sigma_1-\\mu\\rho_1\\sigma_2+\\epsilon\\theta^2\\sigma_2)$;\n\n\\item [] $\\rel{10}:=\\sigma_0^2-\\rho_0^3+\\theta^2(-\\lambda\\rho_0^2+\\rho_0\\rho_1-\n\\mu\\theta^2\\rho_0-\\theta^2\\rho_2-\\nu\\theta^4)$;\n\n\\item [] $\\rel{11}:=\\sigma_0\\sigma_1-\\rho_0^2\\rho_1+\\theta^2(\\theta\\zeta-\n\\lambda\\rho_0\\rho_1)$;\n\n\\item [] $\\rel{12}:=\\sigma_0\\sigma_2-\\rho_0^2\\rho_2+\\theta(-\\rho_0\\zeta+\n\\theta\\rho_1\\rho_2+\\epsilon\\theta^3\\rho_0+\\nu\\theta^3\\rho_1)$;\n\n\\item [] $\\rel{13}:=\\sigma_0\\sigma_3-\\rho_0\\rho_1\\rho_2+\\theta(\\rho_1\\zeta-\n\\mu\\theta^2\\zeta-\\epsilon\\theta\\rho_0^2-2\\nu\\theta\\rho_0\\rho_1-\\mu\\theta\\rho_0\\rho_2\n-2\\lambda\\theta\\rho_1\\rho_2-\\theta\\rho_2^2-\\lambda\\epsilon\\theta^3\\rho_0-\n2\\lambda\\nu\\theta^3\\rho_1-\\nu\\theta^3\\rho_2)$;\n\n\\item [] $\\rel{14}:=\\sigma_1^2-\\rho_0\\rho_1^2-\n\\theta^2\\rho_1(\\lambda\\rho_1+\\rho_2)$;\n\n\\item [] $\\rel{15}:=\\sigma_1\\sigma_2-\\rho_0\\rho_1\\rho_2-\\theta\\rho_1\\zeta$;\n\n\\item [] $\\rel{16}:=\\sigma_1\\sigma_3-\\rho_0\\rho_2^2+\\theta(-\\rho_2\\zeta+\n\\epsilon\\theta\\rho_0\\rho_1+\\lambda\\epsilon\\theta^3\\rho_1+\\epsilon\\theta^3\\rho_2)$;\n\n\\item [] $\\rel{17}:=\\sigma_2^2-\\rho_1^2\\rho_2-\n\\theta^2\\rho_1(\\epsilon\\rho_0+\\nu\\rho_1)$;\n\n\\item [] $\\rel{18}:=\\sigma_2\\sigma_3-\\rho_1\\rho_2^2+\\theta^2(\\epsilon\\theta\\zeta-\n\\nu\\rho_1\\rho_2)$;\n\n\\item [] $\\rel{19}:=\\sigma_3^2-\\rho_2^3+\\theta^2(\\epsilon\\rho_1\\rho_2-\\nu\\rho_2^2-\n\\epsilon^2\\theta^2\\rho_0-\\mu\\epsilon\\theta^2\\rho_2-\\lambda\\epsilon^2\\theta^4)$;\n\n\\item [] $\\rel{20}:=\\sigma_0\\zeta-\\rho_0\\tau_0+\n\\theta(\\theta\\tau_1+\\mu\\rho_1\\sigma_0+\\lambda\\rho_0\\sigma_2+\\rho_0\\sigma_3-\n\\epsilon\\theta^2\\sigma_0-\\nu\\theta^2\\sigma_1)$;\n\n\\item [] $\\rel{21}:=\\sigma_1\\zeta-\\rho_1\\tau_0+\n\\theta\\rho_1(\\mu\\sigma_1+\\lambda\\sigma_2+\\sigma_3)$;\n\n\\item [] $\\rel{22}:=\\sigma_2\\zeta-\\rho_2\\tau_0-\n\\theta\\sigma_1(\\epsilon\\rho_0+\\nu\\rho_1)$;\n\n\\item [] $\\rel{23}:=\\sigma_3\\zeta-\\rho_2\\tau_1+\n\\theta(-\\nu\\theta\\tau_1+\\epsilon\\rho_0\\sigma_2+\\nu\\rho_1\\sigma_2+\\mu\\rho_2\\sigma_2+\n\\lambda\\epsilon\\theta^2\\sigma_2)$;\n\n\\item [] $\\rel{24}:=\\zeta^2-\\rho_1(\\rho_2^2+\\lambda\\rho_1\\rho_2+\\mu\\rho_0\\rho_2\n+\\nu\\rho_0\\rho_1+\\epsilon\\rho_0^2)-\\theta^2\\rho_1(\\lambda\\epsilon\\rho_0+\\lambda\\nu\\rho_1+\n\\nu\\rho_2)$;\n\n\\item [] $\\rel{25}:=\\sigma_0\\tau_0-\\rho_0^2\\zeta+\\theta\\rho_1(\\theta\\zeta-\n\\mu\\rho_0^2-\\lambda\\rho_0\\rho_1-\\rho_0\\rho_2-\\nu\\theta^2\\rho_0)$;\n\n\\item [] $\\rel{26}:=\\sigma_1\\tau_0-\\rho_0\\rho_1\\zeta+\n\\theta\\rho_1(-\\mu\\rho_0\\rho_1-\\lambda\\rho_1^2-\\rho_1\\rho_2+\\epsilon\\theta^2\\rho_0)$;\n\n\\item [] $\\rel{27}:=\\sigma_2\\tau_0-\\rho_1^2\\zeta+\n\\theta\\rho_0\\rho_1(\\epsilon\\rho_0+\\nu\\rho_1)$;\n\n\\item [] $\\rel{28}:=\\sigma_3\\tau_0-\\rho_1\\rho_2\\zeta+\n\\theta(-\\epsilon\\theta\\rho_0\\zeta+\\epsilon\\rho_0\\rho_1^2+\\nu\\rho_0\\rho_1\\rho_2+\n\\lambda\\epsilon\\theta^2\\rho_1^2+\\epsilon\\theta^2\\rho_1\\rho_2)$;\n\n\\item [] $\\rel{29}:=\\sigma_0\\tau_1-\\rho_0\\rho_1\\zeta+\\theta(-\\theta\\rho_2\\zeta+\n\\lambda\\rho_0\\rho_1\\rho_2+\\rho_1^2\\rho_2+\\epsilon\\theta^2\\rho_0\\rho_1+\n\\nu\\theta^2\\rho_1^2)$;\n\n\\item [] $\\rel{30}:=\\sigma_1\\tau_1-\\rho_1^2\\zeta+\n\\theta\\rho_1\\rho_2(\\lambda\\rho_1+\\rho_2)$;\n\n\\item [] $\\rel{31}:=\\sigma_2\\tau_1-\\rho_1\\rho_2\\zeta+\n\\theta\\rho_1(-\\epsilon\\rho_0\\rho_1-\\nu\\rho_1^2-\\mu\\rho_1\\rho_2+\\epsilon\\theta^2\\rho_2)$;\n\n\\item [] $\\rel{32}:=\\sigma_3\\tau_1-\\rho_2^2\\zeta+\\theta\\rho_1(\\epsilon\\theta\\zeta-\n\\epsilon\\rho_0\\rho_2-\\nu\\rho_1\\rho_2-\\mu\\rho_2^2-\\lambda\\epsilon\\theta^2\\rho_2)$;\n\n\\item [] $\\rel{33}:=\\zeta\\tau_0-\\rho_1^2\\sigma_3-\\lambda\\rho_1^2\\sigma_2-\n\\mu\\rho_0\\rho_1\\sigma_2-\\nu\\rho_0\\rho_1\\sigma_1-\\epsilon\\rho_0^2\\sigma_1-\n\\epsilon\\theta^3\\tau_0$;\n\n\\item [] $\\rel{34}:=\\zeta\\tau_1-\\rho_1\\rho_2\\sigma_3-\\lambda\\rho_1^2\\sigma_3-\n\\mu\\rho_1^2\\sigma_2-\\nu\\rho_1^2\\sigma_1-\\epsilon\\rho_1^2\\sigma_0+\n\\theta(\\mu\\rho_1\\tau_1-\\lambda\\epsilon\\theta^2\\tau_0-\\lambda\\epsilon\\theta\\rho_0\\sigma_2-\n\\epsilon\\theta\\rho_1\\sigma_2)$;\n\n\\item [] $\\rel{35}:=\\tau_0^2-\\rho_1(\\rho_1^2\\rho_2+\\lambda\\rho_1^3+\n\\mu\\rho_0\\rho_1^2+\\nu\\rho_0^2\\rho_1+\\epsilon\\rho_0^3)+\\epsilon\\theta^2\\rho_0\\rho_1^2$;\n\n\\item [] $\\rel{36}:=\\tau_0\\tau_1-\\rho_1(\\rho_1\\rho_2^2+\\lambda\\rho_1^2\\rho_2+\n\\mu\\rho_0\\rho_1\\rho_2+\\nu\\rho_0\\rho_1^2+\\epsilon\\rho_0^2\\rho_1)+\n\\theta\\rho_1(-\\mu\\rho_1\\zeta+\\epsilon\\theta^2\\zeta-2\\lambda\\epsilon\\theta\\rho_0\\rho_1\n-\\epsilon\\theta\\rho_0\\rho_2-2\\lambda\\nu\\theta\\rho_1^2-2\\nu\\theta\\rho_1\\rho_2)$;\n\n\\item [] $\\rel{37}:=\\tau_1^2-\\rho_1(\\rho_2^3+\\lambda\\rho_1\\rho_2^2+\n\\mu\\rho_1^2\\rho_2+\\nu\\rho_1^3+\\epsilon\\rho_0\\rho_1^2)+\\epsilon\\theta^2\\rho_1^2\\rho_2$.\n\\end{itemize}\n \\end{prop}\\index{rel'_i@$\\rel{i}$|)}\n\n \\begin{rema}\nThe above polynomials are not all lifts of those listed in \\ref{curvering}: for\ninstance, $\\rel{17}$ is a lift of $s_2^2-r_1^2r_2$ and not of $s_2^2- r_0r_2^2$\n(but, of course, \\ref{curvering} remains true if $s_2^2-r_1^2r_2$ is\nsubstituted by $s_2^2-r_0r_2^2$). The reason for this and other changes was (to\ntry) to keep the $\\rel{i}$ as simple as possible.\n \\end{rema}\n\n\n \\section{Computation of $\\s{E}(\\gph)$}\n\nIf $s\\in\\langle\\theta^2,\\rho_0,\\rho_1\\rangle$, it is not difficult to see that\n$0\\in B\\subset A$ is a singular point. So we can assume that $s=\\rho_2+a\\rho_0+\nb\\rho_1+c\\theta^2\\in\\R'_2$ for some\n$a,b,c\\in\\C$.\\index{a@$a$}\\index{b@$b$}\\index{c@$c$} Notice that, since\n$\\xi^2=s$, in $\\R$ there is the relation\n\\[\\rho_2=\\xi^2-a\\rho_0-b\\rho_1-c\\theta^2.\\]\nThe structure of $\\R$ then follows immediately from \\ref{thetaring}.\n\n \\begin{prop}\\label{canring}\nThe canonical ring $\\R=\\R(\\S,\\ds_{\\S})$ is isomorphic to $Q\/I$, where $Q$ is\nthe polynomial algebra\n\\[Q:=\\C[\\xi,\\theta,\\rho_0,\\rho_1,\\sigma_0,\\sigma_1,\\sigma_2,\\sigma_3,\\zeta,\n\\tau_0,\\tau_1]\\] with $\\deg(\\xi)=\\deg(\\theta)=1$, $\\deg(\\rho_i)=2$,\n$\\deg(\\sigma_i)=\\deg(\\zeta)= 3$, $\\deg(\\tau_i)=4$, and $I$ is the homogeneous\nideal which is minimally generated by the polynomials\n$\\crel{i}$\\index{rel_i@$\\crel{i}$|(} ($1\\le i\\le37$) defined as $\\rel{i}$ in\n\\ref{thetaring}, except that $\\rho_2$ is substituted by $\\xi^2-a\\rho_0-b\\rho_1-\nc\\theta^2$.\n \\end{prop}\n\nWe will write here only the first $\\crel{i}$, which we will use directly later.\n\\begin{itemize}\n\n\\item [] $\\crel{1}=a\\rho_0^2+b\\rho_0\\rho_1+\\rho_1^2-\\xi^2\\rho_0+\n\\theta(-2\\zeta+c\\theta\\rho_0-\\mu\\theta\\rho_1+\\epsilon\\theta^3)$;\n\n\\item [] $\\crel{2}=\\rho_0\\sigma_1-\\rho_1\\sigma_0-\\theta^2\\sigma_2$;\n\n\\item [] $\\crel{3}=\\rho_0\\sigma_2-\\rho_1\\sigma_1+\\theta\\tau_0$;\n\n\\item [] $\\crel{4}=\\rho_0\\sigma_3-\\rho_1\\sigma_2+\\theta(2\\tau_1-\n\\epsilon\\theta\\sigma_0-2\\nu\\theta\\sigma_1-\\mu\\theta\\sigma_2)$;\n\n\\item [] $\\crel{5}=a\\rho_0\\sigma_0+b\\rho_1\\sigma_0+\\rho_1\\sigma_1-\\xi^2\\sigma_0\n+\\theta(-2\\tau_0+c\\theta\\sigma_0+\\mu\\theta\\sigma_1+2\\lambda\\theta\\sigma_2+\n\\theta\\sigma_3)$;\n\n\\item [] $\\crel{6}=a\\rho_0\\sigma_1+b\\rho_1\\sigma_1+\\rho_1\\sigma_2-\\xi^2\\sigma_1\n+\\theta(-\\tau_1+c\\theta\\sigma_1)$;\n\n\\item [] $\\crel{7}=a\\rho_0\\sigma_2+b\\rho_1\\sigma_2+\\rho_1\\sigma_3-\\xi^2\\sigma_2\n+\\theta^2(\\epsilon\\sigma_1+c\\sigma_2)$;\n\n\\item [] $\\crel{8}=\\rho_0\\tau_1-\\rho_1\\tau_0+\\theta(\\mu\\rho_1\\sigma_1+\n2\\lambda\\rho_1\\sigma_2+2\\rho_1\\sigma_3+\\epsilon\\theta^2\\sigma_1)$;\n\n\\item [] $\\crel{9}=\\rho_1\\tau_1+(a\\rho_0+b\\rho_1-\\xi^2+c\\theta^2)\\tau_0+\n\\theta(-2\\epsilon\\rho_0\\sigma_1-2\\nu\\rho_1\\sigma_1-\\mu\\rho_1\\sigma_2+\n\\epsilon\\theta^2\\sigma_2)$;\n\n\\item [] $\\crel{10}=\\sigma_0^2-\\rho_0^3+\\theta^2[-\\lambda\\rho_0^2+\\rho_0\\rho_1+\n(a-\\mu)\\theta^2\\rho_0+b\\theta^2\\rho_1-\\theta^2\\xi^2+(c-\\nu)\\theta^4]$.\n\n\\end{itemize}\\index{rel_i@$\\crel{i}$|)}\n\n \\begin{rema}\\label{regseq}\nIt is clear that $\\R\/(\\theta)$ is isomorphic to the ring $R'\\oplus\\xi' R'$ with\nmultiplication defined by\n\\[(x+\\xi'y)(x'+\\xi'y')=xx'+(r_2+ar_0+br_1)yy'+(xy'+x'y)\\xi'.\\]\nSince $\\xi'$ is a {non--zerodivisor} in $R'\\oplus\\xi' R'$, we see that $(\\theta,\\xi)$ is\na regular sequence in $\\R$.\n \\end{rema}\n\nFrom now on we set $\\w:=(1,1,2,3)$, and we will write (as usual) $\\p$ for\n$\\p(\\w)$, $\\gP$ for $\\gP(\\w)$ and $\\pd_i$ for $\\pd_i(\\w)$.\n\nAssume that $\\gph:\\gS\\to\\gP$ is a good birational weighted canonical projection: $\\gph$ is determined by a morphism of\ngraded rings $\\rh:\\p\\to\\R$, and (up to composing $\\gph$ with an automorphism of\n$\\gP$) we can always assume that\n\\begin{align*}\n& \\rh(x_0)=\\theta, & & \\rh(x_1)=\\xi, & & \\rh(x_2)\\in\\langle\\rho_0,\\rho_1\n\\rangle\\setminus\\{0\\}, & \\rh(x_3)\\in\\R_3\\setminus\\C[\\theta,\\xi,\\rh(x_2)]_3.\n\\end{align*}\n\n \\begin{rema}\nNot every choice of $\\rh$ as above determines a good birational weighted canonical projection. For instance, we must\nhave $\\rh(x_3)\\notin\\C[\\theta,\\xi,\\rho_0,\\rho_1,\\zeta]_3$, otherwise $\\ph$\nwould factor through the involution of $\\S$ which lifts multiplication by $-1$\non $A$ (see \\ref{invar}). On the other hand, one can prove that $\\gph$ is good\nbirational for a general choice of $\\rh$ (see also \\ref{excexi}). Later we will\nprove that $\\gph$ is good birational in a particular case.\n \\end{rema}\n\nSince $\\sw{\\w}=7$, $\\pd_1=2$, $\\pd_2=4$ the vector bundle $\\s{E}(\\gph)$ defined\nin \\ref{Edef} is in our case given by\n\\[\\s{E}(\\gph)=\\so(-2)\\oplus\\sd^1(-1)^2\\oplus\\so(-3)^{\\cre_{-1}(\\gph)}\\oplus\n\\so(-4)^{\\cre_{-2}(\\gph)}\\oplus\\so(-5)^{\\cre_{-3}(\\gph)}.\\]\nIn order to compute the $\\cre_j(\\gph)$ we need some preliminary results.\n\nFirst we study morphisms from $\\sd_{\\gP}^1$ to $\\gph_*\\so_{\\gS}(1)$: the exact\nsequence\n\\[\\sko^{-3}\\mor{\\cdiff{\\sko}^{-3}}\\sko^{-2}\\to\\sd^1\\to0\\]\nshows that $\\Hom_{\\gP}(\\sd^1,\\gph_*\\so_{\\gS}(1))\\cong\n\\ker\\Hom_{\\gP}(\\cdiff{\\sko}^{-3},\\gph_*\\so_{\\gS}(1))$. Remembering the\ndefinition of $\\cdiff{\\sko}^{-3}$, this means that $\\varphi\\in\n\\Hom_{\\gP}(\\sd^1,\\gph_*\\so_{\\gS}(1))$ corresponds to $6$ elements\n\\[\\varphi_{i,j}\\in\\Hom_{\\gP}(\\so(-\\w_i-\\w_j),\\gph_*\\so_{\\gS}(1))\\cong\n\\R_{\\w_i+\\w_j+1}\\]\nfor $0\\le i0$) and $\\tau_i$.\n \\end{proof}\n\n \\begin{rema}\n$\\crel{10}$ also shows that $\\gY=\\im\\gph$ does not contain the points\n$[0,0,1,0]$ and $[0,0,0,1]$, which are (by \\ref{stdmori}) the only non {standard}\npoints of $\\gP$ (and also the only singular points of $\\P$).\n \\end{rema}\n\nBy \\ref{cj} we have $\\s{E}:=\\s{E}(\\gph)=\\so(-2)\\oplus\\so(-3)^2\\oplus\n\\sd^1(-1)^2$, so that the resolution given by \\ref{mthm} is of the form\n\\[0\\to\\so(-8)\\oplus\\so(-6)\\oplus\\so(-5)^2\\oplus(\\sd^2)^2\\mor{\\alpha}\n\\so\\oplus\\so(-2)\\oplus\\so(-3)^2\\oplus\\sd^1(-1)^2\\mor{\\gamma}\\gph_*\\so_{\\gS}\\to0\n\\]\nwith $\\alpha=\\alpha\\dual(-8)$. As a first step to construct $\\alpha$, we choose\n$\\gamma$; $\\Hom_{\\gP}(\\so(-j),\\gph_*\\so_{\\gS})$ will be identified, as usual,\nwith $\\R_j$. Then we take $\\gamma\\rest{\\so}:=1$, $\\gamma\\rest{\\so(-2)}:=\n\\rho_1$, $\\gamma\\rest{\\so(-3)^2}:=(\\sigma_1,\\zeta)$ and\n$\\gamma\\rest{\\sd^1(-1)^2}:=(\\eta,\\eta')$, where $\\eta,\\eta'\\in\n\\Hom_{\\gP}(\\sd^1(-1),\\gph_*\\so_{\\gS})\\cong\\Hom_{\\gP}(\\sd^1,\\gph_*\\so_{\\gS}(1))$\nare as in \\ref{omegamor}. Making the computations (using the relations in $\\R$)\nit turns out that they can be chosen as follows:\n\n$\\eta_{0,1}=a\\sigma_0+b\\sigma_1+\\sigma_2$;\n\n$\\eta_{0,2}=\\xi\\sigma_0$;\n\n$\\eta_{1,2}=-\\tau_0+c\\theta\\sigma_0+\\mu\\theta\\sigma_1+\n(2\\lambda-b)\\theta\\sigma_2+\\theta\\sigma_3$;\n\n$\\eta_{0,3}=\\xi\\rho_0^2$;\n\n$\\eta_{1,3}=-\\rho_0\\zeta-b\\theta^2\\zeta+(c+ab-a\\lambda)\\theta\\rho_0^2+\nb(b-\\lambda)\\theta\\rho_0\\rho_1-b\\theta\\xi^2\\rho_0+\\theta\\xi^2\\rho_1+\n(a^2+bc-a\\mu+\\epsilon)\\theta^3\\rho_0+(ab-b\\mu+\\nu-c)\\theta^3\\rho_1-a\\theta^3\\xi^2+\n(ac-a\\nu+b\\epsilon)\\theta^5$;\n\n$\\eta_{2,3}=\\theta\\xi(\\rho_0\\rho_1-\\lambda\\rho_0^2+(a-\\mu)\\theta^2\\rho_0+\nb\\theta^2\\rho_1-\\theta^2\\xi^2+(c-\\nu)\\theta^4)$;\n\n$\\eta'_{0,1}=a\\sigma_1+b\\sigma_2+\\sigma_3$;\n\n$\\eta'_{0,2}=\\xi\\sigma_1$;\n\n$\\eta'_{1,2}=b\\tau_0+\\tau_1-\\epsilon\\theta\\sigma_0+(c-2\\nu)\\theta\\sigma_1-\n\\mu\\theta\\sigma_2$;\n\n$\\eta'_{0,3}=\\xi\\rho_0\\rho_1$;\n\n$\\eta'_{1,3}=b\\rho_0\\zeta+\\rho_1\\zeta+(a-4b^2+4b\\lambda-\\mu)\\theta^2\\zeta+\n(a\\mu-a^2+2ab^2-2ab\\lambda-\\epsilon)\\theta\\rho_0^2+\n(c+a\\lambda+2b\\mu-2\\nu-3ab+2b^3-2b^2\\lambda)\\theta\\rho_0\\rho_1+\n(2a-2b^2+2b\\lambda-\\mu)\\theta\\xi^2\\rho_0+(3b-2\\lambda)\\theta\\xi^2\\rho_1+\n(a\\nu+c\\mu-2ac-\\lambda\\epsilon+2b^2c-2bc\\lambda)\\theta^3\\rho_0+(2c\\lambda+2b\\nu-3bc-\n2\\lambda\\nu-2b^2\\mu+2b\\lambda\\mu)\\theta^3\\rho_1-\n\\theta\\xi^4+(2c-\\nu)\\theta^3\\xi^2+(2b^2\\epsilon-2b\\lambda\\epsilon+c\\nu-c^2)\\theta^5$;\n\n$\\eta'_{2,3}=\\theta\\xi(\\theta\\zeta-\\lambda\\rho_0\\rho_1)$.\n\n \\begin{lemm}\n$\\gamma:=(1,\\rho_1,\\sigma_1,\\zeta,\\eta,\\eta'):\\so\\oplus\\so(-2)\\oplus\\so(-3)^2\n\\oplus\\sd^1(-1)^2\\to\\gph_*\\so_{\\gS}$ is such that $H^1(\\gP,\\gamma(1))$ and\n$H^0(\\gP,\\gamma(i))$ $\\forall i\\in\\Z$ are surjective.\n \\end{lemm}\n\n \\begin{proof}\nBy \\ref{omegamor} and \\ref{cohomgen} it is clear that\n\\[H^1(\\gP,(\\eta,\\eta')(1)):H^1(\\gP,(\\sd^1)^2)\\cong\\C^2\\to\nH^1(\\gP,\\gph_*\\so_{\\gS}(1))\\cong\\C^2\\] is an isomorphism. As for $H^0$, since\n$H^0(\\gP,\\gph_*\\so_{\\gS}(i))\\cong\\R_i\\ \\forall i\\in\\Z$, it is enough to prove that\n$\\R$ is generated (as graded $\\p$--module) by\n$\\{1,\\rho_1,\\sigma_1,\\zeta,\\eta_{i,j},\\eta'_{i,j}\\}$, and this will follow if\nwe show that $\\R$ is generated by $\\{1,\\rho_1,\\sigma_1,\\sigma_2,\n\\sigma_3,\\zeta,\\tau_0,\\tau_1,\\rho_1\\zeta\\}$ ($\\sigma_2$ comes from\n$\\eta_{0,1}$, $\\sigma_3$ from $\\eta'_{0,1}$, $\\tau_0$ from $\\eta_{1,2}$,\n$\\tau_1$ from $\\eta'_{1,2}$ and $\\rho_1\\zeta$ from $\\eta'_{1,3}$). Now, it is\neasy to see that we can always assume that a monomial in $\\R$ contains at most\none of the variables $\\sigma_i$, $\\zeta$ and $\\tau_i$, and with exponent at\nmost $1$. Moreover, the relation $\\crel{1}$ implies that we can assume that\nalso $\\rho_1$ appears with exponent at most $1$. It follows that $\\R$ is\ngenerated by\n\\[\\{1,\\rho_1,\\sigma_1,\\sigma_2,\\sigma_3,\\zeta,\\tau_0,\\tau_1,\\rho_1\\sigma_1,\n\\rho_1\\sigma_2,\\rho_1\\sigma_3,\\rho_1\\zeta,\\rho_1\\tau_0,\\rho_1\\tau_1\\}.\\]\nMoreover, using $\\crel{3}$, $\\crel{4}$ and $\\crel{7}$ it is immediate to see\nthat $\\rho_1\\sigma_1$, $\\rho_1\\sigma_2$ and $\\rho_1\\sigma_3$ are actually not\nneeded, and similarly for $\\rho_1\\tau_0$ and $\\rho_1\\tau_1$ (using $\\crel{8}$\nand $\\crel{9}$).\n \\end{proof}\n\nThen it follows from \\ref{resdet} that $\\gamma$ is surjective and $\\ker\\gamma\n\\cong(\\so\\oplus\\s{E})\\dual(-8)$. Now we want to construct a morphism $\\beta:\n(\\so\\oplus\\s{E})\\dual(-8)\\to\\so\\oplus\\s{E}$ such that\n\\[0\\to(\\so\\oplus\\s{E})\\dual(-8)\\mor{\\beta}\\so\\oplus\\s{E}\\mor{\\gamma}\n\\gph_*\\so_{\\gS}\\to0\\]\nis exact, and (again by \\ref{resdet}) it is enough to require that $\\gamma\\comp\n\\beta=0$ and $H^0(\\gP,\\beta(i))$ is injective $\\forall i\\in\\Z$. First we need to\nfix some notation. The exact sequence\n\\[0\\to\\sd^1(-1)\\to\\sko^{-1}(-1)=\\so(-2)^2\\oplus\\so(-3)\\oplus\\so(-4)\n\\mor{\\cdiff{\\sko}^{-1}(-1)}\\so(-1)\\to0\\]\ninduces naturally an exact sequence\n\\[0\\to\\so\\oplus\\s{E}\\mor{j}\\tilde{\\s{E}}\\mor{\\tilde{d}}\\so(-1)^2\\to0,\\]\nwhere $\\tilde{\\s{E}}:=\\so\\oplus\\so(-2)\\oplus\\so(-3)^2\\oplus\n(\\so(-2)^2\\oplus\\so(-3)\\oplus\\so(-4))^2$. Denoting $\\dual(-8)$ by $\\tdual$, we\nhave also an exact sequence\n\\[0\\to\\so(-7)^2\\mor{\\tilde{d}\\tdual}\\tilde{\\s{E}}\\tdual\\mor{j\\tdual}\n(\\so\\oplus\\s{E})\\tdual\\to0.\\] It is then clear that giving\n$\\beta:(\\so\\oplus\\s{E})\\tdual\\to\\so\\oplus\\s{E}$ is equivalent to giving\n$\\tilde{\\beta}:\\tilde{\\s{E}}\\tdual\\to\\tilde{\\s{E}}$ such that\n$\\tilde{d}\\comp\\tilde{\\beta}=0$ and $\\tilde{\\beta}\\comp \\tilde{d}\\tdual=0$ (of\ncourse, $\\tilde{\\beta}=j\\comp\\beta\\comp j\\tdual$), and that $\\beta$ is\nsymmetric if and only if $\\tilde{\\beta}$ is. Observe that $\\tilde{\\beta}$ can\nbe identified with a $12\\times12$ matrix of homogeneous polynomials (in $\\p$).\nMoreover, the exact sequence\n\\[0\\to\\sd^2(-1)\\to\\sko^{-2}(-1)=\\so(-3)\\oplus\\so(-4)^2\\oplus\\so(-5)^2\\oplus\n\\so(-6)\\to\\sd^1(-1)\\to0\\]\ninduces naturally an exact sequence\n\\[0\\to\\sd^2(-1)^2\\to\\s{E}'\\mor{p'}\\so\\oplus\\s{E}\\to0,\\]\nwhere $\\s{E}':=\\so\\oplus\\so(-2)\\oplus\\so(-3)^2\\oplus\\sko^{-2}(-1)^2$. While\n$\\beta$ need not factor through $p'$ (because $\\Ext_{\\gP}^1(\\sd^2,\\sd^2(-1))\\ne\n0$, and actually the morphism we are looking for does not factor), $\\beta\\comp\nj\\tdual$ factors through $p'$ (since\n$\\Ext_{\\gP}^1(\\tilde{\\s{E}}\\tdual,\\sd^2(-1))=0$), so that we must have\n$\\tilde{\\beta}=j\\comp p'\\comp\\beta'$ for some $\\beta':\\tilde{\\s{E}}\\tdual\\to\n\\s{E'}$, as we can see from the following diagram\n\\[\\begin{CD}\n0 @>>> \\so(-7)^2 @>\\tilde{d}\\tdual>> \\tilde{\\s{E}}\\tdual @>j\\tdual>>\n(\\so\\oplus\\s{E})\\tdual @>>> 0 \\\\\n@. @. @VV{\\beta'}V @VV{\\beta}V \\\\\n0 @>>> \\sd^2(-1)^2 @>>> \\s{E}' @>p'>> \\so\\oplus\\s{E} @>j>> \\tilde{\\s{E}}\n@>\\tilde{d}>> \\so(-1)^2\n\\end{CD}\\]\nwhich has exact rows, except at $\\so\\oplus\\s{E}$. Then the condition\n$\\tilde{d}\\comp\\tilde{\\beta}=0$ is automatically satisfied, whereas\n$\\tilde{\\beta}\\comp \\tilde{d}\\tdual=0$ is equivalent to\n$p'\\comp\\beta'\\comp\\tilde{d}\\tdual=0$ (because $j$ is injective). Setting\n$\\gamma':=\\gamma\\comp p':\\s{E}'\\to \\gph_*\\so_{\\gS}$ (notice that, by\ndefinition, $\\gamma'=(1,\\rho_1,\\sigma_1,\n\\zeta,\\{\\eta_{i,j}\\},\\{\\eta'_{i,j}\\})$), $\\gamma\\comp\\beta=0$ is clearly\nequivalent to $\\gamma'\\comp\\beta'=0$. Finally, $H^0(\\gP,\\beta(i))$ injective\n$\\forall i\\in\\Z$ translates into the condition that\n\\begin{equation}\\label{gsecseq}\n0\\to H^0(\\gP,\\so(i-7)^2)\\mor{H^0(\\gP,\\tilde{d}\\tdual(i))}\nH^0(\\gP,\\tilde{\\s{E}}\\tdual(i))\\mor{H^0(\\gP,(j\\comp p'\\comp\\beta')(i))}\nH^0(\\gP,\\tilde{\\s{E}}(i))\n\\end{equation}\nis an exact sequence $\\forall i\\in\\Z$ (this is a consequence of the fact that,\n$\\forall i\\in\\Z$, $H^0(\\gP,j(i))$ is injective and $H^0(\\gP,j\\tdual(i))$ is\nsurjective, as $H^0(\\gP,\\so(i-7))=0$).\n\nWe are going to sketch only how it is possible to find such a $\\beta'$. Since\n\\[\\tilde{\\s{E}}\\tdual=\\so(-4)^2\\oplus\\so(-5)^4\\oplus\\so(-6)^5\\oplus\\so(-8),\\]\n$\\beta'$ is given by sections $a_1,a_2\\in H^0(\\gP,\\s{E}'(4))$,\n$b_1,\\dots,b_4\\in H^0(\\gP,\\s{E}'(5))$, $c_1,\\dots,c_5\\in H^0(\\gP,\\s{E}'(6))$,\n$e\\in H^0(\\gP,\\s{E}'(8))$. As we want $\\gamma'\\comp\\beta'=0$, it is clear that\neach of them must be in the kernel of the corresponding $H^0(\\gP,\\gamma'(i))$\n(and must not be mapped to $0$ by $H^0(\\gP,(j\\comp p')(i))$ by\n\\eqref{gsecseq}). So, for instance, denoting by $g(\\rho)$ (where\n$\\rho:\\s{L}\\cong\\so(-i)\\to \\gph_*\\so_{\\gS}$ is one of the components of\n$\\gamma'$) a generator of $H^0(\\gP,\\s{L}(i))\\cong\\C$, it is easy to see that\n$\\ker H^0(\\gP,\\gamma'(4))$ is generated by $-x_1x_3g(1)+g(\\eta_{0,2})$ and\n$x_1g(\\sigma_1)-g(\\eta'_{0,2})$, so that we can choose them as $a_1$ and $a_2$.\nContinuing in this way, it is not difficult to find candidates also for $b_i$,\n$c_i$ and $e$: the $b_i$ must be chosen in such a way that, together with the\n$x_la_i$ ($l=0,1$), they generate $\\ker\nH^0(\\gP,\\gamma'(5))\/H^0(\\gP,\\sd^2(4)^2)$ (notice that before we didn't consider\nthe analogous quotient because $H^0(\\gP,\\sd^2(3))=0$), and similarly for the\n$c_i$ and $e$. It is clear that then $\\gamma'\\comp\\beta' =0$ and the sequences\n\\eqref{gsecseq} are exact, and it is also not hard to get that\n$p'\\comp\\beta'\\comp\\tilde{d}\\tdual=0$ and even that the resulting morphism from\n$(\\sd^2)^2$ to $\\sd^1(-1)^2$ is already symmetric.\n\nOnce we have thus obtained the required $\\beta$ (corresponding to\n$\\tilde{\\beta}=j\\comp p'\\comp\\beta'$), it remains only to symmetrize it. By\n\\ref{resdet} there exists an automorphism $\\delta$ of $(\\so\\oplus\\s{E})\\tdual$\nsuch that $\\alpha:=\\beta\\comp\\delta=\\alpha\\tdual$ and\n\\[0\\to(\\so\\oplus\\s{E})\\tdual\\mor{\\alpha}\\so\\oplus\\s{E}\\mor{\\gamma}\n\\gph_*\\so_{\\gS}\\to0\\] is exact. It is immediate to see that finding such a\n$\\delta$ is equivalent to finding an automorphism $\\tilde{\\delta}$ of\n$\\tilde{\\s{E}}\\tdual$ such that\n$\\tilde{\\alpha}:=\\tilde{\\beta}\\comp\\tilde{\\delta}=\\tilde{\\alpha}\\tdual$ and\n$\\tilde{\\delta}\\comp\\tilde{d}\\tdual=\\tilde{d}\\tdual\\comp s$ for some\nautomorphism $s$ of $\\so(-7)^2$. Regarding $\\tilde{\\beta}$ as a matrix,\ncomposing $\\tilde{\\beta}$ with an automorphism of $\\tilde{\\s{E}}\\tdual$\ncorresponds to making ``homogeneous'' elementary operations on the columns: one\ncan multiply a column by a non zero constant, can exchange two columns\ncorresponding to morphisms from $\\so(i_1)$ and $\\so(i_2)$ with $i_1=i_2$, and\ncan add to a column a multiple of the ``right'' degree of another one. In our\ncase, since the component of $\\tilde{\\beta}$ from $(\\sko^{-1}(-1)\\tdual)^2\\cong\n(\\sko^{-3})^2$ to $\\sko^{-1}(-1)^2$ is already symmetric, we need only to\nmodify the first $4$ columns (those giving\n$\\tilde{\\beta}\\rest{\\so(-8)\\oplus\\so(-6)\\oplus\\so(5)^2}$), which assures that\n$\\tilde{\\delta}\\comp\\tilde{d}\\tdual=\\tilde{d}\\tdual$. After these computations,\nwe eventually obtain the following symmetric matrix\n$\\tilde{\\alpha}=j\\comp\\alpha\\comp j\\tdual$. Denoting by $p_2$, $p_4$, $q_2$,\n$q_4$ the following homogeneous polynomials (subscripts denote degrees)\n\\begin{align*}\np_2 & :=(ab-a\\lambda+c)x_2-bx_1^2+bcx_0^2,\\\\\np_4 & :=(a\\lambda-ab-c)x_2^2+bx_1^2x_2+(a\\mu-a^2-bc-\\epsilon)x_0^2x_2+\nax_0^2x_1^2+(a\\nu-ac-b\\epsilon)x_0^4,\\\\\nq_2 & :=b(\\lambda-b)x_2-x_1^2+(b\\mu-ab+c-\\nu)x_0^2,\\\\\nq_4 & :=\\lambda x_2^2+(\\mu-a)x_0^2x_2+x_0^2x_1^2+(\\nu-c)x_0^4,\n\\end{align*}\nthe first $4$ columns of $\\tilde{\\alpha}$ are\n\n\\begin{tabular}{r|cccc}\n\\multicolumn{2}{c}{} & \\\\\n& $\\so(-8)$ & $\\so(-6)$ & $\\so(-5)$ & $\\so(-5)$ \\\\\n\\hline\n$\\so$ & $0$ & $x_0x_3p_2$ & $x_0p_4$ & $ax_0^2x_3$ \\\\\n$\\so(-2)$ & $x_0x_3p_2$ & $2b(b-\\lambda)x_0x_3$ & $x_0q_2$ & $-x_3$ \\\\\n$\\so(-3)$ & $x_0p_4$ & $x_0q_2$ & $0$ & $x_2+bx_0^2$ \\\\\n$\\so(-3)$ & $ax_0^2x_3$ & $-x_3$ & $x_2+bx_0^2$ & $0$ \\\\\n$\\so(-2)$ & $0$ & $-\\lambda x_0x_1x_2$ & $0$ & $x_0^2x_1$ \\\\\n$\\so(-2)$ & $0$ & $x_2^2+\\lambda x_0^2x_2$ & $-x_3$ & $-x_0^3$ \\\\\n$\\so(-3)$ & $0$ & $-x_1x_2$ & $0$ & $0$ \\\\\n$\\so(-4)$ & $0$ & $0$ & $x_1$ & $0$ \\\\\n$\\so(-2)$ & $x_0x_1q_4$ & $-x_0x_1(x_2+bx_0^2)$ & $0$ & $0$ \\\\\n$\\so(-2)$ & $x_3^2-x_2^3-x_0^2q_4$ & $x_0^2(x_2+bx_0^2)$ & $0$ & $0$ \\\\\n$\\so(-3)$ & $x_1x_2^2$ & $0$ & $0$ & $0$ \\\\\n$\\so(-4)$ & $-x_1x_3$ & $0$ & $0$ & $0$ \\\\\n\\multicolumn{2}{c}{}\n\\end{tabular}\n\nand the last $8$ rows and columns (those giving the component of\n$\\tilde{\\alpha}$ from $(\\sko^{-3})^2$ to $\\sko^{-1}(-1)^2$, which corresponds\nto the component of $\\alpha$ from $(\\sd^2)^2$ to $\\sd^1(-1)^2$) are\n\n\\begin{tabular}{r|cccccccc}\n\\multicolumn{2}{c}{} \\\\\n& $\\so(-6)$ & $\\so(-6)$ & $\\so(-5)$ & $\\so(-4)$ & $\\so(-6)$ & $\\so(-6)$ &\n$\\so(-5)$ & $\\so(-4)$ \\\\\n\\hline\n$\\so(-2)$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $x_3$ & $-x_2$ \\\\\n$\\so(-2)$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\\\\n$\\so(-3)$ & $0$ & $0$ & $0$ & $0$ & $-x_3$ & $0$ & $0$ & $x_0$ \\\\\n$\\so(-4)$ & $0$ & $0$ & $0$ & $0$ & $x_2$ & $0$ & $-x_0$ & $0$ \\\\\n$\\so(-2)$ & $0$ & $0$ & $-x_3$ & $x_2$ & $0$ & $0$ & $0$ & $0$ \\\\\n$\\so(-2)$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\\\\n$\\so(-3)$ & $x_3$ & $0$ & $0$ & $-x_0$ & $0$ & $0$ & $0$ & $0$ \\\\\n$\\so(-4)$ & $-x_2$ & $0$ & $x_0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\\\\n\\multicolumn{2}{c}{}\n\\end{tabular}\n(of course, by symmetry we need not write the first $4$ rows entirely).\n\n \\begin{rema}\nIt is easy to see that the component of $\\alpha$ from $(\\sd^2)^2$ to\n$\\sd^1(-1)^2$ is as expected from \\ref{natmor} (observe that $\\p_1=\\langle\nx_0,x_1\\rangle$ and that the multiplication map $x_i:H^1(\\gP,\\gph_*\\so_{\\gS})\n\\to H^1(\\gP,\\gph_*\\so_{\\gS}(1))$ is $0$ for $i=0$ and is an isomorphism for\n$i=1$).\n \\end{rema}\n\n \\begin{rema}\nThe fact that $\\tilde{\\alpha}=j\\comp\\alpha\\comp j\\tdual$ for some (uniquely\ndetermined) $\\alpha$ is equivalent to the fact that, denoting by\n$\\tilde{\\alpha}^{(i)}$ the i$^{th}$ row of $\\tilde{\\alpha}$, we have (as it\nis immediate to check) $\\sum_{i=0}^3x_i\\tilde{\\alpha}^{(5+i)}=0$ and\n$\\sum_{i=0}^3x_i\\tilde{\\alpha}^{(9+i)}=0$.\n \\end{rema}\n\n \\begin{rema}\nWe checked that, with $\\tilde{\\alpha}$ defined as above, $\\alpha$ actually\nsatisfies the rank condition and its determinant is an irreducible polynomial\n(of degree $24$).\n \\end{rema}\n\n\n\n\n \\begin{appendix}\n\n \\chapter{Abelian categories and derived categories}\n\nIn \\ref{abelquot} we recall the definition and some basic properties about the\nquotient of an abelian category by a localizing subcategory (see \\cite{G} for\nmore details and proofs). In the rest of the appendix, besides proving\na couple of results for which we could not find a reference, we give (without\nproofs) the essential elements of the theory of derived categories. A good\nreference for this subject is \\cite[chapter I]{H2} (where almost all proofs can\nbe found); see also \\cite{V} for more details on triangulated categories and\nderived categories. We refer to \\cite{Bon} for the part on exceptional\nsequences and mutations in a triangulated category.\n\n \\section{Quotient of an abelian category}\\label{abelquot}\n\n \\begin{defi}\nLet $\\cat{A}$ be an abelian category. A full subcategory $\\cat{C}$ of $\\cat{A}$\nis called {\\em thick} if for every short exact sequence $0\\to M'\\to M\\to M''\\to\n0$ in $\\cat{A}$, $M$ is an object of $\\cat{C}$ if and only if $M'$ and $M''$\nare objects of $\\cat{C}$.\n \\end{defi}\n\nGiven an abelian category $\\cat{A}$ and a thick subcategory $\\cat{C}$ of\n$\\cat{A}$, it is possible to define the quotient category $\\cat{A}\/\\cat{C}$,\nwhose objects are the same as those of $\\cat{A}$, and whose morphisms are\ndefined in the following way. Given $M,N$ objects of $\\cat{A}$, the set\n\\[I(M,N):=\\{(M',N')\\st M'\\subseteq M;N'\\subseteq N;M\/M'\\in\\cat{C};N'\\in\n\\cat{C}\\}\\]\nis partially ordered by $(M',N')\\le(M'',N'')$ if and only if $M''\\subseteq M'$\nand $N'\\subseteq N''$. It is easy to prove that $(I(M,N),\\le)$ is a direct set\n(since $\\cat{C}$ is thick), and so we can define\n\\[\\Hom_{\\cat{A}\/\\cat{C}}(M,N):=\\lim_{(M',N')\\in I(M,N)}\\Hom_{\\cat{A}}(M',N\/N')\n.\\]\nAgain using the fact that $\\cat{C}$ is thick, it is not difficult to see that\nthere is a natural way of defining composition of morphisms, so that\n$\\cat{A}\/\\cat{C}$ is indeed a category.\n\n \\begin{prop}(\\cite[prop. 1, p. 367]{G})\nIf $\\cat{C}$ is a thick subcategory of an abelian category $\\cat{A}$, then\n$\\cat{A}\/\\cat{C}$ is an abelian category and the natural functor $T:\\cat{A}\\to\n\\cat{A}\/\\cat{C}$ is exact.\n \\end{prop}\n\n \\begin{defi}\nA thick subcategory $\\cat{C}$ of an abelian category $\\cat{A}$ is called a {\\em\nright localizing} subcategory\\index{localizing subcategory} of $\\cat{A}$ if\nthere exists a functor $S: \\cat{A}\/\\cat{C}\\to\\cat{A}$ which is right adjoint of\n$T:\\cat{A}\\to \\cat{A}\/\\cat{C}$.\n \\end{defi}\n\n \\begin{rema}\nOf course, one can define (in the obvious way) also the notion of left\nlocalizing subcategory.\n \\end{rema}\n\nClearly if $S$ exists it is unique up to isomorphism of functors, and it is\nleft exact (since it is a right adjoint). Moreover, we have the following\nresults (\\cite[prop. 3, p. 371 and prop. 5, p. 374]{G}).\n\n \\begin{prop}\nThe natural transformation (induced by the properties of adjoint functors)\n$T\\comp S\\to{\\rm id}_{\\cat{A}\/\\cat{C}}$ is an isomorphism of functors.\n \\end{prop}\n\n \\begin{prop}\\label{locsubcat}\nLet $F:\\cat{A}\\to\\cat{B}$ be an exact functor between abelian categories, let\n$G:\\cat{B}\\to\\cat{A}$ be a right adjoint of $F$, and assume that the associated\nnatural transformation $F\\comp G\\to{\\rm id}_{\\cat{B}}$ is an isomorphism of\nfunctors. Then, denoting by $\\ker F$ the full subcategory of $\\cat{A}$ whose\nobjects are the $A$ such that $F(A)\\iso0$ and by $T:\\cat{A}\\to\\cat{A}\/\\ker F$\nthe natural functor, $\\ker F$ is a right localizing subcategory of $\\cat{A}$\nand $F$ induces an exact equivalence of categories $F':\\cat{A}\/\\ker F\\to\n\\cat{B}$, whose quasi--inverse is $T\\comp G$.\n \\end{prop}\n\n\n \\section{Triangulated categories}\\label{trcat}\n\nA triangulated category\\index{triangulated category|(} is given by an additive\ncategory $\\cat{C}$ with some additional structures. First there is an additive\nautomorphism $T:\\cat{C}\\to \\cat{C}$, called {\\em translation} (or {\\em shift})\nfunctor; $\\forall n\\in\\Z$ the functor $T^n$ will be usually denoted by $[n]$. A\n{\\em triangle}\\index{triangle} in $(\\cat{C},T)$ is given by three objects $X$,\n$Y$, $Z$ and by three morphisms $u:X\\to Y$, $v:Y\\to Z$, $w:Z\\to X[1]$ of\n$\\cat{C}$, and will be usually denoted by $\\tri{X}{Y}{Z}{u}{v}{w}$. A morphism\nbetween two triangles\\index{morphism of!triangles} $\\tri{X}{Y}{Z}{u}{v}{w}$ and\n$\\tri{X'}{Y'}{Z'}{u'}{v'}{w'}$ is given by three morphisms $f:X\\to X'$, $g:Y\\to\nY'$ and $h:Z\\to Z'$ such that\n\\[\\begin{CD}\nX @>u>> Y @>v>> Z @>w>> X[1] \\\\\n@VVfV @VVgV @VVhV @VVf[1]V \\\\\nX' @>u'>> Y' @>v'>> Z' @>w'>> X'[1]\n\\end{CD}\\]\nis a commutative diagram. Then a structure of triangulated category on\n$(\\cat{C},T)$ is given by a family of triangles (called {\\em distinguished}\ntriangles)\\index{distinguished triangle} which have to satisfy the following\nfour axioms.\n\\begin{description}\n\n\\item [(TR1)] A triangle isomorphic to a distinguished triangle is\ndistinguished.\n\nEvery morphism $u:X\\to Y$ of $\\cat{C}$ is contained in a distinguished triangle\n$\\tri{X}{Y}{Z}{u}{v}{w}$.\n\nFor every object $X$ of $\\cat{C}$ the triangle $\\tri{X}{X}{0}{{\\rm id}_X}{}{}$ is\ndistinguished.\n\n\\item [(TR2)] The triangle $\\tri{X}{Y}{Z}{u}{v}{w}$ is distinguished if and\nonly if the triangle $\\tri{Y}{Z}{X[1]}{v}{w}{-u[1]}$ is distinguished.\n\n\\item [(TR3)] Given two distinguished triangles $\\tri{X}{Y}{Z}{u}{v}{w}$,\n$\\tri{X'}{Y'}{Z'}{u'}{v'}{w'}$ and two morphisms $f:X\\to X'$, $g:Y\\to Y'$ such\nthat $u'\\comp f=g\\comp u$, there exists a morphism $h:Z\\to Z'$ such that\n$(f,g,h)$ is a morphism of triangles.\n\n\\item [(TR4)] Given three distinguished triangles\n\\begin{gather*}\n\\tri{X}{Y}{Z'}{u}{u'}{u''},\\\\\n\\tri{Y}{Z}{X'}{v}{v'}{v''},\\\\\n\\tri{X}{Z}{Y'}{v\\comp u}{w'}{w''},\n\\end{gather*}\nthere exist two morphisms $f:Z'\\to Y'$ and $g:Y'\\to X'$, such that\n$({\\rm id}_X,v,f)$ and $(u,{\\rm id}_Z,g)$ are morphisms of triangles\n\\[\\begin{CD}\nX @>u>> Y @>u'>> Z' @>u''>> X[1] \\\\\n@VV{\\rm id}_XV @VVvV @VVfV @VV{\\rm id}_{X[1]}V \\\\\nX @>v\\comp u>> Z @>w'>> Y' @>w''>> X[1]\\\\\n@VVuV @VV{\\rm id}_ZV @VVgV @VVu[1]V \\\\\nY @>v>> Z @>v'>> X' @>v''>> Y[1]\n\\end{CD}\\]\nand $\\tri{Z'}{Y'}{X'}{f}{g}{u'[1]\\comp v''}$ is a distinguished triangle.\n\\index{triangulated category|)}\n\\end{description}\n\nA functor between two triangulated categories is said to be {\\em\nexact}\\index{exact functor} if it commutes (up to isomorphism) with translation\nfunctors and if it carries distinguished triangles to distinguished triangles.\nA functor $H:\\cat{C}\\to \\cat{A}$ between a triangulated category and an abelian\ncategory is called {\\em cohomological} if for\nevery distinguished triangle $\\tri{X}{Y}{Z}{u}{v}{w}$ of $\\cat{C}$ the sequence\n\\[\\cdots\\to H(X)\\mor{H(u)}H(Y)\\mor{H(v)}H(Z)\\mor{H(w)}H(X[1])\\mor{H(u[1])}\nH(Y[1])\\to\\cdots\\]\nis exact in $\\cat{A}$.\n\n \\begin{prop}\\label{trcatprop}\nA triangulated category $\\cat{C}$ satisfies the following properties:\n\\begin{enumerate}\n\n\\item the composition of two consecutive morphisms in a distinguished triangle\nis zero;\n\n\\item for every object $M$ of $\\cat{C}$, $\\Hom_{\\cat{C}}(M,-):\\cat{C}\\to\n\\cat{Ab}$ and $\\Hom_{\\cat{C}}(-,M):\\cat{C}\\opp\\to\\cat{Ab}$ are cohomological\nfunctors;\\footnote{$\\cat{C}\\opp$ has a natural structure of triangulated\ncategory. $\\cat{Ab}$ is the category of abelian groups.}\n\n\\item for every morphism $u:X\\to Y$ in $\\cat{C}$ a distinguished triangle of\nthe form $\\tri{X}{Y}{Z}{u}{v}{w}$ (which exists by $\\mathbf{(TR1)}$) is not\nunique, but the object $Z$ is uniquely determined up to isomorphism; moreover,\n$u$ is an isomorphism if and only if $Z\\iso0$.\n\\end{enumerate}\n \\end{prop}\n\nFor the rest of this section we will assume that $\\K$ is a field and\nthat $\\cat{C}$ is a $\\K$--linear triangulated category (namely, a\ntriangulated category in which $\\Hom_{\\cat{C}}(X,Y)$ has a structure\nof $\\K$--vector space for every objects $X$ and $Y$ of $\\cat{C}$\nsuch that composition of morphisms is $\\K$--bilinear and the shift\nfunctor is $\\K$--linear).\n\n \\begin{defi}\\label{exc}\nAn object $E$ of $\\cat{C}$ is called {\\em exceptional}\\index{exceptional} if\n$\\Hom_{\\cat{C}}(E,E)\\cong\\K$ and $\\Hom_{\\cat{C}}(E,E[k])=0$ for\n$k\\ne0$.\n\nA sequence $(E_0,\\dots,E_m)$ of objects of $\\cat{C}$ is called an\n{\\em exceptional sequence}\\index{exceptional!sequence} (or {\\em\ncollection}) if each $E_i$ is exceptional and\n$\\Hom_{\\cat{C}}(E_i,E_j[k])=0$ for $i>j$ and $\\forall k\\in\\Z$.\n\nAn exceptional sequence $(E_0,\\dots,E_m)$ is {\\em\nstrong}\\index{strong exceptional sequence} if\n$\\Hom_{\\cat{C}}(E_i,E_j[k])=0$ for $k\\ne0$ and $\\forall i,j$.\n\nAn exceptional sequence $(E_0,\\dots,E_m)$ is {\\em\nfull}\\index{full exceptional sequence} if $\\{E_0,\\dots,E_m\\}$\ngenerates $\\cat{C}$ as a triangulated category (i.e., if every\nstrictly full triangulated subcategory of $\\cat{C}$ containing the\n$E_i$ coincides with $\\cat{C}$).\n \\end{defi}\n\n \\begin{rema}\\label{shiftexc}\nGiven $k_0,\\dots,k_m\\in\\Z$, a sequence $(E_0,\\dots,E_m)$ is\nexceptional (and full) if and only if $(E_0[k_0],\\dots,E_m[k_m])$ is\nexceptional (and full). On the other hand, the property of being\nstrong is not preserved in general, unless $k_0=\\cdots=k_m$.\n \\end{rema}\n\nWe will assume also that\n$\\dim_{\\K}\\bigoplus_{k\\in\\Z}\\Hom_{\\cat{C}}(X,Y[k])<\\infty$ for every\nobjects $X,Y$ of $\\cat{C}$ and we will denote by $L_XY$ the object of\n$\\cat{C}$ defined (up to isomorphism) by the distinguished triangle\n\\[L_XY\\to\\bigoplus_{k\\in\\Z}(\\Hom_{\\cat{C}}(X[k],Y)\\otimes_{\\K}X[k])\\mor{u}Y\n\\to(L_XY)[1]\\]\n(where $u$ is the natural morphism).\n\n \\begin{prop}\nIf $\\sigma=(E_0,\\dots,E_m)$ is a (full) exceptional sequence of $\\cat{C}$, then\nfor $i=0,\\dots m-1$ the sequence\n$(E_0,\\dots,E_{i-1},L_{E_i}E_{i+1},E_i,E_{i+2},\\dots,E_m)$ (called a {\\em left\nmutation}\\index{left!mutation} of $\\sigma$)\\footnote{There is also a similar\nnotion of right mutation, which we will not consider.} is (full) exceptional,\ntoo.\n \\end{prop}\n\nGiven a sequence $(E_0,\\dots,E_m)$, for $0\\le j\\le i\\le m$ we define\ninductively objects $L^{(j)}E_i$ by $L^{(0)}E_i:=E_i$ and\n$L^{(j)}E_i:=L_{E_{i-j}}(L^{(j-1)}E_i)$ for $j>0$.\n\n \\begin{coro}\\label{dualmut}\nIf $\\sigma=(E_0,\\dots,E_m)$ is a (full) exceptional sequence of $\\cat{C}$, then\nthe sequence $(L^{(m)}E_m,L^{(m-1)}E_{m-1},\\dots,L^{(1)}E_1,L^{(0)}E_0=E_0)$\n(called the {\\em left dual}\\index{left!dual} of $\\sigma$) is (full)\nexceptional, too.\n \\end{coro}\n\n\n\n \\section{Derived categories}\\label{dercat}\n\nIf $\\cat{A}$ is an abelian category or, more generally, an additive subcategory\nof an abelian category, $C(\\cat{A})$,\\index{C(A)@$C(\\cat{A})$|(}\n$K(\\cat{A})$\\index{K(A)@$K(\\cat{A})$|(} and\n$D(\\cat{A})$\\index{D(A)@$D(\\cat{A})$|(} will denote the category of complexes,\nthe homotopy category and the derived category of $\\cat{A}$, respectively: we\nrecall the definitions.\n\nThe objects of $C(\\cat{A})$ are just the complexes of $\\cat{A}$ (by a complex\nwe always mean a cohomological complex, that is differentials have degree $+1$)\nand the morphisms are just usual morphisms of\ncomplexes.\\index{C(A)@$C(\\cat{A})$|)} A complex\n\\[\\cdots\\to X^{-1}\\mor{\\cdiff{X}^{-1}}X^{0}\\mor{\\cdiff{X}^{0}}X^{1}\\to\\cdots\\]\nwill be denoted by $\\cp{X}$; similarly, a morphism of complexes from $\\cp{X}$\nto $\\cp{Y}$ given by morphisms $f^i:X^i\\to Y^i$ (with $\\cdiff{Y}^{i+1}\\comp\nf^i=f^{i+1}\\comp\\cdiff{X}^i$) will be denoted by $\\cp{f}$.\\index{$\\cp{}$}\n\nThe objects of $K(\\cat{A})$ are the same as those of $C(\\cat{A})$, whereas the\nmorphisms are the homotopy equivalence classes of morphisms of complexes (two\nmorphisms $\\cp{f},\\cp{g}:\\cp{X}\\to\\cp{Y}$ in $C(\\cat{A})$ are homotopic if\nthere exist morphisms $k^{i}:X^i\\to Y^{i-1}$ such that $f^i-g^i=\\cdiff{Y}^{i-1}\n\\comp k^i+k^{i+1}\\comp\\cdiff{X}^i$).\\index{K(A)@$K(\\cat{A})$|)} By abuse of\nnotation we will denote in the same way a morphism of $C(\\cat{A})$ and its\nequivalence class in $K(\\cat{A})$. Note that if $\\cp{f}:\\cp{X}\\to\\cp{Y}$ is a\nmorphism of $K(\\cat{A})$, then the morphisms in cohomology\n$H^i(\\cp{f}):H^i(\\cp{X})\\to H^i(\\cp{Y})$ are well defined: in particular,\n$\\cp{f}$ is said to be a {\\em quasi--isomorphism}\\index{quasi--isomorphism} if\n$H^i(\\cp{f})$ is an isomorphism $\\forall i\\in\\Z$.\n\nThe objects of $D(\\cat{A})$ are always the complexes of $\\cat{A}$, but the\nmorphisms are obtained from those of $K(\\cat{A})$ by inverting formally the\nquasi--isomorphisms: more precisely, a morphism in $D(\\cat{A})$ from $\\cp{X}$\nto $\\cp{Y}$ is represented by a couple of morphisms (of $K(\\cat{A})$)\n$(\\cp{s}:\\cp{Z}\\to\\cp{X},\\cp{f}:\\cp{Z}\\to\\cp{Y})$ where $\\cp{s}$ is a\nquasi--isomorphism and $\\cp{Z}$ is another object of $D(\\cat{A})$; on these\ncouples one introduces the equivalence relation generated by the direct\nrelation which identifies two couples $(\\cp{s}:\\cp{Z}\\to\\cp{X},\\cp{f}:\\cp{Z}\\to\n\\cp{Y})$ and $(\\cp{t}:\\cp{W}\\to\\cp{X},\\cp{g}:\\cp{W}\\to\\cp{Y})$ if there exists\na morphism $\\cp{a}:\\cp{Z}\\to\\cp{W}$ such that $\\cp{f}=\\cp{g}\\comp\\cp{a}$ and\n$\\cp{s}=\\cp{t}\\comp\\cp{a}$.\\index{D(A)@$D(\\cat{A})$|)} The equivalence class of\na couple $(\\cp{s},\\cp{f})$ will be denoted by $[(\\cp{s},\\cp{f})]$ (or by\n$[(\\cp{s},\\cp{f})]_{\\cat{A}}$ if there can be doubt about $\\cat{A}$); if\n$\\cp{s}={\\rm id}_{\\cp{X}}$, we will simply write $[\\cp{f}]$ instead of\n$[({\\rm id}_{\\cp{X}},\\cp{f})]$.\n\nClearly $C(\\cat{A})$, $K(\\cat{A})$ and $D(\\cat{A})$ are all additive\ncategories, but, while $C(\\cat{A})$ is abelian if $\\cat{A}$ is abelian, this is\nnot the case for $K(\\cat{A})$ and $D(\\cat{A})$. On the other hand, $K(\\cat{A})$\nand $D(\\cat{A})$ have a natural structure of triangulated categories, which we\nare now going to describe briefly.\n\n$\\forall n\\in\\Z$ the shift functor $[n]:C(\\cat{A})\\to C(\\cat{A})$ is defined on\nobjects by\n\\begin{align*}\n& X[n]^i:=X^{n+i}, & \\diff{\\cp{X}[n]}^i:=(-1)^n\\cdiff{X}^{n+i}\n\\end{align*}\nand on morphisms by $f[n]^i:=f^{n+i}$. Since the shift functors preserve\nhomotopic morphisms and quasi--isomorphisms, they induce shift functors\n(denoted again by $[n]$) also on $K(\\cat{A})$ and $D(\\cat{A})$. Clearly in each\ncase $[n]$ is an additive automorphism and $[n]\\comp[m]=[n+m]$ $\\forall n,m\\in\\Z$.\nIn order to be able to say what are the distinguished triangles of $K(\\cat{A})$\nand $D(\\cat{A})$ we need to recall the definition of mapping cone of a morphism\nof complexes.\n\n \\begin{defi}\nThe {\\em mapping cone}\\index{mapping cone} of a morphism of complexes\n$\\cp{f}:\\cp{X}\\to\\cp{Y}$ is the complex $\\mc{f}$\\index{MC(f)@$\\mc{f}$} defined\nby\n\\begin{align*}\n& \\mc{f}^n:=X^{n+1}\\oplus Y^n, &\n\\diff{\\mc{f}}^n:=\\begin{pmatrix}\n-\\cdiff{X}^{n+1} & 0 \\\\\nf^{n+1} & \\cdiff{Y}^n\n\\end{pmatrix}.\n\\end{align*}\nThere are natural morphisms of complexes (inclusion and projection)\n\\begin{align*}\n& \\inc{f}:\\cp{Y}\\to\\mc{f}, & \\pro{f}:\\mc{f}\\to\\cp{X}[1].\n\\end{align*}\n \\end{defi}\\index{j(f)@$\\inc{f}$}\\index{p(f)@$\\pro{f}$}\n\nThen a triangle in $K(\\cat{A})$ is distinguished if it is isomorphic to a\ntriangle of the form $\\tri{X}{Y}{\\mc{f}}{\\cp{f}}{\\inc{f}}{\\pro{f}}$ for some\nmorphism of complexes $\\cp{f}$. A triangle in $D(\\cat{A})$ is distinguished if\nit is isomorphic to the image of a distinguished triangle of $K(\\cat{A})$\nunder the natural functor $Q:K(\\cat{A})\\to D(\\cat{A})$ (which is therefore\nexact by definition). The distinguished triangles of $D(\\cat{A})$ can be\ncharacterized in the following way.\n\n \\begin{prop}\\label{derdist}\nIf $0\\to\\cp{X}\\mor{\\cp{f}}\\cp{Y}\\mor{\\cp{g}}\\cp{Z}\\to0$ is a short exact\nsequence of complexes, then there exists a morphism $\\alpha:\\cp{Z}\\to\\cp{X}[1]$\nin $D(\\cat{A})$ such that $\\tri{\\cp{X}}{\\cp{Y}}{\\cp{Z}}{[\\cp{f}]}{[\\cp{g}]}\n{\\alpha}$ is a distinguished triangle. Conversely, every distinguished triangle\nin $D(\\cat{A})$ is isomorphic to one of this form.\n \\end{prop}\n\nGiven a complex $\\cp{X}$ of $\\cat{A}$ and $n\\in\\Z$, we will denote by\n$X^{\\le n}$ or $X^{n$}\n\\end{cases}, &\n\\diff{X^{\\le n}}^i:=\\begin{cases}\n\\cdiff{X}^i & \\text{if $in-1}$. Then $\\forall n\\in\\Z$\nthere is a short exact sequence of complexes $0\\to X^{\\ge n}\\to\\cp{X}\\to X^{>0$.\nApplying $\\epsilon'_t$ to the triangle $\\tri{X^{>p}}{X^{\\ge p}}{X^p[-p]}{}{}{}$\n(which is distinguished in $K^b(\\cat{A})$ by \\ref{homotdist}), we obtain a\ncommutative diagram with exact rows\n\\[\\begin{CD}\nK'_t(X^{>p}[1]) @>>> K'_t(\\cp{A}[1]) @>>> K'_t(X^{\\ge p}) @>>> K'_t(X^{>p})\n@>>> K'_t(\\cp{A}) \\\\\n@VV{\\epsilon}'_t(X^{>p}[1])V @VV{\\epsilon}'_t(\\cp{A}[1])V\n@VV{\\epsilon}'_t(X^{\\ge p})V @VV{\\epsilon}'_t(X^{>p})V\n@VV{\\epsilon}'_t(\\cp{A})V \\\\\nD'_t(X^{>p}[1]) @>>> D'_t(\\cp{A}[1]) @>>> D'_t(X^{\\ge p}) @>>> D'_t(X^{>p})\n@>>> D'_t(\\cp{A})\n\\end{CD}\\]\n(where $\\cp{A}:=X^p[-p-1]$). By the inductive hypothesis\n$\\epsilon'_t(X^{>p}[1])=\\epsilon'_{t-1}(X^{>p})$ is an isomorphism for $t\\le1$\nand $\\epsilon'_t(X^{>p})$ is an isomorphism for $t\\le0$ and is injective for\n$t=1$. Since $\\forall p'\\in\\Z$\n\\[K'_t(X^p[-p'])\\cong\\begin{cases}\n\\Hom_{\\cat{A}}(X^p,Y^q) & \\text{if $p'=q-t$} \\\\\n0 & \\text{if $p'\\ne q-t$}\n\\end{cases}\\]\nand $D'_t(X^p[-p'])\\cong\\Ext_{\\cat{A}}^{p'-q+t}(X^p,Y^q)$ (by \\ref{Yoneda}),\nthe fact that $\\Ext_{\\cat{A}}^i(X^p,Y^q)=0$ for $0>> K_t(\\cp{B}) @>>> K_t(Y^{\\le q}) @>>> K_t(Y^{>> K_t(\\cp{B}[1]) \\\\\n@VV{\\epsilon}_t(Y^{>> D_t(\\cp{B}) @>>> D_t(Y^{\\le q}) @>>> D_t(Y^{>> D_t(\\cp{B}[1])\n\\end{CD}\\]\n(where $\\cp{B}:=Y^q[-q]$). Now, $\\epsilon_t(Y^{0$.\n \\end{rema}\n\n\\end{appendix}\n\n\n\\backmatter\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe class of asteroidal triple-free (AT-free) graphs is a widely-researched class of graphs. One of the key properties of AT-free graphs is that they have dominating pairs $[3 ]$. It has also been shown that AT-free graphs have dominating pair sets (polar sets) that can be found in linear time. In this paper, we show that every maximal clique within a polar set contains a diametral vertex. This leads to the result that a diametral pair can be found in linear time for AT-free graphs. A wider implication of our result is it reduces time-complexity for computing simplicial vertices in general graphs and recognizing triangle-free graphs. The best known running time for solving both problems is reduced to $O(n^2)$.\n\nThree nodes form an asteroidal triple if the removal of the neighborhood of one disconnects the other two. The class of AT-free graphs $[3 ]$, the class of graphs without asteroidal triples, is a well-known generalization of the interval, permutation, trapezoid, and co-comparability classes of graphs $[7 ]$ that imposes the linearity we see in those graphs. AT-free graphs are intuitively understood by their inability to grow too far in three directions at once and find useful application in critical systems $[6 ]$ due to being both domination preserving (because of their hereditary dominating shortest path property) and distance preserving (because of their convex and isometric properties $[1 ]$). Many problems remain open for AT-free graphs, such as Hamiltonicity and finding a lower-bound for computing diameter.\n\n\\iffalse\nInterestingly, all subgraphs of AT-free graphs have a dominating pair\\textemdash a pair of nodes such that all paths between them are dominating paths. A group of dominating pairs form what is known as a polar pair, though until now they have never been defined for graphs of low-diameter. Later, they discovered the related class of diametral path graphs, superclass of AT-free which may possibly lack a dominating pair $[5 ]$. \n\\fi\n\n\\iffalse\nPr\u017eulj et al. $[10 ]$ characterized certain graphs which are DPG and \\textit{have} an asteroidal triple.\n\\fi\n\nAlthough it has been shown that computing diameter may not be done in subquadratic time for general graphs $[11 ] $ unless the Strong Exponential Time Hypothesis fails, it is a worthy endeavor to find special graph classes for which it can be solved more quickly. A linear-time algorithm for finding the diameter of an AT-free graph has previously eluded researchers. A quadratic time algorithm for finding the diameter on AT-free graphs was proposed in $[2 ]$. The authors believed that diameter on AT-free graphs is unlikely to be solved in less than quadratic time. More recently, an algorithm for finding the diameter of an AT-free $m$-edge graph in $O(m^{3\/2})$-time (that is, subquadratic time) was presented in $[8 ]$. Their approach is based on lexicographical breadth-first search (\\textsc{lbfs}). Then, the authors $[8 ]$ conjectured that a linear-time algorithm using \\textsc{lbfs} is unlikely. To highlight the difficulty of the diameter problem, later in their paper they show that the problem of deciding whether a simplicial vertex exists in a general graph can be reduced in $O(n^2)$-time to finding the diameter on AT-free graphs. However, we prove that if a polar set is given then diameter can be found in linear-time without using \\textsc{lbfs} by exploiting properties of eccentricties within polar sets. The largest graph class for which it is known that polar sets can be found in linear-time is the class of AT-free graphs $[3 ]$, therefore diameter of AT-free graphs is shown to be computable in linear-time.\n\nFinding the diameter of an AT-free graph is important because many problems can be reduced to it. The wider implication of our result is that it re-opens an avenue of research to improve run-time for related theoretical problems, including computing a simplicial vertex in general graphs and finding a triangle in a general graph\\textemdash problems which are listed amongst Kratsch and Spinrad's three basic graph problems $[9 ]$. They state that ``it is natural to to ask whether these problems can be solved... faster (perhaps in linear time), or solved efficiently without using matrix multiplication\" $[9 ]$. Without the use of matrix multiplication, we improve the time-complexity to finding all simplicial vertices in an $n$-vertex general graph to $O(n^2)$, and the time-complexity of finding a triangle in an $n$-vertex general graph is similarly reduced to $O(n^2)$. Thus, we improve the best known running time for several graph theoretical problems.\n\nTo achieve linear-time complexity of diameter on AT-free graphs, we begin by showing that the diameter of a weak dominating pair graph is at most 2 greater than a shortest distance between a dominating pair. Then, we generalize the Polar Lemma $[3 ]$ (described more in section II) to the entire class of graphs with dominating pairs (including those with low diameter). We prove that every maximal clique of dominating pair nodes contains a diametral node. We show the diameter of AT-free graphs can be found in $O(n + m)$-time by an algorithm that works as follows: initially assign all nodes in the polar set $X$ a label and then propagate these labels to the polar set $Y$ on the other end of the graph. Labels are initialized as $\\mathcal L = \\{l_0,l_1,...,l_{|X|}\\}$. After any vertex in $Y$ is reached, we show that any remaining unassigned nodes of $Y$ (a set we will call $Y^*$) contribute at most 1 to the maximum eccentricity of the graph. If any initial label $l_i$ in $\\mathcal L$ is missing from a propagated label in $Y^*$, then $l_i$ can contribute yet another 1 to the maximum eccentricity. Thereby we extend a shortest path in both polar sets, achieving the upper limit of diameter.\n\n\\vspace{.25cm}\n\\section{Preliminaries} Although we will go over some specific graph theoretical terms, additional standard graph theory notation may be found in $[7 ]$.\n\nBy $G = (V,E)$ we denote a connected, simple, and finite graph. $G[S]$ denotes the subgraph of the graph $G$ induced by the vertex set $S \\subseteq V$. We say nodes $x$ and $y$ are incident if $x$ is adjacent to $y$. For any $x \\in V$ let $N(x) = \\{y : \\{x,y\\} \\in E\\}$ be the open neighborhood of $x$ and $N[x] = N(x) \\cup \\{x\\}$ the (closed) neighborhood of $x$. By $N^2[x]$ we refer to the second order neighborhood of $x$. A node is \\textit{pendant} if it has a single neighbor. An \\textit{asteroidal triple} is a set of three nodes s.t. there is a path between any two of them that avoids the neighborhood of the third. A graph that lacks an asteroidal triple is called \\textit{asteroidal triple-free} or simply AT-free.\n\nSet $S$ \\textit{dominates} some set of vertices $T$ if every node in $T$ lies in the neighborhood of $S$. We can perform operations on a graph or set; for example, adding or removing a node is shown by $S + v$ or $S - v$, respectively. A set is called a \\textit{singleton} if it contains only one vertex. A vertex $x$ is \\textit{simplicial} if $N(x)$ induces a complete subgraph. A \\textit{clique} is an induced subgraph of $G$ that is complete. A \\textit{triangle} is a clique on three vertices. A graph is \\textit{triangle-free} if no three vertices form a triangle.\n\nTo identify starting and ending vertices in a path of vertices $P$, path $P$ with endpoints $u$ and $v$ is also called a $u,v$-path or equivalently defined as $P_{u \\rightarrow v}$. By $U,W$-path we mean a path that goes from some vertex in set $U$ to some vertex in set $W$. By $d_G(u,v)$ we denote the \\textit{geodesic distance}, the number of edges along a shortest path, between two vertices $u$ and $v$. The \\textit{diameter} of $G$ is the greatest geodesic distance between any two nodes. The maximum geodesic distance between a node and all other nodes is its \\textit{eccentricity}. We say node $v$ \\textit{achieves the diameter} if $ecc(v) = diam(G)$. A pair of vertices $u,v \\in V$ with $d_G(u,v) = diam(G)$ is called a diametral pair and a shortest path between such a pair is called a \\textit{diametral path}.\n\n\nA dominating pair (DP) is a pair of vertices $(a,b)$ such that any path between them dominates the entire node set of the graph. A \\textit{DP node} is any one node in a DP. It is worth mentioning that $a = b$ is allowed. Given a DP node $a$, we say node $u$ \\textit{extends} $a$ if $u$ has higher eccentricity than $a$. Formally, $u$ extends $a$ if $ecc(u) > ecc(a)$.\n\nDeogun and Kratsch $[6 ]$ generalized several properties of AT-free graphs to define the dominating pair graph (DPG) and weak dominating pair (wDP) graph classes, for which many of our results extend. A graph is a weak dominating pair graph (wDP) if $G$ has a dominating pair. A graph is a dominating pair graph (DPG) if every induced subgraph is wDP. Additional properties of DPGs are discussed in $[10 ]$.\n\n\n\n\\begin{figure}[t]\n \\centering\n\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \\tikzstyle{vertex}=[circle, fill=black!0] \n \\begin{scope}\\Large\n \\node[vertex] (1x) at (0,4+.4) {$X$};\n \\node[vertex] (1y) at (-2.5,-.4) {$X'$};\n \\node[vertex] (1z) at (+2.5,-.4) {$X''$};\n \\node[vertex] (1xz) at (1.25+.5,2) {$Y'$};\n \\node[vertex] (1xy) at (-1.25-.5,2) {$Y''$};\n \\node[vertex] (1yz) at (0,-.4) {$Y$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n \n \\node[vertex] (x) at (0,4) {$ $};\n \\node[vertex] (y) at (-2.5,0) {$ $};\n \\node[vertex] (z) at (+2.5,0) {$ $};\n \\node[vertex] (xz) at (1.25,2) {$ $};\n \\node[vertex] (xy) at (-1.25,2) {$ $};\n \\node[vertex] (a) at (0,2) {$ $};\n \\node[vertex] (b) at (.625,1) {$ $};\n \\node[vertex] (c) at (-.625,1) {$ $};\n \n \\node[vertex] (yz) at (0,0) {$ $};\n \n \\draw(y)edge(z);\n \\draw(z)edge(xz);\n \\draw(xz)edge(a);\n \\draw(a)edge(b);\n \\draw(b)edge(c);\n \\draw(c)edge(xy);\n \\draw(x)edge(xy);\n \n \\draw(y)edge(xy);\n \\draw(yz)edge(b)edge(c);\n \\draw(b)edge(xz);\n \\draw(a)edge(c);\n \\draw(xy)edge(a);\n \\draw(x)edge(xz);\n \n \\draw[thick,gray!60,dashed] (0,4) ellipse (.7cm and .7cm); \n \\draw[thick,gray!60,dashed] (0,0) ellipse (.7cm and .7cm); \n \\draw[thick,gray!60,dashed] (-2.5,0) ellipse (.7cm and .7cm); \n \\draw[thick,gray!60,dashed] (+2.5,0) ellipse (.7cm and .7cm); \n \\draw[thick,gray!60,dashed] (-1.25-.25,2) ellipse (.7cm and .7cm); \n \\draw[thick,gray!60,dashed] (1.25+.25,2) ellipse (.7cm and .7cm); \n \n \\end{tikzpicture}\n \\\\\n \\caption{A DPG where all polar sets are singletons. The set of all possible maximal polar pairs $\\mathscr R = \\{(X,Y),(X',Y'),(X'',Y'')\\}$.}\n\n\\end{figure}\n\n\\iffalse\nAll DPGs are either AT-free or contain an asteroidal triple.\n\\fi\n\n\\textit{Remark.} In this paper, DP refers to dominating pair and \\textit{not} diametral path. For a summary of terms used in this article, refer to the Appendix (Section IX).\n\n\n\n\n\\vspace{.25cm}\n\\section{Updating Polar Sets} \n\nThe Polar Lemma was originally developed to describe the relationship between disjoint sets that contain dominating pairs. The lemma was later generalized to DPG and wDP graphs with high diameter. Theorem 7.1 $[3 ]$ and Theorem 4.1 $[6 ]$ are combined to give the following theorem:\n\n\\vspace{.5cm}\n\\textbf{Polar Lemma.} \\textit{Let $G = (V,E)$ be any AT-free (\\textit{resp. wDP}) with diameter at least 4 (\\textit{resp. 5}). Then there are disjoint sets $X \\subseteq V$ and $Y \\subseteq V$ such that, for all $x,y \\in V$, $(x,y)$ is a dominating pair of $G$ if and only if $x \\in X$ and $y \\in Y$.}\n\n\\vspace{.5cm}\nBoth $X$ or $Y$ are \\textit{polar sets}, and they form the \\textit{polar pair} $(X,Y)$. More simply, a polar pair is a set of DP's, while a polar set is set of DP nodes. To generalize polar sets to any graphs with a DP and of any diameter, we offer an updated definition of polar sets:\n\n\\vspace{.5cm}\n\\textbf{Perfect Polar Lemma.} \\textit{Given a weak dominating pair graph $G = (V,E)$, there exist maximal sets $X$ and $Y$ such that for all $x \\in X$ and all $y \\in Y$, $(x,y)$ is a DP.}\n\n\\vspace{.5cm}\n\\textit{Proof.} The proof is obvious as our new polar sets follow a less-restrictive definition: being neither disjoint nor maximum. $\\square$\n\n\n\n\\begin{figure}[b]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (X) at (-1.7,1) {$A$};\n \\node[vertex] (Y) at (1.7,1) {$B$};\n \n \\node[vertex] (X') at (.3,2-.3) {$A'$};\n \\node[vertex] (Y') at (-.3,.3) {$B'$};\n \\end{scope}\n \\tikzstyle{vertex}=[circle, fill=black]\n \n \\node[vertex] (a) at (-2,0) {$ $};\n \\node[vertex] (b) at (-2,2) {$ $};\n \\node[vertex] (c) at (0,0) {$ $};\n \\node[vertex] (d) at (0,2) {$ $};\n \\node[vertex] (e) at (2,0) {$ $};\n \\node[vertex] (f) at (2,2) {$ $};\n \n \n \\path (a)edge(b)edge(c);\n \\path(d)edge(b)edge(c)edge(f);\n \\path(e)edge(f)edge(c);\n \n \n \\draw[thick,gray!60,dashed] (-2,1) ellipse (.75cm and 1.4cm);\n \\draw[thick,gray!60,dashed] (2,1) ellipse (.75cm and 1.4cm);\n \n \\draw[thick,gray!60,dashed] (0,2-.2) ellipse (.65cm and .65cm); \n \\draw[thick,gray!60,dashed] (0,.2) ellipse (.65cm and .65cm); \n\n\n \\end{tikzpicture}\n \\caption{An AT-free graph where the old definition of polar sets does not hold. Under the new definition, $\\mathscr R = \\{(A,B),(A',B')\\}$.}\n\n \n\\end{figure}\n\n\\vspace{.5cm}\nSo long as there exists at least one dominating pair, maximal cardinality polar pairs certainly exist. Both polar sets are maximal when for all $v \\in V \\setminus X \\setminus Y$, the pairs $(v,x)$ and $(v,y)$ are not DP. \n\nPolar sets $X$ and $Y$ have historically been defined as unique. Under our generalized definition, there may be several configurations of $X$ and $Y$ with respect to the initial choice of DP nodes $x$ and $y$. Consider Fig. 1, which has multiple polar pairs but does \\textit{not} have any single, maximum polar pair. Notice that the polar sets need not be disjoint under our new definition\\textemdash for graphs with low diameter where $x = y$ is possible, polar sets $X$ and $Y$ may overlap. Henceforth, we will define $\\mathscr R$ to be the set of all possible polar pairs in $G$. Evidently, $|\\mathscr R| = 1$ for AT-free graphs where diameter is at least 4.\n\nA linear-time algorithm for finding polar sets, and therefore all DP's, is presented in $[4 ]$. The details of the algorithm, which is based on \\textsc{lbfs}, are not vital for our purposes, but the authors are careful to note that polar sets are not guaranteed to exist for AT-free graphs with diameter less than 4 in accordance to the Polar Lemma defined in $[3 ]$. Given that their algorithm is able to find \\textit{maximum cardinality} polar sets, it stands that a modified function for finding dominating pairs can be created to find one of several possible \\textit{maximal cardinality} polar sets. In $[4 ]$, \\textsc{lbfs} is able to find a dominating pair vertex $x$ in linear time on AT-free graphs. Function \\textsc{dsets}, which depends on the Polar Lemma $[3 ]$, may be adapted to return a maximal polar set $Y$ with respect to the given input DP vertex $x$. Another call to the function with given input $y \\in Y$ will return $X$ to complete the maximal polar pair $(X,Y)$. For our purposes, we do not mind which polar pair is returned, and such an algorithm clearly runs in linear-time. Let's call our function \\textsc{get-polar-pair}.\n\n\n\\vspace{.25cm}\n\\section{Bounds on Diameter of wDP Graphs}\n\nIn this short section, we show that if a dominating pair is found in a wDP graph or any of its subclasses (and that includes AT-free graphs), the diameter is at most 2 greater than the length of a shortest path between the DP nodes. In this section, assume the given graph is wDP. \n\n\\vspace{.5cm}\n\\textbf{Proposition 1.}\n\\textit{Let $x$ be a DP node. No vertex at distance 2 from $x$ can extend $x$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Every vertex is dominated by a shortest path $P = P_{x \\rightarrow y}$. Notice $|P| = d_G(x,y)$. $P$ may not be extended to any vertex in $N^2[y] \\setminus P$ because such a vertex cannot be dominated by every dominating $x,y$-path, a contradiction. $\\square$\n\n\\iffalse\nWe choose some vertex $d \\in N[x]$ such that $\\{d\\} \\cup P$ is a shortest path of higher eccentricity. We modify $P := P + d$. If no such vertex exists, we are done because $a$ is the node of furthest distance from $y$. \n\\fi\n\n\\vspace{.5cm}\nThe following proof is follows directly from Proposition 1 and therefore strengthens the main idea of this section:\n\n\\vspace{.5cm}\n\\textbf{Proposition 2.}\n\\textit{Let $(u,v)$ and $(x,y)$ be dominating pairs in $G$. It must be that $d_G(u,v) \\leq d_G(x,y) + 2$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let $(x,y)$ be a DP whose distance is minimum amongst dominating pairs. We assume that $d_G(u,v) > d_G(x,y) + 2$. Either $u$ or $v$ will not be dominated by a shortest path $P = P_{x \\rightarrow y}$ because such a dominating path cannot cover any shortest path of size $|P| + 3$. Let $u' \\in N(u)$ and $u' \\in P$. We are led to the existence of a path $Q = \\{u,u',...,v\\}$ that is shorter than the assumed length of $d_G(u,v)$, a contradiction. $\\square$\n\n\\vspace{.5cm}\n\n\\textbf{Theorem 1.} \\textit{Given some DP $(x,y)$, the diameter of $G$ is bounded by the following inequality:}\n\\vspace{.25cm}\n\n~~~~~~~~~~~~\\textit{$d_G(x,y) \\leq diam(G) \\leq d_G(x,y) + 2$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} The proof follows from Proposition 1 and Proposition 2. We check whether $x$ can be extended by some vertex in $N[x]$ and similarly check $y$. At most, an $x,y$-path can be extended twice when neither $x$ nor $y$ are diametral nodes. $\\square$\n\n\n\\vspace{.25cm}\n\\section{Partitioning Neighborhoods of DP Node Sets}\n\nHaving determined upper bounds of diameter on wDP graphs, we look more closely at the neighborhoods of dominating pairs. For the following, by $(X,Y)$ we refer to the nodes which make up $X \\cup Y$. By $(x,y) \\in (X,Y)$ we refer to a DP s.t. $x \\in X$ and $y \\in Y$. Sets $X$ and $Y$ may be maximum, disjoint, and unique (when diameter is at least 4) or maximal (when diameter is less than 4). For the latter class of graphs, there is no guarantee that any DP in $(X,Y)$ contains a diametral pair, as demonstrated by Fig. 2.\n\nTo better understand which vertices in the open neighborhood of DP node $x$ are diametral vertices, we will partition $N[x]$ into sets, or \\textit{levels}. Level 1 nodes form the set $L_1(x)$. Level $2$ nodes form the set $L_2(x)$. Level $3$ nodes form the set $L_3(x)$. A level 1 node is any vertex in $N[V \\setminus N[x]]$. Intuitively, level 1 nodes lie on the ``border\" of $N(x)$ and the rest of the graph. A level $2$ node is any vertex in $(N[x] \\setminus L_1(x)) \\cap N(L_1(x))$. Level $2$ nodes may not be adjacent to any nodes outside of $N[x]$. The remaining nodes in $N[x]$ are level $3$ nodes. Level $3$ nodes are all adjacent to level $2$ nodes and may not be adjacent to any level 1 nodes. Level $3$ nodes lie at the furthest extremities of the graph. Obviously, $x$ is always level $2$. Formally:\n\\vspace{.25cm}\n\n$L_1(x) = \\{z \\in N(x) : z$ \\textit{has a neighbor in} $V \\setminus N[x]\\}$\n\\vspace{.25cm}\n\n$L_2(x) = \\{z \\in N[x] \\setminus L_1(x) : z$ \\textit{has a neighbor in} $L_1(x)\\}$\n\\vspace{.25cm}\n\n$L_3(x) = N[x] \\setminus (L_1(x) \n\\cup L_2(x))$\n\n\\vspace{.25cm}\nHence, $\\{L_1(x), L_2(x), L_3(x)\\}$ is a partition of $N[x]$. All the levels can be determined by a simple BFS or by set union and set minus operators after a return from \\textsc{get-polar-pair}.\n\n\n\\begin{figure}[t]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (DP) at (-.3,0) {$x$};\n \\end{scope}\n \\node[vertex] (2*) at (-.3-1,1) {$2$};\n \\node[vertex] (1) at (-.2,1-.2) {$1$};\n \\node[vertex] (1'') at (1+.15 ,-.2) {$1$};\n \\node[vertex] (2) at (-.25,-1+.2) {$1$};\n \\node[vertex] (3) at (-1-.3,-1) {$2^*$};\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (0,1) {$ $};\n \\node[vertex] (c) at (1,0) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (cc) at (2,0) {$ $};\n \\node[vertex] (dd) at (2,1) {$ $};\n \\node[vertex] (ccc) at (3,0) {$ $};\n \\node[vertex] (ddd) at (3,1) {$ $};\n \n \n \\node[vertex] (e) at (-1,1) {$ $};\n \\node[vertex] (f) at (0,-1) {$ $};\n \\node[vertex] (f') at (1,-1) {$ $};\n \\node[vertex] (g) at (-1,-1) {$ $};\n \n \\draw(a)--(b);\n \\draw(a)--(e);\n \\draw(a)--(g);\n \\draw(a)--(f);\n \\draw(a)--(c);\n \\draw(c)--(d);\n \\draw(c)--(cc);\n \\draw(d)--(dd);\n \\draw(cc)--(dd);\n \\draw(cc)--(ccc);\n \\draw(cc)--(d);\n \\draw(dd)--(ddd);\n \\draw(ccc)--(ddd);\n \\draw(e)--(b);\n \\draw(b)--(d);\n \n \\draw(e)--(c);\n \\draw(f')--(c);\n \\draw(f')--(f);\n \\draw(f')--(cc);\n \\draw(g)--(f);\n \\draw(f)--(c);\n \\draw(f)--(e);\n \n \n \n \\draw(f) to[out=60,in=-60] (b);\n \\draw(f') to[out=60,in=-60] (d);\n\n \\end{tikzpicture}\n \\\\\n \\caption{An AT-free graph for which $L_3(x) = \\emptyset$ yet $L_2(x)$ has a $2^*$ node.}\n\n\\end{figure}\n\n\\vspace{.5cm}\n\\textbf{Proposition 3.} \\textit{Every level $3$ node achieves the diameter.}\n\n\\vspace{.5cm}\n\\textit{Proof.} A shortest path from $y$ to a vertex in $L_3(x)$ will certainly traverse $x$\\textemdash call this path $Q$. We can be certain there are no shortest $y,L_3(x)$-paths that traverse $N(x)$ and reach a level 3 node with distance less than $|Q|$. If $N(x)$ can be traversed in one or two hops, then some node in $L_3(x)$ actually lies in $L_2(x)$ or $L_1(x)$, respectively\\textemdash a contradiction. Therefore, nodes in $L_3(x)$ are $d_G(x,y)+2$ away from some node in $N[y]$, assuming $y$ is not a diametral node. This hits the upper limit of diameter as proven in Theorem 1. Otherwise, $y$ is a diametral node, meaning all nodes in $L_3(x)$ are $d_G(x,y)+1$ away from $y$ and can be used to extend $x$. $\\square$\n\n\\vspace{.5cm}\nAdding a node to Fig. 3 that is incident to only $2^*$ and $x$ creates a level $3$ node. Otherwise, it is possible $L_3(x) = \\emptyset$ and the problem of finding the diameter depends more crucially on examining $L_2(x)$. Henceforth, we will partition nodes in $L_2(x)$ into those whose neighborhoods are entirely made of DP nodes and those which are not. We will call the former set of nodes $2^*$ and say that $L_2^*(x) \\subseteq L_2(x)$ where any $z \\in L_2^*(x)$ is a $2^*$ node.\n\n\\vspace{.25cm}\n\n$L_2^*(x) = \\{z \\in N[x] \\setminus L_1(x) : N[z] \\subseteq X\\}$\n\n\\vspace{.25cm}\n\n\n\\vspace{.5cm}\n\\textbf{Proposition 4.} \\textit{Every $2^*$ node achieves the diameter.}\n\n\\iffalse\nA $2^*$ node does not have an edge to any vertex that is not a DP node. Intuitively, any path attempting to reach $2^*$ nodes may not ``skip\" over the DP layer of the graph. Let $s$ be one such $2^*$ node. Now $s \\in N[x]$ where $x$ is a DP node within polar set $X$. Note $x \\neq s$ and that $x$ does not form a diametral pair with any node in polar set $Y$. Also, $x$ is not $2^*$. According to Proposition 1, $x$ can be extended by one. Obviously, $x$ is adjacent to some level 1 node $v$ because $x$ is not $2^*$. However, $s$ is not adjacent to $v$ by the definition of $2^*$. Hence, $d_G(s,v) = 2 > d_G(s,x)$. Every path from nodes in $Y$ to $x$ is shorter than some path from $y \\in Y$ to $s$. Therefore, $s$ extends $x$ because $ecc(s) > ecc(x)$ and $s$ is a diametral node. $\\square$\n\\fi\n\n\\vspace{.5cm}\n\\textit{Proof.} A $2^*$ node does not have an edge to any vertex that is not a DP node. Intuitively, any path attempting to reach $2^*$ nodes may not ``skip\" over the DP layer of the graph. Given a polar pair $(X,Y)$ and some vertex $x \\in X$, let us partition $N[x]$ into three levels. Let $s \\in L_2^*(x)$. Note that $x \\neq s$ and that $x$ does not form a diametral pair with any node in polar set $Y$. Also, $x \\notin L_2^*(x)$. According to Proposition 1, $x$ can be extended by one. Obviously, $x$ is adjacent to some non-DP node $v$ because $x$ is not $2^*$. However, $s$ is not adjacent to $v$ because $s$ \\textit{is} $2^*$. Hence, $d_G(s,v) = 2 > d_G(s,x)$. Every path from nodes in $Y$ to $x$ is shorter than some path from $y \\in Y$ to $s$. Therefore, $s$ extends $x$ because $ecc(s) > ecc(x)$ and $s$ is a diametral node. $\\square$\n\n\\vspace{.5cm} The set $L_2^*(x)$ can be calculated in linear time by the use of set intersection and set minus operations on sets $L_1(x)$ and $L_2(x)$, given any DP node $x$. This can be done in $O(n)$ by employing hash tables. \n\nThe problem of finding a node that achieves the diameter becomes more difficult when $L_3(x) = \\emptyset$ and $L_2^*(x) = \\emptyset$. In this scenario, we cannot determine the diameter by looking only at a polar set $X$. Fig. 5 shows that edge $e$, which lies outside of $N[X]$, is vital in determining the diametral pair. If $e$ was within $N[X]$ instead, then the diametral node could be easily discovered. For the following proof, assume $L_3(x) = \\emptyset$ and $L_2^*(x) = \\emptyset$ for any DP node $x$.\n\n\n\\begin{figure}[b]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (ss) at (0,-.4) {$s$};\n \\node[vertex] (dd) at (1,1.4) {$d$};\n \\node[vertex] (d') at (5,-.4) {$d'$};\n \n \\node[vertex] (v) at (2,1.4) {$v$};\n \\node[vertex] (u) at (3,1.4) {$u$};\n \\node[vertex] (w) at (3,-.4) {$w$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black!30]\n\n \\node[vertex] (s) at (0,0) {$ $};\n \\node[vertex] (a) at (0,1) {$ $};\n \\node[vertex,black] (d) at (1,1) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex,black] (c) at (2,1) {$ $};\n \\node[vertex] (e) at (2,0) {$ $};\n \\node[vertex] (f) at (3,1) {$ $};\n \\node[vertex,black] (g) at (3,0) {$ $};\n \\node[vertex] (h) at (4,1) {$ $};\n \\node[vertex,black] (i) at (4,0) {$ $};\n \\node[vertex] (j) at (5,1) {$ $};\n \\node[vertex,black] (k) at (5,0) {$ $};\n \n \\draw(s)edge(a);\n \\draw(a)edge(d);\n \\draw(d)edge(b);\n \\draw(s)edge(b);\n \\draw(c)edge(e);\n \\draw[black ,dashed](c)--(d);\n \\draw[dashed](e)edge(b);\n \\draw(f)edge(g);\n \\draw(f)edge(c);\n \\draw(g)edge(e);\n \\draw(h)edge(i);\n \n \\draw[ black](g)edge(c);\n \\draw[dashed](f)edge(h);\n \\draw[dashed,black](g)edge(i);\n \\draw(j)edge(k);\n \\draw(j)edge(h);\n \\draw[ black](i)edge(k);\n\n \\end{tikzpicture}\n \\\\\n \\caption{Exemplifying Proposition 5. Path $R$ shown in black.}\n\n\\end{figure}\n\n\\vspace{.5cm}\n\\textbf{Proposition 5.} \\textit{If $G$ is AT-free, some diametral node is a DP node.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Assume otherwise. Let $(d,d')$ be any diametral pair in $G$ while there exists a shortest $d,d'$-path that is not dominating. Let vertex $s$ not be dominated and $s \\in N^2[d]$. If $s$ is not a DP node, then $s$ causes the creation of an asteroidal triple $\\{d,d',s\\}$, a contradiction. Now, assume a shortest $d,d'$-path $D$ is of length $l$ while a shortest $d',s$-path $S$ is of length $l-1$. This implies $l$ is the diameter. By pigeonhole principle, there are not enough nodes along diametral dominating path $D$ for every node to share exactly one edge between all nodes along path $S$. Some node $w$ along $S$ is adjacent to two nodes along $D$. Let $w$ be adjacent to $v$ and $u$ such that $v$ is closer to $d$. The length of $D$ is reduced by 1 due to the existence of path $R = \\{d',...,w,v,...,d\\}$. See Fig. 4. Clearly, $|R|$ equals $|S|$, meaning $s$ becomes a diametral node while also being a DP node when $G$ is AT-free. $\\square$\n\n\\vspace{.5cm}\nOf course, we also must assure ourselves that the set of DP nodes which was returned by \\textsc{get-polar-pair} truly contains a diametral node. In Fig. 2, polar pair $(A',B')$ does not contain a diametral pair. Therefore, there is no guarantee for graphs where $|\\mathscr R| > 1$ that the only discovered polar pair will contain a node that achieves the diameter. Can the polar pair we require be discovered in linear time regardless of the cardinality of $\\mathscr R$?\n\n\n\\begin{figure}[t]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (ee) at (.4,1.7) {$e$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (0,1) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (0,2) {$ $};\n \\node[vertex] (f) at (1,2) {$ $};\n \\node[vertex] (g) at (0,3) {$ $};\n \\node[vertex] (h) at (1,3) {$ $};\n \n \\draw(a)edge(b);\n \\draw(a)edge(c);\n \\draw(b)edge(d);\n \\draw(c)edge(d);\n \\draw(c)edge(e);\n \\draw(d)edge(f);\n \\draw(e)edge(f);\n \\draw(e)edge(g);\n \\draw(f)edge(h);\n \\draw(g)edge(h);\n \n \\draw(c)edge(f);\n \n \\draw[thick,gray!60,dashed] (.5,.5) ellipse (1cm and 1cm);\n\n \\end{tikzpicture}\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (X) at (-1.2,.5) {$N(X)$};\n \\node[vertex] (ee) at (.4,.7) {$e$};\n \\node[vertex] (r) at (1.2,.2) {$v$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (0,1) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (0,2) {$ $};\n \\node[vertex] (f) at (1,2) {$ $};\n \\node[vertex] (g) at (0,3) {$ $};\n \\node[vertex] (h) at (1,3) {$ $};\n \n \\draw(a)edge(b);\n \\draw(a)edge(c);\n \\draw(b)edge(d);\n \\draw(c)edge(d);\n \\draw(c)edge(e);\n \\draw(d)edge(f);\n \\draw(e)edge(f);\n \\draw(e)edge(g);\n \\draw(f)edge(h);\n \\draw(g)edge(h);\n \n \\draw(a)edge(d);\n \n \\draw[thick,gray!60,dashed] (.5,.5) ellipse (1cm and 1cm);\n\n \\end{tikzpicture}\n \n \\caption{$G$ (\\textit{left}) and $H$ (\\textit{right}) have the same diametral node within $N(X)$. Only in $H$ is $v$ a $2^*$ node due to the placement of edge $e$.}\n\n\\end{figure}\n\n\\vspace{.5cm}\n\\textbf{Proposition 6.} \\textit{When $|\\mathscr R| > 1$, the polar pair that contains a diametral node can be found in linear time.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let $G$ be AT-free and let $(X,Y)$ be the polar pair returned by a call to \\textsc{get-polar-pair} and which does not contain a node that achieves the diameter. Also, let $(x,y) \\in (X,Y)$ be a dominating pair. A single run of BFS beginning from some node within $X$ will always find a longer shortest path to some node $b$ outside of $Y$ due to the obvious fact that \\textit{none} of the nodes in $X$ or $Y$ achieve the diameter. By contradiction, let $(a,b) \\in (A,B)$ where $(A,B)$ is a maximal polar pair that does not contain $(x,y)$ and also does not contain the diametral pair $(d,d')$. So, $d_G(d,d') = d_G(a,b) + 1 = d_G(x,y) + 2$. However, path $P = \\{d,a,...,b\\}$ has the same geodesic length as $d_G(d,d')$. We add an edge from $d$ to $v$ where $v \\in P_{a \\rightarrow b}$ to prevent this contradiction. Likewise, $d_G(d',a)$ is equal to $d_G(d,d') + 1$, a contradiction to Theorem 1. We add an edge from $d'$ to $u \\in P_{a \\rightarrow b}$ to resolve this. It must be that $v$ and $u$ are adjacent to $a$ and $b$, respectively, otherwise $d_G(d,d') = d_G(a,b)$. Given also that $x \\in N(d)$ and $y \\in N(d')$ by Proposition 1, we see that the closed neighborhoods of $d$ and $d'$ are a subset of the closed neighborhoods of $a$ and $b$, respectively. Otherwise, assume by contradiction there is some node $v'$ that is incident to $d$ but is not incident to $a$. Assume $u'$ exists and is incident to $d'$ but not to $b$. The path $D = \\{d,v',...,u',d'\\}$ is not completely dominated by path $S = \\{a,v,...,u,b\\}$ unless some node in $S$ is adjacent to two or more nodes in $D$ with reasoning similar to to the pigeonhole principle described in Proposition 5. Hence, when $a$ and $b$ are DP, $d$ and $d'$ are concurrently DP and lie within the same maximal polar pair $(A,B)$. $\\square$\n\n\\vspace{.5cm}\nAll in all, we begin with a call to \\textsc{get-polar-pair}. Regardless of whether it results in a polar pair that contains a diametral pair, we run BFS from some DP node, thereby finding a path of higher eccentricity if it exists\\textemdash call its endpoint $v$. From there, we perform a second call of \\textsc{get-polar-pair} beginning with the removal of $v$.\n\n\\vspace{.5cm}\nIt is not necessary to know the cardinality of $\\mathscr R$. If $|\\mathscr R| = 1$, any call to \\textsc{get-polar-pair} will always return a polar pair that contains a diametral pair. To better understand the neighborhood of our diametral node, we state and prove the following:\n\n\n\\vspace{.5cm}\n\\textbf{Proposition 7.} \\textit{A diametral node lies in every maximal clique of dominating pair nodes.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let $G$ have a polar set $X$ where $X$ contains a diametral node. Arbitrarily choose a DP node $x$ that belongs to two maximal cliques $C_1$ and $C_2$. Both cliques have only DP nodes because $C_1 \\subseteq X$ and $C_2 \\subseteq X$. Let $x' \\in C_1$ and $C_2' = C_2 \\setminus N[C_1]$. If the only diametral node is in $C_2'$, we must extend $x'$ by 2 to find it, a contradiction to Proposition 1. Hence, any maximal clique within a polar set certainly contains a diametral node. $\\square$\n\n\\vspace{.5cm} As implied by Proposition 7, we do not need to consider the entirety of a polar set when searching for the diametral node. Recall the definition of a maximal polar set from Section III. Certainly some vertex within any maximal clique of DP nodes realizes the diameter. Hence, in our examples and the following proofs we will represent polar sets as single maximal cliques of dominating pair nodes, rather than the entirety of a polar set. \n\n\\vspace{.25cm}\n\\section{A Linear Time Algorithm}\n\nFor all following proofs, assume that a second call to \\textsc{get-polar-pair} has returned a polar pair $(X,Y)$ that contains a vertex that achieves the diameter. \n\n\\iffalse\nWe want to develop an algorithm that is able to ``force\" certain level $2$ nodes to become $2^*$ based on the existence of certain edges outside of the neighborhoods of $X$ or $Y$. \n\nWe define an \\textit{unimportant edge} as an edge that can be moved from $G \\setminus N[X]$ to $N[X]$ without changing the diametral nodes of $G$. For any unimportant edge $e$ that has been added to $N[X]$, we require that one endpoint of $e$ be in $X$ while the other endpoint be in $N(X)$. Looking closely at Fig. 5, we argue that, with respect to diameter, $G$ and $H$ are the same graph with a different configuration of unimportant edges. Edge $e$ might as well exist within $X$ and the diametral node does not change. By the placement of edge $e$ in $H$, both endpoints of $e$ become DP nodes which in turn makes $v$ become $2^*$. \n\\fi\n\nLet us assign an initial set of labels $\\mathcal L$ to every node in $X$. By utilizing a modified BFS, we can cause all the labels to ``trickle up\" to $Y$. At every step of this modified BFS, we check every unassigned node in the neighborhood of the recently assigned nodes\\textemdash to each such unassigned node, we assign a set of labels equal to the union of all neighboring assigned labels. Say that it requires $h$ hops of \\textsc{label} to assign any node of $Y$ a label set. Because $Y$ is a clique, it requires one additional hop to propagate a label set to the remaining, unassigned nodes. Let $Y^*$ be a subset of $Y$ that is created by removing all nodes from $Y$ that have been assigned after $h$ hops. Clearly, all nodes in $Y^*$ have greater eccentricity than those in $Y \\setminus Y^*$. If $Y^*$ is empty, then we instead let $Y^* = Y$. \n\nGiven two labels $l_i,l_j \\in \\mathcal L$ that are assigned to nodes $i,j \\in X$, respectively, we say that $l_i$ \\textit{is stopped by} $l_j$ when every label set in $Y^*$ that contains $l_i$ also contains $l_j$.\n\n\\vspace{.5cm}\n\\textbf{Relative Eccentricity Lemma.} \\textit{If label $l_i$ is stopped by $l_j$ then the eccentricity of $i$ is less than or equal to $j$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let node $i$ be assigned label $l_i$ and node $j$ be assigned label $l_j$ s.t. $i,j \\in N(X)$ and $l_i,l_j \\in \\mathcal L$. We wish to prove that $ecc(i) \\leq ecc(j)$. We check whether $l_i$ is always contained in a label set in $Y^*$ that also contains $l_j$\\textemdash if so, then $l_i$ is stopped by $l_j$ because every shortest path that begins at $i$ shares a subpath with every shortest path that begins at $j$.\n\n\\textit{Scenario 1:} Label $l_j$ exists in a label set assigned to node $y \\in Y^*$ that does not contain $l_i$. A shortest path exists that begins at $j$ and reaches $y$ while every shortest path that begins at $i$ will require at least one more step to reach $y$. Node $i$ has higher eccentricity than $j$.\n\n\\textit{Scenario 2:} Alternatively, label $l_j$ exists in a label set if and only if $l_i$ exists in a label set. The stopping relationship between $l_i$ and $l_j$ is symmetric and the eccentricities of both vertices are equal. The eccentricity of $i$ is less than or equal to $j$. $\\square$\n\n\\begin{figure}[b]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (d1) at (-.1,-.35) {$l_0$};\n \\node[vertex] (d2) at (+1-.15,-.25) {$l_1$};\n \\node[vertex] (d3) at (2+.1,-.35) {$l_2$};\n \\node[vertex] (d4) at (1-.3,-1) {$l_3$};\n \n \\node[vertex] (l0) at (-.1,1) {$l_0,l_1$};\n \\node[vertex] (l1) at (1.8,0.9) {$l_1$};\n \\node[vertex] (l2) at (2.5+.5,1.4) {$l_2,l_3$};\n \n \\node[vertex] (l02) at (-.1,2) {$l_0,l_1$};\n \\node[vertex] (l01) at (1.5+.3+.5,2.1) {$l_1,l_2,l_3$};\n \n \\node[vertex] (l02') at (.2-.7,3) {$l_0,l_1,l_2,l_3$};\n \\node[vertex] (l012) at (2.3,3) {$l_1,l_2,l_3$};\n \n \\node[vertex] (l012'top) at (-1-.2,4) {$l_0,l_1,l_2,l_3$};\n \\node[vertex] (l012'top) at (1,4.3) {$l_0,l_1,l_2,l_3$};\n \\node[vertex] (l02''top) at (3,4) {$l_1,l_2,l_3$};\n \n \\node[vertex] (X) at (-.8,-.2) {$X$};\n \\node[vertex] (Y) at (2.3,5) {$Y = Y^*$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,.2) {$ $};\n \\node[vertex] (c) at (2,0) {$ $};\n \\node[vertex] (d) at (1,-1) {$ $};\n \\node[vertex] (e) at (.5,1) {$ $};\n \\node[vertex] (f) at (1.5,1) {$ $};\n \\node[vertex] (g) at (2.5,1.5) {$ $};\n \\node[vertex] (h) at (.5,2) {$ $};\n \\node[vertex] (i) at (1.5,2) {$ $};\n \\node[vertex] (j) at (.5,3) {$ $};\n \\node[vertex] (k) at (1.5,3) {$ $};\n \\node[vertex] (l) at (1,4-.2) {$ $};\n \\node[vertex] (m) at (0,4) {$ $};\n \\node[vertex] (n) at (2,4) {$ $};\n \n \\draw(a)--(b);\n \\draw(b)--(c);\n \\draw(a)--(c);\n \\draw(a)--(d);\n \\draw(b)--(d);\n \\draw(c)--(d);\n \\draw(a)--(e);\n \\draw(b)--(e);\n \\draw(b)--(f);\n \\draw(e)--(f);\n \n \\draw(g)--(c);\n \\draw(g)--(d);\n \n \n \n \\draw(g)--(i);\n \\draw(g)--(e);\n \\draw(g)--(f);\n \n \n \n \n \n \n \n \n \\draw(h)--(i);\n \\draw(h)--(e);\n \\draw(f)--(i);\n \n \\draw(j)--(k);\n \\draw(j)--(e);\n \\draw(k)--(i);\n \n \\draw(l)edge(j);\n \\draw(l)edge(k);\n \\draw(l)edge(m)edge(n);\n \\draw(m)--(j);\n \\draw(m)--(n);\n \\draw(n)--(k);\n \n \n \\draw(i)--(j);\n \n \\draw[thick,gray!60,dashed] (1,-.25) ellipse (1.4cm and 1cm);\n \\draw[thick,gray!60,dashed] (1,4) ellipse (1.4cm and .7cm);\n\n \\end{tikzpicture}\n \\iffalse\n \\begin{tikzpicture}[scale=.9, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (blank) at (3,-2) {$ $};\n \n \\node[vertex] (aa) at (0-.3,-.2) {$l_0$};\n \\node[vertex] (bb) at (1,1.5+.4) {$l_1$};\n \\node[vertex] (cc) at (2+.3,-.2) {$l_2$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,1.5) {$ $};\n \\node[vertex] (c) at (2,0) {$ $};\n \n \n \\draw(a)--node[sloped,font=\\LARGE] {$ > $}(b);\n \\draw(a)--node[sloped,font=\\LARGE] {$ > $}(c);\n \\draw(b) to[out=-15,in=100] node[sloped,font=\\LARGE] {$ < $}(c);\n \n \\draw(b) to[out=-75,in=150] node[sloped,font=\\LARGE] {$ > $}(c);\n\n\n \\end{tikzpicture}\n\n \n \n \\fi\n \n \\caption{Labels are generated by running \\textsc{label}. $\\mathcal L = \\{l_0,l_1,l_2,l_3\\}$. Only initial label $l_0$ is missing in $Y^*$ and the node assigned $l_0$ is the only diametral node in $X$.}\n\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (l0) at (-.4,0) {$l_0$};\n \\node[vertex] (l1) at (1.25,.2) {$l_1$};\n \\node[vertex] (l2) at (2.4,0) {$l_2$};\n \n \\node[vertex] (l0') at (-.4,1) {$l_0$};\n \\node[vertex] (l1') at (1.35,1) {$l_1$};\n \n \\node[vertex] (l0'') at (-.4+.2,2) {$l_0,l_1$};\n \n \\node[vertex] (l0''') at (-.4+.2,3) {$l_0,l_1$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (0,1) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (0+.5,2) {$ $};\n \\node[vertex] (g) at (0+.5,3) {$ $};\n \\node[vertex] (i) at (2,0) {$ $};\n \n \\draw(a) to[out=-37,in=217] (i);\n \\draw(b) to[out=-20,in=200] (i);\n \n \\draw(a)edge(b);\n \\draw(a)edge(c);\n \\draw(b)edge(d);\n \\draw(c)edge(d);\n \\draw(c)edge(e);\n \\draw(e)edge(g);\n \\draw(d)edge(e);\n \n \n\n \\end{tikzpicture}\n ~~~~~~\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \n \n \n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \n \\node[vertex] (X) at (-1,0) {$X$};\n \\node[vertex] (Y) at (1.8,3.8) {$Y = Y^*$};\n \n \\node[vertex] (l0) at (-.4,0) {$l_0$};\n \\node[vertex] (l1) at (1.25,.2) {$l_1$};\n \\node[vertex] (l2) at (2.4,0) {$l_2$};\n \n \\node[vertex] (l0') at (-.4,1) {$l_0$};\n \\node[vertex] (l1') at (1.35,1.2) {$l_1$};\n \\node[vertex] (l2') at (2.4,1) {$l_2$};\n \n \\node[vertex] (l0'') at (-.4+.2,2) {$l_0,l_1$};\n \\node[vertex] (l1'') at (2.5-.3,2) {$l_1,l_2$};\n \n \\node[vertex] (l0''') at (-.4+.2,3) {$l_0,l_1$};\n \\node[vertex] (l1''') at (2.5-.3,3) {$l_1,l_2$};\n \n \\node[vertex] (blank) at (3,3) {$ $};\n \n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (0,1) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (0+.5,2) {$ $};\n \\node[vertex] (f) at (1+.5,2) {$ $};\n \\node[vertex] (g) at (0+.5,3) {$ $};\n \\node[vertex] (h) at (1+.5,3) {$ $};\n \\node[vertex] (i) at (2,1) {$ $};\n \\node[vertex] (j) at (2,0) {$ $};\n \n \\draw(a)edge(b);\n \\draw(a)edge(c);\n \\draw(b)edge(d);\n \\draw(c)edge(d);\n \\draw(c)edge(e);\n \n \\draw(d)edge(f);\n \\draw(e)--(f);\n \\draw(e)edge(g);\n \\draw(f)edge(h);\n \\draw(g)edge(h);\n \n \n \n \n \n \\draw(d)--(e);\n \\draw(i)--(f);\n \\draw(j)--(i);\n \\draw(j)--(b);\n \\draw(i)--(d);\n \n \\draw(a) to[out=-30,in=210] (j);\n \\draw(c) to[out=-30,in=210] (i);\n \n \\draw[thick,gray!60,dashed] (1,0) ellipse (1.7cm and .5cm);\n \\draw[thick,gray!60,dashed] (1,3) ellipse (.8cm and .4cm);\n\n \\end{tikzpicture}\n ~~~~~~\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (l0) at (-.4-1,0) {$l_0$};\n \\node[vertex] (l1) at (1-1-.3,.25) {$l_1$};\n \\node[vertex] (l2) at (2.4-1,0) {$l_2$};\n \n \\node[vertex] (l1') at (1-1-.3,1) {$l_1$};\n \\node[vertex] (l2') at (2.4-1,1) {$l_2$};\n\n \\node[vertex] (l1'') at (2.5-.3-1,2) {$l_1,l_2$};\n \n \\node[vertex] (l1''') at (2.5-.3-1,3) {$l_1,l_2$};\n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (0,1) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (0+.5,2) {$ $};\n \\node[vertex] (g) at (0+.5,3) {$ $};\n \\node[vertex] (i) at (-1,0) {$ $};\n \n \\draw(a) to[out=200,in=-20] (i);\n \\draw(b) to[out=217,in=-37] (i);\n \n \\draw(a)edge(b);\n \\draw(a)edge(c);\n \\draw(b)edge(d);\n \\draw(c)edge(d);\n \\draw(c)edge(e);\n \\draw(e)edge(g);\n \\draw(d)edge(e);\n \n \n\n \\end{tikzpicture}\n \n \\caption{A visual proof of Case 1 in the Missing Label Lemma. Nodes assigned labels $l_2$ and $l_0$ have maximum eccentricity.}\n\n\\end{figure*}\n\n\nFinding the stopping relationship of $\\mathcal L$ is \\textit{not} linear because we have to check all pairs of labels $l_i$ and $l_j$. The number of nodes in $Y$ is bounded by $n$, and each node can have a label set that contains $|\\mathcal L|$ labels where the number of labels is also bounded by $n$. Looping through $i \\leq n$ and $j \\leq n$ demands $O(n^2)$. Fortunately, finding the stopping relationship of all labels is not needed to determine the diameter.\n\n\\vspace{.5cm}\n\\textbf{Missing Label Lemma.} \\textit{After completing \\textsc{label}, node $i$ achieves the diameter if it is missing in any label set of $Y^*$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} It may be that some subsets from $\\{l_j,...,l_x,...,l_k\\} \\setminus \\{l_i\\}$ contain labels which stop one another. Assume a label subset of $Y$ contains $l_x$ but does not contain $l_i$. Let $(x,y)$ be a DP. We will break this proof into three cases based on the number of hops it requires label $l_x$ to reach polar set $Y$.\n\n\\textit{Case 1}: Label $l_x$ requires $h$ hops to reach every node in polar set $Y$. Hence, $Y^* = \\emptyset$, so we set $Y^* = Y$. Let $\\mathscr P$ be the set of all paths from $x$ to all nodes in $Y^*$. Because $l_x$ is incident on all nodes which were assigned labels from $\\mathcal L$, any path in $\\mathscr P$ forms a longer shortest path with $l_i$ as an endpoint. Through any such path, it requires $h+1$ hopes to reach $Y^*$ from $l_i$, which is equal to any other shortest $i,Y$-path that does not traverse $x$. In fact, every path in $\\mathscr P$ is contained in a diametral path with respect to every node in $\\mathcal L$ that is missing in some label set of $Y^*$.\n\n\\textit{Case 2}: Label $l_x$ requires $h$ hops to reach polar set $Y$, but $l_x$ reaches some DP node $y \\in Y$ prior to reaching the remaining DP nodes that make up $N(y)$. Hence, it requires $h + 1$ hops to reach some subset of $Y$. At $h$ iterations of \\textsc{label}, we can easily determine which nodes of $Y$ are yet unassigned\\textemdash we set this subset equal to $Y^*$. The diametral node must be contained in $Y^*$. Next, assume $l_i$ is missing in a label subset of $Y^*$ that contains $l_x$, meaning that node $i$ extends the eccentricity of node $x$ by 1. Hence, total eccentricity between DP $(x,y)$ can be extended in both polar sets, hitting the upper limit of diameter from Proposition 3. If $Y^* = \\emptyset$, we are led to the final case. \n\n\\textit{Case 3}: Label $l_x$ requires $h$ hops to reach $y$, but is unable to reach $N(y) \\subset Y$ because all nodes in $N(y)$ have been assigned other label sets. In other words, $Y^*$ is empty because all nodes in $Y$ have been assigned after $h$ hops of \\textsc{label}. If this remains true for every label, the stopping relationship is empty and all vertices in $X$ are diametral.\n\nUltimately, the stopping relationship of $\\mathcal L$ is not vital to discover. Assuredly, $l_i$ has maximum eccentricity in any case. $\\square$\n\n\\vspace{.5cm}\nFig. 6 is an example of Case 1 in the preceding proof as labels $l_1,l_2,l_3$ reach every node in $Y$ in four hops.\n\nFig. 7 also demonstrates Case 1 because $l_1$ reaches all nodes in $Y$ in three hops. Consider the \\textit{center} graph to be the original graph $G$. The \\textit{left} and \\textit{right} graphs are subgraphs of $G$ that are missing $l_2$ and $l_0$, respectively, from the label sets belonging to $Y^*$. Although $l_1$ stops both $l_0$ and $l_2$, it is evident that stopping relationship does not ultimately matter in finding diameter since we only need to check for a single missing label.\n\nFig. 8 exemplifies Case 2 because $l_0$ reaches $Y$ in one hop (so $h = 1$), earlier than the other labels. After $h + 1$ hops, all nodes in $Y$ have become assigned because $Y$ is restricted to being a clique.\n\n\\vspace{.5cm}\n\n\n\\begin{figure}[b]\n \\centering\n \\begin{tikzpicture}[scale=1.0, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (X) at (3.4,0) {$X$};\n \\node[vertex] (Y) at (-1.4,3) {$Y$};\n \\node[vertex] (Y) at (1.5,3-.2) {$Y^*$};\n \\node[vertex] (l0) at (-.4,0) {$l_0$};\n \\node[vertex] (l1) at (1,-.45) {$l_1$};\n \\node[vertex] (l2) at (2.4,0) {$l_2$};\n \n \\node[vertex] (l1') at (.7,1) {$l_1$};\n \\node[vertex] (l2') at (2.6,1) {$l_1,l_2$};\n \\node[vertex] (l0') at (-.4,3) {$l_0$};\n \n \\node[vertex] (l2'') at (2.8,2) {$l_0,l_1,l_2$};\n \\node[vertex] (l01'') at (2.6,3) {$l_0,l_1$};\n \\node[vertex] (l10'') at (1,3.55) {$l_0,l_1$};\n \n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (1,0) {$ $};\n \\node[vertex] (c) at (2,0) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (2,1) {$ $};\n \\node[vertex] (f) at (2,2) {$ $};\n \n \\node[vertex] (g) at (0,3) {$ $};\n \\node[vertex] (h) at (1,3) {$ $};\n \\node[vertex] (i) at (2,3) {$ $};\n \n \\draw(a)--(g);\n \\draw(g)--(h);\n \\draw(g) to[out=20,in=-200](i);\n \\draw(h)--(i);\n \n \\draw(d)--(h);\n \\draw(d)--(i);\n \\draw(d)--(f);\n \\draw(g)--(f);\n \\draw(f)--(i);\n \\draw(b)--(d);\n \\draw(c)--(e);\n \\draw(b)--(e);\n \\draw(e)--(f);\n \\draw(d)--(g);\n \\draw(e)--(g);\n \n \\draw(a) to[out=-20,in=200] (c);\n \\draw(b)--(c);\n \\draw(a)edge(b);\n \n \\draw[thick,gray!60,dashed] (1,0) ellipse (2cm and .7cm);\n \\draw[thick,gray!60,dashed] (1,3) ellipse (2cm and .8cm);\n \\draw[thick,gray!60,dashed] (1.5,3) ellipse (.75cm and .38cm);\n\n \\end{tikzpicture}\n \n \\caption{Case 2 of the Missing Label Lemma solves the diameter. Only $l_2$ in $X$ is diametral because it is the only missing label in $Y^*$. Diameter is 3.}\n\n\\end{figure}\n\nWe will represent label sets with bitwise arrays. If a label set contains a label $l_f$, the $f$-th index of the array contains a 1, otherwise a 0. For example, for $\\mathcal L = \\{l_0,...,l_2\\}$, a set containing $\\{l_1,l_2\\}$ will be represented as array $[0 | 1 | 1]$. This allows the union of two label sets to be taken in constant time by using the \\textsc{or} operator. In Fig. 7 (\\textit{center}), set $Y$ contains label sets $\\{l_0,l_1\\}$ and $\\{l_1,l_2\\}$ with bitwise arrays $[1 | 1 | 0]$ and $[0 | 1 | 1]$, respectively. Using \\textsc{and} returns $[0 | 1 | 0]$, meaning $l_0$ and $l_2$ are missing labels with respect to some subset in $Y^*$.\n\n\\vspace{.5cm}\n\\textbf{Theorem 2.} \\textit{The time-complexity of calling \\textsc{get-polar-pair}, \\textsc{label} and \\textsc{find-missing} is $O(n + m)$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Finding a polar pair that contains a diametral node may take two calls of \\textsc{get-polar-pair}, requiring $O(2n + 2m)$, and one call of BFS, taking $O(n + m)$, as proven by Proposition 6. We check that there are no level $3$ or $2^*$ nodes in $O(n)$, otherwise diameter is found. We can easily accomplish the task of finding a maximal clique $X$ within a polar set in $\\Theta(n + m)$ by searching the neighborhood of any DP node and comparing edge incidence. The same can be done on the other polar set to find maximal clique $Y$. The nodes of $X$ make up the labels of $\\mathcal L$. Since \\textsc{label} is a modified BFS, computational complexity is $O(n + m)$. We can take the union of all previously assigned labels by using the \\textsc{or} operator on their respective bitwise arrays at every step of \\textsc{label} and on every edge incident to an unassigned node. Propagating labels to all nodes in the graph requires $O(m)$. Once any single label has propagated to $Y$, we let $Y^*$ be the subset of nodes in $Y$ which have not yet been assigned at this point and which require one more hop to become assigned. Creating $Y^*$ can be done in $O(n)$ by simply looping through all nodes of $Y$ prior to the last hop of \\textsc{label} and checking for unassigned nodes. If $Y^* = \\emptyset$, then we let $Y^* = Y$ instead. The procedure for \\textsc{find-missing} is as follows: we find a missing label in $Y^*$ by using the \\textsc{and} operator between all bitwise arrays in $Y^*$\\textemdash requiring $O(n-1)$. If none of the initial labels of $\\mathcal L$ are missing in $Y^*$ then every node in $X$ achieves the diameter because they all \\textit{stop} one another and have equal eccentricities. Running each function as described requires a total time-complexity of $O(8n + 5m)$. Hence, the complete algorithm is $\\Theta(n + m)$. $\\square$\n\n\\vspace{.5cm}\nTo demonstrate the strength of our \\textsc{Label} algorithm against the more traditional \\textsc{lbfs} method, we refer back to an example found in $[2 ]$ where it is stated \"an infinite number of \\textsc{lbfs} sweeps never end at a maximum eccentricity vertex.\"\n\nFig. 9 is the same graph as drawn in $[2 ]$. After finding polar pair $(X,Y)$, and performing \\textsc{Label} to propagate labels $\\{l_0,l_1\\}$ from $X$ to $Y$, we see that the node assigned $l_0$ certainly achieves the diameter.\n\n\\vspace{.5cm}\n\\section{Relationship to Other Problems}\n\nIn this section, we describe the reasoning of how our algorithm improves the best known run-times of a multitude of graph theoretic problems. \n\nTheorem 3 of $[8 ]$ claims that the problem of finding a simplicial vertex in a general graph can be reduced to the diameter problem in AT-free graphs. This reduction is performed by constructing an AT-free graph $H$ from some original graph $G$ where $diam(H) = 4$, and then showing that a vertex $v$ in $H$ is diametral if and only if $v$ is simplicial. The wider implications of our algorithm become self-evident with the following proof:\n\n\n\n\n\n\\vspace{.5cm}\n\\textbf{Fast Simplicial Vertex Theorem.} \\textit{In $H$, any simplicial vertex is in $L_2^*(X)$ or $L_3(X)$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let $H$ be AT-free. Assume there exists a simplicial vertex $v$ that is not $2^*$. Some node $z \\in N(v)$ is not a DP node. A node $u$ exists that is dominated by $v$ but not by $z$. Since $u,z \\in N(v)$ but $u$ and $z$ do not share an edge, that means $N(v)$ does not form a complete, induced subgraph\\textemdash a contradiction. The same holds true if $L_3(X) \\neq \\emptyset$ since any node therein neighbors $L_2(X)$. Recall that all $L_2(X)$ nodes are DP nodes by the definition of polar set $X$. $\\square$\n \n\\vspace{.5cm}\nA polar pair can be found in $H$ by a single call to \\textsc{get-polar-pair} because $|\\mathscr R| = 1$ for AT-free graphs of diameter 4 or greater. If diameter is 3, we can determine this through the algorithm described in Theorem 2 or by verifying that $|\\mathscr R| > 1$ with two calls to \\textsc{get-polar-pair}. Let $X$ be one polar set returned and diameter determined to be 4. We form levels $L_1(X)$, $L_2(X)$, and $L_3(X)$ by using set operators as shown in Section V. Then, $L_2^*(X)$ can be found by utilizing set minus operations. That is: $\\{\\forall x \\in (X \\setminus N[V[H] \\setminus X]) : x \\in L_2^*(X)$ hence $x$ is $2^*\\}$.\n\nSince article $[8 ]$ shows a construction of the AT-free graph $H$ from the general graph $G$, it is worthwhile to revisit several related problems. Article $[9 ]$ states that finding a simplicial vertex can be done in $O(n^2)$ if the cube of the adjacency matrix is given. Due to our result and the reduction from $[8 ]$, we have shown the problem can be solved in $O(n^2)$ without having to compute the cube of the adjacency matrix. \n\n\n\\begin{figure}[t]\n \\centering\n \\begin{tikzpicture}[scale=1.5, every node\/.style={scale=.7}]\n \n \\tikzstyle{vertex}=[circle, fill=white]\n \\begin{scope}\\Large\n \\node[vertex] (X) at (-1.2,.5) {$X$};\n \\node[vertex] (Y) at (4.2,.5) {$Y$};\n \n \\node[vertex] (l0) at (0-.3,1) {$l_0$};\n \\node[vertex] (l1) at (0-.3,0) {$l_1$};\n \n \\node[vertex] (l0') at (3+.4,1) {$l_0,l_1$};\n \\node[vertex] (l1') at (3+.3,0) {$l_1$};\n \n \\node[vertex] (l0'') at (1,1+.3) {$l_0$};\n \\node[vertex] (l1'') at (1,0-.3) {$l_1$};\n \n \\node[vertex] (l0''') at (2,1+.3) {$l_0,l_1$};\n \\node[vertex] (l1''') at (2,0-.3) {$l_1$};\n \n \\node[vertex] (l1'''l) at (.5,.5+.25) {$l_1$};\n \\node[vertex] (l1'''r) at (2.6,.5-.25) {$l_0,l_1$};\n \n \\end{scope}\n \n \\tikzstyle{vertex}=[circle, fill=black]\n\n \\node[vertex] (a) at (0,0) {$ $};\n \\node[vertex] (b) at (0,1) {$ $};\n \\node[vertex] (c) at (1,0) {$ $};\n \\node[vertex] (d) at (1,1) {$ $};\n \\node[vertex] (e) at (2,0) {$ $};\n \\node[vertex] (f) at (2,1) {$ $};\n \\node[vertex] (g) at (3,0) {$ $};\n \\node[vertex] (h) at (3,1) {$ $};\n \n \\node[vertex] (ac) at (.5,.5) {$ $};\n \\node[vertex] (gh) at (2.5,.5) {$ $};\n \n \\draw(a)--(b);\n \\draw(a)--(c);\n \\draw(c)--(d);\n \\draw(b)--(d);\n \\draw(c)--(e);\n \\draw(e)--(f);\n \\draw(f)--(d);\n \\draw(e)--(g);\n \\draw(g)--(h);\n \\draw(h)--(f);\n \n \\draw(a)--(ac);\n \\draw(ac)--(c);\n \\draw(ac)--(d);\n \\draw(ac)edge(e)edge(f);\n \\draw(gh)edge(h)edge(f)edge(e);\n \\draw(gh)edge(c)edge(d);\n \n \n \n \\draw[thick,gray!60,dashed] (0-.3,.5) ellipse (.6cm and 1cm);\n \\draw[thick,gray!60,dashed] (3.3,.5) ellipse (.6cm and 1cm);\n\n\n \\end{tikzpicture}\n \n \\caption{A graph for which \\textsc{lbfs} will never compute diameter.}\n\n\\end{figure}\n\n\n\n\\vspace{.5cm}\n\\textbf{Corollary 1.} \\textit{All simplicial vertices can be computed in general graphs in $O(n^2)$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Let $G = (V,E)$ be a general graph. We perform the reduction in $O(n^2)$ as described in Theorem 3 of $[8 ]$ to create an AT-free graph $H$ in which any simplicial vertex $v$ is exactly a diametral vertex. We return a polar set $X$ that contains $v$ using \\textsc{get-polar-pair} in $O(2n + 2m)$. We obtain the set $L_2^*(X)$ in $O(n)$ using the calculation shown in Section V. If $L_2^*(X) \\neq \\emptyset$ then any node in $L_2^*(X)$ is a node which is simplicial and achieves the diameter by the Fast Simplicial Vertex Theorem and Proposition 4, respectively. In fact, $|L_2^*(X)|$ is equal to the set of all simplicial vertices. Any $2^*$ vertex in $H$ is equal to a simplicial vertex in $G$ according to Theorem 3 of $[8 ]$. Total time-complexity is $T(n) = n^2 + 3n + 2m$. The bottle-neck is the reduction method, meaning time-complexity can more simply be stated as $O(n^2)$. $\\square$\n\n\\vspace{.5cm}\n\nThere are many problems that are at least as hard as deciding a simplicial vertex. For example, determining whether a triangle exists in a general graph with $n$ vertices can be reduced in $O(n^2)$-time to the problem of counting the number of simplicial vertices a $3n$-vertex (so, $O(n)$-vertex) general graph $[9 ]$. We give a final result:\n\n\\vspace{.5cm}\n\\textbf{Corollary 2.} \\textit{Triangle-free graphs can be recognized in $O(n^2)$.}\n\n\\vspace{.5cm}\n\\textit{Proof.} Since the problem of finding a triangle in a general graph can be reduced in $O(n)$-time to finding all diametral nodes (all eccentricities) in an $n$-vertex AT-free graph $[9 ]$, it follows that time-complexity of triangle recognition is reduced to $O(n^2)$. The reduction from the simplicial vertex problem to the diameter problem on AT-free graphs remains the bottle-neck. The diametral nodes in the AT-free graph $H$ can be found in using the same method as described in Corollary 1 because all vertices in $H$ which are simplicial are exactly those vertices which are diametral. $\\square$\n\n\\vspace{.5cm}\nIt is clear now that the following problems are strongly related:\n\\vspace{.25cm}\n\n\\begin{itemize}\n \\item[1.] \\textit{Triangle recognition in a general graph $G$.}\n \\item[2.] \\textit{Finding simplicial vertices in a general graph $G'$.}\n \\item[3.] \\textit{Finding diametral vertices of an AT-free graph $H$.}\n\\end{itemize}\n \n\\vspace{.25cm}\nThere exists a $O(n)$-time reduction $[9 ]$ from $G$ to $G'$. There exists a $O(n^2)$-time reduction $[8 ]$ from $G'$ to $H$. On any AT-free graph, including $H$, our linear-time algorithm (described in Theorem 2) can be performed to find the diametral nodes.\n \n\\vspace{.25cm}\n\\section{Conclusion}\n\n\\subsection{Summary}\n\nWe have made the following contributions. Firstly, lower and upper bounds of diameter with respect to DP distance have been found for the broad class of weak dominating pair graphs. \n\nSecondly, we have shown that polar sets can exist for graphs with low diameter under the Perfect Polar Lemma, and that a maximal clique that contains a diametral node can be efficiently found.\n\nThirdly, we have developed an algorithm that exploits the relative eccentricities of DP nodes to find the vertex that achieves the diameter. This novel algorithm is linear with respect to the number of nodes and edges of a graph (that is, $O(n+m)$-time). Due to reductions from general graphs to AT-free graphs on the problems of triangle recognition and of finding simplicial vertices, we reduce time-complexity of these well-known graph theoretical problems to $O(n^2)$.\n\nIn the future, an improved algorithm for finding maximal polar sets in graphs with asteroidal triples may allow us to solve diameter in linear time for a larger special class of graphs, such as dominating pair graphs. Run-time improvements on the reductions between diameter on AT-free graphs and more fundamental graph theoretic problems will further improve their time-complexities.\n\n\n\\subsection{Acknowledgements}\n\nWe would like to thank Dr. Vinodchandran Variyam from the University of Nebraska-Lincoln for his very helpful comments and suggestions.\n\n\n\\section{Appendix}\n\\vspace{.25cm}\n\n\\begin{tabular}{l l}\n\nAT-free & Asteroidal triple-free.\\\\\n\nDPG & Dominating pair graph.\\\\\n\nwDP & Weak dominating pair graph.\\\\\n\n$d_G(u,v)$ & Geodesic distance between $u$ and $v$.\\\\\n\n$ecc(u)$ & Eccentricity of $u$. \\\\\n\n$diam(G)$ & Diameter of $G$. \\\\\n\n$P_{u \\rightarrow v}$ & A shortest path between $u$ and $v$. \\\\\n\n$u,v$-path & Same as $P_{u \\rightarrow v}$. \\\\\n\nDP & Dominating pair. \\\\\n\nDP node & One node from a DP. \\\\\n\n$X$ & Polar set. It is made up of DP nodes. \\\\\n\n$(X,Y)$ & Pair of polar sets such that any node in \\\\\n~ & $X$ forms a DP with any node in $Y$.\\\\\n\n$\\mathscr R$ & The set of all polar pairs in $G$.\\\\\n\n$L_i(x)$ & Level $i$ ($1 \\leq i \\leq 3)$ partition of $N[x]$ \\\\\n~ & where $x \\in X$. \\\\\n\n$L_2^*(x)$ & Set of $2^*$ nodes in $L_2(x)$. \\\\\n\n$\\mathcal L$ & Labels given to $X$ as part of \\\\\n~ & function \\textsc{Label}.\\\\\n\n$l_x$ & A label assigned to vertex $x$. \\\\\n\n$Y^*$ & A subset of $Y$ that is unassigned after\\\\\n~ & \\textsc{Label} reaches $Y$ in the fewest hops.\\\\\n\n\\end{tabular}\n\\vspace{.5cm}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIt is well known~(see e.g. \\cite{Sha56} and \\cite{Csi73}), that the capacity of a discrete memoryless channel (DMC) is not increased by feedback.\\footnote{According to common practice, we say that feedback is available if the encoder may select the current channel input as a function not only of the message but also of all past channel outputs.} Nevertheless, feedback can help in at least two ways: for a fixed target error probability, feedback can be used to reduce the sender\/receiver complexity and\/or to reduce the expected decoding delay. An example is the binary erasure channel, where feedback makes it possible to implement a communication strategy that is extremely simple and also minimizes the delay. The strategy is simply to send each information bit repeatedly until it is received unerased. This strategy is capacity achieving, results in zero probability of error, and reproduces each information bit with the smallest delay among all possible strategies. \n\nThe reliability function---also called the error exponent---is a natural way to quantify the benefit of feedback. For block codes on channels without feedback the reliability function is defined as\n\\begin{equation}\n E(R)=\\limsup_{T\\to\\infty} -\\frac1T \\ln P_e(\\lceil e^{RT}\\rceil, T),\n\\end{equation}\nwhere $P_e(M,T)$ is the smallest possible error probability of length $T$ block codes with $M$ codewords.\n\n\nThe decoding time $T$ in a communication system with feedback may depend on the channel output sequence.\\footnote{ If the decoding time is not fixed, in the absence of feedback the sender may not know when the receiver has decoded. This problem does not exist if there is feedback. }\nIf it does, the decoding time $T$ becomes a random variable and the notions of rate and reliability function need to be redefined. Following Burnashev~\\cite{Bur76}, in this case we define the rate as\n\\begin{equation}\n R {\\triangleq} \\frac{\\ln M}{\\EX{T}},\n\\end{equation}\nwhere $M$ is the size of the message set. Similarly we define the reliability function as\n\\begin{equation} \n E_f(R) {\\triangleq} \\lim_{t\\to\\infty} -\\frac1{t}\\ln\n P_{e,f}(\\lceil e^{Rt}\\rceil, t) ,\n\\end{equation}\nwhere $P_{e,f}(M,t)$ is the smallest error probability of a variable-length block code with feedback that transmits one of $M$ equiprobable messages by means of $t$ or fewer channel uses on average. As we remark below, the limit exists for all rates from zero to capacity. \n\nBurnashev showed that for a DMC of capacity $C$, the reliability function $E_f(R)$ equals \\begin{equation}\n\\label{eq:bexp}\n E_B(R) = C_1(1-R\/C),\\quad 0\\leq R\\leq C,\n\\end{equation}\nwhere $C_1$ is\ndetermined by the two ``most distinguishable'' channel input symbols as \n\\[ C_1=\\max_{x,x'} D(p(\\cdot|x)\\|p(\\cdot|x')), \\]\nwhere $p(\\cdot|x)$ is the probability distribution of the channel output when the input is $x$, and $D(\\cdot\\|\\cdot)$ denotes the \nKullback-Liebler divergence between two probability distributions. It is remarkable\nthat (\\ref{eq:bexp}) determines the reliability function exactly for all rates. In contrast, the reliability function without feedback is known exactly only for rates above a critical rate. Below the critical rate only upper and lower bounds to the reliability function without feedback are known. For a binary symmetric channel the situation is depicted in Fig.~\\ref{fig:exponents}. \n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{0.8\\textwidth} \\centering \\setlength{\\unitlength}{0.6bp}\n\t \\begin{picture}(220,270)\n\t\t\\put(0,0){\\includegraphics[scale=0.6]{bsc-reliab}}\n\\put(20,0){\\makebox(0,0){$0$}} \\put(84,0){\\makebox(0,0){$R_\\text{crit}$}}\n\\put(200,0){\\makebox(0,0){$C$}}\n\\put(-45,150){\\makebox(0,0){\\rotatebox{90}{$E(R)$}}}\n\\put(0,20){\\makebox(0,0){0}} \\put(-7,145){\\makebox(0,0){0.26}}\n\\end{picture} \\caption{Reliability functions for a binary symmetric channel with\ncrossover probability 0.1. Shown is Burnashev's reliability function for channels with\nfeedback (solid line) and upper and lower bounds to the reliability function for channels\nwithout feedback. The upper bound (dotted line) is given by the \\emph{straight line\nbound} at low rates and by the \\emph{sphere packing bound} at higher rates. The lower bound\n(dot-dashed line) is given by the \\emph{expurgated bound}. The upper and lower bounds coincide above the critical rate, $R_\\text{crit}$.} \\label{fig:exponents}\n\\end{minipage} \n\\end{figure}\n\nBurnashev showed that $E_f = E_B$ by showing that for every communication scheme\n\\begin{equation}\n \\label{eq:whatweprove} E[T]\\geq \\biggl(\\frac{\\ln M}{C} - \\frac{\\ln P_e}{C_1}\\biggr) \\bigl(1-o(1)\\bigr) \n\\end{equation}\nwhere $o(1)$ represents positive terms that tend to zero as $1\/P_e$ tends to infinity,\nand that there exists schemes with\n\\begin{equation}\n \\label{eq:ach} E[T]\\leq \\biggl(\\frac{\\ln M}{C} - \\frac{\\ln P_e}{C_1}\\biggr) \\bigl(1+o(1)\\bigr),\n\\end{equation}\nwhere $o(1)$ now represents positive terms that tend to zero as both $M$ and $1\/P_e$ tend to infinity.\n\nFor a plausibility argument that justifies (\\ref{eq:ach}) it suffices to summarize the achievability construction by Yamamoto and Itoh~\\cite{YamIto79}. Their scheme relies on two distinct transmission phases that we\nshall call the communication and the confirmation phase, respectively. \nIn the communication phase the message is encoded\nusing a fixed-length block code and the codeword is transmitted over the forward channel. The\ndecoder makes a tentative decision based on the corresponding channel output. The encoder knows the channel output and can run the algorithm used by the receiver to determine the tentative decision. If the tentative decision is correct, in the confirmation phase the encoder\nsends ACK. Otherwise it sends NACK. ACKs and NACKs \nare sent via a fixed-length repetition code. (The code consists of two codewords). During the \nconfirmation phase the\ndecoder performs a binary hypothesis test to decide if ACK or NACK was\ntransmitted. If ACK is decoded, the tentative decision becomes final and the transmission\nof the current message ends, leaving the system free to restart with a new message.\nIf NACK is decoded, the tentative decision is discarded\nand the two phase scheme restarts with the same message. \n\nThe overhead caused by retransmissions is negligible if the probability of decoding NACK is small. This is the case\nif both the error probability of the communication phase as well as that of the confirmation phase are \nsmall. Assuming that this is the case, the number of channel uses for the communication phase (including repetitions) is \nslightly above $(\\ln M)\/C$. The probability of error is the probability that NACK is sent and ACK is decoded. In the asymptotic regime of interest this probability is dominated by the probability that ACK is decoded given that NACK is sent. In a straightforward application of Stein's lemma~\\cite{Che56} one immediately sees that we can make this probability to be slightly less than $P_e$ (thus achieve error probability $P_e$) by means of a confirmation code of length slightly above $(-\\ln P_e)\/ C_1$. Summing up, we see that we can make the error probability arbitrarily close to $P_e$ by means of slightly more than $(\\ln M)\/C-(\\ln P_e)\/ C_1$ channel uses on average. This confirms (\\ref{eq:ach}).\n\n\nTo obtain the converse (\\ref{eq:whatweprove}), Burnashev investigated the entropy of the \\latin{a posteriori} probability distribution over the message set. He showed that the average decrease of this entropy due to an additional channel output observation, as well as the average decrease of the logarithm of this entropy, are bounded.\nHe uses these bounds to form two submartingales, one based on the entropy of the \\latin{a posteriori} distribution and the other based on the logarithm of this entropy. He then constructs a single submartingale by patching these two together. Then Doob's optional stopping theorem is applied to this submartingale and the desired bound on the expected decoding time, which is a stopping time, is obtained. Burnashev's proof is an excellent example of the power of martingales, however both the sophistication of the martingale construction and the use of the logarithm of entropy leaves the reader with little insight about some of the terms in the converse bound. While it is easy to see that $(\\ln M)\/C$ channel uses are needed in average, it was not as clear why one needs an additional $(-\\ln P_e)\/ C_1$ channel uses. The connection of the latter term to binary hypothesis testing suggested the existence of an operational justification. The work presented in this paper started as an attempt to find this operational justification.\n\n\nOur converse somewhat parallels the Yamamoto--Itoh achievability scheme. This suggests that that a communication and confirmation phase may be implicit components of any scheme for which the probability of error decreases with the largest possible exponent. Our approach has been generalized by Como, Y\\\"uksel and Tatikonda in \\cite{ComYuk07} to prove a similar converse for variable-length block codes on Finite State Markov Channels.\n\n\n\\section{Channel Model and Variable-Length Codes as Trees}\nWe consider a discrete memoryless channel, with finite input alphabet ${\\cal X}$, finite output alphabet ${\\cal Y}$, and transition probabilities $p(y|x)$. We will denote the channel input and output symbols at time $n$ by $X_n$ and $Y_n$, and denote the corresponding vectors $(X_1,X_2,\\ldots,X_n)$ and $(Y_1,Y_2,\\ldots,Y_n)$ by $X^n$ and $Y^n$, respectively. A perfect causal feedback link is available, i.e., at time $n$ the encoder knows $y^{n-1}$. (Following common practice, random variables are represented by capital letters and their realizations are denoted by the corresponding lowercase letters.)\n \nWe will assume, without loss of generality, that the channel has no ``useless\noutputs symbols'', i.e., no symbols $y$ for which $p(y|x) = 0$ for every $x$.\nNote that for channels for which $C_1$ is infinite, the lower bound to the\nexpected decoding time is a restatement of the fact that feedback does not\nincrease capacity. We will therefore restrict our attention to channels for\nwhich $C_1<\\infty$. For such channels, $p(y|x)>0$ for every $x$ and $y$; if\nnot, there exists an $x$ and $y$ for which $p(y|x)=0$. Since $y$ is reachable\nfrom some input, there also exists an $x'$ for which $p(y|x')>0$. But then\n$D(p(\\cdot|x')\\|p(\\cdot|x))=\\infty$ contradicting the finiteness of $C_1$. The\nfact that both $\\cal X$ and $\\cal Y$ are finite sets lets us further conclude\nthat for the channels of interest to this paper, there is a $\\lambda>0$ for\nwhich $p(y|x)\\geq\\lambda$ for every $x$ and $y$.\n\nA variable-length block code is defined by two maps: the encoder and the decoder. The encoder\\footnote{ For clarity of exposition we will only treat deterministic coding strategies here. Randomized strategies may be included without significant modification to the core of the proof.} functions $f_n(\\cdot,\\cdot): {\\cal W} \\times {{\\cal Y}}^{n-1} \\rightarrow {\\cal X}$, where ${\\cal W} = \\{1,\\ldots,M\\}$ is the set of all possible messages, determine the channel input $X_n=f_n(W,Y^{n-1})$ based on the message $W $ and on past channel outputs $Y^{n-1}$. The decoder function $\\hat{W}(\\cdot): {\\cal Z} \\to {\\cal W}$, where $\\cal Z$ is the receiver observation space until the decoding time $T$, i.e., $Y^{T}$ takes values in ${\\cal Z}$. The decoding time $T$ should be a stopping time\\footnote{A discussion of stopping times can be found in~\\cite[sect.~10.8]{Wil91}.} with respect to the receiver observation $Y^n$ otherwise the decision of when to decode would depend on future channel outputs and the decoder would no longer be causal. We treat the case when $\\EX{T} < \\infty$, and point out that what we are setting out to prove, namely (\\ref{eq:whatweprove}), is trivially true when $\\EX{T} = \\infty$.\n\n\nThe codes we consider here differ from non-block (also called sequential) codes with variable delay, such as those studied in \\cite{Hor63} and~\\cite{Kud79}. In sequential coding, the message (typically as an infinite stream of bits) is introduced to the transmitter and decoded by the receiver in a progressive fashion. Delay is measured separately for each bit, and is defined as the time between the introduction and decoding of the bit. This is in contrast to the codes considered in this paper, where the entire message is introduced to the transmitter at the start of communication, and $T$ measures the duration of the communication. Due to their different problem formulation, sequential codes with feedback have reliability functions that differ from those for variable-length block codes, just as fixed constraint length convolutional codes have reliability functions that differ from those of fixed-length block codes.\n\n\nThe observation space $\\cal Z$ is a collection of channel output sequences and for a DMC with feedback the length of these sequences may vary. (The length of the channel input itself may depend on the channel realization). Nevertheless, these sequences have the property of being prefix-free (otherwise the decision to stop would require knowledge of the future). Thus, ${\\cal Z}$ can be represented as the leaves of a complete $|{\\cal Y}|$-ary tree $\\cT$ (complete in the sense that each intermediate node has $|{\\cal Y}|$ descendants), and has expected depth $\\EX{T} < \\infty$. Note that the decision time $T$ is simply the first time the sequence $Y_1,Y_2,\\dots$ of channel outputs hits a leaf of $\\cT$. Furthermore we may label each leaf of $\\cal T$ with the message decoded by the receiver when that leaf is reached. This way the decoder is completely specified by the labeled tree $\\cal T$. The message statistics, the code, and the transition probabilities of the channel determine a probability measure on the tree $\\cal T$. \n\n\n\\section{Binary Hypothesis Testing with Feedback} \n\\label{sec:prelim} \nThe binary case ($M=2$) will play a key role in our main proof. In this section we assume that the message set contains only two elements. We will arbitrarily denote the two hypotheses by $A$ and $N$ (ACK and NACK, respectively). We denote by $\\PRM{A}$ and $\\PRM{N}$ the corresponding probability distributions on the leaves of $\\cal T$. \n\n\nThe following proposition bounds the Kullback-Leibler divergence $D(\\PRM{A}\\|\\PRM{N})$. It will be used in the main result of this section to bound the error probability of binary hypothesis testing with feedback. The reader familiar with Stein's Lemma will not be surprised by the fact that the Kullback-Leibler divergence $D(\\PRM{A}\\|\\PRM{N})$ plays a key role in binary hypothesis testing with feedback. The steps here closely parallel those in \\cite[Sec. III] {TchTel05} and \\cite[Sec. 2.2]{CsiShi04}.\n\n\\begin{proposition} \\label{prop:expkl} \nFor any binary hypothesis testing scheme\n for a channel with feedback \\[D(\\PRM{A}\\|\\PRM{N})\\leq \\DX \\ECX{T}{A} \\]\n where $T$ is the decision stopping time, $\\EX{T} < \\infty$, and\n $\\ECX{T}{A}$ denotes the expectation of $T$ conditioned on hypothesis $A$.\n\\end{proposition} \n\n\\begin{proof} In the\n\tfollowing, we will denote probability under hypothesis $A$ by $\\PrA(\\cdot)$\n\tand probability under hypothesis $N$ by $\\PrN(\\cdot)$. Let\n\t\\begin{equation} V_n=\\ln\\dfrac{\\PrA(Y_1,\\dots,Y_n)}{\\PrN(Y_1,\\dots,Y_n)},\n\t\\end{equation} so that $ D(\\PRM{A}\\|\\PRM{N})=\\ECX{V_{T}}{A}$, and the\n\tproposition is equivalent to the statement $\\ECX{V_{T} -\\DX T}{A}\\leq 0$.\n\tObserve that \n\\begin{equation} V_n=\\sum_{k=1}^n U_k \\quad\\text{where\n\t $U_k=\\ln\\dfrac{\\PrA(Y_k|Y^{k-1})}{\\PrN(Y_k|Y^{k-1})}$.} \n\\end{equation}\nNote now that \n\\begin{align} \\ECX{U_k}{A,Y^{k-1}} &=\\ECX{\\ln \\frac{\n\t\t\\PrA(Y_k|Y^{k-1})} {\\PrN(Y_k|Y^{k-1})}}{A,Y^{k-1}} \\nonumber\\\\\n\t\t&=\\ECX{\\ln \\frac{\n\t\t\\PrA(Y_k|X_k=f_k(A,Y^{k-1}),Y^{k-1})}{\\PrN(Y_k|X_k=f_k(N,Y^{k-1}),Y^{k-1})}}{A,\n\t\tX_k=f_k(A,Y^{k-1}), Y^{k-1}} \\nonumber\\\\ &=\\sum_{y \\in {\\cal Y}}\n\t\t\\PCX{Y_k=y}{X_k=f_k(A,Y^{k-1})} \\ln\\frac{\n\t\t\\PCX{Y_k=y}{X_k=f_k(A,Y^{k-1})} }{ \\PCX{Y_k=y}{X_k=f_k(N,Y^{k-1})} }\n\t\t\\nonumber\\\\ &\\leq \\DX, \n\\end{align} \nwhere $f_k(\\cdot,\\cdot)$ is the\n\t\tencoder function at time $k$. Consequently, $\\{V_n-n\\DX\\}$ is a\n\t\tsupermartingale under hypothesis $A$. Observe that the existence of a\n\t\t$\\lambda > 0$ for which $p(y|x) > \\lambda$ for all $x,y$ implies that\n\t\t$|U_k| < \\ln \\frac1\\lambda$. We can now use Doob's Optional-Stopping\n\t\tTheorem (see e.g.~\\cite[Sec.~10.10]{Wil91}) to conclude that $\\ECX{V_T-\\DX T}{A}\n\t\t\\leq 0$. \\end{proof}\n\n\nWe can apply Proposition~\\ref{prop:expkl} to find a lower bound on the error\nprobability of a binary hypothesis testing problem with feedback. The bound is\nexpressed in terms of the expected decision time. \n\n\\begin{lemma} \\label{prop:pe-bound} The error probability of a binary\n hypothesis test performed across a DMC with feedback and variable-length\n codes is lower bounded by \\[ \\Pe \\geq \\frac{\\min\\{p_A,p_N\\}}{4}\n e^{-\\CX_1\\EX{T}} \\] where $p_A$ and $p_N$ are the \\latin{a priori}\n probabilities of the hypotheses. \n\\end{lemma}\n\n\\begin{proof} Each decision rule corresponds to a tree where each\n leaf $Y^{T}$ is associated with a decoded hypothesis $\\hat{W}(Y^{T})$. Thus\n we can partition the leaves into two sets corresponding to the two\n hypotheses. \\begin{align*} \\cS &= \\{ y^T : \\hat{W}(y^T) = A \\} \\\\ \\bar{\\cS}\n\t&= \\{ y^T : \\hat{W}(y^T) \\ne A \\} \\end{align*} where $\\cS$ is the decision\n\tregion for hypothesis $A$.\n\nThe log sum inequality \\cite{CovTho91,CsK81} (or data processing lemma for\ndivergence) implies \\begin{equation} \\label{eq:kl-logsum}\n \\KLD{\\PRM{A}}{\\PRM{N}} \\geq \\PRM{A}(\\cS) \\ln\n \\frac{\\PRM{A}(\\cS)}{\\PRM{N}(\\cS)} + \\PRM{A}(\\bar{\\cS}) \\ln\n \\frac{\\PRM{A}(\\bar{\\cS})}{\\PRM{N}(\\bar{\\cS})}. \\end{equation} By\n Proposition~\\ref{prop:expkl}, $\\DX \\ECX{T}{A} \\ge \\KLD{\\PRM{A}}{\\PRM{N}}$,\n thus~(\\ref{eq:kl-logsum}) can be re-arranged to give \\begin{equation}\n\t\\label{eq:depth_conda} \\DX \\ECX{T}{A}\\geq - \\PRM{A}(\\cS) \\ln \\PRM{N}(\\cS)\n\t-\\binh(\\PRM{A}(\\bar{\\cS})), \\end{equation} where $\\binh(\\cdot)$ is the\n\tbinary entropy function. Writing the overall probability of error in terms\n\tof marginal error probabilities yields \\[\\Pe =p_N \\PRM{N}(\\cS)+ p_A\n\t\\PRM{A}(\\bar{\\cS}), \\] which allows us to bound $\\PRM{N}(\\cS)$ as \\[\n\t\\PRM{N}(\\cS)\\leq \\frac{\\Pe}{p_N} \\leq \\frac{\\Pe}{\\min\\{p_A,p_N\\}}. \\] Substituting back\n\tinto~(\\ref{eq:depth_conda}) yields a bound on the expected depth of the\n\tdecision tree conditioned on $A$ just in terms of $\\PRM{A}$ and the\n\t\\latin{a priori} message probabilities \\begin{equation}\n\t \\label{eq:treedeptha} \\DX \\ECX{T}{A} \\geq -\\PRM{A}(\\cS) \\ln \\frac{\\Pe}\n\t {\\min \\{p_A,p_N\\}}- \\binh(\\PRM{A}(\\bar{\\cS})). \\end{equation}\nFollowing identical steps with the roles of $A$ and $N$ swapped yields\n\\begin{equation} \\label{eq:treedepthr} \\DX \\ECX{T}{N} \\geq -\\PRM{N}(\\bar\\cS)\n \\ln \\frac{\\Pe} {\\min \\{p_A,p_N\\}}- \\binh(\\PRM{N}(\\cS)). \\end{equation}\n We can now average both sides of~(\\ref{eq:treedeptha}) and~(\\ref{eq:treedepthr})\n by weighting with the corresponding \\latin{a priori} probabilities. If we do\n so and use the facts that $p_A\\PRM{A}(\\cS)+p_N\\PRM{N}(\\bar\\cS)$ is the\n probability of making the correct decision and $p_A\\PRM{A}(\\bar\\cS)+p_N\\PRM{N}(\\cS)$ \nis the probability of making an error together with the concavity of the \nbinary entropy function, we obtain the following\n unconditioned bound on the depth of the decision tree \\begin{align*} \\DX\n\t\\EX{T} &\\geq -(1-\\Pe) \\ln \\frac{\\Pe}{\\min\\{p_A,p_N\\}}- \\binh(\\Pe) \\\\ &\\geq\n\t-\\ln \\Pe-2\\binh(\\Pe)+\\ln\\min\\{p_A,p_N\\}\\\\ &\\ge -\\ln \\Pe -2\\ln 2 +\n\t\\ln\\min\\{p_A, p_N\\}. \\end{align*}\n\n\n\nSolving for $\\Pe$ completes the proof.\n\\end{proof} It is perhaps worthwhile pointing out why the factor\n$\\min \\{p_A,p_N \\}$ arises: if one of the hypotheses has small \\latin{a\npriori} probability, one can achieve an equally small error probability by\nalways deciding for the other hypothesis, irrespective of the channel\nobservations. \n\n\\section{Expected Tree Depth and Channel Capacity} \n\\label{sec:chan_ca}\nGiven the channel observations $y^n$, one can calculate the \\latin{a posteriori}\nprobability $p_{W|Y^n}(w|y^n)$ of any message $w \\in {\\cal W}$. Recall that a\nmaximum \\latin{a posteriori} (MAP) decoder asked to decide at time $n$ when\n$Y^n=y^n$ will chose (one of) the message(s) that has the largest \\latin{a\nposteriori} probability $p_{\\max}=\\max_w p_{W|Y^n}(w|y^n)$. The probability of\nerror will then be $\\Pe(y^{n})=1-p_{\\max}$. Similarly, we can define the\nprobability of error of a MAP decoder for each leaf of the observation tree\n$\\cal T$. Let us denote by $\\Pe(y^{T})$ the probability of error given the\nobservation $y^{T}$. The unconditioned probability of error is then\n$\\Pe=\\EX{\\Pe(Y^{T})}$. \n\nFor any fixed $\\delta>0$ we can define a stopping time $\\tau$ as the first time that the error\nprobability goes below $\\delta$, if this happens before $T$, and as $T$\notherwise: \\begin{equation}\\label{eq:tau_defn} \\tau = \\inf \\left\\{n:\n \\left(\\Pe(y^n)\\leq \\delta\\right) \\text{ or } ( n = T ) \\right\\} \\end{equation}\n\nIf $\\Pe(Y^{\\tau})$ exceeds $\\delta$, then we are certain that $\\tau = T$, and $\\Pe(Y^n) > \\delta$ for all $0\\le n \\le T$, so the event $\\Pe(Y^\\tau) > \\delta$ is included in the event $\\Pe(Y^T) > \\delta$. (We have inclusion instead of equality since $\\Pe(Y^\\tau) \\leq \\delta$ does not exclude $\\Pe(Y^T) > \\delta$.) Thus\n\\begin{equation}\n \\label{eq:peexdelta} \\PX{\\Pe(Y^{\\tau}) > \\delta} \\leq \\PX{\\Pe(Y^{T})>\\delta}\n \\leq \\frac{\\Pe}{\\delta}, \n\\end{equation}\nwhere the second inequality is an application of Markov's inequality.\n\nGiven a particular realization $y^n$ we will denote the entropy of the \n\\latin{a posteriori} distribution $p_{W|Y^n}(\\cdot|y^n)$ as $\\h{y^n}$. Then $\\HX{n}$ is a random variable\\footnote{Notice that $\\h{y^n}$ is commonly written as $H(W|Y^{n}=y^{n})$. We cannot use the standard notation since it becomes problematic when we substitute $Y^n$ for $y^n$ as we just did.} and \n$\\EX{\\HX{n}} = H(W|Y^n)$. If $\\Pe(y^{\\tau})\\leq \\delta \\leq \\frac12$, then from Fano's inequality it follows that \n\\begin{equation}\n \\label{eq:htau_bound} \\h{y^\\tau} \\leq \\binh(\\delta) +\\delta \\ln M.\n\\end{equation}\nThe expected value of $\\HX{\\tau}$ can be bounded by conditioning on the event\n$\\Pe(Y^{\\tau})\\leq \\delta$ and its complement then\napplying~(\\ref{eq:htau_bound}) and then~(\\ref{eq:peexdelta}) as follows\n\\begin{align*} \\EX{\\HX{\\tau}} &= \\ECX{\\HX{\\tau}}{\\Pe(Y^{\\tau})\\leq\n \\delta}\\PX{\\Pe(Y^{\\tau})\\leq \\delta}+ \\ECX{\\HX{\\tau}}{\\Pe(Y^{\\tau})>\n \\delta}\\PX{\\Pe(Y^{\\tau})> \\delta} \\\\ &\\leq \\left(\\binh(\\delta) +\\delta \\ln M\n \\right)\\PX{\\Pe(Y^{\\tau})\\leq \\delta} + (\\ln M) \\PX{\\Pe(Y^{\\tau})> \\delta)} \\\\\n &\\leq \\binh(\\delta) + \\left( \\delta+\\frac{\\Pe}{\\delta} \\right) \\ln M.\n\\end{align*}\nThis upper bound on the expected posterior entropy at time $\\tau$ can be turned\ninto a lower bound on the expected value of $\\tau$ by using the channel\ncapacity as an upper bound to the expected change of entropy. This notion is\nmade precise by the following lemma, \\begin{lemma} \\label{prop:exp_tau} For any\n $0 <\\delta \\leq \\frac12$ \\[ \\EX{\\tau} \\ge\n \\left(1-\\delta-\\frac{\\Pe}{\\delta}\\right)\\frac{ \\ln\n M}{\\CX}-\\frac{\\binh(\\delta)}{\\CX}. \\] \\end{lemma} \\begin{proof} Observe that\n\t$\\{\\HX{n}+n \\CX\\}$ is a submartingale (an observation already made\n\tin~\\cite[Lemma 2]{Bur76}). To see this, \n\\begin{align*}\n\t \\ECX{\\HX{n}-\\HX{n+1}}{Y^{n}=y^{n}} \n&=I(W;Y_{n+1}|Y^{n}=y^{n})\\\\\n&\\stackrel{(a)}{\\leq}I(X_{n+1};Y_{n+1}|Y^{n}=y^{n})\\\\ \n&\\leq \\CX \n\\end{align*}\n\t where $(a)$ follows from the data processing inequality and the fact that\n\t $W$---$X_{n+1}$---$Y_{n+1}$ forms a Markov chain given $Y^{n}=y^{n}$.\nHence $\\{\\HX{n}+n \\CX\\}$ is indeed a submartingale. Since $\\h{y^n}$ is bounded\nbetween $0$ and $\\ln M$ for all $n$, and the expected stopping time $\\EX{\\tau}\n\\le \\EX{T} < \\infty$, Doob's Optional-Stopping Theorem allows us\nto conclude that at time $\\tau$ the expected value of the submartingale must be\ngreater than or equal to the initial value, $\\ln M$. Hence \n\\begin{align*} \n\\ln M = \\HX{0} &\\leq \\EX{\\HX{\\tau}+\\tau \\CX} \\\\ &= \\EX{\\HX{\\tau}}+\\EX{\\tau} \\CX\\\\ &\\le\n \\binh(\\delta) + \\left( \\delta + \\frac\\Pe\\delta \\right) \\ln M + \\EX{\\tau} \\CX.\n\\end{align*} Solving for $E[\\tau]$ yields \\[ \\EX{\\tau} \\ge\n\\left(1-\\delta-\\frac{\\Pe}{\\delta}\\right) \\frac{ \\ln\nM}{\\CX}-\\frac{\\binh(\\delta)}{\\CX}. \\] \\end{proof}\n\n\n\\section{Burnashev's Lower Bound} \n\\label{sec:burnbound}\n\nIn this section we will combine the two bounds we have established in the\npreceding sections to obtain a bound on the overall expected decoding time.\nLemma~\\ref{prop:exp_tau} provides a lower bound on $\\EX{\\tau}$ as a function of\n$M$, $\\delta$ and $\\Pe$. We will show that a properly constructed binary\nhypothesis testing problem allows us to use Lemma~\\ref{prop:pe-bound} to lower bound\nthe probability of error in terms of $\\ECX{T-\\tau}{Y^{\\tau}}$. This in turn\nwill lead us to the final bound on $\\EX{T}$.\n\nThe next proposition states that a new channel output symbol can not change the\n\\latin{a posteriori} probability of any particular message by more than some constant factor when $\\DX$\nis finite.\n\\begin{proposition} \\label{prop:postent} $C_1 < \\infty$ implies \n\\[\\lambda p\\left(w|y^{n-1}\\right) \\leq p\\left(w|y^n\\right) \\leq \\frac{ p\\left(w|y^{n-1}\\right)}{\\lambda}, \\] \nwhere $0< \\lambda=\\min_{x,y} p(y|x) \\le \\frac12$. \\end{proposition}\n\n\\begin{proof} Using Bayes' rule, the posterior may be written recursively as\n\\[ \np\\left(w|y^n\\right) = p\\left(w|y^{n-1}\\right) \\frac{p\\left(y_n|x_n=f_n(w,y^{n-1})\\right)}{ p(y_n|y^{n-1})}.\n\\]\nThe quotient may be upper and lower bounded using $1 \\ge p\\left(y_n|x_n=f_n(w,y^{n-1})\\right) \\ge \\lambda $ and $1 \\ge p(y_n|y^{n-1}) \\ge \\lambda$, which yields the statement of the proposition. \\end{proof}\n\nOur objective is to lower bound the probability of error of a decoder that decides at time $T$. The key idea is that a binary hypothesis decision such as deciding whether or not $W$ lies in some set ${\\cal G}$ can be made at least as reliably as a decision on the value of $W$ itself.\n\n\nGiven a set ${\\cal G}$ of messages, consider deciding between $W\\in {\\cal G}$ and $W\\not\\in {\\cal G}$ in the following way: given access to the original decoder's estimate $\\hat W$, declare that $W\n\\in {\\cal G}$ if $\\hat W\\in{\\cal G}$, and declare $W \\not\\in{\\cal G}$ otherwise. This binary\ndecision is always correct when the original decoder's estimate $\\hat W$ is\ncorrect. Hence the probability of error of this (not necessarily optimal)\nbinary decision rule cannot exceed the probability of error of the original decoder, for\nany set ${\\cal G}$. Thus the error probability of the optimal decoder deciding at time \n$T$ whether or not $W\\in{\\cal G}$ is a lower bound to the error probability of any \ndecoder that decodes $W$ itself at time $T$. This fact is true even if the set ${\\cal G}$ is chosen at a \nparticular stopping time $\\tau$ and the error probabilities we are calculating \nare conditioned on the observation $Y^{\\tau}$. \n\n\nFor every realization of $Y^{\\tau}$, the message set can be divided into two\nparts, ${\\cal G}{(Y^{\\tau})}$ and its complement ${\\cal W}\\setminus{\\cal G}{(Y^{\\tau})}$, in such a way that both\nparts have an \\latin{a posteriori} probability greater than $\\lambda\\delta$.\nThe rest of this paragraph describes how this is possible. \nFrom the definition of $\\tau$, at time $\\tau-1$ the \\latin{a posteriori} probability of every\nmessage is smaller than $1-\\delta$. This implies\nthat the sum of the \\latin{a posteriori} probabilities of any set of $M-1$\nmessages is greater than $\\delta$ at time $\\tau -1$, and by\nProposition~\\ref{prop:postent}, greater than $\\lambda\\delta$ at time $\\tau$.\nIn particular, $\\Pe(y^{\\tau}) \\geq \\lambda \\delta$. We separately consider the\ncases $\\Pe(y^{\\tau}) \\leq \\delta$ and $\\Pe(y^{\\tau}) > \\delta$. In the first\ncase, $\\Pe(y^\\tau)\\leq \\delta$, let ${\\cal G}{(Y^{\\tau})}$ be the set consisting of only\nthe message with the highest \\latin{a posteriori} probability at time $\\tau$. \nThe \\latin{a posteriori} probability of ${\\cal G}{(Y^{\\tau})}$ then satisfies \n$\\PX{{\\cal G}{(Y^{\\tau})}} \\geq 1-\\delta \\geq 1\/2 \\geq \\lambda\\delta$. As\nargued above, its complement (the remaining $M-1$ messages) also has \\latin{a posteriori}\nprobability greater than $\\lambda\\delta$, thus for this ${\\cal G}{(Y^{\\tau})}$,\n$\\PCX{{\\cal G}{(Y^{\\tau})}}{Y^{\\tau}} \\in [\\lambda\\delta,1-\\lambda\\delta]$. \nIn the second case, namely when $\\Pe(y^\\tau)>\\delta$, the\n\\latin {a posteriori} probability of each message is smaller than $1-\\delta$.\nIn this case the set ${\\cal G}{(Y^{\\tau})}$ may be formed by starting with the empty set and\nadding messages in arbitrary order until the threshold $\\delta\/2$ is exceeded. This\nensures that the \\latin{a priori} probability of ${\\cal G}{(Y^{\\tau})}$ is greater than \n $\\lambda\\delta$. Notice that the threshold will be\nexceeded by at most $1-\\delta$, thus the complement set has an \\latin{a\nposteriori} probability of at least $\\delta\/2 > \\lambda\\delta$. Thus \n$\\PCX{{\\cal G}{(Y^{\\tau})}}{Y^{\\tau}} \\in [\\lambda\\delta,1-\\lambda\\delta]$. \n\n\n\n\n\n\nFor any realization of $Y^{\\tau}$ we have the binary hypothesis testing problem, running from $\\tau$ until $T$, \ndeciding whether or not $W\\in{\\cal G}{(Y^{\\tau})}$. Notice that the \\latin{a priori} probabilities\n of the two hypotheses of this\nbinary hypothesis testing problem are the \\latin{a posteriori} probabilities of\n${\\cal G}{(Y^{\\tau})}$ and ${\\cal W}\\setminus{\\cal G}{(Y^{\\tau})}$ at time $\\tau$ each of which is shown to be greater\nthan $\\lambda \\delta$ in the paragraph above . We apply Lemma~\\ref{prop:pe-bound}\nwith $A={\\cal G}{(Y^{\\tau})}$ and $N={\\cal W}\\setminus{\\cal G}{(Y^{\\tau})}$ to lower bound the probability of error of\nthe binary decision made at time $T$ and, as argued above, we use the result to\nlower bound the probability that $\\hat W\\ne W$. Initially everything is\nconditioned on the channel output up to time $\\tau$, thus \n\\[\\PCX{\\hat{W}\\left(Y^{T}\\right)\\neq W }{Y^{\\tau}} \\geq \\frac{\\lambda\\delta}{4} e^{-\\DX\n\\ECX{T-\\tau}{Y^{\\tau}}}. \\]\nTaking the expectation of the above expression over all realizations of $Y^\\tau$ \nyields the unconditional\nprobability of error \\[\\Pe =\\EX{\\PCX{\\hat{W}\\left(Y^{T}\\right)\\neq W }{Y^{\\tau}}}\\geq \\EX{\n\\frac{\\lambda\\delta}{4} e^{-\\DX \\ECX{T-\\tau}{Y^{\\tau}}} } .\\] Using the\nconvexity of $e^{-x}$ and Jensen's inequality, we obtain\n\\[\\Pe \\geq \\frac{\\lambda\\delta}{4} e^{-\\DX \\EX{T -\\tau}}. \\]\nSolving for $\\EX{T-\\tau}$ yields \\begin{equation} \\label{eq:boundon2} \\EX{T\n -\\tau} \\geq \\frac{-\\ln \\Pe - \\ln 4 +\\ln (\\lambda \\delta) }{\\DX}.\n\\end{equation} Combing Lemma~\\ref{prop:exp_tau} and~(\\ref{eq:boundon2}) yields:\n\\begin{theorem} \n The expected decoding time $T$ of any variable-length block code for\n a DMC used with feedback is lower bounded by \\begin{equation} \\EX{T} \\geq\n\t\\left(1-\\delta-\\frac{\\Pe}{\\delta}\\right)\\frac{ \\ln M}{\\CX} + \\frac{-\\ln\n\t\\Pe}{\\DX} -\\frac{\\binh(\\delta)}{\\CX} + \\frac{\\ln (\\lambda \\delta) - \\ln 4\n\t}{\\DX}, \\end{equation} where $M$ is the cardinality of the message set,\n\t$P_e$ the error probability, $\\lambda = \\min_{x\\in{\\cal X},y\\in{\\cal Y}} p(y|x)$, and\n\t$\\delta$ is any number satisfying $0<\\delta \\leq \\frac{1}{2}$. \\hfill\\QED\n \\end{theorem} \n \n Choosing the parameter $\\delta$ as $\\delta=-\\frac{1}{\\ln P_e}$ achieves the required scaling\n for (\\ref{eq:whatweprove}).\n \n\n\\section{Summary}\\label{sec:conc} We have presented a new derivation of\nBurnashev's asymptotically tight lower bound to the average delay needed for a\ntarget error probability when a message is communicated across a DMC used with\n(channel output) feedback. Our proof is simpler than the original, yet\nprovides insight by clarifying the role played by the quantities that appear in\nthe bound.\nSpecifically, from the channel coding theorem we expect it to take roughly\n$\\frac{\\ln M}{C}$ channel uses to reduce the probability of error of a MAP\ndecision to some small (but not too small) value. At this point we can\npartition the message set in two subsets, such that neither subset has too\nsmall an \\latin{a posteriori} probability. From now on it takes\n(asymptotically) $-\\frac{\\ln P_e}{C_1}$ channel uses to decide with\nprobability of error $P_e$ which of the two sets contains the true message. It\ntakes at least as many channel uses to decide which message was selected and\nincur the same error probability. \n\n\nFor obvious reasons we may call the two phases the communication and the binary\nhypothesis testing phase, respectively. These two phases exhibit a pleasing\nsimilarity to the communication and confirmation phase of the optimal scheme\nproposed and analyzed by Yamamoto and Itoh in \\cite{YamIto79}. The fact that\nthese two phases play a key role in proving achievability as well as in proving\nthat one cannot do better suggests that they are an intrinsic component of an\noptimal communication scheme using variable-length block codes over DMCs with\nfeedback.\n\n\n\n\\section*{Acknowledgement} The authors would like to thank R.~G.~Gallager for\nhis help in pointing out an error in an earlier version of this paper and the\nreviewers for their helpful comments.\n\n\n\\bibliographystyle{ieeetr}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Acknowledgments}\n\nWe thank Vladimir Shiltsev for enlightening discussions. We thank the staffs at Fermilab and collaborating institutions,\nand acknowledge support from the\nDepartment of Energy and National Science Foundation (United States of America);\nAlternative Energies and Atomic Energy Commission and\nNational Center for Scientific Research\/National Institute of Nuclear and Particle Physics (France);\nMinistry of Education and Science of the Russian Federation, \nNational Research Center ``Kurchatov Institute\" of the Russian Federation, and \nRussian Foundation for Basic Research (Russia);\nNational Council for the Development of Science and Technology and\nCarlos Chagas Filho Foundation for the Support of Research in the State of Rio de Janeiro (Brazil);\nDepartment of Atomic Energy and Department of Science and Technology (India);\nAdministrative Department of Science, Technology and Innovation (Colombia);\nNational Council of Science and Technology (Mexico);\nNational Research Foundation of Korea (Korea);\nFoundation for Fundamental Research on Matter (The Netherlands);\nScience and Technology Facilities Council and The Royal Society (United Kingdom);\nMinistry of Education, Youth and Sports (Czech Republic);\nBundesministerium f\\\"{u}r Bildung und Forschung (Federal Ministry of Education and Research) and \nDeutsche Forschungsgemeinschaft (German Research Foundation) (Germany);\nScience Foundation Ireland (Ireland);\nSwedish Research Council (Sweden);\nChina Academy of Sciences and National Natural Science Foundation of China (China);\nand\nMinistry of Education and Science of Ukraine (Ukraine).\n\n\n\\section{Introduction} \n\\label{toc:Intro}\nThe top quark, discovered by the CDF and \\dzero experiments in 1995 \\cite{top_disc1,top_disc2}, is the heaviest of all elementary particles in the standard model (SM). The production of top quark-antiquark pairs (\\ensuremath{t\\bar{t}}\\xspace) at the Fermilab Tevatron Collider is dominated by the quark-antiquark (\\ensuremath{q\\bar{q}}\\xspace) annihilation process. The measurement of the inclusive \\ensuremath{t\\bar{t}}\\xspace production cross section provides a direct test of quantum chromodynamics (QCD), the theory of the strong interaction. Inclusive \\ensuremath{t\\bar{t}}\\xspace production cross sections have been previously measured at the Tevatron \\cite{Publ54_xsec,cdf_incl} and the LHC \\cite{cms_incl,atlas_incl,lhcb_incl}. In this article we present a measurement using a refined analysis technique, which is optimized to be less dependent on the top quark mass. Compared to the previous \\dzero result \\cite{Publ54_xsec} we employ nearly a factor of 2 more data, which allows for higher precision tests of perturbative QCD (pQCD).\n\nThe mass of the top quark has been directly measured with a precision of less than 0.43\\% in a single measurement \\cite{d0mass}. The Tevatron combination currently yields a top quark mass of $174.34 \\pm 0.64$ GeV \\cite{TevatronMassCombo}. The direct measurements employed for the Tevatron combination are based on analysis techniques which use \\ensuremath{t\\bar{t}}\\xspace events provided by Monte Carlo (MC) simulation for different assumed values of the top quark mass $m_t$. Applying these techniques to data yields a mass quantity corresponding to the top quark mass scheme implemented in the MC and we refer to that quantity as the ``MC mass\" or $m_t^{\\mathrm{MC}}$. Theoretical arguments suggest that $m_t^{\\mathrm{MC}}$\\xspace is within about 1 GeV of the well-defined top quark pole mass \\cite{poleToMCmassDiff}. An alternative measurement approach employs the inclusive \\ensuremath{t\\bar{t}}\\xspace cross section to extract the mass of the top quark $m_t$. We assume the SM cross section dependence on $m_t$ as provided by the highest order of pQCD available at this time, namely a next-to-next-to-leading-order (NNLO) calculation. Comparing the dependence of the inclusive cross section on $m_t$, as calculated in pQCD, with the experimental measurement, accounting for the variation of the acceptance with $m_t$, yields a theoretically well-defined top quark pole mass. We employ this approach to extract a top quark pole mass with reduced experimental uncertainties due to our optimized analysis technique.\n\nEvents are selected in the lepton+jets ($\\ell +$jets\\xspace) and dilepton ($\\ell\\ell$\\xspace) top quark decay channels, where the lepton ($\\ell$) refers to either an electron or a muon. These channels correspond to $t\\bar{t} \\rightarrow W^{+}b W^{-}\\bar{b}$ decays, where in the $\\ell +$jets\\xspace channel one of the two $W$ bosons decays leptonically ($W \\rightarrow \\ell \\nu$), while the other $W$ boson decays hadronically ($W \\rightarrow q\\bar{q}'$). In the dilepton decay channel both $W$ bosons decay leptonically. Both decay channels include small contributions from electrons and muons stemming from the decay of $\\tau$ leptons ($t \\rightarrow Wb \\rightarrow \\tau \\nu_{\\tau} b \\rightarrow \\ell \\nu_{\\ell} \\nu_{\\tau} b$). \n\n\\section{Measurement strategy and outline} \n\\label{toc:strategy} \nThis measurement uses various multivariate analysis (MVA) techniques \\cite{mva,mva2,mva3}, as implemented in \\tmva \\cite{tmva}, to measure the inclusive cross section in the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace decay channels. For the dilepton decay channel we use a discriminant solely based on the output distribution of the MVA employed to identify jets that are likely to originate from $b$ quarks ($b$-tagged jets) \\cite{bid_nim}. This method is superior to a simple cut-and-count analysis since each \\ensuremath{t\\bar{t}}\\xspace event contains two $b$-quarks from the decays of top quarks. We refer to this method in the following as ``$b$-ID MVA.\" We construct a combined discriminant for events in the $\\ell +$jets\\xspace decay channel to make the best use of the distinct topological signature of top quark events along with $b$-tagging information. We refer to this method in the following as ``combined MVA.\" We use the entire distribution of the MVA discriminants in each decay channel to build MC templates. We use nuisance parameters to profile systematic uncertainties and to constrain their impact using data. For a combined inclusive \\ensuremath{t\\bar{t}}\\xspace cross section measurement we simultaneously employ the discriminant distribution of the $b$-ID MVA\\xspace in the dilepton decay channel and the combined MVA\\xspace in the $\\ell +$jets\\xspace decay channel in a nuisance-parameter-based profiling method. This combination benefits from the cross-calibration of the two different decay channels, leading to reduced systematic uncertainties.\n\nThis article is organized as follows. In Sec.\\ \\ref{toc:detector} we provide a brief review of the relevant aspects of the \\dzero detector and object reconstruction. A brief description of our event simulation approach, the QCD predictions employed, and a discussion of the assumptions for the modeling of the signal and background contributions follows in Sec.\\ \\ref{toc:generators}. The selection requirements for \\ensuremath{t\\bar{t}}\\xspace events in the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace decay channels are discussed in \\mbox{Sec.\\ \\ref{toc:data_samples}}. The determination of the sample composition in the two decay channels, the resulting event yields, and distributions of the data compared to MC are discussed in Sec.\\ \\ref{toc:sampleComp}. The details of the MVA techniques employed in this measurement are described in Sec.\\ \\ref{toc:methods_intro}. The methodology of the inclusive \\ensuremath{t\\bar{t}}\\xspace production cross section measurement is described in Sec.\\ \\ref{toc:fitting_collie}, and the systematic uncertainties relevant for this measurement are discussed in Sec.\\ \\ref{toc:xsec_sys}. The results of the cross section measurement are presented in Sec.\\ \\ref{toc:results}, followed by the extraction of the top quark pole mass given in Sec.\\ \\ref{toc:tmass}, and we conclude in Sec.\\ \\ref{toc:conclusion}.\n\n\\section{The \\dzero Detector and object reconstruction}\n\\label{toc:detector}\nThe \\dzero detector \\cite{d0detector} consists of several subdetectors designed for identification and reconstruction of the products of \\ensuremath{p\\bar{p}}\\xspace collisions. A silicon microstrip tracker (SMT) \\cite{smt,smt_l0} and central fiber tracker surround the interaction region for pseudorapidities\\footnote{The pseudorapidity $\\eta=-\\ln\\left[\\tan(\\theta\/2)\\right]$ is measured relative to the center of the detector, and $\\theta$ is the polar angle with respect to the proton beam direction. The azimuthal angle $\\phi$ is orthogonal to $\\theta$. The $z$ axis is pointing along the proton beam direction.} $|\\eta| < 3$ and $|\\eta| < 2.5$, respectively. These elements of the central tracking system are located within a superconducting solenoidal magnet generating a 1.9 T field, providing measurements for reconstructing event vertices and trajectories of charged particles. The SMT allows for a precision of $40~\\mathrm{\\mu m}$ or better for the reconstructed primary \\ensuremath{p\\bar{p}}\\xspace interaction vertex (PV) in the plane transverse to the beam direction. The impact parameter of typical charged-particle trajectories relative to the PV is determined with a precision between 20 and 50 $\\mu$m depending on the number of SMT hits and particle momentum. The impact parameter and its measurement uncertainty are key components of the lifetime-based identification of jets containing $b$ quarks. Particle energies are measured using a liquid argon sampling calorimeter that is segmented into a central calorimeter covering $|\\eta| < 1.1$, and two end calorimeters extending the coverage to $|\\eta| = 4.2$. Outside of the calorimetry, trajectories of muons are measured using three layers of tracking detectors and scintillation trigger counters, and an iron toroidal magnet generating a 1.8 T field between the first two layers \\cite{muonPaper}. Plastic scintillator arrays are located in front of the end calorimeter cryostats to measure the luminosity \\cite{lumi_nim,lumiTM}. The trigger and data acquisition systems are designed to accommodate the high luminosities provided by the Tevatron \\cite{triggerSystem}.\n\n\\subsection{Object reconstruction}\n\\label{toc:objectReco}\nThe object reconstruction is based on events identified by the D0 trigger system in which we require at least one lepton or at least one lepton and a jet. Since electrons mostly deposit energy in the electromagnetic (EM) calorimeter, the reconstruction and identification (ID) of electrons \\cite{elec_nim} is based on clusters in the EM calorimeter with an associated track. Such a track, as reconstructed by the central tracking detector, is required to have a minimum transverse momentum, $p_T$, of 5 GeV that points to the EM cluster within a window of $\\Delta \\eta \\times \\Delta \\phi = 0.05 \\times 0.05$. We define an angular separation $\\Delta R = \\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2}$ based on the distance $R = \\sqrt{\\eta^2+\\phi^2}$ in the $\\eta$-$\\phi$ plane. Electron candidates are required to be isolated by only accepting events with $\\Delta R(e,\\mathrm{jet}) > 0.5$ (the definition and reconstruction of a jet is discussed below). Further selection requirements on these electron candidates are applied by means of a multivariate analysis of the calorimeter shower profiles and tracking information. MC efficiencies are adjusted to match data efficiencies measured in electron enriched data samples.\n\nThe identification of muons \\cite{muon_nim} begins with a candidate formed using information from the muon system. Such a candidate is required to have a track, as reconstructed by the central tracking devices, associated with it. This association employs a $\\chi^2$ measure to match muon tracks provided by the muon detector with a track from the central tracking detector, taking into account effects from multiple scattering and energy loss, as well as the inhomogeneous magnetic field. Isolation criteria are applied based on the information from the hadronic and electromagnetic calorimeters and the central tracking devices. MC efficiencies are adjusted to match data efficiencies measured in muon enriched data samples.\n\nJets are reconstructed from energy depositions in the calorimeter using a midpoint cone algorithm \\cite{runIICone} employing a cone size of 0.5. Jets containing a muon within an angular separation of $\\Delta R(\\mu,\\mathrm{jet}) < 0.5$ are considered to originate from a semileptonic $b$-quark decay and are corrected for the momentum carried away by the muon and the neutrino. For this correction, it is assumed that the neutrino carries the same momentum as the muon.\n\nThe jet energy scale (JES) \\cite{jescorrection} corrects the measured energy of the jet to the energy of its constituent particles. The JES is derived using a quark-jet-dominated \\mbox{$\\gamma$ + jet} sample, and corrects data and MC for the difference in detector responses between jets and electromagnetic showers. An additional correction based on the single-particle response accounts for the different characteristics of quark and gluon jets. This correction implements a calibration of the simulated response to single particles inside a jet using data \\cite{jescorrection}. Jets in MC simulations have their transverse momenta smeared so that the simulated resolution matches that observed in data. Calibrations of the jet reconstruction and identification efficiency in MC simulations are determined using \\zplus data events. Jets are required to contain at least two tracks (see Sec.~\\ref{toc:data_samples}), and in MC simulations the corresponding efficiency is adjusted to match that derived in dijet data.\n\nThe presence of a neutrino in the final state of the top quark decay can be detected only from the energy imbalance in the transverse plane, denoted by \\met. This is reconstructed from the vector sum of the transverse energies of all calorimeter cells above a certain threshold. The vector opposite to this total visible momentum vector is denoted the raw missing energy vector. The fully corrected \\met is obtained after correcting for the effects of JES, muon momenta, and muon minimally ionizing deposition in the calorimeter.\n\n\\section{Monte Carlo Simulations and QCD predictions}\n\\label{toc:generators}\nWe use MC simulations to simulate physics processes, to model the reconstruction of the observables, and to estimate systematic uncertainties associated with the measurements. Different MC event generators are used to implement hard scattering processes based on leading-order (LO) and next-to-leading-order (NLO) QCD calculations, and are complemented with parton shower evolution programs. To simulate detector effects, generated events are passed through a detailed simulation of the \\dzero detector based on \\geant~\\cite{geant}. To account for effects from detector noise and additional overlapping \\ensuremath{p\\bar{p}}\\xspace interactions, events are randomly recorded in \\ensuremath{p\\bar{p}}\\xspace collisions and overlaid on the fully simulated MC events with the same instantaneous luminosity distribution as for data.\n\nThe \\ensuremath{t\\bar{t}}\\xspace samples are generated with \\mcatnlo version 3.4 \\cite{mcatnlo} or with \\alpgen version 2.11 \\cite{alpgen}, which both produce only on-shell top quarks. For events generated with \\mcatnlo, the parton showering is performed with \\herwig version 6.510 \\cite{herwig}. Events generated with \\alpgen employ parton showering as implemented by \\pythia version 6.409 \\cite{pythia} or \\herwig. We use the \\alppyt signal sample as our default to measure the \\ensuremath{t\\bar{t}}\\xspace cross section and the alternative \\mcherwig, or \\alpher, signal samples to estimate systematic uncertainties related to effects of NLO corrections or parton showering (see Sec.\\ \\ref{toc:xsec_sys}), respectively. Single top quark production $(q\\bar{q}' \\rightarrow t\\bar{b}, q'g \\rightarrow tq\\bar{b})$ is modeled using \\comphep \\cite{comphep,comphepMan}. For events generated with \\comphep, parton showering is implemented by \\pythia. The choice of the parton distribution functions (PDFs) made in generating MC events is CTEQ6L1 \\cite{cteq6l}, with the exception of \\mcatnlo and \\comphep (for the $t$-channel single top quark production), where CTEQ6M \\cite{cteq6m} PDFs are used. For all the MC simulations involving the generation of top quarks, a top quark mass of $m_t = 172.5$ GeV is used. The difference with the current Tevatron top quark mass combination of 174.34 GeV \\cite{TevatronMassCombo} has negligible impact on the analysis. For the \\ensuremath{t\\bar{t}}\\xspace $\\to $ \\lplus ($\\ell\\ell$\\xspace) decay channel the branching fraction $B$ of $0.342 \\pm 0.004$ ($0.04 \\pm 0.001$) \\cite{pdg} is used. These values include electrons and muons originating from the leptonic decay of $\\tau$ leptons ($\\tau \\rightarrow \\ell \\nu_{\\ell} \\nu_{\\tau}$).\n\nSeveral QCD predictions for inclusive \\ensuremath{t\\bar{t}}\\xspace cross sections have been calculated at higher orders than those included in the MC generators: approximate NNLO \\cite{mochUwer}, fully resummed NNLO \\cite{nnloInclXsec}, and an approximate next-to-NNLO \\cite{kidonakis}. The scale used to calculate the inclusive \\ensuremath{t\\bar{t}}\\xspace cross sections is set to $m_t$. For normalization of our MC events, we employ the approximate NNLO QCD calculation (using $m_t = 172.5$ GeV and the CTEQ6M PDF), which yields $7.48^{+0.48}_{-0.67}\\thinspace(\\mathrm{scale} + \\mathrm{pdf})~\\mathrm{pb}$. The result of this approximate NNLO calculation is close to the fully resummed NNLO QCD calculation (using $m_t = 172.5$ GeV), which finds $\\sigma_{\\mathrm{tot}}^{\\mathrm{res}} = 7.35 ^{+0.23}_{-0.27}\\thinspace(\\mathrm{scale} + \\mathrm{pdf})$ pb. The result of an approximate next-to-NNLO order calculation for \\mbox{$m_t = 173$ GeV} finds $\\sigma_{\\mathrm{tot}}^{\\mathrm{res}} = 7.37 \\pm 0.39\\thinspace(\\mathrm{scale} + \\mathrm{pdf})$ pb, very close to the fully resummed NNLO calculation. Both use the MSTW2008 NNLO PDF \\cite{mstw2008nnlo}.\n\nWe use the fully resummed NNLO QCD calculation as implemented in \\toppp \\cite{toppp_prg} to derive the theoretical \\ensuremath{t\\bar{t}}\\xspace cross section dependence as a function of the top quark mass (see Sec.\\ \\ref{toc:tmass}). The theoretical calculations use $\\sqrt{s}=1.96$ TeV as input parameter which is known at the Tevatron to a precision of 0.1\\%. This beam energy uncertainty yields a negligible 0.3\\% effect on the fully resummed NNLO \\ensuremath{t\\bar{t}}\\xspace cross section value.\n\n\\subsection{Modeling of background contributions in the $\\bm{\\ell+}$jets decay channel}\n\\label{toc:backgroundModel_ljets}\nThe main background to \\ensuremath{t\\bar{t}}\\xspace production in the $\\ell +$jets\\xspace decay channel is the production of \\wplus, including jets originating from heavy quarks. These events are generated with \\alpgen interfaced to \\pythia for showering and hadronization. The \\wplus final state can be split into four subsamples according to parton flavor: $Wb\\bar{b}+\\mathrm{jets}$, $Wc\\bar{c}+\\mathrm{jets}$, $Wc+\\mathrm{jets}$, and $W$ light partons $+$ jets (\\ensuremath{Wlp} + jets\\xspace), where light refers to gluons, $u$, $d$ or $s$ quarks. The additional ``jets'' in these \\wplus final states originate dominantly from gluon radiation. The \\wplus contribution dominates especially at the lower jet multiplicities. The LO \\alpgen cross sections are corrected for NLO effects as provided by \\mcfm \\cite{mcfm}: the $W+\\mathrm{jets}$ cross section is multiplied by 1.30, and the cross sections of $W+$ heavy flavor (WHF) processes are additionally multiplied by a scale factor $s^{\\mathrm{WHF}}$ of 1.47 for $Wb\\bar{b}+\\mathrm{jets}$ and $Wc\\bar{c}+\\mathrm{jets}$ and 1.27 for $Wc+\\mathrm{jets}$. Apart from these theoretical corrections we constrain the absolute background normalization by employing the data as described below in Sec.\\ \\ref{toc:sampleComp}. The $p_T$ distribution of the $W$ boson in MC simulation is reweighted to match the $p_T$ distribution of $Z$ bosons measured in \\dzero data \\cite{zbosonpt} multiplied by the SM ratio of these two distributions calculated at NLO using \\resbos \\cite{resbos}.\n\nThe second most dominant background contribution is due to multijet processes where a jet is misidentified as an electron in the \\eplus channel, or where a muon originating from the semileptonic decay of a heavy hadron appears to be isolated in the \\muplus channel. More details and a brief discussion on the determination of the multijet background are given in Sec.\\ \\ref{toc:sampleComp}.\n\nOther backgrounds include events from \\zplus production, which includes $Z$ bosons and virtual photons ($\\gamma^*$) decaying to electron, muon, or tau pairs. These events are generated with \\alpgen interfaced to \\pythia for showering and hadronization. The LO \\alpgen predictions are corrected using the NLO calculation of \\mcfm. The \\zplus cross section is multiplied by 1.30. The heavy flavor components of the \\zplus cross sections, $Z\/\\gamma^{*}c\\bar{c}+$jets, $Z\/\\gamma^{*}b\\bar{b} +$jets, are multiplied by an additional 1.67 and 1.52, respectively. The simulated $p_T$ distribution of the $Z$ boson is reweighted to match the measured $p_T$ distribution in $Z \\rightarrow \\ell \\ell$ data \\cite{zbosonpt}.\n\nThe single top quark background originates from $s$- and $t$-channel production, which are normalized to the NLO cross sections of 1.04 and 2.26 pb \\cite{singleTopXsec}, respectively. As the single top quark background yields only a few events passing all selection criteria described later, we do not consider the dependence of this background on $m_t$.\n\nDiboson production ($WW$, $WZ$, and $ZZ$ bosons) processes are another source of background and normalized to their NLO cross sections, calculated with \\mcfm, of 11.62 pb, 3.25 pb, and 1.33 pb, respectively.\n\n\\subsection{Modeling of background contributions in the $\\bm{\\ell \\ell}$ decay channel}\n\\label{toc:backgroundModel_ll}\nThe backgrounds in the dilepton decay channel are smaller than in the $\\ell +$jets\\xspace decay channel. The dominant source is \\zplus production, followed by diboson production. For both processes the modeling employs the same implementation as described above for the $\\ell +$jets\\xspace decay channel.\n\nThe third most dominant source of background is multijet events, with the determination summarized in Sec.\\ \\ref{toc:sampleComp}.\n\n\\section{Event Selection}\n\\label{toc:data_samples}\nThis analysis is based upon the full Tevatron data sample recorded by the \\dzero detector at $\\sqrt{s}=1.96$ TeV and, after applying data quality requirements, corresponding to an integrated luminosity of 9.7 fb${}^{-1}$ \\cite{lumi_nim}. The general selection criteria applied to both the $\\ell +$jets\\xspace and dilepton decay channels are summarized in the following:\n\\begin{enumerate}\n\\item \\label{sel:vertexzpv} Accepted events have a PV within $|z_{\\mathrm{PV}}|<60$~cm of the center of the detector along the beam axis.\n\\item \\label{sel:vertextrk} The number of tracks associated with the PV is greater or equal three.\n\\item \\label{sel:jets} After correcting the jet energy to the particle level, only jets with a transverse momentum $p_{T}>20$~GeV and $\\left|\\eta\\right|<2.5$ are selected. \n\\item \\label{sel:jetsVC} Jets which satisfy the $b$-tagging requirement are required to have at least two tracks coming from the PV. More details on jet requirements for the individual decay channels are provided below.\n\\item \\label{sel:leptons} Identified leptons are required to originate from the PV by demanding $|\\Delta z(\\ell, \\mathrm{PV})| < 1~\\mathrm{cm}$. These $z$ values correspond to the point of closest approach to the beam line of these tracks. \n\\item \\label{sel:leptonsdR} To ensure that electrons are isolated, an angular separation in $\\Delta R$ of at least 0.5 between an electron and the closest jet is required.\n\\end{enumerate}\n\nThe measurements in both decay channels employ the $b$-tagging discriminant output distribution as provided by the $b$-ID MVA. The discriminant combines variables that characterize the presence and properties of secondary vertices and tracks within jets \\cite{bid_nim}. We do not impose any requirements on this discriminant; instead we employ the entire distribution to measure the inclusive cross section as described in Sec.~\\ref{toc:methods_intro}. \n\nThe specific selection requirements for $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace events are described below; the requirements are chosen such that the selections are mutually exclusive.\n\n\\begin{figure*}[ht]\n \\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1d.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1e.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1f.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1g.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1h.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig1i.eps}\n\\end{center}\n \\caption{Distributions of (a)--(c) the scalar sum of the $p_T$ values of the lepton and jets, $H_T$, (d)--(f) $\\met$ and (g)--(i) the maximal MVA $b$-ID value of all jets, \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace, for events with a lepton and two, three or four or more jets. The data are compared to the sum of predicted contributions from signal and background processes, using the theoretical value of the inclusive $t\\bar{t}$ cross section of $7.48~\\mathrm{pb}$ \\cite{mochUwer} and $m_t=172.5$ GeV. The highest bin in the histograms includes overflows. The ratios of data to the sum of the signal and all background contributions are shown in the panels below the individual distributions. Only statistical uncertainties of the data are shown and the $\\chi^2\/\\mathrm{ndf}$ values only take statistical uncertainties into account.} \\label{fig:ljets_control}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n \\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2d.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2e.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig2f.eps}\n \\end{center}\n \\caption{Distributions of (a)--(c) the transverse mass of the lepton + \\met system, $m_T^W$, and (d)--(f) $p_{T}^{\\mathrm{lepton}}$ for events with a lepton and two, three or four or more jets. The data are compared to the sum of predicted contributions from signal and background processes, using the theoretical value of the inclusive $t\\bar{t}$ cross section of $7.48~\\mathrm{pb}$ \\cite{mochUwer} and $m_t=172.5$ GeV. The highest bin in the histograms includes overflows. The ratios of data to the sum of the signal and all background contributions are shown in the panels below the distributions. Only statistical uncertainties of the data are shown and the $\\chi^2\/\\mathrm{ndf}$ values only take statistical uncertainties into account.} \\label{fig:ljets_controlO}\n\\end{figure*}\n\n\n\\subsection{Event selection in the $\\bm{\\ell+}$jets decay channel}\nThe selection requirements for the cross section measurement for the $\\ell +$jets\\xspace channel are very similar to the ones described in Ref.~\\cite{diffXsecPaper} and are summarized briefly in the following:\n\n\\begin{enumerate}\n\\item \\label{sel:triggerLJ} The trigger requirement is a logical ``OR\" of the conditions for at least ``one lepton\" and for at least ``a lepton plus a jet\" in an event. Lepton trigger thresholds of 15 or 10 GeV were implemented for the single lepton trigger or the lepton plus jet trigger, respectively. \n\\item \\label{sel:leptonsLJ} Exactly one isolated lepton with a transverse momentum $p_{T}>20$~GeV and $\\left|\\eta\\right|<1.1$ (for electrons) or $\\left|\\eta\\right|<2$ (for muons) is required. Events with more than one lepton satisfying these criteria are rejected.\n\\item \\label{sel:met} We require $\\met>20$~GeV.\n\\item \\label{sel:muonQual} For the \\muplus sample we remove misreconstructed muons by requiring upper limits on the transverse mass of the muon + \\met system, $M_T^W$, of $M_T^W < 250$ GeV and \\met$<250$ GeV. To further remove such events, we employ an additional requirement on the significance of the track curvature described in more detail in Ref.~\\cite{diffXsecPaper}.\n\\item \\label{sel:fakes} To reduce multijet background we require a minimum separation between the direction of the lepton and the direction of the missing momentum \\cite{diffXsecPaper}: \\mbox{$\\Delta \\phi(e, $ \\metNo$) > 2.2 - 0.045 \\cdot \\metNo\/\\mathrm{GeV}$} and \\mbox{$\\Delta \\phi(\\mu, $ \\metNo$) > 2.1 - 0.035 \\cdot \\metNo\/\\mathrm{GeV}$}.\n\\item \\label{sel:jetsLJ} At least two jets are required. To suppress jets from additional collisions, jets are required to contain at least two tracks with a closest approach in $z$ to the PV of less than 1 cm.\n\n\\end{enumerate}\n\n\\subsection{Event selection in the dilepton decay channel}\nIn addition to the general selection requirements discussed in the opening of this section, additional requirements specific to the dilepton channel are made. The selection requirements for this cross section measurement are very similar to those used for the leptonic asymmetry measurements in the dilepton channel published earlier~\\cite{leptonicAfbdilepton} and are summarized briefly in the following list.\n\n\\begin{enumerate}\n\\item \\label{sel:globtrig} In the $e\\mu$ channel, no explicit trigger requirement is applied, whereas in the $ee$ or $\\mu\\mu$ channels single lepton triggers with thresholds at 15 GeV are employed.\n\\item \\label{sel:leptonsLL} Electrons are required to have a transverse momentum of $p_T> 15$~GeV and $|\\eta|<2.5$. We exclude the $1.1 < |\\eta| < 1.5$ region with poor resolution.\n\\item \\label{sel:leptonsLLmu} Muons are selected with $p_T > 15$ GeV and $|\\eta|<2$. To remove misreconstructed muons we require muons to have $p_T<200$~GeV for the dimuon channel.\n\\item \\label{sel:tighte} For the $e\\mu$ channel exactly one electron and one or more muons are required.\n\\item \\label{sel:mu} For the $\\mu\\mu$ channel two or more muons are required.\n\\item \\label{sel:die} For the $ee$ channel two or more electrons are required.\n\\item \\label{sel:emupair} The two selected leptons must have opposite charges. If more than one oppositely charged lepton pair is found, the lepton pair with the largest $p_T$ scalar sum is chosen.\n\\item At least one jet is required in the $e\\mu$ channel and at least two jets are required in the $\\mu\\mu$ and $ee$ channel. \n\\item \\label{sel:leptonsQualLL} Additional quality requirements are imposed to remove background from bremsstrahlung. \n\\item \\label{sel:topological} We further reduce background contributions by imposing the following topological requirements: in the $ee$ channel we require a \\met significance of $\\ge 5$, in the $\\mu\\mu$ channel we require $\\met > 40$~GeV and a \\met significance of $\\ge 2.5$, and in the $e\\mu$ channel we require $H_{T}>110$~GeV, where $H_{T} = p_{T}(\\mathrm{leading~lepton}) + p_{T}(2~\\mathrm{leading~jets})$. More details are described in Ref. \\cite{leptonicAfbdilepton}.\n\\end{enumerate}\n\n\\begin{table*}[htp]\n \\caption{Expected number of events in the $\\ell +$jets\\xspace channel with two, three or $\\ge$ four jets. The sum of signal and background agrees well with the number of data events by construction; uncertainties are statistical and systematic added in quadrature (see Sec.\\ \\ref{toc:sys_sampleComp} for details). Events from \\ensuremath{t\\bar{t}}\\xspace dilepton decays are treated as background and denoted as ``\\ensuremath{t\\bar{t}}\\xspace $\\ell \\ell$.\"} \\label{tab:yields_ljets}\n\\begin{ruledtabular}\n\\begin{tabular}{l d{-1} d{-1} d{-1} d{-1} d{-1} d{-1}}\n\\multicolumn{7}{c}{$\\ell +$jets\\xspace decay channel}\\\\\n\\multicolumn{1}{c}{Process} & \\multicolumn{1}{c}{$e + 2$ jets} & \\multicolumn{1}{c}{$e + 3$ jets} & \\multicolumn{1}{c}{$e + \\ge 4$ jets} & \\multicolumn{1}{c}{$\\mu + 2$ jets} & \\multicolumn{1}{c}{$\\mu + 3$ jets} & \\multicolumn{1}{c}{$\\mu + \\ge 4$ jets} \\\\ \\hline \\T\nMultijet\t& 9160.2350 & 2266.550 & 464.120\t& 1546.$\\hphantom{0}$630 & 418.170 & 99.40\\\\ \\T \nSingle top\t& 471.$\\hphantom{00}$60 & 129.$\\hphantom{0}$20 & 27.$\\hphantom{00}$5 & 331.$\\hphantom{00}$40\t& 92.$\\hphantom{0}$10 \t& 20.$\\hphantom{0}$3 \\\\ \\T \n\\ensuremath{Wlp} + jets\\xspace\t & 37937.\\,\\,^{1350}_{700} & 5544.\\,\\,^{200}_{100} & 850.\\,\\,\\,\\,^{30}_{20} & 32701.\\,\\,^{1150}_{600} & 5313.\\,\\,^{200}_{100} & 835.\\,^{30}_{15} \\\\ \\T \n\\whf\t\t& 6020.\\,\\,^{1000}_{1400} & 1502.\\,\\,^{250}_{350} & 329.\\,\\,\\,\\,^{60}_{80} & 4998.\\,\\,^{850}_{1150} & 1391.\\,\\,^{250}_{300} & 315.\\,^{50}_{70} \\\\ \\T \n\\ensuremath{Z\/\\gamma^{*}lp}+\\rm jets\\xspace\t & 2031.$\\hphantom{0}$400 & 390.$\\hphantom{0}$80 & 57.$\\hphantom{0}$10 & 2557.$\\hphantom{0}$500 & 422.$\\hphantom{0}$80 & 49.10 \\\\ \\T \n\\zhf\t\t & 369.$\\hphantom{00}$70 & 114.$\\hphantom{0}$20 & 24.$\\hphantom{00}$5 & 485.$\\hphantom{0}$100 & 120.$\\hphantom{0}$20 & 21.$\\hphantom{0}$5 \\\\ \\T \nDiboson\t & 1926.$\\hphantom{0}$140 & 338.$\\hphantom{0}$20 \t& 52.$\\hphantom{00}$5 \t& 1417.$\\hphantom{0}$100 \t& 249.$\\hphantom{0}$20 \t& 40.$\\hphantom{0}$5 \\\\ \\T \n\\ensuremath{t\\bar{t}}\\xspace $\\ell \\ell$ & 566.$\\hphantom{00}$30 & 182.$\\hphantom{0}$10 & 31.$\\hphantom{00}$5 & 345.$\\hphantom{00}$20 & 118.$\\hphantom{0}$10 & 22.$\\hphantom{0}$5 \\\\ \\hline \\T \n$\\sum\\,\\mathrm{background}$ & 58479.2900 & 10465.650 & 1834.140 & 44381.1650 & 8123.350 & 1402.80 \\\\ \\T \n\\ensuremath{t\\bar{t}}\\xspace $\\ell +$jets\\xspace & 669.$\\hphantom{00}$30 & 1460.$\\hphantom{0}$70 & 1177.$\\hphantom{0}$60 & 393.$\\hphantom{00}$20 & 1002.$\\hphantom{0}$50 & 909.50 \\\\ \\hline \\T\n$\\sum\\, \\mathrm{(signal + background)}$ & 59148.2900 & 11925.650 & 3011.140 & 44773.1650 & 9125.350 & 2310.80 \\\\ \\T \nData\t & 59122 & 11905\t& 3007 & 44736 & 9098 & 2325 \\\\\n \\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\\section{Sample Compositions}\n\\label{toc:sampleComp}\nWe distinguish between instrumental backgrounds and irreducible backgrounds from processes with final states identical to \\ensuremath{t\\bar{t}}\\xspace. Instrumental backgrounds are due to multijet processes where one or more jets are misidentified as an electron, or where one or more muons originating from the semileptonic decay of a heavy hadron appear to be isolated, and hence fulfill all selection requirements of a lepton stemming from the decay of a top quark. Irreducible backgrounds are for example due to \\wplus or \\zplus processes with the same final state as the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace top quark decay channel. Systematic uncertainties on the determination of the sample composition are discussed in Sec.\\ \\ref{toc:sys_sampleComp}. The following section describes the composition of the input or ``pre-fit\" $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace samples, which are used to extract the \\ensuremath{t\\bar{t}}\\xspace cross section as described in Sec.\\ \\ref{toc:fitting_collie}.\n\n\\subsection{Determination of the $\\ell +$jets\\xspace sample composition}\n\\label{toc:sampleCompLjets}\nThe irreducible background processes are estimated using MC simulations as described in Sec.\\ \\ref{toc:generators}. Compared to the $\\ell\\ell$\\xspace channel, the $\\ell +$jets\\xspace channel has a larger background fraction, each with larger systematic uncertainties than in the $\\ell\\ell$\\xspace channel. Since most of this background arises from \\ensuremath{W+\\ge 2} jets\\xspace production we estimate this contribution following the same approach as for the measurement of the differential \\ensuremath{t\\bar{t}}\\xspace cross section \\cite{diffXsecPaper}. The \\wplus cross section is iteratively scaled for each jet multiplicity bin separately by a \\wplus heavy-flavor scale factor $s_{\\mathrm{fit}}^{\\mathrm{WHF}}$ and \\wplus light-flavor scale factor $s_{\\mathrm{fit}}^{\\mathrm{WLF}}$ to match the number of data events after subtraction of all other instrumental and irreducible background contributions as well as the signal contribution. This approach yields reasonable initial values for the log-likelihood profile fit (introduced in Sec.\\ \\ref{toc:fitting_collie}). The details of the estimation of systematic uncertainties are described in Sec.\\ \\ref{toc:sys_sampleComp}. \n\nData-driven and MC methods are combined in the ``matrix method\" \\cite{matrixMethod, Publ54_xsec}, which is employed to model the instrumental background originating from multijet (MJ) processes in the $\\ell +$jets\\xspace channel. The MJ contribution is determined employing two samples of $\\ell +$jets\\xspace events: one applying the nominal lepton selection requirements and one with looser lepton selection requirements denoted ``loose.\" In addition, an orthogonal data sample is defined by requiring $\\not\\!\\!E_T < 10~\\mathrm{GeV}$ (the nominal requirement is $\\not\\!\\!E_T > 20~\\mathrm{GeV}$) and the above selection criteria for the signal sample. This data sample is enriched in MJ events and any contributions from isolated leptons, as expected from MC, are subtracted from all considered distributions. No real isolated leptons are assumed to be included. We determine the shape and absolute contribution (misidentification rate) of multijet events for different jet multiplicities by comparing this data sample with the data sample containing loose leptons but the same \\met requirement.\n\nFigures \\ref{fig:ljets_control} and \\ref{fig:ljets_controlO} show the modeling of the selected events in the $\\ell +$jets\\xspace sample with the background and signal contributions. The expected composition of the sample after the final selection is given in Table \\ref{tab:yields_ljets}.\n\n\\begin{table*}[htp]\n \\caption{Expected number of events in the $ee + \\ge 2$ jets, $\\mu\\mu + \\ge 2$ jets, $e\\mu + 1$ jets and $e\\mu + \\ge 2$ jets channels due to each process; uncertainties are statistical and systematic added in quadrature (see Sec.\\ \\ref{toc:sys_sampleComp} for details).} \\label{tab:yields_dilepton}\n\\begin{ruledtabular}\n\\begin{tabular}{l z{-1} z{-1} z{-1} z{-1}}\n\\multicolumn{5}{c}{dilepton decay channel}\\\\\n\\multicolumn{1}{c}{Process} & \\multicolumn{1}{c}{$ee + \\ge 2$ jets} & \\multicolumn{1}{c}{$\\mu\\mu + \\ge 2$ jets} & \\multicolumn{1}{c}{$e\\mu + 1$ jets} & \\multicolumn{1}{c}{$e\\mu + \\ge 2$ jets} \\\\ \\hline \\T\nMultijet\t\t & 5.7,\\,\\,^{0.9}_{0.9} & 7.0,\\,\\,^{3.3}_{2.6} & 28.3,^{6.6}_{6.6} & 32.5,^{7.4}_{7.4} \\\\ \\T\n\\ensuremath{Z\/\\gamma^{*} \\rightarrow \\ell\\ell}+\\rm jets\\xspace\t\t&66.6,^{17.9}_{17.2} & 107.6,^{22.1}_{22.0} & 74.6,^{15.8}_{15.8} & 57.5,^{13.8}_{13.4}\\\\ \\T\nDiboson\t \t & 9.9,\\,\\,^{2.4}_{2.2} & 12.6,\\,\\,^{2.8}_{3.0} & 38.5,^{4.6}_{4.2} & 14.7,^{3.7}_{3.5} \\\\ \\hline \\T\n$\\sum\\,\\mathrm{background}$ & 82.2,$\\hphantom{0}$18 & 172.2,$\\hphantom{0}$22 & 141.4,18 & 104.7,15 \\\\ \\T \n\\ensuremath{t\\bar{t}}\\xspace $\\ell \\ell$ \t& 107.7,$\\hphantom{0}$15 & 101.5,$\\hphantom{0}$12 & 86.5,11 & 313.7,38 \\\\ \\hline \\T\n$\\sum\\, \\mathrm{(signal + background)}$ &190,$\\hphantom{0}$23 & 229,$\\hphantom{0}$25 & 228,21 & 418,42 \\\\ \\T\nData\t & 215 & 242 & 236 & 465 \\\\\n \\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\\begin{figure*}[ht]\n \\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig3a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig3b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig3c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig3d.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig3e.eps}\n \\end{center}\n\\caption{Distributions of (a) the scalar sum, $H_{T}$, of the $p_T$ values of the leading lepton and leading and second-leading jets, (b) $\\met$, (c) \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace, (d) the number of jets, and (e) lepton $p_T$ for $\\ell\\ell$\\xspace final states with at least one jet in the $e\\mu$ and at least two jets in the $ee$ and $\\mu\\mu$ channels. The data are compared to the sum of predicted contributions from signal and background processes, using the theoretical value of $7.48~\\mathrm{pb}$ for the $t\\bar{t}$ cross section and $m_t=172.5$ GeV. The highest bin in the histograms includes overflows. The ratios of data to the sum of the signal and all background contributions are shown in the panels below the distributions. Only statistical uncertainties of the data are shown and the $\\chi^2\/\\mathrm{ndf}$ values only take statistical uncertainties into account.} \\label{fig:dilepton_control}\n\\end{figure*}\n\n\n\n\\subsection{Determination of the $\\bm{\\ell \\ell}$ sample composition}\nThe main backgrounds in the dilepton final state originate from $Z\/\\gamma^{*} \\rightarrow$ $\\ell\\ell$\\xspace instrumental backgrounds, and diboson production ($WW$, $WZ$, $ZZ$). The $Z\/\\gamma^{*} \\rightarrow$ $\\ell\\ell$\\xspace and diboson backgrounds are evaluated from MC as described in Sec.~\\ref{toc:backgroundModel_ll}. We use a mixture of MC and data-driven approaches for the instrumental background determination. Similarly to the $\\ell +$jets\\xspace channel, the normalization of events with jets misidentified as electrons is estimated directly from data using the matrix method separately for the $ee$ and $e\\mu$ channels \\cite{leptonicAfbdilepton}. The estimation of the instrumental background from events with jets producing muons is based on events with two leptons of the same charge in the $\\mu\\mu$ and $e\\mu$ channels, where for the latter the contribution from misidentified electrons is subtracted beforehand. The electron misidentification rate for the matrix method is derived from an orthogonal data sample by requiring that the two leptons have the same charge. This sample is selected applying the same selection criteria as for \\ensuremath{t\\bar{t}}\\xspace events, but the final selection on $H_T$ is replaced with requiring \\met significance $<15$ to avoid contribution from $W \\rightarrow e\\nu +$ jets events. The remaining contribution of the instrumental background is small, and we combine the MJ and \\ensuremath{W}+\\rm jets\\xspace components to reduce the statistical uncertainty on the background estimate. The shape of the MJ template is derived using a looser electron selection of only $e\\mu$ events and employed for all $\\ell\\ell$\\xspace channels. Since the MJ contribution in the $ee$ and $\\mu\\mu$ sample is small the difference in shape is not significant. \n\nThe yields, after applying the described selection, are given in Table \\ref{tab:yields_dilepton} for the individual channels.\n\nFigure \\ref{fig:dilepton_control} demonstrates the quality of the modeling of the selected events in the $\\ell\\ell$\\xspace sample with the background and signal contributions, using a theoretical inclusive $t\\bar{t}$ cross section of $7.48~\\mathrm{pb}$ \\cite{mochUwer} and $m_t = 172.5$ GeV. The slight disagreement in the \\met distribution between data and the sum of signal and background contributions is covered by systematic uncertainties related to the JES and jet $p_T$ resolution.\n\n\\section{Multivariate measurement techniques}\n\\label{toc:methods_intro}\nThe inclusive \\ensuremath{t\\bar{t}}\\xspace cross section \\xsttbar is measured using the different MVA techniques introduced in Sec.\\ \\ref{toc:strategy}. We use different discriminant output distributions of decision trees to separate the signal from the background for the \\lplus and $\\ell\\ell$\\xspace final states. To construct the MVA, the event sample is subsequently split into smaller samples until each event is placed in one of a set of distinct nodes. At each splitting point the separation is optimized by employing training samples for the signal and background contributions. The output or discriminant value provides the probability of an event to be signal. In the case of the combined MVA\\xspace applied in the \\lplus channel we use each individual background contribution in the training process and verify that there is no bias due to overtraining of the method. We employ a method called ``boosted decision trees with gradients\" (BDTG) \\cite{bdtgRef}. The BDTG implements additional weights to minimize classification errors in the training sample and improve signal to background separation. To measure the cross section we perform a log-likelihood profile fit of MC simulation templates to the data using a nuisance parameter for every source of systematic uncertainty as described in Sec.\\ \\ref{toc:fitting_collie}. \n\n\\subsection{MVA methods in the $\\bm{ \\ell+}$jets channel}\n\\label{toc:methods_intro_topo}\nEvents in the $\\ell +$jets\\xspace channel are separated into six different samples according to the lepton type and the number of jets, $n_\\mathrm{jet} = 2, 3, \\geq 4$. We studied a further separation according to the number of $b$-tagged jets which gave increased systematic uncertainties and was not used for the final measurement. To build a discriminant, a total of 50 variables were analyzed. The individual distributions are verified to have a good modeling of the data by the MC by means of a Kolmogorov-Smirnov test \\cite{KStest} and a $\\chi^2$ test. We exclude all variables with poor modeling of the data. One variable among those studied, namely the maximal MVA $b$-ID value of all jets in the event, \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace, shows only moderately good modeling of the data by MC [see Figs. \\ref{fig:ljets_control}(g)--\\ref{fig:ljets_control}(i)]. However, the data are reasonably described when considering systematic uncertainties related to $b$-tagging. The \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace variable provides good separation power, and hence we use that variable in the discriminant.\n\nDepending on the number of jets in an event, at least 24 variables are selected as input to the combined MVA\\xspace for each jet multiplicity and lepton type bin (see the Appendix for a detailed variable description). In particular we include various MVA $b$-ID discriminants into the combined MVA discriminant, which allows superior signal-to-background separation, as discussed in the following paragraph. All selected variables are defined in the Appendix. Adding more variables has a negligible effect on the signal-to-background separation of the discriminant. \n\nFigure \\ref{fig:ROC_example} shows the separation of signal from background events in the $\\ell +$jets\\xspace decay channel using as an example events with exactly three jets in the \\eplus channel. In this figure we show the performance of the combined MVA\\xspace discriminant compared to the performance of a similar MVA discriminant that does not include the \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace input variable (topological MVA\\xspace). For comparison we also show the performance of the MVA solely based on \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace. The $b$-ID MVA\\xspace has a higher signal and lower background efficiency than the topological MVA\\xspace for a signal efficiency less than about 80\\%. Above this point the topological MVA\\xspace surpasses the $b$-ID MVA\\xspace. Compared to these two MVA methods the combined MVA\\xspace shows superior behavior, with the area under the curve increased by $6-10$\\%. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=0.975\\columnwidth,angle=0]{fig4.eps}\n \\end{center}\n \\caption{Signal efficiency vs background rejection for different MVA choices for the $e + 3$ jets sample. For details, see text in Sec.\\ \\ref{toc:methods_intro}. The dashed line from $(0,1)$ to $(1,0)$ is shown for reference.} \\label{fig:ROC_example}\n\\end{figure}\n\nFigure \\ref{fig:prefit_topo_emujets} shows the pre-fit MVA output distributions of the combined MVA method using a theoretical \\ensuremath{t\\bar{t}}\\xspace cross section of $7.48$ pb. \n\n\\begin{figure*}[htb]\n\\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5d.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5e.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig5f.eps}\n\\end{center}\n\\caption{Pre-fit output distributions of the combined MVA discriminant using the theoretical \\ensuremath{t\\bar{t}}\\xspace cross section and $m_t=172.5$ GeV for the $e + $ two, three or four or more jets [panels (a) -- (c)], and for the $\\mu + $ two, three or four or more jets [panels (d) -- (f)]. Statistical uncertainties of the data are shown and the pre-fit systematic uncertainties are indicated by the hashed band in the bottom panel of the histogram. The $\\chi^2\/\\mathrm{ndf}$ values take statistical and systematic uncertainties into account.}\n\\label{fig:prefit_topo_emujets}\n\\end{figure*}\n\n\n\\subsection{MVA ${\\bm b}$-jet method in the $\\bm{ \\ell \\ell}$ channel}\n\\label{toc:methods_intro_mvab}\nWe measure the \\ensuremath{t\\bar{t}}\\xspace production cross section in the $\\ell\\ell$\\xspace channel using \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace to separate signal from background. Events in the dilepton channel are separated into samples according to the lepton type and the number of jets. Due to the small background contribution and the size of the signal contribution in the dilepton channel, the separation provided by the \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace is sufficiently good and the combined MVA\\xspace was not employed for the $\\ell\\ell$\\xspace channel.\n\nThe MVA output distributions in the dilepton channel allow one to distinguish between \\ensuremath{t\\bar{t}}\\xspace events dominantly located at high output values and the most dominant \\zplus background contribution typically located at low output values. For the $e\\mu$ channel we split the sample into sub-samples with exactly one and $\\ge$ two jets, whereas for $ee$ and $\\mu\\mu$ only events with two or more jets are used. The \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace distributions are shown in Fig.\\ \\ref{fig:prefit_mva_dilep}. A theoretical \\ensuremath{t\\bar{t}}\\xspace cross section of 7.48 pb is used \\cite{mochUwer}.\n\n\\begin{figure*}[htb]\n\\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig6a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig6b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig6c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig6d.eps}\n\\end{center}\n\\caption{The pre-fit MVA output distributions (using the theoretical \\ensuremath{t\\bar{t}}\\xspace cross section and $m_t=172.5$ GeV) of the $b$-ID MVA\\xspace given by the jet with the maximum value, \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace, for (a) $e\\mu$ events with exactly one jet, (b) $e\\mu$ events with at least two jets, (c) $ee$ events with at least two jets, and (d) $\\mu\\mu$ events with at least two jets. Statistical uncertainties of the data are shown and the pre-fit systematic uncertainties are indicated by the hashed band in the bottom panel of the histogram. The $\\chi^2\/\\mathrm{ndf}$ values take statistical and systematic uncertainties into account.}\n\\label{fig:prefit_mva_dilep}\n\\end{figure*}\n\n\\section{Cross section determination}\n\\label{toc:fitting_collie}\nAs shown in Figs. \\ref{fig:prefit_topo_emujets} and \\ref{fig:prefit_mva_dilep}, the MVA output distributions for the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace channels allow the discrimination of the \\ensuremath{t\\bar{t}}\\xspace signal contribution and the most dominant background sources. We perform a simultaneous fit of MC templates to the data using the software package \\collie (A Confidence Level Limit Evaluator) \\cite{collie} to determine the inclusive \\ensuremath{t\\bar{t}}\\xspace cross section \\xsttbar. \n\nThe combined likelihood includes prior probability densities on systematic uncertainties $\\pi(\\vec{\\theta})$, and is based on the product of likelihoods for the individual channels, each of which is a product over bins of histograms of a particular analysis channel:\n\\begin{equation}\n \\label{eqn:xsecDef}\n{\\cal L}(\\vec{s},\\vec{b}|\\vec{n},\\vec{\\theta}) \\times \\pi(\\vec{\\theta}) = \\prod_{i=1}^{N_C} \\prod_{j=1}^{N_{\\mathrm{bins}}} \\mu_{ij}^{n_{ij}} \\dfrac{e^{-\\mu_{ij}}}{n_{ij}!} \\times \\prod_{k=1}^{n_\\mathrm{sys}} e^{-\\theta^{2}_{k}\/2}.\n\\end{equation}\nThe first product is over the number of channels ($N_C$) and the second product is over histogram bins containing $n_{ij}$ events, binned in ranges of the final discriminants used for the individual analyses. The predictions for the bin contents are $\\mu_{ij} = s_{ij}(\\vec{\\theta}) + b_{ij}(\\vec{\\theta})$ for channel $i$ and histogram bin $j$, where $s_{ij}$ and $b_{ij}$ represent the expected signal and background in the bin. The predictions $\\mu_{ij}$ include effects from limited detector resolution and efficiency, including those from trigger and selection efficiencies and for the kinematic and geometric acceptance.\n\nSystematic uncertainties are parametrized by the dependence of $s_{ij}$ and $b_{ij}$ on $\\vec{\\theta}$. Each of the $n_\\mathrm{sys}$ components of $\\vec{\\theta}$, $\\theta_k$, corresponds to a single independent source of systematic uncertainty scaled by its standard deviation, and each parameter may affect the predictions of several sources of signal and background in different channels, thus accounting for correlations. For the combination of the combined MVA\\xspace in the $\\ell +$jets\\xspace channel with the $b$-ID MVA\\xspace in the $\\ell\\ell$\\xspace channel using \\collie systematic uncertainties are either assumed to be fully correlated or not correlated (see Sec.\\ \\ref{toc:xsec_sys}).\n\n\\collie models nuisance parameters using a Gaussian prior probability density function specified by $\\pm1$ standard deviation of the systematic uncertainty in question [see Eq.\\ (\\ref{eqn:xsecDef})]. For asymmetric uncertainties, two half-Gaussian functions model separately the positive and negative parts of the nuisance parameters. In the log-likelihood profile fit, the nuisance parameters and the cross section are simultaneously fitted. Hence, sources of systematic uncertainties not only contribute to the final cross section uncertainty, but also shift the fitted cross section value. Since the simultaneous fit is constrained by data it also provides a reduction of the impact of the different systematic uncertainty sources.\n\nThe central value of the \\collie fit provides a scale factor that is applied to the expected number of signal events using the theoretical \\ensuremath{t\\bar{t}}\\xspace cross section. The scaled number of signal events, $N^{\\mathrm{signal}}$, is preferred by the data and the systematic uncertainties. To determine $\\sigma_{t\\bar{t}}$ for the full phase space of \\ensuremath{t\\bar{t}}\\xspace production we correct $N^{\\mathrm{signal}}$ for the detector efficiency and acceptance, the branching ratio and the integrated luminosity.\n\n\\section{Systematic Uncertainties}\n\\label{toc:xsec_sys}\nSystematic uncertainties are assessed by varying the values of a specific parameter in the modeling of the data, and determining the effect on the distributions or MC templates of the combined MVA\\xspace or the $b$-ID MVA\\xspace. Compared to the earlier D0 measurement \\cite{Publ54_xsec} we employ a more refined strategy for systematic uncertainties including the newly added hadronization uncertainty. Unless otherwise stated, the magnitude of the parameter modifications is obtained from alternative calibrations of the MC simulation. Each of the modified MVA distributions is used to determine the effect of systematic uncertainties. As described in Sec.\\ \\ref{toc:fitting_collie} all nuisance parameters are fitted simultaneously with the nominal MVA distributions to measure the \\ensuremath{t\\bar{t}}\\xspace production cross section. Systematic uncertainties are constrained by the data and are minimized since we use the full shape information of the MVA templates. A further reduction of correlated systematic uncertainties is achieved when combining the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace decay channels, since systematic uncertainties are then cross-calibrated.\n\nIn total we assign 39 ($\\ell +$jets\\xspace channel), 37 ($\\ell\\ell$\\xspace channel), and 53 (combination) individual systematic uncertainties as discussed below for the decay channels. The pre-fit systematic uncertainties are summarized in Table \\ref{tab:pre_syst_uncorr}, whereas the post-fit effects of the systematic uncertainties are summarized in Table \\ref{tab:syst_uncorr}. We group systematic uncertainties addressing a similar object, e.g.\\ jet-related ones, into a combined source of systematic uncertainty.\n\n\\begin{table}[tp]%\n\\begin{center}\n\\caption {\\label{tab:pre_syst_uncorr} Sources of grouped pre-fit systematic uncertainties for the \\ensuremath{t\\bar{t}}\\xspace cross section measurement assuming the theoretical \\ensuremath{t\\bar{t}}\\xspace cross section of 7.48 pb \\cite{mochUwer} and $m_t = 172.5$ GeV. The systematic uncertainty in pb from each source on the inclusive cross section is given for the $\\ell +$jets\\xspace and the $\\ell\\ell$\\xspace channels. The column denoted as ``Type\" refers to a systematic uncertainty affecting the shape and normalization $S$ or only the normalization $N$ of a MVA distribution. The numbers presented for shape-dependent uncertainties represent averages across the entire set of distributions and all samples. The term ``n.a.\" is used where a systematic uncertainty is not applicable for a given decay channel.}\n\\end{center}\n\\begin{ruledtabular}\n\\begin {tabular}{llclc}\n\\multicolumn{1}{l}{Source of uncertainty} &\n\\multicolumn{1}{c}{$\\delta_{\\ell \\mathrm{+ jets}}$, pb} & \\multicolumn{1}{c}{Type} & \n\\multicolumn{1}{c}{$\\delta_{\\ell \\ell}$, pb} & \\multicolumn{1}{c}{Type} \\\\ \\hline\n\n\\textit{Signal modeling} \t\t\t\t&\t\t\t\t& \\T \\\\\n~~~Signal generator & \t\t\t\t\t\t $\\pm 0.86$\t \t& $S$ & $\\pm 0.28$ & $S$ \\T \\\\\n~~~Hadronization & \t\t\t\t $\\pm 0.59$\t\t & $S$ & $\\pm 0.29$ & $S$ \\T \\\\\n~~~Color reconnection & \t\t\t $\\pm 0.21$\t \t& $S$ & $\\pm 0.08$ & $S$ \\T \\\\\n~~~ISR\/FSR variation &\t\t\t\t$\\pm 0.07$\t\t & $S$ & $\\pm 0.04$ & $S$ \\T \\\\\n\\textit{PDF} & \t\t\t\t\t\t$\\pm 0.20$\t\t & $S$ & $\\pm 0.07$ & $S$ \\T \\\\\n\\textit{Detector modeling}\t&\t & \\T\\\\\n~~~Jet modeling \\& ID & \t\t\t $\\pm 0.33$\t\t & $S$ & $\\pm 0.34$\t& $S$ \\T \\\\\n~~~$b$-jet modeling \\& ID & \t\t\t$\\pm 0.19$\t\t & $S$ & $\\pm 0.56$ & $S$ \\T \\\\\n~~~Lepton modeling \\& ID\t &\t\t $\\pm 0.23$\t \t& $S$ & $\\pm 0.31$ & $N$ \\T \\\\\n~~~Trigger efficiency& \t\t\t $\\pm 0.06$\t\t & $N$ & $\\pm 0.07$ & $N$ \\T \\\\\n~~~Luminosity\\footnote{To prevent constraining the luminosity uncertainty by data, we do not assign the luminosity uncertainty to the \\ensuremath{W}+\\rm jets\\xspace and multijet contribution.} & \t\t\t\t $\\pm 0.32$\t \t& $N$ & $\\pm 0.32$ & $N$ \\T \\\\\n\\textit{Sample Composition} \t\t&\t\t& \t\t& &\\T \\\\\n~~~MC cross sections &\t\t $\\pm 0.02$ \t & $N$ & $\\pm 0.12$ & $N$ \\T \\\\\n~~~$Z \/ W$ $p_T$ reweighting &\t\t$\\pm 0.03$\t \t& $S$ & $\\pm 0.28$ & $S$ \\T \\\\\n~~~Multijet contribution &\t\t\t $\\pm 0.26$ \t& $S$ & $\\pm 0.13$& $S$ \\T \\\\\n~~~\\zplus SF\t\t\t& \t\t\t$\\pm 0.09$ \t & $S$ &$\\pm 0.12$ &$S$ \\T \\\\\n~~~\\wplus HF SF & \t\t\t\t\t$\\pm 0.41$ \t & $S$ & $\\mathrm{n.a.}$ & $\\mathrm{n.a.}$\\T \\\\\n~~~\\wplus LP SF & \t\t\t\t\t$\\pm 0.16$ & $S$ & $\\mathrm{n.a.}$ & $\\mathrm{n.a.}$\\T \\\\\n\\textit{MC statistics} \t\t\t\t&\t$\\pm 0.01$ & $\\mathrm{S}$ & $\\pm 0.10$ & $S$ \\T \\\\ \n\\end {tabular}\n\\end{ruledtabular}\n\\end {table}\n\n\\begin{table*}[tp]%\n\\caption {\\label{tab:syst_uncorr} Sources of grouped post-fit systematic uncertainties for the \\ensuremath{t\\bar{t}}\\xspace cross section measurement. Owing to the complexity of the correlations among the systematic uncertainties, we only show here the symmetrized uncertainties. For the top quark signal and background contributions we assume $m_t=172.5$ GeV. For each group, the systematic uncertainty on the inclusive cross section is given for the $\\ell +$jets\\xspace, $\\ell\\ell$\\xspace, and combined measurement. The last column shows the shift in pb in the combined inclusive cross section due to a particular group. A shift of $0.00$ pb indicates a shift of $0.004$ pb or less. The total uncertainty is provided by the nominal fit, when including all individual sources of systematic uncertainties, and denoted as ``central \\collie\". For comparison only we also provide the combined systematic uncertainty (quadratic sum) of the grouped post-fit systematic uncertainties. Due to correlations between the systematic uncertainties, that value differs from the total systematic uncertainty of the nominal fit.}\n\\begin{ruledtabular}\n\\begin {tabular}{lcccc}\n\\multicolumn{1}{l}{Source of uncertainty} & \n \\multicolumn{1}{c}{$\\delta_{\\ell \\mathrm{+ jets}}$, pb} & \\multicolumn{1}{c}{$\\delta_{\\ell \\ell}$, pb} & \\multicolumn{1}{c}{$\\delta_{\\mathrm{comb,}}$, pb} & \\multicolumn{1}{c}{Shift, pb} \\\\ \\hline\n \\textit{Signal modeling} \t&\t \t & & \t &\\\\\n ~~~Signal generator &\t\t \t\t$\\pm 0.21$ &$\\pm0.05$& $\\pm 0.17$\t & $+0.08$ \\\\ \n ~~~Hadronization & \t\t\t\t$\\pm 0.26$ &$\\pm0.33$& $\\pm 0.25$\t & $+0.12$ \\\\ \n ~~~Color reconnection & \t\t\t$\\pm 0.08$ &$\\pm0.05$& $\\pm 0.09$\t & $+0.02$ \\\\\n ~~~ISR\/FSR variation &\t\t\t\t$\\pm 0.08$ &$\\pm0.04$& $\\pm 0.06$\t & $-0.05$ \\\\ \n \\textit{PDF} & \t\t\t\t\t\t$\\pm 0.04$ &$\\pm0.03$& $\\pm 0.02$\t & $-0.01$ \\\\ \n \\textit{Detector modeling}\t&\t \t & & \t &\\\\\n ~~~Jet modeling \\& ID & \t\t$\\pm 0.11$ &$\\pm0.08$& $\\pm 0.04$\t & $+0.07$ \\\\ \n ~~~$b$-jet modeling \\& ID & \t$\\pm 0.27$ &$\\pm0.26$& $\\pm 0.23$\t & $-0.15$ \\\\ \n ~~~Lepton modeling \\& ID&\t$\\pm 0.20$ &$\\pm0.26$& $\\pm 0.17$\t & $-0.11$ \\\\ \n ~~~Trigger efficiency & \t\t\t\t$\\pm 0.32$ &$\\pm0.08$& $\\pm 0.16$\t & $+0.01$ \\\\ \n ~~~Luminosity & \t\t\t\t\t$\\pm 0.30$ &$\\pm0.30$& $\\pm 0.27$\t & $+0.10$ \\\\ \n \\textit{Sample Composition} \t\t&\t& & \t \t & \\\\\n ~~~MC cross sections \t\t\t\t&$\\pm 0.07$ &$\\pm0.13$& $\\pm 0.09$\t & $+0.01$ \\\\ \n ~~~Multijet contribution &\t\t\t$\\pm 0.11$ &$\\pm0.02$& $\\pm 0.10$\t & $+0.10$ \\\\ \n ~~~\\wplus scale factor & \t\t\t$\\pm 0.21$ &$\\pm0.01$& $\\pm 0.15$\t & $-0.50$ \\\\ \n ~~~\\zplus scale factor & \t\t\t$\\pm 0.07$ &$\\pm0.11$& $\\pm 0.12$\t & $+0.12$ \\\\ \n \\textit{MC statistics} \t\t&\t $\\pm 0.01$ &$\\pm0.01$& $\\pm 0.02$\t & $+0.00$ \\\\ \\hline \n Total systematic uncertainty (quadratic sum) \t& \t\t$\\pm 0.70$ &$\\pm 0.64$&$\\pm 0.60$ & \\\\ \n Total systematic uncertainty (central \\collie) & $\\pm 0.67$ &$\\pm 0.73$&$\\pm 0.55$ & \\T \n\\end {tabular}\n\\end{ruledtabular}\n\\end {table*}\n\n\n\\subsection{The $\\bm{\\ell+}$jets channel}\nIn the following we describe the sources of systematic uncertainties studied in the $\\ell +$jets\\xspace channel. As discussed above, each source of systematic uncertainty yields a modified discriminant distribution, which is parametrized with a nuisance parameter (see Sec.\\ \\ref{toc:fitting_collie}). We assign an uncertainty on the shape, but not on the normalization, of the \\wplus and multijet contribution (see Sec.\\ \\ref{toc:sampleCompLjets}). In particular the trigger and luminosity uncertainties affecting the normalization are not assigned to the \\wplus and multijet contribution, and consequently the luminosity uncertainty cannot be constrained by data.\n\n\\subsubsection{Signal modeling}\nThe effect of an alternative signal model for \\ensuremath{t\\bar{t}}\\xspace production is estimated by comparing \\ensuremath{t\\bar{t}}\\xspace events generated with \\mcherwig to those from \\alpher. Comparing \\alppyt to \\alpher, we estimate the effect of hadronization uncertainties. Additional uncertainties on signal arise from color reconnection (CR), and initial- and final-state radiation (ISR\/FSR) producing additional jets. The effect of CR is determined by comparing identical \\alpgen events interfaced to \\pythia with two different tunes, Perugia 2011 and Perugia 2011NOCR \\cite{perugia}, which either include color reconnection effects (Perugia 2011) or not (Perugia 2011NOCR). The effect of ISR\/FSR is determined by modifying the factorization and renormalization scale implemented in the MC. More details can be found in Ref.\\ \\cite{ljetsMassPRD}.\n\n\\subsubsection{Parton distribution functions} \nThe uncertainty on the cross sections due to the uncertainty on PDFs is estimated following the procedure of Ref.\\ \\cite{cteq6m} by reweighting the MC simulation according to each of the 20 pairs of error eigenvectors of the CTEQ6M PDF.\n\n\\subsubsection{Detector modeling}\nUncertainties on the modeling of the detector include uncertainties on trigger efficiency, lepton identification, and $b$-quark identification. The identification efficiencies for $b$, $c$, light quarks ($u,d,s$), and gluons in MC simulations are calibrated using dijet data \\cite{topmass_sdc}, and variations within the calibration uncertainty are used to determine the systematic uncertainty due to $b$-quark identification. Additional uncertainties arise from track multiplicity requirements on the selected jets in the identification of $b$ quarks.\n\nThe measurement of the \\ensuremath{t\\bar{t}}\\xspace cross section and the subsequent extraction of the top quark pole mass relies on a precise knowledge of normalization uncertainties. Hence, this measurement is the first measurement in D0 employing the reduced systematic uncertainty on the luminosity measurement of 4.3\\% \\cite{lumi_nim,lumiTM}. We use an auxiliary data sample where no cut is made on the primary vertex position in $z$ to verify that negligible uncertainty arises for the $|z_{\\mathrm{PV}}|< 60$ cm requirement used in this analysis. Other instrumental uncertainties from modeling the detector arise from the calibration of the jet energy, resolution, and efficiency.\n\n\\subsubsection{Sample composition} \n\\label{toc:sys_sampleComp}\nUncertainties in the composition of the selected events arise from $s_{\\mathrm{fit}}^{\\mathrm{WHF}}$ and $s_{\\mathrm{fit}}^{\\mathrm{WLF}}$ used for \\wplus events, the assumed \\ensuremath{t\\bar{t}}\\xspace cross section, single top quark and diboson cross sections, and the estimate of the contributions from misidentified leptons. As introduced in Sec.\\ \\ref{toc:sampleComp}, we determine an initial sample composition from a simultaneous fit to the MVA distribution in the \\lplustw, \\lplusth and \\lplusgefo samples. For this initial sample composition we fit $s_{\\mathrm{fit}}^{\\mathrm{WHF}}$ and $s_{\\mathrm{fit}}^{\\mathrm{WLF}}$ assuming an uncertainty of 5\\% on the normalization of the \\ensuremath{t\\bar{t}}\\xspace processes. This initial sample composition is only used to determine a systematic uncertainty on the contribution of \\ensuremath{W}+\\rm jets\\xspace processes. From the fit we derive a systematic uncertainty of $^{+3.5}_{-1.8}$\\% on the normalization of the \\ensuremath{Wlp} + jets\\xspace and $^{+17}_{-23}$\\% on the normalization of the $Wc\\bar{c}+\\mathrm{jets}$ and $Wb\\bar{b}+\\mathrm{jets}$ processes. The statistical uncertainties on these processes are negligible. An uncertainty of 25\\% on the \\zplus cross section is assigned. The uncertainty on the single top quark cross sections is 12.6\\%, taken from varying the factorization and renormalization scales simultaneously by factors of $2$ and $0.5$. An uncertainty of 7\\% on the diboson cross sections is assigned, corresponding to half the difference between the LO and NLO predictions. The uncertainties on the single top quark and diboson contributions are labeled ``MC cross sections\" in the corresponding tables. The uncertainties on the data-driven method of estimating MJ background and its kinematic dependences, mostly due to the uncertainties on the selection rates of true and false lepton candidates, are 40\\% in the \\muplus and 25\\% in the \\eplus sample (including statistical components). These uncertainties are estimated by varying the contribution of $Wc\\bar{c}+\\mathrm{jets}$, $Wb\\bar{b}+\\mathrm{jets}$, $Zc\\bar{c}+\\mathrm{jets}$ and $Zb\\bar{b}+\\mathrm{jets}$ by $\\pm 20\\%$, the \\ensuremath{t\\bar{t}}\\xspace contribution by $\\pm 10\\%$, and then comparing the fake and true signal rates in different variables (quoting the largest difference as additional parametrization uncertainty) \\cite{diffXsecPaper}.\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7d.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7e.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig7f.eps}\n\\end{center}\n\\caption{Post-fit output distributions of the combined MVA discriminant using the measured combined \\ensuremath{t\\bar{t}}\\xspace cross section and $m_t=172.5$ GeV for the $e + $ two, three or four or more jets [panels (a) -- (c)], and for the $\\mu + $ two, three or four or more jets [panels (d) -- (f)]. Statistical uncertainties of the data are shown and the post-fit systematic uncertainties are indicated by the hashed band in the bottom panel of the histogram. The $\\chi^2\/\\mathrm{ndf}$ values take statistical and systematic uncertainties into account.}\n\\label{fig:postfit_topo_emujets}\n\\end{figure*}\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig8a.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig8b.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig8c.eps}\n \\includegraphics[width=0.675\\columnwidth,angle=0]{fig8d.eps}\n\\end{center}\n\\caption{The post-fit MVA output distributions (assuming $m_t=172.5$ GeV) of \\ensuremath{j_{\\mm{b-ID}}^{\\mm{max}}}\\xspace for (a) $e\\mu$ events with exactly one jet, (b) $e\\mu$ events with at least two jets, (c) $ee$ events with at least two jets, and (d) $\\mu\\mu$ events with at least two jets are shown. Statistical uncertainties of the data are shown and the post-fit systematic uncertainties are indicated by the hashed band in the bottom panel of the histogram. The $\\chi^2\/\\mathrm{ndf}$ values take statistical and systematic uncertainties into account.}\n\\label{fig:postfit_mva_dilep}\n\\end{figure*}\n\n\n\\subsection{The $\\bm{ \\ell \\ell}$ channel}\nIn the following we describe the sources of systematic uncertainties studied in the $\\ell\\ell$\\xspace channel, which are mostly similar to those in the $\\ell +$jets\\xspace channel. As discussed above, each source of systematic uncertainty yields a modified discriminant distribution of the $b$-ID MVA\\xspace and a nuisance parameter is used in the fit to determine the \\ensuremath{t\\bar{t}}\\xspace production cross section in the $\\ell\\ell$\\xspace channel. Uncertainties on the sample composition only apply to certain sample contributions. Uncertainties due to common sources are assumed to be 100\\% correlated between the $\\ell +$jets\\xspace channel and the $\\ell\\ell$\\xspace channel unless otherwise specified.\n\n\\subsubsection{Signal modeling}\nThe same sources of systematic uncertainties for the modeling of the signal as in the $\\ell +$jets\\xspace decay channel are considered for the $\\ell\\ell$\\xspace channel as well. \n\n\\subsubsection{Parton distribution functions}\nThe uncertainty on the cross sections due to the uncertainty on PDFs is estimated following the same procedure as in the $\\ell +$jets\\xspace case. \n\n\\subsubsection{Detector modeling}\nThe assigned uncertainties related to the modeling of the detector are the same as the ones assigned in the $\\ell +$jets\\xspace channel and include uncertainties on the efficiencies of electron and muon identification, uncertainties on trigger efficiencies, the uncertainty in jet energy scale, jet energy resolution, jet identification efficiency, and $b$-quark jet tagging efficiency. The $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace event selection ensures that the samples are dominantly orthogonal. Hence, we assume the uncertainties arising from modeling of the trigger to be not correlated between the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace channel. \n\n\\subsubsection{Sample composition}\nWe estimate the uncertainty on the instrumental background contribution in the $\\ell\\ell$\\xspace channel by changing the normalization of that background by $\\pm 1$ standard deviation of its uncertainty. It includes both the statistical uncertainty on the sample used to derive the normalization and the systematic uncertainty in the lepton misidentification rate. Uncertainties from \\zplus and diboson production are taken into account with the same assumptions as in the $\\ell +$jets\\xspace case. Uncertainties arising from the determination of the MJ background are assumed to be not correlated between the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace channels.\n\n\\section{Cross Section Results}\n\\label{toc:results}\nThe result of the measurement in the $\\ell +$jets\\xspace channel using the combined MVA method is\\\\\n\\begin{center}\n$ \\sigma_{\\ttbar} = 7.33 \\pm 0.14\\,(\\mm{stat.})\\,^{+0.71}_{-0.61}\\,(\\mm{syst.})$\\xspace pb,\n\\end{center} with a relative total uncertainty of 9.2\\%. For the $\\ell\\ell$ decay channel we employ the MVA $b$-jet method and measure\\\\\n\\begin{center}\n$ \\sigma_{\\ttbar} = 7.58 \\pm 0.35\\,(\\mm{stat.})\\,^{+0.69}_{-0.58}\\,(\\mm{syst.})$\\xspace pb,\n\\end{center} with a relative total uncertainty of 9.6\\%. \n\nThe combination of the \\ensuremath{t\\bar{t}}\\xspace cross section is carried out by a simultaneous \\collie fit of the combined MVA and the MVA $b$-jet discriminant distributions in the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace channels. For a top quark mass of 172.5 GeV we measure\\\\\n\\begin{center}\n$ \\sigma_{\\ttbar} = 7.26 \\pm 0.13\\,(\\mm{stat.})\\,^{+0.57}_{-0.50}\\,(\\mm{syst.})$\\xspace pb,\n\\end{center} which corresponds to a relative total uncertainty of $7.6\\%$. For the combined $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace \\ensuremath{t\\bar{t}}\\xspace cross section measurement we profile the systematic uncertainties by employing the MVA discriminant distributions simultaneously in both channels. The combined \\ensuremath{t\\bar{t}}\\xspace cross section does not coincide with the weighted average of the individual $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace results, which is attributed to the effect of correlations of systematic uncertainties between both channels.\n\nTable \\ref{tab:syst_uncorr} summarizes the post-fit systematic uncertainties on the \\ensuremath{t\\bar{t}}\\xspace cross section in the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace decay channels and for the combination. The impact of these sources is estimated by removing or including the corresponding ``group\" of individual sources from the fit. The total uncertainty is provided by the nominal fit, when including all individual sources of systematic uncertainties, and denoted as ``central \\collie'' in Table \\ref{tab:syst_uncorr}. For comparison only we also provide the quadratic sum of the groups of systematic uncertainties. Due to correlations being different between the groups and all the individual systematic uncertainties, that value differs from the total systematic uncertainty of the nominal fit. In addition, we provide the ``Shift\" in units of pb, which refers to shifts on the combined inclusive cross section due to a particular source of systematic uncertainty relative to the central value of the combined \\ensuremath{t\\bar{t}}\\xspace cross section.\n\nFigure \\ref{fig:postfit_topo_emujets} shows the post-fit MVA combined discriminant distributions for the \\lplus channel using the combined \\ensuremath{t\\bar{t}}\\xspace cross section. Similarly Fig.~\\ref{fig:postfit_mva_dilep} shows the post-fit MVA $b$-ID discriminant distribution for the $\\ell\\ell$\\xspace channel using the combined \\ensuremath{t\\bar{t}}\\xspace cross section. This result is consistent with and supersedes our earlier measurement using $5.3~\\mathrm{fb^{-1}}$ of data \\cite{Publ54_xsec}. The inclusive \\ensuremath{t\\bar{t}}\\xspace production cross section is in agreement with the fully resummed next-to-next-to-leading logarithm at NNLO QCD calculation (see Sec.\\ \\ref{toc:generators}) of $\\sigma_{\\mathrm{tot}}^{\\mathrm{res}} = 7.35 ^{+0.23}_{-0.27}\\thinspace(\\mathrm{scale} + \\mathrm{pdf})$ pb.\n\n\\section{Top Quark Pole Mass}\n\\label{toc:tmass}\nTable \\ref{tab:mt_values} presents the measured combined inclusive \\ensuremath{t\\bar{t}}\\xspace cross section as a function of the top quark mass. For each top quark mass point shown a separate combined log-likelihood fit of the $\\ell +$jets\\xspace and $\\ell\\ell$\\xspace channel MVA discriminant inputs was performed, as was done for the mass point of 172.5 GeV. The measured \\ensuremath{t\\bar{t}}\\xspace cross section only changes by 0.7\\% for a change of 1 GeV in the assumed top quark mass. Systematic uncertainties of the \\ensuremath{t\\bar{t}}\\xspace contribution are taken from the 172.5 GeV case and assigned to other masses as a relative systematic uncertainty of the same size. \n\n\\begin{table}[htp]\n \\caption{The measured combined inclusive \\ensuremath{t\\bar{t}}\\xspace cross section as a function of the top quark MC mass with statistical and systematic uncertainties given separately.} \\label{tab:mt_values}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{c l}\nTop quark mass [GeV] & Cross section \\xsttbar [pb]\\\\ \\hline \\T\n$150$ & $9.70 \\pm 0.16\\,(\\mathrm{stat.}) ^{+0.73}_{-0.67}\\,(\\mathrm{syst.})$ \\\\ \\T \n$160$ & $8.25 \\pm 0.14\\,(\\mathrm{stat.}) ^{+0.63}_{-0.57}\\,(\\mathrm{syst.})$ \\\\ \\T\n$165$ & $7.46 \\pm 0.13\\,(\\mathrm{stat.}) ^{+0.58}_{-0.51}\\,(\\mathrm{syst.})$ \\\\ \\T\n$170$ & $7.55 \\pm 0.13\\,(\\mathrm{stat.}) ^{+0.58}_{-0.55}\\,(\\mathrm{syst.})$ \\\\ \\T\n$172.5$ &$7.26 \\pm 0.12\\,(\\mathrm{stat.}) ^{+0.57}_{-0.50}\\,(\\mathrm{syst.})$ \\\\ \\T\n$175$ &$7.28 \\pm 0.12\\,(\\mathrm{stat.}) ^{+0.54}_{-0.49}\\,(\\mathrm{syst.})$ \\\\ \\T\n$180$ &$6.86 \\pm 0.12\\,(\\mathrm{stat.}) ^{+0.53}_{-0.47}\\,(\\mathrm{syst.})$ \\\\ \\T\n$185$ &$6.50 \\pm 0.11\\,(\\mathrm{stat.}) ^{+0.50}_{-0.43}\\,(\\mathrm{syst.})$ \\\\ \\T\n$190$ &$6.70 \\pm 0.11\\,(\\mathrm{stat.}) ^{+0.60}_{-0.47}\\,(\\mathrm{syst.})$ \n \\end{tabular}\n\\end{ruledtabular}\n \\end{center}\n\\end{table}\n\nFigure \\ref{fig:topMass_vs_topxsec_NNLO} shows the measured and theoretical mass dependence of the inclusive \\ensuremath{t\\bar{t}}\\xspace production cross section as provided by \\toppp \\cite{toppp_prg}. We parametrize the experimentally measured dependence with a fourth-order polynomial function to the individual cross section measurements at the mass points reported in Table \\ref{tab:mt_values}. There is negligible change if a cubic function is chosen. Uncertainties on the measured values include the statistical and systematic uncertainties discussed in Sec.\\ \\ref{toc:xsec_sys} and are indicated by the dotted lines. Theoretical uncertainties include those from the variation of the renormalization and factorization scales by a factor of 2 and PDF uncertainties \\cite{mstw2008nnlo} taken from the MSTW2008 NNLO PDF set. These are added in quadrature and indicated by the dotted lines surrounding the central theoretical prediction.\n \n\\begin{figure}[htb]\n\\begin{center}\n \\includegraphics[width=0.975\\columnwidth,angle=0]{fig9.eps}\n\\end{center}\n\\caption{The measured \\ensuremath{t\\bar{t}}\\xspace production cross section dependence on the top quark mass (points) parametrized by a quartic function (solid black line) and compared to the dependence provided by the NNLO pQCD calculation \\toppp \\cite{toppp_prg}, which implements pQCD calculations according to Ref.~\\cite{nnloInclXsec}.}\n\\label{fig:topMass_vs_topxsec_NNLO}\n\\end{figure}\nTo determine the top quark pole mass from the inclusive \\ensuremath{t\\bar{t}}\\xspace cross section following the method in Ref.~\\cite{d0_polemass_5fb}, we extract the most probable $m_t$ value and uncertainty by employing a normalized joint-likelihood function, which takes into account the total experimental uncertainty, the theoretical uncertainties on the renormalization and factorization scales, and the PDF uncertainties. Employing the quartic parametrization and the theory predictions at NNLO pQCD we obtain\n\\begin{eqnarray*}\nm_t &=& 172.8 \\pm 1.1\\,(\\mathrm{theo.})\\,^{+3.3}_{-3.1}\\,(\\mathrm{exp.})\\,\\mathrm{GeV}, \\mathrm{or}\\\\\nm_t &=& 172.8 ^{+3.4}_{-3.2}\\,(\\mathrm{tot.})\\,\\mathrm{GeV}\n\\end{eqnarray*}\nThe experimental uncertainties dominate the precision of the determination. The precision of this determination is 1.9\\%, and represents the most precise determination of the top quark pole mass from the inclusive \\ensuremath{t\\bar{t}}\\xspace cross section at the Tevatron. This supersedes our previous determination which had a precision of 3\\% \\cite{d0_polemass_5fb}.\n\n\\section{Conclusions}\n\\label{toc:conclusion}\nThe inclusive \\ensuremath{t\\bar{t}}\\xspace production cross section has been measured combining the lepton+jets and dilepton top quark decay channels based on the full Tevatron data set at $\\sqrt{s} = 1.96$ TeV. We performed a simultaneous log-likelihood fit to profile systematic uncertainties and, for a top quark mass of 172.5 GeV, we measured a combined \\ensuremath{t\\bar{t}}\\xspace cross section of\n\\begin{center}\n$ \\sigma_{\\ttbar} = 7.26 \\pm 0.13\\,(\\mm{stat.})\\,^{+0.57}_{-0.50}\\,(\\mm{syst.})$\\xspace pb,\n\\end{center} which corresponds to a relative uncertainty of $7.6\\%$. This result and the measured inclusive cross sections per decay channel are in good agreement with predictions by QCD. \n\nWe employed the dependence of the theoretical cross section on the top mass, to determine a pole mass of the top quark of \n\\begin{center}\n $m_t = 172.8 ^{+3.4}_{-3.2}\\,(\\mathrm{tot.})\\,\\mathrm{GeV}$.\n \\end{center} \nThe uncertainty corresponds to a precision of 1.9\\% and represents the most precise determination of the top quark pole mass at the Tevatron.\n\n\\input{New_acknowledgement_APS_full_names.tex}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n{\\it Sperner's lemma} is a statement about labellings (colorings) of triangulated simplices ($d$-balls). It is a discrete analog of the Brouwer fixed point theorem.\n\n\n\nLet $S$ be a $d$-dimensional simplex with vertices $v_1,\\ldots,v_{d+1}$. Let $T$ be a triangulation of $S$. Suppose that each vertex of $T$ is assigned a unique label from the set $\\{1,2,\\ldots,d+1\\}$. A labelling $L$ is called {\\it Sperner's} if the vertices are labelled in such a way that a vertex of $T$ belonging to the interior of a face $F$ of $S$ can only be labelled by $k$ if $v_k$ is on $S$. \n\n \\medskip\n\n\\noindent{\\bf (Sperner's lemma \\cite{Sperner})} \n\t{\\it Every Sperner labelling of a triangulation of a $d$-dimensional simplex contains a cell labelled with a complete set of labels: $\\{1,2,\\ldots, d+1\\}$.}\n\n\n\n\n\\medskip\n\n\n\nThe two-dimensional case is the one referred to most frequently. It is stated as follows:\\\\\n{\\it Given a triangle $ABC$, and a triangulation $T$ of the triangle. The set $V(T)$ of vertices of $T$ is colored with three colors in such a way that \n(i) $A, B$ and $C$ are colored 1, 2 and 3 respectively; \n(ii) Each vertex on an edge of $ABC$ is to be colored only with one of the two colors of the ends of its edge. Then there exists a triangle from $T$, whose vertices are colored with the three different colors.}\n\n\n\n\n\\medskip\n\n\n\n\n\n\nThere are several extensions of Sperner' lemma for quadrangulations (cubical decompositions), see \\cite{KyFan60,Kuhn,Shashkin,Wolsey}. In particular, Ky Fan \\cite{KyFan60} proved that any Sperner 0--1 labelling of a $d$-pile with the {neighbourhood property} contains a fully labelled $d$-cube. \nShashkin \\cite{Shashkin} extended Ky Fan's formula for local degrees of simplicial maps in \\cite{KyFan67} to cubical maps. (Actually, he proved it only for small dimensions $d\\le4$.) \n\n\\medskip\n\nIn two dimensions a {\\it quadrangulation} is the division of a surface or plane polygon into a set of quadrangles with the restriction that each quadrangle side either is entirely shared by two adjacent quadrangles or lies on the boundary. \n\nOur main example is the {\\it pile of cubes} $\\Pi_d(n_1,\\ldots,n_d)$, see details in \\cite[Chapter 5]{Ziegler}. It is the polytopal complex formed by all unit cubes with integer vertices in the $d$-box \n$$\nB(n_1,\\ldots,n_d):=\\{x=(x_1,\\ldots,x_d) \\in {\\Bbb R}^d:0\\le x_i\\le n_i \\mbox{ for } 1\\le i \\le d\\}. \n$$\nSo each $k$-cell, $0\\le k\\le d$, of this complex is a $k$-dimensional unit cube.\n\nIn Section 2 we consider Theorem 1 (oriented case) and Theorem 2 (non-oriented case). Both theorems yield the following extension of Sperner's lemma for piles in two dimensions.\n\n\\medskip\n\n\\noindent{\\bf Theorem A.}\n{\\it Given a pile $P:=\\Pi_2(m,n)$ with corners $ABCD$. The set of vertices of $P$ is colored with four colors in such a way that $A, B,C$ and $D$ are colored 1, 2, 3 and 4 respectively; and \neach vertex on an edge of $ABCD$ is to be colored only with one of the two colors of the ends of its edge. Then either there exists a quadrangle from $P$, whose vertices are colored with the four different colors or there exists an edge of $P$ whose two ends are colored with $(1,3)$ or $(2,4)$.}\n\n\\medskip\n\n\\noindent{\\bf Remark.} In fact, for a 2-pile the {\\it neighbourhood property} in \\cite{KyFan60} means that there are no edges whose two vertices are colored with $[1,3]$ or $[2,4]$. So Theorem A first proved by Ky Fan. \n\n\n\n\\medskip\n \n\\medskip\n\n\\begin{center}\n\\begin{picture}(320,160)(-100,-80)\n\\put(-130,-70){Figure 1: Sperner's labelling of $\\Pi_2(4,3)$. One edge is colored with $(1,3)$.}\n\n\\multiput(-70,-40)(60,0){5\n{\\line(0,1){120}}\n\n\\multiput(-70,-40)(0,40){4\n{\\line(1,0){240}}\n\n\\put(-80,-50){$A$}\n\\put(173,-50){$B$}\n\\put(173,80){$C$}\n\\put(-83,80){$D$}\n\n\\put(-66,-36){1}\n\\put(160,-36){2}\n\\put(160,68){3}\n\\put(-66,68){4}\n\n\\put(-66,4){1}\n\\put(-66,44){4}\n\n\\put(160,4){2}\n\\put(160,44){3}\n\n\\put(-6,-36){2}\n\\put(-6,4){1}\n\\put(-6,44){4}\n\\put(-6,68){3}\n\n\\put(54,-36){2}\n\\put(54,4){1}\n\\put(54,44){3}\n\\put(54,68){3}\n\n\\put(114,-36){1}\n\\put(114,4){2}\n\\put(114,44){3}\n\\put(114,68){4}\n\n{\\linethickness{0.7mm}\n\\put(50,0){\\line(0,1){40}}\n}\n \n\\end{picture}\n\\end{center}\n\n\\medskip\n\n\nWe will call a quadrangles (2-cell) whose vertices are colored with the four different colors [1,2,3,4] and edges (1-cells) whose two vertices are colored with $[1,3]$ or $[2,4]$ as {\\it centrally labelled} cells. \n\nThe main reason for this name is the following. \nLet $C$ be a quadrangle whose vertices $V(C):=v_1,v_2,v_3$ and $v_4$ that are labelled with 1, 2, 3 and 4 respectively. Then among all cells with vertices in $V(C)$ there are only three cells: $v_1v_2v_3v_4$ (2-cell), and two diagonals (1-cells) $v_1v_3$ and $v_2v_4$ that contain the center of $C$ inside. \n\n\nWe can say it in another way. We assume that the colors 1, 2, 3, 4 correspond to the labels $(-1,-1), (1,-1),(1,1),(-1,1)$ respectively. Then centrally labelled cells are cells with zero sum of its labels. \n\n\n \nIn two dimensions, the labelling in Theorem A is called Sperner's labelling. \nIn other words, Theorem A states:\n\n\\medskip\n\n\\noindent{\\it Any Sperner labelling of the pile $\\Pi_2(m,n)$ contains at least one centrally labelled cell.}\n\n\\medskip\n\nTheorem A can be extended for all dimensions. Let $C^d=\\Pi_d(1,\\ldots,1)$ denote the unit cube in ${\\Bbb R}^d$. The vertex set of $C^d$ consists of all the $2^d$ binary $d$-tuples, and where two\nvertices are adjacent if the corresponding $d$-tuples differ in exactly one coordinate position. Therefore, there is one-to-one correspondence between the vertex set $V(C^d)$ and the set of binary numbers from 0 to $2^d-1$, i. e. the labelling $c:V(C^d)\\to \\{0,1,\\ldots,2^d-1\\}$ is bijection.\n\nNote that corners of the pile $P:=\\Pi_d(n_1,\\ldots,n_d)$ can be considered as vertices of $C^d$. \nWe say that a labelling $L:V(P)\\to V(C^d)$ is {\\it Sperner's} if the set of corners of $P$ are labelled with the correspondent labels from $V(C^d))$ and vertices on the boundary of $P$ are labelled in such a way that a vertex belonging to the interior of a cell (face) $F$ of $C^d$ can be only labelled with $v$ where $v$ is a vertex of $F$. \n\nIn Section 2 are defined centrally labelled cells for all dimensions, see Definition 2. \n\n\\medskip\n\n{\\bf Theorem B.} {\\it Any Sperner labelling of the pile $\\Pi_d(n_1,\\ldots,n_d)$ contains at least one centrally labelled cell.}\n\n\\medskip\n\nNow extend Theorem A for quadrangulations. We found a weaker assumption than Sperner's coloring. \n\nLet $L:V\\to\\{1,2,3,4\\}$ be a labelling of a set $V:=\\{v_1,\\ldots, v_m\\}$ in a circle such that any two adjacent vertices cannot have labels (1,3) or (2,4). Let $$\\deg([1,2],L):=p_*-n_*,$$ where $p_*$ (respectively, $n_*$) is the number of (ordering) pairs $(v_k,v_{k+1})$ such that $L(v_k)=1$ and $L(v_{k+1})=2$ (respectively, $L(v_k)=2$ and $L(v_{k+1})=1$). \n\n\\medskip\n\nFor instance, let $L=(12212341234211223412341)$. Then $p_*=5$ and $n_*=2$. Thus, $\\deg([1,2],L)=5-2=3.$\n\n\\medskip\n\nNote that, instead of $[1,2]$ we can take $[2,3],\\, [3,4]$ or $[4,1]$. Namely, we have $$\\deg([1,2],L)=\\deg([2,3],L)=\\deg([3,4],L)=\\deg([4,1],L).$$\nThis fact proved in \\cite[Lemma 2.1]{MusArSp}. \n\n\n\nLet $Q$ be a quadrangulation of a polygon $M$. Denote by $\\partial Q$ the boundary of $Q$. Then $\\partial Q$ is a polygonal contour with vertices $v_{1},\\ldots,v_m$ that can be considered as points in a circle. (We assume that these vertices are in counterclockwise order.) \n\nLet $L:V(Q)\\to\\{1,2,3,4\\}$ be a labelling. This labelling implies the labelling $L_0:\\partial Q\\to\\{1,2,3,4\\}$. We consider the set of {\\it neighboring labellings} $\\nl(\\partial Q)$. We write $L\\in\\nl(\\partial Q)$ if any edge on the boundary $\\partial Q$ have no labels (1,3) or (2,4). Then $\\deg([1,2],L_0)$ is well defined. Denote $$\\deg(L,\\partial Q):=\\deg([1,2],L_0).$$\n\n\\medskip\n\n{\\bf Theorem C.} {\\it Let $Q$ be a quadrangulation of a polygon $M$. Suppose $L:V(Q)\\to\\{1,2,3,4\\}$ be a labelling such that $L\\in\\nl(\\partial Q)$. Then $Q$ contains at least $|\\deg(L,\\partial Q)|$ centrally labelled cells.}\n\n\\medskip\n\n\\medskip\n \n\\medskip\n\n\\begin{center}\n\\begin{picture}(320,160)(-100,-80)\n\\put(-130,-70){Figure 2: Since $\\deg(L,\\partial Q)=2$, there are two centrally labelled cells.}\n\n\\multiput(-70,-40)(60,0){5\n{\\line(0,1){120}}\n\n\\multiput(-70,-40)(0,40){4\n{\\line(1,0){240}}\n\n\\put(-70,40){\\line(-1,0){60}}\n\\put(-70,0){\\line(-1,0){60}}\n\\put(-130,40){\\line(0,-1){40}}\n\n\\put(-66,-36){1}\n\\put(160,-36){1}\n\\put(160,68){1}\n\\put(-66,68){1}\n\n\\put(170,80){\\line(1,0){60}}\n\\put(170,40){\\line(1,0){60}}\n\\put(170,0){\\line(1,0){60}}\n\\put(230,0){\\line(0,1){80}}\n\n\\put(220,4){1}\n\\put(220,44){1}\n\\put(220,68){2}\n\n\\put(-126,4){3}\n\\put(-126,28){3}\n\n\\put(-66,4){2}\n\\put(-66,44){2}\n\n\\put(160,4){4}\n\\put(160,44){1}\n\n\\put(-20,-36){2}\n\\put(-20,4){1}\n\\put(-6,44){1}\n\\put(-6,68){1}\n\n\\put(54,-36){3}\n\\put(54,4){4}\n\\put(54,44){1}\n\\put(54,68){3}\n\n\\put(114,-36){4}\n\\put(114,4){1}\n\\put(114,44){1}\n\\put(114,68){2}\n\n{\\linethickness{0.7mm}\n\\put(50,40){\\line(0,1){40}}\n}\n\n\\multiput(0,-30)(10,0){5\n{\\circle*{2}} \n\\multiput(0,-20)(10,0){5\n{\\circle*{2}}\n\\multiput(0,-10)(10,0){5\n{\\circle*{2}}\n\n{\\linethickness{0.5mm}\n\\put(-10,-40){\\line(1,0){60}}\n\\put(-10,0){\\line(1,0){60}}\n\\put(-10,-40){\\line(0,1){40}}\n\\put(50,-40){\\line(0,1){40}}\n}\n \n\\end{picture}\n\\end{center}\n\n\\medskip\n\nTheorem 1 in Section 2 extends Theorem C for all dimensions. Note that if $L$ is a Sperner labeling, then $|\\deg(L,\\partial Q)|=1$. So Theorem C implies Theorem A and from Theorem 1 follows Theorem B.\n\n\n\n\\section{Main results}\n\nHere we consider the main theorem and its corollaries for a very general class of spaces $M$. One of very natural extensions of Theorem C is the case when $M$ is a polytope in ${\\mathbb R}^d$.\n \nIn our papers \\cite{Mus,MusSpT,MusKKM,MusArSp} we studied a more general case when $M$ is a piece-wise linear manifold. \nIn this case, if $M$ is a compact oriented manifold with boundary, then from the one side it extends Theorem C for all dimensions and to a huge class of spaces (even in two dimensions), on the other side, almost all proofs can be easily transfer for this case. \n\nAll results in this section hold for manifolds that admit quadrangulations. The class of such manifolds is called {piece-wise linear (PL) manifolds.} Note that a smooth manifold can be triangulated and quadrangulated, therefore it is also a PL manifold. However, there are topological manifolds that do not admit a triangulation and therefore do not admit a quadrangulation. \n \nEvery PL manifold $M$ admits a {\\it quadrangulation:} that is, we can find a collection of cells $Q$ of dimensions $0, 1,\\ldots, d$, such that (1) any cell of dimension $k$ is homeomorphic to the $k$-cube $C^k$, (2) any face of a cell belonging to $Q$ also belongs to $Q$, (3) any nonempty intersection of any two cells of $Q$ is a face of each, and (4) the union of the cells of $Q$ is $M$. (See details in \\cite{Bryant}.) Actually, a PL--manifold $M$ can be quadrangulated by many ways. \n\nThroughout this paper by word {\\it manifold} we assume a compact PL manifold with or without boundary. \n\nLet $Q$ be a quadrangulation of $M$. The vertex set of $Q$, denoted by $V(Q)$, is the union of the vertex sets of all cells of $Q$. \n\n\\medskip \n\nNow for any labelling $c:V(C^n)\\to \\{0,1,\\ldots,2^d-1\\}$ we define a map $f_L:C^n\\to C^d$. Since we have the bijection $c:V(C^d)\\to \\{0,1,\\ldots,2^d-1\\}$ this labelling can be considered as a map $L:V(C^n)\\to V(C^d)$. \n\n\\begin{lemma} For any labelling $L:V(C^n)\\to V(C^d)$ we may uniquely define a multilinear map $f_L:C^n\\to C^d$ such that for each vertex $v\\in V(C^n)$ we have $f_L(v)=L(v)$. \n\\end{lemma}\n\\begin{proof} A multilinear map is a map of several variables that is linear separately in each variable. Actually, $f=(f_1,\\ldots,f_d)$ is a multilinear map if each function $f_i(x_1,\\ldots,x_n)$ is a multilinear function. A function $F(x_1,\\ldots,x_n)$ is multilinear if \n$$\nF(x_1,\\ldots,x_n)=\\sum\\limits_{0\\le w\\le 2^n-1}{a_wx^w},$$ \n where for $w=(p_1,\\ldots,p_n),$ \\, $p_k=0$ or 1, \n$\nx^w:=x_1^{p_1}\\cdot\\cdot\\cdot x_n^{p_n}. \n$\n\nSo a linear function $F(x)=a_0+a_1x$ and a bilinear \n$$\nF(x_1,x_2)=a_{00}+a_{01}x_1+a_{10}x_2+a_{11}x_1x_2.\n$$\n\nIf a multilinear function $F(x_1,\\ldots,x_n)$ is defined in $V(C^n)$, i. e. for $w\\in V(C^n)$, $F(w)=b_w$ then the coefficients $a_w$ can be found. Indeed, for $n=1$ we have \n$$a_0=b_0, \\; a_1=b_1-b_0.$$\nFor $n=2$: $$a_{00}=b_{00}, \\, a_{01}=b_{01}-b_{00},\\, a_{10}=b_{10}-b_{00}, \\, a_{11}=b_{11}-b_{10}-b_{01}+b_{00}.$$\nBy induction these coefficients can be found for all $a_w$. \n\n Let $u=(u_1,\\ldots,u_n)$ and $w=(w_1,\\ldots,w_n)$ be two binary numbers (0 and 1 strings). We write $u\\preceq w$ if $u_i\\le w_i$ for all $i=1,\\ldots,n$. Denote by $h(u,w)$ the Hamming distance between $u$ and $w$, i. e. the number of digits in positions where they have different digit. Then \n$$\na_w=\\sum\\limits_{u\\in V(C^n):u\\preceq w}{(-1)^{h(u,w)}b_u}. \n$$\nSo all coefficients of the map $f_L:C^n\\to C^d$ can be explicitly found. \n\\end{proof}\n\n\n\\medskip \n\n\\noindent{\\bf Definition 1.} Let $Q$ be a quadrangulation of a $d$-dimensional manifold $M$. Let $L:V(Q)\\to V(C^d)$ be a labelling. Every $n$-cell $\\sigma$ in $Q$ is homeomorphic to $C^n$. So by Lemma 1 $f_L:\\sigma\\to C^d$ is uniquely defined. It defines a piece-wise multilinear map $f_L:Q\\to C^d$. \n\n\\medskip\n\n\\noindent{\\bf Definition 2.} Let $Q$ be a quadrangulation of a $d$-dimensional manifold $M$. Let $L:V(Q)\\to V(C^d)$ be a labelling. \nWe say that a cell $\\sigma\\in Q$ is {\\it centrally labelled} if there is an internal point $x$ in $\\sigma$ such that $f_L(x)=z$, where $z=(1\/2,\\ldots,1\/2)$ is the center of $C^d$. \n\n\\medskip\t\n\n\\noindent{\\bf Examples.} It is easy to see that if $d=2$, then there are two centrally labelled 1-cells with labels in the ends: [(0,0),(1,1)] and [(0,1),(1,0)] and one 2-cell with labels [(0,0),(1,0),(1,1),(0,1)]. \n\nFor $d=3$ the set of centrally labelled cells is more complicated. We have three 1-cells with labels $[u,v]$, where $u+v=(1,1,1)$. There are two types of centrally labelled 2-cells: [(0,0,0),(1,0,0),(1,1,1),(0,1,1)] and [(0,0,0),(1,1,0),(1,0,1),(0,1,1)]. \n\nOne of centrally labelled 3-cells is fully labelled, i. e. when $L:VC^3)\\to V(C^3)$ and $L(w)=w$ for all $w\\in V(C^3)$. There is a non-symmetric example. Let $L(w)=w$ for all $w\\in V(C^3), \\, w\\ne(1,1,1)$, and $L(1,1,1)=(1,1,0)$. It is also a centrally labelled 3-cell. \n\n\\medskip\n\n\nAn {\\it oriented cube} is a cube $C^k$ together with a choice of one of its two possible orientations. \nAn {\\it oriented quadrangulation} $Q$ of an oriented $d$-dimensional manifold $M$ is $Q$ equipped with an orientation on each $d$-cell and such that any two $d$-cells with a common $(d-1)$-face must have the same orientation. \n \n The {\\it degree of a continuous mapping} or {\\it Brouwer's degree} between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. It is well known that the degree is a topological invariant (see, for instance, \\cite{Milnor} and \\cite[pp. 44--46]{Mat}).\n\nLet us define the degree more rigorously. Let $f:M_1\\to M_2$ be a continuous (piece-wise smooth) map, where $M_1$ and $M_2$ are compact $d$-dimensional manifolds without boundary. Let $y\\in M_2$ be a regular value of $f$, that means $f^{-1}(y)=\\{x_1,\\ldots,x_n\\}$ and in a neighborhood of each $x_i$ the map $f$ is a local diffeomorphism. Then \n$$\n\\deg(f,y):=\\sum\\limits_{x\\in f^{-1}(y)}{\\sgn\\,\\det Df(x)},\n$$\nwhere $Df(x)$ is the Jacobi matrix of $f$ in $x$.\nActually, $\\deg(f,y)$ does not depend on $y$ and so the value $\\deg(f)$ is well defined. \n\nIn a similar way, we could define the degree of a map between compact oriented manifolds with boundary. Let $M_i, \\, i=1,2,$ be a compact manifolds with boundary $\\partial M_i$. Suppose that $f:M_1\\to M_2$ is such that $f(\\partial M_1)\\subseteq\\partial M_2$. Denote $f^\\partial:=f|_{\\partial M_1}:\\partial M_1\\to\\partial M_2$. It is well known that in this case \n$$\n\\deg(f)=\\deg(f^\\partial). \n$$ \n(A proof of this fact also can be fined in our paper \\cite{MusArSp}, see the proof of Theorems 2.1 and Theorem 4.1.)\n\n\\medskip\n\n\\noindent{\\bf Definition 3.} Let $Q$ be a quadrangulation of an oriented $d$-dimensional manifold $M$. Let $L:V(Q)\\to V(C^d)$ be a labelling. We say that $L$ is a {\\it neighboring labelling} and write $L\\in\\nl(\\partial Q)$ if $f_L(\\partial Q)\\subseteq \\partial C^d$. \n\nIf $f_L(\\partial Q)\\subseteq \\partial C^d$, then $f_L^\\partial:\\partial Q\\to\\partial C^d$ is well defined. Denote $\\deg(f_L^\\partial)$ by $\\deg(L,\\partial Q)$. If $\\partial Q$ is empty, i. e. $M$ is a manifold without boundary, then we set $\\deg(L,\\partial Q)=0$.\n\n\\begin{theorem} \\label{SpQ} Let $Q$ be a quadrangulation of an oriented $d$-dimensional manifold $M$. Suppose $L:V(Q)\\to V(C^d)$ be a labelling such that $L\\in\\nl(\\partial Q)$. Then $Q$ contains at least $|\\deg(L,\\partial Q)|$ centrally labelled cells.\n\\end{theorem}\n\\begin{proof} Let $z$ denote the center of the cube $C^d$. Then $z=(1\/2,\\ldots,1\/2)$ is an internal point of $C^d$. Since $$\\deg(f_L,z)=\\deg(f_L)=\\deg(f_L^\\partial)=\\deg(L,\\partial Q),$$ the set $f_L^{-1}(z)$ consists of $n\\ge |\\deg(L,\\partial Q)|$ points $\\{x_1,\\ldots,x_n\\}$ in $M$. (Moreover, the assumption $L\\in\\nl(\\partial Q)$ guarantees that these points cannot lie on the boundary $\\partial M$.) \n\t\nSince $x_i$ cannot be a vertex of $Q$, there is a cell $c_i\\in Q$ such that $x_i$ is an internal point of $c_i$. By definition, $c_i$ is centrally labelled. So we have $n$ centrally labelled cells. \n\\end{proof}\n\nActually, Theorem C is a two-dimensional version of Theorem 1. It is easy to see that if $L$ is a Sperner labelling of a pile $P$, then $L\\in\\nl(\\partial P)$ and $|\\deg(L,\\partial P)|=1$. Thus, Theorem 1 yields Theorem B as well as Theorem C implies Theorem A. \n\n\\medskip\n\nNot all manifolds can be oriented. For instance, the M\\\"obius strip is a non-orientable manifold with boundary. Theorem 1 can be extended for the non-orientable case. This extension is based on the concept of the degree of a continuous mapping modulo 2. Let $f:M_1\\to M_2$ be a continuous map between two manifolds $M_1$ and $M_2$ of the same dimension. The degree is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. Then $\\dg2(f)$ (the degree modulo 2) is 1 if this number is odd and 0 otherwise. It is well known that $\\dg2(f)$ of a continuous map $f$ is a topological invariant modulo 2. \n\n\\begin{theorem} \\label{SpQ2} Let $Q$ be a quadrangulation of a $d$-dimensional manifold $M$. Suppose $L:V(Q)\\to V(C^d)$ be a labelling such that $L\\in\\nl(\\partial Q)$ and $\\dg2(L,\\partial Q)\\ne 0$. Then $Q$ contains at least one centrally labelled cell.\n\\end{theorem}\n\n\n\n\\medskip\n\n\\noindent{\\bf Acknowledgment.} I wish to thank Fr\\'ed\\'eric Meunier for references and helpful comments. \n \n\n\n\\medskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrkyj b/data_all_eng_slimpj/shuffled/split2/finalzzrkyj new file mode 100644 index 0000000000000000000000000000000000000000..d501a481c8d869497813815945f010547edd6480 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzrkyj @@ -0,0 +1,5 @@ +{"text":"\n\nFor Misheel\n\nWith thanks to Ben and Sarah at Full Circle Yurts; Sue Kendall; The Brunswick Boys Club, Bootle; Anna McGrath for her coat; Caroline Wheatley at Linacre Primary School, Bootle; Cheryl Mackie at SPACE; Ermek Tleikhan, Batkhishig Luvsanpurev and Tsogtgerel Namsraijav; and Conor Lawrenson, Molly Lawrenson, Lewis O'Brien and Shannonn Byrne.\n\nI hadn't seen this photograph since the day it was taken, until now. Even so, I can tell you anything you want to know about it. The boy on the left is Shocky. The one on the right is Duncan, who used to come to school with biscuits in his pocket. He's married now, inexplicably. The girl on the left is Mimi Toolan and the one on the right is me.\n\nAt the moment the picture was taken, I was mostly wondering whether Mimi would ask me back to hers after school. Mimi's mother let her play with her make-up, which my mother definitely did not, even though I was mature and sophisticated.\n\nI was also thinking Oh. My. Days. Shocky has put his hand on my shoulder! Once, just before Christmas, I had managed to manipulate Shocky into being my partner in a classroom activity. This should have resulted in a moment of physical contact because it was a trust game, only it turned out that Shocky was not to be trusted. And by the time this photograph was taken, Shocky had completed an unbroken run of two hundred and thirty-seven days of failing to notice my existence.\n\nHow do I remember my thoughts so clearly? Because those were the only thoughts I had in the first two terms of Year Six:\n\n**1. Mimi, can I come back to yours?**\n\n**2. Shocky, please notice me.**\n\nAlso, this photograph was taken in the summer term of Year Six. And doesn't everyone remember everything about their last summer in primary school? The sports day. The leavers' trip. The leavers' photograph. The endless discussion of which school you were going to next, the promise to stay friends even though you were going to different schools. Everyone signing their names on everyone else's shirts on the last afternoon. And all the time, you had the feeling that day by day, inch by inch, a door was opening and sunshine was pouring in, and any day now you would be allowed out through that door, laughing and yelling so loud that you wouldn't even hear when it closed behind you, for ever.\n\nI can tell you when it was taken. It was the second week of the summer term. During morning break, Mimi spotted two kids \u2013 one big and one little, the big one holding the little one's hand \u2013 staring through the railings of the playground. The little one was wearing a furry hat and they had identical coats. Mad coats \u2013 long, like dressing-gowns, with fur inside. But any coat would have looked mad. The sun was beating down. The tarmac in the car park was melting. Everyone else was wearing T-shirts.\n\nMimi went over and said, \"What are you two looking at?\"\n\nThe big one put his finger on his lips, shushing her, and said, \"Pay attention to your teacher.\" He pointed at Mrs Spendlove, and the very minute he did, she blew the whistle for the end of break, like he knew she was going to do it.\n\nWhen we were all lined up, somehow these two were standing right behind me. I was looking at the littlest one, who had his hat pulled down right over his eyes. It looked so uncomfortable; I wanted to fix it for him \u2013 but the big one put his hand under my chin and turned my head away. \"Don't look at him,\" he said.\n\nHe was asking for a slap, quite honestly. But before I could do anything about that, Mrs Spendlove was walking us into class. The two boys went straight to the back and the little one made himself at home in what was supposedly my seat. I stood there, staring right at him, thinking he'd take a hint. But no.\n\nMrs Spendlove said, \"Everyone, I'd like you all to say a big hello to a new face in our class. A happy new face, I hope. This is Chingis.\"\n\nEveryone said hello except me. I said, \"What about the other one, Miss? What's he called?\"\n\nShe hadn't noticed the little one until then. \"Oh. Chingis,\" she said, \"I'm afraid your little brother isn't in this class. He's in Miss Hoyle's class just along the corridor.\"\n\n\"No,\" said Chingis, \"my little brother is in this class. Look, he's here next to me.\"\n\nEveryone laughed except Mrs Spendlove. \"Sorry, sorry,\" she said. \"I mean he _belongs_ in Miss Hoyle's class.\" She was flapping her hands at the rest of us to be quiet, mortified because she thought we were laughing at him and it was her fault. But I was standing next to him and I could see he hadn't made a mistake. He was digging in.\n\n\"Julie, would you show Chingis's brother to Miss Hoyle's class?\"\n\nI certainly would. For one thing, I wanted my seat back.\n\nAs soon as I stepped towards the little one, though, the big one put his hand up, right in my face, and said, \"No.\"\n\n\"Excuse me?\"\n\n\"He must stay with me. I am bound to take care of him. Protect him. I must stay with him.\"\n\n\"Well, it doesn't really work like that, Chingis. For one thing, once he's in Miss Hoyle's class, she'll protect him. And besides, he won't really need protecting because...\"\n\nHe wasn't even listening. He just took out some pencils and stuff and settled down to do a bit of drawing.\n\nMrs Spendlove opened up her laptop and poked around for a while. \"Ah,\" she said, talking to the little one directly. \"You need to go to a different class, Kub\u2014\" and started trying to spell out this unbelievable name, syllable by syllable. Before she got to the third syllable, Chingis looked up and said, \"No,\" again.\n\n\"No,\" just like that.\n\nIt was the second time he'd said no to her. Once might be a mistake. Twice was game on. Definitely. We were witnessing a struggle for power.\n\nMrs Spendlove made the first play. \"Excuse me?\" she said.\n\n\"Call him Nergui,\" he said. \"It's easier.\" Which was definitely cheek in her face, telling Mrs Spendlove what to do, telling her she wouldn't be up to the job of pronouncing someone's actual name.\n\nMrs Spendlove slapped that down. \"Well, that's not what I've got here,\" she said, and tried spelling out the long name again.\n\nChingis stood up.\n\nShe looked him in the eye.\n\nHe said, \"Please.\"\n\n_Please_ was good. _Please_ was some kind of stand-down. _Please_ was definitely points to her.\n\nShe closed the laptop really, really slowly. \"OK,\" she said. \"Just for today, you can stay in this class, Nergui.\"\n\nChingis said thank you and sat back down. It looked like victory to Mrs Spendlove. Except that somehow this kid had ended up with everything he wanted \u2013 his little brother was sitting next to him, being called by some unofficial name. Maybe Mrs Spendlove sensed this. Maybe that's why she decided she had to push it.\n\n\"So, if you take your hat off, Nergui,\" she said, \"we can all get started.\"\n\nThe kid didn't move and neither did Chingis. They both just sat there, with _What're you going to do about it?_ faces. Pretending they didn't understand.\n\nShe tried again. \"I'm afraid you have to take your hat off, Nergui.\"\n\n\"No,\" said Chingis.\n\nNow everyone looked at Mrs Spendlove.\n\n\"We can't have people wearing hats in class, Chingis.\"\n\nEveryone looked at Chingis.\n\nThis was like watching high-tension tennis.\n\n\"It will be dangerous to take off my brother's hat.\"\n\n\"How can it be dangerous to take off his hat? Is his head not securely fastened to his neck?\"\n\nShe got a laugh for that. The laugh gave her some edge.\n\n\"Not dangerous for him. Dangerous for you.\"\n\nMrs Spendlove frowned. Was he threatening her?\n\n\"If I take off his hat,\" he continued, \"maybe he will go insane and kill everyone.\"\n\nHe was definitely threatening her. Threatening all of us. With his little brother.\n\n\"Chingis...\"\n\n\"When you need your eagle to be calm, what do you do?\"\n\n\"I don't know.\" She looked around the class. Did anyone know? Why _would_ anyone know?\n\n\"Of course,\" he said, \"you cover its eyes with a hood. When you want the eagle to fly and kill, you take off the hood. My brother is my eagle. With his hood on, he is calm enough. Without his hood, I don't know what he will be like.\"\n\nYear Six. We had been at school for six years and until that moment I thought I had probably learned all I would ever need to learn. I knew how to work out the volume of a cube. I knew who had painted the \"Sunflowers\". I could tell you the history of St Lucia. I knew about lines of Tudors and lines of symmetry and the importance of eating five portions of fruit a day. But in all that time, I had never had a single lesson in eagle-calming. I had never even heard the subject mentioned. I'd had no idea that a person might need eagle-calming skills.\n\nAnd in that moment, I felt my own ignorance spread suddenly out behind me like a pair of wings, and every single thing I didn't know was a feather on those wings. I could feel them tugging at the air, restless to be airborne.\n\nI wanted to talk to the new boy. I wanted to talk about eagles. But Mimi seemed to regard the whole Chingis incident as a minor interruption in the ongoing global cosmetics conversation. Only the boys were interested. At lunchtime, dozens of them crowded round Chingis and Nergui, asking them if they really had eagles, and how big they were, and whether Chingis was a liar or not.\n\n\"Where d'you get eagles from, then? Eagles R Us?\"\n\n\"Everyone has eagles where I come from.\"\n\n\"Where's that, then?\"\n\n\"Mongolia.\"\n\nThey poked and pestered little Nergui, who still had his hat pulled right down, hiding his eyes. They kept telling him to make eagle noises. The kid \u2013 Nergui \u2013 huddled down in his coat, pulled his arms out of his sleeves and crossed them over his chest. His sleeves were flapping loose and he did fully look like a bird.\n\nThen Chingis spotted me over their heads and shouted, \"You. You must come and help me.\"\n\nI didn't know what he expected me to do. But I was fully delighted to be asked. I slid past the boys and then turned on them. \"All right,\" I said. \"Move on. Haven't you seen a pair of Mongolian brothers before?\"\n\n\"No.\"\n\n\"Well, you have now. So move on.\"\n\n\"As if they're Mongolian, anyway.\" It was Shocky. \"Why would they come here from Mongolia? They're probably from Speke.\"\n\nEveryone agreed that the brothers were probably from Speke and then went back to their footie.\n\n\"Please stand still,\" said Chingis. He moved me back a bit and pulled something out of his bag that looked like an old-fashioned radio. When he pressed a button, it made this whirring sound, the top half shot open and a lens popped out.\n\nI know now that it was a Polaroid camera. At the time I think I thought it was some kind of mad, starey cuckoo clock.\n\n\"I need a picture,\" he said, \"so I can remember which one you are. You are to be our Good Guide here. OK?\"\n\nMimi had come over by this point \u2013 she could hear a camera being deployed at five hundred metres. We both did our loveliest smiles, and that would be when Shocky and Duncan came over and tried to get into the picture. Almost as soon as Chingis had clicked the button, a strip of paper rolled out of the front of the camera. He peeled off some kind of label, then waved the paper around in the air, and there we all were. Caught for ever. He wrote something on the photo, which I didn't see at the time.\n\nI saw it for the first time today. He'd written, \"Our Good Guide.\"\n\n\"You will be our Good Guide,\" he said. \"In Mongolia we are nomads. When we come to a new country, we need to find a Good Guide. You will be our Good Guide in this place. Agree?\"\n\nOf course I agreed. No one had ever asked me to be anything before, definitely not anything involving a title. And that was when I stopped thinking about make-up, lips and Shocky. That was when I started walking round the place thinking, Hi, I'm the Good Guide.\n\nI really did want to be a good guide.\n\nThat's not Nergui's coat in the picture. That's Chingis's coat. I saw that coat today for the first time since we all left. I'd heard that they were going to knock the school down this summer. As it was the last day of term and my last chance to take a look, I went over on my way back from work.\n\nMrs Spendlove was still working there, incredibly, and she recognized me right away. Thirty-four years she's taught there. Imagine that. She let me go round with her while she collected her stuff and some souvenirs. The old store cupboards, the cloakroom, her classroom. And there at the back of our old classroom was a big blue plastic tub with **LOST PROPERTY** written on it. Mostly trainers and socks and a few books, a lockable Miffy diary, a couple of _In the Night Garden_ lunchboxes. And the coat.\n\nThe unforgettable coat of Chingis Tuul.\n\nI lifted it out and held it up at arm's length. I wish I could say it looked like a bird, but it was more like a big hairy bat, just hanging there. I went through the pockets, the way you do.\n\nAnd that's how I found these pictures.\n\nI really did take my Good Guide duties seriously. I took Chingis and Nergui to the dining hall every lunchtime and made sure they could sit together \u2013 even though I didn't have dinners myself. I made sure they knew what they had to bring for games and swimming. I told them to lose their weird-looking coats and wear something normal. And when it was our class assembly, I lobbied for it to be \"All About Mongolia\", thinking that Chingis would join in and maybe even be pleased. But it didn't work out that way.\n\nI brought in pictures and looked stuff up on Wikipedia for the first time in my life. He did nothing. Even on the day, he just stood there, looking Mongolian, while I told the school all I had learned about how Mongolia, a landlocked presidential republic in Central Asia, was the most sparsely populated country in the world, where a lot of the people were still nomads who lived in big tents called yurts and the men liked to hunt wolves with eagles. And how there was a city there called Xanadu, which was the fifth Great Khan's summer capital. It had fountains and brooks and meadows and woods that were full of every kind of wild beast, and the Khan went hunting with his eagles there. The palace itself was made of tightly woven bamboo so that it could be taken apart and moved. Inside, it was all painted with birds and animals and trees so that you couldn't really tell if you were inside or out.\n\nWhen someone said, \"Is it really like that where you come from?\" Chingis said, \"Yes. Nothing has changed.\"\n\n\"What did you come to Bootle for, then?\"\n\nWhen everyone sniggered, he just shrugged his shoulders and said, \"We are nomads. We move around.\"\n\nI didn't do all this out of the goodness of my heart. It was part of my plan: I wanted to be asked back to their house. I imagined it would be stuffed with silks, with a horse-head fiddle in one corner and a samovar bubbling in the other. I really had done my homework.\n\nThanks to my obsession with Mimi's make-up, I already knew a bit about getting yourself asked back to places. All you had to do was walk with someone until they were nearly home then say, \"Oh, is this where you live?\" and if that wasn't enough, just say you needed the toilet. Once you were through the door, their mother usually asked you to stay.\n\nThis didn't work with Chingis and Nergui, though, for the simple reason that they seemed to take a different route home every day. One day they'd head left up Hawthorne Road, so the next day I'd go that way and wait for them to catch up. I'd wait for ages and then discover they'd gone off down the Avenue. So the next day I'd go that way, only to see them turn round and walk back the way they had come, heading straight past me. Sometimes they'd disappear into the terraces. Sometimes they'd even slide off into the back alleys.\n\nI gave up trying to follow them, but whenever I was out I would look at the windows of the houses and flats, wondering if one of them was theirs and feeling certain that somewhere in the narrow streets or tower blocks there was a room with the silks and the samovar, like a secret gateway.\n\nSomewhere in Bootle, Xanadu was buried like treasure.\n\nThen one day I went into Savedra's shop for a packet of Monster Munch and a bottle of Sunny Delight, and there they were, the two of them, standing in the doorway looking at me. Chingis said, \"Are they good?\" pointing at the Monster Munch. I offered him some and started walking slowly towards my house.\n\nChingis crunched the Munch. \"Yes, it is good. You can give some to Nergui.\"\n\nWe walked along, with them dipping into my Monster Munch every couple of metres. I subtly changed course whenever they changed theirs. I chugged some Sunny Delight so I could be convincing when I asked for the toilet. But somehow, we ended up outside my house, not theirs.\n\n\"I need the toilet,\" said Chingis. \"And so does Nergui. You have toilets?\"\n\n\"Sure. Come in.\"\n\nAs soon as they came through the door, my mum asked if they wanted to stay to tea.\n\n\"Sure,\" said Chingis.\n\nHe and Nergui went up to the toilet. Mum asked me if I thought they'd like fish fingers. \"Or is that against their religion?\"\n\n\"I'm not sure what religion they are. They eat normal school dinners.\"\n\nWe heard the toilet flush but the boys did not come down. We could hear them walking around upstairs, opening doors and even drawers.\n\n\"That's a bit much, you know. Doors are one thing but I draw the line at drawers.\"\n\nShe couldn't, in fact, draw the line, though, because Chingis walked into the kitchen and said, \"Please, we need to bake something right away. You have flour?\"\n\nSomething in his voice managed to infect Mum with baking panic. Personally, I'd never heard of emergency baking before. But Mum was yanking a mixing bowl out of the cupboard like it was a fire extinguisher. Bags of flour, slabs of butter \u2013 she threw them onto the table like medical supplies.\n\n\"Yeast?\"\n\n\"Yeast?! We don't have yeast!\"\n\nIt seemed that we might all be doomed by lack of yeast and that only Chingis could save us.\n\n\"It's OK,\" he said. \"This time I'll do it without yeast. Stand back, please. And warm the oven.\"\n\nMum more or less ran to the oven and Chingis started throwing stuff into the mixing bowl and bashing the dough about. Nergui stood there watching as though this was heart surgery and it was his heart in the mixing bowl.\n\nIt was only when I said, \"What is it exactly that you're doing?\" that things started to calm down.\n\n\"Yeah,\" said Mum. \"What is it exactly that you're doing?\"\n\n\"You have some raisins?\" asked Chingis.\n\n\"Sure.\" She passed him a bag of raisins. He squeezed the dough into the shape of a little boy and added raisins for teeth and eyes.\n\n\"You know, if we're going to eat this, you should have washed your hands.\"\n\n\"This is not for eating. Not for _us_ to eat, anyway. We need something else to eat. What do you have?\"\n\nMum said that she was thinking of eating fish fingers if it wasn't against their religion.\n\n\"There is a religion in this country that forbids fish fingers?\" asked Chingis.\n\n\"No, I don't think so. I'll put them on. Do you want to phone your mother and tell her you're staying to tea?\"\n\n\"No. We are nomads. She doesn't expect us to come home like children who are not nomads. Maybe we will be staying here for the night.\"\n\n\"Oh, will you, maybe?\" said Mum. \"Maybe you won't, either.\"\n\nChingis glanced at Nergui. Then he looked around the room, as if checking that no one was listening. Mum had this mirror near the back door next to the See How I Grow chart. Chingis went over, took the mirror off the wall and turned it face down on the table. Then he closed the blind.\n\n\"We are telling you something in secret,\" he said. He looked at Nergui again and Nergui nodded. \"My brother believes he is being chased by a certain demon.\"\n\n\"A demon?\" said Mum. \"In Bootle? Are you sure there's only one?!\"\n\n\"We have to take steps to save him. For instance, Nergui is not his name. We never speak his name. Nergui means no one, so if the demon hears us speaking to Nergui, it thinks we are speaking to no one.\"\n\n\"Right,\" said Mum. \"Well. Obviously. Don't know why I didn't think of it myself.\"\n\n\"Also, we take a different route home from school each night so that it can't easily find where we live.\"\n\n\"But it does know where you go to school?\"\n\n\"He saw it in school. Twice.\"\n\n\"So he has actually seen this demon, then?\"\n\n\"Of course. Or how would he know it was following him? We are not people who are afraid for no reason.\"\n\n\"Course not. So what does it look like?\"\n\n\"It's in disguise. It looks like an ordinary man.\"\n\n\"So... how do you know it's a demon?\"\n\n\"Because it wants to make him vanish. It's a demon that makes things vanish.\"\n\n\"Right,\" said Mum.\n\n\"That's why we had to leave Mongolia. This demon was there. It wanted to make us vanish. So we had to leave. We walked along the railway track that led out of our country. We followed the railway for days and days. Until we came to here.\"\n\n\"There's a direct rail link from here to Mongolia?! Really?\"\n\n\"Not direct, no. We make many changes. We do it to confuse the demon. And now I have made this boy out of dough. If we leave him on your doorstep, then maybe if the demon has followed us, it can think that this is Nergui. And maybe that will vanish instead.\"\n\n\"Great plan,\" said Mum. \"Now if you're going to eat fish fingers, go and wash your hands.\"\n\nSo they stayed to supper. Before they ate, they put the dough boy on the doorstep. While we were eating, we watched the door. We couldn't help it. The lights as the cars went by, the voices of passing people, they all seemed like demon-related activities to me.\n\nA few weeks before, I had not known that there was any such thing as a portable bamboo palace. I hadn't even known there was such a person as Chingis Khan, who had been born with a clot of blood grasped in his fist and who had conquered nearly the entire world in hardly any time at all, sweeping over the steppe into Central Asia and right up to the very gates of Europe. I hadn't even known there was such a place as the steppe! The steppe that was flat as a pavement but wide as a sea, with nothing but grass and Great Bustards. Wide as a sea and I hadn't even known it was there. If there were seas of grass and woven palaces in this world, why wouldn't there be demons too? And why wouldn't one of them be crouched on our doorstep in William Morris Road right that minute, munching a boy made of dough?\n\nChingis cleared the plates without being asked. Nergui stayed staring at the door, looking tense.\n\n\"I think that little fella wants his mother,\" said Mum.\n\n\"I'll see if it's safe,\" said Chingis.\n\nHe opened the door and looked down at the step. The dough boy had disappeared.\n\n\"Honestly,\" said Mum, \"you could leave a bucket of nuclear waste on your doorstep round here and it would be gone in five minutes. They really will nick anything.\"\n\n\"We can go home,\" said Chingis. \"The demon has eaten the dough boy. It won't need to eat again tonight.\"\n\nI remember this one too. One day, Mrs Spendlove made this announcement that we were going to do an experiment the next day and we were all supposed to bring in a little canister, like an old-fashioned plastic film canister. \"You are bound to have one hanging around your house,\" she said. \"We're going to make our own rockets.\"\n\nBut when the next day came, no one had brought one in. She seemed disappointed. \"I wanted you all to do this yourselves,\" she said. \"I've only got three so I'm afraid you'll all have to watch instead.\"\n\nAnyway, she did this experiment where she stuck an Alka-Seltzer to the inside of the lid of one of the canisters with Blu-Tack, then she poured some water in the canister, put the lid on and turned it upside down. No one was that interested, until about two minutes later when the canister shot up to the ceiling and hit a light bulb, spraying water everywhere. Suddenly everyone wanted a go but Mrs Spendlove made this big speech about how you only get out what you put in, and how since none of us had gone to the bother of finding a canister, we would all just have to copy out some diagrams about the expansion of gases and propulsion and stuff, instead of having fun.\n\nAs soon as little playtime began, Chingis went to the quiet-corner benches and announced that he had film canisters, Blu-Tack and Alka-Seltzer all available for a pound a time. He opened his bag. He must have had thirty plastic film canisters, a massive slab of Blu-Tack and a catering pack of Alka-Seltzer.\n\n\"You can buy from me your own small rocket,\" he said. \"One film canister, one stick of Blu-Tack and an Alka-Seltzer \u2013 a pound.\"\n\nEveryone wanted one. People were pushing around him, handing over money.\n\nI said, \"You brought these in but you didn't tell Mrs Spendlove? Why?\" I'd felt sorry for her.\n\nHe said, \"Why would I use them in the lesson? Last night, I collected all of these from neighbours to sell.\"\n\n\"You could have sold them to us _before_ the lesson. Then everyone would have had one and she would have been really happy.\"\n\n\"Before she showed how it works, no one wanted one. No one would pay a pound when they don't know what it does. Now Mrs Spendlove explains, everyone wants one. Her lesson was good advertising.\"\n\n\"But you brought Blu-Tack and Alka-Seltzer as well as the canisters \u2013 how did you know?\"\n\n\"Rockets were invented in Mongolia. Propulsion is in our blood.\"\n\nA lad called Backy \u2013 he's in the police now \u2013 said, \"They're my canisters. You can't make me pay for my own canisters.\"\n\nApparently, the night before, Chingis had gone and knocked on every door in his block of flats \u2013 including Backy's \u2013 and asked if they had any film canisters.\n\n\"I give you the canister free,\" said Chingis. \"Blu-Tack is free also. Just one pound for the Alka-Seltzer.\"\n\nPoor old Backy went for that. Honestly, they didn't know they'd been cut in half until they tried to walk away. Chingis got little Nergui to help him count all the cash. Thirty-four quid they made, and all day the school was full of popping sounds and quietly fizzing puddles.\n\nThere is only so much Good Guiding you can do in a primary school playground. Even though I really tried to spin it out, I'd pretty much pointed out every stone and weed by the end of ten playtimes. Mrs Harrison was the dinner lady \u2013 she had the power to give you a plaster if you cut yourself. The big red box held outdoor play equipment \u2013 you could take what you wanted but you had to put it back when you were finished (Mrs Harrison was watching). The numbers and squares painted on the floor were for playing hopscotch but no one ever did, because no one really knew how. The quiet corner \u2013 a little ring of apple trees with the bench in the middle \u2013 had been made in memory of a boy who'd died just before we came to the school. Everyone said he was buried under the trees and there were bits of him in the apples \u2013 but it wasn't true; he was buried in a graveyard like everyone else.\n\nUntil Chingis came, I had mostly spent playtime with Mimi, walking up and down beside the railings, talking about the inevitable future success of our imaginary girl group, the Surfing Eskimos. Until I took on the Good Guide duties, I'd had no idea there was a small tribe of sad-looking Year Five lads who more or less lived in the clump of trees that had been planted to hide the bins. They spent their time playing on their DSes and avoiding Shocky and Duncan. I'd had no idea that Mrs Harrison spoke French because she came from the Congo. I hadn't known that you could find frogs in the big clump of nettles between the playground and the car park. And I hadn't known there were hula hoops in the outdoor play box.\n\nChingis took one out and asked me what it was. I put it over my head and tried to spin it round my waist but it just clattered straight onto the floor.\n\n\"Oh,\" he said, \"it's a hula hoop. You do it like this.\"\n\nHe took it off me, dropped it over his head and, barely moving his hips, made it spin like a blender. He could carry on walking and talking at the same time. Nergui too.\n\nMrs Harrison laughed and said, \"You don't often see boys doing that.\"\n\nChingis stood still. The hoop dropped to the floor. He looked around the playground. \"This place is nearly nothing but girls. Where are the boys?\"\n\n\"This is the red playground,\" I said, \"it's for anyone. If you want to play football, you go on the blue playground. It's through there... Nearly all the boys play football.\"\n\nChingis led Nergui through the little gap between the dining hall and the Portakabin and I followed. The two of them stood there for the rest of break, watching the boys charging up and down the blue playground, yelling and pointing and chasing the ball.\n\nChingis said, \"Does this have rules?\"\n\n\"I think so,\" I said.\n\n\"You need to find them out. We need to play this.\"\n\n\"Right.\"\n\nSo the two of them came back to ours and made me explain the rules of football. I had to borrow a ball from next door and, in the end, I had to borrow the little boy too. Jordan, he was called. He couldn't understand why they didn't know how to play.\n\nI said, \"They're from Mongolia.\"\n\n\"Even so, they must have footie. They have footie everywhere.\"\n\n\"Not in Mongolia.\"\n\n\"No,\" said Chingis. \"Mongolia is just horses, horses, horses. If we want to play with a ball, we get on a horse first. If we want to throw something or shoot something, we get on a horse. If we want to hunt, we get on a horse. If we get married, we take the bride on a horse. Only exception is wrestling. Wrestling is on the ground.\"\n\nJordan had heard of countries that had floods and disease and war but he'd never heard of anywhere so bad that it had no football. He was nearly in tears. He taught them skills, rules and tricks, and after that they sat on the pavement and he taught them the entire history of Liverpool FC.\n\nChingis said, \"In football, it is just as much talking as it is playing.\"\n\nFrom then on, Chingis spent little playtime trying to get me to talk to him about the offside rule, the transfer window and the history of the Champions' League. He didn't play, though. He made Nergui play and took me to watch him. He'd point to him running up and down, barging into people, and say, \"On the ball, yes, he's a bit of a donkey, but off the ball, he's a real workhorse.\" Even when he was talking about football, he was still quite horsey in his thinking.\n\nAny football word he used \u2013 foul, penalty, you've got to be kidding, yes! \u2013 he said with a strong Liverpool accent. I'd been hoping he would turn me into some kind of Mongolian princess but instead he was turning into a Scouser. \"See our kid?\" he said, pointing at the pitch. \"Can you even tell which one he is from here?\"\n\n\"He's the one with his hat pulled down over his eyes.\"\n\n\"That means nothing. Know the best way to hide a needle in a haystack?\"\n\n\"I don't think there is a best way. Once the needle is in the haystack, it's really hard to find. I think that's the point, really.\"\n\n\"What if you've got a metal detector?\"\n\n\"Do demons have metal detectors?\"\n\n\"Don't talk about demons. Don't even mention them. OK? The point is: he looks just like any other kid now.\" And he did look just like any other kid. \"The best way to hide a needle in a haystack is to disguise the needle as a piece of hay.\"\n\nThe truth was: the boys weren't just learning English; they were hiding themselves _inside_ English, burying themselves in footie and insults, swearing and buzz words. They were _learning_ themselves ordinary. And in our school, ordinary boys did not hang out with girls. Soon they didn't need a Good Guide any more, and I found myself back at the railings with Mimi, planning our future fame and deciding what to wear on Own Clothes Day.\n\nOwn Clothes Day was when you were allowed to wear what you liked to school as long as you gave money to charity. I think there'd been a tidal wave or an earthquake. I say you were allowed to wear whatever you liked. In fact, if you were a boy you were allowed to wear an Everton football strip \u2013 and risk being skitted by Liverpool fans \u2013 or a Liverpool football strip \u2013 and risk being skitted by Everton fans \u2013 or something other than a football strip \u2013 and risk being beaten up by everyone. Girls were allowed to wear basically anything they liked as long as it was really, really short and involved a huge bag. For some reason, I decided to go in the boy next door's Everton kit. I probably thought it was my chance to remind Chingis that I was the one who'd taught him about footie.\n\nWhen I got to school, he was waiting with Nergui at the gate. The two of them were standing completely still. They were wearing their mad coats again \u2013 the furry dressing-gown coats they'd worn on their first day. And they were both wearing hats with fur inside. They looked mighty.\n\nChingis looked at me in my Everton kit. \"Perfect.\"\n\nI thought he was paying me a compliment. I said, \"It's just a footie kit. It's not even an official one. It's a rip-off and it's not even a rip-off of this season's kit.\"\n\n\"Please come with us.\"\n\nHe made me wait outside the boys' toilets while they went inside. A few minutes later, he came out carrying Nergui's coat. \"You wear this,\" he said. \"Nergui will have your football kit.\"\n\nIn the girls' toilets, a million tiny hooks of static tried to cling on as I peeled myself out of the nylon Everton top. Then I stepped into Nergui's coat as if I was stepping into another country. The cuffs were frayed and worn \u00ad\u2013 I imagined that was where the eagle used to perch. The corners of the pockets were packed with grit \u2013 probably sand from the Gobi Desert. I was half expecting to find the rest of the desert waiting for me when I opened the toilet door. It wasn't.\n\nChingis was waiting for me, though. And after Nergui had disappeared back into the toilets with the football kit, the two of us stood there in the corridor, letting people look at us. No one said anything bad. We looked too scary.\n\nWhen Nergui came out, it was the first time I'd seen him without his hat. He had long hair.\n\nI said, \"Nergui! You've got long hair! You look like a girl!\"\n\n\"He thinks if he looks like a girl, his demon won't recognize him.\"\n\nNergui said, \"You look like a boy in there. Maybe my demon will take you.\"\n\nIt was the first time I'd heard him talk too. He sounded just like any other boy in our school.\n\nI just said, \"Maybe.\"\n\nObviously I knew they'd only asked me to swap clothes to confuse their demon. Did I care about being used as demon-bait? No, because by then I didn't believe in demons. Plus, wearing that fur coat made me one of a pair with Chingis, one of a pair of swaggering nomads with eagle-calming skills and strings of horses somewhere in the desert.\n\nIn the afternoon, Mrs Spendlove asked Chingis to tell the class a little bit about the clothes we were wearing. I stood up and explained that they had to be thick and warm because it could get so cold on the steppe, even in the desert, at night. Minus fifty. And even though they hadn't asked me, I told them about Chingis Khan \u2013 who Chingis was named after, just like lots of Mongolian boys. Because I knew all about him, I said how when he was dying, he asked to be buried in a secret place with no monument so that the whole of his empire could be his gravestone. After his friends had laid his body in the ground, they stampeded a thousand horses over it to churn up the soil and disguise his grave. Then they all rode home and killed each other so that no one would ever tell the secret.\n\n\"What about you, Chingis? What can you tell us about your country?\" said Mrs Spendlove. Thinking about it now, that may have been a pointed remark as I seemed to be doing all the talking.\n\nHe said, \"Well, there's the desert. And then there are some oasis things in the desert, you know, with trees that look like giant flowers.\" I didn't know about them. \"They're sort of magical. And we have mountains that are made of metal and they are shiny in the sun.\" I didn't know about them, either. \"And if you're in trouble, you can make a pile of stones and maybe put some horse's skull on it, or a prayer flag, and walk around it three times clockwise, and that'll help. It's called an ovoo.\"\n\n\"How will that help?\" someone asked.\n\n\"I'm not sure. If you see one that someone else has built, you should walk round that three times too. I'll show you a photo if you like.\"\n\nAnd he did. He had photos of the desert. Of the magical oasis with the flower-like trees. Of the pile of stones.\n\nTalking about these things made school and Hawthorne Road and all of Bootle feel temporary and little, as though we were just passing through on our way to some indescribable adventure.\n\nI kept that feeling all the way home, through the front door and into the kitchen, where Mum looked up from the washing machine and said, \"What the hell do you think you're wearing?!\"\n\n\"It was Own Clothes Day.\"\n\n\"And those are not your own clothes.\"\n\n\"They're Nergui's \u2013 we swapped.\"\n\n\"Julie, I'm not being held responsible for someone else's coat. You take it round to theirs right now.\"\n\n\"But I don't know where they live.\"\n\nIt took Mum two phone calls to find out what I'd been burning to know for weeks \u2013 where did Chingis and Nergui live? Where was Xanadu? Turns out it was up on the tenth floor of Roberts Tower \u2013 the tower block nearest the flyover.\n\n\"And we're going over there right now,\" she said.\n\nShe packed the coat into a big plastic bag and drove me over to the Tower. When we got there, though, she decided she didn't like the look of the place \u2013 there was a pile of rubble in the middle of the car park, like a lookout tower, with two kids perched on top of it.\n\n\"I'm not leaving the car here with no one in it. They'd have it ransacked by the time we called the lift.\"\n\n\"That's all right. You stay with the car. I'll go in.\"\n\n\"I'm not letting you walk into that place on your own, either.\"\n\n\"So what are you going to do? Drive the car into the lift?\"\n\n\"Maybe leave it till the morning after all.\"\n\n\"No. No, I'm happy to go in.\" I didn't wait for her answer. I was running into Xanadu.\n\nThe lift was working but smelly. As it clanked up to the tenth floor, I pulled the coat out of the bag and put it on. I walked up to the door, thinking, This is it. Their mum will see me in it and ask me in. \"Come in and lounge about on the silks while the samovar is bubbling,\" she'll say. \"Father, give us a tune on your horse-head fiddle.\"\n\nI rang the bell. I heard it ringing somewhere inside the flat. There were some voices. A door opened. Or maybe closed. Then it all went quiet.\n\nI waited.\n\nNothing.\n\nMaybe some whispering.\n\nI knew it wasn't polite to ring twice but I wasn't going to walk away. I'd been searching for this place for weeks. I'd been longing for it for weeks. I rang the bell again.\n\nSilence.\n\nA tense, holding-your-breath type of silence.\n\nAgain.\n\nNothing.\n\nA deep, rasping growl behind me made me jump. For some reason, I was thinking of their demon. But it was the lift. The doors opened and Mum stepped out.\n\nI said, \"I thought you were waiting in the car?\"\n\n\"Of course I'm not waiting in the car. As if I'm going to leave my own daughter to her fate just so I can look after the car. Did you smell that lift? Dear me. Have you rung the doorbell?\"\n\n\"Yes.\"\n\n\"Didn't they answer?\"\n\n\"Not yet.\"\n\n\"So they're not in. So let's get back to the car.\"\n\n\"Shush. Listen.\" We could hear hushed voices arguing and water running from a tap.\n\n\"So they're in but they don't want to answer the door. They're probably having their dinner. Leave it till tomorrow.\"\n\nShe'd already pressed the button for the lift to come back. I said, \"Maybe the bell is broken,\" and I hammered on the door.\n\nThe voices stopped. The tap stopped. I hammered again.\n\nMum grabbed my hand. \"Don't do that,\" she hissed. \"It's rude.\"\n\nBut there were footsteps in the hallway now and someone pulled the door open. A woman...\n\nShe was not wearing the traditional jewelled Mongolian headdress. She was not wrapped in silk. She was not happy to see me. She didn't see me at all to start with.\n\nShe looked at Mum.\n\nMum said, \"I'm Julie's mother. A friend of your Chingis. This is his coat.\" The woman didn't answer. She just stared at Mum and then at me. As if she was trying to work out the answer to a puzzle. I knew she wasn't going to ask us in. She stood and watched me as I struggled out of the coat. I got a glimpse of the flat \u2013 the long, empty corridor, the bare light bulb at the far end and, near the door, a line of bags and suitcases, bulging and fastened, as if the family were about to leave.\n\nI tried to hand back the coat but she didn't take it. Instead, she covered her eyes with her hands and I realized she was crying. I stood there holding the coat while she held her face. Then a door opened at the far end of the corridor and suddenly Chingis was striding towards me. He didn't say anything. He didn't look at me. He just took the coat, threw it over his shoulder with one hand and took hold of his mother's elbow with the other. He steered her back into the flat, then slammed the door in my face.\n\nI thought Mum would go on about how rude they were, or how she was sure the car had been stolen. But she didn't say a word all the way down in the lift and all the way home in the car. It was only when we were safely parked in William Morris Road that she said, \"Well, what was that all about?\"\n\nI knew exactly what it was all about. I didn't know the details or the reasons. But I knew it was all about fear.\n\nI didn't know why. But I knew that everyone in that house was afraid.\nThe next day, Chingis came to school in his coat again and didn't take it off even when it was time to go to class. When I said, \"Hey, it's not Own Clothes Day any more,\" he didn't even look at me.\n\nWhen he sat down, Mrs Spendlove also said, \"Chingis, it's not Own Clothes Day any more.\"\n\n\"Yeah, and these aren't me own clothes, are they?\" said Chingis. \"They're me grandad's, see what I'm saying?\"\n\nHe was doing his pretending-not-to-understand thing again but it wasn't really that convincing now he had a thick Scouse accent. Mrs Spendlove said, \"Just step outside, go to the cloakroom and hang the coat up.\"\n\nChingis slouched out of the room. Mrs Spendlove rolled her eyes and Shocky said something like, \"What d'you expect? He is named after Chingis Khan.\"\n\nI said, \"And what do you know about Chingis Khan, then?\"\n\n\"Yes,\" said Mrs Spendlove. \"And what do you know about Chingis Khan?\"\n\nHe was supposed to say, \"Nothing, Miss,\" but in fact he stood up and delivered this whole lecture. \"Chingis Khan was born in 1100 and something. He had red hair. When he grew up, he conquered and what for? All he was ever interested in was horses. He just went round conquering countries and killing loads of people so he could have more horses. Like, he'd capture a city and they'd say, 'What d'you want to do with it?' And he'd be like, 'I know! Let's knock it down and we'll have a bit more room for our horses.' Or someone would come to him who'd discovered a new country and they'd be like, 'The people there have wings and they can read minds and they've got a city that floats in the air,' or something, and he'd be like, 'Have they got any horses? No? Can't be bothered then.'\"\n\nI said, \"But they were nomads. Horses were very important. They needed horses to survive. And for status. People nowadays, they use their cars to show their status.\"\n\n\"So if Chingis was alive today,\" said Shocky, \"it would be cars instead of horses. He'd kill everyone so he could grab their cars and then he'd tarmac the whole of Europe so he could drive wherever he wanted.\"\n\nIt was hard to believe that I was actually having any kind of conversation with Shocky, let alone one about the history of the Mongol Empire. Afterwards, Mimi put it like this: \"He obviously fancies you \u2013 otherwise why would he look up all that stuff about horses or whatever?\"\n\nMrs Spendlove broke into the Great Mongolian History Debate to ask, \"Where actually _is_ Chingis? He was only supposed to go and hang up his coat. Duncan, go and get him.\"\n\nSo Duncan went to get him. But he'd already gone. His coat was hanging up in the cloakroom but there was no sign of him. Mrs Spendlove said we should just concentrate on the lesson and let him sulk if he wanted to.\n\n\"He's probably gone to kill someone and then have a drink out of his skull. That's what the original Chingis did \u2013 and they're still naming their kids after him! Imagine that. Imagine if Germans called their kids Adolf. And then imagine if people followed those kids home every night and taught them how to play football and swapped coats with them and everything.\" Shocky was bright red by the time he'd finished.\n\nI said, \"At least they're polite enough to talk to people and not just ignore someone for an entire year!\"\n\n\"Excuse me,\" said Mrs Spendlove, \"could you two old married people take your domestic disputes somewhere else?\"\n\nMarried?! What was she on about?\n\nBy lunchtime there was still no sign of Chingis, and when I looked on the blue playground there was no sign of Nergui, either. No one else seemed that worried. Kids did sag from our school quite a bit \u2013 and anyway, Chingis seemed like he could look after himself. Only I had seen those faces at the door the night before. I knew about the fear that was in their house.\n\nSo I went through the pockets of the coat. I was trying to help. I found two tiny plastic chairs and a table, like from a doll's house. I seemed to know them from somewhere but I couldn't think where. There was also a rolled-up notebook, with all his Polaroids stuck inside like Top Trumps cards. I flicked through the Polaroids. I came across the one of the oasis in the Gobi Desert \u2013 the one with the strange, flower-like trees. Under the trees was a table and two chairs, the same chairs and table I was holding in my hand. I looked at the photograph again. They weren't flower-like trees at all. They were just flowers. And I knew which flowers.\n\nI took the pictures, the notebook and the doll's furniture out to the clump of trees where the Lost Tribe of Year Five lived. There were long, skinny flowers growing between the trees. I put the doll's furniture in between them, then crouched down and squinted. It took me about two minutes to find a perfect match. I swear I even found the tiny imprints of the table legs in the soil.\n\nThe magical Mongolian oasis was behind the bins in our school yard.\n\nThe ovoo with the horse's skull \u2013 when I looked again \u2013 was the pile of rubble in the car park of Roberts Tower.\n\nWhat other wonders of the world were actually in Bootle?\n\nThere was also a Polaroid of two railway lines striking out across a flat prairie. Probably this was the railway line they had followed by foot on the epic journey across the steppe and along the Silk Road when they were trying to escape from the terrible demon.\n\nEither that or it was the Merseyrail to Southport, which runs behind our school.\n\nI rolled up the notebook, put it back in the coat pocket and sneaked out through the blue playground. It was the easiest way out. If anyone stopped you, you could say you were going after a stray football. I walked up to the station, stood on the platform and peered down the track. It looked completely different from the photo \u2013 it was all houses and flyover \u2013 except that the rails were identical. So I started to follow them.\n\nI had been crashing through the nettles and broken bottles along the tracks for about five minutes when I heard this strange, high-pitched singing coming from the rails. I looked behind me. There was a blob of bright blue light hovering over the rails. A train was coming. I didn't think about it. I ran back to the platform and jumped on board the train.\n\nFor five minutes there was nothing outside the window but more houses and the rest of the flyover. Then something flashed in the sun. A mountain of scrap metal, towering over the Seaforth dock, shining in the hot afternoon sun. Metal mountains.\n\nAfter that came the fields. I'd had no idea till then how close our house was to cows and horses. How come no one had ever mentioned it? Maybe no one knew. Maybe I was discovering an unknown country that everyone had missed even though it was so near by.\n\nAfter the fields came the trees. I put the Polaroid of the Mongolian forest up against the window as the train clunked into Freshfield Station. They were the same trees. Definitely.\n\nSo I got off. And I walked up the road and into the trees. Someone went by with a dog and smiled at me. Like it was fine for me to be there. A couple went past with a pushchair with massive wheels. They said hello too. Then it was just trees and dark, worrying shadows in among the trees, and sometimes scratching sounds, though I could usually tell that it was just birds. And once I saw a squirrel. I was off the road now, following a sandy path. Every now and then there was a little white post with a number on it. Which made me feel like I was going the right way for something, though I didn't know what...\n\nThen, all of a sudden, Nergui was walking next to me.\n\nI didn't see it happen. I just sort of felt someone there and then I saw him from the corner of my eye. \"Where's Chingis?\" I asked.\n\n\"Just coming. How come you got here before us? We thought we'd have to wait all night for you to catch up.\"\n\n\"I caught the train.\"\n\n\"Oh.\"\n\n\"How did you know I was coming?\"\n\n\"You're our Good Guide. It's your job.\"\n\nWe sat on a log and waited for Chingis.\n\nChingis didn't seem that surprised to see me, either. \"Which way now?\" was all he said.\n\nI was quite buzzed by the way they just expected me to be there, in a forest. No one else has ever expected me to just be there in a forest for them. I liked being the Good Guide so I said, \"This way,\" like I knew what I was doing, and carried on following the numbered posts. Although it looked like we were at the end of the world, I knew there was a train every fifteen minutes that would take us back to Bootle in twenty-three minutes.\n\nThe whole thing reminded me of when a dream gets weird and you're sort of scared but you also somehow know that you could wake yourself up and you'd be in your own bed, so you carry on dreaming just a bit longer. And that's what we did. We carried on walking.\n\nWhen we spotted a pile of logs and twigs up ahead, I pointed at them like I'd arranged for them to be there and said, \"I thought you might want to make an ovoo or something.\"\n\n\"Yeah. Good idea.\"\n\nSo the three of us piled the wood up into a pyramid and Nergui went off and found a long branch to stick in the top like a flagpole. We tied my school jumper to it for a flag \u2013 even though it was getting cold. Then Chingis opened his bag and pulled out a horse skull.\n\n\"Where did you get that?\"\n\n\"From my grandad's horse.\"\n\n\"Right. Well, obviously _that_ would be in your schoolbag.\"\n\nHe put the skull on top of the ovoo.\n\nI said, \"Are we going to walk round it now three times in a clockwise direction?\"\n\n\"Yes,\" said Chingis. \"Which way is clockwise?\"\n\nI showed them, but Nergui was unconvinced. \"Anyway,\" he said, \"it all depends on the clock, doesn't it?\"\n\nI pointed out that all clocks go the same way.\n\n\"Of course they do,\" said Chingis. \"You are so stupid.\"\n\nSo we agreed that all clocks went the same way but none of us could agree which way that was. Even I wasn't sure \u2013 we've only got a digital clock on our computer. So we went round what I thought was clockwise and then we went the other way \u2013 just in case. And then Nergui started to worry that by going one way and then the other, we were undoing what we'd only just done.\n\nI said, \"We could make a fire but it's illegal.\"\n\n\"I don't want to break the law,\" said Nergui.\n\n\"We could dance. Dancing's not illegal.\"\n\nThey both stared at me.\n\n\"You're supposed to light a fire and then dance around it.\"\n\nThey both stared at me even more.\n\nThen Chingis said, \"Let's move on, then, lads.\"\n\nSo we moved on. And I explained to them that you could build a mental ovoo in your head if you wanted, and put all your good memories on it and a mental flag on top.\n\nThe trees grew thinner and then we came to a field. It might have been a cornfield \u2013 some kind of long grass, anyway. I said we definitely weren't supposed to walk through that. You could see that no one else ever had. But Chingis was dead keen. \"It will fully baffle that demon,\" he said. \"You walk next to me and Nergui will walk behind me so it'll look like the tracks of two people, not three. It will sack him off completely.\"\n\nWe trudged through the waist-high grass. Frightened birds flew up around us, whistling and beeping like little fax machines. And the corn rustled like wrapping paper. Chingis pulled the Polaroid camera out of his bag and took a picture of the tracks.\n\nI said, \"Chingis, where did you get that camera?\"\n\n\"Refugee Project Summer Holiday Party in St Anne's, Overbury Street. Won it on the tombola.\"\n\n\"They had a bouncy castle too,\" added Nergui.\n\nSo that was this summer here in Liverpool? So you didn't have that camera when you were in Mongolia? So none of your photos is actually of Mongolia? Are you even from Mongolia? But I didn't say any of that.\n\nChingis shook the Polaroid dry and showed it to me. The funny thing was, it looked like Mongolia, as though he could turn bits of Liverpool into bits of Mongolia just by pointing his camera at them.\n\nWe carried on making our way through the field until we came out the other side. Now it was just sand in front of us, all the way to the sky.\n\n\"The desert,\" said Nergui.\n\n\"We are back in the desert, where we belong,\" said Chingis.\n\nI said, \"I think this is the beach, to be honest.\"\n\n\"If it's the beach, where's the sea?\"\n\n\"Over there, behind the dunes.\"\n\n\"Honestly, this is the desert. Welcome to our desert!\" And he took another picture \u2013 and he turned the beach into the desert with his camera. He gave me this photo and the one of the cornfield.\n\nI made the boys slog up the dunes. The wind was throwing sand in our eyes and that really sharp grass was cutting my legs. I didn't care \u2013 I just wanted to show them that the sea was there and that they were wrong.\n\nBut when we got to the top of the dunes, there was no sea. Nothing. Just miles and miles of sand and mud shining in the sun.\n\n\"See!\" said Chingis. \"The desert.\"\n\nI said, \"The tide is out.\"\n\n\"No tide goes that far out,\" said Chingis.\n\nWe scrambled down the other side of the dunes. They started to walk straight out towards the horizon but the wind was blasting in now, and bit by bit we ended up following the line of the dunes. I pointed out that since the sand was wet and muddy and there were shells and seaweed and even starfish, this was clearly the sea.\n\n\"So where's it gone, then?\"\n\n\"Maybe it's vanished. Maybe your demon made it vanish. That's what it does, isn't it? Make things vanish.\"\n\n\"Will you stop talking about it? Don't you know it can hear you when you talk about it? If it does get us, it'll be your fault.\"\n\n\"If it does get you, I'll be fully surprised. Don't you know it's not real? And people don't just vanish.\"\n\n\"A lot of people just vanish. Practically everyone we know vanished. That's why we had to leave home \u2013 because people kept vanishing.\"\n\nIt was windy on the beach and I wished my jumper wasn't being used as a prayer flag. There was no one around and nothing seemed to be moving. I said, \"Maybe we've already vanished. Maybe this is where you come to when you vanish.\"\n\n\"You'll get used to vanishing,\" said Chingis, who seemed to think he owned the whole beach.\n\nI was worried that the tide would come back in without us noticing and sweep us all out to sea. Also, the wind was cold, even though Chingis said it wasn't cold and went on about how in Mongolia you knew when it was cold because there was frost and snow on the hump of your camel.\n\nI led them back into the dunes, away from the wind and the possibility of sudden drowning. They never asked me where I was going or why. I was the guide and they were following.\n\n\"The less we know,\" said Chingis, \"the less the demon can find out from us.\"\n\nThere was a rough path made from logs laid out on the sand, with gorse and nettles growing up in between the wood and on either side of the path. And poking out of the gorse there were one or two of the numbered wooden posts. Without saying anything, I followed them to the top of a high dune, where we stopped and looked down, and for a minute we didn't say anything, but they each grabbed one of my hands and squeezed it...\n\nDown beneath us, sheltered from the wind, was a cluster of plump Mongolian yurts.\n\n\"How did you do this?\" asked Nergui. \"Are we home? Is this Mongolia?\"\n\nIt definitely looked like Mongolia. I had no idea how I'd done it but didn't want to admit it. I just said, \"Let's see.\"\n\nThe yurts were all empty and there was no sign of life, except that the tents themselves seemed to be breathing as the wind moved in and out of them.\n\n\"Why is there no one here?\" said Nergui.\n\n\"Is it all right to go in one?\" said Chingis.\n\nI had no idea whether it was all right or not, though I told him of course it was.\n\nSo we did. Inside, the air was warm and still, like the air of a different country. There was some kind of bamboo matting on the floor and a huge pile of cushions in one corner. We spread them in a circle on the floor and I noticed a gas heater. It had an ignition button, so I turned it on and we clustered round it. The air started to smell of slightly toasted sand.\n\nNergui said, \"Why have the people who lived here vanished?\"\n\n\"I don't think they've vanished. I think they don't come here till the holidays.\"\n\nChingis had found some kind of storm lantern hanging from the central post. He lit it using the heater and took it outside.\n\n\"Where is he going?\" I asked.\n\n\"Ssssh, watch.\"\n\nChingis set up the lantern on top of a log so that it threw a pool of light onto the side of the tent. Then he stood between the lantern and the tent and made shadows with his hands. Sitting inside looking at the shadows he was casting on the canvas we saw amazing shapes and stories. Chingis reached out his arm so that it looked like a horse's neck stretched out across the canvas, whinnying and neighing. He made a demon shape out of some cardboard he had in his bag.\n\nNergui screamed and told me to go out and make him stop.\n\nI said, \"Don't be thick \u2013 the pretend demon is putting fear in the real demon's face.\"\n\n\"Oh yeah. I never thought of that.\"\n\nWhen Chingis made a girl shape with some grass for hair and did kissing noises, Nergui laughed till I thought he'd hurt himself.\n\nThen Chingis came back inside and we all sat staring at the lantern. Someone's stomach started to complain. I was thinking, These two are nomads. Are they going to use their nomad skills to get food and water for us? I don't think so. In fact we wouldn't even be warm if the stove didn't have an ignition button. In my bag I still had my lunch. I spread out the food \u2013 a packet of Dairylea Lunchables, some Quavers, a Capri-Sun and a ham sandwich. The two of them just stared at it like these were the riches of Kublai Khan. Then I said, \"Go ahead,\" and they dived on it. There was a bit of a discussion about how to share the Lunchables but we worked it out in the end and then just lay there, not chewing, just letting the cheese dissolve in our mouths. We took turns with the Capri-Sun. First we sucked the juice out and then we sucked the air. The noise it made as the plastic collapsed was hilarious for a while. Then it was really quiet. Except for when Nergui said, \"What's that?!\" every time a twig snapped or a pine cone fell outside.\n\nI thought, These two don't know anything about being on your own and soon it'll be dark. Somehow I've come out here with two nomads and they've put me in charge!\n\nI said, \"Back in Mongolia they tell stories round the fire.\"\n\n\"Go on, then.\"\n\n\"OK. I'll tell you a ghost story. Unless you'd be scared.\"\n\n\"Scared?!\" Chingis laughed. \"We wouldn't be scared, would we, Nergui?\"\n\n\"Not ever,\" said Nergui, looking around. \"Where do you think everyone has gone?\"\n\n\"I don't know,\" I said, and started the story. \"There was this old man, in Italy I think it was...\" I was telling them the only story I knew. \"He was an undertaker \u2013 do you know what that is? He buried people. When they were dead. And he was also very greedy. He loved jewellery the most. Anyway, one day, someone brings him the body of an old lady to bury\u2014\"\n\n\"Exactly how scary is this story?\" asked Nergui.\n\n\"Completely scary. Want me to stop?\"\n\n\"No.\"\n\n\"So someone brings him the body of this old lady and she has a fabulous ring on one hand\u2014\"\n\n\"How fabulous?\"\n\n\"Rubies.\"\n\n\"They're unlucky.\"\n\n\"Diamonds.\"\n\n\"OK.\"\n\n\"So he decides it'd be a waste to bury the ring. He's going to try and steal it. Well, he tries to pull it off. Won't come. He rubs her hands with soap. Nothing. Still can't get the ring off. But the more he can't get it off, the more he wants it. So what does he do in the end?\"\n\n\"Cuts off her hand.\" Chingis smiled. \"That's what I'd do.\"\n\n\"And that's what he does \u2013 cuts off her hand. Then he folds her arms so no one will see. Then he sells the ring and buys himself a nice new car. Of course it's very hard to get rid of a hand without someone finding it and making a fuss, so he hides it in the glove compartment of his car.\n\n\"Anyway, one night, he's out driving in that nice new car and he comes to a crossroads. And he sees someone at the crossroads, standing there in the rain. So\u2014\"\n\n\"Oh, he doesn't stop, does he? Never stop at a crossroads,\" said Nergui.\n\n\"Well, _he_ doesn't know that. And it's raining really hard and the person standing there is old. An old woman. So he feels sorry for her and stops. 'Do hop in,' he says. 'I'll take you where you want to go.'\n\n\"This old woman starts to climb in but it is quite a high-up car and not that easy to get into so the man very kindly says, 'Here, give me your hand.'\n\n\"And the old woman says\" \u2013 I did this next bit in a spooky voice \u2013 \"'You already have my hand...'\"\n\nAs soon as I said that, Nergui screamed and ran out towards the woods.\n\nWe ran after him. But outside the tent, everything had changed. The sun had dropped by now. Our shadows stretched out in front of us, like wriggling flags, and the whole landscape seemed to be glowing. Chingis just went charging off, yelling, crunching twigs and scaring birds. I went after him, grabbed him and put my hand over his mouth to shut him up.\n\n\"Listen. We can't see him. So we've got to listen.\"\n\nWe held our breath. We listened. There was a scuffling somewhere off to the left. I picked up the storm lantern Chingis had used for the shadow show earlier and lifted it over my head as we walked slowly towards the sound.\n\nEyes. Bright-green luminous eyes staring into the light. They seemed to be floating in the air.\n\n\"The demon!\" gasped Chingis.\n\n\"A fox,\" I said. I could see it trotting off into the long grass, its head and tail both pointing down.\n\nA cry came from somewhere in the long, golden grass. Something was in there, making the grass wave, sending ripples of gold across the field.\n\nI shouted, \"Nergui, stay still. We're coming.\"\n\nWe trudged through the grass but we couldn't see him.\n\n\"He's vanished,\" said Chingis.\n\n\"Of course he hasn't vanished. Nergui! Shout to us again!\"\n\nHe shouted but the voice seemed to come from nowhere. Or from everywhere.\n\n\"He's vanished. He's just a voice in the air.\"\n\nThen a thought struck me. \"Nergui, are you crouching down?\"\n\n\"Yeah.\"\n\n\"Can you just stand up?\"\n\nAnd there he was, right in front of us.\n\n\"Idiot!\" said Chingis.\n\nI said, \"Right, follow me.\" I could see the yellow of the gorse and it led me to the little white numbered posts. I counted them down \u2013 13-12-11-10 \u2013 leading us back towards the road. Chingis and Nergui never asked where we were going. They just followed. Followed me all the way to the station and onto the train. We got off at Bootle New Strand. And they followed me, still not asking questions, all the way to Roberts Tower, clockwise round the pile of rubble, in at the main door and up in the lift. They just padded along behind me.\n\nBut then when I knocked on their door, Chingis exploded. \"What?!\" he shouted, as if he'd just woken up. \"What're you doing? What've you brought us here for?\"\n\n\"It's where you live. I can't look after you.\"\n\n\"You've cheated. You've cheated us. She's done us. Nergui, quick...\"\n\nNergui spat at me. He was crying and shivering. Chingis was pounding on the lift button. You could barely hear the lift clunking up the shaft over Chingis's shouting. Then the door of the flat opened and he went quiet. His mother walked onto the landing. She didn't even look at me. She picked Nergui up and carried him inside.\n\nThe lift arrived. Its doors creaked open. Chingis stood there staring into it as if he might still step inside. His mother stood in the doorway watching him. Two doors. Two places to go. The lift doors began to close. Chingis turned around and walked through the lighted door, into the flat.\n\nWhen I stepped towards the lift, the doors started to re-open. I looked back at the flat to see if the boys would say anything. Ask me in maybe. But nothing. Their mother closed the door and I stepped into the lift. The doors closed me in.\n\nWhen I got home, I told my mum I'd been to Mimi's.\nOn the way to school the next day I talked to Mimi about it. \"They were idiots. They thought they could run away but Chingis didn't even bring his coat. Hunting with eagles? They were scared of a fox. I saved their lives. They'd've died of hunger without me.\"\n\n\"What were they running away for, anyway?\"\n\nI didn't want to tell her about the demon because it was so stupid, so I just said, \"I don't know.\"\n\n\"Maybe their mum is a bit of a witch. She never comes to get them, does she?\"\n\n\"No. But honestly, they were on the beach and they didn't even know it was the beach. If the tide had come in, they would have drowned. I saved their lives.\"\n\n\"I don't know why you bothered. I don't know why you hang around with them, anyway.\"\n\n\"Well, I won't now, anyway.\"\n\nWhich I didn't.\n\nBecause in the first lesson Mrs Spendlove came in and stood in front of the class without saying anything for ages. She did this so long that in the end, everyone went quiet.\n\n\"I've got some very sad news,\" she said into the unexpected quietness. \"I got a phone call this morning, in the very early hours of the morning. It was from Chingis.\"\n\nEveryone sat up. _Who phones a teacher?!_\n\n\"It's a complicated situation and I don't know all the details but basically, because of things to do with the law, Chingis's family were not supposed to be in this country. They didn't have the right papers, and though they'd been trying to get them, they ran out of time. I'm afraid the police came very early this morning to take them away and send them back to their own country. That's why Chingis rang me. He wanted to say goodbye to you all.\"\n\nAnd that was that. We never saw him again.\n\nI thought about things, talked about things. Mostly with Shocky, funnily enough. He was the only one who seemed to grasp how bad this was.\n\nI think Chingis knew that something was going to happen. Of course he did. Their bags were packed and in the hall. His mother was terrified when there was an unexpected knock on the door. He knew it was going to happen. They were going to come.\n\nI think he had some idea that if he ran away with Nergui then maybe when the police \u2013 or whoever \u2013 came, they wouldn't make their parents leave because the kids were missing. I think he thought that if he could hide out for a while, maybe it would be OK.\n\nBut I found him. The Good Guide. I took him home. And that's where they got him. I'd led him straight back to them.\n\nThe day Mrs Spendlove made the announcement, I waited in the cloakroom after school, wondering what to do. I watched all the coats go one by one, until only one was left.\n\n\"Miss!\" I shouted as she came out of the classroom. \"Look, Chingis's coat. It's still here.\" How could his coat be here and not him? \"They'll have to let him come back for his coat, Miss, won't they, Miss?\"\n\n\"I'm not sure, Julie. No, I don't think so.\"\n\n\"But temperatures reach as low as minus fifty on the steppe in winter. There's frost on the humps of the camels, Miss. How can they send someone there without a coat?\"\n\n\"Maybe someone will give him another one.\"\n\n\"It's a special coat, though, Miss \u2013 for extreme conditions. It's a traditional Mongolian coat.\"\n\n\"I'll take it. How about that? When we find out where they are, I'll post it to him.\"\n\n\"That'd be best, Miss. Can you do that?\"\n\n\"I'll do it first thing.\"\n\nBut she left the coat hanging there when she went. And it was still there the next morning. And here it is now, all these years later, in the lost property box. It was never returned.\n\nI can see now that it wasn't anything like a traditional Mongolian coat. It's some kind of big, ancient hippy coat. An Afghan coat. There's a label inside that says BIBA \u2013 LONDON. They probably got it from a charity shop or the box of donations at the refugee centre.\n\nAnd in the pocket are the photographs. Photographs of a Mongolia cobbled together from bits of Bootle. Chingis's Mongolia was one big mental ovoo.\n\n\"Did you ever hear from him again, Miss?\"\n\n\"Call me Claire, now that you're a grown-up, Julie.\"\n\n\"Yes, Miss.\"\n\n\"No. I don't know how bad things were for them. It really was true that they'd walked out of Mongolia following the railway line. I don't suppose they did that because they were having fun.\"\n\n\"You don't happen to have any pictures of him, do you? All these photos, he took them himself so he's not in any of them.\"\n\n\"No. I'm sorry.\"\n\nThat night, I pick up my little one from the childminder, take her home, feed her, and while she is playing with her Playmobil, I root out the Polaroids he'd given me that day on the beach and put them in Chingis's notebook with the others. In bed later, I flick through the notebook with her and make up a bit of a story to go with the photos, pretending it's a picture book. I examine every photo in case he appears in one of them. But no. How could he? He took them all himself.\n\nThen, wedged in the middle, I find one Polaroid that's completely black...\n\nThe black is the cover you pull off a Polaroid when it's developed. So this one has never been exposed. Maybe he took a picture of himself. I dig into the black with my fingernails and the cover starts to come up. I can peel it off. So I do...\n\nPolaroids don't work like that, though. If you leave them too long, the light turns back into darkness. Polaroids are like people.\n\nIt's only when I close the notebook and see his name written in full on the front page that I realize I could just google him.\n\nIt turns out that Tuul is the third most common surname in Mongolia and every single boy is called Chingis. That's if his name really was Chingis. There were pages and pages of Chingis Tuuls. I'd also forgotten that I didn't read Mongolian. So there were pages and pages of strange-looking letters with his name highlighted every now and then. Maybe he's the president now or the winner of _Mongolia's Got Talent_ or something.\n\nOf course, Julie O'Connor's not an unusual name, either. If he was googling me from Mongolia, what would he find? Pages and pages about a girl who runs a kayak school in California. A woman who makes coats to order in Cleveland. An aromatherapist in Newcastle. Even \"Julie O'Connor + Liverpool\" gives you a dieting blog, a barrister, two grief counsellors and street dance lessons. Even if you just search Facebook there are more than two dozen of us, with only our profile pictures to identify us. And I'm sure none of us look like we did when we were eleven.\n\nAnd anyway, why would he be looking for me? He probably still thinks it was my fault he got deported.\n\nI scan the Polaroids and add them to my Facebook page. I even change my profile picture to that photo of the coat hanging in the cloakroom.\n\nTwo days later, he tries to add me as a friend. Maybe it was a coincidence. Maybe he's been checking all the Julie O'Connors in the world every day for years. I don't know. But he's tried to add me. So I accept the add, say thanks and ask if he wants the coat back.\n\nHe puts \"yes please\" and messages me his address. And he tags me in this photograph...\n\n# Afterword\n\nA few years back, just after my book _Millions_ came out, a teacher asked me to come and visit her primary school. The teacher's name was Sue Kendall and the school was Joan of Arc Primary in Bootle. It was my very first \"author visit\" \u2013 the first time I'd ever heard myself described as an author, the first time I'd ever strolled into an unfamiliar building and had a bunch of strangers sit in a circle and listen while I told a story. Amazing. I felt like Homer.\n\nI've done it hundreds of times since, but it still feels full of ancient magic to me. And I remember pretty much every detail of that first time. There was a big boy there called Christian, who was one of the inspirations for the overgrown Liam in my book _Cosmic_. Mrs Kendall and Joan of Arc Primary are always popping up in my stories. But the thing I remember most is meeting a girl called Misheel. She was a refugee from Mongolia and she just lit up the room. The other children were touchingly proud of her and told me about the time Misheel turned up to the school disco in full Mongolian costume with her elaborate headdress and fabulous robes. They knew all about Mongolia \u2013 its customs and epic landscape \u2013 because of her. Her presence massively enriched their lives. Everyone must have felt the same way because she was chosen to lead the Lord Mayor's procession that year. There's a line in this book about Xanadu being hidden in the heart of Bootle, and that's what she seemed to be \u2013 a wonder of the world living among them.\n\nThen one day the Immigration Authorities came and snatched her and her family in the middle of the night. Misheel managed to get one phone call through to Sue Kendall before one of the officers grabbed her phone. And, of course, she has not been seen since. I don't know much about immigration policy or the politics of our relationship with Mongolia. Maybe there is some complicated reason why a depopulated and culturally deprived area like Bootle shouldn't be allowed generous and brilliant visitors. I do know that a country that authorizes its functionaries to snatch children from their beds in the middle of the night can't really be called civilized.\n\nThe Joan of Arc children were upset, of course, and one of the things that most worried them was that Misheel had left her coat behind. They knew that it was cold in Mongolia and worried about how she would manage without it. That image of the left-behind coat really haunted me. I talked to my friends Carl Hunter and Clare Heney at the time, and we planned to make a documentary in which we took some of the kids to Mongolia to look for Misheel and give her back her coat. But it never happened.\n\nThen last year Jane Davis asked me to write a book for Our Read. I went to meet her with a pocketful of stories \u2013 stories about the Gold Rush, about the future, about ghosts \u2013 all of them thought through and ready to go \u2013 but somehow I found myself talking about Misheel instead. I said, \"But actually that's not a story, that's just something that happened.\" But we both knew that this was the one I wanted to write really. And that Carl and Clare should be working on it with me.\n\nI changed Misheel into a boy for this book. Because this isn't Misheel's story. It's a made-up story. I didn't want to tell Misheel's story because I didn't want that story to be over. Strangely, as I was putting the finishing touches to this book, I bumped into Sue Kendall and she told me that Misheel had \u2013 for the first time ever \u2013 rung her that morning from Mongolia. I wanted to know everything that had happened to her. But that really is a different story...\n\n# \nCarl, Clare and Frank\n\n**Frank Cottrell Boyce** won the 2004 Carnegie Medal for his first children's book, _Millions_. He has since written three novels for children, including _Framed_ , which was shortlisted for the 2005 Carnegie Medal and the Whitbread Children's Book of the Year Award, and _Cosmic_ , which was shortlisted for the 2009 Carnegie Medal, the Guardian Children's Fiction Prize and the Roald Dahl Funny Prize. Frank is an accomplished, successful and well-known screenwriter. He lives with his family in Liverpool.\n\n**Carl Hunter** is a film-maker and also plays bass guitar for Liverpool-based band The Farm. **Clare Heney** is a film\u2011maker and photographer. She and Carl both lecture in media at Edge Hill University.\n\nFrank, Carl and Clare have collaborated on a number of projects, including the film _Accelerate_.\n\nFrank Cottrell Boyce wrote _The Unforgotten Coat_ to support The Reader Organisation, helping them in their aim to bring about a Reading Revolution.\n\nThe Reader Organisation is a pioneering charity, working across the UK and beyond, making it possible for people of all ages, backgrounds and abilities to enjoy and engage with reading. The work they do is driven by a love for great literature and a strong belief that shared reading is a deeply powerful activity that can significantly enrich and improve lives and the communities we live in.\n\nThrough their hundreds of \"read aloud\" Get into Reading groups, Read to Lead training, Our Read campaign, Community Theatre and Reader-in-Residence projects in NHS trusts, care homes, prisons, schools and libraries, The Reader Organisation is transforming society's collective approach to reading \u2013 making literature accessible, available, emotionally rewarding and fun.\n\nThis book is part of the Reading Revolution! Please go out there and share it with someone you know.\n\nVisit www.thereader.org.uk to learn more.\nThis is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or, if real, used fictitiously.\n\nThis Walker edition first published 2011 by Walker Books Ltd \n87 Vauxhall Walk, London SE11 5HJ\n\nText \u00a9 2011 Frank Cottrell Boyce \nPhotographs \u00a9 2011 Carl Hunter and Clare Heney \nPhotograph on page 106 by Chay Heney \nCover photograph \u00a9 2011 Eva Garcia Iba\u00f1ez de Opacua\n\nThe right of Frank Cottrell Boyce, and Carl Hunter and Clare Heney to be identified as author and photographers respectively of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988\n\nAll rights reserved. No part of this book may be reproduced, transmitted or stored in an information retrieval system in any form or by any means, graphic, electronic or mechanical, including photocopying, taping and recording, without prior written permission from the publisher.\n\nBritish Library Cataloguing in Publication Data: a catalogue record for this book is available from the British Library\n\nISBN 978-14063-3561-3 (ePub)\n\nwww.walker.co.uk\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n\n\nProduced by Chris Curnow, Christian Boissonnas and the\nOnline Distributed Proofreading Team at http:\/\/www.pgdp.net\n(This file was produced from images generously made\navailable by The Internet Archive)\n\n\n\n\n\n\n\n\n\n TEN THOUSAND\n WONDERFUL THINGS\n\n COMPRISING\n\n WHATEVER IS MARVELLOUS AND RARE, CURIOUS\n ECCENTRIC AND EXTRAORDINARY\n\n IN ALL AGES AND NATIONS\n\n ENRICHED WITH\n\n _HUNDREDS OF AUTHENTIC ILLUSTRATIONS_\n\n EDITED BY\n\n EDMUND FILLINGHAM KING, M.A.\n\n LONDON\n GEORGE ROUTLEDGE AND SONS, LIMITED\n\n BROADWAY, LUDGATE HILL\n MANCHESTER AND NEW YORK\n\n 1894\n\n\n\n\nSTANDARD WORKS OF REFERENCE.\n\n_UNIFORM WITH THIS VOLUME._\n\n\nLEMPRIERE'S CLASSICAL DICTIONARY.\n\nWALKER'S RHYMING DICTIONARY.\n\nMACKAY'S THOUSAND AND ONE GEMS OF ENGLISH POETRY.\n\nD'ISRAELI'S CURIOSITIES OF LITERATURE.\n\nBARTLETT'S FAMILIAR QUOTATIONS.\n\nCRUDEN'S CONCORDANCE TO THE BIBLE.\n\nTHE FAMILY DOCTOR.\n\n\n\n\nPREFACE.\n\n\nA BOOK OF WONDERS requires but a brief introduction. Our title-page\ntells its own tale and forms the best exposition of the contents of the\nvolume.\n\nEverything that is marvellous carries with it much that is instructive,\nand, in this sense, \"Ten Thousand Wonderful Things,\" may be made useful\nfor the highest educational purposes. Events which happen in the\nregular course have no claim to a place in any work that professes to\nbe a register of what is uncommon; and were we to select such Wonders\nonly as are capable of familiar demonstration, we should destroy their\nright to be deemed wondrous, and, at the same time, defeat the very\nobject which we profess to have in view. A marvel once explained away\nceases to be a marvel. For this reason, while rejecting everything that\nis obviously fictitious and untrue, we have not hesitated to insert\nmany incidents which appear at first sight to be wholly incredible.\n\nIn the present work, interesting Scenes from Nature, Curiosities\nof Art, Costume and Customs of a bygone period rather predominate;\nbut we have devoted many of its pages to descriptions of remarkable\nOccurrences, beautiful Landscapes, stupendous Water-falls, and sublime\nSea-pieces. It is true that some of our illustrations may not be\nbeautiful according to the sense in which the word is generally used;\nbut they are all the more curious and characteristic, as well as\ntruthful, on that account; for whatever is lost of beauty, is gained by\naccuracy. What is odd or quaint, strange or startling, rarely possesses\nmuch claim to the picturesque and refined. Scrape the rust off an\nantique coin, and, while you make it look more shining, you invariably\nrender it worthless in the eyes of a collector. To polish up a fact\nwhich derives its value either from the strangeness of its nature, or\nfrom the quaintness of its narration, is like the obliterating process\nof scrubbing up a painting by one of the old masters. It looks all the\ncleaner for the operation, but, the chances are, it is spoilt as a work\nof art.\n\nWe trust it is needless to say that we have closed our pages against\neverything that can be considered objectionable in its tendency; and,\nwhile every statement in this volume has been culled with conscientious\ncare from authentic, although not generally accessible, sources, we\nhave scrupulously rejected every line that could give offence, and\nendeavoured, in accordance with what we profess in our title-page, to\namuse by the eccentric, to startle by the unexpected, and to astonish\nby the marvellous.\n\n\n\n\nINDEX TO ENGRAVINGS\n\n\n PAGE\n\n ABYSSINIAN ARMS, 509\n\n ---- LADIES, 492\n\n ---- ORNAMENTS OF, 493\n\n ---- LADY TATTOOED, 496\n\n ALTAR-PIECE OF SAN MINIATO, 601\n\n AMULET WORN BY EGYPTIAN FEMALES, 452\n\n AMULET BROTCHE, 332\n\n ANCIENT METHOD OF KEEPING A WASHING ACCOUNT, 3\n\n ---- NUT-CRACKERS, 236\n\n ---- SNUFF-BOXES, 210\n\n ANGLO-SAXONS, SEPULCHRAL BARROW OF THE, 27\n\n APTERYX, THE, OR WINGLESS BIRD, 308\n\n ARCH, A BEAUTIFUL, IN CANNISTOWN CHURCH, 433\n\n ---- OF TRAJAN AT BENEVENTUM, 445\n\n ARCHITECTURE FOR EARTHQUAKES, 324\n\n ARMLET, AN ANCIENT, 425\n\n ARMOUR, ANCIENT, CURIOUS PIECE OF, 341\n\n ASH, THE SHREW, 397\n\n AZTEC CHILDREN, THE, 37\n\n\n BAGPIPES, 505\n\n BANDOLIERS, 560\n\n BANNERS AND STANDARDS, ANCIENT, 584, 585\n\n BASTILLE, STORMING OF THE, 195\n\n BEAU BRUMMELL (A), OF THE SEVENTEENTH CENTURY, 61\n\n BECTIVE ABBEY, 392\n\n BEDESMEN IN THE TIME OF HENRY VII, 593\n\n BELLOWS, A PRIMITIVE PAIR OF, 637\n\n BELL SHRINE, AN ANCIENT, 348\n\n ---- OF SAINT MURA, 412\n\n BIBLE USED BY CHARLES I. ON THE SCAFFOLD, 271\n\n BILLY IN THE SALT BOX, 181\n\n BLACKFRIARS, PARIS GARDEN AT, 465\n\n BLIND GRANNY, 70\n\n ---- JACK, 23\n\n BOAT, A BURMESE, 668\n\n BOOK-SHAPED WATCH, 328\n\n BRACELET, A MAGICIAN'S, 345\n\n BRAMA, THE HINDOO DEITY, 556\n\n BRANK, THE, 2\n\n BRASS MEDAL OF OUR SAVIOUR, 241\n\n BRITANNIA TUBULAR BRIDGE, 173\n\n BROOCH, ANCIENT SCANDINAVIAN, 401\n\n BRICKS OF BABYLON, 613\n\n BRIDGE OVER THE THAMES, THE FIRST, 428\n\n ---- A CHINESE, 440\n\n ---- CROMWELL'S, AT GLENGARIFF, 648\n\n BUCKINGER, MATTHEW, 53\n\n BUCKLER OF THE FIFTEENTH CENTURY, WITH PISTOL INSERTED, 30\n\n BUNYAN'S (JOHN) TOMB, 157\n\n BURMESE PRIEST PREACHING, 266\n\n BUST, AN ANCIENT ETRURIAN, 677\n\n\n CAMDEN CUP, THE, 250\n\n CANDLESTICK, A REMARKABLE, IN FAYENCE, 592\n\n ---- OF MARY, QUEEN OF SCOTS, 436\n\n CARFAX CONDUIT, 333\n\n CARRIAGE, A TURKISH, 656\n\n CASCADE DES PELERINES, 135\n\n CATACOMBS AT ROME, 87\n\n CAVE, PORT , 516\n\n ---- THE TIGER, AT CUTTACK, 361\n\n CHAIR BROUGHT OVER TO AMERICA BY THE PILGRIM FATHERS, 186\n\n ---- DAGOBERT'S, ANCIENT, 421\n\n ---- HENRY VIII.'S, 488\n\n ---- THE DUCHESS OF LAUDERDALE'S, 401\n\n CHAPTER-HOUSE, A, IN THE TIME OF HENRY VII., 600\n\n CHARLEMAGNE, CROWN OF, 377\n\n CHIEFTAIN, ANCIENT SCOTTISH, 500\n\n CHINESE METHOD OF FISHING, 316\n\n ---- PUNISHMENT OF THE KANG, OR WOODEN COLLAR, 134\n\n CHRISTMAS, PROCLAIMING THE NON-OBSERVANCE OF, 19\n\n CISTERN OF MAJOLICA WARE, 597\n\n COFFEE POT, IN STONEWARE, A CURIOUS, 649\n\n COIN, THE FIRST, WITH BRITANNIA ON IT, 468\n\n COLLARS, ANCIENT STONE, 665\n\n COLUMN AT CUSSI, 533\n\n COMB, A CURIOUS INDIAN, 657\n\n CORAL REEFS, 74\n\n CORPSE BEARER DURING THE PLAGUE, 284\n\n COSTUMES, ANCIENT, 18, 71, 78, 86, 212, 213, 220, 296, 297\n\n ---- GERMAN, OF THE 16TH CENTURY, 548\n\n COSTUME, FOREIGN, IN 1492, 543\n\n ---- OF A GERMAN NOBLE, 536\n\n COUTEAU-DE-CHASSE, 633\n\n CRADLE OF MOSS, 325\n\n ---- HENRY V., 416\n\n CROSBY, SIR JOHN, HELMET OF, 520\n\n CROSS OF CONG, 457\n\n ---- MUIREDACH, 369\n\n CUCKING STOOL, 1\n\n CUPID OF THE HINDOOS, THE, 552\n\n CURFEW BELL, THE, 33\n\n CURIOUS FIGURES ON A SMALL SHRINE, 203\n\n\n DAGGER OF RAOUL DE COURCY, 263\n\n ---- AN ANCIENT, 673\n\n DAGOBERT, ANCIENT CHAIR OF, 421\n\n DANCING NATIVES OF NEW SOUTH WALES, 225\n\n DARNEY (JENNY), A HARMLESS ECCENTRIC OF THE YEAR 1790, 187\n\n DERVISHES DANCING, 669\n\n DIAL AND FOUNTAIN IN LEADENHALL STREET, 553\n\n DINNER PARTY IN THE 17TH CENTURY, 609\n\n ---- TABLE, AN EGYPTIAN, 537\n\n DIOGENES IN A PITHOS--NOT TUB, 524\n\n DOG-WHEEL, THE OLD, 101\n\n DRINKING CUP, A CURIOUSLY SHAPED, 413\n\n ---- EARLY GERMAN, 460\n\n ---- VESSEL, A DECORATIVE, 336\n\n ---- GLASS, ANCIENT, 153\n\n DROPPING WELL OF KNARESBOROUGH, 143\n\n DRUID'S SEAT, THE, 464\n\n DUCK-BILLED PLATYPUS, OR ORNITHORYNCHUS PARADOXUS, 273\n\n DYAK WITH HEADS, SKULL HOUSE, AND HOUSE OF SEA DYAKS, 276, 277\n\n ---- WAR BOAT IN BORNEO, 540\n\n DYAKS OF BORNEO, WAR DANCE OF THE, 541\n\n\n EAST INDIA HOUSE, THE FIRST, 206\n\n EDDYSTONE LIGHTHOUSE, 109\n\n EGYPTIAN TOYS IN THE BRITISH MUSEUM, 130\n\n EMBROIDERED GLOVE, PRESENTED BY MARY, QUEEN OF SCOTLAND,\n TO AN ATTENDANT ON THE MORNING OF HER EXECUTION, 263\n\n EXTRAORDINARY CATARACT, 224\n\n ---- SITUATION FOR A TREE, 313\n\n ---- TREE, 183\n\n\n FASHIONABLE DISFIGUREMENT OF THE TIME OF CHARLES I., 213\n\n FAWKES HALL, OLD MANOR HOUSE OF, 380\n\n FETE OF THE FEDERATION OF THE NATIONAL GUARDS OF\n FRANCE, 1790, 289\n\n FIGG (JAMES), THE CHAMPION PRIZE-FIGHTER OF 1733, 113\n\n FISH, SHOOTING, 432\n\n FISHERMAN, BULGARIAN, 497\n\n FLOATING CITY OF BANKOK, 309\n\n FONT AT KILCARN, THE, 417\n\n FRENCH ASSIGNATS, FAC-SIMILE OF THE FORMS IN WHICH THEY\n WERE ISSUED TO THE PUBLIC, 254\n\n FULLERTON'S (COLONEL) DEVICE FOR PASSING A MOUNTAIN TORRENT, 194\n\n FUNEREAL JAR, 481\n\n\n GARDEN, EGYPTIAN, 349\n\n GARRICK'S CUP, 232\n\n GATE, THE, ON OLD LONDON BRIDGE, 561\n\n GAUNTLET OF HENRY, PRINCE OF WALES, 661\n\n GIANT TREE, 229\n\n GLAIVE, A, 504\n\n GRACE KNIVES, 641\n\n GRAVES OF THE STONE PERIOD, 364\n\n GREAT WALL OF CHINA, 233\n\n GREY MAN'S PATH, THE, 528\n\n GUN, A CELEBRATED, 568\n\n GUY, THOMAS, PORTRAIT OF, 605\n\n\n HACKNEY COACHMAN OF THE TIME OF CHARLES II., 258\n\n HACKNEY COACH, THE EARLIEST, 211\n\n HEAD-BREAKER, A, 665\n\n ---- ORNAMENT, ANTIQUE, 393\n\n HEART OF LORD EDWARD BRUCE AND CASE, 246, 247\n\n HELMET, AN EARLY ENGLISH, 632\n\n HELMET OF SIR JOHN CROSBY, 520\n\n HENRY V., CRADLE OF, 516\n\n ---- VII., BEDESMEN IN THE TIME OF, 393\n\n ---- VIII., CHAIR OF, 488\n\n ---- I. (KING) DREAM OF, 26\n\n ---- VIII.'S WALKING STICK, 30\n\n HINDOO ADORATION OF THE SALAGRAM, 588\n\n HOLY-WATER SPRINKLER, 532\n\n HOOPS, LADIES', IN 1740, 6\n\n HUDSON, JEFFERY, THE DWARF OF THE COURT OF CHARLES I., 472\n\n\n IMPLEMENTS USED IN BUDDHIST TEMPLES, 621\n\n INCENSE CHARIOT, AN ANCIENT, 513\n\n INSTRUMENTS OF TORTURE:--THE EXECUTIONER'S AXE; THE BLOCK\n ON WHICH LORDS BALMERINO AND LOVAT WERE BEHEADED; THE\n SCAVENGER'S DAUGHTER; SPANISH BILBOES; MASSIVE IRON\n COLLAR FOR THE NECK; THUMB SCREW. BRAND FOR MARKING\n FELONS: IMPRESSION OF BRAND; PUNISHMENT FOR DRUNKARDS,\n FORMERLY IN USE AT NEWCASTLE-ON-TYNE; THE WHIRLIGIG,\n A MILITARY METHOD OF PUNISHMENT; PILLORY, STOCKS, AND\n WHIPPING-POST FORMERLY ON LONDON BRIDGE, 60, 90\n\n IRRIGATION, TURKISH MACHINE FOR, 681\n\n\n JAMES I., CURIOUS JEWEL WHICH BELONGED TO, 456\n\n ---- II., AND THE CHURCH OF DONORE, 557\n\n JEWEL PRESENTED BY MARY, QUEEN OF SCOTS, TO EARL HUNTLEY, 243\n\n JOHNSON'S (DR. SAMUEL), RESIDENCE IN INNER TEMPLE LANE, 48\n\n ---- OLD STAIRCASE IN, 49\n\n JOY (WILLIAM), THE ENGLISH SAMPSON, 177\n\n\n KING'S STONE, THE, 461\n\n KNIGHT'S COSTUME OF THE 13TH CENTURY, 480\n\n\n LAMPS, ANCIENT ROMAN, 437\n\n LOCOMOTIVE, THE FIRST, 96\n\n ---- THE PRESENT, AND TRAIN, 97\n\n LORD OF MISRULE, 15\n\n LOUIS XII., IVORY SCEPTRE OF, 476\n\n LOUIS XVI., EXECUTION OF, 255\n\n LUTHER'S (MARTIN) TANKARD, 150\n\n LYNCH'S CASTLE, GALWAY, 581\n\n\n MAGICIAN'S MIRROR, 344\n\n ---- BRACELET, 345\n\n MAY-POLES, 101\n\n MAIL, ANCIENT SUIT OF, 484\n\n MARY, QUEEN OF SCOTS', CANDLESTICK, 436\n\n MEDMENHAM ABBEY, 429\n\n MILITARY HATS IN THE OLDEN TIME, 75\n\n MILL AT LISSOY, 469\n\n MIRROR, A MAGICIAN'S, 344\n\n MONSOONS, 180\n\n MONSTROUS HEAD-DRESS OF 1782, 242\n\n MONUMENTS, WAYSIDE, 588\n\n ---- ROCK CUT, OF ASIA MINOR, 444\n\n MORAYSHIRE FLOODS, 126\n\n MOSQUE OF OMAR, 317\n\n ---- ST. SOPHIA, 104\n\n MUMMERS, OR ANCIENT WAITS, 14\n\n MUMMY CASES, 409\n\n MUSICAL INSTRUMENT, HINDOO, 684\n\n ---- ---- A CURIOUS BURMESE, 628, 629\n\n ---- ---- EGYPTIAN, 405\n\n\n NAORA, THE, 636\n\n NEBUCHADNEZZAR, MASK OF, 105\n\n NECKLACE, ANCIENT JET, 529\n\n NELL GWYNNE'S LOOKING-GLASS, 237\n\n NEWTON CHURCH, DOORWAY OF, 473\n\n NEWTON'S (SIR ISAAC) OBSERVATORY, 10\n\n ---- HOUSE, ST. MARTIN'S STREET, 11\n\n NORMAN CAPS, 44\n\n NORTH AMERICAN INDIAN WAR DESPATCH, 45\n\n\n OLD LONDON BRIDGE, GATE ON THE, 561\n\n ---- ---- SIGNS, 120\n\n ORNAMENTS ABYSSINIAN FEMALE, 493\n\n ---- ANTIQUE HEAD, 393\n\n ---- FEMALE, OF THE IRON PERIOD, 400\n\n ---- EGYPTIAN FEMALE, 448\n\n ---- PERSONAL, OF EGYPTIANS, 453\n\n ---- OF FEMALE DRESS IN THE TIMES OF THE ANCIENT BRITONS, 79\n\n\n PAGODA, THE GREAT SHOEMADOO, 572\n\n PAILOOS, CHINESE, 625\n\n PAPYRUS ROLL, FROM A SPECIMEN IN THE BRITISH MUSEUM, 82\n\n ---- SYRIAN, WITH AND WITHOUT FLOWERS, 83\n\n PARIS GARDEN AT BLACKFRIARS, 465\n\n PASS OF KEIM-AN-EIGH, 329\n\n PENN'S (WILLIAM) SILVER TEA SERVICE, 202\n\n PENNSYLVANIA JOURNAL, FAC-SIMILE OF THE HEADING OF\n THE LAST NUMBER, 1765, 63\n\n PEST HOUSE DURING THE PLAGUE IN TOTHILL FIELDS, 573\n\n PETER THE GREAT, HOUSE OF, AT ZAANDAM, 545\n\n PLOUGHING, ANCIENT MODE OF, 66\n\n POISON CUP, THE, 485\n\n PONT DU GARD, THE GREAT AQUEDUCT OF, 312\n\n POPE'S CHAIR, 577\n\n POPULAR AMUSEMENTS IN 1743, 56\n\n PORCELAIN FIGURES, 517\n\n POTTERY IN CHINA, THE ART OF, 321\n\n POWERSCOURT FALL, PHENOMENON AT, 305\n\n PREACHING FRIAR, 221\n\n PRE-ADAMITE BONE CAVERNS, 199\n\n PRIESTS OF SIKKIM, 664\n\n PRINCE RUPERT, HEAD QUARTERS OF, DURING THE SIEGE\n OF LIVERPOOL, IN 1644, 292\n\n PULPIT OF JOHN KNOX AT ST. ANDREW'S, 270\n\n PUNISHMENT, ANCIENT INSTRUMENT OF, 680\n\n PYRAMIDS OF EGYPT, 131\n\n\n QUEEN ELIZABETH'S STATE COACH, 198\n\n ---- ---- SIDE SADDLE OF, 340\n\n\n RAFFAELLE, TOMB OF, 569\n\n RALEIGH'S (SIR WALTER) ANCIENT RESIDENCE AT BLACKWALL, 161\n\n REVOLVER, A, OF THE FIFTEENTH CENTURY, 30\n\n RING, FORMERLY THE PROPERTY OF CHARLES I., 263\n\n RINGS, CALCINATED, 408\n\n ---- SARDONYX, WITH CAMEO HEAD OF QUEEN ELIZABETH, 372\n\n ---- A TOAD STONE, 424\n\n ROCK OF CASHEL, THE, 352\n\n RUINS OF CLONMACNOIS, 612\n\n\n SACK-POT, OLD ENGLISH, 521\n\n SAINT GEORGE, TOMB OF, 281\n\n SAINT GEORGE'S HALL, GIBRALTAR, 7\n\n SALAGRAM, HINDOO ADORATION OF THE, 589\n\n SARDONYX RING, WITH CAMEO HEAD OF QUEEN ELIZABETH, 373\n\n SCEPTRE, IVORY, OF LOUIS XII., 476\n\n SCHOOL, A CHINESE, 525\n\n SCRIPTURAL ANTIQUITIES:--DRUM, OR TIMBREL; DRUM IN USE\n IN THE EAST; HARP; LUTES; INSCRIBED STONE; SANDALS;\n DISTAFF; ROMAN FARTHING; STONEMONEY-WRIGHTS; HAND\n MILL; EASTERN WINE AND WATER BOTTLES, 217\n\n SELKIRK (ALEXANDER) AND THE DANCING GOATS, 22\n\n SEPULCHRAL VASE, 320, 608\n\n SHAKESPEARE'S JUG, 576\n\n SHIELD, ANCIENT DANISH, 420\n\n SHRINE OF ST. SEBALD AT NUREMBERG, 604\n\n SILVER LOCKET IN MEMORY OF THE EXECUTION OF CHARLES I., 263\n\n SNAKE CHARMER, 300\n\n SOUTH STACK LIGHTHOUSE, 240\n\n SPANISH DAGGER OF THE SIXTEENTH CENTURY, 263\n\n SPIDER, THE TRAP-DOOR, 384\n\n ---- NEST OF THE, 385\n\n ST. WINIFRED'S WELL, 304\n\n STAMP, MEDICINE, ANTIQUE ROMAN, 449\n\n STANDARDS, EGYPTIAN, 396\n\n ---- ASSYRIAN, 584, 585\n\n STEAM BOAT, FAC-SIMILE OF THE FIRST, 301\n\n STICKS, OLD WALKING, 388\n\n SWORD BREAKER, ANCIENT, 672\n\n ---- AN EXECUTIONER'S, 676\n\n ---- CURIOUS ANTIQUE, 596\n\n ---- THE HAWTHORNDEN, 353\n\n ---- THE SETON, 357\n\n SUMMERS' MAGNET, OR LOADSTONE, 41\n\n\n TEMPLAR'S BANNER, CALLED BEAUSEANT, 565\n\n TEMPLE AT SIMONBONG, INTERIOR VIEW OF, 620\n\n THRASHING CORN, ANCIENT METHOD OF, 67\n\n TILBURY FORT, WATER-GATE OF, 190\n\n TOILET BOXES, EGYPTIAN, 381\n\n TOMB, ANCIENT GREEK, INTERIOR VIEW OF, 617\n\n ---- A CHINESE, 508\n\n ---- OF RAFFAELLE, 569\n\n TOMB OF CAECILIA METELLA, 477\n\n TOPE, THE SANCHI, 389\n\n TORTURE CHAMBER AT NUREMBERG, 616\n\n TOWER OF THE THUNDERING WINDS, 93\n\n TRAJAN, ARCH OF, AT BENEVENTUM, 445\n\n TREATY STONE AT LIMERICK, 564\n\n TRIPOD, AN ANCIENT, 549\n\n TUMBREL, THE, 2\n\n TUNISIAN TURNER, A, 652\n\n TYRIAN PURPLE, THE SHELL FISH FROM WHICH IT IS OBTAINED, 644\n\n\n UMBRELLA, ANGLO-SAXON, 624\n\n\n VASES, ANCIENT, 337\n\n ---- GREEK, 501\n\n ---- ROMAN, IN BLACK WARE, 372\n\n ---- A SEPULCHRAL, OF ANCIENT EGYPT, 608\n\n VASES TEUTONIC, HUT-SHAPED, 580\n\n VAUXHALL, 380\n\n VESSEL, A CURIOUSLY SHAPED, 376\n\n VESUVIUS, CRATER OF, IN 1829, 165\n\n VISHNU, THE GOD, 645\n\n VOLCANO OF JORULLO, MEXICO, 161\n\n\n WAR CHARIOT OF EGYPT, 365\n\n WATCH, ANTIQUE, 368\n\n ---- PRESENTED BY LOUIS XII. OF FRANCE TO CHARLES I.\n OF ENGLAND, 640\n\n ---- PRESENTED BY MARY OF SCOTLAND TO MARY SKATON, 285\n\n WATER CARRIER OF THE OLDEN TIME, 259\n\n WEAPON, AN ANCIENT, 660\n\n ---- A POISON, 672\n\n WEAVER BIRD, SOCIAL NEST OF, 441\n\n WIGS OF VARIOUS PERIODS, 31\n\n\n\n\nINDEX.\n\n\n PAGE\n\n Abbey Buildings, The Arrangement of, 658\n\n Abraham and Sarah, 101\n\n Abyssinian Ladies, Dress of the, 491\n\n Abyssinian Lady, Tattooed, 495\n\n Advertisement, an American, 111\n\n Advertisements, Curious, 406, 447, 455, 478\n\n ---- in the last Century, 207\n\n ---- of a Dying-speech Book, 116\n\n ---- New Style of, 249\n\n ---- a Pudding as an, 228\n\n ---- of a Fleet Parson, 116\n\n A False Find, 31\n\n A Female Sampson, 62\n\n A Fine Old Soldier, 314\n\n A Floating City, 308\n\n A Funeral appropriately conducted, 235\n\n Aged Persons, instances of many Dying, 283\n\n Ages of Celebrated Men, 102\n\n A Great Marvel seen in Scotland, 138\n\n A Happy Family, 28\n\n A Harmless Eccentric, 186\n\n Albertus Magnus, Receipts from, 91\n\n Ale Too Strong, 267\n\n Alexandria, Pharos at, 274\n\n Algerine Invasion of Ireland, 176\n\n A Last Chance, 103\n\n All Humbugs, 85\n\n A Lucky Find, 6\n\n A Man in a Vault Eleven Days, 69\n\n ---- Carries his House on his Head, 290\n\n ---- Selling his own Body, 95\n\n ---- aged One Hundred Years, 256\n\n A Monster, 287\n\n Ambassador, French, Entry into London, 262\n\n ---- why Held by the Arms, 162\n\n Amphitheatres, 102\n\n Amulets worn by Egyptian Females, 120\n\n ---- Brotche, 332\n\n Amusements in the 15th Century, 254\n\n ---- in 1743, Popular, 56\n\n An apparent Singularity accounted for, 93\n\n An Eccentric Tourist, 139\n\n Ancients, Credulity of the, 144\n\n Anglo-Saxons, Sepulchral Barrow of, 26\n\n Animals, Food of, 24\n\n ---- Communication between, 294\n\n Animation, Suspended, 374\n\n Anne Boleyn, Execution of, 375\n\n Antimony, 570\n\n Antipathies, 391\n\n ---- Unaccountable, 196\n\n Antiquities, Egyptian, 642\n\n Apollo, Oracles of, in France, 675\n\n Arabian Horses, 291\n\n Arabs, Horses of the, 498\n\n Archbishop, an, Washing Feet, 5\n\n Arch, A Beautiful, 433\n\n A remarkable Old Man, 214\n\n Armlet, Ancient, 425\n\n Armour, Ancient, Curious Piece of, 341\n\n Arms, Abyssinian, 509\n\n Artists, Duration of Life amongst, 196\n\n A Sea above the Sky, 81\n\n Ash, the Shrew, 397\n\n Ass, The, 116\n\n Assiduity and Perseverance, 304\n\n Attar of Roses, Origin of, 343\n\n Attar of Roses, 298\n\n A Woman takes the Lighted Match, 40\n\n ---- Defends a Post singly, 52\n\n Authors, some Learned, Amusements of, 137\n\n A Unique Library, 211\n\n Aztec Children, 37\n\n\n Babes of Bethlehem, The, 660\n\n Bagpipes, Irish, 505\n\n Ballot, Origin of the, 673\n\n Bandoliers, 560\n\n Bank, A Mattrass for a, 323\n\n Banner, The Templars', called Beauseant, 564\n\n Banquets of the Ancients, 439\n\n Bara, a Machine used in Sicily, 415\n\n Barbers, 94\n\n Barometer, Incident connected with, 136\n\n Bartholomew Fair in 1700, Handbill of, 148\n\n Bastille of Paris, Storming of the, 194\n\n Bazaar, a Turkish, 614\n\n Bear, a Shaved, 17\n\n Beard, Care of the, 503\n\n Beau Brummell (a) of the 17th Century, 61\n\n Bective Abbey, 392\n\n Bedesmen in the time of Henry VII., 593\n\n Bedford Missal, The, 407\n\n Bee, The Queen, 25\n\n Bees, Obedient to Training, 95\n\n Beggars, Severe Enactment against, 302\n\n ---- selected as Models by Painters, 281\n\n Bell, The Great, of Burmah, 559\n\n ---- of Rouen, 650\n\n Bells, 193\n\n ---- of the Ancients, 279\n\n ---- of St. Mura, 411\n\n Bell-Shrine, an Ancient, 347\n\n Bellows, Primitive Pair of, 637\n\n Bible, 118, 372, 490\n\n ---- Bunyan's, 121\n\n ---- Summary of the, 169\n\n ---- used by Charles I. on the Scaffold, 271\n\n Billy in the Salt-box, 181\n\n Birds, The Ear of, not to be Deceived, 228\n\n Blind Jack, 23\n\n ---- Granny, 70\n\n ---- Workman, 155\n\n Boat, Burmese, 667\n\n Bobart, Jacob, 22\n\n Boiling to Death, 663\n\n Bolton Abbey, Origin of, 273\n\n Bombardier Beetle, The, 68\n\n Bones, Adaptation of to Age, 52\n\n Book-shaped Watch, 328\n\n Boots an object of Honour, 232\n\n Boydell, Alderman, 9\n\n Brama, the Hindoo Deity, 555\n\n Bramins, Philosophy of the, 371\n\n Brank, The, 2\n\n Brass Medal, of our Saviour, 241\n\n Breakfasting Hut in 1745, 158\n\n Bribery, 141\n\n Bricks of Babylon, The, 612\n\n Bridge, Old London, The Gate of, 561\n\n ---- Chinese, 439\n\n ---- Suspension, at Freybourg, 166\n\n Britannia Tubular Bridge, 172\n\n British Islands, Size of the, 245\n\n Brooch, Ancient Scandinavian, 401\n\n Bruce, Lord Edward, Case containing the Heart of, 215\n\n Brunswick, House of, Anecdote of the, 459\n\n Buckinger, Matthew, 53\n\n Buddist Temples, Instruments used in, 621\n\n Bumper, 153\n\n Bunyan's, John, Tomb, 156\n\n Burial Places of Distinguished Men, 390\n\n Burmah, Elephant God of, 537\n\n Bust, Etrurian, An Ancient, 677\n\n Byng, Admiral, Execution of, 182\n\n\n Cader Idris, 118\n\n Cagots, The, 638\n\n Calculation, Interesting, 474\n\n Cambridge Clods, 20\n\n Camden Cup, 250\n\n Camel, as a Scape-Goat, 522\n\n Cameleon, The Eye of the, 479\n\n Candles in the Church, 449\n\n Cannon, Ancient, raised from the Sea, 40\n\n ---- at the Siege of Constantinople, 69\n\n ---- First Iron, 320\n\n Canute, The Discovery of the Body of, 176\n\n Cardinals, Colour of the Hat for, 234\n\n Cards, Games with, in the 16th Century, 618\n\n Carfax Conduit, 333\n\n Carronades, 149\n\n Carrara, Francis, Cruelty of, 504\n\n Carriage, Turkish, 655\n\n Cascade des Pelerines, 135\n\n Cat, Instinct in a, 353\n\n Catacombs at Rome, 87\n\n Cataract, Extraordinary, 223\n\n Cat-Clock, A, 631\n\n Cats, White, 51\n\n ---- with Knotted Tails, 238\n\n Caves, The Hawthornden, 382\n\n Chaffinch Contest, 651\n\n Chalice, Iona, The Golden, 422\n\n Changes of Fortune, 371\n\n Chaplain, Instructions to a, 458\n\n Chapter-House in Henry VIIth's time, 599\n\n Charing Cross, Autobiography of, 128\n\n Charity instead of Pomp, 407\n\n ---- Rewarded by a Mendicant, 257\n\n Charlemagne, Clock presented to, 145\n\n Charles I., Anecdote relative to, 174\n\n ---- II., Privy Purse, Expenses of, 234\n\n Cherry Tree, 458\n\n Chess, in India, How it Originated, 305\n\n Chieftain, Ancient Scottish, 500\n\n Chilcott, the Giant, 71\n\n Child, Test of Courage in a, 132\n\n Children of Aged Parents, 319\n\n China, Origin of the Great Wall of, 233\n\n Chinese Dainties, 91\n\n ---- Ivory Balls, 144\n\n ---- Method of Fishing, 315\n\n ---- Punishment of the Kang, 134\n\n ---- Ladies, Small Feet of, 475\n\n ---- Mirrors, 425\n\n ---- School, 525\n\n ---- Therapeutics, 369\n\n Chocolate, Early use of, 52\n\n Christmas Customs, Bygone, 14, 19\n\n Christening, Novel Mode of Celebrating a, 393\n\n Chronology of Remarkable Events, 218\n\n Church of Donore, James II. and the, 557\n\n Cigars, Extraordinary Fashion in, 274\n\n Circumstance, a Curious, 430\n\n ---- Extraordinary, 15\n\n Cistern of Majolica Ware, 597\n\n Clock at Hernhuth, Watchmen Imitating, 20\n\n ---- Wonderful, 167\n\n Clocks, Early, 171\n\n Clonmacnois, Ruins of, 289\n\n Coachmen of the Time of Charles II., 257\n\n Cock Fighting at Schools, 219\n\n Coffee, 153\n\n Coffee and Tea, 122\n\n Coffee-house in London, the First, 4\n\n ---- Attractions in 1760, 41\n\n Coin, The First, with Britannia on it, 468\n\n Coinage, Variations in the, 650\n\n Coincidences, some Curious, 434\n\n Collars, Stone, Ancient, 665\n\n Column at Cussi, 533\n\n Comb, Curious Indian, 657\n\n Conecte, Thomas, 433\n\n Confectionary Art in 1660, 373\n\n Conjuring, Public Taste for in 1718, 122\n\n Conway Church, Inscription in, 112\n\n Coral Reefs, 73\n\n Coronations, Prices for Seats at, 160\n\n ---- Expenses at, 283\n\n Corpulent Man, 78\n\n Corpulence, Cure for, 80\n\n Cost of Articles in the 14th Century, 330\n\n Costume, Ancient Female, 71, 78\n\n Costumes, 395, 437, 536, 544, 547, 630, 651\n\n Couteau-de-Chasse, Ancient, 633\n\n Cranmer's (Archbishop) Dietary, 137\n\n Credulity, Extraordinary Instance of, 311\n\n Cricket-Matches, Extraordinary, 408\n\n Criminal, a Rich and Cruel, 450\n\n Criminals, Old Custom Relating to, 598\n\n Cromwell's Bridge at Glengariff, 648\n\n Cross of Cong, The, 457\n\n ---- ---- Muiredach, 369\n\n ---- Ordeal of the, 463\n\n Crown of Charlemagne, 377\n\n Cucking-Stool, The, 1\n\n Cupid, The, of the Hindoos, 230\n\n Curious Feats, 181, 239\n\n ---- Law, 8\n\n ---- Manuscript, 214\n\n Curiously-shaped Vessel, 376\n\n Curiously-shaped Drinking Cups, 413\n\n Curiosities, Strange, 457\n\n Custom, Means of attracting, 683\n\n Customs, Singular Local, 653\n\n\n Daffeys' Elixir, 173\n\n Dagger, An Ancient, 673\n\n Dagobert, Ancient Chair of, 421\n\n Dance, Curious Provincial in France, 679\n\n Dances, Fashionable of the last Century, 220\n\n Dancing Rooms, 57\n\n Dead, Fashions for the, 523\n\n Dead Bodies, Preservation of, 251, 280, 638\n\n Death, Boiling to, 663\n\n ---- Lunar Influence in, 346\n\n ---- Pressing to, 515\n\n Decorative Drinking Vessel, 336\n\n Della Robbia Ware, 601\n\n Demons, Bribing the, 531\n\n Dervishes, Dancing, 669\n\n Desolation, Scene of, 329\n\n Destitute Cats, Asylum for, 280\n\n Dial and Fountain in Leadenhall Street, 553\n\n Dilemma, 499\n\n Dinner, an Egyptian, 537\n\n ---- in China, 596\n\n ---- Party in the 17th Century, 609\n\n Diogenes in a Pithos, not Tub, 101\n\n Disorders Cured by Fright, 307\n\n Dispute and appropriate Decision, 140\n\n Dog (A) Extinguishing a Fire, 20\n\n ---- Combination of Instinct and Force, 284\n\n ---- A Sensible, Refusing to Bait a Cat, 76\n\n ---- Persevering, 80\n\n ---- Friendship, 84\n\n ---- A Piscatorial, 367\n\n ---- Sensible, 376\n\n ---- in Japan, 622\n\n ---- Figures of on Ancient Tombs, 682\n\n Dog-wheel, The Old, 101\n\n Dole in consequence of a Dream, 503\n\n Doles, 399\n\n Down among the Dead Men, 185\n\n Dress, Forty years ago, 212\n\n Dress in London, 18, 114, 253, 295\n\n ---- Fastidiousness at an Old Age, 243\n\n ---- of the Ancient Britons, 79\n\n Drinking Bouts in Persia, 547\n\n Drinks, Intoxicating, Antiquity of, 611\n\n Dropping Wells, 142\n\n Druids' Seat, 464\n\n Drunkenness, the Offspring of, 666\n\n Duns in the Mahratta Country, 379\n\n Dyaks of Borneo, 275\n\n\n Ears, Character Indicated by, 65\n\n Earthenware, English, 575\n\n Earthquake Panic, 520\n\n ---- Swallowed up by an, 329\n\n ---- at Lisbon, 200\n\n ---- Nottingham, in 1816, 280\n\n Earthquakes, 398, 432\n\n East India House, the First, 206\n\n Eating for a Wager, 4\n\n Eccentric Englishman, An, 438\n\n Eccentrics, a Couple of, 318\n\n Echo, Extraordinary, 341\n\n Eddystone Lighthouse, 108\n\n Edicts against Fiddlers, 328\n\n Eel, Large, 10\n\n Egypt, 491\n\n ---- Pyramids of, 130\n\n Egyptian Toys in the British Museum, 129\n\n Elephant Detects a Robber, An, 99\n\n Elephants Frightened at Pigs, 9\n\n Energy, A Triumph of, 193\n\n England before the Romans, 86\n\n Englishman, A Fat, 28\n\n Epitaph, an Inculpatory, 268\n\n Etna, Mount, Great Eruption of, 451\n\n ---- Changes of, 406\n\n Europa, Ruins of, 567\n\n Exchequer-bills, Origin of, 676\n\n Execution, in 1793, 84\n\n Extraordinary Tree, 183\n\n Extravagance at Elections, 149\n\n ---- Oriental, 499\n\n Eyam, The Desolation of, 226\n\n\n Fallacy of the Virtues of a Seventh Son, 315\n\n False Accusers, Punishing, 230\n\n Farmers, Illustrious, 304\n\n Fashionable Disfigurement, 213\n\n Fayence, The, of Henry II. of France, 591\n\n Feasts, Anglo-Saxon, 517\n\n Federation, Fete of the, 288\n\n Female Intrepidity, Extraordinary, 248\n\n Ferrers, Earl, Execution of, 107\n\n Figg, Champion, 113\n\n Finger Rings, Porcelain, 486\n\n Fire at Burwell, Cambridgeshire, 293\n\n Fire-arms in the Tower of London, 29\n\n Fire-engines, When first made, 223\n\n Fish, Shooting, 432\n\n ---- High Price of, in London, 312\n\n ---- Extraordinary Ponds and, 561\n\n ---- Tame, 659\n\n ---- Wonderful, 542\n\n Fishermen, Bulgarian, 497\n\n Fleet Marriages, about 1740, 299\n\n Floods, the Morayshire, 126\n\n Flying Coach, 228\n\n Fog of 1783, The Great, 414\n\n Font at Kilcarn, The, 417\n\n Food of the Ancients, 450\n\n Foot-Racing in 1699, 457\n\n Foreigners in London in 1567, 371\n\n Fortune, Change of, 371\n\n Fox Killed by a Swan, 4\n\n Francis I., Funeral Oration of, 363\n\n Franklin's Celebrated Letter to Strahan, 39\n\n Frederick the Great at Table, 579\n\n French Dress, 102\n\n ---- Assignats, the Origin, 253\n\n Friars, Preaching, 221\n\n Frost Fairs, 67\n\n ---- Extraordinary, 209\n\n Funeral, an Eccentric, 395\n\n ---- Jar, 481\n\n ---- Obsequies, Strange, 108\n\n\n Game Preserves at Chantilly, 362\n\n Gamblers, Chinese, Playing for Fingers, 593\n\n Gambling, Legalised, 141\n\n ---- Extraordinary, 359\n\n Gaming, a National Taste for, 267\n\n Gander, an Old, 27\n\n Garden, an Egyptian, 349\n\n ---- at Kenilworth, when in its Prime, 641\n\n ---- Love of, 419\n\n ---- Sacred, 420\n\n ---- The Hanging, of Babylon, 558\n\n Garrick's Cup, 232\n\n Gauntlet of Henry, Prince of Wales, 661\n\n George II., Proclamation for, 200\n\n Georgians as Topers, 511\n\n Giant Tree, 229\n\n Gibraltar, Siege of, 6\n\n Gigantic Bones, 248\n\n Glaives, 504\n\n Glove Money, 503\n\n Gloves, Anne Boleyn's, 600\n\n ---- Origin of \"Pin Money\", 275\n\n Grace Knives, 641\n\n Graham Island, 443\n\n Graves of the Stone Period, 363\n\n Greek Vases, 501\n\n Gretna Green Marriages, 159\n\n Grey Man's Path, The, 528\n\n Grinning for a Wager, 13\n\n Groaning Boards, 66\n\n Groat, a Castle for a, 470\n\n Grotto, Remarkable, and Story connected with it, 625\n\n Guillotine, Decapitation by the, 8\n\n Gun, Celebrated, 568\n\n Gunpowder, Making a Candlestick of, 249\n\n\n Hackney Coach, The Earliest, 211\n\n Hair, Ancient, Quantity and Colour of the, 4\n\n ---- Price of Human, 242\n\n ---- Remarkable Preservation of, 122\n\n ---- Transplantation of, 40\n\n ---- Turned Grey by Fright, 327\n\n ---- Two of the Fathers, on False, 24\n\n Hamster Rat, The, 265\n\n Handbills, Distributing, 178\n\n ---- from Peckham Fair, in 1726, 72\n\n Hanging a Mayor, 140\n\n \"Happy Dispatch\" in Japan, The, 578\n\n Head Breaker, A., 338\n\n Head-dress, Monstrous, 242\n\n ---- Ornament, Antique, 393\n\n Hejirs, The, 222\n\n Helmet, Early English, 632\n\n ---- of Sir John Crosby, 520\n\n Henry I., Dream of, 26\n\n ---- II., Stripped when Dead, 39\n\n ---- V., Cradle of, 416\n\n ---- the VIIIth's Chair, 488\n\n ---- VIII., Curious Extracts from the Household Book\n of Lady Mary, Daughter of, 399\n\n Highlander, A Remarkable, 238\n\n Highwaymen in 1782, 5\n\n Hindoo Computation, 507\n\n ---- Rites, Cruelty of, 627\n\n Historical Anecdote, 156\n\n Holy Water Sprinkler, 532\n\n Homer in a Nutshell, 127\n\n Hooking a Boy Instead of a Fish, 319\n\n Hoops, in 1740, 6\n\n Horse, A, Getting himself Shod, 76\n\n Horse-race, Indenture of a, 52\n\n Horses of the Arabs, 498\n\n Horses, Different Sorts of, in the 16th Century, 634\n\n ---- Feeding one another, 368\n\n ---- Vicious, Novel Way of Curing, 174\n\n Hot Cross Buns, 251\n\n House, Novel Way of Designating a, 539\n\n ---- of Hens' Feathers, 646\n\n Household Rules of the 16th Century, 518\n\n How Distant Ages are Connected, 200\n\n Hudson, Jeffery, the Dwarf of the Court of Charles I., 472\n\n \"Humbug,\" Origin of the Term, 97\n\n Hume, David, on his own Death, 215\n\n Hundred Families' Lock, 435\n\n Hunting Party, a Regal, 391\n\n Husband, Novel way of Purchasing a, 275\n\n Hydra, Extraordinary Reproductive Power of the, 490\n\n\n Ice, Ground, 506\n\n Ignorance and Fear, 290\n\n Impostor, An, 50\n\n Impudence or Candour? Which is it? 239\n\n Incense Chariot, An Ancient, 513\n\n Incremation, Instance of, 353\n\n Indian Jugglers, European Balancing, 293\n\n Inhumanity, Extraordinary Instances of, 436\n\n Innkeeper's Bill in 1762, 431\n\n Insects, Wonderful Formation of the Eye in, 467\n\n Insect Life, Minuteness of, 338\n\n Instinct of Animals, 410\n\n Insurance Agent, Canvass of an, 465\n\n Interesting and Fanciful Relique, 243\n\n Inventors, The Perils of, 141\n\n Irrigation, Turkish Machine for, 349\n\n \"It's much the same Now\", 94\n\n\n James II. and the Church of Donore, 557\n\n James II., Spent by the Corporation of Coventry at the\n Entertainment of, in his Progress through Coventry, 378\n\n Javanese, Superstition of the, 244\n\n Jenny's Whim, 174\n\n Jewel, A Curious, which belonged to James I., 456\n\n Jews, Wealth of the, 359\n\n Johnson, Dr., A Visit to the Residence of, 48\n\n Joy, William, the English Sampson, 176\n\n Judas Iscariot, Legends of, 339\n\n Judges attending Public Balls, 303\n\n ---- Salaries, 446\n\n Jugglers in Japan, 529\n\n ---- of Modern Egypt, 342\n\n\n Kildare, Death of the Earl of, 172\n\n Killed by eating Mutton and Pudding, 73\n\n King Edward I., Household Expenses of, 231\n\n ---- Fine for Insulting a, 149\n\n ---- of Kippen, The, 139\n\n ---- John and Pope Innocent, 463\n\n King-Maker, Warwick the, 527\n\n King's Bed, Ceremonial for Making the, 562\n\n ---- Cock Crower, The, 137\n\n ---- Dishes with the Cook's Name, 235\n\n ---- Stone, The, at Kingston, 461\n\n Kitchen, Spacious, 383\n\n Knight's Costume of the 13th Century, 480\n\n Knives and Forks, 133\n\n Knox, John, The Pulpit of, at St. Andrews, 269\n\n\n Lady, Origin of the Word, 147\n\n Lagmi, and the Use made of it, 623\n\n Lambeth Wells, the Apollo Gardens, 272\n\n Lamps, Roman, 437\n\n Land, Change in the Value of, 196\n\n Landslip at Colebroke, Shropshire, 184\n\n Lantern, Curious, 100\n\n Lauderdale, The Duchess of, 403\n\n Law of the Mozcas, 454\n\n Law and Order in the Streets of London, 131\n\n Laws, a Hundred years ago, Severity of, 234\n\n Leadenhall Street, Old Dial and Fountain in, 553\n\n Legend, A Superstitious, 351\n\n Legends among Savage Nations, 146\n\n Length of Life without Bodily Exercise, 274\n\n Lepers, Treatment of, in England, 493\n\n Leprosy, Lazars, and Lazar Houses, 169\n\n Letter, Extraordinary, 322\n\n Lettsom's (Dr.) Reasons, 71\n\n Lewson, The Eccentric Lady, 221\n\n Life, An Eventful, 427\n\n ---- in Death, 443\n\n Lighting the Streets, Bequests for, 310\n\n Lightning, Calmuc's Opinion of, 63\n\n Living, Style of, among the Nobility of the 15th Century, 533\n\n ---- in the 16th Century, 357\n\n Lizards, Swallowing, 41\n\n Loaf Sugar, 166\n\n Locomotives, the First, 96\n\n Locusts, 151\n\n London Localities in the 16th Century, 526\n\n London Water Carrier in Olden Time, 258\n\n ---- in 1756, State of, 147\n\n London Resorts a Hundred Years Ago, 197\n\n Longevity, 269\n\n Long Meg and her Daughters, 394\n\n Lord Mayor's Feast in 1663, 551\n\n Lotteries, 619\n\n Louis XVI., Execution of, 258\n\n Luther's (Martin) Tankard, 149\n\n Luxury in 1562, 418\n\n Lynch's Castle, Galway, 581\n\n\n Mackarel, Price of, 576\n\n Madness, Sudden Recovery from, 168\n\n Madyn, the Capital of Persia, Magnificence of,\n when invaded by the Saracens A.D. 636, 554\n\n Magic Rain Stone, 168\n\n Magician's Mirror and Bracelet, 344\n\n Magnet, The Summers' or Loadstone, 41\n\n Magnificence of Former Times, 111\n\n Magpie Stoning a Toad, 92\n\n Mahomet, Personal Appearance of, 571\n\n Mail, Ancient suit of, 483\n\n Malady, Extraordinary, 670\n\n Mandrin, the Smuggler, 167\n\n Manners, Ancient, of the Italian, 585\n\n Man without Hands, 77\n\n Manufacture, One of the Effects of, 142\n\n Marat, Funeral of, 375\n\n Marriage Custom, Curious, 543\n\n ---- Lottery, 91\n\n ---- Vow, 419\n\n Mary, Queen of Scots, her First Letter to English, 370\n\n Mary Queen of Scots, her Candlestick, 436\n\n Maternal Affection in a Dumb Woman, 140\n\n May-pole in the Strand, 534\n\n ---- Fate of the Last, in the Strand, 682\n\n May-poles, 100\n\n Mecca, The Black Stone at, 550\n\n Medmenham Abbey, 429\n\n Memento-Mori Watch, 285\n\n Mental Affection, A Curious, 335\n\n Merman, A, 16\n\n Mexican Tennis, 375\n\n Michaelmas-day, Origin of eating Goose on, 198\n\n Military Hats in Olden Time, 75\n\n Mill at Lissoy, 469\n\n Miraculous Escape, 266\n\n Misers, Two, 459\n\n Missal, The Bedford, 163\n\n Mob Wisdom, 294\n\n Monasteries, Libraries of destroyed, 334\n\n Monkeys Demanding their Dead, 415\n\n Monkish Prayers, 383\n\n Monks, Gluttony of the, 347\n\n ---- and Friars, 680\n\n Monsey (Dr.) bequeaths his own Body, 93\n\n Monsoons, 179\n\n Monument, Rock-cut, of Asia Minor, 441\n\n Monuments, Wayside, 587\n\n Mosque of Omar, 316\n\n Mother Mapp, the Bone Setter, 158\n\n Mountains, Height of, 148\n\n Mouth, Character of the, 106\n\n M.P.'s and Mayors, Privateers, 176\n\n Mulgrave, Origin of the House of, 602\n\n Mullet and Turbot, with the Romans, 488\n\n Mummy Cases, 409\n\n Murderess, a Young but Cruel, 392\n\n Music, Effect of, on a Pigeon, 64\n\n ---- of the Hindoos, 683\n\n ---- ---- ---- Sea, 351\n\n Musical Instrument, A Curious, 628\n\n Musical Instruments, Burmese, 629\n\n ---- ---- Egyptian, 404\n\n\n Names, Strange Custom about, 295\n\n Naora, The, 635\n\n Narrow Escape, 121\n\n Nature, Wonderful Provision of, 55\n\n Nebuchadnezzar, Gold Mask of, 105\n\n Necklace, Ancient Jet, 529\n\n , Bill of Sale for a, in 1770, 39\n\n Nell Gwynne's Looking-Glass, 237\n\n Never Sleeping in a Bed, 331\n\n Newspapers, Vacillating, 514\n\n New South Wales, Dances of the Natives of, 225\n\n Newton, A Visit to the Observatory of, 10\n\n New Zealand, The Wingless Bird of, 307\n\n Norman Caps, 44\n\n North American Indian War Dispatch, 45\n\n Nose, Effect of a New, 102\n\n Nostrums, 63\n\n Nun, The First English, 330\n\n Nut Crackers, Ancient, 236\n\n\n Oaks, Extraordinary, 310, 426, 466, 455\n\n ---- Remarkable, 405\n\n Old Age, Dying of, at Seventeen Years, 47\n\n Old Books, 360\n\n Old London Signs, 118\n\n Opera, The First, 567\n\n Opium, Best Position for Smoking, 675\n\n Oraefa Mountain, in Ireland, 356\n\n Ornaments, Personal Antique, 293, 400, 447, 452\n\n Orthography in the Sixteenth Century, 17\n\n\n Pagoda, The Great Shoemadoo, 572\n\n Pailoos, Chinese, 625\n\n Panama, Isthmus of, Passage through, 148\n\n Paper, 619\n\n Papyrus, The, 82\n\n Parental Authority, Too Much, 513\n\n Paris Garden at Blackfriars, 465\n\n Parlour Dogs, 320\n\n Passport, A Traveller's, 679\n\n Pastimes, Popular, 514\n\n Pates de Foies Gras, 142\n\n Peacocks, 366\n\n Pear-Tree, Great, 454\n\n Pearls, British, 363\n\n ---- Fondness of the Romans for, 208\n\n Pedestrian Feat, Wonderful, 327\n\n Peg Tankards, 43\n\n Penn, Tea Service which belonged to, 201\n\n Penny Post, Origin of the, 47\n\n Pennsylvania Journal, 63\n\n Perfumes, 253\n\n Persecution, 430\n\n ---- in the Reign of Queen Mary, 587\n\n Perseverance rewarded by Fortune, 287\n\n Persia, Drinking Bouts in, 547\n\n Personal Charms Disclaimed, 118\n\n Peru, Condor in, 170\n\n Peruvian Bark, 51\n\n Pest-house, during the Plague, in Tothill Fields, 573\n\n Pestilence, The Black, 402\n\n Peter the Great at Zaandam, 544\n\n Physic, A Friend to, 267\n\n Physick for the Poor, Choice Receipts for, 117\n\n Pigeon Catching near Naples, 437\n\n Pig, Roast, Advertisement of, in 1726, 46\n\n Pike, An Old, 667\n\n Pilgrim Fathers, Chair belonging to, 186\n\n Pillory for Eating Flesh in Lent, 68\n\n Plague in England, The, 183\n\n ---- Corpse Bearers during the, 283\n\n Plantagenets, Yellow Hair in the Time, 103\n\n Plate, Use of, in the time of Henry VIII., 523\n\n Platypus, the Duck-billed, 273\n\n Playbill, Curious, 227\n\n ---- in the time of William III., 530\n\n Ploughing and Threshing, Ancient, 66\n\n Poets, English, Fates of the Families of, 471\n\n Pogonias Vocal Fish, 478\n\n Poison Cup, The, 485\n\n Poisoning the Monarch, 12\n\n Police, London, Disgraceful State of, 193\n\n Pont du Gard, Great Aqueduct of, 312\n\n Pope's Chair, 577\n\n Porcelain, Anecdote in, 517\n\n Port Cave, 516\n\n Poet Haste One Hundred Years ago, 182\n\n \"Postman,\" The, Paragraph from, in 1697, 219\n\n Pottery in China, Art of, 321\n\n Powerscourt Fall, Phenomenon at the, 304\n\n Prayers, Unusual Locality for Saying, 171\n\n Praying by Machinery, 314\n\n ---- by Wheel and Axle, 539\n\n Pre-Adamite Bone Caverns, 199\n\n Precocious Children, 64\n\n Presence of Mind--Escape from a Tiger, 330\n\n Priests in Burmah, Knavery of the, 266\n\n ---- of Sikkim, 663\n\n Prince of Wales, Origin of the Crest of the, 115\n\n Prince Rupert, at Everton, 291\n\n Prolific Author, 320\n\n Proteus Anguinus, The, 152\n\n Psalm, Value of a Long, 512\n\n Pterodactylus, The, 360\n\n Pulpit, Refreshments for the, 262\n\n Punishing by Wholesale, 680\n\n Punishment, Ancient Instrument of, 680\n\n ---- Russian, 654\n\n ---- and Torture, Ancient Instruments of, 58, 88\n\n Puritan Zeal, 579\n\n Purple, Tyrian, 643\n\n\n Quackery in the Olden Time, 671\n\n Queen Elizabeth, Banquets of, 414\n\n ---- ---- Dresses of, 501\n\n ---- ---- Old Verses on, 204\n\n ---- ---- saddle of, 340\n\n ---- ---- State Coach of, 128\n\n ---- ----'s Laws, 151\n\n\n Raffaelle, Tomb of, 568\n\n Raffle, A, in 1725, 47\n\n Raleigh, Sir Walter, Residence of, 160\n\n Ranelagh, 204\n\n Ranz des Vaches, 173\n\n Rats, Destructive Force of, 463\n\n Ravilliac, Execution of, 132\n\n Receipts, Quaint, 153\n\n Red Sea, Luminous Appearance of the, 454\n\n Regiments, The Modern Names of, 639\n\n Reichstadt, The Duke de, 435\n\n Relics, 393\n\n ---- A Group of, 261\n\n ---- Rescued, 618\n\n Remarkable Events and Inventions, 145\n\n Revenge, New Mode of, 423\n\n Rheumatism, Strange Cure for the, 201\n\n Rhinoceros, First in Europe, 655\n\n Richardson, the Showman, 251\n\n Ringing the Changes, 192\n\n Rings, Calcinated, 408\n\n Rites, Hindoo, Cruelty of, 627\n\n Roads in 1780, 327\n\n Rock of Cashel, 352\n\n Romans in Britain, Dress of Native Females at that Period, 86\n\n Rouen, The Great Bell of, 650\n\n Royal Touch, The, 42\n\n Royal Giants, Specimens of, 121\n\n ---- Prisoner, Expenses of, 260\n\n\n Sack Pot, Old English, 521\n\n Sacro Catino, The, 608\n\n Sadler's Wells, 112\n\n Saint George, Tomb of, 281\n\n Saint Lawrence, 464\n\n Salagram, Hindoo Adoration of the, 589\n\n Sand Columns in Africa, 610\n\n Sandwiches, Origin of the, 563\n\n Sardonyx Ring, with Cameo Head of Queen Elizabeth,\n in the possession of Rev. Lord Thynne, 373\n\n Scape Goat, Camel as a, 190\n\n Sceptre, Ivory, of Louis XII., 476\n\n School, Chinese, 525\n\n School Expenses in the Olden Time, 427\n\n Science and Perseverance, Triumphs of, 123\n\n Scottish Wild Cattle, 278\n\n Scriptural Antiquities, 215\n\n Sea, Phosphorescence of the, 418\n\n Sea Serpent, Immense, 42\n\n Sea-Urchin, Wonderful Construction of, 475\n\n Second Sight, 65\n\n Seeing Two Generations, 211\n\n Self-Nourishment, 315\n\n Selkirk and the Dancing Goats, 22\n\n Sepulchral Vase from Peru, 320\n\n Sermons, Anecdotes in, 147\n\n Serpent, Anecdote of a, 85\n\n Seven, The Number, 354\n\n Sevres Porcelain, Prices of, 487\n\n Sex, Change of, 189\n\n \"Sforza,\" Origin of the Title, 554\n\n Shakspeare's Jug, 575\n\n Sham Prophets, 319\n\n Sharks, The Queen's, 203\n\n Sheba, The Queen of, 518\n\n Sheep Killer, Hunting a, 268\n\n Shell Fish, in 1675, Price of, 178\n\n Shetland, The Noss in, 324\n\n Shield, Ancient Danish, 420\n\n Shilling, Cutting a Wife off with a, 359\n\n Shocking Depravity, 117\n\n Shoes, Long-toed, Origin of, 646\n\n Shrine, Curious Figures on a, 202\n\n Shrine of St. Sebald at Nuremberg, 271\n\n Simoom, The, 662\n\n Skin, Human, a Drum made of, 398\n\n Slave Advertisements, 25\n\n Slave Trade, Iniquities of the, 175\n\n Slaves, Recent Prices of, 435\n\n Sleep, Protracted, 483\n\n ---- State of the Mind during, 350\n\n Sleeper, An Extraordinary, 28\n\n Smoking, Attachment to, 322\n\n Snake Charmers, 299\n\n Snakes, Power of Fascination in, 64\n\n Snow Storm, Memorable, 327\n\n Snuff Boxes, Ancient, 209\n\n Snuff, Time Wasted in taking, 512\n\n Something like a Feast, 129\n\n Somnambulism, 72\n\n Sound, Phenomena of, 367\n\n Southcottian Delusion, A Phase of the, 230\n\n South-stack Lighthouse, 239\n\n Spain, Wealth of, under the Moors, 235\n\n Spider, Bite of the Tarantula, 13\n\n Spiders Fond of Music, 157\n\n Spirit Drinker, An Aged, 228\n\n Spontaneous Combustion, 431\n\n Sports of the Lower Classes, 155\n\n Sportsman, A Royal, 443\n\n Springs, Intermittent, 455\n\n Stage Coach in 1760, 155\n\n Stag-Hunt in the 16th Century, 511\n\n Stags like Cattle, Driving, 208\n\n Stamps, Antique Roman, 448, 643\n\n Standards, Ancient Banner and, 396, 583\n\n State Coach in 1796, 156\n\n Statue, Metal, the Largest in the World, 454\n\n Steam boat, Facsimile of the First, 301\n\n Stevens's Specific, 50\n\n St. George's Cavern, 421\n\n St. James's Square, 123\n\n St. Paul's, Old, 162\n\n St. Paul and the Viper, 125\n\n St. Winifred's Well, 303\n\n Sticks, Old Walking, 387\n\n Stirrups, 571\n\n Stomach Brush, 55\n\n Stoneware, 649\n\n Strasburg, Curious Custom at, 185\n\n Strength, Feats of, in 1789, 9\n\n Street Cries of Modern Egypt, 401\n\n Stuff Ball at Lincoln, Origin of the, 49\n\n Sultan, City of the, 103\n\n Sun and Moon, Worship of the, 81\n\n Superstition in 1856, 538\n\n ---- Curious, 424\n\n ---- Death caused by, 124\n\n ---- in France, 519\n\n ---- Vitality of, 474\n\n Sweating Sickness, 110\n\n Sweets, Artificial, 579\n\n Sword, Curious Antique, 596\n\n ---- Executioner's, 340\n\n ---- The Hawthornden, 353\n\n ---- The Seton, 356\n\n ---- Fish and Whales, 565\n\n Sword-Breaker, An Ancient, 672\n\n\n Taking a Man to Pieces, 79\n\n Tapestry, The Bayeux, 642\n\n Tar and Feather, Notices to, 38\n\n Taxation, Universality of, 318\n\n Tea, 94\n\n Tea-Drinkers, The First, Puzzled, 532\n\n Teapot, The, 482\n\n Temple of Pou-tou, The, 673\n\n ---- at Simonbong, 620\n\n Temples of Brambanam, 442\n\n Terrier, Anecdote of a, 358\n\n Thames, Frost Fair on the, 106\n\n ---- The First Bridge over the, 428\n\n Thanksgiving Day in 1697, 527\n\n Theatre, Roman, at Orange, 366\n\n Theatres in the Time of Shakespeare, 597\n\n The First Hermits--Why so Called, 125\n\n The Ruling Passion, 32, 188\n\n Theodora de Verdion, 207\n\n Thief Caught in his own Trap, The, 77\n\n ---- Singular Discovery of a, 115\n\n Thugs, The, 574\n\n Tiger Cave at Cuttack, 361\n\n Tilbury Fort, 189\n\n Time, Division of, in Persia, 633\n\n Tobacco, Origin of the Use of, 57\n\n Toilet, Absurdities of the, 536\n\n ---- Boxes, Egyptian, 381\n\n Tomb, Chinese, 508\n\n ---- of Caecilia Metella, 477\n\n ---- of Darius, 560\n\n Tomb of the Emperor Maximilian at Inspruck, 590\n\n \"Too Late,\" quoth Boice, 489\n\n Tope, the Sanchi, 389\n\n Topers, Georgians as, 511\n\n Toping in the Last Century, 314\n\n Torture, 639\n\n ---- Chamber at Nuremberg, 615\n\n Tower of the Thundering Winds, 93\n\n Trajan, Arch of, at Beneventum, 112\n\n Trance, A, 354\n\n ---- at Will, 462\n\n Trap-door Spider, 383\n\n Travelling, Common, 220\n\n ---- in Olden Times, 108,162\n\n ---- in the United States, 208\n\n Treaty-Stone at Limerick, 563\n\n Tree, Extraordinary Situation for a, 313\n\n Trees, Age of, 521\n\n ---- that Grow Shirts, 62\n\n Tripod, Ancient, 549\n\n Trivial Circumstances, A Great Result from, 605\n\n Tumbrel, The, 2\n\n Tunisians, Ingenuity of the, 652\n\n Turban, The, in Arabia, 618\n\n Turkish Mode of Reparation, 326\n\n Twin-Worm, Extraordinary Formation of the, 136\n\n Types, the Invention of, 152\n\n\n Umbrella, Anglo-Saxon, 624\n\n Upas Tree, 123\n\n Useful and the Beautiful, 647\n\n\n Vampire, The Blood-sucking, 417\n\n Varnish-Tree of the Japanese, 615\n\n Vases, Ancient, 337\n\n ---- Greek, 169\n\n ---- Greek, Prices of, 385\n\n ---- Roman, in Black Ware, 373\n\n ---- Sepulchral, of Greek Pottery, 616\n\n ---- Sepulchral, of Ancient Egypt, 607\n\n ---- Teutonic, Hut-shaped, 580\n\n Vauxhall, 380\n\n Venetians, The, 428\n\n Vengeance, Novel Mode of taking, 586\n\n Ventriloquist, a Female, 62\n\n Vesuvius, Crater of, in 1829, 165\n\n Vinegar on the Skin, Effect of, 115\n\n Vishnu, Incarnations of, 645\n\n Volcanic Eruption in Japan, 601\n\n Volcano of Jurullo, Formation of the, 163\n\n Volition, Suspended, 199\n\n Voltaire, English Letter of, 422\n\n Vow, Singular Hindoo, 658\n\n\n Wagers, Curious, 373\n\n Walking-Sticks, Old, 387\n\n Wall, Governor, Execution of, 154\n\n Wallace, the Hero of Scotland, 99\n\n War Boat, A Dyak, in Borneo, 540\n\n ---- Dance of the Dyaks of Borneo, 540\n\n ---- Chariot of Ancient Egypt, 365\n\n Warwick, the King-Maker, 527\n\n Washing Account, Method of Keeping, 3\n\n Washington, 583\n\n Watch, An Antique, 368\n\n ---- presented by Louis XIII. to Charles I. of England, 640\n\n Watches, the First in England, 515\n\n Water for Old London, Supply of, 282\n\n ---- Preservative Power of Coal-pit, 25\n\n ---- Supply of, for London, in Olden Times, 546\n\n ---- Snakes, Battle of, 470\n\n \"We hae been\", 47\n\n Weapon, Ancient, 660\n\n ---- A Poison, 672\n\n Weaver-Bird, The Sociable, 440\n\n Wedding, A, A Hundred Years Ago, 640\n\n Weight, Reducing, 85\n\n Whipping Prisoners, 175\n\n Whitehall, Ceiling of, 121\n\n Whitsuntide, at Durham Cathedral, 8\n\n Why a Man Measures more in the Morning than in the Evening, 75\n\n Wife, Diving for a, 479\n\n Wigs, 17, 31\n\n Will, Eccentric, 209\n\n William the Conqueror, Courtship of, 555\n\n Willow, Weeping, Introduction of the, 148\n\n Wind Mills, The First, 577\n\n Witch-Testing, at Newcastle, in 1649, 21\n\n Wolves in England, 441\n\n Woman, The Hairy, of Burmah, 677\n\n Woman's Cleverness, 260\n\n Women of England, The, 159\n\n ---- in Former Times, 127\n\n Wonderful Escape, 215, 300\n\n Wren's (Sir Christopher) Cost of Churches, 171\n\n ---- ---- ---- Report, 183\n\n Writing Materials, 481\n\n Writings, Terra Cotta, 466\n\n\n Yorkshire Tike, The, 24\n\n Yorkshire in the Last Century, 283\n\n\n\n\nTEN THOUSAND WONDERFUL THINGS.\n\n\nPUNISHMENTS IN PROVINCIAL TOWNS IN THE OLDEN TIME.\n\nThe instruments most in vogue with our ancestors were three--the\ncucking-stool, the brank, and the tumbrel.\n\nThe Cucking-stool was used by the pond in many village greens about one\nhundred years ago or little more, and then deemed the best corrective\nof a scolding woman.\n\n[Illustration: The Cucking-stool.]\n\nBy the sea, the quay offered a convenient spot. The barbican, at\nPlymouth, was a locality, doubtless terrible to offenders, however\ncareless of committing their wordy nuisance of scolding. Two pounds\nwere paid for a cucking-stool at Leicester in 1768. Since that it\nhas been placed at the door of a notorious scold as a warning. Upon\nadmission to the House of Correction at Liverpool, a woman had to\nundergo the severity of the cucking-stool till a little before the year\n1803, when Mr. James Neild wrote to Dr. Lettsom. The pump in the men's\ncourt was the whipping-post for females, which discipline continued,\nthough not weekly.\n\n _Kingston-upon-Thames._ _s._ _d._\n\n 1572, The making of the cucking-stool 8 0\n Iron work for the same 3 0\n Timber for the same 7 6\n Three brasses for the same, and three wheels 4 10\n -----------\n L1 3 4\n\n At Marlborough, in 1625, a man had 4_d._ for his help at the\n cucking of Joan Neal.\n\n _Gravesend._\n\n 1636, The porters for ducking of Goodwife Campion 2 0\n Two porters for laying up the ducking-stool 0 8\n\n[Illustration: The Brank.]\n\nThe Brank, for taming shrews, was preferred to the cucking-stool in\nsome counties, and was used there for the same purpose. The brank was\nin favour in the northern counties, and in Worcestershire, though there\nwere, notwithstanding, some of the other instruments of punishment\nused, called in that county gum-stools.\n\nThe brank was put over the head, and was fastened with a padlock. There\nare entries at Worcester about mending the \"scould's bridle and cords\nfor the same.\"\n\nThe cucking-stool not only endangered the health of the party, but also\ngave the tongue liberty 'twixt every dip. The brank was put over the\nhead, and was fastened with a padlock.\n\n[Illustration: The Tumbrel.]\n\nThe tumbrel was a low-rolling cart or carriage (in law Latin,\n_tumberella_) which was used as a punishment of disgrace and infamy.\nMillers, when they stole corn, were chastised by the tumbrel. Persons\nwere sometimes fastened with an iron chain to a tumbrel, and conveyed\nbareheaded with din and cry through the principal streets of towns.\n\n\n_Court of Hustings Book, 1581._ (_Lyme._)\n\n\"The jury present that the tumbrel be repaired and maintained from time\nto time, according to the statute.\"\n\nIn 1583, Mr. Mayor was to provide a tumbrel before All Saints Day,\nunder a penalty of 10_s._\n\n\nANCIENT METHOD OF KEEPING A WASHING ACCOUNT.\n\nShakerley Marmion, in his \"Antiquary,\" says:--\n\n \"I must rev'rence and prefer the precedent\n Times before these, which consum'd their wits in\n Experiments; and 'twas a virtuous\n Emulation amongst them, that nothing\n Which, might profit posterity should perish.\"\n\n[Illustration: [++] Washing Tablet.]\n\nWithout a full adherence to this dictum, we would nevertheless admit\nthat we are indebted to the past for the germ of many of our most\nimportant discoveries. The ancient washing tablet, although of humble\npretensions to notice, is yet a proof of the simple and effective means\nfrequently adopted in olden times for the economy of time and materials.\n\nA reference to the engraving obviates a lengthened explanation. It\nwill there be seen that if the mistress of a family has fifteen\n_pillow-covers, or so many collars, or so many bands_, to be mentioned\nin the washing account, she can turn the circular dial, by means of\nthe button or handle, to the number corresponding with the rough\nmark at the bottom of the dial, above which is written _sheets_,\n_table-cloths_, &c. This simple and ingenious contrivance, obviates the\nnecessity of keeping a book.\n\nThe original \"washing board,\" from which the engraving is taken, was of\na larger size, and showed the numbers very distinctly. Similar dials\nmay be made of either ivory or metal.\n\n\nTHE HAIR.\n\nThe quality and colour of the hair was a subject of speculative theory\nfor the ancients. Lank hair was considered indicative of pusillanimity\nand cowardice; yet the head of Napoleon was guiltless of a curl!\nFrizzly hair was thought an indication of coarseness and clumsiness.\nThe hair most in esteem, was that terminating in ringlets. Dares, the\nhistorian, states that Achilles and Ajax Telamon had curling locks;\nsuch also was the hair of Timon, the Athenian. As to the Emperor\nAugustus, nature had favoured him with such redundant locks, that no\nhair-dresser in Rome could produce the like. Auburn or light brown\nhair was thought the most distinguished, as portending intelligence,\nindustry, a peaceful disposition, as well as great susceptibility to\nthe tender passion. Castor and Pollux had brown hair; so also had\nMenelaus. Black hair does not appear to have been esteemed by the\nRomans; but red was an object of aversion. Ages before the time of\nJudas, red hair was thought a mark of reprobation, both in the case\nof Typhon, who deprived his brother of the sceptre of Egypt, and\nNebuchadnezzar who acquired it in expiation of his atrocities. Even the\ndonkey tribe suffered from this ill-omened visitation, according to the\nproverb of \"wicked as a red ass.\" Asses of that colour were held in\nsuch detestation among the Copths, that every year they sacrificed one\nby hurling it from a high wall.\n\n\nTHE FIRST COFFEE HOUSE IN LONDON.\n\nCoffee is a native of Arabia, supposed by some to have been the chief\ningredient of the old Lacedemonian broth. The use of this berry was not\nknown in England till the year 1657, at which time Mr. D. Edwards, a\nTurkey merchant, on his return from Smyrna to London, brought with him\none Pasquet Rossee, a Greek of Ragusa, who was used to prepare this\nliquor for his master every morning, who, by the way, never wanted\ncompany. The merchant, therefore, in order to get rid of a crowd of\nvisitants, ordered his Greek to open a coffee-house, which he did in\nSt. Michael's Alley, in Cornhill. This was the first coffee-house\nopened in London.\n\n\nEATING FOR A WAGER.\n\nThe handbill, of which the subjoined is a literal copy, was circulated\nby the keeper of the public-house at which the gluttony was to happen,\nas an attraction for all the neighbourhood to witness:--\n\n\"_Bromley in Kent_, July 14, 1726.--A strange eating worthy is to\nperform a Tryal of Skill on St. James's Day, which is the day of our\n_Fair_ for a wager of Five Guineas,--viz.: he is to eat four pounds of\nbacon, a bushel of French beans, with two pounds of butter, a quartern\nloaf, and to drink a gallon of strong beer!\"\n\n\nFOX KILLED BY A SWAN.\n\nAt Peusey, a swan sitting on her eggs, on one side of the river,\nobserved a fox swimming towards her from the opposite side; rightly\njudging she could best grapple with the fox in her own element, she\nplunged into the water, and after beating him off for some time with\nher wings, at length succeeded in drowning him.\n\n\nHIGHWAYMEN IN 1782.\n\nOn Wednesday, the 9th January, 1782, about four o'clock in the\nafternoon, as Anthony Todd, Esq., Secretary to the Post-office, was\ngoing in his carriage to his house at Walthamstow to dinner, and\nanother gentleman with him, he was stopt within a small distance of his\nhouse by two highwaymen, one of whom held a pistol to the coachman's\nbreast, whilst the other, with a handkerchief over his face, robbed\nMr. Todd and the gentleman of their gold watches and what money they\nhad about them. As soon as Mr. Todd got home all his men-servants were\nmounted on horses, and pursued the highwaymen; they got intelligence\nof their passing Lee-bridge, and rode on to Shoreditch; but could not\nlearn anything farther of them.\n\nThe same evening a gentleman going along Aldermanbury, near the church,\nwas accosted by a man with an enquiry as to the time; on which the\ngentleman pulled out his gold watch. The man immediately said, \"I\nmust have that watch and your money, sir, so don't make a noise.\" The\ngentleman seeing nobody near, he delivered his gold watch and four\nguineas, with some silver. The thief said he was in distress, and\nhoped the gentleman would not take away his life if ever he had the\nopportunity.\n\nSunday, the 13th January, 1782, about twelve o'clock, a man was, by\nforce, dragged up the yard of the French-Horn Inn, High Holborn, by\nsome person or persons unknown, and robbed of his watch, four guineas,\nand some silver; when they broke his arm and otherwise cruelly treated\nhim. He was found by a coachman, who took him to the hospital.\n\n\nAN ARCHBISHOP WASHING THE FEET OF THE POOR.\n\nIn the _Gentleman's Magazine_, we find the following\nobservance:--_Thursday, April 15, 1731_.--Being Maunday-Thursday, there\nwas distributed at the Banquetting-house, Whitehall, to forty-eight\npoor men, and forty-eight poor women (the King's age 48) boiled beef\nand shoulders of mutton, and small bowls of ale, which is called\ndinner; after that, large wooden platters of fish and loaves, viz.,\nundress'd, one large old ling, and one large dry'd cod; twelve red\nherrings, and nineteen white herrings, and four half quartern loaves;\neach person had one platter of this provision: after which was\ndistributed to them shoes, stockings, linnen and woolen cloath, and\nleathern bags, with one penny, two penny, three penny, and four penny\npieces of silver, and shillings: to each about L4 in value. His Grace\nthe Lord Archbishop of York, Lord High Almoner, performed the annual\nceremony of washing the feet of a certain number of poor in the Royal\nChapel, Whitehall, which was formerly done by the Kings themselves, in\nimitation of our Saviour's pattern of humility, &c. James II. was the\nlast King who performed this in person. His doing so was thus recorded\nin the _Chapel Royal Register_.--\"On Maunday Thursday April 16 1685 our\ngracious King James y{e} 2{d} wash'd wip'd and kiss'd the feet of 52 poor\nmen w{th} wonderful humility. And all the service of the Church of\nEngland usuall on that occasion was performed, his Maty being psent all\nthe time.\"\n\n\nA LUCKY FIND.\n\n_Sunday, April 1._--A few days ago, Sir Simon Stuart, of Hartley, in\nHampshire, looking over some old writings, found on the back of one\nof them a memorandum noting that 1,500 broad pieces were buried in a\ncertain spot in an adjoyning field. Whereupon he took a servant, and\nafter digging a little in the place, found the treasure in a pot, hid\nthere in the time of the late civil wars, by his grandfather, Sir\nNicholas Stuart.--_Gentleman's Magazine_, 1733.\n\n\nHOOPS IN 1740.\n\n[Illustration: [++] Ladies Hoops in 1740.]\n\nThe monstrous appearance of the ladies' hoops, when viewed behind, may\nbe seen from the following cut, copied from one of Rigaud's views.\nThe exceedingly small cap, at this time fashionable, and the close\nup-turned hair beneath it, give an extraordinary meanness to the head,\nparticularly when the liberality of gown and petticoat is taken into\nconsideration: the lady to the left wears a black hood with an ample\nfringed cape, which envelopes her shoulders, and reposes on the summit\nof the hoop. The gentleman wears a small wig and bag; the skirts of his\ncoat are turned back, and were sometimes of a colour different from the\nrest of the stuff of which it was made, as were the cuffs and lappels.\n\n\nSIEGE OF GIBRALTAR.\n\nGibraltar had been taken by a combined English and Dutch fleet in 1704,\nand was confirmed as a British possession, in 1713, by the peace of\nUtrecht; but in 1779 it was assailed by the united forces of France\nand Spain, and the siege continued till the 2nd of February, 1783. The\nchief attack was made on the 13th September, 1782. On the part of the\nbesiegers, besides stupendous batteries on the land side, mounting two\nhundred pieces of ordnance, there was an army of 40,000 men, under the\ncommand of the Duc de Crillon. In the bay lay the combined fleets of\nFrance and Spain, comprising forty-seven sail of the line, beside ten\nbattering ships of powerful construction, that cost upwards of L50,000\neach. From these the heaviest shells rebounded, but ultimately two of\nthem were set on fire by red-hot shot, and the others were destroyed to\nprevent them from falling into the hands of the British commander. The\nrest of the fleet also suffered considerably; but the defenders escaped\nwith very little loss. In this engagement 8,300 rounds were fired by\nthe garrison, more than half of which consisted of red-hot balls.\nDuring this memorable siege, which lasted upwards of three years, the\nentire expenditure of the garrison exceeded 200,000 rounds,--8,000\nbarrels of powder being used. The expenditure of the enemy, enormous\nas this quantity is, must have been much greater; for they frequently\nfired, from their land-batteries, 4,000 rounds in the short space of\ntwenty-four hours. Terrific indeed must have been the spectacle as the\nimmense fortress poured forth its tremendous volleys, and the squadron\nand land-batteries replied with a powerful cannonade. But all this\nwaste of human life and of property was useless on the part of the\nassailants; for the place was successfully held, and Gibraltar still\nremains one of the principal strongholds of British power in Europe.\n\n[Illustration: Saint George's Hall, Gibraltar.]\n\nDuring the progress of the siege, the fortifications were considerably\nstrengthened, and numerous galleries were excavated in the solid rock,\nhaving port-holes at which heavy guns were mounted, which, keeping up\nan incessant fire, proved very efficacious in destroying the enemy's\nencampments on the land side. Communicating with the upper tier of\nthese galleries are two grand excavations, known as Lord Cornwallis's\nand St. George's Halls. The latter, which is capable of holding several\nhundred men, has numerous pieces of ordnance pointed in various\ndirections, ready to deal destruction on an approaching enemy.\n\n\nKEEPING WHITSUNTIDE AT DURHAM CATHEDRAL.\n\nThe following curious account of the consumption of provisions in the\ncathedral of Durham, during Whitsun week, in 1347, together with the\nprices of the articles, is taken from the rolls of the cellarer, at\npresent in the treasury at Durham:--six hundred salt herrings, 3s.;\nfour hundred white herrings, 2s. 6d.; thirty salted salmon, 7s. 6d.;\ntwelve fresh salmon, 5s. 6d.; fourteen ling, fifty-five \"kelengs;\"\nfour turbot, 23s. 1d.; two horse loads of white fish, and a \"congr,\"\n5s. 10d.; \"playc,\" \"sparlings,\" and eels, and fresh water fish, 2s.\n9d.; nine carcases of oxen, salted, so bought, 36s.; one carcase and\na quarter, fresh, 6s. 11-3\/4d.; a quarter of an oxe, fresh, bought in\nthe town, 3s. 6d.; seven carcases and a half of swine, in salt, 22s.\n2-1\/4d.; six carcases, fresh, 12s. 9d.; fourteen calves, 28s. 4d.;\nthree kids, and twenty-six sucking porkers, 9s. 7-1\/2d.; seventy-one\ngeese with their feed, 11s. 10d.; fourteen capons, fifty-nine chickens,\nand five dozen pidgeons, 10s. 3d.; five stones of hog's lard, 4s. 2d.;\nfour stones of cheese, butter, and milk, 6s. 6d.; a pottle of vinegar,\nand a pottle of honey, 6-1\/2d.; fourteen pounds of figs and raisins,\nsixteen pounds of almonds, and eight pounds of rice, 3s. 7d.; pepper,\nsaffron, cinnamon, and other spices, 2s. 6d.; one thousand three\nhundred eggs, 15s. 5d.--sum total, L11 4s. Similar consumptions took\nplace during the week of the feast of St. Cuthbert, and other feasts,\namong the monks of Durham, for a long period of years.\n\n\nCURIOUS LAW.\n\nThe following curious law was enacted during the reign of Richard I.\nfor the government of those going by sea to the Holy Land:--\"He who\nkills a man on shipboard, shall be bound to the dead body and thrown\ninto the sea; if the man is killed on shore, the slayer shall be bound\nto the dead body and buried with it. He who shall draw his knife to\nstrike another, or who shall have drawn blood from him, to lose his\nhand; if he shall have only struck with the palm of his hand without\ndrawing blood, he shall be thrice ducked in the sea.\"\n\n\nDECAPITATION BY THE GUILLOTINE.\n\nA gentleman of intelligence and literary attainments, makes, in an\naccount of his travels on the continent, the following most singular\nremarks on an execution he witnessed, in which the culprit was beheaded\nby the guillotine:--\"It appears,\" says he, \"to be the best of all\npossible modes of inflicting the punishment of death; combining the\ngreatest impression on the spectator, with the least possible suffering\nto the victim. It is so rapid, that I should doubt whether there were\nany suffering; but from the expression of the countenance, when the\nexecutioner held up the head, I am inclined to believe that sense and\nconsciousness may remain for a few seconds after the head is off. The\neyes seemed to retain speculation for a moment or two, and there was a\nlook in the ghastly stare with which they stared upon the crowd, which\nimplied that the head was aware of its ignominious situation.\"\n\n\nALDERMAN BOYDELL.\n\nIt was the regular custom of Mr. Alderman Boydell, who was a very early\nriser, at five o'clock, to go immediately to the pump in Ironmonger\nLane. There, after placing his wig upon the ball at the top of it, he\nused to sluice his head with its water. This well-known and highly\nrespected character, who has done more for the British artist than all\nthe print-publishers put together, was also one of the last men who\nwore a three-cornered hat.\n\n\nFEATS OF STRENGTH IN 1739.\n\nApril 21.--The following notice was given to the public:--\"For the\nbenefit of Thomas Topham, the strong man, from Islington, whose\nperformances have been looked upon by the Royal Society and several\npersons of distinction, to be the most surprising as well as curious\nof any thing ever performed in England; on which account, as other\nentertainments are more frequently met with than that he proposes, he\nhumbly hopes gentlemen and ladies, &c., will honour him with their\npresence at the Nag's Head, in Gateshead, on Monday the 23d of this\ninstant, at four o'clock, where he intends to perform several feats of\nstrength, viz.:--He bends an iron poker three inches in circumference,\nover his arm, and one of two inches and a quarter round his neck; he\nbreaks a rope that will bear two thousand weight, and with his fingers\nrolls up a pewter dish of seven pounds hard metal; he lays the back\npart of his head on one chair, and his heels on another, and suffering\nfour men to stand on his body, he moves them up and down at pleasure;\nhe lifts a table six feet in length, by his teeth, with a half hundred\nweight hanging at the further end of it; and, lastly, to oblige the\npublick, he will lift a butt full of water.\" \"Each person to pay one\nshilling.\" This \"strong man\" fell a victim to jealousy, as is proved by\nthe following:--\"August 10th, 1749, died, Mr. Thomas Topham, known by\nthe name of the strong man, master of a publick house in Shoreditch,\nLondon. In a fit of jealousy, he stabbed his wife, then cut his own\nthroat and stabbed himself, after which he lived two days.\"\n\n\nELEPHANTS FRIGHTENED AT PIGS.\n\n\"Then on a tyme there were many grete clerkes and rad of kyng\nAlysaunder how on a tyme as he sholde have a batayle with ye kynge\nof Inde. And this kynge of Inde broughte with hym many olyphauntes\nberynge castelles of tree on theyr backes as the kynde of the is to\nhaue armed knyghtes in ye castell for the batayle, them ne knewe\nAlysaunder the kynge, of the olyphauntes that they drad no thynge more\nthan the jarrynge of swyne, wherefore he made to gader to gyder all\nye swyne that myghte be goten, and caused them to be dryuen as ny the\nolyphauntes as they myghte well here the jarrynge of the swyne, and\nthenne they made a pygge to crye, and whan the swyne herde the pygges a\nnone they made a great jarrynge, and as soone as the olyphauntes herde\nthat, they began to fle eche one, and keste downe the castelles and\nslewe the knyghtes that were in them, and by this meane Alysaunder had\nye vyctory.\"--_Liber Festivalis, printed by W. Caxton in_ 1483.\n\n\nA VISIT TO THE OBSERVATORY OF SIR ISAAC NEWTON.\n\nThe memory of a great and good man is imperishable. A thousand years\nmay pass away, but the fame that has survived the wreck of time remains\nunsullied, and is even brighter with age.\n\n \"The actions of the just\n Smell sweet, and blossom in the dust.\"\n\nIn an age of progress like our own we have frequently to regret the\ndestruction (sometimes necessary) of places associated with the genius\nof the past; but in the case of Sir Isaac Newton we have several\nrelics existing, none of which, perhaps, are more interesting than\nthe house in which he resided, still standing in St. Martin's Street,\non the south side of Leicester Square. The engravings of the interior\nand exterior of this building have been made from drawings made on\nthe spot. The house was long occupied as an hotel for foreigners, and\nwas kept by a M. Pagliano. In 1814 it was devoted to the purposes of\neducation. The Observatory, which is at the top, and where Sir Isaac\nNewton made his astronomical researches, was left in a dilapidated\ncondition until 1824, when two gentlemen, belonging to a committee of\nthe school, had it repaired at their own expense, and wrote a brief\nmemoir of the philosopher, which was placed in the Observatory, with a\nportrait of him.\n\n[Illustration: Interior of Sir Isaac Newton's Observatory.]\n\n[Illustration: House of Sir Isaac Newton, St. Martin's Street,\nLeicester Square.]\n\nIn this house Sir Isaac Newton resided for many years; and it was\nhere, according to his biographer, that he dispensed, under the\nsuperintendence of his beautiful niece, an elegant hospitality. Our\nsketch gives a good idea of the appearance of the exterior of the\nhouse at the present day; the front, it will be seen, has been well\nplastered, which, although clean and pleasant-looking to some eyes,\nseems to us to destroy the character of the building. The old doorway,\nwith a projecting top, has also been removed. The interior of the\nhouse is in excellent repair, and has undergone very little change.\nThe cornices, panelling, and the spacious staircase, are not altered\nsince the days of Newton. The rooms are very large. Tradition states\nit was in the back drawing-room that the manuscript of his work, the\n\"New Theory of Light and Colours,\" was destroyed by fire, caused by a\nfavourite little dog in Sir Isaac's absence. The name of this canine\nincendiary was Diamond. The manner in which the accident occurred is\nthus related:--The animal was wantoning about the philosopher's study,\nwhen it knocked down a candle, and set fire to a heap of manuscript\ncalculations upon which he had been employed for years. The loss was\nirretrievable; but Sir Isaac only exclaimed with simplicity, \"Ah,\nDiamond, Diamond, you little know what mischief you have been doing!\"\n\nPassing upstairs, and looking slightly at the various rooms, which\nare all well panelled, but which do not require particular notice, we\nreached the little observatory shown in the engraving. There, in the\nroom in which Sir Isaac has quietly studied, and in which he may have\nheld conferences with the most distinguished of his contemporaries,\nwe found two shoemakers busily at work, with whom we had some\npleasant conversation. Our artist has represented the interior of\nthe observatory, with its laborious occupants, worthy sons of St.\nCrispin. Shoemakers are well known to be a thoughtful class of men,\nalthough sometimes they unfortunately do not make the best use of their\nknowledge. Brand, the historian and author of the excellent book on\n\"Popular Antiquities,\" was at one time a shoemaker; so was Bloomfield,\nthe poet, who, when working at the \"last\" in Bell Alley, near the Bank,\nstrung together the charming recollection of his plough-boy life. We\ncould give a long list of shoemakers who have been eminent for talents.\n\nWe have not the exact date at which Newton came to reside here, but\ncertainly he was living in this house, at intervals, after 1695, when\nhe was appointed Warder of the Mint, of which establishment he rose to\nbe Master in the course of three years. The emoluments of this office\namounted to L1200 a-year, which enabled him to live in ease and dignity.\n\nIn 1703 he was chosen President of the Royal Society--an honourable\npost, to which he was annually elected until the time of his death.\n\n\nPOISONING THE MONARCH.\n\nAn idea of the popular notions about poisoning in the middle of the\nseventeenth century, may be formed from the following extract from an\nold tract, published in 1652, with the title of \"Papa Patris, or the\nPope in his Colours\":--\"Anno Dom: 1596; one Edward Squire, sometimes a\nscrivener at Grenewich, afterwards a deputy purveyor for the Queene's\nstable, in Sir Francis Drake's last voyage was taken prisoner and\ncarried into Spaine, and being set at liberty, one Walpole, a Jesuite,\ngrew acquainted with him, and got him into the Inquisition, whence\nhe returned a resolved , he persuaded Squire to undertake to\npoyson the pummell of the Queene (Elizabeth's) saddle, and, to make\nhim constant, made Squire receive the Sacrament upon it; he then gave\nhim the poyson, showing that he should take it in a double bladder,\nand should prick the bladder full of hoales in the upper part, when he\nshould use it (carrying it within a thick glove for the safety of his\nhand) should after turne it downward, pressing the bladder upon the\npummell of the Queene's saddle. This Squire confest. Squire is now in\nSpaine, and for his safer dispatch into England it was devised that\ntwo Spanish prisoners taken at Cales should be exchanged for Squire\nand one Rawles, that it might not be thought that Squire came over but\nas a redeemed captive. The Munday sennight after Squire returned into\nEngland, he, understanding the horses were preparing for the Queene's\nriding abroad, laid his hand, and crushed the poyson upon the pummell\nof the Queene's saddle, saying, 'God save the Queene,' the Queene\nrode abroad, and as it should seem laid not her hand upon the place,\nor els received no hurt (through God's goodnesse) by touching it.\nWalpole, counting the thing as done, imparted it to some principall\nfugitives there, but being disappointed of his hope, supposing Squire\nto have been false, to be revenged on him sent one hither (who should\npretend to have stolne from thence) with letters, wherein the plot of\nSquires was contained; this letter was pretended to be stolne out of\none of their studies. Squire, being apprehended, confessed all without\nany rigor, but after denied that he put it in execution, although he\nacknowledged he consented to it in the plot, at length he confessed the\nputting it in execution also.\"\n\n\nGRINNING FOR A WAGER.\n\n_June 9, 1786._--On Whit-Tuesday was celebrated at Hendon, in\nMiddlesex, a burlesque imitation of the Olympic Games. One prize was a\ngold-laced hat, to be grinned for by six candidates, who were placed on\na platform, with horses' collars to exhibit through. Over their heads\nwas printed in capitals,--\n\n Detur Tetriori; or\n The ugliest grinner\n Shall be the winner.\n\nEach party grinned five minutes _solus_, and then all united in a\ngrand _chorus_ of distortion. This prize was carried by a porter to a\n_vinegar_ merchant, though he was accused by his competitors of foul\nplay, for rinsing his mouth with _verjuice_. The whole was concluded\nby a hog, with has tail shaved and soaped, being let loose among nine\npeasants; any one of which that could seize him by the _queue_, and\nthrow him across his shoulders, was to have him for a reward. This\noccasioned much sport: the animal, after running some miles, so tired\nhis hunters that they gave up the chase in despair. A prodigious\nconcourse of people attended, among whom were the Tripoline Ambassador,\nand several other persons of distinction.\n\n\nBITE OF THE TARANTULA SPIDER.\n\nA Neapolitan soldier who had been bitten by a tarantula, though\napparently cured, suffered from an annual attack of delirium, after\nwhich he used to sink into a state of profound melancholy; his face\nbecoming livid, his sight obscure, his power of breathing checked,\naccompanied by sighs and heavings. Sometimes he fell senseless, and\ndevoid of pulsation; ejecting blood from his nose and mouth, and\napparently dying. Recourse was had to the influence of music; and the\npatient began to revive at the sound, his hands marking the measure,\nand the feet being similarly affected. Suddenly rising and laying hold\nof a bystander, he began to dance with the greatest agility during\nan uninterrupted course of four-and-twenty hours. His strength was\nsupported by administering to him wine, milk, and fresh eggs. If he\nappeared to relapse, the music was repeated, on which he resumed his\ndancing. This unfortunate being used to fall prostrate if the music\naccidentally stopped, and imagine that the tarantula had again stung\nhim. After a few years he died, in one of these annual attacks of\ndelirium.\n\n\nBYGONE CHRISTMAS CUSTOMS.\n\n ------------\"Now, too, is heard\n The hapless , tuning through the streets\n His _carol_ new; and oft, amid the gloom\n Of midnight hours, prevail th' accustom'd sounds\n Of wakeful _waits_, whose harmony (composed\n Of hautboy, organ, violin, and flute,\n And various other instruments of mirth),\n Is meant to celebrate the coming time.\"\n\n[Illustration: The Mummers, Or Ancient Waits.]\n\nThe manner in which this period of the year has been observed has often\nvaried. The observances of the day first became to be pretty general\nin the Catholic church about the year 300. By some of our ancestors\nit was viewed in the double light of a religious and joyful season of\nfestivities. The midnight preceding Christmas-day every person went\nto mass, and on Christmas-day three different masses were sung with\nmuch solemnity. Others celebrated it with great parade, splendour, and\nconviviality. Business was superseded by merriment and hospitality;\nthe most careworn countenance brightened on the occasion. The nobles\nand the barons encouraged and participated in the various sports: the\nindustrious labourer's cot, and the residence of proud royalty, equally\nresounded with tumultuous joy. From Christmas-day to Twelfth-day there\nwas a continued run of entertainments. Not only did our ancestors make\ngreat rejoicings on, but before and after Christmas-day. By a law in\nthe time of Alfred, the \"twelve days after the nativity of our Saviour\nwere made festivals;\"[1] and it likewise appears from Bishop Holt, that\nthe whole of the days were dedicated to feasting.\n\n[1] Thus we have the origin of Twelfth-day.\n\nOur ancestors' various amusements were conducted by a sort of master\nof the ceremonies, called the \"Lord of Misrule,\" whose duty it was to\nkeep order during the celebration of the different sports and pastimes.\nThe universities, the lord mayor and sheriffs, and all noblemen and\ngentlemen, had their \"lords of misrule.\" These \"lords\" were first\npreached against at Cambridge by the Puritans, in the reign of James\nI., as unbecoming the gravity of the university.\n\n[Illustration: The Lord of Misrule.]\n\nThe custom of serving boars' heads at Christmas bears an ancient date,\nand much ceremony and parade has been occasionally attached to it.\nHenry II. \"served his son (upon the young prince's coronation) at the\ntable as server, bringing up the _boar's head_ with trumpets before it.\"\n\nThe custom of strolling from street to street with musical instruments\nand singing seems to have originated from a very ancient practice\nwhich prevailed, of certain minstrels who were attached to the king's\ncourt and other great persons, who paraded the streets, and sounded the\nhour--thus acting as a sort of watchmen. Some slight remains of these\nstill exist, but they no longer partake of the authoritative claim\nas they originally did, as the \"lord mayor's music,\" &c. It may not,\nperhaps, be generally known, that even at the present day \"waits\" are\nregularly sworn before the \"court of burgesses\" at Westminster, and act\nunder the authority of a warrant, signed by the clerk, and sealed with\nthe arms of the city and liberty; in addition to which, they were bound\nto provide themselves with a silver badge, also bearing the arms of\nWestminster.\n\nIn the north they have their _Yule log_, or _Yuletide log_, which is\na huge log burning in the chimney corner, whilst the Yule cakes are\nbaked on a \"girdle,\" (a kind of frying-pan) over the fire; little lads\nand maidens assemble nightly at some neighbouring friends to hear the\ngoblin story, and join in \"fortune-telling,\" or some game. There is a\npart of an old song which runs thus:\n\n \"Now all our neighbours' chimneys smoke.\n And _Christmas logs_ are burning;\n Their ovens they with baked meate choke,\n And all their spits are turning.\"\n\nAmong the plants usual to Christmas are the rosemary, the holly, and\nthe mistletoe. Gay says:\n\n \"When _rosemary_ and _bays_, the poet's crown,\n Are bawled in frequent cries through all the town,\n Then judge the festival of Christmas near--\n Christmas, the joyous period of the year.\n Now with bright _holly_ all your temples strow,\n With _laurel_ green and sacred _mistletoe_.\"\n\n\nA MERMAN.\n\n\"The wind being easterly, we had thirty fathoms of water, when at ten\no'clock in the morning a sea monster like a man appeared near our ship,\nfirst on the larboard, where the master was, whose name is William\nLomone, who took a grappling iron to pull him up; but our captain,\nnamed Oliver Morin, hindered him, being afraid that the monster would\ndrag him away into the sea. The said Lomone struck him on the back, to\nmake him turn about, that he might view him the better. The monster,\nbeing struck, showed his face, having his two hands closed as if he\nhad expressed some anger. Afterwards he went round the ship: when he\nwas at the stern, he took hold of the helm with both hands, and we\nwere obliged to make it fast lest he should damage it. From thence he\nproceeded to the starboard, swimming still as men do. When he came\nto the forepart of the ship, he viewed for some time the figure that\nwas in our prow, which represented a beautiful woman, and then he\nrose out of the water as if he had been willing to catch that figure.\nAll this happened in the sight of the whole crew. Afterwards he came\nagain to the larboard, where they presented to him a cod-fish hanging\ndown with a rope; he handled it without spoiling it, and then removed\nthe length of a cable and came again to the stern, where he took hold\nof the helm a second time. At that very moment, Captain Morin got a\nharping-iron ready, and took it himself to strike him with it; but\nthe cordage being entangled, he missed his aim, and the harping-iron\ntouched only the monster, who turned about, showing his face, as he\nhad done before. Afterwards he came again to the fore part, and viewed\nagain the figure in our prow. The mate called for the harping-iron; but\nhe was frightened, fancying that this monster was one La Commune, who\nhad killed himself in the ship the year before, and had been thrown\ninto the sea in the same passage. He was contented to push his back\nwith the harping-iron, and then the monster showed his face, as he had\ndone at other times. Afterwards he came along the board, so that one\nmight have given him the hand. He had the boldness to take a rope held\nup by John Mazier and John Deffiete, who being willing to pluck it out\nof his hands, drew him to our board; but he fell into the water and\nthen removed at the distance of a gun's shot. He came again immediately\nnear our board, and rising out of the water to the navel, we observed\nthat his breast was as large as that of a woman of the best plight.\nHe turned upon his back and appeared to be a male. Afterwards he swam\nagain round the ship, and then went away, and we have never seen him\nsince. I believe that from ten o'clock till twelve that this monster\nwas along our board; if the crew had not been frighted, he might have\nbeen taken many times with the hand, being only two feet distant.\nThat monster is about eight feet long, his skin is brown and tawny,\nwithout any scales, all his motions are like those of men, the eyes\nof a proportionable size, a little mouth, a large and flat nose, very\nwhite teeth, black hair, the chin covered with a mossy beard, a sort of\nwhiskers under the nose, the ears like those of men, fins between the\nfingers of his hands and feet like those of ducks. In a word, he is a\nwell-shaped man. Which is certified to be true by Captain Oliver Morin,\nand John Martin, pilot, and by the whole crew, consisting of two and\nthirty men.\"--_An article from Brest, in the Memoirs of Trevoux._--This\nmonster was mentioned in the Gazette of Amsterdam, October 12, 1725,\nwhere it is said it was seen in the ocean in August, same year.\n\n\nA SHAVED BEAR.\n\nAt Bristol I saw a shaved monkey shown for a fairy; and a shaved\nbear, in a check waistcoat and trousers, sitting in a great chair\nas an Ethiopian savage. This was the most cruel fraud I ever saw.\nThe unnatural position of the beast, and the damnable brutality of\nthe woman-keeper who sat upon his knee, put her arm round his neck,\ncalled him husband and sweet-heart, and kissed him, made it the most\ndisgusting spectacle I ever witnessed! Cottle was with me.--_Southey._\n\n\nTHE ORIGIN OF WIGS.\n\nAs for the origin of wigs, the honour of the invention is attributed\nto the luxurious Sapygians in Southern Italy. The Louvain theologians,\nwho published a French version of the Bible, affected, however, to\ndiscover the first mention of perukes in a passage in the fourth\nchapter of Isaiah. The Vulgate has these words: \"Decalvabit Dominus\nverticem filiarum Sion, et Dominus crinem earum nudabit.\" This, the\nLouvain gentlemen translated into French as follows: \"Le Seigneur\ndechevelera les tetes des filles de Sion, et le Seigneur decouvrira\nleurs perruques;\" which, done into English, implies that \"The Lord will\npluck the hair from the heads of the daughters of Sion, and will expose\ntheir periwigs.\"\n\n\nDRESS IN 1772.\n\n[Illustration: [++] Maccaronies.]\n\nThe year 1772 introduced a new style for gentlemen, imported by a\nnumber of young men of fashion who had travelled into Italy, and\nformed an association called the Maccaroni Club, in contradistinction\nto the Beef-steak Club of London. Hence these new-fashioned dandies\nwere styled Maccaronies, a name that was afterwards applied to ladies\nof the same genus. The accompanying cut delineates the peculiarities\nof both. The hair of the gentleman was dressed in an enormous toupee,\nwith very large curls at the sides; while behind it was gathered and\ntied up into an enormous club, or knot, that rested on the back of\nthe neck like a porter's knot; upon this an exceedingly small hat\nwas worn, which was sometimes lifted from the head with the cane,\ngenerally very long, and decorated with extremely large silk tassels;\na full white handkerchief was tied in a large bow round the neck;\nfrills from the shirt-front projected from the top of the waistcoat,\nwhich was much shortened, reaching very little below the waist, and\nbeing without the flap-covered pockets. The coat was also short,\nreaching only to the hips, fitting closely, having a small turn-over\ncollar as now worn; it was edged with lace or braid, or decorated\nwith frog-buttons, tassels, or embroidery; the breeches were tight,\nof spotted or striped silk, with enormous bunches of strings at the\nknee. A watch was carried in each pocket, from which hung bunches of\nchains and seals: silk stockings and small shoes with little diamond\nbuckles completed the gentleman's dress. The ladies decorated their\nheads much like the gentlemen, with a most enormous heap of hair, which\nwas frequently surmounted by plumes of large feathers and bunches of\nflowers, until the head seemed to overbalance the body. The gown was\nopen in front; hoops were discarded except in full-dress; and the gown\ngradually spread outward from the waist, and trailed upon the ground\nbehind, shewing the rich laced petticoat ornamented with flowers and\nneedlework; the sleeves widened to the elbow, where a succession of\nruffles and lappets, each wider than the other, hung down below the\nhips.\n\n\nCHRISTMAS OBSERVANCES PUT DOWN BY THE PURITANS.\n\nDuring the Commonwealth, when puritanical feelings held iron sway\nover the rulers of the land, and rode rampant in high places, many\nstrong attempts were made to put down what they were pleased to term\nsuperstitious festivals, and amongst these was that of Christmas\nDay. So determined was the Puritan party to sweep away all vestiges\nof evil creeds and evil deeds, that they were resolved to make one\ngrand attempt upon the time-honoured season of Christmas. The Holly\nand the Mistletoe-bough were to be cut up root and branch, as plants\nof the Evil One. Cakes and Ale were held to be impious libations to\nsuperstition; and the Roundheads would have none of it.\n\n[Illustration: Proclaiming the Non-observance of Christmas.]\n\nAccordingly, we learn that, in the year 1647, the Cromwell party\nordered throughout the principal towns and cities of the country, by\nthe mouth of the common crier, that Christmas Day should no longer\nbe observed--it being a superstitious and hurtful custom; and that\nin place thereof, and the more effectually to work a change, markets\nshould be held on the 25th day of December.\n\nThis was attacking the people, especially the country folks, in\ntheir most sensitive part. It was hardly to be expected that they\nwould quietly submit to such a bereavement; nor did they, as the\nstill-existing \"News-letters\" of those days amply testify.\n\n\nTHE MANNER OF WATCHMEN INTIMATING THE CLOCK AT HERRENHUTH IN GERMANY.\n\n VIII. Past eight o'clock! O, Herrenhuth, do thou ponder;\n Eight souls in Noah's ark were living yonder.\n\n IX. 'Tis nine o'clock! ye brethren, hear it striking;\n Keep hearts and houses clean, to our Saviour's liking.\n\n X. Now, brethren, hear, the clock is ten and passing;\n None rest but such as wait for Christ embracing.\n\n XI. Eleven is past! still at this hour eleven,\n The Lord is calling us from earth to heaven.\n\n XII. Ye brethren, hear, the midnight clock is humming;\n At midnight, our great Bridegroom will be coming.\n\n I. Past one o'clock; the day breaks out of darkness:\n Great Morning-star appear, and break our hardness!\n\n II. 'Tis two! on Jesus wait this silent season,\n Ye two so near related, will and reason.\n\n III. The clock is three! the blessed Three doth merit\n The best of praise, from body, soul, and spirit.\n\n IV. 'Tis four o'clock, when three make supplication,\n The Lord will be the fourth on that occasion.\n\n V. Five is the clock! five virgins were discarded,\n When five with wedding garments were rewarded.\n\n VI. The clock is six, and I go off my station;\n Now, brethren, _watch yourselves for your salvation_.\n\n\nA DOG EXTINGUISHING A FIRE.\n\nOn the evening of the 21st February, 1822, the shop of Mr. Coxon,\nchandler, at the Folly, Sandgate, in Newcastle, was left in charge of\nhis daughter, about nine years of age, and a large mastiff, which is\ngenerally kept there as a safeguard since an attempt was made to rob\nthe shop. The child had on a straw bonnet lined with silk, which took\nfire from coming too near the candle. She endeavoured to pull it off,\nbut being tied, she could not effect her purpose, and in her terror\nshrieked out, on which the mastiff instantly sprang to her assistance,\nand with mouth and paws completely smothered out the flame by pressing\nthe bonnet together. The lining of the bonnet and the child's hair only\nwere burnt.\n\n\nCAMBRIDGE CLODS.\n\nAbout sixty years since, two characters, equally singular in their way,\nresided at Cambridge: Paris, a well-known bookseller, and Jackson, a\nbookbinder, and principal bass-singer at Trinity College Chapel in that\nUniversity; these two gentlemen, who were both remarkably corpulent,\nwere such small consumers in the article of bread, that their\nabstemiousness in that particular was generally noticed; but, to make\namends, they gave way to the greatest excess and indulgence of their\nappetites in meat, poultry, and fish, of almost every description. So\none day, having taken an excursion, in walking a few miles from home,\nthey were overtaken by hunger, and, on entering a public-house, the\nonly provision they could procure was a clod of beef, weighing near\nfourteen pounds, which had been a day or two in salt; and this these\ntwo moderate bread consumers contrived to manage between them broiled,\nassisted by a due proportion of buttered potatoes and pickles. The\nlandlord of the house, having some knowledge of his guests, the story\ngot into circulation, and the two worthies were ever after denominated\nthe Cambridge Clods!\n\n\nWITCH-TESTING AT NEWCASTLE IN 1649.\n\nMarch 26.--Mention occurs of a petition in the common council books\nof Newcastle, of this date, and signed, no doubt, by the inhabitants,\nconcerning witches, the purport of which appears, from what followed,\nto have been to cause all such persons as were suspected of that crime\nto be apprehended and brought to trial. In consequence of this, the\nmagistrates sent two of their sergeants, viz.--Thomas Shevill and\nCuthbert Nicholson, into Scotland, to agree with a Scotchman, who\npretended knowledge to find out witches, by pricking them with pins, to\ncome to Newcastle, where he should try such who should be brought to\nhim, and to have twenty shillings a piece, for all he should condemn as\nwitches, and free passage thither and back again. When the sergeants\nhad brought the said witch-finder on horseback to town, the magistrates\nsent their bellman through the town, ringing his bell and crying,\nall people that would bring in any complaint against any woman for a\nwitch, they should be sent for, and tried by the person appointed.\nThirty women were brought into the town-hall, and stripped, and then\nopenly had pins thrust into their bodies, and most of them were found\nguilty. The said reputed witch-finder acquainted Lieutenant-Colonel\nPaul Hobson, deputy-governor of Newcastle, that he knew women whether\nthey were witches or no by their looks; and when the said person\nwas searching of a personable and good-like woman, the said colonel\nreplied, and said, surely this woman is none, and need not be tried,\nbut the Scotchman said she was, and, therefore, he would try her; and\npresently, in the sight of all the people, laid her body naked to the\nwaist, with her cloathes over her head, by which fright and shame all\nher blood contracted into one part of her body, and then he ran a pin\ninto her thigh, and then suddenly let her cloathes fall, and then\ndemanded whether she had nothing of his in her body, but did not bleed!\nbut she being amazed, replied little; then he put his hands up her\ncloathes and pulled out the pin, and set her aside as a guilty person,\nand child of the devil, and fell to try others, whom he made guilty.\nLieutenant-Colonel Hobson, perceiving the alteration of the aforesaid\nwoman, by her blood settling in her right parts, caused that woman to\nbe brought again, and her cloathes pulled up to her thigh, and required\nthe Scot to run the pin into the same place, and then it gushed out of\nblood, and the said Scot cleared her, and said she was not a child of\nthe devil. The witch-finder set aside twenty-seven out of the thirty\nsuspected persons, and in consequence, fourteen witches and one wizard,\nbelonging to Newcastle, were executed on the town moor.\n\n\nALEXANDER SELKIRK AND THE DANCING GOATS.\n\nThe adventures of Alexander Selkirk, an English sailor, who, more than\none hundred and fifty years since, was left alone on the island of Juan\nFernandez are very wonderful.\n\nThis extraordinary man sought to beguile his solitude by rearing kids,\nand he would often sing to them, and dance with his motley group around\nhim. His clothes having worn out, he dressed himself in garments made\nfrom the skins of such as run wild about the island; these he sewed\ntogether with thongs of the same material. His only needle was a long\nslender nail; and when his knife was no longer available, he made an\nadmirable substitute from an iron hoop that was cast ashore.\n\n[Illustration: [++] Alexander Selkirk.]\n\nUpon the wonderful sojourn of this man, Defoe founded his exquisite\ntale of \"Robinson Crusoe,\" a narrative more extensively read and better\nknown than perhaps any other ever written.\n\n\nJACOB BOBART.\n\nA curious anecdote of Jacob Bobart, keeper of the physic garden at\nOxford, occurs in one of Grey's notes to _Hudibras_--\"He made a dead\nrat resemble the common picture of dragons, by altering its head and\ntail, and thrusting in taper sharp sticks, which, distended the skin on\neach side till it resembled wings. He let it dry as hard as possible.\nThe learned immediately pronounced it a dragon; and one of them sent\nan accurate description of it to Dr. Magliabecchi, librarian to the\nGrand Duke of Tuscany; several fine copies of verses were wrote on so\nrare a subject; but at last Mr. Bobart owned the cheat. However, it was\nlooked upon as a masterpiece of the art; and, as such, deposited in the\nMuseum.\"\n\n\nBLIND JACK.\n\n[Illustration: [++] Blind Jack.]\n\nThe streets of London, in the reigns of Queen Anne and Georges I. and\nII., were infested with all sorts of paupers, vagabonds, impostors,\nand common adventurers; and many, who otherwise might be considered\nreal objects of charity, by their disgusting manners and general\nappearance in public places, rather merited the interference of the\nparish beadles, and the discipline of Bridewell, than the countenance\nand encouragement of such persons as mostly congregated around common\nstreet exhibitions. One-eyed Granny and Blind Jack were particular\nnuisances to the neighbourhoods in which the first practised her\nmad-drunk gambols, and the latter his beastly manner of performing on\nthe flageolet. John Keiling, alias _Blind Jack_, having the misfortune\nto lose his sight, thought of a strange method to insure himself a\nlivelihood. He was constitutionally a hale, robust fellow, without\nany complaint, saving blindness, and having learnt to play a little\non the flageolet, he conceived a notion that, by performing on that\ninstrument in a different way to that generally practised, he should\nrender himself more noticed by the public, and be able to levy larger\ncontributions on their pockets.\n\nThe manner of _Blind Jack's_ playing the flageolet was by obtruding\nthe mouthpiece of the instrument up one of his nostrils, and, by long\ncustom, he could produce as much wind as most others with their lips\ninto the pipe; but the continued contortion and gesticulation of his\nmuscles and countenance rendered him an object of derision and disgust,\nas much as that of charity and commiseration.\n\n\nTHE YORKSHIRE TIKE.\n\n Ah iz i truth a country youth,\n Neean us'd teea Lunnon fashions;\n Yet vartue guides, an' still presides,\n Ower all mah steps an' passions.\n\n Neea coortly leear, bud all sincere,\n Neea bribe shall ivver blinnd me,\n If thoo can like a Yorkshire tike,\n A rooague thoo'll nivver finnd me.\n\n Thof envy's tung, seea slimlee hung,\n Wad lee aboot oor country,\n Neea men o' t' eearth booast greter wurth,\n Or mare extend ther boounty.\n\n Oor northern breeze wi' uz agrees,\n An' does for wark weel fit uz;\n I' public cares, an' all affairs,\n Wi' honour we acquit uz.\n\n Seea gret a moind is ne'er confiand,\n Tu onny shire or nation;\n They geean meeast praise weea weel displays\n A leearned iddicasion.\n\n Whahl rancour rolls i' lahtle souls,\n By shallo views dissarning,\n They're nobbut wise 'at awlus prize\n Gud manners, sense, and leearnin.\n\n\nTWO OF THE FATHERS ON FALSE HAIR.\n\nTertullian says, \"If you will not fling away your false hair, as\nhateful to Heaven, cannot I make it hateful to yourselves, by reminding\nyou that the false hair you wear may have come not only from a\ncriminal, but from a very dirty head; perhaps from the head of one\nalready damned?\" This was a very hard hit indeed; but it was not nearly\nso clever a stroke at wigs as that dealt by Clemens of Alexandria. The\nlatter informed the astounded wig-wearers, when they knelt at church\nto receive the blessing, that they must be good enough to recollect\nthat the benediction remained on the wig, and did not pass through\nto the wearer! This was a stumbling-block to the people; many of\nwhom, however, retained the peruke, and took their chance as to the\npercolating through it of the benediction.\n\n\nFOOD OF ANIMALS.\n\nLinnaeus states the cow to eat 276 plants, and to refuse 218; the goat\neats 449, and declines 126; the sheep takes 387, and rejects 141;\nthe horse likes 262, and avoids 212; but the hog, more nice in its\nprovision than any of the former, eats but 72 plants, and rejects 171.\n\n\nSLAVE ADVERTISEMENTS.\n\nThe following announcements are curious, as showing the merchandise\nlight in which the was regarded in America while yet a colony of\nGreat Britain:--\n\n _FRANCIS LEWIS, Has for SALE_,\n\n A Choice Parcel of Muscovado and Powder Sugars, in Hogsheads,\n Tierces, and Barrels; Ravens, Duck, and a Woman and \n Boy.--The Coach-House and Stables, with or without the Garden\n Spot, formerly the Property of Joseph Murray, Esq; in the Broad\n Way, to be let separately or together:--Inquire of said Francis\n Lewis.\n\n _New York Gazette_, Apr. 25, 1765.\n\n * * * * *\n\n This Day Run away from _John M' Comb_, Junier, an Indian Woman,\n about 17 Years of Age, Pitted in the face, of a middle Stature and\n Indifferent fatt, having on her a Drugat, Wastcoat, and Kersey\n Petticoat, of a Light Collour. If any Person or Persons, shall\n bring the said Girle to her said Master, shall be Rewarded for\n their Trouble to their Content.\n\n _American Weekly Mercury_, May 24, 1726.\n\n * * * * *\n\n A Female Child (of an extraordinary good Breed) to be given\n away; Inquire of Edes and Gill.\n\n _Boston Gazette_, Feb. 25, 1765.\n\n * * * * *\n\n _To be Sold, for want of Employ._\n\n A Likely Fellow, about 25 Years of Age, he is an\n extraordinary good Cook, and understands setting or tending a\n Table very well, likewise all Kind of House Work, such as washing,\n scouring, scrubbing, &c. Also a Wench his Wife, about 17\n Years old, born in this City, and understands all Sorts of House\n Work. For farther Particulars inquire of the Printer.\n\n _New York Gazette_, Mar. 21, 1765.\n\n\nPRESERVATIVE POWER OF COAL-PIT WATER.\n\nThe following is extracted from the register of St. Andrew's, in\nNewcastle:--\"April 24th, 1695, wear buried, James Archer and his son\nStephen, who, in the moneth of May, 1658, were drowned in a coal-pit\nin the Galla-Flat, by the breaking in of water from an old waste. The\nbodys were found intire, after they had lyen in the water 36 years and\n11 months.\"\n\n\nTHE QUEEN BEE.\n\nReaumur relates the following anecdote of which he was a witness:--A\nqueen bee, and some of her attendants, were apparently drowned in a\nbrook. He took them out of the water, and found that neither the queen\nbee, nor her attendants were quite dead. Reaumur exposed them to a\ngentle heat, by which they were revived. The plebeian bees recovered\nfirst. The moment they saw signs of animation in their queen, they\napproached her, and bestowed upon her all the care in their power,\nlicking and rubbing her; and when the queen had acquired sufficient\nforce to move, they hummed aloud, as if in triumph!\n\n\nDREAM OF KING HENRY I.\n\nA singular dream, which happened to this monarch when passing over\nto Normandy in 1130, has been depicted in a manuscript of Florence\nof Worcester, in Corpus Christi College, Oxford. The rapacity and\noppressive taxation of his government, and the reflection forced on\nhim by his own unpopular measures, may have originated the vision. He\nimagined himself to have been visited by the representatives of the\nthree most important grades of society--the husbandmen, the knights,\nand the clergy--who gathered round his bed, and so fearfully menaced\nhim, that he awoke in great alarm, and, seizing his sword, loudly\ncalled for his attendants. The drawings that accompany this narrative,\nand represent each of these visions, appear to have been executed\nshortly afterwards, and are valuable illustrations of the general\ncostume of the period. One of them is introduced in this place.\n\n[Illustration: [++] King Henry I. Dreaming.]\n\nThe king is here seen sleeping; behind him stand three husbandmen,\none carrying a scythe, another a pitchfork, and the third a shovel.\nThey are each dressed in simple tunics, without girdles, with plain\nclose-fitting sleeves; the central one has a mantle fastened by a\nplain brooch, leaving the right arm free. The beards of two of these\nfigures are as ample as those of their lords, this being an article\nof fashionable indulgence within their means. The one with the scythe\nwears a hat not unlike the felt hat still worn by his descendants in\nthe same grade: the scroll in his left hand is merely placed there to\ncontain the words he is supposed to utter to the king.\n\n\nSEPULCHRAL BARROW OF THE ANGLO-SAXONS.\n\nThe engraving on the next page is copied from a plate in Douglas's\n_Neniae_ and represents one of the most ancient of the Kentish barrows\nopened by him in the Chatham Lines, Sept. 1779; and it will enable the\nreader at once to understand the structure of these early graves, and\nthe interesting nature of their contents. The outer circle marks the\nextent of the mound covering the body, and which varied considerably in\nelevation, sometimes being but a few inches or a couple of feet from\nthe level of the ground, at others of a gigantic structure. In the\ncentre of the mound, and at the depth of a few feet from the surface,\nan oblong rectangular grave is cut, the space between that and the\nouter circle being filled in with chalk, broken into small bits, and\ndeposited carefully and firmly around and over the grave. The grave\ncontained the body of a male adult, tall and well-proportioned, holding\nin his right hand a spear, the shaft of which was of wood, and had\nperished, leaving only the iron head, 15 inches in length, and at the\nbottom a flat iron stud (_a_), having, a small pin in the centre, which\nwould appear to have been driven into the bottom of the spear-handle;\nan iron knife lay by the right side, with remains of the original\nhandle of wood. Adhering to its under side were very discernible\nimpressions of coarse linen cloth, showing that the warrior was buried\nin full costume. An iron sword is on the left side, thirty-five and\na quarter inches in its entire length, from the point to the bottom\nof the handle, which is all in one piece, the wood-work which covered\nthe handle having perished; the blade thirty inches in length and two\nin breadth, flat, double-edged, and sharp-pointed, a great portion of\nwood covering the blade, which indicates that it was buried with a\nscabbard, the external covering being of leather, the internal of wood.\nA leathern strap passed round the waist, from which hung the knife and\nsword, and which was secured by the brass buckle (_b_), which was found\nnear the last bone of the vertebrae, or close to the os sacrum. Between\nthe thigh-bones lay the iron umbo of a shield, which had been fastened\nby studs of iron, four of which were found near it, the face and\nreverse of one being represented at (_c_.) A thin plate of iron (_d_),\nfour and a half inches in length, lay exactly under the centre of the\numbo, having two rivets at the end, between which and the umbo were\nthe remnants of the original wooden (and perhaps hide-bound) shield;\nthe rivets of the umbo having apparently passed through the wood to\nthis plate as its bracer or stay. In a recess at the feet was placed a\nvase of red earth, slightly ornamented round the neck with concentric\ncircles and zigzag lines.\n\n[Illustration: [++] Anglo-Saxon Sepulchral Barrow.]\n\n\nAN OLD GANDER.\n\nWilloughby states in his work on Ornithology, that a friend of his\npossessed a gander eighty years of age; which in the end became so\nferocious that they were forced to kill it, in consequence of the\nhavock it committed in the barn-yard. He also talks of a swan three\ncenturies old; and several celebrated parrots are said to have attained\nfrom one hundred to one hundred and fifty years.\n\n\nEXTRAORDINARY SLEEPER.\n\nM. Brady, Physician to Prince Charles of Lorraine, gives the following\nparticulars of an extraordinary sleeper:--\n\n\"A woman, named Elizabeth Alton, of a healthful strong constitution,\nwho had been servant to the curate of St. Guilain, near the town\nof Mons, about the beginning of the year 1738, when she was about\nthirty-six years of age grew extremely restless and melancholy. In\nthe month of August, in the same year, she fell into a sleep which\nheld four days, notwithstanding all possible endeavours to awake her.\nAt length she awoke naturally, but became more restless and uneasy\nthan before; for six or seven days, however, she resumed her usual\nemployments, until she fell asleep again, which continued eighteen\nhours. From that time to the year 1753, which is fifteen years, she\nfell asleep daily about three o'clock in the morning, without waking\nuntil about eight or nine at night. In 1754 indeed her sleep returned\nto the natural periods for four months, and, in 1748, a tertian ague\nprevented her sleeping for three weeks. On February 20, 1755, M. Brady,\nwith a surgeon, went to see her. About five o'clock in the evening,\nthey found her pulse extremely regular; on taking hold of her arm it\nwas so rigid, that it was not bent without much trouble. They then\nattempted to lift up her head, but her neck and back were as stiff as\nher arms. He hallooed in her ear as loud as his voice could reach; he\nthrust a needle into her flesh up to the bone; he put a piece of rag\nto her nose flaming with spirits of wine, and let it burn some time,\nyet all without being able to disturb her in the least. At length, in\nabout six hours and a half, her limbs began to relax; in eight hours\nshe turned herself in the bed, and then suddenly raised herself up,\nsat down by the fire, ate heartily, and began to spin. At other times,\nthey whipped her till the blood came; they rubbed her back with honey,\nand then exposed it to the stings of bees; they thrust nails under her\nfinger-nails; and it seems these triers of experiments consulted more\nthe gratifying their own curiosity than the recovery of the unhappy\nobject of the malady.\"\n\n\nA FAT ENGLISHMAN.\n\nKeysler, in his travels, speaks of a corpulent Englishman, who in\npassing through Savoy, was obliged to make use of twelve chairmen. He\nis said to have weighed five hundred and fifty pounds, or thirty-nine\nstone four pounds.\n\n\nA HAPPY FAMILY.\n\nA gentleman travelling through Mecklenburg, some years since, witnessed\na singular association of incongruous animals. After dinner, the\nlandlord of the inn placed on the floor a large dish of soup, and gave\na loud whistle. Immediately there came into the room a mastiff, an\nAngora cat, an old raven, and a remarkably large rat, with a bell about\nits neck. They all four went to the dish, and, without disturbing each\nother, fed together; after which the dog, cat, and rat, lay before\nthe fire, while the raven hopped about the room. The landlord, after\naccounting for the familiarity of these animals, informed his guest\nthat the rat was the most useful of the four; for the noise he made had\ncompletely freed his house from the rats and mice with which it was\nbefore infested.\n\n\nANCIENT FIRE-ARMS IN THE TOWER OF LONDON ARMORY.\n\nWe have just now before us a drawing of an old piece of ordnance,\nformed of bars of iron, strongly hooped with the same material, which\nforms a striking contrast with the finely-wrought cannons which may be\nseen in store at Woolwich Arsenal, and elsewhere, at the present day.\nThe exact date and manner of the introduction of cannon is a matter\nwhich has caused much dispute. The earliest mention of the use of\ncannon on shipboard is in Rymer's \"Foedera.\" It is an order to Henry\nSomer, Keeper of the Private Wardrobe in the Tower, to deliver to Mr.\nGoveney, Treasurer to Queen Philippa, Queen of Sweeden, Denmark, and\nNorway, (who was then sent by her uncle, Henry the Fourth, to her\nhusband, in the ship called the Queen's Hall,) the following military\nstores: 11 guns, 40 petras pro gunnes, 40 tumpers, 4 torches, 1 mallet,\n2 fire-pans, 40 pavys, 24 bows, 40 sheaves of arrows.\n\nAfter the old cannon composed of bars of iron, hooped together, had\nbeen some time in use, hand-cannon, a simple tube fixed on a straight\nstake, was used in warfare, charged with gunpowder and an iron bullet.\nThis was made with trunnions and casabel precisely like the large\ncannon. In course of time, the touch-hole was improved, and the barrel\ncast in brass. This, fixed to a rod, had much the appearance of a large\nsky-rocket. What is now called the stock was originally called the\nframe of the gun.\n\nVarious improvements were from time to time made in the hand-gun,\namongst which was a pan fixed for containing the touch-powder. In rainy\nweather, this became a receptacle for water; to obviate which, a small\npiece of brass made to turn on a pin was placed as a cover. This done,\nthere was a difficulty in preserving the aim in consequence of the\nliability of the eye to be diverted from the sight by the motion of the\nright hand when conveying the lighted match to the priming. This was,\nto a certain extent, prevented by a piece of brass being fixed to the\nbreech and perforated. The improved plan for holding the lighted match\nfor firing the hand-guns is shown in the engraving of the Buckler and\nPistol; it consists of a thin piece of metal something in shape of an S\nreversed, the upper part slit to hold the match, the lower pushed up by\nthe hand when entended to ignite the powder.\n\nAfter the invention of the hand-cannon, its use became general in a\nvery short space of time in most parts of the civilized world.\n\nPhilip de Comines, in his account of the battle of Morat, in 1476, says\nhe encountered in the confederate army 10,000 _arquebusiers_.\n\nThe arquebusiers in Hans Burgmain's plates of the \"Triumph of\nMaximilian the First,\" have suspended from their necks large powder\nflasks or horns, a bullet bag on the right hip, and a sword on the\nleft, while they carry the matchlock in their hands.\n\nHenry the Eighth's Walking-stick, as the Yeomen of Guard at the Tower\ncall it, is a short spiked mace, in the head of which are three short\nguns or pistols, which may be fired at very primitive touch-holes by a\nmatch.\n\nThe Revolver has four barrels, and although clumsy in construction, is\nnot very different in principles from those recently introduced.\n\n[Illustration: 1. Henry the Eighth's Walking-stick. 2. A Revolver of\nthe Fifteenth century. 3. Buckler, with Pistol inserted.]\n\nThe use of the pistol inserted inside the buckler is obvious as the\nlatter affords protection to the person while using the former.\n\n\nWIGS.\n\n[Illustration: [++] Wigs.]\n\nIn 1772 the Maccaronies, as the exquisites of that time were called,\nwore wigs similar to 1, 2, 3, with a large toupee, noticed as early as\n1731, in the play of the _Modern Husband_: \"I meet with nothing but a\nparcel of _toupet_ coxcombs, who plaster up their brains upon their\nperiwigs,\" alluding to the pometum with which they were covered. Those\nworn by the ladies in 1772 are given as 4, showing the rows of curls at\nthe sides. The pig-tails were worn hanging down the back, or tied up in\na knot behind, as in 5. About 1780 the hair which formed it was allowed\nto stream in a long lock down the back, as in 6, and soon afterwards\nwas turned up in a knot behind. Towards the end of the century, the\nwig, as a general and indispensable article of attire to young and old,\nwent out of fashion.\n\n\nA FALSE FIND.\n\nAt Falmouth, some years ago, the sexton found coal in digging a grave;\nhe concluded it must be a mine, and ran with the news and the specimen\nto the clergyman. The surgeon explained that they had stolen a French\nprisoner who died, and filled his coffin with coal that the bearers\nmight not discover its emptiness.\n\n\nBELLS.\n\nAs far back as the Anglo-Saxon times, before the conclusion of the\nseventh century, bells had been in use in the churches of this country,\nparticularly in the monastic societies of Northumbria; and were,\ntherefore, in use from the first erection of parish churches among us.\nThose of France and England appear to have been furnished with several\nbells. In the time of Clothaire II., King of France, and in the year\n610, the army of that king was frightened from the siege of the city of\nSens, by ringing the bells of St. Stephen's Church. They were sometimes\ncomposed of iron in France; and in England, as formerly at Rome, they\nwere frequently made of brass. And as early as the ninth century many\nwere cast of a large size and deep note.\n\nWeever, in his work on funeral monuments, says--\"In the little\nsanctuary at Westminster, King Edward III., erected a clochier, and\nplaced therein three bells, for the use of St. Stephen's Chapel. About\nthe biggest of them were cast in the metal these words:--\n\n \"King Edward made mee thirty thousand weight and three;\n Take me down and wey mee, and more you shall find mee.\"\n\n\"But these bells being taken down in the reign of Henry VIII., one\nwrote underneath with a coal:--\n\n \"But Henry the Eight,\n Will bait me of my weight.\"\n\nThis last distich alludes to a fact mentioned by Stow, in his survey\nof London--ward of Farringdon Within to wit--that near to St. Paul's\nSchool stood a clochier, in which were four bells, called _Jesus'\nbells_, the greatest in all England, against which Sir Miles Partridge\nstaked an hundred pounds, and won them of Henry VIII., at a cast of\ndice.\n\nMatthew Paris observes, that anciently the use of bells was prohibited\nin time of mourning. Mabillon adds, that it was an old practice to\nring the bells for persons about to expire, to advertise the people\nto pray for them--whence our passing-bell. The passing-bell, indeed,\nwas anciently for two purposes--one to bespeak the prayers of all good\nChristians for a soul just departing; the other to drive away the evil\nspirits who were supposed to stand at the bed's foot.\n\nThis dislike of spirits to bells is mentioned in the Golden Legend, by\nWynkyn de Worde. \"It is said, evill spirytes that ben in the regyon\nof thayre, doubte moche when they here the belles rongen; and this is\nthe cause why the belles ben rongen when it thondreth, and when grete\ntempeste and outrages of wether happen; to the ende that the fiends and\nwycked spirytes shold be abashed and flee, and cease of the movynge of\ntempeste.\" Another author observes, that the custom of ringing bells at\nthe approach of thunder is of some antiquity; but that the design was\nnot so much to shake the air, and so dissipate the thunder, as to call\nthe people to church, to pray that the parish might be preserved from\nthe terrible effect of lightning.\n\nWarner, in his history of Hampshire, enumerates the virtues of a bell,\nby translating the lines from the \"Helpe to Discourse:--\n\n \"Men's death's I tell by doleful knell;\n Lightning and thunder I break asunder.\n On Sabbath all to church I call;\n The sleepy head I raise from bed;\n The winds so fierce I doe disperse;\n Men's cruel rage I doe assuage.\"\n\n[Illustration: The Curfew Bell.]\n\nFour of the bells of the ancient Abbey of Hexham were dedicated or\nbaptised; and although the old bells no longer exist, the legends upon\nthe whole six have been preserved, and a free translation given by Mr.\nWright, is as follows:--\n\n 1. Even at our earliest sound,\n The light of God is spread around.\n\n 2. At the echo of my voice,\n Ocean, earth and air, rejoice.\n\n 3. Blend thy mellow tones with mine,\n Silver voice of Catherine!\n\n 4. Till time on ruin's lap shall nod.\n John shall sound the praise of God.\n\n 5. With John in heavenly harmony,\n Andrew, pour thy melody.\n\n 6. Be mine to chant Jehovah's fame,\n While Maria is my name.\n\nThese epigraphs or legends on bells, are not uncommon. The Rev. W. C.\nLukis, in his notices on church bells, read at the Wilts Archaeological\nMeeting, gave the following instances:--\n\nAt Aldbourne, on the first bell, we read, \"The gift of Jos. Pizzie and\nWm. Gwynn.\n\n \"Music and ringing we like so well,\n And for that reason we gave this bell.\"\n\nOn the fourth bell is,--\n\n \"Humphry Symsin gave xx pound to buy this bell,\n And the parish gave xx more to make this ring go well.\"\n\nA not uncommon epigraph is,--\n\n \"Come when I call\n To serve God all.\"\n\nAt Chilton Foliatt, on the tenor, is,--\n\n \"Into the church the living I call,\n And to the grave I summon all.\n Attend the instruction which I give,\n That so you may for ever live.\"\n\nAt Devizes, St. Mary, on the first bell, is,--\n\n \"I am the first, altho' but small.\n I will be heard above you all.\"\n\nAnd on the second bell is,--\n\n \"I am the second in this ring,\n Therefore next to thee I will sing.\"\n\nWhich, at Broadchalk, is thus varied:--\n\n \"I in this place am second bell,\n I'll surely do my part as well.\"\n\nOn the third bell at Coln is,--\n\n \"Robert Forman collected the money for casting this bell\n Of well-disposed people, as I do you tell.\"\n\nAt Bath Abbey, on the tenth bell, is,--\n\n \"All you of Bath that hear me sound,\n Thank Lady Hopton's hundred pound.\"\n\nOn the fifth bell at Amesbury is,--\n\n \"Be strong in faith, praise God well,\n Frances Countess Hertford's bell.\"\n\nAnd, on the tenor,--\n\n \"Altho' it be unto my loss,\n I hope you will consider my cost.\"\n\nAt Stowe, Northamptonshire, and at St. Mary the Virgin, Oxford, we\nfind,--\n\n \"Be it known to all that doth me see,\n That Newcombe, of Leicester, made me.\"\n\nAt St. Michael's, Coventry, on the fourth bell, is,--\n\n \"I ring at six to let men know\n When to and from their work to go.\"\n\nOn the seventh bell is,--\n\n \"I ring to Sermon with a lusty bome,\n That all may come and none can stay at home.\"\n\nOn the eighth bell is--\n\n \"I am and have been called the common bell\n To ring, when fire breaks out to tell.\"\n\nAt St. Peter's-le-Bailey, Oxford, four bells were sold towards\nfinishing the tower, and in 1792 a large bell was put up, with this\ninscription:--\n\n \"With seven more I hope soon to be\n For ages joined in harmony.\"\n\nBut this very reasonable wish has not yet been realized; whereas at St.\nLawrence's, Reading, when two bells were added to form a peal of ten,\non the second we find--\n\n \"By adding two our notes we'll raise,\n And sound the good subscribers' praise.\"\n\nThe occasion of the erection of the Westminster Clock-tower, is said\nto have been as follows:--A certain poor man, in an action for debt,\nbeing fined the sum of 13s. 4d., Radulphus Ingham, Chief Justice of\nthe King's Bench, commiserating his case, caused the court roll to\nbe erased, and the fine reduced to 6s. 8d., which being soon after\ndiscovered, Ingham was amerced in a pecuniary mulct of eight hundred\nmarks, which was employed in erecting the said bell-tower, in which\nwas placed a bell and a clock, which, striking hourly, was to remind\nthe judges in the hall of the offence of their brother. This bell was\noriginally called Edward; \"but,\" says a writer in the \"Antiquarian\nRepertory,\" \"when the Reformation caused St. Edward and his hours to\nbe but little regarded; as other bells were frequently called Tom, as\nfancied to pronounce that name when stricken--that at Lincoln, for\ninstance, and that at Oxford--this also followed the fashion, of which,\nto what I remember of it before it was hung up, I may add another proof\nfrom a catch made by the late Mr. Eccles, which begins--\n\n \"'Hark, Harry, 'tis late--'tis time to be gone,\n For Westminster Tom, by my faith, strikes one.\"\n\nHawkins, in his \"History of Music,\" says,--\"The practice of ringing\nbells in change, or regular peals, is said to be peculiar to England:\nwhence Britain has been termed the _ringing island_. The custom seems\nto have commenced in the time of the Saxons, and was common before the\nConquest. The ringing of bells, although a recreation chiefly of the\nlower sort, is, in itself, not incurious. The tolling of a bell is\nnothing more than the producing of a sound by a stroke of the clapper\nagainst the side of the bell, the bell itself being in a pendant\nposition, and at rest. In ringing, the bell, by means of a wheel and\na rope, is elevated to a perpendicular; in its motion, the clapper\nstrikes forcibly on one side, and in its return downwards, on the\nother side of the bell, producing at each stroke a sound.\" There are\nstill in London several societies of ringers. There was one called the\nCollege Youths (bell-ringers, like post-boys, never seem to acquire old\nage). Of this it is said Sir Matthew Hale, Lord Chief Justice of the\nKing's Bench, was, in his youthful days, a member; and in the life of\nthat upright judge, by Burnet, some facts are mentioned which favour\nthis relation. In England the practice of ringing has been reduced to\na science, and peals have been composed which bear the names of their\ninventors; some of the most celebrated of these were composed about\nfifty years ago by one Patrick. This man was a maker of barometers.\nIn the year 1684, one Abraham Rudhall, of the city of Gloucester,\nbrought the art of bell-founding to great perfection. His descendants\nin succession have continued the business of casting bells; and by a\nlist published by them at Lady Day, 1774, the family, in peals and odd\nbells, had cast to the amount of 3,594. The peals of St. Dunstan's\nin the East, St. Bride's, London, and St. Martin's-in-the-Fields,\nare among the number. The following \"Articles of Ringing\" are upon\nthe walls of the belfry in the pleasant village of Dunster, in\nSomersetshire. They are dated 1787:--\n\n \"1. You that in ringing take delight,\n Be pleased to draw near;\n These articles you must observe,\n If you mean to ring here.\n\n \"2. And first, if any overturn\n A bell, as that he may,\n He forthwith for that only fault\n In beer shall sixpence pay.\n\n \"3. If any one shall curse or swear\n When come within the door,\n He then shall forfeit for that fault\n As mentioned before.\n\n \"4. If any one shall wear his hat\n When he is ringing here,\n He straightway then shall sixpence pay\n In cyder or in beer.\n\n \"5. If any one these articles\n Refuseth to obey,\n Let him have nine strokes of the rope,\n And so depart away.\"\n\n\nBILL OF SALE FOR A IN 1770.\n\n\"Know all Men by these Presents, That I, Elizabeth Treat, of Boston,\nin the county of Suffolk, widow, in consideration of the sum of L25\n13s. 4d. to me in hand, paid before the ensealing hereof by Samuel\nBreck, of Boston aforesaid, merchant, the receipt whereof I do hereby\nacknowledge, have granted, bargained, and sold, and by these presents\ndo fully and absolutely grant, bargain, and sell unto the said Samuel\nBreck, my man named Harry, aged about forty years, with his\napparel, to have and to hold the said man Harry, with his\napparel, unto the said Samuel Breck, his executors, administrators,\nand assigns, to his and their only proper use, benefit, and behoof for\never; And I, the said Elizabeth Treat, for myself, my heirs, executors,\nand administrators, do covenant, that at the time of ensealing, and\nuntil the delivery hereof, I am the true and lawful owner of the said\n man, and that he is free from all former sales, charges, and\nincumbrances whatsoever, and that I will warrant and defend the said\n man unto the said Samuel Breck, his heirs, and assigns for ever,\nagainst the lawful claims and demands of all persons whomsoever.\n\n\"Witness my hand and seal, this tenth day of October, Anno Domini, one\nthousand seven hundred and seventy, in the tenth year of His Majesty's\nreign.\n\n\"Signed, sealed, and delivered in presence of us.\n\n \"THOMAS MELVILLE.\n \"MARY WHITE.\n \"ELIZABETH TREAT.\"\n\n\nTHE AZTEC CHILDREN.\n\n[Illustration: The Aztec Children, As Exhibited in England.]\n\nAmong the animated curiosities which are occasionally exposed to\nthe gaze of the wonder-loving public, we may prominently notice the\nAZTEC CHILDREN--two singular Lilliputians who were recently exhibited\nthroughout the kingdom. Maximo and Bartolo (for by these names the two\nAztec children have been baptized) are by some medical men supposed\nto be of the respective ages of twenty-two and sixteen. Professor\nOwen, stated them to be ten or twelve, and seven or nine in 1853. The\nheight of the boy (the elder is about three feet, and the girl does\nnot reach quite two feet six inches). Their limbs, though slender,\nare proportionate and well formed, and the general development of\ntheir figures is remarkably graceful. The cranium is peculiar, being\nnarrower than that of any other races of beings known to the world;\nand though the face is somewhat prominent, the features are regular\nand the countenances agreeable, and, after a short acquaintance,\nhighly interesting. Each has a beautiful head of jet black hair, which\nflows gracefully in curls. They are lively and intelligent, showing\nconsiderable aptitude for mental training, and have already learned to\ngive utterance to several expressions which can be readily understood\nby visitors.\n\nSince the arrival of these prodigies from the United States, they have\nbeen the objects of curious ethnological speculations. Dr. Latham does\nnot consider them as a new species of the _genus homo_. Professor Owen\nregards them as instances of impeded development, and Dr. Conolly was\nstruck with their resemblance to idiots.\n\n\nNOTICES TO TAR AND FEATHER.\n\nThe original handbills of the committee for Tarring and Feathering\nsubjoined, are of singular interest, as they were the earliest\nemanations of the spirit that led to England's losing her American\ncolonies, and the consequent rise of the United States:--\n\n\n_To the Delaware Pilots._\n\nThe Regard we have for your Characters, and our Desire to promote your\nfuture Peace and Safety, are the Occasion of this Third Address to you.\n\nIn our second Letter we acquainted you, that the Tea Ship was a Three\nDecker; We are now informed by good Authority, she is not a Three\nDecker, but an _old black Ship_, _without a Head_, or _any Ornaments_.\n\nThe _Captain_ is a _short fat_ Fellow, and a little _obstinate_\nwithal.--So much the worse for him.--For, so sure as he _rides rusty_,\nWe shall heave him Keel out, and see that his Bottom be well fired,\nscrubb'd and paid.--His Upper-Works too, will have an Overhawling--and\nas it is said, he has a good deal of _Quick Work_ about him, We will\ntake particular Care that such Part of him undergoes a thorough\nRummaging.\n\nWe have a still _worse Account of his Owner_;--for it is said, the Ship\nPOLLY was bought by him on Purpose, to make a Penny of us: and that\n_he_ and Captain _Ayres_ were well advised, of the Risque they would\nrun, in thus daring to insult and abuse us.\n\n_Captain Ayres_ was here in the Time of the Stamp-Act, and ought to\nhave known our People better, than to have expected we would be so mean\nas to suffer his _rotten_ TEA to be funnel'd down our Throats, with the\n_Parliament's Duty_ mixed with it.\n\nWe know him well, and have calculated to a Gill and a Feather, how much\nit will require to fit him for an _American Exhibition_. And we hope,\nnot one of your Body will behave so ill, as to oblige us to clap him in\nthe Cart along Side of the _Captain_.\n\nWe must repeat, that the SHIP POLLY is an _old black Ship_, of about\nTwo Hundred and Fifty Tons burthen, _without a Head_, and _without\nOrnaments_,--and, that CAPTAIN AYRES is a _thick chunky Fellow_.--As\nsuch, TAKE CARE TO AVOID THEM.\n\n Your Old Friends,\n THE COMMITTEE FOR TARRING AND FEATHERING.\n _Philadelphia, December 7, 1773._\n\n\n_To Capt. Ayres, of the Ship Polly, on a Voyage from London to\nPhiladelphia._\n\n SIR,\n\nWe are informed that you have, imprudently, taken Charge of a Quantity\nof Tea; which has been sent out by the _India_ Company, _under the\nAuspices of the Ministry_, as a Trial of _American_ Virtue and\nResolution.\n\nNow, as your Cargo, on your Arrival here, will most assuredly bring you\ninto hot water; and as you are perhaps a Stranger _to these Parts_,\nwe have concluded to advise you of the present Situation of Affairs\nin _Philadelphia_--that, taking Time by the Forelock, you may stop\nshort in your dangerous Errand--secure your Ship against the Rafts of\ncombustible Matter which may be set on Fire, and turned loose against\nher: and more than all this, that you may preserve your own Person,\nfrom the Pitch and Feathers that are prepared for you.\n\nIn the first Place, we must tell you, that the _Pennsylvanians_\nare, _to a Man_, passionately fond of Freedom; the Birthright of\n_Americans_; and at all Events are determined to enjoy it.\n\nThat they sincerely believe, no Power on the Face of the Earth has a\nRight to tax them without their Consent.\n\nThat in their Opinion, the Tea in your Custody is designed by the\nMinistry to enforce such a Tax, which they will undoubtedly oppose; and\nin so doing, give you every possible Obstruction.\n\nWe are nominated to a very disagreeable, but necessary Service.--To our\nCare are committed all Offenders against the Rights of _America_; and\nhapless is he, whose evil Destiny has doomed him to suffer at our Hands.\n\nYou are sent out on a diabolical Service; and if you are so foolish and\nobstinate as to compleat your Voyage; by bringing your Ship to Anchor\nin this Port; you may run such a Gauntlet, as will induce you, in your\nlast Moments, most heartily to curse those who have made you the Dupe\nof their Avarice and Ambition.\n\nWhat think you Captain, of a Halter around your Neck--ten Gallons of\nliquid Tar decanted on your Pate--with the Feathers of a dozen wild\nGeese laid over that to enliven your Appearance?\n\nOnly think seriously of this--and fly to the Place from whence you\ncame--fly without Hesitation--without the Formality of a Protest--and\nabove all, Captain _Ayres_ let us advise you to fly without the wild\nGeese Feathers.\n\n Your Friends _to serve_\n THE COMMITTEE _as before subscribed_.\n _Philadelphia, Nov. 27, 1773._\n\n\nB. FRANKLIN'S CELEBRATED LETTER TO STRAHAN.\n\nAs a sequel to the foregoing notices, we give Dr. Franklin's celebrated\nletter, written in the actual heat of the first outbreak.\n\n Philadelphia, July 5, 1775.\n\nMr. STRAHAN,--You are a member of Parliament, and one of that majority\nwhich has doomed my country to destruction. You have begun to burn our\ntowns, and murder our people. Look upon your hands! They are stained\nwith the blood of your relations! You and I were long friends; you are\nnow my enemy, and\n\n I am, yours,\n B. FRANKLIN.\n\n\nHENRY II. STRIPT WHEN DEAD.\n\n1189. Immediately upon his death, those that were about him applied\ntheir market so busilie in catching and filching awaie things that\nlaie readie for them, that the king's corps laie naked a long time,\ntill a child covered the nether parts of his body with a short cloke,\nand then it seemed that his surname was fulfilled that he had from his\nchildhood, which was Shortmantell, being so called, because he was the\nfirst who brought short clokes out of Anjou into England.\n\n\nTRANSPLANTATION OF HAIR.\n\nThe Signor Dottore Domenico Nardo addressed a letter to the Academy of\nPadua, in 1826, on the subject of the growth of hair after death, and\neven after its separation from the body. The latter property had been\npreviously observed by Krafft. The Signor Nardo recounts the results of\nexperiments made on his own person in the transplantation of hair, and\nrelates, that by transplanting quickly a hair, with its root, from a\npore of his head, into a pore of his chest, easily to be accomplished\nby widening the pore somewhat with the point of a needle, introducing\nthe root with nicety, and exciting within the pore itself, by friction,\na slight degree of inflammation, the hair takes root, continues to\nvegetate, and grows; in due season changes colour, becomes white, and\nfalls.\n\n\nANCIENT CANNON RAISED FROM THE SEA.\n\nA fisherman of Calais some time since, drew up a cannon, of very\nancient form, from the bottom of the sea, by means of his nets. M. de\nRheims has since removed the rust from it, and on taking off the breech\nwas much surprised to find the piece still charged. Specimens of the\npowder have been taken, from which, of course, all the saltpetre has\ndisappeared after a submersion of three centuries. The ball was of\nlead, and was not oxidized to a depth greater than that of a line.\n\n\nCOFFEE-HOUSE ATTRACTIONS IN 1760.\n\nThe great attraction of Don Saltero's Coffee-house was its collection\nof rarities, a catalogue of which was published as a guide to the\nvisitors. It comprehends almost every description of curiosity, natural\nand artificial. \"Tigers' tusks; the Pope's candle; the skeleton of\na Guinea-pig; a fly-cap monkey; a piece of the true Cross; the Four\nEvangelists' heads cut on a cherry-stone; the King of Morocco's\ntobacco-pipe; Mary Queen of Scot's pincushion; Queen Elizabeth's\nprayer-book; a pair of Nun's stockings; Job's ears, which grew on a\ntree; a frog in a tobacco-stopper;\" and five hundred more odd relics!\nThe Don had a rival, as appears by \"A Catalogue of the Rarities to be\nseen at Adams's, at the Royal Swan, in Kingsland Road, leading from\nShoreditch Church, 1756.\" Mr. Adams exhibited, for the entertainment\nof the curious, \"Miss Jenny Cameron's shoes; Adam's eldest daughter's\nhat; the heart of the famous Bess Adams, that was hanged at Tyburn with\nLawyer Carr, January 18, 1736-7; Sir Walter Raleigh's tobacco-pipe;\nVicar of Bray's clogs; engine to shell green pease with; teeth that\ngrew in a fish's belly; Black Jack's ribs; the very comb that Abraham\ncombed his son Isaac and Jacob's head with; Wat Tyler's spurs; rope\nthat cured Captain Lowry of the head-ach, ear-ach, tooth-ach and\nbelly-ach; Adam's key of the fore and back door of the Garden of Eden,\n&c., &c.\" These are only a few out of five hundred others equally\nmarvellous.\n\n\nA WOMAN TAKES THE LIGHTED MATCH FROM A BOMB.\n\nDuring the siege of Gibraltar, in 1782, the Count d'Artois came\nto St. Roch, to visit the place and works. While his highness was\ninspecting the lines, in company with the Duke de Crillon, they both\nalighted with their suite, and all lay flat upon the ground, to avoid\nthe effects of a bomb that fell near a part of the barracks where a\nFrenchwoman had a canteen. This woman, who had two children in her arms\nat the time, rushed forth with them, and having seated herself, with\nthe utmost _sang-froid_, on the bomb-shell, she put out the match,\nthus extricating from danger all that were around her, many of whom\nwitnessed this courageous and devoted act. His highness rewarded this\nintrepid female by bestowing on her a pension of three francs a day,\nand engaged to promote her husband after the siege; while the Duke de\nCrillon, imitating the generous example of the prince, ensured to her\nlikewise a daily payment of five francs.\n\n\nTHE SUMMERS MAGNET, OR LOADSTONE.\n\n[Illustration: [++] Summers Magnet.]\n\nAmong the great naval officers of Elizabeth's reign must be ranked\nSir George Summers, the discoverer of the Bermudas, often called the\nSummers Islands from that circumstance. Here is a representation given\nof what the descendants of Sir George Summers call the \"Summers magnet,\nor loadstone.\" It is in the possession of Peter Franklin Bellamy, Esq.,\nsurgeon, second son of Dr. Bellamy, of Plymouth. The tradition in the\nfamily is that the admiral before going to sea used to touch his needle\nwith it. The stone is dark-, the precise geological formation\ndoubtful. This curious stone, with armature of iron, was probably an\nancient talisman.\n\n\nSWALLOWING LIZARDS.\n\nBertholin, the learned Swedish doctor, relates strange anecdotes of\nlizards, toads, and frogs; stating that a woman, thirty years of age,\nbeing thirsty, drank plentifully of water at a pond. At the end of a\nfew months, she experienced singular movements in her stomach, as if\nsomething were crawling up and down; and alarmed by the sensation,\nconsulted a medical man, who prescribed a dose of orvietan in a\ndecoction of fumitory. Shortly afterwards, the irritation of the\nstomach increasing, she vomited three toads and two young lizards,\nafter which, she became more at ease. In the spring following, however,\nher irritation of the stomach was renewed; and aloes and bezoar being\nadministered, she vomited three female frogs, followed the next day by\ntheir numerous progeny. In the month of January following, she vomited\nfive more living frogs, and in the course of seven years ejected as\nmany as eighty. Dr. Bertholin protests that he heard them croak in her\nstomach!\n\n\nIMMENSE SEA SERPENT.\n\nA species of sea-serpent was thrown on shore near Bombay in 1819. It\nwas about forty feet long, and must have weighed many tons. A violent\ngale of wind threw it high above the reach of ordinary tides, in which\nsituation it took nine months to rot; during which process travellers\nwere obliged to change the direction of the road for nearly a quarter\nof a mile, to avoid the offensive effluvia. It rotted so completely\nthat not a vestige of bone remained.\n\n\nTHE ROYAL TOUCH.\n\nFor many ages one of the regal prerogatives in this country was to\ntouch for the cure of _regius morbus_, or scrofula; a disease too\nwell known to need any description. At different periods hundreds of\npersons assembled from all parts of the country annually to receive\nthe royal interposition. Lists of the afflicted were published, to\nafford a criterion for determining as to its success; and from Edward\nthe Confessor to the reign of Queen Anne, its efficacy appears to have\nobtained a ready and general belief.\n\nThe ceremony was announced by public proclamations; one of which we\ncopy from \"The Newes,\" of the 18th of May, 1664. \"His Sacred Majesty\"\n(Charles II.) \"having declared it to be his royal will and purpose to\ncontinue the healing of his people for the Evil during the month of\nMay, and then to give over until Michaelmas next, I am commanded to\ngive notice thereof, that the people may not come up to town in the\ninterim, and lose their labour.\"\n\nAn extract from the \"Mercurius Politicus\" affords additional\ninformation. \"Saturday,\" says that paper, \"being appointed by His\nMajesty to touch such as were troubled with the Evil, a great company\nof poor afflicted creatures were met together, many brought in chairs\nand flaskets, and being appointed by His Majesty to repair to the\nbanqueting-house, His Majesty sat in a chair of state, where he stroked\nall that were brought unto him, and then put about each of their necks\na white ribbon, with an angel of gold on it. In this manner His Majesty\nstroked above six hundred; and such was his princely patience and\ntenderness to the poor afflicted creatures, that, though it took up\na very long time, His Majesty, who is never weary of well-doing, was\npleased to make inquiry whether there were any more who had not yet\nbeen touched. After prayers were ended, the Duke of Buckingham brought\na towel, and the Earl of Pembroke a basin and ewer, who, after they\nhad made obeisance to His Majesty, kneeled down, till His Majesty had\nwashed.\"\n\nThis sovereign is said to have touched nearly one hundred thousand\npatients.\n\nWith Queen Anne the practice was discontinued. But so late as the\n28th of February, 1712, little more than two years before her death,\nthe following proclamation appeared in the \"Gazette\":--\"It being Her\nMajesty's royal intention to touch for the Evil on Wednesday, the\n19th of March next, and so to continue weekly during Lent, it is Her\nMajesty's command that tickets be delivered the day before at the\noffice in Whitehall; and that all persons shall bring a certificate\nsigned by the Minister and Churchwardens of their respective parishes,\nthat they have never received the royal touch.\" Dr. Johnson, when\nan infant, was brought, with others, for this purpose; \"and when\nquestioned upon the subject, confessed he had a faint recollection of\nan old lady with something black about her head.\"\n\nA religious service, of which Dr. Heylin, Prebendary of Westminster, in\nhis \"Examen Historicum,\" has given us the particulars, accompanied the\nceremony; which, as a document of pious interest, we transcribe:--\"The\nfirst Gospel is the same as that on the Ascension-day, Mark xvi. 14,\nto the end. At the touching of every infirm person these words are\nrepeated: 'They shall lay their hands on the sick, and they shall\nrecover.' The second Gospel begins with the first of St. John, and ends\na these words: (John i. 14:) 'Full of grace and truth.' At the putting\nthe angel about their necks were repeated, 'That light was the true\nlight, which lighteth every man that cometh into the world.'\n\n\"'Lord, have mercy upon us.'\n\n\"'Christ have mercy upon us.'\n\n\"'Lord have mercy upon us. Our Father, &c.'\n\n\"'_Minister._--O Lord, save thy servants:'\n\n\"'_Response._--Which put their trust in thee.'\n\n\"'_M._--Send unto them help from above:'\n\n\"'_R._--And ever more defend them.'\n\n\"'_M._--Help us, O God, our Saviour!'\n\n\"'_R._--And for the glory of thy name sake deliver us: be merciful unto\nus, sinners, for thy name sake!'\n\n\"'_M._--O Lord, hear our prayer:'\n\n\"'_R._--And let our cry come unto thee.'\n\n\"'_The Collect._--Almighty God, the eternal health of all such as\nput their trust in thee, hear us, we beseech thee, on the behalf of\nthese thy servants, for whom we call for thy merciful help; that they\nreceiving health, may give thanks unto thee in thy holy Church, through\nJesus Christ our Lord! Amen.'\n\n\"'The peace of God,' &c.\"\n\n\nPEG TANKARDS.\n\nThe pegging, or marking the drinking cups, was introduced by St.\nDunstan, to check the intemperate habits of the times, by preventing\none man from taking a larger draught than his companions. But the\ndevice proved the means of increasing the evil it was intended to\nremedy; for, refining upon Dunstan's plan, the most abstemious were\nrequired to drink precisely to a peg or pin, whether they could soberly\ntake such a quantity of liquor or not. To the use of such cups may be\ntraced the origin of many of our popular phrases. When a person is much\nelated, we still say, \"He is in a merry pin;\" and, \"He is a peg too\nlow,\" when he is not in good spirits. On the same principle we talk of\n\"taking a man down a peg,\" when we would check forwardness.\n\n\nNORMAN CAPS.\n\nThere is nothing more amusing to the traveller on the continent, than\nto observe the extraordinary variety of those head-appendages, many\nof them heirlooms for generations in some families, all more or less\nprized according to the richness of materials employed upon them, and\nthe peculiarity of shape. There is no article of dress more important\nto the _Normande_, whatever may be her means, than the cap which so\njauntily and triumphantly asserts the dignity of the wearer. The\nwives of fermieres who can afford such luxuries as expensive lace and\ntrimmings, spend a little income in the decoration of their caps.\nMany cost upwards of three thousand francs for the materials and\nmanufacture; and these, as we have before observed, are handed from\nmother to daughter through successive years, and are highly prized.\n\n[Illustration: [++] Norman Caps.]\n\nIn the primitive villages of Normandy, on some holidays, it is a\npleasing sight to see the dense army of caps, with flaps fanning the\nair, and following the gesticulatory movements of their talkative and\nvolatile owners. When the weather is doubtful, the cap-wearers take\ncare to be provided with a red umbrella of a clumsy construction,\nremarkably heavy, and somewhat similar, perhaps, to the original with\nwhich Jonas Hanway braved the jeers of a London populace in first\nintroducing it.\n\n\nNORTH AMERICAN INDIAN WAR DESPATCH.\n\nThe following is a _facsimile_ of a gazette of a tribe of North\nAmerican Indians, who assisted the French forces in Canada, during the\nwar between France and England:--\n\n[Illustration: [++] Gazette of North American Indians.]\n\n_Explanation of the Gazette, giving an account of one of their\nexpeditions. The following divisions explain those on the plate, as\nreferred to by the numbers_:--\n\n1. Each of these figures represents the number ten. They all signify,\nthat 18 times 10, or 180 American Indians, took up the hatchet, or\ndeclared war, in favour of the French, which is represented by the\nhatchet placed over the arms of France.\n\n2. They departed from Montreal--represented by the bird just taking\nwing from the top of a mountain. The moon and the buck show the time to\nhave been in the first quarter of the buck-moon, answering to July.\n\n3. They went by water--signified by the canoe. The number of huts, such\nas they raise to pass the night in, shows they were 21 days on their\npassage.\n\n4. Then they came on shore, and travelled seven days by\nland--represented by the foot and the seven huts.\n\n5. When they arrived near the habitations of their enemies, at\nsunrise--shown by the sun being to the eastward of them, beginning,\nas they think, its daily course, there they lay in wait three\ndays--represented by the hand pointing, and the three huts.\n\n6. After which, they surprised their enemies, in number 12 times 10, or\n120. The man asleep shows how they surprised them, and the hole in the\ntop of the building is supposed to signify that they broke into some of\ntheir habitations in that manner.\n\n7. They killed with the club eleven of their enemies, and took five\nprisoners. The former represented by the club and the eleven heads, the\nlatter by the figures on the little pedestals.\n\n8. They lost nine of their own men in the action--represented by the\nnine heads within the bow, which is the emblem of honour among the\nAmericans, but had none taken prisoners--a circumstance they lay great\nweight on, shown by all the pedestals being empty.\n\n9. The heads of the arrows, pointing opposite ways, represent the\nbattle.\n\n10. The heads of the arrows all pointing the same way, signify the\nflight of the enemy.\n\n\nRECEIPTS FROM ALBERTUS MAGNUS.\n\n_If thou wylt make a Carbuckle stone, or a thyng shyning in the\nnyght._--Take verye many of the lyttle beastes shyninge by nyghte, and\nput them beaten smale in a bottel of glasse, and close it, and burye\nit in hoate horses doung, and let it tarye xv dayes, afterwarde thou\nshalte destyll water of them Peralembicum, which thou shalt put in a\nvessel of Christal or glasse. It giueth so great clearnesse, that euery\nman may reade and write in a darke place where it is. Some men make\nthis water of the gall of a snale, the gal of a wesel, the gall of a\nferet, and of a water dogge: they burie them in doung and destyll water\nout of them.\n\n_If thou wylt see that other men can not see._--Take the gall of a male\ncat, and the fat of a hen all whyte, and mixe them together, and anoint\nthy eyes, and thou shalt see it that others cannot see.\n\nIf the hart, eye, or brayne of a lapwyng or blacke plover be hanged\nvpon a mans necke it is profitable agaynste forgetfulnesse, and\nsharpeth mans vnderstanding.--\"_Albertus Magnus._\" _Black Letter: very\nold._\n\n\nADVERTISEMENT OF ROAST PIG IN 1726.\n\n\"On Tuesday next, being Shrove Tuesday, there will be a fine _hog\nbarbyqu'd_ whole, at the house of Peter Brett, at the Rising Sun, in\nIslington Road, with other diversions.--_Note._ It is the house where\nthe ox was roasted whole at Christmas last.\"\n\nA hog barbecu'd is a West Indian term, and means a hog roasted whole,\nstuffed with spice, and basted with Madeira wine. Oldfield, an eminent\nglutton of former days, gormandised away a fortune of fifteen hundred\npounds a-year. Pope thus alludes to him,--\n\n \"Oldfield, with more than harpy throat endu'd,\n Cries, 'Send me, O, gods, a whole hog _barbecu'd!_'\"\n\n\nDYING OF OLD AGE AT SEVENTEEN YEARS.\n\nMarch 19th, 1754, died, in Glamorganshire, of mere old age and a\ngradual decay of nature, at seventeen years and two months, Hopkins\nHopkins, the little Welchman lately shown in London. He never weighed\nmore than seventeen pounds, but for three years past no more than\ntwelve. The parents have still six children left, all of whom no way\ndiffer from other children, except one girl of twelve years of age, who\nweighs only eighteen pounds, and bears upon her most of the marks of\nold age, and in all respects resembles her brother when at that age.\n\n\n\"WE HAE BEEN.\"\n\nIn Ayrshire there is a tradition, that the family motto of De\nBruce--\"We have been,\" originated from a lady named Fullarton, married\nto a cadet of the family of Cassilis. They had been gained to favour\nEngland during the chivalrous achievements of Wallace, and still\ncontinued zealous partisans of Edward. Before Bruce avowed his purpose\nto emancipate his country, he came, disguised as a palmer, to acquaint\nhimself how far he could rely on aid from the people. A storm compelled\nhim, and a few faithful adherents, to take shelter on the coast of\nAyrshire. Extreme darkness, and the turbulence of the billows, deprived\nthem of all knowledge where they landed; and as, in those unhappy\ntimes, the appearance of a few strangers would create alarm, the chiefs\ndispersed in different directions. Bruce chanced to go into the house\nof Mr. Kennedy, where the servants treated him with great reverence.\nThe lady had gone to bed, and the prince wished they would not disturb\nher, but permit him to sit by the fire till day; however, one damsel\nhad given her immediate notice of the visitor. He was ushered into her\npresence. She eyed him with scrutinizing earnestness. \"We hae been--we\nhae been fause,\" said she, in the Scottish dialect, \"but a royal ee\ntakes me back to haly loyalty. I seid ye, mes royal de Bruce, I ken\nye weel. We hae been baith untrue to Scotland, but rest ye safe: and\nalbiet a' that's gane, Meg Fullarton wad dee in your cause.\"\n\n\nORIGIN OF THE PENNY POST.\n\nThe penny-post was devised in 1683, by one Mr. David Murray, an\nupholder in Paternoster Row. It soon became an object of attention to\nGovernment; but so low were its profits that one Dockwra, who succeeded\nMurray, had a pension of only L200 a year given him in lieu of it. This\noccurred in 1716.\n\n\nA RAFFLE IN 1725.\n\nMay 8. The following copy of an advertisement, in the _Newcastle\nCourant_ of this date, may be considered curious:--\"On Friday in the\nrace week, being the 28th of May, at the Assembly House, in Westgate,\nwill be raffled for, 12 fine Fans, the highest three guineas, the worst\n5s., at half a Crown per Ticket. Note: the lowest throw is to have the\nsecond best Fan, value L3, the other according to the height of the\nnumbers which shall be thrown. There will be an assembly after for\nthose who raffle.\"\n\n\nA VISIT TO THE RESIDENCE OF DR. JOHNSON, IN INNER TEMPLE LANE, LONDON.\n\nIn one of the dreary, old-fashioned houses leading from the arched\nentrance to the Temple, which almost every passenger through Temple\nBar must have remarked, whether he is a stranger, or a resident in the\nmetropolis, Dr. Johnson, who occupies one of the most distinguished\npositions in the literature of our country, resided for several years.\n\n[Illustration: Dr. Johnson's Residence in Inner Temple Lane.]\n\nIt was in this place that Dr. Johnson became acquainted with his future\nbiographer, Boswell, who thus describes their first meeting:--\n\n\"A few days afterwards I called on Davies, and asked him if he thought\nI might take the liberty of waiting on Mr. Johnson at his chambers\nin the Temple. He said I certainly might, and that Mr. Johnson would\ntake it as a compliment. His chambers were on the first floor of No.\n1, Inner Temple Lane, and I entered them with an impression given me\nby the Rev. Dr. Blair, of Edinburgh, who described his having found\nthe giant in his den. He received me very courteously; but it must be\nconfessed, that his apartment, and furniture, and morning dress, were\nsufficiently uncouth. His brown suit of clothes looked very rusty; he\nhad on a little, old, shrivelled, unpowdered wig, which was too small\nfor his head; his shirt neck and knees of his breeches were loose, his\nblack worsted stockings ill drawn up, and he had a pair of unbuckled\nshoes by way of slippers;--but all these slovenly particulars were\nforgotten the moment he began to talk.\"\n\nThe \"den\" in which the \"giant\" lived, the staircase leading to it, and\nindeed the whole appearance of the locality, has recently undergone\ndemolition, and its interesting features knocked down to the highest\nbidder, to be, let us hope, preserved in some museum or other place of\nsafety.\n\n[Illustration: Old Staircase in the Residence of Dr. Johnson.]\n\nDr. Johnson resided at various times in Holborn, the Strand, and other\nplaces, and died, as it is well known, in No. 8, Bolt Court, Fleet\nStreet, in 1784. His remains were placed in a grave under the statue\nof Shakspere, in Westminster Abbey, and near the resting-place of his\nfriend and companion, David Garrick.\n\n\nORIGIN OF THE STUFF BALL AT LINCOLN.\n\nDuring the want of employment in the manufactories in 1801, Mrs.\nChaplain, of Blankney, in Lincolnshire, formed a patriotic institution\nfor the encouragement of the local trade of the district. A ball was\ngiven at Lincoln for the benefit of the stuff manufactory, at which\nladies were admitted gratis, on their appearance in a stuff gown and\npetticoat, spun, wove, and finished within the county, and producing a\nticket signed by the weaver and dyer at Louth, one of which tickets\nwas delivered with every twelve yards of stuff. The gentlemen were\nrequired to appear without silk or cotton in their dress, stockings\nexcepted. The impulse thus given to trade, was of the most signal\nservice in relieving distress, and at the same time promoting habits of\nindustry.\n\n\nSTEVENS'S SPECIFIC.\n\nIn the reign of Charles II., Dr. Jonathan Goddard obtained 5,000_l._\nfor disclosing his secret for making a medicine, called \"_Guttae\nAnglicanae_.\" And in 1739, the Parliament of England voted 5,000_l._ to\nMrs. Stevens for a solvent for stone.\n\nThe celebrated David Hartley was very instrumental in procuring this\ngrant to Joanna Stevens. He obtained also a private subscription to the\namount of L1,356, published one hundred and fifty-five _successful_\ncases, and, by way of climax to the whole, after eating _two hundred\npounds weight_ of soap! David himself died of the stone.\n\n\nAN IMPOSTOR.\n\nFrom the Testament of Jerome Sharp, printed in 1786:--\"I entered,\"\nsays the narrator, \"with one of my friends, and found a man resembling\nan ourang-outang crouched upon a stool in the manner of a tailor. His\ncomplexion announced a distant climate, and his keeper stated that he\nfound him in the island of Molucca. His body was bare to the hips,\nhaving a chain round the waist, seven or eight feet long, which was\nfastened to a pillar, and permitted him to circulate out of the reach\nof the spectators. His looks and gesticulations were frightful. His\njaws never ceased snapping, except when sending forth discordant cries,\nwhich were said to be indicative of hunger. He swallowed flints when\nthrown to him, but preferred raw meat, which he rushed behind his\npillar to devour. He groaned fearfully during his repast, and continued\ngroaning until fully satiated. When unable to procure more meat, he\nwould swallow stones with frightful avidity; which, upon examination\nof those which he accidentally dropped, proved to be partly dissolved\nby the acrid quality of his saliva. In jumping about, the undigested\nstones were heard rattling in his stomach.\"\n\nThe men of science quickly set to work to account for these feats,\nso completely at variance with the laws of nature. Before they had\nhit upon a theory, the pretended Molucca savage was discovered to\nbe a peasant from the neighbourhood of Besancon, who chose to turn\nto account his natural deformities. When staining his face for the\npurpose, in the dread of hurting his eyes, he left the eyelids\nunstained, which completely puzzled the naturalists. By a clever\nsleight of hand, the raw meat was left behind the pillar, and cooked\nmeat substituted in its place. Some asserted his passion for eating\nbehind the pillar to be a proof of his savage origin; most polite\npersons, and more especially kings, being addicted to feeding in\npublic. The stones swallowed by the pretended savage were taken from\na vessel left purposely in the room full of them; small round stones,\nencrusted with plaster, which afterwards gave them the appearance of\nhaving been masticated in the mouth. Before the discovery of all this,\nthe impostor had contrived to reap a plentiful harvest.\n\n\nPERUVIAN BARK.\n\nIn 1693, the Emperor Kanghi (then in the thirty-second year of his\nreign, and fortieth of his age) had a malignant fever, which resisted\nthe remedies given by his physicians; the emperor recollected that\nTchang-tchin, (Father Gerbillon), and Pe-tsin, (Father Bouret) two\njesuit missionaries, had extolled to him a remedy for intermittents,\nbrought from Europe, and to which they had given the name of chin-yo\n(two Chinese words, which signify \"_divine remedies_;\") and he proposed\nto try it, but the physicians opposed it. The emperor, however, without\ntheir knowledge took it, and with good effect. Sometime afterwards,\nhe experienced afresh several fits of an intermittent, which, though\nslight, made him uneasy; this led him to proclaim through the city,\nthat any person possessed of a specific for this sort of fever, should\napply without delay at the palace, where patients might also apply to\nget cured. Some of the great officers of his household were charged\nto receive such remedies as might be offered, and to administer them\nto the patients. The Europeans, Tchang-tching, (Gerbillon) Hang-jo,\n(Father de Fontenay, jesuit) and Pe-tsin, (Bouret) presented themselves\namong others, with a certain quantity of quinquina, offered it to the\ngrandees, and instructed them in the manner of using it. The next\nday it was tried on several patients, who were kept in sight, and\nwere cured by it. The officers, or grandees who had been appointed\nto superintend the experiment, gave an account to the Emperor of the\nastonishing effect of the remedy, and the monarch decided instantly on\ntrying it himself, provided the hereditary prince gave his consent.\nThe prince, however, not only refused, but was angry with the grandees\nfor having spoken so favourably of a remedy, of which only one\nsuccessful trial had been made; at last, after much persuasion, the\nPrince reluctantly grants his consent, and the emperor takes the bark\nwithout hesitation, and permanently recovers. A house is given by the\nemperor to the Europeans, who had made known the remedy, and through\nthe means of Pe-tsin (Father Bouret) presents were conveyed to the King\nof France, accompanied with the information, that the Europeans (that\nis, the French jesuits) were in high favour.--_Histoire Generale de la\nChine, &c._ tome xi. p. 168, 4to. Paris, 1780.\n\n\nWHITE CATS.\n\nIn a number of \"Loudon Gardener's Magazine,\" it is stated that white\ncats with blue eyes are always deaf, of which extraordinary fact there\nis the following confirmation in the \"Magazine of Natural History,\" No.\n2, likewise conducted by Mr. Loudon:--Some years ago, a white cat of\nthe Persian kind (probably not a thorough-bred one), procured from Lord\nDudley's at Hindley, was kept in a family as a favourite. The animal\nwas a female, quite white, and perfectly deaf. She produced, at various\ntimes, many litters of kittens, of which, generally, some were quite\nwhite, others more or less mottled, tabby, &c. But the extraordinary\ncircumstance is, that of the offspring produced at one and the same\nbirth, such as, like the mother, were entirely white, were, like her,\ninvariably deaf; while those that had the least speck of colour on\ntheir fur, as invariably possessed the usual faculty of hearing.\n\n\nA WOMAN DEFENDS A FORT SINGLY.\n\nLord Kames in his \"Sketches of the History of Man,\" relates an\nextraordinary instance of presence of mind united with courage.\n\nSome Iroquois in the year 1690, attacked the fort de Vercheres, in\nCanada, which belonged to the French, and had approached silently,\nhoping to scale the palisade, when some musket-shot forced them to\nretire: on their advancing a second time they were again repulsed, in\nwonder and amazement that they could perceive no person, excepting\na woman who was seen everywhere. This was Madame de Vercheres, who\nconducted herself with as much resolution and courage as if supported\nby a numerous garrison. The idea of storming a place wholly undefended,\nexcept by women, occasioned the Iroquois to attack the fortress\nrepeatedly, but, after two days' siege, they found it necessary to\nretire, lest they should be intercepted in their retreat.\n\nTwo years afterwards, a party of the same nation so unexpectedly made\ntheir appearance before the same fort, that a girl of fourteen, the\ndaughter of the proprietor, had but just time to shut the gate. With\nthis young woman there was no person whatever except one soldier, but\nnot at all intimidated by her situation, she showed herself sometimes\nin one place, sometimes in another, frequently changing her dress,\nin order to give some appearance of a garrison, and always fired\nopportunely. In short, the faint-hearted Iroquois once more departed\nwithout success. Thus the presence of mind of this young girl was the\nmeans of saving the fort.\n\n\nINDENTURE OF A HORSE-RACE BETWIXT THE EARLS OF MORTON AND ABERCORN AND\nTHE LORD BOYDE.\n\nAs indicating the state of the English language amongst the nobility of\nScotland in 1621, the following is curious:--\n\n\"_Ane Indentour of ane Horse-raise betuix my Lords Mortoun, Abercorne,\nand Boyde._--The erle of Mortoun obleissis himselff to produce George\nRutherfuirdis Barb Naig: The erle of Abercorne obleissis him to produce\nhis gray Naig: My lord Boyd obleissis him to produce his bay horse;\nUpone the conditions following. Thay ar to run the first Thursday\nNovember nixtocum, thrie mett myleis of Cowper raise in Fyff. The\nwaidger to be for euery horse ten dowbill Anegellis. The foirmest horse\nto win the hail thretty. Ilk rydare to be aucht scottis stanewecht.\nAnd the pairtie not comperaud, or refuisand to consigne the waidger,\nsall undergo the foirfaltour of this sowme, and that money foirfaltit\nsalbe additt to the staik to be tane away be the wynner. Forder, we\ndeclair it to be lesum to ony gentilman to produce ane horse and the\nlyk waidger, and thay salbe welcum. Subscrybith with all our handis, at\nHammiltoune the fyfteine day off August 1621. MORTON, ABERCORNE, BOYDE.\"\n\n\nEARLY USE OF CHOCOLATE.\n\nAn advertisement in \"The Public Adviser,\" from Tuesday, June 16th, to\nTuesday, June 23d, 1657, informs us that \"in Bishopsgate-street, in\nQueen's-head-alley, at a Frenchman's House, is an excellent West India\ndrink, called _Chocolate_, to be sold, where you may have it ready at\nany time, and also unmade, at reasonable rates.\"\n\n\nMATTHEW BUCKINGER.\n\n[Illustration: [++] Matthew Buckinger.]\n\nOf all the imperfect beings brought into the world, few can challenge,\nfor mental and acquired endowments, any thing like a comparison to\nvie with this truly extraordinary little man. Matthew Buckinger was\na native of Nuremberg, in Germany, where he was born, June 2, 1674,\nwithout hands, feet, legs, or thighs; in short, he was little more\nthan the trunk of a man, saving two excrescences growing from the\nshoulder-blades, more resembling fins of a fish than arms of a man. He\nwas the last of nine children, by one father and mother, viz. eight\nsons and one daughter; after arriving at the age of maturity, from the\nsingularity of his case, and the extraordinary abilities he possessed,\nhe attracted the notice and attention of all persons, of whatever rank\nin life, to whom he was occasionally introduced.\n\nIt does not appear, by any account extant, that his parents exhibited\nhim at any time for the purposes of emolument, but that the whole of\nhis time must have been employed in study and practice, to attain the\nwonderful perfection he arrived at in drawing, and his performance on\nvarious musical instruments; he played the flute, bagpipe, dulcimer,\nand trumpet, not in the manner of general amateurs, but in the style of\na finished master. He likewise possessed great mechanical powers, and\nconceived the design of constructing machines to play on all sorts of\nmusical instruments.\n\nIf Nature played the niggard in one respect with him she amply repaid\nthe deficiency by endowments that those blessed with perfect limbs\ncould seldom achieve. He greatly distinguished himself by beautiful\nwriting, drawing coats of arms, sketches of portraits, history,\nlandscapes, &c., most of which were executed in Indian ink, with a pen,\nemulating in perfection the finest and most finished engraving. He was\nwell skilled in most games of chance, nor could the most experienced\ngamester or juggler obtain the least advantage at any tricks, or game,\nwith cards or dice.\n\nHe used to perform before company, to whom he was exhibited, various\ntricks with cups and balls, corn, and living birds; and could play\nat skittles and ninepins with great dexterity; shave himself with\nperfect ease, and do many other things equally surprising in a person\nso deficient, and mutilated by Nature. His writings and sketches of\nfigures, landscapes, &c., were by no means uncommon, though curious;\nit being customary, with most persons who went to see him, to purchase\nsomething or other of his performance; and as he was always employed\nin writing or drawing, he carried on a very successful trade, which,\ntogether with the money he obtained by exhibiting himself, enabled\nhim to support himself and family in a very genteel manner. The late\nMr. Herbert, of Cheshunt, in Hertfordshire, editor of \"Ames's History\nof Printing,\" had many curious specimens of Buckinger's writing\nand drawing, the most extraordinary of which was his own portrait,\nexquisitely done on vellum, in which he most ingeniously contrived to\ninsert, in the flowing curls of the wig, the 27th, 121st, 128th, 140th,\n149th, and the 150th Psalms, together with the Lord's Prayer, most\nbeautifully and fairly written. Mr. Isaac Herbert, son of the former,\nwhile carrying on the business of a bookseller in Pall-Mall, caused\nthis portrait to be engraved, for which he paid Mr. Harding fifty\nguineas.\n\nBuckinger was married four times, and had eleven children, viz., one\nby his first wife, three by his second, six by his third, and one by\nhis last. One of his wives was in the habit of treating him extremely\nill, frequently beating and other ways insulting him, which, for a\nlong time, he very patiently put up with; but once his anger was so\nmuch aroused, that he sprung upon her like a fury, got her down, and\nbuffeted her with his stumps within an inch of her life; nor would he\nsuffer her to arise until she promised amendment in future, which it\nseems she prudently adopted, through fear of another thrashing. Mr.\nBuckinger was but twenty-nine inches in height, and died in 1722.\n\n\nWONDERFUL PROVISION OF NATURE.\n\nThe insects that frequent the waters, require predaceous animals to\nkeep them within due limits, as well as those that inhabit the earth;\nand the water-spider (_Argyroneta aquatica_) is one of the most\nremarkable upon whom that office is devolved. To this end, her instinct\ninstructs her to fabricate a kind of diving-bell in the bosom of that\nelement. She usually selects still waters for this purpose. Her house\nis an oval cocoon, filled with air, and lined with silk, from which\nthreads issue in every direction, and are fastened to the surrounding\nplants. In this cocoon, which is open below, she watches for her prey,\nand even appears to pass the winter, when she closes the opening. It\nis most commonly, yet not always, under water; but its inhabitant has\nfilled it for her respiration, which enables her to live in it. She\nconveys the air to it in the following manner: she usually swims on her\nback, when her abdomen is enveloped in a bubble of air, and appears\nlike a globe of quicksilver. With this she enters her cocoon, and\ndisplacing an equal mass of water, again ascends for a second lading,\ntill she has sufficiently filled her house with it, so as to expel all\nwater. How these little animals can envelope their abdomen with an\nair-bubble, and retain it till they enter their cells, is still one\nof Nature's mysteries that has not been explained. It is a wonderful\nprovision, which enables an animal that breathes the atmospheric air,\nto fill her house with it under water, and by some secret art to clothe\nher body with air, as with a garment, which she can put off when it\nanswers her purpose. This is a kind of attraction and repulsion that\nmocks all inquiries.\n\n\nSTOMACH BRUSH.\n\nOne of the Court Physicians, in the reign of Charles II., invented an\ninstrument to cleanse the stomach, and wrote a pamphlet on it; and\nridiculous as a chylopoietic-scrubbing-brush may appear, it afterwards\ngot a place among surgical instruments, and is described as the\n_Excutor Ventriculi_, or cleanser of the stomach; but the moderns not\nhaving _stomach_ for it, have transferred it to the wine merchant, who\nmore appropriately applies it to the scouring the interior of bottles.\nHeister gives a minute description of it, and very gravely enters on\nthe mode and manner of using it: the patient is to drink a draught\nof warm water, or spirit of wine, that the mucus and foulness of the\nstomach may be washed off thereby: then, the brush being moistened in\nsome convenient liquor, is to be introduced into the oesophagus, and\nslowly protruded into the stomach, by twisting round its wire handle.\nWhen arrived in the stomach, it is to be drawn up and down, and through\nthe oesophagus, like the sucker in a syringe, till it be at last\nwholly extracted. Some recommend plentiful drinking in the operation,\nto be continued till no more foulness is discharged. But though this\ncontrivance is greatly extolled, and said to prolong life to a great\nage, especially if practiced once a week, month, or fortnight; yet,\nthere are very few (probably, because tried by very few) instances of\nits happy effects.\n\n\nPOPULAR AMUSEMENTS IN 1743.\n\nIn _Merrie England of the Olden Time_, we find the following copy of a\nhandbill announcing performances:--\n\nBy a company of English, French, and Germans, at Phillips's New Wells,\nnear the London Spa, Clerkenwell, 20th August, 1743.\n\n[Illustration: [++] Rope Dancing.]\n\nThis evening, and during the Summer Season, will be performed several\nnew exercises of Rope-dancing, Tumbling, Vaulting, Equilibres,\nLadder-dancing, and Balancing, by Madame Kerman, Sampson Rogetzi,\nMonsieur German, and Monsieur Dominique; with a new Grand Dance,\ncalled Apollo and Daphne, by Mr. Phillips, Mrs. Lebrune, and others;\nsinging by Mrs. Phillips and Mrs. Jackson; likewise the extraordinary\nperformance of Herr Von Eeckenberg, who imitates the lark, thrush,\nblackbird, goldfinch, canary-bird, flageolet, and German flute; a\nSailor's Dance by Mr. Phillips; and Monsieur Dominique flies through\na hogshead, and forces both heads out. To which will be added The\nHarlot's Progress. Harlequin by Mr. Phillips; Miss Kitty by Mrs.\nPhillips. Also, an exact representation of the late glorious victory\ngained over the French by the English at the battle of Dettingen, with\nthe taking of the White Household Standard by the Scots Greys, and\nblowing up the bridge, and destroying and drowning most part of the\nFrench army. To begin every evening at five o'clock. Every one will be\nadmitted for a pint of wine, as usual.\n\n\nDANCING ROOMS.\n\nDancing rooms were much frequented a century or so ago in London, which\nwas then pretty well supplied with this means of recreation. We find\nthat there were rare dancing doings at the original dancing room\n\n in the year\n at the _field_-end of King-Street, Bloomsbury, 1742\n Hickford's great room, Panton-Street, Haymarket, 1743\n Mitre Tavern, Charing-Cross, 1743\n Barber's Hall, 1745\n Richmond Assembly, 1745\n Lambeth Wells, 1747\n Duke's long room, Paternoster Row, 1748\n Large Assembly Room at the Two Green Lamps, near Exeter Change,\n (at the particular desire of Jubilee Dickey!) 1749\n The large room next door to the Hand and Slippers, Long-lane,\n West Smithfield, 1750\n Lambeth Wells, where a _Penny Wedding_, in the _Scotch_\n manner, was celebrated for the benefit of a young couple, 1752\n Old Queen's Head, in Cock-lane, Lambeth, 1755\n\nand at Mr. Bell's, at the sign of the Ship, in the Strand, where, in\n1755, a _Scotch_ Wedding was kept. The bride \"to be dressed without any\nlinen; all in ribbons, and green flowers, with Scotch masks. There will\nbe three bagpipes; a band of Scotch music, &c. &c. To begin precisely\nat two o'clock. Admission, two shillings and sixpence.\"\n\n\nORIGIN OF THE USE OF TOBACCO.\n\n\"Maister John Nicot, Counsellor to the Kyng, beeyng Embassadour for\nthe Kyng in Portugall, in the yeres of our Lorde, 1559, 60, 61, wente\none daye to see the Prysons of the Kyng of Portugall, and a gentleman\nbeeyng the keeper of the saide Prisons presented hym this hearbe, as\na strange Plant brought from Florida; the same Maister Nicot, hauyng\ncaused the saide hearbe to be set in his garden, where it grewe and\nmultiplied marveillously, was vpon a tyme aduertised, by one of his\nPages, that a young man, a kinne to that Page, made a saye of that\nhearbe bruised, both the herbe and the joice together upon an ulcer\nwhiche he had vpon his cheeke nere vnto his nose, coming of a _Noli\nme tangere_ whiche bega to take roote already at the gristles of the\nNose, wherewith he founde hym self marveillously eased. Therefore the\nsaid Maister Nicot caused the sicke yong man to be brought before hym,\ncausing the said herbe to be continued to the sore eight or tenne\ndaies, this saide _Noli me tangere_, was vtterly extinguished and\nhealed: and he had sent it, while this cure was a working to a certaine\nPhysition of the Kyng of Portugall of the moste fame, for to see the\nfurther workyng and effect of the said _Nicotiane_, and sending for the\nsame yong man at the end of tenne daies, the said Phisition seeyng the\nuisage of the said sicke yong man certified, that the saide _Noli me\ntangere_ was utterly extinguished, as in deede he never felt it since.\nWithin a while after, one of the Cookes of the said Embassadour hauyng\nalmost cut off his Thombe, with a great choppyng knife, the steward\nof the house of the saide gentleman ranne to the saide _Nicotiane_,\nand dresssed him there with fyve or sixe times, and so in the ende\nthereof he was healed: from that time forwarde this hearbe began to\nbee famous throughout all _Lisborne_, where the court of the Kyng of\nPortugall was at that presente, and the vertue of this saide hearbe was\npreached, and the people beganne to name it the Ambassadour's hearbe!\nWherefore there came certaine daies after, a gentleman of the country,\nFather to one of the Pages of the Ambassadour, who was troubled with\nan vlcer in his Legge, hauyng had the same two yeres, and demaunded\nof the saide Ambassadour for his hearbe, and vsing the same in suche\norder as is before written, at the ende of tenne or twelve daies he\nwas healed. From that time fourth the fame of that hearbe encreased in\nsuch sorte, that manye came from all places to have that same herbe.\nEmong all others there was a woman that had her face covered with a\nRingworme rooted, as though she had a Visour on her face, to whom the\nsaide L: Ambassadour caused the herbe to be given her, and told how she\nshould vse it, and at the ende of eight or tenne daies, this woman was\nthoroughleye healed, she came and shewed herself to the Ambassadour,\nshewing him of her healyng. After there came a captain to presente\nhis sonne, sick of the Kinges euill to the saide L: Ambassadour, for\nto send him into France, vnto whom there was saye made of the saide\nhearbe, whiche in fewe daies did beginne to shewe greate signes of\nhealing, and finally was altogether healed of the kinges euill. The\nL: Ambassadour seeing so great effectes proceeding of this hearbe,\nand hauing heard say that the Lady Montigny that was, dyed at Saint\nGermans, of an vlcer bredde in her breast, that did turn to a _Noli\nme tangere_, for which there could never be remedey bee founde, and\nlikewise that the Countesse of Ruffe, had sought for all the famous\nPhisitions of that Realme, for to heale her face, unto whom they\ncould give no remedy, he thought it good to communicate the same into\nFraunce, and did send it to Kyng Fraunces the seconde; and to the\nQueen Mother, and to many other Lords of the Courte with the maner of\ngovernyng the same: and how to applie it vnto the said diseases, even\nas he had found it by experience; and chiefly to the lorde of Jarnac\ngovernour of Rogell, with whom the saide Lorde Ambassadour had great\namitie for the service of the Kyng. The whiche Lorde of Jarnac, told\none daye at the Queenes Table, that he had caused the saide _Nicotiane_\nto be distilled, and caused the water to be dronke, mingled with\nwater _Euphrasie_, otherwise called eyebright, to one that was shorte\nbreathed, and was therewith healed.\"--_Joyfvll News ovt of the newe\nfound worlde, &c._, 1577.--_Black Letter._\n\n\nANCIENT INSTRUMENTS OF PUNISHMENT AND TORTURE IN THE TOWER OF LONDON.\n\nThere are few things among the valuable collection of antiquities\npreserved in the Tower of London, which excite so much interest as the\ngrim-looking objects forming the group figured in the accompanying\nengraving.\n\nWith the executioner's axe, that long list of unfortunates who have met\ntheir fate within the walls of the Tower, or on Tower Hill, since the\ntime of Henry VIII., have been beheaded. Among them may be enumerated\nQueen Anne Boleyn, whom Henry first presented to his people as their\nQueen while standing with her on the Tower Stairs, after she had been\nconveyed thither from Greenwich with every possible pomp. Crowds of\ngilded barges, with gay banners waving at their sterns, then lined the\nstream. The noblest of the land were in the young Queen's train or were\nwaiting to receive her. Loud rounds of cannon, and soft, merry strains,\nannounced her arrival; and the burly King stepped forward to kiss her\nin the sight of the assembled multitude. On the same day, three short\nyears afterwards, she was led forth to execution within the Tower\nwalls. The good Sir Thomas More and the chivalrous Earl of Surrey, Lady\nJane Grey and her young husband, the gallant Raleigh, and a host of\nothers, also perished by that sad symbol of the executioner's office.\n\nThe block is said to be of less ancient date, but is known to have been\nused at the execution of three Scotch lords--the unfortunate adherents\nof the Pretender--a little more than a century ago. On the top part of\nthe block, there are three distinct cuts, two of them very deep and\nparallel, and the other at an angle and less effective.\n\nThe horrible instrument of torture called the \"Scavenger's Daughter,\"\nwas, in the \"good old days,\" used as a means of extorting confession.\nThe head of the culprit was passed through the circular hole at the\ntop, and the arms through those below. The whole of this part of the\nmachine opens in somewhat the same manner as a pair of tongs, the upper\npart being fixed round the neck and arms, and the semicircular irons\nplaced on the legs. The body was then bent, and a strong iron bar was\npassed through the irons connected with the head and arms, and those\nin which the legs were placed. \"The culprit would then,\" as one of the\n\"Beefeaters\" who attends on visitors makes a point of observing, \"be\ndoubled up into very small compass, and made exceedingly uncomfortable.\"\n\nThe Bilboes need little explanation, being only a strong rod of iron,\nwith a nob at one end, on which are two moveable hoops, for the purpose\nof holding the legs; these being fixed, and a heavy iron padlock put on\nthe proper part--the wearer was said to be in a _Bilboe_. Instruments\nof this description were much used on board of ship for the purpose of\nsecuring prisoners of war.\n\nThe Iron Collar is a persuader of a formidable description, for it\nweighs upwards of 14 lbs., and is so made that it can be fixed on the\nneck and then locked. Such a necklace would, we think, be sufficiently\ninconvenient; but it is rendered still more uncomfortable by sundry\nprickles of iron knowingly placed.\n\nThe Thumb-screw, also preserved in the Tower, is a characteristic\nexample of a species of torture at one time much resorted to. The\nengraved example has been constructed so as to press both thumbs;\nnevertheless, it is a convenient little instrument, which might be\neasily carried about in the pocket. We have met with varieties of the\nthumb-screw in several collections--some for the accommodation of one\nthumb only. In the Museum of the Royal Antiquarian Society of Scotland\nthere are some thumb-screws which are said to have been used upon the\nCovenanters.\n\n[Illustration: 1. The Executioner's Axe. 2. The Block on which Lords\nBalmerino, Lovat, &c., were beheaded. 3. The Scavenger's Daughter. 4.\nSpanish Bilboes. 5. Massive Iron Collar for the Neck. 6. Thumb-Screw.]\n\nTimes have changed for the better since the \"Scavenger's Daughter,\" and\nthe other matters represented, were amongst the mildest of the methods\nused for the purposes of punishment and intimidation. The stocks,\nthe public whipping-posts, boilings, and burnings in Smithfield and\nelsewhere, the exhibition of dead men's heads over gateways, the boot,\nthe rack, the pillory, the practice of making men eat their own books\nin Cheapside, drawing on hurdles to the place of execution, and then\nhanging, drawing, and quartering, chopping off hands and ears, and\nother revolting punishments, have gone out of use, and it is gratifying\nto know that we are all the better for it.\n\n\nA BEAU BRUMMELL OF THE 17TH CENTURY.\n\n[Illustration: [++] Picture of an English Anticke.]\n\nThis very curious representation of a first-rate exquisite is copied\nfrom a very rare broadside, printed in 1646, and styled _The Picture\nof an English Anticke, with a List of his ridiculous Habits and apish\nGestures_. The engraving is a well-executed copperplate, and the\ndescription beneath is a brief recapitulation of his costume: from\nwhich we learn that he wears a tall hat, with a bunch of riband on\none side, and a feather on the other; his face spotted with patches;\ntwo love-locks, one on each side of his head, which hang upon his\nbosom, and are tied at the ends with silk riband in bows. His beard on\nthe upper lip encompassing his mouth; his band or collar edged with\nlace, and tied with band-strings, secured by a ring; a tight vest,\npartly open and short in the skirts, between which and his breeches\nhis shirt protruded. His cloak was carried over his arm. His breeches\nwere ornamented by \"many dozen of points at the knees, and above them,\non either side, were two great bunches of riband of several colours.\"\nHis legs were incased in \"boot-hose tops, tied about the middle of the\ncalf, as long as a pair of shirt-sleeves, double at the ends like a\nruff-band; the tops of his boots very large, fringed with lace, and\nturned down as low as his spurres, which gingled like the bells of a\nmorrice-dancer as he walked;\" the \"feet of his boots were two inches\ntoo long.\" In his right hand he carried a stick, which he \"played with\"\nas he \"straddled\" along the streets \"singing.\"\n\n\nPRAYING FOR REVENGE.\n\nIn North Wales, when a person supposes himself highly injured, it is\nnot uncommon for him to go to some church dedicated to a celebrated\nsaint, as Llan Elian in Anglesea, and Clynog in Carnarvonshire, and\nthere to offer his enemy. He kneels down on his bare knees in the\nchurch, and offering a piece of money to the saint, calls down curses\nand misfortunes upon the offender and his family for generations to\ncome; in the most firm belief that the imprecations will be fulfilled.\nSometimes they repair to a sacred well instead of a church.\n\n\nA FEMALE SAMPSON: FROM A HANDBILL.\n\nSeptember 4th, 1818, was shown at Bartholomew Fair, \"The strongest\nwoman in Europe, the celebrated French Female Hercules, Madame\nGobert, who will lift with her teeth a table five feet long and\nthree feet wide, with several persons seated upon it; also carry\nthirty-six weights, fifty-six pounds each, equal to 2,016 lbs., and\nwill disengage herself from them without any assistance; will carry\na barrel containing 340 bottles; also an anvil 400 lbs. weight, on\nwhich they will forge with four hammers at the time she supports it\non her stomach; she will also lift with her hair the same anvil,\nswing it from the ground, and suspend it in that position to the\nastonishment of every beholder; will take up a chair by the hind stave\nwith her teeth, and throw it over her head, ten feet from her body.\nHer travelling caravan, (weighing two tons,) on its road from Harwich\nto Leominster, owing to the neglect of the driver, and badness of the\nroad, sunk in the mud, nearly up to the box of the wheels; the two\nhorses being unable to extricate it she descended, and, with apparent\nease, disengaged the caravan from its situation, without any assistance\nwhatever.\"\n\n\nTREES THAT GROW SHIRTS.\n\n\"We saw on the of the Cerra Dnida,\" says M. Humboldt, \"shirt\ntrees, fifty feet high. The Indians cut off cylindrical pieces two feet\nin diameter, from which they peel the red and fibrous bark, without\nmaking any longitudinal incision. This bark affords them a sort of\ngarment which resembles a sack of a very coarse texture, and without\na seam. The upper opening serves for the head, and two lateral holes\nare cut to admit the arms. The natives wear these shirts of Marina\nin the rainy season; they have the form of the ponchos and manos of\ncotton which are so common in New Grenada, at Quito, and in Peru. As in\nthis climate the riches and beneficence of nature are regarded as the\nprimary causes of the indolence of the inhabitants, the missionaries\ndo not fail to say in showing the shirts of Marina, 'in the forests of\nOroonoko, garments are found ready made upon the trees.'\"\n\n\nA FEMALE VENTRILOQUIST.\n\nA female ventriloquist, named Barbara Jacobi, narrowly escaped being\nburnt at the stake in 1685, at Haarlem, where she was an inmate of\nthe public Hospital. The curious daily resorted thither to hear her\nhold & dialogue with an imaginary personage with whom she conversed\nas if concealed behind the curtains of her bed. This individual, whom\nshe called Joachim, and to whom she addressed a thousand ludicrous\nquestions, which he answered in the same familiar strain, was for some\ntime supposed to be a confederate. But when the bystanders attempted to\nsearch for him behind the curtains, his voice instantly reproached them\nwith their curiosity from the opposite corner of the room. As Barbara\nJacobi had contrived to make herself familiar with all the gossip of\nthe city of Haarlem, the revelations of the pretended familiar were\nsuch as to cause considerable embarrassment to those who beset her with\nimpertinent questions.\n\n\nCALMUC OPINION OF LIGHTNING.\n\nThe Calmucs hold the lightning to be the fire spit out of the mouth of\na dragon, ridden and scourged by evil Daemons, and the thunder they make\nto be his roarings.\n\n\nTHE HEADING OF THE EXPIRING PENNSYLVANIA JOURNAL.\n\nJournalism has had its trials and difficulties in England as well as in\nAmerica; but we do not remember to have ever seen a more quaint last\nNumber, than the subjoined _facsimile_ exhibits:--\n\n[Illustration:\n\n The TIMES are\n Dreadful\n Dismal,\n Doleful\n Dolorous, and\n Dollar-less.\n\n An Emblem of the Effect\n of the Stamp\n Of the Fatal Stamp\n\n Adieu Adieu to the LIBERTY OF THE PRESS\n\n Thursday, October 31. 1765\n\n NUMB. 1195\n\n THE\n\n PENNSYLVANIA JOURNAL;\n\n AND\n\n WEEKLY ADVERTISER.\n\n EXPIRING: In Hopes of a Resurrection to LIFE again.\n\n I am sorry to be obliged to acquaint my Readers, that as The\n STAMP-ACT, is fear'd to be obligatory upon us after the _First of\n November_ ensuing, (the _fatal To-morrow_) the Publisher of this\n Paper unable to bear the Burthen, has thought it expedient TO STOP\n awhile, in order to deliberate, whether any Methods can be found to\n elude the Chains forged for us, and escape the insupportable Slavery;\n which it is hoped, from the last Representations now made against the\n Act, may be effected. Mean while, I must earnestly Request every\n Individual of my Subscribers, many of whom have been long behind Hand,\n that they would immediately Discharge their respective Arrers, that I\n may be able, not only to support myself during the Interval, but be\n better prepared to proceed again with this Paper, whenever an opening\n for that Purpose appears, which I hope will be soon.\n\n WILLIAM BRADFORD.]\n\n\nNOSTRUMS.\n\nUnsuccessful gamesters used formerly to make a knot in their linen, of\nlate years they have contented themselves with changing their chair\nas a remedy against ill-luck. As a security against cowardice, it was\nonce only necessary to wear a pin plucked from the winding sheet of a\ncorpse. To insure a prosperous accouchement to your wife, you had but\nto tie her girdle to a bell and ring it three times. To get rid of\nwarts, you were to fold up in a rag as many peas as you had warts, and\nthrow them upon the high road; when the unlucky person who picked them\nup became your substitute. In the present day, to cure a toothache, you\ngo to your dentist. In the olden time you would have solicited alms in\nhonour of St. Lawrence, and been relieved without cost or pain.\n\n\nPRECOCIOUS CHILDREN.\n\nBaillet mentions one hundred and sixty-three children endowed with\nextraordinary talents, among whom few arrived at an advanced age. The\ntwo sons of Quintilian, so vaunted by their father, did not reach their\ntenth year. Hermogenes, who, at the age of fifteen, taught rhetoric to\nMarcus Aurelius, who triumphed over the most celebrated rhetoricians of\nGreece, did not die, but at twenty-four, lost his faculties, and forgot\nall he had previously acquired. Pica di Mirandola died at thirty-two;\nJohannes Secundus at twenty-five; having at the age of fifteen composed\nadmirable Greek and Latin verses, and become profoundly versed in\njurisprudence and letters. Pascal, whose genius developed itself at ten\nyears old, did not attain the third of a century.\n\nIn 1791, a child was born at Lubeck, named Henri Heinekem, whose\nprecocity was miraculous. At ten months of age, he spoke distinctly;\nat twelve, learnt the Pentateuch by rote, and at fourteen months, was\nperfectly acquainted with the Old and New Testaments. At two years\nof age, he was as familiar with Ancient History as the most erudite\nauthors of antiquity. Sanson and Danville only could compete with him\nin geographical knowledge; Cicero would have thought him an \"alter\nego,\" on hearing him converse in Latin; and in modern languages he was\nequally proficient. This wonderful child was unfortunately carried off\nin his fourth year. According to a popular proverb--\"the sword wore out\nthe sheath.\"\n\n\nEFFECT OF MUSIC ON A PIGEON.\n\nBingley gives a singular anecdote of the effect of music on a pigeon,\nas related by John Lockman, in some reflections concerning operas,\nprefixed to his musical drama of Rosalinda. He was staying at a\nfriend's house, whose daughter was a fine performer on the harpsichord,\nand observed a pigeon, which, whenever the young lady played the song\nof \"Speri-si,\" in Handel's opera of Admetus (and this only), would\ndescend from an adjacent dove-house to the room-window where she sat,\nand listen to it apparently with the most pleasing emotions; and when\nthe song was finished it always returned immediately to the dove-house.\n\n\nPOWER OF FASCINATION IN SNAKES.\n\nSome animals are held in universal dread by others, and not the least\nterrible is the effect produced by the rattle-snake. Mr. Pennant\nsays, that this snake will frequently lie at the bottom of a tree,\non which a squirrel is seated. He fixes his eyes on the animal, and\nfrom that moment it cannot escape: it begins a doleful outcry, which\nis so well known that a passer by, on hearing it, immediately knows\nthat a snake is present. The squirrel runs up the tree a little way,\ncomes down again, then goes up and afterwards comes still lower. The\nsnake continues at the bottom of the tree, with his eyes fixed on the\nsquirrel, and his attention is so entirely taken up, that a person\naccidentally approaching may make a considerable noise, without so\nmuch as the snake's turning about. The squirrel comes lower, and at\nlast leaps down to the snake, whose mouth is already distended for\nits reception. Le Vaillant confirms this fascinating terror, by a\nscene he witnessed. He saw on the branch of a tree a species of shrike\ntrembling as if in convulsions, and at the distance of nearly four\nfeet, on another branch, a large species of snake, that was lying with\noutstretched neck and fiery eyes, gazing steadily at the poor animal.\nThe agony of the bird was so great that it was deprived of the power of\nmoving away, and when one of the party killed the snake, it was found\ndead upon the spot--and that entirely from fear--for, on examination,\nit appeared not to have received the slightest wound. The same\ntraveller adds, that a short time afterwards he observed a small mouse\nin similar agonizing convulsions, about two yards from a snake, whose\neyes were intently fixed upon it; and on frightening away the reptile,\nand taking up the mouse, it expired in his hand.\n\n\nSECOND SIGHT.\n\nAbout the year 1725, the marvellous history of a Portuguese woman\nset the whole world of science into confusion, as will be found by\nreferring to the \"Mercure de France.\" This female was said to possess\nthe gift of discovering treasures. Without any other aid than the keen\npenetration of her eyes, she was able to distinguish the different\nstrata of earth, and pronounce unerringly upon the utmost distances at\na single glance. Her eye penetrated through every substance, even the\nhuman body; and she could discern the mechanism, and circulation of\nall animal fluids, and detect latent diseases; although less skilful\nthan the animal magnetisers, she did not affect to point out infallible\nremedies. Ladies could learn from her the sex of their forthcoming\nprogeny.\n\nThe King of Portugal, greatly at a loss for water in his newly built\npalace, consulted her; and after a glance at the spot, she pointed out\nan abundant spring, upon which his Majesty rewarded her with a pension,\nthe order of Christ, and a patent of nobility.\n\nIn the exercise of her miraculous powers, certain preliminaries were\nindispensable. She was obliged to observe a rigid fast; indigestion, or\nthe most trifling derangement of the stomach, suspending the marvellous\npowers of her visual organs.\n\nThe men of science of the day were of course confounded by such\nprodigies. But instead of questioning the woman, they consulted the\nworks of their predecessors; not forgetting the inevitable Aristotle.\nBy dint of much research, they found a letter from Huygens asserting\nthat there was a prisoner of war at Antwerp, who could see through\nstuffs of the thickest texture provided they were not red. The\nwonderful man was cited in confirmation of the wonderful woman, and\n_vice versa_.\n\n\nCHARACTER INDICATED BY THE EARS.\n\nAccording to Aristotle, large ears are indicative of imbecility; while\nsmall ones announce madness. Ears which are flat, point out the rustic\nand brutal man. Those of the fairest promise, are firm and of middling\nsize. Happy the man who boasts of square ears; a sure indication of\nsublimity of soul and purity of life. Such, according to Suetonius,\nwere the ears of the Emperor Augustus.\n\n\nGROANING BOARDS.\n\nGroaning boards were the wonder in London in 1682. An elm plank was\nexhibited to the king, which, being touched by a hot iron, invariably\nproduced a sound resembling deep groans. At the Bowman Tavern, in\nDrury Lane, the mantel-piece did the same so well that it was supposed\nto be part of the same elm-tree; and the dresser at the Queen's Arm\nTavern, St. Martin le Grand, was found to possess the same quality.\nStrange times when such things were deemed wonderful; even to meriting\nexhibition before the monarch.\n\n\nANCIENT PLOUGHING AND THRESHING.\n\nThe ancient plough was light, the draught comparatively easy; but then\nthe very lightness required that the ploughman should lean upon it with\nhis whole weight, or else it would glide over the soil without making\na single furrow. \"Unless,\" said Pliny, \"the ploughman stoop forward,\nto press down the plough, as well as to conduct it, truly it will turn\naside.\"\n\n[Illustration: Ancient Mode of Ploughing.]\n\nOxen were anciently employed in threshing corn, and the same custom is\nstill retained in Egypt and the east. This operation is effected by\ntrampling upon the sheaves, and by dragging a clumsy machine, furnished\nwith three rollers that turn on their axles. A wooden chair is attached\nto the machine, and on this a driver seats himself, urging his oxen\nbackwards and forwards among the sheaves, which have previously been\nthrown into a heap of about eight feet wide and two in height. The\ngrain thus beaten out, is collected in an open place, and shaken\nagainst the wind by an attendant, with a small shovel, or, as it is\ntermed, a winnowing fan, which disperses the chaff and leaves the grain\nuninjured:--\n\n \"Thus, with autumnal harvests covered o'er,\n And thick bestrewn, lies Ceres' sacred floor;\n While round and round, with never-wearied pain,\n The trampling steers beat out th' unnumber'd grain.\"\n\n HOMER.\n\nHorace further tells us, that the threshing floor was mostly a smooth\nspace, surrounded with mud walls, having a barn or garner on one side;\noccasionally an open field, outside the walls, was selected for this\npurpose, yet uniformly before the town or city gates. Such was the void\nplace wherein the king of Israel, and Jehoshaphat, king of Judah, sat\neach of them on his throne, clothed in his robes, at the entering in of\nthe gate of Samaria, and all the prophets prophesied before them. In\nthe marginal reading we are informed, that this void space was no other\nthan a threshing floor; and truly the area was well adapted for such an\nassemblage, being equally suited to accommodate the two kings and their\nattendants, and to separate them from the populace.\n\n[Illustration: Oxen Threshing Corn.]\n\nEastern ploughshares were of a lighter make than ours, and those who\nnotice the shortness and substance of ancient weapons, among such as\nare preserved in museums, will understand how readily they might be\napplied to agricultural uses.\n\n\nFROST FAIRS.\n\nIn 1788-9, the Thames was completely frozen over below London-bridge.\nBooths were erected on the ice; and puppet-shows, wild beasts,\nbear-baiting, turnabouts, pigs and sheep roasted, exhibited the\nvarious amusements of Bartholomew Fair multiplied and improved. From\nPutney-bridge down to Redriff was one continued scene of jollity during\nthis seven weeks' saturnalia. The last frost fair was celebrated\nin the year 1814. The frost commenced on 27th December, 1813, and\ncontinued to the 5th February, 1814. There was a grand walk, or mall,\nfrom Blackfriars-bridge to London-bridge, that was appropriately\nnamed _The City Road_, and lined on each side with booths of all\ndescriptions. Several printing-presses were erected, and at one of\nthese an orange- standard was hoisted, with \"_Orange Boven_\"\nprinted in large characters. There were E O and Rouge et Noir tables,\ntee-totums, and skittles; concerts of rough music, viz. salt-boxes\nand rolling-pins, gridirons and tongs, horns, and marrow-bones and\ncleavers. The carousing booths were filled with merry parties, some\ndancing to the sound of the fiddle, others sitting round blazing fires\nsmoking and drinking. A printer's devil bawled out to the spectators,\n\"Now is your time, ladies and gentlemen,--now is your time to support\nthe freedom of the press! Can the press enjoy greater liberty? Here you\nfind it working in the middle of the Thames!\"\n\n\nMAGIC RAIN STONE.\n\nThe Indian magi, who are to invoke Yo He Wah, and mediate with the\nsupreme holy fire that he may give seasonable rains, have a transparent\nstone of supposed great power in assisting to bring down the rain, when\nit is put in a basin of water, by a reputed divine virtue, impressed on\none of the like sort, in time of old, which communicates it circularly.\nThis stone would suffer a great decay, they assert, were it even seen\nby their own laity; but if by foreigners, it would be utterly despoiled\nof its divine communicative power.\n\n\nTHE BOMBARDIER BEETLE.\n\nThe bombardier beetle (_Carabus crepitans_) when touched produces a\nnoise resembling the discharge of a musket in miniature, during which\na blue smoke may be seen to proceed from its extremity. Rolander says\nthat it can give twenty discharges successively. A bladder placed near\nits posterior extremity, is the arsenal that contains its store. This\nis its chief defence against its enemies; and the vapour or liquid\nthat proceeds from it is of so pungent a nature, that if it happens to\nbe discharged into the eyes, it makes them smart as though brandy had\nbeen thrown into them. The principal enemy of the bombardier is another\ninsect of the same tribe, but three or four times its size. When\npursued and fatigued it has recourse to this stratagem; it lies down in\nthe path of its enemy, who advances with open mouth to seize it; but on\nthe discharge of the artillery, this suddenly draws back, and remains\nfor a while confused, during which the bombardier conceals itself in\nsome neighbouring crevice, but if not lucky enough to find one, the\nother returns to the attack, takes the insect by the head, and bears it\noff.\n\n\nTHE PILLORY FOR EATING FLESH IN LENT.\n\nEven in this kingdom, so late as the Reformation, eating flesh in\nLent was rewarded with the pillory. An instance of this occurs in\nthe \"Patriot King,\" the particulars of which, quoted in \"Clavis\nCalendaria,\" are somewhat amusing. Thomas Freburn's wife, of\nPaternoster-row, London, having expressed a particular inclination\nfor pig, one was procured, ready for the spit; but the butter-woman\nwho provided it, squeamish as to the propriety of what she had done,\ncarried a foot of it to the Dean of Canterbury. The Dean was at\ndinner, and one of his guests was Freburn's landlord, and Garter King\nat Arms, who sent to know if any of his family were ill, that he ate\nflesh in Lent. 'All well,' quoth Freburn, (perhaps too much of a\nDissenter for the times,) 'only my wife longs for pig.' His landlord\nsends for the Bishop of London's apparitor, and orders him to take\nFreburn and his pig before Stocksly, the Bishop, who sent them both\nto Judge Cholmley; but he not being at home, they were again brought\nback to the Bishop, who committed them to the Compter. Next day, being\nSaturday, Freburn was carried before the Lord Mayor, who sentenced\nhim to stand in the pillory on the Monday following, with one half\nof the pig on one shoulder, and the other half on the other. Through\nCromwell's intercession, the poor man at last gained his liberty by a\nbond of twenty pounds for his appearance. The mischief-making pig was,\nby the order of the Bishop, buried in Finsbury-field, by the hand of\nhis Lordship's apparitor; but Freburn was turned out of his house, and\ncould not get another in four years. Hence we may infer his ruin.\n\n\nHUGE CANNON AT THE SIEGE OF CONSTANTINOPLE.\n\nIn 1432, several kinds of artillery are mentioned, cannons, bombards,\nvulgaires, coulverins. The vulgaires were ordinary artillery. In the\nyear 1460, James II. of Scotland was killed by the accidental bursting\nof a cannon. The artillery of the Turks, in the year 1453, surpassed\nwhatever had yet appeared in the world. A stupendous piece of ordnance\nwas made by them; its bore was twelve palms, and the stone bullet\nweighed about 600 lbs.; it was brought with great difficulty before\nConstantinople, and was flanked by two almost of equal magnitude:\nfourteen batteries were brought to bear against the place, mounting 130\nguns; the great cannon could not be loaded and fired more than seven\ntimes in one day. Mines were adopted by the Turks, and counter-mines\nby the Christians. At this siege, which was in 1453, ancient and\nmodern artillery were both used. Cannons, intermingled with machines\nfor casting stones and darts, and the battering-ram was directed\nagainst the walls. The fate of Constantinople could no longer be\naverted: the diminutive garrison was exhausted by a double attack; the\nfortifications were dismantled on all sides by the Ottoman cannon;\na spirit of discord impaired the Christian strength. After a siege\nof fifty-three days, Constantinople, which had defied the power of\nChosroes, the Chagan, and the Caliphs, was subdued by the arms of\nMahomet II.\n\n\nA MAN IN A VAULT ELEVEN DAYS.\n\n_St. Benedict Fink._--\"1673, April 23, was buried Mr Thomas Sharrow,\nCloth-worker, late Churchwarden of this parish, killed by an accidental\nfall into a vault, in London Wall, Amen Corner, by Paternoster Row,\nand was supposed had lain there eleven days and nights before any\none could tell where he was, _Let all that read this take heed of\ndrink._\"--Truly, a quaint warning!\n\n\nBLIND GRANNY.\n\n[Illustration: [++] Blind Granny.]\n\nThis miserable, wretched, drunken object, who was blind of one eye,\nused to annoy the passengers in the streets of London, while sober,\nwith licking her blind eye with her tongue, which was of a most\nenormous length, and thickness; indeed, it was of such a prodigious\nsize, that her mouth could not contain it, and she could never close\nher lips, or to use a common expression, keep her tongue within\nher teeth. This wonderful feat of washing her eye with her tongue\nwas exhibited with a view of obtaining money from such as crowded\naround her, and no sooner had she obtained sufficient means, but\nshe hastened to the first convenient liquor-shop, to indulge her\npropensity in copious libations, and when properly inspired, would\nrush into the streets with all the gestures of a frantic maniac, and\nroll and dance about, until she became a little sobered, which was\nsometimes accelerated by the salutary application of a pail of water,\ngratuitously bestowed upon her by persons whose doorway she had taken\npossession of, as shelter from the persecuting tormentings of boys and\ngirls who generally followed her.\n\n\nANCIENT FEMALE COSTUME.\n\n[Illustration: [++] costume of a female of the higher classes.]\n\nA good specimen of the costume of a female of the higher classes is\nhere given, from an effigy of a lady of the Ryther family, in Ryther\nchurch, Yorkshire, engraved in Hollis's _Monumental Effigies_. She\nwears a wimple, covering the neck and encircling the head, the hair\nof which is gathered in plaits at the sides, and covered with a\nkerchief, which falls upon the shoulders, and is secured by a fillet\npassing over the forehead. The sleeves of the gown hang midway from\nthe elbow and the wrist, and display the tight sleeve with its rows of\nbuttons beneath. The mantle is fastened by a band of ribbon, secured\nby ornamental studs. The lower part of the dress consists of the wide\ngown, lying in folds, and completely concealing the feet, which have\nbeen omitted, in order to display the upper part of this interesting\neffigy to greater advantage.\n\n\nCHILCOTT, THE GIANT.\n\n1815. Died at Trenaw, in Cornwall, a person known by the appellation\nof Giant Chilcott. He measured at the breast six feet nine inches, and\nweighed four hundred and sixty pounds. One of his stockings held six\ngallons of wheat.\n\n\nDR. LETTSOM'S REASONS FOR DISMISSING A SERVANT.\n\nThe Doctor was in the practice of carrying the produce of his fees\ncarelessly in his coat-pocket. His footman being aware of this, used to\nmake free with a guinea occasionally, while it hung up in the passage.\nThe Doctor, having repeatedly missed his gold, was suspicious of the\nfootman, and took an opportunity of watching him. He succeeded in\nthe detection, and, without even noticing it to the other servants,\ncalled him into his study, and coolly said to him, \"John, art in want\nof money?\" \"No;\" replied John. \"Oh! then, why didst thou make so free\nwith my pocket? And since thou didst not want money, and hast told me a\nlie, I must part with thee. Now, say what situation thou wouldst like\nabroad, and I will obtain it for thee; for I cannot keep thee; I cannot\nrecommend thee; therefore thou must go.\" Suffice it to say, the Doctor\nprocured John a situation, and he went abroad.\n\n\nHANDBILL FROM PECKHAM FAIR IN 1726.\n\nOur ancestors just 133 years ago had but limited opportunities for\ngratifying a taste for Natural History if we may judge from the supply\nof animals deemed sufficient to attract attention in 1726:--\n\n\"_Geo. I. R._\n\n\"To the lovers of living curiosities. To be seen during the time of\n_Peckham Fair_, a Grand Collection of Living Wild Beasts and Birds,\nlately arrived from the remotest parts of the World.\n\n\"1. The _Pellican_ that suckles her young with her heart's blood, from\nEgypt.\n\n\"2. The Noble _Vultur Cock_, brought from _Archangell_, having the\nfinest tallons of any bird that seeks his prey; the fore part of his\nhead is covered with hair, the second part resembles the wool of a\nBlack; below that is a white ring, having a Ruff, that he cloaks his\nhead with at night.\n\n\"3. An _Eagle of the Sun_, that takes the loftiest flight of any bird\nthat flies. There is no bird but this that can fly to the face of the\nSun with a naked eye.\n\n\"4. A curious Beast, bred from a _Lioness_, like a foreign _Wild Cat_.\n\n\"5. The _He-Panther_, from Turkey, allowed by the curious to be one of\nthe greatest rarities ever seen in _England_, on which are thousands of\nspots, and not two of a likeness.\n\n\"6 & 7. The two fierce and surprising _Hyaenas_, Male and Female, from\nthe River _Gambia_. These Creatures imitate the human voice, and so\ndecoy the s out of their huts and plantations to devour them.\nThey have a mane like a horse, and two joints in their hinder leg more\nthan any other creature. It is remarkable that all other beasts are to\nbe tamed, but _Hyaenas_ they are not.\n\n\"8. An _Ethiopian Toho Savage_, having all the actions of the human\nspecies, which (when at its full growth) will be upwards of five feet\nhigh.\n\n\"Also several other surprising Creatures of different sorts. To be seen\nfrom 9 in the morning till 9 at night, till they are sold. Also, all\nmanner of curiosities of different sorts, are bought and sold at the\nabove place by John Bennett.\"\n\n\nSOMNAMBULISM.\n\nSome years ago a Hampshire Baronet was nearly driven to distraction\nby the fact that, every night, he went to bed in a shirt, and every\nmorning awoke naked, without the smallest trace of the missing garment\nbeing discovered.\n\nHundreds of shirts disappeared in this manner; and as there was no fire\nin his room, it was impossible to account for the mystery. The servants\nbelieved their master to be mad; and even he began to fancy himself\nbewitched. In this conjuncture, he implored an intimate friend to\nsleep in the room with him; and ascertain by what manner of mysterious\nmidnight visitant his garment was so strangely removed. The friend,\naccordingly, took up his station in the haunted chamber; and lo! as the\nclock struck one, the unfortunate Baronet, who had previously given\naudible intimation of being fast asleep, rose from his bed, rekindled\nwith a match the candle which had been extinguished, deliberately\nopened the door, and quitted the room. His astonished friend followed:\nsaw him open in succession a variety of doors, pass along several\npassages, traverse an open court, and eventually reach the stable-yard;\nwhere he divested himself of his shirt, and disposed of it in an old\ndung-heap, into which he thrust it by means of a pitchfork. Having\nfinished this extraordinary operation, without taking the smallest heed\nof his friend who stood looking on, and plainly saw that he was walking\nin his sleep, he returned to the house, carefully reclosed the doors,\nre-extinguished the light, and returned to bed; where the following\nmorning he awoke as usual, stripped of his shirt!\n\nThe astonished eye-witness of this extraordinary scene, instead of\napprising the sleep-walker of what had occurred, insisted that the\nfollowing night, a companion should sit up with him; choosing to have\nadditional testimony to the truth of the statement he was about to\nmake; and the same singular events were renewed, without the slightest\nchange or deviation. The two witnesses, accordingly, divulged all they\nhad seen to the Baronet; who, though at first incredulous, became of\ncourse convinced, when, on proceeding to the stable-yard, several\ndozens of shirts were discovered; though it was surmised that as many\nmore had been previously removed by one of the helpers, who probably\nlooked upon the hoard as stolen goods concealed by some thief.\n\n\nKILLED BY EATING MUTTON AND PUDDING.\n\n_Teddington._--\"James Parsons, who had often eat a shoulder of mutton\nor a peck of hasty pudding, at a time, which caused his death, buried\nMarch 7, 1743-4, aged 36.\"\n\n\nCORAL REEFS.\n\nCoral reefs are produced by innumerable small zoophytes, properly\ncalled _Coral-insects_. The Coral insect consists of a little oblong\nbag of jelly closed at one end, but having the other extremity open,\nand surrounded by tentacles or feelers, usually six or eight in number,\nset like the rays of a star. Multitudes of these diminutive animals\nunite to form a common stony skeleton called _Coral_, or _Madrepore_,\nin the minute openings of which they live, protruding their mouths\nand tentacles when under water; but suddenly drawing them into their\nholes when danger approaches. These animals cannot exist at a greater\ndepth in the sea than about ten fathoms, and as the Coral Islands often\nrise with great steepness from a sea more than three hundred fathoms\ndeep, it would seem that a great alteration must have taken place in\nthe depth of the ocean since the time when these little architects\ncommenced their labours. Throughout the whole range of the Polynesian\nand Australasian islands, there is scarcely a league of sea unoccupied\nby a coral reef, or a coral island; the former springing up to the\nsurface of the water, perpendicularly from the fathomless bottom,\n\"deeper than did ever plummet sound;\" and the latter in various stages,\nfrom the low and naked rock, with the water rippling over it, to an\nuninterrupted forest of tall trees.\n\n\"Every one,\" says Mr. Darwin, \"must be struck with astonishment when\nhe first beholds one of these vast rings of coral rock, often many\nleagues in diameter, here and there surmounted by a low verdant island\nwith dazzling white shores, bathed on the outside by the foaming\nbreakers of the ocean, and on the inside surrounding a calm expanse of\nwater, which, from reflection, is of a bright but pale green colour.\nThe naturalist will feel this astonishment more deeply after having\nexamined the soft and almost gelatinous bodies of these apparently\ninsignificant creatures; and when he knows that the solid reef\nincreases only on the outer edge, which, day and night, is lashed by\nthe breakers of an ocean never at rest.\"\n\n[Illustration: [++] Coral Reefs.]\n\nCoral being beautiful in form and colour, is sought after for purposes\nof ornament; and its fishery or gathering gives employment to many\npersons in the Red Sea, the Persian Gulf, the Mediterranean, and other\nplaces. In the Straits of Messina, the rocks which yield coral are from\nabout 350 to 650 feet below the surface of the water. The coral here\ngrows to about the height or length of twelve inches, and requires\neight or ten years to come to perfection. In the general mode of\nfishing for coral, the instrument used consists of two heavy beams of\nwood, secured together at right angles, and loaded with stones to sink\nthem.\n\n\nMILITARY HATS IN OLDEN TIME.\n\n[Illustration:\n\n No. 1, Charles I.\n No. 2, William III.\n No. 3, Nivernois.\n No. 4, Kevenhuller.\n No. 5, Ramilies.\n No. 6, Wellington.]\n\n\nWHY A MAN MEASURES MORE IN THE MORNING THAN IN THE EVENING, &c.\n\nThere is an odd phenomenon attending the human body, as singular as\ncommon: that a person is shorter standing than lying; and shorter in\nthe evening when he goes to bed, than in the morning when he rises.\n\nThis remark was first made in England, and afterwards confirmed at\nParis, by M. Morand, a member of the Royal Academy of Sciences in\nFrance, and by the Abbot Fontana likewise.\n\nThe last-mentioned person found, from a year's experience, that\nordinarily in the night he gained five or six lines, and lost nearly as\nmuch in the day.\n\nThe cause of which effect, so ancient, so common, but so lately\nperceived, proceeds from the different state or condition of the\nintervertebral annular cartilages.\n\nThe vertebrae, or joints of the spine, are kept separate, though joined\nby particular cartilages, every one of which has a spring. These yield\non all sides, without any inflexion on the spine, to the weight of\nthe head and upper extremities; but this is done by very small and\nimperceptible degrees, and most of all when the upper parts of the body\nare loaded with any exterior weight. So that a man is really taller\nafter lying some time, than after walking, or carrying a burthen a\ngreat while.\n\nFor this reason it is that, in the day and evening, while one is\nsitting or standing, the superior parts of the body that weigh or press\nupon the inferior, press those elastic annular cartilages, the bony\njointed work is contracted, the superior parts of the body descend\ntowards the inferior, and proportionably as one approaches the other,\nthe height of the stature diminishes.\n\nHence it was, that a fellow enlisting for a soldier, by being measured\nover-night, was found deficient in height, and therefore refused; but\nby accident being gauged again the next morning, and coming up to the\nstature, he was admitted.\n\nOn the contrary, in the night-time, when the body is laid a-bed, as\nit is in an horizontal situation, or nearly so, the superior parts\ndo not weigh, or but very little, upon the inferior; the spring of\nthe cartilages is unbent, the vertebrae are removed from one another,\nthe long jointed work of the spine is dilated, and the body thereby\nprolonged; so that a person finds himself about half an inch, or more,\nhigher in stature in the morning than when going to bed. This is the\nmost natural and simple reason that can be given, for the different\nheights of the same person at different times.\n\n\nA SENSIBLE DOG REFUSING TO BAIT A CAT.\n\nA dustman of the name of Samuel Butcher, residing at Mile-end, who\nkept a large dog, having taken it into his head to divert himself and\nothers, a few days ago, by the cruel sport of cat baiting, which the\ndog refusing to perform to the satisfaction of his master, was beat by\nhim in a most brutal manner, when the animal at length, in retaliation,\nflew at his unmerciful keeper, and inflicted very severe wounds about\nhis face, limbs, and body, in some instances tearing large mouthfuls of\nhis flesh quite clean out, and at one time clung so fast to the man,\nthat before he disengaged from him the animal's throat was obliged to\nbe cut. The man was promptly conveyed to the London Hospital, and there\ndied of the injuries he received.\n\n\nA HORSE GETTING HIMSELF SHOD.\n\nA horse having been turned into a field by its owner, Mr. Joseph Lane,\nof Fascombe, in the parish of Ashelworth, was missed therefrom the\nnext morning, and the usual inquiries set afoot, as to what could have\nbecome of him. He had, it seems, been shod (all fours) a few days\nbefore, and _as usual_ got pinched in a foot. Feeling, no doubt, a\nlively sense of proper shoeing, and desirous of relieving the cause of\npain, he contrived to unhang the gate of his pasture with his mouth,\nand make the best of his way to the smithy, a distance of a mile and a\nhalf from Fascombe, waiting respectfully at the door until the bungling\n_artist_ got up. The smith relates that he found him there at opening\nhis shed; that the horse advanced to the forge and held up his ailing\nfoot; and that he himself, upon examination, discovered the injury,\ntook off the shoe, and replaced it more carefully, which having done,\nthe sagacious creature set off at a merry pace homewards. Soon after,\nMr. Lane's servants passed by the forge in quest of the animal, and\nupon inquiry, received for answer--\"Oh, he has been here and got shod,\nand is gone home again.\"\n\n\nMAN WITHOUT HANDS.\n\nThe following account is extracted from a letter sent to the Rev. Mr.\nWesley by a person named Walton, dated Bristol, October 14, 1788:--\n\n\"I went with a friend to visit this man, who highly entertained us at\nbreakfast, by putting his half-naked foot upon the table as he sat,\nand carrying his tea and toast between his great and second toe to\nhis mouth, with as much facility as if his foot had been a hand, and\nhis toes fingers. I put half a sheet of paper upon the floor, with a\npen and ink-horn: he threw off his shoes as he sat, took the ink-horn\nin the toes of his left foot, and held the pen in those of his right.\nHe then wrote three lines, as well as most ordinary writers, and as\nswiftly. He writes out all his own bills, and other accounts. He then\nshowed how he shaves himself with a razor in his toes, and how he combs\nhis own hair. He can dress and undress himself, except buttoning his\nclothes. He feeds himself, and can bring both his meat or his broth to\nhis mouth, by holding the fork or spoon in his toes. He cleans his own\nshoes; can clean the knives, light the fire, and do almost every other\ndomestic business as well as any other man. He can make his hen-coops.\nHe is a farmer by occupation; he can milk his own cows with his toes,\nand cut his own hay, bind it up in bundles, and carry it about the\nfield for his cattle. Last winter he had eight heifers constantly to\nfodder. The last summer he made all his own hay-ricks. He can do all\nthe business of the hay-field (except mowing), as fast and as well,\nwith only his feet, as others can with rakes and forks. He goes to the\nfield and catches his horse; he saddles and bridles him with his feet\nand toes. If he has a sheep among his flock that ails anything, he can\nseparate it from the rest, drive it into a corner, and catch it when\nnobody else can. He then examines it, and applies a remedy to it. He is\nso strong in his teeth, that he can lift ten pecks of beans with them.\nHe can throw a great sledge-hammer as far with his feet as other men\ncan with their hands. In a word, he can nearly do as much without, as\nothers can with, their arms. He began the world with a hen and chicken;\nwith the profit of these he purchased an ewe; the sale of these\nprocured him a ragged colt (as he expressed it) and then a better;\nafter this he raised a few sheep, and now occupies a small farm.\"\n\n\nTHE THIEF CAUGHT IN HIS OWN TRAP.\n\nA man having, some years since, stolen a sheep at Mitcham, in Surrey,\ntied its hind legs together, and put them over his forehead to carry it\naway, but in getting over a gate the sheep, it is thought, struggled,\nand, by a sudden spring, slipped its feet down to his throat; for they\nwere found in that posture, the sheep hanging on one side of this gate\nand the man dead on the other.\n\n\nCOSTUME OF THE LADIES IN THE TIME OF THE PLANTAGENETS.\n\nThe ladies' costume may be seen to advantage in the annexed engraving\nfrom the Sloane MSS., No. 3983. A wimple or gorget is wrapped round\nthe neck, and is fastened by pins at the sides of the face, which are\ncovered above the ears; a gown of capacious size, unconfined at the\nwaist and loose in the sleeves, trails far behind in the dirt. The\nunder-garment, which is darker, has sleeves that fit closely; and\nit appears to be turned over, and pinned up round the bottom. The\nunnecessary amount of stuff that was used in ladies' robes rendered\nthem obnoxious to the satirists of that period.\n\n[Illustration: [++] proud woman who wore a white dress.]\n\nIn Mr. Wright's collection of Latin stories, published by the Percy\nSociety, there is one of the fourteenth century, which is so curious an\ninstance of monkish satire, and is so apt an illustration of the cut\nbefore us, that I cannot resist presenting it to my readers. It runs\nthus:--\n\n\"_Of a Proud Woman._--I have heard of a proud woman who wore a white\ndress with a long train, which, trailing behind her, raised a dust as\nfar as the altar and the crucifix. But, as she left the church, and\nlifted up her train on account of the dirt, a certain holy man saw a\ndevil laughing; and having adjured him to tell why he laughed, the\ndevil said, \"A companion of mine was just now sitting on the train of\nthat woman, using it as if it were his chariot, but when she lifted her\ntrain up, my companion was shaken off into the dirt: and that is why I\nwas laughing.\"\n\n\nCORPULENT MAN. NOTTINGHAM, 1819.\n\nNovember 10.--Death of Mr. Henry Bucknall, confectioner,\nChandlers-lane, aged forty-nine. He was excessively corpulent, weighing\nmore than twenty-five stone, and died very suddenly, immediately after\neating a hearty breakfast. In Lord Howe's memorable engagement, on the\n1st of June, 1794, he had served as a marine on board the Brunswick.\nHis interment, at St. Mary's New Burial-ground, on the 14th, drew\ntogether a large concourse of spectators. The coffin was of enormous\nsize, and nearly equalled the body in weight. It was made of excellent\noak, was 6 feet 8 inches in length, and 2 feet 11 inches across the\nbreast; the bottom was 2-1\/2 inches thick, the sides 1-1\/2, and the\nlid 1. The whole, including the body, considerably exceeded five\nhundred-weight.\n\n\nTAKING A MAN TO PIECES AND SETTING HIM UP AGAIN.\n\n\"Don John, of Austria,\" says Staveley, \"Governor of the Netherlands for\nPhilip the 2d of Spain, dying at his camp at Buge (Bouges, a mile from\nNamur), was carried from thence to the great church at Havre, where his\nfuneral was solemnised, and a monument to posterity erected for him\nthere by Alexander Farnese, the Prince of Parma. Afterwards his body\nwas taken to pieces, and the bones, packed in mails, were privately\ncarried into Spain, where being set together with small wires, the\nbody was rejointed again, which being filled or stuffed with cotton,\nand richly habited, Don John was presented to the king entire, leaning\non his commander's staff. Afterwards the corpse being carried to the\nchurch of St. Laurence, at the Escurial, was there buried near his\nfather, Charles V., with a fitting monument for him.\"\n\n\nORNAMENTS OF FEMALE DRESS IN THE TIMES OF THE ANCIENT BRITONS.\n\n[Illustration: [++] Necklace of Beads.]\n\nFig. 1 is a necklace of beads, each bead being cut so as to represent\na group of several, and give the effect of many small round beads to\nwhat are in reality long and narrow ones. Fig. 2 is a necklace of\nsimpler construction, consisting of a row of rudely-shaped beads, its\ncentre being remarkable for containing a rude attempt at representing a\nhuman face, the only thing of the kind Hoare discovered of so ancient\na date in Britain. Fig. 3 is another necklace, consisting of a series\nof curious little shells, like the hirlas horn used by the Britons,\nwhich are perforated lengthways, and thus strung together. Fig. 4 is a\npin of iron, supposed to have been used as a fastening for a mantle; it\nis ornamented with two movable rings. Fig. 5 is a small gold ornament,\ncheckered like a chessboard, and suspended from a chain of beautiful\nworkmanship, which, in taste and execution, bears a striking similarity\nto our modern curb-chains. Fig. 6 is an ear-ring, a bead suspended\nfrom a twisted wire of gold. Fig. 7 is a brass ornament, and Fig. 8 a\nsimilar one of gold: such ornaments are usually found upon the breasts\nof the exhumed skeletons of our barrows, and were probably fastened on\ntheir clothes as ornaments. Their cruciform character might lead to a\ndoubt of their high antiquity, if we were not aware of the fact, that\nthe symbol of the cross was worn, as an amulet or ornament, ages before\nthe Christian era.\n\n\nLARGE EEL.\n\nLately, near Malden, an eel was taken, measuring _five feet six inches\nin length, seventeen in girth_, and weighing _26 pounds_, the largest\nof the species ever caught, or described in natural history.\n\n\nPERSEVERING DOG.\n\nA boast being made of the obedience of a Newfoundland dog in fetching\nand carrying, the master put a marked shilling under a large square\nstone by the road side, and, having ridden on three miles, ordered\nthe dog to go back and fetch it. The dog set off, but did not return\nthe whole day. He had gone to the place, and being unable to turn the\nstone, sat howling by it. Two horsemen came by and saw his distress,\nand one of them alighting removed the stone, and finding the shilling,\nput it in his pocket, not supposing that the dog could possibly be\nlooking for that. The dog followed the horses for upwards of twenty\nmiles, stayed in the room where they supped, got into the bed-room, got\nthe breeches in which the fatal shilling had been put, made his escape\nwith them, and dragged them through mud and mire, hedge and ditch, to\nhis master's house.\n\n\nCURE FOR CORPULENCE.\n\nA few years ago, a man of about forty years of age, hired himself as a\nlabourer, in one of the most considerable ale-breweries in the City: at\nthis time he was a personable man; stout, active, and not fatter than a\nmoderate-sized man in high health should be. His chief occupation was\nto superintend the working of the new beer, and occasionally to set up\nat night to watch the sweet-wort, an employment not requiring either\nactivity or labour; of course, at these times, he had an opportunity of\ntasting the liquor, of which, it appears, he always availed himself;\nbesides this, he had constant access to the new beer. Thus leading a\nquiet inactive life, he began to increase in bulk, and continued to\nenlarge, until, in a very short time, he became of such an unwieldy\nsize, as to be unable to move about, and was too big to pass up the\nbrewhouse staircase; if by any accident he fell down, he was unable to\nget up again without help. The integuments of his face hung down to\nthe shoulders and breast: the fat was not confined to any particular\npart, but diffused over the whole of his body, arms, legs, &c., making\nhis appearance such as to attract the attention of all who saw him. He\nleft this service to go into the country, being a burthen to himself,\nand totally useless to his employers. About two years afterwards he\ncalled upon his old masters in very different shape to that above\ndescribed, being reduced in size nearly half, and weighing little\nmore than ten stone. The account that he gave of himself was, that as\nsoon as he had quitted the brewhouse he went into Bedfordshire, where\nhaving soon spent the money he had earned, and being unable to work,\nhe was brought into such a state of poverty, as to be scarcely able to\nobtain the sustenance of life, often being a whole day without food;\nthat he drank very little, and that was generally water. By this mode\nof living he began to diminish in size, so as to be able to walk about\nwith tolerable ease. He then engaged himself to a farmer, with whom he\nstayed a considerable time, and in the latter part of his service he\nwas able to go through very hard labour, being sometimes in the field\nploughing and following various agricultural concerns, for a whole day,\nwith no other food than a small pittance of bread and cheese. This was\nthe history he gave of the means by which this extraordinary change was\nbrought about. He added, his health had never been so good as it then\nwas.\n\n\nWORSHIP OF THE SUN AND MOON.\n\nThe Sun was first worshipped, probably, as a bright manifestation of\nGod, but soon began to be regarded as the Deity himself. The Moon,\nin the absence of the Sun, and next in splendour, would succeed\nit in superstitious attention. And so we find the Romans, as well\nas the Saxons, dedicating the first and second days of the week\nrespectively to these \"great lights.\" Formerly, festivals were held\non the appearance of a New Moon; and in some parts of England it is\nstill customary to bless it, and in Scotland at the same time to drop\na courtesy. And in times not long past, the influence of the Moon was\nconsidered to be so great as to regulate the growth of air, and the\neffect of medicine, and to cause steeples and other elevated buildings\nto bend from their upright positions.\n\n\nA SEA ABOVE THE SKY.\n\nThis belief is curiously illustrated by two legendary stories preserved\nby Gervase of Tilbury. \"One Sunday,\" he says, \"the people of a village\nin England were coming out of church on a thick cloudy day, when they\nsaw the anchor of a ship hooked to one of the tombstones; the cable,\nwhich was tightly stretched, hanging down from the air. The people were\nastonished, and while they were consulting about it, suddenly they saw\nthe rope move as though some one laboured to pull up the anchor. The\nanchor, however, still held fast by the stone, and a great noise was\nsuddenly heard in the air, like the shouting of sailors. Presently a\nsailor was seen sliding down the cable for the purpose of unfixing the\nanchor; and when he had just loosened it, the villagers seized hold of\nhim, and while in their hands he quickly died, just as though he had\nbeen drowned. About an hour after, the sailors above, hearing no more\nof their comrade, cut the cable and sailed away. In memory of this\nextraordinary event, the people of the village made the hinges of the\nchurch doors out of the iron of the anchor, and 'there they are still\nto be seen.'--At another time, a merchant of Bristol set sail with\nhis cargo for Ireland. Some time after this, while his family were at\nsupper, a knife suddenly fell in through the window on the table. When\nthe husband returned, he saw the knife, declared it to be his own, and\nsaid that on such a day, at such an hour, while sailing in an unknown\npart of the sea, he dropped the knife overboard, and the day and hour\nwere known to be exactly the time when it fell through the window.\nThese accidents, Gervase thinks, are a clear proof of there being a sea\nabove hanging over us.\"--_St. Patrick's Purgatory. By Thos. Wright._\n1844.\n\n\nTHE PAPYRUS.\n\nPaper as we now have it, that is to say, paper made of the pulp of\nfibrous materials, pressed into thin sheets, dried, and, when intended\nfor writing or printing purposes, sized, is of comparatively modern\nintroduction to Europe and Western Asia; although the Chinese appear to\nhave formed paper out of silk pulp, mixed with the inner pith of the\nbamboo, as early at least as 95 A.D.:--not from time immemorial, as\nsome authors have stated, because the circumstance is well attested,\nthat in the time of Confucius, the Chinese wrote with a style on the\ninner bark of trees.\n\n[Illustration: Papyrus Roll, from a Specimen in the British Museum.]\n\nBefore the invention of paper, the surfaces employed for writing upon\nwere numerous. Surfaces of lead or other metal; tables covered with\nwax, skins of animals,--(parchment in fact)--all were used; but no one\nof these was ever so extensively employed as the Egyptian papyrus,\nwhenever the latter material could be obtained. So soon, however, as\nthe Saracens in the seventh century conquered Egypt, the exportation of\npapyrus was at an end; and writing surfaces became so scarce in Europe\nthat many ancient documents of great value were erased in order to\nrender them adapted for being written on once more. Thus perished many\ntreasures of antiquity.\n\nAs the Saracens closed the avenue of supply for the ancient papyrus,\nso they compensated to Europe for this deprivation by discovering the\nmanufacture of ordinary paper--at least paper made in the ordinary\nmodern fashion,--though the material was cotton, not linen. This\ndiscovery was made some time anterior to the year 706 A.D., for at\nthat period a manufactory of paper existed at Samarcand. In the\neighth century the Saracens conquered Spain, and introduced into the\nPeninsula, amongst other arts, that of the manufacture of paper, which\nart was a long time finding its way into other parts of Europe,--in\nItaly not until the eleventh or twelfth century. The vast amount of\npapyrus which must have been employed in Italy, may be inferred\nfrom the number of rolls or _scapi_ of this substance discovered in\nHerculaneum and Pompeii; also from a perusal of many existing documents\nbearing directly or indirectly on this branch of commerce. Even so late\nas the commencement of the sixth century, Cassiodorus congratulated the\nworld on the abolition, by King Theodoric, of the high duty on papyrus\nfrom Egypt; and he spoke in high flown terms of the great utility of\nthe material. The latest papyrus roll known is of the twelfth century,\ncontaining a brief of Pope Paschal II., in favour of the Archiepiscopal\nsee of Ravenna.\n\n[Illustration: Syrian Papyrus Without Flowers.]\n\n[Illustration: Syrian Papyrus With Flowers.]\n\nThe various species of papyrus plants belong to the natural order\n\"Cyperaceae,\" or sedges, of botanists; a main characteristic of which is\na certain triangularity of stem. The method of constructing a writing\nsurface from these stems was as follows:--The available portion being\ncut off (it was seldom more than twelve inches in length), and split,\nor, more properly speaking, unfolded into thin sheets, which were glued\ntogether transversely in such a manner that the original length of the\npapyrus stem became the breadth of the future sheet; the length of\nwhich might be increased at the pleasure of the operator. Frequently\nthe manufactured scrolls were more than thirty feet long. As different\nmethods prevail in the manufacture of our ordinary paper, so in like\nmanner there were different processes of fashioning the papyrus into\nshape. The rudest manufacture appears to have been that of Egypt, and\nthe best papyrus sheets appear to have been made in Rome during the\nAugustine AEra. The preceding sketch represents a papyrus roll, copied\nfrom a specimen in the Egyptian Room of the British Museum.\n\nConsidering the numerous pieces entering into the composition of the\nroll, of which our illustration represents a portion, the lines of\njuncture are remarkably well concealed, only a sort of grain being\nvisible. The surface, moreover, is smoothed, and its colour very\nmuch like that of India paper. The hieroglyphics are as is\nusual, red is the predominant tint, and the colours are no less well\ndemarcated and separate than they would have been on glazed paper.\n\nOur preceding wood-cuts represent the Sicilian or Syrian papyrus,\nhitherto termed _cyperus papyrus_, in two states of development--one\nwith flowers, the other without. In order that inflorescence may take\nplace, the plant requires to be well supplied with water.\n\n\nEXECUTION IN 1733.\n\n_Friday, March 9_--Was executed at Northampton, William Alcock, for the\nmurder of his wife. He never own'd the fact, nor was at all concerned\nat his approaching death; refusing the prayers and assistance of any\npersons. In the morning he drank more than was sufficient, yet sent and\npaid for a pint of wine, which being deny'd him, he would not enter\nthe cart before he had his money return'd. On his way to the gallows\nhe sung part of an old song of \"Robin Hood,\" with the chorus, \"Derry,\nderry, down,\" &c., and swore, kick'd, and spurn'd at every person that\nlaid hold of the cart; and before he was turn'd off, took off his\nshoes, to avoid a well known proverb; and being told by a person in the\ncart with him, it was more proper for him to read, or hear somebody\nread to him, than so vilely to swear and sing, he struck the book out\nof the person's hands, and went on damning the spectators and calling\nfor wine. Whilst psalms and prayers were performing at the tree he did\nlittle but talk to one or other, desiring some to remember him, others\nto drink to his good journey, and to the last moment declared the\ninjustice of his case.\n\n\nDOG FRIENDSHIP.\n\nAt Bishops Stortford there were two dogs, which belonged to nobody, and\nlived upon the quay of the river or canal there. They took the greatest\ndelight in rat hunting, and when the maltsters went about at night to\nsee that all was safe, these dogs invariably followed them. Their mode\nof proceeding was very ingenious. As soon as the door of the malt-house\nwas unlocked, one rushed in and coursed round the warehouse, not\nchasing any rat which might start, but pursuing its way among the malt.\nThe other stood at the door and snapped at the rats as they endeavoured\nto escape. The one standing at the door was known to kill six rats,\nall of which had rushed to the door at the same time. The next room\nthey came to, they would change posts; the one which hunted before,\nnow standing at the door and seizing the prey. By this means the dogs\nkilled in the malting-houses of one maltster alone, upwards of 2,000\nrats in the course of one year. One of them on one occasion killed\nsixty-seven in less than five minutes. They seemed to pursue the sport\nsimply for their amusement.\n\n\nALL HUMBUGS.\n\nJust as a strolling actor at Newcastle had advertised his benefit, a\nremarkable stranger, no less than the _Prince Annamaboo_, arrived, and\nplacarded the town that he granted audiences at a shilling a-head. The\nstroller, without delay, waited on the proprietor of the _Prince_, and\nfor a good round sum prevailed on him to command his Serene Highness\nto exhibit his august person on his benefit night. The bills of the\nday announced that between the acts of the comedy _Prince Annamaboo_\nwould give a lively representation of the _scalping operation_, sound\nthe _Indian war-whoop_ in all its melodious tones, practice the\ntomahawk exercise, and dine _a la cannibal_. An intelligent mob were\ncollected to witness these interesting exploits. At the conclusion of\nthe third act, his _Highness_ marched forward flourishing his tomahawk,\nand shouting, \"_Ha, ha!--ho, ho!_\" Next entered a man with his face\nblacked, and a piece of bladder fastened to his head with gum; the\n_Prince_, with an enormous carving-knife, began the scalping part of\nthe entertainment, which he performed in a truly _imperial_ style,\nholding up the piece of bladder as a token of triumph. Next came the\nwar-whoop, an unearthly combination of discordant sounds; and lastly,\nthe banquet, consisting of raw beef-steaks, which he rolled up into\nrouleaus, and devoured with right royal avidity. Having finished\nhis delicate repast, he wielded his tomahawk in an exulting manner,\nbellowed \"_Ha, ha!--ho, ho!_\" and made his exit. The _beneficiaire_\nstrolling through the market-place the following-day, spied the most\npuissant _Prince Annamaboo_ selling penknives, scissors, and quills,\nin the character of a Jew pedlar. \"What!\" said the astonished _Lord\nTownley_, \"my _Prince_, is it you? Are you not a pretty circumcised\nlittle scoundrel to impose upon us in this manner?\" Moses turned round,\nand with an arch look, replied, \"_Princh_ be d--d! I _vash_ no Princh;\nI _vash_ acting like you. Your troop _vash_ Lords and Ladies last\nnight; and to-night dey vil be Kings, _Prinches_, and Emperor! I _vash\nhumpugs_, you _vash humpugs_, all _vash humpugs_!\"\n\n\nREDUCING WEIGHT.\n\nA gentleman, of great respectability in the mercantile world, who\nweighed thirty-two stone nine pounds, put himself upon a strict diet of\n_four ounces of animal food_, _six ounces_ of bread, and _two pounds\nof liquid_, in twenty-four hours. In one week he lost thirty pounds\nweight, and in six months he was diminished the astonishing quantity of\none hundred and thirty-four pounds. His health and spirits were much\nimproved, and considering his remaining size of twenty-three stone, he\nwas very active.\n\n\nANECDOTE OF A SERPENT.\n\nLord Monboddo relates the following singular anecdote of a serpent:--\"I\nam well informed of a tame serpent in the East Indies, which belonged\nto the late Dr. Vigot, once kept by him in the suburbs of Madras. This\nserpent was taken by the French, when they invested Madras, and was\ncarried to Pondicherry in a close carriage. But from thence, he found\nhis way back again to his old quarters, though Madras was above one\nhundred miles distant from Pondicherry.\"\n\n\nENGLAND BEFORE THE ROMANS.\n\n[Illustration [++] Shoes of Raw Cow-Hide.]\n\nBefore the Roman invasion, the dress of its chieftains consisted of\na close coat or covering for the body, called by Dio a tunic, and\ndescribed as checkered with various colours in divisions. It was open\nbefore, and had long close sleeves to the wrist. Below were loose\npantaloons, called by the Irish _brigis_, and by the Romans _brages_\nand _bracae_; whence the modern term \"breeches.\" Over their shoulders\nwas thrown the mantle or cloak, called by the Romans _sagum_, and\nderived from the Celtic word _saic_, which signified a skin or hide,\nand which was the original cloak of the country. Diodorus tells us\nthat it was of one uniform colour, generally either blue or black, the\npredominating tint in the checkered trousers and tunic being red. On\ntheir heads they wore a conical cap, which derived its name from the\n\"cab,\" or hut of the Briton, which was of similar form. On their feet\nwere shoes made of raw cow-hide, that had the hair turned outward, and\nwhich reached to the ankles. Shoes so constructed were worn within the\nlast few years in Ireland; and we engrave two from specimens in the\nRoyal Irish Academy. One is of cow-hide, and drawn together by a string\nover the foot; and the other has a leather thong, which is fastened\nbeneath the heel inside, and, passing over the instep, draws the shoe\nlike a purse over the foot. It is of untanned leather.\n\n\nROMANS IN BRITAIN--DRESS OF NATIVE FEMALES AT THAT PERIOD.\n\n[Illustration: [++] Female Roman Shoes.]\n\nThe British _gwn_, from whence comes the modern \"gown,\" descended to\nthe middle of the thigh, the sleeves barely reaching to the elbows:\nit was sometimes confined by a girdle. Beneath this a longer dress\nreached to the ancles. The hair was trimmed after the Roman fashion;\nand upon the feet, when covered, were sometimes worn shoes of a\ncostly character, of which we know the Romans themselves to have\nbeen fond. An extremely beautiful pair was discovered upon opening a\nRoman burial-place at Southfleet in Kent, in 1802. They were placed\nin a stone sarcophagus, between two large glass urns or vases, each\ncontaining a considerable quantity of burnt bones. They were of\nsuperb and expensive workmanship, being made of fine purple leather,\nreticulated in the form of hexagons all over, and each hexagonal\ndivision worked with gold, in an elaborate and beautiful manner.\n\n[Illustration: The Catacombs. Rome.]\n\nAmid the ruins of stately temples, and numerous remains of the \"Eternal\nCity,\" there are no objects which have such great and general interest\nas the subterranean churches, dwellings, and places of sepulchre of the\nearly Christians, which perforate, by a network of excavations, the\nneighbourhood of Rome.\n\nThe great increase in the extent and magnificence of Rome during\nthe times of the Republic, led to the formation of quarries in the\nsurrounding parts. The peculiar nature of the soil has caused the\nexcavations to be made in a manner similar to that used in the working\nof coal, iron, stone, lime, &c. The useful material has, in fact, been\ncleared away, leaving long ranges of dark caves and passages. After\nthe stone had been removed from these underground quarries, it was,\nfor many centuries, customary to work out the sand for the purpose of\nmaking cement. Vitruvius has stated that the sand obtained from the\nEsquiline pits was preferable to any other. Ultimately the quarries\nand sandpits extended to a distance of upwards of fifteen miles on\none side of Rome. Parts of this large range of excavations were from\ntime to time used as burial-grounds by such of the Romans as could not\nafford the cost of burning the bodies of their dead relations. And, in\naddition, the Esquiline hills became infested by banditti, and was from\nthese various causes rendered almost impassable.\n\nIn these excavations, it is said, that not only persons, but cattle,\ncontrived to support existence; and although it was well known that\nlarge numbers were lodged in these dismal dwellings, their intricacy\nand numberless entrances rendered them a comparatively secure retreat.\nIt is related that attempts were made to cover the galleries with\nearth, in order to destroy those who were concealed within.\n\n[Illustration: [++] Inscription in the Catacombs.]\n\nIn course of time the catacombs became, with the exception of one or\ntwo, neglected and filled up with rubbish, and remained for a period\nof upwards of one thousand years untouched and almost unknown. In the\nsixteenth century the whole range of the catacombs were reopened, and\nnumerous inscriptions and other matters connected with the struggles\nand hardships of the early Christians brought to light. The annexed\nbrief memorial will show the general style of the lettering.\n\n\nOBSOLETE MODES OF PUNISHMENT.\n\nAnte page 60, we gave representations of some ancient instruments of\npunishment and torture, all more or less terrible in their character,\nthe use of which, for many a long year, has been happily abandoned. As\na companion to this group, we have engraved a few of the instruments\nof punishment by which criminals of a vulgar character were sought\nto be reformed. The first of these is the felon's brand, the mark of\nwhich rendered a man infamous for life. Figure 1, p. 90 represents the\ninstrument itself. Figure 2, the mark branded in, which latter has\nbeen engraved the exact size. The device, which is deeply cut into the\nmetal, is a gallows, such as was used before the invention of the Drop\nand the Wheel for Execution and torture.\n\nThe Stocks and Whipping-post, although long since removed from London\nBridge, may be met with in retired country places. We have noticed\nsome characteristic examples in the Forest of Dean, Gloucestershire,\nwhere some of the may-poles, day-wheels, and other curious relics, may\nstill be seen.[2] In some instances the Stocks and Whipping-posts were\nrichly carved, and clamped with iron work of an ornamental character.\nWe remember seeing the stocks used within the last thirty years, once\nat Newcastle-on-Tyne, and once at Gateshead, the adjoining town. The\nculprit in the one instance was an elector, who, in the excess of zeal\nand beer, during an old-fashioned contested election, rushed into one\nof the churches during the Sunday's service, and shouted out, \"Bell\n(one of the candidates) for ever.\" He was speedily taken hold of,\nand placed for several hours in the stocks in the churchyard; and,\nas the stimulating effect of the strong drink passed away, he looked\na deplorable object, decked as he was with numerous cockades, the\n\"favours\" of the candidate, whose cause he so indiscreetly supported.\n\n[2] A good specimen was demolished at Tottenham not long ago.\n\nThe punishment of the barrel we should think to have been adapted for\ndrunkards who could preserve a perpendicular position.\n\nIn the histories of London, it is mentioned that bakers and other\ndealers caught giving false weight, or in any other ways cheating the\npoor, were exhibited occasionally in this manner; but more frequently\nthey were placed in the parish dung-cart, and slowly drawn through the\nstreets of the district.\n\nThe Whirligig, a circular cage which could be moved swiftly round on\na pivot, was, in bygone days, in use for offenders in the English\narmy. There was another instrument used for the same purpose called\nthe Horse, which was made in rude resemblance of the animal whose\nname it bore. The body was composed of planks of wood, which formed a\nsharp angle along the back. On this the soldier was seated, and his\nlegs fastened below to several heavy muskets. This is said to have\nbeen a very severe and dangerous punishment. In addition to the above,\nand flogging, imprisonment, &c., there were three ancient methods of\npunishment in the English army--viz., beheading, hanging, and drowning.\nThe latter of these, according to Grose, was in use only in the reign\nof Richard I. This author observes that, some centuries ago, capital\npunishment was rare in our army, the men having generally property,\nwhich was confiscated in case of ill conduct. He, however, refers\nto some terrible means which were resorted to for the purpose of\npreserving discipline. Hanging was chiefly confined to spies; who were\ntaken to a tree in sight of the camp, and yet sufficiently distant, and\nthere hung up. In many instances, when a corps or a considerable body\nof men were guilty of crime, for which the established punishment was\ndeath, to prevent too great a weakening of the army, the delinquents,\nGrose says, \"were decimated, that is, only every tenth man was taken. A\nnumber of billets, equal to that of the body to be decimated, were put\ninto a helmet, every tenth billet being marked with the letter D, or\nsome other character signifying death; the helmet was then shaken, in\norder to mix them, and the soldiers, filing off singly from the right,\npassed by the commanding officers, before whom, on a table, stood the\nhelmet; as they passed, each drew a billet and presented it to an\nofficer placed to receive them. If the billet had the fatal mark; the\nsoldier was seized and marched into the rear.\"\n\nThis wholesale method of capital punishment must have been a solemn\naffair. At times, it was customary to punish the man at the right hand\nof companies; without giving them the chance of the billet--on the\nprincipal that these were the most influential persons, and must, from\ntheir companionship with the others, have been acquainted with and have\npossessed the means of checking or giving information, which would\nprevent dangerous offences.\n\n[Illustration: 1. Brand for Marking Felons. 2. Impression of Brand. 3.\nPunishment for Drunkards, formerly in use at Newcastle-on-Tyne. 4. The\nWhirligig, a military method of punishment. 5. Pillory, Stocks, and\nWhipping Post, formerly on London Bridge.]\n\nThe regulations of the English army during the time of Henry VIII., and\nprevious reigns, may be met with in \"Grose's Military Antiquities.\"\n\n\nORIGIN OF THE TERM \"HUMBUG.\"\n\nThis, now, common expression, is a corruption of the word Hamburgh, and\noriginated in the following manner:--During a period when war prevailed\non the Continent, so many false reports and lying bulletins were\nfabricated at Hamburgh, that at length, when any one would signify his\ndisbelief of a statement, he would say, \"You had that from Hamburgh;\"\nand thus, \"That is Hamburgh,\" or Humbug, became a common expression of\nincredulity.\n\n\nMARRIAGE LOTTERY.\n\nIt has often been said figuratively that marriage is a lottery; but we\ndo not recollect to have met with a practical illustration of the truth\nof the simile, before the following, which is a free translation of an\nadvertisement in the Louisiana Gazette:--\"A young man of good figure\nand disposition, unable, though desirous to procure a wife, without\nthe preliminary trouble of amassing a fortune, proposes the following\nexpedient to attain the object of his wishes. He offers himself as\nthe prize of a lottery to all widows and virgins under 32. The number\nof tickets to be 600, at 50 dollars each. But one number to be drawn\nfrom the wheel, the fortunate proprietor of which is to be entitled to\nhimself and the 30,000 dollars.\"\n\n\nCHINESE DAINTIES.\n\nThe common people of the country seem to fare hardly and sparingly\nenough, but one of our envoys praises much of the good cheer he found\nat the tables of the great men. They had pork, fish, and poultry,\nprepared in a great variety of ways, and very nice confectionery in\nabundance. The feasts, moreover, were served up in a very neat and\ncleanly manner. But there was one dainty which much offended their\nnostrils, and nearly turned their stomachs when it was named to them.\nIt was not stewed dog or fricaseed pup. No; it consisted of three\nbowls of _hatched eggs_! When the Englishmen expressed some surprise\nat the appearance of this portion of the repast, one of the native\nattendants observed that hatched eggs formed a delicacy beyond the\nreach of the poor--a delicacy adapted only for persons of distinction!\nOn inquiry, it was found that they cost in the market some thirty\nper cent. more than fresh eggs. It seems that they always form a\ndistinguished part of every great entertainment, and that it is the\npractice, when invitations are sent out, to set the hens to hatch.\nThe feast takes place about the tenth or twelfth day from the issuing\nthe invitations,--the eggs being then considered as ripe, and exactly\nin the state most agreeable and pleasant to the palate of a Chinese\nepicure.\n\n\nRECEIPTS FROM ALBERTUS MAGNUS.\n\n\"Bubo a shrick owle, is a byrd wel inough knowen, which is called\nMagis of the Chaldes, and Hysopus of the Greekes. There bee maruaylous\nvertues of this Fowle, for if the hart and ryght foote of it be put\nupon a man sleeping, hee shall saye anone to thee whatsoever thou shalt\naske of him. And thys hath beene prooued of late tyme of our brethren.\nAnd if any man put thys onder his arme hole, no Dog wyll barke at hym,\nbut keepe silence. And yf these thynges aforesayde ioyned together with\na wyng of it be hanged up to a tree, byrdes wyl gather together to that\ntree.\"\n\n\"When thou wylt that thy wyfe or wenche shewe to thee all that shee\nhath done, take the hart of a Doove, and the heade of a Frog, and drye\nthem both, and braie them vnto poulder, and lay them vpon the brest of\nher sleeping, and shee shall shew to thee all that shee hath done, but\nwhen shee shall wake, wipe it awaye from her brest, that it bee not\nlifted vp.\"\n\n\"Take an Adders skyn, and Auri pigmentum, and greeke pitch of\nReuponticum, and the waxe of newe Bees, and the fat or greace of an\nAsse, and breake them all, and put them all in a dull seething pot full\nof water, and make it to seeth at a slowe fire, and after let it waxe\ncold, and make a taper, and euery man that shall see light of it shall\nseeme headlesse.\"--_The Secreetes of Nature, set foorth by Albertus\nMagnus in Latine, newlye translated into English._ Imprinted at London\nby me Wyllyam Copland. No date. _Black letter_, very old.\n\n\nTHE MAGPIE STONING A TOAD.\n\nThere is a story told of a tame magpie, which was seen busily employed\nin a garden, gathering pebbles, and with much solemnity, and a\nstudied air, dropping them in a hole, about eighteen inches deep,\nmade to receive a post. After dropping each stone, it cried, Currack!\ntriumphantly, and set off for another. On examining the spot a poor\ntoad was found in the hole, which the magpie was stoning for his\namusement.\n\n\nADAPTATION OF BONES TO AGE IN THE HUMAN FRAME.\n\nGrowth produces in the species a somewhat remarkable change in the\nmechanical qualities of the bones. This important part of our organism\nconsists of three constituents--fibre, cartilage, and the earthy matter\nalready mentioned called _phosphate of lime_. From the fibre they\nderive their toughness; from the cartilage their elasticity; and from\nthe lime their hardness and firmness. Nothing can be more admirable\nin the economy of our body than the manner in which the proportion of\nthese constituents adapts itself to the habitudes of age. The helpless\ninfant, exposed by a thousand incidents to external shocks, has bones,\nthe chief constituents of which being gristly and cartilaginous, are\nyielding and elastic, and incur little danger of fracture. Those of the\nyouth, whose augmented weight and increased activity demand greater\nstrength, have a larger proportion of the calcareous and fibrous\nelements, but still enough of the cartilaginous to confer upon the\nsolid framework of his body the greatest firmness, toughness, and\nelasticity. As age advances, prudence and tranquil habits increasing,\nas well as the weight which the bones have to sustain, the proportion\nof the calcareous constituent increases, giving the requisite hardness\nand strength, but diminishing the toughness and elasticity.\n\nWhile the bones thus change their mechanical qualities as age advances,\nthey diminish in number, the frame consequently having fewer joints\nand less flexibility. The bones of a child, whose habits require\ngreater bodily pliability, are more numerous than those of an adult,\nseveral of the articulations becoming ossified between infancy and\nmaturity. In like manner, the bones at maturity are more numerous than\nin advanced age, the same progressive ossification of the joints being\ncontinued.\n\nIt has been ascertained by anatomists that, on attaining the adult\nstate, the number of bones constituting the framework of the human body\nis 198; of which 52 belong to the trunk, 22 to the head, 64 to the\narms, and 60 to the legs.\n\n\nTOWER OF THE THUNDERING WINDS.\n\n[Illustration: [++] Tower of the Thundering Winds.]\n\nThe Great Wall is certainly a wonderful monument of ancient times; but\nit is almost the only one that we read of in China, except a famous\nTemple, or Tower, partly in ruins, which stands on an eminence in\nthe neighbourhood of Hang-chow-foo. It is called the \"Tower of the\nThundering Winds,\" and is supposed to have been built about 2,500 years\nago.\n\n\nDR. MONSEY BEQUEATHS HIS OWN BODY.\n\nThis eccentric person died at the great age of 96, and was for half\na century, physician to Chelsea Hospital. He left his body for\ndissection, and a few days before he died, wrote to Mr. Cruikshanks,\nthe Anatomist, begging him to know, whether it would suit his\nconvenience to do it, as he felt he could not live many hours, and Mr.\nForster, his surgeon, was then out of town. He died as he predicted,\nand his wishes with respect to his body, were strictly attended to.\n\n\nTEA.\n\nA folio sheet of the time of Charles II. entitled \"An Exact Description\nof the Growth, Quality, and Virtues of the Leaf Tea, by Thomas Garway,\nin Exchange Alley, near the Royal Exchange, in London, Tobacconist, and\nSeller and Retailer of Tea and Coffee,\" informs us that \"in England\nit hath been sold in the leaf for six pounds, and sometimes for ten\npounds the pound weight; and in respect of its former scarceness and\ndearness, it hath been only used as a regalia in high treatments, and\nentertainments, and presents made thereof to princes and grandees till\nthe year 1657. The said Thomas Garway did purchase a quantity thereof,\nand first publikely sold the said Tea in leaf and drink, made according\nto the direction of the most knowing merchants and travellers in those\neastern countries: and upon knowledge and experience of the said\nGarway's continued care and industry, in obtaining the best Tea, and\nmaking drink thereof, very many noblemen, physicians, merchants, and\ngentlemen of quality, have ever since sent to him for the said leaf,\nand daily resort to his house, in Exchange Alley, to drink the drink\nthereof.\"\n\n\nIT'S MUCH THE SAME NOW.\n\nThe following lines, from the _Gentleman's Magazine_ of 1733, will give\nus some idea of what fashionable life was at that period:--\n\n_The Town Lady's Answer to_,--\"_What tho' I am a Country Lass_.\"\n\n What tho' I am a London dame,\n And lofty looks I bear, a?\n I carry, sure, as good a name,\n As those who russet wear, a.\n\n What tho' my cloaths are rich brocades?\n My skin it is more white, a\n Than any of the country maids\n That in the fields delight, a.\n\n What tho' I to assemblies go,\n And at the Opera's shine, a?\n It is a thing all girls must do,\n That will be ladies fine, a:\n\n And while I hear Faustina sing,\n Before the king and queen, a\n By Eyes they are upon the wing,\n To see, if I am seen, a.\n\n My Peko and Imperial Tea\n Are brought me in the Morn, a.\n At Noon Champaign and rich Tokay\n My table do adorn, a.\n\n The Evening then does me invite\n To play at dear Quadrille, a:\n And sure in this there's more delight,\n Than in a purling rill, a.\n\n Then since my Fortune does allow\n Me to live as I please, a;\n I'll never milk my father's cow\n Nor press his coming cheese, a.\n\n But take my swing both night and day,\n I'm sure it is no sin, a:\n And as for what the grave ones say,\n I value not a pin, a.\n\n\nBARBERS.\n\nThe barber's pole, one of the popular relics of Merrie England, is\nstill to be seen in some of the old streets of London and in country\ntowns, painted with its red, blue, and yellow stripes, and surmounted\nwith a gilt acorn. The lute and violin were formerly among the\nfurniture of a barber's shop. He who waited to be trimmed, if of\na musical turn, played to the company. The barber himself was a\nnimble-tongued, pleasant-witted fellow. William Rowley, the dramatist,\nin \"A Search for Money, 1609,\" thus describes him:--\"As wee were but\nasking the question, steps me from over the way (over-listning us) a\nnews-searcher, viz. a _barber_: hee, hoping to attaine some discourse\nfor his next patient, left his baner of basons swinging in the ayre,\nand closely eave-drops our conference. The saucie treble-tongu'd knave\nwould insert somewhat of his knowledge (treble-tongu'd I call him, and\nthus I prove't: hee has a reasonable mother-tonger, his barber-surgions\ntongue; and a tongue betweene two of his fingers, and from thence\nproceeds his wit, and 'tis a snapping wit too). Well, sir, hee (before\nhee was askt the question,) told us that the wandring knight (Monsier\nL'Argent) sure was not farre off; for on Saterday-night hee was faine\nto watch till morning to trim some of his followers, and its morning\nthey went away from him betimes. Hee swore hee never clos'd his eyes\ntill hee came to church, and then hee slept all sermon-time; (but\ncertainly hee is not farre afore, and at yonder taverne showing us\nthe bush) I doe imagine hee has tane a chamber.\" In ancient times the\n_barber_ and the _tailor_, as news-mongers, divided the crown. The\nbarber not only erected his _pole_ as a sign, but hung his _basins_\nupon it by way of ornament.\n\n\nBEES OBEDIENT TO TRAINING.\n\nThough it is customary in many rural districts of England, when bees\nare swarming, to make a clanging noise with metal implements, under the\nimpression--an erroneous one we believe--that it will induce the swarm\nto settle, it is not generally supposed that bees are susceptible of\nbeing trained to obey in many respects the orders of their teacher.\nSuch, however, is the fact, and an instance of it occurs in the\nfollowing advertisement, which we have copied from an old newspaper. We\ngive it as we find it, but it is not very clear what locality is meant\nby \"their _proper_ places\":--\n\n\"At the Jubilee Gardens, Dobney's, 1772. Daniel Wildman rides, standing\nupright, one foot on the saddle, and the other on the horse's neck,\nwith a curious mask of bees on his face. He also rides, standing\nupright on the saddle, with the bridle in his mouth, and, by firing a\npistol, makes one part of the bees march over a table, and the other\npart swarm in the air, and return to their proper places again.\"\n\n\nA MAN SELLING HIS OWN BODY.\n\nAnatomists and surgeons have frequently incurred the odium of being\nprecipitate in their post mortem examinations. It has been charged\nupon the illustrious Vessalius, and, in more modern times, on Mons. de\nLassone, and others; nay, credulity has gone so far, as to suppose,\nthat subjects have occasionally been kept till wanted; nor is such\na notion altogether extravant, when we find an article of this kind\noffered to Joshua Brookes, the anatomical lecturer, in the following\nterms:--\n\n\"Mr. Brooke, i have taken it into consideration to send this poor man\nto you, being greatly in distress, hopeing you will find sum employment\nfor him in silling the dead carcases; and if you can find him no\nemployment, the berer of this wishes to sill himself to you, as he is\nweary of this life. And I remain your humble servant,\n\n \"JOHN DAVIS.\"\n\n\nTHE FIRST LOCOMOTIVES.\n\n[Illustration: The First Locomotive.]\n\nIt is little more than thirty years ago, when, on the river Tyne, a\nlarge fleet of peculiarly-formed vessels was to be seen daily employed\nin the carriage of coals to the ships from the \"staiths,\" which\nprojected into the river from the various colliery tramways. At that\nperiod, there was only one very small and ill-constructed steam-packet\nfor the conveyance of passengers between Newcastle and Shields, and\nagainst which so much prejudice existed, that the majority of persons\npreferred the covered wherries, which, for some centuries before,\nhad been in use; yet so slow and uncertain was this means of transit\nbetween the two towns, that persons in a hurry often found it advisable\nto walk the intervening distance, which is about eight miles.\n\n[Illustration: The Present Locomotive and Train.]\n\nThe collieries situated away from the river had tramways of wood let\ninto the ordinary roads, in such a manner as to form wheel-tracks for\ncarriages. These, drawn by horses, were the only means thought of for\nbringing the coals to the river bank. Some of these tramways were\nnearly as old as the times of Queen Elizabeth or James I., when the\nincrease of London and other causes began to overcome the prejudice\nagainst the use of \"sea-coal.\" Many of the tramways passed amid green\nand shadowy woods and other pleasant places, and we have often thought\nwhen wandering through them, of the difficulties that beset travellers\nat that time. Even at a more recent date, in 1673, day coaches were\nconsidered dangerous, and it was suggested that the multitude of them\nin London should be limited, and not more than one be allowed to\neach shire, to go once a week backwards and forwards, and to perform\nthe whole journey with the same horses they set out with, and not to\ntravel more than thirty miles a day in summer, and twenty-five in\nwinter. The arguments advanced in favour of these proposals were, that\ncoaches and caravans were mischievous to the public, destructive to\ntrade, and prejudicial to the land--because, firstly, they destroyed\nthe breed of good horses, and made men careless of horsemanship;\nsecondly, they hindered the breed of watermen, who were the nursery of\nseamen; thirdly, they lessened the revenue.\n\nIn 1703, the road from Petworth to London (less than 50 miles) was so\nbad that the Duke of Somerset was obliged to rest a night on the road.\n\nIn March, 1739 or 1740, Mr. Pennant, the historian, travelled by the\n_stage_, then no despicable vehicle for country gentlemen, and in the\nfirst day, with \"much labour,\" got from Chester to Whitechurch--twenty\nmiles; and, after a \"wondrous effort,\" reached London before the\ncommencement of the sixth night.\n\nWithout entering into an account of the rapid improvement of the\nEnglish roads soon after the time of Pennant, we may mention that,\nat about the date 1765, the colliery tramways underwent considerable\nimprovement, by plating the wooden rails in many parts with iron:\nstone-ways were tried in some instances, but were not found successful;\nand in course of time the old tramways were covered with cast-iron\nrails laid on the old foundations. Inclined planes, with fixed\nsteam-engines, also came into use; and at the same time the idea of\na locomotive engine was attracting attention in various directions.\nIn 1805 a machine was used on a tramway near Merthyr Tydvil, and soon\nafter this the \"Iron Horse,\" shown in the engraving, was placed upon\nthe wagon way of the Wylam Colliery, from Wylam to Newburn, on the\nTyne, near Newcastle, and greatly astonished all who saw it drawing\nalong, at the rate of three miles and a half per hour, from fifteen to\ntwenty wagons of coals, making all the while a horrible and snorting\nnoise, difficult to describe, and sending forth at the same time fire\nand dense clouds of black smoke. George Stephenson was then beginning\nto make way, and had provided several improved locomotives for Heaton\nColliery. In 1816-1817, patents for improvements in locomotives were\ntaken out by George Stephenson, in connexion with Messrs. Dodd and\nLosh; and in 1825 the projection of the Liverpool and Manchester\nRailway afforded a further opportunity for their development. The\nopposition to the use of steam-engines on this line of railway seems\nsingular enough at the present day; still it was very great. The use of\nhorses was, however, found to be too expensive, and George Stephenson\nhaving stated that he could work a locomotive with safety at a rate\nof from six to eight miles an hour (\"I knew,\" said he, \"that if I\ntold them more than that, they would look upon me as more fit for a\nlunatic house than to give evidence in the House of Commons\"), a reward\nof 500_l._ was offered for the best locomotive engine. A trial took\nplace in October, 1829--_only twenty-seven years ago!_--of the steam\nlocomotive engines which were offered in competition. Of these, one\nwas withdrawn at the commencement of the experiment. The \"Novelty,\" by\nBraithwait and Ericsson, met with an accident; and the \"Sanspareil,\" by\nHackworth, attained a velocity of fifteen miles an hour, with a gross\nload of nineteen tons, but at length gave way, owing to an accident;\nthe remaining engine, constructed by Robert Stephenson and Mr. Booth,\nsucceeded in performing more than was stipulated.\n\nThe contrast between the date mentioned at the commencement of our\narticle and the present time is remarkable: the old and clumsy fleet\nhas vanished from the Tyne; a railway carries passengers from Newcastle\nto Shields in a few minutes; numerous steam vessels sail upon the\nriver, some of large size; which travel to various and distant ports.\nOn the colliery railway hundreds of locomotives are at work, and\nhundreds of thousands of miles of iron rails spread over a wide extent\nof the civilized world; and, in addition to other wonders, the electric\ntelegraph will, ere long, outrival the power of Puck, the fairy, and\n\"put a girdle round the world in (less than) forty minutes.\"\n\n\nSIR WILLIAM WALLACE THE HERO OF SCOTLAND.\n\n1305.--This year was marked by the capture of Sir William Wallace.\nIt appears that the King of England had anxiously sought to discover\nhis retreat, and that, tempted by the prospects of the rewards his\nbaseness might earn for him, Ralph de Haliburton, one of the prisoners\ntaken a short time previously at Sterling, had proffered his services\nfor that purpose. Upon being seized, he was conveyed to the castle\nof Dumbarton, and thence to England. He was brought to London, \"with\ngreat numbers of men and women,\" says Stow, \"wondering upon him. He\nwas lodged in the house of William Delect, a citizen of London, in\nFenchurch-street. On the morrow, being the eve of St. Bartholomew, he\nwas brought on horseback to Westminster, John Segrave and Geoffrey,\nknights, the mayor, sheriffs, and aldermen of London, and many others,\nboth on horseback and on foot, accompanying him; and in the great\nhall at Westminster, he being placed on the south bench, crowned with\nlaurel--for that he had said in times past that he ought to bear a\ncrown in that hall, as it was commonly reported--and being appeached\nfor a traitor by Sir Peter Malorie, the king's justice, he answered,\nthat he was never traitor to the king of England, but for other things\nwhereof he was accused, he confessed them.\" These circumstantial and\nminute details, inartificially as they are put together, and homely or\ntrivial as some of them may be thought, are yet full of interest for\nall who would call up a living picture of the scene. Wallace was put to\ndeath as a traitor, on the 23rd of August, 1305, at the usual place of\nexecution--the Elms in West Smithfield. He was dragged thither at the\ntails of horses, and there hanged on a high gallows, after which, while\nhe yet breathed, his bowels were taken out and burnt before his face.\nThe barbarous butchery was then completed by the head being struck\noff, and the body being divided into quarters. The head was afterwards\nplaced on a pole on London-bridge; the right arm was sent to be set\nup at Newcastle, the left arm to Berwick, the right foot and limb to\nPerth, and the left to Aberdeen.\n\n\nAN ELEPHANT DETECTS A ROBBER.\n\nAn officer in the Bengal army had a very fine and favourite elephant,\nwhich was supplied daily in his presence with a certain allowance of\nfood, but being compelled to absent himself on a journey, the keeper of\nthe beast diminished the ration of food, and the animal became daily\nthinner and weaker. When its master returned, the elephant exhibited\nthe greatest signs of pleasure; the feeding time came, and the keeper\nlaid before it the former full allowance of food, which it divided into\ntwo parts, consuming one immediately, and leaving the other untouched.\nThe officer, knowing the sagacity of his favourite, saw immediately the\nfraud that had been practiced, and made the man confess his crime.\n\n\nMAY-POLES.\n\n[Illustration: [++] Village May-pole.]\n\nThe May-pole, decked with garlands, round which the rustics used to\ndance in this month, yet stands in a few of our villages through the\nwhole circle of the year. A May-pole formerly stood in the Strand,\nupon the site of the church by Somerset House, but was taken down in\n1717. The village May-pole we engrave still remains by the ruins of St.\nBriavel Castle, Forest of Dean, Gloucestershire, and forms an object\nof considerable interest to the visitor. Several in the village could\nremember the May-day dancers, and the removal and setting up of the\nMay-pole. No notice whatever of this old English festival has, however,\nbeen taken for some years. The May-pole is about sixty feet high; about\nhalf-way up is the rod to which it was usual to fasten the garlands\nand ribbons. Let us observe, that in many parts of Dean Forest, those\nwho love to trace the remains of old manners and customs will find\nample employment. The people are civil and hospitable; their manner of\naddress reminds us of the wording of the plays of Shakspere's times;\nand in most houses, if a stranger calls, cider and bread are offered,\nas in the olden time.\n\n\nTHE OLD DOG WHEEL.\n\n[Illustration: [++] Old Dog Wheel.]\n\nAbout a century and a half ago, the long-backed \"turnspit\" dog, and\nthe curious apparatus here shown, yclept the \"Old Dog Wheel,\" were\nto be found in most farm houses; simple machinery has, however, now\nbeen substituted for the wheel which the dog was made to turn round,\nlike the imprisoned squirrels and white mice of the present day; and\nnot only the dog wheels, but also the long-backed \"turnspit\" dog have\nalmost disappeared. That which we engrave, however, still exists, and\nmay be seen by the curious, at the Castle of St. Briavel, which stands\non the borders of the Forest of Dean, in Gloucestershire.\n\n\nABRAHAM AND SARAH.\n\nThe Talmudists relate that Abraham, in travelling to Egypt, brought\nwith him a chest. At the custom-house the officers exacted the duties.\nAbraham would have readily paid them, but desired they would not open\nthe chest. They first insisted on the duties for clothes, which Abraham\nconsented to pay; but then they thought by his ready acquiescence that\nit might be gold. Abraham consents to pay for gold. They now suspect\nit might be silk. Abraham was willing to pay for silk, or more costly\npearls--in short, he consented to pay as if the chest contained the\nmost valuable of things. It was then they resolved to open and examine\nthe chest; and, behold, as soon as the chest was opened, that great\nlustre of human beauty broke out which made such a noise in the land\nof Egypt--it was Sarah herself! The jealous Abraham, to conceal her\nbeauty, had locked her up in this chest.\n\n\nAGES OF CELEBRATED MEN.\n\nHippocrates, the greatest physician the world has ever seen, died at\nthe age of one hundred and nine, in the island of Cos, his native\ncountry. Galen, the most illustrious of his successors, reached the\nage of one hundred and four. The three sages of Greece, Solon, Thales,\nand Pittacus, lived for a century. The gay Democritus outlived them\nby two years. Zeno wanted only two years of a century when he died.\nDiogenes ten years more; and Plato died at the age of ninety-four,\nwhen the eagle of Jupiter is said to have borne his soul to heaven.\nXenophon, the illustrious warrior and historian, lived ninety years.\nPolemon and Epicharmus ninety-seven; Lycurgus eighty-five; Sophocles\nmore than a hundred. Gorgias entered his hundred and eighth year;\nand Asclepiades, the physician, lived a century and a half. Juvenal\nlived a hundred years; Pacuvius and Varro but one year less. Carneades\ndied at ninety; Galileo at sixty-eight; Cassini at ninety-eight; and\nNewton at eighty-five. In the last century, Fontenelle expired in\nhis ninety-ninth year; Buffon in his eighty-first; Voltaire in his\neighty-fourth. In the present century, Prince Talleyrand, Goethe,\nRogers, and Niemcewicz are remarkable instances. The Cardinal du Belloy\nlived nearly a century; and Marshal Moncey lately terminated a glorious\ncareer at eighty-five.\n\n\nEFFECT OF A NEW NOSE.\n\nVan Helmont tells a story, of a person who applied to Taliacotius to\nhave his nose restored. This person, having a dread of an incision\nbeing made in his own arm, for the purpose of removing enough skin\ntherefrom for a nose, got a labourer, who, for a remuneration, suffered\nthe skin for the nose to be taken from his arm. About thirteen months\nafter, the adscitious nose suddenly became cold, and, after a few days,\ndropped off, in a state of putrefaction. The cause of this unexpected\noccurrence having been investigated, it was discovered that, at the\nsame moment in which the nose grew cold, the labourer at Bologna\nexpired.\n\n\nFRENCH DRESS.\n\nSigebert was buried in St. Medrad's church, at Soissons, where his\nstatue is still seen in long clothes, with the mantle, which the Romans\ncalled _chlamys_. This was the dress of Colvil's children, whether as\nmore noble and majestic, or that they looked on the title of Augustus\nas hereditary in their family. However it be, long clothes were, for\nseveral ages, the dress of persons of distinction, with a border of\nsable, ermine, or miniver. Under Charles V. it was emblazoned with all\nthe pieces of the coat of arms. At that time, neither ruffs, collars,\nnor bands were known, being introduced by Henry II. 'Till this time\nthe neck of the French king was always quite bare, except Charles\nthe _Wise_, who is everywhere represented with an ermin collar. The\nshort dress anciently worn in the country and the camp, came to be the\ngeneral fashion under Louis XI. but was laid aside under Louis XII.\nFrancis I. revived it, with the improvement of flashes. The favourite\ndress of Henry II. and his children was a tight, close doublet, with\ntrunk hose, and a cloak scarce reaching the waist. The dress of French\nladies, it may be supposed, had likewise its revolutions. They seem\nfor nine hundred years, not to have been much taken up with ornament.\nNothing could require less time or nicety than their head-dress, and\nthe disposition of their hair. Every part of their linen was quite\nplain, but at the same time extremely fine. Laces were long unknown.\nTheir gowns, on the right side of which was embroidered their husband's\ncoat of arms, and on the left that of their own family, were so close\nas to shew all the delicacy of their shape, and came up so high as to\ncover their whole breast, up to their neck. The habit of widows was\nvery much like that of the nuns. It was not until Charles VI. that they\nbegan to expose their shoulders. The gallantry of Charles the VII.'s\nCourt brought in the use of bracelets, necklaces, and ear-rings. Queen\nAnne de Bretagne despised those trinkets; and Catherine de Medicis made\nit her whole business to invent new.\n\n\nA LAST CHANCE.\n\nJohn Jones and Jn. Davis, condemn'd for robberries on the highway,\nwere executed at Tyburn. Davis feign'd himself sick, and desir'd he\nmight not be ty'd in the cart: But when he came to the tree, while the\nhangman was fastening the other's halter, he jumpt out of the cart,\nand ran over two fields; but being knock'd down by a countryman, was\nconvey'd back and hang'd without any more ceremony. Jones confessed\nhe had been confederate in several robberies with Gordon, lately\nexecuted.--_Gentleman's Magazine 1733._\n\nA convict running away over two fields at Tyburn, and then being caught\nby a countryman! How strange this seems, when we look at the streets\nand squares which now cover the locality, and when the only countrymen\nnow seen there are those who come up from the rural districts!\n\n\nYELLOW HAIR IN THE TIME OF THE PLANTAGENETS.\n\nYellow hair was at this time esteemed a beauty, and saffron was used by\nthe ladies to dye it of a colour esteemed \"odious\" by modern ladies.\nElizabeth also made yellow hair fashionable, as hers was of the same\ntint. In the romance of _King Alisaunder_, we are told of Queen\nOlympias:--\n\n \"Hire yellow hair was fair atyred\n With riche strings of gold wyred,\n And wryen hire abouten all\n To hire gentil myddel small.\"\n\n\nTHE CITY OF THE SULTAN.\n\n[Illustration: The Mosque of St. Sophia.]\n\nThe Mosques of Constantinople are the most wonderful objects of\nthat renowned city. More than 300 are picturesquely distributed in\nconspicuous parts, and form a most attractive feature to the eye of\nthe traveller. The city itself is built upon seven gentle hills,\nwhich is the main cause not only of its grandeur of appearance, but\nalso of its salubrity and comparative cleanliness. There are fourteen\nchief or imperial mosques, all lofty, and magnificent in their general\ndimensions, and built from base to dome, of enduring materials, chiefly\nof white marble, slightly tinged with grey. Some of these have two,\nsome four, and one (that of Sultan Achmet) has even six of those light,\nthin, lofty, arrowy, and most graceful towers called minarets. The\nmosque of Santa Sophia was once a Christian cathedral, and is rich in\nhistorical recollections. This mosque ranks as one of the grandest\nedifices. The ridge of the first hill on which the city stands, setting\nout from the north eastern part, is covered by the Serai or palace of\nthe Sultan, behind which, a little on the reverse of the hill, the dome\nof Santa Sophia shows itself. The colleges and hospitals, which are\ngenerally attached to or near the great mosques, offer no striking\narchitectural features; but some of the detached chapels or sepulchres\n(_turbes_), where sultans, viziers, and other great personages repose,\nare handsome.\n\n\nGOLD MASK FROM THE BANKS OF THE EUPHRATES.\n\n[Illustration: Mask of Nebuchadnezzar.]\n\nThis interesting relic of remote antiquity is at present preserved in\nthe Museum of the East India Company. It was found by Colonel Rawlinson\nwhile engaged in prosecuting the discoveries commenced by Layard and\nBotta, at Nineveh and Babylon; and is supposed to have belonged to\nKing Nebuchadnezzar. In exhuming from the mounds of these long-lost\nrival cities, the instructive remains of this once gigantic Power, the\nColonel discovered, in a perfect state of preservation, what is well\nbelieved to be the mummy of Nebuchadnezzar. The face of the rebellious\nmonarch of Babylon, covered by one of those gold masks usually found in\nAssyrian tombs, is described as very handsome--the forehead high and\ncommanding, the features marked and regular. The mask is of thin gold,\nand independent of its having once belonged to the great monarch, has\nimmense value as a relic of an ancient and celebrated people.\n\nThe Arab tribes encamping about Wurka and other great mounds search\nin the loose gravel with their spears for coffins. Gold and silver\nornaments, which have been buried in these graves for centuries, are\nworn by the Arab women of the present day; and many a rare object\nrecovered from them is sold and melted by the goldsmiths of the East.\nThe Arabs mention the discovery, by some fortunate shepherd, of Royal\ntombs, in which were crowns and sceptres of solid gold.\n\n\nFROST FAIR ON THE THAMES.\n\n\"I went crosse the Thames,\" says Evelyn, January 9, 1683-4, \"on the\nice, which now became so thick as to bear not only streetes of boothes,\nin which they roasted meate, and had divers shops of wares, quite\nacrosse as in a towne, but coaches, carts, and horses passed over.\nSo I went from Westminster Stayres to Lambeth, and din'd with the\nArchbishop. I walked over the ice (after dinner) from Lambeth Stayres\nto the Horseferry.\n\n\"The Thames (Jan{y} 16) was filled with people and tents, selling all\nsorts of wares as in a citty. The frost (Jan{y} 24) continuing more and\nmore severe, the Thames before London was still planned with boothes\nin formal streetes, all sorts of trades and shops furnished and full\nof commodities, even to a printing-presse, where the people and ladyes\ntook a fancy to have their names printed on the Thames. This humour\ntook so universally, that 'twas estimated the printer gained L5 a-day,\nfor printing a line only, at sixpence a day, besides what he got by\nballads, &c. Coaches plied from Westminster to the Temple, and from\nseveral other staires to and fro, as in the streetes, sleds, sliding\nwith skeates, a bull-baiting, horse and coach races, puppet playes and\ninterludes, cookes, tipling, and other lewd places, so that it seem'd\nto be a bacchanalian triumph, or carnival on the water.\"\n\n\"It began to thaw (Feb. 5), but froze againe. My coach crossed from\nLambeth to the Horseferry at Millbank, Westminster. The booths were\nalmost all taken down; but there was first a map, or landskip, cut\nin copper, representing all the manner of the camp, and the several\nactions, sports, and pastimes thereon, in memory of so signal a frost.\"\n\n\nTHE CHARACTER OF THE MOUTH.\n\nWe give the following extract from a very old work; not only because\nit contains several shrewd observations, but also because it is a good\nspecimen of the spelling and diction which prevailed in the sixteenth\ncentury, at which period there is internal evidence that the book was\nwritten, though it bears no date on the title page:--\n\n\"The mouth greate and wyde betokeneth wrath, boldnes and warre.\nAnd such men are commonly glottons. A wyde mouth withoute meesure,\nas thought it were cutte and stretched out, sygnifieth ravening\ninhumanitie, wickednes, a warlyke hart and cruell, like unto beastes\nof the sea. Such men are greate talkers, boasters, babblers, enuious,\nlyars, and full of follye. The mouthe that hathe but a lyttle closynge\nand a lyttle openynge, sygnyfyeth a fearful man, quyet, and yet\nunfaithfull. The mouthe that is verye apparent and rounde with thycknes\nof lyppes, sygnyfyeth vnclenlynes, follye, and cruelltye. The mouth\nwhyche hath a quantitie in his sytuation with a lyttle shutting, and\nsmylynge eyes wyth the reste of the face, sygnyfyeth a carnall man,\na lover of daunces, and a greate lyar. When the mouthe turneth in\nspeakinge it is a sygne that it is infected with some catarre or murre\nas is manyfest ynough. The long chynne declareth the man to be very\nlyttle subiecte to anger, and of a good complexion: and yet he is\nsomewhat a babbler and a boaster of hymselfe. They that have a lyttle\nchinne, are much to be avoyded and taken heede of, for besydes all\nvices with the whyche they are fylled they are full of impietye and\nwyckednes and are spyes, lyke unto serpents. If the ende of the chynne\nbe round it is a sygne of feminine maners and also it is a sygne of a\nwoman. But the chynne of a man muste be almoste square.\"--\"_The most\nexcellent, profitable, and pleasant booke of the famous doctour and\nexpert Astrologien Arcandam or Aleandrin._\" * * *. _Now ready turned out\nof French into our vulgare tonge, by Will. Warde. Black letter. No\ndate._ Printed by J. Rowbothum.\n\n\nEXECUTION OF EARL FERRERS FOR MURDER, 1760.\n\nLord Ferrers was hung for the deliberate and cruel murder of his\nsteward, Mr. Johnson, and his execution at Tyburn furnishes a curious\ninstance of the exhibition of egregious vanity in a man who was just\nabout to meet an ignominious death, and of misplaced pride in his\nfamily who could actually decorate the scaffold with the emblems of\nrespectful mourning.\n\nHis lordship was dressed in his wedding-clothes, which were of light\ncolour, and embroidered in silver. He set out from the Tower at nine\no'clock, amidst crowds of spectators. First went a large body of\nconstables, preceded by one of the high constables; next came a party\nof grenadiers and a party of foot; then the sheriff, in a chariot\nand six, the horses dressed with ribbons; and next, Lord Ferrers, in\na landau and six, escorted by parties of horse and foot. The other\nsheriff's carriage followed, succeeded by a mourning-coach and six,\nconveying some of the malefactor's friends; and lastly, a hearse and\nsix, provided for the purpose of taking the corpse from the place of\nexecution to Surgeons' Hall.\n\nThe procession was two hours and three-quarters on its way. Lord\nFerrers conversed very freely during the passage. He said, \"the\napparatus of death, and the passing through such crowds of people,\nare ten times worse than death itself; but I suppose they never saw a\nlord hanged, and perhaps they will never see another.\" He said to the\nsheriff. \"I have written to the king, begging that I might suffer where\nmy ancestor, the Earl of Essex, the favourite of Elizabeth, suffered,\nand was in great hopes of obtaining that favour, as I have the honour\nof being allied to his Majesty, and of quartering part of the royal\narms. I think it hard that I must die at the place appointed for the\nexecution of common felons.\"\n\nThe scaffold was hung with black by the undertaker, at the expense of\nLord Ferrers' family. His lordship was pinioned with a black sash,\nand was unwilling to have his hands tied, or his face covered, but\nwas persuaded to both. On the silken rope being put round his neck,\nhe turned pale, but recovered instantly. Within seven minutes after\nleaving the landau, the signal was given for striking the stage, and in\nfour minutes he was quite dead. The corpse was subjected to dissection.\n\n\nSTRANGE FUNERAL OBSEQUIES.\n\nThe following, taken from an old magazine, is a singular manifestation\nof eccentricity in a person who, from the books he selected to be\nburied with him, was evidently a man of an educated and refined mind:--\n\nDied, May 4, 1733, Mr. John Underwood, of Whittlesea, in\nCambridgeshire. At his burial, when the service was over, an arch was\nturn'd over the coffin, in which was placed a small piece of white\nmarble, with this inscription, \"_Non omnis moriar_, 1733.\" Then the six\ngentlemen who follow'd him to the grave sung the last stanza of the\n20th Ode of the 2d book of Horace. No bell was toll'd, no one invited\nbut the six gentlemen, and no relation follow'd his corpse; the coffin\nwas painted green, and he laid in it with all his cloaths on; under his\nhead was placed Sanadon's \"Horace,\" at his feet Bentley's \"Milton;\"\nin his right hand a small Greek Testament, with this inscription in\ngold letters, \"eimientobaus [Greek: ei mi en to bausa], J. U,\" in his\nleft hand a little edition of \"Horace\" with this inscription, \"_Musis\nAmicus_, J. U.;\" and Bentley's \"Horace\" under his back. After the\nceremony was over they went back to his house, where his sister had\nprovided a cold supper; the cloth being taken away the gentlemen sung\nthe 31st Ode of the 1st Book of \"Horace,\" drank a chearful glass,\nand went home about eight. He left about 6,000_l._ to his sister, on\ncondition of her observing this his will, order'd her to give each of\nthe gentlemen ten guineas, and desir'd they would not come in black\ncloaths. The will ends thus, \"Which done I would have them take a\nchearful glass, and think no more of John Underwood.\"\n\n\nQUICK TRAVELLING IN OLD TIMES.\n\nSaturday, the seventeenth day of July, 1619, Bernard Calvert, of\nAndover, about three o'clock in the morning, tooke horse at St.\nGeorge's Church in Southwarke, and came to Dover about seaven of the\nclocke the same morning, where a barge, with eight oares, formerly sent\nfrom London thither, attended his suddaine coming: he instantly tooke\nbarge, and went to Callice, and in the same barge returned to Dover,\nabout three of the clocke the same day, where, as well there as in\ndiverse other places, he had layed sundry swift horses, besides guides:\nhe rode back from thence to St. George's Church in Southwarke the\nsame evening, a little after eight o'clock, fresh and lusty.--_Stow's\nAnnals._\n\n\nEDDYSTONE LIGHTHOUSE.\n\n[Illustration: The Eddystone Lighthouse.]\n\nAs the arts and sciences improved, so did the construction of\nLighthouses, until one of the greatest accomplishments of engineering\nskill, ever attempted upon such works, was exhibited in the\nconstruction of the Eddystone Lighthouse, which is, indeed, much\nmore entitled than the Pharos of Alexandria to be considered one of\nthe wonders of the world. The rock on which this tower is built is\nplaced about twelve miles south-west of Plymouth, and consists of a\nseries of submarine cliffs, stretching from the west side (which is so\nprecipitous that the largest ship can ride close beside them) in an\neasterly direction, for nearly half a mile. At the distance of about\na quarter of a mile more is another rock, so that a more dangerous\nmarine locality can hardly be imagined. Both these rocks had proved the\ncause of many fatal shipwrecks, and it was at last resolved to make an\nattempt to obviate the danger. In the year 1696, a gentleman of Essex,\nnamed Winstanley, who had a turn for architecture and mechanics, was\nengaged to erect a lighthouse upon the Eddystone rock, and in four\nyears he completed it. It did not, however, stand long, for while some\nrepairs were in progress under his direction in 1703, on the 26th\nNovember, a violent hurricane came on which blew the lighthouse down,\nand Mr. Winstanley and all his workmen perished--nothing remaining of\nthe edifice but a few stones and a piece of iron chain.\n\nIn the spring of 1706 an Act of Parliament was obtained for rebuilding\nthe lighthouse, and a gentleman named Rudyerd, a silk mercer, was the\nengineer engaged. He placed five courses of heavy stones upon the\nrock and then erected a superstructure of wood. The lighthouse on the\nBell Rock, off the coast of Fife, and the one placed at the entrance\nof the Mersey on the Black Rock, are similarly constructed, so that\nthere seemed to be good reason for adopting the principle. Mr. Smeaton\nthought that the work was done in a masterly and effective manner; but\nin 1755 the edifice was destroyed by fire, and he was next retained as\nthe engineer for this important building.\n\nThe result of his labours has justly been considered worthy of the\nadmiration of the world, for it is distinguished alike for its\nstrength, durability, and beauty of form. The base of the tower is\nabout twenty-six feet nine inches in diameter, and the masonry is so\nformed as to be a part of the solid rock, to the height of thirteen\nfeet above the surface, where the diameter is diminished to nineteen\nfeet and a half. The tower then rises in a gradually diminishing curve\nto the height of eighty-five feet, including the lantern, which is\ntwenty-four feet high. The upper extremity is finished by a cornice, a\nbalustrade being placed around the base of the lantern for use as well\nas ornament.\n\nThe tower is furnished with a door and windows, and the whole edifice\noutside bears the graceful outline of the trunk of a mighty tree,\ncombining lightness with elegance and strength. Mr. Smeaton commenced\nhis labours in 1756, and completed the building in four years. Before\ncommencing operations he took accurate drawings of the exterior of the\nrock, and the stones, which were brought from the striking and romantic\ndistrict of Dartmoor, were all formed to fit into its crevices, and so\nprepared as to be dovetailed together, and strung by oaken plugs. When\nput into their places, and then firmly cemented, the whole seemed to\nform, and does indeed constitute, a part of the solid rock.\n\n\nSWEATING SICKNESS.\n\nThe Sweating Sickness first visited England Anno Dom. 1483, and\nrepeated its visitations 1485, 1506, 1517, 1528, and last of all, 1551.\n\nThis epidemic disease raged with such peculiar violence in England,\nand had so quick a crisis, that it was distinguished by the name of\n_Ephemera Britannica_. This singular fever seems to have been of the\nmost simple, though of the most acute kind, and notwithstanding princes\nand nobles were its chief victims, the physicians of the day never\nagreed upon the method of treating it.\n\nThe splendid French embassy, which arrived in England in 1550, found\nthe court-festivities damped by a visitation of that strange and\nterrific malady.\n\n\"This pestilence, first brought into the island by the foreign\nmercenaries who composed the army of the Earl of Richmond, afterwards\nHenry VII., now made its appearance for the fourth and last time in\nour annals. It seized principally, it is said, on males, on such as\nwere in the prime of their age, and rather on the higher than the\nlower classes: within the space of twenty-four hours, the fate of the\nsufferer was decided for life or death. Its ravages were prodigious;\ntwo princes died of it; and the general consternation was augmented,\nby a superstitious idea which went forth, that Englishmen alone were\nthe destined victims of this mysterious minister of fate, which tracked\ntheir steps, with a malice and sagacity of an evil spirit, into every\ndistant country of the earth whither they might have wandered, whilst\nit left unassailed all foreigners in their own.\"\n\n\nAN AMERICAN ADVERTISEMENT.\n\nThe following is an early specimen of that system of poetical\nadvertising which in recent times has become so common. It is always\ninteresting to note the origin of customs with which we subsequently\nbecome familiar:--\n\n_Notice to the Public, and especially to Emigrants, who wish to settle\non Lands._--The Subscriber offers for Sale, several Thousand Acres\nof Land, situated in well settled Front Townships, in Lots to suit\nPurchasers.\n\n Particulars about Location,\n May be known by application.\n For quality of soil, and so forth,\n Buyers to see, on Nag must go forth.\n This much I'll tell ye plainly,\n Of big trees ye'll see mainly.\n 'Bout Butter Nut and Beach,\n A whole week I could preach;\n But what the plague's the use of that?\n The lands are nigh, low, round, and flat.\n There's rocks and stumps, no doubt enough,\n And bogs and swamps, just _quantum-suff_\n To breed the finest of Musquitoes;\n As in the sea are bred Bonitos,\n No lack of fever or of ague;\n And many other things to plague you.\n In short, they're just like other people's,\n Sans houses, pigsties, barns, or steeples\n What most it imports you to know,\n 'S the terms on which I'll let 'em go.\n So now I offer to the Buyer,\n A Credit to his own desire,\n For butter, bacon, bread, and cheese,\n Lean bullocks, calves, or ducks and geese,\n Corn, _Tates_, flour, barley, rye,\n Or any thing but _Punkin-Pie_.\n In three, four years, _Aye, five or six_,\n If that won't do, why let _him_ fix.\n But when once fix'd, if payment's slack,\n As sure as Fate, I'll take 'em back.\n\n THOMAS DALTON.\n\n Kingston Brewery, (Canada,) Nov. 2, 1821.\n\n\nMAGNIFICENCE OF FORMER TIMES.\n\n_Account how the Earl of Worcester lived at Ragland Castle in\nMonmouthshire, before the Civil Wars, which began in 1641._\n\nAt eleven o'clock in the forenoon, the Castle gates were shut, and the\ntables laid; two in the dining-room; three in the hall; one in Mrs.\nWatson's apartment, where the chaplains are, (Sir Toby Mathews being\nthe first;) and two in the housekeeper's room for the lady's women.\n\nThe Earl came into the dining-room attended by his gentlemen. As soon\nas he was seated, Sir Ralph Blackstone, Steward of the house, retired.\nThe Comptroller, Mr. Holland, attended with his staff, as did the\nSewer, Mr. Blackburne; the daily waiters, Mr. Clough, Mr. Selby, and\nMr. Scudamore; with many gentlemen's sons, from two to seven hundred\npounds a year, bred up in the Castle; my Lady's Gentleman Usher, Mr.\nHarcourt; my Lord's Gentlemen of the Chamber, Mr. Morgan and Mr. Fox.\n\nAt the first table sat the noble family, and such of the nobility as\ncame.\n\nAt the second table, in the dining-room, sat Knights and Honourable\nGentlemen, attended by footmen.\n\nIn the hall, at the first table sat Sir Ralph Blackstone, Steward; the\nComptroller, Mr. Holland; the Secretary; the Master of the Horse, Mr.\nDelewar; the Master of the Fish Ponds, Mr. Andrews; my Lord Herbert's\nPreceptor, Mr. Adams; with such Gentlemen as came there under the\ndegree of a Knight, attended by footmen, and plentifully served with\nwine.\n\nAt the second table in the hall, (served from my Lord's table, and with\nother hot meats,) sat the Sewer, with the Gentlemen Waiters and Pages,\nto the number of twenty-four.\n\nAt the third table in the hall, sat the Clerk of the Kitchen, with the\nYeomen Officers of the House, two Grooms of the Chamber, &c.\n\nOther Officers of the Household were, Chief Auditor, Mr. Smith; Clerk\nof the Accounts, Mr. George Wharton; Purveyor of the Castle, Mr.\nSalsbury; Ushers of the Hall, Mr. Moyle and Mr. Croke; Closet Keeper,\nGentleman of the Chapel, Mr. Davies; Keeper of the Records; Master of\nthe Wardrobe; Master of the Armoury; Master Groom of the Stable for the\nWar Horses; Master of the Hounds; Master Falconer; Porter and his man.\n\nTwo Butchers; two Keepers of the Home Park; two Keepers of the Red Deer\nPark.\n\nFootmen, Grooms, and other menial Servants, to the number of 150. Some\nof the footmen were brewers and bakers.\n\nOut Officers.--Steward of Ragland, William Jones, Esq.; the Governor of\nChepstow Castle, Sir Nicholas Kemys, Bart.; Housekeeper of Worcester\nHouse, in London, James Redman, Esq.\n\nThirteen Bailiffs.\n\nTwo Counsel for the Bailiffs to have recourse to.\n\nSolicitor, Mr. John Smith.\n\n\nSADLER'S WELLS.\n\n\"T. G., Doctor in Physic,\" published, in 1684, a pamphlet upon\nthis place, in which he says:--\"The water of this well, before the\nReformation, was very much famed for several extraordinary cures\nperformed thereby, and was thereupon accounted sacred, and called\nHolywell. The priests belonging to the priory of Clerkenwell using to\nattend there, made the people believe that the virtue of the water\nproceeded from the efficacy of their prayers; but at the Reformation\nthe well was stopped, upon the supposition that the frequenting of\nit was altogether superstitious; and so by degrees it grew out of\nremembrance, and was wholly lost until then found out; when a gentleman\nnamed Sadler, who had lately built a new music-house there, and being\nsurveyor of the highways, had employed men to dig gravel in his garden,\nin the midst whereof they found it stopped up and covered with an arch\nof stone.\" After the decease of Sadler, Francis Forcer, a musician of\nsome eminence in his profession, became proprietor of the well and\nmusic-room; he was succeeded by his son, who first exhibited there the\ndiversions of rope-dancing and tumbling, which were then performed in\nthe garden. The rural vicinity of the \"Wells,\" long made it a favourite\nretreat of the pleasure-seeking citizens.\n\n[Illustration: Champion Figg.]\n\nJames Figg, a native of Thame, in Oxfordshire, was a man of remarkable\nathletic strength and agility, and signalized himself greatly over any\nof his country competitors in the art of cudgel-playing, single-stick,\nand other gymnastic exercises. Having acquired a considerable knowledge\nof the broadsword, he came to London, and set up as master in that\nscience, undertaking to teach the nobility and gentry of his day\nthe noble art of self defence; and championed himself against all\ncomers. He took a waste piece of ground, the corner of Wells and\nCastle-streets, Oxford-road, and erected a wooden edifice, which,\nin imitation of the Romans, he denominated an amphitheatre; and\nestablished here a regular academy, to train pupils in the practice of\ncudgeling, broadsword, &c. &c., as well to use it, on fixed occasions,\nfor the exhibition of prizefighting. He had many followers, and we find\nhim commemorated and praised by most of the wits of his time. \"The\nTattler,\" \"Guardian,\" and \"Craftsman,\" have equally contributed to\npreserve his memory, as have several writers. Bramstone, in his \"Man of\nTaste\" tells us:--\n\n \"In Figg the prize-fighter by day delight,\n And sup with Colley Cibber every night.\"\n\nAnother writer notices him in the following lines:--\n\n \"To Figg and Broughton he commits his breast,\n To steel it to the fashionable test.\"\n\nSutton, the pipe-maker of Gravesend, was his rival, and dared the\nmighty Figg to the combat. Twice they fought, with alternate advantage;\nbut, at the third trial, a considerable time elapsed before victory\ndecided for either party; at length the palm of victory was obtained\nby Figg. In short, neither Ned Sutton, Tom Buck, nor Bob Stokes, could\nresist, or stand against his skill and valour. He was never defeated\nbut once, and then by Sutton, in one of their previous combats, and\nthat was generally supposed to have been in consequence of an illness\nhe had on him at the time he fought.\n\nWhen Faber engraved his portrait from a painting by Ellys, he was at\na loss what he should insert, as an appropriate motto, and consulting\nwith a friend what he should put, was answered, \"_A Figg for the\nIrish._\" This was immediately adopted, and the print had a rapid sale.\n\nFigg died in 1734. William Flander a noted scholar of his, fought at\nthe amphitheatre, in 1723, with Christopher Clarkson, from Lancashire,\nwho was called the Old Soldier. The fashion of attending prizefighting\nmatches had attained its highest zenith in Figg's time, and it was\nlooked upon as a very great proof of self-denial in an amateur if he\nfailed a meeting on those occasions.\n\n From Figg's theatre he will not miss a night,\n Though cocks, and bulls, and Irish women, fight.\n\nFigg left a widow and several children; so recently as 1794 a\ndaughter-in-law of his was living, and resided in Charles-street,\nWestminster, where she kept a house, and supported herself very\ndecently by letting lodgings, aided by a very small income.\n\n\nDRESS IN 1573.\n\nThe wardrobe of a country gentleman is thus given from a will, dated\n1573, in the Prerogative Court of Canterbury, in Brayley and Britton's\n_Graphic Illustrator_--\"I give unto my brother Mr. William Sheney\nmy best black gown, garded and faced with velvet, and my velvet cap;\nalso I will unto my brother Thomas Marcal my new shepe gowne,\ngarded with velvet and faced with cony; also I give unto my son Tyble\nmy shorte gown, faced with wolf (skin), and laid with Billements lace;\nalso I give unto my brother Cowper my other shorte gowne, faced with\nfoxe; also I give unto Thomas Walker my night gown, faced with cony,\nwith one lace also, and my ready (ruddy) hose; also I give\nunto my man Thomas Swaine my doublet of canvas that Forde made me, and\nmy new gaskyns that Forde made me; also I give unto John Wyldinge a\ncassock of shepes colour, edged with ponts skins; also I give unto John\nWoodzyle my doublet of fruite canvas, and my hose with fryze bryches;\nalso I give unto Strowde my frize jerkin with silke buttons; also I\ngive Symonde Bisshoppe, the smyth, my other frize jerkyn, with stone\nbuttons; also I give to Adam Ashame my hose with the frendge (fringe),\nand lined with crane- silk; which gifts I will to be delivered,\nimmediately after my decease.\"\n\n\nORIGIN OF THE CREST OF THE PRINCE OF WALES.\n\nThe loss of the French at the battle of Crecy was immense. There fell\n1,200 knights; 1,400 esquires; 4,000 commissioned officers; 30,000 rank\nand file; Dukes of Lorraine and Bourbon; Earls of Flanders, Blois,\nHarcourt, Vaudemont, and Aumale; the King of Bohemia; the King of\nMajorca. The English lost one esquire, three knights, and less than one\nhundred rank and file. Here did they first use field artillery; and\non this battle-field did the young Prince of Wales adopt the ostrich\nplumes and motto of the slain King of Bohemia, who, being blind,\ndesired to be led at a gallop between two knights into the thick of\nthe fight, and thus met death. Those feathers and the two words \"Ich\ndien,\" \"I serve,\" are to this day the heraldic bearings of the Prince\nof Wales, whom God preserve! So much for Crecy or Cressy!\n\n\nSINGULAR DISCOVERY OF A THIEF IN 1822.\n\nOn February 20, as a servant in the employ of J. L. King, Esq., of\nStogumber, was entering a field, his attention was attracted by a\nmagpie, which appeared to have escaped from a neighbouring house. The\nbird spoke so uncommonly plain that the man was induced to follow it.\n\"_Cheese for Marget, Cheese for Marget_,\" was its continual cry, as\nit hopped forward, till it stopped behind a hay-stack, and began to\neat. On inspection, a number of hams, a quantity of cheese, &c., were\ndiscovered, which had been stolen, a short time previously, from Mr.\nBowering, of Williton. The plunder was deposited in sacks, on one of\nwhich was marked the name of a person residing in the neighbourhood,\nwhich led to the apprehension of four fellows, who have been committed\nto Wilton gaol.\n\n\nEFFECT OF VINEGAR ON THE SKIN.\n\nBy the use of vinegar the Spanish General Vitellis, made his skin hang\nabout him like a pelisse; but of the wonderful dilatability of the\nskin, no instance equals the Spaniard who showed himself to Van-Horn,\nSilvius, Piso, and other learned men at Amsterdam. Taking up with his\nleft hand the skin of his right shoulder, he would bring the same up\nto his mouth: again he would draw the skin of his chin down to his\nbreast like a beard, and presently put it upwards to the top of his\nhead, hiding both his eyes therewith; after which, the same would\nreturn orderly and equally to its proper place.\n\n\nADVERTISEMENT OF A DYING SPEECH BOOK IN 1731.\n\nNewgate literature was more popular in the last century than it is now.\nThe following is an advertisement in the _Gentleman's Magazine_ of the\nabove date:--\n\n\"A General History of Executions for the year, 1730. Containing the\nlives, actions, dying speeches, confessions and behaviour, of sixty\nmalefactors executed at Tyburn, and elsewhere; particularly three\nunfortunate young gentlemen, viz., Mr. Goodburn, a Cambridge scholar,\nMr. Johnston, and Mr. Porter, son to the late Lord Mayor of Dublin:\nand of several notorious highwaymen, foot-pads, street-robbers, and\nhousebreakers, as Dalton, Everet, Doyle, Newcomb, &c., and of the five\nyoung highwaymen taken at Windsor, said to have formed a design to rob\nthe Queen there. To which is added, the trial of William Gordon at\nChelmsford for a robbery on the highway; an account of the incendiaries\nat Bristol, and the apprehending John Power, for sending threatening\nletters, and firing Mr. Packer's house; also the life of Col. Ch--s.\nTogether with an alphabetical list of all the persons indicted or\ntried at the Old Bailey, the year past. With the judgment of the court\nrespectively passed upon each, referring to the pages in the session\nbooks for the trials at large. Printed for R. Newton at St. John's\nGate, and sold by the booksellers price bound 2_s._ 6_d._\"\n\n\nADVERTISEMENT OF A FLEET PARSON.\n\nIn the last century, when marriages were allowed to be transacted--we\ncannot say solemnized--in the Fleet Prison, and the adjacent taverns,\nthe profligate wretches who disgraced their sacred profession by taking\npart in such iniquities, were obliged to bid against one another for\ncustom--here is one of their advertisements:--\n\n G. R.\n\n At the true Chapel\n at the old Red Hand and Mitre, three doors from Fleet Lane and\n next Door to the White Swan;\n Marriages are performed by authority by the Reverend Mr. Symson\n educated at the University of Cambridge, and late Chaplain to the\n Earl of Rothes.\n\n N.B. Without Imposition.\n\n\nTHE ASS.\n\nIn all countries, this sure-footed and faithful animal is adopted as\nan emblem of stupidity, from the patience with which it submits to\npunishment and endures privation. A pair of ass's ears is inflicted\nupon a child in reproof of his duncehood; and through life we hear\nevery blockhead of our acquaintance called an ass. Whereas the ass is\na beast of great intelligence; and we often owe our safety to its sure\nand unerring foot beside the perilous precipice, where the steps of the\nman of science would have faltered.\n\nThe Fathers of the Church, and the Disciples of the Sorbonne, persuaded\nof the universal influence of the Christian faith, believed the dark\ncross on the back of the ass to date only from the day on which our\nSaviour made his entry into Jerusalem. The ass of the desert was an\nanimal of great price. Pliny mentions that the Senator Arius paid\nfor one the sum of four hundred thousand sesterces. Naturalists have\nfrequently remarked the extraordinary dimensions of an ass's heart,\nwhich is thought an indication of courage; and it is the custom of the\npeasantry of some countries to make their children wear a piece of\nass's skin about their person. The ass's skin is peculiarly valuable,\nboth for the manufacture of writing-tablets and drums; which may be\nthe reason why a dead ass is so rarely seen. It is too valuable to be\nleft on the highway. In many places, the ass serves as a barometer.\nIf he roll in the dust, fine weather may be expected; but if he erect\nhis ears, rain is certain. Why should not these animals experience the\nsame atmospheric influences as man? Are we not light-hearted in the\nsunshine, and depressed in a heavy atmosphere?\n\n\nCHOICE RECEIPTS FROM \"PHYSICK FOR THE POOR. LONDON, 1657.\"\n\n_To make any one that Sleepeth answer to whatsoever thou ask._--Take\nthe heart of an oul, and his right legg, and put them upon the breast\nof one that sleepeth, and they shall reveal whatsoever thou ask them.\n\n_To know any Man or Woman's minde when they are Asleep._--Take the hart\nof a dove, and the legg of a frog, dry it well, and beat them to powder\nin a morter, put this up in a linnen cloth, with three or four round\npibble stones, as big as wallnuts, then lay this upon the parties pit\nof their stomach, and they shall tell you all things that they have\ndone, if there is anything remarkable that troubles them.\n\n_To make the Nose Bleed._--Take the leaves of yerrow, put it up in thy\nnose; this will make the nose bleed immediately.\n\n_To make a Tooth Drop out._--Mizaldus saith that if you make a powder\nof earth-worms and put it in the hollow of a rotten tooth, it will\nimmediately drop out.\n\nHow strange must have been the education and intelligence of the\nperiod, when people could write, publish, and practice such incredible\ntrash!\n\n\nSHOCKING DEPRAVITY.\n\nThe following account, from an old magazine, affords a strange and\nlamentable instance of a wretch just about to die, being only intent\nwith his latest breath to defame his own mother:--\n\nMary Lynn, condemn'd last Assizes for the County of Norfolk, was burnt\nto ashes at a stake, for being concern'd in the murder of her mistress;\nand Smith, the principal, was hang'd for the same fact. She deny'd her\nbeing guilty, and said Smith could clear her if he would. She behaved\nwith decency, and died penitent. Smith was drunk at the gallows; and\nseem'd to have but little sense either of his crime or punishment;\nhowever, desired all masters to pay their servants' wages on Saturday\nnight, that they might have money to spend, and not run in debt. Said,\n\"My mother always told me I should die in my shoes, but I will make her\na liar;\" so threw them off.\n\n\nPERSONAL CHARMS DISCLAIMED.\n\nIf any human being was free from personal vanity, it must have been\nthe second Duchess d'Orleans, Charlotte Elizabeth of Bavaria. In one\nof her letters (dated 9th August, 1718), she says, \"I must certainly\nbe monstrously ugly. I never had a good feature. My eyes are small, my\nnose short and thick, my lips broad and thin. These are not materials\nto form a beautiful face. Then I have flabby, lank cheeks, and long\nfeatures, which suit ill with my low stature. My waist and my legs\nare equally clumsy. Undoubtedly I must appear to be an odious little\nwretch; and had I not a tolerable good character, no creature could\nendure me. I am sure a person must be a conjuror to judge me by my eyes\nthat I have a grain of wit.\"\n\n\nCADER IDRIS.\n\nOn the very summit of Cader Idris there is an excavation in the solid\nrock, resembling a couch; and it is said that whoever should rest a\nnight in that seat, will be found in the morning either dead, raving\nmad, or endued with supernatural genius.\n\n\nOLD LONDON SIGNS.\n\nSome notion of the houses and shops of old London may be gathered by\na visit to Bell Yard, near Temple Bar; Great Winchester Street, near\nthe Bank; the wooden houses near Cripplegate Church; and a few other\ndistricts which were spared by the Great Fire of 1666. In Bell Yard,\nfor instance, the national feeling for improvement has from time to\ntime effected changes; the lattices of diamond-shaped lead-work, carved\npendants, and the projecting signs of the various tradesmen, have\ndisappeared, and here and there sheets of plate glass have been used,\nto give a somewhat modern appearance to the places of business. Still\nthe projecting and massive wood-work of the shops, and the peculiar\npicturesque appearance of the houses, cannot be altogether disguised;\nand if any of our readers, who may be curious in such matters, will\nwalk up Bailey's Court, on the west side of Bell Yard, he will there\nsee a group of wooden buildings exactly like the great mass which was\ncleared by the fire. In some of the pictures of London of about this\ntime, the shops of the various tradesmen were chiefly unglazed, and\nabove the door of each was suspended the silver swans; the golden\nswans; the chained swans; the golden heads; mitres; bells--black, red,\nwhite, and blue; rising and setting suns; moons of different phases;\nmen in the moon; sceptres; crowns, and many other devices, which, even\nat that time, were necessary to distinguish one shop from another. The\nchequers; St. George and the dragon; royal oaks; king's heads; and\ndouble signs, such as the horseshoe and magpie; bell and crown; bell\nand horns, and such like, were more particularly set apart for the use\nof the various hostelries. Everyone, however, who had a London shop of\nany kind or consequence, had his sign. Many of them were well carved in\nwood, and ornamented with emblazonry and gilding.\n\nNo doubt if it were possible to find at the present time the same\npicturesque architectural displays as were to be met with in London in\nQueen Elizabeth's days, our artistic friends would be able to pick up\nmany a nice subject for their pencils, but in those days there were\nplenty of drawbacks; the pavement was bad, the drainage was worse, and\nfrom the eaves of the houses and pents of the shops, streams of water\nran down in wet weather upon the wayfarers, and, by lodging in the\nthoroughfares, made the London streets something in the same state as\nthose of Agar Town and some other neglected parts of the metropolis.\nWe must not forget that in the days to which we allude there were no\nflagged footpaths, and that the only distinction from the horse and\ncart roads, and that for the foot passengers, was a separation by\nwooden posts, which, in genteel places, were made supports for chains.\nPeople, however, got tired of this bad state of things, and measures\nwere taken to put a stop to the streams of water from the roofs, &c.\nAfter the Great Fire, an enactment was made for an alteration in\nthe spouts, &c.; all barbers poles, and projecting signs, and other\nprojections were to be done away with, and other changes made for the\nbetter. Up to the reign of Queen Anne, we find, by reference to views\nof Cheapside and the neighbourhood of the Monument, that the projecting\nsigns were still in use; and that even at that recent date, many of\nthe London shops in the important neighbourhoods above mentioned were\nwithout glazing, and looked much like some of the greengrocers' sheds\nin use now in Bermondsey and some other places.\n\nSevere measures seem to have been at length taken against the\nprojecting signs, and most of them disappeared, and then it became\na most difficult matter either to address letters, or find a man's\nshop. In Dr. Johnson's day, he and other persons gave the address\n\"over against\" a particular sign, or so many doors from such a sign.\nIn consequence of this uncertainty, many houses in London, which from\ntheir association with eminent men would possess much interest now,\ncannot be pointed out; and it was a wonderful benefit to the metropolis\nwhen the plan of numbering the houses in each street was hit upon. But\nfor this, considering that the population has doubled in the last fifty\nyears, it is difficult to know how the genius of Rowland Hill would\nhave worked his plan of London post-office delivery, or business could\nbe carried on with any kind of comfort.\n\nThe booksellers and publishers seem to have been the last, with the\nexception of the tavern-keepers, to give up the old signs. After the\nGreat Fire, some of the ancient signs which were cut in stone, and\nwhich had escaped the conflagration, were got out of the ruins, and\nafterwards placed in the front of the plain, yet solid, brick buildings\nwhich were erected after that event. Some of these--the \"Chained\nBear,\" the \"Collared Swan,\" the \"Moon and Seven Stars,\" and \"Sun,\" in\nCheapside, and some others which we now engrave--are still preserved.\nThe carved wooden sign of the \"Man in the Moon,\" in Wych Street,\nStrand, is a rare example; and the \"Horse-shoe and Magpie,\" in Fetter\nLane, is one of the last of the suspended signs to be now found in the\nCity.\n\n[Illustration: [++] Painted Signs of London Taverns.]\n\nAmongst the painted signs of London taverns worth notice, is one in\nOxford-street (nearly opposite Rathbone-place), said to have been\npainted by Hogarth. The subject is \"a man loaded with mischief.\" He has\na stout woman on his shoulders, together with a monkey, magpie, etc.\nThe male figure shown in this street picture seems to bear up pretty\nwell under his burden.\n\n\nNARROW ESCAPE.--CALM RELIANCE ON PROVIDENCE.\n\nIn the year 1552, Francis Pelusius, of sixty-three years old, digging a\nwell forty foot deep in the hill of St. Sebastian, the earth above him\nfell in upon him to thirty-five foot depth; he was somewhat sensible\nbefore of what was coming, and opposed a plank, which by chance he had\nwith him, against the ruins, himself lying under it; by this means he\nwas protected from the huge weight of earth, and retained some room\nand breath to himself, by which he lived seven days and nights without\nfood or sleep, without any pain or sorrow, being full of hope, which\nhe placed in God only. Ever and anon he called for help, as being yet\nsafe, but was heard by none, though he could hear the motion, noise\nand words of those that were above him, and could count the hours as\nthe clock went. After the seventh day, he being all this while given\nfor dead, they brought a bier for his corpse, and when a good part of\nthe well was digged up, on a sudden they heard the voice of one crying\nfrom the bottom. At first they were afraid, as if it had been the\nvoice of a subterranean spirit; the voice continuing, they had some\nhope of his life, and hastened to dig to him, till at last, after they\nhad given him a glass of wine, they drew him up living and well, his\nstrength so entire that to lift him out he would not suffer himself to\nbe bound, nor would use any help of another. Yea, he was of so sound\nunderstanding, that, jesting, he drew out his purse and gave them\nmoney, saying _He had been with such good hosts, that for seven days it\nhad not cost him a farthing_.\n\n\nCEILING OF WHITEHALL.\n\nThe celebrated painting on the roof of the Banqueting House, has been\nrestored, re-painted, and refreshed, not fewer than three times. In the\nreign of James II., 1687, Parrey Walton, a painter of still life, and\nthe keeper of the king's pictures, was appointed to re-touch this grand\nwork of art, which had then (as appears by the Privy Council Book)\nbeen painted only sixty years. Walton was paid L212 for its complete\nrestoration, which sum was considered by Sir Christopher Wren, \"as very\nmodest and reasonable.\" It was restored a second time by the celebrated\nCipriani; and for a third time by a painter named Rigaud.\n\n\nBUNYAN'S BIBLE.\n\nJohn Bunyan's Bible (printed by Bill and Barker) bound in morocco, and\nwhich had been his companion during his twelve years' unjustifiable\nconfinement in Bedford gaol, where he wrote his \"Pilgrim's Progress,\"\nwas purchased at the sale of the library of the Rev. S. Palmer, of\nHackney, March, 1814, for the late Samuel Whitbread, Esq., for the\nsum of L21. This Bible, and the \"Book of Martyrs,\" are said to have\nconstituted the whole library of Bunyan during his imprisonment.\n\n\nSPECIMENS OF ROYAL GRANTS.\n\nIn 1206, King John grants to W. de Camville a licence to destroy game\nin any of the royal forests, which proves the origin of the Game Laws.\n\n1238. Henry III. gave 500_l._ to Baldwyn, Emperor of Constantinople.\n\n1342. King Edward III. forgives to the mayor and citizens of London the\nindignation and rancour of mind that he had conceived against them.\n\n1344. The king grants to Adam Thorp, the trimmer of his beard, certain\nlands at Eye, near Westminster. The scrupulous attention which Edward\nIII. paid to that ornament of his face, may be seen in his bronze\neffigy in Westminster Abbey, which was taken from a mask after his\ndeath.\n\n1409. The king settles on Joan of Navarre, his queen, 10,000_l._ per\nannum.\n\n1417. Henry V. grants to Joan Warin, his nurse, an annuity of 20_l._\nduring life.\n\n1422. The jewels which had belonged to King Henry V., and were valued\nat so large a sum as 40,000_l._, were delivered to Sir Henry Fitz Hugh,\nand his other executors, for the payment of his personal debts.\n\n1422. The \"Pysane,\" or great collar of gold and rubies, was pawned by\nthe king to his uncle, Cardinal Beaufort, who is supposed, at the time\nof his death, to have amassed more wealth than any subject in England.\n\n\nCOFFEE AND TEA.\n\nThe bill for attendance at the Dorchester Assizes in 1686 of Mr. John\nBragge, the town-clerk of Lyme, presents this novelty--the article\n_coffee_ is charged 2d. This may have been drunk at a coffee-house.\nCoffee was introduced from Turkey in 1650.\n\nAn advertisement in the \"Mercurius Politicus,\" Sept. 30, 1658,\ninstructs how \"That excellent and by all physitians approved _China_\ndrink, called by the Chineans Tcha, by other nations, _tay_ alias\n_tee_, is sold at the Sultana's Head Coffee-house, in Sweeting's-rents,\nby the Exchange, _London_.--\"\n\nThere was a \"cophee-house\" in St. Michael's-alley, Cornhill, about\n1657. Tea, coffee, and chocolate were placed under the excise. There\nwas no tax upon these commodities when imported, but when made into\ndrink, as tea was, at 8d. a gallon, and sold at these houses.\n\n\nREMARKABLE PRESERVATION OF HUMAN HAIR SINCE THE NORMAN PERIOD.\n\nIn 1839 a coffin was discovered in the abbey church of Romsey, which\nhad originally contained the body of a female of the above early time.\nThe bones had entirely decayed, but the hair, with its characteristic\nindestructibility, was found entire, and appeared as if the skull had\nonly recently been removed from it, retaining its form entire, and\nhaving plaited tails eighteen inches in length. It is still preserved\nin a glass case, lying upon the same block of oak which has been its\npillow for centuries.\n\n\nPUBLIC TASTE FOR CONJURING IN 1718.\n\nOne of the amusements of 1718 was the juggling exhibition of a\nfire-eater, whose name was De Hightrehight, a native of the valley\nof Annivi in the Alps. This tremendous person ate burning coals,\nchewed flaming brimstone and _swallowed_ it, licked a red-hot poker,\nplaced a red-hot heater on his tongue, kindled coals on his tongue,\nsuffered them to be blown, and broiled meat on them, ate melted pitch,\nbrimstone, bees-wax, sealing-wax, and rosin, with a spoon; and, to\ncomplete the business, he performed all these impossibilities five\ntimes _per diem_, at the Duke of Marlborough's Head, in Fleet-street,\nfor the trifling receipts of 2s. 6d., 1s. 6d., and 1s. Master\nHightrehight had the honour of exhibiting before Lewis XIV., the\nEmperor of Germany, the King of Sicily, the Doge of Venice, and an\ninfinite number of princes and nobles--and the Prince of Wales, who had\nnearly lost this inconceivable pleasure by the envious interposition of\nthe Inquisition at Bologna and in Piedmont, which holy office seemed\ninclined to try _their mode of burning_ on his _body_, leaving to him\nthe care of resisting the flames and rendering them harmless; but he\nwas preserved from the unwelcome ordeal by the interference of the\nDutchess Royal Regent of Savoy and the Marquis Bentivoglia.\n\n\nTHE TRIUMPHS OF SCIENCE AND PERSEVERANCE.\n\nDistance seems not to have entered into the calculations of the\nengineers who built those monuments of human skill--carriage-roads\nover the Alps. They were after a certain grade, and they obtained it,\nthough by contortions and serpentine windings that seem almost endless.\nThus the Simplon averages nowhere more than one inch elevation to a\nfoot, and, indeed, not quite that. Thirty thousand men were employed on\nthis road six years. There are six hundred and eleven bridges in less\nthan forty miles, ten galleries, and twenty houses of refuge, while\nthe average width of the road is over twenty-five feet. The Splugen\npresents almost as striking features as the Simplon. From these facts,\nsome idea may be gathered of the stupendous work it must be to carry a\ncarriage-road over the Alps.\n\n\nCHRISTMAS PIE.\n\nThe following appeared in the _Newcastle Chronicle_, 6th January,\n1770:--\n\n\"Monday last was brought from Howick to Berwick, to be shipped for\nLondon, for Sir Henry Grey, bart., a pie, the contents whereof are as\nfollows:--2 bushels of flour, 20 lbs. of butter, 4 geese, 2 turkeys, 2\nrabbits, 4 wild ducks, 2 woodcocks, 6 snipes, 4 partridges, 2 neats'\ntongues, 2 curlews, 7 blackbirds, and 6 pigeons: it is supposed a very\ngreat curiosity, was made by Mrs. Dorothy Patterson, housekeeper at\nHowick. It was near nine feet in circumference at bottom, weighs about\ntwelve stones, will take two men to present it at table; it is neatly\nfitted with a case, and four small wheels to facilitate its use to\nevery guest that inclines to partake of its contents at table.\"\n\n\nTHE UPAS, (POISON) TREE.\n\nWe give here an instance of the extravagancies of ancient travellers,\nthis tissue of falsehoods being taken from \"Foersch's Description of\nJava:\"--\n\nThe _Bohon Upas_ is situated in the Island of Java about twenty-seven\nleagues from Batavia, fourteen from Soulis Charta, the seat of the\nEmperor, and between eighteen and twenty leagues from Tinkjoe, the\npresent residence of the Sultan of Java. It is surrounded on all sides\nby a circle of high hills and mountains; and the country round it,\nto the distance of ten or twelve miles from the tree, is entirely\nbarren. Not a tree, nor a shrub, nor even the least plant or grass is\nto be seen. I have made the tour all around this dangerous spot, at\nabout eighteen miles distant from the centre, and I found the aspect\nof the country on all sides equally dreary. The easiest ascent of the\nhills is from that part where the old Ecclesiastick dwells. From his\nhouse the criminals are sent for the poison, into which the points of\nall warlike instruments are dipped. It is of high value, and produces\na considerable revenue to the Emperor. The poison which is procured\nfrom this tree is a gum that issues out between the bark and the tree\nitself, like the _camphor_. Malefactors, who for their crimes are\nsentenced to die, are the only persons who fetch the poison; and this\nis the only chance they have of saving their lives. After sentence is\npronounced upon them by the Judge, they are asked in Court, whether\nthey will die by the hands of the executioner, or whether they will go\nto the Upas-tree for a box of poison? They commonly prefer the latter\nproposal, as there is not only some chance of preserving their lives,\nbut also a certainty, in case of their safe return, that a provision\nwill be made for them in future by the Emperor. They are also permitted\nto ask a favour from the Emperor, which is generally of a trifling\nnature, and commonly granted. They are then provided with a silver or\ntortoise-shell box, in which they are to put the poisonous gum, and are\nproperly instructed how to proceed while they are upon their dangerous\nexpedition. They are always told to attend to the direction of the\nwind, as they are to go towards the tree before the wind; so that the\neffluvia from the tree is always blown from them. They go to the house\nof the old ecclesiastick who prepares them by prayers and admonitions\nfor their future fate; he puts them on a long leathern cap with two\nglasses before their eyes, which comes down as far as their breast; and\nalso provides them with a pair of leather gloves. They are conducted by\nthe priest, and their friends, and relations, about two miles on their\njourney. The old Ecclesiastick assured me that in upwards of thirty\nyears, he had dismissed above seven hundred criminals in the manner\ndescribed, and that scarcely two out of twenty have returned. All the\nMalayans consider this tree as an holy instrument of the great prophet\nto punish the sins of mankind, and, therefore, to die of the poison of\nthe Upas is generally considered among them as an honourable death.\nThis, however, is certain, that from fifteen to eighteen miles round\nthis tree, not only no human creature can exist, but no animal of _any\nkind_ has ever been discovered, there are no fish in the waters, and\nwhen any birds fly so near this tree that the effluvia reaches them,\nthey drop down dead.\n\n\nDEATH CAUSED BY SUPERSTITION.\n\nIn Hamburg, in 1784, a singular accident occasioned the death of a\nyoung couple. The lady going to the church of the Augustin Friars,\nknelt down near a Mausoleum, ornamented with divers figures in marble,\namong which was that of Death, armed with a scythe, a small piece\nof the scythe being loose, fell on the hood of the lady's mantelet.\nOn her return home, she mentioned the circumstance as a matter of\nindifference to her husband, who, being a credulous and superstitious\nman, cried out in a terrible panic, that it was a presage of the death\nof his dear wife. The same day he was seized with a violent fever,\ntook to his bed, and died. The disconsolate lady was so affected at\nthe loss, that she was taken ill, and soon followed him. They were\nboth interred in the same grave; and their inheritance, which was very\nconsiderable, fell to some very distant relations.\n\n\nST. PAUL AND THE VIPER.--THE CHURCH AT MALTA.\n\nNot far from the old city of Valetta, in the island of Malta, there\nis a small church dedicated to St. Paul, and just by the church, a\nmiraculous statue of the Saint with a viper on his hand; supposed to be\nplaced on the very spot on which he was received after his shipwreck\non this island, and where he shook the viper off his hand into the\nfire, without being hurt by it. At which time the Maltese assure us,\nthe Saint cursed all the venomous animals of the island, and banished\nthem for ever; just as St. Patrick treated those of his favourite isle.\nWhether this be the cause of it or not, we shall leave to divines to\ndetermine, though if it had, St. Luke would probably have mentioned it\nin the Acts of the Apostles; but the fact is certain, that there are no\nvenomous animals in Malta.\n\n\nTHE FIRST HERMITS--WHY SO-CALLED.\n\nHermits, or _Eremites_, (from the Greek [Greek: _eremos_], a desert\nplace,) were men who retired to desert places to avoid persecution;\nthey lodged in caves and cells:--\n\n \"Where from the mountain's grassy side,\n Their guiltless feast they bring;\n A scrip with herbs and fruit supply'd,\n And water from the spring.\"\n\nThe first hermit was Paul, of Thebes, in Egypt, who lived about the\nyear 260; the second, was St. Anthony, also of Egypt, who died in 345,\nat the age of 105.\n\n\nST. JAMES'S SQUARE.\n\nThe author of _A Tour through the Island of Great Britain_ (Daniel\nDefoe), second edition, 1738, gives us the following particulars of\nthis aristocratic locality:--\"The alterations lately made in St.\nJames's Square are entitled to our particular notice. It used to be in\na very ruinous condition, considering the noble houses in it, which are\ninhabited by the first quality. But now it is finely paved all over\nwith heading-stone; a curious oval bason full of water, surrounded with\niron rails on a dwarf wall, is placed in the middle, mostly 7 feet\ndeep and 150 diameter. In the centre is a pedestal about fifteen feet\nsquare, designed for a statue of King William III. The iron rails are\noctagonal, and at each angle without the rails, is a stone pillar about\n9 feet high, and a lamp on the top. The gravel walk within the rails\nis about 26 feet broad from each angle to the margin of the basin.\nIt was done at the expense of the inhabitants by virtue of an act of\nparliament. The house that once belonged to the Duke of Ormond, and\nsince to the Duke of Chandos, is pulled down and makes three noble\nones, besides fine stables and coach-houses behind, and two or three\nmore good houses in the street leading to St. James's Church. This\nnoble square wants nothing but to have the lower part of it, near Pall\nMall, built of a piece with the rest, and the designed statue to be\nerected in the middle of the basin.\n\n\"His Royal Highness the Prince of Wales has taken the Duke of Norfolk's\nhouse, and another adjoining to it, which are now (October, 1737),\nactually repairing for his town residence; Carlton House being too\nsmall for that purpose.\"\n\n\nTHE MORAYSHIRE FLOODS.\n\n[Illustration: [++] The Morayshire Floods.]\n\nIn the month of August, 1829, the province of Moray and adjoining\ndistricts were visited by a tremendous flood. Its ravages were most\ndestructive along the course of those rivers which have their source\nin the Cairngorm mountains. The waters of the Findhorn and the Spey,\nand their tributaries, rose to an unexampled height. In some parts\nof their course these streams rose fifty feet above their natural\nlevel. Many houses were laid desolate, much agricultural produce\nwas destroyed, and several lives were lost. The woodcut in our text\nrepresents the situation of a boatman called Sandy Smith, and his\nfamily, in the plains of Forres. \"They were huddled together,\" says\nthe eloquent historian of the Floods, \"on a spot of ground a few feet\nsquare, some forty or fifty yards below their inundated dwelling. Sandy\nwas sometimes standing and sometimes sitting on a small cask, and, as\nthe beholders fancied, watching with intense anxiety the progress of\nthe flood, and trembling for every large tree that it brought sweeping\npast them. His wife, covered with a blanket, sat shivering on a bit of\na log, one child in her lap, and a girl of about seventeen, and a boy\nof about twelve years of age, leaning against her side. A bottle and\na glass on the ground, near the man, gave the spectators, as it had\ndoubtless given him, some degree of comfort. About a score of sheep\nwere standing around, or wading or swimming in the shallows. Three cows\nand a small horse, picking at a broken rick of straw that seemed to be\nhalf-afloat, were also grouped with the family.\" The account of the\nrescue of the sufferers is given with a powerful dramatic effect, but\nwe cannot afford space for the quotation. The courageous adventurers\nwho manned the boat for this dangerous enterprise, after being\ncarried over a cataract, which overwhelmed their boat, caught hold\nof a floating hay-cock, to which they clung till it stuck among some\nyoung alder-trees. Each of them then grasping a bough, they supported\nthemselves for two hours among the weak and brittle branches. They\nafterwards recovered the boat under circumstances almost miraculous,\nand finally succeeded in rescuing Sandy and his family from their\nperilous situation.\n\n\nTREATMENT AND CONDITION OF WOMEN IN FORMER TIMES.\n\nFrom the subversion of the Roman Empire, to the fourteenth or fifteenth\ncentury, women spent most of their time alone, almost entire strangers\nto the joys of social life; they seldom went abroad, but to be\nspectators of such public diversions and amusements as the fashions of\nthe times countenanced. Francis I. was the first who introduced women\non public days to Court; before his time nothing was to be seen at\nany of the Courts of Europe, but grey-bearded politicians, plotting\nthe destruction of the rights and liberties of mankind, and warriors\nclad in complete armour, ready to put their plots in execution. In\nthe thirteenth and fourteenth centuries elegance had scarcely any\nexistence, and even cleanliness was hardly considered as laudable. The\nuse of linen was not known; and the most delicate of the fair sex wore\nwoollen shifts. In Paris they had meat only three times a week; and\none hundred livres, (about five pounds sterling,) was a large portion\nfor a young lady. The better sort of citizens used splinters of wood\nand rags dipped in oil, instead of candles, which, in those days,\nwere a rarity hardly to be met with. Wine was only to be had at the\nshops of the Apothecaries, where it was sold as a cordial; and to ride\nin a two-wheeled cart, along the dirty rugged streets, was reckoned\na grandeur of so enviable a nature, that Philip the Fair prohibited\nthe wives of citizens from enjoying it. In the time of Henry VIII. of\nEngland, the peers of the realm carried their wives behind them on\nhorseback, when they went to London; and in the same manner took them\nback to their country seats with hoods of waxed linen over their heads,\nand wrapped in mantles of cloth to secure them from the cold.\n\n\nHOMER IN A NUTSHELL.\n\nHuet, Bishop of Avranches, thus writes in his autobiography:--\"When\nhis Highness the Dauphin was one day confined to his bed by a slight\nillness, and we who stood round were endeavouring to entertain him by\npleasant conversation, mention was by chance made of the person who\nboasted that he had written Homer's Iliad in characters so minute,\nthat the whole could be enclosed in a walnut shell. This appearing\nincredible to many of the company, I contended not only that it might\nbe done, but that I could do it. As they expressed their astonishment\nat this assertion, that I might not be suspected of idle boasting, I\nimmediately put it to the proof. I therefore took the fourth part of\na common leaf of paper, and on its narrower side wrote a single line\nin so small a character that it contained twenty verses of the Iliad:\nof such lines each page of the paper could easily admit 120, therefore\nthe page would contain 2400 Homeric verses: and as the leaf so divided\nwould give eight pages it would afford room for above 19,000 verses,\nwhereas the whole number in the Iliad does not exceed 17,000. Thus by\nmy single line I demonstrated my proposition.\"\n\n\nAUTOBIOGRAPHY OF CHARING CROSS AND CHEAPSIDE CROSS.\n\nThe following interesting \"Autobiographies\" of the Old London Crosses,\nare extracted from Henry Peacham's _Dialogue between the Crosse in\nCheap and Charing Cross, confronting each other, as fearing their fall\nin these uncertaine times_, four leaves, 4to. 1641.\n\n\"_Charing Cross._--I am made all of white marble (which is not\nperceived of euery one) and so cemented with mortar made of the purest\nlime, Callis sand, whites of eggs and the strongest wort, that I defie\nall hatchets and hammers whatsoever. In King Henry the Eighth's daies\nI was begged, and should have been degraded for that I had:--Then in\nEdward the Sixe, when Somerset-house was building, I was in danger;\nafter that, in the reign of Queen Elizabeth, one of her footmen had\nlike to have run away with me; but the greatest danger of all I was in,\nwhen I quak'd for fear, was in the time of King James, for I was eight\ntimes begged:--part of me was bespoken to make a kitchen chimney for a\nchiefe constable in Shoreditch; an inn-keeper in Holborn had bargained\nfor as much of me as would make two troughes, one to stand under a\npumpe to water his guests' horses, and the other to give his swine\ntheir meate in; the rest of my poore carcase should have been carried\nI know not whither to the repaire of a decayed stone bridge (as I was\ntold) on the top of Harrow-hill. Our royall forefather and founder,\nKing Edward the First you know, built our sister crosses, Lincolne,\nGranthame, Woburne, Northampton, Stonie-Stratford, Dunstable, Saint\nAlbanes, and ourselves here in London, in the 21st yeare of his raigne,\nin the yeare 1289.\"\n\n\"_Cheapside Cross._--After this most valiant and excellent king had\nbuilt me in forme, answerable in beauty and proportion to the rest,\nI fell to decay, at which time one John Hatherley, maior of London,\nhaving first obtained a licence of King Henry the Sixt, anno 1441, I\nwas repaired in a beautiful manner. John Fisher, a mercer, after that\ngave 600 markes to my new erecting or building, which was finished\nanno 1484, and after in the second yeare of Henry the Eighth, I was\ngilded over against the coming in of Charles the Fift Emperor, and\nnewly then gilded against the coronation of King Edward the Sixt, and\ngilded againe anno 1554, against the coronation of King Philip. Lord,\nhow often have I been presented by juries of the quest for incombrance\nof the street, and hindring of cartes and carriages, yet I have kept my\nstanding; I shall never forget how upon the 21st of June, anno 1581,\nmy lower statues were in the night with ropes pulled and rent down, as\nin the resurrection of Christ--the image of the Virgin Mary, Edward\nthe Confessor, and the rest. Then arose many divisions and new sects\nformerly unheard of, as Martin Marprelate, _alias_ Penrie, Browne, and\nsundry others, as the chronicle will inform you. My crosse should have\nbeen taken quite away, and a _Piramis_ errected in the place, but Queen\nElizabeth (that queen of blessed memory) commanded some of her privie\ncouncell, in her Majesties name, to write unto Sir Nicholas Mosely,\nthen Maior, to have me againe repaired with a crosse; yet for all this\nI stood bare for a yeare or two after: Her Highness being very angry,\nsent expresse word she would not endure their contempt, but expressly\ncommanded forthwith the crosse should be set up, and sent a strict\ncommand to Sir William Rider, Lord Maior, and bade him to respect my\nantiquity; for that is the ancient ensigne of Christianity, &c. This\nletter was dated December 24, anno 1600. Last of all I was marvellously\nbeautified and adorned against the comming in of King James, and\nfenced about with sharp pointed barres of iron, against the rude and\nvillainous hands of such as upon condition as they might have the\npulling me down, would be bound to rifle all Cheapside.\"\n\nIt is scarcely necessary to say that both crosses have long since\ndisappeared, and their sites become uncertain, although the name of\nCharing Cross still distinguishes an important London district.\n\n\nSOMETHING LIKE A FEAST.\n\nLeland mentions a feast given by the Archbishop of York, at his\ninstallation, in the reign of Edward IV. The following is a\nspecimen:--300 quarters of wheat, 300 tuns of ale, 100 tuns of wine,\n1,000 sheep, 104 oxen, 304 calves, 304 swine, 2,000 geese, 1,000\ncapons, 2,000 pigs, 400 swans, 104 peacocks, 1,500 hot venison pasties,\n4,000 cold, 5,000 custards hot and cold. Such entertainments are a\npicture of manners.\n\n\nEGYPTIAN TOYS IN THE BRITISH MUSEUM.\n\nThe truth of the old proverb, that \"there is nothing new under the\nsun,\" will be recognised on an examination of the interesting group\nwhich forms the subject of our engraving. Here are dolls of different\nshapes, some of them for good children, and some, perhaps, for bad;\nfoot-balls, covered with leather, &c., the stitches in parts still\nfirmly adhering; models of fishes and fruit; and round pellets, which\nthe \"small boys\" of the present day would call \"marbles.\" These toys\nhave been played with by little Egyptians who have been dead and buried\nthree or four thousand years.\n\nMany of the toys that hold places in the English and other markets\nare, so far as fashion is concerned, of considerable antiquity, having\nbeen made, without any alteration in pattern, by certain families for\nseveral generations. In the mountainous districts of the Savoy and\nSwitzerland, large numbers, both of children and grown persons, are\nconstantly employed in the manufacture of Noah's-arks, milkmaids &c.\nSome of the animals carved in wood, and sold here for small prices,\nshow considerable skill in the imitation of the forms of nature, and\ncould only be produced at their present cost, owing to the cheapness of\nliving in those districts, and to the systematic division of labour.\n\n[Illustration: [++] Egyptian Toys.]\n\nNear the birthplace of Prince Albert is a very large manufactory of\nmilitary toys, such as drums, trumpets, helmets, &c.; and in parts of\nHolland--\n\n \"----The children take pleasure in making\n What the children of England take pleasure in breaking.\"\n\n\nTHE PYRAMIDS OF EGYPT.\n\nThe Pyramids of Egypt, especially the two largest of the Pyramids of\nJizeh, are the most stupendous masses of building, in stone, that human\nlabour has ever been known to accomplish. The Egyptian Pyramids, of\nwhich, large and small, and in different states of preservation, the\nnumber is very considerable, are all situated on the west side of the\nNile, and they extend, in an irregular line, and in groups, at some\ndistance from each other, from the neighbourhood of Jizeh, in 30 deg. N.\nlat. as far south as 29 deg. N. lat., a length of between 60 and 70 miles.\nAll the Pyramids have square bases, and their sides face the cardinal\npoints.\n\n[Illustration: [++] Pyramids of Jizeh.]\n\nThe Pyramids of Jizeh are nearly opposite to Cairo. They stand on a\nplateau or terrace of limestone, which is a projection from the Libyan\nmountain-chain. The surface of the terrace is barren and irregular,\nand is covered with sand and small fragments of rock; its height,\nmeasured from the base of the Great Pyramids, is 164 feet above the\nNile in its low state, taken at an average of the years 1798 to 1801.\nThe north-east angle of the Great Pyramid is 1700 yards from the canal\nwhich runs between the terrace and the Nile, and about five miles from\nthe Nile itself.\n\nHerodotus was informed by the priests of Memphis that the Great Pyramid\nwas built by Cheops, King of Egypt, about 900 B. C., or about 450 years\nbefore Herodotus visited Egypt. He says that 100,000 men were employed\ntwenty years in building it, and that the body of Cheops was placed\nin a room beneath the bottom of the Pyramid, surrounded by a vault\nto which the waters of the Nile were conveyed through a subterranean\ntunnel. A chamber under the centre of the Pyramid has indeed been\ndiscovered, but it does not appear to be the tomb of Cheops. It is\nabout 56 feet above the low-water level of the Nile. The second Pyramid\nwas built, Herodotus says, by Cephren, or Cephrenes, the brother and\nsuccessor of Cheops; and the third by Mycerinus, the son of Cheops.\n\n\nTEST OF COURAGE IN A CHILD.\n\nIn the education of their children, the Anglo-Saxons only sought to\nrender them dauntless and apt for the two most important occupations\nof their future lives--war and the chase. It was a usual trial of a\nchild's courage, to place him on the sloping roof of a building, and\nif, without screaming or terror he held fast, he was styled a stout\nherce, or brave boy.--_Howel._\n\n\nEXECUTION OF RAVILLIAC, WHO ASSASSINATED HENRY THE FOURTH OF FRANCE.\n\nThe scene is thus described in a volume published in 1728:--\n\n\"This Francis Ravilliac was born in Angoulesme, by profession a lawyer,\nwho, after the committing of that horrid fact, being seized and put\nupon the rack, May 27; the 25th he had sentence of death passed on him,\nand was executed accordingly in the manner following. He was brought\nout of prison in his shirt, with a torch of two pound weight lighted in\none hand, and the knife wherewith he murdered the king chained to the\nother; he was then set upright in a dung-cart, wherein he was carried\nto the greve or place of execution, where a strong scaffold was built;\nat his coming upon the scaffold he crossed himself, a sign that he dyed\na ; then he was bound to an engine of wood made like St. Andrew's\ncross; which done, his hand with the knife chained to it was put into\na furnace, then flaming with fire and brimstone, wherein it was in a\nmost terrible manner consumed, at which he cast forth horrible cries\nyet would he not confess any thing; after which the executioner having\nmade pincers red hot in the same furnace, they did pinch the brawn of\nhis arms and thighs, the calves of his legs, with other fleshy parts of\nhis body, then they poured into the wounds scalding oil, rosin, pitch,\nand brimstone melted together; but to make the last act of his tragedy\nequal in torments to the rest, they caused four strong horses to be\nbrought to tear his body in pieces, where being ready to suffer his\nlast torment, he was again questioned, but would not reveal any thing,\nand so died without calling upon God, or speaking one word concerning\nHeaven: his flesh and joints were so strongly knit together, that\nthese four horses could not in a long time dismember him, but one of\nthem fainting, a gentleman who was present, mounted upon a mighty\nstrong horse, alighted, and tyed him to one of the wretch's limbs, yet\nfor all this they were constrained to cut the flesh under his arms\nand thighs with a sharp razor, whereby his body was the easier torn\nin pieces; which done, the fury of the people was so great, that they\npulled his dismembered carcass out of the executioner's hands, which\nthey dragged up and down the dirt, and, cutting off the flesh with\ntheir knives, the bones which remained were brought to the place of\nexecution, and there burnt, the ashes were cast in the wind, being\njudged unworthy of the earth's burial; by the same sentence all his\ngoods were forfeited to the king. It was also ordained that the house\nwhere he had been born should be beaten down, a recompence being given\nthe owner thereof, and never any house to be built again upon that\nground; that within fifteen days after the publication of the sentence,\nby sound of trumpet in the town of Angoulesme, his father and mother\nshould depart the realm, never to return again; if they did, to be\nhanged up presently: his brethren, sisters, and other kindred were\nforbidden to carry the name of Ravilliac, but to take some other, and\nthe substitute of the king's attorney-general had charge to see the\nexecution of the sentence at his peril.\"\n\n\nKNIVES AND FORKS.\n\n\"In all ancient pictures of Eating, &c. knives are seen in the hands of\nthe guests, but _no Forks_.\"--_Turner's Saxons._\n\n\"Here I will mention a thing,\" says Coryat in his 'Crudities,' \"that\nmight have been spoken of before in the discourse of the first Italian\ntoun. I obserued a custome in all those Italian cities and townes\nthrough which I passed, that is not vsed in any other country that\nI saw in my traules, neither doe I think that any other nation of\nChristendome doth vse it, but only Italy. The Italians, and also most\nstrangers that are commorant in Italy, doe alwaies at their meales vse\na _little forke_ when they cut their meate. For while with their knife,\nwhich they hold in one hand, they cut their meate out of the dish, they\nfasten their forke, which they hold in their other hand, upon the same\ndish, so that whatsoever he be that sitting in the company of others at\nmeate, should vnaduisedly touch the dish of meate with his fingers from\nwhich all at the table doe cut, he will give occasion of offence unto\nthe company, as hauing transgressed the laws of good manners, in so\nmuch that for his error he shall be at the least broue-beaten, if not\nreprehended in words.\n\nThis form of feeding, I vnderstand, is generally vsed in all places of\nItaly, their forkes being for the most part made of yron or steele, and\nsome of siluer; but those are used only by gentlemen. The reason of\nthis their curiosity is, because the Italian cannot by any means indure\nto have his dish touched with fingers, seeing all men's fingers are\nnot alike clean. Hereupon I myself thought good to imitate the Italian\nfashion by this forked cutting of meate, not only while I was in Italy,\nbut also in Germany, and oftentimes in England, since I came home:\nbeing once quipped for that frequent vsing of my forke, by a certain\ngentleman, a familiar friend of mine, one Mr. Laurence Whitaker, who\nin his merry humour doubted not to call me at table _furcifer_, only\nfor vsing a forke at feeding, but for no other cause.\"--_Coryat's\nCrudities_, 1611.\n\nEven when Heylin published his Cosmography, (1652,) forks were still a\nnovelty. See his Third Book, where having spoken of the ivory sticks\nused by the Chinese, he adds, \"The use of silver forks, which is by\nsome of our spruce gallants taken up _of late_, came from thence into\nItaly, and from thence into England.\"--_Antiquarian Repertory._\n\n\nCHINESE PUNISHMENT OF THE KANG OR WOODEN COLLAR.\n\nThe Chinese are very quiet and orderly; and no wonder, because they are\nafraid of the great bamboo stick.\n\n[Illustration: [++] Kang or Wooden Collar.]\n\nThe mandarins (or rulers of towns) often sentence offenders to lie upon\nthe ground, and to have thirty strokes of the bamboo. But the wooden\ncollar is worse than the bamboo stick. It is a great piece of wood with\na hole for a man to put his head through. The men in wooden collars are\nbrought out of their prisons every morning and chained to a wall, where\neverybody passing by can see them. They cannot feed themselves in their\nwooden collars, because they cannot bring their hands to their mouths;\nbut sometimes a son may be seen feeding his father, as he stands\nchained to the wall. There are men also whose business it is to feed\nthe prisoners. For great crimes men are strangled or beheaded.\n\n\nCASCADE DES PELERINES.\n\n[Illustration: [++] Cascade des Pelerines.]\n\nThere is a waterfall in Chamouni which no traveller should omit going\nto see, called the Cascade des Pelerines. It is one of the most curious\nand beautiful scenes in Switzerland. A torrent issues from the Glacier\ndes Pelerines, high up the mountain, above the Glacier du Bossons, and\ndescends, by a succession of leaps, in a deep gorge, from precipice\nto precipice, almost in one continual cataract; but it is all the\nwhile merely gathering force, and preparing for its last magnificent\ndeep plunge and recoil of beauty. Springing in one round condensed\ncolumn out of the gorge, over a perpendicular cliff, it strikes, at\nits fall, with its whole body of water, into a sort of vertical rock\nbasin, which one would suppose its prodigious velocity and weight would\nsplit into a thousand pieces; but the whole cataract, thus arrested,\nat once suddenly rebounds in a parabolic arch, at least sixty feet\ninto the air; and then, having made this splendid airy curvature,\nfalls with great noise and beauty into the natural channel below. It\nis beyond measure beautiful. It is like the fall of divine grace into\nchosen hearts, that send it forth again for the world's refreshment,\nin something like such a shower and spray of loveliness, to go winding\nits life-giving course afterwards, as still waters in green pastures.\nThe force of the recoil from the plunge of so large a body of water, at\nsuch a height, is so great, that large stones, thrown into the stream\nabove the fall, may be heard amidst the din striking into the basin,\nand then are instantly seen careering in the arch of flashing waters.\nThe same is the case with bushes and pieces of wood, which the boys\nare always active in throwing in, for the curiosity of visitors, who\nstand below, and see each object invariably carried aloft with the\ncataract, in its rebounding atmospheric gambols. When the sun is in\nthe right position, the rainbows play about the fall like the glancing\nof supernatural wings, as if angels were taking a shower-bath. If you\nhave \"the head and the legs of a chamois,\" you may climb entirely above\nthis magnificent scene, and look out over the cliff right down into the\npoint where the cataract shoots like the lightning, to be again shot\nback in ten thousand branching jets of diamonds.\n\n\nINTERESTING INCIDENT CONNECTED WITH THE BAROMETER.\n\nIn navigation, the barometer has become an important element of\nguidance, and a most interesting incident is recounted by Capt. Basil\nHall, indicative of its value in the open sea. While cruising off the\ncoast of South America, in the Medusa frigate, one day, when within\nthe tropics, the commander of a brig in company was dining with him.\nAfter dinner, the conversation turned on the natural phenomena of the\nregion, when Captain Hall's attention was accidentally directed to\nthe barometer in the state-room where they were seated, and to his\nsurprise he observed it to evince violent and frequent alteration. His\nexperience told him to expect bad weather, and he mentioned it to his\nfriend. His companion, however, only laughed, for the day was splendid\nin the extreme, the sun was shining with its utmost brilliance, and\nnot a cloud specked the deep blue sky above. But Captain Hall was too\nuneasy to be satisfied with bare appearances. He hurried his friend to\nhis ship, and gave immediate directions for shortening the top hamper\nof the frigate as speedily as possible. His lieutenants and the men\nlooked at him in mute surprise, and one or two of the former ventured\nto suggest the inutility of the proceeding. The captain, however,\npersevered. The sails were furled; the topmasts were struck; in short,\neverything that could oppose the wind was made as snug as possible. His\nfriend, on the contrary, stood in under every sail.\n\nThe wisdom of Captain Hall's proceedings was, however, speedily\nevident; just, indeed, as he was beginning to doubt the accuracy of his\ninstrument. For hardly had the necessary preparations been made, and\nwhile his eye was ranging over the vessel to see if his instructions\nhad been obeyed, a dark hazy hue was seen to rise in the horizon,\na leaden tint rapidly overspread the sullen waves, and one of the\nmost tremendous hurricanes burst upon the vessels that ever seaman\nencountered on his ocean home. The sails of the brig were immediately\ntorn to ribbons, her masts went by the board, and she was left a\ncomplete wreck on the tempestuous surf which raged around her, while\nthe frigate was driven wildly along at a furious rate, and had to scud\nunder bare poles across the wide Pacific, full three thousand miles,\nbefore it could be said that she was in safety from the blast.\n\n\nARCHBISHOP CRANMER'S DIETARY.\n\nIn this curious document, quoted by Warton (Hist. of Poet, iii., 177,\nedit. 1840) an archbishop is allowed to have two swans or two capons in\na dish, a bishop one; an archbishop six blackbirds at once, a bishop\nfive, a dean four, an archdeacon two. If a dean has four dishes in\nhis first course, he is not afterwards to have custards or fritters.\nAn Archbishop may have six snipes, an archdeacon two. Rabbits, larks,\npheasants, and partridges, are allowed in these proportions. A canon\nresidentiary is to have a swan only on a Sunday; a rector of sixteen\nmarks, only three blackbirds in a week.\n\n\nTHE KING'S COCK CROWER.\n\nA singular custom, of matchless absurdity, formerly existed in the\nEnglish Court. During Lent, an ancient officer of the crown, styled the\nKing's Cock Crower, crowed the hour each night within the precincts\nof the Palace. On the Ash Wednesday, after the accession of the house\nof Hanover, as the Prince of Wales (afterwards George II) sat down to\nsupper, this officer abruptly entered the apartment, and in a sound\nresembling the shrill pipe of a cock, crowed past ten o'clock! The\nastonished prince, at first conceiving it to be a premeditated insult,\nrose to resent the affront, but upon the nature of the ceremony being\nexplained to him, he was satisfied. Since that period, this silly\ncustom has been discontinued.\n\n\nCHINESE DELICACIES.\n\nThe Chinese eat, indiscriminately, almost every living creature which\ncomes in their way; dogs, cats, hawks, owls, eagles and storks, are\nregular marketable commodities: in default of which a dish of rats,\nfield-mice, or snakes, is not objected to. Cockroaches, and other\ninsects and reptiles are used for food or for medicine. Their taste\nfor dogs' flesh is quite a passion. Young pups--plump, succulent, and\ntender--fetch good prices at the market-stalls, where a supply is\nalways to be found. A dish of puppies, prepared by a skilful cook,\nis esteemed as a dish fit for the gods. At every grand banquet it\nmakes its appearance as a hash or stew. A young Englishman attached to\nour Canton factory, dining one day with a wealthy Hong merchant, was\ndetermined to satisfy his curiosity in Chinese gastronomy by tasting\nall or most of the numerous dishes which were successively handed\nround. One dish pleased him so well that he ate nearly all that was put\nbefore him. On returning homewards some of his companions asked him\nhow he liked the dinner, and how such and such dishes; and then began\nto imitate the whining and barking of half a dozen puppies. The poor\nyoung man then understood, for the first time, that he had been eating\ndog, and was very angry, and very sick at the stomach. Other Europeans,\nhowever, have been known to declare that they succeeded in conquering a\nprejudice, and that a six weeks old pup, properly fattened upon rice,\nand dressed _a la Chinoise_, was really a _bonne bouche_.\n\n\nA GREAT MARVEL SEEN IN SCOTLAND.\n\nThe following strange and almost incredible account is given by\nLindsay, of Pitscottie:--\"About this time (the beginning of the\nsixteenth century) there was a great marvel seen in Scotland. A bairn\nwas born, reckoned to be a man-child, but from the waist up was two\nfair persons, with all members pertayning to two bodies; to wit, two\nheads, well-eyed, well-eared, and well-handed. The two bodies, the\none's back was fast to the other's, but from the waist down they were\nbut one personage; and it could not be known by the ingene of men\nfrom which of the bodies the legs, &c., proceeded. Notwithstanding\nthe King's Majesty caused great care and diligence on the up-bringing\nof both bodies; caused nourish them, and learn them to sing and play\non instruments of music. Who within short time became very ingenious\nand cunning in the art of music, whereby they could play and sing two\nparts, the one the treble, and the other the tenor, which was very\ndulce and melodious to hear; the common people (who treated them also)\nwondered that they could speak diverse and sundry languages, that is\nto say, Latin, French, Italian, Spanish, Dutch, English, and Irish.\nTheir two bodies long continued to the age of twenty-eight years, and\nthe one continued long before the other, which was dolorous and heavy\nto the other; for which, when many required of the other to be merry,\nhe answered, \"How can I be merry which have my true marrow as a dead\ncarrion about my back, which was wont to sing and play with me: when I\nwas sad he would give me comfort, and I would do the like to him. But\nnow I have nothing but dolour of the having so heavy a burthen, dead,\ncold, and unsavoury, on my back, which taketh all earthly pleasure from\nme in this present life; therefore I pray to God Allmighty to deliver\nme out of this present life, that we may be laid and dissolved in the\nearth, wherefrom we came, &c.\"\n\nBuchanan, who relates the same strange tale, avers that he received it\nfrom \"many honest and credible persons, who saw the prodigy with their\nown eyes.\" He adds that the two bodies discovered different tastes\nand appetites; that they would frequently disagree and quarrel, and\nsometimes would consult each other, and concert measures for the good\nof both; that when any hurt was done to the lower parts, each upper\nbody felt pain; but that when the injury was above the junction, then\none body only was affected. This monster, he writes, lived twenty-eight\nyears, but died wretchedly; one part expiring some days before the\nother, which, half-putrified, pined away by degrees.\n\n\nTHE KING OF KIPPEN.\n\nThe following anecdote is valuable, inasmuch as it gives us an idea of\nthe manners which a King of Scotland could practice without offence to\nhis subjects:--\n\nKing James V. was a very sociable, _debonnaire_ prince. Residing at\nStirling in Buchanan of Arnpryor's time, carriers were very frequently\npassing along the common road with necessaries for the use of the\nking's family. One of these being near Arnpryor's house, and he having\nsome extraordinary occasion, ordered him to leave his load at his house\nand he would pay him for it; which the carrier refused to do, telling\nhim he was the king's carrier, and his load was for his majesty's use.\nTo which Arnpryor seemed to have small regard, compelling the carrier,\nin the end, to leave his load; telling him, if King James was King of\nScotland, he was king of Kippen, so that it was reasonable he should\nshare with his neighbour king in some of these loads so frequently\ncarried that road. The carrier representing this usage, and telling the\nstory as Arnpryor spoke it, to some of the king's servants, it came\nat length to his majesty's ears, who, shortly thereafter, with a few\nattendants, came to visit his neighbour king, who was, in the meantime,\nat dinner. King James having sent a servant to demand access, was\ndenied the same by a tall fellow with a battle-axe, who stood porter at\nthe gate, telling him there could be no access till dinner was over.\nThis answer not satisfying the king, he sent to demand access a second\ntime; upon which he was desired by the porter to desist, otherwise\nhe would find cause to repent his rudeness. His majesty finding this\nmethod would not do, desired the porter to tell his master that the\ngood man of Ballangeich desired to speak with the King of Kippen. The\nporter telling Arnpryor so much, he, in all humble manner, came and\nreceived the king, and having entertained him with much sumptuousness\nand jollity, became so agreeable to King James, that he allowed him to\ntake so much of any provision he found carrying that road as he had\noccasion for; and, seeing he made the first visit, desired Arnpryor in\na few days to return him a second at Stirling, which he performed, and\ncontinued in very much favour with the king, always thereafter being\ntermed King of Kippen while he lived.\n\n\nAN ECCENTRIC TOURIST.\n\nSir Hildebrand Jacob, of Yewhall, in Oxfordshire, died at Malvern\nin 1790. He succeeded his grandfather, Sir John, 1740, his father,\nHildebrand, having died in 1739. He was a very extraordinary character.\nAs a general scholar, he was exceeded by few; in his knowledge of the\nHebrew language he scarcely had an equal. In the earlier part of his\nlife, one custom which he constantly followed was very remarkable. As\nsoon as the roads became pretty good, and the fine weather began to\nset in, his man was ordered to pack-up a few things in a portmanteau,\nand with these his master and himself set off, without knowing whither\nthey were going. When it drew towards evening, they enquired at the\nfirst village they saw, whether the great man in it was a lover of\nbooks, and had a fine library. If the answer was in the negative,\nthey went on farther; if in the affirmative, Sir Hildebrand sent his\ncompliments, that he was come to see him; and there he used to stay\ntill time or curiosity induced him to move elsewhere. In this manner\nSir Hildebrand had, very early, passed through the greatest part of\nEngland, without scarcely ever sleeping at an inn, unless where the\ntown or village did not afford one person in it civilized enough to be\nglad to see a gentleman and a scholar.\n\n\nHANGING A MAYOR.\n\nOn the right of the road leading towards Caergwrle, and about a mile\nfrom Mold, is an old structure, which presents a singular specimen\nof the style of domestic architecture during the ages of lawless\nviolence in which it was erected: it consists of an ancient square\ntower of three stories, and appears to have been designed as a place of\nfortified habitation. During the wars between the houses of York and\nLancaster, it was inhabited by Reinallt ab Gruffydd ab Bleddyn, who was\nconstantly engaged in feuds with the citizens of Chester. In 1495, a\nconsiderable number of the latter came to Mold fair, and a fray arising\nbetween the hostile parties, great slaughter ensued on both sides; but\nReinallt, who obtained the victory, took the mayor of Chester prisoner,\nand conveyed him to his mansion, where he hung him on the staple in\nhis great hall. To avenge this affront, a party of two hundred men\nwas despatched from Chester to seize Reinallt, who, retiring from his\nhouse into the adjoining woods, permitted a few of them to enter the\nbuilding, when, rushing from his concealment, he blocked up the door,\nand, setting fire to the house, destroyed them in the flames; he then\nattacked the remainder, whom he pursued with great slaughter; and such\nas escaped the sword were drowned in attempting to regain their homes.\nThe staple on which the mayor was hung still remains fixed on the\nceiling of the lower apartment.\n\n\nMATERNAL AFFECTION IN A DUMB WOMAN.\n\nMary, Countess of Orkney, was both deaf and dumb; she was married in\nthe year 1753, by signs. Shortly after the birth of her first child,\nthe nurse, with considerable astonishment, saw the mother cautiously\napproach the cradle in which the infant was sleeping, evidently full\nof some deep design. The Countess, having perfectly assured herself\nthat the child really slept, raised an immense stone which she had\nconcealed under her shawl, and, to the horror of the nurse, (who\nwas an Irishwoman, and like all persons of the lower orders in her\ncountry, and indeed in most countries, was fully impressed with an\nidea of the peculiar cunning and malignity of \"dumbies,\") lifted it\nwith an apparent intent to fling it down vehemently. Before the nurse\ncould interpose, the Countess had flung the stone,--not, however, as\nthe servant had apprehended, at the child, but on the floor, where,\nof course, it made a great noise. The child immediately awoke, and\ncried. The Countess, who had looked with maternal eagerness to the\nresult of her experiment, fell on her knees in a transport of joy. She\nhad discovered that her child possessed the sense which was wanting\nin herself. She exhibited on many other occasions similar proofs of\nintelligence, but none so interesting.\n\n\nTHE PERILS OF INVENTORS.\n\nThe dangers which inventors have frequently to encounter are very\ngreat. Among many instances we may mention the following:--\n\nMr. Day perished in a diving bell, or diving boat of his own\nconstruction, at Plymouth, in June, 1774, in which he was to have\ncontinued for a wager, twelve hours, one hundred feet deep in water,\nand probably, perished from his not possessing all the hydrostatic\nknowledge that was necessary. Mr. Spalding was professionally\ningenious in the art of constructing and managing the diving bell,\nhe had practised the business many years with success. He went down,\naccompanied by one of his young men, twice to view the wreck of the\nImperial East Indiaman, at Kish Bank, in Ireland; on descending the\nthird time, in June, 1783, they remained about an hour under water, and\nhad two barrels of air sent down to them, but on the signals from below\nnot being repeated, after a certain time, they were drawn up by their\nassistants, and both found dead in the bell.\n\n\nBRIBERY.\n\nThe triumphant exposure and punishments of corrupt bribe-takers on a\ngrand scale belongs to the close of the seventeenth century. In 1695\nSir John Trevor, the Speaker of the House of Commons, was compelled\nto put the question himself that he should be expelled. A bill for\nsecuring the right application to poor orphans of freemen of London of\nfunds belonging to them could not be carried without purchasing the\nsupport of influential members and of the Speaker himself, at a bribe\nfor the latter of 1,000 guineas!\n\nSir Thomas Cook, the governor of the East India Company, paid L167,000\nin one year for bribes to members of the House, of which Sir Basil\nFirebrace took for his share L40,000. Corruption was universal,\ntherefore deemed venial.\n\n\nLEGALISED GAMBLING.\n\nThe following statement shows the extent to which lotteries encouraged\na spirit of gambling among the people, and we may hence appreciate the\nsoundness of the policy which dictated their suppression:--\n\nThe _Post Boy_ of December 27, says:--\"We are informed that the\nParliamentary Lottery will be fixed in this manner:--150,000 tickets\nwill be delivered out at 10_l._ each ticket, making in all the sum\nof 1,500,000_l._ sterling; the principal whereof is to be sunk, the\nParliament allowing nine per cent. interest for the whole during the\nterm of 32 years, which interest is to be divided as follows: 3,750\ntickets will be prizes from 1,000_l._ to 5_l._ per annum during the\nsaid 32 years; all the other tickets will be blanks, so that there\nwill be 39 of these to one prize, but then each blank ticket will be\nentitled to fourteen shillings a year for the term of 32 years, which\nis better than an annuity for life at ten per cent. over and above\nthe chance of getting a prize.\" Such was the eagerness of the publick\nin subscribing to the above profitable scheme, that Mercers'-hall was\nliterally crowded, and the Clerks were found incompetent to receive the\ninflux of names. 600,000_l._ was subscribed January 21; and on the 28th\nof February, the sum of 1,500,000_l._ was completed.\n\n\nONE OF THE EFFECTS OF MANUFACTURES.\n\nHow greatly does the introduction of a manufacturing establishment into\na town where none previously existed, alter its whole character and\ncondition!\n\nIt is said that the burgh of Lanark was, till very recent times, so\npoor that the single butcher of the town, who also exercised the\ncalling of a weaver, in order to fill up his spare time, would never\nventure upon the speculation of killing a sheep till every part of\nthe animal was ordered beforehand. When he felt disposed to engage\nin such an enterprise, he usually prevailed upon the minister, the\nprovost, and the town-council, to take shares; but when no person came\nforward to bespeak the fourth quarter, the sheep received a respite\ntill better times should cast up. The bellman or _skellyman_, as he\nis there called, used often to go through the streets of Lanark with\nadvertisements such as are embodied in the following popular rhyme:--\n\n \"Bell-ell-ell!\n There's a fat sheep to kill!\n A leg for the provost,\n Another for the priest,\n The bailies and deacons,\n They'll tak the neist;\n And if the fourth leg we connot sell,\n The sheep it maun leeve and gae back to the hill!\"\n\n\nPATES DE FOIES GRAS.\n\nStrasbourg is the great market for _pates de foies gras_, made, as it\nis known, of the livers of geese. These poor creatures are shut up\nin coops, so narrow they cannot turn round in them, and then stuffed\ntwice a day with Indian corn, to enlarge their livers, which have been\nknown to swell till they reached the enormous weight of two pounds and\na half. Garlick, steeped in water, is given them, to increase their\nappetites. This invention is worthy of the French nation, where cooks\nare great as nobles.\n\n\nINSCRIPTION IN CONWAY CHURCH.\n\nHere lyeth the body of Nicholas Hookes, of Conway, gentleman, (who was\nthe forty-first child of his father, Wm. Hookes, Esq., by Alice, his\nwife,) the father of twenty-seven children, who died the 27th day of\nMarch, 1637.\n\n\nDROPPING-WELLS.\n\nIf you journey through Yorkshire, be sure to stop opposite the ruins\nof Knaresborough Castle, because, on the south-west bank of the river\nNidd, you will observe the petrifying spring of Knaresborough,--the\ncelebrated dropping-well--where the peasants and the needy crowd to\nmake their humble fortunes by afterwards retailing small sprigs of\ntrees, such as the elder or ash, or pieces of the elegant geranium, the\nwild angelica, or the lovely violet, turned into \"obdurate stone.\"\n\n[Illustration: [++] The Petrifying Spring of Knaresborough.]\n\nEvery spring does not possess the petrifying properties of that\nof Knaresborough; but there are, doubtless, many dropping-wells\ndistributed over the earth's crust; and some of these are well known\nto possess the property of petrifying various objects submitted to the\naction of their waters. For example: we have seen birds' nests, with\nthe eggs, and delicate sprigs of moss surrounding them, and even the\nfibres of wool turned into stone, aye, and delicate flowers. Whence is\nthis extraordinary power? From the soil over which the waters flow!\nThe limpid streams absorb the silicious particles, and deposit them in\nthe intimate structure of the materials submitted to the action of the\nwaters; and thus we find the materials of which the earth's crust is\ncomposed, always undergoing a change.\n\nTwenty gallons are poured forth every minute from the top of the\nKnaresborough cliff, and the beauty of the scene can only be\nappreciated by those who have stood upon the margin of those \"stony\nwaters\" and beheld the crystal fluid descend from above with metallic\nfall.\n\n\nCHINESE IVORY BALLS.\n\nNothing can afford a greater proof of the patience and perseverance,\nas well as of the taste of a Chinese handicraftsman, than one of these\nelegant baubles, each ball being exquisitely carved, and no two alike\nin pattern. Each of the balls rolls freely within that which encloses\nit, and is visible through apertures; so that however many there be,\nthe beauties of each can be examined, and the number of the whole\ncounted. Much time is spent upon the carving of these toys, for the\ncleverest artist will employ a whole month in the execution of each\nseparate ball; consequently the labour of two years is not unfrequently\nbestowed on the production of a single toy, which is formed out of a\nsolid globe of ivory, and has no junction in any part. The outside\nof this globe is first carved in some very open pattern, and is then\ncarefully cut with a sharp, fine instrument, through the openings, till\na complete coating is detached from the solid part inside, as the peel\nof an orange might be loosened with a scoop from the fruit, without\nbeing taken off. One hollow ball is thus formed, with a solid one\ninside of it. The surface of the inner ball is then carved through the\ninterstices of the outer one, and when finished, is subjected to the\nsame operation as the first; and thus a second hollow ball is produced,\nstill with a solid one of smaller dimensions inside. This process is\nrepeated again and again, the difficulties increasing as the work\nproceeds, till at length only a small ball, of the size of a marble, is\nleft in the centre, which is also ornamented with figures cut upon it,\nand then the ingenious but useless bauble is complete. This process is\nsaid to be performed under water.\n\n\nCREDULITY OF THE ANCIENTS.\n\nThe credulity of even the learned men in the early ages may be judged\nof by the following facts:--\n\nMarcus Varro writeth, that there was a town in Spain undermined with\nrabbits; another likewise in Thessaly by moles or molewharps. In Africa\nthe people were compelled by locusts to leave their habitations;\nand out of Gyaros, an island, one of the Cyclades, the islanders\nwere forced by rats and mice to fly away; moreover in Italy the city\nof Amyclae was destroyed by serpents. In Ethiopia there is a great\ncountry lies waste and desert, by reason it was formerly dispeopled\nby scorpions; and if it be true that Theophrastus reporteth, the\nTreriens were chased away by certain worms called scolopendres.\nAnnius writes, that an ancient city situate near the Volscian Lake,\nand called Contenebra, was in times past overthrown by pismires, and\nthat the place is thereupon vulgarly called to this day, the Camp of\nAnts. In Media, saith Diodorus Siculus, there was such an infinite\nnumber of sparrows that eat up and devoured the seed which was cast\ninto the ground, so that men were constrained to depart from their old\nhabitations, and remove to other places.\n\n\nCLOCK PRESENTED TO CHARLEMAGNE.\n\nThe French historians describe a clock sent to Charlemagne in the\nyear 807, by the famous eastern caliph, Haroun al Raschid, which was\nevidently furnished with some kind of wheelwork, although the moving\npower appears to have been produced by the fall of water. This clock\nwas a rather wonderful affair, and excited a great deal of attention\nat the French court. In the dial of it were twelve small doors forming\nthe divisions for the hours, each door opened at the hour marked by the\nindex, and let out small brass balls, which, falling on a bell, struck\nthe hours--a great novelty at that time. The doors continued open\nuntil the hour of twelve, when twelve figures representing knights on\nhorseback came out and paraded round the dial plate.\n\n\nREMARKABLE EVENTS, INVENTIONS, &c.\n\nMemnon, the Egyptian, invents the letters, in the year 1822, _before_\nChrist.\n\nThe Alexandrian library, consisting of 400,000 valuable books, burnt by\naccident, B. C. 52.\n\nSilk first brought from India, 274: the manufactory of it introduced\ninto Europe by some monks, 551: first worn by the clergy in England, in\n1534.\n\nGlass invented in England by Benalt, a monk, A. C. 400.\n\nThe University of Cambridge founded A. C. 915.\n\nPaper made of cotton rags was in use, 1000; that of linen rags in 1170:\nthe manufactory introduced into England, at Dartford, 1588.\n\nMusical notes invented, 1070.\n\nJustices of the Peace first appointed in England in 1076.\n\nDoomsday-book began to be compiled by order of William, from a survey\nof all the estates in England (and finished in 1086), 1080.\n\nGlass windows began to be used in private houses in England in 1186.\n\nSurnames now began to be used, first among the nobility, in 1200.\n\nThe houses of London and other cities in England, France, and Germany,\nstill thatched with straw in 1233.\n\nTallow candles so great a luxury, that splinters of wood were used for\nlights, 1298.\n\nWine sold by apothecaries as a cordial, 1298.\n\nGunpowder and guns first invented by Swartz, a monk of Cologn, 1340;\nEdward 3rd had four pieces of cannon, which contributed to gain him the\nbattle of Cressy, 1346; bombs and mortars were invented in the same\nyear.\n\nCards invented in France for the king's amusement in 1391.\n\nWindsor Castle built by Edward 3rd, 1386.\n\nGuildhall, London, built 1410.\n\nAbout 1430, Laurentius, of Haarlem, invented the art of Printing,\nwhich he practised with separate wooden types. Guttenburg afterwards\ninvented cut metal types: but the art was carried to perfection\nby Peter Schoeffer, who invented the mode of casting the types in\nmatrices. Frederick Corsellis began to print at Oxford, in 1468, with\nwooden types; but it was William Caxton who introduced into England the\nart of printing with fusile types, in 1474.\n\nShillings first coined in England, 1505.\n\nSilk stockings first worn by the French King, 1543; first worn in\nEngland by Queen Elizabeth in 1561.\n\nTobacco first brought from Virginia into England, 1583.\n\nWatches first brought into England from Germany, in 1597.\n\nRegular Posts established from London to Scotland, Ireland, &c., 1635.\n\nThe Plague rages in London, and carries off 68,000 persons, 1665.\n\nThe great fire of London began, September 2nd, and continued three\ndays, in which were destroyed 13,000 houses, and 400 streets, 1666.\n\nTea first used in England, 1666.\n\nThe Habeas Corpus act passed, 1678.\n\nWilliam Penn, a Quaker, receives a charter for planting Pennsylvania,\n1680.\n\nBank of England established by King William 1693.\n\nThe first public Lottery was drawn same year.\n\nThe first British Parliament, 1707.\n\nThe Cathedral Church of St. Paul, London, rebuilt by Sir Christopher\nWren, in 37 years, at one million expense, by a duty on coals, 1710.\n\nWestminster Bridge, consisting of 15 arches, begun 1738, finished in\n1750, at the expense of 389,000_l._, defrayed by parliament.\n\nCommodore Anson returns from his voyage round the world, 1774.\n\nThe British Museum erected at Montagu House, 1753.\n\n149 Englishmen are confined in the black-hole at Calcutta, in the East\nIndies, by order of the Nabob, and 123 found dead next morning, 1755.\n\n\nLEGENDS AMONG SAVAGE NATIONS.\n\nIt is curious to note how savages endeavour to account for the\nprodigies of nature. In the island of Samoa, one of the Sandwich group,\nthere is the following legend.\n\nMafuie is their god of earthquakes, who was deemed to possess great\npower, but has, according to the Samoans, lost much of it. The way in\nwhich they say this occurred is as follows:--One Talago, who possessed\na charm capable of causing the earth to divide, coming to a well-known\nspot, cried, \"Rock, divide! I am Talago; come to work!\" The earth\nseparating at his command, he went down to cultivate his taro patch.\nHis son, whose name was Tiitii, became acquainted with the charm, and\nwatching his father, saw him descend, and the earth close after him. At\nthe same spot, Tiitii said, \"Rock, divide! I am Talago; come to work!\"\nThe rock did not open, but on repeating the words and stamping his foot\nviolently, the earth separated, and he descended. Being a young man,\nhe made a great noise and bustle, notwithstanding the advice of his\nfather to be quiet, lest Mafuie would hear him. The son then asked,\n\"Who is Mafuie, that I should be afraid of him?\" Observing smoke at\na distance, he inquired the cause of it. Talago said, \"It is Mafuie\nheating his oven.\" Tiitii determined to go and see, notwithstanding\nall the persuasions of his father, and met Mafuie, who inquired who he\nwas, \"Are you a planter of taro, a builder, or a twister of ropes?\" \"I\nam a twister of ropes,\" said Tiitii; \"give me your arm, and I shall\nshow you.\" So, taking the arm of Mafuie, he twisted it off in a moment.\nSuch a practical illustration of his powers soon made Mafuie cry out,\n\"Na fia ola, na fia ola!\"--I desire to live, I desire to live! Tiitii\nthen took pity upon him, and let him go. The natives, on feeling an\nearthquake, exclaim, \"Thanks that Mafuie has but one arm! if he had\ntwo, he would shake the earth to pieces.\"\n\n\nORIGIN OF THE WORD LADY.\n\nIt was the custom at the time of the Plantagenets, and previously, for\nladies of distinction and wealth regularly to distribute money or food\nto the poor. The title of _lady_, is derived from the Anglo-Saxon, and\nliterally signifies _giver of bread_. The purse, with similar meaning,\nwas named as a receptacle for _alms_, and not as an invention for the\npreservation of money.\n\n\nANECDOTES IN SERMONS.\n\nThe fashion which once prevailed of introducing historical anecdotes\ninto addresses from the pulpit, is illustrated by the following extract\nfrom a sermon by the Martyr Bishop Ridley:--\n\nCambyses was a great emperor, such another as our master is; he had\nmany lord-deputies, lord-presidents, and lieutenants under him. It is\na great while ago since I read the history. It chanced he had under\nhim, in one of his dominions, a briber, a gift-taker, a gratifier of\nrich men; he followed gifts as fast as he that followed the pudding,\na hand-maker in his office, to make his son a great man; as the old\nsaying is, \"Happy is the child whose father goeth to the devil.\" The\ncry of the poor widow came to the emperor's ear, and caused him to flay\nthe judge quick, and laid his skin in his chair of judgment, that all\njudges that should give judgment afterward should sit in the same skin.\nSurely it was a goodly sign, a goodly monument, the sign of the judge's\nskin: I pray God we may once see the sign of the skin in England.\n\n\nSTATE OF LONDON IN 1756.\n\nThe state of the police regulations in the metropolis at the above\ndate, is exhibited in the following extract from an old magazine:--\n\n\"At one o'clock this morning (Oct. 4, 1756), the Hon. Captain Brudenel\nwas stopped in his chair, just as it entered Berkeley-square, from\nthe Hay-hill, by two fellows with pistols, who demanded his money; he\ngave them five-sixpences, telling them he had no more, which having\ntaken, they immediately made off. The captain then put his purse and\nwatch under the cushion, got out, drew his sword, and being followed\nby one of the chairmen with his pole, and the watchman, pursued them\nup the hill, where the Hon. Captain West, who was walking, having\njoined them, one of the fellows having got off, they followed the other\ninto Albemarle-mews, where finding himself closely beset, he drew a\npistol, and presented it, upon which the captain made a lunge at him,\nand ran him through the body. The fellow at the same time fired his\npistol, which, the captain being still stooping, went over his head\nand shot the watchman through the lungs; at the instant the pistol was\ndischarged, while the fellow's arm was extended, the chairman struck\nit with his pole and broke it; he was then seized and carried with\nthe watchman to the round-house in Dover-street, where Mr. Bromfield\nand Mr. Gataker, two eminent surgeons, came; but the captain would\nnot suffer the villain to be dressed, till he discovered who he and\nhis confederates were; when he acknowledged they were both grenadiers\nin Lord Howe's company. The poor watchman died in half an hour after\nhe was shot; and the soldier was so disabled by his wound that he was\ncarried in a chair to Justice Fielding, who sent him to New Prison,\nwhere he died.\"\n\n\nFROM A HANDBILL OF BARTHOLOMEW FAIR IN 1700.\n\nThe following extract is worth notice, inasmuch as it shows that in the\nmatter of amusement, the tastes of the lower orders of the present day\nare not much improved since the last century:--\n\n\"You will see a wonderful girl of ten years of age, who walks backwards\nup the sloping rope driving a wheelbarrow behind her; also you will see\nthe great Italian Master, who not only passes all that has yet been\nseen upon the low rope, but he dances without a pole upon the head of a\nmast as high as the booth will permit, and afterwards stands upon his\nhead on the same. You will be also entertained with the merry conceits\nof an Italian scaramouch, who dances on the rope with two children and\na dog in a wheelbarrow, and a duck on his head.\"\n\n\nPASSAGE THROUGH THE ISTHMUS OF PANAMA SUGGESTED THREE HUNDRED TEARS AGO.\n\n_Ancient Globe._--In the Town Library (_Stadt Bibliothek_) of Nuremberg\nis preserved an interesting globe made by John Schoner, professor of\nmathematics in the Gymnasium there, A.D. 1520. It is very remarkable\nthat the passage through the Isthmus of Panama, so much sought after in\nlater times, is, on this old globe, carefully delineated.\n\n\nHEIGHT OF MOUNTAINS.\n\nThe perpendicular height of Snowdon is, by late admeasurements, 1,190\nyards above the level of the sea. This makes it, according to Pennant,\n240 yards higher than Cader Idris. Some state Whernside, in Yorkshire\nto be the highest mountain in South Britain, and more than 4,000 feet.\nHelvellyn is 3,324 feet, Ben Lomond 3,262. Mont Blanc rises 15,680\nfeet; the American Chimborazo is 20,909 feet, the highest ground ever\ntrodden by man; and the mountains of Thibet above 25,000 feet, the\nhighest at present known.\n\n\nINTRODUCTION OF THE WEEPING WILLOW INTO ENGLAND.\n\nThe _Salix Babylonica_, that is the Willow of Babylon, or our English\nweeping Willow, is a native of the Levant, the coast of Persia,\nand other places in the East. The manner of its introduction into\nEngland is curious; the account is as follows: Pope, the celebrated\npoet, having received a present of Turkey figs, observed a twig of\nthe basket, in which they were packed, putting out a shoot. The twig\nhe planted in his garden: it soon became a fine tree, and from this\nstock, all our weeping Willows have descended. This species of Willow\nis generally planted by a still pool, to which it is a beautiful\nappropriate ornament; and when in misty weather, drops of water are\nseen distilling from the extremities of its branches, nothing can be\nmore descriptive than the title it has obtained of _the weeping Willow_.\n\n\nFINE FOR INSULTING A KING.\n\nThe use of gold and silver was not unknown to the Welsh in 842, when\ntheir laws were collected. The man who dared to insult the King of\nAberfraw, was to pay (besides certain cows and a silver rod) a cup,\nwhich would hold as much wine as his majesty could swallow at a\ndraught; its cover was to be as broad as the king's face; and the whole\nas thick as a goose's egg, or a ploughman's thumb-nail.\n\n\nCARRONADES.\n\nThis species of great gun, so much used on board of ships, is generally\naccounted a modern invention, taking its name from the Carron foundry\nwhere they were made. In the patent office, however, will be found\na notice dated September, 1727, to the following effect: \"That his\nMajesty was pleased to grant to Henry Brown, Esquire, a patent for the\nsole use and benefit of his new invention of making cannon and great\nguns, both in iron and brass, which will be much shorter and lighter,\nand with less powder will carry farther than those of equal bore now in\nuse, and which, it is said, will save great expense to the public.\"\n\n\nEXTRAVAGANCE AT ELECTIONS.\n\nOn the death of Sir James Lowther, his son William stood for the shire\nof Cumberland, and entertained 3,650 gentlemen freeholders at a dinner,\nat which were consumed 768 gallons of wine, 1,454 gallons of ale, and\n5,814 bottles of punch. Sir James appears to have been eccentric in\nsome of his habits, for after his decease L30,000 in bank notes were\ndiscovered in a closet, and L10,000 in the sleeve of an old coat.\n\n\nMARTIN LUTHER'S TANKARD.\n\nThis interesting relic of the great Reformer is of ivory, very richly\ncarved, and mounted in silver gilt. There are six medallions on its\nsurface, which consist, however, of a repetition of two subjects. The\nupper one represents the agony in the garden, and the Saviour praying\nthat the cup might pass from Him; the base represents the Lord's\nSupper, the centre dish being the incarnation of the bread. This\ntankard, now in the possession of Lord Londesborough, was formerly in\nthe collection of Elkington of Birmingham, who had some copies of it\nmade. On the lid, in old characters, is the following inscription--\"C.\nM. L., MDXXIIII.\" This drinking vessel, which, independent of its\nartistic merit, was no doubt highly valued as a mere household\npossession, brings to mind many recollections of the life of him who\nraised himself from a very lowly position to one of great power and\nusefulness.\n\n[Illustration: [++] Martin Luther's Tankard.]\n\nMartin Luther, who was the son of John Lotter or Lauther (which name\nour Reformer afterwards changed to Luther) and Margaret Lindenen, was\nborn in the little town of Islebern, in Saxony, on November 10th,\n1483. His father was a miner. Luther died in 1546, and princes, earls,\nnobles, and students without number, attended the funeral of the\nminer's son in the church of Islebern. On this occasion, Melancthon\ndelivered the funeral oration.\n\n\nHOT CROSS BUNS.\n\nHow strange the following reads from an old journal! and how odd the\nstate of things to give rise to such an intimation!\n\n 1793.\n _Wednesday, 27th March._\n ROYAL BUN HOUSE, CHELSEA, GOOD FRIDAY.\n _No Cross Buns._\n\nMrs. Hand respectfully informs her friends, and the public, that in\nconsequence of the great concourse of people which assembled before her\nhouse at a very early hour, on the morning of Good Friday; by which her\nneighbours (with whom she has always lived in friendship and repute,)\nhave been much alarmed and annoyed; it having also been intimated, that\nto encourage or countenance a tumultuous assembly at this particular\nperiod, might be attended with consequences more serious than have\nhitherto been apprehended; desirous, therefore, of testifying her\nregard and obedience to those laws by which she is happily protected,\nshe is determined, though much to her loss, not to sell _Cross Buns_ on\nthat day, to any person whatever; but Chelsea Buns as usual.\n\nMrs. Hand would be wanting in gratitude to a generous public, who, for\nmore than fifty years past, have so warmly patronized and encouraged\nher shop, to omit so favourable an opportunity of offering her sincere\nacknowledgments for their favours; at the same time, to assure them she\nwill, to the utmost of her power, endeavour to merit a continuance of\nthem.\n\n\nLOCUSTS.\n\nThe locusts are remarkable for the hieroglyphic that they bear upon the\nforehead. Their colour is green throughout the whole body, excepting a\nlittle yellow rim that surrounds their head, and which is lost at the\neyes. This insect has two upper wings, pretty solid. They are green,\nlike the rest of the body, except that there is in each a little white\nspot. The locust keeps them extended like great sails of a ship going\nbefore the wind. It has besides two other wings underneath the former,\nand which resemble a light transparent stuff pretty much like a cobweb,\nand which it makes use of in the manner of smack sails, that are along\na vessel. But when the locust reposes herself, she does like a vessel\nthat lies at anchor; for she keeps the second sails furled under the\nothers.\n\n\nQUEEN ELIZABETH'S LAWS.\n\nThe following extract from a very old book is truly curious:--\n\n\"Queene Elizabeth, in the xiiii and xviii yeres of hir gracious rayne,\ntwo Actes were made for ydle vagrante and maisterlesse persons, that\nvsed to loyter, and would not worke, should for the first offence\nhaue a hole burned through the gristle of one of his eares of an\nynch compasse. And for the second offence committed therein, to be\nhanged. If these and such lyke lawes were executed iustlye, treulye,\nand severelye (as they ought to be,) without any respect of persons,\nfavour, or friendshippe, this dung and filth of ydlenesse woulde\neasily be reiected and cast oute of thys Commonwealth, there woulde\nnot be so many loytering ydle persons, so many Ruffians, Blasphemers,\nSwinge-Buckelers, so many Drunkards, Tossepottes, Dauncers, Fydlers,\nand Minstrels, Dice-players, and Maskers, Fencers, Theeves,\nEnterlude-players, Cut-purses, Cosiners, Maisterlesse Seruantes,\nJugglers, Roges, sturdye Beggars, counterfaite Egyptians, &c., as there\nare, nor yet so manye Plagues to bee amongst vs as there are, if these\nDunghilles and filthe in Commonweales were remooued, looked into, and\ncleane caste oute, by the industrie, payne, and trauell of those that\nare sette in authoritie, and haue gouernment.\"--\"_A Treatise against\nDicing, Dauncing, Vaine playes or Enterluds._\" _Black Letter; no date._\n\n\nTHE INVENTION OF TYPES.\n\nThe honour of the invention of movable types has been disputed by\ntwo cities, Haarlem and Mentz. The claims of Haarlem rest chiefly\nupon a statement of Hadrien Junius, who gave it upon the testimony\nof Cornelius, alleged to be a servant of Lawrence Coster, for whom\nthe invention is claimed. The claims of Mentz, which appear to be\nmore conclusive, are in favour of Peter Schaeffer, the assistant and\nson-in-law of John Faust, better known as Dr. Faustus. The first\nedition of the _Speculum humanae salvationis_ was printed by Coster at\nHaarlem, about the year 1440, and is one of the earliest productions of\nthe press of which the printer is known. The celebrated Bible, commonly\nknown as the Mentz Bible, without date, is the first important specimen\nof printing with moveable metal types. This was executed by Gutenberg\nand Faust, or Fust, as it is sometimes spelt, between the years 1450\nand 1455. The secret of the method then becoming known, presses were\nspeedily established in all parts of Europe, so that before the year\n1500 there were printing-offices in upwards of 220 different places in\nAustria, Bavaria, Bohemia, Calabria, the Cremonese, Denmark, England,\nFlanders, France, Franconia, Frioul, Geneva, Genoa, Germany, Holland,\nHungary, Italy, Lombardy, Mecklenburg, Moravia, Naples, the Palatinate,\nPiedmont, Poland, Portugal, Rome, Sardinia, Upper and Lower Saxony,\nSicily, Silesia, Spain, Suabia, Switzerland, Thessalonica, Turkey,\nTuscany, the Tyrol, Venice, Verona, Westphalia, Wurtemberg, &c.\n\nThis vast and rapid extension of the art, combined with the skill which\nthe earlier printers displayed in it, seems to be totally incompatible\nwith the date assigned to the invention, and it is more than probable,\nthat the art having been long practised in private under continued\nattempts at secrecy, it at length broke into publicity after it had\nalready attained a considerable degree of perfection.\n\n\nTHE PROTEUS ANGUINUS.\n\nIt has been satisfactorily proved that the polypus cannot see its\nprey, but is only aware of its presence by the actual agitation of\nthe water, from its remaining altogether passive when a thin piece\nof glass is interposed between them. There are many Monads, which,\nwithout possessing any trace of an eye, are yet susceptible of light.\nAn equally extraordinary phenomenon presents itself in the Proteus\nAnguinus. This singular animal is found in the subterranean lakes of\nthe interminable stalactital caverns in the limestone range of the\nCarniolan Alps, where the author saw it. In appearance it is between a\nfish and a lizard; it is of a flesh-colour, and its respiratory organs,\nwhich are connected with lungs, so as to enable it to breathe above or\nbelow the water, form a red crest round the throat, like a cock's comb.\nIt has no eyes, but small points in the place of them, and light is so\nobnoxious to it, that it uses every effort to exclude it, by thrusting\nits head under stones. It is reported also to exist in Sicily, but is\nknown nowhere else.\n\n\nBUMPER.\n\n[Illustration: [++] A Bumper.]\n\nThe jolly toper is so fond of the thing we call a _bumper_, that he\ntroubles not himself about the name, and so long as the liquor is but\nfine and clear, cares not a farthing in how deep an obscurity the\netymology is involved. The sober antiquarian, on the contrary, being\nprone to etymology, contemplates the sparkling contents of a full glass\nwith much less delight than he does the meaning, the occasion, and the\noriginal of the name. The common opinion is, that the _bumper_ took\nits name from the _grace-cup_; our Roman Catholic ancestors, say they,\nafter their meals, always drinking the Pope's health in this form, _au\nbon Pere_. But there are great objections to this; the Pope was not the\n_bon Pere_, but the _Saint Pere_; amongst the elder inhabitants of this\nkingdom, the attribute of sanctity being in a manner appropriated to\nthe Pope of Rome, and his see. Again, the grace-cup, which went round\nof course, after every repast, did not imply anything extraordinary,\nor a full glass. Drinking-glasses were not in use at the time here\nsupposed, for the grace-cup was a large vessel, proportioned to the\nnumber of the society, which went round the table, the guests drinking\nout of one cup, one after another.\n\n\nCOFFEE.\n\nFrom a number of the \"_Public Advertiser_,\" of May 19 to May 26, 1657,\nwe have 'In Bartholomew-lane, on the back side of the Old Exchange, the\ndrink called _Coffee_ is advertised as to be sold _in the morning_, and\nat _three of the clock_ in the afternoon.'\n\n\nQUAINT RECEIPTS.\n\nThe following Receipts are taken from a work entitled, \"_New\nCuriosities_ in _Art_ and _Nature_, or a _Collection_ of the most\nvaluable _Secrets_ in all _Arts_ and _Sciences_. Composed and\nExperimented by Sieur Lemery, Apothecary to the French King. London:\nJohn King, Little Britain. 1711.\"\n\n_To make one Wake or Sleep._--You must cut off dexterously the head of\na toad alive, and at once, and let it dry, in observing that one eye be\nshut, and the other open; that which is found open makes one wake, and\nthat shut causes sleep, by carrying it about one.\n\n_Preservative against the Plague._--Take three or four great toads,\nseven or eight spiders, and as many scorpions, put them into a pot well\nstopp'd, and let them lye some time; then add virgin-wax, make a good\nfire till all become a liquor, then mingle them all with a spatula, and\nmake an ointment, and put it into a silver box well stopp'd, the which\ncarry about you, being well assured that while you carry it about you,\nyou will never be infected with the plague.\n\nWe give the above as indicating the delusions which prevailed with\nrespect to certain nostrums as late as the year 1711.\n\n\nEXECUTION OF GOVERNOR WALL IN 1802.\n\nAs the following account, by a gentleman who witnessed the scene,\navoids all disgusting details, we give it as containing a description\nof some of the circumstances which attended the execution, at the\ncommencement of the present century, of a criminal of the higher class.\nThe wretched man was hung for murder and barbarity: his victims were\nthe men he had under his charge as Governor of the Island of Goree:--\n\n\"As we crossed the Press-yard, a cock crew; and the solitary clanking\nof a restless chain was dreadfully horrible.\n\n\"The prisoner entered. He was death's counterfeit, tall, shrivelled,\nand pale; and his soul shot so piercingly through the port-holes of\nhis head that the first glance of him nearly petrified me. I said in\nmy heart, putting my pencil in my pocket, God forbid that I should\ndisturb thy last moments! His hands were clasped, and he was truly\npenitent. After the yeoman had requested him to stand up, 'he pinioned\nhim,' as the Newgate phrase is, and tied the cord with so little\nfeeling that the governor, who had not given the wretch the accustomed\nfee, observed 'You have tied me very tight;' upon which Dr. Ford,\nthe chaplain, ordered him to slacken the cord, which he did, but not\nwithout muttering, 'Thank you, sir,' said the governor to the doctor:\n'it is of little moment.' He then observed to the attendant, who had\nbrought in an immense iron shovel-full of coals to throw on the fire,\n'Ay, in one hour that will be a blazing fire,' then turning to the\ndoctor, questioned him: 'Do tell me, sir: I am informed I shall go down\nwith great force; is it so?' After the construction and action of the\nmachine had been explained, the doctor questioned the governor as to\nwhat kind of men he had at Goree:--'Sir,' he answered, 'they sent me\nthe very riff-raff.' The poor soul then joined the doctor in prayer;\nand never did I witness more contrition at any condemned sermon than he\nthen evinced.\n\n\"The sheriff arrived, attended by his officers, to receive the prisoner\nfrom the keeper. A new hat was then partly flattened on his head,\nfor owing to its being too small in the crown, it stood many inches\ntoo high behind. As we were crossing the Press Yard, the dreadful\nexecrations of some of the felons so shook his frame that he observed,\nthe clock had struck; and quickening his pace, he soon arrived at the\nroom where the sheriff was to give a receipt for his body, according to\nthe usual custom. Owing, however, to some informality in the wording of\nthis receipt, he was not brought out as soon as the multitude expected;\nand it was this delay which occasioned a partial exultation from those\nwho betted as to a reprieve, and not from any pleasure in seeing him\nexecuted.\n\n\"After the execution, as soon as I was permitted to leave the prison,\nI found the Yeoman selling the rope with which the malefactor had\nbeen suspended, at a shilling an inch; and no sooner had I entered\nNewgate-street, than a lath of a fellow, passed threescore years\nand ten, who had just arrived from the purlieus of Black Boy Alley,\nwoe-begone as _Romeo's_ apothecary, exclaimed, 'Here is the identical\nrope at sixpence an inch.'\"\n\n\nSTAGE-COACH IN 1760.\n\n_Ayscough's Nottingham Courant_ of this date, contained the following\nadvertisement:--The flying machines on steel springs set off from the\nSwan with Two Necks Inn, Lad-lane, London, and from the Angel Inn in\nSheffield, every Monday and Thursday morning at five o'clock, and lies\nthe first night from London at the Angel Inn in Northampton, the second\nat the Blackmoor's Head Inn, Nottingham, and the third at Sheffield.\nEach passenger to pay 1_l._ 17_s._, and to be allowed fourteen pounds\nof luggage. Performed (if God permit) by John Hanforth and Samuel\nGlanville.\n\n\nBLIND WORKMAN.\n\nA young man, in Greenock, of the name of Kid, who was blind from his\ninfancy, finished the model of a sixty-four gun ship, of about five\nfeet keel, planked from the keel, with carriages for the guns, and\nevery necessary material and apparelling of a ship of that rate,\nwithout any assistance whatever, or other instrument than a small knife\nand hammer.\n\n\nSPORTS OF THE LOWER CLASSES IN 1749.\n\nThe following handbill is curious, on account of the light it sheds on\nwhat was considered attractive to the million a hundred years ago:--\n\n\"_According to Law. September 22, 1749._ On Wednesday next, the 27th\ninst., will be run for by _Asses_ (!!) in _Tothill Fields_, a purse\nof gold, not exceeding the value of Fifty Pounds. The first will be\nentitled to the gold; the second to two pads; the third to thirteen\npence half-penny; the last to a halter fit for the neck of any ass in\nEurope. Each ass must be subject to the following articles:--\n\n\"No person will be allowed to run but _Taylors_ and _Chimney-sweepers_;\nthe former to have a cabbage-leaf fixed in his hat, the latter a\nplumage of white feathers; the one to use nothing but his yard-wand,\nand the other a brush.\n\n\"No jockey-tricks will be allowed upon any consideration.\n\n\"No one to strike an ass but the rider, lest he thereby cause a\nretrograde motion, under a penalty of being ducked three times in the\nriver.\n\n\"No ass will be allowed to start above thirty years old, or under ten\nmonths, nor any that has won above the value of fifty pounds.\n\n\"No ass to run that has been six months in training, particularly above\nstairs, lest the same accident happen to it that did to one nigh a town\nten miles from London, and that for reasons well known to that place.\n\n\"Each ass to pay sixpence entrance, three farthings of which are to be\ngiven to the old clerk of the race, for his due care and attendance.\n\n\"Every ass to carry weight for inches, if thought proper.\"\n\nThen follow a variety of sports, with \"an ordinary of _proper\nvictuals_, particularly for the riders, if desired.\"\n\n\"_Run, lads, run! there's rare sport in Tothill Fields!_\"\n\n\nSTATE COACH AT THE PROROGATION OF PARLIAMENT IN 1796.\n\nNever was a greater assemblage of persons collected together than on\nthis occasion: in the Park and in Parliament-street there were at least\n20,000 people. By the repair of the state coach, which has undergone\nseveral material alterations since the damage it received at the\nopening of the last session, the king is now secluded from the sight.\nHitherto, the upper pannels of it had always been of glass, so that\nthe multitude could see the king in all directions, through the front,\nthrough the sides, as well as through the windows in the doors: it has\nbeen newly glazed, and the whole of the carriage is lined with sheet\ncopper, musket proof; between the crimson lining of the carriage is a\nwadding of fine wool, coated with buffalo skin, the nature of which is\nso close that no bullets can penetrate it.\n\n\nHISTORICAL ANECDOTE.\n\nOn the dollars, stivers, &c., coined at the town of Dordrecht in\nHolland, is the figure of a milk-maid sitting under her cow, which\nfigure is also exhibited in relievo on the water-gate of the place. The\noccasion was as follows: In the noble struggle of the United Provinces\nfor their liberties, the Spaniards detached a body of forces from the\nmain army, with the view of surprising Dordrecht. Certain milkmaids,\nbelonging to a rich farmer in the vicinity of the town, perceived as\nthey were going to milk, some soldiers concealed under the hedges.\nThey had the presence of mind to pursue their occupation without any\nsymptoms of alarm. On their return home they informed their master of\nwhat they had seen, who gave information to the Burgomaster, and the\nsluices were let loose, by which great numbers of the Spaniards were\ndrowned, and the expedition defeated. The States ordered the farmer a\nhandsome revenue for the loss he sustained by the overflowing of his\nlands, rewarded the women, and perpetuated the event in the manner\ndescribed.\n\n\nTOMB OF JOHN BUNYAN.\n\n\"Who has not read the \"Pilgrim's Progress,\" \"that wonderful book,\"\nwrites Mr. Macaulay, \"which, while it obtains admiration from the most\nfastidious critics, is loved by those who are too simple to admire it?\"\nWe can remember our own delight on reading, for the first time, the\nprecious volume. This was in the days of our childhood, when we were\ndeeply imbued with the fairy lore which at that time was so plentifully\nsupplied, and so eagerly devoured.\n\nJohn Bunyan was buried in Bunhill Fields burying-ground, City-road; and\nthe tablet on his tomb, which the engraving very correctly represents\nis as follows:--\"Mr. John Bunyan, author of the 'Pilgrim's Progress,'\nob. 12 Aust. 1688, aet. 60.\" Formerly there were also the following\nlines:--\n\n \"The Pilgrim's Progress now is finished.\n And death has laid him in his earthly bed.\"\n\n[Illustration: [++] Tomb of John Bunyan.]\n\nBunhill Fields burying-ground was opened as a suburban cemetery\nin 1665, in the time of the great plague, and was a favourite\nburying-place with the Dissenters. Here are buried Daniel Defoe; Dr.\nIsaac Watts; Joseph Ritson the antiquary; Dr. Thomas Goodwin, the\nchaplain who attended Cromwell's death-bed; George Fox, the founder\nof the Quakers; the mother of John Wesley; Lieut.-General Fleetwood,\nson-in-law of Oliver Cromwell; Thomas Stothard, R.A., and other eminent\nmen.\n\n\nSPIDERS FOND OF MUSIC.\n\nSpiders hear with great acuteness, and it is affirmed that they are\nattracted by music. Disjonval relates the instance of a spider which\nused to place itself on the ceiling of a room over the spot where a\nlady played the harp, and which followed her if she removed to another\npart; and he also says that the celebrated violinist Berthome, when a\nboy, saw a spider habitually approach him as soon as he began to play,\nand which eventually became so familiar that it would fix itself on his\ndesk, and on his arm. Bettina noticed the same effect with a guitar, on\na spider which accidentally crossed over it as she was playing.\n\n\nBREAKFASTING HUT IN 1745.\n\nThis quaint announcement, in a handbill of the time, shows how cheaply\nthose who lived a century or so past could enjoy suburban pleasures in\nmerrie Islington:--\n\n\"This is to give notice to all Ladies and Gentlemen, at Spencer's\noriginal Breakfasting-Hut, between Sir Hugh Middleton's Head and St.\nJohn Street Road, by the New River side, fronting Sadler's Wells, may\nbe had every morning, except Sundays, fine tea, sugar, bread, butter,\nand milk, at fourpence per head; coffee at threepence a dish. And in\nthe afternoon, tea, sugar, and milk, at threepence per head, with good\nattendance. Coaches may come up to the farthest garden-door next to the\nbridge in St. John Street Road, near Sadler's Wells back gate.--_Note._\nLadies, &c., are desired to take notice that there is another person\nset up in opposition to me, the next door, which is a brick-house,\nand faces the little gate by the Sir Hugh Middleton's, and therefore\nmistaken for mine; but mine is the _little boarded place_ by the river\nside, and my backdoor faces the same as usual; for\n\n I am not dead, I am not gone,\n Nor liquors do I sell;\n But, as at first, I still go on,\n Ladies, to use you well.\n\n No passage to my hut I have,\n The river runs before;\n Therefore your care I humbly crave,\n Pray don't mistake my door.\n\n \"Yours to serve,\n S. SPENCER.\"\n\n\nSPONTANEOUS COMBUSTION.\n\nIn Leroux's Journal de Medicine, is an account of a very fat woman,\ntwenty-eight years of age, who was found on fire in her chamber, where\nnothing else was burning. The neighbours heard a noise of something\nlike frying, and when the body was removed it left a layer of black\ngrease. The doctor conceives that the combustion began in the internal\nparts, and that the clothes were burnt secondarily.\n\n\nMOTHER MAPP THE BONE-SETTER.\n\nShe was the daughter of a man named Wallis, a bone-setter at Hindon, in\nWiltshire, and sister to the celebrated \"Polly Peachem,\" who married\nthe Duke of Bolton. Upon some _family quarrel_, Sally Wallis left\nher professional parent, and wandered up and down the country in a\nmiserable manner, calling herself \"Crazy Sally,\" and pursuing, in her\nperambulations, a course that fairly justified the title. Arriving at\nlast at Epsom, she succeeded in humbugging the worthy bumkins of that\nplace, so decidedly, that a subscription was set on foot to keep her\namong them; but her fame extending to the metropolis, the dupes of\nLondon, a numerous class then as well as now, thought it no trouble\nto go ten miles to see the conjuror, till at length, she was pleased\nto bless the afflicted of London with her presence, and once a week\ndrove to the Grecian Coffee-house, in a coach and six with out-riders!\nand all the appearance of nobility. It was in one of these journeys,\npassing through Kent-street, in the Borough, that being taken for a\ncertain woman of quality from the Electorate in Germany, a great mob\nfollowed and bestowed on her many bitter reproaches, till Madame,\nperceiving some mistake, looked out of the window, and accosted them in\nthis gentle manner, \"Confound you, don't you know me? I am Mrs. Mapp,\nthe _bone-setter_!\" upon which, they instantly changed their revilings\ninto loud huzzas.\n\n\nTWO CERTIFICATES OF GRETNA-GREEN MARRIAGES AT DIFFERENT DATES.\n\n\"This is to sartfay all persons that my be consernid, that A B from the\nparish of C in the County of D and E F from the parish of G and in the\ncounty of H and both comes before me and declayred themseless both to\nbe single persons, and now mayried by the form of the Kirk of Scotland,\nand agreible to the Church of England, and givine ondre my hand, this\n18{th} day of March 1793.\"\n\n \"Kingdom of Scotland\n \"County of Dumfries\n \"Parish of Gretna\n\n\"These are to certify, to all whom it may concern, that John N....\nfrom the parish of Chatham in the County of Kent, and Rosa H.... from\nthe Parish of St. Maries in the County of Nottingham, being both here\nnow present and having declared to me that they are single persons,\nbut have now been married conformable to the Laws of the Church of\nEngland, and agreeable to the Kirk of Scotland. As witness our hands at\nSpringfield this 4th day of October 1822.\n\n \"Witness \"Witness me.\n Jane Rae David Lang.\n John Ainslie.\" John N....\n Rosa H....\"\n\n\nTHE WOMEN OF ENGLAND.\n\nThe women here are generally more handsome than in other places,\nsufficiently endowed with natural beauties, without the addition of\nadulterate sophistications. In an absolute woman, say the Italians,\nare required the parts of a Dutch woman, from the girdle downwards; of\na French woman, from the girdle to the shoulders: over which must be\nplaced an English face. As their beauties, so also their prerogatives\nare greater than any nation; neither so servilely submissive as the\nFrench, nor so jealously guarded as the Italians; but keeping so true a\ndecorum, that as England is termed the Pergatorie of Servants, and the\nHell of Horses, so is it acknowledged the _Paradise_ of _Women_. And it\nis a common by-word amongst the Italians, that _if there were a bridge\nbuilt across the narrow seas, all the women in Europe would run into\nEngland_. For here they have the upper hand in the streets, the upper\nplace at the table, the thirds of their husband's estates, and their\nequal share of all lands; privileges with which other women are not\nacquainted. They were in high esteem in former times amongst foreign\nnations, for the modestie and gravitie of their conversation; but of\nlate so much addicted to the light garb of the French, that they have\nlost much of their ancient honour and reputation amongst knowing and\nmore sober men of foreign countries who before admired them.--_Peter\nHeylin's Cosmographie_, 1652.\n\n\nPRICES FOR SEATS AT CORONATIONS.\n\nOn consulting Stowe, Speed, and other antiquaries, it appears that the\nprice of a good place at the coronation of William the Conqueror was\na _blank_; and probably the same at that of his son William Rufus. At\nthat of Henry I. it was a _crocard_, and at King Stephen's and Henry\nthe Second's a _pillard_. At King Richard's and King John's, it was a\n_fuskin_; and rose at Henry the Third's to a _dodkin_. In the reign of\nEdward I. the coins began to be more intelligible; and we find that for\nseeing his coronation a Q was given, or the half of a _ferling_, or\nfarthing, which was, as now, the fourth part of a _sterling_, or penny.\nAt the coronation of Edward II. it was a farthing; and at that of\nEdward III. a halfpenny, which was very generally given. In the reign\nof Richard II. it was a penny, and continued the same at that of Henry\nIV. But at that of Henry V. it was two pennies, or half of a _grossus_,\nor groat; and the same at that of Henry VI. and of Edward IV.; nor do\nwe find it raised at the coronation of Richard III. or that of Henry\nVII.\n\nAt that of Henry VIII. it was the whole _grossus_, or groat, nor was\nthe price altered at those of Edward VI. and Queen Mary; but at Queen\nElizabeth's it was a _teston_, _tester_, or sixpence. At those of James\nI. and Charles I. a shilling was given; which sum was advanced to half\na crown at the coronations of Charles and James II. At King William's\nand Queen Anne's, it was a crown; and at George the First's the show\nwas seen by many at the same price.\n\nAt the coronation of George II. some gave half a guinea; but at that\nof George III. and Queen Charlotte, anno 1761, curiosity seems to\nhave risen to an amazing height. On this occasion the price given for\nsingle seats were almost incredible; in some houses ten guineas, and\nin ordinary houses five guineas. Great and universal anxiety prevailed\nto see this grand spectacle, from the reflection how improbable\nit was that many who were there could ever have an opportunity of\nwitnessing the like again. As an instance of this extreme anxiety, it\nis confidently related, that a gentleman was prevailed on to take a\nroom for his lady, at the price of one hundred and forty guineas; but\nthe appointment of the solemnity of the coronation falling unluckily at\nthe exact time when she expected to be delivered, she actually further\nprevailed on her husband to let a skilful man-midwife, nurse, &c.,\nattend her, and to hire another room, lest the hurry of the day should\nbring on her labour, when it might be impossible for her to be removed\nwithout endangering her life.\n\n\nANCIENT HOUSE AT BLACKWALL--SAID TO BE THE RESIDENCE OF SIR WALTER\nRALEIGH.\n\nThe house shown in the engraving is interesting from two causes; first,\nthat it was the house in which Sir Walter Raleigh smoked his first\npipe of tobacco in England, and secondly, that it is one of the few\nrelics remaining of those picturesque old houses of the days of Queen\nBess. The house is built of strongly framed timber, which, in recent\nyears, has been plastered over; and the carved heads that ornament the\ngables, and which are good both in design and execution, show that this\nhouse is at least 350 years old.\n\n[Illustration: [++] Ancient House at Blackwall.]\n\nAt the present time a tavern has been built between this house and\nthe river. Formerly, however, there was, no doubt, a trimmed garden\nand terrace towards the Thames, from which the inhabitants may have\nwatched the progress of Queen Elizabeth from the Tower to her palace at\nGreenwich.\n\nIt is singular to notice the fashion of these old houses, arising from\nthe value of space within walled towns; each floor projects over the\nother, so that the upper apartments have more room than the lower.\nWhile, in an artistic point of view, we cannot help regretting the\ndisappearance of the venerable and quaint gables, for sanitary and\nother reasons we must be content with the change.\n\n\nAMBASSADORS--WHY HELD BY THE ARMS AT THE OTTOMAN COURT.\n\nA dervise addressed Bajazet, emperor of the Turks, 1495, for alms, and\nwhile the charitable Sultan searched for his money, the treacherous\nbeggar wounded him with a dagger, and was instantly slain by the\nroyal attendants. This incident is rendered memorable by its having\noccasioned the ungracious restraint under which even the ambassadors of\nChristian powers were subject to in former times when they received an\naudience from the Ottoman Emperor.\n\nThey were held by the arms by two attendants, when they approached the\nthrone, nor were their arms loosed till they had quitted the presence.\n\n\nTRAVELLING IN 1760.\n\nThe nobility and gentry were accustomed to make their long journeys in\nponderous family-carriages, drawn by four horses. These vehicles would\nbe laden at the top with an array of trunks and boxes, while perhaps\nsix or seven persons, with a lapdog, would be stowed within. The danger\nof famine on the road was averted by a travelling larder of baskets of\nvarious condiments; the risk of thirst would be provided against by\nbottles of usquebaugh, black cherry-brandy, cinnamon-water, sack, port,\nor strong beer: while the convoy would be protected by a basket-hilted\nsword, an old blunderbuss, and a bag of bullets and a great horn of\ngunpowder.\n\n\nOLD ST. PAUL'S.\n\nIn the old cathedral was a tower of stone, in height from the ground\n260 feet, on which was a spire of wood, covered with lead, 274 feet\nhigh. In the tower was a celebrated peal of bells; and somewhat above\nthe stone-work was a \"faire dial,\" from which there was order taken in\nthe eighteenth year of Edward III. that the rich chasing and gilding\nshould be always kept in good preservation. On this dial was the figure\nof an angel pointing to the hours of both day and night--a device more\nappropriate than most of the clock-hands in present use. From this\nlofty steeple, which formed such an important feature of old London,\nthe chimes rung merrily on saints' days and holidays; and at times the\nchoristers mounted up aloft and chaunted forth their orisons at dawn\nand sunset--a custom still observed at Durham Cathedral. Before the\nfire of London, the spire of St. Paul's was more than once destroyed or\ndamaged by fire and lightning.\n\nOn Candlemas Eve, 1444, about two o'clock in the afternoon, the\nlightning fired the steeple. The citizens came forth and succeeded in\novercoming the fire; it, however, broke forth again at night, and but\nlittle of the spire was saved. In the year 1561, in the month of June,\nthere fell a prodigious quantity of rain, attended with thunder and\nlightning. St. Paul's steeple was struck within a yard of the top. At\nfirst, a little fire appeared, resembling the light of a torch, and\nin eight minutes the weather-cock fell; and the wind rising high, the\nfire within an hour afterwards destroyed the steeple down to the very\nbattlements, and then, in consequence of the mass of burning timber\nthat fell from the spire, burnt so violently that the iron-work and\nthe bells melted and fell upon the stairs in the church; the east and\nwest roofs catching fire communicated with the north and south, and\ndestroyed them all. Much damage was also done to other parts.\n\nThe spire was again reared, and the damaged bells properly replaced.\nIn addition to the bells in the tower of old St. Paul's there was a\ncommon bell, the property of the city, hung in a suitable building,\nclosely adjoining to the Cathedral, which was rung that the inhabitants\nmight assemble at wardmotes and other important occasions. Another fire\ndamaged the ancient church, and then the great fire of 1666, swept\nsteeples, bells, churches, and all before it.\n\n\nTHE BEDFORD MISSAL.\n\nIn January, 1786, when the Bedford Missal was on sale, with the rest\nof the Duchess of Portland's collection, King George III. sent for\nhis bookseller, and expressed his intention to become the purchaser.\nThe bookseller ventured to submit to his majesty, that the article in\nquestion, as one highly curious, was likely to fetch a high price. \"How\nhigh?\" exclaimed the king. \"Probably two hundred guineas,\" replied the\nbookseller. \"Two hundred guineas for a Missal!\" exclaimed the Queen,\nwho was present, and lifted up her hands with astonishment. \"Well,\nwell,\" said his Majesty, \"I'll have it still; but since the Queen\nthinks two hundred guineas so enormous a price for a Missal, I'll go no\nfurther.\" The biddings for the royal library did actually stop at that\npoint; and Mr. Edwards carried off the prize by adding three pounds\nmore. The same Missal was afterwards sold at Mr. Edwards's sale, in\n1815, and purchased by the Duke of Marlborough, for L637 15s.\n\n\nFORMATION OF THE VOLCANO OF JORULLO.\n\nThe Mexican volcanoes of Orizaba, Popocatepetl, Jorullo, and Colima\nappear to be connected with each other, being placed in the direction\nof a line running transverse to the former, and passing east and west\nfrom sea to sea.\n\nAs was first observed by Humboldt, these mountains are all situated\nbetween north latitude 18 deg. 59' and 19 deg. 12'. In an exact line of\ndirection with the other volcanoes, and over the same transverse\nfissure, Jorullo was suddenly elevated on the 29th of September, 1759.\nThe circumstances attending the production of this volcano are so\nremarkable, that we shall here notice them in some detail.\n\n[Illustration: Volcano of Jorullo, Mexico.]\n\nAn extensive plain, called the Malpays, was covered by rich fields of\ncotton, sugar-cane, and indigo, irrigated by streams, and bounded by\nbasaltic mountains, the nearest active volcano being at the distance\nof eighty miles. This district, situated at an elevation of about 2600\nfeet above the level of the sea, was celebrated for its beauty and\nextreme fertility. In June, 1759, alarming subterranean sounds were\nheard, and these were accompanied, by frequent earthquakes, which were\nsucceeded by others for several weeks, to the great consternation of\nthe neighbouring inhabitants. In September tranquillity appeared to\nbe re-established, when, in the night of the 28th, the subterranean\nnoise was again heard, and part of the plain of Malpays, from three to\nfour miles in diameter, rose up like a mass of viscid fluid, in the\nshape of a bladder or dome, to a height of nearly 1700 feet; flames\nissued forth, fragments of red-hot stones were thrown to prodigious\nheights, and, through a thick cloud of ashes, illumined by volcanic\nfire, the softened surface of the earth was seen to swell up like an\nagitated sea. A huge cone, above 500 feet high, with five smaller\nconical mounds, suddenly appeared, and thousands of lesser cones\n(called by the natives _hornitos_, or ovens,) issued forth from the\nupraised plain. These consisted of clay intermingled with decomposed\nbasalt, each cone being a _fumarolle_, or gaseous vent, from which\nissued thick vapour. The central cone of Jorullo is still burning,\nand on one side has thrown up an immense quantity of scoriaceous and\nbasaltic lavas, containing fragments of primitive rocks. Two streams,\nof the temperature of 186 deg. of Fahrenheit, have since burst through the\nargillaceous vault of the hornitos, and now flow into the neighbouring\nplains. For many years after the first eruption, the plains of Jorullo\nwere uninhabitable from the intense heat that prevailed.\n\n\nCRATER OF VESUVIUS IN 1829.\n\n[Illustration: [++] Crater of Vesuvius.]\n\nThe crater Stromboli, which has been in activity since the most\nancient times, presents at present the same appearances as those which\nwere described by Spallanzani, in 1788. It is constantly filled with\nlava in a state of fusion, which alternately rises and falls in the\ncavity. Having ascended to ten or twelve yards below the summit of the\nwalls, this boiling fluid is covered with large bubbles, which burst\nwith noise, letting enormous quantities of gas escape from them, and\nprojecting on all sides scoriaceous matter. After these explosions, it\nagain subsides, but only to rise again and produce like effects--these\nalternations being repeated regularly at intervals of some minutes.\nIn craters where the lava is less fluid than in that of Stromboli, new\ncones are sometimes formed in the midst of the Crater, which first\nrise in the form of a dome, and then burst out so as to form a small\nactive volcano in the middle of the crater of the great one. This\nphenomenon is often presented within the crater of Vesuvius, and was\nmore particularly witnessed in 1829.\n\n\nLOAF SUGAR.\n\nIn 1553 a sugar-loaf was presented to Mr. Waldron, of Bovey House,\nwhich weighed 7 lbs., at 1s. 1d. per lb. (7s. 7d.)\n\nThe late Lord Rolle married the last of that branch of the Waldron\nfamily. The house remains about ten miles west of Lyme. The sugar-loaf\nwas charged at a high rate, considering the greater value of money\nin Queen Mary's reign. This article began to be highly prized. The\nsugar-cane, which had been grown from the year 1148 in Sicily, had\nbeen imported into Madeira A.D. 1419. About the year 1503 the art\nof refining sugar, before called \"blanch powdre,\" was discovered by\na Venetian; before which the juice, when selected instead of honey\nfor sweetening, was used as it came from the cane. Only twenty-seven\nyears from this date, in 1526, it was imported from St. Lucar in Spain\nby Bristol merchants. Let not the present of the Mayor of Lyme be\nconsidered as a cheap article produced in abundance in the islands of\nthe West Indies. The sugar-cane was not imported thither into Barbadoes\nfrom the Brazils till the year 1641. How surprising the result of\nofficial inquiries in the year 1853 into the consumption of sugar! It\namounted to 7,523,187 cwts., or 30 lbs. each individual of the United\nKingdom.\n\n\nSUSPENSION BRIDGES AT FREYBOURG.\n\nThere are two suspension bridges in Freybourg; one remarkable for its\ngreat length, the other for its extreme beauty. The latter connects\nthe top of two mountains, swinging over a frightful gulf that makes\none dizzy to look down into. There are no buttresses or masonwork in\nsight at a little distance; shafts are sunk in the solid rock of the\nmountains, down which the wires that sustain it are dropped. There\nit stretches, a mere black line, nearly three hundred feet in the\nheavens, from summit to summit. It looks like a spider's web flung\nacross a chasm; its delicate tracery showing clear and distinct against\nthe sky. While you are looking at the fairy creation suspended in\nmid-heaven, almost expecting the next breeze will waft it away, you\nsee a heavy waggon driven on it; you shrink back with horror at the\nrashness that could trust so frail a structure at that dizzy height;\nbut the air-hung cobweb sustains the pressure, and the vehicle passes\nin safety. Indeed, weight steadies it; while the wind, as it sweeps\ndown the gulf, makes it swing under you. The large suspension bridge\nis supported on four cables of iron wire, each one composed of one\nthousand and fifty-six wires. As the Menai bridge of Wales is often\nsaid to be longer than this, I give the dimensions of both as I find\nthem in Mr. Murray:--Freybourg: length, nine hundred and five feet;\nheight, one hundred and seventy-four feet; breadth, twenty eight feet.\nMenai: length, five hundred and eighty feet; height, one hundred and\nthirty feet; breadth, twenty-five feet. A span of nine hundred and five\nfeet, without any intermediate pier, seems impossible at first, and one\nneeds the testimony of his own eyes before he can fully believe it.\n\n\nWONDERFUL CLOCK.\n\nTowards the end of the last century, a clock was constructed by a\nGenevan mechanic named Droz, capable of performing a variety of\nsurprising movements, which were effected by the figures of a ,\na shepherd, and a dog. When the clock struck, the shepherd played six\ntunes on his flute, and the dog approached and fawned upon him. This\nclock was exhibited to the King of Spain, who was highly delighted with\nthe ingenuity of the artist. The king, at the request of Droz, took an\napple from the shepherd's basket, when the dog started up and barked\nso loud that the king's dog, which was in the same room, began to bark\nalso. We are moreover informed that the , on being asked what hour\nit was, answered the question in French, so that he could be understood\nby those present.\n\n\nMANDRIN THE SMUGGLER, 1757.\n\nMandrin was the son of a peasant in Dauphiny who dealt in cattle. His\nfirst employment was buying and selling horses, by which he subsisted\nseveral years. But having on some occasion committed a murder, he was\nobliged to fly from justice, and in his absence was condemned by the\nParliament of Grenoble to be broken on the wheel. Being now a fugitive,\nand destitute of employment, he learned to counterfeit money, and\nby this fraud made considerable gain, till, being discovered, the\nofficers of the Mint at Lyons issued a warrant for apprehending him,\nand he was again obliged to quit the country. While he was wandering\nabout from place to place, and hiding himself in caves and woods,\nhe became acquainted with a gang of smugglers, and associating with\nthem was, after some time, made their captain. As this gang was very\nnumerous, he was less cautious of being seen, and having at length\nlost his sense of fear by habitual danger, he frequently entered towns\nand cities, raised contributions on the king's officers by force, and\nspread the same terror among others that others had brought upon him.\nBut in proportion as he became more formidable he was, in fact, less\nsecure; for the Government found it necessary to detach after him such\na force as he could not resist, and the Farmers-General offered 48,000\nlivres reward for taking him. After many times attacking his party in\na running fight, in which several were cut off, Mandrin, with eight of\nhis men, took shelter in a castle on the frontiers of Savoy. They were\nclosely pursued by several detachments, under the command of Colonel\nde Moliere, who entered the King of Sardinia's territory after him,\nwithout having first obtained leave. Moliere was immediately opposed by\na great number of peasants: whether they were instigated by Mandrin, or\nwhether they were jealous of their privilege, is not known; but all his\nexpostulations being fruitless, and being determined not to relinquish\nhis prey, for whom he hoped to receive so considerable a reward, he\nforced his way against them, killing twelve and wounding many others.\nMandrin waited the issue of this contest in his castle, where he was\nsoon besieged by 150 men, who attacked the place with great vigour.\nMandrin and his partisans defended themselves like men who had nothing\nto fear in a battle equal to being taken alive; and after several of\nthem were killed, and the castle gates burst open, they retreated,\nfighting from chamber to chamber, and from story to story, till,\nreaching the garret, and being able to proceed no further, they were at\nlast overpowered by numbers, having killed twenty of their adversaries,\nand spent all their ammunition. Mandrin, with those that survived of\nhis little party, were carried prisoners to Valence in Dauphiny. * * *\nMandrin was examined every day from the 13th of May to the 25th,\nin order to discover his accomplices. In the mean time several of\nhis associates were put to the torture to discover what they knew of\nhim, and were afterwards broken on the wheel, that death might give a\nsanction to their testimony.\n\nHe himself was subjected to torture, but without eliciting anything\nfurther than he had previously revealed. Throughout he steadfastly\nrefused to betray his comrades, and conducted himself with much dignity\nand heroism. On the day of his execution he received absolution\nfrom Father Gasperini, a Jesuit, who had administered to him the\nconsolations of religion during his confinement.\n\nBefore he was led out of the prison, his shoes and stockings were taken\nfrom him; but, though barefooted, he walked along with great firmness\nand a good grace. When he came to the cathedral to perform the _amende\nhonorable_, he asked forgiveness of the monks and priests for his want\nof respect to their order, and was then conducted to the scaffold. He\nmounted with great composure, and addressed himself in a short and\npathetic exhortation to the spectators, especially the young persons\nof both sexes; he then sat down on the nave of the wheel, and loosened\nthe buttons of his shirt-sleeves himself. Then he entreated pardon of\nthe custom-house officers, whom he had so often and so grossly injured;\nand turning to the penitents who surrounded the scaffold--with his\nconfessor and two other eminent persons of his order--he earnestly\nrecommended himself as the object of their prayer, and immediately\ndelivered himself up to the executioner. He received eight blows on\nhis arms and legs, and one on his stomach, and was intended to have\nbeen left to expire of the wounds; but as the executioner was going\ndown from the scaffold, an order came to strangle him; the bishop\nand all the considerable persons at Valence having interceded for\nthis mitigation of his punishment. Mandrin was twenty-nine years of\nage, about five feet five inches high, well made, had a long visage,\nblue eyes, and sandy chesnut hair; he had something rough in his\ncountenance, and a strong robust port; he was perpetually smoking\ntobacco, with which he drank plentifully of any liquor that was at\nhand, and ate till the last with a good appetite.\n\n\nSUDDEN RECOVERY FROM MADNESS.\n\nThe following extraordinary account is taken from the _Gentleman's\nMagazine_ of 1784:--\"About six years since, a seafaring person was\ntaken into the Asylum for Maniacs at York; during the space of five\nyears and six months he never expressed any desire for sustenance,\nand was fed in the manner of an infant. The servants undressed him at\nnight, and dressed him in the morning; he never spoke, and remained\nwith his body bent all day, and was regarded by all about him as an\nanimal nearly converted into a vegetable. About the middle of May,\n1783, he suddenly astonished the people round him with saying, 'Good\nmorrow to you all.' He then thanked the servants for the care they had\ntaken of him, and appeared perfectly sane. A few days after, he wrote a\nletter to his wife, in which he expressed himself with great propriety.\nOn the 28th of May following he was allowed to leave the hospital, and\nreturn to his family; and has now the command of a ship in the Baltic\ntrade, and is in full enjoyment of perfect health, both in mind and\nbody. This very singular case is attested by Dr. Hunter, F.E.S., of\nYork, in a letter to Dr. Percival, of Manchester, and by the servants\nnow at the Asylum in York.\"\n\n\nSUMMARY OF THE BIBLE.\n\nThe following table is published, as containing accurate particulars of\nthe English version of the Bible:--\n\n _In the Old Testament._| _In the New Testament._| _Total._\n Books, 39 | Books, 27 | Books, 66\n Chapters, 929 | Chapters, 260 | Chapters, 1,189\n Verses, 23,214 | Verses, 7,959 | Verses, 31,173\n Words, 592,493 | Words, 181,253 | Words, 773,746\n Letters, 2,728,100 | Letters, 838,380 | Letters, 3,566,480\n\nThe middle chapter and the shortest in the Bible is the hundred and\nseventeenth Psalm; the middle verse is the eighth of the hundred and\neighteenth Psalm. The twenty-first verse of the seventh chapter of\nEzra, in the English version, has all the letters of the alphabet\nin it. The nineteenth chapter of the second book of Kings and the\nthirty-seventh chapter of Isaiah are alike.\n\n\nTHE LEPROSY.--LAZARS.--LAZAR-HOUSES.\n\nThat loathsome disorder, leprosy, was introduced into England in the\nreign of Henry I., and was supposed to have been brought out of Egypt,\nor perhaps the East, by means of the crusaders. To add to the horror,\nit was contagious, which enhanced the charity of a provision for such\nmiserables, who were not only naturally shunned, but even chased by\nroyal edict, from the society of their fellow-creatures.\n\nLepers, or Lazars, were sick persons removed out of monasteries to\ncells or hospitals, always built out of cities and towns. Their\nusual maintainence was, from liberty allowed them to go upon every\nmarket-day, to the market, where with a dish, called a _clap_ dish,\nthey would beg corn.\n\nTheir sickness and loathsome appearance giving great disgust, many\nwithheld their charity, upon which account they were afterwards\nrestrained from begging at large, but permitted to send the proctor\nof the hospital, who came with his box one day in every month to the\nchurches, and other religious houses, at time of service; and there\nreceived the voluntary charity of the congregations. This custom is\nsaid to be the origin of the present practice of collecting briefs.\n\nThe leprosy was much more common formerly, in this part of the globe,\nthan at present. It is said, that there were in Europe fifteen thousand\nhospitals founded for them. Perhaps near half the hospitals that were\nin England were built for lepers.\n\nLepers were so numerous in the twelfth century, that by a decree of\nthe Lateran Council under pope Alexander III., A.D. 1179, they were\nempowered to erect churches for themselves, and to have their own\nministers to officiate in them. This shows at once how infectious and\noffensive their distemper was.\n\nAnd on this account, \"In England where a man was a leper, and was\ndwelling in a town, and would come into the churches, or among his\nneighbours when they were assembled, to talk to them to their annoyance\nor disturbance, a writ lay De Leproso amovendo.\"--What follows is\nremarkable. The writ is for those lepers \"who appear to the sight\nof all men, they are lepers, by their voice and their sores, the\nputrefaction of their flesh, and by the smell of them.\"\n\nAnd so late as the reign of Edward VI. multitudes of lepers seem to\nhave been in England; for in 1 Edw. 6. c. 3. in which directions are\ngiven for carrying the poor to the places where they were born, &c. we\nread the following clause: \"Provided always, that all _leprous_ and\npoor _bed-red_ creatures may, at their liberty, remain and continue in\nsuch houses appointed for lepers, or bed-red people, as they now be in.\"\n\n1184 to 1191.--The leprosy was at this period, and long after, a cruel\nepidemic in our country, possibly brought by the crusaders from the\nHoly Land, and spread here by filth and bad diet. It was supposed to\nbe infectious, and was shunned as the plague; so that, had it not been\nfor these pious institutions, multitudes must have perished under this\nloathsome disorder.\n\nAmong other wild fancies of the age, it was imagined that the persons\nafflicted with leprosy, a disease at that time (1327, Edward II.) very\ncommon, probably from bad diet, had conspired with the Saracens to\npoison all springs and fountains; and men being glad of any pretence\nto get rid of those who were a burthen to them, many of those unhappy\npeople were burnt alive on the chimerical imputation.\n\nEvery one of the lazar-houses had a person, called a _fore-goer_, who\nused to beg daily for them.\n\n\nTHE CONDOR IN PERU.\n\nDr. Pickering, of the United States Antarctic Expedition of 1839,\nbeing in the vicinity of the Andes, attempted the ascent of one of the\nsummits; by noon he had reached a high elevation, and looking up, he\nespied a huge condor soaring down the valley. He stopped to observe the\nmajestic bird as it sailed slowly along. To his surprise it took a turn\naround him, then a second and a third, the last time drawing so near\nthat he began to apprehend that it meditated an attack. He describes\nhimself as being in the worst possible condition for a fight, his\nstrength being exhausted by climbing, and his right hand having been\nlamed for some days from a hurt. The nature of the ground, too, was\nanything but favourable for defence; but there was nothing left but to\nprepare for a fight, and with this intent he took a seat and drew his\nknife. At the instant, as if intimidated by the sight of the weapon,\nthe bird whirled off in another direction. Dr. Pickering confessed,\nhowever humiliating the acknowledgment, that he was at the time very\nwell satisfied with the condor's determination to let him alone.\n\n\nCOST OF SIR CHRISTOPHER WREN'S CHURCHES.\n\nThe following is an account of what the undermentioned churches cost\nbuilding, the designs for which were furnished by Sir Christopher\nWren:--\n\n L s. d.\n St. Paul's 736,752 2 3-1\/4\n Allhallows the Great 5,641 9 9\n ---- Bread-street 3,348 7 2\n ---- Lombard-street 8,058 15 6\n St. Alban's, Wood-street 3,165 0 8\n St. Anne and Agnes 2,448 0 10\n St. Andrew's, Wardrobe 7,060 16 11\n ---- Holborn 9,000 0 0\n St. Antholin's 5,685 5 10-3\/4\n St. Austin's 3,145 3 10\n St. Benet, Grailchurch 3,583 9 5-1\/4\n ---- Paul's Wharf 3,328 18 10\n ---- Fink 4,129 16 10\n St. Bride's 11,430 5 11\n St. Bartholomew's 5,077 1 1\n Christ Church 11,778 9 6\n St. Clement, Eastcheap 4,365 3 4-1\/2\n ---- Danes 8,786 17 0-1\/2\n St. Dionis Back Church 5,737 10 8\n St. Edmund the King 5,207 11 0\n St. George, Botolph-lane 4,509 4 10\n St. James, Garlick-hill 5,357 12 10\n ---- Westminster 8,500 0 0\n St. Lawrence, Jewry 11,872 1 9\n St. Michael, Basinghall 2,822 17 1\n ---- Royal 7,455 7 9\n St. Michael, Queenhithe 4,354 3 8\n ---- Wood-street 2,554 2 11\n ---- Crooked-lane 4,641 5 11\n ---- Cornhill 4,686 5 11\n St. Martin, Ludgate 5,378 18 8\n St. Matthew, Friday-str 2,301 8 2\n St. Margaret Pattens 4,986 10 4\n ---- Lothbury 5,340 8 1\n St. Mary, Abchurch 4,922 2 4-1\/2\n ---- Magdalen 4,291 12 9-1\/4\n ---- Somerset 6,579 18 1-1\/4\n ---- at Hill 3,980 12 3\n ---- Aldermanbury 5,237 3 6\n ---- le Bow 8,071 18 1\n ---- le Steeple 7,388 8 7-3\/4\n St. Magnus, Lond. bridge 9,579 19 10\n St. Mildred, Bread-street 3,705 13 6-1\/4\n ---- Poultry 4,654 9 7-3\/4\n St. Nicholas Cole Abbey 5,042 6 11\n St. Olav, Jewry 5,580 4 10\n St. Peter's, Cornhill 5,647 8 2\n St. Swithin, Canon-street 4,687 4 6\n St. Stephen, Wallbrook 7,652 13 8\n ---- Coleman-str 4,020 16 6\n St. Vedast, Foster-lane 1,853 15 6\n\n\nEARLY CLOCKS.\n\nThe first clock which appeared in Europe, was probably that which\nEginhard (the secretary of Charlemagne), describes as sent to his royal\nmaster by Abdalla, King of Persia. \"A horologe of brass, wonderfully\nconstructed, for the course of the twelve hours, answered to the\nhourglass, with as many little brazen balls, which drop down on a sort\nof bells underneath, and sounded each other.\"--The Venetians had clocks\nin 872, and sent a specimen of them that year to Constantinople.\n\n\nSINGULAR SPECIMEN OF ORTHOGRAPHY IN THE SIXTEENTH CENTURY.\n\nThe following letter was written by the Duchess of Norfolk to Cromwell,\nEarl of Essex. It exhibits a curious instance of the monstrous\nanomalies of our orthography in the infancy of our literature, when a\nspelling book was yet a precious thing:--\n\n\"My ffary gode lord,--her I sand you in tokyn hoff the neweyer, a\nglasse hoff Setyl set in Sellfer gyld, I pra you tak hit in wort. An hy\nwer babel het showlde be bater. I woll hit war wort a m crone.\"\n\nThus _translated_:--\n\n\"My very good lord,--Here I send you, in token of the new year, a glass\nof setyll set in silver gilt; I pray you take it in worth. An I were\nable it should be better. I would it were worth a thousand crown.\"\n\n\nDEATH OF THE EARL OF KILDARE.\n\nIn 1513, died the most powerful baron and active soldier of his age,\nFitzgerald, Earl of Kildare. He had been, during thirty years, at\ndifferent times, chief governor of Ireland, and was too potent to\nbe set aside, otherwise his strong attachment to the house of York\nwould probably have been his ruin. The untameable spirit of the earl\nsometimes involved him in trouble, from which he was extricated by a\nlucky bluntness; as when once, when charged before Henry VIII. with\nsetting fire to the cathedral of Cashel, \"I own it,\" said the earl,\n\"but I never would have done it had I not believed that the archbishop\nwas in it.\" The king laughed, and pardoned the ludicrous culprit. The\nBishop of Meath was his bitterest foe. He accused him to Henry of\ndivers misdeeds, and closed his accusation with \"Thus, my liege, you\nsee that all Ireland cannot rule the earl.\" \"Then,\" said the perverse\nmonarch, \"the earl shall rule all Ireland,\" and instantly made him\nlord-deputy. The English loved the earl because he was brave and\ngenerous, and because his good humour equalled his valour. Once, when\nhe was in a furious paroxysm, a domestic who knew his temper, whispered\nin his ear, \"My lord, yonder fellow has betted me a fine horse, that\nI dare not take a hair from your lordship's beard; I pray, my lord,\nwin me that wager.\" The earl's features relaxed, and he said to the\npetitioner, \"Take the hair, then, but if thou exceedest thy demand, my\nfist shall meet thy head.\"\n\n\nTHE BRITANNIA TUBULAR BRIDGE.\n\nThis is one of the most remarkable structures in the world, the design\nof the celebrated architect, Sir R. Stephenson. This bridge is on the\nline of the Chester and Holyhead Railway, crossing the Menai Straits,\nwithin sight of Telford's Chain Suspension Bridge. It is made of cast\niron of a tubular form, in the tube of which the railway passes. Four\nof these span the Strait, and are supported by piles of masonry; that\non the Anglesea side is 143 feet 6 inches high, and from the front\nto the end of the wing walls is 173 feet. These wing walls terminate\nin pedestals, on which repose colossal lions of Egyptian character.\nThe Anglesea pier is 196 feet high, 55 feet wide, and 32 feet long.\nIn the middle of the Strait is the Britannia Rock, from which the\nbridge derives its name; on this the Britannia pier is raised. It is\nequi-distant from the Anglesea and Carnarvon piers, being 460 feet in\nthe clear from each, and sustains the four ends of the four long tubes,\nwhich span the distance from shore to shore. There are two pairs of\nshort and two of long tubes, the lengths of these pairs being 250 feet\nand 470 respectively. The Egyptian lions are 25 feet 6 inches long, 12\nfeet 6 inches high, 8 feet wide, and weigh 80 tons. Two thousand cubic\nfeet of stone were required for each lion. The total quantity of stone\nin the bridge is 1,400,000 cubic feet. The weight of malleable iron in\nthe tubes is 10,000 tons; of cast iron, 1,400 tons. The whole length of\nthe entire bridge, measuring from the extreme front of the wing walls,\nis 1,833 feet, and its greatest elevation at Britannia pier, 240 feet\nabove low-water-mark. The total cost of the structure is L601,865. This\nwonderful structure was begun April 13, 1846, and completed July 25,\n1850; opened for traffic Oct. 21, 1850.\n\n[Illustration: [++] The Britannia Tubular Bridge.]\n\n\nDAFFEY'S ELIXIR.\n\nIn the _Postboy_, Jan. 1, 1707-8, is the following curious\nadvertisement:--\"Daffey's famous _Elixir Salutis_ by Catherine Daffey,\ndaughter of Mr. Thomas Daffy, late rector of Redmile, in the valley\nof Belvoir, who imparted it to his kinsman, Mr. Anthony Daffy, who\npublished the same to the benefit of the community and his own great\nadvantage. The original receipt is now in my possession, left to me by\nmy father. My own brother, Mr. Daniel Daffy, apothecary in Nottingham,\nmade the Elixir from the said receipt, and sold it there during his\nlife. Those who know it, will believe what I declare; and those who do\nnot, may be convinced that I am no counterfeit, by the colour, taste,\nsmell, and operation of my Elixir. To be had at the Hand and Pen,\nMaiden-Lane, Covent Garden.\"\n\n\nJENNY'S WHIM.\n\n\"This was a tea garden, situated, after passing over a wooden bridge on\nthe left, previous to entering the long avenue, the coach way to where\nRanelagh once stood. This place was much frequented, from its novelty,\nbeing an inducement to allure the curious, by its amusing deceptions,\nparticularly on their first appearance there. Here was a large garden,\nin different parts of which were recesses; and if treading on a spring,\ntaking you by surprise, up started different figures, some ugly enough\nto frighten you--a harlequin, a Mother Shipton, or some terrific\nanimal. In a large piece of water, facing the tea alcoves, large fish\nor mermaids, were showing themselves above the surface. This queer\nspectacle was first kept by a famous mechanist, who had been employed\nat one of the winter theatres, there being then two.\"--Angelo's _Pic\nNic or Table Talk_, p. 106.\n\nHorace Walpole, more than once alludes to this place of entertainment\nin his Letters; and in 1755 a 4to. satirical tract appeared entitled\n_Jenny's Whim; or a Sure Guide to the Nobility, Gentry, and other\nEminent Persons, in this Metropolis_.\n\n\nANECDOTE RELATIVE TO THE MASKED EXECUTIONER OF CHARLES I.\n\nIt is universally known, that, at the execution of King Charles I., a\nman in a vizor performed the office of executioner. This circumstance\nhas given rise to a variety of conjectures and accounts. In the\nGentleman's Magazine for November, 1767, and January, 1768, are\naccounts of one William Walker, who is said to be the executioner.\nIn the same magazine for June, 1784, it is supposed to be a Richard\nBrandon, of whom a long account is copied from an Exeter newspaper.\nBut William Lilly, in his \"History of his Life and Times,\" has the\nfollowing remarkable passage:--\"Many have curiously inquired who it\nwas that cut off his [the king's] head: I have no permission to speak\nof such things: only thus much I say, he that did it is as valiant and\nresolute a man as lives, and one of a competent fortune.\" To clear up\nthis passage, we shall present our readers with Lilly's examination (as\nrelated by himself) before the first parliament of King Charles II. in\nJune, 1660.\n\n\"At my first appearance, many of the young members affronted me highly,\nand demanded several scurrilous questions. Mr. Weston held a paper\nbefore his mouth; bade me answer nobody but Mr. Prinn; I obeyed his\ncommand, and saved myself much trouble thereby, and when Mr. Prinn put\nany difficult or doubtful query unto me, Mr. Weston prompted me with\na fit question. At last, after almost one hour's tugging, I desired\nto be fully heard what I could say as to the person that cut Charles\nI.'s head off. Liberty being given me to speak, I related what follows,\nviz.:--\n\n\"That the next Sunday but one after Charles I. was beheaded, Robert\nSpavin, Secretary to Lieutenant-General Cromwell at that time, invited\nhimself to dine with me, and brought Anthony Pearson, and several\nothers, along with him to dinner. That their principal discourse all\ndinner-time was only who it was that beheaded the king; one said it was\nthe common hangman; another, Hugh Peters; others were also nominated;\nbut none concluded. Robert Spavin, so soon as dinner was done, took\nme by the hand, and carried me to the south window: saith he. 'These\nare all mistaken; they have not named the man that did the fact; it\nwas Lieutenant-Colonel Joice. I was in the room when he fitted himself\nfor the work, stood behind him when he did it; when done, went in with\nhim again. There is no man knows this but my master, viz., Cromwell,\nCommissary Ireton, and myself.'--'Doth Mr. Rushworth know it?' saith\nI.--'No, he doth not know it,' saith Spavin. The same thing Spavin\nsince has often related to me when we were alone.\"\n\n\nWHIPPING PRISONERS.\n\nMr. Ellesdon, Mayor of Lyme, in 1595, paid for--\n\n _s._ _d._\n Four yards of canvas to make a coat to whip the rogues in 3 0\n\n Making the same 0 6\n\n Whipping of three of the ship boys for stealing of Mr.\n Hassard's salmon fish in the Cobb 1 0\n\n (N.B.--Salmon was plentiful in the west at this epoch.)\n\nThe charge of fourpence made for whipping a boy continued for many\nyears the same. The whipping of a woman who was a stranger was little\nmore costly; but the inflicting such a punishment upon a townswoman\nwas remunerated at a higher rate, as may well be supposed, from a\nconsideration of several circumstances. To take a violent, noisy woman\nfrom her chamber, tie madam to the tumbrel and whip her round the town,\nwas an undertaking that demanded assistance and protection to the\nofficial or hireling that wielded the thong. In the Town Accompt Book\nare found such entries as those which are given in illustration:--\n\n _s._ _d._\n 1625. For whipping William Wynter's boy 0 4\n \" Agnes Abbott twice 2 4\n 1644. Paid two soldiers to attend the whipping of a woman 2 6\n Paid to whipping four women 4 0\n\n\nTHE INIQUITIES OF THE SLAVE TRADE.\n\nWe may form some idea of the temptations which the trade in human\nbeings held out, even to people who held an honourable position in the\nworld, from the fact that the captain of a frigate, within a few years\nbefore the slave trade was abolished, was known to purchase slaves in\nthe West India market, have them entered as able seamen, and compel the\nartificers to teach them a trade; so that when the ship returned each\nwas sold at a high rate as a valuable piece of property. The worst,\nhowever, has to be told. Upon sailing from Portsmouth, some of the\nbest men were sent away upon duty in a ship's boat, in order that they\nmight be returned \"run,\" by which they lost pay and clothes, but made\nroom for the s lately kidnapped, who were entered, though they\ndid no work for the ship, as able seamen! We have all heard of a naval\nofficer who had his pocket picked at a Westminster election, and who\nopenly professed his vow, which he rigidly performed, of flogging every\nLondoner that joined his ship for this act. This, it is said, was no\nidle vow!\n\n\nDISCOVERY OF THE BODY OF CANUTE THE GREAT.\n\nIn June 1766, some workmen who were repairing Winchester Cathedral\ndiscovered a monument, wherein was contained the body of King Canute.\nIt was remarkably fresh, had a wreath round the head, and several other\nornaments of gold and silver bands. On his finger was a ring, in which\nwas set a large and remarkably fine stone; and in one of his hands a\nsilver penny. _Archaeologia_, vol. iii. The penny found in the hand is\na singular instance of a continuance of the pagan custom of always\nproviding the dead with money to pay Charon.\n\n\nM.P.'S AND MAYORS PRIVATEERS.\n\nWilliam Morfote, who represented Winchelsea in Parliament in 1428, was\na privateer with a hundred men under him. He found it necessary to\nobtain the king's pardon in 1435, by the advice of Parliament, there\nbeing a legal difficulty about his having broken prison at Dover Castle.\n\nTwo merchants of Sherborne in Dorsetshire were robbed of their cargo,\nworth L80, A.D. 1322, by Robert de Battyle. This transaction did not\nlose him the good opinion of his townsmen, who chose him Mayor of\nWinchelsea a few years later.\n\n\nALGERINE INVASION OF IRELAND.\n\nThe Algerines landed in Ireland in 1627, killed 50 persons, and carried\noff about 400 into slavery. One vessel captured by them was worth\nL260,000. They made purchases of stores and provisions they wanted in\nthe western parts of Ireland by Baltimore, and in 1631 carried off 100\ncaptives from that town. They landed their poor captives at Rochelle,\nand marched them in chains to Marseilles. Twenty-six children are said\nto have been carried off at one time from Cornwall. In 1633, Lord\nWentworth, appointed lord deputy of Ireland, named noted pirate vessels\noff the coast of Ireland and their captures. Persons in their wills\nused to leave sums of money for redeeming well-known captives from\nbondage in Algiers and other places.\n\n\nWILLIAM JOY, THE ENGLISH SAMPSON.\n\n[Illustration: [++] William Joy.]\n\nWilliam Joy was a native of Kent, and born May 2, 1675, at St.\nLawrence, a small village one mile from Ramsgate, in the Isle of\nThanet. When very young, he distinguished himself among his juvenile\ncompanions and playmates, by his amazing superiority in strength,\nover any antagonist that dare to come in competition with his power,\nwhether in play or earnest When about twenty-four years of age, he\nfirst began to exhibit in public his astonishing feats, in a display of\npersonal prowess inferior to none but the Hebrew champion recorded in\nholy writ. Among many other of this man's extraordinary performances\nmay be recorded:--1. A strong horse, urged by the whip to escape his\npowerful rein, is restrained and kept from escape solely by the check\nof his pull, aided by a strong rope, and this without any stay or\nsupport whatever. 2. Seated upon a stool, with his legs horizontally\nelevated, solely by muscular power, he jumps clearly from his seat.\n3. To prove the agility and flexibility of his joints, he places\na glass of wine on the sole of his foot, and, in an erect posture,\nwithout the least bending of his head or body, raises the glass to\nhis mouth, and drinks the contents, turning his foot with both hands,\nto accommodate his draught. 4. Aided by a strong leather girdle, or\nbelt, and supporting himself by pressing his arms on a railing, he\nlifts from the ground a stone of the enormous weight of 2,240 lbs. 5.\nA rope fastened to a wall, which had borne 3,500 lbs. weight, without\ngiving way, is broke asunder by his amazing strength. The celebrity\nof this man attracted the curiosity of King William III., before whom\nhe exhibited at Kensington Palace; likewise before George, Prince of\nDenmark, and his royal consort, the Princess, afterwards Queen Anne,\nand their son William, Duke of Gloucester, called the Hope of England.\nHe also went through a regular course of performances at the Duke's\nTheatre, in Dorset-gardens, Salisbury-square, which was attended by the\nfirst nobility and gentry in the kingdom.\n\n\nPRICE OF SHELL-FISH IN 1675.\n\nA bill for shell-fish enables us to ascertain the prices paid in\nCharles II.'s reign for these delicacies. Mr. Walter Tucker, mayor of\nLyme, Dorset, paid for the judges, for--\n\n 30 lobsters L1 10 0\n 6 crabs 0 6 0\n 100 scallops 0 5 0\n 300 oysters 0 4 0\n 50 oranges 0 2 0\n -------\n L2 7 0\n\n\nDISTRIBUTING HAND-BILLS.\n\nThe month of July 1736 afforded a singular _popular explosion_,\ncontrived in the following strange manner:--A brown paper parcel,\nwhich had been placed unobserved near the side-bar of the Court of\nKing's-bench, Westminster-hall, blew up during the solemn proceedings\nof the Courts of Justice assembled, and scattered a number of printed\nbills, giving notice, that on the last day of Term five Acts of\nParliament would be publicly burnt in the hall, between the hours of\ntwelve and one, at the Royal Exchange, and at St. Margaret's hill,\nwhich were the Gin Act, the Smuggling Act, the Mortmain Act, the\nWestminster Bridge Act, and the Act for borrowing 600,000_l._ on the\nSinking fund.\n\nOne of the bills was immediately carried to the Grand Jury then\nsitting, who found it an infamous libel, and recommended the offering\nof a reward to discover the author.\n\n\nRANZ DES VACHES.\n\nThe \"Ranz des Vaches,\" which is commonly supposed to be a single air,\nstands in Switzerland for a class of melodies, the literal meaning\nof which is cow-rows. The German word is _Kureihen_--rows of cows.\nIt derives its origin from the manner the cows march home along the\nAlpine paths at milking time. The shepherd goes before, keeping every\nstraggler in its place by the tones of his horn, while the whole herd\nwind along in Indian file, obedient to the call. From its association\nit always creates home-sickness in a Swiss mountaineer, when he hears\nit in a foreign land. It is said, these melodies are prohibited in\nthe Swiss regiments attached to the French army, because it produces\nso many desertions. One of the \"Ranz des Vaches\" brings back to his\nimagination his Alpine cottage--the green pasturage--the bleating of\nhis mountain goats--the voices of the milkmaids, and all the sweetness\nand innocence of a pastoral life; till his heart turns with a sad\nyearning to the haunts of his childhood, and the spot of his early\ndreams and early happiness.\n\nThe Swiss retain their old fondness for rifle-shooting, and there is\nannually a grand rifle match at some of the large towns, made up of the\nbest marksmen in all Switzerland. There are also yearly contests in\nwrestling, called _Zwing Feste_, the most distinguished wrestlers at\nwhich are from Unterwalden, Appenzel, and Berne.\n\n\nMONSOONS.\n\nThese are periodical winds which blow over the Indian Ocean, between\nAfrica and Hindustan for nearly six months from the north-east, and\nduring an equal period from the south-west. The region of the monsoons\nlies a little to the north of the northern border of the trade-winds,\nand they blow with the greatest force and with most regularity between\nthe eastern coast of Africa and Hindustan. When the sun is in the\nsouthern hemisphere a north-east wind, and when it is in the northern\nhemisphere, a south-west wind blows over this sea. The north-east\nmonsoon blows from November to March. It extends one or two degrees\nsouth of the equator. It becomes regular near the coasts of Africa\nsooner than in the middle of the sea, and near the equator sooner than\nin the vicinity of the coasts of Arabia. This wind brings rain on the\neastern coasts of Africa. The south-west monsoon does not extend south\nof the equator, but usually begins a short distance north of it. It\nblows from the latter end of April to the middle of October. Along the\ncoast of Africa, it appears at the end of March; but along the coast\nof Malabar, not before the middle of April; it ceases, however, sooner\nin the former than in the latter region. The rainy season on the west\ncoast of Hindustan commences with the first approach of the south-west\nmonsoon. The monsoons prevail also on the seas between Australia and\nChina.\n\n[Illustration: [++] Mpnsoons.]\n\nThe effect of the struggle which precedes the change in the direction\nof the wind in this part of the world is thus described in \"Forbes's\nOriental Memoirs.\" The author was encamped with the English troops:\n\n\"The shades of evening approached as we reached the ground, and just\nas the encampment was completed, the atmosphere grew suddenly dark,\nthe heat became oppressive, and an unusual stillness presaged the\nimmediate setting-in of the monsoon. The whole appearance of external\nnature resembled those solemn preludes to earthquakes and hurricanes\nin the West Indies, from which the East in general is providentially\nfree. We were allowed very little time for conjecture. In a few minutes\nthe heavy clouds burst over us. I had witnessed seventeen monsoons\nin India, but this surpassed them all in its awful appearance and\ndreadful effects. Encamped in a low situation on the borders of a lake\nformed to collect the surrounding water, we found ourselves in a few\nhours in a liquid plain; tent-pins giving way in a loose soil--the\ntents fell down--and left the whole army exposed to the contending\nelements. It requires a lively imagination to conceive the situation of\na hundred thousand human beings of every description, with more than\ntwo hundred thousand elephants, camels, horses, and oxen, suddenly\noverwhelmed by this dreadful storm in a strange country, without any\nknowledge of high or low ground, the whole being covered by an immense\nlake, and surrounded by thick darkness, which rendered it impossible\nfor us to distinguish a single object except such as the vivid\nglare of the lightning occasionally displayed in horrible forms. No\nlanguage can adequately describe the wreck of a large encampment thus\ninstantaneously destroyed, and covered with water, amid the cries of\nold men and helpless women, terrified by the piercing shrieks of their\nexpiring children, unable to afford them relief. During this dreadful\nnight more than two hundred persons and three thousand cattle perished\nmiserably, and the morning dawn exhibited a shocking spectacle!\"\n\n\nUNUSUAL LOCALITY FOR SAYING PRAYERS.\n\nFrancis Atkins was porter at the palace gate, at Salisbury, from the\ntime of Bishop Burnet to the period of his death in 1761, at the age\nof 104 years. It was his office every night to wind up the clock,\nwhich he was capable of performing regularly till within a year of his\ndecease, though on the summit of the palace. In ascending the lofty\nflight of stairs, he usually made a halt at a particular place and said\nhis evening prayers. He lived a regular and temperate life, and took a\ngreat deal of exercise; he walked well, and carried his frame upright\nand well balanced to the last.\n\n\nBILLY IN THE SALT BOX.\n\nPolitical caricatures are generally well worth preserving, they\nfamiliarize us with the features and peculiarities of celebrated men,\nand they tell us what was the popular feeling of the day. We regret\nthat in general they are too large for our pages, but now and then we\nmeet with a small one which we are glad to present to our readers.\n\n[Illustration: [++] Billy in the Salt Box.]\n\nMr. Pitt's budget of 1805 was not allowed to pass without severe\nremarks, and a heavily increased duty on salt excited general\ndissatisfaction. People said that the grand contriver of taxes had\nvisited every corner of the house above stairs, and that he had now\ndescended into the kitchen; and the annexed caricature, by Gilray,\nwhich was published at this period, represents the premier alarming the\npoor cook by popping his head out of the salt-box, with the unexpected\nsalutation--\"How do you do, cookey?\" The person thus apostrophised\ncries out in consternation, \"Curse the fellow, how he has frightened\nme!--I think, on my heart, he is getting in everywhere!--who the deuce\nwould have thought of finding him in the salt-box?\"\n\n\nDANGEROUS FEAT.\n\nAn extraordinary instance of the rash feats which men with cool\nheads and courageous hearts will sometimes perform, was witnessed at\nNottingham on January 22, 1789.--The vane at the top of St. Peter's\nspire, which was placed there in 1735, and measured thirty-three inches\nin length, having become insecure, the parish officers agreed with Mr.\nRobert Wooton, of Kegworth, to take it down and reinstate it.\n\nThis venturous man, henceforth known as \"_the steeple climber_,\"\ncommenced his undertaking by placing a ladder against the steeple, and\nsecuring it to the wall with tenters: he then mounted that with another\non his shoulder, which he fastened above it in like manner; and so on\ntill he reached the top. To prevent himself falling, he was girded\nround with belts, which he connected with the ladders by means of\nhooks. In this manner he replaced the vane and cock, and rebuilt four\nyards of the steeple.\n\nThe celerity with which the man placed the ladders was remarkable. He\nbegan to affix the first at eleven in the morning, and brought the\nvane down in triumph by two in the afternoon. The bells were then set\na-ringing, the congregation of people became very great, and Wooton\nre-ascended the spire, to exhibit his daring. He extended himself on\nits summit, only thirteen inches in diameter, and spread out his arms\nand legs. He afterwards balanced himself on the uppermost stave of\nthe top ladder, and for a quarter of an hour capered about in every\nimaginable posture, the admiring crowd beneath expecting momentarily to\nwitness his descent in a manner much less agreeable than precipitate.\n\nSubsequently, when his undertaking was accomplished, to excite\nadmiration and obtain money, he again balanced himself on the apex\nof the spire, beat a drum, and drank a bottle of ale, in the sight\nof thousands of people, on a market-day; but the reprobation of the\nman's temerity so far preponderated over public approval, as in a\nconsiderable degree to diminish his expected reward.\n\n\nPOST-HASTE ONE HUNDRED YEARS AGO.\n\nGlasgow is now within one minute of London; in the last century it was\nscarcely within a fortnight of it. It is a positive fact that when the\npost arrived there a hundred years ago, the firing of a gun announced\nits coming in. The members of the clubs who heard it tumbled out of\nbed, and rushed down to the club-room, where a tankard of hot herb ale,\nor a beverage which was a mixture of rum and sugar, was ready for them\nbefore breakfast. How forcibly do these things bring before us the size\nof Glasgow at that time, and the habits of its citizens.\n\n\nEXECUTION OF ADMIRAL BYNG.\n\nThe horrid details of the execution of criminals are wholly unfitted\nfor our pages, but Admiral Byng was not a criminal; his life was\nsacrificed to party spirit and party interests, and an account of his\nmurder--for such it really was--is therefore highly interesting, as it\nenables us to see the dauntless manner in which a brave man can meet\na dreadful fate, which he knew to be wholly undeserved. The execution\ntook place on board the \"St. George,\" man-of-war in Portsmouth harbour,\non the 14th of March, 1757. The Admiral, accompanied by a clergyman who\nattended him during his confinement, and two gentlemen, his relations,\nwalked out of the great cabin to the quarterdeck, where he suffered, on\nthe larboard side, a few minutes before twelve o'clock. He was dressed\nin a light grey coat, white waistcoat, and white stockings, and a large\nwhite wig, and had in each hand a white handkerchief. He threw his hat\non the deck, kneeled on a cushion, tied one handkerchief over his eyes,\nand dropped the other as a signal, on which a volley from six marines\nwas fired, five of whose bullets went through him, and he was in an\ninstant no more. The sixth went over his head. From his coming out\nof his cabin could not be two minutes till he fell motionless on his\nleft side. He died with great resolution and composure, not showing\nthe least sign of timidity. The _Ramillies_, the ship the admiral had\nin the Mediterranean, was riding at her moorings in the harbour, and\nabout half an hour before he suffered, she broke her mooring chain, and\nonly held by her bridle, which is looked on as a wonderful incident by\npeople who do not consider the high wind at that time.\n\n\nEXTRAORDINARY TREE.\n\n[Illustration: [++] Banyan Tree.]\n\nThe Samoan group of islands in the South Sea lies between the latitudes\nof 13 deg. 30' and 14 deg. 30' S, and the longitudes of 168 deg. and 173 deg. W. In\nsome of these islands there is a most remarkable tree which well\ndeserves a place in our roll of extraordinary productions. It is a\nspecies of banyan (_Ficus religiosa_), and is called by the natives\nOhwa. Our sketch gives a good idea of some of these trees. The pendant\nbranches of many of them take root in the ground to the number of\nthousands, forming stems from an inch to two feet in diameter,\nuniting in the main trunk more than eighty feet above the ground, and\nsupporting a vast system of horizontal branches, spreading like an\numbrella over the tops of the other trees.\n\n\nTHE PLAGUE IN ENGLAND.\n\nThe Register of Ramsay, in Huntingdonshire, mentions 400 people who\ndied there of the plague, in or about February 1665, and that it\nwas introduced into the place by a gentleman, who first caught the\ninfection by wearing a coat, the cloth of which came from London: the\ntailor who made the coat, with all his family, died, as did no less\nthan the number above mentioned.\n\nBut the ravages made by the plague in _London_, about 1665, are well\nknown: it was brought over from Holland, in some Levant goods, about\nthe close of the year 1664: its progress was arrested, in a great\ndegree, by a hard frost which set in in the winter; but as the spring\nof 1665 advanced, its virulence advanced. Infected houses were shut\nup and red crosses painted on the doors, with this inscription, \"Lord\nhave mercy upon us.\" Persons going to market took the meat off the\nhooks themselves, for their _own_ security, and for the _Butcher's_,\ndropped their money into pans of vinegar; for it was supposed that even\ntheir provisions were tainted with the infection. In the months of\nAugust and September the greatest mortality occurred; for the deaths\nof one week have been estimated at 10,000! It may be supposed, that no\ngreat accuracy existed in the Registers, to afford a correct estimate;\nfor, in the parish of Stepney, it is said they lost, within the year,\n116 sextons, grave-diggers and their assistants; and, as the disorder\nadvanced, the churchyards were incapable of holding more bodies, and\nlarge pits were therefore dug in several parts, to which the dead\nwere brought by cartloads, collected by the ringing of a bell and the\nmournful cry of \"Bring out your dead.\" Add to this, that these carts\nworked in the night, and no exact account was kept, as the clerks\nand sextons were averse to a duty exposing them to such dangerous\nconsequences, and often carried off before such accounts as they had\ntaken were delivered in. All the shops were shut up, grass grew in the\nmost public streets, until about December 1665, when the plague abated,\nand the citizens who had left their abodes for the country, crowded\nback again to their residences. The computation is, that this horrible\ndisease carried off 100,000 persons in London: it is singular, that the\nonly parish quite exempt from infection was St. John the Evangelist, in\nWatling Street.\n\n\nLANDSLIP AT COLEBROOK, SHROPSHIRE.\n\nA most remarkable circumstance happened there in the morning of the\n27th of May, 1773, about four o'clock. Near 4,000 yards from the river\nSevern stood a house, where a family dwelt; the man got up about three\no'clock, heard a rumbling noise, and felt the ground shake under him,\non which he called up his family. They perceived the ground begin to\nmove, but knew not which way to run; however, they providentially and\nwonderfully escaped, by taking an immediate flight, for just as they\ngot to an adjacent wood, the ground they had left separated from that\non which they stood. They first observed a small crack in the ground\nabout four or five inches wide, and a field that was sown with oats\nto heave up and roll about like waves of water; the trees moved as\nif blown with wind, but the air was calm and serene; the Severn (in\nwhich at that time was a considerable flood) was agitated very much,\nand the current seemed to run upwards. They perceived a great crack\nrun very quick up the ground from the river. Immediately about thirty\nacres of land, with the hedges and trees standing (except a few that\nwere overturned), moved with great force and swiftness towards the\nSevern, attended with great and uncommon noise, compared to a large\nflock of sheep running swiftly. That part of the land next the river\nwas a small wood, less than two acres, in which grew twenty large oaks;\na few of them were thrown down, and as many more were undermined and\noverturned; some left leaning, the rest upright, as if never disturbed.\nThe wood was pushed with such velocity into the channel of the Severn\n(which at that time was remarkably deep), that it forced the waters up\nin columns a considerable height, like mighty fountains, and drove the\nbed of the river before it on the opposite shore, many feet above the\nsurface of the water, where it lodged, as did one side of the wood;\nthe current being instantly stopped, occasioned a great inundation\nabove, and so sudden a fall below, that many fish were left on dry\nland, and several barges were heeled over, and when the stream came\ndown were sunk, but none were damaged above. The river soon took its\ncourse over a large meadow that was opposite the small wood, and in\nthree days wore a navigable channel through the meadow. A turnpike road\nwas moved more than thirty yards from its former situation, and to all\nappearance rendered for ever impassable. A barn was carried about the\nsame distance, and left as a heap of rubbish in a large chasm; the\nhouse received but little damage. A hedge that was joined to the garden\nwas removed about fifty yards. A great part of the land was in confused\nheaps, full of cracks, from four inches to more than a yard wide.\nSeveral very long and deep chasms were formed in the upper part of the\nland, from about fourteen to upwards of thirty yards wide, in which\nwere many pyramids of earth standing, with the green turf remaining on\nthe tops of some of them. Hollows were raised into mounts, and mounts\nreduced into hollows. Less than a quarter of an hour completed this\ndreadful scene.\n\n\nCURIOUS CUSTOM AT STRASBOURG.\n\nAt Strasbourg they show a large French horn, whose history is as\nfollows:--About 400 years ago, the Jews formed a conspiracy to betray\nthe city, and with this identical horn they intended to give the enemy\nnotice when to attack.\n\nThe plot, however, was discovered; many of the Jews were burnt alive,\nthe rest were plundered of their money and effects, and banished the\ntown; and this horn is sounded twice every night from the battlements\nof the steeple in gratitude for the deliverance.\n\nThe Jews deny the fact of this story, except the murdering and\npillaging their countrymen. They say the whole story is fabricated to\nfurnish a pretext for these robberies and murders, and assert that the\nsteeple of Strasbourg, as has been said of the Monument of London,--\n\n \"Like a tall bully lifts the head and lies.\"\n\n\nDOWN AMONG THE DEAD MEN.\n\nThe following is an extraordinary instance of the recklessness of\nsailors when in the pursuit of what they call pleasure. In the year\n1779, a Mr. Constable, of Woolwich, passing through the churchyard\nthere at midnight, heard people singing jovially. At first he thought\nthey were in the church, but the doors were locked, and it was all\nsilent there:--on looking about he found some drunken sailors who had\ngot into a large family vault, and were regaling with bread, cheese,\ntobacco, and strong beer. They belonged to the Robust, man of war,\nand having resolved to spend a jolly night on shore, had kept it up in\na neighbouring alehouse till the landlord turned them out, and then\nthey came here to finish their evening. They had opened some of the\ncoffins in their dare-devil drunkenness and crammed the mouth of one\nof the bodies with bread, and cheese, and beer. Constable, with much\ndifficulty, prevailed on them to return to the ship. In their way one\nfell down in the mud, and was suffocated, as much from drunkenness as\nthe real danger. The comrades took him on their shoulders, and carried\nhim back to sleep in company with the honest gentlemen with whom he had\npassed the evening.\n\n\nCHAIR BROUGHT OVER TO AMERICA IN THE MAYFLOWER BY THE PILGRIM FATHERS.\n\n[Illustration: [++] John Carver's Chair.]\n\nHow frequently do we obtain, from the ordinary articles of domestic\nlife which they were accustomed to use, a correct idea of the habits\nand tastes of whole communities which have long since passed away.\nA striking instance of this is the chair, of which the above is a\ncorrect sketch. It belonged to John Carver, who was one of the band of\nsingle-hearted men who constituted the Pilgrim Fathers, and who after\nfirst setting out from Holland, eventually sailed from Plymouth in\nEngland, in August, 1620. They landed in Cape Cod Harbour, New England,\non the 9th of November following. Carver, was one of the chief spirits\nof the band, and the chair which we have sketched was one of his best\narticles of furniture, which he took with him in the Mayflower. He\nwas elected the first governor of the community, and died in the year\nfollowing his election. How forcibly does it show the simplicity of\ntaste, and the freedom from pomp and vanity which characterised the\ndevoted and fearless men who left their native shores, and sought\n\"freedom to worship God\" in a land to them unknown, that they should\nhave selected as their first governor, an individual, the best chair in\nwhose house was the homely article which we have here depicted.\n\n\nA HARMLESS ECCENTRIC.\n\n[Illustration: [++] Jenny Darney, a Harmless Eccentric.]\n\nThe annexed cut represents a singular character who was well known\nabout the year 1790 in the southern part of the county of Cumberland.\nHer appearance is thus described by a correspondent of the Gentleman's\nMagazine of that date:--\"Though I have seen her at various times, and\nfrequently conversed with her, for these 20 years, I have never been\nable to learn any particulars respecting her family, friends, or name.\nThe country people know her by the appellation of Jenny Darney, from\nthe manner, I presume, in which she used to mend her clothes. Her\npresent garb is entirely of her own manufacture. She collects the small\nparcels of wool which lie about the fields in sheep farms, spins it on\na rock and spindle of her own making; and as she cannot find any other\nmethod of making the yarn into cloth, she knits it on wooden needles,\nand by that means procures a warm comfortable dress. In the lifetime of\nthe late Charles Lutwidge, Esq., of Holm Rook, she took possession of\nan old cottage, or rather cow-house, on his estate, in which she has\never since been suffered to continue. Her intellects seem at certain\ntimes greatly deranged, but her actions are harmless, and her language\ninoffensive. On that score she is caressed by all the villagers, who\nsupply her with eatables, &c., for money she utterly refuses. She\nseems a person in her lucid intervals, of much shrewdness, and her\nunderstanding is above the common level. This has also been improved\nby a tolerable education. Her appearance has been much the same for\nthese 20 years, so that she must now be nearly 90 years of age; but of\nthis, as well as her family and name, she is always silent. She seems\nto have chosen out the spot where she now lives, to pass the remainder\nof her days unknown to her friends, and in a great measure from a\ndistaste of a wicked world, to 'prepare herself,' as she often in her\nquiet hours says, 'for a better.'\"\n\n\nTHE RULING PASSION.\n\nA remarkable instance of the irresistible strength of the ruling\npassion was to be seen a few years ago in a Londoner, who had kept are\nretail spirit-shop, and retired into the adjoining county when he had\nmade a fortune, to enjoy himself. This man used to amuse himself by\nhaving one puncheon filled with water, and measuring it off by pints\ninto another. There was also another retired cit who used every day to\nangle in his round wash-hand-basin sized fish-pond for gold-fish. One\nfish he knew, because it had once lost its eye in being caught--and he\nused to say \"Confound that fellow, this is the fifth, sixth, &c., time\nthat I have caught him this season.\" It used to provoke him.\n\n\nINTERESTING REPORT WRITTEN BY SIR CHRISTOPHER WREN.\n\nIn the history of public buildings and monuments, it is always\ncurious to note the original plans of those who designed them, and to\nmark the different proposals and suggestions which were taken into\nconsideration. On this account our readers will no doubt be gratified\nby perusing the following Report of Sir Christopher Wren, on the\nornament which it would, in his opinion, be most desirable to place on\nsummit of the Monument, on Fish Street-hill. The Report was drawn up\nfor the use of the Committee of City Lands:--\n\n\"In pursuance of an Order of the Comittee for City Landes, I doe\nheerwith offer the several designes which some monthes since I showed\nHis M{tie}. for his approbation; who was then pleased to thinke a\nlarge Ball of metall, gilt, would be most agreeable, in regard it would\ngive an Ornament to the Town at a very great distance; not that His\nM{tie}. disliked a statue; and if any proposall of this sort be more\nacceptable to the City, I shall most readily represent the same to His\nM{tie}.\n\n\"I cannot but comend a large Statue, as carrying much dignitie with it,\nand that w{ch} would be more valewable in the eyes of Forreiners and\nstrangers. It hath been proposed to cast such a one in Brasse, of 12\nfoot high for L1,000. I hope (if it be allowed) wee may find those who\nwill cast a figure for that mony of 15 foot high, w{ch} will suit the\ngreatnesse of the pillar, & is (as I take it) the largest at this day\nextant, and this would undoubtedly be the noblest finishing that can be\nfound answerable to soe goodly a worke in all men's judgements.\n\n\"A Ball of Copper, 9 foot diameter, cast in severall peeces with the\nFlames and gilt, may well be don with the iron worke and fixing for\n350lb., and this will be most acceptable of any thing inferior to a\nstatue, by reason of the good appearance at distance, and because one\nmay goe up into it, & upon occasion use it for fireworkes.\n\n\"A Phoenix was at first thought of, & is the ornament in the wooden\nmodell of the pilar w{ch} I caused to be made before it was begun; but\nupon second thoughtes I rejected it, because it will be costly, not\neasily understood at that highth, and worse understood at a distance,\nand lastly dangerous, by reason of the sayle, the spread winges will\ncarry in the winds.\n\n\"The Belcony must be made of substantial well forged worke, there being\nnoe need at that distance of filed worke, and I suppose (for I cannot\nexactly guesse the weight) it may be well performed and fixed according\nto a good designe for fourscore & ten poundes, including painting, All\nw{ch} is humbly submitted to your consideration.\n\n\"July 28, 1675.\n\n \"CHR. WREN.\"\n\n\nCHANGE OF SEX.\n\nConnected with the plumage of birds is an extraordinary problem which\nhas baffled all research, and towards the solution of which not the\nslightest approach has been made. Among certain of the gallinaceous\nbirds, and it has been observed in no other family, the females\noccasionally assume the male plumage. Among pheasants in a wild state,\nthe hen thus metamorphosed, assumes with the livery a disposition\nto war with her own race, but in confinement she is spurned and\nbuffeted by the rest. From what took place in a hen pheasant in the\npossession of a lady, a friend of the late Sir Joseph Banks, it would\nseem probable that this change arises from some alteration in the\ntemperament at a late period of the animal's life. This lady had paid\nparticular attention to the breeding of pheasants. One of the hens,\nafter having produced several broods, moulted, and the succeeding\nfeathers were exactly those of a cock. This animal never afterwards\nlaid an egg. The pea-hen, has sometimes been known to take the plumage\nof the cock bird. Lady Tynte had a favourite pea-hen, which at eight\nseveral times produced chicks. Having moulted when about eleven years\nold, the lady and her family were astonished by her displaying the\nfeathers peculiar to the other sex, and appearing like a pied peacock.\nIn this process the tail, which was like that of the cock, first\nappeared. In the following year she moulted again, and produced similar\nfeathers. In third year she did the same, and then had also spurs\nresembling those of the cock. The bird never bred after this change of\nher plumage.\n\n\nTILBURY FORT.\n\nThe chief fame of Tilbury rests on the formation of the camp here, in\nthe reign of Queen Elizabeth, to defend London against the Spanish\ninvasion. Although it is unnecessary to recount the well-known\ncircumstances which led to the formation of the Tilbury camp, it may\nnot be out of place to give the famous speech of Queen Elizabeth on the\noccasion of her visit:--\n\n\"My loving People,--We have been persuaded by some that are careful of\nour safety, to take heed how we trust ourselves to armed multitudes\nfor fear of treachery; but assure you I do not desire to live to\ndistrust my faithful and loving people. Let tyrants fear. I have always\nso behaved myself that under God I have placed my chiefest strength\nand safeguard in the loyal hearts and goodwill of my subjects; and\ntherefore I am come among you at this time, not as for my recreation or\nsport, but being resolved, in the midst and heat of the battle, to live\nor die amongst you all--to lay down for my God, and for my kingdom, and\nfor my people, my honour and my blood even in the dust. I know that I\nhave but the body of a weak and feeble woman, but I have the heart of a\nking, and a king of England too; and I think foul scorn that Parma or\nSpain, or any prince of Europe, should dare to invade the borders of\nmy realms to which, rather than any dishonour shall grow by me, I will\nmyself take up arms--I myself will be your general, judge, and rewarder\nof every one of your victories in the field.\"\n\nThe most full description of Elizabeth's reception at Tilbury is\nprinted in a sort of doggrel poem, headed, \"Elizabetha Triumphans,\nbriefly, truly, and effectually set forth, declared, and handled by\nJames Aske.\"\n\nThe poem mentions, that when about 20,000 well-appointed men had\narrived at Tilbury, orders were sent to the various shires to cause the\ntroops in each to remain until further notice; and so great was the\ndesire to meet the enemy, that one thousand men of Dorsetshire offered\nL500 to be allowed to march to the camp at Tilbury.\n\nThe alarm of the Spanish invasion was, however, not the last to\nthreaten the Londoners, and direct attention to Tilbury.\n\nOn the 8th of June, 1667, Ruyter, the Dutch admiral, sailed out of\nthe Texel with fifty ships, and came to the mouth of the Thames, from\nwhence he detached Vice-Admiral Van Ghent, with seventeen of his\nlightest ships and some fire-ships. Van Ghent in the same month sailed\nup the Medway, made himself master of the fort of Sheerness, and,\nafter burning a magazine of stores to the value of L40,000, blew up\nthe fortifications. This action alarmed the City of London; so that\nto prevent similar mischief, several ships were sunk, and a large\nchain put across the narrowest part of the Medway. But by means of an\neasterly wind and a strong tide, the Dutch ships broke through the\nchain, and sailed between the sunk vessels. They burnt three ships,\nand carried away with them the hull of the \"Royal Charles,\" besides\nburning and damaging several others. After this they advanced as far as\nUpnor Castle, and burnt the \"Royal Oak,\" the \"Loyal London,\" and the\n\"Great James.\" Fearing that the whole Dutch fleet would sail to London\nBridge, the citizens caused thirteen ships to be sunk at Woolwich, and\nfour at Blackwall, and platforms furnished with artillery to defend\nthem were raised in several places. The consternation was very great,\nand the complaints were no less so. It was openly said the king, out\nof avarice, had kept the money so generously given to him to continue\nthe war, and left his ships and subjects exposed to the insults of\nthe enemy. After this exploit, Ruyter sailed to Portsmouth, with a\ndesign to burn the ships in that harbour; but finding them secured,\nhe sailed to the west, and took some ships in Torbay. He then sailed\neastward, beat the English force before Harwich, and chased a squadron\nof nineteen men-of-war, commanded by Sir Edward Spragg, who was obliged\nto retire into the Thames. In a word, he kept the coasts of England in\na continual alarm all July, till he received news of the conclusion of\npeace.\n\n[Illustration: Water-gate of Tilbury Fort.]\n\nThis daring attack was no doubt the cause of Tilbury Fort being made\nto assume its present form. It is now a regular fortification, and\nmay be justly looked upon as the key to the City of London. The plan\nof the building was laid out by Sir Martin Beckman, chief engineer to\nCharles II., who also designed the works at Sheerness. The foundation\nis laid upon piles driven down, two on end of each other, till they\nwere assured they were below the channel of the river, and that the\npiles, which were pointed with iron, entered into the solid chalk rock.\nOn the land side, the works are complete; the bastions are faced with\nbrick. There is a double ditch, or moat, the innermost of which is 180\nfeet broad, with a good counterscarp, and a covered way marked out\nwith ravelins and tenailles. There are some small brick redoubts; the\nchief strength, however, of this part of the fort consists in being\nable to lay the whole level under water, and, by that means, make it\nimpossible for an enemy to carry on approaches that way. On the river\nside is a very strong curtain, with the picturesque water-gate shown\nin our engraving in the middle. Before this curtain is a platform, in\nthe place of a counterscarp, on which are planted cannon of large size.\nThese completely command the river, and would no doubt the\nships of an enemy attempting to pass in this direction. A few years ago\nthere were placed on the platform 106 cannon, carrying from 24 to 46\npounds each, besides smaller ones planted between them. The bastions\nand curtains are also planted with guns.\n\nThe circular tower shown in the engraving was in existence in the time\nof Queen Elizabeth, and was called the Block-house.\n\n\nRINGING THE CHANGES.\n\nIt is curious to note the number of changes which may be rung on\ndifferent peals. The changes on seven bells are 5,040; on twelve\n479,001,600, which it would take ninety-one years to ring at the rate\nof two strokes in a second. The changes on fourteen bells could not\nbe rung through at the same rate in less than 16,575 years: and upon\nfour-and-twenty, they would require more than 117,000 billions of years.\n\n\nDISGRACEFUL STATE OF THE LONDON POLICE IN 1724.\n\nThat notorious burglar, Jack Sheppard, finished his disgraceful career\nat Tyburn in the year 1724, and we notice the event, not with the view\nof detailing the disgusting particulars of an execution, but because\nthe outrages which were allowed to take place after the dreadful scene\nwas over, exhibit in a striking light the miserable police regulations\nwhich existed at that period, and the manner in which the mob were\nallowed to have it nearly all their own way. The Sheriff's officers,\naware of the person they had to contend with, thought it prudent to\nsecure his hands on the morning of execution. This innovation produced\nthe most violent resistance on Sheppard's part; and the operation was\nperformed by force. They then proceeded to search him, and had reason\nto applaud their vigilance, for he had contrived to conceal a penknife\nin some part of his dress. The ceremony of his departure from our\nworld passed without disorder; but, the instant the time expired for\nthe suspension of the body, an undertaker, who had followed by his\nfriends' desire with a hearse and attendants, would have conveyed it to\nSt. Sepulchre's churchyard for interment; but the mob, conceiving that\nsurgeons had employed this unfortunate man, proceeded to demolish the\nvehicle, and attack the sable dependants, who escaped with difficulty.\nThey then seized the body, and, in the brutal manner common to those\nwretches, beat it from each to the other till it was covered with\nbruises and dirt, and till they reached Long-acre, where they deposited\nthe miserable remains at a public-house called the Barley-mow. After\nit had rested there a few hours the populace entered into an enquiry\nwhy they had contributed their assistance in bringing Sheppard to\nLong-acre; when they discovered they were duped by a bailiff, who\nwas actually employed by the surgeons; and that they had taken the\ncorpse from a person really intending to bury it. The elucidation of\ntheir error exasperated them almost to phrensy, and a riot immediately\ncommenced, which threatened the most serious consequences, The\ninhabitants applied to the police, and several magistrates attending,\nthey were immediately convinced the civil power was insufficient to\nresist the torrent of malice ready to burst forth in acts of violence.\nThey therefore sent to the Prince of Wales and the Savoy, requesting\ndetachments of the guards; who arriving, the ringleaders were secured,\nthe body was given to a person, a friend of Sheppard, and the mob\ndispersed to attend it to the grave at St. Martin's in the fields,\nwhere it was deposited in an elm coffin, at ten o'clock the same night,\nunder a guard of soldiers, and with the ceremonies of the church.\n\n\nA TRIUMPH OF ENERGY.\n\nAfter the accession of Tippoo Saib to the throne of Mysore in 1782,\nthe English made overtures for a termination of the war which had been\ncommenced by his father; but flushed by the possession of a large\narmy, a well-filled treasury, a passion for war, and an inordinate\nsense of his own importance, Tippoo refused all terms of pacification,\nand left the English no alternative but to battle against him as they\ncould. Lord Macartney, who was at that time the Governor of Madras,\non becoming acquainted with the determination of Tippoo, resolved to\nprosecute hostilities with the greatest vigour, and having placed Col.\nFullerton at the head of his force, he provided him with an army,\ncollected from various parts, of 16,000 good troops, and afforded that\nexcellent officer all available assistance in carrying the war into\nTippoo's territory. Fullerton laid his plans with considerable skill;\nhe encouraged the natives to bring and sell provisions to him on his\nmarch, effectually checked devastation and plundering, scrupulously\nrespected the religious opinions of the Hindus, consolidated and\nimproved the mode of march, and availed himself of the subtle cunning\nand nimble feet of the natives to establish a remarkably complete\ncourier-system, whereby he could receive and communicate intelligence\nwith a rapidity never before attained by any European officer in\nIndia. He had to choose between two systems of strategy--either to\nmarch through the Mysore territory, and frustrate Tippoo in his\nsiege of Mangalore; or boldly to attack Seringapatam, in order to\ncompel Tippoo to leave Mangalore as a means of defending his own\ncapital. The colonel decided on the adoption of the latter course, as\npromising more fruitful results. Being at Daraporam, 200 miles south\nof Seringapatam, Fullerton resolved to divert the route, and take a\ncircuit nearer the western coast, where the capture of the strong\nfort of Palagatcherry would afford him a valuable intermediate depot,\ncommanding one of the chief roads from the Malabar to the Coromandel\ncoasts. On the 18th of October he started. After capturing a few small\nforts, he ascended to high ground, where dense forests, deep ravines,\nand tortuous water courses embarrassed every yard of his progress:\nto fill up the ravines before he could drag his artillery over them,\nto throw trees across them where the depth was too great for filling\nup, to clear gaps through forests with the axe, to contend against\ntremendous rains--were only part of the difficulties he had to meet;\nbut he met them like a skilful commander, reached Palagatcherry on\nthe 5th of November, and captured the fort on the 15th, obtaining\nwith it a welcome supply of money, grain, guns, powder, shot, and\nmilitary stores. When the difficulties which Colonel Fullerton had\nto encounter, and the triumphant manner in which he overcame them,\nare taken into consideration, it will be readily admitted, we think,\nthat his enterprise is well deserving of being recorded as a striking\nexample of what may be accomplished by a union of professional skill\nand invincible energy. Our engraving represents one of the devices\nwhich Colonel Fullerton employed for the purpose of enabling his forces\nto pass over a mountain torrent.\n\n[Illustration: [++] Bridge over Mountain Torrent.]\n\n\nSTORMING OF THE BASTILLE AT PARIS.\n\n[Illustration: [++] The Bastille.]\n\nThe great Revolution in France, at the close of the last century, was\nfull of wonderful events, many of which might be appropriately recorded\nin our pages. One of the most striking among them was the storming and\ncapture of the Bastille, a vast state-prison which was begun to be\nbuilt in 1369 by Charles V., and finished by his successor in 1383.\nThe demolition of this fortress was the first triumph of the armed\npopulace of Paris, and it rendered the progress of the revolution\nirresistible. As the day closed in on the evening of Monday, the 14th\nof July, 1789, a reckless multitude of rioters, after seizing 30,000\nmuskets and several pieces of artillery at the Hotel des Invalides,\nrushed in wild excitement to the Bastille, rendered hateful to the\npeople by the political imprisonment of many hapless men in past times,\nalthough less frequently applied to similar purposes under the milder\nrule of Louis XVI. An armed mob of at least 100,000 men, aided by\ntroops who joined them in whole regiments at a time, had not long to\ncontend against the old fortress. The governor, De Launay, made such a\ndefence as a brave officer might at such a juncture; but his few troops\nwere bewildered and wavering; he received orders from the Hotel de\nVille which he knew not whether to obey or resist, but no instructions\nfrom the court or the ministers; and the military aid to the mob became\nstronger than any force he could bring to bear against them. The chains\nof three drawbridges were broken by hatchets; straw, wood, oil, and\nturpentine were brought and kindled, to burn down the gates; and after\nmany volleys from the mob had been answered by a few from the fortress,\nDe Launay, seeing no hope of succour, resolved to blow up the place\nrather than yield. In this he was prevented by the Swiss guards, who\nformed a part of the small garrison, and who, after a parley with\nthe insurgents, opened the gates, and surrendered. The Bastille was\ntaken. The ruffians, heeding nothing but their own furious passions,\ndisregarded the honourable rules of capitulation; they beheaded De\nLaunay in a clumsy and barbarous manner, and putting his head on a\nspike, carried it through the streets shouting, laughing, and singing;\nthey were prevented only by an accidental interruption from burning\nalive a young lady whom they found in one of the court-yards; they\nhung or maltreated many of the Swiss and invalid soldiers; and they\nfearfully hacked the bodies of three or four officers in the endeavour\nto decapitate them. The prisoners within, only seven in number, were\nliberated, and treated with a drunken revel; while the Chatelet and\nother prisons became scenes of renewed disorders. The sketch which we\ngive above, of the attack on the Bastille, is taken from a medallion by\nAndrieu.\n\n\nDURATION OF LIFE AMONG ARTISTS.\n\nIn Gould's Dictionary of Artists, published in 1839, the names, with\nthe ages, of 1,122 persons are given; which furnish the following\nremarkable facts as to the longevity of this class of men. Died under\n60 years old, 474; 60 years and under 70, 250; 70 years and under 80,\n243; 80 years and under 90, 134; 90 years and under 100, 19; above 100,\n1. The mean age at death of the whole number being 55 years; from which\nit would appear that the pursuit of the fine arts has a tranquilizing\neffect upon the spirits, and a tendency to moral refinement in the\nhabits and manners of its professors extremely favourable to the\nprolongation of life.\n\n\nCHANGE IN THE VALUE OF LAND.\n\nAt Brighton, within the present century, a spot of ground was offered\nto a hair-dresser in fee, upon condition of shaving the possessor for\nlife. The terms were declined, and the land soon became of immense\nvalue.\n\n\nUNACCOUNTABLE ANTIPATHIES.\n\nThe following are a few of the more striking manifestations of that\nunaccountable feeling of antipathy to certain objects, to which so many\npersons are subject, and with instances of which--in a modified form\nperhaps--most people are acquainted with:--\n\nErasmus, though a native of Rotterdam, had such an aversion to fish,\nthat the smell of it threw him into a fever.\n\nAmbrose Pare mentions a gentleman, who never could see an eel without\nfainting.\n\nThere is an account of another gentleman, who would fall into\nconvulsions at the sight of a carp.\n\nA lady, a native of France, always fainted on seeing boiled lobsters.\nOther persons from the same country experienced the same inconvenience\nfrom the smell of roses, though they were particularly partial to the\nodour of jonquils or tuberoses.\n\nJoseph Scaliger and Peter Abono never could drink milk.\n\nCardan was particularly disgusted at the sight of eggs.\n\nUladislaus, king of Poland, could not bear to see apples.\n\nIf an apple was shown to Chesne, secretary to Francis I., he bled at\nthe nose.\n\nA gentleman, in the court of the emperor Ferdinand, would bleed at the\nnose on hearing the mewing of a cat, however great the distance might\nbe from him.\n\nHenry III. of France could never sit in a room with a cat.\n\nThe Duke of Schomberg had the same aversion.\n\nM. de Lancre gives an account of a very sensible man, who was so\nterrified at seeing a hedgehog, that for two years he imagined his\nbowels were gnawed by such an animal.\n\nThe same author was intimate with a very brave officer, who was so\nterrified at the sight of a mouse, that he never dared to look at one\nunless he had his sword in his hand.\n\nM. Vangheim, a great huntsman in Hanover, would faint, or, if he had\nsufficient time, would run away at the sight of a roasted pig.\n\nJohn Rol, a gentleman in Alcantara, would swoon on hearing the word\n_lana_, wool, pronounced, although his cloak was woollen.\n\nThe philosophical Boyle could not conquer a strong aversion to the\nsound of water running through a pipe.\n\nLa Mothe le Vayer could not endure the sound of musical instruments,\nthough he experienced a lively pleasure whenever it thundered.\n\nThe author of the Turkish Spy tells us that he would rather encounter\na lion in the deserts of Arabia, provided he had but a sword in his\nhand, than feel a spider crawling on him in the dark. He observes, that\nthere is no reason to be given for these secret dislikes. He humorously\nattributes them to the doctrine of the transmigration of the soul; and\nas regarded himself, he supposed he had been a fly, before he came into\nhis body, and that having been frequently persecuted with spiders, he\nstill retained the dread of his old enemy.\n\n\nLONDON RESORTS A HUNDRED YEARS AGO.\n\nIn addition to the regular theatres, there were many places of\namusement, such as the Vauxhall and Ranelagh Gardens, the site of the\nlatter being now occupied by the houses that hem in Chelsea College;\nthe Rotunda, famous for its music, its gardens, and its piece of water;\nBell-size House and Gardens on the Hampstead Road, where tea, coffee,\nand other refreshments could be had, together with music, from seven in\nthe morning,--with the advantage of having the road to London patrolled\nduring the season by twelve \"lusty fellows,\" and of being able to ride\nto Hampstead by coach for sixpence a-head; Perrot's inimitable grotto,\nwhich could be seen by calling for a pot of beer; Jenny's Whim, at the\nend of Chelsea Bridge, where \"the royal diversion of duck-hunting\"\ncould be enjoyed, \"together with a decanter of _Dorchester_\" for\nsixpence; Cuper's Gardens, in Lambeth, nearly opposite Somerset House,\nthrough which the Waterloo Road was ruthlessly driven; the Marble\nHall, at Vauxhall, where an excellent breakfast was offered for one\nshilling; Sadler's Wells, celebrated both for its aquatic and its\nwire-dancing attractions; the Floating Coffee-House, on the river\nThames, the Folly House at Blackwall, Marybone Gardens, the White\nConduit House, and a multitude of others, to enumerate which would be\ntedious and unprofitable. On Sunday, we are told, the \"snobocracy,\"\namused themselves by thrusting their heads into the pillory at Georgia,\nby being sworn at Highgate, or rolling down Flamstead Hill in Greenwich\nPark. Some regaled their wives and families with buns at Chelsea and\nPaddington; others indulged in copious draughts of cyder at the Castle\nin the pleasant village of Islington; while the undomestic cit, in\nclaret- coat and white satin vest, sipped his beer and smoked\nhis pipe at Mile End, or at the \"Adam and Eve\" in Pancras, or \"Mother\nRed Cap's\" at Camden.\n\n\nQUEEN ELIZABETH'S STATE COACH.\n\n[Illustration: [++] Queen Elizabeth's State Coach.]\n\nThe accompanying engraving is taken from a very old print representing\nthe state procession of Queen Elizabeth on her way to open Parliament\non 2nd April, 1571. This was the first occasion on which a state coach\nhad ever been used by a Sovereign of England, and it was the only\nvehicle in the procession; the Lord Keeper, and the Lords Spiritual and\nTemporal, all attending on horseback. It was drawn by two palfreys,\nwhich were decked with trappings of crimson velvet; and, according\nto an old authority, the name of the driver was William Boonen, a\nDutchman, who thus became the first state coachman.\n\n\nTHE ORIGIN OF EATING GOOSE ON MICHAELMAS DAY.\n\nQueen Elizabeth, on her way to Tilbury Fort on the 29th of September,\n1589, dined at the ancient seat of Sir Neville Umfreville, near that\nplace; and as British Bess had much rather dine off a high-seasoned\nand substantial dish than a simple fricassee or ragout, the knight\nthought proper to provide a brace of fine geese, to suit the palate\nof his royal guest. After the Queen had dined heartily, she asked for\na half-pint bumper of Burgundy, and drank \"Destruction to the Spanish\nArmada.\" She had but that moment returned the glass to the knight who\nhad done the honours of the table, when the news came (as if the Queen\nhad been possessed with the spirit of prophecy) that the Spanish fleet\nhad been destroyed by a storm. She immediately took another bumper,\nin order to digest the goose and good news; and was so much pleased\nwith the event, that she every year after, on that day, had the above\nexcellent dish served up. The Court made it a custom, and the people\nthe same, ever since.\n\n\nPRE-ADAMITE BONE CAVERNS.\n\nAmong the wonders of the world, the bone caves of the pre-Adamite\nperiod deserve a prominent place. It is to this period that the\nextensive remains of Mammiferae found in the strata of the Pampas\nof Buenos Ayres, and in the caverns which are scattered in such\nvast numbers over the continents of Europe and America, and even in\nAustralia, are to be ascribed. We regret that we can find room for a\ndescription of only one of these caverns, but it is a most extensive\none, and among the first which attracted attention. It is situated\nat Baylenreuth, in Franconia, and the engraving which we here give\nrepresents a section of it.\n\n[Illustration: [++] Pre-Adamite Bone Cavern.]\n\nThe entrance of this cave, about seven feet in height, is placed on\nthe face of a perpendicular rock, and leads to a series of chambers\nfrom fifteen to twenty feet in height, and several hundred feet in\nextent, in a deep chasm. The cavern is perfectly dark, and the icicles\nand pillars of stalactite reflected by the torches present a highly\npicturesque effect. The floor is literally paved with bones and fossil\nteeth, and the pillars and corbels of stalactite also contain osseous\nremains. Cuvier showed that three-fourths of the remains in this and\nlike caverns were those of bears, the remainder consisting of bones of\nhyenas, tigers, wolves, foxes, gluttons, weasels, and other Carnivora.\n\n\nHOW DISTANT AGES ARE CONNECTED BY INDIVIDUALS.\n\nMr. Robert Chambers, in a curious and interesting chapter in the\n\"Edinburgh Journal,\" entitled \"Distant Ages connected by Individuals,\"\nstates, in 1847, \"There is living, in the vicinity of Aberdeen, a\ngentleman who can boast personal acquaintance with an individual who\nhad seen and conversed with another who actually had been present at\nthe battle of Flodden Field!\" Marvellous as this may appear, it is not\nthe less true. The gentleman to whom allusion is made was personally\nacquainted with the celebrated Peter Garden, of Auchterless, who died\nin 1775, at the reputed age of 131, although there is reason to believe\nthat he was several years older. Peter, in his young days, was servant\nto Garden, of Troup, whom he accompanied on a journey through the north\nof England, where he saw and conversed with the famous Henry Jenkins,\nwho died 1670, at the age of 169. Jenkins was born in 1501, and was of\ncourse twelve years old at the period of the battle of Flodden Field;\nand, on that memorable occasion, bore arrows to an English nobleman\nwhom he served in the capacity of page. \"When we think of such things,\"\nadds Mr. Chambers, \"the ordinary laws of nature seem to have undergone\nsome partial relaxation; and the dust of ancient times almost becomes\nliving flesh before our eyes.\"\n\n\nTHE EARTHQUAKE AT LISBON.\n\nOn the 1st of November, 1755, a few minutes before 10 a.m. the\ninhabitants of Lisbon were alarmed by several violent vibrations of\nthe ground which then rose and fell several times with such force that\nhundreds of houses came toppling into the streets, crushing thousands\nof people. At the same time the air grew pitchy dark from the clouds of\ndust that rose from the crumbling edifices. Many persons ran down to\nthe river side, in the hope of escaping to the shipping; but the water\nsuddenly rose some yards perpendicularly, and swept away everything\nbefore it. The quay, with nearly 200 human beings standing on it,\nall at once disappeared. Large ships, which were lying high and dry,\nfloated off, and were dashed against each other or carried down the\nriver. In every direction the surface of the water was overspread with\nboats, timber, casks, household furniture and corpses. The scene on\ndry land was yet more horrifying. Churches, government buildings, and\nprivate houses, were all involved in the same ruin. Many thousands\nof trembling fugitives had collected in the great square, when it\nwas discovered that flames were spreading in every quarter. Taking\nadvantage of the universal panic and confusion, a band of miscreants\nhad fired the city. Nothing could be done to stay the progress of\nthe flames, and for eight days they raged unchecked. Whatever the\nearthquake had spared fell a prey to this new calamity. \"It is not to\nbe expressed by human tongue,\" writes an eye-witness, \"how dreadful\nand how awful it was to enter the city after the fire was abated; and\nlooking upwards, one was struck with horror in beholding dead bodies,\nby six or seven in a heap, crushed to death, half buried and half\nburnt; and if one went through the broad places or squares, nothing was\nto be met with but people bewailing their misfortunes, wringing their\nhands, and crying, 'The world is at an end.' If you go out of the city,\nyou behold nothing but barracks, or tents made with canvass or ship's\nsails, where the poor inhabitants lye.\"\n\nAnother eye-witness is still more graphic. \"The terror of the people\nwas beyond description: nobody wept,--it was beyond tears;--they ran\nhither and thither, delirious with horror and astonishment--beating\ntheir faces and breasts--crying '_Misericordia_, the world's at\nan end;' mothers forgot their children, and ran about loaded with\ncrucifixed images. Unfortunately, many ran to the churches for\nprotection; but in vain was the sacrament exposed; in vain did the poor\ncreatures embrace the altars; images, priests, and people, were buried\nin one common ruin. * * * The prospect of the city was deplorable.\nAs you passed along the streets you saw shops of goods with the\nshopkeepers buried with them, some alive crying out from under the\nruins, others half buried, others with broken limbs, in vain begging\nfor help; they were passed by crowds without the least notice or sense\nof humanity. The people lay that night in the fields, which equalled,\nif possible, the horrors of the day; the city all in flames; and if\nyou happened to forget yourself with sleep, you were awakened by the\ntremblings of the earth and the howlings of the people. Yet the moon\nshone, and the stars, with unusual brightness. Long wished-for day at\nlast appeared, and the sun rose with great splendour on the desolated\ncity. In the morning, some of the boldest, whose houses were not burnt,\nventured home for clothes, the want of which they had severely felt in\nthe night, and a blanket was now become of more value than a suit of\nsilk.\"\n\n\nSTRANGE CURE FOR RHEUMATISM.\n\nBridget Behan, of Castle-waller, in the county of Wicklow, Ireland,\nretained the use of all her powers of body and mind to the close of\nher long life, 110 years, in 1807. About six years preceding her death\nshe fell down stairs, and broke one of her thighs. Contrary to all\nexpectation, she not only recovered from the effects of the accident,\nbut actually, from thence, walked stronger on this leg, which,\npreviously to the accident, had been a little failing, than she had\ndone for many years before. Another remarkable circumstance relating\nto this fracture was, that she became perfectly cured of a chronic\nrheumatism of long standing, and from which on particular occasions she\nhad suffered a good deal of affliction. A short while before her death\nshe cut a new tooth.\n\n\nSILVER TEA SERVICE WHICH BELONGED TO WILLIAM PENN.\n\n[Illustration: [++] Silver Tea Service.]\n\nArticles of ordinary use, however small may be their intrinsic value,\nwhich have once been the property of men who have been good and\ngreat--how rare the conjunction!--are always invested with a peculiar\ninterest. They often afford a clue to the tastes of those who once\npossessed them. On this account we have great pleasure in laying\nbefore our readers a representation of the silver tea-service which\nbelonged to the celebrated William Penn, the founder and legislator\nof Pennsylvania, whom Montesquieu denominates the modern Lycurgus.\nHe was the son of Admiral Penn, was born at London in 1644, and was\neducated at Christchurch, Oxford. At college he imbibed the principles\nof Quakerism, and having endeavoured to disseminate them by preaching\nin public, he was thrice thrown into prison. It was during his first\nimprisonment that he wrote \"_No Cross, no Crown_.\" In March, 1680-81,\nhe obtained from Charles II. the grant of that territory which now\nbears the name of Pennsylvania. In 1682 he embarked for his new\ncolony; and in the following year he founded Philadelphia. He returned\nto England in 1684, and died in July, 1718. He was a philosopher, a\nlegislator, an author, the friend of man, and, above all, a pious\nChristian. In addition to the reasons above given, the sketch of\nthe tea-service is an object of curiosity, as showing the state of\nsilversmith's work in England, at the close of the seventeenth century,\nfor articles of domestic use.\n\n\nCURIOUS FIGURES ON A SMALL SHRINE.\n\n[Illustration: [++] Soldiers Watching the Body of Our Lord.]\n\nThe figures here given are copied from a curious little bronze,\nstrongly gilt, which was engraved in the \"Gentleman's Magazine\" for\n1833, accompanied with a description, by A. J. Kempe, Esq., the author\nof the letterpress to \"Stothard's Monumental Effigies,\" whose intimate\nknowledge in these matters enables him to well authenticate dates; and\nhe considers this relic may safely be attributed to the early part of\nthe twelfth century; it was discovered in the Temple Church, and had\noriginally formed a portion of a pyx, or small shrine, in which the\nconsecrated host was kept. Our engraving is more than half the size of\nthe original, which represents the soldiers watching the body of Our\nLord, who was, in mystical form, supposed to be enshrined in the pyx.\nThey wear scull-caps of the Phrygian form, with the nasal like those in\nthe Bayeux Tapestry; and the mailles or rings of the hauberk appear,\nas in the armour there, sewn down, perhaps, on a sort of gambeson, but\nnot interlaced. They bear kite-shaped shields, raised to an obtuse\nangle in the centre, and having large projecting bosses: the third of\nthese figures is represented beside the cut in profile, which will\nenable the reader more clearly to detect its peculiarities. On two of\nthese shields are some approaches to armorial bearings; the first is\nmarked with four narrow bendlets; the second is fretted, the frets\nbeing repeated in front of his helmet, or _chapelle de fer_: all the\nhelmets have the nasal. A long tunic, bordered, and in one instance\nornamented with cross-lines, or chequered, appears beneath the tunic.\nThe sword is very broad, and the spear carried by the first figure,\nobtuse in the head,--a mark of its antiquity. The shoes are admirable\nillustrations of that passage of Geoffry of Malmesbury, where,\nrepresenting the luxury of costume in which the English indulged at the\ntime when Henry I. began his reign, he says: \"Then was there flowing\nhair, and extravagant dress; and then was invented the fashion of shoes\nwith curved points: then the model for young men was to rival women in\ndelicacy of person, to mimic their gait, to walk with loose gesture,\nhalf-naked.\" The curvature of the points of the shoes in the little\nrelic before us, in conformity with the custom censured by Malmesbury,\nis quite remarkable. One turns up, another down; one to the left,\nanother to the right; and scarcely any two in the same direction.\n\n\nTHE QUEEN'S SHARKS.\n\nThe harbour of Trincomalee swarms with gigantic sharks, and strange to\nrelate, they are all under British protection; and if any one is found\nmolesting or injuring them, the fine is L10, or an imprisonment! How\nthis ridiculous custom originated, it is hard to say; but we are told,\nthat in the early days of British conquest in the East, sailors were\napt to desert, and seek refuge in the then inaccessible wilds of the\ninterior; and of later years, when civilisation has unbarred the gates\nof Cingalese commerce to all nations of the world, the soldiers of\nthe regiment stationed at Trincomalee, discontented with their lot in\nlife, were wont to escape from the thraldom of the service, by swimming\noff to American and other foreign vessels, preferring chance, under a\nstrange flag, to a hard certainty under their own. Thus the Queen's\nsharks are duly protected as a sort of water-police for the prevention\nof desertion both from the army and navy.\n\n\nOLD VERSES ON QUEEN ELIZABETH.\n\nThe following quaint and curious verses are taken from a very old\nvolume, entitled _A Crowne Garland of Goulden Roses, Gathered out of\nEngland's Royall Garden, &c., &c. By Richard Johnson_.\n\nA SHORT AND SWEET SONNET MADE BY ONE OF THE MAIDES OF HONOR UPON THE\nDEATHE OF QUEENE ELIZABETH, WHICH SHE SOWED UPON A SAMPLER IN RED SILKE.\n\n_To a new tune, or \"Phillida flouts me.\"_\n\n Gone is Elizabeth,\n Whom we have lov'd so deare;\n She our kind mistres was\n Full foure and forty yeare.\n\n England she govern'd well,\n Not to be blamed;\n Flanders she govern'd well\n And Ireland tamed.\n\n France she befrended,\n Spaine she hath foiled,\n s rejected,\n And the Pope spoyled.\n\n To princes powerfull,\n To the world vertuous,\n To her foes mercifull,\n To her subjects gracious.\n\n Her soule is in heaven,\n The world keeps her glory,\n Subjects her good deeds,\n And so ends my story.\n\n\nRANELAGH.\n\nRanelagh, of which no traces now remain, was situated on part of\nChelsea Hospital garden, between Church Row and the river, to the east\nof the Hospital. It takes its name from a house erected in 1691, by\nViscount Ranelagh. This house, in which the Viscount had resided from\nthe period of its being built, was sold in 1733 to an eminent builder\nnamed Timbrell for L3,200, who advertised it for sale in the following\nyear, as a freehold with garden, kitchen garden, and offices, and a\nsmaller house and garden with fruit trees, coach-houses, &c., &c. These\nwere the first vicissitudes of Ranelagh, preparatory to its conversion\ninto a place of public amusement.\n\nWalpole, in one of his entertaining letters to Mann, April 22nd, 1742,\nthus speaks of the gardens, which were then unfinished:--\n\n\"I have been breakfasting this morning at Ranelagh Garden; they have\nbuilt an immense ampitheatre, with balconies full of little alehouses;\nit is in rivalry to Vauxhall, and cost above twelve thousand pounds.\nThe building is not finished, but they got great sums by people going\nto see it and breakfasting in the house: there were yesterday no less\nthan three hundred and eighty persons, at eighteen-pence a piece.\"\nAgain, under the date May 26th, 1742, he writes to his friend as\nfollows:--\n\n\"Two nights ago, Ranelagh Gardens were opened at Chelsea; the prince,\nprincess, duke, much nobility, and much mob besides were there. There\nis a vast ampitheatre, finely gilt, painted, and illuminated; into\nwhich everybody that loves eating, drinking, staring, or crowding, is\nadmitted for twelve pence. The building and disposition of the gardens\ncost sixteen thousand pounds. Twice a week there are to be ridottos at\nguinea tickets, for which you are to have a supper and music. I was\nthere last night, but did not find the joy of it. Vauxhall is a little\nbetter, for the garden is pleasanter, and one goes by water.\"\n\n\"The only defect in the elegance and beauty of the ampitheatre at\nRanelagh,\" says the _London Chronicle_ for August, 1763, \"is an\nimproper and inconvenient orchestra, which, breaking into the area\nof that superb room about twenty feet farther than it ought to do,\ndestroys the symmetry of the whole, and diffuses the sound of music\nwith such irregular rapidity, that the harmonious articulations escape\nthe nicest ear when placed in the most commodious attitude; it also\nhurts the eye upon your first entry.\n\n\"To remedy these defects, a plan has been drawn by Messrs. Wale\nand Gwin, for adding a new orchestra, which being furnished with a\nwell-proportioned curvature over it, will contract into narrower bounds\nthe modulations of the voice, and render every note more distinctly\naudible. It will, by its form, operate upon the musical sounds, in the\nsame manner as concave glasses affect the rays of light, by collecting\nthem into a focus. The front of this orchestra being planned so as\nto range parallel to the balustrade, the whole area also will be\ndisencumbered of every obstruction that might incommode the audience\nin their circular walk. There is likewise provision made in this plan\nfor a stage capable of containing 30 or 40 performers, to officiate as\nchorus-singers, or otherwise assist in giving additional solemnity on\nany extraordinary occasion.\"\n\n\"At Ranelagh House, on the 12th of May, 1767,\" says the _Gentleman's\nMagazine_, \"were performed (in the new orchestra) the much admired\ncatches and glees, selected from the curious collection of the Catch\nClub; being the first of the kind publickly exhibited in this or any\nother kingdom. The entertainments consisted of the favourite catches\nand glees, composed by the most eminent masters of the last and present\nage, by a considerable number of the best vocal and instrumental\nperformers. The choral and instrumental parts were added, to give the\nthe catches and glees their proper effect in so large an amphitheatre;\nbeing composed for that purpose by Dr. Arne.\"\n\nThe Rotunda, or amphitheatre, was 185 feet in diameter, with an\norchestra in the centre, and tiers of boxes all round. The chief\namusement was promenading (as it was called) round and round the\ncircular area below, and taking refreshments in the boxes while the\norchestra and vocalists executed different pieces of music. It was a\nkind of 'Vauxhall under cover,' warmed with coal fires. The rotunda\nis said to have been projected by Lacy, the patentee of Drury Lane\nTheatre. \"The _coup d'oeil_,\" Dr. Johnson declared, \"was the finest\nthing he had ever seen.\"\n\nThe last great event in the history of Ranelagh was the installation\nball of the knights of the Bath, in 1802, shortly after which the place\nwas pulled down.\n\n\nTHE FIRST EAST INDIA HOUSE.\n\n[Illustration: [++] The First East India House.]\n\nThe tradition is, that the East India Company, incorporated December\n31st, 1600, first transacted their business in the great room of the\nNag's Head Inn, opposite St. Botolph's Church, Bishopsgate Street.\nThe maps of London, soon after the Great Fire of 1666, place the\nIndia House on a part of its present site in Leadenhall Street. Here\noriginally stood the mansion of Alderman Kerton, built in the reign of\nEdward VI., rebuilt on the accession of Elizabeth, and enlarged by its\nnext purchaser, Sir W. Craven, Lord Mayor in 1610. Here was born the\ngreat Lord Craven, who, in 1701, leased his house and a tenement in\nLime Street to the Company at L100 a year. A scarce Dutch etching, in\nthe British Museum, of which the annexed engraving is a correct copy,\nshows this house to have been half timbered, its lofty gable surmounted\nwith two dolphins and a figure of a mariner, or, as some say, of the\nfirst governor; beneath are mecrhant ships at sea, the royal arms,\nand those of the Company. This grotesque structure was taken down in\n1726, and upon its site was erected the old East India House, portions\nof which yet remain; although the present stone front, 200 feet long,\nand a great part of the house, were built in 1798 and 1799, and\nsubsequently enlarged by Cockerell, R.A., and Wilkins, R.A.\n\n\nADVERTISEMENTS IN THE LAST CENTURY.\n\nThe following strange advertisements have been culled at random from\nmagazines and newspapers _circa_ 1750. They give us a good idea of the\nmanners and tastes of that period:--\n\n\"Whereas a tall young Gentleman above the common size, dress'd in a\nyellow-grounded flowered velvet (supposed to be a Foreigner), with a\nSolitair round his neck and a glass in his hand, was narrowly observed\nand much approved of by a certain young lady at the last Ridotto. This\nis to acquaint the said young Gentleman, if his heart is entirely\ndisengaged, that if he will apply to A. B. at Garaway's Coffee House in\nExchange Alley, he may be directed to have an interview with the said\nyoung lady, which may prove greatly to his advantage. Strict secresy on\nthe Gentleman's side will be depended on.\"\n\n\"A Lady who had on a Pink- Capuchin, edged with Ermine, a black\nPatch near her right eye, sat in a front seat in the next Side Box but\none to the Stage on Wednesday night at Drury Lane Playhouse; if that\nLady is single and willing to treat on terms of honour and generosity\nof a married state, it would be deemed a favour to receive a line\ndirected to C. D., at Clifford's Inn Old Coffee House, how she may be\naddress'd, being a serious affair.\"\n\n\"To be seen this week, in a large commodious room at the George Inn, in\nFenchurch-street, near Aldgate, the Porcupine Man and his Son, which\nhas given such great satisfaction to all that ever saw them: their\nsolid quills being not to be numbered nor credited till seen; but give\nuniversal satisfaction to all that ever saw them; the youth being\nallowed by all to be of a beautiful and fine complexion, and great\nnumbers resort daily to see them.\"\n\n\"A Bullfinch, that pipes 'Britons rouse up your great magnanimity,' at\ncommand, also talks, is to be sold at the Cane Shop facing New Broad\nstreet, Moorfields; likewise to be sold, two Starlings that whistle and\ntalk extremely plain.\n\n \"Great variety of fine long Walking Canes.\"\n\n\nTHEODORA DE VERDION.\n\nThis singular woman was born in 1744, at Leipsic, in Germany, and\ndied at her lodgings, in Upper Charles-street, Hatton Garden, London,\n1802. She was the only daughter of an architect of the name of Grahn,\nwho erected several edifices in the city of Berlin, particularly the\nChurch of St. Peter's. She wrote an excellent hand, and had learned\nthe mathematics, the French, Italian, and English languages, and\npossessed a complete knowledge of her native tongue. Upon her arrival\nin England she commenced teaching of the German language, under the\nname of Dr. John de Verdion.\n\nIn her exterior, she was extremely grotesque, wearing a bag wig, a\nlarge cocked hat, three or four folio books under one arm, and an\numbrella under the other, her pockets completely filled with small\nvolumes, and a stick in her right hand. She had a good knowledge of\nEnglish books; many persons entertained her for her advice relative\nto purchasing them. She obtained a comfortable subsistence from\nteaching and translating foreign languages, and by selling books\nchiefly in foreign literature. She taught the Duke of Portland the\nGerman language, and was always welcomed to his house, the Prussian\nAmbassador to our Court received from her a knowledge of the English\nlanguage; and several distinguished noblemen she frequently visited\nto instruct them in the French tongue; she also taught Edward Gibbon,\nthe celebrated Roman Historian, the German language, previous to his\nvisiting that country. This extraordinary female has never been known\nto have appeared in any other but the male dress, since her arrival in\nEngland, where she remained upwards of thirty years; and upon occasions\nshe would attend court, decked in very superb attire; and was well\nremembered about the streets of London; and particularly frequent in\nattending book auctions, and would buy to a large amount, sometimes a\ncoachload. Here her singular figure generally made her the jest of the\ncompany. Her general purchase at these sales was odd volumes, which she\nused to carry to other booksellers, and endeavour to sell, or exchange\nfor other books. She was also a considerable collector of medals\nand foreign coins of gold and silver; but none of these were found\nafter her decease. She frequented the Furnival's Inn Coffee-house,\nin Holborn, dining there almost every day; she would have the first\nof every thing in season, and was as strenuous for a large quantity,\nas she was dainty in the quality of what she chose for her table.\nAt times, it is well-known, she could dispense with three pounds of\nsolid meat; and we are very sorry to say, she was much inclined to the\ndreadful sin of drunkenness. Her death was occasioned by falling down\nstairs, and she was, after much affliction, at length compelled to make\nherself known to a German physician, who prescribed for her, when the\ndisorder she had, turned to a dropsy, defied all cure, and finished the\nlife of so remarkable a female.\n\n\nDRIVING STAGS LIKE CATTLE.\n\nBuried at Disley, Cheshire, June 2nd, 1753, Mr. Joseph Watson, in the\n105th year of his age. He was born at Moseley Common, in the parish of\nLeigh, in the county of Lancaster; and married his wife from Etchells,\nnear Manchester, in the said county. They were an happy couple 72\nyears. She died in the 94th year of her age. He was park-keeper to\nthe late Peter Leigh, Esq., of Lime, and his father used to drive and\nshow red deer to most of the nobility and gentry in that part of the\nkingdom, to the general satisfaction of all who ever saw them; for he\ncould have driven and commanded them at his pleasure, as if they had\nbeen common horned-cattle. In the reign of Queen Anne, Squire Leigh was\nat Macclesfield, in Cheshire, in company with a number of gentlemen,\namongst whom was Sir Roger Mason, who was then one of the members for\nthe said county; they being merry and free, Squire Leigh said his\nkeeper should drive 12 brace of stags to the Forest of Windsor, a\npresent to the Queen. Sir Roger opposed it with a wager of 500 guineas,\nsaying that neither his keeper, nor any other person, could drive 12\nbrace of red deer from Lime Park to Windsor Forest on any account. So\nSquire Leigh accepted the wager from Sir Roger, and immediately sent\na messenger to Lime for his keeper, who directly came to his master,\nwho told him he must immediately prepare himself to drive 12 brace\nof stags to Windsor Forest, for a wager of 500 guineas. He gave the\nSquire, his master, this answer, that he would, at his command, drive\nhim 12 brace of stags to Windsor Forest, or to any part of the kingdom\nby his worship's direction, or he would lose his life and fortune. He\nundertook, and accomplished this most astonishing performance, which\nis not to be equalled in the annals of the most ancient history. He\nwas a man of low stature, not bulky, of a fresh complexion, pleasant\ncountenance, and he believed he had drank a gallon of malt liquor a\nday, one day with another, for above sixty years of his time.\n\n\nECCENTRIC WILL.\n\nThe following will, as an exhibition of strange eccentricity, is not\ninappropriate to our pages. Mr. Tuke, of Wath, near Rotherham, who died\nin 1810, bequeathed one penny to every child that attended his funeral\n(there came from 600 to 700); 1s. to every poor woman in Wath; 10s.\n6d. to the ringers to ring one peal of grand bobs, which was to strike\noff while they were putting him into the grave. To seven of the oldest\nnavigators, one guinea for puddling him up in his grave. To his natural\ndaughter, L4 4s. per annum. To his old and faithful servant, Joseph\nPitt, L21 per annum. To an old woman who had for eleven years tucked\nhim up in bed, L1 1s. only. Forty dozen penny loaves to be thrown from\nthe church leads at twelve o'clock on Christmas day for ever. Two\nhandsome brass chandeliers for the church, and L20 for a set of new\nchimes.\n\n\nEXTRAORDINARY FROST.\n\nAs an instance of great rarity in England of the severity of a frost,\nit is worth notice, that in January, 1808, the rain froze as it fell,\nand in London the umbrellas were so stiffened that they could not be\nclosed. Birds had their feathers frozen so that they could not fly, and\nmany were picked up as they lay helpless on the ground.\n\n\nANCIENT SNUFF-BOXES.\n\nThese ancient snuff-boxes furnish proof of the love of our ancestors\nfor the titillating powder. An admiring writer of the last century,\nreflecting on the curious and precious caskets in which snuff was then\nimprisoned, asks--\n\n \"What strange and wondrous virtue must there be,\n And secret charm, O snuff! concealed in thee,\n That bounteous nature and inventive art,\n Bedecking thee thus all their powers exert.\"\n\n[Illustration: [++] Ancient Snuff Boxes.]\n\nBut every age, since snuff was in use, appears to have cherished\ngreat regard for the beauty and costliness of its snuff boxes, and\neven at the present time, the snuff box is the recognised vehicle of\nthe highest honour a corporation can bestow. Those here represented\nare not so much boxes as bottles. They are richly and elaborately\nornamented with sporting subjects, and no doubt once belonged to some\nfamous personage. Judging of their very antique form and figures, we\nare inclined to think they must have been in use earlier than it is\ngenerally supposed that snuff was introduced into this country.\n\n\nSEEING THE FIRST AND THE LAST OF TWO GENERATIONS.\n\nFrances Barton, of Horsley, Derbyshire, died 1789, aged 107. She\nfollowed the profession of a midwife during the long period of eighty\nyears. Her husband had been sexton of the parish seventy years; so that\nthis aged pair frequently remarked, that _she_ had twice brought into\nthe world, and _he_ had twice buried, the whole parish. Her faculties,\nher memory in particular, were remarkably good, so that she was enabled\nwell to remember the Revolution in 1688, and being present at a merry\nmaking on that glorious occasion.\n\n\nTHE EARLIEST HACKNEY-COACH.\n\n[Illustration: [++] The Earliest Hackney-Coach.]\n\nThe above is a correct representation of one of the earliest forms in\nwhich coaches for hire were first made. They were called Hackney, not,\nas is erroneously supposed, from their being first used to carry the\ncitizens of London to their villas in the suburb of Hackney, but from\nthe word \"hack,\" which signifies to offer any article for sale or hire.\nHackney coaches were first established in 1634, and the event is thus\nmentioned in one of _Strafford's Letters_, dated April in that year:--\n\n\"One Captain Bailey hath erected some four _Hackney-coaches_, put\nhis men in livery, and appointed them to stand, at the May-pole in\nthe Strand, giving them instructions at what rates to carry men into\nseveral parts of the town, where all day they may be had. Other\nhackney-men seeing this way, they flock to the same place, and perform\ntheir journeys at the same rate. So that sometimes there is twenty of\nthem together, which disperse up and down; that they and others are to\nbe had everywhere, as watermen are to be had by the water-side. * * *\nEverybody is much pleased with it.\"\n\n\nA UNIQUE LIBRARY.\n\nA singular library existed in 1535, at Warsenstein, near Cassel; the\nbooks composing it, or rather the substitutes for them, being made of\nwood, and every one of them is a specimen of some different tree. The\nback is formed of its bark, and the sides are constructed of polished\npieces of the same stock. When put together, the whole forms a box;\nand inside of it are stored the fruit, seed, and leaves, together with\nthe moss which grows on the trunk, and the insects which feed upon the\ntree; every volume corresponds in size, and the collection altogether\nhas an excellent effect.\n\n\nDRESS FORTY YEARS AGO.\n\nCaricature, even by its very exaggeration, often gives us a better idea\nof many things than the most exact sketches could do. This is more\nespecially the case with respect to dress, a proof of which is here\ngiven by the three caricatures which we now lay before our readers.\nThey are copied from plates published at the period to which they\nrefer, and how completely do they convey to us a notion of the fashions\nof the day!\n\n[Illustration: [++] Caricature of Dress.]\n\nWith the peace of 1815 commenced a new era in English history; and\nwithin the few years immediately preceding and following it, English\nsociety went through a remarkably rapid change; a change, as far as\nwe can see, of a decidedly favourable kind. The social condition of\npublic sentiment and public morals, literature, and science, were all\nimproved. As the violent internal agitation of the country during the\nregency increased the number of political caricatures and satirical\nwritings, so the succession of fashions, varying in extravagance,\nwhich characterised the same period, produced a greater number of\ncaricatures on dress and on fashionable manners than had been seen at\nany previous period. During the first twelve or fifteen years of the\npresent century, the general character of the costume appears not to\nhave undergone any great change. The two figures here given represent\nthe mode in 1810.\n\nA few years later the fashionable costume furnished an extraordinary\ncontrast with that just represented. The waist was again shortened, as\nwell as the frock and petticoat, and, instead of concealment, it seemed\nto be the aim of the ladies to exhibit to view as much of the body\nas possible. The s of 1819 and 1820 received the name of dandies,\nthe ladies that of dandizettes. The accompanying cut is from a rather\nbroadly caricatured print of a dandizette of the year 1819. It must\nbe considered only as a type of the general character of the foppish\ncostume of the period; for in no time was there ever such a variety of\nforms in the dresses of both sexes as at the period alluded to.\n\nWe give with the same reservation, a figure of a dandy, from a\ncaricature of the same year. The number of caricatures on the dandies\nand dandizettes, and on their fopperies and follies, during the years\n1819, 1820, and 1821, was perfectly astonishing.\n\n[Illustration: Dandizette.]\n\n[Illustration: Dandy.]\n\n\nFASHIONABLE DISFIGUREMENT.\n\n[Illustration: [++] Patching of the Face.]\n\nThe extent to which people may be led to disfigure themselves by a\nblind compliance with the fashion of the day, was never more strikingly\ndisplayed than in the custom of dotting the face with black patches of\ndifferent patterns. It might easily be supposed that the annexed sketch\nis a caricature, but such is not the case; it is a correct likeness of\na lady of the time of Charles the First, with her face in full dress.\nPatching was much admired during the reign of that sovereign, and for\nseveral succeeding years. Some authors think that the fashion came\noriginally from Arabia. No sooner was it brought to England and France,\nthan it became an absolute _fureur_. In the former country, old and\nyoung, the maiden of sixteen and the grey-haired grandmama, covered\ntheir faces with these black spots, shaped like suns, moons, stars,\nhearts, crosses, and lozenges; and some even, as in the instance\nbefore us, carried the mode to the extravagant extent of shaping the\npatches to represent a carriage and horses.\n\n\nA REMARKABLE OLD MAN.\n\nMr. Ingleby, of Battle Abbey, Sussex, died 1798, aged 117. He had\nbeen for upwards of ninety-five years a domestic in the family of\nLady Webster. The following narrative of this remarkable man is by a\ngentleman who visited him in the autumn of 1797:--\n\n\"To my great surprise,\" he says, \"I found Mr. Ingleby in a situation\nvery far removed from the luxuries of life, or the place which might\nbe deemed necessary for his years. He was in an antique outbuilding,\nnear the Castle Gate, where his table was spread under an arched roof;\nnearly the whole of the building being filled with billet-wood, and\nscarcely affording room for the oaken bench on which this wonder of\nlongevity was reclining by the fire. His dress was a full-bottomed\nwig, and a chocolate- suit of clothes with yellow buttons. His\nair and demeanour was pensive and solemn; though there was nothing\nin his look which impressed the mind with the idea of a person more\nthan fourscore years old, except a slight falling of the under jaw,\nwhich bespoke a more advanced age. We were introduced by a matron, who\nserved as a sort of interpreter between us--Mr. Ingleby's deafness\nnot permitting any regular conversation. When the nurse explained\nour errand, he replied, in a very distinct but hollow voice, 'I am\nmuch obliged to the gentlemen for the favour they do me; but I am not\nwell, and unable to converse with them.' He then turned his face to\nthe higher part of the bench on which he reclined, and was silent.\nIn each of his withered hands he held a short, rude, beechen walking\nstick, about three feet high, by the help of which he was accustomed\nnot only to walk about the extensive premises in which he passed the\nmost part of his life, but also to take his little rambles about the\ntown; and once (for, occasionally, the old gentleman was irascible,)\nhe set out on a pedestrian excursion to Hastings, _to inquire for\nanother situation in service_, because his patroness desired him to\nbe more attentive to personal neatness. It is but justice to the lady\nalluded to, to add, that the uncouth abode in which Mr. Ingleby dwelt\nwas the only one in which he could be persuaded to reside, and which\nlong familiarity had rendered dear to him. The choice appeared very\nextraordinary; but such persons, in their conduct, are seldom governed\nby the fixed and settled rules by which human life is ordinarily\nregulated.\"\n\n\nCURIOUS MANUSCRIPT.\n\nA very curious manuscript was presented to the Antiquarian Society\nof Yorkshire in 1828. It contains sundry rules to be observed by\nthe household of Henry the 8th, and enjoins the following singular\nparticulars:--\"None of his Highness's attendants to _steal_ any\nlocks, or keys, tables, forms, cupboards, or other furniture, out\nof noblemen's, or gentlemen's, houses where he goes to visit. No\nherald, minstrel, falconer, or other, to bring to the Court any boy or\n_rascal_; nor to keeps lads or rascals in Court to do their business\nfor them. Master cooks not to employ such scullions as shall go about\n_naked, or lie all night on the ground_ before the kitchen fire. Dinner\nto be at _ten_, and supper at _four_. The Knight Marshal to take\ncare that all such unthrifty and common women as follow the Court be\nbanished. The proper officers are, between six and seven o'clock every\nmorning, to make the fire in and _straw_ his Highness's Privy Chamber.\nOfficers of his Highness's Privy Chamber to keep secret every thing\nsaid or done, leaving hearkening or inquiring where the King is or\ngoes, be it early or late, without grudging, mumbling, or talking of\nthe King's past time, late or early going to bed, or any other matter.\nCoal only allowed to the King's, Queen's, and Lady Mary's Chambers.\nThe Queen's Maids of Honour to have a chet loaf, a manchet, a _gallon\nof ale_, and a chine of beef, for their _breakfasts_. Among the fishes\nfor the table is a porpoise, and if it is too big for a _horse-load_, a\nfurther allowance is made for it to the purveyor.\" The manuscript ends\nwith several proclamations. One is \"to take up and punish strong and\nmighty beggars, rascals, and vagabonds, who hang about the Court.\"\n\n\nWONDERFUL ESCAPE.\n\nIn 1809, a barge was going along the new cut from Paddington with\ncasks of spirits and barrels of gunpowder. It is supposed that one of\nthe crew bored a hole in a powder-barrel by mistake, meaning to steal\nspirits; the gimlet set fire to the powder, and eleven other barrels\nwere driven to the distance of 150 yards; but only the single barrel\nexploded.\n\n\nDAVID HUME ON HIS OWN DEATH.\n\nThe letter which we here lay before our readers was addressed by David\nHume to the Countess de Boufflers, and is supposed to be the last\nthat was ever written by that great historian, as he died only five\ndays afterwards, August 25th. With what calmness did that illustrious\nphilosopher contemplate the rapid approach of his own death!\n\nThe letter was torn at the places where the words are printed in\nitalics:\n\n \"Edinburgh, 20th of August, 1776.\n\n\"Tho' I am certainly within a few weeks, dear Madam, and perhaps within\na few days, of my own death, I could not forbear being struck with the\ndeath of the Prince of Conti, so great a loss in every particular. My\nreflection carried me immediately to your situation in this melancholy\nincident. What a difference to you in your whole plan of life! Pray,\nwrite me some particulars; but in such terms that you need not care, in\n_event_ of decease, into whose hands your letter may fall.\n\n\"_My_ distemper is a diarrhoea, or disorder in my bowels, which has\n_been_ gradually undermining me these two years; but within these six\nmonths has been visibly hastening me to my end. I see death approach\ngradually without any anxiety or regret. I salute you with great\naffection and regard for the last time.\n\n \"DAVID HUME.\"\n\n\nSCRIPTURAL ANTIQUITIES.\n\n[Illustration: A. Drum, or Timbrel, of Baked Potter's Clay.--AA. Drum\nin use in the East.--B. Harp.--C. Lutes.--D. Inscribed Stone.--E.\nSandals.]\n\nThe rude musical instruments here represented, have been collected by\nmodern travellers, and are but little changed from the ancient forms.\nThe drum or timbrel marked A, is made of thin baked clay, something in\nthe shape of a bottle, with parchment stretched over the wider part. On\nbeing struck with the finger, this instrument makes a remarkably loud\nsound. These relics are lodged in the London Scriptural Museum, and are\nall ticketed with the texts they serve to illustrate. This arrangement\nis very judicious, and gives a great additional interest to the sacred\nobjects while under inspection.\n\n[Illustration: 1. Distaff.--2. Roman Farthing.--3. Stone Money\nWeights.--4. Hand Mill. 5. Eastern Wine and Water Bottles.]\n\nThe distaff was the instrument which wrought the materials for the\nrobes of the Egyptian Kings, and for the \"little coat\" which Hannah\nmade for Samuel; by it, too, were wrought the cloths, and other fabrics\nused in Solomon's temple. By reference to the above engraving, it will\nbe seen that nothing can be more simple than this ancient instrument,\nwhich is a sort of wooden skewer, round which the flax is wrapped; it\nis then spun on the ground in the same manner as a boy's top, and the\nthread wrought off, and wound upon a reel shown in the foreground of\nthe picture. \"Querns,\" or stone hand-mills of various sizes, similar\nto that represented in our engraving, have been repeatedly found\nin connection with Roman, Saxon, and other ancient remains in this\ncountry. They are still to be met with in constant use over the greater\npart of India, in Africa, and also those districts of the East which\nare more particularly associated with Holy Writ. It may be worth while\nto mention that this description of mill is an improvement upon the\nmethod of simply crushing the corn laid on a flat stone with another\nheld in the hand. The \"Quern\" is a hard stone roughly rounded, and\npartly hollowed, into which another stone, which has a handle, is\nloosely fitted. The corn required to be ground is placed in the hollow\nreceptacle, and the inner stone is moved rapidly round, and, in course\nof time, by immense labour, the wheat &c. is ground into flour. The\nScripture prophecies mention that of two women grinding at the mill,\none shall be left, and the other taken--the two-handled mill will\nexplain the meaning of this passage.\n\n\nCHRONOLOGICAL TABLE OF REMARKABLE EVENTS.\n\nThe following curious table is extracted _literatim_ from Arthur\nHopton's _Concordancie of Years_, 1615:--\n\n1077.--A blazing star on Palm Sunday, nere the sun.\n\n1100.--The yard (measure) made by Henry I.\n\n1116.--The moone seemed turned into bloud.\n\n1128.--Men wore haire like women.\n\n1180.--Paris in France, and London in Englande, paued, and\n thatching in both left, because all Luberick was spoiled\n thereby with fire.\n\n1189.--Robin Hood and Little John lived. This yeare London\n obtained to be gouerned by sheriffes and maiors.\n\n1205.--By reason of a frost from January to March wheate was sold\n for a marke the quarter, which before was at 12 pence. _Anno\n Regni_ 6. John.\n\n1209.--London bridge builded with stone; and this yeare the\n citizens of London had a grant to choose them a maior.\n\n1227.--The citizens of London had libertie to hunt a certain\n distance about the citie, and to passe toll-free through\n England.\n\n1231.--Thunder lasted fifteen daies; beginning the morrow after\n St. Martin's day.\n\n1233.--Four sunnes appeared, beside the true sunne, of a red colour.\n\n1235.--The Jews of Norwich stole a boy and circumcised him,\n minding to have crucified him at Easter.\n\n1247.--The king farmed Queene-hiue for fifty pounds per annum, to\n the citizens.\n\n1252.--Great tempests upon the sea, and fearful; and this year the\n king (Henry III.) granted, that wheretofore the citizens of\n London were to present the maior before the king, wheresoeuer\n he were, that now barons of the exchequer should serue.\n\n1292.--The Jewes corrupting England with vsury, had first a badge\n giuen them to weare, that they might be knowne, and after were\n banished to the number of 150,000 persons.\n\n\n1313.--This yeare the king of France burned all his leporous and\n pocky people, as well men as women: for that he supposed they\n had poysoned the waters, which caused his leprosie. About this\n time, also, the Jews had a purpose to poyson all the\n Christians, by poysoning all their springs.\n\n1361.--Men and beasts perished in diuers places with thunder and\n lightning, and fiends were seene speake unto men as they\n trauelled.\n\n1372.--The first bailiffes, in Shrewsbury.\n\n1386.--The making of gunnes found; and rebels in Kent and Essex, who\n entred London, beheaded all lawyers, and burnt houses and all\n bookes of law.\n\n1388.--Picked shooes, tyed to their knees with siluer chaines were\n vsed. And women with long gownes rode in side-saddles, like\n the queene, that brought side-saddles first to England; for\n before they rode astrid.\n\n1401.--Pride exceeding in monstrous apparrell.\n\n1411.--Guildhall in London begun.\n\n1417.--A decree for lantherne and candle-light in London.\n\n1427.--Rain from the 1st of Aprill to Hollontide.\n\n1510.--St. John's College in Cambridge being an ancient hostell, was\n conuerted to a college by the executors of the Countesse of\n Richmond and Derby, and mother of Henry VII., in this yeare, as\n her will was.\n\n1552.--The new service book in English.\n\n1555.--The first use of coaches in England.\n\n1606.--The cawsies about London taken down.\n\n1610.--Britaines Bursse builded. Hix Hall builded. Aldgate builded\n new. Sutton's Hospitall founded. Moore fields new railed and\n planted with trees. Westminster palace paued.\n\n\nCOCK-FIGHTING AT SCHOOLS.\n\nMany years ago the scholars at our large schools had regular\ncock-fights, which would appear to have been an affair of the school,\nrecognised by the masters, and the charges for which were defrayed\nby them, to be afterwards paid by the parents, just as some innocent\nexcursions and festivities are managed now a days. The credit of the\nschool was, without doubt, often involved in the proper issue of the\nfight.\n\nSir James Mackintosh, when at school at Fortrose in 1776-7, had this\nentry in his account, in which books were charged 3s. 6d.:--\n\n To cocks'-fight dues for 2 years, 2s. 6d. each, 5s.\n\nAssociated are three months' fees at the dancing-school, minuet,\ncountry-dances, and hornpipe, &c. Cock-fighting up to the end of\nthe last century was a very general amusement, and an occasion for\ngambling. It entered into the occupations of the old and young.\nTravellers agreed with coachmen that they were to wait a night if\nthere was a cock-fight in any town through which they passed. A battle\nbetween two cocks had five guineas staked upon it. Fifty guineas, about\nthe year 1760, depended upon the main or odd battle. This made the\ndecision of a \"long main,\" at cock-fighting an important matter. The\nchurch bells at times announced the winning of a \"long main.\" Matches\nwere sometimes so arranged as to last the week. When country gentlemen\nhad sat long at table, and the conversation had turned upon the\nrelative merits of their several birds, a cock-fight often resulted, as\nthe birds in question were brought for the purpose into the dining-room.\n\n\nCOMMON TRAVELLING.\n\n[Illustration: [++] Common Travelling Dress.]\n\nWe have here the common travelling dress in use at the commencement\nof the 12th century, _tempus_ Henry I. and Stephen. The original is\nintended for the Saviour meeting the two disciples on the road to\nEmmaus. The Saviour wears an under tunic, and his mantle, fastened by a\nnarrow band across the chest, is held up by the right hand. The figures\nof the disciples are, however, the most curious, the central one\nparticularly so, as he would seem to wear a dress expressly invented\nfor travelling: his large round hat, with its wide brim, seems to be\nthe original of the pilgrim's hat so well known in later times, and\nwhich formed so distinguishing a mark in their costume. His short green\ntunic, well adapted for journeying, is protected by a capacious mantle\nof skin, provided with a \"capa\" or cowl, to draw over the head, and\nwhich was frequently used instead of a hat. He wears white breeches\nornamented with red cross-stripes; they end at the ankle, where they\nare secured by a band or garter, the foot being covered by close shoes.\nHis companion wears the common cap so frequently met with, and he has\nhis face ornamented to profusion by moustaches and beard, each lock\nof which appears to be most carefully separated and arranged in the\nnicest order. He has an under-tunic of white, and an upper one of red,\nand a white mantle bordered with gold; he also wears the same kind of\nbreeches, reaching to the ankle, but he has no shoes, which frequently\nappears to have been the case when persons were on a journey.\n\n\nFASHIONABLE DANCES OF THE LAST CENTURY.\n\nThe style of dancing which was fashionable at the latter part of the\nlast century, may be seen from the following advertisement from a\ndancing-master, which we have copied from a newspaper of the year\n1775:--\n\n\"At Duke's Long Room, in Paternoster Row, Grown Gentlemen or Ladies\nare taught a Minuet, or the Method of Country Dances, with the modern\nMethod of Footing; and that in the genteelest, and most expeditious,\nand private Manner. And for the greater expedition of such gentlemen,\nas chuse to dance in company, there's a complete Set of Gentlemen\nassembled every Monday and Wednesday evening for the said purpose.\nGentlemen or Ladies may be waited on at their own Houses by favouring\nme with a line directed as above. Likewise to be had at my House, as\nabove, a Book of Instructions for the figuring part of Country Dances,\nwith the Figure of the Minuet annex'd thereon, drawn out in Characters,\nand laid down in such a Manner, that at once casting your Eye on it,\nyou see the Figure directly form'd as it is to be done; so that a\nperson, even that had never learnt, might, by the help of this book,\nsoon make himself Master of the figuring Part. Such as reside in the\nCountry, I doubt not, would find it of immediate Service, as they have\nnot always an Opportunity of having Recourse to a Dancing Master. Price\n10s. 6d. N. Dukes, Dancing Master.\"\n\n\nPREACHING FRIARS.\n\n[Illustration: [++] Preaching Friar.]\n\nIn the romance of \"St. Graal,\" executed in the fourteenth century, we\nhave this representation of one of these preaching friars in his rude\nportable pulpit. From the contrast afforded by their mendicancy, and\nenthusiasm in teaching, to the pride and riches of the higher clergy,\nand their constant mixing with the people, they became excessively\npopular. The preacher in the cut has a crowded and attentive audience\n(though one lady seems inclined to nap); the costume of the entire\nbody, who are all seated, after a primitive fashion, on the bare\nground, is worthy of note, and may be received as a fair picture of the\ncommonalty of England about the year 1350.\n\n\nTHE ECCENTRIC LADY LEWSON.\n\nMrs. Jane Lewson, widow, of No. 12, Coldbath Square, London, died 1816,\naged 116. Mrs. Lewson, from the very eccentric style of her dress,\nwas almost universally recognised as _Lady_ Lewson. She was born in\nEssex Street, Strand, in the year 1700, during the reign of William\nand Mary; and was married at an early age to a wealthy gentleman then\nliving in the house in which she died. She became a widow at the early\nage of 26, having only one child, a daughter, living at the time. Mrs.\nLewson being left by her husband in affluent circumstances, though\nshe had many suitors, preferred to remain in a state of widowhood.\nWhen her daughter married, being left alone, she became very fond of\nretirement, and rarely went out or permitted the visits of any person.\nFor the last thirty years of her life she had kept no servant, except\none old female, who died in 1806; she was succeeded by the old woman's\ngranddaughter, who was married about 1813; and she was followed in\nthe situation by an old man, who attended the different houses in\nthe square to go on errands, clean shoes, &c. Mrs. Lewson took this\nman into her house, and he acted as her steward, butler, cook, and\nhousemaid; and with the exception of two old lap-dogs and a cat, he was\nher only companion. The house she occupied was elegantly furnished,\nbut after the old style; the beds were kept constantly made, although\nthey had not been slept in for about fifty years. Her apartment was\nonly occasionally swept out, but never washed; the windows were so\nencrusted with dirt that they hardly admitted a ray of light to pass\nthrough them. She had used to tell her acquaintances that if the rooms\nwere wetted, it might be the occasion of her taking cold; and as to\ncleaning the windows, she observed that many accidents happened through\nthat ridiculous practice; the glass might be broke, and the person\nwounded, when the expense of repairing the one, and curing the other,\nwould both fall upon her. A large garden at the rear of the house was\nthe only thing connected with her establishment to which she really\npaid attention. This was always kept in good order; and here, when\nthe weather permitted, she enjoyed the air, or sometimes sat and read\nby way of pastime; or else chatted on times past with any of the few\nremaining acquaintances whose visits she permitted. She seldom visited\nany person except Mr. Jones, a grocer at the corner of the square,\nwith whom she dealt. She was so partial to the fashions prevailing in\nher youthful days, that she never changed the manner of her dress from\nthat worn by ladies in the reign of George the First. She always wore\npowder with a large _toupee_ made of horsehair on her head, nearly\nhalf a foot high, over which her front hair was turned up; a cap over\nit, which knotted under the chin, and three or four curls hanging down\nher neck. She generally wore silk gowns, the train long with a deep\nflounce all round, a very long narrow waist, very tightly laced up to\nher neck, round which was a ruff or frill. The sleeves of her gown, to\nwhich four or five large ruffles were attached, came below the elbow;\na large straw bonnet, quite flat, high-heeled shoes, a full-made black\nsilk cloak trimmed round with lace, and a gold-headed cane, completed\nher every-day costume for the last eighty years of her life, and in\nwhich habiliments she occasionally walked round the square, when she\nwas uniformly spoken of by all spectators as _Lady Lewson_. She never\npractised ablutions of any kind, or hardly in any degree, because, as\nshe alleged, those persons who washed themselves were always taking\ncold, or laying the foundation of some dreadful disorder. Her method\nwas to besmear her face and neck all over with hog's lard, because\nthat was soft and lubricating; and then, because she required a little\ncolour in her cheeks to set off her person to advantage, she had used\nto paint them with rose-pink. Her manner of living was so methodical,\nthat she would not take her tea out of any other than a favourite\ncup. She was equally particular with respect to her knives, forks,\nplates, &c. At breakfast she arranged, in a particular manner, the\nparaphernalia of her table: at dinner she always observed a particular\nrule as to the placing of the two or three empty chairs, by which the\ntable was surrounded, but herself always sat in one favourite chair.\nShe constantly enjoyed an excellent state of health; assisted at all\ntimes in regulating the affairs of her household; and never, until a\nlittle previous to her decease, had an hour's illness. She entertained\nthe greatest aversion to medicine; and, what is remarkable, cut two\nnew teeth at the age of 87, and was never troubled with the toothache.\nTowards the close of her life her sight failed her. She lived in five\nreigns, and was believed to be the most faithful living chronicler of\nthe age. A few days previous to her decease, an old lady who was her\nneighbour died suddenly, which had such an effect upon her that she\nfrequently said her time was also come, and she should soon follow.\nShe enjoyed the use of all her faculties till that period, when she\nbecame weak and took to her bed; but steadily refused all medical aid.\nHer conduct to a few relations was extremely capricious; and she would\nnever see any of them; and it was not until a few hours before her\ndissolution that any relaxation in her temper was manifested. She was\ninterred in Bunhill Fields burying-ground.\n\n\nWHEN FIRE ENGINES WERE FIRST MADE.\n\nThe Phoenix was the first fire-office established, in 1682. There were\nused, in towns, squirts or syringes, for extinguishing fire, which\ndid not exceed two or three feet in length. These yielded to the Fire\nEngine, with leathern pipes, which was patented in 1676. Water-tight,\nseamless hose was made in Bethnal Green in 1720. About this date--\n\n L s. d.\n A fire engine and pipe for Lyme cost 6 0 0\n A square pipe, 23 feet long 1 18 0\n 12 leather fire-buckets 2 3 3\n\nA Fire Engine was considered an appropriate present for an aspirant to\na borough. At Lewes, in 1726, T. Pelham, Esq., gave one, and having\nbeen chosen representative in 1731, he presented a second.\n\n\nEXTRAORDINARY CATARACT.\n\n[Illustration: [++] Cataract in Pulo Penang.]\n\nIn the Island of Pulo Penang, in the Straits of Malacca, there is a\ncataract which is surpassed by very few in the four quarters of the\nearth. It is rarely visited, and, therefore, has been but seldom\ndescribed; but those who have been fortunate enough to witness it all\nagree in the opinion that it forms one of the wonders of the world.\nThe stream which supplies it is of considerable volume, and after\ntraversing a long tract of comparatively level country, is suddenly\nprecipitated almost without a break into a ravine nearly two hundred\nfeet below the summit of the fall. The annexed engraving gives an\nexcellent representation of the scene. The stream descends with a\nmighty roar, and rushes on with a lightning speed. If you take the\ntrouble of bringing a small looking-glass in your pocket, and come\nhere about an hour before noon, you will be able to produce some very\nbeautiful artificial rainbows. But, whatever you do, never attempt to\nclamber to the top of the rocks; for though, doubtless, the scenery\nis very sublime up there, the pathway is slippery and dangerous in\nthe extreme; and the guides can tell how two hapless youths, officers\nbelonging to a regiment stationed here some twenty years ago, clambered\nup that hill, and how they shouted with triumph on reaching yon summit,\nand waved their handkerchiefs bravely; but they can also tell the\ngloomy and disastrous end of all this; how the wild screams echoed far\nand wide, as both slipped and fell headlong into the surging torrent,\nand the sun shone brightly upon the bright red uniforms as they were\nhurried over the precipice, and dashed from rock to rock; and, whilst\nyet the horror-stricken spectators gazed with speechless agony and\nterror, the bodies of the poor young men were borne away and hid by the\nblood-stained waters from human recovery.\n\n\nDANCES OF THE NATIVES IN NEW SOUTH WALES.\n\n[Illustration: [++] Festival Dance.]\n\nThe manners and customs of the uncivilized are always legitimate\nobjects of wonder and curiosity to the civilized. It is on this account\nthat we give the above sketch of one of the festival dances of the\nnatives of Australia.\n\nThese dances are not only the usual close of their combats, but are\nfrequent in time of peace. They appear almost necessary to stir up\ntheir blood; and under the excitement they produce, the whole nature\nof the people seems to be changed. To a spectator the effect of one of\nthese exhibitions almost equals that of a tragic melo-drama.\n\nA suitable place for the performance is selected in the neighbourhood\nof their huts. Here a fire is built by the women and boys, while\nsuch of the men as are to take a share in the exhibition, usually\nabout twenty in number, disappear to arrange their persons. When\nthese preparations are completed, and the fire burns brightly, the\nperformers are seen advancing in the guise of as many skeletons. This\neffect is produced by means of pipe clay, with which they paint broad\nwhite lines on their arms and legs, and on the head, while others of\nless breadth are drawn across the body, to correspond to the ribs. The\nmusic consists in beating time on their shields, and singing, and to it\nthe movements of the dancers conform. It must not be supposed that this\nexhibition is a dance in our sense of the word. It consists of violent\nand odd movements of the arms, legs, and body, contortions and violent\nmuscular actions, amounting almost to frenzy. The performers appear\nmore like a child's pasteboard supple-jack than anything human in their\nmovements.\n\nThis action continues for a time, and then the skeletons, for so they\nappear to be, since they truly resemble them, suddenly seem to vanish\nand reappear. The disappearance is effected by merely turning round,\nfor the figures are painted only in front, and their dusky forms are\nlost by mingling with the dark background. The trees, illuminated by\nthe fire, are brought out with some of the figures in bold relief,\nwhile others were indistinct and ghost-like. All concurs to give an\nair of wildness to the strange scene. As the dance proceeds, the\nexcitement increases, and those who a short time before appear only\nhalf alive, become full of animation, and finally are obliged to stop\nfrom exhaustion.\n\n\nA PUDDING AS AN ADVERTISEMENT.\n\nThe following fact is interesting, inasmuch as it gives us an insight\ninto the popular tastes of the period, and the power of mob-law:--\n\nIn 1718, James Austin, inventor of the Persian ink powder, invited his\ncustomers to a feast. There was a pudding promised, which was to be\nboiled fourteen days, instead of seven hours, and for which he allowed\na chaldron of coals. It weighed 900 pounds. The copper for boiling it\nwas erected at the Red Lion in Southwark Park, where crowds went to\nsee it; and when boiled, it was to be conveyed to the Swan Tavern,\nFish Street Hill, to the tune of \"What lumps of pudding my mother gave\nme.\" The place was changed to the Restoration Gardens in St. George's\nFields, in consequence of the numerous company expected, and the\npudding set out in procession with banners, streamers, drums, &c., but\nthe mob chased it on the way and carried all off.\n\n\nTHE DESOLATION OF EYAM.\n\nThe ancient custom of hanging a garland of white roses, made of writing\npaper, and a pair of white gloves over the pew of the unmarried\nvillagers who die in the flower of their age, prevailed up to the year\n1837 in the village of Eyam, and in most other villages and little\ntowns in the Peak of Derbyshire. In the year 1665, the plague was\nconveyed to this unfortunate village, which for a time had been chiefly\nconfined to London. The infection, it appears, was carried in a box of\nwoollen clothes; the tailor, to whom they were directed was, together\nwith his family, the immediate victims of this fatal importation;\nand a few days sufficed to confirm the fact, that the entire hamlet\nwas deeply infected. A general panic ensued, the worthy and truly\nchristian Rector, the Rev. William Mompesson, at this eventful and\nawful crisis, summoned the parish, and after energetically stating the\ncase, and declaring his decided intention of remaining at his post,\ninduced his hearers to adopt the measures he was about to propose, if\nnot for their own preservation, at least for the more important cause,\nthe preservation of the surrounding country. Eyam, from this moment,\nlike a besieged city, was cut off from the living world, and to the\nzeal and fidelity of this ever-to-be-respected minister was confided\nthe present, as well as eternal welfare of those who were about to\nprove to posterity, that devotion to their country, as well as to\ntheir God, was combined in the truly christian creed taught them by\nthis reverend man. But alas! it was the will of the Almighty that the\nranks of this devoted flock should be rapidly thinned, though Mr. and\nMrs. Mompesson had been hitherto spared; but in August, the latter\nwas carried off by the fatal disease, in the 27th year of her age;\nher monument may still be seen at no great distance from the chancel\ndoor. A number of grave-stones, bearing date 1666, in the churchyard,\nshow that for a time, at least, the dead had been deposited there in\nthe usual manner. Soon after the death of Mrs. Mompesson, the disorder\nbegan to abate, and in about two months might be said to have entirely\nceased. The pious and amiable Rector was graciously preserved.\n\n\nCURIOUS PLAY BILL.\n\nThe following remarkable theatrical announcement is worth preservation,\ninasmuch as it forms a curious effusion of vanity and poverty, in the\nshape of an appeal to the taste and feelings of the inhabitants of a\ntown in Sussex:--\n\n (_Copy._)\n\nAt the old theatre in East Grinstead, on Saturday, May 5th, 1758, will\nbe represented (by particular desire, and for the benefit of Mrs. P.)\nthe deep and affecting Tragedy of Theodosius, or the Force of Love,\nwith magnificent scenes, dresses, &c.\n\nVaranes, by Mr. P., who will strive, as far as possible, to support the\ncharacter of this fiery Persian Prince, in which he was so much admired\nand applauded at Hastings, Arundel, Petworth, Midworth, Lewes, &c.\n\nTheodosius, by a young gentleman from the university of Oxford, who\nnever appeared on any stage.\n\nAthenais, by Mrs. P. Though her present condition will not permit her\nto wait on gentlemen and ladies out of the town with tickets, she\nhopes, as on former occasions, for their liberality and support.\n\nNothing in Italy can exceed the altar, in the first scene of the play.\nNevertheless, should any of the Nobility or Gentry wish to see it\nornamented with flowers, the bearer will bring away as many as they\nchoose to favour him with.\n\nAs the coronation of Athenais, to be introduced in the fifth act,\ncontains a number of personages, more than sufficient to fill all\nthe dressing rooms, &c., it is hoped no gentlemen and ladies will be\noffended at being refused admission behind the scenes.\n\nN.B. The great yard dog, that made so much noise on Thursday night,\nduring the last act of King Richard the Third, will be sent to a\nneighbour's over the way; and on account of the prodigious demand for\nplaces, part of the stable will be laid into the boxes on one side, and\nthe granary be open for the same purpose on the other.\n\n _Vivat Rex._\n\n\nTHE EAR OF BIRDS NOT TO BE DECEIVED.\n\nThe sense of hearing in birds is singularly acute, and their instinct\nleads them instantly to detect the slightest variation in the song\nof those of their own kind. The following is a laughable instance of\nthis:--\n\nA bird-catcher, wishing to increase his stock of bullfinches, took\nout his caged bird and his limed twigs, and placed them in such a\nsituation of hedge and bush as he judged favourable to his success. It\nso happened that his own bird was one of education, such as is usually\ntermed a piping bullfinch. In the first instance a few accidentally\nthrown out natural notes, or calls, had attracted three or four of his\nkindred feather, which had now taken their station not far distant\nfrom the cage. There they stood in doubt and curiosity, and presently\nmoving inch by inch, and hop by hop towards him and the fatal twigs,\nthey again became stationary and attentive. It was in this eager and\nsuspended moment that the piping bullfinch set up the old country-dance\nof \"Nancy Dawson.\" Away flew every astounded bullfinch as fast as wings\ncould move, in such alarm and confusion as bullfinches could feel and\nthey only can venture to describe.\n\n\nFLYING COACH.\n\nIf the _Exeter Flying Stage_ arrived from London at Dorchester in two\ndays, and at Exeter at the end of the third day, about 1739, the speed\nmust have been considered surprising. Those who made use of such a\nconveyance were doubtless looked upon as presumptuous, neck-or-nothing\nmortals.\n\nThere was a \"Devizes chaise\" from London at this time which took a\nroute through Reading, Newbury, and Marlborough.\n\nThere is a good house at Morcomb Lake, east of Charmouth, now no longer\nin the road, owing to this having been diverted. This was a road-side\ninn, where the judges slept. The Fly Coach from London to Exeter\n_slept_ there the fifth night from town. The coach proceeded the next\nmorning to Axminster, where _it_ breakfasted, and there a woman barber\n_shaved the coach_.\n\n\nAN AGED SPIRIT DRINKER.\n\nDaniel Bull M'Carthy, of the county of Kerry, Ireland, died 1752, aged\n111. At the age of eighty-four he married a fifth wife, a girl little\nmore than fourteen years of age, by whom he had twenty children--one\nevery subsequent year of his life. It was remarked that he was scarcely\never seen to expectorate; nor did any extent of cold ever seem to\naffect him. For the last seventy years of his life, when in company, he\ndrank plentifully of rum and brandy, which he always took neat; and, if\nin compliance with solicitations he took wine or punch, always drank\nan equal sized glass of rum or brandy, which he designated _a wedge_.\nThe temperature of his body was generally so hot that he could bear but\nlittle clothing, either by day or night upon his person.\n\n\nGIANT TREE.\n\n[Illustration: [++] Giant Tree in Pulo Penang.]\n\nThere are few trees in the world like the giant tree in the island\nof Pulo Penang, of which the annexed engraving is a correct\nrepresentation. It is one of the various kinds of palm, and some idea\nmay be formed of its height from the fact that it is twice as tall,\nand quite as straight, as the mainmast of a line-of-battle ship; there\nare no branches, no twigs anywhere to be seen, save just at the very\nsummit, and here they bend over gracefully, something like what one\nwould imagine a large-sized palm-tree to be if gazed at through Lord\nRosse's telescope. It is a only specimen of its kind to be met with in\nthe whole island.\n\n\nPUNISHING FALSE ACCUSERS.\n\nWisdom may sometimes be learned at a Quarter Sessions, and it would\nbe advantageous if we occasionally took a hint from our ancestors.\nThe magistrates at sessions in Charles the First's reign could and\ndid address themselves to questions arising between parties moving in\nhumble life, very important to them, and who could now-a-day in vain\nseek redress in the same quarter. A modern Bridget might continue\nto charge men with a breach of promise of marriage without legal\nmeasures being available against her. This was not so in 1626. Her\ncase was considered, and her injurious conduct and mode of life were\nduly estimated, with what result we shall learn from the following\nentry in the minute book of a quarter sessions in Devonshire of that\ndate:--\"Forasmuch as it hath appeared unto this Court that Bridget\nHowsley of Langton, spinster, liveth idly and lewdly at home, not\nbetaking herself to any honest course of life, and hath lately falsely\nand scandalously accused one [left blank in the original] of Honiton,\nin Devon, challenging a promise of marriage from him, which tended\nmuch to his disgrace, and that she is a continual brawler and sower\nof strife and debate between her neighbours, inhabitants of Langton\naforesaid, this court doth therefore think fit and order that the said\nBridget Howsley be forthwith committed to the House of Correction,\nthere to be set on work and remain for the space of six whole months,\nand from thenceforth until she shall find very good sureties for her\nappearance at the next Sessions, after the said six months shall be\nexpired, or until she shall procure a master that will take her into\nservice.\"\n\n\nA PHASE OF THE SOUTHCOTTIAN DELUSION.\n\nOne of the most remarkable cases on record of combined knavery,\ncredulity, and superstition, is the belief which so extensively\nprevailed about fifty years ago in the mission and doctrines of Joanna\nSouthcott, and of which, strange to say, some traces remain even to\nthe present day. Is it not astonishing that so recently as the year\n1814, August 3rd, the following paragraph--which we believe gives a\ncorrect statement of the facts--should have appeared in the _Courier_\nnewspaper? \"Joanna Southcott has lately given out that she expects in\na few weeks to become the mother of the true Messiah. She is nearly\nseventy years of age. A cradle of most expensive and magnificent\nmaterials has been bespoken by a lady of fortune for the accouchement,\nand has been for some days exhibited at the warehouse of an eminent\ncabinet maker in Aldersgate-street. Hundreds of genteel persons of both\nsexes have been to see this cradle, in which her followers believe the\ntrue Messiah is to be rocked. The following has been given us as a\ncorrect description: 'A child's crib, three feet six inches, by two\nfeet, of satin wood, with brass trellis, side and foot board; turned\nfeet, carved and gilt, on castors; a swing cot, inside caned, to swing\non centre; at each end gilt mouldings, top and bottom for gold letters;\na canopy cover, with blue silk; carved and gilt under it, a gold ball,\nand dove, and olive branch; green stars at each corner, gilt; blue silk\nfurniture; an embroidered celestial crown, with Hebrew characters, gold\nletters; a lambs'-wool mattress, with white fustian down bed, down\npillow, and two superfine blankets.'\"\n\n\nHOUSEHOLD EXPENSES OF KING EDWARD THE FIRST.\n\nEdward the First kept three Christmasses at Rhuddlan castle, in\nFlintshire; and it is a fact not generally known, that his queen\nEleanor, exclusively of the young prince Edward, born at Caernarvon,\nwas delivered of a princess there in 1283. This shows that his entire\nhousehold must have been transferred into Wales, at the time his\npolicy was directed to complete the annexation of the principality of\nWales to that of England. In an ancient record in the tower of London,\ndated 1281-2, and translated by Samuel Lysons, Esq., is a curious roll\nof Edward's expenses when at Rhuddlan. It consists of four sheets,\ncontaining the particulars, under proper heads, of the sums of money\npaid for the maintenance of his household. The sum of the expenses in\nthis roll is L1,395 10s., which sum, with the expenses of the other\nroll of the queen's household is L2,220 2s. 10-1\/2d. The roll is very\ncurious, but too long to be inserted here. We append the following as a\nspecimen of the various items it contains:--\n\n Paid on the day of the queen's churching in oblations to\n mass L0 3 0\n The queen's gift to divers minstrels attending her churching 10 0 0\n The queen's gift to a female spy 0 1 0\n A certain female spy, to purchase her a house as a spy 1 0 0\n For the brethren at the hospital at Rhuddlan 0 1 1\n For a certain player as a gift 0 8 0\n For the celebration of mass for the soul of William de Bajor 0 1 10\n For the messenger carrying letters to the king at London,\n to be sent to the court of Rome, for his expenses 0 1 0\n Paid sundry bailiffs at the castle 0 4 10\n For the carriage of 80 casks of wine from the water to\n the castle 0 22 0\n For a cart bringing lances and cross bows from Ruthlan\n to Hope 0 1 4\n For the carriage of L3,000 from the king's wardrobe to\n the queen's wardrobe 0 10 5\n For 600 turves, to place about the queen's stew pond in\n the castle 0 1 0\n Carriage of figs and raisins to Aberconway 0 0 1\n Paid wages for 1,060 archers at twopence, with 53 captains\n at fourpence, with 10 constables of cavalry at 12d.\n a day 68 8 6\n Paid the same for 1,040 archers, &c. &c. 67 4 0\n\n\nGARRICK'S CUP.\n\n[Illustration [++] Garrick's Cup.]\n\nThis celebrated Shakspearean relic was presented to David Garrick,\nby the Mayor and Corporation of Stratford-upon-Avon, in September,\n1769, at the Jubilee which he instituted in honour of his favourite\nBard. It measures about 11 inches in height. The tree from which it\nis carved was planted by Shakspeare's own hand, in the year 1609,\nand after having stood 147 years, was, in an evil hour, and when at\nits full growth and remarkably large, cut down, and cleft to pieces\nfor fire-wood, by order of the Rev. Francis Gastrell, to whom it had\nbecome an object of dislike, from its subjecting him to the frequent\nimportunities of travellers. Fortunately, the greater part of it fell\ninto the possession of Mr. Thomas Sharp, a watchmaker of Stratford,\nwho, \"out of sincere veneration\" for the memory of its immortal\nplanter, and well knowing the value the world set upon it, converted\nthe fragments to uses widely differing from that to which they had been\nso sacrilegiously condemned. Garrick held this cup in his hand at the\nJubilee, while he sung the beautiful and well-known air, which he had\ncomposed for the occasion, beginning\n\n \"Behold this fair goblet, 'twas carved from the tree,\n Which, O my sweet Shakspeare, was planted by thee;\n As a relic I kiss it, and bow at the shrine,\n What comes from thy hand must be ever divine!\n All shall yield to the Mulberry tree,\n Bend to thee,\n Blest Mulberry;\n Matchless was he\n Who planted thee,\n And thou like him immortal be!\"\n\n\nQUICK WORK.\n\nMr. John Coxetter, of Greenham Mills, Newbury, had two South down sheep\nshorn at his factory exactly at five o'clock in the morning, from the\nwool of which, after passing its various processes, a complete damson\n coat was made, and worn by Sir John Throckmorton, at a quarter\npast six in the evening, being two and three-quarter hours within the\ntime allotted, for a wager of 1,000 guineas. The sheep were roasted\nwhole, and a sumptuous dinner given by Mr. Coxetter.\n\n\nORIGIN OF THE GREAT WALL OF CHINA.\n\n[Illustration [++] Great Wall of China.]\n\nAs has been invariably the case in the early history of all the leading\nnations of the earth, great confusion and civil discord existed in\nthe empire of China in its first stages. It was divided into petty\nprincedoms, each prince striving to outwit the other, and all anxiously\naiming at the supreme power of the land, till the Emperor Chi-hoang-ti,\nwho came to the throne about three hundred years before the Christian\nera, conquered the whole of the jealous petty princes, and united\ntheir states into one vast empire. But no sooner had he achieved this,\nthan the Tartars began to be troublesome, and, hoping effectually\nto exclude their invasions, this emperor caused to be constructed\nthe often-read-of great wall of China, a stupendous work of masonry,\nextending from the sea to the western province of Shensee and carried\nover a tract of fifteen hundred miles, comprising high mountains, deep\nvalleys, and broad rivers, the wall being supported over the latter\nby gigantic arches. Fortified towers were erected at every hundred\nyards, and its summit admitted of six horsemen riding abreast. This\nsovereign is said to be the founder of the Hau dynasty. The wall proved\nan insignificant barrier to the Huns or Tartars, who harassed the\nprinces of the Hau dynasty, and were a very scourge to the farmers of\nthe frontier provinces. About the year 264, the Hau dynasty gave way\nto the Tsin, which latter was founded by a lineal descendant, through\nmany generations, of the builder of the great wall. In the sketch which\nwe have given, our chief object has been to show the extraordinary\ninflexibility of the Chinese in carrying their wall strictly along\ntheir frontier line, in spite of the stupendous obstacles which,\nintervened in the shape of mountains and valleys.\n\n\nPRIVY PURSE EXPENSES OF CHARLES II.\n\nMalone, the well known editor of Shakespeare, possessed a curious\nvolume--an account of the privy expenses of Charles II, kept by Baptist\nMay. A few extracts from this MS., taken from Malone's transcripts, are\nhere offered:--\n\n L s. d.\n\n My Lord St. Alban's bill 1,746 18 11\n Lady Castlemaine's debts 1,116 1 0\n Sir R. Viner, for plate 850 0 0\n For grinding cocoa-nuts 5 8 0\n Paid Lady C., play money 300 0 0\n For a band of music 50 0 0\n To the footman that beat Teague 5 7 6\n To Mr. Pears, for the charges of a body\n dissected before the king 5 1 0\n Lady C., play money 300 0 0\n To the Morrice Dancers at Ely 1 1 0\n Lady C., play money 300 0 0\n Mr. Knight for bleeding the king 10 10 0\n For a receipt of chocolate 227 0 0\n Mr. Price, for milking the asses 10 0 0\n To one that showed tumblers' tricks 5 7 6\n For weighing the king 1 0 0\n Paid Hall for dancing on the rope 20 0 0\n The Queen's allowance 1,250 0 0\n Paid Lord Lauderdale for ballads 5 0 0\n To a bone-setter attending the Duchess\n of Monmouth 10 0 0\n Paid Terry for waiting on the king swimming 10 0 0\n For 3,685 ribbons for the healing 107 10 4\n Mrs. Blague, the king's valentine 218 0 0\n Nell Gwyn 100 0 0\n Lost by the king at play on Twelfth-night 220 0 0\n Paid what was borrowed for the Countess of\n Castlemaine 1,650 0 0\n\n\nCOLOUR OF THE HAT FOR CARDINALS.\n\nInnocent IV. first made the hat the symbol or cognizance of the\ncardinals, enjoining them to wear a _red_ hat at the ceremonies and\nprocessions, _in token of their being ready_ to spill their blood for\nJesus Christ.\n\n\nSEVERITY OF THE LAWS A HUNDRED YEARS AGO.\n\nTwo lads were hanged for stealing a purse containing two shillings\nand a brass counter. Of ten criminals convicted at one sessions,\nfour were hanged and six transported. Very often half a dozen were\nsentenced to death at a single sessions. On the 17th March, 1755,\neight malefactors were hanged together at Tyburn. It was recorded as a\nmatter of surprise, that, \"only six convicts received sentence of death\nat Gloucester Assizes.\" One of these was a woman named Anne Ockley,\nwho was executed on the following day, on the charge of murdering an\nillegitimate child. To the last she denied her guilt, except in not\nhaving called in medical advice for her infant after a bad fall. She\ntook the Sacrament, and begged for more time to prepare herself for the\nchange; this favour being denied, she remained praying for two hours on\nthe drop before she would give the signal.\n\n\nMARKING THE KING'S DISHES WITH THE COOK'S NAMES.\n\nKing George II. was accustomed every other year to visit his German\ndominions, with the greater part of the officers of his household, and\nespecially those belonging to the kitchen. Once on his passage at sea,\nhis first cook was so ill with the sea-sickness, that he could not\nhold up his head to dress his majesty's dinner; this being told to the\nking, he was exceedingly sorry for it, as he was famous for making a\nRhenish soup, which his majesty was very fond of; he therefore ordered\ninquiry to be made among the assistant-cooks, if any of them could make\nthe above soup. One named Weston (father of Tom Weston, the player)\nundertook it, and so pleased the king, that he declared it was full\nas good as that made by the first cook. Soon after the king's return\nto England, the first cook died; when the king was informed of it, he\nsaid, that his steward of the household always appointed his cooks, but\nthat he would now name one for himself, and therefore asking if one\nWeston was still in the kitchen, and being answered that he was, \"That\nman,\" said he, \"shall be my first cook, for he makes most excellent\nRhenish soup.\" This favour begot envy among all the servants, so that,\nwhen any dish was found fault with, they used to say it was Weston's\ndressing: the king took notice of this, and said to the servants, it\nwas very extraordinary that every dish he disliked should happen to be\nWeston's; \"In future,\" said he, \"let every dish be marked with the name\nof the cook that makes it.\" By this means the king detected their arts,\nand from that time Weston's dishes pleased him most.\n\nThis custom was kept up till late in the reign of George III.\n\n\nPARLOUS DAYS.\n\nBloodletting, considered during the last century to be necessary for\nevery one in health or not, at spring and fall, was an operation\nperformed by the country surgeons on the labourers on a Sunday morning,\nat a charge of 6d. each. Bleeding in bed by a barber was, in the reign\nof Charles II., sometimes charged, for a lady, so high as 10s., and\nfor a gentleman, 1s. and 2s. 6d. The operator perhaps barboured the\npatient at an additional charge. Barbouring by the year was charged\n16s. Superstition had marked certain days in each month as dangerous\nfor bloodletting, which were called _parlous_ days. In July, the 1st,\n7th, 13th, 12th, 25th, and 20th were of the above kind.\n\nAs the whole population had recourse to bloodletting twice a year,\nbleeders or barbers were in constant demand.\n\n\nA FUNERAL APPROPRIATELY CONDUCTED.\n\nDuring the year 1700, the minister of a parish in Kent was interred\nat the age of 96 years; the gentleman who preached his funeral sermon\nwas 82; he who read the service 87; the clerk of the parish was the\nsame age; the sexton was 86; in addition to which list of aged persons,\nthere were several present from the adjacent parishes 100 years old\neach, and upwards.\n\n\nANCIENT NUT-CRACKERS.\n\n[Illustration [++] Ancient Nut-Crackers.]\n\nThe two quaint instruments pictured in our engraving, of about the\ntime of Charles I. or II., are made of hard wood rather rudely carved;\nand look as if in their time they had seen good service. The grotesque\nheads, with the mouth, affording the means of cracking the nuts, are\nexamples of the fitness of design for a particular purpose, which\ncharacterize many of the objects in domestic use in the middle ages,\nand up to the reign of Queen Anne, after which ornamental art for\nhousehold uses seems almost to have been disused. Even in the time of\nGeorge III., our chairs, tables, side-boards, &c., were made heavy,\nvery ugly, and without any attempt at appropriate pattern.\n\n\nNELL GWYNNE'S LOOKING-GLASS.\n\n[Illustration [++] Nell Gwynne's Looking-Glass.]\n\nThis glass is in the possession of Sir Page Dicks, of Port Hall. It\nbears the likeness of Nell Gwynne and King Charles, which are modelled\nin wax; and also the supporters, or crest, which Nell assumed, namely,\nthe lion and the leopard. The whole is curiously worked in \nglass beads, and the figures, with the dresses, made to project in very\nhigh relief; indeed, they are merely attached to the groundwork. In the\nupper compartment is Charles in his state dress; and the bottom one,\nthat of Nell Gwynne, in her court dress--the pattern of which is very\ntasteful. On the right is Charles in his hunting dress. The beads have\nretained their colours, which are very appropriate to the subject, and\nmust have been a work of considerable time and patience; but whether\ndone by Nell or not, there is no record.\n\n\nA REMARKABLE HIGHLANDER.\n\nIn August, 1827, John Macdonald expired in his son's house, in the\nLawnmarket, Edinburgh, at the advanced age of one hundred and seven\nyears. He was born in Glen Tinisdale, in the Isle of Skye, and, like\nthe other natives of that quarter, was bred to rural labour. Early\none morning in his youth, when looking after his black cattle, he was\nsurprised by the sight of two ladies, as he thought, winding slowly\nround a hill, and approaching the spot where he stood. When they came\nup, they inquired for a well or stream, where a drink of water could be\nobtained. He conducted them to the \"Virgin Well,\" an excellent spring,\nwhich was held in great reverence on account of its being the scene of\nsome superstitious and legendary tales. When they had quenched their\nthirst, one of the ladies rewarded Macdonald with a shilling, the first\nsilver coin of which he was possessed. At their own request he escorted\nthem to a gentleman's house at some distance, and there, to his great\nsurprise and satisfaction, he learned that the two \"ladies\" were Flora\nMacdonald and Prince Charles Stewart.\n\nThis was the proudest incident in Macdonald's patriarchal life; and,\nwhen surrounded by his Celtic brethren, he used to dilate on all the\nrelative circumstances with a sort of hereditary enthusiasm, and more\nthan the common garrulity of age. He afterwards turned joiner, and\nbore a conspicuous part in the building of the first Protestant church\nwhich was erected in the island of North Uist. He came to Edinburgh\ntwenty-three years before his death, and continued to work at his trade\ntill he was ninety-seven years of age.\n\nMacdonald was a temperate, regular-living man, and never paid a\nsixpence to a surgeon for himself, nor had an hour's sickness in the\nwhole course of his life. He used to dance regularly on New-year's day,\nalong with some Highland friends, to the bagpipe. On New-year's day,\n1825, he danced a reel with the father, the son, the grandson, and\ngreat-grandson, and was in more than his usual spirits. His hearing was\nnothing impaired, and till within three weeks of his demise he could\nhave threaded the finest needle with facility, without glasses.\n\n\nCATS WITH KNOTTED TAILS.\n\nWe extract the following paragraph from the narrative of a voyager in\nthe Indian Ocean, because it contains an account of a rarity in natural\nhistory with which few, we believe, are acquainted.\n\n\"The steward is again pillowed on his beloved salt fish, and our\nonly companion is a Malacca cat, who has also an attachment for the\nsteward's pillow. Puss is a tame little creature, and comes rubbing\nherself mildly against our shoes, looking up in our faces, and mewing\nher thoughts. Doubtless she is surprised that you have been so long\nlooking at her without noticing the peculiarity in her tail, which\nso much distinguishes her from the rest of the feline race in other\nquarters of the globe. Take her up in your lap, and see for yourself.\nDid you ever observe such a singular knot--so regular, too, in its\nformation? Some cruel monster must have tied it in a knot whilst puss\nwas yet a kitten, and she has outlived both the pain and inconvenience.\nBut here comes a kitten, all full of gambols and fun, and we find\nthat her tail is in precisely the same condition. So, then, this is\na remarkable feature amongst the whole race of Malayan cats, but\nfor which, no one we meet with, is able to give us a satisfactory\nexplanation.\"\n\n\nCURIOUS FEATS.\n\nIn 1553, the following extraordinary exhibition was performed\nin the presence of Queen Mary, in her passage through London to\nWestminster.--It is thus described by Holinshed, in his \"Chronicle,\"\nprinted 1577:--\"When shee didd come to Sainte Paule's churchyarde,\nMaister Haywood sat in a pageant under a vine, and made to her an\noration in Latine; and then there was one Peter, a man of Holland,\nwho didd stand upon the weathercocke of St. Paule's steeple, holdyng\na streamer in his handes of five yardes long, and waving thereof. Hee\nsometimes stood on one foot and shock the other, and then hee kneeled\non his knees to the verie grate marvel of al the people. Hee hadd\nmade two scaffolds under him--one above the cross, having torches\nand streamers sett upon it, and another over the ball of the cross,\nlikewise sett with streamers and torches which could not burne, the\nwind was so greate.\" Our chronicler further informs us, that \"Peter\ndidd have xvi pounds xiii shillings and iii pence given to him by the\ncitie of London for his costes and pains, and for all his stuffe.\"\n\n\nIMPUDENCE OR CANDOUR, WHICH IS IT?\n\nThe following advertisement appeared in the _St. James's Chronicle_ of\n1772. \"Wanted immediately, fifteen hundred, or two thousand pounds,\nby a person not worth a groat; who, having neither houses, land,\nannuities, or public funds, can offer no other security than that\nof simple bond, bearing simple interest, and engaging the repayment\nof the sum borrowed in five, six, or seven years, as may be agreed\non by the parties. Whoever this may suit, (for it is hoped it will\nsuit somebody), by directing a line for A. Z. in Rochester, shall be\nimmediately replied to, or waited on, as may appear necessary.\"\n\n\nTHE SOUTH STACK LIGHTHOUSE.\n\n[Illustration [++] South Stack Lighthouse.]\n\nThough not so celebrated as the Eddystone, the South Stack Lighthouse\nis unquestionably one of the marvels of science, and as such may be\nappropriately described in our pages. It is erected on the summit of\nan isolated rock, three or four miles westward from Holyhead, and\nseparated from the main land by a chasm ninety feet in width. This\nsplendid structure was raised in the year 1808. The elevation of the\nsummit of the rock on which it is erected is 140 feet above the level\nof the sea at high-water mark; the height of the tower, from the base\nto the gallery, is sixty feet; and the lantern is twelve feet high\nfrom the gallery; making the total elevation of the light 212 feet\nabove high-water mark. The light is produced by twenty-one brilliant\nlamps, with powerful reflectors, placed on a revolving triangular\nframe, displaying a full-faced light every two minutes, which, in clear\nweather, is distinctly visible at a distance of ten leagues. Latterly\nthere has been an addition of three red lights placed at the rock,\nwhich are more distinctly visible in foggy weather than the lighthouse\nlights. The rough sea caused by the strong tides about the head\nrendered the communication by boat very precarious. In order to obviate\nthe danger, a passage was contrived by means of two ropes thrown across\nthe gulf, along which the individual was drawn in a box or cradle, by\nthe assistance of pulleys affixed at each end. This plan was superseded\nby a bridge of ropes, which was used some years after, though always\nconsidered unsafe, on account of the constant wear of the ropes. In\n1827, a modern suspension chain-bridge was thrown over the sound, the\nspan of which is 110 feet, the chains being firmly bolted in the rock\non each side, and carried over two massive stone pillars erected for\nthe purpose. The chain supports a platform of timber five feet wide,\nand seventy feet above high-water mark. The bridge is attained by\ndescending the Holyhead mountain in a zigzag direction by a flight of\n380 steps.\n\n\nBRASS MEDAL OF OUR SAVIOUR.\n\nIn 1702, the late Rev. H. Rowlands, author of _Mona Antiqua_, while\nsuperintending the removal of some stones, near Aberfraw, Wales, for\nthe purpose of making an antiquarian research, found a beautiful\nbrass medal of our Saviour, in a fine state of preservation, which he\nforwarded to his friend and countryman, the Rev. E. Llwyd, author of\nthe _Archeologiae Britannica_, and at that time keeper of the Ashmolean\nlibrary at Oxford.\n\nThis medal, of which an engraving is subjoined, has on one side the\nfigure of a head exactly answering the description given by Publius\nLentulus of our Saviour, in a letter sent by him to the emperor\nTiberius and the senate of Rome. On the reverse side, it has the\nfollowing legend or inscription, written in Hebrew characters, \"This\nis Jesus Christ, the Mediator or Reconciler;\" or \"Jesus, the Great\nMessias, or Man Mediator.\" And being found among the ruins of the chief\nDruids resident in Anglesea, it is not improbable that the curious\nrelic belonged to some Christian connected with Bran the Blessed, who\nwas one of Caractacus's hostages at Rome from A.D. 52 to 59, at which\ntime the Apostle Paul was preaching the gospel of Christ at Rome. In\ntwo years afterwards, A.D. 61, the Roman General Suetonius extirpated\nall the Druids in the island. The following is a translation of the\nletter alluded to, a very antique copy of which is in the possession of\nthe family of Kellie, afterwards Lord Kellie, now represented by the\nEarl of Mar, a very ancient Scotch family--taken from the original at\nRome:--\n\n\"There hath appeared in these our days, a man of great virtue, named\nJesus Christ, who is yet living among us, and of the Gentiles is\naccepted as a prophet, but his disciples call him 'the Son of God.' He\nraiseth the dead, and cures all manner of diseases; a man of stature\nsomewhat tall and comely, with very reverend countenance, such as the\nbeholders both love and fear; his hair the colour of chesnut, full\nripe, plain to his ears, whence downwards it is more orient, curling,\nand waving about his shoulders. In the midst of his head is a seam or a\npartition of his hair after the manner of the Nazarites; his forehead\nplain and very delicate; his face without a spot or wrinkle, beautified\nwith the most lovely red; his nose and mouth so formed that nothing\ncan be reprehended; his beard thickish, in colour like his hair,\nnot very long but forked; his look, innocent and mature; his eyes,\ngrey, clear, and quick. In reproving, he is terrible; in admonishing,\ncourteous and fair spoken; pleasant in conversation, mixed with\ngravity. It cannot be remarked that any one saw him laugh, but many\nhave seen him weep. In proportion of body, most excellent; his hands\nand arms most delicate to behold. In speaking, very temperate, modest,\nand wise. A man, for his singular beauty, surpassing the children of\nmen!\"\n\n[Illustration [++] Brass Medal of Our Saviour.]\n\nThe representation of this sacred person which is in the Bodleian\nlibrary, somewhat resembles that of the print of this medal, when\ncompared together. It was taken from a likeness engraved in agate, and\nsent as a present from the sultan for the release of his brother, who\nwas taken prisoner. There is a well-executed drawing of this at the\nMostyn library, much worse for age.\n\n\nMONSTROUS HEAD-DRESS.\n\n[Illustration [++] Head-Dress of 1782.]\n\nAt no period in the history of the world was anything more absurd in\nhead-dress worn than that here depicted, which was in vogue with the\nfashionables of 1782. The body of this erection was formed of tow, over\nwhich the hair was turned, and false hair added in great curls, bobs,\nand ties, powdered to profusion; then hung all over with vulgarly-large\nrows of pearls, or glass beads, fit only to decorate a chandelier;\nflowers as obtrusive were stuck about this heap of finery, which was\nsurmounted by broad silken bands and great ostrich-feathers, until the\nhead-dress of a lady added three feet to her stature, and the male\nsex, to use the words of the _Spectator_, \"became suddenly dwarfed\nbeside her.\" To effect this, much time and trouble was wasted, and\ngreat personal annoyance was suffered. Heads, when properly dressed,\n\"kept for three weeks,\" as the barbers quietly phrased it; that they\nwould not really \"keep\" longer may be seen by the many recipes they\ngive for the destruction of insects which bred in the flour and pomatum\nso liberally bestowed upon them. The description of \"opening a lady's\nhead,\" after a three weeks' dressing, given in the magazines of this\nperiod, it would be imagined, would have taught the ladies common\nsense; but fashion could reconcile even the disgust that must have been\nfelt by all.\n\n\nPRICE OF HUMAN HAIR.\n\nLong flaxen hair was bought from the head at 10s. the ounce, and any\nother fine hair at 5s. or 7s. the ounce in 1662.\n\nWithin the present century the heads of hair of whole families in\nDevonshire were let out by the year at so much rent per poll. An Exeter\nperriwig maker went round periodically, cut the locks, and oiled the\nnumskull of each thus left in stubble.\n\n\nINTERESTING AND FANCIFUL RELIQUE.\n\n[Illustration [++] Enamelled Jewel.]\n\nThe enamelled jewel, of which we give an engraving, was presented by\nMary, Queen of Scots, to George Gordon, fourth Earl of Huntley. The\nprecise period at which the gift was made is not now known, though\nthe time was not improbably during the residence of the Queen in\nFrance, when the Order of St. Michael was conferred on the Duke of\nChatelherault, the Earl of Huntley, and several other Scottish nobles,\nabout 1548. The lock of Mary's hair which is attached to the small\nivory skull, is of a light auburn, inclining to a gold-colour; and if\nallowance be made for some fading in the course of years, and for the\nhair of the Queen having generally become darker as she advanced in\nlife, the accuracy of Melvil will be confirmed, when, in speaking of\nher after her return to Scotland, he says, \"her hair was light auburn;\nElizabeth's more red than yellow.\" In this particular little reliance\ncan be placed upon the portraits of Queen Mary; since it is well known,\nthat in the latter part of her life, it was a fashionable practice to\nwear false hair of various hues, though in some of her pictures the\ncolour of the locks is nearly similar to the hue of that represented in\nthe present. The skull, from which it issues is connected by a twisted\nskein of silk with the figure of a Cupid shooting an arrow, standing on\na heart enamelled red, transfixed with a dart. On one side the heart is\na setting for a precious stone, now vacant; and, on the other, in white\nletters, the words \"Willingly Wounded.\" From the point of the heart is\na pendant, containing on one side a small ruby, and having the other\nenamelled blue with an ornament in white. Our engraving represents one\nside of the jewel, of the exact size of the original.\n\n\nFASTIDIOUSNESS IN DRESS AT AN OLD AGE.\n\nJonn Benbow, of Northwood, in the parish of Prees, Salop, died 1806,\naged 107. His occupation was that of a maker of clocks and watches.\nHis steadiness of hand, clearness of intellect, and complete command\nof all his faculties, were such that, till within a very few years of\nhis decease, he was enabled to execute the most intricate and delicate\nmanipulations connected with his business. He lived in three centuries;\nand, at the time of his decease, had a son, a grandson, and several\ngreat-grandchildren, living in the house with him. He was remarkable\nfor industry, sobriety, early rising, and soon retiring to rest,\nand was universally respected for his integrity and ingenuity. His\nfavourite beverage was \"small beer\" brewed of molasses. To the very\nclose of his life he was remarkable for his extreme attention to his\ndress and everything relating to his personal appearance, as will be\nseen by the following anecdote. About three years before his death, his\ntailor brought him home a new coat; on examining which he discovered\nthat the man, either through not being provided with the necessary\nmaterial or inadvertence, had substituted a cloth collar for a velvet\none, which he was accustomed to have added to his garment. Mortified at\nthis circumstance, and learning that the tailor had not velvet of the\nnecessary quality by him, he took up his walking-stick and straitway\nwent off to Whitchurch, a distance of seven miles, to purchase the\nmaterials proper to make a new collar, and, to the astonishment of all\nhis family, returned home in a few hours.\n\n\nSUPERSTITION OF THE JAVANESE.\n\nNowhere has superstition a greater power over the human mind than among\nthe inhabitants of Java.\n\nWhen the proper chord is touched, there is scarcely anything too gross\nfor the belief of these islanders. Mr. Crawfurd relates that some\nyears since, it was almost accidentally discovered, that the skull of\na buffalo was superstitiously conveyed from one part of the island to\nanother. The point insisted upon was, never to let it rest, but to keep\nit in constant progressive motion. It was carried in a basket, and no\nsooner was one person relieved from the load than it was taken up by\nanother; for the understanding was, that some dreadful imprecation was\ndenounced against the man who should let it rest. In this manner, the\nscull was hurried from one province to another, and after a circulation\nof many hundred miles, at length reached the town of Samarang, the\nDutch governor of which seized it and threw it into the sea, and thus\nthe spell was broken. The Javanese expressed no resentment, and nothing\nfurther was heard of this unaccountable transaction. None could tell\nhow or where it originated.\n\nThe same writer relates a still more extraordinary instance of\ninfatuation. During the occupation of Java by the English, in the\nmonth of May 1814, it was unexpectedly discovered, that, in a remote\nbut populous part of the island, a road, leading to the top of the\nmountain of Sumbeng, one of the highest in Java, had been constructed.\nAn enquiry being set on foot, it was discovered that the delusion which\ngave rise to the work had its origin in the province of Banyunas, in\nthe territories of the Susunan, and that the infection had spread\nto the territory of the Sultan, and thence extended to that of the\nEuropeans. On examination a road was found constructed twenty feet\nbroad, and from fifty to sixty miles long, and it was wonderfully\nsmooth and well made. One point which appears to have been considered\nnecessary, was, that this road should not cross rivers, and in\nconsequence it wound in a thousand ways. Another point as peremptorily\ninsisted on was, that its straight course should not be interrupted by\nany private rights; and in consequence trees and houses were overturned\nto make way for it. The population of whole districts, occasionally\nto the amount of five or six thousand labourers, were employed on the\nroad, and, among a people disinclined to active exertion the laborious\nwork was nearly completed in two months--such was the effect of the\ntemporary enthusiasm with which they were inspired. It was found in\nthe sequel that the whole work was set in motion by an old woman, who\ndreamt, or pretended to have dreamt, that a divine personage was about\nto descend from heaven on the mountain in question. Piety suggested\nthe propriety of constructing a road to facilitate his descent; and\nit was rumoured that divine vengeance would pursue the sacrilegious\nperson who refused to join in the meritorious labour. These reports\nquickly wrought on the fears and ignorance of the people, and they\nheartily joined in the enterprise. The old woman distributed slips of\npalm-leaves to the labourers, with magic letters written upon them,\nwhich were charms to secure them against sickness and accidents. When\nthis strange affair was discovered by the native authorities, orders\nwere issued to desist from the work, and the inhabitants returned\nwithout a murmur to their wonted occupations.\n\n\nSIZE OF THE BRITISH ISLANDS.\n\nThe exact size of our own country is a legitimate object of curiosity.\nWe believe the following will be found strictly accurate:--\n\n The area of England is estimated at 31,929,340 acres.\n \" Wales 4,320,000 \"\n \" Scotland 16,240,000 \"\n \" S. Isles adjacent to the coast 1,055,080 \"\n \" W. Isles 851,200 \"\n \" Orkneys 153,606 \"\n \" Shetlands 643,840 \"\n\n\nCASE CONTAINING THE HEART OF LORD EDWARD BRUCE.\n\nLord Edward Bruce was eldest son of Sir Edward, baron of Kinloss,\nso created by James I. in 1603, to whom the king gave the dissolved\nabbey of Kinloss, in Ayrshire, after he had been instrumental in his\nsuccession to the crown of England; whither accompanying the king, he\nwas made master of the Rolls in 1604, died in 1610, and was buried\nin the Rolls chapel. His son, the lord Edward, killed in duel by Sir\nEdward Sackville in 1613, was succeeded by his brother, who was created\nEarl of Elgin in 1633, and an English baron in 1641.\n\nSir Edward Sackville, by whose hand the Lord Edward Bruce fell, was\nyounger brother to Richard Sackville, Earl of Dorset, on whose death\nhe succeeded to the title. He was lord president of the council, a\njoint lord keeper, and filled several other distinguished offices under\nCharles I., to whom he adhered, by whose side he fought at the battle\nof Edge-hill, and whose death he took so much to heart, that he never\nafterwards stirred out of his house in Salisbury-court, but died there\non the 17th of July, 1652.\n\n[Illustration [++] Case Containing the Heart of Lord Edward Bruce.]\n\nBetween these noblemen there arose a quarrel, which terminated in their\nduel; and all that is, or probably can be known respecting it, is\ncontained in the following correspondence, preserved in a manuscript in\nQueen's college library, Oxford.\n\n _A Monsieur, Monsieur Sackvile._\n\n\"I that am in France, hear how much you attribute to yourself in this\ntime, that I have given the world leave to ring your praises; and for\nme, the truest almanack, to tell you how much I suffer. If you call\nto memory, when as I gave you my hand last, I told you I reserved the\nheart for a truer reconcilliation. Now be that noble gentleman, my love\nonce spoke, and come and do him right that could recite the tryals you\nowe your birth and country, were I not confident your honour gives\nyou the same courage to do me right, that it did to do me wrong. Be\nmaster of your own weapons and time; the place wheresoever, I will wait\non you. By doing this, you shall shorten revenge, and clear the idle\nopinion the world hath of both our worths.\n\n \"ED. BRUCE.\"\n\n\n _A Monsieur, Monsieur Baron de Kinloss._\n\n\"As it shall be always far from me to seek a quarrel, so will I always\nbe ready to meet with any that is desirous to make tryal of my valour,\nby so fair a course as you require. A witness whereof yourself shall\nbe, who, within a month, shall receive a strict account of time, place\nand weapon, where you shall find me ready disposed to give honourable\nsatisfaction, by him that shall conduct you thither. In the mean time,\nbe as secret of the appointment, as it seems you are desirous of it.\n\n \"E. SACKVILE.\"\n\n\n _A Monsieur, Monsieur Baron de Kinloss._\n\n\"I am at Tergose, a town in Zeland, to give what satisfaction your\nsword can render you, accompanied with a worthy gentleman for my\nsecond, in degree a knight. And, for your coming, I will not limit you\na peremptory day, but desire you to make a definite and speedy repair,\nfor your own honour, and fear of prevention; at which time you shall\nfind me there.\n\n_Tergose, 10th of August, 1613._\n\n \"E. SACKVILE.\"\n\n\n _A Monsieur, Monsieur Sackvile._\n\n\"I have received your letter by your man, and acknowledge you have\ndealt nobly with me; and now I come, with all possible haste, to meet\nyou.\n\n \"E. BRUCE.\"\n\nThe combat was fierce, and fatal to Lord Bruce.\n\nIt has always been presumed that the duel was fought under the walls of\nAntwerp; but the combatants disembarked at Bergen-op-Zoom, and fought\nnear that town, and not Antwerp.\n\n[Illustration [++] Silver Case Shaped Like a Heart.]\n\nIn consequence of a tradition, that the heart of Lord Edward Bruce had\nbeen sent from Holland, and interred in the vault or burying-ground\nadjoining the old abbey church of Culross, in Perthshire, Sir Robert\nPreston directed a search in that place in 1808, with the following\nresult:--Two flat stones, without inscription, about four feet in\nlength and two in breadth, were discovered about two feet below the\nlevel of the pavement, and partly under an old projection in the wall\nof the old building. These stones were strongly clasped together\nwith iron; and when separated, a silver case, or box, of foreign\nworkmanship, shaped like a heart, was found in a hollow or excavated\nplace between them. Its lid was engraved with the arms and name \"Lord\nEdward Bruse;\" it had hinges and clasps; and when opened, was found to\ncontain a heart, carefully embalmed, in a brownish liquid.\nAfter drawings had been taken of it, as represented in the present\nengravings, it was carefully replaced in its former situation. There\nwas a small leaden box between the stones in another excavation; the\ncontents of which, whatever they were originally, appeared reduced to\ndust.\n\nSome time after this discovery, Sir Robert Preston caused a delineation\nof the silver case, according to the exact dimensions, with an\ninscription recording its exhumation and re-deposit, to be engraved on\na brass plate, and placed upon the projection of the wall where the\nheart was found.\n\nIt is a remarkable fact, that the cause of the quarrel between Lord\nBruce and Sir Edward Sackville has remained wholly undetected,\nnotwithstanding successive investigations at different periods. Lord\nClarendon, in his \"History of the Rebellion,\" records the combat as an\noccurrence of magnitude, from its sanguinary character and the eminence\nof the parties engaged in it. He does not say any thing respecting the\noccasion of the feud, although Lord Bruce's challenge seems to intimate\nthat it was a matter of public notoriety.\n\nThe exact day of the duel is not known, but it was certainly in 1613,\nand most probably in August from the date of one of the above letters.\n\n\nEXTRAORDINARY FEMALE INTREPIDITY.\n\nEarly on the 24th of January, 1822, the turnpike-house, about four\nmiles from Basingstoke, on this side of Overton, was attacked, with\nintent to enter, by two men, who had taken off some tiles at the\nback part of the premises (the roof being very low) to effect their\npurpose. These villains knew, it would appear, that a lone woman, Mrs.\nWhitehouse, received the tolls at this gate, and that her husband\nattended a gate as far distant as Colebrook. Mrs. Whitehouse, however,\nvery fortunately possessed three loaded pistols, one of which she\nfired--then a second, and a third, without effect. These determined\nruffians (notwithstanding being thrice fired at) were, it appears,\nresolved not to depart without accomplishing the projected robbery.\nMrs. Whitehouse's little boy, only 11 years of age, in the mean time\nhad re-loaded a brace of pistols, one of which Mrs. Whitehouse fired,\nand wounded one of the desperadoes full in the face--he fell, and the\nblood flowed profusely; yet, strange to relate, the accomplice had\nhardihood enough to drag away the wounded robber! On observing this,\nMrs. Whitehouse fired the fifth pistol at them, but missed them. The\nfellow who received the contents of the fourth pistol being supposed to\nhave been killed, and some persons residing at a considerable distance\nfrom the spot having heard of the circumstance, assembled, and made\ndiligent search at daybreak to discover the body of the deceased; but,\nalthough the blood could be traced some distance from the house, the\nbody could not be found; nor were those concerned in the attack ever\nfound out. The successful resistance, however, deserves to be recorded.\n\n\nGIGANTIC BONES.\n\nWhenever any bones of unusual magnitude were discovered, it was\ninvariably the custom to ascribe them to some giant. This was always\nso up to recent years, and no wonder it was intensely the case at the\nearly period of 1660. About that period, when the brook or rivulet\nfrom which the town of Corbridge, in the north of England, derives its\nname, had been worn away by some impetuous land-flood, a skeleton,\nsupposed to be that of a man of extraordinary and prodigious size,\nwas discovered. The length of the thigh bone was nearly six feet, and\nthe skull, teeth, and other parts proportionably monstrous, so that\nthe length of the whole body was computed at twenty-one feet. It is\nconjectured, by the more enlightened men of modern times, that these\nstrange bones belonged to some large animal that had been sacrificed\nby the Romans at the altar dedicated to Hercules, which was found\nhere some years ago. Notwithstanding that the superstition of our\nforefathers has lost nearly all its credit and influence, a singularly\nlarge bone found here is now exhibited in the Keswick Museum as the rib\nof the giant Cor.\n\n\nNEW STYLE OF ADVERTISEMENT.\n\nThe following editorial announcement is taken from the Philadelphia\n_Weekly Mercury_, of November 30, 1752, because it forms a complete\nnovelty in its way, and also affords us an insight into the degree of\ncommunication which existed at that period between the large towns\nand the provinces in America. It is, moreover, a curious jumble of\ninformation, strangely mixing up the starting of the stage coach with\nthe news of the day:--\n\nOn _Monday_ next the Northern Post sets out from _New-York_, in order\nto perform his Stage but once a Fortnight, during the Winter Quarter;\nthe Southern Post changes also, which will cause this Paper to come\nout on _Tuesdays_ during that Time. The Colds which have infested the\nNorthern Colonies have also been troublesome here, few Families having\nescaped the same, several have been carry'd off by the Cold, among whom\nwas _David Brintnall_, in the 77th Year of his Age; he was the first\nMan that had a Brick House in the City of _Philadelphia_, and was much\nesteem'd for his just and upright dealing. There goes a Report here,\nthat the Lord _Baltimore_ and his Lady are arrived in _Maryland_,\nbut the Southern Post being not yet come in, the said Report wants\nConfirmation.\n\n\nMAKING A CANDLESTICK OF GUNPOWDER.\n\nA marvellous escape from destruction is related in the MS. Life of\nAlderman Barnes.--\"One of his brother-in-law's (Alderman Hutchinson's)\napprentices, stepping up into the back-lofts to fetch somewhat he\nwanted, in his heedlessness and haste, stops his candle into a\nbarrel of gunpowder whose head was struck off, to serve instead of a\ncandlestick. But the man reflecting what he had done, was struck with\naffrightment, his heart failed him, nor durst he stay any longer,\nbut running down stairs, leaves the candle burning in the gunpowder\ncask, and with horror, trembling, and despair, tells the family what\nindiscretion he had committed; they were all immediately as their\nwits' end, and well they might, for the lofts were three stories\nhigh, very large, and stowed full with whatever is combustible, as\nbrandy, oil, pitch, tar, rosin, flax, alum, hops, and many barrels of\ngunpowder. Had the candle fallen to one side, or had the least spark\nfallen from the snuff into the cask, the whole town had been shaken,\nand the whole of the house immediately blown up and in a blaze; but\none of the labourers, a stout fellow, ran forthwith into the loft, and\njoining both his hands together, drew the candle softly up between\nhis middlemost fingers, so that if any snuff had dropped, it must\nhave fallen into the hollow of the man's hand, and by this means was\nNewcastle saved from being laid in ashes.\" This must have happened\nabout the year 1684.\n\n\nTHE CAMDEN CUP.\n\nThe subjoined engraving represents the Silver-gilt Standing Cup\nand Cover bequeathed by the celebrated historian, William Camden,\nClarenceux King at Arms, to the Worshipful Company of Painter\nStainers'. Camden's will is recorded in the Prerogative Court\nof Canterbury (in the register designated III Swann 3, probate\ngranted November 10, 1623), and it has been printed by Hearne in\nhis _Collection of Curious Discourses_, Ox. 1720. After directing\nthe sum of eight pounds to be given \"to the poore of that place\n(Chislehurst) when it shall please God to call me to his mercie,\"\nCamden continues--\"I bequeath to Sir Foulke Greville, Lord Brooke,\nChancellor of the Exchequer, who preferred me gratis to my Office, a\npeece of plate of ten pounds; Item, to the Company of Painter-Stainers\nof London, to buy them a peece of plate in memoriall of mee, sixteene\npounds;\" the inscription upon which is directed to be--\"_Guil. Camdenus\nClarenceux, filius Sampsonis, Pictoris Londinensis, dono dedit_.\"\n\n[Illustration [++] The Camden Cup.]\n\nThis stately and richly-decorated cup and cover is used on Corporation\nFestivals, in memory of the illustrious donor. In height, it is\naltogether twenty-three inches and a quarter, the cover only being\neight inches and three-quarters; and the cup, independent of the\nstand, five inches and a-half, its greatest diameter being five inches\nand a-half. The inscription encircles the upper rim of the cup; and\ndirectly under it is an engraved escutcheon of Camden's arms; _Or_, a\nfess engrailed, between six cross crosslets fitchee, _Sable_. The cover\npresents an object of much elegance, a richly ornamented open pyramid,\nbased on the heads of birds, the breasts bending gracefully with\ncartouche ornaments: the pinnacle of the pyramid surmounted by a female\nfigure, the right hand resting on a shield, charged with the same arms\nas shown on the side of the cup. The birds' heads have apparently a\nreference to the phoenix heads in the second and third quarters of the\narmorial ensigns, and to the crest of the Company of Paper-Stainers.\n\n\nRICHARDSON, THE SHOWMAN.\n\nThis eccentric individual, who died in 1836, left behind him upwards of\nL20,000. He was born in the workhouse of Marlow, Bucks, but ran away\nfrom that place in order to seek his fortune in London. After various\nvicissitudes, he became the landlord of the Harlequin public-house, in\nDrury-lane, where he saved some money, which he embarked in fitting up\na portable theatre, and was known for forty years as the \"Prince of\nShowmen,\" and used frequently to boast that Edmund Kean and several\nother eminent actors were brought out by him. His property, after\nvarious legacies to the itinerant company which had attended him for\nmany years, descended to two nephews and a niece, and he desired by\nhis will to be buried in Marlow churchyard, in the same grave as his\nfavourite \"spotted boy,\" a lad who, some years before, was exhibited\nby him, and attracted great notice in consequence of the extraordinary\nmanners in which he was marked on various parts of his body. Some years\nsince the scenery, dresses, and decorations of Richardson's theatre\nwere exposed for auction by Mr. George Robins, and L2,000 were bid\nfor them. They were bought in; the \"old man,\" as he was technically\ndenominated, considering them to be worth at least L3,000.\n\n\nPRESERVATION OF DEAD BODIES.\n\nThere is an arched vault, or burying-ground, under the church of\nKilsyth, in Scotland, which was the burying-place of the family\nof Kilsyth, until the estate was forfeited, and the title became\nextinct in the year 1715; since which it has never been used for that\npurpose, except once. The last Earl fled with his family to Flanders,\nand, according to tradition, was smothered to death about the year\n1717, along with his lady and an infant child, and a number of other\nunfortunate Scottish exiles, by the falling in of the roof of a house\nin which they were assembled. What became of the body of the Earl is\nnot known, but the bodies of Lady Kilsyth and her infant were emboweled\nand embalmed, and soon afterwards sent over to Scotland. They were\nlanded, and lay at Leith for some time in a cellar, whence they were\nafterwards carried to Kilsyth, and buried in great pomp in the vault\nabove mentioned. In the spring of 1796, some rude regardless young men,\nhaving paid a visit to this ancient cemetery, tore open the coffin of\nLady Kilsyth and her infant. With astonishment and consternation, they\nsaw the bodies of Lady Kilsyth and her child as perfect as in the hour\nthey were entombed. For some weeks this circumstance was kept secret,\nbut at last it began to be whispered in several companies, and soon\nexcited great and general curiosity.\n\n\"On the 12th of June,\" says the Minister of the parish of Kilsyth,\nin a letter to J. Garnet, M.D., \"when I was from home, great crowds\nassembled, and would not be denied admission. At all hours of the\nnight, as well as the day, they afterwards persisted in gratifying\ntheir curiosity. I saw the body of Lady Kilsyth soon after the coffin\nwas opened; it was quite entire. Every feature and every limb was as\nfull, nay, the very shroud was as clear and fresh, and the colours of\nthe ribbons as bright, as the day they were lodged in the tomb. What\nrendered this scene more striking and truly interesting was, that the\nbody of her son and only child, the natural heir of the title and\nestates of Kilsyth, lay at her knee. His features were as composed as\nif he had been only asleep. His colour was as fresh, and his flesh as\nplump and full, as in the perfect glow of health; the smile of infancy\nand innocence sat on his lips. His shroud was not only entire, but\nperfectly clean, without a particle of dust upon it. He seems to have\nbeen only a few months old. The body of Lady Kilsyth was equally well\npreserved; and at a little distance, from the feeble light of a taper,\nit would not have been easy to distinguish whether she was dead or\nalive. The features, nay the very expression of her countenance, were\nmarked and distinct; and it was only in a certain light that you could\ndistinguish anything like the ghastly and agonizing traits of a violent\ndeath. Not a single fold of her shroud was decomposed nor a single\nmember impaired.\n\n\"Let the candid reader survey this sketch; let him recal to mind the\ntragic tale it unfolds; and say, if he can, that it does not arrest the\nattention and interest the heart. For my own part, it excited in my\nmemory a thousand melancholy reflections; and I could not but regret\nthat such rudeness had been offered to the ashes (remains) of the dead,\nas to expose them thus to the public view.\n\n\"The body seemed to have been preserved in some liquid, nearly of\nthe colour and appearance of brandy. The whole coffin seemed to have\nbeen full of it, and all its contents saturated with it. The body\nhad assumed somewhat the same tinge, but this only served to give it\na fresher look. It had none of the ghastly livid hue of death, but\nrather a copper complexion. It would, I believe, have been difficult\nfor a chemist to ascertain the nature of this liquid; though perfectly\ntransparent; it had lost all its pungent qualities, its taste being\nquite vapid.\n\n\"The head reclined on a pillow, and, as the covering decayed, it was\nfound to contain a collection of strong-scented herbs. Balm, sage,\nand mint were easily distinguished; and it was the opinion of many,\nthat the body was filled with the same. Although the bodies were thus\nentire at first, I confess I expected to see them crumble into dust;\nespecially as they were exposed to the open air, and the pure aromatic\nfluid had evaporated; and it seems surprising that they did not. For\nseveral weeks they underwent no visible change, and had they not been\nsullied with dust and drops of grease from the candles held over them,\nI am confident they might have remained as entire as ever; for even a\nfew months ago (many months after), the bodies were as firm and compact\nas at first, and though pressed with the finger did not yield to the\ntouch, but seemed to retain the elasticity of the living body. Even the\nshroud, through torn by the rude hands of the regardless multitude, is\nstill strong and free from rot.\n\n\"Perhaps the most singular phenomenon is, that the bodies seem not to\nhave undergone the smallest decomposition or disorganization. Several\nmedical gentlemen have made a small incision into the arm of the\ninfant; the substance of the body was quite firm, and every part in\nits original state.\" To the above remarkable instance we may add the\nfollowing:--The tomb of Edward the First, who died on the 7th July,\n1307, was opened on the 2nd of January, 1770, and after the lapse of\n463 years, the body was found not decayed; the flesh on the face was a\nlittle wasted, but not putrid.\n\nThe body of Canute the Dane, who got possession of England in the year\n1017, was found very fresh in the year 1766, by the workmen repairing\nWinchester Cathedral. In the year 1522, the body of William the\nConqueror was found as entire as when first buried, in the Abbey Church\nof St. Stephen, at Caen; and the body of Matilda, his wife, was found\nentire in 1502, in the Abbey Church of the Holy Trinity in the same\ncity.\n\nNo device of art, however, for the preservation of the remains of the\ndead, appears equal to the simple process of plunging them over head\nand ears in peat-moss.\n\nIn a manuscript by one Abraham Grey, who lived about the middle of\nthe 16th century, now in the possession of his representative, Mr.\nGoodbehere Grey, of Old Mills, near Aberdeen, it is stated, that\nin 1569, three Roman soldiers in the dress of their country, fully\nequipped with warlike instruments, were dug out of a moss of great\nextent, called Kazey Moss. When found, after a lapse of probably about\nfifteen hundred years, they \"were quite fresh and plump.\"\n\n\nPERFUMES.\n\nSo perfect were the Egyptians in the manufacture of perfumes, that some\nof their ancient ointment, preserved in an alabaster vase in the Museum\nat Alnwick, still retains a very powerful odour, though it must be\nbetween 2,000 and 3,000 years old.\n\n\nFRENCH ASSIGNATS--THEIR ORIGIN.\n\nExtraordinary devices for raising money are legitimate subjects for\nour pages. Of these devices, the French Assignats are not the least\nremarkable. They originated thus--in the year 1789, at the commencement\nof the great Revolution in France, Talleyrand proposed in the National\nAssembly a confiscation of all church property to the service of the\nstate. The Abbe Maury opposed this project with great vehemence, but\nbeing supported by Mirabeau, it received the sanction of the Assembly\nby an immense majority on the 2nd of November. The salaries fixed\nfor the priesthood were small, and, moreover, were not sufficiently\nguaranteed; whence originated much misery to all classes of priests,\nfrom the archbishops down to the humble cures; and as monastic\ninstitutions were treated in the same way, monks and nuns were suddenly\nplaced in precarious circumstances regarding the means of subsistence.\nHere, however, an unexpected difficulty sprang up; the National\nAssembly were willing to sell church property, but buyers were wanting;\nconscience, prudence, and poverty combined to lessen the number of\nthose willing to purchase; and thus the urgent claims of the treasury\ncould not be satisfied. Applications for loans were not responded to;\ntaxes had been extinguished; voluntary donations had dwindled almost to\nnothing; and 400,000,000 of livres were necessary for the vast claims\nof the year 1790. The municipalities of Paris and other cities sought\nto ameliorate the state of affairs by subscribing for a certain amount\nof church property, endeavouring to find private purchasers for it, and\npaying the receipts into the national exchequer. This, however, being\nbut a very partial cure for the enormity of the evils, the National\nAssembly fell upon the expedient of creating state-paper or bank-notes,\nto have a forced currency throughout the kingdom. Such was the birth\nof the memorable assignats. Four hundred millions of this paper were\nput in circulation; and a decree was passed that church property to\nthat amount should be held answerable for the assignats. Our sketch\nrepresents several of the different forms in which the Assignats were\nissued to the public.\n\n[Illustration [++] French Assignats.]\n\n\nEXECUTION OF LOUIS XVI.\n\nThe judicial murder of Louis XVI. was the climax of the Revolution in\nFrance. The Convention voted his death at three o'clock on the morning\nof the 20th January, 1793, and he was taken to execution in twenty-six\nhours afterwards.\n\n[Illustration [++] Execution of Louis XVI.]\n\nThe guillotine was erected in the middle of the Place Louis XV.,\na large open square, having the Champs Elysees on one side, and\nthe gardens of the Tuileries on the other. The Place bristled with\nartillery, and every street and avenue leading to it was crowded with\ntroops and armed multitudes, who had cannon with them charged with\ngrape-shot; while the carriage was surrounded by picked men, who\nhad orders to despatch the king with their carbines in case of any\nrescue being attempted. At about half-past ten, the king, who had been\nengaged in prayer during the ride, arrived at the spot; he descended\nfrom the coach, and his confessor followed him. Three executioners\napproached to remove his upper garments, but he put them back, and\nperformed that simple office for himself. He resisted somewhat the\nindignity of having his hands tied, and only yielded on the entreaty\nof his confessor; and had also to yield on the subject of cutting off\nhis back hair. He ascended the steps that led to the platform with\na firm bearing, still followed by M. Edgeworth. When on the top, he\nmade a sudden movement towards the edge of the scaffold, and exclaimed\nwith a loud and firm voice: \"Frenchmen, I die innocent; it is from\nthe scaffold, and when about to appear before my God, that I tell\nyou so. I pardon my enemies; I pray that France\"----Here Santerre, on\nhorseback, raised his right hand, and cried: \"Drums! Executioners, do\nyour duty!\" Several drummers immediately began by their noise to drown\nthe sound of the king's voice: and six executioners brought him to the\ncentre of the scaffold. He exclaimed again: \"I die innocent; I ever\ndesired the good of my people;\" but his voice could be heard only by\nthe executioners and the priest. He then knelt down, in order to place\nhis head in the appointed spot; the confessor, bending over him said:\n\"Son of St. Louis, ascend to heaven!\" The spring of the machine was\ntouched, the heavy axe descended in its grooves, and the once royal\nhead was severed from the body. Samson, the chief executioner, took\nup the bleeding head by the hair, and walked three times round the\nscaffold, holding it up at arm's-length to show it to the people. The\ntroops and the spectators shouted: \"Vive la Republique!\" put their hats\nand caps upon their bayonets and pikes, and waved them in the air,\nwith prolonged and re-echoing cries of \"Vive la Republique!\" \"Vive la\nNation!\" \"Vive la Liberte!\" Many of the savage men standing near the\nscaffold dipped their pike-heads into the king's blood, and others\ntheir handkerchiefs--not as a sacred memento, but as a symbol of the\ndownfall of all kings; they even paraded these gore-stained objects\nbefore the windows of the Temple, that perchance the queen and her\nchildren might see them. The headless trunk of Louis was put into a\nlarge wicker-basket, placed in the coach, and carried to the cemetery\nof La Madeleine; where, without coffin or shroud, it was thrown into a\ndeep pit, partly filled up with quicklime. On that same morning, one\nBenoit Leduc, a tailor, who had on some occasions worked for Louis,\npresented a petition to the Convention, praying to be allowed, at his\nown expense, to bury the body of the king by the side of his father,\nLouis XV., and under the monument raised to that prince by the city\nof Sens; but the Convention rejected his petition, and ordered the\nexecutive council to see that Louis was buried like other criminals.\n\n\nA MAN AGED ONE HUNDRED YEARS CLAIMING A BOTTLE OF WINE.\n\nJohn Bull, of London, stock-broker, died 1848, aged 100 years. When\nat the age of about 93, and in the employ of Messrs. Spurling,\nstock-brokers, he left by mistake in the office of the accountant of\nthe Bank of England, a large number of bank notes. On discovering\nhis loss, after diligently searching for the missing parcel, he went\nback to the accountant's office, partly to acquaint Mr. Smee with the\ncircumstance, and partly as a last hope that he might there find the\nmissing treasure. To his great joy he found the parcel safe in the\naccountant's possession, whom he earnestly implored to keep the secret,\nlest his employers should think his faculties were failing. Mr. Smee\nof course gave him the required assurance, and goodnaturedly added,\nthat when Mr. Bull should attain the age of 100 years, he would treat\nhim to the finest bottle of wine in his cellar. Some time before his\nbecoming a centenarian, he was pensioned off by his employer, and Mr.\nSmee had, in all probability, quite forgotten the affair; when, true\nto the engagement, the venerable, but still active old clerk, made his\nappearance at the bank on the important day, and claimed the promised\nbottle of wine. The claim was promptly allowed; and the last birthday\nof the aged official was one of the happiest among his friends of the\nlong list of such events which had been its precursor. After continuing\nvigorous and active, and almost free from indisposition up to this\ntime, he, along with many other aged persons, fell a victim to that\nfatal influenza which prevailed so extensively throughout the country,\nand more especially in London and its suburbs, during the autumn of\n1847 and the winter of 1848.\n\n\nCHARITY REWARDED BY A RICH AND LIBERAL MENDICANT.\n\nWithin the present century, a beggar in Moorfields used daily to\nhave a penny given him by a merchant on his way to the Exchange. The\npenny was withheld, and the appearance of the merchant manifested his\nembarrassment and distress. The beggar at length spoke to him, offered\nhim a loan of L500, and another of the same sum if it were required. It\nre-established his affairs.\n\n\nHACKNEY COACHMAN OF THE TIME OF CHARLES II.\n\nThe print from which the engraving on next page is taken, is one of\na set published by Overton, at the sign of the \"White Horse\" without\nNewgate; and its similarity to the figures given by Francis Barlow in\nhis _AEsop's Fables_, and particularly in a most curious sheet-print\netched by that artist, exhibiting Charles the Second, the Duke of\nYork, &c., viewing the Races on Dorset Ferry, near Windsor, in 1687,\nsufficiently proves this Hackney Coachman to have been of the reign of\nthat monarch.\n\nThe early Hackney Coachman did not sit upon the box as the present\ndrivers do, but upon the horse, like a postillion; his whip is short\nfor that purpose; his boots, which have large open broad tops, must\nhave been much in his way, and exposed to the weight of the rain. His\ncoat was not according to the fashion of the present drivers as to the\nnumerous capes, which certainly are most rational appendages, as the\nshoulders never get wet; the front of the coat has not the advantage of\nthe present folding one, as it is single breasted.\n\nHis hat was pretty broad, and so far he was screened from the weather.\nAnother convincing proof that he rode as a postillion is, that his\nboots are spurred. In that truly curious print representing the very\ninteresting Palace of Nonsuch, engraved by Hoefnagle, in the reign of\nQueen Elizabeth, the coachman who drives the royal carriage in which\nthe Queen is seated, is placed on a low seat behind the horses, and has\na long whip to command those he guides. How soon, after Charles the\nSecond's time, the Hackney Coachmen rode on a box we have not been able\nto learn, but in all the prints of King William's time the coachmen are\nrepresented upon the box, though by no means so high as at present;\nnor was it the fashion at the time of Queen Anne to be so elevated as\nto deprive the persons in the carriage of the pleasure of looking over\ntheir shoulders.\n\nIn 1637, the number of Hackney Coaches in London was confined to 50,\nin 1652 to 200, in 1654 to 300, in 1662 to 400, in 1694 to 700, in\n1710 to 800, in 1771 to 1,000, and in 1802 to 1,100. In imitation of\nour Hackney Coaches, Nicholas Sauvage introduced the Fiacres at Paris,\nin the year 1650. The hammer-cloth is an ornamental covering of the\ncoach-box. Mr. S. Pegge says, \"The coachman formerly used to carry a\nhammer, pincers, a few nails, &c., in a leather pouch hanging to his\nbox, and this cloth was devised for the hiding of them from public\nview.\"\n\n[Illustration [++] Hackney Coachman.]\n\nIt is said that the sum of L1,500, arising from the duty on Hackney\nCoaches, was applied to part of the expense in rebuilding Temple Bar.\n\n\nA LONDON WATER-CARRIER IN OLDEN TIMES.\n\nThe conduits of London and its environs, which were established at\nan early period, supplied the metropolis with water until Sir Hugh\nMiddleton brought the New River from Amwell to London, and then the\nconduits gradually fell into disuse, as the New River water was by\ndegrees laid on in pipes to the principal buildings in the City, and,\nin the course of time, let into private houses.\n\nWhen the conduits afforded a supply, the inhabitants either carried\ntheir vessels, or sent their servants for the water as they wanted it;\nbut we may suppose that at an early period there were a number of men\nwho for a fixed sum carried the water to the adjoining houses.\n\nThe figure of a Water-carrier in the following engraving, is copied\nfrom one of a curious and rare set of cries and callings of London,\npublished by Overton, at the \"White Horse\" without Newgate. The figure\nretains the dress of Henry the Eighth's time; his cap is similar to\nthat usually worn by Sir Thomas More, and also to that given in the\nportrait of Albert Durer, engraved by Francis Stock. It appears by this\nprint, that the tankard was borne upon the shoulder, and, to keep the\ncarrier dry, two towels were fastened over him, one to fall before him,\nthe other to cover his back. His pouch, in which we are to conclude he\ncarried his money, has been thus noticed in a very curious and rare\ntract, entitled, _Green's Ghost, with the merry Conceits of Doctor\nPinch-backe_, published 1626: \"To have some store of crownes in his\npurse, coacht in a faire trunke flop, like a boulting hutch.\"\n\n[Illustration [++] Water-Carrier.]\n\n\nEXPENSES OF A ROYAL PRISONER TWO HUNDRED YEARS AGO.\n\nThe following curious document is a return, by the Parliamentary\nCommittee of Revenue, of the expenses of Charles the First and\nhis retinue, during a residence of twenty days, at Holdenby, in\nNorthamptonshire, in the year 1647, commencing February the 13th and\nending March the 4th inclusive. Sir Christopher Hatton had built a\nsplendid mansion at Holdenby in the time of Queen Elizabeth, and\nto it King Charles was conveyed a prisoner by the Parliamentary\nCommissioners, after he had been given up to them by the Scottish\narmy:--\n\n His Majestie's diet of xxviij dishes at xxxl. per diem L700\n The Lords' diet of xx days 520\n For the Clarke of the green cloth, kitchen, and spicery,\n a messe of vij dishes 40\n Dyetts for the household and chamber officers, and the guard 412\n Board wages for common houshold servants, pott and scourers,\n and turnbroaches 36\n Badges of Court and riding wages 140\n For linnen for his Majestie's table, the lords and other diets 273\n For wheat, wood, and cole 240\n For all sorts of spicery store, wax-lights, torches,\n and tallow-lights 160\n For pewter, brasse, and other necessaries incident to all\n officers and carriages 447\n\n\nWOMAN'S CLEVERNESS.\n\nIt is a singular fact that on one occasion the lives of thousands,\nprobably, of the Irish Protestants, were saved by a clever device,\nwhich the unaided wit and presence of mind of a woman enabled her to\nplan and execute.\n\nAt the latter end of Queen Mary's reign, a commission was signed for\nthe purpose of punishing the heretics in that kingdom, and Dr. Cole,\nDean of St. Paul's, was honoured with this _humane_ appointment, to\nexecute which, he set off with great alacrity. On his arrival at\nChester, he sent for the mayor to sup with him, and in the course of\nconversation related his business; then going to his cloak-bag, he took\nout the box containing the commission, and having shewn it, with great\njoy exclaimed, 'This will lash the heretics of Ireland.' Mrs. Edmonds,\nthe landlady, overheard this discourse, and having several relations\nin Ireland, who were Protestants as well as herself, resolved to put a\ntrick upon the doctor; and while he went to attend the magistrate to\nthe door, took the commission out of the box, and in its room placed a\npack of cards, with the knave of clubs uppermost. The zealous doctor,\nsuspecting nothing of the matter, put up his box, took shipping,\nand, arriving safe in Dublin, went immediately to the Viceroy. A\ncouncil was called; and, after a speech, the doctor delivered his box,\nwhich being opened by the secretary, the first thing that presented\nitself was the knave of clubs. This sight surprised the Viceroy and\nthe council, but much more the doctor, who assured them that he had\nreceived a commission from the Queen, but what was come of it, he could\nnot tell. 'Well, well,' replied the Viceroy, 'you must go back for\nanother, and we will shuffle the cards in the mean time.' The doctor\naccordingly hastened across the channel; but at Holyhead he received\nthe intelligence of the Queen's death, and the accession of Elizabeth,\nwho settled on Mrs. Edmonds a pension of forty pounds a year, for\nsaving her Protestant subjects in Ireland.\n\n\nDRESS IN THE PROVINCES IN 1777.\n\nIn the days when mail-coaches had not begun to run, and when railroads\nand telegraphs had not entered into the imagination of man, the style\nof dress in the provinces was often very different to what it was in\nLondon, and on this account the following paragraph is deserving of\nrecord. We have taken it from a copy of the _Nottingham Journal_, of\nSeptember 6, 1777, where it is headed \"Ladies undress.\"--\"The ladies'\nfashionable undress, commonly called a _dishabille_, to pay visits in\nthe morning, also for walking in the country, on account of its being\nneat, light, and short, consists of a jacket, the front part of which\nis made like a sultana; the back part is cut out in four pieces; the\nmiddle part is not wider at the bottom than about half an inch; the\nsides in proportion very narrow. The materials most in vogue are, white\nmuslins with a printed border chintz pattern, printed on\npurpose, in borders about an inch deep. The silks, which are chiefly\nlutestrings, are mostly trimmed with gauze. The gauze is tuckered upon\nthe bottom of the jacket, and edged with different- fringes.\nThe petticoat is drawn up in a festoon, and tied with a true lover's\nknot, two tassels hanging down from each festoon. A short gauze apron,\nstriped or figured, cut in three scollops at the bottom, and trimmed\nround, with a broad trimming closely plaited; the middle of the apron\nhas three scollops reversed. The cuffs are puckered in the shape of a\ndouble pine, one in the front of the arm, the other behind, but the\nfront rather lower. To complete this dress for summer walking, the most\nelegant and delicate ladies carry a long japanned walking-cane, with\nan ivory hook head, and on the middle of the cane is fastened a silk\numbrella, or what the French call 'a parasol,' which defends them from\nthe sun and slight showers of rain. It opens by a spring, and it is\npushed up towards the head of the cane, when expanded for use. Hats,\nwith the feathers spread, chiefly made of chip, covered with fancy\ngauze puckered, variegated artificial flowers, bell tassels, and other\ndecorations, are worn large.\"\n\n\nA GROUP OF RELICS.\n\nThe Dagger of Raoul de Courcy, of which a representation is included\nin the cut over leaf, is an interesting relic, and its authenticity\ncan be relied upon. Raoul de Courcy, according to the old French\nchroniclers was a famous knight, the lord of a noble castle, built\nupon a mountain that overlooks the Valee d'Or, and the descendant of\nthat haughty noble who took for his motto: \"Neither king, nor prince,\nnor duke, nor earl am I, but I am the Lord of Courcy\"--in other words,\ngreater than them all. He fell in love with the wife of his neighbour,\nthe Lord of Fayel, and the beautiful Gabrielle loved him in return.\nOne night he went as usual to meet her in a tower of the Chateau of\nFayel, but found himself face to face with her lord and master. Raoul\nescaped, and Gabrielle was ever after closely guarded. Still they found\nthe opportunity for numerous interviews, at which they interchanged\ntheir vows of love. At length, Raoul, like a true knight, set out to\nfight beneath the banner of the Cross, for the possession of the Holy\nSepulchre. Ere he went, at a stolen meeting, he bade the fair Gabrielle\nadieu, giving to her \"a silken love-knot, with locks of his own hair\nworked in with the threads of silk.\" She gave him a costly ring,\nwhich she had always worn, and which he swore to wear till his last\nbreath. What tears were shed--what kisses were exchanged at this last\nmeeting!--for the Holy Land was very far from France in the Middle Ages.\n\nOn his arrival in Syria, Ralph de Courcy became known as the \"Knight of\nGreat Deeds,\" for it seems he could only conquer his love by acts of\ndaring valour. After braving every danger, he was at length wounded in\nthe side by an arrow, at the siege of Acre. The king of England took\nhim in his arms with respect, and gave him the kiss of hope, but the\narrow was a poisoned one, Raoul felt that he had little time to live.\nHe stretched out his arms towards France, exclaiming, \"France, France!\nGabrielle, Gabrielle!\"\n\nHe resolved to return home, but he was hardly on board the ship that\nwas to waft him there, ere he summoned his squire, and begged of him\nafter he was dead, to carry his heart to France, and to give it the\nLady Fayel, with all the armlets, diamonds, and other jewels which he\npossessed, as pledges of love and remembrance.\n\nThe heart was embalmed, and the squire sought to deliver his precious\nlegacy. He disguised himself in a mean dress, but unluckily met with\nthe Lord of Fayel, and, not knowing him, applied to him for information\nas to how admittance into the chateau could be gained. The Lord of\nFayel at once attacked and disarmed the poor squire, who was wounded in\nthe side with a hunting-hanger. The precious packet was soon torn open,\nand the heart discovered. The Lord of Fayel hastened home, and, giving\nit to his cook, desired that it might be dressed with such a sauce as\nwould make it very palatable.\n\nRaoul's heart was served up at table, and the fair Gabrielle partook of\nit. When she had finished eating, the Lord of Fayel said--\"Lady, was\nthe meat you eat good?\" She replied, that the meat was good. \"That is\nthe reason I had it cooked,\" said the Castellan; \"for know that this\nsame meat, which you found so good, was the heart of Raoul de Courcy.\"\n\n\"Lord of Fayel,\" said Gabrielle, \"the vengeance you have taken\ncorresponds with the meanness of your soul; you have made me eat his\nheart, but it is the last meat I shall ever eat. After such noble food\nI will never partake of any other.\"\n\nShe fainted, and only recovered her consciousness a few minutes before\ndeath. Such is the history of Raoul de Courcy and the Lady Gabrielle,\nas told in the language of the old chroniclers.\n\n[Illustration: 1. Dagger of Raoul de Courcy. 2. Embroidered Glove,\npresented by Mary Queen of Scotland, on the Morning of her Execution,\nto one of her Attendants. 3. Spanish Dagger of the Sixteenth Century.\n4. Ring, with Inscription, \"Behold the End,\" formerly the Property of\nCharles I. 5. Silver Locket, in Memory of the Execution of Charles I.]\n\nThe glove shown in the engraving is said to have been presented by\nthe unfortunate Queen Mary, on the morning of her execution, to a\nlady of the Denny family. The embroidery is of tasteful design, and\nmay be useful as a contrast with many of the patterns for needlework\nat present in fashion. Moreover, the sight of this memorial brings\nto recollection a few particulars in connection with this somewhat\nimportant part of both male and female costume.\n\nThe ancient Persians wore gloves, and the Romans, towards the decline\nof the empire, began to use them. In England they seemed to have been\nintroduced at a very early period. In the Anglo-Saxon literature we\nmeet with _glof_, a covering for the hand, and in the illuminated MSS.\nof that period the hands of bishops and other dignitaries are shown\nencased in gloves which, in many instances, were ornamented with costly\nrings; while on the tombs of kings and queens, &c., the hands are shown\nalmost invariably covered.\n\nIt is related of the patron Saint of Brussels, who lived in the sixth\ncentury, that she was famous for only two miracles: one consisted\nin lighting a candle by means of her prayers, after it had been\nextinguished; the other happened in this way--the fair saint being in a\nchurch barefooted, a person near, with respectful gallantry, took off\nhis gloves and attempted to place them under her feet. This comfort she\ndeclined; and, kicking the gloves away, they became suspended at some\nheight in the church for the space of an hour.\n\nOn opening the tomb of Edward the First, some years ago, in Westminster\nAbbey, the antiquaries assembled on that occasion were surprised to\nfind no traces of gloves. It has been suggested that in this instance\nlinen or silk gloves had been used at the burial of the king, but which\nare supposed to have perished with age.\n\nThe practice of throwing down a glove as a challenge, is mentioned by\nMatthew Paris as far back as 1245; and a glove was worn in the hat or\ncap as a mistress's favour, as the memorial of a friend, and as a mark\nto be challenged by an enemy.\n\nAt a time when the Borders were in a state of incessant strife, Barnard\nGilpin, who has been so justly called \"the Apostle of the North,\"\nwandered unharmed amid the confusion. On one occasion, entering a\nchurch (we believe that of Rothbury, Northumberland,) he observed a\nglove suspended in a conspicuous place, and was informed that it had\nbeen hung up as a challenge by some horse-trooper of the district. Mr.\nGilpin requested the sexton to remove it; who answered, \"Not I sir, I\ndare not do it.\" Then Gilpin called for a long staff, took down the\nglove, and put it in his bosom, and in the course of his sermon, said,\n\"I hear that there is one among you who has even in this sacred place\nhung up a glove in defiance;\" and then producing it in the midst of\nthe congregation, he challenged them to compete with him in acts of\nChristian charity.\n\nGloves, in former times, were common amongst other gifts offered to\nfriends at the new year; and they were received without offence by\nthe ministers of justice. It is related that Sir Thomas More, as Lord\nChancellor, decreed in favour of Mrs. Crooker against the Earl of\nArundel. On the following New-year's day, in token of her gratitude,\nshe presented Sir Thomas with a pair of gloves containing forty angels.\n\"It would be against good manners,\" said the chancellor, \"to forsake\nthe ladies' New-year's gift, and I accept the gloves; the lining you\nmay bestow otherwise.\"\n\nThe custom of the presentation by the sheriff of a pair of white gloves\nto the judge on the occasion of a maiden assize is still in vogue; and,\njudging from the reports in the newspapers, such presents appear to be\nof frequent occurrence.\n\n\"Gloves, as sweet as damask roses,\" were highly prized by Queen\nElizabeth, and, in her day, formed such an important item of a lady's\nexpenses, that a sum was generally allowed for \"glove money.\"\n\nThe old fashioned gloves have now a considerable value amongst the\ncurious. At the sale of the Earl of Arran's goods in 1759, the gloves\ngiven by Henry VIII. to Sir Anthony Denny, sold for 38_l._ 17s.; those\ngiven by James I. to Edward Denny, sold for 22_l._ 4s.; and the mitten\ngiven by Queen Elizabeth to Sir Edward Denny's lady, for 25_l._ 4s.\n\nSome of the English towns which formerly were famous for the\nmanufacture of gloves, still keep up their character. Amongst these\nWoodstock, Yeovil, Leominster, Ludlow, and Worcester may be mentioned.\n\nThe Spanish dagger formerly belonged to a governor of Castile, in the\nsixteenth century, as is shown by the perforated fetter-lock on the\nblade; and although the initials are engraven there also, we have\nnot been able to discover any particulars of the original owner. The\nworkmanship and style of the dagger are of great beauty.\n\nThe little ring with the inscription \"Behold the end,\" was once the\nproperty of Charles I., and was presented by him to Bishop Juxon on the\nmorning of his execution. The silver lockets, on which are the emblems\nof death, were extensively manufactured and sold after the execution of\nCharles I. They generally bore the date of the king's death.\n\n\nTHE HAMSTER RAT.\n\nThere are various kinds of rats, and one of these is the Hamster, of\nthe genus Cricetus of Cuvier. Though rare in Europe to the west of\nthe Rhine, it is widely spread from that river to the Danube on the\nsouth-west, and north-easterly through a vast extent of country into\nSiberia. We notice it in our pages on account of its extraordinary\nhabits. Its life appears to be divided between eating and fighting.\nIt seems to have no other passion than that of rage, which induces it\nto attack every animal that comes in its way, without in the least\nattending to the superior strength of its enemy. Ignorant of the art\nof saving itself by flight, rather than yield, it will allow itself to\nbe beaten to pieces with a stick. If it seizes a man's hand, it must\nbe killed before it will quit its hold. The magnitude of the horse\nterrifies it as little as the address of the dog, which last is fond of\nhunting it. When the hamster perceives a dog at a distance, it begins\nby emptying its cheek-pouches if they happen to be filled with grain;\nit then blows them up so prodigiously, that the size of the head and\nneck greatly exceed that of the rest of the body. It raises itself on\nits hind legs, and thus darts upon the enemy. If it catches hold, it\nnever quits it but with the loss of its life; but the dog generally\nseizes it from behind, and strangles it. This ferocious disposition\nprevents the hamster from being at peace with any animal whatever. It\neven makes war against its own species. When two hamsters meet, they\nnever fail to attack each other, and the stronger always devours the\nweaker. A combat between a male and a female commonly lasts longer\nthan between two males. They begin by pursuing and biting each other,\nthen each of them retires aside, as if to take breath. After a short\ninterval, they renew the combat, and continue to fight till one of them\nfalls. The vanquished uniformly serves for a repast to the conqueror.\n\n\nKNAVERY OF THE PRIESTS IN BURMAH.\n\n[Illustration: Burmese Priest Preaching.]\n\nThe manner in which an uncivilized people will calmly submit to be\nduped by the extortionate rascality of their priests, is strongly\nexhibited in the kingdom of Burmah. The people who are there held in\nthe highest estimation are the priests. Any one who pleases may be\na priest. The priests pretend to be poor, and go out begging every\nmorning with their empty dishes in their hands; but they get them well\nfilled, and then return to their handsome houses, all shining with\ngold, in which they live together in plenty and in pride. They are\nexpected to dress in rags, to show that they are poor; but not liking\nrags, they cut up cloth in little pieces, and sew the pieces together\nto make their yellow robes; and this they call wearing rags. They\npretend to be so modest, that they do not like to show their faces, and\nso hide them with a fan, even when they preach; for they do preach in\ntheir way, that is, they tell foolish stories about Buddha. The name\nthey give him is Guadama, while the Chinese call him Fo. They have\nfive hundred and fifty stories written in their books about him; for\nthey say he was once a bird, a fly, an elephant, and all manner of\ncreatures, and was so good whatever he was, that at last he was born\nthe son of a king. Is it not marvellous that a whole people should, for\ngeneration after generation, not only submit to be thus scandalously\ncheated, but should also hold those who cheat them in the highest\nesteem? A curious fact, indeed, in the history of mankind.\n\n\nMIRACULOUS ESCAPE.\n\nOne of the most singular circumstances occurred a few years ago that\never came within our observation. Mr. Charlton, surgeon, of Wylam, near\nNewcastle-on-Tyne, having at a late hour been called upon in haste\nto give his attendance at Ovingham, borrowed a spirited horse of a\nfriend, that he might proceed with the least possible delay. He had\nnot gone above half a mile when he perceived his horse stumble, and\nhe immediately threw himself from the saddle. It was fortunate he did\nso, for the next instant his horse had fallen down a precipice of near\nseventy feet; and, incredible as it may seem, the animal sustained no\ninjury, but immediately dashed into the Tyne, and swam to the opposite\nside. Search was made after him, and hearing his master's voice, he was\nheard to neigh even across the water in token of recognition, and was\nultimately restored without speck or blemish.\n\n\nA NATIONAL TASTE FOR GAMING.\n\nIt is a remarkable fact that a taste for gaming appears in some cases\nto pervade a whole people, and to become one of the chief national\ncharacteristics. No where is this more manifest than among the\ninhabitants of the Asiatic Islands.\n\nGames of hazard are the favourites of these islanders. Some of them\nthey have learned of the Chinese, the most debauched of gamesters, and\nothers of the Portuguese. The only game of hazard, of native origin,\namong the Javanese consists in guessing the number of a certain kind of\nbeans which the players hold in their hands.\n\nBut of all the species of gaming that to which the Indian islanders\nare most fondly addicted is betting on the issue of the combats of\npugnacious animals, and particularly the cock. The breed in highest\nestimation is the produce of Celebes. The people of Java fight their\ncocks without spurs; but the Malays and natives of Celebes with an\nartificial spur, in the shape of a small scythe, which, notwithstanding\nits barbarous appearance, is in reality less destructive than the\ncontrivance employed among ourselves.\n\nQuail fighting also is extremely common in Java. The most famous breed\nof this bird is found in the island of Lombok; and it is a singular\nfact, that the female is used in these bitter but bloodless combats,\nthe male being comparatively small and timid. Neither do the Javanese\nhesitate to bet considerable sums on a battle between two crickets,\nwhich are excited to the conflict by the titillation of a blade of\ngrass judiciously applied to their noses. They will likewise risk their\nmoney on the strength and hardness of a nut, called _kamiri_; and much\nskill, patience and dexterity, are exercised in the selection and\nthe strife. At other times two paper kites decide the fortune of the\nparties; the object of each in this contest being to cut the string of\nhis adversary. On a favourable day fifty or sixty kites, raised for\nthis purpose, may sometimes be seen hovering over a Javanese city.\n\n\nA FRIEND TO PHYSIC.\n\nMr. Samuel Jessup, who died at Heckington, Lincolnshire, in 1817,\nwas an opulent grazier and of pill-taking memory. He lived in a\nvery eccentric way, as a bachelor, without known relatives, and at\nhis decease was possessed of a good fortune, notwithstanding a most\ninordinate craving for physic, by which he was distinguished for the\nlast thirty years of his life, as appeared on a trial for the amount of\nan apothecary's bill, at the assizes at Lincoln, a short time before\nMr. Jessup's death, wherein he was defendant. The evidence on the trial\naffords the following materials for the epitaph of the deceased, which\nwill not be transcended by the memorabilia of the life of any man. In\ntwenty-one years (from 1791 to 1816) the deceased took 226,934 pills\n(supplied by a most highly respectable apothecary and worthy person\nof the name of Wright, who resided at Bottesford), which is at the\nrate of 10,806 pills a year, or 29 pills each day; but as the patient\nbegun with a more moderate appetite, and increased it as he proceeded,\nin the last five years preceding 1816, he took the pills at the rate\nof 78 a-day, and in the year 1814, he swallowed not less than 51,590.\nNotwithstanding this, and the addition of 40,000 bottles of mixture,\nand juleps and electuaries, extending altogether to fifty-five closely\nwritten columns of an apothecary's bill, the deceased lived to attain\nthe advanced age of sixty-five years.\n\n\nAN INCULPATORY EPITAPH.\n\nThe following epitaph at West Allington, Devon, is deserving a place in\nour record of curiosities, inasmuch as it appears to be a successful\nattempt in making a monumental stone, both a memorial of the deceased,\nand also a means of reproving the parson of the parish:--\n\n \"Here lyeth the Body of\n Daniel Jeffery the Son of Michael\n Jeffery and Joan his Wife he\n was buried y{e} 22 day of September\n 1746 and in y{e} 18{th} year of his age.\n This Youth When In his sickness lay\n did for the minister Send + that he would\n Come and With him Pray + But he would not ate{nd}\n But When this young man Buried was\n The minister did him admit + he should be\n Caried into Church + that he might money geet\n By this you See what man will dwo + to geet\n money if he can + who did refuse to come\n pray + by the Foresaid young man.\"\n\n\nHUNTING A SHEEP KILLER.\n\nIt has been remarked, that when once a dog acquires wild habits, and\ntakes to killing sheep, he does far more mischief than a wild beast,\nsince to the cunning of the tamed animal he adds the ferocity of the\nuntamed. A remarkable case of this sort is mentioned in the following\nparagraph, which we have copied from the _Newcastle Courant_ of the\nyear 1823. It is also curious to note the account of the chase, and of\nthe joy which the whole country-side seems to have manifested at the\nslaughter of the animal.--September 21--A few days ago a dog of a most\ndestructive nature infested the fells of Caldbeck, Carrock, and High\nPike, about sixteen miles south of Carlisle. Little doubt remains of\nits being the same dog which has been so injurious to the farmers in\nthe northern parts of Northumberland, as no less than sixty sheep or\nupwards have fallen victims to its ferocity. It was thought proper to\nlose no time in attempting to destroy it, and Tuesday last was fixed\nupon. Sir H. Fletcher, Bart., of Clea Hall, offered his pack of hounds,\nand several other dogs with about fifty horsemen set out from Hesket\nNew-market. Several persons with firearms were stationed at different\nparts. The dog was descried upon an eminence of Carrock-fell, and on\nsight of the pursuers set off by way of Hesket New-market, Stocklewath,\nand Barwick-field, then returned by Cowclose, Castle Sowerby, and\nattempted to gain the fells again, when Mr. Sewell, farmer at Wedlock,\nlying in ambush at Mossdale, fired, and succeeded in shooting him. He\nappears to be of the Newfoundland breed, of a common size, wire-haired,\nand extremely lean. During the chase he frequently turned upon the dogs\nwhich were headmost, and so wounded several as obliged them to give up\nthe pursuit. The joy manifested on this occasion was uncommon, insomuch\nthat on the day following about thirty persons sat down to a dinner\nprovided at Mr. Tomlinson's, Hesket New-market. Upon the most moderate\ncomputation, excluding the various windings, the chase could not be\nless than thirty miles, and occupied no less than six hours.\n\n\nLONGEVITY.\n\nHenry Jenkins, of Ellerton-upon-Swale, Yorkshire, died 1670, aged 169.\nHe remembered the battle of Flodden Field, fought between the English\nand the Scotch, September 9, 1513, when he was about twelve years old.\nHe was then sent to Northallerton with a cartload of arrows, but an\nolder boy was employed to convey them to the army. At Ellerton there\nwas also living, at the same time, four or five other old men, reputed\nto be of the age of one hundred years and thereabouts, and they all\ntestified that Jenkins was an elderly man when first they knew him.\nJenkins was once butler to Lord Conyers; he perfectly remembered the\nAbbot of Fountain's Dale before the dissolution of the monasteries.\nIn the last century of his life he was a fisherman, and often swam in\nthe river after he was a hundred years old. In the King's Remembrancer\nOffice in the Exchequer, there is a record of a deposition in a cause,\ntaken April, 1665, at Kettlewell, Yorkshire, where Henry Jenkins, of\nEllerton-upon-Swale, labourer, aged 157 years, was produced, and made\ndeposition as a witness. He was buried at Bolton, Yorkshire. In 1743,\na monument, with a suitable inscription, was erected to perpetuate his\nmemory.\n\n\nTHE PULPIT OF JOHN KNOX AT ST. ANDREW'S.\n\nJohn Knox, the great precursor of the Protestant Reformation, having\nbeen driven from Edinburgh by the threats of his opponents, reluctantly\nwithdrew to St. Andrew's, in the county of Fife, where he continued\nwith undiminished boldness to denounce the enemies of the reformed\nfaith. It was in that place that he had first discoursed against the\ndegeneracy of the Church of Rome, and there he occupied the Pulpit\nrepresented in the accompanying engraving; and the following curious\nand characteristic anecdote connected with his preaching in it, is\nrelated in the Manuscript Diary of James Melville, then a student at\nthe college of St. Andrew's, and subsequently Minister of Anstruther.\n\"Of all the benefits I haid that year (1571) was the coming of that\nmaist notable profet and apostle of our nation, Mr. Jhone Knox, to St.\nAndrew's: who, be the faction of the Queen occupying the castell and\ntown of Edinburgh, was compellit to remove therefra, with a number\nof the best, and chusit to come to St. Andrew's. I heard him teache\nthere the Prophecies of Daniel that simmer, and the winter following;\nI haid my pen and my little buike, and tuk away sic things as I could\ncomprehend. In the opening up of his text he was moderat the space of\nan half houre; but when he onterit to application, he made me so to\n_grew_ (thrill) and tremble, that I could not hold a pen to wryt. He\nwas very weak. I saw him every day of his life go _hulie and fear_\n(hoolie and fairly--slowly and warily) with a furring of marticks,\n(martins) about his neck, a staffe in the ane hand, and gud godlie\nRichard Ballanden, his servand, haldin up the uther _oxier_ (arm-pit),\nfrom the Abbey to the Parish-Kirk; and be the said Richart and another\nservant lifted up to the Pulpit, whar he _behovit_ (was obliged) to\nlean at his first entry: bot er he had done with his sermone he was sa\nactive and vigourous, that he was lyk to _ding the pulpit in blads_\n(beat it into shivers) and flie out of it.\"\n\n[Illustration [++] Pulpit of John Knox at St. Andrew's.]\n\nThe interesting relique commemorated in this curious extract, is\nof that stately style of carving which was introduced towards the\nclose of the sixteenth century in Protestant preaching-places; and\ncontinued, though of a more heavy character, throughout the whole of\nthe succeeding century. A scroll-bracket remaining on the preacher's\nleft hand, and some broken pieces at the top of the back, appear to\nindicate that it was once more extended, and had probably a canopy or\nsounding-board.\n\n\nTHE BIBLE USED BY KING CHARLES THE FIRST ON THE SCAFFOLD.\n\n[Illustration [++] Bible Used by Charles the First on the Scaffold.]\n\nThere is so much external evidence of the genuineness of this very\nbeautiful and interesting relique, that no doubt can exist as to its\nperfect authenticity, though the circumstance of the King having a\nBible with him on the scaffold, and of presenting it to Dr. Juxon, is\nnot mentioned in any contemporaneous account of his death. The only\nnotice of such a volume, as a dying gift, appears to be that recorded\nby Sir Thomas Herbert, in his narrative, which forms a part of the\n_Memoirs of the last Two Years of the Reign of that unparalleled Prince\nof ever-blessed memory, King Charles I._ London, 1702, 8vo, p. 129, in\nthe following passage:--\"The King thereupon gave him his hand to kiss:\nhaving the day before been graciously pleased under his royal hand,\nto give him a certificate that the said Mr. Herbert was not imposed\nupon him, but by his Majesty made choice of to attend him in his\nbed-chamber, and had served him with faithfulness and loyal affection.\nHis Majesty also delivered him his Bible, in the margin whereof he\nhad with his own hand, written many annotations and quotations, and\ncharged him to give it to the Prince so soon as he returned.\" That this\nmight be the book represented in our engraving, is rendered extremely\nprobable, by admitting that the King would be naturally anxious, that\nhis son should possess that very copy of the Scriptures which had been\nprovided for himself when he was Prince of Wales. It will be observed\nthat the cover of the volume is decorated with the badge of the\nPrincipality within the Garter, surmounted by a royal coronet in silver\ngilt, inclosed by an embroidered border; the initials C. P. apparently\nimproperly altered to an R., and the badges of the Rose and Thistle,\nupon a ground of blue velvet: and the book was therefore bound between\nthe death of Prince Henry in 1612, and the accession of King Charles\nto the throne in 1625, when such a coronet would be no longer used by\nhim. If the Bible here represented were that referred to by Herbert,\nthe circumstance of Bishop Juxon becoming the possessor of it might\nbe accounted for, by supposing that it was placed in his hands to be\ntransmitted to Charles II. with the George of the Order of the Garter\nbelonging to the late King, well known to have been given to that\nPrelate upon the scaffold, January 30th, 1648-9.\n\n\nLAMBETH WELLS, THE APOLLO GARDENS, AND FINCH'S GROTTO.\n\nAmong the numerous public places of amusement which arose upon the\nsuccess of Vauxhall Gardens, which were first opened about 1661, was\none in Lambeth Walk, known as Lambeth Wells. This place was first\nopened on account of its mineral waters, which were sold at a penny\nper quart. The music commenced at seven o'clock in the morning, and\nthe price of admission was three pence. A monthly concert under the\ndirection of Mr. Starling Goodwin, organist of St. Saviour's Church\nSouthwark, was afterwards held here, and Erasmus King, who had been\ncoachman to the celebrated Dr. Desaguliers, read lectures and exhibited\nexperiments in natural philosophy, the price of admission being raised\nto sixpence.\n\nThis place was open before 1698, and existed as late as 1752, when \"A\nPenny Wedding after the Scotch fashion, for the benefit of a young\ncouple,\" was advertised to be kept there.\n\nLambeth Wells at length becoming a public nuisance, the premises were\nshut up, and ultimately let as a Methodist Meeting-house. The music\ngallery was used as a pulpit; but the preacher being greatly disturbed\nin his enthusiastic harangues, he was obliged to quit, when the\npremises were converted to various purposes, except the dwelling, which\nis now known by the sign of the Fountain public-house.\n\nOn the site of Messrs. Maudslay's factory, in the Westminster Road,\nformerly stood the Apollo Gardens. This place of amusement was opened\nin 1788, by an ingenious musician named Clagget, who published, in\n1793, a small quarto pamphlet, entitled \"Musical Phenomena: An Organ\nmade without Pipes, Strings, Bells, or Glasses; the only Instrument in\nthe world that will never require to be re-tuned. A Cromatic Trumpet,\ncapable of producing just Intervals, and regular Melodies in all Keys,\nwithout undergoing any change whatever. A French Horn answering the\nabove description of the Trumpet.\"\n\nThe Apollo Gardens had one spacious room elegantly fitted up, and\ndecorated in taste suitably to its intention. The gardens consisted\nof a number of elegant pavilions or alcoves, well adapted for the\naccommodation of different companies; they were ornamented chiefly with\na succession of paintings, relating to romantic histories, particularly\nthe different adventures of Don Quixote. It had a fine orchestra\nerected in the centre of the gardens. The place being ultimately\nconverted into a receptacle for loose and dissolute characters, the\nmagistracy very properly suppressed it about the year 1799.\n\nIn Gravel Lane, Southwark, was Finch's Grotto, a public garden and\nplace of amusement, so named from William Finch, the proprietor. The\nGrotto was opened to the public in 1770 upon the plan of Vauxhall\ngardens. An orchestra and a band of musicians, added to the rural\ncharacter of the place, and drew a numerous body of visitors.\n\nVery little is known about the Grotto, but it is supposed to have been\nclosed early in the present century.\n\n\nTHE DUCK-BILLED PLATYPUS, OR, ORNITHORYNCHUS PARADOXUS.\n\n[Illustration [++] Duck-billed Platypus.]\n\nOf the genus _Ornithorynchus_ only one species--the _Paradoxus_--has\nyet been discovered in the whole world, and it is, therefore, one of\nthe great curiosities of animal life. It appears to be a union of a\nquadruped and a bird, and is only to be found in New Holland, where it\ninhabits the reeds by the side of rivers. Our engraving represents it\nvery accurately. It is about twenty inches long, having a flattened\nbody, somewhat like the otter, and is clothed with a dark soft fur.\nThe elongated nose very much resembles the beak of a duck, like which\nthese animals feed upon water insects, shell-fish, and aquatic plants.\nThe feet are five-toed and webbed, and in the fore-feet this membrane\nextends beyond the nails: the male is armed with a spur on each hind\nleg. This curious animal, in which a duck's beak is united to the body\nof a quadruped, rolls itself up like a hedgehog, when it sleeps in its\nburrows on the banks of the streams whence its food is derived.\n\n\nORIGIN OF BOLTON ABBEY.\n\nAbout midway up the Vale of Bolton, amidst the gloomy recesses of\nthe woods, the Wharfe, which is otherwise a wide and shallow river,\nis suddenly contracted by two huge rocks, which approach each other\nso nearly, that the country folk, or rather the villagers, call it\nthe _Strid_, because adventurous people stride or leap from one rock\nto the other. In ancient days, the whole of this valley belonged to\nBaron Romillie, whose eldest son having died, left a younger brother,\nof the name of EGREMONT, sole heir of the domains and inheritance of\nthis family. One day, however, when this young man, familiarly called\nthe \"Boy of Egremont,\" was returning from hunting with the hounds in\nthe _leash_, he, as he had done many times before, was going to leap\nthe _Strid_, when, just as he had attempted it, the hounds held back,\nand precipitated him headlong into the deep and awful chasm, which the\nimpetuous fall of water (thus produced by the sudden contraction of the\nriver) had worn in the base of the two rude rocks, and he was never\nseen afterwards. The Baron, being now left childless, built the Abbey,\nand endowed it with the domains of Bolton.\n\n\nLENGTH OF LIFE WITHOUT BODILY EXERCISE.\n\nThe Rev. William Davies, Rector of Staunton-upon-Wye, and Vicar of\nAll Saints, Hereford, died 1790, aged 105. The life of this gentleman\ndisplays one of the most extraordinary instances of departure from all\nthose rules of temperance and exercise, which so much influence the\nlives of the mass of mankind, that is, probably to be found in the\nwhole records of longevity. During the last thirty-five years of his\nlife, he never used any other exercise than that of just slipping his\nfeet, one before the other, from room to room; and they never after\nthat time were raised, but to go down or up stairs, a task, however, to\nwhich he seldom subjected himself. His breakfast was hearty; consisting\nof hot _rolls well buttered_, with a plentiful supply of tea or coffee.\nHis dinner was substantial, and frequently consisted of a variety of\ndishes. At supper he generally eat hot roast meat, and always drank\nwine, though never to excess. Though nearly blind for a number of\nyears, he was always cheerful in his manners, and entertaining in his\nconversation, and was much beloved by all who knew him. He had neither\ngout, stone, paralysis, rheumatism, nor any of those disagreeable\ninfirmities which mostly attend old age; but died peaceably in the\nfull possession of all his faculties, mental and corporeal, save his\neyesight. Like most long livers he was very short of stature.\n\n\nEXTRAORDINARY FASHION IN CIGARS.\n\nA taste for tobacco in some form or other seems to extend over the\nwhole inhabitable globe. In this respect it matters not whether nations\nare civilized or uncivilized; and however completely they may differ\nfrom each other in everything else, they all agree in a fondness for\n\"the weed.\" In the mode, however, of indulging in the luxury, there is\nthe greatest diversity, and no where is this more strikingly manifested\nthan in the Philippine Islands.\n\n\"It is not till evening that the inhabitants of the higher class begin\nto stir; till that time they are occupied in eating, sleeping, and\nsmoking tobacco, which is no where more general than on the island of\nLuzon; for children, before they can walk, begin to smoke segars. The\nwomen carry their fondness for it to a greater height than the men;\nfor, not content with the usual small segars, they have others made\nfor them, which are a foot long and proportionably thick. These are\nhere called the women's segars, and it is a most ludicrous sight to see\nelegant ladies taking their evening walk, with these burning brands in\ntheir mouths.\"\n\nHow widely does the fashion in Luzon differ from the fashion at Paris!\n\n\nNOVEL WAY OF PURCHASING A HUSBAND.\n\nThe following paragraph, which we have copied from a magazine of 1790,\nnot only gives us a curious instance of female determination in the\npursuit of a husband, but tells us of the price which human hair was\nworth at the period when ladies wore such monstrous head-dresses of\nfalse curls.\n\n\"An Oxfordshire lass was lately courted by a young man of that country,\nwho was not willing to marry her unless her friends could advance\n50_l._ for her portion; which they being incapable of doing, the lass\ncame to London to try her fortune, where she met with a good chapman\nin the Strand, who made a purchase of her hair (which was delicately\nlong and light), and gave her _sixty pounds_ for it, being 20 ounces\nat 3_l. an ounce_; with which money she joyfully returned into the\ncountry, and bought her a husband.\"\n\n\nGLOVES.--ORIGIN OF \"PIN MONEY.\"\n\nGloves were very common as New Year's gifts. For many hundreds of years\nafter their introduction into England in the 10th century, they were\nworn only by the most opulent classes of society, and hence constituted\na valuable present. They are often named in old records. Exchange of\ngloves was at one period a mode of investiture into possession of\nproperty, as amongst the ancient Jews was that of a shoe or sandal; and\n\"glove-money\" is to this day presented by High Sheriffs to the officers\nof their courts, upon occasion of a maiden assize, or one in which\nno cause is tried. Pins, which at the commencement of the sixteenth\ncentury displaced the wooden skewers previously in use, became a\npresent of similar consequence; and at their first introduction were\nconsidered of so much importance in female dress, that \"pin-money\" grew\ninto the denomination of dower, which, by the caution of parents, or\njustice of a consort, was settled upon a lady at her marriage.\n\n\nHABITS AND HABITATIONS OF THE DYAKS OF BORNEO.\n\nIt is impossible to appreciate properly the courage, determination,\nand skill which have been displayed by the gallant Sir James Brooke,\nunless we make ourselves acquainted with the character and habits of\nthe extraordinary race of men over whom he triumphed. The Dyaks are a\nsavage people who inhabit Borneo. They lived there before the Malays\ncame, and they have been obliged to submit to them. They are savages\nindeed. They are darker than the Malays; yet they are not black; their\nskin is only the colour of copper. Their hair is cut short in front,\nbut streams down their backs; their large mouths show a quantity of\nblack teeth, made black by chewing the betel-nut. They wear but very\nlittle clothing, but they adorn their ears and arms, and legs, with\nnumbers of brass rings. Their looks are wild and fierce, but not\ncunning like the looks of the Malays. They are not Mahomedans; they\nhave hardly any religion at all. They believe there are some gods, but\nthey know hardly anything about them, and they do not want to know.\nThey neither make images to the gods, nor say prayers to them. They\nlive like the beasts, thinking only of this life; yet they are more\nunhappy than beasts, for they imagine there are evil spirits among the\nwoods and hills, watching to do them harm. It is often hard to persuade\nthem to go to the top of a mountain, where they say evil spirits dwell.\nSuch a people would be more ready to listen to a missionary than those\nwho have idols, and temples, and priests, and sacred books.\n\n[Illustration: Dyak With Heads.]\n\n[Illustration: Head of a Dyak.]\n\nTheir wickedness is very great. It is their chief delight to get the\nheads of their enemies. There are a great many different tribes of\nDyaks, and each tribe tries to cut off the heads of other tribes. The\nDyaks who live by the sea are the most cruel; they go out into the\nboats to rob and bring home, not _slaves_, but HEADS!! And how do they\ntreat a head when they get it? They take out the brains, and then they\ndry it in the smoke, with the flesh and hair still on; then they put a\nstring through it, and fasten it to their waists. The evening that they\nhave got some new heads, the warriors dance with delight,--their heads\ndangling by their sides;--and they turn round in the dance, and gaze\nupon their heads,--and shout,--and yell with triumph! At night they\nstill keep the heads near them; and in the day they play with them,\nas children with their dolls, talking to them, putting food in their\nmouths, and the betel-nut between their ghastly lips. After wearing the\nheads many days, they hang them up to the ceilings of their rooms.\n\nNo English lord thinks so much of his pictures, as the Dyaks do of\ntheir heads. They think these heads are the finest ornaments of their\nhouses. The man who has _most_ heads, is considered the _greatest_ man.\nA man who has _no heads_ is despised! If he wishes to be respected, he\nmust get a head as soon as he can. Sometimes a man, in order to get a\nhead, will go out to look for a poor fisherman, who has done him no\nharm, and will come back with his head. When the Dyaks fight against\ntheir enemies, they try to get, not only the heads of men, but also\nthe heads of women and children. How dreadful it must be to see a poor\nbaby's head hanging from the ceiling! There was a Dyak who lost all\nhis property by fire, but he cared not for losing anything, so much as\nfor losing his precious heads; nothing could console him for his loss;\nsome of them he had cut off himself, and others had been cut off by his\nfather, and left to him!\n\n[Illustration: House of Sea Dyaks.]\n\n[Illustration: Skull House.]\n\nPeople who are so bent on killing, as these Dyaks are, must have many\nenemies. The Dyaks are always in fear of being attacked by their\nenemies. They are afraid of living in lonely cottages; they think it a\nbetter plan for a great many to live together, that they may be able\nto defend themselves, if surprised in the night. Four hundred Dyaks\nwill live together in one house. The house is very large. To make it\nmore safe, it is built upon very high posts, and there are ladders to\nget up by. The posts are sometimes forty feet high; so that when you\nare in the house, you find yourself as high as the tall trees. There\nis one very large room, where all the men and women sit, and talk,\nand do their work in the day. The women pound the rice, and weave the\nmats, while the men make weapons of war, and the little children play\nabout. There is always much noise and confusion in this room. There are\na great many doors along one side of the long room; and each of these\ndoors leads into a small room where a family lives! the parents, the\nbabies, and the girls sleep there, while the boys of the family sleep\nin the large room, that has just been described.\n\nThe Hill Dyaks do not live in houses quite so large. Yet several\nfamilies inhabit the same house. In the midst of their villages, there\nis always one house where the boys sleep. In this house all the heads\nof the village are kept. The house is round, and built on posts, and\nthe entrance is underneath, through the floor. As this is the best\nhouse in the village, travellers are always brought to this house to\nsleep. Think how dreadful it must be, when you wake in the night to see\nthirty or forty horrible heads, dangling from the ceiling! The wind,\ntoo, which comes in through little doors in the roof, blows the heads\nabout; so that they knock against each other, and seem almost as if\nthey were still alive. This is the Dead-house. Such are the men whom\nthe Rajah Brooke subdued!\n\n\nSCOTTISH WILD CATTLE.\n\nThe wild white cattle, a few of which are still to be found in\nChatelherault Park, belonging to the Duke of Hamilton, in Lanarkshire,\nare great objects of curiosity, inasmuch as they are identical with the\nprimitive source of all our domestic cattle.\n\nThe following description of their habits is abridged from an article\nby the Rev. W. Patrick, in the Quarterly Journal of Agriculture:--\n\n\"I am inclined to believe that the Hamilton breed of cattle is the\noldest in Scotland, or perhaps in Britain. Although Lord Tankerville\nhas said they have 'no wild habits,' I am convinced, from personal\nobservation, that this is one of their peculiar features. In browsing\ntheir extensive pasture, they always keep close together, never\nscattering or straggling over it, a peculiarity which does not belong\nto the Kyloe, or any other breed, from the wildest or most inhospitable\nregions of the Highlands. The white cows are also remarkable for their\nsystematic manner of feeding. At different periods of the year their\ntactics are different, but by those acquainted with their habits they\nare always found about the same part of the forest at the same hour of\nthe day. In the height of summer, they always bivouac for the night\ntowards the northern extremity of the forest; from this point they\nstart in the morning, and browse to the southern extremity, and return\nat sunset to their old rendezvous; and during these perambulations they\nalways feed _en masse_.\n\n\"The bulls are seldom ill-natured, but when they are so they display\na disposition more than ordinarily savage, cunning, pertinacious,\nand revengeful. A poor bird-catcher, when exercising his vocation\namong the 'Old Oaks,' as the park is familiarly called, chanced to be\nattacked by a savage bull. By great exertion he gained a tree before\nhis assailant made up to him. Here he had occasion to observe the\nhabits of the animal. It did not roar or bellow, but merely grunted,\nthe whole body quivered with passion and savage rage, and he frequently\nattacked the tree with his head and hoofs. Finding all to no purpose,\nhe left off the vain attempt, began to browse, and removed to some\ndistance from the tree. The bird-catcher tried to descend, but this\nwatchful Cerberus was again instantly at his post, and it was not till\nafter six hours' imprisonment, and various bouts at 'bo-peep' as above,\nthat the unfortunate man was relieved by some shepherds with their\ndogs. A writer's apprentice, who had been at the village of Quarter on\nbusiness, and who returned by the 'Oaks' as a 'near-hand cut,' was also\nattacked by one of these savage brutes, near the northern extremity of\nthe forest. He was fortunate, however, in getting up a tree, but was\nwatched by the bull, and kept there during the whole of the night, and\ntill near two o'clock the next day.\n\n\"These animals are never taken and killed like other cattle, but are\nalways shot in the field. I once went to see a bull and some cows\ndestroyed in this manner--not by any means for the sake of the sight,\nbut to observe the manner and habits of the animal under peculiar\ncircumstances. When the shooters approached, they, as usual, scampered\noff in a body, then stood still, tossed their heads on high, and\nseemed to snuff the wind; the manoeuvre was often repeated, till they\ngot so hard pressed (and seemingly having a sort of half-idea of the\ntragedy which was to be performed), that they at length ran furiously\nin a mass, always preferring the sides of the fence and sheltered\nsituations, and dexterously taking advantage of any inequality in\nthe ground, or other circumstances, to conceal themselves from the\nassailing foe. In their flight, the bulls, or stronger of the flock,\nalways took the lead! a smoke ascended from them which could be seen at\na great distance; and they were often so close together, like sheep,\nthat a carpet would have covered them. The cows which had young, on the\nfirst 'tug of war,' all retreated to the thickets where their calves\nwere concealed; from prudential motives, they are never, if possible,\nmolested. These and other wild habits I can testify to be inherent\nin the race, and are well known to all who have an opportunity of\nacquainting themselves with them.\"\n\n\nBELLS OF THE ANCIENTS.\n\nBells were known in the earliest ages of which we have any certain\naccount. But the bells of the ancients were very small in comparison\nwith those of modern times, since, according to Polydore Virgil, the\ninvention of such as are hung in the towers, or steeples of Christian\nchurches, did not occur till the latter end of the fourth, or beginning\nof the fifth century; when they were introduced by Paulinus, Bishop of\nNola. The Jews certainly employed bells, since they are spoken of in\nScriptures; and the mention of them by Thucydides, Diodorus Siculus,\nSuidas, Aristophanes, and other ancient writers, proves that they\nwere used in Greece; while Plautus, Ovid, Tibullus, Statius, and a\nvariety of Latin authors, speak of bells as in use among the Romans.\nBut these bells of the ancients were all made for the hand; or were\nof a size to be affixed to other musical instruments, like those\nwhich were occasionally appended to the drum. Whether, when detached\nfrom other instruments, they were used on other occasions, or only in\nparticular ceremonies, or as signals, is not known; nor have we any\nclue by which to guess whether they were tuned in concordance with any\nscale, or whether they were unisons to each other, or not formed to\nany particular pitch, but merely used as sonorous auxiliaries to other\ninstruments, without any regard to their agreement of tone, either with\none another, or with the instruments they accompanied.\n\n\nEARTHQUAKE AT NOTTINGHAM IN 1816.\n\nEarthquakes are providentially occurrences of great rarity in England.\nThe one which took place on the 17th of March, 1816, was one of the\nmost dangerous that has ever been experienced in this kingdom. It\nextended over a vast area of country, and in some localities its\neffects were felt very severely. As a proof of this, we have copied the\nfollowing paragraph from a Nottingham paper of the day:--\n\nNottingham, in common with a great part of the North Midland district,\nexperienced a smart shock of an earthquake. It was felt at half-past\ntwelve p.m., and as Divine service, it being Sunday, was not over\nat the churches, great alarm was expressed by the congregations. At\nSt. Peter's and St. Nicholas's, the consternation was so great, that\nservice had to be suspended for a few seconds, and one lady was borne\nout in a state of insensibility. The pillars supporting St. Mary's\ntower shook very visibly, but, fortunately, the attention of the\ncrowded congregation was so engrossed by the eloquence of the sheriff's\nchaplain, and the presence of the Judge and his retinue, that the alarm\nwas but slight, or the rush and loss of life might have been great.\nIn various parts of the town and neighbourhood, glasses were shaken\noff of shelves, articles of domestic use displaced, window-casements\nthrown open, and other indications manifest of the influence of the\nsubterranean movement.\n\n\nSINGULAR STATE OF PRESERVATION OF A DEAD BODY.\n\nAccording to a statement in Holinshed, in 1495, while digging for a\nfoundation for the church of St. Mary-at-hill, in London, the body of\nAlice Hackney was discovered. It had been buried 175 years, and yet the\nskin was whole, and the joint pliable. It was kept above ground four\ndays without annoyance, and then re-interred.\n\n\nASYLUM FOR DESTITUTE CATS.\n\nOf all the curious charitable institutions in the world, the most\ncurious probably is the Cat Asylum at Aleppo, which is attached to\none of the mosques there, and was founded by a misanthropic old Turk,\nwho being possessed of large granaries, was much annoyed by rats and\nmice, to rid himself of which he employed a legion of cats, who so\neffectually rendered him service, that in return he left them a sum\nin the Turkish funds, with strict injunctions that all destitute\nand sickly cats should be provided for, till such time as they took\nthemselves off again. In 1845, when a famine was ravaging in all\nNorth Syria--when scores of poor people were dropping down in the\nstreets from sheer exhaustion and want, and dying there by dozens per\ndiem before the eyes of their well-to-do fellow creatures, men might\ndaily be encountered carrying away sack loads of cats to be fed up and\nfeasted on the proceeds of the last will and testament of that vagabond\nold Turk, whilst fellow creatures were permitted to perish.\n\n\nTOMB OF SAINT GEORGE.\n\nThe tomb of Saint George, England's patron-saint, is situated in the\nBay of Kesrouan, between the Nahr-et-Kelb and Batroun, surrounded by\nluxuriant gardens and groups of romantic-looking villages and convents.\nThe Arabs venerate St. George, whom they style Mar Djurios, and point\nto a small ruined chapel (as in our engraving), originally dedicated to\nhim to commemorate his victory over the dragon, which, they say, took\nplace near to the spot. The tradition is, that the dragon was about to\ndevour the king of Beyrout's daughter, when St. George slew him, and\nthus saved the lady fair; and the credulous natives point to a kind of\nwell, upwards of sixty feet deep, where they stoutly affirm that the\ndragon used to come out to feed upon his victims.\n\n[Illustration [++] Tomb of Saint George.]\n\nAll this is very curious, inasmuch as it gives an Arabian interest to\nthe career of the patron saint of England, whose portrait, in the act\nof slaying the dragon, constitutes the reverse of most English coin,\nand is regarded as the embodiment of English valour.\n\n\nBEGGARS SELECTED AS MODELS BY PAINTERS.\n\nMichael Angelo Buonarotti often drew from beggars; and report says,\nthat in the early part of his life, when he had not the means of paying\nthem in money, he would make an additional sketch, and, presenting it\nto the party, desire him to take it to some particular person, who\nwould purchase it. Fuseli, in his life of Michael Angelo, says that \"a\nbeggar rose from his hand the patriarch of poverty.\" The same artist,\nin one of his lectures, delivered at the Royal Academy, also observes,\nthat \"Michael Angelo ennobled his beggars into Patriarchs and Prophets,\nin the ceiling of the Sistine Chapel.\"\n\nAnnibal Caracci frequently drew subjects in low life. His _Cries of\nBologna_, etched by Giuseppe Maria Mitelli: pub. 1660, in folio, are\nevidently from real characters. It will also be recollected, that some\nof the finest productions of Murillo, Jan Miel, and Drogsloot, are\nbeggars. Callot's twenty-four beggars are evidently from nature; and\namong Rembrandt's etchings are to be found twenty-three plates of this\ndescription.\n\nSir Joshua Reynolds frequently painted from beggars, and from these\npeople have originated some of his finest pictures, particularly\nhis \"Mercury as a Pickpocket,\" and \"Cupid as a Link-boy.\" His Count\nUgolino was painted from a paviour, soon after he had left St. George's\nHospital, from a severe fever. Mr. West painted the portrait of a\nbeggar, on the day when he became a hundred years old; and considered\nhim as a pensioner for several years afterwards. The same person was\nused also as a model, by Copley, Opie, &c. Who can forget the lovely\ncountenance of Gainsborough's \"Shepherd's Boy,\" that has once seen\nEarlom's excellent engraving from it? He was a lad, well known as a\nbeggar to those who walked St. James's-street seventy years ago. The\nmodel for the celebrated picture of the \"Woodman,\" by the same artist,\ndied in the Borough, at the venerable age of 107.\n\nMr. Nollekens, in 1778, when modelling the bust of Dr. Johnson,\nwho then wore a wig, called in a beggar to sit for the hair. The\nsame artist was not equally fortunate in the locks of another great\ncharacter; for on his application to a beggar for the like purpose, the\nfellow declined to sit, with an observation that three half-crowns were\nnot sufficient for the trouble.\n\n\nSUPPLY OF WATER FOR OLD LONDON.\n\nLeaden pipes conveyed spring water to London city from Tyburn in 1236;\nand in 1285 the first great conduit of lead was begun there. In 1442\nHenry VI. granted to John Hatherley, Mayor, license to take up 200\nfother of lead. The pipes from Highbury brought in the water in 1483.\nWe may learn how much was thought of this useful work by the fact that\nthe Lord Mayor, Aldermen, and many worshipful persons used to ride and\nview the conduit heads at Tyburn; and after dinner there, somewhat\ndifferent from recent sportsmen, they hunted a fox.\n\nThe water-works at London Bridge were established in 1512. In 1534,\ntwo-fifteenths were granted by the Common Council for defraying the\nexpense of bringing water from Hackney to Aldgate to a conduit. But\nPeter Morris did not bring his supply of water to the highest parts of\nLondon till the year 1569, and Sir Hugh Middleton's far-famed New River\nwas only rendered available in 1618, that is, a space of sixty-eight\nyears after the introduction of a stream of pure water into the western\nparts of the town of Lyme in Dorset.\n\n\nCOMBINATION OF INSTINCT AND FORCE OF HABIT IN A DOG.\n\nA dog which had been accustomed to go with his master regularly for\nsome time to Penkridge church, still continued to go there by himself\nevery Sunday for a whole year, while the edifice was under repair, and\ndivine service was not held. Whenever he could, he would get into the\nfamily pew and there pass the proper time. His instinct enabled him to\nperceive the occasion, and to measure the regular time, but it could\ncarry him no further. A remarkable exemplification of the difference\nbetween instinct and reason.\n\n\nYORKSHIRE IN THE LAST CENTURY.\n\nAnecdotes which are apparently trifling in themselves, are often of\nimportance, as exhibiting in a striking light the dialect and social\ncondition of the people, and the period they refer to. An instance of\nthis is the following, which has been recorded as the bellman's cry at\nRipon, on the occasion of a great frost and fall of snow, about 1780:--\n\n\"I is to gie notidge, that Joanie Pickersgill yeats yewn to neit, to\nmoarn at moarn, an to moarn at neit, an nea langer, as lang as storm\nhods, 'cause he can git na mare eldin.\"\n\n _The Translation._\n\nI am to give notice, that John Pickersgill heats his oven to-night,\nto-morrow morning, and to-morrow at night, and no longer as long as the\nstorm lasts, because he can get no more fuel.\n\n\nINSTANCE OF MANY AGED PERSONS DYING ABOUT THE SAME DATE.\n\nThe following is taken from a copy of Nile's \"Weekly Register,\"\npublished at Baltimore, in the month of January, 1823. It is the list\nof deaths which had been notified to the paper within one week, and we\ngive it, as a singular instance of the decease of so many persons above\none hundred years old being announced in the same paragraph.\n\n\"In Franklin co. Pennsylvania, Elizabeth Campbell, aged 104--several\nof her relatives had reached 100.--At Troy, N. Y., Ann Fowler,\n100.--At Tyngsboro', N. Y., Abigail Hadlock, 104.--At Somers, N. Y.,\nMichael Makeel, 103.--At Rutland, Oswego, N. Y., Mrs. Buroy, 110.--At\nBrunswick, Maine, Gen. James W. Ryan, 107--his wife is yet living,\naged 94; they were married together 75 years before his death.--At\nGeorgetown, Col. Yarrow, a Moor, (supposed) 135!--At the city of New\nYork, a woman, a native of St. Domingo, 106. At Sargus, Mass., Mrs.\nEdwards, 101.--In Edgecomb county, N. C., William Spicer, aged about\n112.--In Boston, William Homer, 116.\"\n\n\nCORPSE BEARERS DURING THE PLAGUE.\n\nOf all the calamities with which a great city is infested, there can\nbe none so truly awful as that of a plague, when the street doors of\nthe houses that were visited with the dreadful pest were padlocked up,\nand only accessible to the surgeons and medical men, whose melancholy\nduty frequently exposed them even to death itself; and when the\nfronts of the houses were pasted over with large bills exhibiting red\ncrosses, to denote that in such houses the pestilence was raging, and\nrequesting the solitary passenger, to pray that the Lord might have\nmercy upon those who were confined within. Of these bills there are\nmany extant in the libraries of the curious, some of which have borders\nengraved on wood printed in black, displaying figures of skeletons,\nbones, and coffins They also contain various recipes for the cure of\nthe distemper. The Lady Arundel, and other persons of distinction,\npublished their methods for making what was then called plague-water,\nand which are to be found in many of the rare books on cookery of\nthe time; but happily for London, it has not been visited by this\naffliction since 1665, a circumstance owing probably to the Great Fire\nin the succeeding year, which consumed so many old and deplorable\nbuildings, then standing in narrow streets and places so confined, that\nit was hardly possible to know where any pest would stop.\n\n[Illustration [++] Corpse Bearer.]\n\nEvery one who inspects Agas's Plan of London, engraved in the reign of\nElizabeth, as well as those published subsequently to the rebuilding\nof the City after the fire, must acknowledge the great improvements\nas to the houses, the widening of the streets, and the free admission\nof fresh air. It is to be hoped, and indeed we may conclude from the\nvery great and daily improvements on that most excellent plan of\nwidening streets, that this great city will never again witness such\nvisitations.\n\nWhen the plague was at its height, perhaps nothing could have been more\nsilently or solemnly conducted than the removal of the dead to the\nvarious pits round London, that were opened for their reception; and it\nwas the business of Corpse Bearers, such as the one exhibited in the\npreceding engraving, to give directions to the carmen, who went through\nthe city with bells, which they rang, at the same time crying \"Bring\nout your Dead.\" This melancholy description may be closed, by observing\nthat many parts of London, particularly those leading to the Courts of\nWestminster, were so little trodden down, that the grass grew in the\nmiddle of the streets.\n\n\nA MEMENTO-MORI WATCH.\n\nThe curious relic, of which we herewith give an engraving, was\npresented by Mary, Queen of Scots, to her Maid of Honour, Mary Seaton,\nof the house of Wintoun, one of the four celebrated Maries, who were\nMaids of Honour to her Majesty.\n\n \"Yestreen the Queen had four Maries,\n The night she'll hae but three;\n There was Marie Seaton, and Marie Beaton,\n And Marie Carmichael and me.\"\n\n[Illustration [++] Memento-Mori Watch.]\n\nThe watch is of silver, in the form of a skull. On the forehead of the\nskull is the figure of Death, with his scythe and sand-glass; he stands\nbetween a palace on the one hand, and a cottage on the other, with\nhis toes applied equally to the door of each, and around this is the\nlegend from Horace \"_Pallida mors aequo pulsat pede pauperum tabernas\nRegumque turres_.\" On the opposite, or posterior part of the skull, is\na representation of Time, devouring all things. He also has a scythe,\nand near him is the serpent with its tail in its mouth, being an\nemblem of eternity; this is surrounded by another legend from Horace,\n\"_Tempus edax rerum tuque invidiosa vetustas_.\" The upper part of the\nskull is divided into two compartments: on one is represented our first\nparents in the garden of Eden, attended by some of the animals, with\nthe motto, \"_Peccando perditionem miseriam aeternam posteris meruere_.\"\nThe opposite compartment is filled with the subject of the salvation\nof lost man by the crucifixion of our Saviour, who is represented as\nsuffering between the two thieves, whilst the Mary's are in adoration\nbelow; the motto to this is \"_Sic justitiae satisfecit, mortem superavit\nsalutem comparavit_.\" Running below these compartments on both sides,\nthere is an open work of about an inch in width, to permit the sound to\ncome more freely out when the watch strikes. This is formed of emblems\nbelonging to the crucifixion, scourges of various kinds, swords, the\nflagon and cup of the Eucharist, the cross, pincers, lantern used in\nthe garden, spears of different kinds, and one with the sponge on its\npoint, thongs, ladder, the coat without seam, and the dice that were\nthrown for it, the hammer and nails, and the crown of thorns. Under all\nthese is the motto, \"_Scala caeli ad gloriam via_.\"\n\nThe watch is opened by reversing the skull, and placing the upper\npart of it in the hollow of the hand, and then lifting the under jaw\nwhich rises on a hinge. Inside, on the plate, which thus may be called\nthe lid, is a representation of the Holy Family in the stable, with\nthe infant Jesus laid in the manger, and angels ministering to him;\nin the upper part an angel is seen descending with a scroll on which\nis written, \"_Gloria excelsis Deo, et in terra pax hominibus bonae\nvolu----_\" In the distance are the shepherds with their flocks, and one\nof the men is in the act of performing on a cornemuse. The works of\nthe watch occupy the position of the brains in the skull itself, the\ndial plate being on a flat where the roof of the mouth and the parts\nbehind it under the base of the brain, are to be found in the real\nsubject. The dial plate is of silver, and it is fixed within a golden\ncircle richly carved in a scroll pattern. The hours are marked in large\nRoman letters, and within them is the figure of Saturn devouring his\nchildren, with this relative legend round the outer rim of the flat,\n\"_Sicut meis sic et omnibus idem_.\"\n\nLifting up the body of the works on the hinges by which they are\nattached, they are found to be wonderfully entire. There is no date,\nbut the maker's name, with the place of manufacture, \"Moyse, Blois,\"\nare distinctly engraven. Blois was the place where it is believed\nwatches were first made, and this suggests the probability of the\nopinion that the watch was expressly ordered by Queen Mary at Blois,\nwhen she went there with her husband, the Dauphin, previous to his\ndeath. The watch appears to have been originally constructed with\ncatgut, instead of the chain which it now has, which must have been\na more modern addition. It is now in perfect order, and performs\nwonderfully well, though it requires to be wound up within twenty-six\nhours to keep it going with tolerable accuracy. A large silver bell, of\nvery musical sound, fills the entire hollow of the skull, and receives\nthe works within it when the watch is shut; a small hammer set in\nmotion by a separate escapement, strikes the hours on it.\n\nThis very curious relic must have been intended to occupy a stationary\nplace on a _prie-dieu_, or small altar in a private oratory, for its\nweight is much too great to have admitted of its having been carried in\nany way attached to the person.\n\n\nA MONSTER.\n\nIt is almost incredible that such a monster, as the one we are about\nto describe should have been allowed to continue his wicked career for\nsome years, in a civilized country like France, little more than a\nhundred years ago, but the following paragraph is copied from a Paris\njournal of that period--1755, January the 17th--and there is every\nreason to believe that it is strictly correct. \"What was his fate we\ndo not know, but can hardly doubt.--The Marquis de Plumartin, whose\nexecrable crimes are known over all France, has at last been taken\nin his castle, by 300 men of the King's Own regiment of foot, and\ncarried to Poitiers, loaded with irons. The king is going to appoint\na commission to try him. This monster turned away his wife some years\nago, and became the terror of Poitou. Neither woman nor man durst\nappear in the neighbourhood. Having one day lost a cause in one of the\nking's courts, he caused the usher and his man, who came to intimate\nthe sentence to him, to be burnt alive. Some days after, having drawn\nsix of his creditors into his castle, where he had shut himself up with\nseveral of his crew, he ordered some of his people to drag them into\na pond, tied to the tails of horses, and afterwards fastened them to\na stake near a great fire, where three expired, and the other three\ndied a few days after. Thirty of the Marshalsea guards, who were sent\nto apprehend him, having beset his castle, he barricaded the doors\nand fired on them from the garret window, killing the commanding\nofficer and five others. After which he left the kingdom, but absurdly\nimagining that his crimes were forgot, he lately returned.\"\n\n\nPERSEVERANCE REWARDED BY FORTUNE.\n\nWe have copied the following paragraph from the pages of a local\nhistorian, because it gives us a striking instance of what perseverance\nand good fortune will accomplish, in raising a man to comparative\ndistinction from the humblest walks of life.\n\nAugust 26, 1691--Sir John Duck, bart., departed this life, being\nWednesday at night, and was buried upon the Monday after, being the\n31st of August. The wealthiest burgess on the civic annals of Durham.\nOf Sir John's birth, parentage, and education, the two first have\nhitherto remained veiled in impenetrable obscurity; as to the third,\nhe was bred a butcher under John Heslop, in defiance of the trade and\nmystery of butchers, in whose books a record still exists, warning\nJohn Heslop that he forbear to sett John Ducke on worke in the trade\nof a butcher. John Duck however grew rich, married the daughter of his\nbenefactor, and was created a baronet by James II. He built a splendid\nmansion in Silver-street, where a panel still exists recording his\nhappy rise to fortune. The baronet, then humble Duck, cast out by\nthe butchers, stands near a bridge in an attitude of despondency; in\nthe air is seen a raven bearing in his bill a piece of silver, which\naccording to tradition fell at the feet of the lucky John, and was\nnaturally calculated to make a strong impression on his mind. He bought\na calf, which calf became a cow, and which cow being sold enabled John\nto make further purchases in cattle, and from such slender beginnings,\nto realise a splendid fortune. On the right of the picture is a view of\nhis mansion in Silver-street, and he seems to point at another, which\nis presumed to be the hospital he endowed at Lumley. He died without\nissue, and was buried at St. Margaret's, where his wife, Pia----\nPrudens---- Felix, lies buried beside him.\n\n On Duck the Butchers shut the door;\n But Heslop's Daughter Johnny wed:\n In mortgage rich, in offspring poor,\n Nor son nor daughter crown'd his bed.\n\n\nTRAVELLING IN THE UNITED STATES EXACTLY ONE HUNDRED YEARS AGO.\n\nThe American advertisement, of which we here give a literal copy,\nis deserving of preservation on account of the quaintness of the\ninn-signs, the peculiarity of the spelling and diction, the \"shifting\"\nof the passengers which it announces, and the general idea it gives us\nof the way in which travelling was performed in America at the time\nwhen it was issued.\n\n Philadelphia STAGE-WAGGON, and New-York STAGE BOAT\n performs their Stages twice a Week.\n\nJOHN BUTLER, with his waggon, sets out on Mondays from his House,\nat the Sign of the Death of the Fox, in Strawberry ally, and drives\nthe same day to Trenton Ferry, when Francis Holman meets him, and\nproceeds on Tuesday to Brunswick, and the passengers and goods being\nshifted into the waggon of Isaac Fitzrandolph, he takes them to the\nNew Blazing-Star to Jacob Fitzrandolph's the same day, where Rubin\nFitzrandolph, with a boat well suted, will receive them, and take\nthem to New-York that night. John Butler returning to Philadelphia\non Tuesday with the passengers and goods delivered to him by Francis\nHolman, will again set out for Trenton Ferry on Thursday, and Francis\nHolman, &c. will carry his passengers and goods, with the same\nexpedition as above to New-York.\n\n _Weekly Mercury._\n\n March 8, 1759.\n\n\nFETE OF THE FEDERATION. PARIS 1790.\n\n[Illustration [++] Fete of the Federation.]\n\nThe leading events of the great Revolution in France, may be fairly\nclassed with the marvellous, and among our \"Ten Thousand Wonderful\nThings\" there will be found few more wonderful than the civic festival\nof the general federation of the National Guards of France, which took\nplace on the 14th of July, 1790, and of which the above is a correct\nrepresentation, taken from a view by Duplessis Bertaux. The proceedings\nof that memorable day had in them a mixture of religious celebration\napparently singular among a people who had lately so much trampled on\nreligion; but as this celebration was more pagan than Christian in\nits character, the singularity becomes less marked. On the preceding\nevening, a _Hierodrame_ was performed at the cathedral of Notre Dame--a\nkind of sacred drama, made up by M. Desaugiers of scraps from the\nBible mixed with other matter, and set to music; it professed to tell\nthe story of the taking of the Bastille, and to typify the sadness,\ntrouble, confusion, joy, and alarm of the Parisians. Then succeeded a\n_Te Deum_, chanted in presence of some of the principal federal and\nmunicipal bodies. Early in the morning of the 14th, amid dense clouds\nand heavy rain, the National Guards from all the eighty-three\ndepartments of France, together with deputations from the state army\nand navy, began to assemble, and speedily formed an immense line from\nthe Porte St. Antoine to the Porte St. Martin; whence they marched,\nwith bands playing and colours flying, to the Champ de Mars, regaled\nand cheered by the Parisians on the route. On reaching the great square\nof the Tuileries, the procession was headed by the municipality of\nParis and the members of the National Assembly, and followed by a body\nof gray-headed veterans. The procession traversed the Seine by one of\nthe bridges, greeted by salvos of artillery drawn up on the quays, and\nentered the Champ de Mars under a triumphal arch almost hidden by flags\nand patriotic inscriptions. One o'clock had arrived before the various\nbodies forming the procession had taken their destined places in the\nenclosed parallelogram, surrounded by nearly 300,000 spectators on\nthe raised terraces, most of whom were by this time drenched by the\ncontinuous rain. In the centre of the area was a lofty altar, half\npagan, half Catholic in its adornments; and around this altar the\nprovincial National Guards danced and sang in very excited fashion.\nThe royal family appeared at three o'clock. In an immense gallery near\nthe altar, the National Assembly were seated, with the king and the\npresident on two chairs of state exactly equal in height and richness,\nand the queen and the rest of the court seated behind--a significant\ninterpretation of the decree just announced. At the instant of the king\ntaking his seat, the air was rent with cries of _Vive le Roi! Vive la\nNation!_ The banners were unfurled; 1,800 musicians burst forth with\njubilant strains; cannon poured out continuous volleys; Talleyrand,\nas bishop of Autun, assisted by sixty chaplains of the Paris National\nGuards, performed mass at the altar; and the banners were blessed by\nsprinkling with holy-water. Then Lafayette, dismounting from his white\ncharger, received from the hands of the king a written form of oath;\nhe swore to this oath at the altar, and with his raised arm gave a\nsignal for the countless host to do likewise--every one raising his\nright hand, and saying _Je le jure!_ The king took the oath prescribed\nto him; and the queen held up the dauphin in her arms, as if to denote\nthat he also, poor child, had sworn to defend the national liberties.\nAt five o'clock the royal family retired, and the crowd began to\nleave the Champ de Mars. Twenty-five thousand federates or provincial\ndeputies went to a royal chateau about a mile distant, where a dinner\nhad been prepared for them by order of the municipality of Paris,\nwith Lafayette as chairman of the banquet. At night all Paris was\nilluminated; and for three or four days the feastings, reviews, and\ncelebrations were numerous, including a grand dance on the site of the\ndemolished Bastille. On the 18th, Lafayette reviewed the provincial or\nfederate National Guards, and on the 19th they were reviewed by the\nking. Paris was intoxicated for an entire week, each man displaying at\nonce his delight and his vanity.\n\n\nA MAN CARRIES HIS HOUSE ON HIS HEAD.\n\nSimeon Ellerton, of Craike, Durham, died 1799, aged 104. This man,\nin his day, was a noted pedestrian, and before the establishment of\nregular \"Posts,\" was frequently employed in walking commissions, from\nthe northern counties to London and other places, which he executed\nwith singular fidelity and despatch. He lived in a neat stone cottage\nof his own erecting; and what is remarkable, he had literally carried\nhis house on his head; it being his constant practice to bring back\nwith him from every journey which he undertook, some suitable stone, or\nother material for his purpose, and which, not unfrequently, he carried\n40 or 50 miles on his head.\n\n\nIGNORANCE AND FEAR.\n\nIn the year 1712, Whiston predicted that the comet would appear on\nWednesday, 14th October, at five minutes after five in the morning, and\nthat the world would be destroyed by fire on the Friday following.\nHis reputation was high, and the comet appeared. A number of persons\ngot into boats and barges on the Thames, thinking the water the safest\nplace. South Sea and India stock fell. A captain of a Dutch ship threw\nall his powder into the river, that the ship might not be endangered.\nAt noon, after the comet had appeared, it is said that more than one\nhundred clergymen were ferried over to Lambeth, to request that proper\nprayers might be prepared, there being none in the church service.\nPeople believed that the day of judgment was at hand, and acted some\non this belief, more as if some temporary evil was to be expected.\nThere was a prodigious run on the bank, and Sir Gilbert Heathcote, at\nthat time the head director, issued orders to all the fire offices in\nLondon, requiring them to keep a good look out, and have a particular\neye upon the Bank of England.\n\n\nARABIAN HORSES.\n\nIt is a singular circumstance, that it is to the Arabian that England\nis indebted for her improved, and now unrivalled, breed of horses for\nthe turf, the field, and the road.\n\nThe Arabian horses are divided into two great branches; the Kadischi\nwhose descent is unknown, and the Kochlani, of whom a written genealogy\nhas been kept for 2000 years. These last are reserved for riding\nsolely, they are highly esteemed and consequently very dear. They are\nsaid to derive their origin from King Solomon's studs. However this\nmay be they are fit to bear the greatest fatigues, and can pass whole\ndays without food. They are also said to show uncommon courage against\nan enemy. It is even asserted, that when a horse of this race finds\nhimself wounded and unable to bear his rider much longer, he retires\nfrom the fray, and conveys him to a place of security. If the rider\nfalls upon the ground, his horse remains beside him, and neighs till\nassistance is brought. The Kochlani are neither large nor handsome but\namazingly swift. The whole race is divided into several families, each\nof which has its proper name. Some of these have a higher reputation\nthan others on account of their more ancient and uncontaminated\nnobility.\n\nWe may not believe, perhaps, all that is told us of the Arabian. It has\nbeen remarked that there are, on the deserts which his horse traverses,\nno milestones to mark the distance, or watch to calculate the time;\nand the Bedouin is naturally given to exaggeration, and most of all\nwhen relating the prowess of the animal which he loves as dearly as\nhis children; yet it cannot be denied that at the introduction of the\nArabian into the European stables, there was no other horse comparable\nto him.\n\n\nHEAD-QUARTERS OF PRINCE RUPERT AT EVERTON, DURING THE SIEGE OF\nLIVERPOOL, IN 1644.\n\nPrince Rupert, assisted by the Earl of Derby, having taken Bolton\nby storm, and refreshed his army there for some days, advanced on\nLiverpool, where the Parliament had a strong garrison under the command\nof Colonel More, of Bank-hall; and finding on his approach to the\ntown, the high ground near it favourable to his design, compared it\nto a crow's nest, probably imagining it would be taken with as little\ndifficulty; but the resistance he met with, induced him to declare it\nwas more like an eagle's nest, or a den of lions.\n\n[Illustration [++] Head-Quarters of Prince Rupert.]\n\nThe siege began about the 2nd of June, and the view exhibits his\nhead-quarters from that time till the reduction of the place. His main\ncamp was established round the beacon, about a mile from the town,\nand his officers were placed in the adjoining villages, from whence a\ndetachment marched every day, being relieved every twenty-four hours,\nto open trenches and erect batteries. From these advances Prince\nRupert frequently attacked the besieged and their works in the way of\nstorm, but was constantly repulsed with great slaughter of his men. At\nlength, Colonel More, finding the town must of necessity surrender, and\ndesirous of ingratiating himself with the Prince, for the preservation\nof his house and effects at Bank Hall, gave such orders for his\nsoldiers to retire, that the works on the enemy's side were abandoned,\nand the royalists entered the town at three o'clock in the morning of\nJune 26, putting to the sword all they met with, till they arrived\nat the High Cross, which then stood on the site where the Exchange\nnow stands. Here the soldiers of the Castle, drawn up in line, beat a\nparley, and demanded quarter, which, on their submitting as prisoners\nof war, and surrendering the Castle to the Prince, was granted. The\nsoldiers were then sent to the tower, St. Nicholas's Church, and other\nplaces of security; but the Parliament-army, soon after the siege,\nrepossessed themselves of the Castle, and appointed Col. Birch, as\ngovernor.\n\n\nFIRE AT BURWELL, CAMBRIDGESHIRE. 1727.\n\nSome strollers brought down a puppet-show, which was exhibited in a\nlarge thatched barn. Just as the show was about to begin, an idle\nfellow attempted to thrust himself in without paying, which the people\nof the show preventing, a quarrel ensued. After some altercation,\nthe fellow went away, and the door being made fast, all was quiet;\nbut the same man, to gain admittance privately, got over a heap of\nhay and straw, which stood near to the barn, and accidentally set it\non fire. The spectators of the show, alarmed by the flames, which\nhad communicated to the barn, rushed to the door; but it happened\nunfortunately that it opened inwards, and the crowd pressing violently\nagainst the door, there could be no escape. Thus the whole company,\nconsisting of more than 160 persons, were kept confined till the roof\nfell in, and covered them with fire and smoke: six only escaped with\nlife; the rest, among whom were several young ladies of fortune, were\nreduced to one undistinguishable heap of mangled bodies, totally\ndisfigured. The friends of the dead, not knowing which were the remains\nthey sought, caused a large hole to be dug in the churchyard, and all\nthe bodies were promiscuously interred together, and a tablet erected\nin the church to perpetuate this most melancholy event.\n\n\nAN APPARENT SINGULARITY ACCOUNTED FOR.\n\nIt is generally well known that birds are very active agents in the\nextension of vegetation, and that fruit and flowers are, to a great\nextent, rendered prolific by the insects which visit their blossoms;\nbut few people are aware of the means through which fish are formed\nin lakes and ponds, which are not connected with other waters. Here,\nalso, an insect is the principal agent. The large water-beetle, which\nis in the habit of feeding upon the spawn of fish, occasionally in\nthe evening climbs up the stems of rushes, &c. out of the water,\nsufficiently high to enable it to take wing; in these circumstances it\nhas been caught, and, putting it into water, has been found to give out\nthe spawn with which it had gorged itself previous to taking flight,\nboth in a digested and undigested state; so that, on trial, it has been\nfound that it produced fish of various kinds.\n\n\nEUROPEAN BALANCING EQUAL TO THE INDIAN JUGGLERS.\n\nThe astonishing dexterity of the Indian jugglers is known to all,\nbut many years ago a Spaniard named Cadenas made himself equal, if\nnot superior to them. He may be truly said to be superior to them,\ninasmuch as several of his feats have never been attempted by them. Don\nCadenas extended himself flat on his back on a large table. He then\nelevated his legs until they were at right angles with his body; he\nwas assisted in keeping this position by a sort of pyramidal cushion,\nwhich was placed under him, a little below the lower end of his back.\nHis feet and ankles were covered with boots, on which were many small\ncastanets and little bells. The tranca, which is a round piece of\nwood, about 8 feet long and five inches in diameter, handsomely\npainted, was then laid horizontally on the soles of his feet, his legs\nbeing perpendicular. Having exactly balanced the tranca, he alternately\nstruck his feet against it, the castanets, &c., keeping time with the\nmusic. In proportion to the strength with which he struck the tranca,\nwith one foot or both feet, was the height to which he elevated it,\nalways catching it, in its descent, with great accuracy, on the soles\nof his feet. Sometimes by bending his knees and then striking out with\nhis limbs, he threw the tranca several feet into the air, catching it,\nin its descent, on his feet, with as much neatness and more certainty\nthan the Indian jugglers used to catch the brass balls in their hands.\nHe concluded the performance with the tranca, by exactly balancing it\non the sole of his left foot, and then by repeated strokes of his right\nfoot set it rapidly in motion like a horizontal fly-wheel.\n\n\nMOB-WISDOM.\n\nA singular instance of a mob cheating themselves by their own headlong\nimpetuosity, is to be found in the life of Woodward, the comedian. On\none occasion, when he was in Dublin, and lodged opposite the Parliament\nHouse, a mob who were making the members swear to oppose an unpopular\nbill, called out to his family to throw them a Bible out of the window.\nMr. W. was frightened, for they had no such book in the house, but he\nthrew out a volume of Shakespere, telling the mob they were welcome to\nit. They gave him three cheers, swore the members upon this book, and\nafterwards returned it without discovering its contents.\n\n\nCOMMUNICATION BETWEEN ANIMALS.\n\nThe means by which animals contrive to communicate their ideas to each\nother is a phenomenon which has never been satisfactorily explained.\nThe two following instances of it are very curious. A gentleman who\nwas in the habit of occasionally visiting London from a distant county\nperformed the journey on horseback, accompanied by a favourite little\nterrier dog, which he left at an inn at some distance from London\ntill his return. On one occasion on calling for his dog the landlady\ntold him that it was lost; it had had a quarrel with the great house\ndog, and had been so worried and bit that it was thought he would\nnever recover, but at the end of a few days he crawled out of the\nyard, and no one saw him for almost a week, when he returned with\nanother dog bigger than his enemy, on whom they both fell and nearly\ndestroyed him. This dog had actually travelled to its own home at\nWhitmore in Staffordshire, had coaxed away the great dog in question,\nwhich followed him to St. Alban's to assist in resenting the injury\nof his friend. The following story is related of a little spaniel\nwhich had been found lame by a surgeon at Leeds. He carried the poor\nanimal home, bandaged up his leg, and after two or three days turned\nhim out. The dog returned to the surgeon's house every morning till\nhis leg was perfectly well. At the end of several months, the spaniel\nagain presented himself in company with another dog, which had also\nbeen lamed; and he intimated, as well as piteous and intelligent looks\ncould intimate, that he desired the same assistance to be rendered to\nhis friend as had been bestowed upon himself. The combination of ideas\nin this case, growing out of the recollection of his own injury, and\nreferring that to the cure which had been performed; the compassion he\nhad for his friend to whom he communicated the occurrence, and induced\nto seek relief under his guidance, together with the appeal to the\nhumane surgeon, is as extraordinary a piece of sagacity as can be found\nin all the annals of animals.\n\n\nSTRANGE CUSTOM ABOUT NAMES.\n\nThe following anecdote forcibly illustrates the absurd custom which\nprevailed many years ago in America, of giving children names, made up\nof Scripture sentences. We record the anecdote as being descriptive\nof a curious local custom. About the beginning of the present\ncentury a New England sea captain having some business at a public\noffice, which required him to sign his name, was rather tedious in\nperforming the operation, which did not escape the observation of\nthe officer, who was a little impatient at the delay, and curious\nwithal to see what sort of a name it could be that required so long\na time to spread it upon paper. Perhaps the captain had a long\nstring of titles to grace it, such as honorable, esquire, colonel of\nmilitia, selectman of the town of ----, &c., which he chose to make\nan ostentatious parade of; or perhaps it was his whim to subscribe\nthe place of his nativity and that of his residence, together with\nhis age, height, and complexion. He was mistaken; for the captain\nhad subscribed nothing but simply his name, which, when he had\ndone, the officer, after some trouble in decyphering, found to read\nthus:--Through-Much-Tribulation-We-Enter-Into-the-Kingdom-of-Heaven\nClapp. \"Will you please to tell me, Captain Clapp,\" said he, with as\ndemure a face as his violent inclination to indulge in a hearty laugh\nwould allow him to put on, \"what might your mother have called you\nin your infancy, to save herself the trouble of repeating a sermon\nwhenever she had occasion to name her darling?\" \"Why, sir,\" replied\nCaptain Clapp, with laughable simplicity, \"when I was little they used\nto call me Tribby, for shortness.\"\n\n\nDRESS IN LONDON DURING THE LAST CENTURY.\n\nThe seven illustrations which accompany this article represent\nthe progress of dress in London from 1690 to 1779. They speak for\nthemselves, and tell their own tale far better than any description\nin words could tell it for them. The scale in society to which the\npersons depicted in the engravings belong, is what may be called the\nupper middle class, and we thus obtain a more correct idea of the\ngeneral style of dress, than we should have done had we confined our\nobservations solely to the higher ranks.\n\n[Illustration: Dress 1690-1715.]\n\n[Illustration: Dress 1721.]\n\n[Illustration: Dress, 1735--common Life.]\n\n[Illustration: Dress, 1738.]\n\n[Illustration: Dress, 1752.]\n\n[Illustration: Dress Circa 1773, 1778.]\n\n[Illustration: Dress, 1779.]\n\nIt is, however, very curious to notice the value placed upon dress\nduring the period indicated; and how frequently its loss is recorded.\nThus we find it mentioned that Lady Anderson, whose house was robbed\nat a fire in Red Lion Square in 1700, lost a gown of orange damask,\nlined with, striped silk. The family of George Heneage, Esq., at the\nsame time, and by the same casualty, lost \"_a head_, with very fine\nlooped lace of very great value, a Flanders' laced hood, a pair of\ndouble ruffles and tuckers, two laced aprons, one edged with point\nlace, and a large black scarf embroidered with gold.\" At the same\nperiod the ladies wore Holland petticoats, embroidered in figures with\ndifferent silks and gold, with broad orrice at the bottom.\nIn 1702 diamond stomachers adorned the ladies; they were composed of\nthat valuable stone set in silver, and sewed in a variety of figures\nupon black silk. The men imported the Champaign wig from France. They\nwere made very full, curled, and eighteen inches in length to the\npoint, with drop locks. In the _Post Boy_, of November 15, 1709, there\nwere advertised as stolen, \"A black silk petticoat, with red and white\ncalico border, cherry- stays, trimmed with blue and silver, a\nred and dove- damask gown, flowered with large trees; a yellow\nsatin apron, trimmed with white Persian, and muslin head-clothes, with\ncrow-foot edging; a black silk furbelowed scarf, and a spotted hood.\"\nBlack and beaver hats for ladies were advertised in 1719, faced with\n silks, and trimmed with gold and silver lace. A man of fashion\nin 1720 wore the full flowing curled wig, which fell in ringlets\nhalf-way down his arms and back, a laced coat, straight, formal, with\nbuttons to the very bottom, and several on the pockets and sleeves; his\nshoes were square at the toes, had diminutive buckles, a monstrous flap\non the instep, and high heels, a belt secured the coat and supported\nthe sword. Perukes were a highly important article of dress in 1734.\nFans were much used, ladies seldom appeared without this useful\nornament in their hands. The hoop underwent many important changes;\nsometimes it projected at the sides only, or, like its ancestor, the\nfardingale, it spread itself all round in imposing majesty. High-heeled\nshoes maintained their place. In 1740 tight sleeves with full ruffles,\nsmall pointed waists, enclosed in whalebone, loose gowns, called\nsacques, and cloaks with hoods, named cardinals, were _la grande\nmonde_. Among the gentlemen's costumes, the most striking was the\n_Ramilies_ tail, which was a plaited tail to the wig, with an immense\nbow at the top and one at the bottom. Claret clothes were\nconsidered as handsome; and light blue with silver button-holes, and\nsilver garters to the knees, was very fashionable between 1740 and\n1751. The change to wearing the natural hair instead of wigs took place\nabout 1765. From that date the female dress altered by degrees: the cap\nwas enlarged to an enormous size, and the bonnet swelled in proportion.\nHoops were entirely discontinued. Hats and bonnets of straw, chip, and\nbeaver, became well proportioned, and velvet pelisses, shawls and silk\nspencers were contrived to improve rather than injure the form. The\nmale dress also insensibly changed from formality to ease, and thus, by\ndegrees, the fashion became what our illustrations represent it to have\nbeen in 1779.\n\n\nATTAR OF ROSES.\n\nLieutenant Colonel Polier gives a full history of extracting this\nessential oil, in vol i. p. 332, of the _Asiatic Researches_. The roses\ngrow, cultivated near Lucknow, in fields of eleven acres each. The\noil is procured by distillation; the petals of the flowers only are\nused; and in that country no more than a quantity of about two drachms\ncan be procured from an hundred-weight of rose leaves, and even that\nin a favourable season, and by the process being performed with the\nutmost care. The oil is by accident of different colours; of a bright\nyellow, of a reddish hue, and a fine emerald. It is to the mother of\nMebrul Nessa Begum, afterwards called Nourjehan Begum, or, _Light\nof the World_, that the fair sex is indebted for this discovery. On\nthis occasion the emperor of Hindostan rewarded the inventress with a\nstring of valuable pearls. Nourjehan Begum was the favourite wife of\nJehangir, and her game the fiercest of India. In a hunting party she\nkilled four tigers with a matchlock, from her elephant, and her spouse\nwas so delighted at her skill, that he made her a present of a pair of\nemerald bracelets, valued at a lack of rupees, and bestowed in charity\na thousand mohurs.\n\n\nFLEET MARRIAGES ABOUT 1740.\n\nMany of the early Fleet weddings were _really_ performed at the\nchapel of the Fleet; but as the practice extended, it was found more\nconvenient to have other places within the Rules of the Fleet, (added\nto which the Warden was compelled by act of parliament not to suffer\nthem,) and thereupon many of the Fleet parsons and tavern-keepers in\nthe neighbourhood fitted up a room in their respective lodgings or\nhouses as a chapel. The parsons took the fees, allowing a portion to\nthe plyers, &c., and the tavern-keepers, besides sharing in the fees,\nderived a profit from the sale of liquors which the wedding party\ndrank. In some instances the tavern-keepers kept a parson on their\nestablishment at a weekly salary of twenty shillings; while others,\nupon a wedding-party arriving, sent for any clergyman they might please\nto employ, and divided the fee with him. Most of the taverns near the\nFleet kept their own registers, in which (as well as in their own\nbooks,) the parsons entered the weddings.\n\n\nEFFECTS OF THE GREAT EARTHQUAKE AT LISBON.\n\nThe earthquake happened on November the 1st, 1755, and its sphere of\naction embraced many cities and states. St. Ubes was totally destroyed.\nAt Cadiz the sea broke down the outer wall, flooded the town, and\ndrowned some hundreds of persons. The Cathedral of Seville was\nseriously damaged, several houses overthrown, and many persons injured.\nThe shock was felt, indeed, throughout the whole of Spain, except in\nCatalonia, and also in Germany. In many parts of Great Britain the\nwater in lakes and ponds was violently upheaved, and ebbed and flowed\nover the banks. A solemn Fast was consequently commanded to be observed\non the 6th of February next ensuing, in the hope to avert, by prayer\nand penitence, a similar calamity from this country. A ship at sea,\n100 leagues to the westward of Lisbon, had her cabin windows shattered\nto fragments, and many vessels in deep water quivered as if they had\nstruck against a rock. In Morocco the effects of the shock were most\ndisastrous. In Mequinez two-thirds of the houses were destroyed, and\nabove 300 in Fez. A caravan of 200 persons going along the coast\nfrom Sallee to Morocco were overwhelmed by the sea, and a still more\nnumerous caravan was swept away by the sudden rise of the inland\nrivers. In France and Holland earthquakes were repeatedly felt during\nthe entire month of November, and occasionally even in December.\n\n\nSNAKE-CHARMERS.\n\nIn the East Indies, the Pambatees, or snake-charmers, come from the\nmountains called the Ghauts. They make a trade of catching serpents,\ntraining them and exhibiting them for money. These reptiles are\ncommonly the _cobra-di-capello_, the hooded or spectacle serpent, and\nof other similar species. A Pambatee will sometimes carry eight or\nmore of them in a low round basket, in which the serpents lie coiled\nround one another.\n\nAs soon as the lid is removed from the basket, the serpent creeps\nout of it. The master plays on an instrument somewhat resembling the\nbagpipe, and the snakes are taught to mark the cadence by the motion\nof their heads, till at length they fall asleep. In order to rouse\nthem, the Pambatee suspends his music and shakes a ring round his arm\nto which a piece of red cloth is fastened. The irritated serpent darts\nat the ring; but as the master has taken care to extract the pouch\ncontaining the poison, and to file his teeth, he can do no harm.\n\n[Illustration [++] Snake-Charmer.]\n\nThe musical instrument just mentioned is called _magootee_. It is\ncomposed of a hollow calebash, to one end of which is fitted a\nmouthpiece similar to that of the clarinet. To the other extremity is\nadapted a tube perforated with several holes, which are successively\nstopped by the fingers, like those of the flute, while the player blows\ninto the mouthpiece. In the middle of the instrument is a small mirror,\non which the serpents fix their eyes while dancing. The above engraving\nwill convey a correct idea of the Pambatee and his instrument.\n\n\nWONDERFUL ESCAPE.\n\nIn 1785, at Winster, in Derbyshire, a show being exhibited at a\npublic-house, some gunpowder being scattered on the floor of an upper\nchamber, took fire, and communicated to the remainder of a barrel,\nby which the whole upper part of the house was blown up; about sixty\npersons were below, and not one hurt.\n\n\nFAC-SIMILE OF THE FIRST STEAM BOAT.\n\n[Illustration [++] First Steam Boat.]\n\nThe triumph of steam navigation is one of the wonders of science; and,\ntraversed in all directions as the navigable waters of the earth now\nare, by vessels propelled by steam, it is not a little curious to look\nat the first rude effort, and to examine the attempt which has been\nfollowed by such extraordinary success.\n\nThe world stands indebted, not for the discovery, but for the\nsuccessful application of steam power to navigation, to Robert Fulton,\nwho was born in Pennsylvania in 1765, being the son of a poor Irish\nlabourer who had emigrated to America. He came to London in 1786,\nand subsequently, in the character of an inventor and projector,\nproceeded to Paris, where, however, he did not meet with much success\nor encouragement. It is evident, from the following letter to a friend,\nthat while residing in the French capital, that his attention was even\nthen turned to the subject of propelling vessels by mechanical power:--\n\n Paris, the 20th of September, 1802.\n\n TO MR. FULNER SKIPWITH.\n\nSir,--The expence of a patent in France is 300 livers for three years,\n800 ditto for ten years, and 1500 ditto for fifteen years; there can\nbe no difficulty in obtaining a patent for the mode of propelling a\nboat which you have shewn me; but if the author of the model wishes\nto be assured of the mirits of his invention before he goes to the\nexpence of a patent, I advise him to make the model of a boat, in which\nhe can place a clock spring which will give about eight revolutions;\nhe can then combine the movements so as to try oars, paddles, and the\nleaves which he proposes; if he finds that the leaves drive the boat a\ngreater distance in the same time than either oars or paddles, they\nconsequently are a better application of power. About eight years ago\nthe Earl of Stanhope tried an experiment on similar leaves in Greenland\nDock, London, but without success. I have also tried experiments on\nsimilar leaves, wheels, oars, paddles, and flyars similar to those of\na smoak jack, and found oars to be the best. The velocity with which a\nboat moves, is in proportion as the sum of the surfaces of the oars,\npaddles, leaves, or other machine is to the bow of the boat presented\nto the water, and in proportion to the power with which such machinery\nis put in motion; hence, if the sum of the surfaces of the oars is\nequal to the sum of the surfaces of the leaves, and they pass through\nsimilar curves in the same time, the effect must be the same; but oars\nhave this advantage, they return through air to make a second stroke,\nand hence create very little resistance; whereas the leaves return\nthrough water, and add considerably to the resistance, which resistance\nis increased as the velocity of the boat is augmented: no kind of\nmachinery can create power; all that can be done is to apply the manuel\nor other power to the best advantage. If the author of the model is\nfond of mechanics, he will be much amused, and not lose his time, by\ntrying the experiments in the manner I propose, and this perhaps is the\nmost prudent measure, before a patent is taken.\n\n I am, Sir, with much respect, yours,\n\n ROBT. FULTON.\n\nIn the following year, 1803, he appears to have made an experiment in\nFrance of propelling a vessel by mechanism, and though it failed in\nconsequence of the timbers of the boat being too weak, it served to\nconvince him so completely of ultimate success, that he immediately\ngave instructions to Watt and Boulton to prepare a suitable steam\nengine for him, and send it to New York. Having returned to that city\nin 1806, he set about building a boat, and having received the engines\nhe had ordered, he successfully started the first steam-boat in the\nworld on her trial trip to Albany from New York in August, 1807.\nHer name was the \"_Clermont_,\" and the above engraving is a correct\nrepresentation of her. She was in length 133 feet, in depth 7, and in\nbreadth 18.\n\n\nSEVERE ENACTMENT AGAINST BEGGARS.\n\nAt the commencement of the reign of Edward VI., a most severe and\nextraordinary statute was made for the punishment of vagabonds and\nrelief of poor persons. It does not appear who were the contrivers\nof this instrument, the preamble and general spirit of which were\nmore in accordance with the tyrannical and arbitrary measures of the\npreceding reign, than with the mild and merciful character of the\ninfant sovereign, who is well known to have taken a very active part\nin the affairs of government. It repeals all the former statutes on\nthis subject, and enacts, that if any beggar or other person, not being\nlame or impotent, and after loitering or idly wandering for the space\nof three days or more, shall not offer himself to labour, or being\nengaged in any person's service, shall run away or leave his work, it\nshall be lawful for the master to carry him before a justice of peace,\nwho, on proof of the offence, shall cause the party to be marked with\na hot iron with the letter V on the breast, and adjudge him to be his\nmaster's slave for the space of two years, who shall feed him \"on bread\nand water, or at his discretion, on refuse of meat, and cause the said\nslave to work by beating, chaining, or otherwise in such work or labour\n(how vile soever it be) as he shall put him unto.\" If the slave should\nrun away or absent himself for a fortnight without leave, the master\nmay pursue and punish him by chaining or beating, and have his action\nof damage against any one who shall harbour or detain him. On proof\nbefore the justice of the slave's escape, he is to be sentenced to be\nmarked on the forehead or ball of the cheek with a hot iron with the\nletter S, and adjudged to be his master's slave for ever; and for the\nsecond offence of running away, he is to be regarded as a felon and\nsuffer death. The children of beggars to be taken from them, and, with\nother vagrant children, to be apprenticed by the magistrate to whoever\nwill take them; and if such children so apprenticed run away, they are\nto be retaken, and become slaves till the age of twenty in females,\nand twenty-four in males, with punishment by chains, &c., and power\nto the master to let, sell, or bequeath them, as goods and chattels,\nfor the term aforesaid. If any slave should maim or wound the master,\nin resisting correction, or conspire to wound or murder him, or burn\nhis house or other property, he is to suffer death as a felon, unless\nthe master will consent to retain him as a slave for ever; and if any\nparent, nurse, or bearer about of children, so become slaves, shall\nsteal, or entice them away from the master, such person shall be liable\nto become a slave to the said master for ever, and the party so stolen\nor enticed away restored. If any vagrant be brought to a place, where\nhe shall state himself to have been born, and it shall be manifest that\nhe was not so born there, for such lie he shall be marked in the face\nwith an S, and become a slave to the inhabitants or corporation of the\ncity for ever. Any master of a slave may put a ring of iron about his\nneck, arm, or leg, for safe custody, and any person taking or helping\nto take off such ring, without consent of the master, shall forfeit the\nsum of ten pounds.\n\nThis diabolical statute, after remaining for two years, was repealed,\non the ground that, from its extreme severity, it had not been enforced.\n\n\nJUDGES IN THEIR ROBES ATTENDING PUBLIC BALLS.\n\nThat the ideas of good taste and propriety which now prevail are\ngreatly in advance of those which our ancestors entertained, is\nstrikingly manifested by the fact, that the dreadful scenes which\nfollowed the last business of a county assize did not prevent a festive\nbeginning of the same. On the commission day at each county town was\nheld an assize ball. The judges attended in black silk gowns with band\nand two-curl bob-wig. They did not dance, but usually played at whist.\nWhat would be thought now-a-days of judges who went to a public ball\nroom on commission day, and played at whist in their robes?\n\n\nST. WINIFRED'S WELL.\n\nThe most copious spring in Great Britain is St. Winifred's Well, near\nthe town of Holywell, in Flintshire. The well is an oblong square,\nabout twelve feet by seven. The water passes into a small square court\nthrough an arch; it has never been known to freeze, and scarcely ever\nvaries in quantity either in drought or after the greatest rains. The\nwater thrown up is not less than eighty-four hogsheads every minute.\n\n[Illustration: St. Winifred's Well.]\n\nThis sacred well is the object of many pilgrimages, even in the present\nday, and several modern miracles are related of the influence of its\nwaters. Pope Martin V. especially enjoined such pilgrimages, and the\nmonks of Basingwerk were furnished with pardons and indulgences to sell\nto the devotees. James the 2nd visited the well in 1686, and Leopold,\nKing of the Belgians, in 1819. Apart from all superstitious notions,\nits waters doubtless possess many curative properties.\n\nOver the well, Queen Margaret, the mother of Henry VII., erected a\nbeautiful chapel, whose elegantly fretted roof, and graceful columns\nand arches, are generally admired as examples of good architecture. Our\nengraving is a correct representation of the interior.\n\n\nINSTANCE OF ASSIDUITY AND PERSEVERANCE.\n\nThe Rev. Wm. Davy, a Devonshire curate, in the year 1795, begun a\nmost desperate undertaking, viz., that of printing himself twenty-six\nvolumes of sermons, which he actually did, working off page by page,\nfor fourteen copies; and continuing this almost hopeless task for\ntwelve years, in the midst of poverty! Such wonderful perseverance\nalmost amounts to a ruling passion.\n\n\nPHENOMENON AT THE POWERSCOURT FALL.\n\n[Illustration [++] Powerscourt Fall.]\n\nThe Powerscourt Fall, of which the annexed is an engraving, is formed\nby the river Dargle, and is situated in the county of Wicklow. When\nthe river is full, it presents a very grand appearance. The stream\nprecipitates itself over a nearly perpendicular cliff, 300 feet in\nheight, and falls into a natural basin or reservoir, encircled by\nrocky masses of considerable magnitude, whilst the whole scene is\nbacked by mountains. This fall exhibits rather a singular phenomenon,\nin the different degrees of velocity with which the water descends\nin different parts of the cascade. Thus, on one side, the water may\nbe observed to pour down with considerable velocity; while, on the\nother side, the fall, in the upper part, presents the appearance of\na continued stream of frothy foam, gliding slowly down the face of\nthe cliff, though the lower part moves with greater velocity. This\ncircumstance is, however, readily accounted for; being, in fact, mainly\nattributable to the comparatively small body of water which forms the\ncascade. The water, on the one side, that which descends with the\ngreater velocity (and this forms by far the larger portion of the\ncascade) meets with no interruption in its descent, but falls, almost\nfrom the top, to the bottom in an unbroken sheet. On the other side,\nhowever, the cliff in the upper part deviates from the perpendicular,\nand the consequence is, that, owing to the or inclination of\nthe rock over which it flows, the progress of the water is checked\nin that particular part, though lower down, where the cliff is again\nperpendicular, it regains its velocity. If the body of water in this\ncascade were greater, this phenomenon would not occur.\n\n\nHOW CHESS ORIGINATED IN INDIA.\n\nBy the unanimous consent of all nations, chess holds the first place\namong social amusements. The history of this game has exercised\nmany able pens. According to Sir William Jones, it is decidedly\nof Hindoo invention. \"If,\" says he, in a learned memoir on this\nsubject inserted in the second volume of the _Asiatic Researches_,\n\"evidence were required to prove this fact, we may be satisfied with\nthe testimony of the Persians, who, though as much inclined as other\nnations to appropriate the ingenious inventions of a foreign people,\nunanimously agree that the game was imported from the west of India\nin the sixth century of our era. It seems to have been immemorially\nknown in Hindoostan by the name of _Cheturanga_, the four _angas_, or\nmembers of an army, which are _elephants_, _horses_, _chariots_, and\n_foot-soldiers_; and in this sense, the word is frequently used by epic\npoets in their description of real armies. By a natural corruption\nof the pure Sanscrit word, it was changed by the old Persians into\n_chetrang_; but the Arabs, who soon after took possession of their\ncountry, had neither the initial nor the final letter of that word in\ntheir alphabet, and consequently altered it farther into _shetranj_,\nwhich presently found its way into the modern Persian, and at length\ninto the dialects of India, where the true derivation of the name is\nknown only to the learned. Thus has a very significant word in the\nsacred language of the Brahmins been transformed by successive changes\ninto _axedrez_, _scacchi_, _echecs_, chess, and by a whimsical\nconcurrence of circumstances given birth to the English word _check_,\nand even a name to the _exchequer_ of Great Britain.\"\n\nOf the origin of this game various accounts are given. Some Hindoo\nlegends relate, that it was invented by the wife of Ravanen, king of\nLanca, or Ceylon, to amuse her husband with an image of war, when\nRama, in the second age of the world, was besieging his capital. The\nhigh degree of civilization which the court of Ravanen had attained\nat so remote a period is worthy of notice. An ancient Hindoo painting\nrepresents his capital regularly fortified with embattled towers. He\nthere defended himself with equal skill and valour, whence he and\nhis subjects were denominated magicians and giants. Ravanen seems to\nhave been the Archimedes of Lanca; and his science must have appeared\nsupernatural to the invader, Rama, and his wild horde of mountaineers,\nwho were termed in derision satyrs or apes, whence the fable of the\ndivine Hanooman.\n\nAccording to another account, the occasion of this invention was as\nfollows:--Behub, a young and dissolute Indian prince, oppressed his\npeople in the most cruel manner. Nassir, a Brahmin, deeply afflicted\nby his excesses, and the lamentations of his subjects, undertook to\nrecal the tyrant to reason. With this view he invented a game, in which\nthe king, impotent by himself, is protected only by his subjects, even\nof the lowest class, and frequently ruined by the loss of a single\nindividual.\n\nThe fame of this extraordinary invention reached the throne, and the\nking summoned the Brahmin to teach him the game, as a new amusement.\nThe virtuous Brahmin availed himself of this opportunity to instil\ninto the mind of the young tyrant the principles of good government,\nand to awaken him to a sense of his duties. Struck by the truths which\nhe inculcated, the prince conceived an esteem for the inventor of the\nnew game, and assured him of his willingness to confer a liberal\nremuneration, if he would mention his own terms. Nassir demanded as\nmany grains of wheat as would arise from allowing one for the first\nsquare, two for the second, four for the third, and so on, doubling\nfor each square of the sixty-four on the chessboard. The king, piqued\nat the apparently trivial value of the demand, desired him somewhat\nangrily to ask a gift more worthy of a monarch to bestow. When,\nhowever, Nassir adhered to his first request, he ordered the required\nquantity of corn to be delivered to him. On calculating its amount, the\nsuperintendents of the public granaries, to their utter astonishment,\nfound the demand to be so enormous, that not Behub's kingdom only, but\neven all Hindoostan would have been inadequate to the discharge of it.\nThe king now admired the Brahmin still more for the ingenuity of his\nrequest than for the invention, appointed him his prime-minister, and\nhis kingdom was thenceforward prosperous and happy.\n\nThe claim of the Hindoos to the invention of chess has been disputed in\nfavour of the Chinese; but as they admit that they were unacquainted\nwith the game till 174 years before Christ, and the Hindoos\nunquestionably played it long before that time, the pretensions of the\nlatter must naturally fall to the ground.\n\n\nDISORDERS CURED BY FRIGHT.\n\nFabritius makes mention of a gentleman, with whom he was familiar,\nwho, being unjustly suspected, was tortured upon the rack, and, when\nreleased, found himself quite cured of the gout, which was, _before_\nthis violent remedy, rather troublesome. Again, we have instances of\ndisorders being cured by fright. We find, in the Journal de Henri IV.,\nthat, \"On Friday, June the 9th, 1606, as Henry IV. of France, and his\nQueen, were crossing the water in the ferry-boat of Neuilly, the Duke\nof Vendome being with them, they were all three in great danger of\nbeing drowned, especially the queen, who was obliged to drink a great\ndeal more than was agreeable to her; and had not one of her footmen,\nand a gentleman called La Chatagnieraie, who caught hold of her hair,\ndesperately thrown themselves into the water to pull her out, she\nwould have inevitably lost her life. This accident cured the king of a\nviolent toothache; and, after having escaped the danger, he diverted\nhimself with it, saying he had never met with so good a remedy for that\ndisorder before, and that they had ate too much salt meat at dinner,\ntherefore they had a mind to make them drink after it.\"\n\n\nTHE WINGLESS BIRD OF NEW ZEALAND.\n\nOne of the chief wonders of the world of Ornithology is the Apteryx,\na bird which is found only in New Zealand, and even there, is rapidly\nbecoming extinct. It is a creature so strange, that no imagination\ncould have fancied a bird without wings or tail, with robust legs,\nand with claws which are suited for digging, and are actually used\nin forming excavations, in which this singular bird lays its eggs,\nand hatches its young. If the Apteryx were to become extinct, and all\nthat remained of it, after the lapse of one or two centuries, for the\nscrutiny of the naturalist were a foot in one Museum, and a head in\nanother, with a few conflicting figures of its external form, the real\nnature and affinities of this most remarkable species would be involved\nin as much obscurity and doubt, and become the subject of as many\nconflicting opinions among the ornithologists of that period, as are\nthose of the Dodo in the present day.\n\nThe Apteryx is not larger than a full-grown fowl, and has only a\nrudimentary wing, so covered with the body feathers as to be quite\nconcealed; the terminating slender claw may, however, be discerned on\nexamination.\n\n[Illustration: The Wingless Bird.]\n\nThe bill is long and slightly curved, having the nostrils at the\nextremity; its feathers, the sides of which are uniform in structure,\ndo not exceed four and a-half inches in length, and are much prized as\nmaterial for mantles or cloaks by the chiefs. It is a nocturnal bird,\nusing its long bill in search of worms, upon which it principally\nfeeds; it kicks with great power, and burrows at the root of the rata,\nat the base of which tree is also found the extraordinary Sphaeria\nRobertsia, a species of vegetating caterpillar. Retaining the form\nof the caterpillar, the fungus pervades the whole body, and shoots\nup a small stem above the surface of the ground, the body of the\ncaterpillar being below the earth in an erect position. The Apteryx\nfrequently leans with its bill upon the earth--one of its chief\ncharacteristics--and thus, when viewed from a distance, appears to be\nstanding on three legs.\n\nBy the natives of New Zealand, these birds are called Kiwis, from the\ncry they utter, and they are frequently caught by a cunning imitator of\ntheir tone, who, when they approach, dazzles and frightens them with\na light previously concealed, and throwing his blanket over them thus\nsecures them.\n\n\nA FLOATING CITY.\n\n[Illustration: Floating City of Bankok.]\n\nOne of the most wonderful cities in the world is Bankok. It is the\ncapital of Siam, and is situated on--or rather in--the great river\nMeinam. Our engraving represents a portion of this unique metropolis,\nand we find the following graphic account of it in a volume of recent\ntravels--\"The capital of Siam! Did you ever witness such a sight in\nyour life? On either side of the wide, majestic stream, moored in\nregular streets and alleys, and extending as far as the eye can reach,\nare upwards of seventy thousand neat little wooden houses, each house\nfloating on a compact raft of bamboos; and the whole intermediate space\nof the river presents to our astonished gaze one dense mass of ships,\njunks, and boats, of every conceivable shape, colour, and size. As we\nglide along amongst these, we occasionally encounter a stray floating\nhouse, broken loose from its moorings, and hurrying down the stream\nwith the tide, amidst the uproar and shouts of the inhabitants and\nall the spectators. We also observe that all the front row of houses\nare neatly painted shops, in which various tempting commodities are\nexposed for sale; behind these again, at equal distances, rise the\nlofty and elegant porcelain towers of the various watts and temples.\nOn our right-hand side, far away as we can see, are three stately\npillars, erected to the memory of three defunct kings, celebrated for\nsome acts of valour and justice; and a little beyond these, looming\nlike a line-of-battle ship amongst a lot of cockle-shells, rises the\nstraggling and not very elegant palace of the king, where his Siamese\nMajesty, with ever so many wives and children, resides. Right ahead,\nwhere the city terminates, and the river, making a curve, flows\nbehind the palace, is a neat-looking-fort, surmounted with a tope of\nmango-trees, over which peep the roofs of one or two houses, and a tall\nflag-staff, from which floats the royal pendant and jack of Siam--a\nflag of red groundwork, with a white elephant worked into the centre.\nThat is the fort and palace of the prince Chou Fau, now king of Siam,\nand one of the most extraordinary and intellectual men in the East.\nOf him, however, we shall see and hear more, after we have bundled\nour traps on shore, and taken a little rest. Now, be careful how you\nstep out of the boat into the balcony of the floating house, for it\nwill recede to the force of your effort to mount, and if not aware of\nthis, you lose your balance and fall into the river. Now we are safely\ntranshipped, for we cannot as yet say landed; but we now form an item,\nthough a very small one, of the vast population of the city of Bangkok.\n\nWe take a brief survey of our present apartments, and find everything,\nthough inconveniently small, cleanly and in other respects comfortable.\nFirst, we have a little balcony which overhangs the river, and is about\ntwenty yards long by one and a half broad. Then we have an excellent\nsitting-room, which serves us for parlour, dining-room, and all; then\nwe have a little side room for books and writing; and behind these,\nextending the length of the other two, a bed-room. Of course we must\nbring or make our own furniture; for, though those houses inhabited\nby the Chinese are pretty well off on this score, the Siamese have\nseldom anything besides their bedding materials, a few pots and pans\nto cook with, a few jars of stores and fishing-net or two. Every house\nhas a canoe attached to it, and no nation detests walking so much as\nthe Siamese; at the same time they are all expert swimmers, and both\nmen and women begin to acquire this very necessary art at a very early\nage. Without it a man runs momentary risk of being drowned, as, when a\ncanoe upsets, none of the passers-by ever think it necessary to lend\nany aid, supposing them fully adequate to the task of saving their\nown lives. Canoes are hourly being upset, owing to the vast concourse\nof vessels and boats plying to and fro; and, owing to this negligence\nor carelessness in rendering assistance, a Mr. Benham, an American\nmissionary, lost his life some twelve years ago, having upset his own\ncanoe when it was just getting dusk, and though surrounded by hundreds\nof boats, not one deemed it necessary to stop and pick the poor man up.\"\n\n\nBEQUESTS FOR LIGHTING THE STREETS.\n\nThere cannot be a greater contrast than between the present and the\nancient mode of lighting the streets of London. What a picture do the\ntwo following bequests present to us of the state of things a hundred\nyears ago!\n\nJohn Wardall, by will, dated 29th August, 1656, gave to the Grocers'\nCompany a tenement called the White Bear, in Walbrook, to the intent\nthat they should yearly, within thirty days after Michaelmas, pay to\nthe churchwardens of St. Botolph, Billingsgate, L4, to provide a good\nand sufficient iron and glass lantern, with a candle, for the direction\nof passengers to go with more security to and from the water-side, all\nnight long, to be fixed at the north-east corner of the parish church\nof St. Botolph, from the feast-day of St. Bartholomew to Lady-Day; out\nof which sum L1 was to be paid to the sexton for taking care of the\nlantern. This annuity is now applied to the support of a lamp in the\nplace prescribed, which is lighted with gas.\n\nJohn Cooke, by will, dated 12th September, 1662, gave to the\nchurchwardens, &c., of St. Michael's, Crooked Lane, L76, to be laid out\nto the most profit and advantage, for various uses, and amongst them,\nfor the maintenance of a lantern and candle, to be eight in the pound\nat least, to be kept and hanged out at the corner of St. Michael's\nLane, next Thames Street, from Michaelmas to Lady-Day, between the\nhours of nine and ten o'clock at night, until the hours of four or five\nin the morning, for affording light to passengers going through Thames\nStreet, or St. Michael's Lane.\n\n\nEXTRAORDINARY INSTANCE OF CREDULITY.\n\nTo the honour of the lords of the creation, there are _some_ husbands\nwho so grieve at the death of their partners, that they will not part\nwith them when actually dead; and even go so far as to wish, and try\nhard, for their resurrection; witness Sir John Pryse, of Newtown,\nMontgomeryshire, who married three wives, and kept the first two\nwho died, in his room, one on each side of his bed; his third lady,\nhowever, declined the honour of his hand till her defunct rivals\nwere committed to their proper place. Sir John was a gentleman of\nstrange singularities. During the season of miracles worked by Bridget\nBostock, of Cheshire, who healed all diseases by prayer, faith, and an\nembrocation of fasting spittle, multitudes resorted to her from all\nparts, and kept her salivary glands in full employ. Sir John, with a\nhigh spirit of enthusiasm, wrote to this wonderful woman to make him\na visit at Newtown Hall, in order to restore to him his third and\nfavourite wife (above mentioned), now dead. His letter will best tell\nthe foundation on which he built his strange hope, and very uncommon\nrequest:--\n\n _Purport of Sir J. Pryse's letter to Mrs. Bridget Bostock, 1748._\n\nMadam,--Having received information, by repeated advices, both public\nand private, that you have, of late, performed many wonderful cures,\neven where the best physicians have failed, and that the means used\nappeared to be very inadequate to the effects produced, I cannot but\nlook upon you as an extraordinary and highly-favoured person; and why\nmay not the same most merciful God, who enables you to restore sight to\nthe blind, hearing to the deaf, and strength to the lame, also enable\nyou to raise the dead to life? Now, having lately lost a wife, whom\nI most tenderly loved; my children an excellent step-mother, and our\nacquaintances a very dear and valuable friend, you will lay us all\nunder the highest obligations; and I earnestly entreat you, for God\nAlmighty's sake, that you will put up your petitions to the Throne of\nGrace, on our behalf, that the deceased may be restored to us, and\nthe late dame Eleanor Pryse be raised from the dead. If your personal\nattendance appears to you to be necessary, I will send my coach and\nsix, with proper servants, to wait on you hither, whenever you please\nto appoint. Recompense of any kind, that you could propose, would be\nmade with the utmost gratitude; but I wish the bare mention of it is\nnot offensive to both God and you.\n\n I am, madam, your obedient, &c.\n JOHN PRYSE.\n\n (_Pennant's Wales_, vol. 3, p. 190.)\n\n\nHIGH PRICE OF FISH IN LONDON.\n\nIt is on record that on January 4, 1809, there being only four cod-fish\nin Billingsgate, a fishmonger gave fourteen guineas for them, and\nsalmon soon after was sold at a guinea a pound!\n\n\nTHE GREAT AQUEDUCT OF PONT DU GARD.\n\n[Illustration [++] Acqueduct of Pont du Gard.]\n\nThe remains of Roman aqueducts, of great extent and massiveness,\noccur in various parts of Europe, over which the Roman dominion once\nextended. Among these, the most celebrated are the Pont du Gard,\nnear Nismes, in the Department du Gard, in the south of France; the\naqueduct over the Moselle, near Metz; and the aqueduct of Segovia, in\nOld Castile. The Pont du Gard (of which we here give an engraving)\nwas designed to convey the waters of the fountain of Aure to the town\nof Nismes, the ancient Nemausus. This aqueduct crosses the beautiful\nvalley, and the stream of the river Gardon, uniting two steep hills, by\nwhich the valley is bounded at this place. It consists of two tiers of\nlarge arches, the lower of which are eighty feet in span, and a third\ntier of small arches, which support the trunk of the aqueduct. The\nchannel for the water is above four feet wide, and five deep, and is\nlined with cement three inches thick, and covered with a thin coating\nof red clay. The whole work, with the exception of the above-mentioned\nchannel for the water, is built without mortar or any other cement;\nand its elevation above the bed of the river Gardon, is not less than a\nhundred and fifty feet. The extremities of this splendid structure are\nin a dilapidated condition, but the remainder is in a very good state\nof preservation.\n\n\nEXTRAORDINARY SITUATION FOR A TREE.\n\n[Illustration [++] Tree in Abbey of Muckross.]\n\nThe Lower and Middle Lakes at Killarney are separated by a peninsula,\nupon which stands the ruin of the Abbey of Muckross, which was founded\nin 1440, and re-edified in 1602. The ruin, which consists of parts\nof the convent and church, is not remarkable either for extent, or\nfor beauty of workmanship, but its preservation, seclusion, beauty\nof situation, and accompanying venerable trees, render it one of the\nmost interesting abbey remains in Ireland. The entire length of the\nchurch is about 100 feet, its breadth 24. The cloister, which consists\nof twenty-two arches, ten of them semicircular, and twelve pointed,\nis the best preserved portion of the abbey. In the centre grows a\nmagnificent yew-tree, as represented in our engraving, which covers\nas a roof the whole area; its circumference is thirteen feet, and its\nheight in proportion. It is more than probable that the tree is coeval\nwith the abbey, and that it was planted by the hands of the monks who\nfirst inhabited the building. It is believed by the common people that\nany person daring to pluck a branch, or in any way attempting to injure\nthis tree, will not be alive on that day twelvemonth.\n\n\nPRAYING BY MACHINERY.\n\nMr. Moorcroft informs us, in his \"Journey to Lake Manasawara, in Undes,\na province in Little Thibet,\" that the inhabitants used the following\nmost extraordinary way of saying their prayers:--It is done by motion,\nwhich may be effected by the powers of steam, wind, or water. A large\nhollow cylinder, like a drum, is erected, within which is inclosed all\nthe written prayers the people choose to offer, and then it is set\ngoing, by being whirled round its own axis; thus saving the trouble\nof repeating them. Mr. Turner, whose travels in Thibet are before the\npublic, corroborates the account of these whirligigs. They are common,\nalso, among the Monguls, the Calmucs, and the Kalkas; so that the\nengineers for these pious wheels must have a tolerably extensive trade,\nas this national mode of worship is naturally liable to wear out.\nBut even this mode is innocence itself, compared with that of a set\nof savages, who _pray people to death_; for Lisiansky, in his Voyage\nround the World, gives us an account of an extra-religious sect, in\nthe Sandwich Islands, who arrogate to themselves the power of praying\npeople to death. Whosoever incurs their displeasure, receives notice\nthat the homicide-litany is about to begin; and such are the effects of\nimagination, that the very notice is frequently sufficient, with these\nweak people, to produce the effect, or to drive them to acts of suicide.\n\n\nTOPING IN THE LAST CENTURY.\n\nAt a Somersetshire hunt dinner, seventy years since, thirteen toasts\nused to be drunk in strong beer; then every one did as he liked. Some\nmembers of the hunt occasionally drank a glass of wine at the wind up,\nwho were not themselves previously wound up. In country towns, after a\ndinner at one o'clock P.M., friends used to meet to discuss the local\nnews over their glasses of strong beer, the merits of which furnished\na daily theme. At Bampton one knot of gentlemen took four times the\nduration of the Trojan war, and even then failed to settle which of the\nparty brewed the best beer.\n\n\nA FINE OLD SOLDIER.\n\nJeremiah Atkins, of the Scar, near Bromyard, Herefordshire, died in\n1796, aged 102. He had been a soldier through all the earlier periods\nof his manhood, and had seen much service; was present at the taking\nof Martinico, and at the Havannah; and, on one occasion, being taken\nprisoner by the Indians of North America, was very near being scalped,\nas he was only rescued at the moment they were about to perform the\noperation. He was likewise at the taking of Crown Point, in America,\nand in the battle of Fontenoy with the Duke of Cumberland, whom he also\naccompanied in his resistance to the advance of the Scotch rebels,\nbeing in several of the skirmishes and battles fought on that occasion.\nHe afterwards went again to America, and took part in the storming of\nQuebec, when Wolfe was killed. The last battle in which he was engaged\nwas that of Tournay, in Flanders. This extraordinary man retained the\nfull use of all his natural faculties, save hearing, to the very close\nof his life.\n\n\nPOPULAR FALLACY OF THE VIRTUES OF A SEVENTH SON.\n\nIt is believed that a seventh son can cure diseases, but that a\nseventh son of a seventh son, and no female child born between, can\ncure the king's evil. Such a favoured individual is really looked\non with veneration. An artist visiting Axminster in 1828, noticing\nthe indulgence granted to one urchin in preference to others, and\nseeing something particular in this child, addressed his mother as\nfollows:--\"This little man appears to be a favourite: I presume he is\nyour little Benjamin.\" \"He's a seventh son, sir,\" said the mother.\nAffecting an air of surprise, I expressed myself at the instant as\nbeing one very anxious to know what a _seventh_ son could do? The\nmother, a very civil woman, told me that \"she did think, to cure all\ndiseases, should be the seventh son of a seventh son; but _many folk\ndo come to touch my son_.\" In April, 1826, a respectable looking woman\nwas engaged in collecting a penny from each of thirty young women,\nunmarried; the money to be laid out in purchasing a silver ring, to\ncure her son of epileptic fits. The money was to be freely given,\nwithout any consideration, or else the charm would have been destroyed.\nThe young women gave their pence, because it would have been a _pity_\nfor the lad to continue afflicted _if_ the charm would cure him.\n\n\nSELF-NOURISHMENT.\n\nThat animals may sometimes be kept alive for a long time solely on\nnourishment supplied from their own bodies, is evident from the fact\nthat after a great fall of earth on one occasion from the cliff at\nDover, which buried a whole family, a hog was found alive five months\nand nine days after it had thus been buried! It weighed about seven\nscore when the accident happened, and had wasted to about thirty\npounds, but was likely to do well.\n\n\nCHINESE METHOD OF FISHING.\n\nThere is nothing more extraordinary in the history of the different\nnations of the world than the ingenuity of the Chinese. They are the\nmost handy people on the face of the earth, and the lower orders are\njust as clever as the higher. A proof of this may be seen at a fishing\nvillage which is contiguous to the town of Victoria, in Hong Kong. It\nremains in much the same state as that in which it existed prior to\nthe British occupation of the island. Old worn-out boats, and torn\nmat-sails, bamboos and dried rushes,--these are the principal materials\nemployed in the construction of their domiciles. The fishing boats are\nmost ingeniously built. Each of these has a long projecting bamboo,\nwhich is rigged out from the stem in the form of a bowsprit, only\nworking on a pivot. From the extremity of this outrigger, a strong\nrope communicates with a balance-board, that exactly poises the bamboo\noutrigger, when the net is immersed in water, and the fisherman has\nonly to walk up and down this plank to raise the net and let it drop\nagain in the water. But opposite to the island, and on many of the\nlittle insular rocks which constitute the \"ten thousand isles,\" of\nwhich the emperor of China, amongst his vast pretensions to titles,\nlays claim to be lord, fishing is conducted on a larger scale, though\nworked upon the same principles. Huge poles are driven into the ground\nwhere the water is comparatively shallow, and leading ropes, which pass\nover a block-wheel inserted in the tops of these poles, communicate at\none end with large circular nets, (constructed somewhat in the shape\nof a funnel, the upper rim being attached to floats, whilst from the\ncentre are pendant weights,) the other end being fastened on shore to a\nbalance plank, which the weight of one man suffices to work.\n\n[Illustration: Chinese Method of Fishing.]\n\n\nMOSQUE OF OMAR.\n\nThe opposite engraving represents the Great Mosque at Jerusalem. It is\nbuilt on the exact site of Solomon's Temple, and takes its name from\nits original founder, the Caliph Omar. It is a Turkish edifice, and is\ndevoted to the worship of Mahomet.\n\nTitus having taken Jerusalem in the second year of Vespasian's reign,\nnot one stone was left upon another of that Temple where Christ\nhad done such glorious things, and the destruction of which he had\npredicted. When the Caliph Omar took Jerusalem, in 636 A.D., it appears\nthat the site of the Temple, with the exception of a very small part,\nhad been abandoned by the Christians. Said-Eben-Batrick, an Arabian\nhistorian, relates that the Caliph applied to the Patriarch Sophronius,\nand enquired of him what would be the most proper place at Jerusalem\nfor building a mosque. Sophronius conducted him to the ruins of\nSolomon's Temple. Omar, delighted with the opportunity of erecting a\nmosque on so celebrated a spot, caused the ground to be cleared, and\nthe earth to be removed from a large rock, where God is said to have\nconversed with Jacob. From that rock the new mosque took its name\nof Gameat-el-Sakhra, and became almost as sacred an object to the\nMussulmans, as the mosques of Mecca and Medina. The Caliph El-Oulid\ncontributed still more to the embellishment of El-Sakhra, and covered\nit with a dome of copper, gilt, taken from a church at Balbeck. In the\nsequel, the crusaders converted the Temple of Mahomet into a sanctuary\nof Christ; but when Saladin re-took Jerusalem, he restored this edifice\nto its original use.\n\n[Illustration: Great Mosque at Jerusalem.]\n\nThe form is an octagon, either side being seventy feet in width; it is\nentered by four spacious doors, the walls are white below, intermingled\nwith blue, adorned with pilasters, but above, it is faced with glazed\ntiles of various colours. The interior is described as paved with grey\nmarble, the plain walls are covered with the same material in white.\nIt contains many noble columns, in two tiers. The dome is painted, and\ngilt in arabesque, whence depend antique vessels of gold and silver;\nimmediately beneath it stands a mass of limestone, reported to have\nfallen from heaven when the spirit of prophecy commenced. On this sat\nthe destroying angel, during the slaughter caused by David's numbering\nthe people. From this Mahomet ascended to heaven. Within the storied\nwalls, moreover, are the scales for weighing the souls of men, the\nshield of Mahomet, and other relics, besides the entrance to the\ninfernal regions; seventy thousand angels ever guard the precious stone.\n\nEntrance to this hallowed edifice has been gained only by two or three\nEuropeans; indeed, the Turks will not allow infidels to approach the\nsacred enclosure around it, which measures about sixteen hundred feet\nin length, by one thousand in width, and is adorned with fountains,\norange, cypress, and other trees.\n\nThe mosque itself is esteemed the finest piece of Saracenic\narchitecture in existence, far surpassing St. Sophia in beauty. Its\nview, combined with the distinguished monuments in the City of the\nSultan, in Egypt, Greece, and Italy, strongly induces a belief in the\naccuracy of an able article in the _Quarterly Review_, in which the\norigin of the five predominant styles of architecture throughout the\nworld, viz., the Byzantine, Chinese, Egyptian, Grecian, and Gothic are\nassigned respectively to the convex and concave curves, to the oblique,\nhorizontal, and perpendicular lines.\n\n\nA COUPLE OF ECCENTRICS.\n\nMr. Day, the eccentric founder of Fairlop fair, had a housekeeper, who\nhad lived with him for thirty years, and was equally eccentric. She had\ntwo very strong attachments; one to her wedding-ring and garments, and\nthe other to tea. When she died, Mr. Day would not permit her ring to\nbe taken off; he said, \"If that was attempted, she would come to life\nagain;\" and directed that she should be buried in her wedding-suit, and\na pound of tea in each hand; and these directions were literally obeyed.\n\n\nTHE UNIVERSALITY OF TAXATION.\n\nThe following extract, from the _Edinburgh Review_, is not\ninappropriate to our pages, inasmuch as it is both a rare specimen\nof effective composition, and also serves to show us what the state\nof taxation was in England even within the last forty years.--Taxes\nupon every article which enters into the mouth, or covers the back,\nor is placed upon the feet--taxes upon every thing which it is\npleasant to see, hear, feel, smell, or taste--taxes upon warmth,\nlight, and locomotion--taxes on everything on earth, and the waters\nunder the earth--on every thing that comes from abroad, or is grown at\nhome--taxes on the raw material--taxes on every fresh value that is\nadded to it by the industry of man--taxes on the sauce which pamper's\nman's appetite, and the drug that restores him to health--on the ermine\nwhich decorates the judge, and the rope which hangs the criminal--on\nthe poor man's salt, and the rich man's spice--on the brass nails of\nthe coffin, and the ribands of the bride at bed or board, _couchant_\nor _levant_, we must pay;--the schoolboy whips his taxed top--the\nbeardless youth manages his taxed horse, with a taxed bridle, on a\ntaxed road:--and the dying Englishman, pouring his medicine, which has\npaid seven per cent., into a spoon that has paid fifteen per cent.,\nflings himself back upon his chintz bed, which has paid twenty-two\nper cent.--makes his will on an eight-pound stamp, and expires in the\narms of an apothecary, who has paid a license of an hundred pounds\nfor the privilege of putting him to death. His whole property is then\nimmediately taxed from two to ten per cent. Besides the probate, large\nfees are demanded for burying him in the chancel; his virtues are\nhanded down to posterity on taxed markle; and he is then gathered to\nhis fathers--to be taxed no more.\n\n\nSHAM PROPHETS.\n\nWilliam Hackett, a fanatic of the sixteenth century, after a very ill\nlife, turned prophet, and signified the desolation of England. He\nprophesied at York and at Lincoln; where, for his boldness, he was\nwhipped publicly, and condemned to be banished. He had an extraordinary\nfluency of speech, and much assurance in his prayers; for he said,\nthat if all England should pray for rain, and he should pray to the\ncontrary, it should not rain. Hackett had two brother-prophets joined\nwith him, Edward Coppinger, named the prophet of mercy, and Henry\nArthington, the prophet of judgment. Coppinger, the merciful prophet,\ndeclared that Hackett was the sole monarch of Europe; and at length\nthey proclaimed him, July 16, 1592. On the 28th of the same month,\nhowever, the monarch of the whole earth, who had also personated\ndivinity, was hanged and quartered. Coppinger famished himself in\nprison, and Arthington was pardoned. Fitz Simon relates, that in a\nquarrel Hackett had at Oundle, \"He threw down his adversary, and\nbit off his nose; and, instead of returning it to the surgeon, who\npretended to set it on again, while the wound was fresh, ate it.\"\nHackett, on the scaffold, made a blasphemous prayer, which is recorded\nby Fitz Simon and Camden, too horrid to be repeated. He hated Queen\nElizabeth, and tried to deprive her of her crown; he confessed to\nthe judges that he had stabbed the effigies of this princess to the\nheart, with an iron pin; and a little before he was hanged, being an\naccomplished swearer, he cursed her with all manner of imprecations.\n\n\nHOOKING A BOY INSTEAD OF A FISH.\n\nAbout five and thirty years ago, as Mr. George Moor was fishing in the\nriver Tyne at Pipewellgate, Gateshead, he espied something in the water\nwhich seemed like a drowned dog, but the day being clear, and the sun\nshining, he thought he perceived a face, upon which he threw his line\nto it (which had but three hairs at the hook) and hooked a coat, by\nwhich he found it was a boy, but the hook loosing hold, he again cast\nhis line and struck him in the temple and drew him to the shore, and in\nless than quarter of an hour he revived.\n\n\nCHILDREN OF AGED PARENTS.\n\nMargaret Krasiowna, of the village of Koninia, Poland, died 1763, aged\n108. The following extraordinary circumstances are stated, by Eaton,\nas connected with the life of this woman:--\"At the age of ninety-four\nshe married her third husband, Gaspard Raycolt, of the village of\nCiwouszin, then aged one hundred and five. During the fourteen years\nthey lived together she brought him two boys and a girl; and, what\nis very remarkable, these three children, from their very birth, bore\nevident marks of the old age of their parents--their hair being grey,\nand a vacuity appearing in their gums, like that which is occasioned\nby the loss of teeth, though they never had any. They had not strength\nenough, even as they grew up, to chew solid food, but lived on bread\nand vegetables, they were of a proper size for their age, but their\nbacks were bent, their complexions sallow, with all the other external\nsymptoms of decrepitude. Though most of these particulars,\" he adds,\n\"may appear fabulous, they are certified by the parish registers. The\nvillage of Ciwouszin is in the district of Stenzick, in the palatinate\nof Sendonier. Gaspard Raycolt, the father, died soon after, aged 119.\"\n\n\nSEPULCHRAL VASE FROM PERU.\n\n[Illustration [++] Sepulchral Vase of Peru.]\n\nThe vessel of which the annexed is an engraving, was taken from the\ntomb of one of the ancient inhabitants of Peru; the subjects of the\nIncas, or princes who ruled over that country before it was conquered\nby the Spaniards. Vases of this sort were probably placed in the\nsepulchres of the Peruvians to contain the ashes of the dead, or\nofferings to their disembodied spirits;--usages which are familiar\nto us through the frequent allusions to them which we meet with in\nthe works of the poets of ancient Rome, and the discovery of urns and\nlachrymatories in Roman tombs which have been in our own and other\ncemeteries. The specimen which we have engraved is quadruple, but forms\none vessel.\n\n\nFIRST IRON CANNON.\n\nThe first cannon was cast in Sussex in 1535. In after years bonds were\ntaken in L1,000 from the owners of the charcoal furnaces, that none\nshould be sold till a license for the sale or issue of the ordnance had\nbeen procured. Fears were entertained that the enemy would purchase\nthem.\n\n\nPROLIFIC AUTHOR.\n\nNo one need despair, after the following instance, of shining in\nquantity, if not in quality:--\"Hans Sacks was a Nuremberg shoemaker,\nborn there in 1494; he was instructed, by the master-singers of those\ndays, in the praiseworthy art of poetry; he, therefore, continued to\nmake verses and shoes, and plays and pumps, boots and books, until\nthe seventy-seventh year of his age; when he took an inventory of his\npoetical stock in trade, and found, according to his narrative, that\nhis works filled thirty folio volumes, all written with his own hand;\nand consisted of four thousand two hundred mastership songs, two\nhundred and eight comedies, tragedies, and farces (some of which were\nextended to seven acts), one thousand seven hundred fables, tales, and\nmiscellaneous poems, and seventy-three devotional, military, and love\nsongs; making a sum total of six thousand and forty-eight pieces, great\nand small.\" Out of these, we are informed, he culled as many as filled\nthree massy folios, which were published in the year 1558-61; and,\nanother edition being called for, he increased this three volumes folio\nabridgement of his works, in the second, from his other works. None but\nLope de Vega exceeded him in quantity of rhyme-making.\n\n\nTHE ART OF POTTERY IN CHINA.\n\n[Illustration: 1. 2. 3.]\n\nThe Chinese traditions carry back the practice of the potter's art to a\nvery remote epoch. Father Entrecolles, a French missionary, resided in\nChina at the beginning of the last century, and his letters published\nin Paris, in 1741, supply some curious and interesting information\non this subject. Writing in 1712, he says that at that time ancient\nporcelain was very highly prized, and bore large prices. Articles were\nextant which were reputed to have belonged to the Emperors Yao and\nChun, two of the most ancient mentioned in the Chinese annals. Yao\nreigned in 2357 and Chun in 2255 before Christ. Other authorities place\nthe reign of Chun in 2600 before Christ. It appears from the researches\nof M. Stanislaus Julian that, from the time of the Emperor Hoang-ti,\nwho reigned 2698 to 2599 before Christ, there had always existed a\npublic officer bearing the title of the Intendant of Pottery, and that\nit was under the reign of Hoang-ti that the potter's art was invented\nby Kouen-ou. It is also certain that porcelain, or fine pottery, was\ncommon in China in the time of the Emperors Han, 163 B.C.\n\nIn digging the foundations of the palaces, erected by the dynasties of\nHan and Thang, from 163 B.C. to 903 A.D. great quantities of ancient\nvases were found which were of a pure whiteness, but exhibited little\nbeauty of form or fabrication. It was only under the dynasty of Song,\nthat is to say, from 960 to 1278 A.D., that Chinese porcelain began to\nattain a high degree of perfection.\n\nFurther evidence of the antiquity of the potter's art in China, as well\nas of the existence of intercommunication between that country and\nEgypt, is supplied by the discoveries of Rossellina, Wilkinson, and\nothers, who found numerous vases of Chinese fabrication, and bearing\nChinese inscriptions, in the tombs at Thebes. Professor Rossellini\nfound a small vase of Chinese porcelain with a painting of a flower\non one side, and on the other Chinese characters not differing much\nfrom those used at the present day. The tomb was of the time of the\nPharaohs, a little later than the eighteenth dynasty.\n\nThis vase, with its Chinese inscription, is represented in Fig. 1, from\nan exact cast made by Mr. Francis Davis.\n\nAnother of the Chinese vases, found in the Theban tombs, is represented\nin Fig. 2. This is preserved in the Museum of the Louvre. The shape of\nthe vase is that of a flat-sided flask. A side view is given in Fig. 3.\n\nThese flasks are very small. The engravings represent them of their\nproper dimensions. Mr. Wilkinson thinks it probable that they were\nbrought to Egypt from India, the Egyptians having had commercial\nrelations with that country at a very remote epoch, and that they came\nnot as pieces of porcelain, but as vessels containing some articles of\nimportation.\n\n\nSTRONG ATTACHMENT TO SMOKING.\n\nThe following is a curious case of extreme fondness for smoking in a\nvery poor and very old man. In the year 1810, there died in Dartford\nworkhouse, aged 106, one John Gibson. He had been an inmate of the\nhouse for ten years, and till within two months of his death used daily\nto perambulate the town. His faculties were entire to the last. He was\nso much attached to smoking, that he requested his pipe, together with\nhis walking-stick, might be placed in his coffin, which request was\ncomplied with.\n\n\nEXTRAORDINARY LETTER.\n\nThe following strange and curious epistle, we are assured, was sent to\na surgeon of eminence by a malefactor who had been sentenced to death.\nIt has a degree of character and quaintness about it which is rarely\nfound in the letters of convicts. Whether or not the surgeon complied\nwith his request we do not know.\n\n\"Sir,--Being informed that you are the only surgeon in this county, in\nthe habit of dissecting dead bodies--being very poor, I am desirous of\npassing what remains to me of life, with as much comfort as my unhappy\ncondition admits of. In all probability I shall be executed in the\ncourse of a month; having no friend to intercede for me, nor even to\nafford me a morsel of bread, to keep body and soul together till the\nfatal moment arrives, I beg you will favour me with a visit; I am\ndesirous of disposing of my body, which is healthy and sound, for a\nmoderate sum of money. It shall be delivered to you on demand, being\npersuaded that on the day of general resurrection, I shall as readily\nfind it in your laboratory, as if it were deposited in a tomb. Your\nspeedy answer will much oblige your obedient servant,\n\n JAMES BROWN.\"\n\n\nA MATTRESS FOR A BANK.\n\nIn the month of April, 1822, Mrs. Motley, broker, Bedford-street,\nNorth Shields, purchased an old mattress for 2s. from a shipowner,\nwho was going to reside with his daughter; in arranging some papers a\nfew days ago, he found a document in the hand-writing of his deceased\nwife, not intended for his perusal, but that of her son by a former\nhusband, in which it was stated that property to a considerable amount\nwas deposited in the said mattress. His daughter in consequence waited\non Mrs. Motley, and offered her a few shillings to return it. Mrs.\nM. naturally supposed that this seeming generosity was not without\na cause, but having sold it to a Mrs. Hill for 3s., for a small\nconsideration she regained possession of the prize, but on entering her\nhouse the original proprietor and a constable were ready to receive\nher, and without ceremony cut open the mattress, when a purse, said to\ncontain 100gs., two gloves filled with current silver coin, several\nvaluable rings, trinkets, silver spoons, &c., were discovered. Mrs.\nHill had considerably reduced the mattress to fit a small bedstead\nwithout finding the hidden treasure.\n\n\nARCHITECTURE FOR EARTHQUAKES.\n\nSumatra is one of the largest islands in the Indian Archipelago, and\nthe houses of the inhabitants are deserving of notice, inasmuch as\nthey furnish a correct and curious specimen of the style of building,\nwhich the frequent occurrence of earthquakes renders the safest in the\ncountries where such visitations are common.\n\nThe frames of the houses are of wood, the under-plates resting on\npillars six or eight feet high, which have a sort of capital, but no\nbase, and are wider at top than at bottom. The people appear to have no\nidea of architecture as a science, though much ingenuity is often shown\nin working up their materials. The general appearance of their houses\nis accurately represented in the annexed plate. For the floorings they\nlay whole bamboos, four or five inches in diameter, close to each\nother, and fasten them at the ends to the timbers. Across these are\nlaid laths of split bamboo, about an inch wide and of the length of the\nroom, which are tied down with filaments of the rattan, and over these\nare usually spread mats of different kinds. This sort of flooring has\nan elasticity alarming to strangers when they first tread on it.\n\nThe sides of the houses are generally closed in with bamboo, opened\nand rendered flat by notching or splitting the circular joints on the\noutside, chipping away the corresponding divisions within, and laying\nit to dry in the sun pressed down with weights. This is sometimes\nnailed to the upright timbers or bamboos, but in the country parts\nit is more commonly interwoven or matted in breadths of six inches,\nand a piece or sheet formed at once of the size required. In some\nplaces they use for the same purpose the inner bark procured from some\nparticular trees. When they prepare to take it, the outer bark is first\ntorn or cut away; the inner is then marked out with a proper tool to\nthe requisite size, usually three cubits by one; it is afterwards\nbeaten for some time with a heavy stick to loosen it from the stem, and\nbeing peeled off, laid in the sun to dry, care being taken to prevent\nits warping. The bark used in building has nearly the texture and\nhardness of wood; but the pliable and delicate bark of which clothing\nis made is procured from a bastard species of the bread-fruit.\n\n[Illustration [++] Sumatran House.]\n\nThe most general mode of covering houses is with the leaf of a kind of\npalm called _nipah_. These, before they are laid on, are formed into\nsheets about five feet long, and as deep as the length of the leaf will\nadmit, which is doubled at one end over a slip or lath of bamboo. They\nare then disposed on the roof so that one sheet shall lap over the\nother, and are tied to the bamboos which serve for rafters.\n\n\nTHE NOSS IN SHETLAND.\n\nOff Bressay is the most remarkable of the rock phenomena of Shetland,\nthe Noss, a small high island, with a flat summit, girt on all sides\nby perpendicular walls of rock. It is only 500 feet in length, and\n170 broad, and rises abruptly from the sea to the height of 160 feet.\nThe communication with the coast of Bressay is maintained by strong\nropes stretched across, along which a cradle or wooden chair is run, in\nwhich the passenger is seated. It is of a size sufficient for conveying\nacross a man and a sheep at a time. The purpose of this strange\ncontrivance is to give the tenant the benefit of putting a few sheep\nupon the Holm, the top of which is level, and affords good pasture.\nThe animals are transported in the cradle, one at a time, a shepherd\nholding them upon his knees in crossing.\n\n[Illustration: Cradle of Noss.]\n\nThe temptation of getting access to the numberless eggs and young\nof the sea-fowl which whiten the surface of the Holm, joined to the\npromised reward of a cow, induced a hardy and adventurous fowler, about\ntwo centuries ago, to scale the cliff of the Holm, and establish a\nconnexion by ropes with the neighbouring main island. Having driven\ntwo stakes into the rock and fastened his ropes, the desperate man was\nentreated to avail himself of the communication thus established in\nreturning across the gulf. But this he refused to do, and in attempting\nto descend the way he had climbed, he fell, and perished by his\nfoolhardiness.\n\n\nSWALLOWED UP BY AN EARTHQUAKE AND THROWN OUT AGAIN.\n\nA tombstone in the island of Jamaica has the following inscription:--\n\n\"Here lieth the body of Lewis Galdy, Esq., who died on the 22nd of\nSeptember, 1737, aged 80. He was born at Montpellier, in France, which\nplace he left for his religion, and settled on this island, where, in\nthe great earthquake, 1672, he was swallowed up, and by the wonderful\nprovidence of God, by a second shock was thrown out into the sea,\nwhere he continued swimming until he was taken up by a boat, and thus\nmiraculously preserved. He afterwards lived in great reputation, and\ndied universally lamented.\"\n\n\nCUSTOMS OF THE BORDER BETWEEN ENGLAND AND SCOTLAND.\n\nIn the courts held by the lords wardens of the Marches, a jury was\nestablished: the English lord chose six out of Scotland, and the Scotch\nsix out of England. The defendant, upon the trials, was acquitted\nupon his own oath; these oaths are singular: we transcribe them.--1.\nJUROR'S OATH. You shall clean no bills worthy to be fouled: you shall\nfoul no bills worthy to be cleaned; but shall do that which appeareth\nwith truth, for the maintenance of truth, and suppressing of attempts.\nSo help you God.--2. PLAINTIFF'S OATH. You shall leile (little) price\nmake, and truth say, what your goods were worth at the time of their\ntaking, to have been bought and sold in the market, taken all at\none time, and that you know no other recovery but this. So help you\nGod.--3. DEFENDANT'S OATH. You shall swear, by heaven above you, hell\nbeneath you, by your part in Paradise, by all that God made in six days\nand seven nights, and by God himself, you are whart and sackless, of\nart, part, way, witting, ridd, kenning, having, or reciting, of any of\nthe goods and chattles named in this bill. So help you God. These oaths\nand proceedings arose from the frequent incursions of both Scotch and\nEnglish, on both sides the wall, to where they had no right.\n\n\nTURKISH MODE OF REPARATION.\n\nOn April 25th, 1769, at Constantinople, the Turks were removing the\nstandard of Mahomet, making a grand procession through the city; all\nChristians, upon this occasion, were forbid to appear in the streets or\nat their windows. But the wife and daughter of the Imperial minister,\nbeing excited by curiosity, placed themselves at a secret window to\nobserve the procession; which was no sooner discovered by the Turks,\nthan they attacked the ambassador's house, and endeavoured to force an\nentrance. But the servants of the minister opposing them, well-armed,\na dreadful fray ensued, in which no less than one hundred persons lost\ntheir lives, and the ambassador's lady was very severely treated.\nSome of the rioters dragged her down into the court-yard, and made\npreparations to strangle her; when a party of Janissaries, who were\ndespatched to her assistance by an aga in the neighbourhood, happily\ncame and preserved her. Upon complaint being made of this outrage, by\nher husband, to the grand vizier, that minister expressed great sorrow\nfor the insult that had been offered, and assured him he should have\nall the reparation it was possible to procure. A few hours after the\nvizier sent the Imperial minister a rich present of jewels for his\nlady, _and a bag, which was found to contain the heads of the three\nprincipal rioters_.\n\n\nHAIR TURNED GREY BY FRIGHT.\n\nThere is an interesting anecdote of a boy, in one of the rudest parts\nof the County of Clare, in Ireland, who, in order to destroy some\neaglets, lodged in a hole one hundred feet from the summit of a rock,\nwhich rose four hundred feet perpendicular from the sea, caused himself\nto be suspended by a rope, with a scimitar in his hand for his defence,\nshould he meet with an attack from the old ones; which precaution was\nfound necessary; for no sooner had his companions lowered him to the\nnest, than one of the old eagles made at him with great fury, at which\nhe struck, but, unfortunately missing his aim, nearly cut through the\nrope that supported him. Describing his horrible situation to his\ncomrades, they cautiously and safely drew him up; when it was found\nthat his hair, which a quarter of an hour before was a dark auburn, was\nchanged to grey.\n\n\nMEMORABLE SNOW-STORM.\n\nThe following characteristic account is taken _literatim_ from the\nparish register of the village of Youlgrave in Derbyshire:--\"This year\n1614-5 Jan. 16 began the greatest snow which ever fell uppon the earth,\nwithin man's memorye. It cover'd the earth five quarters deep uppon the\nplayne. And for heapes or drifts of snow, they were very deep, so that\npassengers, both horse and foot, passed over yates hedges and walles.\nIt fell at ten severall tymes, and the last was the greatest, to the\ngreate admiration and fear of all the land, for it came from the foure\np{ts} of the world, so that all c'ntryes were full, yea, the south\np'te as well as these mountaynes. It continued by daily encreasing\nuntil the 12{th} day of March, (without the sight of any earth, eyther\nuppon hilles or valleys) uppon w{ch} daye, being the Lordes day, it\nbegan to decrease; and so by little and little consumed and wasted\naway, till the eight and twentyth day of May, for then all the heapes\nor drifts of snow were consumed, except one uppon Kinder-Scout, w{ch}\nlay till Witson week.\"\n\n\nROADS IN 1780.\n\nA squire from the neighbourhood of Glastonbury, journeying to Sarum in\nhis carriage, about 1780, took care that his footman was provided with\na good axe to lop off any branches of trees that might obstruct the\nprogress of the vehicle.\n\n\nWONDERFUL PEDESTRIAN FEAT.\n\nCaptain Cochrane, who set out from St. Petersburg in May, 1820, to walk\nthrough the interior of Russia to the east of Asia, with a view of\nascertaining the fact of a north-east cape, travelled at the rate of\n_forty-three miles a day for one hundred and twenty-three successive\ndays_. He afterwards walked upwards of four hundred miles without\nmeeting a human being. Wherever he went he seems to have accommodated\nhimself to the habits of the people, however rude and disgusting.\nWith the Kalmucks, he ate horse-flesh, elks, and wolves; and with\nthe Tchutski he found as little difficulty in pasturing upon bears,\nrein-deer, and _raw frozen fish_, the latter of which he considered a\ngreat delicacy.\n\n\nBOOK-SHAPED WATCH.\n\n[Illustration [++] Book-Shaped Watch.]\n\nThe unique curiosity, of which the annexed is an accurate\nrepresentation, was one of the choicest rarities of the Bernal\ncollection, and is, therefore, highly appropriate to our pages. It once\nbelonged to, and was made for, Bogislaus XIV., Duke of Pomerania, in\nthe time of Gustavus Adolphus. On the dial-side there is an engraved\ninscription of the Duke and his titles, with the date 1627, and the\nengraving of his armorial bearings; on the back of the case there\nare engraved two male portraits, buildings, &c.; the dial-plate is\nof silver, chased in relief; the insides are chased with birds and\nfoliage. This watch has apparently two separate movements, and a large\nbell; at the back, over the bell, the metal is ornamentally pierced\nin a circle, with a dragon and other devices, and the sides are\npierced and engraved in scrolls. It bears the maker's name, \"Dionistus\nHessichti.\"\n\n\nTHE RULING PASSION.\n\nMr. Henry Stribling, farmer, who died at Goodleigh, near Barnstaple,\nAugust 1st, 1800, in the eightieth year of his age, was one of the\ngreatest fox-hunters in Devonshire, and had collected such a number\nof foxes pads, all of which he had himself cut off when in at the\ndeath, that they entirely covered his stable door and door-posts. At\nhis own particular request, a pad was placed in each of his hands\nin his coffin, and he was attended to the grave by the huntsmen and\nwhippers-in of the packs with which he had hunted.\n\n\nEDICTS AGAINST FIDDLERS.\n\nAn idea may be formed of the strictness with which all popular\namusements were prohibited when the Puritans had the ascendancy, from\nthe fact that in 1656-7 Oliver Cromwell prohibited all persons called\nfiddlers or minstrels from playing, fiddling, or making music in any\ninn, alehouse, or tavern, &c. If they proffered themselves or offered\nto make music, they were to be adjudged to be rogues, vagabonds, and\nsturdy vagabonds, and were to be proceeded against as such.\n\n\nSCENE OF DESOLATION.\n\n[Illustration [++] Pass of Keim-an-eigh.]\n\nThe pass of Keim-an-eigh is one of the numerous wonders of nature.\nIt is situated on the road from Macroom to Bantry, in the county of\nCork, and winds through a deep and narrow rocky defile, about two\nEnglish miles in length. Its name means, in Irish, \"The Path of the\nDeer.\" Perhaps, in no part of the kingdom, is there to be found a\nplace so utterly desolate and gloomy. A mountain has been divided by\nsome convulsion of nature, and the narrow pass is overhung on either\nside, as seen in our engraving, by perpendicular cliffs clothed in wild\nivy and underwood, with, occasionally, a stunted yew-tree or arbutus\ngrowing among them. At every step advance seems impossible--some huge\nrock jutting out into the path, or sweeping round it, seeming to\nconduct only to some barrier still more insurmountable; while from\nall sides rush down the \"wild fountains,\" and forming for themselves\na rugged channel, make their way onward, the first tributary to the\ngentle and fruitful Lee. Nowhere has Nature assumed a more apalling\naspect, or manifested a more stern resolve to dwell in her own\nloneliness and grandeur, undisturbed by any living thing; for even the\nbirds seem to shun a solitude so awful, and the hum of bee or chirp of\ngrasshopper is never heard within its precincts.\n\n\nTHE FIRST ENGLISH NUN.\n\nFace, widow of Edwin, king of Northumberland, is said to have been the\nfirst English nun; and the first nunnery in England appears to have\nbeen at Barking, in Essex, which was founded by Erkenwald, Bishop of\nLondon, wherein he placed a number of Benedictine or black nuns. The\nmost rigid nuns are those of St. Clara, of the order of St. Francis,\nboth of which individuals were born and lived in the same town: the\nnuns are called poor Clares, and both they and the monks wear grey\nclothes. Abbesses had formerly seats in parliament. In one, held in\n694, says Spelman, they sat and deliberated, and several of them\nsubscribed the decrees made in it. They sat, says Ingulphus, in a\nparliament held in 855. In the reigns of Henry III. and Edward I. four\nof them were summoned to a national council, viz. those of Shaftesbury,\nBarking, Winchester, and Wilton.\n\n\nPRESENCE OF MIND--ESCAPE FROM A TIGER.\n\nIn 1812, a party of British naval and military officers were dining in\na jungle at some distance from Madras, when a ferocious tiger rushed in\namong them, seized a young midshipman, and flung him across his back.\nIn the first emotion of terror, the other officers had all snatched\nup their arms, and retired some paces from their assailant, who stood\nlashing his sides with his tail, as if doubtful whether he should seize\nmore prey, or retire with that which he had already secured. They knew\nthat it is usual with the tiger, before he seizes his prey, to deprive\nit of life, by a pat on the head, which generally breaks the skull;\nbut this is not his invariable practice. The little midshipman lay\nmotionless on the back of his enemy; but yet the officers, who were\nuncertain whether he had received the mortal pat or not, were afraid\nto fire, lest they should kill him together with the tiger. While in\nthis state of suspense, they perceived the hand of the youth gently\nmove over the side of the animal, and conceiving the motion to result\nfrom the convulsive throbs of death, they were about to fire, when, to\ntheir utter astonishment, the tiger dropped stone dead; and their young\nfriend sprung from the carcass, waving in triumph a bloody dirk drawn\nfrom the heart, for which he had been feeling with the utmost coolness\nand circumspection, when the motion of his hand had been taken for a\ndying spasm.\n\n\nCOST OF ARTICLES IN THE FOURTEENTH CENTURY.\n\nThe following article is taken from Martin's _History of Thetford_.\nIt is copied from an original record in that borough, when John le\nForester was mayor, in the tenth year of Edward the Third, A.D. 1336.\nIt is so far curious, as it exhibits an authentic account of the value\nof many articles at that time; being a bill, inserted in the town book,\nof the expenses attending the sending two light-horsemen from Thetford\nto the army, which was to march against the Scots that year.\n\n L s. d.\n To two men chosen to go into the army against Scotland 1 0 0\n\n For cloth, and to the tailor for making it into\n two _gowns_ 0 6 11\n\n For two pair of gloves, and a stick or staff 0 0 2\n\n For two horses 1 15 0-1\/2\n\n For shoeing these horses 0 0 4\n\n For two pair of boots for the light-horsemen 0 2 8\n\n Paid to a lad for going with the mayor to Lenn (Lynn),\n to take care of the horses (the distance between\n Thetford and Lynn is 53 miles) 0 0 3\n\n To a boy for a letter at Lenn (viz., carrying\n it thither) 0 0 3\n\n Expenses for the horses of two light-horsemen for four\n days before they departed 0 1 0\n\n\nLAW AND ORDER IN THE STREETS OF LONDON IN 1733.\n\nWhat an extraordinary state of things does the following extract from\nthe _Weekly Register_ of December 8th, 1733, disclose! The stages and\nhackney-coaches actually made open war upon private carriages. \"The\ndrivers,\" says the paragraph, \"are commissioned by their masters to\nannoy, sink, and destroy all the single and double horse-chaises they\ncan conveniently meet with, or overtake in their way, without regard\nto the lives or limbs of the persons who travel in them. What havoc\nthese industrious sons of blood and wounds have made within twenty\nmiles of London in the compass of a summer's season, is best known by\nthe articles of accidents in the newspapers: the miserable shrieks of\nwomen and children not being sufficient to deter the villains from\ndoing what they call their duty to their masters; for besides their\ndaily or weekly wages, they have an extraordinary stated allowance for\nevery chaise they can reverse, ditch, or bring by the road, as the term\nor phrase is.\" Verily, we who live in the present day have reason to\nrejoice that in _some_ things there is a decided improvement upon \"the\ngood old times.\"\n\n\nNEVER SLEEPING IN A BED.\n\nChristopher Pivett, of the city of York, died 1796, aged 93. He was\na carver and gilder by trade; but during the early part of his life\nserved in the army, and was in the retinue of the Duke of Cumberland,\nunder whose command he took part in the battle of Fontenoy, as he did\nat the battle of Dettingen under the Earl of Stair; he was likewise\nat the siege of Carlisle, and the great fight of Culloden. His house,\nafter he had settled at York, being accidentally burnt down, he formed\nthe singular resolution of never again sleeping in a bed, lest he\nshould be burned to death whilst asleep, or not have time sufficient,\nshould such a misfortune again befall him, to remove his property; and\nthis resolution he rigidly acted upon during the last forty years of\nhis life. His practice was to repose upon the floor, or on two chairs,\nor sitting in a chair, but always with his clothes on. During the\nwhole of this period he lived entirely alone, cooked his own victuals,\nand seldom admitted any one into his habitation: nor would he ever\ndisclose to any the place of his birth, or to whom he was related. He\nhad many singularities, but possessed, politically as well as socially,\na laudable spirit of independence, which he boldly manifested on\nseveral trying occasions. Among other uncommon articles which composed\nthe furniture of his dwelling, was a human skull, which he left strict\ninjunctions should be interred with him.\n\n\nAMULET BROTCHE.\n\nThe subjoined engraving represents an ancient Gaelic Brotche, which was\nmade in the reign of Queen Elizabeth, and once belonged to a Highland\nChief, Maclean of Lochbuy in the Isle of Mull, being formed of silver\nfound on his estate. It is of circular form, scolloped, and surrounded\nby small upright obelisks, each set with a pearl at top; in the centre\nis a round crystalline ball, considered a magical gem; the top may be\ntaken off, showing a hollow, originally for reliques. On the reverse\nside of the brotche are engraved the names of the three kings of\nCologne, with the word _consummation_. It was probably a consecrated\nbrotche, and worn not only for the purpose of fastening the dress, but\nas an amulet.\n\n[Illustration [++] Gaelic Brotche.]\n\n\nTHE GOLYNOS OAK.\n\nThis wonderful tree grew on the estate from which it takes its name,\nabout four miles from Newport, Monmouth. It was purchased by Thomas\nHarrison, Esq., in the year 1810, for 100 guineas, and was felled and\nconverted by him the same year. Five men were twenty days stripping\nand cutting it down; and a pair of sawyers were employed 138 days in\nits conversion. The expense of stripping, felling, and sawing was L82.\nThe trunk of the tree was 9-1\/2 feet in diameter, and no saw could\nbe found long enough to cut it down; two saws were therefore brazed\ntogether. The rings in its butt being reckoned, it was discovered that\nthis tree had been improving upwards of 400 years! and, as many of its\nlateral branches were dead, and some broken off, it is presumed it must\nhave stood a century after it had attained maturity. When standing\nit overspread 452 square yards of ground, and produced 2,426 feet of\ntimber. When all its parts were brought to market they produced nearly\nL600.\n\n[Illustration [++] Golynos Oak.]\n\n\nCARFAX CONDUIT.\n\n[Illustration [++] Carfax Conduit.]\n\nIn the grounds at Nuneham Courtenay, near Oxford, belonging to Mr.\nHarcourt, on one of the s that ascend directly from the river\nThames, stands the ancient and far-famed Carfax Conduit, which formerly\nstood as a kind of central point to the four principal streets of\nOxford. Certain alterations requiring its removal, it was, with the\nmost perfect propriety, presented to the Earl Harcourt.\n\nIt was built in 1610, by Otho Nicholson--a liberal and enterprising\ngentleman--in order to supply the city with pure water, brought from\na hill above North Hinksey; and although the conduit is removed, the\npipes still remain, and afford a partial supply that will be superseded\nby the new City Waterworks. It is a square, decorated in accordance\nwith the taste of the time--mermaids holding combs and mirrors, and\ndragons, antelopes, unicorns, being scattered about, while the Empress\nMaude is introduced riding an ox over a ford, in allusion to the\nname of the city. The letters O. N., the initials of the founder,\nare conspicuous; while above the centres of the four arches are the\ncardinal virtues--Justice, Temperance, Fortitude, and Prudence.\n\nCarfax is from a Bishop of that name, who presided over the diocese of\nTours in France, and died in the year 399. He was canonized, and is the\ntutelar saint of Carfax, or St. Martin's church, in the city of Oxford.\n\n\nDESTRUCTION OF LIBRARIES IN THE TIME OF HENRY VIII., AT THE DISSOLUTION\nOF THE MONASTERIES.\n\nIt is a circumstance well known, to every one at all conversant in\nEnglish history, that the suppression of the lesser monasteries by that\nrapacious monarch Henry the Eighth took place in 1536. Bishop Fisher,\nwhen the abolition was first proposed in the convocation, strenuously\nopposed it, and told his brethren that this was fairly shewing the\nking how he might come at the great monasteries. \"And so my lords,\"\ncontinued he, \"if you grant the king these smaller monasteries, you do\nbut make him a handle whereby he may cut down all the cedars within\nyour Lebanon.\" Fisher's fears were borne out by the subsequent act\nof Henry, who, after quelling a civil commotion occasioned by the\nsuppression of the lesser monasteries, immediately abolished the\nremainder, and in the whole suppressed six hundred and forty-five\nmonasteries, of which twenty-eight had abbots who enjoyed seats in\nParliament. Ninety colleges were demolished; two thousand three hundred\nand seventy-four charities and free chapels, and one hundred and ten\nhospitals. The havoc that was made among the libraries cannot be better\ndescribed than in the words of Bayle, Bishop of Ossory, in the preface\nto Leland's \"New Year's Gift to King Henry the Eighth.\"\n\n\"A greate nombre of them whyche purchased those superstychouse mansyons\n(monesteries) reserved of those librarye bookes, some to serve theyr\njokes, some to scoure thyr candlestyckes, and some to rubbe theyr\nbootes. Some they solde to the grossers and sope-sellers, and some\nthey sent over see to the book bynders, not in small nombre, but at\ntymes whole shyppes full to the wonderynge of foren nacyons: yea ye\nuniversytes of thys realme are not alle clere in this detestable fact,\nbut cursed is that bellye whych seketh to be fedde with suche ungodlye\ngaynes, and so depelye shameth hys natural conterye. I knowe a merchant\nmanne whyche shall at thys tyme be namelesse, that boughte ye contentes\nof two noble lybraryes for forty shyllinges pryce: a shame it is to be\nspoken: Thys stuffe hath he occupyed in the stede of grey paper by the\nspace of more than these ten yeares and yet he hath store ynoughe for\nas manye yeares to come. A prodygyouse example is thys to be abhorred\nof all men whych love thyr nacyon as they shoulde do. The monkes kept\nthem undre dust, ye ydle headed prestes regarded them not, theyr latter\nowners have most shamefully abused them, and ye covetouse merchantes\nhave solde them awaye into foren nacyons for moneye.\"\n\n\nCURIOUS MENTAL AFFECTION.\n\nSingular faculties have been developed during somnambulism in the\nmental condition. Thus a case is related of a woman in the Edinburgh\ninfirmary who, during her paroxysm, not only mimicked the manner\nof the attendant physicians, but repeated correctly some of their\nprescriptions in Latin.\n\nDr. Dyce, of Aberdeen, describes the case of a girl, in which this\naffection began with fits of somnolency, which came upon her suddenly\nduring the day, and from which she could at first be roused by shaking\nor by being taken into the open air. During these attacks she was in\nthe habit of talking of things that seemed to pass before her like\na dream, and was not at the time sensible of anything that was said\nto her. On one occasion she repeated the entire of the baptismal\nservice, and concluded with an extempore prayer. In her subsequent\nparoxysms she began to understand what was said to her, and to answer\nwith a considerable degree of consistency, though these replies were\nin a certain measure influenced by her hallucination. She also became\ncapable of following her usual employment during her paroxysm. At one\ntime she would lay out the table for breakfast, and repeatedly dress\nherself and the children, her eyes remaining shut the whole time. The\nremarkable circumstance was now discovered, that, during the paroxysm,\nshe had a distinct recollection of what had taken place in former\nattacks, though she had not the slightest recollection of it during the\nintervals. She was taken to church during the paroxysm, and attended\nthe service with apparent devotion, and at one time was so affected\nby the sermon that she actually shed tears; yet in the interval she\nhad no recollection whatever of the circumstance, but in the following\nparoxysm she gave a most distinct account of it, and actually repeated\nthe passage of the sermon that had so much affected her. This sort of\nsomnambulism, relating distinctly to two periods, has been called,\nperhaps erroneously, a _state of double consciousness_.\n\nThis girl described the paroxysm as coming on with a dimness of sight\nand a noise in the head. During the attack, her eyelids were generally\nhalf shut, and frequently resembled those of a person labouring under\namaurosis, the pupil dilated and insensible. Her looks were dull and\nvacant, and she often mistook the person who was speaking to her. The\nparoxysms usually lasted an hour, but she often could be roused from\nthem. She then yawned and stretched herself like a person awakening\nfrom sleep, and instantly recognised those about her. At one time, Dr.\nDyce affirms, she read distinctly a portion of a book presented to her,\nand she would frequently sing pieces of music more correctly and with\nbetter taste than when awake.\n\n\nDECORATIVE DRINKING VESSEL.\n\n[Illustration [++] German Decorative Vessel.]\n\nThe above represents a German decorative drinking vessel of the early\npart of the seventeenth century. It is a stork bearing in its beak an\ninfant; in accordance with the old German nursery tale that the king\nof the Storks is the bringer and protector of babies. It is of silver,\nchased all over; the eyes are formed of rubies; and one wing takes off\nthat liquid may be placed in the body, and imbibed through the neck,\nby a hole in the crown of the bird. It was probably a quaint fancy for\nsome German noble nursery.\n\n\nEXAMPLES OF ANCIENT VASES.\n\nThe Vases which are grouped in the annexed engraving are highly\ndeserving of a place in our collection of curiosities, inasmuch as they\nare truly unique and beautiful specimens of the degree of perfection\nto which the art of glass-making had been carried at the period when\nRome was mistress of the world. They all belong to that period, and in\nelegance of form and skill of workmanship they equal--we had almost\nsaid, surpass, the most artistic productions of the present day.\n\n[Illustration [++] Ancient Vases.]\n\nFigure 1 is that celebrated vase which for more than two centuries was\nthe principal ornament of the Barberini palace at Rome. It was thence\ngenerally known as the \"Barberini Vase;\" but having been purchased\nby Sir W. Hamilton, and then sold by him to the Duchess of Portland,\nit was at her death munificently presented by her son, the Duke of\nPortland, to the British Museum, where it has ever since remained as\none of its choicest gems, and is now known as the \"Portland Cinerary\nVase.\" It was found about the middle of the sixteenth century, enclosed\nin a marble sarcophagus, within a sepulchral chamber under the Monte\ndel Grane, two miles and a half from Rome, on the road to Frascati. The\ntomb is believed to have been that of the Emperor Alexander Severus,\nand his mother Mammaea. The vase is made of purple glass, ornamented\nwith white opaque figures in bas-relief. The execution of the design\nis most admirable. In the first place, the artist must have had the\naptitude to blow in purple glass a beautiful form of vase, with\nhandles attached: and, even thus far, this is considered in our day a\nmasterpiece of skill at our best glass-houses. Secondly, with the oxide\nof tin forming an opaque white glass, the artist managed to cover the\nwhole of the purple vase with this white opaque glass, to at least\nthe thickness of a quarter of an inch. The artist then, in the manner\nof cutting a cameo on the onyx stone, cut the opaque glass away,\nleaving the white figures and allegory embossed upon the purple. The\nfigures in relief are in two groups: in the former of these, a female\nis represented in a recumbent posture, with a cupid hovering above her\nhead, and a serpent in her lap; a young man on one side supporting her\nstretched out arm, and on the other a bearded personage of more mature\nage, attentively regarding her. The latter group, on the opposite side\nof the vase, consists of a female reclining on a pile of tablets, with\nher right hand placed on her head, and holding in her hand a lighted\ntorch with the flame downwards--a young man being seated on a pile on\none side of her, and a female, holding a rod or staff in the right\nhand, sitting on the other. The subject of the bas-relief has created\nmuch difference of opinion, but it is generally supposed to have\nreference to the birth of Severus. A few years ago this vase was broken\nby a madman, but it has since been repaired in a most artistic manner.\n\nFigure 2 is the \"Alexandrian Vase,\" of the Museo Borbonico, Naples.\n\nFigure 3 is the \"Pompeii Vase,\" also of the Museo Borbonico. It was\ndiscovered in a sepulchre of Pompeii in 1839, and is of the same\ncharacter in the colours and quality of the glass as the Portland Vase,\nbut of a more recent date. It is probably the production of Greek\nartists working in Rome.\n\nFigure 4 is the \"Aldjo Vase,\" which was found in 1833 at Pompeii, in\nthe house of the Fauna. The ground of the vase is of a deep sapphire\nblue, on which, in opaque white glass, the ornaments are cut. It was\nfound broken. Part is in the possession of Mr. Auldjo; the other in the\nBritish Museum. The shape of this vase is elegant, the handle and lip\nof exquisite form, and the taste and execution of the ornamental work\nin the purest style.\n\n\nMINUTENESS OF INSECT LIFE.\n\nAs the telescope enables the eye of man to penetrate into far-distant\nspace, and reveals to him myriads of suns and systems which otherwise\nwould have remained for ever hidden from his natural sight, so\nthe microscope opens up a world of life everywhere around us, but\naltogether unsuspected, astounding us as much by the inappreciable\nminuteness of its discoveries, as the former by the stupendous\nmagnitude and remoteness of the objects. If we go to any ditch or pool\nwhich the summer sun has covered with a mantle of stagnant greenness,\nand lift from it a minute drop of water, such as would adhere to\nthe head of a pin, we shall find it, under a high magnifying power,\nswarming with living beings, moving about with great rapidity, and\napproaching or avoiding each other with evident perception and will.\n\n\"Vain would it be,\" observes Professor Jones, \"to attempt by words\nto give anything like a definite notion of the minuteness of some\nof these multitudinous races. Let me ask the reader to divide an\ninch into 22,000 parts, and appreciate mentally the value of each\ndivision: having done so, and not till then, shall we have a standard\nsufficiently minute to enable us to measure the microscopic beings upon\nthe consideration of which we are now entering. Neither is it easy to\ngive the student of nature, who has not accurately investigated the\nsubject for himself, adequate conceptions relative to the numbers in\nwhich the _Infusoria_ sometimes crowd the waters they frequent; but\nlet him take his microscope, and the means of making a rough estimate,\nat least, are easily at his disposal. He will soon perceive that the\nanimalcule-inhabitants of a drop of putrid water, possessing, as many\nof them do, dimensions not larger than the 2,000th part of a line, swim\nso closely together, that the intervals separating them are not greater\nthan their own bodies. The matter, therefore, becomes a question for\narithmetic to solve, and we will pause to make the calculation.\n\n\"The _Monas termo_, for example--a creature that might be pardonably\nregarded as an embodiment of the mathematical point, almost literally\nwithout either length, or breadth, or thickness--has been calculated to\nmeasure about the 22,000th part of an inch in its transverse diameter;\nand in water taken from the surface of many putrid infusions, they are\ncrowded as closely as we have stated above. We may therefore safely\nsay, that, swimming at ordinary distances apart, 10,000 of them would\nbe contained in a linear space one inch in length, and consequently\na cubic inch of such water will thus contain more living and active\norganized beings than there are human inhabitants upon the whole\nsurface! However astounding such a fact may seem when first enunciated,\nnone is more easily demonstrated with the assistance of a good\nmicroscope.\"\n\nThe term _Infusoria_ has been by some naturalists applied to these\ndiminutive animals, because they are invariably found in the infusions\nof vegetable or animal substances. They can thus be obtained at all\ntimes, by simply steeping a little hay, or chaff, or leaves or stems of\nany plant, in a vessel of water, and placing the infusion in the sun\nfor a week or ten days.\n\n\nLEGENDS OF JUDAS ISCARIOT.\n\nIt was believed in Pier della Valle's time, that the descendants of\nJudas Iscariot still existed at Corfu, though the persons who suffered\nthis imputation stoutly denied the truth of the genealogy.\n\nWhen the ceremony of washing the feet is performed in the Greek Church\nat Smyrna, the bishop represents Christ, and the twelve apostles are\nacted by as many priests. He who personates Judas must be paid for it,\nand such is the feeling of the people, that whoever accepts this odious\npart, commonly retains the name of Judas for life (Hasselquist, p. 43).\n\nJudas serves in Brazil for a Guy Faux to be carried about by the\nboys, and made the subject of an auto-da-fe. The Spanish sailors hang\nhim at the yard arm. It is not long since a Spaniard lost his life\nat Portsmouth, during the performance of this ceremony, by jumping\noverboard after the figure.\n\nThe Armenians, who believe hell and limbo to be the same place, say\nthat Judas, after having betrayed our Lord, resolved to hang himself,\nbecause he knew Christ was to go to limbo, and deliver all the souls\nwhich he found there, and therefore he thought to get there in time.\nBut the Devil was cunninger than he, and knowing his intent, held him\nover limbo till the Lord had passed through, and then let him fall plum\ninto hell. (Thevenot.)\n\n\nQUEEN ELIZABETH'S SIDE-SADDLE.\n\nIn a retired part of the county of Essex, at a short distance from the\nroad, in a secluded and lovely spot, stands the picturesque residence\ncalled Horeham Hall. The mansion is in the parish of Thaxted, and is\nabout two miles south-west of the church. It was once in the possession\nof the important family of the De Wauton's; it afterwards belonged to\nSir John Cutts, and eventually it became the property of Sir W. Smijth,\nof Hill Hall, in whose family it has remained up to the present time.\n\nOf the learned Sir Thomas Smijth, the secretary to King Edward VI.\nand Queen Elizabeth, there is still preserved an ancient portrait on\npanel, which is let into a circle over the carved fire-place of one of\nthe parlours. It is remarkable as being one of the very few portraits\npainted by Titian.\n\n[Illustration [++] Queen Elizabeth's Side Saddle.]\n\nAnother interesting relic is represented in the annexed cut. It is\npreserved in the Great Hall, and is the side-saddle of Queen Elizabeth;\nthe pommel is of wrought metal, and has been gilt; the ornament upon\nit is in the then fashionable style of the Renaissance; the seat of\nvelvet is now in a very ruinous condition; but it is carefully kept\nbeneath a glass case, as a memento of the Queen's visits to this place.\nWhen princess, Elizabeth retired to Horeham as a place of refuge during\nthe reign of her sister Mary; the loveliness of the situation and its\ndistance from the metropolis rendered it a seclusion befitting the\nquietude of one anxious to remain unnoticed in troublous times. A room\non the first floor in the square tower is shown as that in which Queen\nElizabeth resided. She found the retirement of Horeham so agreeable,\nthat often after she had succeeded to the throne she took a pleasure in\nre-visiting the place.\n\n\nTHE WINFARTHING OAK, IN NORFOLK.\n\nA writer in the \"Gardener's Magazine\" gives the following account of\nthis remarkable tree:--\"Of its age I regret to be unable to give any\ncorrect data. It is said to have been called the 'Old Oak' at the\ntime of William the Conqueror, but upon what authority I could never\nlearn. Nevertheless, the thing is not impossible, if the speculations\nof certain writers on the age of trees be at all correct. Mr. South,\nin one of his letters to the Bath Society (vol. x.) calculates that an\noak tree forty-seven feet in circumference cannot be less than fifteen\nhundred years old; and Mr. Marsham calculated the Bentley Oak, from its\ngirting thirty-four feet, to be of the same age. Now, an inscription on\na brass plate affixed to the Winfarthing Oak gives us the following\nas its dimensions:--'This oak, in circumference, at the extremities of\nthe roots, is seventy feet; in the middle, forty feet, 1820.' Now, I\nsee no reason, if the size of the rind is to be any criterion of age,\nwhy the Winfarthing should not, at least, equal the Bentley oak; and\nif so, it would be upwards of seven hundred years old at the Conquest;\nan age which might very well justify its then title of the 'Old Oak.'\nIt is now a mere shell, a mighty ruin, bleached to a snowy white; but\nit is magnificent in its decay. The only mark of vitality it exhibits\nis on the south side, where a narrow strip of bark sends forth a few\nbranches, which even now occasionally produce acorns. It is said to\nbe very much altered of late; but I own I did not think so when I saw\nit about a month ago (May 1836); and my acquaintance with the veteran\nis of more than forty years' standing: an important portion of _my_\nlife, but a mere span of its own.\"\n\n\nCURIOUS PIECE OF ANCIENT ARMOUR.\n\n[Illustration [++] Bascinet.]\n\nThe above engraving represents a helmet, of the time of Richard II.,\nwhich was termed by ancient armourers a bascinet. This extremely rare\nspecimen was obtained from Her von Hulshoff, at his castle, near\nMunster, in Westphalia. The visor lifts upward on a hinge, and its\nposition may be further regulated by the screw which slips in the\ngroove above it. The row of holes on the lower edge of the bascinet was\nmade to secure the _camail_, or tippet of chain-mail which covered the\nneck of the wearer.\n\n\nEXTRAORDINARY ECHO.\n\nBeneath the suspension-bridge across the Menai Strait in Wales, close\nto one of the main piers, is a remarkably fine echo. The sound of a\nblow on the pier with a hammer, is returned in succession from each\nof the cross beams which support the roadway, and from the opposite\npier, at a distance of 576 feet; and in addition to this, the sound is\nmany times repeated between the water and the roadway. The effect is\na series of sounds, which may be thus described:--The first return is\nsharp and strong from the roadway overhead, the rattling which succeeds\ndies rapidly away; but the single repercussion from the opposite pier\nis very strong, and is succeeded by a faint palpitation, repeating the\nsound at the rate of twenty-eight times in five seconds, and which,\ntherefore, corresponds to a distance of 180 feet, or very nearly the\ndouble interval from the roadway to the water. Thus it appears, that\nin the repercussion between the water and the roadway, that from the\nlatter only affects the ear, the line drawn from the auditor to the\nwater being too oblique for the sound to diverge sufficiently in that\ndirection. Another peculiarity deserves especial notice,--viz., that\nthe echo from the opposite pier is best heard when the auditor stands\nprecisely opposite to the middle of the breadth of the pier, and\nstrikes just on that point. As it deviates to one or the other side,\nthe return is proportionably fainter, and is scarcely heard by him when\nhis station is a little beyond the extreme edge of the pier, though\nanother person stationed on the same side of the water, at an equal\ndistance from the central point, so as to have the pier between them,\nhears it well.\n\n\nJUGGLERS OF MODERN EGYPT.\n\nPerformers of sleight-of-hand tricks, who are called _hhowa'h_ (in the\nsingular, _hha'wee_) are numerous in Cairo. They generally perform in\npublic places, collecting a ring of spectators around them; from some\nof whom they receive small voluntary contributions during and after\ntheir performances. They are most frequently seen on the occasions of\npublic festivals; but often also at other times. By indecent jests and\nactions, they attract as much applause as they do by other means. The\nhha'wee performs a great variety of tricks, the most usual of which\nwe will here mention. He generally has two boys to assist him. From a\nlarge leather bag, he takes out four or five snakes, of a largish size.\nOne of these he places on the ground, and makes it erect its head and\npart of its body; another he puts round the head of one of the boys,\nlike a turban, and two more over the boy's neck. He takes these off,\nopens the boy's mouth, apparently passes the bolt of a kind of padlock\nthrough his cheek, and locks it. Then, in appearance, he forces an\niron spike into the boy's throat; the spike being really pushed up\ninto a wooden handle. He also performs another trick of the same kind\nas this. Placing the boy on the ground, he puts the edge of a knife\nupon his nose, and knocks the blade until half its width seems to have\nentered. The tricks which he performs alone are more amusing. He draws\na great quantity of various- silk from his mouth, and winds\nit on his arm; puts cotton in his mouth, and blows out fire; takes\nout of his mouth a great number of round pieces of tin, like dollars;\nand, in appearance, blows an earthen pipe-bowl from his nose. In most\nof his tricks he occasionally blows through a large shell (called the\nhha'wee's zoomma'rah), producing sounds like those of a horn. Most\nof his sleight-of-hand performances are nearly similar to those of\nexhibitors of the same class in our own and other countries. Taking a\nsilver finger-ring from one of the bystanders, he puts it in a little\nbox, blows his shell, and says, \"'Efree't change it!\" He then opens the\nbox, and shows, in it, a different ring: shuts the box again; opens\nit, and shows the first ring: shuts it a third time: opens it, and\nshows a melted lump of silver, which he declares to be the ring melted,\nand offers to the owner. The latter insists upon having his ring in\nits original state. The hha'wee then asks for five or ten fud'dahs to\nrecast it; and having obtained this, opens the box again (after having\nclosed it, and blown his shell), and takes out of it the perfect ring.\nHe next takes a larger covered box; puts one of his boy's skull-caps\nin it, blows his shell, opens the box, and out comes a rabbit: the cap\nseems to be gone. He puts the rabbit in again; covers the box; uncovers\nit, and out run two little chickens. These he puts in again, blows his\nshell, uncovers the box, and shows it full of fatee'rehs (or pancakes),\nand koona'feh (which resembles vermicelli): he tells his boys to eat\nits contents; but they refuse to do it without honey. He then takes a\nsmall jug, turns it upside-down, to show that it is empty; blows his\nshell, and hands round the jug full of honey. The boys, having eaten,\nask for water to wash their hands. The hha'wee takes the same jug, and\nhands it filled with water, in the same manner. He takes the box again,\nand asks for the cap; blows his shell, uncovers the box, and pours out\nfrom it, into the boy's lap (the lower part of his shirt held up), four\nor five small snakes. The boy, in apparent fright, throws them down,\nand demands the cap. The hha'wee puts the snakes back into the box;\nblows his shell, uncovers the box, and takes out the cap. Another of\nhis common tricks is to put a number of slips of white paper into a\ntinned copper vessel (the tisht of a seller of sherbet), and to take\nthem out dyed of various colours. He pours water into the same vessel;\nputs in a piece of linen; then gives to the spectators, to drink, the\ncontents of the vessel, changed to sherbet of sugar. Sometimes he\napparently cuts in two a muslin shawl, or burns it in the middle, and\nthen restores it whole. Often he strips himself of all his clothes,\nexcepting his drawers; tells two persons to bind him, hands and feet,\nand put him in a sack. This done, he asks for a piaster; and some one\ntells him that he shall have it if he will put out his hand and take\nit. He puts out his hand free; draws it back, and is then taken out of\nthe sack, bound as at first. He is put in again, and comes out unbound,\nhanding to the spectators a small tray, upon which are four or five\nlittle plates filled with various eatables; and, if the performance be\nat night, several small lighted candles placed round. The spectators\neat the food.\n\n\nORIGIN OF ATTAR OF ROSES.\n\n\"In the Histoire Generale de l'Empire du Mogol, (_T._ 1, _p_, 327,)\ncompiled by Catrou the Jesuit, from Manouchi's papers, this perfume is\nsaid to have been discovered by accident. Nur-Jahan, the favorite wife\nof the Mogul Jahan-Ghur, among her other luxuries, had a small canal of\nrose water. As she was a walking with the Mogul upon its banks, they\nperceived a thin film upon the water,--it was an essential oil made by\nthe heat of the sun. They were delighted with its exquisite odour, and\nmeans were immediately taken for preparing by art a substance like that\nwhich had been thus fortuitously produced.\"\n\n\nA MAGICIAN'S MIRROR AND BRACELET.\n\nA strange blending of pure science and gross superstition is remarkably\nillustrated in the history of the celebrated Dr. Dee. Born in London\nin 1527, John Dee raised himself at an early age to a great reputation\nfor his learning, in the mathematical sciences especially, in the most\ncelebrated universities in his own country and of the continent. He is\nsaid to have imbibed a taste for the occult sciences while a student at\nLouvain, but there was evidently in his temper much of an enthusiastic\nand visionary turn, which must have given him a taste for such\nmysterious pursuits, without the necessity of an external impulse. One\nof the oldest and most generally credited of magical operations, was\nthat of bringing spirits or visions into a glass or mirror, a practice\nwhich has continued to exist in the East even to the present day, and\nwhich prevailed to a very considerable extent in all parts of Western\nEurope during the sixteenth century. The process was not a direct one,\nfor the magician did not himself see the vision in the mirror, but\nhe had to depend upon an intermediate agent, a sort of familiar, who\nin England was known by the name of a _skyrer_, and whose business\nit was to look into the mirror and describe what he saw. Dr. Dee's\nprincipal skyrer was one Edward Kelly, and during his connexion with\nhim, Dee kept an exact diary of all his visions, a portion of which was\nprinted in a folio volume by Merio Casaubon in 1659. In this journal\nmore than one magical mirror is evidently mentioned, and that which we\nhere engrave is believed to have been of the number. It is now in the\ncollection of Lord Londesborough.\n\n[Illustration [++] Magician's Mirror.]\n\nIt is a polished oval slab of black stone, of what kind we have not\nbeen able to ascertain, but evidently of a description which was not\nthen common in Western Europe, and Dr. Dee, who died in 1608, may have\nconsidered it as extremely precious, and as only to be obtained by\nsome extraordinary means. It was one of the ornaments of the museum\nof Horace Walpole at Strawberry Hill; and Walpole has attached to it\na statement of its history in his own hand-writing, from which we\nlearn that it was \"long\" in the possession of the Mordaunts, earls\nof Peterborough, in whose catalogue it was described as \"the black\nstone into which Dr. Dee used to call his spirits.\" It passed from\nthat collection to Lady Elizabeth Germaine, from whom it went to\nJohn Campbell, Duke of Argyll, whose son, Lord Frederick Campbell,\npresented it to Horace Walpole. This interesting relic was bought at\nthe Strawberry Hill sale for the late Mr. Pigott; and at the more\nrecent sale of that gentleman's collection, it passed into the hands\nof Lord Londesborough. Its history and authenticity appear, therefore,\nto be very well made out. The family of the Mordaunts held a prominent\nplace in English history during the whole of the seventeenth century,\nand it is hardly probable that they would have received an object like\nthis without having good reason for believing that its history was\nauthentic. It is believed that Butler alluded to this identical stone\nin his well-known lines:--\n\n \"Kelly did all his feats upon\n The devil's looking-glass or stone,\n When, playing with him at bo-peep,\n He solv'd all problems ne'er so deep.\"\n _Hudibras._ Part II. Canto 3.\n\n[Illustration [++] Magician's Bracelet.]\n\nThe regular fitting out of the magician at this period was a\ncomplicated process. He required his implements of various kinds, and,\nin addition to these, various robes, made especially for the occasion,\nwith girdles and head-pieces, and magical rings and bracelets. A\nvery curious example of the last-mentioned article of the magician's\naccoutrements, is represented in the preceding cut, about one-third\nthe size of the original. It was purchased by Lord Londesborough in\n1851, and had formerly been in the possession of Charles Mainwaring,\nEsq., of Coleby, near Lincoln. It is of silver, the letters of the\ninscription round the bracelet being engraved and filled with niello.\nThis inscription may be distinctly read as follows:--\n\n + IONA + IHOAT + LONA + HELOI + YSSARAY + || +\n MEPHENOLPHETON + AGLA + ACHEDION + YANA +\n BACHIONODONAVALI M[*] ILIOR + || BACHIONODONAVALI M[**] ACH +\n\nSome explanation of this mysterious inscription might, no doubt, be\nobtained by a diligent comparison of some of the numerous works on\nmagic compiled in the age of Dr. Dee, and in the seventeenth century.\nThe bracelet has had four pendants on it, of which three still remain,\nwith the silver setting of the fourth. One of the pendants which remain\nis a brownish pebble, secured by three flat bands of silver; another is\nan oval cage of strong silver wire, containing a nut of some kind and\nsome other vegetable substance; the third has on one side a circular\nconvex pebble set in silver, and on the back three smaller pebbles.\n\n\nLUNAR INFLUENCE IN DEATH.\n\nMany modern physicians have stated the opinions of the ancients as\nregards lunar influence in diseases, but none have pushed their\ninquiries with such indefatigable zeal as the late Dr. Moseley; he\naffirms that almost all people in extreme age die at the new or at full\nmoon, and this he endeavours to prove by the following records:--\n\n Thomas Parr died at the age of 152, two days after the full moon.\n Henry Jenkins died at the age of 169, the day of the new moon.\n Elizabeth Steward, 124, the day of the new moon.\n William Leland, 140, the day after the new moon.\n John Effingham, 144, two days after full moon.\n Elizabeth Hilton, 121, two days after the full moon.\n John Constant, 113, two days after the new moon.\n\nThe doctor then proceeds to show, by the deaths of various illustrious\npersons, that a similar rule holds good with the generality of mankind:\n\n Chaucer, 25th October, 1400, the day of the first quarter.\n Copernicus, 24th May, 1543, day of the last quarter.\n Luther, 18th February, 1546, three days after the full.\n Henry VIII., 28th January, 1547, the day of the first quarter.\n Calvin, 27th May, 1564, two days after the full.\n Cornaro, 26th April, 1566, day of the first quarter.\n Queen Elizabeth, 24th March, 1603, day of the last quarter.\n Shakspeare, 23rd April, 1616, day after the full.\n Camden, 2nd November, 1623, day before the new moon.\n Bacon, 9th April, 1626, one day after last quarter.\n Vandyke, 9th April, 1641, two days after full moon.\n Cardinal Richelieu, 4th December, 1642, three days before full moon.\n Doctor Harvey, 30th June, 1657, a few hours before the new moon.\n Oliver Cromwell, 3rd September, 1658, two days after full moon.\n Milton, 15th November, 1674, two days before the new moon.\n Sydenham, 29th December, 1689, two days before the full moon.\n Locke, 28th November, 1704, two days before the full moon.\n Queen Anne, 1st August, 1714, two days after the full moon.\n Louis XIV., 1st September, 1715, a few hours before the full moon.\n Marlborough, 16th June, 1722, two days before the full moon.\n Newton, 20th March, 1726, two days before the new moon.\n George I., 11th June, 1727, three days after new moon.\n George II., 25th October, 1760, one day after full moon.\n Sterne, 13th September, 1768, two days after new moon.\n Whitfield, 18th September, 1770, a few hours before the new moon.\n Swedenburg, 19th March, 1772, the day of the full moon.\n Linnaeus, 10th January, 1778, two days before the full moon.\n The Earl of Chatham, 11th May, 1778, the day of the full moon.\n Rousseau, 2nd July, 1778, the day after the first quarter.\n Garrick, 20th January, 1779, three days after the new moon.\n Dr. Johnson, 14th December, 1784, two days after the new moon.\n Dr. Franklin, 17th April, 1790, three days after the new moon.\n Sir Joshua Reynolds, 23rd February, 1792, the day after the new moon.\n Lord Guildford, 5th August, 1722, three days after the full moon.\n Dr. Warren, 23rd June, 1797, a day before the new moon.\n Burke, 9th July, 1797, at the instant of the full moon.\n Macklin, 11th July, 1797, two days after full moon.\n Wilkes, 26th December, 1797, the day of the first quarter.\n Washington, 15th December, 1790, three days after full moon.\n Sir W. Hamilton, 6th April, 1803, a few hours before the full moon.\n\nThe doctor winds up this extract from the bills of mortality by the\nfollowing appropriate remark: \"Here we see the moon, as she shines on\nall alike, so she makes no distinction of persons in her influence:\n\n \"----aequo pulsat pede pauperum tabernas,\n Regumque turres.\"\n\n\nGLUTTONY OF THE MONKS.\n\nKing John, pointing to a fat deer said, \"See how plump he is, and yet\nhe has never heard mass!\" John might have alluded to the gluttony of\nthe monks, which was notorious in his days; for Giraldus Cambrensis\nsays, that from the monks of St. Swithin's, Winchester, Henry II.\nreceived a formal complaint against the abbot for depriving his priests\nof three out of thirteen dishes at every meal. The monks of Canterbury\nexceeded those of St. Swithin; they had seventeen dishes every day, and\neach of these cooked with spices and the most savoury and rich sauces.\n\n\nANCIENT BELL-SHRINE.\n\nThe annexed engraving represents one of the most valuable and curious\necclesiastical relics of the early Christian Period that has ever been\ndiscovered. It consists of a bronze bell-shrine and bell, found about\nthe year 1814, on the demolition of the ruined wall at Torrebhlaurn\nfarm, in the parish of Kilmichael-Glassrie, Argyleshire, and now one of\nthe most valued treasures in the Museum of the Scottish Antiquaries.\n\n[Illustration [++] Bronze Bell-Shrine and Bell.]\n\nThat it must have been deposited in the wall where it was found,\nfor the purpose of concealment at a period of danger and alarm,\nseems abundantly obvious; but of the occasion of this concealment no\ntradition has been preserved. Within the beautiful case is a rude\niron bell, so greatly corroded that its original form can only be\nimperfectly traced; yet this, and not the shrine, was obviously the\nchief object of veneration, and may, indeed, be assumed, with much\nprobability, to be some centuries older than the ornamental case in\nwhich it is preserved. Whether it shall be thought to have been an\nancient reliquary or a mass-bell, or whatever else may be conjectured\nof its nature and use, it may fairly be presumed to have remained\nin the neglected spot in which it was found since the subversion\nof the Roman Catholic worship in the sixteenth century, when the\nfavoured objects of external adoration and reverence, under the\nformer superstition, came to be regarded with impatient contempt and\nabhorrence.\n\nIt is deserving of attention that the figure of our crucified Saviour\nin invested with a regal crown, and not with a crown of thorns, as\nis usually the case. The brass chain or collar, of rude workmanship,\nabout three feet six inches long, now attached to the case, and the\nextremities of which are connected with a small cross of the same\nmetal, was discovered at the same time, not far from the case.\n\n\nEGYPTIAN GARDEN.\n\nThe diagram which accompanies this article is an Egyptian sketch of\nan Egyptian garden; and it is expressly curious, both as an example\nof the pictorial art of the period, and as giving us an idea of the\npleasure-gardens of Egypt in its most flourishing days.\n\n[Illustration [++] Egyptian Garden.]\n\nThe garden here represented stood beside a canal of the Nile, with\nan avenue of trees between it and the bank, on which side was the\nentrance. It was surrounded by an embattled wall, through which a noble\ngateway gave access to the garden. The central space was occupied by\nthe vineyard, surrounded by its own wall, in which the vines were\ntrained on trellises supported by slender pillars. At the further end\nof the vineyard was a building of three storeys, the windows from which\nopened over the luxurious foliage and purple clusters, regaling the\nsenses both of sight and smell. Four large tanks of water kept the\nvegetation well supplied with nutritive moisture; and, with the smooth\nand verdant turf which borders them, the water-fowl that sported over\nthe surface, and the lotus-flowers that sprang from their clear depths,\nadded a new beauty to the scene. Near the tanks stood summer-houses,\noverlooking beds of various flowers, and sheltered from the sun by\nsurrounding trees. Two enclosed spaces between the tanks, being filled\nwith trees, were probably devoted to some species of particular rarity,\nor remarkable for the excellence of their fruit. Rows of date trees and\nTheban palms, alternating with other trees, bordered the whole garden,\nand environed the vineyard wall.\n\nThe very numerous allusions to gardens in the Sacred Scriptures show\nthat the Hebrews inherited the same taste as the Egyptians. In these\nallusions we find the same characteristics that are so observable in\nthose depicted on the monuments; such as the absolute necessity of\nwater, the custom of having pools in them, the advantage of a situation\nby the side of a river, the practice of enclosing them from intrusion,\nand appropriation of enclosures to particular productions.\n\nWith the early Egyptians the love of flowers seems to have been almost\na passion; they appear to have been in constant request in offerings to\nthe gods, and as ornaments of the person, as decorations of furniture;\nas graceful additions to several entertainments, they occur at every\nturn. Flowers were painted on walls, furniture, dresses, chairs, boxes,\nboats, and, in short, on whatever was wished to be ornamental. Wreaths\nand chaplets were likewise in common use among the Egyptians, and\nartificial flowers were not uncommon.\n\n\nSTATE OF THE MIND DURING SLEEP.\n\nThe following is an instance of phantasms being produced by our\nassociations with bodily sensations, and tends to show how alive our\nfaculties continue during sleep to the highest impressions:--\n\nThe subject of this observation was an officer in the expedition to\nLouisburg in 1758, who had this peculiarity in so remarkable a degree,\nthat his companions in the transport were in the constant habit of\namusing themselves at his expense. They could produce in him any kind\nof dream by whispering in his ear, especially if this was done by a\nfriend with whose voice he had become familiar. One time they conducted\nhim through the whole progress of a trial, which ended in a duel; and\nwhen the parties were supposed to have met, a pistol was put into\nhis hand, which he fired, and was awakened by the report. On another\noccasion they found him asleep on the top of a locker in the cabin,\nwhen they made him believe he had fallen overboard, and exhorted him to\nsave himself by swimming. They then told him that a shark was pursuing\nhim, and entreated him to dive for his life. He instantly did so, and\nwith so much force as to throw himself from the locker upon the cabin\nfloor, by which he was much bruised, and awakened of course. After\nthe landing of the army at Louisburg, his friends found him one day\nasleep in his tent, and evidently annoyed by the cannonading. They\nthen made him believe that he was engaged, when he expressed great\nfear, and showed an evident disposition to run away. Against this they\nremonstrated, but at the same increased his fears by imitating the\ngroans of the wounded and the dying; and when he asked, as he often\ndid, who was hit, they named his particular friends. At last they\ntold him that the man next himself in his company had fallen, when he\ninstantly sprung from his bed, rushed out of the tent, and was only\nroused from his danger and his dream by falling over the tent-ropes. A\nremarkable thing in this case was, that after these experiments he had\nno distinct recollection of his dreams, but only a confused feeling of\noppression or fatigue, and used to tell his friends that he was sure\nthey had been playing some trick upon him. It has been observed that\nwe seldom feel courageous or daring in our dreams, and generally avoid\ndanger when menaced by a foe, or exposed to any probable peril.\n\n\nMUSIC OF THE SEA.\n\nThe mysterious music that is heard in the bay at West Pascagoula,\nis described by those who have listened to it as being singularly\nbeautiful. \"It has, for a long time,\" says Mrs. Child, an American\nauthoress, \"been one of the greatest wonders of the south-west.\nMultitudes have heard it, rising, as it were, from the water, like the\ndrone of a bagpipe, then floating away, away, away, in the distance,\nsoft, plaintive, and fairy-like, as if AEolian harps sounded with richer\nmelody through the liquid element; but none have been able to account\nfor the beautiful phenomenon. There are several legends touching these\nmysterious sounds; but in these days few things are allowed to remain\nmysterious.\" These strange sounds, which thus assume the beauty and\nthe harmony of regular music, are stated to proceed from the cat-fish.\nA correspondent of the _Baltimore Republican_ thus explains the\nphenomenon:--\"During several of my voyages on the Spanish main, in the\nneighbourhood of Paraguay and San Juan de Nicaragua, from the nature of\nthe coast, we were compelled to anchor at a considerable distance from\nthe shore; and every evening, from dark to late night, our ears were\ndelighted with AEolian music, that could be heard beneath the counter of\nour schooner. At first I thought it was the sea-breeze sweeping through\nthe strings of my violin (the bridge of which I had inadvertently left\nstanding); but after examination I found it was not so. I then placed\nmy ear on the rail of the vessel, when I was continually charmed with\nthe most heavenly strains that ever fell upon my ear. They did not\nsound as close to us, but were sweet, mellow, and aerial, like the soft\nbreathings of a thousand lutes, touched by fingers of the deep sea\nnymphs, at an immense distance. Although I have considerable \"music in\nmy soul,\" one night I became tired, and determined to fish. My luck,\nin half-an-hour, was astonishing. I had half filled my bucket with\nthe finest white cat-fish I ever saw; and it being late, and the cook\nasleep, and the moon shining, I filled my bucket with water, and took\nfish and all into my cabin for the night. I had not yet fallen asleep,\nwhen the same sweet notes fell upon my ear; and, getting up, what\nwas my surprise to find my cat-fish discoursing sweet sounds to the\nsides of my bucket! I examined them closely, and discovered that there\nwas attached to each lower lip an excrescence, divided by soft wiry\nfibres. By the pressure of the upper lip thereon, and by the exhalation\nand discharge of breath, a vibration was created, similar to that\nproduced by the breath on the tongue of the Jews' harp.\"\n\n\nTHE ROCK OF CASHEL.\n\nAny work which professed to be a record of what is rare and curious,\nwould surely be incomplete if it did not contain an account of the\ncelebrated Rock of Cashel; for the venerable buildings which crown\nits summit are, from their number, variety, preservation, and site,\ndecidedly the most interesting ruins in the Emerald Isle, and, to\nuse the words of Sir Walter Scott, \"such as Ireland may be proud\nof.\" Cashel, which is distant about one hundred miles from Dublin,\nappears to be a place of high antiquity, and was long the residence\nof the kings of Munster; but as its early history is involved in much\nobscurity, it is uncertain at what period it became a diocesan site. It\nis stated that previous to the year 1101 the buildings on the Rock were\noccupied as a royal residence, and that in that year the hitherto royal\nseat was dedicated solely to ecclesiastical uses.\n\n[Illustration [++] Rock of Cashel.]\n\nThe buildings consist of a round tower, Cormack's chapel, cathedral,\ncastle and monastery; the latter is a few yards detached, and the least\nremarkable of the number; all the former are closely connected. The\nRound Tower, the date and uses of which are in common with those of all\nother similar structures involved in much obscurity, raises its tall\nand yet scarce dilapidated head far above its younger and more decaying\ncompanions. It is fifty-six feet in circumference, and ninety feet\nin height. Cormack's Chapel, which, with the exception of the Round\nTower, is the most ancient structure of the group, was built by Cormack\nM'Carthy, king of Munster, in 1136. It is roofed with stone, and in its\ncapitals, arches, and other features and details, the Norman style is\ndistinctly marked. The numerous ornaments, grotesque heads, and other\ncurious sculptures, which adorn the arches, columns, and pilasters,\nare all in uniformity of style. The building altogether is a perfect\ngem, and the architectural antiquary and the artist will find in it\na most valuable addition to their studies. The cathedral is a noble\nremnant of what is usually termed the pointed Gothic, and contains many\ninteresting relics.\n\nThe rock, which is here presented as it appears from the plain below,\nhas the buildings we have just mentioned on its very summit; it rises\nabruptly from a widely extended fertile country, to a considerable\nheight above the town, and from many parts at a distance it forms a\nvery striking object. On the top of the rock, and around the ruins,\nan area of about three acres has been enclosed, which is open to the\npublic.\n\n\nINSTANCE OF INCREMATION.\n\nLast night (26th September, 1769), say the chronicles of the day, the\nwill of Mrs. Pratt, a widow lady, who lately died at her house in\nGeorge Street, Hanover Square, was punctually fulfilled, by the burning\nof her body to ashes in her grave, in the new burying-ground adjoining\nto Tyburn turnpike.\n\n\nTHE HAWTHORNDEN SWORD.\n\nThe great antiquity of the Scottish claymore is proved by its being\nfigured in the sculptures both of Iona and Oronsay, with considerable\nvariety of details. In some the blade is highly ornamented, and the\nhandle varies in form, but all present the same characteristic, having\nthe guards bent back towards the blade. A curious variety of this\npeculiar form is seen in a fine large two-handed sword preserved at\nHawthornden, the celebrated castle of the Drummonds, where the Scottish\npoet entertained Ben Johnson during his visit to Scotland in 1619. It\nis traditionally affirmed to have been the weapon of Robert Bruce,\nthough little importance can be attached to a reputation which it\nshares with one-half the large two-handed swords still preserved. Our\nengraving is a correct representation of it.\n\n[Illustration [++] Hawthornden Sword.]\n\nThe handle appears to be made from the tusk of the narwhal, and it\nhas four reverse guards, as shown in the cut. The object aimed at by\nthis form of guard, doubtless, was to prevent the antagonist's sword\nglancing off, and inflicting a wound ere he recovered his weapon, and,\nin the last example especially, it seems peculiarly well adapted for\nthe purpose.\n\n\nINSTINCT IN A CAT.\n\nThe following anecdote almost places the cat on a level with the\ndog:--\"A physician of Lyons was requested to inquire into a murder\nthat had been committed on a woman of that city. In consequence of\nthis request he went to the habitation of the deceased, where he\nfound her extended lifeless on the floor, weltering in her blood. A\nlarge white cat was mounted on the cornice of a cupboard, at the far\nend of the apartment, where he seemed to have taken refuge. He sat\nmotionless, with his eyes fixed on the corpse, and his attitude and\nlooks expressing horror and affright. The following morning he was\nfound in the same station and attitude, and when the room was filled\nwith officers of justice, neither the clattering of the soldiers' arms,\nnor the loud conversation of the company, could in the least degree\ndivert his attention. As soon, however, as the suspected persons were\nbrought in, his eyes glared with increased fury, his hair bristled, he\ndarted into the middle of the apartment, where he stopped for a moment\nto gaze at them, and then precipitately retreated under the bed. The\ncountenances of the assassins were disconcerted, and they were now, for\nthe first time, abandoned by their atrocious audacity.\"\n\n\nA TRANCE.\n\nMrs. Godfrey, sister to the Duke of Marlborough, had nearly been\nburied alive; the physicians all declaring that the breath of life was\nirrecoverably gone. Her husband, Colonel Godfrey, had, however, the\npleasure to see her revive, seven days after (that day week, and same\nhour), and what is more, she never knew till the day of her death the\nlength of her trance, or sleep.\n\n\nTHE NUMBER SEVEN.\n\nThe number is composed of the first two perfect numbers, equal and\nunequal, 3 and 4; for the number 2, consisting of repeated unity, which\nis no number, is not perfect, it comprehends the primary numerical\ntriangle or trine, and square or quartile conjunction, considered by\nthe favourers of planetary influence as of the most benign aspect.\nIn six days creation was completed, and the 7th was consecrated to\nrest. On the 7th day of the 7th month, a holy observance was ordained\nto the children of Israel, who feasted 7 days, and remained 7 days\nin tents; the 7th year was directed to be a Sabbath of rest for all\nthings; and at the end of 7 times 7 years commenced the grand jubilee.\nEvery 7th year the land lay fallow; every 7th year there was a general\nrelease from all debts, and all bondmen were set free. From this law\nmay have originated the custom of our binding young men to 7 years'\napprenticeship, and punishing incorrigible offenders by transportation\nfor 7, twice 7, and three times 7, years. Every 7 years the law was\nto be read to the people. Jacob served 7 years for the possession of\nRachael; and also other 7. Noah had 7 days' warning of the flood, and\nwas commanded to take the fowls of the air in by 7, and the clean\nbeasts by 7. The ark touched ground on the 7th month; and in 7 days the\ndove was sent out, and again in 7 days after. The 7 years of plenty,\nand 7 years of famine were foretold in Pharaoh's dream by 7 fat and 7\nlean beasts, and the 7 full and 7 blasted ears of corn. Nebuchadnezzar\nwas 7 years a beast; and the fiery furnace was 7 times hotter to\nreceive Shadrach, &c. A man defiled was, by the Mosaic law, unclean 7\ndays; the young of both animals was to remain with the dam 7 days, and\nat the end of the 7th was to be taken away. By the old law, man was\ncommanded to forgive his offending brother 7 times; but the meekness\nof the revealed law extended his humility to 70 times 7: if Cain shall\nbe avenged 7 times, truly Lamech 70 times 7. In the destruction of\nJericho, 7 priests bore 7 trumpets 7 days; on the 7th they surrounded\nthe wall 7 times; after the 7th, the walls fell. Balaam prepared 7\nyears for a sacrifice; and 7 of Saul's sons were hanged to stay a\nfamine. Laban pursued Jacob 7 days' journey. Job's friends sat 7 days\nand 7 nights, and offered 7 bullocks and 7 rams, as an atonement for\ntheir wickedness. In the 7th year of his reign, King Ahazuerus feasted\n7 days, and on the 7th deputed his 7 chamberlains to find a queen,\nwho was allowed 7 maidens to attend her. Miriam was cleansed of her\nleprosy by being shut up 7 days. Solomon was 7 years in building the\nTemple, at the dedication of which he feasted 7 days; in the Temple\nwere 7 lamps; 7 days were appointed for an atonement upon the altar,\nand the priest's son was ordained to wear his father's garments 7\ndays. The children of Israel eat unleavened bread 7 days. Abraham gave\n7 ewe-lambs to Abimelech, as a memorial for a well. Joseph mourned 7\ndays for Jacob. Naaman was cleansed of his leprosy by bathing 7 times\nin Jordan. The Rabbins say that God employed the power of this number\nto perfect the greatness Of Samuel, his name answering the value of\nthe letters in the Hebrew word, which signifies 7; whence Hannah his\nmother, in her thanksgiving, says, the barren hath brought forth 7. In\nScripture are enumerated 7 resurrections: the widow's son, by Elias;\nthe Shunamite's son, by Elisha; the soldier who touched the bones of\nthe prophet; the daughter of the ruler of the synagogue; the widow's\nson of Nain; Lazarus, and our Lord. The apostles chose 7 deacons.\nEnoch, who was translated, was the 7th from Adam; and Jesus Christ was\nthe 77th in a direct line. Our Lord spoke 7 times on the cross, on\nwhich he was 7 hours; he appeared 7 times; and after 7 times 7 days\nsent the Holy Ghost. In the Lord's prayer are 7 petitions, contained in\n7 times 7 words, omitting those of mere grammatical connexion; within\nthis number are concealed all the mysteries of apocalypse revealed\nto the 7 churches of Asia. There appeared seven golden candlesticks\nand 7 stars in the hand of him that was in the midst; 7 lambs before\nthe 7 spirits of God; the book with 7 seals; the lamb with 7 horns\nand 7 eyes; 7 angels with 7 trumpets; 7 kings; 7 thunders; 7,000 men\nslain. The dragon with 7 heads and 7 crowns; and the beast with 7\nheads; 7 angels bearing 7 plagues, and 7 vials of wrath. The vision of\nDaniel was of 70 weeks and the elders of Israel were 70. There were\nalso 7 heavens, 7 planets (query), 7 stars, 7 wise men, 7 champions\nof Christendom, 7 notes in music, 7 primary colours, 7 deadly sins,\nand 7 sacraments in the Catholic church. The 7th son was considered\nas endowed with pre-eminent wisdom; and the 7th son of a 7th son is\nstill thought to possess the power of healing diseases spontaneously.\nPerfection is likened to gold 7 times purified in the fire; and we\nyet say you frightened me out of my 7 senses. The opposite sides of a\ndice make 7, whence the players at hazard make 7 the main. Hippocrates\nsays the septenary number, by its occult virtues, tends to the\naccomplishment of all things, to be the dispense of life, and fountain\nof all its changes; and, like Shakespeare, he divided the life of man\ninto 7 ages; for as the moon changes her phases every seven days,\nthis number influences all sublunary beings. The teeth spring out on\nthe 7th month, and are shed and renewed in the 7th year, when infancy\nis changed into childhood; at twice 7 years puberty begins; at three\ntimes 7 the faculties are developed, and manhood commences, and we are\nbecome legally competent to all civil acts; at four times 7 man is in\nfull possession of all his strength; at five times 7 he is fit for the\nbusiness of the world; at six times 7 he becomes grave and wise, or\nnever: at 7 times 7 he is in his apogee, and from that time decays;\nat eight times 7 he is in his first climacterick; at nine times 7, or\n63, he is in his last or grand climacterick, or year of danger; and\nten times 7, or three score and ten, has, by the royal prophet, been\npronounced the natural period of human life.\n\n\nSUPERSTITIOUS LEGEND.\n\nWe are told that when St. Helena, of pious memory, had discovered the\ntrue Cross of Christ, she permitted various fragments to be taken from\nit, which were encased, some in gold, and some in gems, and conveyed to\nEurope, leaving the principal or main part of the wood in the charge\nof the Bishop of Jerusalem, who exhibited it annually at Easter, until\nChosroes, king of Persia, plundered Jerusalem in the reign of the\nemperor Phocas, and took away this holy relic.\n\nBefore this fatal event we are taught to believe, by Rigordus, an\nhistorian of the thirteenth century, that the mouths of Christians used\nto be supplied with 30, or in some instances, no doubt according to\ntheir faith, with 32 teeth; but that _after_ the Cross was stolen by\nthe infidels no mortal has ever been allowed more than 23!\n\n\nORAEFA MOUNTAIN IN ICELAND.\n\nThis mountain, which is the loftiest in Iceland, has been rendered\ncelebrated by an eruption which took place about a century ago. Nothing\ncan be more striking than the account given of this calamity by the\naged minister of the parish. He was in the midst of his service on\nthe Sabbath, when the agitation of the earth gave warning that some\nalarming event was to follow. Rushing from the church, he saw a peak of\nthe neighbouring mountain alternately heaved up and sinking; till at\nlast, the stone, of which this portion of the mountain was composed,\nran down in a melted state into the plain, like melted metal from a\ncrucible, filling it to such a height, that no more of the mountain,\nwhich formerly towered to such a height, remains, than about the size\nof a bird; volumes of water being in the meantime thrown forth in a\ndeluge from the crater, and sweeping away whatever they encountered in\ntheir course. The Oraefa then broke forth, hurling large masses of ice\nto a great distance; fire burst out in every direction from its side;\nthe sky was darkened by the smoke and ashes, so that the day could\nhardly be distinguished from the night. This scene of horror continued\nfor more than three days, during which the whole region was converted\ninto utter desolation.\n\n\nTHE SETON SWORD.\n\n[Illustration [++] Seton Sword.]\n\nThe two-handed sword, which was introduced later than the claymore,\nthough still so familiar to us, is perhaps the most interesting, in an\narchaeological point of view, of all the military relics pertaining\nto the Medieval Period. The huge, ponderous, and unwieldy weapon,\nseems the fittest emblem that could be devised, of the rude baron of\nthe thirteenth century, who lived by \"the good old rule\" of physical\nforce, and whose hardy virtues, not unsuited to an illiterate age--are\nstrangely mistaken for a chivalry such as later ages have not seen.\nCalmly reasoning from this characteristic heirloom, we detect in it the\nevidence of just such hardy, skilless, overbearing power, as history\ninforms us was the character of the medieval baron, before the rise of\nthe burgher class readjusted the social balance by the preponderance\nof rival interests. The weapon figured here is a remarkably fine and\nunusually large specimen of the old Scottish two-handed sword, now in\nthe possession of George Seton, Esq., representative of the Setons of\nCariston. It measures forty-nine inches in the blade, five feet nine\ninches in entire length, and weighs seven and a half pounds. But the\nchief interest of this old relic arises from the well-authenticated\nfamily traditions which associate it with the memory of its first\nknightly owner, Sir Christopher Seton of that Ilk, from whom some of\nthe oldest scions of the Scottish peerage have been proud to trace\ntheir descent. He was married to Christian, sister of King Robert\nthe Bruce, whom he bravely defended at the battle of Methven. He was\nshortly after taken prisoner by Edward I., and basely hanged as a\ntraitor.\n\n\nSTYLE OF LIVING IN THE SIXTEENTH CENTURY.\n\nThe most perfect notion of the living and domestic arrangements of\nthe old English nobility and gentry will be found in the entries of\nwhat were called the Household Books of the times. One of the most\ncelebrated of these records is the _Northumberland Household Book_,\nbeing the regulations of the establishment of the fifth earl of\nNorthumberland, at his castles of Wrenill and Lekinfield, in Yorkshire,\nbegun in 1512. No baron's family was on a nobler or more splendid\nfooting. It consisted of one hundred and sixty-six persons, masters\nand servants; fifty-seven strangers were reckoned upon every day; on\nthe whole two hundred and twenty-three. During winter they fed mostly\non salt meat and salt fish; and with that view there was a provision of\none hundred and sixty gallons of mustard per year; so that there cannot\nbe any thing more erroneous than the magnificent ideas formed of \"the\nroast beef of _Old_ England.\" On flesh days, (that is, when meat was\nnot forbidden by the Catholic religion), through the year, breakfast\nfor my lord and lady was a loaf of bread, two manchets, a quart of\nbeer, a quart of wine, half a chine of mutton, or a chine of beef\nboiled. On meagre days (or when meat was forbidden), a loaf of bread,\ntwo manchets, a quart of beer, a quart of wine, a dish of butter, a\npiece of salt fish, or a dish of buttered eggs. During Lent, a loaf\nof bread, two manchets, a quart of beer, a quart of wine, two pieces\nof salt fish, six baconed herrings, four white herrings, or a dish of\nsprats. There was as little variety in other meals, except on festival\ndays; and this way of living was, at the time, high luxury. There were\nbut two cooks to dress victuals for two hundred persons; and fowls,\npigeons, plovers, and partridges were prohibited as delicacies, except\nat my lord's table. The table-cloth was washed about once a month; no\nsheets were used; and only forty shillings were allowed for washing\nthroughout the year. The family rose at six in the morning, dined at\nten, and supped at four in the afternoon; and the castle gates were\nshut at nine. Mass was said in the chapel at six o'clock, that all the\nservants might rise early. The earl passed the year at three country\nseats, but he had furniture only for one: he carried every thing along\nwith him, beds, tables, chairs, kitchen utensils; and seventeen carts\nand one waggon conveyed the whole: one cart sufficed for all his\nkitchen utensils, cooks' beds, &c. There were in the establishment\neleven priests, besides seventeen persons, chanters, musicians, &c.,\nbelonging to the chapel. No mention is made of plate, but only of the\nhiring of pewter vessels. Wine was allowed in abundance for the lord's\ntable, but the beer for the hall was poor indeed, only a quarter of\nmalt being allowed for two hogsheads. The servants seem all to have\nbought their own clothes from their wages. Every thing in the household\nwas done by order, with the pomp of proclamation; and laughable as it\nmay now seem, an order was issued for the right making of mustard,\nbeginning \"It seemeth good to us and our council.\"\n\n\nANECDOTE OF A TERRIER.\n\nA terrier, known to Professor Owen, was taught to play at hide and seek\nwith his master, who summoned him, by saying \"Let us have a game;\" upon\nwhich the dog immediately hid his eyes between his paws, in the most\nhonourable manner, and when the gentleman had placed a sixpence, or a\npiece of cake in a most improbable place, he started up and invariably\nfound it. His powers were equalled by what was called a fox-terrier,\nnamed , who would hide his eyes, and suffer those at play with\nhim to conceal themselves before he looked up. If his play-fellow\nhid himself behind a window-curtain, would, for a certain time,\ncarefully pass that curtain, and look behind all the others, behind\ndoors, etc., and when he thought he had looked long enough, seize the\nconcealing curtain and drag it aside in triumph. The drollest thing,\nhowever, was to see him take his turn of hiding; he would get under a\nchair, and fancy that he was not seen; of course, those at play with\nhim pretended not to see him, and it was most amusing to witness his\nagitation as they passed. When he was ill he had been cured by some\nhomoeopathic globules, and ever after, if anything were the matter with\nhim, he would stand near the medicine box, and hold his mouth open.\n\n\nCUTTING A WIFE OFF WITH A SHILLING.\n\nIn the year 1772, died at Lambeth, J---- G----e, Esq. In his will was\nfound the following remarkable clause:--\"Whereas, it was my misfortune\nto be made very uneasy by Elizabeth G----, my wife, for many years,\nfrom our marriage, by her turbulent behaviour; for she was not content\nwith despising my admonitions, but she contrived every method to make\nme unhappy; she was so perverse in her nature, that she would not\nbe reclaimed, but seemed only to be born to be a plague to me; the\nstrength of Sampson, the knowledge of Homer, the prudence of Augustus,\nthe cunning of Pyrrhus, the patience of Job, the subtlety of Hannibal,\nand the watchfulness of Hermogenes, could not have been sufficient to\nsubdue her; for no skill or force in the world would make her good;\nand, as we have lived several years separate, and apart from each other\neight years, and she having perverted her son to leave and totally\nabandon me; therefore I give her one shilling only.\"\n\n\nWEALTH OF THE JEWS.\n\nAbout the year 1707, the Jews offered Lord Godolphin, Minister of Queen\nAnne, to pay L500,000, (and they would have made it a million,) if the\ngovernment would allow them to purchase the town of Brentford, with\nleave of settling there entirely, with full privileges of trade, &c.\nLord Godolphin did not comply with the request, and a curious reason\nis assigned by Dean Lockier, because it would provoke two of the most\npowerful bodies in the nation, the clergy and the merchants. The Jews\nhad better success with Oliver Cromwell: they offered him L60,000 to\nhave a synagogue in London. He took the money, and they had their\ntemple.\n\n\nGAMBLING EXTRAORDINARY.\n\nThe following instance of frantic or drunken gambling appeared in the\n_Times_ of April 17, 1812:--\n\n\"On Wednesday evening an extraordinary investigation took place at\nBow Street. Croker, the officer, was passing the Hampstead Road; he\nobserved at a short distance before him two men on a wall, and directly\nafter saw the tallest of them, a stout man about six feet high,\nhanging by his neck from a lamp-post, attached to the wall, being that\ninstant tied up and turned off by the short man. This unexpected and\nextraordinary sight astonished the officer; he made up to the spot with\nall speed, and just after he arrived there, the tall man who had been\nhanged, fell to the ground, the handkerchief with which he had been\nsuspended having given way. Croker produced his staff, said he was\nan officer, and demanded to know of the other man the cause of such\nconduct; in the mean time the man who had been hanged recovered, got\nup, and on Croker interfering, gave him a violent blow on the nose,\nwhich nearly knocked him backward. The short man was endeavouring to\nmake off; however, the officer procured assistance, and both were\nbrought to the office, when the account they gave was, that they worked\non canals. They had been together on Wednesday afternoon, tossed up\nfor money, and afterwards for their clothes, the tall man who was\nhanged won the other's jacket, trowsers and shoes; they then tossed\nup which should hang the other, and the short one won the toss. They\ngot upon the wall, the one to submit, and the other to hang him on\nthe lamp-iron. They both agreed in this statement. The tall one who\nhad been hanged said, if he won the toss, he would have hanged the\nother. He said, he then felt the effects on his neck at the time he\nwas hanging, and his eyes was so much swelled that he saw double. The\nmagistrates expressed their horror and disgust, and ordered the man who\nhad been hanged to find bail for the violent and unjustifiable assault\nupon the officer, and the short one for hanging the other. Not having\nbail, they were committed to Bridewell for trial.\"\n\n\nOLD BOOKS.\n\nThe Pentateuch and the history of Job are the most ancient books in\nthe world; and in profane literature the works of Homer and Hesiod.\nThe first book known to have been written in our own vernacular was\n\"The Confessions of Richard, Earl of Cambridge,\" _temp._ 1415; and the\nearliest English ballad is supposed to be the \"Cuckoo Song,\" which\ncommences in the following style:--\n\n \"Sumer is icumen in\n Lhude sing cuccu,\n Groweth sed, and bloweth med,\n And springth ye wede nu:\n Singe cuccu.\"\n\n\nFOSSIL REPTILE; THE PTERODACTYLUS.\n\nThe pterodactylus was a flying animal. It had the wings of a bat, and\nthe structure of a reptile; jaws with sharp teeth, and claws with long\nhooked nails. The power which it had of flying was not by means of its\nribs, nor by wings without fingers, as in birds, but by wings supported\nby one very elongated toe, the others being short and furnished with\nclaws. The remains of this animal were brought under examination by M.\nCollini, director of the Museum of the Elector Palatine at Manheim.\nThere was at first some discussion as to the actual character of the\nanimal. M. Blumenbach supposed it to be a bird, and M. de Soemmering\nclassed it among the bats. M. Cuvier, however, maintained that it was\na reptile, and showed that all its bones, from the teeth to the claws,\npossessed the characters which distinguish that class of animals. But\nstill it differed from all other reptiles in possessing the capability\nof flying. It is probable that it could at pleasure fold up its wings\nin the same manner as birds, and might suspend itself on branches\nof trees by its fore toes, though it possessed the power of sitting\nupright on its hind feet. This is the most anomalous of all the fossil\nreptiles.\n\n\nTIGER CAVE AT CUTTACK.\n\nThe geographical distribution of the rock-cut caves of the Buddhists in\nIndia is somewhat singular, more than nine-tenths of those now known\nbeing found within the limits of the Bengal Presidency. The remainder\nconsist of two groups, those of Behar and Cuttack, neither of which\nare important in extent, in Bengal; one only, that of Mahavellipore,\nin Madras; and two or three not very important groups which have been\ntraced in Afghanistan and the Punjaub.\n\nOne of the most remarkable of these caves is that at Cuttack, which is\ncalled the Tiger cave--being in fact a large mass of rock, carved into\na form intended to represent the head of that animal, whose extended\njaws form the verandah leading into a small apartment excavated in the\ninterior of the skull: our engraving is a correct representation of it.\n\n[Illustration [++] Tiger Cave at Cuttack.]\n\nGenerally speaking, these single cells have a porch of two pillars\nto protect the doorway, which leads into a small room, 10 or 12 ft.\nsquare, constituting the whole cave. Buildings on precisely the same\nplan are still very common in India, except that now, instead of being\nthe abode of a hermit, the cell is occupied by an image of some god or\nother, and is surmounted by a low dome, or pyramidal spire, converting\nit into a temple of some pretensions. The lower part, however, of these\nsmall temples is very similar to the rock-cut hermitages of which we\nare speaking.\n\n\nTHE JEWS IN ENGLAND.\n\nWilliam the Conqueror permitted great numbers of Jews to come over from\nRouen, and to settle in England in the last year of his reign. Their\nnumber soon increased, and they spread themselves throughout most of\nthe cities and capital towns in England where they built synagogues.\nThere were fifteen hundred at York about the year 1189. At Bury, in\nSuffolk, is a very complete remain of a Jewish synagogue of stone in\nthe Norman style, large and magnificent. Hence it was that many of the\nlearned English ecclesiastics of those times became acquainted with\ntheir books and their language. In the reign of William Rufus, the Jews\nwere remarkably numerous at Oxford, and had acquired considerable\nproperty; and some of their Rabbis were permitted to open a school\nin the university, where they instructed not only their own people,\nbut many Christian students in Hebrew literature, about the year\n1094. Within 200 years after their admission or establishment by the\nConqueror, they were banished from the kingdom. This circumstance was\nhighly favourable to the circulation of their learning in England. The\nsuddenness of their dismission obliged them for present subsistence,\nand other reasons, to sell their moveable goods of all kinds, among\nwhich were large quantities of all Rabbinical books. The monks\nin various parts availed themselves of the distribution of these\ntreasures. At Huntingdon and Stamford there was a prodigious sale of\ntheir effects, containing immense stores of Hebrew manuscripts, which\nwere immediately purchased by Gregory of Huntingdon, Prior of the abbey\nof Ramsey. Gregory speedily became an adept in the Hebrew, by means\nof these valuable acquisitions, which he bequeathed to his monastery\nabout the year 1250. Other members of the same convent, in consequence\nof these advantages, are said to have been equal proficients in the\nsame language, soon after the death of Prior Gregory, among whom were\nRobert Dodford, Librarian of Ramsey, and Laurence Holbech, who compiled\na Hebrew Lexicon. At Oxford a great number of their books fell into the\nhands of Roger Bacon, or were bought by his brethren the Franciscan\nfriars of that university.\n\n\nGAME PRESERVES AT CHANTILLY.\n\nThe establishment at Chantilly, which formerly belonged to the great\nfamily of Conde, included 21 miles of park, and 48 miles of forest. The\nhorses, when the family were at that place, were above 500. The dogs,\n60 to 80 couple: the servants, above 500. The stables the finest and\nbest in Europe. We shall now present to the sporting and un-sporting\nreader, for both will lift up their eyes, a list of game killed, year\nby year, through a series of thirty-two years--beginning with the year\n1748, ending with the year 1779:--\n\n _List of the Game._\n 54,878 24,029 37,209 19,932\n 37,160 27,013 42,902 27,164\n 58,712 26,405 31,620 30,429\n 39,892 33,055 25,994 30,859\n 32,470 50,812 18,479 25,813\n 39,893 40,234 18,550 50,666\n 32,470 26,267 26,371 13,304\n 16,186 25,953 19,774 17,566\n\nNow let us give (of birds and beasts) their bill of mortality; that\nis the numbers, in detail, of each specific description, registered\nas below, and detailed to have been killed at Chantilly, in the\nabove-mentioned series of years. Hares, 77,750; rabbits, 587,470;\npartridges, 117,574; red ditto, 12,426; pheasants, 86,193; quails,\n19,696; rattles (the male quail), 449; woodcocks, 2,164; snipes, 2,856;\nducks, 1,353; wood-piquers, 317; lapwings, 720; becfique (small birds\nlike our wheatear), 67; curlews, 32; oyes d'Egypte, 3; oyes sauvage,\n14; bustards, 2; larks, 106; tudells, 2; fox, 1; crapeaux, 8; thrushes,\n1,313; guynard, 4; stags, 1,712; hinds, 1,682; facons, 519; does,\n1,921; young does, 135; roebucks, 4,669; young ditto, 810; wild boars,\n1,942; marcassins (young boars), 818. A magnificent list of animal\nslaughter, carefully and systematically recorded as achievements.\n\n\nBRITISH PEARLS.\n\nThe river Conway, in North Wales, was of considerable importance,\neven before the Roman invasion, for the pearl mussel (the _Mya\nMargaritifera_ of Linnaeus) and Suetonius acknowledged that one of his\ninducements for undertaking the subjugation of Wales was the pearl\nfishery carried forward in that river. According to Pliny, the mussels,\ncalled by the natives _Kregindilin_, were sought for with avidity by\nthe Romans, and the pearls found within them were highly valued; in\nproof of which it is asserted that Julius Caesar dedicated a breastplate\nset with British pearls to Venus Genetrix, and placed it in her temple\nat Rome. A fine specimen from the Conway is said to have been presented\nto Catherine, consort of Charles II., by Sir Richard Wynne, of Gwydir;\nand it is further said that it has since contributed to adorn the regal\ncrown of England. Lady Newborough possessed a good collection of the\nConway pearls, which she purchased of those who were fortunate enough\nto find them, as there is no regular fishery at present. The late Sir\nRobert Vaughan had obtained a sufficient number to appear at Court with\na button and loop to his hat, formed of these beautiful productions,\nabout the year 1780.\n\n\nFUNERAL ORATION OF FRANCIS THE FIRST.\n\nPierre Duchatel, in a funeral oration on the death of Francis I.,\npublished 1547, took upon himself to affirm, that the soul of the king\nhad gone _direct to Paradise_. This passing over of purgatory gave\noffence to the doctors of the Sorbonne, who sent a deputation to warn\nhim of his error. The prelate being absent, one of his friends received\nthem, and, in reply, gaily said--\"Be not uneasy, gentlemen, every one\nknows that the late king, my master, never stopped long in any one\nplace, however agreeable. Supposing, then, that he went to purgatory,\nbe assured that his stay would be very short.\" This pleasantry disarmed\nthe severity of the doctors, and the affair went no farther.\n\n\nGRAVES OF THE STONE PERIOD.\n\nStone Chambers, which once formed places of interment, are frequently\ndiscovered within large barrows of earth raised by the hands of man.\nThey are to be referred to the period of the Danish Invasion, which\nis generally termed among antiquaries the \"Stone Period,\" because the\nuse of metals was then in a great measure unknown; and while a few are\nto be found in Great Britain, there are many more of them in Denmark.\nThese tombs, which are covered with earth, have most probably contained\nthe remains of the powerful and the rich. They are almost all provided\nwith long entrances, which lead from the exterior of the mound of\nearth to the east or south side of the chambers. The entrances, like\nthe chambers, are formed of large stones, smooth on the side which\nis turned inwards, on which very large roof-stones are placed. The\nchambers, and even the entrances, which are from sixteen to twenty\nfeet in length, are filled with trodden earth and pebbles, the object\nof which, doubtless, was to protect the repose of the dead in their\ngraves, and the contents which are found in them consist of unburnt\nhuman skeletons (which were occasionally placed on a pavement of flat\nor round stones), together with implements and weapons, and tools of\nflint or bone, ornaments, pieces of amber, and urns of clay. In some\ncases smaller chambers have been discovered, annexed to one side of the\npassage which leads to the larger chamber, and one of these smaller\nchambers we have engraved as a specimen of the sort of tombs we are now\ndescribing.\n\n[Illustration [++] Stone Chamber in a Barrow.]\n\nThe above sketch represents a chamber which was discovered in a barrow,\nsituated near Paradis, in the parish of the Vale, in the island of\nGuernsey. On digging into the mound, a large flat stone was soon\ndiscovered; this formed the top, or cap-stone, of the tomb, and on\nremoving it, the upper part of two human skulls were exposed to view.\nOne was facing the north, the other the south, but both disposed\nin a line from east to west. The chamber was filled up with earth\nmixed with limpet-shells, and as it was gradually removed, while the\nexamination was proceeding downwards into the interior, the bones\nof the extremities became exposed to view, and were seen to greater\nadvantage. They were less decomposed than those of the upper part;\nand the teeth and jaws, which were well preserved, denoted that they\nwere the skeletons of adults, and not of old men. The reason why the\nskeletons were found in this extraordinary position it is impossible to\ndetermine. Probably the persons who were thus interred were prisoners,\nslaves, or other subordinates, who were slain--perhaps buried alive--on\noccasion of the funeral of some great or renowned personage, who was\nplaced in the larger chamber at the end of the passage; and this view\nof the case is considerably strengthened by the fact that the total\nabsence of arms, weapons, or vases, in the smaller chamber, denotes\nthat the quality of the persons within it was of less dignity or\nestimation.\n\n\nWAR CHARIOT OF ANCIENT EGYPT.\n\nThis chariot, which is mentioned in various parts of Scripture, and\nmore especially in the description of the pursuit of the Israelites\nby Pharaoh, and of his overthrow in the Red Sea, was a very light\nstructure, consisting of a wooden framework strengthened and adorned\nwith metal, and leather binding, answering to the descriptions which\nHomer has given of those engaged in the Trojan war.\n\n[Illustration [++] War Chariot.]\n\nThe sides were partly, and the back wholly open; and it was so low that\na man could easily step into it from behind; for there was no seat, the\nrider always standing in war or hunting, though when wearied he might\noccasionally sit on the sides, or squat, in eastern fashion, on his\nheels. The body of the car was not hung on the axle _in equilibrio_,\nbut considerably forward, so that the weight was thrown more upon the\nhorses. Its lightness, however, would prevent this from being very\nfatiguing to them, and this mode of placing it had the advantage of\nrendering the motion more easy to the driver. To contribute further\nto this end, the bottom or floor consisted of a network of interlaced\nthongs, the elasticity of which in some measure answered the purpose of\nmodern springs.\n\nThe Egyptian chariots were invariably drawn by two horses abreast,\nwhich were richly caparisoned; it is, perhaps, to the extreme elegance\nand magnificence of their trappings, no less than to their own beauty,\nthat allusion is made in the Song of Songs (1-9), where the royal\nbridegroom addresses his spouse thus: \"I have compared thee, O my love,\nto a company of horses in Pharaoh's chariots.\" The chariot of Egypt\nordinarily carried two persons, one of whom acted as the warrior,\nthe other as the charioteer. Occasionally we find three persons in\na chariot, as when two princes of the blood, each bearing the royal\nsceptre, or flabellum, accompanying the king in a state procession,\nrequiring a charioteer to manage the reins.\n\n\nPEACOCKS.\n\nIndia, says Mr. Pennant, gave us peacocks, and we are assured by Knox,\nin his \"History of Ceylon,\" that they are still found in the wild\nstate, in vast flocks, in that island and in Java. So beautiful a bird\ncould not be permitted to be a stranger in the more distant parts; for\nso early as the days of Solomon (1 Kings, x. 22) we find among the\narticles imported in his Tarshish navies, apes and peacocks. A monarch\nso conversant in all branches of natural history, would certainly not\nneglect furnishing his officers with instructions for collecting every\ncuriosity in the country to which they made voyages, which gave him\na knowledge that distinguished him from all the princes of his time.\nAElian relates that they were brought into Greece from some barbarous\ncountry, and that they were held in such high estimation that a male\nand female were valued at Athens at 1,000 _drachmae_, or L32 5s. 10d.\nTheir next step might be to Samos, where they were preserved about the\ntemple of Juno, being the birds sacred to that goddess; and Gellius, in\nhis \"_Noctes Atticae_\" commends the excellency of the Samian peacocks.\nIt is, therefore probable that they were brought there originally for\nthe purposes of superstition, and afterwards cultivated for the uses of\nluxury. We are also told, when Alexander was in India, he found vast\nnumbers of wild ones on the banks of the Hyarotis, and was so struck\nwith their beauty as to appoint a severe punishment on any person that\nkilled them.\n\nPeacocks' crests, in ancient times, were among the ornaments of the\nkings of England. Ernald de Aclent (Acland) paid a fine to King John\nin a hundred and forty palfries, with sackbuts, _lorains_, gilt spurs,\nand peacocks' crests, such as would be for his credit.--Some of our\nregiments of cavalry bear on their helmets, at present, the figure of a\npeacock.\n\n\nROMAN THEATRE AT ORANGE.\n\nOne of the most striking Roman provincial theatres is that of Orange,\nin the south of France. Perhaps it owes its existence, or at all events\nits splendour, to the substratum of Grecian colonists that preceded the\nRomans in that country. Its auditorium is 340 ft. in diameter, but much\nruined, in consequence of the princes of Orange having used this part\nas a bastion in some fortification they were constructing.\n\nThe stage is tolerably preserved. It shows well the increased\nextent and complication of arrangements required for the theatrical\nrepresentations of the age in which it was constructed, being a\nconsiderable advance towards the more modern idea of a play, as\ndistinguished from the stately semi-religious spectacle in which the\nGreeks delighted. The noblest part of the building is the great wall at\nthe back, an immense mass of masonry, 340 ft. in extent, and 116 ft. in\nheight, without a single opening above the basement, and no ornament\nexcept a range of blank arches, about midway between the basement and\nthe top, and a few projecting corbels to receive the footings of the\nmasts that supported the velarium. Nowhere does the architecture of\nthe Romans shine so much as when their gigantic buildings are left to\ntell their own tale by the imposing grandeur of their masses. Whenever\nornament is attempted, their bad taste comes out. The size of their\nedifices, and the solidity of their construction, were only surpassed\nby the Egyptians, and not always by them; and when, as here, their mass\nstands unadorned in all its native grandeur, criticism is disarmed,\nand the spectator stands awe-struck at its majesty, and turns away\nconvinced that truly \"there were giants in those days.\" This is not, it\nis true, the most intellectual way of obtaining architectural effect,\nbut it is the easiest and the most certain to secure the desired result.\n\n\nA PISCATORIAL DOG.\n\nMr. Jukes, in his \"Excursions in and about Newfoundland,\" speaks of a\ndog which appeared to be of the pure breed, and which he thought to\nbe more intelligent than the mixed race. This animal caught his own\nfish, for which purpose he sat on a projecting rock, beneath a fish\nstage, on which the fish were laid to dry, watching the water, the\ndepth being from six to eight feet, and the bottom quite white with\nfish-bones. On throwing a piece of cod-fish into the water, three or\nfour heavy, clumsy-looking fish, called in Newfoundland _sculpins_,\nwould swim to catch it. The instant one turned his broadside towards\nhim, he darted down, and seldom came up without the fish in his mouth.\nHe regularly carried them as he caught them to a place a few yards off,\nwhere he deposited them, sometimes making a pile of fifty or sixty in\nthe day. As he never attempted to eat them, he appeared to fish for his\namusement.\n\n\nPHENOMENA OF SOUND.\n\nIn the gardens of Les Rochas, once the well-known residence of Madame\nde Sevigne, is a remarkable echo, which illustrates finely the\nconducting and reverberating powers of a flat surface. The Chateau\ndes Rochas is situated not far from the interesting and ancient town\nof Vitre. A broad gravel walk on a dead flat conducts through the\ngarden to the house. In the centre of this, on a particular spot, the\nlistener is placed at the distance of about ten or twelve yards from\nanother person, who, similarly placed, addresses him in a low and, in\nthe common acceptation of the term, inaudible whisper, when, \"Lo! what\nmyriads rise!\" for immediately, from thousands and tens of thousands\nof invisible tongues, starting from the earth beneath, or as if every\npebble was gifted with powers of speech, the sentence is repeated with\na slight hissing sound, not unlike the whirling of small shot passing\nthrough the air. On removing from this spot, however trifling the\ndistance, the intensity of the repetition is sensibly diminished, and\nwithin a few feet ceases to be heard. Under the idea that the ground\nwas hollow beneath, the soil has been dug up to a considerable depth;\nbut without discovering any clue to the solution of the mystery.\n\n\nANTIQUE WATCH.\n\n[Illustration [++] Antique Silver Watch Shaped Like a Duck.]\n\nThe above engraving represents a fancy silver watch of the time of\nQueen Elizabeth. It is shaped like a duck; the feathers chased. The\nlower part opens, and the dial plate, which is also of silver, is\nencircled with a gilt ornamental design of floriated scrolls and\nangels' heads. The wheels work on small rubies. It has no maker's\nname. It is preserved in the original case of thin brass, covered with\nblack leather, and ornamented with silver studs, as represented in the\nwoodcut below. It forms one of the curiosities in the Museum of Lord\nLondesborough.\n\n[Illustration [++] Watch in Original Case.]\n\n\nHORSES FEEDING ONE ANOTHER.\n\nM. de Bossanelle, captain of cavalry in the regiment of Beauvilliers,\nrelates in his \"Military Observations,\" printed in Paris, 1760, \"That,\nin the year 1757, an old horse of his company, that was very fine and\nfull of mettle, had his teeth all on a sudden so worn down, that he\ncould not chew his hay and corn; and that he was fed for two months,\nand would still have been so had he been kept, by two horses on each\nside of him, that ate in the same manger. These two horses drew hay\nfrom the rack, which they chewed, and afterwards threw before the\nold horse; that they did the same with the oats, which they ground\nvery small, and also put before him. This (adds he) was observed and\nwitnessed by a whole company of cavalry, officers and men.\"\n\n\nCROSS OF MUIREDACH.\n\n[Illustration [++] Cross of Muiredach.]\n\nFrom the rude pillar-stone marked with the symbol of our\nfaith, enclosed within a circle, the emblem of Eternity, the\nfinely-proportioned and elaborately-sculptured crosses of a later\nperiod are derived. In the latter, the circle, instead of being simply\ncut on the face of the stone, is represented by a ring, binding, as it\nwere, the shaft, arms, and upper portion of the cross together. There\nare two beautiful specimens of this style of cross at Monasterboice,\nnear Drogheda, about thirty-five miles from Dublin. The smaller, more\nbeautiful, and more perfect of these we here engrave. The figures and\nornaments with which its various sides are enriched appear to have been\nexecuted with an unusual degree of artistic skill. It is now almost as\nperfect as it was when, nearly nine centuries ago, the artist, we may\nsuppose, pronounced his work finished, and chiefs and abbots, bards,\nshanachies, warriors, and ecclesiastics, and, perhaps, many a rival\nsculptor, crowded round this very spot full of wonder and admiration\nfor what they must have considered a truly glorious, and, perhaps,\nunequalled work. An inscription in Irish upon the lower part of the\nshaft, desires \"A prayer for Muiredach, by whom was made this cross,\"\nand there is reason for assigning it to an abbot of that name who died\nin the year 924. Its total height is exactly fifteen feet, and it is\nsix in breadth at the arms. The shaft, which at the base measures in\nbreadth two feet six inches, and in thickness one foot nine inches,\ndiminishes slightly in its ascent, and is divided upon its various\nsides by twisted bands into compartments, each of which contains either\nsculptured figures, or tracery of very intricate design, or animals,\nprobably symbolical.\n\n\nCHINESE THERAPEUTICS.\n\nIn the treatment of disease, the Chinese, so fond of classification,\ndivide the medicinal substances they employ into heating, cooling,\nrefreshing, and temperate: their _materia medica_ is contained in\nthe work called the _Pen-tsaocang-mou_ in fifty-two large volumes,\nwith an atlas of plates; most of our medicines are known to them and\nprescribed; the mineral waters, with which their country abounds,\nare also much resorted to; and their emperor, Kang-Hi, has given an\naccurate account of several thermal springs. Fire is a great agent, and\nthe _moxa_ recommended in almost every aliment, while acupuncture is in\ngeneral use both in China and Japan; bathing and _champooing_ are also\nfrequently recommended, and bloodletting is seldom resorted to.\n\nChina has also her animal magnetisers, practising the _Cong fou_, a\nmysterious manipulation taught by the bonzes, in which the adepts\nproduce violent convulsions.\n\nThe Chinese divide their prescriptions into seven categories:\n\n1. The great prescription.\n\n2. The little prescription.\n\n3. The slow prescription.\n\n4. The prompt prescription.\n\n5. The odd prescription.\n\n6. The even prescription.\n\n7. The double prescription.\n\nEach of these receipts being applied to particular cases, and the\ningredients that compose them being weighed with the most scrupulous\naccuracy.\n\nMedicine was taught in the imperial colleges of Pekin; but in every\ndistrict, a physician, who had studied six years, is appointed to\ninstruct the candidate for the profession, who was afterwards allowed\nto practise, without any further studies or examination; and it is\nsaid, that, in general, the physician only receives his fee when\nthe patient is cured. This assertion, however, is very doubtful, as\nthe country abounds in quacks, who, under such restrictions as to\nremuneration, would scarcely earn a livelihood. Another singular, but\neconomical practice prevails amongst them--a physician never pays\na second visit to a patient unless he is sent for. Whatever may be\nthe merits of Chinese practitioners both in medicine and surgery, or\ntheir mode of receiving remuneration, it appears that they are as much\nsubject to animadversion as in other countries:--A missionary having\nobserved to a Chinese, that their medical men had constantly recourse\nto fire in the shape of moxa, red-hot iron, and burning needles; he\nreplied, \"Alas! you Europeans are carved with steel, while we are\nmartyrized with hot iron; and I fear that in neither country will the\nfashion subside, since the operators do not feel the anguish they\ninflict, and are equally paid to torment us or to cure us!\"\n\n\nMARY QUEEN OF SCOTS TO SIR FRANCIS KNOLLYS, FROM BOLTON, SEPT. 1ST,\n1568: HER FIRST LETTER IN ENGLISH.\n\n(MS. Cotton. Calig. C. I. fol. 161 b. _Orig._)\n\nMester Knoleis, y heuv har (I have heard) sum neus from Scotland; y\nsend zou the double off them y vreit (wrote) to the quin (queen) my\ngud Sister, and pres (pray) zou to du the lyk, conforme to that y\nspak zesternicht vnto zou, and sut hesti ansur y refer all to zour\ndiscretion, and wil lipne beter in zour gud delin (dealing) for mi,\n(me) nor y kan persuad zou, nemli in this langasg (language) excus\nmy ivil vreitin (writing) for y neuver vsed it afor, and am hestit\n(hasted). Ze schal si my bel (bill) vhuilk (which) is opne, it is sed\nSeterday my unfrinds wil be vth (with) zou, y sey nething bot trests\nweil, and ze send oni to zour wiff ze mey asur schu (she) wald a bin\nweilcom to apur (poor) strenger hua (who) nocht bien (not being)\naquentet vth her, wil nocht bi ouuer bald (bold) to vreit bot for the\naquentans betuix ous (us: _i_. _e_. herself and Sir Francis Knolles). Y\nwil send zou letle tokne (token) to rember (remember) zou off the gud\nhop y heuu (have) in zou guef (gif--if) ze fend (find) a mit (meet)\nmesager y wald wish ze bestouded (bestowed) it reder (rather) apon her\nnon (than) ani vder; thus effter my commendations y prey God heuu zou\nin his kipin.\n\n \"Zour asured gud frind.\n\n \"MARIE R.\n\n \"Excus my ivel vreitin thes furst tym.\"\n\n\nPHILOSOPHY OF THE BRAMINS.\n\nThe order of creation, which is described in the Institutes of Menu\n(c. 1, pp. 75-8), is remarkable. \"First emerges the subtle ether, to\nwhich philosophers ascribe the quality of conveying sound: from ether,\neffecting a transmutation in form, springs the pure and potent air,\na vehicle of all scents; and air is held endued with the quality of\ntouch: then from air, operating a change, rises light, or fire, making\nobjects visible, dispelling gloom, spreading bright rays; and it is\ndeclared to have the quality of figure: but from light, a change being\neffected, comes water, with the quality of taste: and from water is\ndeposited earth, with the quality of smell; such were they created in\nthe beginning.\" This passage bears at least as strong a resemblance to\nthe chemical philosophy of our days, as certain parts of the Hindoo\nfables bear to the mysteries of the Christian religion. But it is more\ndifficult to account for the philosophy, (if, indeed, it be any thing\nmore than mere theory,) than to explain how the distorted traces of\nChristianity found their way into the fables of Hindostan.\n\n\nFOREIGNERS IN LONDON IN 1567.\n\n\"We learn from the Bishop of London's certificate, that, in December,\n1567, there were then in London and its immediate vicinity, or places\nwhich are now included in the word 'London,' 3838 Dutchmen; 720\nFrenchmen; 137 Italians; 14 Venetians; 56 Spaniards; 25 Portuguese; 2\nGrecians; 2 Blackamores; 1 Dane; and but 58 Scots! making a total of\n4851 foreigners.\"\n\n\nCHANGES OF FORTUNE.\n\nIn 1454, Sir Stephen Forster was Lord Mayor of London. He had been long\nin prison and penury, on account of his inordinate profuseness. It\nchanced that a most fantastical widow, who knew not how to get rid of\nher immense wealth, saw him begging at the gate; she admired his fine\nperson, learnt his history, paid his debts, and married him; asking\nof him only this one favour, that he would lavish away her fortune as\nfast as he could. Forster, probably from perverseness, became a sober\nhusband and a prudent manager, and only expended large sums in adding\na chapel and other advantageous appendages to Ludgate, where he had\nsuffered so many hardships.\n\n\nROMAN VASES IN BLACK WARE.\n\nThe principal subjects represented on vases of ancient Roman pottery\nof black ware are hunting scenes--such as dogs chasing stags,\ndeer, hares,--also, dolphins, ivy wreaths, and engrailed lines;\nand engine-turned patterns. In a few instances men with spears are\nrepresented, but in a rude and debased style of art. The principal form\nis the cup of a jar shape, sometimes with deep oval flutings, as on one\nfound at Castor; but dishes, cups, plates, and mortars are not found in\nthis ware.\n\n[Illustration [++] Roman Vase in Black Ware.]\n\nSome of the vases of this ware have ornaments, and sometimes letters\npainted on them in white slip upon their black ground, as represented\nin our engraving. They are generally of a small size, and of the nature\nof bottles or cups, with inscriptions, such as AVE, hail! VIVAS, may\nyou live! IMPLE, fill; BIBE, drink; VINVM, wine; VIVA, life; VIVE BIBE\nMVLTIS; showing that they were used for purposes purely convivial. Such\nare the vases found at Etaples, near Boulogne, the ancient Gessoriacum,\nand at Mesnil.\n\nSome rarer and finer specimens from Bredene, in the department of Lis,\nhave a moulding round the foot. Great quantities are found in England,\nHolland, Belgium, and France. It is found on the right bank of the\nRhine. A variety of this ware has been lately found at a spot called\nCrockhill, in the New Forest, together with the kilns in which it was\nmade, and a heap of potter's sherds, or pieces spoilt in the baking.\nThe paste was made of the blue clay of the neighbourhood, covered with\nan alkaline glaze of a maroon colour, perhaps the result of imperfect\nbaking; for the pieces when submitted again to the action of the fire,\ndecrepitated and split. They were so much vitrified as to resemble\nmodern stone ware, yet as all of them have proofs of having been\nrejected by the potters, it is probable that this was not the proper\ncolour of the ware. Almost all were of the pinched up fluted shape, and\nhad no bas-reliefs, having been ornamented with patterns laid on in\nwhite colour. The kilns are supposed to be of the third century of our\nera, and the ware was in local use, for some of it was found at Bittern.\n\n\nFRENCH BIBLE.\n\nThere was a French Bible, printed at Paris in 1538, by Anthony\nBonnemere, wherein is related \"that the ashes of the golden calf which\nMoses caused to be burnt, and mixed with the water that was drank by\nthe Israelites, stuck to the beards of such as has had fallen down\nbefore it; by which they appeared with gilt beards, as a peculiar mark\nto distinguish those which had worshipped the calf.\" This idle story\nis actually interwoven with the 32nd chapter of Exodus. And Bonnemere\nsays, in his preface, this French Bible was printed in 1495, at the\nrequest of his most Christian Majesty Charles VIII.; and declares\nfurther that the French translator \"has added nothing but the genuine\ntruths, according to the express terms of the Latin Bible; nor omitted\nanything but what was improper to be translated!\" So that we are to\nlook upon this fiction of the gilded beards as matter of fact; and\nanother of the same stamp, inserted in the chapter above mentioned,\nviz., that, \"Upon Aaron's refusing to make gods for the Israelites,\nthey spat upon him with so much fury and violence that they quite\nsuffocated him.\"\n\n\nSARDONYX RING WITH CAMEO HEAD OF QUEEN ELIZABETH, IN THE POSSESSION OF\nREV. LORD THYNNE.\n\n[Illustration [++] Sardonyx Ring.]\n\nThis is said to be the identical ring given by Queen Elizabeth to\nEssex, and so fatally retained by Lady Nottingham. It has descended\nfrom Lady Frances Devereux, Essex's daughter, in unbroken succession\nfrom mother and daughter to the present possessor. The ring is gold,\nthe sides engraved, and the inside of blue enamel; the execution of the\nhead of Elizabeth is of a high order, and whether this be _the_ ring or\nnot, it is valuable as a work of art.\n\n\nCURIOUS WAGERS.\n\nThere have been travelling wagers, and none of the least singular of\nsuch was that of Mr. Whalley, an Irish gentleman (and who we believe\nedited Ben Johnson's works), who, for a very considerable wager (twenty\nthousand pounds, it was said,) set out on Monday the 22nd of September,\n1788, to walk to Constantinople and back again in one year. This wager,\nhowever whimsical, is not without a precedent. Some years ago a baronet\nof good fortune (Sir Henry Liddel) laid a considerable wager that he\nwould go to Lapland, bring home two females of that country, and two\nrein-deer, in a given time. He performed the journey, and effected\nhis purpose in every respect. The Lapland women lived with him about\na year, but desiring to go back to their own country, the baronet\nfurnished them with means and money.\n\n\nCONFECTIONERY ART IN 1660.\n\nThe following is extracted from a work on Cookery, by Robert May,\npublished in 1660. It is entitled the \"_Accomplisht Cook, &c., &c._\n\n\"Triumphs and Trophies in Cookery, to be used in Festival Times, as\nTwelfth Day, &c.:--Make the likeness of a ship in pasteboard with flags\nand streamers, the guns belonging to it of kickses, bind them about\nwith pack-thread and cover them with paste proportionable to the\nfashion of a cannon with carriages; lay them in places convenient, as\nyou see them in ships of war, with such holes and trains of powder that\nthey may all take fire. Place your ships firm in a great charger; then\nmake a salt round about it, and stick therein egg-shells full of sweet\nwater; you may by a great pin take out all the meat out of the egg by\nblowing, and then fill it with rose-water. Then in another charger have\nthe proportion of a stag made of coarse paste, with a broad arrow in\nthe side of him, and his body filled up with claret wine. In another\ncharger at the end of the stag have the proportion of a castle with\nbattlements, percullices, gates, and drawbridges, made of pasteboard,\nthe guns of kickses, and covered with coarse paste as the former; place\nit at a distance from the ship to fire at each other. The stag being\nplaced betwixt them, with egg-shells full of sweet water (as before)\nplaced in salt. At each side of the charger wherein is the stag, place\na pie made of coarse paste, in one of which let there be some live\nfrogs, in the other live birds; make these pies of coarse paste, filled\nwith bran, and yellowed over saffron, or yolks of eggs: gild them over\nin spots, as also the stag, the ship and castle; bake them, and place\nthem with gilt bay leaves on the turrets and tunnels of the castle and\npies; being baked make a hole in the bottom of your pies, take out the\nbran, put in your frogs and birds, and close up the holes with the\nsame coarse paste; then cut the lids neatly up to be taken off by the\ntunnels. Being all placed in order upon the table, before you fire the\ntrains of powder, order it so that some of the ladies may be persuaded\nto pluck the arrow out of the stag; then will the claret wine follow,\nas blood running out of a wound. This being done with admiration to\nthe beholders, after some short pause, fire the train of the castle,\nthat the pieces all of one side may go off; then fire the trains of\none side of the ship as in a battle; next turn the chargers, and by\ndegrees fire the trains of each other side, as before. This done, to\nsweeten the stink of the powder, the ladies take the egg-shells full of\nsweet waters, and throw them at each other, all dangers being seemed\nover, and by this time you may suppose they will desire to see what\nis in the pies; when lifting first the lid off one pie, out skip some\nfrogs, which makes the ladies to skip and shriek; next after the other\npie, whence comes out the birds; who by a natural instinct flying at\nthe light, will put out the candles; so that what with the flying\nbirds and skipping frogs, the one above, the other beneath, will cause\nmuch delight and pleasure to the whole company: at length the candles\nare lighted and a banquet brought in, the music sounds, and every one\nwith much delight and content rehearses their actions in the former\npassages. These were formerly the delights of the nobility, before good\nhouse-keeping had left England, and the sword really acted that which\nwas only counterfeited in such honest and laudable exercises as these.\"\n\n\nSUSPENDED ANIMATION.\n\nDavid Beck, the celebrated portrait painter, and pupil of Vandyke,\ntravelling through Germany, was suddenly taken ill, and to all\nappearance died, and was laid out as a corpse. His servants, sitting\nround the bed, grieved heartily for the loss of so good a master; and,\nas grief is thirsty, drank as heartily at the same time. One of them,\nbecoming more fuddled than the rest, then addressed his companions\nthus: \"Our master when alive was fond of his glass, let us now, out of\ngratitude, then give him one now he is dead.\" Assent was given, the\nhead of the dead painter was raised up, and some wine poured down or\nspilt about, the fragrance or spirit of which caused Beck to open his\neyes; upon which the servant, who, being drunk, half forgetting his\nmaster was dead, forced down the remainder of the glass. The painter\ngradually revived, and thus escaped a living interment.\n\n\nFUNERAL OF MARAT.\n\nThe funeral of Marat was celebrated at Paris, July 17th, 1793, with the\ngreatest pomp and solemnity. All the sections joined the procession.\nAn immense crowd of people attended it. Four women bore the bathing\nmachine in which Marat was standing when he was assassinated; his\nshirt, stained with blood, was carried by a fury, in the shape of a\nwoman, at the top of a pike. After this followed a wooden bedstead,\non which the corpse of Marat was carried by citizens. His head was\nuncovered, and the gash he had received could be easily distinguished.\nThe procession was paraded through several streets, and was saluted on\nits march by several discharges of artillery.\n\n\nEXECUTION OF ANNE BOLEYN.\n\nIn Houssaie's \"Memoirs,\" Vol. I. p. 435, a little circumstance is\nrecorded concerning the decapitation of the unfortunate Anne Boleyn,\nwhich illustrates an observation of Hume. Our historian notices that\nher executioner was a Frenchman of Calais, who was supposed to have\nuncommon skill; it is probable that the following incident might\nhave been preserved by tradition in France, from the account of the\nexecutioner himself. Anne Boleyn being on the scaffold, would not\nconsent to have her eyes covered with a bandage, saying that she had\nno fear of death. All that the divine who assisted at her execution\ncould obtain from her was, that she would shut her eyes. But as she\nwas opening them at every moment, the executioner could not bear their\ntender and mild glances. Fearful of missing his aim, he was obliged to\ninvent an expedient to behead the queen. He drew off his shoes, and\napproached her silently; while he was at her left hand, another person\nadvanced at her right, who made a great noise in walking, so that this\ncircumstance drawing the attention of Anne, she turned, her face from\nthe executioner, who was enabled by this artifice to strike the fatal\nblow without being disarmed by that pride of affecting resignation\nwhich shone in the eyes of the lovely Anne Boleyn.\n\n\nMEXICAN TENNIS.\n\nThe Mexicans had one singular law in their play with the ball. In the\nwalls of the court where they played certain stones, like mill-stones\nwere fixed, with a hole in the middle, just large enough to let the\nball pass through; and whoever drove it through, which required great\nskill, and was, of course, rarely effected, won the cloaks of the\nlookers-on. They, therefore, took to their heels to save their cloaks,\nand others pursued to catch them, which was a new source of amusement.\n\n\nCURIOUSLY-SHAPED VESSEL.\n\n[Illustration [++] Vessel in the Shape of a Lion.]\n\nThere is a singular class of Northern relics, of the Christian Period,\nof which analogous types have been found in Scotland, which well\ndeserve our attention. The relics of which we speak consist of a\ncurious variety of vessels, presumed to have been designed for holding\nliquors, but invariably made in the form of some animal or monstrous\nhybrid. The annexed figure represents one of these, in the collection\nof Charles Kirkpatrick Sharp, Esq., and found by him among a hoard of\nlong-forgotten family heirlooms, in a vault of his paternal mansion\nof Hoddam Castle, Dumfriesshire. Of its previous history nothing is\nknown. It is made of bronze. The principal figure is a lion, without a\ntail, measuring fourteen inches in length, and nearly fourteen inches\nin greatest height. On the back is perched a nondescript animal, half\ngreyhound, half fish, apparently intended for a handle to the whole,\nwhile from the breast projects a stag's head with large antlers. This\nhas a perforation in the back of the neck, as if for the insertion of\na stop-cock, and it appears probable was designed for running off the\nliquid contained within the singular vessel to which it is attached. A\nsmall square lid on the top of the lion's head, opening with a hinge,\nsupplies the requisite aperture for whatever liquor it was designed to\nhold. A similar relic, possessed by Sir John Maxwell, Bart., was dug up\na few years since on the Pollock estate; and another, in the collection\nof the late E. W. A. Drummond Hay, Esq., was also in the form of a lion.\n\n\nA SENSIBLE DOG.\n\nProfessor Owen was walking with a friend, the master of the dog,\nby the side of a river, near its mouth, on the coast of Cornwall,\nand picked up a small piece of seaweed. It was covered with minute\nanimals, and Mr. Owen observed to his companion, throwing the weed\ninto the water,--\"If this small piece afforded so many treasures, how\nmicroscopically rich the whole plant would be! I should much like to\nhave one!\" The gentleman walked on; but hearing a splashing in the\nwater, turned round and saw it violently agitated. \"It is Lion!\" both\nexclaimed. \"What can he be about? He was walking quietly enough by our\nside a minute ago.\" At one moment they saw his tail above the water,\nthen his head raised for a breath of air, then the surrounding element\nshook again, and at last he came ashore, panting from his exertions,\nand laid a whole plant of the identical weed at Mr. Owen's feet. After\nthis proof of intelligence, it will not be wondered at, that when Lion\nwas joyfully expecting to accompany his master and his guest on an\nexcursion, and was told to go and take care of and comfort Mrs. Owen,\nwho was ill, that he should immediately return to the drawing-room, and\nlay himself by her side, which he never left during the absence of his\nowner; his countenance alone betraying his disappointment, and that\nonly for a few minutes.\n\n\nTHE CROWN OF CHARLEMAGNE.\n\nAs the emblem of sovereignty which once adorned the brows of one of\nearth's mightiest men, and as a unique specimen of the state at which\nthe goldsmith's art had arrived as early as the ninth century, we here\npresent our readers with an engraving of the crown of Charlemagne.\n\n[Illustration [++] Crown of Charlemagne.]\n\nThis great man was the eldest son of Pepin the Short, and grandson of\nCharles Martel, and was born at the castle of Ingelheim, near Metz,\nin the year 742. His father dying in 768 he succeeded to the crown in\nconjunction with his brother Carloman, whose death in 771 left him sole\nmonarch of the Franks. By his alliances, negociations, and principally\nby his numerous and glorious wars, he so enlarged his dominions, that\nat length they extended from the Ebro to the mouth of the Elbe, from\nthe Atlantic to the mountains of Bohemia and the Saal, and from the\nBritish Channel to the Volturno. In the year 800 he was crowned at\nRome, as Emperor of the West, by Pope Leo III., and died of a pleurisy\nin 814, at Aix-la-Chapelle, in the cathedral of which city he was\nburied with extraordinary magnificence. Equally illustrious in the\ncabinet and in the field, a wise legislator, and a great warrior, the\npatron of men of letters, and the restorer of learning, Charlemagne\nhas united in his favour the suffrages of statesmen and soldiers, and\nof ecclesiastics, lawyers, and men of letters, who have all vied with\none another in bestowing the homage of their praise on the celebrated\nfounder of the Western Empire.\n\nThe crown of this illustrious man, of which our engraving is a correct\nrepresentation, is now preserved at Vienna in the Imperial Treasury.\nIt is composed of eight plates of gold, four large and four small,\nconnected by hinges. The large ones, studded with precious stones, form\nthe front, the back, and the intermediate points of the crown; the\nsmall ones, placed alternately with these, are ornamented with enamels\nrepresenting Solomon, David, King Hezekiah seated on his throne,\nand Christ seated between two flaming seraphim, such as the Greeks\nusually represent them. The costume of the figures resembles that of\nthe Emperors of the Lower Empire, and although the inscriptions which\naccompany the figures are in Latin, the whole bears the impress of\nGreek workmanship. The ground of the figures is formed by the metal\nitself, which has been hollowed out to receive the enamel; but all the\ndetails of the design are traced out with fine fillets of gold. The\nflesh-tints are in rose- enamel; the colours employed in the\ndraperies and accessories are deep and light blue, red, and white. The\ncrown has unquestionably been retouched at various periods, but yet\nthere is nothing to invalidate the tradition which assigns the more\nancient portions to the time of Charlemagne. The enamels must belong to\nthe same early period.\n\n\nSPENT BY THE CORPORATION OF COVENTRY AT THE ENTERTAINMENT OF KING JAMES\nII. IN HIS PROGRESS THROUGH COVENTRY, 1687.\n\n(Mr. Richard Haywood, Treasurer.)\n\n L s. d.\n Gave a gold cup 171 17 6\n Mr. Septimus Butt, mayor, for sweetmeats 27 17 0\n Meat 13 14 0\n Wine 21 12 6\n Homage fee 41 6 8\n King's cook 10 0 0\n City cook 9 8 6\n Steward Fielding, for making a speech to his Majesty 5 7 6\n For linen spoiled, borrowed of Mrs. Smith, Spon-street 2 12 6\n The aldermen that went to Worcester to invite him 3 18 9\n Several companies for waiting on the King 27 9 4\n Alderman Webster, for meat 3 6 0\n Alderman Bradney for corn 3 5 6\n His Majesty's clerk of the market 1 1 6\n The King's trumpeters 2 0 0\n Richard Howcott, for carrying the city streamer 0 7 0\n The city bailiff's bill for fish, fowl, and wine 88 18 2\n -------------\n L434 2 9\n\n\nTRAVELLING EXPENSES IN THE THIRTEENTH CENTURY.\n\nOf travelling expenses in the thirteenth century, a roll is in\nexistence, and is too interesting to be passed over. It contains\na steward's accompts of the daily expenses of a person of rank in\nthe reign of Edward I, on a journey from Oxford to Canterbury,\nand during his sojourn in London, about the year 1289; while the\nrecord throws much light upon the mode of our ancestors' living,\nat a period concerning which we have very few similar memorials.\nOne day's expenses are as follow: \"In bread, sixpence. Two gallons\nof wine, a gift of hospitality from the rector of Berton. Item in\nbread, sixpence. Two gallons of wine, a gift of hospitality from the\nrector of Mistern. Beer, sixpence. Herrings, threepence. Stockfish,\nfourpence. Porpoise and fish, fourpence. Perch and roach, seven-pence.\nLarge eels, seven-pence. Vegetables, threepence farthing. Figs and\nraisins, twopence. Fuel, five-pence. A bed for two nights, twopence.\nHay for seven horses, seven-pence. A bushel of oats, twenty-pence.\nApples, a halfpenny. Sum, six shillings and eight-pence halfpenny.\"\nThe most expensive day in the roll is on a Sunday, \"in expenses of\nmy lord at Westminster, when he held a breakfast there for knights,\nclerks, and squires. Bread, two shillings. Beer, twelve-pence. Wine,\nthree shillings and eight-pence. Half a salmon, for the standard,\nwith the chine, three shillings and eight-pence. A fresh conger eel,\nthree shillings. Three fat pikes, five fat eels, and twenty-seven fat\nroaches, twelve shillings and fourpence. Half a hundred lamprorns,\ntwelve-pence. Oysters, threepence. Vegetables, twopence. The hire of\na boy to prepare the breakfast, one penny. Fare to Westminster, one\npenny. A basket, one penny farthing. On the same day at the inn: bread,\nfive-pence farthing. Beer from the store. Two gallons of beer for the\nboys, twopence. Fish from the store. Candles, a halfpenny. Fuel, a\nhalfpenny. Hay bought, five-pence three farthings. Straw, sixpence.\nTwo bushels of oats, eight-pence. Two pair of shoes for my lord,\ntwelve-pence. Sum, thirty shillings and threepence farthing.\"\n\n\nDUNS IN THE MAHRATTA COUNTRY.\n\nThe Mahratta mode of recovering debts is curious. When the creditor\ncannot get his money, and begins to see the debt as rather desperate,\nhe sits _dhurna_ upon his debtor; that is, he squats down at the door\nof the tent, and becomes, in a certain mysterious degree, the master\nof it. No one goes in or out without his approbation. He neither eats\nhimself, nor suffers his debtor to eat; and this famishing contest is\ncarried on till the debt is paid, or till the creditor begins to _feel_\nthat want of food is a greater punishment than the want of money. This\ncurious mode of enforcing a demand is in universal practice among the\nMahrattas; Scindiah himself, the chieftain, not being exempt from it.\nThe man who sits the _dhurna_, goes to the house, or tent, of him\nwhom he wishes to bring to terms, and remains there till the affair\nis settled; during which time, the one under restraint is confined to\nhis apartment, and not suffered to communicate with any persons but\nthose whom the other may approve of. The laws by which the _dhurna_\nis regulated are as well defined and understood as those of any other\ncustom whatever. When it is meant to be very strict, the claimant\ncarries a number of his followers, who surround the tent, sometimes\neven the bed of his adversary, and deprive him altogether of food;\nin which case, however, etiquette prescribes the same abstinence to\nhimself: the strongest stomach, of course, carries the day. A custom of\nthis kind was once so prevalent in the province and city of Benares,\nthat Brahmins were _trained_ to remain a long time without food. They\nwere then sent to the door of some rich individual, where they made a\nvow to remain without eating, till they should obtain a certain sum of\nmoney. To preserve the life of a Brahmin is so absolutely a duty, that\nthe money was generally paid; but never till a good struggle had taken\nplace, to ascertain whether the man was staunch or not; for money is\nthe life and soul of all Hindoos.\n\n\nVAUXHALL.\n\n[Illustration [++] Vauxhall.]\n\nThe trees seen above the houses at the foot of the Surrey side of\nWestminster Bridge are those of Vauxhall Gardens, the site of which\nwill soon be covered with buildings. These grounds were once the glory\nof English pleasure-gardens, frequented by the highest in the land\nfrom the gay days of Charles II. to those of \"the Regency,\" and were\ncelebrated in musical history for talent of the highest kind here\nintroduced. In the old orchestra, whose towering summit may be seen\nfrom the Thames, the greatest musical celebrities have sung. Handel,\nDr. Arne, and Hook superintended its concerts; and Hogarth decorated\nits walls with paintings. It obtained its name from a very old mansion\nthat once stood near it. This old manor-house of Fawkes Hall, as it\nexisted in the reign of Charles I., is shown in our engraving; at that\ntime it was described as a \"fair dwelling-house, strongly built, of\nthree stories high, and a pier staircase breaking out from it nineteen\nfeet square.\" This staircase occupied one of the towers, in accordance\nwith the ancient plan, and the house was a curious specimen of the old\ntimber houses of the gentry in the sixteenth century.\n\nIt appears to have obtained its name from Foukes de Breut, who married\nthe heiress of the manor, the Countess of Albemarle, sister to Baldwin,\nArchbishop of Canterbury; and it was granted by the name of the manor\nof Foukeshall, by Edward III. to his favourite Hugh le Despenser. In\n1615 the records of the Duchy of Cornwall prove the premises known as\nVauxhall Gardens to have been the leasehold property of Jane Vaux,\nwidow of John Vaux, citizen and vintner of London, and a benefactor\nto the parish of Lambeth. It has always remained, with the manor of\nKennington, as the property of the crown, and belongs to the Prince of\nWales as part of his Duchy of Cornwall. Vauxhall Gardens closed for\never on July 25th, 1859, with an _al fresco fete_.\n\n\nEGYPTIAN TOILET BOXES.\n\n[Illustration [++] Egyptian Toilet Boxes.]\n\nThe ladies of ancient Egypt were very fond of having their apartments\nset off with a profusion of knick-knacks, and among other articles of\nthat sort, they usually had several different kinds of toilet-boxes\non their dressing-tables. The above engraving represents a group of\nthem. They have been found in considerable numbers among the ruins\nof the palaces, and they form interesting objects among the Egyptian\ncuriosities in many of our museums. They were made of wood, or of\nivory, often inlaid, and always elaborately carved. Sometimes they\npartook of the nature of spoons, the containing part being shallow,\nat the end of a long solid handle; the handle was carved into the\nmost fanciful forms--a grotesque human figure, a woman, a fox, or a\nfish--and the spoon part was generally covered with a lid, which turned\non a pivot. In one of those in the engraving, the spoon takes the\nform of a fish, the cover being carved to resemble its scales, while\nanother, also in the form of a fish, has two cavities, the one covered,\nthe other permanently open. Sometimes the body of a goose formed the\nbox, either trussed for the table, or in the posture of life, and other\nforms were devised from the fancy of the artist. Some of these shallow\nboxes are supposed to have been used for holding small quantities of\nointments and cosmetics upon the toilet-table.\n\n\nSPACIOUS KITCHEN.\n\nOne of the most spacious kitchens in England is that of Raby Castle,\nthe magnificent seat of the Duke of Cleveland. It is a square of thirty\nfeet, having three chimneys, one for the grate, a second for stoves,\nand the third, (now stopped up,) for the great cauldron. The roof\nis arched, with a small cupola in the centre: it has likewise five\nwindows, from each of which steps descend, but only in one instance to\nthe floor; and a gallery runs round the whole interior of the building.\nThe ancient oven is said to have allowed a tall person to stand upright\nin it, its diameter being fifteen feet. It has since been converted\ninto a wine cellar, the sides being divided into ten parts, and each\nholding a hogshead of wine in bottles. Vast as is this kitchen, it must\nhave been but suitable to the hospitality of former ages: for, in one\nof the apartments of Raby Castle, seven hundred knights are stated to\nhave been entertained at one time.\n\n\nTHE HAWTHORNDEN CAVES.\n\nIn almost every country on the earth there are natural or artificial\ncaves, which have supplied hiding-places, retreats for anchorites, and\neven permanent native dwellings. Such caves abound in Scotland, and\nespecially along the coast, but in general their interest arises rather\nfrom the associations of popular traditions, than from any intrinsic\npeculiarity of character pertaining to them. Few such retreats are more\nremarkable, either for constructive art, or historic associations,\nthan the well-known caves beneath the old tower of Hawthornden, near\nEdinburgh. They have been hewn, with great labour and ingenuity, in\nthe rocky cliff which overhangs the river Esk. No tradition preserves\nthe history or date of their execution, but concealment was evidently\nthe chief design of the excavators. The original entrance is most\ningeniously made in the shaft of a very deep draw-well, sunk in the\ncourt-yard of the castle, and from its manifest utility as the ordinary\nand indispensable appendage of the fortress, it most effectually\nconceals its adaptation as a means of ingress and communication with\nthe rock chambers beneath. These are of various forms and sizes, and\none in particular is pierced with a series of square recesses, somewhat\nresembling the columbaria of a Roman tomb, but assigned by popular\ntradition as the library of its later owner, Drummond, the Scottish\npoet. Whatever was the purpose for which these were thus laboriously\ncut, the example is not singular. A large cave in Roxburghshire, hewn\nout in the lofty cliff which overhangs the Teviot, has in its sides\nsimilar recesses, and from their supposed resemblance to the interior\nof a pigeon-house, the cavern has received the name of the _Doo-cave_.\nAuthentic notices of the Hawthornden caves occur so early as the reign\nof David II., when a daring band of Scottish adventurers made good\ntheir head-quarters there, while Edward held the newly-fortified castle\nof Edinburgh, and the whole surrounding district. In the glen of the\nlittle river Ale, which falls into the Teviot at Ancrum, extensive\ngroups of caves occur, all indicating, more or less, artificial\nadaptation as human dwellings; and in many other districts similar\nevidences may be seen of temporary or permanent habitation, at some\nremote period, in these rude recesses. Along the coast of Arran there\nare several caves of various dimensions, one of which, at Drumandruin,\nor Drumidoon, is noted in the older traditions of the island as the\nlodging of Fin M'Coul, the Fingal of Ossian, during his residence in\nArran. Though low in the roof, it is sufficiently capacious for a\nhundred men to sit or lie in it. In this, as in other examples, we find\nevidences of artificial operations, proving its connexion with races\nlong posterior to those with whose works we have chiefly to do in this\nsection of archaeological inquiry. In the further end a large detached\ncolumn of rock has a two-handed sword engraved on it, surmounted by\na deer, and on the southern side of the cave a lunar figure is cut,\nsimilar in character to those frequently found on the sculptured\npillars and crosses which abound in Scotland. It is now more frequently\nstyled the king's cave, and described as the retreat of Robert the\nBruce, while he lurked as a fugitive in the Western Isles; but, like\nmany other traditions of the Bruce, this seems to be of very recent\norigin. Other caves in the same island are also of large dimensions,\nand variously associated with popular traditions, as, indeed, is\ngenerally the case where subterranean retreats of any considerable\nextent occur. Some are the supposed dwellings of old mythic chiefs,\nwhose names still live in the traditional songs of the Gael. Others\nare the retreats which the primitive confessors of Scotland excavated\nor enlarged for their oratories or cells. Of the latter class are\nthe caves of St. Molio, on the little island of Lamlash, or the Holy\nIsle, on the east coast of Arran; of St. Columba and St. Cormac, on\nthe Argyleshire coast; of St. Ninian, in Wigtonshire; of St. Serf,\nat Dysart, on the Fifeshire coast; and the celebrated \"ocean cave of\nSt. Rule, in Saint Andrew's Bay.\" This last oratory consists of two\nchambers hewn out of the sandstone cliffs of that exposed coast. The\ninner apartment is a plain cell, entered from the supposed oratory of\nthe Greek saint. The latter is nearly circular, measuring about ten\nfeet in diameter, and has a stone altar hewn in the solid rock on its\neastern side.\n\n\nMONKISH PRAYERS.\n\nThe Monks used to pray heartily, or rather say their prayers no less\nthan seven times in the twenty-four hours. We will give their names:--\n\n 1st.--Nocturnal, at cock-crowing, or two o'clock in the morning.\n 2nd.--Matins, at six o'clock in the morning.\n 3rd.--Tierce, at nine o'clock in the morning.\n 4th.--Sext, at twelve o'clock at noon.\n 5th.--None, at three o'clock in the afternoon.\n 6th.--Vespers, at six o'clock in the afternoon.\n 7th.--Compline, soon after seven.\n\nQuarles has a neat epigram on the subject:--\n\n For all our prayers th' Almighty does regard\n The judgment of the _balance_, not the _yard_;\n He loves not words, but matter; 'tis his pleasure\n To buy his wares by _weight_, and not by _measure_.\n\n\nTHE TRAP-DOOR SPIDER.\n\n[Illustration: Trap-door Spider.]\n\nThere are few insects of such extraordinary habits as the Trap-door\nSpider, and the following account of it by Professor Jones is so\ninteresting, that we are glad to extract it from his excellent work on\ninsect Architecture:--\n\nIn the Ionian islands, and also in the West Indies [as well as\nin the south of France, and in Corsica], there are found certain\nspiders (_Cteniza_) commonly known as Trap-door Spiders, which make a\ncylindrical nest in the earth, and cover the entrance with a door of\ntheir own construction, framed of alternate layers of silk and earth,\nand fastened to the opening by a hinge of stout silk. These spiders\nalso line their nests throughout with numerous layers of silken web\nto the thickness of stout cartridge paper, and finish it with the\ngreatest care. This beautiful lining is yet further strengthened in\nparticular parts, where the nest is likely to be exposed to danger. But\nthe greatest amount of skill and care is bestowed upon the trap-door\nand its silken hinge. The door is about the eighth of an inch thick,\nrough on the outside, not much unlike an oyster-shell, which it also\nresembles in being thick and strong near the hinge, but thinner towards\nthe circumference. The breadth of this hinge is various, but sometimes\nit is very considerable, as shown in the figure accompanying. It also\npossesses great elastic force, so that, on being opened, it closes\nagain of itself. This is principally accomplished by a fold or doubling\nof the web, at each end of the hinge, which permits the door to be\nopened nearly to a right angle with the aperture, but no further,\nunless violence be used. The underside of the door is perfectly smooth\nand firm, being shaped so as to fit accurately, and yet to offer no\nresistance when pushed open by the insect.\n\n[Illustration: Section of Nest.]\n\n[Illustration: Nest of Trap-door Spider.]\n\n[Illustration: Trap-door Opening by a Lever.]\n\nAs might be expected, there are varieties in the shape and size of\nthese nests. Some specimens found in the island of Zante had the silken\nlayers of the lid extended into a sort of handle, or lever, just above\nthe hinge, on pressing which, in ever so slight a degree, the trap-door\nopened. From this it would appear, that the entrance to such a nest\ncould be effected as easily by the enemies of the spider as by the\nspider itself; this, however, is not the case; for repeated observation\nhas shown that the spider keeps guard at the entrance, and actually\nholds the door with her fore-feet and palpi, while the hind-feet are\nextended down the side of the nest, and the mandibles are thrust into\nthe opposite side near the door. By this means the insects gets such\npower as to resist with considerable force the opening of the door. If\nit be asked how this is known, we are able to refer to the experiments\nof careful observers, who extracted a number of nests from the ground,\nand opening them at the lower end, looked up, and saw the spider so\noccupied. A section view of the nest will show that the curved form of\nthe cover, and the shape of the side walls, must favour this method of\nkeeping the door shut. In some cases, small hollows were formed round\nthe interior edge of the lid, into which the spider thrust its feet\nwhen keeping guard. It is a curious fact, that when several of these\nspiders enclosed in their nests were kept as a matter of curiosity\nin a box of earth, and the doors frequently opened to examine their\nproceedings, one or two of them, as if wearied at these repeated\ninterruptions, effectually closed their doors by weaving a piece of\nsilken tapestry, which was spread over the interior of the opening, and\nrounded like the inside of a thimble. This was so strongly attached to\nthe door and to the side walls, that no opening could be made without\ndestroying the nest.\n\n\nPRICES OF GREEK VASES.\n\nIn the ancient times of Rome the vases of Greek pottery bore a high\nvalue, and sold for enormous sums to connoisseurs, which has also\nbeen the case in modern times. Cleopatra spent daily, on the fragrant\nor flowery ware of Rhossus, a Syrian town, six minae. Of the actual\nprices paid for painted vases, no positive mention occurs in classical\nauthorities, yet it is most probable that vases of the best class, the\nproducts of eminent painters, obtained considerable prices. Among the\nGreeks, works of merit were at all times handsomely remunerated, and it\nis probable that vases of excellence shared the general favour shown to\nthe fine arts. For works of inferior merit only small sums were paid,\nas will be seen by referring to the chapter on inscriptions, which\nwere incised on their feet, and which mentioned their contemporary\nvalue. In modern times little is known about the prices paid for these\nworks of art till quite a recent period, when their fragile remains\nhave realised considerable sums. In this country the collections of\nMr. Townley, Sir W. Hamilton, Lord Elgin, and Mr. Payne Knight, all\ncontained painted vases. A sum of L500 was paid in consideration of\nthe Athenian vases in Lord Elgin's collection, which is by no means\nlarge when the extraordinary nature of these vases is considered, as\nthey are the finest in the world of the old primitive vases of Athens.\nL8,400 were paid for the vases of the Hamilton collection, one of the\nmost remarkable of the time, and consisting of many beautiful specimens\nfrom southern Italy. The great discoveries of the Prince of Canino, in\n1827, and the subsequent sale of numerous vases, gave them, however,\na definite market value, to which the sale of the collection of Baron\nDurand, which consisted almost entirely of vases, affords some clue.\nHis collection sold in 1836 for 313,160 francs, or about L12,524. The\nmost valuable specimen in the collection was the vase representing\nthe death of Croesus, which was purchased for the Louvre at the price\nof 6,600 francs, or L264. The vase with the subject of Arcesilaus\nbrought 1,050 francs. Another magnificent vase, now in the Louvre,\nhaving the subject of the youthful Hercules strangling the serpents,\nwas only secured for France after reaching the price of 6,000 francs,\nor L240: another, with the subject of Hercules, Dejanira, and Hyllus,\nwas purchased for the sum of 3,550 francs, or L142. A _crater_, with\nthe subject of Acamas and Demophoon bringing back AEthra, was obtained\nby M. Magnoncourt for 4,250 francs, or L170. A Bacchic amphora, of the\nmaker Execias, of the archaic style, was bought by the British Museum\nfor 3,600 francs, or L142 in round numbers. Enough has, however, been\nsaid to show the high price attained by the most remarkable of these\nworks of art. The inferior vases of course realised much smaller sums,\nvarying from a few francs to a few pounds; but high prices continued\nto be obtained, and the sale by the Prince of Canino in 1837, of some\nof his finest vases, contributed to enrich the museums of Europe,\nalthough, as many of the vases were bought in, it does not afford a\ngood criterion as to price. An _oenochoe_, with Apollo and the Muses,\nand a _hydria_, with the same subject, were bought for 2,000 francs,\nor L80 each. A _cylix_, with a love scene, and another with Priam\nredeeming Hector's corpse, brought 6,600 francs, or L264. An amphora\nwith the subject of Dionysius, and a cup with that of Hercules, sold\nfor 8,000 francs, or L320 each. Another brought 7,000 francs, or L280.\nA vase with the subject of Theseus seizing Helen, another with the\narming of Paris, and a third with Peleus and Thetis, sold for 6,000\nfrancs, or L240. Nor can the value of the finest specimens of the art\nbe considered to have deteriorated since. The late Mr. Steuart was\noffered 7,500 francs for a large _crater_, found in southern Italy,\nornamented with the subject of Cadmus and the dragon; 3,000 francs, or\nL120, were paid by the British Museum for a fine _crater_ ornamented\nwith the exploits of Achilles: 2,500 francs, or L100, for an amphora of\nApulian style, with the subject of Pelops and OEnomaus at the altar of\nthe Olympian Zeus. For another vase, with the subject of Musaeus, 3,000\nfrancs, or L120 were paid, and 2,500 francs, or L100, for the Athenian\nprize vase, the celebrated Vas Burgonianum, exhumed by Mr. Burgon. At\nMr. Beckford's sale, the late Duke of Hamilton gave L200 for a small\nvase, with the subject of the Indian Bacchus.\n\nThe passion for possessing fine vases has outstripped these prices\nat Naples; 2,400 ducats, or L500, was given for the vase with gilded\nfigures discovered at Cumae. Still more incredible, half a century back,\n8,000 ducats, L1,500, was paid to Vivenzio for the vase in the Museo\nBorbonico representing the last night of Troy; 6,000 ducats, or L1,000,\nfor the one with a Dionysiac feast; and 4,000 ducats, or L800, for\nthe vase with the grand battle of the Amazons, published by Shultz.\nBut such sums will not be hereafter realised, not that taste is less,\nbut that fine vases are more common. No sepulchre has been spared when\ndetected, and no vase neglected when discovered; and vases have been\nexhumed with more activity than the most of precious relics.\n\n\nOLD WALKING STICKS.\n\nIt would seem that at the present time the fashion of carrying\nwalking-sticks has to a considerable extent \"gone out.\" So great\nis the bustle in our city thoroughfares, that the use of a staff,\nexcept by those who are lame, is seldom adopted by business people.\nProfessional men still affect the custom, however; and your City man,\nalthough he may repudiate the use of a walking-stick in town, straps\na good sapling to his portmanteau whenever he has a chance of getting\namongst the woods and green fields. About a century and a-half ago\neverybody carried a cane. Dr. Johnson, Oliver Goldsmith, and a host\nof others, considered a good stick as necessary as a coat; and a\ncollection of these staves would, if they could be had at the present\nday, be valuable, not only as relics, but also as an indication of the\ncharacters of the owners, perhaps.\n\nIn former times, a golden-mounted stick or staff was commonly used by\nboth the male and female heads of families. Queen Elizabeth carried one\nof these towards the end of her life. They were then more frequently\nused, however, as a sign of authority than for any other purpose.\n\nThe staff was a weapon long before flint-headed arrows and such-like\ninstruments were invented. Sheriffs, and others high in authority, have\nwands or staffs borne before them on important occasions; the bishops'\npastoral staff is as old as episcopal authority.\n\nIn former times the running footmen, who, in a body of half-a-dozen,\non each side of a carriage ran to alarm robbers and to assist the\nlumbering vehicle out of the ruts, were well armed with stout\nstaves. At the present time they are still carried by the Plush\nfamily, although the use of them is not so clear. In the royal state\nprocessions, the footmen with their staves walk as in former days, and\nwe should be sorry were these little bits of ceremony dispensed with,\ninasmuch as they bring to recollection a former condition of things,\nwhich makes us feel comfortable by comparison.\n\nThe monstrous sticks shown in the engraving are drawn from specimens\nwhich have been preserved by dealers in London, and put as a sort of\nsign at the doors of umbrella and walking-stick dealers. These were,\nhowever, a century ago, common enough, and might have been seen by the\nhundred together, borne by tall footmen behind ladies dressed in the\nold hooped dresses which we are trying now to imitate. At that time\nthere was also a taste for various kinds of monsters, in China, wood,\nand other materials. Monkeys and pug-dogs were made pets of, and the\nsticks of the footmen fashioned into such ugly forms as no modern bogey\never dreamed of.\n\nThese clubs, sticks, maces, or whatever they may be called, were about\nsix feet high, and were in parts painted and gilt. The centre one is\nan elm-sapling, and the natural bumps have been taken advantage of by\nthe artist to model a sort of Moorish head, with ornamental covering;\nlower down, the knobs are fashioned into terrible heads, in which are\nmounted glass eyes of various and impossible colours.\n\n[Illustration [++] Old Walking Sticks.]\n\nNo doubt before long these staffs, which might be necessary for\nthe protection of the ladies from the \"Mohawks\" of the time, will\nhave disappeared, and people will look with curiosity at Hogarth's\nrepresentation of them. Perhaps good specimens of such objects, which\nhave passed out of use, would be worthy of a place in our national\nmuseum. One of the old-fashioned tinder-boxes would be a curiosity\nthere now. Although but a few years have passed since the introduction\nof lucifer matches, it is no easy matter to get one of those\nold-fashioned machines.\n\n\nTHE SANCHI TOPE.\n\nUnder the name of topes are included the most important class of\nBuddhist architecture in India. They consist of detached pillars,\ntowers, and tumuli, all of a sacred or monumental character. The word\nis a corruption of the Sanscrit _sthupa_, meaning a mound, heap, or\ncairn.\n\nBy far the finest as well as the most perfect tope in India is that\nof Sanchi, the principal one of those opened near Bilsah, in Central\nIndia. It is uncertain whether it ever contained relics or not, as it\nhad been dug into in 1819 by Sir Herbert Maddock, since which time it\nhas remained a ruin, and may have been plundered by the natives. At any\nrate it must have been a spot of peculiar sanctity, judging both from\nits own magnificence, and from the number of subordinate topes grouped\naround it. In fact there are a greater number of these monuments on\nthis spot, within a space not exceeding 17 miles, than there are, so\nfar at least as we now know, in the whole of India from the Sutlej to\nCape Comorin.\n\n[Illustration [++] Sanchi Tope.]\n\nThe general appearance of the Sanchi Tope will be understood from the\nannexed view of it. The principal building consists of a dome somewhat\nless than a hemisphere, 106 feet in diameter, and 42 feet in height,\nwith a platform on the top 34 feet across, which originally formed the\nbasis of the _tee_ or capital, which was the invariable finish of these\nmonuments.\n\nThe dome rests on a sloping base, 14 feet in height by 120 in\ndiameter, having an offset on its summit about 6 feet wide. This,\nif we may judge from the representations of topes on the sculptures,\nmust have been surrounded by a balustrade, and was ascended by a\nbroad double ramp on one side. It was probably used for processions\nencircling the monument, which seem to have been among the most common\nBuddhist ceremonials. The centre of this great mound is quite solid,\nbeing composed of bricks laid in mud; but the exterior is faced with\ndressed stones. Over these was laid a coating of cement nearly 4 inches\nin thickness, which was, no doubt, originally adorned either with\npainting or ornaments in relief.\n\nThe fence by which this tope is surrounded is extremely curious. It\nconsists of stone posts 8 ft. 8 in. high, and little more than 2 ft.\napart. These are surmounted by a plain architrave, 2 ft. 4 in. deep,\nslightly rounded at the top. So far this enclosure resembles the outer\ncircle at Stonehenge; but between every two uprights three horizontal\ncross-pieces of stone are inserted of an elliptical form, of the same\ndepth as the top piece, but only 9 in. thick in the thickest part. This\nis the only _built_ example yet discovered of an architectural ornament\nwhich is found _carved_ in every cave, and, indeed, in almost every\nancient Buddhist building known in India. The upright posts or pillars\nof this enclosure bear inscriptions indicating that they were all given\nby different individuals. But neither these nor any other inscriptions\nfound in the whole tope, nor in the smaller topes surrounding it\n(though there are as many as 250 inscriptions in all), contain any\nknown name, or any clue to their age.\n\nStill more curious, however, than even the stone railing are the four\ngateways. One of these is shown in our view. It consists of two square\npillars, covered with sculptures, with bold elephant capitals, rising\nto a height of 18 ft. 4 in.; above this are three lintels, slightly\ncurved upwards in the centre, and ending in Ionic scrolls; they are\nsupported by continuations of the columns, and three uprights inserted\nin the spaces between the lintels. They are covered with elaborate\nsculptures, and surmounted by emblems. The total height is 33 ft. 6 in.\nOne gateway has fallen, and if removed to this country would raise the\ncharacter of Indian sculpture, as nothing comparable to it has yet been\ntransported from that part of the world to Europe.\n\n\nBURIAL PLACES OF DISTINGUISHED MEN.\n\nChaucer was buried in the cloisters of Westminster Abbey, _without_\nthe building, but removed to the south aisle in 1555; Spenser lies\nnear him. Beaumont, Drayton, Cowley, Denham, Dryden, Howe, Addison,\nPrior, Congreve, Gray, Johnson, Sheridan, and Campbell, all lie within\nWestminster Abbey. Shakspeare, as every one knows, was buried in the\nchancel of the church at Stratford, where there is a monument to his\nmemory. Chapman and Shirley are buried at St. Giles'-in-the-Fields;\nMarlow, in the churchyard of St. Paul's, Deptford; Fletcher and\nMassinger, in the churchyard of St. Saviour's, Southwark; Dr. Donne, in\nOld St. Paul's; Edward Waller, in Beaconsfield churchyard; Milton, in\nthe churchyard of St. Giles', Cripplegate; Butler, in the churchyard\nof St. Paul's, Covent Garden; Otway, no one knows where; Garth, in the\nchurchyard at Harrow; Pope, in the church at Twickenham; Swift, in St.\nPatrick's, Dublin; Savage, in the churchyard of St. Peter's, Dublin;\nParnell, at Chester, where he died on his way to Dublin; Dr. Young, at\nWelwyn, in Hertfordshire, of which place he was the rector; Thomson,\nin the churchyard at Richmond, in Surrey; Collins, in St. Andrew's\nChurch, at Chichester; Gray, in the churchyard at Stoke-Pogis, where\nhe conceived his \"Elegy;\" Goldsmith, in the churchyard of the Temple\nChurch; Falconer, at sea, with \"all ocean for his grave;\" Churchill,\nin the churchyard of St. Martin's, Dover; Cowper, in the church at\nDereham; Chatterton, in a churchyard belonging to the parish of St.\nAndrew's, Holborn; Burns, in St. Michael's churchyard, Dumfries; Byron,\nin the church of Hucknall, near Newstead; Crabbe, at Trowbridge;\nColeridge, in the church at Highgate; Sir Walter Scott, in Dryburgh\nAbbey; Southey, in Crosthwaite Church, near Keswick.\n\n\nA REGAL HUNTING PARTY.\n\nThe following is an account of the destruction of game in Bohemia, by\na hunting party of which the Emperor Francis made one, in 1755. There\nwere twenty-three persons in the party, three of whom were ladies;\nthe Princess Charlotte of Lorraine was one of them. The chase lasted\neighteen days, and during that time they killed 47,950 head of game,\nand wild deer; of which 19 were stags, 77 roebucks, 10 foxes, 18,243\nhares, 19,545 partridges, 9,499 pheasants, 114 larks, 353 quails, 454\nother birds. The Emperor fired 9,798 shots, and the Princess Charlotte\n9,010; in all, there were 116,209 shots fired.\n\n\nANTIPATHIES.\n\nCertain antipathies appear to depend upon a peculiarity of the senses.\nThe horror inspired by the odour of certain flowers may be referred to\nthis cause. Amatus Lusitanus relates the case of a monk who fainted\nwhen he beheld a rose, and never quitted his cell when that flower\nwas blooming. Scaliger mentions one of his relations who experienced\na similar horror when seeing a lily. In these instances it is not the\nagreeableness or the offensive nature of the aroma that inspires the\nrepugnance; and Montaigne remarked on this subject, that there were men\nwho dreaded an apple more than a musket-ball. Zimmerman tells us of a\nlady who could not endure the feeling of silk and satin, and shuddered\nwhen touching the velvety skin of a peach. Boyle records the case of\na man who felt a natural abhorrence to honey. Without his knowledge,\nsome honey was introduced in a plaster applied to his foot, and the\naccidents that resulted compelled his attendants to withdraw it. A\nyoung man was known to faint whenever he heard the servant sweeping.\nHippocrates mentions one Nicanor who swooned whenever he heard a\nflute: our Shakspeare has alluded to the effects of the bagpipe. Julia\ndaughter of Frederick, king of Naples, could not taste meat without\nserious accidents. Boyle fainted when he heard the splashing of water;\nScaliger turned pale at the sight of water-cresses; Erasmus experienced\nfebrile symptoms when smelling fish; the Duke d'Epernon swooned on\nbeholding a leveret, although a hare did not produce the same effect.\nTycho Brahe fainted at the sight of a fox, Henry the Third of France\nat that of a cat, and Marshal d'Albert at a pig. The horror that whole\nfamilies entertain of cheese is generally known. Many individuals\ncannot digest, or even retain certain substances, such as rice, wine,\nvarious fruits, and vegetables.\n\n\nA YOUNG BUT CRUEL MURDERESS.\n\nOn the 3d of July, 1772, was executed at Lisbon, pursuant to her\nsentence, Louisa de Jesus, for the murder of the thirty-three infants,\nthat were at different times committed to her care by the Directors\nof the Foundling Hospital at Coimbra; for which (as appears by the\nsentence published) she had no other inducement but six hundred reals\nin money, a coverda of baize, and a cradle, that she received with each\nof them. She was but twenty-two years of age when executed. Going to\nexecution, she was pinched with hot irons, and at the gallows her hands\nwere struck off; she was then strangled, and her body burnt.\n\n\nBECTIVE ABBEY.\n\nBective Abbey, the ruins of which form the subject of the annexed\nengraving differs in its general arrangement from every other monastic\nstructure in the kingdom. It was, in fact, a monastic castle, and,\nprevious to the use of artillery, must have been regarded as a place of\ngreat strength. It is for this reason that we select it as one of our\n\"Wonderful Things.\"\n\n[Illustration [++] Bective Abbey.]\n\nThe ruins are in the immediate neighbourhood of Trim, and about thirty\nmiles from Dublin.\n\nThe ruins combine a union of ecclesiastical with military and domestic\narchitecture in a remarkable degree. Their chief feature is a strong\nbattlemented tower, the lower compartment of which is vaulted, placed\nat the south-west corner of the quadrangular space occupied by the\nvarious buildings, and in the centre of which the cloisters remain\nin excellent preservation. The cloister arches are late in the first\npointed style, and are cinque-foiled. The featherings are mostly\nplain, but several are ornamented with flowers or leaves, and upon\none a hawk-like bird is sculptured. A fillet is worked upon each of\nthe clustered shafts, by which the openings are divided, and also\nupon their capitals. The bases, which are circular, rest upon square\nplinths, the angles of which are ornamented with a leaf, as it were,\ngrowing out of the base of the moulding.\n\nOf the church there are scarcely any remains. As the northern wall\nof the cloister is pierced with several windows, which have now the\nappearance of splaying externally, it is extremely probable that it\nalso served as the south wall of the church, no other portion of which\ncan at present be identified. Those buildings which were for the most\npart devoted to domestic purposes are for the most part situated upon\nthe east side of the quadrangle. Their architectural details are of a\ncharacter later than those of the tower and of the other portions, but\nadditions and alterations have evidently been made.\n\n\nNOVEL MODE OF CELEBRATION.\n\nUpon the occasion of the christening of the 21st child of Mr. Wright,\nof Widaker, near Whitehaven, by the same woman, in the year 1767, the\ncompany came from 21 parishes, and the entertainment consisted of 21\npieces of beef, 21 legs of mutton and lamb, 21 gallons of brandy, three\ntimes 21 gallons of strong ale, three times 21 fowls, roasted and\nboiled, 21 pies, &c.\n\n\nANTIQUE HEAD ORNAMENT.\n\n[Illustration [++] Bronze Head Ornament.]\n\nThe annexed engraving represents an exceedingly beautiful bronze\nrelic, apparently of the class of head rings, in the collection of the\nSociety of Antiquaries of Scotland, which was discovered in the year\n1747, about seven feet below the surface, when digging for a well, at\nthe east end of the village of Stitchel, in the county of Roxburgh.\nIt bears a resemblance in some respects to relics of the same class\nin the Christiansborg Palace, yet nothing exactly similar to it has\nyet been found among Scandinavian relics; while some of its ornamental\ndetails closely correspond to those which characterize the British\nhorse furniture and other native relics of this period. One of its most\nremarkable peculiarities is, that it opens and shuts by means of a\nhinge, being clasped when closed by a pin which passes through a double\ncatch at a line intersecting the ornament; and so perfect is it that\nit can still be opened and secured with ease. It is probable that this\nalso should rank among the ornaments of the head, though it differs\nin some important respects from any other object of the same class.\nThe oval which it forms is not only too small to encircle the head,\nbut it will be observed from the engraving that its greatest length is\nfrom side to side, the internal measurements being five and nine-tenth\ninches by five and one-tenth inches.\n\n\nRELICS.\n\nAt the commencement of the seventeenth century there was a crucifix\nbelonging to the Augustine friars at Burgos in Spain, which produced a\nrevenue of nearly seven thousand crowns per annum. It was found upon\nthe sea, not far from the coast, with a scroll of parchment appended\nto it, descriptive of the various virtues it possessed. The image\nwas provided with a false beard and a chesnut periwig, which its holy\nguardians declared were natural, and they also assured all pious\nvisitors that on every Friday it sweated blood and water into a silver\nbasin. In the garden of this convent grew a species of wheat, the grain\nof which was peculiarly large, and which its possessors averred was\nbrought by Adam out of Paradise. Of this wheat they made small cakes\ncalled pancillos, kneaded with the aforesaid blood and water, and\nsold them to the credulous multitude for a quartillo a piece. These\ncakes were an infallible remedy for all disorders, and over those\nwho carried them the devil had no power. They sold also blue ribands\nof the exact length of the crucifix, for about a shilling each, with\nthis inscription in silver letters, \"La madi del santo crucifisco de\nBurgos.\" These ribands were a sovereign cure for the headache.\n\n\nLONG MEG AND HER DAUGHTERS.\n\nAs there is something remarkable or out of the way in this family of\nheavy stone, we present it to the reader. This venerable Druidical\nmonument, which is by the country-people called Long Meg and her\nDaughters, stands near Little Salkeld, in the county of Cumberland.\nIt consists of 67 massy stones, of different sorts and sizes, ranged\nin a circle of nearly 120 paces diameter; some of these stones are\ngranite, some blue and grey limestone, and others flint; many of them\nare ten feet high, and fifteen or sixteen feet in circumference:\nthese are called Long Meg's Daughters. On the southern side of this\ncircle, about seventeen or eighteen paces out of the line, stands the\nstone called Long Meg, which is of that kind of red stone found about\nPenrith. It is so placed, that each of its angles faces one of the\ncardinal points of the compass; it measures upwards of eighteen feet\nin height, and fifteen feet in girth; its figure is nearly that of a\nsquare prism; it weighs about sixteen tons and a half. In the part\nof the circle the most contiguous, four large stones are placed in a\nsquare form, as if they had been intended to support an altar; and\ntowards the east, west, and north, two large stones stand a greater\ndistance from each other than any of the rest, seemingly to form the\nentrances into a circle. It is remarkable that no stone-quarry is to\nbe found hereabouts. The appearance of this circle is much hurt by\na stone wall built across it, that cuts off a considerable segment,\nwhich stands in the road. The same ridiculous story is told of these\nstones, as of those at Stonehenge, _i. e._, that it is impossible to\ncount them, and that many persons who have made the trial, could never\nfind them amount twice to the same number. It is added, that this was\na holy place, and that Long Meg and her Daughters were a company of\nwitches transformed into stones, on the prayers of some saint, for\nventuring to prophane it; but when, and by whom, the story does not\nsay. Thus has tradition obscurely, and clogged with fable, handed down\nthe destination of this spot, accompanied with some of that veneration\nin which it was once undoubtedly held, though not sufficiently to\nprotect its remains from the depredations of avarice; the inclosure\nand cultivation of the ground bidding fair to destroy them. These\nstones are mentioned by Camden, who was either misinformed as to, or\nmis-reckoned their number; unless, which seems improbable, some have\nbeen taken away. \"At Little Salkeld, (says he,) there is a circle of\nstones seventy-seven in number, each ten feet high; and before these,\nat the entrance, is a single one by itself, fifteen feet high. This the\ncommon people call Long Meg, and the rest her Daughters; and within\nthe circle, are two heaps of stones, under which they say there are\ndead bodies buried; and, indeed, it is probable enough that this has\nbeen a monument erected in memory of some victory.\" The history of the\nBritish Druidical Antiquities having been thoroughly investigated,\nsince Camden's time, these circles are now universally agreed to have\nbeen temples and places of judgment, and not sepulchral monuments.\nIndeed his editor has, in some measure, rectified his mistake, by the\nfollowing addition: \"But, as to the heaps in the middle, they are no\npart of the monument, but have been gathered off the ploughed lands\nadjoining; and (as in many other parts of the county) thrown up here in\na waste corner of the field; and as to the occasion of it, both this,\nand the Rolrick stones in Oxfordshire, are supposed by many, to have\nbeen monuments erected at the solemn investiture of some Danish Kings,\nand of the same kind as the Kingstolen in Denmark, and Moresteen in\nSweden; concerning which, several large discourses have been written.\"\n\n\nCURIOUS PARTICULARS RELATIVE TO DRESS AT THE COMMENCEMENT OF THE\nFIFTEENTH CENTURY.\n\nCloth of gold, satin, and velvet, enriched by the florid decorations\nof the needle, were insufficient to satisfy the pride of nobles; robes\nformed of these costly materials were frequently ornamented with\nembroidery of goldsmiths' work, thickly set with precious stones;\nand the most absurd and fantastic habits were continually adopted,\nin the restless desire to appear in new inventions. John of Ghent is\nrepresented in a habit divided straight down the middle, one side\nwhite, the other half dark blue; and his son, Henry IV., on his\nreturn from exile, rode in procession through London in a jacket of\ncloth-of-gold, \"after the German fashion.\" The dukes and earls who\nattended his coronation wore three bars of ermine on the left arm, a\nquarter of a yard long, \"or thereabouts;\" the barons had but two: and\nover the monarch's head was borne a canopy of blue silk, supported by\nsilver staves, with four gold bells, \"that rang at the corners.\" \"Early\nin the reign of Richard II. began,\" says Stowe, \"the detestable use of\npiked shoes, tied to the knees with chains of silver gilt; also women\nused high attire on their heads with piked horns and long training\ngowns. The commons also were besotted in excesse of apparel; in wide\nsurcoates reaching to their loines; some in a garment reaching to their\nheels, close before and sprowting out at the sides, so that on the\nbacke they make men seeme women, and this they call by a ridiculous\nname--_gowne_. Their hoodes are little, and tied under the chin.\"\n\n\nECCENTRIC FUNERAL.\n\nMr. John Oliver, an eccentric miller of Highdown Hill, in Sussex, died,\naged eighty-three, the 27th of May, 1793. His remains were interred\nnear his mill, in a tomb he had caused to be erected there for that\npurpose, near thirty years ago; the ground having been previously\nconsecrated. His coffin, which he had for many years kept under his\nbed, was painted white; and the body was borne by eight men clothed in\nthe same colour. A girl about twelve years old read the burial service,\nand afterwards, on the tomb, delivered a sermon on the occasion, from\nMicah 7, 8, 9, before at least two thousand auditors, whom curiosity\nhad led to see this extraordinary funeral.\n\n\nEGYPTIAN STANDARDS.\n\nThe engraving which we here lay before our friends, represents a group\nof Egyptian standards, as they were used in the army in the time of\nPharaoh.\n\n[Illustration [++] Egyptian Standards.]\n\nEach regiment and company had its own peculiar banner or standard,\nwhich were therefore very numerous, and various in their devices. A\nbeast, bird, or reptile, a sacred boat, a royal name in a cartouche,\nor a symbolic combination of emblems, were the most common forms. As\nthey appear to have been objects of superstitious veneration that were\nselected for this purpose, they must have contributed greatly to the\nenthusiasm so highly valued in battle; and instances are common in\nall history of desponding courage revived, and prodigies of valour\nperformed, on behalf of those objects which were so identified with\nnational and personal honour.\n\nAllusions to standards, banners, and ensigns are frequent in the Holy\nScriptures. The four divisions in which the tribes of Israel marched\nthrough the wilderness had each its governing standard, and tradition\nhas assigned to these ensigns the respective forms of the symbolic\ncherubim seen in the vision of Ezekiel and John--that of Judah being\na lion, that of Reuben a man, that of Ephraim an ox, and that of Dan\nan eagle. The post of standard-bearer was at all times of the greatest\nimportance, and none but officers of approved valour were ever chosen\nfor such a service; hence Jehovah, describing the ruin and discomfiture\nwhich he was about to bring on the haughty King of Assyria, says, \"And\nthey shall be as when a standard-bearer fainteth.\"\n\n\nTHE SHREW ASH.\n\n[Illustration [++] Shrew Ash.]\n\nAt that end of Richmond Park where a gate leads to Mortlake, and near\na cottage in which resides one of the most estimable gentlemen of the\nage--Professor Owen--there still lives and flourishes a tree that has\nbeen famous for many ages: it is the Shrew Ash, and the above is a\ncorrect engraving of it. It stands on rising ground, only a few yards\nbeyond the pond which almost skirts the Professor's lawn. White, in\nhis Natural History of Selborne, describes a shrew-ash as an ash whose\ntwigs or branches, when gently applied to the limbs of cattle, will\nimmediately relieve the pains which a beast suffers from the running\nof a shrew-mouse over the part affected; for it is supposed that a\nshrew-mouse is of so baleful and deleterious a nature, that wherever it\ncreeps over a beast, be it horse, cow, or sheep, the suffering animal\nis afflicted with cruel anguish, and threatened with the loss of the\nuse of the limb. Against this evil, to which they were continually\nliable, our provident forefathers always kept a shrew-ash at hand,\nwhich, when once medicated, would maintain its virtue for ever. A\nshrew-ash was made potent thus:--Into the body of a tree a deep hole\nwas bored with an auger, and a poor devoted shrew-mouse was thrust in\nalive, and plugged in, no doubt with several quaint incantations, long\nsince forgotten. The shrew-ash in Richmond Park is, therefore, amongst\nthe few legacies of the kind bequeathed to their country by the wisdom\nof our ancestors.\n\nOur readers will perceive that across the hollow of the tree near the\ntop there is a little bar of wood. The legend runs that were this\nbar removed every night, it would be replaced in the same spot every\nmorning. The superstition is, that if a child afflicted with what the\npeople in the neighbourhood call \"decline,\" or whooping-cough, or any\ninfantine disease, is passed nine times up the hollow of that tree, and\nover the bar, while the sun is rising, it will recover. If the charm\nfails to produce the desired effect, the old women believe that the sun\nwas too far up, or not up enough. If the child recovers, of course, the\nfame of the tree is whispered about. There is a sort of shrew-mother to\nevery shrew-ash, who acts as guide and teacher to any young mother who\nhas an afflicted child and believes in the charm. The ash in Richmond\nPark is still used, and still firmly believed in.\n\n\nA DRUM MADE OF HUMAN SKIN.\n\nJohn Zisca, general of the insurgents who took up arms in the year 1419\nagainst the Emperor Sigismund, to revenge the deaths of John Huss,\nand Jerome of Prague, who had been cruelly burnt to death for their\nreligious tenets, defeated the Emperor in several pitched battles. He\ngave orders that, after his death, they should _make a drum of his\nskin_; which was most religiously obeyed, and those very remains of\nthe enthusiastic Zisca proved, for many years, fatal to the Emperor,\nwho, with difficulty, in the space of sixteen years, recovered Bohemia,\nthough assisted by the forces of Germany, and the terror of Crusades.\nThe insurgents were 40,000 in number, and well disciplined.\n\n\nEARTHQUAKE IN JAMAICA.\n\nThe Earthquake of Jamaica, in 1692, is one of the most dreadful that\nhistory has to record. It was attended with a hollow rumbling noise\nlike that of thunder, and in less than a minute all the houses on one\nside of the principal street in the town of Port Royal sank into a\nfearful gulf forty fathoms deep, and water came roaring up where the\nhouses had been. On the other side of the street the ground rose up\nand down like the waves of the sea, raising the houses and throwing\nthem into heaps as it subsided. In another part of the town the street\ncracked along all its length, and the houses appeared suddenly twice\nas far apart as they were before. In many places the earth opened\nand closed again, so that several hundred of these openings were to\nbe seen at the same time; and as the wretched inhabitants ran out of\ntheir tottering dwellings, the earth opened under their feet, and in\nsome cases swallowed them up entirely; while in others, the earth\nsuddenly closing, caught them by the middle, and thus crushed them\nto death. In some cases these fearful openings spouted up cataracts\nof water, which were attended by a most noisome stench. It is not\npossible for any place to exhibit a scene of greater desolation than\nthe whole island presented at this period. The thundering bellowing\nof the distant mountains, the dusky gloom of the sky, and the crash\nof the falling buildings gave unspeakable horror to the scene. Such\nof the inhabitants as were saved sought shelter on board the ships in\nthe harbour, and remained there for more than two months, the shocks\ncontinuing with more or less violence every day. When, at length, the\ninhabitants were enabled to return, they found the whole face of the\ncountry changed. Very few of the houses which had not been swallowed\nup were left standing, and what had been cultivated plantations were\nconverted into large pools of water. The greater part of the rivers had\nbeen choked up by the falling in of detached masses of the mountains,\nand spreading over the valleys, they had changed what was once fertile\nsoil into morasses, which could only be drained by cutting new channels\nfor the rivers; while the mountains themselves had changed their shapes\nso completely, that it was conjectured that they had formed the chief\nseat of the earthquake.\n\n\nCURIOUS EXTRACTS FROM THE HOUSEHOLD BOOK OF LADY MARY, DAUGHTER OF THE\nKING, IN VARIOUS YEARS, FROM THE 28TH TO THE 36TH OF HENRY VIII. ROYAL\nMSS. BRIT. MUS.\n\n\"Item, geven to George Mountejoye drawing my Layde's Grace to his\nValentine, xl{s}.\n\n\"Item, geven amongs the yeomen of the King's guard bringing a Leke to\nmy Lady's Grace on Saynt David's Day, xv{s}.\n\n\"Item, geven to Heywood playeng an enterlude with his children before\nmy Lady's Grace, xl{s}.\n\n\"Item, payed for a yerde and a halfe of damaske for Jane the fole,\nvij{s}.\n\n\"Item, for shaving of Jane fooles hedde, iiij{d}.\n\n\"Payed for a frountlet lost in a wager to my Lady Margaret, iiij{li}.\n\n\"Item, payed for a brekefast lost at bolling by my Lady Mary's Grace,\nx{s}.\"\n\n\nGIVING DOLES.\n\nA bishop of Durham, in the reign of Edward III, had every week\neight quarters of wheat made into bread for the poor, besides his\nalms-dishes, fragments from his table, and money given away by him\nin journeys. The bishop of Ely, in 1532, fed daily at his gates two\nhundred poor persons, and the Lord Cromwell fed the same number.\nEdward, earl of Derby, fed upwards of sixty aged poor, besides all\ncomers, thrice a week, and furnished, on Good Friday, two thousand\nseven hundred people with meat, drink, and money. Robert Winchelsey,\narchbishop of Canterbury, gave, besides the daily fragments of his\nhouse, on Fridays and Sundays, to every beggar that came to his door,\na loaf of bread of a farthing value; in time of dearth he thus gave\naway five thousand loaves, and this charity is said to have cost his\nlordship five hundred pounds a year. Over and above this he gave on\nevery festival day one hundred and fifty pence to as many poor persons,\nand he used to send daily meat, drink, and bread unto such as by age\nand sickness were not able to fetch alms from his gate; he also sent\nmoney, meat, apparel, &c., to such as he thought wanted the same, and\nwere ashamed to beg; and, above all, this princely prelate was wont to\ntake compassion upon such as were by misfortune decayed, and had fallen\nfrom wealth to poor estate. Such acts deserve to be written in letters\nof gold.\n\n\nFEMALE ORNAMENT OF THE IRON PERIOD.\n\n[Illustration [++] Beaded Torc.]\n\nOne of the most beautiful neck ornaments of the Teutonic or Iron Period\never found in Scotland is a beaded torc, discovered by a labourer\nwhile cutting turf in Lochar Moss, Dumfriesshire, about two miles to\nthe north of Cumlongan Castle; and exhibited by Mr. Thomas Gray, of\nLiverpool, at the York meeting of the Archaeological Institute. We here\nannex an engraving of it. The beads, which measure rather more than\nan inch in diameter, are boldly ribbed and grooved longitudinally.\nBetween every two ribbed beads there is a small flat one formed like\nthe wheel of a pulley, or the vertebral bone of a fish. The portion\nwhich must have passed round the nape of the neck is flat and smooth\non the inner edge, but chased on the upper side in an elegant incised\npattern corresponding to the ornamentation already described as\ncharacteristic of this period, and bearing some resemblance to that on\nthe beautiful bronze diadem found at Stitchel in Roxburghshire, figured\non a subsequent page. The beads are disconnected, having apparently\nbeen strung upon a metal wire, as was the case in another example\nfound in the neighbourhood of Worcester. A waved ornament, chased\nalong the outer edge of the solid piece, seems to have been designed\nin imitation of a cord; the last tradition, as it were, of the string\nwith which the older necklace of shale or jet was secured. Altogether\nthis example of the class of neck ornaments, to which Mr. Birch has\nassigned the appropriate name of beaded torcs, furnishes an exceedingly\ninteresting illustration of the development of imitative design, in\ncontradistinction to the more simple and archaic funicular torc, which,\nthough continued in use down to a later period, pertains to the epoch\nof primitive art.\n\n\nCURIOUS LANTERN.\n\nIn 1602, it is related that Sir John Harrington, of Bath, sent to James\nVI King of Scotland, at Christmas, for a new year's gift, a dark\nlantern. The top was a crown of pure gold, serving also to cover a\nperfume pan; within it was a shield of silver, embossed, to reflect\nthe light; on one side of which were the sun, moon, and planets, and\non the other side, the story of the birth and passion of Christ, as\nit was engraved by David II King of Scotland, who was a prisoner at\nNottingham. On this present, the following passage was inscribed in\nLatin--\"Lord, remember me when thou comest into thy kingdom.\"\n\n\nANCIENT SCANDINAVIAN BROOCH.\n\n[Illustration [++] Shell-Shaped Brooch.]\n\nThe characteristic and beautiful ornament, usually designated the\nshell-shaped brooch, and equally familiar to Danish and British\nantiquaries, belongs to the Scoto-Scandinavian Period. In Scotland many\nbeautiful examples have been found, several of which are preserved\nin the Museum of Scottish Antiquaries. From these we select the one\nrepresented in the annexed engraving, as surpassing in beauty of design\nand intricacy of ornament any other example of which we are aware. It\nconsists, as usual, of a convex plate of metal, with an ornamental\nborder, surmounted by another convex plate of greater depth, highly\nornamented with embossed and perforated designs, the effect of which\nappears to have been further heightened by the lower plate being gilded\nso as to show through the open work. In this example the gilding\nstill remains tolerably perfect. On the under side are the projecting\nplates, still retaining a fragment of the corroded iron pin, where it\nhas turned on a hinge, and at the opposite end the bronze catch into\nwhich it clasped. The under side of the brooch appears to have been\nlined with coarse linen, the texture of which is still clearly defined\nof the coating of verd antique with which it is now covered. But its\npeculiar features consist of an elevated central ornament resembling a\ncrown, and four intricately-chased projections terminating in horses'\nheads. It was found in September, 1786, along with another brooch of\nthe same kind, lying beside a skeleton, under a flat stone, very near\nthe surface, above the ruins of a Pictish house or burgh, in Caithness.\nIt measures nearly four and a half inches in length, by three inches\nin breadth, and two and two-fifth inches in height to the top of the\ncrown. Like many others of the same type, it appears to have been\njewelled. In several examples of these brooches which we have compared,\nthe lower convex plates so nearly resemble each other, as to suggest\nthe probability of their having been cast in the same mould, while the\nupper plates entirely differ.\n\n\nSTREET CRIES OF MODERN EGYPT.\n\nThe cries of the street hawkers in Egypt at the present day are very\nsingular, and well deserve a place in our repertory of curiosities.\nThe seller of _tir'mis_ (or lupins) often cries \"Aid! O Imba'bee!\naid!\" This is understood in two senses: as an invocation for aid\nto the sheykh El-Imba'bee, a celebrated Moos'lim saint, buried at\nthe Imba'beh, on the west bank of the Nile, opposite Cairo; in the\nneighbourhood of which village the best tir'mis is grown; and also as\nimplying that it is through the aid of the saint above mentioned that\nthe tir'mis of Imba'beh is so excellent. The seller of this vegetable\nalso cries, \"The tir'mis of Imba'beh surpasses the almond!\" Another\ncry of the seller of tir'mis is, \"O how sweet are the little children\nof the river!\" This last cry, which is seldom heard but in the country\ntowns and villages of Egypt, alludes to the manner in which the tir'mis\nis prepared for food. To deprive it of its natural bitterness, it is\nsoaked, for two or three days, in a vessel full of water; then boiled,\nand, after this, sewed up in a basket of palm-leaves (called _furd_),\nand thrown into the Nile, where it is left to soak again, two or three\ndays; after which, it is dried, and eaten cold, with a little salt. The\nseller of sour limes cries, \"God make them light [or easy of sale]!\nO limes!\" The toasted pips of a kind of melon called '_abdalla'wee_,\nand of the water-melon, are often announced by the cry of \"O consoler\nof the embarrassed! O pips!\" though more commonly, by the simple cry\nof \"Roasted pips!\" A curious cry of the seller of a kind of sweetmeat\n(hhala'wee), composed of treacle fried with some other ingredients,\nis, \"For a nail! O sweetmeat!\" He is said to be half a thief: children\nand servants often steal implements of iron, &c., from the house in\nwhich they live, and give them to him in exchange for his sweetmeat.\nThe hawker of oranges cries, \"Honey! O oranges! Honey!\" and similar\ncries are used by the sellers of other fruits and vegetables; so that\nit is sometimes impossible to guess what the person announces for sale;\nas, when we hear the cry of \"Sycamore-figs! O grapes!\" excepting by\nthe rule that what is for sale is the least excellent of the fruits,\n&c., mentioned; as sycamore-figs are not so good as grapes. A very\nsingular cry is used by the seller of roses: \"The rose was a thorn:\nfrom the sweat of the Prophet it opened [its flowers].\" This alludes\nto a miracle related of the Prophet. The fragrant flowers of the\nhhen'na-tree (or Egyptian privet) are carried about for sale, and the\nseller cries, \"Odours of paradise! O flowers of the hhen'na!\" A kind of\ncotton cloth, made by machinery which is put in motion by a bull, is\nannounced by the cry of \"The work of the bull! O maidens!\"\n\n\nTHE BLACK PESTILENCE.\n\nThe black pestilence of the fourteenth century caused the most terrific\nravages in England. It has been supposed to have borne some resemblance\nto the cholera, but that is not the case; it derived its name from\nthe dark, livid colour of the spots and boils that broke out upon\nthe patient's body. Like the cholera, the fatal disease appeared to\nhave followed a regular route in its destructive progress; but it did\nnot, like the cholera, advance westward, although, like that fearful\nvisitation, it appears to have originated in Asia.\n\nThe black pestilence descended along the Caucasus to the shores of the\nMediterranean, and, instead of entering Europe through Russia, first\nappeared over the south, and, after devastating the rest of Europe,\npenetrated into that country. It followed the caravans, which came\nfrom China across Central Asia, until it reached the shores of the\nBlack Sea; thence it was conveyed by ships to Constantinople, the\ncentre of commercial intercourse between Asia, Europe, and Africa. In\n1347 it reached Sicily and some of the maritime cities of Italy and\nMarseilles. During the following year it spread over the northern part\nof Italy, France, Germany, and England. The northern kingdoms of Europe\nwere invaded by it in 1349, and finally Russia in 1351--four years\nafter it had appeared in Constantinople.\n\nThe following estimate of deaths was considered far below the actual\nnumber of victims:--\n\n Florence lost 60,000 inhabitants\n Venice \" 10,000 \"\n Marseilles \" in one month 56,000 \"\n Paris \" \" 50,000 \"\n Avignon \" \" 60,000 \"\n Strasburg \" \" 16,000 \"\n Basle \" \" 14,000 \"\n Erfurth \" \" 16,000 \"\n London \" \" 100,000 \"\n Norwich \" \" 50,000 \"\n\nHecker states that this pestilence was preceded by great commotion\nin the interior of the globe. About 1333, several earthquakes and\nvolcanic eruptions did considerable injury in upper Asia, while in the\nsame year, Greece, Italy, France, and Germany suffered under similar\ndisasters. The harvests were swept away by inundations, and clouds of\nlocusts destroyed all that floods had spared, while dense masses of\noffensive insects strewed the land.\n\nAs in the recent invasion of cholera, the populace attributed this\nscourge to poison and to the Jews, and these hapless beings were\npersecuted and destroyed wherever they could be found. In Mayence,\nafter vainly attempting to defend themselves, they shut themselves up\nin their quarters, where 1,200 of them burnt to death. The only asylum\nfound by them was Lithuania, where Casimir afforded them protection;\nand it is, perhaps, owing to this circumstance that so many Jewish\nfamilies are still to be found in Poland.\n\n\nTHE DUCHESS OF LAUDERDALE.\n\nFew mansions are more pleasantly situated than Ham House, the dwelling\nof the Tollemaches, Earls of Dysart. It stands on the south bank of\nthe Thames, distant about twelve miles from London, and immediately\nopposite to the pretty village of Twickenham. It was erected early in\nthe seventeenth century; the date 1610 still stands on the door of\nthe principal entrance. Its builder was Sir Thomas Vavasour, and it\nsubsequently came into the possession of Katherine, daughter of the\nEarl of Dysart, who married first Sir Lionel Tollemache, and for her\nsecond husband Earl, afterwards Duke, of Lauderdale.\n\nThe Duchess of Lauderdale was one of the \"busiest\" women of the busy\nage in which she lived. Burnet insinuates that, during the life time\nof her first husband, \"she had been in a correspondence with Lord\nLauderdale that had given occasion for censure.\" She succeeded in\npersuading him that he was indebted for his escape after \"Worcester\nfight\" to \"her intrigues with Cromwell. She was a woman,\" continues\nthe historian, \"of great beauty, but of far greater parts. She had\na wonderful quickness of apprehension, and an amazing vivacity in\nconversation. She had studied, not only divinity and history, but\nmathematics and philosophy. She was violent in everything she set\nabout,--a violent friend, but a much more violent enemy. She had\na restless ambition, lived at a vast expense, and was ravenously\ncovetous, and would have stuck at nothing by which she might compass\nher ends.\" Upon the accession of her husband to political power after\nthe Restoration, \"all applications were made to her. She took upon her\nto determine everything; she sold all places; and was wanting in no\nmethod that could bring her money, which she lavished out in a most\nprofuse vanity.\"\n\nThis Duchess of Lauderdale--famous during the reigns of four\nmonarchs--the First and Second James, and the First and Second Charles,\nand through the Protectorship of Cromwell--refurnished the house at\nHam, where she continued to reside until her death at a very advanced\nage.\n\nAmong other untouched relics of gone-by days, is a small ante-chamber,\nwhere, it is said, she not only condescended to receive the second\nCharles, but, if tradition is to be credited, where she \"cajoled\"\nOliver Cromwell. There still remains the chair in which she used to\nsit, her small walking cane, and a variety of objects she was wont to\nvalue and cherish as memorials of her active life, and the successful\nissue of a hundred political intrigues.\n\n[Illustration [++] Chair of the Duchess of Lauderdale.]\n\n\nMODERN EGYPTIAN MUSICAL INSTRUMENTS.\n\n[Illustration: Sa'ga't (1), Ta'r (2), and Dar'abook'keh (3).]\n\nThe durwee'shes, who constitute a sort of religious mendicant order in\nEgypt, often make use of, in their processions and in begging, a little\ntubl, or kettle-drum, called _ba'z_; six or seven inches in diameter;\nwhich is held in the left hand, by a little projection in the centre\nof the back, and beaten by the right hand, with a short leather strap,\nor a stick. They also use cymbals, which are called _ka's_, on similar\noccasions. The ba'z is used by the Moosahh'hhir, to attract attention\nto his cry in the nights of Rum'ada'n. Castanets of brass, called\n_sa'ga't_ are used by the public female and male dancers. Each dancer\nhas two pairs of these instruments. They are attached, each by a loop\nof string, to the thumb and second finger, and have a more pleasing\nsound than castanets of wood or ivory. There are two instruments which\nare generally found in the hharee'm of a person of moderate wealth,\nand which the women often use for their diversion. One of these is\na tambourine, called _ta'r_, of which we insert an engraving. It is\neleven inches in diameter. The hoop is overlaid with mother-of-pearl,\ntortoise-shell, and white bone, or ivory, both without and within, and\nhas ten circular plates of brass attached to it, each two pairs having\na wire passing through their centres. The ta'r is held by the left or\nright hand, and beaten with the fingers of that hand and by the other\nhand. The fingers of the hand which holds the instrument, striking only\nnear the hoop, produce higher sounds than the other hand, which strikes\nin the centre. A tambourine of a larger and more simple kind than that\nhere described, without the metal plates, is often used by the lower\norders. The other instrument alluded to in the commencement of this\nparagraph is a kind of drum, called _dar'abook'keh_. The best kind is\nmade of wood, covered with mother-of-pearl and tortoise-shell, &c. One\nof this description is here represented with the ta'r. It is fifteen\ninches in length, covered with a piece of fishes' skin at the larger\nextremity, and open at the smaller. It is placed under the left arm;\ngenerally suspended by a string that passes over the left shoulder; and\nis beaten with both hands.\n\n\nREMARKABLE OAKS.\n\nThe oaks most remarkable for their horizontal expansion, are, according\nto Loudon, the following:--\"The Three-shire Oak, near Worksop, was\nso situated, that it covered part of the three counties of York,\nNottingham, and Derby, and dripped over seven hundred and seventy-seven\nsquare yards. An oak between Newnham Courtney and Clifton shaded a\ncircumference of five hundred and sixty yards of ground, under which\ntwo thousand four hundred and twenty men might have commodiously\ntaken shelter. The immense Spread Oak in Worksop Park, near the white\ngate, gave an extent, between the ends of its opposite branches, of\nan hundred and eighty feet. It dripped over an area of nearly three\nthousand square yards, which is above half an acre, and would have\nafforded shelter to a regiment of nearly a thousand horse. The Oakley\nOak, now growing on an estate of the Duke of Bedford, has a head of an\nhundred and ten feet in diameter. The oak called _Robur Britannicum_,\nin the Park, at Rycote, is said to have been extensive enough to cover\nfive thousand men; and at Ellerslie, in Renfrewshire, the native\nvillage of the hero Wallace, there is still standing 'the old oak\ntree,' among the branches of which, it is said, that he and three\nhundred of his men hid themselves from the English.\"\n\n\nCURIOUS ADVERTISEMENT.\n\nA few years ago the following actually appeared in one of the London\npapers: certainly a most economical speculation for the use of soul and\nbody:--\n\n\"Wanted, for a family who have bad health, a sober, steady person,\nin the capacity of doctor, surgeon, apothecary, and man-midwife. He\nmust occasionally act as butler, and dress hair and wigs. He will\nbe required sometimes to read prayers, and to preach a sermon every\nSunday. A good salary will be given.\"\n\n\nCHANGES OF MOUNT ETNA.\n\nSignor Maria Gemmellario has given, from a meteorological journal kept\nat Catania, a very interesting view of the successive changes of Mount\nEtna, at a period in which it was in the phase of moderate activity;\nand no description could convey so accurate a conception of the\never-changing phenomena.\n\nOn the 9th of February, 1804, there was a sensible earthquake. Etna\nsmoked ninety-seven days, but there was no eruption nor any thunder.\n\nOn the 3rd of July, 1805, there was an earthquake. Etna smoked\nforty-seven days, and emitted flame twenty-eight days. There was an\neruption in June, but no thunder.\n\nThere were earthquakes on the 27th of May and 10th of October, 1806.\nThe mountain smoked forty-seven days, flamed seven, and detonated\ntwenty-eight: little thunder.\n\nOn the 24th of February and 25th of November, 1807, there were\nearthquakes. Etna smoked fifty-nine days: little thunder.\n\nIn August, September, and December, 1808, earthquakes were frequent.\nEtna smoked twelve days, flamed one hundred and two, and often\ndetonated. Thunder storms were frequent.\n\nFrom January to May, and during September and December, 1809, there\nwere thirty-seven earthquakes. The most sensible shock was on the 27th\nof March, when the mountain ejected lava on the western side. This\neruption lasted thirteen days, and part of the Bosco di Castiglione was\ninjured. The mountain smoked one hundred and fifty-two days, flamed\nthree, and detonated eleven. Little thunder.\n\nOn the 16th and 17th of February, 1810, there were four earthquakes.\nOn the 27th of October, Etna was in a state of eruption on the eastern\nside, and the lava flowed into the Valle del Bue. There were about\ntwenty thunder storms.\n\n1811, no earthquakes, but the mountain continued until the 24th of\nApril to eject lava from the east. At this time the Mount St. Simon was\nformed. No thunder.\n\nEarthquake on the 3rd and 13th of March, 1813. The mountain smoked\ntwenty-eight days. On the 30th of June, and on the 5th of August, St.\nSimon smoked. There were twenty-one thunder storms.\n\nOn the 3rd of November, 1814, there was an earthquake, preceded by a\ndischarge of sand from that part of the mountain called Zoccolaro.\nThere were twelve thunder storms.\n\nOn the 6th of September, 1815, there was an earthquake. The mountain\nsmoked forty-two days, and there were eleven thunder storms. On the\n6th, 7th, and 11th of January the lightning was tremendous.\n\n1816, no earthquakes. On the 13th of August a part of the interior side\nof the crater fell in. Ten thunder storms.\n\nThere was an earthquake on the 18th of October, 1817. The mountain\nsmoked twenty-two days. There were eight thunder storms.\n\nDuring 1818 there were twenty-five earthquakes. The most violent was\nin the neighbourhood of Catania, on the 20th of February. The mountain\nsmoked twenty-four days. No thunder.\n\n\nCHARITY INSTEAD OF POMP.\n\nAccording to the \"Annual Register\" for August, 1760, there were\nexpended at the funeral of Farmer Keld, of Whitby, in that year, one\nhundred and ten dozen of penny loaves, eight large hams, eight legs\nof veal, twenty stone of beef (fourteen pounds to the stone), sixteen\nstone of mutton, fifteen stone of Cheshire cheese, and thirty ankers of\nale, besides what was distributed to about one thousand poor people,\nwho had sixpence each in money given them.\n\n\nTHE BEDFORD MISSAL.\n\nOne of the most celebrated books in the annals of bibliography, is the\nrichly illuminated Missal, executed for John, Duke of Bedford, Regent\nof France, under Henry VI.; by him it was presented to that king, in\n1430. This rare volume is eleven inches long, seven and a-half wide,\nand two and a-half thick; contains fifty-nine large miniatures, which\nnearly occupy the whole page, and above a thousand small ones, in\ncircles of about an inch and a-half diameter, displayed in brilliant\nborders of golden foliage, with variegated flowers, etc.; at the bottom\nof every page are two lines in blue and gold letters, which explain the\nsubject of each miniature. This relic, after passing through various\nhands, descended to the Duchess of Portland, whose valuable collection\nwas sold by auction, in 1786. Among its many attractions was the\nBedford Missal. A knowledge of the sale coming to the ears of George\nIII., he sent for his bookseller, and expressed his intention to become\nthe purchaser. The bookseller ventured to submit to his majesty the\nprobable high price it would fetch. \"How high?\" exclaimed the king.\n\"Probably, two hundred guineas,\" replied the bookseller. \"Two hundred\nguineas for a Missal!\" exclaimed the queen, who was present, and\nlifted her hands up with astonishment. \"Well, well,\" said his majesty,\n\"I'll have it still; but since the queen thinks two hundred guineas\nso enormous a price for a Missal, I'll go no further.\" The biddings\nfor the Royal Library did actually stop at that point; a celebrated\ncollector, Mr. Edwards, became the purchaser by adding three pounds\nmore. The same Missal was afterwards sold at Mr. Edwards' sale, in\n1815, and purchased by the Duke of Marlborough, for the enormous sum of\nL637 l5s. sterling.\n\n\nCALICINATED RINGS.\n\n[Illustration [++] Calicinated Ring.]\n\nThere is a particular class of antique gold ornaments, belonging to\nthe Bronze Period, which is deserving of especial attention, from\nthe circumstance that the British Isles is the only locality in\nwhich it has yet been discovered. These ornaments consist of a solid\ncylindrical gold bar, beat into a semicircle or segmental arc, most\nfrequently tapering from the centre, and terminated at both ends with\nhollow cups, resembling the mouth of a trumpet, or the expanded calix\nof a flower. A remarkable example of these curious native relics is\nengraved in the \"Archaeological Journal.\" The cups are formed merely by\nhollows in the slightly dilated ends; but it is further interesting\nfrom being decorated with the style of incised ornaments of most\nfrequent occurrence on the primitive British pottery. It was dug up at\nBrahalish, near Bantry, county Cork, and weighs 3 oz. 5 dwts. 6 grs.\nIn contrast to this, another is engraved in the same journal, found\nnear the entrance lodge at Swinton Park, Yorkshire, scarcely two feet\nbelow the surface. In this beautiful specimen the terminal cups are\nso unusually large, that the solid bar of gold dwindles into a mere\nconnecting-link between them. The annexed figure of a very fine example\nfound by a labourer while cutting peats in the parish of Cromdale,\nInverness-shire, somewhat resembles that of Swinton Park in the size of\nits cups. It is from a drawing by the late Sir Thomas Dick Lauder, and\nrepresents it about one-half the size of the original. Similar relics\nof more ordinary proportions have been brought to light, at different\ntimes, in various Scottish districts.\n\n\nEXTRAORDINARY CRICKET MATCHES.\n\nEvery day in summer wagers are made at Lord's cricket ground, upon\nmatches there to be played; but there have been more extraordinary\nmatches elsewhere relative to this exercising game; for a cricket match\nwas played on Blackheath, in the year 1766, between eleven Greenwich\npensioners who had lost each an arm, and eleven others who had lost\neach a leg. The former won with ease. And again, on the 9th of August,\n1796, a cricket match was played by eleven Greenwich pensioners with\none leg, against eleven with one arm, for one thousand guineas, at the\nnew cricket ground, Montpelier gardens, Walworth. At nine o'clock\nthe men arrived in three Greenwich stages; about twelve the wickets\nwere pitched, and they commenced. Those with but one leg had the first\ninnings, and got 93 runs; those with but one arm got but 42 runs\nduring their innings. The one-leg commenced their second innings, and\nsix were bowled out after they had got 60 runs; so that they left off\none hundred and eleven more than those with one arm. Next morning the\nmatch was played out; and the men with one leg beat the one-arms by\none hundred and three runs. After the match was finished the eleven\none-legged men ran a sweep-stakes of one hundred yards distance for\ntwenty guineas, and the three first had prizes.\n\n\nMUMMY CASES.\n\n[Illustration [++] Mummy Cases.]\n\nThe annexed engraving represents a set of Egyptian mummy cases, several\nof which were used for the interment of one body, the smaller one being\nenclosed within the larger. On the death of a king in Egypt, \"three\nscore and ten days\" was the period that intervened from his departure\nto the termination of the embalming operations; the earlier and more\nimportant of which, exclusive of the soaking in natron, occupied forty\ndays. The coffin, or wooden case, in which the embalmed body of Joseph\nwas preserved, till at the exodus it was carried from Egypt, was,\ndoubtless, of such a form and appearance as those with which we are\nfamiliar at our museums. An account of some specimens of these, and\nof the internal shells which were considered requisite for persons of\nrank, will be read with interest.\n\nBefore the better kind of mummies were put into their wooden cases,\nthey were placed in a shell in the following manner:--Nine thick layers\nof hempen or linen cloth were well gummed together, so as to make a\nstrong flexible kind of board, something like a piece of papier mache.\nThis was formed into the shape of the swathed mummy, which was inserted\nin it by means of a longitudinal aperture on the under side, reaching\nfrom the feet to the head. The two sides of this long aperture were\nthen drawn together by a coarse kind of stitching, done with a large\nneedle and thin hempen cord. The inside of this hempen case was covered\nwith a thin coating of plaster, and the outside was also covered with\na similar sort of plaster, on which were painted rude figures of\nbeetles, ibides, &c., &c., apparently with ochrous earths tempered\nwith water; they could be easily rubbed off with the finger, except\nwhere they were fixed by an outer coating of gum. On the upper part of\nthis case a human face was represented, and for the purpose of giving\nadditional strength and firmness to that part of the hempen covering, a\nconsiderable quantity of earth and plaster was stuck on the inside, so\nthat it would be more easy to mould the material on the outside, while\nstill flexible, into a resemblance of the human form. The face was\ncovered with a strong varnish, to keep the colour fixed. The outer case\nwas generally made of the Egyptian fig-sycamore wood, and the parts of\nit were fastened together with wooden pegs. This wood was used by the\nEgyptians for a variety of purposes, as we find even common domestic\nutensils made of it. The pegs of the sycamore cases were not always of\nthe sycamore wood, which, when cut thin, would hardly be so suitable as\nsome more closely-grained wood; the pegs, therefore, of the inner cases\nwere of a different wood, generally of cedar. Bodies embalmed in the\nhighest style of fashion, had, in addition to the inner coffin which we\nhave described, an outer wooden box, such as Herodotus mentions, with\na human face, male or female, painted on it. Some of these cases were\nplain, and others highly ornamented with figures of sacred animals, or\nwith paintings representing mythological subjects.\n\nThe wooden case which contained the body was sometimes cut out of\none piece of wood, and the inside was made smooth, and fit for the\nreception of the painted figures, by laying on it a thin coat of fine\nplaster. This plaster was also used as a lining for the wooden cases\nwhich were not made of a single piece. There was often a second wooden\ncase, still more highly ornamented and covered with paintings secured\nby a strong varnish. These paintings were intended to embody the ideas\nof the Egyptians as to the state of death, the judgment or trial which\npreceded the admission into the regions below, and other matters\nconnected with the ritual of the dead and the process of embalming.\n\nThe upper part of each of the wooden cases was made to represent a\nhuman figure, and the sex was clearly denoted by the character of\nthe head-dress, and the presence or absence of the beard. Both the\nhead-dress and the ornaments about the neck, as far as the bosom, were\nexactly of the same character as those which we see on the sculptures\nand paintings. The brief remark of Herodotus, that the friends put the\nswathed mummy \"into a wooden figure made to resemble the human form,\"\nis amply borne out.\n\n\nINSTINCT OF ANIMALS.\n\nGall and various observers of animals have fully ascertained that the\nattention of dogs is awakened by our conversation. He brought one\nof these intelligent creatures with him from Vienna to Paris, which\nperfectly understood French and German, of which he satisfied himself\nby repeating before it whole sentences in both languages. A recent\nanecdote has been related of an old ship-dog, that leaped overboard\nand swam to shore on hearing the captain exclaim, \"Poor old Neptune! I\nfear we shall have to drown him!\" and such was the horror which that\nthreat inspired, that he never afterwards would approach the captain\nor any of the ship's company, to whom he had previously been fondly\nattached. It must, however, be observed that in the brute creation, as\nin ours (sometimes more brutal species), peculiar attributes, that do\nnot belong to the race, distinguish individuals gifted with what in\nman we might call a superior intellect, but which in these animals\nshows a superiority of what we term instinct. Spurzheim relates an\ninstance of a cow belonging to Mr. Dupont de Nemours, which, amongst\nthe whole kindred herd, was the only one that could open the gate\nleading to their pastures; and her anxious comrades, when arriving\nat the wished-for spot, invariably lowed for their conductor. It is\nalso related of a hound, who, unable to obtain a seat near the fire\nwithout the risk of quarrelling with the dozing occupants that crowded\nthe hearth, was wont to run out into the court-yard barking an alarm\nthat brought away his rivals in comfort, when he quietly re-entered\nthe parlour, and selected an eligible stretching-place. This animal\ndisplayed as much ingenuity as the traveller who, according to the\nwell-known story, ordered oysters for his horse for the purpose of\nclearing the fireside.\n\n\nBELL OF ST. MURA.\n\n[Illustration [++] Bell of St. Mura.]\n\nThis curious relic, engraved over leaf, two-thirds the size of the\noriginal, is remarkable as a work of art, as well as a genuine\nrelic of the most venerable antiquity; it was formerly regarded\nwith superstitious reverence in Ireland, and any liquid drunk from\nit was believed to have peculiar properties in alleviating human\nsuffering; hence, the peasant women of the district in which it was\nlong preserved, particularly used it in cases of child-birth, and a\nserious disturbance was excited on a former attempt to sell it by its\nowner. Its legendary history relates that it descended from the sky\nringing loudly; but as it approached the concourse of people who had\nassembled at the miraculous warning, the tongue detached itself and\nreturned towards the skies; hence it was concluded that the bell was\nnever to be profaned by sounding on earth, but was to be kept for\npurposes more holy and beneficent. This is said to have happened on\nthe spot where once stood the famous Abbey of Fahan, near Innishowen\n(County Donegal), founded in the seventh century by St. Mura, or\nMuranus, during the reign of Abodh Slaine. For centuries this abbey\nwas noted as the depository of various valuable objects, which were\nheld in especial veneration by the people. Amongst these were several\ncurious manuscripts written by St. Mura, his crozier, and this bell;\nwhich ultimately came into the possession of a poor peasant residing\nat Innishowen, who parted with it to Mr. Brown, of Beaumaris, at whose\nsale in 1855 it was purchased by Lord Londesborough. The material of\nthe bell is bronze, and its form quadrangular, resembling other ancient\nIrish bells, and leading to the conclusion that it is the genuine\nwork of the seventh century. The extreme feeling of veneration shown\ntowards it in various ages is proved by the ornament with which it is\nencased. By the accidental removal of one portion of the outer casing,\na series of earlier enrichments were discovered beneath, which were\nmost probably placed there in the ninth century. The portion disclosed\n(the lower right hand corner) consists of a tracery of Runic knots\nwrought in brass, and firmly attached to the bell by a thin plate of\ngold;--whether the remainder of these early decorations, now concealed,\nbe similar, cannot be determined without removing the outer plates.\nThese exterior ornaments consist of a series of detached silver plates\nof various sizes diversely embossed in the style known to have\nprevailed in the eleventh century. The centre is adorned with a large\ncrystal, and smaller gems have once been set in other vacant sockets\naround it, only one of amber remaining. The two large spaces in front\nof the arched top were also most probably filled with precious stones,\nas the gold setting still remains entire. The best workmanship has been\ndevoted to these decorations; the hook for suspending the bell is of\nbrass, and has been covered with early bronze ornament which has been\nfilled in with niello, the intervening space being occupied by silver\nplates ornamented like the rest of the later decorations which cover\nits surface. From the absence of any traces of rivets on the back\nor sides of the bell, the decoration it has received may have been\nrestricted to the casing of the handle and the enrichment of the front\nof this venerated relic.\n\n\nCURIOUSLY-SHAPED DRINKING CUP.\n\nDrinking cups of a fantastic shape were very much in vogue in the\nsixteenth century. Sometimes they assumed the shape of birds, sometimes\nof animals. In general it is the head that takes off, and serves as\na lid or cover; but sometimes the orifice is in another part of the\nbody, as, for example, on the back. The specimen now before us is from\nLord Londesborough's collection.\n\n[Illustration [++] Drinking Cup in the Shape of a Stag.]\n\nThe stag is of silver, gilt all over; the collar set with a garnet.\nSilver bands encircle this curious figure, to which are appended\nmany small silver escutcheons engraved with the arms and names of\ndistinguished officers of the Court of Saxe Gotha, the latest being\n\"Her Von Maagenheim, Camer Juncker und Regierung Assessor in Gotha,\nd. 15 Augusti, A{o} 1722.\" It has probably been a prize for shooting,\nsuccessively won by those persons whose arms decorate it.\n\n\nBANQUETS TO QUEEN ELIZABETH.\n\nFew English sovereigns were so well acquainted with their dominions\nas was Queen Elizabeth: she may be said to have visited every corner\nof her empire, and in these royal journeys or \"progresses,\" as they\nare called, her loyal subjects strove to outvie each other in the\nsplendour of their receptions. Nothing could surpass the magnificence\nof the entertainments thus planned for the queen's gratification,\neither as respects the splendour of show, or the costliness of the\nmore substantial banquet. These occasions are too numerous to mention;\nand we can only notice one of the queen's visits to the palace at\nGreenwich, as described by a German, who travelled in England in 1598.\nIt was Sunday, and after attending service in the chapel, the queen\nprepared for dinner. A gentleman entered the room bearing a rod, and\nwith him another bearing a table-cloth, which, after they had both\nkneeled three times, he spread upon the table, and after kneeling\nagain, they both retired: then came two others, one with the rod\nagain, the other with a salt-cellar, a plate, and bread, which, after\nkneeling, they also placed on the table: then came an unmarried and a\nmarried lady, bearing a tasting-knife, and having stooped three times\ngracefully, they rubbed the table with bread and salt. Then came the\nyeomen of the guard, bringing in, at each time, a course of dishes,\nserved in plate, most of it gilt; these dishes were received by a\ngentleman, and placed upon the table, while the lady-taster gave to\neach guard a mouthful to eat of the particular dish he had brought, for\nfear of any poison. During the time that this guard (which consisted\nof the tallest and stoutest men that could be found in all England,\nbeing carefully selected for this service) were bringing dinner, twelve\ntrumpets and two kettle-drums made the hall ring for half an hour\ntogether. After this a number of unmarried ladies appeared, who lifted\nthe meat from the table, and conveyed it to the queen's inner and more\nprivate chamber, where, after she had chosen for herself, the rest was\nsent to the ladies of the court. The queen dined and supped alone, with\nvery few attendants.\n\n\nTHE GREAT FOG OF 1783.\n\nIt prevailed over the adjoining continent, and produced much fear that\nthe end of all things was at hand. It appeared first at Copenhagen on\nthe 29th of May, reached Dijon on the 14th June, and was perceived\nin Italy on the 16th. It was noticed at Spydberg, in Norway, on the\n22nd, and at Stockholm two days later; the following day it reached\nMoscow. On the 23rd it was felt on the St. Gothard, and at Buda. By\nthe close of that month it entered Syria; and on the 18th of July,\nreached the Altai Mountains. Before its appearance at these places the\ncondition of the atmosphere was not similar; for in this country it\nfollowed continued rains; in Denmark it succeeded fine weather of some\ncontinuance; and in other places it was preceded by high winds. The sun\nat noon looked rusty-red, reminding one of the lines of Milton. The\nheat was intense during its continuance, and the atmosphere was highly\nelectric. Lightnings were awfully vivid and destructive. In England\nmany deaths arose from this cause, and a great amount of property was\nlost. In Germany public edifices were thrown down or consumed by it;\nand in Hungary one of the chief northern towns was destroyed by fires,\ncaused by the electric fluid, which struck it in nine different places.\nIn France there were hailstones and violent winds. In Silesia there\nwere great inundations. The dry fogs of 1782-83 were accompanied by\ninfluenza; at St. Petersburgh 40,000 persons were immediately attacked\nby it, after the thermometer had suddenly risen 30 degrees. Calabria\nand Sicily were convulsed by earthquakes; in Iceland a volcano was\nactive, and about the same time one sprung out of the sea off Norway.\nThe co-existence of dry fogs with earthquakes and volcanic eruptions\nhad been previously observed--_e.g._, in the years 526, 1348, 1721; and\nsince then, in 1822 and 1834.\n\nA somewhat similar fog overspread London before the cholera of 1831,\nand the influenza of 1847. Hecker (\"Epidemics of the Middle Ages\")\nhas collected notices of various phenomena of this kind, which\nhave preceded the great continental plagues, and have often been\ncharacterised by offensive odours.\n\n\nMONKEYS DEMANDING THEIR DEAD.\n\nMr. Forbes tells a story of a female monkey (the Semnopithecus\nEntellus) who was shot by a friend of his, and carried to his tent.\nForty or fifty of her tribe advanced with menacing gestures, but stood\nstill when the gentleman presented his gun at them. One, however, who\nappeared to be the chief of the tribe, came forward, chattering and\nthreatening in a furious manner. Nothing short of firing at him seemed\nlikely to drive him away; but at length he approached the tent door\nwith every sign of grief and supplication, as if he were begging for\nthe body. It was given to him, he took it in his arms, carried it away,\nwith actions expressive of affection, to his companions, and with them\ndisappeared. It was not to be wondered at that the sportsman vowed\nnever to shoot another monkey.\n\n\nBARA.\n\nMr. Howel, in his descriptive travels through Sicily, gives a\nparticular account of the magnificent manner in which the festival of\nthe Assumption of the Virgin is kept by the Sicilians under the title\nof Bara, which, although expressive of the machine he describes, is\nalso, it appears, generally applied as a name of the feast itself.\nAn immense machine of about 50 feet high is constructed, designing\nto represent Heaven; and in the midst is placed a young female\npersonating the Virgin, with an image of Jesus on her right hand;\nround the Virgin 12 little children turn vertically, representing so\nmany Seraphim, and below them 12 more children turn horizontally,\nas Cherubim; lower down in the machine a sun turns vertically, with\na child at the extremity of each of the four principal radii of his\ncircle, who ascend and descend with his rotation, yet always in an\nerect posture; and still lower, reaching within about 7 feet of the\nground, are placed 12 boys, who turn horizontally without intermission\naround the principal figure, designing thereby to exhibit the 12\napostles, who were collected from all corners of the earth, to be\npresent at the decease of the Virgin, and witness her miraculous\nassumption. This huge machine is drawn about the principal streets by\nsturdy monks; and it is regarded as a particular favour to any family\nto admit their children in this divine exhibition.\n\n\nCRADLE OF HENRY V.\n\nMost of our readers have probably seen, in the illustrated newspapers\nof the day, sketches of the magnificently artistic cradles which\nhave been made for the children of our good Queen, or for the Prince\nImperial of France. It will be not a little curious to contrast with\nthose elaborately beautiful articles the cradle of a Prince of Wales in\nthe fourteenth century. We here give a sketch of it.\n\n[Illustration [++] Cradle of Henry V.]\n\nIt was made for the use of Henry Prince of Wales, afterwards King\nHenry V, generally called Henry of Monmouth, because he was born in\nthe castle there in the year 1388. He was the son of Henry IV of\nBolingbroke, by his first wife Mary de Bohun. He was educated at\nQueen's College, Oxford, under the superintendence of his half uncle,\nthe great Cardinal Henry Beaufort. On the accession of his father to\nthe throne, he was created Prince of Wales, and, at the early age\nof sixteen, was present at the battle of Shrewsbury, where he was\nbadly wounded in the face. After having greatly distinguished himself\nin the war against Owen Glendour, he spent some years idleness and\ndissipation, but on his coming to the throne, by the death of his\nfather, April 20, 1413, he threw off his former habits and associates,\nchose his ministers from among those of tried integrity and wisdom in\nhis father's cause, and seemed everywhere intent on justice, on victory\nover himself, and on the good of his subjects. After a short but\nglorious reign of ten years, in which the victory of Agincourt was the\nprincipal event, he expired at the Bois de Vincennes, near Paris, on\nthe last day of August, 1422, in the thirty-fourth year of his age. He\nwas engaged at the time in a war with the Dauphin of France. His heart\nwas warm as his head was cool, and his courage equal to his wisdom,\nwhich emboldened him to encounter the greatest dangers, and surmount\nthe greatest difficulties His virtues were not inferior to his\nabilities, being a dutiful son, a fond parent, an affectionate brother,\na steady and generous friend, and an indulgent master. In a word,\nHenry V., though not without his failings, merits the character of an\namiable and accomplished man, and a great and good king. Such was the\nsovereign, for whose infant years the plain, but still not tasteless,\ncradle was made, which we have here engraved, as it is preserved in the\ncastle of Monmouth, his birthplace.\n\n\nTHE FONT AT KILCARN.\n\nThe venerable old church at Kilcarn, near Navan, in the county of\nMeath, contains a font of great rarity, and we have selected it as a\nfitting object for our work, inasmuch as it is a striking instance of\nthe union of the beautiful with the curious.\n\n[Illustration [++] Font at Kilcarn.]\n\nPlaced upon its shaft, as represented in the cut, it measures in height\nabout three feet six inches; the basin is two feet ten inches in\ndiameter, and thirteen inches deep. The heads of the niches, twelve in\nnumber, with which its sides are carved, are enriched with foliage of\na graceful but uniform character, and the miniature buttresses which\nseparate the niches are decorated with crockets, the bases resting upon\nheads, grotesque animals, or human figures, carved as brackets. The\nfigures within the niches are executed with a wonderful degree of care,\nthe drapery being represented with each minute crease or fold well\nexpressed. They are evidently intended to represent Christ, the Virgin\nMary, and the twelve apostles. All the figures are seated. Our Saviour,\ncrowned as a King, and holding in his hand the globe and cross, is in\nthe act of blessing the Virgin, who also is crowned, the \"Queen of\nHeaven.\" The figures of most of the apostles can easily be identified:\nSaint Peter by his key; Saint Andrew by his cross of peculiar shape;\nand so on. They are represented barefooted, and each holds a book in\none hand.\n\n\nTHE BLOOD-SUCKING VAMPIRE.\n\nCaptain Stedman, who travelled in Guiana, from 1772 to 1777, published\nan account of his adventures, and for several years afterwards it\nwas the fashion to doubt the truth of his statements. In fact, it\nwas a general feeling, up to a much later period than the above,\nthat travellers were not to be believed. As our knowledge, however,\nhas increased, and the works of God have been made more manifest,\nthe reputation of many a calumniated traveller has been restored,\nand, among others, that of Captain Stedman. We shall, therefore,\nunhesitatingly quote his account of the bite of the vampire:--\"On\nwaking, about four o'clock this morning, in my hammock, I was\nextremely alarmed at finding myself weltering in congealed blood,\nand without feeling any pain whatever. Having started up and run to\nthe surgeon, with a firebrand in one hand, and all over besmeared\nwith gore, the mystery was found to be, that I had been bitten by the\nvampire or spectre of Guiana, which is also called the flying dog\nof New Spain. This is no other than a bat of monstrous size, that\nsucks the blood from men and cattle, sometimes even till they die;\nknowing, by instinct, that the person they intend to attack is in a\nsound slumber, they generally alight near the feet, where, while the\ncreature continues fanning with his enormous wings, which keeps one\ncool, he bites a piece out of the tip of the great toe, so very small\nindeed, that the head, of a pin could scarcely be received into the\nwound, which is consequently not painful; yet, through this orifice he\ncontrives to suck the blood until he is obliged to disgorge. He then\nbegins again, and thus continues mucking and disgorging till he is\nscarcely able to fly, and the sufferer has often been known to sleep\nfrom time into eternity. Cattle they generally bite in the ear, but\nalways in those places where the blood flows spontaneously. Having\napplied tobacco-ashes as the best remedy, and washed the gore from\nmyself and my hammock, I observed several small heaps of congealed\nblood all around the place where I had lain upon the ground; upon\nexamining which, the surgeon judged that I had lost at least twelve or\nfourteen ounces during the night. Having measured this creature (one of\nthe bats), I found it to be, between the tips of the wings, thirty-two\ninches and a-half; the colour was a dark brown, nearly black, but\nlighter underneath.\"\n\n\nLUXURY IN 1562.\n\nThe luxury of the present times does not equal, in one article at\nleast, that of the sixteenth century. Sir Nicholas Throckmorton, the\nQueen's ambassador at Paris, in a letter to Sir Thomas Chaloner, the\nambassador at Madrid, in June, 1562, says,\n\n\"I pray you good my Lord Ambassador sende me two paire of parfumed\ngloves, parfumed with orrange flowers and jacemin, th'one for my\nwives hand, the other for mine owne; and wherin soever I can pleasure\nyou with any thing in this countrey, you shall have it in recompence\nthereof, or els so moche money as they shall coste you; provided\nalwaies that they be of the best choise, wherein your judgment is\ninferior to none.\"\n\n\nSINGULAR PHENOMENON--PHOSPHORESCENCE OF THE SEA.\n\nThe sea has sometimes a luminous appearance, a phenomenon that has\nbeen observed by all sailors, who consider it the forerunner of windy\nweather. It is said to occur most frequently in the summer and autumn\nmonths, and varies so much in its character, as to induce a doubt\nwhether it can always be attributed to the same cause. Sometimes the\nluminous appearance is seen over the whole surface of the water, and\nthe vessel seems as though floating upon an ocean of light. At other\ntimes, the phosphorescence is only seen immediately around the ship.\nA portion of water taken from the sea does not necessarily retain its\nluminous appearance, but its brilliance will generally continue as long\nas the water is kept in a state of agitation. Some naturalists imagine\nthe phosphorescence of the sea to arise from the diffusion of an\nimmense number of animalculae through the medium, and others attribute\nit to electricity. Dr. Buchanan has given an account of a very\nremarkable appearance of the sea, observed by him during a voyage from\nJohanna to Bombay. About eight o'clock in the evening of the 31st of\nJuly, 1785, the sea had a milk-white colour, and upon it were floating\na multitude of luminous bodies greatly resembling that combination of\nstars known as the milky way, the brightest of them representing the\nlarger stars of a constellation. The whiteness, he says, was such as\nto prevent those on board from seeing either the break or swell of\nthe sea, although, from the motion of the ship and the noise, they\nknew them to be violent, and the light was sufficiently intense to\nilluminate the ropes and rigging. This singular phenomenon continued\ntill daylight appeared. Several buckets of water were drawn, and in\nthem were found a great number of luminous bodies, from a quarter of an\ninch to an inch and a half in length, and these were seen to move about\nas worms in the water. There might be, he said to Dr. Buchanan, four\nhundred of these animals in a gallon of water. A similar appearance had\nbeen observed before in the same sea by several of the officers, and\nthe gunner had seen it off Java Head, in a voyage to China.\n\n\nMARRIAGE VOW.\n\nThe matrimonial ceremony, like many others, has undergone some\nvariation in the progress of time. Upwards of three centuries ago,\nthe husband, on taking his wife, as now, by the right hand, thus\naddressed her:--\"I. N. _undersygne_ the N. for my wedded wyfe, for\nbetter, for worse, for richer, for porer, yn sickness, and in helthe,\ntyl dethe us departe, (not \"do part,\" as we have erroneously rendered\nit, the ancient meaning of \"departe,\" even in Wickliffe's time, being\n\"separate\") as holy churche hath ordeyned, and thereto I plygth the my\ntrowthe.\" The wife replies in the same form, with an additional clause,\n\"to be buxom to the, tyl dethe us departe.\" So it appears in the first\nedition of the \"Missals for the use of the famous and celebrated Church\nof Hereford, 1502,\" fol. In what is called the \"Salisbury Missal,\" the\nlady pronounced a more general obedience: \"to be bonere and buxom in\nbedde and at the borde.\"\n\n\nLOVE OF GARDENS.\n\nLouis XVIII., on his restoration to France, made, in the park in\nVersailles, the _facsimile_ of the garden at Hartwell; and there\nwas no more amiable trait in the life of that accomplished prince.\nNapoleon used to say that he should know his father's garden in Corsica\nblindfolded, by the smell of the earth! And the hanging-gardens\nof Babylon are said to have been raised by the Median Queen of\nNebuchadnezzar on the flat and naked plains of her adopted country,\nto remind her of the hills and woods of her childhood. We need not\nspeak of the plane-trees of Plato--Shakspeare's mulberry-tree--Pope's\nwillow--Byron's elm? Why describe Cicero at his Tusculum--Evelyn at\nWotton--Pitt at Ham Common--Walpole at Houghton--Grenville at Dropmere?\nWhy dwell on Bacon's \"little tufts of thyme,\" or Fox's geraniums? There\nis a spirit in the garden as well as in the wood, and the \"lilies of\nthe field\" supply food for the imagination as well as materials for\nsermons.\n\n\nANCIENT DANISH SHIELD.\n\n[Illustration [++] Ancient Danish Shield.]\n\nIn Asia, from whence the greater number, probably all, of the European\nnations have migrated, numerous implements and weapons of copper\nhave been discovered in a particular class of graves; nay, in some\nof the old and long-abandoned mines in that country workmen's tools\nhave been discovered, made of copper, and of very remote antiquity.\nWe see, moreover, how at a later period attempts were made to harden\ncopper, and to make it better suited for cutting implements by a slight\nintermixture, and principally of tin. Hence arose that mixed metal\nto which the name of \"bronze\" has been given. Of this metal, then,\nthe Northmen of \"the bronze period\" formed their armour, and among\nnumerous other articles, three shields have been discovered which\nare made wholly of bronze; and we here give a sketch of the smallest\nof them, which is about nineteen inches in diameter, the other two\nbeing twenty-four. These shields are formed of somewhat thin plates\nof bronze, the edge being turned over a thick wire metal to prevent\nthe sword penetrating too deeply. The handle is formed of a cross-bar,\nplaced at the reverse side of the centre boss, which is hollowed out\nfor the purpose of admitting the hand.\n\n\nSACRED GARDENS.\n\nThe origin of sacred gardens among the heathen nations may be traced\nup to the garden of Eden. The gardens of the Hesperides, of Adonis, of\nFlora, were famous among the Greeks and Romans. \"The garden of Flora,\"\nsays Mr. Spence (Polymetis, p. 251), \"I take to have been the Paradise\nin the Roman mythology. The traditions and traces of Paradise among the\nancients must be expected to have grown fainter and fainter in every\ntransfusion from one people to another. The Romans probably derived\ntheir notions of it from the Greeks, among whom this idea seems to have\nbeen shadowed out under the stories of the gardens of Alcinous. In\nAfrica they had the gardens of the Hesperides, and in the East those of\nAdonis, or the _Horti Adonis_, as Pliny calls them. The term _Horti\nAdonides_ was used by the ancients to signify gardens of pleasure,\nwhich answers to the very name of Paradise, or the garden of Eden, as\n_Horti Adonis_ does to the garden of the Lord.\"\n\n\nANCIENT CHAIR OF DAGOBERT.\n\n[Illustration [++] Chair of Dagobert.]\n\nThe chair which we here engrave claims to be regarded as a great\ncuriosity, on two separate grounds: it is the work of an artist who\nwas afterwards canonized, and it was used by Napoleon I. on a most\nimportant occasion. Towards the close of the sixth century the artists\nof France were highly successful in goldsmith's work, and Limoges\nappears to have been the principal centre of this industry. It was at\nthis time that Abbon flourished--a goldsmith and mint-master, with whom\nwas placed the young Eloy, who rose from a simple artizan to be the\nmost remarkable man of his century, and whose virtues were rewarded by\ncanonization. The apprentice soon excelled his master, and his fame\ncaused him to be summoned to the throne of Clotaire II., for whom\nhe made two thrones of gold, enriched with precious stones, from a\nmodel made by the king himself, who had not been able to find workmen\nsufficiently skilful to execute it. The talents and probity of St.\nEloy also gained him the affection of Dagobert I., who entrusted him\nwith many important works, and among them, with the construction of\nthe throne, or chair of state which is the subject of this article. It\nis made of bronze, carved and gilded, and is a beautiful specimen of\nworkmanship. The occupant of the chair would sit upon a cloth of gold\nsuspended from the two side bars. For a long time it was preserved in\nthe sacristy of the royal church of St. Denis, at Paris; but it was\nsubsequently removed to the Great Library, where it now is. It was upon\nthis chair that Napoleon I., in August, 1804, distributed the crosses\nof the Legion of Honour to the soldiers of the army assembled at\nBoulogne for the invasion of England. Napoleon caused the chair to be\nbrought from Paris for the express purpose.\n\n\nST GEORGE'S CAVERN.\n\nNear the town of Moldavia, on the Danube, is shown the cavern where St.\nGeorge slew the Dragon, from which, at certain periods, issue myriads\nof small flies, which tradition reports to proceed from the carcass of\nthe dragon. They respect neither man nor beast, and are so destructive\nthat oxen and horses have been killed by them. They are called the\nGolubacz's fly. It is thought when the Danube rises, as it does in\nthe early part of the summer, the caverns are flooded, and the water\nremaining in them, and becoming putrid, produces this noxious fly. But\nthis supposition appears to be worthless, because, some years ago, the\nnatives closed up the caverns, and still they were annoyed with the\nflies. They nearly resemble mosquitoes. In summer they appear in such\nswarms as to look like a volume of smoke; and they sometimes cover a\nspace of six or seven miles. Covered with these insects, horses not\nunfrequently gallop about until death puts an end to their sufferings.\nShepherds anoint their hands with a decoction of wormwood, and keep\nlarge fires burning to protect themselves from them. Upon any material\nchange in the weather the whole swarm is destroyed thereby.\n\n\nENGLISH LETTER BY VOLTAIRE.\n\nThe subjoined letter is copied literally from the autograph of\nVoltaire, formerly in the possession of the Rev. Mr. Sim, the editor of\nMickle's Poems:--\n\n \"Sir,\n\n\"j wish you good health, a quick sale of y{r} burgundy, much latin, and\ngreeke to one of y{r} Children, much Law, much of cooke, and littleton,\nto the other. quiet and joy to mistress brinsden, money to all. when\nyou'll drink y{r} burgundy with m{r} furneze pray tell him j'll never\nforget his favours.\n\nBut dear john be so kind as to let me know how does my lady\nBollingbroke. as to my lord j left him so well j dont doubt he is so\nstill. but j am very uneasie about my lady. if she might have as much\nhealth as she has Spirit and witt, sure She would be the strongest body\nin england. pray dear s{r} write me Something of her, of my lord, and\nof you. direct y{r} letter by the penny post at m{r} Cavalier, Belitery\nSquare by the R. exchange. j am sincerely and heartily y{r} most humble\nmost obedient rambling friend\n\n \"VOLTAIRE.\n\n \"to\n \"john Brinsden, esq.\n \"durham's yard\n \"by charing cross.\"\n\n\nTHE GOLDEN CHALICE OF IONA.\n\nA chalice, as used in sacred ceremonies, is figured on various early\nScottish ecclesiastical seals, as well as on sepulchral slabs and other\nmedieval sculptures. But an original Scottish chalice, a relic of\nthe venerable abbey of St. Columba, presented, till a very few years\nsince, an older example of the sacred vessels of the altar than is\nindicated in any existing memorial of the medieval Church. The later\nhistory of this venerable relic is replete with interest. It was of\nfine gold, of a very simple form, and ornamented in a style that gave\nevidence of its belonging to a very early period. It was transferred\nfrom the possession of Sir Lauchlan MacLean to the Glengarry family,\nin the time of AEneas, afterwards created by Charles II. Lord Macdonell\nand Arross, under the circumstances narrated in the following letter\nfrom a cousin of the celebrated Marshal Macdonald, Duke of Tarentum,\nand communicated by a clergyman (Rev. AEneas M'Donell Dawson), who\nobtained it from the family of the gentleman to whom it was originally\naddressed:--\n\n\"The following anecdote I heard from the late bishop, John Chisholm,\nand from Mr. John M'Eachan, uncle to the Duke of Tarentum, who died at\nmy house at Irin Moidart, aged upwards of one hundred years:--\n\n\"Maclean of Duart, expecting an invasion of his lands in Mull, by\nhis powerful neighbour the Earl of Argyll, applied to Glengarry for\nassistance. AEneas of Glengarry marched at the head of five hundred men\nto Ardtornish, nearly opposite to Duart Castle, and crossing with a few\nof his officers to arrange the passage of the men across the Sound of\nMull, Maclean, rejoicing at the arrival of such a friend, offered some\nchoice wine in a golden chalice, part of the plunder of Iona. Glengarry\nwas struck with horror, and said, folding his handkerchief about the\nchalice, 'Maclean, I came here to defend you against mortal enemies,\nbut since, by sacrilege and profanation, you have made God your enemy,\nno human means can serve you.' Glengarry returned to his men, and\nMaclean sent the chalice and some other pieces of plate belonging\nto the service of the altar, with a deputation of his friends, to\npersuade him to join him; but he marched home. His example was followed\nby several other chiefs, and poor Maclean was left to compete,\nsingle-handed, with his powerful enemy.\"\n\nSuch was the last historical incident connected with the golden\nchalice of Iona, perhaps, without exception, the most interesting\necclesiastical relic which Scotland possessed. Unfortunately its\nlater history only finds a parallel in that of the celebrated Danish\ngolden horns. It was preserved in the charter-chest of Glengarry,\nuntil it was presented by the late Chief to Bishop Ronald M'Donald, on\nwhose demise it came into the possession of his successor, Dr. Scott,\nBishop of Glasgow. Only a few years since the sacristy of St. Mary's\nRoman Catholic Church in that city, where it was preserved was broken\ninto, and before the police could obtain a clue to the depredators,\nthe golden relic of Iona was no longer a chalice. Thus perished, by\nthe hands of a common felon, a memorial of the spot consecrated by\nthe labours of some of the earliest Christian missionaries to the\nPagan Caledonians, and which had probably survived the vicissitudes\nof upwards of ten centuries. In reply to inquiries made as to the\nexistence of any drawing of the chalice, or even the possibility of a\ntrustworthy sketch being executed from memory, a gentleman in Glasgow\nwrites:--\"I have no means of getting even a sketch from which to make a\ndrawing. Were I a good hand myself, I could easily furnish one, having\noften examined it. It was a chalice that no one could look on without\nbeing convinced of its very great antiquity. The workmanship was rude,\nthe ornamental drawings or engravings even more hard than medieval ones\nin their outlines, and the cup bore marks of the original hammering\nwhich had beaten it into shape.\"\n\n\nNEW MODE OF REVENGE.\n\nMonkeys in India are more or less objects of superstitious reverence,\nand are, consequently, seldom or ever destroyed. In some places they\nare even fed, encouraged, and allowed to live on the roofs of the\nhouses. If a man wish to revenge himself for any injury committed\nupon him, he has only to sprinkle some rice or corn upon the top of\nhis enemy's house, or granary, just before the rains set in, and the\nmonkeys will assemble upon it, eat all they can find outside, and then\npull off the tiles to get at that which falls through the crevices.\nThis, of course, gives access to the torrents which fall in such\ncountries, and house, furniture, and stores are all ruined.\n\n\nCURIOUS SUPERSTITION.\n\nThe ring of which we here give a sketch has been selected by us as\na subject for engraving and comment, because it embodies a curious\nsuperstition which was very prevalent in England in the fifteenth and\nsixteenth centuries.\n\n[Illustration [++] Fifteenth Century Ring.]\n\nThe setting is of silver, and the jewel which it carries is called\na toadstone. This stone was popularly believed to be formed in the\nheads of very old toads, and it was eagerly coveted by sovereigns,\nand by all persons in high office, because it was supposed to have\nthe power of indicating to the person who wore it the proximity of\npoison, by perspiring and changing colour. Fenton, who wrote in 1569,\nsays--\"There is to be found in the heads of old and great toads a\nstone they call borax or stelon;\" and he adds--\"They, being used as\nrings, give forewarning against venom.\" Their composition is not\nactually known; by some they are thought to be a stone--by others,\na shell; but of whatever they may be formed, there is to be seen in\nthem, as may be noticed in the engraving, a figure resembling that of\na toad, but whether produced accidentally or by artificial means is\nnot known, though, according to Albertus Magnus, the stone always bore\nthe figure on its surface, at the time it was taken out of the toad's\nhead. Lupton, in his \"1000 Notable Things,\" says--\"A toadstone, called\ncrepaudina, touching any part envenomed, hurt, or stung with rat,\nspider, wasp, or any other venomous beast, ceases the pain or swelling\nthereof.\" The well known lines in Shakespeare are doubtless in allusion\nto the virtue which Lupton says it possesses:--\n\n \"Sweet are the uses of adversity;\n Which like a toad, ugly and venomous,\n Wears yet a precious jewel in his head.\"\n\nBen Jonson also in the _Fox_, has,--\n\n \"Were you enamoured on his copper rings,\n His saffron jewel, with the loadstone in't?\"\n\nAnd Lyly, in his _Euphues_--\n\n \"The foule toad hath a faire stone in his head.\"\n\nThe ring we have engraved is a work of the fifteenth century; it forms\none of the many rare curiosities of the Londesborough Collection, and\nis considered to be a very perfect specimen.\n\n\nANCIENT ARMLET.\n\nIn May, 1840, some workmen were employed at Everdale, near Preston,\nin carrying earth to replace the soil which had been washed away from\nbehind a wall formerly built to protect the banks of the river Ribble.\nIn digging for this purpose, they discovered, at a distance of about\nforty yards from the banks, a great number of articles, consisting\nof ingots of silver, a few ornaments, some silver armlets, and a\nlarge quantity of coins. An attentive examination of all these, and\nespecially of the coins, leads to the conclusion that this mass of\ntreasure was deposited about the year 910, and the articles must be\nconsidered such as were worn at the time of King Alfred, or perhaps\nsomewhat earlier.\n\n[Illustration [++] Ancient Armlet.]\n\nThe armlets, which were all of silver, vary in breadth from a quarter\nof an inch to an inch and a quarter, and perhaps more. They are\ngenerally ornamented, and almost all the ornaments are produced by\npunching with tools of various forms. The patterns are numerous, but\nthe forms of the punches are very few, the variations being produced by\ncombining the forms of more punches than one, or by placing the same\nor differently-formed punches at a greater or less distance from each\nother, or by varying their direction. In the specimen which we have\nhere engraved the punch has had a small square end, and the ornament is\nformed by a series of blows in transverse or oblique lines. Patterns of\nthe period and localities to which these ornaments belong are scarcely\never found finished by casting or chasing. It would appear, also that\nthe use of solder to unite the various parts of objects was either\nlittle known or little practised; for the ends of these ornaments are\ntied together, and, upon other occasions where union is necessary,\nrivets are employed.\n\n\nCHINESE MIRRORS.\n\nThere is a puzzling property in many of the Chinese mirrors which\ndeserves particular notice, and we may give it, together with the\nsolution furnished by Sir David Brewster:--\"The mirror has a knob in\nthe centre of the back, by which it can be held, and on the rest of\nthe back are stamped in relief certain circles with a kind of Grecian\nborder. Its polished surface has that degree of convexity which gives\nan image of the face half its natural size; and its remarkable property\nis, that, when you reflect the rays of the sun from the polished\nsurface, the image of the ornamental border and circles stamped upon\nthe back, is seen distinctly reflected on the wall,\" or on a sheet of\npaper. The metal of which the mirror is made appears to be what is\ncalled Chinese silver, a composition of tin and copper, like the metal\nfor the specula of reflecting telescopes. The metal is very sonorous.\nThe mirror has a rim (at the back) of about 1-4th or 1-6th of an inch\nbroad, and the inner part, upon which the figures are stamped, is\nconsiderably thinner.\n\n\"Like all other conjurors (says Sir David Brewster), the artist has\ncontrived to make the observer deceive himself. The stamped figures on\nthe back are used for this purpose. The spectrum in the luminous area\n_is not an image of the figures on the back_. The figures are a copy\nof the picture which the artist has _drawn on the face of the mirror_,\nand so concealed by polishing, that it is invisible in ordinary lights,\nand can be brought out only in the sun's rays. Let it be required,\nfor example, to produce the dragon as exhibited by one of the Chinese\nmirrors. When the surface of the mirror is ready for polishing, the\nfigure of the dragon may be delineated upon it in extremely shallow\nlines, or it may be eaten out by an acid much diluted, so as to remove\nthe smallest possible portion of the metal. The surface must then be\nhighly polished, not upon pitch, like glass and specula, because this\nwould polish away the figure, but upon cloth, in the way that lenses\nare sometimes polished. In this way the sunk part of the shallow lines\nwill be as highly polished as the rest, and the figure will only be\nvisible in very strong lights, by reflecting the sun's rays from the\nmetallic surface.\"\n\n\nTHE CADENHAM OAK.\n\nAmongst the many remarkable trees in the New Forest in Hampshire, is\none called the Cadenham Oak, which buds every year in the depth of\nwinter. Gilpin says, \"Having often heard of this oak, I took a ride to\nsee it on the 29th of December, 1781. It was pointed out to me among\nseveral other oaks, surrounded by a little forest stream, winding round\na knoll on which they stood. It is a tall straight plant, of no great\nage, and apparently vigorous, except that its top has been injured,\nfrom which several branches issue in the form of pollard shoots. It\nwas entirely bare of leaves, as far as I could discern, when I saw it,\nand undistinguishable from the other oaks in its neighbourhood, except\nthat its bark seemed rather smoother, occasioned, I apprehended, only\nby frequent climbing. Having had the account of its early budding\nconfirmed on the spot, I engaged one Michael Lawrence, who kept the\nWhite Hart, a small alehouse in the neighbourhood, to send me some of\nthe leaves to Vicar's Hill, as soon as they should appear. The man,\nwho had not the least doubt about the matter, kept his word, and sent\nme several twigs on the morning of the 5th of January, 1782, a few\nhours after they had been gathered. The leaves were fairly expanded,\nand about an inch in length. From some of the buds two leaves had\nunsheathed themselves, but in general only one. One of its progeny,\nwhich grew in the gardens at Bulstrode, had its flower buds perfectly\nformed so early as the 21st of December, 1781.\n\n\"This early spring, however, of the Cadenham oak, is of very short\nduration. The buds, after unfolding themselves, make no further\nprogress, but immediately shrink from the season and die. The tree\ncontinues torpid, like other deciduous trees, during the remainder of\nthe winter, and vegetates again in the spring, at the usual season.\nI have seen it in full leaf in the middle of the summer, when it\nappeared, both in its form and foliage, exactly like other oaks.\"\n\nDean Wren, speaking of this tree, says, \"King James could not be\ninduced to believe the [Greek: to toi] (_reason_) of this, till\nBishop Andrewes, in whose diocese the tree grew, caused one of his\nown chaplaines, a man of known integritye, to give a true information\nof itt, which he did; for upon the eve of the Nativitye he gathered\nabout a hundred slips, with the leaves newly opened, which he stuck in\nclaye in the bottom of long white boxes, and soe sent them post to the\ncourte, where they deservedly raised not only admiration, but stopt the\nmouth of infidelitye and contradiction for ever. Of this I was both\nan eye-witness, and did distribute many of them to the great persons\nof both sexes in court and others, ecclesiastical persons. But in\nthese last troublesome times a divelish fellow (of Herostratus humour)\nhaving hewen itt round at the roote, made his last stroke on his own\nlegg, whereof he died, together with the old wondrous tree; which now\nsproutes up againe, and may renew his oakye age againe, iff some such\nenvious chance doe not hinder or prevent itt; from which the example of\nthe former villaine may perchance deterr the attempt. This I thought\nto testifie to all future times, and therefore subscribe with the same\nhand through which those little oakye slips past.\"\n\n\nSCHOOL EXPENSES IN THE OLDEN TIME.\n\nOf the expenses incurred for schoolboys at Eton early in the reign of\nElizabeth, we find some curious particulars in a manuscript of the\ntime: the boys were sons of Sir William Cavendish, of Chatsworth, and\nthe entries are worth notice, as showing the manners of those days.\nAmong the items, a breast of roast mutton is charged ten-pence; a\nsmall chicken, fourpence; a week's board, five shillings each; besides\nthe wood burned in their chamber; to an old woman for sweeping and\ncleaning the chamber, twopence; mending a shoe, one penny; three\ncandles, nine-pence; a book, Esop's Fables, fourpence; two pair of\nshoes, sixteen-pence; two bunches of wax lights, one penny; the sum\ntotal of the payments, including board paid to the bursars of Eton\nCollege, living expenses for the two boys and their man, clothes,\nbooks, washing, &c., amounts to twelve pounds twelve shillings and\nseven-pence. The expense of a scholar at the university in 1514 was but\nfive pounds annually, affording as much accommodation as would cost\nsixty pounds, though the accommodation would be far short of that now\ncustomary at Eton.\n\n\nAN EVENTFUL LIFE.\n\nIt is much to be feared that on the field of battle and naval actions\nmany individuals, apparently dead, are buried or thrown overboard. The\nhistory of Francois de Civille, a French captain, who was missing at\nthe siege of Rouen, is rather curious. At the storming of the town he\nwas supposed to have been killed, and was thrown, with other bodies, in\nthe ditch, where he remained from eleven in the morning to half-past\nsix in the evening; when his servant, observing some latent heat,\ncarried the body into the house. For five days and five nights his\nmaster did not exhibit the slightest sign of life, although the body\ngradually recovered its warmth. At the expiration of this time, the\ntown was carried by assault, and the servants of an officer belonging\nto the besiegers, having found the supposed corpse of Civille, threw it\nout of the window, with no other covering than his shirt. Fortunately\nfor the captain, he had fallen upon a dunghill, where he remained\nsenseless for three days longer, when his body was taken up by his\nrelatives for sepulture, and ultimately brought to life. What was still\nmore strange, Civille, like Macduff, had \"been from his mother's womb\nuntimely ripp'd,\" having been brought into the world by a Caesarean\noperation, which his mother did not survive; and after his last\nwonderful escape he used to sign his name with the addition of \"three\ntimes born, three times buried, and three times risen from the dead by\nthe grace of God.\"\n\n\nFIRST BRIDGE OVER THE THAMES.\n\n[Illustration [++] First Bridge over the Thames.]\n\nThe humble village bridge which we here engrave is well deserving\nof a place in our pages as being the first of that grand series of\nbridges whose last member is London-bridge. What a contrast between\nthe first bridge over the Thames and the last! Thames Head, where\nthe river rises, is in the county of Gloucester, but so near to its\nsouthern border, that the stream, after meandering a mile or two,\nenters Wiltshire, near the village of Kemble. On leaving this village,\nand proceeding on the main road towards the rustic hamlet of Ewen,\nthe traveller passes over the bridge which forms the subject of our\nwoodcut. It has no parapet, and is level with the road, the water\nrunning through three narrow arches. Such is the first bridge over the\nmighty Thames.\n\n\nTHE VENETIANS.\n\nThe Venetians were the first people in Italy who had printed books.\nThey originated a Gazette in the year 1600, and the example was\nfollowed at Oxford in 1667, and at Vienna in 1700. They also undertook\nthe discovery of America, and the passage to India by the Cape of Good\nHope.\n\n\nMEDMENHAM ABBEY.\n\nOn descending the river Thames, from Henley, after passing Culham Court\nand Hambledon Lock, the adjacent country become exceedingly beautiful,\nvaried by alternate mills, islands, meadows, and hills, with every\nnow and then ornamental forest trees hanging over the stream, and\ngiving pleasant shade to the current on its downward flow. The wood\nof Medmenham soon comes in sight; the ruined Abbey is seen among the\ntrees, and close beside it is a pretty ferry, with the pleasant wayside\ninn of Mrs. Bitmead--a domicile well known to artists, her frequent\nguests, one of whom, who has since become famous, painted a sign-board\nwhich hangs over the door, and is of so good a quality that it might\ngrace the exhibition of the Royal Academy. The Abbey has been pictured\na hundred times, and is a capital subject seen from any point of view;\nthe river runs close beside it; there is a hill adjacent--Dane's Hill;\ndark woods and green meadows are at hand; gay boats and traffic barges\nare continually passing; the ferry is always picturesque, and the\nartist is constantly supplied on the spot with themes for pictures;\nespecially he has before him the venerable ruin--\"venerable,\" at least,\nas far as the eye is concerned. Time has touched it leniently; some of\nits best \"bits\" are as they were a century ago, except that the lichens\nhave given to them that rich clothing of grey and gold which the\npainter ever loves, and added to it, here and there, a green drapery of\nivy.\n\n[Illustration [++] Manor of Medmenham.]\n\nThe manor of Medmenham was, in the reign of King Stephen, given by\nits lord, Walter de Bolebec, to the Abbey of Cistercian Monks he had\nfounded at Woburn in Bedfordshire; and in 1204 the monks placed some\nof their society here, on this pleasant bank of the Thames. Here arose\na small monastery, being rather--as the writers of the order express\nthemselves--\"a daughter than a cell to Woburn.\" In 1536 it was annexed\nto Bisham. At the Dissolution, according to returns made by the\ncommissioners, \"the clear value of this religious house was 20_l._ 6s.;\nit had two monks designing to go to houses of religion; servants, none;\nwoods, none; debts, none; its bells worth 2_l._ 1s. 8d.; the value of\nits moveable goods 1_l._ 3s. 8d.; and the house wholly in ruin.\" It\nmust have undergone considerable repair early in the sixteenth century,\nand probably very little of the original structure now exists, although\nrelics of antiquity may be traced in many of its remains. That portion\nwhich fronts the Thames is kept in proper repair, and a large room is\nused for the convenience of pleasure parties. The property belongs to\nthe Scots of Danesfield, a mansion that crowns a neighbouring hill.\nMedmenham derives notoriety from events of more recent date than the\noccupation of its monks, without goods and without debt. Here, about\nthe middle of the last century, was established, a society of men of\nwit and fashion, who assumed the title of the Monks of St. Francis,\nand wore the habit of the Franciscan order. Although it is said\nthat the statements contained in a now forgotten but once popular\nnovel--\"Chrysal; or the Adventures of a Guinea,\"--were exaggerated,\nthe character which the assumed monks bore in the open world was\nsufficiently notorious to justify the worst suspicions of their acts\nin this comparative solitude. Their principal members were Sir Francis\nDashwood (afterwards Lord Le Despencer), the Earl of Sandwich, John\nWilks, Bubb Doddington, Churchill, and Paul Whitehead, the poet. The\nmotto--\"Fay ce que voudras,\" indicative of the principle on which the\nsociety was founded--still remains over the doorway of the Abbey House.\nTradition yet preserves some anecdotes illustrative of the habits of\nthe \"order,\" and there can be little doubt that this now lonely and\nquiet spot was the scene of orgies that were infamous.\n\n\nPERSECUTION.\n\nGrotius, an historian celebrated for moderation and caution, has\ncomputed that in the several persecutions promoted by Charles V., no\nless than a hundred thousand persons perished by the hands of the\nexecutioner. In the Netherlands alone, from the time that his edict\nagainst the reformers was promulgated, he states that there had been\nfifty thousand persons hanged, beheaded, buried alive, or burned, on\naccount of their religion. Indeed, during the reign of Philip the\nSecond, the Duke of Alva boasted that in the space of nine years he\nhad destroyed, in the Low Countries, 36,000 persons by the hands of\nthe executioner alone. At the massacre of Paris, on the feast of St.\nBartholomew, King Charles the Ninth of France assisted in person, and\nboasted that he had sacrificed in one night 10,000 of his subjects;\nfor that massacre the Pope had \"Te Deum\" sung in the chapel of the\nVatican and issued a bull for a jubilee to be celebrated throughout\nFrance on the 7th December, 1512, in commemoration of what he termed\nthe _happy success of the king against his heretic subjects_, and\nconcluded by writing with his own hand a letter to Charles the Ninth,\nexhorting him to pursue this salutary and blessed enterprise. In the\nshort reign of Queen Mary, there were in this realm burned at the stake\none archbishop, four bishops, twenty-one ministers, and nearly three\nhundred persons of all classes, of whom fifty-five were women, and four\nwere children, one of whom sprang from its mother's womb while she was\nconsuming, and was flung into the flames by the spectators. In 1640\nthe same spirit of papal bigotry occasioned in Ireland the butchery of\n40,000 Protestants, under circumstances of aggravated atrocity which a\nChristian will shudder to peruse. Lewis XIV., the most Christian king\nand eldest son of the church, starved a million Huguenots at home, and\nsent another million grazing in foreign countries.\n\n\nINNKEEPER'S BILL IN 1762.\n\nThe following innkeeper's bill was sent in to the Duke de Nivernois,\nwho supped and breakfasted at an inn in Canterbury, in 1762; and\nconsidering the value of money at that time, must be deemed extremely\nmoderate:--\n\n L s. d.\n\n Tea, coffee, and chocolate 1 4 0\n Supper for self and servant 15 10 0\n Bread and beer 3 0 0\n Fruit 2 15 0\n Wine and punch 10 8 8\n Wax candles and charcoal 3 0 0\n Broken glass and china 2 10 0\n Lodging 1 7 0\n Tea, coffee, and chocolate 2 0 0\n Chaise and horse, for next stage 2 16 0\n\nThere were only twelve persons in the whole company.\n\n\nSPONTANEOUS COMBUSTION.\n\nJoseph Battaglia, a surgeon of Ponte Bosio, relates the following case:\nDon. G. Maria Bertholi, a priest of Mount Valerius, went to the fair of\nFiletto, and afterwards visited a relation in Fenilo, where he intended\nto pass the night. Before retiring to rest, he was left reading his\nbreviary; when, shortly afterwards, the family were alarmed by his\nloud cries and a strange noise in his chamber. On opening the door, he\nwas lying prostrate on the floor, and surrounded by flickering flames.\nBattaglia was immediately sent for, and on his arrival the unfortunate\nman was found in a most deplorable state. The integuments of the arms\nand the back were either consumed or detached in hanging flaps. The\nsufferer was sufficiently sensible to give an account of himself.\nHe said that he felt, all of a sudden, as if his arm had received a\nviolent blow from a club, and at the same time he saw scintillations\nof fire rising from his shirt-sleeves, which were consumed without\nhaving burned the wrists; a handkerchief, which he had tied round his\nshoulders, between the shirt and the skin, was intact. His drawers were\nalso sound; but, strange to say, his silk skull-cap was burnt while his\nhair bore no marks of combustion. The unfortunate man only survived\nthe event four days. The circumstances which attended this case would\nseem to warrant the conclusion that the electric fluid was the chief\nagent in the combustion.\n\n\nSHOOTING FISH.\n\n[Illustration: Horned Chaetodon.]\n\nOur shores have produced a few specimens of a richly- fish\ncalled Ray's Sea Bream (_Brama Rayi_), interesting because it\nrepresents a family, almost confined to the tropical seas, of very\nsingular forms and habits. The family is named _Chaetodontidae_, from\nthe principal genus in it. They are very high perpendicularly, but\nthin and flattened sidewise; the mouth in some projects into a sort\nof snout, the fins are frequently much elevated, and send off long\nfilaments. They are generally adorned with highly-contrasted colours,\nwhich run in perpendicular bands. They are often called scaly-finned\nfishes, because the dorsal and anal are clothed, at least in part, with\nscales, so as not to be distinguished from the body. The tubular snout\nof some, as of a little species which we here represent, is applied to\nan extraordinary use, that of shooting flies! The fish approaches under\na fly which it has discovered, resting on a leaf or twig, a few feet\nabove the water, taking care not to alarm it by too sudden a motion;\nthen, projecting the tip of its beak from the surface, it shoots a\nsingle drop at the insect with so clever an aim, as very rarely to\nmiss it, when it falls into the water and is devoured. Being common\nin the Indian seas, it is often kept by the Chinese in vases, as we\nkeep golden-fish, for the amusement of witnessing this feat. A fly is\nfastened at some distance, at which the fish shoots, but, disappointed\nof course, and wondering that its prey does not fall, it goes on to\nrepeat the discharge for many times in succession, without seeming to\ntake in a fresh stock of ammunition, and scarcely ever missing the\nmark, though at a distance of three or four feet.\n\n\nEXTRAORDINARY EARTHQUAKES.\n\nAround the Papandayang, one of the loftiest mountains in Java, no less\nthan forty villages were reposing in peace. But in August 1772, a\nremarkable luminous cloud enveloping its top aroused them from their\nsecurity. But it was too late; for at once the mountain began to sink\ninto the earth, and soon it had disappeared, with the forty villages\nand most of the inhabitants, over a space fifteen miles long and six\nbroad. Still more extraordinary, the most remarkable on record was an\neruption in Sumbawa, one of the Molucca islands, in 1815. It began on\nthe fifth day of April, and did not cease till July. The explosions\nwere heard in one direction nine hundred and seventy miles, and in\nanother seven hundred and twenty miles. So heavy was the fall of ashes\nat the distance of forty miles that houses were crushed and destroyed.\nThe floating cinders in the ocean, hundreds of miles distant, were\ntwo feet thick, and vessels were forced through with difficulty. The\ndarkness in Java, three hundred miles distant, was deeper than the\nblackest night; and, finally, out of the twelve thousand inhabitants of\nthe island, only twenty-six survived the catastrophe.\n\n\nBEAUTIFUL ARCH.\n\nOne of the rarities of architecture is the beautiful arch in the choir\nof Cannistown Church, not far from Bective, near Trim, in Ireland. Down\nto the very latest period of Gothic architecture, the original plan of\na simple nave, or nave and chancel, was followed, and the chief or only\ndifference observable in churches of very late date, from those of the\nsixth and seventh centuries, consists in the form of the arch-heads,\nthe position of the doorway, the style of the masonry, which is usually\nmuch better in the more ancient examples, and the use of bell-turrets,\nthe cloigeteach, or detached round tower, having answered this purpose\nduring the earlier ages. A beautiful and highly characteristic example\nof an early pointed church is that at Cannistown. As usual, it consists\nof a nave and chancel, and there are the remains of a bell-turret upon\nthe west gable, the usual position. The choir arch is represented in\nthe annexed cut.\n\n[Illustration [++] Arch in the Choir of Cannistown Church.]\n\nThere are numerous examples of churches of this style scattered\nover Ireland, but they are usually plain, and the choir arch is\ngenerally the plainest feature in the building. As example, we can\nrefer our readers to the churches of Kilbarrack, Dalkey, Kinsale, and\nRathmichael, all in the immediate neighbourhood of Dublin.\n\n\nTHOMAS CONECTE.\n\nThere was a Carmelite friar, Thomas Conecte, who, previous to his\nbeing burnt as a heretic at Rome, in 1434, excited the admiration of\nall Flanders by his vehement sermons against the luxury of the women.\nHis satire was chiefly levelled against their head-dresses, which rose\nto so enormous a height, that the most exalted head-dresses of a late\nday were but dwarfs to them. Juvenal des Ursins, who lived at that\nperiod, declares that, notwithstanding the troubles of the times, the\nmaidens and married ladies rose to prodigious excess in their attire,\nand wore hair of a surprising height and breadth, having on each side\ntwo ears of so unaccountable a size, that it was impossible for them\nto pass through a door. Their dresses were the hennins of Flanders,\nwhich the worthy Carmelite was so inveterate against. He made them\ndress themselves in a more modest manner. But, alas no sooner had Friar\nThomas left the country than the head-dresses shot up to a greater\nheight than ever. They had only bowed their heads like bullrushes\nduring the storm. Poor Thomas attacked the infallible church itself,\nand they, in default of better arguments, burnt him.\n\n\nCURIOUS COINCIDENCES.\n\nOn the 21st of April, 1770, Lewis XVI. was married.\n\n21st of June, 1770, fifteen hundred people were trampled to death at\nthe _fete_.\n\n21st of January, 1782, _fete_ for the birth of the Dauphin.\n\n21st of June, 1791, the flight to Varennes.\n\n21st of September, 1792, the abolition of royalty.\n\n21st of January, 1793, the unfortunate monarch's decapitation.\n\n\nAMPHITHEATRES.\n\nThe deficiency of theatres erected by the Romans is far more than\ncompensated by the number and splendour of their amphitheatres, which,\nwith their baths, may be considered as the true types of Roman art. It\nseems almost certain that they derived this class of public buildings\nfrom the Etruscans. At Sutri there is a very noble one cut out of\nthe tufa rock, which was no doubt used by that people for festal\nrepresentations long before Rome attempted anything of the kind. It is\nuncertain whether gladiatorial fights or combats of wild beasts formed\nany part of the amusements of the arena in those days, though boxing,\nwrestling, and contests of that description certainly did; but whether\nthe Etruscans actually proceeded to the shedding of blood and slaughter\nis more than doubtful.\n\nEven in the remotest parts of Britain, in Germany, and Gaul, wherever\nwe find a Roman settlement, we find the traces of their amphitheatres.\nTheir soldiery, it seems, could not exist without the enjoyment of\nseeing men engage in doubtful and mortal combats--either killing one\nanother, or torn to pieces by wild beasts. It is not to be wondered at\nthat a people who delighted so much in the bloody scenes of the arena\nshould feel but very little pleasure in the mimic sorrows and tame\nhumour of the stage. It fitted them, it is true, to be a nation of\nconquerors, and gave them the empire of the world, but it brought with\nit feelings singularly inimical to all the softer arts, and was perhaps\nthe great cause of their debasement.\n\nAs might be expected, the largest and most splendid of these buildings\nis that which adorns the capital; and of all the ruins which Rome\ncontains, none have excited such universal admiration as the Flavian\namphitheatre. Poets, painters, rhapsodists, have exhausted all the\nresources of their arts in the attempt to convey to others the\noverpowering impression this building produces on their own minds.\nWith the single exception, perhaps, of the Hall at Karnac, no ruin has\nmet with such universal admiration as this. Its association with the\nancient mistress of the world, its destruction, and the half-prophetic\ndestiny ascribed to it, all contribute to this. Still it must be\nconfessed that\n\n \"The gladiator's bloody circus stands\n A noble wreck in ruinous perfection,\"\n\nand worthy of all or nearly all the admiration of which it has been the\nobject. Its interior is almost wholly devoid of ornament, or anything\nthat can be called architecture--a vast inverted pyramid. The exterior\ndoes not possess one detail which is not open to criticism, and indeed\nto positive blame. Notwithstanding all this, its mass, its form, and\nits associations, all combine to produce an effect against which the\ncritic struggles in vain.\n\nThe length of the building, measured along its greatest diameter, is\n620 ft., its breadth 513, or nearly in the ratio of 6 to 5, which may\nbe taken as the general proportion of these buildings, the variations\nfrom it being slight, and apparently either mistakes in setting out\nthe work in ancient times, or in measuring it in modern days, rather\nthan an intentional deviation. The height of the 3 lower stories is\n120 ft.; the total height as it now stands, 157 ft. The arena itself\nmeasures 287 ft. in length by 180 in breadth, and it is calculated that\nthe building would contain 80,000 spectators; 50,000 or 60,000 would be\nmuch nearer the truth, at least according to the data by which space is\ncalculated in our theatres and public places.\n\n\nHUNDRED FAMILIES' LOCK.\n\nA common Chinese talisman is the \"hundred families' lock,\" to procure\nwhich a father goes round among his friends, and, having obtained from\na hundred different parties a few of the copper coins of the country,\nhe himself adds the balance, to purchase an ornament or appendage\nfashioned like a lock, which he hangs on his child's neck, for the\npurpose of locking him figuratively to life, and making the hundred\npersons concerned in his attaining old age.\n\n\nTHE DUKE DE REICHSTADT.\n\nAt the Imperial Palace of Schoenbrun, about five English miles from\nVienna, is shown the window fractured by the bullet of the enthusiastic\nstudent who shot at Napoleon while he was reviewing the Imperial Guard,\nand also the apartment he occupied when he made this his head-quarters,\ninstead of entering the city. An additional interest is imparted to the\nplace, by the circumstance of the Duke de Reichstadt having, when taken\nill, chosen the identical chamber and spot in which his father Napoleon\nhad slept, to close his mortal career: and by a singular coincidence,\nthe remains of the young prince were subjected to a post-mortem\nexamination upon the same table at which the Emperor had held his\ncouncils. In imitation of the military hardihood of his sire, the young\nduke was in the habit of exposing himself to all weathers, and keeping\nguard during successive nights, a practice which often called forth\nfrom his surgeon, Dr. Malfati, the expressive words, '_Rappelez vous,\nmon Prince, que vouz avez un Coeur de Fer dans un Corp de Verre_.'\n\n\nMARY QUEEN OF SCOTS' CANDLESTICK.\n\nAlmost every article, however trifling its intrinsic value, and however\nhomely its appearance, which once belonged to a celebrated individual,\nis always regarded as an object of interest, and we have, therefore,\nno hesitation in presenting our readers with the annexed engraving\nof one of a pair of candlesticks which were once the property of the\nunfortunate Mary, Queen of Scots.\n\n[Illustration [++] Mary Queen of Scots' Candlestick.]\n\nThey are made of brass, each of them of eleven and a-half inches in\nheight. They are of French manufacture; the sunk parts are filled\nup with an inlay of blue, green, and white enamel, very similar to\nthat done at Limoge. These extremely elegant and curious articles are\nthe property of Lord Holland, and are preserved at Holland House,\nKensington.\n\nHolland House is associated \"with the costly magnificence of Rich,\nwith the loves of Ormond, the councils of Cromwell, and the death\nof Addison.\" It has been for nearly two centuries and a-half the\nfavourite resort of wits and beauties, of painters and poets, of\nscholars, philosophers, and statesmen. In the lifetime of the late Lord\nHolland, it was the meeting-place of \"the Whig Party;\" and his liberal\nhospitality made it \"the resort, not only of the most interesting\npersons composing English society--literary, philosophical, and\npolitical, but also to all belonging to those classes who ever visited\nthis country from abroad.\"\n\n\nEXTRAORDINARY INSTANCES OF INHUMANITY.\n\nIn 1534, in the wars of Edward III. with France, Fordun relates that a\nFrenchman purchased from the Scots several English prisoners, and that\nhe beheaded them to avenge the death of his father. This sentimental\ncruelty can perhaps be paralleled by that of Coccinas, who, at the\nmassacre of Paris, bought many Huguenots, that he might torture them\nto death for his private satisfaction. Philip Galeas Visconti, Duke of\nMilan, was a man of a nature so timid, that thunder threw him into\nagonies; yet was he so inhuman, that he could _enjoy the shrieks of\na female stretched upon a rack_. Wenceslaus, the German Emperor, say\nMezeray, Voltaire, and others, _roasted his cook alive_, for dressing\nhis dinner amiss; and never had so intimate a friend in Prague as the\ncommon executioner; and even _him_ he put to death at last, for not\ntaking him at his word, when he once had bid him cut his head off, and\nactually knelt down to receive the stroke.\n\n\nANCIENT ROMAN LAMPS.\n\nThe earliest lamps fabricated by the potters of ancient Rome have an\nopen circular body, with a curved projecting rim to prevent the oil\nfrom spilling, and occur both in terra-cotta, and also in the black\nglazed ware found in the sepulchres of Nola. Many have a projecting\nhollow pipe in the centre, in order to fix them to a stick on the top\nof a candelabrum. These lamps have no handles. They may have been\nplaced in the sacella or lararia, and were turned on the potter's wheel.\n\nThe shoe-shaped is the most usual, with a round body, a projecting\nspout or nozzle having a hole for the wick, and a small annular handle,\nwhich is more or less raised.\n\n[Illustration [++] Ancient Roman Lamp.]\n\nA singular variety of lamp, well adapted for a table, was fitted\ninto a kind of small altar, the sides of which were ornamented with\nreliefs. Several however, from their unusual shape, maybe considered\nas fancy ware, the upper part, or the whole lamp, being moulded into\nthe resemblance of some object. Such are lamps in the British Museum in\nthe shape of a female head surmounted by a flower, or of the head of a\n or Nubian with open jaws, through which the wick was inserted.\n\nMost of these lamps appear to have been made between the age of\nAugustus and that of Constantine. The style, of course best at the\nearlier period of the empire, degenerates under the later emperors,\nsuch as Philip and Maximus, and becomes at last Byzantine and bad.\n\nMost lamps had only one wick, but the light they afforded must have\nbeen feeble, and consequently some have two wicks, the nozzle for\nwhich project beyond the body of the lamp. In the same manner were\nfabricated lamps of three, five, and seven wicks. If more were required\nthe nozzles did not project far beyond the body of the lamp, which\nwas then moulded in a shape adapted for the purpose, and especially\nthe favourite one of a galley. Sometimes a conglomeration of small\nlamps was manufactured in a row, or in a serrated shape, which enabled\nthe purchaser to obtain what light he required; still the amount of\nillumination must have been feeble. As many as twenty wicks have been\nfound in some lamps.\n\nThe greater number average from three to four inches long, and one\ninch high; the walls are about one-eighth of an inch thick, and the\ncircular handles not more than one inch in diameter. Some of the larger\nlamps, however, are about nine inches or a foot long, with handles\neight or nine inches high.\n\n\nAN ECCENTRIC ENGLISHMAN.\n\nMr. Henry Hastings, a most singular character, and genuine sportsman\nlived in the time of James and Charles I. Mr. Hastings was second son\nto the Earl of Huntingdon; and inherited a good estate in Dorsetshire\nfrom his mother. He was one of the keepers of New Forest, Hampshire;\nand resided in the lodge there during a part of every summer season.\nBut his principal residence was at Woodlands, in Dorsetshire, where\nhe had a capital mansion. One of his nearest neighbours, was the\nLord Chancellor Cooper, first Earl of Shaftesbury. Two men could not\nbe more opposite in their disposition and pursuits. They had little\ncommunication therefore; and their occasional meetings were rendered\nmore disagreeable to both from their opposite sentiments in politics.\nLord Shaftesbury, who was the younger man, was the survivor; and the\nfollowing account of Mr. Hastings is said to have been the production\nof his pen. \"Mr. Hastings was low of stature, but very strong, and\nvery active; of a ruddy complexion, with flaxen hair. His clothes were\nalways of green cloth. His house was of the old fashion; in the midst\nof a large park, well stocked with deer, rabbits, and fish-ponds. He\nhad a long narrow bowling-green in it; and used to play with round\nsand-bowls. Here, too, he had a banqueting-room built, like a stand\nin a large tree. He kept all sorts of hounds, that ran buck, fox,\nhare, otter, and badger; and had hawks of all kinds, both long and\nshort-winged. His great hall was commonly strewed with marrow-bones;\nand full of hawk-perches, hounds, spaniels, and terriers. The upper\nend of it was hung with fox-skins of this and the last year's killing.\nHere and there a pole-cat was intermixed; and hunter's poles in great\nabundance. The parlour was a large room, completely furnished in the\nsame style. On a broad hearth, paved with bricks, lay some of the\nchoicest terriers, hounds, and spaniels. One or two of the great chairs\nhad litters of cats in them, which were not to be disturbed. Of these,\nthree or four always attended him at dinner; and a little white wand\nlay by his trencher, to defend it, if they were too troublesome. In\nthe windows, which were very large, lay his arrows, cross-bows, and\nother accoutrements. The corners of the room were filled with his best\nhunting and hawking poles. His oyster-table stood at the lower end of\nthe room, which was in constant use twice a day, all the year round;\nfor he never failed to eat oysters both at dinner and supper; with\nwhich the neighbouring town of Poole supplied him. At the upper end of\nthe room stood a small table with a double desk; one side of which held\na church Bible; the other, the Book of Martyrs. On different tables of\nthe room lay hawks' hoods; bells, old hats with their crowns thrust in,\nfull of pheasants' eggs, tables, dice, cards, and a store of tobacco\npipes. At one end of this room was a door, which opened into a closet,\nwhere stood bottles of strong beer and wine, which never came out\nbut in single glasses, which was the rule of the house; for he never\nexceeded himself, nor permitted others to exceed. Answering to this\ncloset was a door into an old chapel, which had been long disused for\ndevotion; but, in the pulpit, as the safest place, was always to be\nfound a cold chine of beef, a venison pasty, a gammon of bacon, or a\ngreat apple-pie with thick crust, well baked. His table cost him not\nmuch, though it was good to eat at. His sports supplied all but beef\nand mutton, except on Fridays, when he had the best of fish. He never\nwanted a London pudding; and he always sang it in with, \"_My part lies\ntherein-a._\" He drank a glass or two of wine at meals; put syrup of\ngilly-flowers into his sack; and had always a tun-glass of small-beer\nstanding by him, which he often stirred about with rosemary. He lived\nto be an hundred; and never lost his eyesight, nor used spectacles. He\ngot on horseback without help; and rode to the death of the stag, till\nhe was past fourscore.\"\n\n\nPERFUMED BANQUETS OF THE ANCIENTS.\n\nA very remarkable peculiarity in the banquets of the ancients was,\ntheir not confining the resources of the table to the gratification of\none sense alone. Having exhausted their invention in the confection\nof stimulants for the palate, they broke new ground, and called in\nanother sense to their aid; and by the delicate application of odours\nand richly-distilled perfumes, these refined voluptuaries aroused the\nfainting appetite, and added a more exquisite and ethereal enjoyment\nto the grosser pleasures of the board. The gratification of the sense\nof smelling (a sense held by us in very undeserved neglect, probably\non account of its delicacy) was a subject of no little importance\nto the Romans. However this may be, it is certain that the Romans\nconsidered flowers as forming a very essential article in their festal\npreparations; and it is the opinion of Bassius, that at their desserts\nthe number of flowers far exceeded that of fruits. When Nero supped\nin his Golden House, a mingled shower of flowers and odorous essences\nfell upon him; and one of Heliogabalus' recreations was to smother\nhis courtiers with flowers, of whom it may be said, they \"died of a\nrose in aromatic pain.\" Nor was it entirely as an object of luxury\nthat the ancients made use of flowers; they were considered to possess\nsanative and medicinal qualities. According to Pliny, Athenaeus, and\nPlutarch, certain herbs and flowers were of sovereign power to prevent\nthe approaches of ebriety, or, as Bassius less clearly expresses it,\nclarify the functions of the brain.\n\n\nCHINESE BRIDGES.\n\nOf Chinese bridges, some have been very much exaggerated in the\naccounts by Du Halde and the missionaries, as it appears from the\nlater reports concerning the bridge at Foo-chow-foo, visited during\nthe unsuccessful commercial voyage of the ship \"Amherst,\" in 1832, and\nsince the war become familiar to our countrymen. This same bridge,\nwhich proved a very poor structure after all, had been extolled by\nthe Jesuits as something quite extraordinary. A bridge of ninety-one\narches, being in fact a very long causeway, was passed by Lord\nMacartney between Soo-chow and Hang-chow, and near the Lake called\nTae-hoo. The highest arch, however, was supposed to be between twenty\nand thirty feet in height, and the whole length of the causeway half\na mile. It was thrown across an arm of the lake, on the eastern side\nof the canal. The late Sir George Staunton observed a bridge between\nPeking and Tartary, built across a river which was subject to being\nswelled by mountain floods. This was erected upon caissons of wattles\nfilled with stones. It appeared to have been built with expedition,\nand at small cost, where the most solid bridge would be endangered by\ninundations. The caissons were fixed by large perpendicular spars, and\nover the whole were laid planks, hurdles, and gravel. It was only in\nKeang-nan that solid bridges were observed to be thrown over the canal,\nbeing constructed of coarse grey marble, or of a reddish granite. Some\nof the arches were semicircular, others the transverse section of an\nellipse, and others again approached the shape of a horseshoe, or\nGreek [Greek: capital Omega], the space being widest at top. In the\nornamental bridges that adorn gardens and pleasure-grounds, the arch is\noften of height sufficient to admit a boat under sail, and the bridge\nis ascended by steps.\n\n[Illustration [++] Chinese Bridge.]\n\nAll the stones of a Chinese arch are commonly wedge-shaped, their\nsides forming radii which converge towards the centre of the curve.\nIt is observable that, according to the opinion of Captain Parish,\nwho surveyed and made plans of the Great Wall, no masonry could be\nsuperior to it. The arched and vaulted work was considered by him as\nexceedingly well turned. The Chinese, therefore, must have understood\nthe construction and properties of the arch long before the Greeks and\nRomans, whose original and most ancient edifices consisted of columns,\nconnected by straight architraves, of bulk sufficient to support the\nincumbent pressure of solid masonry.\n\n\nSOCIABLE WEAVER-BIRD.\n\nThere are some birds whose social instinct impels them to live in\ncompany, and to unite their powers in the construction of a common\nedifice: in this respect resembling the Beaver among quadrupeds, and\nthe Bee among insects. Among these we may mention the Ani (_Crotophaga\nani_) of the West Indies; the Pensile Grosbeak (_Loxia pensilis_)\nof West Africa; and the Bottle-nested Sparrow of India: but more\nremarkable than any of these is the Sociable Grosbeak (_Loxia\nsocialis_) of South Africa, whose habits are described by Le Vaillant.\n\n\"Figure to yourself,\" says this enterprising traveller, \"a huge,\nirregular, sloping roof, with all the eaves completely covered with\nnests, crowded close together, and you will have a tolerably correct\nidea of these singular edifices.\" The birds commence this structure by\nforming the immense canopy of a mass of grass, so compact and firmly\nbasketed together as to be impenetrable to the rain. This sometimes\nsurrounds a large tree, giving it, but for the upper branches, somewhat\nthe form of a mushroom. Beneath the eaves of this canopy the nests\nare formed; the upper surface is not used for this purpose, but as it\nis sloping, with a projecting rim, it serves to let the rain-water\nrun off, and preserves each little dwelling from the wet. Le Vaillant\nprocured one of these great shelters, and cut it in pieces with a\nhatchet: the chief portion consisted of Boshman's grass, so compact\nas to be impenetrable by rain. Each nest is three or four inches in\ndiameter, which is sufficient for the bird; but, as they are all close\ntogether around the eaves, they appear to the eye to form but one\nbuilding, and, in fact, are distinguishable from each other only by a\nlittle external aperture, which serves as an entrance to the nest. This\nlarge nest contained 320 inhabited cells.\n\n[Illustration: Nest of Sociable Weaver Bird.]\n\n\nWOLVES IN ENGLAND.\n\nKing Edward the First commissioned Peter Corbet to destroy the wolves\nin the counties of Gloucester, Worcester, Hereford, Salop, and\nStafford; and ordered John Gifford to hunt them in all the forests of\nEngland.\n\nThe forest of Chiltern was infested by wolves and wild bulls in the\ntime of Edward the Confessor. William the Conqueror granted the\nlordship of Riddesdale, in Northumberland, to Robert de Umfraville,\non condition of defending that part of the country against enemies and\nwolves. King John gave a premium of ten shillings for catching two\nwolves.\n\nIn the reign of King Henry the Third Vitalis de Engaine held the manors\nof Laxton and Pitchley, in the county of Northampton, by the service\nof hunting the wolf, whenever the king should command him. In the\nreign of Edward the First, it was found by inquisition that John de\nEngaine held the manor of Great Gidding, in the county of Huntingdon,\nby the service of hunting the hare, fox, wild cat, and wolf, within the\ncounties of Huntingdon, Northampton, Buckingham, Oxford, and Rutland.\nIn the reign of Edward the Third, Thomas de Engaine held certain manors\nby the service of finding, at his own proper cost, certain dogs for the\ndestruction of wolves, foxes, martins, and wild cats in the counties of\nNorthampton, Rutland, Oxford, Essex, and Buckingham.\n\n\nTEMPLES OF BRAMBANAM.\n\nIn the island of Java, and not far from the ruins of Boro Buddor,\nare situated the Buddhist temples of Brambanam; certainly one of the\nmost extraordinary groups of buildings of its class, and very unlike\nanything we now find in India; though there can scarcely be a doubt but\nthat the whole is derived from an Indian original now lost.\n\nThe great temple is a square building above 45 ft. square, and 75 ft.\nhigh, terminating upwards in an octagonal straight-lined pyramid. On\neach face of this is a smaller temple of similar design joined to the\ngreat one by corridors; the whole five thus constituting a cruciform\nbuilding. It is raised upon a richly ornamented square base. One of\nthe smaller temples serves as an entrance-porch. The building itself\nis very curiously and richly ornamented with sculpture; but the most\nremarkable feature of the whole group is the multitude of smaller\ntemples which surround the central one, 239 in number. Immediately\nbeyond the square terrace which supports the central temple stand 28\nof these, forming a square of 8 on each side, counting the angular\nones both ways. Beyond these, at a distance of 35 ft., is the second\nsquare, 44 in number; between this and the next row is a wide space of\nabove 80 ft., in which only 6 temples are situated, two in the centre\nof the north and south faces, and one on each of the others. The two\nouter rows of temples are situated close to one another, back to back,\nand are 160 in number, each face of the square they form being about\n525 ft. All these 239 temples are similar to one another, about 12 ft.\nsquare at the base, and 22 ft. high, all richly carved and ornamented,\nand in every one is a small square cell, in which was originally placed\na cross-legged figure, probably of one of the Jaina saints, though\nthe drawings which have been hitherto published do not enable us to\ndetermine whom they represent--the persons who made them not being\naware of the distinction between Buddhist and Jaina images.\n\nThe date given to these monuments by the natives is about the 9th or\n10th century, at which time the Jains were making great progress at\nGuzerat and the western parts of India; and if the traditions are to\nbe relied upon, which bring the Hindu colonists of Java from that\nquarter, it is almost certain that they would have brought that\nreligion with them. If the age, however, that is assigned to them be\ncorrect, they are specimens of an earlier date and form than anything\nwe now find in India, and less removed from the old Buddhist type than\nanything that now remains there.\n\n\nGRAHAM ISLAND.\n\nThe most recent instance of subaqueous eruption, with which we are\nacquainted is that which produced Hotham or Graham Island, in the year\n1831. This island was thrown up in the Mediterranean, between the\nsouth-west coast of Sicily and the African coast, in latitude 37 deg. 8'\n30\" north, and longitude 12 deg. 42' 15\" east. The eruption seems to have\nbeen first observed by John Corrao, the captain of a Sicilian vessel,\nwho passing near to the spot on the 10th of July, observed an immense\ncolumn of water ejected from the sea to the height of sixty feet, and\nabout eight hundred yards in circumference.\n\nOn the 16th of July, Corrao again passed the same spot, and he found\nthat a small island had been formed, twelve feet high, with a crater in\nthe centre, from which immense columns of vapour and masses of volcanic\nmatter were ejected.\n\nThe island was afterwards visited by several scientific gentlemen,\nand is said to have been two hundred feet high, and three miles in\ncircumference, on the 4th of August. But from this time the island\ndecreased in size; for being composed of loose scoriae and pumice, it\nwas rapidly acted upon by the water; and on the 3rd of September, when\ncarefully measured by Captain Wodehouse, was only three-fifths of a\nmile in circumference, and one hundred and seven feet high. At the end\nof October the island had entirely disappeared, except one small point\ncomposed of sand and scoriae. Captain Swinburne examined the spot in the\nbeginning of the year 1832, and found an extensive shoal to occupy the\nplace where the island had once been. In 1833 there was a dangerous\nreef, of an oval form, three-fifths of a mile in circumference.\n\n\nA ROYAL SPORTSMAN.\n\nWhen the King of Naples (the greatest sportsman in Europe) was in\nGermany, about the year 1792, it was said in the German papers, that\nin the different times he had been shooting in Austria, Bohemia, and\nMoravia, he had killed 5 bears, 1,820 wild boars, 1,968 stags, 13\nwolves, 354 foxes, 15,350 pheasants, 1,121 rabbits, 16,354 hares, 1,625\nshe-goats, 1,625 roebucks, and 12,435 partridges.\n\n\nLIFE IN DEATH.\n\nThe wife of the consul of Cologne, Retchmuth, apparently died of the\nplague, in 1571; a ring of great value, with which she was buried,\ntempted the cupidity of the grave-digger, and was the cause of many\nfuture years of happiness. At night the purloiner marched to his\nplunder, and she revived. She lived to be the mother of three children,\nand, when deceased in reality, was re-buried in the same church,\nwhere a monument was erected, reciting the particulars above stated\nin German verse. A woman of Poictiers, being buried with four rings,\ntempted the resurrection-man, who _awoke_ the woman in the attempt, as\nhe was rather rude in his mode of possessing them. She called out; he,\nbeing frightened, fled. The lady walked home, recovered, and had many\nchildren afterwards.\n\n\nROCK-CUT MONUMENTS OF ASIA MINOR.\n\nThe engraving below represents an example of rock-cut monuments which\nare found at Doganlu, in Asia Minor. They are placed on the rocky side\nof a narrow valley, and unconnected apparently with any great city or\ncentre of population. Generally they are called tombs, but there are\nno chambers nor anything about them to indicate a funereal purpose,\nand the inscriptions which accompany them are not on the monuments\nthemselves, nor do they refer to such a purpose. Altogether, they are\ncertainly among the most mysterious remains of antiquity, and, beyond\na certain similarity to the rock-cut tombs around Persepolis, it is\nnot easy to point out any monuments that afford even a remote analogy\nto guide us in our conjectures. They are of a style of art clearly\nindicating a wooden origin, and consist of a square frontispiece,\neither carved into certain geometric shapes, or prepared apparently\nfor painting; at each side is a flat pilaster, and above a pediment\nterminating in two scrolls. Some, apparently the more modern, have\npillars of a rude Doric order, and all indeed are much more curious\nthan beautiful. When more of the same class are discovered, they may\nhelp us to some historic data: all that we can now say of them is,\nthat, judging from their inscriptions and the traditions in Herodotus,\nthey seem to belong to some Indo-Germanic race from Thessaly, or\nthereabouts, who had crossed the Hellespont and settled in their\nneighbourhood; and their date is possibly as far back as 1000, and most\nprobably before 700 B.C.\n\n[Illustration [++] Rock-Cut Monument.]\n\n\nARCH OF TRAJAN AT BENEVENTUM.\n\nTriumphal arches were among the most peculiar forms of art which the\nRomans borrowed from those around them, and used with that strange\nmixture of splendour and bad taste which characterises all their works.\n\n[Illustration [++] Arch of Trajan at Beneventum.]\n\nThese were in the first instance no doubt borrowed from the Etruscans,\nas was also the ceremony of the triumph with which they were ultimately\nassociated. At first they seem rather to have been used as festal\nentrances to the great public roads, whose construction was considered\nas one of the most important benefits a ruler could confer on his\ncountry. There was one erected at Rimini in honour of an important\nrestoration of the Flaminian Way by Augustus; another at Susa in\nPiedmont, to commemorate a similar act of the same Emperor. Trajan\nbuilt one on the pier at Ancona, when he restored that harbour, and\nanother at Beneventum, when he repaired the Via Appia, represented in\nthe woodcut here given. It is one of the best preserved as well as\nmost graceful of its class in Italy. The arch of the Sergii at Pola\nin Istria seems also to have been erected for a like purpose. That of\nHadrian at Athens, and another built by him at Antinoe in Egypt, were\nmonuments merely commemorative of the benefits which he had conferred\non those cities by the architectural works he had erected within their\nwalls. By far the most important application of these gateways, in Rome\nat least, was to commemorate a triumph which may have passed along the\nroad over which the arch was erected beforehand, for the triumphal\nprocession to pass through, of which it would remain a memorial.\n\n\nJUDGES' SALARIES.\n\nIn the reign of Henry III. the King's Justices enjoyed a salary of ten\nmarks per annum, which, in the twenty-third year of that King, was\naugmented to twenty pounds, and soon after to more. Under Henry IV. the\nChief Justices of the King's Bench and Common Pleas had forty pounds,\nand one of the judges of Common Pleas had fifty-five marks. In 1466,\nthe salary of Thomas Littleton, judge of the King's Bench, amounted to\nL136 13s. 4d. modern money; besides about L17 7s. for his fur-gown,\nrobes, &c.\n\n\nEXTRAORDINARY OAK.\n\nGilpin, in his \"Forest Scenery,\" says, \"Close by the gate of the\nwater-walk at Magdalen College, in Oxford, grew an oak, which perhaps\nstood there a sapling when Alfred the Great founded the university.\nThis period only includes a space of nine hundred years, which is no\ngreat age for an oak. It is a difficult matter indeed to ascertain the\nage of a tree. The age of a castle or abbey is the object of history;\neven a common house is recorded by the families that built it. All\nthese objects arrive at maturity in their youth, if I may so speak.\nBut the tree, gradually completing its growth, is not worth recording\nin the early part of its existence. It is then only a common tree;\nand afterwards, when it becomes remarkable for its age, all memory of\nits youth is lost. This tree, however, can almost produce historical\nevidence for the age assigned to it. About five hundred years after the\ntime of Alfred, William of Wainfleet, Dr. Stukely tells us, expressly\nordered his college to be founded near the Great Oak; and an oak could\nnot, I think, be less than five hundred years of age to merit that\ntitle, together with the honour of fixing the site of a college. When\nthe magnificence of Cardinal Wolsey erected that handsome tower which\nis so ornamental to the whole building, this tree might probably be\nin the meridian of its glory, or rather, perhaps, it had attained a\ngreen old age. But it must have been manifestly in its decline at that\nmemorable era when the tyranny of James gave the fellows of Magdalen so\nnoble an opportunity of withstanding bigotry and superstition. It was\nafterwards much injured in Charles the Second's time, when the present\nwalks were laid out. Its roots were disturbed, and from that period it\ndeclined fast, and became reduced by degrees to little more than a mere\ntrunk. The oldest members of the university can scarcely recollect it\nin better plight. But the faithful records of history have handed down\nits ancient dimensions. Through a space of sixteen yards on every side\nfrom its trunk, it once flung its boughs, and under its magnificent\npavilion could have sheltered with ease three thousand men, though in\nits decayed state it could for many years do little more than shelter\nsome luckless individual whom the driving shower had overtaken in his\nevening walk. In the summer of 1788, this magnificent ruin fell to the\nground, alarming the college with its rushing sound. It then appeared\nhow precariously it had stood for many years. Its grand tap-root was\ndecayed, and it had hold of the earth only by two or three roots, of\nwhich none was more than a couple of inches in diameter. From a part of\nits ruins a chair has been made for the President of the College, which\nwill long continue its memory.\"\n\n\nECCENTRIC ADVERTISEMENT.\n\nThe following strange advertisement is copied from the Harleian MSS.:\n\"_In Nova fert Animus_. These are to give notice, (for the benefit of\nthe public,) that there is newly arrived from his travels, a gentleman,\nwho, after above forty years' study, hath, by a wonderful blessing on\nhis endeavours, discovered, as well the nature as the infallible cure\nof several strange diseases, which (though as yet not known to the\nworld) he will plainly demonstrate to any ingenious artist, to be the\ngreatest causes of the most common distempers incident to the body of\nman. The names of which take as follow:\n\n The strong fives\n The marthambles\n The moon-pall\n The hockogrocle.\n\n\"Now, though the names, natures, symptoms, and several cures of these\ndiseases, are altogether unknown to our greatest physicians, and the\nparticular knowledge of them would (if concealed) be a vast advantage\nto the aforesaid person; yet, he well knowing that his country's\ngood is to be preferred to his private interest, doth hereby promise\nall sorts of people, a faithful cure of all or any of the diseases\naforesaid, at as reasonable rates as our modern doctors have for that\nof any common distemper.\n\n\"He is spoken with at the ordinary hours of business, at the Three\nCompasses, in Maiden-lane.\"\n\n\nMODERN EGYPTIAN FEMALE ORNAMENTS.\n\n[Illustration [++] Ckoo'r.]\n\nAmong the many ornaments which the women of Egypt in modern times\nare so fond of wearing, none is more curious or more generally worn\nthan the _Ckoo'r_. It is a round convex ornament, commonly about five\ninches in diameter, of which there are two kinds. The first that we\nshall describe, and which is the only kind worn by ladies, or by the\nwives of tradesmen of moderate property, is the _ckoor's alma's_, or\ndiamond ckoor's. This is composed of diamonds set generally in gold;\nand is open work, representing roses, leaves, &c. The diamonds are\ncommonly of a very poor and shallow kind; and the gold of this and all\nother diamond ornaments worn in Egypt is much alloyed with copper. The\nvalue of a moderately handsome diamond ckoor's is about a hundred and\ntwenty-five, or a hundred and fifty pounds sterling. It is very seldom\nmade of silver; and I think that those of gold, when attached to the\ndeep red turboo'sh, have a richer effect, though not in accordance\nwith our general taste. The wives even of petty tradesmen sometimes\nwear the diamond ckoor's: they are extremely fond of diamonds, and\ngenerally endeavour to get some, however bad. The ckoor's, being of\nconsiderable weight, is at first painful to wear; and women who are in\nthe habit of wearing it complain of headache when they take it off:\nhence they retain it day and night; but some have an inferior one\nfor the bed. Some ladies have one for ordinary wearing, another for\nparticular occasions, a little larger and handsomer; and a third merely\nto wear in bed. The other kind of ckoor's, _ckoor's dah'ab_ (or, of\ngold), is a convex plate of very thin embossed gold, and almost always\na false emerald (a piece of green glass), not cut with facets, set in\nthe centre. Neither the emerald nor the ruby are here cut with facets:\nif so cut, they would generally be considered false. The simple gold\nckoor's is lined with a thick coat of wax, which is covered with a\npiece of paper. It is worn by many women who cannot afford to purchase\ndiamonds; and even by some servants.\n\n\nANTIQUE ROMAN MEDICINE STAMP.\n\n[Illustration [++] Roman Medicine Stamp.]\n\nBy far the most remarkable of the recently discovered remains of\nthe Roman occupants of Scotland is a medicine stamp, acquired by\nthe Society of Antiquaries of Scotland, along with a very valuable\ncollection of antiquities, bequeathed to them by E. W. A. Drummond\nHay, Esq., formerly one of the secretaries of the society. From his\nnotes it appears that it was found in the immediate vicinity of Trenent\nChurch, East Lothian, in a quantity of _debris_, broken tiles, and\nbrick-dust, which may not improbably have once formed the residence\nand laboratory of Lucius Vallatinus, the Roman oculist, whose name\nthis curious relic supplies. It consists of a small cube of pale\ngreen stone, two and three-fifth inches in length, and engraved on\ntwo sides as in the annexed woodcut; the letters being reversed for\nthe purpose of stamping the unguents or other medicaments retailed by\nits original possessor. The inscriptions admit of being extended thus\non the one side: L. VALLATINI EVODES AD CICATRICES ET ASPRITUDINES,\nwhich may be rendered--The evodes of Lucius Vallatinus for cicatrices\nand granulations. The reverse, though in part somewhat more obscure,\nreads: L. VALLATINI A PAL{O} CR{O}CODES AD DIATHESES--The crocodes, or\npreparation of saffron, of L. Vallatinus, of the Palatine School,\n(?) for affections of the eyes. Both the Euodes and the Crocodes are\nprescriptions given by Galen, and occur on other medicine stamps.\nSeveral examples have been found in England, and many in France and\nGermany, supplying the names of their owners and the terms of their\npreparations. Many of the latter indicate their chief use for diseases\nof the eye, and hence they have most commonly received the name of\nRoman oculists' stamps. No example, however, except the one figured\nhere, has ever occurred in Scotland; and amid legionary inscriptions,\nmilitary votive altars, and sepulchral tablets, it is peculiarly\ninteresting to stumble on this intelligent memento, restoring to us the\nname of the old Roman physician who ministered to the colonists of the\nLothians the skill, and perchance also the charlatanry, of the healing\nart.\n\n\nCANDLES IN THE CHURCH.\n\nIn the formulae of Marculphus, edited by Jerome Bignon, he tells us,\nwith respect to lights, that the use of them was of great antiquity\nin the church; that the primitive Christians made use of them in the\nassemblies which they held before day out of necessity; and that\nafterwards they were retained even in daylight, as tokens of joy, and\nin honour of the Deity. Lactantius says, speaking of the absurdities\nof the wax lights in Romish churches, \"They light up candles to God,\nas if he lived in the dark; and do they not deserve to pass for madmen\nwho offer lamps and candles to the author and giver of light?\" It is\nreally astounding to our ideas that wax candles as long as serjeants'\npikes should be held as necessary in the worship of God. That it is so\nheld, and that by a large class of Christians, every one must allow,\nfor they may have occular demonstration of the singular fact. The show\nis however extremely imposing. Thirty-five thousand seven hundred and\nfifty pounds of wax lights were burned every year, for nine hundred\nmasses said in the castle of Wittemburgh! Philip Melancthon speaks of a\nJesuit who said that \"he would not extinguish one taper, though it were\nto convert all the Huguenots\" (Protestants).\n\n\nA RICH AND CRUEL CRIMINAL.\n\nJohn Ward, Esq. of Hackney, Member of Parliament, being prosecuted by\nthe Duchess of Buckingham, and convicted of forgery, was first expelled\nfrom the House, and then stood on the pillory on the 17th of March,\n1727. He was suspected of joining in a conveyance with Sir John Blount,\nto secrete L50,000 of that director's estate, forfeited to the South\nSea Company by Act of Parliament. The Company recovered the L50,000\nagainst Ward; but he set up prior conveyances of his real estate to his\nbrother and son, and concealed all his personal, which was computed\nto be L150,000. These conveyances being also set aside by a bill in\nchancery, Ward was imprisoned, and hazarded the forfeiture of his life,\nby not giving in his effects till the last day, which was that of his\nexamination. During his confinement, his amusement was to give poison\nto dogs and cats, and see them expire by slower or quicker torments. To\nsum up the _worth_ of this man, at the several eras of his life; at his\nstanding in the pillory, he was worth above L200,000; at his commitment\nto prison, he was worth L150,000.\n\n\nFOOD OF THE ANCIENTS.\n\nThe diversity of substances which we find in the catalogue of articles\nof food is as great as the variety with which the art or the science\nof cookery prepares them. The notions of the ancients on this most\nimportant subject are worthy of remark. Their taste regarding meat\nwas various. Beef they considered the most substantial food: hence\nit constituted the chief nourishment of their athletae. Camels' and\ndromedaries' flesh was much esteemed, their heels most especially.\nDonkey-flesh was in high repute: Maecenas, according to Pliny, delighted\nin it; and the wild ass, brought from Africa, was compared to venison.\nIn more modern times we find Chancellor Dupret having asses fattened\nfor his table. The hog and the wild boar appear to have been held in\ngreat estimation; and a hog was called \"animal propter convivia natum;\"\nbut the classical portion of the sow was somewhat singular--\"vulva\nnil dulcius ampla.\" Their mode of killing swine was as refined in\nbarbarity as in epicurism. Plutarch tells us that the gravid sow was\nactually trampled to death, to form a delicious mass fit for the gods.\nAt other times, pigs were slaughtered with red-hot spits, that the\nblood might not be lost. Stuffing a pig with assafoetida and various\nsmall animals, was a luxury called \"porcus Trojanus;\" alluding, no\ndoubt, to the warriors who were concealed in the Trojan horse. Young\nbears, dogs, and foxes, (the latter more esteemed when fed upon\ngrapes,) were also much admired by the Romans; who were also so fond of\nvarious birds, that some consular families assumed the names of those\nthey most esteemed. Catius tells us how to drown fowls in Falernian\nwine, to render them more luscious and tender. Pheasants were brought\nover from Colchis, and deemed at one time such a rarity, that one of\nthe Ptolemies bitterly lamented his never having tasted any. Peacocks\nwere carefully reared in the island of Samos, and sold at such a high\nprice, that Varro informs us they fetched yearly upwards of L2,000 of\nour money.\n\n\nTHE EARLIEST ENGLISH BIBLE.\n\nThe first translation of any part of the Holy Scriptures into English\nthat was committed to the press was the New Testament, translated from\nthe Greek, by William Tyndale, with the assistance of John Foye and\nWilliam Roye, and printed first in 1526, in octavo.\n\nTyndale published afterwards, in 1530, a translation of the Five Books\nof Moses, and of Jonah, in 1531, in octavo. An English translation of\nthe Psalter, done from the Latin of Martin Bucer, was also published\nat Strasburgh in 1530, by Francis Foye, in octavo. And the same book,\ntogether with Jeremiah and the Song of Moses, were likewise published\nin 1534, in duodecimo, by George Joye, sometime Fellow of Peter-House\nin Cambridge.\n\nThe first time the whole Bible appeared in English was in the year\n1535, in folio. The translator and publisher was Miles Coverdale,\nafterwards Bishop of Exeter, who revised Tyndale's version, compared\nit with the original, and supplied what had been left untranslated\nby Tyndale. It was printed at Zurich, and dedicated to King Henry\nthe Eighth. This was the Bible, which by Cromwell's injunction of\nSeptember, 1536, was ordered to be laid in churches.\n\n\nGREAT ERUPTION OF MOUNT ETNA.\n\nOne of the most remarkable eruptions of this mountain was that which\noccurred in the year 1669, which was so violent that fifteen towns\nand villages were destroyed, and the stream was so deep that the lava\nflowed over the walls of Catania, sixty feet in height, and destroyed\na part of the city. But the most singular circumstance connected with\nthis eruption was the formation of a number of extensive fissures,\nwhich appeared as though filled with intumescent rock. At the very\ncommencement of the volcanic excitement, one was formed in the plain\nof St. Lio, twelve miles in length and six feet broad, which ejected\na vivid flame, and shortly after five others were opened. The town of\nNicolosi, situated twenty miles from the summit of Etna, was destroyed\nby earthquake; and near the place where it stood two gulfs were formed,\nfrom which so large a quantity of sand and scoriae was thrown, that\na cone, called Mount Rossi, four hundred and fifty feet high, was\nproduced in about three months.\n\n\nAMULETS WORN BY MODERN EGYPTIAN FEMALES.\n\n[Illustration [++] Amulet.]\n\nOne of the most remarkable traits in modern Egyptian superstition is\nthe belief in written charms. The composition of most of these amulets\nis founded upon magic, and occasionally employs the pen of almost\nevery village schoolmaster in Egypt. A person of this description,\nhowever, seldom pursues the study of magic further than to acquire\nthe formulae of a few charms, commonly consisting, for the greater\npart, of certain passages of the Koran, and names of God, together\nwith those of spirits, genii, prophets, or eminent saints, intermixed\nwith combinations of numerals, and with diagrams, all of which are\nsupposed to have great secret virtues. The amulet thus composed, or\n_hhega'b_, as it is called, is covered with waxed cloth, to preserve\nit from accidental pollution, and enclosed in a case of thin embossed\ngold or silver, which is attached to a silk string, or a chain, hung\non the right side, above the girdle, the string or chain being passed\nover the left shoulder. Sometimes these cases bear Arabic inscriptions,\nsuch as \"Ma'sha-lla'h\" (\"God's will\") and \"Ya'cha'dee el-hhaga't\" (\"O\ndecreer of the things that are needful!\") We here insert an engraving\nof three hhega'bs of gold, attached to a string, to be worn together.\nThe central one is a thin, flat case, containing a folded paper: it is\nabout a third of an inch thick; the others are cylindrical cases, with\nhemispherical ends, and contain scrolls: each has a row of burck along\nthe bottom. Hhega'bs such as these, or of a triangular form, are worn\nby many children, as well as women; and those of the latter form are\nattached to a child's head-dress.\n\n\nPERSONAL ORNAMENTS OF THE ANCIENT EGYPTIANS.\n\nThe passion of the Egyptians for decorative jewellery was indeed\nexcessive. Men as well as women delighted thus to adorn themselves;\nand the desire was not confined to the higher ranks, for though the\nsubordinate classes could not afford the sparkling gems and precious\nmetals which glowed upon the persons of their superiors, their vanity\nwas gratified by humbler imitations, of bronze, glass, and porcelain.\n\n\"Costly and elegant ornaments,\" observes Professor Rosselini, \"abounded\nin proportion as clothing in general was simple and scarce among the\nEgyptians. Girdles, necklaces, armlets, ear-rings, and amulets of\nvarious kinds suspended from the neck, are found represented in the\npainting, and in fact still exist on the mummies. Figures of noble\nyouths are found entirely devoid of clothing, but richly ornamented\nwith necklaces and other jewels.\"\n\nAn immense number of these \"jewels of silver and jewels of gold\"\nhave been found in the tombs, and on the persons of mummies, and are\ndeposited in profusion in every museum. The accompanying engravings\nwill give an idea of the style and form of some of them.\n\n[Illustration [++] Personal Ornaments.]\n\nThe ear-rings generally worn by the ladies were large, round, single\nhoops (as _a_) from 1-1\/2 inches to 2-1\/3 inches in diameter; and\nfrequently of a still greater size; or made of six rings soldered\ntogether (as _b_); sometimes an asp, whose body was of gold, set with\nprecious stones, was worn by persons of rank as a fashionable caprice.\nFigures _c_, _d_, of gold bear the heads of fanciful animals; _e_,\nalso of gold, is remarkable for its singularity of form, and for the\ndelicacy of its workmanship; and _f_ for its carrying two pearls and\nbeing double in its construction.\n\n[Illustration [++] Ear-Rings.]\n\nBracelets, armlets, and anklets were worn by men as well as by women;\nthey were usually of gold, frequently set with precious stones, or\ninlaid with enamel. The one marked _a_ in the annexed cut is now in the\nLeyden Museum: it is of gold, 3 inches in diameter, and 1-1\/2 inches in\nheight, and is interesting, because it belonged to the Pharaoh whom we\nconclude to have been the patron and friend of Joseph, Thothmes III.,\nwhose name it bears. The armlet _b_ is of gold, and represents a snake;\nthe other, _c_, is of bronze. Rings were worn in profusion, gold being\nthe material chiefly selected. Some resemble watch seals of the present\nday--sometimes the stone having four flat sides, all engraved, turned\non a pivot, like some seals seen at present. One of this character,\nwhich Sir J. G. Wilkinson estimates to contain 20_l._ worth of gold,\nis represented at _d_ in the above engraving. It consists of a massive\nring of gold, bearing an oblong plinth of the same metal, an inch in\nlength, and more than half an inch in its greatest width. On one side\nis engraved the hieroglyphic name of Storus, the successor of Amunoph\nIII.; the three others contain respectively a scorpion, a crocodile,\nand a lion.\n\n\nGREAT PEAR TREE.\n\nThe most remarkable pear tree in England stands on the glebe of the\nparish of Holme Lacy, in Herefordshire. When the branches of this\ntree, in its original state, became long and heavy, their extremities\ndrooped till they reached the ground. They then took root; each branch\nbecame a new tree, and in its turn produced others in the same way.\nEventually it extended itself until it covered more than an acre of\nground, and would probably have reached much further if it had been\nsuffered to do so. It is stated in the church register, that \"the great\nnatural curiosity, the great pear tree upon the glebe, adjoining to the\nvicarage-house, produced this year (1776) fourteen hogsheads of perry,\neach hogshead containing one hundred gallons.\" Though now much reduced\nin size, it is still healthy and vigorous, and generally produces from\ntwo to five hogsheads. The liquor is not of a good quality, being very\nstrong and heating. An idea of the superior size of this tree, when in\nits prime, over others of the same kind, may be formed from the fact,\nthat in the same county, an acre of ground is usually planted with\nthirty trees, which, in a good soil, produce annually, when full grown,\ntwenty gallons of perry each. So large a quantity as a hogshead from\none tree is very unusual. The sorts principally used for perry are such\nas have an austere juice.\n\n\nLAW OF THE MOZCAS.\n\nA very remarkable law prevailed among the Mozcas, one of the tribes\nof the Nuevo Reyno de Granada. There, as among more advanced nations,\nthe king could do no wrong; but the subordinate chiefs could. These\nchiefs were men, the people reasoned, like themselves; they could not\nbe punished by their vassals, for there would be a natural unfitness\nin that; the king, it seems, was not expected to interfere, except\nin cases of state offences; the power of punishment, therefore, was\nvested in their wives; and a power it was, says Piedrahita, which they\nexercised famously whenever it fell to them to be judges of their poor\nhusbands. The conqueror Quesada calling one morning upon the chief of a\nplace called Suesca, found him under the hands of his nine wives, who\nwere tying him, and having done so, proceeded, in spite of Quesada's\nintercession, to flog him one after the other. His offence was, that\nsome Spaniards the night before had lodged in his house, and he had\npartaken too freely of their Spanish wine. Drunkenness was one of the\nsins which fell under the cognizance of his wives: they carried him\nto bed that he might sleep himself sober, and then awoke him in the\nmorning to receive the rigour of the law.\n\n\nLARGEST METAL STATUE IN THE WORLD.\n\nArona is an island on the Lago Maggiore, and has a strong castle. Upon\nan eminence is a statue of bronze to St. Charles Borromeo, from whom\nthe hill is called, Monte di S. Carlo. The statue was erected by the\nPope in 1624, in memory of the Saint, who was Archbishop of Milan.\nThe pedestal of the statue is thirty-six feet high. It is the largest\nmetal statue in existence; and the height of the statue itself is\nseventy-two feet, making a total of 108 feet. Fifteen persons may get\ninto the saint's head, which will also accommodate four persons and a\ntable on which they can dine. The cost is said to have been one million\none hundred Milanaise livres.\n\n\nTHE OAK OF MAMRE.\n\nIn one remarkable instance the Jews, the Christians, and the pagan\nArabs united in religious feelings. This was in their reverence for\nthe Oak of Mamre, where the angels appeared to Abraham: for Abraham's\nsake the Jews held the place holy; the Arabs for the angels'; the\nChristians, because, in their ignorance of their Scriptures, they\naffirmed that the Son of God had accompanied those angels to destroy\nSodom and Gomorrah. An annual fair was held there, and every man\nsacrificed after the manner of his country; nor was the meeting ever\ndisgraced by any act of intemperance or indecency. Nothing had been\ndone to injure the venerable antiquity of the place. There was nothing\nbut the well which Abraham had dug, and the buildings which he had\ninhabited, beside the oak. These remains were destroyed by order of\nConstantine, in abhorrence of the _impious_ toleration exhibited\nthere! A church was built upon the spot, and Mamre, so interesting to\nthe poet, the philosopher, and the pious man, became a mere den of\nsuperstition.\n\n\nSTRANGE ADVERTISEMENT.\n\nThe following appeared in the _Evening Post_, May 23rd, 1730:--\n\n\"I, Elizabeth, duchess dowager of Hamilton, acknowledge I have for\nseveral months been ill in my health, but never speechless, as certain\npenny authors have printed; and so, to confute these said authors\nand their intelligence, it is thought by my most intimate friends,\n_it is the very last thing that will happen to me_. I am so good an\nEnglishwoman, that I would not have my countrymen imposed upon by\npurchasing false authors; therefore, have ordered this to be printed\nthat they may know what papers to buy and believe, that are not to be\nbribed by those who may have private ends for false reports. The copy\nof this is left in the hands of Mr. Berington, to be shown to any body\nwho has a curiosity to see it signed with my own hand.\n\n \"E. HAMILTON.\"\n\n\nINTERMITTENT SPRINGS.\n\nOne of the most remarkable of these is at Bolder-Born in Westphalia.\nAfter flowing for twenty-four hours, it entirely ceases for the\nspace of six hours. It then returns with a loud noise, in a stream\nsufficiently powerful to turn three mills very near its source. Another\nspring of the same nature occurs at Bihar in Hungary, which issues many\ntimes a day, from the foot of a mountain, in such a quantity as in a\nfew minutes to fill the channel of a considerable stream.\n\nThe Lay Well near Torbay, ebbs and flows sixteen times in an hour: and\nin Giggleswick Well in Yorkshire, the water sometimes rises and falls\nin ten or fifteen minutes.\n\nSt. Anthony's Well, on Arthur's Seat, near Edinburgh, has a similar\nmovement, but on a smaller scale.\n\nIn Savoy, near the lake of Bourget, is another spring of this kind, but\nit differs from those which have been already mentioned in being very\nuncertain in its intervals.\n\n\nCURIOUS JEWEL WHICH BELONGED TO JAMES I.\n\nIn former times it was a common practice with princes and nobles to\nhave elaborate articles of jewellery constructed in such forms as had\na religious and emblematical signification. An inventory of the Dukes\nof Burgundy, made in 1396, speaks of a _fleur-de-lys_ which opened,\nand contained inside a picture of the Crucifixion. In 1416, the Duke\nof Berri had \"a fair apple,\" which opened, and contained within on one\nside the figure of Christ, and on the other that of the Virgin. Among\nthe jewels of the Dukes of Burgundy in 1392 there were two pears of\ngold, enamelled, each containing an image of Our Lady. We find similar\nentries in the other different inventories of the Dukes of Burgundy:\nAn apple of silver, enamelled, containing in the inside a picture of\nSt. Catherine, in 1400; a pine-apple of gold, which contained figures\nof the birth of Christ, and of the three kings, in 1467; and, in the\nsame year, two apples of gold, one containing, on the opposite halves,\nOur Lady and St. Paul, the other, St. Peter and St. Paul--the latter\nsuspended by three small chains. These kinds of devices continued in\nfashion till a much later period, and a very curious example, from the\ncollection of Lord Londesborough, which appears to have belonged to\nKing James I., is here engraved.\n\n[Illustration [++] Small Skull.]\n\n[Illustration [++] Apple.]\n\nThe whole is of silver, and the leaves appear to have been painted\ngreen. On opening it we find in the inside the small skull here\nrepresented above the apple. The top of the skull opens like a lid,\nand inside are two small paintings, representing the Creation and the\nResurrection, with the inscription, \"_Post Mortem, vita eternitas_.\"\nThe external inscription is not gallant. To give the apple externally\na more natural appearance, there are marks of two bites on the side\nopposite that here represented, showing a large and small set of teeth.\n\n\nSTRANGE CURIOSITIES.\n\nIn the Anatomy Hall of Leyden is a drinking cup of the skull of a Moor,\nkilled in the beleaguring of Haerlem. Also a cup made of a double brain\npan. We observe also that No. 51 is the skin of a woman, and No. 52 the\nskin of a woman, prepared like leather; No. 53 the skin of a Malacca\nwoman, above 150 years old, presented by Richard Snolk, who probably\nhad her flayed.\n\n\nTHE CROSS OF CONG.\n\nThe cross, of which the following is a correct representation,\npossesses eminent claims to a place among our curiosities, since it\nconstitutes the gem of the Museum of the Royal Irish Academy.\n\n[Illustration [++] Cross of Kong.]\n\nThis cross was made at Roscommon, by native Irishmen, about the year\n1123, in the reign of Turlogh O'Connor, father of Roderick, the last\nmonarch of Ireland, and contains what was supposed to be a piece of the\ntrue cross, as inscriptions in Irish, and Latin in the Irish character,\nupon two of its sides record. The engraving affords a correct idea of\nthe original, as the extremely minute and elaborate ornaments with\nwhich it is completely covered, and a portion of which is worked in\npure gold, could not possibly be expressed on so reduced a scale. The\nornaments generally consist of tracery and grotesque animals fancifully\ncombined, and similar in character to the decorations found upon\ncrosses of stone of about the same period. A large crystal, through\nwhich a portion of the wood which the cross was formed to enshrine is\nvisible, is set in the centre.\n\n\nFOOT-RACING IN 1699.\n\nA remarkable foot-race was run about the year 1699, which is thus\ndescribed in the manuscript journal of a lady who was one of the\nspectators:--\"I drove through the forest of Windsor to see a race run\nby two footmen, an English and a Scotch, the former a taller bigger man\nthan the other. The ground measured and cut even in a round was about\nfour miles; they were to run it round so often as to make up twenty-two\nmiles, which was the distance between Charing Cross and Windsor Cross,\nthat is, five times quite round, and so far as to make up the odd\nmiles and measure. They ran a round in twenty-five minutes. I saw them\nrun the first three rounds and half another in an hour and seventeen\nminutes, and they finished it in two hours and a half. The Englishman\ngained the start the second round, and kept it at the same distance the\nfive rounds, and then the Scotchman came up to him and got before him\nto the post. The Englishman fell down within a few yards of the post.\nMany hundred pounds were won and lost about it. They ran both very\nneatly, but my judgment gave it to the Scotchman, because he seemed to\nsave himself to the last push.\"\n\n\nTHE CHERRY TREE.\n\nThe Cherry Tree was introduced into Great Britain before A.D. 53. The\nearliest mention of the fruit being exposed to sale by hawkers in\nLondon is in Henry the Fifth's reign, 1415. New sorts were introduced\nfrom Flanders, by Richard Haines, Henry the Eighth's fruiterer, and\nbeing planted in Kent were called \"Flanders,\" or \"Kentish Cherries,\" of\nwhich Gerard (1597) says, \"They have a better juice, but watery, cold,\nand moist.\" Philips says, \"There is an account of a cherry-orchard of\nthirty-two acres in Kent, which, in the year 1540, produced fruit that\nsold in those early days, for 1,000_l._; which seems an enormous sum,\nas at that period good land is stated to have let at one shilling per\nacre.\" Evelyn tells us, that in his time (1662) an acre planted with\ncherries, one hundred miles from London, had been let at 10_l._ During\nthe Commonwealth (1649), the manor and mansion of Henrietta Maria,\nQueen of Charles I., at Wimbledon, in Surrey, were surveyed previously\nto being sold, and it appears that there were upwards of two hundred\ncherry trees in the gardens. Since that time the cherry tree has found\nuniversal admission into shrubberies, gardens, and orchards.\n\n\nINSTRUCTIONS TO A CHAPLAIN.\n\nThe following, and we believe they are unique, are Sir John Wynne of\nGwedir's instructions to his chaplain, the Rev. John Pryce. \"First, you\nshall have the chamber I showed you in my gate, private to yourself,\nwith lock and key, and all necessaries. In the morning, I expect you\nshould rise, and say prayers in my hall, to my household below, before\nthey go to work, and when they come in at night, that you call before\nyou all the workmen, specially the youth, and take account of them\nof their belief, and of what Sir Meredith taught them. I beg you to\ncontinue for the most part in the lower house: you are to have onlye\nwhat is done there, that you may inform me of any disorder there. There\nis a bayliff of husbandry and a porter, who will be commanded by you.\nThe morning after you be up, and have said prayers, as afore, I would\nyou to bestow in study on any commendable exercise of your body. Before\ndinner you are to come up and attend grace or prayers, if there be any\npublicke; and to sit up if there be not greater strangers above the\nchyldren, who you are to teach in your own chamber. When the table\nfrom half downwards is taken up, then you are to rise and to walk in\nthe alleys near at hand until grace time, and to come in then for that\npurpose. After dinner, if I be busy, you may go to bowles, shuffel\nbord, or any other honest, decent recreation, until I go abroad. If\nyou see me void of business, and go to ride abroad, you shall command\na gelding to be made ready by the grooms of the stable, and to go with\nme. If I go to bowles or shuffel bord, I shall lyke of your company,\nif the place be not made up with strangers. I would have you to go\nevery Sunday in the year to some church hereabouts, to preache, giving\nwarnynge to the parish, to bring the yowths at after noon to the church\nto be catechysed; in which poynt is my greatest care that you should\nbe paynfull and dylygent. Avoyd the alehouse, to sytte and keepe\ndrunkard's company ther, being the greatest discredit your function can\nhave.\"\n\n\nTWO MISERS.\n\nIn the year 1778 died, at a village near Reading, John Jackson,\naged ninety-three, and James Jackson, aged eighty-seven. These two\nbrothers were old bachelors, and afforded a striking instance of the\ninsufficiency of wealth to create happiness. Though these old men had\nbeen blest with great riches ever since they were twenty years of age,\nthey absolutely denied themselves the common necessaries of life;\nand lived in the village for fifty years past as poor men, and often\naccepted of charity from rich persons who resided near them. They never\nsuffered any woman or man to come into their apartment (which was only\none shabby room), and were both taken ill, and languishing a short\ntime, they expired on the same day, within one hour of each other. It\nis computed, by the writings left behind them, that they died worth\nL150,000.\n\n\nANECDOTE OF THE HOUSE OF BRUNSWICK.\n\nThe following anecdote relating to the august House of Brunswick\nis taken from the \"Annual Register\" of 1765:--\"The late Duchess of\nBlakenburgh, great grandmother to the hereditary prince, who died\nsome years since in a very advanced age, had the singular happiness\nto reckon amongst her posterity, sixty-two princes and princesses;\n(fifty-three of whom she saw at one time alive;) and amongst them three\nemperors, two empresses, two kings, and two queens; a circumstance\nthat, probably, no sovereign house but that of Brunswick ever produced\nanything like it.\"\n\n\nAMUSEMENTS OF SOME LEARNED AUTHORS.\n\nTycho Brahe polished glass for spectacles, and made mathematical\ninstruments. D'Andilly delighted, like our Evelyn, in forest-trees;\nBalzac, with the manufacturing of crayons; Pieresc, with his medals and\nantiques; the Abbe de Marolles, with engravings; Rohault's greatest\nrecreation was in seeing different mechanics at their labour; Arnauld\nread the most trashy novels for relaxation; as did our Warburton, the\nlate Lords Loughborough and Camden; Montaigne fondled his cat; Cardinal\nRichelieu, in jumping and leaping. Grumm informs us that the Chevalier\nde Boufflers would crow like a cock, and bray like an ass; in both of\nwhich he excelled, not metaphorically but literally.\n\n\nEARLY GERMAN DRINKING CUP.\n\n[Illustration: [++] Early German Drinking Cup.]\n\nThe above, taken from the Londesborough collection, is a good example\nof the German drinking cups of fanciful shape, which were so much in\nfashion in that country in the sixteenth and seventeenth centuries. The\nspecimen before us is of silver, and dated 1619. The mill and scroll\nornament on the cup are gilt. It was held in the hand to be filled, and\ncould not be set down until emptied; the drinker, blowing through the\ntube into the mill, set the sails in motion, and reversed the cup on\nthe table.\n\n\nTHE KING'S STONE.\n\n[Illustration: [++] The King's Stone.]\n\nKingston-on-Thames is among the oldest of English towns; and is said\nto have been \"the metropolis of the Anglo-Saxon kings:\" certainly it\nwas a famous place when the Romans found and conquered the Britons in\nthis locality: there are indeed arguments for believing that the \"ford\"\nwhich Caesar crossed was here, and not at Walton; and indications of\nbarrows, fosses, and ramparts of Roman origin, are to be found in many\nplaces in the neighbourhood. It is more than probable that a bridge was\nconstructed by the Romans here, and that a fortress was erected for its\nprotection. The Saxons followed in due course, and here they had many\ncontests with their enemies the Danes; but A.D. 838, Egbert convened at\nKingston an assembly of ecclesiastics and nobles in council, and here,\nundoubtedly, many of the Saxon kings were crowned: \"The townish men,\"\nsays Leland, \"have certen knowledge that a few kinges were crounid\nafore the Conqueste.\" Its first charter was from King John, and many\nsucceeding sovereigns accorded to it various grants and immunities.\nDuring the war between Charles I. and the Parliament, Kingston was the\nscene of several \"fights,\" being always on the side of the king. The\ntown is now populous and flourishing, although without manufactures of\nany kind. Since the establishment of a railway, villa residences have\nlargely increased in the neighbourhood; and the two suburbs, Surbiton\nand Norbiton, are pretty and densely-crowded villages of good houses.\nThe church has suffered much from mutilation and restoration; it is a\nspacious structure, and was erected about the middle of the fourteenth\ncentury, on the site of an earlier edifice. Amongst the monuments is a\nfine brass, to a civilian and his wife, of the year 1437. Of existing\nantiquities there are but few: county historians, however, point out\nthe sites of the ancient Saxon palace, \"the castle,\" the Jews' quarter,\nand the Roman town, Tamesa; and the game of \"foot-ball,\" it is said, is\nstill practised by the inhabitants on Shrove Tuesday, in commemoration\nof the feats of their ancestors, by whom the head of a king-assassin\nwas \"kicked\" about the town. But perhaps the most interesting object\nnow to be found in Kingston is \"THE KING'S STONE.\" It had long\nremained neglected, though not unknown, among disregarded heaps of\n_debris_ in \"the new court-yard,\" when it occurred to some zealous and\nintelligent antiquaries that so venerable a relic of remote ages was\nentitled to some show of respect. It was consequently removed from its\ndegraded position, planted in the centre of the town, and enclosed by\na \"suitable\" iron railing. It is now, therefore, duly and properly\nhonoured, as may be seen by the preceding engraving.\n\n\nTRANCE AT WILL.\n\nColonel Townsend possessed the remarkable faculty of throwing himself\ninto a trance at pleasure. The heart ceased apparently to throb at\nhis bidding, respiration seemed at an end, his whole frame assumed\nthe icy chill and rigidity of death; while his face became colourless\nand shrunk, and his eye fixed, glazed, and ghastly. His mind itself\nceased to manifest itself, for during the trance it was as utterly\ndevoid of consciousness as his body of animation. In this state he\nwould remain for hours, when these singular phenomena wore away, and\nhe returned to his usual condition. Medical annals furnish no parallel\nto this extraordinary case. Considered whether in a physiological or\nmetaphysical point of view, it is equally astonishing and inexplicable.\n\n\nDESTRUCTIVE FORCE OF RATS.\n\nThe amount of destructive force possessed by rats cannot be better\nexemplified than in the report given to the French Government, relating\nto the removal of the horse slaughter-houses, situated at Montfaucon,\nto a greater distance from Paris; one great objection being the\ndisastrous consequences which might accrue to the inhabitants of the\nneighbourhood, if these voracious creatures were suddenly deprived of\ntheir usual sustenance. It is well known that the mischief which they\noccasion is not confined to what they eat; but they undermine houses,\nburrow through dams, destroy drains, and commit incalculable havoc in\nevery place and in everything.\n\nThe report states, that the carcases of horses killed one day, and\namounting to thirty-five, would be found the next morning with the\nbones picked clean. A person of the name of Dusaussois, belonging\nto the establishment, made this experiment. A part of his yard was\nenclosed by solid walls, at the foot of which, several holes were made\nfor the entrance and exit of the rats. Into this enclosure he put the\nbodies of three horses, and in the middle of the night he stopped up\nall the holes as quietly as he could; he then summoned several of his\nworkmen, and each, armed with a torch and a stick, entered the yard,\nand carefully closed the door. They then commenced a general massacre;\nin doing which, it was not necessary to take aim, for wherever the blow\nfell it was sure to knock over a rat, none being allowed to escape by\nclimbing over the walls. This experiment was repeated at intervals of\na few days, and at the end of a month, 16,050 rats had been destroyed.\nIn one night they killed 2,650; and yet this cannot give an entirely\nadequate idea of their number, for the yard in question did not cover\nmore than a twentieth part of the space allotted to killing horses. The\nrats in this place have made burrows for themselves, like catacombs;\nand so great is their number, that they have not found room close by\nthe slaughter-houses. They have gone farther; and the paths to and from\ntheir dwellings may be traced across the neighbouring fields.\n\n\nORDEAL OF THE CROSS.\n\nWhen a person accused of any crime had declared his innocence upon\noath, and appealed to the cross for its judgment in his favour, he\nwas brought into the church before the altar. The priest previously\nprepared two sticks exactly like one another, upon one of which was\ncarved a figure of the cross. They were both wrapped up, with great\ncare and many ceremonies, in a quantity of fine wool, and laid upon the\naltar, or on the relics of the saints. A solemn prayer was then offered\nup to God, that he would be pleased to discover, by the judgment of\nhis holy cross, whether the accused person were innocent or guilty.\nA priest then approached the altar, and took up one of the sticks,\nand the assistants unswathed it reverently. If it was marked with the\ncross, the accused person was innocent; if unmarked, he was guilty.\nIt would be unjust to assert, that the judgments just delivered were\nin all cases erroneous; and it would be absurd to believe that they\nwere left altogether to chance. Many true judgments were doubtless\ngiven, and, in all probability, most conscientiously; for we cannot but\nbelieve that the priests endeavoured beforehand to convince themselves\nby strict inquiry and a strict examination of the circumstances,\nwhether the appellant were innocent or guilty, and that they took up\nthe crossed or uncrossed stick accordingly. Although, to all other\nobservers, the sticks, as enfolded in the wool, might appear exactly\nsimilar, those who enwrapped them could, without any difficulty,\ndistinguish the one from the other.\n\n\nKING JOHN AND POPE INNOCENT.\n\nWhen Cardinal Langton was made Archbishop of Canterbury, by the\nintrigues of the Pope, whose creature he was, in despite of King John,\nto appease the latter, his Holiness presented him with four gold rings,\nset with precious stones, and enhanced the value of the gift (mark\nthat, jewellers!) by informing him of the many mysteries implied in it.\nHe begged of him (John) to consider seriously the _form_ of the rings,\ntheir _number_, their _matter_, and their _colour_. Their _form_, he\nsaid, being round, shadowed out eternity, which had neither beginning\nnor end, and he ought thence to learn the duty of aspiring from\nearthly objects to heavenly, from things temporal to things eternal.\nThe _numbers_ four, being a square, denoted steadiness of mind, not\nto be subverted either by prosperity or adversity, fixed for ever in\nthe basis of the four cardinal virtues. _Gold_, which is the matter,\nbeing the most precious of metals, signified wisdom, which is the\nmost precious of all accomplishments, and justly preferred by Solomon\nto riches, power, and all exterior attainments. The blue _colour_ of\nthe sapphire represented faith; the verdure of the emerald hope; the\nrichness of the ruby charity; and the splendour of the topaz good works.\n\n\nDRUID'S SEAT.\n\n[Illustration: [++] Druid's Judgment Seat.]\n\nThe singular pile of stones which we have sketched here is popularly\ncalled the \"Druid's Judgment Seat,\" and stands near the village of\nKilliney, not far from Drogheda, near the Martello Tower. It was\nformerly enclosed within a circle of great stones and a ditch. The\nformer has been destroyed, and the latter so altered that little\nof its ancient character remains. The \"Seat\" is composed of large,\nrough, granite blocks, and if really of the period to which tradition\nrefers it, an unusual degree of care must have been exercised for its\npreservation. The following are its measurements: Breadth, at the\nbase, eleven feet and a half; depth of the seat, one foot nine inches;\nextreme height, seven feet.\n\n\nBOOTS AN OBJECT OF HONOUR.\n\nAmong the Chinese no relics are more valuable than the _boots_ which\nhave been worn by an upright magistrate. In Davis's interesting\ndescription of the empire of China, we are informed, that whenever\na judge of unusual integrity resigns his situation, the people all\ncongregate to do him honour. If he leaves the city where he has\npresided, the crowd accompany him from his residence to the gates,\nwhere his boots are drawn off with great ceremony, to be preserved\nin the hall of justice. Their place is immediately supplied by a new\npair, which, in their turn, are drawn off to make room for others\nbefore he has worn them five minutes, it being considered sufficient to\nconsecrate them that he should have merely drawn them on.\n\n\nSAINT LAWRENCE.\n\nIn the south aisle of the church at Tuxford, beneath a flowery arch is\na very rude relief of St. Lawrence placed on the gridiron. By him is a\nfellow with a pair of bellows, blowing the fire, and the executioner\ngoing to turn him. The zealous Fox, in his \"Martyrology,\" has this very\nthought, and makes the martyr say, in the midst of his sufferings,\n\"This side is now roasted; turn me, O tyrant dear.\"\n\n\nPARIS GARDEN AT BLACKFRIARS.\n\n[Illustration: [++] Paris Garden.]\n\nThe Blackfriar's Road now passes over the site of Paris Garden where,\nin the sixteenth century, bear and bull-baiting rejoiced the citizens,\nthe gala days being usually Sundays. Our cut is copied from the rare\nwoodcut map in the time of Henry VIII., in the library at Guildhall,\nand exhibits in the foreground the kennels for the dogs, and the tanks\nin which they were washed. A graphic description of the place has\nbeen left by Paul Hentzner, a German, who visited it in 1598. He says\nit was \"built in the form of a theatre, for the baiting of bulls and\nbears: they are fastened behind, and then worried by great English\nbull-dogs; but not without great risk to the dogs, from the horns of\nthe one and the teeth of the other; and it sometimes happens they are\nkilled upon the spot: fresh ones are immediately supplied in the place\nof those that are wounded or tired. To this entertainment there often\nfollows that of whipping a blinded bear, which is performed by five or\nsix men, standing circularly with whips, which they exercise upon him\nwithout any mercy, as he cannot escape from them because of his chain.\nHe defends himself with all his force and skill, throwing down all who\ncome within his reach, and are not active enough to get out of it,\nand tearing the whips out of their hands, and breaking them. At these\nspectacles, and everywhere else, the English are constantly smoking\ntobacco. Fruits, such as apples, pears, and nuts, according to the\nseason, are carried about to be sold, as well as ale and wine.\"\n\n\nCANVASS OF AN INSURANCE AGENT.\n\nThe Manchester agent of an Insurance Company, gives the following\ncurious results of a personal canvass at 1,349 houses, in seventy\nstreets in the district of Hulme and Charlton, chiefly rentals from\nL12 to L24 per annum. The inquiry showed that there were 29 insured; 8\npersons too old; 11 who never heard of life assurances, and who were\nanxious to have it explained to them; 471 who had heard of it, but did\nnot understand it; 419 who were disinclined to assure; 19 favourable,\nif their surplus incomes were not otherwise invested; 89 persons who\nhad it under consideration, with a view to assure, as soon as their\narrangements were completed, and who appointed times for the agent to\ncall again; 21 refused the circulars, or to allow an explanation; 175\ndoors not answered; 102 houses empty; 3 had sufficient property not to\nrequire it; 1 favourable, but afraid of litigation; 1 preferred the\nsaving's bank; 1 used abusive language; 2 would trust their families to\nprovide for themselves; and 1 had been rejected by an office, although\nhe never was unwell, and was consequently afraid to try again, although\nvery anxious.\n\n\nTERRA-COTTA WRITINGS.\n\nThe Assyrians, unlike any other nation of antiquity, employed pottery\nfor the same objects, and to the same extent as papyrus was used in\nEgypt. Thus bulletins recording the king's victories, and even the\nannals of his reign, were published on terra-cotta cylinders, shaped\nlike a rolling-pin, and usually hollow, and on hollow hexagonal\nprisms. These are of a remarkably fine material, sometimes unpolished\nor unglazed, and at others covered with a vitreous siliceous glaze,\nor white coating. On the cylinders the inscriptions are engraved\nlengthwise; on the prisms they are in compartments on each face.\nEach wedge is about one-eighth of an inch long, and the complicity\nwith which the characters (a cuneiform writing-hand) are arranged is\nwonderful, and renders them extremely difficult for a tyro to read.\nThose hitherto published or known, contain the annals of the reign of\nSennacherib, and the precis of the reign of another king.\n\nThere are the Shergat cylinder, containing the History of Tiglath\nPileser; a cylinder of Sargon; Sennacherib's cylinders; Esarhaddon's\ncylinder.\n\nSales of land and other title-deeds were also incised on pieces of this\npolished terra-cotta, and, in order to prevent any enlargement of the\ndocument, a cylinder was run round the edges, leaving its impression\nin relief; or if the names of witnesses were affixed, each impressed\nhis oval seal on the wet terra-cotta, which was then carefully baked\nin the kiln. The celebrated cylinders of carnelian, chalcedony, and\nother substances, were in fact the official or private seals by which\nthe integrity of these documents was attested. These title-deeds are\nportable documents of four or five inches square, convex on each side,\nand occasionally also at the edges. Their colour varies, being a bright\npolished brown, a pale yellow, and a very dark tint, almost black. The\npaste of which they are made is remarkably fine and compact. The manner\nin which the characters were impressed on the terra-cotta barrels and\ncylinders is not known; those on the bricks used for building were\napparently stamped from a mould, but those on the deeds and books\nwere separately incised, perhaps with a prismatic stick, or rod, or,\nas others have conjectured, with the edge of a square rod of metal.\nIn some instances, where this substance was used for taking accounts,\nit seems just possible that the moist clay, rolled up like paste, may\nhave been unrolled and incised with rods. The characters are often so\nbeautifully and delicately made, that it must have required a finely\nconstructed tool to produce them.\n\nSome small fragments of a fine reddish-grey terra-cotta which have been\nfound among the ruins, appear to contain calculations or inventories,\nwhilst others are perhaps syllabaries or vocabularies, to guide the\nAssyrian readers of these difficult inscriptions. A large chamber, or\nlibrary, of these archives, comprising histories, deeds, almanacks, and\nspelling-books, was found in the palace of Sennacherib at Kouyunjik.\nIt is supposed that altogether about 20,000 of these clay tablets\nor ancient books of the Assyrians, containing the literature of\nthe country, have been discovered. Some of the finer specimens are\ncovered with a pale straw-covered engobe, over which has been thrown a\nglaze. Some horoscopes have been already found on stone, and careful\nexamination has now detected the records of some astronomer royal of\nBabylon or Nineveh inscribed on a brick. Thus, while the paper and\nparchment learning of the Byzantine and Alexandrian schools has almost\ndisappeared after a few centuries, the granite pages of Egypt, and the\nclay leaves of Assyria, have escaped the ravages of time and the fury\nof barbarism.\n\nIn Egypt some receipts and letters have been discovered written on\nfragments of tile, and on the fine porcelain of the Chinese are\noften found extracts of biographical works, snatches of poetry, and\neven whole poems; but the idea of issuing journals, title-deeds,\ninventories, histories, prayers, and poems, not from the press, but\nfrom the kiln, is startling in the nineteenth century.\n\n\nWONDERFUL FORMATION OF THE EYE IN INSECTS.\n\nThe perfection which is bestowed on the organs of sense in insects,\nespecially when we consider their minuteness, is calculated to fill us\nwith adoring admiration of the skill of \"the Great Workmaster.\" Take an\nexample from the _eyes_, which are of several kinds, evidently designed\nfor distinct modes of vision, of which we, who have but one sort of\neyes, can form no adequate notion. The bee and many other insects have\non the crown of the head a number, usually three, of simple glassy\neyes, set like \"bull's-eyes\" in a ship's deck; and besides these a\ngreat compound eye on each side, consisting of a multitude of lenses\naggregated together upon the same optic nerve. The microscope reveals\nto us that the compound eye of an ant contains fifty lenses; that of\na fly, four thousand; that of a dragon-fly, twelve thousand; that of\na butterfly, seventeen thousand; and that of a species of _Mordella_\n(a kind of beetle), the amazing number of twenty-five thousand. Every\none of these regular, polished, and many-sided lenses; is the external\nsurface of a distinct eye, furnished with its own iris and pupil, and\na perfect nervous apparatus. It will thus be seen that each hexagonal\nfacet forms a transparent horny lens, immediately behind which is a\nlayer of pigment diminishing to a point in the centre, where it forms\na pupil; that behind this a long six-sided prism, answering to the\ncrystalline and vitreous humours in the human eye, extends, diminishing\nto its lower extremity, where it rests upon the retina, or network\nexpansion of the optic nerve. Some of the minuter details of this\nexquisite organisation are still matters of conflicting opinion; but\nthese we omit, as our purpose is rather to convey to our readers a\ngeneral idea of the structure of this complex organ of vision. \"This\nalso cometh forth from the Lord of Hosts, which is wonderful in counsel\nand excellent in working.\"\n\n\nFIRST COIN WITH BRITANNIA ON IT.\n\n[Illustration: [++] Roman Coins.]\n\nIn process of clearing away the foundations of Old London Bridge many\nantiquities were discovered; it had been the great highway over the\nThames from the Roman era, and numerous relics were obtained, varying\nin date from that period to our own. We here engrave such specimens\nof Roman coins that were found as belong to the Britannic series. The\nlarge central coin is one struck by Hadrian, and remarkable for the\nfigure of Britannia, the first time impersonated as an armed female\nseated on a rock. It is the prototype of the more modern Britannia,\nreintroduced by Charles II., and which still appears on our copper\nmoney. The smaller coins are such as were struck, during the reign of\nConstantine the Great, in the City of London, and are marked with the\nletters P. LON., for \"Pecunia Londinensis,\" money of London.\n\n\nEXTRAORDINARY FORMATION OF THE TWIN-WORM.\n\nAn extraordinary creature was discovered by Dr. Nordman, infesting the\ngills of one of our commonest river fishes--_Cyprinus Brama_--and to\nwhich he gave the appropriate appellation of the Twin-worm (_Diplozoon\nparadoxum_). It is not more than one-fourth of an inch in length, but\nconsists of two bodies, precisely resembling each other, united by\na central band, exactly in the manner of the Siamese youths, whose\nexhibition excited so much attention in England and America a few\nyears ago. We might have supposed that, like the human monstrosity\nin question, the Twin-worm was formed by the accidental union of two\nindividuals, if abundant observation had not proved that this is the\ncommon mode of life belonging to the species.\n\nEach portion of the animal is complete in all its organs and economy;\npossessing its own sets of suckers, its own mouth, its own digestive\ncanal, with its tree-like ramifications, its own perfect generative\nsystem, and its own elaborate series of vascular canals,--every organ\nor set of organs in the one-half finding its exact counterpart in the\nother.\n\nIt scarcely detracts from the marvellous character assumed by this\n\"Twin-worm,\" that, according to recent observations, the two halves\nhave already enjoyed a phase of existence as distinct individuals. The\norganic union, or \"fusion\" of two such individuals, is necessary to\nthe development of the generative system, which, up to that event, is\nwanting in each constituent half.\n\n\nMILL AT LISSOY.\n\n[Illustration: [++] Mill at Lissoy.]\n\nThe above picturesque sketch represents the \"busy mill\" at Lissoy,\nbetter known as \"Sweet Auburn--loveliest village of the plain\"--the\nscene of Goldsmith's beautiful poem of the \"Deserted Village.\" Lissoy,\nabout six miles from Athlone, stands on the summit of a hill at the\nbase of which is the mill that forms the subject of our sketch. The\nwheel is still turned by the water of a small rivulet, converted,\nnow and then, by rains, into a sufficient stream. The mill is a mere\ncountry cottage, used for grinding the corn of the neighbouring\npeasantry, and retains many tokens of age. Parts of the machinery are,\nno doubt, above a century old, and are probably the very same that left\ntheir impress on the poet's memory.\n\n\nA CASTLE BUILT FOR A GROAT.\n\nThe castle of Monkstown, near Cork, is reported by popular tradition to\nhave been built in 1636, at the cost of only a groat. To explain the\nenigma, the following story is told:--Anastatia Goold, who had become\nthe wife of John Archdeken, determined, while her husband was abroad,\nserving in the army of Philip of Spain, to give him evidence of her\nthrift on his return, by surprising him with a noble residence which he\nmight call his own. Her plan was to supply the workmen with provisions\nand other articles they required, for which she charged the ordinary\nprice; but, as she had made her purchases wholesale, upon balancing her\naccounts, it appeared that the retail profit had paid all the expenses\nof the structure except fourpence! The Archdekens were an Anglo-Irish\nfamily, who \"degenerating\" became \"Hibernices quam Hiberniores\"--more\nIrish than the Irish themselves--and assumed the name of Mac Odo, or\nCody. They \"forfeited,\" in 1688, having followed the fortunes of James\nII.\n\n\nBATTLE OF WATER-SNAKES.\n\nThe following story is narrated by Mr. St. John, in his \"Letters of\nan American Farmer.\" After describing the size and strength of some\nhemp-plants, around which a wild vine had formed natural arbours,\nhe thus proceeds:--\"As I was one day sitting, solitary and pensive,\nin this primitive arbour, my attention was engaged by a strange\nsort of rustling noise at some paces distance. I looked all around\nwithout distinguishing anything, until I climbed up one of my great\nhemp-stalks; when, to my astonishment, I beheld two snakes of a\nconsiderable length, the one pursuing the other with great celerity\nthrough a hemp-stubble field. The aggressor was of the black kind, six\nfeet long; the fugitive was a water snake, nearly of equal dimensions.\nThey soon met, and in the fury of their first encounter, appeared in an\ninstant firmly twisted together; and whilst their united tails beat the\nground, they mutually tried, with open jaws, to lacerate each other.\nWhat a fell aspect did they present! Their heads were compressed to a\nvery small size; their eyes flashed fire; but, after this conflict had\nlasted about five minutes, the second found means to disengage itself\nfrom the first, and hurried towards the ditch. Its antagonist instantly\nassumed a new posture, and, half-creeping, half-erect, with a majestic\nmien, overtook and attacked the other again, which placed itself in\na similar attitude, and prepared to resist. The scene was uncommon\nand beautiful; for, thus opposed, they fought with their jaws, biting\neach other with the utmost rage; but, notwithstanding this appearance\nof mutual courage and fury, the water snake still seemed desirous of\nretreating towards the ditch, its natural element. This was no sooner\nperceived by the keen-eyed black one, than, twisting its tail twice\nround a stalk of hemp, and seizing its adversary by the throat, not by\nmeans of its jaws, but by twisting its own neck twice round that of the\nwater snake, he pulled it back from the ditch. To prevent a defeat,\nthe latter took hold likewise of a stalk on the bank, and, by the\nacquisition of that point of resistance, became a match for his fierce\nantagonist. Strange was this to behold; two great snakes strongly\nadhering to the ground, mutually fastened together by means of the\nwrithings which lashed them to each other, and stretched at their full\nlength; they pulled, but pulled in vain; and in the moments of greatest\nexertion, that part of their bodies which was entwined seemed extremely\nsmall, while the rest appeared inflated, and now and then convulsed\nwith strong undulations rapidly following each other. Their eyes\nappeared on fire, and ready to start out of their heads. At one time\nthe conflict seemed decided; the water snake bent itself into great\nfolds, and by that operation rendered the other more than commonly\noutstretched; the next minute the new struggles of the black one gained\nan unexpected superiority; it acquired two great folds likewise, which\nnecessarily extended the body of its adversary in proportion as it\nhad contracted its own. These efforts were alternate; victory seemed\ndoubtful, inclining sometimes to one side, sometimes to the other,\nuntil at last the stalk to which the black snake was fastened suddenly\ngave way, and, in consequence of this accident, they both plunged into\nthe ditch. The water did not extinguish their vindictive rage, for by\ntheir agitations I could still trace, though I could not distinguish,\ntheir attacks. They soon reappeared on the surface, twisted together,\nas in their first onset; but the black snake seemed to retain its\nwonted superiority, for its head was exactly fixed above that of the\nother, which it incessantly pressed down under the water, until its\nopponent was stifled, and sank. The victor no sooner perceived its\nenemy incapable of further resistance, than, abandoning it to the\ncurrent, it returned to the shore and disappeared.\"\n\n\nFATES OF THE FAMILIES OF ENGLISH POETS.\n\nIt is impossible to contemplate the early death of Byron's only child\nwithout reflecting sadly on the fates of other females of our greatest\npoets. Shakspeare and Milton, each died without a son, but both left\ndaughters, and both names are now extinct. Shakspeare's was soon so.\nAddison had an only child--a daughter, a girl of some five or six\nyears at her father's death. She died, unmarried, at the age of eighty\nor more. Farquhar left two girls, dependant on the friendship of his\nfriend Wilkes, the actor, who stood nobly by them while he lived. They\nhad a small pension from the Government, and having long outlived\ntheir father, and seen his reputation unalterably established, both\ndied unmarried. The son and daughter of Coleridge both died childless.\nThe two sons of Sir Walter Scott died without children--one of two\ndaughters died unmarried, and the Scotts of Abbotsford and Waverley\nare now represented by the children of a daughter. How little could\nScott foresee the sudden failure of male issue? The poet of the \"Fairie\nQueen\" lost a child when very young by fire, when the rebels burned his\nhouse in Ireland. Some of the poets had sons and no daughters. Thus we\nread of Chaucer's son,--of Dryden's sons,--of the sons of Burns,--of\nAllan Ramsey's son,--of Dr. Young's son,--of Campbell's son,--of\nMoore's son,--and of Shelley's son. Ben Johnson survived all his\nchildren. Some, and those amongst the greatest, died unmarried--Butler,\nCowley, Congreve, Otway, Prior, Pope, Gay, Thompson, Cowper, Akenside,\nShenstone, Collins, Gray, Goldsmith, and Rogers, who lately died. Some\nwere unfortunate in their sons in a sadder way than death could make\nthem. Lady Lovelace has left three children--two sons and a daughter.\nHer mother is still alive to see, perhaps, with a softened spirit, the\nshade of the father beside the early grave of his only child. Ada's\nlooks, in her later years--years of suffering, borne with gentle and\nwomanly fortitude--have been happily caught by Mr. Henry Phillips,\nwhose father's pencil has preserved to us the best likeness of Ada's\nfather.\n\n\nJEFFERY HUDSON, THE DWARF OF THE COURT OF CHARLES I.\n\nThe celebrated dwarf of whom we here give a sketch, was born at Oakham\nin Rutlandshire in 1619, and about the age of seven or eight, being\nthen but eighteen inches high, was retained in the service of the\nDuke of Buckingham, who resided at Burleigh-on-the-Hill. Soon after\nthe marriage of Charles I., the king and queen being entertained at\nBurleigh, little Jeffery was served up at table in a cold pie, and\npresented by the duchess to the queen, who kept him as her dwarf.\nFrom seven years of age till thirty he shot up to three feet nine\ninches, and there fixed. Jeffery became a considerable part of the\nentertainment of the court. Sir William Davenant wrote a poem on a\nbattle between Jeffery and a turkey cock, and in 1638 was published\na very small book, called a \"New Year's Gift,\" presented at court by\nthe Lady Parvula to the Lord Minimus (commonly called Little Jeffery)\nher Majesty's servant, &c. &c., written by Microphilas, with a little\nprint of Jeffery prefixed. Before this period Jeffery was employed on\na negotiation of great importance; he was sent to France to fetch a\nmidwife for the queen; and on his return with this gentlewoman and her\nmajesty's dancing-master, and many rich presents to the queen from her\nmother, Mary de Medicis, he was taken by the Dunkirkers. This was in\n1630. Besides the presents he was bringing for the queen, he lost to\nthe value of L2,500 that he had received in France on his own account\nfrom the queen-mother and ladies of that court.\n\n[Illustration: [++] Jeffery Hudson.]\n\nJeffery thus made of consequence, grew to think himself really so.\nHe had borne with little temper the teasing of the courtiers and\ndomestics, and had many squabbles with the king's gigantic porter.\nAt last, being provoked by Mr. Crofts, a young gentleman of family,\na challenge ensued; and Mr. Crofts coming to the rendezvous armed\nonly with a squirt, the little creature was so enraged that a real\nduel ensued; and the appointment being on horseback, with pistols,\nto put them more on a level, Jeffery, with the first fire, shot his\nantagonist dead on the spot. This happened in France, whither he had\nattended his royal mistress in the troubles.\n\nHe was again taken prisoner by a Turkish rover, and sold into Barbary.\nHe probably did not long remain in slavery; for at the beginning of\nthe civil war, he was made a captain in the royal army, and in 1644\nattended the queen to France, where he remained till the restoration.\n\nAt last, upon suspicion of his being privy to the Popish Plot, he was\ntaken up in 1682 and confined in the Gate-house, Westminster, where he\nended his life in the sixty-third year of his age.\n\n\nCHURCH AT NEWTON, IRELAND.\n\n[Illustration: [++] Doorway of the Church at Newton, Ireland.]\n\nThe ancient doorway, of which, on account of its singular beauty, we\ngive a sketch, belongs to the church which was built by the first of\nthe Montgomeries at Newtown in Ireland. Though the church is a fine\nand beautiful example of architecture, no attempt whatever has been\nmade to preserve it from sinking into ruin. The Montgomeries, ancient\nlords of this district, were the descendants of that Montgomery who\naccidentally killed Henry II., of France, at a tournament. Some years\nafter the sad event, which was confessedly a mischance, he was taken\nby Catherine of Medicis, put to the torture and beheaded; with the\nadditional penalty of having his children degraded to villeinnage; on\nhis way to execution, he pronounced this noble and memorable sentence,\nin reference to the punishment inflicted on his children, \"If they\nhave not the virtue to raise themselves again, I consent to their\ndegradation.\"\n\n\nINTERESTING CALCULATION.\n\nSome years ago, an eminent zoologist gave the following table as\nhis estimate of the probable number of existing species of animals,\ndeduced from facts and principles then known. Later discoveries tend to\nincrease rather than to diminish the estimate.\n\n Quadrupeds 1,200\n Birds 6,800\n Reptiles 1,500\n Fishes 8,000\n Insects 550,000\n Worms 2,500\n Radiata 1,000\n Polypes, &c. 1,530\n Testacea 4,500\n Naked Testacea 600\n\nmaking an aggregate of 577,600 species.\n\n\nVITALITY OF SUPERSTITION.\n\nIn the \"Annual Register\" for 1760, an instance of the belief in\nwitchcraft is related, which shows how superstition lingers. A dispute\narose in the little village of Glen, in Leicestershire, between two\nold women, each of whom vehemently accused the other of witchcraft.\nThe quarrel at last ran so high that a challenge ensued, and they both\nagreed to be tried by the ordeal of swimming. They accordingly stripped\nto their shifts--procured some men, who tied their thumbs and great\ntoes together, cross-wise, and then, with a cart-rope about their\nmiddle, suffered themselves to be thrown into a pool of water. One of\nthem sank immediately, but the other continued struggling a short time\nupon the surface of the water, which the mob deeming an infallible sign\nof her guilt, pulled her out, and insisted that she should immediately\nimpeach all her accomplices in the craft. She accordingly told them\nthat, in the neighbouring village of Burton, there were several old\nwomen \"as much witches as she was.\" Happily for her, this negative\ninformation was deemed sufficient, and a student in astrology, or\n\"white-witch,\" coming up at the time, the mob, by his direction,\nproceeded forthwith to Burton in search of all the delinquents. After\na little consultation on their arrival, they went to the old woman's\nhouse on whom they had fixed the strongest suspicion. The poor old\ncreature on their approach locked the outer door, and from the window\nof an upstairs room asked what they wanted. They informed her that\nshe was charged with being guilty of witchcraft, and that they were\ncome to duck her; remonstrating with her at the same time upon the\nnecessity of submission to the ordeal, that, if she were innocent, all\nthe world might know it. Upon her persisting in a positive refusal to\ncome down, they broke open the door and carried her out by force, to a\ndeep gravel-pit full of water. They tied her thumbs and toes together\nand threw her into the water, where they kept her for several minutes,\ndrawing her out and in two or three times by the rope round her middle.\nNot being able to satisfy themselves whether she were a witch or no,\nthey at last let her go or more properly speaking, they left her on\nthe bank to walk home by herself, if she ever recovered. Next day they\ntried the same experiment upon another woman, and afterwards upon a\nthird; but fortunately, neither of the victims lost her life from this\nbrutality. Many of the ringleaders in the outrage were apprehended\nduring the week, and tried before the justices at quarter-sessions. Two\nof them were sentenced to stand in the pillory and to be imprisoned for\na month; and as many as twenty more were fined in small sums for the\nassault, and bound over to keep the peace for a twelvemonth.\n\n\nSMALL FEET OF THE CHINESE LADIES.\n\nThe compression of ladies' feet to less than half their natural size is\nnot to be regarded as a mark, or as a consequence, of the inferiority\nof the sex; it is merely a mark of gentility. Various accounts are\ngiven of the origin of this custom. One is, that an emperor was jealous\nof his wife, and to prevent her from gadding abroad, put her feet in\niron stocks. Another is, that a certain empress, Tan-ke (B.C. 1100),\nwas born with club-feet, and that she caused the emperor to issue an\nedict, adopting her foot as the model of beauty, and requiring the\ncompressing of female infants' feet so as to conform to the imperial\nstandard. While a third account is, that the Emperor Le-yuh (A.D. 961)\nwas amusing himself one day in his palace, when the thought occurred\nto him that he might improve the appearance of the feet of a favourite\nconcubine. He caused her feet to be so bent as to raise the instep\ninto an arch, to resemble the new moon. The figure was much admired\nby the courtiers, who soon began to introduce it into their families.\nIt is said that another emperor, two hundred years later, placed a\nstamp of the lotus-flower (water-lily) on the sole of the small shoe\nof his favorite concubine, so that at every step she took she left on\nthe ground the print of the flower; hence girls with small feet are\ncomplimented at the present day as \"the golden lilies.\" The operation\nof bandaging and compressing the feet is very painful; children cry\nvery much under it. Mortification of the feet has been known to result\nfrom the cruel practice. Custom, however, imposes it as a necessary\nattraction in a woman. An old gentleman at Canton, being asked the\nreason why he had bandaged his daughter's feet, replied, that if she\nhad large feet she could not make a good marriage.\n\n\nWONDERFUL CONSTRUCTION OF THE SEA-URCHIN.\n\nProfessor Forbes informs us that in a moderate-sized Urchin there are\nsixty-two rows of pores in each of the ten avenues, and as there are\nthree pairs of pores in each row, the total number of pores is 3,720;\nbut as each sucker occupies a pair of pores, the number of suckers is\n1,860. He says, also, that there are above three hundred plates of one\nkind, and nearly as many of another, all dovetailing together with the\ngreatest nicety and regularity, bearing on their surfaces above 4,000\nspines, each spine perfect in itself, and of a complicated structure,\nand having a free movement in its socket. \"Truly,\" he adds, \"the\nskill of the Great Architect of Nature is not less displayed in the\nconstruction of a Sea-urchin than in the building up of a world!\"\n\n\nIVORY SCEPTRE OF LOUIS XII.\n\n[Illustration: [++] Ivory Sceptre.]\n\nThe above engraving represents an ivory sceptre, or Main de Justice,\nwhich was made at the early part of the sixteenth century for Louis\nXII., King of France. The three parts 1, 2, 3, screw together and form\nthe sceptre. Fig. 4 is the hand on the top of the sceptre, given on\na larger scale, showing the ring set with a small pearl, worn on the\nthird finger. Fig. 5 is the inscription on the sceptre; it is engraved\nin relievo upon three of the convex decorations, and commences on the\nlowest one.\n\n\nTOMB OF CAECILIA METELLA.\n\n[Illustration: [++] Tomb of Caecilia Metella.]\n\nOf the tombs of Consular Rome nothing remains except perhaps the\nsarcophagus of Scipio; and it is only on the eve of the Empire that we\nmeet with the well-known one of Caecilia Metella, the wife of Crassus,\nwhich is not only the best specimen of a Roman tomb now remaining to\nus, but the oldest building of the imperial city of which we have\nan authentic date. It consists of a bold square basement about 100\nft. square, which was originally ornamented in some manner not now\nintelligible. From this rose a circular tower about 94 ft. in diameter,\nof very bold masonry, surmounted by a brace of ox-skulls with wreaths\njoining them, and a well-profiled cornice: 2 or 3 courses of masonry\nabove this seem to have belonged to the original work; and above this,\nalmost certainly, in the original design rose a conical roof, which has\nperished. The tower having been used as a fortress in the middle ages,\nbattlements have been added to supply the place of the roof, and it\nhas been otherwise disfigured, so as to detract much from its beauty\nas now seen. Still we have no tomb of the same importance so perfect,\nnor one which enables us to connect the Roman tombs so nearly with the\nEtruscan. The only addition in this instance is that of the square\nbasement or podium, though even this was not unknown at a much earlier\nperiod, as for instance in the tomb of Aruns. The exaggerated height\nof the circular base is also remarkable. Here it rises to be a tower\ninstead of a mere circular base of stones for the earthen cone of the\noriginal sepulchre. The stone roof which probably surmounted the tower\nwas a mere reproduction of the original earthen cone.\n\n\nPOGONIAS.\n\nThese vocal fish differ from the umbrinas in having their jaws tagged\nlaterally with many, in place of carrying but one barbel at the\nsymphysis. Schoeff reports of them that they will assemble round the\nkeel of a vessel at anchor, and serenade the crew; and Mr. John White,\nlieutenant in the navy of the United States, in his voyage to the\nseas of China, relates to the same purpose, that being at the mouth\nof the river of Cambodia, the ship's company were \"astonished by some\nextraordinary sounds which were heard around the bottom of the vessel.\nThey resembled,\" he says, \"a mixture of the bass of the organ, the\nsound of bells, the guttural cries of a large frog, and the tones which\nimagination might attribute to an enormous harp; one might have said\nthe vessel trembled with it. The noises increased, and finally formed\na universal chorus over the entire length of the vessel and the two\nsides. In proportion as we went up the river the sounds diminished, and\nfinally ceased altogether.\" As the interpreter told Captain White, the\nship had been followed by a \"troop of fish of an oval and flattened\nform,\" they were most probably pogonias. Humboldt met with a similar\nadventure in the South Sea, but without suspecting its cause. \"On\nFebruary 20th, 1803, at seven P.M., the whole crew was astounded by a\nvery extraordinary noise, resembling drums beaten in the air; we at\nfirst attributed it to the breakers; speedily it was heard all over\nthe vessel, especially towards the poop, and was like the noise which\nescapes from fluid in a state of ebullition; we began to fear there\nmight be some leak in the bottom. It was heard synchronously in all\nparts of the vessel, but finally, about nine P.M., ceased altogether.\"\nHow these fish manage to _purr_ in the deep, and by means of what organ\nthey communicate the sound to the external air, is wholly unknown. Some\nsuppose it to proceed from the swim-bladder; but if that be the drum,\nwhat is the drumstick that beats upon it? And cushioned as it is in an\nobese envelope and without issue, the swim-bladder cannot be a bagpipe\nor wind instrument.\n\n\nCURIOUS ADVERTISEMENT.\n\nThe following appeared in the public papers of January 24th,\n1737:--\"Whereas Frances, wife of the Right Honourable the Lord Viscount\nVane, has, for some months past, absented herself from her husband,\nand the rest of her friends:--I do hereby promise to any person or\npersons who shall discover where the said lady Vane is concealed, to\nme, or to Francis Hawes, esq. her father, so that either of us may\ncome to the speech of her, the sum of L100, as a reward, to be paid by\nme on demand at my lodgings in Piccadilly. I do also promise the name\nof the person, who shall make such discovery, shall be concealed, if\ndesired. Any person concealing or lodging her after this advertisement,\nwill be prosecuted with the utmost rigour; or, if her ladyship will\nreturn to me, she may depend upon being kindly received. She is about\ntwenty-two years of age, tall, well-shaped, has light-brown hair, is\nfair-complexioned, and has her upper teeth placed in an irregular\nmanner. She had on, when she absented, a red damask French sacque, and\nwas attended by a French woman, who speaks very bad English.\n\n \"VANE.\"\n\n\nTHE EYE OF THE CHAMELEON.\n\nA most extraordinary aspect is communicated to chameleons by the\nstructure and movements of their eyes. In the first place, the head is\nenormous, and, being three-sided, with projecting points and angles,\nmakes a sufficiently uncouth visage; but the eyes which illuminate\nthis notable head-piece must, indeed, to borrow for the nonce the\nphraseology of Barnum, \"be seen to be appreciated.\" There is on\neach side an immense eye-ball, full and prominent, but covered with\nthe common shagreened skin of the head, except at the very entre,\nwhere there is a minute aperture, corresponding to the pupil. These\ngreat punctured eye-balls roll about hither and thither, but with\nno symmetry. You cannot tell whether the creature is looking at you\nor not; he seems to be taking what may be called a general view of\nthings--looking at nothing in particular, or rather, to save time,\nlooking at several things at once. Perhaps both eyes are gazing\nupwards at your face; a leaf quivers behind his head, and in a moment\n_one eye_ turns round towards the object, while the other retains its\nupward gaze: presently a fly appears; one eye rapidly and interestedly\nfollows all its movements, while the other leisurely glances hither and\nthither, or remains steady. Accustomed as we are to see in almost all\nanimals the two eyes move in unison, this want of sympathy produces an\neffect most singular and ludicrous.\n\n\nDIVING FOR A WIFE.\n\nIn many of the Greek islands, the diving for Sponge forms a\nconsiderable part of the occupation of the inhabitants, as it has\ndone from the most remote antiquity. Hasselquist says:--\"Himia is a\nlittle, and almost unknown island directly opposite Rhodes. It is worth\nnotice, on account of the singular method the Greeks, inhabitants of\nthe island, have to get their living. In the bottom of the sea the\ncommon Sponge is found in abundance, and more than in any other place\nin the Mediterranean. The inhabitants make it a trade to fish up this\nSponge, by which they get a living far from contemptible, as their\ngoods are always wanted by the Turks, who use an incredible number of\nSponges at their bathings and washings. A girl in this island is not\npermitted by her relations to marry before she has brought up a certain\nquantity of Sponges, and before she can give proof of her agility by\ntaking them from a certain depth.\" In other islands the same custom\nprevails, but with reversed application, as in Nicarus, where the\nfather of a marriageable daughter bestows her on the best diver among\nher suitors,--\"He that can stay longest in the water, and gather the\nmost Sponges, marries the maid.\"\n\n\nKNIGHT'S COSTUME OF THE THIRTEENTH CENTURY.\n\nThe engraving represents a knight's costume of the year 1272,\ntaken from the library of MSS. at Paris. It is that of a Count\nHohenschwangen, of the family of Welf, and depicts the wearer in a\nlong sleeveless, dark blue surcoat, with his armorial device; a white\nswan on a red field with a light red border. Under his coat he wears\na _cap-a-pie_ suit of mail. The helmet is original, very like the\nGreek, with the furred mantle as we see it in the seal of Richard King\nof England, of the date of 1498. This helmet does not appear to be a\ntilting helmet, which usually rests upon the shoulders; but this kind\nof helmet would be fastened, like the vizor with the mailed hood, by\nan iron throat-brace, and a leather thong. Upon the covered helmet he\nwears the swan as a crest. The sword-hilt is of gold, the sheath black,\nthe girdle white, the furred mantle is red, lined with white.\n\n[Illustration: [++] Knight's Costume of 1272.]\n\nChivalry began in Europe about A.D. 912. From the twelfth to the\nfifteenth century it had considerable influence in refining the manners\nof most of the nations of Europe. The knight swore to accomplish the\nduties of his profession as the champion of God and the ladies. He\ndevoted himself to speak the truth, to maintain the right, to protect\nthe distressed, to practise courtesy, to fulfil obligations, and\nto vindicate in every perilous adventure his honour and character.\nChivalry, which owed its origin to the feudal system, expired with it.\nThe origin of the title of knight, as a military honour, is said to be\nderived from the siege of Troy, but this solely depends on a passage or\ntwo in Homer, and the point is disputed by several learned commentators.\n\n\nCURIOUS CIRCUMSTANCE.\n\nButton, in his Life, tells us of one of his ancestors, a trooper, who,\nseeing a young girl at the river-side, lading water into her pail,\ncast a large stone with design to splash her; but not being versed\nin directing a stone so well as a bullet, he missed the water, and\nbroke her head; he ran off. Twelve years afterwards, he settled at\nDerby, courted a young woman, and married her. In the course of their\nconversations he proved to be the very man who had cast the stone, and\nshe the girl with the broken head.\n\n\nFUNEREAL JAR.\n\nThe term \"funereal\" has been erroneously applied to all pottery\nfound in tombs, even where the utensils have no relation to funereal\npurposes, but were probably in common use. There have been found,\nhowever, in Corsica vessels of earthenware, which may strictly be\ncalled \"funereal.\"\n\nThough the precise period of the fabrication of the funereal vessels\nfound in Corsica is not ascertained, they must be considered of very\nancient date. These vessels, when found entire, at first appear\ncompletely closed up, and no trace of joining can be discovered. But\nit has been ascertained that they are composed of two equal parts, the\nend of one fitting exactly into the other, and so well closed that\nthe body, or at least the bones which they contain, appear to have\nbeen placed within them before they were baked upon the kiln. Diodorus\nSiculus, in speaking of the usages of the Balearic Isles, states that\nthese people were in the habit of beating, with clubs, the bodies of\nthe dead which, when thus rendered flexible, were deposited in vessels\nof earthenware. This practice of the Corsicans coincides singularly\nwith that of the Coroados Indians, who inhabit a village on the Paraiba\nriver, near Campos, in the Brazils. They use large earthen vessels,\ncalled _camucis_, as funereal urns. The bodies of their chiefs, reduced\nto mummies, are placed in them in a bent posture, decked with their\nornaments and arms, and are then deposited at the foot of the large\ntrees of the forest.\n\n[Illustration: [++] Funereal Jar.]\n\nThe cut which we here give speaks for itself. It represents the\nfunereal jar containing the chief as described; the animal at his feet\nappears to be a panther or tiger cat.\n\n\nWRITING MATERIALS.\n\nThe materials used for writing on have varied in different ages and\nnations. Among the Egyptians slices of limestone, leather, linen, and\npapyrus, especially the last, were universally employed. The Greeks\nused bronze and stone for public monuments, wax for memorandums and\npapyrus for the ordinary transactions of life. The kings of Pergamus\nadopted parchment, and the other nations of the ancient world chiefly\ndepended on a supply of the paper of Egypt. But the Assyrians and\nBabylonians employed for their public archives, their astronomical\ncomputations, their religious dedications, their historical annals,\nand even for title-deeds and bills of exchange, tablets, cylinders,\nand hexagonal prisms of terra-cotta. Two of these cylinders, still\nextant, contain the history of the campaign of Sennacherib against\nthe kingdom of Judah; and two others, exhumed from the Birs Nimroud,\ngive a detailed account of the dedication of the great temple by\nNebuchadnezzar to the seven planets. To this indestructible material,\nand to the happy idea of employing it in this manner, the present age\nis indebted for a detailed history of the Assyrian monarchy; whilst\nthe decades of Livy, the plays of Menander and the lays of Anacreon,\nconfided to a more perishable material, have either wholly or partly\ndisappeared amidst the wreck of empires.\n\n\nCURIOUS DISPUTE AND APPROPRIATE DECISION.\n\nFuller, in his 'Holy State,' p. 170, gives a very _apposite_ story; a\npoor man in Paris, being very hungry, went into a cook shop, and staid\nthere so long, (for the master was dishing-up meat,) that his appetite\nbeing lessened by the steam, he proposed to go without his meal; the\ncook insisted upon payment all the same. At length, the altercation was\nagreed to be referred to the first person that passed the door; that\nperson happened to be a notorious idiot. Having heard the complaint, he\ndecreed that the poor man's money should be placed between two empty\ndishes, and that the cook should be recompensed with the jingling of\nhis cash, as the other was with the fumes of the meat; and this little\nanecdote is literally matter of fact.\n\n\nTHE TEA-POT.\n\nNo specimen of the ceramic art possesses greater variety of form than\nthe tea-pot. On none has the ingenuity of the potter been more fully\nexercised, and it is worthy of remark, that the first successful\nproduction of Boettcher in hard porcelain was a tea-pot. The so-called\nElizabethan tea-pots must be of a later date, for tea was not known in\nEngland until the time of Charles II; but it is interesting to trace\nthe gradual increase in the size of the tea-pot, from the diminutive\nproductions of the Elers, in the time of Queen Anne and George I., when\ntea was sold in apothecaries' shops, to the capacious vessel which\nsupplied Dr. Johnson with \"the cup that cheers but not inebriates.\"\n\nMr. Croker, in his edition of Boswell's Life, mentions a tea-pot that\nbelonged to Dr. Johnson which held two quarts; but this sinks into\ninsignificance compared with the superior magnitude of that in the\npossession of Mrs. Marryat, of Wimbledon, who purchased it at the sale\nof Mrs. Piozzi's effects at Streatham. This tea-pot, which was the one\ngenerally used by Dr. Johnson, holds more than three quarts. It is\nof old Oriental porcelain, painted and gilded, and from its capacity\nwas well suited to the taste of one \"whose tea-kettle had no time to\ncool, who with tea solaced the midnight hour, and with tea welcomed\nthe morn.\" George IV. had a large assemblage of tea-pots, piled in\npyramids, in the Pavilion at Brighton. Mrs. Elizabeth Carter was also\na collector of tea-pots, each of which possessed some traditionary\ninterest, independently of its intrinsic merit; but the most diligent\ncollector of tea-pots was the late Mrs. Hawes. She bequeathed no less\nthan three hundred specimens to her daughter, Mrs. Donkin, who has\narranged them in a room appropriated for the purpose. Among them are\nseveral formerly belonging to Queen Charlotte. Many are of the old\nJapan; one with two divisions, and two spouts for holding both black\nand green tea; and another of a curious device, with a small aperture\nat the bottom to admit the water, there being no opening at the top,\natmospheric pressure preventing the water from running out. This\nsingular Chinese toy has been copied in the Rockingham ware.\n\n\nPROTRACTED SLEEP.\n\nOne of the most extraordinary instances of excessive sleep is that\nof the lady at Nismes, published in 1777, in the \"Memoirs of the\nRoyal Academy of Sciences at Berlin.\" Her attacks of sleep took place\nperiodically, at sunrise and about noon. The first continued till\nwithin a short time of the accession of the second, and the second\ncontinued till between seven and eight in the evening--when she awoke,\nand continued so till the next sunrise. The most extraordinary fact\nconnected with this case is, that the first attack commenced always\nat daybreak, whatever might be the season of the year, and the other\nalways immediately after twelve o'clock. During the brief interval of\nwakefulness which ensued shortly before noon, she took a little broth,\nwhich she had only time to do when the second attack returned upon\nher, and kept her asleep till the evening. Her sleep was remarkably\nprofound, and had all the character of complete insensibility, with\nthe exception of a feeble respiration, and a weak but regular movement\nof the pulse. The most singular fact connected with her remains to be\nmentioned. When the disorder had lasted six months, and then ceased,\nthe patient had an interval of perfect health for the same length of\ntime. When it lasted one year, the subsequent interval was of equal\nduration. The affection at last wore gradually away; and she lived,\nentirely free of it, for many years after. She died in the eighty-first\nyear of her age, of dropsy, a complaint which had no connexion with her\npreceding disorder.\n\n\nANCIENT SUIT OF MAIL.\n\nThe two figures depicted on next page represent Henry of Metz receiving\nthe oriflamme from the hands of St. Denis, derived from a painted\nwindow in the church of Notre Dame de Chartres. The oriflamme was a\nred banner attached to a staff, and cut in the manner shown in our\nengraving. Henry of Metz was Marshal of France, and is here represented\nin a complete suit of mail, his hood being thrown back upon his\nshoulders. This suit is perfect, even to the extremities; and it is to\nbe remarked that the defence for the hands is divided in the manner of\na common glove. Over the mail is worn a loose surcoat, on which is\nemblazoned the cross, traversed by a red baton--the type of his high\noffice.\n\n[Illustration: [++] Ancient Suit of Mail.]\n\n\nTHE POISON CUP.\n\n[Illustration: [++] The Poison Cup.]\n\nIn the time of James I. poison was too frequently resorted to,\nespecially on the Continent, as a means of getting rid of individuals\nwho had rendered themselves obnoxious to certain parties who were\nprosecuting their own private ends; and so extensively did this\ninfamous practice prevail that there was a class of persons who\nwere known to have studied the art of secret poisoning, and whose\nservices could be engaged for a high reward. In order to counteract the\noperations of the poisoners, various devices were employed, and among\nthem was the art which the pretended magicians of those days professed\nto have discovered, of making a kind of glass which would fly in pieces\nif poison was poured into any vessel that was formed of it. The cut at\nthe head of our article represents a tankard of this sort, in which\nthe glass is mounted in silver gilt arabesque and silver filagree. It\nwas believed that the large crystal which is seen standing out at the\ncentre of the lid would become discoloured at the approach of poison.\nThe tankard is a work of the sixteenth century, and was presented to\nClare Hall, Cambridge--where it is still preserved--by Dr. William\nButler, an eminent physician in the time of James I.\n\n\nPORCELAIN FINGER-RINGS.\n\nThe porcelain finger-rings of ancient Egypt are extremely beautiful;\nthe band of the ring being seldom above one-eighth of an inch in\nthickness. Some have a plate on which, in bas-relief, is the god _Set_,\nor _Baal_, full face, or playing on the tambourine, as the inventor of\nMusic; others have their plates in the shape of the right symbolical\neye, the emblem of the Sun; of a fish, of the perch species; or of\na scarabaeus, which is said to have been worn by the military order.\nSome few represent flowers. Those which have elliptical plates with\nhieroglyphical inscriptions, bear the names of Amen-Ra, and of other\ngods and monarchs, as Amenophis III., Amenophis IV., and Amenanchut,\nof the Eighteenth and Nineteenth dynasties. One of these rings has\na little bugle on each side, as if it had been strung on the beaded\nwork of a mummy, instead of being placed on the finger. Blue is the\nprevalent colour, but a few white and yellow rings, and some even\nornamented with red and purple colours are found. It is not credible\nthat these rings, of a substance finer and more fragile than glass,\nwere worn during life. Neither is it likely that they were worn by the\npoorer classes, for the use of the king's name on sepulchral objects\nseems to have been restricted to functionaries of state. Some larger\nrings of porcelain of about an inch diameter, seven-eighths of an inch\nbroad, and one-sixteenth of an inch thick, made in open work, represent\nthe constantly repeated lotus flowers, and the god Ra, or the Sun,\nseated, and floating through the heaven in his boat. Common as these\nobjects were in Egypt, where they were employed as substitutes for\nthe hard and precious stones, to the Greeks, Etruscans, and Italian\nGreeks they were articles of luxury, just as the porcelain of China\nwas to Europeans some centuries ago. The Etruscans set these bugles,\nbeads, and amulets in settings of their exquisite gold filigree work,\nintermixed with gold beads and precious stones. Strung as pendants they\nhung round the necks of the fair ones of Etruria. In one of the tombs\nalready alluded to at the Polledrara, near Vulci, in Italy, was found\na heap of annular and curious Egyptian bugles, which had apparently\nformed a covering to some bronze objects, but the strings having given\nway, the beads had dropped to pieces. These, as well as the former,\nhad been obtained from some of the Egyptian markets, like that at\nNaucratis: or from the Phoenician merchants, in the same manner as the\nflasks. One of the most remarkable of these personal ornaments is a\nbracelet, composed of small fish strung together and secured by a clasp.\n\n\nPIGEON CATCHING NEAR NAPLES.\n\nBetween La Cava and Naples, about half a mile from the town, are\ncertain Bluebeard-looking towers, several centuries old, erected for\nthe purpose of snaring wood-pigeons; with which view the gentlemen of\nthe neighbourhood, who are generally expert and practised slingers,\nassemble and man the towers in May. A long line of nets, some quarter\nof a mile in circuit, held up in a slanting position by men concealed\nin stone sentry-boxes placed here and there along the _enceinte_, is\nspread in front. As the pigeons are seen advancing (the time of their\napproach is generally looked for at early dawn, when they are making\nfor the woods), the nearest slingers commence projecting a succession\nof white stones in the direction of the nets. These the birds no sooner\nbehold, then attracted, or alarmed (for the motive does not certainly\nappear), they swoop down upon them, and when sufficiently near to fall\nwithin reach of the nets, the persons holding let go, rush from their\nambush, and secure the covey. Thousands of wood-pigeons are thus, we\nhave been told by a proprietor, annually taken, and transmitted for\npresents to distant friends; as we used to send out game, before the\nsale of it was legalised. Thus birds, as well as fish, and fish as well\nas man, often get entangled and caught in their headlong pursuit of a\npleasure that still eludes them.\n\n\nFRAME REQUISITE TO SUPPORT THE DRESS.\n\nJames I., and his subjects who wished to clothe themselves loyally,\nwore stupendous breeches. Of course the \"honourable gentleman\" of\nthe House of Commons were necessarily followers of the fashion. But\nit led to inconveniences in the course of their senatorial duties.\nIt was an old mode revived; and at an earlier day, when these nether\ngarments were ample enough to have covered the lower man of Boanerges,\nthe comfort of the popular representative was thus cared for:--\"Over\nthe seats in the parliament-house, there were certain holes, some two\ninches square, in the walls, in which were placed posts to uphold a\nscaffold round about the house within, for them to sit upon who used\nthe wearing of great breeches stuffed with hair like wool-sacks, which\nfashion being left the eighth year of Elizabeth, the scaffolds were\ntaken down, and never since put up.\" So says Strutt; but doubtless the\ncomforts of the members were not less cared for when the old fashion\nagain prevailed.\n\n\nPRICES OF SEVRES PORCELAIN.\n\nAs one of the curiosities of luxury and taste it is worth while to note\nthe high prices for which some portions of the very choice collection\nof Sevres porcelain at Stowe were sold:--A small coffee cup, which\nweighed scarcely three ounces, realised 46 guineas; and another,\nsimilar, but somewhat inferior, sold for 35 guineas. A chocolate cup\nand saucer, Bleu de Roi, with beautiful miniatures of two ladies of\nthe Court of Louis XV., and four paintings of Cupids, though slightly\ninjured during the view, realised 45 guineas. The prices obtained for\nmost of the cups and saucers were from 10 to 12 guineas. A beautiful\nspecimen of a Bleu de Roi cup, saucer and cover, jewelled in festoons,\ncameos, and imitation of pearls, sold for L35 10s.; and another,\nsomewhat inferior, for 21 guineas. A salver, mounted in a table with\normolu ornaments, sold for 81 guineas; the companion piece for L100.\n\n\nHENRY THE EIGHTH'S CHAIR.\n\nIn the earlier half of the sixteenth century a large proportion of\nthe furniture used in this country, as well as of the earthenware and\nother household implements during the greater part of that century, was\nimported from Flanders and the Netherlands. Hence, in the absence of\nengravings at home, we are led to look at the works of the Flemish and\nGerman artists for illustrations of domestic manners at this period.\nThe seats of that day were termed joint (or joined) stools and chairs.\nA rather fine example of a chair of this work, which was, as was often\nthe case, three-cornered, is preserved in the Ashmolean museum, at\nOxford, where it is reported to have been the chair of Henry VIII. We\nhere annex a sketch of it.\n\n[Illustration: [++] Henry the Eighth's Chair.]\n\n\nMULLET AND TURBOT WITH THE ROMANS.\n\nThe Romans were enthusiastic for the mullet. It was for them _the_\nfish, _par excellence_. It was sometimes served up six pounds in\nweight, and such a fish was worth L60 sterling. It was cooked on the\ntable, for the benefit and pleasure of the guests. In a glass vessel\nfilled with brine made from water, the blood of the mackerel, and salt,\nthe live mullet, stripped of all its scales, was enclosed; and as its\nfine pink colour passed through its dying gradations, until paleness\nand death ensued, the _convives_ looked on admiringly, and lauded the\nspectacle.\n\nThe turbot was next in estimation, but as, occasionally, offending\nslaves were flung into the turbot preserves for the fish to feed upon,\nsome gastronomists have affected to be horror-stricken at the idea\nof eating a _turbot a la Romaine_; quite forgetting that so many of\nour sea-fish, in their domain, feed largely on the human bodies which\naccident, or what men call by that name, casts into the deep.\n\n\n\"TOO LATE,\" QUOTH BOICE.\n\nThe history of the ancient castle of Maynooth is one of much interest;\nabounding in incidents akin to romance. In the reign of Henry the\nEighth, during the rebellion of \"Silken Thomas,\" one of the bravest\nand most heroic of the Geraldines, it was taken by treachery. In the\nabsence of its lord, the governorship was entrusted to \"Christopher\nParese,\" his foster-brother. This \"white-livered traitor resolved to\npurchase his own security with his lord's ruin;\" and therefore sent\na letter to the lord-deputy, signifying that he would betray the\ncastle, on conditions; \"and here the devil betrayed the betrayer,\nfor in making terms for his purse's profit, he forgot to include his\nperson's safety.\" The lord-deputy readily accepted his offer, and,\naccordingly, the garrison having gained some success in a sally, and\nbeing encouraged by the governor in a deep joyous carouse, the ward\nof the tower was neglected--the traitorous signal given, and the\nEnglish scaled the walls. They obtained possession of the strong-hold,\nand put the garrison to the sword--\"all except two singing men, who,\nprostrating themselves before the deputy, warbled a sweet sonnet called\n_dulcis amica_, and their melody saved their lives.\" Parese, expecting\nsome great reward, with impudent familiarity presented himself before\nthe deputy, who addressed him as follows:--\"Master Parese, thou\nhast certainly saved our lord the king much charge, and many of his\nsubjects' lives, but that I may better know to advise his highness\nhow to reward thee, I would ascertain what the Lord Thomas Fitzgerald\nhath done for thee?\" Parese, highly elevated at this discourse,\nrecounted, even to the most minute circumstance, all the favours that\nthe Geraldine, even from his youth up, had conferred on him, to which\nthe deputy replied, \"And how, Parese, couldst thou find it in thy heart\nto betray the castle of so kind a lord? Here, Mr. Treasurer, pay down\nthe money that he has covenanted for; and here, also, executioner,\nwithout delay, as soon as the money is counted out, cut off his head!\"\n\"Oh,\" quoth Parese, \"had I known this, your lordship should not have\nhad the castle so easily.\" Whereupon Mr. Boice, a secret friend of\nthe Fitzgerald, a bystander, cried out \"Auntraugh,\" _i. e._ \"too\nlate,\" which occasioned a proverbial saying, long afterwards used in\nIreland--\"Too late, quoth Boice.\" The castle is said by Archdall to\nhave been erected by John, the sixth Earl of Kildare, early in the\nfifteenth century; but in that case it must have been preceded by some\nother defensive structure; for it is certain that the Kildare branch of\nthe Geraldines resided at Maynooth at a much earlier period. The first\nEarl of Kildare, John Fitz Thomas, was created by patent, dated 14th\nMay, 1316.\n\n\nSUPPRESSED BIBLES.\n\n1538.--An English Bible, in folio, printed at Paris, unfinished.\n\n1542.--Dutch Bible by Jacob Van Leisvelt. The sixth and best edition\ngiven by Leisvelt, and famous as being the cause of this printer being\nbeheaded.\n\n1566.--French Bible by Rene Benoist, Paris, 1566, folio, 3 vols.\ncompleted.\n\n1622.--Swedish Bible, printed at Lubeck, in 4to., very defective.\n\n1666.--A German Bible, printed at Helmstedt, in part only, 4to.\n\n1671.--A French Bible, by Marolles, in folio, containing only the books\nof Genesis, Exodus, and the first twenty-three chapters of Leviticus.\n\n\nEXTRAORDINARY REPRODUCTIVE POWER OF THE HYDRA.\n\nOne of the fresh-water Polypes, from its power of perpetual\nreproduction, has received the name of _Hydra_, by which it is known\namong naturalists: as if it realised the ancient monster of fabulous\nstory, whose heads sprouted anew as fast as they were cut off by\nHercules.\n\nMost curious monstrosities were produced by the experiments of\nphilosophers on these animals, especially by partial separations. If a\npolype be slit from the summit to the middle, one will be formed having\ntwo heads, each of which will capture and swallow food. If these again\nbe slit half-a-dozen times, as many heads will be formed surmounting\nthe same body. If now all these be cut off, as many new ones will\nspring up in their place, while each of the severed heads becomes a\nnew polype, capable of being, in its turn, varied and multiplied _ad\ninfinitum_;--so that in every respect our little reality exceeds its\nfabulous namesake.\n\nThe polypes may be grafted together. If cut-off pieces be placed in\ncontact, and pushed together with a gentle force, they will unite and\nform a single one. The head of one may be thus planted on the trunk of\nanother.\n\nAnother method of uniting them, perhaps still more wonderful, is by\nintroducing one within the other; the operator forced the body of the\none into the mouth of the other, pushing it down so that the heads\nwere brought together. After forcibly keeping it for some time in this\nstate, the two individuals at length united, and a polype was formed,\ndistinguishable only by having twice the usual number of tentacles.\n\nThere is one species which can actually be turned inside out like a\nglove, and yet perform all the functions of life as before, though\nthat which was the coat of the stomach is now the skin of the body and\n_vice versa_. If it should chance that a polype so turned had young\nin the act of budding, these are, of course, now within the stomach.\nIf they have arrived at a certain degree of maturity, they extend\nthemselves towards the mouth of the parent, that they may thus escape\nwhen separated. But those which are less advanced turn themselves\nspontaneously inside out, and thus place themselves again on the\nexterior of the parent.\n\nA multitude of other variations, combinations, and monstrosities, have\nbeen, as it were, created by the ingenuity of philosophers; but these\nare sufficient to give a notion of the extraordinary nature of these\nanimals; and to account for the wonder with which they were regarded.\n\n\nEGYPT.\n\nEgypt was the land visited by Abraham in search of food, when there\nwas a famine in his own country;--the land to which Joseph was carried\nas a slave, and which he governed as prime minister. From Egypt, Moses\nled the Israelites through the waters of the Red Sea. Here Jeremiah\nwrote his Lamentations. Here Solon, Pythagoras, Plato, and many other\nGreek philosophers, came to study. Here Alexander the Great came as\nconqueror; and here the Infant Saviour was brought by his parents to\navoid the persecution of Herod. Egyptian hieroglyphics, in which the\ncharacters are taken from visible objects, are the earliest form of\nwriting; and the Hebrew and Greek alphabets were both borrowed from\nthem. Egypt taught the world the use of paper--made from its rush,\nthe papyrus. In Egypt was made the first public library, and first\ncollege of learned men, namely, the Alexandrian Museum. There Euclid\nwrote his Elements of Geometry, and Theocritus his Poems, and Lucian\nhis Dialogues. The beauty of Cleopatra, the last Egyptian Queen, held\nJulius Caesar, and then Marc Antony, captive. In Egypt were built the\nfirst monasteries; the Christian fathers, Origen and Athanasius, lived\nthere. The Arien and Athanasian controversy began there.\n\nThe buildings which now remain are the oldest buildings in the world,\nand the largest in the world. On the banks of its great river may be\nseen the oldest arch, and the oldest column. Up this noble river sailed\nHerodotus, the most entertaining of travellers, and Strabo, the most\njudicious. Indeed, as the country is little more than the narrow strip\nthat is watered by the Nile's overflow, from the river may be seen\nalmost all its great cities and temples.\n\n\nABYSSINIAN LADIES.\n\nThe women of Abyssinia are dressed quite as decently as any women in\nthe world, without having a particle of the trouble of the ladies of\nmore civilised nations. There is a distinguishing costume for young\ngirls, and for those who, from being married or otherwise, are no\nlonger considered as such. The dress of the former is indeed rather\nslight, though far more picturesque than that of the latter. In one\npart of the country (about Shire) the girls merely wear a piece of\ncotton stuff wrapped round the waist and hanging down almost to the\nknee, and another (or the end of the former, if it be long enough)\nthrown over the left shoulder, so as to leave the right arm and breast\nexposed. In other parts of Tigre, a black goat-skin, ornamented with\ncowries, is often substituted for this latter. An ordinary woman\nwears a large loose shirt down to the feet, with sleeves made tight\ntowards the wrist. This, with a \"quarry\" similar to those of the\nmen, but worn rather differently, and a parasol when out of doors,\nis a complete suit. A fine lady, however, as in our engraving, has a\nsplendid \"mergeff quarry,\" and her shirt is made probably of calico\nfrom Manchester, instead of the country fabric, and richly embroidered\nin silk of divers colours and various patterns round the neck, down the\nfront, and on the cuffs. She will also, of course, own a mule; and then\nmay choose to wear (alas, that it should be so, even in Abyssinia!) the\ninexpressibles. These are made of calico, and rather loose, but getting\ngradually tighter at the ankle, where they are embroidered like the\nshirt.\n\n[Illustration: [++] Abyssinian Ladies.]\n\nThe fair sex all over the world are fond of ornaments. In Abyssinia\nthey wear a profusion of silver, in the shape of chains, bracelets,\n&c., or, to be more explicit, a well-dressed lady will hang three or\nfour sets of amulets about her neck, as well as her blue cord, and a\nlarge flat silver case (purporting to contain a talisman, but more\noften some scented cotton) ornamented with a lot of silver bells\nhanging to the bottom edge of it, and the whole suspended by four\nchains of the same metal. Three pairs of massive silver and gilt\nbracelets are on her wrists, and a similar number of \"bangles\" on her\nankles; while over her insteps and to her heels are a quantity of\nlittle silver ornaments, strung like beads on a silk cord. Her fingers\n(even the upper joints) are covered with plain rings, often alternately\nof silver and silver-gilt, and a silver hair-pin, something similar\nto those now worn by English ladies, completes her decoration. Women\nof the poorer class, and ladies on ordinary occasions, wear ivory\nor wooden pins neatly carved in various patterns, and stained red\nwith henna-leaves. The Abyssinian ladies, like those of most Eastern\nnations, stain their hands and feet with henna, and darken their\neyelids with antimony.\n\n[Illustration: 1, 2. Hair-pins made of hard wood, and stained with\nhenna. 3. Ditto, of silver and fil-et-grain work. (About one-half usual\nsize.)]\n\n\nTREATMENT OF LEPERS IN ENGLAND.\n\nAccording to the tenor of various old civil codes and local enactments,\nwhen a person became affected with leprosy, he was looked upon as\nlegally and politically dead, and lost the privileges belonging to\nhis right of citizenship. By the laws of England, lepers were classed\nwith idiots, madmen, outlaws, &c., as incapable of being heirs. But it\nwas not by the eye of the law alone that the affected was looked upon\nas defunct, for the church also took the same view, and performed the\nsolemn ceremonials of the burial of the dead over him, on the day on\nwhich he was separated from his fellow creatures, and confined to a\nlazar house. The various forms and ceremonies which were gone through\non this occasion are described by French authors; but it is highly\nprobable that the same observances were common in our own country.\n\nA priest, robed with surplice and stole, went with the cross to the\nhouse of the doomed leper. The minister of the church began the\nnecessary ceremonies, by exhorting him to suffer, with a patient and\npenitent spirit, the incurable plague with which God had stricken him.\nHe then sprinkled the unfortunate leper with holy water, and afterwards\nconducted him to the church, the usual burial services being sung\nduring their march thither. In the church, the ordinary habiliments of\nthe leper were removed; he was clothed in a funeral pall, and, while\nplaced before the altar, between _trestles_, the libera was sung,\nand the mass for the dead celebrated over him. After this service he\nwas again sprinkled with holy water, and led from the church to the\nhouse or hospital destined for his future abode. A pair of clappers, a\nbarrel, a stick, cowl, and dress, &c., were given him. Before leaving\nthe leper, the priest solemnly interdicted him from appearing in\npublic without his leper's garb,--from entering inns, churches, mills,\nand bakehouses,--from touching children, or giving them ought he had\ntouched,--from washing his hands, or any thing pertaining to him, in\nthe common fountains and streams,--from touching, in the markets, the\ngoods he wished to buy, with any thing except his stick,--from eating\nand drinking with any others than lepers,--and he specially forbade\nhim from walking in narrow paths, or from answering those who spoke to\nhim in roads and streets, unless in a whisper, that they might not be\nannoyed with his pestilent breath, and with the infectious odour that\nexhaled from his body,--and last of all, before taking his departure,\nand leaving the leper for ever to the seclusion of the lazar house,\nthe official of the church terminated the ceremony of his separation\nfrom his living fellow-creatures, by throwing upon the body of the poor\noutcast a shovelful of earth, in imitation of the closure of the grave.\n\nAccording to the then customary usage, Leper Hospitals were always\nprovided with a cemetery for the reception of the bodies of those who\nhad died of the malady.\n\n\nLUMINOUS APPEARANCE OF THE RED SEA.\n\nAll who have frequented the Red Sea, have observed the luminous\nappearance or phosphorescence of its waters. \"It was beautiful,\" says\na picturesque writer, who sailed from Mocha to Cosseir, \"to look down\ninto this brightly transparent sea, and mark the coral, here in large\nmasses of honeycomb-rock, there in light branches of a pale red hue,\nand the beds of green seaweed, and the golden sand, and the shells, and\nthe fish sporting round the vessel, and making colours of a beauty to\nthe eye which is not their own. Twice or thrice we ran on after dark\nfor an hour or two; and though we were all familiar with the sparkling\nof the sea round the boat at night, never have I seen it in other\nwaters so superlatively splendid. A rope dipped in it and drawn forth\ncame up as a string of gems; but with a life, and light, and motion,\nthe diamond does not know.\" Those sea-lights have been explained by\na diversity of causes; but the singular brilliancy of the Red Sea\nseems owing to fish spawn and animalculae, a conjecture which receives\nsome corroboration from the circumstance that travellers who mention\nit visited the gulf during the spawning period--that is, between the\nlatter end of December and the end of February. The coral-banks are\nless numerous in the southern parts. It deserves notice, that Dr. Shaw\nand Mr. Bruce have stated--what could only be true, so far as their own\nexperience went--that they observed no species of weed or flag; and the\nlatter proposes to translate Yam Zuph \"the Sea of Coral\"--a name as\nappropriate as that of Edom.\n\n\nRECENT PRICES OF SLAVES.\n\nPrices of course vary at Constantinople according to the vigilance\nof Russian cruisers, and the incorruptibility of Russian agents at\nTrebizond, Samsoon, and Sinope. The following is the average price in\nCircassia:--\n\n A man of 30 years of age, L10\n \" 20 \" 10 to L30\n \" 15 \" 30 \" 70\n \" 10 \" 20 \" 50\n \" 5 \" 10 \" 30\n A woman of 50 years of age, L10 to L30\n \" 40 \" 30 \" 40\n \" 30 \" 40 \" 70\n \" 20 to 25 \" 50 \" 100\n \" 14 \" 18 \" 50 \" 150\n \" 8 \" 12 \" 30 \" 80\n \" 5 \" 20 \" 40\n\n\nTATTOOED ABYSSINIAN LADY.\n\n[Illustration: [++] Tattooed Abyssinian Lady.]\n\nThe annexed cut is a sketch of an Abyssinian lady, tattooed in the\nheight of the fashion. The following extract from that interesting\nwork \"Parkyns's Abyssinia\" gives a good account of the custom as it\nprevails in the larger cities there, and of the manner in which the\noperation is performed. \"The men seldom tattoo more than one ornament\non the upper part of the arm, near the shoulder, while the women cover\nnearly the whole of their bodies with stars, lines, and crosses, often\nrather tastefully arranged. I may well say nearly the whole of their\npersons, for they mark the neck, shoulders, breasts, and arms, down to\nthe fingers, which are enriched with lines to imitate rings, nearly\nto the nails. The feet, ankles, and calves of the legs, are similarly\nadorned, and even the gums are by some pricked entirely blue, while\nothers have them striped alternately blue and the natural pink. To see\nsome of their designs, one would give them credit for some skill in\nthe handling of their pencil; but, in fact, their system of drawing\nthe pattern is purely mechanical. I had one arm adorned; a rather\nblind old woman was the artist; her implements consisted of a little\npot of some sort of blacking, made, she told me, of charred herbs; a\nlarge home-made iron pin, about one-fourth of an inch at the end of\nwhich was ground fine; a bit or two of hollow cane, and a piece of\nstraw; the two last-named items were her substitutes for pencils.\nHer circles were made by dipping the end of a piece of a cane of the\nrequired size into the blacking, and making its impression on the skin;\nwhile an end of the straw, bent to the proper length, and likewise\nblackened, marked all the lines, squares, diamonds, &c., which were to\nbe of equal length. Her design being thus completed, she worked away\non it with her pin, which she dug in as far as the thin part would\nenter, keeping the supply of blacking sufficient, and going over the\nsame ground repeatedly to ensure regularity and unity in the lines.\nWith some persons, the first effect of this tattooing is to produce\na considerable amount of fever, from the irritation caused by the\npunctures; especially so with the ladies, from the extent of surface\nthus rendered sore. To allay this irritation, they are generally\nobliged to remain for a few days in a case of vegetable matter, which\nis plastered all over them in the form of a sort of green poultice.\nA scab forms over the tattooing, which should not be picked off, but\nallowed to fall off of itself. When this disappears, the operation\nis complete, and the marks are indelible; nay more, the Abyssinians\ndeclare that they may be traced on the person's bones even after death\nhas bared them of their fleshy covering.\"\n\n\nBULGARIAN FISHERMEN.\n\nThe following interesting account of the Bulgarian fishermen on the\nshores of the Black Sea is taken from the translation of a narrative of\na boat excursion made in 1846 by M. Xavier Hommaire, along part of the\nnorthern coast of the Black Sea:--\n\n[Illustration: [++] Bulgarian Fishermen.]\n\n\"The fishermen are, almost without exception, Bulgarians--a population\nat once maritime and agricultural, very closely resembling, in race\nand costume, the Bretons of France--and they enjoy a monopoly of all\nthe fisheries in the Bosphorus and the adjacent parts of the Black\nSea. Their elegant barks appeared on stated days and hours, shooting\nalong with extraordinary rapidity through the waters of the Gulf of\nBuyuk Dereh, which appears to be their head-quarters, and sustaining\nthe test of comparison even with the famous caiques of Constantinople.\nThe most important object of their fishery is a delicious kind of\nsmall thunny, called palamede. They are Bulgarians, also, who own\nthe singular fisheries which form such admirable subjects for the\nartist's sketch-book. They are found throughout the Bosphorus, from\nBechiktusch and Scutari to the lighthouses of Europe and Asia. They\nmight be called dog-kennels, but rickety and worm-eaten with antiquity,\nand are suspended by means of cords, pegs, and tatters to the top\nof an indescribable framework of props. There on high, petrified in\nmotionless and uninterrupted silence, in company with some old pots of\nmignionette (where will not the love of flowers find a home!), a man,\nwith the appearance of a wild beast or savage, leans over the sea, at\nthe bottom of which he watches the passage of its smallest inhabitants,\nand the capricious variations of the current. At a certain distance\nis arranged, in the form of a square, a system of nets, which, at\nthe least signal from the watcher, fall on the entire shoal of fish.\nA contrivance yet more primitive than these airy cells, if not so\npicturesque, was that of simple posts, which we encountered some time\nbefore in the channel of the Bosphorus, rising about fifteen feet above\nthe surface of the water. Half-way up is perched, crouching (one cannot\nsee how), something having the human form, and which is found to be a\nBulgarian. For a long time I watched them without being able to make\nthem out, either pole or its tenant; and often have I seen them in the\nmorning, and observed them again in the evening, not having undergone\nthe least change of posture.\n\n\"On returning to our encampment, the commandant of the fort, to whom\nwe paid a visit, gave us a very different report of the fishermen of\nthe morning, whom he described as an assemblage of all the vagabonds of\nthe neighbourhood. Convinced even that the fact of their having fallen\nin with us must have inspired them with the project of coming to prowl\nby night round our camp, he wished us to accept some of the men in his\ngarrison as a guard.\"\n\n\nHORSES OF THE ARABS.\n\nArabs make intimate friends of their horses, and so docile are these\ncreatures that they are ridden without a bit, and never struck or\nspurred. They share their owner's diet, and are as well cared for as a\nchild. They divide their horses, however, into two kinds: The one they\ncall kadischi, that is, horses of an unknown birth; the other, they\ncall kochlani, that is, horses whose genealogy is known for thousands\nof years. They are direct descendants, so they say, of the stud of\nSolomon. The pedigree of an Arabian horse is hung round his neck soon\nafter his birth, which is always properly witnessed and attested.\n\nThe following is the pedigree of a horse purchased by a French officer\nin Arabia:--\"In the name of God, the merciful and compassionate, and of\nSaed Mahomed, agent of the high God, and of the companions of Mahommed,\nand of Jerusalem. Praised be the Lord, the Omnipotent Creator. This\nis a high-bred horse, and its colt's tooth is here in a bag about\nhis neck, with his pedigree, and of undoubted authority, such as no\ninfidel can refuse to believe. He is the son of Rabbamy, out of the dam\nLabadah, and equal in power to his sire of the tribe of Zazhalah; he is\nfinely moulded, and made for running like an ostrich. In the honours\nof relationship, he reckons Zuluah, sire of Mahat, sire of Kallac, and\nthe unique Alket sire of Manasseh, sire of Alsheh, father of the race\ndown to the famous horse, the sire of Lahalala; and to him be ever\nabundance of green meat, and corn, and water of life, as a reward from\nthe tribe of Zazhalah; and may a thousand branches shade his carcass\nfrom the hyaena of the tomb, from the howling wolf of the desert;\nand let the tribe of Zazhalah present him with a festival within an\nenclosure of walls; and let thousands assemble at the rising of the\nsun in troops hastily, where the tribe holds up under a canopy of\ncelestial signs within the walls, the saddle with the name and family\nof the possessor. Then let them strike the bands with a loud noise\nincessantly, and pray to God for immunity for the tribe of Zoab, the\ninspired tribe.\"\n\n\nDILEMMA.\n\nProtagoras, an Athenian rhetorician, had agreed to instruct Evalthus\nin rhetoric, on condition that the latter should pay him a certain sum\nof money if he gained his first cause. Evalthus when instructed in all\nthe precepts of the art, refused to pay Protagoras, who consequently\nbrought him before the Areopagus, and said to the Judges--\"Any verdict\nthat you may give is in my favour: if it is on my side, it carries\nthe condemnation of Evalthus; if against me, he must pay me, because\nhe gains his first cause.\" \"I confess,\" replied Evalthus, \"that the\nverdict will be pronounced either for or against me; in either case I\nshall be equally acquitted: if the Judges pronounce in my favour, you\nare condemned; if they pronounce for you, according to our agreement,\nI owe you nothing, for I lose my first cause.\" The Judges being unable\nto reconcile the pleaders, ordered them to reappear before the Court a\nhundred years afterwards.\n\n\nORIENTAL EXTRAVAGANCE.\n\nMr. Forbes has given a curious picture of the kind of magnificence\naffected by Asuf ul Dowlah, who succeeded his father on the throne\nof Oude. This nabob was fond of lavishing his treasures on gardens,\npalaces, horses, elephants, European guns, lustres, and mirrors. He\nexpended annually about L200,000 in English manufactures. He had more\nthan one hundred gardens, twenty palaces, one thousand two hundred\nelephants, three thousand fine saddle horses, one thousand five hundred\ndouble-barrel guns, seventeen hundred superb lustres, thirty thousand\nshades of various forms and colours; seven hundred large mirrors,\ngirandoles and clocks. Some of the latter were very curious, richly set\nwith jewels, having figures in continual movement, and playing tunes\nevery hour; two of these clocks only, cost him thirty thousand pounds.\nWithout taste or judgment, he was extremely solicitous to possess all\nthat was elegant and rare; he had instruments and machines of every\nart and science, but he knew none; and his museum was so ridiculously\narranged that a wooden cuckoo-clock was placed close to a superb\ntimepiece which cost the price of a diadem; and a valuable landscape of\nClaude Lorraine suspended near a board painted with ducks and drakes.\nHe sometimes gave a dinner to ten or twelve persons, sitting at their\nease in a carriage drawn by elephants. His jewels amounted to about\neight millions sterling. Amidst this precious treasure, he might be\nseen for several hours every day handling them as a child does his toys.\n\n\nANCIENT SCOTTISH CHIEFTAIN.\n\n[Illustration: [++] Scottish Costume.]\n\nAnnexed is a Scottish costume of the eighth or ninth century, after\na drawing on parchment, extracted from an old book, which, according\nto the characters on the back, appears to have been written in\nGaelic or Erse. According to the assertion of the possessor, this\nCaledonian document was brought to Germany in the year 1596, during the\ndevastating Reformation in Scotland, when all cloisters and religious\nendowments were destroyed, and a perfect victory obtained over the\nepiscopacy, so that many persons took refuge with their treasures,\non the Continent, where the Scottish monks possessed many religious\nhouses; some being at Nuremberg. Our figure represents a Highland\nchief, whose dress is picturesque and extremely beautiful. The Scottish\ntunic or blouse, checkered or striped in light and dark green, with\nviolet intermixed, and bordered with violet stripes, is covered with a\nsteel breastplate, accompanied by a back-piece, judging from the iron\nbrassarts--positively a bequest of the Romans, by whom the Scots were\nonce subjugated; this, indeed, is also attested by the offensive weapon\nthe javelin; the sword, however, must be excepted, for it is national\nand like that of the present time. The strong shield may also have\ndescended from the Romans, as well as the helmet, which is decorated\nwith an eagle's wing; these, together with the hunting-horn, give to\nthe figure a very imposing appearance. The national plaid is wanting,\nthis was borne by attendants or squires. We are involuntarily reminded\nof the heroes of Fingal and Ossian, and we might almost think that\nthis figure belonged to the time of the Scottish king, Kenneth the\nSecond, grandson of King Achaias, and the sister of the Pictish king,\nHang.\n\n\nGREEK VASES.\n\nVases of various shapes have been found in the sepulchres of Greece,\nsuch as the _oenochoe_, or jug; the _askos_, or wine-skin; the _phiale\nomphalotos_, or saucer having a boss in the centre; _rhyta_, or jugs,\nimitated from the _keras_, or horn, as well as some moulded in the\nshape of the human bust. Vases of this class, however, occur more\nfrequently in Italy than in Greece. Some are of remarkable shape. One\nin the Durand collection has its interior receded, and in the centre\na medallion of the Gorgon's head; at the edge is the head of a dog or\nfox, and to it is attached a long handle terminating in the head of an\nanimal. Similar handles are often found. Another vase from Sicily, also\nin the same collection, with a conical cover, is ornamented externally\nwith moulded subjects of wreaths, heads of Medusa, &c., painted and\ngilded.\n\n[Illustration: [++] Greek Vase.]\n\nMany of the vases intended for ornamental purposes are covered with a\nwhite coating, and painted with colours of the same kind as those used\non the figures before described, but with few and simple ornaments,\nplain bands, maeanders, chequered bands and wreaths. A vase found at\nMelos affords a curious example. We here annex a sketch of it. It\nconsists of a number of small vases united together and arranged in a\ndouble circle round a central stand. This kind of vase is supposed to\nbe the _kernos_, used in the mystic ceremonies to hold small quantities\nof viands. By some persons, however, it is thought to have been\nintended for eggs or flowers. It is covered with a white coating of\nclay, and the zigzag stripes are of a maroon colour. Such vases might\nhave been used for flower-pots, and have formed small temporary gardens\nlike those of Adonis, or have been employed as lamps.\n\n\nQUEEN ELIZABETH'S DRESSES.\n\nThe list of the Queen's wardrobe, in 1600, shows us that she had then\nonly 99 robes, 126 kirtles, 269 gowns (round, loose, and French), 136\nforeparts, 125 petticoats, and 27 fans, not to mention 96 cloaks, 83\nsave guards, 85 doublets, and 18 lap mantles.\n\nHer gowns were of the richest materials--purple, gold tissue, crimson\nsatin, cloth of gold, cloth of silver, white velvet, munsy cloth,\ntawney satin, horse-flesh satin, Isabella satin dove\n velvet, lady blush satin, drake satin, and [...]\n satin.\n\nThe cloaks are of perfumed leather, black taffety; the petticoats of\nblue satin; the jupes of orange satin; the doublets of straw\n satin; the mantles of white blush, striped with red swan's\ndown.\n\nThe most characteristic dresses are the following:--\n\nA frock of cloth of silver, checquered with red silk, like bird's eyes,\nwith demi sleeves, a cut of crimson velvet twisted on with silver,\nlined with crimson velvet.\n\nA mantle of white lawn, cut and turned in, embroidered all over with\nworks of silver, like pomegranates, roses, honeysuckles, and acorns.\n\nOne French kirtle of white satin, cut all over, embroidered with loops,\nflowers, and clouds of Venice gold, silver, and silk.\n\nOne round kirtle of white satin, embroidered all over with the work\nlike flames, peascods, and pillars, with a border likewise embroidered\nwith roses.\n\nThe stomacher (fore part) of white satin, embroidered very fair with\nborders of the sun, moon, and other signs and planets of Venice gold,\nsilver, and silk of sundry colours, with a border of beasts beneath,\nlikewise embroidered.\n\nOther gowns we find adorned with bees, spiders, flies, worms, trunks of\ntrees, s, oak leaves, and mulberries; so that \"Bess\" must have\nlooked like an illustrated edition of _AEsop's Fables_.\n\nIn one case she shines in rainbows, clouds, flames of fire, and suns;\nin another, with fountains and trees, snakes, and grasshoppers; the\nbuttons themselves, in one instance, assume the shape of butterflies,\nin another of birds of Paradise.\n\nThe fans were of white and feathers, with gold handles set\nwith precious stones, or of crystal and heliotrope; one of them\ncontained a looking-glass, another Leicester's badge of the bear and\nragged staff. Her swords had gilt handles and blood-stone studs; her\nponiards were gold and ivory, ornamented with tassels of blue silk; her\nslippers of cloth of silver, and of orange- velvet, embroidered\nwith seed pearls; her parasol was of crimson velvet damask, striped\nwith Venetian gold and silver lace, the handle mother-of-pearl.\n\nHer jewels were both numerous and curious: the head ornaments\nresembling a white lion with a fly on his side, a golden fern-branch\nwith a lizard, ladybird, and a snail upon it, an Irish dart of gold set\nwith diamonds, a golden rose with a fly and spider upon it, a golden\nfrog set with jewels, a golden daisy, and emerald buttons, gown studs\nof rubies and pearls, and a chain of golden scallop shells, with chains\nof agate and jet. A sumptuous magnificence was the characteristic of\nthe costume of this reign. When Elizabeth visited the Earl of Hertford,\nat Elvetham, that nobleman met her with 3,000 followers, with black and\nyellow feathers in their hats, and most of them wearing gold chains.\nWhen she visited Suffolk, 200 bachelors in white velvet, with as many\nburghers in black velvet coats and gold chains, and 1800 serving-men\nreceived on horseback. For the French ambassador's amusement, in 1559,\n1400 men-at-arms, clad in velvet, with chains of gold, mustered in\narms in Greenwich Park; and on another occasion there was a tournament\non Midsummer (Sunday) Night at the palace of Westminster, between ten\nknights in white, led by the Earl of Essex, and ten knights in blue,\nled by the Earl of Rutland.\n\n\nCARE OF THE BEARD.\n\nThe Mahometans are very superstitious touching the beard. They bury\nthe hairs which come off in combing it, and break them first, because\nthey believe that angels have charge of every hair, and that they gain\nthem their dismissal by breaking it. Selim I. was the first Sultan who\nshaved his beard, contrary to the law of the Koran. \"I do it,\" said\nhe apologetically to the scandalized and orthodox mufti, \"to prevent\nmy vizier leading me by it.\" He cared less for it than some of our\nancestors, two centuries ago, did for their own. They used to wear\npasteboard covers over them in the night, lest they should turn upon\nthem and rumple them in their sleep!\n\nThe famous Raskolniki schismatics had a similar superstition to the\nMahometan one mentioned above. They considered the divine image in man\nto reside in the beard.\n\n\nDOLE IN CONSEQUENCE OF A DREAM.\n\nAt Newark-upon-Trent, a curious custom, founded upon the preservation\nof Alderman Clay and his family by a dream has prevailed since the days\nof Cromwell. On 11th March every year, penny-loaves are given away to\nevery one who chooses to appear at the Town Hall and apply for them,\nin commemoration of the Alderman's deliverance, during the siege of\nNewark by the Parliamentary forces. This gentleman, by will, dated 11th\nDecember, 1694, gave to the Mayor and Aldermen, one hundred pounds, the\ninterest of which was to be given to the Vicar yearly, on condition\nof his preaching an annual sermon. Another hundred pounds were also\nappropriated for the behoof of the poor, in the way above-mentioned.\nThe origin of this bequest is singular. During the bombardment of\nNewark by Oliver Cromwell's forces, the Alderman dreamed three nights\nsuccessively that his house had taken fire, which produced such a vivid\nimpression upon his mind, that he and his family left it, and in a few\ndays the circumstances of his vision actually took place, by the house\nbeing burned down by the besiegers.\n\n\nGLOVE MONEY.\n\nGloves were popular new-year's gifts, or sometimes \"glove-money\" in\nplace of them; occasionally, these gloves carried gold pieces in them.\nWhen Sir Thomas More was Chancellor, he decided a case in favour\nof Mrs. Croaker against Lord Arundel; the former, on the following\nnew-year's day, gratefully presented the judge with a pair of gloves\nwith forty angels in them. \"It would be against good manners,\" said\nthe Chancellor, \"to forsake a gentlewoman's new-year's gift, and I\naccept the gloves. The _lining_ you will elsewhere bestow.\"\n\n\nGLAIVES.\n\nThe glaive was derived from the Celtic custom of placing a sword with\na hollow handle at the end of a pole, called by the natives of Wales\n\"llavnawr\"--_the blade weapon_, and takes its name from the Cleddyv,\nor Gleddyv, of the Welsh. In an abstract of the grants of the 1st\nof Richard III., among the Harleian MSS., No. 443, is a warrant to\nNicholas Spicer, authorising him to impress smiths for making 2,000\nWelsh glaives; and 20s. 6d. are charged for 30 glaives, with their\nstaves, made at Abergavenny and Llanllolved. In the romaunt of Guy,\nEarl of Warwick, by Walter of Exeter, written in the time of Edward\nII., also in the Harleian Library of the British Museum, they are\ncalled gleves; thus--\n\n \"Grant coupes de gleves trenchant\n Les escurs ne lur vailut gans.\"\n\n \"Such powerful strokes from cutting gleves,\n That the shields were not worth a glove.\"\n\n[Illustration: [++] Glaive.]\n\nThey were also in frequent use on the Continent, and the \"Chronicle of\nFlanders\" mentions an instance of the cavalry having armed themselves\nwith glaives, which they ornamented with pennoncels. The specimen which\nwe have here engraved is one which was made for the Doge of Venice,\nduring the time that the Emperor Charles V. had the command there, in\ncompliment to whom the centre ornament is the Austrian eagle. Upon this\nthe arms of the succeeding Doge, Francisco Veneri, who held the office\nfrom 1554 to 1556, have been deeply incised, no doubt to commemorate\nthe expulsion of the Germans. The pole, at the top of which the weapon\nwas fixed, is omitted in our engraving.\n\n\nCRUELTY OF FRANCIS CARRARA.\n\nFrancis Carrara, the last Lord of Padua, was famous for his cruelties.\nThey shew (at Venice) a little box for a toilette, in which are six\nlittle guns, which are so ordered with springs, and adjusted in such a\nmanner, that upon the opening of the trunk, the guns fired and killed\nthe lady to whom Carrara sent it for a present. They show also with\nthis, some little pocket cross-bows and arrows of steel, with which he\ntook pleasure to kill those he met, so secretly, that they could hardly\neither perceive the blow, or him that gave it.\n\n\nIRISH PIPES.\n\nThe accompanying figures represent the Irish bagpipes in their\nprimitive and improved forms. We have here the earliest pipes,\noriginally the same as the Scotch, as appears from a drawing made in\nthe sixteenth century, and given in Mr. Bunting's work; but they now\ndiffer, in having the mouthpiece supplied by the bellows A, which,\nbeing filled by the motion of the piper's arm, to which it is fastened,\nfills the bag B; whence, by the pressure of the other arm, the wind is\nis conveyed into the chanter C, which is played on with the fingers,\nmuch like a common pipe. By means of a tube, the wind is conveyed into\ndrones _a_, _a_, _a_, which, tuned at octaves to each other, produce\na kind of cronan, or bass, to the chanter. The second cut represents\nthe improved, or union pipes, the drones of which, tuned at thirds and\nfifths by the regulator, have keys attached to them, which not only\nproduce the most delightful accords, but enable the player to perform\nparts of tunes, and sometimes whole tunes, without using the chanter\nat all. Both drones and chanter can be rendered quiescent by means of\nstops.\n\n[Illustration: Common Bagpipes.]\n\n[Illustration: Union Pipes.]\n\nThe pipers were at one period the \"great originals\" of Ireland. The\nrace is gradually departing, or at least \"sobering\" down into the ranks\nof ordinary mortals; but there was a time when the pipers stood out\nvery prominently upon any canvas that pictured Irish life. Anecdotes\nof their eccentricities might be recorded that would fill volumes. For\nmany years past their power has been on the wane; temperance committed\nsad havoc on their prospects; and at length the introduction of \"brass\nbands\" effectually destroyed the small balance that remained to them of\nhope.\n\n\nNOVEL WAY OF CURING VICIOUS HORSES.\n\nBurckhardt tells us of a strange mode of curing a vicious horse. He has\nseen, he says, vicious horses in Egypt cured of the habit of biting by\npresenting to them, while in the act of doing so, a leg of mutton just\ntaken from the fire. The pain which the horse feels in biting through\nthe hot meat causes it to abandon the practice.\n\n\nGROUND ICE.\n\nEvery one who has watched the freezing of a lake or pond, or any other\ncollection of still water, must be well aware that the ice begins to\nform on the surface in thin plates or layers, which on the continuance\nof the frost gradually become thicker and more solid, until the water\nis affected in a downward direction, and becomes, perhaps, a solid\nmass of ice. This is universally the case in stagnant water, but\nit has been repeatedly proved that in rapid and rugged streams the\nprocess of freezing is often very different. In direct opposition,\nas it would seem, to the laws of the propagation of heat, the ice in\nrunning water frequently begins to form at the bottom of the stream\ninstead of the top; and this fact, while it is received with doubt by\nsome, even among the scientific, is frequently attested by those whose\nbusiness leads them to observe the phenomenon connected with rivers.\nMillers, fishermen, and watermen find that the masses of ice with which\nmany rivers are crowded in the winter season rise from the bottom or\nbed of the stream. They say that they have seen them come up to the\nsurface, and have also borne them up with their hooks. The under part\nof these masses of ice they have found covered with mud or encrusted\nwith gravel, thus bearing plain marks of the ground on which the ice\nhad rested. The testimony of people of this class in our country agrees\nwith that of a similar class in Germany, where there is a peculiar term\nmade use of to designate floating ice, i. e. _grundeis_ (ground-ice).\n\nA striking example of the formation of ground-ice is mentioned by the\nCommander Steenk, of Pillau. On the 9th of February, 1806, during a\nstrong south-east wind, and a temperature a little exceeding 34 deg. Fahr.,\na long iron chain, to which the buoys of the fair-way are fastened,\nand which had been lost sight of at Schappeiswrack in a depth of from\nfifteen to eighteen feet, suddenly made its appearance at the surface\nof the water and swam there; it was, however, completely encrusted with\nice to the thickness of several feet. Stones, also, of from three to\nsix pounds' weight, rose to the surface; they were surrounded with a\nthick coat of ice. A cable, also, three and a half inches thick, and\nabout thirty fathoms long, which had been lost the preceding summer in\na depth of thirty feet, again made its appearance by swimming to the\nsurface; but it was enveloped in ice to the thickness of two feet. On\nthe same day it was necessary to _warp_ the ship into harbour in face\nof an east wind; the anchor used for that purpose, after it had rested\nan hour at the bottom, became so encrusted with ice, that it required\nnot more than half the usual power to heave it up.\n\nM. Hugi, president of the Society of Natural History at Soleure,\nobserved, in February, 1827, a multitude of large icy tables on the\nriver Aar. These were continually rising from the bottom, over a\nsurface of four hundred and fifty square feet, and the phenomenon\nlasted for a couple of hours. Two years afterwards he witnessed a\nsimilar occurrence. On the 12th of February, 1829, at sunrise, and\nafter a sudden fall in the temperature, the river began to exhibit\nnumerous pieces of floating ice, although there was no sign of freezing\non the surface, either along the banks, or in shady places where the\nwater was calm. Therefore it could not be said that the floating masses\nwere detached from the banks. Nor could they have proceeded from any\nlarge sheet of ice farther up the river, because, higher up, the river\nexhibited hardly any ice. Besides, flakes of ice commenced soon to rise\nup above the bridge; towards mid-day, islands of ice were seen forming\nin the centre of the river; and by the next day these were twenty-three\nin number, the largest being upwards of two hundred feet in diameter.\nThey were surrounded with open water, resisting a current which flowed\nat the rate of nearly two hundred feet in a minute, and extending over\na space of one-eighth of a league. M. Hugi visited them in a small\nboat. He landed, examined them in every direction, and discovered that\nthere was a layer of compact ice on their surface a few inches in\nthickness, resting on a mass having the shape of an inverted cone, of a\nvertical height of twelve or thirteen feet, and fixed to the bed of the\nriver. These cones consisted of half-melted ice, gelatinous, and much\nlike the spawn of a frog. It was softer at the bottom than at the top,\nand was easily pierced in all directions with poles. Exposed to the\nopen air, the substance of the cones became quickly granulated, like\nthe ice that is formed at the bottom of rivers.\n\nIn the same year the pebbles in a creek of shallow water, near a\nvery rapid current of the Rhine, were observed to be covered with a\nsort of transparent mass, an inch or two in thickness, and which, on\nexamination, was found to consist of icy spicula, crossing each other\nin every direction. Large masses of spongy ice were also seen in\nthe bed of the stream, at a depth of between six or seven feet. The\nwatermen's poles entered these with ease, and often bore them to the\nsurface. This kind of ice forms most quickly in rivers whose bed is\nimpeded with stones and other foreign bodies.\n\n\nHINDOO COMPUTATION.\n\nThe Hindoos call the whole of their four ages a _divine age_; a\nthousand divine ages form a _calpa_, or one of Brahma's days, who,\nduring that period, successively invested fourteen _menus_, or holy\nspirits, with the sovereignty of the earth. The _menu_ transmits his\nempire to his posterity for seventy-one divine ages, and this period\nis called _manawantara_, and as fourteen _manawantara_ make but nine\nhundred and ninety-four divine ages, there remain six, which are the\ntwilight of Brahma's day. Thirty of these days form his month; twelve\nof these months one of his years; and one hundred of these years the\nduration of his existence. The Hindoos assert that fifty of these years\nhave already elapsed, so that we are in the first day of the first\nmonth of the fifty-first year of Brama's age, and in the twenty-eighth\ndivine age of the seventh _manawantara_. The first three human ages of\nthis age, and five thousand years of the fourth are past. The Hindoos\ntherefore calculate that it is 131,400,007,205,000 years since the\nbirth of Brahma, or the beginning of the world.\n\n\nCHINESE TOMB.\n\nLike all people of Tartar origin, one of the most remarkable\ncharacteristics of the Chinese is their reverence for the dead, or, as\nit is usually called, their ancestral worship. In consequence of this,\ntheir tombs are not only objects of care, but have frequently more\nornament bestowed upon them than graces the dwellings of the living.\n\n[Illustration: [++] Chinese Tomb.]\n\nTheir tombs are of different kinds; but the most common arrangement is\nthat of a horseshoe-shaped platform, cut out of the side of a hill,\nas represented in our engraving. It consequently has a high back,\nin which is the entrance to the tomb, and s off to nothing at\nthe entrance to the horseshoe, where the wall generally terminates\nwith two lions or dragons, or some fantastic ornament common to\nChinese architecture. When the tomb is situated, as is generally the\ncase, on a hillside, this arrangement is not only appropriate, but\nelegant. When the same thing is imitated on a plain, it is singular,\nmisplaced, and unintelligible. Many of the tombs are built of granite,\nfinely polished, and carved with a profusion of labour that makes us\nregret that the people who can do such things should have so great a\npredilection for ephemeral wooden structures, when capable of employing\nthe most durable materials with such facility.\n\n\nABYSSINIAN ARMS.\n\n[Illustration: [++] Abyssinian Arms.]\n\nThe above engraving represents a group of Abyssinian arms. The sword,\nspear, and shield are essentially the weapons of the Abyssinians,\nfirearms being only of comparatively recent introduction, and not\ngenerally used. The shields are round, and nearly a yard in diameter;\nthey are very neatly made of buffalo's hide, and of the form most\ncalculated to throw off a lance-point; namely, falling back gradually\nfrom the boss or centre (which protrudes) to the edges. At the centre,\nin the inside, is fixed a solid leather handle, by which the shield is\nheld in the hand when fighting, or through which the arm is passed\nto the elbow, for convenience of carrying on a journey. The edge is\nperforated with a number of holes, through which leather loops are\npassed, and by these it is hung up in the houses. The face of the\nshield is often ornamented in various ways, according to the wealth\nor fancy of the owner. Some have simply a narrow strip of lion's skin\non each side of the boss, but crossing each other above and below\nit, the lower ends being allowed to hang at some length; others have\na large broad strip of the mane down the centre of the shield, and\nhanging several inches below it. This is, of course, usually made of\ntwo or three pieces stitched together, as it would be difficult to get\na single piece of sufficient length and beauty of fur. Others to this\nadd a lion's paw or tail, fastened on the left side of the mane, and\noften highly adorned with silver. The beautiful long black and white\nfur of a sort of monkey, called \"goreza,\" occasionally supplies the\nplace of that of the nobler yet scarcely so beautiful animal. A shield\nalmost completely covered with plates and bosses of silver, is usually\nthe mark of the chief of some district. Those similarly plated in\nbrass were likewise formerly used only by chiefs, though now they are\ncarried by every soldier who can afford to buy them. The plated shield\nis called \"tebbora.\" Those in brass are not much approved of, as they\nusually cover a bad skin; for a man possessed of a good handsome shield\nwould never think of thus hiding its intrinsic beauties.\n\nIn former times a beautiful crooked knife was used in Tigre, the sheath\nand handle of which were profusely enriched with silver and gilt.\nThese, however, are never worn now, the long \"shotel\" in Tigre, and the\nEuropean-shaped sword among the Amhara and most of the soldiers, have\nentirely superseded them.\n\nThe \"shotel\" is an awkward-looking weapon. Some, if straight, would be\nnearly four feet long: they are two-edged, and curved to a semicircle,\nlike a reaper's sickle. They are principally used to strike the point\ndownwards over the guard of an adversary, and for this the long curved\nshape is admirably adapted. It is, however, a very clumsy weapon to\nmanage. The sheath is of red morocco leather, its point being often\nornamented with a hollow silver ball, called \"lomita,\" as large as\na small apple. Many of the swords used are made in Europe, and are\nsuch as would be carried by the light cavalry, though lighter than\nours. Being, however, cheap, showy articles, they are apt to break,\nand therefore the Abyssinians are getting tired of them, preferring\nthose made of soft iron in their own country. These they make also\nwith the faible considerably broader than the forte, to give force to\nthe blow. Of course, they bend on the least stress; but, in defence\nof this failing, their owners say that, if a sword breaks, who is\nto mend it?--while, if it bends, you have only to sit on it, and it\ngets straight again. The handles of both this and the \"shotel\" are\nmade of the horn of the rhinoceros. They are cut out of the horn at\ngreat loss of material, and hence they fetch a good price. It should\nbe remembered that the heart of the horn is black, outside of which\nthere is a coating, not quite an inch thick, of a semi-transparent\nwhite colour. To make a sword-handle, a piece of horn of the requisite\nlength is first sawn off. This is then re-sawn longitudinally into\nthree pieces, of which the inner one only is eligible for handles. This\npiece is about an inch and a half thick, four or five inches broad at\nthe broader extremity, and three at the narrower. As it lies sawn flat\nbefore us we can distinctly see the black stripe in the centre, with\nthe white on each side. Next, a nearly semicircular piece is cut out\nat each side, leaving only four points of the white as four corners,\nand the grasp black. The handle is then finished, bored for the shank\nof the blade, and polished. The shank is usually clinched over a\nhalf-dollar beaten convex; a _fil-et-grain_ boss, called \"timbora,\" is,\nhowever, sometimes substituted. A sword-hilt thus made is obviously a\nvery clumsy one to handle, as the points are parallel to the edge, and\nthose farthest from the blade are longest.\n\n\nGEORGIANS AS TOPERS.\n\nIt is as unsurpassable topers, as well as for their military qualities,\nwhich have always been acknowledged, that the Georgians have acquired\nnotoriety. At their frequent drinking parties it is said they will pass\nseveral days and nights, almost without intermission, in quaffing the\nproductions of the vineyards of Kakheti, a district in the mountains\neast of Tiflis. This wine is by no means of bad quality; it is of a\ndeep red colour, so deep that one fancies it has been tinged with some\ndye to produce so intense a hue. They are said to consume incredible\nquantities of wine on these occasions, and in a fashion that would\nput to shame the drinking triumphs of Ireland, recorded by Sir Jonah\nBarrington, in days of old, when intoxication was the standard of\nspirit. The drinking vessel is a cow's horn, of considerable length,\nand the point of honour is to drain it at a draught. The brethren and\nconvivial rivals of the Georgians in the neighbouring provinces of\nImeretia and Mingrelia, instead of a horn, use a delicately-hollowed\nglobe of walnut tree, with a long narrow tube at the orifice. It holds\nfully a pint, and like its companion, the horn, the contents are\nconsumed at a single gulp. How these globes are hollowed is as great\na marvel as the construction of the ingenious Chinese puzzle of ball\nwithin ball.\n\n\nSTAG-HUNT IN THE SIXTEENTH CENTURY.\n\nThe following vivid picture of a stag-hunt is taken from the page of\nan old author, and refers to the days of the unfortunate Mary Queen\nof Scots:--\"In the year 1567, the Earl of Athol, a prince of the\nblood royal, had, with much trouble and vast expense, a hunting-match\nfor the entertainment of our most illustrious and gracious queen.\nOur people called this a royal hunting. I was then a young man, and\npresent on that occasion. Two thousand Highlanders, or wild Scotch,\nas you call them, were employed to drive to the hunting-ground all\nthe deer from the woods and hills of Athol, Badenoch, Marr, Murray,\nand the counties about. As these Highlanders use a light dress, and\nare very swift of foot, they went up and down so nimbly, that in less\nthan two months' time, they brought together two thousand red deer,\nbesides roes and fallow deer. The queen, the great men, and a number\nof others, were in a glen when these deer were brought before them.\nBelieve me, the whole body moved forward in something like battle\norder. The sight delighted the queen very much; but she soon had cause\nfor fear. Upon the earl--(who had been accustomed from his early days\nto such sights)--addressing her thus:--'Do you observe that stag who\nis foremost of the herd? There is danger from that stag; for if either\nfear or rage should force him from the ridge of that hill, let every\none look to himself, for none of us will be out of the way of harm;\nfor the rest will follow this one, and having thrown us under foot,\nthey will open a passage to this hill behind us.' What happened a\nmoment after confirmed this opinion; for the queen ordered one of the\nbest dogs to be let loose on one of the deer: this the dog pursues;\nthe leading stag was frighted; he flies by the same way he had come\nthere; the rest rush after him, and break out where the thickest body\nof Highlanders are; they had nothing for it but to throw themselves\nflat on the heath, and allow the deer to pass over them. It was told\nthe queen that several of the Highlanders had been wounded, and that\ntwo or three had been killed outright; and the whole body had got off,\nhad not the Highlanders, by their skill in hunting, fallen upon a\nstratagem to cut off the rear from the main body. It was of those that\nhad been separated that the queen's dogs and those of the nobility made\nslaughter. There were killed that day three hundred and sixty deer,\nbesides some roes.\"\n\n\nTIME WASTED IN TAKING SNUFF.\n\nA vast quantity of valuable time is wasted by the votaries of tobacco,\nespecially by the smokers; and that the devotees of snuff are not\ngreatly behind in this respect, will be shown by the following singular\ncalculation of Lord Stanhope:--\n\n\"Every professed, inveterate, and incurable snuff-taker,\" says his\nlordship, \"at a moderate computation, takes one pinch in ten minutes.\nEvery pinch, with the agreeable ceremony of blowing and wiping the nose\nand other incidental circumstances, consumes one minute and a half.\nOne minute and a half out of every ten, allowing sixteen hours to a\nsnuff-taking day, amounts to two hours and twenty-four minutes out of\nevery natural day, or one day out of ten. One day out of every ten,\namounts to thirty-six days and a half in a year. Hence, if we suppose\nthe practice to be persisted in forty years, two entire years of the\nsnuff-taker's life will be dedicated to tickling his nose, and two more\nto blowing it. The expense of snuff, snuff-boxes, and handkerchiefs,\nwill be the subject of a second essay, in which it will appear that\nthis luxury encroaches as much on the income of the snuff-taker as it\ndoes on his time; and that by proper application of the time and money\nthus lost to the public, a fund might be constituted for the discharge\nof the national debt.\"\n\n\nVALUE OF A LONG PSALM.\n\nFormerly a psalm was allowed to be sung at the gallows by the culprit,\nin case of a reprieve. It is reported of one of the chaplains to\nthe famous Montrose, that being condemned in Scotland to die, for\nattending his master in some of his glorious exploits, and being\nupon the ladder, ordered to set out a psalm, he expecting a reprieve,\nnamed the 119th Psalm (with which the officer attending the execution\ncomplied, the Scotch Presbyterians being great psalm-singers): and it\nwas well for him he did so, for they had sung it half through before\nthe reprieve came: any other psalm would have hanged him.\n\n\nANCIENT INCENSE CHARIOT.\n\n[Illustration: [++] Ancient Incense Chariot.]\n\nThe implement which we have engraved was found in a tomb at Cervetri\nin Etruria, and unquestionably belongs to a very remote date of the\narchaic period. It was used in the ritual services of the ancients,\nand seems to have been destined for burning incense. The perfume was,\nno doubt, placed in the concave part, and the fact of the whole being\nmounted upon four wheels proves that it was intended to be moved about,\nwhich, in religious services, may have been a great convenience. The\nborders are adorned by a row of flower-shaped ornaments, the graceful\nforms of which will be appreciated in the side-view we have given of\nit. It must be confessed, indeed, that this monument, which is marked\nby the stamp of an antiquity so exceedingly remote, displays within the\nlimits of its archaic character much elegance, conveying the idea of a\nhighly refined taste, suitable to a person of dignified position, as\nthe priest or king may be supposed to have been, to whom the article\nbelonged.\n\n\nTOO MUCH PARENTAL AUTHORITY.\n\nAll the world over, the current of natural affection flows strongly\ndownwards to posterity. Love for children, in most nations, seems to\nbe stronger than the love for parents. But in China, the current of\nnatural affection is thrown back towards parents with undue strength.\nThe love of posterity is in danger of being checked and weakened by\ntheir excessive veneration for parents. The father has absolute power,\neven the power of life and death, over his children. A few years ago,\na Chinese father said to his wife, \"What shall we do with our young\nson? He is undutiful and rebellious, and will bring disgrace on our\nfamily name; let us put him to death.\" Accordingly, having tied a cord\nround the boy's neck, the father pulled one end of it, and the mother\nthe other, and thus they strangled their son. The magistrates took no\nnotice of the occurrence. A wealthy Chinese gentleman at Ningpo shut\nup one of his orphan grandchildren and starved her to death. He could\nnot be troubled rearing her up. Another man at the same place, having\ncommanded two of his sons one day to follow him, entered a boat, and\nrowed out to the middle of the stream. He then deliberately tied a\nstone to the neck of one of his sons, and threw him into the river. The\nother lad was compelled to assist his father in the cruel proceeding.\nThese facts are well known to the missionaries at that place. They\nheard the cries of the poor girl, and rescued her sister from a similar\nfate, and they saw the youth drowned by his father. But the authorities\nnever thought of interfering.\n\n\nPOPULAR PASTIMES.\n\nThe popular pastimes of the time of James the First are enumerated in\nthe following lines, in a little work entitled \"The Letting of Humour's\nBlood in the Head-vaine; with a New Morisco daunced by seven Satyres\nupon the bottome of Diogenes' tubbe:\" 8vo, Lond. 1611.\n\n \"Man, I dare challenge thee to THROW THE SLEDGE,\n To jump or LEAPE over ditch or hedge,\n To WRASTLE, play at STOOLEBALL, or to RUNNE:\n To PITCH THE BARRE, or to SHOOTE OFF A GUNNE:\n To play at LOGGETS, NINE HOLES, or TEN PINNES:\n To try it out at FOOT-BALL by the shinnes:\n At TICKTACKE, IRISH NODDIE, MAW, and RUFFE,\n At HOT-COCKLES, LEAP-FROG, or BLINDMAN-BUFFE;\n To drinke halfe-pots, or deale at the whole can:\n To play at BASE, or PEN-AND-YNKHORNE SIR JUAN;\n To daunce the MORRIS, play at BARLEY-BREAKE,\n At all exploytes a man can thinke or speake;\n At SHOVE-GROATE, VENTER-POYNT, or CROSSE & PILE,\n At BESHROW HIM THAT'S LAST AT YONDER STYLE;\n At LEAPING O'ER A MIDSOMMER-BON-FIER,\n Or at the DRAWING DUN OUT OF THE MYER:\n At any of those, or all these presently,\n Wagge but your finger, I am for you, I!\"\n\n\nVACILLATING NEWSPAPERS.\n\nThe newspapers of Paris, submitted to the censorship of the press, in\n1815, announced in the following terms, Bonaparte's departure from the\nIsle of Elba, his march across France, and his entry into the French\nCapital:--9th March--The Cannibal has escaped from his den. 10th--The\nCorsican ogre has just landed at Cape Juan. 11th--The Tiger has arrived\nat Gap. 12th--The Monster has passed the night at Grenoble. 13th--The\nTyrant has crossed Lyons. 14th--The Usurper is directing his course\ntowards Dijon, but the brave and loyal Burgundians have risen in a\nbody, and they surround him on all sides. 18th--Bonaparte is sixty\nleagues from the Capital; he has had skill enough to escape from the\nhands of his pursuers. 19th--Bonaparte advances rapidly, but he\nwill never enter Paris. 20th--To-morrow, Napoleon will be under our\nramparts. 21st--The Emperor is at Fontainebleau. 22nd--His Imperial and\nRoyal Majesty last evening made his entrance into his Palace of the\nTuileries, amidst the joyous acclamations of an adoring and faithful\npeople.\n\n\nPRESSING TO DEATH, AND PRAYING AND FASTING.\n\nIn a number of Oliver Cromwell's Newspaper, \"The Perfect Account of\nthe Daily Intelligence,\" dated April 16th, 1651, we find this horrid\ninstance of torture:--\n\n\"Mond. April 14th.--This session, at the Old Bailey, were four men\npressed to death that were all in one robbery, and, out of obstinacy\nand contempt of the court, stood mute and refused to plead; from whence\nwe may perceive the exceeding great hardness some men are grown unto,\nwho do not only swerve from instructions, exhortations, and goodnesse,\nbut become so lewd and insolent that they render themselves the proper\nsubjects for whom severe laws were first invented and enacted.\"\n\nThe very next paragraph in the paper is to the following effect:--\n\n\"Those of the congregate churches, and many other godly people in\nLondon and parts adjacent, have appointed Friday, the 25th instant, as\na day of solemn fasting and prayer, for a blessing upon the armies at\nland, the fleet at sea, and negociations abroad.\"\n\n\nTHE FIRST WATCHES IN ENGLAND.\n\nIn 1584 watches began to come from Germany, and the watchmaker soon\nbecame a trader of importance. The watches were often of immense size,\nand hung in a rich case from the neck, and by s wound up with great\ngravity and ceremony in Paul's or at the ordinary dinner. Catgut\nmainsprings must have been slightly affected by changes of weather, and\nsometimes a little out of time in wet Novembers; but, Sessa, let the\nworld live! An early specimen of the watch that we have seen engraved\nwas, however, not larger than a walnut, richly chased, and enclosed\nin a pear-shaped case. It had no minute hand, but was of beautiful\nworkmanship. Country people, like Touchstone, sometimes carried pocket\ndials, in the shape of brass rings, with a slide and aperture, to be\nregulated to the season.\n\n\nEXTRAORDINARY CIRCUMSTANCE.\n\nJesse, in his interesting \"Gleanings in Natural History,\" gives\nthe following remarkable instance of an extraneous substance being\nfound imbedded in the solid timber of an ash:--\"A person on whose\naccuracy and veracity I can place every reliance, informed me that\nhearing from some of his brother workmen, that in sawing up the butt\nof a large ash-tree, they had found a bird's nest in the middle of\nit; he immediately went to the spot, and found an ash cut in two\nlongitudinally on the saw-pit, and the bird's nest nearly in the centre\nof the tree. The nest was about two-thirds of a hollow globe, and\ncomposed of moss, hair, and feathers, all seemingly in a fresh state.\nThere were three eggs in it, nearly white and somewhat speckled. On\nexamining the tree most minutely with several other workmen, no mark\nor protuberance was found to indicate the least injury. The bark was\nperfectly smooth and the tree quite sound.\" In endeavouring to account\nfor this curious fact, we can only suppose that some accidental hole\nwas made in the tree before it arrived at any great size, in which a\nbird had built its nest, and forsaken it after she had laid three eggs.\nAs the tree grew larger, the bark would grow over the hole, and in\nprocess of time the nest would become embedded in the tree.\n\n\nPORT CAVE.\n\n[Illustration: [++] Port Cave.]\n\nThe above is a sketch of a cave which well deserves a place among our\ncollection of Wonders. It is called Port Cave, and is in the\nline of rocks near the Giants' Causeway. It may be visited either by\nsea or by land. Boats may row into it to the distance of a hundred\nyards or more, but the swell is sometimes dangerous; and although\nthe land entrance to the cave is slippery, and a fair proportion of\nclimbing is necessary to achieve the object, still the magnificence of\nthe excavation, its length, and the formation of the interior, would\nrepay greater exertion; the stones of which the roof and sides are\ncomposed, and which are of a rounded form, and embedded, as it were,\nin a basaltic paste, are formed of concentric spheres resembling the\ncoats of an onion; the innermost recess has been compared to the side\naisle of a Gothic cathedral; the walls are most painfully slimy to the\ntouch; the discharge of a loaded gun reverberates amid the rolling of\nthe billows, so as to thunder a most awful effect; and the notes of a\nbugle, we are told, produced delicious echoes.\n\n\nANECDOTE IN PORCELAIN.\n\n[Illustration: [++] Count Bruehl's Tailor and His Wife.]\n\nThe finest specimens of Dresden porcelain were undoubtedly made\npreviously to the Seven Years' War, when no expense was spared, and\nwhen any price might be obtained. Count Bruehl, the profligate minister\nof Augustus III., whose splendid palace and terrace is the great\nornament of Dresden, was importuned by his tailor to be allowed to\nsee the manufactory, admission to which was strictly prohibited. At\nlength he consented, and the tailor upon his entrance was presented\nwith the two last new pieces made, which were--one a grotesque figure,\na portrait of himself mounted upon a he-goat, with the shears, and all\nhis other implements of trade; and the other, his wife upon a she-goat,\nwith a baby in swaddling clothes. The poor tailor was so annoyed with\nthese caricatures, that he turned back without desiring to see more.\nThese pieces, known as Count Bruehl's Tailor and his Wife, are now much\nsought after, from their historical interest. They were made in 1760,\nby Kaendler.\n\n\nANGLO-SAXON FEASTS.\n\nIt is a mark of Anglo-Saxon delicacy, that table-cloths were features\nat Anglo-Saxon feasts; but, as the long ends were used in place of\nnapkins, the delicacy would be of a somewhat dirty hue, if the cloth\nwere made to serve at a second feast. There was a rude sort of display\nupon the board; but the order of service was of a quality that would\nstrike the \"Jeameses\" of the age of Victoria with inexpressible\ndisgust. The meat was never \"dished,\" and \"covers\" were as yet unknown.\nThe attendants brought the viands into the dining-hall on the spits,\nknelt to each guest, presented the spit to his consideration; and,\nthe guest having helped himself, the attendant went through the same\nceremony with the next guest. Hard drinking followed upon these same\nceremonies; and even the monasteries were not exempt from the sins\nof gluttony and drunkenness. Notwithstanding these bad habits, the\nAnglo-Saxons were a cleanly people; the warm bath was in general use.\nWater, for hands and feet, was brought to every stranger on entering a\nhouse wherein he was about to tarry and feed; and, it is said that one\nof the severest penances of the church was the temporary denial of the\nbath, and of cutting the hair and nails.\n\n\nHOUSEHOLD RULES IN THE SIXTEENTH CENTURY.\n\nFrom Sir J. Harrington's (the translator of Ariosto) rules for\nservants, we obtain a very clear conception of the internal government\nof a country gentleman's house in 1566.\n\nA servant who is absent from prayers to be fined. For uttering an oath,\n1d.; and the same sum for leaving a door open.\n\nA fine of 2d., from Lady Day to Michaelmas, for all who are in bed\nafter six, or out after ten.\n\nThe same fine, from Michaelmas to Lady Day, for all who are in bed\nafter seven, or out after nine.\n\nA fine of 1d. for any bed unmade, fire unlit, or candle-box uncleaned\nafter eight.\n\nA fine of 4d. for any man detected teaching the children obscene words.\n\nA fine of 1d. for any man waiting without a trencher, or who is absent\nat a meal.\n\nFor any one breaking any of the butler's glass, 12d.\n\nA fine of 2d. for any one who has not laid the table for dinner by\nhalf-past ten, or the supper by six.\n\nA fine of 4d. for any one absent a day without leave.\n\nFor any man striking another, a fine of 1d.\n\nFor any follower visiting the cook, 1d.\n\nA fine of 1d. for any man appearing in a foul shirt, broken hose,\nuntied shoes, or torn doublet.\n\nA fine of 1d, for any stranger's room left for four hours after he be\ndressed.\n\nA fine of 1d. if the hall be not cleansed by eight in winter and seven\nin summer.\n\nThe porter to be fined 1d. if the court-gate be not shut during meals.\n\nA fine of 3d. if the stairs be not cleaned every Friday after dinner.\n\nAll these fines were deducted by the steward at the quarterly payment\nof the men's wages. If these laws were observed, the domestic\ndiscipline must have been almost military in it.\n\n\nTHE QUEEN OF SHEBA.\n\nBelkis, according to the Arabs, was the famous Queen of Sheba or Saba,\nwho visited, and afterwards married, Solomon, in the twenty-first\nyear of her reign. Tabari has introduced her story with such gorgeous\nembellishments as to resemble a fairy tale rather than episode in\nserious narrative. She is said to have been subdued by the Jewish\nmonarch, who discovered her retreat among the mountains, between Hejaz\nand Yemen by means of a lapwing, which he had despatched in search\nof water during his progress through Arabia. This princess is called\nNicolaa by some writers. The Abyssinians claim the same distinction for\none of their queens; and have preserved the names of a dynasty alleged\nto have been descended from her union with Solomon.\n\n\nSUPERSTITION IN FRANCE.\n\nIn France, superstition at this day is even more prevalent than it\nis in England. Garinet, in his history of Magic and Sorcery in that\ncountry, cites upwards of twenty instances which occurred between the\nyears 1805 and 1818. In the latter year no less than three tribunals\nwere occupied with trials originating in this humiliating belief: we\nshall cite only one of them. Julian Desbourdes, aged fifty-three, a\nmason, and inhabitant of the village of Thilouze, near Bourdeaux,\nwas taken suddenly ill, in the month of January 1818. As he did not\nknow how to account for his malady, he suspected at last that he was\nbewitched. He communicated this suspicion to his son-in-law Bridier,\nand they both went to consult a sort of idiot, named Boudouin, who\npassed for a conjuror or _white-witch_. This man told them that\nDesbourdes was certainly bewitched, and offered to accompany them to\nthe house of an old man named Renard, who, he said, was undoubtedly\nthe criminal. On the night of the 23rd of January all three proceeded\nstealthily to the dwelling of Renard, and accused him of afflicting\npersons with diseases by the aid of the devil. Desbourdes fell on his\nknees and earnestly entreated to be restored to his former health,\npromising that he would take no measures against him for the evil he\nhad done. The old man denied in the strongest terms that he was a\nwizard; and when Desbourdes still pressed him to remove the spell from\nhim, he said he knew nothing about the spell, and refused to remove\nit. The idiot Boudouin, the _white-witch_, now interfered, and told\nhis companions that no relief for the malady could ever be procured\nuntil the old man confessed his guilt. To force him to confession they\nlighted some sticks of sulphur which they had brought with them for the\npurpose, and placed them under the old man's nose. In a few moments he\nfell down suffocated and apparently lifeless. They were all greatly\nalarmed; and thinking that they had killed the man, they carried him\nout and threw him into a neighbouring pond, hoping to make it appear\nthat he had fallen in accidentally. The pond, however, was not very\ndeep, and the coolness of the water reviving the old man, he opened his\neyes and sat up. Desbourdes and Bridier, who were still waiting on the\nbank, were now more alarmed than before, lest he should recover and\ninform against them. They therefore waded into the pond, seized their\nvictim by the hair of the head, beat him severely, and then held him\nunder water till he was drowned.\n\nThey were all three apprehended on the charge of murder a few days\nafterwards. Desbourdes and Bridier were found guilty of aggravated\nmanslaughter only, and sentenced to be burnt on the back, and to work\nin the galleys for life. The _white-witch_ Boudouin was acquitted on\nthe ground of insanity.\n\n\nHELMET OF SIR JOHN CROSBY.\n\n[Illustration: [++] Helmet of Sir John Crosby.]\n\nWe here present our readers with a sketch of the helmet of Sir John\nCrosby, as it originally appeared when suspended over his tomb in St.\nHelen's Church, Bishopsgate. He was an eminent merchant of London; but\nis represented upon his tomb in a full suit of armour. He died in 1475.\nThe extreme height of the crown of the helmet resembles that on the\ntomb of the Earl of Warwick, in the Beauchamp Chapel at Warwick; and\nwas intended to support the crest of the wearer, the holes for affixing\nit being still visible.\n\n\nEARTHQUAKE PANIC.\n\nA panic terror of the end of the world seized the good people of Leeds\nand its neighbourhood in the year 1806. It arose from the following\ncircumstances. A hen, in a village close by, laid eggs, on which were\ninscribed the words, \"_Christ is coming_.\" Great numbers visited the\nspot, and examined these wondrous eggs, convinced that the day of\njudgment was near at hand. Like sailors in a storm, expecting every\ninstant to go to the bottom, the believers suddenly became religious,\nprayed violently, and flattered themselves that they repented them of\ntheir evil courses. But a plain tale soon put them down, and quenched\ntheir religion entirely. Some gentlemen, hearing of the matter, went\none fine morning and caught the poor hen in the act of laying one of\nher miraculous eggs. They soon ascertained beyond doubt that the egg\nhad been inscribed with some corrosive ink, and cruelly forced up\nagain, into the bird's body. At this explanation, those who had prayed,\nnow laughed, and the world wagged as merrily as of yore.\n\n\nOLD ENGLISH SACK-POT.\n\n[Illustration: [++] Sack-Pot.]\n\nSack was such a national beverage of the jolly old England of the\nseventeenth century, that we are sure our readers will thank us for\ngiving them an idea of the vessel in which it was commonly used. The\nbottle here engraved, and inscribed \"Sack,\" was found in Old Tabley\nHall, Cheshire, and is a veritable specimen of the sort of vessel from\nwhich the topers of the \"good old times\" poured into their cups the\ndrink with which they so loved to warm their heart-strings. It is of a\ndull-white, with blue letters, and it is in the possession of the Hon.\nRobert Curzon, jun., author of the interesting work on the Monasteries\nof the Levant. Two old English bottles of similar character, one\nlettered Sack, the other Claret, dated 1646, were sold at Strawberry\nHill.\n\n\nAGE OF TREES.\n\nMr. Twining was engaged, in the year 1827, in measuring and inspecting\na large lot of hemlock timber cut from the north-eastern of East\nRock, New Haven (America), and destined for the foundation of a wharf.\nWhile thus employed he took particular notice of the successive layers,\neach of which constitutes a year's growth of the tree, and which in\nthat kind of wood are very distinct. These layers were of various\nbreadths, and plainly showed that in some seasons the trees made a\nmuch greater advance than in others, some of the layers being five or\nsix times broader than others. Every tree had thus preserved a _record\nof the seasons_ for the period of its growth, whether thirty years or\ntwo hundred--and what was worthy of notice, _every tree told the same\nstory_. Thus, by beginning at the outer layer of two trees, the one\nyoung the other old, and counting back twenty years, if the young tree\nindicated, by a full layer, a _growing season_ for that kind of timber,\nthe other tree indicated the same.\n\n\"I had then before me,\" (says this intelligent observer) \"two or three\nhundred _meteorological tables_, all of them as unerring as nature; and\nby selecting one tree from the oldest, and sawing out a thin section\nfrom its trunk, I might have preserved one of the number to be referred\nto afterwards. It might have been smoothed on the one side by the\nplane, so as to exhibit its record to the eye with all the neatness and\ndistinctness of a drawing. On the opposite side might have been minuted\nin indelible writing the locality of the tree, the kind of timber, the\nyear and month when cut, the soil where it grew, the side and point\nwhich faced the north, and every other circumstance which can possibly\nbe supposed ever to have the most remote relation to the value of the\ntable in hand. The lover of science will not be backward to incur such\ntrouble, for he knows how often, in the progress of human knowledge,\nan observation or an experiment has lost its value by the disregard of\nsome circumstance connected with it, which at the time was not thought\nworthy of notice. Lastly, there might be attached to the same section\na written meteorological table compiled from the observations of some\nscientific person, if such observations had been made in the vicinity.\nThis being done, why, in the eye of science, might not this _natural_,\n_unerring_, _graphical_ record of seasons past deserve as careful\npreservation as a curious mineral, or a new form of crystals?\"\n\n\nTHE CAMEL AS A SCAPE-GOAT.\n\nA very singular account of the use to which a camel is sometimes put,\nis given by the traveller Bruce. He tells us that he saw one employed\nto appease a quarrel between two parties, something in the same way as\nthe scape-goat was used in the religious services of the Jewish people.\nThe camel being brought out was accused by both parties of all the\ninjuries, real or supposed, which belonged to each. All the mischief\nthat had been done, they accused this camel of doing. They upbraided\nit with being the cause of all the trouble that had separated friends,\ncalled it by every opprobious epithet, and finally killed it, and\ndeclared themselves reconciled over its body.\n\n\nSUSPENDED VOLITION.\n\nA young lady, an attendant of the Princess ----, after having been\nconfined to her bed for a great length of time with a violent nervous\ndisorder, was at last, to all appearance, deprived of life. Her lips\nwere quite pale, her face resembled the countenance of a dead person,\nand the body grew cold.\n\nShe was removed from the room in which she lay, was put in a coffin,\nand the day of her funeral fixed on. The day arrived, and, according to\nthe custom of the country, funeral songs and hymns were sung before the\ndoor. Just as the people were about to nail on the lid of the coffin, a\nkind of perspiration was observed to appear on the surface of her body.\nIt grew greater every moment, and at last a kind of convulsive motion\nwas observed in the hands and feet of the corpse. A few minutes after,\nduring which time fresh signs of returning life appeared, she at once\nopened her eyes and uttered a most pitiable shriek. Physicians were\nquickly procured, and in the course of a few days she was considerably\nrestored.\n\nThe description which she gave of her situation is extremely\nremarkable, and forms a curious and authentic addition to psychology.\n\nShe said it seemed to her, as if in a dream, that she was really dead;\nyet she was perfectly conscious of all that happened around her in\nthis dreadful state. She distinctly heard her friends speaking, and\nlamenting her death, at the side of her coffin. She felt them pull on\nthe dead-clothes, and lay her in them. This feeling produced a mental\nanxiety which is indescribable. She tried to cry, but her soul was\nwithout power, and could not act on her body. She had the contradictory\nfeeling as if she were in her body, and yet not in it, at one and the\nsame time. It was equally impossible for her to stretch out her arm or\nto open her eyes, or to cry, although she continually endeavoured to do\nso. The internal anguish of her mind was, however, at its utmost height\nwhen the lid of the coffin was about to be nailed on. The thought that\nshe was to be buried alive was the one that gave activity to her soul,\nand caused it to operate on her corporeal frame.\n\n\nFASHIONS FOR THE DEAD.\n\nThe following advertisement appeared in a Glasgow paper about the\nmiddle of the last century. \"James Hodge, who lives in the first close\nabove the Cross, on the west side of the street, Glasgow, continues to\nsell burying Crapes ready made; and his wife's niece, who lives with\nhim, dresses dead Corpses at as cheap a rate as was formerly done by\nher aunt, having been educated by her, and perfected at Edinburgh, from\nwhence she is lately arrived, and has all the newest and best fashions.\"\n\n\nCOMMON USE OF PLATE IN THE TIME OF HENRY VIII.\n\nA writer in the early part of the sixteenth century tells us that in\nhis time, in the reign of Henry the Eighth, the luxury of the table\nhad descended even to citizens, and that there were few whose tables\nwere not daily provided with spoons, cups, and a salt-cellar of silver.\nThose of a higher sphere affected a greater profusion of plate; but the\nquantity accumulated by Cardinal Wolsey, though the precious metals\nare now so copious, still continues to excite our surprise. At Hampton\nCourt, where he feasted the French ambassadors and their splendid\nretinue in 1528, two cupboards, extending across the banquet chambers,\nwere piled to the top with plate and illuminated; yet, without\nencroaching on these ostentatious repositories, a profuse service\nremained for the table. Two hundred and eighty beds were provided for\nthe guests; every chamber had a bason and ewer of silver, beside other\nutensils.\n\n\nDIOGENES IN A PITHOS, NOT TUB.\n\nA pithos is a description of earthen vessel or jar, distinguished from\nthe amphora by its large mouth, and comparatively flattened base. Its\nshape was more that of a gourd, or pot; its size large enough to have\nrendered it applicable to the purposes of a cistern, or water butt.\nSuch, indeed, appear in some instances to have been its dimensions,\nthat it has long been a matter of dispute amongst the learned whether,\nif Diogenes dwelt in a tub at all (a point by no means settled), his\nhumble habitation were of wood or earthenware. Brougniart adopts the\nlatter opinion, and has illustrated it by a partial copy from a print\nin Winckelmann. In the original, the philosopher is shown holding his\nwell-known chat with Alexander the Great, at the gate of the Metroum,\nor Temple of the Mother of the Gods at Athens; but his tub has there\nthe addition of a dog lying on the outside, above his master's head,\nevidently on the watch to defend him, if necessary, against any\nattack from the royal warrior. Winckelmann's engraving, which we here\npresent, is taken from a bas-relief discovered in the Villa Albani;\nin which the cynic's tub is clearly of earthenware, having a large\nfracture on one side, which has been repaired with some other material\ndovetailed across the crack. This, Winckelmann concludes to have been\nlead (commesso col piombo), simply, however, upon the authority of the\nfollowing lines in Juvenal:--\n\n \"Si Fregeris, altera fiet\n Cras domus, aut eadem plumbo commissa manebit.\"\n\n _Sat._ xiv 310.\n\n[Illustration: [++] Diogenes in a Pithos.]\n\nBe all this, however, as it may, the controversy is not without its\nvalue in connexion with the ceramic productions of the period. If\nthe \"dolia\" and \"[Greek: pithakne].\" of the ancients had not been of\nsufficient capacity, however kennel-like, to have served as a dwelling,\nor shelter, for the philosopher, the tale would hardly have existed.\nNor does it seem probable that Juvenal, in allusion to the story,\nwould have used the term _testa_ (testa cum vidit in illa magnum\nhabitatorem), or have dwelt upon their fragility, or have said that\nthey would not burn (dolia nudi non ardent Cynici), if vessels of the\nsort had not been commonly of earthenware. These vessels, both ancient\nand modern, have a thickness and strength which enables them to be\nrolled on a ladder to and from the top of the kiln, where they are\nbaked, without injury.\n\n\nCHINESE SCHOOL.\n\n[Illustration: [++] Native Chinese School.]\n\nThe annexed engraving is a curiosity both in itself and in what it\nrepresents. It is taken from a sketch by a native Chinese artist, and\ndepicts the internal arrangements of a native Chinese school. The\nextraordinary nature of the Chinese language renders it impossible\nfor a schoolmaster to instruct more than a very few scholars at\na time, since the meaning of the words actually depends on their\ncorrect intonation. Every vocable in the language is capable of being\npronounced in six different tones of voice, and of conveying six\nmeanings, totally different from each other, according to the tone\ngiven to it. Pronounced in one tone, it conveys one meaning, and is\nrepresented by one written character; pronounced in another tone, it\nconveys an entirely distinct meaning, and is represented in writing\nby another character altogether different. The correct and distinct\nenunciation of these tones is the chief difficulty in learning to\nspeak the language. These tones are stereotyped and fixed, and must\nbe learned, as part of the word, at the same time that its form and\nsignification are mastered. Moreover, they are all arranged upon\nsystem, like the notes in a gamut, and when thoroughly mastered, the\ntheory of the tones is really beautiful. If a wrong tone, then, is\ngiven to a word in reading or in conversation, it grates upon a Chinese\near like a false note in playing the fiddle. Further, if the voice be\nnot correctly modulated, and the words correctly intoned, not only is\na jarring note pronounced, but actually a wrong word is uttered, and a\ndifferent meaning conveyed from what was intended. A missionary to the\nChinese, therefore, should be possessed of a musical ear. Without this,\nthe acquisition of the spoken language will be attended by very arduous\nlabour; and, perhaps, after years of toil, he will find that he still\nfrequently fails in correctly conveying his meaning.\n\n\nLONDON LOCALITIES IN THE SIXTEENTH CENTURY.\n\nAt Ludgate was a gaol, where the prisoners clamoured for alms at the\nbarred grate; and it was here that Sir Thomas Wyatt had been repulsed.\nThe city wall that joined this gate to its other fellow gates ran\nfrom the Tower through the Minories to Aldgate, Houndsditch, and\nBishopsgate, through Cripplegate to Aldersgate, and so past Christ's\nHospital by Newgate and Ludgate to the Thames.\n\nPimlico was a country place where citizens used to repair to eat\n\"pudding pies\" on a Sunday, as they did to Islington or Hogsden to take\ntobacco and drink new milk; as Islington was famous for its dairy,\nwhere Sir Walter Raleigh is said to have lived in an old house still\nstanding, so Holloway was famous for its cheese cakes; and it is these\npeculiarities that, after all, confer immortality upon a place. Chelsea\nwas the mere village of Chelsea, known from Sir Thomas More's house,\nwhere Henry VIII. had walked with his arm round that great statesman's\ndoomed neck; as Holborn was then a country road leading to the pleasant\nvillage of St. Giles, and trending on to the way that led to Oxford and\nto fatal Tyburn, so called from its burn or brook, then well known to\npatient city anglers. The triple tree or gallows stood at the corner\nof the present Edgware Road. The same Oxford Street led also, if you\nturned up one side of the Hampstead Road, to the Tottenham Court,\nwhich stood there alone far in the country, and Primrose Hill was an\nuntrodden hillock, surrounded by wide paths and ditches between this\ncourt and Hampstead.\n\nA cheerful little stream, known by the pleasant name of the Fleet, rose\nnear Hampstead Hill, and joined by the Old Bourne and recruited by\nsparkling Clerken Well, emptied itself in the Thames. Though even then\nmerely a sewer, it was open, and had four bridges of its own, while the\nThames had but one; and these were known as Holborn Bridge, Fleet-lane\nBridge, Fleet Bridge, and Bridewell Bridge.\n\nSpitalfields was a grassy open space, with artillery grounds and a\npulpit and cross, where fairs were held and sermons preached. There\nwere also Tothill Fields, and Finsbury Fields, and Moor Fields, just\noutside the city walls, laid out in walks, and planted, as far as\nHoxton. Round these squares there were windmills and everything equally\nrural. As for Piccadilly, it was everywhere known as a road to Reading,\nand by many herbalists, as harbouring the small wild foxglove in its\ndry ditches.\n\nOutside Temple Bar, before the wooden gatehouse was built, lay the\nStrand, the road leading from the city to the houses of Court. This\nriver bank was the chosen residence of the nobility, whose gardens\nstretched to the edge of the then undefiled river. The sky then was\npure and bright, for our ancestors burnt wood fires, and the water was\ngay with thousands of boats. Each house had its terrace, its water\nstairs, and garden. The street houses were so scattered that the river\ncould be seen between them, and there were three water courses there\ntraversed by bridges, besides two churches and a maypole. Here stood\nYork House, where Bacon was born, and Durham Place, where Raleigh\nlived, with his study in a turret overlooking the river; there also\nwere Arundel House and Essex House, where great men pined and plotted.\n\nAt Whitehall stood Wolsey's Palace, enlarged by Henry VIII., and\nElizabeth's favourite residence when not at Nonsuch in Surrey, Windsor,\nGreenwich, or Richmond. The tilt-yard stood where the Horse Guards\nnow stands. St. James's Palace, also built by Henry VIII., where the\nQueen's melancholy-bigot sister had died, was seldom inhabited by the\nCourt; but the park was even then existing. As for the old palace of\nRichard III. (Baynard's Castle), that had been let to the Earl of\nPembroke, and the same king's dwelling of Crosby Hall had fallen into\nthe hands of an alderman.\n\n\nWARWICK THE KING-MAKER.\n\nOn the right-hand side of Newgate-street are various streets and\ncourts leading into Paternoster-row. Of these, Warwick and Ivy lanes,\nPanyer-alley, and Lovel's-court, merit the attention of the lover of\nliterary and historical antiquities. Warwick-lane, now the abode of\nbutchers and tallow-chandlers, took its name from the inn or house of\nthe celebrated Warwick, the king-maker.\n\nStow mentions his coming to London in the famous convention of 1458,\nwith 600 men, all in red jackets, embroidered, with ragged staves,\nbefore and behind, and was lodged in Warwick-lane; \"in whose house\nthere was often six oxen eaten at a breakfast, and every taverne was\nfull of his meate, for hee that had any acquaintance in that house,\nmight have there so much of sodden and roaste meate, as he could pricke\nand carry upon a long dagger.\"\n\nThe memory of the earl was long preserved by a small stone statue,\nplaced in the side front of a tobacconist's, at the corner of this\nlane; and there is a public-house which has the earl's head for its\nsign.\n\n\nTHANKSGIVING DAY IN 1697.\n\nThe following is an extract from the \"Post Boy\" of the above date:--\n\n\"Thursday, December 2, 1697. Thursday being appointed for the day of\nThanksgiving, the same was ushered in with ringing of bells; the king\nwent to the Chapel Royal, where, &c., and at night we had bonfires and\nilluminations. The fine fireworks in St. James's Square were lighted\nafter this manner:--About twelve o'clock, the Foot Guards lined the\navenues; the rockets and all things being fixed on the rails the day\nbefore: a little after six, the king, attended by his guards, came to\nthe Earl of Romney's house, from whence soon after a signal was given,\nby firing a rocket, for the fireworks to go off, which were immediately\nlighted; the performance was extraordinary fine, and much applauded;\nthe same continued somewhat better than half an hour, and there were\ndivers sorts of fireworks; some had the king's name, others the arms\nof England; in a word, they were very curious. There was a man and a\nwoman unfortunately killed, and divers others hurt by the falling down\nof sticks. About half an hour after, His Majesty went to St. James's\nthere being a fine ball.\"\n\n\nTHE GREY MAN'S PATH.\n\n[Illustration: [++] The Grey Man's Path.]\n\nThe annexed sketch depicts a scene in the coast rocks at Fairhead,\nnear Ballycastle in Ireland. Fhir Leith, or \"The Grey Man's Path,\" (a\nfissure in the precipice,) viewed either from land or sea, is never to\nbe forgotten: it seems as though some supernatural power, determined to\nhew for itself a pathway through the wonderful formations that tower\nalong the coast--so that it might visit or summon the spirits of the\ndeep, without treading a road made by mortal hands--had willed the\nfearful chasm that divides the rocky promontory in two. The singular\npassage, in its narrow part, is barred across by the fragment of a\npillar, hurled, as it were, over the fissure, and supported on both\nsides at a considerable elevation. If you descend, you perceive the\npassage widens, and becomes more important; its dark sides assume\ngreater height, and a more wild and sombre magnificence; and at last\nthey extend upwards, above 220 feet, through which the tourist arrives\nat the massive _debris_ which crowd the base of the mighty promontory,\nwhere the northern ocean rolls his threatening billows. From the\ncragsmen and boatmen of this wild coast you hear no tales of Faery, no\nhints of the gentle legends and superstitions collected in the south,\nor in the inland districts of the north; not that they are a whit less\nsuperstitious, but their superstition is, as the superstition of the\nsea kings, of a bold and peculiar character; their ghosts come from out\nthe deep, before or after the rising of the moon, and climb, or rather\nstalk up the rocks, and, seated upon those mysterious pillars, converse\ntogether; so that, in the fisherman's huts, they say, \"it thunders.\"\nEven mermaids are deemed too trifling in their habits and manners\nfor this stupendous scenery, where spirits of the gigantic world\ncongregate, and where the \"Grey Man\" of the North Sea stalks forth,\nsilently and alone, up his appropriate path, to witness some mighty\nconvulsion of nature.\n\n\nANCIENT JET NECKLACE.\n\n[Illustration: [++] Ancient Jet Necklace.]\n\nVarious interesting ornaments, belonging to the Archaic, or Bronze\nperiod in Scotland, are preserved in the Museum of Scottish\nAntiquaries, and one set in particular, found enclosed in an urn within\na rude stone cist, on the demolition of a tumulus near the Old House of\nAssynt, Rossshire, in 1824, we here engrave. They include a necklace\nof irregular oval jet beads, which appear to have been strung together\nlike a common modern string of beads, and are sufficiently rude to\ncorrespond with the works of a very primitive era. The other ornaments\nwhich are represented here about one-fourth the size of the original,\nare curiously studded with gold spots, arranged in patterns similar\nto those with which the rude pottery of the British tumuli are most\nfrequently decorated, and the whole are perforated with holes passing\nobliquely from the back through the edge, evidently designed for\nattaching them to each other by means of threads.\n\n\nJUGGLERS IN JAPAN.\n\nThe perfection of jugglery in Japan entitles it to be ranked amongst\nthe fine arts. An eye-witness thus describes the performance of a\nJapanese juggler. \"Here are some of his feats:--No. 1. He took an\nordinary boy's top, spun it in the air, caught it on his hand, and then\nplaced it (still spinning) upon the edge of a sword, near the hilt.\nThen he dropped the sword point a little, and the top moved slowly\ntowards it. Arrived at the very end, the hilt was lowered in turn, and\nthe top brought back. As usual, the sword was dangerously sharp. No. 2\nwas also performed with the top. He spun it in the air, and then threw\nthe end of the string back towards it with such accuracy that it was\ncaught up and wound itself all ready for a second cast. By the time it\nhad done this it had reached his hand, and was ready for another spin.\nNo. 3 was still performed with the top. There was an upright pole, upon\nthe top of which was perched a little house, with a very large front\ndoor. The top was spun, made to climb the pole, knock open the said\nfront door, and disappear. As well as I remember, the hand end of the\nstring was fastened near the door, so that this was almost a repetition\nof the self-winding feat. But feat No. 4 was something even more\nastonishing than all this. He took two paper butterflies, armed himself\nwith the usual paper fan, threw them into the air, and, fanning gently,\nkept them flying about him as if they had been alive. 'He can make\nthem alight wherever you wish! Try him!' remarked the Kami (Prince),\nthrough the interpreter. Mr. H---- requested that one might alight\nupon each ear of the juggler. No sooner expressed than complied with.\nGentle undulations of the fan waved them slowly to the required points,\nand there left them comfortably seated. Now, whether this command over\npieces of paper was obtained simply by currents of air, or by the power\nof a concealed magnet, Mr. H---- could not tell or ascertain. One\nthing, however, was certain, the power was there.\"\n\n MAY-FAIR PLAY BILL IN THE TIME OF WILLIAM III.\n WILLIAM REX.\n MAY-FAIR.\n MILLER'S,\n OR THE LOYAL ASSOCIATION BOOTH,\n AT THE UPPER END OF\n BROOK-FIELD MARKET,\n NEAR HYDE PARK CORNER.\n DURING THE TIME OF MAY-FAIR, WILL BE PRESENTED\n AN EXCELLENT DROLL, CALLED\n KING WILLIAM'S HAPPY DELIVERANCE\n AND GLORIOUS TRIUMPH OVER HIS ENEMIES,\n OR THE CONSULTATION OF THE\n POPE, DEVIL, FRENCH KING, AND THE GRAND TURK,\n WITH THE WHOLE FORM OF THE SIEGE OF NAMUR,\n AND THE HUMOURS OF A RENEGADE FRENCH MAN\n AND BRANDY JEAN,\n WITH THE CONCEITS OF SCARAMOUCH AND HARLEQUIN,\n TOGETHER WITH THE BEST SINGING AND DANCING THAT WAS\n EVER SEEN IN A FAIR, ALSO A DIALOGUE SONG.\n VIVAT REX.\n\n\nBELLS.\n\nBells were formerly a prolific source of superstition. There is a\nvalley in Nottinghamshire, where a village is said to have been\nswallowed up by an earthquake, and it was the custom on Christmas Day\nmorning for the people to assemble in this valley and listen to the\nfancied ringing of the church bells underground. At Abbot's Morton\nthere is a tradition that the silver bells belonging to the abbot are\nburied in the site of his old residence there. At Ledbury, a legend\nrelates that St. Katharine had a revelation that she was to travel\nabout, and not rest at any place, till she heard the bells ringing of\ntheir own accord. This was done by the Ledbury bells on her approaching\nthat town. When the church at Inkberrow was rebuilt on a new site in\nancient days, it was believed that the fairies took umbrage at the\nchange, as they were supposed to be averse to bells; they accordingly\nendeavoured to obstruct the building, but, as they did not succeed, the\nfollowing lamentation was occasionally heard by the startled rustics:\n\n \"Neither sleep, neither lie,\n For Inkbro's ting-tangs hang so nigh.\"\n\nMany years ago the twelve parish churches in Jersey each possessed a\nbeautiful and valuable peal of bells; but during a long civil war, the\nstates determined on selling these bells to defray the heavy expenses\nof their army. The bells were accordingly collected, and sent to\nFrance for that purpose; but, on the passage, the ship foundered, and\neverything was lost, to show the wrath of Heaven at the sacrilege.\nSince then, before a storm, these bells ring up from the deep; and,\nto this day, the fishermen of St. Ouen's Bay always go to the edge of\nthe water before embarking, to listen if they can hear \"the bells upon\nthe wind;\" and, if those warning notes are heard, nothing will induce\nthem to leave the shore; if all is quiet they fearlessly set sail. As a\ngentleman, who has versified the legend, says:\n\n \"'Tis an omen of death to the mariner,\n Who wearily fights with the sea;\n For the foaming surge is his winding sheet,\n And his funeral knell are we:\n His funeral knell our passing bells beat,\n And his winding sheet the sea.\"\n\n\nBRIBING THE DEMONS.\n\nThe rich inhabitants of the Celestial Empire, it is almost needless\nto say, make an exorbitant display at funerals. They invite as many\nrelations and friends as they can, in order to muster an imposing\nprocession, and the mourning dresses worn by the whole party are at\nthe cost of the family of the deceased, who are also bound to provide\nthem for several days together with splendid repasts. A great number\nof musicians are hired for the occasion, and also of _weepers_, for\nthough most people in China are pretty well skilled in the art of\nshedding tears, there exist mourners by profession, who have carried it\nto still greater perfection, and are absolutely inimitable at sobs and\ngroans. They follow the coffin in long white robes, hempen girdles, and\ndishevelled hair; and their lamentations are accompanied by the beating\nof gongs, by the sharp and discordant sounds of rude instruments of\nmusic, and the discharge of fireworks. The sudden explosion and the\nsmell of the powder are supposed to be efficacious in frightening away\nthe demons, and hindering them from seizing on the soul of the defunct,\nwhich never fails to follow the coffin; and as these malevolent spirits\nhave also the reputation of being extremely covetous and fond of money,\npeople endeavour to get on their weak side. They let fall, for this\npurpose, all along the road, sapecks and bank-notes, that the wind\ncarries away in all directions; and as the demons in China are by no\nmeans so cunning as the men, they are taken in by this device, and\nfall into the trap with charming simplicity, though, the supposed\nbank-notes are in fact only bits of white paper. Whilst they are\nengaged in pursuing these deceitful appearances of riches, the soul of\nthe defunct proceeds quietly and comfortably after its coffin without\nany danger of its being stopped by the way.\n\n\nHOLY-WATER SPRINKLER.\n\nTo sprinkle the holy water was, in ancient times, the cant phrase for\nfetching blood, which will account for the appellation of a certain\nclass of weapons, as there is no resemblance whatever between them and\nthe aspergillum used by Roman Catholics. The specimen we have here\nsketched is a demi holy-water-sprinkler--to speak in the language of\nthe time--\"with gonnes at the ende.\" This awkward weapon, prior, in\npoint of date, to the invention of the matchlock, and, therefore, not\nlater than the time of Edward IV., was made to hang at the saddle-bow\ninstead of a mace. The iron cap at the end is furnished with a\nspear-like blade, and opens on an hinge, or is held in its place by\na hook. It contains four short barrels, each of which is fired by a\nmatch, and its touch-hole is protected by a sliding piece of wood.\n\n[Illustration: [++] Holy Water Sprinkler.]\n\nIn using this weapon the intention was first to fire at the enemy with\nthe \"gonnes at the ende,\" and then to club him on coming to close\nquarters. To effect all this, however, in a satisfactory manner, much\ntime must have been lost, and many accidents, no doubt, were liable to\nhappen to the person who used such a weapon as this, which was almost\nas dangerous to the man who possessed it, as to the enemy against whom\nhe directed it. The lid at the top must first have been opened, and\nnot only so, but must have been kept open all the time the weapon was\nused as a gun, and then, previously to closing with the foe, it must\nhave been necessary to secure it, lest, in brandishing the instrument\nas a club, the open lid should strike against the head of the man who\nwielded it. No wonder that this dangerous compound of club and gun soon\nwent out of fashion, and survived its invention only a very few years.\n\n\nFIRST TEA-DRINKERS PUZZLED.\n\nThe first brewers of tea were often sorely perplexed with the\npreparation of the new mystery. \"Mrs. Hutchinson's great grandmother\nwas one of a party who sat down to the first pound of tea that ever\ncame into Penrith. It was sent as a present, and without directions how\nto use it. They boiled the whole at once in a bottle, and sat down to\neat the leaves with butter and salt, and they wondered how any person\ncould like such a diet.\"\n\n\nCOLUMN AT CUSSI.\n\n[Illustration: [++] Column at Cussi.]\n\nThe great object of the erection of pillars of victory was to serve\nas vehicles for sculpture; though, as we now see them, or as they are\ncaricatured at Paris and elsewhere, they are little more than instances\nof immense labour bestowed to very little purpose. In the original\nuse of these pillars, they were placed in small courts surrounded by\nopen porticos, whence the spectator could at two, or perhaps at three\ndifferent levels examine the sculpture at his leisure at a convenient\ndistance, while the absurdity of a pillar supporting nothing was not\napparent, from its not being seen from the outside. A good specimen of\nthis class is that at Cussi, near Beaune, in France. It is represented\nin the annexed cut. It probably belongs to the time of Aurelian, and\nno doubt was first erected within a court; but it is not known either\nby whom it was erected, or what victory it was designed to celebrate.\nStill that it is a pillar of victory is certain, and its resemblance\nto pillars raised with the same object in India is quite striking.\nThe arrangement of the base, serving as a pedestal for eight statues,\nis not only elegant, but appropriate. The ornament which covers the\nshaft takes off from the idea of its being a mere pillar, and, at the\nsame time, is so subdued as not to break the outline or interfere with\nconstructive propriety. The capital of the Corinthian order is found\nin the neighbourhood, used as the mouth of a well. In its original\nposition it no doubt had a hole through it, which being enlarged\nsuggested its application to its present comparatively ignoble purpose,\nthe hole being no doubt intended either to receive or support the\nstatue or emblem that originally crowned the monument, but of that no\ntrace now remains.\n\n\nSTYLE OF LIVING AMONG THE NOBILITY OF THE FIFTEENTH CENTURY.\n\nThe ordinary meals were now increased to four a day--breakfast at\nseven in the morning, dinner at ten, supper at four in the afternoon,\nand \"liveries,\" which were taken in bed, between eight and nine at\nnight. These latter, as well as the breakfast, were of no light or\nunsubstantial character, consisting of good beef and mutton (or salt\nfish in Lent), with beer and wine in the morning; and of a loaf or\ntwo, with a few quarts of mulled wine and beer, at nights. At dinner\nthe huge oaken table, extending the whole length of the great hall,\nwas profusely covered with joints of fresh and salt meat, followed by\ncourses of fowl, fish, and curious made-dishes. The Lord took his seat\non the dais or raised floor at the head; his friends and retainers were\nranged above or below the salt, according to their rank. As forks were\nnot yet in use, the fingers were actively employed, whilst wine and\nbeer in wooden or pewter goblets were handed round by the attendants.\nOver head the favourite hawks stood upon their perches, and below the\nhounds reposed upon the pavement.\n\nThe dinner generally lasted for three hours, and all pauses were filled\nup by the minstrels, jesters, or jugglers, or by the recitation of\nsome romance of chivalry. At the end of each course they sometimes\nintroduced a dish called _subtlety_, composed of curious figure in\njellies or confectionery, with a riddling label attached for the\nexercise of social wit. The monasteries were especially noted for\ntheir good dinners, and the secular clergy, not to be outdone in their\nhospitality invented _glutton-masses_ in honour of the Virgin. These\nwere held five times a year in the open churches, whither the people\nbrought food and liquor, and vied with each other in this religious\ngormandizing. The general diet of the common people continued, however,\nto be coarse and poor, and severe famines not unfrequently occurred.\n\n\nORIGIN OF THE TITLE \"SFORZA.\"\n\nJames Sforza, the father of Francis the first duke, was the founder of\nthe house of Sforza, which gave six dukes to Milan, and was allied with\nalmost every sovereign in Europe during the fifteenth and sixteenth\ncenturies. He was born in 1369, at Catignuola, near Faenza; his\nfather, according to tradition, was a day labourer, and to others, a\nshoemaker, but probably wrought as both. Perceiving some soldiers pass,\nhe was struck with the desire of bearing arms. \"I will go,\" said he to\nhimself, \"and dart my hatchet against that tree, and if it stick fast\nin the wood I will immediately become a soldier.\" The hatchet stuck\nfast, and because, says the Abbot of Choisi, he threw the axe with all\nhis force, he assumed the supposed fortunate name of Sforza, as his\nreal name was Giacomuzzo, or James Attendulo.\n\n\nMAY-POLE IN THE STRAND.\n\nDuring the austere reign of the Puritans, when theatres were closed,\nand every sort of popular amusement was considered sinful, the\nMay-poles fell into disrepute, and were pulled down in various parts\nof London. Among the rest, the famous May-pole in the Strand came to\nthe ground. With the restoration of the monarchy, the people saw the\nrestoration of their ancient sports; and on the very first May-day\nafter the return of Charles II., the May-pole in the Strand was set\nup again, amid great popular rejoicing. The following account of the\nceremony is taken from a rare tract of the times, entitled \"The Citie's\nLoyaltie displayed. London, 4to., 1641,\" and quoted in the first volume\nof Hone's \"Every-Day Book,\" page 557:--\n\n\"Let me declare to you the manner in general,\" says the loyal author,\n\"of that stately cedar erected in the Strand, 134 feet high, commonly\ncalled the May-pole, upon the cost of the parishioners there adjacent,\nand the gracious consent of His Sacred Majesty, with the illustrious\nPrince the Duke of York. This tree was a most choice and remarkable\npiece, 'twas made below bridge, and brought in two parts up to Scotland\nYard, near the King's Palace, and from thence it was conveyed, April\n14th, to the Strand, to be erected. It was brought with a streamer\nflourishing before it, drums beating all the way, and other sort of\nmusic. It was supposed to be so long that landsmen, as carpenters,\ncould not possibly raise it. Prince James, the Duke of York, Lord High\nAdmiral of England, therefore commanded twelve seamen to come and\nofficiate the business; whereupon they came, and brought their cables,\npulleys, and other tackling, with six great anchors. After these were\nbrought three crowns, borne by three men bareheaded, and a streamer\ndisplaying all the way before them, drums beating, and other music\nplaying, numerous multitudes of people thronging the streets, with\ngreat shouts and acclamations all day long.\n\n\"The May-pole then being joined together, and hooped about with bands\nof iron, the crown and vane, with the King's arms, richly gilded, was\nplaced on the head of it: a large top, like a balcony, was about the\nmiddle of it. This being done, the trumpets did sound, and in four\nhours' space it was advanced upright; after which being established\nfast in the ground, again great shouts and acclamations did the\npeople give, that rang throughout all the Strand. After that came a\nmorris-dance, finely decked with purple scarfs, in their half shirts,\nwith a tabor and pipe, the ancient music, and danced round about the\nMay-pole, and after that danced the rounds of their liberty. Upon\nthe top of this famous standard is likewise set up a royal purple\nstreamer, about the middle of it are placed four crowns more, with the\nKing's arms likewise. There is also a garland set upon it, of various\ncolours, of delicate rich favours, under which are to be placed three\ngreat lanthorns, to remain for three honours, that is, one for Prince\nJames, Duke of York, Lord High Admiral of England; the other for the\nVice-Admiral; the third for the Rear-Admiral. These are to give light\non dark nights, and to continue so as long as the pole stands, which\nwill be a perpetual honour for seamen. It is placed as near hand as\nthey could guess in the very same pit where the former stood, but far\nmore glorious, higher, and bigger, than ever any one that stood before\nit; and the seamen themselves do confess that it could not be built\nhigher, and there is not such an one in Europe besides, which doth\nhighly please His Majesty and the illustrious Prince, Duke of York.\nLittle children did much rejoice, and ancient people did clap their\nhands, saying that golden days began to appear. I question not but it\nwill ring like melodious music throughout every county in England when\nthey read this story exactly penned. Let this story satisfy for the\nglories of London, that other loyal subjects may read what we here do\nsee.\"\n\n\nCOSTUME OF A GERMAN NOBLE.\n\nThe annexed cut represents the dress of a young noble of the year 1443,\nfrom the extremely interesting genealogical history of the baronial\nfamily of Haller von Halleostein. The figure is that of Franz Haller\nvon Halleostein, who died unmarried in the above year. He wore an open\njerkin of a greenish colour, and very finely plaited chemisette. The\njerkin has a white silk trimming with a black border throughout, and\nis held together by fine white silk ribbons, beneath which appears the\nwhite shirt. The sword-couple and sheath, are black, hilt and mountings\nare of the colour of steel. The stockings are vermilion, and on the\nright leg is a white and yellow stripe. The shoes are black, turned\nwith white. The hair is long, and over it is worn a neat cap with\nlappets and a golden agraffe and love-knot, to support the hair.\n\n[Illustration: [++] Dress of a Young Noble in 1443.]\n\nAt the period of this costume very great attention was bestowed by\nthe German nobility to their dress. The sums they expended on it\nwere enormous, and in many instances families were reduced to ruin\nby the extravagant decorations of their person. Jewellery, furs,\nsilks, and laces, all of which were far more expensive and difficult\nto be obtained than they are now, were used in reckless profusion,\nand one nobleman vied with another in the magnificence, novelty, and\nexpensiveness of their attire. The illustrated books of that period\nabound in sketches of the most beautiful costumes, and are a fund of\ninterest to those who are curious in such matters.\n\n\nABSURDITIES OF THE TOILET.\n\nThe ladies of Japan are said to gild their teeth, and those of the\nIndies to paint them red, while in Guzerat the test of beauty is to\nrender them sable. In Greenland, the women used to colour their faces\nwith blue and yellow. The Chinese must torture their feet into the\nsmallest possible dimensions--a proof positive of their contracted\nunderstandings. The ancient Peruvians, and some other Indian tribes,\nused to flatten their heads: and among other nations, the mothers, in a\nsimilar way, maltreat the noses of their offspring.\n\n\nAN EGYPTIAN DINNER.\n\nThe complicated, and, at first sight, somewhat incomprehensible sketch\nwhich we here lay before our readers, was taken from an interior wall\nof a palace in Egypt. It is, of course, by Egyptian artists, and the\nsubject of it is no other than an Egyptian dinner-table set out and\nadorned for a banquet.\n\n[Illustration: [++] Egyptian Dinner-Table.]\n\nAt a dinner in ancient Egypt, small and low circular tables were used,\nstanding on a single pillar, with a dilated base; sometimes one of\nthese was apportioned to every guest, the viands being brought round\nby the servants successively, from a larger pillar-table which had\nbeen brought in readily set out by two men. The accompanying engraving\nshows a table thus laid out, requiring, however, a little allowance for\nthe lack of perspective. Round and oblong cakes of bread flattened and\npricked in patterns, a goose, a leg of a kid or antelope, baskets of\nfigs and other fruit, are crowned by a huge bunch of the lotus-lily.\nUnder the table are bottles of wine placed on stands in a series, and\ncrowned with a lotus-garland, upon which is thrown a long withe of what\nseems from the tendrils a vine, loaded with clusters of grapes, as well\nas thickly set with foliage.\n\n\nELEPHANT-GOD OF BURMAH.\n\nA white elephant is a great rarity, and whenever one is caught, the\nBurmese treat it as a god and pay worship to it. Captain Yule thus\ndescribes the white elephant of 1855, and his palace at Amarapoora, the\ncapital of Burmah:--\n\n\"In the area which stretches before the Hall of Audience are several\ndetached buildings. A little to the north is the \"Palace,\" or state\napartment, of the Lord White Elephant, with his highness's humbler\nevery-day residence in rear. To the south are sheds for the vulgar\nherd of the same species, and brick godowns in which the state\ncarriages and golden litters (the latter massive and gorgeous in great\nvariety of design) are stowed away. Temporary buildings, used as\nbarracks and gunsheds, run along the wall. The present white elephant\nhas occupied his post for at least fifty years. I have no doubt he\nis the same as Padre San-germano mentions as having been caught in\n1806, to the great joy of the King, who had just lost the preceding\nincumbent, a female, which died after a year's captivity. He is a very\nlarge elephant, close upon ten feet high, with as noble a head and pair\nof tusks as I have ever seen. But he is long-bodied and lanky, and not\notherwise well made as an elephant. He is sickly and out of condition,\nand is, in fact, distempered during five months of the year, from April\nto August. His eye, the iris of which is yellow with a reddish outer\nannulus, and a small clear black pupil, has an uneasy glare, and his\nkeepers evidently mistrust his temper. We were always warned against\ngoing near his head. The annulus round the iris of the eye is pointed\nout as resembling a circle of the nine gems. His colour is almost\nuniform all over; nearly the ground-tint of the mottled or freckled\npart of the trunk and ears of common elephants, perhaps a little\ndarker. He also has pale freckles in the same parts. On the whole, he\nis well entitled to his appellation of white. His royal paraphernalia,\nwhich are set out when visitors are expected, are sufficiently\nsplendid. Among them was a driving-hook about three feet long, the stem\nof which was a mass of small pearls, girt at frequent intervals with\nbands of rubies, and the hook and handle of crystal tipped with gold.\nHis headstall was of fine red cloth, plentifully studded with fine\nrubies, and near the extremity having some valuable diamonds. To fit\nover the two bumps of the forehead were circles of the nine gems, which\nare supposed to be charms against evil influences. When caparisoned he\nalso wore on the forehead, like other Burmese dignitaries, including\nthe King himself, a golden plate inscribed with his titles, and a\ngold crescent set with circles of large gems between the eyes. Large\nsilver tassels hung in front of his ears, and he was harnessed with\nbands of gold and crimson set with large bosses of pure gold. He is a\nregular \"estate of the realm,\" having a woon or minister of his own,\nfour gold umbrellas, the white umbrellas which are peculiar to royalty,\nwith a suite of attendants said to be thirty in number. The Burmese\nwho attended us removed their shoes before entering his 'Palace.' The\nelephant has an appanage or territory assigned to him 'to eat,' like\nany other dignitary of the empire. I do not know where his estate is at\npresent, but in Burney's time it was the rich cotton district of Taroup\nMyo.\"\n\n\nSUPERSTITION IN 1856.\n\nIn April, 1856, a poor woman, residing in a village about three miles\nfrom Pershore, acting upon the advice of her neighbours, brought her\nchild, who was suffering from whooping cough, to that town, for the\npurpose of finding out a married couple answering to the names of\nJoseph and Mary, and soliciting their interference on behalf of her\nafflicted child, as she had been informed that if two married persons\nhaving those names could but be induced to lay their hands on her\nchild's head, the whooping cough would be immediately cured. After\nscouring the town for a considerable time in search of \"Joseph and\nhis fair lady,\" they were at length discovered in the persons of a\nrespectable tradesman and his wife residing in Bridge Street, to whom\nthe poor silly woman made known her foolish request, which at first\nexcited a smile from the good woman of the house, but was quickly\nfollowed, not by \"the laying on of hands,\" but by good advice, such as\nmothers only know how to give in these matters. The poor mother then\nthankfully departed a wiser woman.\n\n\nPRAYING BY WHEEL AND AXLE.\n\nThe Japanese, like the inhabitants of Thibet, are not content with\ndevout prayers, pilgrimages, prostrations, offerings to the gods\nin order to secure blessings here and hereafter; they also pray by\nmachine, by _wheel and axle_. There is a square post, nearly eight feet\nin length, and near the centre, at a convenient height to be reached\nby the hand, is fixed vertically a wheel, which moves readily on an\naxle passed through the post. Two small rings are strung upon each of\nthree spokes of the wheel. Every person who twists this instrument in\npassing is supposed to obtain credit in heaven for one or more prayers\ninscribed on the post, the number being graduated according to the\nvigour of the performer's devotion, and the number of revolutions\neffected. The jingle of the small iron rings is believed to secure\nthe attention of the deity to the invocation of the devout, and the\ngreater the noise, the more certain of its being listened to. Some of\nthe inscriptions on this post are worth remembering:--\"The great round\nmirror of knowledge says, 'wise men and fools are embarked in the same\nboat;' whether prospered or afflicted, both are rowing over the deep\nlake; the gay sails lightly hang to catch the autumnal breeze; then\naway they straight enter the lustrous clouds, and become partakers of\nheaven's knowledge.\"\n\n\"He whose prescience detects knowledge says:--'As the floating grass is\nblown by the gentle breeze, or the glancing ripples of autumn disappear\nwhen the sun goes down, or as the ship returns home to her old shore,\nso is life: it is a smoke, a morning tide.'\"\n\n\"Others are more to the point--as to the machine--'Buddha himself\nearnestly desires to hear the name of this person (who is buried), and\nwishes he may go to life.'\"\n\n\nNOVEL WAY OF DESIGNATING A HOUSE.\n\nIn the \"New View of London,\" published in 1708, it is mentioned as a\nremarkable circumstance attaching to the history of Prescott Street,\nnear the Strand, that instead of signs, the houses were distinguished\nby numbers, as the stair-cases in the Inns of Court, and Chancery. The\nfollowing advertisement, taken from newspapers a century and a half\nold, is interesting at this distance of time, as it shows the shifts\nto which advertisers were reduced, to point out their houses to their\ncustomers:--\n\n\"Doctor James Tilborgh, a German doctor, states that he liveth at\npresent over against the New Exchange, in Bedford Street, at the sign\nof the 'Peacock,' where you shall see at night two candles burning\nwithin one of the chambers before the balcony; and a lanthorn with a\ncandle in it upon the balcony: where he may be spoke with all alone,\nfrom 8 in the morning till 10 at night.\"\n\n\nDYAK WAR-BOAT IN BORNEO.\n\nThe Malay war-boat, or _prahu_, is built of timber at the lower part;\nthe upper is of bamboo, rattan, and kedgang (the dried leaf of the Nepa\npalm). Outside the bends, about a foot from the water line, runs a\nstrong gallery, in which the rowers sit cross-legged. At the after-part\nof the boat is a cabin for the chief who commands, and the whole of\nthe vessel is surmounted by a strong flat roof, upon which they fight,\ntheir principal weapons being the kris and spear, both of which, to be\nused with effect, require elbow-room.\n\n[Illustration: [++] Dyak War-Boat.]\n\nThe Dyak war-boat, as represented in the annexed sketch, is a\nlong-built canoe, more substantially constructed than the prahu of the\nMalays, and sufficiently capacious to hold from seventy to eighty men.\nThis also has a roof to fight from. They are generally painted, and the\nstern ornamented with feathers.\n\nBoth descriptions of war-boats are remarkably swift, notwithstanding\nsuch apparent top-weight.\n\n\nWAR-DANCE OF THE DYAKS OF BORNEO.\n\nAlmost every savage nation has its peculiar war-dance, and the\ndifferent steps, movements, and cries, in each depict different stages\nin the supposed fight. An account of the various kinds of dances\nwould form an interesting work, and as a contribution to it we here\ncall attention to the following description of a war-dance which was\npractised for the entertainment of the officers of the Semarang, on\nthe occasion of their visiting a Dyak Chief. It is taken from Captain\nMarryat's \"Borneo:\"--\n\n[Illustration: [++] Dyak War-Dance.]\n\n\"A space was cleared in the centre, and two of the oldest warriors\nstepped into it. They were dressed in turbans, long loose jackets,\nsashes round their waists descending to their feet, and small bells\nwere attached to their ankles. They commenced by first shaking hands\nwith the rajah, and then with all the Europeans present, thereby giving\nus to understand, as was explained to us, that the dance was to be\nconsidered only as a spectacle, and not to be taken in its literal\nsense, as preparatory to an attack upon us, a view of the case in which\nwe fully coincided with them.\n\n\"This ceremony being over, they rushed into the centre, and gave a\nmost unearthly scream; then poising themselves on one foot, they\ndescribed a circle with the other, at the same time extending their\narms like the wings of a bird, and then meeting their hands, clapping\nthem and keeping time with the music. After a little while the music\nbecame louder, and suddenly our ears were pierced with the whole of\nthe natives present joining in the hideous war-cry. Then the motions\nand screams of the dancers became more violent, and every thing was\nworked up to a state of excitement, by which even we were influenced.\nSuddenly, a very unpleasant odour pervaded the room, already too\nwarm, from the numbers it contained. Involuntarily we held our noses,\nwondering what might be the cause, when we perceived that one of the\nwarriors had stepped into the centre, and suspended round the shoulders\nof each dancer a human head in a wide-meshed basket of rattan. These\nheads had been taken in the late Sakarron business, and were therefore\nbut a fortnight old. They were encased in a wide network of rattan, and\nwere ornamented with beads. Their stench was intolerable, although,\nas we discovered upon after examination, when they were suspended\nagainst the wall, they had been partially baked and were quite black.\nThe teeth and hair were quite perfect, the features somewhat shrunk,\nand they were altogether very fair specimens of pickled heads; but our\nworthy friends required a lesson from the New Zealanders in the art of\npreserving. The appearance of the heads was the signal for the music\nto play louder, for the war-cry of the natives to be more energetic,\nand for the screams of the dancers to be more piercing. Their motions\nnow became more rapid, and the excitement in proportion. Their eyes\nglistened with unwonted brightness. The perspiration dropped down\ntheir faces, and thus did yelling, dancing, gongs, and tom-toms become\nmore rapid and more violent every minute, till the dancing warriors\nwere ready to drop. A farewell yell, with emphasis, was given by\nthe surrounding warriors; immediately the music ceased, the dancers\ndisappeared, and the tumultuous excitement and noise was succeeded by\na dead silence. Such was the excitement communicated, that when it was\nall over we ourselves for some time remained panting to recover our\nbreath. Again we lighted our cheroots, and smoked for a while the pipe\nof peace.\"\n\n\nWONDERFUL FISH.\n\nThe Greek Church of Baloukli contains an extraordinary instance of the\ncredulity of superstition. Some wonderful fish are there preserved,\nwhich are thus described by Mr. Curzon in his admirable book on the\n\"Monasteries of the Levant:\"--\n\n\"The unfortunate Emperor Constantine Paleologus rode out of the city\nalone to reconnoitre the outposts of the Turkish army, which was\nencamped in the immediate vicinity. In passing through a wood he found\nan old man seated by the side of a spring, cooking some fish on a\ngridiron for his dinner; the emperor dismounted from his white horse,\nand entered into conversation with the other; the old man looked up at\nthe stranger in silence, when the emperor inquired whether he had heard\nanything of the movement of the Turkish forces: 'Yes,' said he, 'they\nhave this moment entered the city of Constantinople.' 'I would believe\nwhat you say,' replied the emperor, 'if the fish which you are broiling\nwould jump off the gridiron into the spring.' This, to his amazement,\nthe fish immediately did, and, on his turning round, the figure of the\nold man had disappeared. The emperor mounted his horse and rode towards\nthe gate of Silivria, where he was encountered by a band of the enemy,\nand slain, after a brave resistance, by the hand of an Arab or a .\n\n\"The broiled fishes still swim about in the water of the spring, the\nsides of which have been lined with white marble, in which are certain\nrecesses in which they can retire when they do not wish to receive\ncompany. The only way of turning the attention of these holy fish to\nthe respectful presence of their adorers is accomplished by throwing\nsomething glittering into the water, such as a handful of gold or\nsilver coin: gold is the best; copper produces no effect; he that sees\none fish is lucky, he that sees two or three goes home a happy man;\nbut the custom of throwing coins into the spring has become, from its\nconstant practice, very troublesome to the good monks, who kindly\ndepute one of their community to rake out the money six or seven times\na day with a scraper at the end of a long pole. The emperor of Russia\nhas sent presents to the shrine of Baloukli, so called from the Turkish\nword Balouk, a fish. Some wicked heretics have said that these fishes\nare common perch: either they or the monks must be mistaken; but of\nwhatever kind they are, they are looked upon with reverence by the\nGreeks, and have been continually held in the highest honour from the\ntime of the siege of Constantinople to the present day.\"\n\n\nCURIOUS MARRIAGE CUSTOM.\n\nAt Petze, in the department of Finisterre, in France, the following\nsingular marriage custom still prevails:--\"On an appointed day, the\n_paysannes_, or female pretenders to the holy state of matrimony,\nassemble on the bridge of the village, and, seating themselves upon the\nparapet, there patiently await the arrival of the intended bridegrooms.\nAll the neighbouring cantons contribute their belles to ornament this\nrenowned bridge. There may be seen the peasant of _Saint Poliare_, her\nruddy countenance surrounded by her large muslin sleeves, which rise\nup and form a kind of framework to her full face; by her may be seated\nthe heavy _Touloisienne_, in her cloth _caline_, or gown; the peasant\nof _la Leonarde_, in a Swiss boddice, bordered with different \nworsted braid, and a scarlet petticoat, may next appear, presenting\na gaudy contrast to her neighbour from _Saint Thegonnec_, in her\nnun-like costume. On one side extends _la coulie de Penhoat_, bordered\nwith willows, honeysuckles, and the wild hop; on the other, the sea,\nconfined here like a lake, between numerous jets of land covered with\nheath and sweet broom; and below the bridge, the thatched town, poor\nand joyous as the beggar of _Carnouailles_. The bay is here so calm,\nthat the whole of this gay scene is reflected in its still waters; and\na few scenes of rural festivity present a more animated or diverting\npicture.\n\n\"The arrival of the young men, with their parents, is the signal for\nsilence among the candidates for a husband. The gentlemen advance, and\ngravely parade up and down the bridge, looking first on this side,\nand then on that, until the face of some one of the lasses strike\ntheir fancy. The fortunate lady receives intimation of her success\nby the advance of the cavalier, who, presenting his hand, assists\nher in descending from her seat, making at the same time a tender\nspeech; compliments are exchanged, the young man offers fruit to his\nintended bride, who remains motionless before him, playing with her\napron strings. In the mean while the parents of the parties approach\neach other, talk over the matter of their children's marriage, and if\nboth parties are agreeable they shake hands, and this act of friendly\ngratulation is considered a ratification of the treaty between them,\nand the marriage is shortly afterwards celebrated.\"\n\n\nFOREIGN COSTUME IN 1492.\n\n[Illustration: [++] Count Eberhard the Elder.]\n\nThe nobleman portrayed here is Count Eberhard the elder, first Duke\nof Wurtemberg, in a festival habit at Stuttgardt, in the year 1492,\non the occasion of his receiving the order of the Golden Fleece, the\nfirst which Austria instituted for herself (King Maximilian inherited\nit from Burgundy) and which he received together with King Henry VIII.\nof England. His costume is taken from an old illumination which, in\nthe year 1847, was copied for King William of Wurtemberg, and which is\nnow preserved in his private library at Stuttgardt. This exemplifies\nthe quilled doublet, made of a kind of damasked black velvet, which\nappears to have been worn over the defensive armour improved by King\nMaximilian. Upon the black surcoat appear the orders of the Golden\nFleece and the Holy Sepulchre. According to contemporary statues and\nmonuments, Georg von Ekingen and Heinrich von Waellwerth, officers of\nthe court of Eberhard, wore this kind of doublet. The former, according\nto a portrait, of a red colour; the latter authority is in the\nWaellwerth Chapel, in the cloister of Lorch near Schw. Gmuend.\n\n\nPETER THE GREAT AT ZAANDAM.\n\nWe learn from authentic records that Peter the Great, Czar of Russia,\nentered himself, in the year 1697, on the list of ship's carpenters at\nthe Admiralty Office of Amsterdam, in Holland. This is true; but before\nPeter so enrolled himself, he had made an attempt to fix his abode, for\nthe purpose of study, at Saardam, or Zaandam, a little town situated\non the river Zaan, about half an hour's voyage, by steam, from the\npopulous and wealthy city of Amsterdam.\n\nZaandam, though then, as now, one of the most primitive, original\nlittle towns in Europe, had for some time held important commercial\nintercourse with Russia; and Peter had long seen the advantage to be\nderived from studying at its head-quarters the art which he felt sure\nwould elevate his country in an extraordinary way. He therefore opened\na private correspondence with some trusty friends in Holland, and set\nforth, with his hand of intelligent companions, early in the summer of\n1697; in the autumn of the same year he disembarked at Zaandam, and,\nalone and unattended, sought an humble lodging from a man of the name\nof Gerrit Kist, who had formerly been a blacksmith in Russia, and who,\nas may well be imagined, was astonished at the \"imperial apparition;\"\nindeed he could not believe that Peter really wished to hire so humble\nan abode. But the Czar persevered, and obtained permission to occupy\nthe back part of Kist's premises, consisting of a room and a little\nshed adjoining, Kist being bound to secresy as to the rank of his\nlodger: Peter's rent amounted to seven florins (about eleven shillings)\na week.\n\n[Illustration: [++] The Maisonnette of Peter the Great.]\n\nThe _maisonnette_, or hut, of Peter the Great now stands alone, and has\nbeen encased in a strong wooden frame in order to preserve it. It is in\nmuch the same state as when occupied by the Czar. The chief apartment\nis entered by the door you see open, the projecting roof covers the\nroom probably occupied by Peter's servant, and on the left of the\nlarger room is the recess or cupboard in which Peter slept. Formerly\nthe rear of this abode was crowded with inferior buildings; it is now\nan airy space, with trees waving over the wooden tenement, and a garden\nfull of sweet-scented flowers embalms the atmosphere around it. A civil\nold Dutchwoman is the guardian of the property, which is kept up with\nsome taste, and exquisite attention to cleanliness.\n\nThe _maisonnette_ has but one door. In Zaandam the old Dutch custom of\nclosing one entrance to the house, except on state occasions, is still\nkept up; the purpose of the other, the _porte mortuaire_, or _mortuary\nportal_, is sufficiently explained by its name.\n\nAfter Peter's departure, his dwelling passed from hand to hand, and\nwould have fallen into oblivion had not Paul the First of Russia\naccompanied Joseph the Second of Austria and the King of Sweden to\nZaandam, on purpose to visit the Czar's old abode. After this it became\na sort of fashion to make pilgrimages to the once imperial residence;\nand it acquired a still greater celebrity when the Emperor Alexander\nvisited it in 1814, and made a great stir in the waters of the Zaan\nwith a fleet of three hundred yachts and innumerable barges, gaily\ndecked with flying pennons. In 1818, William the First of Holland\npurchased the property, and gave it to his daughter-in-law, the\nPrincess of Orange and a royal Russian by birth: it is to her care the\nbuilding owes its present state of preservation. Her royal highness\nappointed a Waterloo invalid as first guardian of the place.\n\nBonaparte brought Josephine here in 1812. Poor Josephine had no idea of\nold associations; she jumped from the sublime to the ridiculous at once\non entering the \"mean habitation,\" and startled the then proprietor by\na burst of untimely laughter.\n\nMany royal and illustrious names may be read on the walls of the\nprincipal chamber, and in the book in which the traveller is\nrequested to write his name. Verses and pictures challenge, somewhat\nimpertinently, the attention of the wayfarer; but as we sat down in\nthe triangular arm-chairs, and turned from the dark recess in which\nPeter slept, to the ingle-nook of the deep chimney, and from the ingle\nto the dark recess again, we could realize nothing but Peter in his\nworking dress of the labours of the day. There he was in the heat of an\nautumnal evening still at work, with books and slates, and instruments\nconnected with navigation, before him on the rude deal table, and he\nplodding on, as diligently as a common mechanic, in pursuit of that\nknowledge by which nations are made great.\n\n\nSUPPLY OF WATER FOR LONDON IN OLDEN TIMES.\n\nIn 1682 the private houses of the metropolis were only supplied with\nfresh water twice a-week. Mr. Cunningham, in his \"Handbook of London,\"\ninforms us that the old sources of supply were the Wells, or Fleet\nRiver, Wallbrook and Langbourne Waters, Clement's, Clerk's, and Holy\nWell, Tyburn, and the River Lea. Tyburn first supplied the city in the\nyear 1285, the Thames not being pressed into the service of the city\nconduits till 1568, when it supplied the conduit at Dowgate. There were\npeople who stole water from the pipes then, as there are who steal gas\nnow. \"This yere\" (1479), writes an old chronicler of London, quoted\nby Mr. Cunningham, \"a wax charndler in Flete-stre had bi craft perced\na pipe of the condite withynne the ground, and so conveied the water\ninto his celar; wherefore he was judged to ride through the citee with\na condite upon his hedde.\" The first engine which conveyed water into\nprivate houses, by leaden pipes, was erected at London-bridge in 1582.\nThe pipes were laid over the steeple of St. Magnus; and the engineer\nwas Maurice, a Dutchman. Bulmer, an Englishman, erected a second engine\nat Broken Wharf. Previous to 1656, the Strand and Covent Garden, though\nso near to the river, were only supplied by water-tankards, which were\ncarried by those who sold the water, or by the apprentice, if there\nwere one in the house, whose duty it was to fill the house-tankard at\nthe conduit, or in the river. In the middle of the seventeenth century,\nFord erected water-works on the Thames, in front of Somerset House; but\nthe Queen of Charles II.--like the Princess Borghese, who pulled down\na church next to her palace, because the incense turned her sick, and\norgan made her head ache--ordered the works to be demolished, because\nthey obstructed a clear view on the river. The inhabitants of the\ndistrict depended upon their tankards and water-carriers, until the\nreign of William III., when the York-buildings Waterworks were erected.\nThe frequently-occurring name of Conduit-street, or Conduit-court,\nindicates the whereabout of many of the old sources whence our\nforefathers drew their scanty supplies.\n\n\nDRINKING BOUTS IN PERSIA.\n\nIn their drinking parties the Persians are reported, among even the\nhighest classes, to exceed all bounds of discretion. Half a dozen\nboon companions meet at night. The floor is covered with a variety\nof stimulating dishes to provoke drinking, for which no provocation\nwhatever is required; among these are pickles of every possible\nvariety, and salted prawns or cray-fish from the Persian Gulf--a\nfood which ought to be an abomination to a true Sheeah. Singers and\ndancing-boys enliven the scene. A Persian despises a wine-glass; a\ntumbler is his measure. He has an aversion to \"heeltaps,\" and he\ndrains his glass to the dregs, with his left hand under his chin to\ncatch the drops of wine, lest he should be detected next morning\nin respectable society by the marks on his dress. They begin with\npleasant conversation, scandal, and gossip; then they become personal,\nquarrelsome, abusive, and indecent, after the unimaginable Persian\nfashion. As the orgies advance, as the mirth waxes fast and furious,\nall restraint is thrown aside. They strip themselves stark naked,\ndance, and play all sorts of antics and childish tricks. One dips his\nhead and face into a bowl of curds, and dances a solo to the admiring\ntopers; while another places a large deeg, or cooking-pot, on his head,\nand display his graces and attitudes on the light fantastic toe, or\nrather heel.\n\n\nGERMAN COSTUMES OF THE SIXTEENTH CENTURY.\n\nThe costume-sketch which we give on next page, is taken from an\noriginal drawing, having the following superscription:--\n\n \"Varium et mutabile semper foemina\n Faec suo quem amat scripsit.\n Georgius Wolfgang Von Kaltenthal. 1579.\"\n\n[Illustration: [++] German Costumes of the Sixteenth Century.]\n\nThe group represents the above-named young knight, with his youthful\nwife, taking a ride. She wears a blue silken dress, with a boddice of\ngold brocade, trimmed with fur, and a rose- silk scarf; the\nhead-dress is quite plain, the hair being fastened with a golden dagger\nset in jewels. The knight's dress consists of a light green doublet,\nwith dark green stripes; slashed hose, edged with white; yellowish\nleather surcoat without sleeves, riding boots of untanned leather,\nand grey felt hat with red and white plume, dagger, and sword. The\naccoutrements of the horse are simply black, with some metal ornaments.\nThe young lady is the beautiful Leonora Caimingen, who was at that time\na great favourite of the Court at Wurtemberg. In travelling thus (which\nwas at that time the only mode), females of the higher rank only were\naccustomed to make use of masks, or veils, for the preservation of\ntheir complexions, that custom being generally unusual. The ancestral\ncastle of the knights of Kaltenthal was situated between Stuttgardt and\nBoeblingen, on the summit of a rock overhanging the valley of Hesslach.\nIt exists no longer.\n\n\nANCIENT TRIPOD.\n\n[Illustration: [++] Ancient Tripod.]\n\nTripods are, next to vases, the most ancient furniture in the world;\nthe imagination of the ancients invested them with fanciful forms,\nand we meet with designs which, although very simple, show already\nthe power exercised by the re-productive faculties of the mind upon\nthe objects surrounding these ancient nations. Representations of\nthe kind were, however, exceedingly rare till the last forty years,\nand it must be considered an especial piece of good fortune that the\nexcavations made in several parts of Etruria, have afforded more than\none example of this description. The specimen engraved was found in\nthe Gailassi Regulini tomb of Cervetri, in Etruria, and in it we see a\nlarge vessel placed on the tripod, from the edge of which five lions'\nheads start forth with hideous expression. These monsters lend to\nthe whole that fanciful aspect distinguishing objects of the archaic\nperiod. When we imagine to ourselves this kettle boiling, and these\ncruel animals wreathed and enveloped in smoke, we can understand how\nthe fancy of superstitious worshippers, who were wont to make use of\nthese implements in their religious ceremonies, may have found in\nthem an allusion to the spirits of the victims whose remains were\nexposed to the destructive fire glowing underneath. To us, at least,\nthis representation may illustrate the terrific but grand passage of\nHomer, where the bodies of the slaughtered sun bulls become once more\ninstinct with life, demanding vengeance with fearful cries: Odyssey,\nBook xii, verse 395.\n\n \"The skins began to creep, and the flesh around the spits bellowed,\n The roasted as well as the raw. And thus grew the voice of\n the oxen.\"\n\nThe careful construction of the three-legged mechanism which lends\na firm support to this fire-stand, has been restored according to\nthe indication of some fragments found on the spot. It presents a\ngraceful aspect, and forms, in some respects, a remarkable contrast\nto the heavy character of the vessel occupying so lofty a position,\nas the proportions of the legs are exceedingly slender, and the feet\nthemselves, instead of being broad and shapeless, are all composed of a\ngreat many fine articulations.\n\n\nFONDNESS OF THE ROMANS FOR PEARLS.\n\nOf all the articles of luxury and ostentation known to the Romans,\npearls seem to have been the most esteemed. They were worn on all parts\nof the dress, and such was the diversity of their size, purity, and\nvalue, that they were found to suit all classes, from those of moderate\nto those of the most colossal fortune. The famous pearl ear-rings of\nCleopatra are said to have been worth about L160,000, and Julius Caesar\nis said to have presented Servilia, the mother of Brutus, with a pearl\nfor which he had paid above L48,000; and though no reasonable doubt\ncan be ascertained in regard to the extreme exaggeration of these\nand similar statements, the fact that the largest and finest pearls\nbrought immense prices is beyond all question. It has been said that\nthe wish to become master of the pearls with which it was supposed to\nabound, was one of the motives which induced Julius Caesar to invade\nBritain. But, though a good many were met with in various parts of the\ncountry, they were of little or no value, being small and ill-.\nAfter pearls and diamonds, the emerald held the highest place in the\nestimation of the Romans.\n\n\nTHE BLACK STONE AT MECCA.\n\nNear the entrance of the Kaaba at Mecca, at the north-eastern corner,\nis the famous Black Stone, called by the Moslems _Hajra el Assouad_,\nor Heavenly Stone. It forms a part of the sharp angle of the building,\nand is inserted four or five feet above the ground. The shape is an\nirregular oval, about seven inches in diameter. Its colour is now a\ndeep reddish brown, approaching to black; and it is surrounded by a\nborder of nearly the same colour, resembling a cement of pitch and\ngravel, and from two to three inches in breadth. Both the border\nand the stone itself are encircled by a silver band, swelling to a\nconsiderable breadth below, where it is studded with nails of the same\nmetal. The surface is undulated, and seems composed of about a dozen\nsmaller stones, of different sizes and shapes, but perfectly smooth,\nand well joined with a small quantity of cement. It looks as if the\nwhole had been dashed into many pieces by a severe concussion, and then\nre-united--an appearance that may perhaps be explained by the various\ndisasters to which it has been exposed. During the fire that occurred\nin the time of Yezzid I. (A.D. 682), the violent heat split it into\nthree pieces; and when the fragments were replaced, it was necessary\nto surround them with a rim of silver, which is said to have been\nrenewed by Haroun el Raschid. It was in two pieces when the Karmathians\ncarried it away, having been broken by a blow from a soldier during the\nplunder of Mecca. Hakem, a mad sultan of Egypt, in the 11th century,\nendeavoured, while on the pilgrimage, to destroy it with an iron club\nwhich he had concealed under his clothes; but was prevented and slain\nby the populace. Since that accident it remained unmolested until 1674,\nwhen it was found one morning besmeared with dirt, so that every one\nwho kissed it returned with a sullied face. Though suspicion fell on\ncertain Persians, the authors of this sacrilegious joke were never\ndiscovered. As for the quality of the stone, it does not seem to be\naccurately determined. Burckhardt says it appeared to him like a\nlava, containing several small extraneous particles of a whitish and\nyellowish substance. Ali Bey calls it a fragment of volcanic basalt,\nsprinkled with small-pointed crystals, and varied with\nred feldspar upon a dark black ground like coal, except one of its\nprotuberances, which is a little reddish. The millions of kisses and\ntouches impressed by the faithful have worn the surface uneven, and to\na considerable depth. This miraculous block all orthodox Mussulmans\nbelieve to have been originally a transparent hyacinth, brought from\nheaven to Abraham by the angel Gabriel; but its substance, as well\nas its colour, have long been changed by coming in contact with the\nimpurities of the human race.\n\n\nPARAGRAPH FROM THE \"POSTMAN\" IN 1697.\n\n\"Yesterday being the day of thanksgiving appointed by the\nStates-General for the peace, His Excellency, the Dutch ambassador,\nmade a very noble bonfire before his house in St. James's Square,\nconsisting of about 140 pitch barrels placed perpendicularly on seven\nscaffolds, during which the trumpets sounded, and two hogsheads of wine\nwere kept continually running amongst the common people.\"\n\n\nLORD MAYOR'S FEAST IN 1663.\n\nPepys gives a curious account of a Lord Mayor's dinner in 1663. It was\nserved in the Guildhall, at one o'clock in the day. A bill of fare\nwas placed with every salt-cellar, and at the end of each table was a\nlist of the persons proper there to be seated. Here is a mixture of\nabundance and barbarism. \"Many were the tables, but none in the hall,\nbut the Mayor's and the Lords' of the Privy Council, _that had napkins\nor knives_, which was very strange. I sat at the merchant-stranger's\ntable, where ten good dishes to a mess, with plenty of wine of all\nsorts; but it was very unpleasing that we had no napkins, nor change\nof trenchers, and drank out of earthen pitchers and wooden dishes. The\ndinner, it seems, is made by the Mayor and two Sheriffs for the time\nbeing, and the whole is reckoned to come to L700 or L800 at most.\"\nPepys took his spoon and fork with him, as was the custom of those days\nwith guests invited to great entertainments. \"Forks\" came in with Tom\nCoryat, in the reign of James I.; but they were not \"familiar\" till\nafter the Restoration. The \"laying of napkins,\" as it was called, was\na profession of itself. Pepys mentions, the _day before_ one of his\ndinner-parties, that he went home, and \"there found one laying of my\nnapkins against to-morrow, in figures of all sorts, which is mighty\npretty, and, it seems, is his trade, and he gets much money by it.\"\n\n\nTHE CUPID OF THE HINDOOS.\n\nAmong the Hindoo deities _Camdeo_, or Manmadin differs but little from\nthe Cupid of the ancients. He is also called _Ununga_, or, without\nbody; and is the son of Vishnu and Lacshmi. Besides his bow and\narrows, he carries a banner, on which is delineated a fish: his bow\nis a sugar-cane; the cord is formed of bees; the arrows are of all\nsorts of flowers; one only is headed, but the point is covered with\na honeycomb--an allegory equally just and ingenious, and which so\ncorrectly expresses the pleasures and the pangs produced at one and the\nsame time by the wounds of love. Manmadin is represented, as in the\nannexed plate, riding on a parrot.\n\n[Illustration: [++] Manmadin Riding on a Parrot.]\n\nOne day, when Vishnu, to deceive Sheeva, had assumed the figure of a\nbeautiful young female, Manmadin discharged an arrow, which pierced\nthe heart of the formidable deity, and inflamed it with love of the\nnymph. The latter fled, and at the moment when Sheeva had overtaken\nher, Vishnu resumed his proper form. Sheeva, enraged at the trick\nplayed upon him, with one flash of his eyes burned and consumed the\nimprudent Manmadin, who hence received the name of _Ununga_. He was\nrestored to life by a shower of nectar, which the gods in pity poured\nupon him: but he remained without body and is the only Indian deity who\nis accounted incorporeal. Camdeo is particularly worshipped by females\ndesirous of obtaining faithful lovers and good husbands.\n\n\nOLD DIAL AND FOUNTAIN IN LEADENHALL-STREET.\n\n[Illustration: [++] Old Dial and Fountain.]\n\nThe above sketch is taken from an old work on astronomy and geography\nby Joseph Moxon, and printed by him, and sold \"at his Shop on\n_Cornhill_, at the signe of _Atlas_, 1659.\" We cannot do better than\ngive Moxon's own words with reference to the dial:--\"To make a dyal\nupon a solid ball or globe, that shall show the hour of the day without\na gnomon. The equinoctial of this globe, or (which is all one) the\nmiddle line must be divided into 24 equal parts, and marked with 1,\n2, 3, 4, &c., and then beginning again with 1, 2, 3, &c. to 12. Then\nif you elevate one of the poles so many degrees above an horizontal\nline as the pole of the world is elevated above the horizon in your\nhabitation, and place one of the twelves directly to behold the north,\nand the other to behold the south, when the sun shines on it, the globe\nwill be divided into two halfs, the one enlightened with the sunshine,\nand the other shadowed; and where the enlightened half is parted from\nthe shadowed half, there you will find in the equinoctial the hour of\nthe day, and that on two places on the ball, because the equinoctial\nis cut in two opposite points by the light of the sun. A dyal of this\nsort was made by Mr. John Leak and set up on a composite columne at\nLeadenhall Corner, in London, in the majoralty of Sir John Dethick,\nknight. The figure whereof I have inserted because it is a pretty peece\nof ingenuity, and may, perhaps, stand some lover of the art in stead\neither for imitation or help of invention.\"\n\n\nMAGNIFICENCE OF MADYN, THE CAPITAL OF PERSIA, WHEN INVADED BY THE\nSARACENS, A.D. 626.\n\nThe invaders could not express their mingled sensations of surprise\nand delight, while surveying in this splendid capital the miracles\nof architecture and art, the gilded palaces, the strong and stately\nporticoes, the abundance of victuals in the most exquisite variety and\nprofusion, which feasted their senses, and courted their observation\non every side. Every street added to their astonishment, every chamber\nrevealed a new treasure; and the greedy spoilers were enriched beyond\nthe measure of their hopes or their knowledge. To a people emerging\nfrom barbarism, the various wonders which rose before them in all\ndirections, like the effect of magic, must have been a striking\nspectacle. We may therefore believe them when they affirm, what is\nnot improbable, that the different articles of merchandise--the\nrich and beautiful pieces of manufacture which fell a prey on this\noccasion--were in such incalculable abundance, that the thirtieth part\nof their estimate was more than the imagination could embrace. The gold\nand silver, the various wardrobes and precious furniture, surpassed,\nsays Abul-feda, the calculation of fancy or numbers; and the historian\nElmacin ventured to compute these untold and almost infinite stores at\nthe value of 3,000,000,000 pieces of gold.\n\nOne article in this prodigious booty, before which all others seemed\nto recede in comparison, was the superb and celebrated carpet of\nsilk and gold cloth, sixty cubits in length, and as many in breadth,\nwhich decorated one of the apartments of the palace. It was wrought\ninto a paradise or garden, with jewels of the most curious and costly\nspecies; the ruby, the emerald, the sapphire, the beryl, topaz, and\npearl, being arranged with such consummate skill, as to represent, in\nbeautiful mosaic, trees, fruits, and flowers, rivulets and fountains;\nroses and shrubs of every description seemed to combine their fragrance\nand their foliage to charm the sense of the beholders. This piece of\nexquisite luxury and illusion, to which the Persians gave the name\nof _Baharistan_ or the mansion of perpetual spring, was an invention\nemployed by their monarchs as an artificial substitute for that\nloveliest of seasons. During the gloom of winter they were accustomed\nto regale the nobles of their court on this magnificent embroidery,\nwhere art had supplied the absence of nature, and wherein the guests\nmight trace a brilliant imitation of her faded beauties in the\nvariegated colours of the jewelled and pictured floor. In the hope that\nthe eyes of the Caliph might he delighted with this superb display of\nwealth and workmanship, Saad persuaded the soldiers to relinquish their\nclaims. It was therefore added to the fifth of the spoil, which was\nconveyed to Medina on the backs of camels. But Omar, with that rigid\nimpartiality from which he never deviated, ordered the gaudy trophy to\nbe cut up into small pieces, and distributed among the chief members\nof the Mohammedan commonwealth. Such was the intrinsic value of the\nmaterials, that the share of Ali alone, not larger than the palm of\na man's hand, was afterwards sold for 20,000 drachms (L458 6s. 8d.),\nor, according to others, for as many dinars (L9,250). Out of this vast\nstore the Caliph granted pensions to every member of his court in\nregular gradation, from the individuals of the Prophet's family to the\nlowest of his companions, varying from L275 to L4 11s. per annum.\n\nThe military part of the booty was divided into 60,000 shares, and\nevery horseman had 12,000 dinars (L5,550); hence, if the army consisted\nof 60,000 cavalry, their united shares would amount to the incredible\nsum of L333,000,000 sterling.\n\n\nCOURTSHIP OF WILLIAM THE CONQUEROR.\n\nThe following extract from the life of the wife of the Conqueror,\nis exceedingly curious, as characteristic of the manners of a\nsemi-civilized age and nation:--\"After some years' delay, William\nappears to have become desperate; and, if we may trust to the evidence\nof the 'Chronicle of Ingerbe,' in the year 1047 way-laid Matilda in\nthe streets of Bruges, as she was returning from mass, seized her,\nrolled her in the dirt, spoiled her rich array, and, not content with\nthese outrages, struck her repeatedly, and rode off at full speed.\nThis Teutonic method of courtship, according to our author, brought\nthe affair to a crisis; for Matilda, either convinced of the strength\nof William's passion, by the violence of his behaviour, or afraid of\nencountering a second beating, consented to become his wife. How he\never presumed to enter her presence again, after such a series of\nenormities, the chronicler sayeth not, and we are at a loss to imagine.\"\n\n\nBRAMA, THE HINDOO DEITY.\n\n_Brama_, _Birmah_, or _Brouma_, is one of the three persons of the\nIndian Trinity, or rather the Supreme Being under the attribute of\n_Creator_. Brama, the progenitor of all rational beings, sprung from a\ngolden egg, sparkling like a thousand suns, which was hatched by the\nmotion imparted to the waters by the Supreme Being. Brama separated\nthe heavens from the earth, and placed amid the subtle ether the eight\npoints of the universe and the receptacle of the waters. He had five\nheads before Vairevert, one of Sheeva's sons, cut off one of them.\nHe is delineated floating on a leaf of the lotus, a plant revered in\nIndia. The Bramins relate, that the fifteen worlds which compose the\nuniverse were each produced by a part of Brama's body. At the moment\nof our birth he imprints in our heads, in characters which cannot be\neffaced, all that we shall do, and all that is to happen to us in life.\nIt is not in our power, nor in that of Brama himself, to prevent what\nis written from being fulfilled.\n\n[Illustration: [++] Brama.]\n\nBrama, according to the vulgar mythology, takes but little notice of\nhuman affairs. Identified with the sun, he is adored by the Bramins\nin the _gayatri_, the most sacred passage of the _vedas_ (or sacred\nbooks), which is itself ranked among the gods, and to which offerings\nare made. One of the most important attributes of Brama is that of\nfather of legislators; for it was his ten sons who diffused laws and\nthe sciences over the world. He is considered as the original author of\nthe _vedas_, which are said to have issued from his four mouths; though\nit was not till a later period, that is, about fourteen hundred years\nbefore Christ, that they were collected and arranged by Vyasa, the\nphilosopher and poet. The laws which bear the name of Menu, the son of\nBrama, and the works of the other _richeys_, or holy persons, were also\nre-copied, or perhaps collected from tradition, long after the period\nwhen they are said to have been published by the sons of Brama.\n\nBrama, the father of the legislators of India, has a considerable\nresemblance to the Jupiter of the Greek poets, the father of Minos,\nwhose celebrated laws were published in the very same century that\nVyasa collected the _vedas_. Jupiter was worshipped as the sun, by\nthe name of _Anxur_ or _Axur_, and Brama is identified with that\nluminary. The most common form in which Brama is represented, is that\nof a man with four heads and four hands; and it is remarkable that the\nLacedaemonians gave four heads to their Jupiter. Lastly, the title of\nFather of Gods and Men is equally applicable to Brama and to Jupiter.\n\nBrama is delineated, as in the engraving, holding in one hand a ring,\nthe emblem of immortality; in another, fire, to represent force; and\nwith the other two writing on _olles_, or palm-leaves, the emblem of\nlegislative power.\n\n\nJAMES II. AND THE CHURCH OF DONORE.\n\n[Illustration: [++] Church of Donore.]\n\nThe annexed engraving represents a celebrated locality. It is the ruin\nof the little church on the hill at Donore, in the county of Meath,\nthe spot where James II. was stationed when he beheld the overthrow\nof his army and the ruin of his cause at the battle of the Boyne,\nTuesday, July 1st, 1690. The Boyne is a very beautiful and picturesque\nriver; it winds through the fertile valleys of Meath, and from its\nrichly-wooded banks the hills rise gradually; there are no lofty\nmountains in the immediate neighbourhood. The depth, in nearly all\nparts, is considerable, and the current, consequently, not rapid; its\nwidth, near the field of battle, varies little, and is seldom less\nthan fifty or sixty yards. James had the choice of ground, and it was\njudiciously selected. On the south side of the river, in the county of\nMeath, his army was posted with considerable skill: on the right was\nDrogheda; in front were the fords of the Boyne, deep and dangerous,\nand difficult to pass at all times; the banks were rugged, lined by a\nmorass, defended by some breastworks, with \"huts and hedges convenient\nfor infantry;\" and behind them was an acclivity stretching along the\nwhole of \"the field.\" James fixed his own tent upon the summit of a\nhill close to the little church of Donore, now a ruin; it commanded\nan extensive view of the adjacent country, and the opposite or south\nside of the river--the whole range, indeed, from Drogheda to Oldbridge\nvillage--and looked directly down upon the valley, in which the battle\nwas to be fought, and the fords of the Boyne, where there could be no\ndoubt the troops of William would attempt a passage. From this spot,\nJames beheld his prospering rival mingling in the thick of the _melee_,\ngiving and taking blows; watched every turn of fortune, as it veered\ntowards or against him; saw his enemies pushing their way in triumph,\nand his brave allies falling before the swords of foreigners--a safe\nand inglorious spectator of a battle upon the issue of which his throne\ndepended. The preceeding night he had spent at Carntown Castle, from\nwhence he had marched, not as the leader, but as the overseer, of the\nIrish army; having previously given unequivocal indications of his\nprospects, his hopes, and his designs, by despatching a commissioner to\nWaterford, \"to prepare a ship for conveying him to France, in case of\nany misfortune.\"\n\n\nHANGING GARDENS OF BABYLON.\n\nWhen Babylon the Great was in the zenith of her glory, adjoining the\ngrand palace, and within the general enclosure, the Hanging Gardens\nwere constructed by the king to gratify his wife Amytis, who being a\nnative of Media (she was the daughter of Astyages, the king of Media),\ndesired to have some imitation of her native hills and forests.\n\n \"Within the walls was raised a lofty mound,\n Where flowers and aromatic shrubs adorn'd\n The pensile garden. For Nebassar's queen,\n Fatigued with Babylonia's level plains,\n Sigh'd for her Median home, where nature's hand\n Had scooped the vale, and clothed the mountain's side\n With many a verdant wood: nor long she pined\n Till that uxorious monarch called on Art\n To rival Nature's sweet variety.\n Forthwith two hundred thousand slaves uprear'd\n This hill--egregious work; rich fruits o'erhang\n The sloping vales, and odorous shrubs entwine\n Their undulating branches.\"\n\nThese gardens, as far as we learn from ancient accounts, contained\na square of above 400 feet on each side, and were carried up in the\nmanner of several large terraces, one above the other, till the height\nequalled that of the walls of the city. The ascent from terrace to\nterrace was by stairs ten feet wide. The whole pile was sustained\nby vast arches, raised on other arches one above another, and was\ndefended and condensed by a wall, surrounding it on every side, of\ntwenty-two feet in thickness. On the top of the arches were first laid\nlarge flat stones, sixteen feet long and four broad; over these was a\nlayer of weeds mixed and cemented with a large quantity of bitumen, on\nwhich were two rows of bricks closely cemented together with the same\nmaterial. The whole was covered with thick sheets of lead, on which lay\nthe mould of the garden. And all this floorage was so contrived as to\nkeep the moisture of the mould from running away through the arches.\nThe earth laid thereon was so deep that large trees might take root\nin it: and with such the terraces were covered, as well as with the\n[...] plants and flowers proper to adorn an eastern pleasure-garden.\nThe trees planted there are represented to have been of various kinds.\nHere grew the larch, that, curving, flings its arms like a falling\nwave; and by it was seen the grey livery of the aspen; the mournful\nsolemnity of the cypress and stately grandeur of the cedar intermingled\nwith the elegant mimosa; besides the light and airy foliage of the\nsilk-tasselled acacia, with its vast clusters of beauteous lilac\nflowers streaming in the wind and glittering in the sun; the umbrageous\nfoliage of the chesnut, and ever-varying verdure of the poplar; the\nbirch, with its feathered branches light as a lady's plumes--all\ncombined with the freshness of the running stream, over which the\nwillow waved its tresses.--\n\n \"And the jessamine faint, and the sweet tuberose,\n The sweetest flower for scent that blows;\n And all rare blossoms from every clime\n Grew in that garden in perfect prime.\"\n\nAll these varied delights of nature were ranged in rows on the\nside of the ascent as well as on the top, so that at a distance it\nappeared to be an immense pyramid covered with wood. The situation of\nthis extraordinary effort of human skill, aided by human wealth and\nperseverance, adjoining the river Euphrates, we must suppose that in\nthe upper terrace was an hydraulic engine, or kind of pump, by which\nthe water was forced up out of the river, and from thence the whole\ngardens were watered, and a supply of the pure element furnished to the\nfountains and reservoirs for cooling the air. In the spaces between\nthe several arches, on which the whole structure rested, were large\nand magnificent apartments, very lightsome, and commanding the most\nbeautiful prospects that even the glowing conceptions of an eastern\nimagination could dream to exist.\n\n\nTHE GREAT BELL OF BURMAH.\n\nAt a temple in the environs of Amarapoora, the capital of Burmah, there\nis an enormous bell, which is thus described by Captain Yule:--\"North\nof the temple, on a low circular terrace, stands the biggest bell in\nBurmah--the biggest in the world, probably, Russia apart. It is slung\non a triple beam of great size, cased and hooped with metal; this beam\nresting on two piers of brickwork, enclosing massive frames of teak.\nThe bell does not swing free. The supports were so much shaken by the\nearthquake, that it was found necessary to put props under the bell,\nconsisting of blocks of wood carved into grotesque figures. Of course\nno tone can now be got out of it. But at any time it must have required\na battering-ram to elicit its music. Small ingots of silver (and some\nsay pieces of gold) may still be traced, unmelted, in the mass, and\nfrom the inside one sees the curious way in which the makers tried to\nstrengthen the parts which suspend it by dropping into the upper part\nof the mould iron chains, round which the metal was run. The Burmese\nreport the bell to contain 555,555 viss of metal (about 900 tons). Its\nprincipal dimensions are as follow:--External diameter at the lip,\n16 feet 3 inches; external diameter 4 feet 8 inches above the lip,\n10 feet; interior height, 11 feet 6 inches; exterior ditto, 12 feet;\ninterior diameter at top, 8 feet 6 inches. The thickness of metal\nvaries from six inches to twelve, and the actual weight of the bell\nis, by a rough calculation, about eighty tons, or one-eleventh of the\npopular estimate. According to Mr. Howard Malcolm, whose authority was\nprobably Colonel Burney, the weight is stated in the Royal Chronicle\nat 55,500 viss, or about ninety tons. This statement is probably,\ntherefore, genuine, and the popular fable merely a multiplication of it\nby ten.\"\n\nThis monster Burmese bell is, therefore, fourteen times as heavy as\nthe great bell of St. Paul's, but only one-third of that given by the\nEmpress Anne to the Cathedral of Moscow.\n\n\nBANDOLIERS.\n\n[Illustration: [++] Bandoliers.]\n\nWe here engrave a set of bandoliers, a species of weapon much in vogue\nabout the close of the sixteenth century. The specimen before us\nconsists of nine tin cases covered with leather, with caps to them,\neach containing a charge of powder, and suspended by rings from a cord\nmade to pass through other rings. The caps are retained in their places\nby being contrived so as to slip up and down their own cords. Two\nflaps of leather, on each side, are intended to protect the bandoliers\nfrom rain, and attached to one of these may be perceived a circular\nbullet-purse, made to draw with little strings. This specimen was\nbuckled round the waist by means of a strap; others were worn round the\nbody and over the shoulder. The noise they made, agitated by the wind,\nbut more especially the danger of all taking fire from the match-cord,\noccasioned their disuse, as Sir James Turner tells us, about the year\n1640.\n\n\nTOMB OF DARIUS.\n\nAmong the most remarkable tombs of the ancients, may be noticed the\nsepulchre carved out of the living rock, by order of Darius, the\nwarrior and conqueror king of Persia, for the reception of his own\nremains; and which is existing to this day at Persepolis, after a\nduration of twenty-three centuries.\n\nThe portico is supported by four columns twenty feet in height, and\nin the centre is the form of a doorway, seemingly the entrance to the\ninterior, but it is solid; the entablature is of chaste design. Above\nthe portico there is what may be termed an ark, supported by two rows\nof figures, about the size of life, bearing it on their uplifted hands,\nand at each angle a griffin--an ornament which is very frequent at\nPersepolis. On this stage stands the king, with a bent bow in his hand,\nworshipping the sun, whose image is seen above the altar that stands\nbefore him, while above his head hovers his ferouher, or disembodied\nspirit. This is the good genius that in Persian and Ninevite sculpture\naccompanies the king when performing any important act. On each side\nthe ark are nine niches, each containing a statue in bas-relief. No\nother portion of the tomb was intended to be seen, excepting the\nsculptured front; and we must, therefore, conclude that the entrance\nwas kept secret, and that the avenues were by subterranean passages, so\nconstructed that none but the privileged could find their way. We are\ntold by Theophrastus, that Darius was buried in a coffer of Egyptian\nalabaster; and also that the early Persians buried their dead entire,\npreserving their bodies with honey or wax.\n\n\nTHE GATE ON OLD LONDON BRIDGE.\n\n[Illustration: [++] Gate on Old London Bridge.]\n\nIn the reign of Queen Elizabeth, a strongly embattled gate protected\nthe entrance from Southwark to Old London Bridge, and it was usually\ngarnished with traitors' heads in \"rich abundance,\" as may be seen\nin the accompanying cut, which is copied from Visscher's view, in\n1579. The bridge was at that period covered with houses, a narrow\nroad passing through arcades beneath them, and they abutted on props\nover the river on either side. The bridge was proudly spoken of by\nour ancestors. Thus, in the translation of Ortelius, published by J.\nShaw, in 1603, he says of the Thames:--\"It is beautified with statelye\npallaces, built on the side thereof; moreover, a sumptuous bridge\nsustayned on nineteen arches, with excellent and beauteous housen built\nthereon.\" Camden, in his great work, the \"Britannica,\" says, \"It may\nworthily carry away the prize from all the bridges in Europe,\" being\n\"furnished on both sides with passing faire houses, joining one to\nanother in the manner of a street.\"\n\n\nEXTRAORDINARY PONDS AND FISH.\n\nThe ponds in the department of Ain in France are 1667 in number. The\nindustry and ingenuity of man have converted the marshes into fertile\nplains and productive ponds, by constructing s from one hill to\nthe other, for the plateaux are covered with small hills. When the\nproprietor of one of these ponds wishes to cultivate it, he draws off\nthe water into the attached to it. Wheat, barley, and oats are\nthen sown, and the seed thus fertilised by the slime produces a crop\ndouble that produced by the land in the vicinity. After the harvest\nis collected, the water is permitted to return to its former bed, and\ncarp, tench, and roach are then thrown into it. Some of these ponds\nwill support 100,000 of carp, and 100 pounds of little tench and roach.\nIn the course of two years these carp, which weighed only one ounce\nand a-half, will have attained the size of two pounds and a half. The\nfishing begins in April, and is continued until November. The increase\nof the fish is as one to five.\n\n\nTHE CEREMONIAL OF MAKING THE KING'S BED.\n\nThe following account of the old ceremony of making the King's bed in\nthe time of Henry the Eighth, was sent to the Society of Antiquaries,\nin 1776, by Mr. J. C. Brooke, of the Heralds' College, F.S.A. &c. In a\nletter to the president, he says,--\n\n\"It is extracted from an original manuscript, elegantly written,\nbeautifully illuminated, and richly bound, which was some time in the\nlibrary of Henry, Duke of Norfolk, earl marshal of England, to whom it\ncame by descent from Thomas, the great Duke of Norfolk, beheaded in\nthe reign of Queen Elizabeth; who married Mary, daughter and coheir\nof Henry Fitz-Alan, Earl of Arundel, lord chamberlain to King Henry\nthe Eighth. It contains the whole duty of the lord chamberlain, and\nof the officers in his department; is the original copy kept for the\ninformation of that earl; and had been compiled by order of, and\napproved by, the King himself in council.\"\n\n\"_The oolde ordre of Makynge the Kynges Bedd not to used nor done, but\nas Hys Grace woll comaund and apoynte from tyme to tyme herafter._\n\n\"_Furste_, a groome or a page to take a torche, and to goo to the\nwarderobe of the kynges bedd, and bryng theym of the warderobe with\nthe kynges stuff unto the chambr for makyng of the same bedde. Where\nas aught to be a gentylman-usher, iiii yomen of the chambr for to make\nthe same bedde. The groome to stande at the bedds feete with his torch.\nThey of the warderobe openyng the kinges stuff of hys bedde upon a\nfayre sheete, bytwen the sayde groome and the bedds fote, iii yeomen,\nor two at the leste, in every syde of the bedde; the gentylman-usher\nand parte commaundyng theym what they shall doo. A yoman with a dagger\nto searche the strawe of the kynges bedde that there be none untreuth\ntherein. And this yoman to caste up the bedde of downe upon that,\nand oon of theym to tomble over yt for the serche thereof. Then they\nto bete and tufle the sayde bedde, and to laye oon then the bolster\nwithout touchyng of the bedd where as it aught to lye. Then they of\nthe warderobe to delyver theym a fustyan takyng the saye therof. All\ntheys yomen to laye theyr hands theroon at oones, that they touch not\nthe bedd, tyll yt be layed as it sholde be by the comaundement of the\nussher. And so the furste sheet in lyke wyse, and then to trusse in\nboth sheete and fustyan rownde about the bedde of downe. The warderoper\nto delyver the second sheete unto two yomen, they to crosse it over\ntheyr arme, and to stryke the bedde as the ussher shall more playnly\nshewe unto theym. Then every yoman layeing hande upon the sheete, to\nlaye the same sheete upon the bedde. And so the other fustyan upon or\nii with such covervnge as shall content the kynge. Thus doon, the ii\nyomen next to the bedde to laye down agene the overmore fustyan, the\nyomen of the warderobe delyverynge theym a pane sheete, the sayde yoman\ntherewythall to cover the sayde bedde. And so then to laye down the\novermost sheete from the beddes heed. And then the sayd ii yomen to lay\nall the overmost clothes of a quarter of the bedde. Then the warderoper\nto delyver unto them such pyllowes as shall please the kynge. The sayd\nyoman to laye theym upon the bolster and the heed sheete with whych the\nsayde yoman shall cover the sayde pyllowes. And so to trusse the endes\nof the sayde sheete under every ende of the bolster. And then the sayd\nwarderoper to delyver unto them ii lyttle small pyllowes, werwythall\nthe squyres for the bodye or gentylman-ussher shall give the saye to\nthe warderoper, and to the yoman whych have layde on hande upon the\nsayd bedde. And then the sayd ii yomen to lay upon the sayde bedde\ntoward the bolster as yt was bifore. They makyng a crosse and kissynge\nyt where there handes were. Then ii yomen next to the feete to make the\nfeers as the ussher shall teche theym. And so then every of them sticke\nup the aungel about the bedde, and to lette down the corteyns of the\nsayd bedde, or sparver.\n\n\"Item, a squyer for the bodye or gentylman-ussher aught to sett the\nkynges sword at hys beddes heed.\n\n\"Item, a squyer for the bodye aught to charge a secret groome or page,\nto have the kepynge of the sayde bedde with a lyght unto the time the\nkynge be disposed to goo to yt.\n\n\"Item, a groome or page aught to take a torche, whyle the bedde ys yn\nmakyng, to feche a loof of brede, a pott wyth ale, a pott wyth wine,\nfor them that maketh the bedde, and every man.\n\n\"Item, the gentylman-ussher aught to forbede that no manner of man\ndo sett eny dysshe upon the kynge's bedde, for fere of hurtying of\nthe kynge's ryche counterpoynt that lyeth therupon. And that the sayd\nussher take goode heede, that noo man wipe or rubbe their handes uppon\nnone arras of the kynges, wherby they myght bee hurted, in the chambr\nwhere the kynge ys specially, and in all other.\"\n\n\nORIGIN OF SANDWICHES.\n\nTo the memory of \"Lord Sandwich\" belongs the name of that edible.\nBeing, during his administration (as was very usual with him), at a\ngambling-house, he had, in the fascination of play, for more than five\nand twenty hours forgotten fatigue and hunger, when suddenly, feeling\ndisposed to break his fast, though still riveted to the table, he\ncalled to bid some one bring anything that was to be had to eat, which\nhappened to prove a slice of beef, and two pieces of bread. Placing\nthem together for the sake of expedition, he devoured them with the\ngreatest relish. The most ecstatic encomiums published his discovery,\nand giving it his name, bequeathed it as a memento to his country, as\none of the most important acts of his administration.\n\n\nTHE TREATY-STONE AT LIMERICK.\n\nThe city of Limerick is very famous in history. Before it, in 1651,\nIreton \"sate down;\" there he continued to \"sit\" for six months; and\nunderneath its walls the fierce republican died of plague. Greater\ncelebrity, and higher honour, were, however, obtained by Limerick in\n1690. Early in August, William summoned it to surrender; the French\ngeneral, Boileau, who commanded the garrison--\"rather for the King\nof France than the King of England\"--returned for answer, that \"he\nwas surprised at the summons, and thought the best way to gain the\ngood opinion of the Prince of Orange was to defend the place for his\nmaster King James.\" The siege was at once commenced. The flower of the\nIrish army were within its walls, or in its immediate neighbourhood;\nthe counties of Clare and Galway were open to them, from which to\ndraw supplies; and a French fleet rode triumphantly in the Shannon.\nThe garrison, however, were little disposed to act in concert: the\njealousy of the commanders of the French and Irish had spread to their\ntroops; and they cherished feelings of contempt or hatred towards each\nother, that argued ill for their success in opposing the steady and\ndisciplined forces of William.\n\n[Illustration: [++] Treaty-Stone at Limerick.]\n\nYet the Irish succeeded; the siege was raised on the 30th of August.\nBut, in the autumn of 1691, it endured a second, which occupied about\nsix months; when the garrison, wearied of a struggle from which they\ncould derive nothing but glory, on the 23rd of September, a cessation\nof hostilities took place; an amicable intercourse was opened between\nthe two armies; and articles of capitulation were, after a few brief\ndelays, agreed upon. The \"violated treaty\" was signed on the 3rd of\nOctober, 1691; it consisted of two parts, civil and military. It is\nsaid to have been signed by the several contracting parties on a large\nstone, near to Thomond Bridge, on the county of Clare side of the\nriver. The stone remains in the position it occupied at the period, and\nis an object of curiosity to strangers, as well as of interest to the\ncitizens of Limerick. We, therefore, thought it desirable to procure a\ndrawing of the relic, which retains its name of \"the Treaty Stone.\"\n\n\nTHE TEMPLARS' BANNER CALLED BEAUSEANT.\n\nWhen Constantine the Great was on the eve of a battle with Maxentius,\nwe are told that a luminous standard appeared to him in the sky with a\ncross upon it, and this inscription:--\"_In hoc signo vinces_--By this\nsign you shall conquer;\" and that this sign so encouraged Constantine\nand his soldiers that they gained the next day a great victory.\n\nWhen Waldemar II. of Denmark was engaged in a great battle with the\nLivonians in the year 1219, it is said that a sacred banner fell from\nheaven into the midst of his army, and so revived the courage of his\ntroops, that they gained a complete victory over the Livonians; and\nin memory of the event, Waldemar instituted an order of knighthood,\ncalled \"St. Danebrog,\" or the strength of the Danes, and which is\nstill the principal order of knighthood in Denmark. Now, taking these\nlegends for as much as they are worth, and no more; what do they prove?\nNot that this miraculous standard and cross came to the assistance\nof Constantine; not that this miraculous banner came to the aid of\nWaldemar; but they prove that such was the paramount importance\nattached to the sacred banner among the forces, that wherever it was\npresent, it was a great means of inspiriting the men with increased\nconfidence and courage, and so contributed to the victory.\n\n[Illustration: [++] Beauseant.]\n\nThe great importance attached to the banner in the middle ages is not\nto be wondered at, when we consider that it was a kind of connecting\nlink between the military and the clergy; it was a religious symbol\napplied to a military purpose, and this was the feeling which animated\nthe Crusaders and the Templars in their great struggle against the\nenemies of Christianity. The contest then was between the crescent and\nthe cross--between Christ and Mahomet. The Knights Templars had a very\nremarkable banner, being simply divided into black and white, the white\nportion symbolising peace to their friends, the black portion evil to\ntheir enemies, and their dreaded war cry, \"BEAUSEANT.\"\n\n\nSWORD-FISH _v._ WHALES.\n\nSo boundless is the sword-fish's rage and fury against whales in\nparticular, that many observers imagine his sallies against rocks and\ntimber to originate in an error of judgment, that all these lunges are\nintended to punish leviathan, and are only misdirected in consequence\nof the imperfect vision which prevents this scomber, like many of\nhis family, from accurately distinguishing forms. Whenever a supposed\nwhale is descried, our savage _sabreur_ rushes forward to intercept\nhis progress, and suddenly flashing before his victim, either alone\nor in conjunction with some other unfriendly fish, instantly proceeds\nto the attack. Relations of such sea-fights, attested by credible\neye-witnesses, are not uncommon; we content ourselves with the citation\nof one of unimpeachable accuracy. Captain Crow, cited by Mr. Yarrell,\nrelates that in a voyage to Memel, on a calm night, just off the\nHebrides, all hands were called up to witness a strange combat between\nsome thrashers (carcharias vulpes) and a sword-fish leagued together\nagainst a whale; as soon as the back of the ill-starred monster was\nseen rising a little above the water, the thrashers sprang several\nyards into the air, and struck him with their descending tails, the\nreiterated percussions of which sounded, we are told, like a distant\nvolley of musketry. The sword-fish meanwhile attacked the whale from\nbelow, getting close under his belly, and with such energy and effect\nthat there could be little doubt of the issue of a fray, which the\nnecessity of prosecuting their voyage prevented the crew from watching\nto its close. The sword-fish is not less remarkable for strength than\npugnacity, the depot of its great physical powers being, as in most\nscombers, in the tail.\n\n\nWEALTH OF SPAIN UNDER THE MOORS.\n\nThe Moors, whose conquest and expulsion were attended with such\natrocities, and such triumphs to the Catholic church, were by far the\nmost industrious and skilful part of the Spanish population, and their\nloss was a blow to the greatness and prosperity of that kingdom from\nwhich it has never recovered. The literary activity and commercial\nenterprise of the Arabs, which the wise policy of their Caliphs\nencouraged, contributed both to enrich and adorn their adopted country.\nCordova, the seat of the Ommiades, was scarcely inferior, in point of\nwealth and magnitude, to its proud rival on the banks of the Tigris.\nA space of twenty-four miles in length, and six in breadth, along the\nbanks of the Guadalquiver, was occupied with palaces, streets, gardens,\nand public edifices; and for ten miles the citizens could travel by\nthe light of lamps along an uninterrupted extent of buildings. In the\nreign of Almansor it could boast of 270,000 houses, 80,455 shops,\n911 baths, 3,877 mosques, from the minarets of which a population of\n800,000 were daily summoned to prayers. The seraglio of the Caliph, his\nwives, concubines, and black eunuchs, amounted to 6,300 persons; and he\nwas attended to the field by a guard of 12,000 horsemen, whose belts\nand scimitars were studded with gold. Granada was equally celebrated\nfor its luxury and its learning. The royal demesnes extended to the\ndistance of twenty miles, the revenues of which were set apart to\nmaintain the fortifications of the city. Of the duty on grain, the\nking's exchequer received about L15,000 yearly, an immense sum at that\ntime, when wheat sold at the rate of sixpence a bushel. The consumption\nof 250,000 inhabitants kept 130 water-mills constantly at work in the\nsuburbs. The population of this small kingdom under the Moors is\nsaid to have amounted to 3,000,000, which is now diminished perhaps\nto one-fifth of that number. Its temples and palaces have shared the\nsame decay. The Alhambra stands solitary, dismantled, and neglected.\nThe interior remains of the palace are in tolerable preservation, and\npresent a melancholy picture of the romantic magnificence of its former\nkings. Seville, which had continued nearly 200 years the seat of a\npetty kingdom, enjoyed considerable reputation as a place of wealth\nand commerce. The population in 1247 was computed at 300,000 persons,\nwhich, in the sixteenth century, had decreased one-third. It was one\nof the principal marts for olives in the Moorish dominions; and so\nextensive was the trade in this article alone that the _axarafe_, or\nplantations round the suburbs, employed farm-houses and olive-presses\nto the amount of 100,000, being more than is now to be found in the\nwhole province of Andalusia.\n\n\nTHE FIRST OPERA.\n\nThe first composer who tried his hand at setting an opera to music was\nFrancisco Bamirino, an Italian artist; and the piece to which he lent\nthe charm of a melodious accompaniment, was the \"Conversion of St.\nPaul,\" which was brought out at Rome in 1460.\n\n\nRUINS OF EUROPA.\n\nLady Sheil, in her \"Life in Persia,\" thus describes some wonderful\nruins which she saw about thirty miles from Tehran:--\n\n\"From near Verameen a most remarkable antiquity still survives the\nlapse of twenty centuries, that is, if what we hear be true. It\nconsists of an immense rampart, twenty or thirty feet in height, and\nof proportional thickness, including a space of about half a mile in\nlength and nearly the same in breadth. It is in the form of a square;\nthe rampart is continuous, and at short intervals is strengthened by\nbastions of prodigious size. The whole is constructed of unbaked bricks\nof large dimensions, and is in a state of extraordinary preservation.\nThe traces of a ditch of great size, though nearly filled up, are\nevident in front of the rampart. No buildings are found inside, where\nnothing is visible excepting a few mounds,--not a single habitation or\nhuman being. The solitude of this striking vestige of antiquity adds\nto its solemnity. It stood alone; Elboorz, distant only a few miles,\ngazing down on its hoary walls, with Demawend, in its garments of\nsnow, to complete the scene. From no place have I had a finer view of\nthis grand mountain, which seemed to lie exactly to the north. I am\ninformed that these magnificent ruins represent Europa, a city built\nby Seleucus, which, if true, would make it upwards of two thousand\nyears old. On seeing the perfect state of the ruins, and the materials\nof which they are composed, one feels no hesitation in crediting so\nvenerable an antiquity. Seleucus chose the spot well. The district of\nVerameen is renowned for its fertility, though not at this period for\nthe salubrity of its climate. The surrounding country is covered with\nearthen mounds, denoting former edifices, which, if explored, might\nreveal objects worthy of the erudition and intellect of even Sir Henry\nRawlinson.\"\n\n\nCELEBRATED GUN.\n\n[Illustration: [++] Gun of Charles IX. of France.]\n\nThe gun, of which the annexed is a sketch, is one of the many\ncuriosities of the Londesborough Museum. It once formed part of the\ncollection of Prince Potemkin, and was originally the property of\nCharles IX. of France; it is traditionally reported to have been the\ngun he used in firing on his Huguenot subjects, from one of the windows\nof the Louvre, during the massacre of St. Bartholomew. The barrel is\nrichly chased in high relief, with a stag-hunt amid foliage. The stock\nis inlaid with ivory, sculptured into a series of hunting scenes,\nknights on horseback.\n\nThe dreadful massacre of Saint Bartholomew commenced at Paris on the\nnight of the festival of that saint, August 24th, 1572. Above 500\npersons of rank, and 10,000 of inferior condition, perished in Paris\nalone, besides those slaughtered in the provinces. The king, who had\nbeen persuaded that the destruction of the Huguenots to the last man\nwas necessary to the safety of his throne, beheld the slaughter from a\nwindow, and being carried away by the example of those whose murderous\ndoings he witnessed, ordered some long arquebusses to be brought,\nand on their being loaded, and handed to him one after another, he\nfor some time continued to fire on the unfortunate fugitives as they\npassed, crying at the same time with a loud voice, \"Kill, kill.\" He\nafterwards went and inspected the bodies of the slain, and expressed\nhis satisfaction at the effective manner in which his orders had been\nexecuted.\n\n\nTOMB OF RAFFAELLE.\n\nThe great painter Raffaelle died at Rome, April 7th 1520, at the early\nage of thirty-seven. He was buried in the Pantheon, in a chapel which\nwas afterwards called Raffaelle's Chapel. For more than a century and\na half his tomb had only a plain epitaph, but Carlo Maratti desired to\nplace a more striking memorial of Raffaelle's resting-place than the\nsimple inscription, and accordingly, in the year 1764, a marble bust\nof the painter, executed by Paolo Nardini, was placed in one of the\noval niches on each side of the chapel. The epitaph to Maria Bibiena\n(Raffaelle's betrothed) was removed to make way for Maratti's new\ninscription; and it was currently believed that the skull of Raffaelle\nwas removed; at least such was the history given of a skull shown\nas the painter's, religiously preserved by the Academy of St. Luke,\nand descanted on by phrenologists as indicative of all the qualities\nwhich \"the divine painter\" possessed. But scepticism played its part;\ndoubts of the truth of this story led to doubts of Vasari's statement\nrespecting the exact locality of Raffaelle's tomb. Matters were brought\nto a final issue by the discovery of a document proving this skull to\nbe that of Don Desiderio de Adjutorio, founder of the society called\nthe Virtuosi, in 1542. Thereupon, this society demanded the head of its\nfounder from the Academy of St. Luke; but they would neither abandon\nthat, nor the illusion that they possessed the veritable skull of the\ngreat artist. Arguments ran high, and it was at length determined to\nsettle the question by an examination of the spot, which took place on\nthe 13th of September 1833, in the presence of the Academies of St.\nLuke and of Archaeology, the Commission of the Fine Arts (including\nOverback and others), the members of the Virtuosi, the governor of Rome\n(Monsignor Grimaldi), and the Cardinal Zurla, the representative of the\npope.\n\n[Illustration: [++] Tomb of Raffaelle.]\n\nThe result will be best given in the words of an eye-witness, Signor\nNibby (one of the Commission of Antiquities and Fine Arts), who thus\ndescribed the whole to M. Quatremere de Quincy, the biographer of\nRaffaelle:--\"The operations were conducted on such a principle of exact\nmethod as to be chargeable with over nicety. After various ineffectual\nattempts in other directions, we at length began to dig under the altar\nof the Virgin itself, and taking as a guide the indications furnished\nby Vasari, we at length came to some masonry of the length of a man's\nbody. The labourers raised the stone with the utmost care, and having\ndug within for about a foot and a half, came to a void space. You can\nhardly conceive the enthusiasm of us all, when, by a final effort, the\nworkmen exhibited to our view the remains of a coffin, with an entire\nskeleton in it, lying thus as originally placed, and thinly covered\nwith damp dust. We saw at once quite clearly that the tomb had never\nbeen opened, and it thus became manifest that the skull possessed by\nthe Academy of St. Luke was not that of Raffaelle. Our first care was,\nby gentle degrees, to remove from the body the dust which covered it,\nand which we religiously collected, with the purpose of placing it in\na new sarcophagus. Amongst it we found, in tolerable preservation,\npieces of the coffin, which was made of deal, fragments of a painting\nwhich had ornamented the lid, several bits of Tiber clay, formations\nfrom the water of the river, which had penetrated into the coffin by\ninfiltration, an iron stelletta, a sort of spur, with which Raffaelle\nhad been decorated by Leo X, several _fibulae_, and a number of metal\n_anelli_, portions of his dress.\" These small rings had fastened the\nshroud; several were retained by the sculptor Fibris, who also took\ncasts of the head and hand, and Camuccini took views of the tomb and\nits precious contents; from one of these our cut is copied.\n\nOn the following day the body was further examined by professional\nmen: the skeleton was found to measure five feet seven inches, the\nnarrowness of the coffin indicated a slender and delicate frame. This\naccords with the contemporary accounts, which say he was of a refined\nand delicate constitution; his frame was all spirit; his physical\nstrength so limited that it was a wonder he existed so long as he did.\nThe investigation completed, the body was exhibited to the public from\nthe 20th to the 24th, and then was again placed in a new coffin of\nlead, and that in a marble sarcophagus presented by the pope, and taken\nfrom the antiquities in the Museum of the Vatican. A solemn mass was\nthen announced for the evening of the 18th of October. The Pantheon was\nthen illuminated, as for a funeral; the sarcophagus, with its contents,\nwas placed in exactly the same spot whence the remains had been taken.\nThe presidents of the various academies were present, with the Cavalier\nFabris at their head. Each bore a brick, which he inserted in the\nbrickwork with which the sepulchre was walled in. And so the painter\nawaits \"the resurrection of the just,\" and the fellowship of saints\nand angels, of which his inspired pencil has given us the highest\nrealisation on earth.\n\n\nANTIMONY.\n\nThe origin of the use of _anti-moine_, or antimony, is a remarkable\ncircumstance. Basil Valentin, superior of a college of religionists,\nhaving observed that this mineral fattened the pigs, imagined that it\nwould produce the same effect on the holy brotherhood. But the case was\nseriously different; the unfortunate fathers, who greedily made use of\nit, died in a short time, and this is the origin of its name, according\nto the pure French word. In spite of this unfortunate beginning,\nParacelsus resolved to bring this mineral into practice; and by mixing\nit with other preparations make it useful. The Faculty at Paris were\non this occasion divided into two parties, the one maintaining that\nantimony was a poison; the other affirmed that it was an excellent\nremedy. The dispute became more general, and the Parliament and\nthe College of the Sorbonne interfered in the matter; but sometime\nafterwards people began to judge rightly concerning this excellent\nmineral; and its wonderful and salutary effects have occasioned the\nFaculty to place it among their best medicines.\n\n\nPERSONAL APPEARANCE OF MAHOMET.\n\nFor the personal appearance and private life of Mahomet, we must rely\non the Arabian writers, who dwell with fond and proud satisfaction on\nthe graces and intellectual gifts with which nature had endowed him. He\nwas of a middle stature, of a clear, fair skin, and ruddy complexion.\nHis head and features, though large, were well proportioned; he had\na prominent forehead, large dark-brown eyes, an aquiline nose, and\na thick bushy beard. His mouth, though rather wide, was handsomely\nformed, and adorned with teeth white as pearls, the upper row not\nclosely set, but in regular order--which appeared when he smiled, and\ngave an agreeable expression to his countenance. He had a quick ear,\nand a fine sonorous voice. His dark eyebrows approached each other\nwithout meeting. His hair fell partly in ringlets about his temples,\nand partly hung down between his shoulders. To prevent whiteness, the\nsupposed effect of Satanic influence, he stained it, as the Arabs\noften do still, of a shining reddish colour. His frame was muscular\nand compact--robust rather than corpulent. When he walked, he carried\na staff, in imitation of the other prophets, and had a singular\naffectation of being thought to resemble Abraham. The assertion of\nthe Greeks and Christians, that he was subject to epilepsy, must be\nascribed to ignorance or malice.\n\n\nSTIRRUPS.\n\nFrom every information we have been able to collect, we believe that\nthe appendage of stirrups were not added to saddles before the sixth\ncentury. It is said, that previous to the introduction of stirrups,\nthe young and agile used to mount their horses by vaulting upon them,\nwhich many did in an expert and graceful manner; of course, practice\nwas essential to this perfection. That this should be afforded, wooden\nhorses were placed in the Campus Martius, where this exercise was\nperformed of mounting or dismounting on either side; first, without,\nand next with arms. Cavalry had also occasionally a strap of leather,\nor a metallic projection affixed to their spears, in or upon which the\nfoot being placed, the ascent became more practicable. Respecting the\nperiod of this invention, Montfaucon has presumed that the invention\nmust have been subsequent to the use of saddles; however, opposed to\nthis opinion, an ingenious argument has been offered, that it is\npossible they might have been anterior to that invention; because, it\nis said, they might have been appended to a girth round the body of the\nhorse. Both Hippocrates and Galen speak of a disease to which the feet\nand ancles were subject, from long riding, occasioned by suspension\nof the feet without a resting-place. Suetonius, the Roman, informs us\nthat Germanicus, the father of Caligula, was wont to ride after dinner,\nto strengthen his ancles, by the action of riding affording the blood\nfreer circulation in the part.\n\n\nTHE GREAT SHOEMADOO PAGODA.\n\n[Illustration: [++] Great Shoemadoo Pagoda.]\n\nThe Buddhist temple of which we here give an engraving is the great\nShoemadoo Pagoda at Pegu. Among other things it is interesting as\nbeing one of the earliest attempts at that class of decoration, which\nconsists in having at the base of the building a double range of small\npagodas, a mode of ornamentation that subsequently became typical in\nHindu architecture; their temples and spires being covered, and indeed\ncomposed of innumerable models of themselves, clustered together so as\nto make up a whole.\n\nThe building stands on two terraces, the lower one about 10 ft. high,\nand 1391 ft. square: the upper one, 20 ft. in height, is 684 ft.\nsquare; from the centre of it rises the pagoda, the diameter of whose\nbase is 395 ft. The small pagodas are 27 ft. high, and 108 or 110 in\nnumber; while the great pagoda itself rises to the height of 331 ft.\nabove its terrace, or 361 ft. above the country, thus reaching a height\nnearly equal to St. Paul's Cathedral; while the side of the upper\nterrace is only 83 ft. less than that of the great Pyramid.\n\nTradition ascribes its commencement to two merchants, who raised it to\nthe height of 12 cubits at an age slightly subsequent to that of Buddha\nhimself. Successive kings of Pegu added to this from time to time, till\nat last it assumed its present form, most probably about three or four\ncenturies ago.\n\n\nPEST HOUSE DURING THE PLAGUE IN TOTHILL FIELDS.\n\n[Illustration: [++] Pest House in Tothill Fields.]\n\nTothill Fields, a locality between Pimlico and the Thames, was\nanciently the manor of Tothill, belonging to John Maunsel, chancellor,\nwho in 1256, entertained here Henry III. and his court at a vast feast\nin tents and pavilions. Here were decided wagers of battle and appeals\nby combat. Necromancy, sorcery and witchcraft were punished here; and\n\"royal solemnities and goodly jousts were held here.\" In Culpeper's\ntime the fields were famous for parsley. In 1642 a battery and\nbreastwork were erected here. Here also were built the \"Five Houses,\"\nor \"Seven Chimneys,\" as pest-houses for victims of the plague. One of\nthese pest-houses is given in the above engraving, taken from an old\nprint. In the plague time of 1665, the dead were buried \"in the open\nTuttle Fields.\" In Queen Anne's reign here was William Well's head\ngarden on the site of Vincent-square. The Train Bands were drawn out\nhere in 1651. In the last century the fields were a noted duel-ground,\nand here, in 1711, Sir Cholmeley Deering, M.P., was killed by the first\nshot of Mr. Richard Thornhill, who was tried for murder and acquitted,\nbut found guilty of manslaughter and burnt in the hand.\n\n\nTHE THUGS.\n\nThe following account of these horribly extraordinary men is taken from\nDr. Hooker's Himalayan Journals; writing at Mirzapore, he says:--\"Here\nI had the pleasure of meeting Lieutenant Ward, one of the suppressors\nof Thuggee (_Thuggee_, in Hindostan, signifies a deceiver; fraud,\nnot open force, being employed). This gentlemen kindly showed me the\napprovers, or king's evidence of his establishment, belonging to those\nthree classes of human scourges, the Thug, Dakoit, and Poisoner. Of\nthese the first was the Thug, a mild-looking man, who had been born and\nbred to the profession: he had committed many murders, saw no harm in\nthem, and felt neither shame nor remorse. His organs of observation and\ndestructiveness were large, and the cerebellum small. He explained to\nme how the gang waylay the unwary traveller, enter into conversation\nwith him, and have him suddenly seized, when the superior throws his\nown girdle round the victim's neck and strangles him, pressing the\nknuckles against the spine. Taking off his own girdle, he passed it\nround my arm, and showed me the turn as coolly as a sailor once taught\nme the hangman's knot. The Thug is of any caste, and from any part of\nIndia. The profession have particular stations, which they generally\nselect for murder, throwing the body of their victim into a well.\n\n\"Their origin is uncertain, but supposed to be very ancient, soon\nafter the Mahommedan conquest. They now claim a divine original, and\nare supposed to have supernatural powers, and to be the emissaries\nof the divinity, like the wolf, the tiger, and the bear. It is only\nlately that they have swarmed so prodigiously--seven original gangs\nhaving migrated from Delhi to the Gangetic provinces about 200 years\nago, from whence all the rest have sprung. Many belong to the most\namiable, intelligent, and respectable classes of the lower and even\nmiddle ranks: they love their profession, regard murder as sport, and\nare never haunted with dreams, nor troubled with pangs of conscience\nduring hours of solitude, or in the last moments of life. The victim\nis an acceptable sacrifice to the goddess Davee, who by some classes\nis supposed to eat the lifeless body, and thus save her votaries the\nnecessity of concealing it.\n\n\"They are extremely superstitious, always consulting omens, such as\nthe direction in which a hare or a jackal crosses the road; and even\nfar more trivial circumstances will determine the fate of a dozen of\npeople, and perhaps of an immense treasure. All worship the pickaxe,\nwhich is symbolical of their profession, and an oath sworn on it binds\ncloser than on the Koran. The consecration of this weapon is a most\nelaborate ceremony, and takes place only under certain trees. The\nThugs rise through various grades: the lowest are scouts; the second,\nsextors; the third, are holders of the victim's hands; the highest,\nstranglers.\n\n\"Though all agree in never practising cruelty, or robbing previous\nto murder--never allowing any but infants to escape (and these are\ntrained to Thuggee), and never leaving a trace of such goods as may be\nidentified--there are several variations in their mode of conducting\noperations: some tribes spare certain castes, others none; murder of\nwoman is against all rules; but the practice crept into certain gangs,\nand this it is which led to their discountenance by the goddess Davee,\nand the consequent downfall of the system. Davee, they say, allowed the\nBritish to punish them, because a certain gang had murdered the mothers\nto obtain their daughters to be sold to prostitution.\n\n\"Major Sleeman has constructed a map demonstrating the number of\n'bails,' or regular stations for committing murder, in the kingdom of\nOude alone, which is 170 miles long by 100 broad, and in which are 274,\nwhich are regarded by the Thug with as much satisfaction and interest\nas a game preserve is in England; nor are these 'bails' less numerous\nthan in other parts of India. Of twenty assassins who were examined,\none frankly confessed to having been engaged in 931 murders, and the\nleast guilty of the number in 24. Sometimes 150 persons collected into\none gang, and their profits have often been immense, the murder of six\npersons on one occasion yielding 82,000 rupees, upwards of L8,000.\"\n\n\nENGLISH EARTHENWARE AND SHAKSPEARE'S JUG.\n\nMuch uncertainty exists regarding the period when the manufacture\nof fine earthenware was first introduced into England. Among the\ndocuments in the Foedera, occur various lists of articles, ordered to\nbe purchased in England for several foreign potentates, and permitted\nto be exported for their use without paying the Custom duties. One of\nthese lists, dated in 1428, enumerates many objects as then shipped\nfor the use of the King of Portugal and the Countess of Holland, among\nwhich are \"six silver cups, each of the weight of six marks (or four\npounds), a large quantity of woollen stuffs, and 2000 plates, dishes,\nsaucers, and other vessels of _electrum_.\"\n\nAs these articles were, no doubt, the produce of the country, it\nwould appear that utensils for domestic use were then made of metal,\nand not of pottery; and it was not till some time afterwards that\nthe latter was introduced by the Dutch, whose manufactory at Delft\nprobably existed as early as the fifteenth century, and who sent large\nquantities of their ware to England. The skill and excellence of the\nEnglish artizans consisted in the manufacture of silver and other\nmetals. Of this, instances are recorded in the correspondence of La\nMothe Fenelon, the French ambassador at the Court of Queen Elizabeth;\nand in the travels of Hentzner, who visited England in 1598. Both\ndescribe in glowing colours the silver plate which adorned the buffets,\nas well as the magnificent furniture and decorations of the palaces of\nthat sumptuous queen.\n\nStill Elizabeth, who so highly prided herself upon the state and\nsplendour of her establishment, and who was in constant intercourse\nwith the Court of France and the Low Countries, was not likely to\nhave remained altogether satisfied without possessing, among the\nmanufactures of her own kingdom, something similar to the fine Fayence\nthen in use in every foreign court. Though it is probable that Delft\nware procured from Holland was first used, it may reasonably be\npresumed that the ware called by her name was afterwards manufactured,\nunder her immediate patronage, for the use of the court and the\nnobility; and although there is no record of the fact, it is supposed\nthat Stratford-le-Bow was the site of the manufactory.\n\n[Illustration: [++] Shakspeare's Jug.]\n\nShakspeare's Jug, of which we here give an engraving, which has been\ncarefully preserved by the descendants of the immortal bard since the\nyear 1616, is, perhaps, the most remarkable example of the Elizabethan\npottery now existing. The shape partakes very much of the form of the\nold German or Dutch ewer, without, however, the usual top or cover;\nthe one now attached to the jug being a modern addition of silver,\nwith a medallion bust of the poet in the centre, beautifully executed\nand inscribed \"WM. SHAKSPEARE, AT THE AGE OF FORTY.\" It is about ten\ninches high, and sixteen inches round at the largest part, and is\ndivided lengthwise into eight compartments, having each a mythological\nsubject in high relief. All of these, although executed in the quaint\nstyle of the period, possess considerable merit. Some of them, indeed,\nmanifest much masterly grouping of both human figures and animals; and\nsuch is the admirable state of preservation of this very interesting\nold English relic, that as correct a judgment may be formed of its\nworkmanship, as in the days of its first possessor; at all events, as\nregards the degree of perfection to which English Pottery had attained\nin the Elizabethan age; an inspection of this jug will justify the\npresumption, that her Court was not less tastefully provided in that\nrespect than those of the Continent, notwithstanding the obscurity\nin which the precise locality and extent of the manufactory is\nunfortunately involved.\n\n\nPRICE OF MACKAREL.\n\nThe price of mackarel, in May, 1807, in the Billingsgate market, was\nas follows:--Forty guineas for every hundred of the first cargo, which\nmade the fish come to seven shillings apiece! The next supplies were\nalso exorbitant, though much less so than the first, fetching thirteen\npounds per hundred, or two shillings apiece. The very next year the\nformer deficiencies were more than made up, for it appears that during\nthe season 1808, mackarel were hawked about the streets of Dover, at\nsixty for a shilling, or five for a penny; whilst they so blockaded\nthe Brighton coast that on one night it became impossible to land the\nmultitudes taken, and at last both fish and nets went to the bottom\ntogether.\n\n\nPOPE'S CHAIR.\n\n[Illustration: [++] Alexander Pope's Chair.]\n\nIn one of the rooms at that stately and picturesque baronial hall,\nAudley End, the seat of Lord Braybrooke, there is preserved the\ninteresting relic which forms the subject of the annexed engraving.\nIts history is thus told on a brass plate inserted in the back--\"This\nchair, once the property of Alexander Pope, was given as a keep-sake\nto the nurse who attended him in his illness; from her descendants\nit was obtained by the Rev. Thomas Ashley, curate of the parish of\nBinfield, and kindly presented by him to Lord Braybrooke, in 1844,\nnearly a century after the poet's decease.\" It is apparently of Flemish\nworkmanship, and of rather singular design; in the centre medallion is\na figure of Venus holding a dart in her right hand, and a burning heart\nin her left. The narrow back and wide-circling arms give a peculiarly\nquaint appearance to this curious relic of one of our greatest poets.\n\n\nFIRST WIND-MILLS.\n\nMabillon mentions a diploma of the year 1105, in which a convent in\nFrance is allowed to erect water and windmills, _molendina ad ventum_.\n\nBartolomeo Verde proposed to the Venetians in 1332, to build a\nwind-mill. When his plan had been examined, he had a piece of ground\nassigned him, which he was to retain if his undertaking succeeded\nwithin a specified time. In 1373, the city of Spires caused a wind-mill\nto be erected, and sent to the Netherlands for a person acquainted\nwith the method of grinding by it. A wind-mill was also constructed at\nFrankfort, in 1442; but it does not appear to have been ascertained\nwhether there were any there before.\n\nAbout the twelfth century, in the pontificate of Gregory, when both\nwind and water-mills became more general, a dispute arose whether mills\nwere titheable or not. The dispute existed for some time between the\npersons possessed of mills and the clergy; when neither would yield. At\nlength, upon the matter being referred to the pope and sacred college,\nthe question was (as might have been expected when interested persons\nwere made the arbitrators) determined in favour of the claims of the\nchurch.\n\n\nTHE \"HAPPY DISPATCH\" IN JAPAN.\n\nThe _Hari-kari_, or \"Happy Dispatch,\" consists in ripping open their\nown bowels with two cuts in the form of a cross--after the artistic\ndissector's fashion. Officials resort to it under the fear of the\npunishment which they may expect; for it is a leading principle that\nit is more honourable to die by one's own hand than by another's.\nPrinces and the high classes receive permission to rip themselves up\nas a special favour, when under sentence of death: their entire family\nmust die with the guilty. Sometimes, by favour, the nearest relative\nof the condemned is permitted to perform the function of executioner\nin his own house. Such a death is considered less dishonourable than\nby the public executioners, aided by the servants of those who keep\ndisreputable houses.\n\nBut the Japanese, for the most part, always ask permission to rip\nthemselves; and they set about it with astonishing ease, and not\nwithout evident ostentation. The criminal who obtains this favour\nassembles all his family and his friends, puts on his richest apparel,\nmakes an eloquent speech on his situation, and then, with a most\ncontented look, he bares his belly, and in the form of a cross rips\nopen the viscera. The most odious crimes are effaced by such a death.\nThe criminal thenceforward ranks as a brave in the memory of men. His\nfamily contracts no stain, and his property is not confiscated.\n\nIt is curious that the Romans and the Japanese should hit upon\ncrucifixion as a mode of punishment. These coincidences often startle\nus in reviewing the manners and customs of men. Vainly we strive to\nconjecture how such a mode of punishment could have suggested itself\nto the mind of man. The _in terrorem_ object scarcely accounts for it.\nConstantine abolished it amongst the Romans, in honour of Him who was\npleased to make that mode of dying honourable in the estimation of men.\n\nThe Hari-kari, or happy dispatch, is still more incomprehensible.\nWe shudder at the bare idea of it. To commit suicide by hanging, by\ndrowning, by poison, by firearms, by a train in rapid motion--all these\nmodes are reasonable in their madness; but to rip open our bowels!--and\nwith _two_ cuts! We are totally at a loss to imagine how such a mode\nof self-murder could have been adopted; we cannot but wonder at the\nstrength of nerve which enables it to be accomplished: but we feel no\ndoubt of the everlasting force of national custom--especially amongst\nthe Orientals--in the continuance of this practice. Montesquieu said,\n\"If the punishments of the Orientals horrify humanity, the reason is,\nthat the despot who ordains them feels that he is above all laws. It\nis not so in Republics, wherein the laws are always mild, because he\nwho makes them is himself a subject.\" This fine sentiment, thoroughly\nFrench, is evidently contradicted by the institutions of Japan, where\nthe Emperor himself, the despot, is a subject: besides, Montesquieu\nwould have altered his antithesis had he lived to see the horrors of\nthe Reign of Terror in the glorious French Republic.\n\n\nPURITAN ZEAL.\n\nThe following is a copy of the order issued by Government for the\ndestruction of Glasgow Cathedral:--\"To our traist friendis,--Traist\nfriendis, after most hearty commendacion, we pray you fail not to pass\nincontinent to the kirk, (of Glasgow, or elsewhere, as it might be) and\ntak down the hail images thereof, and bring furth to the kirk-zyard,\nand burn them openly. And sicklyke cast down the altaris, and purge the\nkirk of all kynd of monuments of idolatrye. And this ze fail not to do,\nas ze will do us singular emplesure; and so commitis you to protection\nof God.\n\n (Signed)\n\n AR. ARGYLE.\n\n JAMES STEWART.\n\n RUTHVEN.\n\n_From Edinburgh the XII. of Aug. 1560._\n\n\"Fail not, but ze tak guid heyd that neither the dasks, windows,\nnor duris, be ony ways hurt or broken, uthe glassin wark, or iron\nwark.\"\n\n\nFREDERICK THE GREAT AT TABLE.\n\nThe table of the great Frederic of Prussia was regulated by himself.\nThere were always from nine to a dozen dishes, and these were brought\nin one at a time. The King carved the solitary dish, and helped the\ncompany. One singular circumstance connected with this table was,\nthat each dish was cooked by a different cook, who had a kitchen to\nhimself! There was much consequent expense, with little magnificence.\nFrederic ate and drank, too, like a boon companion. His last work,\nbefore retiring to bed, was to receive from the chief cook the bill\nof fare for the next day; the price of each dish, and of its separate\ningredients, was marked in the margin. The monarch looked it cautiously\nthrough, generally made out an improved edition, cursed all cooks\nas common thieves, and then flung down the money for the next day's\nexpenses.\n\n\nARTIFICIAL SWEETS.\n\nProfessor Playfair, in an able lecture delivered in the Great\nExhibition, and since published, has raised a curtain, which\ndisplays a rather repulsive scene. He says, the perfume of flowers\nfrequently consists of oils and ethers, which the chemist can compound\nartificially in his laboratory. Singularly enough these are generally\nderived from substances of an intensely disgusting odour. A peculiarly\nfetid oil, termed the \"fusel\" oil, is formed in making brandy. This\nfusel oil distilled with sulphuric acid and acetate of potass, gives\nthe oil of pears (?). The oil of apples is made from the same fusel,\nby distillation with the same acid and chromate of potass. The oil\nof pineapples is obtained from the product of the action of putrid\ncheese on sugar! or by making a soap with butter. The artificial\noil of bitter almonds is now largely employed in perfuming soap\nconfectionary; extracted by nitric acid and the fetid oil of gas tar.\nMany a fair forehead is damped with _eau de mille fleurs_ without the\nknowledge that its essential ingredient is derived from the drainage of\ncow-houses!\n\n\nTEUTONIC HUT-SHAPED VASES.\n\n[Illustration: [++] Teutonic Hut-Shaped Vases.]\n\nSome remarkable sepulchral urns, of which we give a sketch, resembling\nthose of the early inhabitants of Alba Longa, in Italy, have been\nfound in Germany, and are distinctly Teutonic. They occur in the\nsepulchres of the period when bronze weapons were used, and before the\npredominance of Roman art. One found at Mount Chemnitz, in Thuringen,\nhad a cylindrical body and conical top, imitating a roof. In this was\na square orifice, representing the door or window, by which the ashes\nof the dead were introduced, and the whole then secured by a small\ndoor fastened with a metal pin. A second vase was found at Roenne; a\nthird in the island of Bornholm. A similar urn exhumed at Parchim had a\nshorter body, taller roof, and door at the side. Still more remarkable\nwas another found at Aschersleben, which has its cover modelled in\nshape of a tall conical thatched roof, and the door with its ring still\nremaining. Another, with a taller body and flatter roof, with a door\nat the side, was found at Klus, near Halberstadt. The larger vases\nwere used to hold the ashes of the dead, and are sometimes protected\nby a cover, or stone, or placed in another vase of coarser fabric. The\nothers are the household vessels, which were offered to the dead filled\nwith different viands. Some of the smaller vases appear to have been\ntoys.\n\nExtraordinary popular superstitions have prevailed amongst the German\npeasantry as to the origin and nature of these vases, which in some\ndistricts are considered to be the work of the elves,--in others, to\ngrow spontaneously from the ground like mushrooms--or to be endued\nwith remarkable properties for the preservation of milk and other\narticles of food. Weights to sink nets, balls, discs, and little rods\nof terra-cotta, are also found in the graves.\n\n\nLYNCH'S CASTLE, GALWAY.\n\n[Illustration: [++] Lynch's Castle.]\n\nThe house in the town of Galway, still known as \"Lynch's Castle,\"\nalthough the most perfect example now remaining, was at one period by\nno means a solitary instance of the decorated habitations of the Galway\nmerchants. The name of Lynch, as either provost, portreve, sovereign,\nor mayor of Galway, occurs no fewer than ninety-four times between the\nyears 1274 and 1654; after that year it does not appear once. The house\nhere pictured was the residence of the family for many generations.\nIt had, however, several branches, whose habitations are frequently\npointed out by their armorial bearings, or their crest, a lynx, over\nthe gateway. One of its members is famous in history as the Irish\nJunius Brutus. The mere fact is sufficiently wonderful without the aid\nof invention; but it has, as may be supposed, supplied materials to a\nhost of romancers. The story is briefly this:--\n\nJames Lynch Fitzstephen was mayor or warden of Galway in 1493; he\ntraded largely with Spain, and sent his son on a voyage thither to\npurchase and bring back a cargo of wine. Young Lynch, however, spent\nthe money entrusted to him, and obtained credit from the Spaniard,\nwhose nephew accompanied the youth back to Ireland to be paid the debt\nand establish further intercourse. The ship proceeded on her homeward\nvoyage, and as she drew near the Irish shore, young Lynch conceived the\nidea of concealing his crime by committing another. Having seduced,\nor frightened, the crew into becoming participators, the youth was\nseized and thrown overboard. The father and friends of Lynch received\nthe voyager with joy; and the murderer in a short time became himself\na prosperous merchant. Security had lulled every sense of danger,\nand he proposed for a very beautiful girl, the daughter of a wealthy\nneighbour, in marriage. The proposal was accepted; but previous to the\nappointed day, one of the seamen became suddenly ill, and in a fit of\nremorse summoned old Lynch to the dying-bed, and communicated to him a\nfull relation of the villany of his only and beloved son. Young Lynch\nwas tried, found guilty, and sentenced to execution--the father being\nhis judge. The wretched prisoner, however, had many friends among the\npeople, and his relatives resolved with them that he should not die a\nshameful death. They determined upon his rescue. We copy the last act\nof the tragedy from \"Hardiman's History of Galway.\" \"Day had scarcely\nbroken when the signal of preparation was heard among the guards\nwithout. The father rose, and assisted the executioner to remove the\nfetters which bound his unfortunate son. Then unlocking the door, he\nplaced him between the priest and himself, leaning upon an arm of each.\nIn this manner they ascended a flight of steps lined with soldiers,\nand were passing on to gain the street, when a new trial assailed\nthe magistrate for which he appears not to have been unprepared. His\nwretched wife, whose name was Blake, failing in her personal exertions\nto save the life of her son, had gone in distraction to the heads of\nher own family, and prevailed on them, for the honour of their house,\nto rescue him from ignominy. They flew to arms, and a prodigious\nconcourse soon assembled to support them, whose outcries for mercy to\nthe culprit would have shaken any nerves less firm than those of the\nmayor of Galway. He exhorted them to yield submission to the laws of\ntheir country; but finding all his efforts fruitless to accomplish the\nends of justice at the accustomed place, and by the usual hands, he, by\na desperate victory over parental feeling, resolved himself to perform\nthe sacrifice which he had vowed to pay on its altar. Still retaining\na hold of his unfortunate son, he mounted with him by a winding stair\nwithin the building, that led to an arched window overlooking the\nstreet, which he saw filled with the populace. Here he secured the end\nof the rope--which had been previously fixed round the neck of his\nson--to an iron staple, which projected from the wall, and after taking\nfrom him a last embrace, he launched him into eternity. The intrepid\nmagistrate expected instant death from the fury of the populace; but\nthe people seemed so much overawed or confounded by the magnanimous\nact, that they retired slowly and peaceably to their several dwellings.\nThe innocent cause of this sad tragedy is said to have died soon after\nof grief, and the unhappy father of Walter Lynch to have secluded\nhimself during the remainder of his life from all society except that\nof his mourning family. His house still exists in Lombard Street,\nGalway, which is yet known by the name of 'Dead Man's Lane;' and over\nthe front doorway are to be seen a skull and cross-bones executed in\nblack marble, with the motto, 'Remember Deathe--vaniti of vaniti, and\nall is but vaniti.'\"\n\nThe house in which the tragedy is said to have occurred was taken down\nonly so recently as 1849; but the tablet which contains the \"skull\nand cross-bones\" bears the date 1624--upwards of a century after the\nalleged date of the occurrence.\n\n\nWASHINGTON.\n\nIt is something singular, that Washington drew his _last_ breath, in\nthe _last_ hour, of the _last_ day, of the _last_ week, of the _last_\nmonth, of the _last_ year, of the _last_ century. He died on Saturday\nnight, twelve o'clock, December 31st, 1799.\n\n\nANCIENT BANNERS AND STANDARDS.\n\nBanners have been in use from the earliest ages. Xenophon gives us the\nPersian standard as a golden eagle, mounted on a pole or spear. We\nfind banners very early in use among the nations of Europe. In this\ncountry the introduction of banners was clearly of a religious origin.\nVenerable Bede says, that when St. Augustin and his companions came to\npreach Christianity in Britain in the latter part of the sixth century,\nand having converted Ethelbert, the Bretwalda of the Anglo-Saxons (his\nQueen Bertha had already embraced the Christian faith) the monk and\nhis followers entered Canterbury in procession, chanting, \"We beseech\nthee O Lord, of thy mercy, let thy wrath and anger be turned away from\nthis city, and from thy Holy Place, for we have sinned. Hallelujah.\"\nAnd they carried in their hands little banners on which were depicted\ncrosses. The missionaries were allowed to settle in the Isle of Thanet,\nand Canterbury became the first Christian church.\n\nThe raven has been regarded from very early ages as an emblem of God's\nprovidence, no doubt from the record in Holy Writ of its being employed\nto feed Elijah the Prophet, in his seclusion by the brook Cherith;\nand it was the well-known ensign of the Danes, at the time of their\ndominion in this country. In the year 742, a great battle was fought\nat Burford, in Oxfordshire, and the Golden Dragon, the standard of\nWessex, was victorious over Ethelbald, the King of Mercia. The banners\nof several of the Saxon kings were held in great veneration, especially\nthose of Edmund the Martyr, and of Edward the Confessor. The latter\nking displayed as an ensign a cross flory between five martlets gold,\non a blue field, and which may still be seen on a very ancient shield\nin the south aisle of Westminster Abbey. When William the Norman set\nout to invade England he had his own ensign, the two lions of Normandy,\ndepicted on the sails of his ships; but on the vessel in which he\nhimself sailed, besides some choice relics, he had a banner at the\nmast-head with a cross upon it, consecrated by the Pope, to give\nsanctity to the expedition. Indeed it has been the practice in every\nage for the Pope to give consecrated banners wherever he wished success\nto any enterprise, numerous instances of which might be cited in very\nrecent times. And in our own army down to the present day, whenever any\nregiment receives new banners (or colours, as the modern term is), the\nregiment is drawn out in parade, the colours are then blessed by the\nprayers of several clergymen of the Church of England, and afterwards\npresented to the regiment by the fair hand of a lady of rank.\n\n[Illustration: [++] Banner.]\n\nCaesar has recorded a fine example of patriotism, to the credit of one\nof his own officers, when he attempted to land his Roman forces on our\nshores, and meeting with a warmer reception than they anticipated from\nthe Britons, considerable hesitation arose among his troops; but the\nstandard-bearer of the Tenth Legion, with the Roman eagle in his hand,\ninvoking the gods, plunged into the waves, and called on his comrades\nto follow him, and do their duty to their general and to the republic;\nand so the whole army made good their landing.\n\n[Illustration: [++] Caesar's Banner.]\n\nWe have in the Nineveh sculptures some highly interesting specimens\nof the ancient Assyrian standards, consisting principally of two\nvarieties, which are here given. The principal archer appears to be\ndrawing his bow, while the standard-bearer elevates the standard in\nfront of the chariot.\n\n\nANCIENT MANNERS OF THE ITALIANS.\n\nAbout the year 1238, the food of the Italians was very moderate, or,\nrather scanty. The common people had meat only three days a week.\nTheir dinner consisted of pot-herbs, boiled with meat; their supper,\nthe cold meat left from dinner. The husband and wife eat out of the\nsame dish; and they had but one or two cups in the house. They had\nno candles made of wax or tallow; but, a torch, held by one of the\nchildren, or a servant, gave them light at supper. The men, whose chief\npride was in their arms and horses, wore caps made with iron scales,\nand cloaks of leather, without any other covering. The women wore\njackets of stuff, with gowns of linen, and their head-dresses were very\nsimple. Those who possessed a very small sum of money, were thought\nrich; and the homely dress of the women required only small marriage\nportions. The nobles were proud of living in towers; and thence the\ncities were filled with those fortified dwellings.\n\n\nAMUSEMENTS OF THE LOWER ORDERS IN THE FIFTEENTH CENTURY.\n\nThe most popular amusements of the lower orders were wrestling,\nbowling, quoit and ninepin playing, and games at ball. In wrestling the\nCornwall and Devonshire men excelled, and a ram, or sometimes a cock,\nwas the prize of the victor. Bowling alleys were commonly attached\nto the houses of the wealthy, and to places of public resort. Among\nthe games at ball we find tennis, trap-ball, bat and ball, and the\nballoon-ball, in which a large ball filled with air was struck from one\nside to the other by two players with their hands and wrists guarded by\nbandages. Archery was now on the decline, owing to the introduction of\nfirearms; nor could all the legislative enactments of the day revive\nits constant use. The quarter-staff was also a favourite weapon of\nsportive fence, which was a staff about five or six feet long, grasped\nin the middle with one hand, while the other slid up and down as it was\nrequired to strike or to ward a blow.\n\nThe citizens of London enjoyed themselves in winter by skating on\nthe Thames, (the old shankbones of sheep having now been superseded\nby regular skates, probably introduced from the Netherlands,) and\nin summer with sailing and rowing. Dice and cards, prisoner's base,\nblind man's buff, battledoor and shuttlecock, bull-baiting, and\ncock-fighting, a rude species of mumming, the dancing of fools at\nChristmas, and other games, completed the gratifications of the\npopulace.\n\n\nNOVEL MODE OF TAKING VENGEANCE.\n\nThe Chinese have a book entitled _Si-yuen_, that is to say, \"The\nWashing of the Pit,\" a work on medical jurisprudence, very celebrated\nall over the empire, and which should be in the hands of all Chinese\nmagistrates. It is impossible to read the Si-yuen without being\nconvinced that the number of attempts against life in this country\nis very considerable, and especially that suicide is very common.\nThe extreme readiness with which the Chinese are induced to kill\nthemselves, is almost inconceivable; some mere trifle, a word almost,\nis sufficient to cause them to hang themselves, or throw themselves\nto the bottom of a well; the two favourite modes of suicide. In other\ncountries, if a man wishes to wreak his vengeance on an enemy, he tries\nto kill him; in China, on the contrary, he kills himself. This anomaly\ndepends upon various causes, of which these are the principal:--In\nthe first place, Chinese law throws the responsibility of a suicide\non those who may be supposed to be the cause or occasion of it. It\nfollows, therefore, that if you wish to be revenged on an enemy, you\nhave only to kill yourself to be sure of getting him into horrible\ntrouble; for he falls immediately into the hands of _justice_, and will\ncertainly be tortured and ruined, if not deprived of life. The family\nof the suicide also usually obtains, in these cases, considerable\ndamages; so that it is by no means a rare case for an unfortunate man\nto commit suicide in the house of a rich one, from a morbid idea of\nfamily affection. In killing his enemy, on the contrary, the murderer\nexposes his own relatives and friends to injury, disgraces them,\nreduces them to poverty, and deprives himself of funeral honours,\na great point for a Chinese, and concerning which he is extremely\nanxious. It is to be remarked also, that public opinion, so far from\ndisapproving of suicide, honours and glorifies it. The conduct of a man\nwho destroys his own life, to avenge himself on an enemy whom he has no\nother way of reaching, is regarded as heroic and magnanimous.\n\n\nPERSECUTION IN THE REIGN OF QUEEN MARY.\n\nThe total number of persons who perished in the flames for their\nreligion during this reign has been variously reckoned at 277 and 288,\namongst whom were 5 bishops, 21 divines, 8 gentlemen, 84 artificers,\n100 husbandmen, servants, and labourers, 26 wives, 20 widows, 9\nunmarried women, 2 boys, and 2 infants, of which last one was whipped\nto death by the savage Bonner, and the other, springing out of its\nmother's womb, at the stake, was mercilessly thrown back into the\nfire. The number of those that died in prison was also very great. Yet\nEngland may be considered as comparatively free from persecution during\nthis period, for all over the continent the victims of bigotry were\nreckoned, not by hundreds, but by thousands, and in the Netherlands\nalone 50,000 persons are said to have lost their lives in the religious\nwars of the Spaniards.\n\n\nWAYSIDE MONUMENTS.\n\nThe sketch on next page represents a curious custom which still\nprevails in the neighbourhood of Cong, near Oughterard in Ireland.\nIt is well described in the following account of their tour by Mr.\nand Mrs. S. C. Hall:--\"On the way to Joyce's Country we saw heaps of\npiled-up stones on either side of the road; these heaps continuing\nfor above a mile, after their commencement a short distance from the\nwestern entrance to the town. The artist may convey a better notion\nof their peculiar character than any written description can do. We\nleft our car to examine them minutely; and learned they were monuments\nto the memory of \"deceased\" persons, \"erected\" by their surviving\nfriends. Upon death occurring, the primitive tumulus is built,--if\nthat may be called building which consists in placing a few large\nstones upon a spot previously unoccupied. Each relative of the dead\nadds to the heap; and in time it becomes a \"mountain\" of tolerable\nsize. Each family knows its own particular monument; and a member of,\nor a descendant from it, prays and leaves his offering only at that\nespecial one. The custom has endured for many generations: some of the\nheaps bore tokens of great age; and one was pointed out to us of which\nthere were records, in the transferred memories of the people, for at\nleast 500 years. The bodies are in no instance buried here--it is not\nconsecrated earth; the monuments are merely memorials, and no doubt\noriginated at a period when a Roman Catholic was, according to the\nprovisions of a law equally foolish and cruel, interred, without form\nor ceremony, in church ground--the ground that had been the property\nof their ancestors. None of these stone cairns have any masonwork, and\nthey are generally of the rudest forms, or rather without any form,\nthe stones having been carelessly cast one upon another. Upon one of\nthem only could we discover any inscription--this one is introduced\ninto the print; it is built with far more than the usual care; it\ncontained an inscription; \"Pray for y{e} soule of John Joyce, & Mary\nJoyce, his wife, died 1712;\" some of them, however, seem to have been\nconstructed with greater care than others, and many of them were topped\nwith a small wooden cross. We estimated that there were at least 500\nof these primitive monuments--of all shapes and sizes--along the road.\nIn each of them we observed a small hollow, which the peasants call\na \"window;\" most of these were full of pebbles, and upon inquiry we\nlearned that when one of the race to whom the deceased belonged kneels\nby the side of this record to his memory and offers up a prayer for\nthe repose of his soul, it is customary to fling a little stone into\nthis \"cupboard;\" the belief being that gradually as it fills, so,\ngradually, the soul is relieved from punishment in purgatory; when\ncompletely full the soul has entered paradise. We have prolonged our\ndescription of this singular and interesting scene, because it seems to\nhave been altogether overlooked by travellers, and because we believe\nthat nothing like it is to be met with in any other part of Ireland;\nalthough similar objects are to be found in several other places about\nConnemara, none of them, however, are so extensive as this which\nadjoins Cong.\"\n\n[Illustration: [++] Primitive Monuments.]\n\n\nHINDOO ADORATION OF THE SALAGRAM.\n\n[Illustration: [++] Hindoo Adoration of the Salagram.]\n\nAmong the many forms which Vishnu is believed by his Hindoo worshippers\nto have assumed is that of the Salagram--an ammonite-stone, found in\nthe river Gandaka and other streams flowing from the Himalayas. The\nreason for the worship of this is stated in one of the sacred books.\n\"Vishnu created the nine planets to preside over the fates of men.\nSani (Saturn) proposed commencing his reign by taking Brahma under his\ninfluence for twelve years. The matter was referred to Vishnu, who\nbeing equally averse to be placed under the inauspicious influence of\nthis planet, requested him to call the next day. The next day Saturn\ncould nowhere discover Vishnu, but perceived that he had united himself\nto the mountain Gandaka; he entered the mountain in the form of a\nworm called Vajrakita (the thunderbolt worm). He continued to afflict\nthe mountain-formed Vishnu for twelve years, when Vishnu assumed his\nproper shape, and commanded that the stones of this mountain should be\nworshipped, and become proper representatives of himself; adding that\neach should have twenty marks in it, similar to those on his body, and\nthat its name should be Salagram.\"\n\nThe Salagram is usually placed under a tulasi-tree, which is planted\non the top of a pillar in the vicinity of a temple of Vishnu, or near\na house. Tulasi, a female, desired to become Vishnu's wife, but was\nmetamorphosed by Lakshmi into a tree, a small shrub, called therefore\n_Tulasi_, or holy basil (Ocymum Sanctum). Vishnu, however, promised\nto assume the form of a Salagram, and always continue with her. The\nVaishnaya priests, therefore, keep one leaf of the shrub under and\nanother over the Salagram, and thus pay their adorations to the stone\nand the tree. In the evening a lamp is placed near it. In the month\nof May it is watered from a pot suspended over it, as appears in the\nengraving, which represents a person engaged in the worship at this\nsingular shrine.\n\n\nTOMB OF THE EMPEROR MAXIMILIAN AT INSPRUCK.\n\nThis majestic tomb is placed in the centre of the middle aisle of the\nchurch, upon a platform approached by steps of red marble. The sides\nof the tomb are divided into twenty-four compartments, of the finest\nCarrara marble, on which are represented, in bas-relief, the most\ninteresting events of the emperor's warlike and prosperous career. The\nworkmanship of the tablets is exquisite; and, taken in connexion with\nthe lofty deeds they record, they form the most princely decorations\never seen. Each of the tablets contributing to this splendid\nlithobiography is in size 2 feet 4 inches by 1 foot 8 inches; and every\nobject contained therein is in the most perfect proportion, while the\nexquisite finish of the heads and draperies requires a magnifying glass\nto do it justice. The tomb is surmounted by a colossal figure in bronze\nof the emperor, kneeling in the act of prayer; and around it are four\nallegorical figures, of smaller size, also in bronze.\n\nBut, marvellous as is the elaborate beauty of this work, it is far from\nbeing the most remarkable feature of this imperial mausoleum. Ranged in\ntwo long lines, as if to guard it, stand twenty-eight colossal statues\nin bronze, of whom twenty are kings and princes, alliances of the house\nof Hapsburg, and eight their stately dames. Anything more impressive\nthan the appearance of these tall dark guardians of the tomb, some clad\nin regal robes, some cased in armour, and all seeming animated by the\nmighty power of the artist, it would be difficult to imagine.\n\nIn the death-like stillness of the church, the visitor who, for the\nfirst time, contemplates this tomb and its gloomy guard, is struck by\na feeling of awe, approaching to terror. The statues, with life-like\nindividuality of attitude and expression--each solemn, mournful,\ndignified, and graceful; and all seeming to dilate before the eye into\nenormous dimensions, and, as if framed to scare intruders, endowed by\na power more than mortal, to keep watch and ward round the mighty dead.\nThey appear like an eternal procession of mourners, who, while earth\nendures, will cease not to gaze on, mourn over, and protect the relics\nof him who was the glory of their noble, long since fallen race.\n\n\nTHE FAYENCE OF HENRY II. OF FRANCE.\n\nThe earliest known fabric of this earthenware is that mysterious and\nunique manufacture of the \"Renaissance,\" the fine Fayence of Henry\nII. The manufacture of this ware, which was at once carried to a high\ndegree of perfection, seems to have been suddenly and unaccountably\nlost, without leaving any record of where or by whom it was produced.\nBy many it is supposed to be of Florentine manufacture, and to have\nbeen sent by some of the relations of Catherine de Medicis as a present\nto Henry II.; but it differs too essentially from Italian Majolica,\nboth in the paste of which it is composed, and in the style in which\nit is decorated, to warrant such a conjecture. Italy does not possess\nin her museums a single specimen of this ware, and of the thirty-seven\npieces extant, twenty-seven have been traced as coming from Touraine\nand La Vendee. Many antiquaries, therefore, infer that the manufacture\nwas at Thouars, in Touraine, although the Fayence may have been the\nwork of an Italian artist.\n\nBut if the place of its manufacture is unknown, the pieces extant\nclearly attest the period of its fabrication. The Salamander, and\nother insignia of Francis I., are met with on the earlier specimens of\nthis pottery; but upon the majority of pieces, upon those more pure\nin design and more beautiful in execution than the preceding, we find\nthe arms of Henry II., with his device, the three crescents, or his\ninitial H, interlaced with the two D's of the Duchesse de Valentinois.\nIndeed, so constantly do her emblems appear upon the pieces, that the\nware, though usually designated as \"Faience de Henri II.,\" is sometimes\nstyled \"Faience de Diane de Poitiers.\" Even her widow's colours, black\nand white, are the two which are employed in some of the finest pieces.\nThey were the fashionable colours of the court, Henry wore no others\nduring his life, and was attired in them in the fatal tournament in\nwhich he fell. Her _impresa_, the crescent of Diana, is conspicuous on\nhis palaces, and he even caused it to be engraved upon his coins. From\nthese circumstances we must, therefore, conclude that the manufacture\nof this ware began at the end of the reign of Francis I., was continued\nunder that of Henry II., and, as we find upon it the emblems of these\ntwo princes only, we may naturally infer that it is of French origin.\n\nThe paste of which this Fayence is composed is equally distinct from\nMajolica and Palissy ware. The two latter are both soft, whereas this,\non the contrary, is hard. It is a true pipeclay, very fine, and very\nwhite, so as not to require, like the Italian Fayence, to be concealed\nby a thick enamel, and the ornaments with which it is enriched are\nsimply covered with a thin, transparent, yellowish varnish.\n\nThe style of decoration in this ware is unique. Patterns or arabesques,\nare engraved on the paste, and the indentures filled with \npastes, so as to present an uniform, smooth surface, of the finest\ninlaying, or resembling, rather, a model of Cellini's silver work,\nchiselled and worked in niello. Hence it is sometimes styled \"Faience\na niellure.\" These patterns are sometimes disposed in zones of yellow\nochre, with borders of dark brown, sometimes of a pink, green, violet,\nblack, or blue; but the dark yellow ochre is the predominant colour.\n\n[Illustration: [++] Fayence Candlestick.]\n\nThe collection of the late M. Preaux was the richest in the world in\nthe most beautiful examples of Fayence; it was disposed of by auction\nabout twelve years ago, in consequence of the death of the proprietor,\nand the choicest specimen in it was the candlestick, of which we give\na figure, and which was purchased by Sir Anthony de Rothschild for\nabout L220, duty included. The surface is exquisitely enriched with\narabesque patterns, either in black upon a white ground, or in white\nupon a black. The form is monumental, and in the finest style; three\nfigures of genii support escutcheons, bearing the arms of France,\nand the double D. These genii stand upon masks, which are united by\ngarlands enamelled in green. The top of the candlestick terminates\nin the form of a vase, and bears inscribed the fleurs-de-lys and the\nmonogram of our Saviour. This piece, for delicacy of detail and beauty\nof execution, is unequalled by any specimen known of this exquisite\nFayence. Sir Anthony de Rothschild also purchased at M. Preaux's sale a\nsmall cup, decorated in the same style, with the descents interlaced,\nfor which he gave 1300 francs. He, therefore, now is fortunate in\nhaving the finest collection known of this ware, as, in addition to\nthe specimens already mentioned, he possesses two exquisite ewers of\nthe Henry II. Fayence. One he purchased at the sale of the Comte de\nMonville for 2300 francs; the other, with a curious handle of elaborate\nworkmanship, he bought for nineteen guineas at Strawberry Hill, where\nhe also purchased a tripod salt-cellar, supported with scroll ornaments\nfor L21.\n\n\nREFRESHMENTS FOR THE PULPIT.\n\nIn the books of Darlington parish church, the following items appear,\nshowing that, in the olden time, provision was made for comforting the\ninner man: \"Six quarts of sack to the minister who preached when he had\nno minister to assist, 9s. For a quart of sack bestowed on Jillett,\nwhen he preached, 2s. 6d. For a pint of brandy when Mr. George Bill\npreached here, 1s. 4d. For a stranger who preached, a dozen of ale.\nWhen the Dean of Durham preached here, spent in a treat in the house,\n3s. 6d.\" This would hardly be considered orthodox at the present day.\n\n\nBEDESMEN IN THE TIME OF HENRY VII.\n\n[Illustration: [++] Bedesmen.]\n\nMost of the monasteries in former times had hospitals of poor men and\nwomen attached to them; generally either within the precincts or near\nadjoining. Thus, at St. Edmund's Bury, there was St. John's Hospital,\nor God's House, without the South Gate, and St. Nicholas' Hospital\nwithout the East Gate, and St. Peter's Hospital without the Risby Gate,\nand St. Saviour's Hospital without the North Gate,--all founded by\nabbots of St. Edmund's. At Reading there was the Hospital of St. Mary\nMagdalene for twelve leprous persons and chaplains, and the Hospital\nof St. Lawrence for twenty-six poor people, and for the entertainment\nof strangers and pilgrims, both founded by abbots of Reading. One at\nthe gate of Fountains' Abbey for poor persons and travellers; one\nat Glastonbury, under the care of the almoner, for poor and infirm\npersons. Thirteen was a favourite number for the inmates of a hospital.\nFrom the initial letter of a deed in the British Museum (Harl. 1498),\nby which King Henry VII. founded a fraternity of thirteen poor men in\nWestminster Abbey, who were to be under the governance of the monks,\nwe take the accompanying illustration, which represents the abbot and\nmonks before the king, with a group of the king's bedesmen, each of\nwhom has the royal badge, a rose surmounted by a crown, on the shoulder\nof his habit.\n\n\nCHINESE GAMBLERS PLAYING FOR FINGERS.\n\nThe following strange account is taken from Hue's \"Chinese Empire:--\n\n\"The Chinese are industrious and economical, but their cupidity, their\nimmoderate love of lucre, and their decided taste for stockjobbing\nand speculation, easily tempts them to gambling, when they are not\nengaged in traffic. They seek eagerly for strong excitements, and when\nonce they have got into the habit of gambling they seldom or never\nrecover from it. They cast aside every obligation of station, duty,\nand family, to live only for cards and dice; and this fatal passion\ngains such an empire over them, that they proceed even to the most\nrevolting extremities. When they have lost all their money they will\nplay for their houses, their land, and their wives even, whose destiny\noften depends on a cast of the dice. Nay, the Chinese gambler does not\nstop here, for he will stake the very clothes he has on for one game\nmore, and this horrible custom gives rise to scenes that would not be\ncredible, did we not know that the passions always tend to make men\ncruel and inhuman.\n\n\"In the northern provinces, especially in the environs of the Great\nWall, you may sometimes meet, during the most intense cold of winter,\nmen running about in a state of complete nudity, having been driven\npitilessly from the gaming-houses when they had lost their all. They\nrush about in all directions like madmen to try and save themselves\nfrom being frozen, or crouch down against the chimneys, which in those\ncountries are carried along the walls of the houses, on a level with\nthe ground. They turn first one side towards the warmth, then the\nother, while their gambling companions, far from trying to help them,\nlook on with ferocious and malignant hilarity. The horrible spectacle\nseldom lasts long, for the cold soon seizes the unfortunate creatures,\nand they fall down and die. The gamblers then return to their table,\nand begin to play again with the most perfect composure. Such facts\nas these will appear fabulous to many persons, but having resided\nseveral years in the north of China, we can testify to their perfect\nauthenticity.\n\n\"These excesses seem surprising enough, but the truth is, that Chinese\ngamblers have invented still more extraordinary methods of satisfying\ntheir passion, which is really carried to absolute madness. Those who\nhave nothing more to lose will collect round a table and actually\nplay for _their fingers_, which they will cut off reciprocally with\nfrightful stoicism. We had thought to pass over these revolting\nparticulars, for we do not like to put the confidence of our readers\nto too great a trial. We have a strong objection to relating things\nthat, although we know them to be strictly true, have an improbable\nappearance. But these facts concerning Chinese gamblers were known, and\ncommented upon, by the Arab travellers in the ninth century. Here is a\npassage on the subject from the 'Chain of Chronicles,' from which we\nhave already quoted more than once:--\n\n\"'Amongst men of a volatile and boastful character, those who belong\nto the lower classes, and who have no money, will sometimes play for\nthe fingers of their hands. During the game, they keep by them a vase\ncontaining nut, or sesame oil, for olive oil is not known in this\ncountry. A fire is kept burning under it, and between the two players\nis placed a small but very sharp hatchet. The one who wins then takes\nthe hand of the loser, places it on a stone and cuts off one of his\nfingers with the hatchet; the piece falls, and the vanquished party\nimmediately dips his hand into the hot oil, which cauterises the wound.\nThis operation does not prevent the players from beginning again. Some\nwill take a match, dip it in oil, place it on their arms, and set fire\nto it; the match burns, and you can smell the odour of the consuming\nflesh, but the man goes on with his game, and exhibits no sign of\npain.'\"\n\n\nENTRY OF THE FRENCH AMBASSADOR INTO LONDON, IN 1698.\n\nThe following is an extract from the \"Flying Post,\" of May 17, 1698:--\n\n\"Yesterday, (Monday, May 16,) in the afternoon, Count Tallard, the\nFrench Ambassador, made his public entry. The Earl Marshal's men came\nfirst, then followed the Earl of Macclesfield's footmen, after them\ntwenty of the Ambassador's footmen, in red liveries with gold lace;\nthen came two of the Ambassador's gentlemen and six pages on horseback;\nnext came two heralds before His Majesty's coach, in which His\nExcellency the Ambassador, the Earl of Macclesfield, and some others\nof quality: after them came three of His Royal Highness the Prince of\nDenmark's coaches, and next, three of the Ambassador's coaches, the\nfirst of them very rich, and drawn by eight horses; then followed His\nGrace the Duke of Norfolk's coach, with about forty-seven more, drawn\nby six horses each. There was a splendid entertainment prepared for His\nExcellency at Ossulston House, in St. James's Square.\"\n\n\nEXPENSES AT CORONATIONS.\n\nThe quantity of provisions consumed at the feasts given by some of our\nearly Kings, was extraordinarily great. For that of King Edward I.\nFebruary 10th, 1274, the different Sheriffs were ordered to furnish\nbutcher meat at Windsor, in the following proportions:--\n\n Oxen. Swine. Sheep. Fowls.\n Sheriff of Gloucester, 60 101 60 3,000\n \" Bucks and Bedford 40 66 40 2,100\n \" Oxford 40 67 40 2,100\n \" Kent 40 67 40 2,100\n \" Surrey and Sussex 40 67 40 2,100\n \" Warwick and Leicester 60 98 40 3,000\n \" Somerset and Dorset 100 176 110 5,000\n \" Essex 60 101 60 3,160\n ---- ---- ---- ------\n Total, twelve counties 440 743 430 22,560\n\nIn the year 1307, King Edward II. issued an order to the seneschal\nof Gascony, and constable of Bordeaux, to provide a thousand pipes\nof good wine, and send them to London, to be used at the approaching\ncoronation. The purchase and freight were to be paid by a company\nof Florentine merchants, who farmed the revenues of Gascony. The\ncoronation oath was first taken by Ethelred II., A.D. 979; that now\nused in 1377. It was amended in 1689. The first coronation sermon was\npreached in 1041. The following statement of the prices given for\nseats, to obtain a view of passing objects during the coronations of\nformer times, may, perhaps, prove interesting:--\n\nThe price of a good place at the coronation of William the Conqueror,\nwas a _blank_; at that of his son, William Rufus, the same. At Henry\nI's., it was a _crocard_; at Stephen's and Henry II.'s, it was a\n_pollard_. At Richard's, and King John's, it was a _fuskin_. It rose\nat the 3d. Henry's, to a _dodkin_. In the reign of Edward, the coin\nbegins to be more intelligible; and we find that, for a seat, to view\nhis coronation, a _Q_ was given, or the half of a ferling, or farthing,\nthe fourth part of a sterling, or penny. At the 2d. Edward's, it was a\nfarthing; and at his son's, Edward III. a halfpenny. At Richard II.'s\nit was a penny, and continued the same to that of Henry IV. inclusive.\nAt the 5th Henry's, it was _two pennies_; and similar prices were\npaid at the coronations of Henry VI., Edward IV., Edward V., Richard\nIII., and Henry VII. At that of Henry VIII. it was a _grossus_, or\ngroat; and the same was paid at that of Edward VI. and Queen Mary's.\nAt Queen Elizabeth's, it rose to _a testoon_, or _tester_. At those of\nJames I. and Charles I. _a shilling_ was given; which was advanced to\n_half-a-crown_, at those of Charles II. and James II. At King William's\nand Queen Anne's it was _a crown_, and the same at that of George I.\nAt George II.'s _half-a-guinea_, and, afterwards, at George III.'s _a\nguinea_ was the common charge. But, at that of George IV, as high as\n_forty guineas_ were given for a single seat.\n\n\nCURIOUS ANTIQUE SWORD.\n\nThe engraving which accompanies this article is a sketch of the upper\npart of an antique Danish sword, which was found, together with several\nother weapons, by the labourers who were engaged in the construction of\nthe railway from Dublin to Cashel.\n\n[Illustration: [++] Antique Danish Sword.]\n\nThe discovery of the weapons was made at a locality called Island\nBridge, and many of them were fortunately secured for the Museum of the\nRoyal Irish Academy, where they may now be seen. The swords are long\nand straight, formed for cutting as well as thrusting, and terminate in\npoints formed by rounding off the edge towards the back of the blade.\nThe hilts are very remarkable in form, and in one or two instances,\nlike the example we have engraved, are highly ornamented. The mountings\nare generally of a kind of brass, but several richly plated with silver\nwere found, and it is said that one of them had a hilt of solid gold.\nThe spears are long and slender, and similar in form to the lance-heads\nused in some of the cavalry corps.\n\nAll these weapons, with one exception, are composed of a soft kind of\niron. Many of the swords were found doubled up, a circumstance for\nwhich it is difficult to assign a reason, as they had evidently been\npurposely bent. The sword we have represented in our engraving, is\nremarkable for the unusual degree of ornament which appears upon its\nhilt, and also for its material, steel.\n\n\nDINNER IN CHINA.\n\nIt is certain that a real Chinese dinner would be a very odd thing in\nthe eyes of a stranger, especially if he were one of those who think,\nas some people do, that there is only one way of living. To begin\ndinner with the dessert, and end it with the soup; to drink the wine\nsmoking hot, out of little china cups, and have your food brought to\nyou ready cut up into small pieces, and to be presented with a couple\nof sticks, instead of a knife and fork, to eat it with; to have,\ninstead of napkins, a provision of little bits of silk paper by the\nside of your plate, which, as you use, the attendants carry off; to\nleave your place between the courses, to smoke or amuse yourself;\nand to raise your chop-sticks to your forehead, and then place them\nhorizontally upon your cup, to signify that you have finished your\ndinner;--all these things would doubtless seem very odd, and create the\ncuriosity of Europeans. The Chinese, on the other hand, can never get\nover their surprise at our way of dining. They ask how we can like to\ndrink cold fluids, and what can have put it into our heads to make use\nof a trident to carry food to our mouths, at the risk of pricking our\nlips or poking our eyes out. They think it very droll to see nuts put\non the table in their shell, and ask why our servants cannot take the\ntrouble to peel the fruit, and take the bones out of the meat. They are\nthemselves certainly not very difficult in the nature of their food,\nand like such things as fried silkworms and preserved larvae, but they\ncannot understand the predilection of our epicures for _high_ game, nor\nfor cheese that appears to belong to the class of animated beings.\n\n\nCISTERN OF MAJOLICA WARE.\n\n[Illustration: [++] Cistern of Majolica Ware.]\n\nWe have engraved the annexed, as it affords at once both a beautiful\nspecimen of the potter's art, and also an example of the taste and\nluxury of the present day in articles of expensive ornament. It is a\ncistern made of Majolica, or the enamelled pottery of Italy, the most\nbeautiful specimens of which were made in the sixteenth century. The\none before us came to England from the collection of the Borghese\nPalace; and at the great sale at Stowe, the seat of the Duke of\nBuckingham, was disposed of by auction for sixty-four guineas, and this\nalthough it was much broken.\n\n\nTHEATRES IN THE TIME OF SHAKSPEARE.\n\nIn Blackfriars was a theatre, the memory of which with the one or the\nother shore of the river at Bankside, enjoys the honour of having been\nused for the first representations of many of Shakspeare's plays, and\nwhere the bard himself performed in them. The whole district becomes\nclassic, from the remembrance. The following interesting description\nof the theatres in London at that time, and which applies to the\nBlackfriars' theatre as we well as the rest, is taken from a short\nmemoir of Shakspeare, by the Rev. Alexader Dyce, prefixed to the Aldine\nedition of Shakspeare's poems: \"Nearly all these buildings, it is\nprobable, were constructed of wood. Those which, for some undiscovered\nreason, were termed private theatres, were entirely roofed in from\nthe weather, while the public theatres were open to the sky, except\nover the stage and galleries. On the outside of each was exhibited\na sign indicative of its name; and on the roof, during the time of\nperformance, was hoisted a flag. The interior arrangements resemble\nthose of the present day. There were tiers of galleries or _scaffolds_;\nbeneath these the boxes or _rooms_, intended for persons of the higher\nclass, and which at the private theatres were secured with locks,\nthe keys being given to the individuals who engaged them; and there\nwas the centre area, (separated, it seems, from the stage by pales),\nat the private theatres, termed the _pit_, and furnished with seats;\nbut at the public theatres, called the _yard_, and affording no such\naccommodation. Cressets, or large open lanterns, served to illuminate\nthe body of the house; and two ample branches, of a form similar to\nthose now hung in churches, gave light to the stage. The band of\nmusicians, which was far from numerous, sat, it is supposed, in an\nupper balcony, over what is now called the stage box: the instruments\nchiefly used were trumpets, cornets, hautboys, lutes, recorders, viols,\nand organs. The amusements of the audience previous to the commencement\nof the play, were reading, playing at cards, smoking tobacco, drinking\nale, and eating nuts and apples. Even during the performance it was\ncustomary for wits, critics, and young gallants, who were desirous of\nattracting attention, to station themselves on the stage, either lying\non the rushes or seated on hired stools, while their pages furnished\nthem with pipes and tobacco. At the third sounding, or flourish of\ntrumpets, the exhibition began. The curtain, which concealed the\nstage from the audience, was then drawn, opening in the middle, and\nrunning upon iron rods. Other curtains, called _traverses_, were used\nas a substitute for scenes. At the back of the stage was a balcony,\nthe platform of which was raised about eight or nine feet from the\nground; it served as a window, gallery, or upper chamber. From it\na portion of the dialogue was sometimes spoken, and in front of it\ncurtains were suspended to conceal, if necessary, those who occupied\nit, from the audience. The internal roof of the stage, either painted\nblue or adorned with drapery of that colour, was termed the _heavens_.\nThe stage was generally strewed with rushes, but on extraordinary\noccasions was matted. There is reason to believe that, when tragedies\nwere performed, it was hung with black. Moveable painted scenery there\nwas assuredly none. A board, containing the name of the place of\naction in large letters, was displayed in some conspicuous situation.\nOccasionally, when some change of scene was necessary, the audience\nwas required to suppose that the performers, who had not quitted the\nboards, had retired to a different spot. A bed thrust forth showed that\nthe stage was a bed-chamber; and a table, with pen and ink, indicated\nthat it was a counting-house. Rude contrivances were employed to\nimitate towers, walls of towns, hell-mouths, tombs, trees, dragons, &c.\nTrap-doors had been early in use; but to make a celestial personage\nascend to the roof of the stage was more than the machinists of the\ntheatre could always accomplish. The price of admission appears to have\nvaried according to the rank and estimation of the theatres. A shilling\nwas charged for a place in the best boxes; the entrance-money to the\npit and galleries was the same--sixpence, twopence, and a penny. The\nperformance commenced at three in the afternoon.\"\n\n\nOLD CUSTOM RELATING TO CRIMINALS.\n\nThe custom of offering doomed criminals a last earthly draught of\nrefreshment is undoubtedly one of considerable antiquity. The right\nof offering wine to criminals, on their passage to the scaffold, was\noften a privilege granted to religious communities. In Paris, the\nprivilege was held by the convent of Filles-Dieu, the nuns of which\nkept wine prepared for those who were condemned to suffer on the gibbet\nof Montfaucon. The gloomy procession halted before the gate of the\nmonastery, the criminal descended from the cart, and the nuns, headed\nby the Lady Abbess, received him on the steps with as much, perhaps\nmore, heartfelt ceremony than if he had been a king. The poor wretch\nwas led to a crucifix near the church door, the feet whereof he humbly\nkissed. He then received, from the hands of the Superior, three pieces\nof bread (to remind him of the Trinity), and _one_ glass of wine\n(emblem of Unity). The procession then resumed its dread way to the\nscaffold.\n\n\nALE TOO STRONG.\n\nA memorial signed by nineteen inhabitants of Bayton, in Worcestershire,\nwas sent to the Sessions in the year 1612, setting forth \"that John\nKempster and John Byrd do not sell their ale according to the law,\nbut doe sell a pynte for a penny, and doe make ytt soe extraordynarye\nstrong that itt draweth dyvers ydle p'sons into the said alehouses,\nby reason whereof sondrye assaults, affrayes, blodshedds, and other\nmisdeameanors, are there daylie comytted by idle and dronken companie\nwhich doe thither resort and there contineue in their dronckenes three\ndays and three nights together, and also divers men's sonnes and\nservants do often resort and contineue drinking in the said houses day\nand night, whereupon divers disorders and abuses are offered to the\ninhabitants of Bayton aforesaid, as in pulling down styles, in carrying\naway of yertes, in throwing men's waynes, plowes, and such like things,\ninto pooles, wells, and other bye places, and in putting their yokes\nfor their oxen into lakes and myery places, &c.\" A nice picture of\nyoung England in the seventeenth century.\n\n\nA CHAPTER-HOUSE IN THE TIME OF HENRY VII.\n\nIn abbey-churches of the olden time the Chapter-house was always on the\neast side of the court. In establishments of secular canons it seems\nto have been always multisided, with a central pillar to support its\ngroining, and a lofty, conical, lead-covered roof. In these instances\nit is placed in the open space eastward of the cloister, and is usually\napproached by a passage from the east side of the cloister court. In\nthe houses of all the other orders the chapter-house is rectangular,\neven where the church is a cathedral. Usually, then, the chapter-house\nis a rectangular building on the east side of the cloister, and\nfrequently its longest apsis is east and west--at Durham it has an\neastern apsis. It was a large and handsome room, with a good deal of\narchitectural ornament; often the western end of it is divided off as a\nvestibule or ante-room; and generally it is so large as to be divided\ninto two or three aisles by rows of pillars. Internally, rows of stalls\nor benches were arranged round the walls for the convent; there was a\nhigher seat at the east end for the abbot or prior, and a desk in the\nmiddle from which certain things were read. Every day after the service\ncalled Tierce, the convent walked in procession from the choir to the\nchapter-house, and took their proper places. When the abbot had taken\nhis place, the monks descended one step and bowed; he returned their\nsalutation, and all took their seats. A sentence of the rule of the\norder was read by one of the novices from the desk, and the abbot, or\nin his absence, the prior, delivered an explanatory or hortatory sermon\nupon it; then, from another portion of the book was read the names\nof brethren, and benefactors, and persons who had been received into\nfraternity, whose decease had happened on that day of the year; and the\nconvent prayed a _requiescat in pace_ for their souls, and the souls\nof all the faithful departed this life. Then members of the convent\nwho had been guilty of slight breaches of discipline confessed them,\nkneeling upon a low stool in the middle, and on a bow from the abbot,\nintimating his remission of the breach, they resumed their seats. If\nany had a complaint to make against any brother, it was here made and\nadjudged. Convent business was also transacted. The woodcut gives an\nexample of the kind. Henry VII. had made grants to Westminster Abbey,\non condition that the convent performed certain religious services on\nhis behalf; and in order that the services should not fall into disuse,\nhe directed that yearly, at a certain period, the chief justice, or\nthe king's attorney, or the recorder of London, should attend in\nchapter, and the abstract of the grant and agreement between the king\nand the convent should be read. The grant which was thus to be read\nstill exists in the British Museum; it is written in a volume superbly\nbound, with the royal seals attached in silver cases; it is from the\nilluminated letter at the head of one of the deeds that our woodcut is\ntaken. It rudely represents the chapter-house, with the chief-justice\nand a group of lawyers on one side, the abbot and convent on the other,\nand a monk reading the grant from the desk in the midst.\n\n[Illustration: [++] Chapter-House in the time of Henry VII.]\n\n\nANNE BOLEYN's GLOVES.\n\nAnne Boleyn was marvellously dainty about her gloves. She had a nail\nwhich turned up at the side, and it was the delight of Queen Catharine\nto make her play at cards, without her gloves, in order that the\ndeformity might disgust King Hal. The good Queen Bess was extravagant,\nfastidious, and capricious in the extreme, about her gloves. She used\nto display them to advantage in playing the virginal, and gloves at\nthat time were expensive articles.\n\n\nDELLA ROBBIA WARE.\n\nLuca della Robbia, born in 1388, was an eminent sculptor in marble and\nbronze, and worked both at Florence and at Rimini. Having abandoned his\noriginal employment for that of modelling in terra cotta, he succeeded,\nafter many experiments, in making a white enamel, with which he coated\nhis works, and thus rendered them durable. Vasari writes of him, \"che\nfaceva l'opere di terra quasi eterne.\" His chief productions are\nMadonnas, Scripture subjects, figures, and architectural ornaments:\nthey are by far the finest works ever executed in pottery. He adorned\nthe Italian churches with tiles, as well as with altar-pieces, in terra\ncotta enamelled; and he is the founder of a school which produced works\nnot much inferior to his own. The \"Petit Chateau de Madrid,\" in the\nBois de Boulogne, near Paris, received the appellation of \"Chateau de\nFayence,\" from having been ornamented with enamelled tiles, the work\nof an Italian artist, named Girolamo della Robbia, a grand nephew of\nLuca, whom Francis I. brought from Italy. This chateau is now wholly\ndestroyed. The tiles seem to have been introduced into portions of the\narchitectural composition, rather as accessory ornaments than as a\n\"lining\" or revetement of the walls. Analogous ornaments, the work of\nLuca de Maiano, 1521, were to be seen in the old gate, Whitehall, and\nat Hampton Court.\n\n[Illustration: [++] Della Robbia Ware.]\n\nLuca della Robbia sometimes, though rarely, used a instead\nof white enamel in his compositions. The above cut represents the\naltar-piece of San Miniato, near Florence, by him. The ground is blue,\nthe figures white, the fruits, &c., gold colour, and the garlands green.\n\n\nVOLCANIC ERUPTION IN JAPAN.\n\nThe peninsula of Wountsendake, and the greater part of Kewsew,\nbristle with volcanic mountains, some extinct, others still acting as\nsafety-valves to the incomprehensible excitements of mother Earth; but\nof all the manifestations of her internal throes and torment, and their\nconsequent desolation inflicted on the habitations of her children,\nthat of 1792 was the most terrible for ages before.\n\n\"On the eighteenth day of the first month of that year,\" says the\n_Annals of Japan_, \"the summit of the mountain was seen to crumble\nsuddenly, and a thick smoke rose in the air. On the sixth of the\nfollowing month there was an eruption in a spur on the eastern \nof the mountain. On the second of the third month an earthquake shook\nthe whole island. At Simabara, the nearest town to the mountain, all\nthe houses were thrown down, amidst a general terror and consternation,\nthe shocks following each other with frightful rapidity. Wountsendake\nincessantly sent forth a hail-storm of stones, showers of ashes, and\nstreams of lava, which devastated the country for many leagues round.\nAt length, on the first day of the fourth month, there was a new\ncommotion, which increased in intensity from moment to moment.\n\n\"Simabara was now a vast heap of ruins. Enormous blocks of rock,\ntumbling from the top of the mountain, crushed and ground to atoms all\nbeneath them. Thunder rolled overhead, and dreadful sounds rumbled\nbeneath the feet at one and the same time. All of a sudden, after an\ninterval of calm, when men thought the scourge had passed over, the\nnorthern spur of Wountsendake, the Moikenyamma, burst forth with a\ntremendous detonation. A vast portion of that mountain was blown into\nthe air. Colossal masses fell into the sea. A stream of boiling water\nrushed forth foaming from the cracks of this new volcano, and sped to\nthe ocean, which at the same time advanced and flooded the land.\"\n\nThen was seen a sight never seen before, intensifying the terror of\nthe innumerable witnesses of that terrible day, which might well seem\na Day of Judgment come. From the conflict of the boiling waters of\nthe volcano with the cold waters of the tempestuous ocean, suddenly\nmingled, there arose waterspouts which ravaged the land in their\ndevouring gyrations.\n\nThe disasters caused by this accumulation of catastrophes, earthquakes,\nvolcanic eruptions, waterspouts, inundations, united together, exceed\nbelief. Not a single house of Simabara and its environs was spared:\nonly the citadel remained, whose Cyclopean walls were formed of\ngigantic blocks of stone. The convulsions of nature on that day so\nchanged the coast-line, that the most experienced mariners could not\nrecognise its once familiar shape and bendings.\n\nFifty-three thousand persons perished on that fatal day.\n\n\nORIGIN OF THE HOUSE OF MULGRAVE.\n\nThe first diving bell was nothing but a very large kettle, suspended\nby ropes, with the mouth downwards, and planks to sit on fixed in\nthe middle of its concavity. The Greeks at Toledo, in 1588, made an\nexperiment before the Emperor Charles V. with it, when they descended\nwith a lighted candle to a considerable depth. In 1683 William Phipps,\nthe son of a blacksmith, formed a project for unloading a rich Spanish\nship, sunk at Hispaniola; Charles II. gave him a ship, with every\nnecessary for the undertaking; but being unsuccessful, Phipps returned\nin great poverty. He then endeavoured to procure another vessel,\nbut failing, he got a subscription, to which the Duke of Albemarle\ncontributed. In 1687, Phipps set sail in a ship of 200 tons, having\npreviously engaged to divide the profits according to the twenty\nshares of which the subscription consisted. At first all his labours\nproved fruitless, but at length, when he seemed almost to despair, he\nwas fortunate enough to bring up so much treasure that he returned to\nEngland with L200,000 sterling. Of this sum he got about L20,000, and\nthe Duke of Albemarle L90,000. Phipps was knighted by the king, and\nlaid the foundation of the present house of Mulgrave.\n\n\nSHRINE OF ST. SEBALD AT NUREMBURG.\n\nThe city of Nuremberg--the birthplace of Albert Durer--is enriched with\nmany works of high art. The most remarkable is the bronze shrine of St.\nSebald, the work of Peter Vischer and his five sons, which still stands\nin all its beauty in the elegant church dedicated to the saint. The\nsketch on next page is a correct representation of it.\n\n[Illustration: [++] Shrine of St. Sebald.]\n\nThe shrine encloses, amid the most florid Gothic architecture, the\noaken chest encased with silver plates, containing the body of the\nvenerated saint: this rests on an altar decorated with basso-relievos,\ndepicting his miracles. The architectural portion of this exquisite\nshrine partakes of the characteristics of the _Rennaissance_ forms\nengrafted on the mediaeval, by the influence of Italian art. Indeed,\nthe latter school is visible as the leading agent throughout the\nentire composition. The figures of the Twelve Apostles and others\nplaced around it, scarcely seem to belong to German art; they are\nquite worthy of the best _Transalpine_ master. The grandeur, breadth,\nand repose of these wonderful statues, cannot be excelled. Vischer\nseems to have completely freed his mind from the conventionalities of\nhis native schools: we have here none of the constrained, \"crumpled\ndraperies,\" the home studies for face and form so strikingly present in\nnearly all the works of art of this era, but noble figures of the men\nelevated above the earthly standard by companionship with the Saviour,\nexhibiting their high destiny by a noble bearing, worthy of the solemn\nand glorious duties they were devoted to fulfil. We gaze on these\nfigures as we do on the works of Giotto and Fra Angelico, until we feel\nhuman nature may lose nearly all of its debasements before the \"mortal\ncoil\" is \"shuffled off,\" and that mental goodness may shine through and\nglorify its earthly tabernacle, and give an assurance in time present\nof the superiorities of an hereafter. Dead, indeed, must be the soul\nthat can gaze on such works unmoved, appealing, as they do, to our\nnoblest aspirations, and vindicating humanity from its fallen position,\nby asserting its innate, latent glories. Here we feel the truth of the\nscriptural phrase--\"In his own image made he them.\"\n\nThe memory of Peter Vischer is deservedly honoured by his townsmen.\nThe street in which his house is situated, like that in which Durer's\nstands, has lost its original name, and is now only known as Peter\nVischer's Strasse; but these two artists are the only ones thus\ndistinguished. Vischer was born in 1460, and died in 1529. He was\nemployed by the warden of St. Sebald's, and magistrate of Nuremberg,\nSebald Schreyer, to construct this work in honour of his patron saint;\nhe began it in 1506, and finished it in 1519. Thirteen years of labour\nwere thus devoted to its completion, for which he received seven\nhundred and seventy florins. \"According to this tradition, Vischer\nwas miserably paid for this great work of labour and art; and he\nhas himself recorded, in an inscription upon the monument, that 'he\ncompleted it for the praise of God Almighty alone, and the honour of\nSt. Sebald, Prince of Heaven, by the aid of pious persons, paid by\ntheir voluntary contributions.'\" The elaboration of the entire work is\nmarvellous; it abounds with fanciful figures, seventy-two in number,\ndisposed among the ornaments, or acting as supporters to the general\ncomposition. Syrens hold candelabra at the angles; and the centre has\nan air of singular lightness and grace. It is supported at the base\nby huge snails. At the western end there is a small bronze statue of\nVischer; he holds his chisel in his hand, and in his workman's dress,\nwith capacious leather apron, stands unaffectedly forth as a true,\nhonest labourer, appealing only to such sympathies as are justly due to\none who laboured so lovingly and so well.\n\n\nA GREAT RESULT FROM TRIVIAL CIRCUMSTANCES.\n\nThat magnificent institution of active benevolence, Guy's Hospital,\nis one among a numerous list of instances where trifling events have\nproduced most disproportionate consequences.\n\n[Illustration: [++] Thomas Guy.]\n\nThomas Guy, of whom the above is a sketch, taken from an old print, was\nthe son of Thomas Guy an Anabaptist, lighterman and coal-dealer, in\nHorsleydown, Southwark. He was put apprentice in 1660 to a bookseller\nin the porch of Mercer's Chapel, and set up trade with a stock of about\ntwo hundred pounds, in the house that forms the angle between Cornhill\nand Lombard-street. The English Bibles being at that time very badly\nprinted, Mr. Guy engaged with others in a scheme for printing them in\nHolland and importing them; but this being put a stop to, he contracted\nwith the University of Oxford for their privilege of printing them,\nand carried on a great Bible trade for many years to considerable\nadvantage. He thus began to accumulate money, and his gains rested in\nhis hands, for being a single man, and very penurious, his expenses\nwere very trifling. His custom was to dine on his shop counter, with\nno other table-cloth than an old newspaper; he was also as little nice\nin regard to his dress. The bulk of his fortune, however, was acquired\nby the less reputable purchase of seamen's tickets during Queen Anne's\nwars, and by the South Sea stock in the memorable year 1720.\n\nIn proof of what we said at the outset, it is a fact that the public\nare indebted to a most trifling incident for the greatest part of\nhis immense fortunes being applied to charitable uses. Guy had a\nmaid-servant whom he agreed to marry; and preparatory to his nuptials\nhe had ordered the pavement before his door to be mended as far as a\nparticular stone which he marked. The maid, while her master was out,\ninnocently looking on the paviours at work, saw a broken place they had\nnot repaired, and mentioned it to them; but they told her that Mr. Guy\nhad desired them not to go so far. \"Well,\" says she, \"do you mend it;\ntell him I bade you, and I know he will not be angry.\" It happened,\nhowever, that the poor girl presumed too much on her influence over\nher wary lover, with whom the charge of a few shillings extraordinary\nturned the scale against her, for Guy, enraged to find his orders\nexceeded, renounced the matrimonial scheme, and built hospitals in\nhis old age. In 1707 he built and furnished three wards on the north\nside of the outer court of St. Thomas's Hospital, and gave one hundred\npounds to it annually for eleven years preceding the erection of his\nown hospital.\n\nSometime before his death he erected the stately gate with the large\nhouses on each side, at the expense of about three thousand pounds.\nHe was seventy-six years of age when he formed the design of building\nthe hospital near St. Thomas's, which bears his name. The charge of\nerecting this vast pile amounted to L18,793, besides L219,499 which he\nleft to endow it, and he just lived to see it roofed in.\n\nHe erected and endowed an almshouse and library at Tamworth, the place\nof his mother's nativity, and which he represented in Parliament. It\ncontains fourteen poor men and women, and the fund provides also for\nthe apprenticing of poor children. He also bequeathed four hundred\npounds a-year to Christ's Hospital.\n\nMr. Guy died December 17th, 1724 in the eighty-first year of his age,\nand his will bears date September 4th, in the same year.\n\n\nPHAROS AT ALEXANDRIA.\n\nTo render the harbour safe of approach at all times, Ptolemy Soter,\nwho, on the death of Alexander, obtained the government of Egypt,\ndetermined on erecting a lighthouse on the eastern extremity of the\nisle of Pharos, the celebrity of which has given the same name to all\nother lighthouses.\n\nThis \"pharos\" was in height 450 feet, and could be seen at a distance\nof 100 miles. It was built of several stories, decreasing in dimension\ntowards the top, where fires were lighted in a species of lantern. The\nground-floor and the two next above it were hexagonal; the fourth was a\nsquare with a round tower at each angle; the fifth floor was circular,\ncontinued to the top, to which a winding staircase conducted. In the\nupper galleries some mirrors were arranged in such a manner as to show\nthe ships and objects at sea for some considerable distance. On the top\na fire was constantly kept, to direct sailors into the bay, which was\ndangerous and difficult of access.\n\nThe whole of this masterpiece of art was exquisitely wrought in stone,\nand adorned with columns, balustrades, and ornaments, worked in the\nfinest marble. To protect the structure from the ocean storms, it was\nsurrounded entirely by a sea wall. Ancient writers say the building of\nthis tower cost 800 talents, which is equivalent to L165,000, if Attic\ntalents; but if Alexandrian, double that sum.\n\nThe building was not completed during the reign of the first Ptolemy,\nbut was finished in the reign of his son Ptolemy Philadelphus, who put\nthis inscription upon it:--\n\n \"King Ptolemy, to the Gods the Saviours, for the benefit of\n sailors.\"\n\nSostratus the architect, wishing to claim all the glory of the\nbuilding, engraved his own name on the solid marble, and afterwards\ncoated it with cement. Thus, when time had decayed the mortar Ptolemy's\nname disappeared, and the following inscription became visible:--\n\n \"Sostratus the Cnidian, to the Gods the Saviours, for the benefit\n of sailors.\"\n\nOf this remarkable tower not a vestige remains, and history gives us\nno further information than we have here: of its gradual decay or of\nits violent destruction we have no record; but that such a structure\nas described stood there, there can be not a shadow of doubt, from the\nfact that all buildings for like purposes among the Greeks and Romans\nderive their designation from this.\n\n\nSEPULCHRAL VASES OF ANCIENT EGYPT.\n\n[Illustration: [++] Sepulchral Vase.]\n\nIn ancient Egypt terra-cotta pottery was extensively made use of for\nvases or jars to hold the entrails of the dead. In order to preserve\nthe body effectually, it was necessary to remove the softer portions,\nsuch as the thoracic and abdominal viscera, and these were embalmed\nseparately. In some instances they were returned into the stomach,\nwith wax models of four deities, commonly called the four genii of\nthe Ament or Hades. It was, however, usual in the embalmment of the\nwealthier classes to soak them carefully in the requisite preparations,\ntie them up in neat cylindrical packets, and deposit them in vases\nhaving the shape of the four genii. The bodies of these deities, which\nwere usually represented as mummied, formed the bodies of the vases,\nand were cylindrical below and rounded above. The mouths of the jars\nwere sometimes countersunk to receive the lower part of the covers\nwhich fitted into them like a plug. The jar of the first genius, whose\nname was _Am-set_, \"the devourer of filth,\" held the stomach and large\nintestines, and was formed at the top like a human head. This genius\ntypified, or presided over the southern quarter of the compass. He\nwas the son of Osiris or of Phtha Socharis Osiris, the pygmean god\nof Memphis. The second vase of the series was in the shape of the\ngenius Hapi, the \"concealed.\" Its cover was shaped like the head of a\ncynocephalus, and it held the smaller viscera. This genius presided\nover the north, and was also the son of Osiris. The third vase was that\nof the genius Trautmutf, \"the adorer of his mother.\" We here annex an\nengraving of it. It had a cover in shape of the head of a jackal, and\nheld the lungs and heart. This genius presided over the East, and was\nbrother of the preceding. The last was that of the genius Kebhsnuf,\nthe refresher of his brethren. It had a cover shaped like the head\nof a sparrow-hawk, and held the liver and gall-bladder. This genius\npresided over the west, and was also brother of the preceding. Three\nvases of a set, in the British Museum, have all human-shaped heads, and\nare provided with handles at the sides of the bodies. Specimens of a\nvery unusual kind are also to be found in the same collection, having\nthe whole body formed without a cover, in the shape of a dome above,\nand surmounted by a rudely modelled figure of a jackal, couchant upon\na gateway, formed of a detached piece. The entrails were introduced by\nthe rectangular orifice in the upper part. In some other instances the\ncovers appear to have been secured by cords passing through them to\nthe body of the vase. When secured, the vases were placed in a wooden\nbox, which was laid on a sledge and carried to the sepulchre, where\nthey were often taken out and placed, two on each side of the coffin.\nIt was only the poorer classes that used pottery for these purposes.\nThe viscera of high officers of state were embalmed in jars of fine\nwhite limestone, and the still more valuable oriental alabasters or\narragonite, obtained from the quarries of Tel El Amarna, or the ancient\nAlabastron.\n\n\nTHE SACRO CATINO.\n\nThe celebrated \"Sacro Catino,\" part of the spoil taken by the Genoese\nat the storming of Cesarea, which was believed to be cut from a single\nemerald, and had, according to tradition, been presented by the Queen\nof Sheba to Solomon, was for ages the pride and glory of Genoa, and an\nobject of the greatest devotional reverence at the yearly exhibitions,\nwhich were attended with great pomp and ceremony. Such was the opinion\nof its intrinsic value, that on many occasions the republic borrowed\nhalf a million of ducats upon the security of this precious relic.\nWhen the French armies, during the first Revolution, plundered Italy\nof its treasures, it was sent with other spoils to Paris. Upon\nexamination, it was, instead of emerald, proved to be composed of\nglass, similar to that found in the Egyptian tombs, of which country it\nwas, no doubt, the manufacture. At the Restoration the Sacro Catino was\nreturned in a broken state, and now lies shorn of all its honours, a\nmere broken glass vessel, in the sacristy of the Church of San Lorenzo.\n\n\nDINNER PARTY IN THE SEVENTEENTH CENTURY.\n\n[Illustration: [++] Dinner Party in the Seventeenth Century.]\n\nThe cut which we here present to our readers is taken from the English\nedition of the Janua Linguarum of Comenius, and represents the forms of\ndining in England under the Protectorate. It will be best described by\nthe text which accompanies it in the book, and in which each particular\nobject is mentioned. \"When a feast is made ready,\" we are told, \"the\ntable is covered with a carpet and a table-cloth by the waiters, who,\nbesides, lay the trenchers, spoons, knives, with little forks, table\nnapkins, bread, with a salt-cellar Messes are brought in platters, a\npie in a plate. The guests being brought in by the host, wash their\nhands out of a laver or ewer, over a hand-basin, or bowl, and wipe\nthem with a hand towel: they then sit at the table on chairs. The\ncarver breaketh up the good cheer, and divideth it. Sauces are set\namongst roste-meat in sawsers. The butler filleth strong wine out of a\ncruse, or wine-pot, or flagon, into cups or glasses, which stand on a\ncupboard, and he reacheth them to the master of the feast, who drinketh\nto his guests.\" It will be observed here that one salt-cellar is here\nplaced in the middle of the table. This was the usual custom; and, as\none long table had been substituted for the several tables formerly\nstanding in the hall, the salt-cellar was considered to divide the\ntable into distinct parts, guests of more distinction being placed\nabove the salt, while the places below the salt were assigned to\ninferiors and dependents. This usage is often alluded to in the old\ndramatists. Thus, in Ben Jonson, it is said of a man who treats his\ninferiors with scorn, \"he never drinks _below the salt_, _i. e._ he\nnever exchanges civilities with those who sit at the lower end of the\ntable.\" And in a contemporary writer, it is described as a mark of\npresumption in an inferior member of the household \"to sit above the\nsalt.\"\n\n\nSAND-COLUMNS IN AFRICA.\n\nOf this remarkable phenomenon, we extract the following interesting\naccount from the Rev. N. Davis's \"Evenings in my Tent\";--\n\n\"The heat, during the last day or two, has been intense. The\nthermometer in my tent, during day and night, has been almost\nstationary at 100 degrees. My men have done, and still do, everything\nin their power to keep the tent cool, by erecting a high palm-branch\nfence around it, and by a constant immersion of the ground, but all\nthis to very little effect. The wind, during this day, has been as\nhot as the flames issuing from a furnace; and the clouds of sand it\nraised, and carried along in its furious march, have been immense.\nIn the distance could be seen numbers of sand columns; but these did\nnot retain their form any considerable length of time. A contrary\nblast brought them in collision with each other; and these, blending\ntheir contents, raised a complete and dense barrier between us and the\ncountry beyond. I am no lover of danger; but, I must confess, I had\nan inward desire to see this phenomenon--one of the horrors of the\ndesert--in greater perfection. I believe Bruce witnessed one of the\nmost stupendous exhibitions of sand columns or sand spouts, caused by\ncircular or whirl-winds, on record. In his journey through the desert\nof Senaar, his attention was attracted to a number of prodigious\npillars of sand, at different distances, moving at times with great\ncelerity, at others, stalking on with majestic slowness: at intervals,\nhe thought they were coming in a very few minutes to overwhelm him and\nhis companions. Again they would retreat, so as to be almost out of\nsight, their tops reaching to the very clouds. There the tops often\nseparated, from the bodies; and these, once disjoined, dispersed in\nthe air, and appeared no more. Sometimes they were broken near the\nmiddle, as if struck with a large cannon-shot. About noon, they began\nto advance with considerable swiftness upon them, the wind being very\nstrong at north. Eleven of these awful visitors ranged alongside of\nthem, at about the distance of three miles. The greatest diameter of\nthe largest appeared to him, at that distance, as if it would measure\nten feet. They retired from them, with a wind at south-east, leaving an\nimpression upon the mind of our intrepid traveller to which he could\ngive no name, though he candidly admits that one ingredient in it was\nfear, with a considerable deal of wonder and astonishment. He declares\nit was in vain to think of flying; the swiftest horse, or fastest\nsailing ship, could be of no use to carry them out of this danger,--and\nthe full persuasion of this riveted him to the spot where he stood.\nNext day they were gratified by a similar display of moving pillars,\nin form and disposition like those already described, only they seemed\nto be more in number, and less in size. They came several times in\na direction close upon them; that is, according to Mr. Bruce's\ncomputation, within two miles. They became, immediately after sunrise,\nlike a thick wood, and almost darkened the sun, his rays, shining\nthrough them for near an hour, gave them an appearance of pillars of\nfire. At another time they were terrified by an army of these sand\npillars, whose march was constantly south, a number of which seemed\nonce to be coming directly upon them, and, though they were little\nnearer than two miles, a considerable quantity of sand fell around\nthem. On the 21st of November, about eight in the morning, he had a\nview of the desert to the westward as before, and saw the sands had\nalready begun to rise in immense twisted pillars, which darkened the\nheavens, and moved over the desert with more magnificence than ever.\nThe sun shining through the pillars, which were thicker, and contained\nmore sand apparently than any of the preceding ones, seemed to give\nthose nearest them an appearance as if spotted with stars of gold. A\nlittle before twelve, the wind at north ceased, and a considerable\nquantity of fine sand rained upon them for an hour afterwards.\"\n\n\nANTIQUITY OF INTOXICATING DRINKS.\n\nIt is a common belief that wine was the only inebriating liquor known\nto antiquity, but this is a mistake. Tacitus mentions the use of ale\nor beer as common among the Germans of his time. By the Egyptians,\nlikewise, whose country was ill adapted to the cultivation of the\ngrape, it was employed as a substitute for wine. Ale was common in\nthe middle ages, and Mr. Park states that very good beer is made, by\nthe usual process of brewing and malting, in the interior of Africa.\nThe favourite drink of our Saxon ancestors was ale or mead. Those\nworshippers of Odin were so notoriously addicted to drunkenness,\nthat it was regarded as honourable rather than otherwise; and the\nman who could withstand the greatest quantity was looked upon with\nadmiration and respect: whence the drunken songs of the Scandinavian\nscalds: whence the glories of Valhalla, the fancied happiness of\nwhose inhabitants consisted of quaffing draughts from the skulls of\ntheir enemies slain in battle. Even ardent spirit, which is generally\nsupposed to be a modern discovery, probably existed from a very early\nperiod. It is said to have been first made by the Arabians in the\nmiddle ages, and in all likelihood may lay claim to a still remoter\norigin. The spirituous liquor called arrack has been manufactured in\nthe island of Java, as well as in the continent of Hindostan, from\ntime immemorial. Brandy was made in Sicily at the commencement of the\nfourteenth century. As to wine, it was so common in ancient times as\nto have a tutelar god appropriated to it; Bacchus and his companion\nSilenus are as household words in the mouths of all, and constituted\nmost important features of the heathen mythology. We have all heard\nof the Falernian and Campanian wines, and of the wines of Cyprus and\nShiraz. Indeed, there is reason to believe that the ancients were in\nno respect inferior to the moderns in the excellence of the vinous\nliquors, whatever they may have been in the variety. Wine was so common\nin the eastern nations that Mahomet, foreseeing the baleful effects\nof its propagation, forbade it to his followers, who, to compensate\nthemselves, had recourse to opium. The Gothic or dark ages seem to have\nbeen those in which it was the least common; in proof of this it may\nbe mentioned that, so late as 1298, it was vended as a cordial by the\nEnglish apothecaries. At the present day it is little drunk, except by\nthe upper classes, in those countries which do not naturally furnish\nthe grape. In those that do, it is so cheap as to come within the reach\nof even the lowest.\n\n\nRUINS OF CLONMACNOIS.\n\n[Illustration: [++] Ruins of Clonmacnois.]\n\nA few miles south of Athlone are the famous ruins of Clonmacnois, the\nschool where, according to Dr. O'Connor, \"the nobility of Connaught had\ntheir children educated, and which was therefore called Cluan-mac-nois,\n'the secluded recess of the sons of nobles.'\" It was also, in ancient\ntimes, a renowned cemetery of the Irish kings; and for many centuries\nit has continued a favourite burial-place, the popular belief enduring\nto this day, that all persons interred here pass immediately from\nearth to heaven. The abbey is said to have been founded by St. Kieran\nabout the middle of the sixth century, and soon became \"amazingly\nenriched,\" so that, writes Mr. Archdall, \"its landed property was so\ngreat, and the number of cells and monasteries subjected to it so\nnumerous, that almost half of Ireland was said to be within the bounds\nof Clonmacnois.\" The ruins retain marks of exceeding splendour. In the\nimmediate vicinity there are two \"Round Towers.\" The above engraving\nrepresents one of the many richly-carved stone crosses that are\nscattered in all directions among the ruins.\n\n\nTHE BRICKS OF BABYLON.\n\n[Illustration: [++] Brick of Babylon.]\n\nBesides sun-dried bricks, remains of kiln-baked or burnt bricks are\nfound in all the principal ruins of ancient Babylonia, and were used\nfor the purpose of revetting or casing the walls. Like the sun-dried\nbricks they are made of clay mixed with grass and straw, which have,\nof course, disappeared in the baking, leaving, however, traces of the\nstalks or stems in the clay. Generally they are slack-burnt, of a pale\nred colour, with a slight glaze or polish. The finest sort, according\nto Mr. Rich, are white, approaching more or less to a yellowish cast,\nlike our Stourbridge, or fire-brick; the coarsest are red, like our\nordinary brick. Some have a blackish cast, and are very hard. The\nfinest are those which come from the ruins of the Akerkuf. The general\nmeasurement of the kiln-dried bricks, at the Birs Nimrud, is 1 ft. 1\nin. square, and 3 in. thick. Some are submultiples, or half of these\ndimensions. A few are of different shapes for particular purposes, such\nas rounding corners. Those at the Akerkuf measured a trifle less, or\n12-1\/2 in. square, and 2-3\/4 in. thick, and are placed at the base of\nthe monument. The bricks of Al Hymer, on the eastern bank, measure 14\nin. long, 12-3\/4 in. broad, 2-1\/2 in. thick, and are of fine fabric.\nThere are bricks of two dimensions at this ruin of the Birs Nimrud;\nthose on the northern brow, a little way down it, measure 12 in.\nsquare, and 3-1\/4 in. thick; they are of a pale red colour, and used\nfor revetting the monument. Lower down to the east of this, they are\n4-3\/4 in. broad, and 12-3\/4 in. long. Similar bricks were found at the\nMujellibe, and in one place was an entire wall of them 60 feet thick.\nThe whole plain here is covered with masses of brick work, and on one\nof the mounds the bricks are so red, that it looks one bright gleaming\nmass. The bricks from the Mujellibe or Kasr are described as very hard,\nand of a pale yellow colour; and this edifice presents a remarkable\nappearance of freshness. We have seen only one fragment of a brick from\nNiffer; it is of a white, or rather yellowish white colour, and sandy,\ngritty texture. This spot, it will be remembered, is supposed to be the\nsite of old Babylon. All these bricks are made by the same process as\nthose of Assyria, namely, stamped out of a wooden or terra-cotta mould,\nand are also impressed with several lines of cuneiform character. This\nimpression is always sunk below the superficies, rectangular, and\noften placed obliquely on the brick, with that disregard to mechanical\nsymmetry which is so usual on works of ancient art. The stamp is\ngenerally about 6 inches long, by 4 inches wide, and the number of\nlines varies from three to seven: an arrangement quite different from\nthat observed on the bricks of Assyria, and rather resembling that\nadopted by the brick-makers of Egypt. The engraving on previous page\nis of a brick stamped with the name of Nebuchadnezzar, which is now in\nthe possession of the Royal Society of Literature. The inscriptions\nsometimes commence with the figure of a lion, a bull, or what may be\nintended for an altar. These read, according to Sir H. Rawlinson,--\n\n [of] Nebuchadnezzar,\n the king of Babylon,\n founder of Beth Digla, or Saggalu,\n and of Beth Tzida\n son of Nebopalasar [I am].\n\n\nA TURKISH BAZAAR.\n\nA Turkish bazaar is one of the most wonderful sights in the world, and\nwell deserves a place in our record of curiosities. We cannot do better\nthan quote the description which Mr. Albert Smith gives of one of these\nextraordinary places in his \"Month at Constantinople:\"--\n\n\"Smyrna had, in some measure, prepared me for the general appearance\nof an oriental bazaar; but the vast extent of these markets at\nConstantinople created a still more vivid impression. To say that the\ncovered rows of shops must altogether be miles in length--that vista\nafter vista opens upon the gaze of the astonished stranger, lined with\nthe costliest productions of the world, each collected in its proper\ndistrict--that one may walk for an hour, without going over the same\nground twice, amidst diamonds, gold, and ivory; Cashmere shawls, and\nChinese silks; glittering arms, costly perfumes, embroidered slippers,\nand mirrors; rare brocades, ermines, Morocco leathers, Persian\nnick-nacks; amber mouthpieces, and jewelled pipes--that looking along\nthe shortest avenue, every known tint and colour meets the eye at once,\nin the wares and costumes, and that the noise, the motion, the novelty\nof this strange spectacle is at first perfectly bewildering--all\nthis, possibly, gives the reader the notion of some kind of splendid\nmart, fitted to supply the wants of the glittering personages who\nfigure in the Arabian Nights' Entertainments; yet it can convey but\na poor idea of the real interest which such a place calls forth,\nor the most extraordinary assemblage of treasures displayed there,\namidst so much apparent shabbiness. No spot in the world--neither the\nParisian Boulevards, nor our own Regent-street--can boast of such\nan accumulation of valuable wares from afar, as the great bazaar at\nConstantinople. Hundreds and thousands of miles of rocky road and sandy\ndesert have been traversed by the moaning camels who have carried\nthose silks and precious stones from Persia, with the caravan. From\nthe wild regions of the mysterious central Africa, that ivory, so\ncunningly worked, in the next row, has been brought--the coal-black\npeople only know how--until the Nile floated it down to Lower Egypt.\nThen those soft Cashmere shawls have made a long and treacherous\njourney to Trebizond, whence the fleet barks of the cold and stormy\nEuxine at last brought them up the fairy Bosphorus to the very water's\nedge of the city. From the remote active America; from sturdy England;\nfrom Cadiz, Marseilles, and all along the glowing shores of the\nMediterranean, safely carried over the dark and leaping sea, by brave\niron monsters that have fought the winds with their scalding breath,\nthese wares have come, to tempt the purchasers, in the pleasant, calm,\nsubdued light of the bazaars of Stamboul.\"\n\n\nVARNISH-TREE OF THE JAPANESE.\n\nThe _urusi_ or varnish-tree, of which they make so extensive a use, is\na noble tree when grown to its full size. On incision it yields a rich,\nmilky, glutinous juice, out of which the Japanese make the celebrated\nvarnish, known by the name of _Japan_. With this varnish they cover\nand coat all their household furniture, all their dishes and plates,\nand all their drinking-vessels, whether made of wood or of paper. The\nuse of plate, or porcelain, or glass appears to be very limited, and\nis probably interdicted by some rule of nationality or religion: from\nthe emperor down to the meanest peasant, all make use of the light\nvarnished or japanned cups and dishes, the inner substance of which is\nwood or paper, or what we term papier-mache.\n\nAnother tree, called _forasi_, renders a varnish of an inferior quality.\n\n\nTORTURE-CHAMBER AT NUREMBERG.\n\nNuremberg, being a \"free city,\" was governed by its own appointed\nmagistrates, having independent courts of law. The executive council\nof state consisted of eight members, chosen from the thirty patrician\nfamilies, who, by the privilege granted to them from the thirteenth\ncentury, ruled the city entirely. In process of time these privileges\nassumed the form of a civic tyranny, which was felt to be intolerable\nby the people, and occasionally opposed by them. The fierce religious\nwars of the sixteenth century assisted in destroying the monopoly of\npower still more; yet now that it is gone for ever, it has left fearful\ntraces of its irresponsible strength. All who sigh for \"the good old\ntimes,\" should not moralise over the fallen greatness of the city,\nand its almost deserted but noble town-hall; but descend below the\nbuilding into the dark vaults and corridors which form its basement;\nthe terrible substructure upon which the glorious municipal palace of\na free imperial self-ruled city was based in the middle ages, into\nwhose secrets none dared pry, and where friends, hope, life itself,\nwere lost to those who dared revolt against the rulers. There is no\nromance-writer who has imagined more horrors than we have evidences\nwere perpetrated under the name of justice in these frightful vaults,\nunknown to the busy citizens around them, within a few feet of the\nstreets down which a gay wedding procession might pass, while a true\npatriot was torn in every limb, and racked to death by the refined\ncruelty of his fellow-men. The heart sickens in these vaults, and an\ninstinctive desire to quit them takes possession of the mind, while\nremaining merely as a curious spectator within them. The narrow steps\nleading to them are reached through a decorated doorway, and the\npassage below receives light through a series of gratings. You shortly\nreach the labyrinthine ways, totally excluded from external light and\nair, and enter, one after another, confined dungeons, little more than\nsix feet square, cased with oak to deaden sounds, and to increase\nthe difficulty of attempted escape. To make these narrow places even\nmore horrible, strong wooden stocks are in some, and day and night\nprisoners were secured in total darkness, in an atmosphere which seems\neven now too oppressive to bear. In close proximity to these dungeons\nis a strong stone room, about twelve feet wide each way, into which\nyou descend by three steps. It is the torture-chamber, which we here\nengrave.\n\n[Illustration: [++] Torture-Chamber at Nuremberg.]\n\nThe massive bars before you are all that remain of the perpendicular\nrack, upon which unfortunates were hung with weights attached to their\nankles. Two such of stone, weighing each fifty pounds, were kept\nhere some years back, as well as many other implements of torture\nsince removed or sold for old iron. The raised stone bench around the\nroom was for the use of the executioner and attendants. The vaulted\nroof condensed the voice of the tortured man, and an aperture on one\nside gave it freedom to ascend into a room above, where the judicial\nlisteners waited for the faltering words which succeeded the agonising\nscreams of their victim.\n\n\nSEPULCHRAL VASES OF GREEK POTTERY.\n\nThe number of these vases deposited in the great public museums of\nEurope is very large, and from calculations derived from catalogues,\nor from observations made on the spot, may be stated in round numbers\nas follows:--The Museo Borbonico, at Naples, contains about 2,100;\nthe Gregorian Museum in the Vatican, about 1,000; Florence has about\n700; and at Turin there are 500. On the side of the Alps, the Imperial\nMuseum of Vienna possesses about 300; Berlin has 1,690; Munich about\n1,700; Dresden, 200; Carlsruhe, 200; the Louvre, at Paris, about\n1,500; while 500 more may be found in the Bibliotheque Imperiale.\nThe British Museum has about 2,600 vases of all kinds. Besides the\npublic collections, several choice and valuable specimens of ancient\nart belong to individuals. The most important of these private\ncollections are those of the Duc de Luynes, the Duc de Blacas, the\nCount de Pourtales-Gorgier, the Jatta collection, that belonging to\nM. St. Angela at Naples, and a fine and choice one belonging to the\nMarquis Campana at Rome. In England, the collections of Mr. Hope, of\nMr. Jekyll, of the Marquis of Northampton, and of Mr. Hertz, contain\nseveral interesting examples. In addition to these, several thousand\nmore vases are in the hands of the principal dealers, as S. Barone, of\nNaples; and the heirs of S. Basseggio, Capranesi and Messrs. Sotheby,\nin London. The total number of vases in public and private collections\nprobably amounts to 15,000 of all kinds.\n\n[Illustration: [++] Sepulchral Vases.]\n\nAll these were discovered in the sepulchres of the ancients, but the\ncircumstances under which they were found differ according to locality.\nIn Greece, the graves are generally small, being designed for single\ncorpses, which accounts for the comparatively small size of the vases\ndiscovered in that country. At Athens, the earlier graves are sunk\ndeepest in the soil, and those at Corinth, especially such as contain\nthe early Corinthian vases, are found by boring to a depth of several\nfeet beneath the surface. The early tombs of Civita Vecchia and Caere,\nor Cervetri, in Italy, are tunnelled in the earth; and those at Vulci\nand in the Etruscan territory, from which the finest and largest vases\nhave been extracted, are chambers hewn in the rocks. In Southern\nItaly, especially in Campania, they are large chambers, about 5-1\/4\npalms under the surface.\n\nThe engraving on previous page will convey an idea of the manner in\nwhich the vases are arranged round the bodies of the dead in the tombs\nof Veii, Nola, and Cumae.\n\nThe tomb there represented is constructed of large blocks of stone,\narranged in squared masses, called the Etruscan style of wall, in\ncontradistinction to the Cyclopean. The walls are painted with\nsubjects, the body is laid upon the stone floor, and the larger vases,\nsuch as the _oxybapha_ and _craters_ are placed round it. The jugs are\nhung upon nails round the walls.\n\n\nGAMES WITH CARDS IN THE SIXTEENTH CENTURY.\n\nCards were used by every one. The game of Gleek was played by three\npersons. The dealer dealt twelve cards and left eight on the table for\nstock, seven were bought, and the ace turned up for the dealer; if it\nwas Tiddy (four of trumps) such player gave four to the dealer. The ace\nwas called Tib, the knave Tim, the fifth Towser, and the sixth Tumbler.\nThe players then begin bidding for the stock in hopes of bettering\ntheir game, the buyer taking in seven cards and putting out seven. If\nTib was turned up, it counted fifteen to the dealer. The players then\npicked for Ruff, the one having most of a suit winning it--unless any\none had four aces, which always carried it. The first then said, \"I'll\nvie the Ruff;\" the next, \"I'll see it:\" the third, \"I'll see it, and\nrevie it;\" the first again, \"I'll see your revie;\" and the middle,\n\"I'll not meddle with it.\" They then showed their cards, and he that\nhad most of a suit won six of him that held out longest, and forty of\nhim who said he could see it, and then refused to meddle with it.\n\nOmbre, Basset, Whist, Costly Colours, and Five Cards, were, we believe,\nof later introduction. Of our period, are Ruff, Bone, Ace, Pult. The\ngreat game in the West of England was Post and Pair, as All Fours was\nin Kent, and Five Cards in Ireland. In Post and Pair, the ace of trumps\nwas the best card; at Post the best cards were one and two, but a\npair of court cards one. The daring of the game consisted in the vye,\nor the adventuring upon the goodness of your hand to intimidate your\nantagonist.\n\n\nRESCUED RELICS.\n\nThe following is a list--translated from the original in the chartulary\nof the University of Glasgow; of the relics which were carried\naway from Glasgow Cathedral, by the Archbishop, before the work of\ndemolition began, in 1560:--\n\nThe image of Christ in gold, and those of the twelve apostles in\nsilver, with the whole vestments belonging to the church.\n\nA silver cross, gilt in the upper part, and adorned with precious\nstones in the lower part, with a small portion of the cross of our\nSaviour!\n\nAnother silver cross, adorned with precious stones, with several other\nportions of the cross of Christ!\n\nA silver casket, gilt, containing the hair of the blessed Virgin!\n\nA square silver coffer, containing several of the scourges of St.\nKentigern, and St. Thomas of Canterbury, and a portion of the hair\ngarment worn by the former saint!!\n\nAnother silver casket, gilt, containing part of the skin of\nBartholomew, the apostle!!\n\nA silver casket containing a bone of St. Ninian!\n\nA silver casket, containing part of the girdle of the Virgin Mary!!\n\nA crystal case, containing a bone of some saint and of St. Magdalene!!\n\nA small vial of crystal, containing the milk of the blessed Virgin, and\npart of the manger of Christ!!!\n\nA small phial of a saffron colour, containing the fluid which formerly\nflowed from the tomb of St. Mungo!\n\nA phial, containing several of the bones of St. Eugene, and of St.\nBlaze!\n\nA phial, containing a part of the tomb of St. Catherine the virgin!\n\nA small hide, with a portion of the cloak of St. Martin!\n\nA precious hide, with portions of the bodies of St. Kentigern and St.\nThomas of Canterbury!!\n\nSome other hides, with bones of saints and other relics!\n\nA wooden chest, containing many small relics!\n\nTwo linen bags, with the bones of St. Kentigern, St. Thanew, and other\ndeceased saints!!\n\n\nPAPER.\n\nWith respect to the paper now in use, Dr. Blair says, the first\npaper-mill (in England, we suppose) was erected at Dartford, in the\nyear 1588, by a German of the name of Speillman; from which period we\nmay, perhaps, date its manufacture in this country.\n\nIt appears, however, that it was known in the East much earlier; it\nbeing observed that most of the ancient manuscripts in Arabic and other\nOriental languages, were written upon cotton paper, and it is thought\nthe Saracens first introduced it into Spain.\n\nAnderson, in his \"History of Commerce,\" says that, till the year 1690,\nthere was scarcely any paper made in England but the coarse brown\nsort. Paper was previously imported from France, Genoa, and Holland.\nHowever, the improvement of this article in England, in consequence of\nthe French war, produced a saving to the country of L100,000 annually,\nwhich had been paid to France for paper alone.\n\n\nLOTTERIES.\n\nIf the antiquity of a practice could justify its existence, lotteries\nmight claim peculiar reverence. The Romans, we are told, used to\nenliven their Saturnalia with them, by distributing tickets, all of\nwhich gained some prize. Augustus instituted lotteries, that consisted,\nhowever, of things of little value. Nero also established lotteries,\nfor the people, in which 1,000 tickets were daily distributed, and\nseveral of those who were favoured by fortune got rich by them. The\nfirst lottery of which we find any record in our annals, was in the\nyear 1659, which, according to Stow, consisted of 40,000 lots, at 10s.\neach. The prizes were plated; and the profits were to be applied to\nthe purpose of repairing the havens of the kingdom. This lottery was\ndrawn at the west door of St. Paul's cathedral; and began on the 11th\nJuly, 1569, and continued incessantly, day and night, till the 6th May\nfollowing. The tickets were three years in being disposed of. In the\nyear 1612, King James granted a lottery to promote the plantation of\nEnglish colonies in Virginia, which was also drawn at St. Paul's.\n\n\nTEMPLE AT SIMONBONG.\n\n[Illustration: [++] Lepcha Temple at Simonbong.]\n\nThe above is a correct representation of the great Lepcha temple at\nSimonbong, in Sikkim, a district of India near Thibet. We take the\nfollowing account of it from the Journal of Dr. Hooker, who visited it\nin 1848:--\"Simonbong is one of the smallest and poorest goompas, or\ntemples, in Sikkim, being built of wood only. It consists of one large\nroom, raised on a stone foundation, with small sliding shutter windows,\nand roofed with shingles of wood; opposite the door a wooden altar was\nplaced, rudely chequered with black, white, and red; to the right and\nleft were shelves, with a few Tibetan books, wrapped in silk; a model\nof Symbonath temple in Nepal, a praying-cylinder, and some implements\nfor common purposes, bags of juniper, English wine-bottles and glasses,\nwith tufts of _Abies Webbina_, rhododendron flowers, and peacock's\nfeathers, besides various trifles, clay ornaments and offerings, and\nlittle Hindoo idols. On the altar were ranged seven little brass cups,\nfull of water; a large conch-shell, carved with the sacred lotus; a\nbrass jug from Lhassa, of beautiful design, and a human thigh-bone,\nhollow, and perforated through both condyles.\n\n\"Facing the altar was a bench and a chair, and on one side a huge\ntambourine, with two curved iron drumsticks. The bench was covered with\nbells, handsomely carved with idols, and censers with juniper-ashes;\nand on it lay the _dorge_, or double-headed thunderbolt. Of all these\narticles, the human thigh-bone is by much the most curious; it is very\noften that of a Lama, and is valuable in proportion to its length.\nAs, however, the Sikkim Lamas are burned, these relics are generally\nprocured from Tibet, where the corpses are cut in pieces and thrown to\nthe kites, or thrown into the water.\"\n\n\nIMPLEMENTS USED IN BUDDHIST TEMPLES.\n\n[Illustration: [++] Implements Used in Buddhist Temples.]\n\nThe above sketch places before us the implements generally used in the\nBuddhist temples of India:--a praying cylinder in stand, another to be\ncarried in the hand, cymbals, bell, brass cup, three trumpets (one of\nthem made of a human thigh-bone), conch, and dorje, or double-headed\nthunderbolt, which the Lama, or high-priest, holds in his hand during\nservice. The praying cylinder is made to revolve by means of an axle\nand string, and a projecting piece of iron strikes a little bell at\neach revolution. Within such cylinders are deposited written prayers,\nand whoever pulls the string properly is considered to have said his\nprayers as often as the bell rings. The worshippers, on entering the\ntemple, walk up to the altar, and, before or after having deposited\ntheir gifts, they lift both hands to the forehead, fall on their\nknees, and touch the ground three times with head and hands. They then\nadvance to the head Lama, _kotow_ similarly to him, and he blesses\nthem, laying both hands on their heads, and repeating a short formula.\nSometimes the dorje is used in blessing, as the cross is in Europe,\nand when a number of people request a benediction, the Lama pronounces\nit from the door of the temple with outstretched arms, the people all\nbeing prostrate, with their foreheads touching the ground.\n\n\nPROCLAMATION FOR THE PERSON OF GEORGE II.\n\nOn the young Pretender landing in Scotland, Government issued a\nproclamation, offering a reward of L30,000 for his head, alive or dead.\nIn opposition to this, the following curious paper was issued by the\nPrince and his council, which, Mr. Beloe says, \"is so rare, that I\nnever heard of any other than that which accident lately deposited in\nthe British Museum.\"\n\n \"Charles, Prince of Wales, &c.\n\n Regent of the Kingdoms of Scotland, France, and Ireland, and the\n Dominions thereunto belonging,\n\nWhereas, we have seen a certain scandalous and malicious paper,\npublished in the style and form of a proclamation, bearing date the 1st\ninstant, wherein, under pretence of bringing us to justice, like our\nRoyal Ancestor, King Charles I. of blessed memory, there is a reward of\nL30,000 sterling promised to those who shall deliver us into the hands\nof our enemies, we could not but be moved with a just indignation at\nso insolent an attempt; and though, from our nature and principles,\nwe abhor and detest a practice so unusual among Christian Princes, we\ncannot but, out of just regard to the dignity of our person, promise a\nlike reward of L30,000 sterling to him, or those, who shall seize and\nsecure till our further orders, the person of the Elector of Hanover,\nwhether landed, or attempting to land, in any part of his Majesty's\ndominions. Should any fatal accident happen from hence, let the blame\nbe entirely at the door of those who first set the infamous example.\n\n \"CHARLES, P. R.\n\n \"Given at our Camp, at Kinlockeill, August 22, 1745.\n\n \"By his Highness's Command.\n\n \"JOHN MURRAY.\"\n\n\nDOGS IN JAPAN.\n\nDogs or common curs they have, and in superfluous numbers. These\ndogs are as much the pest of the towns of Japan as they are of\nConstantinople and the other foul cities and towns of the Ottoman\nEmpire. This vast increase of the canine species, and the encouragement\nand immunity accorded to it, arose (according to the popular account)\nout of a curious superstition and an extravagant imperial decree. An\nEmperor who reigned at the close of the eighteenth century chanced to\nbe born under the Sign of the Dog, the Dog being one of the twelve\ncelestial signs of the Japanese Zodiac. For this reason the Emperor had\nas great an esteem for dogs as the Roman Emperor Augustus is reported\nto have entertained for rams. When he ascended the throne, he willed\nand ordained that dogs should be held as sacred animals; and, from\nthat time, more puppies saw the light, and were permitted to live in\nJapan than in any other country on the face of the earth, Turkey,\nperhaps, excepted. These dogs have no masters, but lie and prowl\nabout the streets, to the exceeding great annoyance of passengers,\nespecially if they happen to be foreign travellers, or Christians in\nChristian dresses. If they come round you in packs, barking, snarling,\nand showing their teeth; nay, even if they fall upon you and bite you,\nyou must on no account take the law into your own hands, and beat them\noff or shoot them. To kill one of them is a capital crime, whatever\nmischief the brute may have done you. In every town there are Guardians\nof the Dogs, and to these officers notice must be given in case of any\ncanine misdemeanour, these guardians alone being empowered to punish\nthe dogs. Every street must keep a certain number of these animals,\nor at least provide them with victuals; huts, or dog-hospitals,\nstand in all parts of the town, and to these the animals, in case of\nsickness, must be carefully conveyed by the inhabitants. The dogs that\ndie must be brought up to the tops of mountains and hills, the usual\nburying-places of men and women, and there be very decently interred.\nOld Kaempfer says:--\"The natives tell a pleasant tale on this head. A\nJapanese, as he was carrying the carcase of a dead dog to the top of\na steep mountain, grew impatient, grumbled, and cursed the Emperor's\nbirthday and whimsical command. His companion bid him hold his tongue\nand be quiet, and, instead of swearing, return thanks to the gods that\nthe Emperor was not born under the Sign of the Horse, for, in that\ncase, the load would be heavier.\"\n\n\nLAGMI, AND THE USE MADE OF IT.\n\nMohammed, we are told, prohibited the use of wine, owing to a drunken\nquarrel among the chiefs of his army, which produced great disorder\nand confusion in his affairs, and almost caused the prophet's death in\none of his daring military engagements. He, therefore, addressed his\nfollowers in these words: \"The devil desires to sow dissensions among\nyou, through wine and games of chance, to divert you from remembering\nGod, and praying to him. Abandon wine and games of chance. Be obedient\nto God and the prophet, his apostle, and take heed unto yourselves.\"\nBut the prophet, who could so minutely delineate the furniture of\nheaven, and the instruments of torture of hell--who could describe the\nmysterious occurrences before the creation was formed into its present\nshape, and predict stupendous events to happen in thousands of years\nto come--could not forsee that man would stupify himself by any other\nbeverages besides \"wine.\" The believers in the Koran at Tozar, a city\nnear the Great Desert, in Africa, certainly abstain from wine, and thus\nobey the prophet's precept, but then they indulge freely in _lagmi_,\nor the juice of the palm-tree, which, when fermented, is as pernicious\nin its effect, when taken in excess, as the wine possibly can be. This\njuice is easily obtained, and more easily still prepared. An incision\nis made in the tree, just beneath the branches, and a jar so fastened\nthat it receives every drop of liquid flowing out. During a night they\nprocure from a tree \"in a producing condition\" (in which it is not\nalways) from a quart to three pints of _lagmi_. When drunk immediately\nit tastes like _genuine_ rich milk, and is perfectly harmless; but\nwhen allowed to stand one night, or, at most, twenty-four hours, it\npartakes (with the exception of the colour, which is whitish,) of the\nquality and flavour of champagne, and that of a far superior sort than\nis usually offered in the British markets. This date-tree wine, (for so\nit may be called,) procured at so little trouble and expense, is to be\nfound in every house, and has its victims reeling through the streets\nof Tozar, just as the stupifying porter has in the streets of English\ncities. But the curious part in connexion with this is, that \"the\nfaithful\" persist in their justification that they do not transgress\ntheir prophet's precept! \"_Lagmi_ is not wine,\" they say, \"and the\nprophet's prohibition refers to wine.\"\n\n\nANGLO-SAXON UMBRELLA.\n\n[Illustration: [++] Anglo-Saxon Umbrella.]\n\nIn Anglo-Saxon times the traveller always wore a covering for his\nhead, which, though in various shapes, in no instance resembled our\nhat, though it was characterised by the general term _haet_. He seems\nto have been further protected against the inclemency of the weather\nby a cloak or (_mentel_). One would be led to suppose that this outer\ngarment was more varied in form and material than any other part\nof the dress from the great number of names which we find applied\nto it, such as--_basing_, _haecce_, _haecla_, or _hacela_, _poell_,\n_pylca_, _scyccels_, _waefels_, &c. The writings which remain throw no\nlight upon the provisions made by travellers against rain; for the\ndictionary makers who give _scur-scead_ (shower-shade) as signifying\nan umbrella are certainly mistaken. Yet that umbrellas were known to\nthe Anglo-Saxons is proved beyond a doubt by a figure in the Harleian\nMSS. which we have engraved above. A servant or attendant is holding an\numbrella over the head of a man who appears to be covered at the same\ntime with the cloak or mantle.\n\n\nTHE HEJIRA.\n\nThe Hejira, Hegira, or Hejra. The flight of Mohammed from Mecca to\nMedina is the epoch of the Mohammedan nations. Omar, the second Caliph,\ninstituted the Hegira in imitation of the Christians, who counted their\nyears from their persecution by Diocletian, (A.D. 284,) and who called\nit the era of the martyrs. Thus the Mohammedans wished to commence\ntheir calculation of time from the period of the most memorable\npersecution they had suffered. The learned Mohammedan astronomers have\nbeen divided in opinion on the exact year of the Christians which\ncorresponds with the Hegira. But the generality of writers place\nthis epoch on Friday, the 16th of July, A.D. 622. The ancient Arabs\ncounted time by solar months; these months always returned in the same\nseason, and their names correspond with the employments which the\nseasons rendered necessary. Since the epoch of the Hegira was fixed the\nMohammedans count time by lunar months, the Arabian year consisting of\n354 days, eight hours, and forty-eight minutes. The intercalary days\nare adjusted by a cycle of thirty lunar years, of which nineteen are\nof 354 days, and eleven of 355 days. The years of excess are in the\nfollowing order:--2, 5, 7, 10, 13, 15, 18, 21, 24, 26, 29.\n\n\nCHINESE PAILOOS.\n\n[Illustration: [++] Chinese Pailoos.]\n\nThe Pailoos, or, as they are commonly but erroneously called, triumphal\narches, form an object of Chinese architecture which, from its constant\nrecurrence in views of Chinese scenery, is almost as familiar to us\nas the pagoda. They are, in fact, monuments to deceased persons of\ndistinction, generally of widows who have not married a second time,\nor of virgins who have died unmarried. The smaller and less important\nones consist merely of two upright posts of wood or granite, supporting\na flat board with an inscription, like, both in purpose and design, to\nthe wooden rails which are used as substitutes for tombstones in some\ndistricts in England. The more important Pailoos have three openings,\nsupported by several boards, with more or less ornament and carving.\nSometimes they are wholly of wood; in others no material is used but\nstone, generally granite; and these two materials are combined in\nvarious proportions in other examples. Sometimes they are raised on\nplatforms as in the annexed example, from a peculiarly graceful one\nnear Canton.\n\nAt other times they are placed on the ground, and even across roads,\nso as to form arches, if they may be called, though certainly not\ntriumphal ones.\n\n\nREMARKABLE GROTTO, AND STORY CONNECTED WITH IT.\n\nNear Lunel, in France, on the eastern bank of the river Herault, is\nthe grotto, known in this part of the country as _la Baume de las\nDonmaisellas_, or _des Fees_. This grotto consists of many large, deep\napartments, some of which are indeed inaccessible; the second (and\nthey are all one below the other), presents to the eye of the beholder\nfour beautiful pillars, about thirty feet high, terminating at the top\nlike palm trees; they are detached from the roof, which is only to be\naccounted for by supposing that the _bottom_, or _floor_, has, in some\nconcussion of nature, sunk from its original level: the third chamber,\nstill descending, and like the former only to be reached by ropes and\nladders, presents, at the farther end, one vast curtain of crystal, to\nwhich the lights, carried on such occasions, give the appearance of all\nmanner of precious stones. Some of the stalactites of this apartment\nare solid and white as alabaster, some clear and transparent as glass;\nthey are of every fantastic form and description, as well as displaying\nperfect representations of cascades, trees, festoons, lances, pillars,\nfruits, flowers, and even the regular arrangement of architecture in\na cathedral. The fourth chamber is a long gallery covered with fine\nsand: beyond this three great pillars present themselves, and behind,\nthere is a lake of thick muddy water. All these grottoes have been\nlong known to the peasantry, but another was lately penetrated, in\nwhich every former variety of stalactite was seen, but, in addition\nto these was found an altar, white, like fine china, having regular\nsteps to it, of the same material: it is composed apparently of layers\nof the opaque stalactite, of a dazzling white and exquisite polish:\nfour twisted columns, of a yellow colour and transparent, whose height\nis lost in the vast roof; an obelisk, perfectly round, of a reddish\ncolour, of a great height, and a colossal figure of a woman, holding\ntwo children in her arms, and placed upon a pedestal, completed the\nastonishment of the daring explorers of this subterraneous cavern. But\nalas! this astonishment was changed into feelings of a more melancholy\ndescription, when they recalled the circumstance, still current in\nthe neighbourhood, that, during the religious wars, a family (whether\nProtestant or Catholic is not ascertained), consisting of a father and\nmother and one or two children, sought refuge in these subterraneous\ngrottoes from the persecution of their enemies, and there preserved a\nmiserable existence, far from the cruelty of\n\n Man, whom Nature formed of milder clay,\n With every kind emotion in his heart,\n And taught alone to weep.\n\nFor some years they supported themselves with berries, and now and\nthen they were seen endeavouring to secure a stray kid or goat for\nfood. The solitude and silence of their almost inaccessible dwelling,\nimbued them and their fate with an awful character; and from being\nobjects of _pity_, they became at length objects of _terror_, to the\nneighbouring peasantry, who told strange stories of the unfortunate\nbeings thus consigned to cold and hunger, and compelled to seek a\nwretched home within the bowels of the earth. Their spare forms, their\npale countenances, their tattered garments waving in the breeze, all\nthrew a mystic feeling over their appearance, and they were transformed\ninto fairies and spectres. The shepherds fled when they appeared, and\nthe children, as they clung affrighted to their parents, with strained\neyes and parted lips, followed the rapid movements of the mountaineers,\nas they in their turn, alarmed at the sight of their fellow-creatures,\nfled from height to height, until they gained their rocky asylum. Such\nan accumulation of suffering and misery was not, however, calculated\nto prolong existence: terror and fear destroyed the mind, as hunger\nand cold destroyed the body, and after the lapse of a few years, one\nby one, these _spectres_ disappeared: but still they figure in all the\nlocal stories and traditions peculiar to the neighbourhood, under the\nform of witches, fairies, and sorcerers. The question is, whether the\naltar and the figure are not the work of these unfortunate beings, who\nmight find in this employment a transitory solace for their misery.\n\n\nCRUELTY OF HINDOO RITES.\n\nWe extract the following account from \"The Land of the Veda,\" as\nit affords an extraordinary instance of the lengths to which the\nfanaticism of a gross superstition will induce men to proceed:--\n\n\"To satisfy ourselves of the sanguinary character of some of the Hindoo\ndeities, and of the influence they exert over the deluded victims of\nsuperstition, we must witness some of the cruel practices which the\npopular goddess, Kali, imposes on her worshippers. The most remarkable\nfestival is the one called _Charak Puja_.\n\n\"This festival derives its name from _chakra_, a wheel or discus; in\nallusion to the circle performed in the act of rotating, when suspended\nfrom the instrument of this horrible superstition. Being desirous of\nwitnessing the ceremony in all its parts, I went to the spot where one\nof these ceremonies was about to take place. An upright pole, twenty\nor thirty feet in height, was planted in the ground, across the top of\nwhich, moving on a pivot, a long pole was placed. From one end of this\ntransverse beam a long rope was suspended and left to hang loosely,\nwhilst a shorter rope was attached to the other end, bearing a couple\nof strong iron hooks. A good-looking man, perhaps thirty years of age,\ncame from the midst of the crowd, and doing obeisance beneath the\ninstrument of torture, presented himself as a candidate for the honour\nhe aspired to. The attendant, before whom he stood erect, struck a\nsmart blow on the small of the back, and fixed one of the hooks in the\nflesh, and then did the same on the other side. The man then laid hold\nof the rope just above the hooks and held it, whilst certain persons\nin the crowd, seizing the loose rope, pulled him up, by depressing the\nother end of the beam. As he rose he relinquished his hold of the rope\nby which he was suspended, and resigned himself to the rotary motion,\nby which he was whirled round and round in mid air, suspended by the\nflesh of his own body. Whilst he was thus enduring the torture incident\nto this horrid service, at once gratifying the cruel goddess Kali and\nthe crowd of admiring spectators, he drew from his girdle fruits and\nflowers, which he scattered among the attendants. These were picked\nup by the crowd, with the greatest eagerness, as precious relics that\nmight avail as charms in cases of personal or domestic extremity. This\nwretched dupe of a foul superstition remained in the air at least a\nquarter of an hour, and, of course, in his own estimation and in that\nof the spectators, gained by this brief infliction a large amount of\nmerit, and consequent title to certain rewards to be reaped in a future\nstate of being. No sooner had he descended, than another was ready for\nthe ceremony. These cruel practices are carried on in various parts of\nthe native town, from day to day, as long as the festival lasts. It\nnot unfrequently happens that the ligaments of the back give way, when\nthe man, tossed to an immense distance, is dashed to pieces. In such\ncases, the inference is, that the victim of such accident, by virtue\nof demerit in a former state of existence, was not merely unworthy of\nthe privileges attached to this privileged ceremonial, but destined to\nexpiate his evil deeds by this dreadful accident.\"\n\n\nCURIOUS MUSICAL INSTRUMENT.\n\nThe musical instrument which we engrave below, is used in the Burman\nempire, and is thus described by Captain Yule, in his \"Mission to Ava,\"\nwriting from the town of Magwe, in Burmah. The Captain says;--\n\n[Illustration: [++] Drum-Harmonicon.]\n\n\"This evening the members of the mission made their first acquaintance\nwith the Burmese drama; an entertainment which from this time would\noccupy a very large place in the daily history of our proceedings if\nall were registered.\n\n\"The Governor had provided both a puppet play and a regular dramatic\nperformance for our benefit, and on this first occasion of the kind the\nEnvoy thought it right that we should visit both.\n\n\"Each performance was attended by a full Burmese orchestra. The\nprincipal instruments belonging to this are very remarkable, and, as\nfar as I know, peculiar to Burmah.\n\n\"The chief instrument in size and power is that called in\nBurmese _pattshaing_, and which I can only name in English as a\ndrum-harmonicon. It consists of a circular tub-like frame about thirty\ninches high and four feet six inches in diameter. This frame is formed\nof separate wooden staves fancifully carved, and fitting by tenon into\na hoop which keeps them in place. Round the interior of the frame\nare suspended vertically some eighteen or twenty drums, or tom-toms,\ngraduated in tone, and in size from about two and a-half inches\ndiameter up to ten. In tuning the instrument the tone of each drum is\nmodified as required by the application of a little moist clay with a\nsweep of the thumb, in the centre of the parchment. The whole system\nthen forms a sort of harmonicon, on which the performer, squatted in\nthe middle, plays with the natural plectra of his fingers and palms,\nand with great dexterity and musical effect.\"\n\n\nBURMESE MUSICAL INSTRUMENTS.\n\n[Illustration: [++] Burmese Musical Instruments.]\n\nThe two Burmese musical instruments which we here engrave are thus\ndescribed by Captain Yule in his \"Mission to Ava:\"--\n\n\"The bamboo harmonicon or staccato is a curious example of the\nproduction of melody by simple and unexpected means. Its use, though\nunknown in India, extends throughout the Eastern Archipelago; and\nsomething similar is possessed, I believe, by the slaves in\nBrazil. Eighteen to twenty-four flat slips of bamboo, about an inch and\na half broad, and of graduated length, are strung upon a double string\nand suspended in a catenary over the mouth of a trough-like sounding\nbox. The roundish outside of the bamboo is uppermost, and whilst the\nextremities of the slips are left to their original thickness, the\nmiddle part of each is thinned and hollowed out below. The tuning is\naccomplished partly by the regulation of this thinning of the middle\npart. The scale so formed is played with one or two drumsticks, and\nthe instrument is one of very mellow and pleasing tone. Though the\nmaterials are of no value, a good old harmonicon is prized by the\nowner, like a good old Cremona, and he can rarely be induced to part\nwith it.\n\n\"There was one example at the capital, of a similar instrument formed\nof slips of iron or steel. It was said to have been made by the august\nhands of King Tharawadee himself, who, like Louis Seize, was abler as\na smith than as a king. The effect was not unpleasing, and strongly\nresembled that of a large Geneva musical box, but it was far inferior\nin sweetness to the bamboo instrument.\n\n\"Another instrument used in these concerts is a long cylindrical guitar\nof three strings, shaped like an alligator and so named. It is placed\non the ground before the performer.\"\n\n\nDRESS REGULATED BY ACT OF PARLIAMENT.\n\nThe foreign knights and visitors who came to Windsor in Edward\nthe First's reign, and brought with them a continual succession\nof varying fashions, turned the heads of the young with delight,\nand of the old with disgust. Douglas, the monk of Glastonbury, is\nespecially denunciative and satirical on this point. He says that in\nthe horrible variety of costume,--\"now long, now large, now wide, now\nstraight,\"--the style of dress was \"destitute and devert from all\nhonesty of old arraye or good usage.\" It is all, he says, \"so nagged\nand knibbed on every side, and all so shattered and also buttoned, that\nI with truth shall say, they seem more like to tormentors or devils in\ntheir clothing, and also in their shoying and other array, than they\nseemed to be like men.\" And the old monk had good foundation for his\ncomplaint; and the Commons themselves having, what the Commons now have\nnot, a dread of becoming as extravagant as their betters in the article\nof dress, actually sought the aid of Parliament. That august assembly\nmet the complaint by restricting the use of furs and furls to the royal\nfamily and nobles worth one thousand _per annum_. Knights and ladies\nworth four hundred marks yearly, were permitted to deck themselves in\ncloths of gold and silver, and to wear certain jewellery. Poor knights,\nsquires, and damsels were prohibited from appearing in the costume of\nthose of higher degree. As for the Commons themselves, they could put\non nothing better than unadorned woollen cloth; and if an apprentice or\na milliner had been bold enough to wear a ring on the finger, it was in\nperil of a decree that it should be taken off,--not the finger, but the\nring,--with confiscation of the forbidden finery.\n\nThe consequence was that the Commons, being under prohibition to put\non finery, became smitten with a strong desire to assume it; and much\ndid they rejoice when they were ruled over by so consummate a as\nRichard of Bordeaux. All classes were content to do what many classes\njoyfully do in our own days,--dress beyond their means; and we find in\nold Harding's \"Cronicle\" that not only were\n\n \"Yemen and gromes in cloth of silk arrayed,\n Sattin and damask, in doublettes and in gownnes.\"\n\nbut that all this, as well as habits of \"cloth of greene and\nscarleteen,--cut work and brodwar, was all,\" as the Chronicler\nexpresses it, \"for unpayed;\" that is, was _not paid for_. So that\nvery many among us do not so much despise the wisdom afforded us by\nthe example of our ancestors as didactic poets and commonplace honest\nwriters falsely allege them to do. And those ancestors of Richard\nthe Second's time were especially given to glorify themselves in\nparti- garments of white and red, such being the colours of the\nKing's livery (as blue and white were those of John of Gaunt); and they\nwho wore these garments, sometimes of half-a-dozen colours in each, why\nthey looked, says an old writer, \"as though the fire of St. Anthony, or\nsome such mischance,\" had cankered and eaten into half their bodies.\nThe long-toed shoes, held up to the knee by a chain and hook, were\ncalled _crackowes_, the fashion thereof coming from Cracrow in Poland.\nThe not less significant name of \"devil's receptacles\" were given to\nthe wide sleeves of this reign, for the reason, as the Monk of Evesham\ntells us, that whatever was stolen was thrust into them.\n\n\nA CAT-CLOCK.\n\nThe following curious incident is to be found in Huo's \"Chinese\nEmpire:--\n\n\"One day when we went to pay a visit to some families of Chinese\nChristian peasants, we met, near a farm, a young lad, who was taking\na buffalo to graze along our path. We asked him carelessly, as we\npassed, whether it was yet noon. The child raised his head to look at\nthe sun, but it was hidden behind thick clouds, and he could read no\nanswer there. \"The sky is so cloudy,\" said he; \"but wait a moment;\" and\nwith these words he ran towards the farm, and came back a few minutes\nafterwards with a cat in his arms. \"Look here,\" said he, \"it is not\nnoon yet;\" and he showed us the cat's eyes, by pushing up the lids with\nhis hands. We looked at the child with surprise, but he was evidently\nin earnest; and the cat, though astonished, and not much pleased at the\nexperiment made on her eyes, behaved with most exemplary complaisance.\n\"Very well,\" said we; \"thank you;\" and he then let go the cat, who made\nher escape pretty quickly, and we continued our route.\n\nTo say the truth, we had not at all understood the proceeding; but we\ndid not wish to question the little pagan, lest he should find out that\nwe were Europeans by our ignorance. As soon as ever we reached the\nfarm, however, we made haste to ask our Christians whether they could\ntell the clock by looking into the cat's eyes. They seemed surprised\nat the question; but as there was no danger in confessing to them our\nignorance of the properties of the cat's eyes, we related what had\njust taken place. That was all that was necessary; our complaisant\nneophytes immediately gave chase to all the cats in the neighbourhood.\nThey brought us three or four, and explained in what manner they might\nbe made use of for watches. They pointed out that the pupil of their\neyes went on constantly growing narrower until twelve o'clock, when\nthey became like a fine line, as thin as a hair, drawn perpendicularly\nacross the eye, and that after twelve the dilation recommenced.\n\nWhen we had attentively examined the eyes of all the cats at our\ndisposal, we concluded that it was past noon, as all the eyes perfectly\nagreed upon the point.\n\nWe have had some hesitation in speaking of this Chinese discovery, as\nit may, doubtless, tend to injure the interest of the clock-making\ntrade, and interfere with the sale of watches; but all considerations\nmust give way to the spirit of progress. All important discoveries\ntend in the first instance to injure private interests, and we hope,\nnevertheless, that watches will continue to be made, because, among\nthe number of persons who may wish to know the hour, there will, most\nlikely, be some who will not give themselves the trouble to run after\nthe cat, or who may fear some danger to their own eyes from too close\nan examination of hers.\"\n\n\nEARLY ENGLISH HELMET.\n\n[Illustration: [++] Twelfth Century English Helmet.]\n\nThe above is a correct representation of a helmet of the latter part\nof the twelfth century, resembling those seen on the great seals of\nRichard I. The _aventaille_, or moveable grating for covering the face,\nhas been lost, but the hinges, staples, and other means of fastening it\nstill remain. Its form may be seen on the great seals of Henry III. and\nEdward I.\n\n\nILLUSTRIOUS FARMERS.\n\nAdam was a farmer while yet in Paradise, and after his fall was\ncommanded to earn his bread by the sweat of his brow. Job, the honest,\nupright, and patient, was a farmer, and his firm endurance has passed\ninto a proverb. Socrates was a farmer, and yet wedded to the glory of\nhis immortal philosophy. Cincinnatus was a farmer, and the noblest\nRoman of them all. Burns was a farmer, and the Muse found him at his\nplough, and filled his soul with poetry. Washington was a farmer, and\nretired from the highest earthly station to enjoy the quiet of rural\nlife, and present to the world a spectacle of human greatness. To these\nnames may be added a host of others, who sought peace and repose in the\ncultivation of their earth. The enthusiastic Lafayette, the steadfast\nPickering, the scholastic Jefferson, the fiery Randolph, all found an\nEl Dorado of consolation from life's cares and troubles, in the green\nand verdant lawns that surrounded their homestead.\n\n\nANCIENT COUTEAU-DE-CHASSE.\n\n[Illustration: [++] Couteau-de-Chasse.]\n\nAs the chase was regarded as the honourable and most instructive\noccupation of an age in which warlike prowess was deemed the principal\nobject of emulation and applause, every respectable mansion had, in\nformer times, its hall decorated with hunting implements. One of these\nwe here present to our readers. It is a couteau-de-chasse of the time\nof William III. The left-hand figure represents it in its sheath, which\nis highly ornamented; the other figures represent the blade drawn, and\nthe three knives, fork, and bodkin, which the sheath also contains. The\nform is precisely like those engraved in the \"Triumph of Maximilian,\"\nwhich shows that no variation had taken place since the commencement of\nthe sixteenth century. Erasmus, in his \"Praise of Folly,\" thus alludes\nto this weapon, Kennet translating it \"a slashing hanger.\" Speaking of\nthose engaged in the chase, he says, \"When they have run down their\ngame, what strange pleasure they take in cutting it up! Cows and sheep\nmay be slaughtered by common butchers, but what is killed in hunting\nmust be broke up by none under a gentleman, who shall throw down his\nhat, fall devoutly on his knees, and drawing a slashing hanger (for\na common knife is not good enough), after several ceremonies, shall\ndissect all the parts as artistically as the best skilled anatomist;\nwhile all that stand round shall look very intently and seem to be\nmightly surprised with the novelty, though they have seen the same an\nhundred times before; and he that can but dip his finger and taste of\nthe blood shall think his own bettered by it.\"\n\n\nDIVISION OF TIME IN PERSIA.\n\nTime is of no value in Persia, from which reason it must be that\nso complicated a system has been maintained as that of counting by\nsolar time, lunar time, and the Toork cycle. The first is observed by\nastronomers, and was in general use in Persia until it was superseded\nby Mahommed's lunar year. It consists of twelve months of thirty days\neach, with the required number of intercalary days. The second, which\nis now in general use, consisting of three hundred and fifty-four days,\nis therefore perpetually changing: an event commemorated in one year\nwill come round ten days earlier the succeeding year. The third is a\ncurious method of counting introduced by the Toorks into Persia, but\nwhich we are told has been forgotten in Turkey. They divide time into\ncycles of twelve years, each year having a separate name, but they\nhave no designation for the cycles. Thus, if they wanted to describe\nan event which happened sixty-five years ago, they could only mention\nthe name of the fifth year. These years are solar, and are thus\ndesignated:--\n\n Sichkan eel Year of the Mouse.\n Ood eel \" Bull.\n Bars eel \" Leopard.\n Tavishkan eel \" Hare.\n Looee eel \" Crocodile.\n Eelan eel \" Snake.\n Yoont eel \" Horse.\n Kooree eel \" Ram.\n Beechee eel \" Monkey.\n Tekhakoo eel \" Cock.\n Eet eel \" Dog.\n Tenkooz eel \" Hog.\n\nIt seems strange their number should be twelve, as if there were a\nzodiac of years, instead of months.\n\nThis method of marking time is preserved only in government documents,\nsuch as firmans, grants, &c. No one seems able to account for its\norigin, excepting that, according to tradition, the Toorks of old\nbrought it from Tartary.\n\n\nDIFFERENT SORTS OF HORSES IN THE SIXTEENTH CENTURY.\n\nThe different sorts of horses in use among the nobility and others,\nmay be collected from the following entry in the Northumberland\nhousehold-book, first printed in the year 1768. It is entitled\nthe regulations and establishment of Algernon Percy the Earl of\nNorthumberland, 1512.\n\n\"This is the ordre of the chequir roule of the nombre of all the horsys\nof my lordis and my ladys, that are apoynted to be in the charge of the\nhous verely, as to say gentill hors, palfreys, hobys, naggis, clothsek\nhors.\n\n\"First, gentill hors, to stand in my lordis stable, six. Item, palfreys\nof my ladys, to wit, oone for my lady, and two for her gentill-women,\nand oone for her chamberer. Four hobys and naggis for my lordis oone\nsaddill, viz. oone for my lorde to ride, oone to led for my lorde,\nand oone to stay at home for my lorde. Item, chariot hors to stand in\nmy lordis stable yerely: Seven great trottynge hors to draw in the\nchariott, and a nagg for the chariott-man to ryde, eight. Again, hors\nfor my lorde Percy, his lordis sonne. A great doble trottynge hors to\ntravel on in winter. Item, double trottynge hors, called a curtal, for\nhis lordship to ryde on out of townes. Another trottynge gambaldyn\nhors, for his lordship to ryde upon when he comes into townes. An\namblynge hors, for his lordship to journey on daily. A proper amblyng\nlittle nag, for his lordship when he gaeth on hunting or hawkin. A gret\namblyng gelding to carry his male.\"\n\nThe _gentill_ horse was one of superior breed, so called in contrast to\nsuch as were of ordinary extraction.\n\n_Palfreys_, were an elegant and easy sort of horses, used upon common\noccasions by knights, and others, who reserved their great and managed\nhorses for battle and the tournament.\n\n_Hobys_, were strong, active horses, of rather a small size. They are\nsaid to be originally natives of Ireland.\n\n_Nags_ were of the same description.\n\n_Clothseck_, was a cloak-bag horse; as a _male horse_ was one that\ncarried the portmanteau. Horses to draw the _chariots_, were waggon\nhorses; from the French word _charrette_, whence, the English word\n_cart_.\n\n_A great double trottynge_ horse, was a tall, broad horse, whose best\npace was the trot, being too unwieldly to be able to gallop.\n\n_A curtail_, was a horse whose tail was cut, or shortened.\n\n_A gambaldynge_ horse, was one of shew and parade; a managed horse.\n\n_An amblynge_ horse, received this appellation, from the ease and\nsmoothness of its pace. In former times almost all saddle horses were\nbroke to perform it.\n\n\nTHE NAORA.\n\nThe Oasis of Tagius or Wodian, in the Desert of Sahara, in Africa,\ncomprehends these villages--D'kash, Krees, Wozorkan, Owlad, Majed,\nSedadah, Zowiat Elarab, and Sidy Bohlan.\n\nThese villages are situated at short distances from each other,\nnumbering together a population of between 25,000 and 30,000, whose\nchief employment consists in cultivating the palm, or date tree. At\nKreez they have an excellent spring, but which does not suffice to\nwater all their plantations, and hence they are forced to have recourse\nto the _naora_, so common on the coast. The naora is the name given\nto the rude, though ingenious contrivance, by means of which, through\nthe agency of either a camel, a mule, or a horse, water is raised\nfrom a deep well in earthen jars, which, as soon as they have emptied\ntheir contents into a wooden trough, descend for fresh supplies. The\nwater from the trough is then conducted by the planters into channels\nand trenches, as occasion requires. These are again easily diverted,\nand as soon as it is considered that the trees in one particular\ndirection have had a sufficient supply, fresh trenches are opened in\nanother direction, and in this manner the whole plantation receives the\nrequisite moisture and nourishment. We here engrave the naora.\n\nThe pain and labour which the inhabitants of such an oasis take with\ntheir vast date plantations are immense, but their toil is amply repaid\nby the \"lord of the vegetable world.\" Independent of its picturesque\nappearance, grateful shade, luscious fruit, and agreeable beverage, it\nsupplies them with fuel, and wood for the construction of their houses.\nFrom its leaves they manufacture baskets, ropes, mats, bags, couches,\nbrushes, brooms, fans, &c. From the branches they make fences, stools,\nand cages. The kernels, after being soaked in water for two or three\ndays, are eagerly eaten by camels.\n\nEvery palm-tree shoots forth a number of suckers, which are removed at\nthe proper season and transplanted. With care, these will produce fruit\nin about ten years, whereas those raised from kernels will only yield\ndates when they reach to the age of twenty. The tree reaches its vigour\nat thirty, and continues so till a hundred years old, when it begins\nto decline, and decays about the end of its second century. During its\nvigorous years, a good tree will produce between twenty and thirty\nclusters, each weighing about thirty pounds.\n\n[Illustration: [++] Naora.]\n\nMr. Morier relates an anecdote, which greatly illustrates how highly\nthe date-tree is appreciated by those who are from their infancy taught\nto value it. An Arab woman who had been in England, and who returned in\nthe suite of the English ambassador to Persia, on her reaching home,\ntold her countrywomen of the riches and beauty of the country she had\nvisited, and described the roads, the carriages, the scenery, the\nsplendour of the cities, and the fertility of the well-cultivated soil.\nHer audience were full of admiration, and had almost retired in envy,\nwhen she happened to mention that there was but one thing wanting to\nmake the whole almost a Paradise. \"And what is that?\" said they. \"Why,\nit has not a single date-tree. All the time that I was there, I never\nceased to look for one, but I looked in vain.\" The charm was instantly\nbroken; the Arabs turned away in pity for men, who, whatever might be\ntheir comforts, or their magnificence, were doomed to live in a country\nwhere there are no date-trees.\n\n\nPRIMITIVE PAIR OF BELLOWS.\n\n[Illustration: [++] Bellows.]\n\nAtmospheric denudation and weathering have produced remarkable effects\non the lower part of the Nonkreem valley, in the Khasia mountains,\nin India, which is blocked up by a pine-crested hill, 200 feet high,\nentirely formed of round blocks of granite, heaped up so as to resemble\nan old moraine; but, like the Nunklow boulders, these are not arranged\nas if by glacial action. The granite is very soft, decomposing into a\ncourse reddish sand, that colours the Boga-panee. To procure the iron\nsand, which is disseminated through it, the natives conduct water over\nthe beds, and as the lighter particles are washed away, the remainder\nis removed to troughs, where the separation of the ore is completed.\nThe smelting is very rudely carried on in charcoal fires, blown by\nenormous double-action bellows, worked by two persons, who stand on the\nmachine, raising the flaps with their hands, and expanding them with\ntheir feet, as shown in our cut. There is neither furnace nor flux used\nin the reduction. The fire is kindled on one side of an upright stone\n(like the head-stone of a grave), with a small arched hole close to the\nground: near this hole the bellows are suspended: and a bamboo tube\nfrom each of its compartments meets in a larger one, by which the draft\nis directed under the hole in the stone to the fire. The ore is run\ninto lumps as large as two fists, with a rugged surface: these lumps\nare afterwards cleft nearly in two to show their purity.\n\n\nPRESERVATION OF DEAD BODIES.\n\nAbout a mile distant from Palermo in Sicily, is a celebrated Monastery\nof Capuchins, in which there is a vault made use of as a receptacle for\nthe dead. It consists of four wide passages, each forty feet in length,\ninto which the light is admitted by windows, placed at the ends. Along\nthe sides of these subterraneous galleries are niches, in which the\nbodies are placed upright, and clothed in a coarse dress, with their\nheads, arms, and feet bare. They are prepared for this situation by\nbroiling them six or seven months upon a gridiron, over a slow fire,\ntill all the fat and moisture are consumed. The skin which looks\nlike pale- leather, remains entire, and the character of the\ncountenance is, in some degree preserved.\n\n\nTHE CAGOTS.\n\nIn the Department of the Hautes Pyrenees in France is sometimes to be\nmet with a creature about four feet high, with an enormous head, stiff,\nlong hair, a pale countenance, a dead-looking eye, legs that have the\nappearance of being in the last stage of a dropsy, and an enormous\n_goitre_ on the neck, which sometimes hangs down below the stomach.\nThis unhappy being begs for charity by extending his hand, smiling\nvaguely, and by uttering inarticulate sounds or suppressed cries,\nwhich his desolate and degraded situation alone interprets. These\n_Cagots_, for so they are here called, live isolated from the rest of\nthe world; twenty years ago, if any one of these unfortunate beings\nleft his hut, and ventured into the towns or villages, the children\nwould exclaim--_Cagot! Cagot!_ and this cry would bring the smith from\nhis forge, the shopkeeper from his counter, the private individual\nfrom his fireside; and, if the poor being did not hasten his flight,\nand slow was his progress, he not unfrequently lost his life by the\nstones that were flung after him. There was, however, one day in the\nweek--Sunday, the Lord's day--and one asylum--the church, the Lord's\nhouse--that was free to them; yet man there made a distinction between\nhim and his fellow man. A narrow door, through which no one passed\nbut the _Cagots_, a chapel, which no one entered but these unhappy\n_Cagots_, was reserved for their sole use, where they offered up their\nimperfect prayers, without seeing or being seen by any one. Even in\nthese days, they are still considered an outcast race; and an alliance\nof a peasant girl of the plains with a _Cagot_, would excite as much\ncommotion among the inhabitants of the valleys of the Pyrenees, as\nthe famed one between Idamore and Neala, in M. Delavigne's celebrated\ntragedy of the Paria. Yet it is strange that these deformities do not\nshow themselves until a child has passed the age of six or seven: he\nis before this period like other healthy children; his complexion is\nfresh, his eye lively, and his limbs in proportion; but at twelve, his\nhead has increased prodigiously, his complexion has become sallow,\nhis teeth have lost their whiteness, his eye its fire. Three years\nlater his skin is shrivelled, his teeth open with difficulty, and he\npronounces all the consonants with a whistling indistinctness, that\nrenders his language unintelligible to strangers. His mind partakes of\nthe deformity and weakness of his body, for he is, at fifteen, little\nbetter than an idiot. Such are the _Cagots_ of the _Pyrenees_.\n\n\nDISCONTINUANCE OF TORTURE.\n\nTorture had been applied, down to the close of Elizabeth, to the\ninvestigation of all kinds of crime; but after that time it was chiefly\nconfined to state offences. Its favourite instrument was the dreadful\nrack, or break, traditionally said to have been introduced under\nHenry VI. by John, Duke of Exeter, constable of the Tower, whence it\nwas called the Duke of Exeter's daughter. A _milder_ punishment was\ninflicted by Skevington's gyves, which compressed the victim closely\ntogether, whilst the rack distended his whole frame in the most painful\nmanner. In 1588 the manacles were introduced, and soon became the\nmost usual mode of torture, but their precise character is not well\nunderstood. A variety of instruments of torture are still shown in\nthe Tower, taken, it is said, out of the Spanish Armada, but at all\nevents admirably suited to the gloomy dungeon wherein they appear, and\nin which half-starvation, and the horrid cells called Little Ease and\nRat's Dungeon (the latter placed below high water mark, and totally\ndark, so that the rats crowded in as the tide rose,) added to the\nsufferings of the poor victim when released for a brief space from\nthe fell grasp of the prison-ministers. Torture was not abolished\nin Scotland till 1708; in France till 1789; in Russia till 1801; in\nBavaria and Wurtemberg till 1806; in Hanover till 1822; nor in the\nGrand Duchy of Baden till 1831.\n\n\nTHE MODERN NAMES OF REGIMENTS.\n\nThe modern names of regiments were first given to them in the reign\nof Charles II., the Coldstreams or Foot Guards being formed in 1660,\nwhen two regiments were added to one raised about ten years before by\nGeneral Monk at Coldstream on the borders of Scotland; to these were\nadded the 1st Royal Scots, brought over from France at the Restoration.\nThe Life Guards were raised in 1661, with the Oxford Blues (so called\nfrom the first commander, Aubrey, Earl of Oxford); and also the 2nd\nor Queen's Foot. The 3rd or Old Buffs were raised in 1665, and the\n21st Foot or Scotch Fusileers (from their carrying the fusil, which\nwas lighter than the musket), in 1678. In that year the Grenadiers (so\nnamed from their original weapon, the hand grenade) were first brought\ninto our service, and in 1680 the 4th or King's Own were raised. James\nII. added to the cavalry the 1st or King's Regiment of Dragoon's\nGuards, and the 2nd or Queen's ditto in 1685; and to the infantry, in\nthe same year, the 5th and 7th, or Royal Fusileers; and in 1688 the\n23rd or Welsh Fusileers.\n\n\nWATCH PRESENTED BY LOUIS THE THIRTEENTH OF FRANCE TO CHARLES THE FIRST\nOF ENGLAND.\n\n[Illustration: [++] Watch Made for Charles I.]\n\nThe annexed engraving represents the watch which was made for Louis\nXIII. to present to King Charles I. It is of silver, richly gilt, the\nornaments covered with transparent enamel in white, red, green, blue,\nand yellow. The numbers are on a band of deep blue; the wheel-like\nornament in the centre on a ruby ground. The back is chased in high\nrelief with a figure of St. George conquering the Dragon; the horse is\ncovered with white enamel; the flesh tints on St. George are also of\nenamel; his tunic is red, and his scarf blue. On the side of the watch\nis the motto of the Order of the Garter; the _fleurs-de-lys_ above and\nbelow it on a ruby ground. The interior of the case is enriched by a\ndelicately executed arabesque filled with black enamel upon a dotted\nground. The entire works take out of the case, being secured thereto by\nsprings, and are all more or less decorated with engraving, the whole\ninterior being chased and gilt. The maker's name is S. Vallin.\n\n\nA WEDDING A HUNDRED YEARS AGO.\n\nOn the 7th June, 1750, was married at Rothbury, Mr. William Donkin, a\nconsiderable farmer, of Tosson, in the county of Northumberland, to\nMiss Eleanor Shotten, an agreeable young gentlewoman, of the same\nplace. The entertainments on this occasion were very grand, there\nbeing provided no less than one hundred and twenty quarters of lamb,\nforty quarters of veal, twenty quarters of mutton, a large quantity\nof beef, twelve hams, with a suitable number of chickens, which was\nconcluded with eight half ankers of brandy made into punch, twelve\ndozen of cider, and a great many gallons of wine. The company consisted\nof five hundred ladies and gentlemen, who were diverted with the music\nof twenty-five fiddlers and pipers; and the evening was spent with the\nutmost unanimity.\n\n\nGRACE KNIVES.\n\nThere is a curious class of knives, of the sixteenth century, the\nblade, of which have on one side the musical notes to the benediction\nof the table, or grace before meat, and on the other the grace after\nmeat. We here engrave a specimen.\n\n[Illustration: [++] Grace Knife.]\n\nThe set of these knives usually consisted of four. They were kept in\nan upright case of stamped leather, and were placed before the singer\naccording to the adaptation of each part to the voice indicated upon\nthem.\n\n\nGARDEN AT KENILWORTH WHEN IN ITS PRIME.\n\nGossiping Laneham is very eloquent about the Kenilworth Garden, at\nwhich he took a timid and surreptitious peep. It was an acre or more in\nextent, and lay to the north of the stately castle: a pleasant terrace,\nten feet high, and twelve feet broad, even under foot and fresh with\ntrim grass, ran beside it along the castle wall. It was set with a\ngoodly show of obelisks and spheres, and white bears of stone, raised\nupon goodly bases. At each end was a fine arbour, redolent with sweet\ntrees and flowers. The garden-plot near had fair alleys of turf, and\nothers paved with smooth sand, pleasant to walk on as the sea-shore\nwhen the wave has just retired. The enclosure was divided into four\neven quarters: in the midst of each, upon a base of two feet square,\nrose a porphyry square pilaster, with a pyramidical pinnacle fifteen\nfeet high, pierced and hollowed, and crowned with an orb. All around\nwas covered with redolent herbs and flowers, varied in form, colour,\nand quantity, and mixed with fruit trees.\n\nIn the midst, opposite the terrace, stood a square aviary, joined to\nthe north wall, in height twenty feet, thirteen long, and fourteen\nbroad; it had four great windows, two in front and two at each end,\nand each five feet wide. These windows were arched, and separated by\nflat pilasters, which supported a cornice. The roof was of wire net,\nof meshes an inch wide; and the cornice was gilded and painted with\nrepresentations of precious stones. This great aviary had also eaves\nin the wall, for shelter from sun and heat, and for the purpose of\nbuilding. Fair holly trees stood at each end, on which the birds might\nperch and pounce. They had a keeper to attend to their seeds and water,\nand to clean out their enclosure. The birds were English, French, and\nSpanish. Some were from America; and Laneham is \"deceived\" if some were\nnot from the Canary Islands.\n\nIn the centre of this miniature Paradise stood a fountain with an\noctagonal basin rising four feet high; in the midst stood the figures\nof two Athletes, back to back, their hands upholding a fair marble\nbowl, from whence sundry pipes distilled continual streams into the\nreservoir. Carp, tench, bream, perch, and eel disported in the fresh\nfalling water; and on the top of all the ragged staff was displayed;\non one side Neptune guided his sea-horses with his trident, on another\nstood Thetis with her dolphins. Here Triton and his fishes, there\nProteus and his herds, Doris and her daughter, and half the Nereids,\ndisported in sea and sand, surrounded by whales, sturgeons, tunnies,\nand conch shells, all engraven with exquisite device and skill. By the\nsudden turn of a tap, the spectator could be drenched at the pleasure\nof any wit.\n\n\nEGYPTIAN ANTIQUITIES.\n\nIt appears from a paper recently read in the Academy of Archaeology, at\nRome, that Father Secchi has found a new interpretation of the Egyptian\nhieroglyphics, which enables him to declare, that most of them are not\nmere tombstone inscriptions, as is generally assumed, but poems. He\nhas given several of his readings, which display great ingenuity, and\nprofesses to be able to decipher the inscriptions on the Obelisk of\nLuxor, at Paris.\n\n\nTHE BAYEUX TAPESTRY.\n\nThe cathedral at Bayeux is a gothic building, dedicated to the Virgin.\nThe portal and three belfries, which belong to it, are objects of\ncuriosity. It is in this cathedral that the celebrated tapestry,\ndenominated _of Bayeux_, is kept. Its length is one hundred and\nthirty-two feet; its breadth, seven and a half. \"I had,\" says Dr.\nDucarel, \"the satisfaction of seeing that famous piece of furniture,\nwhich, with great exactness, though in _barbarous needlework_,\nrepresents the history of Harold, King of England; and of William,\nDuke of Normandy; from the embassy of the former to Duke William, at\nthe command of Edward the Confessor, to his overthrow and death, at\nthe battle fought near Hastings. The ground of this piece of work is a\nwhite linen cloth, or canvas. The figures of men, horses, &c. are in\ntheir proper colours, worked in the manner of the samplers, in worsted,\nand of a style not unlike what we see upon the China and Japan ware;\nthose of the men, particularly, being without the least symmetry or\nproportion. There is a small border, which runs at the top and the\nbottom of the tapestry; with several figures of men, beasts, flowers,\nand even fables, which have nothing to do with the history, but are\nmere ornaments. At the end of every particular scene there is a tree,\nby way of distinction; and over several of the principal figures there\nare inscriptions, but many of them obliterated. It is annually hung\nup on St. John's day, and goes round the nave of the church, where\nit continues eight days; and at all other times it is carefully kept\nlocked up in a strong wainscot press, in a chapel on the south side of\nthe cathedral, dedicated to Thomas a Becket. By tradition it is called,\n_Duke William's toilet_, and is said to be the work of Matilda, his\nqueen, and the ladies of her court, after he had obtained the crown of\nEngland.\" Mr. Strutt, in his \"Complete View of the Dresses and Habits\nof the People of England,\" affirms, that it is the work of half a\ncentury later than the time of the Conqueror.\n\n\nROMAN STAMP.\n\nThis curiosity is preserved in the British Museum. It is the very\nearliest specimen we possess of printing, by means of ink or any\nsimilar substance. It is made of metal, a sort of Roman brass; the\nground of which is covered with a green kind of verdigris rust, with\nwhich antique medals are usually covered. The letters rise flush up to\nthe elevation of the exterior rim which surrounds it. Its dimensions\nare, about two inches long, by one inch broad. At the back of it is a\nsmall ring for the finger, to promote the convenience of holding it. As\nno person of the name which is inscribed upon it is mentioned in Roman\nHistory, he is therefore supposed to have been a functionary of some\nRoman officer, or private steward, and who, perhaps, used this stamp to\nsave himself the trouble of writing his name. A stamp somewhat similar,\nin the Greek character, is in the possession of the Antiquarian\nSociety, of Newcastle-on-Tyne.\n\n\nTYRIAN PURPLE.\n\nThe shell-fish portrayed on next page is that from which the Tyrian\npurple dye is obtained. The ancients were very devoid of chemical\nknowledge; their list of adjective dye-stuffs was therefore restricted,\nand all the most celebrated dyes of antiquity belonged to the\nsubstantive division, of which Tyrian purple was undoubtedly the chief.\nThe purple dye of Tyre, which admits with great propriety of being\nincluded amongst the dyes of Greece and Rome, was discovered about\nfifteen centuries before the Christian era, and the art of using it\ndid not become lost until the eleventh century after Christ. It was\nobtained from two genera of one species of shell-fish, the smaller of\nwhich was denominated _buccinum_, the larger _purpura_, and to both the\ncommon name murex was applied. The dye-stuff was procured by puncturing\na vessel in the throat of the larger genus, and by pounding the smaller\nentire. Having been thus extracted, salt was added, also a certain\namount of water. The whole was then kept hot about eight or ten days in\na vessel of lead or tin, the impurities as they rose being assiduously\nskimmed off. The dye-stuff was now ready to receive the texture to\nbe dyed (wool, universally), and the operation of dyeing was simple\nenough; nothing further being required than the immersion of the whole\nfor a sufficient time, when, at the expiration of a certain period, the\nwhole of the colouring matter was found to have been removed, and to\nhave combined with the textile fabric.\n\nThe tints capable of being imparted by this material were\nvarious--representing numerous shades between purple and crimson.\nAmongst these a very dark violet shade was much esteemed, but the\nright imperial tint, we are informed, was that resembling coagulated\nblood. The discovery of Tyrian purple dye is referred to the fifteenth\ncentury before Christ. That it was known to the Egyptians, in the\ntime of Moses, is sufficiently obvious from the testimony of more\nthan one scriptural passage. Ultimately, in later ages, a restrictive\npolicy of the eastern emperors caused the art to be practised by only\na few individuals, and at last, about the commencement of the twelfth\ncentury, when Byzantium was already suffering from attacks without,\nand dissensions within, the secret of imparting the purple dye of Tyre\nbecame lost.\n\n[Illustration: [++] Shell-Fish From Which Tyrian Purple Is Extracted.]\n\nThe re-discovery of Tyrian purple as it occurred in England was made\nby Mr. Cole of Bristol. About the latter end of the year 1683, this\ngentleman heard from two ladies residing at Minehead, that a person\nliving somewhere on the coast of Ireland, supported himself by marking\nwith a delicate crimson colour the fine linen of ladies and gentlemen\nsent him for that purpose, which colour was the product of some liquid\nsubstance taken out of a shell-fish. This recital at once brought\nto the recollection of Mr. Cole the tradition of Tyrian purple. He,\nwithout delay, went in quest of the shell-fish, and after trying\nvarious kinds without success, his efforts were at length successful.\nHe found considerable quantities of the buccinum on the sea-coast\nof Somersetshire, and the opposite coast of South Wales. The fish\nbeing found, the next difficulty was to extract the dye, which in its\nnatural state is not purple, but white, the purple tint being the\nresult of exposure to the air. At length our acute investigator found\nthe dye stuff in a white vein lying transversely in a little furrow or\ncleft next to the head of the fish.\n\n\nTHE INCARNATIONS OF VISHNU.\n\nThere is a part of the mythology of India which seems to be blended\nwith the history of that country. It relates to the different _avatars_\nof Vishnu, or his incarnations and appearances on earth.\n\n[Illustration: [++] Incarnation of Vishnu.]\n\nThe first of these _avatars_ has reference to that general deluge of\nwhich all nations have preserved some traditions. Vishnu, we are told,\nmetamorphosed himself into a fish.\n\nThe second incarnation is that of _Kourma_, or the tortoise. The gods\nand the giants, wishing to obtain immortality by eating _amourdon_,\ndelicious butter, formed in one of the seven seas of the universe,\nwhich the Indians call sea of milk, transported, by Vishnu's advice,\nthe mountain of Mandreguivi into that sea: they twisted round it the\nserpent Adissechen, and alternately pulling, some by his hundred heads,\nothers by the tail, they made the mountain turn round in such a manner,\nas to agitate the sea and to convert it into butter; but they pulled\nwith such rapidity, that Adissechen, overcome with weakness, could no\nlonger endure it. His body shuddered; his hundred trembling mouths made\nthe universe resound with hisses; a torrent of flames burst from his\neyes; his hundred black pendent tongues palpitated, and vomited forth a\ndeadly poison, which immediately spread all around. The gods and giants\nbetook themselves to flight. Vishnu, bolder than the rest, took the\npoison, and with it rubbed his body, which became quite blue. It is in\nmemory of this event, that this colour is given to his image in almost\nall the temples.\n\nThe gods and the giants, encouraged by Vishnu's example, fell to\nwork again. After they had laboured a thousand years, the mountain\nwas on the point of sinking in the sea, when Vishnu, in the form of\na tortoise, quickly placed himself beneath, and supported it. At\nlength they saw the cow Camadenu, the horse with seven heads, and the\nelephant with three trunks, coming out of the sea of milk; also the\ntree _calpaga vrutcham_; Lacshmi, goddess of riches, wife of Vishu;\nSaraswadi, goddess of the sciences and of harmony, married to Brama;\nMondevi, goddess of discord and misery, whom nobody would have, and who\nis represented riding on an ass, and holding in her hand a banner, on\nwhich a raven is delineated; and, lastly, Danouvandri, the physician,\ncarrying a vessel full of _amourdon_, which the gods instantly\nseized, and greedily devoured, without leaving a morsel. The giants,\ndisappointed in their expectations, dispersed over the earth, prevented\nmankind from paying worship to the gods, and strove to obtain adoration\nfor themselves. Their insolence occasioned the subsequent incarnations\nof Vishnu, who endeavoured to destroy this race, so inimical to the\ngods. He is adored in this second metamorphosis, by the name of\n_Kourma Avatara_. The followers of Vishnu believe that this god,\nthough omnipresent, resides more particularly in the _vaicondom_, his\nparadise, amidst the sea of milk, reclined, in contemplative slumber,\non the serpent Adissechen, which serves him for a throne: in this state\nhe is called _Siranguan_. In all the temples of Vishnu is to be seen\nthe figure of this god; but as the serpent on which he lies cannot be\nrepresented with his hundred heads, he is delineated with only five.\n\nThere are altogether ten incarnations of Vishnu; nine of these have\nalready been fulfilled, and one is yet to be manifested, it is\nexpected about ninety thousand years hence. The account of many of the\ntransformations is exceedingly extraordinary, but we have room for no\nmore than the one we have given.\n\n\nORIGIN OF LONG-TOED SHOES.\n\nLong-toed shoes were invented by Fulk, Count of Anjou, to hide an\nexcrescence on one of his feet. These toes were so long as to be\nfastened to the knees with gold chains, and carved at the extreme point\nwith the representation of a church window, a bird, or some fantastic\ndevice.\n\n\nTHE HOUSE OF HEN'S FEATHERS.\n\nThere exists at Pekin a phalanstery which surpasses in eccentricity\nall that the fertile imagination of Fourier could have conceived. It\nis called Ki-mao-fan--that is, \"House of the Hen's Feathers.\" By dint\nof carrying out the laws of progress, the Chinese have found means\nto furnish to the poorest of the community a warm feather-bed, for\nthe small consideration of one-fifth of a farthing per night. This\nmarvellous establishment is simply composed of one great hall, and the\nfloor of this great hall is covered over its whole extent by one vast\nthick layer of feathers. Mendicants and vagabonds who have no other\ndomicile come to pass the night in this immense dormitory. Men, women,\nand children, old and young, all without exception, are admitted.\nCommunism prevails in the full force and rigour of the expression.\nEvery one settles himself and makes his nest as well as he can for\nthe night in this ocean of feathers; when day dawns he must quit the\npremises, and an officer of the company stands at the door to receive\nthe rent of one sapeck each for the night's lodging. In deference no\ndoubt to the principle of equality, half-places are not allowed, and a\nchild must pay the same as a grown person.\n\nOn the first establishment of this eminently philanthropic and moral\ninstitution, the managers of it used to furnish each of the guests\nwith a covering, but it was found necessary to modify this regulation,\nfor the communist company got into the habit of carrying off their\ncoverlets to sell them, or to supply an additional garment during the\nrigorous cold of winter. The shareholders saw that this would never do,\nand they should be ruined, yet to give no covering at all would have\nbeen too cruel, and scarcely decent. It was necessary therefore to find\nsome method of reconciling the interests of the establishment with the\ncomfort of the guests, and the way in which the problem was solved was\nthis. An immense felt coverlet, of such gigantic dimensions as to cover\nthe whole dormitory, was made, and in the day time suspended from the\nceiling like a great canopy. When everybody had gone to bed, that is to\nsay, had lain down upon the feathers, the counterpane was let down by\npulleys; the precaution having been previously taken to make a number\nof holes in it for the sleepers to put their heads through, in order\nto escape the danger of suffocation. As soon as it is daylight, the\nphalansterian coverlet is hoisted up again, after a signal has been\nmade on the tam-tam to awaken those who are asleep, and invite them to\ndraw their heads back into the feathers, in order not to be caught by\nthe neck and hoisted into the air with the coverlet. This immense swarm\nof beggars is then seen crawling about in the sea of dirty feathers,\nand inserting themselves again into their miserable rags, preparatory\nto gathering into groups, and dispersing about the various quarters of\nthe town to seek by lawful or unlawful means their scanty subsistence.\n\n\nTHE USEFUL AND THE BEAUTIFUL.\n\nThe tomb of Moses is unknown; but the traveller slakes his thirst at\nthe well of Jacob. The gorgeous palace of the wisest and wealthiest\nof monarchs, with cedar, and the gold, and ivory, and even the great\nTemple of Jerusalem, hallowed by the visible glory of the Deity\nhimself, are gone; but Solomon's reservoirs are as perfect as ever. Of\nthe ancient architecture of the Holy City, not one stone is left upon\nanother, but the Pool of Bethsaida commands the pilgrim's reverence, at\nthe present day. The columns of Persepolis are mouldering into dust;\nbut its cistern and aqueducts remain to challenge our admiration. The\ngolden house of Nero is a mass of ruins, but the Aqua Claudia still\npours into Rome its limpid stream. The Temple of the Sun, at Tadmore,\nin the wilderness, has fallen, but its fountain sparkles in its rays,\nas when thousands of worshippers thronged its lofty colonnades. It\nmay be that London will share the fate of Babylon, and nothing be\nleft, to mark it, save mounds of crumbling brickwork. The Thames will\ncontinue to flow as it does now. And if any work of art should rise\nover the deep ocean, time, we may well believe, that it will be\nneither a palace nor a temple, but some vast aqueduct or reservoir;\nand if any name should flash through the mist or antiquity, it would\nprobably be that of the man, who in his day, sought the happiness of\nhis fellow men, rather than glory, and linked his memory to some great\nwork of national utility or benevolence. This is the true glory which\noutlives all others, and shines with undying lustre from generation\nto generation, imparting to works some of its own immortality, and in\nsome degree rescuing them from the ruin which overtakes the ordinary\nmonument of historical tradition or mere magnificence.\n\n\nCROMWELL'S BRIDGE AT GLENGARIFF.\n\n[Illustration: [++] Cromwell's Bridge at Glengariff.]\n\nThe village of Glengariff, near Bantry Bay, consists of but a few\nhouses. The only \"antiquity\" in the immediate neighbourhood is the\nold bridge, now a picturesque ruin, which, in ancient times, was on\nthe high road to Berehaven; it is called \"Cromwell's Bridge.\" It is\naccurately represented in the above engraving. History being silent as\nto the origin of the name, we must have recourse to tradition. When\nOliver was passing through the glen, to \"visit\" the O'Sullivans, he had\nso much trouble in getting across the narrow but rushing river, that he\ntold the inhabitants, if they did not build him a bridge by the time he\nreturned, he would hang up a man for every hour's delay he met with.\n\"So the bridge was ready agin he come back,\" quoth our informant; \"for\nthey knew the ould villian to be a man of his word.\"\n\n\nTHE TURBAN IN ARABIA.\n\nA fashionable Arab will wear fifteen caps one above another, some of\nwhich are linen, but the greater part of thick cloth or cotton. That\nwhich covers the whole is richly embroidered with gold, and inwrought\nwith texts or passages from the Koran. Over all there is wrapped a sash\nor large piece of muslin, with the ends hanging down, and ornamented\nwith silk or gold fringes. This useless encumbrance is considered a\nmark of respect towards superiors. It is also used, as the beard was\nformerly in Europe, to indicate literary merit; and those who affect\nto be thought men of learning, discover their pretensions by the size\nof their turbans. No part of Oriental costume is so variable as this\ncovering for the head. Niebuhr has given illustrations of forty-eight\ndifferent ways of wearing it.\n\n\nSTONEWARE.\n\nStoneware was made at a very early period in China, and is much used as\na basis on which a paste of porcelain is laid, to save the expenditure\nof the latter material, as well as to give strength and solidity to the\npiece. Most of the larger pieces of Oriental production are found to be\nthus formed. The red Japan ware is a very fine unglazed stoneware, and\nhas raised ornaments, which are sometimes gilt. A curious coffee-pot\nof this ware, imitating a bundle of bamboo canes, and not unlike the\nChinese musical instrument called a mouth-organ, from the collection of\nthe late Mr. Beckford, is here represented.\n\n[Illustration: [++] Stoneware Coffee-Pot.]\n\nStoneware is supposed to have been made at a very early period in\nEngland by Dutch and German workmen; and from this circumstance it\nis almost impossible to distinguish the earlier fabrics of these\nrespective countries. The discovery, in 1690, of an economical process\nof glazing this ware by means of common salt, which made it impermeable\nto liquids, soon brought it into general use, and displaced all the\nmanufactures of the Delft and soft paste fabrics. A mottled-brown\nstoneware, known to collectors, is stated to be the manufacture of the\nage of Edward VI., in consequence of some of the specimens having a\nsilver mounting of the make and fashion of the period of Elizabeth's\nreign. There is also a large flagon in the Museum of Economic Geology,\nornamented with the royal arms of Elizabeth in relief, with the date\n1594. These specimens cannot, however, be deemed conclusive of so early\na manufacture in England. The first-mentioned specimens, though the\nmounting is English, may have been of German manufacture, as pieces\nof similar description of ware are to be seen in various collections\nof German pottery abroad. The latter specimen may either have been\nmade at Cologne for the use of the Queen's household, or if of English\nmanufacture, it must, in the opinion of a very eminent manufacturer,\nhave been made at a much later period than the date upon it. In a\nletter received, he states \"that it is a common practice even now among\npotters to use moulds of all dates and styles, which have been got up\noriginally for very different kinds of ornamental work, and that he\nis strongly inclined to think that the mould from which the devices\non this vessel have been pressed, was modelled many years before the\nvessel was made, and that the vessel itself is comparatively modern.\"\nStoneware, ornamented with devices in white clay, was made in the\nseventeenth century at Fulham, also at Lambeth, and subsequently at\nStaffordshire; but there is no satisfactory evidence of any earlier\nmanufactory in England.\n\nTowards the end of the seventeenth century, some specimens of red Japan\nware were imported into Europe. Both Dutch and English manufacturers\nattempted to imitate them, but failed for want of the proper clay.\nAbout this period, two brothers of the name of Elers, from Nuremberg,\ndiscovered at Bradwell, only two miles distant from Burslem, a bed\nof fine compact red clay, which they worked in a small manufactory,\nestablished in a retired situation upon the bed itself. They took\nevery precaution to prevent any one seeing their process or learning\ntheir secret. They went so far as to employ none but the most ignorant\nand almost idiot workmen they could find. Astbury, the elder, had\nthe talent to counterfeit the idiot, and, moreover, the courage to\npersevere in this character for some years during which he continued\nin their employ. From memory he made notes of the processes, and\ndrawings of the machinery used. In consequence of the secret being\nthus discovered, numerous establishments arose in competition with\nthat of the Elers, and, owing to the general prejudice against them\nas foreigners, they were finally compelled, in 1720, to quit their\nestablishment. They retired to the neighbourhood of London, and, it is\nsupposed, contributed by their skill and industry to the establishment\nof the Chelsea Porcelain Manufactory.\n\n\nGREAT BELL OF ROUEN.\n\nThe grand entrance to the cathedral of Rouen is flanked by two towers,\nthe one was erected by St. Romain; the expense for constructing the\nother, which bears the whimsical name of _Tour-de-beurre_, was raised\nby the product arising from permissions granted to the more wealthy\nand epicurean part of the inhabitants of the city, to eat butter in\nLent. It was in this tower that the celebrated bell, the largest in\nthe world, was erected; it weighed 40,000 lbs.; it was converted into\ncannon in the year 1793. The founder of this bell died of joy on seeing\nits completion. It went by his name, that of George D'Amboise, and\nround it was the following distich in gothic characters:--\n\n \"Je suis nomme George d'Amboise,\n Qui bien trente-six-mille poise,\n Et celui qui bien me pesera,\n Quarante mille trouvera.\"\n\n\nVARIATIONS IN THE COINAGE.\n\nHenry VIII. greatly debased both his gold and silver coins, which he\nalloyed with copper to a great extent. The proportions of the pound,\nindeed, in 1546, amounted to 8 oz. of alloy to 4 oz. of silver, which\nconstituted, a positively base coin, the old allowance having been\nbut 18 pennyweights of alloy to 11 oz. and 2 pennyweights of silver.\nHis depreciations were equally daring, for out of the pound of silver\nhe now coined 576 pennies or 48s. The gold coins of this monarch were\nsovereigns, half-sovereigns or rials, half and quarter rials, angels,\nhalf and quarter angels, George nobles, and forty-penny pieces. In\nthis reign the immemorial privileges of the sees of Canterbury, York,\nand Durham, for coining small money, was abandoned, the last Bishop\nthat used it being Wolsey's successor, Edward Lee.\n\nEdward VI. carried both depreciation and debasement still farther; but\ntowards the close of his reign he was obliged to restore the currency\nto something like the ancient standard. He was the first that issued\ncrowns, half-crowns, and sixpences. Little alterations were made by\nMary, beyond striking coins with her husband's head as well as her own;\nbut under Elizabeth the coinage was, at length, completely recovered\nfrom its debasement, the old proportion of 18 pennyweights of alloy\nbeing restored, which has continued to the present day. The number of\nshillings struck out of a pound of silver was not lessened, however,\nfor it continued to be sixty, as in the preceding reign, till 1601,\nwhen it was increased to sixty-two, at which rate it went on to 1816,\nwhen it was raised to sixty-six, at which it now remains. Her gold\ncoins are much the same as before, but are distinguished by having\nthe edges milled for the first time. Shortly before her death she had\nintended to coin farthings and other small pieces of copper, a metal\nwhich had not yet been made use of in this country.\n\n\nCHAFFINCH CONTEST.\n\nAt the town of Armentieres, in France, there is a _fete du pays_,\ncalled _hermesse_, or _ducasse d'Armentieres_, in which the chaffinch\nand its fellows are the chief actors and objects of attraction. Numbers\nof these birds are trained with the greatest care, and no small share\nof cruelty, for they are frequently blinded by their owners, that\ntheir song may not be interrupted by any external object. The point\nupon which the amusement, the honour, and the emolument rests, is, the\nnumber of times which a bird will repeat his song in a given time. A\nday being fixed, the amateurs repair to the appointed place, each with\nhis bird in a cage. The prize is then displayed, and the birds are\nplaced in a row. A bird-fancier notes how many times each bird sings,\nand another verifies his notes. In the year 1812, a chaffinch repeated\nhis song seven hundred times in one hour. Emulated by the songs of each\nother, they strain their little \"plumed throats,\" as if conscious that\nhonour was to result from their exertions.\n\n\nEXPENSIVENESS OF DRESS IN THE TIME OF JAMES I.\n\nDress, indeed, must have swallowed up almost every thing at a time when\nJames and his courtiers set the fashion of appearing in a new garb\nalmost every day. When the Duke of Buckingham was sent to France to\nbring over Henrietta Maria, he provided, amongst others, one suit of\nwhite uncut velvet, and a cloak set all over with diamonds, valued at\nL80,000; besides a feather made of great diamonds, and sword, girdle,\nhat-band, and spurs, thick set with the same. Another suit of purple\nsatin, embroidered all over with pearls, was valued at L20,000. At the\nmarriage of the Princess Elizabeth with the Palatine, Lady Wotton wore\na gown profusely ornamented with embroidery that cost L50 a yard; and\nLord Montague spent L1,500 on the dresses of his two daughters for that\noccasion. By this account it would seem that the ladies were, at all\nevents, not more expensive in their attire than gentlemen.\n\n\nINGENUITY OF THE TUNISIANS.\n\nA stranger visiting a city like Tunis, cannot but be struck with the\nvarious peculiarities, which present themselves to his view, wherever\nhe turns. In their government, mercantile pursuits, professions and\ntrades, the Tunisians are centuries behind. But, with all their\ndisadvantages, the traveller, in traversing their crowded _sooks_\n(market places) and serpentine streets, finds numerous illustrations of\nthe proverb, \"Necessity is the mother of invention.\" In every workshop\nsome tool, or implement, presents itself, which is as curious in its\nformation as it is strange to see the peculiar use for which it is\nintended, and the manner in which it is employed. We may illustrate\nthis by a sketch of a turner.\n\n[Illustration: [++] Ingenuity of a Turner.]\n\nThe extraordinary ingenuity here exhibited by the remarkable use which\nthe artisan makes of his feet and toes, as well as of his hands, cannot\nfail to attract attention; and the display of his lathe and tools is\nequally curious.\n\n\nSHANAR MUSICAL INSTRUMENT.\n\nTwo acts seem essential to the demon worship of the Shanars of\nTin-nevelly (a portion of the aborigines of India)--dancing and bloody\nsacrifices. They have no priest. The person who conducts the ceremony,\nwhich is undertaken from choice, is called the rotator of the demon.\nThe head man of the village, or any other person, male or female, may\nofficiate. The dress is grotesque, consisting of a sort of coat of\nvarious colours, a cap, and other vestments, arranged so as to strike\nthe spectators with their comic appearance. In this service several\nmusical instruments are used, but the most notable among them is one\ncalled a _bow_. It consists of a bow strung and ornamented with bells.\nThis is placed on a brazen vessel of a globular form. The bow is struck\nwith a plectrum, and the bass is produced by the application of an\ninstrument to the brazen pot, another person keeping time by playing a\npair of cymbals, as seen in the annexed cut.\n\n[Illustration: [++] Shanar Musical Instrument.]\n\nThe jarring, discordant, uproarious and cacophonous character of this\nmusical accompaniment exceeds description, and when the parties are\nvieing with each other for pre-eminence, it is indeed the most horrid\ndin that can be produced. At first the movements of the dancer may\nbe slow, but as the music waxes louder and takes effect, he becomes\ngradually more excited, urging himself to phrenzy by striking himself\nviolently, and applying his mouth to the neck of the decapitated\nsacrificial victim, he drinks its blood, and possibly a potation\nof ardent spirits. The afflatus thus acquired, its effects become\nvisible in the frantic glare and the convulsive gesticulations of\nthe possessed. This is greeted by the spectators with the loudest\nacclamations. The dancer is now deified or demonized, and he is\nconsulted by the eager and delighted worshippers who do him homage.\nEach one puts his questions as his fancy or his needs may dictate.\nThe possessed or demonized dancer, being more like a maniac than\naught else, and subject to various contortions of body, utters his\noracles with much indistinctness, rendering it necessary that some\none initiated into these mysteries should interpret his wild and\nincoherent utterances. His ambiguous sayings and curious innuendos are\nso indefinite as to need interpretation.\n\n\nSINGULAR LOCAL CUSTOMS.\n\nIn the department of the Hautes Alpes of France, in the commune\nof _Guillaume-Perouse_, at the village of _Andrieux_, where the\ninhabitants are deprived during one hundred days of the bright beams\nof the sun, there is a fete, called _Le retour du soleil_, on the\n10th of February. At the dawn of day, four shepherds announce, to the\nsound of fifes and trumpets, the commencement of this joyous day.\nEvery cottager having prepared an omelette, the eldest inhabitant of\nthe village, to whom the title of _Venerable_ is given, leads the way\nto the square; here they form a chain and dance the _ferandola_ round\nhim: after the dance is concluded, he leads the way to a stone bridge\nat the entrance of the village, the shepherds playing upon their rural\ninstruments the while. Every one having deposited his omelette on the\nstone coping, they repair to a neighbouring meadow, where the dancing\nre-commences and continues until the first rays of the sun gleam\nathwart the velvet turf: the dance then instantly ceases, each one\nhastens for his pancake, and holding it up, presents it as an offering\nto the god of day; the _Venerable_ holds his up with both his hands.\nAs soon as the sun shines upon the village the procession returns to\nthe square, where the party separates, and every one repairs to his\nown home, to eat his pancake with his family. This ceremony cannot\nfail to recal the heathen mythology to the reader, who must see in it\nthe offerings made to Apollo; or, perhaps, it may be the remains of\nsome Druidical superstition, as the Druids paid particular devotion to\nthe sun; at any rate, it is a curious vestige of some religion long\nsince gone by. In some of the communes of this department the dead\nare wrapped in a winding-sheet, but are not inclosed in a coffin. In\nthe valleys of _Queyras_ and of _Grave_, the dead are suspended in a\nbarn during five months in the winter, until the earth be softened by\nthe sun's rays, when the corpse is consigned to its native element.\nAll funereal ceremonies are closed by eating and drinking. In some\ncommunes the people carry a flagon of wine to the churchyard; and on\nthe return of the guests to the home of the deceased, it becomes a\nscene of bacchanalian revels, in which the groans and sighs of the\nmourners mingle with the songs and jests of the inebriated guests.\nAt _Argentiere_, after the burial, the tables are set out round the\nchurchyard; that of the curate and the mourning family over the grave\nitself. The dinner concluded, the nearest relation takes a glass; his\nexample is followed by the rest, repeating with him, _A la sante du\npauvre mort_.\n\n\nSEVERITY OF RUSSIAN PUNISHMENTS.\n\nThe Russians are remarkable for the severity and variety of their\npunishments, which are both inflicted and endured with a wonderful\ninsensibility. Peter the Great used to suspend the robbers upon the\nWolga, and other parts of his dominions by iron hooks fixed to their\nribs, on gibbets, where they writhed themselves to death, hundreds,\nnay thousands, at a time. The single and double knoute were lately\ninflicted upon ladies, as well as men of quality. Both of them are\nexcruciating, but in the double knoute, the hands are bound behind\nthe prisoner's back; and the cord being fixed to a pulley, lifts him\nfrom the ground, with the dislocation of both his shoulders, and then\nhis back is in a manner sacrificed by the executioner, with a hard\nthong, cut from a wild ass's akin. This punishment has been so often\nfatal, that a surgeon generally attends the patient to pronounce the\nmoment that it should cease. Another barbarous punishment practised in\nRussia is, first boring the tongue of the criminal through with an hot\niron, and then cutting it out: and even the late Empress Elizabeth,\nthough she prohibited capital punishments, was forced to give way to\nthe necessity of those tortures. From these particulars, many have\nconcluded that the feelings of the Russians are different from those of\nmankind in general.\n\n\nFIRST RHINOCEROS IN EUROPE.\n\nThe first rhinoceros ever seen in Europe was that of which Pliny speaks\nas having been presented by Pompey to the Roman people. According to\nDion Cassius, Augustus caused another to be killed in the Roman circus,\nwhen celebrating his triumph over Cleopatra. Strabo states that he saw\none at Alexandria, and he has left a description of it. All these were\nof the one-horned species. At a later period the two-horned species\nwere introduced, as appears from medals bearing their effigies struck\nin the reign of Domitian. During the time known as the dark ages,\ninvestigations in natural history and every other department of science\nand learning were utterly neglected, and the rhinoceros was as mythical\nto Europe as the phoenix or the salamander. On the revival of letters,\nhowever, and the extension of maritime discovery, a lively interest was\nmanifested in the productions of foreign countries. In 1513 the king\nof Portugal presented the Roman Pontiff with a rhinoceros captured in\nIndia; but, unfortunately, the ship was wrecked on its way to Italy:\nthe pope lost his present, and the rhinoceros his life. All that was\npreserved was a rough sketch, engraved by Albert Durer; and down to a\nvery recent date, nearly all our representations were taken from this\nrough draft.\n\nIn 1685 a rhinoceros was captured and brought to England. In 1739 and\n1741 two others were exhibited in various parts of Europe. In 1800 a\nyoung one was brought from India, intended for a menagerie at Vienna,\nbut died at London on the way, and was dissected by Mr. Thomas, who\npublished the results of his investigations, and thus gave the public a\nbetter idea of the animal than they ever had before.\n\n\nTURKISH CARRIAGE.\n\nThe curiously-shaped vehicle which we have engraved on the next page,\nis a Turkish _araba_, a carriage chiefly used by ladies. An account of\none of them is pleasantly introduced by Mr. Albert Smith in his \"Month\nat Constantinople\" when describing the visit of the Sultan to one of\nthe mosques:--\n\n\"Every Friday the Sultan goes to mosque publicly. It is not known until\nthe very morning which establishment he means to patronise; but your\ndragoman has secret channels of information, and he always informs you\nin time to 'assist' at the ceremony.\n\n\"The first time I went, Abdul Medjid had selected for his devotions the\nmosque of Beglerbeg, a village on the Asiatic side of the Bosphorus,\nthe temple of which stands in the same relation and bearing to St.\nSophia--to use a very familiar simile--as Rotherhithe Church does to\nSt. Paul's. It was a perfect English morning--foggy and cold (Oct. 7)\nwith muddy streets and spitting rain. I crossed into Asia--one learns\nto speak of Asia, at Constantinople, as he would do of the borough--in\na two-oared caique, and on landing went up to the mosque, which is\nclose to the shore.\n\nA crowd of people, consisting principally of females, had collected\nbefore the mosque, and a square space was kept by the soldiers. Some\nlittle courtesy was shown to visitors, as the Franks were permitted to\ncross this enclosure to a corner close to the door, by which the Sultan\nwas to enter.\n\n[Illustration: [++] Turkish Carriage.]\n\nHe was not very punctual to his time, but there was enough to amuse\nthe visitors; more especially in the arrival of the women, who came up\nas near as they could to the building, in all sorts of odd vehicles.\nSeveral were like those I had seen on the bridge at Pera, but one\nwas very fine indeed. It was more like a waggon than a carriage, and\npainted bright blue, with red wheels and awning. In it were five ladies\nof the Sultan's harem, very gaily dressed, and laughing loudly as the\nvehicle shook them about over the rugged road. It was drawn by two\nbuffaloes, and they had a singular arrangement of worsted tufts over\ntheir heads, of various bright colours. This was the first waggon of\nthe kind I had seen, but I afterwards found them very common. Other\nwomen were on foot, and a number of these had collected upon a hillock\nunder a tree, where they talked and quarrelled incessantly. One very\npale and handsome girl arrived alone, in a car, preceded by two or\nthree attendants: and, whilst trying to pass a narrow thoroughfare\namongst the other vehicles, the wheel of her own got smashed to pieces.\nShe was then close to the Frank visitors, and, as she appeared likely\nto be overturned, two or three gentlemen from Misseri's hotel, ran\nforward to offer their assistance. In a minute they were put back by\nthe attendants, who could not think of allowing their mistress to be\ntouched, even from chance, by a Christian. The carriage was propped\nup, as well as it could be; and its inmate, who had remained perfectly\ntranquil during the accident, fixed her large eyes on the enclosure,\nand never moved them again, to the right or left.\"\n\n\nCURIOUS INDIAN COMB.\n\n[Illustration: [++] Indian Comb.]\n\nAt the foot of the Himalayas, and not far from the European station\nof Darjeeling, there is a tract of country which is still inhabited\nby a tribe of very ancient origin, called the Mechs; they are rapidly\ndegenerating, and indeed may be said to be even now almost worn out as\na distinct tribe. They are but rarely visited by Europeans; but Dr.\nHooker inspected their district in 1850, and gives the following brief\ndescription of its appearance:--\n\n\"We arrived on the third day at the Mechi river, to the west of which\nthe Nepal Morung begins, whose belt of Sal forest loomed on the\nhorizon, so raised by refraction as to be visible as a dark line, from\nthe distance of many miles. It is, however, very poor, all the large\ntrees having been removed. We rode for several miles into it, and\nfound the soil dry and hard, but supporting a prodigious undergrowth\nof gigantic harsh grasses that reached to our heads, though we were\nmounted on elephants. Tigers, wild elephants, and the rhinoceros are\nsaid to be found here; but we saw none.\n\n\"The old and new Mechi rivers are several miles apart, but flow in the\nsame depression, a low swamp many miles broad, which is grazed at this\nseason, and cultivated during the rains. The grass is very rich, partly\nowing to the moisture of the climate, and partly to the retiring\nwaters of the rivers; both circumstances being the effects of proximity\nto the Himalayas. Hence cattle (buffaloes and the common humped cow\nof India) are driven from the banks of the Ganges 300 miles to these\nfeeding grounds, for the use of which a trifling tax is levied on each\nanimal. The cattle are very carelessly herded, and many are carried off\nby tigers.\"\n\nWe give a sketch on previous page of a pocket-comb which Dr. Hooker\nobtained from one of the natives: it is, at all events, much more\ntasteful in its form and ornamentation than the usual run of English\npocket-combs.\n\n\nSINGULAR HINDOO VOW.\n\nThe following extraordinary vow is performed by some of the Hindoo\nat their festival of _Charak Puja_:--Stretching himself on the earth\non his back, the devotee takes a handful of moist earth, and placing\nthis on his under lip, he plants in it some mustard-seed, and exposes\nhimself to the dews of the night and the heat of the day till the seed\ngerminates. In this posture the man must lie in a fixed motionless\ncondition, without food or drink, till the vegetable process liberates\nhim, which will generally be about the fourth day.\n\n\nTHE ARRANGEMENT OF ABBEY BUILDINGS.\n\nAt the dissolution of the Abbeys in England, under King Henry VIII.,\n190 were dissolved, of from L200 to L35,000 a year; amounting to an\naggregate sum of L2,853,000 per annum. The principal buildings of\nan Abbey, were, first, the church, differing little from one of the\ncathedrals of the present day. Attached to one side of the nave,\ncommonly the southern, was, secondly, the great cloister, which had\ntwo entrances to the church, at the eastern and western ends of the\naisles of the nave, for the greater solemnity of processions. Over the\nwestern side of the cloister, was, thirdly, the dormitory of the monks;\na long room, divided into separate cells, each containing a bed, with\na mat, blanket, and rug, together with a desk and stool, and occupied\nby a monk. This apartment had a door, which opened immediately into\nthe church, on account of midnight offices. Attached to the side of\nthe cloister, opposite to the church, was fourthly, the refectory,\nwhere the monks dined; near to which, was the locutorium, or parlour,\nan apartment answering to the common room of a college, where in the\nintervals of prayer and study, the monks sat and conversed. Beyond,\nwas the kitchen and its offices; and, adjoining to it, the buttery,\n&c. On the eastern side of the cloisters was, in the centre, the\nchapter-house, where the business of the Abbey was transacted; and near\nit, the library, and scriptorium, where the monks employed themselves\nin copying books. On this side, also, was the treasury, where the\ncostly plate and church ornaments were kept. The abbot and principal\nofficers of the convent, had all separate houses, to the eastward of\nthe cloister; in which part of the building, were usually the hostelry\nand question hall--rooms for the entertainment of strangers; and, also,\nthe apartment of novices. Westward of the cloister was an outward\ncourt, round which was the monks' infirmary, and the almery. An\nembattled gatehouse led to this court, which was the principal entrance\nof the Abbey. The whole was surrounded with a high wall, including\nin its precincts, gardens, stables, granary, &c. Some of the great\nAbbeys--as Glastonbury, and Furness--covered sixty acres of ground. The\nsituation chosen for the site of an Abbey was as different from that of\nthe castle as the purpose to which it was applied. The one meant for\ndefence stands boldly on the hill; the other, intended for meditation,\nis hid in the sequestered valley. The abbots were originally laymen and\nsubject to the bishop.\n\n\nTAME FISH.\n\nIn sailing down the river Irawadi, in the neighbourhood of Amarapoora,\nthe capital of the empire of Burmah, Captain Yule met with some tame\nfish, which he thus describes:--\n\n\"Having gone over the little island, I returned to my boat, where a\nsight awaited me, that I confess astonished me more than anything I\nhave ever seen before.\n\n\"On nearing the island as we descended the river, the headman in the\nboat had commenced crying out _tet-tet! tet-tet!_ as hard as he could,\nand on my asking him what he was doing, he said he was calling the\nfish. My knowledge of Burmese did not allow me to ask him further\nparticulars, and my interpreter was in the other boat, unwell. But, on\nmy coming down to the boat again, I found it surrounded on both sides\nwith large fish, some three or four feet long; a kind of blunt-nosed,\nbroad-mouthed dog-fish. Of these there were, I suppose, some fifty.\nIn one group, which I studied more than the others, there were ten.\nThese were at one side of the boat, half their bodies, or nearly half,\nprotruded vertically from the water, their mouths all gaping wide. The\nmen had some of the rice prepared for their own dinners, and with this\nthey were feeding them, taking little pellets of rice, and throwing\nthese down the throats of the fish. Each fish, as he got something to\neat, sunk, and having swallowed his portion, came back to the boatside\nfor more. The men continued occasionally their cry of _tet-tet-tet!_\nand, putting their hands over the gunnel of the boat, stroked the fish\non the back, precisely as they would stroke a dog. This I kept up for\nnearly half an hour, moving the boat slightly about, and invariably the\nfish came at call, and were fed as before. The only effect which the\nstroking down or patting on the back of the fish seemed to have, was to\ncause them to gape still wider for their food. During March, I am told,\nthere is a great festival here, and it is a very common trick for the\npeople to get some of the fish into the boat, and even to gild their\nbacks by attaching some gold leaf, as they do in the ordinary way to\npagodas, &c. On one of these fish remains of the gilding were visible.\nI never was so amused or astonished. I wished to have one of the fish\nto take away as a specimen, but the people seemed to think it would be\na kind of sacrilege, so I said nothing more on the point. The Phoongyis\nare in the habit of feeding them daily, I was informed. Their place of\nabode is the deep pool formed at the back of the island, by the two\ncurrents meeting round its sides. And it is, it appears, quite a sight,\nwhich the people from great distances come to see, as well as to visit\nthe Pagoda, which is said to be very ancient and much venerated.\"\n\n\nANCIENT WEAPON.\n\nThe formidable weapon which we here engrave, is a concealed ranseur\nof the time of Henry VIII., from Genoa. It forms one long instrument,\nbut our limits have compelled us to divide into three parts. 1, is\nthe butt: 2, the middle; and 3, the point. The upper part is an iron\ncylinder, with a cap on the top. This is opened by touching the bolt\nseen a little below it in front, and then, by giving the weapon a jerk\nforwards, the blades fly out, and produce the form of the partisan.\nUpon those, on each side, is written, \"Al Segno Del Cor\"--\"To the mark\nof the heart.\" When in the state seen in the engraving, the blades are\nheld so firmly that they cannot be thrust back; and the only mode of\nreturning them into the cylinder is by striking the butt end against\nthe ground, when they instantly fall in.\n\n[Illustration: [++] Concealed Ranseur.]\n\nThis weapon, we apprehend, must have been more formidable in appearance\nthan useful in action. Once let a man get a fair thrust with it at his\nenemy, and, it is true, the effect of that one stroke would be fatal,\nbut in battle it would most probably prove fatal also to the man who\nwielded the weapon, for before he could have time to draw it back, a\ncomrade of the wounded man would have plenty of opportunity to rush in\nand cut the striker of the blow down. On seeing this and other clumsy\nweapons which were so much in vogue in former times, we cannot be\nsurprised that none of them have continued in use to the present day.\nWeapons such as the one we here engrave, have long been thrown aside,\nand short weapons are now only used for all hand to hand encounters.\n\n\nTHE BABES OF BETHLEHEM.\n\nIt is an ancient custom at Norton, near Evesham, Worcestershire, on the\n28th of December (Innocents' Day) to ring a muffled peal, in token\nof sorrow for the slaughter of the hapless \"babes of Bethlehem,\" and,\nimmediately afterwards, an unmuffled peal, in manifestation of joy for\nthe deliverance and escape of the infant Saviour.\n\n\nGAUNTLET OF HENRY PRINCE OF WALES.\n\n[Illustration: [++] Gauntlet of Henry Prince of Wales.]\n\nThe highly interesting relic of which we here give a sketch is of a\nrusset colour, engraved and gilt, the ornamental parts being sunk\nlower than the surface. The initials of the owner, surmounted by a\ncoronet, occur in two places, as do also the rose and thistle. Henry\nwas born on the 19th of February, 1594 and was nine years of age when\nhis father ascended the throne of England. When seven, he commenced\nthe acquirement of martial exercises--as the use of the bow, pike,\nfirearms, and the art of riding; and at ten applied to Colonel Edmondes\nto send him a suit of armour from Holland. On the discovery of the\nGunpowder Plot, Lord Spencer made him a present of a sword and target;\nand, in 1607, Louis, the Dauphin, son of Henry IV. of France, sent him\na suit of armour, well gilt and enamelled, together with pistols and\na sword of the same kind, and the armour for a horse. His martial and\nromantic disposition displayed itself on the occasion of his being\ncreated Prince of Wales in 1610, when he caused a challenge to be given\nto all the knights in Great Britain, under the name of Maeliades, Lord\nof the Isles; and on the day appointed, the Prince, assisted only by\nthe Duke of Lenox, the Earls of Arundel and Southampton, Lord Hay, Sir\nThomas Somerset, and Sir Richard Preston, who instructed his Highness\nin arms, maintained the combat against fifty-six earls, barons,\nknights, and esquires. Henry himself gave and received thirty-two\npushes of the pike, and about three hundred and sixty strokes of\nthe sword, not being yet sixteen years of age. From the size of the\ngauntlet, the initials H. P., and a prince's coronet, if not made on\nthis occasion, it could not have been much anterior; and, from most\nof his armour being sent from abroad, the impression would be that it\nis of foreign manufacture. Yet there is in the State Paper Office an\noriginal warrant ordering the payment of L200, the balance of L340,\nfor a rich suit of armour made for Henry Prince of Wales, dated July\n11, 1614, he having died on the 6th of November, 1612. This document\nis directed by King James I. to the Commissioners for the exercise of\nthe office of High Treasurer of England, and states that, \"Whereas\nthere was made, in the office of our armory of Greenwich, by William\nPickeringe, our master workman there, one rich armour with all peeces\ncompleate, fayrely gilt and graven, by the commaundement of our late\ndeere sonne Prince Henry, which armour was worth (as we are informed)\nthe somme of three hundred and forty poundes, whereof the said William\nPickeringe hath receaved of our said late deere sonne the somme of one\nhundred and forty poundes only, soe as there remayneth due unto him the\nsomme of two hundred poundes\"--therefore they are ordered to discharge\nthe same forthwith.\n\n\nTHE SIMOOM.\n\nArabia is frequently visited by the terrible simoom, called by the\nnatives _shamiel_, or the wind of Syria, under whose pestilential\ninfluence all nature seems to languish and expire. This current\nprevails chiefly on the frontiers, and more rarely in the interior.\nIt is in the arid plains about Bussora, Bagdad, Aleppo, and in the\nenvirons of Mecca, that it is most dreaded, and only during the intense\nheats of summer. The Arabs, being accustomed to an atmosphere of great\npurity, are said to perceive its approach by its sulphureous odour,\nand by an unusual redness in the quarter whence it comes. The sky, at\nother times serene and cloudless, appears lurid and heavy; the sun\nloses his splendour, and appears of a violet colour. The air, saturated\nwith particles of the finest sand, becomes thick, fiery, and unfit for\nrespiration. The coldest substances change their natural qualities;\nmarble, iron, and water, are hot, and deceive the hand that touches\nthem. Every kind of moisture is absorbed; the skin is parched and\nshrivelled; paper cracks as if it were in the mouth of an oven. When\ninhaled by men or animals, the simoom produces a painful feeling as of\nsuffocation. The lungs are too rarefied for breathing, and the body is\nconsumed by an internal heat, which often terminates in convulsions\nand death. The carcases of the dead exhibit symptoms of immediate\nputrefaction, similar to what is observed to take place on bodies\ndeprived of life by thunder, or the effect of electricity.\n\nWhen this pestilence visits towns or villages, the inhabitants shut\nthemselves up, the streets are deserted, and the silence of night\neverywhere reigns. Travellers in the desert sometimes find a crevice\nin the rocks; but if remote from shelter, they must abide the dreadful\nconsequences. The only means of escaping from these destructive blasts,\nis to lie flat on the ground until they pass over, as they always move\nat a certain height in the atmosphere. Instinct teaches even animals to\nbow down their heads, and bury their nostrils in the sand. The danger\nis most imminent when they blow in squalls, which raise up clouds of\nsand in such quantities, that it becomes impossible to see to the\ndistance of a few yards. In these cases the traveller generally lies\ndown on the lee side of his camel; but as the desert is soon blown\nup to the level of its body, both are obliged frequently to rise and\nreplace themselves in a new position, in order to avoid being entirely\ncovered. In many instances, however, from weariness, faintness, or\nsleepiness, occasioned by the great heat, and often from a feeling\nof despair, both men and animals remain on the ground, and in twenty\nminutes they are buried under a load of sand. Caravans are sometimes\nswallowed up; and whole armies have perished miserably in these\ninhospitable deserts.\n\n\nBOILING TO DEATH.\n\nOne Rouse, who had attempted to poison Fisher, Bishop of Rochester,\nwho was afterwards murdered in his 77th year, (by Henry VIII.)--was\nactually boiled to death in Smithfield, for his offence. The law which\nthus punished him, was afterwards repealed.\n\n\nSIKKIM PRIESTS.\n\nThe Sikkim country is situated on the frontiers of Thibet and Nepal\nand on a portion of the Himalayas. Dr. Hooker, who visited it a few\nyears ago, gives the following account in his Journal of some of its\nscenery:--\"January 1st, 1849.--The morning of the new year was bright\nand beautiful, though much snow had fallen on the mountains; and we\nleft Sunnook for Pemiongchi, situated on the summit of a lofty spur on\nthe opposite side of the Ratong.\n\n\"The ascent to Pemiongchi was very steep, through woods of oaks,\nchesnuts, and magnolias, but no tree-fern, palms, pothos, or planntain,\nwhich abound at this elevation on the moister outer ranges of Sikkim.\nThe temple is large, eighty feet long, and in excellent order, built\nupon the lofty terminal point of the great east and west spur, that\ndivides the Kulhait from the Ratong and Rungbee rivers; and the great\nChangachelling temple and monastery stands on another eminence of the\nsame ridge, two miles further west.\n\n\"The view of the snowy range from this temple is one of the finest in\nSikkim; the eye surveying at once glance the vegetation of the tropics\nand the poles. Deep in the valleys the river beds are but 3,000 feet\nabove the sea, and are choked with fig-trees, plantains, and palms;\nto these succeed laurels and magnolias; and still higher up, oaks,\nchesnuts, birches, &c.; there is, however, no marked line between the\nlimits of these two last forests, which form the prevailing arboreous\nvegetation between 4,000 and 10,000 feet, and give a lurid hue to the\nmountains. Fir forests succeed for 2,000 feet higher, when they give\nplace to a skirting of rhododendron and barberry. Among these appear\nblack naked rocks, between which are gulleys, down which the snow now\ndescended to 12,000 feet. The mountain flanks are much more steep and\nrocky than those at similar heights on the outer ranges, and cataracts\nare very numerous, and of considerable height, though small in volume.\n\n\"Pemiongchi temple, the most ancient in Sikkim, is said to be 400 years\nold; it stands on a paved platform, and is of the same form and general\ncharacter as that of Tassisuding. Inside, it is most beautifully\ndecorated, especially the beams, columns, capitals, and architraves,\nbut the designs are coarser than those of Tassisuding. The square end\nof every beam in the roof is ornamented either with a lotus flower, or\nwith a Tibetan character, in endless diversity of colour and form, and\nthe walls are completely covered with allegorical paintings of Lamas\nand saints with glories round their heads, mitred, and holding the\ndorje and jewel.\n\n\"The principal image is a large and hideous figure of Sakya-thoba in a\nrecess under a blue silk canopy, contrasting with a calm figure of the\nlate Rajah, wearing a cap and coronet.\n\n\"Pemiongchi was once the capital of Sikkim, and called the Sikkim\nDurbar: the Rajah's residence was on a curious flat to the south of\nthe temple, and a few hundred feet below it, where are the remains of\n(for this country) extensive walls and buildings. During the Nepal\nwar, the Rajah was driven east across the Teesta, whilst the Ghorkas\nplundered Tassisuding, Pemiongchi, Changachelling, and all the other\ntemples and convents to the west of that river. It was then that the\nfamous history of Sikkim, compiled by the Lamas of Pemiongchi, and kept\nat this temple, was destroyed, with the exception of a few sheets,\nwith one of which Dr. Campbell and myself were each presented. We were\ntold that the monks of Changachelling and those of this establishment\nhad copied what remained, and were busy compiling the rest from oral\ninformation, &c.: whatever value the original may have possessed,\nhowever, is irretrievably lost. A magnificent copy of the Buddhist\nScriptures was destroyed at the same time; it consisted of 400 volumes,\neach containing several hundred sheets of Daphne paper.\"\n\n[Illustration: [++] Sikkim Priests.]\n\nOf the figures given in our article, the one on the extreme left is a\nLama, or Sikkim priest, having in his hand a _dorge_, or double-headed\nthunderbolt; next to him, a monk; next to the monk, a priest, with a\npraying cylinder; and at the extreme right, another monk.\n\n\nA HEAD-BREAKER.\n\n[Illustration: [++] Head-Breaker.]\n\nWith many savage nations it is a custom when prisoners have been\ncaptured in war, to keep them in confinement for some time, till the\npreparations for a grand festival have been completed, and then to put\nthem to death in the presence of the great men and chief priests of the\ncountry. They were slaughtered, sometimes as offerings to the gods,\nsometimes as sacrifices to the spirits of those slain in the war in\nwhich they were captured, and at other times as incentives to the young\nwarriors who were to be the future defenders of the nation. In all\nthese cases, appropriate and peculiar ceremonies were prescribed, and\nthe victims were generally despatched by a particular official, whose\nespecial duty it was to perform the bloody deed. A particular weapon\nwas also used, and one of these is sketched at the head of our article.\nIt was used by one of the tribes which inhabit the shores of Nootka\nSound. It is intended to represent the sacred bird of their nation, and\nis made of wood, inlaid with mother-of-pearl, with a blade of basalt.\nThe lower end is hollow for the insertion of a handle.\n\n\nANCIENT STONE COLLARS.\n\nPerhaps the most singular relics of that Pagan period in Scotland\nwhen the use of metals was in a great measure unknown, are two stone\ncollars, found near the celebrated parallel roads of Glenroy, and now\npreserved at the mansion of Tonley, Aberdeenshire. We here give an\nengraving of them.\n\n[Illustration: [++] Stone Collars.]\n\nThey are each of the full size of a collar adapted to a small Highland\nhorse; the one formed of trap or whin-stone, and the other of a\nfine-grained red granite. They are not, however, to be regarded as\nthe primitive substitutes for the more convenient materials of later\nintroduction; on the contrary, a close imitation of the details of a\nhorse collar of common materials is attempted, including the folds, the\nleather, nails, buckles, and holes for tying particular parts together.\nThey are finished with much care and a high degree of polish, and\nare described as obviously the workmanship of a skilful artist. Mr.\nSkene, who first drew attention to these remarkable relics, suggests\nthe peculiar natural features of Glenroy having led to the selection\nof this amphitheatre for the scene of ancient public games, and that\nthese stone collars might commemorate the victor in the chariot race,\nas the tripods, still existing, record the victor in the Choragic games\nof Athens. But no circumstances attending their discovery are known\nwhich could aid conjecture either as to the period or purpose of their\nconstruction.\n\n\nTHE OFFSPRING OF DRUNKENNESS.\n\nFrom an interesting lecture on drunkenness, and on popular investments,\nrecently delivered by the Rev. J. B. Owen, M.A., of Bilston, we\nselect this impressive enumeration of the crimes mainly springing\nfrom drunkenness. Drink was the desolating demon of Great Britain.\nThey had spent in intoxicating drinks during the present century as\nmuch as would pay the national debt twice over! There were 180,000\ngin drinkers in London alone, and in that city three millions a year\nare spent in gin! In thirteen years 249,006 males and 183,921 females\nwere taken into custody for being drunk and disorderly. In Manchester\nno less than a million a-year were spent in profligacy and crime. In\nEdinburgh there were 1,000 whisky shops--160 in one street--and yet\nthe city contained only 200 bread shops. Of 27,000 cases of pauperism,\n20,000 of them were traceable to drunkenness. In Glasgow the poor rates\nwere L100,000 a-year. \"Ten thousand,\" says Alison, \"get drunk every\nSaturday night--are drunk all day Sunday and Monday, and not able to\nreturn to work till Tuesday or Wednesday.\" Glasgow spends L1,200,000\nannually in drink, and 20,000 females are taken into custody for being\ndrunk. And what were some of the normal results of such appalling\nstatistics? insanity, pauperism, prostitution, and crime. As to the\ninsanity affiliated on drink, the Bishop of London stated, that of\n1,271 maniacs, whose previous histories were investigated, 649, or\nmore than half of them, wrecked their reason in drinking. As to its\npauperism, it is estimated that not less than two-thirds of our paupers\nwere the direct or indirect victims of the same fatal vice. As to\nits prostitution, its debauching influence was remotely traceable in\nthe 150,000 harlots of London, and in their awful swarms in all our\nlarge towns and cities. Its relation to crime was equally conclusive.\nIn Parkhurst prison, it was calculated, that 400 out of 500 juvenile\nprisoners, were immured there, as the incidental results of parental\ndebauchery. The Chaplain of the Northampton County Gaol, lately\ninformed the lecturer, that, \"of 302 prisoners in this gaol, during the\nlast six months, 176 attributed their ruin to drunkenness; 64 spent\nfrom 2s. 6d. to 10s. a week in drink; 15 spent from 10s. to 17s.; and\n10 spent all their savings. Is it not remarkable,\" he added, \"that\nout of 433 prisoners in this gaol, I have not had one that has had\none sixpence in a saving's bank, nor above six that ever had sixpence\nin one? On the contrary, I have many members of friendly societies,\nof course of unsound ones, which with two or three exceptions, all\nmet at public houses; and there they learned to drink, and became\nfamiliarised with crime.\" Judge Erskine declared at the Salisbury\nAssizes in 1844, that 96 cases out of every 100 were through strong\ndrink. Judge Coleridge added, at Oxford, that he never knew a case\nbrought before him, which was not directly or indirectly connected with\nintoxicating liquors; and Judge Patteson capped the climax, at Norwich,\nby stating to the grand jury, \"If it were not for this drinking, you\nand I should have nothing to do!\" Of the 7,018 charges entered at Bow\nStreet Police Office, in the year 1850, half of them were for being\ndrunk and incapable; and if they added to these the offences indirectly\ninstigated by intoxication, the proportion rose at least to 75 per cent.\n\n\nAN OLD PIKE.\n\nIn the year 1497 a giant \"Jack-killer\" was captured in the vicinity\nof Mannheim, with the following announcement in Greek appended to\nhis muzzle:--\"I am the first fish that was put into this pond by the\nhands of the Emperor Frederic the Second, on this 3rd day of October,\n1262.\" The age of the informant, therefore, if his lips spoke truth\n(and the unprecedented dimensions of the body left little doubt on that\npoint), was more than two hundred and thirty-five years. Already he\nhad been the survivor of many important changes in the political and\nsocial world around, and would have swam out perhaps as many more had\nthe captors been as solicitous to preserve his life as they were to\ntake his portrait. This, on the demise of the original, was hung up in\nthe castle of Lautern, and the enormous carcase (which, when entire,\nweighed three hundred and fifty pounds, and measured nineteen feet)\nwas sent to the museum at Mannheim, where, deprived of its flesh, and\ncaparisoned _de novo_, it hung, and haply yet hangs, a light desiccated\nskeleton, which a child might move.\n\n\nBURMESE BOAT.\n\nThe curious boat which is here depicted in full sail is one of those\nwhich is used by the Burmese on the river Irawadi. They are called\n_hnau_, and Captain Yule gives the following description of them in his\n\"Mission to Ava:\"--\n\n\"The model is nearly the same for all sizes, from the merest dinghy\nupwards. The keel-piece is a single tree hollowed out, and stretched\nby the aid of fire when green, a complete canoe, in fact. From this,\nribs and planking are carried up. The bow is low with beautiful hollow\nlines, strongly resembling those of our finest modern steamers. The\nstern rises high above the water, and below the run is drawn out fine\nto an edge. A high bench or platform for the steersman, elaborately\ncarved, is an indispensable appendage. The rudder is a large paddle\nlashed to the larboard quarter, and having a short tiller passing\nathwart the steerman's bench.\n\n\"The most peculiar part of the arrangement of these vessels is in the\nspars and rigging. The mast consists of two spars; it is, in fact,\na pair of shears, bolted and lashed to two posts rising out of the\nkeel-piece, so that it can be let down, or unshipped altogether, with\nlittle difficulty. Above the mainyard the two pieces run into one,\nforming the topmast. Wooden rounds run as ratlines from one spar of the\nmast to the other, forming a ladder for going aloft.\n\n\"The yard is a bamboo, or a line of sliced bamboos, of enormous length,\nand, being perfectly flexible, is suspended from the mast-head by\nnumerous guys or halyards, so as to curve upwards in an inverted bow.\nA rope runs along this, from which the huge mainsail is suspended,\nrunning on rings like a curtain outwards both ways from the mast. There\nis a small topsail of similar arrangement.\n\n[Illustration: [++] Burmese Boat.]\n\n\"The sail-cloth used is the common light cotton stuff for clothing.\nOf any heavier material it would be impossible to carry the enormous\nspread of sail which distinguishes these boats. At Menh'la one vessel\nwas lying so close to the shore that I was enabled to pace the length\nof the half-yard. I found it to be 65 feet, or for the length of the\nwhole spar, neglecting the curve, 130 feet. The area of the mainsail in\nthis case could not have been very much less than 4,000 square feet, or\none-eleventh of an acre.\n\n\"These boats can scarcely sail, of course, except before the wind. But\nin ascending the Irawadi, as on the Ganges during the rainy season,\nthe wind is almost always favourable. A fleet of them speeding before\nthe wind with the sunlight on their bellying sails has a splendid\nthough fantastic appearance. With their vast spreading wings and\nalmost invisible hulls, they look like a flight of colossal butterflies\nskimming the water.\"\n\n\nDANCING DERVISHES.\n\n[Illustration: [++] Dancing Dervishes.]\n\nThe Dancing Dervishes at Constantinople are a remarkable instance of\nthe lengths to which superstition and credulity will proceed. The\nsaltatory ceremony which they perform at their religious services\nis thus admirably described by Mr. Albert Smith in his \"Month at\nConstantinople:\"--\n\n\"I have said it was Friday; and so, on my return, I had an opportunity\nof seeing the Dancing Dervishes at Pera. They exhibit--for it is rather\na sight than a solemnity--on this day, as well as on Tuesday, in every\nweek. Their convent is facing the scrap of burying-ground on the road\nfrom Galata to Pera, and any one may witness their antics. Having put\noff our shoes, we entered an octagonal building, with galleries running\nround it, and standing places under them, surrounding the railed\nenclosure in which the Dervishes were to dance, or rather spin. One\ndivision of this part of the building was put aside for Christians, the\nothers were filled with common people and children. When I arrived, one\nold Dervish, in a green dress, was sitting at one point of the room,\nand twenty-four in white, were opposite to him. A flute and drum played\nsome very dreary music in the gallery. At a given signal they all fell\nflat on their faces, with a noise and precision that would have done\nhonour to a party of pantomimists; and then they all rose and walked\nslowly round, with their arms folded across their breasts, following\nthe old green Dervish, who marched at their head, and bowing twice\nvery gravely to the place where he had been sitting, and to the spot\nopposite to it. They performed this round two or three times. Then the\nold man sat down, and the others, pulling off their cloaks, appeared\nin a species of long petticoat, and one after the other began to spin.\nThey commenced revolving precisely as though they were waltzing by\nthemselves; first keeping their hands crossed on their breast, and then\nextending them, the palm of the right hand and the back of the left\nbeing upwards. At last they all got into play, and as they went round\nand round, they put me in mind of the grand party we have seen on the\ntop of an organ, where a _cavalier seul_ revolves by himself, and bows\nas he faces the spectators.\n\n\"They went on for a long time without stopping--a quarter of an hour,\nperhaps, or twenty minutes. There was something inexpressibly sly and\noffensive in the appearance of these men, and the desire one felt to\nhit them hard in the face became uncomfortably dominant. At the end of\ntheir revolutions they made another obeisance to the old man, and all\nthis time the players in the orchestra howled forth a kind of hymn.\nThis ceremony was repeated three or four times, and then they all sat\ndown again and put their cloaks on, whilst another Dervish, who had\nwalked round and round amongst the dancers, whilst they were spinning,\nsang a solo. During this time their faces were all close to the ground.\nThis done, they rose and marched before the old green Dervish once\nmore, kissing his hand as they passed, and the service concluded,\noccupying altogether about three-quarters of an hour.\"\n\n\nEXTRAORDINARY MALADY.\n\nDigne, the principal town in the department of the Basses Alpes\nin France might be passed by the traveller without exciting one\nobservation, its walks and its warm mineral waters being the only\nobjects worthy of notice. Its inhabitants do not now exceed 3,500;\nbut, in the year 1629, 10,000 industrious citizens followed their\nnumerous avocations within its precincts. At that period, however, an\nextraordinary plague broke out, in the month of June, which lasted till\nOctober, committing the most awful ravages, so that in that short space\nof time the wretched inhabitants were reduced to the number of 1,500,\namong whom six only had escaped this very singular malady, the effects\nof which are thus described by a French writer:--\"This malady strangely\naffected the invalids; some fancied they could fly; others, climb\nfrom one object to another like squirrels; some sunk into a profound\nlethargy, even for so long a time as six days; and one young woman who\nhad been hastily interred in a vineyard, rose three days afterwards,\nfor the grave-diggers were content just to cover the bodies. During\nthese four months the town was covered with a thick fog: the heat was\nsuffocating, accompanied by frequent and dreadful storms; and in order\nto complete the horrors of such a situation, the parliament forbade any\nof the inhabitants to quit the city, or the small territory belonging\nto it. Guards placed upon the _Bleonne_ fired upon those who attempted\nto escape. The magistrates abandoned their functions; the clocks no\nlonger sounded the hours; the neighbouring springs dried up, so that\nthe mills could not work; and famine began to add its fearful horrors\nto the miseries which already desolated the city, now become a living\nsepulchre, for the dead bodies lay in the streets unburied, and the\nfew remaining persons who still paraded the streets appeared more\nlike the spectres of those departed than living beings. Many persons\nnot only prepared but put on the habiliments of death, and quietly\nawaited the approach of the _king of terrors_. A new edict condemned\nthe pestilential city to the flames; but this inhuman decree was\ncountermanded, after the destruction of one country house, with all\nits inhabitants. The disease having somewhat abated in the surrounding\nvillages, humanity at length dictated the necessity of making some\nefforts to save the remaining few, who had escaped the contagion,\nfrom the no less frightful evil of famine. The scene that presented\nitself was appalling; several little children, whose parents were\ndead, were found sucking goats; in short, the desolation was so great\nthat, although two centuries have passed away since this fatal scourge\ndevastated the country, _Digne_ has never recovered its effects.\"\n\n\nQUACKERY IN THE OLDEN TIME.\n\nIn the reign of Henry VIII. many of the medical practitioners were\nmere horse-farriers. A distinguished patient, the great Lord Burghley,\nsecretary of state to Queen Elizabeth, was addressed by one Audelay, on\na certain occasion, in this wise, \"Be of goode comfort, and plucke up\na lustie, merrie hearte, and then shall you overcome all diseases: and\nbecause it pleased my good Lord Admiral lately to praise my physicke,\nI have written to you such medicines as I wrote unto him, which I have\nin my boke of my wyffe's hand, _proved upon herselfe and mee both_:\nand if I can get anything that may do you any goode, you may be well\nassured it shall be a joye unto me to get it for you.\" \"A good medicine\nfor weakness or consumption:--Take a pig of nine days olde, and slaye\nhim, and quarter him, and put him in a skillat, with a handfull of\nspearment, and a handfull of red fennell, a handfull of liverwort,\nhalf a handfull of red neap, a handfull of clarge, and nine dates,\ncleaned, picked, pared, and a handful of great raisins, and picke out\nthe stones, and a quarter of an ounce of mace, and two stickes of\ngoode cinnamon, bruised in a mortar, and distill it with a soft fire,\nand put it in a glass, and set it in the sun nine days, and drinke\nnine spoonfulls of it at once when you list!\" \"A compost:--item--take\na porpin, otherwise called an English hedgehog, and quarter him in\npieces, and put the said beast in a still, with these ingredients:\nitem--a quart of redde wyne, a pinte of rose-water, a quarter of a\npound of sugar--cinnamon and two great raisins.\" \"If thore be any\nmanner of disease that you be aggrieved withal, I pray you send me\nsome knowledge thereof, and I doubt not but to send you an approved\nremedie. Written in haste at Greenwiche, y{e} 9 of May, 1553, by your\ntrewe heartie friend, JOHN of AUDELAY.\"\n\n\nA POISON WEAPON.\n\n[Illustration: [++] Poison Weapon.]\n\nThe instrument sketched forms one of the curiosities in the splendid\nmuseum of the late Sir S. R. Meyrick, and is a singular instance of\nthat refinement of cruelty which is too prominent a characteristic of\nthe sixteenth century. It is a weapon for throwing poisoned needles\namong a crowd. Where the lid at the top is seen lifted up, is the\nchamber in which the needles are kept stuck into a cork at the bottom.\nOn the opposite side a needle is seen put through a hole in a strong\nspring, held in its place by a catch above, which, when pressed by the\nthumb disengages it and ejects the needle with considerable force. As\nthe fore-finger goes through the centre ring, and the thumb is at the\ntop, the weapon is almost entirely concealed by the hand. The spring\ncan be adjusted by a screw at the side. This cruel instrument was\nused by men on horseback, or from a window, and as the needles were\npoisoned, many painful injuries must have been inflicted without the\nsufferers being able to discover by whom their wounds were caused.\n\n\nANCIENT SWORD-BREAKER.\n\n[Illustration: [++] Sword-Breaker.]\n\nThe immense two-handed swords of former times were most fearful\nweapons, and far more easily used than the appearance of them would\nlead us to suppose. They were admirably poised, and the position in\nwhich they were held may be learned from various writers of their\ntimes. One hand was placed close to the cross-bar, while the other\nheld the pommel. De Grassi, in 1594, tells us that those who use them\ncontrive to \"amase with the furie of the sword, and deliver great\nedge blows down-right and reversed, fetching a full circle or compass\ntherein with exceeding great swiftness, staying themselves upon one\nfoote, sometimes on the other, utterlie neglecting to thrust, and\npersuading themselves that the thrust serveth to amaze one man onlie,\nbut those edge blows are of force to incounter many. The hand towards\nthe enemie must take hold fast of the handle neere the crosse and\nunderneath, the other hand above and near the pomell.\"\n\nSilver, in his \"Paradox\" gives the following as the proportions of a\ntwo-handed sword in his day: \"The perfect length of your two-handed\nsword is the blade to the length and hilt of your single sword.\"\n\nThe instrument which we have sketched on previous page, was used in\nthe time of Henry VIII., for the purpose, not only of defence against\none of those \"great edge-blows down-right\" but of catching the blade\nbetween the teeth, and then breaking it by a sharp turn of the wrist.\n\n\nORIGIN OF THE BALLOT.\n\nThe origin of electing members by balls may be traced to the Grecians.\nWhen a member was to be elected, every one threw a little pellet of\nbran, or crumb of bread into a basket, carried by a servant on his head\nround the table, and whoever dissented flattened their pellet at one\nside.\n\n\nANCIENT DAGGER.\n\nThe weapon which forms the subject of the woodcut is a dagger of the\ntime of Philip and Mary, ornamented with engraving. After being thrust\ninto a person, by pulling a little catch, it is made to open within\nhim, and the prolongation of the blade allows means for a second blow.\nThe two small hooks at the inner side of the two blades would admit of\nthe dagger being thrust deeper in, but would prevent its being drawn\nout.\n\n[Illustration: [++] Ancient Dagger.]\n\nAt the period these daggers were most in vogue, personal combats were\nvery sanguinary and determined, seldom terminating without the death of\none, and in some cases of both, of the parties engaged. They first used\nthe long sword, and when that weapon was broken, they closed with one\nanother, and used their daggers by stabbing at the most mortal part of\ntheir foe they could manage to reach.\n\n\nTHE TEMPLE OF POU-TOU.\n\nPou-tou is an island of the great archipelago of Chusan, on the coasts\nof the province of Tche-kiang. More than 100 monasteries, more or less\nimportant, and two of which were founded by Emperors, are scattered\nover the sides of the mountains and valleys of this picturesque and\nenchanting island, which nature and art have combined to adorn with\ntheir utmost magnificence. All over it you find delightful gardens,\nfull of beautiful flowers,--grottoes cut in the living rock, amidst\ngroves of bamboo and other trees, with aromatic banks. The habitations\nof the Bonzes are sheltered from the scorching rays of the sun by\numbrageous foliage, and scattered about in the prettiest situations\nimaginable. Thousands of winding paths cross the valleys in various\ndirections, and the brooks and rivulets, by means of pretty bridges of\nstone or painted wood, and for the communications between the scattered\ndwellings. In the centre of the island rise two vast and brilliant\nedifices--Buddhist temples--the yellow bricks of which announce that\ntheir construction is due to imperial munificence. The religious\narchitecture of the Chinese does not at all resemble ours. They have no\nidea of the majestic, solemn, and perhaps somewhat melancholy style,\nthat harmonizes so well with the feelings which ought to be inspired\nby a place devoted to meditation and prayer. When they wish to build a\npagoda, they look out for the most gay and smiling site they can find\non the declivity of a mountain or in a valley; they plant it with great\ntrees of the evergreen species; they trace about it a number of paths,\non the sides of which they place flowering shrubs, creeping plants, and\nbushes. It is through these cool and fragrant avenues you reach the\nbuilding, which is surrounded by galleries, and has less the air of a\ntemple than of a rural abode charmingly situated in the midst of a park\nor garden.\n\nThe principal temple of Pou-tou is reached by a long avenue of grand\nsecular trees, whose thick foliage is filled with troops of crows\nwith white heads; and their cawings and flapping of wings keep up a\ncontinual clamour. At the end of the avenue is a magnificent lake,\nsurrounded with shrubs that lean over its waters like weeping willows.\nTurtle and gold-fish gleam through them; and mandarin-ducks, in their\ngaily- plumage, play over their surface, amidst the splendid\nwater-lilies whose rich corollas rise majestically upon tender green\nstalks spotted with black. Several bridges of red and green wood are\nthrown over this lake, and lead to flights of steps, by which you\nascend to the first of the temple buildings--a kind of porch, supported\nupon eight enormous granite columns. On the right and left are\nstationed, like sentinels, four statues of colossal size, and two side\ngates lead to the vestibule of the principal nave, where is enthroned a\nBuddhist Trinity, representing the Past, the Present, and the Future.\nThese three statues are entirely gilt, and, although in a crouching\nposture, of gigantic dimensions--at least twelve feet high. Buddha is\nin the midst, his hands interlaced, and gravely placed on his majestic\nabdomen. He represents the Past, and the unalterable and eternal quiet\nto which it has attained; the two others, which have the arm and the\nright hand raised, in sign of their activity, the Present and Future.\nBefore each idol is an altar covered with little vases for offerings,\nand cassolets of chiselled bronze, where perfumes are constantly\nburning.\n\nA crowd of secondary divinities are ranged round the hall, the\nornaments of which are composed of enormous lanterns of painted paper\nor horn--square, round, oval--indeed, of all forms and colours; and the\nwalls are hung with broad strips of satin, with sentences and maxims.\n\nThe third hall is consecrated to _Kouang-yu_, whom the greater number\nof accounts of China persist in regarding as a goddess of porcelain,\nand sometimes also of fecundity. According to the Buddhist mythology,\nKouang-yu is a person of the Indian Trimourti, or Triune God,\nrepresenting the creative power.\n\nFinally, the fourth hall is a pantheon, or pandemonium, containing\na complete assortment of hideous idols, with ogres' and reptiles'\nfaces. Here you see, huddled together pell-mell, the gods of heaven and\nearth; fabulous monsters, patrons of war, of the silk manufacture, of\nagriculture, and of medicine; the images of the saints of antiquity,\nphilosophers, statesmen, warriors, literary men--in a word, the most\nheterogeneous and grotesque assembly conceivable.\n\n\nORACLES OF APOLLO IN FRANCE.\n\nTowards the frontiers of Auvergne and Velay, upon the high rock of\nPolignac, there was formerly a temple of Apollo, famous for its\noracles. The time of its foundation ascends to the first years of the\nChristian era, since, in the year 47, the Emperor Claudius came hither\nin great pomp, to acknowledge the power of the god; and he left proofs\nof his piety and munificence. The debris and mysterious issues that\nare found even now upon the rock, in the heart of its environs, reveal\nthe secret means employed by the priests to make their divinities\nspeak, and to impose upon the people. At the bottom of the rock was\nan aedicula: it was on this spot that the pilgrims took up their\nfirst station, and deposited their offerings and made their vows. A\nsubterranean passage communicated from this aedicula to the bottom of\na great excavation, pierced, in the form of a tunnel, from the base\nto the summit of the rock. It was by this enormous opening that the\nvows, the prayers and questions, pronounced in the very lowest voice\nby the pilgrims, reached instantly the top of the rock, and were there\nheard and collected by the college of priests; the answers were then\nprepared, while the believers, by a sinuous and long path, slowly\narrived at the end of their pilgrimage. The answers being ready, the\npriests commissioned to transmit them repaired to profound and deep\napartments, contiguous to a well, the orifice of which terminated in\nthe temple. This well, crowned by an altar, being enclosed by a little\nhemispherical roof, supported in its external parts the colossal figure\nof Apollo; the mouth of this statue being half open, in the middle of\na large and majestic beard, appeared always ready to pronounce the\nsupreme decrees. It was also through this opening, by the means of a\nlong speaking-trumpet, that the priests at the bottom of this den of\nmystery and superstition made known those famous oracles so imposing\nand so powerful in their effects upon the human soul as to impede for\ncenturies the substitution of the more pure and holy precepts of the\ngospel.\n\n\nBEST POSITION FOR SMOKING OPIUM.\n\nOpium is not smoked in the same manner as tobacco. The pipe is a tube\nof nearly the length and thickness of an ordinary flute. Towards one\nend of it is fitted a bowl of baked clay or some other material, more\nor less precious, which is pierced with a hole communicating with the\ninterior of the tube. The opium, which before smoking is in the form\nof a blackish viscous paste, is prepared in the following manner:--A\nportion, of the size of a pea, is put on a needle, and heated over a\nlamp until it swells and acquires the requisite consistence. It is\nthen placed over the hole in the bowl of the pipe, in the form of a\nlittle cone that has been previously pierced with a needle so as to\ncommunicate with the interior of the tube. The opium is then brought\nto the flame of the lamp, and after three or four inspirations the\nlittle cone is entirely burnt, and all the smoke passes into the\nmouth of the smoker, who then rejects it again through his nostrils.\nAfterwards the same operation is repeated, so that this mode of smoking\nis extremely tedious. The Chinese prepare and smoke their opium lying\ndown, sometimes on one side, sometimes on the other, saying that this\nis the most favourable position; and the smokers of distinction do not\ngive themselves all the trouble of the operation, but have their pipes\nprepared for them.\n\n\nEXECUTIONER'S SWORD.\n\n[Illustration: [++] Executioner's Sword.]\n\nThe weapon engraved below forms one of the curiosities in the superb\ncollection of ancient armour belonging to the late Sir Samuel R.\nMeyrick, at Goodrich Court, Herefordshire. It is the sword of an\nexecutioner, having on it the date 1674. The blade is thin, and\nexceeding sharp at both edges; and engraved on it is a man impaled,\nabove which are the words, in German, of which the following is a\ntranslation:--\n\n \"Let every one that has eyes\n Look here, and see that\n To erect power on wickedness\n Cannot last long:\"\n\na man holding a crucifix, his eyes bandaged, and on his knees; the\nexecutioner, with his right hand on the hilt, and his left on the\npommel, is about to strike off his head; above is written--\n\n \"He who ambitiously exalts himself,\n And thinks only of evil,\n Has his neck already encompassed\n By punishment.\"\n\nOn the other side, a man broken on the wheel; over which is--\n\n \"I live, I know not how long;\n I die, but I know not when:\"\n\nand a man suspended by the ribs from a gibbet, with the inscription--\n\n \"I move, without knowing whither;\n I wonder I am so tranquil.\"\n\n\nORIGIN OF EXCHEQUER BILLS.\n\nIn the year 1696 and 1697, the silver currency of the kingdom being by\nclipping, washing, grinding, filing, &c., reduced to about half its\nnominal value, Acts of Parliament were passed for its being called in\nand recoined, and whilst the recoinage was going on, Exchequer bills\nwere first issued to supply the demands of trade.\n\n\nANCIENT ETRURIAN BUST.\n\n[Illustration: [++] Ancient Etrurian Bust.]\n\nIf we look backwards to the most remote times of Greek industry, we\nfind that long before fire-casting became customary, almost every kind\nof work was carried out by the simple means of the hammer and tongs,\nwielded by skilful hands. Even products of art were created in this\nmanner; and as statues, vases, and the like could not be put together\nby the process of soldering, nails were used for the purpose, as we\nlearn not only from ancient writers, but even from monuments which have\nlately been discovered in Etruria, and the most important specimens\nof which are now possessed by the British Museum. In one of the tombs\nbelonging to the vast necropolis of Vulci were discovered, about twenty\nyears ago, a great many bronzes of this very ancient workmanship; one\nof them represents a bust placed on a basement covered with thin copper\nplates, and adorned by a row of figures, which are likewise chased;\nlong curls fall down over the neck and shoulders, and these parts\nespecially are formed in the most simple manner: one would be tempted\nto call it child-like, did not the whole composition show a certain\ncharacter which enables the experienced eye of the art-philosopher\nto distinguish in these rude attempts at plastic metal work the very\ngerm of those wonderfully-styled productions of a later period. The\nengraving here annexed, giving a side view of this remarkable, and as\nyet unique monument, is intended to show the arrangement of the hair,\nwhich, in spite of its simple treatment, presents as a whole some\ntrace of grace, and principles of fine proportions. We perceive that\nthe curls are formed by rolling and twining together small strips of\nbronze plate, connected with the head itself by the mechanical means\nwe have alluded to. There is no trace of soldering; and we may be sure\nthat we possess in this figure a good specimen of those hammer-wrought\nsculptures of old which were spoken of by the Greeks themselves as\nbelonging to a fabulous period.\n\n\nTHE HAIRY WOMAN OF BURMAH.\n\nThe following account of this remarkable freak of nature is taken from\nCaptain Yule's \"Mission to Ava.\" Writing from the city of Amarapoora,\nthe capital of Burmah, the Captain says:--\n\n\"To-day we had a singular visitor at the residency. This was Maphoon,\nthe daughter of Shwe-maong, the \" hirsutus\" described and depicted\nin Crawfurd's narrative, where a portrait of her, as a young child,\nalso appears. Not expecting such a visitor, one started and exclaimed\ninvoluntarily as there entered what at first-sight seemed an absolute\nrealization of the dog-headed Anubis.\n\n\"The whole of the Maphoon's face was more or less covered with hair.\nOn a part of the cheek, and between the nose and mouth, this was\nconfined to a short down, but over all the rest of the face was a thick\nsilky hair of a brown colour, paling about the nose and chin, four or\nfive inches long. At the alea of the nose, under the eye, and on the\ncheek-bone, this was very fully developed, but it was in and on the\near that it was most extraordinary. Except the extreme upper tip, no\npart of the ear was visible: all the rest was filled and veiled by a\nlarge mass of silky hair, growing apparently out of every part of the\nexternal organ, and hanging in a dependent lock to a length of eight\nor ten inches. The hair over her forehead was brushed so as to blend\nwith the hair of the head, the latter being dressed (as usual with\nher countrywomen) _a la Chinoise_. It was not so thick as to conceal\naltogether the forehead.\n\n\"The nose, densely covered with hair so as no animal's is that I know\nof, and with long fine locks curving out and pendent like the wisps of\na fine Skye terrier's coat, had a most strange appearance. The beard\nwas pale in colour, and about four inches in length, seemingly very\nsoft and silky.\n\n\"Poor Maphoon's manners were good and modest, her voice soft and\nfeminine, and her expression mild and not unpleasing, after the first\ninstinctive repulsion was overcome. Her appearance rather suggested the\nidea of a pleasant-looking woman masquerading than that of anything\nbrutal. This discrimination, however, was very difficult to preserve\nin sketching her likeness, a task which devolved on me to-day in Mr.\nGrant's absence. On an after-visit, however, Mr. Grant made a portrait\nof her, which was generally acknowledged to be most successful. Her\nneck, bosom, and arms appeared to be covered with fine pale down,\nscarcely visible in some lights. She made a move, as if to take off\nher upper clothing, but reluctantly, and we prevented it. Her husband\nand two boys accompanied her. The elder boy, about four or five years\nold, had nothing abnormal about him. The youngest, who was fourteen\nmonths old and still at the breast, was evidently taking after his\nmother. There was little hair on the head, but the child's ear was\nfull of long silky floss, and it could boast a moustache and beard of\npale silky down that would have cheered the heart of many a cornet. In\nfact, the appearance of the child agrees almost exactly with what Mr.\nCrawford says of Maphoon herself as an infant. This child is thus the\nthird in descent exhibiting this strange peculiarity; and in this third\ngeneration, as in the two preceding, this peculiarity has appeared only\nin one individual. Maphoon has the same dental peculiarity also that\nher father had--the absence of the canine teeth and grinders, the back\npart of the gums presenting merely a hard ridge. Still she chews pawn\nlike her neighbours.\n\nMr. Camaretta tells some story of an Italian wishing to marry her and\ntake her to Europe, which was not allowed. Should the great Barnum hear\nof her, he would not be so easily thwarted.\n\nAccording to the Woundouk, the King offered a reward to any man who\nwould marry her, but it was long before any one was found bold enough\nor avaricious enough to venture. Her father, Shwe-maong, was murdered\nby robbers many years ago.\"\n\n\nA TRAVELLER'S PASSPORT.\n\nThe following document, included among the rolls, is dated 1680, from\nWhitehall:--\n\n\"Dame Mary Yate, having asked his majesty's permission to pass beyond\nthe seas, for the recovery of her health, his majesty was most\ngraciously pleased to grant her request, under the usual clauses\nand provisoes, according to which ye said Dame Mary Yate having\ngiven security not to enter into any plott or conspiracy against his\nmajesty or his realms, or behave herself in any such manner as may\nbe prejudicial to his majesty's government, or the religion here by\nlaw established, and that she will not repaire to the city of Roome,\nor return unto this kingdome without first acquainting one of his\nmajesty's principal secretaries of state, and obtaining leave for the\nsame, in pursuance of his majesty's commands in council hereby will and\nrequire you to permit and suffer the said Dame Mary Yate to imbarque\nwith her trunkes of apparel and other necessaries not prohibited at\nany port of this kingdom, and from thence to pass beyond the seas,\nprovided that shee departe this kingdom within 14 days after the date\nhereof.\"--April 14.\n\nIf the above refers to the celebrated Lady Mary Yate (a daughter of the\nhouse of Pakington) who is commemorated on a monument in Chaddesley\nChurch, Worcestershire, as having died in 1696, at the age of 86, she\nmust have been 70 years old when these precautions were taken by the\nGovernment against the poor old lady attempting to invade the country,\nor to comfort the Pope with her presence and support. Dame Mary Yate\nwas no doubt a Roman Catholic, and the permission above referred to was\ngranted under the seventh section of the statute 3rd James I, chap. 5,\nwhich was virtually repealed by the statute 33rd George III, chap. 30,\nwhich exempted Roman Catholics from all the penalties and restrictions\nmentioned and enjoined in the older acts, if in one of the Courts at\nWestminster or at the Quarter Sessions they made a declaration which to\nthem was unobjectionable.\n\n\nCURIOUS PROVINCIAL DANCE IN FRANCE.\n\nThe inhabitants of Roussillon are passionately fond of dancing; they\nhave some dances peculiar to themselves. The men generally commence\nthe country dance by a _contre-pas_, the air of which is said to be of\nGreek origin; the women then mingle in the dance, when they jointly\nperform several figures, passing one among the other, and occasionally\nturning each other round. At a particular change in the air, the male\ndancer must dexterously raise his partner and place her on his hand in\na sitting posture. Accidents sometimes happen upon these occasions,\nand the lady falls to the ground amidst the jokes and laughter of her\ncompanions. One of these dances, called _lo salt_, is performed by four\nmen and four women. At the given signal, the cavaliers simultaneously\nraise the four ladies, forming a pyramid, the caps of the ladies\nmaking the apex. The music which accompanies these dances consists of\na _lo flaviol_, a sort of flageolet, a drum, two hautboys, prima and\ntenor, and the _cornemuse_, called in the country _lo gratla_: this\ninstrument, by its description, must somewhat resemble the bagpipes.\nThe dance called _Segadilles_ is performed with the greatest rapidity:\nat the end of every couplet, for the airs are short and numerous, the\nfemale dancers are raised, and seated on the hands of their partners.\n\n\nANCIENT INSTRUMENT OF PUNISHMENT.\n\n[Illustration: [++] Whip of Steel.]\n\nThe instrument which we here engrave is a whip of steel that was made\nand used as an engine of punishment and torture about the middle of\nthe sixteenth century. It is composed of several truncated cones,\ngrooved with sharp edges, and held in opposite directions, so as to\ngive sufficient oscillation without rising so far as to strike the hand\nof the executioner. It seems to have been held by a strap; but its\nbarbarity is evident.\n\n\nPUNISHING BY WHOLESALE.\n\nHenry VIII. is recorded, in the course of his reign, to have hanged no\nfewer than 72,000 robbers, thieves, and vagabonds. In the latter days\nof Elizabeth scarcely a year passed without 300 or 400 criminals going\nto the gallows. In 1596, in the county of Somerset alone, 40 persons\nwere executed, 35 burnt in the hand, and 37 severely whipped.\n\n\nMONKS AND FRIARS.\n\nThere was a distinction between the Monks and Friars, which caused the\nlatter to become the object of hatred and envy. Both the monastic, or\nregular, and parochial clergy, encouraged the attacks made upon them.\nThe Monks were, by most of their rules, absolutely forbidden to go out\nof their monasteries, and, therefore, could receive only such donations\nas were left to them. On the contrary, the Friars, who were professed\nmendicants, on receiving notice of the sickness of any rich person,\nconstantly detached some of their members, to persuade the sick man\nto bequeath alms to their convent; thus often, not only anticipating\nthe Monks, but, likewise the parochial clergy. Besides, as most of\nthem were professed preachers, their sermons were frequently compared\nwith those of the clergy, and in general, not to the advantage of the\nlatter. In these sermons, the poverty and distress of their order,\nwere topics that, of course, were neither omitted, nor slightly passed\nover. Considering the power of the Church, before the Reformation, it\nis not to be supposed that any of the Poets, as Chaucer, &c., would\nhave ventured to tell those rediculous stories of the Friars, with\nwhich their works abound, had they not been privately protected by the\nsuperior clergy.\n\n\nCURIOUS TURKISH CONTRIVANCE.\n\n[Illustration: [++] Curious Turkish Contrivance.]\n\nWonderful are the appliances by which ingenuity contrives to supply the\nevasions of idleness. We give one of them, as described by Mr. Albert\nSmith, in his \"Month at Constantinople.\"\n\n\"Passing some cemeteries and public fountains, we came to the outskirts\nof the city, which consist chiefly of gardens producing olives,\noranges, raisins and figs, irrigated by creaking water-wheels worked\nby donkeys. To one of these the droll contrivances which attracted our\nnotice was affixed. The donkey who went round and round was blinded,\nand in front of him was a pole, one end of which was fixed to the axle\nand the other slightly drawn towards his head-gear and there tied: so\nthat, from the spring he always thought somebody was pulling him on.\nThe guide told us that idle fellows would contrive some rude mechanism\nso that a stick should fall upon the animal's hind quarters at every\nround, and so keep him at work whilst they went to sleep under the\ntrees.\"\n\n\nFIGURES OF DOGS ON ANCIENT TOMBS.\n\nIn attempting to assign a reason for the frequent occurrence of dogs\nat the feet of tombs, we shall most probably be right if we simply\nattribute the circumstance to the affection borne by the deceased for\nsome animal of that faithful class. That these sculptured animals were\nsometimes intended for likenesses of particular dogs is evident. Sir\nBryan Stapleton, on his brass at Ingham, Norfolk, rests one foot on\na lion, the other on a dog; the name of the latter is recorded on a\nlabel, _Jakke_. Round the collar of a dog at the feet of an old stone\nfigure of a knight, in Tolleshunt Knight's Church, Essex, letters were\nformerly traced which were supposed to form the word _Howgo_.\n\nIn a dictionary of old French terms, we find that the word _Gocet_\nmeans a small wooden dog, which it was customary to place at the foot\nof the bed. Now it has been thought that something of this kind was\nintended in the representation of dogs on tombs, and that this support\nof the feet merely indicates the old custom of having that sort of\nwooden resting-place for the feet when in a recumbent position. But\nour first supposition appears the more natural, and is supported by\nthe fact that a large proportion of these sculptured dogs, instead of\nbeing placed beneath the feet, are seated on the robe or train, looking\nupwards with the confidence of favourite animals. Judith, daughter of\nthe Emperor Conrad, is represented on her tomb (1191) with a little dog\nin her right hand.\n\nOn the tomb of Sir Ralph de Rochford, in Walpole Church, Norfolk, his\nlady is by his side, dressed in a reticulated head-dress and veil,\na standing cape to her robe, long sleeves buttoned to her wrists, a\nquatrefoil fastens her girdle, and a double necklace of beads hangs\nfrom her neck. At her feet is a dog looking up, and another couchant.\nIn the chancel at Shernborne, Norfolk, the figure of Sir Thomas\nShernborne's lady (1458) has at the right foot a small dog sitting,\nwith a collar of bells.\n\nOn a large antique marble in the chancel at Great Harrowden,\nNorthamptonshire, are the portraits of a man in armour, and his wife in\na winding sheet. The man stands on a greyhound, and the woman has at\nher feet two little dogs looking upwards, with bells on their collars.\nThis monument is that of William Harwedon and Margery, daughter of Sir\nGiles St. John of Plumpton. She died in the twentieth year of Henry VI.\n\n\nTHE FATE OF THE LAST MAY-POLE IN THE STRAND.\n\nThe May-pole, which had been set up in 1641, having long been in a\nstate of decay, was pulled down in 1713, and a new one, with two\ngilt balls and a vane on the top of it, was erected in its stead.\nThis did not continue long in existence; for, being in 1718 judged\nan obstruction, to the view of the church then building, orders were\ngiven by the parochial authorities for its removal. Sir Isaac Newton\nbegged it of the parish, and it was conveyed to Wanstead Park, where it\nlong supported the largest telescope in Europe, belonging to Sir Isaac\nNewton's friend, Mr. Pound, the rector of Wanstead. It was 125 feet\nlong; and presented to Mr. Pound by Mr. Huson, a French member of the\nRoyal Society.\n\n\nMEANS OF ATTRACTING CUSTOM.\n\nBefore houses were numbered, it was a common practice with tradesmen\nnot much known, when they advertised, to mention the colour of their\nnext neighbour's door, balcony, or lamp, of which custom the following\ncopy of a handbill will present a curious instance:--\n\n\"Next to the GOLDEN DOOR, opposite Great Suffolk Street, near Pall\nMall, at the Barber's Pole, liveth a certain person, Robert Barker, who\nhaving found out an excellent method for sweating or fluxing of wiggs;\nhis prices are 2s. 6d. for each _bob_, and 3s. for every _tye wigg_ and\n_pigtail_, _ready money_.\"\n\n\nMUSIC OF THE HINDOOS.\n\nAmong the fine arts of India, music holds a distinguished place; and\nalthough its cultivation has declined, and but few are now found\nwho have attained to eminence either in the science or art of this\nunequalled source of recreation, refinement, and pleasure, yet no\npeople are more susceptible of its charms than the Hindoos. Reading is\nwith them invariably, as with the Arabians and other Eastern nations, a\nspecies of _recitativo_, a sort of speaking music, delivered in dulcet\nthough not measured tones. The recitation of lessons in a school or\nacademy always takes this form. The man at the oar, women beating lime,\nthe labourer engaged in irrigation, alike accompany their toil with\nsong.\n\nThe word _sangita_, symphony, as applied to music by the Hindoos,\nconveys the idea of the union of _voices_, _instruments_, and _action_.\nMusical treatises accordingly treat of _gana_, _vadya_, _uritya_, or\n_song_, _percussion_, and _dancing_; the first comprising the measures\nof poetry; the second, instrumental sounds; and the third, theatrical\nrepresentation. The ancient dramas of the Hindoo exhibited the union of\nthese in their unequalled poetry, modulated with the accompaniments of\nvoice, and instruments, and the attractions of appropriate scenery.\n\nThe music of the Hindoos includes eighty-four modes, each supposed\nto have a peculiar expression, capable of moving some particular\nsentiment or affection. The modes take their denomination from the\nseasons, or from the hours of day or night. Musical composition is\nsupposed capable of adaptation to the different periods of the day,\nand therefore its provisions are regulated by the hours. The ideas of\nthe Hindoos on music, as promoting the pleasures of imagination, may\nbe inferred from the names applied by ancient authors to their musical\ntreatises. One is called _Ragarnava_, the Sea of the Passions; another,\n_Ragaderpana_, the Mirror of Modes; and a third, _Sobhavinoda_, the\nDelight of Assemblies; a fourth, _Sangitaderpana_, the Mirror of Song;\nand another, _Ragavibodha_, the Doctrine of Musical Modes. Some of\nthese works explain the law of musical sounds, their divisions and\nsuccession, variations of scales by temperament, and the enunciation\nof modes; besides a minute description of the different _vinas_ (lute),\nand the rules for playing them. This is a fretted instrument of the\nguitar kind, usually having seven wires or strings, and a large gourd\nat each end of the finger-board. Its extent is two octaves, and its\ninvention is attributed to Nareda, the son of Brahma. There are many\nvarieties, named according to the number of their strings. Of one of\nthem we give an engraving below.\n\nMusic, like everything else connected with India, is invested with\ndivine attributes. From the sacred Veda was derived the Upaveda, or\nsubsidiary Veda of the Gandharbas, the heavenly choristers. The art\nwas communicated to mortals by Sarasvati, the consort of Brahma. She,\nas before stated, is the patroness of the fine arts, the goddess of\nspeech. Their son, an ancient lawgiver and astronomer, invented the\nVina. The first inspired man, Bherat, invented the Drama.\n\n[Illustration: [++] Hindoo Guitar.]\n\n PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND DECCLES.\n\n\n\n\nROUTLEDGE'S\n\n=BOOKS FOR THE PEOPLE\n\n70 VOLUMES.=\n\n_Crown 8vo, Cloth, 2\/- each. Crown 8vo, Paper Covers, 1\/6 each._\n\n\n =Macaulay's Essays and Lays of Ancient Rome.= Complete Edition.\n 924 pages.\n =The Spectator.= Complete Edition, reproducing the original\n text. Edited by Prof. H. MORLEY. 944 pages.\n =Carlyle's French Revolution.= Complete Edition.\n =---- Sartor Resartus, Heroes, and Hero Worship, Past and Present.=\n =---- Oliver Cromwell's Letters and Speeches.=\n =Carleton's Traits and Stories of the Irish Peasantry.= Complete\n Edition.\n =Anne Bowman's New Cookery Book. 635 pages.= (Cloth limp, =18.=)\n =Cruden's Concordance to the Old and New Testaments.=\n =Wise Sayings of the Great and Good.=\n =Book of Humour, Wit, and Wisdom.=\n =Foxe's Book of Martyrs.= Abridged from Milner's Large Edition.\n =Robinson Crusoe.= With 52 Plates by J. D. WATSON.\n =Gulliver's Travels into several Remote Regions of the World.=\n SWIFT.\n =Arabian Nights' Entertainments.=\n =Adventures of Don Quixote.= JARVIS'S Translation.\n =Southey's Life of Nelson.=\n =Josephus.= Translated by WHISTON.\n =Book of Epigrams.= W. D. 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HOLMES.\n =AEsop's Fables.= Illustrated by HARRISON WEIR.\n =Milman's History of the Jews.=\n =Lord Lytton's Pamphlets and Sketches.=\n =The Waverley Anecdotes.=\n =Lord Lytton's England and the English.=\n =Rabelais' Works.= Edited by HENRY MORLEY.\n =Prescott's History of the Reign of Ferdinand and Isabella.= 3 vols.\n =---- History of the Conquest of Mexico.= 3 vols.\n =---- History of the Conquest of Peru.= 3 vols.\n =---- History of the Reign of Philip II.= 3 vols.\n =---- History of the Reign of Charles V.= 2 vols.\n =---- Critical and Historical Essays.=\n =The Pickwick Papers.= By CHARLES DICKENS.\n =Principles of Political Economy.= By JOHN STUART MILL.\n =Popular Astronomy.= By O. M. 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Crown 8vo, Paper Covers, 1\/6 each._\n\n =Shakspere.= Edited by CHARLES KNIGHT, with Portrait and\n 63 page plates by Sir JOHN GILBERT, R.A. 832 pages.\n =Longfellow's Poetical Works.= Complete Edition, with 126\n Copyright Poems.\n =Scott's Poetical Works.= 640 pages.\n =Byron's Poetical Works.= 752 pages.\n =Burns's Poetical Works.= 512 pages.\n =The Ingoldsby Legends.= 512 pages, with 22 Illustrations from Designs\n by CRUIKSHANK and LEECH.\n =Homer's Iliad and Odyssey.= Pope's Translation.\n =Moore's Poetical Works.= With Memoir by CHARLES KENT.\n =Bret Harte's Poetical Works.=\n =Mrs. Hemans' Poetical Works.=\n =Hood's Poetical Works, Serious and Comic.= 528 pages.\n =Shelley's Poetical Works.= With Memoir by W. B. SCOTT.\n =Shakspere Gems=: Extracts, specially designed for Youth.\n =Cowper's Poetical Works.=\n =Milton's Poetical Works.= From the Text of Dr. NEWTON.\n =Sacred Poems: Devotional and Moral.=\n =Choice Poems and Lyrics.= Beautiful pieces from 130 Poets.\n =Chaucer's Canterbury Tales.=\n =Book of British Ballads.= Edited by S. C. HALL.\n =Holmes's Poetical Works.=\n =Lowell's Poetical Works.=\n =Willis's Poetical Works.=\n =Marlowe's Faustus and Goethe's Faust.= Complete.\n =Poems.= By ELIZABETH BARRETT BROWNING.\n =Doubtful Plays of Shakspeare.= Edited by HAZLITT.\n =Byron's Don Juan.=\n =Lord Lytton's King Arthur.=\n =---- ---- The New Timon.=\n =Dante's Divine Comedy.= LONGFELLOW'S Translation with Notes.\n =Whittier's Poetical Works.=\n =Bryant's Poetical Works.=\n =Campbell's Poetical Works.=\n =Coleridge's Poetical Works.=\n =Dodd's Beauties of Shakspeare.=\n =Herbert's Poetical Works.=\n =Keats' Poetical Works.=\n =Poe's Poetical Works.=\n =Pope's Poetical Works.=\n =Wordsworth's Poetical Works.=\n =Schiller and Horace.= Translated by LORD LYTTON.\n =Percy's Reliques of Ancient English Poetry.=\n =Keble's The Christian Year.=\n =Virgil's AEneid.= Translated by DRYDEN.\n =Butler's Hudibras.=\n =Herrick's Poems.=\n =Comic Poets of the Nineteenth Century.=\n =Spenser's Faerie Queene.=\n =Dryden's Poetical Works.=\n =Southey's Poetical Works.=\n =The Book of Familiar Quotations.=\n =Poets' Corner=: A Manual for Students in English Poetry.=\n =Sheridan Knowles' Dramatic Works.=\n\n\n\n\nFIFTY VOLUMES HAVE NOW BEEN ISSUED OF =SIR JOHN LUBBOCK'S HUNDRED\nBOOKS.=\n\nPublished on the 10th and 25th of each Month.\n\nCrown 8vo, Red Cloth, uncut edges; or Blue Cloth, cut edges.\n\n\n_ORDER OF PUBLICATION._\n\n 1. HERODOTUS. Literally Translated from the Text of BAEHR, by\n HENRY CARY, M.A. 3s. 6d.\n\n 2. DARWIN'S VOYAGE OF A NATURALIST IN H.M.S. \"BEAGLE.\" 2s. 6d.\n\n 3. THE MEDITATIONS OF MARCUS AURELIUS. Translated from the Greek\n by JEREMY COLLIER. 1s. 6d.\n\n 4. THE TEACHING OF EPICTETUS. Translated from the Greek, with\n Introduction and Notes, by W. T. ROLLESTON. 1s. 6d.\n\n 5. BACON'S ESSAYS, 1s. 6d.\n\n 6. MILL'S POLITICAL ECONOMY. 3s. 6d.\n\n 7. CARLYLE'S FRENCH REVOLUTION. 3s. 6d.\n\n 8. SELF-HELP. By SAMUEL SMILES. 6s.\n\n 9. WHITE'S NATURAL HISTORY OF SELBORNE. Edited by SIR WILLIAM\n JARDINE, Bart. With Illustrations. 3s. 6d.\n\n 10. THE PICKWICK PAPERS. By CHARLES DICKENS. With Illustrations by\n \"PHIZ.\" 3s. 6d.\n\n 11. THE SHI KING: the Old \"Poetry Classic\" of the Chinese.\n Translated by WILLIAM JENNINGS, M.A. 3s. 6d.\n\n 12. HOMER'S ILIAD AND ODYSSEY. Translated by ALEXANDER POPE. 3s.\n 6d.\n\n 13. VIRGIL'S AENEID. Translated by JOHN DRYDEN. 1s. 6d.\n\n 14. 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EMERSON'S ESSAYS. 3s. 6d.\n\n=London: George Routledge & Sons, Limited.=\n\n\n * * * * *\n\n\n\n TRANSCRIBER'S NOTE:\n\n Minor typographical errors have been corrected without note.\n\n Punctuation and spelling were made consistent when a predominant\n preference was found in this book; otherwise they were not changed.\n\n Ambiguous hyphens at the ends of lines were retained.\n\n Mid-paragraph illustrations have been moved between paragraphs and\n some illustrations have been moved closer to the text that\n references them. The Index to Engravings paginations were not\n corrected.\n\n Superscripts are enclosed in brackets.\n\n [++] indicates a caption added by the transcriber.\n\n\n VARIANTS UNCHANGED:\n\n -- Pages xiii and 201: Colebrook and Colebroke.\n\n -- Page 51: Tchang-tchin and Tchang-tching.\n\n -- Pages 108, 390 and others: Southwark and Southwarke.\n\n -- Pages 498 and 571: Mahomed, Mahommed and Mahomet.\n\n -- Pages 298, 306 and others: Hindostan and Hindoostan.\n\n -- Pages 322 and 452: Rossellina, Rosselini and Rossellini.\n\n -- Pages 370 and 371: ivil and ivel.\n\n -- Pages 432, 537 and others: Brama and Brahma.\n\n -- Pages 416 and 422: Bolinbroke and Bollinbroke.\n\n -- Pages 634 and 635: Clothseck and clothsek; Gambaldyn and gambaldynge.\n\n -- Page 635: Krees and Kreez.\n\n -- Multiple pages: Shakespere\/Shakspeare\/Shakspere.\n\n\n OTHER NOTES:\n\n -- Pages v and xi: Removed entry \"Architecture for Earthquakes\"\n as no such engraving or description could be found.\n\n -- Page 85: beneficiare changed to beneficiaire.\n\n -- Page 128: Hatherly changed to Hatherley.\n\n -- Page 246: Sackvile changed to Sackville.\n\n -- Page 250: Clarencieux changed to Clarenceux.\n\n -- Page 330: Shaftsbury to Shaftesbury.\n\n -- Page 339: Hasselquiet changed to Hasselquist.\n\n -- Page 346: + IONA + IHOAT + IONA + HELOI + YSSARAY + 11 +\n MEPHENOLPHETON + AGLA + ACHEDION + YANA +\n BACHIONODONAVALI M ILIOR + 11 BACHIONODONAVLI M ACH +\n\n changed to\n\n + IONA + IHOAT + LONA + HELOI + YSSARAY + || +\n MEPHENOLPHETON + AGLA + ACHEDION + YANA +\n BACHIONODONAVALI [M*] ILIOR + || BACHIONODONAVALI [M**] ACH +\n\n [M*] denotes counter-clockwise rotated M and [M**] clockwise\n rotated M. Also rendered in other texts as the letter Z or the\n number 3. Sources: Thomas R. Forbes, \"Verbal Charms in British\n Folk Medicine.\" Proceedings of the American Philosophical\n Society, Vo. 115, August 20, 1971, p. 308. And Frederick W.\n Fairholt, Ancient, Medieval, and Renaissance Remains in the\n Possession of Lord Londesborough. Drawn, Engraved and Described\n by Frederick W. Fairholt. London: Chapman and Hall, 1857, p. 83.\n\n -- Page 451: Strasburgh changed to Strasburg.\n\n -- Page 502: \"drake satin, and [illegible word] \n satin.\" Changed to \"drake satin, and [...] \n satin.\"\n\n -- Page 546: \"into his elar;\" changed to \"into his celar.\n\n -- Pages 549-550: \"the slaughtered sun hulls become....\" \"hulls\"\n changed to \"bulls.\"\n\n -- Page 550: \"according the the indication\" changed to \"according\n to the indication.\"\n\n -- Page 558: \"as with the [illegible word]\" changed to \"as with\n the [...].\"\n\n -- Page 630: \"It said to have been made....\" changed to \"It was said\n to have been made....\"\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Ten Thousand Wonderful Things, by\nEdmund Fillingham King\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n# \n\n# \nThis personalized essay collection is based upon the memories of its contributors. All the events in this story are as accurate and truthful as possible. Many names and places have been changed to protect the privacy of others. Mistakes, if any, are caused solely by the passage of time.\n\nCopyright \u00a9 2015 by Amy Phillips Penn\n\nPhotograph credits appear near photographs\n\nAll rights reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018.\n\nSkyhorse Publishing books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.\n\nSkyhorse\u00ae and Skyhorse Publishing\u00ae are registered trademarks of Skyhorse Publishing, Inc.\u00ae, a Delaware corporation.\n\nVisit our website at www.skyhorsepublishing.com.\n\n10 9 8 7 6 5 4 3 2 1\n\nLibrary of Congress Cataloging-in-Publication Data is available on file.\n\nCover design by Erin Seaward-Hiatt\n\nCover photo by Jessica Burstein\n\nPrint ISBN: 978-1-63220-272-7\n\nEbook ISBN: 978-1-63450-007-4\n\nPrinted in China\nContents\n\nPreface\n\nBY LIZ SMITH\n\nIntroduction\n\nSo . . . Who Was Elaine Kaufman?\n\nBY AMY PHILLIPS PENN\n\nThere's No Elaine's Without Elaine\n\nBY AMY PHILLIPS PENN\n\nTake Four: Customer, Publicist, Columnist, And \"Let's Do Lunch, Elaine\"\n\nBY AMY PHILLIPS PENN\n\n\"Marry Me, Marry Me\" At Elaine's\n\nBY DAVID BLACK\n\nElaine's Empties the Stanley Cup\n\nBY PER BJURMAN\n\n\"Give Him Table Six, He Says He Has Money.\"\n\nBY STEVE MCPARTLIN\n\nWhen Elaine Socked Me\n\nBY CHINA GIRARD\n\nHow Elaine Became My Photo Agent\n\nBY JESSICA BURSTEIN\n\nTravels With Elaine\n\nBY JESSICA BURSTEIN\n\nMy Last Night At Elaine's\n\nJODI GARDNER\n\n\"I'm a Fucking Icon.\"\n\nBY ELAINE MADSEN\n\nElaine Was My Dating Coach\n\nBY KEN PLISKA\n\nMy Home Away From Home\n\nBY DANA MOYLES\n\nThe Good Penny\n\nBY CURT BLOCK\n\n\"Elaine's Was Really Three Different Places.\"\n\nBY ASH BENNINGTON\n\nWoody Allen On Elaine's\n\nA Q&A BY AMY PHILLIPS PENN\n\nTop Ten Reasons Your New Hangout Will Never Replace Elaine's\n\nBY CHARLES KIPPS\n\nShooting Elaine's (But Not Elaine)\n\nBY RON GALELLA\n\nWhen Spinelli Met Plimpton\n\nAS TOLD TO AMY PHILLIPS PENN\n\n\"Not Only Did You Share A Meal Or A Drink, But You Shared Yourself.\"\n\nBY MARK ROSSINI\n\nElaine's, A Tribute\n\nPHOTOGRAPHS BY LARRY FINK\n\nDabney Coleman Does The Godfather Gig At Elaine's. Just Guess Who's Listening.\n\nBY DABNEY COLEMAN\n\nElaine Kaufman: Friend and Investor\n\nBY ESTHER MARGOLIS\n\n\"I Was Married To My Restaurant; Elaine Was Married To Elaine's.\"\n\nBY STEVE (PALLY) MCFADDEN\n\nElaine Was An Interesting, Tough Lady. (I saw her punch out a guy.)\n\nBY KEN MORAN\n\nThe Era Is Over\n\nBY AL SAPIENZA\n\nStill \"Anchored\" At Elaine's\n\nBY BOB DRURY\n\n\"Who's Da Poet?\"\n\nBY TAKI THEODORACOPULOS\n\nElaine As Mama Earth\n\nBY FRED MORTON\n\nThere's \"Nothing Like Gedempte Flaische.\"\n\nBY STEVE WALTER\n\nPeople Soup\n\nBY TONY HENDRA\n\nWho's on First? (In Elaine's Bathroom)\n\nBY RICHARD JOHNSON\n\n\"I Would Not Be the Person I Am Today if I Had Not Gone Into Elaine's.\"\n\nBY LIBBY SCHOETTLE\n\nAfterword\nPreface\n\nLIKE ALL FAMOUS folks who do something really unusual and special in this old world, it's important to understand Elaine Kaufman in a historical context. In the movie Casablanca, they say, \"Everyone goes to Rick's.\"\n\nIn recent New York, everyone tried to go to Elaine's! Everyone tried to be accepted, tried to get the okay of the owner.\n\nRick says, famously, \"Of all the gin joints in the entire world, you had to walk into mine!\"\n\nElaine said to me when I was writing about her in a book titled Dishing: \"Liz, don't say I ever threw anyone out; it makes me sound so tough.\"\n\nHaving once been ejected from Elaine's for coming in with a DuPont heiress, which annoyed Don (Donald Ward, Elaine's then-factotum), I was highly amused at this. Don told me that unescorted women weren't welcome; it must have been before Gloria Steinem.\n\n\"Unescorted women aren't welcome?\" I asked. \"I thought they were the best kind.\"\n\nElaine found this vastly amusing.\n\nShe herself was the original unescorted woman. She never needed an escort, though she certainly liked men more than she liked women.\n\nAnd men loved Elaine. One of my handsome brothers came to New York just for the purpose of bedding Elaine. So we were more or less \"related.\"\n\nIn her time, Elaine's did turn out to be a bit like Rick's. Deals were made . . . controversies spawned.\n\nDramatic things happened here; here were the beginnings of beautiful friendships and terrible hatreds.\n\nI first met Elaine in the 1950s at Portofino in Greenwich Village. She was then in love with the charming owner, Alfredo Viazzi.\n\nElaine was a damned good waitress, but she soon bested her Italian lover and became bigger and better known herself, against all odds. Eventually, Elaine's was full of Pulitzer Prize and Nobel Prize winners with a lot of writers eating on the cuff.\n\nShe couldn't resist talented people, adored writers, and was the inheritor of traditions from places like that haunt of the old Herald Tribune newspaper, Bleeks Artists & Writers Restaurant, where James Thurber would play the Match Game against Lucius Beebe, who was wearing white tie and tails.\n\nIn time, Elaine's inherited remarkable literary stars, much like those from the heyday of the old Algonquin Roundtable and from the confines of 21, a place grown glitzy & expensive, as it changed from being a speakeasy after Prohibition. The elite fled to Elaine's, even though the address was inconvenient.\n\nElaine's was a legend on its own terms, ranking her as a host of the grandeur of John Perona of El Morocco, Sherman Billingsley of the Stork Club, and Toots Shor, who once said of Hamlet, \"I bet I'm the only bum here who don't know how this turns out.\"\n\nElaine revered the likes of Tim Costello and creative hamburger guys like Danny Levezzo of PJ Clarke's. You couldn't buy a hamburger at Elaine's. She sneered at people who asked for one.\n\nElaine made up for any lacks. She had stars like Elaine Stritch and Jackie Gleason, who vied for the privilege of serving drinks behind the bar.\n\nShe was big-hearted, a closet intellectual in her way, a pushover, but also a hard taskmaster for phonies, fakes, and wannabes\u2014the \"civilians,\" she called the greater public.\n\nShe put her stamp on New York caf\u00e9 life in an utterly focused career. Though she learned from caf\u00e9 & saloon forebearers, there has never been anyone even vaguely like her. Texas Guinan, are you kidding? If Elaine was your friend, you hardly needed a lot of other ones. And in spite of a cynical veneer of sophistication, she was a pussycat, softhearted to a large extent.\n\nYes, she played favorites and sometimes she could be unreasonable.\n\nOne had to mind one's manners in Elaine's. You couldn't be demanding, you couldn't bother and pester the famous, you couldn't ask for autographs.\n\nIn 2003, Elaine Kaufman was made a \"Living Landmark\" by the distinguished New York Landmarks Conservancy.\n\nShe was childishly impressed by this honor and she supported this charity, which tries to save New York, ever after, attending every event until her death.\n\nWe of the Conservancy were also stunned to discover that Elaine Kaufman's becoming a Living Landmark had sold more tickets and more tables than anyone else except for George Steinbrenner.\n\nWhile other honorees often accept becoming Living Landmarks\u2014some of them making fun of it all, some bringing others more famous to laud them, and a few singing, dancing, reciting poems or offering films of themselves, and other self-aggrandizements\u2014I shall never get over Elaine's acceptance speech.\n\nI introduced her to giant fanfare, for there was only one Elaine Kaufman and one Elaine's, and her fans were there. She stepped up to the microphone, looked around, and then said simply: \"Thank you. I accept this for the late George Plimpton.\"\n\nIt was heaven. It was priceless. It was classy.\n\nIt said everything there was to say about art and artists, about culture, taste, good writers, vulgarity, tradition, New York City, indeed, about class, sass, and pizzazz.\n\n\u2014Liz Smith\nIntroduction\n\nA WRITER'S JOURNEY is an epic one that brakes for \"banal.\" While it's soothing to believe that you have control over your own odyssey, these wise words linger and crescendo: \"If you want to make God laugh, tell him your plans.\"\n\nI am honored and intrigued that Elaine Kaufman and I are reunited in this book.\n\nElaine's path rose from cosmetician to waitress and ultimately to a restaurateur of colossal fame. Playwright Jack Richardson suggested that Elaine invite writers to her uptown, out of the safe-and-surveyed New York restaurant\/tavern turf. He recommended that she use round tables to encourage intimacy and conversation.\n\nElaine would come to mentor, mother, encourage, and feed her writers.\n\nAs I interviewed many of her regulars, the common denominator wrote itself. Many, many regulars and even occasional visitors miss Elaine and Elaine's. They often feel lost, as if their home was uprooted and can never be rebuilt along with its mystique.\n\nMy re-entwinement with Elaine's life began on an unassuming day.\n\nOne morning I decided to start a Facebook group. I had never ever even considered spearheading an entourage on social media, but there it was: a simple seed spurned my own frustration on what was becoming of the America I knew and deeply cherished.\n\nReminiscing about all the innovative and magical components that make America \"America,\" I founded a group and named it \"Write on America.\" I invited my ever-sprouting group of Facebook and three-dimensional friends to join me.\n\nMy first post was a YouTube of Arlo Guthrie singing \"City of New Orleans.\"\n\nThe first seed sunk in, ready to sprout on, be smoked, or blown away. The immediate response came from two \"real\" friends who remembered the writer of the song, Steve Goodman, strumming his version at J.P.'s, a bar they worked at in the 1970s\u2014not too far a cry from the birth of creativity at Elaine's.\n\nSasha Tcherevkoff, a Facebook friend who was working for a site named NewYorkNatives.com contacted me. He liked the name of my group and thought that the names might enhance each other or perhaps merge in some way. We made a phone date.\n\nSasha asked if I was a native New Yorker. I replied that \"I was born in Doctor's Hospital.\"\n\nSo began my column, dubbed by NewYorkNatives.com as \"Vintage Gossip.\"\n\nMy first column was about Mayor Koch, who had just passed. I had been to Gracie Mansion, met the Mayor, and relished the chance to reminisce in wobbly Yiddish and New York English.\n\nMy column grew into a weekly ritual.\n\nI wrote about Elaine's. I told tales of my times there: first as a guest, then a publicist, and ultimately as a columnist and Elaine's occasional lunch companion.\n\nWhen I received an email from Nicole Frail, an editor at Skyhorse Publishing, my mind was already on my next column, the wild wave that the Surf Club rode. Nicole inquired if I'd be interested in writing a book about Elaine's.\n\nA writer's life is packed and unpacked, the contents scattered almost everywhere but where we discard or carefully fold them. Here I was in California, with a polo pony and three rescue dogs, and I was writing a column for a New York site, and now a book on Elaine's was on the best table that Elaine could offer.\n\nCuriouser and curiouser.\n\nWhy was Nicole interested in a book on Elaine's? She replied that it evolved from a discussion that she'd had with her editorial director about New York restaurants in another time. He told her about Elaine's and when he paused, she said, \"I wonder if that's a book.\" A bit of research led her my way. My answer was affirmative: I missed Elaine, and Elaine Kaufman was an intriguing, seemingly complicated, and infinitely controversial woman.\n\nSo began the process of researching my subject.\n\nI read every article I could find, A.E. Hotchner's book, Everyone Comes to Elaine's, and bartender-turned-journalist Brian McDonald's Last Call at Elaine's. Raymond Lindie, who also bartended there, wrote a play titled Elaine's Paradisio and a memorable story, \"The Beefeater Martini.\"\n\nI read on: columns and excerpts from Gay Talese, Liz Smith, and New York Times editor Peter Khoury, who I consider a new friend in the making. David Black, a screenwriter and the subject of Jessica Burstein's \"The Kiss,\" has also approached that status: potential new friends with treasured memories.\n\nOne door opens another\u2014if it doesn't hit you in your bifocals first.\n\nAdd on Susan Morse, who contacted Woody Allen for me. To my childlike delight, he actually replied to my questions. Jessica Burstein, Elaine's official photographer, with whom I had worked back in our Studio 54 days, was essential to introducing me to the members of Table 4, which honors Elaine; along with detectives, actors, writers, and assorted other regulars. She has contributed her own funny, loving, vintage Elaine Kaufman diary excerpt, along with several of her excellent photographs.\n\nSo many fascinating people entered the doors of this book that I can only beam a seismically never-ending thank-you to everyone I interviewed for this technicolor, touching memory of Elaine's.\n\nI have always missed Elaine's, but now I miss it even more. I wish that I had known Elaine better. The words loyal friend are continually chorused in her eulogies.\n\nWhen you read this book, imagine that we are meeting at Elaine's, in a neighborhood that once was shaky, but shook the trembling and roared into unexpected careers for many into nights that spanned from intellectually intriguing conversations to watching Elaine tossing garbage cans at the paparazzi.\n\nHer inimitable language boisterously beckons asterisks, while her quotes are beyond quotable: who else would dub someone \"half a whore\"?\n\nCome inside, and share the lore, the inspiration, the tears, and the legend that was Elaine's.\n\nWelcome in.\n\nPhoto credit: Susan Hathaway.\nSo . . . Who Was Elaine Kaufman?\n\nAmy Phillips Penn\n\nOR AS ELAINE might say: Who the fuck was Elaine?\n\n\"Yes, I am a fucking icon,\" proclaimed Elaine Kaufman, the erstwhile, controversial proprietress of New York's celebrity hub, Elaine's.\n\n\"Everyone tells me that. What is that? What did you do that is so earth-shattering? Just survive, and you know about that. I guess you get points for surviving,\" she says in the documentary I Know a Woman Like That.\n\nA fucking New York icon she was, remains, and you can put your roulette chip on \"the icon goes legendary.\"\n\n\"If you ask me who is Elaine Kaufman, I'd say she's the big mama of them all . . . I'm somebody who is still in the womb, a mother and a motherfucker . . . While I'm waiting to find out how it all turns out, I'll be having a good time,\" said Elaine.\n\nElaine Kaufman, Bronx-born, overweight, feisty, and not exactly adored by everyone, soared into the ownership of one of New York's most legendary celebrity restaurants in spite of a prickly stem that should have come with its own app.\n\nElaine's opened uptown in what was then an \"iffy\" (think dangerous) section of Manhattan, Yorkville, near the corner of Eighty-eighth Street and Second Avenue.\n\nWith a New York attitude that could be \"abrasive,\" she kowtowed to no one, no matter how indulgent, celebrated, or powerful they were. Where photographers were concerned, they had to keep their distance and earn her respect\u2014a tough prize to win.\n\n\"You're too close to my front door,\" she screamed at celebrity paparazzi Ron Galella as she hurled a slew of garbage can lids at him. Just one click and an East Side garbage can lid became famous, in a Warhol-esque way.\n\nThe woman warrior hurled the garbage top at Galella and hit a parked limousine by mistake. She was not amused. In time, she and Galella put their garbage can lids aside and moved on, though.\n\n\"She liked the press I got her,\" Ron concludes.\n\nElaine's preference for men was well documented.\n\n\"She called men when she wanted fun; she called women when she wanted something,\" notes Peter Khoury, a New York Times editor.\n\nHad this one-time nearly anorexic nighttime cosmologist planned to be the hostess and powerhouse behind one of New York's most seductive celebrity hangouts? Not likely.\n\nDo the math: the restaurant business is one of the best known entrepreneurial gambles, swimming with and against unforeseen tsunamis and fickle clients. Elaine remained relatively calm, cultivated, and rewarded loyalty.\n\nStruggling as a waitress in a restaurant in Greenwich Village, followed by a breakup with its owner, Elaine dug into her very own eponymous digs.\n\nWhen she built it, they did arrive and imbibe, often running up years of unpaid tabs, and occasionally dishing the dishes: the food at Elaine's was sneered at by those who chose to sneer. But they all came: writers, celebrities, artists, politicians, socialites, athletes, detectives, wannabes, and those she befriended, offered business backing to, hosted weddings for, punched, socked, verbally abused or 86'd.\n\nShe has been called everything from a bitch to a hypocrite to a loyal friend.\n\nSo . . . how did lightning strike? One tab at a time.\n\nElaine's opened its doors in Yorkville in 1963. The early sixties were an electrically fascinating time in New York and the world. We lost a president, gained The Beatles, marched against wars, contemplated the life of a hippy, dressed in minis, maxis, and even midis, or emulated Jackie Kennedy.\n\nWriters, singers, and painters had much to explore. Warhol's inundation of repeated images from Marilyn Monroe to the Kennedy assassination entered our collective conscious.\n\nWomen were starting to compete with men in the workplace, but were not quite comfortable going solo into a restaurant without a man and making an entrance with a girlfriend was still a bit of an ouch.\n\nElaine partnered up with Donald Ward to start Elaine's. In a metaphorical coin toss, the name was either going to land on Donald or Elaine. Elaine's it was. The story goes that Donald made his exit after he tried to keep Truman Capote out of Elaine's simply because Capote was gay. Elaine would have none of that, so Elaine's became Elaine's.\n\nAs Elaine's grew in popularity, it and its owner were immortalized in song and film. Billy Joel referred to Elaine's in his song \"Big Shot\" (1978): \"they were all impressed with your Halston dress and the people that you knew at Elaine's.\" Stephin Merritt of The Magnetic Fields mentioned Elaine's in \"Love is Like a Bottle of Gin\" from the record 69 Love Songs: \"You can find it on the Bowery \/ Or you can find it at Elaine's \/ It makes your words more flowery \/ It makes the sun shine, makes it rain.\" Woody Allen, who was an Elaine's ultra-regular, featured Elaine's in scenes from his films Manhattan (1979) and Celebrity (1998). Even the film Morning Glory (2010) showed Elaine Kaufman at the bar in Elaine's. The restaurant was also a constant in Stone Barrington novels by Stuart Woods; the first chapter always cites Elaine's as the location of the opening scene.\n\nElaine's continues to be a subject of conversation many years after its owner's passing. On May 10, 2014, The Moth Radio Hour featured anecdotes about Elaine's. George Plimpton recalled introducing Jerry Spinelli to writers, editors, and director Woody Allen at Elaine's, while Jos\u00e9 Torres recounted a story he'd shared at Elaine's about his first time facing a white man in the boxing ring.\n\nAlex Gregg and Elaine.\n\nPhoto provided by Alex Gregg.\nThere's No Elaine's Without Elaine*\n\nAmy Phillips Penn\n\nTHE FIRST TIME I met Elaine she scared the shit out of me, and I don't scare easily. I'm a New Yorker, after all.\n\nThe culture at Elaine's jumpstarted the Studio 54 mentality: Can you get a good table (or in Studio's case, get in at all)? And if you do, can you please get a gander at who's making these decisions?\n\nMichael M. Thomas, who has written many a New York column, found Elaine refreshing. \"I got a kick out of her. She was who she was. She learned the art of sucking up to people by insulting them. Elaine was the Toots Shor of the writing world.\"\n\nToots was known as \"the master of the needle jibe.\" Take it from there.\n\nElaine Kaufman was born in New York and was raised in Queens and the Bronx. One of her first jobs was as a night cosmetician. See? Scary.\n\nMs. Kaufman entered the New York restaurant biz in 1959\u2014no easy odyssey, even back then\u2014when she started running Portofino along with her then boyfriend, Alfredo Viazzi. The artsy crowd gathered and re-gathered: publishers, theatre lovers, writers.\n\nFour years later, Kaufman launched the eponymous Elaine's on East Eighty-eighth Street, where she mothered her chosen tribe: writers. The literati's tabs were known to lengthen and linger. No problem\u2014if Elaine liked you, that is.\n\nElaine and Norman \"don't-fuck-with-me\" Mailer dug into a doozy of a drag out that ended with Mailer writing Elaine a letter in which he vowed never to return to her restaurant. Elaine scribbled the word BORING on his pages and mailed them right back to him. A few nights later, Mailer showed up again.\n\nLoyalty doesn't always pay off, but in Elaine's case it did. Vonnegut, Mailer, Plimpton, and Talese were all Elaine's regulars. Woody Allen could usually be seen holding court at his round table. If you sauntered to the loo, you were elbow to elbow with Woody's linguine. How private can you get?\n\nAnd how did you get to the ladies\/men's room? Just ask Elaine. \"Take a right at Michael Caine,\" she might say.\n\nCelebrities celebrated her. Billy Joel immortalized her in the song \"Big Shot,\" and Woody Allen paid homage by shooting a scene of Manhattan at Elaine's.\n\nBooking a table at Elaine's was no easy feat. Getting a good table at Elaine's required Herculean-strong connections. And if you weren't going to be in the front room, preferably near a table where Elaine hopped to and from, why even bother? Sitting in \"Siberia,\" does not a New York image make.\n\nElaine's was center stage for people watching: celebrities, writers, politicians, athletes, and the who's who of the week, whose attendance would surely be noted by Page Six.\n\nOne torrential night, I went into Elaine's as part of my public relations gig to coordinate a story for a New York news station. I asked Elaine what her priorities were.\n\n\"Do what your boss told you to do,\" she said, and then she ran to Woody's table as if it were on fire to escort him and his guests out so their privacy would remain intact. Her priorities were clear.\n\nHow did she treat the press? This from former New York Post photographer Adam Scull.\n\n\"Elaine was loud, pushy, boisterous, tough-as-nails, took shit from NO ONE, including us photographers. She almost always kept us out on the street cooling our heels whilst the likes of Woody Allen, Robert De Niro, and every other famous actor and author kept coming back for dinner night after night. Elaine's was the place to go mostly because the authors and actors were protected vociferously by Elaine. 'Screw the photographers' I used to hear her say, time and time again, only occasionally allowing just a select few of us in from time to time. Elaine Kaufman was the doyenne of protective restaurateurs. She caved to no one.\"\n\nWhy did Elaine scare me? Who wants to mess with that?\n\nIn time, Elaine and I warmed up to each other, one New York edgy step at a time. I had my own column by then.\n\n\"Thanks for the plug,\" she would say as I walked in.\n\nElaine slipped into my \"like\" list surprisingly, but surely.\n\nWhen she was going through a slow time, I had a date with a producer. His secretary called me and told me that I could pick \"any restaurant in town.\"\n\n\"Elaine's,\" I replied, although the numbers 21 were adding up in my head.\n\n\"You could go anywhere in New York and you picked Elaine's?\" she said.\n\nWhen we walked in that night, Elaine didn't say anything, but I knew that she recorded the gesture. We became friends.\n\nAfter Elaine's death in 2010, Diane Becker, the heir to Elaine's, made a decision. Elaine's was closing.\n\n\"There's no Elaine's without Elaine,\" she conceded.\n\nWhat choice do we have but to agree?\n\n* This is the article I wrote for New York Natives that caught my publisher's attention. Reprinted here with permission from New York Natives.\nTake Four : Customer, Publicist, Columnist, and \"Let's do lunch, Elaine\"\n\nAmy Phillips Penn\n\nI ALWAYS LOVED going to Elaine's.\n\nI don't remember the exact date that I went there, but it was sometime in the late 1960s or early 1970s. Glory Days.\n\nI didn't meet Elaine until years later, which was fine with me. She sat with her favorites, changed tables when the mood struck, and had no idea that I was alive. She didn't radiate \"warm and fuzzy,\" and I was happy to keep a respectful distance.\n\nWhen I first went to Elaine's, it was as someone's \"date.\" Whether it was at Elaine's, 21, aka. the numbers, or Le Club, most of the women took an invisible note as to where they were seated. It was a reflection on their date's allure, not theirs.\n\nThere were unspoken rules at Elaine's, such as \"please do not feed or interrupt the celebrities,\" but I was oblivious.\n\nI broke that rule before I knew it was a rule.\n\nOn my way out of Elaine's, dressed in a maxi raincoat and floppy brown suede hat, and feeling very Annie Hall, I went over to Woody Allen's table and announced that I had a screenplay that I wanted him to see.\n\nI was in my twenties and believed that New York held no barriers especially when it came to writers and artists and lovers of New York like the brilliant Woody Allen.\n\nArtists of all kinds seemed so approachable, then. The art world was booming into pop, op, and anything goes, punched with price tags as bold as a Jackson Pollock.\n\nAt museum openings, casual Saturday New York gallery going, art auctions at Sotheby's and parties in the East Hampton, artists and writers mingled with the rest of us, flaunting no attitudes of superiority for the most part. They had all dreamed the dream once.\n\nWoody was extremely gracious.\n\nHe told me where to send the script.\n\n\"Are you sure that I can't send it to you?\"\n\nI was twenty, and felt that you could ask for the moon, and that it just might RSVP favorably. Everyone needs a break, and writers and artists know better than to miss their moment. Embarrassment is a small dividend for having your words published, or echoed on screen.\n\nThe screenplay had evolved in my parent's New York kitchen. My mother had painted the walls an electric yellowish orange, which by proxy vote, we had all declared frightening. She and I had visited a famous psychic, who had described the love of my life, who I had (and have) yet to meet.\n\nWe decided to write a satirical screenplay about it. The heroine is looking for her prince-to-be, and who could be funnier in that role than Woody Allen? Since he doesn't even know that I exist, the mother and daughter decide to stage a fake kidnapping and leave them on an island for a few days. Naturally they fall in love. The best laid plans . . . But they manage to get kidnapped for real. No one pays the ransom, because they think it's part of the plan. The two lovers are shot. The credits read: \"Better Life Next Life\" or \"The Girl Who Cried Love,\" on the chalk outlines of their bodies.\n\nJust another New York slice of black humor.\n\nSometime in the late seventies, I was working for a small publicity firm named Gifford-Wallace. It was run by a husband-and-wife team, Michael (the wife) and her husband, Ed Gifford. They were a small, but powerful team. Their clients had included: Hair, Godspell, Studio 54 (from day one), Metromedia TV, and other prime accounts.\n\nMy assignment was to go into Elaine's with a Metromedia camera crew and organize a segment on Elaine's. The Giffords had called ahead to Elaine to let her know that we were coming.\n\nIt was a torrential night. I arrived drenched, but on time. The camera crew was there.\n\nIn my best New York girl school manners, I politely asked Elaine where we should set up. She roared at me, a lioness predator in high hormonal gear.\n\n\"Do what your boss told you to do.\"\n\nNo, \"please, thank you, or would you like anything to drink?\"\n\nWhat stratosphere was I in and who did you have to cozy up to get out?\n\nShimmying, in a loose Mumu style dress, this more than ample woman, all but gathered up Woody Allen and a few other customers. Like a mother cat, she was saving her kittens from getting wet by throwing them into the downpour escorted by a chauffeur and waiting limo.\n\nAfter we finished our assignment, we left without saying goodbye to Elaine. The crew from Metromedia gave me a ride home in their truck. Someone passed around a joint, and soon I was home, the gruff, dismissive woman I had tried to make friends with was soon forgotten.\n\nSometime in the late seventies and early eighties, I freelanced as an assistant to the world renowned society\/fashion columnist Eugenia Sheppard. I covered parties, movie screenings, younger Manhattan, and pretty much anything I chose that did not conflict with Eugenia's priorities.\n\nWhen Eugenia died, I was honored to carry on her column with my own byline. I had become a syndicated New York society columnist. I gleaned instant respect, nonstop invitations to everywhere and anything, and a parcel of power, New York style.\n\nI don't know if Elaine and I were ever officially introduced, but I do remember her next words to me after I had written about Elaine's in my column.\n\n\"Thanks for the plug,\" she said when I came in for dinner.\n\nThe roar was reserved for elsewhere.\n\nI have arrived.\n\nElaine feels free to plop someone down at our table. I'm sitting at a table at Elaine's with my friend Patrick Shields.\n\nPatrick was the six-foot-seven director of Le Club, a private dinner club, which hosted Caroline and John Jr. Kennedy's birthday party on the same night.\n\nJackie O wrote Patrick a handwritten thank-you note. I wouldn't be surprised if he was buried with it.\n\nCan you stand it?\n\nOne night, when I was sitting with my friend Harvey Kirk, a publicist for the Giffords and Studio 54, Elaine came over to the table.\n\nJohn Lennon wants to know if he can get into Studio 54.\n\n\"I'll take care of it,\" said Harvey, and went back to his pasta.\n\nDavid Black and Barbara Weisberg's wedding. Alec Baldwin is the best man.\n\nPhoto provided by David Black.\n\"Marry Me, Marry Me\" At Elaine's\n\nDavid Black\n\nDavid Black's awards runneth over, as a journalist, novelist, screenwriter, and producer. Black received the Edgar Allan Poe Writer's Guild Award nomination three times: for best fact crime book Murder at the Met and \"Happily Ever After\" and \"Carrier\" (episodes for Law & Order). Then there's the National Endowment of the Arts Grant in fiction, Playboy's Best Article of the Year Award, and yes, a Pulitzer Prize nomination for The Plague Years, and there's so much more. Black has taught writing at Mt. Holyoke College and Harvard. All this, and when he talks about Elaine, he tears up. Black is the vice president of the Executive Committee for the Table 4 Writers Foundation, \"continuing the tradition of Elaine Kaufman.\" We welcome him in.\n\nTHE FIRST TIME I went into Elaine's, it must have been 1968 or something like that. I was writing for one of the underground newspapers and I got an invitation to go to Norman Mailer's birthday party at Elaine's. It was wonderful.\n\nI was a kid from Massachusetts and it was my dream of what every night in New York must have been like\u2014because everyone in New York must have been there.\n\nIn 1982 or 1983, I was commuting two days a week from Mount Holyoke, where I was the writer in residence. After my class, I would go directly to Elaine's. I usually sat at the family table with Jack Richardson and we became very close friends. I loved Jack; I actually twelve-stepped him. He was a drinker and a coker, so was I.\n\nThat's when I really started going a lot. I met some amazing people and had some crazy times at Elaine's.\n\nThere was Bob Datilla, who did all his literary agenting out of Elaine's like it was his office. And I saw Dick Wolf quite a bit. One night, John Henry brought a fire eater into Elaine's and when somebody took out his cigar, she pulled fire from her mouth. Another night, Michael Disend came in with a woman he was dating who was part of a live sex act. She told us all these stories. She had bravado. She said that once on stage she had taken a bowling pin up her ass. Now, Elaine's is usually very noisy, but the room suddenly went silent when she said that.\n\nWhen I got divorced, I told Pepe, the maitre d', \"If I come in here with a date and I look like I'm not having a good time, get me out of here fast.\"\n\nHe said, \"What if you look like you're having a good time?\"\n\nI replied, \"Well, get me out of here faster.\"\n\nWhen I met my current wife, Barbara Weisberg, I knew the minute that I saw her\u2014it was like going home\u2014it was love at first sight.\n\nI was going to ask her immediately to marry me. I said, \"You're the one. I'm not looking anymore because it's you or no one.\" I couldn't say it on the first date because I'd cut myself shaving, and if I leaned over, I would bleed all over her. So I waited for the second date and asked her to marry me. She looked deep into my eyes and said. \"Are you out of your f'ing mind?\"\n\nBut she agreed to keep going out with me. It would take me two years to woo her.\n\nFor the fourth date, I took her to Elaine's and we kissed. Jessica Burstein was taking pictures that night. Two weeks later, that picture appeared in a New York Magazine article about Elaine's.\n\nAnd over the next years and up until Elaine's closed, Jessica made it a point to photograph our kissing. She's always threatened to put on an exhibit that would show our relationship progressing through seven years by how we're kissing in all these pictures.\n\nWhen it came time for us to finally get married, we talked to Elaine about our plans. We wanted to have the party and wedding at Elaine's. Elaine turned to Barbara and asked her if this was the first time she'd been married, and Barbara said yes.\n\nElaine whacked me on the side of the head and said, \"What's the matter? Do it right.\"\n\nSo she planned the entire wedding for us, from flowers to a chuppa. And it was a great wedding. We had a rabbi who looked like a leprechaun officiate. Alec Baldwin was my best man and we had a ball.\n\nI love Elaine's and one of the reasons why I\u2014and everyone else\u2014loved her is that when I walked in for the first time, I was an unpublished novelist. A no one with no platform. She talked to me for a few minutes and immediately adopted me.\n\nElaine didn't much care who you were, where you came from, what your merit badges were. If you were family, you were family. Over the years when I was broke, she'd feed me for nothing. She'd write on my check, \"Tip the waiter.\" As I continued to work, she'd introduce me to people like Marty Bregman and say, \"You gotta hire this guy.\"\n\nWhen I became successful, she'd do the same thing with me. She'd say to me, \"You gotta hire this guy.\" As I was making more money, I would pay twice as much for my meals at Elaine's to make sure the next guy who was broke was covered.\n\nElaine paid medical bills, she sent clothes to my home for my kids, and she did that for many, many people\u2014and that was why she was greatly loved.\n\nThe best tribute to Elaine is to keep her memory alive, the memory alive of a woman who was extraordinarily special. It matters to many, many of us.\nElaine Empties the Stanley Cup\n\nPer Bjurman\n\nPer Bjurman became Sweden's leading music writer and was voted The Most Influential Critic by the leading national music industry publication three years in a row. In 2005, he was offered the position as US Correspondent for Aftonbladet, which he gladly accepted and moved to New York. The following years, he covered everything from elections\u2014not least the historical US presidential election in 2008\u2014to natural disasters, big trials, and space shuttle launches. He is still in New York, working for Aftonbladet, but now mostly covers hockey in the NHL, a league in which more than sixty Swedes play.\n\nI CAME TO New York in January 2005 as the correspondent for the Swedish daily newspaper Aftonbladet and immediately decided Elaine's was going to be my home away from home.\n\nI had been there before, when I was just a guest in the city. I loved the atmosphere, the crowd, the exhilaration, and the feeling that anybody could walk through the door at any moment. For almost a year, I dined by myself at the two-person table under a wonderful photograph of JFK and LBJ at Griffith Stadium in Washington, taken by the photographer and Elaine's regular, Neil Leifer.\n\nEventually, I asked Tony, the great Elaine's waiter, if he would be willing to introduce me to Elaine. \"Sure,\" he said, and walked me over to the Michael Caine's table where Elaine was parked for the moment. Before I even had the chance to tell her who I was, she just growled, \"Yeah, yeah . . . you're that Swedish journalist. Sit down.\"\n\nPer Bjurman and Elaine.\n\nPhoto provided by Per Bjurman.\n\nPhoto provided by Per Bjurman.\n\nNobody came into Elaine's joint that often without Elaine knowing exactly who the hell they were.\n\nFrom then on, I was part of the family\u2014I always had a seat in \"The Line,\" and, for the next five years, I enjoyed more unforgettable nights under the warm glow of her Art Deco light sconces than most Swedes experience in a lifetime. But it was more than just fun, excitement, and the thrill of having a perfect dry martini placed on the table before I sat down. Elaine became my surrogate mother in New York. She took me in, introduced me to people she knew I would love, and made sure that New York became my new home. It meant everything to me as a shy foreigner, far away from friends and family. There is no way I can repay that kind of life-changing favor, but I did my best in a weekly column I had, and still have, in Aftonbladet.\n\nI wrote extensively about the nights on Eighty-eighth Street and Second Avenue. Not so much about the celebrities\u2014though, now and again, I couldn't resist bragging about sitting at tables next to Arnold Schwarzenegger, Robert Altman, Joan Rivers, Larry Hagman, and Richard Dreyfuss\u2014but rather about Elaine herself, her magnificent staff, and the wonderful characters that really were her closest family and also became members of my own New York family: Ash Bennington, Pete Khoury, Helene Gresser, Father Pete, Jessica Burstein, Josh Gaspero, and the incomparable Joey, the unknown Lenny Bruce of the Bronx. After a while, Swedish tourists who read those columns started coming in\u2014and paying for dinner. Elaine really liked that.\n\nThroughout the years, I also brought in a lot of Swedish hockey players. I cover the NHL for my newspaper and have gotten to know my countrymen in the league. Henrik Lundqvist, the New York Rangers superstar goalie, came for dinner a couple of times. We, meaning me and the bartenders Craig and Duffy, both devoted Ranger's fans themselves, made sure Lundqvist got to see the picture of Elaine drinking from the Stanley Cup. The photo was taken late at night when the 1994 championship team, lead by Mark Messier, came into Elaine's to celebrate, and it hung behind the bar for twenty-five years. Lundqvist looked at the picture quietly for a long time. You'd be excused for thinking that was the moment Henrik Lundqvist decided that he some day was going to win that cup, and drink from it exactly as Elaine had.\n\nAnother night, the Detroit Red Wings were in town. A bunch of injured Swedish players, with friends and family, joined me at Ms. Kaufman's saloon. It was one of those magical nights when everyone was hopping tables for hours, while the check somehow followed you around. The night ended at the bar across the street at 4:30 in the morning\u2014with me dancing with one of the Red Wing stars' crutches. That kind of thing could only happen during a night at Elaine's.\n\nWhen my mother visited me from Sweden, we of course went to Elaine's. My mother doesn't speak much English, but she and Elaine found a way to communicate anyway. At the end of the night, Elaine put one of her big hands on my mother's wrist, smiled and said: \"Don't worry, I'll take care of the kid for you.\"\n\nShe sure did. I'm still around and I still love New York, not least thanks to the friends Elaine introduced me to. But it's not the same anymore. Life itself is less exciting without my surrogate mother on the Upper East Side.\n\nI miss her terribly.\n\"Give Him Table Six, He Says He Has Money.\"\n\nSteve McPartlin\n\nA former bartender, Steve McPartlin made the fantasy move into reality, as he segued into TV and radio broadcasting. McPartlin was one of the original correspondents for Current Affair, a host for Inside Edition, and a radio sportscaster for ABC Sports.\n\nIN THE EARLY 1970s, there was a joint on Second Ave up the street from Elaine's named Cavanaugh's. It was run by a great saloon guy named Lou Cavanaugh and his lovely wife Maureen.\n\nI would frequent their establishment often even though I lived and worked tending bar on the north end of the island in Inwood.\n\nOne day I get a call from Lou, who told me he hurt his ankle and he needed someone to tend bar. All of his regular bartenders were otherwise indisposed\u2014a fact I would attribute to the fact that there was going to be a severe snowstorm that night. Chomping at the bit to work \"downtown,\" I accepted.\n\nIt was a slow, slow night and, at about 1:30, Lou said we should leave, so we started to close up with not a soul in sight.\n\nAll of a sudden, the door flies open and this two-hundred-and-fifty-pound woman starts screaming and swearing, \"Those motherfuckers! What a bunch of fucking thieves!\" I looked at Lou, who had a reputation for being a tough guy, and said, \"This is all yours, brother.\" Lou calmed her down and sat her at the bar. It turned out a cab driver wanted twenty bucks for a five-dollar fare to take her home and she was having none of that.\n\nLou said, \"Steve, this is Elaine Kaufman. She owns the joint down the street. Give her a drink.\" Now, of course I knew who Elaine was because I can read and she was always in the papers in those days. I was studying acting and dating a singer, so I was well aware of the glitterati and literati who frequented her restaurant.\n\nWe three had a few drinks and Lou said we were going to drive her home. When we got to her place, I walked her through the snow up to the door and said, \"Elaine. I am dating this girl who sings at Joe's Pier 52 and it's her birthday next week. Can I bring her to your restaurant for dinner?\"\n\nElaine looked at me like I was a freak. \"Do you have money?\"\n\nI said yes.\n\nShe replied, \"Well, it's open to the fucking public, ya know.\"\n\nThe birthday came and we went to Elaine's. My girlfriend was a bit on the skeptical side that on this, her birthday night, I was taking her to a place where we would be treated like interlopers and placed somewhere near the bathroom and kitchen.\n\nWe were greeted by Aldo, a huge guy who had the personality of a Cossack. When I told him I wanted a table for two, he was ready to lead us to Siberia when I spotted Elaine at the end of the bar doing the checks. I said hi and reintroduced myself. She turned to Aldo and said, \"Give him table six, he says he has money.\"\n\nAfter an hour or so, Elaine sat with us and told us stories and made fun of me treating the place like it was a temple.\n\nWe became fast friends and, for the next forty years, I never sat in the back. She always gave me a table up front and she always took care of me.\n\nNot long after that, I gave up the bar for a career in radio and TV sports\/news and, whenever I would walk in by myself, she would beckon me to join a group she thought either would be good for me to know or people I could entertain. If you were a journo of any kind, the people she would introduce you to were invaluable, from the famed writers and actors to cops and robbers and assorted members of the clergy.\n\nAnything you ever needed to know about what was going on in New York, you could find out at Elaine's.\n\nThe only name dropping I will do: Once I walked in and she grabbed me at the door and said, \"Sit with me, I want you to meet Polly Bergen. She likes young guys.\"\n\nI was fifty at the time so it was a great compliment and I complied.\n\nMy last conversation with her was toward the end. It was just the two of us at a table on a Sunday night in the summer. She was lamenting about how business had fallen off. I, trying to put things into perspective, said something to the effect of \"Geez, baby, after all these years, does that really matter that much? You own half the block.\"\n\nMy dear friend Elaine looked at me and said in her own very special way, \"Hey, asshole, count your own fucking money.\"\n\nAnother lesson learned from my dear friend.\nWhen Elaine Socked Me\n\nChina Girard\n\nChina Girard is a former Ford model who, along with three other Blue-Blood friends formed The What Four, aka \"the first all-girl rock band of the sixties.\" Columbia Records signed them; they recorded, toured, and were included in an album that boasted two Grammy nominations. Ooh-la-la meets yeah, yeah, yeah. Of course, China went to Elaine's.\n\nIN 1966, I was sitting with songwriter Jerry Leiber and Paris Review editor George Plimpton, having a conversation about the music business, mostly because I had just started an all-girl rock band called the What Four.\n\nSuddenly a drunk and loud Norman Mailer intruded, demanding Jerry engage in an arm wrestling contest with him. Jerry politely said no thank you and tried to continue eating his spaghetti.\n\nWell, Norman was deaf to the word no, so he plunked himself down in the chair next to me and leaned toward Jerry, grabbing his wrist. Now, I was in the unfortunate position of being in the middle. I scooted back just as Jerry yanked his arm away from Norman's gasp. Then Norman pushed Jerry's hand in his spaghetti, making Jerry furious. George was now on his feet, pleading with Norman to back off, at which point Norman pushed Jerry into the wall. Then hell broke loose. Jerry took a swing at Norman, who kept yelling for Jose Torres to help him. Jose, being a champion boxer and whose hands were licensed as lethal weapons, could not assist and pretty much backed away, causing Norman to get even angrier.\n\nCarmen Capalbo, China Girard, and Peter Dinkle.\n\nPhoto provided by China Girard.\n\nMeanwhile, I was only feet away at this point when Elaine marched up and socked me in the jaw, thinking that I had started it all. She said, \"Men are always fighting over you.\" Even if that exaggeration was so, it was no reason to punch me. I probably should have punched her back.\n\nBy then, the whole bar was in chaos. Norman was escorted out the door by Jose. I took my seat at the table, as did Jerry and George, who said, \"Well, that was an interesting episode.\"\n\nOne thing that we could all count on was George's wonderful humor.\nHow Elaine Became My Photo Agent\n\nJessica Burstein\n\nElaine Kaufman has mingled magic and success into the lives of many. If Elaine was your friend, or simply believed in you, she would introduce you to the world, support you financially, and allow writers to run up unpaid tabs for years. Often, a writer would arrive at Elaine's years, or even decades, later to repay Elaine thousands of dollars in bar and pasta booty. Often, they returned to their lair with Pulitzer Prizes and bestsellers.\n\nJessica Burstein was a New York photographer who was driving along a challenging\u2014possibly catastrophic\u2014road when Elaine made an emphatic entrance in her life that would change Jessica's destiny.\n\nI KNEW ELAINE Kaufman for a long time before she really got to know me. Although I'd spent time at her restaurant since the mid-seventies, and she knew me well enough to, for example, often try to \"pimp me out\" to guys whom she thought would like me for my looks, Elaine didn't take me seriously until she saw my photographic work at a gallery exhibition in 1990. It was a defining moment in what would become a rather complicated personal relationship\u2014mother-daughter, friend (and very often foe), and ultimately, the person who became my photo agent.\n\nSomehow Elaine made her way down to Soho, in New York City, for the opening of my exhibition and immediately decided that a triptych of three portraits I'd taken of Truman Capote would look perfect on the wall of \"the line\" in the front room of her restaurant. She also decided that she'd like to get her paws on another Capote print\u2014this one of him sitting with a stuffed Cobra, which looked as if it were coming out of his groin. In fact, it was Truman's favorite photograph of himself\u2014or at least, that's what he'd told me. Of course, in lieu of money, Elaine offered my gallery dealer the great opportunity to have my work prominently displayed in her restaurant. After she stared him down (and with a little help from me), she got the prints as a gift. Within a day, they were on her walls. Just as quickly, she suddenly viewed me as a woman worthy of her serious attention.\n\nIn 1992, during a particularly dry work spell, I was having second thoughts about continuing as a photographer. While in the restaurant, I told Elaine that I was thinking of going to law school.\n\n\"Are you nuts?\" she yelled at me. \"Quit? Kid, you're better than Avedon! You got no place to shoot,\" she continued, \"then come here to do it.\"\n\nThat moment opened to the door for the next twenty years\u2014twenty years of my life I spent as a photographer. Shooting at the restaurant began long before digital photography existed, and so, between the cost of film, processing, paper and printing, shooting at Elaine's was to be the most expensive job I ever had. But it was not without its rewards. In the beginning, no matter what I photographed there, Elaine would have the prints framed and placed on the walls. There was actually a \"Jessica Wall,'' with only my work on it. But it wasn't easy. There were, often, problems dictated by Elaine's moods when she'd decide that I couldn't shoot something that really interested me. That drove me nuts, and we'd get into some heavy fights.\n\nJessica Burstein with Chris Noth.\n\nThroughout it all, however, Elaine talked me up to anyone and everyone who could possibly give me assignments. Within a year of shooting at the restaurant, she'd introduced me and my work to, among others, television producer Dick Wolf, which resulted in twenty years of work on the Law & Order franchise, a published book about the show's crime scenes, and a resulting solo museum exhibition of this work at the famed George Eastman House International Museum of Photography and Film in Rochester, New York. She was also instrumental in getting George Steinbrenner to see my work, which resulted (with the help of the Yankee's President, Randy Levine) in my being commissioned as the fine art photographer for the building of the new Yankee Stadium.\n\nElaine took every opportunity to try to coach me about how to run business\u2014a subject that has always eluded me. Her efforts failed, because business literally makes me sick. To the end of her life, Elaine tried in every way to get me to understand the importance of business in the arts. She concurred with my father, who'd often told me: \"It's not art if it stays in the drawer. The work has to be seen.\"\n\nAfter noting, once again, that I was hopeless about business, one of the last things Elaine said to me was \"I won't be here, forever. You've got to get an agent.\"\n\nI'm still, admittedly, terrible about business, but one thing that I have made a point of \"getting out of the drawer\" is my collection of photographs from Elaine's, only made possible because of Elaine Kaufman's unending belief in my talent. I will forever be grateful.\nTravels With Elaine\n\nJessica Burstein\n\nELAINE LIKED TO travel with me because, frankly, she liked fun and she knew that I'd deliver. Of the various trips we took together, the highlight was our 2005 trip to France.\n\nI preface this story by noting that Elaine was always torn about leaving her restaurant and in particular, as she got older, her trips away became shorter. Generally, she'd last a few days, while endlessly calling the restaurant, and then decide that she needed to go back to Manhattan.\n\nI was planning to go alone, in earlier September, to Vichy, France, when Elaine asked me if I'd accompany her, prior to that time, to the Deauville Film Festival.\n\nOur prep for the trip included:\n\nA. Asking me to push back my planned vacation to Vichy, France, to go with her to the Deauville Film Festival.\n\nB. Speaking with her pulmonary doctor about her ability to travel with breathing problems.\n\nC. Each of us buying a seat in coach (no wild airline expenditure for Elaine and no allowing me my mileage upgrade to Business Class.).\n\nD. My calling the hotel in Deauville at least five times to ensure that her room (not mine), as had been promised, was comp-ed.\n\nE. Calling Hertz to reserve a Citro\u00ebn, preferably automatic, with enough room for the two of us and . . .\n\nF. For the largest Tumi suitcase, which Elaine had spent days packing.\n\nDay 1:\n\nThree p.m. arrival at JFK, making certain that there's a wheelchair for Elaine. The TSA insists on patting her down. At the time, men were still allowed to do the pat-downs on women. I am outraged. She giggles and says, \"Go get me a sandwich.\" In coach, I'm crushed in the window seat, praying that Elaine has to get up fairly often so that I can move. Every time she does get up for bathroom breaks, I run around the aisles like a lunatic and get back to my seat before she gets back to hers. The flight attendant can't get Elaine's tray table down properly. Elaine tells the flight attendant, \"Just rest it on my tits.\" I stay awake for the full eight-and-a-half hours, while between bathroom breaks and food, Elaine sleeps.\n\nDay 2:\n\nSeven a.m. arrival at Roissy. Check that there's a wheelchair for Elaine. Leave the plane, assuming she'll soon be along. Wait twenty minutes at Customs. No Elaine. Worried, I question every official in sight. Nobody knows anything or seems to care. An hour passes. Frantic, I threaten to sue the airline for losing a passenger. I make up every possible horrible thing that could happen to these people if they don't find Elaine. Another half hour and suddenly . . . Elaine appears, in a wheelchair, a big grin on her face. She tells me that the airline left her in her seat. I ask why she said nothing. Her reply: \"You know that I'm shy.\"\n\nArrive at Hertz to find that since we are more than two hours late, the Citro\u00ebn is gone. But, they have one car, a \"tr\u00e8s belle\" stick shift Mini-Cooper (pronounced 'mini-coop-ear' in France) available. I am livid. Elaine insists that we look at it, which we do, since obviously we have no other choice. It's Royal Blue and a convertible and it's love at first sight for Elaine. By some miracle, we fit in\u2014luggage and all. Exhausted, on Paris's p\u00e9reph\u00e9rique, I get lost for two hours, driving in circles before I figure out the correct \"Porte\" to Deauville. Elaine is giggling the entire time. I'm going to kill her. Just as I think it's smooth riding all the way to the hotel (for my planned nap), Elaine wants to stop to eat.\n\nAt 4 p.m., in a tiny French village, there is no where to eat. I find one small bakery barely open, beg, flash some Euros, and by another miracle, get Elaine a ham sandwich. By the time we get to the hotel, the idea of a nap is a memory, since we are expected at the Film Festival's opening dinner gala and Elaine needs me to unpack for her and help her to get dressed. Our dinner table is filled with men. Elaine's not just in Deauville, she's in heaven.\n\nDays 3\u20136\n\nAwakened by Elaine's raspy voice on the phone: \"You missed breakfast. Havin' lunch?\" Seems Elaine is not at all interested in the films at the festival. Her interests included food; reading; the pool; lots of rides in the beloved mini-coop-ear; a visit to the Normandy WWII museum; my translating for her French to English, and vice-versa; endlessly having me call her restaurant to make sure that there are no problems; fiddling with my hair; bragging (to her credit) about a photo of mine prominently displayed in the Festival's catalogue; cocktail parties; dinner parties; and . . . bumping into Roman Polanski. Polanski appears at the hotel's cocktail lounge where Elaine and I are sitting. Having recently won his libel suit against Vanity Fair for reporting remarks ostensibly made at Elaine's restaurant, he jubilantly tells Elaine, \"I won.\"\n\n\"Big deal,\" she replies, \"You got no real money.\"\n\nPolanski's reply of \"That's not the point,\" falls on Elaine's deaf ears. I watch, fascinated by how much he wants her approval.\n\nDay 7\n\nFree at last! I'm off, alone, to Vichy. Wait, wait . . . Elaine tells me that she wants to go with me. I can't believe my ears. This woman, who can't normally bear to leave her restaurant, is now extending her trip away from it. She's being insistent and so, I relent. I pack up Elaine's things and my own, and by late afternoon, we stuff ourselves in the mini-coop-ear.\n\nPhoto credit: Jessica Burstein.\n\nMy plan to drive directly to Vichy, a trip of less than four hours, is suddenly sidetracked when Elaine announces that she wants to fly out of Paris. This means that we have to stay overnight at Orly, because there are no night flights to Clermont-Ferrand. From there, it's an hour's drive to Vichy. Again, I relent. I make reservations for a flight and for the Orly Hilton. The car route to Vichy is virtually a straight line, but going to Orly means a trip back through the dreaded p\u00e9riph\u00e9rique and, sure enough, I get mixed up and find myself coming out of a tunnel onto the Champs-\u00c9lys\u00e9es.\n\nElaine has been reading and loudly chewing gum throughout the entire ride and only when I mention that we're somehow on the Champs-\u00c9lys\u00e9es does she bother to look up. She laughs . . . and laughs. And I'm ready to, once again, kill her. Instead, I continue on the route to Orly. It's at this point, when I reach the airport, when the real trouble rears its head. The route to the Orly Hilton is designed for travelers coming from the airport itself. But, if you're coming from outside the airport, it's tricky. I'd arrived at the airport around 7 p.m. By the time I found the entrance to the Hilton, it was 10 p.m. Had we driven from Deauville, straight to Vichy, we would have been there long ago. Instead, Elaine and I were driving around for hours and in some really creepy places, and she thought it was funny. I had to keep the inside light on while she read, but she met every desperate sigh from me with a chuckle. Finally, at the Hilton, we each had vodka to toast what she thought a great adventure.\n\nDay 8\n\nQuelle Journ\u00e9e! Early in the morning, after making certain that our airplane tickets were in order and that a wheelchair was in place for Elaine to get on the plane, I returned the mini-coop-ear. I ran back to the airport gate to find Elaine, expecting to see her in the wheelchair. Instead, she was sitting on a regular chair. The flight was boarding and she had no rational explanation as to why she wasn't in the wheelchair. It turns out that she was \"too shy\" to insist on the wheelchair. In consequence, we missed the flight, at which point Elaine wanted to hire a car and driver to take us to Vichy. In my head, I added up what this was costing in terms of money and time. And I knew that I was putting up with this craziness for Elaine. I agreed to the car and driver at a fee that was exorbitant\u2014half of which I paid. To make matters worse, throughout the ride, Elaine would repeat, in a mantra-like tone, \"Now, this is the way to travel!\" After what she'd put me through, I was a hair's breadth away from strangling her. Finally, arriving at the hotel in Vichy, we went to the hotel's spa to set up appointments for \"Les Soins\"\u2014massages, exercise classes, general well-being and \"taking the waters.\" We had an early dinner at which she confessed that she was always worried about money. She said that she didn't really know how to make it, but that she was very clear about saving it. And indeed, she has been very good at that, refusing to exchange her dollars for Euros (\"a bother\"). When it's necessary to use cash for payment, yours truly remained the banker.\n\nDay 9\n\nTwo friends of mine came to the hotel for a pre-planned surprise welcome for Elaine. We called her to the lobby, where they told her that they were inviting her for dinner at the \"Elaine's of Vichy\"\u2014Brasserie du Casino. Apart from the fact that they knew about Elaine's, she was immediately taken by one of my friends\u2014Bruno Pinard Legry. It was to be the beginning of a beautiful friendship (reference the Vichy bottle, Bogart, and Claude Rains in Casablanca). We spent the day at the spa, but for me, the day was spent primarily waiting around to see that Elaine was okay. There were a couple of hitches during Elaine's massages, but none for publication. It was a great night at the Brasserie, where the owner introduced Elaine to the applause of all the patrons.\n\nDay 10\n\nHave to keep my eye on Elaine. In one of the spa's pools, Elaine got into an argument with a Saudi prince over what she called his \"harem\"\u2014a group of women who were there with him. Got a call in my room, asking me to get over to the spa, ASAP. Arrive to hear Elaine name-calling the prince, who wasn't doing a bad job getting back at her. Both parties use the word fat. Hopeless. With help, I get Elaine out of the pool. Later, she tells me that she's proud that she stood up for women\u2014the irony of that remark escaping someone who is, in fact, known for being mean to them. I say nothing. But, in the meantime, I speak to the prince, having already met his beefy, gangster-ish bodyguards. I tell him she was sorry to have offended him. At the concierge's desk, a gift from the prince\u2014a Burberry scarf is left for me. I give it to Elaine, who immediately throws it in the garbage. Things improve that night as Bruno takes Elaine on a solo \"date\" at the Brasserie, the \"Elaine's of Vichy.\"\n\nDay 11\n\nI'm tired. I love Elaine, but it's a lot of work taking care of her. I haven't had a real vacation in three years and I need Elaine to leave. Don't know how to ask her. We take a cab ride through the city because she's unable to walk it and I want her to see as much as possible. Turns out that I don't have to ask her when she's leaving. She tells me that she has to get back to the restaurant, that this is the longest trip she has ever taken and plans to leave tomorrow. I spend the afternoon interviewing drivers because I want her to leave happy. This means finding a young, handsome, and charming male driver to take her on the three-hour drive to the airport, with food stops adding time along the way. I find him. Tonight, we go for a farewell dinner at the Brasserie, the only restaurant where Elaine wants to be.\n\nDay 12\n\nTen a.m. Elaine meets her driver. She's ready to dump me in half a second and get going with Mr. Young, Handsome, and Charming. Bruno and friends come to say goodbye. I'm oddly feeling a little separation anxiety, as are my friends. Elaine is a major presence and something will be lost with her departure. We stand on the street, feeling small, waving like little children to a parent saying goodbye. Tonight, we will go to the Brasserie to toast her with all the patrons, who've become \"family\" at the \"Elaine's of Vichy.\"\n\nAddendum\u2014Two weeks later:\n\nI arrive at Elaine's and people virtually bow down to me. Elaine calls me over and says, \"Thank you. I've told everyone that this was the best trip I've had in my life.\" She's the kindest that she has ever been to me. I'm not hungry and she's not making me eat! Unheard of! It feels way too uncomfortably weird, but I needn't have worried, because . . .\n\nThe following night:\n\nElaine is back to being the not so \"shy\" Elaine of her restaurant. There, I'm immediately greeted with the growl: \"You betta be havin' dinna!\"\n\nCandace Bushnell, Gay Talese (behind), John Scribner, Jr. (at her foot), and Richard Behar (with cigar) celebrate the Oscars.\n\nPhoto credit: Jessica Burstein.\nMy Last Night At Elaine's\n\nJodi Garner\n\nJodi Garner is the author of \"The 45-Year Old Intern\" published in The 52 Weeks by Karen Amster-Young and Pam Godwin (Skyhorse Publishing 2014). Jodi is also the producer of The Smart Kids Guide to Grownups, Off-Broadway, in production; associate producer, Disaster! A 70s Disaster Movie Musical, Broadway, in production; associate producer, The Jazz Singer, Off-Broadway, in production; producer Danny Boy, 2005 New York Fringe Festival Audience Favorite; and associate producer, Stephen Schwartz's Captain Louie, Off-Broadway. Jodi is also a board member of The W. Eugene Smith Memorial Fund for Humanistic Photography. Jodi lives in New York City with her fourteen-year-old twins and was at Elaine's on the last night before it closed in May 2011.\n\nB, R, AND our friend A, who had also joined us many a night at Elaine's from 2004 through the end, and all of whom always stayed way later than me, were on hand for the last night at Elaine's. B made all the arrangements for us to get in that night and she, R, and A got there early. I had to get the babysitter settled so I arrived later. By the time I got there, a crowd had gathered, the likes of which I hadn't before seen in front of Elaine's. TV crews and cameras were on hand to document this momentous occasion. While there was a festive air in Elaine's that night, a tribute to when Elaine's was Elaine's, there was also a sadness. Elaine's was a refuge; a community. For the regulars, holdovers from the go-go years, and the more recent additions like myself when almost anyone could get a table along the wall, there was almost a nervous merriment. What would happen to us all when Elaine's was gone? Where would we all go to be connected, to be a part of the community?\n\nI wasn't particularly sad that last night. Elaine and Elaine's had come into my life for a season and a reason. Elaine's was there when I needed it and Elaine Kaufman was Elaine's.\n\nWithout Elaine, there just couldn't be anymore Elaine's.\n\"I'm a Fucking Icon.\"\n\nElaine Madsen\n\n\"You have to interview me. I'm a fucking icon,\" Elaine announced to Emmy Award\u2013winning director Elaine Madsen while Madsen was working out the details for her film I Know a Woman Like That (in which Elaine Kaufman's interview appears). The film has been recognized by the Rhode Island International Film Festival with their \"Helping Hand International Humanitarian Award,\" given annually to a film that inspires social change, community outreach, and a better understanding of the world in which we live. When the film screened at the Chicago International Film Festival, Roger Ebert cited it as \"transformative and extraordinary.\"\n\nELAINE HAD NO varnish. That's what I remember most about her. Within minutes of her sitting next to us, the greeting turned into a conversation about something\u2014something you cared to talk about. I am not famous, I brought nothing to the table but myself; her famousness never sat down with her.\n\nShe found out I was a writer and asked to see my book which I didn't have with me.\n\nWhen I came again, she asked after it right away. She really wanted to see it!\n\nElaine Madsen interviews Elaine Kaufman.\n\nPhoto credit: T. J. Rosza.\n\nWhen my daughter, Virginia, told Elaine I was doing a documentary, which Virginia was producing, Elaine immediately said: \"You have to interview me, I'm a fucking icon!\"\n\nAnd she grandly repeated that memorable quote for the interview. She knew we had no big sponsors, no big budget and for sure she knew she was doing far more good for our story than any good we could conceivably bring to her.\n\nIn our interview, she spoke to me about her childhood and her voracious reading habit\u2014a childhood characteristic we shared. When the interview concluded we shared a glass of wine and she suggested it would be good for our film if I captured a night at Elaine's. I wouldn't have asked. But, shortly thereafter, I got a call and an invitation to let us film her forty-fifth birthday party; for certain a highlight of the film.\n\nWe were very happily surprised that our cameraman was the only one filming the event.\n\nShortly thereafter, my daughters and I came to Elaine's to celebrate my birthday. She surprised me with the most elaborate birthday cake I'd ever seen and a beautiful rose. As Diane set the cake in front of me, she whispered in my ear, \"She doesn't do this, you know.\"\n\nDid all of us have moments with Elaine that she \"doesn't do\"? That was one of ours.\n\nCelebrating Elaine Madsen's birthday at Elaine's with her daughters. L to R: Virginia Madsen, Elaine Madsen, Cheri Borowiec.\n\nPhoto credit T. J. Rosza.\nElaine Was My Dating Coach\n\nKen Pliska\n\nKen Pliska moved to New York City from Scranton, Pennsylvania in 1987. A graduate of Pennsylvania State University with a degree in architectural engineering, his early career was in the architecture and engineering field. In 2000, he started working in real estate development and consulting as an owner's representative. Currently, he is partners in the firm Second River Station, LLC. His wife of eighteen years, Kimberly McGovern Pliska, passed away in 2008.\n\nKen made many friends at Elaine's, from celebrities to the average local. He hung at the bar but also sat at tables. Because he was a middle-of-the-road kind of guy, he was able to start the \"All the People that You Knew at Elaine's\" Facebook page and was invited to be on the board of the Table 4 Writers Foundation.\n\nElaine's was a saloon he called home. It was a place that helped him heal during the most difficult period of his life. Elaine once informed him that he was \"a slab of meat on a hook.\" And so began her work as his dating coach.\n\nMY FIRST STOP in Elaine's was in 1987. I had just moved to NYC and was checking out restaurants and bars on Second Avenue. I knew nothing about Elaine or the place, but it looked fun and I popped in. I had a few drinks but it wasn't near my apartment, so I didn't make it back for a few years.\n\nMy wife and I moved to the Yorkville neighborhood in 1992. Elaine's became our local spot for a special dinner. Since we were not regulars, we typically got seated in the rear of the restaurant. As the years passed, we went from one to two visits a year to about eight to ten. This was still not regular status, but Elaine knew our faces, as this went on for more than ten years.\n\nIn the fall of 2008, my wife had an accident and passed away. My dear friend, Joni Loeffler, was a regular who had been going to Elaine's for decades. Joni insisted that I spend more time in Elaine's. She said, \"It will help you heal.\"\n\nI found myself in Elaine's about four to five nights a week. Shortly after this change to my schedule, Elaine took notice and asked Gianni, the head waiter, \"What's the big guy's story?\" Gianni filled her in and she asked me to join her and tell my story. From that moment, Elaine and I were friends.\n\nI did heal in Elaine's. I met many new friends and Elaine was always on the lookout for the proper business and social connections. She was a master of the art of networking. After about a year, she also became my dating coach.\n\nIt was December 2009. I was sitting at the bar on a quiet evening. After Elaine came in and got settled, she joined me at the bar. In the middle of our conversation, she stopped and asked, \"Are you ready yet?\" I had no idea what she was talking about.\n\nI said, \"Ready for what?\"\n\nElaine replied, \"Are you ready to start dating again?\"\n\nI said \"Sure, I guess so.\"\n\nHer response was, \"Good, because you're a slab of meat on a hook.\" I guess this was an Elaine-styled compliment.\n\nWhile I was a late part of the Elaine's family, the last few years of the restaurant meant the world to me. I will forever miss the comfort of her saloon.\nMy Home Away From Home\n\nDana Moyles\n\nDana N. Moyles has been in the commercial real estate industry since 1993. She has worked in both NYC and Chicago. She has represented owners, landlords, and tenants. She has worked as an Asset Manager and Leasing Agent. She currently runs her own company and is focusing on Tenant Representation in NYC and Chicago. Clients include Chicago Cut Steakhouse, The Local Chicago, Bobby Van's, Cibo, NIF Services of New York and Satori Investment Partners. Ms. Moyles graduated from Lehigh University in 1994 and is highly active in the alumni community. She is a licensed real estate broker in New York and Illinois.\n\nHOW DO I explain the impact and the history Elaine's has had in my life? I am not famous or literary; I am just a woman who works hard in her life for her career, family, and friends. So this is a tribute from someone who was lucky to find Elaine's and have it become part of her world.\n\nFrom the first moment I stepped in the door in my evening gown at twenty-four, to my last drink there a couple of years ago, it had been my home away from home, my safe place to go. I grew up at Elaine's.\n\nAnd I was blessed to become someone who can be called a regular even after I moved to Chicago.\n\nThere are so many memories. Duffy, the bartender, wanting my keys! Birthdays celebrated for me and my father. Elections won and lost (thanks to Alex, the other bartender, for being my ally). Sporting events survived! Being the lone Mets fan in the Subway World Series\u2014it was Elaine who protected me from being harassed.\n\nBoys dated and dumped (thanks Duffy to for saving me from the Hand!). And one tragically lost on September 11. Through that loss and the immediate aftermath, I vividly recalling sitting outside crying and Duffy rescuing me. And always taking care to make sure I was okay.\n\nThrough my father's illnesses and my mother's death, Elaine's was the safe place to go. There were people to support me and always a hug from a wonderful woman who granted me her kindness. I am so lucky.\n\nAnd, of course, all the friendships started over the years. The bar people and table people. How that's all become part of my history. People who have become part of the fabric of my life especially Rosemary, Tommy, Duffy, Alex, Ray, Malcolm, Frank, and Johnny (sorry the lottery numbers never panned out, Johnny). And the other friends who were in from out of town and became friends like Rich Cooper.\n\nI loved knowing Elaine was so happy for me when I moved and that she embraced me when I came home every time. She encouraged me, told me I looked beautiful and happy. And she always knew that if I was in NYC, I was coming to see her. We made deals about my future. I am grateful that she knew and liked John. She met the man I love who even my mother never got to meet.\n\nAnd then there were the best New Year's Eves ever. With Rosemary, Chris, Kathleen, Fraser, Traci, Darci, Kurt, and John, to name a few. What can ever compare? Certainly not New Year's Eve in Chicago.\n\nThese are just some of my memories. I cry as I write this, knowing I wasn't there for the final night. But I post and send this in honor of the woman and the legacy of Elaine Kaufman. Bricks and mortar may change hands but the impact of you personally on my life can never be altered.\nThe Good Penny\n\nCurt Block\n\nCurt Block couldn't resist contacting me with this role-reversal story. His date, after trying to make nice to Elaine with dismissive responses, finally told Elaine to \"fuck off.\" Tables were turned, and none were in Siberia.\n\nCurt Block worked for twenty-four years in the NBC Press and Publicity Department in New York with responsibility for entertainment, news, and sports. Block also spent five years as a sportswriter at United Press International and worked on many other impressive media accounts.\n\nHe's not quite sure if he visited Elaine's after this night.\n\nMOST OF MY earliest visits to Elaine's were with Jessica Burstein, a close friend for many years. I had been introduced to Elaine several times but never had the feeling that I'd had made any lasting impression on her\u2014or, in truth, any impression.\n\nOne night, in the late 1990s, I visited the restaurant, sans Jessica, with a stunning young woman named Penny. She was aware of the restaurant, but had never been there. To my surprise, our meal was better than any other than I'd had at Elaine's. Truly unexpected. Preparing to leave, I asked my date if she would like an after-dinner drink at the bar. It was relatively early so there was plenty of room there.\n\nI suddenly noticed Elaine seated alone at a table directly opposite the bar and whispered to Penny that I thought I should say hello to the proprietor. I approached the table to ask if I could join her. She gave me a quick once over and said: \"Suit yourself\"\u2014clearly not a warm greeting, but still, not a rejection, which I'd often seen with so many others who'd approached her.\n\nHaving twice tried to compliment Elaine, Penny had received no reaction or eye contact and she looked at me as if to say \"Get me out of here.\" Within a minute, however, she'd had enough, turned to Elaine, and in a voice loud enough for anyone in the restaurant to clearly hear, said: \"Fuck this!\" and bolted from the table.\n\nHaving more invested in Penny than Elaine, I quickly paid the bar bill and looked for Penny, who was by this time already outside, on Second Avenue, flagging a cab.\n\nThis story has stayed with me, because knowing how intimidated most people were of her over the years, I've wondered how many customers' actual last words to Elaine were to \"fuck off.\"\n\nAsh Bennington and Elaine.\n\nPhoto credit Alex Rass.\n\"Elaine's Was Really Three Different Places.\"\n\nAsh Bennington\n\nElaine's was not only \"three different places,\" but it could easily be described as three different times. Ash Bennington was there for the final chapters.\n\nAsh Bennington is a former CNBC reporter with fifteen years of financial services experience. His work has appeared in the Christian Science Monitor, TheStreet.com, BusinessInsider, ZeroHedge, and numerous other publications. He is currently senior editor at Roubini's Edge, a subsidiary of the economics consultancy Roubini Global Economics.\n\nPEOPLE OFTEN TALK about Elaine's during the glory days\u2014in the sixties, seventies, and eighties\u2014but Elaine's was magnificent until the moment Elaine died. I started going to the restaurant regularly around 2006, about five years before it closed. Elaine, and Elaine's, changed my life in ways I still don't understand. When I started going to Elaine's, I was working in banking, but wanted very much to be a writer. Somehow, Elaine's made that possible. I got my first two writing jobs standing at the bar, though I still don't really know how it happened.\n\nElaine's was an insomniac's paradise. Elaine knew that writers lead solitary lives, and that they crave comradeship at night. (There's a lot of overlap between the categories of \"writer\" and \"insomniac.\") I was at the restaurant perhaps four or five nights a week, usually after midnight. There were other kinds of people at Elaine's besides writers and reporters\u2014show business people, Wall Street types, FBI agents\u2014but they all loved the nighttime.\n\nLate at night, the lighting at Elaine's was an oasis. Warm light is no small matter for refugees from sleep. I once mentioned this unusual quality to one of the regulars, and was told that Elaine had bought the fixtures from a funeral home.\n\nThe first night I attempted to go to Elaine's by myself, also late at night, I didn't make it through the front door. There was a writer standing in front of the restaurant\u2014a writer I had studied years earlier in college\u2014with a blonde woman on one arm and a brunette on the other\u2014smoking a cigarette. I was so nervous that instead of going into Elaine's I had dinner at the Midnight Express Diner next door.\n\nYes, there were movie stars too. Elaine's was closed four days a year: Christmas, New Year's Day, Fourth of July, and Oscar night, when she threw the Entertainment Weekly party. One night, someone snapped a picture of Arnold Schwarzenegger on their iPhone. When Schwarzenegger came over to kiss Elaine goodnight, I showed the photo to him. Schwarzenegger said in his dense Teutonic accent, \"You should put it up on the Twitter.\"\n\nThe celebrities at Elaine's were like fine art hanging on a wall: They were always around\u2014and every once in awhile you remembered their presence with awe\u2014but then you returned to the conversation at hand. Elaine's was about conversation\u2014with other regulars, with the staff, and, of course, with Elaine herself\u2014all blown together through her door by the night wind.\n\nAnother night, I saw Peter O'Toole come in for dinner with the legendary publicist Bobby Zarem. Walking across the terra cotta tiles, in the electric glow of the funeral lights, O'Toole cut no less an impressive figure than he had walking across the Al-Nefud desert in Bedouin robes.\n\n\"Table hopping,\" floating from one group to another, was the norm at Elaine's. Groups dissolved and merged over the course of nights that often did not end until nearly dawn. For my thirty-fifth birthday party at Elaine's, I invited some friends I used to work with at a bank. One of the guys from the IT department tapped me on the shoulder and said, \"Is that Jack Nicholson at your birthday party?\"\n\nI looked up and saw Nicholson, talking with someone I'd met at Elaine's a few months earlier. Of course, Jack Nicholson wasn't at my birthday party\u2014but, owing to the odd nature of Elaine's, he wasn't quite not at my birthday party, either.\n\n\"To tell you the truth, I'm not sure,\" I said.\n\nThat was the way Elaine's was.\n\nIf there was a secret to Elaine's, other than the magnetism of its proprietress, it was that Elaine's was really three different places.\n\nWhile the restaurant got most of the attention, the bar was a great New York saloon. The bar fed the restaurant new blood, and kept it from acquiring the claustrophobic feel of a private club. The fact that anyone could walk in off the street for a drink at the bar always kept the game interesting.\n\nThe side room of the restaurant, called Siberia by the regulars and staff, ran on reservations. It was the kind of room where a group of businessmen from, say, Chicago could come for a veal chop, a martini, and a healthy dose of New York night life while they were in town.\n\nBut when most people talk about Elaine's they are referring to The Line\u2014where the table hopping happened. The Line was the row of round tables along the wall where the regulars, and Elaine herself, held court. In the course of a good night, a regular might sit at six or eight different tables, joining and rejoining groups for hours on end. It was a phenomenon I have not seen before or since. (Side note: Whenever one of the customers from Siberia asked, \"What's the best table?\" Elaine replied, invariably, \"The one I'm sitting at.\")\n\nThe Line is where the writers sat. One night, I joined a party of four at a table with Elaine. After a few drinks, I leaned close to Elaine and said \"We are the only two people at this table who don't have Pulitzer Prizes.\" Elaine just shook her head; to her, they were simply two old friends.\n\nThe night Kurt Vonnegut died, I saw Peter Khoury, an editor at the New York Times, as I was walking to my table. Peter said to me, \"Kurt Vonnegut has just left us.\" Before his meaning became clear, I turned toward the window to look for the great novelist\u2014whom I mistakenly assumed had just left the restaurant. As the night drew on, a number of other writers arrived. They, too, it seemed, felt sorrow at the loss of a legend\u2014and were somehow drawn to gather among their compatriots at Elaine's.\n\nOf all the conversations I had with writers at Elaine's, the one that stands out most in my memory lasted barely fifteen words. I'd been introduced to Gay Talese that night, and the person introducing us had told Talese that I was working on a novel. Talese shook my hand and said \"Finish it.\" I nodded and said \"Yes.\" Gay pumped my arm slowly, stared into my eyes, and said, \"No. Finish it. Do you understand? Finish It.\" I believe I replied, \"Yes, Mr. Talese.\" His analysis was spot on\u2014but, since his advice went unheeded, I'm still a little anxious about meeting him again.\n\nBut what I miss most is having dinner alone with Elaine. The honor of eating dinner with Elaine alone meant that the house rules of conversation would be suspended. There was no need to fill the air with words. It was like sitting inside the eye of a hurricane, as everything around you spun. Eventually, you'd be swept up into the whirlwind of the night.\n\nWoody Allen and Elaine.\n\nPhoto credit Ron Galella.\nWoody Allen On Elaine's\n\nNo story about Elaine's would be complete without Woody Allen. Many, many thanks to Susan Morse, an outstanding film editor, who collaborated with Woody Allen for many years, for the connection.\n\nDo you think that Elaine's personified New York?\n\nAt the time, Elaine's did not necessarily personify New York to me, but in retrospect it seems like a good question.\n\nWhat was your most memorable time at Elaine's?\n\nMy most memorable time is almost eating there every night for ten years.\n\nAny great romantic moments at Elaine's?\n\nNothing particularly romantic occurred and no single funny incident although I had a million laughs there in dinner conversations with friends.\n\nIf Elaine was cast in a movie, who do you think should play her?\n\nKathy Bates is one person who could play Elaine.\n\nWhat drew you to Elaine's?\n\nElaine's was like a club. One great thing about Elaine's was that while there was a lot of action going on all around one had complete privacy at one's table because everyone there was so famous that there was no celebrity ogling, no autographs asked for and it was quite a private dinner. I simply ate, enjoyed looking around, and rarely socialized but was always friendly if anyone said hello. I never ever wore a hat at Elaine's. Also I shot several scenes at Elaine's and not just Manhattan. It was in Manhattan Murder History as well and Celebrity and maybe even more that I am forgetting.\nTop Ten Reasons Your New Hangout Will Never Replace Elaine's\n\nCharles Kipps\n\nCharles Kipps has won Emmy, Peabody, and Edgar Awards and is the author of several books. Kipps' TV scripts include Law & Order and Colombo. He has written and produced music for Gladys Knight, Aretha Franklin, and the Temptations. Impressed?\n\nCharles Kipps wrote a Top Ten List of \"Things I Have Heard Elaine Say\" for A.E. Hotchner's book, Everyone Comes to Elaine's.\n\nHere's his custom-made Top Ten List for us.\n\n10. They close down if there's a blizzard.\n\n9. For some reason, the tables in the bar are considered undesirable.\n\n8. If you haven't been there for a few days they don't make you feel guilty.\n\n7. You can mention other restaurants and not be thrown out.\n\n6. The owner doesn't sit with you and eat off your plate.\n\n5. If you stop in after attending an event that included dinner, they don't make you order food.\n\n4. It's okay to send back an entr\u00e9e if there's something wrong with it.\n\n3. The patrons all seem disturbingly normal.\n\n2. You can go home whenever you want.\n\n1. There will never be another Elaine.\n\nWarner Leroy and Elaine once discussed diets at an event.\n\nPhoto provided by Bridget Leroy.\nShooting Elaine's (But Not Elaine)\n\nRon Galella\n\nRon Galella is the paparazzo who pursued and then pursued some more. Galella doesn't \"do\" no, especially from Jackie O. Elaine threw garbage cans at him; Marlon Brando allegedly punched him in the jaw. Galella has been in and out of courtrooms, defending his right to click on. He is the subject of Smash his Camera and the author of Jackie: My Obsession.\n\nDURING MY \"GOLDEN years\" of shooting\u2014in the 1960s and 1970s\u2014when there were freedom and opportunity, I covered Elaine's Restaurant. Elaine's was a late-night hangout for celebrities, especially writers, and was, of course, Woody Allen's favorite. I usually made it my last stop of the night on my way home to Westchester. One of my many great takes was on September 14, 1978, when I photographed Woody Allen and Caroline Kennedy. Not together, of course, but each out with friends. That night Woody did something unusual: he decided to walk home going west on Eighty-eighth Street. I was behind him, with his back facing me, and I yelled to him, \"On the count of three, turn around.\" I counted, and to my surprise, he actually turned around! That photo was used for an artist's rendering, which was published on the cover of the April 30, 1979, issue of Time magazine. I was paid the cover fee, $750!\n\nWoody and Mia Farrow had a hot new romance at the time, so my wife Betty and I waited to catch them leaving Elaine's one night. Unfortunately, they ran out separately to Woody's Rolls Royce, but Betty and I surprised them shortly thereafter at a traffic light on Central Park West. They tried to dodge the cameras by hiding in the backseat, but we were able to catch them by each shooting through the back windows. Betty actually wound up getting the better shot, and those were the first photos of them together!\nWhen Spinelli Met Plimpton\n\nAn Elaine's Legend As told to Amy Phillips Penn\n\nGEORGE PLIMPTON, THE legendary editor who competed in sporting events, acting, and comedy in Las Vegas, all for the power of a hands-on, pen's-on story, was a regular at Elaine's.\n\nOne tale that has been told and retold and, like a good game of \"operator,\" invites literary salt and paprika, was the night when Jerry Spinelli, an aspiring writer-to-be, found his way to Elaine's via George Plimpton. Jerry's wife, Eileen, had all but decimated their savings to buy an auction promise of \"A Night with George Plimpton.\"\n\nAnd so it was off to Elaine's.\n\n\"Table after table was filled with literary bigwigs: Kurt Vonnegut, Jill Krementz, Irwin Shaw, Peter Stone, Dan Jenkins,\" Plimpton recalled in Hotchner's Everyone Comes to Elaine's. At Elaine's, though, it was understood by all that you never, ever interrupt Woody Allen. However, Plimpton dared to brave it in the name of the up-and-coming Spinelli.\n\n\"Woody,\" he said, \"forgive me. This is Jerry Spinelli, the writer from Philadelphia.\"\n\n\"Yes,\" Allen said evenly. \"I know.\"\n\nWhile Spinelli embraces the memory of his one and only night at Elaine's, he says that the Woody Allen introduction never happened . . .\n\nTo date, Jerry Spinelli has published close to thirty books, both for children and adults. He recounts the story of his night at Elaine's often at lectures, and in less formal settings.\n\n\"George Plimpton contacted me before he published a magazine article on our night at Elaine's. He suggested that he might embellish a bit. The Woody Allen story is the only embellishment I remember, as it simply did not happen.\"\n\nWhat did happen was a cab ride to remember after dinner at Elaine's.\n\n\"It was not long after our night at Elaine's with George Plimpton that I had my first novel accepted for publication by Little, Brown. I can't say there was a direct link between the two events, but this is for sure: a recounting of that night has entertained audiences I've spoken to for more than three decades.\n\n\"We left Elaine's that night in a taxi. The cab first dropped Plimpton off at his place. We said our goodbyes and headed for the train station to return home. As we were stopped at a traffic light, the driver turned to us and said, \"I couldn't help overhearing your conversation. Are you writers?\"\n\n\"Yes,\" they replied.\n\n\"Well\u2014\" he said, and he reached down to the front seat and when his hand reappeared it was holding a stack of paper that looked familiar: a manuscript. \"I happen to have a book of my own here,\" he said. \"I wonder if you would mind having a look at it.\"\n\n\"We had to tell him we were not qualified to do so\u2014not yet, anyway\u2014but his heartfelt request brought the evening around to a perfect circle. The reason Eileen had bid our life savings in the first place for a Night on the Town with George Plimpton was to give my manuscript an audience with a famous writer, trusting that publication would surely follow.\"\n\"Not Only Did You Share A Meal Or A Drink, But You Shared Yourself.\"\n\nMark Rossini\n\nMark T. Rossini became an FBI Special Agent in July 1991, and after new agent training at Quantico, Virginia, he reported to the New York field office. Mr. Rossini worked on complex white-collar crime cases for the first six years of his career, concentrating mostly in the areas of public corruption, financial fraud, and corporate malfeasance. Years later, Mr. Rossini extensively traveled on sensitive counterterrorism operations, working with foreign law enforcement and intelligence agencies.\n\nMr. Rossini has lectured at the FBI Academy, the CIA University, and many government and private institutions regarding the PATRIOT ACT, FISA, and the operational duties and responsibilities of the FBI, CIA, and the NCT.\n\nI FIRST STARTED going to Elaine's around 1995. I became an FBI agent in 1991 and worked mostly in Manhattan and The Bronx. I had always heard of Elaine's, having grown up in New York, and having an apartment just a few blocks away from its fabled location, but I was too busy with something or another and never had an opportunity to go in. I started to go there regularly with my boss John P. O'Neill after he transferred in from FBIHQ. I have to say it was an amazing place. The mixture of personalities and professions was incomparable. From the Police Commissioner to the titans of Hollywood and finance, Elaine's was the place where everyone gathered and \"shared.\" Not only did you share a meal or a drink, but you shared yourself. You were able to cross those visible and invisible lines that separate classes and professions, in an atmosphere that essentially was a private club managed by an indomitable presence and personality. Elaine was the power and force at the center, and you better heed to her instructions and advice on anything and everything. From her menu suggestions to telling someone (and I mean anyone no matter who they were) to sit with you and talk if she thought it would do either one of you good. We all dutifully listened like children. Many of us referred to her as Mamma. I would call another habitu\u00e9, Steve Levy, on the phone and say, \"See you at Mamma's tonight.\" We never said \"Elaine's.\"\n\nI think that's what hurt most about her passing. It was like losing your mother. Elaine's was like your home, and you were always welcome. Sunday's were referred to as \"family night.\" After I would see my parents up in Westchester County, I would stop there and have some quiet time with the rest of the \"family.\" I remember one Sunday night that changed my life. My mom had a stroke on September 11, 2000 (one year before the event that changed our lives). After her stroke, she deteriorated to the point where she was in a wheelchair and becoming atrophied.\n\nIn the fall of 2004, I was having a family night meal with Elaine and Dr. Richie Saitta, another loyal and faithful habitu\u00e9, and Elaine asked me how my mom was doing. I put my head down and said that she was in a very bad state and that we were looking for homes to put her in. Dr. Richie asked after condition and I told him the story. Without hesitation, he said, \"She has to see Michael.\" Michael being Dr. Michael Kaplitt, MD, another loyal patron and the number 2 of neurology at New York Hospital Weill Cornell Medical Center.\n\nDr. Richie called Michael right there, and the road to my mom's miracle recovery started. Turns out my mom was misdiagnosed with Parkinson's. What she had actually was hydrocephalus or \"water on the brain\" as a result of the emergency craniotomy that was done on the night of the stroke, which all other doctors failed to see except Michael. After he operated on her, she walked out of the hospital and continued to walk and walk and walk . . . up to nine miles a day until she died recently of cancer.\n\nMy other boss in the FBI was John Miller, who is now the Deputy Police Commissioner for Intelligence at the NYPD. John was fond of saying that there \"wasn't anything that couldn't be solved or fixed at Elaine's.\" And he was right.\nElaine's, A Tribute\n\nPhotographs by Larry Fink\n\nDabney Coleman Does The Godfather Gig At Elaine's. Just Guess Who's Listening.\n\nDabney Coleman\n\nDabney Coleman is an actor who can make you laugh even on a long-distance phone call. He came to Elaine's via his role in Tootsie; most of the crew ate, partied, and revived themselves at Elaine's, along with the film's director Sydney Pollack. Pollack donated a piano to Elaine's: Dabney Coleman did duets with and without the piano. Among Dabney's sixty-plus roles, he is remembered for his villainous take of the sexist\/tyrant boss in 9 to 5. Gotta love him.\n\nTHE FIRST TIME that I went to Elaine's was when I was in Tootsie. I'd never heard of the place. Somehow a whole group of us decided on it; Dustin (Hoffman), Jessica (Lange), I don't know whether Geena Davis was there or not. That was the first time I heard of it. Elaine was very cordial. It was a very celebrity-oriented place, a hot place forty years ago and it remained hotter.\n\nWoody Allen was there virtually every night, it was a fun place. I mean politicians, cops, bad guys, mafia guys, athletes, writers, press people, hot lawyers, and other restaurateurs.\n\nThe way that Elaine set it up was quite unique. She was a very large woman, a very powerful woman. If your table wasn't ready, she'd grab somebody and say, \"Norman Mailer, this is Dabney Coleman, you guys talk to each other and your table will be ready in a minute.\"\n\nSometimes you might talk for a minute or two, and sometimes you might stay right there, where she put you, and consider yourself to be very lucky to be talking to Norman Mailer, or the Mayor of New York, Ed Koch.\n\nYou never knew what the luck of the draw would be and that was the magic of it. And she would add to it from time to time by coming over and joining you. She was a fun person, knew what great combinations were, and was a great catalyst. Paul Hornung was there, the golden boy of football, so you might sit with him, which I did for an hour or half, or right next to Ralph Connor, homerun champion in the 1940s or 1950s, or guys like Joe Pepitone, Cal Rifkin, or some gangster.\n\nElaine had said of one such person, \"Well, he's going to jail tomorrow for fifteen years, this is his goodbye party.\"\n\nThese firemen were there, maybe eight of them at a long table, and I saluted them when I knew who they were. They were from Firehouse 10, which is a pretty notorious firehouse; they lost almost everybody on 9\/11.\n\nSydney Pollack was a great aficionado; of course Woody Allen. I went in with Louise Lasser a couple of times, and Gianni Uzielli who had his own restaurant, Uzi's. He's gone unfortunately, but we became very good friends.\n\nYou just would see this very rich combination of people at Elaine's. One night, I'm looking over at this large table, obviously all Mafia with one bodyguard and one guy with his arm folded, keeping track of who's coming in the door. I started doing my Godfather impersonation. They can't hear me, but they can see me and it was very clear who I was doing.\n\nThe bodyguard watches me for a few minutes, deadpan in the beginning, and slowly this look crept across his face and he can barely contain himself. He whispers in this guy's ear, and the guy turns around and looks at me and he hits the guy next to him and it goes all the way down the table and they're leaning over motionless and soundless.\n\nSo, now I raise my voice a little bit and know I've got them. Some of them had this noncommittal grin and some of them did not. They were all riveted. But not necessarily smiling at the end, I gave this backhanded wave, and said \"Buona Sera\" and they all applauded. Stuff like that would just happen at Elaine's.\n\nDo you know the song \"Why, oh why, oh why, oh, did I ever leave Ohio?\" I'm one of the few people who can do the harmony to it. It's a very pretty, sweet little song.\n\nOne night at Elaine's, I catch Edie Adams, who was in the show it came from. I stopped Edie and said: \"Excuse me, what if I told you that I knew the harmony to 'Why oh why oh?' How'd you like to sing it with me?\"\n\nSo we tuned it up and we started singing this song. The whole place went dead-ass silent.\n\nWe get a standing ovation.\n\nAnd lo and behold if it wasn't in Page Six in the New York Post the next day.\n\nPhoto provided by Susan Hathaway.\nElaine Kaufman: Friend and Investor\n\nEsther Margolis\n\nElaine was Esther's first investor, and unsolicited, at that. So began Esther Margolis's phenomenal company: Newmarket Publishing and Communications, and Newmarket Press. An ingenious publicist of the literary world and their books at Bantam, Margolis worked with Maya Angelou, Gore Vidal, Gail Sheehy, and other illustrious writers. Her dazzling creative know-how was key to the global success of the Guinness Book of World Records.\n\nI GO BACK with Elaine to the late sixties, a long time\u20141966 or 1967, when it was really becoming known as a hangout for authors and journalists. I was at Bantam books, as head of publicity and marketing there.\n\nOne of the men I worked with suggested that we go to Elaine's. It was uptown at Germantown, and I didn't know anyone there.\n\nI started going more frequently in the late 1960s and early 1970s, because I knew a lot of journalists and I had been friends with a few men at the New York Times and one of the few women, Lacey Fosburgh, who worked at the Times.\n\nShe happened to be quite a gorgeous girl. She was related to the Whitney family, which was only important in that it gave her access to people. (After she left the New York Times, she covered the Patty Hearst Symbionese Liberation Army and was one of the few journalists privileged to interview the reclusive J. D. Salinger.)\n\nLacey was quite terrific and amazing. That's who I knew, and Sidney Zion, David Halberstam, Gay Talese, Warren Hinckle, and that whole crowd.\n\nThat was fantastic for me, because I was doing publicity at a paperback publishing house. I'd only been in New York for six or seven years and it was exciting for me, because there was a lot going on the sixties, and Elaine's became my dining room.\n\nElaine's would go on till four in the morning. There were backgammon games and gambling. I was single, I was free, and I would go down there till late and still make a meeting at Bantam at 7:30 in the morning.\n\nDuring the making of the film Tootsie, which Sydney Pollack was making in New York, I was doing the book of The Making of Tootsie. My first two film-related books were Tootsie and Gandhi. That was in my first year.\n\nThey gave me full access to the Tootsie set. Pollack would do the dailies at night and come into Elaine's after; he was at Elaine's practically every night. He wound up giving Elaine a piano as a gift after Tootsie.\n\nElaine's was wonderful, wonderful experience for me.\n\nThere was a time when I wasn't going there as often as I had been. I was reluctant to go there as a woman on my own, as much because I was on my own as because I was a woman and it was the seventies.\n\nIf I had a business lunch with a male, I would get to whichever restaurant early and have them run the credit card to make sure that the check was taken care of. It was clear that men at that time were embarrassed if women were picking up the check. Elaine's didn't take credit cards at that time, though, which made this trick difficult.\n\nI would often see Elaine at publishing parties. When Elaine saw me at these various functions, she made a point of asking why I wasn't coming in.\n\n\"Well, you know, I'm not seeing so and so anymore.\"\n\nShe said, \"No, you come; I'm setting up a house account for you.\"\n\nFrom that point on, I was more comfortable going there and taking men for business dinners.\n\nPeople were surprised that I was so comfortable. I invited Nora Ephron for dinner there one night and she was somewhat taken aback.\n\nShe said, \"The two of us?\" and I said sure. I'm sure it was the first time Nora was there with another woman.\n\nIt got easier as I got older, and people were more accustomed to seeing women in the professional world. I went there a lot as Bantam was growing as a company. It wasn't a place that you had to put on airs and obviously it was interesting. Elaine was very good at maintaining your privacy, often suggesting that you might want to meet someone, as she also maintained contacts. She would whisper in my ear, \"So and so is over there and has a new book. Would you like to meet him?\"\n\nOne time when Woody Allen was sitting with Mia Farrow, I was sitting opposite them. Elaine came over and said, \"Do you know who is over there?\" Looking at a table of two women, one I knew, a journalist, but the other one I didn't recognize.\n\nShe said, \"That's Simone de Beauvoir.\"\n\nI said, \"Oh, really?\"\n\nWoody Allen actually got up and walked over to her table to meet her, which he apparently never ever did. And they conversed for several minutes. The woman with Simone waved me over and I went and talked to them for ten or fifteen minutes. It was an exciting moment.\n\nElaine had high regard for me, which was wonderful, and she became a really good friend. Socially, when I would date anyone seriously, I would take them to Elaine's and she would make a point of coming over. She had a way of assessing people and, within three minutes, she would whisper in my ear something either good or bad. The first time that I took my late husband Stan to Elaine's, she came and sat with us about five minutes, leaned over, and said: \"Keep this one.\"\n\nWe were married for over thirty years.\n\nWe had a regular table at Elaine's.\n\nThere was a time that Elaine tried to build up a lunch trade in the 1980s that didn't work.\n\nStan worked nearby and he would come in and have lunch with her. He was a psychologist who had a specialty in self-hypnosis.\n\nThere were certain courtesies that you would maintain at Elaine's. I would call to say I was coming and make sure it was okay. I recognized most of the waiters' voices.\n\nElaine followed my career. I was at Bantam until 1980. When the Agnelli group had basically bought Bantam, I took a few of the Italian group there and they loved it and made it their regular spot before they sold the company.\n\nWhen I decided that I would be leaving Bantam, Elaine was very encouraging that I start something on my own. I wanted to have an equity-based company in publishing and consulting.\n\nOne night, Elaine put something in my hand. It was a check for a considerable amount. She said, \"Whatever you do with it, kid, I know you're going to do it right. Here: Figure it out.\"\n\nI hadn't even gone out to raise any money yet, so she was my first investor.\n\nHer early reputation was that she didn't like women, or that she only gave her attention to the men who came through the door, but that wasn't true. She had a lot of women friends: Rona Jaffe came in and Carol Higgins Clark was a regular.\n\nElaine was really good at respecting your privacy, if that was what was really called for. I came in one time for a business dinner, and the maitre'd, I think it was Elio, said that he was so glad that I was here and asked who I was meeting. He never asked that kind of question, so I asked why and he said, \"Because I have to see who to place at that table back there. . . . I have to be careful who I put back there, that's why I'm asking you.\"\n\nI said, \"Well, who's there?\"\n\nHe whispered, \"Elizabeth Taylor.\"\n\nI said, \"You can put me back there; I'll be good, I promise.\"\n\nI made sure that my chair was facing hers; I wasn't going to give up that moment. She was lovely. She was there with two bodyguards. It was between her marriages to Burton, she was single at the time, and she was gorgeous. They had to stop me from following her into the ladies room. I'm exaggerating, but it was tempting, I have to say.\n\nElaine's was that kind of place where they were concerned that people were not bothered.\n\nElaine was knocked a lot for the food there, but you just had to know what to order. You didn't go there for the food, anyway.\n\"I Was Married To My Restaurant; Elaine Was Married To Elaine's.\"\n\nSteve (Pally) McFadden\n\nSteve McFadden partnered up in 1979 to launch New York's famous McFadden's restaurant on Forty-second Street and Second Avenue. His customers included Daily News reporters, U.N. dignitaries, and those who simply loved to party. Then it was on to Elaine's.\n\nELAINE WAS A very good pal of mine. I went in there when I was a young kid just out of school and it first opened, then I didn't go again for a while. I have a place called McFadden's on Forty-second and Second that opened in 1979.\n\nIn those days, we had all the press people and people like that McAlary, Drury and Andrew Cuomo. We would all hang out at McFadden's and then we would all go up to Elaine's and close the joint. It wasn't every night, but it was on a fairly consistent basis, and that's how I became a pal of hers and that was how she became a real good pal of mine.\n\nShe did a lot of good things for a lot of good people. She was gruff in her demeanor, but strong when it came to taking care of her people.\n\nWhen she opened up, she may have been one of the first women to own a restaurant or saloon. She found a very strange location, when she opened up in 1963. In those days, it was all men, usually three or four buddies who went to school together. She was a pioneer in that respect, ahead of her time; she was like the ultimate feminist. She was as tough as any bar owner in New York City, getting exactly what she wanted in terms of the d\u00e9cor and the clientele.\n\nShe was totally hands on.\n\nShe did it very well for a very long time, she had boatloads of friends, but they loved her tough demeanor. If she had been a sweet little lady, they wouldn't have enjoyed the place. And it was like when she said something rude to you; it was almost like a badge of honor. Yes! Elaine yelled at me. Get out of that seat and move over.\n\nOne time Geraldo (Rivera) was in there. This guy\u2014I think his name was Michael\u2014he was a reporter for Newsday, I believe, was in there. Out of the blue, this creep, this bozo, Geraldo started picking on him. This guy's a little guy and Geraldo was an ex-boxer in his day. Elaine interceded. A lot of f-bombs were thrown and out the door went Geraldo.\n\nThe next day she says to me, \"Pally, do you believe this thing?\"\n\nIt was a big wreath of yellow roses, and looked liked a mobster's funeral piece, from Geraldo, saying he was sorry, that sort of thing. I don't think he ever came in again. It didn't matter.\n\nIt would have been easy to take his side of the thing because he was a celebrity. Elaine could have stuck up for the celebrity, but she stuck up for who was in the right, she wasn't worried about who was better known: that was sort of like the integrity part of her. That's why Geraldo apologized with the flowers and I guess he felt like a jerk.\n\nElaine wasn't really petty about things, she'd say what was on her mind. It was part of a day's work. People can be an idiot one day and the next day they apologize and you forget about it. I've been in the saloon biz for more than forty years. If someone acts up one, you forget about it. That's the way life is, people have a few drinks, and even without drinks, this one isn't talking to that one.\n\nElaine had her favorites; they were people who interested her. If someone spent a lot of money, she wasn't upset about it by any means, but she was more interested in their conversation and what they brought to the table than what family they came from.\n\nThe Elaine's saga is a really noble one, believe me. She really ran things. She loved to laugh, she loved to have fun, and she loved to come into her place every night. If she had a cold, it would drive her crazy to stay in. She'd be calling the place five or six times a night\u2014who's coming? Who's leaving? Sometimes, she even came in at eleven if she was sick. She would be there every night. It was her place. She took a few days off to go to the Deauville Film Festival; and she closed only two days a year on Thanksgiving and Christmas. She never stayed home, though. She would go to P.J. Clarke's.\n\nShe could be a girlie girl when she wanted to be. She was very complex.\n\nI had a cancer operation. I was in Sloan and I get this thing of flowers and it's from Elaine's. People asked if I knew her. So I go, yes, she's a pal of mine. I went home after three or four days, and I got a delivery of flowers and with it was a container of chicken soup\u2014and it was hot, that was the thing. She'd call and say, \"Did you have the fucking soup? You feeling better now?\"\n\nWhen she got older, she really loved when young people would come in and when people would bring their sons and daughters. She wanted to stay young all her life. She thought young, and as life went on, she lost a lot of close friends. Elaine was a person who looked at tomorrow; that's the secret of aging.\n\nThen it was her time. An awful lot of laughter went out of the placed just like that.\n\nYou could say about Elaine, like the line from The Last Hurrah, \"How do you thank a woman for a million laughs?\"\n\nShe provided that. People miss her to this day. There'll never be another place like that in New York. We're lucky we caught it.\nElaine Was An Interesting, Tough Lady. (I Saw Her Punch Out A Guy.)\n\nKen Moran\n\nKen Moran started out on the lobster shift at the New York Post, where he talked a woman out of committing suicide on the phone. Moran became a sportswriter for the New York Post, and celebrated his first front page, which had little to do with fun and games, at Elaine's.\n\nTHERE WERE DIFFERENT groups at Elaine's, but we were the second generation. The first generation was 1960s and early 1970s, Gay Talese and all those people.\n\nBut then Elaine's had a down period. We\u2014myself, John Miller, Michael McAlary, and a whole bunch of newspaper media people\u2014started her up again in the mid 1980s. We kind of revived the bar energy and what was going on there.\n\nIt was just a great playground.\n\nElaine was an interesting, tough lady. (I saw her punch out a guy.)\n\nI got close to the bartenders, especially Tommy Carney, and I got to meet a lot of great people including Joey Heatherton, who jumped on me the first time we met. I looked at her and said, \"I don't know who you think I am, but I'm not him.\"\n\nShe had thought I was Derek Sanderson, who played a lot of great hockey for the Rangers. Apparently, I looked like him. And as it turned out, later on that evening she was sitting by herself, and I sat down by her and I ended up dating her for about a year.\n\nThere was a famous night when I was in there with Paul Hornung and Keith Hernandez. The three of us were the only ones in the place aside from Tommy and some models, and Paul was so drunk that he fell right off the barstool three times. This is a guy who won the Super Bowl three times. We had to pick him up and put him back in the stool. I was amazed at how many times he did this. Most of us would have been dead.\n\nAt Elaine's, I was always amazed at who I would run into in the men's room. For the most part we were all very friendly, like Arnold Schwarzenegger and what's his name\u2014Rambo (Stallone). All of a sudden you'd be in the middle of a conversation with people in the middle of doing your business. It became quite natural after a while.\n\nEverybody had a book party at Elaine's. The most poignant one for me was Mike McAlary, a columnist for the Daily News. He had a number of his book parties there. He was about as good a friend as I had. His last book party was right after he was diagnosed with cancer, and everyone showed up in the newspaper business because they realized how sick he was. I was in tears. He wrote some great stuff in a book that he gave me. I had to keep walking away, so he wouldn't see me in tears.\n\nAfter fifteen years of that place, I'm lucky I'm still here. I love the fact that Elaine would sit down and talk to you and always try to help you with your career and push you for better things.\n\nI was a sportswriter, but there was a time I was upstate and there was a murder in the Catskills, and I was up there, playing golf. So, I get a call from the desk and I covered the story. It was the first time I ever had the front page. On my way home, I stopped in at Elaine's, and she had all these people there and I got a standing ovation.\n\nI became one of her boys, and she threw birthday parties for me a couple of times. I would always have one of the prime tables up front and people would look at me like, Who the hell are you?\n\nErica Jong, Stuart Woods, and Elaine.\n\nPhoto credit: Harry Benson.\nThe Era Is Over\n\nAl Sapienza\n\nAl Sapienza is well-known for his roles in The Sopranos and House of Cards. Betwixt and between, he sang Beatles duets at Elaine's. He is the over-the-moon father of a little girl who is affectionately known as Lambchop.\n\nI WAS ON tour with Beatlemania in 1977 till around 1980. I would go to Elaine's now and then. When I got The Sopranos, I started going to Elaine's once a week.\n\nI'm a real wanderer and traveler. I work between New York, L.A., and Toronto and I always go to dinner by myself. You could go into Elaine's by yourself. She kept several tables for herself and her friends, regulars and celebrities. I'd sit at one of those tables and have these incredible conversations with people who had interesting jobs and an incredible influence on our culture.\n\nI'd have a political conversation with a guy from 60 Minutes, and three days later I'd be sitting with a Broadway actress, and then I'd be talking to Ray Kelly (the former police commissioner) about law enforcement.\n\nOnce I brought a friend, a CEO of an airline, to Elaine's. I ordered a steak and saved that part near the tip for the end. Then Elaine leaned over my plate, cut it off, and ate it with her fingers before I could.\n\nThat same night, my friend says to her, \"What you are doing for the holidays?\"\n\nAnd she said, \"What the hell you think I'm doing? I'm going to be here. I'm working.\"\n\nI met a bunch of women there. Elaine was a major matchmaker for me. But when I was ready to get married and have a child, I looked elsewhere. I didn't need to meet the mother of my child at a restaurant bar.\n\nI actually went on the Internet to find a wife. I wanted to meet someone outside my entertainment world, someone much more grounded than myself. I met the perfect person; she's a psychiatrist at Mount Sinai\u2014Dr. Michelle Widlitz. Elaine liked her the second she met her; she looked at me like \"you're a smart guy.\"\n\nAnytime that I'd go to Elaine's with Michelle, Elaine would sit at our table for at least a half hour and we'd have these great conversations about everything: politics, Bloomberg, and whatever else was going on.\n\nElaine would always offer us a round of drinks when we were ready to leave and didn't want to eat or drink anymore; never in the beginning when we were going to drink them.\n\nWhen we decided to get married, I had my bachelor's party and my rehearsal dinner at Elaine's. My wedding was at the Essex House. I invited Elaine to come out of respect, but I didn't think she'd show. I really wanted her to come, though. And she did.\n\nFor the centennial, I sang at Elaine's with Uncle Junior, my partner in The Sopranos. We sang a bunch of Beatles songs on New Year's Eve in 1999. It was the greatest place on Earth as far as I'm concerned.\n\nWhen that place closed, that stopped. I was friends with Fred Gallo, who designed practically every Broadway set, and other Elaine's regulars. I'm too busy to call those guys and they're too busy to call me. There's no place anymore where everyone goes. The era is over.\n\nEvery time I walked in, either Josh (Gasparo), Dr. Joe, or Tony Danza would be there to sit with. There's no restaurant in New York like that. It's just over. The era is over.\n\nPhoto credit: Walter Bernard.\nStill \"Anchored\" At Elaine's\n\nBob Drury\n\nIn the mega macho days of reporting, Bob Drury personified the game. Drury began as a copy boy, then a sportswriter, then crime reporter and eventually as an editor of a well-groomed men's magazine. He is the author\/co-author editor of nine books.\n\nELAINE'S WAS AN anchor. It always was; it was the place I would go.\n\nElaine's served so many purposes. When I was a police reporter for New York Newsday in the mid 1980s, if I needed to get something out of a detective, I would take them to Elaine's. This was in the middle of the crack wars.\n\nI would call Elaine's and I said \"the usual.\" I'd come in and she'd make a big deal out of whomever I was with and offer us a front table. I'd say, \"Elaine, do you mind? Can you hold that table for us? I'd like to go in the back and talk to Detective Shapiro here.\"\n\nOf course, his eyes were all sparkling because she had made such a big deal over us. This had been all planned. We'd go in the back and talk for a few hours and then she'd put us in the front. I swear, it worked like a charm with every homicide detective I ever interviewed. This was pre-planned.\n\nShe liked me.\n\nWe went back to the early eighties. I had a literary agent, Jay Acton. He had an old Yankee ballplayer, Ryne Duren, and Ryne was an obstreperous type. People knew him as \"the man with the coke bottle glasses.\" He was blind and half drunk when he played and everyone knew it. His warm-up pitches were a hundred miles an hour. He never knew where they were going.\n\nHe'd gotten sober and he'd gotten God. So Jay said, \"Listen.\" (And I was a kid, I think I was a sports columnist for the New York Post.) \"Could you go live with this guy for a month in Minnesota and polish his book? I've got most of it.\"\n\nI did. Anyway, the book comes out eight months later. Jay calls me and says he has a check for twenty-two grand, something that made my eyes bulge at the time. Jay said, \"Why don't you come up? We're having a book party at Elaine's tomorrow, before she opens.\"\n\nElaine's! At this point, I'd never been to Elaine's.\n\nSo I go there, and there's Jay standing at the door and he pulls out the check. I knew no one there. Tommy was at the bar. I asked for a Bud. I took out some money and he said, \"It's a book party, pal. Don't you know anything?\" I left a two-dollar tip on the bar.\n\nThis went on for a few hours. I didn't know anyone there, so I just kept drinking.\n\nI noticed that there was this kind of daunting woman sitting at the bar and she was kind of watching me. I said, \"Hi, I'm Drury, I'm one of Jays clients.\"\n\nShe said, \"I know who you are. I saw the size of the check you got. I've been watching you give two-dollar tips to the bartender every time he gives you free drinks. You're fucking welcome in here.\"\n\nI said, \"Who are you?\"\n\nShe said, \"I'm fucking Elaine.\"\n\nIt was the beginning of a beautiful relation.\n\nWhen she hit that time in the late 1980s when celebs weren't really coming in any more, things were emptying out, and I think the newspaper guys, especially those who covered cops, really kind of buoyed her.\n\nWhen she got hot again in the 1990s with the celebs, she never forgot me, Esposito, McAlary, and Moran. She always had a special place for us, because we maybe bridged a hollow period for her.\n\nEveryone used to compare Elaine to Toots Shor with that whole irascible thing. Was that an act? I don't think so. I think she was pretty irascible, but she was also loyal. She was like a mean old junkyard dog, but once you got her on your side, she was loyal. I can't think of another place that had such a strong female owner. She knew her stuff about journalism; she treated the early SNL guys as her own. I'd heard that men had fucked her over in her life, but once you had given her loyalty, she gave it right back. She was really mean to women, she was jealous of women, especially pretty women.\n\nLike it or leave it, she liked me; I used to be a big handsome strappy guy. I don't have a clubhouse in the city any more, and that was our clubhouse.\n\nI entered it one toe at a time.\n\nShe was always good to me.\n\"Who's Da Poet?\"\n\nTaki Theodoracopulos\n\nThe last conversation I had with Taki was about having \"fuck-you money.\" We were at a party in some New York townhouse. Perhaps it was his.\n\nTaki's column High Life has been in The Spectator since1977. In 1984, Taki took time out from all the glam and did penance in Pentonville Prison for possession of cocaine. In 2002, he founded The American Conservative Magazine with Pat Buchanan and Scott McConnell. Yes, he's controversial, from racial slurs to his own \"fuck-you money,\" but never, ever boring.\n\nTHERE WAS ALWAYS a place for me because she made a special compensation for drunk writers or journalists. She always had a good table for me.\n\nI always had editorial meetings on Sunday nights at Elaine's. I had a section called Taki's Top Drawer in the New York Press. So we used to meet, five, six seven of us to make the plans for the week's issue right there at Elaine's.\n\nShe used to join us and put in her two cents\u2014more than two cents, the whole dollar, mostly. She used to tell us who the good guys were, who the bad guys were . . . It was fun to have editorial meetings there. She used to advertise and never pay us. She ever took something off the enormous bill. It was a unique to have the meetings there because people were actually inspired.\n\nI started going there in the 1960s and early 1970s along with Norman Mailer and Nigel Dempster. When Clay Felker brought myself, Anthony Haden-Guest, and Nigel Dempster over from England, the place to meet up was Elaine's. There was always a good table, especially for me, because unlike the rest of the British hacks, I paid my own way.\n\nI think that the only time that she got angry at me was the time that I came up from the Village. There were these two guys outside just begging for money, and I brought them in for a drink (I was drunk). And she just lost her temper.\n\n\"Never do that again.\"\n\nOne funny incident that I remember: there was a very Romeo and Juliet note that I send to girls so I quickly wrote that thing on Spectator paper, the magazine that I worked for, and sent it to a nearby table of girls and guys. I was sitting with Elaine, by the way, because she used to come and plunk herself down. And after about ten minutes, this wonderful actor who I had never seen before, Joe Pesci, he came up and said, in a very loud voice, \"Who's da poet?\"\n\nSo we all looked around and said, \"What you are talking about?\" and tried to dismiss him. Then he whips the note I'd written to the girls.\n\n\"Who said this?\" he asked.\n\nI said, \"I did.\"\n\nHe said, \"Listen, kid. I wouldn't do this if I were you. Those guys are all gangsters.\"\n\nI remember Elaine saying, \"Nice going, kid.\"\n\nShe was always very funny because nothing would ever shock her.\n\nElaine with director Roman Polanski.\n\nPhoto credit: Jessica Burstein.\nElaine As Mama Earth\n\nFred Morton\n\nFrederic Morton has contributed columns to the New York Times, Esquire, and Playboy. He is the author of several books including The Rothschilds: Portrait of a Dynasty. Not too shabby for the son of a blacksmith. The Austrian-born Jewish\u2013American writer immigrated to America in 1940 and started his career as a baker before studying literature. Watch him go Freudian.\n\nWHAT I LIKE about Elaine's is that it was a home away from home; it had the same appeal that it had to most of its habitu\u00e9s, and that was all Elaine's personality. She was not a restaurateur, she was a den mother.\n\nIf she'd had children, the restaurant would not have been nearly as attractive. If you went there three, four, or five times, and you weren't utterly personally repellent, you became one of her children, which meant that she started yelling at you instead of being polite.\n\nIf you were doing something that displeased her or she thought was bad for you, she was yelling at you. The people of course were all in show business or media places where you feel on one hand you were part of the glamour group, on the other, these people were all loners, including, me. You want Mama, no matter how old you are or how successful you are, and she was Mama, and part of the Mama is the yelling, at least if you were brought up in a Jewish family as I was.\n\nA gathering for Dr. Joe Platania's birthday at Elaine's.\n\nPhoto provided by Susan Hathaway.\n\nIt's very strange. I think her size contributed to it, that enormous body that suggested Mother Earth, that she was capable of having an infinite number of children. The ducklings kept circling around the mother duck. If you didn't come back that week or at least the week after, that was a betrayal and that betrayal, once she voiced it, it upset me, but somehow after the initial upset, there's a warmth after. \"She missed me, I mean my God, she missed me.\"\n\nShe conveyed to you that you were not the restaurants' customer, but that you were a beloved child. That was more important than the food, which was not as bad as it was reported to be. I'm lucky because I have such an obtuse palette that it didn't make too much difference. The restaurant looked sort of sloppy and messy, and you felt that you didn't have to dress up. It was like home. At home you could throw away your tie and come as you are, which is basically naked and sloppy\u2014not naked in the literal sense.\n\nThe downside of all this was the ego confrontations among those who were the well-knowns and those who wanted to be known. All that was complicated by the overall warmth once you were part of her brood\u2014and that included people who were not well-known. She was just as human and vain as all of us. She liked the idea that she had all these names in her heyday, which lasted a long time, but basically it was the idea of having a family evening.\n\nShe once told me it was expensive for her. It's not nearly as expensive as Le Cirque or other expensive restaurants, the average time which people spend at a meal there is two hours. At Elaine's, the experience lasts four hours or more because people start table hopping. And even if they order more drinks, it doesn't make up for the loss of turnover. \"But it means financially a loss for me, and the worse thing is that I like it that way. Because it means they love being here,\" she said.\n\n\"In a blood sense, I don't have a family, but what are you going to do if you're in a family? I have it here. There are really terrific people to talk to every night. I prefer my Elaine's family because a family is always loaded with problems, which is really boring. There has never been a night there when I've been bored. Not with the people who come to Elaine's.\" (Everyone Comes to Elaine's, Hotchner.)\nThere's \"Nothing Like Gedempte Flaische.\"\n\nSteve Walter\n\nSteve Walter is an owner of the Cutting Room, a live music venue, restaurant, and bar since 1999. He graduated from Berklee College of Music, played in bands, and taught guitar in New Jersey. Versatile in style, he was a woman's coat manufacturer for more than a decade. And the beat rocks on.\n\nAFTER THE FIRST time I went into Elaine's, I became addicted. You met so many nice people there and Elaine would never let you be alone. It was like the old school.\n\nI've met so many incredible people there. If she could help you in any way, she'd call and say \"Tell Stevie to come down.\"\n\nIf she thought she could introduce you to someone romantically, she would. If you ate with Elaine, she'd stick her fork into what you were eating. But you had to eat, she didn't want a saloon. She'd rather you ate fifty dollars' worth of food than drink one hundred dollars' of liquor. She'd say \"nothing like gedempte flaische,\" which is Yiddish for potted meat.\n\nShe was sharp as could be and always knew what was going on in every inch in that place. She was there, always making sure that you were having a good time and never alone. If there was no one for you to eat with, she'd eat with you. She had great stories.\n\nShe was a class act all the way, if she liked you. You know, she didn't like women too much, but she did have a few women friends. She loved dirty jokes; she did that with Father Pete: she cursed like sailor and she had that smoker's voice. She was a big woman.\n\nShe was full of advice, too. I remember once I was telling her that I was comping some celebrity and she said, \"I beg you. You can't comp them. You gotta teach them.\" I've learned so much from her.\n\nOne Sunday night, it was a dead night and a group of about four people came in dressed in black tie. The woman was with CNN, I think, and her husband was a big-time lawyer.\n\n\"We just got back. We did our mother's eightieth birthday at 21.\"\n\nWell, that's all she needed to hear\u2014that they ate somewhere else. She said, \"Sit down and eat.\" They were scared at this point and they did what they were told.\n\nShe ordered a filet mignon and a couple of bottles of Veuve Clicquot. She said, \"Here, have some.\" She put the food on our plates and poured Veuve Clicquot in our glasses, as well. These people had just met us, but she knew they were rich.\n\n\"Come on, get another bottle,\" she told them. We had close to six bottles and they had quite a tab.\n\nWhen I saw her the next night, she said, \"That was fun; I almost got them to get six bottles. You gotta get it from those who have it.\"\n\nI get her point.\n\nI was there for the election, the Super Bowl, and for 9\/11.\n\nOn 9\/11, you went there for the sense of community. The TV was on, people were at tables, you felt a safety, like your home. It was just a warm feeling. I heard that Louie Garcia was in there, doing some shots. He was the chief fire marshal. She said, \"Louie have something to eat,\" and she took care of him.\n\nJoan Rivers played the Cutting Room for four years and she once said, \"Elaine gives you a free drink whenever you're ready to leave.\" This turned out to be true. Joan stopped going to Elaine's for a while because a close friend of hers who went there frequently had died. When I told the reason to Elaine, she responded, \"Tell her to grow up.\"\n\nThis was part of her business philosophy.\n\nOne of my favorite lines of hers was, \"Someone's gotta pay the real estate. You gotta get asses in the seats.\"\nPeople Soup\n\nTony Hendra\n\nTony Hendra has been described by The Independent of London as \"one of the most brilliant comic talents of the post-war period\" and by the New York Times as \"legendary.\" He began his career in Cambridge University Footlights as the comedy partner of Graham Chapman, later of Monty Python, appeared multiple times on the Ed Sullivan Show, was one of the original editors of National Lampoon, and produced and directed the Lampoon's off-Broadway hit Lemmings, which gave John Belushi, Chevy Chase and Christopher Guest their first starring roles.\n\nIn addition to being a bestselling nonfiction author, he is founder and CEO of the online satirical hub TheFinalEdition.com and co-producer of The Final Edition Radio Hour, and as of 2015, he assumed a new role as CCO (Chief Creative Officer) of the newly resurgent National Lampoon brand.\n\nI don't remember exactly when I first went to Elaine's\u2014hey, it was the sixties\u2014but I know it was a consequence of patronizing a splendid Irish bar called Himself on East Eighty-eighth Street. At Himself, I made my first close New York friend, Malachy McCourt, who held court there with a spectrum of Irish geniuses from Paul O'Dwyer to Richard Harris. Himself was a grand bar\u2014until midnight when you had to eat something to sop up all the booze. True to Irish form, all the place had to offer was an early, unsuccessful version of fusion cuisine: American Grade-A hamburger, cooked by an Irishman to the consistency of an eight-ball, served on an English muffin.\n\nTony Hendra and Elaine.\n\nPhoto credit: Lucy Hendra.\n\nLuckily 'round the corner on Second Avenue was a bar called Elaine's that actually banned hamburgers. It had the same dark, delicious fug as Himself, but also toothsome food, decent wine, tables you could linger at all night\u2014and writers. Back then, I was a modestly successful comedian (my partner and I headlined for a couple years at Julius Monk's Plaza 9), but writing was in my blood. At Elaine's you could hang around the likes of Lewis Lapham, Gay Talese, Bruce Jay Friedman, Jack Richardson, Warren Hinckle, George Plimpton, Sidney Zion, Herb Sargent . . . Writers and\/or editors who talked books, and publishing, writers and articles, magazines and newspapers\u2014anything from low gossip to high-flown debate\u2014till sun-up. Eventually you would even get invited to sit at their tables.\n\nI've lost count of the special moments that happened for one reason or another with one set of people or another in the forty or so years I frequented the dark bar with the yellow awning. But a couple stand out . . .\n\nI was returning from a printing plant in Toledo, Ohio at nine o'clock one chilly October night in 1978, carrying a stack of copies fresh off the press, of Not The New York Times. The Times was then going through a major strike and a bunch of us\u2014a very Elaine's bunch: George Plimpton, Chris Cerf, Rusty Unger (wife of Tom Guinzberg), and myself\u2014had decided to publish a parody of it, written by eminent or notorious New York literati, to slake the jones of the city's Times readers, deprived for months of their daily fix.\n\nPushing through the battered doors of Elaine's, I saw several large tables in the front of the restaurant packed with my fellow editors and contributors. This was the big moment: had their wildly funny collective vision actually come to fruition? Apparently, yes. As copies were passed round, the front tables exploded with laughter. Soon the whole place was rocking. Elaine sat in the midst of the happy mayhem, the candles on the tables glinting off her glasses, nodding and smiling. It was one of the greatest nights of my life.\n\nFlash-forward almost thirty years to my eldest son's sixteenth birthday. My wife and I had brought him to Elaine's for dinner. Nick was a tall strong kid, a basketball player whose ambition was to go to a Division I school on a full ride. (He achieved this goal.) That night, though, he was sullen because none of the sports or movie stars we said might be at the bar were not in attendance. Elaine joined us. She picked up the vibe at the table and started chatting with Nick. Sports, being a teen, his school, parents . . . Soon they were chuckling away together. I realized I'd never seen Elaine talk to anyone younger than 25 or 30 unless they were a celebrity and she had to. But she and Nick were clicking big time. Eventually the moment came for him to go to the real party downtown with his teammates and homies. He got up, told Elaine quite warmly (for him) that it was great to meet her, and turned to leave. Wait, she said, and produced a roll of c-notes the size of an office Rolodex from her skirts. She peeled one off and tucked it in his shirt pocket. \"Don't listen to them,\" she said, indicating us. \"Have a good time tonight.\"\n\nI made life-long friendships at Elaine's\u2014Lapham and Plimpton were just two\u2014though it has to be said that there was such a thing as The Elaine's Friend. Someone you only ever met at Elaine's, who greeted you cordially and shared your table\u2014someone who debated with you fiercely, made you laugh like an idiot, and whose favorite bibendum you knew by heart, but with whom you almost never socialized elsewhere. Over the forty years I went to Elaine's, there were scores upon scores of these friends, kindred spirits of unique charm or staggering talent or both, whom I thought of with affection and respect . . . but only partied or commiserated with, in an unprepossessing, book-lined Second Avenue joint with uncomfortable chairs and wobbly tables.\n\nPeople have accused Elaine's\u2014perhaps because of The Elaine's Friend Syndrome\u2014of superficiality. But I believe there was simply a curious magic to the place that drew me back year after year for decades; I knew there would always be a few of those kindred spirits, heads turning, arms waving me over to join them as I walked through the door. And that was Elaine's doing.\n\nThe night I left Himself and walked into Elaine's, I had no idea why I'd been admitted to the far-flung company of people who always got a table and a quick sit-down from the Big Lady herself. But for some reason, I did. Thousands of others did, too\u2014people who were like me: not especially brilliant or beautiful or famous or fabulous. And they probably don't know why they had been admitted either.\n\nI miss Elaine's and Elaine to this day, terribly. But what I miss most of all is her miraculous, mysterious, and never-to-be replicated recipe for people soup.\nWho's on First? (In Elaine's Bathroom)\n\nRichard Johnson\n\nRichard Johnson, a handsome New Yorker, is less than six degrees of pages or separation from the New York Post's infamous Page Six. Actually, Johnson is Page Six and has been for quite some time. He has seen or heard it all and then some\u2014and then inked it in for the rest of us to wake up to: the zing-a ling a-ling wohza-powza of many a New York\/Hollywood \"scandal.\" Who can resist ripping through a paper with headlines like \"Headless Body Found in Topless Bar\"? And we're just getting warmed up.\n\nWE ASKED RICHARD Johnson for a few of his favorite Elaine's memories. Here goes:\n\nThe best story ever was the one I wrote on Page Six about the night that Gianni Uzielli's girlfriend was caught in flagrante with Mets star Keith Hernandez.\n\nThe two had disappeared after many drinks, and Uzielli finally left the table to look for them. He found her performing fellatio on Hernandez. Then came an operatic scene as the cuckolded Uzielli screamed at the woman, who sobbed.\n\nAs a reporter, the big issue for me the next day was determining whether it had happened in the ladies room, as I had been told, or in the men's room, as Hernandez maintained. This issue being, who followed whom? Who instigated this act of betrayal?\n\nI'll also never forget the night George Plimpton and I debated the JFK assassination and the Warren Commission report. Plimpton believed in the single gunman and the magic bullet. I was and remain convinced Lee Harvey Oswald was set up by co-conspirators.\n\nThen there was the night Bo Dietl and I got into a discussion of Donald Manes's suicide in 1986. Dietl said, \"What a coward. What kind of man leaves his family like that?\"\n\nI remarked that he didn't take the easy way out with pills or carbon monoxide.\n\nHe stabbed himself in the heart with a kitchen knife. That isn't easy.\n\nGood times.\n\"I Would Not Be the Person I Am Today if I Had Not Gone Into Elaine's.\"\n\nLibby Schoettle\n\nElizabeth Schoettle (who often goes by Libby) is a New York City\u2013based mixed\/media collage artist who uses Polaroids as well as found photographs. Libby collects images of women, and girls, specifically in search of ones who bear resemblance to her body (and her mood) as a way to identify with how she is feeling. Libby had her first solo mixed-media art show at Meredith Ward Fine Art NY in 2007 and is currently working on her first memoir-based novel \"revealing\" her life as an artist. As well, she is the subject of a feature documentary film, directed by filmmaker Jyll Johnstone, about her life as this artist. Libby lives and works in New York City.\n\nWHEN I FIRST moved to New York, I lived above a restaurant, where I met Gianni Uzielli (former owner of Uzi's restaurant). We responded to each other in a very past-life kind of way; we were instant friends. I was about twenty-two, it was around 1996. He was probably in is fifties.\n\nGianni took me to Elaine's.\n\nIt was a very powerful experience, going there was an immediate connection for me in terms of the energy and ambiance; I thought, \"This is New York.\"\n\nThere are so many places in New York and Elaine's did not exist anywhere else. Sitting in a front table at Elaine's was not something very common; I just took it for granted. People would look at you, like, who is this person sitting at a door table? I liked the show-star atmosphere and there were all sorts of types in there; you felt like you wanted to be them or know them.\n\nIt was a tough ambiance to recreate. Now people are depressed. There's no club like Elaine's to go into. It was a very mature, comfortable atmosphere and you felt like you belonged there.\n\nI didn't understand Elaine when I first met her, but she had a serious personality and I liked it. I had never met a woman like that. I would see the way she was with her restaurant, the way she managed it, the way she looked around, and which people she let in and which ones she didn't. She was like a man. She didn't back down and she really owned it. I really respected it and really enjoyed her presence. There was something really strong about her.\n\nEveryone liked her or they didn't like her. There was no in-between. I went to Elaine's over a period of fourteen years. At first, she made comments, like, \"What's she doing here?\" It never bothered me because she was so blatant, so honest.\n\nI got to know her as best as I could. We got to a point where she said. \"Let's go shopping. I'll pick you up and we'll go fabric shopping.\" I loved her style for what she was. She was big and she pulled off her clothing so well, and she wanted to take me to her dressmaker. It never happened. I felt kind of important. I mean, we're talking about shopping.\n\nI loved the ladies' room, the posters. You could feel the energy of the dead people who moved on. There was something poetic and romantic about that.\n\nI went to the party Elaine's after Gianni's funeral. I found it very sad. (N.B. Elaine and Gianni had a falling out when he was opening a new restaurant. She felt that he was deliberately taking customers away from her, and their friendship ruptured.)\n\nI felt the emotion between them and they were really close. I felt that when he left, she was never the same. I really felt like she missed him. He was an incredible spark in her life. He was so funny and there was no one like him. There was no one like either of them; their banter was wonderful. They were both very stubborn people. They both thought they were always right. Theirs was a serious falling out. I really felt like she missed him, but couldn't let him back in.\n\nEverywhere I went with Gianni, everyone knew him. I thought it was absolutely miraculous . . . I would not be the person I am today if I had not gone into Elaine's with him. It made me aware of what my taste is. I was so fresh to New York. It was magical to me. It was a perfect match when I walked in there.\nAfterword\n\n\"What really knocks me out is a book that, when you're all done reading it, you wish the author that wrote it was a terrific friend of yours and you could call him up on the phone whenever you felt like it. That doesn't happen much, though.\"\n\n\u2014J.D. Salinger, The Catcher in the Rye\n\nTO MANY OF her friends and onlookers, Elaine Kaufman was the author, without the title, that they wanted to know.\n\nHer language wasn't always the King's English: she was often \"short, but not so sweet;\" funny, irascible, New York sarcastic, observant, abusive and better informed than most newscasters and journalists.\n\nShe sat down to enjoy life with her favorite authors, politicians, artists, journalists: those she considered family.\n\nAt times she shared their food, without asking, and billed them for it. No one seemed to mind. It often seemed like the thing to do to criticize Elaine's food. Perhaps that was because that there little else to criticize if you were nestled or ruffled in her ample nest, and longed to stay there.\n\nElaine's was different. Elaine felt comfortable, moving from table to table, and making room at tables for those who were on their own. Some of us have had that opportunity, others never will.\n\nSo many of her friends miss her deeply and know that Elaine and Elaine's shall never be replaced. She leaves behind writers, photographers, and other aspiring souls whose lives she has scrupulously sculpted. Their knowledge, anecdotes, skills, and introductions are welcomed by the next generation.\n\nSo long, Elaine.\n\nYour legend lives on in books, both dog-eared and yet to be written.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nLetters by Leonard Bernstein \u00a9 Amberson Holdings LLC. Used by permission of The Leonard Bernstein Office, Inc.\n\nAll other letters are the copyright of their respective owners. All rights, including the right of further reproduction or transmission, are reserved.\n\nIntroductory and editorial material copyright \u00a9 2013 Nigel Simeone\n\nAll rights reserved. This book may not be reproduced in whole or in part, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press) without written permission from the publishers.\n\nFor information about this and other Yale University Press publications, please contact:\n\nU.S. Office: sales.press@yale.edu www.yalebooks.com\n\nEurope Office: sales@yaleup.co.uk www.yalebooks.co.uk\n\nSet in Arno Pro by IDSUK (DataConnection) Ltd\n\nPrinted in Great Britain by TJ International Ltd, Padstow, Cornwall\n\nLibrary of Congress Cataloging-in-publication Data\n\nBernstein, Leonard, 1918\u20131990. \n[Correspondence. Selections] \nThe Leonard Bernstein letters \/ edited by Nigel Simeone. \npages cm \nIncludes bibliographical references and index. \nISBN 978\u20130\u2013300\u201317909\u20135 (hardback) \n1. Bernstein, Leonard, 1918\u20131990\u2014Correspondence. 2. Musicians\u2014United States\u2014Correspondence. 3. Composers\u2014United States\u2014Correspondence. 4. Conductors (Music)\u2014United States\u2014Correspondence. I. Simeone, Nigel, 1956\u2013 editor. II. Title. \nML410.B566A4 2013 \n780.92\u2014dc23 \n[B]\n\n2013033122\n\nA catalogue record for this book is available from the British Library.\n\n10 9 8 7 6 5 4 3 2 1\nFor \nMark Horowitz, \nLauren Doughty, \nand \nJasmine Simeone\nContents\n\n_List of Illustrations_\n\n_Introduction and Acknowledgments_\n\n**1** Early Years, 1932\u201341 (Letters 1\u201389)\n\n**2** First Successes: From Tanglewood to _On the Town_ , 1941\u20134 (Letters 90\u2013185)\n\n**3** Conquering Europe and Israel, 1945\u20139 (Letters 186\u2013294)\n\n**4** Marriage, Passport Problems, and Italy, 1950\u201355 (Letters 295\u2013358)\n\n**5** _West Side Story_ , 1955\u20137 (Letters 359\u2013409)\n\n**6** The New York Philharmonic Years, 1958\u201369 (Letters 410\u2013544)\n\n**7** Triumphs, Controversies, Catastrophe, 1970\u201378 (Letters 545\u2013591)\n\n**8** Final Years, 1979\u201390 (Letters 592\u2013650)\n\n_Appendix One: Arthur Laurents (with Leonard Bernstein): Outline for Romeo_ sent to _Jerome Robbins_\n\n_Appendix Two: Bernstein's Letters and Postcards to Mildred Spiegel_\n\n_Bibliography_\n\n_Index of Compositions by Leonard Bernstein_\n\n_General Index_\nIllustrations\n\nEndpapers. Letter from Bernstein to his mother, Israel, November 1948, illustrated by Yossi Stern.\n\n 1. Leonard Bernstein at the piano, 1936.\n\n 2. Leonard Bernstein conducting Stravinsky's _Soldier's Tale_ at Tanglewood, 1940. Photo: _Berkshire Eagle_.\n\n 3. The Revuers, 1940s.\n\n 4. Leonard Bernstein, 24 May 1944. Photo: Carl van Vechten.\n\n 5. Leonard Bernstein in Hollywood, 1944, with his portable typewriter.\n\n 6. Leonard Bernstein and Lukas Foss, 1944.\n\n 7. Leonard Bernstein with Jerome Robbins, Betty Comden and Adolph Green working on _On the Town_ , 1944.\n\n 8. Leonard Bernstein with Aaron Copland at Bernardsville, NJ, August 1945.\n\n 9. Leonard Bernstein studying the vocal score of Stravinsky's _Oedipus Rex_. Photo: William P. Gottlieb.\n\n10. Two photographs of Leonard Bernstein in his West 10th Street apartment, c. 1946. Photo: William P. Gottlieb.\n\n11. A series of photographs of Bernstein conducting in Carnegie Hall, c. 1946. Photo: William P. Gottlieb.\n\n12. Bette Davis, c. 1940s.\n\n13. David Diamond, c. 1945.\n\n14. James M. Cain.\n\n15. Serge Koussevitzky with Leonard Bernstein and Lukas Foss, 1949. Photo: Whitestone Photo, Heinz-H. Wasserstein.\n\n16. Leonard Bernstein with David Oppenheim.\n\n17. Leonard Bernstein at a kibbutz in Israel, festival of Shavuot, 22 May 1950. Photo: Behr, Tel-Aviv.\n\n18. Martha Gellhorn with Ernest Hemingway at the Stork Club, New York, 1941.\n\n19. Leonard Bernstein with Stephen Sondheim and Jerome Robbins working on _West Side Story_. Photo: Alfred Eisenstaedt, published in _Life_ , 7 January 1957.\n\n20. Arthur Laurents, Leonard Bernstein and Jerome Robbins being presented with the Key to Washington D.C. by Commissioner Robert E. McLaughlin, 31 August 1957. Photo: Nate Fine.\n\n21. Leonard and Burton Bernstein skiing, 1958.\n\n22. Felicia Bernstein as Joan in Honegger's _Joan of Arc at the Stake_ , 1958. Photo: Carl van Vechten.\n\n23. Lukas Foss.\n\n24. Sid Ramin, on the back cover of his LP _Love is a Swinging Word_. RCA Victor, released in 1959.\n\n25. Leonard and Felicia Bernstein with Boris Pasternak, Moscow, 1959.\n\n26. Leonard Bernstein and Aaron Copland with the score of _El Sal\u00f3n M\u00e9xico_ , c. 1960.\n\n27. Leonard Bernstein at the MacDowell Colony, working on _Kaddish_ , 1962. Photo: Bernice Perry.\n\n28. Leonard Bernstein: canon written for Humphrey Burton and Erik Smith, 25 March 1977. Humphrey Burton, private collection.\n\n29. Jacqueline Kennedy with Leonard and Felicia Bernstein and their children Alexander and Jamie at the Theatre De Lys, New York, 28 June 1965. Photo: Bettmann\/Corbis.\n\n1 Leonard Bernstein at the piano in 1936.\n\n2 Leonard Bernstein conducting Stravinsky's _Soldier's Tale_ at Tanglewood in 1940 in a version with Bernstein's own words (see Letter 57). Serge Koussevitzky and his wife Natalie are looking down from the balcony. The inscription reads \"For Helen [Coates]: Le commencement de 'L'Histoire d'un \u00e9l\u00e8ve' \u2013 with much love. Lenny, Aug. 1940\".\n\n3 The Revuers (left to right): Adolph Green, John Frank, Betty Comden, Alvin Hammer, Judy Holliday, 1940s.\n\n4 Leonard Bernstein, 24 May 1944.\n\n5 Leonard Bernstein in Hollywood with his portable typewriter in 1944.\n\n6 Leonard Bernstein and Lukas Foss in 1944.\n\n7 Leonard Bernstein with Jerome Robbins, Betty Comden and Adolph Green working on _On the Town_ in 1944.\n\n8 Leonard Bernstein with Aaron Copland at Bernardsville, NJ, in August 1945.\n\n9 Leonard Bernstein studying the vocal score of Stravinsky's _Oedipus Rex_ , a work he conducted with the New York City Symphony in November 1946.\n\n10 Two photographs of Leonard Bernstein in his West 10th Street apartment, c. 1946. The signed photograph on the grand piano is inscribed \"To my very dear Lenushka with all my faith, hope and love, Serge Koussevitzky\".\n\n11 A series of photographs of Bernstein conducting in Carnegie Hall, c. 1946.\n\n12 Bette Davis in the 1940s.\n\n13 David Diamond c. 1945.\n\n14 James M. Cain, with whom Bernstein corresponded about a proposed setting of Cain's novel _Serenade_ in 1947.\n\n15 Serge Koussevitzkywith Leonard Bernstein and Lukas Foss, celebrating Koussevitzky's 74th birthday in 1949.\n\n16 Leonard Bernstein with David Oppenheim.\n\n17 Leonard Bernstein at a kibbutz in Israel1 festival of Shavuot1 22 May 1950.\n\n18 Martha Gellhorn with Ernest Hemingway at the Stork Club, New York, in 1941.\n\n19 Leonard Bernstein with Stephen Sondheim and Jerome Robbins working on _West Side Story_.\n\n20 Arthur Laurents, Leonard Bernstein and Jerome Robbins being presented with the Key to Washington D.C. by Commissioner Robert E. McLaughlin, 31 August 1957.\n\n21 Leonard and Burton Bernstein skiing in 1958.\n\n22 Felicia Bernstein as Joan in Honegger's _Joan of Arc at the Stake_ , 1958.\n\n23 Lukas Foss\n\n24 Sid Ramin, on the back cover of his LP _Love is a Swinging Word_ (RCA Victor, released in 1959). To the right of Ramin's photo is an endorsement by Bernstein: \"I have known Sid Ramin since we were both thirteen. I was impressed with his great musicality then, and have continued to be more and more impressed ever since. His work with me (and Irv Kostal) on _West Side Story_ was invaluable \u2013 sensitive, strong, and facile. Long may he wave!\"\n\n25 Leonard and Felicia Bernstein with Boris Pasternak, Moscow, 1959.\n\n26 Leonard Bernstein and Aaron Copland with the score of _El Sal\u00f3n M\u00e9xico_ , c. 1960.\n\n27 Leonard Bernstein at the MacDowell Colony in 1962, working on _Kaddish_.\n\n28 Leonard Bernstein: canon written for Humphrey Burton and Erik Smith, 25 March 1977. The text reads: \"Humphrey Burton is forty-six, so is Erik Smith. Erik Smith is forty-six, so is Humphrey Burton.\" Burton recalls the occasion on which it was written: \"We dined with LB at the Garrick Club, which is where he produced his composition and we all lustily joined in an impromptu performance, so far as I am aware never repeated\".\n\n29 Jacqueline Kennedy with Leonard and Felicia Bernstein and their children Alexander and Jamie at the Theatre De Lys, New York, on 28 June 1965. The occasion was the opening night of _Leonard Bernstein's Theatre Songs_ , a revue featuring songs from shows for which Bernstein composed the music.\nIntroduction and Acknowledgments\n\nThe Beginning of the Project\n\nIn early 2010, just after finishing a book about _West Side Story_ , I was in the Performing Arts Reading Room at the Library of Congress, talking to Mark Horowitz about possible future projects. Mark's position as a Senior Music Specialist in the Music Division includes responsibility for the Leonard Bernstein Collection \u2013 so he knows this enormous archive better than anyone. In the course of one of our frequent chats, Mark made an apparently straightforward suggestion: \"Why don't you do a book of the correspondence?\" Those words lodged in my mind and the idea quickly began to take root.\n\nOne reason _not_ to do the letters was their sheer bulk: many tens of thousands of them, grouped in different series: Personal Correspondence, Writings (which include few but wonderful letters), Fan Mail, and Business Papers \u2013 taking up hundreds of linear feet. But the temptation of working with Bernstein's correspondence was far too exciting a challenge to let these statistics \u2013 however daunting \u2013 get in the way. Betty Comden wrote to Bernstein back in 1950 about how he saved \"every scrap of correspondence [...] from Koussevitzky's pages on life, music, and your career \u2013 to Auntie Clara's hot denunciations of meat\" (Letter 301). How right she was. I already knew some of the letters from earlier research, and a trawl through the general correspondence was enough to demonstrate what an engrossing project this could be.\n\nBut how best to approach the task? The Bernstein Collection, used in conjunction with others in the Library of Congress, offered an enticing option: to present correspondence both to and from Bernstein. This was also made possible thanks to the efforts of Charlie Harmon at the Leonard Bernstein Office: after Bernstein's death in 1990, Harmon contacted significant people in Bernstein's life requesting photocopies of the letters they had received from him, and these copies were integrated into the folders in the Bernstein Collection. The Library of Congress already had Aaron Copland's and Serge Koussevitzky's papers, and it acquired David Diamond's in the course of my research. I drew up a preliminary selection of letters in early 2010 and set to work on the process of transcription and annotation. By the end of that year, the selection needed major revision \u2013 for the best of reasons. At the end of 2010, the Bernstein estate decided that a substantial group of letters sealed after his death should be made available, and added to the Bernstein Collection in the Library of Congress. Many of these \"new\" letters turned out to be enthralling: personal, funny, and revealing. As work progressed, still more letters came to light (thanks to the generosity of the recipients, or their heirs), and I was in a position to make a final selection \u2013 acutely aware, of course, that more Bernstein letters will emerge in the future.\n\nThe selection of correspondence in _The Leonard Bernstein Letters_ is necessarily a personal choice: there were some very difficult decisions to be made in terms of what to leave out, and there is scope \u2013 and more than enough correspondence \u2013 for several further volumes. To give just a couple of examples: I have omitted most of the correspondence with his sister Shirley (including a large number of letters, mostly undated, written while she was a student at Mount Holyoke College) and from his brother Burton. A book of Bernstein family letters could make for fascinating reading. Many of them, however, concern family matters, and I had already decided that my principal focus for this book should be on correspondence that told us something about Bernstein himself, and particularly his life as a musician. It is for a similar reason that I have omitted most of the letters from Martha Gellhorn \u2013 many of them have little to say about music \u2013 though I have included a splendid letter about _West Side Story_ and a most revealing one about her marriage to Ernest Hemingway.1\n\nIlluminating a Musical Life\n\nAnyone interested in Bernstein has the great advantage of Humphrey Burton's superbly researched and beautifully written biography. Twenty years after its first publication, it remains definitive as well as enthralling, and subsequent writers on Bernstein owe Humphrey Burton a great debt of gratitude. The present book would have been unthinkable without his work, but it sets out to do something different. While Burton quotes from a good deal of correspondence, his main purpose is to tell a life story. In _The Leonard Bernstein Letters_ I have aimed to allow the letters mostly to speak for themselves, rather than to be woven into a linear biographical narrative. In addition, a number of letters have emerged or become available since Burton's book was published in 1994.\n\nOne of the delights of the Bernstein Collection is its astonishing breadth: there's extensive correspondence not only with those working in music, but also with writers, politicians, film stars, artists, journalists \u2013 and long-standing friends who offered Bernstein support at times when he needed it. I have tried to reflect something of the range of these friendships. Even so, it is as a gigantic _musical_ personality that Bernstein is remembered, and this has been my primary criterion for choosing the letters to include in what is the first published volume devoted to Bernstein's correspondence. In terms of other composers, Bernstein was in very close contact with Aaron Copland from the end of the 1930s onwards, and he also had an extensive correspondence with David Diamond stretching over five decades \u2013 a group of letters sometimes marked by tetchiness on Bernstein's part, and by a tendency to over-sensitivity on Diamond's. Such is the volume of the correspondence that I have had to omit letters from other close musical friends such as Paul Bowles and Irving Fine. These deserve to see the light of day in a future publication. In addition to correspondence with composers and performers, I have also aimed to include letters that chart the genesis of Bernstein's compositions. Two of his first big successes were collaborations with Jerome Robbins: a ballet ( _Fancy Free_ ) and a musical ( _On the Town_ ). In the case of _Fancy Free_ , much of it was conceived and composed while Robbins and Bernstein were working away from home. As a result, there was detailed discussion by letter. It's frustrating that Robbins' letters to Bernstein about this ballet seem not to have survived. (Bernstein was constantly moving house at the time \u2013 and it was just before the arrival on the scene of his assistant, Helen Coates, who ensured that everything thereafter was carefully saved.) However, Bernstein's letters to Robbins constitute a fascinating chronicle of the work's composition. _On the Town_ is a very different case: a collaboration where those concerned were working in the same place at the same time. As a result, there is no substantial correspondence about it with any of the collaborators (Bernstein, Robbins, Betty Comden, and Adolph Green) \u2013 indeed, in 1945, just after the show had opened on Broadway Comden mentioned in a letter to Bernstein that it was the first time she had written him since 1941.\n\n_West Side Story_ presents a more complex case \u2013 partly because it took so long to get started. A fascinating letter from Arthur Laurents, undated, but probably written in April 1949, raises some detailed points responding to what had evidently been a difficult phone conversation with Bernstein. A follow-up letter from Laurents (Letter 283) reveals that Bernstein considered pulling out of the project altogether. In 1955, the collaboration was revived, with greater determination on all sides to see it through. Again, there are some revealing letters, especially one from Robbins in which he responds in detail to Laurents and Bernstein about a draft scenario. Stephen Sondheim joined the creative team just after this, but there is no correspondence with him about the show until the opening night on Broadway (26 September 1957; Letter 402); then, a few weeks later, a marvelous description of the sessions for the original cast recording and the trials and tribulations of the show early in its run. Sondheim and Bernstein were not only in the same town but often in the same room while they were working on _West Side Story_ , so the lack of correspondence during its creation shouldn't come as a surprise. That it was a happy creative partnership from the start, we learn from letters sent to other people: Bernstein wrote to his brother Burton in October 1955 that \" _Romeo_ proceeds apace, with a new young lyricist named Steve Sondheim, who is going to work out wonderfully\" (Letter 363). The final stages of _West Side Story_ are described by Bernstein in an engrossing series of letters to his wife Felicia, full of interesting details as well as his excitement, frustration, exhaustion, and optimism.\n\nBernstein's descriptions of his concerts reveal some recurring tensions. He often wrote (without irony) of his \"triumphs\" on the podium, but his phenomenal public success in the United States, in Europe, and in Israel was often tempered by an underlying frustration: after describing yet another acclaimed performance, Bernstein would sometimes declare that he was going to do less and less conducting, in order to devote time to what really mattered to him \u2013 composition.\n\nIt was conducting that gave him the opportunity to travel extensively, and Bernstein wrote some memorable letters home describing the places he visited. From being a young man who told his Harvard friends that he wasn't sure whether European travel was for him, he became not only a globe-trotting maestro but also an unusually observant traveler, writing about the sights and sounds of Prague, London, and Paris in the years after the Second World War, of months spent in Italy in 1955, of South America, Japan, and \u2013 most touching of all, perhaps \u2013 the accounts of his long visit to Israel during the 1948 war.\n\n\"Every one I love, I love passionately\"2\n\nMusic was Bernstein's greatest and most constant passion. But his love life was an essential part of his make-up, and his letters allow us to form a fuller picture of an emotional life that was full of twists and turns \u2013 neatly summarized by the conducter Marin Alsop in 2010: \"Clearly, he was comfortable with being sexual in many different ways and yet he wanted a traditional life, with a wife and children to whom he was devoted. He was a complex, complex man, and complex people have complex personal lives.\"3 Intriguing as the letters are from those (usually men) with whom Bernstein had relationships during the 1940s, I have chosen instead to focus on Bernstein's own attitude to his sexuality, and its implications for his career. In correspondence with Copland and David Oppenheim in particular, and in some letters to his sister Shirley and to Diamond, he explores his sexual identity, often revealing a state of confusion and inner conflict. On the one hand, his background inculcated traditional values and relationships \u2013 ultimately marriage; on the other, his preferences in the 1940s were usually for men. Once his college studies were over, he began a process of self-exploration with the psychoanalyst he called the \"Frau\" \u2013 Marketa Morris. As we can see from their letters, he shared the same analyst with Oppenheim (with whom Bernstein had a close, surely intimate relationship in the early 1940s; their friendship was lifelong).\n\nIt's no surprise that Bernstein remained silent on the subject of his sexuality in letters to Koussevitzky \u2013 until, that is, he proudly announced his first engagement to Felicia in December 1946, suggesting a picture of his sexuality that was at best incomplete. Bernstein himself was anxious that his sex life might have a damaging impact on his employment prospects, fearing he could have difficulty finding a job as a conductor if it became known that he was gay.\n\nIt's worth pausing for a moment to consider the cultural and social context that gave Bernstein such concern about how others might view his sexuality. Many American psychoanalysts in the 1930s and 1940s considered homosexuality to be a mental illness that could respond to \"treatment\". The research by Alfred Kinsey and others published in 1948 as _Sexual Behavior in the Human Male_ (the first \"Kinsey Report\") attempted to codify degrees of homosexual, heterosexual, and asexual behavior in men with the \"Kinsey Scale\", aiming to demonstrate that men did not fit into neat and exclusive categories.4\n\nThere was a predictably violent reaction to Kinsey's findings: among others, J. Edgar Hoover, director of the FBI, was quick to denounce the findings in the pages of _Reader's Digest_ : \"Man's sense of decency declares what is normal and what is not. Whenever the American people, young or old, come to believe that there is no such thing as right or wrong, normal or abnormal, those who would destroy civilization will applaud a major victory over our way of life.\"5 In other words, homosexuality, like communism, was \"Un-American\". Two years later, in December 1950, the austerely named Senate Committee on Expenditures in the Executive Departments issued a report on the \"Employment of Homosexuals and Other Sex Perverts in Government,\" coming to the hair-raising conclusion that \"homosexuals and other sex perverts are not proper persons to be employed in Government for two reasons; first, they are generally unsuitable, and second, they constitute security risks.\"6\n\nBernstein was not, of course, seeking employment in the government, but he craved acceptance. There's little solid evidence to suggest that conductors were not appointed to particular positions because of their homosexuality in the 1940s and 1950s, though Dimitri Mitropoulos apparently believed he had been victimized. But several of the most highly regarded figures in the arts were homosexuals, not least Aaron Copland, who had, by the mid-1940s, become the most popular and distinctive voice in American classical music. Bernstein, however, aspired to be the music director of a major American orchestra and felt\u2013 rightly or wrongly \u2013 that he needed to demonstrate he was a conventional, traditional family man. Despite Bernstein's frequent protestations that he craved the more private life of a composer (where his sexuality would not have been an issue), he could never let go of conducting as an essential part of his career.\n\nWhat he didn't need to worry about as much was the possible impact his sexuality might have on his marriage \u2013 at least not as far as his chosen partner, Felicia Montealegre, was concerned. She knew what she was committing herself to: just after they married, she wrote: \"you are a homosexual and may never change [...] I am willing to accept you as you are, without being a martyr or sacrificing myself on the L.B. altar (Letter 320).\"\n\nAfter a shaky start (mainly due to Bernstein's initial tendency to regard marriage as a kind of experiment), the relationship of Leonard and Felicia blossomed \u2013 particularly after Jamie, the first of their children, was born in 1952. An exceptionally bright child, it's clear from Bernstein's letters home how much he adored her. The same love shines through in Bernstein's comments on all his children (Jamie, Alexander, and Nina); and his absolute devotion to Felicia is apparent in many letters from the early 1950s until the mid-1970s. It was a relationship that had its rocky moments, but only with the crisis of 1976 and their \"trial separation\" did it threaten to fall apart. At the end of his life, Bernstein joked to Jonathan Cott that \"you need love, and that's why I have ten thousand intimate friends which is unfair to them because I can't give any one of them everything\".7 But for a quarter of a century, Felicia was the exception: she was unquestionably the greatest love of his life.\n\nEditorial Method\n\nOriginal spellings have been preserved (except where stated otherwise), as have ampersands and punctuation in the main texts of letters, though opening salutations have been standardized to be followed by a comma. Names have sometimes been added in square brackets for the sake of clarification. Titles of works that would normally be italicized in a printed text ( _West Side Story, Fancy Free, The Age of Anxiety_ ) have been italicized. In the original letters they appear in a variety of styles \u2013 in double quotation marks, in single quotation marks, underlined, in capital letters, in plain text. For the sake of consistency, I decided to standardize their presentation. Words underlined in letters have been italicized. Dates of letters are presented in a standardized day-month-year format, the form usually preferred by Bernstein himself. Where a date (or part of a date) is uncertain, or speculative, or deduced from the content of a letter, it is given in square brackets. Addresses have been standardized, and for the sake of avoiding ambiguity, those sent from outside the United States include the country. Those sent from within the United States include the standard two-letter state codes (NY, MA, CA, and so on). In rare cases where a word is unreadable, this has been noted in square brackets. Most letters are presented complete, but where cuts have been made, or where only an extract has been included, these are shown by an ellipsis in square brackets, thus: [...]. In many cases the letters speak for themselves, but occasionally clarification or further explanation is necessary, and those letters can have quite extensive notes. I have also included short notes about all the correspondents (at the end of the first letter to or from the person concerned). In the case of a particularly long or complex document such as Bernstein's 1953 affidavit, I thought it useful to include an explanatory note exploring the context in greater detail.\n\nAcknowledgments\n\nMy largest debt of gratitude is to Mark Horowitz of the Library of Congress. I am immensely grateful to him for planting the idea for the project in the first place, and for all his subsequent help and advice, his constant support and encouragement, and his friendship. During many visits to the Performing Arts Reading Room of the Library, every single member of the staff I've encountered has been helpful and as done a great deal to make my research easier.\n\nMarie Carter, Vice-President of Licensing and Publishing at the Leonard Bernstein Office, Inc., has been encouraging from the start, and extremely helpful throughout. I am deeply grateful for her patience in answering my numerous queries and the wisdom of her replies, and for allowing me access to newly released correspondence at the earliest possible opportunity.\n\nMervyn Cooke offered a number of invaluable suggestions after reading an early draft of the text. His wisdom and experience have done much to improve the book.\n\nSophie Redfern shared the fruits of her own research on Bernstein's early ballets with overwhelming generosity, and also read the text from start to finish with a most careful and discriminating eye. I am enormously grateful to her.\n\nFor various acts of kindness \u2013 large and small \u2013 there are many people I need to thank, including Mark Audus, Peter and Mary Bacon, Stephen Banfield, Adam Binks, Humphrey Burton, Marius Carney, William Crawford, Lauren Doughty, Barry Irving, Libby Jones, Barbara Kelly, John McClure, Dominic McHugh, Richard Marshall, Gary O'Shea, Tom Owen, Robert Pascall, Caroline Rae, Catherine C. Rivers, Reggie and Josephine Simeone, M\u00e1ire Taylor, John Tyrrell, and, most importantly, my extraordinary wife Jasmine.\n\nThe editor and publisher gratefully acknowledge the permission granted to reproduce copyright material in this book. Every effort has been made to trace copyright holders and to obtain their permission for the use of copyright material. The publisher apologizes for any errors or omissions, and would be grateful for any corrections, which will be incorporated in future reprints or editions of this book. For kind permission to quote letters, I thank the following individuals and institutions: The Leonard Bernstein Office, Inc., Ellen Adler, Marin Alsop, the Richard Avedon Foundation, the Britten-Pears Foundation, Humphrey Burton, Victor Cahn, the Aaron Copland Fund for Music, Inc., Christopher Davis (Marc Blitzstein), Sam Elliott (David Diamond) Martin Fischer-Dieskau Cornelia Foss (Lukas Foss), Very Rev. Nicholas Frayling (Walter Hussey), David Grossberg (Alan Jay Lerner), the Barbara Hogensen Agency (Thornton Wilder), Janis Ian, Pat Jaffe (David Oppenheim), Jay Julien (Farley Granger), Caroline Kennedy, Marko Kleiber, Alexandra Laederich (Centre international Nadia et Lili Boulanger), Maureen Lipman, Sandy Matthews (Martha Gellhorn), Michael Merrill (Bette Davis), Laurie Miller, Phyllis Newman (Adolph Green), Tom Oppenheim, Christopher Pennington (Robbins Rights Trust), Shirley Gabis Perle, Eddie Pietzak (Elia Kazan), Menahem Pressler, Andr\u00e9 Previn, Harold Prince (Saul Chaplin), Sid Ramin, Mary Rodgers, Isabella de Sabata, Gunther Schuller, Anthony and Andrea Schuman, Lady Valerie Solti, Stephen Sondheim, Stockhausen Stiftung f\u00fcr Musik, John Stravinsky, Margaret Styne, the Music Division, Library of Congress, Washington, D.C.; the Beinecke Rare Book and Manuscript Library, Yale University; the New York Public Library for the Performing Arts; and Stanford University Library.\n\nAt Yale University Press, my proposal for this book was taken up with the sort of enthusiasm that would warm any writer's heart. When I first presented the project to Robert Baldock and Malcolm Gerratt, their eagerness did much to spur me on, and Malcolm has calmly nurtured the book throughout. Tami Halliday's eagle-eyed reading during the book's final stages was of the greatest assistance. Candida Brazil has overseen the editing of my unwieldy manuscript with kindness and skill. Steve Kent devised the attractive layout and design of the book. Thanks are also due to Lauren Doughty for compiling the index. The text has been improved beyond recognition by the copy-editing of Richard Mason and the proof-reading of Vanessa Mitchell. All its faults, however, are mine.\n\nNigel Simeone\n\nRushden, Northamptonshire\n\nJune 2013\n\n1 Other letters from Gellhorn to Bernstein are to be found in Moorehead 2006, pp. 265, 277\u20139, 280\u20132, 290, 292\u20133, 317\u201318, 323\u20134, 351\u20132, 413\u201314, 438, and 482\u20133. The letter about Hemingway is not included in Moorehead 2006.\n\n2 Leonard Bernstein to Mark Adams Taylor, quoted in Burton 1994, p. 507.\n\n3 Dougary 2010.\n\n4 Kinsey, Alfred C., Wardell B. Pomeroy, and Clyde E. Martin (1948): _Sexual Behavior in the Human Male_. Philadelphia: W. B. Saunders.\n\n5 J. Edgar Hoover, contribution to \"Must We Change our Sex Standards?\", _Reader's Digest_ , June 1948, p. 6.\n\n6 This report is reprinted in Foster, Thomas A., ed. (2013): _Documenting Intimate Matters: Primary Sources for a History of Sexuality in America_. Chicago: University of Chicago Press, pp. 144\u20137. According to an editorial note (p. 144): \"More homosexuals than communists were fired from federal jobs in this period [the 1950s].\"\n\n7 Cott 2013, p. 77.\n1\n\nEarly Years\n\n1932\u201341\n\nLeonard Bernstein was born on 25 August 1918, the first child of Jennie and Samuel Bernstein, in Lawrence, Massachusetts, 25 miles north of Boston. He attended the William Lloyd Garrison Elementary School in Roxbury, 35 miles from Lawrence, then, from 1929 to 1935, the prestigious Boston Latin School \u2013 founded in 1635. The oldest public school in the United States, its distinguished alumni included five Founding Fathers of the United States (among them Benjamin Franklin), the author Ralph Waldo Emerson, and the Puritan preacher Cotton Mather. The most famous musician to attend Boston Latin School before Bernstein was Arthur Fiedler (1894\u20131979), conductor of the Boston Pops for half a century. It was here that Bernstein's interest in languages and literature began to flourish, but what already obsessed him as a teenager was music. His first piano lessons (in 1928) were from Frieda Karp, the daughter of a neighbor, who charged $1 an hour for a lesson. Bernstein remembered her as \"unbelievably beautiful and exotic looking,\"1 and his musical progress under her tutelage was swift. By 1930, he was taking lessons from Susan Williams at the New England Conservatory of Music, and in 1932 he auditioned with a former pupil of Theodor Leschetizky, Heinrich Gebhard, a distinguished soloist and the most sought-after piano teacher in Boston at the time. Gebhard believed that there was still fundamental technical work to be done, so he suggested Bernstein first take lessons with his assistant, Helen Coates. Bernstein's first communication with Miss Coates \u2013 who became his devoted secretary in 1944 until her death in 1989 \u2013 is also the earliest letter in this book. She taught him until 1935, when she sent him on to Gebhard, but by then they had become firm and devoted friends. Other friends and contemporaries with whom Bernstein corresponded regularly during his years at Boston Latin School, and later Harvard, included Sid Ramin, Beatrice Gordon, and Mildred Spiegel. Bernstein's letters to Sid Ramin are overflowing with shared enthusiasm for new musical discoveries \u2013 and talk of girlfriends \u2013 while to Beatrice Gordon he is passionate, self-revealing, and poetic. With very few exceptions, Bernstein's correspondence with Mildred Spiegel (later Mildred Zucker) has not been made public, but as this book goes to press the Library of Congress anticipates adding these letters to its collection shortly. They document an important and lasting friendship. Descriptions of this correspondence can be found in Appendix Two.\n\nBernstein mentions difficulties with his father in a number of his letters from the 1930s. A one-page essay written by Bernstein on 11 February 1935 entitled \"Father's Books\" begins: \"My father is a very complicated human being. A man of irregular temperament and unusual convictions, he is a rare combination of the shrewd businessman and ardent religionist.\" He was also an implacable opponent of Bernstein's pursuit of a career in music, and relations between father and son were often strained. His mother, by contrast, provided a warm, supportive household in which her son's ambitions flourished.\n\nIt was while studying music at Harvard University (1935\u20139) that Bernstein made some of his most important friendships: three of them in 1937. In January that year, he met the conductor Dimitri Mitropoulos, an encounter that left a deep impression on him. Then, as a music counselor at Camp Onota near Pittsfield, Massachusetts in the summer, Bernstein instantly formed a close bond with Adolph Green, who was to give him some of his first paid work (as pianist for The Revuers, nightclub performers of songs and comedy material, including Betty Comden, Green, and Judy Holliday) and who collaborated with him on two Broadway shows ( _On The Town_ and _Wonderful Town_ ). Finally, on 14 November, during a chance encounter at a dance recital in New York, Bernstein met Aaron Copland \u2013 father figure, confidant, and the closest Bernstein came to having a composition teacher.\n\nThough it was as a pianist that Bernstein first attracted the attention of the local press, he confided to some of his closest friends that his real interest was conducting. In 1936 he wrote to Beatrice Gordon about auditioning to be assistant conductor of Harvard's Pierian Sodality (founded in 1808, and now known as the Harvard\u2013Radcliffe Orchestra); at Camp Onota in 1937 he was photographed for the local paper conducting a group of children. In 1939, during his Senior Year at Harvard, Bernstein appeared for the first time as a composer\u2013-conductor (directing his incidental music for a production of Aristophanes' _The Birds_ ), and he directed Marc Blitzstein's musical _The Cradle Will Rock_ from the piano.\n\nAfter graduating from Harvard, Bernstein was uncertain about his future. He spent the summer of 1939 looking for a job in New York (sharing an apartment with Adolph Green), and explored the possibility of studying conducting at the Juilliard School (but he had missed the deadline). His only realistic option was to audition for the Curtis Institute in Philadelphia \u2013 specifically for the conducting class taught by Fritz Reiner \u2013 and he was admitted. From 1939 to 1941, he studied with teachers who were all at the top of their respective fields: conducting with Reiner, the piano with Isabelle Venegerova, orchestration with Randall Thompson, counterpoint with Richard St\u00f6hr, and score-reading with Ren\u00e9e Longy Miquelle.\n\nFinding Philadelphia a grim and dirty place, Bernstein would escape to New York for weekends at the slightest opportunity. His years at Curtis were marked by some important firsts, including his earliest professional recordings. These demonstrated his versatility, playing improvised incidental music and song accompaniments for _The Girl with the Two Left Feet_ by The Revuers, and recording a Prelude and Fugue by David Diamond (less than five minutes of music about which Bernstein received long, anguished letters from the composer while preparing for the recording). In the summer of 1940 \u2013 midway through his studies at Curtis \u2013 Bernstein attended the inaugural summer course of the Berkshire Music Center at Tanglewood, to study conducting with the legendary Serge Koussevitzky. Mentor and pupil quickly became friends, and that summer Bernstein conducted the Second Symphony by Randall Thompson. Before the end of his studies in Philadelphia, Bernstein's first musical publication had also appeared in print: his solo piano transcription of Copland's _El Sal\u00f3n M\u00e9xico_. He received his conducting diploma from Curtis in May 1941 \u2013 not a moment too soon, as he had been desperate to get away from the stifling atmosphere of Philadelphia.\n\nAt the first opportunity, Bernstein headed to Boston, where his years of study came full circle: he returned to Harvard to conduct the new incidental music that he had composed for a production of _The Peace_ by Aristophanes. With war raging in Europe, it was a poignant choice. Back in January 1941, one of Bernstein's closest friends at Harvard \u2013 his room-mate, Al Eisner \u2013 had died in his early twenties. Eisner's letters from Hollywood are among the funniest and the most brilliant of all Bernstein's correspondents during his time at Curtis, while there was also a lively correspondence with Kenneth Ehrman, another Harvard friend, with whom Bernstein shared hopes, fears, and doubts about what his future in music might be.\n\n1. Leonard Bernstein to Helen Coates2\n\n8 Pleasanton Street, Roxbury, MA\n\n15 October 1932\n\nDear Miss Coates,\n\nI recently had an interview with Mr. Gebhard at his home. He was very encouraging in his remarks, and referred me to you as a teacher, with an occasional lesson from himself.\n\nHaving talked the matter over at home, I have decided to study with you, taking one lesson every two weeks. Would you please let me know by mail or phone when it would be convenient for you to give me my first lesson?3\n\nHoping to have the pleasure of studying with you soon,\n\nSincerely yours,\n\nLeonard Bernstein\n\n2. Leonard Bernstein to Sid Ramin4\n\n17 Lake Avenue, Sharon, MA\n\n26 June 19335\n\nDear Sid,\n\nI couldn't possibly write to you on newspaper (which was all the stationery we had in the house). I didn't until, a couple of days ago, I bought a box of stationery. So here I am and I have so darned much to tell you I don't know where to begin. Let's see...\n\nFirst, I don't know if 40 is the right number Walnut Ave, but I'll take a chance. But I've got much more important news. _Turn over and see!_\n\nI bought _Bolero_!!!\n\nWell, well! You see, I didn't know it was arranged for 1 piano, but I happened to see it in Homeyer's window. Of course dad gave me the necessary $0.80 as he is so enthused about the piece. So for the past week it's been nothing but _Bolero_. My mother says I'm boleroing her head off. But am I in heaven! It's all written in French, and it's all repeats. In the original orchestral score, they repeat four times, but I repeat only once \u2013 which is enough because it gets boresome on the same instrument all the time, and repeating once takes 10 minutes anyway. And I can't get over it. Of course it doesn't come up to the way the orchestra plays it, but it's marvelous anyway. And the ending! Speaking of cacophony!! Boom! Crash! Discord! Sock! Brrrr-rr!! (down the scale).\n\nWell now that I've got that off my chest I feel better. Oh you have got to hear it soon. But my piano is so lousy that one note doesn't play \u2013 but it serves the purpose.\n\nI'll write you again soon and tell you a convenient time to come to Sharon, etcetera, and so forth, Amen.\n\nBut first write me \u2013 immediately \u2013 please don't forget. I'm dying to see and hear from you. Answer soon \u2013 meanwhile\n\nLenny\n\nIs waiting.\n\nP.S. I'm starting to teach my mother jazz! Heh! Heh!\n\nP.P.S. I arrived home at 3.00 this A.M. Some time.\n\nP.P.P.S. Write soon. Sincerely, L.B.\n\n3. Leonard Bernstein to Sid Ramin\n\n17 Lake Avenue, Sharon, MA\n\n14 July 1933\n\nDear Sid,\n\nI'm going to fool you twice. First \u2013 I'm not following your pattern on the envelope \u2013 you know, the \"US\" stuff. Know why? You couldn't guess in a million years. The post office complained about your exalted style \u2013 and \"hope it shall be discontinued in the future.\" Imagine! So... But don't let it worry you.\n\nThe second way I'm fooling you is that I _did_ hear Fray and Braggiotti6 Tues. night. Were they swell! I was praying you were listening too. Will we have plenty to try over when you come. I hope it's lousy.\n\nListen, you probably know that the Chicago Civic Opera is putting on _Aida_ \u2013 open-air \u2013 at Braves Field the 20th of this month. It looks like my father might take me. Wanna come? I'll be in town Mon. to get tickets. So expect a call from me Monday morning and tell me whether or not you're coming so I'll buy you a ticket. It'll be swell \u2013 a real big production \u2013 so try and come \u2013 I'm dying to see you anyway. So be ready Monday to say \"Yes\".\n\nGosh, I'm not in a letter-writing mood today, as you can probably see \u2013 this letter is a flop. But I'm tired from over-sleep. About 12 hours a night. I'll have to stay up all night tonight to make it up.\n\nListen, you come to see _Aida_ with me, and we'll discuss all about your coming out here \u2013 in a week or two, I think.\n\nWell a kid just called for me to go swimming \u2013 so I'll close here.\n\nSay \u2013 write longer letters; that last one was no answer for my 7-page letter.\n\nWrite soon.\n\nExpect call Mon.\n\nRegards to all.\n\nSincerely,\n\nLenny\n\nP.S. Try to come next Thurs.\n\nP.P.S. Fray & Braggiotti also played _Espa\u00f1a_.\n\n4. Leonard Bernstein to Sid Ramin\n\n17 Lake Avenue, Sharon, MA\n\n18 July 1933\n\nDear Sid,\n\nYou didn't receive a call from me Mon. morning as we are not going to see _Aida_ \u2013 that is, my father isn't, so that's where the \"we\" comes in. That's my whole card. Much as I hate to waste the rest of the card's worth, I have nothing more to write \u2013 so I must.\n\nSo long,\n\nLenny\n\nLetter following.\n\n5. Leonard Bernstein to Sid Ramin\n\n[Sharon, MA]\n\n25 July, 11:05 a.m. [1933]\n\nDear Sid,\n\nI have a letter of letters in store for you (if I can get this pen to write).\n\nThere \u2013 that's better. I have so much news to write that it would take a telephone book to write it all. So I thought that it's as good an excuse as any to invite you, and you can come any time you want between now and Christmas. Only drop me a card letting me know when so that I can expect you. But make it darn soon. Tomorrow isn't soon enough.\n\nWell, little Lenny has turned chauffeur! In the past week I have driven (in the old Chrysler) some 90 miles. Yesterday I did 60 [miles] an hour to and from Newton on the new road. What a life! My mother calls me \"a good driver but a little reckless.\" But who could resist 60 on that road? We went to Newton to pick out colors for the new home. You should see that place! It's bigger, I think, than the 2-family house I lived in last year. A regular Colonial. It is beautiful.\n\nYou know, I'm making $1.00 every day I go in town and work for my father. And do I work! Last week I worked 3 days \u2013 $3.00. It's not so bad. So between that and working on these grounds I'm kept pretty busy.\n\nListen! Guess who's coming out here to visit someone across the street. Phil Saltman, who plays over the radio! You know him. I met his sister at a dance last Sat. night and she told me all about him. Am I excited! By the way, did you hear _Bolero_ played by the Goldman Band last Sun. night? \"Swunderful\".\n\nNow this letter is also going to be very private correspondence. So guard it in your \"iron frame\". First, you're not the only one who's met a nice girl. There are a couple of girls who keep pestering us, but we don't pay any attention to them. But last night a crowd of us went for a moonlight swim (it was wonderful! \u2013 till it began to thunder and lightning) \u2013 and I met her \u2013 and \u2013 well, we're kinda interested in each other. I['ll] let you know of further developments.\n\nSecondly, I'm on a \"no cigarette\" campaign. I'm trying my darndest not to smoke. But you know the old psychology, \"If you want to break one habit you must substitute something else for it.\" So I'm trying the old pipe. And it seems to be working OK. You know, a pipe is a much healthier smoke than a cigarette \u2013 so I hope it works. Did you see Eddie Schnaub? How does he look. Does he speak like a New Yorkite?\n\nListen, don't answer this letter. Just drop a card, as follows:\n\nLenny:\n\nWill be out on __________________. Sid.\n\nThat's all \u2013 and come as soon as possible. If you have no way to come, write me first the same and I pick you up in Roxbury coming home from town. Forget not.\n\nSo that's that. Make sure you come. That's the main point to this letter.\n\nExpecting you soon,\n\nLenny\n\nP.S. What to bring? About a week's supply of stockings, handkerchiefs, a couple of shirts, a sweater, bathing suit, tooth-brush, comb, a couple of pair of pants \u2013 one old and one new \u2013 _and_ expect to be talked to death and driven by me up a lamp post.\n\nHeh Heh!\n\nL.B.\n\nSee you sooner than soon.\n\n6. Sid Ramin to Leonard Bernstein\n\nPostmark Revere, MA\n\n28 August 1933\n\nDear Len,\n\nI just heard the _Creole Rhapsody_ written by Duke Ellington. It was also played by him and his orchestra. It's written on the same scale as _my_ Rhapsody in Blue and you ought to hear the big discords. Wow! It's written in two parts and it has a very pretty melody running throughout. Listen to it. Yes, it's nice. I've _only_ heard it about six times.\n\nSyd\n\nP.S. Say, answer my letter!\n\n7. Leonard Bernstein to Sid Ramin\n\n17 Lake Avenue, Sharon, MA\n\n2 September 1933\n\nDear Sid,\n\nI plead for every pardon in not writing to you before \u2013 _but_ I can fall back on the old, substantial excuse \u2013 no stationery \u2013 and I couldn't get any until I went into town yesterday and bought some of Kresge's famous 10\u00a2 'Evon' stationery. (Can't you recognize it?)\n\nBy the way, I heard the Fred Waring version of _Bolero_ \u2013 and it was sort of heavenly. But too much was cut out.\n\nAnd to think you used a whole postcard just to inform me of the existence of the _Creole Rhapsody_! Thanks. I haven't heard it yet.\n\nListen, I'm thinking seriously about meeting you in town. Is this OK?\n\nTime: Wednesday, Sept 6 between 9 and 10 at\n\nPlace: my father's office, 48 Wash[ington] St, Boston.\n\nIf I'm not there, wait! If you're not, I will. Bring some dough \u2013 we'll see a show, have lunch, etc. etc. Please try to make it. I'm counting on it. If you can't, well, just do anyway. It'll be one of the last times I'll see you before I move to Newton. I was in town yesterday and we moved. Is it beautiful in Newton! Our house couldn't be gorgeouser than it is. And guess what!! I'm getting an organ for Newton!!!! Don't ask, now you'll have to come out and visit us.\n\nI haven't written half the things I had in mind, so I'll tell you Wed. morning. Please try to come. I'll be expecting you.\n\nSincerely,\n\nLenny.\n\nCome Wed.\n\nP.S. Eddie R[yack]7 just went home. I think he had a nice time.\n\nPlease come Wed.\n\n8. Leonard Bernstein to Sid Ramin\n\n17 Lake Avenue, Sharon, MA\n\n7 September 1933\n\nDear Sid,\n\nIt's just as well you didn't come Wed., because at the last minute my father told me I couldn't go. So that's that. I prayed that my letter wouldn't reach you in time.\n\nWrite me by return mail how long you'll be in Revere, and also your new address. We'll be in Sharon until the middle of Oct.\n\nLenny\n\nP.S. We were in Winthrop the other day but I didn't have time to look you up. (Write soon.) L.\n\n9. Leonard Bernstein to Helen Coates\n\n17 Lake Avenue, Sharon, MA\n\n13 July 1935\n\nDear Miss Coates,\n\nI was awaiting the opportunity of receiving my Board returns before writing you, and that event has just transpired \u2013 with explosive definiteness. Following is the glad news:\n\nEnglish: the highest mark in the school, 90%. That makes me one of the very fractionary percentage of candidates who were allowed to receive 90% or over. (I believe the percent is .1%.)\n\nFrench: 90%\n\nPhysics: one of the highest marks in the school, 70%\n\nHistory, my nemesis: 60%. Which is excellent considering my heavy doubts and serious lack of knowledge in the field.\n\nAll of which makes me an 82% man; and with the fine recommendations I have received, I should be accepted into Harvard. I shan't know until about the end of this month.\n\nBefore I forget myself and write an \"I\" letter, I want to wish you a very pleasant summer. I surmise that you are now basking in the sun-pure, orange-sweet air of California.\n\nMy summer has so far been so full I haven't had time to waste. I'm in perfect health, have gained weight, and grown bodily and mentally. But there is more than that. I intend to offer the public another Bernstein operatic production such as _Carmen_ last year. We intend to use _Rigoletto_ or possibly _Faust_.\n\nI know how interested you are in my friends and associations, and so I feel I must tell you what a wonderful friend I have just made. Last week a girl I know here in Sharon introduced me to another boy she knows. His name is Laurie Bearson,8 and he is the epitome of intelligence and artistic sympathy. We became very close friends in the past week. It is as though we were soul-mates; there is a perfect understanding between us. He is intensely interested in dramatic work, and has been doing Sunday night broadcasts for some time. He is four years older than I, but that seems such an insignificant factor when we talk together. Of course there is always an interference; and in this case it is that he is going to New York to work. He left this morning and it feels as though a mountain has collapsed. But we shall correspond regularly.\n\nMy Sunday evening broadcasts are finished, and with apparent success as far as \"Avol\"9 is concerned. I am to play next week for Mischa Tulin's10 program. I have begun to do some earnest practicing, and with the help of our mutual friend Mildred [Spiegel], hope to keep it up.\n\nI should love to hear from you in the near future. Write me and tell me how you are enjoying your vacation.\n\nVery sincerely,\n\nLeonard Bernstein\n\nBy the way, one of the themes I wrote on the English board was based on genius. Being allowed to draw from unliterary material, I used my musical knowledge, and that probably secured for me my 90 \u2013\n\nL.B.\n\n10. Leonard Bernstein to Beatrice Gordon11\n\n9 September 1935\n\nEnvelope addressed: \"For one in whom I cannot distinguish the Pitti-Sing from the Beatrice.\"\n\nTo Beatrice, on the occasion of the 17th anniversary of her birth.\n\nI.\n\nI sometimes think of you as a Beethoven who frowns;\n\nAnd wastes his passion, eloquently labored\n\nOn clowns.\n\nII.\n\nI sometimes think that you are Palestrina, who measures;\n\nAnd sets an irrevocable, Bach-like standard\n\nFor pleasures.\n\nIII.\n\nI one time thought that you were a Godiva \u2013 shameless;\n\nWho flaunts her unconventionalities,\n\nBlameless.\n\nIV.\n\nAnd ofttimes you are Miniver,12 who mourns each passing day,\n\nBecause it carries him from Renaissance\n\nFurther away.\n\nV.\n\nI sometimes think of you as Amy Lowell;13 \u2013 \"Old Lace\" \u2013\n\nToo delicate to touch, and yet to[o] stern\n\nTo face.\n\nThere are, you see, two youths to every life;\n\nThe first, the ten and seven years just past,\n\nIn which the phantasies of you engage in strife;\n\nThe next, which till your dying day will last,\n\nWill harbor all these phantasies again,\n\nBut bring them into concord, free from pain,\n\nTo make the complex _you_ , sans blush, sans feign...\n\nBest of luck!\n\nLeonard Bernstein\n\nSept 9, 1935\n\n11. Leonard Bernstein to Beatrice Gordon\n\nEliot G-41, Harvard University, Cambridge, MA\n\n1 October [1936]\n\nDear Verne,14\n\nNever before in the history of mankind have such great and impossible steps been taken by an individual to improve his native resources and induce foreign ones. I have resolved myself to a year of work & study \u2013 imagine, a complete cutting down of \u2013 I shouldn't say complete \u2013 I haven't the courage to resolve completely \u2013 but cutting out, shall one say, of most social life, of a great deal of outside playing, of innumerable other time-wasters. Diametrically opposed as that realization is to my character, I have little doubt of the results, but there is no harm in hoping. And I am going to practice! For instance, three hours of it tomorrow in the very romantic tower room. Under the expert guidance of my roommate who does everything by systems & budgets, I shall perhaps prosper.\n\nI don't know when I can see Dixon (properly spelled [Harry Ellis?] Dickson) but I think I can next Friday morning.\n\nAmong other things [...] there is a quasi-contest here sponsored by the conductor of the Harvard Orchestra (Pierian Sodality to you) for an apt candidate for associate conductor with opportunity to conduct rehearsals & \"study conducting with Malcolm Holmes\" (Pfui*). Tryouts next week.\n\nCourses? Two in Music \u2013 Harmony & historical survey \u2013 a complete Shakespeare (perfectly thrilling) and a course in types of Philosophy given by Hosking. Later on, advanced Italian (which means Dante, Tasso, Ariosto, Castiglione etc.) Wonderful?\n\nWrite soon, & let me know of your studies.\n\nAffectionately,\n\nLen\n\n*Consult Oxford Dictionary\n\n12. Leonard Bernstein to Beatrice Gordon\n\nEliot G-41, Harvard University, Cambridge, MA\n\n30 January 1937\n\nYour note, dear Verne, was most charming and thoughtful. In fact it was really a lamb's ear.15\n\nI'm so sorry that I couldn't come over sooner or let you know why, whether, or when; but, you see, your note appeared on a scene of great confusion, it being (and still is) exam period, and in the midst of that still greater emotional upheaval of which you have no doubt heard.16 I must tell you all the details some time. It's the most fascinating, occult, hair-raising fairy story you could conceive of. You know, something one reads or dreams about \u2013 not experiences.\n\nMy very dear Beatrice, I'm so anxious to see you & will do my very best to be down as soon as I can after the examinations.\n\nWhat are you doing to fill your time? I'm very anxious to hear about your 50-cent pupil. Give him (or her, or it) my very best personal regards and recite to her the following after each lesson:\n\nSome folks think they get a lot\n\nBy paying huge recompense;\n\nBut I know one who gets the best\n\nBy paying fifty cents.\n\nAlso, if you have time:\n\nIt's a funny thing about strip-girls,\n\nThey give you so much, no more;\n\nThey never go below a certain point\n\nExcept Tuesday which have twenty-four.\n\n(A metrical masterpiece).\n\nTake care, & my love to your mother and all concerned.\n\nFrom\n\nLeonard\n\nwith affectionate January.\n\n13. Leonard Bernstein to Beatrice Gordon\n\nEliot G-41, Harvard University, Cambridge, MA\n\n28 February 1937\n\nIt's a curious thing, Beatrice \u2013 I'm being quite frank \u2013 or maybe I'm a curious being, I know not which. But I can pass months at a time, blindly busy with the immediacies which engage me, always unconscious of time and space, barred in by momentary emotions and reactions. Then I may chance upon something quite without this fettered up little circle, and be quite startled. That happened to me today, when I saw you. It occurred quite suddenly to me that I hadn't seen you for a very long time, and that I really was interested in seeing you. And it's doubly curious when I think that I _have_ been communicating with you, hearing your name mentioned, even occasionally struck by a thought of you; and always taking the thing so amazingly without cerebral deflection. And for no reason \u2013 tho your picture has been here on my desk, and tho your hair had collapsed \u2013 I remember that here you were, and I hadn't seen you.\n\nAll this must sound fairly obviously like the product of dementia, and Lord knows I don't know why I'm writing it. I merely felt a moment ago that I should like to talk to you. I have nothing to say \u2013 I'm too tired \u2013 yet this. I am suddenly aware of you.\n\nThat's all: there's nothing to say.\n\nGood night, Beatrice.\n\n14. Leonard Bernstein to Sid Ramin\n\nEliot G-41, Harvard University, Cambridge, MA\n\n12 March 1937\n\nDear Syd,\n\nI'm glad to see that you've decided to study with me. I think you'll get a lot out of it. You see, even if I don't have the professional training that the \"Miss Jewels\" have, I can give you a comprehensive outlook on harmony \u2013 which is the most important of the three \u2013 with an eye ever cast in the direction of jazz. What I can give you will always be directly applicable to jazz, and there will be nothing superfluous, and, I hope, nothing neglected.\n\nIt seems to me that Fridays at about 3:30 would be ideal; if you can't make it so early perhaps 4:00 would do, but not later. In fact, the earlier the better; if 3:00 it's even better. See if you can't make it next Friday at 3:00. If you can't, drop me a card and we'll make other arrangements. If I don't hear from you, I'll take it that you're coming.\n\nBest of everything,\n\nLen\n\nIncidentally, voici les particuliers of how to get here:\n\nGo to Summer Street Station and walk thru to a Cambridge train. Get off the latter at the last stop \u2013 Harvard Square, which is like this:\n\nWalk down Dunster Street _as far as you can_ ; it will take you right in the back door of Eliot House, leading you to a courtyard into which all the entries face. Go to G (gee) entry, walk up to Room 41 (all doors are marked) and knock vigorously. Voil\u00e0!\n\nThen I'll either see you next Friday or hear from you sooner.\n\nGood luck,\n\nLen\n\n15. Leonard Bernstein to Helen Coates\n\nCamp Onota, Pittsfield, MA\n\n4 August 1937\n\nDear Miss Coates,\n\nI hope that you are now fully recovered from your operation. I was so sorry to hear of it, but I'm sure that you are glad to be over & above it by now.\n\nI'm having a splendid time here at camp,17 though I get very little time for myself. But I guess that a good vacation is as important as work; and I am trying to rearrange my schedule to allow for practice.\n\nI hope that you're enjoying a very pleasant summer. I should love to hear from you.\n\nAs always,\n\nLeonard Bernstein\n\n16. Leonard Bernstein to Beatrice Gordon\n\nEliot G-41, Harvard University, Cambridge, MA\n\n8 September 1937\n\nBeatrice \u2013 Dear Beatrice!\n\nHad I known that you were not invited to the party I should have taken definite measures. I understood that you had been asked \u2013 I cannot imagine any possible reason for such an oversight. I've argued and argued with my sister, and she cannot possibly find any reason for forgetting you. She just did, as any child of her age is apt to do, and as she did forget some other people. So please forgive her. But I cannot understand why you didn't come up after I had asked you.\n\nAt any rate, let's pass over it \u2013 there's not much to be done now.\n\nI expected to see you in Sharon, but today you suddenly disappeared. Sunday I was busy with my horde of guests from N.Y \u2013 Monday & Tuesday were horrible holidays, & when I finally sought you Tuesday evening, lo! you had awayed to the movies. I did want very much to see you \u2013 there would have been much to say.\n\nI'm really very happy over your new (or is it already old?) job, & I wish you all success in it.\n\nAs for me, I shall spend until the opening of Harvard College gently fed sleep by a rosebeam.\n\nI do hope to see you soon.\n\nMy best to all at home.\n\nTake care of yourself.\n\nLamb's Ear\n\n17. Leonard Bernstein to Beatrice Gordon\n\nEliot G-41, Harvard University, Cambridge, MA\n\n[9 September 1937]\n\nRosebeam:\n\nStill the years go on,\n\nAnd still you're two weeks behind me \u2013\n\nPerhaps as years go on\n\nYou will really catch up and find me;\n\nBut until you do\n\nMay Fortune be our brother;\n\nAnd may we in joy march thru the years\n\nTwo weeks apart from each other.\n\nAnd one more little wish:\n\n(For me as well as you) \u2013\n\nMay nothing but those two little weeks\n\nEver come between us two.\n\nLamb's Ear\n\nN.B.1: I make up, you notice, for your omitting to send me the customary poem by addressing the gist of the above trivial, tho very sincere, masterpiece to the both of us.\n\nN.B.2: Lose not another minute before reading Gabriele D'Annunzio's _The Flame of Life_. Quick! It's incredible.\n\n_Happiest of Birthdays!_\n\n18. Leonard Bernstein to Mildred Spiegel18\n\n[October 1937]\n\nAnnouncement for a concert at the Sanders Theatre, Cambridge, in which Bernstein played the Ravel Piano Concerto with the State Symphony Orchestra, amended by Bernstein:\n\n19. Dimitri Mitropoulos19 to Leonard Bernstein\n\nMinneapolis, MN\n\n5 February 1938\n\nMy dear, dear boy,\n\nBelieve me, your letter touched me very deeply. I never forget you. I was only really very busy, all this past year and now just the same. But now I feel you more near and that gives me more courage to write you.\n\nThen, dear friend, is that so, is that true, that you believe so much in me? Have I really failed to you, have I really left you a void after our last meeting? This thought makes me crazy, and so happy that I dare not believe it. Nobody else has ever written me such a thing! In any way even, that you thought to write it makes me happy.\n\nDear boy, if you only could know how alone I am, all my life is a complete devotion to my art. Beyond this I am living like an ascetic.\n\nThere are many people probably who love me and are my friend, but it fails me, this unique one to whom I can believe with all my heart and soul. I am so full of the necessity to give my love, I am so full of love, that I am always spending it to every human being. Your letter was really a great gift for me and I thank you, thank you so much for this your unexpected gift.\n\nNow let me tell you what I am thinking about your last interest on modern American dance music. I can't say I know it well, but in any way I advise you to be careful and don't forget that even the American dance music is always a dance form and that this kind of music form is not the most interesting and useful form to exercise oneself on it. I feel sorry if the most part of your composing is devoted to such a poor form of music. Of course I agree that we may release from time to time doing easy and light things, to amuse ourselves, but not too much. We must train ourselves to [do] difficult things, to surpass ourselves, not to leave even a moment of your life without to be anxious to do it. In any way, to avoid to sleep too much on a very soft bed! I hope you will understand me. I had the impression that you are a very deep feeling boy and I hope that this your last sympathy with dance music is momentary. Perhaps you needed to relax, but excuse me in your age you don't need to relax before [you] have done your duty towards your art. If it is only for a pleasure, good, but not too much. We must keep ourselves as pure as possible.\n\nNow tell me dear boy, do you wish to spend some holidays (about a week for instance if you can be free) and come to me here. I am inviting you in any way and I shall take care of _all your expenses_. Will you?\n\nI shall be very happy if there was a possibility to see you again.\n\nWith all my sincerest sympathy.\n\nYours,\n\nD. Mitropoulos20\n\n20. Leonard Bernstein to Aaron Copland21\n\nEliot E-51, Harvard University, Cambridge, MA\n\n[received 22 March 1938]\n\nGod damn it, Aaron,\n\nWhy practice Chopin Mazurkas? Why practice even the Copland _Variations_? The week has made me so sick, Aaron, that I can't breathe any more. The whole superfluousness of art shows up at a time like this, and the whole futility of spending your life in it. I take it seriously \u2013 seriously enough to want to be with it constantly till the day I die. But why? With millions of people going mad \u2013 madder every day because of a most mad man strutting across borders \u2013 with every element that we thought had refined human living and made what we called civilization being actively forgotten, deliberately thrown back like railroad tracks when you look hard enough at them \u2013 what chance is there? Art is more than ever now proved entertainment \u2013 people, we thought, were ready, after two thousand years of refining Christianity, to look for entertainment as such; to look for things that come out of the category of vital necessity! And so we were willing to spend our lives creating that entertainment. Aaron, it's not feasible; it's a damned dirty disappointment.\n\nThen came the climax of the week. Cara Verson \u2013 whoever she is; to me she looks like an enlarged porcupine \u2013 had advertised for weeks that she was going to give in the Jordan Hall here, a whole program of modern music. I was all excited; it was unprecedented, and very courageous of her in this dead city, etc., etc. And I put so much hope in that damned concert. It came: and I find it difficult to talk about it. It was a tremendous program \u2013 Malipiero, Kod\u00e1ly, Hindemith \u2013 and \u2013 joy of joys! \u2013 the Copland _Variations_.22 That, I guess, was the premiere in Boston. Well, to get to the point, I don't know whether you knew it was going to be played here, but if you did, how did you allow it?\n\nIn short, she gave really no performance at all. I can stand a bad performance, but not _no_ performance. She began the thing wrong, played about two measures, skipped some variations, got lost again, skipped about 5 pages, played a few measures out of tempo \u2013 entirely without any discernment, without any idea of rhythm \u2013 and kept this up (playing little measures from choice variations) until she reached the coda. Then she played about half of it and called it a day. I was purple \u2013 I wish I could let you know how incredibly bad it was. It was the work of an imbecile. I left then and broke dishes in the Georgian cafeteria.\n\nDo you see what that farce meant, Aaron? The few people that were there thought she was wonderful \u2013 such a _touch!_ (!!!) They tried to look intellectually intelligent about the music when the whole performance was one of bafflement! The one little chance that this little town gets to hear some modern piano stuff \u2013 (nobody dares to do it at a recital) \u2013 we find instead the complete distortion of the whole art, a perversion of these people's attitude when we need every resource to show them the right thing, correctly done. And where did this foul woman get press notices for her folder? Aaron, find that woman and have her put away. She's fatal.\n\nExcuse this outburst, Aaron, but the whole concatenation of rotten, destructive things has made me very angry and disappointed. At Harvard the situation is aggravated by these horrible musical dolls who infest the place. I find it almost impossible to stand. Thank God for you. Our last hope is in the work you are doing.\n\nLeonard Bernstein\n\n21. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n23 March 1938\n\nDear Leonard,\n\nWhat a letter! What an \"outburst\"! Hwat a boy! It completely spoiled my breakfast. But it couldn't spoil the weather, so thank Marx for that. The sun has been shining in a way to defy all wars and dictators, and there's nothing to be done about it.\n\nThat \"female\" you tell of [Cara Verson]. I've never seen her, but I had reports of her at a time when she played the _Variations_ here, which I studiously avoided attending. I see that did no good, since she continues to \"play\" them. But what can a poor composer do? I know of know way of stopping her once the piece is published, do you? Think what people do to the three B's etc. and nothing can be done about _that_. As for your general \"disappointment\" in Art, Man and Life I can only advise perspective, perspective, and yet more perspective. This is only 1938. Man has a long time to go. Art is quite young. Life has its own dialectic. Aren't you always curious to see what tomorrow will bring?\n\nOf course, I understand exactly how you feel. At 21, in Paris, with Dada thumbing its nose at art, I had a spell of extreme disgust with all things human. What's the use \u2013 it can't last, and it didn't last. The next day comes, there are jobs to do, problems to solve, and one gets gradually inured to things. At my advanced age (37) I can't even take a letter like yours completely seriously. But I'm glad you wrote it, if only to let off steam. Write some more!\n\nNow it's definite that I'm not due to be up in Boston. I've been bought off with the promise of a performance at the coming Berkshire Festival. (Don't mention this around, will you.) I'm vaguely thinking of a trip to England in May. Sir Adrian Boult is to conduct my _Sal\u00f3n M\u00e9xico_ at the I.S.C.M. Festival in June, and previously on the BBC on April 20th; also here when he conducts the NBC orchestra in May. I hope you're coming to New York soon. I always enjoy seeing you.\n\nAlways,\n\nAaron\n\n22. Aaron Copland to Leonard Bernstein\n\nBox 15, Princeton, NJ\n\nSat. [April 1938]\n\nDear Leonard,\n\nI've come out to Princeton for a few weeks to try to finish that book23 by June. It's lovely here \u2013 reminds me of my spring in Cambridge (unfortunately Cambridge is 5 hrs from N.Y. and this only 1 hr.).\n\nI got your Minn[eapolis] card. Wish I could hear more about your trip.\n\nI had a letter from the WPA24 orchestra in Boston the other day saying they had programmed my _Dance Symphony_ for Apr 26 at Sanders Theatre. If you have time to go will you write me your impressions of the performance? And send me any reviews that appear? (Don't forget \u2013 it's an early work!)\n\nHow are you in general? Is Bennington decided upon? I haven't seen Norman Lloyd25 since I talked with Davis, but I'll mention you to him when I do.\n\nAny chance of your being at the Berkshire Fest. this year? I'm vaguely tempted to go. By the way, Adrian Boult is supposedly broadcasting my Mexican piece on May 14th over NBC at 10 p.m.\n\nRemember me to Davis,\n\nYours,\n\nAaron\n\n23. Dimitri Mitropoulos to Leonard Bernstein\n\nMinneapolis Symphony Orchestra, Minneapolis, MN\n\n4 May 1938\n\nMy dear, dear boy,\n\nYes, you are right to be worried about me. I couldn't answer your first letter; you were asking me too much. If you remember, you wished to know more about me; but I think it is better that you look at me as you wish to \u2013 put at me your own imagination, your ideal. Who knows? \u2013 otherwise you would be disappointed. And, dear boy, I need your appreciation, your respect, your love! It is of great importance in my life. I should be happy to see you again before I leave. I am beginning the rehearsals in New York the 24th. The concert is the 28th of May, and I leave the 8th of June. May I ask a small picture of you to be my companion on my Europe trip?\n\nWith affection,\n\nD. Mitropoulos\n\n24. Dimitri Mitropoulos to Leonard Bernstein\n\nThe Biltmore, at Grand Central Terminal, New York, NY\n\n7 June 1938\n\nDearest friend,\n\nThank you! I was so unhappy this last time! But now everything is again all right. I was so stupid to think that you didn't care so much about me. Wasn't I stupid?\n\nYour picture is so good, I like it, God bless you!\n\nYou see, my dear boy, sometimes I am so sad, and I need so few, just a little to be happy, and this little sometimes nobody gives me, it seems to avoid me.\n\nCan you imagine for a moment, I thought I lost your love, and then, I was asked me, perhaps I am not right to ask anything, to expect anything, from anybody, that my destiny is to be alone with myself and my art.\n\nBut you my dear friend, tell me, it is not so, I am something for you, yes... don't forget me.\n\nGoodbye dear,\n\nDimitri\n\n25. Leonard Bernstein to Aaron Copland\n\nEliot E-51, Harvard University, Cambridge, MA\n\n[received 20 October 1938]\n\nDear Aaron,\n\nIt's going to be hard to keep this from being a fan letter. The concert was gorgeous \u2013 even the Dvo\u0159\u00e1k.26 I still don't sleep much from the pounding of:\n\nin my head.27 In any event, it's a secure feeling to know we have a master in America. I mean that too (don't pooh-pooh). I sat aghast at the solid sureness of that construction of yours. Timed to perfection. Not an extra beat. Just long enough for its material. Orchestral handling plus. Invention superb. And yet, with all that technique, it was a perfect rollercoaster ride. And it's not the exhaustible kind of cleverness (like Fran\u00e7aix, or his ilk).\n\nI want seriously to have the chance to study with you soon. My heart's in it. Never have I come across anyone capable of such immediate absorption of musical material, possessing at the same time a fine critical sense _with_ the ability to put that criticism into words successfully. This is not rot. The little demonstration you gave with those early things of mine proved it to me conclusively.\n\nSaw the Group Theatre bunch today and they all asked for and about you. [Clifford] Odets,28 true to form, thinks the _Sal\u00f3n M\u00e9xico_ \"light\", also Mozart except the G minor Symphony. That angers me terrifically. I wish these people could see that a composer is just as _serious_ when he writes a work, even if the piece is not defeatist (that Worker word again) and Weltschmerzy and misanthropic and long. Light piece, indeed. I tremble when I think of producing something like the _Sal\u00f3n_.\n\nCasting is a wretched business.29 It's slow but sure. And so tiring. (What word from the Marc [Blitzstein]?) But I think we'll have a fine show.\n\nLet me hear soon. As Dame Fortune said to you backstage last Saturday night, \"On to bigger & nobler things.\"\n\nAlways,\n\nLenny\n\nP.S. I hope you're really haunted by:\n\nMaybe not _convincing_ , but maybe _haunting_.\n\n26. Aaron Copland to Leonard Bernstein\n\nInternational Society for Contemporary Music, United States Section, New York, NY\n\n[October or November 1938]\n\nDear Lenny,\n\nOf course you're crazy! I'm sorry if you felt a \"strained feeling\" that Saturday. The moral being \u2013 you mustn't be so sensitive. I remember Victor [Kraft] was acting strangely and I was embarrassed at not being able to invite you to the ballet30 \u2013 but that's nothing to have \"omens\" about. Anyhow \u2013 remember this \u2013 I feel much too friendly and sympathetic to you for anything I can possibly imagine making our relations \"strained\".\n\nAs a peace offering, I'm sending you a copy of the _Second Hurricane_31 which is just out.\n\nAffectionately,\n\nAaron\n\n27. Leonard Bernstein to Aaron Copland\n\nEliot E-51, Harvard University, Cambridge, MA\n\n[received 19 November 1938]\n\nDear Aaron,\n\nIn the midst of ten million other things I'm writing a thesis for honors. I think it's interesting \u2013 certainly it is potentially interesting. The subject is Nationalism in American music \u2013 presumably a nonentity but on the whole a vital problem. We've talked about it once or twice. You said, \"don't worry \u2013 just write it \u2013 it will come out American.\"\n\nThe thesis tries to show how the stuff that the old boys turned out (Chadwick, Converse, Shepherd, Gilbert, MacD[owell], Cadman etc.) failed utterly to develop an American style or school or music at all, because their material (Negro, American Indian, etc.) was not common \u2013 the old problem of America the melting pot. Having ruthlessly revealed the invalidity of an Indian tune surrounded by Teutonic development, etc., I will try to show that there is something American in the newer music, which relies not on folk material, but on a native spirit (like your music, and maybe Harris' & Sessions' \u2013 I don't know), or which relies on a new American form, like Blitzstein's. Whether this is tenable or not, it is my thesis and I'm sticking to it.\n\nNow how to go about it? It means going through recent American things, finding those that sound, for some reason, American, and translate that American sound into musical terms. I feel convinced that there is such a thing, or else why is it that the _Variations_ sound fresh and vital and not stale and European and dry?\n\nThis is where you can help, if you would. What music of what other composers in America would support my point, and where can I get hold of it? Would the music of Harris? or Ives? or Schuman? or Piston? or [Nicolai] Berezowsky? You see, I know and hear so little American stuff. This is my great opportunity to get to know it well, and find out something about it. I feel more and more that there's something to all this, and that it can be told in terms. I'll be infinitely thankful for any suggestions.\n\nAgain, thousands thanks for the _Second Hurricane_ \u2013 it's just swell.\n\nAlways,\n\nLenny\n\n28. Leonard Bernstein to Kenneth Ehrman32\n\nEliot E-51, Harvard University, Cambridge, MA\n\n6 December 1938\n\nIt's wonderful, Ken, the way you lack enthusiasm when it's more or less expected of you. Hello. A dull thud \u2013 that's the way your letter read. 500th crossing, and all that \u2013 trip is really as can be expected.\n\nBut it was very good to hear from you, all wrapped up in your mauve-lined Parisian envelope. [...]\n\nEverything is almost O.K. You ask after the _Advocate_33 competition \u2013 I won. I am now proud possessor of all manner of records. As for the _Birds_ \u2013 I won that too, and am busy as a bee on the composition and orchestration thereof. As for courses, I am fuguing and advance orchestrating and thesising, and another half next half (what am I saying). [Tillman] Merritt hates me, but Mother loves me. [Walter] Piston doubts me, but Copland encourages me. I hate the Harvard Music Department. You can quote that. You can even print it if you want. I hate it because it is stupid & highschoolish and \"disciplinary\" and prim and foolish and academic and stolid and fussy. I want to go home.34 [...]\n\nToujours \u2013\n\nBernstein\n\n29. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n7 December 1938\n\nDear Lenny,\n\nI know I'm late in answering but I've been swamped with things to do and your letter asked so many questions!\n\nAren't you coming down to N.Y. during Christmas holydays? And since it would be so much better to do this _viva voce_ than by letter, could it wait till then for my grandfatherly advice?\n\nYou sound as if you were very much on the right track anyhow both as to ideas and composers' names. Don't make the mistake of thinking that _just_ because a Gilbert used Negro material, there was therefore nothing American about it. There's always the chance it might have an \"American\" quality despite its material. Also, don't try to prove _too_ much. Composing in this country is still pretty young no matter how you look at it.\n\nGood luck and let's hear if you're coming down.\n\nConsider yourself missed.\n\nA.\n\n30. Leonard Bernstein to Kenneth Ehrman\n\nEliot E-51, Harvard University, Cambridge, MA\n\n17 January 1939\n\nDear Kenuel,\n\nI had a mad inspiration to write you several days ago when the enclosed suddenly popped up in one of my pockets. But put it off again and again I did, and so it was after all your letter which evoked this little opus. Mad Shirley tells me that our Stark (\"Watch that boy\") played the lead in the school production of _Tovarich_ some time ago, and walked off with the public-speaking prize. Are we proud!\n\nLife has been going on, as it has a way of doing. Just a series of minor catastrophes of varying kinds. Most noteworthy: I left a valuable manuscript of Copland's plus another printed piece of his plus a valuable manuscript book of mine plus a valuable fountain pen plus _all my thesis notes_ over which I had theoretically slaved (!) in New York on the train coming back from that City of Sin. The infallible New Haven Railroad is unable to find these things, which means that I must start my thesis all over again at double speed, and type this letter, faut d'un stylo, and be generally upset at having lost Aaron's manuscript for him. He of course took it as only he could take it \u2013 with a philosophical phrase. Good old Aaron: if it had been anyone else but he I should long ago have gone into voluntary exile.\n\nAaron, by the way, could never understand my lack of desire for going abroad. In his day, says he, there was never a composer who would not insist on Paris first. I have always been inclined to pooh-pooh the idea, on the grounds that a composer can go through his \"Paris\" period here in America as well as abroad. But lately little ideas have been creeping around in the brain. The thought of Paree attracts me mightily these days. And your letter has set me off on another gush. It would be wonderful, I am convinced. Of course, a great deal depends on funds. But there's always the possibility of a fellowship (please Goddy Woddy); and then my father almost bowled me over last night with the statement that he would be responsible for me for one more year after college to the extent of the equivalent of what it takes to keep me in college one year. I may go abroad with it if I choose (sounds impossible to me), or to a music school. I still don't know whether to take him seriously or not. But that added to a fellowship would be simply de trop, as Mary Boland says.\n\nYou see, Ken, the sudden surge toward la vie transatlantique is due, I suppose, to an equally sudden horror of what is to come here next year. I planned for a while on a few months in Mexico. It would be swell, but, in the final analysis, pretty unconstructive. I also toyed with the idea of California (I'm still a C\u2014\u2014\u2014iac35) for a year or so. But I know all the while that I am not yet prepared to settle down somewhere and write music. I still have so much to study. In America there is but one person I am interested in working with after college, and that's Aaron. Now (how mathematical this discussion is) working with Aaron involves being in New York. Maybe it's just that I've recently returned therefrom and had a bellyful, but more and more I do not want to go there \u2013 at least not yet. The people of the \"artistic world\" that I encountered on this last trip revolt me in every way. I have been made sick by the depravity of the Greenwich Villagers, the totally degenerate homosexuals, the equally degenerate heterosexuals, the foolish and destructive attitudes, and the frantic attempts to preserve the atmosphere of postwar bohemianism. Oh, there are some who are all right, of course. They are far and few enough, God knows. And the thing I am really afraid of is that I could so easily fall prey to that sort of thing. You may remember my chief weakness \u2013 my love for people. I need them all the time \u2013 every moment. It's something that perhaps you cannot understand: but I cannot spend one day alone without becoming utterly depressed. Any people will do. It's a terrible fault. And in New York, the people who would fill that place with me would inevitably be those wretched people who haunt the Village Vanguard by night, and each other's studios by day, and act positively in only one way \u2013 as a destructive and retarding force in their societies. This, by the way, is not bitter or dramatic in any way. But it is this great horror of taking my place with these people, and becoming an \"artist\" that half kills me. There are two rebuttals, of course. One, you should say that I ought to be strong enough to resist all this; but sometimes I am much afraid that it wouldn't take too much effort on my part before I would be like them. I always absorb my surroundings \u2013 but to a degree! Second rebuttal: Paris holds the same kind of crowd and the same lack of healthy art. Well that's for you to tell me. I don't know. Let me know the lowdown soon. I am thinking very seriously now of going abroad.\n\nStrange how I miss you. Perhaps I'm not so \"universal\" or promiscuous as I thought. You were very important to me last year. So steadying for me; and you helped me over many a rough place without perhaps knowing it. No, the afterthought is not after all, \"He meant well. Good old Ken.\"\n\n_The Birds_ comes along slowly and unsteadily. The whole score is finished but the Finale. The chorus doesn't show up for rehearsals. The orchestra is still being slowly assembled. But it's fun.\n\nThe [ _Harvard_ ] _Advocate_ has a new Board, all Socialites. So the new Music Editor is a Socialite. No more records for little Lenny.\n\nI have already reset the ribbon on this machine twice, which means that this is a very long and probably silly letter, and I ought to stop.\n\nMy best to Aunt Mattie, whom etc., and to all the Messings, Truebloods, Bluebloods, Peggrams, etc. that you come across.\n\nHappy skiing.\n\nAlways,\n\nLen\n\n[...]\n\n31. Dimitri Mitropoulos to Leonard Bernstein\n\nMinneapolis Symphony Orchestra, Minneapolis, MN\n\n18 February 1939\n\nMy dearest friend,\n\nAt last a letter from you. I was completely despaired and beginning to think that you completely forgot me. For a while I thought it was because I sent you by mistake a letter without signing it, and that you were offended; but fortunately your letter came to prevent my disappointment.\n\nI am very happy to hear that you are working hard, but I am sorry to see that you neglect your piano, which could be a great help to your career.36 I see you too come to the position now to have problems: musical, artistic, social and spiritual \u2013 and the worst of all, sexual. Unfortunately I am too far away to help \u2013 to give you good advice. But I hope you are a clever boy and that you realize the great responsibility toward yourself, its importance. As far as the conditions of my personal life are concerned, actually, I must tell you that neither my life nor my Weltanschauung has undergone any change; but it has improved, I think, in wisdom and in self-concentration.\n\nMay I tell you an agreeable thing. I am invited to conduct the Boston orchestra in the middle of next January for two weeks. I look forward to have some moments again in your inspiring and friendly company.\n\nWith my best wishes,\n\nD. Mitropoulos\n\n32. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n[April 1939]\n\nDear Lenny,\n\nI've been meaning to write and tell you what a swell host you make and how pleasant in retrospect my little \"week-end\" was, and now it has to be accompanied by this regretful letter from La Holm37 which of course spoils my effect. But don't fret \u2013 something _must_ turn up sooner or later.\n\nWhen I got back here I learned that _Quiet City_ has been cancelled. The Group [Theatre] wasn't satisfied so my career in the theatre has been a flop \u2013 obviously. Nothing left to do but write a Piano Sonata.38 Or perhaps something special called \"The Beach at Revere\" or \"The Birches at Sharon\".\n\nWell anyway \u2013 I hope something will drive you to N.Y. soon. D[imitri] M[itropoulos]'s Greek concert, or the World's Fair, or my ballet or sumpin!\n\nBest to you,\n\nA\n\n33. Leonard Bernstein to Kenneth Ehrman\n\nEliot E-51, Harvard University, Cambridge, MA\n\nApril [1939]\n\nGreat & Good Kenneth,\n\nTo think!\n\na) No fellowship\n\nb) No job at Mills College this summer. They're reducing their staff, and cutting out the production end entirely, reserving only [for] educational.\n\nc) I'm dead on my feet. I just handed in the thesis, having stayed up all night, and I'm just beginning to recover. I have [no] control over myself, so excuse typing, et al.\n\nd) No prospects for the summer or next year. Maybe a job with a dancer next year. Maybe a job on _Modern Music_. Maybe Mexico this summer. Maybe Sharon (God forbid!) Maybe anything. The prospect is lousy. Any suggestions?\n\ne) Everybody's lousy. People are always getting divorces from nice people like An or being impotent with nice girls or flunking exams or vomiting over the European situation. And for God's sake keep out of Greece and concentration camps. Of all times to be where you are!39\n\nf) I never recovered the lost thesis notes. (This is irrelevant.) (But so's Margaret.40 I told [David] Prall41 about that particular fetish and he beamed all over.)\n\ng) _The Birds_ finally comes off next week.42 It should be good.\n\nGod, Ken, it's a dull and wretched state I'm in. No practicing, no composing, no plans, no money, no ideas. Static. Tired all over. I'll be all right tomorrow. I've met a wonderful girl. I'm about to have a sex life again. That's encouraging.\n\nHave a swell trip, and be careful. [...]\n\nI long for California, and the peaches and ladyfingers and artichokes. I long for so many things. Why can't I stop this silly wailing. Really, I'm just tired.\n\nAgain, be careful, stay alive, write. With devotion, affection, greetings, warmth, cordiality, sincerity, verytrulyness, blessings,\n\nLenny\n\n34. Adolph Green43 to Leonard Bernstein\n\nHotel Astor, Times Square, New York, NY\n\n[27 May 1939]\n\nDollink Lennie,\n\nFirst of all \u2013 good luck with _The Cradle_ tonight.44 I hope Blitzstein is there. He told me about it, & I was going to come up, but I can't get Saturday off. You see, I'm working every night except Monday & Tuesday. We started on April 6th or so doing Saturday Shows, just about the time I was planning to come up for your Aristophanes.45\n\nLennish, I'm doing famously. So help me. Its all \u2013 quotes \u2013 like a dream \u2013 unquotes. Exciting as hell. But I'm not in the dough yet. We've been signed by the Wm. Morris Agency but they haven't found us any good spots yet. And there's Radio & Television yet & everything and all in the offing. Nothing materialized. But, Gott zei dank, I'm making a little salary. I've lost about 25 pounds, so I'm no longer rolly-polly-Adolph, just a flabby Adolph.\n\nIt all began about 2\u00bd months ago, when we did a guest booking for an Actors' Party at Caf\u00e9 Society. Herman Shumlin & Arthur Kober & others were there & they went nuts about us. So since then everything happened. Publicity \u2013 all kinds \u2013 _New Yorker, N.Y. Times, Post, Journal-American_ , [ _The New_ ] _Masses_ , [...] [ _Daily_ ] _Worker_ , all over. Everyone in theatre has been to see us. And composers \u2013 ay, ay \u2013 Blitzstein, Copland, Paul Bowles, Jerry Moross, etc., etc. They're swell people, too. By the way, I gypped a Satie composition from Bowles' house for you.\n\nSo enough of this self-indulgent bloating.\n\nI would like, is it possible, to hitch-hike up to Boston on Monday & stay for a day or so. Hmmm? Is it possible? With me, I would have a young boy, Julian Claman,46 a nice charming lad, who is our lighting man at the Vanguard.47 Would that be possible? If so, send a telegram to A. Green, 835 Riverside Drive. If not, also send a telegram. You can send it collect _if you want to_. Heh, Heh, Heh.\n\nListen, are you coming to the City this summer? Will you have a place to live? If not, I've found a terrific 2-room also with kitchen apt. \u2013 with 3 beds \u2013 to be sub-letted for the summer. It's got swell furnishings, it's got a phonograph-radio, it's got a good Grand Piano, it's large, comfortable, on 55th St near 6th Ave, and Julian and I want to take it. With 3 of us at about $50 a month, it would amount to $3.50 a week each. We'll talk about it.\n\nSo let me know about Monday.\n\nGive my love to S[hirley] B[ernstein].\n\nGive my love to Blitzstein if he's there.\n\nAdolph\n\nP.S. Lenny, my love, I love you.\n\nCopland told me your Aristophanes music was a remarkably fine work.\n\nChrist, what an incoherent letter. Forgive, forgive.\n\nP.S. I've seen Weil48 once in the last 6 months. Is my heart bleeding? Not precisely.\n\n35. Leonard Bernstein to Aaron Copland\n\n17 Lake Avenue, Sharon, MA\n\n26 June 1939\n\nAaron,\n\nPatiently I have waited. The rains have come and gone. The sun, the moon, have seen another cycle. Millennia have elapsed. I have graduated with honors. I have been to all kinds of class days, commencements, baccalaureate services. I have grown old. And no word about cabins. Have you investigated? Are there cabins? Are they good cabins? Are they in America? Have you forgotten me? Is there something wrong? Do you hesitate? These, and other thoughts, as Kipling would say, are my constant companions.\n\nI am madly trying to recover my lost ergs. According to laws of nature they must be conserved somewhere, but I'm having a time finding where. I frequently stand up only to fall down. I sleep very easily (a bad sign for one who has always slept not too well). I have subtle little pains in my back. I have become positively hypochondriac. I live in waiting to hear from you. Please \u2013 before I rot in the provinces, let me know the outcome. At the above address.\n\nLetter from Blitzstein says that he likes Yaddo [artists' community] again and is about to convince himself that he ought to get to work.\n\nI have ideas for a piano and fiddle sonata, but I can't work on them here. I have begun to practice again. It is a strange feeling. Fingers slowly begin to move again.\n\nFor God's sake, Harvard, I got an A in that Government exam!\n\nTired, and so to bed, silently, alone.\n\nI wait with renewed vigor...\n\nLenny\n\nThere is talk among some people I know of a cabin for me or us or two cabins for us in Scituate, Mass. (on the South Shore) \u2013 very nice. Promise of seclusion. Would you be interested? Cheap...\n\nSay, I thought you were leaving the Empire on May 30?\n\n36. Leonard Bernstein to Kenneth Ehrman\n\n61 East 9th Street, New York, NY\n\n13 July 1939\n\nCher Kenuel,\n\nI know it's been overlong, and I've been lax, and I deserve one thing and another, but so much has happened \u2013 all relatively unimportant \u2013 that I've saved it all up to this minute.\n\nA. I've graduated with a bang. An incredible A in the Government course, and a cum laude. A great class day skit which I performed to a roaring crowd through a mike, and got in some parting cracks [...] at the old school and its officials.\n\nB. The inevitable letdown. Bringing us to\n\nC. What now? So I came to New York, where I've subletted an apartment with two friends of mine in the nightclub entertainment business. Here I am trying to establish that most horrible of words _contacts_ (insurance man's term). I have made a few, but can as yet report no results. All kinds of troubles with the union. You have to belong, and they won't take you because they have more unemployed than they can take care of, so they are charging an outrageous price for admission.\n\nD. I'll stay yet awhile, and see what happens. If nothing, well. But oh for something! Any old kind of job that would pay a living wage. You see, I still don't really know quite what I want to do. Conduct, compose, piano, produce, arrange, etc. I'm all of these and none of them. The Big Boys here (and thank God they're rooting for me) have it all decided that I am to become America's Great Conductor. They need an Apostle for their music. Or else they want to keep a rival composer out of the field. At any rate, they've been fine to me, and there is even talk of a scholarship at Juilliard in the fall. But nothing happens here until it happens, so only waiting. They also wait who serve the meat.\n\nE. How are you? Al [Eisner] and Austin, as you have probably heard, are in Hollywood. Al has this incredible job at $50 per with MGM, and Austin is living with him or on him.\n\nF. Complaints about the piano playing here may mean our eviction. That would be at least interesting.\n\nGod bless you, son, and my best to all the lads. Flights of angels wherever you may be, Greece or Naziland or Turkey. Write write away and right rite away. I hope maybe some day to see you, you vagrant wretch.\n\nAlaways,\n\nLenny\n\n37. Leonard Bernstein to Aaron Copland\n\n61 East 9th Street, New York, NY\n\n30 July 1939\n\nAstral Aaron,\n\nI. Many blessings for the Stadium tickets. I'll see that they're used.\n\nII. Last night I almost wrote you a very desperate letter. I was desperate. Having to do with those in the songwriting industry is no fun. And no success. I now know what it is to be rebuffed by the Beeg Ceety. But today I'm more cheerful. Probably because I've forgotten about songwriting. The crowning disappointment, though, was when I went up to the Juilliard School to see about conducting fellowship, and found I was a month too late for application. Can something be done? Or do I turn in desperation to the possibility of Curtis?\n\nIII. I saw Rolf Kaltenborn today, and he says I should see Davidson Taylor about employment.49 You can do me a _great_ favor by writing Taylor a letter about me right away so that I can see him soon, loaded down with press clippings & scores (?). Please write him \u2013 it means much to me \u2013 and can you let me know as soon as you've written him so that I can plan an attack? Thanks, thanks.\n\nIV. I saw _The City_ and it's a knockout. The opening is too long. But the NYC sequence is swell. And the music all one could ask for.50\n\nV. [...] You've been wonderful and terribly helpful, and I'm more bewildered than ever. I think in the last analysis, it's all up to me. [...]\n\nEver,\n\nLenny\n\nPlease \u2013 about Taylor. I'm so in need of a job \u2013 especially psychologically. Write me soon.\n\nL\n\n38. Leonard Bernstein to Aaron Copland\n\n[New York, NY]\n\n9 August 1939\n\nDear Aaron,\n\nMany thanks for all endeavors. You've been superb. I spoke with Davidson Taylor and he held out little hope, but I felt hopeful for the first time. He was the first interview I've had in NYC where I wasn't greeted by a patronizing attitude. He says he'll try and see what can be done, but oh! _Experience!_ Where does one buy or rent radio experience. You have to start somewhere.\n\n[Roy] Harris was very nice. He is writing Hutcheson51 at Chautauqua about me, and may be able to get me in. He also speaks very seriously about my changing my name. Something Anglosaxon like Roy Harris, no doubt. He thinks I might thereby ride in on the crest of the wave of reaction against the foreign artist craze, which reaction he thinks is due for the next twenty years. Mind, he is not referring to the Jewish question (!).\n\nI saw Reiner at the Stadium last night (bless you for the tickets), and he was matteroffact. Then I mentioned the aspiration towards conducting, he threw up his hands and yelled BAD! But write to Curtis, says he, and he'll give me an examination at the end of September. Purely routine. He asked me to tell you, by the way, that it is very urgent that he have the score of the _Outdoor Overture_ immediately. Address: Westport, Conn.\n\nAll else is the same. O tempora.\n\nWrite soon. The address is still 61 E 9[th Street], and will be probably for another week. Then \u2013 shux.\n\nBest,\n\nLenny\n\n39. Leonard Bernstein to Aaron Copland\n\n17 Lake Avenue, Sharon, MA\n\n29 August 1939\n\nAaron,\n\nTwo momentous things.\n\na) I've just finished the _Sal\u00f3n M\u00e9xico_ reduction \u2013 it's fine.\n\nb) I've just finished my Hebrew song for mezzo-sop. and ork. I think it's my best score so far (not much choice). It was tremendous fun.\n\nUnder separate cover, as they say, I'm sending the _Lamentation_ for your dictum. Please look at it sort of carefully \u2013 it actually means much to me. Of course, no one will ever sing it \u2013 it's too hard, and who wants to learn all those funny words? Eventually the song will become one of a group, or a movement of a symphony for voice and orch., or the opening of a cantata or opera, unless you give a very bad verdict.\n\nI'm not sending the _Saloon_ , as you couldn't read my hasty script, anyway.\n\nYour card was forwarded to me days after you sent it. I'm awfully sorry to have left NYC. Can't we see each other soon?\n\nBest,\n\nLenny\n\nMy best to [Benjamin] Britten.52 Did he get his Concerto (?) back?53\n\nNo jobs. No future.54\n\n40. Aaron Copland to Leonard Bernstein\n\nWoodstock, NY\n\n30 August [1939]\n\nDear L,\n\nI sent off the paper to Curtis [the] day after it got here \u2013 with a heated recommendation at the bottom.\n\nWhen there was no answer to my card I assumed you had left NY. I actually did broadcast again at the Stadium.\n\nNice that you've been able to finish pieces in these troublous times. (I haven't been able to do much more than read newspapers). If you sent the _Lamentation_ by express, it's waiting for me over in W. Hurley. I'll try to write you about it, tho I hate writing \"critiques\" \u2013 much prefer telling you viva voce! (Things one says with a glance look so awful on paper!)\n\nTry to make an ink copy of the _Saloon_ as I'm keen to see what you've done.\n\nAre you going back to NY during Sept? If so, maybe you could stop off here on your way down. It's only 60 miles from Albany \u2013 down the river. (It's even nearer Poughkeepsie \u2013 if there's any way of going there directly from Boston.) Anyway keep it in mind and keep me posted with where you are.\n\nB. Britten left Woodstock a week ago. He never mentioned anything about his Concerto coming back. He was all flustrated about the war news when he left.\n\nI've been reading about the phenomenal success of the Vanguard children.55 Nice!\n\nMet a friend of yours at the Stadium last time \u2013 a pupil of Prall's about to leave for Mexico \u2013 an athletics scholar \u2013 name Szatmary (?) or something like that.56 Also exchanged words with young Kaltenborn about you.\n\nGlad you took me out to Sharon that day. Now I can visualize where you are.\n\nAmiti\u00e9s.\n\nAaron\n\n41. Helen Coates to Leonard Bernstein\n\nBarbizon Plaza Hotel, New York, NY\n\n14 October 1939\n\nDear Lenny,\n\nI was very disappointed not to see you while you were home, tho' I appreciate the effort you made. Mother was happy to see you, as always. I phoned your mother Wednesday evening to see what had happened to you, and had a nice talk with her. She seemed quite reassured to hear that I thoroughly approve of your specializing in conducting.\n\nI'm thrilled to know that you got the scholarship at Curtis and hope it will fulfill all your desires for _serious study_ this year. I do hope you'll get some lessons with [Rudolf] Serkin, too.\n\nKnowing my passion for details, you'll be prepared for an avalanche of questions about your work, living, etc.\n\nFirst of all, how are you working out your expenses? Just what does the _scholarship_ give you, and did you decide to accept D.M.'s (as I shall call your Minn. friend)57 offer to finance you this year? I shall, of course, keep this confidential. So where are you living and are you able to practice regularly at Curtis? And what kind of people do you find studying at Curtis? Tell me about the new friends you make.\n\nI came down from Boston last night, and had a little time before I began to teach, so thought I'd answer your welcome card. I'm hoping you'll be coming to New York on one of my weekends here, so we can have a real visit and I can find out how everything is going with you. I'll be here again on _November 4th_ and also on _Nov. 25th_. Let me know in _plenty of time_ if you are to be in N.Y. on either of those dates.\n\nI know you'll make the most of your time this year, and I do hope you'll live a regular sort of life for the most part and get _plenty of sleep_. You have a wonderful opportunity ahead of you, if you work hard and fulfill the hopes D.M. (and I) have for you.\n\nDo write me soon \u2013 a real letter \u2013 and answer all my questions. I'm always interested in anything that vitally concerns you.\n\nMother would want to send you her love with this.\n\nMy love and best of wishes for a truly wonderful year \u2013 and may the little scores be a real inspiration and a reminder of\n\nYour affectionate friend,\n\nHelen\n\nSorry I mixed up these pages!\n\n42. Leonard Bernstein to Helen Coates\n\n408 South 22nd Street, Philadelphia, PA\n\n17 October 1939\n\nDear Helen,\n\nThanks for the lovely Barbizon Plaza letter \u2013 even without it I know you're with me every inch of the way.\n\nThings couldn't be better. The Institute is surpassing. The staff is perhaps the _only_ bureaucratic setup that I have ever seen so helpful and considerate. Randall Thompson (director) is all a director should be, and the staff of teachers (I should say faculty!) is the best.\n\nAs to conducting, it looks like a long uphill climb, but I proceed nothing daunted, despite all the venomous attacks I hear on all sides against Mr. Reiner, with whom I am studying. As a matter of fact, in the one class we've had (only 3 students) he was gentle as a lamb. Maybe only lamb's clothing. There are several supplementary courses (students can take the whole curriculum if they wish \u2013 it's all open & free) \u2013 such as transposition and score-reading, orchestration, etc. The first two are with Madame Miquelle (daughter of Longy) and the third with Thompson. There is also a course in formal analysis with Dr. St\u00f6hr, an incredible old German with walrus mustache & Van Dyke & all. A wonderful man, though given to a high-school-teacher manner.\n\nAh! The piano! I passed my audition for Serkin, but am not studying with him. First, he's not yet in the country; second, I seem to be _over age_ (!). They've tentatively allotted him a chap of 16 & a girl of 13, so that he can _mould_ them. At the same time, I had impressed the worthy jury (and am consequently majoring _both_ in piano & conducting). So they asked Mme. Vengerova,58 who, I am told, is the greatest piano teacher in America, better than Serkin a million times, etc., etc. I don't know if she would take me. They decided they couldn't give me to a supplementary teacher. She accepted (hesitantly at first at being second fiddle to Serkin, & having me \"palmed off\" on her), but since her contract hours were full, they increased her contract & I am now studying with the greatest teacher Curtis offers. As a result of which, the school moved a Steinway grand into my room this morning \u2013 just an old Curtis custom \u2013 since Madame _insists_ I have just that to practice on. It's a dream of a piano. As I say, things couldn't be better.\n\nI work & work & work (practice about 3 to 5 hrs a day), & do nothing else except sleep plenty. No social life \u2013 no friend to speak of.\n\nI have accepted D[imitri] M[itropoulos]'s offer. Not in desperation \u2013 in joy. My mind has changed drastically \u2013 I want nothing more than to have obligations to him.\n\nYou wanted details \u2013 you got them. It's rather nice & easy to write details that are pleasant.\n\nBest of all: I _may_ be in NYC on Nov 4 \u2013 I'll write later.\n\nMeanwhile write again & much affection from this city of dust & grit & horror.\n\nAlways,\n\nLenny\n\n43. Alfred Eisner59 to Leonard Bernstein\n\nMetro-Goldwyn-Mayer Pictures, Culver City, CA\n\n[?October 1939]\n\nLennie,\n\nIn haste and I've mislaid your address so it will take longer, so for the love of God answer pronto and even faster if possible. They've dug up a babe named Helen Gilbert who is beautiful enough to make St. Anthony tear his beard in frustrated rage and who also happens to be a magnificent cellist, yes indeed, and I'm doing one for her and I need a little information, to wit: some technical language that might be expected to pass between a teacher of cello and his pupil: you know, bowing, arpeggio, crap like that. Also, technical language that might be expected to pass between a teacher of voice and a pupil: breathing, or what have you. Casting the babe opposite a young baritone they're grooming to replace Nelson Eddy. This will definitely not be shit: the girl's a refugee from Vienna (in the picture) and her old man, a violinist, gets his hands burned with acid even as did the great Feuermann, and the setting is to be the Paris Conservatory (after the escape from Austria) and for chrissake get the dope to me as fast as fast, because if this job turns out as well and gets as much attention as my last you'll have refuge indeed in times of stress and that in style.\n\nAnd why the hell don't you write to a guy: Too busy. Yeah, sure.\n\nAl\n\n44. Alfred Eisner to Leonard Bernstein\n\nMetro-Goldwyn-Mayer Pictures, Culver City, CA\n\n31 October 1939\n\nLenny mine,\n\nNot a word did I know, no, not a soup\u00e7on, of thy good fortune until but three days gone when I walked into Aaron [Copland] outside my office door (like the Raven). He lost no time in acquainting me with the vagaries of my sainted room-mate. Liebchen, you cannot imagine what being put au courant of your goings and comings did to this brassed over heart, this spirit worn thin in the wasteland. The doubtful privilege of three thousand miles gives me the right to vicarious thrills, and second hand glowings. I was, and am, so goddamned glad for you. Aaron tells me you went to town on a competitive exam, which makes it even better. Baby, look down dat lonesome road: do you see the shining city, the purple hill? If I am over sentimental and Tom Wolfish, forgive me; it's because I miss you so hellishly much. Yes, that vaunted individualist, that brave soul: Eisner, pines. It would be a boon indeed in his young life to pass a quip once again with you. Ah, Steinbern, would I could climb again on those collines de autre fois, crunch through les neiges d'antan, eheu, eheu, miserere mihi (does miserere take the dative?) But 'twont be so very long: I return to God's country as soon after the 19 December as is humanly possible, to remain for a month and then once again must I take the flinted trail back and earn those golden objects which, it is my fond hope, will give me a measure of independence next year.\n\nIntelligence note: Pratiner celebrates his nuptials shortly. A bitch name of June Herbert. Excuse: If he marries her maybe he can take her for granted and get some work done. R.I.P.\n\nInterruption Goddammit. Some damn fool wants me to collaborate on a picture that will offset _Mr. Smith Goes To Washington_. In the kishkas I'll give him.\n\nMy personal little river flows along. I finished the script about the cellist to much critical acclaim, which is probably all the attention the damn thing will ever get. A wonderful job, but too much this, not enough that, etc. I don't care. If the acclaim will get me $300 per _week_ that is sufficient, thank you. And I think I'll get it. Jesus, what a laugh. Well, I'm worth 300 as much as some of these mockies are woith 1500. Note silly manifestation of conscience.\n\nPolitical note: purged myself from the Party. Recent events, coupled with tortured casuistry of _Daily Worker_ did for me.\n\nNot much else in the way of news. I languish most of the time. This job is absolutely the snappiest snap ever invented by the mind of man. One of the boys has produced a chess set and now no one even talks about working. Before that it was all day crap games. I see a lot of pictures in private projection booths and shit like that. I seldom go on the stages anymore. I don't feel particularly well. Climate seems to be enervating. I sleep a lot, read some, work a little. Don't get around much anymore. The testament of a man in exile. [...]\n\nWhat of Ken [Ehrman]? Did he join the French Foreign Legion or something?\n\nAgain and toujours, write.\n\nDe profundis\n\nAl\n\n45. Betty Comden60 to Leonard Bernstein\n\n29 West 65th Street, New York, NY\n\n1 February 1940\n\nDear Lennie,\n\nJust a line to let you know, you darling, that we all miss you \u2013 but that we are glad your work for the moment is not being interrupted by the intrusion of anybody's two left feet.61 To get to the sordid facts \u2013 the machine is still in a state of collapse \u2013 necessitating the checkup up of every inch of wiring in the place. This little investigation will take at least another week or so \u2013 and we have chalked it up to a stroke of \"Revuers luck\". The phrase is much talked of nowadays.\n\nI'll keep writing and who knows, some time this year we may get the old opera recorded yet. We've added some new music and attendant stuff, but nothing too frightening. Oh \u2013 _Pursuit of Happiness_ on Feb. 18.62\n\nAre you coming in at all? \u2013 business or no? Again, I miss you. The house is an empty shell, reverberating with your memory.\n\nBetty\n\n46. Betty Comden to Leonard Bernstein\n\n29 West 65th Street, New York, NY\n\n9 February 1940\n\nLennie dear,\n\nAll is not lost \u2013 well, not quite all. We have set a date for the recording \u2013 February 24 and 25 \u2013 but is that convenient for you? It is more likely that this time The Revuers and not the machine will break down \u2013 but I, personally, am sticking to my cod-liver oil \u2013 and hoping that all will go well. Also \u2013 do you think you might be able to come on Friday \u2013 the 23rd? \u2013 It's been so long, and also there may be some new music, and vigorous rehearsal would be a great idea.\n\nListen \u2013 if you can bear it \u2013 on the 18th to _The Pursuit_ ( _of Happiness_ ) \u2013 and write to me as soon as you get this \u2013 or wire \u2013 about your coming to New York. Will the 24, 25 & 26 fit into the Bernstein schedule?\n\nWish you very much love from Betty \u2013 and the other bright young people.\n\n47. Leonard Bernstein to Kenneth Ehrman\n\n408 South 22nd Street, Philadelphia, PA\n\n[20 February 1940]\n\nO never-to-be-forgotten one!\n\nThe only news I've had from you is the one solitary fact that a letter you sent to me this summer was returned to you, address unknown. I live from moment to moment in expectancy of your return, capture by the Germans, or some such rot. Why don't you let a fellow who remembers a brown bag that sags to the ground, & will never forget Mr. Bowie? (If there are _New Yorkers_ in Paris, cf. Thurber cartoon of recent date captioned \"Every day is Arbor Day to Mr. \u2013\")\n\nThings are progressing at a great rate here. I can almost play the piano again, was the only student to get an A in conducting from Reiner, have had pieces broadcast, am receiving ghastly label of Pennsylvania composer. You would like our director, Randall Thompson, one of U. of Calif.\n\nSo is Margaret.63\n\nKenneth, believe me, I perish to see you. I get these fits from time to time \u2013 it was studying Brahms' Third for Reiner tonight that did it. Please, I beg, write endless letters, or better still, come home. I don't know whether to envy you, or be perplexed, or simply be angry: once in a while la moutarde me monte au nez. From what I can gather, you're doing nothing in France. And all these Hearst papers in America.\n\nNext year looks wonderful for me. Great things are brewing, & I can't say anything now. But, as you've already guessed, it's mostly the fault of Dimitri. A most blessworthy man.\n\nAre you married? Do you still like music? Have you read Spengler? (I am now). Is tertiary syphilis curable? (Jot that down, Miss Wilson). Is Gordon still Messing? Is Bob still Wernick?64 Tell all to yr still truest \u2013\n\nLove that transcendeth death,\n\nLenny\n\n48. Betty Comden to Leonard Bernstein\n\n[New York, NY]\n\n1 March 1940\n\nLennie dear,\n\nThere's little left to say by this time. If your patience has not worn too thin (I'm suffering too, you know \u2013 it's not us if I'm enjoying all this, you know) \u2013 whether next weekend is divinely free for you \u2013 and whether you could possibly come in on Friday \u2013 even if it is late. There has been such a lapse of time since we last battled through that score \u2013 and some new to learn (!!!). The more time we have for it the better. Yes \u2013 I know April is far off and we needn't rush into this blindly, but Lennie \u2013 the age of miracles is not yet dead \u2013 we may yet make those there new records _March 9th and 10th_.\n\nI listened last Sunday \u2013 and your music was by far the most distinguished on the program, no matter what you may say. Hearing your name gave me a warm glow.\n\nWrite me Len \u2013 and if you're free, come anyway. It has been _much_ too long, too long.\n\nLove,\n\nB\n\n49. Betty Comden to Leonard Bernstein\n\n[New York, NY]\n\n6 March 1940\n\nLennie!!!!\n\nYou didn't like _Grapes of Wrath_!!!! But no matter \u2013 it alters not the \"us\" that stands triumphant and invulnerable above all petty differences. I don't care if you didn't like _Winter Carnival_. Come to dinner Friday anyway, and I shall try to break bread with you without a hard word about the Joads. Come at six-ish? We have an appointment with that unmentionable recording outfit at _eleven_ \u2013 Saturday morning. That leaves us Friday evening to run over what is now a rusty better-forgotten memory to us all. In fact, it probably leaves us the whole weekend if I know Musicraft. But no, this time they said \"positively\".65\n\nAll the news is exciting about you, Lenny. Mme Miquelle said you were only _talented_?\n\nTill Friday,\n\nBetty\n\n50. Leonard Bernstein to David Diamond66\n\n408 South 22nd Street, Philadelphia, PA\n\n18 April 1940\n\nDear David,\n\nForgive me if I emote. Today has been horrible, and I have been missing you strangely. April is the cruelest month. I received a wire from Dimitri that knocked my world completely to hell. I have had queer forebodings about the next year affair in Minneapolis all along \u2013 and you, sympathetic to the supernormal, can certainly understand them. Today saw their manifestation. I quote from Western Union:\n\n\"Don't leave your class for next season \u2013 some real difficulties here because of my engagement in N.Y. and one month of orchestra tour and some guest conductors \u2013 it is not wise to stop studying for a doubtful season for you here \u2013 am very awfully sorry. Dimitri.\"\n\nNow take a breath & put yourself here with me at 408 [South 22nd Street]. I have been staggering & pale green all day, fighting with my lifeblood that wants to stop coursing.\n\nWhat happens to one's Weltanschauung when the curtains fall together again? When one has been living & working according to a quasi-divine plan, setting aside certainties and doubts into one category that depends from the arm of a world-force \u2013 succumbing to \u00e9lan vital \u2013 letting oneself be driven by a reversed future? And suddenly all the bases, the category itself \u2013 the whole reason for doing all this is wiped out \u2013 an instant \u2013 equivalent to a career. Time recedes. The instant equals the career. All of Time is upon me now; I can hardly bear it.\n\nDon't think I am carrying on, please. The prospect of next year, prefaced by a summer in the Koussevitzky class, was for me the one, single motive of my activity now; every move, every note studied, project rejected, person loved, hope ignored, was a direct preparation for next year. From the scores I chose to study to the sexual life which I have abandoned \u2013 all.\n\nIt is as hard to write this as it must have been for Dimitri to send the wire.\n\nThis is all meaningless to you: but I have to think to someone else in order to arrive at any conclusions. I'm much clearer now \u2013 I think I know what I shall write Dimitri.\n\nAfter all, I could still go as pianist, unless he is really regretting a hasty decision. And again the possibility of _2_ seasons hence is not removed. One must have faith & be able to make these efforts at adjustment. One day to another is one adjustment to another. So please say nothing of this to anyone. I had to think aloud to you: somehow the rapport I feel with you is uncanny.\n\nI received yr. letter & music today just before the telegram. You will understand that I have not had a moment of open mind to look at your music. I'll do it now.\n\nWrite to me, & flights of angels attend yr Kouss [Koussevitzky] reading. I love you.\n\nLenny\n\n51. Leonard Bernstein to David Diamond\n\n408 South 22nd Street, Philadelphia, PA\n\n23 April 1940\n\nDear David,\n\nMatters are worse, & more complex. It all came out, in a letter from D[imitri] M[itropoulos] this morning that it's the union that put the monkey wrench in the works. All sorts of complicated things involving firing and hiring & local talent as opposed to imported same etc. etc. The maestro will be in NYC around the 15th so I'm going to delay returning to Boston til after that. Ergo, no need to hang around Boston. I'll try to get in touch with you in the Beeg City.\n\nPlease don't think those thoughts re Dimitri. He's no false promiser. He has an integrity that is sans pareil. He's simply up against a strong machine. I too have heard the story of the Woltmann $1000 affair (minus the ten-dollar bill aspect), & I'm inclined to believe it.67 He does things like that, & has faith in Woltmann. The man is incredible that way.\n\nIt's difficult for me even to think about our relationship (yours & mine, I mean), harder to discuss it. I too must be very careful for many reasons, all of which I will burden you with when I see you in New York.\n\nGod bless Rochester.\n\nAnd you.\n\nLenny\n\nI hope you will send me programs & reviews. Even if they don't make sense. It will fortify your presence here. Alles gutes to alle Guten, especially Roy.\n\n52. Alfred Eisner to Leonard Bernstein\n\nMetro-Goldwyn-Mayer Pictures, Culver City, CA\n\n10 June [1940]\n\nBernsteinell,\n\nReturned in triumph from San Francisco to be promptly assigned to a western for Beery68 which is directly responsible for the half dozen heads I've bitten off in the past week. All very sad and making me want all kinds of out, preferably Mexico. Great exodus to Mex. from here and the Eisner feet itch _\u00e0 faire peur_. If I could get a few bobs ahead I could do a little thinking with guts about such things \u2013 but good broke.\n\nMad about Frisco: a city of character, albeit some hilly. I walked up the California St hill and so have reason to be bitter. Squandered sixty bucks in four days and not a solitary sou regretted. Ate expansively if expensively \u2013 and sometimes not so expensively. Stumbled on a place on Front St. where you eat for two bits, and second helpings yet; as yet no ptomaine. Really, if such phrases as \"feeling, character, spirit\" have any meaning when applied to cities, Frisco is all of that. I called Ken (your letter didn't reach me until return) and Ehrman _m\u00e8re_ took a deep breath and _drove_ \u2013 quelle talker. Ken, it appears, is supposed to have taken ship at Marseilles last week, but in his last letter home (some months before) he had hinted darkly at joining the ambulance service and hanging around for the duration. So Mrs. Ehrman didn't know and wouldn't I come out? All in all great time containing proper ingredients of culture, gastronomy, debauchery and lechery \u2013 and how is your sex life.\n\nNo, not married. Lived for a couple of weeks with a would-be actress in a house (small) in the hills with all of God's California spread like a burned over rug outside the bedroom window. Temperaments clashed and I'm back in an apartment which looks, fittingly enough, like a monastery. Actress, so help me God, had a glass eye and came from Brooklyn. And the most luscious body ever I rubbed against. Something, mon vieux, to see that eye come out. No, I didn't fuck her in the eye socket, although I imagine the procedure would have had a tang.\n\nOtherwise, but little. I may hit _story_ again soon. They're making up their flighty little minds now. Been writing quite some. Started my first novel. The \"To Own The World\" epic I worked on will be released soon under the title _We Who Are Young_.69 Has been sneaked and reactions very whoopsifying. (Glossary: Whoops \u2013 to puke, to vomit under strong emotional stress due to violent nausea, colloq.) Remember when your head is being held, only about 20% of the clotted pool about your feet is my fault.\n\nWould like very much to get out and away, but the studio has a viselike grip about my economic balls. The prospect is not encouraging: you look about you and realize that literary New York has descended _en masse_ on Hollywood demanding bread and butter. Even that old die-hard, Maltz,70 is in town looking for a job and, incidentally, not getting it. They're letting them out in droves and starvation marches with banners along Hollywood Blvd. Heard of a writer last week who was reduced to living in a house with a view on only one side and no swimming pool. Poor lamb. I get around quite a bit now and think nothing of going to parties where you put ashes in the drinks of Dorothy Parker or Sinclair Lewis etc. etc. Everybody is out here. Sometimes you actually tell yourself you're having an exciting time of it. I intend to get out in the fall. I'll have some dollars five hundred, and we'll see.\n\nDistressing about your next year crumbling about your hapless head. But methinks next year is going to crumble about everybody's hapless head and plans and dreams and whims and hopes and lives will be all one lovely goulash thanks to the Stukas. Some guy I know out here leaves next week for France and the volunteer ambulance service. Wanted me to go along. I laughed for the first time in weeks.\n\nAnd so at the coming of the great noon tide my phone rings and a producer wants to talk to me about a scene for Beery that even Beery can't play, and I am suddenly sick with the thought of talking about it. Weep for me lost among the lost and write.\n\nAl\n\n53. Leonard Bernstein to David Diamond\n\n86 Park Avenue, Newton, MA\n\n[14 June 1940]\n\nDear David,\n\nTerribly sorry about that bit of automatic watercolor \u2013 it was God trying to express himself. As I remember it, the chief substance thereof was simply the cheerful fact that I had sat waiting by the phone from day to day, so to speak, and to this moment have had no word from D[imitri] M[itropoulos]. A strange silence, which is difficult for me to understand. It cannot be that the thing is over. I wrote him several days ago asking _why_ and as yet have no answer. I'm sure that when I feel there's something wrong or misunderstood that I'm not simply making excuses to myself. These things simply cannot present themselves, naked and unexplained; the human quality would then become utterly cheap, like the stationery I'm writing you on.\n\nAs things go, however, I shall be in New York next week, probably from Wednesday to Saturday, for a television job with The Revuers. There's a sizeable reward, and I can well use it this summer. The Berkshire people have agreed to pay half my living expenses as well as the scholarship, and this job, I think, will just make up the rest. Ergo, if you're anywhere around the city next week, perhaps you might call me. I'll be staying at 29 W 65; the name is Comden; the phone is Trafalgar 7\u20139719, I think.\n\nRen\u00e9e Miquelle tells me of your piece, which she liked, & of doing the Village with you. Was it fun? She is terribly upset about France, more than all of us, because her home town & mother are involved. A ghastly business.\n\nIt was Heaven when Aaron came to Boston.\n\nI want an reste, I've been working intermittently on my Violin Sonata, which I like, & which you probably won't, I'm afraid. These labors, Catiline, what with learning new scores & practising _Tombeau de Couperin_ & assorted small subjects, & trying to see all the people I should when I have no real desire to, and trying to awake from a general depression, have all kept me fairly tied up. La vie marche. La mort approche. La naissance reste.\n\nI really regret not having seen you \u2013 I wanted to \u2013 and perhaps you'll have some reason to be in the City next week. I hope so. Write me there if you can't come.\n\nBest always.\n\nLenny\n\n54. Betty Comden to Leonard Bernstein\n\n[New York, NY]\n\nWednesday, 26 June [1940]\n\nDear Lenny,\n\nEnclosed please find an impressive looking letter \u2013 and my love. I'm glad New Hampshire was on hand to welcome you and it sounds nice up there. You may or may not know that we all crashed Leonard Lyons' column in an item about Aaron [Copland] being the distinguished page-turner at the telecast.71 Adolph has spoken to Aaron and he is amused and amazed \u2013 and not the least bit angry.\n\nThere is a slight lull. Judy [Holliday] is away. The rest of us are doing some work and hoping that by the end of the week we'll be able to take a day or two off too. A dull letter, I know \u2013 but I'll write soon again \u2013 and I hope I'll have something to _enclose_ as well.\n\nLove,\n\nB\n\nDo you want those lovely slacks? \u2013 and jacket?\n\n55. Leonard Bernstein to Ren\u00e9e Longy Miquelle72\n\nHanover, NH\n\n1 July 1940\n\nCh\u00e8re Mme,\n\nLunch is calling, and I have but a moment to write \u2013 the television venture being over, I am safely ensconced in this charming but dull college town with the Silvermans73 (or should it be Silvermen?) and working, actually. I see nobody, but lead a quiet, useful and pleasant existence. I've already learned Beethoven's 4th and _Scheherazade_. Starting Copland's _Music for the Theatre_ today. Practicing. Composing. The fiddle Sonata almost pr\u00eat. Leaving for the Cranwell School, Lenox (my next address: please write) on the 5th, probably. Aaron told me that I might have to conduct Randall Thompson's Symph. (No. 2, naturellement) the first week, since that's when Randy will be there. Kouss, in searching around for the person to do it, suddenly said to Copland (so goes the tale) \" _your_ Bernstein!\" I don't know.\n\nNothing to do but pray. And perk up about the abroad situation. It's getting exciting now; Russia, it seems, is going to have a lot to say about what Germany does, soon, and forcibly. Again, we can only pray. [...]\n\nLenny\n\n56. Betty Comden to Leonard Bernstein\n\n17 July 1940\n\nLenny dear,\n\nKouss may have been impressed by your conducting \u2013 but his feelings cannot compare with mine. I'm thrilled at the thoughts of your concerts and I absolutely will make it my business to get up to see you somehow this summer. But actually conducting! And after all that silly fretting over whether or not you'd memorize _Scheherazade_ in time. [...] It's wonderful about the conducting and the summer sounds magnificent for you. [...]\n\n57. Leonard Bernstein to Kenneth Ehrman\n\n17 Lake Avenue, Sharon, MA\n\n[August 1940]\n\nKen,\n\nI was very desolated that your visit was so short \u2013 it hit me afterwards that I hadn't really seen you outside of some small talk & some big talk forced in somewhere to give your return significance. I wanted so to reestablish us again \u2013 & then you left. I find you, thank God (?) very much the same Ken, the most pleasant article to be with I've ever encountered, cool, and unimpressed by most superficial things, more impressed than he will admit by the basicker things. I was touched by your reaction to the hundreds of busy little Tanglewood bees: if it caused the slightest stirring up in yr creative being, I feel a Messiah, indirectly. Hast du ein eingiges Wort geschrieben? Is Palo Alto? Does your family intrigue you any more? You should keep away from it. It's a kind of monstrosity anyway, as you will admit \u2013 but you can't live in a sideshow.\n\nCome east, where I can see you often.\n\nTanglewood was a complete success. Where did you leave? \u2013 yes, at the Bach, which was done standing up & [Putnam] Aldrich playing the harpsichord. The performance was an ode to Viola Wasterlain. _Scheherazade_ was wonderfully exciting, despite some bad slips from the solos, & there followed the Haydn _Symphonie concertante_ , the Brahms _Haydn Variations_ , Copland's _Outdoor Overture_ (at the Allies Benefit!), & a performance I wish you could have seen & heard of Stravinsky's _Histoire du Soldat_ with my own words (local color) served up as a surprise for Koussie on his terrace at a tea he gave for the school. A hit. Kouss is greatly impressed wants me to study with him in Boston this winter, if he can get an orchestra for me to work with. I'll know in a few weeks. Write, & spare no gory details. I'm with you til the plane wheels flaming to the Japanese (Chinese, I suppose it shd be) soil \u2013\n\nLen\n\n58. Leonard Bernstein to Aaron Copland\n\n17 Lake Avenue, Sharon, MA\n\n[August or early September 1940]\n\nAaron, foremost of men,\n\nWhere are you? And if so, why no word? You said you'd write, according to Green. Not seeing you is something of a shock, you understand. The summer was a revelation in that regard. Neither of us (I hope) tired of the other (I had feared you might) and I came, in fact, to depend in many ways on you. I've never felt about anyone before as I do about you. Completely at ease, & always comforted with you. This is not a love letter, but I'm quite mad about you.\n\nMight Yaddo on Sept. 7 & 8 be interesting?74 Are you planning to go? I was thinking of upping to Lenox next week or so to see the Kouss. Perhaps I could combine both. Write fast & let me hear. Best to Victor.\n\nLove,\n\nLenny\n\nP.S. I've finished the Fiddle Sonata, &, by God, there's something about the ending that's wonderful \u2013 almost mature. I want you to see the whole thing now \u2013 I like it better.\n\n59. Leonard Bernstein to Serge Koussevitzky75\n\n17 Lake Avenue, Sharon, MA\n\n[before 5 September 1940]\n\nDear Dr. Koussevitzky,\n\nWords are a remote enough medium of expression for any musician, but it is especially difficult for me to find words for this letter. Let it be brief.\n\nThis summer to me was beauty \u2013 beauty in work, and strength of purpose, and cooperation. I am full of humility and gratitude for having shared so richly in it. These last six weeks have been the happiest and most productive of my life. I have been able, for the first time, to concentrate completely on my main purpose, with a glorious freedom from personal problems.\n\nIt was a renaissance for me \u2013 a rehabilitation of the twisted and undefined Weltanschauung with which I came to you.\n\nFor your creative energy, your instinct for truth, your incredible incorporation of teacher and artist, I give humble thanks. Seeing in you my own concepts matured is a challenge to me which I hope to fulfill in your great spirit.\n\nI am now at home, resting with my family. I hope to be in Lenox within the next few weeks, and I should very much like to see you and talk with you. Can you let me know when this would be best for you?\n\nPlease give my very warm greetings to Madame Koussevitzky, and to Miss Naumoff.\n\nIn devotion, and in gratitude,\n\nLeonard Bernstein\n\n60. Serge Koussevitzky to Leonard Bernstein\n\nLenox, MA\n\n5 September 1940\n\nDear Leonard,\n\nThank you for your letter.\n\nNothing could have made me happier than to know that your work this summer has really given you beauty and strength and a better understanding of the gifts with which nature has endowed you.\n\nI shall be glad to see you sometime during the middle of this month, let us say Tuesday, the 17th, or Wednesday, the 18th, \u2013 and I shall look forward to your coming to Lenox.\n\nMy best wishes are with you always.\n\nSerge Koussevitzky\n\n61. Leonard Bernstein to Serge Koussevitzky\n\n86 Park Avenue, Newton, MA\n\n30 September 1940\n\nDear Dr. Koussevitzky,\n\nAs I sit and wait for the outcome of your plan, in a kind of Proustian twilight state between knowing and not knowing, between sleeping and waking \u2013 in the midst of all this I have had an inspiring idea. It would have to have \u2013 and I pray it will have \u2013 your support.\n\nI have met one or two of the people who have been conducting small orchestras in Greater Boston, and I have been singularly unimpressed \u2013 or rather, singularly _im_ pressed with their lack of equipment. It occurred to me that if they can get orchestras of young people, perhaps I could. And with your support, almost certainly.\n\nIf you are unable to establish connections with the representatives of Backward Boston, don't you think it would be wise for me to attempt the organization of a young orchestra? I am sure there are many instrumentalists in Boston who would be glad of orchestral experience; if you liked the idea, we might even establish it as a kind of training orchestra for the Boston Symphony. If these young people knew you were behind it, I am sure they would rally to the cause.\n\nThe problem for me is to make contact with these people. Again, if you could speak to the men of your orchestra, they might be willing to send their pupils to this orchestra. I realize the responsibility I would be shouldering, but I do it only under the influence of your spirit which still hovers around me. I could then work with an orchestra (which would derive great benefit from their association with you) and still be here to work with you this season. Please don't think me presumptuous; I am just making a great effort to be practical.\n\nPlease try to get some rest before the season. I am sorry to intrude on your privacy even with this letter; but I am made bold by my recent reading of Nietzsche, who teaches me that I must be somewhat bolder if I, like his Zarathustra, shall ever face \"the great Noon-Tide.\"\n\nIn eternal devotion,\n\nLeonard\n\nWarmest greetings to Madame Koussevitzky and Miss Naumoff.\n\n62. Leonard Bernstein to David Diamond\n\n2122 Walnut Street, Philadelphia, PA\n\n11 October 1940\n\nDear David,\n\nI am (O wrest the power from the powerful) in Philly. Nay, living here. There has been a commotion in the diplomatic heavens76 and I, O fearful pawn, was set with a sharp click in Rittenhouse Square. I'll tell you all anon.\n\nWhich means that I shan't be in Boston when you are. God bless the Sat. night concert, & have a good burlesque show.\n\nI'd be incredibly happy to do the NMQR recordings. I _am_ \"serious\" about it, and very flattered that you should still want me to do it. So write, & set some dates, voice the stipulations, & I'll pop out as from a pigeon-hole. Mais l'important, que tu m'\u00e9crives, et cela bient\u00f4t.\n\nLenny\n\n63. Leonard Bernstein to Kenneth Ehrman\n\n2122 Walnut Street, Philadelphia, PA\n\n15 October 1940\n\nCher Kenneth,\n\nIncredible that you put up with what you put up with what you put up (with?). Everything that constitutes a means of improvement for you is at your disposal, except, apparently, the initial desire. That's available, to, at a reasonable fee. You're not lazy. Why don't you investigate?\n\nI couldn't imagine what you were doing at Lafayette, Ind., but I wrote you there, & no response. Of course, I wrote you again at Palo Alto (first sending it to Box 817, Palo Alto, 'steada Menlo Park). By the way, did you know there was Menlo Park, New Jersey?\n\nBut having forgotten what I said to you in that Odyssean letter, I may be guilty of repeating myself. En tous cas, comme tu aper\u00e7ois, sans doute, ci-haut (that sounds wrong) I am in Philly. I don't know; I never asked to be here. Something makes it inevitable. I may have told you that Kouss had great plans for me which involved my staying in Boston but they were given the K.O. by the Hon. Curtis Institute, which objected strenuously at (read: to) student-swiping, & vowed to discontinue all relationship with Kouss & his school if I didn't return. Matter of ethics, don't you know, setting a precedent, don't you know, etc. Reiner was furious. He had seen me referred to in printed items as Kouss' pupil \u2013 no mention of _him_ (antecedents again; means Reiner). So me voici, & lucky to be, O misery me, ta da ta da. But I've got a magnificent room with a double bed & massive mahogany furniture, & the school is doing almost all the supporting. Therefore, easier. Therefore, out goes Mexico, & Kouss, & you, & Cambridge, & California. Nothing left for you to do but come here. Please try; there are some things you might profit by that you haven't yet seen. By \"here\" I mean East, not Philly: I'm in this city only because Koussie realized that he'd be losing an A-1 customer in Mrs. Bok if he didn't kowtow. So I became a fearful pawn ([Edna St Vincent] Millay) in the hands of wily diplomats. Write me very soon.\n\nI to my naked spinet in your corner.\n\nLenny the pawn, & Penny the prawn,\n\nand Henny the lawn, & Jenny the spawn, & Renny the griswoldforlawn.\n\n64. Alfred Eisner to Leonard Bernstein\n\nMetro-Goldwyn-Mayer Pictures, Culver City, CA\n\n[?October 1940]\n\nLennie, old mole,\n\nI have just finished reading Hemingway's new novel from the galley proofs and am still a little breathless.77 A master stroke managing to get hold of those proofs, involving larceny of some proportion. But she's a novel, old gum drop, a book of infinite insight and agony and the soul of a man hovering halo fashion over the brow of his body while he kills with cold savagery. Writing that positively _gooses_ you! Hemingway saw Spain cameo-clear; and his book is just two years \u2013 two? Four! \u2013 too late and its anger will cause not a ripple in the hysteria of warmaking. 'Twill be just a good novel by Hemingway about something out of ancient history. The people who will realize what it is he is saying will already know, and there it will end.\n\nWhere have you been, old sausagefoot? Aeons have whirled their course since that card from Tanglewood. Not a word. I suppose by this late date you have a devoted slave in the Kouss. I thought as much. Indirect word of you from Austin, who described you as \"forging ahead\" and deciding with some acumen, I think, that he (A.) and Kenneth needed a kick in the pants. Back at Curtis? What? All the days crammed to the brim that I must know about. Successes. Quiff. Friends. Plans. Already written, in the writing, to be written. Prospects. Aaron. On and on, and I want to hear about it all. So to it, Rosinante, to the road again; get thee pants and write me a letter that will consume at least a morning of MGM's time in the reading. Trust me to kill the afternoon.\n\nI won't even mention the draft. It's just too goddamn funny to even talk about. One thing: what are you going to do? Me, I can't wait until the army makes a man of me. Can't decide between the air force and fighting my war toying with some secretary's breasts. Simply _can't_ decide. We live in parlous times, halvah, old boy, very. Dies irae, dies illa, solvet saeclum in favilla. But yes.\n\nHattie arrives in some two weeks to take over not only my care and feeding but also the not inconsiderable job of delivering me from the financial toils. I make enough money for three men, and am a pauper, yea, I sit among the ashes and for garments sackcloth. I'm tooting madly about looking at houses. To date, nothing I like. _Must_ find something within the week.\n\nOf my tenuous existence, trivia. The Eisner soul grows small and curls inward like the anemone to confess that he has ghosted two (I must be very tired or something) jobs for a quickie studio, for the gain of silver.78 One, a mouse called _Thirty Boys and a Girl_ (subtitle: The Gangfuck) has just been finished. Shooting time: six days. Budget: 10 grand. So you can imagine. No, you can't imagine. But the banks honor their checks, if you worry them into giving it to you. Working here at Metro with a couple of German Jew writers whose English is even worse than their script which is negligible. We communicate in mangled French, English and German. Nice people though: one an outspoken anarchist. All for the assassination of L.B. Mayer (after the expiration of his contract, of course). The other is a gourmet who drags me about to restaurants of stature and flies into tantrums if the b\u00e9arnaise sauce is not up to his idea of sauce b\u00e9arnaise. One night he insisted on going to the kitchen and bawling out the chef. I was sure he was going to get a meat cleaver in his head. Nip and tuck there for a minute.\n\nTaking a course in the novel given by the League of American Writers school here, more as discipline to make me sit down more often to an already started novel, than in the hope of learning anything. Some very wacky people in that class: I expect murder before the month is out. And River79 ( _The Torguts_ ) to have nervous prostration. Also a new story of mine will appear in _The Clipper_ , a literary monthly put out by the League pretending to some excellence. Be in excellent company anyway: Dreiser, Meyer Levin, Belfrage, others. I think every writer in America is either here or on his way. You meet them all.\n\nIntended coming home this month, but Mother Metro like the python has me in her coils. In the spring. Letter from Ann and Nathan today: they desire word of you. And remember, it's _Keats_. So heavyhearted, I go home to my cell and it's another day. Write, sluggard, hear the voice crying in the wilderness and write.\n\nCon brio,\n\nAl\n\n65. David Diamond to Leonard Bernstein\n\nYaddo, Saratoga Springs, NY\n\n29 October 1940\n\nDear Lenny,\n\nHenry Cowell80 forwarded your little \"critique\" to me, asking me to get some definite plan worked out for the recording. Naturally, I don't feel the way you do about the 3rd fugue and neither do I approve of a series of preludes. If I had wanted that, I would have written them without the fugues. And Lenny, perhaps your kind of musicianly temperament will be the kind that succeeds best because it turns hot and cold easiest, but in the long run you will find your own way of treating music (the way John Kirkpatrick does by the way, but instead of saying \"dull\", he says, \"quasi-baroque\") to be the merest surface glazing. It seems incredible to me, that in this short time, you can already pronounce so dark a verdict on the several fugues, when I'm sure, knowing your high-pressure endocrine system, that you could hardly have spent much concentrated effort on them since Cowell got the music back to you. I can only say that I believe in the 3rd Prelude and Fugue whole, that if the fugue to you seems dull, it is like X telling me much of the _Art of Fugue_ and the _Great Fugue_ is dry and paper-music. The art of counterpoint is a true art. It has to be realized before one can say things pro or con. And to realize the 16th century masters, much of late Bach and Beethoven, the Stravinsky of _Persephone_ and the _Symphonie de Psaumes_ , the fugue from Bart\u00f3k's _Music for Strings, Percussion and Celesta_ , we must first know the notes so well, the line so accurately and the nuances so perfectly and rightly proportioned, we should be able to reconstruct the works ourselves. You will say, but I am no sixteenth century master, Bach, Beethoven, Stravinsky, or Bart\u00f3k \u2013 but my aims and purposes are the same, my deep compassion for the past the same and my belief in the future the same. Lots of Reger may be dull, but not this baby. I've learned when to stop the machine in time of crisis! When you employ the word _dull_ to the 3rd fugue, you are simply failing to unmask the secret character behind the piece.\n\nAs Cowell seems very pressed, I promised I would write at once ill or not. I've been in bed over a week with a serious streptococcus infection. This is really my first day up. But back to the recording: no, I don't OK the prelude idea \u2013 sounds too easy \u00e0 la Bernstein. The original plan is the one I hold to. If you feel the matter much too taxing to go through because the fugue is dull, just write and tell Cowell so. I'm getting more and more used to this kind of thing. There are fewer and fewer kindred spirits left each year. By the time this war is over, there will be none. If you take yourself by both your shoulders, for a change, work the notes carefully, you'll find the fugue will grow quite rapidly inside of you. Let me know what you decide. And if you go ahead, I'll send you $2 with which to make a test recording for me as I will not be able to come to NY. Make it in Philadelphia at a reliable recording place and send it to me, and I'll write right back about tempo etc. All good things ever, and let Cowell know at once what you've decided.\n\nDavid\n\n66. Leonard Bernstein to David Diamond\n\n2122 Walnut Street, Philadelphia, PA\n\n[after 29 October 1940]\n\nDear David,\n\nI was shocked by your letter. I'm afraid you misunderstood. I intended no criticism of the music _per se_ , but simply referred to the probable reaction of a record audience. I suggested that the 3rd fugue, unpianistic and unrelieved as it is, might be an unfortunate choice with which to introduce yourself via recording. If you want it, then certainly I will be glad to do it.81\n\nBut why the maleure? I understand that you've been ill, and down, and probably out, and kind of out of the world, but, Lord, David, \u2013 \"too easy \u00e0 la Bernstein\" \u2013 \"take yourself by your shoulders for a change\", etc. etc. And I thought you knew me better than to intimate that I would make superficial dicta about music. Believe me, I know what the fugue is worth. I can list for you all the fine points \u2013 your achievements in it. But there are \"stains\"; your second stretto, for instance, is anticlimactic because it is a four-measure stretto, whereas the preceding one was a _one_ -measure stretto. This is especially true of a subject in unrelieved half-notes. Again, you speak of nuances to be mastered thoroughly \u2013 but you haven't _one_ in the piece except the opening _ff_! From my point of view there must be a dynamic growth \u2013 involving especially a drop to _piano_ in the 17th measure, to rise to the final climactic stretto, & possibly the same thing again (modified) before the second stretto. Write me what you think of this. And is [ = 63 strictly to be maintained throughout?\n\nI shall make a test record as soon as possible & send it to you. Let me know about the above very soon. And please keep well, & somewhat happier.\n\nBest,\n\nLenny\n\n67. David Diamond to Leonard Bernstein\n\nYaddo, Saratoga Springs, NY\n\n5 November 1940\n\nDear Lennie,\n\nI am happy that you have decided upon the fugue too. As I look over the letter you sent to Mr. Cowell, it is true that you were thinking of recording purposes, but all the same _that_ word is there thus: \"The third prelude is good but the fugue is _dull_...\" etc. \u2013 and to me that still pertains to the music, recording or not. I won't argue the point further, for we seem to be beyond that now. Furthermore, I think you should have realized by now that I'm pretty stupid on consideration to be given the purely commercial aspects of music. Where public relations are concerned I simply don't function nor want to. And whether one makes the right start in recording works by choosing the best to represent the composer on discs doesn't interest me either since no one cares how a composer is introduced or in what order. It's the music that counts in the long run. Where Aaron for example should have made his bow on discs with say, _Music for the Theater_ or _Hear Ye, Hear Ye_ , he made it with the _Piano Variations_ and _Vitebsk_ and that would seem wrong but nevertheless that is how he began and it makes no difference to anyone. People still hate the _Variations_ and eat up the _Sal\u00f3n M\u00e9xico_ , alors? So don't worry your pretty head about the introduction part of it; \"c'est la musique qui compte\" as [Nadia] Boulanger always said to me when I would pose such questions as you have brought up. Truly, dear Lennie, I don't care a double fuck for anything dehors de la musique. You know that, why worry about it, no one else does, time takes care of all the necessary sifting out.\n\nNow about the music: I don't agree about the drop to a _piano_ at the 17th measure, lawsy no! After the full strong half note pulse with accents and the general _ff_ , it would only appear as an affectation! I should have right in the first measure put _sempre_ after the _ff_. It should be _ff_ throughout and always a quarter to 63 with the natural push forward towards 70 circa which happens by itself. It must be very sustained, very bold, the notes will provide the necessary _espressivo_ I believe. The chord quality at measure 17 is sufficient to stir emotionally anyone capable of being touched by the chord without changing the dynamics. You mistake my meaning of the word, nuance; it does not mean dynamics \u2013 I'd rather it meant conception and feeling for the natural flow and modulation of the whole musical line, _chiaro_? I'm not worried in the least, I know you'll do a beauty of a job. And if you can keep a secret, a venture is afoot which will mean much for both of us if it works out with Victor. I enclose $2 for the test recording and as soon as I've heard it, I shall let you know my feelings.\n\nWhy let us start a series of polemics? You say I should know you by now, I can hardly agree with that. You never allowed me to, ever. I tried in so many ways to know you better, looked forward to seeing you in NY, but you didn't come last spring, came to Tanglewood to see you but got a cold shoulder [...] At Yaddo [...] I was simply hurt by the strain you forced to exist in not being open enough about your true feelings. It would have been far better to have said, \"David, understand that there can be no physical relationship between us, therefore I'd rather not stay with you tonight.\" [...] It's all right though now. Nothing matters much right now.\n\nAlways,\n\nDavid\n\n68. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n6 November [1940]\n\nDear LLLLLLL,\n\nYour latest burned the end of my fingers. The next one will probably burn up en route. All I can say is that you've made rapid progress. All that's missing is for you to fall in love with a hermaphroditic goat... or sumpin'.\n\nI vaguely remember the Mr. Nelson you mentioned. It was two years ago on the way to France. It was his great moment \u2013 just at the age when the world seems too too wonderful, and he [was] sought after by all the most desirable creatures, a different bed every night, etc., etc. That type is the worst after the first fresh glow of youth is gone. All his stories are rot, of course.\n\nThe enclosure by D[avid] D[iamond] is also sumpin! The less said the better. There is a frighteningly dumb and humorless streak in the boy.\n\nI'm up to my neck in fussing over the new loft. V[ictor Kraft]'s dark room is sumptuous, but all my things are strewn to the four corners of the joint. It's awful \u2013 I'll never move again.\n\nDon't forget to listen in on Sat. at 9:35.82\n\nLove,\n\nA\n\n69. Harold Shapero83 to Leonard Bernstein\n\n\"XI\u2013?\u20131940\" [November 1940]\n\nDear heart,\n\nYou're full of shit. You're full of shit because you're exuberant, and exuberance is not especially welcome at this moment. The world and I don't need exuberance, what we need is revolution. Especially the world. Especially me. Especially me. And the world too.\n\nFirst, I have to say, and not with any obligation, but with sincere co-jubilation, that I'm glad for you and all the successes you mentioned. I'm glad for every every weentzy one of 'em. You're doing the right things in a big way. (Funny, that's what you always told me.) And you're making money too. _'Sis dir gut!_\n\nWalter84 thinks Diamond stinks, and what's good enough for Walter is good enough for me. It's always been a mystery to me (on the basis of the stuff I've seen, and I've seen enough to get an indication) how the hell Diamond's gotten as far as he's gotten. Honest to God, Lenny dear, David Diamond is a bust as far as I'm concerned. For his own sake I hope he's better than I think he is.\n\nTell me about how Charlie Demuth was a tragic figger.85 I don't really know his biography. Nobody seems to.\n\nWhat was your draft number? 800086 I hope I hope I hope.\n\nI saw a Georges Rouault show at Boston's hoi polloi Institute of Modern Art. The guy's a great, awfully great, painter, even though he believes in God, and is fanatic about it. To be corny, the sheer magic of the man's textures (great gobs of paint) and the magic of the man's facility for expressionism, Jeez, terrific.\n\nThe curious tone at the beginning of this letter wasn't for nothing. I will now list mes calamit\u00e9s:\n\n1) Do you remember the slow movement of the quartet? Well Walter, and finally I, agreed that it stunk. So, with Walter egging me on I kept going. It still stunk. It has now been thrown away.\n\n2) I had to copy my Overture (why, I don't know) and as I copied I got an awful feeling. You, Bernstein, don't realize how silly and false that piece is.87\n\n3) I started a fast quartet movement that was gonna be great stuff. I took a look at it two weeks ago: the sterilest, lousiest, beatest, etc.\n\n4) Since I passed in my thesis title one (1) day late the Committee on Honors of Harvard Univ. has refused to let me write a thesis. So: I can't get honors. So: I can't get a fellowship.88 So: I have not written a note (except harmony & strict cpt. exercises) for a month and I'm not going to for at least a year. Probably more. So: I'm enrolling in the Museum School after graduation and I'm gonna learn how to paint & draw. At least if I can't use my head I can use my hands.\n\n5) I'm in love with a New Yorker who's in love with someone else. And I fell in love with her because she said things like \"Jeez, this guy's (me) got the terrificest vocabulary, all the way from A to Beat.\"\n\n6) I could go on forever, I got millions of 'em.\n\nGoodbye, be good, have a Scotch on me.\n\nSonny\n\n70. Alfred Eisner to Leonard Bernstein\n\nMetro-Goldwyn-Mayer Pictures, Culver City, CA\n\n[?1940]\n\nDear Lennie, mon vieux, mon vieux,\n\nAt long last opportunity offers itself to look up from this verdammter desk and inform the friends de ma jeunesse that I am definitely not dead. Have been working day and night and night and day on a yarn scheduled for Bob Montgomery having to do with the Jack the Ripper slashings and a guy who achieves a change of personality as a result of suspicion accidentally falling his way: a very choice assignment and the first story I have anything to do with that excited me at all. Picture to be produced by one [Seymour] Nebenzal, a refugee cinemateer who did _M_ and _Mayerling_ among others. A very intelligent gimmick and we have an orgy making ourselves understood in mangled English, French and German. Believe it or not, progress manifests. Neby is of the old German school of thought that reasons something like this: if I can work 20 hours a day, why, please tell me why you can't work 15? So you work 15 or better. I whacked out an 85 page treatment in exactly four days and nights, severing myself from all matters earthly and living the life of a hermit, yea, a veritable anchorite. Maybe this one will pay off. Please God.\n\nItem: have a new car, a most spectacular 1933 Plymouth convertible coupe that is doing its very able best to bleed me to death. To date: new brake re-line, clutch plates, oil filter, new floormat, motor tuneup \u2013 and I've had the car only about a month. Really, it's in grand shape and a swell buy and I drive it all over hell and gone desertward, seaward, and mountainward. Eisner discovering California. However, I borrowed dough from the studio to pay for the car and they nick me every week for a payment, bills, expenses, so for a change I'm eternally broke. But I live, and not too badly. At long last beginning to make friends, good people, and I don't go as batty from loneliness as I did. Getting quite a lot of work done of all descriptions. Have a new girl: a rabbi's daughter, praise Gawd. Hi-ho-methusalem, etc. Fucks like a jackrabbit, and cooks wonderful goulash, a duality of accomplishment not nearly as unimportant as it sounds.\n\nOf else, but little. Time and tide and flux. Much rain: the Pacific in perpendicular lines, hills washing down, flood. California. Write of yourself and that without delay.\n\nEwig,\n\nAl\n\n71. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\nFriday [?1940]\n\nDear Pupil,\n\nWhat terrifying letters you write: fit for the flames is what they are. Just imagine how much you would have to pay to retrieve such a letter forty years from now when you are conductor of the Philharmonic. Well it all comes from the recklessness of youth, that's what it is. Of course I don't mean that you mustn't write such letters (to me, that is), but I mustn't forget to burn them.\n\nYou were right about the chuckle \u2013 but it was a very sympathetic one. Actually, when I opened your letter I was worried that something had gone wrong. If it's any consolation, things like that incident can sometimes turn out very wrong indeed. (That's Lecture No. VI.) However, I reluctantly admit that they sometimes turn out very well. You takes yer chances \u2013 but I'm not sure you're in the proper mood. Anyway, I should have liked to have taken the first train down there to investigate the \"situation\", but I controlled myself. [...]\n\nI'm a little busy with the new loft \u2013 fixing it up, etc. When are you going to find a pretext for another visit. The last one made a deep impression.\n\nRegards to the blue hat. But be careful!\n\nA\n\n72. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\nFri [Autumn 1940]\n\nDear L,\n\nGlad you're settled OK. I'll remember the invitation!\n\nNothing new here \u2013 except Virgil was made critic of the _Tribune_ which is positively flabbergastious.89\n\nSaw Paul Bowles. He says he is going to Phillie for 2 weeks when _12th Night_ goes there, which should be in another few weeks. So look him up. Added 2 more ink pages to the Sonata.\n\n[Robert] Weatherly90 came over to try out _Quiet City_. He played it OK but it still should be changed.\n\nI'm lecturing in Boston on Dec 12. Will you be there then or is that too soon for Xmas Holydays?\n\nI dream about you frequently.\n\nA\n\n73. Leonard Bernstein to Aaron Copland\n\n2122 Walnut Street, Philadelphia, PA\n\n[?December 1940]\n\nDear Aaron,\n\nChi[cago] is vergangen! Wieder zu Hause, & a little bit glad to get back to even a semi-normal life. All-night vigils both ways on Chicago trains are not my speed. But the week was fine. Thompson A-1, expenses almost nil, much opera, rehearsals & performances, ballet. [...] I missed _Billy_ [ _the Kid_ ] by a week. The Kurt Weill was really exciting. _Rosenkavalier_ is puffed up, but has extraordinarily beautiful passages. Reiner is a genius. Music is a hard profession. All this have I gleaned, O richer I, from a week in Chicago! [...]\n\nTo work aussit\u00f4t que possible on the _Saloon_. I reel at the thought of royalties. (Isn't that split, by the way, another typical gesture?) I accept $25 of course, beggars can't etcetera, but should they know about the royalties? If not I'll shut up. Am I being a pig in taking them (assuming there will be any?).\n\nI'm afraid I'll still be in Philly on the 12th, but my love to Koussy\n\n& to you,\n\nLenny\n\n[...]\n\n74. Leonard Bernstein to Aaron Copland\n\n2122 Walnut Street, Philadelphia, PA\n\n10 December 1940\n\nDear Aaron,\n\nShipped off the _Saloon_ to you today to dispose of at your convenience to Heinsheimer (sorry, but I'd lost the Boo[sey] & Haw[kes] address). Hope it's good enough. Look especially at the rather turgid & theatrical _ossia_ at the end of the slow middle section, & if it gives pain simply cross it out. I did it only because there had to be some theatrical interest at that point (which is, I'm afraid, a bit dull even in the orch.) Don't take it too hard.\n\nI'm desolated that I can't be in Boston with you, but I'll ring up en passant through NYC.\n\nAt a cute performance of _Boh\u00e8me_ tonight I ran upon my painter friend Zeil (recall?) who, it seems, has been sick with a housemaid's knee variety of arthritis. Gruesome. I am to see him this week. I quail; I suspect syphilis.\n\nHow do you do these days? It seems aeons since I heard from you. Literally. Write soon, very.\n\nLove,\n\nLenny\n\nVolevi dire, bella come un tramonto...91 very nice libretto.\n\n75. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\nMon. [16 December 1940]\n\nDear Lennypenny,\n\nI suppose we'll be crossing letters. But anyway \u2013\n\nThe piece came and I've been sweating my whatchacallits off ever since trying to put it in shape. Your idea of a manuscript \"ready for the printer\" is to weep. I'm preparing one of my best lectures for you on said subject. But when are you coming through?? We need a couple of hours to talk over several points. What I'm doing now is mostly crossing the T's and dotting the I's. But I don't want to hand it over the Heins[heimer] until I've seen you.\n\nWas up Boston way and saw that screwball Shapero. Also met John Lessard92 there, who turns out to be as nice as his Piano Sonata.\n\nPlan to stay over a day or two so that I can get a good look at you.\n\nIt just occurs to me \u2013 _of course_ you'll be coming for Dimitri's debut on Thursday.93 Or do I err? [...] If I guess right, let's have supper zusammen, just to start things properly.\n\nA\n\n76. Leonard Bernstein to Aaron Copland\n\n[Philadelphia, PA]\n\n[December 1940]\n\nAaron Excelsus,\n\nNo, our letters didn't cross, because I couldn't decide whether to cut the President's (Mrs. Bok's) Christmas Party on Friday night, plus a class or two, to be in New York Thurs. night. Lord knows I would have loved to. I had planned so hard on it. But I've been persuaded, & duties is duties, and I must stay. I'll be in NYC Saturday afternoon, probably around 2 or 3 o'clock. I'll call immediately (if not OK let me know like mad) \u2013 we can operate on _Saloon_ on Saturday & I can hear Dimitri on Sunday, & be home Monday. Damn the Christmas Party anyway.\n\nYou sadden me infinitely about the _Saloon_. I thought I had it done. Heinsheimer has been clamoring. I tremble at the thought of your lecture. See you Sat.\n\nLove,\n\nLenny\n\n77. Kiki Speyer94 to Leonard Bernstein\n\nWednesday [?December 1940 or January 1941]\n\nMon pauvre petit chou,\n\nI was so sorry to receive such a sad letter from you \u2013 and I wish I could come to Philly for a few days to cheer you up. However this is impossible as I have very recently come back from New York. I went with the B.S.O. and had a simply wonderful time. I saw your sweet friend Dimitri (what a marvelous person). Also Aaron, Chasins, Hindemith, Szigeti etc. I was wined and dined to my heart's content and came back to Boston no longer quite so depressed. Friend Stresemann95 I also saw quite often!!\n\nWhat a shame you are unable to go off on a \"toot\". You need it, Leonard dear, and I mean it seriously when I say that you drive yourself too hard. At least if you keep yourself as busy as you did in Tanglewood and here when you were on \"vacation\". Do be careful \u2013 you've had one bad cold and now la grippe est everywhere, so button up your stunning new overcoat!\n\nThere is little news \u2013 concerts, rehearsal, lessons and practising keep me fairly well occupied. A few dates sprinkled in add zest \u2013 yet like you I am sad, and, my pet, it is the war. I think of it so often although to what avail??\n\nKouss is on vacation for three weeks in the Berkshires. He had a cold (yours no doubt!!) and felt rotten in New York. His Shostakovitch had a marvelous ovation in Carnegie.96 Even I was so thrilled when I heard it \u2013 the chills ran up and down my spine! And Haydn...\n\nWords of sympathy come to my pen with difficulty but be advised that I fully realize your loss.97 I often think of you and hope that we will see you soon. Mother sends her love and a big kiss. Dad and Andr\u00e9 also send their best.\n\nLove,\n\nKiki\n\n78. Leonard Bernstein to Aaron Copland\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n8:45 p.m. [January 1941]\n\nAaron,\n\nI don't know quite what I'm saying \u2013 but just a word to tell you I didn't elope or get killed last night \u2013 just drunk. In fact, I slept in the Empire from 7:00 to 10:00. Didn't you get a message to call me?\n\nI missed the funeral & spent all day chez les Eisner.98 I am at the moment a half-crazed mystic. Forgive last night's disappearance act.\n\nExpecting you in Philly. Write exactly when.\n\nLove,\n\nLenny\n\n79. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\nTuesday [January 1941]\n\nDear L,\n\nAll terribly dramatic wasn't it? I felt so sorry that your vacashe turned out so hectic. I'll be curious to hear the full story of the last night and day (I got a message to call 511 \u2013 Mr. Green \u2013 never dreaming you were there \u2013 and I just waited too late before taking the pain to call.\n\nI'll be down on Monday. I'll take the one o'clock train. Will you be home after 2:30? If so we can have an hour together before I start my duties: Around 3:30 at the Free Library, at 5 for interviews, and at 7 for dinner at the Art Alliance. (Ain't it awful.) I won't be staying over \u2013 so may I use your room as a place to change in?\n\nHeinsheimer says the proofs of _El Sal\u00f3n_ are being sent me \u2013 so I may bring them with me if there are any changes needed.\n\nI feel even closer to you after all that's happened.\n\nLove,\n\nA\n\n80. Leonard Bernstein to Helen Coates\n\n[10 January 1941]\n\nDear Helen,\n\nForgive me for not writing sooner \u2013 I have been in a messy state of mind. Last Saturday I lost one of my closest friends \u2013 Alfred Eisner. I'm sure you remember him as my roommate for two years. A malignant tumor on the brain \u2013 and it was all over. Just as he was about to reach the top \u2013 with the world before him, wonderful jobs in his hands, and 24 years old. I've been completely numb ever since: something seems to die in you, & refuses to accept a fact as cruel and unjust as that. Coming so soon after the death of David Prall, it makes me very leery of world values \u2013 so much so that at the present I find it difficult to consider anything important.\n\nPlease write soon \u2013 I know you will understand the brevity of this letter.\n\nLove,\n\nLenny\n\n81. Helen Coates to Leonard Bernstein\n\n66 The Fenway, Boston, MA\n\n13 January 1941\n\nLenny dear,\n\nI found your letter here less than an hour ago but I'm so shocked over the death of your close friend, Alfred Eisner, that I must write you at once. My heart goes out to you in the tragic loss of your very good friend, and I grieve to think that such a brilliant young person should be taken from this world. Tho I never met him, I, of course, do remember him as your roommate, and often heard you speak of him. And then when we had dinner together in New York in late November, you went on that evening to the hospital to see him. I never thought to inquire of him afterward as at the time you thought he had some sort of a throat infection, nothing serious.\n\nDeath is appalling enough at any time (except in the cases of very old people or those who are seriously incapacitated) but at your age the death of anyone so close to you strikes a particularly hard blow. I feel the keenest sympathy for you, for I had so many experiences, beginning when I was just a little older than you, that just knocked all support from under me and very nearly disillusioned me completely. All sense of values goes, and as you say, you \"find it difficult to consider anything important.\" There's nothing very encouraging I can say to you at such a time. Each one has to battle his own way thru the fog and find something to hold on to. I can tell you this however: that, as one gets older, the blows from any tragic experience seem to strike a little less deeply. I suppose one's senses are not quite so keen, and one's emotional nature not so easily unbalanced.\n\nIt is cruel that you have lost two such wonderful friends in so short a time. Last week in N.Y. I saw a friend from the west whom I hadn't seen for 10 years \u2013 and discovered to my amazement that he and his twin brother (whom I did not see that day) were old and very good friends of David Prall. Robert and Frederick Schlick (from Portland, Oregon) graduated from the Univ. of California and studied there with David Prall. They were very good friends \u2013 often went together to Carmel to stay with other friends. Later when the twins were in Paris and David Prall was at Oxford, he went over to stay with them in Paris. I'd like so much to have you and the Schlick twins know each other. (Their mother and my mother are very old friends, and were friends before I was born.) Frederick is a playwright. He's had one play produced on Broadway a number of years ago,99 and he's been writing for Paramount. Robert is a poet and has a very unique plan about his life's work.100\n\nIf you and I can hit upon another weekend in N.Y. this winter (they are there just for 4 months) I'll arrange to have you meet them (they're just 35).\n\nPhila. must have seemed particularly gloomy after such a tragic personal experience. Somehow, I hate to have you come so close to tragedy. I'd like to protect you from such heartbreaks and disillusionment. But, unfortunately, no matter how fond we are of another, we cannot do that for anyone. The only solace I've ever found when the world looked so black I didn't see how I'd ever go on, was _work_ and _more work_ , and \u2013 _the passing of time_ which somehow eases the keen edge of suffering.\n\nWrite me soon.\n\nMuch love,\n\nHelen\n\n82. Leonard Bernstein to Aaron Copland\n\n2122 Walnut Street, Philadelphia, PA\n\n[January 1941]\n\nAaron, best of all,\n\nPlease don't think of me as a fool \u2013 I was just knocked out by the death of my friend: I didn't quite know what I was doing. I tried to get you on your return from Allentown, but apparently you didn't revisit the loft or the hotel.\n\nI'm back, and suffering. I can't tell you how numb I feel. As if part of me had died, refusing to accept the fact. The phenomenon of music on the brain, which has always been with me (you know that) has stopped. I have no tune to sing. My head feels like dry, brown, cracking wood. I took a piano lesson tonight, & having nothing to play, did the _Variations_ ,101 badly, & Vengerova didn't like them. Thursday night will be good. Strange \u2013 they seem like a different piece now. I see them more in formal outline.\n\nHow was the Whittemore & Lowe affair?102\n\nI'm sending a pseudo-bill to Boosey.\n\nLet me know when you arrive in Philly.\n\nHeard Serkin in dull Reger tonight.\n\nSaw my 16-yr-old girl103 \u2013 I don't know.\n\nMuch love, & apology,\n\nLenny\n\nP.S. Just got yr. letter \u2013 of course I'll be on hand after 2.30 Monday. Just taxi up to the door. And you can change or anything else you want to do here (I can't decide whether to come to NYC Sunday next).\n\n83. Oliver Smith104 to Leonard Bernstein\n\n7 Middagh Street, Brooklyn, NY\n\n6 February 1941\n\nDear Leonard,\n\nThe Stravinsky last Saturday was very swell; I want you to know how much I appreciated it, as well as the tremendous concert a week earlier. I would have written sooner, but I have been in the process of moving. I am now living in Brooklyn Heights, which is the nicest part of the city. It seems completely dominated by the bridge, the river with the endless chain of boats, and nice old streets, very quiet and full of nostalgia. I don't know why I sound so dopey. Perhaps it's because I'm freezing in this new house which is marvellous. Auden, Geo. Davis, Gypsy Rose Lee, Benj. Brittle105 live here. Miss Lee has since vacated. I think perhaps Paul and Jane [Bowles] will move in. I have two rooms, no furniture, and lots of nice drawings on the walls, a Picasso, several B\u00e9rards, a Chirico, and a Smith. Downstairs are empty rooms with rat holes, an enormous piano \u2013 I think you would sound very well on it \u2013 and books. The atmosphere is completely surrealist. I enjoyed meeting you very much, and I hope I will see you again some time.\n\nSincerely,\n\nOliver Smith\n\n84. Leonard Bernstein to Aaron Copland\n\n2122 Walnut Street, Philadelphia, PA\n\n6 February 1941\n\nDear A,\n\nThere's a good chance of my doing _Quiet City_ at a _private_ concert here on Feb. 20. I want very much to do it. (Orch. from Curtis.) How much wd. we have to pay, if any? Could I get the parts & score before Feb. 18? Trumpet part sooner? Please make it go thru \u2013\n\nLenny\n\nWrite writeaway\n\n85. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street,\n\nNew York, NY\n\nFriday\n\n[February 1941]\n\nDear L,\n\nHeinsheimer says you can have _Quiet City_. Write to him direct. I forgot to ask him whether there would be any charge. The proofs of the score are to be ready Monday. I have a sort of a trumpet part I could let you have. OK?\n\nJust finished _Billy_!! Mills was a big help.\n\n_El Saloon_ is being held up on account of they say they want a picture on the cover.\n\n_Outdoor Overture_ is being done today in Manchester by the London Philharmonic \u2013 Germans permitting.\n\nI _still_ think on you lada dia.\n\nSee you Thursday.\n\nA\n\n\"Unless love is love is love love?\" Paul Frederic Bowles.\n\n86. Leonard Bernstein to Kenneth Ehrman\n\n2122 Walnut Street, Philadelphia, PA\n\n13 February 1941\n\nKen,\n\nWhat's happened to you? You have a most unnerving way of dropping out of people's lives. If you live, reply!\n\nThere's much to write, but I'm taking no chances with disappearers. Reply, reply, softly, and with brittle brads.106\n\nI've just earned a small but cozy nest-egg & am thinking _O so seriously_ of an exotic trip in May \u2013 Mexico or Cuba, or maybe Palo Alto. Why don't you come with me? Please. Almost anywhere you say. Costa Rica. Guatemala. Reply, reply, gravely and with Jesuit joy.\n\nI can't make myself say any more until I hear from you, embodied and calm.\n\nA more than ludicrous transition, but I take it you've been informed of the wretched and tragic death of Al Eisner. Not another word about it. You understand.\n\nPlease write immediately if you can. And keep that trip seriously in mind.\n\nAlways the best,\n\nLenny\n\n87. Benjamin Britten107 to Leonard Bernstein\n\n7 Middagh Street, Brooklyn, NY\n\n28 April 1941\n\nDear Lenny,\n\nPlease forgive the lateness of this \u2013 but I have been working all days & nights for the last three weeks on the score of the operetta108 & I haven't had a moment for letters.\n\nI was very, very pleased that you liked the _Sinf_ [ _onia_ ] _da Req_ [ _uiem_ ].109 Judging by your remarks you certainly \"got\" what I wrote, & it was extremely nice of you to take the trouble to write & say so. I am sure that it's the \"best so far\" \u2013 and as it's the last, that is as it should be. I might argue with you about one or two of your remarks about my earlier masterpieces \u2013 but may be there is something in what you say. The only thing is, may be those particular vices are less vicious than some others I can think of \u2013 such as inhibitions, sterility, self-conscious ideas of originality \u2013 but we won't go into that now!\n\nHow are you? I saw you were conducting on the radio on Saturday \u2013 how did it go? When do you come to New York? I shall be around until June 1st or thereabouts. Give me a call when you get here. How are your chamber concerts going? As you probably know, the Bowlesesses departed for Mexico.\n\nThe operetta is chaotic. [Max] Goberman is not doing it \u2013 Hugh Ross has taken it over \u2013 & although he has the right mentality for training choruses (entre nous) he is not so hot on orchestras. However \u2013 we shall see.\n\nThank you again for your note. You ask how the others liked the symphony \u2013 all the ones I respect were pleased \u2013 including Aaron, Ch\u00e1vez, Colin [McPhee], Lincoln [Kirstein] et. all \u2013\n\nBest of luck,\n\nYours ever,\n\nBenjy B\n\n88. Leonard Bernstein to Shirley Gabis110\n\n86 Park Avenue, Newton, MA\n\n[May 1941]\n\nHello, you Galatea!\n\nTime out (2 minutes) from orchestrating music for a Harvard production of _The Peace_ of Aristophanes (a new headache I've contracted). My hand is numb from writing score; & to make matters worse, I bruised my metacarpal (!) playing baseball this afternoon. All of which makes good for concerto-playing the 25th! To say nothing of Scriabine-playing the 17th. And conducting the _Peace_ music the 23rd & 24th. Life, dear one, is hectic plus \u2013 I really need your steadying hand on mine now. It's amazing to look back & see that it really was a steadying hand. Phenomenal effect for an adolescent Galatea to have! But then, you're you.\n\nOf course Bill [Saputelli]111 has told you of our Atlantic City escapade (mostly gabbing with Curtisites). And now you're doing algebra, & going to Ivy Balls [...] & putting your hair up and down according to your escort, & eating chez Saputelli, \u2013 I've completely left your mind. See? I told you so.\n\nBut make an effort anyway, & write me all \u2013 a bright moment of letter-opening in all this muddle of a bustle. And darling, take care of yourself.\n\nAll my love, to split with Rae.112\n\nLenny.\n\nIsn't it daring to do _The Peace_ at a time like this? I love it \u2013 the music is good too.113\n\n89. Leonard Bernstein to Shirley Gabis\n\n86 Park Avenue, Newton, MA\n\n[after 25 May 1941]\n\nSweets,\n\nIt's all over & I breathe again. Quel week! The Greek show brought the house down both nights & my score was universally beloved. But at the price of awaking on Concerto day with a fine feverish cold. Hence the concerti were under par, but very exciting. Good reviews, too. Vital, vastly impressive, etc. But could have been perfect but for that damned fever.\n\nNow I'm home nursing this lovely cold, sort of good-for-nothing & let down. No trip for me, I think \u2013 I'll spend the dough on records, raquets & phonographs. God, it's good to be home.\n\nWhen do you graduate? Or do you?\n\nMiquelle came for the concert \u2013 in fact she's dropping round this afternoon, as one might expect.\n\nAll kinds of congrats to Mel & Dot. May they grow fine grapes together.\n\nNothing now 'til Tanglewood; & I plan to go up next week to [...] see Kouss. Will you be up this summer?\n\nI think I'll write an orchestral piece.\n\nHave you read Henry James? Read _The Turn of the Screw_ , one of 2 stories in a book called _The Two Magics._\n\nWrite soon.\n\nLove,\n\nLenny\n\n1 Burton 1994, p. 11.\n\n2 Helen Coates (1899\u20131989) was Bernstein's piano teacher from the end of 1932 until 1935, and they became close friends. She kept a maternal eye on her prot\u00e9g\u00e9 during his student years, and gave up piano teaching to become his secretary in 1944. She remained so until 1989. Her unspoken role in this book is of the greatest importance. It was Miss Coates who assembled and maintained Bernstein's scrapbooks, filed all his correspondence, labeled his photographs, organized his papers, and, as a consequence, did much to arrange the comprehensive documentation of his life that later formed the basis of the Leonard Bernstein Collection in the Library of Congress.\n\n3 At the top of this letter Helen Coates has written the date of their first lesson: \"Sat. at 1:00 Oct. 22\".\n\n4 Sid Ramin (b. 22 January 1919), American composer, songwriter, arranger, and orchestrator. His credits as a Broadway orchestrator (in collaboration with Irwin Kostal, Robert Ginzler, or Hershy Kay) include _West Side Story_ (1957), _Gypsy_ (1959), _A Funny Thing Happened on the Way to the Forum_ (1962) and _1600 Pennsylvania Avenue_ (1976). He won an Academy Award for the new orchestrations he made with Kostal for the film of _West Side Story_ (1961) and wrote the original score for _Stiletto_ (1969). As a songwriter, Ramin is best known for \"Music to Watch Girls By,\" originally written for a Diet Pepsi commercial, and subsequently widely recorded.\n\n5 This is the first of several letters sent by Bernstein to Ramin during the summer of 1933. Bernstein turned 15 on 25 August 1933 and Ramin was five months younger. Both had attended the William Lloyd Garrison Elementary School; when Bernstein went to the Boston Latin School (1929\u201335), Ramin went to Roxbury Memorial High School, but the two remained close. Ramin characterized their lifelong friendship as \"a relationship of teacher and student that never changed throughout the years. Lenny was a born teacher; I was a born listener. [...] That relationship really was one that lasted until Lenny passed away. Always, it was teacher and student\" (quoted in Oja and Shelemay 2009, p. 13).\n\n6 Jacques Fray and Mario Braggiotti were a celebrated piano duo. They began their career at Le B\u0153uf sur le Toit, the Parisian cabaret, and became friends with George Gershwin when he visited Paris in 1928. They first came to New York in 1929 and quickly achieved immense popularity through regular appearances on the radio, and extensive tours of the United States.\n\n7 Eddie Ryack was a school friend, and co-inventor with Bernstein of Rybernian, the imaginary language they devised.\n\n8 Lawrence Bearson, who later worked as a writer for the Federal Theatre Project of the Works Progress Administration (WPA), c. 1937\u20139.\n\n9 AVOL were sponsors of a series of Sunday radio broadcasts on WBZ, Boston in 1935. These gave the young Bernstein some early experience of playing on the radio.\n\n10 Mischa Tulin (d. 1957) was a Russian pianist and Theremin virtuoso who emigrated to America at the time of the 1917 Russian Revolution. His teachers included Glazunov and Busoni.\n\n11 Beatrice Gordon (1918\u201383) lived in Roxbury, MA. She was described by Bernstein as \"the love of my life\" at the time (Burton 1994, p. 22). She performed in his teenage productions of _Carmen_ (1934) \u2013 a spoof version, performed in drag (she appeared as Don Jos\u00e9 opposite Bernstein's Carmen), and as Pitti-Sing in _The Mikado_ (1935), a more ambitious staging put on by \"The Sharon Players.\" Burton summarized their relationship as follows: \"His attachment to Beatrice Gordon, who sang Don Jos\u00e9 in matador pants at Sharon, went well beyond music. They were both romantics, enamoured of poetry and words. He called her \"Tiger on Brocade\" (an early example of his obsession with anagrams) and \"Rosebeam\"' (Burton 1994, p. 29).\n\n12 Probably a reference to \"Miniver Cheevy\" (who \"loved the Medici\"), by the American poet Edwin Arlington Robinson (1869\u20131935).\n\n13 Amy Lowell (1874\u20131925), American poet who was awarded a Pulitzer Prize posthumously in 1926.\n\n14 Verne was Beatrice Gordon's middle name.\n\n15 In some of his letters to Beatrice Gordon, Bernstein signs himself \"Lamb's Ear\".\n\n16 A reference to Bernstein's first encounter with Dimitri Mitropoulos in January 1937, which was evidently a torrid occasion.\n\n17 On 31 July 1937, a few days before writing this letter, Bernstein had been musical director for a Camp Onota production of Gilbert and Sullivan's _The Pirates of Penzance_ , with Adolph Green as the Pirate King. The first known photograph of Bernstein as a conductor \u2013 conducting the Camp Onota Rhythm Band \u2013 also dates from the summer of 1937.\n\n18 Mildred Spiegel (b. 1916), American pianist. One of Bernstein's closest friends in the 1930s and early 1940s, she became his regular partner on two pianos and they were both pupils of Heinrich Gebhard. Bernstein wrote his Piano Trio for Spiegel's Madison Trio (active from 1935 to about 1940). Its members were Mildred Spiegel (piano), Dorothy Rosenberg (violin), and Sarah Kruskall (cello). According to the violist Raphael Hillyer (1914\u20132010), Spiegel also played the Piano Trio with him (violin) and Jesse Ehrlich (cello) at Harvard in about 1939 (see Derrick Wang: Raphael Hillyer Interview, ). In 1951, Spiegel married Zevi Harry Zucker (1921\u20132012).\n\n19 Dimitri Mitropoulos (1896\u20131960), Greek conductor. Bernstein first encountered him in January 1937 and the two immediately became very close. At some time in the late 1930s they almost certainly became involved in a relationship.\n\n20 Some of Mitropoulos' spelling has been silently corrected.\n\n21 Aaron Copland (1900\u201390), American composer whose music, and whose friendship, wisdom, and advice, made the deepest impression on Bernstein. Always regarded by Bernstein as his most important musical mentor, Copland's admiration for Bernstein was more nuanced: he had the highest regard for him as a conductor but sometimes expressed reservations about his compositions. From the start of Bernstein's career, Copland gave him the fullest support. On 17 March 1940, he wrote the following recommendation to Mrs Grant for Bernstein to be admitted to Koussevitzky's conducting class at Tanglewood: \"In my opinion, Mr. Bernstein is an extraordinarily gifted young musician. I have seldom met his equal for sheer musicianship. His musical memory is remarkable, and so is his ability to sight-read both scores and piano music. He is besides a first-rate pianist. He possesses the type of temperament which I believe is particularly sympathetic to Dr. Koussevitzky. His practical experience as a conductor is very slight, but he has had a year's training at the Curtis Institute under Fritz Reiner. Randall Thompson told me that Mr. Reiner considered Bernstein one of the best students he had ever had. Needless to say, I think it important that Bernstein gain an individual seance with Dr. Koussevitzky when the proper moment arrives. I'd appreciate it if that could be arranged.\"\n\n22 A work Bernstein knew very well. He had played to Copland when they first met, on 14 November 1937 \u2013 Copland's birthday.\n\n23 Presumably Copland's _What to Listen for in Music_ which was published in 1939.\n\n24 The Works Progress Administration.\n\n25 Copland's mention of Bennington is presumably a reference to Bennington College in Vermont. Norman Lloyd (b. 1914), American actor, a member of the Mercury Theatre founded by Orson Welles and John Houseman.\n\n26 Koussevitzky conducted the Boston Symphony Orchestra on 14 and 15 October in a programme that included Copland's _El Sal\u00f3n M\u00e9xico_ , Mozart's Divertimento K287, and Dvo\u0159\u00e1k's \"New World\" Symphony.\n\n27 Copland's _El Sal\u00f3n M\u00e9xico_.\n\n28 Clifford Odets (1906\u201363), American playwright and screenwriter. He was a founding member of the Group Theatre.\n\n29 For Bernstein's Harvard production of Blitzstein's _The Cradle Will Rock_.\n\n30 Probably _Billy the Kid_ , first performed on 16 October 1938.\n\n31 The piano\u2013vocal score of _The Second Hurricane_ was first published by C. C. Birchard of Boston in 1938.\n\n32 Kenneth Ehrman was a friend of Bernstein's in Eliot House at Harvard. He graduated in 1938 and Bernstein corresponded with him regularly during his own senior year at Harvard and his time at the Curtis Institute. Ehrman recalled one of his earliest encounters with Bernstein in a letter to Humphrey Burton dated 23 October 1991 (copy in the Library of Congress): \"He had been playing the piano in the Eliot House Common Room after dinner, and it was a beautiful warm spring night and we walked around Cambridge for several hours. He was talking non stop about his plans \u2013 the composing, playing, writing, teaching (he never mentioned conducting in those days, at least to me). About the [...] musical world, and how he would be such a big part of it. Suddenly he stopped, looked at me and said, 'Who do I think I am, everybody?'\"\n\n33 _The Harvard Advocate_ is Harvard's undergraduate journal of poetry, fiction, art, and criticism.\n\n34 This is part of a longer letter, but its main interest is Bernstein's scathing attack on the Harvard Music Department.\n\n35 Presumably \"Californiac\".\n\n36 Mitropoulos specialized in directing concertos from the keyboard, notably Prokofiev's Piano Concerto No. 3.\n\n37 Hanya Holm (1893\u20131992), German-born American choreographer and dance educator.\n\n38 Copland did so. His Piano Sonata was composed between 1939 and 1941. The earliest sketches date from about June 1939, shortly after he sent this letter.\n\n39 Ehrman was living in Paris at the time, on his way to Italy and Greece.\n\n40 Margaret Prall, David Prall's wife.\n\n41 David Prall (1886\u20131940), professor of aesthetics at Harvard whose teaching had a profound influence on Bernstein.\n\n42 _The Birds_ was first performed on 21 April 1939.\n\n43 Adolph Green (1914\u20132002), American lyricist, playwright, and scriptwriter whose writing partnership with Betty Comden produced several successful Broadway shows including _On the Town_ and _Wonderful Town_ , both with Bernstein. In Hollywood, Comden and Green's greatest successes were at MGM, notably _Singin' in the Rain_. Green first met Bernstein at Camp Onota in the summer of 1937, where Bernstein put on a production of Gilbert and Sullivan's _Pirates of Penzance_ and Green sang the part of the Pirate King. They instantly became firm friends through a shared passion for music, and within a couple of years Bernstein was playing the piano for Comden and Green's comedy troupe, The Revuers.\n\n44 Bernstein's production of _The Cradle Will Rock_ opened on 27 May 1939. The _Harvard Crimson_ (23 May 1939) announced the event as follows: \"The Student Union will present Marc Blitzstein's opera _The Cradle Will Rock_ at Sanders Theatre, Saturday evening at eight-thirty o'clock. The production is under the direction of Leonard Bernstein '39, and Arthur Szathmary, 2G. This Student Union presentation will be the first Boston production of Blitzstein's proletarian music drama. Featured in the cast are Donald Davidson, '39, William Whitcraft, '39, Rupert Pole, '40, and Myron Simons, '40, all of Eliot House. The piano, which furnishes all the music, will be played by Bernstein.\"\n\n45 The production of _The Birds_ with Bernstein's incidental music.\n\n46 Julian Claman (1918\u201369) sometimes appeared with The Revuers and became a writer and playwright. His 1955 play, _A Quiet Place_ , starred Tyrone Power, but closed out of town, before reaching Broadway. The title and title song (sung by Power in the show) were taken from _Trouble in Tahiti_. In 1953, Claman married the actress Marian Seldes; they divorced in 1961.\n\n47 Claman has written in the margin: \"Tentative love, Julian\".\n\n48 Probably Robert Weil, mentioned in Letter 538.\n\n49 Davidson Taylor worked as musical supervisor for the CBS radio network at the time. It was one of several positions he held in the company. In 1938, Taylor had been the executive producer of Orson Welles' _The War of the Worlds_. He later became a vice president at CBS.\n\n50 Made for the 1939 New York World's Fair, _The City_ is a documentary film notable for its close integration of spoken narration (written by Lewis Mumford), cinematography (by Ralph Steiner and Willard Van Dyke), and Copland's score. This score led directly to Copland's invitation to work in Hollywood the following year on _Our Town_.\n\n51 Ernest Hutcheson (1871\u20131951) was an Australian pianist and composer. He was President of the Juilliard School, and Director of the Chautauqua School of Music at the Chautauqua Institution, in western New York State. In 1925, Hutcheson had arranged for Gershwin to be given secluded accommodation at the Chatauqua Institution so that he could complete the Concerto in F.\n\n52 Britten came to America in April 1939, staying until April 1942 when he returned to England. He first met Copland at the 1938 ISCM Festival in London (where Copland's _El Sal\u00f3n M\u00e9xico_ and Britten's _Variations on a Theme of Frank Bridge_ were performed in the same concert); \"Britten thought _El Sal\u00f3n_ 'really beautiful and exhilarating' and the 'brightest piece in the festival,' while Copland admired 'the technical adroitness and wizardry' of the _Variations_ \" (Pollack 1999, p. 72). Copland spent a weekend with Britten after the festival, and a warm friendship developed during Britten's time in the United States. On 17 June 1950, Pears and Britten gave the world premiere of the first set of Copland's _Old American Songs_ at the Aldeburgh Festival, and subsequently made the first recording for HMV. For Britten's links with Bernstein see note to Letter 87.\n\n53 Britten and Pears spent six weeks of the summer at Woodstock, NY, as neighbors of Copland's; they left Woodstock a few days before Bernstein wrote this letter. The \"Concerto\" was Britten's Violin Concerto, which he worked on throughout the summer and completed at the end of September.\n\n54 Bernstein spent the summer of 1939 in New York, looking for a job, without success. But it was a productive stay: there was a grand piano in the apartment and he spent a good deal of time playing and composing, as well as enjoying evenings with Green and the other members of The Revuers, including Judy Holliday and Betty Comden. Most importantly, Bernstein was starting to think seriously for the first time about a conducting career, and getting down in earnest to composition. The Lamentation he sent to Copland did indeed become \"a movement of a symphony.\" In the programme note for the 1944 premiere of the _Jeremiah_ Symphony, Bernstein wrote: \"In the summer of 1939 I made a sketch for a Lamentation for soprano and orchestra. This sketch lay forgotten for two years, until in the spring of 1942 I began a first movement of a symphony. I then realized that this new movement, and the scherzo that I had planned to follow it, made logical concomitants with the Lamentation. Thus the symphony came into being.\" Almost forty years later, at a concert for Bernstein's 60th birthday on 25 August 1978, Copland conducted the Lamentation from _Jeremiah_.\n\n55 Presumably a reference to The Revuers whose regular venue was the Village Vanguard in Greenwich Village.\n\n56 Arthur Szathmary, who co-directed _The Cradle Will Rock_ with Bernstein at Harvard.\n\n57 Dimitri Mitropoulos.\n\n58 Isabelle Vengerova (1877\u20131956), pianist and teacher who studied with Theodor Leschetitzky. One of the founding teachers of the Curtis Institute, her pupils included Samuel Barber, Lukas Foss, Gary Graffman, Abbey Simon, Gilbert Kalish, and Jacob Lateiner, as well as Bernstein.\n\n59 Alfred (Al) Eisner was Bernstein's room-mate in Eliot House at Harvard. After graduation, he went to Hollywood, where he worked as a scriptwriter at MGM. He died from a brain tumor on 4 January 1941, in his early twenties. The third of Bernstein's _Seven Anniversaries_ is entitled \"In Memoriam: Alfred Eisner (Jan. 4, 1941)\".\n\n60 Betty Comden (1917\u20132006), American lyricist, writer, and performer, the writing partner of Adolph Green for numerous successful Broadway shows and Hollywood films. Bernstein came to know Comden in 1939 through Green, when both of them were members of The Revuers. Bernstein made one of his first recordings with The Revuers in March 1940 ( _The Girl with the Two Left Feet_ ), and they subsequently collaborated on two Broadway triumphs: _On The Town_ and _Wonderful Town_. Betty Comden was to remain a lifelong friend. Like Green, she was passionate about serious music, and was as interested in Bernstein's conducting and his concert compositions as she was in his work for Broadway.\n\n61 A reference to _The Girl with the Two Left Feet_ by The Revuers, with improvised music by Bernstein. A proposed recording is the subject of Comden's letter.\n\n62 _The Pursuit of Happiness_ was a CBS radio show broadcast for one season (1939\u201340; 30 episodes in all). Directed by Norman Corwin and Brewster Morgan, and hosted by Burgess Meredith, it was aired immediately after the Sunday afternoon broadcast from the New York Philharmonic. The Revuers appeared in two episodes, on 12 November 1939 and 18 February 1940.\n\n63 Probably Margaret Prall.\n\n64 Gordon Messing and Robert Wernick were friends from Harvard.\n\n65 The Revuers and Bernstein did record _The Girl with the Two Left Feet_ for Musicraft in March 1940. This was Bernstein's first commercial recording. He plays an improvised score between accomp-anying the songs performed by The Revuers. This delightful recording was released on CD by Pearl in the set _Leonard Bernstein \u2013 Wunderkind_ (GEMS 0005).\n\n66 David Diamond (1915\u20132005), American composer. He quickly became one of Bernstein's closest friends, though their friendship was often stormy and disputatious. Their correspondence over almost half a century is extremely lively, and sometimes volatile. Bernstein provided financial assistance for Diamond from time to time, while Diamond was an enthusiastic supporter of Bernstein's own compositions. Diamond studied with Bernard Rogers at the Eastman School, with Roger Sessions, and with Nadia Boulanger. Bernstein performed several of Diamond's major works and recorded the Fourth Symphony for Columbia in 1958. Diamond recalled his first encounter with Bernstein in an interview with Paul Remington for _Cosmik Debris_ (No. 21, February 1997):\n\nI had heard about him from Aaron Copland and Marc Blitzstein. They told me about this extraordinary pianist that was at Harvard studying with Walter Piston, Edward Burlingame Hill and Randall Thompson. And, one weekend I was going up the stairs to thank Koussevitzky for such a wonderful performance after a Boston Symphony concert. They performed Ravel's Left Hand Piano Concerto. Of course, he knew I had known Ravel, so he was so pleased that I came back and that I was so moved by the performance of the work. And there at the top of the stairs was this very good-looking young man. I remember he was looking down at me. He said, \"I know you!\" I said, \"Who are you?\" He said, \"I'm Leonard Bernstein.\" I evidently reacted to that, and he said, \"and you're David Diamond!\" He said, \"You must come out to the Curtis sometime and spend a weekend with me there.\" He had enrolled in the Curtis Institute and had a full scholarship there. So, I got him to record some of my Preludes and Fugues. It was the first professional recording he had made.... I would go out to visit him almost every other weekend. I helped him with his counterpoint, I remember.... He was still a conducting student of Fritz Reiner's while at the Curtis Institute. He was composing at that time and had written theater music while he was at Harvard. But, he was working on a clarinet sonata, or maybe it was a violin and piano sonata that became a clarinet sonata. I didn't know him at all as a composer. But, he was a phenomenal pianist. From an orchestral score, he read through my 1st Symphony that way. He just knocked me out as a musician. He was just phenomenal. And so, as the years went on, he saw that I was being performed a great deal. Then he made that amazing debut with the New York Philharmonic, substituting very quickly for Bruno Walter. Then, suddenly, he was on the map as a conductor. Then he was given the City Center Orchestra, which was an orchestra that was put together for him. He wasn't paid a salary, but that orchestra that he built up had marvelous programs. The second year he had that orchestra he did my 2nd Symphony, after he had heard Koussevitzky do it. Then, almost every other year he would perform a work of mine. And then he began to compose a lot. But, I guess I was the only one of the friends that felt he was a gifted composer. Copland didn't think he really had it as a composer. He thought he was very good for Broadway, but he didn't care for his composing. He didn't like _Jeremiah_ whatsoever. Now, I thought _Jeremiah_ was extraordinary.\n\n67 In 1940, Mitropoulos conducted _The Coliseum at Night_ by Frederick Woltmann.\n\n68 The actor Wallace Beery (1885\u20131949).\n\n69 _We Who Are Young_ was released on 19 July 1940. Produced by Seymour Nebenzal, the cast included Lana Turner in one of her first major Hollywood roles.\n\n70 Albert Maltz (1908\u201385), author and screenwriter, one of the \"Hollywood Ten\" blacklisted in 1947 for refusing to answer questions from the House Un-American Activities Committee.\n\n71 This telecast was a \"special\" from NBC's experimental studio, featuring The Revuers. Bernstein was the pianist and Copland turned the pages (see Burton 1994, p. 72). In Lyons' article, Bernstein is described only as \"the accompanist\": \"Saturday Night, when The Revuers appeared on NBC's full-hour television program, the unbilled stranger who turned the pages for the accompanist was Aaron Copland, the noted American composer\" ( _New York Post_ , 25 June 1940).\n\n72 Ren\u00e9e Longy Miquelle (1897\u20131979), French-born pianist, theorist, and teacher. She moved to the United States when her father, Georges Longy, became principal oboist of the Boston Symphony Orchestra. Her first teaching post was at the Longy School of Music, which her father founded in 1915. Subsequently, she taught at the Curtis Institute (where she was Bernstein's score-reading teacher), then at the Peabody Conservatory in Baltimore, the University of Miami, and the Juilliard School in New York. Her other pupils included the pianist Jacob Lateiner, cellist Leonard Rose, and several members of the New York Philharmonic including flutist Julius Baker and oboist Harold Gomberg.\n\n73 Raphael Silverman, later known as Raphael Hillyer (1914\u20132010), was a graduate student at Harvard where he often performed with Bernstein. He led the orchestra in the 1939 production of _The Birds_ with Bernstein's incidental music. After a spell in the Boston Symphony Orchestra as a violinist, Hillyer switched to the viola in 1946 and became a founding member of the Juilliard String Quartet, with which he played for 23 years. He was the dedicatee of Bernstein's Violin Sonata, referred to in this letter.\n\n74 A weekend of the Yaddo Music Festival comprising four concerts of contemporary American music given at the Yaddo artists' community in Saratoga Springs, NY. The programme for the weekend was printed in _The New York Times_ on 7 September 1940 and listed works by (among others) Paul Bowles, Henry Cowell, Paul Creston, David Diamond, Roy Harris, Mary Howe, Charles Ives, Gail Kubik, Otto Luening, Paul Pisk, and Quincy Porter.\n\n75 Serge Koussevitzky (1874\u20131951), Russian-born conductor who served as Music Director of the Boston Symphony Orchestra from 1924 to 1949. Along with Copland, Koussevitzky was one of Bernstein's most important mentors, initially through conducting classes at Tanglewood, then as an enthusiastic promoter of Bernstein's career as a conductor, pianist, and a composer of \"serious\" music: he strongly disapproved of Bernstein's activities as a Broadway composer, and this may well be one of the reasons why, after _On the Town_ , Bernstein wrote no Broadway shows until after Koussevitzky's death.\n\n76 Presumably, the \"commotion\" was Fritz Reiner's furious reaction to Koussevitzky's suggestion that Bernstein should study with him in Boston rather than with Reiner.\n\n77 Hemingway's _For Whom the Bell Tolls_ was published in October 1940. It is likely that Eisner's letter was written shortly before then. The draft mentioned later in Eisner's letter was the Selective Training and Service Act, which came into force on 16 September 1940, the first peacetime conscription in American history.\n\n78 Much of this sentence has been deleted and retyped.\n\n79 Walter Leslie River (1902\u201381), American novelist and screenwriter.\n\n80 Henry Cowell (1897\u20131965), American composer and founder of the periodical _New Music_ for which Diamond's piece was being recorded.\n\n81 Bernstein's first solo recording, made for New Music Recordings in January 1941, was of David Diamond's Prelude and Fugue No. 3. The tetchy correspondence about it was symptomatic of many of the letters Bernstein and Diamond exchanged over the next half century.\n\n82 The first performance of the Suite from _Billy the Kid_ was broadcast on Saturday, 9 November 1940, by the NBC Symphony Orchestra under William Steinberg.\n\n83 Harold Shapero (1920\u20132013), American composer. A near contemporary of Bernstein's at Harvard (Shapero graduated in 1941), he subsequently taught at Brandeis University for 37 years. Bernstein recorded Shapero's _Symphony for Classical Orchestra_ in 1953.\n\n84 Walter Piston (1894\u20131976), American composer and teacher who taught both Shapero and Bernstein at Harvard.\n\n85 Charles Demuth (1883\u20131935), American artist whose friends included the poet William Carlos Williams, the artist Marsden Hartley, and the photographer Alfred Stieglitz. He bequeathed many of his paintings to Georgia O'Keeffe. A childhood illness left Demuth with a pronounced limp for the rest of his life, and his health was always precarious. According to Robert Hughes: \"Demuth was not a flaming queen, in fact he was rather a discreet gay, but if he could not place his deepest sexual predilections in the open, he could still make art from them\" (Robert Hughes, _American Visions: The Epic History of Art in America_ , New York: Knopf, 1997, p. 380).\n\n86 Probably a reference to the draft papers drawn up in Washington, D.C. in 1940, which had the numbers 1 to 7,836 printed on them.\n\n87 Shapero is being hard on himself here. The _Nine-Minute Overture_ won the American Prix de Rome in 1941. His success was reported in the _Harvard Crimson_ on 9 June 1941: \"Harold S. Shapero '41 of Newton has won the annual Prix de Rome in Music, it was announced yesterday by Howard Barlow, conductor of the Columbia Symphony Orchestra. The work, _A Nine-Minute Overture_ , was played during Barlow's regular Sunday afternoon radio program. Instead of the customary privilege of studying at the American Academy in Rome, Shapero will receive $1,000 outright. Recently awarded the Knight Prize for composition by the Music Department, Shapero is the first Harvard undergraduate to win the Prix de Rome. The winning piece is his first attempt at writing for orchestra.\"\n\n88 The award of the Prix de Rome and Harvard's George Arthur Knight Prize (for composition) enabled Shapero to study with Nadia Boulanger after graduation.\n\n89 Virgil Thomson (1896\u20131989) began writing for the _New York Herald Tribune_ in 1940.\n\n90 Robert Weatherly (1921\u20132005) was a student at Juilliard when Copland sent this letter. He later became principal trumpet of the St. Louis Symphony Orchestra.\n\n91 A quotation from _La Boh\u00e8me_.\n\n92 John Lessard (1920\u20132003), American composer.\n\n93 Mitropoulos made his debut with the New York Philharmonic on Thursday, 19 December 1940.\n\n94 Kiki [Jacqueline] Speyer was the daughter of Louis Speyer, who had played under Pierre Monteux in the premiere of Stravinsky's _Sacre du printemps_ and was the cor anglais player of the Boston Symphony from 1919 until 1965.\n\n95 Possibly Wolfgang Stresemann, then a young conductor.\n\n96 Koussevitzky conducted Shostakovich's Fifth Symphony with the Boston Symphony in Carnegie Hall in November 1940 and again in January 1941. The November concert also included a Haydn symphony.\n\n97 Possibly a reference to the death of Alfred Eisner.\n\n98 Alfred Eisner died on 4 January 1941.\n\n99 Frederick Schlick's _Bloodstream_ ran on Broadway in 1932 for less than a month.\n\n100 Robert Schlick was a gay poet who married the London-born artist and illustrator Pamela Bianco in 1930.\n\n101 Copland's _Piano Variations_.\n\n102 The two-piano duo Arthur Whittemore and Jack Lowe gave a recital at Town Hall, New York. It was reviewed by Howard Taubman ( _The New York Times_ ), who mentions that the concert included an unspecified work by Copland.\n\n103 Probably a reference to Shirley Gabis, who was born in 1924.\n\n104 Oliver Smith (1918\u201394) was a designer and producer whose brilliant sets were an important part almost all Bernstein's theatrical works, including _Fancy Free, On The Town, Facsimile, Candide, West Side Story_ , and _Mass_. This letter appears to be their earliest contact. The house at 7 Middagh Street, where Smith was living at the time, was an old brownstone that was home to a remarkable group of creative artists in 1941. It was owned by George Davis, and among the residents were W. H. Auden, Britten (\"Brittle\"), Peter Pears, Louis MacNeice, Carson McCullers and \u2013 shortly after Smith wrote \u2013 Paul and Jane Bowles. Britten and Pears stayed for just a few months, finding the atmosphere too Bohemian.\n\n105 A humorous reference to Benjamin Britten.\n\n106 Small wire nails or prickly spikes. Bernstein is referring to a phrase in William Francis Hooker's _The Prairie Schooner_ (1918): \"For hours a kindly bullwhacker helped me pluck the sharp and brittle brads from my back.\"\n\n107 Benjamin Britten (1913\u201376), English composer, already established as one of the most brilliant figures in British music by the time he went to the USA in April 1939. It was in America that Britten wrote his first work for the stage, the operetta _Paul Bunyan_. Bernstein conducted several of his works: the American premiere of _Peter Grimes_ at Tanglewood in 1946; the _Spring Symphony_ (in 1963) and the _Sinfonia da Requiem_ (in 1968) with the New York Philharmonic; and in April 1976, the US premiere of Britten's _Suite on English Folk Tunes: A Time there Was_ with the New York Philharmonic. Bernstein included the _Four Sea Interludes_ in his final concert, with the Boston Symphony Orchestra at Tanglewood, on 19 August 1990.\n\n108 _Paul Bunyan_ , first performed at Columbia University on 5 May 1941.\n\n109 The world premiere of the _Sinfonia da Requiem_ took place in Carnegie Hall on 29 March 1941, with John Barbirolli conducting the New York Philharmonic.\n\n110 Shirley Gabis (b. 1924), American pianist. She became Bernstein's closest friend at the Curtis Institute and the two remained friends until his death. She later became Shirley Rhoads, then Shirley Perle when she married the composer and Berg scholar George Perle. The first of Bernstein's _Thirteen Anniversaries_ , dated July 1981, is entitled \"For Shirley Gabis Rhoads Perle (b. April 7, 1924).\"\n\n111 William Saputelli (1916\u20132001) was a friend of Bernstein's at the Curtis Institute. A cellist, he joined the Philadelphia Orchestra in 1952 and remained with it until 1988.\n\n112 Rae was Shirley Gabis' mother.\n\n113 \"Playgoer\" in the _Harvard Crimson_ (23 May 1941) reported: \"The play is by Aristophanes and three thousand years old, but the production to be offered at Sanders tonight and tomorrow by the Student Union Theatre is as timely as the latest headline, and as diverting as the brightest Broadway revue. Even in dress rehearsal disarray (which is when we caught it) _Peace_ gave every indication of being the most stimulating theatrical event around Cambridge this season. Of course, the plot \u2013 the attempt of a group of Athenians to bring Peace back to their city \u2013 is a natural for Student Union parallel-parable making, but even the most ardent Bundle for Britain will hardly object to swallowing this socially-significant pill, sugar-coated as it is with distinctively modern music by Leonard Bernstein, clever lyrics by William Abrahams, a colorful abstract set by Howard Turner and John Holabird, and a cast that is not merely capable but alive. And all of these elements have been brought together skillfully and with a refreshing lack of pretension by director Robert Nichols. There are, to be sure, flaws; but what this Student Union Theatre group may lack in slickness, it more than makes up for in spontaneity. These people are obviously having a good time, and their enthusiasm communicates itself to the audience. They are immeasurably helped by the Aristophanic tradition which is one of rowdy fun, rather than self-conscious artiness, and within the limits of the tradition this company is almost wholly successful. The production lasts little over an hour, the admission is sensibly low, and anyone should have fun. For those who don't, there is always the advice of the concluding couplet in the conga finale: 'If you don't like venery, Get thee to a nennery.'\" The music for the Conga was reused by Bernstein in _Wonderful Town_. Another section, the \"Sacrificial March,\" became the chase music in _On the Town_ (see Massey 2009, pp. 80 and 81).\n2\n\nFirst Successes: \nFrom Tanglewood to _On the Town_\n\n1941\u20134\n\nAfter receiving his diploma in conducting from the Curtis Institute in May 1941, Bernstein went to Harvard to conduct his incidental music for _The Peace_ before spending the summer at Tanglewood, where his conducting was widely admired, especially a performance of William Schuman's _American Festival Overture_. Bernstein then fled to Key West at the southern tip of Florida, to escape a complicated romantic entanglement (with Kiki Speyer) and to compose. It was a productive stay: he started the Clarinet Sonata and an unfinished ballet called _Conch Town_ that was to provide a rich harvest of musical ideas for subsequent works, including _Fancy Free_ and _West Side Story_. After Key West, Bernstein returned to Boston in need of a job. He set up a studio to teach piano and musical analysis in December 1941, but attracted depressingly few pupils. The year 1942 saw some early successes: the first performance of the Clarinet Sonata in April (by David Glazer and Bernstein) and, the following month, Bernstein conducted Copland's _Second Hurricane_ (subtitled a \"play-opera for high schools\") in Boston, repeating it a month later. Throughout this time he received constant encouragement from Copland, from Ren\u00e9e Longy Miquelle, and from Betty Comden. To earn a living, Bernstein took a job at a music publisher (part of Warner Bros.), working as an assistant and occasional arranger, often under the name Lenny Amber.\n\nIn the summer of 1942, Bernstein met David Oppenheim at Tanglewood. He was a young clarinetist studying at the Eastman School of Music, and the two quickly became very close. The correspondence between them in 1943 is often absorbing, with letters that are by turns passionate, funny, full of career worries (and, in Oppenheim's case, his military service) and musical questions, above all because of Oppenheim's performances of the Clarinet Sonata. Though David Glazer had given the premiere, it was Oppenheim who introduced the work in New York and who made the first recording (both with Bernstein at the piano), and he appears as its dedicatee on the first edition. Oppenheim and Bernstein also shared a fascination with psychoanalysis and the interpretation of dreams \u2013 they went to the same therapist (Marketa Morris, \"The Frau\") and had similar obsessions. While the correspondence with Copland is warm, funny, and loving, with some entertaining anecdotes from Copland in Hollywood and musical news of the East Coast from Bernstein, it is with Oppenheim that Bernstein shared some of his innermost thoughts.\n\nThe year 1943 was a crucial one for every aspect of Bernstein's career: in February he played Copland's Piano Sonata in New York, and in March he made his New York conducting debut (Paul Bowles' _The Wind Remains_ ). Bernstein played his own music too: a performance of the Clarinet Sonata with Oppenheim led quickly to its recording and publication. Adolph Green's long letter to Bernstein in September 1943 paints a funny and richly detailed portrait of Hollywood viewed through the eyes of a native New Yorker who seemingly finds himself in a weird and alien country.\n\nThe end of 1943 brought the most spectacular successes: in September he became Assistant Conductor of the New York Philharmonic \u2013 his first conducting appointment. Such a junior position usually involves quite menial tasks, but in Bernstein's case good fortune struck two months into the job. On 14 November 1943, Bruno Walter was due to conduct the Sunday Philharmonic concert at Carnegie Hall, but his sudden indisposition meant that Bernstein had to take over. The result (broadcast nationally on the radio) was a triumph \u2013 and a major news story: the front page of the next day's _New York Times_ was dominated by news of the war in Europe and the Pacific, but it also reported Bernstein's debut under the headline: \"Young Aide Leads Philharmonic. Steps In When Bruno Walter Is Ill\". Two months later, on 28 January 1944, Bernstein conducted the world premiere of his _Jeremiah_ Symphony No. 1 in Pittsburgh \u2013 at the invitation of his old conducting teacher Fritz Reiner.\n\nAt the same time Bernstein was hard at work on the ballet _Fancy Free_. Because Jerome Robbins and Bernstein were both away from New York for weeks on end while the ballet was being written, this is one of the very few Bernstein collaborations where musical matters are discussed in considerable detail by letter: those from Robbins to Bernstein have not survived, but the letters from Bernstein to Robbins are a fascinating chronicle of work in progress. On 18 April 1944, _Fancy Free_ triumphed at the Metropolitan Opera House. Bernstein's first collaboration with Robbins was instantly acclaimed \u2013 the headline of John Martin's review in the _New York Times_ (19 April) read: \"Ballet by Robbins Called Smash Hit\". Of the score, Martin wrote: \"The music by Leonard Bernstein utilizes jazz in about the same proportion that Robbins' choreography does. It is not in the least self-conscious about it, but takes it as it comes. It is a fine score, humorous, inventive and musically interesting. Indeed the whole ballet, performance included, is just exactly ten degrees north of terrific.\" Robbins and Bernstein turned at once to their next collaboration \u2013 a Broadway show, with Bernstein's old friends Comden and Green brought in to write the book and lyrics. With the support of the vastly experienced George Abbott as director, the result was another huge success: _On the Town_ opened at Broadway's Adelphi Theatre on 28 December 1944 to rave reviews. But Bernstein's brilliant achievement with his first musical brought conflicts too: Koussevitzky was already uneasy about Bernstein's balancing act between conducting and composing, and he lost no time telling his prot\u00e9g\u00e9 that writing for Broadway was a waste of his talents. The criticism hit home: Bernstein did not write another musical until after Koussevitzky's death in 1951.\n\n90. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nCranwell School, Lenox, MA\n\n15 July 1941\n\nCh\u00e8re Ren\u00e9e,\n\nTanglewood again \u2013 and as wonderful as ever. It never fails to impress me \u2013 as much each morning as the first time I set foot here. The _esprit de corps_ just got going (I conducted my first rehearsal this morning \u2013 the Billy Schuman [ _American Festival_ ] Overture!) I was supposed to open the series last Friday night with Billy's piece, but I had to go back to Boston to receive my award _and conduct The Esplanade!_ Had you heard? I did the _Meistersinger_ prelude \u2013 22,000 people! Tr\u00e8s exciting. I would have let you know, but it was all in such an unsettled state & I wasn't sure until very shortly before the concert. So Billy is this weekend (Friday night), & he's coming up for the performance \u2013 Kouss liked the rehearsal today, but insisted that I looked like un Moulin qui va avec le vent.\n\nFirst tragedy of the season \u2013 Gundersen1 (1st fiddle, BSO) died last night. Heart failure. Great sorrow.\n\nIf you haven't heard of this tragedy, hold your seat. This is really heartbreaking, I shall simply state it and not say another word. M. [Gaston] Dufresne2 has an assistant here \u2013 Miss Kathryn Wolf. Don't ask me another thing about it, I'm completely nonplussed.\n\nAny hope? Any news from La Bok?3 How's Claude4 & regards from Shanelian. Rest, & really summerize.\n\nBien \u00e0 toi,\n\nLenny\n\n91. Leonard Bernstein to Shirley Gabis\n\nCranwell School, Lenox, MA\n\n[after 15 August 1941]\n\nDear Gabe, babe,\n\nI'm limp. I've just written Alvin.5 Quel effort! I just took off a morning, & canceled everything at Tanglewood, & stayed home, & wrote letters. Otherwise impossible. Life here is hectic \u2013 but hectic. Tremendous successes in conducting the past two weeks: I did William Schuman's _American Festival Overture_ , & it knocked everyone for a bingo.6 Really brought it down. And last week,7 I did Lambert's _Rio Grande_ , with chorus, &c. Tr\u00e8s brilliant, & terrific hit. This week I'm stuck with the Brahms B\u266d Concerto,8 but it's only an interregnum of rest. Ain't you never coming up?\n\nJust heard from the Quashens,9 & they'll be here Thursday. With Anna Sokolov.\n\nWhatsamatter with you? Are you a step child?\n\nGod, I pity you in Philly! I'd perish, personally. You, of course, are of hardier stock!\n\nI got my questionnaire.\n\nLove to Rae \u2013 & let's hear.\n\nLove,\n\nLenny\n\n92. Samuel Barber10 to Leonard Bernstein\n\nThe Hermit, Pocono Lake Preserve, PA\n\n24 August 1941\n\nDear Leonard,\n\nI suppose the Berkshire performance was one of the most exciting evenings of my life: nevertheless in retrospect I felt as if I had returned from a political convention in which there were nine thousand people too many. It narrows down to individuals, and in the end I cannot remember many of them. But I should like to continue the brief acquaintance you and I began there.\n\nI hope the army will not get you \u2013 there are too few conductors who can beat legato (even in Lambert); anyone can hurl _sfs_ at brass instruments. I shall be here and in New York the next few weeks, and could put you up either place. Life in the woods here is rather solitary but pleasant and it is only 3 hours by train from New York. Is there any chance of seeing you in either place?\n\nLet me know. One meets few people who promise something in their own right or as friends. We might become the latter.\n\nBest greetings and luck.\n\nYours, not misanthropically,\n\nreally tr\u00e8s bien dispos\u00e9.\n\nSam Barber\n\n93. Samuel Barber to Leonard Bernstein\n\nThe Hermit, Pocono Lake Preserve, PA\n\nMonday [25 August or 1 September 1941]\n\nDear Leonard,\n\nCurious, our letters crossed. I wrote you c\/o Berkshire Music Centre. Why don't you stop in here? Are you driving? I can put you up anyway for three or four days, but cannot say definitely for longer, as my plans are uncertain after the middle of next week. But I may very possibly stay on here. The only hotels are very dull and expensive (Buck-Hill, Sky-Top) but life here in the woods is quiet and pleasant. I've no servant, at present, but we eat in a nearby dining camp. The woods are beautiful, it is about 2,000 ft high (certainly higher than Bethlehem, a vile place) and I, who am also allergic to rag-weed (not cats) rarely remember sneezing here at all, except from the cold \u2013 the nights are very cool. If you come, bring some books and music \u2013 there is nothing much to do except swim or tennis, if you like. But I think you might like it, and I should enjoy having you.\n\nYou can get me by phone by calling Pocono Lake Preserve and they page me (1\/2 hour wait for you) or telegraph. One drives here by Port Jarvis, Stroudsberg, Pocono Pines. Train is more complicated. I am afraid you would have to come from New York (3 hrs) on the Delaware & Lackawanna to Pocono Summit, where I could meet you. Bring your most decrepit clothes, it is just backwoods.\n\n[Gian Carlo] Menotti and a poet are here at present, working feverishly on the translation of his new opera, but they are departing at the end of this week, probably Sunday, after which I'm alone. There is always room, anyway. Do come.\n\nBest greetings,\n\nSam Barber\n\nP.S. Sorry not to be able to recommend a place to stay here, but there is really nothing very attractive that I know. S.\n\n94. Leonard Bernstein to Serge Koussevitzky\n\n17 Lake Avenue, Sharon, MA\n\n[August or September 1941]\n\nDear Dr. Koussevitzky,\n\nAgain it is my privilege to be able to thank you for another summer of glorious and inspiring study. I feel humble and grateful in the face of the added responsibility that comes with each new advance in my work.\n\nI am rather in doubt as to how to continue that advance now. As you know, I have already received a questionnaire from the army; and, as far as I know, I am perfectly eligible, except for a siege of asthma and hay fever that I am now undergoing. It is therefore difficult to formulate any winter plans; for I cannot be given a responsible position while there is the probability of my being suddenly taken away from it by the army. Secondly, my formal schooling, I believe, is reasonably complete, embracing nineteen years and three diplomas! And, in general, once in the army I should have to forget completely about my work, and begin all over again, God willing, in some uncertain future year.\n\nIn the light of world events, however, I want _least_ of all to shirk my responsibility to my country; and I therefore wonder if I might be of service to the U.S.O.,11 where I could simultaneously serve national defense, and remain in my field of endeavor. Do you agree with this attitude? I am registered, of course, in Philadelphia (Local Board no. 9, in the Land Title Building). Please let me know how you feel about this question, as I want to do the right thing morally and practically; and I feel that I can rely completely on your guidance.\n\nIn devotion and gratitude,\n\nLeonard Bernstein\n\n95. Leonard Bernstein to Shirley Gabis\n\nTamiami Champion (East Coast) New York\u2013Palm Beach\u2013Miami [on board train]\n\n\"August something or other\" [1941]\n\nDear Chipmunk,\n\nI passed Philly last night, but the train didn't stop long enough for me to phone. Besides, you were probably in Hanover, unless I miss my guess.\n\nThe secret is that I'm on my way to Key West for at least a week of escape.\n\n(Do they have NYA12 orchs in Key West?) Key West for a rest, the rest can be guessed, the pest to be blest, the best for a guest, a rest in Key West. And maybe a stealthy boat trip over to Havana on the side. All alone. No one to phone. Sounds like fone, no?\n\nI hope the Tanglewood evils have blown over, and all is in clover. 'Twas all so freaky with Kiki. And all so bleaky. And cheeky.\n\nI'll write you again, with a better pen. The train sways madly, so I write so badly. My love to Rae, & all the rest; I'm on my way to old Key West. (For a much-needed rest.) Have you ever seen a letter that naturally rhymed better?\n\nFour hours of sleep in a small coach seat, you arise in a heap unable to eat.\n\nMuch love to Rae\n\nAnd to Shirley Ga\u2013\n\n(Bis-mark-Antony-and Cleopatra-with asps on her breast \u2013 Oh Mother Nature \u2013 I'm coming!)\n\nLenny\n\n96. Betty Comden to Leonard Bernstein\n\nSummit Lodge, Fort Thomas, KY\n\n9 September 1941\n\nDear Lennie,\n\nWe have a slight business proposition: we have to have our music rearranged. That is, we have arrangements of our numbers, but there are many changes that have been made since, and the music is all marked up and hard to follow. Also the stuff is for 17 pieces, and we want it arranged for 9. Also the music belongs to NBC. In a word \u2013 will you please get in touch with us at Judy's (write to her \u2013 Judith Tuvim,13 226 West 58th St) and tell us whether or not you would undertake the job. There are about eight numbers that need such treatment. Could you figure out a price and quote it to us? And also tell us when you can come into New York to talk about it \u2013 and about how long it would take. We will be back in N.Y. on Sept 19, so if you can write us here before the 18, do that. Otherwise wait till the 22nd and write to Judy in the city.\n\nBy the way \u2013 we're in Kentucky! \u2013 right across the river from Cincinnati! \u2013 and we're doing well! \u2013 and hope you are the same!\n\nWith much love,\n\nThe Revuers\n\nP.S. This is terribly important to us, Lennie, because the bad orchestrations make our work much harder \u2013 so please write and inform us as soon as you can. Again, thanks.\n\n97. Leonard Bernstein to Kenneth Ehrman\n\n17 Lake Avenue, Sharon, MA\n\n17 September 1941\n\nDear Ken,\n\nGod it's been years. Are you still around? One said you were in the army; one said in Hearst; one said in sin. I have much eagerness to see you. Really. Write me now.\n\nI've just returned from Key West, brown & asthmatic \u2013 a new factor in my life, all bound up with September & hay fever. Ghastly \u2013 no sleep.\n\nDefinite career-point reached \u2013 thru with school for good; but plans from now on very vague. I'm off to see Koussie this weekend to discuss same. Perhaps some NYA14 orch., if possible. I need an orch. so badly \u2013 know of any?\n\nI've always had a great yearning to live in San Francisco, as you know. Do you think me mad to have the idea of going, Horatio-Alger-like15 to S.F & climb there? Is it untapped & fertile? How old is Monteux (hm!)? Diable que je suis! But I consider it well & seriously. How's the NYA there?\n\nOf course, all this is thought of with no consideration of\/for the Draft, by which I have already been _questioned_. And you? Please write to Sharon & let me know all \u2013\n\nLove,\n\nLenny\n\n98. Betty Comden to Leonard Bernstein\n\n15 October 1941\n\nDear, dear Lenny,\n\nThe news about the army is terrific! I am so very happy for you. Now you can go ahead and make the plans you want to make, instead of having that hopeless feeling. I send my love and my very best wishes to you.\n\nI am also sending, under very separate cover, the music. Hoping you will still have time.16 Write and tell me if you have. We would never have waited so long had our plans been more definite. But things have been sort of vague and ghastly.\n\nI have tried to explain some of this battered road map as best I can. My suggestion is that you look through it all first \u2013 just the piano parts and the attendant remarks on yellow paper \u2013 and see if it makes sense. If _anything_ at all seems puzzling, please, please ask \u2013 by phone or any way \u2013 because it would be foolish to go ahead with any mistakes. Will you be in town at all? That of course would be perfect \u2013 an hour or so with you once you have looked over the stuff would clear up any questions I am sure.\n\nCould you drop me a line as soon as you get the stuff? Just so I'll know where it is.\n\nMuch love \u2013\n\nMany thanks \u2013\n\nBetty.\n\n_The Banshees_ is the most complicated.\n\n99. Leonard Bernstein to Aaron Copland\n\n17 Lake Avenue, Sharon, MA\n\n[Autumn 1941]\n\nAaron Copland!\n\nYou didn't get my letter from Key West? Clearly addressed to the American Consul in Lima, & you should have coincided, according to the itinerary with same. A long and passionate epistle it was, too, and full of Key West Weltschmerz. Now there's so much to tell that you should know already.\n\n1. I went to K[ey] W[est] to get away from people, & Kiki [Speyer] who came back to Sharon with us, & I went back to Lenox with her, & to see Kouss, who was yachting, so I fled south. Hot & lovely & wonderful down there. Beautiful & tragic, & how I longed for you! I never thought it possible to miss anyone so.\n\n2. All the while suffering wretchedly from asthmatic hay fever, & returned home with same to face \u2013\n\n3. The draft situation. Kouss had written an imploring letter to Mrs. Bok in Philly about it, but that was all rendered unnecessary by my complete rejection in Boston by the Medical Advisory Board! God, I have a lucky star! Not so much the asthma, either (tho that was the legal excuse) as the fact that the particular doctor who examined me insisted on preserving the cultural foundations of the USA, not killing all the musicians. And so I am in class IV! Go, attend to your career, said the great M.D., and that will be yr greatest service. Osanna in excelsis!\n\n4. Since then in Boston, or rather Sharon, except for a spell in NYC where who should I meet in a shady 8th Street bar but [blanked out]17 with whom now very friendly & sentimental! God, the curves of life! And all in secret, too!\n\n5. Have been directly under Kouss' wing for the past month. He keeps me on a string re: some surprise which never seems to come thru. But I'm very happy because he wants me to play with him (concertos, yours possibly), and \u2013\n\n6. There's a great possibility of a guest appearance with the New York NYA Orch now that Mahler's out. Looks very good indeed, due to a good thick letter from Kouss and\n\n7. I'm probably going to play the Ch\u00e1vez concerto with him (Ch\u00e1vez) when he is guest in Boston in February! Isn't that terrific? It just happened & everything is settled but the contract from the management.\n\n8. Tomorrow night Mozart & Ravel concertos in New Bedford, of all places, with Fiedler and the Boston NYA. A slow beginning, this.\n\n9. Then to NYC to confer with Stanley Stevens about the NYA possibility. Pray for me in Rio [...].\n\n10. So much to tell you \u2013 God! This week most theatrical, since the Evans company is in town with _Macbeth_ & I met them all thru Alex Courtnay who was the charming boy in the Tanglewood box office last summer, & now he's in _Macbeth_ , & in my heart, & Evans is coming to my Fiedler rehearsal tomorrow, & it's all so mixed up because\n\n11. I confessed all, like a ghoul, to Kiki, explaining the whole summer fiasco, & now it's all normal again, & she wants to marry me anyway, and accept the double life, or try for my recovery. And Alex blows in on all this! It's such a confused week! But all my weeks are, as you well know. Why can't you be here, & tell me what I should know in such cases? Aaron, I miss you so that I could scream. Write long and hard & soon. There's much more to tell, but I forget, & I must be up early in the morning. I'm waiting for your answer already, so please \u2013\n\nLove, love,\n\nLenny\n\nOn second thoughts, there's more.\n\n12. Delighted at yr great success in the south. Can't wait to hear the 3rd mov't of the Sonata. Have a nice, robust sex life in Rio.\n\n13. A great to do about Kouss doing Billy Schuman's 3rd Symph. K[ouss] was sehr disappointed, & wanted to call it off, & called me in for advice, & I made him do it all, with great cuts which Billy sanctioned wholeheartedly, & it was a great success & now Kouss adores the work. Ach, Gott, my life is full of Kouss & Kiki and Kiki & Kouss & Kiki & Kiki and Kouss & Alex & Olga and Ted and memories of you.\n\nHow do you like my manuscript pen? Look [illustrates with the first four notes of Beethoven's Fifth Symphony]\n\n100. Aaron Copland to Leonard Bernstein\n\nThe Francis Marion Hotel, Charleston, SC\n\n13 December [1941]\n\nLenny, Lenny Lenny!\n\nJust as I was about to arrive back home by air we were grounded in Charleston! I'll be home ingloriously by train tomorrow. South America is all over now \u2013 but I think it was worth it. I saw more than 60 composers and looked at their stuff \u2013 I made about 25 public appearances as lecturer, performer, or on the air. I played the Piano Sonata in B[uenos] A[ires], Rio, and Havana. But most fun was conducting in Santiago de Chile. Musically speaking, i.e. the rest can wait!...\n\nOf course I got your letter in Rio and it certainly sounded 100% like you. Hecticness personified. But now I feel all out of touch again and wish you would write me to the Empire and bring me up to date.\n\nIs there any chance of your coming down the week that Mitrop[oulos] does the _Statements_?18 (I'm assuming that Mitrop. is still planning to do them.)\n\nI haven't any news much \u2013 spent 3 weeks in Rio which is all it's cracked up to be \u2013 was very palsy-walsy with Villa-Lobos and [Francisco] Mignone \u2013 had very good Portuguese lessons (took your advice) \u2013 stopped off for a day in Bel\u00e9m where there is an incredible fort19 and zoo, stopped off in Trinidad and saw Rudi who is stationed there \u2013 spent 10 days in Havana which is wonderful as always (renewed acquaintance with the younger set) \u2013 and now I'm in Charleston, South Carolina.\n\nIt's all been quite wonderful \u2013 but I'm glad to be back \u2013 and keen keen keen to see you.\n\nLove,\n\nAaron\n\n101. Leonard Bernstein to Aaron Copland\n\n295 Huntington Avenue, Boston, MA\n\n[late 1941 or early 1942]\n\nAaron, Liebchen,\n\nWon't you come to Boston for _Quiet City_ , which sends its quiet message to all loving Boston hearts this Fri & Sat? Also the Harris 3rd [Symphony]. Oh I know you've heard it before, but what a good excuse it provides! Are you too busy? No.\n\nI never thought it possible to miss one person so thoroughly as I have you. And if you come \u2013 please come \u2013 won't you bring a copy of the Buenos Aires Sonata? My studio hungers for your blessing.\n\nLove,\n\nLenny\n\n102. Kiki Speyer to Leonard Bernstein\n\n37 Addington Road, Brookline, MA\n\n[late 1941 or early 1942]\n\nDear Leonard,\n\nMy message of wishing good fortune and happiness is long overdue for your new venture. I didn't feel the need to tell you in so many words \u2013 hoping you might feel them even when they were left unsaid. I realize, however, that perhaps now is a good time to tell you that I feel with all my heart that your success is near \u2013 and your studio20 is the first step toward it. It won't fail if you have a little patience, for faith in your tremendous gifts exists in many hearts.\n\nThis letter may seem strange to you \u2013 but I sensed sadness and a little feeling of defeat in seeing you the other day. You have much to give \u2013 but don't be too generous, even to the \"first pupil\" \u2013 same for the rest that are to come.\n\nYour past successes have perhaps been easier \u2013 this one will be really worthwhile for the comfort you will receive and the certain peace for which I think you are searching.\n\nI've never told you that I've always felt you were a grand person with such a beautiful mind! You'll be a great musician and the best conductor ever \u2013 just you watch and see \u2013 you need help from yourself only.\n\nYours,\n\nKiki\n\nP.S. That has killed the bad taste... so chalk it up as your first fan letter.\n\n103. Judy Holliday21 to Leonard Bernstein\n\n[?early 1942]\n\nDear Lenny,\n\nI'm sorry not to have written you in such a long time but many upsetting things have been happening. This morning, after a long siege of distemper, the dog died. It seems weak to be sad and weepy over a dog's death \u2013 as many people have assured me \u2013 when so many worse things are happening in the world, but after all it's just a question of relative importance, or something. What am I saying. I do feel completely lousy. Well, other things too are coming to a close I'm afraid. Nothing has been said yet between Eddie and me, but she becomes more unhappy every day, and as much as I try to bring things back to where they were, I only succeed in resenting her more for suffering because of me.\n\nExcuse me Lenny. When things are all right I can write amusing letters. Just now I can't even achieve a coherent sentence. The strain of the past few weeks has told on me in the form of a hugely swollen cheek \u2013 from wisdom tooth. [...]\n\nAbout Adolph [Green] I've had to tell him a few things about you which may not be strictly the truth. You ought to know about it.\n\nAs of course you know, he's crazy about you and he felt rotten that this business with Lizzie [Reitell]22 should have implicated you in any way. You probably didn't, as I didn't, know the extent to which she went in describing her feelings for you. She threw it in his face that she was very much in love with you and had spent every moment with you in Boston. Naturally the whole thing has been preying on his mind. He was very shocked when I told him I had told you something about the way Lizzie felt and he wanted to keep you out of it. But he worried nevertheless about how you felt about Lizzie. I suppose it was a mixture of being hurt himself and not wanting you to be hurt by Lizzie, and not wanting your friendship to be hurt. However, I told him that you not only were not in the least bit in love with Lizzie but that the idea of any contact with her horrified you. Maybe I shouldn't have stretched the truth quite so much but it gratified and consoled him to hear it so much that I wasn't sorry I had put it so strongly. I just felt that you ought to know what I've been up to. I think it would probably be just as well if you never brought up the subject, which you wouldn't want to anyhow. The thing seems well ironed out now and Adolph has succeeded in making a sort of monster out of Lizzie for his own calm, and since it's helping him to get over the hurt, it seems just as well.\n\nIf you can make anything out of this letter, write me and tell me. I would love to see you. We're leaving the Park Central next Tuesday and Adolph said something about going up and joining you. If he does, don't tell him I messed about in this, please, because I suppose I shouldn't have.\n\nWrite me soon though I don't deserve it, and have a successful time.\n\nMuch love,\n\nJudy\n\n104. Leonard Bernstein to Shirley Gabis\n\n295 Huntington Avenue, Boston, MA\n\n[?early 1942]\n\nDear Gabeling,\n\nThanks, thanks, thanks. And I really didn't mean to cause a furore about the ankle, which, by the way, is prospering beautifully. It's now mauve.\n\nI want to send you a thankyou present. Don't shriek \u2013 I don't usually, but now I do. Something in the Brillo price range. What do you want? Halvah? Defense stamps?\n\nOf course there's really no use in trying to apologize away the situation that arose in Phil[adelphia], like going to the opera with A[dolph] and a slight Liz session. You understand & know. I worry greatly about the Adolph-Liz marriage.23 From what she says, it won't work much longer. God \u2013 what a blow to Adolph that will be! I spent the whole night trying to change her mind. Did Adolph mention anything to you?\n\nWrite soon, work hard, & give my best to all who hate me.\n\nLove, & to Rae,\n\nLenny\n\n105. Leonard Bernstein to Aaron Copland\n\n[early 1942]\n\nAaron darling,\n\nI had the feeling (silly but que faire?) of having perpetuated a misdemeanor. Lousy have I felt about it since, tho I know you understand. Or do you? The last days, nay moments, of Hyatt, ere the fateful return of the Vanishing Virginian. You must know. The V[anishing] V[irginian] returned, 4:00 a.m. Sat. morning, & called & I shall see him tonight, & it won't be easy to take all that barrage of Boston brashness after the dulcet quietness of Dick [Hyatt]. And so the answer is still to be found \u2013 I'm still searching. Which is as it should be, 23 years & everything, & a late start, considered.\n\nDick H. is convinced that nothing matters to me but Copland. I gave him a chronological recital of you last night. He was thrilled, & his extravagant remark is almost completely true. When everything is said & done & over, I guess the core of the whole thing will still be Aaron, & there's nothing to be done about it.\n\nI want an _Hurricane_ , much talk & activity is afoot for a second performance, probably in Sanders Theatre, & possibly for Russian Relief. At this moment I am waiting for the phone on same. Collier is really excited this time about the publicity & anything can happen. Pray. Any news from [Alfred] Wallenstein?\n\nI want you to have this _Monitor_ review, as an example of the worst possible tripe by a fuckfool who obviously didn't attend the performance. Note especially the Gertie Stein paragraph.\n\nTake care of Jean & advise him right. He's had more than his share of trouble already.\n\nWhen do you leave for the Hills?\n\nAaron, Aaron, Aaron. My God.\n\nL\n\n106. Leonard Bernstein to Aaron Copland\n\n[before 21 April 1942]\n\nAaronchen, Liebchen,\n\nHow I was furious and raging at this little frustration! I did not spend Wednesday night _beim_ studio, & found your wire only upon returning the next morning. What to do? My head spins crazily. Cleveland! He's been out of Cleveland for ages now.24 And at that moment came the second wire. The shame, and disappointment of it all!\n\nIt would have been wonderful to see you. God, yes. On our first beautiful spring day. And we would have walked in all Boston's parks and spoken long, quietly & with the heart. Such gab. Can't you come anyway? We must have a session on that Copland youth opera, you know. The master's interpretation. Hell, I miss you so.\n\nVoici le printemps, et moi sans amour.\n\nLife has only just stopped being hectic. Since I've seen you last I have sped through a series of great triumphs involving a heavy social life, and all ending in complete nothingness. I've turned down jobs, gotten into a row with Paul Lukas, who was all ready to wire Hollywood to give me a job (big talker), miffed a good chance with Irving Caesar, etc. I should have come to NYC this year, dammit. Even Kouss has shown his usual cooling process \u2013 not a word all month about playing with the BSO. I suppose I haven't approached his model for me sufficiently. I haven't changed my name, or learned to schmoos, or become a dignified continental. The hell with it.\n\nEt voici le printemps, et moi sans amour. But wait.\n\nLovely letter from David, who seems to be supported by one Aaron Sapiro, & has won Stravinsky's heart. Very mysteriously interested in my summer plans, but can't divulge why.\n\nWe must also meet to plan _our_ course \u2013 or is it still on?\n\nWhat a year. It's hard to keep calling it transitional & let it go at that. Nothing happening but the Institute Concerts, & the happy prospect of the _2nd Hurricane_. I've been dying to give a Sonata recital, including yours, & Schumann, & Scriabin, & Scarlatti, & I can't find anyone to rent me the hall. God dammit. But I'm learning the Sonata (3rd mov't is a bone in the throat), & I'm working very hard on my Key West piece, which ought to turn into a ballet.25\n\nSorry you saw [David] Glazer. You weren't supposed to know that the Clarinet Sonata was being done!26 Direct defiance of your orders. But, hell, I've got to hear it. You will condone, won't you?\n\nDo write, or come to Boston. Love to everyone & kiss Jean27 for me. (The worst part of all that wire business was that I _was_ free on Thursday from noon on!)\n\nMuch love, & Goddammit,\n\nLenny\n\n107. Leonard Bernstein to Aaron Copland\n\n[Boston, MA]\n\n[early 1942]\n\nDear Mr. Copland, dear Aaron,\n\nA more sober word today. Shirley is sending you the reviews \u2013 all good, and one even _interesting_. I hope she remembers to get the _Monitor_ this afternoon.\n\nI'm rather serious suddenly about making that piano reduction of _Billy_ [ _the Kid_ ]. It would be a \"suite from the suite\", maybe using a connecting motive & only a few numbers. The orch. suite is terrific in performance, but I think all that return of slow music at the end is a bit letting-down after the excitement of the middle. It occurred to me that the suite would be a real killer if it ended with the macabre dance, in C major:\n\nAre you shocked? Of course it's too late now, but the piano suite could well do that. (My writing suddenly looks just like yours.)\n\nAaron \u2013 this is important. I was at the Peabody Playhouse this morning trying to sell the _2nd Hurricane_. They'll let me know, they say, after discussing it with the big shots. _You must_ send a letter of recommendation for me (they're wary about whom they entrust their kids to), saying that I'm the ideal one to do it, that I've had all this experience, am Koussy's prize, etc. etc. Can handle children, etc. I didn't think I'd have to sell myself, but so it seems. Write _as soon as possible_ to:\n\nMiss Hyek, Peabody Playhouse, 357 Charles Street, Boston.\n\nMake it good. It means much for both of us, I hope. I'm so sleepy. And rather useless. But there's a certain _\u00e9lan vital_ left, pretty latent now. I love Jean [Middleton],\n\nAnd you.\n\nLenny\n\n108. Aaron Copland to Leonard Bernstein\n\nHotel Empire, Broadway at 63rd Street, New York, NY\n\n26 May 1942\n\nLensky,\n\nI loved having your letter, even though, as you guessed, the first paragraph about \"misdemeanors\" was completely unnecessary. Silly is a better word for it. You could talk till doomsday but I could never hope to make competition with the D[ick] H[yatt]s of this world, in my own mind, I mean. You've convinced me that I have my little niche ( _big_ niche, really), and on my own \"acceptance\" theory, that's what I accept. As Edwin [Denby] so prettily puts it in this number of _M_ [ _odern_ ] _M_ [ _usic_ ]: \"It is the thrill of needing, not the delight of having.\" But in accordance with the theory, I'm right ready to always be on either side of Edwin's comma. If this is too much literature, it's your own doings, so there!\n\nI'm still talking about you and the _Hurricane_. Kouss was here \u2013 and we spent all Sunday afternoon together raving about one Leonard. You ought to be very proud \u2013 quietly proud \u2013 to have two such supports. Just as I am quietly certain that if you hang on it is bound to end someplace good. In the meantime I called Wallenstein and of course he was all set with rehearsals, assistants, etc., \u2013 which is not so good.\n\nI'm off to Stockbridge on Monday. I'm hunting frantically for a cook-houseworker. I've had a few nibbles \u2013 but haven't taken anyone on. All the young uns are seemingly in the Navy. This will have to be solved by Monday, come what may.\n\nJust to give you my plans: I'll be down to NY of the 11th for the broadcast of the _S_ [ _econd_ ] _H_ [ _urricane_ ] and down again on the 17th for conducting the _Outdoor Overture_ with the Goldman Band. When you know your own plans better we can fix up a time for a quiet interlude for us both in the Boikshires.\n\nKouss was fighting mad about the Festival, but anyhow I think we'll have a school. He says it's all to be decided tomorrow (Wed.)\n\nSince I got back I've had an offer of a ballet commission from the Monte Carlo company, but we are still dickering about a subject. If it should work out, I'll have to do it fast in June. All this is just as you prophesized, but keep it under your hat.\n\nI can't seem to connect with Jean. I'm always out or he's always out. Donald [Fuller] says he wants J. to live with him in NY this summer. He (DF) played me 1st movement of his Symphony. Terribly complicated it is.\n\nDid you see the _Times_ and our joint letter?28 What say.\n\nGood bye and be good. If I said what I felt the paper would melt. (Poetry).\n\nA\n\n109. Leonard Bernstein to Aaron Copland\n\nSharon, MA\n\n[8 June 1942]\n\nAaron darling,\n\nThe second _Second Hurricane_ is over,29 and I am a limp rag drying out in the Sharon sun. And a lovely sun it is. The performance was, as I guess you know, carried out with piano only, and it was apparently even more successful than the first. No reviews, naturally, unfortunately. We covered all expenses, doing it all on our own. We did our own publicity, management, printing, etc, & came out better than the Institute of Modern Art has, with all its fancy pretensions. And so this is a brief but lovely period of smug nose-thumbing at the decaying aristocracy. WCOP here gave a whole day to plugging the thing, & I had an interview during which I gave quite a lecture on your history. And Boston is all agog and all aware. What a team! _You write 'em, kid, & I'll do 'em._\n\nAnd now a month intervenes before the Koussevitzky Memorial Foundation, Inc. (sic!) takes over. I suppose you know that setup \u2013 it's staggering. There are five big shots: [Howard] Hanson (why?), Olin Downes (God!) [Gregor] Piatigorsky (I guess he has money connections), [Richard] Burgin & Kouss selbst. And private money. Quite a boy, ce Serge-ci.\n\nDuring this month I would like to make a visit with you. When? And seriously, do you suppose I could live in your house this summer? I'd adore it. Think it through, from the points of view of the _exec. & admin. depts_. Have you an extra room? Did you get a cook? Is it gorgeous & quiet up there? I miss you terribly.\n\nDid you see me publicly announced as a two-fold assistant in the _Times_? I'm rather glad of that announcement. It gives me dignity.\n\nRen\u00e9e [Longy Miquelle] told me of her Dallas idea & your rather discouraging reply. I think however that this might really be something. Jacques Singer, the draftee, was very young too. There _is_ a precedent. And what an opportunity! Perhaps Kouss should write a letter. God, I want an orch. Can you help?\n\nMuch love \u2013 & write to 295 Hunt[ington Avenue] now. [...]\n\nLenny\n\nTry this one on your piano:\n\nSi ces seize cent soixante six sensus-ci sont sur son sein sans susser son sang, s\u00fbrement ces seize cent soixante six sensus-ci seront sans succ\u00e8s, c'est s\u00fbr.\n\nIt looks like no local broadcast. Maybe later.\n\nWhat of Downes' retort in yesterday's _Times_? Sort of leaves the whole thing back where it started.30\n\n110. Leonard Bernstein to Aaron Copland\n\n40 Charlton Street, New York, NY\n\n[?June 1942]\n\nDear Aaron Copland, Earth-Scorcher, Location-Adorner,\n\nCharming, charming to get your letter. I know I've been remiss, as they say in elegant diction, but so have you, and I've been moving, for a change. Look! This is my fourth address this year, and a few more are coming up soon. I can hardly keep track of myself. I find myself getting off the subway at 23rd instead of Houston, which is now my locale (and not a bad one, either, if you like that sort of thing). It's always very difficult when you move so much: you have to spend all your time making sure that someone who wants to get in touch with you has found the correct forwarding address, which they haven't, of course, since the Chelsea is so remiss, as they say, about giving out forwarding addresses. For instance, Henry Simon, who wants me to do the piano parts of the operas for his new book on operas (and for $2500 yet!) had to get in touch with me via Sharon! Today I was back there complaining away in a loud bitter voice and found to my astonishment a dozen or so letters that had never been forwarded. And yours among them, just so as not to bore you. And a letter from Edwin Franko Goldman saying that there would be a rehearsal of the Band _yesterday_ \u2013 I tell you, it's infuriating. Oh yes, I forgot: I'm conducting the _Outdoor Overture_ with the Goldman Band June 19 in Prospect Park and June 20th in Central Park. It's good fun seeing the old notes again; though I'm completely nonplussed by all the fancy instruments, their incomprehensible arrangement on the page, and especially on the stage. And I must memorize the damn thing, since your lovely big manuscript score won't fit on the bandstand. All in all quite a job. Goldman, in fact, had asked me to be his assistant this summer, and then pulled a long face, saying that the budget would not allow an assistant. Sounds like a typical Bernstein, doesn't it?\n\nTo make sure that you'll keep reading this, I'll start a new paragraph. It must be a strain.\n\nWho(m) do you think called me up the other day from his house in Westport?31 And he wants to do my Symphony in Pittsburgh next fall, and he loves it, and he wants me to conduct a program anyway, and maybe to do the Symph myself! Lovely lovely news. But he is most anxious for a fourth movement, insists it's all too sad and defeatist. Same criticism my father had; which raises Pop in my estimation no end. I really haven't the time or energy for a fourth movement. I seem to have had my little say as far as that piece is concerned, and I want to get on with something else. And parts have to be made. Real young composer tsurus.32 Apropos of which, I saw Marion Bauer33 t'other day, and she insists on a young composers' committee meeting for next season, of which I shall be one. Feels sort of like a composer. But my real function, I find, is to be the middle-man between all the pairs of antagonists and antipathetic little cliques here. They're all my friends, and hence none of them is really my friend. I go around justifying Berger to Schuman, and Schuman to Bowles, and Thomson to Schuman and Bowles to Diamond, and I'm always having dinner with all of them but none of them ever has dinner with any other one of them. Good Lord, I'm lucid and articulate tonight! Must be the invigorating air (of the _Outdoor Overture_ , I mean, not, certainly, of Charlton Street).\n\nIt just looks as though I'll never see you. Though, b'God, if that Simon job comes through (and there's a competition factor, with old Szirmay)34 I'll take the dough and fly to Mexico to see you. Nothing can stop me, once there is dough. Really, Aaron, I don't understand how and why I get along at all with you away so long. And here's what I mean:\n\nThe Frau-sessions have borne some fruit.35 Little green fruit, of course, but fruit. The main thing being that I can't kid myself any more. Kid myself, that is, into thinking that I have a closeness with someone when it is all really wishful thinking, or induced, or imagined, or escape from being alone with myself, etc. And so, one by one, all the old relationships tend to fall away; and I find that I'm not at all interested in seeing anybody \u2013 really \u2013 whereas I used to run and see anybody at the drop of a hat. This all makes the trouble harder, of course; since I still hate being alone, and yet don't want anyone in particular. And that's where you come in; cause you're the only one that persists and persists, come hell or high water. And I love you and miss you as much as I did the first month I knew you, and always will. Believe that, Earth-Scorcher, it's so real. And then this wish for closeness always manifests itself in a sexual desire, the more promiscuous the better \u2013 giving rise to experiences like being taken (by Pfb [Bowles], of course) to a Bain Turc (or is it Turque?) and seeking out the 8th Street bars again. But I'm not attracted any more to any one I find there, and it's just as horrible as if I hadn't gone at all. One of those unpleasant stages forward.\n\nI'm living, of all places, in a high school! I have a whole school to myself; but I really live in the apartment atop the Little Red Schoolhouse High School in the Village. It's quite nice, but I shall have to move again in July when the real tenant returns (a lovely lady who is unfortunately in love with me) and then my troubles start over again. The Staten Island venture seems to grow more dismal all the time. It's so complicated without a city apartment, and I'll have to have one of those. I won't trouble you with that problem: it involves too much Bowles.\n\nDavid Diamond is going to study the piano with me! And don't let him kid you into that \"outdone me\" stuff; the soldier (what a boy!) would rather have come with me, but DD had done all the _work_ , and the soldier was afraid of a scene. As it turned out, he came to my room the next night, full of love and amusement.\n\nOne final experience \u2013 and then we close, with love to Victor.36 Last night I resolved to stay home for a change and cook my own dinner and study. Which I did; but just as I had begun to work, there was a blackout. I went up on the roof to see it all (I have a marvelous roof) and found a young soldier there, in the blackout, who, it turns out, lives with the housekeeper downstairs. Sure enough, he knew me, had attended my concerts, worshipped me; and there was fun. _Until_ the lights went on; and he turned out to be so fat that I could hardly stand it; and now I'm in a Bernsteinian pickle, with an adoring fatman and no wish to see him and life in a high school is hard. Moral: if you need sex, don't go searching everywhere \u2013 look in your own back yard. Which does _not_ necessarily apply to you! All kinds of love, and write soon.\n\nL\n\n111. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nc\/o Welsh, Box 411, Lenox, MA\n\n11 July 1942\n\nDear R,\n\nI heard tell that Baudex had called you. Qu'est-ce qui est arriv\u00e9? By this time, something must have happened. If not, you had better come out and replace K[athryn] W[olf] by brute force.\n\nTanglewood is all different this year. Not half the spirit or the excitement of previous seasons. And I, as Aaron put it, have been \"kicked out upstairs\". Because I have been elevated to \"assistant\" I never get a chance with the orchestra, since all the time is taken up with the conductors' preparation for their concert. I am doing jackal's work \u2013 but that's great experience for one who is too easily a lion.\n\nWalter [Hendl]37 is apparently in a bad state. His mother seems a real problem, and he doesn't appear to be able to concentrate on his work. He is terribly worried about his concert on Saturday.\n\nSo \u2013 he is in a bad state, you are in a bad state, I am in a bad state. Let's get together and form a secessionary State Confederacy!\n\nAnd I've been rather stupidly ill all week with a bad stomach. \u00c7a va mieux maintenant, mais il me faut s'endormir. I can't be well except when I'm _too_ busy. Remember a certain _Hurricane_?\n\nWrite & tell me all.\n\nLove\n\nL\n\nThanks for the \"Philadelphia Story\" \u2013 I gave it to all those interested.\n\n112. Leonard Bernstein to David Oppenheim38\n\n158 West 58th Street, New York, NY\n\npostmark 22 September 1942\n\nAre you dead?39\n\n113. Leonard Bernstein to David Diamond\n\n295 Huntington Avenue, Boston, MA\n\n[September\u2013October 1942]40\n\nDavid,\n\nThank you, thank you, thank you. For so many things. For housing me. For leading me straight into the arms of a great and quiet and radiant joy. For making one sacrifice after another. For the burst of temper, which showed that you still had enough feeling & respect for me to worry about my weaknesses. For being such a remarkable host. For showing me at last what a terrific talent you have. That I now know, & understand.\n\nAnd for every one of those things, there is a corresponding apology. None of which I need itemize. You're really very tolerant, and you know very well what you know.\n\nShirley is very, very fond of you, despite her first confusion. And that's an indication of something phenomenal.\n\nPaul du Pont41 has his clothes, & I hope to see _Porgy_ this week. [...]\n\nBless you \u2013 & please write.\n\nLenny\n\n114. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\n158 West 58th Street, New York, NY\n\npostmark 22 October 1942\n\nDear R,\n\nI think something may be breaking (if the union doesn't make complications). Next Wednesday night, Oct 28 at 10:30, CBS, I'm conducting a show called _The Man Behind the Gun_.42 It will be swell fun, & 50 dollars, & if I'm good it's a steady job! Pray for me and listen (music by Diamond).\n\nLove,\n\nLenny\n\n115. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nAdvanced Music Corporation, RCA Building, Rockefeller Center, New York, NY\n\npostmark 8 December 1942\n\nMy dear Madame Miquelle (!)\n\nJust wanted you to see my gorgeous new official stationery. Here I sit, at a desk, important as a bookworm, at $25 a week, doing little or nothing, waiting until the great bosses decide what my function here should be. This is the great musical industry of Warner Brothers Pictures, and is called the Music Publishers' Holding Company (impressive, n'est-ce pas?) and includes four publishers, Harms, Remick, Witmark, and Advanced, where I am. It's all tr\u00e8s d\u00e9lass\u00e9 et doux, and I must be here from 10 to 5:30, just being around en causant, fumant, causant, etc. And I have my pupils come here for their coaching. In fact, Bobby is coming here in an hour to play my pieces for me. He's broadcasting them today. It's great fun.\n\nAnd then the Riobamba Club opens Thursday,43 and I've written the title song, \"The Riobamba\" which will be plugged and plugged, and may even be a success, if you keep your fingers crossed.44 That keeps me plenty busy with rehearsals. This song is all that is left of the once hopeful Key West Piece [ _Conch Town_ ], but I think it makes a better popular song. And speaking of such matters, I finished orchestrating the first movement of _Jeremiah_ , and it's being copied, and I have no time to devote to the second movement, but I swear it will be done, if only as a gift to you.45\n\nAnd tomorrow night I do Aaron's two-piano piece46 with him in Town Hall. And that has me tied up in knots. I wish you could be there to hear it. Don't you think I'm really very lucky, for a young Boston yokel only three months in New York?\n\nI'm flabbergasted at the fire in Boston,47 and I'm told that Bob Lubell's sister was a victim. It's too shocking to believe. Anyone else we know?\n\nTake care of yourself. And don't write Edys [Merrill] [a] card, and leave me in the cold! I know I deserve such treatment, but be merciful, and come to NYC soon.\n\nMuch love,\n\nLenny\n\n116. Samuel Barber to Leonard Bernstein\n\n166 East 96th Street, New York, NY\n\n[?1942]\n\nDear Leonard,\n\nAwfully busy with unimportant things, Curtis opening etc., firing Filipinos, looking for a little house in the country to run off to. I can imagine what you are going through with the draft question: ever since I've heard that any use of musical gifts in the army or U.S.O. is discouraged, I'm all for getting out of it \u2013 at least for the present. With some imagination you ought to be able to develop an impassioned asthmatic wheeze which would send them rolling. Let me know what happens. Was Koussevitzky of any real help?\n\nWhen are you coming down to New York? I've told no-one that I am back, and it is very pleasant. It was raining very hard on my terrace; I spent the morning in bed \u2013 there was a Baudelaire on the shelf and I sank into a lazy stupor and felt as sinful as if I were 16 again. The Filipino came in with new headlines about the Russian war. I felt worthless but happy. The next time I shall have a box of chocolates! Interesting life, eh? Now the sun is out again, damn it.\n\nBest to you,\n\nSam B.\n\n117. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\npostmark 16 December 1942\n\nDear D,\n\nVery comforting to know that you exist. Of course come down to the city before Draftuary the first.48 Let me know when. Above is my new address, apartment extraordinaire: the rest is PL-5-2966. Thanks for the letter.\n\nBest always,\n\nLenny\n\n118. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\npostmark 15 January 1943\n\nDave,\n\nIt appears that WNYC wants the Clarinet Sonata on their Festival of American Music sometime between 12 and 22 [February]. Could you possibly do it, via a Lincoln's Birthday week or some such device? Probably not, but I'd love it if you could, needless to say. Otherwise I shall have to get a NY guy, maybe Eric Simon with whom I'm rehearsing these days. Even he would be second choice, obviously. You're the top 5\/8 man in these parts [and you] know it.\n\nIn the latter regrettable case, you'd have to send the score so that it can be used, or copied, and more scores made. Depending, in turn, on when you plan to do it in Rochester, if you still do. Do you? You see, I tend to lose track of your vibrations when you don't write.\n\nAnd if you \"can't write\" as you put it, at least write saying you can't write.\n\nLet me know Jack's49 address, the story of the Sonata, and what leprechauns (or have they become gremlins by now?) are goading you through these tough days.\n\nLove,\n\nL\n\nI tripped across Whitman again yesterday.\n\nI hope you trip too.\n\nRead the Calamus poems again \u2013 they may have a new angle these realistic days.50\n\n119. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\n[before 19 January 1943]\n\nDear D,\n\nIn the first place, that Complexion Soap with Lanolin, Macy's 15 cakes for something or other, is _not_ the same as our soap, which is clearly labeled \"Lightfoot's\", and far superior.\n\nIn the second place, I have two more pupils and have become violently busy.\n\nThe score (a beautiful job of quasi-legible gray ink) was two days late. I was all for jumping off Harvard Bridge when I heard the New England bitchvoice of Elizabeth Allen saying \"It's an inflexible ryule! I'm afraid I cahwn't accept it. Good-bah.\" At which moment [...] Alex Thiede, a prominent Boston semi-conductor (really semi-Boston too, being a good guy) who called influential folk, explained how wars made trains late and unpredictable (lucky if they're going in the right direction), & the thing is now before the board of trustees, who must decide if they can abrogate the rule relating to deadlines, and suffer the score to enter the contest. (There being such a plethora of American talent on the boards, of course, that they can afford, the bastards, to be snooty about what scores they accept, the idiots, when the issue is a day's lateness, the lice.)\n\nIt is to be prayed for, and keened over. I charge you to organize a wake in Rochester, with Wing-of-Angel Avshalomoff as chief crier and you as bartender.\n\nThis empty prattle is induced by the extraordinary emptiness and prattledom of this office, where I still have nothing to do, and grow weary just pretending to be absorbed. For the last hour it has been the London _Times_ Crossword Puzzle \u2013 something you'd go mad about. I must admit I'm writing a song, stolen heartlessly from the Double Concerto of Brahms. Mighty purty. Has to have words about love, and I'm stymied. Me, primus amoris filius! It's my last vestige of self-consciousness in the matter, being afraid I'll reveal too much. I'm sorry to hound you, but I love you very much. You will, of course, destroy this letter. Unless you some day give up the clarinet in favor of blackmail.\n\nThe office grows gradually madder, each songwriter fighting for the phonograph to demonstrate his latest horror. There's a new one about to be plugged \u2013 a real nightmare \u2013 called \"Each Time I Puff On My Cigarette\". Watch for it, if only in order to avoid it.\n\nIf you think your mad dreams about mixed up sexes are confusing, listen to this little job of mine that I cooked up in Boston last weekend. I was due at a small legendary village on Long Island to give a lecture. I arrived at a completely deserted little station \u2013 _end of the line_ \u2013 called Arnold Park. (Related, among more obvious things, to Ozone Park, on my way to Rockaway, where I lectured this autumn.)51 No one was there to meet me, but I waited a few minutes and was suddenly confronted by the most gorgeous girl imaginable, and she loved me and I loved her. At which point some dope called up to inquire about a rehearsal. I enclose 2 & 9 for a complete interpretation by Zolar Oppenheim. It is understood that this coupon puts me under no obligation whatsoever.\n\nBless you for all the yeoman work you did last weekend. I could never have done it without you [...] especially the last five pages of the Scherzo,52 which came out very strangely primitive. Also interestin'. And my best to Jack who was truly noble.\n\nAnd to you \u2013 \u00e7a va sans dire.\n\nLenny\n\n120. David Oppenheim to Leonard Bernstein\n\n19 January 1943\n\nDear Len,\n\nOf course I'll come down to N.Y.C.! Did you think I'd let an outsider play it? I can be down any time between the 14th and the 22nd. On the 13th is our \"big dance\" and of course I have to go, duty calls. I like dancing but not \"big dances\". Maybe this time will be different. You see I'm working on dozens of theories about myself, all more or less inspired by the writing of our friend Karen Horney.53 And I've really had some results. Tell you about the whole business in N.Y.C.\n\nI've been working hard since I saw you last. I'm at school at 7:30 a.m. and leave at 5:30. And I average 3 hrs a night of reading etc. Sunday is a field day for work. And I feel fine for it. The important thing to me is that I want to do it. None of the horrible neurotic impatience of yesteryear. If my progress in self-analysis keeps up I will be a new man and better, one day.\n\nI didn't get your letter or your card until yesterday. The fuel ration board thought we could heat the joint [for] a month with the amt. of oil we ordinarily use for a week. So now I have a room in a small \"hotel\" (8 people live here). It is atop Rochester's best restaurant and it is warm, private and _alone_. I can really be by myself now more than ever. It is not a neurotic compulsion either. I just have enuf to do and enuf to think about so that I don't need anyone. Pardon the digression.\n\nMy address is: 33 Chestnut, c\/o Belvedere Hotel.\n\nI really had no idea (?why) you would write me. If you keep writing I'm sure I will. I am going to have a lot to tell you I think. So \u2013\n\nWhen I do come to N.Y.C. I want a couple of appointments with the Frau if possible \u2013 for the purpose of seeing if I am on the right track.\n\nSeeing mother was terrible. A tight-stomached unrelaxed experience full of hate or something very strong and uncomfortable. Good that it was short. That phase of my life is still unilluminated \u2013 a few ideas but not much evidence. Wish I understood it. I'm sure it is significant. [...]\n\nA plague on Lizzy Allen, and bless Alex T[hiede]. They had damn well better accept the score or I will personally pluck Liz's pubic hairs out one by one with ice tongs.\n\nWe are all \"dovening\"54 for you here at Rabbi Hanson's55 Schule.\n\nI have had no repercussions, at least not negative ones, about anything that happened in N.Y Lenny. None, do you hear me \u2013 none. I never felt better, see.\n\nToday I rendered at the console \u2013 piano exam \u2013 McHose56 \u2013 Head of department decided to wean me on Haydn symphonies. So my first score reading at the pianoforte begins. Hope I am up to it. I dazzled him with a Haydn sonata \u2013 worked up a fever pitch in 7 days. Also I am to learn accompaniments to clarinet repertoire (pronounced repertwa with a little soft palate thrown in). An excellent idea. Horn players learn horn accompaniments etc. So when we teach we can drown out our little aspiring bastards \u2013 if there is a piano in our attic.\n\nYou can contact Jack c\/o Eastman School of Music. I can't recall his home address now.\n\nI am in a strange relation to Mad now \u2013 very complex and to be told about in detail with you in N.Y.C. and incidentally, I refuse to copy a note or draw a single line in N.Y. Understand!!! I have a copying pen and some ink and I practice on every orchestration paper I turn in. I am doing lots better. You would be proud, Leannish.\n\nHow is Edys?57 My best to her. In the short time I knew her I became very attached to her.\n\nIt was awfully good hearing from you, son.\n\nDave\n\n[Musical quotation enclosed on a slip of manuscript paper]:\n\nWhat is this? \u2013 been on my mind for weeks.\n\n121. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\n[before 14 February 1943]\n\n\"Dear Len \u2013 Terrible and wonderful things have happened to me since I saw you in NY.\"\n\nMe too. And almost the identical symptoms, for the identical length of time. Fantastic. More about it in a separate letter.\n\nThanks, anyway, for the case history. I was getting worried. The Frau has heard it all.\n\nThe 6\/8 theme (the \"lousy theme\") is, i'faith, from a Mendelssohn Trio, I think. Check on it. D minor probably.58\n\nAnd _will_ you be in NY for the 14th? Just say it, so I'll know.\n\nAnd did you take the 6 Pieces back to the Northwoods? [...] They are nowhere to be found. Are you copying them? Is the Cl. Sonata being copied? Sorry to trouble you with these worldly problems now, but they're _reality_ aren't they?\n\nThe Frau says she now has a superficial report from the Rohrschach (first time I ever wrote it) test, but that the main, detailed job lies ahead. She says that it's full of revealing and fascinating ideas and facts. She is drawing up a paper for you \u2013 prefers not to write you \u2013 so you must come down and hear about it.\n\nI can never tell you on paper what I went through yesterday. It was the most formidable day of my life. I'll wait till you come. Bless you in your emotional trials. Best to Mad.\n\nAnd love to you,\n\nL\n\n122. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\n[early 1943]\n\nDave,\n\nThis is one of those letters that the OPA59 or WXQR60 would list as \"non-essential\". But I'm all full of strange mixed feelings, and you've been elected to receive them, willy-nilly. Brace up, boy, they're wild! All having to do with petty ridiculousnesses (!) like having had to sit with a fuming but empty lyric writer named Eddie DeLange, while he burst his stolid head trying to get a lyric to the tune I stole from the [Brahms] Double Concerto. He finally came through with a little horror called \"Exactly As You Left Me\" (That's How I'll Beee When You Retoin), which nauseated me so that I tore homeward to find a little peace & do a little work, only to find that the apartment had been invaded by a small army of plumbers, wreckers, carpenters, et al, who have orders to remodel the apartment, break down walls, rip out waterpipes, put in walls, put in doors, etc., for a whole week, without any warning. Christ. The noise. The dirt. The lack of walls. And of water. It's a panic. All of which leaves with only one resort \u2013 to write a non-essential letter to Uncle Dave, and get some Peace by Proxy.\n\n_Are you serene?_\n\nDo you feel jittery when walls fall around you?\n\nDo you get peevish when you see the bathtub on its side?\n\n_Take_ Oppenheim's _Little Love Tablets_.\n\nDelicious to chew slowly!\n\nNibble your Neuroses away!\n\nBut tonight is better. I played a wow concert at the Stage Door Canteen,61 and they screamed for more, and I gave autographs, and was f\u00eated by three Australian airmen who represent a fresh, new beauty in the world. I wish you could have talked with them, heard their eternal-young speech, seen them getting drunk on milk. All with a tranquil and unconscious bravery that surpasses heroism \u2013 they leave for the raiding-grounds tomorrow.\n\nBe good to yourself: work hard, stay healthy, and God, man, keep away from that Army \u2013 wonderful Dave.\n\nL\n\n123. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\n[February 1943]\n\nL\u2013P\u2013,62\n\nNow I'm in _your_ class. I have a desk, a phone, stationery, \u2013 even a Steinway baby grand. The metamorphosis is awful sudden. Actually I'm back on the old lot near the big gas tank where I wrote _Our Town_. Everyone seems very pleased to have me around \u2013 even including the Big Boss, Goldwyn himself. I may even be writing some songs with Ira Gershwin as lyricist. (All subject to change, of course.)63\n\nAnyway, I'm here. Holy Wood is surprisingly the same as when I was here almost 3 years ago \u2013 outwardly, I mean. There are even the same extraordinary young men with wavy hair and impeccable complexions who used to be on Hollywood Blvd \u2013 still around. I wonder how they escaped the Army. And there's plenty of the Army too. Well, it's a fantastic city.\n\nI'm not settled yet \u2013 just in a hotel. So maybe it's a little early to invite you for a weekend or a year. I must say it's relaxing to get away from N.Y. \u2013 even tho my conscience pains me no end when I think of all my skipped responsibilities.\n\nI hope you're writing me voluminously. But don't forget who your real pal is \u2014\u2014> ME\n\n124. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Hollywood, CA\n\n13 February 1943\n\nDear Lensky,\n\nThe blow has fallen, they are not letting me go back to NY for the concert. I'm naturally disappointed as hell, but I got myself into this jam so I might as well take it like a man (if possible). I don't know what Kenneth Klein, Heinsheimer and Saidenberg will decide to do \u2013 maybe call off the whole show, but if not I've suggested you play the Sonata instead of me. You'll probably be hearing from them, and if you can't wait, call Heinsheimer. In the meantime I feel like something between a heel and a fraud. However, it's interesting to know what a bad conscience feels like for a change. I suppose way in the back of my little head I had been trusting to my usual good luck to get me out of this mess. Nice to think that Dave [Oppenheim] is arriving in time for the concert. Nice also to think about you \"walking streets\" because Hollywood Blvd is a continual temptation which is bloody hard to resist. I'm still not settled in a place of my own, nor have I looked up anyone except Jerry Moross.\n\nWell, there's nothing for me to do but sit and wait for the dawn of the 18th.64\n\nAs always, you slave,\n\nMe\n\n125. Leonard Bernstein to David Oppenheim\n\n15 West 52nd Street, New York, NY\n\n[February 1943]\n\nDear D,\n\nJust received word from WNYC65 that the date is definitely set for Sunday afternoon, Feb. 21, from 6:30 to 7:00 p.m. I get the full half-hour, to boot, so that I can do, or have done, the 6 Piano Pieces.66 It ought to be good fun, and I'd like to have the whole thing on records for all time.\n\nWhy don't you come down for the whole week from Sunday the 14th to Sunday the 21st? That would seem to work well \u2013 assuming, of course, that you have some sort of vacation then. The Copland Forum will take place in Town Hall on Wednesday of that week, and you surely want to hear that. Aaron is supposed to play his Sonata; but he has just left for Hollywood to do the Lillian Hellman picture,67 and it's possible he won't be able to be back for the forum, in which case I would play the Sonata, which would also be fun. _Music for the Theatre_ is to be done as well, Saidenberg conducting, me on piano.\n\nA Rochester composer named Burnall Phillips just called, and is coming over this afternoon to spiel me his works. Do you know him? He seems like a nice guy.\n\nWhere are your letters? Must I give two to your one?\n\nLove,\n\nLenny\n\n126. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\n15 West 52nd Street, New York, NY\n\n20 February 1943\n\nCh\u00e8re Madame,\n\nAs you may have read in the New York papers, I have suddenly made a totally unexpected Town Hall d\u00e9but. Aaron was to have played his Sonata at the Music Forum at Town Hall last Wed., but couldn't escape the clutches of Sam Goldwyn in Hollywood. I was asked at the last minute, and practiced madly for a day, bought a suit of tails, played the Sonata, very successfully, answered questions with \"adroit wit\" ( _N.Y. Post_ ), and got lovely reviews. \"Superbly interpreted\" ( _Herald-Trib_.), \"great facility and remarkably complete understanding\" ( _Sun_ ). And _P.M._68 was great: \"L.B. played the Sonata with all the devotion and skill the composer himself was unable to bring to it.\" And Virgil Thomson, on the stage, publicly acclaimed me. What a d\u00e9but! Especially since I had to play it all over again at the end of the program. I wish you could have been there. It was really exciting. And the second time was much better than the first, which was full of errors. But then, no time to practice, no right notes. Vengerova insists that I send La Bok a program and clippings!\n\nWonderful that you have a job \u2013 but terrible that it's so dull. Tell me more. And try to get to NYC (impossible) Sunday, the 21st, at 6:30. Clarinet Sonata & 6 pieces for piano. And write me.\n\nLove,\n\nL\n\n127. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\nSunday [21 February 1943]\n\nDear Lensky,\n\nAs far as I can judge Wed.'s concert seems to have been a real triumph for you. You apparently were the Rob't Shaw of the occasion! Why even Minna Lederman was won over. What surprised me more than the \"superb\" performance of the Sonata was the report of how good you were in the forum. The new Movie Suite seems to have been put completely in the shade. What's it really like?\n\nI think Heinsheimer was appalled at the criticisms. They certainly were stinkeroos. But then what can one expect from Noel Strauss or J[erome] Bohm. (Of course Virgil's sending Bohm to both the _Danz\u00f3n_ concert and Wed.'s event, knowing in advance he was giving me the ax \u2013 while sending P[aul] B[owles] to cover his own concert \u2013 is purest bitchery. Well, anyhow, now you see what it's like \u2013 being a composer, I mean.)\n\nOf course, what really interests me is what went on at 15 W 52 _after_ the Wittenbergs! I want a play by play description. You're a pretty smart fellow. (Where was D[avid] O[ppenheim] all this time? As soon as I dropped the phone I regretted not having asked.)\n\nHollywood is dull dull dull. I've written a guerrilla song that everyone says is good (32 meas[ures]! Oh no \u2013 it's 36!!) \u2013 even Mr. Sam Goldwyn. Wish you could have seen us playing the Internationale to him in his office \u2013 (He said: It's a \"steering\" tune.)\n\nYou deserve some kind of medal \u2013 but I'd rather wait till I can pin it on myself. Anyway, I'm proud of you.\n\nLove,\n\nMe\n\nP.S. Just heard the Strav. Symphony on the air.69\n\n128. Leonard Bernstein to David Oppenheim\n\nNew York, NY\n\npostmark 25 February 1943\n\nDave,\n\nExcuse this ridiculous card70 long enough to be warned that at risk of life, limb & name, I have succeeded in getting the Cl. Sonata on the League program Mar. 14. Don't fail now.\n\nLove,\n\nL\n\n129. David Oppenheim to Leonard Bernstein\n\n[Rochester, NY]\n\n2 March 1943\n\nDear Len,\n\nWhen I got back from N.Y. I played the Sonata records71 for Rogers72 and showed him the piano pieces (which I took quite by accident). He was impressed but screamed Copland all thru your last movement \u2013 5\/8 etc.\n\nRogers: \"Did he study with Copland?\"\n\nMe: \"No, but they are very close friends.\"\n\nR: \"Is he that way \u2013 you know \u2013 Copland isn't normal. Is _he_ normal?\"\n\nMe: \"Perfectly.\"\n\nWe talked for about an hour. I gathered that he thinks Copland distant and impossible to get to know if you aren't his type. He cited his own case of 15 yrs acquaintanceship without any familiarity at all. He also thinks C.'s music intellectual and unlyrical and believes Aaron hasn't fulfilled the promise he showed 10 yrs ago. R[ogers] admires Aaron's fight for music in Amer. and has general positiveness and seemed apologetic but firm in his criticism. Insists no bitterness exists. Thinks the L[eague] of C[omposers] a narrowing element. Thinks Harris is awkward & not at all graceful. Thinks Boulanger not what she is cracked up to be. He studied wit h her for three months \u2013 with Bloch for three years.\n\nHe is a good guy tho \u2013 I like him. The comp. students around here idolize him & his music. I haven't heard enough to say.\n\nDream No. 89625436 \u2013 I dreamt of cigarettes in sugar bowls filled with sugar. [...] I heard or read a story about a guy who fucked his secretary: \"He put a cigarette in her monkey (vagina) to make it smoke.\"\n\nInterpretation \u2013 maybe my cigarette phobia finds here its genesis. I jumped 3 feet off the chair when I had it. It seemed right. I haven't had time to see if it had any effect on me yet. [...]\n\nYour letter was your most optimistic utterance to date and a good thing. Sounds wonderful. With the progress I have been making I think I will be able to be just as optimistic before long. Many things must happen first tho. Incidentally, K[aren] H[orney]'s _Neurotic Personality of Our Time_ is a fine book for you \u2013 better than the other two I believe. Read it!\n\n1. The clarinet is better than ever.\n\n2. I am working harder & better than ever.\n\n3. I feel better than ever.\n\n4. I miss you.\n\nLove,\n\nDave\n\nDon't get the impression Rogers doesn't like C[opland] because I don't think that is true.\n\nHello to E[dys] M[errill].\n\n130. Leonard Bernstein to David Oppenheim\n\nNew York, NY\n\npostmark 5 March 1943\n\nMy Dear Mr. Oppenheim,\n\nListening to the recording of the Clarinet Sonata, I am more and more impressed and moved by your performance. Especially the \"high, controlled\" part of the last movement. It is, in all seriousness, some of the finest, flutiest playing I've ever heard. Bless you. Those records are giving you quite a reputation in the Big City.\n\nAs to Lukas Foss, I have already asked him about sending you his Konzert, but he has only one copy and has to hold on to it. Perhaps you can see it while you're here. Are you coming, I ask again, at the risk of being an utter bore.\n\nI had a lovely surprise the other day, which you will please keep a dead secret. I was called into the Arthur Judson office by Bruno Zirato (manager of the Philharmonic), who told me that Rodzinski73 is very interested in me, and wants me to hang around all Philharmonic rehearsals etc., next year, and finally conduct a concert. I had no idea he even remembered me, but apparently he does, very clearly. Some Brahms performance or other at Tanglewood.\n\nI had a terrific night last night. I had my \"attack\" again at _Shadow of a Doubt_74 and thought I was going insane. All of which provoked some stunning analysis, and had me sitting up late, putting on cards all the elements of the story of my life. I think I'm pulling it all together now, and will soon have a working basis for active self-analysis. It's a good, active sign.\n\nBut why these attacks of panic and insanity-implications? Is it the state of nerves under analysis? I have been this way since you left. It is as though a layer of skin had been removed, figuratively, so that each little emotion, resentment, etc., inflicts a real registering of panic, instead of the customary vague heart-throb. It's a kind of crystallization of all psychic processes. I'm unprotected, and have been laid bare, baby. I think perhaps it signifies progress, somehow or other. It is Joe Id's way of informing me what's really going on. And the root of the whole thing is definitely lack of aggressiveness, out of fear, fear of being active, hostile, retaliative. It's all fairly clear now.\n\nI believe in your cigarette analysis. It sounds quite right. Keep up the mental life, keep sending those lovely senseless picture postals, and I miss you too.\n\nLove,\n\nL\n\n131. Leonard Bernstein to Aaron Copland\n\n15 West 52nd Street, New York, NY\n\n[late February or early March 1943]\n\nDearest Aaron,\n\nNow that it's all died down, and everyone has written you all the details, including Edwin Denby & [Kenneth] Klein & [Minna] Lederman (she came back & spoke to me \u2013 endlessly!) \u2013 I would like to add only that it was a real experience to feel that I really _had_ redeemed the Sonata; that I played it marvelously, full of errors through lack of preparation (and nervousness), that it was thrilling to see & hear Virgil [Thomson] so impressed with it, to have Jim Fassett call up & tell me how he had always disliked it, and now loved it \u2013 to have such a lovely accidental d\u00e9but, such good reviews \u2013 and especially to hear Virgil say that I seemed to be composing it as I played it. That's always my feeling with your notes, my love. It's in the books. There's much more to say, but I must wait til Dave Oppenheim leaves (he's sitting here reading _Serenade_75 right now. Am I a masochist!) He came down to do the Clar. Sonata \u2013 which was lovely. I also did the Six Pieces, & the piece I wrote at your house in Stockbridge (now called _Dedication to Aaron Copland_ )76 and a new one-minute clar & piano affair called _Extension of a Theme by Adolph Green_.77 I have it all on five records, for you to hear \u2013 ah, but when?\n\nAnd I've seen Victor [Kraft], & talked with him for the first time, & I've been confusing him & Dave [Oppenheim] in my deep down mind. What's it mean?\n\nAnd I was swamped by Ted Colombo in a bar, & _that_ was _interestin._\n\nAnd life is most peculiar. What are you doing?\n\nLove, love, love,\n\nYour slave,\n\nL\n\n132. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\nSunday [March 1943]\n\nDear Disciple,\n\nWhat a letter! I had a wonderful time with it \u2013 better than any novel. But now I want to read the next chapter. Thing that surprises me most \u2013 you always look and act so conscious that I can't imagine the inner psyche doing its own imaginings. It made me wonder if I too had an inner psyche doing funny things without my knowing it. But I guess it's just envy \u2013 sounds like so much more fun. The identification of D[avid] & V[ictor] seems so natural \u2013 the part that seems curious is that you should want to be me. (I'm so glad you're not!) I always want you to be you, so that I can go on feeling about you the way I do. _Please_ write the next sequel soon.\n\nI've got a little grey home in the west now. It's tiny but cute. A sun porch, a big piano, a eucalyptus tree, some new books \u2013 everything except a companion. It's better than living in a hotel room as I had been \u2013 but still I'm not like that Louisiana tree \u2013 and standing alone in the Hollywood desert gets oppressive at times. Any suggestions?\n\nBumped into guess who in the blvd \u2013 Jesse Ehrlich78 and wife. We are all going to hear the W. Coast premiere of the Sonata tomorrow night. (Any report as to how J. Sykes performed it?) J & J are settled here during the period when Warner's are filming the army show. They seem to have living with them a delicate young negro boy from Katherine Dunham's dance group. Nice looking kid. (Did I say anything?)\n\nI heard Roy [Harris]'s 5th Symph broadcast yesterday. Decided my chapter on him79 was triply just. What a pity \u2013 with all that good material he can't pull it all together and make it go places. Still, the personality is so strong that it may make up for the lack of intellectual grasp of the material. I'd be surer of this if he didn't repeat himself so much in general mood and formulas. His music shows no signs whatever of reactions to outward events. Well, you know all this.\n\n(By the way, as your \"only\" musical influence, I'd like to know what the influence consists of \u2013 in one or twenty sentences.)\n\nI've been here four weeks now and accomplished practically nothing. Most of the time goes in gab fests. Shooting starts tomorrow and lasts 2 months. I'm doing a 2 minute Russian peasant dance with Lichine as choreographer. It's practically impossible at this stage to figure out what the whole thing will add up to, musically speaking. [...]\n\nIf only you had experience in pictures we could bring you out to conduct. I've tried them \u2013 they won't take a guy who has never before done a picture. So there. Now will you come to Hollywood?\n\nLove you too,\n\nMe\n\n133. Leonard Bernstein to Aaron Copland\n\n15 West 52nd Street, New York, NY\n\n[14 March 1943]\n\nDear A,\n\nI haven't heard from you yet, and by all precedents you don't deserve a letter. But I have some confessing to do; and after all, I can't betray you without letting you know!\n\nFirst, I betrayed you by playing the Clarinet Sonata today at the League Concert (Library)80 instead of the Six Pieces, as you had wished. I felt the need to present my first League composition as a piece with a slightly larger form than just six germs for large pieces, and the Sonata does approach, at least, a big form. Besides, it was more fun than playing alone, and \u2013 biggest point \u2013 it provided a lovely excuse for having D[avid] O[ppenheim] come down. So you will forgive me, won't you? Amusing part, of course, is the great secret twixt Marion Bauer & me, since neither she nor any \"committee\" ever saw the Sonata first; but she trusted me, &, as it turned out, liked it, & didn't think the 1st movement was Hindy81 at all!\n\nThe reviews aren't out yet. Paul [Bowles] covered it for the _Tribune_ ; & Virgil's article today on the French approach to music is something of a masterpiece.\n\nI've betrayed you further by deciding to urge Victor to go to Hollywood. It's so hard for him to get his mind on a goal & set about reaching it. And here he piles up hate on resentment for analysis. In L.A. he might lose that, & under your aegis, even try it. He'd be so much happier there than torturing himself here (and getting drunk, & forgetting the simplest obligations & duties). And especially _you_ would be so much happier. So why not? We've played squash together somewhat, & I know him better, & thus decided on betrayal. As for my feelings, I can be awful controlled sometimes. I'm a good disciple, no, my love?\n\nSaw _Lady in the Dark_ tonight,82 & loved it, especially seeing it with D[avid] O[ppenheim] who is consulting the Frau madly these days. It is, as you say, slick \u2013 over-slick \u2013 but I'm no critic, being an analysand (!).\n\nThanks so much for the Lincoln piece:83 it looks marvelous, & I wish the Kostelanetz ban was off it. I love you. And, oh, I resented Lukas [Foss] telling me that he'd got one too. Crazy frankness, but that's the sort of irrational habit analysis gets me into. Out with the resentments, Bernstein!\n\nI finished a fifth _Kid_ song, completing the cycle,84 & it's beautiful, if a little on the Copland side. I have to make a change to the \"Indian\" one, & it will be done: everyone loves them.\n\nHave I told you how things have been popping? Like Herman Starr placed _our_ Brahms song in a Warner Bros picture? (Don't worry, you'll get the 10%!) And how there's a possibility for me to be ass't conductor of the Goldman Band this summer? And that I've had a _nibble_ in Hollywood, but I don't like it so good, so I'm waiting still. And how I've been offered a teaching job for next season at the Little Red Schoolhouse? And that Rodzinski wrote me, asking me what my plans were for next year; and I hear from the Judson office that he wants me to conduct a Philharmonic concert? Have I told you? Cause I'd hate to repeat myself.\n\nWrite me of the progress & the Hollywood life, & Jesse & June [Ehrlich] (give them my love) and I hear you have a Hauserish85 cook, and I hope V[ictor] comes out soon to cheer you up, & I only wish it could be me.\n\nAll my love,\n\nL\n\nP.S. Just got the _Tribune_ \u2013 have a review! Hindy not mentioned!86 I love you.\n\n134. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\n25 March 1943\n\nQueridissimo L\u2013P,\n\nI think of you every day, particularly when I don't write to you. It seems a long time since I last did \u2013 tho for no special reason. Life goes on placidly enough out here. With my little songs and choruses written there's nothing much to do but leisurely visit the shooting on the back lot \u2013 see the daily rushes the next day \u2013 and generally keep Mr. Goldwyn happy.\n\nBest news of the week was a wire from Kouss telling me he had managed to get Kostie87 to unban the Lincoln piece. He's doing 7 performances \u2013 isn't that wonderful? I'll be listening to the broadcast from Boston on the 10th. Maybe you'll write me the impression in a concert hall, and what Kouss does to it. (Why, oh why, am I in Hollywood??)\n\n_So_ \u2013 you played the Clar[inet] Sonata! It's still full of Hindemith, because I say so. (And don't forget who sat next to D[avid] O[ppenheim] at the Lenox Town Hall and practically arranged the original performance.) I want to hear about your writing a song that has no Copland, no Hindemith, no Strauss, no Bloch, no Milhaud and no Bart\u00f3k. Then I'll talk to you.\n\nThe mysterious check enclosed is for \u00bd the royalties I collected for the 1 piano _Sal\u00f3n_. They sold 250 copies (a suspiciously round number) and I get 8\u00be\u00a2 on each copy. So you get 4 \u215c\u00a2 on each copy. Of course, the point is to prick your conscience so that I collect 10% on the Brahms song, which should amount to 20,000,000,000,000,000.00. In the meantime spend the 10.94 in good health!\n\nTickled to hear about the Rodzinski letter you got. He also wrote me \u2013 but I'm not saying nothin' \u2013 mostly because you repeat everything you hear (you also leave letters lying about.) Also tickled that you \"resented\" my sending Lukas a score. That's fine \u2013 shows you really care about me. Also tickled with your Goldman possibility. Hollywood nibble, and Red school house \"job\". However, don't forget you're a conductor waiting for an orchestra.\n\nDid I tell you about the hero of our picture? Just 17 and doing his first film \u2013 Farley Granger by name. Sensitive as a flower. It would be very easy for somebody to do a _Death in Venice_ on him.\n\nStravinsky invited me to dinner! Cordial as could be. Made me big compliments about _Rodeo_ , of all things. [George] Antheil88 was there and we played the _Symphony in C_ in the Stokowski version, with Strav. singing all the tempi as they should have been. He's coming to N.Y. so you'll probably meet him.\n\nYou're a good disciple \u2013 but an angelic love \u2013\n\nMe\n\n135. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nAdvanced Music Corporation, RCA Building, Rockefeller Center, New York, NY\n\npostmark 29 March 1943\n\nDear Ren\u00e9e,\n\nDesol\u00e9 that I didn't get a chance to see you after all. I had to catch such an early train the next morning that it seemed really silly to go all the way out to Rosl[indale] that night, especially after I'd been invited to start at the Copley-Plaza with Bill Schuman (just around the corner from your fabrique \u2013 you should have dropped in for a rye). Anyway, I'm desol\u00e9.\n\nThese days are all full of the Museum of Modern Art & the Bowles opera (which goes on tomorrow night, & is really a mess) and the Marquis[e] de la Casafuerte, & much talking in French & pidgin Spanish. I'm also conducting Revueltas' _Homage a Garcia Lorca_ \u2013 my first conducting in NYC89 \u2013 & my orchestra is wretched \u2013 really dumb trombone & oboe & harp \u2013 and no amount of shrieking helps \u2013 and there's no time for rehearsal, & they hammer up scenery during the rehearsal \u2013 and it's a mess. It will probably end up either a _fiasco_ or a _succ\u00e8s fou_ \u2013 & the Marquise Yvonne is probably going \"in jail\", as she says, because she put an ad in the _Times_ , and it's forbidden by the Museum Charter. Shades of a former opera and a former Museum. Rather the same caliber of people \u2013 rather more neurotic \u2013 the same bungling and disorganization. Virgil is largely the bungler (destroy this letter!) and the director is wacky. Anyway, it's an orchestra.\n\nRumors float around like crazy. That Rodzinsky plans for me to do 2 weeks with the Philharmonic next year! That I may get the Goldman Band this summer (assistant, of course). Etcetera. Nothing _real_ to report. Finished the \"Kid Songs\" & they're universally loved. I want you to hear them. They may be published; & I've had an offer to publish the Clarinet Sonata.\n\nWell, now you tell me. And soon.\n\nMuch love.\n\nL\n\n136. Leonard Bernstein to David Oppenheim\n\nAdvanced Music Corporation, RCA Building, Rockefeller Center, New York, NY\n\npostmark 29 March 1943\n\nDave,\n\nWhwhwhwhwheeeeere have you been? Pas un mot. Have you had a post-NYC reaction? That would seem rather silly. Let's have a sign of life.\n\nDays and nights are full of the Bowles opera: rehearsals \u2013 and rehearsals \u2013 and a really wretched orch \u2013 stupid & unmusical people \u2013 some of them Philharmonic, too, and _not_ fired by Rodzinski \u2013 and we do need you! I'm exhausted.\n\nI've had an offer to publish the Clarinet Sonata. Something called the Hargail Press or something. They publish recorder music & heard your performance, & want to print it. Do I have your consent?\n\nWhat happens in Rochester? When is the army?\n\nLove, as usual,\n\nLenny\n\nSome hours later, I still have the feeling for writing to you. Things that were left unsaid:\n\nLike: Indolence in correspondence will not be tolerated. I, no. 1 indolent in correspondence, expect prompt remission.\n\nItem: Had a glorious long talk with Shirley last night \u2013 I'm sure it bore lovely fruit \u2013 she is growing up like crazy. And sends you her warmest, no doubt.\n\nLike: What is Jack's address, so we can send him a reward of some sort?\n\nOr: How was Detroit and mother? Did you have enough money? Was the meeting bleak or short enough to be gay. How are your post-New York, post-neurotic neuroses?\n\nItem: What has happened to your resolution to study? And investigate scores? What goes with Mad. (to whom my best).\n\n(Seizing on anything for paper)\n\nThe other unsaid things are almost unsayable things. Be good, unconfused, hardworking, and write me everything you feel.\n\nAgain,\n\nL\n\n137. Leonard Bernstein to David Oppenheim\n\n[after 30 March 1943]\n\nDear Dave,\n\nGiven any situation at all that I happen to fall into, it inevitably becomes involved. I told you of this here Hargail Music Co. that wants very much to publish the Sonata (the owner heard us do it at the Library). He makes me an excellent offer, and moreover wants to issue a commercial recording simultaneously! (Of course, with you & me as soloists.) All that is quite wonderful, provides a lovely pretext for another visit to NYC, and would once and for all finish this Clarinet Sonata business.\n\nThe thing gets involved, tho. I had to report this as a matter of course and courtesy, to Warner Bros., and now they're all excited about it (stupid competitive instincts). They say they can do so much more with it \u2013 that a small outfit like Hargail could never sell what they could, etc. And at that moment, Frank [Campbell-] Watson, who works as editor in the \"standard\" dep't, as they call their _classical_ dep't, came in, said he had just been up to Eastman, and had heard great things about the Sonata from Fennell!90 Now I'm up a tree; and have to show them the score, & play the records. I personally trust the Hargail man more, and am especially attracted by the idea of an authentic recording, with you on the clarinet. At any rate, send me the score _immediately_ , really immediately, insured, etc, since this must all be done fast. (You do have it, don't you?)\n\nIt is also important to know exactly when you may be called by the army, so that we can make this recording, if we do, before you go (that dreaded moment).\n\nLet me know all this very soon.\n\nThe Bowles, etc. concert was a knockout, with a real whopperoo of a rave review from Virgil (picture and all) and other lovely ones. I'm exhausted now, and trying to recover.\n\nA pupil is zooming up the stairs \u2013 so addio \u2013 and be very good & send the manuscript presto.\n\nI love you, as usual \u2013\n\nL\n\n138. Leonard Bernstein to Aaron Copland\n\n[after 30 March 1943]\n\nAaron, my love,\n\nThe Third Serenade91 reached actuality and by some miracle or other, was a success. Paul's music is, I think, universally loved, tho what it has to do with that fantastic slop of words I shall never know. Paul's OK. He should be in Hollywood \u2013 can't you get him there? \u2013 doing scores for things like the _Human Comedy_. Stothart, indeed!92\n\nPoisonally, I came off very well. The _Times_ was lovely, & the enclosed whopperoo from Voigil is about the best yet.93 I feel good \u2013 and tired. Those last few days of trying to get an opera & concertst\u00fcck together in no time at all were terrific. I was exhausted _before_ the concert, and ready for the Bellevue _after_ it, but after that Askew party \u2013 ay! All night affair, and I very drunk, and Constance94 lovingly making me play all night, which I did \u2013 she's fun \u2013 and Diamond being an uninvited unwelcome guest, & really _getting_ it from Constance, and being dramatic, and V[ictor] very drunk, and everyone missing you, and I found a new boyfriend [...] (married, Goddam) [...] the one who sang Paul's opera. Looks a bit serious, but not to worry. My French has picked up enormously, what with the Marquise [de Casa Fuerte], & Mme Alphand, and my new friendy-wendy, Prince George Chavchavadze.95 All very confusing, & I still love D[avid] O[ppenheim]. What to do? I know, marry my new girlfriend \u2013 she's lovely \u2013 my dentist's daughter.96\n\nA thing called Hargail Music Co., run by a Harold Newman, which publishes recorder music (!) wants to publish my Clarinet Sonata! _And_ make a record of it \u2013 commercially. Sounds good. And Schirmer's may bite at the Kid Songs97 [ _I Hate Music!_ ]. Life is full and empty by turns \u2013 the latter mostly cause you're away.\n\nBless you for the pretty _Saloon_ check. It's lovely.\n\nI'm going to the _Lincoln Portrait_ this weekend. Kouss says it's your masterpiece. He's swimming in ecstasy. My love to Farley Granger. Can you fix us up? Write soon. I love you.\n\nL\n\n139. Leonard Bernstein to Aaron Copland\n\nThe Park Savoy, 158 West 58th Street, New York, NY98\n\n[after 3 April 1943]\n\nDear A,\n\nJust a few words a propos of the _Lincoln Portrait_ performance. (Look at the stationery I discovered \u2013 remember?)\n\nI suppose everyone has written you all about it. How Will Geer99 completely disregarded your foreword in the score, and drawled away in a pseudo-Lincolnesque performance that was truly embarrassing. (He made up to look a little like Abe, and sat on this chair like the Lincoln Memorial in Washington. I'm told the whole thing was Koussie's idea, which is certainly plausible.)\n\nKouss didn't do as well by it as I had expected. The opening was too agitated because he divided his beats \u2013 there was no feeling of tranquility and space. He missed a few cues (or passed them up), like \"disen _thrall_ ourselves\"; and usually simply waited for the end of a speech, & then played music. But the sum total was good \u2013 the impression was very exciting, & the audience was charmed. Why must you be in Hollywood? Although after the parade of bowing composers (Billy & Barber) it was very distinguished of you not to be present \u2013 sort of specialness, you know.100\n\nIt's a fine piece, my love, despite all the repetition, and Vernon Duke's verdict that it's \"just 20th Century Fox\". He's insufferable anyway.\n\nMad party at Arthur Berger's last night \u2013 Jean [Middleton] & Victor [Kraft] & Paul [Bowles] & Virgil [Thomson] & David [Diamond], etc., etc., etc., oh, & Colin [McPhee] & Paul Morrison. David was drunk & cutting up, & being dramatic again. I left with Paul Morrison, & it was like old times (remember the Boston incident?) Jean was pretty dramatic too, & left early. What a good world to stay away from! And I'm confused as ever, what with my new friend, and my new girl-friend, whom I am afraid to involve unfairly, and Edys'101 & my decision to part company, which raises all sorts of problems, like which one of us is to move; and if she does whom should I get to share the rent here \u2013 since I ought to but can't afford to live alone. It popped into my mind that Victor might be a sensible idea for it, if he wants to stay here (in N.Y.), but, as he says, can't stand the loft alone. What do you think? Is it crazy as hell? Of course I realize it would be the talk of the town. But that's such fun \u2013 makes such good memoirs, as you would say.\n\nJust finished Gide's memoirs, by the way ( _If I Die_ ) \u2013 & had an excellent time thinking of you. I miss you like my right arm.\n\nWrite me more than you do \u2013\n\nLove,\n\nL\n\n140. Aaron Copland to Leonard Bernstein\n\n[Los Angeles, CA]\n\n11 April 1943\n\nDear Honeychile,\n\nThere are two letters of yours here that are still \"unanswered\". I'm beginning to lose contact with your every thought. I have a sense of your having had a great triumph out of the Bowles opus,102 followed by a kind of let-down which is natural enough, complicated by several new personal adventures, new boy and girl friends, who are nothing but names to me. Hadn't you better expand a bit? So's I know where I am.\n\nV[ictor] wrote a full description of the Askew party that is a classic of its kind. Bill Schuman wrote a description of the Bowles opus that tore it to shreds. Anyway I get the impression I know just what it was like. After your own description of Koussie's _Lincoln_ [ _Portrait_ ] I heard it on the air yesterday at Ira Gershwin's house, surrounded by Harold Arlen, Harry Warren, Yip Harburg, Earl Robinson, Arthur Kober and other noted worthies. (Get the company I'm in!) Of course I didn't approve of Geer's way of doing it, but more than that, why did Kouss take the middle part so fast?? It made it seem superficial. It lacked charm and bite done that way. What we need is _Amurkian_ conductors. In the meantime, however, I'll take Kouss.\n\nI don't know what to write you about because nothing much has happened out here. I've been to a couple of musicians' parties for Sanrom\u00e1103 and P[aul] Whiteman. I've stopped going to the studio in the mornings because there's nothing to do. The picture is growing each day, but I have no over-all idea of what it's like yet. Next week I do a short dance sequence with Lichine. But nothing of background music as yet. I dawdle a lot, and fuss with themes of my own, and the unfinished 1st mov't of the violin piece, and argue with Mr. Goldwyn. And that's about it.\n\nSpent an evening with G. Antheil104 who played me his 2nd, 3rd and 4th Symphonies. They're hard to describe. He's in a Mahler\u2013Shostakovich period, and everything comes out of there in great unwashed gobs of sound that billow you about until it's all over and you're not sure what you heard. Some of it is very effective, and it all has a typical Antheil drive, but somehow when it's all over, one doesn't give a damn. That's the sort of thing that's hard to tell a composer.\n\nI read a lot, mostly to make up for the lack of any warm relationships here. I get a great sense of luxury out of buying all the books I want. I spend whole nights in book stores making up my mind. Victor is sending me a two volume affair105 that had to be ordered via Dr. Safford. Do you remember my telling you about it? I originally spied it in a store in Rio, and now I've tracked it down. I'll save it for you.\n\nThis is the end of my tenth week. I have ten more to go, and then a big question mark. It would be a much more exciting life if you were here.\n\nLove and all,\n\nA\n\nP.S. Monday. Just had word that V[ictor] is driving out. [Margaret] Bourke-White is coming to take stills.\n\n141. David Oppenheim to Leonard Bernstein\n\n17 April 1943\n\nDear Len,\n\n\"Publishing-pains\" indeed. Seems to me like the worst publishing pains a piece could have is no publishing at all. And you wound up with not one, but two publishers, get the biggest plus the tempting part of the smaller publisher's offer \u2013 recording \u2013 and you can say P[ublishing] P[ains]. Nonsense, Bernstein, nonsense. At any rate I'll be down in NYC next Friday nite. May I stay with you or will there be complications. Also I will be there until May 2nd. OK? Better be!\n\nSunday a.m. is awfully soon for the recording in as much as I am just recovering from a two day sojourn at the Hospital where I was treated for flu & an infection on my face. If you can with ease and safety stall it off for a few days so much the better except that I never get any work done in N.Y. anyhow.\n\nFive years with W[arner] B[ros.] is a long time m'boy. Are you sure you want that. Would that tie you up. Could you travel, conduct etc. I suppose the decision has already been made & I am just making things harder. Sorry, but somehow I hate to see you do it.\n\nI see you have learned to spell Rorschach.106 I'm glad about the new seriousness. Soon I'll see you & we can talk about it. I'm very anxious to see you.\n\nGene Shamalter, the guy who wanted to see me in N.Y. but couldn't & who wrote me at your place was here this week. As I suspected, he is as I suspected. Remember. But complete, never felt otherwise. He is alternately resigned and unhappy about it, wants to renounce sex altogether. A most sensitive chap too.\n\n1. I've slept with Mad.\n\n2. Army day is in May I think.\n\n3. I'm 21 now.\n\nLove,\n\nDave\n\n142. Leonard Bernstein to Aaron Copland\n\n15 West 52nd Street, New York, NY\n\n[April 1943]\n\nAaron darling,\n\nSince I never hear from you any more, I suppose I'll have to write, & wring a letter from you. May I burden you with my many little present conflicts?\n\nFirst, though, I think it's great that V[ictor] is finally going west. He needs it, you need it, I need it; and the best, of course, is that he made a decision at all. Treat him right now \u2013 get him going as a man. He's really such a foetus! (Look who's talking!)\n\nViz: the little Hargail Music Co. (mostly recorder music) wanted to publish the Clarinet Sonata. Out of professional courtesy I showed it first to Warner's, & they knocked me over by loving it, & insisting on publishing it. I was downhearted, since Hargail wanted to make a commercial recording of it. Now Hargail is offering me all sorts of fantastic royalty rates if I'll give it to them, & says that they will make the recording anyway! A labor of love, if I ever heard one. But Warner's points out that they, as a large firm (Witmark will be the publisher) can do so much more for it than can a little thing like Hargail. What you do think? Matters are now suspended by a hair.\n\nViz (2): Warner's presented me with a five year contract! I'm taking it to a lawyer today to find out what it says. It looks like my life that I'm signing away. But it adds to my little old salary a substantial weekly advance on future mythical royalties, which increases each year. What do you think?\n\nViz (3): I've got to move, and there's the biggest problem. I want the sort of apartment I can't afford alone. And I have a wonderful guy to live with \u2013 which is quite a story. He's [...]107 the big, beautiful, brilliant 20-yr-old [...] & I want to help him (but how can a poet earn a living?) I really want desperately to help him \u2013 although I have doubts about whether he can stick it out. [...] The Frau won't hear of it. I have to be alone, & suffer, & break through the pain of loneliness, even if it means living the summer in a hot place, which is unbearable in New York. God, I'm perplexed! What do you think?\n\nViz (4): \u2013 a possibility of giving a recital in Town Hall next month for the Little Red Schoolhouse,108 which is mad but exciting, and I don't know if I should do it. I do get diffused. And Jacoby really wants me bad in his new night club (the Blue Angel), & wants to build me, etc., comme impresario. And I had another nice letter from Rodzinski. Am I diffused?\n\nSo I should talk about Victor, yet!\n\nPlease write, my love \u2013 I miss you like mad.\n\nEwig, ewig, ohne End!\n\nL\n\n143. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\n29 April 1943\n\nDarlingest L.P.\n\nI know you're probably feeling awfully neglected, or maybe you're all absorbed in your new friend and have forgotten all about me.\n\nAnyway, from my angle \u2013 not hearing from me doesn't mean a thing. I got tied up writing a Russian number for Lichine's choreography. It was the last of the pre-recorded stuff \u2013 and now I don't have much to do but wait around until they finish shooting the picture, which looks as if it would be around June 15th. Of course, I get ideas and whole sections while waiting. I wonder what it will all add up to. [...]\n\nV[ictor] arrived in Posh109 (now renamed Poshalopy) yesterday, so a new chapter in my Hollywood life begins. He seemed pleased with my house, my office, my secretary, etc. And I'm very pleased to have him here.\n\nHe brought reports of your signing the Warner's contract. Is it true? I wish I could have read it. I was out on the Warner lot the other night watching Adolph Deutsch scoring a picture with an orchestra of 80. When I think how you could fit in over there I get noivous.\n\nIs the Clarinet Sonata being published?\n\nAnd are you alone \u2013 like the Frau wants?\n\nI never heard the sequel to your last letter which was full of problems.\n\nBy the way, that book I once wrote you [about] arrived \u2013 and it contains the most wonderful Glossary of specialized slang you have ever seen or ever can hope to see. I can't wait to show it to you. The rest of the book is H. Ellis' case histories brought up to date \u2013 and I recognize a little bit of you in each of the 300 cases!110\n\nHow are you? That is, hello. And what's the summer look like? All signs point to me being out here until August 1st. After that I hope Mexico or New Mexico. Wouldn't it be lovely if we could somehow connect up sometime somewhere.\n\nTell Pfb [Bowles], if you see him, that his _M_ [ _odern_ ] _M_ [ _usic_ ] phrase about \"harp vomit\" has become famous among Hollywood orchestrators. I met a number of them at a party the other night \u2013 they're my principal public out here. Seems that Max Steiner's wife is a harpist \u2013 which they say explains the featuring.\n\nMargaret Bourke-White is on the lot, photographing us. We just carried out the \"scorched earth\" policy on the back lot set \u2013 and boy did we make a mess of that. Beautiful set. Farley Granger gets more simpatico every day. I've promised him the _Saloon_ records. Do you think that will do it??\n\nStill I love _you_.\n\nA\n\n144. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nHotel Chelsea, New York, NY\n\n14 May 1943,\n\nCh\u00e8re Ren\u00e9e,\n\nCe sont des jours tellement fran\u00e7ais, ces jours-ci. Je viens de lire Gide en fran\u00e7ais. Je vais jouer tous les dimanches-soirs chez \"Le Bleu Angel\", un nouveau club en quelque sorte Parisien (comme l'ancien _B\u0153uf sur le Toit_ , ou le _Ruban Bleu_ ); le clique des Concerts _S\u00e9r\u00e9nades_ (ton amie la Marquise, etc.); mon monde semble aujourd'hui tout \u00e0 fait fran\u00e7ais. Et alors, que faire? Rien que d'\u00e9crire un mot \u00e0 la premi\u00e8re Fran\u00e7aise de toutes Fran\u00e7aises. Tu me crois enivr\u00e9? C'est point l'ivresse \u2013 c'est l'amiti\u00e9.\n\nLa cause imm\u00e9diate de cette lettre, c'est Mme Claude Alphand, chanteuse extraordinaire au _Blue Angel_.111 Chaque fois qu'elle chante \"Les Moules marini\u00e8res\" ou \"La Belle Journ\u00e9e\" ou \"Tu m'as voulu, tu m'as eu\",112 je me souviens violentement de \"Mon Mari est bien malade\".113\n\nQu'est-ce que tu fait ces jours? Pas encore l'assembly-line, j'esp\u00e8re! En tous cas, je serai \u00e0 Boston la semaine prochaine, et j'insiste de te revoir. Notre ancien ami, L'Institut de l'Art Moderne (zut) m'a invit\u00e9 \u00e0 jouer la-haut le jeudi soir. Eh bien, quelques dollars, et un voyage pay\u00e9 \u00e0 Boston! Mais quelle existence! Le mardi, j'ai un lecture-recital tr\u00e8s important \u00e0 Town Hall; le mercredi, j'ai un lecture \u00e0 L'Art Alliance \u00e0 Philly!!!! Et le jeudi \u00e0 Boston! Je reserve toutes les nouvelles pour ton oreille, pas ton oeil....\n\nLenny.\n\nEt voil\u00e0 celui qui a gagn\u00e9 le Prix Paderewski! Gardner Read! _Effrayant_.114\n\n145. Leonard Bernstein to Serge Koussevitzky\n\nHotel Chelsea, New York, NY\n\n29 May 1943\n\nDear Doctor,\n\nEvery once in a while I am appalled at the idea that I never see you \u2013 and I feel that I must write you, or talk to you, if for no other reason than my constant warmth of affection for you. No matter how much time elapses without seeing you, you are always with me, guiding my work, providing the standards by which I measure my progress in our art. And today I feel simply that I must communicate with you, out of love and friendship \u2013 that is all.\n\nReading your letter to the [ _New York_ ] _Times_115 made me think of the wonderful Tanglewood days when we discussed your wonderful plan together. I became inspired all over again; and I was very happy to find that the general reaction to your idea is so favorable and understanding. But who can resist an idea at once so bold and so simple?\n\nOf course I am desolated that there is no Tanglewood this year for the first time in many a year. The summer holds no attraction for me. I am searching for a little farmhouse on Staten Island, where I can be alone and work during the summer months. What are you planning to do? I have heard reports that you may go West! That would be a grand idea, if the traveling were not too difficult. There is nothing on earth quite like the Far West of our country.\n\nAs for me, I am still in an undecided state. I hear rumors, all the time, about my coming connection with the Philharmonic \u2013 sometimes they reach crazily exaggerated proportions \u2013 but I have still had no definite word from Rodzinski. But I am used to this kind of delay \u2013 it is rather typical of my life. The one moment I still anticipate eagerly next year is my conducting my symphony with the Boston orchestra. That will be a _real_ moment!\n\nMeanwhile, I go on doing my horrible chores for Warner Brothers in order to live. It is dull beyond belief, and takes much too much time; but I feel that somehow better things must be coming for me.\n\nI have given up my apartment, and live temporarily at the Chelsea Hotel, until I find my summer house. Please let me hear from you and Olga, for it may be a long time until I see you again.\n\nWarmest greetings to Olga, and to you, the same love and sincerity,\n\nLeonard\n\n146. Aaron Copland to Leonard Bernstein\n\n\"about 30 miles outside Hollywood\"\n\n3 June 1943\n\nL P (you dawg),\n\nDon't worry, I haven't lost my job. It's just that we are on location \u2013 about 30 miles outside Hollywood, in heavenly rolling hills, dotted with cattle. Pure William Bonney116 country. I come out each day with about 250 people. They are about to film 2 of my songs, only the sun won't stay out as it's supposed to in Cal[ifornia] \u2013 so 250 people sit around at old Goldwyn's expense \u2013 and I get a chance to write the letter I've been thinking for weeks on end.\n\nWe're on the 4th month of the picture's shooting and still no end in sight. My contract, which was to have ended on June 19th, will have to be continued indefinitely or there'll be no score. As things stand now, I can't imagine being free of the place until Aug 1\u201315. Because so far there _is_ no score, except for a few songs and a dance number. Isn't it amazing? Most composers get 2\u20133 weeks to write their music. And look at me, sitting pretty in my 18th week!\n\nThe really good thing would be if I could tell you I'd been working on the side all that time. But I ain't! Hollywood affects me as it does everybody else \u2013 not creative country... (except when you're paid).\n\nOf course, you're a villain and a wretch for letting weeks and weeks go by with nary a word. And your letter \u2013 tho I ate it up \u2013 was scrappy. I put it all down to the evil genii of the Chelsea Hotel. Watch out for those guys. You can listen, but don't touch.\n\nD[avid] D[iamond] sent a triumphant paragraph of how he had outdone you one Sat. night. It gave me visions of a promiscuity \"sans bornes\", and I tremble for you. I expect to return and find nothing but a pulpy dismembered jellyfish. Awful!\n\nI went up to Oakland (Cal!) last weekend and delivered me of a lecture at Mills College and spent some charming hours with the Milhauds and saw much of Sandy Jones. I even played the Piano Sonata. I can't tell you what a nice person Darius is. He played me the opening page of _Bolivar_ , and presented me with a manuscript of his Lily Pons songs. I wish something could be done about getting his bigger works put on more regularly. Another job for young conductors!\n\nRead a biography of Hart Crane by Horton. Very touching book. Did you ever read it? Also I've been reading Latin-American poetry, my first Lorca plays, more Henry Miller, the Fausset Whitman biography, and good old Hindy's117 _Unterweisung_ in translation.\n\nV[ictor] was busy with M. Bourke-White when she was here and is now \"recovering\". Looks quite \"Hollywood\".\n\nTell Pf [Bowles] I have his letter and will answer soon.\n\nAntonio writes wild letters from Mexico. Why can't we all meet at Ch\u00e1vez's Festival of Modern Music Oct 22\u201329? Simple idea, what?\n\nLead the good life.\n\nAlways your viejo,\n\nA\n\n147. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\n3 July 1943\n\nLenny-Pen,\n\nYou write the most wonderful letters \u2013 just the kind I love to get: the \"I miss you I adore you\" kind, the while sailors and marines flit through the background in a general atmosphere of moral decay.\n\nWell, the fact is I miss you too, and there aren't any sailors in the background either. In fact, there isn't anybody \u2013 because V[ictor] has gone to Mexico, and I've been alone for a week. So the scene is set for a wonderful reunion \u2013 the only hitch being that you'd have to come here. How about it? How about just hopping [on] a plane and coming here for two weeks or two months or whatever. I know it's a wild idea \u2013 but it's fun to contemplate. I have a tiny house with a concert grand that fills it up completely. There's a little porch where one sunbathes, and a big eucalyptus tree that covers all. With a whirlwind like you around the neighbors will suffer, but that's their lookout. I even have a small kitchen where you could demonstrate the culinary art. And it's never hot \u2013 just pleasantly warm. Oh yes, and it wouldn't cost you anything once you got here (just a minor detail!) and of course you'd write reams of music, \u2013 and good music, it being my house. What do you say.\n\nThe idea is probably full of complications for you. Your draft board, your job, your frau, your things, your etc., etc. I dread thinking about the fit of confused brainstorms this letter will bring on. But I just can't resist the temptation of suggesting the whole thing and living in hopes for a couple of days. Maybe you'd better wire me collect as soon as you know anything. The two week plan couldn't be so complicated, could it? Anyhow, even if nothing comes of it, I've had the pleasure of asking you, and it makes me feel less of a wretch in abandoning you all these many months.\n\nTruth is I'll probably be stuck here until Sept. 1st. That's why V decided to go. There wasn't much for him to do here, and it looked as if he would just be hanging around, and not even get his trip to Mexico in. So I encouraged him to make the break though it was as hard as getting caramel out of your teeth.\n\nThe picture is now in the cutting stage. In another week or two they'll be dropping it in my lap and screaming for the music in a hurry. In the meantime I finished the first movement of the Violin Sonata, and started a ballet for Martha Graham.118 And when Hollywood is over I am still hoping to fly to Mexico for a short stay before coming home.\n\nI had Virgil out to dinner the other night and he gave me a few details of the winter music temporada. He also launched into a full scale attack on all psychoanalysts that took me by surprise. He says all that deep down stuff is better left unstirred. He sees no harm in talking about yourself for a few months, but insists that the new science never cured anybody. It also seemed to annoy him that it cost so much in most cases. What sayest thou?\n\nYou wanna hear what's fun? Stokie wrote and asked to see the _Short Symphony_! Just ten years after abandoning the performance the first time. David says he won't play it anyhow, but I was amused to think he hadn't forgotten it.119 Did you happen to hear that [Alexander] Smallens version of _Rodeo_ at the Stadium? I suspect it was murder.\n\nI know you want me to be amazed at your successes as composer but nothing that happens to you can ever surprise me. Isn't that too bad. Least of all your triumphs as composer. But I am pleased that Reiner wants you to conduct in Pittsburgh. Koussie will be jealous that he didn't get you first. Maybe you can start a career as our first native guest conductor.\n\nWhatever \"news\" I had I must have written to D[avid] D[iamond] who must have told it all to you, so I won't make the mistake of repeating myself. How I would like to sit in on one of those \"piano\" lessons. They must be the most original lesson periods given anywhere.\n\nWell, anyway, if you are coming here, no need to go on. Let me know sumpin' _soon_.\n\nLove,\n\nMe\n\nP.S. My home address is 8663 Holloway Plaza Drive. Tel. Crestview 1-0432. Just in case I miss you at the airport!\n\n148. Leonard Bernstein to David Oppenheim\n\n40 Charlton Street, New York, NY\n\npostmark 12 July 1943\n\nDear Dave,\n\nHere I am with two letters from you, and not the faintest idea of what's going on inside that newly militarized brain of yours. Is it all censorable? Or haven't you collected your _real feelings_ (big chord)? It's good to know that you've been classified as a musician, which seems to augur well for your future life-span. Does your uniform fit you? Have you got a weird little snapshot? How much vibrato do you apply to the sax?\n\nI suddenly find myself with a lovely call for an Army physical tomorrow night. God knows what will happen. I sometimes have a strong wish to go and get it over with and be calm and unresponsible. Then I see how easy that way out of a mess is, and the old realities, like career, and so on, crop up, and I want to stay as far away from it as I can.\n\nMy plans now call for two more Goldman Band concerts this month, finishing up Warner Bros. chores, a visit to Kouss in Lenox, to make plans for a performance of my symphony in the fall, then to Boston to conduct a pair of cute concerts with the 25 or so first-desks of the Boston Symphony (very good chance, and all modern pieces) and then to light out to Hollywood for a month of rest with Aaron, then perhaps to Mexico for a very short visit, then back here to become assistant to Rodzinsky. This is, of course, all the ideal way, and probably none of it will pan out, as it depends first of all on what the Army physical turns out to decide, then on whether I get a job collaborating on a book with Henry Simon, which I have been promised, and which would net me several thousand bucks over the summer, and finally on whether Rodzinsky ever makes up his mind. I found out, by the way, who the other two conductors are that Rodzinsky has asked to be his assistants \u2013 Max Goberman and Danny Saidenberg. What a trio we would make! But no real competition, you'll admit. Or won't you?\n\nReiner has set the date for my conducting of my symphony in Pittsburgh \u2013 probably some time in January. It's to be a three-ring circus for Bernstein \u2013 I'm to solo in the Beethoven Triple Concerto, then to conduct my (our) _Jeremiah_ (which seems more beautiful every time I correct another page of score) and then to finish up the program conducting some big work. Isn't that fun? And it really doesn't sound like Reiner to allow all that, does it? Wish you were around to take the clarinet solos. When Kouss heard that Reiner loved the piece so much he got all pepped up again, and asked me to come and play it for him again (his reaction last time was tired, you may recall, and there were so many people in the room, etc.). Truth is, it takes him a while to grasp a piece, as he himself will admit: so this time when I play it for him in Lenox it ought to be a real hit. He wants me to do that same sort of three-ring circus in Boston.\n\nApropos of which, I've run into conductor trouble about the first performances. Reiner demanded that his January date be the first (first performance), and I had to consent. Then when I had breakfast with Kouss at the St Regis the other day (!) he was a bit hurt and put out, as you can imagine, and ruefully suggested that he had a November date in mind for me. What an act. But he's still awfully sweet. It occurs to me now and then that with my idiotic way of handling these situations I may well wind up without either performance. It would be typical Bernstein. At the moment everything is passably under control. And Warner's is starting to make me the parts etc. (you know, they took the symphony) so maybe we'll have a piece yet. The Clarinet Sonata should be out any week now. As for the records, who knows the mysterious ways of Petrillo.120\n\nI feel good that I solved the double acrostic puzzle in the _Times_ this morning.\n\nAnd I wrote a new song called \"The Nicest Time of Year\".121\n\nMrs. Landeck has come back, bringing another married woman with her, and I am stranded here in sin with both of them, since I can't find another apartment, and anyway it seems silly to take one at this mixed up point. It creates quite an interesting triangle.\n\nI had the final Frau session, and left with conflicting feelings of regret and relief. She'll be back in September, and believe me, I really welcome the breathing spell. But there's no denying that she's done wonders, or at least somebody has. I can almost look at the problem now as a problem instead of as a Fascist enemy ready to strike. I have some amazingly tranquil moments. I almost married Rhoda [Saletan] last night; but stopped when I saw Judy [Holliday], and decided to marry _her_. What do you hear from Mad?\n\nAnd what in Heaven's name happened during those ten days in the psychopathic ward? It must be a luscious tale. Sit down when you're bored some evening (are you ever bored?) and put it all down on a piece of stationery. There's no postage necessary, after all.\n\nYes, Avshalomoff came to see me at my last Goldman Band concert, and we had a fine time. We saw each other again at David Diamond's. He's a good boy, and very sincere, and I wonder if he writes good music too. That would be almost too much.\n\nLove, and write me the ganze Geschichte.\n\nL\n\nWhat means all that nonsense in your address? It's the most fantastic one to date.122\n\n149. Leonard Bernstein to Aaron Copland\n\n40 Charlton Street, NNNYYYCCC [New York, NY]\n\n[July 1943]\n\nDear and Wonderful Aaron,\n\nWhen I got your insidious invitation to go west I dismissed it immediately as a real wacky idea. As time goes by I find it becoming more and more a real possibility. The only thing is that I couldn't possibly come now, especially since I want very much to conduct that little concert in Boston. I did write you about it, didn't I? It's a pair of concerts in a little series whipped up by the first desks and associates of the BSO, to relieve the monotony. I've been asked to conduct and if possible play the piano at the same time. I plan to do the _Cr\u00e9ation du Monde_ , thank God, the _Dumbarton Oaks_ (in _two_ rehearsals, forgive me) and a suite from PFB's [Paul Bowles'] opera. It seems to me that I could stick around here for the rest of July, finish up my chores at Warner's, do another Goldman Band pair ( _Outdoor Overture_ again, and Billy's [Schuman's] _Newsreel_ ), spend a few days with Kouss in Lenox, play him the Symph again, go to Boston, do the concert, and streak out like a wild one for Hollywood. Does that sound reasonable or not? Very simple. And then Mexico in September where we could meet Pfb who is dying to go there in September, and then home to become, God willing, an assistant conductor of the Philharmonic. Sort of a nice way to become 25 years old. Is it still possible with you, or has V. suddenly come home to roost? Of course I completely disregard the situation of his absence that makes my visit possible; I don't usually go in for being 2nd fiddle, but with you it looks good. (Or on you.) I'm all for it.\n\nOf course a lot of this depends on whether that job that Henry Simon has offered me comes through or not. Remember? The book on opera he's writing, and I'm to do the musical part of it, for 2 or 3 thousand bucks. Which would make the trip feasible. I ought to know in a few days.\n\nAnd the other catch is that I've received another call for an Army physical on Monday night. If my asthma is anything then like it is at this moment, they'll toss me out on my ear. I've really got it bad today, suddenly. Maybe unconscious preparation for the exam, as any analysand would say.\n\nHad the final Frau session today, which is rather thought-provoking about what has been accomplished. Answer: much. The rest is up to me now, as you might imagine. And God knows what evil deterrent influences may befall me in sinful California. But come what may, I'm ready to try.\n\nFound out who the other two conductors are who have been asked for assistantships by Rodzinsky. Guess. Max Goberman and Danny Saidenberg. What a trio we'd make \u2013 Saidenberg, Bernstein and Goberman, the three prides of Goebbels. No competition though, really, you must admit.\n\nHad breakfast with Kouss yesterday at the St. Regis. Lovely time, he's a lovely man, it's a lovely hotel. He asked me up to Lenox for a few days, admitting that he hadn't really heard the Symphony very well when I played it to him in Boston, since there were so many people in the room, and he was tired. I think Reiner's enthusiasm kind of pepped him up. You were right about the jealousy though (you're always right). He was a little hurt that Fritz had copped the first performance. Added rather ruefully, \"Well I have planned a date for you in November (when I told him that Reiner's date was in January) but I suppose now we'll have to make it for the second half of the season.\" What an act. We'll see. I have a feeling I'm getting into a deep well full of hot water with this first performance racket. I really don't quite know how to handle the boys. But just think how well I'll understand someday when some young composer from Podunkville won't know how to handle me!\n\nWent to a long cocktail party last night with Pfb [Paul Bowles], home of Paul Peters and Herr von Auw!!! Occasion: triumphant return to these parts of one [John] LaTouche. Great to-do. It went on all night, mostly me and Touche and Paul, and wound up at Peter Monro Jacks' (horrid) and we finally left Touche there in his four o'clock cups. He's a terror if there ever was one. I sort of like him in a weird way, especially when he's sentimental in that mountebank manner. It's a wonderful aggressiveness.\n\nJust thought: what will happen to poor David's piano lessons if I leave?\n\nMy asthma is really kicking up. I'm going to bed, [words blanked out] and creep guiltily back to some semblance of normalcy.\n\nMorton Gould completely ruined _Billy the Kid_ at the Stadium last week, to say nothing of _Newsreel_ , which isn't so hot to begin with, to say nothing of Roy's _Ode to Truth_ , which \u2013 well, I can't even describe it, except as an eternal measure of turdlike notes that will never be counted out. It's a piece with his three tricks in it, and it does make a very unpleasant sound. What else? Seems like I've given out with a lot of news. I'll have nothing to say when I arrive. Write me right away, and tell me you approve of my little plans.\n\nI love you, as if I had to point _that_ out!\n\nL\n\nThe _Salon_ cover is awful pretty.\n\n150. Aaron Copland to Leonard Bernstein\n\nSamuel Goldwyn Studios, Los Angeles, CA\n\n16 July 1943\n\nDear Second Fiddle Black Magic,\n\nToo bad, too bad \u2013 this was the perfect moment for you to come to Hollywood. I dread to think what life will be like in August. I may not have a house by then \u2013 it's only a sub-let and my lady threatens to come back \"sometime in August\". Houses are scarcer than Filipinos out here, and _they've_ practically disappeared. I will be doubled up with notes \u2013 right in the midst of it in August \u2013 with hardly a moment to eat, no less take care of a bombshell like thou. (There's an hour's worth of music to be written \u2013 I just finished calculating it. Where will all those notes come from, I wonder?)\n\nOn the other hand, I saw the announcements of the Boston concerts \u2013 it all looks quite impressive and naturally you're right to be staying and conducting. So of course come in August if you still think it's insidiously attractive \u2013 but consider yourself warned that you may have to sleep on park benches, and converse with a guy that has noten indigestion. Whatever you decide to do, you'd better make reservations now, because trains and planes are full-up. As you point out, the Army may step in and end this little pipe dream.\n\nI was surprised you had a \"final session\" with the Frau. In my innocence I thought those things went on for years. You mean you're done? Finished? How extraordinario.\n\nAnd exactly, may I ask, what is an assistantship to Rodzinsky? Is anything guaranteed? Or even promised? I'd like to know what's really up the old boy's sleeve. His monkey business with American composers in Cleveland makes me suspicious of the purity of his motives. But don't forget to neglect to mention my reaction to him.\n\nHeinsheimer says that Reiner has programmed the _Sal\u00f3n_ with the Philharmonic for Aug. 8th. Did you have a hand in that?\n\nSpent an awful pleasant evening with Jesse and June [Ehrlich] in their little house perched precariously on top of a Hollywood mountain. They're so gentle and \"different\" \u2013 sort of poetic people. They had a soldier guest for me, who kept spewing venom at Irving Berlin all evening. Oh yes, and I met Leonard Posner out here. Remember him? He played that there violin piece of yours on WNYC. And also I met a guy who says he knows you \u2013 pretty cute looking too. A Mr. Pole. He was in the Army, but they let him out after 4 months \u2013 lung trouble. Anyway, that's what he says. Oh yeah, and I had dinner with Gail Kubik123 in a uniform. Quite a lot of boyish charm despite the bald head. And the Van Eycks \u2013 Ruth and Goetz (renamed Peter for the movies) have parties all the time. Harold Clurman came out and made me less lonely. (You lush thing you.) And thus and so.\n\nLove\n\n151. Leonard Bernstein to Serge Koussevitzky\n\n17 Lake Avenue, Sharon, MA\n\n[August 1943]\n\nDear Dr. Koussevitzky,\n\nAn amazing thing has happened! Thursday night I was deferred for all time from the army. As you know, my recent siege of asthmatic hay fever had caused me to be sent for reexamination to the Medical Advisory Board, where I fell into the hands of one Dr. Wesselhoeft, who is in charge. He is a firm believer in the British policy of leaving as intact as possible the cultural foundations of our country, even \u2013 or rather, especially \u2013 in time of war. He was therefore _happy_ to disqualify me on medical grounds; and put me in Class IV, where, he assured me, \"nothing can interfere with your career.\"\n\nI am therefore free to pursue my work through its next channel, whatever that may be; and I am happy to say that because of the spirit of the Medical Board, I feel no guilt whatsoever at my deferment.\n\nDevotedly as always,\n\nLeonard\n\n152. Leonard Bernstein to Serge Koussevitzky\n\nHotel Chelsea, New York, NY\n\n[September 1943]\n\nDear Serge Alexandrovich,\n\nHow I would love to be with you now to share my great joy with you! I am still so excited I can hardly write this letter. Everything seems to be going so well.\n\nI finally had my talk with [Arthur] Judson and [Bruno] Zirato this morning. They were very nice indeed, and extremely authoritative. I realized immediately that they had the situation in hand, and that I was simply being told their terms, not asked my own. All of which was perfectly all right with me, since I feel so strongly about doing this job, and doing it as well as possible, that I would probably, in my enthusiasm, accept if there were no salary at all.\n\nThe first thing is that there is apparently to be no contract at all. As Zirato pointed out, he doesn't believe in them, and never had one with, for instance, [Mishel] Piastro. I am to receive $125 a week. I realize that this is not tremendous, and that there are only 28 weeks in the season. But I am very contented with it, especially insofar as my publishers have raised my weekly royalty advance to $50 a week, which will continue all year. I simply felt that until I have proved myself to the Philharmonic and to the public, I have no real right to make any demands. On the other hand, the absence of a contract has its advantages, because I can be free for the summer, or for any occasion that may arise. It makes finances a little bit unsure, of course, but believe me, I am very happy in spite of that. I hope you can understand the situation in which I found myself; in fact I am _sure_ that you will understand it. I simply could not ask $12,000 or any other sum for a job which thousands of conductors in this country would gladly pay to have. Once I have shown that I am of real value to the Society, then there is time enough for me to make demands. I am perfectly willing to seem na\u00efve now; as long as I know myself that I am seeming na\u00efve. The main thing is to do my job; if I can do that well enough, and if I can bear all the huge responsibilities that come with it, the rest will come by itself, I am sure. Believe me, I tried very hard to feel like Koussevitzky while I was in the Judson office, but I was only Leonard Bernstein, and I had to act as I did. Don't you think it is for the best?\n\nAnd in the middle of all this, I only have to look at your picture in my room, and I am perfectly contented, knowing that there is one supreme friend that I have, who will understand whatever I do, mistakes included. I hope to see you very soon; meanwhile take good care of your health, and know that my love is with you always.\n\nLeonard\n\n153. Fritz Reiner124 to Leonard Bernstein\n\nRambleside, Westport, CT\n\n4 September 1943\n\nMy dear Leonard,\n\nNeedless to say that I am very happy about the news!125 It is a great chance and I do not doubt for one moment that you are going to make the best of it. You have the talent and the tenacity to put it over. I hope that the years spent at the Curtis with me will bear fruit.\n\nAs to your appearance in Pittsburgh on Jan. 28 & 30th, you will feel relieved to know that you are not expected to play the Triple or any other concerto. You will only conduct your Symphony and another work at the end of the program about which I would like to have your suggestions. Also \u2013 please let me know the name of the lady who is to sing the Symphony in Boston with you. Maybe we could use her also in Pittsburgh.\n\nI shall be at the Hampshire House from Tuesday the 7th until Thursday the 9th in case that you want to get in touch with me.\n\nHeartiest congratulations once more, in which Mrs. Reiner joins me.\n\nSincerely yours,\n\nFritz Reiner\n\n154. Adolph Green to Leonard Bernstein\n\n[Hollywood, CA]\n\n[September 1943]\n\nDollink Leonard,\n\nI'm writing, I'm writing, I can't believe it. My pen is tracing figures on paper, making bold, masculine markings indicative of a strong character and a willful mind plus creative ability, yet with a strange strain of tenderness withal and a slight indication of liver trouble.\n\nForgive me for not writing sooner. As always was the case with the Revuers, we have been through parlous times. I'll give you a brief resum\u00e9.\n\nThe day we arrived, our agent, Kurt Frings, told us that _Duffy's Tavern_126 was off. The varied producers had quarreled \u2013 but, said Kurt, this was a good thing because now we were free to receive really good offers. We opened at the New Trocadero & were sensational, so sensational that the owner let us go after 4 weeks because he figured that now that he was doing a landslide business, it would continue so without us. At this point we realized that no movie company wanted us. Too smart, they said. First M.G.M. turned us down. Then everybody else turned us down all the way down the line, including dinky little Universal, who screen-tested us and said we stank. We were in despair. Then \u2013 our agent Kurt Frings (who, by the way, is a sensational agent, plus a Viennese gentleman) got us an audition at 20th Century Fox, right on the lot. It seems that 20th was the one company that hadn't caught us at the Trocadero. Oh God, we said, an audition, how horrible. We went to the audition. First, in walked high and mighty Lew Schreiber, Darryl Zanuck's chief assistant. We trembled. Then all the producers and directors \u2013 Lubitsch,127 Schambitsch,128 Perlberg,129 LeBaron130 etc., etc. And finally the great Darryl himself in simple slacks & polo shirt. We started doing our numbers. For four numbers no one smiled. We noticed that a number of them wanted to laugh, but had to stifle it, because Zanuck didn't look happy or pleased. Then suddenly it happened. D.Z. grinned. HE GRINNED!! Then he chuckled. CHUCKLED!! From that second on we were in. Everybody there roared & rolled & clutched their sides with helpless laughter. We did number after number & they screamed. That very day we were signed for a super-duper all-dancing all-technicolor, all-Alice Faye131 picture with a minimum guarantee of 6 weeks at a very fine figure indeed. The picture doesn't start till the middle of October, & we don't quite know what we're doing till then. Maybe a two or three week engagement at the Mark Hopkins132 in San Francisco.\n\nThe picture by the way is called _Greenwich Village_ , and we're going to do _Bazooka_ in it, plus a new spot which should be something stupid about the Vie de Boh\u00e8me of Greenwich Village. We'll have to build it around a song by Nacio Herb Brown \u2013 in march tempo \u2013 sample lyrics as follows (lyrics _not_ by us):\n\nIt's all for art's sake\n\nIt really is\n\nWhatever we do we really do\n\nFor art's sake.\n\nThere's that lady there\n\nA very mysterious gypsy\n\nBut honest folks,\n\nShe's really from Poughkeepsie.\n\nPlus another undetermined spot, plus parts in the picture itself. All in all, the set-up looks very good for us.\n\nHollywood is the weirdest country in the world. I'm only afraid you would love it here. One day Aaron [Copland] & I were envisioning the way you might take to it \u2013 a mad swirl of parties and gatherings, with you the life of the [party]. Then you awakening in the morning with a hangover \u2013 or fluff on your lungs, a fly on your tongue, etc., etc. \u2013 and filled with remorse. \"My God, I'm not getting any work done \u2013 Oh God, what the hell am I doing \u2013 it's fantastic, I'm not accomplishing anything. Oh, my God!\"\n\nOf course, I should not talk. Almost everyone I've ever known is out here and everyone is rich as Croesus, and life for me has been that self-same swirl \u2013 not terribly mad but the liquor and the thick steaks flow. It's a terribly unreal life out here, if you're with prosperous people who've decided you're a comer & sort of take you up. At first your conscience bothers you that these swimming pools & groaning boards exist while the whole world is starving and dying, & generally tightening its belt. After a while, you relax & enjoy it. After that, you suddenly become horribly bored with it. It's really meaningless & stupid & everyone out here is bored & screaming for some kind of diversion. You see the movie people out here never exchange anything resembling ideas. Most of them are stupid to begin with & impossibly spoiled by all their money, & the more intelligent minority are just _afraid_ to exchange any intelligent remarks. Nothing is secret out here, and even the most casual statement might drift back to the wrong person, & shit, you just mustn't offend anybody. The first thing you know, you'll be out on your ear. A good friend of mine is a movie director, Frank Tuttle. He is prosperous now & back in the dough, through having discovered & put across LADD: an Alan. But he was black-listed for almost 4 years because of having openly expressed sympathy for the Loyalist cause back in 1937.\n\nI'll write more about H\u2013wood later. Oy, I've seen so many movie faces and know them all, all the sad little extras & bit players. As a matter of fact, I've scared the hell out of a lot of people with my well-stocked memory.\n\nIncidentally, Tuttle took us out to visit Charlie Chaplin last Sunday. Quelle disappointment! Charlie is now a fattish, ageing man, and he insists on being the life of the party. He was bounding around all afternoon, clowning, grimacing, putting on native Balinese and Hindu phonograph records & dancing madly to them. This sounds charming, I know, & had you been there we might have had some fun with it \u2013 but somehow Chaplin was a little more frightening than amusing, mainly I think because there was more of an air of desperation than joie de vi[vr]e in his cutting up. The guy just didn't look cute and I kept thinking, \"Who does this mincing fat-necked little fellow think he is, imitating Charlie Chaplin?\"\n\nI've seen Uncle Aaron a few times since I've been here. He's been working furiously finishing up his film133 \u2013 and the scoring was completed last night. Aaron let me come to the studio to watch. They had only five small scenes to complete, but the music sounded fine. There's an especially cute little theme for Walter Brennan who seems to be portraying a crusty, lovable old peasant.134 It's sort of a cross between a Slavonic dance & Schumann's \"Jolly Farmer\". Anyway, that's the mood.\n\nThe orchestra that recorded the score was largely made up of the Warner Bros. musicians' crew and I never saw musicians as excited & enthusiastic over anything as they were over Aaron's score. It was just miraculous to them after all the Steiner\u2013Korngold crap they've been playing. Imagine a composer who not only does not have the hero & heroine do their big kiss to the accompaniment of surging strings, & bl-w-l-anging harps in great Straussian release, but cuts out the music entirely at that point.\n\nDo you see Billy Schuman? Give him & Frankie my love & tell him that I have seen much of his old friend & co-partner, Frankie Loesser, plus his wife. Loesser, it seems, is an old admirer of ours, a hysterical admirer, in fact, and he & his wife have been most generous to us \u2013 many dinners, parties, etc. He is a typical Hollywood case \u2013 horribly prosperous and a back-slapping one of the boys. He is a very nice guy, though, and really talented at writing lyrics. His new movie _Thank Your Lucky Stars_ has some very nice stuff in it, which he wrote with Arthur Schwartz, and any song he touches these days is a sure hit. But a typical example of Hollywood in what he said about Billy Schuman. If you repeat this to Billy I will loathe you to my dying day. I was having a nice conversation with him the first time I met him. Here is a man, I said to myself, who hasn't gone Hollywood. Then I mentioned Billy \u2013 \"Yeah, he's a swell guy\", says Loesser, \"but you know Adolph, where the hell is he today? That long-haired stuff doesn't get you anywhere. O.K. he's teaching & turning out that symphony stuff & he's got a wonderful wife & a home in Westchester \u2013 but what the hell, he's going to end up on the shit end of the stick. He ain't on that gravy train, Adolph. There's no dough on the Icky express\" etc., etc. \u2013\n\nFor great Horowitz' sake don't tell this to Billy. It just might get back to Frankie L. And besides Loesser really loves him. He was just giving out with the Hollywood jive that only strong men don't succumb to the lure of.\n\nLoesser told me a cute story of him & Billy when young. Loesser is quite a small guy about 5 foot 4. One day he & Billy were walking down a street. Suddenly, out of the clear blue sky Billy turned to Frank & said, \"What the hell, I'm bigger than you\", and proceeded to wallop the shit out of him.\n\nEnough of all this crap!\n\nAll the Revuers are fine, Betty [Comden] is ecstatically happy. Lizzie [Reitell]135 is here on a 16 day leave. Little Alvin [Hammer] is soon sending for his wife & child. Judy [Holliday] is this moment on her way to New York for 2 weeks. Why don't you call her at mama's around Monday. SU-7-6229. She'll be able to tell you in detail of what's been going on.\n\nWrite me everything that's been happening to you, at once, do you hear, at once!!!!!!!!!\n\nI'll write you more later.\n\nI think I've been pretty happy here so far, and I look staggeringly better than I have my whole life.\n\nI have a grizzly feeling that we've really got a future in this place, Lord help me, even if we do just do one picture, we won't be back in town before January.\n\nSo please write!\n\nLove\n\nShrdlu\n\nP.S. I hate people who go to Hollywood.\n\nP.P.S. Heard about your appointment with the Philharmonic. Nice goin' kid. Congrats & all that. But strictly between us, where's all that long-haired stuff going to get you? You don't want to end up on the shit end of the stick. You'd better get on that gravy train, son. But nice goin' kid.\n\nWrite, write write.\n\nI love you\n\nI miss you.\n\nRegards to everybody.\n\nWhat happened to your draft board?\n\nPaul Bowles?\n\nDavid Diamond?\n\nYour love life?\n\nYour Boston concerts?\n\nWarner Bros?\n\nRhoda Saletan?\n\nYour symphony? Reiner? Koussevitzky?\n\nSee, it's wonderful about the Philharmonic. It's thrilling!! It's marvelous!! I can't wait for the Copland festival!!\n\nHave you seen Jesse Ehrlich136 & weib? His wife is _not_ the big fat colored woman you dreamt of. [...]\n\nWe have a wonderful apartment here. It's more of a house than an apartment. 6 rooms, 2 baths, 3 radios, piano, bamboo liquor bar, roof terrace. We got it by a sheer miracle, because you know the housing conditions here are impossible. Viola Essen & her mother found it for us etc. etc. There's an extra bed waiting for you. Come out!!\n\nLove,\n\nAdolph\n\n155. Randall Thompson137 to Leonard Bernstein\n\nDivision of Music, University of Virginia, Charlottesville, VA\n\n16 September 1943\n\nDear Leonard,\n\nAfter many years of hard work \u2013 under grueling taskmasters; in the face of tyrannies, rivalries, pedantries, rebuffs \u2013 there came to the young musician a first-class opportunity worthy of his powers. Notwithstanding a mercurial temperament which was wont to raise him into the Empyrean at moments and drag him, at others, into a deep and brooding melancholy, historians are generally agreed that this sharp upswing in his musical career brought him, inwardly, a steady and deep feeling of satisfaction, security and happiness. And they are equally agreed that this feeling was fully matched by what his many (and varied) friends felt on hearing of his new appointment. Not the most eminent nor yet the least devoted among them is known to have been\n\nHis ever sincerely,\n\nRandall Thompson\n\n156. Leonard Bernstein to Aaron Copland\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\nFriday [?September 1943]\n\nMy Dear Mr. Copland,\n\nI'm saving all my talk for your very own personal ears; but I just wanted to show you this super-authentic stationery with its _free_ stamped envelope. Ah, the life of an assistant conductor.\n\nIt all seems to be working out beautifully. Rodzinsky, of all things, turns out to be a fine gentleman.\n\nI'm off to dinner with Kouss (we've already spoken very seriously of reviving your _Ode_!)\n\nThen the weekend with Bill Schuman & Frankie who is _pregnant as hell_!\n\nBut all this is mere substitute for the real thing \u2013 the week of Oct 4 is all yours (except that it's the first week of the Philharmonic season). I can't wait. Speed the day.\n\nI love you.\n\nL\n\nLost my temper with D[avid] D[iamond] t'other night, & left him in a rage in a bar. That's just _one_ delicious bit of gossip, a sample of the horrors in store for you. Hurry, hurry, hurry.\n\n157. Leonard Bernstein to David Oppenheim\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n5 October 1943\n\nDear David,\n\nI just ran across, in moving, your ink copy of a little known piece of mine called \"Two\" which set all kind of memories, delicious and otherwise, in motion. I have a tremendous desire to see you again. Is there any possibility? Where are you? (I'm taking a chance on your last address, as of last summer.) Why did your fertile crop of letters from the army suddenly stop? You never answered mine, you know; or did you never receive it?\n\nSo much has happened since our last contact that it is impossible even to begin to deliver information. Life has been marvelous, hectic, and unreally beautiful since my fantastic appointment, of which you must have read somewhere. It was a real shock to me, since I had had no inkling of it, beyond a rumor that I might become one of _three_ assistants. _And_ I had never met Rodzinski (who turns out to be a swell and honest guy). The position is unprecedented for one such as me, and a really historic step in terms of other young conductors. But I must see you to tell you, as the Frau says, \"what is really going on.\"\n\nI have a fine large apartment _in_ Carnegie Hall (address: Carnegie Hall, Apt 803, NYC) from which I can literally walk on to the stage. It's quite beautiful, and I'm having a very quaint experience furnishing it, and it has an extra bed for you, and my _own_ bathroom & kitchenette. If you are within 100 miles of New York at any time, please let me know, & come to town. In fact, let me know where you are in any case.\n\nThe day before my appointment was revealed to me, I was rejected might and main by the army for asthma.\n\nAaron returns next week. Write, & spend your furloughs here.\n\nLove,\n\nLenny\n\n158. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n9 October 1943\n\nDear [musical notation: D-A-E, i.e. Re-La-Mi],\n\nWell, the opening is over, with a bang-up reception at the St. Regis & your nice letters to both of us, and I a great hit with Mrs. Lytel [Lytle] Hull, & Myrna Loy was there, & Frank Sinatra (who Oscar Levant says is the image of _me_ ) and Bruno Walter & Fabien Sevitzky & Marshall Field & and & and & and. You would have loved it. The concert was less exciting than the reception but maybe it will all pick up soon.\n\nThe other news is that Steinway has just moved a piano into my room. The same color as yours, same shape & size, & they're standing together now side by side like two beautiful horses in the meadow. But one is more in tune than the other (guess which?). Now \u2013 what is the action to be taken on _Baby_ Steinway? Or will I have _Babies_ Steinway? Do let me know.138\n\nWhat the hell is Edgewood Road [Longy's address in Baltimore, Maryland]? Liberty 6510 sounds like Boston. Is it fun? Any nice people \u2013 are you branded a Jew-lover yet? Any good students? When do you come to NYC?\n\nThese and many other things, write, and make no bones about it.\n\nLove\n\nLenny\n\nMy job is marvelous \u2013 29 hours a day.\n\n159. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n15 October 1943\n\nDear R,\n\nA lovely series of illegible postals have been arriving daily. You say nothing about your work, your students, your milieu. Is it bearable?\n\nAs to the furniture, sure I could use it: especially did you say, a chest of drawers or a bureau? Or is that French for desk? The bench sounds swell: the desk is not essential, but could be useful, I suppose. Could it be used as a chest of drawers? That's what's really on my mind \u2013 my shirts are all in suitcases. Would it be easy to ship? In storage? In NYC?\n\nThe \"dame merveilleuse\" is writing you today and sending the first payment.\n\nGlad you saw Randall. He's lovely.\n\nI haven't actually conducted yet. Monday the 18th is the first time, & I will do readings of Diamond's 2nd Symph., Haieff's Symph.,139 & Charles Mills' Symph.140 And probably all the Chaikovsky rehearsals too.\n\nYes, we often rehearse on Saturdays. The orchestra is already greatly improved, & Rodzinski is a fine guy, & a very conscientious (if not always over-profound) conductor. Secrecy, please.\n\nLet me know how it goes. I'm very happy. Very. Aaron is back, & all is right again. The _North Star_ (his movie) is fine enough, & I've been fighting with Oscar Levant again. I may get a commission to do a ballet for the Ballet Theatre!141\n\nAs I say, let me know...\n\nLove,\n\nL\n\n160. Leonard Bernstein to David Oppenheim\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY, Carnegie Hall 803\n\npostmark 22 October 1943\n\nDear Dave,\n\nDelighted to hear you sounding much the same (it's even fun to hear you're still neurotic). But it's not so nice to hear that you're so unsettled. Why can't the army discover you and use you properly? It seems a shame that you should languish this way in what you call the plague state.\n\nI'm in a big hurry \u2013 as always these days \u2013 since the JOB is all-consuming. It's quite glorious & exciting, & I wish you were here to be in on it a little bit.\n\nMy best to Mad, & love to you \u2013\n\nL\n\nKeep out of those suicidal depths.\n\nThe recording depends on Petrillo's release.142\n\nThe Sonata is off the press today. Slight delay because of editor's sickness. Will send you a copy.\n\nI guess Detroit is off on both our sides.\n\nAlso, Frauistically speaking, I have never been better.\n\nWhy don't you communicate with her directly?\n\nAddress: Marketa Morris, 562 W. 113th St NYC.143\n\n161. Ren\u00e9e Longy Miquelle to Leonard Bernstein\n\n3901 Edgewood Road, Baltimore, MD\n\nLundi soir [?October 1943]\n\nDarling Spookietchka!\n\nIt's such fun getting a really happy-sounding letter, for yours actually exudes happiness and in turn I am truly heureuse for you.\n\nBut after these many years you certainly should have learned to read my so called illegible handwriting... after all it's not baritone clefs.\n\nI wrote you a postal this morning; I guess I knew you were conducting then. Your letter which I just found this minute confirms that \"je ne sais quoi\" which I felt earlier in the day.\n\nWhat have you been fighting with Oscar \u2013 \"again\" you say \u2013 didn't know you did before. By the way, if Sinatra looks anything like the picture I saw of him in _Time_ magazine, Oscar L. is completely \"dingo\" to say the two of you look alike.\n\nNow on a \"terestrial digression\" (or is it terrestrial? poor French me don't know) \u2013 those odd pieces of furniture are in storage in Philly. They very likely could be picked up by some moving truck, as a fill-in load and brought to you. A bureau is French for _desk_. It's a flat top desk, mahogany finish, with three drawers on the right hand side and another shallower one across the knee-hole [...] The bench you sat on when you had those cotelettes de veau \u00e0 la cr\u00e8me, and sat on it many another time afterwards. Sorry the bureau does not happen to be une commode, ni un chiffonier. When I go to Philly, I shall check up on all these things and let you know what there is. If you still want it, well and good.\n\nSo you have \"abandoned\" our baby Steinway... a fine unpaternal person you've become.\n\nIf I say nothing about my work, students and milieu it's because there is nothing to relate about it. Work? Same as before, interesting to some extent, the students fair (ni lard, ni cochon, honest, conscientious, not at all exciting). Milieu? non existent as yet; although it looks as though it might become so, fairly soon.\n\nHave not been branded a Jewess or a Jew-lover yet, although am sharing the apartment of the mother of a former Curtis pupil and they are Jews. Edgewood Road is a bit like the Newtons, but a more recent development; suburbanish. It's a bit far from town, and certainly adds up to an expensive transportation item, let alone the expense of energy (from 50 to 60 minutes to, and same from...) Am biding my time, and looking around for something closer to things.\n\nSaw Lester Englander last Friday (he has a job here as cantor, you know, and he comes down every week). He told me that Leo Luskin is back in Philly... wonder if he is as literal as ever.\n\nI must stop this so to have it in the mail for the one and only collection around these parts.\n\nMy love to Aaron when you see him.\n\nTo you... beaucoup d'affection.\n\nR.\n\n162. Leonard Bernstein to David Oppenheim\n\nCarnegie Hall 803, New York, NY\n\npostmark 4 November 1943\n\nDear Dave,\n\nItem: You should receive any day one (1) copy of the Bernstein\u2013Oppenheim Clarinet Sonata, prepaid, and with my love.\n\nItem: You owe the Frau exactly $26.00, plus $10.00 for the Rohrschach job.\n\nItem: You're very slow on answering letters.\n\nThings are fine, O.K, progressing rapidly.\n\nLet me hear from you.\n\nLove,\n\nL\n\n163. Leonard Bernstein to David Oppenheim\n\nCarnegie Hall 803, New York, NY\n\npostmark 9 November 1943\n\nDave,\n\nThinking of you daily, I sometimes get the thought that you are desperately needed by symphonic organizations. Isn't there some sort of honorable discharge that would fit your case? Isn't a \"line\" (and a livelihood) one of your big needs?\n\nLove,\n\nL\n\nDid you get the Licorice St\u00fccke?144\n\n164. Leonard Bernstein to Jerome Robbins145\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\nTuesday [late 1943]\n\nDear Jerry,\n\nI've been a stinker not to have written sooner, but I guess you know what has been going on with this baby. I have hardly breathed in the last two weeks. Nothing but reporters & photographers, & calls & mail & rehearsals, & I'm conducting this week (listen on Sunday!), & my scores pile up mercilessly. My Symphony parts lie uncorrected, & my \u2013 _our_ \u2013 ballet lives only in the head \u2013 only one scene on paper. But _it is_ on paper (not legible, but I'll make it so as soon as I can).146 That should cheer you. Fear not: somehow I'll get it done, though it's a fancy challenge.\n\nThe scene that's almost done is from the Entrance of Girl I through to the Entrance of Girl II, & the Pas de Deux. Everyone seems to be quite mad about it \u2013 I hope you will be. Of course it's all only 3 or 4 minutes \u2013 but that leaves only 16 more!! God, what a race with destiny!\n\nI now to my naked bed to regain all those vanished ergs. All success to you & I really will do my best.\n\nLove\n\nL\n\nBy the way, I have written a musical double-take when the sailor sees Girl #2 \u2013 has that ever been done before? And the rhythm of your pas de deux is something startling \u2013 hard at first, but oh so danceable with the pelvis!\n\n165. Leonard Bernstein to Jerome Robbins\n\nSunday [December 1943]147\n\nDear Jerry,\n\nThis to announce that I've really finished numbers III & IV. I'm not quite sure about the exact timing, but I have a bit of a suggestion. What do you think of the idea of having a _part_ of a regular commercial song sung (by the bartender, or the jukebox, or something) during the pause from the exit of the 2 sailors with Girl #1 to the entrance of Girl #2? You see, the _pas de deux_ between you & Girl #2 is based on a popular song style, but rather a complicated variation of same; and I think it might also have a bit more (a lot more) meaning if the song \u2013 a part of it \u2013 had already been heard in a purely nonchalant, commercial way. It might also prove to be a success song \u2013 which would help the ballet's career, & yours, & mine. And it would also provide an increased suspense \u2013 during a welcome lull. And add time\u2013weight to the whole work.\n\nI realize that there are all sorts of handicaps, like paying a singer (what about bartender?) \u2013 or getting sound equipment for a jukebox. What do you think? The song itself is very blue, intimate, sexy and naive, but unusual formally.\n\nI'm really doing my best to have it ready in time. It's a battle, but everything's a battle that ever turns out to be good.\n\nRan across Agnes de Mille last night, & she's really rooting for you & the ballet. Contract is signed, I've received initial payment. (I've been rooked out of all bounds, but I don't care.)\n\nGood success to your tour and let me hear soon. Be good.\n\nLove,\n\nL\n\n166. Leonard Bernstein to Jerome Robbins\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n[December 1943]\n\nDear Jerry,\n\nYour description of the state of the Ballet Theatre sounds gruesome. Don't take it too seriously, and get some sleep. (I should talk.)\n\nI've sent the prints of III & IV off to you. I finally got them in ink: but I still am not satisfied with the end of III. It will have to be changed eventually, but it won't make it too hard for you, will it?\n\nI'm working on the second (relaxed) section. I think finally I've got the idea. It's rather suspended in feeling, with little interruptions of staccato rhythms punctuating slow woodwind phrases. Also, the little dotted tune in No. III (\"Much faster\") will be introduced in this section, as a kind of hot pants feeling. So, you will have, in general, a slow, slightly tense, slightly serious section, but lyrical, perfumed by these little rhythmic urgings in the balls.\n\nI know you can do fine with it. Does it fit with your plans. (It's all thematically related to the opening dance.) Incidentally, I'm stretching the opening dance a little by adding a little section of about 15 seconds. It makes the form more telling.\n\nI've also decided to give the piano quite a solo role. It grows more & more important all the time (it never remains alone for more than a few measures); and seems to be the auditory key to the ballet \u2013 since a piano gives the feeling of percussion, brazenness, hardness, brightness, honky-tonkness, clarity, and intimacy. Don't you agree? How does a solo piano sound at the Met?\n\nI heed heartily your pleas for simplicity. The score actually is very simple \u2013 only the rhythms have to be concentrated upon like fury. There's no simplifying the rhythms \u2013 they're there, & they're the essence & basis of the whole score \u2013 but the notes will be very easy. I think it can be done. We can only pray. Nay, I know it can be done.\n\nWe'll be good. If we worry enough, it's got to be a smash. Wish you were back in N.Y. It would all go much faster.\n\nAbout that radio. It's a problem. You see, the song on which IV is based cannot under any circumstances come just before the pas de deux on the radio. The song itself and its completion in IV must be separated (preferably by the transition before it). Can we have the song sung at the end of III (the radio having been turned on during III, as indicated in the print), and proceed directly to the transition? I mistakenly indicated the Radio _after_ the transition, which is just where I _don't_ want it. It seems simple enough. Is it OK?\n\nNo completed Variations yet. Soon, soon. Grace of God, Moses, & The Societ\u00e0 Filharmonica. I'm to meet Toscanini on New Year's Day, & I'm being very Italian these days. Bless you, & write.\n\nLove,\n\nLenny\n\n167. Shirley Gabis to Leonard Bernstein\n\n[?January 1944]\n\nDear Lenny,\n\nWill you be a sweet boy and send me Adolph's address in Cal.? have a sudden yen to write to him. I heard the Clarinet Sonata on records and feel moved to give you the following criticism \u2013 unasked for though it may be \u2013 as an honest human being and a conscientious composer it would seem that you should not be satisfied with your music until it has a little bit of the real Lenny in it \u2013 and not just a rehash of Hindemith and Copland.148\n\nOf course I haven't the vaguest idea of what has happened to the real Lenny and even if he exists any more.\n\nBut if you would only be your own severest critic and not let anything you write go out for public consumption until you are sure way down deep that what you have written is truly worthy of you and that is music that is really music and not trash.\n\nMy god Lenny why don't you listen to a few Beethoven quartettes \u2013 and perhaps you will find in them the true meaning of artistic integrity.\n\nI hope I don't seem too harsh \u2013 but it makes me sore as hell that there is perhaps something deep down inside you that is honest, sincere, and good that the you outside ignores. And Lenny, believe me when I tell you that although you are headed for a brilliant career, it will never be a great one. Your driving ambition to be the most versatile creature on earth will kill any possibility of you becoming a truly great artist in any one of the talents you possess.\n\nThink hard, Lenny, bore way down deep into yourself and find there the courage to be honest. Is your mission in life to be the greatest of all dilettantes??\n\nIf you have a real contribution to make Lenny, you must find out now what it is. Concentrate and work and make it a great thing \u2013 and don't write clarinet sonatas that make any serious musician think you an utter fool \u2013 it's not fair to yourself because you're not really an utter fool.\n\nAnd I say again, listen to lots of late Beethoven, play the sonatas often. Understanding Beethoven can teach you more about the things you must learn than anyone can possibly tell you.\n\nOf course there is no reason why anything I think should impress you greatly, and when I tell you that at present I have little respect for you as a person and an artist, you can think \"she's a little fool\" and with perfect right \u2013 but Lenny, despite all I still have a vague sentimental attachment for you, and remember good things about you that seem completely lost now \u2013 and I do wish the very best for you, and sincerely hope that somehow, some way, those good things in you might aid you in becoming something truly worthwhile.\n\nAnd please send Adolph's address for which I thank you in advance.\n\nShirley\n\nP.S. Good luck for Pittsburgh.\n\n168. Leonard Bernstein to Serge Koussevitzky\n\nHotel Schenley, Pittsburgh, PA\n\n[28 January 1944]\n\nDear Serge Alexandrovitch,\n\nHere I am, finally, in Pittsburgh. I am excited beyond words at hearing my symphony. I must say that to me it sounds just as I thought it would. The orchestra is rough, but in a way like the Tanglewood orchestra \u2013 full of spirit, young, and cooperative to a great degree. I am having a marvelous experience here. Even the scherzo is _almost_ perfectly played \u2013 but for a real performance we must wait for the Boston performance. I have had to make surprisingly few changes. And the _Firebird_ is a real, fiery, Tanglewood performance.\n\nHave you been reading about the mess in New York since I left? They phoned me here Monday to return for the Tuesday rehearsal, but it was impossible. Then Byrnes [Harold Byrns] rehearsed _Rosenkavalier_ , and made a mess of it; so it came off the program \u2013 and Hans [William] Steinberg will conduct. I derive great satisfaction from it all \u2013 it feels wonderful to have the Philharmonic really dependent on me.\n\nJennie Tourel is here, and sends love and kisses. She sings the symphony like an angel. It's really heartbreaking.\n\nReiner is being very kind and helpful. He sends you his warmest greetings; and he suggests that the greatest triumph over Mr. Rodzinski would be to bring the symphony to New York with the _Boston Symphony Orchestra_!! (He specifically asked me to tell you this!)\n\nMy love to you; and keep well until I see you again.\n\nLove to Olga.\n\nLenushka\n\n169. Leonard Bernstein to Jerome Robbins\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n[early 1944]\n\nDear Jerry,\n\nII is finished, and should reach you very soon. It's much shorter than we had hoped, but it doesn't sustain. It's difficult to be bored for a long time on stage, & not bore the audience too. (This is _musical_ talk again.) At any rate, it provides all the chances you need for pantomime, gum, drinks, or peanuts. And it is constructed so that it can be easily changed around to suit your timing. When you get it, cross out the last measure (unfinished) of page 10, & proceed to page 11. You will notice that II is really part of I. I think the connection to III is a knockout. I changed my mind from dying out of II to a recurrence at the end of it of the excited opening material, giving the feeling of Let's _Do Something!!_ And at the least expected moment the girl arrives. I think you'll agree when you hear it. I don't think that the introducing of \"Much faster\" in II will hurt: when it appears in III it's all different. I'd like to cut a few measures of it, tho', in III; and have changed the tempo mark to \"Somewhat faster.\" God I wish you were here. It's so hard to write about these things. Be good, though, & happy, & like the stuff as much as I do.\n\nLove,\n\nL\n\n(There's a phrase most Aaron-like in II \u2013 I hope you don't mind. It's so pretty I can't remove it.)\n\n170. Leonard Bernstein to Jerome Robbins\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n[early 1944]\n\nDear Jerry,\n\nIt's awfully hard for us to keep up with each other. I've had a flood of special deliveries that appall me. Jerry \u2013 _everything_ has gone out to you from #1 to #4 inclusive! Why don't you have it? And I sent two copies for you, keeping three, but apparently [J. Alden] Talbot kept one en route. Henceforth I'll send three \u2013 one for Talbot.\n\nI'll do everything possible to get recordings to you pronto. It's next to impossible to find the time, but I'll do my best. I can certainly sympathize about the pianist; it's hard to play anyway, apart from the special jazzy style. Wait till he tries number I!\n\nKeep up the spirit. It's gonna be fine. And I'm in the thick of the Variations. And actually with ideas. _And_ please like #II!\n\nLove,\n\nLenny\n\n171. Leonard Bernstein to Jerome Robbins\n\nPhilharmonic-Symphony Society of New York, Steinway Building, New York, NY\n\n[28 February 1944]\n\nDear Jerry,\n\nSlow, slow, but sure. Number 5 is done and being shipped. It's true enough connective tissue, \u00e0 la movies, and ought to come off. It's more or less all development of the competition motive, with some of the music of Girl #1 to use (during the rotation) for the conspiring of the two girls.\n\nThe pauses at the beginning indicate nabbing the exiting pas-de-deux couple, and indecision about the situation. (Rotation.) Then the competition comes in earnest, developing, through a scherzo-like section to a \"dancy\" section, by which time I imagine them on the dance floor! In the last measure a snare drum rolls, fortissimo, continuing beyond the music as a lead-in to the first variation (\u00e0 la circus). I think it works.\n\nMy only worry is that it may all be too short (don't forget the repeat! It lasts about 2 minutes). If you need more build up, let me know. I'm leaving tonight (Feb. 28) for Montreal, Canada, where I conduct the Montreal Symphony. I stay until Mon. 8th. Write me at _The Hotel Windsor_ , Montreal.\n\nAt last, it begins to take shape. I can't wait to finish it. I plan to begin orchestrating in Montreal. Has it occurred to you (as it has to lots of composers, etc., for whom I've played it) that this could be a wow for two pianos alone (with maybe percussion?). It would save much time, rehearsal worries, difficulty of performance, and so on. Billy Schuman thinks it's a natural, & predicts that the orchestration will be very difficult. I only _suggest_ this \u2013 you can throw it out of the window. But let me know.\n\nI've just spoken to Hurok. The dates are settled \u2013 April 18, 22, 24. We'll talk about rehearsal schedules later. I hope it will be orchestrated in time! Pray for me.\n\nThese are hectic times. When I return from Montreal I have three concerts with the Philharmonic, including my symphony (alas, no broadcast!). Then, boom, the ballet. I'll be a wreck, but I hope a happy one.\n\nAnd you take care of yourself too. We need one healthy guy in this project.\n\nLove,\n\nLenny\n\nThere's much more, of course. To answer your letter:\n\n1) I'm making the rhythms as simple as I can. The music more or less depends on them, and they can't be scored any more simply. It has to be \u2013 and I think it's all feasible enough. (A word for _two pianos_!)\n\n2) I love your description of #1. It surely can be played a little more slowly, if you need it. I raced it in recording.\n\n3) About the two extra bars on page 6: if you need them, OK; but it makes awful music. Maybe we can straighten it out in N.Y. at the last minute.\n\n4) About the ending of #2: Throw it out. It's not necessary, and I see your point. It's very easily fixed; and I used it at the beginning of #5, where it works much better. Do you want the revised ending immediately?\n\n5) I don't see how #3's beginning can be extended. Is it _absolutely_ necessary? It gets so dull. If worse comes to worse, OK; but try to avoid it.\n\n6) In \"much faster\" (changed to \"somewhat faster\") it's perfectly OK to extend those contrapuntal bars. Just double them. And a slower pace is OK.\n\n7) On the pas-de-deux, repeat anything you want. (But use the extra bar as it was used at the beginning.)\n\n8) On Variation 2 (which turned out very pretty, and as Aaron says, \"dancy\") the feeling is mostly sweet plus cocky. Your idea, I think. And change the last chord (it's too sour). The left hand should be:\n\nThis chord represents a sort of ballet-ish _bow_ (or male equivalent of curtsey).\n\nThere remains only your variation, which I'm saving for last, the Lindy hop & fight (oy!) and the gag finale. Not too bad. I hope to have it. I hope!! I hope!!!! Again, pray, & be good in the bad West.\n\nLove again\n\nL\n\n172. Leonard Bernstein to Jerome Robbins\n\nNew York, NY\n\nSaturday [11 March 1944]\n\nDear Jerry,\n\nJust a fast one. I've been in Canada for almost two weeks conducting the Montreal Symphony Orch., and returned to find a batch of wires, letters, etc. from you which I have _yet_ to untangle. Your last is the one from British Columbia (hello, fellow-Canadian!).\n\nI must dash to conduct a concert at West Point tonight. It's a fearfully frantic year, & I'm almost crazy. But the ballet comes along fine. Have you received No. 5? I hope it's O.K. And I brought the Finale (everything up to the final gag, including a wow Boogie Dance) to the blueprint today.\n\nDidn't you receive my letter to Los Angeles? Judging from your writing, you didn't; and that was a long one, full of discussion. I hope it catches up with you. I told you then that it was OK to cut the end of #2 \u2013 you're absolutely right. I'm enclosing the new ending (beginning as if the first measure of p. 12).\n\n(I posed in that letter the problem of whether it might be better to use 2 pianos & percussion, only. It's wild, and only a suggestion.)\n\nNow \u2013 Variation 2. I swear to you, Jerry \u2013 it's not melancholy at all \u2013 not a whit! It must have been played ridiculously. It's whimsical, _very dancy_ , a little poignant in the harmony, full of a lyrical jazziness. The main thing is _sweet_ and _sympathetic_. The last chord represents a gracious ballet _bow_.\n\nThese things are really impossible to discuss like this. For God's sake, get home! I need you!\n\nI'll record the new stuff as soon as I can, but I haven't a minute now for days. Bear with me.\n\nLove,\n\nL\n\nLove to Mitropoulos, if you see him.\n\n173. Aaron Copland to Leonard Bernstein\n\n988 Memorial Drive, Cambridge, MA149\n\nTuesday [25 April 1944]\n\nDear El Bee,\n\nGot your Friday note today. Poor thing \u2013 you don't even know my address \u2013 988 Memorial Drive Cambridge. (Tel. \u2013 for you \u2013 Kirkland 3042).\n\nI'm tickled about all the excitement, but wish it didn't mean Boston gets a measly two days of you, instead of two weeks. Now that _Rodeo_ and _Fancy Free_ are hopelessly married,150 I'd better watch out that people don't say my new one151 shows Bernstein influences! It's amusing to ruminate on where it will all end \u2013 but right now it makes a question mark as big as your piano.\n\nI wrote to Phillie for the _Our Town_ parts. Hope they arrive in time!\n\nYou must have all the Cambridge news from Helen Coates, so I won't repeat. Our Sanders Theatre concert is tonight.152 Irving Fine plays the _Danz\u00f3n_ [ _cubano_ ] fine. Apparently all Cambridge Society plans to attend in force.\n\nV[ictor] wired from Miami. He should be in these parts in about five days. (Remember him?)\n\nBe a good boy \u2013 take care of yourself \u2013 and don't forget your one and only begetter.\n\nA\n\nGive my best to Jerry [Robbins]. He's a sweetheart.\n\n174. Joseph Szigeti153 to Leonard Bernstein\n\nPalos Verdes Estates, CA\n\n16 May 1944\n\nDear Bernstein (if you'll allow a grandfather to drop the \"Mister\"!)\n\nThanks for your letter to which a hurried reply as my wife's _eye_ blood-vessels have been burned by a criminally negligent doctor who gave her infra-red and other \"rayons\" for her _hand_ arthritis! He didn't shield her eyes and she is now suffering untold torture (she was taken to LA to a specialist & will be away for at least 48 hrs).\n\nAs I played Mozart A major, Tartini, Prokof[iev] & Chausson in Chicago in _March_ my program had to avoid these works, naturally! Your programs will have Beeth[oven] the first night154 and Mozart D major No. IV with Bart\u00f3k's Rhapsody No. I (dedicated to me) (9\u00bd min.) the second night,155 with which choice I feel sure you'd be \"d'accord\". The Bart\u00f3k record is no longer on sale but some of our mutual friends are sure to have it (John Hammond's wife? Goddard Lieberson? perhaps Serly? or the boys at Record Collectors Exchange?)\n\nIf we can't get a _cimbalom_ player who can follow your beat (gypsies often cannot!) we'll have to give the cimbalom part to a piano (with paper fixing between strings). Kuyper or the personnel manager should enquire at Blue Danube Restaurant in Chicago (the pianist there is Harmati) about cimbalom.\n\nLooking forward to our working together.\n\nVery cordially yrs,\n\nJoseph Szigeti\n\n175. Aaron Copland to Leonard Bernstein\n\nDepartment of Music, Harvard University, Cambridge, MA\n\n20 May 1944\n\nDear Lensky,\n\nI think I solved the _Our Town_ problem. It repeats the waltz as you like, and removes entirely the \"ties that bind\" section. Some fancy juggling of tonalities on the way, of course. Heinsheimer says he will copy a set of parts \"special\" if you intend to do it three or four times. OK? And how soon would the part have to be ready for the Stadium, if that's to be the first one?\n\nI'll never forget that face of yours in Reuben's.156 I've never seen you look so sad. Why you wouldn't even look me in the eye! What on earth were you thinking at that moment??? And the contrast with everyone else seeming so pleased and happy. And me worrying about my lost organ (the voice, I mean). You seemed in a mood to make the most out of any disapproving remark I let drop in your direction. Aren't you idiotical \u2013 you know very well I have you hopelessly under my skin. But I'll always watch you like a hawk \u2013 that goes without saying.\n\nHow does the dinner seem in retrospect \u2013 I mean aside from your personal feelings.\n\nYou see you owe me a letter.\n\nAlles \u2013 but _alles_\n\nA\n\n176. Harold Newman to Leonard Bernstein\n\nHargail Records, 299 Madison Avenue, New York, NY\n\n1 June 1944\n\nDear Lenny,\n\nHere's the long promised royalty statement.157 Am I supposed to charge for any of the sets sent out for you? I don't know. [annotation by Helen Coates: (No \u2013 all gratis)]\n\nSorry, AmMus refused permission re the K[oussevitzky] Concerto158 \u2013 it is reserved for Koussevitzky.\n\nGlad Lukas is appearing with you at the Stadium.159\n\nGood luck,\n\nHarold\n\n177. Herschel, Janice, and Lois Levit160 to Leonard Bernstein\n\nCamden ( _sob_ ), NJ ( _sob_ )\n\n3 July 1944\n\nDear Lennie of the Lenap\u00e9s,161\n\nMy tongue hung low and drippy over your program for this week at Ravinia Pk. Oi!!! We noted, with a certain degree of pride, that information as seen in the NY _Times_. How marvelous that your symphony can be heard again \u2013 but where do I come in? I have yet to hear it, really. If it is at all possible, please send us a copy of any decent recording you've made of it. Do you think you'll ever do it in Philly? Conduct your symphony, I mean. At any rate, you'd better send the recording or I'll get Lois' boy friend after you. _Then_ what will you do?\n\nMitropoulos conducted the first two weeks at The Dell \u2013 we heard [Arthur] Rubinstein do the Brahms B\u266d \u2013 as an encore he played the Rhapsodie in B minor, some encore! He & Mitropoulos were pretty terrific. Dimitri made a big hit with his playing & conducting of the Prokofieff 3rd. Tonight we hear [Nathan] Milstein do a couple of jobs: the Mendelssohn & the Lalo.\n\nWill you be back with the Philharmonic in the fall or are you going to freelance? It certainly would be great if you can get to the Academy to do your symphony.\n\nBest wishes,\n\nHerschel & Janice & Lois\n\nP.S. Dear Lennie, Incidentally, for the summer we are at the address on the back envelope. The house is strictly from modern, as a matter of fact Herschel calls it the \"Museum of Modern Art\" for short.162 If you get the chance come visit us \u2013 there's a Steinway and you can have your own room & bath. Let us know when. J\n\n178. Leonard Bernstein to Serge Koussevitzky\n\n33 West 67th Street, New York, NY\n\n26 July 1944\n\nDear Serge Alexandrovich,\n\nI am sending you, with my love and deepest congratulations, a few notes on your birthday, which form a small sketch for the piece I hope soon to have for you. Life is so complicated and busy not that I cannot set any really definite date when I expect the composition to be finished, but I am trying to make it as quickly as possible without sacrificing any quality: \u2013 I want this to be as fine as I can make it, since it is for you, who represent quality itself to me.\n\nPlease accept this little sketch now, and let us hope it grows into a composition worthy of your greatness.163\n\nShirley joins me in wishing you long life and happiness.\n\nDevotedly,\n\nLeonard\n\n179. Leonard Bernstein to Serge Koussevitzky\n\n2074 Watsonia Terrace, Hollywood, CA\n\n[August 1944]\n\nDear Serge Alexandrovich,\n\nHere in Hollywood one sometimes loses perspective on symphonic music; the commercial aspects are so important to everyone here \u2013 and the quality is reserved for the mediocre. So it is a great pleasure to contemplate my program with the Boston Symphony \u2013 it is a thought like a spring in the desert.\n\nIf the entire first half is to consist of the Brahms [First] Piano Concerto, I thought that the second half might offer _Verkl\u00e4rte Nacht_ of Schoenberg (this is Schoenberg's anniversary year), followed by a suite from my ballet _Fancy Free_. This seemed well rounded to me, and my ballet suite (about 15 minutes) might make a good closing piece.164\n\nIf you feel that _Fancy Free_ should come after intermission, then the program could close with a more usual fin-de-concert piece like the _Firebird_.\n\nDon't you think there should be a short overture, or the equivalent \u2013 perhaps Mozart \u2013 before the Brahms? Or perhaps the radio requirements prevent this?\n\nWhat I should really like is to have ready the composition I am planning for you \u2013 but I guess I just can't be finished in time. Did you receive the little birthday sketch I sent you?165\n\nI hope these program ideas are satisfactory. I would be so grateful for any advice you would give me on this concert. I'm very excited about it, and I want it to be good.\n\nI think of you every day, and send you my love \u2013\n\nLenushka\n\n180. Sid Ramin to Leonard Bernstein\n\nHeadquarters, 84th Inf. Div., Special Service Office,166 A.P.O. 84, c\/o Postmaster, New York\n\n10 September 1944\n\nDear Len,\n\nJust a note to let you know that I'm leaving for overseas shortly.\n\nIn the past, I've been able to tell you where I was and a bit about what I was doing, but the following address makes that a little difficult; however, I can tell you that I'm able to get to New York very frequently \u2013 so there's your clue.\n\nI was home on furlough and heard some of your recordings \u2013 the side dedicated to Shirley brought back some wonderful memories and the William Schuman bit delighted me.167 By the way, I saw a copy of _I Hate Music_ (Five Pieces for Children) displayed very prominently in a large music store in St. Louis.\n\nSince mail is being censored, I find it hard to write to the folks at home \u2013 they're thirsting for news and I'd like to tell them what's happening but I can't.\n\nMy pleasures are very simple now and eating at a nice restaurant or club and seeing a musical is just about the ultimate in enjoyment for me at this stage in the game. Yesterday I saw _One Touch of Venus_ for the second time, ate at the Kungsholm and had a couple of drinks at a good bar \u2013 it all made for a pleasant day. The dance routines (Agnes de Mille) in _Venus_ impressed me tremendously \u2013 but, then again, maybe four years in the army have warped my sense of what's good and what isn't. I expect to be in New York again several times and would like to see you for a minute if you're available and have a minute.\n\nMail is important in the life of a soldier whether he be overseas or in this country. Overseas, though, the importance of mail cannot be overemphasized and getting a letter from you will mean very much to me.\n\nEach day must be exciting and busy for you and I often think of the contrast between an average day now as compared to the days when we could ramble through the park in Roxbury and while away the hours.\n\nHow are your folks? Do you ever run into Harold Shapero?\n\nBest always,\n\nSid\n\n181. Leonard Bernstein to Aaron Copland\n\n33 West 67th Street, New York, NY\n\n[received 28 September 1944]\n\nDearest, wonderful A,\n\nI'm a dawg, a dawg, a dawg not to have done this before. You can always check on me because your last letter was a birthday letter one full month ago. But I've reread it so many times, because it brings back something that you think everything is O.K. without, and then suddenly you find it isn't at all, and somehow something's got to be done about the Aaron Copland side of one's life,168 which is always turning out to be a major side. If that sentence makes sense, especially in Tepoztlan, you're a genius. Just a new way, of course, this being the 57936th letter I've written you, of saying that I adore you.\n\nFor the rest, I'm back in town, with problems, being busy, all of which is boring old fluff for you by now. One realistic chimera is that I've got to move by Saturday (Three Days) and I have no apartment. I have thought of temporizing in your loft, for maybe a week or so, if Helen (Coates) still can't find me a place;169 but I hesitate to ask, and maybe I will find one. I spend the weekends in Detroit now (horreur) conducting broadcasts of the Detroit Symphony (three). Next week is in Boston with the Ballet, so I'm really running out on Helen and leaving the whole problem to her. I'm looking at a really wonderful place on East 57th (Kostelanetz' building), but it's a cooperative and you have to buy it for $5000 and then pay $175 a month upkeep, and then the Board of Directors has to vote on me, and decide whether I look rich enough and dependable enough and presentable enough to be accepted. They've looked into forty references besides my bank balance, and I resent it and stand for it only because it's such a dream of an apartment and I can play music there and I want to live in it.\n\nThe show [ _On the Town_ ] is a wild monster now which doesn't let me sleep or eat or anything; in fact the world seems to be composed of the show the show the show, and little else, except a _Verkl\u00e4rte Nacht_ or a Schumann symphony here and there. Maybe it will be a great hit, and maybe it will lay the great _egg_ of all time. It's an enormous gamble.\n\nYou're sweet to dedicate the _Our Town_ piece to me; I could do with a dedication or two in these grim loveless days.\n\nGod how I wish I were with you. How long do you plan to stay? Forever? Indefinitely? It can't be. At least come back for the opening of the show, which will be, with the grace of God, and if we get George Abbott to direct which looks likely, around Christmas time. And please write a lot, and find me someone nice to bring home as a present, and write a great piece, and give my love to Victor, and I had lunch with P[aul] Bowles and [Yvonne de la] Casafuerte today and got drunk. Bless you, I love you.\n\nL\n\n182. Aaron Copland to Leonard Bernstein\n\nTepoztlan, Morelos, Mexico\n\n9 October 1944\n\nDear Dawg,\n\nYou're in the delightful position of getting a letter from me without having to answer it, because I'm coming home before you can safely get one here. I have just two more weeks in Tepoztlan (till the 23rd), then to Mexico City for two days, and then I fly to Washington (the 25th). Martha Graham is supposedly doing a ballet of mine that week-end.170 No one has seen any public announcement of the event \u2013 nor have I \u2013 but [Harold] Spivacke keeps writing it will take place, so I go on faith. I ought to be back in my hole on 63rd St by Oct. 30th or Nov. 1st. Then we can turn on the \"AC side of your life\", whatever that is.\n\nIn retrospect the summer seems frightfully short. I never got settled until August, and didn't get a piano until the middle of the month. Also, I've been alone a lot of the time. V[ictor] has been away a good deal, doing odd jobs or seeing odd females up in Mexico DF. I find I don't particularly thrive on solitude. So, altho the setting was perfect, inspiration has been spurty. I guess I'd better come home.\n\nSometimes Mexico seems too much to bear. In Sept. the rain got out of hand, and it poured like hell for three weeks. Then we had a small cyclone that sounded very big in our valley. Then suddenly the water works of the town, being all above ground, goes kaput. Then comes the news of a minor rebellion two miles away and the government sends troops. Then the cook announces the horrifying news that eggs have just gone up to 4 cents apiece. Then suddenly the sun comes out, there's water in the excusado, the troops go away, the eggs go down to 3\u00bd cents and Mexico seems like heaven on earth. It's a peculiar country.\n\nI'm tickled pink that the show isn't going to come off until Christmas.171 That means I can hang around the theatre during rehearsal period, than which I adore nothing better \u2013 as you know. I hope you found a perfect place to live in, but _not_ in the Far East 50s. (Too far away.)\n\nTill velly soon.\n\nLove,\n\nA\n\n183. Leonard Bernstein to Philip Marcuse172\n\n40 West 55th Street, New York, NY\n\n25 October 1944\n\nDear Phil,\n\nThanks very much for your recent letter, and for the Shostakovich173 records. The latter arrived yesterday, but I have not yet had the chance to hear them yet. I'm curious to hear how it all sounds, even to the thumping of my irrepressible foot. Who knows, hearing this thumping on the record may cure me of that bad habit. Anyway, it will be good for me to hear just how it sounds to the radio audience. [...]\n\nThe show174 goes into rehearsal in early November, and I'm working night and day to have the score ready by then.\n\nPlease give my thanks to the person who made the recording, and, again, my thanks to you for sending them to me.\n\nSincerely yours,\n\nLenny Bernstein\n\n184. Philip Marcuse to Leonard Bernstein\n\nStockwell & Marcuse Advertising, 2026 National Bank Building, Detroit, MI\n\n26 December 1944\n\nDear Leonard,\n\nHere's hoping that _On the Town_ will be a tremendous success. I read _Variety_ 's preliminary review of it, and I gather that it was damned good even though the reviewer didn't know how to say it.\n\nAlthough we haven't communicated for a couple of months, I have been following your progress and enjoying your radio appearances. We saw _Fancy Free_ two weeks ago and found it truly terrific. So did everyone else.\n\nWe look forward to seeing you in New York three weeks hence, and once again, our best wishes for your newest musical triumph.\n\nSincerely,\n\nPhil Marcuse\n\n185. David Oppenheim to Leonard Bernstein\n\nGermany\n\n27 December 1944\n\nAre You Happy???\n\nDo You Stay Awake Nights??\n\nWould your autobiography be entitled: \"My Life as a Mole\" or \u2013 \"Digging my way thru Europe\"??\n\nDo you fall flat on your face when someone whistles [...] Have a _crush_ on screaming music? Burpgunitis etc. etc.??\n\nTry Bernstein's Little Battle Pills \u2013 Delicious to chew slowly.\n\nNibble your neurosis away!!!175\n\nN.B. For adventurers only \u2013 my ass!!\n\nI've been watching _N_ [ _ew_ ] _Y_ [ _ork_ ] _er and_ no _On The Town_ yet. [...] Do I get score & book over here? And are you gonna write?\n\nA letter I wrote to you to California came back \u2013 you had moved and not left a forwarding address.\n\nNow \u2013 what are you doing \u2013 how is Marketa [Morris]? Adolph [Green], Judy [Holliday], Betty [Comden] \u2013 I'd sure like to have Judy here to line my foxhole.176 She is 20th Century Foxing?? I saw a bad picture of her in a movie magazine some time ago.\n\nWill you conduct the N.Y. orch this year? Or is Art177 still mad?\n\nLove,\n\nDave\n\n1 Robert Gundersen (1895\u20131941).\n\n2 Gaston Dufresne taught solf\u00e8ge and played double bass in the Boston Symphony Orchestra. Presumably Bernstein was hoping to find work for Ren\u00e9e Longy Miquelle at Tanglewood.\n\n3 Mary Louise Curtis Bok, founder of the Curtis Institute.\n\n4 Miquelle's son.\n\n5 Probably Alvin Ross, an artist friend.\n\n6 The composer himself was among those \"knocked for a bingo.\" Schuman wrote in his diary: \"A most remarkable performance \u2013 Bernstein should develop into the first sensational American conductor. He has everything. Koussevitzky so excited by Bernstein performance \u2013 he walked up to the stage & kissed us both in public!\" (Swayne 2011, p. 120).\n\n7 Bernstein conducted _The Rio Grande_ at Tanglewood on 15 August 1941.\n\n8 The soloist in the Brahms Piano Concerto No. 2 was Carlos Moseley, later the Managing Director of the New York Philharmonic.\n\n9 Probably the family of Ben Quashen who studied with Hindemith at Tanglewood.\n\n10 Samuel Barber (1910\u201381), American composer. Bernstein was never particularly enthusiastic about Barber's music, nor did they get on in later life. Bernstein recorded the Violin Concerto (with Isaac Stern) and the Adagio for Strings. A performance of the _Second Essay_ from 1959 was published on CD in _Bernstein Live_ (New York Philharmonic NYP2003).\n\n11 The United Service Organization, which provided (and still provides) entertainment for United States troops.\n\n12 The National Youth Administration (NYA) was originally set up under the New Deal as part of the WPA. It organized educational and cultural opportunities for Americans between the ages of 16 and 25, including the establishment of orchestras for young musicians.\n\n13 Judith Tuvim was the real name of Judy Holliday.\n\n14 The National Youth Administration. See note to Letter 95.\n\n15 Horatio Alger (1832\u201399) was the author of numerous rags-to-riches tales for children. He travelled to California to gather material for his stories.\n\n16 Presumably the music for songs by The Revuers referred to in Letter 96.\n\n17 In some of Bernstein's letters to Copland, names were blanked out at a later date.\n\n18 Mitropoulos conducted the world premiere of Copland's _Statements_ with the New York Philharmonic on 7 January 1942.\n\n19 Presumably the fortified Bel\u00e9m Tower.\n\n20 Bernstein opened a studio for teaching the piano and musical analysis at 295 Huntington Avenue, Boston, in December 1941.\n\n21 Judy Holliday (1921\u201365), American actress. Born Judy Tuvim, she was a member of The Revuers. She later married David Oppenheim. One of her greatest Broadway successes was her Tony-winning performance as Ella in _Bells are Ringing_ , written by Comden and Green with music by Jule Styne and choreography by Jerome Robbins.\n\n22 Lizzie (Elizabeth) Reitell (1921\u20132001) was Adolph Green's first wife. She later had an affair with Dylan Thomas in the last months of his life.\n\n23 They married in 1941, but as Bernstein gloomily predicted, it didn't last.\n\n24 Probably a reference to Artur Rodzinski, who was Music Director of the Cleveland Orchestra from 1933 to 1943, when he moved to the New York Philharmonic.\n\n25 A reference to _Conch Town_ , which Bernstein almost completed in a version for two pianos and percussion, but never finished. He later harvested it for several works including _Fancy Free_ (the \"Danz\u00f3n\") and _West Side Story_ (\"America\").\n\n26 The first performance of Bernstein's Clarinet Sonata was given at the Institute of Modern Art in Boston on 21 April 1942 by David Glazer, with Bernstein at the piano, in a concert promoted by Young American Musicians. A lukewarm review by Winthrop P. Tryon in the next day's _Christian Science Monitor_ described it as \"well-constructed in general, and in particular, too, as far as the mere music went. Whether it was inevitably an ensemble composition for wind and keyboard instruments might be questioned. Both the clarinet and the piano have plenty of business allotted to them, and both chime together all right, if for its own sake chiming counts greatly. But the problem of treatment of two qualities of sound seemed more or less ignored. The music turned out to be abstract in a way not usually intended in studies of the sort. But, for all anybody knows, that may be one of the very purposes of Young American Musicians, to get things free of the old tracks.\"\n\n27 Probably Jean Middleton, who was a pianist and a composition pupil of Arthur Berger's, and one of Bernstein's longer romantic relationships during the 1940s.\n\n28 On 24 May 1942, _The New York Times_ published a letter signed by 11 composers (Aaron Copland, Arthur Berger, Edward T. Cone, Henry Cowell, David Diamond, Anis Fuleihan, Alexei Haieff, Frederick Jacobi, J. B. Middleton, Harold Morris, and William Schuman) who had been performed in a concert of works chosen by members of the Music Critics Circle. The letter was a furious protest against Olin Downes' scathing review of this event. A short extract gives a flavor of the composers' mood: \"Mr. Downes does not hesitate to lambaste savagely and hold up to ridicule ten of the twelve works carefully selected in the first place by their original performers, and chosen in the second place for rehearing by his own fellow-critics. This leaves Mr. Downes in a position of lonely grandeur from which he can survey the stupidity of every one concerned but himself. Perhaps we American composers are as puerile-minded as Mr. Downes gloatingly proclaims, but we are not so half-witted as to be led into a discussion of the merits of our compositions with the pontifically minded Mr. Downes. Others may do so if they care to. We shall continue to write our compositions, they will continue to be played, and Mr. Downes will, no doubt, continue to survey the field from his isolated post.\"\n\n29 After conducting a successful performance of _The Second Hurricane_ under the auspices of the Institute of Modern Art in Boston on 21 May 1942 (attended by Copland), Bernstein conducted a second performance on 5 June at the Sanders Theatre in Cambridge, MA.\n\n30 Downes' response to the protest letter by Copland and others appeared in _The New York Times_ on 7 June 1942, under the title \"Critic's Duty. Further Examination of Reaction to Circle Concerts of American Works.\" Downes wrote a lengthy and exhaustively argued defense of his position.\n\n31 Fritz Reiner.\n\n32 A Yiddish word (sometimes spelt \"tsuris\") for trouble or difficulty.\n\n33 Marion Bauer was a leading figure in the League of Composers.\n\n34 Albert Sirmay. See note to Letter 395.\n\n35 \"The Frau\" was Bernstein's nickname for his psychoanalyst, Marketa Morris.\n\n36 Victor Kraft, \"the most important romantic relationship of [Copland's] life\" (Pollack 1999, p. 239).\n\n37 Walter Hendl (1917\u20132007), American conductor who studied at the Curtis Institute with Fritz Reiner, and at Tanglewood with Koussevitzky (when he met Bernstein). He was Music Director of the Dallas Symphony Orchestra 1949\u201358 and in 1958 became Associate Conductor to Fritz Reiner at the Chicago Symphony. From 1964 to 1972 he was Director of the Eastman School of Music.\n\n38 David Oppenheim (1922\u20132007), American clarinetist, record producer, television producer, and academic administrator. Oppenheim and Bernstein were extremely close friends in the 1940s, and remained on very affectionate terms for the rest of Bernstein's life. They first met at Tanglewood in 1942. Oppenheim appears as the dedicatee on the 1943 publication of Bernstein's Clarinet Sonata and he made the first recording of the work with Bernstein in 1943 for Hargail Records. In 1948 he married Judy Holliday, who had worked with Bernstein as a fellow member of The Revuers (Oppenheim and Holliday divorced in 1957). He was hired by Goddard Lieberson to work for Columbia Masterworks and served as director of the label from 1950 to 1959, producing recordings by artists such as Bruno Walter and George Szell, before moving to television, first at PBS and then at CBS as a producer of arts documentaries, several of which involved Bernstein, the most famous of which was probably _Inside Pop: The Rock Revolution_ in which Bernstein discussed the Beatles, the Monkees, Bob Dylan, and Janis Ian whose \"Society's Child\" became a hit after being featured in the broadcast (see Letters 527 and 533). In 1969, Oppenheim began the last and most successful part of his career when he became Dean of the School of Arts at New York University. According to his obituary in _The New York Times_ (3 December 2007), he \"transformed NYU's arts programs into a major institution\" and among his enduring achievements was to secure a huge donation from the Tisch brothers that enabled NYU to build its Tisch School of Arts. He retired from the University in 1991.\n\n39 With this pithy, three-word postcard, written in pencil, Bernstein re-established contact with David Oppenheim, a few weeks after meeting him for the first time at Tanglewood.\n\n40 From the mention of the Broadway revival of _Porgy and Bess_ (13 September\u20132 October) it is possible to date this letter. Bernstein's reference to Diamond's \"terrific talent\" was probably the result of seeing some orchestral works, including the First Symphony that Diamond recalled Bernstein playing from the full score; it had been given its premiere by Mitropoulos and the New York Philharmonic on 21 December 1941.\n\n41 Paul du Pont was the costume designer for the 1942 revival of _Porgy and Bess_.\n\n42 According to John Dunning (Dunning 1998, pp. 430\u20131), _The Man Behind the Gun_ was a war drama series broadcast from 7 October 1942 until 4 March 1944. The show's regular music staff included Bernard Herrmann and Nathan Van Cleave, and the writers included the young Arthur Laurents. David Diamond's incidental music for the program was apparently first used on 14 October 1942 (the show's second episode) conducted by Herrmann (see Kimberling 1987, p. 127).\n\n43 The Riobamba Club, at 151 East 57th Street, opened on 10 December 1942, a glittering social event reported at length in the _New York Evening Post_ the following day. This makes no mention of Bernstein's contribution, but the singer Jane Froman topped the bill and may well have introduced Bernstein's \"Riobamba\" on this occasion.\n\n44 The most lasting success this tune had was as the \"Danz\u00f3n\" in _Fancy Free_.\n\n45 The reason for the \"gift\" was that Miquelle had lent Bernstein the Steinway on which he composed _Jeremiah_. A month later \u2013 as a sign of how grateful he was to her, and of how much he valued their friendship \u2013 Bernstein gave Miquelle the complete autograph short score of _Jeremiah_ (inscribed on the first page \"To Ren\u00e9e with love & gratitude, Lenny, Jan. 5 1943\"). This was returned to Bernstein after Miquelle's death in 1979. In a note to Helen Coates concerning the manuscript, Bernstein wrote: \"HC \u2013 This is very precious \u2013 the entire symphony in _final_ sketch! I had no idea where it was all these years, but of course I had given it to Ren\u00e9e in thanks for her having loaned me her Steinway on which the symphony was composed (52nd St.). I am so happy to have it back.\"\n\n46 _Danz\u00f3n cubano_.\n\n47 The fire at Boston's Cocoanut Grove nightclub on 28 November 1942, which killed almost 500 people.\n\n48 In other words, the date for Oppenheim to join the army.\n\n49 \"Jack\" is mentioned in several of the Bernstein\u2013Oppenheim letters. He is probably the composer Jacob (Jack) Avshalomoff (1919\u20132013), who studied with Bernard Rogers at the Eastman School and was a classmate of Oppenheim's.\n\n50 The \"Calamus poems\" are a group of poems in Walt Whitman's _Leaves of Grass_ that celebrate \"the manly love of comrades.\" They are Whitman's clearest published declaration of his ideas about homosexual love.\n\n51 Related, too, to Arnold Park, Rochester, where Oppenheim lived until December 1942.\n\n52 Almost certainly a reference to Oppenheim's help with copying sections of Bernstein's _Jeremiah_ Symphony, and specifically the end of the Scherzo (\"Profanation\").\n\n53 Karen Horney (1885\u20131952), German psychoanalyst who settled in the United States. Her book _The Neurotic Personality of Our Time_ (1937) was a bestseller.\n\n54 Praying (from Yiddish).\n\n55 A reference to Howard Hanson, Director of the Eastman School.\n\n56 Allen McHose (1902\u201386) was chair of the Theory Department at the Eastman School.\n\n57 Edys Merrill, with whom Bernstein shared an apartment. She was the dedicatee of _I Hate Music! 5 Kid Songs_ , and she had inspired the title: when Bernstein's playing became too much for her, she would go round the apartment singing \"I hate music! But I like to sing \u2013 La dee da da dee.\"\n\n58 Though Oppenheim wrote this theme out in 6\/8 and C major, it is in 3\/4 and A major, and comes from the first movement of Mendelssohn's D minor Piano Trio.\n\n59 The Office of Price Administration, a regulatory authority that could control prices and ration scarce supplies. The OPA banned \"non-essential driving\" in 1943 to save gasoline.\n\n60 WXQR is a classical radio station in New York City, on air since 1939.\n\n61 The basement caf\u00e9 of the 44th Street Theatre, demolished in 1945 (the last show to run there was _On the Town_ ).\n\n62 \"Lenny-Penny\" or \"Lennypenny\", a nickname Copland occasionally used in letters to Bernstein (see, for example, Letter 75).\n\n63 Copland was in Hollywood to write the score for _The North Star_. The cast included Anne Baxter, Dana Andrews, Walter Huston, Erich von Stroheim, Farley Granger, and Walter Brennan, and it was directed by William Wyler. Copland's extensive score includes songs with lyrics by Ira Gershwin.\n\n64 The Town Hall Music Forum in New York devoted to Copland took place on 17 February 1943. Bernstein played Copland's Piano Sonata, and Daniel Saidenberg conducted _Music for the Theatre_ and the first performance of _Music for the Movies_.\n\n65 WNYC is a public radio station in New York City, on air since 1924.\n\n66 Bernstein originally conceived a group of six pieces, but by the time of publication by Witmark in 1944, he had added \"Dedication to Aaron Copland\" to make _Seven Anniversaries_.\n\n67 _The North Star_ was based on a story by Lillian Hellman, with a screenplay by her. It was made by Samuel Goldwyn at the request of President Franklin D. Roosevelt to help boost support for America's alliance with the Soviet Union against Germany. The \"North Star\" of the title is a farming collective in Ukraine, a community whose life is shattered by a brutal Nazi occupation.\n\n68 _P.M._ was a short-lived left-leaning newspaper published in New York between 1940 and 1948.\n\n69 The NBC Symphony Orchestra conducted by Leopold Stokowski gave a broadcast performance of Stravinsky's _Symphony in C_ on 21 February 1943. Bernstein's reply ended with a plea for Copland to put in a word for him, and his reaction to Stravinsky's symphony: \"Can anything be done about me? Do they need Sonata-players in Hollywood? I heard Igor's symphony too. What a fine first mov't! A little long, but so good. Main criticism: sounds too much like Harold Shapero. I just live for the moment when you pin that medal on me. I love you. L.\"\n\n70 The card depicts three \"Skyscrapers of New York City\" (the Empire State Building, the Rockefeller Center \u2013 where the Advanced Music Corporation had its address \u2013 and the Chrysler Building).\n\n71 A recording of the radio broadcast given by Oppenheim and Bernstein on 21 February.\n\n72 Bernard Rogers (1893\u20131968) was a composition pupil of Nadia Boulanger and Ernest Bloch. He taught at the Eastman School during Oppenheim's time as a student there.\n\n73 Artur Rodzinski (1892\u20131958), Polish conductor who was appointed Music Director of the New York Philharmonic, 1943\u20137. Bernstein became the orchestra's Assistant Conductor in September 1943 (see Letter 152).\n\n74 _Shadow of a Doubt_ was directed by Alfred Hitchcock with a screenplay co-written by Thornton Wilder. It was released in January 1943.\n\n75 _Serenade_ by James M. Cain. A few years later, Bernstein contemplated a musical setting of this novel. See Letters 262\u2013265.\n\n76 Collectively these pieces \u2013 the _Six Anniversaries_ and \"Dedication to Aaron Copland\" \u2013 became the _Seven Anniversaries_ , published in 1944.\n\n77 This short piece was subsequently orchestrated as \"Variation 2 (Waltz)\" in _Fancy Free_ (1944); the ballet is dedicated to Adolph Green. I am grateful to Sophie Redfern for helping to clarify this, and for showing me the relevant pages in the sketches for _Fancy Free_. These include sketch pages for clarinet and piano headed \"Extension by Leonard Bernstein\" (in Green's hand) \"of a theme by Adolph Green\" (in Bernstein's hand).\n\n78 The cellist Jesse Ehrlich was one of Bernstein's friends from Harvard (he played the cello in the orchestra for _The Birds_ ), and he was a roommate at Tanglewood in 1940.\n\n79 Copland wrote about Harris in _Our New Music_ (1941).\n\n80 According to Paul Bowles' review published on 15 March 1943 in the _New York Herald Tribune_ , the League of Composers Concert at New York Public Library on 14 March included the _Pastoral_ for viola and piano by Elliott Carter, a String Quartet by Vincent Persichetti, songs by Beatrice Laufer, Lukas Foss' Duo for cello and piano, a group of songs by Van Vactor, Wilde, Bacon, Bricker, and John Cage, and Bernstein's Clarinet Sonata.\n\n81 Paul Hindemith (1895\u20131963), German composer and violist. He emigrated to the United Status in 1940, returning to Europe in 1953.\n\n82 _Lady in the Dark_ ran on Broadway in 1941\u20132, and returned there in February 1943 with Gertrude Lawrence reprising her starring role as Liza Elliott.\n\n83 Copland's _Lincoln Portrait_.\n\n84 _I Hate Music! A Cycle of 5 Kid Songs_.\n\n85 Probably a reference to Caspar Hauser (1812\u201333), the mysterious German youth of reputedly noble origin who inspired a poem by Verlaine and is mentioned by Herman Melville in _Billy Budd_ and by Hans Christian Andersen in _Beauty of Form and Beauty of Mind_ , as well as being the subject of Jakob Wassermann's 1908 novel _Caspar Hauser oder Die Tr\u00e4gheit des Herzens_.\n\n86 Paul Bowles wrote in his review that the Clarinet Sonata \"had something which is at a premium in contemporary music: meaty, logical harmony. It was also alive, tough, and integrated. The idiom was a happy combination of elements from both east and west of the Rhine, but only indirectly from that far away. There were stronger hints of what goes on north and south of the Rio Grande, these perhaps more directly via Copland. Through most of this (the andante seemed less real) ran a quite personal element: a tender, sharp, singing quality which would appear to be Mr. Bernstein's most effective means of making himself articulate. The work was expertly performed by David Oppenheim, with the composer at the piano.\"\n\n87 Andr\u00e9 Kostelanetz (1901\u201380), the Russian-born American conductor who had commissioned Copland's _Lincoln Portrait_ and had exclusive performance rights at the time.\n\n88 George Antheil (1900\u201359), American composer whose career began in Europe as an experimental composer of works inspired by technology ( _Airplane Sonata_ , _Ballet m\u00e9chanique_ ). In Paris he met the likes of Erik Satie, Ezra Pound, James Joyce, Virgil Thomson, and Ernest Hemingway. He went to Hollywood in 1936 and subsequently worked regularly as a film composer while continuing to write concert works.\n\n89 Bernstein's New York conducting debut took place the day after he wrote this letter. On 30 March 1943 he conducted the premiere of Paul Bowles' one-act zarzuela _The Wind Remains_ (after Lorca) at the Museum of Modern Art. The choreography was by Merce Cunningham, and the sets were designed by Oliver Smith. At the same event he conducted _Homenaje a Federico Garc\u00eda Lorca_ (1936) by Silvestre Revueltas.\n\n90 Frederick Fennell (1914\u20132004), American conductor who studied with Koussevitzky at Tanglewood in 1942 when he was a classmate of Bernstein, Lukas Foss, and Walter Hendl. Fennell later made numerous recordings for Mercury with the Eastman Wind Ensemble.\n\n91 The \"Third Serenade\" was presented by the Museum of Modern Art; it was one of five \"Serenades of rare music ancient and modern on alternate Tuesday evenings beginning March 2, 1943,\" so-called because they were modeled on the concerts given by _La S\u00e9r\u00e9nade_ in Paris before the outbreak of the Second World War, which had been organized by the Marquise Yvonne de Casa Fuerte, co-organizer of the Museum of Modern Art \"Serenades\" with Virgil Thomson.\n\n92 Herbert Stothart (1885\u20131949), American composer and arranger who spent the last 20 years of his life working for MGM. His credits included _A Night at the Opera_ for the Marx Brothers, _The Wizard of Oz_ for which his background score won an Oscar, and _Mrs. Miniver_ , which Bowles himself described in an article for _Modern Music_ (November\u2013December 1942) as \"the regular, overstuffed, plush tonality of Hollywood.\"\n\n93 This is less surprising given that Virgil Thomson was one of the organizers of the concert. But he did single out Bernstein for praise, describing his conducting as \"superb and musicianly.\"\n\n94 Constance Askew (1895\u20131984) was a generous patron of artists, writers, and musicians, including Virgil Thomson. She was married to the art dealer Kirk Askew. John Houseman described her as \"a New England woman of means, of broad cultural experience and striking beauty.\" The arresting portrait of her by Pavel Tchelitchew (1938) now hangs in the Wadsworth Athenaeum in Hartford, CT.\n\n95 George Chavchavadze (1904\u201362), Russian pianist.\n\n96 Rhoda Saletan.\n\n97 The songs were published by Witmark.\n\n98 This is the address on the headed paper, but by the time he wrote this letter Bernstein was living at 15 West 52nd Street.\n\n99 Will Geer (1902\u201378), American actor and activist. Following his university studies in Botany, Geer began his acting career in the late 1920s. In 1934, he joined the Communist Party and toured government work camps with Woody Guthrie and Burl Ives as well as working as a classical actor. He appeared regularly as a member of The Group Theatre at its summer home in Pine Brook Country Club, Nichols, CT. Blacklisted in the 1950s for his refusal to testify before the House Un-American Activities Committee, he renewed his interest in botany, setting up the Theatrum Botanicum, an outdoor theater for blacklisted actors with a garden in which every plant mentioned by Shakespeare was grown. Geer later achieved fame as Grandpa Zebulon Walton in _The Waltons_. At his deathbed, Geer's family sang Woody Guthrie's _This Land is Your Land_ and recited poetry by Robert Frost. His ashes were buried in his own Shakespeare Garden.\n\n100 Koussevitzky performed Copland's _Lincoln Portrait_ several times in March and April 1943 in Boston and New York. At Carnegie Hall on 3 April it appeared on the same program as Schuman's _A Free Song_ and Barber's _Essay for Orchestra No. 1_ (\"Billy & Barber\"), with Beethoven's Fifth Symphony in the second half.\n\n101 Edys Merrill.\n\n102 _The Wind Remains_.\n\n103 The pianist Jes\u00fas Mar\u00eda Sanrom\u00e1 (1902\u201384).\n\n104 See note 87 to Letter 134.\n\n105 Probably the two-volume edition of Havelock Ellis' _Studies in the Psychology of Sex_ published by Random House in 1940.\n\n106 Bernstein often spelt it \"Rohrschach\"; Oppenheim's \"Rorschach\" is correct.\n\n107 Name blacked out.\n\n108 On 18 May 1943, Bernstein took part in an evening presented by The Little Red Schoolhouse (a fund-raiser to buy scientific equipment for the progressive school in Greenwich Village) at Town Hall, with Virgil Thomson as master of ceremonies and Bernstein as commentator and pianist, to \"illustrate the influence of folk music and jazz on the contemporary composer.\" Bernstein illustrated his points by playing his piano transcription of Copland's _El Sal\u00f3n M\u00e9xico_. The event was reviewed by Paul Bowles in the _New York Herald Tribune_ on 19 May 1943.\n\n109 Copland's nickname for his car.\n\n110 In Letter 140, Copland called it a \"two volume affair\".\n\n111 The Blue Angel was a nightclub founded by Herbert Jacoby. An article in _Time_ magazine from 26 April 1943 \u2013 just a couple of weeks before Bernstein wrote this letter to Ren\u00e9e Longy \u2013 evokes Claude Alphand's singing at the club:\n\nThe De Gaullist movement has found its loveliest voice. She sang last week at a new Manhattan cabaret, the Blue Angel, opened by balding, long-nosed, toothy Herbert Jacoby, ex-secretary to France's imprisoned ex-Premier Leon Blum. Chic as a Paris bandbox, its jet-black walls garnished with white lilies and orchids, the Blue Angel gave off more than a suggestion of the smarter mortuaries. But it ceased to be funereal when a swarm of De Gaullist refugees and friends produced an opening-night crush of such confusion that New York Daily News Columnist Danton Walker, for one of the few times in his professional life, was presented with his own check.\n\nMany of the throng went especially to hear Claude Alphand sing. She is a beautiful, blonde, rather waxwork-like Frenchwoman who accompanies her balladry on the guitar. Rated by many as the best French chanteuse since Yvette Guilbert and Lucienne Boyer, she sings with a feline throatiness and great stylistic elegance. Her favorite song is: Prenez le temps d'aimer [...] \u2013 Alphand's delivery of such sentiments makes her worth $750 a week to the Blue Angel's Jacoby.\n\nMme. Alphand has only recently turned professional. Before the war she was prominent in Paris society; she is the wife of Herv\u00e9 Alphand, former Treasury attach\u00e9 of the Vichy Government in Washington. Her father, Robert Raynaud, founded La D\u00e9p\u00eache Marocaine, the first French daily newspaper in Morocco. When the Alphands arrived in the US three years ago, Herv\u00e9 Alphand said: \"In France now there are only two things to do: to work and to be silent. I have come here to work and to be silent.\" But he did not stay silent long. Less than a year after his arrival, he announced his disagreement with Vichy policy, resigned, went to England where he joined the De Gaullist fighting forces.\n\nMme. Alphand had to find a way to earn her living. Her friends had long admired her repertory of some 200 salty popular songs. Helped by a group of them (Lady Mendl, Henry Bernstein, Elsa Maxwell), she began appearing at a French hangout called Le Petit Palais. Among Manhattan's Francophile intelligentsia, her nostalgic music was sensational. Manhattan's Liberty Music Shop issued an album of Alphand recordings, quickly sold 1,000 copies.\n\nToday, though Manhattan's swankest pub-crawlers flock to hear her, Mme. Alphand is already tired of professional life. Says she, with a Gallic shrug: \"If I am not to sing, then I must sew, I must make hats or something.\" But she admits that she is not doing badly in the new world, says: \"Heaven was very charming to me.\"\n\n112 Three popular French _chansons_.\n\n113 A French folk song.\n\n114 Bernstein's original French text is included out of interest. The following is an English translation:\n\nEverything is so French these days. I've just read Gide in French. I go to play every Sunday evening at The Blue Angel, a new club of a Parisian sort (like the old Ox on the Roof, or the Blue Ribbon); the clique of the Serenade Concerts (your friend the Marquise, etc.); my world seems these days to be completely French. So what to do about it? Nothing but to write a word to the first Frenchwoman among all Frenchwomen. D'you think I'm drunk? It's not drunkenness at all \u2013 it's friendship.\n\nThe immediate cause of this letter is Mme Claude Alphand, the extraordinary singer at The Blue Angel. Each time she sings \"Les Moules marini\u00e8res\" or \"La Belle Journ\u00e9e\" or \"Tu m'as voulu, tu m'as eu,\" I am forcefully reminded of \"Mon Mari est bien malade.\"\n\nWhat are you doing these days? Not still the assembly line, I hope! In any case, I will be in Boston next week and I insist on seeing you again. Our old friend, the Institute of Modern Art (damn!) has invited me to play up there on Thursday evening. Well, it's a few dollars and a trip to Boston paid for! But what an existence! On the Tuesday, I've a very important lecture-recital at Town Hall; and on the Wednesday, a lecture at the Art Alliance in Philly!!!! And Thursday it's Boston! I'll keep all the news for your ears, not your eyes....\n\nLenny.\n\nAnd look who has won the Paderewski Prize! Gardner Read! _Frightful_.\n\n115 Koussevitzky's letter to _The New York Times_ published on 16 May 1943 was headed \"Justice to Composers,\" and was a plea to support creative musicians: \"What is being done for the composer of our day? [...] It is time to wake up to our responsibility toward the composer and to repay the debt long standing that we owe him. [...] We musicians must be first to stand by the composer because we owe him most. We have ripened to this consciousness. Therefore I say the time is ripe to act.\" He goes on to propose the setting up of a fund to support the work of composers, initially by a donation of $1 from each professional performing musician in the country. This would, he argues, \"go a long way toward establishing a composers' fund. A far-reaching and wise plan must be worked out for a proper distribution of the fund [...] For that purpose an organizing committee must be formed without delay. Whatever action we take now will lay the groundwork for the impelling and just cause of the composer. Embracing that cause, we shall ascend to new heights, we shall gain in confidence, in self-esteem and in fortitude.\"\n\n116 Billy the Kid.\n\n117 Hindemith's.\n\n118 This became _Appalachian Spring_.\n\n119 Stokowski conducted the America premiere of Copland's notoriously difficult _Short Symphony_ with the NBC Symphony Orchestra on 9 January 1944 (though according to Copland himself it was an \"extremely inadequate reading\").\n\n120 James Petrillo (1892\u20131984), the powerful leader of the American Federation of Musicians. In July 1942, Petrillo imposed a ban on American musicians making commercial recordings for major American companies because of a dispute over royalty payments. The union settled with Decca and Capitol in October 1943, and with RCA and Columbia in November 1944. During the strike, Petrillo had to authorize the release of new recordings.\n\n121 An autograph sketch and a fair copy of _The Nicest Time of Year_ (both with a slightly different title, \"The Nicest Time of Day\") are in the Leonard Bernstein Collection. The tune was used for \"Lucky To Be Me\" (\"What a day, Fortune smiled and came my way,\" etc.) in _On the Town_. But, as indicated in this letter, it was composed as a single song a year before Bernstein started working on the show. My thanks to Sophie Redfern for drawing my attention to this manuscript.\n\n122 The envelope is addressed to \"Pvt. David Oppenheim, A.S.N. 12208749, 1633rd S.V., Brk. 130, Co. A, Camp Grant, Illinois.\"\n\n123 Gail Kubik (1914\u201384), American composer who studied at Harvard with Walter Piston and with Nadia Boulanger. During the Second World War he was Music Director of the Motion Picture Bureau of the Office of War Information.\n\n124 Fritz Reiner (1888\u20131963), Hungarian-born conductor, and Bernstein's teacher at the Curtis Institute. One of the most inspiring (and feared) conductors working in America, Reiner was Music Director of the Pittsburgh Symphony Orchestra (1938\u201348), conducted regularly at the Metropolitan Opera, and became Music Director of the Chicago Symphony Orchestra in 1953.\n\n125 Bernstein's appointment as Assistant Conductor of the New York Philharmonic.\n\n126 _Duffy's Tavern_ was a popular radio comedy show that ran from 1941 to 1951 and often featured guest stars.\n\n127 Ernst Lubitsch.\n\n128 Possibly a name invented by Adolph Green for this list.\n\n129 William Perlberg.\n\n130 William LeBaron, the producer who went on to make _Greenwich Village_.\n\n131 The eventual star of _Greenwich Village_ was Carmen Miranda. Adolph Green, Betty Comden, Judy Holliday, and Alvin Hammer all appeared in the film.\n\n132 The _Top of the Mark_ cocktail lounge at the Mark Hopkins Hotel.\n\n133 _The North Star_.\n\n134 Brennan plays the part of the pig-farmer Karp in the film.\n\n135 Elizabeth Reitell, Adolph Green's first wife.\n\n136 See note 77 to Letter 132.\n\n137 Randall Thompson (1899\u20131984), American composer. He taught Bernstein orchestration at the Curtis Institute. Thompson's Second Symphony was one of the works Bernstein conducted during his first year at Tanglewood (1940), and he remained extremely fond of the piece, playing it in 1959 and 1968 in New York Philharmonic concerts, and recording it for Columbia Records in 1968. An undated note in Bernstein's hand (a draft reply to the Thompson scholar Byron McGilvray) reads as follows: \"Randall was a real friend, right from the beginning. At Curtis we shared the joys of both orchestration (which I studied with him) and the London _Times_ crossword puzzle, of which we were both secret fans. Beyond this, we shared a common conviction that Curtis should be reconceived, & turned from a conservatory-factory into a real place of learning. (We were both academically orientated, as Harvard men should be.) Randall did not exactly succeed in this, and we both left together (as we had entered together), he as dismissed director, and I with my diploma.\"\n\n138 \" _Baby_ Steinway\" is the piano Ren\u00e9e Longy Miquelle had loaned Bernstein on which he composed the _Jeremiah_ Symphony. See note 45 to Letter 115.\n\n139 Alexei Haieff (1914\u201394), American composer.\n\n140 Charles Mills (1914\u201382), American composer who played in jazz bands from the age of 17. He was commissioned by Mitropoulos to compose a work for the New York Philharmonic in 1951. His output includes six symphonies, and some of his compositions involve jazz groups.\n\n141 Bernstein did get the commission, and the result was _Fancy Free_.\n\n142 During the \"Petrillo Ban\" of 1942\u20134, even a small company such as Hargail Records had to obtain a release from James Petrillo of the Associated Federation of Musicians before a recording could be issued.\n\n143 The identity of the \"Frau\" was shrouded in mystery until the emergence of this letter in 2013. For Marketa Morris' letters to Bernstein, see Letters 197, 256, 260, and 261.\n\n144 i.e. the Clarinet Sonata.\n\n145 Jerome Robbins (1918\u201398), American dancer, choreographer, and director. A temperamental and intensely demanding genius, he was without doubt the person who forged the most productive creative relationship with Bernstein: their first collaboration was the ballet _Fancy Free_ , followed by _On the Town_ , _Facsimile_ , _West Side Story_ , and _Dybbuk_.\n\n146 Bernstein is discussing the earliest stages of his work on _Fancy Free_. Progress on the score is documented in several further letters. See Letters 165, 166, 169, 170, 171, and 172.\n\n147 I am most grateful to Sophie Redfern for establishing the chronology of the undated letters from Bernstein to Robbins about _Fancy Free_. She has kindly provided the following information: the contract for the ballet (mentioned in Letter 165) is dated 17 November 1943; the \"song\" described in Letter 165 was the precursor to the \"radio\" mentioned in Letter 166, as can be seen in Bernstein's sketches for the ballet.\n\n148 Shirley Gabis Perle wrote about this letter on 26 January 2013 (by email): \"I remember writing that letter (what nerve), but who knew, as his father famously said, that he would become Leonard Bernstein. His response came on a post card that I didn't save \u2013 he was very annoyed by my criticism. [...] I don't suppose I could have written it if it were not for the depth of the connection between us \u2013 a connection that remained throughout our lives. I subsequently played the Philadelphia premiere of the Sonata with Stanley Drucker \u2013 to make restitution? The piece, after all, does have Lenny's vitality and charm \u2013 I was obviously a stickler for profundity in my youth.\"\n\n149 Copland was a visiting lecturer at Harvard University in 1944.\n\n150 A week before Copland sent this letter, _Fancy Free_ had its triumphant first performance on 18 April 1944, by Ballet Theatre at the Metropolitan Opera House, conducted by Bernstein. The Ballet Russe de Monte Carlo season at City Center in April 1944 included Copland's _Rodeo_. As for the two ballets being \"hopelessly married,\" they were often mentioned together in press reports.\n\n151 Copland's \"new one\" was his \"Ballet for Martha,\" the work that became _Appalachian Spring_. The sketches are dated \"June 1943\u2013June 1944, Hollywood, New York, Cambridge.\" It was first performed on 30 October 1944.\n\n152 This concert took place on 25 April 1944. The program included works by Walter Piston (Violin Sonata and Piano Trio) and Copland (Piano Sonata played by the composer, Violin Sonata, and _Danz\u00f3n cubano_ played by Copland and Irving Fine).\n\n153 Joseph Szigeti (1892\u20131973), Hungarian violinist, pioneer of twentieth-century repertoire, and a friend of Bela Bart\u00f3k: Szigeti is dedicatee of the First Rhapsody and the _Contrasts_ (with Benny Goodman).\n\n154 4 July 1944, Bernstein's Ravinia debut. Reviewing the concert in the _Chicago Daily Tribune_ (5 July 1944), Claudia Cassidy commented that Szigeti \"had an off night, almost as if gremlins rode malevolently on his usually silken bow,\" but that \"the eye and ear inevitably gravitated to the slight young figure on the podium [...] A fascinating fellow, this Bernstein, dynamic, emotional, yet under complete control.\"\n\n155 8 July 1944. Szigeti was evidently back on good form. Though Claudia Cassidy in the _Chicago Daily Tribune_ (9 July 1944) noted that the Bart\u00f3k suffered from \"obvious skimpiness of rehearsal,\" she praised Szigeti's Mozart: \"played with his usual patrician serenity and with a special grace and verve for the Rondo, and the orchestra had a touch of the Mozart fire that warms rather than consumes. Mr. Bernstein conducted quietly and carefully, getting his effects more simply than before, but with no less ardor. He has what it takes to learn as he goes along.\"\n\n156 Probably Reuben's restaurant and deli at 6 East 58th Street in New York.\n\n157 Written on a royalty statement for sales from January to May 1944 of the Hargail Records discs (set MW-501) of Bernstein's Clarinet Sonata. A total of 457 copies sold in that period, with a royalty of 4\u00a2 per set, making a total of $18.28. An annotation by Helen Coates at the top of the page reads: \"Thanks for statement. Send royalty to D[avid] Opp[enheim], P.F.C. 12208749, 413th Infantry, Camp Carson, Colo[rado], U.S. Army.\" Sales of this recording seem to have been livelier than those of the sheet music: a royalty statement from M. Witmark for the three months ending 25 November 1944 lists sales of a mere nine copies of the Clarinet Sonata.\n\n158 Koussevitzky's own Concerto for Double Bass, Op. 3.\n\n159 Lukas Foss appeared with Bernstein at the New York Philharmonic's Lewisohn Stadium concerts on 14 July 1944, conducting Ravel's Piano Concerto in G major, with Bernstein as the soloist \u2013 probably the first of their many appearances together in New York.\n\n160 Herschel and Janice Levit (and their daughter Lois) became friends of Bernstein while he was studying at the Curtis Institute in Philadelphia \u2013 they were introduced in 1940 by Ren\u00e9e Longy Miquelle. The Levits lived near Bernstein's student rooms and he became a regular visitor (often calling round to take a bath). It was in the Levits' apartment on South 22nd Street that Bernstein finished his piano arrangement of _El Sal\u00f3n M\u00e9xico_. Humphrey Burton reports that Bernstein said their upright piano was \"just right,\" \"like a Mexican bar room piano\" (Burton 1994, p. 84).\n\n161 A reference to the Lenni Lenape tribe of American Indians who lived along the banks of the Delaware River in Pennsylvania.\n\n162 The address (noted in pencil by Bernstein on the first page of this letter) was 628 Stetson Rd, Elkins Park, PA, and it is indeed \"modern\" \u2013 in fact it's a magnificent house designed in 1940 by Louis Kahn for his friend Jesse Oser.\n\n163 The sketch enclosed with this letter is inscribed \"A happy birthday and many more glorious years for Serge Alexandrovich: with love, Lenushka, N.Y.C, July 26, 1944.\" It is an early version of what became, with very small modifications, the start of the _Age of Anxiety_ Symphony. The tempo marking is _Andante contemplativo_ (changed to _Lento moderato_ in the Symphony), otherwise the music is largely the same apart from minor changes; no instrumentation is given \u2013 in the Symphony it is played by two clarinets. It is interesting to find this idea so fully formed in Bernstein's mind as early as 1944: W. H. Auden's poem _The Age of Anxiety_ \u2013 the inspiration for the work as a whole \u2013 was not published until July 1947, three years after this sketch. The eventual work was not only first performed by the Boston Symphony Orchestra conducted by Koussevitzky __(on 8 April 1949), but dedicated \"To Serge Koussevitzky, in tribute\".\n\n164 In the end, the concert, given on 24 and 25 November 1944, consisted of just two works: Brahms' First Piano Concerto with Jes\u00fas Mar\u00eda Sanrom\u00e1 as the soloist, and Shostakovich's Fifth Symphony.\n\n165 See Letter 178.\n\n166 Ramin served as a Corporal (20120408) in the 84th Infantry Division. From the end of August 1944, the 84th started to arrive at Camp Kilmer in New Jersey, in preparation for departure to Europe. The Division sailed on 20 September 1944 and arrived in England for training on 1 October, landing on Omaha Beach in Normandy a month later, before taking part in the Ardennes Offensive (the Battle of the Bulge). Sidney N. Ramin of Roxbury, Mass., is listed in the _Roster of Officers and Enlisted Men, 84th Infantry Division, European Theatre of Operations \u2013 World War II_ (Viking Press, 1946). During his time in the Army (in which Ramin was in Special Services), he also found time to arrange the music and conduct the orchestra in a revue called _It's All Yours_ , performed at the Stadt Theater in Heidelberg and in Paris (personal communication from Sid Ramin).\n\n167 Two of the _Seven Anniversaries_.\n\n168 When Bernstein wrote this letter he was hard at work with _On the Town_ , but the need to do something about the \"Aaron Copland side\" of his life and career was ever-present.\n\n169 An indication of the growing importance of Helen Coates in managing Bernstein's domestic affairs, as well as the difficulties of finding an apartment in New York City.\n\n170 The first performance of _Appalachian Spring_ took place in the Coolidge Auditorium at the Library of Congress on Monday, 30 October 1944.\n\n171 _On the Town_ opened at Broadway's Adelphi Theatre on 28 December 1944.\n\n172 This was one of the earliest contacts between Bernstein and his Detroit friends Philip and Barbara Marcuse. In the 1950s, the Marcuses provided a kind of model of stability for the newly married Bernsteins, and offered them warmth, advice, and support.\n\n173 Probably a broadcast of Shostakovich's Fifth Symphony, which Bernstein conducted in Detroit during the 1944\u20135 season.\n\n174 _On the Town_.\n\n175 This recalls a much earlier Bernstein letter to Oppenheim (see Letter 122).\n\n176 Oppenheim and Judy Holliday were married in 1948.\n\n177 Artur Rodzinski.\n3\n\nConquering Europe and Israel\n\n1945\u20139\n\nThe post-war years saw Bernstein's conducting career flourish, not only in the United States but also as the first American-born conductor to develop an extremely successful career in Europe. His letters home from London, Prague, Paris, and elsewhere are fascinating evocations of great cities recovering from war. These were also the years during which Bernstein composed some of his most serious orchestral scores: _Facsimile_ \u2013 a ballet with Jerome Robbins \u2013 and _The Age of Anxiety_ Symphony, composed for Serge Koussevitzky and the Boston Symphony. Despite the encouragement of George Abbott and Betty Comden, Bernstein did not immediately follow up the Broadway success of _On the Town_. Koussevitzky, an inspiration as well as a mentor, gave Bernstein regular opportunities to work with the Boston Symphony Orchestra, but their master\u2013pupil relationship was not without its difficulties, as the tense exchange of letters in December 1946 reveals: Koussevitzky objected strongly to Bernstein's proposal to program his own music in concerts with the orchestra, and Bernstein's only option was capitulation, in order to restore amicable relations. The Koussevitzky connection was not only important personally but also professionally. _The Age of Anxiety_ was commissioned by Koussevitzky, who conducted its first performance on 8 April 1949. But it was Bernstein who gave the first American performance of Britten's _Peter Grimes_ on 6 August 1946, and on 2 December 1949 the world premiere of Messiaen's _Turangal\u00eela-Symphonie_ (a work to which he never returned after the first three performances) \u2013 both of which were commissioned by the Koussevitzky Music Foundation in memory of Natalie Koussevitzky. Bernstein also conducted the first European performance of Copland's Third Symphony, another Koussevitzky Foundation commission, in Prague on 25 May 1947. In other words, Bernstein's reputation for playing large-scale works that were recently composed was nurtured to a significant extent on repertoire that Koussevitzky had commissioned.\n\nIn February 1946, at a party given by the Chilean pianist Claudio Arrau, Bernstein met Felicia Montealegre. In the course of the year they grew ever closer \u2013 a relationship Bernstein chronicled in his letters to Helen Coates \u2013 and at the end of December the couple were engaged in Hollywood. Though the engagement was broken off in September the following year, they were eventually married four years later, in September 1951 \u2013 a union that both parties entered into in the full knowledge of its potential difficulties, the most significant being Bernstein's sexuality. Something of his turmoil about this is revealed in letters from Marketa Morris (the \"Frau\"), whom he consulted from the early 1940s onwards, and Ren\u00e9e Nell, another psychoanalyst Bernstein consulted in the later 1940s.\n\nBernstein's visits to Israel were to become a central part of his career, and they did much to define his Jewish identity. His letters from 1948 to his mother and sister, to Koussevitzky, and to Copland reveal something of the profound impact the country and its people had on Bernstein, the warmth and passion of his commitment to the Israel Philharmonic Orchestra, and the joy he drew from the experience of working with these musicians. For Bernstein, conducting always had to be \"fun\" \u2013 in other words a genuinely rewarding experience \u2013 if it was to be worth doing at all, especially when he could never find enough time for composition. In the Israel Philharmonic he found an orchestra with which he was usually at his happiest, even when \u2013 as on his 1948 visit \u2013 he was confronted with an astonishingly punishing schedule, and concerts that were often interrupted by bombing raids; he was there, after all, during the Arab\u2013Israeli war. On 14 May 1948, David Ben-Gurion had declared the establishment of a Jewish state to be called the State of Israel. War broke out the next day, and was at its height when Bernstein arrived to work with an orchestra that was not only a cultural symbol, but a potent national one as well.\n\n186. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\n40 West 55th Street, New York, NY\n\n3 January 1945\n\nDear Ren\u00e9e,\n\nHow sweet of you to remember, and send me the Michelangelo. I adore it. I hope it was you, because it contained no card, no greeting of any sort, and I am at a loss as to what occasion it represents. New Years? Christmas? Some obscure but meaningful anniversary? The opening of the show? Paul Bowles' birthday?\n\nWell, the show [ _On the Town_ ] has opened and is a phenomenal hit, in spite of all. The reviews are fantastic raves, especially the _Times_ and _PM_ , and the _Hollywood Reporter_ , which called it the greatest musical ever produced! It's thrilling, and I would be a rich man, except that whatever money I get goes back to Uncle Samovitch for taxes. But it's nice to feel that you've earned a stupendous sum, even if you hold it only for a week.\n\nNow I am bleary with a throat infection, and a general let-down collapse, and struggling to get back into my beard (long-hair) and study Brahms' First for Pittsburgh next week. It will be fun to be back there, and this time with a whole program including _Fancy Free_ , the Ravel Concerto, _Euryanthe_ and Brahms' First. I stretch long and loud, yawn, smile, toss my mangy curls, and close with love, to get back to the _Partitur_.1 Let me hear how everything goes with you. When do you come around again?\n\nLove, and thanks again.\n\nLenny\n\nSpookietchka\n\n187. George Abbott2 to Leonard Bernstein\n\nThe Town House, Los Angeles, CA\n\n20 February 1945\n\nDear Lennie,\n\nAccording to the calendar you must now be back in New York where I shall not arrive for another five weeks, by which time you will probably be waving your baton in some distant city.\n\nI hope soon, however, that we shall all find ourselves together again discussing _life_ and _integration_.\n\nI'll postpone that for the moment and take up the subject of opera. I have constituted myself an authority on the subject because I don't like opera; also I have seen very few operas. I find myself moved by the sheer beauty of the sound that assails me, and occasionally by the visual effect, but never by the story. I cannot get from it the feeling of being carried away (to quote from a recent musical comedy hit). Plays, movies, symphonies, novels seem to me to be artistic wholes. Operas seem magnificent anachronisms. So, when I talk of opera, in re [my] interest, I am talking about a new form which does not now exist: I am talking about something which I expect you to create. It will have integration all right, but it will be unhampered by tradition, it will use picture techniques, top dancing or any other feature that adds up to excitement \u2013 and it will ruthlessly eliminate the _ridiculous_.\n\nAs far as _On The Town_ is concerned, please don't let yourself be distressed by minor criticism from some of your pals. It is a wonderful score \u2013 a bit too profligate perhaps, too many fresh melodies thrown in where developments of existing ones would have done. That, however, was not your fault \u2013 except as you share the responsibility for an original structure that wasn't very practical \u2013 but the result of changes done in a hurry. The final result may not have accorded with the ideal upon which it was based, but it is good. And, what is more, you should feel proud that you have proved yourself so adaptable. Had you been the inflexible type, you could have gummed the whole works. In my opinion you should congratulate yourself that you have had the experience and learned so much of practical theatre matters without going through a disaster to pay for it.\n\nI read _Self-Analysis_3 on the train. I gave myself the works. But I'm afraid my subconscious is an almost empty cellar. The book more reposes in the hands of one whose subconscious is boiling away. [...] We were on the train coming out and there's a fellow who not only needs it, but who knows he needs it. We deduced that he hated his mother \u2013 we also staged _King Lear_ \u2013 it was quite a pleasant trip.\n\n_Kiss & Tell_4 is going to be a good picture. Everything is going substantially as I would have it.\n\nThe only flaw in my California life is that I have strained a muscle in my side and have to give up tennis for a few days. Since tennis is practically the rock upon which my local life is founded, I am very sulky about the matter.\n\nGive my best to Betty [Comden] and Adolph [Green] when you see them. I wish I were there. But I soon will be.\n\nYours, as always,\n\nGeorge\n\n188. Leonard Bernstein to Helen Coates\n\nThe Windsor [Hotel], Montreal, Canada\n\n28 February 1945\n\nDear Helen,\n\nCould you reserve two tickets for Arthur Rubinstein (for _On the Town_ , natch) for Friday eve., March 9th. And _prepaid_ , please. He's a swell guy. The concerts are great.\n\nLove,\n\nL\n\n189. Betty Comden to Leonard Bernstein\n\n29 April 1945\n\nDear Lenny,\n\nThis must be the first letter I have written you since those apologetic little notes of Revuer days, airily explaining why no check was enclosed \u2013 yet. No \u2013 a quiet nine piece orchestration is not what is on my mind right now. I'm spoiled forever by the sound of thirty pieces anyway.\n\nWe've been meeting with Paul [Feigay], we've been meeting with Oliver [Smith] \u2013 we've been meeting with George [Abbott] \u2013 and the last has been, needless to say, the most fun of all. And the history of these meetings is briefly this, some of which you know already: We had originally intended to get an idea and do a show with George, leaving one for P[aul] and O[liver] Inc. for some other time. But in the mean time through Bill we were approached with the idea of _The Greeks Had a Word for It_ \u2013 made into a musical about the twenties, and involving Gypsy R. Lee. After the reading of the play, which is good, we realized that it is not typical of the twenties at all \u2013 it was just done at that time \u2013 so why bother with it? Better to make up out of whole cloth and our fevered brains \u2013 a story of our own, using the twenties as a background. Much as we wanted to ignore the whole thing and concentrate on something for G.A. we couldn't help being intrigued by the period, its color and significance \u2013 and against our will got some ideas. These were enthusiastically received by everyone, and they seemed eager to plunge at once. Extremely un-anxious to face as much terrifying vagueness as we had been subjected to before, and dying to do something \u2013 anything \u2013 with George, we said we would do the thing only if he were involved. We told George all about it, and he got really excited about what we've worked out so far \u2013 has met with the Fitelson Gang, and it looks as though something will work out \u2013 a co-producing venture plus, of course, George's directorship. Being a very smart man, and not liking the idea of a \"star\" type deal (possibly involving 10% of the show \u2013 and I mean Gypsy of course), and not liking the idea of having to write a show \"around\" anyone \u2013 George doesn't think Miss Lee is essential to the show, but thinks we should get to work and write it, and if there's a part for her, fine, and if not \u2013 fine again.\n\nThings are hardly in what you would call a definite state just yet, but we have, naturally, been talking about someone who might be equipped to knock out a coupla tunes to go along with all this \u2013 and your name just happened to come up. Not having talked to you, Lenny, since our \"opera\" meeting, I have no real idea what you are planning. But word filtered down to us from Bill and Shirley, that you like the idea, but (this from the latter) that you didn't feel too keen about working with Feigay and Smith again. As you can see from the above we had a qualm or two ourselves. But now George is in the picture, very much so. And we are sort of stimulated by the prospect. Having him makes all the difference in the world.\n\nI am not going to send you a card saying \"I will\/will not do the show\" \u2013 and expect you to underline and return. But we would so love to hear from you, since talking to you is impossible for another week. Musically of course the show has terrific possibilities \u2013 and the period and theme are surprisingly significant plus allowing for lots of beauty and general appeal, as well.5 It would be wonderful to do this show with you, Lenny \u2013 of that we are sure. Please write.\n\nMuch love,\n\nBetty\n\n190. Izso (Isadore) Glickstein6 to Leonard Bernstein\n\n7 May 19457\n\nDear Leonard,\n\nPlease accept my heartfelt thanks for the wonderful gift you sent me. I enjoy immensely playing and singing your finale of _Jeremiah_.8\n\nYou have brought more dignity to the Jewish people than anyone else I know.\n\nI had breakfast with your father today and he told me you may be at Burton's Bar-Mitzvah. I hope you will.\n\nGod bless you!\n\nI. G. Glickstein\n\n191. Jerome Robbins to Leonard Bernstein\n\n13 May 1945\n\nDear Lenny,\n\nIt was rather nice seeing you the other night at Al and Dick's.9 What with your letter and our chance meeting, I thought that maybe (as you said) we could get \"that old show out of our heads\" and do some work. However I have just finished reading the interviews in _Dance Magazine_.10 Have you seen them? Well, in yours, you talk about the trilogy idea based on _Fancy Free_ , even mentioning _Bye Bye Jackie_11 by name and describing the material. And somehow, Leonard, it all sounds like your idea, and to boot my name isn't even connected with my own registered play.12\n\nNow it all might be the fault of the interviewer, and if it is, too bad, because it makes you appear to be dishonest. But if it is something you did yourself, it is a low, dirty trick \u2013 and I wouldn't try it again. Fortunately the majority of people in the dance world already know about _Bye Bye Jackie_ and the Theatre Guild and others have read it, so that if and when it's done, it won't seem that you have supplied my material and ideas.\n\nI don't like writing a letter like this. But I thought it best we get straightened out on things like this. We are well suited to work together as far [as] talents are concerned, and it would be good if we could manage to do some more ballets. But this kind of business is not a good gesture either as a friend or business associate. So let's have no more of it.\n\nSincerely,\n\nJerry\n\n192. Leonard Bernstein to Shirley Bernstein\n\nMexico City, Mexico\n\n24 June 1945\n\nQueridisima Hermanita,\n\nI'll be home very soon. I leave tomorrow. I love this place madly, but I can't wait to get back. I go to Hollywood tomorrow, where [Irving] Rapper has arranged a dinner for me which includes my following fans, believe it or not: Bette Davis, Cary Grant, Van Johnson, Ethel Barrymore, Judy Garland, Dana Andrews, _and others_. Want to join me there?\n\nM\u00fc la d\u00fc,\n\nLad\u00fcm\n\nBought you a _fantastic_ opal!\n\n193. Bette Davis13 to Leonard Bernstein\n\nRiver Bottom, [Glendale, CA]\n\n[late June 1945]\n\nLeonard,\n\nThis is to say hello, that I will be listening Sunday, and any other time I know you will be on the air \u2013 to say I hope so very much \u2013 opportunity will often present itself so I can see you again, and to say \u2013 you and your music came along when I needed them desperately. I had hit a new low, and since being exposed to your mighty talents, I am on high \u2013 there is probably nothing in the world so encouraging for the future of the world as a super talent in someone \u2013 it is the only true inspiration and help in believing the world is really worthwhile. I feel privileged to have spent an evening with you \u2013 and thank you so much for playing for me when I know you didn't feel like it. It is an evening I will always remember with the most enormous pleasure. It is hard to have all this _said_ to one, I know that. Thus the _written_ expression of my delight at meeting you.\n\nIrving14 has very nicely been willing to be messenger boy for me \u2013 wish I were my own messenger boy and could be there to _see_ you conduct. You won't mind if I become a 1945 version of Madame von Meck, as regards you.15 There are some changes in the script already. We have seen each other \u2013 that is not according to style \u2013 and financially there is no similarity. The only resemblance:\n\n1. I am older.\n\n2. I adore your music.\n\n3. I like you.\n\nBye for now.\n\nBette\n\nP.S. If you ever have time, send me an autographed photograph, would you:\n\n\"A real fan I am \u2013 Leonard Bernstein.\"16\n\n194. Bette Davis to Leonard Bernstein\n\nRiver Bottom, [Glendale, CA]\n\n[after 1 July 1945]\n\nDear Piotr,\n\nThat is not the right spelling \u2013 but you get the idea. Your wire I adored. A letter I wrote you for Irving to take to you in San Francisco is still waiting for him to pick up \u2013 he went a day too early. When I have recovered it, will send it on.\n\nI am in bed with a cold, and loved my hour with you and your music from San Francisco. Only one thought: if they have you must they inflict upon us Mr. S.17 \u2013 a duller violinist never lived in my opinion. If you disagree we will talk about that later. The rubles I am saving up to bring in person. Anyway you were very sweet to wire \u2013 I so hope I see you soon. Your _Fancy Free_ music is \u2013 well \u2013 it is Bernstein, and musically I can give you no greater compliment in my opinion. My letter written a few days ago still goes \u2013 the Baroness Von Davis adores your music and likes you \u2013 till later.\n\nBette.\n\nI sent your stockings off to be framed today. We created great gossip, all because you asked me to sit beside you at the party while you were playing. Such simple basic people the Hollywood person is. I am flattered \u2013 hope you are \u2013 all jealousy and I don't blame them. Will drink a toast to you every Wednesday night, in memory of the Wednesday last week when I first heard you play \u2013 I am only furious about one thing, that all the time you were here last summer I didn't have the chance to meet you \u2013 it is so silly.\n\nBye again.\n\nB\n\n195. Bette Davis to Leonard Bernstein\n\n[Summer 1945]\n\nDear Lennie,\n\nIrving and I are working together. The set \u2013 so Irving says \u2013 is very like your apartment.18 If so how you must love it. I was most happy to contribute for your composer \u2013 and hope his career proves to be all you feel it is.\n\nRead about you so very often in the _New York Times_ \u2013 my bible of New York \u2013 and it is so wonderful to know that your triumphs continue. It must be a great satisfaction to you. Your Madame Von Meck \u2013 me \u2013 is very proud \u2013 only feels slighted that no requests for funds have been forthcoming \u2013 can't I get you into my debt somehow? You are not living up to your predecessor \u2013 he was much more helpless. Thank you for your message about my marriage.\n\nThat was as far as I got last week. In the meantime, I banged my head on the windshield and was away for a couple of days, but a hard Yankee head is able to take it, thank god. Our picture progresses very well today \u2013 maybe my head was injured after all \u2013 anyway my best to you Lennie \u2013 and here's to our next meeting. It was so nice to hear from you. Love,\n\nB. Von Meck.\n\nBette\n\n196. David Oppenheim to Leonard Bernstein\n\n7 August 1945\n\nLensky,\n\nDid I dream that you called me the \"laziest guy I know\" or did you actually say it? Let me know. Future relations may depend on it. The actual quote was: \"Besides your other faults you are the laziest guy I know\". My impression is that you merely stated the thing as a fact without anger. A good thing.\n\nThe new atomic bomb is the most frightening thing that has happened ever ever.19 Why am I the only one I know to realize this? The whole future of mankind depends on its future use. Imagine what could happen if a few unscrupulous men were to control the principle. Or what a counter-revolutionary device it is. Or how it makes a war with Russia more possible. And when you figure it is still in its primitive stages.\n\nOf course the good it can do should not be underestimated, & really referring to the principle of atomic power, not the bomb. Men are knocking at the portals of heaven (you may quote me).\n\nWhen mother said there was a call from Montreal this morning I was a bit scared that something was the matter. I'm glad you called though. After you left Monday I felt very blue. Hearing you cheered me up. [...] I hope to hell you do get out to Frisco. My time would be so limited. You might have to meet me someplace to make it all worthwhile. We'll see later.\n\nI have started _The Well of Loneliness_20 but I have little patience for anything just now, being worn out, nervous as a cat and as irritable. Reaction to 30 days of hard living & no sleep. In fact after you called I had breakfast, read the paper and went to sleep for 6 more hours.\n\nMother sends regards to you and to Helen.\n\nI have to stop now before I fidget right off the chair.\n\nPlease don't be lonesome. Be happy, serene and effective & write!\n\nLove,\n\nDave\n\n197. Marketa Morris21 to Leonard Bernstein\n\nCountry Club House, Tannersville, NY\n\n21 August [1945]22\n\nMy dear Lenny,\n\nI just read an article about the possibility of your going to Hollywood. Shall I congratulate you?\n\nIt seems that your decision of not continuing with our work is now a definite one \u2013 so that a few words of mine won't disturb your decision. Of course I would like to get some information about your trip to Mexico, your plans for the next future and last [but] not least how you feel mentally!\n\nIf you feel that writing or talking to me could still influence you and that you are sort of uncomfortable or afraid of it \u2013 skip it please and let's wait.\n\nNot only am I personally interested but I also would like to have your report for my records. But \u2013 I am repeating \u2013 _only_ if it _doesn't interfere with your emotions_.\n\nI am enclosing something which might interest you.\n\nHow do you feel about the Peace?\n\nSincerely yours,\n\nMarketa M.\n\nTo the 5 of September: Tannersville N.Y., Country Club House.\n\n198. Leonard Bernstein to Aaron Copland\n\nNew Hotel Jefferson, St. Louis, MO\n\nSaturday [1 September 1945]\n\nAaron darling,\n\nWell, the monstrous experiment of juxtaposing you and Ch\u00e1vez turned out to be just that \u2013 a monstrous experiment \u2013 but such fun! [...] Everyone loves your piece \u2013 and, strangely, the _Sinfonia India_ was a big hit. One of those Bernstein surprises.\n\nI love you & miss you \u2013 and here are the reviews. It gets lonely: won't you send a slight letter? Too many people & dinners & dullards here. Nice \u2013 but what happens after midnight.\n\nSee you velly soon.\n\nLove\n\nL\n\nI'm here til next Sunday (9th).\n\n199. Leonard Bernstein to Aaron Copland\n\nNew Hotel Jefferson, St. Louis, MO\n\nThurs [September 1945]\n\nMy darling A,\n\n[...] I wish you were here. How about dropping in on me suddenly, like a visit? I have a suite, and it's not far. We'd really have a vacation. I've banged my head around over these programs for Paris (I hope it's still on) and come up with these:\n\nI.\n\nSchuman: Prayer\n\nV. Thomson: Five Portraits\n\nBarber: Violin Concerto\n\n\u2014\n\nBernstein: Jeremiah\n\nGershwin: Am[erican] in Paris\n\nII.\n\nPiston: Concerto\n\nHarris: Symph no. 3\n\nBlitzstein: Freedom Slop23\n\n\u2014\n\nRandall Thompson: Scherzo from Symph II\n\nSessions: Adagio from Symph\n\nCopland: Lincoln or Billy\n\nThat seems to take care of everybody (except D[avid] D[iamond]), and the programs are a little long, but not too much so. And boy, they're hard! Most of the pieces are short, no? I don't think either program is over an hour and a half. What do you think? Couldn't I play the Ravel Concerto on the European program, and your Piano Sonata on the chamber music?\n\nI offer no cheery word on the subject of your father, since I know you're the best one in the world at that.24 But I hope you're better, and back in stride: I refuse to take _arthritis_ seriously in you! It doesn't go at all.\n\nI love you, and hope you'll write soon, if you don't drop around personally; my love to Victor & Ted.\n\nL\n\nThe St. Louis Jazz Society is taking me on a tour of old Southern jazz haunts tonight!\n\n200. David Oppenheim to Leonard Bernstein\n\n19 September 1945\n\nDear Len,\n\nI read in the paper and Shirl tells me that you are up to the neck in auditions. How does it look? Many vacancies to fill? How does your clarinet section shape up? If I were to get out could you hire me for next season or can't you fire a man merely because you have someone else you would rather have play in his place? What does 1st clarinet pay in your orch? Naturally that is the prime consideration. Gotta compare your offer with dozens of others. If I am not let out in time and Local 66 doesn't reinstate me, we will have union trouble.\n\nAll the above assuming you don't run in to a great virtuoso of the licorice st\u00fcchel to replace hypothetical me. Don't!\n\nI've been getting down to L.A. to my Aunt Pauline's for some quiet outdoor weekends, complete [with] charcoal broiled breakfast \u2013 eggs, bacon et al.\n\nMy cousin Judy is here now. She is the one (do you remember) who introduced herself and a hundred other kids to you as my cousin in Detroit. Wonderful charming 15 yr old colt \u2013 the only girl cousin on mother's side of the family \u2013 my pet. She saw you at the Berkshire concert this year.\n\nNothing much to say. I'm grand, as Shirl puts it, except for the old army disease, lackanookie. And not a thing to do about it here. Well, can't keep me in for ever. Congress would raise too big a stink.\n\nLove,\n\nDave\n\n_Seymour's address?_\n\n201. David Oppenheim to Leonard Bernstein\n\nUnited States Army\n\n20 September 1945\n\nBoy, are you lucky \u2013 the second letter from me in as many days. But I just got your note. And it was so good hearing from you that I just had to write.\n\nI had no idea that your season was just 12 weeks. I did imagine though that the wages would be low, being paid by the city etc. Too bad you got going so late. Still, 12 weeks is a short season and it kills the season for a musician. I can see where it is hard getting the right players. What is the net result now? How do you stock up and can you whip them into something that satisfies you and that can play modern things? I think you can.\n\nHere is my setup. Quite a while ago it was announced that men with 45 points up to May 12th and\/or 50 up to Sept 2nd would not be sent overseas. I had 47 on the first count, 52 on the second. OK.\n\nToday an order came out restricting to the camp area all men with between 45 and 59 points. These men (me) will be shipped someplace by the 24th \u2013 Monday (this being Thursday). So almost by the time you receive this I will be entrained for my new station. What, what, what for, or why for, I don't know. Maybe training cadre for new recruits or processing cadre for dischargees. I wish I knew. I don't, but I'll let you know along with my new address. Pray that I am shipped a few thousand miles _East_.\n\nI was supposed to go to Frisco, Sunday through Thursday, for some kind of parade, but now it is off. I was planning to wire you tonight for Seymour's address.\n\nAfter a month, a good month too, of feeling in contact and in tune and in love with Mad, I suddenly find myself out of contact. I wish I understood why. It has happened before, but without much reason, that has to do with thinking and dreaming too much about her. I don't know why this should have the effect it did, if it did. But that is the way it is. Here, away from you all, your memories remain clear and bright for a while, then suddenly dim and it leaves me very much alone. But I am stronger now than ever and I don't feel unduly depressed. _I miss you all, and it all_.\n\nI haven't slept well either. I sleep, yes. But not the timeless death sleep I used to sleep. When I wake up now I am aware of every hour, the full 8. The way it used to be, I felt as though I had just closed my eyes when I got up. Now it is like I feel when I am sleeping with someone, e.g., slightly conscious because of the fear of rolling too much and waking them up. And I dream now too. At my aunt Pauline I practically floated off the couch two nights in a row. And who was the lucky girl? Aunt Pauline. Aren't I awful though.\n\nI'm looking for a score to _Daphnis_. Can you help without troubling too much.\n\nDon't pills help your sleeping? Or don't you want to try that? When the season or the rehearsals begin you will be better. Meanwhile try doing one thing at a time, forgetting the rest and stop having emotional orgasms. Why you have done all you can, that is enough. Forget it and go to sleep, relax, play some game etc. Really, Lensky, you may be the world's best musician, but you need some looking after \u2013 and some common sense. After all what is _really_ important. You, or whatever it is you are beating your head against the wall about?\n\nAnd don't dare say I don't understand how much I don't understand how much it means to have the right kind of players etc. But a little horse sense, or you won't be conducting anything after 40. You'll have ulcers, diabetes, angina and coronary thrombosis etc. etc. etc. Christ man. Take it easy. Do I have to desert to see that you do?\n\nTell Shirley I'll write her soon. And hello to Helen please.\n\nLove,\n\nDave\n\n202. Mildred Spiegel to Leonard Bernstein\n\n[Boston, MA]\n\n25 September 1945\n\nDear Pete,25\n\nI was asked by an organization in Boston to write to you for them. The United Order of True Sisters (in existence for 70 years and founded to benefit crippled children) would like to know if you could play for them \u2013 from 30 to 45 minutes \u2013 at a luncheon and meeting at the Copley Plaza \u2013 Tuesday afternoon, Jan. 29th. They want to pay you $200 plus all expenses. It is to be their biggest affair of the season and they naturally want a big attraction to help them raise a lot of money. They maintain rooms in the Children's Hospital and work in conjunction with the Harvard Commission on infantile paralysis.\n\nOf course I told them about the busy year you have ahead of you, etc., and that the date might conflict, etc., but I'd write you anyway.\n\nI am in no way connected with this group so please don't feel that a refusal would in any way affect me personally \u2013 just do as you like about it. If you can make it, they'd like lighter stuff \u2013 some Chopin, your Liszt Rhapsodie, some boogie-woogie and you can talk too if you don't want to play all the time.\n\nI know you're terribly busy these days, so have Helen drop me a note so that I can tell them your answer.\n\nThings are getting gradually better all the time \u2013 had my first Hershey bar in years today \u2013 also had lunch with Re La Mi.26\n\nGonna see your ballet again next week \u2013 the same night as _Undertow_ and _The Gift of the Magi_ \u2013 Lukas conducts.\n\n[...]\n\nBest of luck for the orchestra. I know it will be a brilliant yearrrr. Will hear you in February. Take care of yourself and get some sleep some time.\n\nMuch love to you and Helen from\n\nMildred\n\n203. Joseph Szigeti to Leonard Bernstein\n\nPalos Verdes Estates, CA\n\n1 October 1945\n\n[Telegram]\n\nJust learned with shock of Bart\u00f3k's death.27 Would it be possible for us to give part of your program pair December 31 January 1 or failing that January 21, 22nd as Bart\u00f3k memorial with one orchestral work and the Portrait in D major and Rhapsody #1 dedicated to me. Naturally no fee involved. These two works take only 19 minutes together. Am doing Berg memorial with Metropolis [Mitropoulos] General Motors December 30. Rhapsody which we played at Ravinia would have first New York performance.\n\nJoseph Szigeti\n\n204. David Oppenheim to Leonard Bernstein\n\n4 October 1945\n\nDear Len,\n\nJust a few interestin' facts about some interestin' people we know.\n\nI saw Seymour [Meyerson] in Frisco. Both he and the town are fine. I like him very much. We had a good time roller-coasting etc. [...] That we missed you was obvious in the way the conversation kept turning back to you. He is a great guy, Len. An amazing combination of artiness and common everydayness, plus a healthy objective interest in people and things that make him nice to be with.\n\nAnd the town is wonderful. I like the hills, water, fog, crooked angling streets, cable cars and the variety of good bars and restaurants. Lots of music too. If my aunt weren't here in L.A. I would go to S.F. on these 3 day passes. But I enjoy even more than S.F. a place to stay and vegetate. I hate the eternal search for rooms and walking lonely streets in search of companionship, that goes with being in a big town minus connections. And I could never get there enough to make any lasting ones.\n\nI thought that your _P.M._ interview was swell, esp. the statement on government sponsorship of music and art etc. But tell me more. How would it work out in relation to other private endeavors as regards comparative wages and as to who is used etc. It could be another political football. I am definitely interested in the idea though.\n\nThat is all! You don't deserve more.\n\nLove,\n\nDave\n\nTell Shirl that I will write, and soon.\n\n_Hello Helen!_\n\n205. Leonard Bernstein to David Oppenheim\n\n1239 Broadway, New York, NY\n\npostmark 8 October 1945\n\nDave Dollink,\n\nYou dope. I've been delaying writing you in hopes of receiving your new address. Your last letter said you were on your way \u2013 and here you are, exactly as before. Anyway, it's a pleasure to have word again.\n\nI'm awfully happy that you and Seymour [Meyerson] hit it off. He's a top-notch kid, and so are you, of course, so it's a natural. I did receive your little recordings, but they wouldn't make [i.e. work] on my machine. Finally (yesterday) I hit on a plan for making them go, and was finally able to distinguish some highly hysterical (or drunk) carrying-on about roller-coasters & trolleys. But you both talk simultanacklach, so most of it is lost to posterity. Excellent spirit, though.\n\nTonight's my big, big night.28 I'm a nervous wreck, but the orchestra is so fabulous and excited and young and interested and in tune and precise and enthusiastic, etc., etc., that if it's not a hit tonight I won't understand it. (Aaron's _Outdoor Overture_ , Shostakovich 1st [Symphony], Brahms 2nd [Symphony].) And the latter is a joy.\n\nYou really ought to hear it. Aaron was at the rehearsal this morning, and said \u2013 \"At last we have American Brahms \u2013 lyrical, like a popular song.\" I'm happy.\n\nI miss you like the devil. So what _is_ happening to you? Don't be so vague.\n\nEwige Liebe,\n\nLenny.\n\nYour remarks on government subsidy are to the point. It's a matter of reconciling this with the popular interpretation of the \"democratic method\". I'm thinking of agitating for a public referendum in NYC. Much love \u2013 and if you come across the article on me in _Collins'_ disregard half of it as pure invention.\n\n206. Rosalyn Tureck29 to Leonard Bernstein\n\n448 Riverside Drive, New York, NY\n\n5 November 1945\n\nDear Leonard,\n\nAt long last, here is your \"personality analysis\". I cannot take these things seriously but they are wonderful fun especially since the person who did it does not know to whom the doodling belongs.\n\nAccording to the analysis it looks as tho you must face the fact that you definitely fit into the genius category. As far as I am concerned from what I heard in your conducting the other night, you do.\n\nI hope to see you at the party after my recital on the 12th.\n\nRegards,\n\nRosalyn\n\n[Enclosed with this letter is the \"Personality Profile\" for Bernstein:]\n\nThis person's character shows a peculiar and great singleness of purpose. The sex development is practically nil and the personality which might have started to assert itself at one stage in the man's development has become completely absorbed by career.\n\nThe career is complex. Its division is almost geometric and the line of demarcation, very clear. For each phase of the career, there is a well thought-out and deliberate development. The dark areas indicate the creative and the white areas the mechanical. The mechanical seems to dominate the subject and he is more curious about the development of it at this stage than he is about his creative development. There is one point about the career, which seems to come early in the middle life, which indicates the great peak of success. The subject will have attained a very happy balance of creation and mechanics.\n\nThe sex symbol is interesting in that the line \u2013 the only line connecting it and the rest of the personality chart \u2013 extends right to the career symbol. This indicates that the subject's development is completely concentrated in his career. His personality symbol shows the same direction. There is no embellishment, no additions to it, there is no sign that any development of self has been accomplished. The sign connecting it with the career is merely two extensions from the sex symbol.\n\nIt is interesting to note that, in spite of the fact that the sex symbol is not developed as a physical unit, it is present and the aesthetic aspects of it will be found in this man's career creations later in life.\n\nThis man may not be a good mathematician, but he has an excellently organized mind. It is well disciplined as demonstrated by the complete lack of extraneous matter. It is also the mind of a purist.\n\nThis man has great ego-maniacal tendencies and will often go to bizarre ends to gain a point. By nature though, he is retiring and socially shy. His great ego, however, serves as a shield against society.\n\nA fruitful creative life is indicated, but an extremely lonely social life will be his lot.\n\n207. Joseph Szigeti to Leonard Bernstein\n\nPalos Verdes Estates, CA\n\n27 October 1945\n\nDear Lenny,\n\nI am delighted that I have at last succeeded in wresting [Harl] McDonald's30 permission to play the Bart\u00f3k pieces with you on March 4th and 5th \u2013 that is, a fortnight before my appearance with the Philadelphia Orchestra in Carnegie Hall. I am intensely looking forward to doing the Bart\u00f3ks with you; you know how much he means to me.\n\nI hope that Elkan Vogel of Philadelphia has already sent you the Bart\u00f3k _Portrait_ Opus Five [No. 1]. It was at Bart\u00f3k's express wish that I revived this forgotten piece of his in Budapest in 1939. He was present at the rehearsals and of course at the concert too, and admitted that this forgotten youthful work of his still meant a great deal to him. (I think it must have been written around 1905 or 6.)\n\nIn the score he gave me, he made a few slight retouches which I will show you in December when I am playing in New York with Mitropoulos. He cuts out the two trombones on page 13, and the first harp \u2013 i.e. one of the two harps \u2013 on page 15.\n\nAs to rehearsals, here is my schedule immediately before our concert: February 26th, Baltimore; 27th, Washington; March 1st afternoon, Baltimore, Peabody; March 3d Sunday afternoon, New York City, Frick Museum. So you can dispose of my time, say on Saturday afternoon, March 2d, and Monday and Tuesday morning.\n\nAll good wishes, and au revoir in December.\n\nYours ever,\n\nJ\u00f3ska\n\n208. Leonard Bernstein to David Oppenheim\n\n1239 Broadway, New York, NY\n\n11 November [1945]\n\nArmistice Day. Ha.\n\nDollink Dovidl,\n\nI'm a swine. I haven't written a word in almost a month. Which is not to say that I wasn't thrilled at this seeming prelude to a discharge \u2013 or is it? And which is not to say that I miss you like crazy, & wish you home, and in my magnificent orchestra. Of course there would be union difficulties, since I don't suppose you're in this local. That's a big headache. Same is true of Jesse Ehrlich, who is getting out next week. I have huge hopes & plans for next season \u2013 all my favorite guys in the orchestra, 24 weeks at least, a commercial radio sponsorship, which looks very likely, and a greatly increased Victor recording contract, which I've already got.\n\nSeymour [Meyerson] has _not_ been here, but seems to be arriving this week \u2013 just when I'm about to leave for a week in your favorite city of Rochester. I'm doing _Jeremiah_ there for the first time, with a mezzo soprano named Zelda Goodman, a student at the school. She's sweet, and it ought to be fun. Last week was Cincinnati \u2013 great triumph \u2013 and this week I had my 3rd pair with my _own_ \u2013 do you hear? \u2013 my own orchestra. They're the best yet. So much love for music makes them sound as no other organization. And I could still use you in it. And how.\n\nI've just returned from a run-through of Adolph & Betty's new show.31 It's a killer, a beast, and slumps only in regard to the score, which is fine but dullish, like so much Gould.32 If that boy had an iota of real _personal personality_ he'd be the best one around. But, alas.\n\nBe a good guy and don't copy my delay in writing. Aren't you ever coming back \u2013\n\nAll my love,\n\nL\n\n209. Ren\u00e9e Longy Miquelle to Leonard Bernstein\n\n814 Cathedral Street, Baltimore, MD\n\nJour de la Ste Ren\u00e9e, 12 November 1945\n\nCher P'tit Kietchka,\n\nHow to begin... I don't know! \u2013 because last week and now the next few days make my heart very full of you, for many \"anniversaries\" are now crowding in together with a very pleasant feeling of pride and well-being while basking in and relishing your musical doings.\n\nIn two days it will be two years that you had your \"big chance\" \u2013 as they say \u2013 and as all of us who know you _knew_ , you've kept right on.\n\nI need not gush and certainly do not want to, for my appreciation of your talents and all attendant accessories is more deeply and dignifyingly rooted than that.\n\nHowever, since it seems a total impossibility to get to you or see you unhurriedly and en \"t\u00eate \u00e0 t\u00eate amical\" I _must_ write and tell you how completely satisfying it is to know you, and to find you growing and maturing all the time; even though there is still more to be done and reached.\n\nMy second visit to your orchestra rehearsal last Monday pleased me far more than the first, mainly because you did so much that was really beautiful with the Mozart, and what is more you put it across.33\n\nYou know what my criticism of the last movement's tempo was (even though Helen said to me beamingly that it was a real Koussy clip, therefore marvelous) \u2013 I don't agree. Should that be a criticism? Koussy always takes _El Sal\u00f3n_ at a more deliberate pace than it is meant, or so I've heard, yet _you_ don't play it that way.\n\nOf course we all know what a temptation it is to play faster than necessary or slower in slow movements; but again, Lenny, _you_ of all people must not give in to that sort of thing. Your sense of tempi is usually so completely right, please don't lose your balance and throw us off ours. But, the rest of the Mozart was so good and so very elegant and nicely wrought that it was a real delight to hear.\n\nWhy did you not tell me that you are doing _La Cr\u00e9ation du monde_ at your next concert? Lucky I got the N.Y. paper yesterday for the first time in months!! So I'll be at rehearsal next Monday together with a young friend who cannot attend the concerts, since she works from 3 or 4 on till 11 p.m. (Please tell Helen to put me and Geraldine Viti on the list for admittance.)\n\nSomehow, Randall's symphony let me down. I had a more exalted recollection of it... very likely because it had been your first conductorial vehicle and I must have been more eyes, heart and anxiety than ears then.\n\nLast night, more nostalgia besieged me... you will know why when I tell you I went to see _Of Mice and Men_ again. Five years have elapsed; and it is just as new, strong and right as it was. It very likely will be so for many years. Aaron's music is so right for it too. It is a very poignant experience. Thanks for all time Lenny for having introduced me to Aaron's music.\n\nWhy did you not warn me and tell me Marc was back and not looking so well? I could not recognize him \u2013 a dreadful feeling \u2013 only when that marvelous smile of his burst forth did I know who it was... Oh! but what infinite sadness in that smile, darling \u2013 I was shocked. What's happened?\n\nWhen are you to conduct in Boston? And where will you be on November 23, 24, 25?\n\nAll my love,\n\nRen\u00e9e\n\nP.S. Please remember to bring your cuff-links to rehearsal on Monday. You know I want to have them engraved.\n\n210. Leonard Bernstein to David Oppenheim\n\nNew Hotel Jefferson, St. Louis, MO\n\npostmark 27 November 1945\n\nDave,\n\nIf it weren't for the fact that you'll probably be out of the army and living around the corner from me by the time I return to NYC, I should be all wails and drama at this moment. Such a teaser as last week with you was! It hit me most clearly as the train pulled out of Penn Station, and I realized that we hadn't even said hello. Maybe we'll never really have to say hello again. Wide-eyed idealism.\n\nHave yourself a time in New York, and be very good to Seymour [Meyerson]. You're so good together, and I like to think of you as reciprocally understanding, even complementary. Singly you're vastly superior guys. Together you're practically God. Let me hear from you in this bleak, foggy place, where, of all things, a charming southern thunderstorm is now raging. The streets are very dark and full of lonely faces. The hotel is very bright and full of lonely faces.\n\nMuch love,\n\nL\n\n211. Aaron Copland to Leonard Bernstein\n\nLimestone Road, Ridgefield, CT\n\nTuesday [?4 December 1945]\n\nMuy querido chatito,\n\nHere is your \"slight letter\". It's being written from the top of a bleak looking ridge on which sits perched a little house in which sits writing little me to you. I've been here since Sunday. Whether I stay the winter or not depends on whether I can get old Posh34 up and down the garage without falling off the cliff. The main assets to the house are 1) a nice work room with Steinway grand and 2) a lovely view out the Steinway window. But the ground is snow covered and makes me think longingly of Cuba.\n\nI missed Leinsdorf's concert35 \u2013 being here \u2013 and couldn't even get the broadcast. But last week I was in Cambridge listening to Faur\u00e9 for two days36 \u2013 and discussing tangled Tanglewood problems with Judd & Koussie. (Did you know that Rodzinski offered his place for sale in the _Times_ \u2013 for $40,000? Maybe you'd like to buy it \u2013 in which case I could rent a room from you at $6 per week. Otherwise maybe I'd better write to Mrs. MacSomething Furniss for a house.) Kouss empowered me as ass't problem unraveller to invite you on the faculty \u2013 _and_ \u2013 aside from _Grimes_ being at his exercise, there is a good possibility \u2013 says Kouss \u2013 that he will invite you to conduct a Festival concert of your own. Maybe you'd like to dash off the text for a pageant for the Music & Culture people, which also seems to be on Kouss' mind. Well \u2013 sounds like a busy summer. (Stravinsky says maybe he'll come, but no definite answer until Jan 15.)\n\nI've tried to imagine what the [ _Sinfonia_ ] _India_ and _Sal\u00f3n_ sound like juxtaposed \u2013 but the mind rebels (Antheil in his book says I composed the _Sal\u00f3n_ in the Hollywood-Franklin Hotel, which he recommended to me. How do you like that for hanging on to the skirts of fame?) Anyway St. Louis seems to have accorded the familiar L.B. triumph.\n\nDave O[ppenheim] came to see me. He's a sweetheart. And so are you \u2013 (but for very different reasons).\n\nL[ove],\n\nA\n\nP.S. My phone no. Ridgefield 637 \u2013 Ring one three.\n\nP.P.S. Thanks to you I'm now a member of the Baldwin family.\n\nP.P.P.S. I'm writing a Symphony \u2013 just in case you forgot.37\n\n212. Seymour Meyerson38 to Leonard Bernstein\n\nCamp John T. Knight, Oakland, CA\n\n5 December 1945\n\nThat you, Lennie!\n\nMy return to the Oakland Army Base was heralded by the collective \"Did you have a good time?\" and made official when everyone in our section walked me over to the PX39 for gooey ice-cream sundaes and lots of cakes.\n\nFlying time was approximately 22 hours, with far too many waits for re-fuelling and what seemed like stupid and pointless conversations between our pilots and the Commanding Officers of the different fields. Navy goes in for a helluva lot of tradition which makes Army routine seem much more sensible simply by contrast.\n\nAt any rate, Dave [Oppenheim] and I have solved our transportation problem, and will probably return the same way. [...]\n\nThe night Helen [Coates] invited me to see _Der Rosenkavalier_ with her at the Met, we had dinner at the Damascus Gardens, a small Armenian restaurant on 32nd Street. The conversation as we ate was highly personal, and I tried to avoid a good many of her questions. The thing that continually surprised me was not how much she knew about your sex life (in itself kind of \"shocking\") but how she accepted it, and sought to discover what satisfactory arrangements could be made for you in order to [be] assured that your career would not suffer. You can imagine how perplexed and embarrassed I felt, but since she was so frank I thought it was all right to listen, and also to see just how far her knowledge ran. The greatest shock for her was the idea that you would one day marry and have a family. The slightest mention of this idea caused her to tremble. I was very much caught by the look in her eyes, that expression which inferred an end to her way of life with you, should some \"other\" woman enter your life. Maybe that's why she can afford to be so tolerant towards your perversity?\n\nSeeing David this coming Saturday. Do you think February will really come?\n\nMuch love,\n\nSeymour\n\n213. Leonard Bernstein to David Oppenheim\n\n1239 Broadway, New York, NY\n\npostmark 18 December 1945\n\nDovidl,\n\nNu? The last I heard was from Seymour \u2013 that he was in doubt as to whether you ever reached camp by 8:00 a.m. after your bout with a stray damozel in a S.F. apartment. Watch it, baby: no point in fucking up your chances for immediate discharge. And I hear you are now being groomed for Wall Street.\n\nI am collapsed at the moment with a tough concert tonight,40 and very little energy. Christmas will be nothing but sleep, eat and you know. My sun lamp helps somewhat.\n\nSt. Louis was a joy. What a _La Mer_! And an immaculate 5th Brandenburg.\n\nGreat love to you.\n\nCome back.\n\nL\n\n214. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nRidgefield, CT\n\npostmark 31 December 1945\n\nDear R\u00e9lami,\n\nYour invitation came just a little bit too late for me to change my commitments. Funny you should have asked me and Aaron, because here I am at his sweet snowbound little country house, one hour from New York, and without that slightest indication that New York exists even one hundred miles from here. Anyway it was sweet as hell of you to ask us, and I hope you have a real good rest. To say nothing of a very happy and prosperous Nouvelle Ann\u00e9e.\n\nWhat do you think of our friend Hendl?41 Were you there by any chance? I was out of town conducting, unfortunately, and missed the excitement, but I'm sure it was swell.\n\nWhen does Claude get back?\n\nAaron sends his best, and I my love.\n\nSpook\n\n215. Leonard Bernstein to David Oppenheim\n\n1239 Broadway, New York, NY\n\npostmark 19 January 1946\n\nDear Davrelink,\n\nThere's a limit to this silence routine. You _are_ supposed to be out, you know. And then comes word from Seymour [Meyerson] that you're very much _in_ , and that he's just seen you in your new layout. What, may I ask, is giving? What is the Hotel Vanderbilt?42 Was your letter written from there implying that you lived there? Are you still dully employed in financeering? Of course I wouldn't dream of anyone else's playing the Klarinetten piece \u2013 but I've had to assign it tentatively to Hoffman, since you said you'd be out in January and here it is almost February. Tell all, and very quickly since time wastes fast, and anyway I miss hearing from you.\n\nYour letter was wonderful but so N!43 I couldn't answer sooner because I've been tearing around the globe again \u2013 but now it's serious. Let me know.\n\nI've been in Rochester a second time (playing the piano too) and seeing no sign of orchestral resentment. Who told you that anyway? They love you there, and your licorice44 teacher even expected you might play Ea on our tour in March. On the other hand Bill Schuman loved you too. (You really impressed him, you charm-monger.) But Juilliard is overcrowded to the bursting point. What will you do? Tell all, my love.\n\nLove, my love,\n\nL\n\nI leave Wednesday for Cincinnati, so write immediately. My love to Seymour & San Fran.\n\n216. Jerome Robbins to Leonard Bernstein\n\nLate Monday night [21 January 1946]45\n\nLen,\n\nIt was a really wonderful concert tonight. I'd never heard you really play anything but _Fancy_ and _Town_ \u2013 & the Bach was quite an exciting experience. Then the Stravinsky was new to me \u2013 & God! What an experience. 3 Bravos for that alone. I rode along on the _Don Juan_ nicely anticipating the Variations \u2013 & then I sat & chuckled & gurgled & beamed & nodded & emphasized & had a wonderful time. They sounded marvelously \u2013 & the only complaint was a little something on the encore of Harold's dance, trying to picture him keeping up with it. But it sounded wonderfully unsaddled by dancers.\n\nSo thank you for a very special evening of music. Good luck & continuous success.\n\nJerry\n\n217. Joseph Szigeti to Leonard Bernstein\n\nThe King Cotton [Hotel], Greensboro, NC\n\n2 February 1946\n\nDear Lennie,\n\nI was so sorry not to have been able to call you last week but the only time I had was on the day of your _Symphonie de Psaumes_ and I didn't want to bother you _then_! I listened in on Monday and was greatly impressed by your performance.\n\nAs to the order of the two Bart\u00f3k pieces I know it would be more orthodox to play _Portrait_ and then _Rhapsody_ but I have the inescapable feeling that the reverse order would be more _right_! The sturdy, \"typical\" Bart\u00f3k first and then this unexpected \"horizontal\" piece... Especially as it is in a way a \"memorial\", the transcendent ending of _Portrait_ seems more appropriate than the brusque (and not very effective) ending of _Rhapsody_.\n\nIn haste, all good things to you!\n\nYours ever,\n\nJ\u00f3ska\n\n218. Leonard Bernstein to Carlos Moseley46\n\n1239 Broadway, New York, NY\n\n23 February 1946\n\nDear Carlos,\n\nThis Prague thing is very exciting indeed: and if I guess correctly, my profound gratitude is due one Moseley.\n\nThe programs have been tentatively settled, & cabled to Prague, as follows:\n\nI.\n\nSchuman \u2013 Am. Fest. Overture\n\nHarris \u2013 Symph. #3\n\nGershwin \u2013 Rhapsody (Eugene List)\n\n\u2014\u2014\n\nMe \u2013 Jeremiah Symph.\n\nII.\n\nRandall Thompson \u2013 Symph. #2\n\n\u2014\u2014\n\npseudo-Czech group:\n\nDvo\u0159\u00e1k \u2013 Husitska Overture\n\nBart\u00f3k \u2013 Rhapsody #1, Portrait in D \u2013 Szigeti\n\n\u2014\u2014\n\nBarber \u2013 Essay #1\n\nCopland \u2013 El Saloon\n\nI think they're swell programs, and I hope you agree. Will we see each other soon?\n\nWhen do I go in order to rehearse sufficiently?\n\nAre new injections required?\n\nDo send me details.\n\nAffectionately,\n\nLenny Bernstein\n\n219. Paul Feigay47 to Leonard Bernstein\n\n137\u2013145 West 48th Street, New York, NY\n\n17 April 1946\n\nDear Mr. Bernstein,\n\nAs you know the business of _On The Town_ on the road has been very disappointing everywhere in spite of the terrific notices. Up to date we have personally lost over $50,000.00, in getting the show ready for the road and the losses on the road. The first week in Chicago the loss was $6,500.00, and then last week we lost $1,600.00, and that was due to the fact that we did not charge off any but cash bills.\n\nWe must appeal to you for help. Full royalties have been paid with the exception of the last few weeks in New York. We ask that royalties be waived retroactive to the opening of the run in Chicago and until such time as we start again operating at a profit and royalties should be paid out of each week's profit up until such payments equal 80% of the royalties due.\n\nUnless we can secure urgent and immediate cooperation from all persons receiving royalties we will be forced to close the run in Chicago immediately, regardless of the fact that business is on the upswing there.\n\nA copy of this has been sent to each person receiving royalties. Please sign the enclosed letter under \"agreed to\" and return it to me as soon as possible.\n\nVery truly yours,\n\nPaul Feigay\n\n220. Leonard Bernstein to David Oppenheim\n\nOrly Field Airport, Paris, France\n\npostmark 7 May 1946\n\nDearest Dave (Dearer than Crockett, Diamond, Jones, the King, Glazer),48\n\nAt this very moment life is a horror. I developed a stiff neck and a stinking cold during my first day in Paris. It would have disappeared, but there's no rest. One spends most of one's days in ATC offices, bureaus, Embassies, and most of one's nights driving to and from the airports. The flying racket is grand, but always involves the wrong times of day.\n\nAll this notwithstanding, the city is so fantastically beautiful that one cannot but be excited. The French are very depressed, and as they say here, on vive tr\u00e8s mal. But my flight to Prague has just been called \u2013 so bless you & all my love,\n\nL\n\n221. Leonard Bernstein to Helen Coates\n\nPrague, Czechoslovakia\n\n9 May 1946\n\nDear Helen,\n\nThings are beginning to pick up. Thank God! So far all the wonderful things Europe holds have offered themselves to me as dim visions, due to the fact that I caught a monstrous cold in Paris on my very first day there, and it's still with me. I've had horrible stiff muscles and aches, and sinus blowups. But now it begins to abate, and I'm beginning to be able to receive all this fabulous wonder.\n\nThis is the greatest day in Czech history. As you remember one year ago on May 5th, with Patton's army 20 miles away and the Russians at the East door, the people of Prague made a revolution against the Nazis. They just couldn't wait. The next day they were liberated by the Red Army. So this whole week is festival \u2013 the first anniversary of liberty. And are they celebrating, as no American would ever dare to do. Outside in the streets the whole town is dancing \u2013 to miked-up records of boogie-woogie and Strauss waltzes! People have come from all the provinces \u2013 Moravia, Slovakia \u2013 in their heavenly national peasant costumes, and the gaiety is beyond description. This morning there was a great parade and celebration in the huge Masaryk stadium, where generals of all the Allies spoke, including the great Konev, McNarney, and chiefs of staff from France, England, Yugoslavia, etc. It was a super-colossal demonstration, with tanks, planes, and the works. Last night there were fireworks on the Moldau, and up in the great Hrad\u010dany Castle. It is the only place on earth to be this week.\n\nOf course, the people look on the Russians as their liberators, but all the Hearst talk of the Red Terror here and the iron grip of Russia is nonsense.49 The Czechs are free as much as men can be, with joy in their reconstruction. It is rather in Paris where the spirit is way down, where the elections bore no fruit, where everyone is pessimistic and wretched (I think probably as much from guilt at their self-defeat as from _la vie dure_ ). The Czechs are happy and look to the future. They are the sweetest people on earth, and I'm going to have a marvelous week.\n\nMy love goes to everyone \u2013 please give it to them, and let this letter go to them all. There's so little time to write. More later.\n\nLove,\n\nL\n\n222. Leonard Bernstein to Shirley Bernstein\n\nHyde Park Hotel, London, England\n\n9 June 1946\n\nDarling,\n\nIt's all a mess. I didn't want to go to England. The plane trip was ghastly and a full day overdue, always stopping to fix the foolish crate. The hotel is dreary beyond description. The food is inedible, what there is of it. The English are very down, except for Victory Day (yesterday) when 10,000,000 people went berserk in London. In an ogrish way. (Why do I always hit the parades?)\n\nAnd worst of all, I'm stuck with horrible programs. I can't fix them \u2013 it's too late. All Ford-Hour stuff, masses of Wagner excerpts, with and without Marjorie Lawrence, and waltzes & polonaises by the score. The one help is _Appalachian Spring_. What a dream of a piece.\n\nI have rarely felt so lonely. I don't really know why, but I react to everything with big, soggy depressions. And H[elen] C[oates] is no help there. How I regret not bringing you instead.\n\nAre you a stage-manager yet? What is the state of your maidenhood? A letter from you would help a lot. Soon, please.\n\nFirst rehearsal tomorrow morning. First concert the day after. ( _Jeremiah_ , of course, is out.) If I hold up through this I'll be extremely grateful. My love to all around. And many kisses to you. I miss you terribly.\n\nL\n\nDon't forget to phone my best to the family.\n\n223. Leonard Bernstein to David Oppenheim\n\nHyde Park Hotel, London, England\n\npostmark 14 June 1946\n\nTo the Royal Husbandman,\n\nBuilder of the House,\n\nDecorator of the interior,\n\nDefender of the Faith:\n\nGREETING.\n\nThis is the dullest yet. Of course, I hit Victory Day again, with parades, illuminations (fireworks to us), and 10,000,000 mad folk releasing their repressions in a \"frightfully gay\" holiday. Now it's over, and it's still dull. Crowds hanging around Buckingham Palace, waiting for Royalty to appear. The sun came out for twenty minutes today, and everyone is grateful.\n\nI'm not happy. The programs are a mess, and there's nothing I can do to change them. I'm tired, usually depressed, and have little if any clarity of mind. I sleep when I have nothing pressing, and try to ignore the dreariness of this hotel and all of London.\n\nI envy you in the excitement of building up your new home. Let me hear about it. This was the time you _were_ going to write, remember?\n\nAll my love,\n\nLenny\n\n224. David Oppenheim to Leonard Bernstein\n\nTuesday a.m., Intermission\n\n[June 1946]\n\nLenushka,\n\nThis is the time I _did_ write. Remember!\n\nThe news here is good. We have had much success in your absence. Felicia [Montealegre] has a lead role in _Swan Song_ , the Ben Hecht\u2013C. MacArthur affair and seems to be doing well in it considering she's a nervophysical wreck and can't swallow food any more, what with 3 days notice on her role. But H[elen] Hayes came up to her dressing room to tell her how much she liked her (I was there) and how much she liked her clothes \u2013 so I guess she will live on that for a while. I didn't even know it was Hayes until she had gone. She calls herself Mrs. MacArthur.50 Who am I to know? She's not a girl any more.\n\nAs for your favorite schizophrenic (looks wrong) he has played for Laszlo H[alasz]51 & will continue to do so for the duration of next season. I signed something or other Monday. You must have primed Fallioni like crazy. He welcomed me into his office like a lost brother\u2013old sweetheart combined \u2013 gave me the parts to about seven operas \u2013 told me to call him \"when ready & if I needed advice.\"\n\nAt the audition I played _Carmen_ & _Traviata_ & _Butterfly_. I wasn't great \u2013 but I was OK, I guess. After I had played the _Traviata_ solo (which I loathe) Laszlo said there was no question about my tone now for some notes. Then the beginning and about 3 pages of Butterballs52 & I'm sticking to him like glue thru a million Puccini rubatos.\n\nHe then conceded that I had \"mastered the instrument. Have you the courage to play 1st?\" To which I replied modestly \u2013 with much Frauentruth \u2013 that I didn't have the courage to play anything else. So he said to Fallioni \u2013 \"Good boy\" \u2013 & that's it.\n\nFallioni says over again \"I think this will make Lenny very happy, eh.\"\n\nThe house I live in now, not mine to own. We accept each other, with all our faults.\n\nI have made a down payment of $5.00 on a small 5\u20326\" porcelain bathtub which when I raise the sufficient capital \u2013 $13.00 \u2013 I will install in my bathroom.\n\nCheer up friend Lenny. Soon you will be back with those who love you and in the [...] Berkshires. Not long from now. We miss you and I miss you.\n\nLove,\n\nDave\n\nBest to Helen\n\n225. Leonard Bernstein to Shirley Bernstein\n\nHyde Park Hotel, London, England\n\n17 June 1946\n\nDarling,\n\nYour letter came this morning \"bringing hope & cheer.\" As a matter of fact, I've been feeling infinitely better since writing you last, when skies were really greymalkin. I had my _L\u014dondonshein_ d\u00e9but yesterday, and apparently it was a huge success, though the _Times_ critic is still back in 1905, worrying about the question of a baton or no baton. Incredible country. The state of music is a shambles, the programs are embarrassing, the standard of performance abysmal.\n\nI am constantly saying, if you were only here! What fun all this nonsense would be! I saw _three_ shambles in a row that would have thrown you. First, a concert conducted by a wildman named de Sabata, who makes Mitrop[oulos] look like a sissy. He beats his head and jumps in the air & the bloody British public screams in delight. Then, the theatre: _The First Gentleman_ with wonderful Robert Morley & heavenly Wendy Hiller, but a play to make one wince (long scenes of dying in childbirth), and the British public screams. Then, the ballet. That was the end. Three ballets in a row at Covent Garden, one more lamentable than the last. And corn! And no imagination, and the audience screamed. So much for the British public.\n\nPeter Lawrence is here, to be followed by the whole crowd. Wait til Nora Kaye gets brought here by two liveried footmen! That will be the take [talk] of the season.\n\nGod, I should have brought you. I miss you [...] and love you so much. But, as you say, you're here with me, and people often catch me giggling to myself, when I am making Rybernian conversation with you on the latest British idiocy.\n\n[...]\n\nAll my love \u2013 be well \u2013 and somehow that Schirmer deal smells bad. I hope it all works out.\n\nIf Kouss lets me, I may stay on here till July 4th to conduct opening night _Fancy Free_ at Covent Garden. I'll let you know.\n\nBless you,\n\nL\n\n226. Leonard Bernstein to Serge Koussevitzky\n\nHyde Park Hotel, London, England\n\n22 June 1946\n\nDearest Serge Alexandrovich,\n\nI have wanted to write you every day since your generous permission to stay an extra week in England arrived by cable \u2013 but I have not been able because I _could_ not decide to stay and give up even one week of Tanglewood. You know so well what that paradise means to me \u2013 there are only six weeks in all, and every one is so important. Besides, I did not want to upset your plans if anything depended on me in the first week. But two things have finally made me take the decision to stay. First, I have had a serious throat infection this week, and had to cancel one concert (Leinsdorf substituted for me!) \u2013 and I need a week's rest in the countryside to get strong and well again for the great work this summer.\n\nThe second reason is that I am to make records here on July 1st,53 and since that is so important, I suppose I must remain, and then conduct my ballet on the 4th. But I promise, when I return I will feel healthier, and be able to work harder; and I _will_ take the _first_ plane after July 4th!\n\nMy concerts are all over here, and they have been very successful, although the programs could not be materially altered. For example, last night was my final concert, in sold-out Albert Hall (over 5,000 people in the audience) \u2013 but the program! Handel's _Water Music_ , Grieg's Piano Concerto (with a bad pianist named Eileen Joyce) and Tchaikovsky's Fifth. At least I had a chance to play a major symphony, and it was a great experience to play the Fifth for the first time. Some people in the audience came back and told me that it was better than _Nikisch!_ (That, of course, meant it was in the Koussevitzky tradition, but not so good.) Isn't it strange how in this small world all the lines of history and destiny come around eventually in perfect circles? I think that gives me more hope than anything else.\n\n_Appalachian Spring_ has had a great success here. I have played it also in the provinces, where they love it, almost more than in London! But the greatest joke is that the _Times_ called me [a] \"real Wagnerian conductor\" after my _Tristan_ and _G\u00f6tterd\u00e4mmerung_ with Marjorie Lawrence! I had never done _G\u00f6tterd\u00e4mmerung_ before, and my whole Wagner repertoire has been almost nothing! So much for the critics.\n\nMy greatest love to you, and all my blessings for the greatest Tanglewood season so far, and for many, many more. I cannot wait until I join you there.\n\nDevotedly,\n\nLeonard\n\n_Kisses for Olga_\n\n227. Leonard Bernstein to David Oppenheim\n\nHyde Park Hotel, London, England\n\npostmark 24 June 1946\n\nDear Solo Cl[arine]t,\n\nFelicitations and bravo, and all the best for the best of seasons with the best of orchestras under the \u2013 what am I saying? I'm very happy. Don't rub it in about Fabbioni. Everything's impartial as a chessmatch. Can I help it if we're both white knights?\n\nAnd kiss Felicia from me. I think it's swell, but I hear _Swan Song_ isn't so swell. Does it last? And what of Gloucester?54 I wish I could see her in it.\n\nAs for bathtubs, I could put two of yours into the one I have here, but I'd give anything to be in yours instead. England has begun to pall. I gave my final concert last night (great success) to a sold-out Albert Hall. Tchai[kovsky]'s 5th. Quite an experience, first time.\n\nI've just come through a strep throat. Canceled one concert. Penicillin worked miracles. Now I'm off to Glyndebourne for the week, to see Britten, & the rehearsals of his new opera _The Rape of Lucrece_ ,55 and to rest in the country. Maybe a side jaunt to Brighton (beim-sea).\n\nThen back here to record the old Ravel Concerto, with crack boys (including [Reginald] Kell, Brahan,56 etc.).\n\nThen to conduct opening night of the Ballet Theatre _Fancy Free_ July 4th. Then home. (Kouss has sent special permission to let me miss first week in Tanglewood, which I loathe to do. But no can help. ( _British ink! Diluted shit!_ )57 When do you go up? Where will you live? Is it all set with the Union? I hope you haven't forgotten (or delayed) to act on that.\n\nAlso (not also, but _also_ , Deutsch) I will be joining you in the Boiks [Berkshires] around July 8th. I shall miss that opening week terribly. To say nothing of missing you. But hold on. I'm coming.\n\nI've been to a Sadler's Wells party tonight. What a bloody bore. And I've had a siege of singing \"Barney Google\"58 with big Andrews Sisters codas.\n\n... with his goo, goo, googly eyes \u2013\n\n_I mean_ \u2013\n\n_Goo\u2013 goo\u2013 goo\u2013 gle\u2013 y\u2013_\n\n_eye.....s!!!_\n\nAll my love,\n\nL\n\n228. Felicia Montealegre59 to Leonard Bernstein\n\n[Summer 1946]\n\nLennie dear!\n\nI'm in the middle of a rehearsal \u2013 I'll try my best to talk to you between cues \u2013 my God what a life! I don't get time to eat any more. I'm playing Raina in _Arms and the Man_ which we will open with the first of July [...]\n\nI was so happy to receive your \"note\" \u2013 I was just a little hurt at your not saying good bye \u2013 you see dear, even though I know you are terribly busy and \"confused\" I still halfheartedly hope you'll remember my existence without me forever reminding you.\n\nAre you happy in London \u2013 do you like it? I hope you finally caught up with your sleep, that you're rested and enjoying everything as much as you can. [...]\n\nWhen I see you again (I wonder how long it'll be before I do) I must have a long talk with you. I've been thinking ( _actually!_ ) and there's a lot \u2013 but a lot \u2013 I want to say. I'll probably have to get Helen Coates to make an appointment for me \u2013 but I intend to have my say! Oh darling, you can be _so_ silly sometimes \u2013 life isn't that serious, honest it isn't! I know I shouldn't take some things too seriously, specially where you are concerned, as for example I haven't even looked at another man since I met you (well, perhaps one or two \u2013 but that's all!) and I'm not exactly beating my head against the wall \u2013 I'm training myself just beautifully, but I must confess that it's rather difficult sometimes!\n\nI never found out how long you were to stay in England. Do you think you might find time to write me again \u2013 a postcard maybe? I'll be leaving for Gloucester on the 17th \u2013 the address is: Bass Rocks Summer Theatre, Gloucester, Mass.\n\nGoodbye darling \u2013 please be good and by the way, why don't you marry Helen off to a retired English Colonel who'll take her off to live in Sussex \u2013 well, it's a good idea anyway!\n\nPip pip, old boy. I do love you rather. A kiss to you with a whiff of K.Y.!\n\n[Felicia]\n\n229. Felicia Montealegre to Leonard Bernstein\n\n[Summer 1946]\n\nDarling!\n\nIt's nearly three but I can't go to sleep \u2013 I'm feeling particularly lonely tonight; it's rather dreadful getting home late after the show, opening the door, putting on the light and being faced with the most acute aloneness \u2013 but then as you once remarked, New York is full of people like me (no consolation at all!).\n\nI should be very happy \u2013 and, of course, there are moments when I am. This job has been a wonderful stroke of luck for me, and a grand beginning.60 It's such a satisfaction to be actually _working_ and not just studying and preparing \u2013 you know \u2013 it's feeling that you belong at last to the \"something\" you were striving for.\n\nI miss you so much. I wonder why an ocean in between should make such a difference but somehow it does \u2013 there's something so irrevocable about it. I talked to Shirley today and I was told you had postponed your return \u2013 well, at least it means you're staying for something worthwhile. I felt so badly when I heard how miserable and disappointed you were.61 Cheer up my love and think of Tanglewood, you'll be happy there I know!\n\nIt's hot as hell \u2013 mierda! mierda! David [Oppenheim] was here this afternoon \u2013 his usual wonderful refreshing self \u2013 I was soundly kissed in your name, merci monsieur! Twas nice... The same to you \u2013 many of them, with love.\n\nFelicia\n\n230. Samuel Barber to Leonard Bernstein\n\nCapricorn, Mount Kisco, NY\n\n[August 1946]\n\nDear Lenny,\n\nI return the Berlioz with thanks. It is all I had to wear in Boston. Whether it was due to _Peter Grimes_ , the Benedictine or the very pleasant evening I spent with you, I do not know: but on arriving at 625 Park, clutching the Berlioz, I allowed the taxi to drive off with tuxedo, diamond studs et al, and nothing has been returned. Perhaps I should wear khaki after all.\n\nEnclosed also my favorite critic which will be good for your ego, especially two \"wiederums\".\n\nSam\n\n231. Leonard Bernstein to Barbara Marcuse\n\n\"Wednesday, but which one?\" [4 September 1946]\n\nMy dear Barbara and m\u00e9nage,\n\nNothing, absolutely nothing, could please me more than a shot at Charlevoix [northern Michigan] the Beautiful. But Charlevoix the Unattainable it must be. The Six Weeks were over, true, leaving a gray and bewildered and _Grimes_ -weary Bernstein to plunge headlong into a new ballet for production this fall. (Ballet Theatre, of course, and J. Robbins). Roughly two weeks of mad note-jostling, and it's not finished yet, and tomorrow it's back to the fiery furnace (32 West 10th St., by the way, a 4-flight walk-up! The times!) and on to opening night in two weeks.62 So that's me at the moment, as usual, and goodbye to all this autumnal glory in the Berkshires, and even to the chance of seeing you. But don't you ever come to New York? You really must, you know. A few mad hours by plane, that's all. And I do miss you \u2013 there always arrives that moment when I recall the quiet security I sometimes borrowed from the Marcuse \"set-up\". Alas.\n\nShirley is \"between shows\", as they say. Dave [Oppenheim] is on the verge of becoming 1st clarinet in my orchestra; England was ghastly; Tanglewood was hectic but rewarding; I have _four weeks_ with the Boston Symphony this winter; and I never said that S[hostakovich]'s 9th was a bore. Tell Phil to take his favorite mag and send it back to Russia.\n\nAll my love,\n\nLenny (Hruba) Nonadjustable\n\nI think I'm going to Palestine, Vienna, Paris, and maybe (shh!) Russia in the spring!\n\nBlessings on Ann and small Ph.63\n\n232. Leonard Bernstein to Serge Koussevitzky\n\n32 West 10th Street, New York, NY\n\n6 October 1946\n\nDearest Serge Alexandrovich,\n\nIt sometimes seems that the courage necessary for living simply and clearly in this world is all but a superhuman quality. This has been a week of shadows and misunderstandings which has left me tired and not a little depressed. There has been this incredible muddle, to begin with, over the appearance of my orchestra in Boston. Everyone involved has apparently used subterfuge in one sense or another, leaving me exposed and guilty of an offense I have never committed. I have finally extracted a promise from Miss Canterbury, which she assured me would be put into writing to you, me, and Mr. [George] Judd, that all mention of my name would henceforth be left out of all her publicity. She will also enclose a slip in her program material stating that I will not appear in November. They are trying to negotiate for another conductor (possibly Stokowski). In any case, it will be a mess; but I am rid of it, I think. I have written to Adams, stating that I will not appear in Boston. What more can I do? The Canterbury woman has been promising since August that she will cancel the engagement, only to proceed with publicity and newspaper advertising behind our backs. Her excuse is that her own Board of Directors will not allow her to break the engagement, no matter how much she _personally_ would like to.\n\nWell, _enfin_ , whom does one trust in this world? I don't know what to believe of all these commercial people; but when you told me that _you_ had doubted _me_ , I was really grief-stricken. If there is no trust between us, dearest Serge Alexandrovich, there is no trust in the world! I believe with all my heart in our bond and in our beautiful relationship; and I am sure that you do. Something so strongly based and real in its love cannot be injured by the meddling of these meaningless interferers!\n\nTo make things even sadder, I have had a miserable weekend with poor David Diamond, who has more or less cracked up emotionally. He has always been subject to hysterical actions, but this time it was a real collapse, brought on by the death of a girl who was very dear to him. I truly think that we shall have to convince him to have proper treatment and care in a qualified psychiatric hospital; otherwise there is only a bad end ahead for this talented and affectionate boy who has lost control of his emotional processes.\n\nLet us talk about happier things. I have been trying to revise our Boston programs to include the Bart\u00f3k work; but the only place where it will fit would be along with the Mahler. Also, since Ruth Posselt must play in New York also, we cannot bring either the Bart\u00f3k or Mahler to New York. Is it definite that Ruth plays in New York? If so, the programs might look like this:\n\nBoston and New York\n\nI. Gluck \u2013 Alceste \nStravinsky \u2013 Le Sacre \nSchubert \u2013 [Symphony] #7\n\nII. Bernstein \u2013 Facsimile (ballet) \n(or an American overture) \nHindemith \u2013 [Violin] Concerto (Posselt) \nBeethoven \u2013 [Symphony] #7\n\nBoston only\n\nIII. Bart\u00f3k \u2013 Music for strings, etc. \nMahler \u2013 [Symphony] #7\n\nBart\u00f3k and Mahler is a very heavy combination, I think, and our old plan of a Mozart Symphony with the Mahler is certainly better; but this was the only place I could find for the Bart\u00f3k. What do you think? I would also leave my ballet flexible until we can hear it performed this month. I have been looking through the Mahler score, which has marvelous things in it, and is also very long. (I have not been able to find out the exact timing.) I would still love to do it, if you wish. Would it be interesting to give the _Shostakovich #7_?\n\nThe concerts here are going marvelously, and it is only this great activity, and all this heavenly music that keeps me going through a week such as I have had. We did the Mozart \"Linz\" Symphony last night, which more than atoned for the agonies of the weekend. It _is_ a marvel! Perhaps I can do that with the Mahler?\n\nI wait to hear from you, with my love and faith strong and intact as always.\n\nDevotedly,\n\nLeonard\n\n233. Leonard Bernstein to Solomon Braslavsky64\n\nNew York, NY\n\n10 October 1946\n\nDear Professor,\n\nI have been suddenly inspired to write you a note (if you can pardon the lack of \"a propos\"). Having listened through a Kol Nidre service, and again the next day to a Yom Kippur service in a completely different kind of synagogue, I have come to realize what a debt I really owe to you \u2013 personally \u2013 for the marvelous music at Mishkan Tefila services. They surpass any that I have ever heard; and the memories I have of them are so bright, strong, and dear, that I shall probably never be able to estimate the real influence those sounds exerted on me. And please tell Cantor Glickstein that he is still my favorite cantor.\n\nAll good luck,\n\nLeonard Bernstein65\n\n234. Solomon Braslavsky to Leonard Bernstein\n\n133 Elm Hill Avenue, Roxbury, MA\n\n16 October 1946\n\nDear Lennie,\n\nI don't know how to thank you for your letter. After all, a note of praise from your pen means much not only for myself, but for the entire Mishkan Tefila. I, therefore, hope you will not scold me for making your letter public in the _Jewish Advocate_.66\n\nA propos (no lack of it on my side) the services at Mishkan Tefila. Another great musician attended our services the first day of Rosh Hashonoh, Dr. [Hugo] Leichentritt. He, too, never heard as beautiful a musical setting as here. [...] We, of course, know the reason, but why discussed here? I wish I could say it to the \"group\" in New York who condemn [Salomon] Sulzer, [Louis] Lewandovsky and others, and are advocating Gretchaninoff (!), Dessau, Milhaud, etc., as proponents of _real_ Jewish music. They would not even mention my name in the _Manuals_.\n\nI am trying to clear my decks for November 13 at Symphony Hall. I should like very much to show you my Symphony or, at least, the four short items of Synagogal music which are published now by the McLaughlin and Reilly Society.67 Will you be able to find a half hour for me during your stay in Boston? I hope, yes.\n\nMany thanks and best wishes for a real happy and successful New Year, with great achievements,\n\nS. Braslavsky\n\n235. Paul Wittgenstein68 to Leonard Bernstein\n\n310 Riverside Drive, New York, NY\n\n16 October 1946\n\nDear Mr. Bernstein,\n\nI wanted to say \"good-bye\" and \"thank you\" after the concert yesterday.69 But when Beethoven and the tremendous applause was finished and I wanted to go to the stage and see you, I found the door closed.\n\nI can only repeat what I have already said: the Ravel concerto is almost as difficult for the orchestra as it is for the soloist, and a success depends upon the conductor as well as upon me, therefore should be divided 50 and 50! I know this out of experience and appreciate with thanks what you have been doing.\n\nThe performance of the concerto on Monday was, I think, really excellent! Please let the fact that it wasn't quite as good on Tuesday not spoil your remembering of it!\n\nDon't reply to this letter.\n\nSincerely & friendly,\n\nYours,\n\nPaul Wittgenstein\n\n236. Leonard Bernstein to Serge Koussevitzky\n\n32 West 10th Street, New York, NY\n\n[25 October 1946]70\n\nDearest Serge Alexandrovich,\n\nI don't know quite how to say this, but I am in the position of having to come to Boston. We have tried everything \u2013 Reiner, Stokowski, [Jos\u00e9] Iturbi (lesser names they do not accept) \u2013 [Manuel] Rosenthal (they never heard of him) \u2013 Morton Gould (he is what they call \"box-office poison\" in Boston) \u2013 everything has failed. We even concocted an idea to send a concert of operatic excerpts with our friend [Laszlo] Halasz, but he would do it only if I took over _Onegin_ , which I am not up to at this moment. It would be disastrous to do _Onegin_ without sufficient preparation; and knowing the situation at the Center as I do, I cannot accept that responsibility. Especially since the premiere is scheduled for the very next night after the Boston visit!\n\nI know that you will understand, better than anyone on earth, what my situation is, and what my obligations are. There is no alternative left; and I do it with a heavy heart, I assure you. I can only hope that the entire incident will be soon forgotten; for it has been the most difficult moment of my short career.71 The last thing in the world I ever expected to do was to do something against your wish; and here Fate has plunged me into this _impasse_.\n\nI think I mentioned the proposed program to you:\n\nPurcell \u2013 Fantasy on One Note\n\nWalton \u2013 Portsmouth Point Overture\n\nBritten \u2013 Violin Concerto\n\n\u2014\u2014\n\nElgar \u2013 Enigma Variations\n\nI have thought of substituting Dvo\u0159\u00e1k's Second Symphony for the Elgar (I did it here this week, and fell in love with it.) If you have any objection to this, or to any other piece on the program, please let me know, and I will change it immediately.\n\nAll the reports of your performance of Aaron's Symphony are superb. I regret so much that I could not hear an actual performance; but the rehearsal that I did hear was a great experience.\n\nI had my ballet premiere last night, and all the reactions seem very favorable for the score. I think it would make a nice little concert piece; and I look forward with so much eagerness to hearing it with the Boston Symphony. It will be a real joy for me.\n\nAgain, dear Serge Alexandrovich, I know you will be sympathetic to my predicament with the City Symphony; and please, please forgive me for ever having caused such an unpleasant experience for you.\n\nIn all devotion,\n\nLeonard\n\n237. Lukas Foss72 to Leonard Bernstein\n\nSymphony Hall, Boston, MA\n\n9 November 1946\n\nDear Lenny,\n\nI received your invitation to participate in the B'nai B'rith contest. I am eager to do so. The only unperformed piece I have is _Song of Anguish_. Even though my neoclassic friends prefer my later biblical venture, I agree with Aaron who felt that _Song of Anguish_ is as good as anything I've done. I would love to see you and perhaps Tod Duncan (who has learned the part) premiere it in March, but there is one difficulty concerning my entering it in the contest: it lasts 18 minutes instead of the demanded 15. May I enter it anyway?\n\nGood luck with your Boston concert on the 13th. I am unfortunately in Baltimore on that day. But I have already bought tickets for _Facsimile_ on the 18th. The best to Helen.\n\nAs always,\n\nLukas\n\nP.S. As I now approach the mature age of 25 this contest is one of very few I can still have a crack at. It's funny not to be \"too young\" anymore but \"too old\" for many of these things. No more prodigy stuff. Actually I feel good about it.\n\nL. F.\n\n238. Leonard Bernstein to Shirley Bernstein\n\n22220 Saticoy Street, Canoga Park, CA\n\n7 December 1946\n\nMine deeoah,\n\n_Mine_ God! and Dine God! and I miss you terribly. This would, as usual, be your meat. But Felicia is really working out. It's a little early to tell anything, but I sense a possible future. How can I tell you in a letter? It must mostly wait. But I can send you signposts, sort of, as we go along.\n\nThis week has been Coma-week. Doing zero, absolutely. When the sun is out it's divine; otherwise freezauk\u00fcd\u00fc. Many fireplaces and scotch, and our own bungalow make it the luxury of all time. Beautiful horses (and can Felicia outride me!), and a pool, when possible.\n\nI've gotten a great idea on the picture and Lester seems sold on it. If it works it will be sensational \u2013 a Hollywood _first_ \u2013 and a really significant vehicle for my debut. The writers are dopes \u2013 but this has nothing to do with them.\n\nTwo recurrent refrains, which I shall not repeat, dominate this letter: _I miss you_ , and _Any signs of a job_? I pray for you. I'm sure you'd do marvelously out here, if we could get you a test. I'm sort of working on it.\n\nI hear _If The Shoe_ is a fiasco,73 goody two-shoes.\n\nAdolph, Betty & Adolph are doing fine, looking great, and send you love. They say Elizabeth Taylor at MGM is your type, but you're prettier.\n\nWrite me great gobs of stuff. We're taking a little Ford convertible to Mexico next week. Wish you could come. Shit.\n\nSunday there's a party here. Dinah Shore & George Montgomery & Buddy Rogers & Billy Wilder & Ad. & Betty & Ad., and so many more. Then Sunday night Rapper is threatening one with Gregory Peck & Lana [Turner] and everything. But that's all. I do nothing but rest. I'm tant fort and grand.\n\nM\u00fc la d\u00fc! and Felicia sends hers & will write soon. I miss you horribly. M\u00fc la d\u00fc,\n\nLadim\n\n239. Leonard Bernstein to Helen Coates\n\n12 December 1946\n\nDear H,\n\nThings are beginning to move again. These last two weeks have really done wonders; I have rarely felt so rested. Felicia and I have just returned from a little side trip through Tijuana, Mexico and Palm Springs \u2013 and it was a gorgeous little interlude. The desert is just perfect now, and the mountains beat everything. We ride and sleep and eat and gain weight and become brown.\n\nAs I say, things are beginning to move again. I've had an idea which may make this picture74 a phenomenal thing for Hollywood. It would involve me in four ways \u2013 composer, conductor, actor and writer. (I've already done a little writing on the screen play.) In other words, really my picture. If it's to be done at all, it must be begun now, while I'm here, so that there's a musical basis before I leave. Then I can finish it in June, assuming that the Soviet invitation doesn't go through, which I expect will be the case. Even if I have to dash back for two weeks after Tanglewood for final touches or retouches, it would be worth it. Financially it would be more than worth it, and artistically it will be a great satisfaction. I am beginning negotiations now, and hope to complete them soon. Of course, I will call Kouss first and try to avoid another hitch with him. I think you would be very excited about this idea.\n\nAs to the _Facsimile_ , I hope to have the ending in a very few days, along with program notes. When I send it, get it to [Arnold] Arnstein, and tell him to make parts. The cost should be stood by Harms, who would have to have scores & parts in their rental library anyway. (They have to pay _some_ copyist. And they'll rook me for it in the end, as usual. Ah well, the way of the world.) Cowan, by the way, has an idea on my publisher problem. We'll see.\n\nFelicia and I grow closer all the time. She's an angel, and a wonderful companion. I shouldn't be surprised if it worked out beautifully in the end.\n\nLove,\n\nL\n\n240. Leonard Bernstein to Helen Coates\n\n22220 Saticoy Street, Canoga Park, CA\n\n22 December 1946\n\nDear H,\n\nLife moves on apace. I'm toying with the notion of becoming engaged to Felicia. No marriage yet \u2013 she must stay here and do a movie contract, and I must travel. We think of it for June, and it's an exciting and somewhat confusing prospect, as you must imagine. But it's good \u2013 I know that, and there's no harm in trying. The pursuit of happiness, you know, is one of our few human rights. Listen to Winchell Sunday night. If I decide, he will probably announce it.\n\nYour notes on the _Facsimile_ ending are exactly right. The new ending replaces the old ballet ending after the long pause. I've sent it out, and I think it's good. I spoke to Kouss the other day, and he has me in circles. He wants me to marry immediately, he says movies are fine, \"but not now\", whatever \"now\" is, and he doesn't want me to play _Facsimile_! This after many calls & wires to Burk, and head-splitting figuring on broadcasts. I don't know what to say to him in argument \u2013 he's so self-contradictory. What a difficult relationship!\n\n[John N.] Burk75 told me you had a cold. Do watch out; everyone in the East seems to have one, Kouss, Blum, etc. Always on vacation one gets a cold! I have so far been spared. The climate makes me sleepy, but the horses and much sleep make up for it. I am now just ripe for a _real_ vacation \u2013 just beginning to be relaxed. Well \u2013 perfection never did exist anywhere.\n\nOn the movie \u2013 don't worry. I won't sign any contract until after I return. Negotiations are proceeding now, and I'm composing a little, and having story conferences \u2013 but that's all. I don't think it would hurt my conducting career, and it seems like my old friend \"fun\".\n\n[...]\n\nI'll call you from Cincinnati (next week already!) Meanwhile get well and rested. Have you been able to see your _friend_ at all during your rest-month?\n\nLove,\n\nL\n\n241. Serge Koussevitzky to Leonard Bernstein\n\nBoston Symphony Orchestra, Symphony Hall, Boston, MA\n\n23 December 1946\n\nDear Leonard,\n\nMy last talk over the telephone with you left a very disturbing impression. And these are the reasons:\n\nSpeaking of your programs you stubbornly insisted on the performance of your own composition, even for the broadcast. Do you realize that you are invited as a guest conductor, to show your capacity as interpreter of great musical works? May I ask you: do you think that your composition is worthy of the Boston Symphony Orchestra and the Boston organization? Can it be placed on the same level as Beethoven, Schubert, Brahms, Stravinsky, Prokofieff, Bart\u00f3k or Copland?... You may answer my question saying that I often perform also works of lesser value and scope. But you must not forget that I am the permanent conductor, that I stand at the head of this organization to further and develop the musical culture of this country, and, therefore, have the obligation to help young composers. Thus, my responsibilities are very different from yours, or other guest conductors.\n\nI also want you to know my reaction to your \"assurance\" that Thor Johnson was invited as conductor of the Cincinnati Orchestra, following your recommendation. Do you believe that the Directors of an old musical organization, such as the Cincinnati Symphony Orchestra, would be so na\u00efve, to consider the recommendation of even a very influential lady in the choice of a permanent conductor? Or do you believe that your influence is so great in this country that a word from you would be enough to bring about immediate decisions? If so you are profoundly mistaken. The Trustees of the Cincinnati Orchestra were exceptionally thoughtful and serious regarding this question. I do not want to go into details now, but can tell you that the engagement of Thor Johnson had nothing to do with your recommendation.\n\nI am writing in this direct manner because I consider it superfluous to talk to you as if you were a \"spoiled child\". You are fully grown up and have to realize that you are responsible for every word you say and all of your actions, especially responsible on account of your gifts and the position you are beginning to occupy.\n\nThink it over, and I hope you will understand the motive which dictates this letter.\n\nYours,\n\nSerge Koussevitzky\n\n242. Leonard Bernstein to Serge Koussevitzky\n\nCanoga Park, CA\n\n27 December 1946\n\nDear Serge Alexandrovich,\n\nI have been deeply grieved all day on account of your last letter. I immediately sent you a telegram, trying to explain the misunderstanding, but I canceled it, realizing that it was not a thorough clarification. I must write you instead, because of my love for you, and my need for you to understand.\n\nWhy do these misunderstandings happen? Is there an evil element in my nature that makes me do and say immoral things? Is it that I say one thing and mean another? Or is it that communication between two people who are as close to each other is so difficult? If so, then life is too difficult; something is missing in the human constitution.\n\nYou must realize that I never meant to suggest that Thor's appointment had anything to do with my recommendation. I was simply reporting an interesting coincidence \u2013 that I had been discussing him with Mrs. Wyman a short time before. And I was so happy to hear that it had come true! But certainly not through _my_ efforts.\n\nAnd you know I am happy to play only what you suggest and approve in my Boston concerts. Whenever I conduct in Boston I am conducting for _you_ , deep inside, and whatever I may do well is a tribute to you. My main concern is to make you proud of me, and justified in all your efforts for me. So when you asked me suddenly on the phone to take off my own piece, I was surprised, and merely questioned, why? Certainly I believe in my music, or else I would not have written it \u2013 not on a level with Beethoven and Bart\u00f3k, naturally, but in its own smaller terms. But if you feel it is wrong to play it, I will certainly follow what you say, and gladly.\n\nI have had a very difficult year trying to adjust myself to the conventions of my profession. The r\u00e9clame means absolutely nothing to me \u2013 in fact, it only complicates further my already complicated life. Managers, agents, public charm, the terrifying sense of competition in other conductors \u2013 the whole desperate race with time would be worth nothing if it were not for the magical joy of music itself. And this joy is bound up tightly with you, who are my only \"spiritus genitor\". That is why I become so depressed when misunderstandings come between us.\n\nForgive me, and\n\nBelieve me,\n\nLeonard\n\n243. Leonard Bernstein to Serge Koussevitzky\n\nMontgomery, OH\n\n2 January 1947\n\nDear Serge Alexandrovich,\n\nIn my excitement about answering your last letter I completely forgot to tell you the most exciting news of all \u2013 that I have become engaged to a wonderful girl from South America named Felicia Montealegre. We plan to be married in June, when I return from Europe. I am very happy, and I hope you will be.\n\nThis news seems to have leaked out into the papers already76 \u2013 and I did want to be the first to tell you, before you might read it. But this has been impossible, since I have been stranded for five days all over the Southwest due to bad flying weather. My plane kept stopping and stopping; so I spent one night in Texas, another in Arkansas, another sitting up on bad trains, and so on. But I am finally here in Cincinnati, tired but happy, and I have had my first two rehearsals today. The orchestra needs a great deal of work, but they respond well, and I hope for an exciting concert.\n\nI am staying with friends who have a wonderful big farm where I can rest a day or two before going to Chicago. Everyone here seems very satisfied with Thor [Johnson]'s appointment, including most of the orchestra men. It was a wonderful move on their part, and a difficult one to make because of the natural opposition to the appointment of a young American. It took courage, and I am sure Thor's success will justify the appointment in every way.77\n\n244. Farley Granger78 to Leonard Bernstein\n\nFriday, 3 [January] 1947\n\nDear Lenny,\n\nI just received your very sweet letter. It was wonderful to hear from you. Lenny, we all miss you very much. Ethel and Saul79 talk of you all the time and last night we played the _Jeremiah_ twice, after which we decided to build you a city.\n\nI am so glad we became friends, and I hope I see a lot more of you, though it seems no one gets that privilege for long.\n\nFelicia told us about your horrible New Year's. I wish you could have stayed. We all had such a wonderful time.\n\nI am having dinner with Felicia tonight. She is a great girl, and I'm sure loves you very much.\n\nYour letter was late because it went to the studio first.\n\nLenny thank you for being so kind, and please write when you can.\n\nRurally yours,\n\nFarley\n\n245. Leonore Goldstein80 to Leonard Bernstein\n\n1141 Hampton Park Drive, Richmond Heights, MO\n\n30 January 1947\n\nDear Leonard,\n\nThere are three reasons which impel me to send these lines to you \u2013 and I don't know which to name first \u2013 yes \u2013 I think I do \u2013 it's your engagement to be married.\n\nMrs. [Kate] Ratcliffe confirmed what I had read in the _Times_. I am never certain when such items appear in the daily press, but I do _believe_ Mrs. Ratcliffe so here are the very heartiest good wishes for your, and your sweetheart's, happiness \u2013 and may it continue to the end of time. I know she is lovely or you would not have chosen her.\n\nThanks over and over for the fine _Leonore III_ Overture. I sat here along with my best concert concentration and listened reverently and so admiringly for your really wonderful Beethoven VIIth. No one could have conducted it better: it was a delight and I wish I could have joined the Providence audience in applauding.\n\nI fear I shall not hear the next radio concert on Tuesday, for I shall be with relatives in New York and Heaven only knows what my granddaughter has planned for that evening.\n\nWill you conduct the Bostonians in New York? If so, I simply _must_ get into the hall \u2013 but how? I know the concerts are sold out. Has the conductor enough influence to melt Mr. Judd's icy heart and have him sell me a seat anywhere in the Hall? I shall be in New York from Feb 3 to Feb 17th and I am hoping that your New York concerts come during these dates.\n\nThat you should be conducting that great orchestra for a month is a real triumph \u2013 and you \u2013 a youngster \u2013 under thirty.\n\nMy thanks for the pleasure you gave last Tuesday, and much love from\n\nLeonore III\n\n246. Felicia Montealegre to Leonard Bernstein\n\n[Hollywood, CA]\n\nThursday night, 6 February [1947]\n\nLenny, my darling, my darling!\n\nFirst of all, today is my birthday \u2013 I am a quarter of a century old, a very frightening fact! It is also a year ago that I sat at your feet, a little drunk but terribly exhilarated at the Arraus'. All day I've been wanting to telephone you and it took a lot of self control not to do so \u2013 it seemed rather silly. However...\n\nJust came back from the Kellys [Gene and Betsy] where I saw \u2013 yes \u2013 _Of Mice & Men_ \u2013 I feel drained and weak, it's so _great_! It's difficult to believe that it came out of Hollywood. I kept thinking of you all through it and listening attentively to Aaron's fabulous score, and remembering the things you had mentioned and loved.\n\nThe biggest thing that has happened to me since you left is the most beautiful, warm, affectionate black cocker spaniel puppy in the world who is at this moment biting at my pen while I write. She's mine, my very own! I bought her a week ago having fallen in love at first sight (pedigree papers and all for $35). Stanley Donen81 christened her Nebish, which she isn't at all but the name stuck. I didn't know whether I'd be able to keep her at first (as a matter of fact it still isn't certain). I've gone through tortures because of David, who I am convinced now is verging on insanity. He decided to cross Joanie who was with me when I bought her and loved the idea of a puppy; he reacted in such a strange way and in a Captain Bligh fashion, told me I must \"get rid of her\" cause she would interfere with his _work!_ She's been such a good girl though (of course I'm constantly taking her out) and I haven't found anyone who'll take care of her for me until my life is settled \u2013 he hasn't brought the subject up again, but I'm living with a Damocles' sword precariously swinging over my head. I refuse to give her up \u2013 we've become inseparable. You'd love her \u2013 everybody does (except David of course). A propos, I could talk to you about him for hours \u2013 he really worries me. I think he's quite sick.\n\nOh sweetie \u2013 your concert! I finally heard it last Tuesday. It's useless to even try to tell you how magnificent it was \u2013 j'\u00e9tais tellement fi\u00e8re de toi! I'm dying to hear the next \u2013 it's so frustrating to know you've already played it and we don't hear it till next week!\n\nStanley gave me a party at his apartment last night \u2013 I didn't know it was for me until midnight when for no apparent reason the lights were turned off and in comes Gene with a lighted cake which Betsy baked herself, and everybody singing the \"happy, happy\". I fought back the usual tears but could have bawled for hours \u2013 aren't they wonderful? Tonight Joan had a special dinner for me \u2013 just us and Adolph [Green] and Allyn [Ann McLerie] \u2013 with another cake and more candles to blow and wishes granted. It was sweet, but I could have felt more cheerful. Sometimes I miss you so much I actually feel sick to my stomach.\n\nYour father gave me the surprise of my life \u2013 such a nice affectionate letter \u2013 I just loved him for it.\n\nThank you for the check. The great Montealegre career is at a complete standstill \u2013 I am seriously contemplating going back, defeated but healthy! The only trouble is though that I won't get a job in New York either! Oh _shit_.\n\nI will have you know for further reference that I have a learner's permit and will have a license next week. I drive alone all over the place, up hill and down dale, heavy traffic and all \u2013 and I'm _great_! So there!!\n\nWhat's with you? You never really tell me how you feel \u2013 is that so difficult? I don't believe it. Are you still terribly worried and depressed about things, or have you decided not to think about it at all. I keep wondering about you and wishing I could be with you \u2013 you always sound so happy and communicative over the telephone, but unfortunately your letters leave me \"con gusto a poco!\" I know, you warned me.\n\nWhen are you due back in New York \u2013 please write me soon, this place is bad enough without your making things worse.\n\nMy love to your family \u2013 of course it's good to be home again. I rather envy you. Where's Shirley and what are her plans?\n\nBoss darling \u2013 good night \u2013 much much love and many many kisses.\n\nFely\n\n247. Bette Davis to Leonard Bernstein\n\nButternut, Sugar Hill, Franconia, NH\n\n[?February 1947]82\n\nLennie,\n\nHow really thoughtful of you to write me about my attempt to be a pianist in _Deception_. From you it meant very much, as you can imagine. You were conducting in Boston the other day \u2013 and here I am with my project sitting not far away \u2013 and unable to come down and hear you. I would have so much loved it. You are doing wonderful things and I am always somewhere else. I read about it though and always the things said are complimentary \u2013 and of course I think that is as it should be. Your signed photograph is now on my wall \u2013 and a proud possession. It is really exciting about the baby \u2013 and so unbelievable to me \u2013 but at this point there is definite evidence so I must believe it.\n\nReally so many thanks for your letter \u2013 I beamed for days.\n\nLove,\n\nBette\n\n248. Leonard Bernstein to Helen Coates\n\nRochester, NY\n\n26 February 194783\n\nDear H,\n\nI. Testimonial for [Menahem] Pressler: _A genuine pianistic poet._\n\nII. The Palestine program, as I recall, is:\n\nSchumann #2\n\n_Jeremiah_\n\nRavel Concerto.\n\nIII. Brussels: Everything seems to be by Aaron or me! I would like to divide the program into three parts, for instance:\n\n_Movies_ : Copland (both suites) appr. 20 min.\n\n_Ballet_ : Fancy Free (appr. 25 min.)\n\n_Theatre_ : Gershwin\u2013Bennett Porgy (appr. 14\u2032)\n\nThis would make a nice program, but short (roughly an hour). If satisfactory, this should be it. Otherwise I'll have to cast about for another ballet & play _On the Town_ with _Porgy_. _Undertow_ is a possibility, but I don't like it. Furthermore, to be really representative about American film music, I _should_ play something like the _Spellbound_ score,84 but I can't stand that kind of Hollywood stuff. Which leaves only Aaron, which precludes a Copland ballet. Would you explain all this to the man and get his reaction?\n\nIV. The PCA85 business is cleared. All is set for March 25th, I believe.\n\nV. Diamond. I think someone should be put in charge of a fund \u2013 not us: we've had our share. Won't Margaret do it? Or, if not, Olga or Mrs. Hirschman. I think musicians should be tapped first; Rodzinsky, Szell, Monteux, Kouss, Dimitri, also Smallens etc. \u2013 all the conductors who have ever played D's music. Also composers: Alec Wilder, Hanson, Goddard Lieberson, etc., etc. I think _chief_ contributor should be Alice Berezowsky.86 I'm serious: she started it.\n\nVI. We are staying overnight in or around Vassar after the concert, leaving next day for Providence. I could go back to N.Y. with you if you can get me a flight to Prov[idence] next day. Although it makes a lot of extra traveling. Don't you think it's simpler to stay over?\n\nLove,\n\nL\n\n249. Adolph Green to Leonard Bernstein\n\n[Hollywood, CA]\n\n[February 1947]\n\nDollink Lennie,\n\nWhat is there to say?\n\nYou are brilliant, brash, you (28) \u2013 I am fat, old (49 \u00bd)87 and feeble. In short, what is there to say?\n\nBetty & I are Hollywood successes, it seems. _Good News_ is about to go before the camera, causing a minor revolution at M.G.M. No picture has ever before been done there without 4 years of preparation, 15 scripts, etc., etc.\n\nNow we are grappling with the horns of an Arthur Freed dilemma. We've been offered a second picture, at mightily increased salaries, and a plum it is by M.G.M. standards: a G-R-E-E-A-A-T-T SCREEN CREDIT. _Easter Holiday_88 \u2013 with J. Garland, G. Kelly, F. Sinatra, K. Grayson.89 All songs by Irving Berlin & we would have to work with Irving B. \u2013 naturally we won't stay in Hollywood but Freed has told us that he can arrange for us to write the bulk of it in N.Y. We are therefore tempted.\n\nMaybe we'll write this thing & be finished just in time for us to write a big, successful show together next winter.\n\nGlad that I didn't marry A.A.90 out here \u2013 double, double doubts \u2013 but _away with them!!_\n\nI feel completely sure now that I want to marry her, and I'm very anxious to get back and do it.91\n\nWill you be around at the end of March? We'll make it then. You must be on hand.\n\nLove to everyone,\n\nAdolph\n\nThere's an item in the Sunday _Times_ about the Hammerstein Rodgers show _Allegro_. Rodgers says \"it will combine dance, music and drama as an integrated unit\".92\n\nAbout time too. It's an unprecedented notion, very daring, never before attempted \u2013 Ahhhh! \u2013 shit, fuck, balls!!!\n\nP.S. I love you.\n\nOur associate producer, one Roger Edens,93 is a great fan of yours, but very displeased with your Boston program. He complains about that Beethoven & Schubert junk and feels it's ruining your career to have to play it. I'm just passing along this opinion for what it's worth. Stick to the moderns, he says.\n\n250. Leonard Bernstein to Ren\u00e9e Longy Miquelle\n\nLehigh Valley Railroad [on board train]\n\n3 March 1947\n\nCh\u00e8re Re-La-Mi [written in musical notation: D-A-E, i.e. Re-La-Mi],\n\nThinking of you on this ghastly train (ghastly pen!), even to the point of recalling your address. So many things made me think of you \u2013 a man from Baltimore next to me at lunch, _Homemade pies_ on the menu; Claude's appearance backstage in Symphony Hall \u2013 etc. etc.94\n\nI'm beginning a week's tour with the Rochester orch. and I'm up to my neck in penicillin. Overtired, I guess, and inevitable consequences. I'll live.\n\nWill I see you before I go to Europe April 9th?\n\nLove,\n\nSpooky\n\n251. Sid Ramin to Leonard Bernstein\n\n32 West 53rd Street, New York, NY\n\n[March 1947]\n\nDear Lenny,\n\nJust a note to congratulate you on your engagement and to wish you continued happiness and success. I guess I ought to apologize for not using your \"secret\" phone number and letting you know how I'm making out, but I guess you know me by now.\n\nIncidentally, your engagement came as quite a surprise (I don't know why it should) and now you've got me thinking about whether I ought to be next.95\n\nThe recordings I was so proud of several months ago, and wanted you to hear, have since lost a lot of the original appeal (as far as my wanting to play them for people). I guess that's a good sign, although I still think there are some good spots worth listening.\n\nHowever, if I have to bring some recording to use as an excuse to see you and your new phonograph, I think I can manage to pick a few better ones, twenty or thirty!!\n\nReally, Lenny, I'd like to see you and ask a bit of advice in regards to quite a few things that are popping here; also keep you posted on what the past year and a half in N.Y. has held for me. And then, of course, I'd like to see you without having the meet through lots of people, for a change. So...\n\nOf course, I don't know where this letter will find you, but when you get back to N.Y. (if you're away right now), how about a letter and an appointment?96\n\nBest regards to everyone,\n\nSid\n\n252. Leonard Bernstein to Serge Koussevitzky\n\nPark Hotel, Tel Aviv, Israel\n\n25 April 1947\n\nDear Serge Alexandrovich,\n\nIf you ever wanted to be involved in a historical moment, this is it.97 The people are remarkable; life goes on in spite of bombs, police, everything. There is a strength and devotion in these people that is formidable. They will never let their land be taken from them; they will die first. And the country is beautiful beyond description. It is a real tropical vacation for me, with the wonderful Mediterranean and the sweet, warm spring.\n\nThe orchestra is fine, and I am having a great success. [Charles] Munch has just finished his weeks here, and we finally met: he came to my rehearsal this morning, and was so excited that he wants to arrange a big concert for me in Paris. Tant mieux.\n\nI miss you, and can't wait to return to Tanglewood. Please don't be worried about me; the bombs fly, but the newspapers exaggerate.\n\nMy love to Olga, and to you my deepest devotion,\n\nLeonard\n\n253. Leonard Bernstein to Aaron Copland\n\nHotel Castiglione, Paris, France\n\n27 May 1947\n\nOld Charmer!\n\nIt's done. Fait. The Symphony's98 been heard. Two days ago in Prague.\n\nFirst I must say it's a wonderful work. Coming to know it so much better I find in it new lights and shades \u2013 and new faults. Sweetie, the end is a sin. You've got to change. Stop the presses! We must talk \u2013 about the whole last movement, in fact.\n\nThe reactions were mixed. Too long, said some. Too eclectic, said Shostakovich (he should talk!). It lacks a real Adagio, said Kubel\u00edk. Not up my street, said Wee Willie Walton. And everyone found Chaikovsky's Fifth in it, which only proves their inanity. I haven't seen the press yet, but I think it will be good. It just wasn't a wow, that's all; it was solid, it was serious. The orchestra was exhausted (end of the festival), and the rehearsals were nightmares. (We had six!) But at the concerts they played marvelously. Even to catching our private rubatos in the third movement\n\nwhich, by the way, is my favorite part. That's the real inspiration \u2013 the real Aaronchen. I could make out fine _anti_ cases for mov'ts I & II (and of course IV) but not III. That's my personal wow.\n\nBy the way, I do it awfully well, and I'd love to do it in the States. Maybe Tangle[wood] \u2013 well, maybe the City Center.\n\nThere is much to say. Letters are impossible. But won't you write me and tell about May and Harvard and the Virgil [Thomson] Opera and where you are and Koussie and Victor [Kraft] and everyone? And D[avid] D[iamond]?\n\nIf you write me to Holland I'll be sure to get it. I'm there June 8\u201313. Write now, and they'll hold it for me. G. de Koos, Noordeinde 62A, Den Haag, Holland.\n\nPalestine was a real thrill. More later. Will you be in NYC for my stadium week?\n\nLove, Love,\n\nL\n\n254. Aaron Copland to Leonard Bernstein\n\nMexico, D.F.\n\n4 June 1947\n\nYoung Charmer!\n\nJust received your forwarded Paris letter and I'm dashing a hasty reply on the chance that it will reach you in Holland. It was fun to read the various reactions to the Symph \u2013 including your own. I've decided that it's a tough job to write an almost 40 min. piece which is perfect throughout. That's about all I'll concede for the moment! You were an angel to struggle with rehearsals at the tail-end of a Festival. The part of your letter I liked best, of course, was your saying you'd like to do it zum States.\n\nDidn't I tell you I was coming to Mexico to conduct the Symph myself? I've had 3 rehearsals already and the concert is still 2 weeks away. My main trouble is giving cues for entrances. Well, anyhow it's very good experience and I'm getting a kick out of it. (Kouss said to me before I left \"If you ruin _my_ Symph I vil keel you.\")99\n\nAll my N.Y. news is probably stale for you by now. Virgil's opera100 was original-looking on the stage \u2013 no one has ever seen anything quite like it. But I thought there was more music in _Four Saints_ [ _in Three Acts_ ]. It's as if a new musical idea hadn't occurred to him in 10 years. The prosody, as per usual, is superb \u2013 but then it's easy to have good prosody if you have nothing else on your mind (I'm quoting myself).\n\nD[avid] D[iamond] looked much improved when last I saw him. I suppose Helen Kates told you of our financial crisis which was solved until August.\n\nBob Shaw did a bee-utiful job with my new chorus.101 Most people seemed to like it, but the press was only mildly interested. I can't imagine how you'll react to it. Any ho you won't have to conduct it \u2013 since there's nothing but voices. (I decided that Bob's conducting technique derives from the football cheerleader. Or did you say that already?)\n\nI was in Cuba the night you played the Symph in Prague. Mexico seems so naively serious by comparison. I see _Jeremiah_ on display here in the record shops. Ch\u00e1vez spoke of asking you to come to conduct a week in August. Did he wire you? And just before getting your letter I was talking about you (I seem to be always talking about you!) with de Spirito102 and Carrington103 at lunch. You'll be glad to hear that San Juan de Letran still thrives and that I live one block away.\n\nI'll be at the Stadium concerts. And thanks Gawd for Tanglewood so's we can talk \u2013 finally. I've lectures all prepared for you about your City Center programs \u2013 completely disinterested since I leave for Brazil on August 14. Your ex-mentor sends you an abrazo muy fuerte \u2013\n\nA.\n\n255. Lena Horne104 to Leonard Bernstein\n\n16 July 1947\n\nDear Leonard Bernstein,\n\nI would very much like to have you as my guest at a party at Jerome Robbins' home, 421 Park Ave., on Thursday night July 24th from 8:00 to 1:00 (corner of 56th Street).105\n\nWe will share delightful drinks and entertainment for the benefit of the fighting veterans organization, United Negro and Allied Veterans of America.106\n\nYours sincerely,\n\nLena Horne\n\n256. Marketa Morris to Leonard Bernstein\n\nContinental House, Stamford, NY\n\n23 July 1947\n\nLenny,\n\nYour letter stirred up lots of problems.\n\nTo go into them adequately would require an elaborate paper \u2013 and that does not agree with my vacations. I try a compromise. I have to be honest in the first place. Honest and short means usually: it hurts! I have to rely on your perspicacity and your English to translate my thoughts into a good, nice, considerate English. Will you?\n\nI don't think that our work will be finished in five months. But there is even some risk of your feeling worse after this period since many problems may have come into the open without finding a solution.\n\nUnder the given circumstances I would want to start only if you are taking the responsibility for such a possible outcome.\n\nOf course there is a chance that we may come to some essential clarification. No way to deny it. It's fifty fifty \u2013 and you have to know it.\n\nIn your dreams there is confusion, you are not able to go where you have to go: two _simultaneous engagements_ or dates and so on. You are seeing Felicia and the day she leaves you _have_ to see a boy.\n\nThe same old pattern. You can't give up. Very eager to resume analysis but the queer fish resistance is as big a fish as your drive to get well.\n\nIf you could give up Europe for the solution of your problems, you would have solved quite some of them and we had the most promising start. But would I make it a condition, which, I have to confess was very tempting \u2013 I am sure it wouldn't work out, since you would use it against me, that is, against our work.\n\nI don't quite understand your dream involving your parents. What does Rochester mean? Did you intend to leave alone by plane (which you missed!). Could it indicate that it is a \"force majeur\" and not your own incapacity that you can't separate yourself from them. Being inside still a child as you say (giving up childhood).\n\nYou are toying around with the possibility of being a dull and uninteresting talent \u2013 or losing your place in the score (Koussevitzky).\n\nRemember that you wanted to challenge people and find out whether they would still love you.\n\nIt's all very sketchy, I know. But I still hope that you can pick out something of help for you.\n\nI did intend to go to Tanglewood, indeed, but it did not materialize.\n\nI had an interesting letter from George today who himself is going to the Berkshires.\n\nLenny, I hope very much that you understand what I really want to convey to you! Do you?\n\nI am back in town between the 5\u201310th of Sept.\n\nSincerely.\n\nM\n\n257. Richard Adams Romney (\"Twig\")107 to Leonard Bernstein\n\n34 Beekman Place, New York, NY\n\n25 July 1947\n\nMy dear Lenny,\n\nI can't resist writing you, even though I know that the mood I am in should prompt me to be still. I feel sad \u2013 and alone in the way only a neurotic can feel alone.\n\nThe idea of living with someone else came crashing down around my knees last night just as hurriedly as it had spent itself in the sky the day before. The thought of having to be responsible to an irresponsible degree for someone else's living condition makes me balk like I have heard men do just before they take a wife. Living in this little box has a security that is that of a desperate grasping squeeze... I'm reminded of the Steig drawing of the man in the box who thinks \"people are no damn good\"! I tripped over a stone yesterday. This was it: the nearer my time came to report to the VA108 for my chance for psycho-analytical treatment, the more tense I noticed myself to be. I translated it as a natural resistance \u2013 for certainly, beyond the chance of talking about myself, hashing over my imaginary reflections, and being an \"actress\", there is a deep conviction that my locked doors must not be opened. The fear of falling with my faults is funny \u2013 for I believe that by uncovering one recovers, yet recognition has not healed me of the failings lately (like Christian Science, for instance, has led me to believe). Why, when one sees an error of premise doesn't that seeing dispel its tenacity? Is it the old self-authority commanding \"thus far and no farther\"?\n\nWell, I sweat through the interview, with the best manners I could muster \u2013 by that I mean without personal messiness, and found I have to go through another interview \u2013 also not an MD or Psychiatrist \u2013 which will determine whether I am in worth[y] condition to take the time of the Psychiatrist. Then it is probable that psychiatric survey and not psycho-analytical treatment will be granted. So \u2013 with all my resistance to being analysed, I am depressingly disappointed that a possibility of being treated is quite improbable. Another contradiction! Perhaps I should take a part-time job and see if I can earn my analysis \u2013 even on through my school-time-days to come. (The analyst Bruce Knight is arranging for me to see is named Berkeley \u2013 and lives near you on 11th Street, I believe. Have you ever heard of her? I wish she wasn't a her.)\n\nHelen wrote me a lovely thank you card \u2013 and it delighted me a good deal. She mentioned your great success with _Appalachian Spring_ \u2013 but I knew of it the day before by my own conviction. I went into Liberty and listened to the Koussevitzky recordings and was disappointed at his interpretation. There is a lovely excitement in the way you play the \"saddle\" part \u2013 is it Part III? \u2013 Koussy rushes it, instead of syncopating it \u2013 and the way you end it is moving. Oh, butch, you're wonderful!\n\nThis morning's book list from Holliday advertises Auden's new thing along with a new one of Edith Sitwell's. When I get round to send you [ _The Age of_ ] _Anxiety_ I will also include _The Shadow of Cain_ which the book store insists \"reflects more directly the tragic impact of contemporary events on an acutely sensitive and perceptive nature.\" Then they add, \"In their likeness and unlikeness, these two books are an absorbing study.\"\n\nI have been to Lewisohn [Stadium] twice this week. Hans Schwieger is the poorest conductor I have ever heard with a first rate orchestra. He conducts as if he were leading a hofbrau band.[...] The Philharmonic must realize that many people take to concerts in the summer because of the outdoor \"tranquility\", and it is their initiation to the best music \u2013 therefore they could increase their winter subscription lists with new enthusiasts, but there is a vital danger in sandwiching the very best with the slip-shod. Evidently you set the standard for their summer concerts extremely high. I have heard from many diversified tastes that your concerts were electric. Mr. Schwieger has been a sad let-down.\n\nDo I sound mean, Pappie. Hating people again?\n\nYou set me such a fine example of living with other people, I am more aware than ever of my anti-social side. Please don't let my untidy sick mind bore you away from me (I want you to feel a teensy bit responsible so that you won't give me up as unworthy of your good affection). You've touched me deeply \u2013 honestly you have.\n\nI'm going to close \u2013 hoping you can scribble a card when you're squatting in that hammock. Kisses to Helen. (Gobbles to you.)\n\nTwig\n\nTell Helen to get _The Gallery_109 from her lending library (it's already out of stock).\n\nWhy don't you try a tone poem of _Anxiety_?110 The four themes \u2013 their inter-relationship, pairing-off drama \u2013 etc. might make a good thing. And you could do it! Name it _T_ he _W_ anderer _I_ n _G_ reenwood after me (forgive that!) [...]\n\n258. Richard Adams Romney (\"Twig\") to Leonard Bernstein\n\n34 Beekman Place, New York, NY\n\n29 July 1947\n\nDear Lenny,\n\nForgive all of the overheatedness in my past letters, but know it is only because I know you don't want me to be sillily overboard that I ask your forgiveness. You made me very very happy in those few days, and I have had to try to know what it was about your example and good heart that made me wish I could get right against you.\n\nHere is a snap of me taken on the wharf of Hamilton, Bermuda that I hope you will look at when you forget what I look like. I think it quite good.\n\nWhat do you think of the _Anxiety_ idea? There is so much musical-subtlety in it, and those various metres brought about by the different roads the couples take and their differing means of transportation, to say nothing of the moods, and the separateness that becomes oneness under alcohol and\/or libidinal urges. You mentioned it being good ballet material, yes, but I think, first, it should be composed as music by itself and therefore protect it from being too obvious program music, and then if some clever choreographer can put the musical composition to work, with what added quality good music may give to the themes and material, well and good.111 I would rather have \"it\" in the concert hall, where it can be less \"handled\" than in the ballet school where many talents brush it up. It's too good a thing for many hands.\n\nI am beginning to suffer more for lack of occupation, but until the VA makes some kind of a decision regarding their award of psycho treatment, I don't think I can obligate my time just yet. Do I ring self-excused and lazy? Hummm. If the VA vetoes treatment, I am determined to work it out on my own but taking temporary work during my school days, and immediately before school begins.\n\nThe _Tribune_ had a high compliment of your first night.112 I wish I could have seen and heard the performance. I read it at a friend's house \u2013 and will get you a copy if you weren't able to grab it in Pittsfield.\n\nSpeaking of my friend, she is Anne Gibson Clark \u2013 and she is toying with the idea of going home to Grand Rapids to pick up her convertible, and then taking me to Tanglewood one week-end to hear some more music. What you do think of that?\n\nKiss Helen for me and tell her I will write her the letter I want to, in answer to her friendly card, very soon. She couldn't have been kinder to me and I thank her every time I think of her \u2013 which is every day.\n\nLots of love,\n\nTwigling\n\n259. Leonard Bernstein to \"Twig\" Romney\n\nBox 102, Lenox, MA\n\n1 August 1947\n\nDear Twig, fighting to the last,\n\nThis should be five or six letters by now. There are so many things to say, and the super-varied contents of your three letters call for all kinds of discussion. But I'm not in my \"analyst\" mood right now, having an uncomfortable back condition these days, and having just finished a long and difficult lecture. So let me just not be \"Pappy\" now, and send you my love and thanks for all the three books (which I keep trying to find a minute to crack) and to tell you how often I think of you.\n\nThe concerts here have been tops; and mine have given me the utmost satisfaction. The reactions have been marvelous, and Kouss was ecstatic. I wish you could have been here: \u2013 can't you get up for my next one on Aug 7th?\n\nI don't quite understand the rise and fall of Bruce Knight. What _really_ happened? And have you met Miss Berkeley?\n\nI have to dash off for a diathermy on my poor aching back. Write more and often, and better still, come (though I can't promise you space here for a while) and I love your photograph.\n\nBless you, and don't let your resistance interfere with going through with the analysis.\n\nAll the affection you need \u2013\n\nL\n\nThe enclosed sheds much light!113\n\n260. Marketa Morris to Leonard Bernstein\n\nContinental House, Stamford, NY\n\n28 August 1947\n\nLenny,\n\nLet me be very brief. I feel in your letter that some part in you expects my support for the cancellation of Palestine! That you dare not to see it, but that you would want to do something completely radical \u2013 for your Resurrection!\n\nThe only thing you can do: try to feel whether that is what _you_ want. Not what I want!\n\nPlease call me up after Labor day (or even before) best between 9\u201312 a.m. [on] 4751 because I would want to arrange for our first session in New York. Will you? It has technical reasons.\n\nI don't quite understand why you were pleased not to feel the necessity to thank me for my time? No obligation for conventional feelings? That's okay? But how about some genuine, warm feeling of gratefulness? Could you imagine?\n\nIn N.Y. I refused to take a brother in law of a patient of mine who wanted to come to St[amford] and have daily sessions by saying: not even if he would pay $25 a session.\n\nI am mentioning it deliberately to show you that it is not only you who has to give up and make sacrifices \u2013 but that I am willing as well to do so, if necessary. I even proposed to see you once more, if you would have wished. I understand perfectly that you did not since you were so busy making so important decisions.\n\nWe'll talk about it more in N.Y.\n\nHave a nice time \u2013 a productive time first of all!\n\nYours,\n\nMarketa\n\nFor the sake of order: I am charging (since last year) $10 a session. We had one in N.Y. and 3 in Stamford.\n\nYou had no dream? How come after all these important events?\n\nM\n\n261. Marketa Morris to Leonard Bernstein\n\n562 West 113th Street, New York, NY\n\n[?1947]\n\nLenny,\n\nI got your dream letter. You know that it is quite impossible to give a written interpretation to a dream \u2013 and more so a dream without interpretation.\n\nWhy am I living in Brooklyn?\n\nJimmy's Restaurant in Greenwich V[illage]\n\nWhy another cab to go to Brooklyn?\n\nWhat's about 289?\n\nIt's getting dark at four o'clock in the afternoon?\n\nSwitches putting on lights _upstairs_ and not _downstairs_? What's the difference between up and downstairs in this beautiful, big, expensive house?\n\nWhat about the two girls blocking the exit from behind your desk?\n\nWrite me if you feel like \u2013 besides the dreams! F[or] i[nstance] why cannot you relax and just simply _not_ compose? Remember, _you had_ the idea that adjustment to homosexuality could facilitate heterosexuality! Couldn't adjustment to relaxation constitute a capacity of creative work? Of course not _pretending_ to relax only.\n\nMarketa\n\nI could see you Monday at 12 (noon) or at 7 p.m. Tuesday at 11 a.m. OK?\n\n262. James M. Cain114 to Leonard Bernstein\n\n666 South Carondelet Street, Los Angeles, CA\n\n1 October 1947\n\nDear Mr. Bernstein,\n\nTwo proposals have been made to me, one by a leading playwright and a reputable producer, the other by the most successful operetta composer we have, hooked up with a highly successful librettist, to put _Serenade_ on the stage;115 but I am bound to report that in spite of a high personal regard for all of these various gentlemen, all I could detect in their ideas was the most obvious theatrical claptrap, and accordingly I did my best to discourage them. _Serenade_ , unfortunately, as seems to be the case with most of my stories, has problems that don't yield to a socko waltz tune, and I am not sure they yield at all. You, though, might be able to get somewhere with it. I mean, I have followed your work & think it might suit your gifts.\n\nAs for my doing your libretto, I can only say I never did one, and have a suspicion that at my age I shouldn't try to learn.116 The rights, for your information, are in the clear, that is the dramatic rights; I own them, and while the publisher cuts in for 25% of anything paid on account of performances, I make the deal, and naturally would be reasonable. My suggestion would be to get in the market for a poet, or poetess, and I would think that the _New Yorker_ , which is in touch with every poet there is, may be of some help, if you were to write them, or better still go in there. Katherine White, I understand, is still with them, and could no doubt think of somebody. After that you are in the lap of chance, but no worse with a _New Yorker_ nominee than with me.\n\nTo elucidate the rights thing, which on re-reading doesn't seem wholly clear. A contract leasing you the right to produce an opera based on the book would be one thing, and would be made by me, either with you or with your producer. Royalties paid me would be split with Alfred A. Knopf, Inc., my publisher. Your libretto & score would be another contract, made by producers, publishers, etc., with you and your librettist, lyricist, etc. If I did the book, I would be involved in it, but as I hesitate, that complication most likely won't arise.\n\nI should naturally be delighted if you undertake the job, and wish you all luck with it. It has a _theme_ , as the picture people found out, as horrendous as the Motif of Sulphur Yellow Truth in Mencken's concert program; but no doubt you know all about that, and let us hope, what to do with it, or how to get rid of it.117 Many thanks for your felicitations. The lady,118 as you may have heard, has yodeled quite a bit of opera herself.\n\nSincerely,\n\nJ. M. Cain\n\n263. Leonard Bernstein to James M. Cain\n\n9 October 1947\n\nDear Mr. Cain,\n\nThank you very much for your kind letter of October 1 and for your kind comments about me and my gifts. I am happy to learn that the rights are clear, but to be frank, I am not reconciled to the fact that you would not be able to write the libretto from your work. However, I will follow your suggestion and look about for a collaborator though at this writing I have some ideas myself as to the book and lyrics.\n\nUnfortunately, I will not be able to give the matter my entire time immediately although the first act is fully formed in my mind both dramatically and musically. I will be occupied with my conducting here until at least the end of the year and then I have foreign commitments to conduct, which will consume the first months of the coming year. However, then I shall have free time and nothing will delight me more than to concentrate on the work.\n\nAt the moment I have no producer and therefore I am proceeding on my own insofar as the work is concerned and what I would like to have from you is the right to dramatize the book for musical purposes until September, 1948, which I believe will give me ample time to finish what I have in mind.\n\nI understand that in the event of our adopting a writer of the book and a lyric writer, we will have to arrange for royalties to be divided, among all of us.\n\nMy sincerest thanks for your kindness and encouragement to proceed in the matter and if you will be good enough to send me a little note regarding the rights, I shall be grateful. It is difficult at long range to discuss these matters from a practical standpoint or try to get together on the book and lyrics and on more technical business, but I believe we understand each other thoroughly and I assure you I shall give the matter my sincerest cooperation.\n\nSincerely yours,\n\nLeonard Bernstein\n\n264. Leonard Bernstein to James M. Cain\n\n32 West 10th Street, New York, NY\n\n8 November 1947\n\nDear Mr. Cain,\n\nI liked your letter very much, not only because of your encouraging words, but also because it was a real Cain special.\n\nI am not completely convinced that you're not my librettist; but, of course, that has to be your decision. May I hope you will keep thinking about this possibility?\n\nWhat I would like from you right now is an option on the rights to dramatize the book which would extend to December 31, 1948. I think this would give me enough time to make real inroads into the work. So much of this coming year is to be taken up by conducting engagements, here and abroad, that I feel a good year is necessary. I can let you know fairly soon about the producer arrangements.\n\nIf you'll be good enough to send me a note regarding these rights, I shall be very grateful. I am sure then that we can get together via attorney or what-have-you on the more technical business.\n\nDon't you ever come east? I do wish we could have a chance to talk about it. For instance, do you belong to the school that believes in the complete deletion of the theme, as you call it, for operatic purposes, or do you agree with me that it could be handled intelligently and without offense?\n\nMany thanks, and best wishes.\n\nSincerely,\n\nLeonard Bernstein\n\n265. James M. Cain to Leonard Bernstein\n\n666 South Carondelet Street, Los Angeles, CA\n\n2 December 1947\n\nDear Mr. Bernstein,\n\nA thousand pardons for not answering your note sooner, and forwarding the reservation requested, but I have been down under the ice winding up a novel, and I know of nothing that claims so much of one's time, and leaves so little juice for anything, except possibly, but not in no way probably, the tying of one's shoes.\n\nThe enclosed letter will do it, I think.119 I haven't consulted lawyers or agents, as they commonly scream for a quid pro quo, as I don't think we are to that stage yet, or that it is even much of a stage, financially speaking, with regard to anything operatic. However, I have had a good many things of the same kind before, and think this will do it, if it covers what you are concerned about.\n\nAbout the libretto: I try to picture myself pulling this off, but have no faintest notion how to go about it, either to block it out by scenes, or what kind of writing to put in it, or anything. I still think you need a poet, and still think Katherine White of the _New Yorker_ should steer you towards a suitable one. She knows every poet in the world, together with how much he drinks before dinner, whether he does it before breakfast, whether he can write iambic hexameters or free verse, and all relevant things. I used to know her quite pleasantly and think she would like to cooperate on that account, as well as being terrifically impressed by you. No, I have no objection to the damned theme, but think as a practical matter it is the most unsuitable to stage use, though it has been tried now and then with no great objection by the police. I merely think it is unpleasant. But if it were used symbolically, a sort of inverted Faust idea, with addiction to _man_ standing for cerebral, cold, and sterile things, and _woman_ pulsing with all those hot, life-giving elements, it might do. Personally, I still say you'd do a hell of a sight better to put the time in on Aaron Burr120 or someone like that. There's a good book out on him, by the way, by Holmes Alexander.\n\nWill be in New York, in any event, around 28th or 30th of this month, or shortly thereafter, & will ring you then. Until then, all luck with it, and I confess the greatest curiosity as to what in the name of God you have in mind.\n\nYours,\n\nJ. M. Cain\n\n266. Leonard Bernstein to Helen Coates\n\nHotel Castiglione, Paris, France\n\n3 May 1948\n\nDear H,\n\nThis is it \u2013 le jour de d\u00e9part to be carried off by the Orient Express into the wilderness of Europe. Nobody knows what will happen. Everyone secretly expects war, but is afraid to say so to himself. It is obvious here that the entire war-plan is being instigated and manufactured in America. There is no need for it \u2013 but I have a horrible feeling that America will have its way. God forbid.\n\nThese last days in Paris have been pure heaven. I haven't begun to see everyone I wanted to see \u2013 but I've spent much time with Ellen [Adler] (who sends you all her love) and Fran\u00e7ois [Val\u00e9ry] and my wonderful shipboard roommate, and my darling Comtesse Marie-Blanche [de Polignac] (and her big Sunday evening last night to which I took Ellen, and where I sang them the _Bonne Cuisine_ ) and Nadia [Boulanger] and Otis Bigelow. It's been exciting and warm, & Paris is breathtaking.\n\nThe traveling plans became complicated at one point \u2013 (Milan to Budapest) \u2013 of course \u2013 because a train would take me through Yugoslav territory, & no Yugoslav permit is faintly possible. I finally found a way to take a train to Zurich, then fly (the next day) to Prague, then to Budapest. This means a Czech permit, & four hours of waiting around the Czech Consulate, plus many grey & pink and mauve cards for the Russian Zone, where I may still be held. But I've made good friends at the American Embassy and they've helped a lot. At least I'll have a night in Switzerland.\n\nThe big news is that I may have _two_ concerts in Paris \u2013 the other with the Conservatoire (and Neveu) on the 30th! Sudden cancellation. I'll know in a few days. It would make an awful scramble (8 rehearsals in 3 days), but I'm game, if it means a public concert. You know, I suppose, that the Radio concert is again without public.\n\nWell \u2013 I'm off to the hinterland, and I'll make every effort to come back.\n\nLove to all & you,\n\nL\n\n267. Leonard Bernstein to Aaron Copland\n\nMilan, Italy\n\n16 May 1948\n\nDear A,\n\nThese should either thrill, amuse or infuriate you. Imagine making an Italian debut with _Appalach_ e! [i.e. _Appalachian Spring_ ] I love the one that wonders what a \"Balletto per Marta\" is, and the one that says it reminds them of American movies, and the one that thinks I'm 6 feet tall.\n\nAnyway, it was a wild success, as was Munich (which was a real international problem), and now on to Budapest. It's all exciting as hell, and I'll write you at length later. My train leaves for Zurich in half an hour.\n\nLove love love,\n\nL\n\nBest to V[ictor] K[raft].\n\nIf you don't want to keep these notices, send them to Helen C., please?\n\n268. Leonard Bernstein to Helen Coates\n\nBudapest, Hungary\n\n20 May 1948\n\nDear H,\n\nThey say there hasn't been such a scene in a Budapest concert hall since Toscanini was here. The audience stamped & shouted, & especially for Bart\u00f3k, which they say has never really been heard until I did it, although it's always being played here. Well \u2013 another one under the belt. Everyone is wild with excitement. I've never known what success was before this. [...]\n\nThe _Times_ story on Munich is garbled, but good to see.121 I think the Tanglewood programs are perfect. I'm eating like a king. Of course, I'm a bit tired and rushed ( _three_ rehearsals yesterday!) but so happy. Budapest is ugly & beautiful together.\n\nIf Shirley wants to join me in Holland or Paris do have her come. Just get her on a plane or something. She seems to need it, and I would love to see her here.\n\nThe bobby-soxers tonight beat everything I've ever seen. I'm exhausted. Off to bed; tomorrow on to Vienna. I'm crossing my fingers for _that_ one.\n\nLove,\n\nL\n\nWhat iron curtain? It's only cellophane.\n\n269. Leonard Bernstein to Helen Coates\n\nParis, France\n\n29 May 1948\n\nDear H,\n\nSo much has happened that I don't know where to begin. I didn't write from Vienna since everything is censored from & to that city (all your clippings and notes bore the stamp of the censor!) \u2013 and besides I had quite a hectic time. It was the toughest city of all to conquer \u2013 a chauvinistic, provincial, nationalistic town, convinced that only Viennese can do anything at all, and that all Americans are fools. There was a mess over the program (I suddenly found myself saddled with a violin soloist) \u2013 and the orchestra was exhausted (five different concerts that one week, with 5 conductors and 12 rehearsals!) and besides they were all very antipathetic. It's the first time it ever took me 3 rehearsals to overcome the natural hostility of an orchestra, but we made it! Love & music conquered all; and the concert turned out a great triumph. I had been forewarned that nobody ever goes to concerts any more in Vienna for lack of money (even Bruno Walter's second concert was only half full \u2013 for the Mahler!!)122 \u2013 but still I had an almost sold-out house. Of course, the violinist helped \u2013 he's been there before. And he turned out to be good, despite the fact that we had to do the Dvo\u0159\u00e1k Concerto! The final trouble was that I had to cancel _Jeremiah_ \u2013 it would have been impossible in the rehearsals and with such a hostile, exhausted orchestra. It's just as well: the house came down, and, as the manager said, \"Ganz Wien in ein' Schlag!\"123 The political situation is a horror: more of that later. I have learned and learned and become very sad. It looks fairly hopeless at the moment.\n\nI was _furious_ at the _Times_ announcement that I had accepted the C[ity] C[enter]. What crust! How dare they! Mortie [Newbold Morris] should be strung up. What will Palestine think \u2013 that I accepted the orchestra just for a gag? It's psychologically horrible for them \u2013 and I do hope you'll make it clear to the Palestine people that this story is false. It's the last straw.\n\nThe orchestra here is angelic. They learned _Jeremiah_ in one rehearsal, and are so fast and good and in love with me that I actually cancelled today's rehearsal! First time I ever felt I could dispense with a rehearsal in Europe. Jennie [Tourel] is greater than ever.\n\nI see Ellen [Adler] a lot & she sends her love. Fran\u00e7ois [Val\u00e9ry] and Nadia [Boulanger] are coming to Holland for my concerts! It just angers me that this second year I still don't have a debut in Paris \u2013 after all the ego-building of these triumphs; to do just a broadcast here is an anticlimax. Well, what to do? Paris operates in terms of press agents, and [Charles] Kiesgen is asleep. The Conservatoire concert didn't work out after all \u2013 a matter of finance & intrigue with the Radio. It's a shame, but it would have been too hectic anyway.\n\nI've had to call off the Prague concert too [...] a wire from Bruno Z[irato] practically threatened excommunication if I went. So it's off. Another shame.\n\nWhat's this with your Hebrew? Are you studying? I'm terribly impressed. But you should learn to write script, not print! [...]\n\nNow that the book contract is signed, I have qualms. The old question \u2013 When?\n\nI guess it's just as well for Shirley to stick around in America. She wouldn't be much happier in Holland, though I probably would. (A propos, Vienna paid me in greenbacks, & I travel about with a bulging wallet of 20s and 50s!)\n\nI did write you from Budapest didn't I? I hope so, but if not \u2013 it was a tremendous experience, and they asked me to stay on a month, giving daily concerts \u2013 and they wouldn't be able to meet the demand! That's how wild the success was. Especially, of all things, the Bart\u00f3k!\n\nNow, for Philadelphia, damn them. I hardly know what to plan (I'm not in a planning frame of mind). But I _must_ play a concerto. Maybe Mendelssohn _Italian_ , Ravel Concerto and Shosty #5. Or Mozart _Linz_ first. Or Copland 3rd and Beethoven Concerto. Or (maybe this is best) Mendelssohn, _Jeremiah_ , and Beethoven Concerto. Yes, this is best. Try & push it.\n\nI guess that brings [me] up to date. I'll write next week from Holland. Give Shirley all my love, & tell her it happens to all of us, and it had a great deal of joy in it, after all; & these things are always made of joy and pain together.\n\nLove,\n\nL\n\n270. Leonard Bernstein to David Oppenheim\n\nScheveningen, Netherlands\n\n10 June 1948\n\nDear D,\n\nThe sun is out, I'm waiting for horses to be brought, I live like a king, I have screaming audiences and flowers at my concerts, and even a lover. There are a few other facts: the sea outside my window is a wonder; the Dutch word for cunt is kunt; there is a performer here named Cilli Wang; and I've bought you a sweet hand-painted chess-set in Budapest.\n\nI send all my love to you & Judy.\n\nL\n\n271. Leonard Bernstein to \"Twig\" Romney\n\nScheveningen, Netherlands\n\n20 June 1948\n\nMy dear T,\n\nWell, it's over. The big swing around this beautiful messy continent is at an end and I am swollen with success, lush with living, loving and learning.\n\nGermany and Austria were fabulous, filthy, Nazi, exciting. Budapest was grim and gay. Milano was the greatest. Paris a joy, as ever, and Holland a comfort, where I've soaked up milk and lobsters and sun, and been horseback riding every day. I'm happy, and a little bit drunk, to celebrate the end of the tour. I don't lift an arm until July 1st! (I sail the 22nd for New York, arriving the 29th, when you will probably be leaving.)\n\nNo sign of Thad anywhere. He missed a thrilling concert \u2013 not just musically, but politically. More anon.\n\nMore anon also about the Russian regions. It's not so good, to put it mildly.\n\nMaybe we can have a moment before we leave New York. As they say here, _tis te hopen_.\n\nAll the best, dear Twig, and if I don't see you, I hope you will love Europe as much as I have. I have a strange lack of nostalgia for America. I could go on here for ever.\n\nLove,\n\nLenny\n\n272. Leonard Bernstein to Aaron Copland\n\nc\/o Israel Philharmonic Orchestra, Tel Aviv, Israel\n\n29 September 1948\n\nDear A,\n\nA word to say me voici, and Gott sei Dank, and it's all beautiful beyond words.\n\nMarc [Blitzstein] cheered me up immensely the day after you brought me down so about the _Age of A_ [ _nxiety_ ] \u2013 he went into positive fits of joy. Who knows?\n\nHow would you like some of the enclosed for a present? It's just for you, it's terribly blond & Swedish, & it will be in America (Philadelphia!) by December. Mmm. It was lovely.\n\nI've already started working like a dog. Thirty-five concerts in the next two months! Somehow in Israel one finds strength for everything.\n\nMy suite & garden are a joy. Real Garden-of-Eden stuff: palms, mimosa, cactus, & what a sea! You must try Israel one day.\n\nHow's by the movie?\n\nYou old bastard, I love you.\n\nL\n\nEveryone awaits the 3rd Symphony with bated breath. Just to take _you_ down a bit, _Billy the Kid_ was _not_ played here, only the Celebration Dance, alone & only, and that had to be repeated. You old encore composer!\n\n273. Leonard Bernstein to \"Twig\" Romney\n\nTel Aviv, Israel\n\n20 October 1948\n\nDear Twig,\n\nI tried hard to call you during my three days in New York last month \u2013 between a wonderful Wyoming visit and my flight to Israel. You had gone, of course, that very day. But it's good to have your note, and to know that you are now ensconced in the sheltered halls of old V\u2013a, studying, and, I presume, making love like a beaver.\n\nThis is such a beautiful experience that I can hardly write of it. Truly I feel I never want to leave, despite all the tragedy and difficulty. I sit here in this charming city in a blackout, with the fucking Egyptians raising hell to the south, my beloved Jerusalem without water, and in siege. But the concerts go on \u2013 dozens of them \u2013 never one missed \u2013 with huge and cheering audiences \u2013 sometimes accompanied by shells and machine guns outside. And Haifa is certainly one of the fabulous beauty-spots on earth. Life is hectic, but pleasant beyond words: the orchestra is the most intelligent and responsive I've known: and I think I've fallen in love.\n\nAdd it up: do you blame me for having no nostalgia for the States? This is a most miraculous people with a heroism and devotion I have never before seen. I know: \u2013 I visited the front in Jerusalem. I could weep with the inspiration of it. Everyone is young, inspired, beautiful in this new Army, and everyone is truly alive in this new State. So they slander and babble in Paris, but these people will never be downed.\n\nI think of you often and with affection, Twig. I shall be back Dec. 9 or so. But I hear that the Philly Orchestra has cancelled the season! Otherwise \u2013 it would be lovely to see you.\n\nAll the best,\n\nL\n\n274. Leonard Bernstein to Serge Koussevitzky\n\nTel Aviv, Israel\n\n29 October 1948\n\nDear Sergei Alexandrovich,\n\nHow to begin? Which of all the glorious facts, faces, actions, ideals, beauties of scenery, nobilities of purpose shall I report? I am simply overcome with this land and its people. I have never so gloried in an army, in simple farmers, in a concert public. I am in perfect health, and very happy \u2013 only a little tired from the fantastic schedule we have here: 40 concerts in 60 days, here, in Haifa, in Jerusalem, Rehovoth, and so on. The concerts are a marvelous success, the audiences tremendous and cheering, the greatest being special concerts for soldiers. Never could you imagine so intelligent and cultured and music-loving an army!\n\nAnd Jerusalem \u2013 what shall I say of my beloved Jerusalem, tragic, under constant Arab fire, without water (only a pail a day) \u2013 with machine guns outside accompanying our performances of Beethoven symphonies! I have visited the fronts, entered Notre Dame, where we held out a few paces only from Arab\u2013British guns, inspected the strategic heights around the city and the Palmach bases. I have played piano in hospitals for the new wounded of the Negev, and in camps for soldiers and \"Kibbutzim\" people. I have been decorated with the Jerusalem Defense medal and the Palmach insignia. I have almost grown to be part of all these wonderful people and history-making days. Believe me, it will end well: there is too much faith, spirit, and will to be otherwise.\n\nWhile in Jerusalem I took a side-trip to a formerly Arab village called Ain-Karem where the captain of the Commandos turned out to be named Moshe Koussevitzky! He tells me he is your nephew, son of your brother, and is a charming, warm person, doing a great and heroic piece of work. I photographed him, and am sending you a print enclosed. I hope this makes you very happy.\n\nI am holding auditions for young conductors next week (I have reason to believe there are real talents here). Our good friend Mrs. Frank Cohen advises me that she wishes to donate a scholarship for an Israeli conducting student at Tanglewood next summer. Would you be willing to accept one, as an active pupil if possible, if we find a very talented one? Do please let me know as soon as you can, so that I can make arrangements while I am still here.124\n\nI hope the season goes as wonderfully as always \u2013 though I am sure this one will surpass all top standards set before. We all love you and pray for you here, and all your dear friends send you warmest wishes, including the President Weizmann, our beloved MacDonald [Ambassador James McDonald] (who is doing a fine job), and the members of the orchestra, who wait only for your visit one day.\n\nAll my love to you and Olga.\n\nLeonard\n\nI feel that I shall spend more and more time here each year. It makes running around the cities of America seem so unimportant \u2013 as if I am not really needed there, while I am really needed here!\n\nBy the way, I met Moshe Koussevitzky just next to the spring of John the Baptist!\n\n275. Leonard Bernstein to Aaron Copland\n\nTel Aviv, Israel\n\n8 November 1948\n\nDearest A,\n\nWell, we've gone and done it. The [Third] Symphony seems to be a success! Of course we could have used much more rehearsal (our schedule is unbelievable), but after the fourth performance it has begun to sound, and quite magnificent at that. It's really a fantastic piece! I must confess I have made a sizeable cut near the end125 (after the second performance) and believe me it makes a whole lot of difference. Hope you enjoy the enclosed. Most people loved it (the symph.).\n\nIn the midst of all this marvelous history & miracle-land & excitement of life, I have miraculously fallen in love. It's the works; and I can't quite believe that I should have found _all_ the things I've wanted rolled into one. It's a hell of an experience \u2013 nervewracking & guts-tearing and wonderful.126 It's changed everything. The Swedish is all yours now.\n\nHope the picture goes beautifully, & my love to all the nice people in H'wood. I can't wait to hear the B[enny] G[oodman] concerto. (By the way, B.G. here means Ben-Gurion. Strange world, no?)\n\nMuch love,\n\nL\n\nHope very much to be able to do the May 4, but I can't be sure til I get back to the States, which will be early in Dec. Where will you be?\n\n276. Leonard Bernstein to Jennie Bernstein\n\n[Israel]\n\n[November 1948]\n\nA wonderful hot day. We left our grand pre-fabricated house, Helen, as you see, in the lead, followed by our friend Katya, m'self and artist Yosi.127 Driving to Ain-Harod we passed so close to Nazareth that we couldn't resist having lunch there. It's a marvelous town; and while Helen ran around photographing everything and everyone in sight, I had a glorious Arab meal, with kh'umus and t'hina, and a fine Arab lad shined my boots. Bought rosaries (blessed by the church) for various Cath[olic] friends, and headed for Ain-Harod, the largest kibbutz in the land.\n\nOf course, first thing I had to have a horse. The gent in charge, a real kibbutz lion, took me out. I had a sort of Palomino beauty which I soon discovered I couldn't handle. So the lion-gent exchanged horses with me (I took his dray-horse), and it turned out he couldn't handle him either. Then the stable-man was called in, and he didn't do better, and since it became a bore to go only in circles we called it a day and went swimming in a marvelous pool in the middle of nowhere. The best swim-sun-and-air I think I've ever had \u2013 or thought I had until I went to Elat, but more of that anon. Meanwhile, there is a concert coming up for all the kibbutznicks of the vicinity, 5000 strong.\n\nJennie T[ourel] sang like an angel. The audience was, as always, the most attentive and appreciative in the world, although they don't know the conventions of clapping, so that Jennie T. lost an encore or two that had been planned. I finally played ye _Rhapsody in Blue_ , and we adjourned to a huge party where we danced and sang and drank and made with the Hora until Godknowswhen A.M. To bed, in a real guest-house (a fantastic achievement for a kibbutz) \u2013 then up betimes and on to Acre (old Arab city which Napoleon couldn't take) and with the military governor of Galilee went off to visit an Arab village. The road up to it was, as you can see, a real reducer. Since we were with the governor, whom the Arabs adore and fear, they staged for us what is known as a \"Fantasia\", with guns going off, music, dancing and nineteen lunches, coffee sessions, etc. Whole lambs are brought, torn to pieces by the host (who never sits with the guests but waits until they are through, then with his pals dives into the leavings. When they are through the women pounce on _their_ leavings, then the children, then the dogs. Such is the hierarchy.) Then, already sick with so much food, we proceeded to mount the local camels, who are nasty, haughty, dirty beasts. Jennie T., who will do anything for a photograph, allowed herself to be ruptured on one. Accompanied by the elders of the village (Druses, and splendid figures they are) we jolted back to Haifa for a concert \u2013 one of the worst I've ever given. Arabic burps punctuated the Mahler, which was worse in Jennie's case than in mine. Next morning a great Oriental dancer named Yardena Cohen128 performed for us \u2013 then quick to Lydda airport for the big climax \u2013 the trip to Elat. This is the newly-won spot on the Red Sea, southernmost Negev, across from Aqaba (Transjordan) on the Gulf. A beautiful flight (we were flown by the army in a Dakota with bucket seats) and landed in a wonderful Arizona-like wilderness, dry and windy and awesome. After a marvelous swim in the Red Sea (which is the bluest thing you ever saw) and a hard-tack dinner we drove up into the hills and entertained the soldiers stationed there. Jennie sang Carmen, of course \u2013 and this place at night really knocked me out. If you can imagine an intimate desert, where every rock and dune seems familiar, this is it. Yosi and I wandered afterwards for hours through the hills. I never wanted to leave, and did everything to miss the plane the next morning. But no soap: they waited for me. So sadly back to T[el] A[viv], and concerts and parties and god-damned professional life, which is driving me mad. But I leave tomorrow for Holland, and my one nostalgia, besides Jerusalem, will be Elat.\n\nLove, Lenny\n\n277. Leonard Bernstein to Philip Marcuse\n\nHotel Schenley, Pittsburgh, PA\n\n26 January 1949\n\nPhil,\n\nMany thanks for the clipping. It makes me very angry, and doubly so to think of all the fuss that's made over the Furtw\u00e4ngler & Gieseking business,129 while no attention at all is paid to this _echt_ Fascism in our own cities. How did H[enry] R[eichhold] ever let himself be quoted in such out-and-out Nazi lines? Have they set the date for burning the library in Cadillac Square?\n\nMuch love to you and Babs.\n\nL\n\n278. Ren\u00e9e Nell130 to Leonard Bernstein\n\n470 West 24th Street, New York, NY\n\n30 January 1949\n\nDear Lenny,\n\nThank you for your nice letter and poem to which I have this to answer: \"When the real animus and the real anima web, you can get married and take your wife to bed.\"\n\nSome short remarks on your dream: when you are unconscious (\"taking a nap, sleeping\"), you find that your rather undifferentiated feeling is playing tricks on you, bringing people into your psychology whom you do not want to have in there. Rather than finding out what these people really want from you, or why they were invited, you get angry at that side of yourself who played the trick on you. You get in touch with that side by hurting it, then you regret. You would know more if you would try to make her understand why you don't want these people anymore. Then, when you do get away from the unwanted collective, you get into an even less desirable one, a very pedestrian collective (street). Being alone now, without anything but yourself, you are eager to make contact with some other side, contact in the usual average pedestrian way \u2013 sex \u2013 which is the substitute for human relationship. When you find that that is impossible you are caught in some very dull, past aspect of your own bourgeois-side. That shows very nicely why you are so eagerly seeking homosexual contact in reality, it seems the way out or the escape from the fear of being caught in bourgeois patterns, and seems to symbolize the free and non-bourgeois life. They talk about your work in the dream; your fear always seems to be that being a conductor and being set in a profession is the same as being dully married and leading a middle-class life. I am sure it could be that way, but must not be that way, and will stop to look to you that way the moment you get some real color into your life; then you can give up to the so-called \"colorful life\" you are leading.\n\nFreud's definition: Id \u2013 subconscious; Ego \u2013 conscious; Super-Ego \u2013 conscience. Ego is the whole of consciousness. Jung: has the same concept of the Ego, he terms it the center of consciousness, the difference between F[reud] and J[ung] is in the way [the] use and function of the Ego are seen. With F. it is the censor and adaptor to reality. With J. it is understood as the channel for the forces that want to flow from the inside to the outside, and vice versa, it has a consciously screening function and serves the forces of the Self or the unconscious. With F. it is supposed to master them. To F. the Ego is the human being as such, therefore it has a very high value; to J. it is an aspect of the human, subordinated to the Self, which means the unspoiled essence of the human being. The Self is to J. the highest value in a human being. I hope that does not confuse you more.\n\nI wonder if you have enough contact with my way of analysis yet that the long distance dream-interpretation means anything to you. Generally it is difficult to get anything out of such answers in such an early stage of work; later when one is more attuned to each other it is easier. Let me know.\n\nI hope you have a fairly good time, not too many tensions.\n\nKindly,\n\nRen\u00e9e\n\n279. Leonard Bernstein to Howard Hoyt131\n\n32 West 10th Street, New York, NY\n\n8 February 1949\n\nDear Howard,\n\nFor a period of one year from the date hereof I hereby engage you, and you hereby accept such engagement, to act as my exclusive adviser and representative, to assist me in securing employment and negotiating contracts for my services as a performer in radio, television and motion pictures, as a composer of scores and incidental music for motion pictures, and in negotiating a contract for the production of _Operation Capulet_ , a dramatico-musical composition based on _Romeo and Juliet_ , for which I am currently composing the score and some of the lyrics.132 Notwithstanding anything to the contrary herein, it is clearly understood that my services as a conductor are excluded from the scope of this contract.\n\nIn consideration of the above services, I shall pay you a sum equal to ten percent (10%) of the gross compensation served or received by me in connection with any employment and contracts in the above fields, which contracts are entered into or negotiated for during the terms hereof, and upon extensions, additions, renewals and substitutions of such contracts and employment. Notwithstanding anything to the contrary herein, it is understood that I shall pay you no compensation with respect to my ASCAP royalties.\n\nVery truly yours,\n\nLeonard Bernstein\n\nAccepted and agreed to by Howard Hoyt\n\n280. Leonard Bernstein to Hans Heinsheimer133\n\nHotel Schenley, Pittsburgh, PA\n\n9 February 1949\n\nDear Hans,\n\nGood news for a change!\n\na) Kouss has promised, finally and definitely, the premiere of _The Age of Anxiety_ for April. Originally it was to be the 22nd and 23rd, but my benefit for Weizmann and Truman on the 23rd interferes. He has promised to find some other week in April.134 Will you check officially with Leslie Rogers at Symphony Hall?\n\nb) I've just this minute finished scoring the first movement (and, to quote Bill Schuman, is it beautiful!!). I'm mailing it to you, registered, and you can begin extracting at once. I hope to finish the second movement by next week. It's a mad race, and I'm exhausted, but it's challenging: and I hope to have it all scored by the beginning of March. (Please God!)\n\nPlease let me know right away how the print comes out: I've been using a Parker 51 pen on it, just as an experiment. If it's no good \u2013 I have the other ink with me, and can change.\n\nI've set two Rilke poems for Jennie Tourel (when? I don't know!) and she will do one or both at her March 13th Town Hall recital. I hope to have more of them soon. They make a fine group: \"Lovesongs\" by Rilke.\n\nAnd love to you \u2013\n\nLenny B\n\n281. Leonard Bernstein to Helen Coates\n\nColumbus, OH\n\nValentine's Day [14 February] 1949\n\nDear H,\n\nThis is a blessed day for a tired guy. Quiet and raining here, and peaceful enough to orchestrate like mad (page 67 already!).135 Concert tonight in Zanesville, O., then return here, and I can work til three tomorrow afternoon, when we leave for Dayton.\n\n[...]\n\nIt's been a grim weekend: I've been overtired again, and had the jumps at the concerts. I played a very inferior Mozart Concerto, but the Shost[akovich] was great. Too much farewell party. Also, Reiner has left a trail of hostility against me in the orchestra136 and around it, which makes it still very difficult to establish _rapport_ with the men. It's the first time I've had this problem, and it does remove the element of fun to an alarming degree. (The critic Lissfelt,137 whom you've no doubt been reading, is one of the Reinerites who is simply basically antagonistic. What a sourpuss!)\n\nI think I'll manage all right on tour, if I can get enough rest. I've set myself a deadline of having the score finished by the time I return to N.Y. for the Rutgers concerts. Let's hope & pray.\n\nLove,\n\nL\n\nIzler Solomon138 just called & we may have a drink tonight. Tossy S[pivakovsky]139 is here playing Bart\u00f3k with him.\n\n282. Arthur Laurents140 to Leonard Bernstein\n\n8227 Lookout Mountain, Hollywood, CA\n\n[?April 1949]141\n\nDear Lenny,\n\nI'm sending a copy of this to Jerry [Robbins] so that there won't be any wheels within wheels and we'll all be abreast of what is what.\n\nQuite frankly, I was disturbed by our phone conversation Monday night. Unless I seriously misinterpreted you, this is what I understood you to say: You conceive of the show's script as being written in an almost purely poetic style; you have doubts (understandable) whether I can write in that style without forcing; and \u2013 most important \u2013 if my writing cannot be that poetic, then you would not have complete faith in the project and, therefore, would rather abandon it before spending any actual working time.\n\nIt is this last point which I found disturbing. I myself have enormous enthusiasm for and belief in the show. I spent a good deal of time and money in New York talking about it and would not have worked on it there _and_ all these weeks here had I not, quite naturally, assumed the thing was settled: we were going to do it. It is difficult to work on something you feel may explode in your face at any moment. But difficult or not, I cannot turn back. I am \"in\" it, and _want_ to be \"in\" it; I must proceed. True, the possibility of chaos does superimpose an unfortunate level to plow through before working. But I'm working. As Jerry knows too well, I was in a vaguely similar situation myself before \u2013 when I first started in analysis, incidentally. But even then, I had spent time on the project and had worked on it, and then pulled out (for reasons, which I now regretfully admit, were erroneous and foolish).\n\nBe that as it may, the phone call brought up several problems I had not realized existed. First, the possibility that, even at this date, you might not do the show. Second \u2013 the conception. What I have tried to do is make the scenes between the lovers poetic in contrast to the violence of the world they live in. I do not think the rest should be ultra-realistic but I am doubtful whether I think it should be as poetic as you apparently do. Almost pure poetry is competing with Shakespeare on his own play. I can't stand up to that, the show can't stand up to that, no one today, unfortunately, could stand up to that. That is realistic fact. Plus the fact that I don't completely see the show that way. Also \u2013 and this would naturally follow your concept \u2013 you conceive, as I do, of the lovers' scenes as being almost completely musical but the others as being almost completely musical, too. I admit I do see less music and certainly \"songs\". I don't mean this as a cut-and-dried rule. As you will see, the last scene in the first act calls for music almost throughout, for example.\n\nPerhaps both you and Jerry disagree with me about the concept: perhaps the difference is merely one of degree (which I fervently hope). I am not adamant or absolute in my feelings; they can be modified. But a musical show (and I see this as a musical show or whatever you want to call it, but _not_ as an opera or even a modified opera) is a collaboration. A collaboration calls for compromise. Certainly, I don't ask either of you, any more than you ask me, to compromise to the point where you disbelieve in what the final outcome will be. That is foolish and makes for bad working conditions and, more important, bad work. We all must create as we feel and believe. I would not ask you to write a pop tune any more than I would ask Jerry to create a Berlin Ballet M\u00e9canique. You might be able to do it, but it's doubtful whether it would be very good. So you would have to do _your_ conception of a pop tune, and Jerry, his conception of the Ballet M\u00e9canique (which, in his hands, might be very funny, come to think of it). I think we all know each other's work, we all can concede and adjust, and there can be a meeting point.\n\nI don't feel that the current lack of exact agreement is a crisis nor do I mean to provoke one. The only serious point in my mind \u2013 and forgive if I repeat this \u2013 is the one I have stressed. Namely, we agreed to work on the show, to do it, and now \u2013 again, unless I misunderstood \u2013 you are unsure whether you will or will not do it \u2013 unless the conception is the way you visualize it. This stand leaves me on a high wire and I don't know how to walk a tight rope although I am willing to learn. Mind you, no word, line or scene I've written or will write is sacred. I agree some of it should be more poetic. In the very first scene, for example, I was counting on the initiation-ritual being sung in contrast to the sharp violence of the opening (and thus to set the pattern of the show). You told me on the phone that you did not think there should be any music in the scene (although if you look at the graph you yourself drew up in New York, I'm sure you will find music indicated there). But I wonder if we do not disagree about the quantity as well as the quality of poetic language and the quantity of the music. And those points must, of course, be settled soon.\n\nI'm continuing on the script, whipping the rest of Act One into shape so that I can send it off to you and Jerry by, I hope, the week-end. After you read it, you will be able to judge whether our concepts are so far apart after all. Please let me know \u2013 and as soon as you conveniently can. The high wire is awfully thin. I'm still planning, too, on coming to New York sometime toward the end of the second week in May.\n\nBest to you both,\n\nArthur\n\n283. Arthur Laurents to Leonard Bernstein\n\n8227 Lookout Mountain, Hollywood, CA\n\n[?April 1949]\n\nDear Lenny,\n\nI'm sorry you've decided not to do the show, sorrier still because of the main reason which led you to drop out. I think I understand. And feeling as you do, your decision is undoubtedly all for the best.\n\nOne thing I'd like to make clear \u2013 and I trust you will believe me. I did not start out with \"a priori prejudices\". Rather, I started out with admiration for your work and eagerness to become a friend. Along the line, hostility popped up. But since I felt it from you, just as you say you felt it from me, and made the exact same efforts you did, I can only conclude we were both projecting a little and, possibly, were further impaired by the occasional whispers which are ever present.\n\nIt's a pity we had such a short run. However, now that the tension is gone, there is probably a much better chance that we can become friends. I certainly hope so.\n\nWhatever show or venture you embark on next, please know I'll be rooting for you.\n\nArthur\n\n284. Ellen Adler142 to Leonard Bernstein\n\n[Paris]\n\n[?Spring 1949]\n\nDearest little Lennie,\n\nOnce I wrote you and never mailed the letter and now I can't even find it. I have heard all about you from Bobby Lewis who was here, and now from Harold143 who arrived a few days ago. I know that you are writing a musical and that you arrived at 161 West 54144 with several other talented young men, each of you with agents flanking you, all of which amused my mother greatly.\n\nI am well. By this time I speak French, I look French and I am assimilated into Paris, all of which makes me miss New York the more. I see your friends around from time to time, once [Fran\u00e7ois] Val\u00e9ry,145 at a caf\u00e9, who told me he was about to write a play about Noah's ark. Now and again I see Nadia [Boulanger] who looks more and more like a Grant Wood146 and is indeed a lurid creature. And then there is, of course, Marie Blanche147 always seen sweeping into an enormous automobile, and leaning, almost toppling into the arms of her tall, blonde painter friend. But I suppose Bigelow148 is the funniest. At first this lad was strictly a right-bank spectacle and very dashing, now he is to be seen in the leftest of left-bank hangouts and always sporting a pair of blue jeans. So there, you see, is the evolution of Paris. I started with the left-bank and shall end up on the right, Bigelow started with the right and was doomed for the left.\n\nI have heard wonderful, wonderful things about your last tour and am happy that everything goes on so very well.\n\nParis is magical, romantic, perfumed and a city of miracles. A spring such as has never been before is here, and it is quite elegant as only Paris can be, and I am very happy. I think of you often, Lennie. Take care of yourself and give my love to all the people about you such as Helen [Coates], Shirley [Bernstein], D[avid] D[iamond] and A[aron] C[opland]. In fact, everyone but Burtie149 because he's a kid and I want him to take me flying one day.\n\nAnd kisses to you,\n\nEllen\n\n285. Leonard Bernstein to Peter Gradenwitz150\n\n1025 Park Avenue, New York, NY\n\n11 May 1949\n\nDear P,\n\nJust time for a few notes:\n\n1) A great joy to receive _our_ book, & many thanks for the sweet dedication.151\n\n2) _The Age of Anxiety_ was a walloping success, & I thought of you often in preparing it.152 Everyone adored it, which amazed me.153\n\n3) Doesn't look so good for you & Tanglewood this summer. I have very little to do with it this year \u2013 just a token appearance, and I've taken a farm 45 miles from it! Have you heard from them?\n\n4) I'd love to see the \"Bernstein article\". The Variations you saw remain about the same.\n\nMy love to you & Rosi & the kids, & from Helen \u2013 sorry this is too short \u2013 I'm rushed with plans for a new show.\n\nLenny\n\n286. Farley Granger to Leonard Bernstein\n\n8227 Lookout Mountain, Hollywood, CA\n\n15 May 1949\n\nMy visible verb, my very Dear,\n\nIt's Sunday morning, I am playing _Jeremiah_ and loving it, and you very, very much. It was so wonderful talking to you the other night. I am sorry that I woke you again, and Shirley too. I hope so very much she really doesn't dislike me. But I have a feeling we two (Shirley and I) will work out well.\n\nLast night I wore the cuff links you gave me. They are so beautiful, and I thank you always. But the thing with me is that I don't feel that I have to wear something you gave me, to be reminded how much I love you. Because [e]very kind, and happy, and warm thing that happens to me I share with you. You are in the lovely adolescent green of the spring trees, you are in the breath of the warm breeze. You surround me with a rare and God given feeling \u2013 called love (you are also making a bit of a poet out of me).\n\nMy picture is still going very well and I am most happy in my work. Tell Betty and Adolph that for a night club scene in the picture the studio reproduced the Village Vanguard.154 I'm getting a kick out of working in the same place the Revuers did.\n\nI was asked if I would like to do a picture in China, a thing called _Rickshaw Boy_ which sounds very exciting but will not come into being for a while. How about arranging a few concerts for the Far East.\n\nSaw _Ball Game_ again and agree with you, there's a good deal of ham in our boy Gene.155\n\nPlease thank Helen for me, that was very sweet of her.\n\nWrite me as much as you can. I will call again soon, but will not make it a late one next time. You are a wonderful man and I love you.\n\nF\n\n287. Leonard Bernstein to William Schuman156\n\nSinging Brook Farm, Charlemont, MA\n\n15 July 1949\n\nDear Bill,\n\nI love it when excuses pop up for me to write you \u2013 it seems the only way we get to communicate these days. There are three such at the moment: and may I preface them with my warmest hello to you & Frankie.\n\n1) I take it you've received a copy of the enclosed from [James] McDonald in Israel. I wish I could say that I'd heard the boy, but I don't recall any such experience, although I do remember hearing good things about him. Of course, I'm all for this Israeli stream into Juilliard on general principles, since Israel is the most musical country I've ever seen, and Juilliard is the best school I know for developing that native talent.\n\n2) At the risk of seeing the Juilliard School turned into a little Tel-Aviv, I make so bold as to propose another Israeli for the student body \u2013 a young conductor who is now here at Tanglewood \u2013 Elyakum Shapirra \u2013 who won a competition over there & was selected to be one of Koussy's chosen \"Three\". He arrived displaying great musicality, integrity, etc., & a lack of knowledge caused by his years in the Israeli Army. He desperately needs schooling, is wild to go to Juilliard; & I think [Jean] Morel would be marvelous for him. The kid's a natural, but such a primitive! He needs Kulture. It would be a great blessing to him.\n\n3) A strong recommendation for Ralph Berkowitz, whom, I hear, you are considering for an additional post at Schirmer's. A nicer guy you couldn't find, nor a harder-working, or more knowledgeable.\n\nThere! Now I can go back to Mann's _Dr. Faustus_ which sits particularly well in this divine isolation 40 miles from Tanglewood. I'm at last finding a few days for myself \u2013 to read, to sleep: it is a most welcome novelty.\n\nLet me hear from you.\n\nLove,\n\nLenny\n\n288. Leonard Bernstein to Shirley Bernstein\n\n[Wednesday] 20 July 1949\n\nMain,\n\nNo need to voice my reaction to the rumor heard the other day of Alan J. [Lerner]'s plan to do a show with Loewe, nor to the item confirming it in today's _Times_.157 The bastard. A fine double-x. Destroy this letter.\n\nNo word yet from Adolph and Betty and Adolph. I am really irked at the Lerner thing. Rest and think, main foot!\n\nThe enclosed came to Tangle for you.\n\nI wormed the Gabe personally each night, but he still drags his arms to the ground. Well, so many of us do, and nobody worries.\n\nIt was lovely seeing you for an extra hour at the Shelton, even under such trying conditions.\n\nFarfel's mad, and the world is full of apes. I suppose you're as disappointed in Lerner as I am. Maybe you should have snapped up Michael Dreyfuss.158\n\nLoveoah,\n\nDein\n\n289. Menahem Pressler159 to Leonard Bernstein\n\nTel Aviv, Israel\n\n15 August 1949\n\nDear Mr. Bernstein,\n\nAlthough much in arrears (for which I am only very little to blame, having had so many appearances and marriage to boot), I still could not deprive myself of the pleasure of writing to you and telling you how much I enjoyed playing your _Anniversaries_.\n\nOn the occasion of American Independence Day I was invited to play on the radio an American programme. For this programme I also chose seven of your _Anniversaries_.\n\nAll the time (for it is ever fresh in my memory) I wanted to write to you and tell you how much I liked them on account of their quaintness, and rare and thrilling harmonizations so strangely appealing to the senses and the intellect. These dedications are real masterpieces of modern art and I admired them very much.\n\nWith my best regards to you and your sister. I am admiringly yours,\n\nMenahem Pressler\n\n290. Olivier Messiaen160 to Leonard Bernstein\n\n13 Villa du Danube, Paris, France\n\n5 October 1949\n\nCher Ma\u00eetre et ami,\n\nI hope you have had the time to look carefully and to work on the very difficult score of my _Turangal\u00eela-Symphonie_.161 Take great care of the dark photocopy that I gave you: it's the only copy that it is possible for your work \u2013 if you lose it, that will be a catastrophe! You will have realized that this work is very long (1#fr1\/2> hours) and very difficult for the whole orchestra. You will decide for yourself how many rehearsals are necessary. The work requires, in addition, the presence of 2 soloists: an Onde Martenot solo and a piano solo. The only possible Ondiste is Ginette Martenot. The only possible pianist is Yvonne Loriod.162 Ginette Martenot is leaving in any case for New York (being invited for a different event) and Boston, and she will be there from 28 October to 1 January. The Boston Symphony Orchestra thus does not need to concern itself with her travel expenses.\n\nOn the other hand, the piano solo part of my symphony is of a difficulty such that only Yvonne Loriod can play it with the special techniques that it requires (triple notes, modes, very complex rhythms, birdsong, lightness, power, passion, etc.), and to play it _by heart_. Yvonne Loriod has a matchless technique, she is a composer, she is the best French pianist for ultra-modern music, and finally she is the specialist in my music, having played my piano works for the last seven years in all the great cities of Europe. She has already been working for a year on the piano part of my symphony, and not only does she know it by heart, but she understands the whole orchestral score perfectly. The [piano] part is considerable in the work on account of its difficulty and importance, and no one on earth can play it with such brilliant eloquence.\n\nMoreover, I also think that my presence would be quite useful at the rehearsals, for giving exact tempos, and the balance of timbres. It would be a great joy for me to hear my work.\n\nI therefore ask the Boston Symphony Orchestra:\n\n1. To engage Yvonne Loriod and Ginette Martenot to play the parts for piano solo and Onde solo of my symphony and to send them their contracts (date of the concerts, total fees, etc.)\n\n2. To kindly pay the expenses of Yvonne Loriod and myself for about 25 days \u2013 a total of $524.\n\n3. You told me that you would give the _Turangal\u00eela-Symphonie_ in the last week of November 1949 in Boston and the first week of December 1949 in New York (Boston Symphony Orchestra). I remind you that this will be the world premiere in Boston, and the second ever performance in New York. Will you be very kind and specify for me the _exact dates_ of the two concerts?\n\n4. _The orchestral material of the Turangal\u00eela-Symphonie is ready_. It belongs to my publisher Durand et Cie, 4 place de la Madeleine, Paris (8e). When should Durand send it? And to whom? Would you prefer that I bring it myself?\n\n5. I recommend especially your studying my work. Serge Koussevitzky commissioned it from me in 1944, which is to say that I have worked on it for a long time. Of all my works, it is the most accomplished and the most original. I am 41 years old, and I put into my symphony all my powers of love, of hope, and of research. But I know that you are a brilliant man and that you will conduct it as I feel it.\n\nI await a prompt reply and ask you to believe my feelings of warm admiration.\n\nOlivier Messiaen163\n\n291. Arturo Toscanini164 to Leonard Bernstein\n\n15 October 1949\n\nMy dear Bernstein,\n\nI compared the Victor recording of the Love Scene from Berlioz's _Romeo_ with the broadcast and confirmed the fact that the Victor is much faster.\n\nAnd I confirmed also another fact \u2013 namely that every man, no matter the importance of his intelligence, can be from time to time a little stupid. So is the case of the old Toscanini.\n\nYour kind visit and dear letter made me very happy. I felt myself forty years younger.\n\nI hope to see you very soon and it will give me a great amount of pleasure.\n\nMost cordially, believe me, dear Bernstein.\n\nYours ever,\n\nArturo Toscanini\n\n292. Leonard Bernstein to Burton Bernstein\n\nNew York, NY\n\n1 November 1949\n\nM'nape,\n\nJust spent an evening with Irwin Edman165 (Dept. Philosophy, Columbia) who made me nostalgic for the shelter of academic life, & made me therefore think of you. Also there was Henry Simon, who submitted our \"Conversation\" (with Spendoah on Beethoven) to his editors who were mad for it & want more and thought the character of YB (Younger Brother) was a masterstroke.166 I'll send you a copy as soon as I make some.\n\nI saw your North Country Epic to Hi-lee,167 and loved it. But I never hear any real news from you, appraisal of your courses, directions you may be finding \u2013 all the thousand new things that happen in a freshman year. Come on, a good long one.\n\nAs for me, only two weeks or so remain before my season starts, and in them I have yet to finish the Herman piece (Woody, remember?) which is on the home stretch; to do music (incidental) for a production of _Peter Pan_ (if they raise all the money); write the book, prepare the Messiaen,168 read scores & concerti for the season, etc. It's a grind. I've been going to parties \u2013 lots of them \u2013 a strange new occupation for me. [...] Life is very pleasant.\n\n_Regina_169 was panned by the morning papers, loved by the afternoon ones. It has a fighting chance. By the way, it was a marvelously exciting opening night.\n\nI await that long letter.\n\nLove,\n\nL\n\n293. Olivier Messiaen to Leonard Bernstein\n\n13 Villa du Danube, Paris, France\n\n6 November 1949\n\nCher Ma\u00eetre et ami,\n\nThank you for your fine letter.\n\nYou must have been looking at the score of _Turangal\u00eela-Symphonie_ and noting that it is a gigantic and very difficult work. I thank you deeply for conducting my work, since I know (having seen you in _The Rite of Spring_ ) that you will do it in a way that is marvelous and brilliant.\n\nYou will understand now that _Turangal\u00eela_ is the work of my life. It's why I've been so insistent about it being presented under the best possible conditions.\n\nI have sent to Mr. Judd a very detailed analytical note for the program of _Turangal\u00eela_ , asking him to make an English translation.\n\nFurthermore, Mr. Judd has telegraphed me that you will conduct _Turangal\u00eela_ 3 times: 2 and 3 December and 7 December. He did not tell me in which cities: I think it's in Boston each time.\n\nThere are some difficulties being raised about the arrival of Ginette Martenot and Yvonne Loriod into the USA by the union of orchestral musicians. I have said to them that neither Ginette Martenot nor Yvonne Loriod are orchestral musicians, but that they are playing in _Turangal\u00eela_ as soloists. And this must be put on the program and on the posters: _Turangal\u00eela-Symphonie_ for piano solo, Onde Martenot solo and large orchestra.\n\nSoloists:\n\nPiano solo: Yvonne Loriod\n\nOnde Martenot solo: Ginette Martenot.\n\nLukas Foss (who is generosity and kindness personified) agrees with me about this.\n\nThe difficulties are thus overcome now.\n\nYvonne Loriod and I will take the liner _\u00cele de France_ on 10 November, disembarking at New York on 16 November in the afternoon. We will meet Ginette Martenot there. I will be attending two concerts (17 and 18 November) when Leopold Stokowski conducts my _Liturgies_ , and then we leave at once for Boston.\n\n_So, Yvonne Loriod, Ginette Martenot and myself will all three arrive in Boston on 24 November. I will have all the orchestral parts for Turangal\u00eela with me._ (You already have the score.) It will be necessary to start the rehearsals on my arrival, owing to the extreme difficulty of the work. Yvonne Loriod knows _Turangal\u00eela_ completely by heart. She can thus assist you rehearsing with certain musicians separately. I can do the same (especially with the percussion which is substantial and very difficult). Finally, I am completely at your disposal \u2013 as is Yvonne Loriod \u2013 to rehearse the work with you at the piano and to demonstrate some of the rhythmic features, tempos, etc. I will attend all the rehearsals and will do everything in my power to be useful to you (balance of timbres, nuances, etc.).\n\nThese concerts will be the greatest joy of my career. See you soon! Believe always in my total admiration, my gratitude, and my friendship.\n\nOlivier Messiaen170\n\n294. George Abbott to Leonard Bernstein\n\n630 Fifth Avenue, New York, NY\n\n25 November 1949\n\nDear Lennie,\n\nJust after you left, Bob Fryer171 came into my office inquiring if I would be interested in doing a musical comedy version of _A Tree Grows in Brooklyn_. As soon as I had a chance to think it over, I was most enthusiastic. It seemed positively inspirational. I am now entering into contracts to do same, and this is to find out if you can take a minute out of your busy life to consider whether you would like to go in on it.\n\nI am going to write the book with Betty Smith.172 And I am going to try to get your old friend, Jerry Robbins, to do the dance, and beyond that, there are no thoughts at the present moment.\n\nIt will be a love story of the married couple rather than a tragic matter, and quite gay, albeit poignant, I hope.\n\nI talked to your secretary today. She told me that you were conducting this afternoon, so I knew I could not reach you by phone.\n\nI thought maybe if you were interested in this we could get Betty and Adolph to do the lyrics.\n\nLet me hear from you as soon as you can tear yourself away from all those woodwinds.173\n\nLove,\n\nGeorge\n\n1 The German word for an orchestral score.\n\n2 George Abbott (1887\u20131995), American theater director, producer, and writer, who also had a successful career in Hollywood. By the time he collaborated with Bernstein, Robbins, Comden, and Green \u2013 all making their Broadway debuts \u2013 in _On the Town_ in 1944, Abbott had already had decades of experience: first as an actor (his debut was as Second Yeoman in the 1915 revival of Gilbert and Sullivan's _Yeomen of the Guard_ ), then as a writer, producer, and director. His earlier Broadway musicals included _On Your Toes_ (1936; book), _The Boys from Syracuse_ (1938; book, producer, and director), and _Pal Joey_ (1940; producer and director).\n\n3 _Self-Analysis_ by Karen Horney, first published in 1942.\n\n4 _Kiss and Tell_ was a 1945 comedy starring Shirley Temple as the American teenager Corliss Archer. It was based on the Broadway play of the same name (both were written by F. Hugh Herbert, 1897\u20131958). The stage play had been produced and directed by George Abbott: it opened at the Biltmore Theatre on 17 March 1943 and ran for a total of 956 performances, closing on 23 June 1945.\n\n5 The show became _Billion Dollar Baby_. Bernstein wasn't able to write the score, so Comden and Green turned to Morton Gould. The choreography was by Jerome Robbins, Oliver Smith designed the sets, and Max Goberman was the musical director \u2013 all of them later involved in _West Side Story_ \u2013 and George Abbott directed. It ran for 220 performances, from 21 December 1945 to 29 June 1946.\n\n6 Izso G. Glickstein (1891\u20131947) was the Russian-born cantor at Temple Mishkan Tefila, where Bernstein had his formative musical experiences. He described Glickstein as \"a fabulous cantor who was a great musician and a beautiful man, very tall, very majestic [...] and he had a tenor voice of such sweetness and such richness\" (Burton 1994, p. 8). Glickstein was indeed an outstanding cantor, as can be heard on his recordings of \"Baroish Hashonu\" and \"Yaale tachnunenu miarev,\" made for Victor in 1925 (Victor 68710).\n\n7 Dated \"7\u20135\u20131945,\" so possibly 5 July 1945.\n\n8 Bernstein must have sent Glickstein the arrangement of \"Lamentation\" for voice and piano or organ version (adapted by F. Campbell-Watson), published by Harms in 1945.\n\n9 A restaurant that was a favorite with the _On the Town_ company.\n\n10 Francis A. Coleman, \"Composer Teams with Choreographer,\" _Dance Magazine_ (May 1945), pp. 12\u201313.\n\n11 _Bye Bye Jackie_ , subtitled a \"ballet play,\" was written by Robbins in 1944 and was proposed to several composers including Aaron Copland (see Pollack 1999, p. 486) and Paul Bowles, as well as Bernstein (Burton 1994, p. 140). In an interview by Anna Kisselgoff for _The New York Times_ (29 May 1994), Robbins recalled the project, intended as a way of explaining the background of one of the sailors in _Fancy Free_ : \"It was about Jackie, a boy living in Brooklyn, who's getting letters from his brother in some foreign place... The kids on the block begin horsing around, and Jackie can't take it. He sees that everyone in the background he's caught in is going off. So he enlists in the Navy, and his girlfriend says, Bye-bye, Jackie. It was a mood piece that went in and out of reality.\" Robbins explained that the work was never choreographed because Bernstein didn't want to write the score. Their next collaboration was _Facsimile_ , in 1946.\n\n12 Robbins' irritation is understandable. Coleman's interview with Bernstein includes the following remarks: \"For some time, Leonard Bernstein has considered the composition of a trilogy of ballets to be built around _Fancy Free_. Mr. Bernstein explains that the trilogy would consist of three one-act ballets with a connecting link in the story which would enable them to be presented as a complete evening's entertainment. The opening work, to be called _Bye, Bye Jackie_ , is to furnish a picture of adolescence, in a Brooklyn setting, of tender and emotional quality. _Fancy Free_ would become 'something akin to the scherzo movement of a symphony', and carry the story of the ballet through its middle section. Following it, the third work, yet to be planned, is to furnish the fitting climax to this vignette of American life.\"\n\n13 Bette Davis (1908\u201389), American actress, and an idol of Bernstein's. As her delightful letters demonstrate, their admiration was mutual.\n\n14 Irving Rapper (1898\u20131999), British-born film director. He was friend of Bernstein's and director of several of Bette Davis' most important films, including _Now Voyager_ (1942), _The Corn is Green_ (1945), and _Deception_ (1946).\n\n15 Nadezhda von Meck (1831\u201394), Tchaikovsky's great patron who stipulated that they should never meet. Coincidentally, one of the reasons Bernstein visited Hollywood in the summer of 1945, en route to San Francisco, was to discuss a possible role in a film to be directed by Irving Rapper: a biopic in which Bernstein would play Tchaikovsky opposite Greta Garbo as Madame von Meck (see Burton 1994, p. 142).\n\n16 Bette Davis received the signed photograph she had requested and sent a telegram on 14 August 1945: \"Please forgive delay in acknowledging photograph. Madame von Meck is very grateful and loves it. Bette.\"\n\n17 Evidently, Bette Davis wasn't a fan of Joseph Szigeti, who appeared with Bernstein at a Summer Promenade concert of the San Francisco Symphony on 1 July 1945. Szigeti played the Mendelssohn Violin Concerto and the program also included a suite from _Fancy Free_ and Shostakovich's Fifth Symphony.\n\n18 Though the letter is undated, the first paragraph refers to _Deception_ (1946). Rapper directed this _film noir_ with a cast led by Bette Davis, Paul Henried, and Claude Rains. It tells the tempestuous story of Christine (Davis) and her stormy relationship with a composer (Rains) and a cellist (Henried). Erich Wolfgang Korngold wrote the Cello Concerto by \"Hollenius\" performed at the film's climax.\n\n19 This letter was written the day after the bombing of Hiroshima on 6 August 1945, and two days before the bombing of Nagasaki on 9 August.\n\n20 The famous lesbian novel by Radclyffe Hall, first published in 1928, which had been the subject of an obscenity trial in England and legal challenges in the United States.\n\n21 Marketa Morris (1889\u20131965) was a psychoanalyst, known to Bernstein and his closest friends as the \"Frau\". A brief article announcing her death appeared in _The New York Times_ on 26 May 1965: \"Milwaukee, May 25. Mrs. Marketa Theiner Morris, a psychoanalyst who practiced in New York from 1942 to 1958, died here Sunday at the age of 76. She was the wife of Prof. Rudolph E. Morris of Marquette University. Mrs. Morris, who taught child psychology to teachers in Prague from 1936 to 1938, practiced here for six years until her retirement in 1964.\" The letters from Marketa Morris to Bernstein provide some insights into Bernstein's innermost thoughts from the psychoanalyst he consulted most regularly in the 1940s, including comments on his dreams and on his sexuality. It is apparent from the letters that while Morris saw Bernstein on a number of occasions, she was frustrated by his schedule, which made it impossible for him to see her on a regular basis.\n\n22 There is no year on this letter, but it can be securely dated 1945: not only did Bernstein spend some time in Hollywood that year, but on 15 August 1945, a few days before it was written, the Japanese surrender ended the Second World War in the Pacific, the \"Peace\" to which Morris refers.\n\n23 i.e. _Freedom Morning_ , composed in 1943.\n\n24 Copland's father, Harris Copland, died in 1945 (see Pollack 1999, p. 15).\n\n25 Mildred Spiegel's pet-name for Bernstein.\n\n26 Ren\u00e9e Longy Miquelle.\n\n27 Bart\u00f3k died in New York on 26 September 1945.\n\n28 The opening concert of the New York City Symphony season. In _The New York Times_ (9 October 1945), Olin Downes waxed enthusiastic: \"Leonard Bernstein, with an orchestra materially improved over that of last season, conducted a concert of exceptional brilliancy last night.\" The Shostakovich was a highlight: \"For vividness, conviction, imagination we do not expect soon to hear this performance surpassed.\" Downes also enjoyed the high spirits of what Copland called \"American Brahms\": it was \"a reading of high excellence. We believe Mr. Bernstein is now in a good place, with an orchestra of young musicians like himself to work with, and a repertory to mature in. Here is a conductor.\"\n\n29 Rosalyn Tureck (1913\u20132003), American harpsichordist and pianist noted for her Bach playing, though she made her Carnegie Hall debut playing the Theremin.\n\n30 Harl McDonald (1899\u20131955), general manager of the Philadelphia Orchestra (1939\u201355) as well as a teacher at the University of Pennsylvania and a composer.\n\n31 _Billion Dollar Baby_ opened on 21 December 1945.\n\n32 Morton Gould (1913\u201396), American composer and conductor. Bernstein never had a high opinion of his music.\n\n33 The Mozart symphony that Bernstein conducted on this occasion was No. 39 in E flat major K543, a work he performed many times subsequently, and recorded with the New York Philharmonic (1961) and Vienna Philharmonic (1981). Olin Downes reviewing the concert in _The New York Times_ (6 November 1945) had some of the same criticisms of this early performance as Longy. He commented that it was played with \"vigor and clarity\" but that Bernstein \"was inclined to drive rather than release song from the instruments.\" Paul Bowles in the _New York Herald Tribune_ was more enthusiastic: \"The high point of sonority in last night's concert came with the Mozart Symphony. Here the orchestra showed that it was no longer 'good, considering', but good, period. The audience responded with rounds of applause.\"\n\n34 Copland's nickname for his car.\n\n35 Possibly a reference to one of Erich Leinsdorf's concerts with the Cleveland Orchestra on 11 and 13 October 1945, which included _Appalachian Spring_.\n\n36 At the end of November 1945, the Harvard Music Department put on four days of concerts to celebrate the centenary of Faur\u00e9's birth. On 25 November, Copland wrote an article in _The New York Times_ previewing what he described as \"a shrine for Faur\u00e9 devotees.\"\n\n37 Copland was working on his Third Symphony.\n\n38 Seymour Meyerson was a close friend of Bernstein and of David Oppenheim, but otherwise he remains a mystery. He is _not_ the same Seymour Meyerson who served in the Army Signals Corps, became a scientist specializing in mass spectrometry (and also co-authored a booklet called _Folk Dancing for Fun_ , which helped pay his way through university). I am most grateful to this Seymour Meyerson for taking the time to explain that he wasn't the one who knew Bernstein.\n\n39 An abbreviation for \"post exchange,\" a type of store operated at US Army bases, which in turn generates income to support recreation, sports, and entertainment.\n\n40 It was an unusual program, including Beethoven's Op. 131 String Quartet arranged by Mitropoulos for string orchestra, Ravel's _Sh\u00e9h\u00e9razade_ (with Jennie Tourel as the soloist) and _Alborada del gracioso_.\n\n41 Walter Hendl was Rodzinski's 28-year-old assistant at the New York Philharmonic. When Rodzinski was taken ill, Hendl took the concert on 8 December (the Overture and Scherzo from Mendelssohn's _Midsummer Night's Dream_ , Schubert's \"Great\" C major Symphony, and Rachmaninov's Second Piano Concerto) at very short notice. On 9 December 1945, _The New York Times_ reported that Hendl's debut \"offered a striking parallel to that of Leonard Bernstein, who first attracted wide attention, when at the last moment he was called upon to conduct the Philharmonic.\"\n\n42 The Hotel Vanderbilt in San Francisco was used by the Army's Officer Pay Section, where Oppenheim was working.\n\n43 It's unclear what Bernstein means by \"so N\".\n\n44 Clarinet.\n\n45 On 21 January, Bernstein conducted the City Symphony Orchestra in Stravinsky's _Symphony of Psalms_ , Strauss' _Don Juan_ , Bach's Brandenburg Concerto No. 5, and Three Variations from _Fancy Free_.\n\n46 Carlos Moseley (1914\u20132012) was to have a long association with Bernstein. In 1941, Moseley was the soloist in Brahms' Second Piano Concerto with Bernstein conducting, at Tanglewood. In 1946 he was working at the State Department, promoting American music abroad, the subject of this letter. In 1955 he joined the New York Philharmonic Orchestra, first as press officer, then associate manager in 1959 (in time for the orchestra's tour to Russia), managing director from 1961 to 1970, president from 1970 to 1978, and finally chairman.\n\n47 Paul Feigay (1918\u201383), American theater and television producer. Feigay's first Broadway credit was as co-producer with Oliver Smith of _On the Town_ , and his subsequent career included producing the television series _Omnibus_ with Bernstein in the 1950s. _On the Town_ ended its successful Broadway run on 2 February 1946, and the arrival of the tour in Chicago (including Nancy Walker and Adolph Green in the cast) was greeted enthusiastically by Claudia Cassidy in the _Chicago Daily Tribune_ in her review published on 2 April. Feigay's letter is an interesting snapshot of the vicissitudes of producing a Broadway show, even one as ostensibly successful as _On the Town_.\n\n48 David Glazer (1913\u20132001) had given the first performance of the Clarinet Sonata with Bernstein on 21 April 1942. From 1951 until his retirement he played in the New York Woodwind Quintet.\n\n49 Bernstein's optimism is heartening, but the situation in Czechoslovakia was volatile, and deteriorated sharply over the next two years. On 26 May 1946, two weeks after Bernstein wrote this letter, the first Czech post-war general election had a voter turnout of 93.9%. The result was a victory for the Communist Party of Czechoslovakia. Edvard Bene\u0161 continued as president, and Jan Masaryk, son of the founding father of Czechoslovakia, continued as foreign minister. The Communist Klement Gottwald became prime minister. The Communists controlled only a minority of ministries, but these included some of the most important, notably Information, Finance, and the Interior (including control of the police). Through their position of power in these ministries, the Communists were able to establish a solid base from which to launch the Soviet-backed coup in February 1948, beginning four decades of Communist rule that ended with the Velvet Revolution of 1989.\n\n50 The famous actress Helen Hayes (1900\u201393), who was married to Charles MacArthur. Something of a legend in the American theater, she is one of a select group to have won an Emmy, an Oscar, a Grammy, and a Tony. She had two Broadway theaters named after her. In 1955 the former Fulton Theatre was renamed the Helen Hayes Theatre; after that was demolished in 1982, the nearby Little Theatre was renamed in her honor.\n\n51 Laszlo Halasz (1905\u20132001) was the first Music Director of New York City Opera, from 1943 to 1951. He then became Recording Director for Remington Records, as well as a conducting teacher at the Peabody Conservatory and Eastman School of Music.\n\n52 i.e. _Madama Butterfly_.\n\n53 The recording session on 1 July 1946 was for Ravel's G major Piano Concerto, in Bernstein's dual capacity as soloist and conductor, made with the Philharmonia Orchestra.\n\n54 Felicia was appearing at the Bass Rocks Summer Theatre in Gloucester, MA.\n\n55 Britten's _Rape of Lucretia_ was first performed at Glyndebourne on 12 July 1946.\n\n56 Presumably Bernstein means Dennis Brain, principal horn of the Philharmonia Orchestra.\n\n57 The ink has become progressively fainter on the page and here Bernstein refills his pen.\n\n58 The song \"Barney Google (with the Goo-Goo-Googly Eyes)\" by Billy Rose and Con Conrad.\n\n59 Bernstein first met Felicia Montealegre Cohn (1922\u201378) in February 1946, at a party given by Claudio Arrau after he had played the Brahms D minor Piano Concerto with Bernstein conducting the New York City Symphony. Felicia was not only a beautiful and gifted actress, but had been a piano pupil of Arrau's. During the autumn of 1946 she and Bernstein saw each other regularly and grew increasingly close. Bernstein took Felicia with him to Hollywood in December, and it was there that their engagement was celebrated by a party. Leonard Lyons reported in his \"Times Square Tattle\" ( _The New York Times_ , 8 January 1947): \"This is how Leonard Bernstein's engagement to Felicia Montealegre, the Chilean actress, was announced: Lester Cowan, producer of _The Beckoning Fair One_ in which Bernstein will costar, conduct and compose the musical score, gave a hoe-down for them at his ranch. Sinatra sang, Gene Kelly danced and John Garfield donned boxing gloves. Then came a song written by Ann Ronell, author of 'Willow Weep for Me', 'Big Bad Wolf', etc. The tune was a blending of Haydn's 'Surprise' Symphony, Mendelssohn's Wedding March and Bernstein's _Fancy Free_ , _On the Town_ and _Jeremiah_. The lyrics ended with the announcement: 'This party has been staged, Because they got engaged. Len & Felicia, Are now officia\u2013lly Two.' \" In the summer of 1947, Felicia spent time with the Bernstein clan at Tanglewood, and Humphrey Burton wrote of the tensions: \"Felicia came to Tanglewood for two long spells that summer [...] Life was not easy for her despite her official status as Leonard's fianc\u00e9e. There was rivalry with Shirley, ostensibly about such mundane matters as who should sit next to Leonard at meals. Helen Coates was also fighting to maintain her old position. Felicia said later that her self-confidence was undermined as Leonard constantly found fault with her\" (Burton 1994, p. 166). By September 1947, the couple had decided to call off the engagement and the _Journal-American_ on 11 September reported that \"Leonard Bernstein's matrimonial plans have been cancelled.\" Four years later, on 12 August 1951, the Associated Press announced the couple's second engagement, made by Mrs. Serge Koussevitzky at a supper for the faculty of the Berkshire Music Center at Tanglewood, where Bernstein had been teaching and conducting during the summer. This time, the engagement was followed by their marriage a month later, on 9 September 1951.\n\n60 During the summer of 1946, Felicia performed in the Broadway production of _Swan Song_ by Charles MacArthur and Ben Hecht at the Booth Theatre.\n\n61 A reference to Bernstein's unhappiness during his visit to London in June 1946.\n\n62 The new Bernstein-Robbins ballet _Facsimile_ was first performed on 24 October 1946 at the Broadway Theatre, New York, by Ballet Theatre, with Bernstein conducting.\n\n63 Philip and Barbara Marcuse's children, Ann and Philip.\n\n64 Solomon Braslavsky (1887\u20131975) was born in Ukraine and studied in Vienna, where he was subsequently appointed professor at the Jewish Theological Seminary and conducted Jewish choirs and orchestras before moving to Boston to become Director of Music at Temple Mishkan Tefila in 1928 (see Sarna 2009, p. 39, for details of Braslavsky's early career). Bernstein was overwhelmed by the music he heard at Temple Mishkan Tefila, and wrote to Braslavsky in 1973 that he would \"never forget the tremendous influence you and your music made on me when I was a youngster.\"\n\n65 The text of this letter is taken from the version published in the _Jewish Advocate_ on 17 October 1946, p. 6.\n\n66 Braslavsky wasted no time doing so: the letter appeared in the _Jewish Advocate_ on 17 October 1946, just a week after Bernstein wrote to Braslavsky.\n\n67 The Preface for this publication was written by Hugo Leichentritt, who pointed out the similarities between Hebrew sacred music and Gregorian chant, and writes that \"Mr. Braslavsky's arrangement is distinguished by a close acquaintance with the peculiar style of this old religious music, and by the skill and beauty of its harmonic treatment.\" A review by Jules Wolfers appeared in the _Jewish Advocate_ (25 September 1947): \"Four extremely interesting Hebrew chants arranged for four part mixed voices and organ [...] have recently been published by McLaughlin and Reilly, Boston. That a Catholic publishing house is the medium through which these chants are issued is in itself indicative of the all-over worth of this music. The day is past when Jewish music was of interest only to Jews. [...] For Jewish choirs and choral groups their chants are obviously a must. In addition, any person interested in Jewish music will probably wish to acquire this set. Publication of four traditional Jewish chants by a firm named McLaughlin and Reilly must make some sort of publishing history. This is a commendable and heart-warming gesture.\"\n\n68 Paul Wittgenstein (1887\u20131961), Austrian-born pianist who commissioned a number of important new works for piano left hand after he lost his right arm during the First World War. The composers he commissioned included Britten, Hindemith, Korngold, Prokofiev, Franz Schmidt, Richard Strauss, and Alexandre Tansman. The most famous Wittgenstein commission was Ravel's Piano Concerto for the Left Hand, the subject of this letter.\n\n69 Wittgenstein performed the Ravel concerto with Bernstein conducting the New York City Symphony on 14 and 15 October, in a program that also included Ravel's _Le Tombeau de Couperin_ and Beethoven's _Eroica_ Symphony. Olin Downes, reviewing the Monday performance in _The New York Times_ (15 October) praised Wittgenstein's playing: \"in a most authoritative way, he interpreted the music... He has a singing tone as well as five fingers with well-nigh the virtuosity of ten, and he is a colorist who understands not only the piano part but every detail of the orchestration. Mr. Bernstein, conducting, supplied a spirited accompaniment and both men acknowledged the long applause.\"\n\n70 Though this letter is undated, it must have been written on 25 October 1946, the day after the \"ballet premiere\" that Bernstein mentions (the first performance of _Facsimile_ ). The concert to which the letter refers was given by the New York City Symphony on a visit to Boston on 13 November 1946. Though Bernstein had included the _Enigma Variations_ in his New York programme on 11 November, the _Christian Science Monitor_ announced the change to Dvo\u0159\u00e1k's Second [Seventh] Symphony in a short article published on 12 November.\n\n71 It's difficult to see what Koussevitzky's problem was with Bernstein bringing his own New York City Symphony to Boston, but clearly the whole episode distressed Bernstein. Relations between the two continued to deteriorate in the last weeks of 1946.\n\n72 Lukas Foss (1922\u20132009), German-born American composer, conductor, and pianist. Foss' friendship with Bernstein lasted fifty years, from the time of their first meeting at Tanglewood in 1940 until Bernstein's death in 1990. He consistently supported Bernstein's compositions and often appeared as the piano soloist in _The Age of Anxiety_. He conducted the first performance of the _Symphonic Dances from West Side Story_.\n\n73 It certainly was. _If The Shoe Fits_ (with a score by David Raksin) opened on Broadway on 5 December 1946 and closed after just twenty-one performances.\n\n74 This refers to a project that was never realized. Bernstein is referring to the proposed film _The Beckoning Fair One_ in which he was to co-star and for which he was to compose and conduct the score (see Letters 228 and 238).\n\n75 Program annotator for the Boston Symphony Orchestra.\n\n76 The Associated Press broke the news of the engagement in Hollywood on 31 December 1946 and it was quickly reported by the East Coast press.\n\n77 Thor Johnson (1913\u201375) was appointed Music Director of the Cincinnati Symphony Orchestra in 1947, announced as the youngest native-born American conductor to lead a major American orchestra. He remained in Cincinnati for eleven years. During his tenure of the orchestra he conducted the premieres of 120 American and European works, many of which he had commissioned.\n\n78 Farley Granger (1925\u20132011), American actor best known for his roles in two Alfred Hitchcock films: _Rope_ and _Strangers on a Train_. His first Hollywood appearance had been as Damian Simonov in _The North Star_ (he met Aaron Copland during the filming). According to Granger's memoirs ( _Include Me Out_ ), he subsequently had a two-night fling with Bernstein in the late 1940s.\n\n79 Ethel Schwartz and Saul Chaplin were married at the time. They divorced in 1949.\n\n80 Leonore Goldstein (1875\u20131971) was on the Board of Directors of the St. Louis Symphony for over sixty-five years. Bernstein often called her \"Leonore III\" after Beethoven's overture. She was the wife of Dr. Max Goldstein, founder of the Central Institute of the Deaf in St. Louis.\n\n81 Stanley Donen (b. 1924), American director of some of the most famous Hollywood musicals. His credits include _On the Town_ , _Singin' in the Rain_ , and _It's Always Fair Weather_.\n\n82 This letter is undated, but must date from early 1947. Bernstein had a four-week guest engagement with the Boston Symphony Orchestra in February 1947, and Bette Davis was pregnant at the time: her daughter Barbara was born on 1 May 1947.\n\n83 Place and date added by Helen Coates.\n\n84 The score for _Spellbound_ was composed by Mikl\u00f3s R\u00f3zsa.\n\n85 Progressive Citizens of America. In Bernstein's FBI file, an Office Memorandum dated 2 March 1949, to the Director from D. A. Ladd, indicates (p. 10) that \"Buffalo informant [redacted] advised that on March 25, 1947, Bernstein had been the principal speaker at a meeting of the Progressive Citizens of America in Buffalo, New York.\"\n\n86 Alice Berezowsky was a friend of Koussevitzky, wife of the composer Nicolai Berezowsky, and author of the book _Duet With Nicky_ (1943).\n\n87 Adolph Green was 32 at the time of writing this letter (b. 2 December 1914).\n\n88 Evidently, this was a working title for _Easter Parade_ , released in 1948. Comden and Green were not involved.\n\n89 _Easter Parade_ , with a score by Irving Berlin, starred Judy Garland and Fred Astaire. Gene Kelly, Frank Sinatra, and Kathryn Grayson were not in the film.\n\n90 Allyn Ann McLerie (b. 1926), Canadian-born actress. Her Broadway debut was in Kurt Weill's _One Touch of Venus_ (1943), and she performed in the ensemble of _On the Town_ before replacing Sono Osato as Ivy Smith.\n\n91 Adolph Green married Allyn Ann McLerie on 21 March 1947. They divorced in 1953 and she married the actor George Gaynes the same year.\n\n92 Lewis Funke: \"News and Gossip Gathered on the Rialto,\" _The New York Times_ , 16 February 1947. Concerning _Allegro_ , Funke wrote that although Rodgers and Hammerstein were \"standing guard over it like a couple of Fort Knox sentries, Mr. Rodgers admitted the other afternoon that it would be a departure from the conventional. In a mood of convivial candor he even breathed, 'experimental', and said it would combine dance, drama and music as an integrated unit \u2013 something the avant garde has been talking about for a long time.\"\n\n93 Roger Edens (1905\u201370) was a composer and producer in Hollywood, an important member of Arthur Freed's team at MGM. Edens is perhaps best known for nurturing the talent of the young Judy Garland (they became lifelong friends). He was Associate Producer on a string of successful MGM musicals. Comden and Green's greatest Hollywood success was _Singin' In The Rain_ (1952), for which they wrote both the story and the screenplay. Most of the songs were by Freed and Nacio Herb Brown, but the new song for which Comden and Green wrote lyrics was \"Moses Supposes,\" with music by Roger Edens.\n\n94 In spring 1947, Bernstein was listed as a member of the Ren\u00e9e Longy Miquelle anniversary committee, set up to celebrate her 50th birthday. It is a sign of how widely admired she was that the members of the committee included Samuel Barber, Olin Downes, Gian-Carlo Menotti, Fritz Reiner, and Randall Thompson. Its aim was to encourage donations from friends, colleagues, and former pupils to pay the outstanding mortgage on her cottage on Cape Cod to \"insure her future security\" in honor of her \"long-standing devotion to the profession.\"\n\n95 Sid Ramin married Gloria Breit on 9 January 1949.\n\n96 Note in Helen Coates' hand: \"Tried to call him but he was in Boston.\"\n\n97 April 1947 was a crucial time in the history of Palestine, soon to become Israel. On 2 April the British government referred the problem of the future of Palestine to the United Nations, and on 13 May the UN appointed a Special Committee to examine the question of Palestine.\n\n98 Bernstein conducted the European premiere of Copland's Third Symphony with the Czech Philharmonic Orchestra on 25 May 1947, at the Prague Spring Festival.\n\n99 Koussevitzky had commissioned Copland's Third Symphony, and gave the first performance in Boston on 18 October 1946.\n\n100 Thomson's _The Mother of Us All_ was first performed on 7 May 1947 at Columbia University. The cast included Teresa Stich-Randall as Henrietta, her operatic debut.\n\n101 _In the Beginning_ was first performed on 2 May 1947 at Harvard Memorial Chapel by the Collegiate Chorale, conducted by Robert Shaw.\n\n102 Romolo de Spirito (sometimes given as \"di Spirito\"), a tenor who specialized in the performance of music by American composers. In his New York debut recital (27 February 1944) he included songs by Paul Bowles, David Diamond, and Virgil Thomson.\n\n103 Carrington Welch, who was Romolo de Spirito's regular accompanist.\n\n104 The legendary singer and actress Lena Horne (1917\u20132010) was closely involved in the civil rights movement. She fought institutional racism in Hollywood in the 1940s (her scenes were customarily shot so that they could be removed for distribution to states in the South), she refused to sing for segregated audiences of troops, and she worked with Eleanor Roosevelt to pass anti-lynching laws. She was named as a Communist sympathizer \u2013 along with the likes of Bernstein, Marc Blitzstein, Aaron Copland, Judy Holliday, Langston Hughes, Burl Ives, Zero Mostel, Dorothy Parker, Pete Seeger, and Artie Shaw \u2013 in the infamous _Red Channels: The Report of Communist Influence in Radio and Television_ (1950), and she was blacklisted by Hollywood. After her death in 2010, the president of the NAACP \u2013 America's oldest and largest civil rights organization \u2013 described Horne as \"an outstanding, groundbreaking entertainer and a staunch civil rights activist who stood on the side of justice and equality. Lena Horne won the hearts of millions of Americans of all backgrounds as a glamorous and graceful actress and singer. She broke many color barriers and fought valiantly to bring down the institutionalized racism that plagues our society and prevents all Americans from an equal opportunity to pursue the American dream.\"\n\n105 This may have been the occasion referred to by the television personality Ed Sullivan \u2013 \"at the very least a facilitator, if not an informant, for the FBI and the House Un-American Activities Committee\" (Vaill 2007, p. 171) \u2013 when he put pressure on Jerome Robbins in 1950 to disclose \"the names of people who had been at a cause party for Soviet\u2013American friendship he'd allowed the singer Lena Horne to give at his apartment\" (Vaill 2007, p. 172). At that point Robbins did not name names, though he did so when he testified in public to the HUAC in May 1953.\n\n106 This was one of many organizations supporting African-American causes that came under suspicion from the government. FBI records reveal that the United Negro and Allied Veterans of America was described by US Attorney General Tom Clark on 4 December 1947 as \"subversive and among the affiliates and committees of the Communist Party, U.S.A. which seeks to alter the form of government of the United States by unconstitutional means.\"\n\n107 Richard Adams Romney (1918\u20132009) was often known to his friends as \"Twig\". A collection of letters to him from Bernstein, Christopher Isherwood, Osbert Sitwell, Pavel Tchelitchew, John Van Druten, and others is to be found in Yale University Library (Beinecke Library, Gen Mss 462). Romney's obituary published in the _Albany Times Union_ (19 July 2009) includes the following information: \"Richard Adams Romney, born July 15, 1918 in Salt Lake City, Utah, died in Troy, N.Y. on July 15, 2009. Mr. Romney had been a resident of the Van Rensselaer Manor since September 2001 where he received wonderful care and made many friends. A veteran of the US Coast Guard, he served in the North Sea, receiving an honorable discharge in 1944. Mr. Romney was a resident of Manhattan's Upper East Side from 1945 to 1997 where he worked in the real estate and insurance industries. From 1950 to 1954, he was a gallery assistant at the Betty Parsons Gallery, Manhattan, the first home of artists like Ellsworth Kelly, Jackson Pollock, and Mark Rothko. He was the original owner of Pollock's _Number 3, 1949_. The great relationship of his life was that with the American heiress and supporter of the arts, Alice De Lamar. Their correspondence for half a century resides in the Beinecke at Yale. There are no surviving family members.\"\n\n108 The United States Department of Veterans Affairs. Romney had served as a Coast Guard in the Second World War.\n\n109 _The Gallery_ by John Horne Burns (1916\u201353) was published in the summer of 1947 and acclaimed by the likes of Gore Vidal, Edmund Wilson, and John Dos Passos. Burns, a Harvard graduate, served as a US intelligence officer in North Africa and Italy during the war, and his book is one of the first to explore gay life in the army.\n\n110 This letter indicates that the suggestion for using Auden's _The Age of Anxiety_ as an inspiration came initially from Romney, who developed his idea in the letter of 29 July 1947. Romney sent Bernstein a copy of the poem very soon after it was first published in July 1947. In Bernstein's reply of 1 August he thanks Romney for the books, but after that there is no mention of _The Age of Anxiety_ in his letters to Romney until May 1950.\n\n111 Romney predicts the early performance history of _The Age of Anxiety_ with uncanny accuracy. After its first concert performances in 1949, the work was used for a ballet by Jerome Robbins in 1950.\n\n112 Presumably a reference to Bernstein's concert at Tanglewood on 27 July 1947 when he conducted the Boston Symphony Orchestra in Mozart's _Magic Flute_ Overture, Schubert's \"Great\" C major Symphony, and Stravinsky's _Rite of Spring_ , the first time anyone other than Koussevitzky had conducted the Boston Symphony at Tanglewood. _The New York Times_ reported that the concert was attended by 8,500 people and that \"Dr Koussevitzky listened from his box in the center of the music shed, and later appeared on the stage to congratulate Mr. Bernstein.\"\n\n113 Whatever was \"enclosed\" has not survived.\n\n114 James M. Cain (1892\u20131977) was an American writer and journalist whose novels included _The Postman Always Rings Twice_ (1934), _Mildred Pierce_ (1941) and _Double Indemnity_ (1943) as well as _Serenade_ (1937). Though Cain disliked the label, he was one of the leading writers of \"hardboiled\" crime fiction.\n\n115 The \"most successful operetta composer we have\" and the \"highly successful librettist\" who had approached Cain (in 1940) were Sigmund Romberg and Oscar Hammerstein II (see Hoopes 1982, p. 366), so Bernstein had reason to be flattered by Cain's positive response to his request for permission to base an opera on _Serenade_.\n\n116 This project was reported in _The New York Times_ more than a year after the exchange of letters between Bernstein and Cain. On 5 December 1948 an article headed \"Opera Projects\" stated: \"Leonard Bernstein has asked James M. Cain for permission to base an opera on _Serenade_. He also asked Mr. Cain to write the libretto for him. The author declined the job, suggesting the composer was competent to write his own book. He promised, though, that he would give no one else prior operatic rights to the novel before the end of the year. He himself is skeptical about the project, for he wrote his agent, Harold Ober: 'I know that anyone who undertakes any stage work based on this book is letting himself in for a thousand headaches.' It was in April that Mr. Bernstein gained his promise from Mr. Cain. Since the composer\u2013conductor will not be back from Palestine till some time this week, it is not known how far he has gone with his plans.\" In fact, this project went back to the autumn of 1947, as we see in the correspondence between Cain and Bernstein.\n\n117 The \"theme\" of _Serenade_ referred to in the letters between Cain and Bernstein needs some explanation. In the original novel, the opera singer John Howard Sharp loses his voice, ostensibly as a consequence of the trauma of his gay relationship with a famous conductor. His voice is restored when he falls in love with a young Mexican prostitute. Cain explained his premise in a letter to his old friend (and erstwhile colleague on the _Baltimore Sun_ ), H. L. Mencken: \"The lamentable sounds that issue from a homo's throat when he sings are a matter of personal observation.... But the theme demanded the next step, the unwarranted corollary that heavy workouts with a woman would bring out the stud horse high notes\" (see Paul Skenazy, _James M. Cain_ , New York: Continuum, 1989, p. 54). In the end, Bernstein abandoned his _Serenade_ project, but a few years later plans were made for a Broadway musical based on the same story. Louis Calta reported in _The New York Times_ on 11 November 1954 that \"The musical stage rights to _Serenade_ , James M. Cain's earthy and highly successful novel of 1937, have been purchased [...] Arthur Laurents [...] has agreed to do the adaptation. Shortly the producers hope to announce the composer and lyricist for the musical venture.\" Stephen Sondheim was auditioned as a potential lyric writer, and Bernstein was asked whether he wanted to compose the score (see his letter of 6 May 1955 to Felicia, Letter 353). This project, too, came to nothing. The 1956 film adaptation of _Serenade_ , starring Mario Lanza, differs wildly from Cain's novel.\n\n118 Cain's new wife, Florence Macbeth, was an opera singer.\n\n119 Cain attached a formal agreement, reserving for Bernstein the dramatic rights to _Serenade_ until 31 December 1948.\n\n120 Burr was Vice-President of the United States under Thomas Jefferson. In 1804, he challenged Alexander Hamilton (former Secretary to the Treasury) to a duel in which Hamilton was mortally wounded.\n\n121 A report of the concert appeared in _The New York Times_ on 11 May 1948: \"at the close of the performance, the audience stood on its feet and applauded [Bernstein] for more than ten minutes in repeated curtain calls, amid a storm of 'bravos'. After the first half dozen bows, Bernstein returned to the podium and with the orchestra, repeated the final portion of the [Ravel] concerto.\"\n\n122 Bruno Walter conducted Mahler's _Resurrection_ Symphony with the Vienna Philharmonic in the Musikverein on 15 May 1948. It's extraordinary to think that this legendary \u2013 and marvelous \u2013 performance (with Maria Cebotari and Rosette Anday as the soloists) was given to a half-full house. A recording of it has been issued on CD by Sony Japan (SICC 92\u20133) and others.\n\n123 \"All Vienna in one go,\" but Bernstein's Viennese debut was also his one and only concert with the Vienna Symphony Orchestra. The program consisted of Schumann's Second Symphony, the Dvo\u0159\u00e1k Violin Concerto (with Gerhard Taschner), and the Ravel G major Piano Concerto with Bernstein directing from the piano. Bernstein's report of an enthusiastic audience may be true, but the concert wasn't a critical success: according to Burton (1994, p. 178), \"several Viennese critics disliked Bernstein's conducting style intensely [...] he did not work again with the Vienna Symphony and it was nearly twenty years before he overcame his prejudice and accepted another Viennese conducting engagement.\"\n\n124 Koussevitzky sent a telegram on 9 November 1948: \"Deeply moved your letter authorize you select outstanding student conductor. Heartiest greetings to all and orchestra. Love Serge Koussevitzky.\"\n\n125 Bernstein was never afraid to make cuts in recent pieces, even ones as substantial and significant as Copland's Third Symphony.\n\n126 This latest love may well have been Yossi Stern, the Hungarian-born Israeli artist (1923\u201392), who illustrated Letter 276.\n\n127 This letter, describing Bernstein's experiences in Israel, is illustrated on every page by Yossi Stern.\n\n128 Yardena Cohen (1910\u20132012), Israeli dancer, choreographer, and teacher. In the 1940s this legendary figure in the dance history of Israel created dance dramas and pageants for kibbutzim. Many of these featured female characters in the Old Testament as the central roles. Cohen opened her Haifa dance studio in 1933 and ran it for the next seventy years. She died on 23 January 2012, at the age of 101.\n\n129 The \"Furtw\u00e4ngler & Gieseking business\" is a reference to the banning of Walter Gieseking from a concert tour of the United States at the time Bernstein sent this letter, and of Wilhelm Furtw\u00e4ngler from returning to conduct the Berlin Philharmonic the same month, both because of concerns about their Nazi past. The \" _echt_ Fascism\" that so enraged Bernstein was taking place in Detroit. According to a report in _The New York Times_ on 21 January 1949, the Detroit Symphony Orchestra \"had been warned by Mr. [Henry] Reichhold that every man would be fired if that were necessary to weed out disloyalty to the conductor [Karl Krueger].\" Reichhold, the orchestra's president added: \"I think a shake-up and good housecleaning is just what the Detroit orchestra needs.\" Georges Miquelle, the orchestra's principal cellist (married to Ren\u00e9e Longy Miquelle in 1919; they later divorced) was fired in public by Reichhold during a rehearsal on the grounds that he had apologized to the violinist Erica Morini about the orchestra's poor accompaniment for her \u2013 an apology that both Miquelle and Morini denied was ever made.\n\n130 Ren\u00e9e Nell (1910\u201394) was a Jungian psychoanalyst. In 1938 she escaped from Berlin to Switzerland where she studied with Carl Jung at the University of Zurich, before moving to the United States and setting up her practice in New York City. A pioneer in work with young offenders, she later established The Country Place in Litchfield, CT, describing it as \"a residential community for the psychologically disturbed adult who has more insight than he or she can use, who knows how he or she should act but withdraws from action.\" Humphrey Burton identifies her as the \"Frau\" (Burton 1994, p. 108), but the letter from Bernstein to David Oppenheim on 22 October 1943 confirms that the \"Frau\" was in fact Marketa Morris. Bernstein later became disenchanted with Nell's analysis, writing to his sister Shirley on 26 April 1950: \"My feeling is one totally apart from analysis: I want only to cope, and through my own powers, without aid \u2013 especially of the indulgent, personal sort that was forthcoming from Miss Nell.\"\n\n131 Howard Hoyt was a theatrical agent and manager who had previously worked as Eastern story editor for MGM.\n\n132 The only known use in Bernstein's papers of _Operation Capulet_ as an early title for what became _West Side Story_. Jerome Robbins first put the idea of the show to Bernstein on 6 January 1949. Robbins, Bernstein, and Arthur Laurents met for the first time to discuss the project on 10 January, and a month later Bernstein signed this agreement with Howard Hoyt.\n\n133 Hans Heinsheimer (1900\u201393) was a legendary figure in the world of music publishing. He first worked in the opera department of Universal Edition in Vienna (1923\u201338), then went to New York to take up a position at Boosey & Hawkes, promoting new works by Copland, Bart\u00f3k, Stravinsky, and Britten. He was fired by Ralph Hawkes in 1947 for writing his memoirs ( _Menagerie in F Sharp_ ), since Hawkes wanted a worker rather than an author. Heinsheimer was immediately hired by Schirmer, where he worked closely with Bernstein, Samuel Barber, and others.\n\n134 The first performance was on 8 April 1949.\n\n135 Bernstein was working on his Second Symphony, _The Age of Anxiety_.\n\n136 The Pittsburgh Symphony Orchestra, with whom Bernstein was conducting a series of concerts. Reiner left as Music Director in 1948.\n\n137 J. Fred Lissfelt, music critic of the Pittsburgh _Sun-Telegraph_.\n\n138 Izler Solomon (1910\u201387), American conductor. He was Music Director of the Columbus Philharmonic Orchestra (1941\u20139) and the Indianapolis Symphony Orchestra (1956\u201376).\n\n139 Tossy Spivakovsky (1906\u201398), Russian-born violinist whose performances of modern concertos such as those of Bart\u00f3k, Menotti, and Sessions were particularly admired.\n\n140 Arthur Laurents (1917\u20132011), American playwright, screenwriter, and stage director. His Broadway credits included the books for _West Side Story_ , _Gypsy_ , and _Anyone Can Whistle_ , and he directed the original production of _La Cage aux Folles_.\n\n141 This letter is undated, but the evidence points to some time in April 1949, when Bernstein began to have doubts about the viability of the project that would eventually become _West Side Story_. In Bernstein's \"Excerpts from a West Side Log\" (Bernstein 1957, p. 47), he included an entry on 15 April 1949, while conducting in Columbus, Ohio: \"Just received the draft of first four scenes. Much good stuff. But this is no way to work. Me on this long conducting tour, Arthur between New York and Hollywood. Maybe we'd better wait until I can find a continuous hunk of time to devote to the project. Obviously this show can't depend on stars, being about kids; and so it will have to live or die by the success of its collaborations; and this remote-control collaboration isn't right. Maybe they can find the right composer who isn't always skipping off to conduct somewhere. It's not fair to them or to the work.\" Bernstein was plainly uneasy about committing himself and was quoted by Craig Zadan as saying: \"I remember receiving about a dozen pages and saying to myself that this is never going to work. [...] I had a strong feeling of staleness of the East Side situation and I didn't like the too-angry, too-bitchy, too-vulgar tone of it\" (Zadan 1974, p. 15).\n\n142 Ellen Adler (b. 1927) is the daughter of fabled acting teacher Stella Adler. Ellen's close friends included Marlon Brando and Ren\u00e9 Leibowitz as well as Bernstein. In 1957 she married David Oppenheim (they divorced in 1976). Ned Rorem was bewitched by her \"dizzying black-tiger beauty\" (Ned Rorem, _Knowing When To Stop: A Memoir_ , New York: Simon and Schuster, 1994, p. 578).\n\n143 Harold Clurman, Ellen Adler's stepfather.\n\n144 161 West 54th Street is an imposing apartment building near the intersection with Seventh Avenue in Manhattan.\n\n145 Fran\u00e7ois Val\u00e9ry, son of the poet Paul Val\u00e9ry.\n\n146 Adler \u2013 delightfully \u2013 likens Nadia Boulanger's austere appearance to the stern faces in paintings such as Grant Wood's _American Gothic_.\n\n147 Marie-Blanche de Polignac (1897\u20131958) was a soprano, pianist, patron of the arts, and heiress to the Lanvin fashion fortune founded by her mother, Jeanne Lanvin. She was an intimate friend of Francis Poulenc, who dedicated several works to her.\n\n148 Otis Bigelow (1920\u20132007), actor, dancer, writer, and later theatrical agent, who spent a year in Paris in 1948\u20139.\n\n149 Burton Bernstein (b. 1932) is the younger brother of Leonard and Shirley. He studied at Dartmouth College and the Columbia Graduate School of Journalism, and from 1957 to 1992 he was a staff writer for _The New Yorker_. He is the author of a biography of James Thurber as well as books about the Bernstein family ( _Family Matters: Sam, Jennie, and the Kids_ ) and about his elder brother ( _Leonard Bernstein: American Original_ , with Barbara B. Haws).\n\n150 Peter Gradenwitz (1910\u20132001), German-born Israeli musicologist and music critic, and an astute commentator on Bernstein's music and his use of jazz idioms. The two became friends when Bernstein visited Israel in 1948 to conduct the Israel Philharmonic. Gradenwitz wrote extensively on Bernstein and his music, from contemporary reports of Bernstein's Israeli concerts for _The New York Times_ to the book _Leonard Bernstein: unendliche Vielfalt eines Musikers_ (Zurich: Atlantis Verlag, 1984), published in an English edition in 1987.\n\n151 Gradenwitz's _The Music of Israel: Its Rise and Growth Through 5000 Years_ was published by Norton in 1949 and dedicated \"To Leonard Bernstein as a token of friendship and sincere appreciation\".\n\n152 Bernstein worked on his new symphony while in Israel, and the first performance of any part of _The Age of Anxiety_ was given there. The \"Dirge\" was played at a Gala Soir\u00e9e in aid of the Israel Philharmonic Pension Fund on 28 November 1948 in Tel Aviv. In his notes for this concert, Peter Gradenwitz wrote of \"the first performance anywhere of a Dirge for piano and orchestra composed by Leonard Bernstein during the few leisure hours left to him on his crowded Tel-Aviv days and completed in full score just in time for tonight's concert \u2013 this is a most expressive song of lament showing the composer's style developed on distinctly novel lines.\"\n\n153 The world premiere of _The Age of Anxiety_ took place in Boston on 8 April 1949, with the Boston Symphony Orchestra conducted by Serge Koussevitzky, and Bernstein himself playing the solo piano part.\n\n154 Probably a reference to _Side Street_ (released in 1950), a thriller set in New York. Granger's most celebrated film role came the following year when he played Guy Haines in Hitchcock's _Strangers on a Train_.\n\n155 _Take Me Out to the Ball Game_ was a 1949 MGM musical starring Gene Kelly, Frank Sinatra, and Esther Williams.\n\n156 William Schuman (1910\u201392), American composer. He formed a dance band while still in high school and was soon collaborating with Frank Loesser, a neighbor who was also at the start of his career (Loesser's first publication, _In Love With A Memory of You_ , has music by William Schuman. With typical modesty Schuman later said \"Frank Loesser has written hits with Hoagy Carmichael, Burton Lane, Jule Styne and other Hollywood grand dukes, but I have the distinction of having written a flop with him.\"). Schuman's subsequent career was as one of America's most distinguished symphonists, president of the Juilliard School and Lincoln Center.\n\n157 A story in _The New York Times_ on Wednesday, 20 July 1949 reported that Alan Jay Lerner and Fredrick Loewe were \"at work on a new musical [...] that is set in the United States in the nineteenth century\" (the show that became _Paint Your Wagon_ ). From Bernstein's angry reaction in this letter, it seems Lerner must have suggested a collaboration. A quarter of a century later these two Harvard graduates did work together on _1600 Pennsylvania Avenue_.\n\n158 Michael Dreyfuss (1928\u201360), American actor and director.\n\n159 The pianist Menahem Pressler (b. 1923) fled Nazi Germany to Palestine. He made his American debut with Eugene Ormandy and the Philadelphia Orchestra in 1947. Olin Downes reviewed their Carnegie Hall performance of the Schumann Concerto in _The New York Times_ , describing Pressler as \"one of the few of the young pianists who consider his instrument the agent of glamorous song and not merely a contraption of wires and keys. This, indeed, was the playing of a free artist.\" Pressler later achieved renown as pianist of the Beaux Arts Trio. Regarding his relationship with Bernstein, Pressler has written: \"I always had a fine relation with Mr. Bernstein. Although he invited me to play with him and the City Center Orchestra to make my debut coming from Israel, Mr. Judson, my manager, insisted that I do it with the Philadelphia Orchestra and Ormandy. But I played his pieces and met him many times in different places, even here in Bloomington. Recently I played his Clarinet Sonata with [Richard] Stoltzman\" (email from Menahem Pressler, 12 January 2013).\n\n160 Olivier Messiaen (1908\u201392), French composer. His first major international commission was from Koussevitzky, for the _Turangal\u00eela-Symphonie_.\n\n161 Bernstein conducted the world premiere of Messiaen's _Turangal\u00eela-Symphonie_ with the Boston Symphony Orchestra on 2 December 1949.\n\n162 Yvonne Loriod (1924\u20132010), French pianist. A pupil of Messiaen at the Paris Conservatoire, she went on to give the first performances of all his major works featuring the piano. In 1961 she became Messiaen's second wife.\n\n163 Written in French; English translation by the editor.\n\n164 Arturo Toscanini (1867\u20131957), Italian conductor, and one of the most celebrated performing musicians of the twentieth century. This letter was sent after Bernstein had visited him and asked about the different speeds in Toscanini's broadcast and studio recordings of Berlioz's _Romeo and Juliet_ (see Burton 1994, p. 196).\n\n165 Irwin Edman (1896\u20131954), American philosopher.\n\n166 A reference to the \"Imaginary Conversation\" subtitled \"Why Beethoven?\" that was published in Bernstein's _The Joy of Music_ (New York: Simon and Schuster, 1959), pp. 21\u201339, in which \"L.B.\", \"L.P.\" (\"Lyric Poet,\" described as a \"poet's poet from Britain\" \u2013 Stephen Spender), and \"Y.B.\" (\"Younger Brother\") converse (see B. Bernstein 1982, pp. 179\u201380).\n\n167 A letter to Shirley Bernstein (\"Hi-Lee\"), presumably a \"North Country Epic\" since it had been sent from Dartmouth College in Hanover, New Hampshire, where Burton was a student.\n\n168 Messiaen's _Turangal\u00eela-Symphonie_.\n\n169 _Regina_ is an opera by Marc Blitzstein based on Lillian Hellman's _The Little Foxes_. It opened at the 46th Street Theatre on Broadway on 31 October 1949, the day before Bernstein wrote this letter.\n\n170 Written in French; English translation by the editor.\n\n171 Robert Fryer (1921\u20132000), theatrical producer whose first Broadway show was _A Tree Grows in Brooklyn_. He later had a string of successes including _Wonderful Town_ , _Sweet Charity_ , _Chicago_ , and _Sweeney Todd_.\n\n172 Betty Smith (1896\u20131972) was the author of the novel _A Tree Grows in Brooklyn_. She collaborated with George Abbott on the stage adaptation.\n\n173 According to a note in Helen Coates' hand, Bernstein replied to Abbott by phone on 12 December, turning down the project. The score for _A Tree Grows in Brooklyn_ (which ran for 257 performances in 1951) was composed by Arthur Schwartz, with lyrics by Dorothy Fields. Robbins was involved informally (uncredited show-doctoring); the musical director was Max Goberman, whose greatest Broadway successes were _On the Town_ and _West Side Story_.\n4\n\nMarriage, Passport Problems, and Italy\n\n1950\u201355\n\nBy 1950, Bernstein had decided that he needed Felicia in his life \u2013 though he seemed unable to tell her directly: two letters sent from Israel to his sister Shirley suggest that he wanted her to be a kind of intermediary \u2013 wooing her back by proxy. Strange as this may seem, it worked, and they were married (after the shortest of second engagements) in September 1951. Extended trips abroad meant that it wasn't only his personal life that was being run by remote control. The production of _Peter Pan_ , for which Bernstein wrote delightful incidental music, was in rehearsal while he was in Israel, and Marc Blitzstein took on responsibility for overseeing things \u2013 describing them in lively detail to the absentee composer. The now-familiar tensions were building: Bernstein the composer was being sent ideas for a new musical by Betty Comden, while Bernstein the conductor was meeting Wilhelm Furtw\u00e4ngler at the Holland Festival before setting off to other European destinations with his brother and sister. The result is a charming series of letters to Bernstein's parents in August\u2013September 1950. That relationship took an odd turn the following year when \"Sam\" and \"Jennie\" became the dysfunctional principals in the one-act opera _Trouble in Tahiti_ : writing to Shirley from Mexico in 1951, Bernstein wrote that \"the two characters, by the way, have gotten themselves called Sam and Jennie, and I think you'll see why.\" That same letter raises a serious issue for Bernstein \u2013 and many of his friends \u2013 in the early 1950s: the \"Red Scare\" and the witch-hunting activities of Senator Joseph McCarthy and others. Aware that others were already coming under scrutiny, Bernstein says it's time to prepare \"our blazing orations now,\" adding that \"I hope I'm as brave as I sound from this distance when it catches up with me.\"\n\nMeanwhile, there was the more private matter of his marriage to Felicia. Bernstein's letters to friends and family suggest that he regarded the first few months of their life together almost as a kind of social experiment, while an undated letter from Felicia underlines the strength of her love for him, but that she was under no illusions: \"you are a homosexual and may never change \u2013 you don't admit to the possibility of a double life, but if your peace of mind, your health, your whole nervous system depend on a certain sexual pattern what can you do?\" It was the birth of Jamie, their first child, in 1952 which transformed their marriage from a slightly uneasy alliance to something that could endure.\n\nPlans for new projects came and went, some of them fascinating, such as an idea discussed with Blitzstein in 1952 for an opera on the life of Eva (Evita) Per\u00f3n. But conducting engagements in Europe and Israel were piling up, and finding time for composition was becoming increasingly difficult. In 1953 the U.S. State Department became a significant player in Bernstein's future \u2013 and a thorn in his side. These were perilous times for Americans with liberal sympathies, especially those involved in film, theater, radio, television, and music. Bernstein was one of the 151 names in the entertainment industry whose \"Communist\" associations were chronicled in _Red Channels_ , published in June 1950, and he came under suspicion throughout the fifties. While he was never summoned by the House Un-American Activities Committee (Copland, Robbins, and Diamond \u2013 among many others \u2013 weren't so fortunate), Bernstein was required to provide exhaustive details about his former associations with groups regarded as suspect, to renounce Communism, and to pledge his loyalty to the country \u2013 all in order to have his passport renewed. The State Department demanded sworn testimony, and the result is Bernstein's long and grimly absorbing affidavit (Letter 328). He wrote to his brother Burton, and to David Diamond, about the humiliation of this episode, but at least he was able to travel again, and in 1954 and 1955 Bernstein wrote a series of long, warm, and often funny letters to Felicia \u2013 including the bizarre tale of the disturbed young man who pursued Bernstein around Italy, threatening to blackmail him. He finished one important new work, the _Serenade_ for violin and orchestra, but another was to prove much more intractable: _Candide_ was to give Bernstein a lot of trouble, and took years of precious composing time, but he stuck with it. On the domestic front, things were a great deal happier, with Felicia expecting a second baby whose nickname _in utero_ was \"Fink\".\n\n295. Marian MacDowell1 to Leonard Bernstein\n\nHillcrest, Peterborough, NH\n\n4 March 1950\n\nDear Mr. Bernstein,\n\nI have just written Lukas [Foss] that I could not help feeling a deep personal pride in your programme with the Philharmonic last Sunday. I think Lukas did his part extraordinarily well, but I don't know when I have been so stirred and excited as I was during the playing of your composition. A splendid piece of work! It came over very well on the radio but I wish I might have been there.2\n\nEvery now and then I most particularly wish I were not ninety-two for I can't do as many things that I would like to do. As for instance, when you are here, leading the local Philharmonic, I am afraid I dare not take the risk of the fatigue which would be entailed should I attempt to go to the concert.\n\nI am really very well but with too little strength to do the things I want to do. I would beg you to come and see me but I know you are going to be overly busy and I dare not hope for it. All the same I would love to.\n\nI have just one regret \u2013 that while Lukas had been a devoted Colonist you should never have been there.3\n\nLet me thank you for the very kind letter you wrote me a couple of weeks ago. When I heard you converse in the Green Room of the New York Philharmonic giving an outline of what you had done this season and what you are going to do it seemed incredible.\n\nWith every warm good wish and deep admiration.\n\nMost sincerely yours,\n\nMarian MacDowell\n\n296. Marc Blitzstein4 to Leonard Bernstein\n\n4 East 12th Street, New York, NY\n\n16 April 1950\n\nLenny dear,\n\nYour casual throwaway phrase: \"If you run into trouble on lyrics (in _Peter Pan_ ), consult Marc \u2013 he's my deputy\" \u2013 has borne all kind of fruit: raw, ripe, rotten.5 First, they wanted \"Dream With Me\" revised to make Wendy virginal but grown-up, intimate but not \"commercial\", etc. So I re-did it. Then it was decided that what the last few minutes of the play needed was a reprise, not a new song. Scurryings about, searching for a spot to introduce \"Dream With Me\" before the end; no luck. So \"Who Am I?\" took its place in the finale, with new lyrics by M.B. More stuff was needed for the mermaids' song, to fill in a spot-in-one \u2013 while they changed sets from the Pirate Ship (thirty-one) to the Nursery (now thirty-three). At this moment that set-change is so complicated that even more insert-music-and-lyrics would seem to be required; and so on. At this moment, two days before the first preview, the production seems generically right (if you like _Peter Pan_ at all), but specifically right almost nowhere.6 That will change for the better; the tricks are cute, adding up to what seems a Hippodrome extravaganza. Hershy [Kay] has done a fine brilliant transparent job of orchestration; Trude [Rittmann] remains the best person in the field for \"incidentals\";7 Miss Arthur gets curiouser and curiouser, which may even help her performance, if she gets over opening-night jitters. [John] Burrell8 is sound theatrically, but lacking in inventiveness. Great hulking crises in matters of flying equipment, set-troubles (Alswang9 is fine but should be pruned down, too many busy gadgets on stage), and musicians'-union bickerings. [Ben] Steinberg10 is almost (not quite) a loss; he refuses to bang down a beat for the poor unmusical performers. The seat of the trouble, I'm afraid, is Peter Lawrence's ineptitude as a producer; much as I like him personally, he's in a deep-sea fog, or appears to be. Who knows? It will probably turn out to be the hit of the century.\n\nI cabled you re Ballo of Radio-Roma, because I had a radiogram from him, asking me to send _Regina_ for consideration at the Venice Festival; and I thought you might talk it up. I hope I didn't shackle you with needless problems during your overworked stay in Italy.\n\nPeter Frye wants to do _Cradle_ in Israel this summer \u2013 Chamber Theatre in Tel Aviv (Guttmann-Bartov, manager). I'd love it, and I'd love to come. Could I send the score to Helen, with instructions not to release it until everything is settled?\n\nThe new opera is nearly half-finished! I seem to be working mightily, but need a couple of weeks' rest badly. One song, lied-like, is the best I've ever written. And I have sunk my teeth into a translation of the whole of the _Dreigroschenoper_ as a sort of memorial to Kurt [Weill]. Folks (Cheryl [Crawford], Lee Strasburg, Gadg [Elia] Kazan, etc.) are wildly enthusiastic at the seven songs already completed; and it may turn out to be a production.11\n\nThis has got to stop somewhere. Love. Have you seen Kit?\n\nMarc\n\n297. Leonard Bernstein to Burton Bernstein\n\nTurin, Italy\n\n18 April 1950\n\nDear Frent,\n\n\u2013 Then they lost my tails at the Naples hotel half an hour before the concert (which was at six, & rehearsals had lasted til four, with many chills and fevers and a raw cold hall) and the piano broke down mitten der Ravel in Milano, and every city I come to there's always a mess. But such fun & fury! And big successes, natch. Speaking of natch, they called Schoenberg's _Verkl\u00e4rte Nacht_ \"Verkl\u00e4rte Natch\" in Naples. I have just come to Turin by sheer dint, having been in Milan on Sunday (which is 2 hours from here) but had to go to Rome yesterday for a meeting with [Sol] Hurok & Kouss. So I told the Turin people I'd be late, took a 7 a.m. plane this morning, Rome to Milan, hired a private car in Milan (15,000 lire!!), sped to Turin, arrived at this glad-rags hotel, called the orchestra people \u2013 and there's nobody there who knows anything \u2013 in fact there's nobody there, & they're all eating & who am I anyway, and \"Vabbene, Maestro, vabbenissimo.\"\n\nThen there was the time the bed fell on grandfather...12\n\nLoved your letter. You're growing it up, growing it up, M\u00fc la d\u00fc. Be sure to be in N.Y. when I arrive mid-July.\n\nLad\u00fcm\u00fc\n\n298. Leonard Bernstein to Shirley Bernstein\n\nSharon Hotel, Herzlia on Sea, Israel\n\n26 April 1950\n\nSweedie,\n\nWithout even waiting for cable news of the _Peter Pan_ opening, I hasten to write you apropos your letter which I received Monday in Rome. I arrived in Jewland yesterday, a hulk of a once proud ship, ridden with intestinal bugs of some sort. The three weeks in Italy were screaming successes, but a nightmare of impossible schedules, cold rainy weather, and diarrhea. Everyone became my doctor; I used fourteen different medicines indiscriminately and somehow got through. Once in Israel, having two days before my first rehearsal, I decided to avoid Tel-Aviv, and ensconced myself here for two days of quiet & sun and swimming. First, no rooms: so I took one in a house down the road, which is rather nice, but eat, etc. in the hotel. Then \u2013 no sun: in fact, a good old New England gale just begun to blow up. I sleep a lot, read a lot ( _Male & Female_13 is more than fascinating, though often in unintelligible prose), and to outgrow this great weakness which follows on these three Italian weeks of superhuman effort. By the way, the _A_ [ge] _of A_ [ _nxiety_ ] brought the house down in Turin, despite a wretchedly nervous and unprepared performance. At the last minute I had to conduct from the piano, since the designated conductor proved incompetent. That is a chore! But at least I know that it can be done, given enough rehearsal, and omitting the famous last piano chord. I shall try the same method here in Israel.\n\nThis hotel is most impressive, full of Miami-type rich American Jews, but too small. The food is good, though meat is almost non-existent \u2013 which is OK, since I am on a rather lean diet \u2013 perf\u00f6\u00f6rce. (By the way, Italians are always saying \"per forza!\" which makes me think of you.) The parents were to come here on Friday next & stay here, but a cable from Dad says the boat is late & they will arrive only Monday, May 1st \u2013 so I'll grab their room Friday to Monday, on which day I move into my little house (occupied until then by the Parays).14 You have heard that [Paul] Paray is finished here, including the American tour, & that Kouss & I will share the tour, with some assistance from [Eleazar de] Carvalho.\n\nBut all this is not what is uppermost in my mind \u2013 and psyche too. How strange that you should have written just now of Felicia! Ever since I left America she has occupied my thoughts uninterruptedly, and I have come to a fabulously clear realization of what she means \u2013 and has always meant \u2013 to me. I have loved her, despite all the blocks that have consistently impaired my loving-mechanism, truly & deeply from the first. Lonely on the sea, my thoughts were only of her. Other girls (and\/or boys) meant nothing. Even the automatic straining toward general sexuality of the moment \u2013 which had always carried a big stick with me, was of no importance. I have been consistently aware of the great companionship of this girl \u2013 seen clearly and independent of the damnable tensions that discolored it, the fears melting into thin air. I fret, for the first time in my life, jealousy \u2013 a growing resentment of her current affair, and a certain knowledge that D[ick] H[art] was horribly wrong for her. Over all this, a real knowledge that she and I were made for each other, then as now: that we have everything to give each other. Just as right is my feeling that it would have been wrong to marry when we planned in '47, in struggle with the complex tensions of both our young lives then. I would marry her tomorrow, sight unseen, ignorant of all she has lived through these two years or so, willing to learn, insatiably eager to learn.\n\nOn the boat I was seized by these feelings \u2013 and more: a grave intuition that she was in trouble and needed someone. I prayed it might be me she needed. So strong was this conviction (though I admitted to myself that intuitive deductions are all too easy in mid-Atlantic) that I wrote her a letter explaining my urge. I felt humble writing it, vastly apologetic for the indifferent treatment I had afforded her during her troubled time in California, and in fact all through our \"engagement\". After mailing it, I was afraid that I had been guilty of bad manners, of possibly trying to disrupt what may have been a good relationship with Hart, of possibly yielding to the impulse of a moment of loneliness. Now I know, weeks later, how sincere and deep the impulse was. I have had no answer, and have thought that my worst fears were justified. Of course I sent it to Washington Place; she may not have gotten it; it may have been intercepted; or she may have reacted only with anger at my interference. How your letter gives me a renewed hope.\n\nI would write all this directly to her, but the unknown fate of my first letter to her gives me pause. I don't mean to use you as a go-between \u2013 I know you understand that as deeply as you do my desire to have her know my feelings. So many things become clear when abroad \u2013 so many cow-webs [ _sic_ ] are cleared away: the tongues of dear friends persuading me that she was wrong for me, etc., the psychiatric womb wherein one is safe from the need to cope with sexual adjustment, etc. My feeling is one totally apart from analysis: I want only to cope, and through my own powers, without aid \u2013 especially of the indulgent, personal sort that was forthcoming from Miss Nell.\n\nHow is Felicia? Did [Eva] Gabor leave, as I hear she might, and did Felicia replace her? Is she still in the show? How does she feel about her career? Is her health OK? These things, of course, I would love to hear directly from her; but if that is not possible, let me hear them from you. I am thrilled that you are close again: that should always be: you have so much for each other.\n\nOnly one thing more: last night I dreamed at length that I had found her and solved our problems together. It was a hard dream, but full of richness. And, on awakening, I was desolate at the thousands of miles that still lay between us, and the grayness of doubt and not-knowing. My day-dreams are of her flying to Israel, and our being married in Jerusalem. Ren\u00e9e [Nell], of course, would be the uninvited fairy who would pronounce the curse. Strangely, though, I think she'd be delighted. I was not at all surprised at your news of Ren\u00e9e: I had always seen these things, but had always diminished their importance in the light of her values and of my affection for her. Of course, I have no intention of returning to her, or, I hope, to anyone, if I can begin really to live my life (as I can now) and not only live on the circumference of it. And, willy-nilly, Ren\u00e9e has helped to that point \u2013 a point where my world changes from one of abstractions and public-hungry performance to one of reality, a world of creativity, of Montealegre-Cohn, of Spanish & French and travel and rest and love and warmth and intimacy. I've never felt so strongly as these weeks in Italy how through I am with the conductor-performer life (except where it really matters) and how ready I am for inner living, which means composing and Felicia. I'll probably never stop conducting completely, but it will never again be in intensity and emphasis what these last seven years have been.\n\nMy interest in _Peter Pan_ grows strangely neutral; I feel that the basic defection of Peter Lawrence, and the manifestations thereof in choosing [John] Burrell & [Ben] Steinberg,15 has (or have?) robbed it, prima facie, of the real life it could have had. If it's a hit, so much the better. But thanks for all your news: and I'm very grateful to Marc. I am shocked by the idea of my name in lights on this show! But these things pass. You are right: I shall never again operate in such a way.\n\nMany thanks for the Mitrop[oulos] profile \u2013 I'd love to see the rest (you stopped in mid-paragraph). And the [Wanda] Landowska picture is a gem. Two things were missing from your letter: there was no word of Gabey, and especially no word of Hi-lee [Shirley]. I miss you terribly, and a long letter from you with no personal word in it makes me worry a little: you will please rectify this situation.\n\nTremendous love, and I await impatiently your reply on all counts. The gale is now over, the sun is out, and I wish you and Felicia were both here.\n\nLovingly,\n\nL\n\n299. Leonard Bernstein to \"Twig\" Romney\n\nKing David Hotel, Jerusalem, Israel\n\n9 May 1950\n\nDear Twig,\n\nYour news is good: and I hope your sense of future is as strong and high as that to be found here. Pretty inspiring stuff, and fairly original, I'd say, in a world of passivity and backsliding.\n\nTonight I give a gala concert here celebrating the 25th anniversary of the Hebrew University. Irony: the University buildings lie in the unspeakably touching view from my window here, across the Old City, but they are unused, unavailable, in Arab hands. Such angering nonsense. But the University functions well in temporary borrowed buildings here, and will celebrate doubly hard tonight.\n\nI am tired, but, as always, goaded on to any amount of activity by the passion in this air. I will be in Holland the first two weeks in July, then flying to America for Tanglewood, then leaving again in mid-August for the Edinburgh Festival. We should be able to see each other there (I shall be in Scotland about Aug 20 to 25): then I return to Holland to finish the Festival. Do let's meet.\n\nThe _Age_ [ _of Anxiety_ ] is already recorded by Columbia, & should be released in the fall. Glad you like it \u2013 I do too.16\n\nTill Scotland, then, be good and tranquil. Helen sends a kiss.\n\nL\n\n300. Leonard Bernstein to Hans Heinsheimer\n\nTel Aviv, Israel\n\n11 May 1950\n\nDear Hans,\n\nThe good news about _Peter Pan_ continues to pour in from all sides. What a pleasure. Now to do something really important musically. I can see more and more clearly that the conducting side of my life will diminish rapidly, and the writing side augment. I feel myself less and less a performer. Maybe you'll have a Bernstein catalogue after all. We have to follow our insides, not our externals: was soll'n wir machen? It is a tide.\n\nI was horrified to read about Weill.17 This must have been a great shock to you. We will all miss him.\n\nGood news about the recordings. How did Goddard [Lieberson] ever agree to a non-Brigitta album?\n\nHow is the \"plugging\" situation?\n\nI was very much pleased with the published songs, but furious at the word-changes in \"Peter Peter\". They may have been necessary for purposes of the show \u2013 but certainly not for general consumption. I find the new lines almost unsingable, awkward and without meaning. Can something be done before more copies are made (if there is a demand)?\n\nThe _Age_ [ _of Anxiety_ ] was a wild hit in Torino, and a great success here with the musicians, though the public seems a bit puzzled. I have decided that the ending is all wrong (don't scream!). It is only a shame that the recording is already made and the 2-piano version published. But I am still going to change that Epilogue. After all, Bart\u00f3k published two endings now and then.\n\nIs it possible that I can see _Peter Pan_ before the recording takes place, or is it to be done immediately? I'd love to have something to say about it. But I suppose not.\n\nMy best to Nat [Broder]: and thank him for all his efforts. And to you, warm greetings from this marvelous land. I hope we meet somewhere in Europe.\n\nFondly,\n\nLenny B\n\n301. Betty Comden to Leonard Bernstein\n\n350 East 69th Street, New York, NY\n\n15 May 1950\n\nDearest Lenny,\n\nKnowing, as I now do, that you save every scrap of correspondence you get, from Koussevitzky's pages on life, music, and your career \u2013 to Auntie Clara's hot denunciations of meat, I write this letter with the full burden of realizing that it must top my incomparable \"Musicraft\", or \"Sam [Paul] Puner\" file of a few years back. As if this were not enough, I have the added load of trying to tell you what has been happening these last weeks with you so far away \u2013 and successfully bridging the gap of time and miles. Need I add that when I say \"I\", I am really referring to a certain dark fellow [Adolph Green] as well as myself \u2013 although somehow, through some odd trick of fate, it is I, only myself, who is stuck with actually writing the letter. Anyway \u2013\n\nI should begin by saying that the show we all want so much to happen \u2013 is not truly an existent entity yet, although we are still working at it, and very hard, too. When you tore off in that hot flush of enthusiasm \u2013 we started meeting, and, as you know, met with Jerry [Robbins]. He got interested in the idea \u2013 and had some general \"feelings\" about it \u2013 but nothing very specific, and since then \u2013 we have met \u2013 just ourselves \u2013 and a few times with Oliver [Smith]. We talked a great deal about the house on Middagh Street \u2013 and the many people who lived there, and what they wanted and how they behaved \u2013 and tried like fury to settle on some main characters, and a line of action. What kept happening is that the feeling of the thing kept veering from what we consider one bad extreme to another \u2013 either it acquired that \"young hopeful\" quality of something like _Look, Ma_ or _Stage Door_ \u2013 being about young artistic people trying to \"get ahead\", or it became about eccentric characters too removed from audience sympathy and identification \u2013 too bizarre \u2013 too enfant terriblish \u2013 too personal. We tried staying in the House for the whole show. We tried getting out of it for most of the show. To arrive at some main story and characters we attempted, finally, a kind of modern _Boh\u00e8me_ \u2013 the girl a smart 1950 tramp and the guy a writer or musician, involved in the House \u2013 which led us out of the House altogether to thinking of just \"a modern love story\" \u2013 (Does this strike a familiar chord?) \u2013 who needs the eccentrics except as background people \u2013 let's write that contemporary big relationship story. This ended nowhere \u2013 after spending a full day and evening with Oliver. Back to the House idea. We even went to Brooklyn with the boy from Middagh Street himself18 \u2013 and roamed the Brooklyn Heights streets \u2013 and saw where the house originally stood \u2013 and looked at the breathtaking view of Manhattan \u2013 and got all inspired, and came back \u2013 and still have not captured what we want on paper. But what has disturbed us particularly is that we get no musical thoughts whatever. We find it hard to think of \"number\" \u2013 the musical expression of what the show should be. We cannot hear it. We sometimes feel it is more a play \u2013 and not suited to musical theatre at all. Have you had any time to think of it at all? \u2013 and if so, what does it seem to mean to you, musically? All of which brings us to how we wish you had not sailed away before anything could be talked out and either concretized \u2013 or discarded. All we have to go on is your parting enthusiasm. That evening should have been followed up by a week of intensive meeting together, but it was impossible. It just feels like no way to work together at all \u2013 being so far apart, and not being able really to talk things out. We feel so far removed from what you may be thinking \u2013 and even wonder if we have been proceeding at all along the lines that you are. Or are you? We know that outside of being frantically busy you have been sick \u2013 but if you have had any fleeting thoughts or impressions about the Idea \u2013 or any concretizing of your own feelings about it \u2013 please enclose them in return mail.\n\nTo add to our situation, we are, as you can tell from the postmark, still in the East \u2013 having run into that situation in Hollywood we all thought might happen. Not the right property. We _could_ be doing _The Life of Sigmund Romberg_ \u2013 but as the creators of _Bazooka_19 we do not feel we could do the subject justice. MGM is out buying us things \u2013 but it has dragged on this long, and may a little longer \u2013 and what does that do to the date department you may well ask. All we know is \u2013 we will do the picture soon.\n\nWe want so much for the show to be. We are keeping at it \u2013 and in fact have a date to see Jerry about it today or tomorrow. I just felt we had to apprise you of the thoughts we have been having \u2013 and also to ask you what has been churning in your head these many weeks. Maybe this will be the first of a series of letters which in a few months will add up to S-H-O-W \u2013 a show that will open next spring. Please write \u2013 we will again soon. Onwards and upwards!\n\nMuch love,\n\nBetty\n\nIt's so wonderful about _Peter Pan_!\n\n302. Leonard Bernstein to Shirley Bernstein\n\nTel Aviv, Israel\n\n19 May 1950\n\nShalom Shvestoah!\n\nSince I couldn't sleep last night after the concert I have made this a do-nothing day: relaxo profundo on the wonderful beach for three or more hours, followed by relaxo in bed, followed by relaxo with Irwin Shaw's novel. Concert coming up again in an hour and a half \u2013 but I want to begin a letter anyway: there is so much to say. Relaxo always brings on tension for me \u2013 I'm so unused to any period without release of some sort. I have been engaged in an imaginary life with Felicia, having her by my side on the beach as a shockingly beautiful Yemenite boy passes \u2013 inquiring into that automatic little demon who always springs into action at such moments \u2013 then testing: if Felicia were there, sharing with me that fantastic instant when the Khamsin is suddenly gone, and a new wind, west from the sea, comes in to cancel the heat with its almost holy approach \u2013 and the test works. It's surprising how true some of the old saws out of the analysis book can be: \"to establish a good relation to yourself is the prerequisite to any other relation.\" This self-relation is what I have begun to find: I have discovered the core in myself of human relationship (words, words til now) \u2013 the core of a sunburst of quiet energy, and always apropos of Felicia. This, after years of compulsive living, of driving headlong down alleys of blind patterns, dictated by God knows what vibrations \u2013 this is a revelation. Not that the demon absents himself: he still pokes me when his occasions arise \u2013 the French horn player, the artist in Jerusalem: but the old willingness to follow him, blind to any future, blind to the inner knowledge of the certain ensuing meaninglessness \u2013 that is gone. So the demon diminishes.\n\nHow can I tell you how touched I was by your letter? You have grown so \u2013 you are more than understanding: you understand so many things I didn't say. And you've become articulate in a way I never expected. Your control problem is all too familiar to me \u2013 an old Bernstein custom. We can work through it: only it can't be planned, like a Bar-Mitzvah, for a certain date. One day \u2013 boom! I feel you're on the road.\n\nI'm happy about Felicia's reaction \u2013 especially by the apparent lack of ingrown hostility. We are both in such a strange condition: she with her double-edged sword, I with pure waiting. Waiting is a salutary state for me: heretofore it wasn't possible without tension. But now I feel such a certainty about us \u2013 I know there's a real future involving a great comradeship, a house, children, travel, sharing, and such a tenderness as I have rarely felt. I want to comfort her for all her heavy wandering, and to make it right. Only one thing: why does she insist on prolonging the suffering? Is she as sure as you that her present life is not her future? I sure hope she is: \u2013 I know from some almighty source that Dick was created for other things. And Felicia is for me, because the thought of her makes me strong enough to deserve her. This is by now repetitious, probably the product of having no contact with her except through you, and I must stop this proxy-relationship. You're a darling, and you have been good and loving in your handling of a delicate position.\n\nThanks for the idea about Aaron's concerto: I've written him & Dave [Oppenheim], although I suspect Ormandy will get it.\n\nWhat of Kay Brown? Never a word from her. And what motivates your dark words about Betty & Adolph? You scare me. As for _Tree_20 I thought Alex North was to do it. I'd hate to be strung up again with a show \u2013 that would be too much after last year. What of Charlie Feldman & _Streetcar_?\n\nThe parents are furious that you haven't written. Neither has Burtie. What is with that journalistic tycoon? I miss him terribly. The aforesaid parents dash about madly: mother is a regular Marco Polo, & Daddy, in his bungling way, is on top of the world. Helen flies about with seventy friends, busy and protective as ever. Since our housekeeper speaks only French and Bulgarian, Helen has to deal with her in her own mad French, which is mostly a series of those old American whinnies plus endless strings of Oui Oui Oui Oui \u2013 replacing the former yes yes yes yes. In French, these strings begin to approximate Rybernian. Weep weep.\n\nIt seems I've been here forever & there's still over a month to go. It's partly, I guess, not seeing you (which had become a pleasant lifelong habit) & partly the waiting game with Felicia. I don't feel the same permanence & in-living here that I felt last trip, during the war. Of course there was Azariah [Rapport] then \u2013 it all seems so impossible and of a different life. I realize now that he personified for me the war, the incredible bravery of these people, the beauty of their vision. Now, in an American double-breasted plaid \u2013 zero.\n\nGive Fel. my dearest love, & tell her not to suffer any longer than absolutely necessary. I treasure the page from _Cue_ [Magazine].\n\nBe well; remind Gabey that I exist, & write more often. A letter takes ages (your last took 14 days, what with return for additional postage, etc.). And if Felicia feels she can write \u2013 even a line \u2013 it would make me so happy!\n\nBless you, dear Hilee,\n\nL\n\n303. Leonard Bernstein to Aaron Copland\n\nc\/o Israel Philharmonic, Tel Aviv, Israel\n\n21 May 1950\n\nShalom Aaron,\n\nSo much time goes by: I don't know you any more \u2013 not even through your music, which is not exactly forthcoming. I played your violin sonata with a fine fiddler here the other day, and had a real old-fashioned nostalgic kick. Those faraway days when the C# was holy and the form so surprisingly right. _Where's your music?_ God knows we need it. There hasn't been a real exciting American premiere in years.\n\nI fought with Kouss valiantly over the Clarinet Concerto, to no avail. Benny & Tanglewood don't mix in his mind. Shirley had a good idea the other day: to do it with Dave O[ppenheim] at the NY Philh. next Feb. Then someone tells me he read that you've disposed of it to Ormandy. What is the story? I'd love to do it in New York with Dave. Let me know.\n\nWhat's up? Movies? [Emily] Dickinson? A piano concerto? Eric [Erik]21 & Victor? _You?_\n\nIsrael is lovely, weather delightful, concerts fine, [Jascha] Heifetz in top form, Roy Harris 3rd successful, the people gay & forward-looking. I miss you.\n\nLove,\n\nL\n\n304. David Oppenheim to Leonard Bernstein\n\n26 May 1950\n\nDear Len,\n\nOf course I'd love to do Aaron's piece22 with you next year. Of course Metro is beckoning with many $s and all kinds of offers, but I guess I'd rather stay our number one clarinettist.\n\nYesterday, tho', I heard a rather dispiriting rumor that Mitropoulos was promising the piece in the fall with McGinnis23 but I have no confirmation and it may be untrue. It certainly seems possible though. I'll call Aaron. Meanwhile see if you can set it with the Philharmonic, because even if it is already programmed it will be good for me to be mentioned.\n\n1. House is being plumbed\n\n2. Judy24 is in Hollywood doing B.Y.25\n\n3. Mrs. M. is the only nice person I know\n\nI may tour for a month with a woodwindy quintet this summer, or I may go to Aspen, Colorado, to play at the festival, or both.\n\nSee you in Tanglewood, a weekend, or week middle maybe.\n\nEnjoy,\n\nDave\n\n305. Leonard Bernstein to Helen Coates\n\nScheveningen, Netherlands\n\n13 July 1950\n\nDear H,\n\nEnd of Round Seven \u2013 these concerts are thankfully over, and I leave tonight (earlier schedule, luckily) for New York. I've never been so anxious to get back \u2013 not for the country, particularly (I'm always chez-moi anywhere) but for people I desperately miss. The concerts have been gratifying, and the Mahler was a sensation in Amsterdam. I have been weak again, and accepted a week with the Concertgebouw the first days of September (7th & 9th). I thought \u2013 as long as I am here anyway until the 2nd I couldn't resist the offer. I had looked forward to it for so long anyway! They are now attacking for me to stay the whole month, and give [Eduard] van Beinum a rest (he is quite tired and ill). But I doubt that I shall.\n\nThe _Age_ [ _of Anxiety_ ] went well last night. Fine response: van Otterloo is a very intelligent and hard-working guy.26\n\nBut best of all \u2013 only 3 concerts in 2 weeks! And cold, cold air, a grim northern sea, angry clouds \u2013 all this is a blessing after months of relentless blue and gold.\n\nI feel much better, though not up to par at all: but I'm prepared for the grind ahead. I hope you've rested, that the Kibbutzim don't prove insufferably hot, and that your trek eastwards proves rewarding. Don't load up with too many 16 \u00d7 16 rugs in Damascus!\n\nTonight I shall hear Furtw\u00e4ngler before the plane.27 He was at my concert last night, and seemed very happy. A nice man: who can judge?\n\nI am more than depressed by the news: not so much out of anxiety as out of a great disappointment in man \u2013 a heavy realization that people really don't want peace, or aren't simple or strong enough simply to decide to have it. It makes all planning seem a little bit ridiculous.\n\nHowever, we go on. Round Eight...\n\nL\n\n306. Betty Comden and Adolph Green to Leonard Bernstein\n\nM.G.M., Culver City, CA\n\n23 July 1950\n\nDearest Lenny,\n\nWe have delayed writing only to try to find out more definitely just what our dates here will be. Talking to you and hearing you sound so low was not as pleasant as getting your marvelous letter, and we wished we could all have been more close \u2013 about our plans, but apparently both you and we are planning on a somewhat later date than September 1. You have taken a couple of more weeks in Holland, and it looks as though we will have to be here until nearly the end of September. Certainly, as far as your staying on in Holland for a rest and vacation goes \u2013 if you must, then do it. On the other hand \u2013 by that time, we should be winding up here, so if you could come out for those last weeks, we could undoubtedly accomplish something. Feasible? \u2013 ou non?\n\nWe are still exploding with the desire to do a show. As we wrote you, the Middagh Street opus bore no edible fruit, and we are happy to hear you are not what they call married to same. To make a further stab at an idea, we came out here by train, as you may know \u2013 and spent three days closeted with ourselves and our heads \u2013 digging \u2013 and did get an idea we liked \u2013 except that present conditions in this frightened and frightening world seem to have ruined its practicability.\n\nRoughly, it was a post-war theme, capturing, we hoped, some of that Age of Anxiety feeling: Four guys who had been together in the war in the same outfit \u2013 pals, they thought never to part \u2013 who naturally drifted apart as soon as peace disbanded them \u2013 and who try to have a reunion.28 They had something together during the war \u2013 a warmth, a companionship, a sharing, a true friendship \u2013 which they try to recapture \u2013 but the great \"levelling\" is not there any more. They are from utterly different walks of life now \u2013 maybe an artist, a rich advertising fellow, an elevator man, a stage carpenter. They either meet by accident or plan the evening, and at some point in the night their great comradeship reasserts itself \u2013 and although at the end they know they won't ever see each other again, they realize that they had a little glimpse of what life could be \u2013 and why should that special kind of togetherness exist only when people have their backs to the wall as in a war or crisis. Or \u2013 possibly the story could have been told through one of the protagonists, the guy who has lived on this memory, and for this day, who looks up to the others and has his memory realized, and then sees it finish when they say goodbye.\n\nAnyway, this idea interested us and we did a little work on it, but it doesn't seem at all possible now.\n\nSince then we have been up to our earmuffs in this Picture we are writing.29 But, as stated earlier, we feel if you could come out toward the end, to vacation \u2013 at the same time, we might get something done.\n\nPlease write, and forgive our delay in answering you.\n\nAll our love,\n\nB and A\n\n307. Leonard Bernstein to Sam and Jennie Bernstein\n\nParis\n\n17 August 1950\n\nDear Sweet Parents,\n\nWe have had a magnificent stay here these few days, in spite of a tough rehearsal schedule, complications with travel accommodations, etc. \u2013 plus the fact that we all three arrived so tired after the plane trip. But we've slept a lot (especially Shirley & Burt, who slept through my rehearsals) and feel really happy and well. Paris is of an unbelievable beauty now \u2013 cool, invigorating, and dewy; and the fact that so many Parisians are away for the summer kind of leaves the city open & clear for the tourists. We've fallen in love with it again, all over: and Burtie is already an old Parisian, with a beret and all.\n\nTomorrow we leave for London with the orchestra (by train and boat): we will stay in London two nights, see the shows, do the town \u2013 and then proceed to Edinburgh by plane Sunday noon, the 20th. My concerts are the 21st and 23rd: we go to Holland directly on the 24th, and will stay there until September 9th. Why don't you write us there \u2013 to the _Kurhaus, Scheveningen, Holland_.\n\nFrom there we go to Ireland (to this great castle): and Burtie will fly home from Shannon on the 18th, arriving the 19th, while we fly back to Paris, where we will stay til the 24th at the St. James Hotel; then we plan to drive south with Peggy Riley and her husband (wonderful girl) and Harry Kurnitz,30 the Hollywood writer, arriving in Rome about a month later.\n\nSounds great, doesn't it? It should be a real rest, fun, and inspiring.\n\nDaddy, hope you're caring for your health: & Mamma, have fun. We wish you were both along! Kisses from us all \u2013\n\nLenny\n\nI'm writing to Israel about the Apes' refrigerator.\n\n308. Leonard Bernstein to Sam and Jennie Bernstein\n\nLondon [headed paper of British European Airways]\n\n24 August 1950\n\nDear Apes,\n\nWe've become the most seasoned travelers on the globe by now. At the moment we're sitting at Northolt Airport waiting for our plane to Amsterdam. We left Edinburgh this morning in a blaze of glory (but at 7:00 a.m.!). The final concert last night was a triumph, with a stamping, screaming ovation. I never conducted better. But even more exciting was Scotland itself. We all fell in love with the Scots \u2013 a great, friendly, proud people: and Burtie is considering skipping his next term at school to go back there and shoot grouse \u2013 to say nothing of Italy and Israel. I think it would be the most wonderful thing for him to visit Israel, don't you?\n\nWe're all very well \u2013 not a cold in a carload \u2013 but a little tired. In Holland for the next two weeks we should gain weight and get a good rest (at least the kids will). The food will be great \u2013 and the horses \u2013 and the sea. And two weeks in one place \u2013 what a joy!\n\nWe have had no word from you yet \u2013 and we want to know how you are. Don't forget to write us \u2013 every bargain has two sides \u2013 and we've been writing steadily.\n\nAdolph [Green] may come over to Europe, & we may vacation together with Allyn Ann [McLerie]31 and some other friends. It would be such fun (Allyn is dancing here with the Ballet Theatre). We may even take a house in Southern France, after all. I have two whole months free \u2013 no dates except Milan in November. Isn't it wonderful?\n\nWrite us to Holland \u2013 the Kurhaus, Scheveningen.\n\nLove,\n\nLenny\n\n309. Leonard Bernstein to Sam and Jennie Bernstein\n\nScheveningen, Netherlands\n\n1 September 1950\n\nDear Jen & Sam,\n\nMy first performance of Beethoven's Ninth was a triumph!32 I have been very worried about this event \u2013 the big test in every conductor's life. But it was so exciting \u2013 the solo quartet was the best I've ever heard \u2013 the chorus was marvelous \u2013 the orchestra never played better. What a tremendous experience it is to do this work! Like tearing your guts out. The public went crazy. Tonight I repeat it. Then tomorrow we are taking a wonderful car to Germany, of all places, for two days. We can get about as far as Cologne, and then come back in time to move to Amsterdam Sunday night. (Monday morning is my first rehearsal there.) Then it's all rest & swimming & sleeping until November!\n\nWe're all getting fat on milk and herring and butter and lobster and never felt better. Burtie sleeps at least 12 hours a night, and Shirley about the same. Only old Lenny gets up early to make the money.\n\nI have an interview now \u2013 so cheerio for the moment \u2013 and we'll write you all about Germany in a few days.\n\nLove & kisses from us all,\n\nLenny\n\n310. Leonard Bernstein to Sam and Jennie Bernstein\n\nExcelsior Hotel Ernst, Cologne, Germany\n\n2 September 1950\n\nDear Apes,\n\nWe have had a most dramatic day driving a little Skoda (a Czech car loaned us by a friend) into the Rhineland, down the Rhine, through Arnhem, D\u00fcsseldorf, and over Hitler's Autobahn to Cologne. We are amazed at the wonderful food, the thriving big city, the luxury of this hotel. It's been raining all day, and Germany looks twice as tragic and ruined and dramatic through the rain. Tomorrow we drive to Bonn \u2013 where Beethoven was born \u2013 and Belgium and back to Holland \u2013 it's all like a wonderful dream for the kids \u2013 Burtie is being the mighty American conqueror here \u2013 and now we're going to investigate Cologne nightlife. We've just finished a dinner of Wiener Schnitzel which Burtie says ranks with Mrs. Hathaway's.\n\nWrite us to Amsterdam \u2013 c\/o Concertgebouw.\n\nLove,\n\nLenny\n\n311. Aaron Copland to Leonard Bernstein\n\nAmerican Academy in Rome, Rome, Italy\n\n16 February 1951\n\nDear Lensk,\n\nTho' I think of you every day I couldn't figure out where you were in space, until a letter from Helen today tipped me off. I think of you as dashing madly about from one triumph to another, with orchestras vaguely in the background, while here _I_ sit in your ex-apt on the Janiculum while Asunta sings your praises. Somehow it all seems as per usual.\n\nI hear you're ditching us at Ta\u2013foot33 this year. I'll miss you \u2013 it was nice to have you put for some weeks of the year. Well you want to do it \u2013 so do it.\n\nAmerica seems a million miles away. Rome feels like a haven for some reason \u2013 perhaps because we have so little contact with what goes on under the surface here. I haven't been anywhere as yet \u2013 just got my Morris-Minor34 (Moyshe to you) last week \u2013 and began at the beginning with the Coliseum at midnight.\n\nHaven't heard a word as to how the Israel Symph. has been received. I'm in Tel Aviv Apr 5 and [Ben-Zion] Orgad is in charge of rounding up 30 composers for me. We are to live together for 5 days in that there Art Colony. And Pesach [Passover] is to be out in a Kibbutzim. If you've got any good advice, send it.\n\nWhat plans have you for the future? What do you see? What do you know?\n\nJust think \u2013 I have a whole Quartet35 you don't know. I'm writing something that I think will be commissioned, but if not, I'm not writing it. In any case I could only write pretty music in this villa.\n\nMake good concerts.\n\nLove,\n\nA\n\nP.S. Erik sends his best.\n\nP.P.S. V[ictor] is in Brazil \u2013 taking jungle pictures, etc.\n\n312. Nadia Boulanger36 to Leonard Bernstein\n\n36 rue Ballu, Paris, France37\n\n19 February 1951\n\nDear Leonard,\n\nForgive my silence \u2013 you have plunged me into an abyss of perplexity.\n\nI like your idea \u2013 but to come to N[ew] Y[ork] to play American works superbly played by great conductors \u2013 what na\u00efvet\u00e9 and what folly on my part.\n\nI offer you:\n\n1. Cantata: Igor Mark\u00e9vitch (or if you prefer, a cantata by Bach).\n\n2. Ask Copland, Piston and a young unknown American to write a Triptych for the occasion.\n\n3. A group of Monteverdi (I guarantee their effectiveness).\n\n4. Works by my sister.\n\nI need a program within my means because \u2013 I am only what I am.\n\nIf you only knew how much I am moved by your affection \u2013 all so simple and so generous.\n\nThank you.\n\nNadia B38\n\n313. Leonard Bernstein to Shirley Bernstein\n\n\"Privada del etc.\", Cuernavaca, Mexico\n\n16 May [1951]\n\nDear Ape,\n\nSo fine to get your last letter: you sound so much better and busier and everything despite the job-problem. Perhaps by now something is a-happen, as they say here? I have never had an answer to my last letter (since our letters cross continually, and you are a stinking correspondent) in which I talked about all other kinds of work from those we know so well. Were you at all reached by this or was it just bilge? What of the Engel sessions?\n\nBut it is still wonderful to read a cheerful letter from you, even if it is stuffed with problems about Bob C. and company. I know that one will work out all right: you sound so sensible about it, and self-knowing. It is high time that you could have a relationship with a guy that was not confined necessarily to either heaven or boredom. Just a nice relationship, with warmth and affection and companionship and even passion. And I fully understand the traumatic situation with him and me and D.O. [David Oppenheim] I know you'll lick it. Meanwhile, please, no compromises with Felicia or anyone else. I find myself missing her, though, although, Goddammit, not half so much as I do you.\n\nI wrote Adolph and of course have no answer. What is with his opus? Bob Rossen39 arrived here again last night and just left for Mexico.40 He is lonely and restless and beset with all the problems of a martyr and hero without being either of them. He is wretchedly lost, having lost Hollywood, and makes braver speeches than he feels. I think he may go back and face the Committee41 next week. A man like him (only slightly sensitive and bright) can hold up only so long without his work or his family, and the absence of both together leaves him functionless. He talks by turns of settling here with his family, of going to Israel to live, of Rome, of England. He feels finished in the States. It's a mess, and I am very sorry for him. It can also happen to all of us, so we had better start preparing our blazing orations now. Maltz42 is also living here (Cuernavaca has become a great haven for these poor guys) and regaled us the other night with tales of his lovely year in jail. I lost my dinner. It is utterly incredible that a man of his solemnity and innocuousness and faith in Jeffersonian democracy should be put away with a raft of moonshiners in a West Virginia jail. Dimitryk43 has certainly made a ghoul of himself: and the boy in the biggest jam right now seems to me to be Garfield.44 He will wind up in a great perjury mess if he doesn't watch out. It may already be too late. Actually, I suppose, there is nothing to be done when your life and career are attacked but strike back with the truth and go honestly to jail if you have to. This dandling about to save a career can neither save the career no make for self-respect. I hope I'm as brave as I sound from this distance when it catches up with me.\n\nIn any case Ross Evans45 walked in the other day with a six-week old puppy, rescuee of an accident between a boxer and a gloomy Airedale bitch, the pup is a dream, a _Boxdale_ (copyright term) and I have named him Machito because he is so macho and he has lovely fleas and will probably wind up on West 55th Street like all the rest, but I love him. The house is bedazzled with pee and shit of all colors, and the whole cycle recommences: Outside! On the paper! Not here! Who's the funniest little Ape? Who's the little Fellowuhss? Well if it isn't the little...! I am just at the crisis of deciding whether to return him or brave it through. He is always guilty and doesn't know why he is punished, and I am dienetically re-experiencing a whole lifetime of European hotels and trains and the carpet in the Israel house. Wish you were here.\n\nBy the way, might you come? Bob Presnell46 writes that he is coming after all, probably next week; and that should be pleasant, if I can work with someone around. The show is going great guns,47 and I keep wishing for you to sing it at, and get the right answers and responses. There are so many aspects of it that only you could understand, libretto-wise as well as musically, and it's a bore to have to figure everything out for myself. The two characters, by the way, have gotten themselves called Sam and Jennie, and I think you'll see why. I have about four and a half scenes sketched out of the seven, and am amazed. It should be roughly finished in a couple of weeks, and then I'll send you a sort of libretto. It's real fun to write music. You may quote. I am ruthlessly turning down offers, still. I finally refused that Harp Concerto commission for Rosenbaum in Philly, and turned down the newly-reforming Detroit Symphony, as well as a three-month stay in Australia. It's much too pleasant just to sit, and sitting here in Cuernavaca is in itself an occupation. I don't know what happens to time here: it gets destroyed, it works on a life of its own, and the days rush by so frighteningly fast that it is tiring only to think of the date. Martha [Gellhorn] and I are talking about a Caribbean jaunt in the fall. But plans in general are hard to make. And it is now time for us to go forth in her jeep to the tennis courts. So bless you and all my love. And write a guy, for cry-eye. Love from Machito, the diarrhetic darling.\n\nX\n\nTell me more about the Chodorov play. Is it just a play or is there chance for real music? The idea is fine.\n\nCome to think of it, why don't you come down for a visit? Maybe drive with Presnell? It would be free! His address: 126 E 56.\n\nX\n\n314. Felicia Montealegre to Leonard Bernstein\n\nheaded paper of the National Broadcasting Company, Inc. [New York, NY]\n\n[before 9 August 1951]\n\nDarling!\n\nNo word from you \u2013 and I _don't_ wonder why! Let's let it lie for the nonce \u2013 this is hardly a step to take with such great vacillation \u2013 someday it may become the most natural and longed for event \u2013 if that blissful state of mind should take over (oh happy day!) and I'm still not yet wed, we'll just get on with it and be miserable for ever after!\n\nI miss you terribly \u2013 you couldn't have been sweeter or more tender. Both of you helped me so wonderfully through what would have been otherwise a very tough time. I love you for it. However it has made N.Y. seem so grim by comparison \u2013 thank God for Hilee [Shirley]!\n\nRehearsals are ginger-peachy. I have a reading for a play tomorrow \u2013 was interviewed for a movie (that old story!) which will be shot here in the East. I had my first real \"sortie\" with Bert the Card and Claude at Sardi's, ran into the Davids \u2013 the Diamond one looking frighteningly thin and peaked and rather cold towards me (judge of my joy!) \u2013 had a pleasant lunch with Goddard [Lieberson] (?) \u2013\n\nI feel very well, have had no dire accidents \u2013 that is, _yet_ \u2013 and life goes on in its own plodding way \u2013 but I am strangely happy though it could be just an overwhelming sense of relief!\n\nI'm afraid I won't be able to get there for the _Missa_48 \u2013 it breaks my heart, but I just have to take this TV thing seriously.49 I hope it's going well and that it's a big smasheroo \u2013 (just don't bow this time!)\n\nOh sweetie!\n\nCall me any way \u2013 I kiss you long and sweet.\n\nFeloo\n\nP.S. She's not yours \u2013 follow me!\n\n315. Leonard Bernstein to Philip Marcuse\n\n127 Wolcott Road, Brookline, MA\n\n5 September 1951\n\nDear Fil,\n\nBoth your letters have lain, screaming for answer, for months, and everything conspires to prevent same. Tanglewood swamped me this summer, and just before the end of the season Felicia and I decided to marry, inducing further activity.\n\nIt's wonderful, & I'm deeply happy about the marriage. I've kind of rediscovered this lovely girl, and believe that we will have a fine time of it. The wedding is Sunday the 9th, & we leave straight for California to see her family, then on to Mexico for the winter, returning probably in March or April.\n\nI'm afraid that this will rule out the 9th Symphony in Detroit. I have been so pleased with the idea, & attracted by the notion of another \"special\" in Detroit, that I have hesitated to write you a yes or a no. But I fear that the no triumphs. This will be a composing year. I may make an exception & do a little festival of 3 weeks at the City Center in N.Y. in the spring, but I've not yet decided. This is a crucial year, & much will be crystallized. I'll be in touch with you throughout.\n\nWon't you & Babs say some special little prayers for us? Let me hear from you soon, via Helen Coates, 155 E. 96th St., New York.\n\nMuch love to both of you.\n\nLenny\n\n316. Leonard Bernstein to Burton Bernstein\n\nRemount Ranch, Cheyenne, WY\n\n18 September 1951\n\nBaud\u00fcm\u00fc,\n\nEvery turn on this ranch makes us think of you. We almost thought of going up to Sheridan, but this took precedence \u2013 and it beats everything. We've been here two days & have to leave today and hate to. Such beauty & luxury: & the Knoxes are real pals.\n\nEvery day marriage gets better.50 It may take a lot of days, but I think the big crisis is over. (That took place in Detroit, but Phil & Barbara [Marcuse] were so great, as you know, & they really helped enormously.)\n\nNow get this:\n\nLast night _we went lionhunting_.\n\nWe really did. Mountain lions. Didn't find one. But the deer abounded. It's infinitely better than Irish deerstalking. It's civilized. We piled into a Chevvy convertible, top down, in the freezing high air, wrapped in heavy winter clothes & earmuffs and gun in hand. You drive slowly around the hills, shining a strong torch into the trees & rocks and sipping Dewar's White Label. Now there's a plan. It was marvelous, despite the lack of lion, but we did get one shot at a porcupine and missed. All in all, _we went lionhunting_.\n\nBaud\u00fcm\u00fc, be smart at school this year, and be serious and learn and become the wonderful guy you are to become. Can't wait til Xmas.\n\nFelicia loves you & sends a big hug*, & so do I.\n\nApe-husband-hunter,\n\nLennuhtt\n\n*She adds _undying_ and _passionate_.\n\n317. Leonard Bernstein to Philip and Barbara Marcuse\n\nHumboldt 53, Cuernavaca, Mexico\n\n9 October 1951\n\nDear Gentle People,\n\nThere it is, the address I mean, and now let's have words from you, lots of them. This is not a moment for lots of words from us: it is a moment of getting installed in a new house, new life, new everything. The tensions (do you recall this word?) accumulate still, are fought, lived through. Every once in a while a state of comparative ease is reached which promises well for the future. And now we have a glorious grand huge house with a huge garden and a huge pool, and tomorrow there ought to be a huge piano and then there should be some huge work. I am not quite sure what F[elicia] is going to do all winter while I am at the piano, and in my own world, so to speak. But most of that, of course, depends on what security she will manage to find in a marriage contracted in insecurity. We hope and we pray and we wait.\n\nWhat did you think of the Cadillac idea? Me, I think it's great; and it is a real come-on for me. The Buick is in fine shape (got us here safely, sans accident, sans blowout, sans being pinched).\n\nLet's have a glorious letter!\n\nLove,\n\nLenny\n\nFelicia adores you both and is as grateful as I am for your presence on earth, and sends her warmest.\n\n318. Leonard Bernstein to Aaron Copland\n\nHumboldt 53, Cuernavaca, Mexico\n\n18 October 1951\n\nDear Sorelymissed A,\n\nFirst off, thanks for the delectable wire on the wedding day. If this thankyou is a bit late, f\u00edjate, no m\u00e1s, the enormous automobile trip we have made from Boston to San Francisco (stopping for days at ranches on the way) to L.A. and thence to Cuernavaca. It took about a month; and we are slowly settling now into a glorious house and garden and pool. The piano has arrived, I have written an extra aria for Captain Hook (what shit!) to grace the new road production of _Peter Pan_ , and am now starting on the long hard road of writing some real things. I have decided, coute que coute, to finish my little opry [ _Trouble in Tahiti_ ] and then write a few more little opries. There may be some stray notes \u2013 like even a piano sonata, and a new idea for an orchestra piece; but the main stem is still that old devil theatre, and I have to see just what my connection with it is.\n\nI still haven't seen the score of the Piano Quartet, and long to. Isn't there something you can do?\n\nWrite to me about your life in Boston. Did everything work out well for a house for you and E[rik]? Give him my best, and thank him also for the wire. What word from Victor? Does he find marriage as fascinating as I do (what a word for marriage)? Actually it is the most interesting thing I have ever done, though there are times when one's interest must be that of a person in an audience, or one would go mad. It is full of compensations and rewards, and reveals more to me about myself than anything else ever has, including a spotty array of analysts.\n\nAs I say, write, long and lovingly, and give my love to Irving [Fine] and Verna [Fine] and Lukas [Foss] and Tillmange.\n\nDear old Judgenose, I miss you.\n\nL\n\n319. Leonard Bernstein to Aaron Copland\n\nHumboldt 53, Cuernavaca, Mexico,\n\n14 November 1951\n\nDearest Aa,\n\nToday is the fourteenth of November, which makes four reasons to write you. 1) that you have just written me; 2) that it's your birthday, whether you like to be reminded or not; 3) that we met on this day about fourteen years ago; and 4) that Bruno Walter celebrated your birthday most spectacularly eight years ago by getting sick.51 So we owe each other this here Martini on this here day. With all my love.\n\nAlso, we bought a phonograph, and there in the same shop was the Clarinet Concerto and the Quartet, on both of which I am therefore now an expert.52 I am disappointed in the Concerto, and I think it may be a little on account of the performance. I remember it as being so much fun when you struggled with it on the piano (of course everything is more fun when _you_ do it on the piano with your apologetic grin), and Benny [Goodman]'s performance is ghastly and student-like, I think. But there is also something that does not quite satisfy about the score, despite its evident beauties. The opening is still ravishing, and I find to my dismay that it is in places less like Satie than like the _Rosenkavalier_ trio. Tant mieux, say I, though surprised not a little. Strange, in fact, how many touches of Strauss there are; there is even a slight _Don Quixote_ feeling here and there. I still disapprove of the cadenza, finding it cute but arbitrary; but it is the last part that disappoints me because the last part was so much fun. Of all things, the form. Doesn't seem to work.\n\nBut the Quartet, ah, there is another matter. I rejoice particularly in the scherzo, because I think it is the longest sustained piece of continuity you have written in a long time, and it is really continuous, yes, really, and it goes and goes in a remarkably convincing way. I feel rather close to the tonal way in which you are handling tone-rows (I've done it too, here and there); and I find that this movement is a real triumph. The last movement is beautiful too in a way which has already become awfully familiar to Coplandites, so that it is not such a thrill as the second. And the first is lovely, but I never did go for you and fugues, especially here where the opening is so reminiscent of the third Hindemith Quartet. Imagine, Hindemith! Who'da thunk it? But it makes a fine piece, especially for records, because you want to hear it again and again (of course with two or three mambos in between); and I still think you are a marvelous composer.\n\nThat's the good news for today, and I seem to have started a piano sonata right in the middle of all my stage-operations, nolens volens, as Kouss used to say.\n\nThe lectures look awfully inviting, and I wish I were around. Instead I've got to start thinking up some of my own for the Brandeis festival. Let me know how they come out.\n\nAs for T'wood, I am not surprised, though a bit beset by wonderment. Yes, I suppose I'll be coming back, but don't say anything yet: I just _might_ have another brainstorm. Sometimes I think I don't really understand about T'wood any more. More of this anon.\n\nAre you really thinking of coming here in May? As far as I know, this house is available then. I think you would love it, though it's very large and expensive (2000 pezozzees a month!) But maybe now that you're rich... Let me know. Of course I'd much rather have you around Boston in June to help us out at Brandeis, but I realize that's too much to ask.\n\nSo you're a Village weekender, just like all the Harvard boys? I find it very chic, and send Eric [Erik] my best.\n\nDear Aa, it was a real joy to have your letter and I miss you and hope you will continue to write zillions more like that Piano Quartet.\n\nMuch love,\n\nL\n\n320. Felicia Bernstein to Leonard Bernstein\n\n[?late 1951 or 1952]\n\nDarling,\n\nIf I seemed sad as you drove away today it was not because I felt in any way deserted but because I was left alone to face myself and this whole bloody mess which is our \"connubial\" life. I've done a lot of thinking and have decided that it's not such a mess after all.\n\n_First_ : we are not committed to a life sentence \u2013 nothing is really irrevocable, not even marriage (though I used to think so).\n\n_Second_ : you are a homosexual and may never change \u2013 you don't admit to the possibility of a double life, but if your peace of mind, your health, your whole nervous system depend on a certain sexual pattern what can you do?\n\n_Third_ : I am willing to accept you as you are, without being a martyr or sacrificing myself on the L.B. altar. (I happen to love you very much \u2013 this may be a disease and if it is what better cure?) It may be difficult but no more so than the \"status quo\" which exists now \u2013 at the moment you are not yourself and this produces painful barriers and tensions for both of us \u2013 let's try and see what happens if you are free to do as you like, but without guilt and confession, please!\n\nAs for me \u2013 once you are rid of tensions I'm sure my own will disappear. A companionship will grow which probably no one else may be able to offer you. The feelings you have for me will be clearer and easier to express \u2013 our marriage is not based on passion but on tenderness and mutual respect. Why not have them?\n\nI know now too that I need to work. It is a very important part of me and I feel incomplete without it. I may want to do something about it soon. I am used to an active life, and then there is that old ego problem.\n\nWe may have gotten married too soon and yet we needed to get married and we've not made a mistake. It is good for us even if we suffer now and make each other miserable \u2013 we will both grow up some day and be strong and unafraid either together or apart \u2013 after all we are both more important as individuals than a \"marriage\" is.\n\nIn any case my dearest darling ape, let's give it a whirl. There'll be crisis (?) from time to time but that doesn't scare me any more. And let's relax in the knowledge that neither of us is perfect and forget about being HUSBAND AND WIFE in such strained capital letters, it's not that awful!\n\nThere's a lot else I've got to say but the pill has overpowered me. I'll write again soon. My wish for the week is that you come back guiltless and happy.\n\nF\n\n321. Marc Blitzstein to Leonard and Felicia Bernstein\n\nFerris Hill Road, New Canaan, CT\n\n19 July 1952\n\nDear L and F,\n\nSitting in the kitchen with Cheryl [Crawford] and Ruth [Norman], eating raspberries for breakfast, it comes all over me what a fine time I have been having recently. The _Regina_ concert started, really sparked, the sense of well-being; then Brandeis and the _Threepenny_ [ _Opera_ ]; then Mina's;53 then Tanglewood and you-all. Not a great deal of work to show for it; maybe something just as good for me now, and for which I have apparently been hungry: well-being, that says it. It comes over me that for a long time since the Broadway _d\u00e9b\u00e2cle_ of _Regina_ I have been slowly withering on the vine. How one needs these vanity-assuagements!\n\nYou are a lovely host-and-hostess, did you know? It goes so smoothly, the guest feels he isn't in the way \u2013 that, I suppose, is the stumbling-block of most visits. And if this note betrays a contented kind of stupor, then it does. I have little guilt about it.\n\nWill you let me know about _Trouble_ [ _in Tahiti_ ]? What you decided, how it came off, how you feel after the performance. _Reuben_ calls, and I return to grappling with it. One small session with [Lotte] Lenya first; then a week-end on [Edward T.] Cone's boat at Water Island; and off to Brigantine and Jo [Davis]'s and the garage. I promise to be more cooperative in the matter of showing you the opera next time.\n\nMy love and thanks to Helen. And to you two, you chuckle-birds. Write me simply: care Davis, Brigantine, New Jersey.\n\nMarc\n\n322. Leonard Bernstein to David Diamond\n\nSunset Farms, Lee, MA\n\n21 July 1952\n\nDear David,\n\nReports fly in from all sides that you are really happy at last, and this makes me happy. Apparently Rome has been a joy for you, work-wise and heart-wise; and I cannot resist writing to tell you how glad I am.\n\nFelicia and I live in a constant thrilled expectancy of the child (due at the end of August or so): and it is all an experience to be cherished. Tanglewood proceeds beautifully; the same breathless six weeks. My little opera was a dud at Brandeis, due mostly to the half-baked state in which it found itself at premi\u00e8re time; but now the revisions are almost finished, and a new (the true) ending composed; and I look forward to a more reasonable and telling production here at T'wood on 10 August. Then the baby; then a full season in New York (imagine!) without a single conducting date: nothing but composing to my heart's content. God, I have waited long for that.\n\nBest to you, & write of your plans and your new works.\n\nLenny\n\n323. Marc Blitzstein to Leonard Bernstein\n\nBox 74, Brigantine, NJ\n\n15 August 1952\n\nYour letter is full of good spirits. The modest expression of what must have been an enormous satisfaction, in the success of _Trouble in Tahiti_ , does you credit. I'm hoping you have copies of the improvements, so I may snag one for my files. And I am happy to see that the Rialto-code-word for the work is not \"Trouble\" but \"Tahiti\".\n\nOf course Evita P\u00e9ron (I can't remember if the accent should fall on the \"o\")54 can make a fine operatic subject: her end, of cancer, while young, doesn't fill the picture of tragedy, really seems senseless (except for the saw that \"Death is democratic\"). But she has glamor, power, evil, and a saint's fa\u00e7ade \u2013 all qualities which, incidentally (aside from the \"good works\") characterized Regina Hubbard. That makes me wonder if Lillian [Hellman] is the precisely right librettist. Once the Latin color is snagged, and the expanded picture of power-area registered, will she (L) not find herself treading well-worn paths? Then you could call the opera \"The Bigger Foxes\" or \"The Same Old Part of Another Forest\".55 But I'm happy you've found a subject.56 Watch out for Fleur Cowles, who has a first-hand interview book on Eva; she may try to claim ownership of what is surely public material. Or, on a second thought, perhaps involve her (Cowles).\n\n_Reuben_ really goes well. More later.\n\nI shall probably be back in town after the first. How is Felicia bearing up these last weeks before the event?57 My love to her and Helen. And remember, we have a September date to do the records of \"3d Opera\".\n\nHelen probably told you that Mina's mother died. She might relish a short note. (Mrs. Mina K. Curtiss, Williamsburg, Mass.)\n\nI hug you.\n\nMarc\n\n324. Leonard Bernstein to Solomon Braslavsky\n\n205 West 27th Street, New York, NY\n\n19 September 1952\n\nDear Brasy,\n\nI want to thank you for _both_ your beautiful letters. You are a warm and good soul!\n\nThe baby is beautiful beyond works (her English name is _Jamie_ ), & the mother is just as beautiful. We are slowly getting settled in a new apartment, and should have a wonderful home-like year.\n\nBest to you & your family, and a Happy New Year.\n\nLenny Bernstein\n\n325. George Abbott to Leonard Bernstein\n\nHotel Carter, Cleveland, OH\n\n23 April 1953\n\nDear Lenny,\n\nWhat a joy it is to work with agreeable people!58 Next time let's put in the contract that any job writers have to submit to a psychiatric test to be sure we get a fairly congenial type. Anyhow Rodgers and Hammerstein would pass the test.\n\nThe show is beginning to take form.59 The scenery never quite works, but it will be when we get where there's a better crew.\n\nThank you for the wire. Give my love to the beautiful Felicia.\n\nI hope next year doesn't get by without our being together on some exciting effort.\n\nYours for diminished sevenths.\n\nLove,\n\nGeorge\n\n326. Arthur Miller60 to Leonard Bernstein\n\nRoxbury, CT\n\n[?June 1953]\n\nDear Lenny,\n\nI especially appreciated your note because as it happened I was about to go in and shake up the _Crucible_ production myself. It is done now. I've removed all the sets, and as much of the fakery as possible, and tried to make it look as much like my work as possible.61 Some night soon when you are at a loss for what to do I wish you'd go in again and let me know what you think. It is my first try at \"directing\", and although I could not, under the circumstances, do half of what I would had I started from the beginning, I think you'll get a weird feeling as well as a sense of horror at what production can do to a piece of writing.\n\nI don't know what you do in the summer but if your mind ever turns North give me a call and come up with your wife and baby. We have plenty of room and eat often. I mean it.\n\nSincerely,\n\nArt Miller\n\n327. Francis Poulenc62 to Leonard Bernstein\n\n24 July [1953]63\n\nMy dear Bernstein,\n\nI've tried, in vain, to obtain your address in the United States. It's promised to me, but I'm still waiting. So I am sending my letter to Salabert USA (I think, after all, that Schirmer is your publisher). I want to thank you, from the bottom of my heart, for all that you've done for my dear _Mamelles_ [ _de Tir\u00e9sias_ ]! I have received some press cuttings and I know that it went very well. With you I was calm, since you played the piece by heart, better than me. As I have a great weakness for this score, you've given me true pleasure, and you have all my gratitude.\n\nAt the moment I am composing a big opera based on the _Dialogues des Carm\u00e9lites_ for La Scala. I hope you'll like that too. Let me know when you are coming to France, Danton 52\u201323, 5 rue de M\u00e9dicis, Paris.\n\nUntil then, I say again thank you, thank you, and I embrace you.\n\nFrancis Poulenc\n\n328. Leonard Bernstein: sworn affidavit for passport application64\n\nDistrict of Columbia\n\n3 August 1953\n\nLeonard Bernstein, being duly sworn deposes and says:\n\nI am a citizen of the United States and was born on August 25, 1918 at Lawrence, Massachusetts.\n\nI attended Harvard University and was graduated in 1939 with a bachelor of arts degree. While at college I majored in music. Thereafter I attended the Curtis Institute of Music from 1939 until 1941.\n\nAfter these studies I became assistant to Serge Koussevitzky at the Berkshire Music Center in 1942 and Assistant Conductor of the New York Philharmonic Orchestra in 1943 and 1944. Commencing in 1945, I was Music Director of the New York City Symphony Orchestra for three succeeding years.\n\nIn May 1946, I was honored to be selected as the representative of American conductors at the International Festival at Prague and conducted two concerts of American music. I also conducted at Prague again in 1947 as well as in other European cities.\n\nIn addition to my work as a conductor and musician I have also composed two symphonies and other musical compositions including the score for several musical shows and two ballets.\n\nI cite the details listed not as a record of achievement, but rather as the briefest kind of summary to indicate that my life and interest have been devoted almost exclusively to the world of music.\n\nThe practice of my profession, which is also my livelihood, necessarily entails frequent travel abroad and any restriction on such travel would be a most serious impairment of my right to engage in my chosen profession and an interference with the right to earn a living. Since travel was resumed after the war I have fulfilled engagements in Europe and elsewhere almost every year and my pending application for a passport is for the same purpose.\n\nIn connection with engagements to conduct concerts in Brazil, Italy, Israel and at other places I applied to the State Department for a renewal of my passport in April, 1953 but to date the travel document has not yet been issued. I was informally advised that the application is being considered in connection with Regulations of the Department of State having to do with the \"Limitation on issuance of passports to persons supporting [the] Communist movement\".\n\nAlthough I have never, to my knowledge, been accused of being a member of the Communist Party, I wish to take advantage of this opportunity to affirm under oath that I am not now nor at any time have ever been a member of the Communist Party or the Communist Political Association. I have never (to paraphrase the language of the Regulation) knowingly engaged in activities which supported the Communist movement under circumstances which would warrant the conclusion that I engaged in such activities as a result of direction, domination or control exercised over me by the Communist movement.\n\nI have not adhered to the so-called Communist Party line or followed it \"on a variety of issues and through shifts and changes of that line.\"\n\nThe Regulations referred to are said to have a possible application to my case by reason of the fact that my name has been linked in various ways with a number of organizations which have been denominated as subversive by the Attorney General in connection with the government employee program.\n\nI wish to state generally as to all the organizations involved that my connection, if any, with them has been of a most casual and nebulous character. Almost without exception, their very names are practically unknown by me except in the vaguest kind of way. In fact, in now attempting to recall them and my connection with them, I have had to rely on the memory of my secretary and refer to old scrapbooks, clippings, etc. Needless to say I never knew their real character as they were later denominated by the Attorney General of the United States. I never could claim any exact knowledge as to their objectives or purpose except the humanitarian, benign or cultural one mentioned at the moment I was accosted by some person or scanned the letter or other sugar-coated communication soliciting funds, the use of my name or my services as a musician and conductor.\n\nI wish to emphasize that the name and real purpose of the organizations to which my name became linked through a charitable and well intended impulse, and obviously without the probing deliberation required, are hardly more than a blur in my memory. The link, if any, was on \"paper\" and not a personal one. Besides my ignorance of their underlying purpose, I have no recollection or knowledge of ever having really _joined_ any of them which had a membership roll in the true sense. I do not recall the address of most of them or the years or the city in which they functioned or the names of the officers or principals. I have no knowledge or recollection of ever having attended an organizational meeting of any one of them. It is my general recollection that my name or sponsorship was usually requested in connection with some public function or activity such as a banquet, benefit or concert.\n\nThe letter of enticement would frequently excuse me from any personal participation of a non-political character. I did not thereby espouse or intend to espouse the concealed and ulterior purposes of such groups. I did not possess the requisite suspicion and caution to probe the devious and subversive objectives of those by whom I and too many others were innocently exploited. And, needless to state we did not have the facilities to make the proper determination.\n\nLike almost every other person who has achieved some prominence, I have received hundreds and perhaps thousands of letters in the past ten years soliciting my assistance in one form or another for some charitable, cultural or liberal cause which is always made to appear worthwhile and which would appeal to many good Americans. The great majority calling for my services or appearance on a particular date would have to be declined because of prior engagements in this country or abroad. Those asking for the use of my name as a sponsor would usually be accepted on the basis of the prominence of the person soliciting me or the fact that other well known persons, not known by me to be suspect, had previously agreed to the use of their own names. The fallacy of this procedure in evaluating an organization has been demonstrated in too many cases involving others to require elaboration. In the event I did grant the use of my name to an organization, my secretary would usually advise it by telephone or in the letter of acceptance that I could not take part in its activities or undertake any responsibility as a sponsor. The letter of solicitation and my reply usually comprised the entire extent of my connection with or contribution to the organization.\n\nThe character of my sponsorship was usually of an innocuous character. In the case of the _American Committee of Yugoslav Relief_ I recall it was Quentin Reynolds who asked me to lend my name to a Town Hall American folk song and jazz concert. The _National Negro Congress_ asked me to serve on an audition board at a negro talent try-out at Town Hall. It is my recollection that I could not attend. The _American Youth for Democracy_ tendered me a dinner and a scroll. I did not attend the dinner and received the scroll in the mail. I assume that this honor was bestowed on me because I had been designated by the United States Junior Chamber of Commerce and others at various times as one of America's \"outstanding young men\". While I consented to the use of my name on the letter head of the _American Council for a Democratic Greece_ , I have no present recollection that I ever attended or participated in its functions or met any of the persons active in it. I have no present recollection of the basis for linking my name with _Action Committee to Free Spain Now_ or the _American Committee for Spanish Freedom_ but I assume some nominal connection did exist.\n\nI recall that I did permit the use of my name by the _Civil Rights Congress_ under the mistaken impression that it was identical with the American Civil Liberties Union. In fact, I only learned of this mistake within the past few days. The only connection with the _Council on African Affairs_ I can recall is a small contribution made in response to the representation it would be used to buy food for starving negroes in South Africa.\n\nWith respect to the _Joint Anti-Fascist Refugee Committee_ I recall being tendered a dinner by the Boston chapter in the latter part of 1944 and attending one or two similar events perhaps in New York or San Francisco. I also believe I made a contribution to a hospital in Mexico City which was being sponsored or assisted by this group.\n\nThe _Jefferson School of Social Science_ has asked me to lecture on music subjects on several occasions but I do not recall complying with these requests. My secretary recalls I made a small contribution to it. During the war, I recall I had some slight association with the Music Committee of the _National Council of American-Soviet Friendship_ which had the support of many outstanding Americans. I recall that my teacher, Dr. Serge Koussevitzky, Conductor of the Boston Symphony Orchestra, was chairman of the Music Committee at the time.\n\nI am advised that my name appears among more than 100 others in a paid advertisement inserted in the March 3, 1945 issue of the _New York Times_ advocating support for Representative John M. Coffee's Resolution H.R. 100 which recommended severance of our relations with Spain. Among the other signers were Quentin Reynolds, James Montgomery Flagg, Franklin P. Adams, Hon. Stanley M. Isaacs and the Hon. Joseph E. Davies. The _Veterans of the Abraham Lincoln Brigade_ is listed as the organization sponsoring the advertisement. I recall no connection with this organization and believe that Paul Robeson communicated with me about the use of my name on this occasion. I met Mr. Robeson one time while we were both backstage during a concert.\n\nThe last occasion on which my name was probably used in what may be described as a controversial setting was in early 1949. The world famous Russian composer Dmitri Shostakovich was expected to arrive in the United States. I recall the _New York Times_ music critic solicited outstanding American musicians to sign a cable of greeting. According to the _New York Times_ clipping this cable read as follows:\n\nWe are delighted to learn of your forthcoming visit to the United States and welcome you as one of the outstanding composers of the world. Music is an international language and your visit will serve to symbolize the bond which music can create among all peoples. We welcome your visit also in the hope that this kind of cultural interchange can aid understanding among our people and thereby make possible an enduring peace.\n\nIn this manner, I suspect I became, with other conductors, composers and musicians, a member of the welcoming committee. The arrival of Shostakovich and other musicians was apparently exploited as a propaganda event by means of a conference and formal dinner at the Waldorf Astoria Hotel under the auspices of an organization, which I am informed, is not on the Attorney General's list.\n\nIn any event, I did not attend the dinner or the conference and did nothing to promote them either than second a welcome to a fellow composer who came to the United States with the permission of this government.\n\nThe controversy which followed the Waldorf Astoria conference in March, 1949 brought unfavorable publicity to myself and others and shortly thereafter my name was included among a list of \"prominent people who, wittingly or not, associate themselves with a Communist-front organization and thereby lend it glamor, prestige or the respectability of American liberalism.\" This article, which appeared in the April 4, 1949 issue of _Life_ Magazine convinced me that my name and my good intentions were being improperly exploited by cleverly camouflaged organizations which concealed their true objectives and Communist aims behind a plausible and appealing front. Since that time I have attempted to be most circumspect about permitting the use of my name to organizations in general.\n\nPerhaps one saving grace with respect to my response to organizational appeals is that during the very same period which found me linked to groups later declared subversive, I was also lending my name to activities completely opposed to communism. I have been most active in the cause of Jewish philanthropy and the promotion of Israel as an independent state free from Soviet domination. I have been honored to accept invitations to preach the sermon in Jewish temples in Boston, Chicago and Houston. My religious training and belief would necessarily make me a foe of communism.\n\nI have contributed to the American Red Cross, made radio appeals on its behalf and given concerts in hospitals under its auspices. I made contributions to the Salvation Army, the Boy Scouts of America, the Riverdale Children's Camp, the Al Smith Memorial Hospital Fund, the Harvard Scholarship Fund, the Greenwich House, the National Urban League, the National Federation of Infantile Paralysis, the Irvington House, the United Jewish Appeal, the United Unitarian Appeal, B'Nai B'Rith, American Friends Services Committee, Hebrew Sheltering and Immigrant Aid Society, United World Federalists, National Association for the Advancement of Colored People, Grace Congregational Church, American Friends of Hebrew University, the Y.M.C.A., New York Guild for the Jewish Blind, the Order of the Purple Heart and many others.\n\nI have permitted the use of my name to organizations having no possible Communist-front implications such as The Nation Association, the Planned Parenthood campaign, European Friends of ORT, the Riverside Children's Association, the American Fund for Palestinian Institutions, the Brooklyn Philanthropic League, the Exhibitions of Palestinian Art in America, the American Christian Palestine Committee, the American Red Magen David for Palestine (similar to American Red Cross), American Arts Committee for Palestine, the Hebrew Union College, the United Jewish Appeal (musicians group), Golden Anniversary of the City of New York, the National War Fund, the Serge Koussevitzky Music Foundation and the World Festivals of Friendship, among others.\n\nI have donated my professional services to a number of organizations and for a number of causes. These would include War Bond drives, the National War Fund, the Music Box Canteen, the American Theatre Wing War Service, Veterans' Administration Hospitals, New York Stage Door Canteen and many others. I have participated in hundreds of benefit performances in the past ten years for worthwhile charitable and cultural purposes.\n\nWhile abroad on previous occasions, I have no reason to believe that my work or activities were a source of possible embarrassment to the government of the United States. Every impression I received, from both the foreign press and other observers, was to the contrary. I had reason to believe sincerely that I was making a small but genuine contribution to international good will and understanding. In this connection, I would like to point out that I have received official and semi-official letters of commendation from members of the Foreign Service of the State Department following my appearances at concerts in European cities. I have also been invited to return on most occasions. The State Department has asked me on several occasions to submit material of a musical nature for dissemination in connection with its information programs abroad. As recently as July 9, 1952 I acceded to a State Department request for the use of my music from the production of _On the Town_ for a Voice of America program. In 1947 I was invited by the State Department to appear on the Austrian section of its International Broadcasting Division and in 1948 I was invited to serve as a visiting lecturer at cultural centers maintained by the State Department in South America.\n\nIn retrospect, perhaps the most profound effect made upon my life, philosophy and thoughts was by the celebrated conductor Serge Koussevitzky under whom I studied and who was, among other outstanding achievements, initiator and director of the Berkshire Music Center and the conductor of the Boston Symphony Orchestra from 1924 until several years before his death in 1951. Koussevitzky, already famous, and twice honored by the Czar as early as 1903, left Russia with the advent of Communism and in my close and long time association with him I came to know and share his strong antipathy for Soviet Communism and its evils. While he detested Communism he loved Russia and its people deeply and always looked forward to the day when they could join the people living in a free world.\n\nMost of the events which have been discussed occurred when I was a young man and while I do not desire to take refuge under any claim of immaturity, I was in fact, inexperienced in the realm in which I dabbled. I have, since reaching my majority, voted almost exclusively in New York City where the voter, during this period has had the opportunity to vote for candidates of political parties other than the two major political parties. I have never, during this time, voted for or otherwise supported any candidates except those of the Democratic and Republican parties.\n\nI have in the past spoken out against the inhibitions imposed upon creative artists, particularly composers, under the Soviet regime. I have frequently and publicly stated this viewpoint, in my lectures on 20th-century Operas and Symphonies at Brandeis University at Waltham, Massachusetts, commencing this year. In February 1948 I publicly expressed somewhat the same viewpoint in the Theatre Arts Magazine in which I commented on the proletarianization of Russian art and music. In this statement I referred to the Soviet effort to force its serious composers to write music limited by mass appeal and described one of the resultant products as \"dreadful\". I added that in the case of the ballet suite in question \"the Soviet idea of music for the masses seems to have reached an all time low\", and further, \"there comes a point beyond which simplicity of thought can become infantile\".\n\nIn conclusion, I am informed that the United States Supreme Court has stated, in the case of an avowed official of the Communist Party itself, that many organizations have several purposes or objectives, some good and some bad, that persons join such organizations for different purposes and that the evidence of membership to fulfill the bad purpose should be \"clear\", \"unequivocal\" and \"convincing\". What then should be the judgment in a case where membership is practically non-existent, the true purpose is unknown and the connection with the organization is of the most nominal and tenuous character?\n\nI have attempted in the foregoing pages to set forth some of the facts which may be of assistance to the State Department in the consideration of my passport application. I express the hope that they may place in better perspective the allegation which has given rise to the long delay in passing upon the problem.\n\nI realize now that I might have made the task of the Passport Division of the State Department an easier one if in 1949, when I came to realize that I had been imposed upon, I had made a public disavowal of the harmful significance which had been attached to the use of my name and prestige by questionable organizations. Unfortunately I did not do so and confined my efforts to advising only my friends and associates of the true situation. I recall that in 1949, in discussing the matter in correspondence with Mr. Edward A. Norman, President of the American Fund for Israel Institutions, that I stated in part as follows:\n\nLet me state for you and for the record that I am not and never have been a member of the Communist Party, nor have I ever subscribed to communist doctrine or ideology. I have been away from the United States for many months on an extended concert tour, and I have not seen a copy of _Red Channels_ ,65 and I do not, therefore, know to what associations or organizations you refer, but I can say unequivocally that I have never belonged or subscribed to any cause which I understood committed me to support of a subversive doctrine, either communist or fascist.\n\nIt is possible that, in the turmoil of life to which an artist is subject, and particularly during those days when the necessity of war created many temporary alliances, that I did sponsor activities without closely examining their affiliations. If so, and if any of these have been investigated and found subversive by the government of my country, I disown and disavow them completely.\n\nI assure you again I have only one allegiance to one flag and to one country, the United States of America, whose democratic form of life I cherish and will defend at any time.\n\nI wish to conclude this affidavit by repeating in the most solemn way the affirmation of loyalty to the United States and opposition to Soviet Communism which I expressed at that time.\n\nLeonard Bernstein\n\nSubscribed and sworn before me this 3rd day of August, 1953.\n\nMaud T. Sauss, Notary Public.\n\n329. Leonard Bernstein to Burton Bernstein\n\nRFD66 #2, Hillsdale, NY67\n\n17 August 1953\n\nCher Baud\u00fcm\u00fc,\n\nIt is hard to know where to write it up on you: are you still in France? Did you get set back by the strikes? Will this letter get to American Express? Are the trains running? These and other questions beset me. But better to write & not reach you than never to write at all.\n\nThis morning I am drawing my first breath after a long haul: Brandeis \u2192 Stadium \u2192 Tanglewood, without interruption. I am d\u00e4dt. We have changed our plans brilliantly (Feloo's decision): to remain here blessedly until the first of Sept, & loaf & compose and swim. Then I to Brazil on the 5th, alone (Chile wasn't such a good idea after all), then on to Iz [Israel]. Felicia, barring a great starring role, will join me in Iz (maybe with baby) around mid-October, & continue on through Italy with me.\n\nRemember our rehearsed Washington investigation in the Napoleon bar in Boston? Well, it came true. Not a subpoena: but since my passport was not to be seen I finally went down to Washington & had to have a hearing with an ape at the State Department, & _got it_!! The great experience of it all was my lawyer whom I was insanely lucky to get \u2013 Jim McInerney,68 formerly heard of Criminal Investigation in the Dept of Justice \u2013 an old Commie-chaser \u2013 just the right person to have on my side. And what a great person he is. It was worth the whole ghastly & humiliating experience just to know him, as well as the $3500 fee. Yes, that's what it costs these days to be a free American citizen.\n\nAll too depressing, but at least it's settled. I am told that the other things will be cleared as a result: the Committee files, & even _Red Channels_. McI[nerney] knows all these people on first-name terms: he's a great & valuable ally. But it's shameful that one needs such an ally to retain 1st class citizenship.\n\nWhere are you & where are you going? Dubrovnik?\n\nLet us hear!?!\n\nLove from all of us.\n\nLad\u00fcm\u00fc (Lennuht)\n\n330. Leonard Bernstein to David Diamond\n\nRFD #2, Hillsdale, NY\n\n17 August 1953\n\nDear D,\n\nThis is the first moment for a breath in some months now. It has been an uninterrupted madness \u2013 the Brandeis festival, the two crazy weeks at the Stadium (recording each night from midnight on after the concert),69 and boom, Tanglewood. It's over now, & here come a blessed two weeks up here in the hills to loaf & compose & swim. I thought it would never happen. I am off to Brazil on 5 Sept. until the end of the month. Then all of Oct in Israel (why don't you hop over?) \u2013 then from Nov 5 on in Italy (Scala, _Florence_ , & Rome). Isn't it nice \u2013 concerts in Florence too? Felicia will be with me from mid-October on, and we can make up for lost time.\n\nYour letters all sound so ecstatic that I'm beginning to think you should never return to this country. Apparently you have everything you want now (except more performances: and they're coming). Let's hope it sustains: and why shouldn't it?\n\nI have had a to-do in Washington getting my passport renewed. Ghastly & humiliating & expensive experience. But it worked out, largely due to a great lawyer named McInerney; and I can travel. But it's a pretty pass one has come to when one has to suffer so much embarrassment & costliness to retain one's first-class citizenship! I sometimes tremble for my country.\n\nWe have had the best Tanglewood summer so far: only a dearth of new works and\/or American works. But it has been smooth & highly successful on the academic level. Ch\u00e1vez was here, & a charmer he is. I did Sibelius 4th & stupefied the audience, poor kids, who didn't know what to make of it.\n\nNow these two weeks I have to cram in a violin & orchestra piece & an opera (begin them, at least) \u2013 since it will be steady conducting until December. Very hard to know how to balance one's life & work. And you \u2013 are you turning exclusively literary? Libretto \u2013 and I hope music too.\n\nBest to you and Ciro.70 It's not long til November.\n\nLove,\n\nLenny\n\n331. Lillian Hellman to Leonard Bernstein\n\n[?Autumn 1953]\n\nLennie dear,\n\nThis time I think I have it. I don't know, but maybe Voltaire's _Candide_. I think it could make a really wonderful combination of opera \u2013 prose \u2013 songs. It's so obviously right that I wonder nobody has done it before, or have they? I am very excited by it, but I want to read it two or three times more, think about it, and not decide until \u2013 Anyway, please reread it quickly and let me know what you think, if you are free when you come back, etc. I think done right, it could have real style & wit, and great importance. Write quick. Much love to you, Madame, child & Italians.\n\nLillian\n\nI wouldn't want to do the song lyrics. So if you like the idea \u2013 and I still do by the time you write \u2013 who would be good? Maybe a good _poet_?\n\nAnd it would have to be written with kind of doll-like fairy tale scenery.\n\n332. Betty Comden to Leonard Bernstein\n\n350 East 69th Street, New York, NY\n\n26 September 1953\n\nDearest Lennie,\n\nYour many questions from exotic parts I shall answer in a moment, but I must start with some news which we hope you will like and which might affect your plans \u2013 nay, _will_. George A[bbott]. is feverishly anxious to revive _On the Town_ \u2013 and to do it at once. Both Adolph and I love the idea, and so does Jerry, and George will talk to Oliver about it. The idea, although not worked out at all yet \u2013 would be to have the six of us possibly getting all the money ($100,000 George thinks) together among ourselves, so we would own the thing. Ideally, we should open in January \u2013 and George thinks a week of prevues in N.Y. and no out-of-town session. There are of course some workings on it we would all like to do. G.A. for some reason wondered whether we'd write another song for it. I don't think anything that drastic need be done \u2013 but of course we can't do this without you \u2013 nor would it be any fun to do it without you. I told George I'd write and get your ideas and schedule. This would of course eliminate your going on with your tours to any of those other cities beyond beginning of December. Coincidentally, two nights ago Chris showed the movies he had filmed of the performance of _On the Town_ \u2013 and they looked simply marvelous and whetted our desire to see the wonderful thing running again. George is sure it will be a smash. We think it will be, too. What do _you_ think? Please come home December as planned hmm?\n\nAs far as working on other shows in December we are still faced with no Hollywood date in Feb. We have made no, but I mean _no_ , show plans \u2013 meeting daily though we have been. True, I have been getting settled here with help problems that cripple the creative spirit \u2013 but things are straightening out now.\n\nThe other questions: Yes, Atkinson on _Carnival_ [ _in Flanders_ ]71 was great, and as you know the show has moved to the elephant graveyard along with _Hazel Flagg_.72 We are still the warmest ticket on B'way73 \u2013 with _Can-Can_74 breathing hotly on our necks. The performances have been excellent. As for road company, Roz's75 plans, and the theatre TV deal, we have had three or four meetings since you left, and they have all been as thoroughly unresolved and unsatisfying as the one you attended. Roz won't say if she's staying or going, and we are on the same old tenter-type hooks.\n\nAs for other departments, Susanna is back at school \u2013 Alan is big and plump. Steve is fine. We had a party last night for Lena [Horne] and Lenny H[ayton], small but musical and late \u2013 and I merely stayed up a few minutes longer to give Alan a bottle at 5:45. I could pretend that sleepiness makes me write this way \u2013 but you know better \u2013 having plowed through the same hieroglyphs on Baker's pudding and other vital topics.\n\nAnyway \u2013 isn't it exciting to think of _O_ [ _n_ ] _T_ [ _he_ ] _T_ [ _own_ ] again! And please tell us when to expect you.\n\nBest on the tour, which sounds wonderful \u2013 and much, much love.\n\nB\u00e4ddim\n\n333. Felicia Bernstein to Leonard Bernstein\n\n\"Monday night 2 a.m.\"\n\n[September 1953]\n\nDarling,\n\nYour wire was waiting when we got back from several movies last night \u2013 I had spent the whole day having visions of you crashing in the jungle somewhere and the whole _Handful of Dust_ bit. Helene, la Belle, has been advised though by now I'm sure you've received everything \u2013 by carrier pigeon if necessary.\n\nHad dinner with Bob and then went to the Anna Russell opening. She didn't use any of the material you described and was, I'm afraid, not very funny or professional (two fs?) We then met Harold C. [Clurman] at Sardi's \u2013 the usual were there. Just took Henry out and he shicked it up.\n\nAfter seeing you off I came home and spent the day in bed. I felt really sick from tiredness and I suppose prospective loneliness. I'm going to miss you mine ape.\n\nYesterday I had Jamie to myself and it was delicious. We went to the park where she carries on like a soap-box orator and stops traffic with her beauty. I was proud, Lennuhtt.\n\nExactly a year ago this minute I started having labor pains \u2013 the best thing we ever did was to get married \u2013 you bet \u2013 and me la\u00fc d\u00fc too.\n\nAs you can gather by now there is no news _at all_. This is just so you won't \"hacer el rid\u00edculo\" at Amer. Express.\n\nI kiss you wildly and passionately.\n\nF\n\n334. Felicia Bernstein to Leonard Bernstein\n\n[before 7 January 1954]\n\nMy darling,\n\nFirst of all thank you for the sweetest telegram you ever sent, and which I received with rather mixed emotions \u2013 it's awful to think you'll be away for that long, it's wonderful you are finally having such a well deserved vacation, it's terrible that I'm not there (we've never had a joyful relaxed holiday together), it's good that you'll be on your own and away from me for a while \u2013 and that's how it goes and will always go, I guess, being the ape that I am.\n\nI couldn't write before this \u2013 at first, just after you left, I was feeling so numb I would not have been able to coordinate my thoughts \u2013 and then I started working and had to dedicate _all my time_ to it, but I will go into that later.\n\nWhat I have to say is hard \u2013 before I start I want you to accept the possibility that most of what I say is true. I know that I tend to dwell on things till they get way out of proportion, but not now.\n\nI was happier in Italy than I have ever been with you \u2013 we had fun, we shared everything, we were truly relaxed for once (I am sorry I ever suggested we come home \u2013 I needed to see Jamie but I should have waited). Here in New York all the old problems and tensions seemed to be lying in wait \u2013 plus the whole Bernstein clan. I love Burt and I love Shirley but they are _your_ brother and _your_ sister. There is no wall keeping me out, but there is blood and a shared past between you \u2013 they are, with Sam & Jennie, your family. I have no family really apart from you and Jamie \u2013 and this is all I need. This place is our home \u2013 yours and mine \u2013 it is beautiful because we have made it so and both our personalities are blended in it \u2013 but all of a sudden it becomes so \"Bernstein\" that I have a hard time keeping in touch with myself, but mostly keeping in touch with you. I can not change this, it is the way things are, but put yourself in my place and admit that it can be a little wearing. You will probably say that all this is a sign of possessiveness \u2013 it isn't. My objecting to Jamie being called \"Jamela\" comes from the same source \u2013 it isn't our way of calling her, it is the Bernstein way \u2013 something quite foreign to me, something I cannot share in which perhaps does smack a bit of the ghetto to me \u2013 it's possible.\n\nPlease don't brood about all this \u2013 it will explain a little my strange behavior before you left. I was also, may I say, terrified about your flying that day and kept cursing Waldner all through the day and sleepless night!\n\nI have never worked so hard on a show before. I've given it all my time and concentration. We've managed to rewrite the whole thing \u2013 it is less obvious and trivial but still dreadfully mediocre. I do have the satisfaction of having created a real character and that has been fun \u2013 the director is exciting and between us we've done really good work.\n\nJamie manages to keep her _joie de vivre_ in spite of constant falls, bumps and cracks \u2013 however she cannot seem to live without music specially \"Sandy the Sandman\" and it is driving us all crazy.\n\nRosalia has arrived and all is well \u2013 everything is clean and in its place, the books are right side up, Miss M. likes her. What could be better?\n\nHow wonderful that _Medea_ triumphed again \u2013 it would have been so anti-climactic otherwise.\n\nDearest, dearest Lennhutt I love you so.\n\nFelicia\n\nPlease give me news of Nancy.\n\n335. Leonard Bernstein to Felicia Bernstein\n\nPalace Hotel, St Moritz, Switzerland\n\n7 January 1954\n\nDarling Goody,\n\nWhen I finally got your letter today it seemed that I had waited for it so long I have already composed it myself. Did you think that I was unaware of all that \"bad trouble\" you were going through? That arrival home must have been one of the worst, with all things conspiring to exaggerate your feeling of left-outness: first Miss Marx vying with you for the role of mater-familias; then your feeling that Jamie was being presented to me rather than to you; and then all the \"clan\" business. All at once. Each one of these is soluble and understandable enough by itself, I suppose, but all three at once must have been too much. I don't think it will ever again be like that. This was our first time away and first time returning; it was a crisis (Waldner must have got his dates mixed) and I hope the hard work on the show provided the necessary means of getting through it. I hope you were great in the show, and that all New York is clamoring for you again. I hope Miss Marx has quieted down in her enthusiasm for showing you what she has done for us while we were away (which again is understandable, however irritating it must have been for you). I hope you have been sleeping and having fun and success, and that you have changed the whole dining room into magenta and beaten gold. I hope Jamie can say mama as well as nana now, and that you really understand that you mustn't take it so hard. And I hope you and Shirley and Burtie can exist again on a relaxed level. There is so little I can do to prevent that particular tension: I had missed them both a lot on my long trip (and had not seen Burt for six months), and I was conscious every second we were together that I must not display too much affection or invoke the past overmuch. That was as hard for me as it was for you, and it seems silly to deprive us all of a warm, easy relationship. You wouldn't want that, I know, especially since all tensions between S[hirley] and B[urton] and myself only provoke more tensions between you and me, as well as between you and them. I don't really think it will ever again be so hard as it was this last time, with everything hitting you in the face at once. At least let's hope so, lovely Goody; and please be happy. We have so much to be happy and grateful for: let's both try not to injure it.\n\nEveryone misses you tremendously in Italy, and they all speak of you in tones of hushed wonder. I received some photos of us at the Scala, and people all said: \"Molto pi\u00f9 bella nella natura.\" I miss you mightily here: it is a lovely place, though I've had only two days of it (after the last _Medea_ ), then had to return to Milan yesterday for a fifth _Medea_ matinee, which was a glorious farewell, and only late this afternoon have returned here after a long snowy tortuous drive with Maria Ricordi; so, to put it in the old terms, I'm still dead tired. Now I have again two days, and then back to Milan and on to Genoa to catch the boat. Not very much rest in all, but even the slight amount is a boon, and it's glorious to be on skis again, no matter how awkwardly. Tomorrow I shall really make a try at getting better: up to now I've had to be monstrously careful because of the Scala performances; now I can relax and spend more time at it. Nancy is here, looking much better, and skating her head off. I wrote you from Milan about her operation (did you get all those letters with other letters enclosed, a check from your mother, etc.?), and we have run into the whole smart international set, wild mad playboys and playgirls (mostly lonely, once-beautiful women, unhappily married or getting divorced or already divorced, accompanied by huge dogs, and wild queers who are amusing and repulsive, and I seem to be the toast of the bar. That is, for one night, the first, which was bar night \u2013 molto dancing and club fun \u2013 and no more. It's all too easy not to rest up here, and I'm resting. There also appeared Hakim (Rafael) who went up and down screaming how beautiful you were, and where were you, and why did you ever pick me instead of him. And a pretty blonde named Jenny who will probably turn up in New York. I returned today to find that Ruggiero had broken his arm skiing yesterday. And I became real good friends with von Karajan, whom you would (and will) adore. My first Nazi. Had dinner last night at Fosca Crespi's with Wally, who sends you his dearest love, as does Nancy and Maria Ricordi and Ruggiero (he really does) and Isabel who finally did appear for _Medea_ with Letizia Boncompagni and husband, but without Laurence, and all the servit\u00f9 of the Duomo and all the folks of the Scala. And that's my social news for the night, Marvin.\n\nI've decided to go along with Lillian on _Candide_ , imagine, after having written her a letter saying no and tearing it up, I think it will be more feasible than the David piece this spring, and will allow me to do other things as well, like the violin piece, and maybe refurbishing _Peter Pan_ for Edwin Lester, who is thinking of doing it with Mary Martin. I'm dying to do David, but for next year. I got lots of ideas, or at least a clearer idea, about the libretto coming over on the plane, and it now looks much more like a big three-act opera with chorus and ballet, which nixes it for this season. I've also decided to give Finzi my general representation in Europe, which was a good decision I think, even if she is always rushing about, because she is young and energetic and will work hard for me, and her assistant Paola is clear-headed for the menial tasks. There is also a lot of talk about London in the winter, and Karajan has asked me to Vienna, etc. So we should have fun next year too. The way it looks now, if you agree, is Europe in May, lit and kiboodle; then Rome (Academy) till Xmas, say; then some real European conducting for two months or so, then home. Almost a year abroad! What do you think? Do you think this plan has any beneficial bearing on paragraph #1 of this letter?\n\nI love you so much, and want you not to worry, ever, or be unhappy, when we must be very happy always. Be good to yourself, and make Liz Arden take away your eye-circles, and help her do it from inside.\n\nAll my love,\n\nL\n\n336. Leonard Bernstein to Cheryl Crawford76\n\nVineyard Haven, MA\n\n6 July 1954\n\nDear Cheryl Crawford,\n\nI received your roundabout request for a short overture to _Tahiti_ and have given it some thought. I would love to oblige you, but I can't for several reasons. One, that the opening trio number is itself a prelude and its function in the opera should be just that. Two, that the only material suitable for an overture (outside of writing a whole new special piece) would be the prelude itself, which would cause repetitiousness. Three, I am so rushed in the writing of _Candide_ that I couldn't begin to think of writing a special new piece to precede the opera. I am sure you know how it is, when a piece is two years behind you, to attempt to make any sizeable change. I hope the production is going well and I would love to be kept informed of your progress. Can you send me your itinerary? Please give my love to Alice77 and David78 and all the cast.\n\nVery sincerely,\n\nLeonard Bernstein\n\n337. Elia Kazan79 to Leonard Bernstein\n\nWarner Bros Pictures Inc., Burbank, CA\n\n14 July 1954\n\nIllustrious Maestro,\n\nI have sent the letter to Bob Anderson who will send it to his agent who will no doubt despatch the necessary information to the Dearest Maestro at La Scala.\n\nI am almost through here. And I ought to leave in three weeks. Don't ask me how the picture is. You never know. Everybody always likes the rushes. They don't mean anything. If my basic story is good I guess I'll have a good picture. Certainly my actors are fresh and real. In fact I don't think anybody has ever seen any of them except their mothers and that's the way I like it.80\n\nI'm still kind of punchy from my Hoboken episode,81 but in a punchy way I'm having a lot of fun. I lack some of my usual doggedness and tenacity but I guess it will all come back if I live long enough. In a word, I'm tired.\n\nWhen I get through with this, I'll come back east and sit out front and enjoy your work. How's it coming? I hope it will be wonderful. When you get all done with it I want to talk with you about a project for us both. My idea, in a sense, is to take a novel and dramatize it entirely as a series of musical numbers with hardly anything in between. You might call it an opera except for the fact that it's not one at all and derives from a much more native source, musical comedy. I'll be starting to think about it.\n\nBetty [Lauren] Bacall misses you. This I know for a fact. What other emotional havoc you brought on out here there is no record of, but she misses you.\n\nSwim a lot. The Pacific is a dirty, cold ocean. You've got the good one there.\n\nLove and kisses,\n\nGadg\n\n338. Leonard Bernstein to Aaron Copland\n\nVineyard Haven, MA\n\n29 July 1954\n\nDear Aa,\n\nI miss you. That's the long and short of it. I don't miss Berlioz or the crowds or the pewpils or the scenery or the meetings on the green furniture of Seranak,82 or even the hot crowded Monday forums, I miss you, ecco. And Lukas.\n\nI want to hear about things like how the Piano Concerto (yours) went, and the immortal problem of Farrand\u2013Roth, and dirty gossip, and what of next year, and how is _Tender Land_ going.\n\nMe, I stay put on this heavenly island, intending never to leave except to Venice for a week in Sept. to conduct my new piece with Isaac [Stern].83 It's finished, imagine, & all orchestrated except for the finale. Man, I need you around for some solid criticism. I could use it. _Candide_ crawls along: it's the hardest thing I ever tried, and \u2013 you won't believe this \u2013 it's very hard trying to be eclectic. I am raising the unwilling ghosts of H\u00e9rold and Auber. A new wrinkle.\n\nLove, & write. A big hug to Lukas, & give my best to [Charles] Munch & [Jean] Morel & Olga [Koussevitzky] & all that sort of thing.\n\nAs I said, I miss you.\n\nL\n\nP.S. [John La] Touche sends all the best, as does Lillian.\n\n339. Darius Milhaud84 to Leonard Bernstein\n\nAspen Institute, Aspen, CO\n\n17 August 1954\n\nDear Lenny,\n\nI am _overjoyed_ at the idea that you will conduct _David_ at the Scala85 \u2013 Ghiringhelli86 hopes very much you can stay until January 13th and from Dec 1st. I will certainly fly to Milano!!\n\nAs for the orchestra score that you asked now I really think the _best_ would be to have one done at the Library of Congress Kouss Foundation as it costs only 5 cents a page. The orchestra score would not be more than 45 dollars I think and you would have your own score that you could keep.\n\nMay be the foundation would be willing to pay for it, otherwise I will be delighted to give it to you as a present and as a proof of my great admiration. Just let me know. The piano score will be soon ready in printing.\n\nVery affectionately,\n\nMilhaud\n\nI _missed_ you in Jerusalem. Everything was OK (orchestra, choir, soloist) but George Singer87 was _horrible_ , hysteric and _can't hold a tempo_.88\n\n340. Leonard Bernstein to Frankie and William Schuman\n\nVineyard Haven, MA\n\n30 August 1954\n\nDear Frankie and Bill,\n\nYour joint letter gave me much food for thought (and sickroom sympathy) \u2013 and, well, you talked yourselves right out of an Italian premi\u00e8re.89 With regret I bowed to your dark warnings, after another look at the score (which I love); and you will be supplanted by the far simpler (and far less interesting) Piston's 4th. Now aren't you sorry to be such Cassandras?\n\nI've finished my _Serenade_ (blithely attributed to the authorship of William Schuman in a _Sunday Times_ squib a few weeks ago) and it looks awfully pretty on paper, at least. The Italian critics will hate it; but I like it a lot.\n\nThe Charaks (?) tell me you are much better, & that they are building you a permanent Vineyard residence. You can't do better than this extraordinary, passionate island. We've loved it this summer. Shall we start making dinner plans _now_? I'll be back from Europe around the 17th.\n\nLove to you all from us all, & be well.\n\nLenny.\n\nThink of it: I was 36 last Wednesday. As a friend put it, 3 and 6 are 9, which is the cube [ _recte_ square] of 3; & 3 times 6 are 18, the digits of which add up to 9; and 3 from 6 leaves 3 which is the cube [ _recte_ square] root of 9. Which means in short, I'm getting old.\n\n341. William Schuman to Leonard Bernstein\n\n241 Elk Avenue, New Rochelle, NY\n\n3 September 1954\n\nDear Lenny,\n\nThanks very so much for your note with the terrible news that you have taken my advice and are not performing my 6th [Symphony] in Venice. You can appreciate that it was very difficult advice for me to have given but I feel that it would have been unfair of me to have been less than candid. Until you find an appropriate moment to program the Symphony, I will be nourished by your stated love of it which, as I think you know, means a great deal to me.\n\nI also noticed in the _New York Times_ that authorship of your _Serenade_ for violin was credited to me. Am I to understand that you are now denying that I wrote the piece and claiming it for yourself? Naturally, I assumed that you were giving me this work because I am sick and old and so disappointed that you could not perform the 6th. In fact, in my mind, it already took the place of the second favorite composition of my authorship, the first being \"The Happy Farmer\", and now I am patiently awaiting the first royalty checks from my _Serenade_. Incidentally, if there is an extra copy around, please send it. I am dying to see it.\n\nI am quite well aware that you are now 36 years of age because last Wednesday I tuned in the radio and heard my name followed by the piece of brilliant piano music which I recognized as your anniversary present to me of several years ago. The whole program was devoted to your music and I enjoyed it immensely, the clarinet piece and all (they played the movements out of order). One of these days you simply must take time off and write a great big opera. It is good to know that you will be back the 17th and we look forward to seeing you.\n\nLove to your house from ours.\n\nBill\n\nP.S. Dictated by phone and signed by Miss Martin so that you will get the note without delay.\n\n342. Leonard Bernstein to Felicia Bernstein\n\nHotel Bauer-Gr\u00fcnwald, Venice, Italy\n\n11 September 1954\n\nDarling,\n\nI just had a cable from George which said you were having a second hurricane \u2013 unbelievable! I tried to get through by phone, but no soap. He assured me everything was OK, but I can't help worrying about it. I'll try and call you from London Monday night. Be safe!\n\nI've missed you terribly \u2013 you would love Venice: and it's been a charming week. Ciro [Cuomo] showed me a picture of you that he carries, & I nearly broke down. You're so lovely, & such a _terrific_ actress (I just rediscovered this last week at Woodstock) and I love you more than I can tell you.\n\nIsaac [Stern] plays the _Serenade_ like an angel \u2013 and everyone adores it, Diamond included. If it goes well tomorrow it should be a knockout. The weather is hot & fine, & the Lido is a joy \u2013 though there's too little time to enjoy it. (Last night we rehearsed til 2 a.m.!)\n\nI've bought you gorgeous things. I can't wait to show them to you. All my love and a fountain of kisses for Jamie. I pray you're all OK. Love to Allyn.\n\nL\n\n343. Darius Milhaud to Leonard Bernstein\n\nMills College, Oakland, CA\n\n23 September [1954]\n\nDearest Lenny,\n\nLovely to have your letter.\n\nI answer your questions:\n\n1. Cuts. _Not one note_. I know it is a long opera. _But so it is_. 2 hours 50 \u2013 we must not lose a _second_ between changing scenes.\n\n2. George Singer. _David_ is easy to prepare \u2013 choirs are easy to sing. Solo parts have no problem. They have in all Italian operas good people to rehearse with. Why take this terrible George Singer, who will prepare all wrong. We need a young conductor from the Scala who teaches choir and soloists notes, solfege, and articulation. Someone who makes _precise work_ , and not this frightful hysteric lunatic.\n\nNow if the Scala wanted him absolutely, and wanted to ruin the preparation, and that you will have to rebuild everything, then they should not have ask[ed] me first.\n\nThat is my sentiment from the bottom of my heart.\n\nAnd I admire _you_ and have a great affection for _you_.\n\nDarius\n\n344. Leonard Bernstein to Barbara and Philip Marcuse\n\n205 West 57th Street, New York, NY\n\n26 November 1954\n\nDearest Babs and Fil,\n\nI've let you down terribly, I know, and I'm taking advantage of the fact that today is Thanksgiving, a season of joy and forgiveness and blessing-counting, to try to make reparation. Your nice Mrs. Gilbert phoned some weeks ago with your messages, and I was sorry she had no time to come and visit with us.\n\nI'm taking this time out of the simple family joys (it seems the whole family is here, both sides) to send you our love and greetings. These last months have been overcast with relentlessly gloomy activity. How can activity be gloomy, you ask? It can. We have had big lyricist trouble in _Candide_ , and have only now, this minute (two weeks ago, that is) made a final and utter break with Mr. LaTouche. At the point of the break the show was less than half-finished. So, here we are. It should have been finished by end of summer. Nothing wasted of course. I did get off a little 34-minute thing for violin and orch. called _Symposium_ which Isaac Stern played like an angel with me at the Venice Festival in September. Didn't read about that in _Variety_ , eh? Small wonder. Then, a long stretch of trying to eke out _Candide_ with Touche, and not getting much of anywhere. Then the last two weeks of searching desperately for a new lyricist, in vain. But other things interspersed: an article published in the _Atlantic Monthly_ (cover piece too, imagine) \u2013 did you read it? And last week, no, two weeks, who knows \u2013 a TV stint on _Omnibus_ that seems to have knocked the national press for a loop. Did you happen to catch it? If not, see last week's _Variety_. I count on that mag as our personal intermediator. Also this week's _Life_ , a ghastly inadequate account of the proceedings, but still, _Life_ Mag. I haven't appeared in that one since '48, when a glorious picture of me as one of America's 50 leading \"Super-Dupes\" graced their pages. Then black silence. So this must mean something, a toe in the door.\n\nSo then, good news. I have postponed my Scala trip to February instead of leaving Sunday (God!) when I was supposed to. I had royally screwed up my schedule for this season. But now things are straightening out. Lil[ian Hellman] and I have decided to do what I've been screaming for since the beginning: namely to write the lyrics90 ourselves. It's so natural and right: what were we futzing with Touche for all this time? So now I feel creative and set-up again, and ready for a two-month creative dash; and this time _Candide_ will get finished by Feb. 1st, when I leave. I'll be in Milan for three solid months; then all of May in Florence, for that festival; then some concerts with the Israel Orch. in Italy; then the Holland festival in June; then Tanglesberg in July and August; then, Godwilling, the production of _Candide_. Then, maybe a vacation.\n\nItem: among the greater things for which to give thanks on this Thanxgiving Day: we're going to have another baby! Next July. Which is also a problem, because it makes me have to think of cancelling the Holland thing in June, maybe. In any case, not a word: because Felicia won't get any jobs if it is known she is preggy. Isn't it wonderful?\n\nBack now to the family joys. Jamie is beyond any description beautiful and wise: and Felicia is blooming all over in her joyful condition. Did you catch her last night on Kraft Cheese?91 A bad adaptation of Jane Austen's _Emma_ , but she was fabulous in the high comedy style.\n\nThere's the news up to this moment. Tune in again, etc. Do you get to New York this ensuing period? Let me hear from you, sooner than you heard from me.\n\nLove from us both.\n\nLenny\n\nLooking this over, I am struck by its hectic tone. Are you? But for a change, it's _nice_ hectic.\n\n345. Leonard Bernstein to Felicia Bernstein\n\nGrand Hotel Duomo, Milan, Italy\n\n4 February 1955\n\nDarling,\n\nImagine, I've been here three days and no sign of sinuses or bronchs or the trots or anything. And today, actually, the sun came out, and it was pure spring. It's a joy to be here: deep in rehearsals again \u2013 the cast in the afternoon, & the chorus in the morning, and conferences at night with Luchino [Visconti], who is marvelous to work with. I've gotten all steamed up about _Sonnambula_ , as with _Medea_ , getting wild ideas for cutting & staging and tempi. It's going to be a dream, I think. Luchino has planned a small production, perfect in every stylistic detail, just as I have planned a small orchestra, with emphasis on buoyancy & youth. I wish you could come in time to see it. We open the 19th, & the last one will be around the 25th. Callas is greater than ever. She has shrunk to a pinpoint, & is positively beautiful, even offstage. She has ash-blonde hair, and dresses much better \u2013 and sings like a doll. Last night I heard her as Maddalena in _Andrea Chenier_ , and she was a divine coquette of 17 or so, completely believable! We had our first reading of _Sonnambula_ today, & she made me cry.\n\nI've hunted everywhere for an apartment, but they are all too expensive, or too crowded, or too something. And besides, an apartment is a bore to take care of, whereas at the Duomo I have everything at the touch of a button. So I have decided to stay here, especially since they came down to 5,000 lire a day for this old duplex of ours, which is already reasonable. It's just the same: the same brass bar loose on the stair rug, & hot as hell upstairs, cold downstairs, and all modernistic & hideous \u2013 but I've come to love it, & think of it as home, even. Once I unpack, & get a piano & chairs & table & cushions in, it will be OK, & just waiting for you.\n\nI miss you my darling. Everyone asks for you (I saw the Ricordis tonight) & waits for you. Write me all about everything, right away. Isn't it glorious to be free of _Samarkand_?92 A big kiss to Jamie, and make her say _Daddy_ at least once a day.\n\nAll my love,\n\nL\n\n346. Leonard Bernstein to Felicia Bernstein\n\nGrand Hotel Duomo, Milan, Italy\n\n11 February 1955\n\nMy darling,\n\nI miss you terribly, and love your letters. They carry a whiff of something warm and familiar and joyful. Imagine \u2013 after three years: joyful! Is it wonderful: home has always been the spot in which I happened to be: and now it is _a place_ , with all that one place connotes. The dining-room one apologizes for, & my studio where you get blind with cigarette smoke; and the two \"modern\" chairs you hate in the library, and the marvelous sala, and the hall wallpaper you can't stand, and our country bedroom, and the loud canary, and Jamie spreading her presence like a marigold, and the difficulties below-stairs, and Bill with his weather, and all the problems and tensions and joy and noise and quiet. Home. A new experience.\n\nHere all is up in the air. Callas is still abed with her farunculo, being a real old-fashioned prima donna, suffering, pale, Violetta. Of course we will have to postpone the prima of _Sonnambula_ , which was scheduled for the 19th, to I don't know when: and it is a mess, with my having to go to London for the prima of _Wonderful Town_ on the 23rd. Very complicated indeed. But pazienza: it will all work out. They love me at the Scala, & do everything to help. I've had some fine rehearsals already, & have fallen for the score, hook, line, etc. It never stales for some reason: fresh & noble & pure.\n\nYou won't believe it, but Charlie Roth93 just arrived, & I'll have to see him tomorrow. God, what a burden he is.\n\nAll the Ricordis are waiting for you to come, as is Wally, & Luchino (a doll), and Ghiringhelli & the works. It is fun here, really. You won't be bored. Wally is giving a Ballo in Maschera next Tuesday: that sort of thing. Ghiringhelli sent you a telegram of love on your birthday, & still awaits an answer, kind of hurt & pouting. Do send him one.\n\nI visited the sartoria of the Scala the other day \u2013 an incredible experience. On the outskirts of Milan \u2013 a huge warehouse with 40,000 costumes, and thousands of new ones constantly being made, all by hand, & such materials & designs & care of work! (I'm going to have tails made for me there!) Ghiringhelli wants to sell about 10,000 of them, since the constant procession of new productions creates a surplus of costumes they cannot house. I promised him I would ask you if you would ask Eaves or Brooks94 if they'd be interested in buying such a number of really marvelous costumes. Do let me or him know.\n\nWhat is this with _Bourgeois Gentilhomme_? What part? Who How What? It sounds glorious \u2013 tell me more. And of course, I want to hear all the news of the opening of _Samarkand_. There must be a lot to tell.\n\nI'm so relieved you've been given the OK by all your MD's. I hope you're blossoming in spite of your kepepelt [cold]. You sounded so strange on the phone: remote & unexcited; and I couldn't decide if it was the kepep or 7:00 a.m. (which was ungodly, but I couldn't help it, as I'd been trying for days, & had to take the call when it got through) or just plain old _abstractness_. I'm planning something with an emerald when you get here (this _must_ have something to do with the foregoing) \u2013 and Callas is going to wear moonstones in _Sonnambula_ , imagine. I'm wandering.\n\nI've never worked so close to a show: really into everything: painting the sets, & spending one hour arguing about the color of the cuff of a sleeve of one costume for one chorus-lady, and kind of co-directing with Luchino, & planning out every second. I'm learning, learning. It's a glorious theatre. Tonight is Nanni's birthday, so a party: & first I'm going with Wally & Luchino to a revue with an incredible sequin-blonde dame who calls herself Dorian Gray. I'm trying to get out of _Boh\u00e8me_ (which is an old production, & therefore a bore for me) & do instead _Traviata_ with Callas & Luchino & a fabulous d\u00e9cor de Lila di Nobile (do you know of her?) \u2013 but [Victor] de Sabata is supposed to do it, but his heart is bad, & it's therefore indefinite, but how can one be sure, & as you see \u2013 it's all very complex.\n\nI hope you can get here in time to see _Sonnambula_ at least once: it will be a sweet production, slightly campy, with the stage apron advanced way out, so that Callas sings in the middle of Scala, & sings her last fabulous aria with all the houselights up, & flowers flying from the boxes and \u2013 you must see it.\n\nMy darling lady, I love you & miss you & wait for your letters with real impatience. Love to everyone, & a big hug & millions of kisses to you & Jamie & Fink.95\n\nL\n\n347. Leonard Bernstein to Felicia Bernstein\n\nGrand Hotel Duomo, Milan, Italy\n\n28 February 1955\n\nMy Darling,\n\nI am so happy that you got out of that Samarkand affair when you did. Of course, it would have been a different show with you in it, but maybe not with Schneider.96 As they say here, meno male.\n\nI have just had two days of that nameless Milan disease again, fever and bed. No symptoms, no diarrhea, nothing, just fever and bed. It's becoming a bore. Maybe I have something glorious and important like hepatitis. Now begins the Settimana Santa, with dress rehearsals and the whole bit you know so well. I'm not quite up to it, as usual, and I can't seem to memorize the recitativi, but I suppose it will all be fine in the end. I spent three days in London (how I wished you were with me) and it was if anything grayer and colder than Milano, with whipping rain and wind and snow Londonness. The show was a smash,97 and the Rag98 was encored, and the people screamed (tears running down, not quite, not in London town) and the press was as good as the London press ever is. The experts forecast a year's run at least.99 Which would pay for Fink, and save our lives.\n\nThe black fairy100 will not give up. He arrived in Milano, and I bought him lunch and listened to his drivel and sent him home. He wanted to become my assistant, imagine, and co-conduct all tours with me, etc. I told him to knock it off and go back to Vienna. Since then there has been arriving a series of letters, really unbelievable, in which he is threatening me: blackmail, no less! Unless, he says, I make him the world's top conductor (\"And when I walk on to the podium, God himself will sing\") he will send around copies he has made of all my letters to him in the past, and ruin everybody's lives. The idiot. Of course, there is nothing in any of those letters, nor has there ever been any relationship between us of any kind: but it's still annoying, _aburridisimo_ , and all the rest, to think of that ugly little maniac running around making trouble. I've been sending back his letters and _parcels_ (whatever they may be) by the ton, unopened. Crazy people always scare me, and especially this one.\n\nSpeaking of such things, the Diamonds are here, as is to be expected, and very sweet they are, faithful to the end. They want us to stay with them in Florence, but no.\n\nI don't quite understand about your big trip here with Helena. You mean you would wait until April to come? Que lata! What does Karish say? I know it would be lovely to make the trip with Helena, and I'd love you to, but can't she come earlier too? Your traveling tourist wouldn't make no never-mind. But I hope you don't expect to start with her a pension too! Of course I'll find a nice inexpensive place, as soon as you tell me definitely that she's coming and when. I hope you sent an answering cable to Ghiringhelli.\n\nI have to dash off to rehearsal. God, I wish I could have a week of sun and rest. When and where? Please tell me when you will be coming so I can try to promote a week off from the Scala to fit your arrival. Then maybe we can go to Sevilla for Holy Week, or something glorious like that.\n\nI miss Jamie and you terribly. Everyone writes that you are both in the pink, and that helps. But do come in person!\n\nAll my love, darling Bubbles.\n\nL\n\nCan Jamie say \"Sonnambula\" yet?\n\n348. Betty Comden to Leonard Bernstein\n\n350 East 69th Street, New York, NY\n\n17 March 1955\n\nDearest Leonard,\n\nShortly after the arrival of your letter, for which many thanks, came photostats of the London reviews101 which cheered us considerably \u2013 particularly the one which said our score was full of \"toe-tapping tunes\" and the one about the cab driver who will soon be whistling his fares home with \"Ohio\" and the \"Wrong Note Rag.\" We won't rest until some New York cabbie whistles _us_ home with \"A Million Kids Just Like You etc.\" For the further popularization of the above and others we have completed a new venture \u2013 an album. And it is of this that I am now writing you \u2013 in the main, that is.\n\nOne Bob Israel, head of Heritage Records called us about making an album \u2013 having heard of us from Harold Rome with whom he just made two highly successful albums \u2013 one \"retrospective\" (anyway back to _Pins and Needles_ ) and the other _Fanny_ (also rear-view if you like). Herbie Harris, drummer-boy extraordinaire also urged this lad to get in touch with us. Although cool at first \u2013 and seemingly not at all familiar with us \u2013 Israel blossomed, after one meeting, into the most hysterically devoted fan we ever had \u2013 and after a few rehearsals with a pianist named Milton Greene,102 and the above mentioned Herb Harris, we cut a platter of some nineteen items one rainy afternoon \u2013 and it is about to be released to the unsuspecting public. At least each and every Sadvorousky will buy one apiece.\n\nHere's what is crammed into a twelve-incher:\n\n_On the Town_\n\nNew York New York\n\nLonely Town\n\nTaxi Song\n\nSome other Time\n\nCarried Away\n\n_Billion Dollar Baby_\n\nBad Timing\n\nBroadway Blossom\n\n_Good News_ (!?X!?X!!!)\n\nFrench Lesson\n\n_Two From the Aisle_\n\nIf\n\nHow Will He Know\n\nCatch Our Act at the Met\n\n_Peter Pan_\n\nDistant Melody\n\nCaptain Hook's Waltz\n\nMysterious Lady\n\n_Wonderful Town_\n\nOhio\n\nIt's Love\n\nQuiet Girl\n\nWrong Note Rag\n\nThe last two shows were in the above order \u2013 not chronological as all else above \u2013 because _W_ [ _onderful_ ] _T_ [ _own_ ] was a better ending. You may wonder at our not doing much in the way of _material_ -ish things in _WT_. This was so decided because of our burning desire to get the _songs_ on records \u2013 somehow. Maybe someone might hear one and make a record? It seemed more important than getting the special stuff recorded. This boy is now so excited anyway he wants to make yet _another_ album. He is convinced it is a smash. We've heard it and are timidly pleased with it \u2013 hoping it will sound to others as good as we blushingly admit it sounds to us. He (Bob) has played it for a few outsiders who echo his enthusiasm. Steve [Kyle] thinks it's terrific.\n\nHere comes the somewhat embarrassing part. Adolph and I would be thrilled and honored if you would write a little something about us for the record jacket. The thing will be called, I believe, simply _Comden and Green_ and our pictures will adorn the front. The back is wide open for a paean of praise \u2013 say, two\u2013three hundred words long.103 Gene O'Kelly may say a word or two about us because of his movie connections with our lives \u2013 but we more than anything hope you will send us something printable about us \u2013 if you can and want to, and have the time to.\n\nOther news \u2013 we're in Hollywood two and a half weeks to come up with outline for new picture to write this summer. Hope to write show \u2013 maybe. Family all well. Adolph well. Miss you very very much.\n\nSend very very much love,\n\nBetty\n\n349. Leonard Bernstein to Marc Blitzstein\n\nGrand Hotel Duomo, Milan, Italy\n\n20 March 1955\n\nDear Marc,\n\nYou're right to be boiling mad, and I would be too; and you're also right in knowing that _Regina_ has never slipped my mind, to put it mildly. It is very hard to get definite word out of these people: there is a kind of compulsion to let things simmer for ages before taking any real step. I don't know if you can feel this ambiente at the Scala; it is such a world of its own, and so convinced that almost nothing else exists or happens on earth, that the time-continuum becomes different from the normal one, and all the time ordinarily allotted to all worldwide decisions is gathered together for Scala purposes only. And this combined with the enormous amount of work and number of hourly problems of work make delays inevitable.\n\nI have brought up the subject on several occasions. Ghiringhelli seems to take it for granted that the work will be done next year, every time I mention it: but it is really up to de Sabata. The latter, a very fine gent, loves the piece, and readily quotes and sings from it (especially \"Watching my gal watch me\", which he performs by heart at the piano) but he has worries about the translation. He has told me that he feels that the \"tough\" quality of the English has no decent equivalent in Italian; and at the same time he feels that the work will not mean much to the Scala public in the original English. I have suggested a kind of bilingual version, wherein all the Negro parts (not affecting plot: i.e. Chinkypin, spirituals, etc.) could be done in English and the \"white\" parts in Italian. How does this strike you? With a pick-axe? Each time we discuss this (between acts of _Sonnambula_ or other interruptions) he always ends by saying \"We must have a long talk about this soon\". And that's why I've been delaying writing to you \u2013 until there was something definite to write. But I'm writing anyway to tell you that the work is very much on my mind, and that I love you and it. What suggestions do you have? I tried to think of _Regina_ in English here, watching _Porgy_ having such a big success: but _Porgy_ is different, obviously, the story being so apparent and drawn in huge choral strokes. But maybe you feel that _Regina_ would work just as well in English. There is also a slight worry about the amount of spoken stuff: the Scala is not well adapted for talk, and the Italians don't understand too well the Opera-Comique idea. Karajan did _Carmen_ here in a sort of horrid version of the \"talk-version\", and it was a large bomb. I don't know what all this adds up to in your mind, but I would love to know soon what you think. I shall try tonight to corner de Sabata (he's been ill) and talk it out. We have had, for example, a tentative date for a month to talk about my Violin _Serenade_ which he likes, and haven't yet said a word. You get the picture?\n\nStill, with all this being true, I still apologize for not having written. It has been a wild rush, as you know; and this is now the first breathing spell, now that _Boh\u00e8me_ is finally on (and I alternate operas almost daily). The critics murdered me for _Boh\u00e8me_ , which is natural, since they think they \"know\" this one; but the public loves it. It's really not a very good production, and I have little to do with the stage, which is ridiculous; but the orch. is divine. The cast is mediocre, the sets likewise, the direction horrid. All this in contrast to _Sonnambula_ , which was closely worked out from the beginning by Luchino and me, and shows it. I learn, I learn, all the time.\n\nNow that I have a little time free from rehearsals, etc., I can begin to do the things I have been postponing for six weeks: get _Regina_ settled, give some real thought to the problems of _Candide_ (which are multitudinous) and all the rest. Let me hear from you about the translation business, and about all the other news that your ire prevented you from sending this time.\n\nIt's wonderful to have Felicia here, and we are taking off the next couple of days, probably driving to Florence to see David [Diamond] and Ciro [Cuomo].\n\nLove from us both,\n\nLenny\n\n350. Leonard Bernstein to Marc Blitzstein\n\nGrand Hotel Duomo, Milan, Italy\n\n28 March 1955\n\nDearest Marc,\n\nI've been through hell over _Regina_. I am furious and disgusted, and what is worse, helpless. I now understand the run-around I've gotten: and I've spent the better part of three days discussing it with de Sabata. The latter is genuinely thrilled with the music, there is no doubt. But since the score did not supply the full dialogue, he was waiting for the arrival of the libretto before putting to rest his qualms about the subject matter. After reading the libretto, he finds he cannot allow it to be done at the Scala. He says that the public will not accept such a theme: it is too sadistic, cruel (he even doubts whether censors here would pass it, which is nonsense), and non-operatic in nature. The talk of money is not for the Scala, nor is the spectacle of a woman letting her husband die, etc. etc. etc. I talked it all out with him, tried to explain the values of the play, that it was internationally famous and loved,104 etc. etc. etc. To no avail. I dug up a copy of Ghiringhelli's letter to me last fall saying that it was _definite_ for next season: he admitted that there was a misunderstanding. I have one card left to play: that I will take it to Florence for the Maggio [Musicale] next year. That might make him react. And I will talk to them in Florence about it. I am now angry about it, especially after your last letter with all your suggestions and willingness to change and set spoken lines and all the rest.\n\nThe worst is that de Sabata admires you so very much, and wants to see everything you've done: and is worried that _Regina_ might ruin you here at the outset. Gian-Carlo [Menotti] is here for the _Saint_ [ _of Bleecker Street_ ], and while not agreeing with de S. does agree that it is a dangerous work for the Scala public. Screw them, say I, and do it anyway. That's the duty of an opera house, in whatever city.\n\nI'm going to have one last fling with de Sabata on it, and then bring up the subject in Florence.\n\nDear Marc, I'm sorry. What can I do? _Tahiti_ is off in Florence also, through Italian bungling of another sort. And the headlines scare me. And Burtie sends a miserable letter from Puerto Rico. It is all in all a depressing day, and the sun is shining away as though it were really spring.\n\nMuch love from us both.\n\nLenny\n\n351. Elia Kazan to Leonard Bernstein\n\n7 April 1955\n\nDear Lennie,\n\nI felt terrible about their passing you by.105 Tiomkin's speech told the whole story. I agree with him.106 You wrote one of the really original scores and I felt terrible about what happened. Anyway, I hope you feel that you want to do more pictures because I'm going to come at you with another one.\n\nBudd and I really have a very good idea.\n\nI'm going to Greece and Turkey. I'll be back on June 2. Where will you be this summer? I'll be driving around a lot. Where will you be? I'll be headquartered at Box 25, Sandy Hook, Connecticut. I'd like to see you. Let me know your plans.\n\nLove,\n\nGadg\n\n352. Betty Comden to Leonard Bernstein\n\n350 East 69th Street, New York City, NY\n\n21 April 1955\n\nDearest Lenny,\n\nA short note about a number of things. First, many many thanks for your beautiful words about us. Second, we are still wildly furious about the stupid Academy and could not hear the next ten minutes of the telecast because of our disappointment. Third, saw _Trouble in Tahiti_ 's opening and was re-thrilled, re-inspired, and cried like \u2013 like an idiot. It was a triumph.\n\nTo get back to the album, one of the things Adolph and I hoped to accomplish with it was getting our _songs_ recorded \u2013 so that maybe _someone_ would hear one and decide to do something with or about it. And that has already happened. Lena H[orne] and Lenny H[ayton] were here at a small gathering \u2013 and Adolph brazenly put on the record, and comes \"It's Love\" both L[ena] and L[enny] got excited and asked about a verse \u2013 and getting hold of it \u2013 and then at the _Tahiti_ opening, there they were and Lena said she's been rehearsing it like mad for nearly a week \u2013 and it's great!! Incidentally we'll ship you an album as soon as we have one \u2013 in about two weeks.\n\nAs for _Trouble in Tahiti_ and my crying \u2013 it was beautifully performed by Alice Ghostley, and it is a truly remarkable piece of work \u2013 and I had that feeling of closeness, and desire for us to be working with you, and the sadness of the pressure of time, and everything all at once overwhelming me.\n\nThe \"personal prayer\" \u2013 your notes about us \u2013 is our prayer too. Again, our thank yous.\n\nLove,\n\nBetty\n\n353. Leonard Bernstein to Felicia Bernstein\n\nVia Salaria 366, Rome, Italy\n\n6 May 1955\n\nDarling Bubbles,\n\nRome is beautiful as I have never seen it: blue blue sky, cool air, and the warm sea, and everything blooming away luxuriantly. I am glad I came; but I miss you here. Perverty it is, of course, as is to be expected, but it is not all so, and I am holding my own very well indeed. I have seen a lot of [Gina] Lollobrigida, a gem of a girl: I went to her singing lesson, and watched her work on her current film, and all with _Serenade_ in mind. Remember that one? It's all in the works again, since they fired their Doggyinthewindow composer; Bob Joseph arrived in Milano with Henry Margolies (a charming man, actually, rich and quiet and sensitive) and we have been talking _Serenade_.107 The indicated combination would be Lolly and Luchino a marvelous combo, and both are very eager about it. I am sure that with the right activity Luchino can get to America. I realize not that Reiner spoke through her ass when she said that Luchino could not be gotten over; she is a big speaker with no authority for one half of what she says. I am sure something will work out. And I would be happy for a postponement of a year on _Candide_ : I'd like to have that time to let it cook, and see what should really be done with it. It's wrong the way it is now, that's all I know. The tape has not arrived yet, so I know nothing of the audition sound. I wait every day for word from Lil about May, and I could go to Paris or wherever, but no word from her. Soon it will be too late. I'm a bit sick of the whole thing, and would adore to get going on _Serenade_ and the M\u00fcnch symphony.108\n\nI will leave here Saturday to go back to Milano for the opening of the _Saint_ [ _of Bleecker Street_ ]: then I have to go to Geneva, of all places, to rehearse the Israel orch there (13th\u201315th), then a concert in Trieste (16th\u201318th), then to Florence (19th\u201322nd) then to Genoa to pick up the Israel gang and make the tour. That's the restful prospect at the moment. I wish you were here to do it all with me. Who's going to writhe out front during my Mozart concerto? But meanwhile I am basking in the Roman sun, and having a sort of social life, and thinking about nothing at all. Lunched with the Roberts yesterday, who sent you dearest love. Tonight cocktail party for me at Ruggiero's followed by dinner at the Chisholms. Thrilling. I spent an afternoon at the American Academy hearing the young music, very exciting indeed. Made me want to compose. Saw a revue, ghastly intellectual type, heard Bricktop, saw La Brignone [Lilla Brignone] act in a bad play of [Curzio] Malaparte, etc. I am sneezy with spring blossoms, but not too much so. The cats are all in the garden (say _that_ in six languages) and this is a beautiful house with divine food and comforts and all and all.\n\nI loved both your letters, and ate up every word, and read them over and over. All is as I feared about _Tahiti_. I realized from the notice that the opera had been reduced to a comic aria by Ghostley. You are right in everything you say about it: the same was true this summer in Westport. I am sure that a recording with Ghostley would be ill-advised, despite her talent: because on a record the vocal aspect only is heard, and that won't do for the touching parts. Why won't people realize that it is the touching parts that the opera is about? The rest is only either comment or diversion. I'd love to see you direct it one day. Thanks for not letting them put in the Trio before the office scene: what a cheap amateur notion! The office set simply _has_ to be ready, and that's all. I do hope Heinsheimer will not be high-handed again about the recording. If it is done I should conduct it, and only if it is well planned out with the right cast. He's one for getting things on at all costs.\n\nI've just had word that the _Candide_ tape has arrived in Milano, so I will hear it shortly.\n\nPlease engage a suite for Charlie Roth at Bellevue.109 I can't wait to break his four-time-fixed nose.\n\nI'm terribly worried about Burtie \u2013 no word in over a month, and nothing since my answer to his desperate letter. I wrote him again begging for a word, but nothing. Maybe you should send him a telegram, and insist on some word from him. I really am very troubled about him. Please did [i.e. do] it.\n\nGive my congrats to Marian and Julian (so glad theirs is ugly) and to Pat and Roald.110 Yesterday a lady told me to rub my left little finger on her nose and then said we were sure to have a boy. No end to the ways of determining sex. Are you bigger and biggering? I hope it's not too hot in NY, and that you stay comfortable through these months. I'm dying to hear about the summer house. Write me care of Finzi, Via Manzoni 5.\n\nYes I _am_ interested in a '52 Jaguar! Is it openable? Let's get it, but make sure someone looks carefully at it to see if it's in good condition, etc. And it really _must_ be a convertible, don't you think? Art Stanton was a darling and acted quickly, but I have changed my mind. Too much trouble for too little worth.\n\nI miss you my darling, and long to come home and see what the hell has happened to that little stranger Jamie, and just have a nice long endless booze hour. Keep well and happy, and un sacco di love. Di nuovo,\n\nLennuhtt.\n\nWhat do you mean, Shirley quit her job? What will she do instead?\n\nI can't tell you what the light is like at this moment on this terrace. And the birds sing in a glorious way. Luchino sends you dear love.\n\nThanks for the _Nation_ puzz!\n\n354. Leonard Bernstein to Felicia Bernstein\n\nGrand Hotel Duomo, Milan, Italy\n\n10 May 1955\n\nDarling Madrina,\n\nI long for the sight of you, belly and all, and I'll push myself fast through these next three weeks of conductifying to get home fast. Rome was a delight: weather such as I have rarely beheld, and uninterruptedly, day after day. I got a lot of sun and garden-lolling, and got to the sea three times. There was also a great series of parties, nice people, mostly theatre kids, and it's the same all over. We played the snapping 123 game for hours and sang old Rodgers songs, figurati. I saw a lot of Lollo[brigida] and of Henry Margolies and ultimately of Arlene Francis and Marty Gable (the latter I don't like atall atall, but he's also a producer of _Serenade_ ) and it looks as if things are really cooking for _Serenade_ with la Lollobrigida! I had her sing various times, and when she relaxes it is really lovely, though too small a voice for theatre: but she still has a year to train and make the voice grow. And now Luchino is involved as a possible director which would be great with Gina (and also great for the show, I think) and _Candide_ can go fly a kite. I am not going to waste another year nohow.\n\nNo word at all from Lillian [Hellman] or Ethel [Linder Reiner]: I sat waiting in Rome for a cable which would make me go to Paris, but nothing came, and thank goodness, since I got a bit of a rest. I don't understand what is happening. I began to take the Royal Jelly cure (Queen-bee) and it seems to be doing something: I don't need so much sleep, for example. But it's still too early to tell whether it works or not.\n\nI'm back at the Duomo for two days, just to clean up some items, and then I got to Geneva for a few days (I leave Thursday, stay until Saturday night, when I take a sleeper to Trieste). Address in Geneva: c\/o de Toledo, 46 Quai Gustave Ador. Address in Trieste: Hotel Excelsior (15th, 16th, 17th). From the 18th through the 22nd I'm in Florence (Grand Hotel). The rest Helen has.\n\nI just got the _Candide_ tapes (what a mess at the customs!) and I'm off to the Ricordis to hear them. Did I tell you that Maria [Callas] had an operation? You might drop her a note. She's fine now. Address: Corso Porta Nuova 10. Remember?\n\nI have to run: I'll keep writing. Much love to the _kids_ , fijate, and a sacco to you.\n\nL\n\n355. Leonard Bernstein to Barbara and Philip Marcuse\n\n[En route from Trieste to Florence, Italy]\n\nheaded paper of the Excelsior Palace Hotel, Trieste\n\n18 May 1955\n\nDearest Marcusi,\n\nHere I sit on a train from Trieste to Florence, at the moment stuck forever in the station of Venice, with apparently no intention of continuing, and I am writing you on stationery from the hotel as you can see, and I am thinking about you very hard. I keep wondering how Phil is (and what he is, now that he has become so fascinating and mysterious with the sort of problems I thought only I had); and I keep hoping you really will show up at Tanglefoot. Felicia has been with me for five weeks (up to about three weeks ago) and I am now doing the last lap by myself. Scala is over (it was a mistake to give 3 months to it: I lost my Milanese glamor) and I am currently doing concerts: last night in Trieste (they loved me in Trieste) and now Florence, and then I pick up the Israel orchestra, which is now touring Europe, and do concerts with them in Genoa, Florence again, Naples, Perugia, Bologna, and finally Milano; and then I go home, having dutifully cancelled Holland as per your instructions, to witness the birth of my newest human bean. I should be home the first week in June, and the Bean is expected about mid-June, and we go to the Berkshires beginning of July. Of course none of these dates will work out accurately, and it will be a typical Bernstein confusion; but somewhere in it all I hope to find word from you about your summer plans and your combined _stati d'animo_ , as we say here. I love you as much as ever and miss you, and hope to see you. I was sorry to have had to turn down the very exciting proposals of Kellman, etc., for a Detroit Festival; but there must be a limit somewhere. I discovered in an old book of Chinese divination that my problem is one of self-limitation, and I am therefore working hard on it (as if I didn't know before out of my own private little Jewish divination what my problem was). Some day, preferably soon, I simply must decide what I'm going to be when I grow up.\n\nLove,\n\nLenny\n\nHugs to the kids.\n\n356. Leonard Bernstein to Felicia Bernstein\n\nGrand Hotel, Florence, Italy\n\n23 May 1955\n\nDarling Bubb,\n\nVera brought me your note, which was another breath of spring in primaveral Florence, as were Jamie's kisses. From Vera's description of Jamie and of her talk I don't know what to expect; apparently she has grown and changed so that I won't recognize her. It's a bit frightening to contemplate: I can't wait to see her again. To say nothing of you. Everyone at the parties and the green rooms is again, as before, stating with certainty that it will be a boy this time. Must mean it will surely be a girl. I'm prepared.\n\nThe concerts in Trieste and Florence have both been extraordinary wows, like the great old days. The Prokofieff grows with each performance,111 and I have been playing a hell of a piano these days, I don't know why. Certainly I haven't practiced. At least now the piano-playing is over, and I now begin to concentrate on the _Serenade_ and Isaac [Stern], which is less nerve-wracking, but harder work, because there is a great shortage of rehearsal time with the Jew orchestra. Off to Genoa tomorrow, Sterns in hand. David D[iamond] gave me a charming party last night: he is in fine shape. Titi has been around a lot, and sends you her love, as do all the other Florentines.\n\nYou won't believe this, but the black fairy112 is in again, having covered half the earth from Vienna to New York, he promptly set forth again to Florence, with what money I don't know, and has been beating at the gates to see me (active at T'wood, assistant on the Israel tour, all the old crap) and I have consistently refused to see him. Luckily another pupil of mine, [Piero] Bellugi, has been around, and I got him to take charge of Roth and keep him away from me. According to Bellugi, Roth is madder than ever, and more dangerous. He is now threatening to expose us \"all\" as communists, the idiot. I am just afraid that he will cause some sort of stink through making a scene, with the consulate here, or somehow. He appeared in my dressing room last night after the concert, and I threw him out, and he left with an air of Well That's It, Now I Do My Worst. I'm really scared both for him and of him. He has no money to get back to the US, and will have to go to the Consulate to be shipped home, and that will cause talk. I alternate between saying Ho-Hum and Jesus-Christ. What a horror he is.\n\nTremendous news: I've got a kepepelt [cold], at last. I only hope that I hold out these next 10 days. Lillian has been constantly on the phone, and I may have to go to Paris for two days either after Genoa or after Milano on the 3rd. I hope not, I really do. I'm also a bit embarrassed about all this peddling of the script to all these directors \u2013 it seems almost everyone in the world has been approached short of the Chinese Theatre, and that does not make good talk. In London she is seeing Rex Harrison and [John] Gielgud and [Garson] Kanin and God knows who else, and in Paris [Ren\u00e9] Clair and [Julien] Duvivier and [Jean] Renoir and God knows who else, and for me there is really no work to show that really adds up. It is a situation I don't like at all. I had forgotten what a charm Lil has: speaking to her over the phone reminded me of why one sticks to her through thick: she has a real attractiveness in spite of everything, and a kind of combination of power and helplessness that in a woman is irresistible.\n\n_I want to come home!_ Just think, in addition to coming home to you and Jamie, there will also be Fink \u2013 and a Cadillac! The latter is a slight worry \u2013 do we have the money? If we do, let's get it, and sell the Olds[mobile] in the fall if we have to. There's always the great idea of the two Nashes!\n\nAll my love,\n\nL\n\nVera says you look glorious!\n\nI spent four hours yesterday looking at emeralds for you, and the only decent one I found cost 21,000,000!!! So, let's wait until Lollobrigida makes us a fortch and then you'll _swim_ in emeraldi.\n\n357. Leonard Bernstein to Felicia Bernstein\n\nHotel Colombia, Genoa, Italy\n\n27 May 1955\n\nDarling,\n\nI am just about to leave Genoa, which was after all great fun, in spite of the exhaustion. I had to rehearse the orch. here for the forthcoming tour, and it was a job and a half, teaching them the _Serenade_ and the Berlioz, neither of which they knew, and at a time when they were so tired they could barely read the notes. Then [...] the hall was not available for rehearsal, etc. We finally did it, by sheer dint, rehearsal all afternoon the day of the concert, after which I rushed around Genoa looking for a set of chimes, of all things, which Maria [Callas] helped me to find through Ricordi's [...] and then there was the concert, and the report is that never before has Genoa seen such a success. Imagine, with my funny modern music and unpopular Berlioz! I had feared for the size of the audience as well as for their applause, and was surprised delightfully on both counts. The papers are raves, and Isaac [Stern] played better than ever, and the orchestra really did miracles, everything considered. Then they just went on to Rome, while I stayed here, thank God, for two days with the Sterns [...] A lovely afternoon yesterday at the home of Maria's old parents; today we rented a car and drove to Portofino, but the sun went in bang and we had to come back. But at least it was a breathing spell. Now on to Bologna, and the rest of the one-night-stands, for six days. _Then home_!\n\nListen: Lillian, from London, asks that I stop off in Paris for a day en route home, and I really can't refuse; but I worry so that you'll spring the Fink while I'm not looking that I hesitate. Keep me informed every minute about how things look and feel; if OK by you, I'll go to Paris, and be home probably on the 5th or 6th. I can't wait, really I can't; and you can't scare me with Jamie's tantrums. I've been expecting them all along. I'm dying to see her.\n\nI loved your letter, and the Caddylacky sounds like a dream and I wish you were here to hear the _Serenade_. My, how pretty it is. Terrible about Agee:113 I was prepared for it.\n\nDarling girl, it won't be long now. Please stop yourself up with adhesive tape or corks till I get home!\n\nLOVE\n\nX\n\n358. Aaron Copland to Leonard Bernstein\n\nL'Orangerie, Le Cannet, Alpes-Maritimes, France\n\n17 July 1955\n\nPapa Lensk!\n\nAren't you both smart! Take a _big_ double bow. Hope Felicia and the Knabe are doing fine.114\n\nI'm sitting on top of a mountain in a villa overlooking Cannes and the Mediterr[anean]. Picked me a real nice spot (very different from Lago di Garda). It's work in the morning and work in the evening, and the beach in the afternoon (very different from Tanglewood).\n\nWhich reminds me \u2013 how is the old girl? (Tanglewood). Has anyone noticed anything missing?\n\nAfter I left you in Rome I conducted in Paris and London. Watch out, I'm gettin' good.\n\nAnd what, may I ask, has been happening to you? (Did you see the Amer[ican] issue of _The Score_ , with _2_ articles on Bernstein? If not, they'll tell you.)\n\nGuess who I met in Cannes? \u2013 Kiki Speyer and son.\n\nI hear you had to abandon _Canticle_ [ _of Freedom_ ] in Hollywood for lack of a chorus. Too bad. Tell Edie to send me Tanglewood school programs. Want to see what you're up to.\n\nLove,\n\nA\n\n1 Marian MacDowell (1857\u20131956), pianist and widow of the composer Edward MacDowell. Though well into her nineties when she wrote this letter, Mrs. MacDowell remained an indefatigable enthusiast for new music. In 1896 she had bought Hillcrest, a farm in Peterborough, New Hampshire, as a quiet place for her husband, Edward MacDowell, to compose. After his death in 1908, this idyllic location became home to the MacDowell Colony, where composers, writers, and artists could work side by side in peace and quiet. Mrs. MacDowell lived to the age of 98. Colonists, and the works they created at Hillcrest, included Copland ( _Billy the Kid_ ), Virgil Thomson ( _The Mother of Us All_ ), Thornton Wilder ( _Our Town_ ), James Baldwin ( _Another Country_ ), and Du Bose and Dorothy Heywood ( _Porgy and Bess_ ).\n\n2 Mrs. MacDowell is writing to express her enthusiasm for the _Age of Anxiety_ Symphony which she had heard broadcast from New York on 26 February 1950.\n\n3 Bernstein himself spent time at the Colony only after Mrs. MacDowell's death, but his stays in 1962, 1970, and 1972 were all productive, as he recalled in 1987 when accepting the 29th MacDowell medal: \"All of those times I was writing works which had, at least in intent, a vastness; which were dealing with subjects of astronomical if not mystical and astrological dimension. The first time was _Kaddish_. The second time was _Mass_. The last time was to write the six lectures that I later gave at Harvard known as the 'Norton Lectures'. This vastness is inherent somehow in this place. The air smells higher here, and sweeter, and closer to the vastness.\"\n\n4 Marc Blitzstein (1905\u201364), American composer: a brilliant and innovative musician and a committed Communist. He met a violent death, murdered by three Portuguese sailors in Martinique. Bernstein's student production of _The Cradle Will Rock_ in May 1939 took place less than two years after the famous Broadway opening of the original staging by Orson Welles and John Houseman (recounted in gripping detail in Houseman 1972, pp. 245\u20139 and 254\u201378). Bernstein was a dedicated advocate of Blitzstein's music (recording the _Airborne Symphony_ twice), as well as a close friend. Blitzstein was godfather to Jamie Bernstein, and the two younger Bernstein children are named after characters in Blitzstein's stage works: Alexander after Alexandra in _Regina_ , and Nina after the heroine in _Reuben Reuben_. In 1964, Bernstein led a performance of _The Cradle Will Rock_ from the piano as part of the Blitzstein Memorial Concert at Carnegie Hall. The cast included some from the original 1937 production.\n\n5 Bernstein was conducting in Italy and Israel during the preparations and opening of this revival of _Peter Pan_ , for which he had written new songs and incidental music.\n\n6 The opening night of _Peter Pan_ at Broadway's Imperial Theatre was on 24 April 1950.\n\n7 This letter gives a glimpse into the preparations for Bernstein's score of _Peter Pan_ , and the involvement of not only Blitzstein but also Hershy Kay and Trude Rittmann.\n\n8 John Burrell directed the production.\n\n9 Ralph Alswang, the set and lighting designer.\n\n10 Ben Steinberg was the show's conductor.\n\n11 Blitzstein was clearly already at work in 1950 on his English version of _The Threepenny Opera_. Bernstein conducted the first performance of this in 1952 at Brandeis University in Walthau, Massachusetts, with Lotte Lenya leading the cast and Blitzstein providing narrations.\n\n12 A reference to James Thurber's short story _The Night the Bed Fell_ (1933).\n\n13 _Male and Female_ by Margaret Mead, published in 1949.\n\n14 The conductor Paul Paray and his wife Yolande. Paray was briefly Music Director of the Israel Philharmonic in 1949\u201350.\n\n15 See notes 8 and 10 to Letter 296.\n\n16 This is the only mention of _The Age of Anxiety_ in Bernstein's letters to Romney, who had sent Bernstein a copy of Auden's poem when it was first published, with suggestions for its musical treatment (see Letters 257 and 258).\n\n17 Kurt Weill died on 3 April 1950.\n\n18 Oliver Smith, who had lived there.\n\n19 _The Baroness Bazooka_ (1942) is a delicious send-up of operetta by Comden and Green.\n\n20 The score for _A Tree Grows in Brooklyn_ was eventually written by Arthur Schwartz.\n\n21 Erik Johns (1927\u20132001). He became Copland's secretary in 1948 and they had a romantic relationship. He was later the librettist of _The Tender Land_ (under the pseudonym Horace Everett).\n\n22 The Clarinet Concerto.\n\n23 Robert McGinnis (1910\u201376), principal clarinet of the New York Philharmonic from 1948 to 1960. He was succeeded by Stanley Drucker.\n\n24 Judy Holliday was married to David Oppenheim at the time.\n\n25 _Born Yesterday_ , directed by George Cukor. Judy Holliday won an Academy Award for Best Actress for her portrayal of Emma \"Billie\" Dawn.\n\n26 Bernstein's _Age of Anxiety_ was given at the Holland Festival in Scheveningen on 12 July 1950 by the Hague Residentie Orchestra conducted by Willem van Otterloo, with Bernstein at the piano.\n\n27 On 13 July 1950, Furtw\u00e4ngler conducted the Concertgebouw Orchestra in Beethoven's Symphony No. 1 and _Leonore_ No. 3 Overture, and Brahms' Symphony No. 1. The whole concert was broadcast and has been issued on CD by Tahra (Furt 1012\u201313).\n\n28 This is very similar to the story and screenplay that Comden and Green wrote for _It's Always Fair Weather_ (1955), with a score by Andr\u00e9 Previn, directed by Stanley Donen and Gene Kelly.\n\n29 The next Comden and Green film to be made was their greatest Hollywood success, _Singin' In The Rain_ (1952).\n\n30 Harry Kurnitz (1908\u201368), American playwright and screenwriter whose Hollywood credits included _Witness for the Prosecution_ and _How to Steal a Million_. He collaborated with No\u00ebl Coward on _The Girl Who Came to Supper_. In the 1930s he had also worked as a music critic for the _Philadelphia Record_.\n\n31 See note 88 to Letter 249.\n\n32 Bernstein's first performance of Beethoven's Ninth Symphony was given on 30 August 1950 in the Kurzaal, Scheveningen. It was performed by the Residentie Orchestra of the Hague, the Hague Toonkunstkoor, and the soloists Corry Bijster (soprano), Annie Hermes (contralto), Frans Vroons (tenor), and Willem Ravelli (bass). In the first half, Bernstein doubled as soloist and conductor in the Piano Concerto No. 1. An unsigned review appeared in _De Tijd_ on 31 August 1950. The critic was lukewarm about some aspects of the performance, complaining of \"sensationalized tempi [...] superficiality and lack of nobility in the expression,\" but the overall impression of Bernstein's Beethoven was \"very handsome and very lively, and the applause was exuberant.\"\n\n33 Tanglewood.\n\n34 A popular British car, made from 1948 to 1972.\n\n35 Copland's Piano Quartet was composed in 1950.\n\n36 Nadia Boulanger (1887\u20131979), French teacher, conductor, and composer. On 5 December 1974, Bernstein recalled his first meeting with Nadia Boulanger in a letter to Sylvia Vickers: \"I first visited Paris after the war (1947?), conducting the Radio Orchestra. I believe it was Fran\u00e7ois Val\u00e9ry (son of Paul) who took me at that time to the house of Marie-Blanche de Polignac, the beauteous Countess who had the great 'salon' of those years. I believe they were Sunday evenings, and since Marie-Blanche was a charming singer and music-lover (& patron) her salon was filled with the likes of Poulenc and B\u00e9rard and Val\u00e9ry and, I think, Cocteau. It was there that I met Nadia, and have adored her from that day to this. I never studied with her, but I _feel_ that I have since everything she said impressed me so profoundly. (Besides, so many composers who are close to me, such as Copland, _did_ study with her.) I have almost never returned to Paris without visiting with Nadia, or at least speaking to her on the telephone. She is a super-faithful correspondent and has thus filled in the long gaps between Paris visits, if only with a few always moving lines. [...] She is to this day so terribly aware of time passing, of missed contacts, of the need to be near those we love during every troubled moment. Only last week I had another note from her, in her own shaky but still legible hand, imagine, at her age and in her near-blindness. May she live forever.\"\n\n37 Written on the headed writing paper of the \u00c9coles d'art am\u00e9ricaines, with Boulanger's personal address printed at the foot of the page.\n\n38 Written in French; English translation by the editor.\n\n39 Robert Rossen (1908\u201366), American film director who won an Academy Award for _All the King's Men_ (1949).\n\n40 Presumably Mexico City, since Bernstein wrote this letter while staying in Cuernavaca, Mexico.\n\n41 The House Un-American Activities Committee (HUAC), set up in 1938 to investigate subversive activities or Communist links of American citizens.\n\n42 Albert Maltz (1908\u201385), American author and screenwriter. One of the Hollywood Ten who were blacklisted in 1947 for alleged involvement with the Communist Party.\n\n43 Edward Dimitryk [Dmytryk] (1908\u201399), Canadian-born film director. Another one of the Hollywood Ten.\n\n44 John Garfield (1913\u201352), American actor and a friend of Bernstein's who had been among the guests at the party for his engagement to Felicia in 1947. Garfield appeared before HUAC on 23 April 1951 (a few weeks before Bernstein wrote this letter). When he refused to name any names, Garfield's life quickly disintegrated, and a year later, on 21 May 1952, he died of a heart attack at the age of 39.\n\n45 Ross Evans was Dorothy Parker's secretary and sometime lover.\n\n46 Robert Presnell (1914\u201386) was a screenwriter, married to the actress Marsha Hunt (b. 1917). Both were friends of Bernstein, and both were blacklisted in Hollywood during the \"Red Scare.\"\n\n47 Bernstein was composing _Trouble in Tahiti_.\n\n48 Bernstein conducted Beethoven's _Missa solemnis_ at Tanglewood on 9 August, in memory of Serge Koussevitzky who had died in Boston on 4 June 1951.\n\n49 Felicia was getting regular work in TV drama series. In August 1951 she appeared in \"Death Sabre,\" an episode of _Suspense_ , with the young Leslie Nielsen.\n\n50 Felicia and Leonard had been married nine days earlier, on 9 September.\n\n51 Bruno Walter's illness gave Bernstein the chance to make his spectacular debut with the New York Philharmonic on 14 November 1943.\n\n52 The first recordings of Copland's Clarinet Concerto (Benny Goodman, Columbia String Orchestra, conducted by Copland) and the Piano Quartet (New York Quartet \u2013 Mieczyslaw Horszowski, Alexander Schneider, Milton Katims, and Frank Miller) were released on the same disc by Columbia Records (ML 4421).\n\n53 Mina Kirstein Curtiss (1896\u20131985), writer, translator, and biographer of Georges Bizet. Her younger brother was Lincoln Kirstein, one of the most important figures in the development of ballet in the United States.\n\n54 It should be on the \"o\". Her full name was Eva Mar\u00eda Duarte de Per\u00f3n.\n\n55 Hellman's plays included _The Little Foxes_ and _Another Part of the Forest_.\n\n56 This project for an opera on Eva Per\u00f3n came to nothing, but it's clear from this letter that Bernstein was contemplating it within days of her death (on 28 July 1952). The theatrical potential of her story was famously explored quarter of a century later in _Evita_ by Andrew Lloyd Webber and Tim Rice.\n\n57 Jamie Anne Maria Bernstein was born on 9 September 1952.\n\n58 _Wonderful Town_ had opened at Broadway's Winter Garden Theatre on 25 February 1953, directed by George Abbott. The show had a book and lyrics by Comden and Green, and a score (composed in four weeks) by Bernstein.\n\n59 Abbott was directing _Me and Juliet_ by Rodgers and Hammerstein, which had a pre-Broadway tryout at the Hanna Theatre in Cleveland.\n\n60 Arthur Miller (1915\u20132005), American playwright. _The Crucible_ was widely perceived at the time as an attack on McCarthyism. Miller was summoned to testify before HUAC in June 1956, following a routine request for a passport renewal (a parallel with Bernstein's situation in 1953; see Letter 328). For refusing to name names, Miller was found guilty of contempt of Congress in 1957, a conviction that was reversed a year later.\n\n61 _The Crucible_ opened at the Martin Beck Theatre on 22 January 1953. In June 1953, Miller made several revisions and recast some of the roles. Brooks Atkinson wrote in _The New York Times_ (2 July 1953) that in this revised version, \" _The Crucible_ has acquired a certain human warmth that it lacked amid the shrill excitements of the original version. The hearts of the characters are now closer to the surface than their nerves.\"\n\n62 Francis Poulenc (1899\u20131963), French composer. Bernstein conducted _Les Mamelles de Tir\u00e9sias_ at the Brandeis University Festival of the Creative Arts in June 1953. For Columbia he recorded Poulenc's Concerto for Two Pianos (with Arthur Gold and Robert Fizdale), the _Gloria_ , and \u2013 as pianist \u2013 three songs with Jennie Tourel. Bernstein commissioned Poulenc's _Sept R\u00e9pons des T\u00e9n\u00e8bres_ for the New York Philharmonic. Shortly after Poulenc's death, Benny Goodman and Bernstein gave the world premiere of the Clarinet Sonata, on 10 April 1963.\n\n63 Written in French; English translation by the editor.\n\n64 This lengthy document, a kind of \"loyalty oath,\" is a chilling reminder of its time: McCarthyism was at its height, and thousands of Americans were investigated for alleged Communist sympathies. A number of Bernstein's friends had been ordered to appear before the House Un-American Activities Committee (HUAC). Jerome Robbins testified on 5 May 1953 and named several names (see Vaill 2007, pp. 215\u201320). Aaron Copland appeared as a witness on 25 May 1953. He wrote a darkly amusing account of his grilling by Senator McCarthy (printed in Copland and Perlis 1992, pp. 193\u20135): \"My impression is that McCarthy had no idea who I was or what I did, other than the fact I was part of the State Department's exchange program at one time... It occurred to me... as McCarthy entered that it was similar to the entrance of Toscanini \u2013 half the battle won before it begins through the power of personality.\" David Diamond was also summoned by HUAC, and according to Howard Pollack (Pollack 1999, p. 191), a nervous Diamond asked his mentor for advice: \"What if I'm asked a question about Lenny?\" Copland's sage reply was, \"You say what you feel you have to say.\"\n\nInstead of being called to HUAC (which was well aware of his alleged associations with \"subversive\" groups), Bernstein endured a different kind of torture: \"He was not subpoenaed to appear before HUAC or Senate committees but was instead drawn into living hell in July 1953 by the US State Department's refusal to renew his passport\" (Seldes 2009, p. 69). The State Department was entitled to use its regulations to refuse or revoke passport applications if it believed that an applicant had Communist sympathies or associations (a practice halted in 1958, at least in theory, by the Supreme Court's landmark judgment on the right to travel in the case of Rockwell Kent et al. v. John Foster Dulles). Bernstein's sworn affidavit, printed here in its entirety, was a comprehensive document, \"a humiliating confession of political sin\" according to Seldes (2009, p. 70). It had the desired short-term effect, since he received his renewed passport a few days later, on 12 August 1953.\n\nBut this document was to haunt Bernstein for years to come. In 1954 it was presented to the American Legion for its approval, so that Bernstein could be allowed to work in Hollywood to compose the score for _On the Waterfront_ (see Seldes 2009, p. 71) \u2013 a film directed by Elia Kazan, with a screenplay by Budd Schulberg and including Lee J. Cobb among its stars: all three had named names to HUAC. That all three should have been involved in the creation of a film about the shame and danger of informing is bitterly ironic.\n\nThe submission of this affidavit didn't end Bernstein's problems. His declassified FBI files make for absorbing reading (they are available online at vault.fbi.gov\/leonard-bernstein) and show not only that he continued to come under suspicion, but that this affidavit was often used as a document that indicated the need for further investigation. In 1954\u20135 the FBI compiled a report for William F. Tompkins, Assistant Attorney General in charge of the Internal Security Division in Eisenhower's administration. On 1 October 1954, Tompkins wrote to the Director of the FBI with the subject: \"Leonard Bernstein. Security Matter \u2013 C[ommunist]. Fraud Against Government.\" The alleged \"fraud\" was based on apparent contradictions between Bernstein's 1953 affidavit and the information gathered by the FBI about his political affiliations. A memorandum was sent to the New York office by the FBI Director on 12 October with instructions to start an investigation, adding that \"this matter should be handled immediately.\" The reports subsequently sent to Tompkins reveal that informants against Bernstein in the past were recontacted for this investigation. Almost a year later, on 11 August 1955, J. Edgar Hoover wrote to Tompkins that the Bureau had \"forwarded additional information for your consideration with regard to a possible violation on the subject's part of the Fraud Against the Government Statutes. It is requested that you advise this Bureau of any decision reached by you relative to this matter.\" After reviewing this \"additional information,\" Tompkins replied that \"the only available evidence linking the subject with the Communist Party is based on hearsay rather than personal knowledge. As such it is insufficient to warrant prosecution of the subject under Title 18, United States Code, Section 1001 [relating to making \"materially false, fictitious, or fraudulent statement or representation\"] and Section 1541 [relating to fraudulent issue of a passport], for his denial of Communist Party membership or having ever knowingly engaged in activities connected with the Communist Party movement.\" That investigation was closed in September 1955.\n\nBut Bernstein was the subject of other security (rather than fraud) investigations by the FBI between 1951 and 1958, catalogued in an Office Memorandum dated 31 August 1959 from G. H. Scatterday to Alan Belmont (then Assistant Director of the Domestic Intelligence Division). This memorandum also states that \"During the security investigation of Bernstein, Washington Field Office reviewed HCUA [House Committee on Un-American Activities] files which were found to be replete with information concerning Bernstein's connection with C[ommunist] P[arty] front organizations.\"\n\nDuring the Kennedy years, Bernstein's past was raked over yet again. On 1 September 1961, Hoover wrote to Kenneth O'Donnell, Kennedy's Special Assistant, responding to a request for \"name checks concerning eighty individuals in connection with the Advisory Committee on the Arts\" and attaching a rehashed summary of the FBI's findings on Bernstein since the 1940s. In 1962 the New York Office of the FBI sent the Washington Field Office a photograph of Bernstein, with a bizarre memorandum headed \" _Unsub_ : American musician alleged to be a Soviet agent.\" During the early 1960s, several concerned cranks wrote directly to Hoover, one of the oddest being a nun from the Sisters of St. Joseph in Brooklyn, NY (as with the names of all informants, her name has been redacted in the released Bernstein files). In a letter postmarked 9 March 1963, she wrote: \"It has been brought to my attention that Leonard Bernstein, the noted conductor of the new Lincoln Center in New York City, has Communistic tendencies. For this reason I am writing to you with the hope that you will be able to enlighten my Community (two thousand Sisters of St Joseph) and me with the truth. His performances are listed among our very limited number of programs which may be seen. I do not know if the Bureau is permitted to disclose any findings of Mr. Bernstein's past life [...] May God bless you for your wonderful work.\" J. Edgar Hoover replied personally on 13 March, thanking his correspondent (\"My dear Sister\") for \"the kind sentiments you expressed concerning my efforts as Director of the FBI,\" explaining that information in FBI files is \"confidential and available for official use only pursuant to regulations of the Department of Justice,\" and ending: \"I trust you will not infer either that we do or do not have information regarding Mr. Leonard Bernstein.\" A note attached to Hoover's reply states that \"Leonard Bernstein was placed on the Security Index 5\u20132\u201351, and was canceled 3\u201318\u201352, when the Prominent Individuals Subdivision of the Security Index was discontinued.\"\n\nAs late as April 1966, when Bernstein applied for his passport to be renewed so that he could go to Vienna to work with the Vienna Philharmonic, an FBI Memorandum recycled much of its earlier material. By 1967, an internal memorandum added Bernstein's support for civil rights organizations to an otherwise familiar summary: \"Bernstein has been active in the civil rights movement and in 1965 Harry Bellafonte organized a group of musical and literary artists to take part in the Selma to Montgomery, Alabama, march. Bernstein was one of the artists who made up this delegation.\" That additional information was communicated to Mrs. Mildred Stegall (aide to President Lyndon B. Johnson) at the White House on 5 March 1968 after she requested a routine security check.\n\nIn the available files, two further episodes attracted the FBI's attention. The first, unsurprisingly, was the event held in the Bernstein apartment on 14 January 1970 to raise funds for the legal defence of the Black Panthers. Through off-the-record press briefings and leaks to friendly journalists, the FBI sought to discredit Bernstein and his wife. Ten years later, after some of the FBI files relating to this event had been made available under the Freedom of Information Act, a furious Bernstein was quoted in _The New York Times_ (22 October 1980): \"I have substantial evidence now available to all that the F.B.I. conspired to foment hatred and violent dissension among blacks, among Jews and between blacks and Jews. My late wife and I were among many foils used for this purpose, in the context of a so-called 'party' for the Panthers in 1970 which was neither a party nor a 'radical chic' event for the Black Panther Party, but rather a civil liberties meeting for which my wife had generously offered our apartment. The ensuing FBI-inspired harassment ranged from floods of hate letters sent to me over what are now clearly fictitious signatures, thinly-veiled threats couched in anonymous letters to magazines and newspapers, editorial and reportorial diatribes in _The New York Times_ , attempts to injure my long-standing relationship with the people of the state of Israel, plus innumerable other dirty tricks. None of these machinations has adversely affected my life or work, but they did cause a good deal of bitter unpleasantness.\"\n\nThe last event covered in the available FBI files is one that shows the organization at its most paranoid. Bernstein's _Mass_ was written for the inaugural event of the John F. Kennedy Center in Washington, D.C., on 8 September 1971. On 9 July and on 16 August 1971 memoranda were sent to Charles D. Brennan, then Assistant Director of the Domestic Intelligence Division, with the subject \"Proposed plans of antiwar elements to embarrass the United States Government.\" The second described a \"plot by Leonard Bernstein, conductor and composer, to embarrass the President [Nixon] and other Government officials through an antiwar and anti-Government musical composition to be played at the dedication of the Kennedy Center for the Performing Arts [several lines redacted]. The purpose of this action was to embarrass high Government officials, possibly even the President who might be present.\" It also cited Bernstein's visits to discuss the _Mass_ with the priest and peace activist Daniel Berrigan while he was in Danbury Jail (he was on the FBI's 10 Most Wanted list for his anti-Vietnam campaigning). The FBI memorandum of 16 August also describes an attempted visit that was thwarted: \"On 7-14-71, Bernstein attempted to visit Berrigan at Danbury but was denied admission by prison officials after consulting Bureau of Prisons in Washington, D.C.\" On the day of the first performance, 8 September, Brennan received another memorandum (for information only) summarizing the situation and citing a report in _Human Events_ that \"Bernstein intended to embarrass the President with an antiadministration bombshell,\" but reminding Brennan that Nixon had already announced he would not be present out of courtesy to Jacqueline Kennedy Onassis, stating that the formal opening \"should really be her night.\" Nixon's views on Bernstein were robustly antagonistic. While he will certainly have known of Bernstein's alleged Communist associations in the past, Nixon also regarded him as a dangerous musical modernist. In a White House memorandum to Bob Haldeman dated 26 January 1970, Nixon offered these thoughts: \"As you, of course, know those who are on the modern art and music kick are 95 percent against us anyway. I refer to the recent addicts of Leonard Bernstein and the whole New York crowd. When I compare the horrible monstrosity of Lincoln Center with the Academy of Music in Philadelphia I realize how decadent the modern art and architecture have become. This is what the Kennedy\u2013-Shriver crowd believed in and they had every right to encourage this kind of stuff when they were in. But I have no intention whatever of continuing to encourage it now.\"\n\nOne final request for a security \"name check\" for Mr. and Mrs. Leonard Bernstein was made during the Ford administration, with the FBI submitting the following on 18 November 1974: \"Mr. Bernstein, who you advised is a conductor [...] has been the subject of various security-type investigations conducted by the FBI since the early 1950s based on information that he had affiliated with or supported in some manner 15 organizations cited as communistic or subversive. Leonard Bernstein [...] has been active in the civil rights movement. [...]. On May 12, 1971, Leonard Bernstein and his wife hosted a fund-raising party in support of Philip F. Berrigan and five co-defendants charged with conspiring to kidnap Henry Kissinger and blow up heating systems in Federal buildings in Washington, D.C.\" While this document describes an alleged \"conspiracy,\" it makes no mention of the humiliating defeat suffered by the government when its case failed to secure any convictions on major charges at the trial of the so-called \"Harrisburg Seven.\" Perhaps this isn't such a surprise in the context of Bernstein's FBI file, which for more than twenty years reveals that officials repeated hearsay allegations of his \"Communist\" associations without ever, it seems, making any serious attempt to discover whether there was a shred of truth in them. The \"Red Scare\" is generally thought of as a phenomenon of America in the 1950s. Bernstein's FBI files reveal that for the security services, at least, it was still an active issue in the 1970s, with campaigning for civil rights, and against the Vietnam War, being added to lists of \"subversive\" activities.\n\n65 _Red Channels: The Report of Communist Influence in Radio and Television_ was published by the right-wing magazine _Counterattack_ on 22 June 1950, and named 151 actors, writers, musicians, journalists, and others as Communist sympathizers, giving what purported to be details of their affiliations with suspect organizations. Bernstein was included, along with a number of his friends and colleagues such as Marc Blitzstein, Aaron Copland, Olin Downes, Lillian Hellman, Judy Holliday, Lena Horne, John LaTouche, and Arthur Laurents. Eric Barnouw prints a complete list of those named, and describes them as \"151 of the most talented and admired people in the industry \u2013 mostly writers, directors and performers. They were people who had helped make radio an honored medium, and who were becoming active in television. Many had played a prominent role in wartime radio, and had been articulators of American war aims. In short, it was a roll of honor\" (Barnouw 1990, pp. 122 and 124).\n\n66 The Rural Free Delivery Service of the United States Postal Service.\n\n67 Address added by hand, on the headed writing paper of the Berkshire Music Center, Tanglewood.\n\n68 James McInerney (1905\u201363), American lawyer. He joined the FBI in 1935 and was responsible for investigating internal security cases at the Department of Justice during the Second World War. In 1950, he was appointed Assistant Attorney General in charge of the Criminal Division by President Truman. With the change of administration in 1953, McInerney left the Justice Department and returned to private practice in Washington until his death in a car accident in October 1963. His most famous clients were the Kennedy family, for whom he handled many delicate matters.\n\n69 Bernstein and the New York Stadium Symphony Orchestra recorded four symphonies for American Decca on 22, 24, 26, 29, and 30 June 1953: Beethoven's _Eroica_ , Brahms' Fourth, Schumann's Second, and Tchaikovsky's _Path\u00e9tique_. A month later, on 28 July, they recorded Dvo\u0159\u00e1k's _New World_.\n\n70 Ciro Cuomo was Diamond's Italian secretary-companion who became his devoted friend.\n\n71 _Carnival in Flanders_ was a dismal flop, opening on 8 September and closing four days later. Set in Flanders in 1616, the cast included Dolores Gray and John Raitt. It had music by Jimmy Van Heusen and lyrics by Johnny Burke. The sets were by Oliver Smith.\n\n72 _Hazel Flagg_ , with a score by Jule Styne and lyrics by Bob Hilliard, ran on Broadway from 11 February to 19 September 1953.\n\n73 _Wonderful Town_ began its successful Broadway run of 559 performances on 25 February 1953.\n\n74 _Can-Can_ , with music and lyrics by Cole Porter, opened on 7 May 1953 and ran for 892 performances.\n\n75 Rosalind Russell (as Ruth Sherwood) was the star of _Wonderful Town_.\n\n76 Cheryl Crawford (1902\u201386), American theater producer. Bernstein was to encounter her again two years later when he was working on _West Side Story_ (she withdrew before the show opened). In this letter he responds to Crawford's request for an overture to accompany a summer tour of _Trouble in Tahiti_.\n\n77 Alice Ghostley (1926\u20132007), American singer and actor. She sang the role of Dinah in _Trouble in Tahiti_ on tour, and again when it arrived on Broadway as part of a triple bill called _All In One_ (alongside dances by Paul Draper and Tennessee Williams' _27 Wagons Full of Cotton_ ), described by Brooks Atkinson in _The New York Times_ (20 April 1955) as \"an evening of superb theatre art.\"\n\n78 David Brooks (1915\u201399), American singer, actor, producer, and director. He directed _Trouble in Tahiti_.\n\n79 Elia Kazan (1909\u20132003), American film director and co-founder of the Actors Studio. As a student at Williams College he was known as \"Gadget\", shortened to \"Gadg\". His notorious appearance as a \"friendly\" witness at the HUAC hearings made him very unpopular among his more liberal friends and colleagues, but his gifts were such that Stanley Kubrick called Kazan, \"without question, the best director we have in America.\" Much of this letter is about _On the Waterfront_ , for which Bernstein wrote his only score composed specially for Hollywood. The majority of the music had been already been recorded in Hollywood, on 24, 27, and 28 April 1954 (see Burlingame 2003, pp. 130\u20131). Bernstein's _Symphonic Suite from \"On the Waterfront\"_ was made in 1955 and first performed on 11 August 1955 at Tanglewood. It was dedicated to Alexander Bernstein, who had been born on 7 July.\n\n80 Kazan is exaggerating a little, since Marlon Brando and Rod Steiger had appeared in earlier films, but _On the Waterfront_ was their first major success; it was a screen debut for Eve Marie Saint.\n\n81 Some of _On the Waterfront_ was filmed in Hoboken, NJ.\n\n82 Serenak was the home of Serge Koussevitzky in the grounds of Tangleword.\n\n83 Bernstein's _Serenade after Plato's Symposium_ for violin and orchestra.\n\n84 Darius Milhaud (1892\u20131974), French composer whose music, with its elements of polytonality and jazz, appealed strongly to Bernstein. One of Bernstein's first recordings (in November 1945) was of Milhaud's _La Cr\u00e9ation du Monde_ , a work he conducted regularly.\n\n85 Bernstein was due to conduct the stage premiere of Milhaud's opera _David_ at La Scala in January 1955. At short notice, he canceled in order to devote time to _Candide_ , and Nino Sanzogno took over conducting duties for _David_.\n\n86 Antonio Ghiringhelli, _sovrintende_ (general manager) of La Scala.\n\n87 George Singer (1908\u201380) conducted the world premiere of Milhaud's _David_ in Jerusalem on 1 June 1954.\n\n88 Milhaud was more diplomatic about Singer in his autobiography, _My Happy Life_ : \"It was George Singer who took on the job of conducting David and he needed all the patience he could muster [...] in the end the singers, the Orchestra of the Jerusalem Radio reinforced by the brass from the Police Band, the Jerusalem Radio Chorus and the Students' Choir of the Conservatoire [...] gave my work a rousingly ardent reading. It was, after all, their piece\" (Milhaud 1995, p. 228).\n\n89 Bernstein originally planned to play Schuman's Sixth Symphony in his Italian concerts.\n\n90 For _Candide_.\n\n91 Felicia appeared in ten episodes of _Kraft Television Theatre_ between 1949 and 1956. _Emma_ was broadcast on 24 November 1954, and Felicia played the title role of Emma Woodhouse in a cast that also included Roddy McDowall as Mr. Elton.\n\n92 Felicia appeared opposite Louis Jourdan in the out-of-town tryouts for _Tonight in Samarkand_ , a play by Jacques Deval presented in Princeton and Boston. She was not in the cast of the short Broadway run that followed.\n\n93 Charles Roth was \"a young conducting student Bernstein taught in 1950 and 1951 at Tanglewood\" (Burton 1994, p. 245). Roth was a disturbed individual who threatened to blackmail Bernstein, or to make public Bernstein's letters to him. He is referred to in several letters to Felicia from 1955 as the \"Black Fairy\".\n\n94 Eaves and Brooks were the two leading theatrical costume companies in New York.\n\n95 Bernstein's pet-name for Alexander before he was born.\n\n96 _Tonight in Samarkand_ was directed by Alan Schneider, whose later Broadway credits included the original (1962) production of Edward Albee's _Who's Afraid of Virginia Woolf?_\n\n97 The London production of _Wonderful Town_ opened on 25 February 1955 at the Prince's Theatre.\n\n98 \"The Wrong-Note Rag\".\n\n99 A slightly optimistic forecast: the London production of _Wonderful Town_ did well, but only ran for 207 performances.\n\n100 \"The Black Fairy\" was Charles Roth. See note 92 to Bernstein's letter (346) to Felicia of 11 February 1955.\n\n101 For _Wonderful Town_.\n\n102 Milton Greene (1913\u20132000) was a conductor, arranger, and pianist whose most important Broadway credit was as the original conductor of _Fiddler on the Roof_.\n\n103 Bernstein did as Comden requested and wrote a charming tribute for the record jacket:\n\nEver since we first met, there has been a beloved object in my life called Betty-and-Adolph. This prodigy, apart from being two very dear people, has for many years supplied me with pure, profound laughter, as it has so many others; and it has a way of turning every hearer into a doting fan. But Betty-and-Adolph is not only a thing of laughter. With the years, it has grown in warmth, understanding, theatrical mastery, subtlety and appeal, always making its personal, sweet-sour comment on the follies and lovable sentimentalities of American life. Betty-and-Adolph as performers represent something complete and exquisite. In recent years I have watched them as creators going forward with increasing power, in a fluid state of development that may lead to any number of new forms. It is my belief that this joyous Cerberus will eventually furnish us with what will one day be known as American opera. Nowhere else in America is there to be found this combination of musical instinct and knowledge, theatrical perfection and literate immediacy. It has been my joy to work with them on two shows, and it is my hope to continue forever.\n\n104 Blitzstein's _Regina_ is based on Lillian Hellman's _The Little Foxes_ (1939), an extremely successful play that was made into a film starring Bette Davis in 1941 before Blitzstein adapted it for his opera. Regina was first performed on Broadway on 31 October 1949, conducted by Maurice Abravanel.\n\n105 _On the Waterfront_ won eight Academy Awards in 1955, including Best Picture, Best Actor (Marlon Brando), Best Supporting Actor (Eve Marie Saint), and Best Director (Kazan). Though Bernstein's score was nominated, it didn't win.\n\n106 The winner of the Academy Award for Best Score was Dimitri Tiomkin, for _The High and the Mighty_.\n\n107 In 1955, the idea of a musical version of James M. Cain's _Serenade_ \u2013 which had so appealed to Bernstein in 1947 \u2013 was explored once again, with Arthur Laurents writing the book. As this letter shows, Bernstein was approached to compose the score.\n\n108 A very early mention of the work that became the _Kaddish_ Symphony.\n\n109 Bellevue Hospital in New York is famous for its psychiatric facilities.\n\n110 Patricia Neal and Roald Dahl. Their first child, Olivia, was born on 20 April 1955. Olivia's tragic death from measles encephalitis seven years later left Dahl \"destroyed\" according to Patricia Neal. He never spoke of her, but on the twentieth anniversary of her death he dedicated _The BFG_ to Olivia's memory.\n\n111 Bernstein programmed Prokofiev's Fifth Symphony for the concert in Florence, along with Mozart's Symphony No. 39 and the Ravel G major Concerto directed from the piano.\n\n112 Charles Roth. See note 92 to Bernstein's letter of 11 February 1955.\n\n113 The writer James Agee died of a heart attack on 16 May 1955.\n\n114 Alexander Bernstein was born on 7 July 1955.\n5\n\nWest Side Story\n\n1955\u20137\n\nIt was in the summer and autumn of 1955 that _West Side Story_ started to take shape as a viable project. Jerome Robbins, Arthur Laurents, and Bernstein had quietly set the idea to one side since first discussing it in 1949, but quite suddenly it was on again, thanks to the impetus of reports in the news about gang warfare. The story told in Bernstein's 1957 _West Side Log_ is that the moment of discovery happened in Hollywood when he had a meeting with Laurents in August 1955, but a letter written to him a month earlier by Laurents suggests that it was more complicated than that. Still, it was the outcome that mattered: a new outline, introducing Puerto Rican immigrants, and, with them, the Latin American musical elements that would be so crucial to the score's colors. As well as refining the plot, a second vital factor was bringing Stephen Sondheim into the project to write the lyrics. Over the next two years, the four collaborators were mostly able to discuss the project in the same place and at the same time \u2013 so there's not much correspondence between them about the show \u2013 but before they got down to serious work, Robbins responded in detail to the new Laurents\u2013Bernstein outline, giving important clues about how he wanted the show to develop. It was not only as a creative genius that Robbins' role was of such fundamental importance in the evolution of _West Side Story_. On a personal level, he was one of the very few people who could get complete commitment from Bernstein, and who could insist on \u2013 and inspire \u2013 music of the highest quality. Robbins' letters to Bernstein are often brusque and brutally honest, but the respect they had for each other is unmistakable.\n\nPreparations for the troublesome _Candide_ kept Bernstein from devoting himself exclusively to what was still known as \"the Romeo show\" in mid-1956. And on top of all that, and a busy conducting schedule, there were also television projects. For _Omnibus_ , Bernstein's \"Introduction to Modern Music\" prompted Gunther Schuller to write a long, eloquently argued letter about how to present a balanced account of recent developments in music: he questioned Bernstein's stance on Schoenberg and his complete omission of Webern. (An intriguing sideline: while it's tempting to read too much into the impact of a single letter, can it be a coincidence that a year after Schuller wrote, Webern's _Six Pieces for Orchestra_ were included on a Bernstein program with the New York Philharmonic, and in one of the Young People's Concerts?)\n\nWhile Bernstein was working feverishly on the last stages of _West Side Story_ , Felicia took Jamie and Alexander to visit her family in Chile, partly to give him some peace and quiet, and partly to escape what must have been an increasingly intense atmosphere. The consequence is an exchange of letters between Bernstein and Felicia that chronicle the final weeks of composing the show, the rehearsals in New York and Washington, D.C., the changes Bernstein was forced to accept (reluctantly in some cases \u2013 but judging from the manuscript evidence of earlier versions of some numbers, the instincts of Robbins, Laurents, and Sondheim were unerringly right), and the euphoria of the first night of the out-of-town try-out at the National Theatre in Washington. Felicia was delighted to hear Bernstein's exciting (and excited) news, and wrote back with plenty of her own, above all some delightful vignettes of the children.\n\n_West Side Story_ opened at Broadway's Winter Garden Theatre on 26 September 1957 \u2013 and the day after, Bernstein and Felicia flew to Israel for the inaugural concerts of the Mann Auditorium in Tel Aviv. Bernstein wasn't in New York for the cast recording of _West Side Story_ , but Sondheim was, and he sent Bernstein a wonderfully detailed account of the sessions. Parallel to _West Side Story_ , there was another major development in Bernstein's career: two weeks before the show opened in Washington, Bernstein signed the contract to become Music Director of the New York Philharmonic: over the next decade, the inevitable consequence of this appointment was a shift in the focus of Bernstein's activities, and it was as a conductor that he would need to concentrate most of his energies. _West Side Story_ was a ground-breaking hit, certainly \u2013 but it was to be Bernstein's last triumph on Broadway.\n\n359. Arthur Laurents to Leonard Bernstein\n\nDune Road, Quogue, NY\n\n19 July [1955]\n\nDear Lenny,\n\nI'm sending a copy of this letter and the enclosed to Jerry. Obviously, it is the barest of skeletons \u2013 but it is on the line we worked out and agreed on. And will, I hope, be some sort of basis for all of us to do some thinking on before we meet again.\n\nI don't know whether you've been so busy that you've missed all the juvenile gang war news.1 Not only is it all over the papers every day, but it is going to be all over the movie screens. Arthur Miller, or so I read, is doing an original drama on the subject for the movies.2\n\nBy accident, then, we have hit on an idea which is suddenly extremely topical, timely, and just plain hot. For this reason, I hope we can get to serious work on it as early as we planned. But more than that, if there is any way of getting the thing done this season, I hope we can find it. To my way of thinking, it would be perfect timing to present this on Broadway early in the spring. I don't know if it's possible but with all this splurge of interest in the subject, I think we would be missing a big opportunity if we didn't capitalize on it.\n\nIncidentally, I hope you noticed I didn't say \"East Side Story\".3 This was because of our mutual feeling that the locale should not be specific or definitely placed in any specific city.\n\nLove to Felicia, your brood and yourself.\n\nArthur\n\n360. Arthur Laurents to Leonard Bernstein\n\n[Beverly Hills, CA]\n\nMonday [?Summer or Autumn 1955]4\n\nDear Leonard,\n\nThis, frankly, is an all-out attempt to get you to come out here for a week. And Jerry and I are going to phone you later this week. I would not write this if I did not really think it was important. I wouldn't be staying out here myself, for that matter, no matter how pool-lush the life is. Let me explain.\n\nThe Comden\u2013Green\u2013Styne show5 has bogged down. For many reasons; maybe you know them. Beginning with Leland [Hayward]'s exit, with Jule's [Styne's] productions, with Betty and Adolph's movie work. They don't know how much more work they will have to do on their latest, when they can get back to their show, etc. Furthermore, they spoke to Jerry about doing the lyrics for ours. I understand you spoke to them, about the lyrics. Jerry told them you and I planned to try ourselves. If you and he feel they are right and would like to have them join the project, I am, you know, willing to go along.6 I don't know how right they are but Jerry seemed to think they could be guided and helped and even pushed \u2013 as, he says, they never really have been into doing a good job.\n\nBut that is secondary. The point is that Jerry's decks are clear _at this moment_ for the spring. He and I have been working very well together. By the end of this week, we should have a pretty good outline to go over with you. His concern about the spring is: Will the show be ready? You know him and how he wants his commitments committed well in advance. If you came out, I am absolutely convinced a) the show would be ready and b) we could convince him that it would be ready and thus he would commit himself for a spring production. He would also take it as an evidence of your faith and more, we could actually wind up all the preliminary work we have all talked about and that is so damn necessary. It would not take more than a week. The fare is inexpensive now and you could stay here. There is plenty of room, he has a man who cooks and cleans (lush life), a big pool (lush life), you and I could work during the day and hash it over with him at night. I admit: no social life. One big party, Jerry says. Unless you make your social life quite late at night or in the afternoons. But what are the words, what can I say to urge you to please come out? None I suppose beyond the fact that I \u2013 again \u2013 am convinced that the outline would be set and the show would be set for spring. Which is really what we all want. Please think very seriously about all this and then make your reservation to come out Sunday night and start work Monday. One week, that's all.\n\nAs for production contracts, that's no part of this really. Except Bob Joseph has really been shooting off his wild mouth rather stupidly, and what he hasn't mucked up, Arnold Weissberger has done, and brilliantly. The plan is now for a collaboration agreement between you, Jerry and me to be drawn up, and _then_ proceed from there. But I suppose David Hocker, whom we just spoke to, will or has informed you about all this.\n\nLenny \u2013 again and again: please do come out for the one week. It is terribly important if you want a spring production. I know I do.\n\nMy love to Felicia and your brood and, obviously, to you.\n\nArthur\n\n361. Jack Gottlieb7 to Leonard Bernstein\n\n806 California Avenue, Urbana, IL\n\n2 October 1955\n\nDear Lenny,\n\nI saw _It's Always Fair Weather_ last night; it was very disappointing. True, there were many funny moments, but they didn't add up. For me it was a slick pastiche without any integration. But more than this, what actually disturbed me was the New York-ese evocation. I should think that after _On the Town_ and _Wonderful Town_ Comden and Green had had their say on the subject. It seems that this gold mine doesn't exhaust itself for them. Don't they ever have pangs about using the same kind of subject matter? \u2013 three soldiers instead of three sailors, the New York stereotypes, etc. Besides repeating themselves, I also felt that at several points they veered dangerously close to _Guys and Dolls_. As for the music \u2013 [Andr\u00e9] Previn may be a great pianist, but really now!\n\nWhat this is leading up to is _East Side Story_. If I'm not wrong, this is supposed to be a Bronxite version of _Romeo and Juliet_. The basic idea is novel (as was _Carmen Jones_ ); so was _Fair Weather_. When I think of what might have really been done with the idea of three war buddies meeting again after ten years \u2013 what a truly great drama ( _not_ musical) could have come out of it. Similarly, _East Side Story_ \u2013 are you sure this is stuff for a _musical_ , and, if it is, are the now familiar gimmicks of the floating crap game, chewing-gum drawl, subway rendezvous ( _S_ [ _ain_ ] _t of Bleak_ [ _Bleecker Street_ ]), etc., ad nauseam, going to be rehashed again? I hope not. Enough metropolitana! Also, don't forget that there are historical connotations involved in _Romeo and Juliet_. When Blacher's version was done at Tanglewood (e.g.) from the historical orientation, how fully could one accept a Negro as Romeo and a white girl as Juliet \u2013 even though the whole medium is, to begin with, artificial?\n\nFor the past two weeks I have been taking my Doctorate entrance exams \u2013 harmonic and formal analysis, ear training and dictation, counterpoint, history, and an English exam and the Miller Analogy Test. I am exhausted! If nothing else, these exams point up how really ill-equipped I am! I still can't write a fugue \u00e0 la Gedalge; my dictation is abominable. It's quite sickening. Actually, I am nothing but an animal in music. I work and respond with emotions; the craft is negligible. One other person is taking the degree with me \u2013 Kenneth Gaburo \u2013 a Gershwin award winner who has just finished a Fulbright in Italy. He is about 33 and is certainly far advanced in relation to myself. If there were ten of us, I wouldn't mind, but it is so easy to make comparisons when only two are involved. He won the Gershwin award along with James Dalgliesh, who has since died.8\n\nIt's a good thing that no one else but myself eats my food. I make enough spaghetti for ten people, pot roast like rubber, stuffed pepper like the stones. For anyone else it may be amusing; for me it's just a pain. At least I keep the pots clean \u2013 so there's [no] chance of my developing dysentery. If I could only cook \u2013 !\n\nThree times a day \u2013 morning, noon, night \u2013 I journey back and forth to school. That is the extent of my contact with the world. From 8 a.m. to 1 p.m. I keep busy at the desk and piano. I practice about 5 hours every night, plus accompany dance classes 1 hour per day. I knock myself out, so that there will be no time to wallow in self-pity. The only misery comes when I try to fall asleep \u2013 it's so damn difficult. I enjoy my students, but, at the same time, I resent their drain upon my time which could be spent composing \u2013 preparing lessons, correcting assignments and tests, private coaching sessions; ugh! I'm sorry to say that the main trouble is that my body is in Urbana, Illinois, but my heart and mind are on W. 57th St in Gotham. If there were only someone to talk to \u2013\n\nLove,\n\nJack\n\n362. Jerome Robbins to Arthur Laurents and Leonard Bernstein\n\n18 October 1955\n\nDear Arthur and Lenny,\n\nI am dictating this in 15 minutes I have free during lunch, so I'll get right to it. Excuse the directness, but it's the only way I can get this off to you.\n\nIt concerns the outline, but before I tell you my objections I want you to know that I think it's a hell of a good job and very much on the right track, and that these differences are incidental to the larger wonderful job you are both doing.9\n\nI don't agree with the 3 act division. I feel strongly that this negates the time pressure connected with the whole show and mitigates against the tenseness of the story being crammed into 2 or 3 days. Moreover, there's not sufficient material in Act II or III to stand up by themselves. And it's a serious mistake to let the audience out of our grip for 2 intermissions.10\n\nAct I, Scene 2. Would like to suggest that the meeting between Romeo and Juliet be more abrupt rather than an observing of each other from a distance at first. In general, suddenness of action is something we should strive for,11 beginning with the tempo key in which we establish Scene 1. Its violence and excitement should cue us for all our dramatic moments; i.e. the suddenness and horror of the murders at the end of the rumble, the discovery of love, etc. etc.\n\nAct I, Scene 5. You are away off the track with the whole character of Anita. She is the typical downbeat blues torch-bearing 2nd character (Julie of _Showboat_ , etc.) and falls into a terrible clich\u00e9.12 The audience will know that somewhere a \"my man done left me\" blues is coming up for her. Furthermore, this puts the girl above the age limit and experience that the gang should have and completely disturbs the adolescent quality.13 If she's \"an-older-girl-kicked-by-love-before-experiencing-the-worst\" (and I'm quoting you) she's much too experienced for the gang, or else is sick, sick, sick to be so attached emotionally and sexually to a younger boy of a teen age gang. I can't put the above strongly enough and at the risk of offending you, Arthur, forget Anita and start writing someone who is either older (like Tante) or younger with the same emotional timber of the rest of the gang.\n\nAct I, Scene 6. The jitterbug dance should finish completely and then start again as an encore with Bernardo entering,14 otherwise we kill the hand, the dance and the audience's pitch. You might consider the reading of the headlines here because this will tie in with second drugstore scene after murders have been committed. (See later note in Act III, Scene 5.)\n\nAct II, Scene 1. I again object to Anita's downbeat note, the \"oh-God-am-I-suffering\".15\n\nAct II, Scene 2. I wonder about \" _children_ playing games and marrying themselves\". Just as long as it doesn't become cute, coy or silly, okay.\n\nAct II Scene 3. There are extremely wrong things here. First, it's another reprise of the Bernardo-Romeo-Juliet scene of Act I, Scene 2, the first time they all meet: and either the fight scene must be provoked immediately or else we're boring the audience and stalling.16 Don't understand why Bernardo doesn't plunge into the scene that follows with his provoking Romeo to a fight. In other words, rather than heightening the following scene, I feel it lessens it and robs it. The only thing I like is the character color of Juliet's strength. I thought the version of Romeo steering someone away from the rumbles was a better idea.\n\nAct III, Scene 2. I am starting to feel we're in serious trouble with the so-called love ballet. (See note on dancers at end.)\n\nAct III, Scene 3. Want to know why Juliet doesn't go with Romeo immediately.17\n\nAct III, Scene 5. There are a couple of things I can't adjust here. The boys are jitterbugging to avert suspicion from the police \u2013 but what has happened about the death of their beloved Mercutio? In other words, how do you make compatible the effect of the murders on the boys with what you have written?18 I think this _can_ be one, but isn't indicated at all in the outline. Here's where the newspaper headlines, with references to the teen-age gang war and murders could be used. The \"hey that's me\" effect.\n\nAct III, Scene 6. From the outline I'm inclined to feel that it's all a little too goofy. Juliet becomes Ophelia with the reeds and flowers and is playing a \"crazy\" scene.19 I had to read the whole thing a couple of times to find out why Romeo died20 and I also think it's too right on the head placing it back in the bridal shop.\n\nAs for the all-over picture, we're dead unless the audience feels that all the tragedy can and could be averted, that there's _hope_ and a wish for escape from that tragedy, and a tension built on that desire. We must always hold out the tantalizing chance of a positive ending. Romeo and Juliet particularly must feel this and be sure of it. It's another reason why I dislike qvetchy Anita so much. Let's not have anyone in the show feel sorry for themselves.21\n\nAbout the dancing. It will never be well incorporated into the show unless some of the principals are dancers. I can see, easily, why Romeo and Juliet must be singers, but Mercutio has to be a dancer, maybe Anita, and for sure some of the prominent gang members, otherwise, if any of the dance sequences do take place over the stage, your principals will move to the side and a terrible separation happens.22 Practically, it's easier to rehearse with separate units, but with all the experience I've had it's by far most beneficial to the unity of the show to have the principals do everything. It's a sorry sight and a back-breaking effort, and usually an unsuccessful one, to build the numbers around some half-assed movements of a principal who can't move. Think it over.\n\nI'm sending this off as fast as possible, so please excuse the abruptness. Let me hear from you both.\n\nLove,\n\nJerry\n\n363. Leonard Bernstein to Burton Bernstein\n\n205 West 57th Street, New York, NY\n\n29th October 1955\n\nDarling Baud\u00fcm\u00fc,\n\nHere we have been complaining for weeks at having had no word from you, and here today I find a letter you sent immediately, and I am contride, contride. You also sent a pretty photo, which we all thought looked exactly like me, and it does, and we miss you more and more each day. As you say, it's all downhill from here on, and maybe no more than just a nice warm winter in the tropics. But, to judge from the letter to the folks, which they showed us yesterday, you are bored. Are you bored? Is there a promotion coming up for you? And a pay increase?\n\nWe just returned from Boston where the _Lark_23 had its premiere and it seems to be a large hit. Raves, and the audience lapped it up. My music sounded good as hell, with marvelous voices (on tape: and cheap [Kermit] Bloomgarten wouldn't rent good enough equipment, so that it grizzled a bit) but still it sounded pretty. I think there's the kernel of a short Mass there, and I may expand it into one for the Juilliard commission (two birds technique, as of old).24\n\nI'm sorry you missed my last _Omnibus_ show, On Jazz, and it seems to have had an ecstatic reaction all over the country. Columbia wants to record it, and all the others in a series, and there is serious talk of filming it (and others) for commercial release in theatres. Goodness. Next one coming up in about three weeks, on Bach, about whom I know nothing about whom.\n\n_Romeo_ proceeds apace, with a new young lyricist named Steve Sondheim, who is going to work out wonderfully. I still have high hopes of a spring production. No deal reached as yet with Bob Joseph and Co. We may have to give it to other producenicks. I wrote a spic song called \"Maria\" which may finally bring me to jukeboxes, who knows. And one called \"Cool\" which will never see a jukebox.\n\n_Candide_ is a certainty for next fall. [Tyrone] Guthrie is signed, and a big rewrite job must take place next late spring and summer. So no T'wood. We're toying with the idea of a huge house in Englewood. Imagine, with a pool and huge grounds. It's one of those monstrosities, but we could make it cheerful and fine. We just may buy it, if we can get it cheaply.\n\nThe first Symph of Air concert comes up on the 9th, with Aaron's new _Canticle of Freedom_ , and Mahler 2nd. Wish you were here.\n\nSo, as you see, life is busier than ever, and everything goes swimmingly except that Felicia just doesn't seem to get work. She is depressed, and I don't blame her. Alexander is a heartthrob, a wonder, and so is Jamie. We long for you. I'm going to try to get to P[uerto] R[ico] for some local research on the Romeo show: any excuse, but I may get away with it. Let's hope. When can you next have a leave?\n\nAll our loves; and do write soon, not following my detestable example.\n\nLennuhtt\n\n364. Marc Blitzstein to Leonard Bernstein\n\n30 December 1955\n\nDear Lenny,\n\nYou have written, and so I answer, although I had planned to do nothing, but let the thing sink into the impassive past. But I am a bum actor; and seeing you both at Eddie Albert's25 was perhaps too soon. My belligerence to Felicia (poor girl, it wasn't her fault I was left alone with her) _was_ Reuben,26 _was_ Bourbon, to the extent I can't remember what I said, but it was probably nonsense \u2013 but it was really your outrageous treatment in subjecting me to a private meeting in your drawing-room with the worm Robbins.27 I have met him at parties, or could; one can't help that. But that it should come from you, who have been over the whole story with me, have once been furious with him (\"slap his face\" and \"my stomach turns at the thought of working with him again\" were two of your phrases),28 and with the passage of time have let that fury cool and dim and at length yield to other considerations \u2013 this is hard to take. I do apologize to Felicia. About you and me \u2013 I don't know what to do about our friendship, or my continuing fondness for you. A few more body-blows \u2013 put it down to your ego, your thoughtlessness, I don't care \u2013 might make any intimacy permanently untenable.\n\nNo, I did not send Jamie a gift.29 Right now, with a sudden unexpected lawsuit against me for \"pilfering\" _Threepenny Opera_ , I was too poor to afford any gifts, including mother, Jo, etc. But she has my love, as does Lex. I don't send cards.\n\nWe'll probably meet at Lillian's, New Year's Eve. Let's not get into this talk; I should only become \"quarrelsome\". Let's let it ride a bit.30\n\nMarc\n\n365. Robert Shaw to Leonard Bernstein\n\nHotel Georgian Terrace, Atlanta, GA\n\n14 February 1956\n\nDear Lennie,\n\nPlease accept my apologies for not contacting you before we left New York concerning your wonderful choruses in _The Lark_ and my attempt to program them through the Spring Tours.\n\nIn the last few hectic days of rehearsals, I discovered \u2013 as I have every season for the past eight years \u2013 that the program already was too long, and simply couldn't see my way to getting them installed and rehearsed adequately (which latter was the principal item).\n\nI do think they are absolutely (and variously) captivating and exciting pieces \u2013 and I continue to hope that we may be able to perform them in some suite-form in the near future.\n\nAgain \u2013 many thanks. It was a delight to see you again.\n\nAll good wishes,\n\nBob Shaw\n\n366. Leonard Bernstein to David Diamond\n\nVineyard Haven, MA\n\n25 May 1956\n\nDear Dov,\n\nThe news is sad indeed. I had no idea your mother was in so bad a state. Poor Sabina [Diamond's sister] \u2013 what that girl has been through, what with one thing and another. And what it must be doing to you \u2013 breaking up your household, your work, your peace, and all for such a rush trip home. Mightn't it be possible to bring your mother to you in Italy, and have her spend her last days in all that beauty and quiet, and with you? It would save you so much and keep you together (all of you) \u2013 but I suppose that travel is just too much for her now.\n\nI've been searching my brain and my acquaintances for a couple to take your villetta, but in vain. I do hope it all works out well. We must see you of course, when you come, even if only for a bit: so do call us when you arrive, or write us here. The phone is Vineyard Haven 1396.\n\nIt's all like a d\u00e9j\u00e0 vu. Here we are again, Lillian and us, back at the Vineyard, exactly like two summers ago, with a T'wood leave of absence, but this time with Tyrone Guthrie, who will direct, and incidentally will save the show, I think. Other change: new lyricist, Richard Wilbur, a marvelous young poet who has never written a lyric in his life, and is already doing wonders. I now have hope for _Candide_ , for the first time in ages. We go into rehearsal after Labor Day.31 Meanwhile I am concurrently writing the other show (the Romeo one, with Arthur Laurents and Robbins and a charming gifted boy named Steve Sondheim) and it begins to look like something. Both at once. It's a wild situation, even for me. And, again, concurrently, I must do something about the Boston Symph commission (not a note yet). Plus concerts this summer, in order to support my expanding brood, plus writing TV shows for the fall, plus plus plus. It doesn't look like a relaxing summer, to say the least, but there is much sky and water and air and beauty here, and I love it. I've been here for a week all alone (Felicia follows with brood and company in a few days). It is cold and sunny and bracing, and I am having fun cooking for myself and feeding the furnace and making the fireplaces and watching the late spring come hesitantly forth.\n\nWe are all well and send our love to you and Ciro [Cuomo]. Let us hear of your arrival.\n\nLove,\n\nLenny\n\n367. Leonard Bernstein to Aaron Copland\n\nVineyard Haven, MA\n\n13 June 1956\n\nDear Aa,\n\nAh, the pressures. When I think that I have thanked you in my mind for the score of _Tender Land_ a hundred times & I have never had a second to write it, I am aghast. I guess I kept thinking I had written, so often had I thought you thanks.\n\nThe more I look at the score the more beautiful & special it becomes. And though I still have grave reservations about the libretto (which led you into unavoidable dangers) I find that it has a great theatrical value after all, due largely to the marvelous tonal world created by the music. It has its own _ambiente_ , & its own authentic \"world\", as every work should: & that is the important thing, hoe-downs to the contrary. It's always the work of a master, no denying. So thank you, dear Aa, very much.\n\nWork here on the island goes in five directions, & I hope gets somewhere. Anyway the island is a dream.\n\nDid you know that David D[iamond] is arriving soon?\n\nAre you coming this way, & if so can we have lobsters-in-the-basket all together like a d\u00e9j\u00e0-vu? This whole summer is a kind of d\u00e9j\u00e0-vu, what with _Candide_ & Lillian & the same house, \u2013 nothing changed. Have fun at T'wood. Best to Jack.\n\nLove,\n\nLenny\n\nGuess who else is arriving \u2013 tomorrow, I think \u2013? Charlie Roth, that gangster maniac. Prends garde. Sauve qui peut.\n\nL\n\n368. Richard Rodgers32 to Leonard Bernstein\n\n488 Madison Avenue, New York, NY\n\n9 October 1956\n\nDear Leonard,\n\nWe were committed many weeks ago to see Judy Garland this past Sunday night and there was no way to cancel it. However, I have had reports from innumerable people about your performance and the infinitely kind things you had to say about Oscar and me.33 I am writing to Paul Feigay to see if I can get a kinescope of the broadcast. I can't tell you what it means to us to have your friendship and enthusiasm.\n\nAll fondest to Felicia and you.\n\nAs ever,\n\nDick\n\n369. Leonore Goldstein to Leonard Bernstein\n\n4615 Lindell Boulevard, St. Louis, MO\n\n17 October 1956\n\nDear Leonard,\n\nI cannot begin to tell you what a glow of pleasure and satisfaction came over me as I read a notice in last evening's _Post Dispatch_ over your appointment to the conductorship of the Philharmonic \u2013 an avalanche of thoughts came to mind. First, your and Felicia's joy at the great reward for your fine work, Miss Coates' pride that \"her boy\" had been so honored and also the happiness of your dear parents. Do you realize that you are the very first American to have reached this exalted position, and at such a young age! I am as excited as though you were of my own kith and kin. How I wish myself in Carnegie Hall when you walk on the stage this winter for your first concert as guest, and as the real boss for the next season. What music you will make with that magnificent organization.\n\nOur first concert will be given this week. Harry Farbman34 will conduct before a group of guests are to appear and then Golschmann35 remains for 10 weeks until the close of the season.\n\nI have also noted that _Candide_ will soon be produced and I am sure that will be another feather in your cap \u2013 if it can hold any more.\n\nI must stop \u2013 your minutes are scarce but be assured that I share your great honor and congratulate you warmly.\n\nAffectionately,\n\nLeonore III\n\n370. Solomon Braslavsky to Leonard Bernstein\n\nTemple Mishkan Tefila, Boston, MA\n\n18 October 1956\n\nMy dear Lenny,\n\nYesterday I read the good news about your appointment as co-conductor of the New York Philharmonic Orchestra for the next season.\n\nSo, you finally made it. Naturally, this is only the beginning. But I am mighty proud of you and everything that happened for two reasons.\n\nFirst, my prophecy became reality. I always predicted to you, your father and to all of our mutual friends that you will land a big position as conductor. It is true, you are a great composer and you rank high among American composers. But few, if any, of the great composers, past and present, are good conductors, not to speak of great conductors [...] (I remember Arnold Schoenberg conducting Beethoven's 9th Symphony. It was pitiful indeed). _But you are_ and you will remain a great composer, but above all you will make history as a great conductor.\n\nThe second reason is that you reached your goal by your _own_ merits (no string[s], no politics) and with your _own name_. It is neither BERNini, nor STEINkovsky. It is what you always were, what you are and what you always will be.\n\nCongratulations and best wishes for great success. May the Lord bless and keep you in good health and happiness.\n\nWith kindest regards to Felicia and the two little geniuses, I hope,\n\nCordially yours,\n\nSolomon\n\n371. Lukas Foss to Leonard Bernstein\n\nUniversity of California, Los Angeles, CA\n\n25 November 1956\n\nDear Lenny,\n\nMailed to you today a full score of my _Psalms_.36\n\nThis is one time I am really curious to hear what it'll sound like \u2013 never had so many instrumental\u2013texture\u2013ideas before. Will you have a chance to glance at it? I wish _Candide_ luck and all your other projects \u2013 have not yet congratulated you about the N.Y. Philharmonic directorship. My feeling there was mostly \"congratulating the Philharmonic\".\n\nIf N.Y. is going to be that exciting musically we better leave our sunny West and come back.\n\nWe will be in N.Y. actually from Jan 14th to February 5th. See you then I hope. Love to Felicia.\n\nLukas\n\n372. Stephen Sondheim37 to Leonard Bernstein\n\n11 East 80th Street, New York, NY\n\nFriday [1956]\n\nDear Lunnit (sp.?),\n\nYou have the distinct privilege of being the first person in these Continental United States to receive correspondence typed on my new and not completely paid-for IBM Electric Typewriter. How about these margins?\n\nApart from showing off this latest acquisition, I do want to reiterate how much I like the _Candide_ music. And, though I was hesitant to say so last evening, I also want to reiterate the offer I made earlier this year \u2013 if at any time you want help on the lyrics (or even the music) you need only ask.38\n\nSee you soon.\n\nLove,\n\nSteve\n\n373. Gunther Schuller39 to Leonard Bernstein\n\n14 January 1957\n\nDear Lennie,\n\nCongratulations on your stunning exploration of the why's and wherefore's of modern music on _Omnibus_ last Sunday. You are \u2013 it goes without saying \u2013 uniquely qualified for the task. I am very confident that you made many converts with your lucid and detailed explanations which, although of necessity assumed that the viewer knew very little of the composer's vocabulary and method, nevertheless at no point talked down to the audience. How few composers would be able to do this with your naturalness, conciseness and virtuosity!\n\nHaving said that much, however, I also felt moved to point out a serious flaw in your delineation, namely an unfortunate intrusion \u2013 conscious or subconscious, it is hard to say \u2013 of your own personal viewpoint of the subject in the latter half of the program.\n\nAfter a very _factual_ exploration of how atonality developed logically from chromaticism, you allowed your own _qualitative_ feelings about Wagnerian heaviness and\/or excessive emotionalism creep in to such an extent that it became quite apparent what \"camp\" you were in. This is all the more a shame since the actual statements you made (juxtaposing the new objectivity of Satie and Hindemith with post-Wagnerian romanticism and expressionism) were mostly valid statements per se \u2013 in cold print they would appear quite sound \u2013 but the slightly sarcastic coloring & inflection you gave these thoughts made it too obvious which way you wanted the listener to be swayed. In other words, you seemed to momentarily abandon at the crucial point the very \"objectivity and clarity\" you mentioned so often.\n\nGiving the picture this slant was a little unfair. The further we get into the middle of our century the more objectively we see some of the highly controversial and heated arguments of earlier decades (you indicated this yourself when you said it seems that the two camps were coming closer together, that a kind of synthesis may be in the making). It has thus also become clear that Debussyan chromaticism, which you had in the anti-Wagner camp, is a lot closer to late Wagner than anybody including Debussy was for a long time willing to admit, and that the important works of the Impressionists were to an until recently greatly underestimated extent responsible for not only the break-through to atonality but the instrumental sound and coloring of early Sch\u00f6nberg and almost all of Webern.\n\nThere is not such a big jump, after all, from Debussy's _Jeux_ or parts of _Pell\u00e9as_ to _Parsifal_ in one direction, and to _Erwartung_ or _Pierrot lunaire_ in the other direction, as was first thought to be the case. Is the mysticism of _Pell\u00e9as_ , the lushness of the _Firebird_ really so much closer to the \"objectivity & clarity\" of Satie than the expressionism of _Pierrot_?\n\nThe role of Debussy in this whole development has only lately been correctly assessed. His own explorations into \u2013 or _almost_ into \u2013 the regions of atonality, his experiments with rhythmic counterpoint and irregular rhythms, and above all his concept of the break-up of textures and lines have been only belatedly evaluated, and their important influence on Webern is still all but unappreciated.\n\nThat brings me to the subject of Webern. Since most of the young generation of European composers, certainly the important ones, are greatly under his influence (much more so than Sch\u00f6nberg's), omitting him in your portrait of modern music tilts the argument heavily to one side. Mind you, I appreciate the problems involved. It would be hard, on a program directed primarily at a nationwide audience of laymen, to spend time talking about a composer almost totally unknown \u2013 even as a name \u2013 in America. More than that, if you _had_ decided to talk about Webern and objectively place him within the present situation, your whole original point about the \"two camps\" would have had to go, and from a viewer's or layman's point of view your picture of 2 opposing factions is a much more attractive one. Yet should that have been allowed to govern your decisions? Obviously for the sake of fairness and objectivity, you should have taken the chance of beclouding the issue a little \u2013 of making the situation less black & white.\n\nThe point about Webern, of course, is that he _was_ able to cut his ties with romanticism much more thoroughly than Sch\u00f6nberg or Berg. Sch\u00f6nberg's whole unsuccessful struggle to pour atonal expressionistic ideas into classical forms was avoided from the start by Webern. The \"objectivity, clarity & simplicity\" which Sch\u00f6nberg couldn't attain (and which you saw only in the music of the other camp) is certainly Webern's most important contribution to contemporary music. What \u2013 except neo-classic Stravinsky \u2013 could be less verbose, less heavy, less square, less involved with these and any other Wagnerian attributes you care to name than Webern? The trouble is that at this short range we still blame the relative weaknesses & discrepancies of Sch\u00f6nberg's art on the 12-tone system, or on the Wagner-influence, or on Mahler etc., etc. \u2013 rather than on Sch\u00f6nberg himself. I am convinced that Sch\u00f6nberg's music, _if_ it is \"neurotic\", \"lacking in humor\", \"subjective\" or what have you (and all these points are debatable), it is so because Sch\u00f6nberg's _personality_ \u2013 and not the 12-tone system or atonality per se \u2013 was such as to cause this. He would have written (and did write) the same under another system. What different and opposite musical concepts & styles are possible within atonality or 12-tone is becoming increasingly obvious. By this serious omission, therefore, you failed to present a complete picture of the 12-tone side, and thus slanted the argument considerably in one direction.\n\nIn this connection you may be interested in the following account by a composer friend of mine (Boulez) of an evening in Paris 3 or 4 years ago. It was told as proof of the genuineness of Stravinsky's conversion to the serial technique. Stravinsky was in the company of a group of young composers (incl. Boulez, Nono, Stockhausen, etc.) and one old school-chum of the master. They were discussing problems in contemporary music, drinking quite a bit; and Stravinsky, as you know, when he does drink gets very sad & nostalgic. At one point he turned to his old friend and, almost in tears, said (not verbatim, but in effect): \"You know, of all of us (meaning himself, Sch\u00f6nberg, Bart\u00f3k, Hindemith etc.), the only one who went in the right direction was Webern. I've been composing wrong all my life\". A pathetic & touching story. It is hard to say whether Stravinsky actually meant _all_ of that, but his continuing adoption of 12-tone thinking \u00e0 la Webern (not Sch\u00f6nberg!!) would seem to indicate that he meant it to be quite an extent, and if he did, what does this do to your \"two camps\"?\n\nI write you with all this because I like to discuss subjects close to my heart with people I respect & admire. I need not emphasize that I appreciate & admire your absolutely unique combination of abilities. So it is not in the sense of hostile criticism that I write you this letter, but rather in a spirit of friendly discussion.\n\nWith best wishes,\n\nSincerely yours,\n\nGunther Schuller\n\nP.S. Have a piece about jazz in the Jan 12th issue of _Sat. Review of Literature_. I think it might interest you.\n\n374. Aldous Huxley40 to Leonard Bernstein\n\n3276 Deronda Drive, Los Angeles, CA\n\n4 April 1957\n\nDear Mr. Bernstein,\n\nAs a very busy man with a large correspondence, I can well understand your annoyance at receiving yet another letter from a perfect stranger. But, at the risk of being a bore, I am writing to ask if you would be at all interested in reading a dramatic version of my novel _Brave New World_ , which I have recently made, with a view to a musical setting. (I envisage the piece as a play with music and dancing, rather than a conventional \"musical\".) The story calls for a very resourceful composer, who can run the gamut from the primitive dances of the Indian Reservation to the music of the hypothetical future. So I naturally thought of you41 and am hopefully writing this on the off chance that you may have the time and the inclination to consider such a project.\n\nYours sincerely,\n\nAldous Huxley\n\n375. Felicia Bernstein to Leonard Bernstein\n\n[Santiago, Chile]\n\n\"Thurs. and Fri.\" [10\u201311 July 1957]\n\nDearest darling Lennuhtt,\n\nIt's all most peculiar, wonderful, strange and yet most familiar. There has been a constant stream of friends \u2013 about _twenty_ at the airport last night \u2013 and all day today \u2013 exhausting but so heart-warming and nice. I never realized I was loved that much, or that my coming would be such an event \u2013 I'm truly overwhelmed by it all! The children are a smash \u2013 Jamie has taken over the Alessandri household already \u2013 what a delight it is to see all those children together! I'm so sorry that you can't see it too \u2013 the squealing, the giggling, the mixture of languages and in the midst of it all Jamie is the queen, the glamorous beautiful imperious pixie and they are at her feet ready for her slightest whim. Alexander spent the afternoon there and the way he and Jamie flung into each other's arms was one of those rare and beautiful moments \u2013 they really adore one another.\n\nThe trip was a _nightmare_! The Miami bit in spades \u2013 it was about 100 degrees, no sign of anybody from the Chilean airline, nobody knew where it was and when we finally found it they didn't know when the plane was leaving \u2013 we spent five hours in that fucking airport. The only air-conditioned place was the restaurant where we went twice but couldn't linger because the children got so restless. Anyway we finally were called in and found that part of the seating space was taken up with cargo! The flight itself wasn't bad except between Panama and the next stop we ran into a terrific storm and poor Jamie got sick. [...] They both were marvelous though, never cried and were perfect lambs. Alexander once in a while would cry out \"vamos a la calle\" out of sheer desperation! It's just _too long_ \u2013 absolutely the end of the world \u2013 plus they made more stops than were bargained for so we arrived bedraggled, weary and worn at _midnight_ \u2013 imagine!!\n\nI think of you constantly and love you more than ever \u2013 Jamie talks about you all the time and said yesterday that she must write because poor Daddy was all alone. She looked out the window in the plane and said she saw a map below. [...]\n\nChita's house is small but adorable with divine food which I've been gobbling up. Yesterday I had a full two course lunch then tea with all the trimmings and then went to Madeleine's for dinner at ten and gorged \u2013 it must be something in the air \u2013 it's nippy, but clear and sunny.\n\n_Following night_. Thank you for your darling cable which I got this morning. As you can see there's no chance of writing in the daytime \u2013 I took the children to Mamita this morning and they went crazy with the chickens, ducks, rabbits, turkeys, a dog and oranges which they plucked from the trees and ate on the spot. She is the darling of all time \u2013 so full of love and goodness, thrilled by the children, supplies them with fresh eggs every day and things that she grows \u2013 and she's getting old and sick and it breaks my heart.\n\nAbout the \"girls\" I'll tell you in my next \u2013 they're marvelous. Madeleine especially has taken a great turn for the better \u2013 details later cause I'll never finish.\n\nMy darling do write \u2013 don't work too hard \u2013 tell me how everything is going with the show \u2013 if Grace is taking care of you.\n\nAs for me I'm bewildered and miss you so that it hurts. I think it's the incredibly depressing distance between us.\n\nI do love you.\n\nFelicia\n\nMy love to Helen and the Kats.\n\n376. Leonard Bernstein to Felicia Bernstein\n\n19 July 1957\n\nDarling,\n\nThe main news is that I love you and miss you, more than I could ever have known. It's all very well to talk about the salutary effects of periodic separations, and all that; but it's lonely in that big apartment upstairs, and everything looks different without my _people_ there. Booze-hour isn't the same with anyone else, sleeping is particularly strange in one of two beds with the other unoccupied. I've managed not to eat alone once so far: that's no problem. But it's different, that's all. In fact, as I've discovered, it's the main difference there can be in living.\n\nBut there's little if any time to think about any of the above. The work grinds on, relentlessly, and sleep is a rare blessing. Jerry continues to be \u2013 well, Jerry: moody, demanding, hurting. But vastly talented. We start on the book Monday, trepidation in hand; and the score is still not completed. At the moment the Problem is the usual one of the 2nd act ballet, which is finished, and will probably not work at all and be yanked and we'll have to manufacture a new one. It's going to be murder from here on in. My nights are all spent on work, so no fun at all. The only relief is dinner. Once at Ofra's (all goes swimmingly, and Shirley is still in the dark) \u2013 once with Lukas [Foss] (who missed you by a day, sends you great love, and was intuitive enough to ask Burtie \"Are you in love? You seem dreamy and different.\") and once with Steve [Sondheim] and once with Debbie and once with George Sch\u00fctz and once with [Kenneth] Shermerhorn, etc. etc. And once at the Ricordis. Last weekend was all work. No Stony Point. Maybe this weekend.\n\nLast night was Martha [Gellhorn] night. She finally made it: and we talked in our customary natter for hours. How she loves you and knows you! And how she knows and loves our love (yours and mine). She finds my life ridiculous, of course, but finds me in better shape than ever, all of which she attributes to you, and rightly of course. I was telling her what a marvelous girl you are, how beautiful and bright and witty and wise, and she said: \"But how did it take you so long to find out what is perfectly obvious?\" Well, I always knew it; I didn't have to find it out; I just suddenly became _aware_ of it, found myself able to experience it and share it, and just be plain grateful for it. I'm endlessly lucky in you, and lucky that you've been strong enough to stick out the bad times. You wonderful girl, you, with whom I am recently in love all over again.\n\nI must run: Big Daddy calls. Big hugs & kisses for my two angels \u2013 and love to all the family. Write more & lots! You have time, I haven't! Have a glorious time \u2013 I kiss you.\n\nL\n\n19 July\n\nToday is Helen's birthday. Do cable her. Grace quit \u2013 I have a fine new maid. Greetings to Rosalia & Julia.\n\n377. Leonard Bernstein to Felicia Bernstein\n\n26 July 1957\n\nMy darling,\n\nI loved your letter (I guess we've established the usual pattern of crossing letters so that nothing ever gets properly answered) \u2013 but it's wonderful just to hear from you \u2013 except to hear that you're sick \u2013 what a bore. I was afraid that abrupt change in climate might do something like that. And of course it was also only natural that the usual reaction to Chita would set in \u2013 too much of a good thing. But you're used to that: don't let it get you down.\n\nI can imagine what a trial it is to be in a place where everything is at a premium \u2013 like the old days in Israel \u2013 & where medicine is backwuhts and the trunks don't arrive \u2013 very Israeli, all that. Don't you dare stop smoking \u2013 you're absolutely right!\n\nA propos Israel: I sent a long nagging cable about _Jeanne_ [ _d'Arc au b\u00fbcher_ ] & just had a letter this morning saying that it's absolutely impossible to get a chorus to prepare it. I'm furious but helpless. There just isn't a chorus that operates in the summer, & they claim they'd need 8 months to learn it, etc. etc. Shit.\n\nSo that's out, my darling; but don't let it discourage you from coming to Is. anyway.\n\nStill no word from Buenos Aires!\n\nMilton Goldman has been calling about a part for you in a play version of _Diabolique_. Interested? And Tony Mines called for you today. Gave him your address.\n\nThe show \u2013 ah, yes. I am depressed with it. All the aspects of the score I like best \u2013 the \"big\", poetic parts \u2013 get criticized as \"operatic\" \u2013 & there's a concerted move to chuck them. What's the use? The 24-hour schedule goes on \u2013 I am tired & nervous & apey. You wouldn't like me at all these days. _This is the last show I do_. The Philharmonic approved the contract yesterday & all is set. I'm going to be a conductor, after all!\n\nNo news on the Burtie\u2013Ofra\u2013Shirley front,\n\nWeather: good \u2013 coolish, fair. I don't get to see it much; & my air-conditioned studio saves my life.\n\nDarling I love you & miss you \u2013\n\nL\n\nDined with Marc last night \u2013 sends fondest love.\n\n378. Leonard Bernstein to Felicia Bernstein\n\nSun a.m. [28 July 1957]\n\nDarling,\n\nGuess where I am \u2013 on a plane. Guess my destination \u2013 Miami. And as always with any flight involving Miami (as you know too well) there's shit to spare. I was to have left last night: arrived breathless at Idlewild to be told that the flight had been cancelled \u2013 only nobody had taken the trouble to inform me of same. No other flights available, except one that would have landed me in Miami at 5 a.m. or so. So back to the Osborne, heckle & peckle, & dinner with Burt, Ofrah, & back to Idlewild this monank\u00fcd\u00fc, & here I am. Now, to be consistent, there should be nobody to meet me at the airport, the convention is over, forced landing at Palm Beach, or something worthy of the tradition.\n\nOh, I didn't tell you why I'm going. Columbia Records is having its annual convention, imagine, & it will be fun & games at the Americana Hotel (this year's hotel). I dread it. Home tomorrow, in time (I hope, barring airport Miamisms) for a _run-thru_ of Act One! Imagine \u2013 already! Where does the time all go to? In a minute it will be August & off to Washington \u2013 & people will be looking at _West Side Story_ in public, & hearing my poor little mashed-up score. All the things I love most in it are slowly being dropped \u2013 too operatic, too this & that. They're all so scared & commercial success means so much to them. To me too, I suppose \u2013 but I still insist it can be achieved with pride. I shall keep fighting.\n\nI miss you all terribly \u2013 especially you who have come to mean something miraculous to me. You reside at the very core of my life, my darling. I hope your kepepelt [cold] is better, & that the fun goes on. Ofrah bets you won't stick out the two months. She's probably right. But if you come home, what would you find? I'd be no good to you \u2013 & you'd hang around the show & get sick of it, & my whining, etc. etc. And then, if I have to go to B[uenos] A[ires] after all \u2013 what's the fun without you there? Anyway, don't make any rash decisions yet.\n\nThere's Palm Beach down there, looking hot, damp and sunless. We'll be landing soon \u2013 & I'll probably drop you a line on Americana stationery, which I am sure is pure gold-leaf.\n\nBless you my love.\n\nL\n\nAbrazos to all.\n\n_Dere Jamie and Alejito:_\n\n_I love you so mucho!_\n\n_Dady_\n\n379. Leonard Bernstein to Felicia Bernstein\n\nSat. night, 3 August 1957\n\nDarling,\n\nTwo big events:\n\n1) I've gotten out of Buenos Aires! The agent, one Uhlfelder, was here in town, & came to see me, & it's settled, & wow, what a relief. So you can return in peace, & I can see my two angels before Israel \u2013 & we can leave from here, after seeing the opening on the 26th, _like a Mensch_. Look: we go to the opening, & then we wait up for the papers, & before you know it it's time for the plane, and Scheu, we're off.\n\n2) I signed the Philharmonic contract. Big moment. Bruno [Zirato] arrived at 10:30 a.m., contract in one hand & a big chilled bottle of Brut in the other, & much emotion (he couldn't write his name for the shaking of his hands) & I'm in \u2013 like for life. I made a coup: the lawyers had fallen out so far that the contract was up to 20-odd pages, & growing: & the disputes were growing correspondingly. So I scotched it by tearing up the whole thing, & writing a one-page letter that said I was engaged for such a period for so much money, sincerely yours. They loved it. Simple, & trusting. We'll settle the details as they come along.\n\nOther events \u2013 nothing but the show. We ran through today for the first time, & the problems are many, varied, overwhelming, but we've got a show there, & just possibly a great one. Jerry is behaving (in his own way) & Arthur is doing well. But the work is endless: I never sleep. Everything gets rewritten every day: & that's my life at the moment. And imagine, we open two weeks from Monday.\n\nSome beautiful shots of you & the kids arrived (taken by the hi-fi man at the vineyard \u2013 remember) \u2013 & they melted me. I miss you so!\n\nI loved your last letter. Did you get mine from Miami?\n\nI hope the trunks are there, & all is in order. My love, & have fun, dear lovely one.\n\nL\n\n380. Igor Markevitch42 to Leonard Bernstein\n\nL'Aiglerie, Villars-sur-Ollon, Switzerland\n\n3 August 1957\n\nMy dearest Lennie,\n\nFirst of all I want to tell you how delighted I am at the project of _Icare_.43 I have carefully looked at the dates, and if nothing unexpected happens until then, I will be able to come from Montreal two days before the concert44 in order to follow the last rehearsals, as you had asked me to do.\n\nThis revival is going to be a great event for us. To help you prepare it, I am sending you by the same mail a record which was made during one of my executions. Unfortunately there are whole passages where certain instruments are completely lost, but as one says elegantly: \"It is better than a spit in the eye.\"\n\nHere is a letter that Bart\u00f3k wrote concerning _Icare_ which I am sending you in French:\n\nCher M. Markevitch, Permettez \u00e0 un coll\u00e8gue qui n'a pas l'honneur de vous \u00eatre connu, de vous remercier de votre merveilleux _Icare_. J'ai n\u00e9cessit\u00e9 du temps pour \u00e9tudier et comprendre toute la beaut\u00e9 de votre partition, et je pense qu'il faudra beaucoup d'ann\u00e9es pour qu'on l'apprecie. Je veux vous dire ma conviction, qu'un jour on rendra justice avec s\u00e9rieux \u00e0 tout ce que vous apportez. Vous \u00eates la personnalit\u00e9 la plus frappante de la musique contemporaine, et je me r\u00e9jouis, Monsieur, de profiter de votre influence. Avec ma respectueuse admiration, B\u00e9la Bart\u00f3k.\n\nThe letter is from the autumn 1933, and had neither date nor address, reason for which I didn't answer it. I add, as I already told you, that I didn't know yet the name of Bart\u00f3k. I would be very pleased if you would send me a word to let me know when you receive the record.\n\nI wish you every possible luck for your new show, my dear Lennie, and I also remind you to keep the promise you made me, to kiss the whole of Israel for me. In the meantime it is Topazia and me who do it with you sending you our most affectionate thoughts.\n\nYours,\n\nIgor\n\n381. Leonard Bernstein to Felicia Bernstein\n\n\"8 Aug already!\" [1957]\n\nDarling,\n\nI had a real scare with the news of Asian flu \u2013 & when your letter came about how you were all down with it I got scareder. But your cable made me feel better \u2013 please be careful! I can't bear the thought of you all sick.\n\nI missed you terribly yesterday. We wrote a new song for Tony that's a killer, & it just wasn't the same not playing it first for you. It's really going to save his character \u2013 a driving 2\/4 in the great tradition (but of course fucked up by me with 3\/4s and what not) \u2013 but it gives Tony balls \u2013 so that he doesn't emerge as just a euphoric dreamer.\n\nThese days have flown so \u2013 I don't sleep much; I work every \u2013 literally every \u2013 second (since I'm doing four jobs on this show \u2013 composing, lyric writing, orchestrating, & rehearsing the cast). It's murder, but I'm excited. It may be something extraordinary. We're having our first run thru for _people_ on Friday. Please may they dig it! And f\u00edjate, I leave for Washington on Tues. the 13th \u2013 so soon, so soon. It's all rushed by like a cyclone.\n\nOf course we're way behind on orchestration etc. \u2013 but that's the usual hassle.\n\nHow are you? You don't say. Are you fatter from eating? (Me: I'm a bit skinnier.) Do you smoke? (I do, lots.) Have you skied? (I haven't). Do you love me?\n\nBless you & _be well_.\n\nLove,\n\nL\n\nI adored Jamie's letter, especially the lentils.\n\n382. Felicia Bernstein to Leonard Bernstein\n\n[Santiago, Chile]\n\n\"Tues. & Wed.\" [August 1957]\n\nMy own blessed wonderful darling,\n\nI refuse to write about \"latas\"45 so I'll skip the whole miserable bit and only tell you that we're on the mend. Jamie has been up for two days and is fine again and Rosalia got up today.\n\n_Too incredible_! Just got your cable this minute. I suppose there must have been something in the papers about this monstrous epidemic \u2013 thank God it wasn't anything worse, just la grippe but in spades. I now read in the paper that it has spread all over the world and that even you may get it \u2013 if there is a vaccine handy take it!\n\nI bless the day you couldn't come \u2013 I went to Klecky's [i.e. Paul Kletzki's] concert last week and oh Lennuhtt! What a shampepuhls... Granted a lot of the musicians were sick, but still! Klecky made \"Un grand scandal\" \u2013 stopped the orchestra in the middle of Beethoven's Fourth and screamed, ranted and then apologized to the audience and started again. We later had dinner at the house of a Canta Maya46 type lady with the upper echelons of Chilean music, Vincente Salas Via, Juan Orrego etc. and he let them have it! I wanted to crawl under the groaning b\u00f6wewehd \u2013 it was ghastly and embarrassing, said he didn't want to finish out his contract etc, etc. But he's right \u2013 like everything else in this insane place it is a hit or miss affair (mostly miss), no musicianship or love of music, no discipline (since they can't be fired) and a devil take it attitude which is hardly conducive to good music. So, with a heavy heart I must say don't did it!\n\nThat Catholic priest you made friends with in Washington called Father Woolen is here \u2013 we met this afternoon and a charming fellow he is. He adores you and talks about you with enormous respect and enthusiasm. He's been all over South America giving concerts of all sorts \u2013 organ recitals, chamber music, piano concertos with different orchestras. Madeleine is in charge of him here (part of her job) and is giving him a small dinner party tomorrow.\n\nLast night I had dinner at an old friend's very old rich family \u2013 ancient Milan type mausoleum house \u2013 much marble, Aubussons, great food, old retainers and in the midst of all this, a passion for jazz! He's an authority, has a fabulous collection of old and recent records among them your _Omnibus_ which he says is the greatest thing to come into his life since the birth of his son \u2013 we listened to it in religious silence, eight of us, and it is really so _wonderful_. You can't imagine what hearing your voice did to me \u2013 so much of your personality and warmth comes through it and then the clarity of your mind and your articulateness. I was so proud of mine Lennuhtt!\n\nAlexander woke up in the middle of the night and carried on a long telephone conversation with you \u2013 he's managed to avoid the flu but has a cold and a cough and can't go out. Jamie is up and about and looks beautiful but is also coughing as is everybody else. Honestly what shit luck. We haven't been able to do a thing or go anywhere \u2013 oh well!\n\nI adore your letters and literally live from one to the next \u2013 please don't stop or I perish.\n\nM\u00ef la\u00fc d\u00fc\n\nTia\n\nFely\n\n383. Leonard Bernstein to Felicia Bernstein\n\n[New York, NY]\n\n11 August 1957\n\nDearest Fely,\n\nIt's 4:00 a.m. & I've just finished another of those incredible nights (which begin in the early morning) \u2013 work & work & work. I'm so sleepless by now I'm punchy. And I'm beginning to miss you _seriously_. And I feel a little bit uncomfortable about you all in that place with no milk and the economy of post-war Germany, & medieval plagues rampant. And then \u2013 you PO\u00d6AH \u2013 the Valparaiso episode, which I just heard about, an accident, yet, and no trunks, & \u2013 God, what are you clothing yourselves with? What are you eating? What a ghastly way to have a restful summer! Why not just chuck it all, as soon as you're all well again, and come home? I'm sure the trunks will be here by Xmas. Why feel you _must_ stick it out just by way of meeting a challenge, a whatever? You've had a good sound month of what sounds like sheer penance. Basta!\n\nBurtie and Ofra are weeking it up in Stony Point, & Shirley weekends with them and _still knows nothing_! Incredible. We had our run-through for People yesterday, & it was a smash. But I'm worried: there is so much that doesn't work \u2013 for me \u2013 & I'm sure for [Walter] Kerr & others. But there's a great show there. Darling, think seriously about coming home. I love you & miss you.\n\nL\n\n384. Burton Bernstein to Felicia Bernstein\n\n[early August 1957]\n\nDearest Feloo,\n\nBy far, your last letter was the grimmest single document I've perused since the publication of the famous Abe Miller letters.47 Even the Abe Miller letters failed in many respects to equal the bathos and pain \u2013 nay, the utter desperation \u2013 of yours. Needless to say, you have my pity, sympathy and concern; in fact you have everyone's. Now, obviously, all that's left to do is come home, and quickly \u2013 even if your trunks haven't arrived in Valparaiso yet. Sergeant, you have done your duty \u2013 now get the hell out of there... and come back to New York in time for the annual Influenza Festival which is about to begin here any day now. (Chuck [Solomon] claims to have a new vaccine ready in three weeks time.) Really, do what Lennuhtt says and come home. The Chile country is \"basheert\" by you \u2013 I've never heard such a tale of woe. I hope everyone is better by now. Has Alejito gotten it yet?\n\nThe news from here is, to say the least, markedly better than your dispatches. Firstnik, _West Side Story_ is going to be a large hit and lives up to our highest expectations. Steve even went as far as to say, \"It's so good even Felicia will like it.\" The run-through I saw was before an ideal audience of theatre folk, so one shouldn't really gauge it by audience reaction \u2013 but still, it was quite exciting. The strange thing (something I've never experienced before) is that B. Lennuhtt comes off as second best: the show actually is a monster ballet (a jot repetitious in spots) where no one is actually directed but choreographed instead. It's too much of a good thing, if you know what I mean (you know what I mean?). There's so much balletics going on on stage that the music is shunted into second place by the sheer physical force of arms, legs and torsos. But on the whole (and you know how I hate to say this about a run-through from past experience) the show is in frighteningly good condition and looks like a sure thing. I hope that by the time I see it with sets and costumes and everything in Washington next week Jerry will have been convinced that there's too much ballet for the show's own good \u2013 or am I being na\u00efve? Anyway, there's very little to fix outside of Lennuhtt's nose... So come home already.\n\n[...]\n\nThe _New Yorker_ 's still fun and I got a raise, or did I write you that already? I've really been accepted there as a permanent fixture; the senior editors talk to me in terms of five or ten years from now. Good luck.\n\nI've been doing a lot of flying and I'm well on my way towards getting my private license back. Just a question of time.\n\nSo come home. Get well. Abrazos for all.\n\nMu lau du,\n\nBB\n\n385. Felicia Bernstein to Leonard Bernstein\n\n[Santiago, Chile]\n\n\"Monday\" [12 August 1957]\n\nMy darling,\n\nTomorrow you're in Washington \u2013 with my meager and distorted sense of our capital I see you all rehearsing in a gleaming white Grecian temple with marble steps to spare just across the street from the White House \u2013 what I'd give to be there with you, misery and all! After all somebody has to cart in the sandwiches or would I be homing in on weepy Sylvia's domain? By the time this reaches you the sets will be up \u2013 with the accompanying gestures. Oliver [Smith] and Jerry [Robbins] won't be speaking. Irene [Sharaff] will arrive (avec la petite Japonaise) and you will keep calling her Gloria. The dancers will find they can't move in their costumes etc. etc. \u2013 from way down here it sounds like sheer heaven. But you must be so tired! Who's helping with the orchestration \u2013 and why do you always have to do everything yourself? Are you living on dexamils48 \u2013 what else?!\n\nTony's new song sounds delicious \u2013 but can he sing it? I thought you were going to change the name \u2013 have you made any cast changes? Someone sent me the pictures of the Sunday Times \u2013 it all looks so exciting! Oh Lennuhtt \u2013 maybe maybe \u2013 I don't dare hope. What theatre, what hotel \u2013 nobody tells me these things!\n\nJamie calls me Tia Fely \u2013 what can you do? She is going to an English kindergarten [...] in the mornings. She has an incredible capacity for enjoying herself \u2013 at least one member of this safari is having a whale of a good time! As for \"muh\" I am bored senseless \u2013 my friends are all sick in bed, it's been raining and cold, no activities are possible and I ain't got nothing to read! Faut[e] de mieux I'm reading _Mario the Magician_ and _Death in Venice_. I miss the _New Yorker_ \u2013 I read _Time_ mag. avidly from cover to cover, even the financial section!\n\nYour Father Woolen turned out to be a _pest_ \u2013 plus there's something about a priest who is so unpriestlike which is rather off-putting.\n\nHad a rather Mexican-type afternoon \u2013 a dear old maid of ours who lives with Mamita was like everyone else sick with the flu but it developed into pneumonia and there wasn't a hospital bed to be had or an ambulance free in all the city! We finally got a hospital to put in an extra cot in a large room and waited till 10 p.m. for the ambulance \u2013 she could have died and there was nothing one could do.\n\nI was asked to give an informal talk on theatre and TV at the Catholic University but it had to be postponed because there were no pupils. I also went to a rehearsal of the Teatro Experimental and the same thing \u2013 no director, no leading lady, no nothing! Most frustrating as you can imagine \u2013 but, c'est la vie! God, I hope it is nice and warm in Israel (they've had the \"Japan 305\" there already). I will lie on the beach with [...] [Zvi] Haftel and live it up! I hear Sam isn't going \u2013 do I hear sighs of relief?\n\nI've just taken my _first_ sleepenk\u00fcd\u00fc and I'm beginning to feel a deliciousness \u2013 I know that from now on there'll hardly be time for you to brush a tooth but please, please let me know how it goes.\n\nEveryone is thinking of you.\n\nP.S. I love you I love you I love you pshh, pshh, pshh.\n\n386. Leonard Bernstein to Felicia Bernstein\n\nJefferson Hotel, Washington, D.C.\n\n15 August 1957\n\nDear Beauty,\n\nWell, look-a me. Back to the nation's capital, & right on the verge. This is Thurs. We open Mon. Everyone's coming, my dear, even [Richard] Nixon and 95 admirals.49 Senators abounding, & big Washington-hostessy type party afterwards in Lennuhtt's honor. See what you miss by going away. Then next Sunday, which is my birthday, there is the Jewish version \u2013 a big party for me, but admission is one Israel bond. All helps the show. We have a 75 thou. advance, & the town is buzzing. Not bad. I have high hopes. I also have a new pen, as you can see, which I adore, because it writes every time, & without a ball-point.\n\nIf I sound punchy it's because I am.\n\nUp all night trying to put together an overture of sorts, to carry us through until I do a real good prelude. Orchestra reading all day yesterday \u2013 a thrill. We have surprisingly good men, who can really play this terribly difficult stuff (except one or two of them) \u2013 the orchestrations have turned out brilliant. I tell you, this show may yet be worth all the agony. As you can see, I'm excited as hell \u2013 oh so different from _Candide_.\n\nNow \u2013 how about my plan that you all come home?\n\nLou Silverstein is getting us (cast, authors, orchestra) all Asian-flu shots \u2013 black market, of course \u2013 so we won't conk out. I don't know what we do in this show for understudies (we have them \u2013 but...) or for substitutes in the orchestra. If the guitarist gets sick, it takes a week for another to learn the part. Same for all the winds. It's a tough show.\n\nWhat else is new? The show the show the show...\n\nI want to see my three Kats!!\n\nLove, Daddy\n\n387. Carol Lawrence50 to Leonard Bernstein\n\nWashington, D.C.\n\n19 August 1957\n\n[Telegram]\n\nI'm so lucky to be me, to have the privilege of singing your music and to know you. I shall always be a most ardent fan. I thank you for letting me be Maria.\n\nLove,\n\nCarol\n\n388. Felicia Bernstein to Leonard Bernstein\n\n[Santiago, Chile]\n\n\"Tues.\" [20 August 1957]\n\nSweetness!\n\nOh joy oh bliss oh rapture. Your cable with the frabgeous news has just arrived \u2013 thank you! I'd been desperate for some word all day long. Congratulations to one and all \u2013 how happy, how marvelously happy you must be.51 As for me I'm busting with pride and frustration \u2013 of all the moments to miss sharing! Last night I spent the evening looking at my watch and imagining the proceedings \u2013 Oh God! How exciting it must have been! Were you very nervous \u2013 did you sit through it or pace? Did anyone go to Washington to hold your hand? Is there still a great deal to be done apart from an Overture? Did you get my wire? I was miserable cause I didn't have your address or name of the theatre but your letter arrived in the nick.\n\nWhy haven't we left? I sure had a good excuse. Jane Broder cabled about a _Studio One_ ,52 rehearsals beginning this week and CBS called me on the telephone no less! But though sorely tempted I just couldn't do it to Chita \u2013 she looked so crushed and depressed. I simply had to wait for the sick to get well, the sun to come out and give everyone the chance to show us a good time \u2013 the epidemic was such a blow to them all and they felt so guilty and miserable. You will understand I know. Anyway things _are_ better. I spent the day in the country which I'd been pining to do since I arrived. It was more beautiful even than I remembered \u2013 a glorious jewel of a day, trees blossoming in profusion, lambs grazing, a soft smell of eucalyptus in the air and for the first time I wished desperately that you could see it, because _that_ is Chile \u2013 the earth, the smells, the snow-covered mountains, the country bread baked in outdoor ovens \u2013 all this I will yearn for always just as I don't care if I ever see Santiago again.\n\nEven the orchestra managed to sound like something \u2013 the last concert was quite good. _Firebird_ , no less, and a [Goffredo] Petrassi piece which I rather liked. Klecki [Kletzki] is good but that's about all I can say for him \u2013 pas tr\u00e8s grande chose.\n\nThen there are the parties being given and planned and we hope to get to do some skiing this weekend with the children. Their noses have stopped running and they're blooming once more. Alexander is so beautiful and naughty and funny I could eat him! Jamie has become so Chilean it's disconcerting \u2013 her Spanish is incredible, full of imagineses de repentes and Por Dioses! They are both really frightfully clever \u2013 the Alessandris think Jamie is a genius \u2013 God forbid!\n\nDarling, I hope this reaches you for your \"Jewish\" birthday and that it is a delicious one. I will get you a great present (I wonder where?). In the nonce my boundless love will have to do \u2013 and many many kisses of all sorts. I do so long for you.\n\nTia,\n\nF.\n\n389. Leonard Bernstein to Felicia Bernstein\n\nJefferson Hotel, Washington, D.C.\n\n23 August 1957\n\nDarling,\n\nIt's all too exciting. I never dreamed it could be like this \u2013 reviews such as one would write for oneself \u2013 the whole town up and doing about the show \u2013 the delicious long lines at the box office \u2013 morale high \u2013 dignitaries every night \u2013 the Senate practically in toto \u2013 parties \u2013 hot newspapers \u2013 all the atmosphere of a mid-season opening \u2013 gala-emeralds, furs \u2013 the works. Only thing missing \u2013 _you_. How I longed to have you there & share the excitement! Of course, as they say, it's only Washington, not New York \u2013 don't count chickens. But it sure looks like a smash, & all our experiments seem to have worked. The book works, the tragedy works, the ballets shine, the music pulses & soars, & there is at least one history-making set. It's all too good to be true.\n\nI've just got lunched at the White House \u2013 no m\u00e1s. Invited by Sherman Adams53 & the whole gang. Again \u2013 you should have been there! What a beautiful place \u2013 such credenzas, such breakfronts. I really felt \"in\". Adams & Rabb & Gen. Snyder \u2013 all were talking of nothing but _West Side Story_ \u2013 I think the whole government is based on it. Jim Hagerty (Ike's press secretary) turns out to be a fan of mine! It's all so crazy and unexpected. Even Adams turns out to be an amateur musician.\n\nNow listen! When are you coming home? I have a constant feeling you're about to turn up any minute \u2013 but look, the time is drawing near. Only 10 days to Labor Day and the summer's over. What I hope is that you'll be back for the Philly opening (Monday the 9th) which is our anniversary, for Chrissake \u2013 or even for Jamie's birthday on Sunday. Please try to manage it, huh? Why stick around that plag[u]ey place, bored as you are, after Labor Day? Let me know right away when you plan to return, and darling, hurry home. I can't stand not seeing the children, and I need my girl!\n\nI love you,\n\nL\n\nI'm 39 in 2 days!!\n\n390. Cole Porter54 to Leonard Bernstein\n\nLos Angeles, CA\n\n23 August 1957\n\n[Telegram]\n\nDear Lenny,\n\nI hear glowing reports about your new show.55 All my congratulations to you and Jerry.\n\nBest,\n\nCole Porter\n\n391. Felicia Bernstein to Leonard Bernstein\n\n[Santiago, Chile]\n\n\"Montag\" [Sunday, 25 August and Monday, 26 August 1957]\n\nDearest Lennuhtt,\n\nAnd today is the Baroness' birthday and I'm not yet there yet! Like the idiot, fool, ridiculous ass that I am I made a special trip at 2 a.m. to send you a cable, address it to the theatre so you'd be sure to get it \u2013 forgetting completely that it is Sunday and probably the only day off you've all had for months!\n\nAnyway, it is sure to be a happy birthday \u2013 such reviews! My God! I carry them around with me to read over again in my free moments. How has Arthur taken it all \u2013 one of the reviews gave the book to Jerry \u2013 oh my! I hear from B[urton] B[ernstein] that there's little to fix apart from your nose.\n\n_Next day_ \u2013 why does one ever telephone \u2013 such a frustrating five minutes! What do you mean Flora Robson?\n\nAnyway I'm waiting to hear from Pan American \u2013 your \"new found\" success gave me courage and we should be off _in style_ a week from tomorrow. Rosalia will follow later by Cinta. My plan is to go straight to Washington (the plane stops there anyway). The children can stay overnight so you can see them and then take a train to New York with Julia. I will have Helen send a suitcase on to Wash[ington] with summer clothes so that I won't need to go to New York at all. How's that for peanuts?\n\nOf course, this last week I've been having a _very good_ time \u2013 dinner parties galore, lunches in terraces, trips to the country and the mountains \u2013 the moment one leaves Santiago it is breathtakingly beautiful! And spring really is here, the trees are abloom, the sun is warm and it is nice \u2013 actually _very nice_. I am glad to have stayed on, it would have been too sad to leave with such a memory of misery!\n\nSince there is _nothing_ to buy, there will be an enormous lack of regalitos.56 I've been wracking my brain and everybody else's thinking of an adequate and fun birthday present for you \u2013 as yet notens\u00fcd\u00fc! I almost got you a race-horse cause it was named after me \u2013 but then, we lost Alfred Vanderbilt who could have kept it for you. Then, of course, there are copper mines, cattle farms, vineyards etc. and that's no[t] any good.\n\nDarling, next time you hear from me t'will be in my own Flora Robson tones. I'll wire arrival time etc. Please remember _not_ to mention my hair which no doubt will look \"desperate\" after the trip! At the moment Zorina57 would die of envy, it is so silky and _straight_. I do look awfully well and my _beauty_ is toasted to in the chicest circles, my dear!\n\nListen, please love me still when I get back. I may not be such a raving beauty there but I love _you_.\n\nF\n\n392. Felicia Bernstein to Helen Coates\n\n[Santiago, Chile]\n\n\"Tuesday\" [27 August 1957]\n\nDearest Helen,\n\nBless you for sending me the Washington reviews \u2013 I'm still groggy from the impact! Overwhelmingly wonderful. I can only imagine how blissfully happy Lenny must be, bless his heart!\n\nAnyway, the time has come to pack up and leave. I hope to get a Pan American flight the 3rd of Sept which leaves me directly in Washington. I will take the children and Julie with me and Rosalia will take Chita back alone from here. Lenny can then see the children and they can take a train later to New York. Now, if you can send my summer clothes to Washington in a suitcase I can simply stay on with Lenny for a while and dispense with New York entirely. [...]\n\nAll this would need to be sent at once though, cause as you know I have nothing but winter clothes with me! I only hope it won't be difficult and a bore for you. I'm sure Ofra would be happy to help \u2013 she knows my clothes rather well.\n\nWe've been having a very nice time this last week \u2013 the weather is heavenly! We took the children to the mountains and they had a marvelous time! They are looking beautiful!\n\nEvery one sends love to you and Marie Grace. See you soon.\n\nLove & Kisses,\n\nFelicia\n\n393. Goddard Lieberson58 to Leonard Bernstein\n\nheaded paper of CBS Television, New York, NY\n\n30 August 1957\n\nLen,\n\nI know you're up to ears in unsolicited praise at this point, but I've found myself so full of this project of yours all this week that I need to let you know what I think of it \u2013 if only as a sort of palliative which will allow me to get back into the heart of hackwork, which is my domain right now.\n\nWe've glutted the language with so many all-words and non-words that by now a term like \"fine\" has little meaning. But it has meaning for me, and it's that word which I now find myself stuck with when I think about the way your talent has combined with your material this time.\n\nI mistrust my initial impression a little \u2013 I really can't believe that any one thing can be the best thing of its sort in my memory \u2013 all I know is the way I felt on Saturday afternoon was very like the experience of _Anne Frank_ a couple of seasons ago59 \u2013 I kept telling myself afterwards that there must have been something wrong with me for those three hours \u2013 something wrong that enabled the play to work on me the way it did.\n\nI must have been something like a perfect piece of audience on Saturday because I've never known anything in musical theater to do me in the way _West Side Story_ did. I'm usually a pain in the neck about those things \u2013 thinking all the while, officiously, how it ought to be better done and what I'd see thrown out. But Saturday it just threw me around and that's about the end of it.\n\nPeople muttered at times about [Gian Carlo] Menotti \u2013 in a good way \u2013 saying that it's the most moving thing since _The Consul_ \u2013 all that lobby crap \u2013 for me Menotti's always been somewhere over the fence because I don't like his use of language and I don't think the whole thing is comfortable \u2013 I never have learned to believe it \u2013 I don't know _The Consul_ , but the rest of it falls the way I've described \u2013 which is a failing in me, I guess, but nonetheless a fact. If it doesn't just happen, then it isn't right.\n\nI haven't ever seen a production which held together the way yours does \u2013 in which the units of work produced by different people fitted so well into a whole \u2013 I don't know how it really was. I know you must have had your problems \u2013 but none of them show up as scars.\n\nFor me, it has terrific power \u2013 terrific unity \u2013 excellent individual work, though I think the women are by and large better than the men \u2013 I can't say enough for Carol Lawrence this time \u2013 I've always liked her \u2013 and [Chita] Rivera is beautifully inside the work \u2013 which I didn't think she'd really be \u2013 she doesn't splash out of the production as I figured she might. This probably isn't going to make her, but it's great for the show. I think the men are at least good and will improve for sure \u2013 [Larry] Kert particularly. He seemed a little nervous still \u2013 but that's like a headcold and can be got rid of.\n\nThe amazing thing to me is that everything seems to work so well \u2013 with the possible exception of the 2d Act ballet, you never get the half-vision of what was attempted against the way it comes off \u2013 I know if you tried to tell somebody why the Balcony Scene works, it'd sound wrong \u2013 yet it's the most moving single musical sequence I've seen since the park bench scene in _Carousel_ \u2013 which I think is a great, great moment.\n\nThe Anita\u2013Maria duet is almost on a level with the Balcony Scene for me \u2013 just amazing.\n\nI hope Kert gets his first two songs into shape \u2013 because they're terrific \u2013 \"Maria\" particularly \u2013 and it isn't quite happening \u2013 or anyhow, didn't on Saturday.\n\nThere's such a fabric there \u2013 and such a flow \u2013 I wouldn't have believed that it'd be possible on this earth at the end of the first week out of town. I still find it a little tough to be sure about \u2013 and yet I know how it was.\n\nFor God's sake, change the last five minutes \u2013 you don't need to say all that crap, because everything you've been saying since 8:40 has been saying it for you. And much better than any single invention could do, I think.\n\nLynn, that sponge, cried for ten minutes afterwards and the glue and mascara ran all over the National lobby.\n\nI won't go on \u2013 there's nothing duller than praise, when you've had a surfeit of it, I imagine \u2013 and you're the last person in the world I'd be caught writing fan mail to \u2013 I only had to say, you really did it, man.\n\nDo you know that when you do a work like this you give a boost to everyone who spades around in the same field? It's kind of like knowing that those productive currents and vibrations are still in the air, if you'll only work and reach and not forget.\n\nSo, thanks for it.\n\nG\n\n394. William Schuman to Leonard Bernstein\n\nMartha's Vineyard Island, MA\n\n1 September 1957\n\nDear Lenny,\n\nIt's just wonderful news that we hear of the show, and Frankie and I are so happy for you and with you. We send love and all the best for the NY opening where we hope to be present. Excuse this fast note but the sun is out on this beautiful island and I can't afford to miss a minute of these last precious days. We missed you up here and you'll have to arrange next year sans summer rehearsals.\n\nLove from the 4\n\nSchumans\n\nI'm delighted you're doing the 6th [Symphony] \u2013 just heard.\n\n395. Albert Sirmay60 to Leonard Bernstein\n\nChappell & Co., Inc., New York, NY\n\n11 September 1957\n\nDear Lenny,\n\nI'm afraid that I'm too repetitious and maybe annoying in expressing so many times my admiration for your score of _West Side Story_. However I do not want to miss to say that it gives me immense satisfaction to read in the article of the _Evening Bulletin_ the same words that I used to you on the opening night in Washington and last night in Philadelphia. It was quite a thrill to see my own words \"theatrical history\" and \"milepost\" repeated by a professional critic.\n\nWell, my dear Lenny, I didn't need any printed words to make me aware of the greatness of your score. So bear with me and accept again from me my warmest and most enthusiastic congratulations.\n\nLove,\n\nSirmay\n\n396. Leonard Bernstein to David Diamond\n\nWarwick Hotel, Philadelphia, PA\n\n12 September 1957\n\nDear David,\n\nIt seems impossible that I haven't written you in all this time \u2013 and yet it's true. Six months or so have elapsed since you left \u2013 six months of one straight monochromatic labor on this show, which is now thankfully open, and provisionally a smash hit.61 The three weeks in Washington were phenomenal \u2013 sell-outs, raving press & public. Now simile in Philly. It really does my heart good \u2013 because this show is my baby, my tragic musical comedy, whatever that is; and if it goes in New York as it has been on the road we will have proved something very big indeed, & maybe changed the face of the American musical theatre.\n\nBut the fact remains that I haven't written a letter in all this time \u2013 hardly to anyone \u2013 and what makes it worse, I have had this news for you for at least two months now that I _will_ be doing the 4th Symph \u2013 with the Philharmonic next year. I should have let you know long ago \u2013 but it's just been impossible to do anything but work on the show. Now I breathe a little \u2013 it's on, & going; & it's not all I want, nor will it ever be; but it's good, & I'm proud, & I can write a letter again. I forget the exact date fixed for your Mercury Symph., but it will be in January \u2013 with, I am almost positive, a Columbia recording (I've almost talked the Kouss Foundation into it). And Mischa Elman playing Mendelssohn on the same program will insure the public. Only hitch: it won't fit into the Sunday broadcast. What an impossible business, making programs in terms of split minutes! But that's a small sacrifice, especially if we get the recording. I'll be happy to renew acquaintance with it again. 10 years!\n\nThe family is home again after a long summer in Chile (I don't know what happened to the summer!) and it's such a joy to be with them again. F & I leave for Israel the 27th (the day after the N.Y. opening of _West Side Story_ ) & will be there through the month of October. Then home again. Can you make it?\n\nYou _were_ marvelous in New York, in spite of all the tragic train of events, & I was proud of you. Give my love & Felicia's to Ciro [Cuomo], & many congratulations. Let me hear what you're doing.\n\nLove,\n\nLenny\n\n397. Margot Fonteyn62 to Leonard Bernstein\n\nHampshire House, New York, NY\n\n24 September 1957\n\nDear Mr. Bernstein,\n\nThank you more than I can say for arranging seats for us for _West Side Story_ on 3rd October. We are longing to see it and would never have got in without your help. It is so kind of you & I am sure you are overwhelmed with requests.\n\nThe Met. Opera House are holding 4 seats should you wish them for tomorrow night. I wish I were dancing, but we are so lazy in the ballet, we only dance now and then!\n\nYours sincerely,\n\nMargot Arias\n\n398. Ren\u00e9e Longy Miquelle to Leonard Bernstein\n\n24 September 1957\n\nDearest Spookietchka,\n\nI have been meaning to write you long since but now I can't delay any further since I have\n\n1) to tell you how very much I like _West Side Story_\n\n2) how marvelous it was to see you, talk and reminisce, and\n\nLastly to wish you the very best Broadway opening and a splendid journey to Israel.\n\nThank you for being you \u2013 for your friendship \u2013 you enjoy a very large place in my heart as you know. God speed to both you & Felicia, and all my love.\n\nRelami [Ren\u00e9e]\n\nIf you have a recent picture of yourself (not a snapshot) please send it me \u2013 sort of a delayed birthday present! The last picture (and only one) I have dates back to Xmas 1945.\n\nAre you giving any exciting new works in Israel? Where can you be addressed there? Till when?\n\n399. Lauren Bacall63 to Leonard Bernstein\n\nLos Angeles, CA\n\n25 September 1957\n\n[Telegram]\n\nIt was worth all the Dexamyl. It's a smash, you're a smash and I'm thrilled for you.\n\nBlessings and love,\n\nBetty\n\n400. Betty Comden, Steven Kyle, and Adolph Green to Leonard Bernstein\n\nNew York, NY\n\n26 September 1957\n\n[Telegram]\n\nDear Lenny,\n\nWe can but echo the deathless words of Samuel Hochman: So what's wrong with the hair business? But seriously folks we know tonight will be everything you hoped it to be.\n\nMuch love,\n\nBetty, Steve and Adolph\n\n401. Albert Sirmay to Leonard Bernstein\n\nNew York, NY\n\n26 September 1957\n\n[Telegram]\n\nDear Lenny,\n\nFor the masterwork which you have created in your score for _West Side Story_ my unlimited admiration and my heartfelt wishes for a long-lasting success which you so well deserve with your immense talent.\n\nAlbert Sirmay\n\n402. Stephen Sondheim to Leonard Bernstein\n\n11 East 80th Street, New York, NY\n\n26 September 195764\n\nDear Lenny,\n\nYou know \u2013 only too well \u2013 how hard it is for me to show gratitude and affection, much less to commit them to writing. But tonight I feel I must. _West Side Story_ means much more to me than a first show, more even than the privilege of collaborating with you and Arthur and Jerry. It marks the beginning of what I hope will be a long and enduring friendship. Friendship is a thing I give and receive rarely, but for what it's worth I want you to know you have it from me always.\n\nI don't think I've ever said to you how fine I think the score is, since I prefer kidding you about the few moments I don't like to praising you for the many I do. _West Side Story_ is as big a step forward for you as it is for Jerry or Arthur or even me and, in an odd way, I feel proud of you.\n\nMuch as I want to write music, I'm not sure I like the idea of doing another show without you.\n\nI will, of course, and I'll play it for you, and you'll criticize it, and I'll be hostile and sarcastic about your criticism. But I look forward to that criticism and I hope you'll give it freely.\n\nMy gratitude and affection, then (in token of which I offer the enclosed unusual portrait of L. Bernstein in a moment away from J. Robbins), and also my best wishes for good luck to our little divertissement. May _West Side Story_ mean as much to the theater and to people who see it as it has to us.\n\nSteve\n\n403. Roger L. Stevens65 to Leonard Bernstein\n\nNew York, NY\n\n26 September 1957\n\n[Telegram]\n\nDear Lenny,\n\nThanks for your graciousness in remembering the dim dark days when it looked like everything was off. My faith was simple because with so many remarkable tunes the production just had to work.\n\nRoger\n\n404. Stephen Sondheim to Leonard Bernstein\n\n[New York, NY]\n\n23 October 1957\n\nDear Lenny or Lennie,\n\nI have one of the most unusual and authentic excuses ever proffered for not writing you before this: my electric typewriter blew a fuse. Started to print words like thhhisss, and made interpretation even more difficult than blotted longhand. But I hope this reaches you before you leave for London. Helen tells me that you've been receiving almost none of the mail she's sent you. In a way, I hope it's true, because otherwise my news may be redundant to you.66\n\nFirst of all, your questions about the show67 could not have arrived at a more opportune time. Ten minutes ago, the phone rang \u2013 it was Martin Charnin, informing me that Stephanie Augustine suddenly had to go on for Carol [Lawrence]. This was at four-thirty, so my guess is that something happened to her during the Kaleidoscope (the ballet starts around four-fifteen \u2013 by the way, today is matinee day, in case you're hopelessly confused by my incoherence). Charnin couldn't fill me in on the details, since he had to go on for Krupke (the number, not the actor), but I'm going to the show to see how Mrs. Hyman68 does. [...]\n\nOn the whole, the cast has been healthier than I expected \u2013 a few cases of flu (Al De Sio, who had refused the shot, and whose name outside the theater has been spaced so as to read Aldesio; Charnin, who had received the shot, and one other, whose name escapes me at the moment), a few broken ankles, torn ligaments and sprained wrists (Calin, Roll, Grover Dale, Lynn Ross) [...] But there have been no serious crises until today.\n\nI don't know if Helen sent you all the notices, but knowing her maternal instincts, I suspect she omitted Harold Clurman's in _The Nation_. I'm afraid it's the end of Frank Lewis and his ground rules69 for us, friend, since I'm cancelling my subscription immediately and trust you will do the same. Clurman's was a nasty, personally antagonistic (why? that's the question on a hundred lips; what did you do to him?) piece accusing us of base motives in writing the show. He called it a \"phoney\" and said we were \"intellectuals slumming\" for the purpose of making money. The review was framed by statements to the effect that the show would run a year, but no longer. The only personal pan in the review was directed at me, but he obviously was offended by the whole thing.70 His, however, is the only real blast we got ( _Time_ and _The New Yorker_ weren't great, but they weren't more than ordinarily unkind) [...]\n\nFrom all indications, however, the show is a smash. We're sold out through February, even though tickets haven't yet been placed on sale beyond December 21st (the extra months are the result of mail orders). The License Commissioner has demanded an investigation of the box office; this, Hal [Prince] assures me, is the sign of a genuine blockbuster \u2013 it happened on _My Fair Lady_ and _South Pacific_. The question is how long will the situation last. Part of the cause for the unavailability of seats, even at the brokers', is the continual flow of theater parties \u2013 every matinee and evening for the next six weeks except for six performances or so. There will be an Actors' Benefit performance November 24th, so jot it down \u2013 it should be very exciting. Anyway, Art or no, we'll be making money for a while.\n\nSomeone named Ullman (Abe, I believe) at Schirmer's keeps calling me for house seats and telling me apocryphal news of sheet music sales and forthcoming recordings. So far, only one has been out on the market: Jill Corey singing \"I Feel Pretty\" [...]\n\nRosemary Clooney has recorded \"Tonight\", but I haven't heard it yet. So has Vera Lynn, an English thrush, as we say in _Variety_. Sammy Davis wants to do \"Cool\", \"Something's Coming\" and \"Tonight\", but no action yet. Mickey Calin got a recording contract with something called Teen-o-Rama Records (they told him that he'd get his picture in all the teenage fan magazines, so he signed) and is recording \"Cool\" tomorrow. (He's also gotten numerous movie offers, which should be a surprise to no one.) Incidentally, _Variety_ named the Corey version of \"Pretty\" as one of its weekly best bets (Best Bets, that is). Another one that week was a song by Mary (Rodgers) and Sammy Cahn called \"I Love You Whoever You Are\".\n\nAs for the cast recording, I was amazed at Goddard [Lieberson]'s efficiency and dispatch, as well as his efforts at maintaining quality. You will probably be displeased with the record for reasons stated below as well as dozens of others, but on the whole I think it's pretty good \u2013 at least, by show album standards. It was recorded simultaneously for stereophonic tape (to be released in November as the first show so recorded) and sounds much better than the record. Some of the balancing isn't all it could be, but most of the trouble we had was due to lack of time \u2013 time on the record and time in the recording studio. As we had suspected, the amount of music was way overlong. Someone had goofed on the pre-recording timing, claiming that the Balcony Scene (starting with the singing) was 2:40, whereas it turned out to be 5:10. I don't have time to go into all the suggestions for remedying this, but the only one that worked was to cut out the best part \u2013 namely, the dialogue. Consequently, to our ears, the scene has been emasculated, going straight from the second chorus to the sung \"Goodnight\"s, with four hurried lines spoken over the bridge between. Thus the first \"goodnight\" has to start on the fifth instead of the second, which ruins it, because the second doesn't fit in with the harmony. It's too bad, but I assure you there was no other way out \u2013 at least, none that occurred to us. You are also likely to be disturbed by the following (I tell you these not to ruin your final three weeks before you hear them, but to soften whatever shocks you may get when you're finally back at the Osborne).\n\nFLASH! Hal just called to say that Carol's singing voice gave out at the top of her range, though her low notes and speaking voice remained. She's at the doctor's right now, will probably not go on tonight, but will tomorrow, if all is well. He saw Stephanie from \"I Have A Love\" on, says that she's okay [...] Jerry wanted to fire her immediately (without giving her a chance to do a whole performance, in effect), but he's in a bad mood anyway \u2013 he started his new ballet for the City Center on Monday, and it's apparently not going well.\n\nDrawbacks in the recording, cont'd: A very fast tempo for the prologue, not so much to save time as to make it more interesting. Without the accompanying action, it tended toward monotony. Incidentally, we included street noises and shouts throughout the album, which works very well for the most part, though they tend to drown out the music in The Rumble. 2) Larry [Kert]'s voice on \"Something's Coming\" gets a little froggy in a few places and he sang the wrong rhythm for \"come on, deliver\", but it was by far the best of the takes, because the feeling was right. Unfortunately, it was the next to last song recorded and he was very tired, having been at the session for nine hours. His best is \"Maria\", which was the first number he recorded. 3) Frank Green took Larry's part in The Rumble (shouting \"Riff, don't!\") and came in about ten bars too late \u2013 just before the stabbing \u2013 but the orchestra played it so well, that we didn't try another take (it was already the third). Also, they forgot to blow the police whistle at the climax. (By the way, the orchestra was increased to 37 men for the recording.) 4) \"America\" and \"I Feel Pretty\" don't sound any better on the record than they do on the stage. 5) A trumpet player goofed badly on the change of key in the final procession. Oddly enough, nobody heard it until it was too late. I was out getting five minutes' sleep during it (I also slept during \"America\", since the session lasted from 10 a.m. to 1 a.m.). There will be a hundred other subtle and unsubtle goofs that will probably anger you, but the general reaction to the record so far (it came out last Friday \u2013 first order being 46,000 copies \u2013 is that good?) has been wonderful. _Variety_ raved, and Douglas Watt in the _News_ gave it a good notice (where he objected to anything, it was to the material, not the recording). The singers were not at their best, Lenny, but they were tired. One thing you ought to like: Goddard insisted that the final chorus of \"Krupke\" be played very slow with a heavy vaudeville beat. Jerry must have had conniptions. Another sidelight: Irv and Sid71 put a major cadence at the end of \"I Have A Love\". I had conniptions, so it was changed back to the relative minor. I presume you didn't want it changed. I certainly didn't.\n\nSo all is very cheery at the Winter Garden, although benefit audiences have dampened the general atmosphere. They seem to like the show, though, and word-of-mouth is good. Only in the applause for individual numbers and scenes can that intramural too-many-martinis feeling be noticed. Sometimes the hand for \"America\" hardly covers the scene change, and sometimes Lee [Becker] has no trouble getting the audience to stop at the end of \"Krupke\". \"Something's Coming\" and \"The Jet Song\" still get weak hands, but the applause at the end of the ballet is constant \u2013 or almost so. Benefit audiences don't really get with it till the rumble, and sometimes not until \"Krupke\". But, as I say, they end up liking the show, so don't worry. I haven't seen a complete performance in two weeks, so I don't know how sloppy they're getting, but Jerry's going to do some rehearsing tomorrow.\n\nI have much more to tell you, but I wanted to send this letter off this afternoon so that it would have a chance to slip in to you between Jewish holidays. I'll write you again, but if I don't stop now \u2013 I have to go out \u2013 I won't finish it for another four days. Love to Felicia [...]\n\nMiss you. Come back on the seventh and no excuses. The groaning board72 is set up and I've peeked at all the letters.\n\nLove,\n\nSteve\n\n405. Paul Tortelier73 to Leonard Bernstein\n\n13 October 1957\n\nMy dear Leonard,\n\nI regret not to have been able to see you after our last performance of the _Schelomo_ [by Bloch]; I wanted so much to thank you again and to tell you \"au revoir\"!\n\nHow is it going now in the formidable Frederick Mann Hall? I often think of you and Isaac [Stern] after the dreamlike days in Israel that we love.\n\nWith my son Pascal (10 years old) we are going to hear [Arthur] Rubinstein tomorrow night in his Paris recital; that will be a great joy to hear him again and \u2013 perhaps \u2013 to talk few minutes with him after the concert.\n\nWell, I must take the coach to the airport now and consequently must leave you quicker than I would like.\n\nMy love to Isaac.\n\nMes hommages \u00e0 votre charmante femme avec la reconnaissance et amiti\u00e9 de votre\n\nPaul Tortelier\n\nP.S. Also my best remembrance to the orchestra if it is not asking too much.\n\n406. Leonard Bernstein to Aaron Copland\n\n16 November 1957\n\nDear A,\n\nI tried valiantly to call on your birthday, but no soap. Also yesterday, idem. So, _per forza_ , a letter \u2013 Happy birthday. Just got back from Izzyland and London, and first thought was \u2013 14 November: there's something terribly familiar about that. Then I realized it was my debut date in 1943, and your birthday (in 1938). (Do you realize that next year on your birthday it will be 20 \u2013 twenty \u2013 XX \u2013 _Twenty_ \u2013 vingt years we know each other?!!)\n\nAnd then I realized that I've missed you very much \u2013\n\nAs I say, happy birthday\n\n& love\n\nLenny\n\n407. Lukas Foss to Leonard Bernstein\n\nUniversity of California, Los Angeles, CA\n\n22 November 1957\n\nDear Lenny,\n\nNow that it is official let me again congratulate you and New York and music in America. All I can say is I shall be much more homesick than ever for New York.\n\nMy own life almost took a turn which would have made it look a little more like yours (or, let us say, like a provincial version of yours). Thor Johnson is leaving Cincinnati and I was offered his job. Cincinnati is not like New York, but it compares to my present UCLA orchestra much like NY compares to Cincinnati. In fact it would feel heavenly to have a _real_ orchestra for a change. Having had considerable repertoire experience over the last 5 years, I was tempted, but finally decided against it. They demanded that I conduct 80 concerts in 28 weeks including pops, junior high, neighborhood concerts, tours etc. Aaron thought I should do it for 2 years \u2013 almost changed my mind. I wonder what you would have advised. Felt like giving you a ring, then felt silly. Now the dice have fallen. I am staying here.\n\nI've got a new piece to play for you when I come in January. All my love to Felicia and you, from both of us \u2013\n\nAnd again: Congrats\n\nLukas\n\n408. Leonard Bernstein to Goddard Lieberson\n\n21 November 1957\n\nDear Goddard,\n\nMy thanks to you are so overdue by now that your beautiful gift-plant makes it imperative that I rise from my bed of pain to thank you. You did a wonderful job on the _W_ [ _est_ ] _S_ [ _ide_ ] _S_ [ _tory_ ] album. It must have been a hectic session, from all I hear, with split-second decisions to be made all the time \u2013 and you did a heroic job.\n\nAs to the plant \u2013 it is lovely, & we thank you & Brigitte with great warmth.\n\nAs to the bed of pain, I'm going to the hospital tomorrow for some check-ups on this ridiculous back of mine, as well as some enforced sleep. S-L-E-E-P, glorious word.\n\nLove, Lenny\n\n409. Joshua Logan74 to Leonard Bernstein\n\n22 November 1957\n\nDear Leonard,\n\n_West Side Story_ is one of the most moving events of my theatergoing experience. Your music is such a part of the whole I can't find out where it began or ended. I was just involved & became the slave of all of you from the moment the curtain went up. Thanks for giving us all such a beautiful experience.\n\nAlso congratulations on your new job with the Philharmonic.\n\nBest always,\n\nJosh\n\n1 It is worth noting that Laurents wrote this letter more than a month before the meeting between Laurents and Bernstein when they discussed the idea of warring gangs of Puerto Rican and white American teenagers.\n\n2 Nothing seems to have come of this idea, but Miller writes at length about his observations of gang warfare in Brooklyn during the summer of 1955 in his autobiography _Timebends_ (1987), pp. 360\u20139.\n\n3 _East Side Story_ was one of the several titles for what became _West Side Story_. For years the collaborators called it simply _Romeo_. One short-lived early title was _Operation Capulet_ (see Letter 279). A much later title, announced in the press as late as June 1957, was _Gangway!_ \u2013 an idea that was quickly (and mercifully) abandoned.\n\n4 This undated letter urges Bernstein to go out to Hollywood (where Laurents, Robbins, and Bernstein all worked at various times in 1955) in order to make progress on _West Side Story_. It's probable that this letter was written before the meeting between Bernstein and Laurents on 25 August 1955 (when Robbins was in New York). But it's also possible \u2013 from Laurents' reference to \"all the preliminary work that we've all talked about\" \u2013 that he wrote the letter after the meeting to request a week's visit from Bernstein to consolidate ideas already discussed, especially as Robbins arrived in Hollywood in mid-September to work on the film of _The King and I_ (he was clearly in California when Laurents wrote this letter). The meeting on 25 August in Beverly Hills was described by Bernstein: \"Had a fine long session with Arthur today, by the pool. (He's here for a movie; I'm conducting at the Hollywood Bowl.) We're fired again by the Romeo notion; only now we have abandoned the whole Jewish\u2013Catholic premise as not very fresh, and have come up with what I think is going to be it: two teen-age gangs as the warring factions, one of them newly-arrived Puerto Ricans, the other self-styled 'Americans.' Suddenly it all springs to life. I hear rhythms and pulses, and \u2013 most of all \u2013 I can sort of feel the form\" (Bernstein 1957, p. 47). However, Laurents had already raised this idea a month earlier, in his letter to Bernstein of 19 July 1955 (Letter 359). The eventual outcome was the radically altered outline that introduced the idea of conflict between white and Puerto Rican gangs (printed in Appendix One) to which Robbins responded on 18 October 1955 (Letter 362).\n\n5 _Bells Are Ringing_.\n\n6 Comden and Green were tentatively considered for _West Side Story_ , and during September 1955 they were apparently contemplating taking it on; they were unable to do so because of existing Hollywood commitments. By October 1955, Stephen Sondheim had joined the show's creative team to write the lyrics.\n\n7 Jack Gottlieb (1930\u20132011), American composer who began work as Bernstein's musical assistant in 1958 and subsequently became publications director of Amberson Enterprises, Bernstein's publishing company.\n\n8 In 1954, Kenneth Gaburo and James Dalgliesh were joint winners of the ninth annual George Gershwin Memorial Contest (organized by the B'nai B'rith) for the best orchestral composition by a young American composer.\n\n9 The outline by Arthur Laurents and Bernstein (see Appendix One) and Robbins' detailed reply in this letter are documents of great importance in the genesis of _West Side Story_. Laurents and Bernstein had evolved the new outline after discussing the project in California in August 1955. Newspaper reports of recent gang violence made them rethink the dramatic outline of a show on which progress had been virtually stalled since Robbins had first proposed the idea in January 1949. Bernstein has written marginal annotations on Robbins' letter that reveal some of his reaction to Robbins' criticisms. These are described in the notes that follow.\n\n10 Marginal note by Bernstein: \"good points.\"\n\n11 Marginal note by Bernstein: \"I thought it _was_ sudden.\"\n\n12 Marginal note by Bernstein: \"True, but maybe a slightly diff[erent] angle.\"\n\n13 Marginal note by Bernstein: \"How much older?\"\n\n14 Marginal note by Bernstein: \"Good point. Both things can happen.\"\n\n15 Marginal note by Bernstein: \"Not necessarily bad \u2013 but avoid clich\u00e9!\"\n\n16 Marginal note by Bernstein: \"True. But there are 2 scenes, so how to achieve the flow like one scene?\"\n\n17 Marginal note by Bernstein: \"?\"\n\n18 Marginal note by Bernstein: \"Don't see any incompatibility here.\"\n\n19 Marginal note by Bernstein: \"Why not?\"\n\n20 Marginal note by Bernstein: \"True. Maybe he doesn't know \u2013 but we know he's doomed.\"\n\n21 Marginal note by Bernstein (with a line beside the whole paragraph): \"All true.\"\n\n22 Marginal note by Bernstein: \"Right.\"\n\n23 Lillian Hellman's adaptation of Jean Anouilh's _The Lark_ , for which Bernstein wrote the incidental music.\n\n24 According to Jack Gottlieb's note in the score of Bernstein's _Missa Brevis_ , it was Robert Shaw who suggested this idea when he attended a performance. In 1988, to mark Shaw's retirement as Music Director of the Atlanta Symphony Orchestra, Bernstein produced the \"short Mass\" that he had contemplated more than thirty years earlier. See also Letter 365.\n\n25 Eddie Albert played Reuben in the original production of Blitzstein's _Reuben Reuben_.\n\n26 Subtitled an \"urban folk opera,\" _Reuben Reuben_ opened in Boston on 10 October 1955 and closed quickly, before reaching Broadway. Blitzstein was still reeling from the disappointment when he wrote this letter.\n\n27 Blitzstein never forgave Robbins for naming names when he testified to the HUAC.\n\n28 This is a fascinating comment; Bernstein's own correspondence contains no remarks about working with Robbins after his HUAC performance, but it would be surprising if Bernstein and Arthur Laurents had not been angered and disappointed by Robbins' conduct. And yet, only a few months before Blitzstein sent this letter, Bernstein and Laurents had got back to serious work on _West Side Story_ , and brought Stephen Sondheim into the creative team.\n\n29 Blitzstein was her godfather.\n\n30 This was not a lasting rift. As Eric Gordon put it, \"Lenny and Felicia Bernstein remained loyal to Marc. They had already made their first child, Jamie, Marc's godchild. Their son Alexander's name is the male form of Alexandra from _Regina_. And their third child they named for Marc's heroine in _Reuben Reuben_ \u2013 Nina\" (Gordon 1989, p. 405).\n\n31 In December 1956, Diamond played as a violinist in the pit orchestra for Bernstein's _Candide_ , though this much-needed paying engagement was interrupted when he was subpoenaed by the HUAC. There's at least one irony in the timing of Diamond's subpoena: _Candide_ was partly intended by Hellman and Bernstein as a polemic against HUAC and Joseph McCarthy's Senate hearings.\n\n32 Richard Rodgers (1902\u201379), American composer, best known for his collaborations with Lorenz Hart and Oscar Hammerstein. These creative partnerships resulted in more than thirty Broadway musicals, including several established classics of the American musical theatre.\n\n33 The episode of _Omnibus_ referred to by Rodgers is \"The American Musical Comedy,\" first broadcast on 7 October 1956.\n\n34 Harry Farbman was Associate Conductor of the St Louis Symphony Orchestra.\n\n35 Vladimir Golschmann was the orchestra's Music Director.\n\n36 Foss' _Psalms_ (for chorus and orchestra) were first performed in May 1957 at the New York Philharmonic in a concert conducted by Dimitri Mitropoulos that also included Kod\u00e1ly's _Psalmus Hungaricus_ , Walton's _Belshazzar's Feast_ , and the premiere of Nils Viggo Bentzon's _Variazioni brevi_.\n\n37 Stephen Sondheim (b. 1930), American composer and lyricist. He first met Bernstein in October 1955: according to Bernstein's datebook, Laurents and Sondheim had their first meeting with him on 18 October. Sondheim adds some interesting details: he heard about _West Side Story_ from Arthur Laurents at the opening-night party for Ugo Betti's _Isle of Goats_ on 4 October, and played for Bernstein the following day, on 5 October: \"I auditioned for him without Arthur, and it was the day after the _Isle of Goats_ opening. Remember, Lenny and Arthur had to wait for a week till Betty and Adolph knew whether they could get out of _Winter Wonderland_ (I think that was the title). I was put on hold for that week, which is when I went to consult Oscar [Hammerstein]. My first official meeting after accepting the job was with Arthur alone, the next with both of them\" (Sondheim, personal communication). Bernstein and Sondheim had more than forty meetings between November 1955 and February 1956, and by the time Sondheim celebrated his 26th birthday on 22 March 1956, the score of Act I was starting to take shape \u2013 with _Romeo_ still as the working title.\n\n38 Seventeen years later, Sondheim provided more than just help: for the 1973 revival of _Candide_ , he wrote several new lyrics including \"Life is Happiness Indeed\" (a replacement for Bernstein and Parker's \"The Venice Gavotte\"), \"This World,\" \"The Sheep Song,\" and half of \"Auto Da Fe.\"\n\n39 Gunther Schuller (b. 1925), American composer, conductor, writer on music, and jazz historian. Two works by Schuller were performed by Mitropoulos in the 1956\u20137 New York Philharmonic season, and his father Arthur Schuller was a violinist in the orchestra for more than forty years. Bernstein relished the kind of discussion prompted by this letter, and Schuller's enthusiasm may well have encouraged him to program Webern's music. In January 1958, Bernstein included Webern's _Six Pieces_ Op. 6 in his Philharmonic concerts, and in December 1965 he conducted Webern's Symphony on the same program as Mahler's Seventh. In 1964 Schuller and Bernstein collaborated on Schuller's _Journey Into Jazz_ , composed specially for the _Young People's Concerts_ , where it was conducted by the composer and narrated by Bernstein. The following is part of Bernstein's spoken introduction to the performance: \"These days [...] there is a new movement in American music actually called the 'third stream' which mixes the rivers of jazz with the other rivers that flow down from the high-brow far-out mountain peaks of twelve-tone, or atonal music. Now the leading navigator of this third stream \u2013 in fact the man who made up the phrase \u2013 is a young man named Gunther Schuller. He is one of those total musicians, like Paul Hindemith [...] only he's American. Mr. Schuller writes music \u2013 all kinds of music \u2013 conducts it, lectures on it, and plays it. Certainly he owes some of his great talent to his father, a wonderful musician who happens to play in our orchestra. We are very proud of Arthur Schuller. But young Gunther Schuller \u2013 still in his thirties \u2013 is now the center of a whole group of young composers who look to him as their leader, and champion. And so I thought that the perfect way to begin today's program about jazz in the concert hall would be to play a piece by Gunther Schuller \u2013 especially this one particular piece which is an introduction to jazz for young people.\"\n\n40 Aldous Huxley (1894\u20131963), British author of _Point Counter Point_ and _Brave New World_ who settled in the United States in 1937. One of Huxley's warmest and most enduring friendships in Los Angeles was with Stravinsky.\n\n41 He had already proposed the idea to Stravinsky who rejected it, as did Bernstein. See Joseph 2001, p. 31.\n\n42 Igor Markevitch (1912\u201383), Ukrainian-born conductor and composer.\n\n43 _Icare_ was originally conceived as a ballet for Serge Lifar in 1932, but it was not staged and Markevitch subsequently reworked it as a concert piece.\n\n44 Bernstein conducted _Icare_ with the New York Philharmonic on 10, 11, 12, and 13 April 1958. The performance from 13 April has been released on CD in _Bernstein Live_ (New York Philharmonic NYP 2003).\n\n45 A Spanish colloquialism for \"annoying things\".\n\n46 Probably a reference to the actress Canta Maya, who appeared in the 1946 film _Bailando en las nubes_ ( _Dancing in the Clouds_ ).\n\n47 Abe Miller, born Abraham Malamud, was Sam Bernstein's cousin and he was eventually employed by the Samuel Bernstein Hair Company. Miller and Sam Bernstein \"had corresponded over the years, Sam convincing [Abe] that his future lay in the New World\" (Burton Bernstein 1982, p. 62). In 1921 he escaped from Korets (Ukraine) via Warsaw, Danzig, and Cuba (where he survived for a year hawking his wares from a wooden box), before eventually arriving in the United States.\n\n48 Dexamyl was a drug introduced in 1950. It contained amphetamine to elevate mood, and barbiturate to counteract the side effects of the amphetamine.\n\n49 _West Side Story_ opened at the National Theatre in Washington D.C. on 19 August 1957.\n\n50 Carol Lawrence (b. 1932) created the role of Maria in _West Side Story_ ; 19 August was the date of the opening night in Washington, D.C.\n\n51 Felicia's letter reacting to the very enthusiastic reception of _West Side Story_ in Washington, D.C., was written the day after it had opened at the National Theatre.\n\n52 _Studio One_ was a long-running television drama series. Felicia appeared in eleven episodes between 1949 and 1956.\n\n53 Sherman Adams (1899\u20131986), White House Chief of Staff for President Dwight D. Eisenhower.\n\n54 Cole Porter (1891\u20131964), American composer and lyricist. After studying at both Yale and Harvard, Porter went to Paris where he took orchestration lessons from Charles Koechlin. On his return to America, Porter became hugely successful on Broadway with shows such as _Anything Goes_ (1934), _Kiss Me, Kate_ (1948), and _Can-Can_ (1953).\n\n55 _West Side Story._\n\n56 Spanish for \"gifts\" or \"treats\".\n\n57 Vera Zorina (1917\u20132003) was the stage name of Brigitta Lieberson, wife of Goddard Lieberson. From 1938 to 1946 she had been married to George Balanchine. Zorina was a dancer and actress who specialized in playing the title role in Honegger's _Jeanne d'Arc au b\u00fbcher_.\n\n58 Goddard Lieberson (1911\u201377), English-born record producer who became president of Columbia Records (1956\u201371 and 1973\u20135). After studying composition at the Eastman School, he joined the classical division of Columbia Records. He took a leading role in the introduction of the long-playing record. As well as overseeing important classical recording projects (such as Stravinsky's recordings of his own work), Lieberson also produced many of the most successful Broadway cast recordings, including _West Side Story_ , which was recorded a month after this letter, on 29 September 1957 at Columbia's 30th Street Studio.\n\n59 A stage adaptation of _The Diary of Anne Frank_ , by Frances Goodrich and Albert Hackett, opened on Broadway on 5 October 1955 and ran for 717 performances.\n\n60 Albert Sirmay (1880\u20131967; originally Szirmai), Hungarian operetta composer who moved to New York in 1926 where he took a job with Chappell & Co., becoming music editor for the likes of Gershwin, Porter, and Jerome Kern. He is credited as the editor of the piano-vocal scores of Rodgers and Hammerstein's musicals: _Allegro_ , _Carousel_ , _Flower Drum Song_ , _The King and I_ , _Me and Juliet_ , _Oklahoma!_ , _Pipe Dream_ , _The Sound of Music_ , and _South Pacific_ ; and he also edited the piano-vocal score of Weill's _Lady in the Dark_. He later became a great Bernstein enthusiast, particularly _West Side Story_ in which he invested $500 as one of the show's original backers (see Simeone 2009, pp. 30 and 113).\n\n61 _West Side Story_ opened for try-outs in Washington on 19 August 1957, then in Philadelphia on 10 September, before the Broadway opening on 26 September.\n\n62 Dame Margot Fonteyn de Arias (1919\u201391), ballet dancer who became _prima ballerina assoluta_ of the Royal Ballet, London. She married the Panamanian diplomat Dr. Roberto Arias in 1955, and was appointed a Dame of the British Empire in 1956.\n\n63 Lauren Bacall (b. 1924, as Betty Joan Perske), American actress. Her idol was Bette Davis and both were friends with Bernstein. Bacall married Humphrey Bogart in 1945. When Bogart became ill in 1956, she wrote a touching letter to Bernstein. (\"So sweet of you to take time out to write, and so lovely to hear from you as always. Bogie is coming along, still terribly weak from the treatments and still twenty-five pounds under weight. But in about three weeks it will all be over, thank God, and we can start fattening him up and getting him really well. He's had a time of it but has been saintly throughout.\") Bogart died on 14 January 1957. In 1988, Bacall made a memorable appearance at Bernstein's 70th birthday gala at Tanglewood, singing \"The Saga of Lenny\" (Stephen Sondheim's witty parody of \"The Saga of Jenny\" by Kurt Weill and Ira Gershwin).\n\n64 Sondheim wrote this letter on the day of the Broadway opening of _West Side Story_.\n\n65 Roger L. Stevens (1910\u201398), American theater producer and real-estate magnate. Stevens remained loyal to the production of _West Side Story_ when Cheryl Crawford withdrew \u2013 the \"dark days\" to which he refers. He later became Chairman of the Kennedy Center for the Performing Arts in Washington, D.C. Stevens was a larger-than-life figure. He was a property developer by profession and someone who relished the big gesture \u2013 none bigger than in 1951, when he led a syndicate to buy the Empire State Building. Stevens gave enthusiastic support to _West Side Story_ , and this extended to organizing the opening-night party in New York. His continued involvement resulted in rather a convoluted formula for the original production credits: \"Robert E. Griffith and Harold S. Prince (by arrangement with Roger L. Stevens).\"\n\n66 Sondheim's detailed and amusing account of _West Side Story_ early in its Broadway run is a mine of information, as is his discussion of the one-day recording session for the Columbia Records cast recording made on Sunday, 29 September 1957. Bernstein was unable to be at the recording as he had flown to Israel for the inaugural concerts in the Frederick Mann auditorium straight after the opening night of _West Side Story_.\n\n67 The original cast members of _West Side Story_ mentioned in this letter played the characters shown in parentheses: Stephanie Augustine (standby for Maria); Lee Becker (Anybodys); Mickey Calin (Riff); Martin Charnin (Big Deal); Grover Dale (Snowboy); Al De Sio (Luis); Larry Kert (Tony); Carol Lawrence (Maria); Eddie Roll (Action); Lynn Ross (Estella).\n\n68 Stephanie Augustine was married to Joseph Hyman.\n\n69 Frank Lewis compiled the cryptic crosswords in _The Nation_. \"Ground rules\" was the example given by Lewis in a note at the bottom of the puzzle as a potential clue for \"lures\". Sondheim and Bernstein shared an enthusiasm for fiendish cryptic crosswords.\n\n70 Harold Clurman's review of _West Side Story_ ( _The Nation_ , 12 October 1957) was a rather bitter attack on the show and its authors, and Sondheim's quotations from the review are exactly as they appear in Clurman's original. The only number he seemed to enjoy was \"Gee, Officer Krupke\".\n\n71 Irwin Kostal and Sid Ramin, the orchestrators of _West Side Story_.\n\n72 This is a phrase that crops up occasionally in Bernstein's correspondence. Traditionally, a \"groaning board\" was a table weighed down by an abundance of food.\n\n73 Paul Tortelier (1914\u201390), French cellist. He played in Koussevitzky's Boston Symphony Orchestra (1937\u201340) before embarking on a very successful solo career. Though not a Jew, he was very sympathetic to the State of Israel, and he stayed regularly with his family on a kibbutz.\n\n74 Joshua Logan (1908\u201388), American theater and film director. Logan's Broadway credits included _Annie Get Your Gun_ and _South Pacific_ , for which he also co-wrote the book and shared a Pulitzer Prize with Rodgers and Hammerstein. Logan had known Bernstein for several years by the time of _West Side Story_. A telegram from Logan dated 17 November 1955 reads: \"Dear Leonard, I know this will be an exciting evening and the more so because of you. Josh Logan\" (sent on the opening night of Lillian Hellman's adaptation of Jean Anouilh's _The Lark_ , for which Bernstein wrote the incidental music).\n6\n\nThe New York Philharmonic Years\n\n1958\u201369\n\nIn 1940, Aaron Copland had joked with Bernstein about the time \"forty years from now when you are conductor of the Philharmonic.\" In fact it was just eighteen years later that Bernstein became Music Director of the orchestra, and over the next decade he was to take it on tour all over the world, to make hundreds of recordings, and to give a staggering number of concerts: in 1971 he conducted his 1,000th concert with the Philharmonic, and plenty more followed (his last concerts with the orchestra were in October 1988). In 1958, the press, particularly the _New York Times_ , was often critical of playing standards in the orchestra, but Bernstein soon lifted both the morale of the musicians and the quality of their performance. Howard Taubman was chief music critic of the _Times_ from 1955 to 1960 and wrote enthusiastically about Bernstein: he warmly welcomed his appointment and was generally positive about his concerts. Taubman's place was taken by Harold Schonberg, who grumbled for years about Bernstein's showmanship and often questioned the value of his musical interpretations. It is sometimes true that a hostile critic can ruin the career of a music director, but Bernstein's popularity was such that even Schonberg's most acidic notices made little impact.\n\nThis was the decade where Bernstein had the most regular contact with other composers: commissioning a great deal of new music (something for which he doesn't always get the credit he deserves), arranging events like the celebrations for Aaron Copland's sixtieth birthday at the Philharmonic, encouraging Stravinsky to come and conduct the orchestra, and corresponding with a startling range of composers about their work: the likes of Stockhausen, Xenakis, Feldman, Cage, and Carter, alongside Poulenc, Messiaen, and friends such as Copland, Foss, Diamond, and Bernstein's erstwhile orchestration teacher Randall Thompson.\n\nTours with the Philharmonic resulted in some remarkable personal encounters, and one of the most memorable came early in Bernstein's tenure, when the orchestra traveled to the Soviet Union in 1959. During this visit he met Boris Pasternak, at a time when the author had been publicly denounced by the Soviet authorities for _Doctor Zhivago_ , and a year before his death. For the rest of his life Bernstein would treasure Pasternak's letters \u2013 and their meetings at the author's _dacha_ and in the green room at one of the concerts. When Bernstein took the orchestra to Japan, Felicia stayed at home, and a long letter he wrote to her is a wonderfully evocative description of the sights and sounds of that country.\n\nBack in the United States, Bernstein was becoming an ever more public figure. Euphoric about the election of President Kennedy in November 1960, he was involved in the ball for Kennedy's Inauguration the following January, and was quite a regular visitor to the White House during the Kennedy years. When the president was assassinated in November 1963, Bernstein was quick to pay tribute to the death of a leader who had become a friend, conducting a televised performance of Mahler's \"Resurrection\" Symphony. Five years later, when Robert Kennedy was assassinated, it was Jacqueline Kennedy who took care of the funeral arrangements, and she asked Bernstein to be in charge of the music. Her moving letter of thanks is eloquent testimony to the warm friendship between the two of them.\n\nIn the 1964\u20135 season, Bernstein took a sabbatical year in order to compose. He conducted just one concert (of his own music) right at the end of the season, and otherwise limited his activities to four Young People's Concerts. The largest project that presented itself at the start of this year was a new musical based on Thornton Wilder's _The Skin of Our Teeth_ , a collaboration with two of Bernstein's best friends, Betty Comden and Adolph Green. By January 1965, this had foundered, and Bernstein wrote to David Diamond about \"a dreadful experience, the wounds still smarting. I am suddenly a composer without a project, with half of that golden sabbatical down the drain.\" The friendship with Comden and Green survived this unhappy episode, but the \"golden sabbatical\" threatened to produce no new music whatsoever. What saved the day was a commission from Rev. Walter Hussey of Chichester Cathedral on England's south coast for a set of Psalms. Some of the music originally composed for _The Skin of Our Teeth_ was quickly recycled in the _Chichester Psalms_ (the opening movement and the lyrical theme of the second). The correspondence with Hussey contains no mention of this, but it does show Bernstein laying out his preliminary thoughts about the work in some detail, the decisive moment when he decided that setting the Psalms in Hebrew was something that excited him, and the circumstances of the first British performance on 31 July 1965 \u2013 seemingly a rather idyllic visit, at least by Bernstein's standards, with all the family able to travel with him to England. Two weeks before the Chichester performance, these same _Chichester Psalms_ had featured as the new work in his only New York Philharmonic concert of the season \u2013 a programme that also included the _Serenade_ and _The Age of Anxiety_. In short, the sabbatical didn't produce the new Broadway show that was hoped for, but it did result in one of Bernstein's most popular concert works. The _Chichester Psalms_ was one of just two substantial pieces to be composed during his years at the Philharmonic \u2013 the other was the _Kaddish_ Symphony, finished in time for its premiere in Israel in December 1963, but not without a struggle. As the most searching and musically advanced expression of Bernstein's Jewish faith, the work required of him a large emotional investment. The dedication to the memory of President Kennedy was, of course, only added at the last moment, and by the time the symphony was first played in the United States (by the Boston Symphony Orchestra under Charles Munch), Bernstein's friend Marc Blitzstein had also been murdered in Martinique. Bernstein's use of the word \"Kaddish\" refers to its specific meaning as a prayer of mourning: praising God in spite of personal loss. Thus, Bernstein wrote ruefully to his sister Shirley in January 1964 that \"It's an open season on Kaddish, all right. The President. Marc.\"\n\nBoth works from the 1960s were described by Bernstein during his interviews with John Gruen in 1967: \"I've written two works in the last 10 years, can you imagine, since I took the Philharmonic, which was at the point when I finished _West Side Story_. Since then I've written two works, neither of them for the theatre [...] one was _Kaddish_ and one is the _Chichester Psalms_ \u2013 they're both biblical in a way. So obviously something keeps making me go back to that book.\"1\n\nIn spite of his commitment to the New York Philharmonic, Bernstein continued to work in Europe and Israel, including his first visit to Vienna since 1948. On that occasion he had worked with the Vienna Symphony Orchestra, but now he was conducting concerts and recordings with the Vienna Philharmonic. His relationship with this orchestra was to flourish over the next quarter century, but initially Bernstein was profoundly disturbed by the anti-semitism within it. Georg Solti, another Jewish conductor with extensive experience of working with the orchestra, wrote not to allay Bernstein's concerns, but to counsel a spirit of forgiveness. Solti added that Helmut Wobisch \u2013 the former SS man who was the orchestra's manager and one of its trumpeters \u2013 was \"despite everything [...] probably one of the few trustworthy members of that orchestra.\" Neither Solti nor Bernstein found it easy to work with the Vienna Philharmonic, despite its fabled past, but both managed to establish a musical relationship that became increasingly close over the years.\n\nMeanwhile the Bernstein family continued to grow: in 1962, Felicia and Leonard's third child, Nina, was born \u2013 and, as he had with Jamie and Alexander, Bernstein waxes lyrical and adoring about their new sister. Bernstein himself reached a personal milestone in August 1968: his 50th birthday. One of his oldest friends, Adolph Green \u2013 who had known Bernstein since 1937 \u2013 wrote a funny, and deeply affectionate letter summarizing what their long friendship meant and, above all, how their first meeting had made such a lasting impact: \"What am I wheezily, puffily, floridly trying to tell you??? The simple fact that suddenly there was meaning in my life. I felt _alive_.\"\n\n410. Jule Styne2 to Leonard Bernstein\n\nNew York, NY\n\n2 January 19583\n\n[Telegram]\n\nI need a fella who can play in a publisher's office for singers, who can write a ballet and play the dance rehearsals and then orchestrate it, who can write the music for a new musical comedy and then orchestrate it, and who can write a ballet and play for Agnes de Mille and Michael Kidd and Jerome Robbins too, and who can take this ballet and orchestrate it the hard way with the orchestra sitting the wrong way and the horns pointing into the trombone player's ear; who can also do a tour of one night stands, lecture on why the oboe is a double reed instrument, also, what's going on in the world of music in five continents. This fellow must also be able to orchestrate the telephone book; also this fellow must be able to conduct practically every major symphony orchestra in the world like Adolph Green. Besides playing piano in publisher's offices, and writing ballets, this fellow must also be able to have the possibility of becoming the world's most famous conductor and musician and pianist. He must be a member of the union and must be available to open with the Philharmonic January 2, 1958. Do you know of such a fellow? I do. Good luck. Can you cook?\n\nJule Styne\n\n411. Jerome Robbins to Leonard Bernstein\n\n154 East 74th Street, New York, NY\n\n[January 1958]4\n\nLenny,\n\nIt was a terrific concert!!! I heard it Sunday \u2013 I'd been sick all week so missed both rehearsals and the Thursday evening premier[e]. I learned so much about _Sacre_ [ _du Printemps_ ] by watching you. _That_ certainly ain't no \"divertissement\". Wow! Boy I hope I can make the stage version have as much stature. Bravo again and again, and also for the Webern which was beautifully played.\n\nJulius Rudel of NYC Opera called me about staging _Trouble in Tahiti_. Wanted to speak to you about it and couldn't get you today. So will you call me about it? Also I would like to talk more about your jazz piece etc. etc. and about ballets, shows, operas etc. So do call.\n\nAnd again congratulations on the concert. You should be _very_ proud. Bask well in all the praise being heaped upon you \u2013 you deserve it all.\n\nLove,\n\nJerry\n\n412. Jerome Robbins to Leonard Bernstein\n\n154 East 74th Street, New York, NY\n\n13 February 1958\n\nDear Lenny,\n\nIt was good to see you last night at the opening. I hope your back feels better. I can't think how a skiing trip is going to help your back any more than it would my leg, but anyway, good luck.\n\nI checked with Edith [Weissmann] this morning. She had returned your call and got as far as the desk as I had reported to you.\n\nI asked you if you had time for any composition because I could use _Fugue with Riffs_ as part of a larger piece of yours, in the fashion we had always talked about \u2013 you know, some crazy pas de deux perhaps, and enough to make at least a twenty minute work. Maybe there's a possibility of selecting some of the dance music from _On the Town_ and adding them to it. What do you think?\n\nEnclosed are two interviews which struck me with as much force and excitement as anything I've read in many, many years. The potential of using this first interview as a basis of an examination of the Beat Generation and their search, pain, drives, ecstasies, depressions and astonished puzzlement could make a wonderful theatre piece.5 My instinct is to use a protagonist who would answer the questions, but open up all the \"meanings\" and possibilities of those answers. I'm sure you'll see the immediate places such as \"visions\", \"motorcycle rides\", \"jazz\", \"dope\", \"Paradise\", \"Heaven\", \"tremendous\", \"empty phantoms\" and most of all the very very painful last line. I can visualize a lot of these things being episodic experiences which well up and take over the protagonist and leave him with everything inside him and a cool, knowing exterior. In a way these are our _W_ [ _est_ ] _S_ [ _ide_ ] _S_ [ _tory_ ] kids a little older. Do you know any of these Beat people? They're scary, and what's most frightening is that it _isn't_ an act or an adopted attitude and fa\u00e7ade to deal with life, but a real living thing.\n\nI'd appreciate hearing from you on this as soon as possible. Drop me a note and give me at least your immediate reaction, and if you are interested, when do you think you will be able to have a talk on it.\n\nAll the best,\n\nJerry\n\n413. Lukas Foss to Leonard Bernstein\n\nUniversity of California, Los Angeles, CA\n\n16 March 1958\n\nCher Lenny,\n\nNews just reached me (via Siegfried Hearst) that you want me to do Mozart C major on a program on which you plan to do my Symph. Donnerwetter!!! What a festive occasion that will be for me.6 You will be amused to know that what _elates_ me most is the fact that you schedule my new piece sight unseen. Makes me feel that I have come into my own as composer. Thanks _ever so_ much for the confidence implied (that something I put out is worth playing \u2013 and this coming from the most \"knowing\" of musicians). Seems the days of Lukas selling his music at the piano are over and gone. Just played it for a few musicians though, who think that it is my \"most\"; curious what _you_ will say. I finished the score day before yesterday and sent it off to you yesterday.\n\nYou will see that it is a virtuoso piece for orchestra, the kind where the difficulty makes for brilliance (I hope). It's no \"sight-reading\" piece, and a half hour long. Here is hoping you didn't schedule [Stravinsky's] _Agon_ , Aaron's _Short Symphony_ & Boulez's _Polyphonie X_ to go with it. (You could probably pull it off, too.)\n\nThe Mozart will be splendid \u2013 my favorite slow movement.\n\nHow was your skiing trip? It was wonderful having Felicia with us.\n\nMuch, much love to both of you.\n\nLukas\n\n414. Felicia Bernstein to Leonard Bernstein\n\n[New York]\n\nThurs. night [1 May 1958]\n\nMy darling Lennuhtt,\n\nAm dying to hear how it's going \u2013 the reports so far sound like a nightmare! I hope that at least your sense of humor has not abandoned you \u2013 think of the pastry makers it will all make!\n\nChildren are wonderful \u2013 I am better \u2013 the first days after you left I utterly collapsed. I looked a mottled green with cold sores on my mouth etc. Have nothing sensational to report \u2013 just got back from dinner at the Oppenheims with Glen[n] Gould, Eugene Istomin and Vladimir Golschmann. Glen[n] and Gene sat at the piano and loudly and eternally played four hands \u2013 sometimes I hate music!7\n\nWent to the opening of _The First Born_8 last night \u2013 a crasher if there ever was one, a truly abysmal evening! Your \"score\" is never heard I'm afraid \u2013 the girl sings a little snatch of something for no good reason that I could see, and your triumphant finale is utterly drowned in applause since the curtain comes down simultaneously \u2013 and there you are!\n\nHave a reservation for the twelfth, will arrive the thirteenth in Lima. The fan mail keeps pouring in \u2013 my ego is having a field day!\n\nPlease write a few lines \u2013 it's unbearably lonely around here without you. Kisses from the three of us and very special love from me,\n\nFelicia\n\n415. Leonard Bernstein to Felicia Bernstein\n\nHotel Tamanaco, Caracas, Venezuela\n\n2 May 1958\n\nDarling,\n\nWow, what a three days. First: we're the all time smash. Second, it's all wildly exhausting, but such fun & stimulating. Venezuela is really one of a kind \u2013 stinking with money, progressive architecture, happy, low on culture, & lower on education. It's mostly a billionaire top stratum & penniless others. But there is arising a middle class, & they love music, to say the least. I've had two concerts in a magnificent hall called the Aula Magna9 \u2013 sheer perfection with mobile baffles by [Alexander] Calder, what else? And both sold out to the rafters. In between we gave a special outdoor concert for 7,000 middle classers in an equally perfect Concha Acoustica: they ate up Copland and Roy Harris. So all is not lost. I'm the local hero, I'm loaded with presents; tons of records of folk music, [and] a liki-liki, which is a great beige linen national garment, very handsome with a high collar held by gold links.\n\nI've met the new president of the Junta,10 most charming, loved the Schuman 6th, fijate. Such houses, such jewels, such luxury I've never ever seen. And, again, of course, everyone knows you & loves you. The American Ambassador (at whose gorgeous residence I just had a party) is named Sparks; and his wife is a lovely lady who knows you of old \u2013 Andr\u00e9e van der Brengen. Do you remember her? An angel. And dozens of others whose names go thru my ears like wind. They've all been expecting you, & are sorely disappointed. But we'll be back \u2013 I've become a national institution, & you'll even like it, you'll see.\n\nTomorrow ploughing on to Maracaibo, to conduct in my liki-liki, then on to Colombia, where I hear a big revolution is fermenting. Just in time for some fireworks. This should be a peppery tour. I hear Paraguay should just about be ready with their revolution when we arrive (May 1st was pretty tense here, too).\n\nBut it seems we're doing good \u2013 much more, according to all concerned, than Nifty Nixon, who has flubbed his whole mission in spades so far.11 Can't wait to hole up with him in Quito.\n\nSo when are you coming already? You're missing a great trip \u2013 much fun; & I'm missing you. Try & come as early as you can. I hope you're resting, & that the back is a thing of the past. You must get all healthy again for the trip. So far, I've been well, though sleepless & oversmoked. But that's usual. Your Joan12 still haunts me \u2013 how did you like _Time_ on the subject?13 I thought the picture was great. Brigitta14 will _fry_.\n\nThanks for the cable. How long for those two guatoncitos!\n\nNicolas is not very bright, pobrecito, but he's trying hard, & he's very sweet. Things are never quite in order, & it always seems as though it was easier before when I knew where everything was. Trouble with a valet is that if he leaves you for a second you're lost & helpless. Also he's developing an old-maidish protectiveness worthy of la Belle.15 He scolds people when something is not done for me that should have been done \u2013 like water in the dressing room, or whatever \u2013 & makes long, indignant Theta-ridden speeches about the great maestro, & how he should be treated. Ah well, it's all so familiar. Love to her, by the way, & to Mither [Mother] & to all the Kats. Do try to come early, like Quito, & let me hear _soon_!\n\nAll my love, dearest one.\n\nL\n\n416. David Oppenheim to Leonard Bernstein\n\nColumbia Records, 799 Seventh Avenue, New York, NY\n\n7 May 1958\n\nDear Lenny,\n\nBelieve it or not, I am still going to the office day after day even though you are not here, and so are a lot of other people. Life is tending to go on despite the wonderful reports we are getting back in the _New York Times_ about your concerts.16 I am delighted things are going so well and I suppose by now the Ravel Concerto must be almost ready to record. This is something we must consider when you return.\n\nAlso, while you were away, Felicia came to dinner with Glenn [Gould], Eugene I[stomin] and Vladimir Golschmann, and I am afraid it was not too stimulating an evening. To begin with, an enormous competition between the two young pianists corralled them out of the party, except for some horrendous noises they made on an untuned Baldwin in a room without a rug and without much furniture. Vladimir turned pages, I smiled encouragingly, and Ellen17 and Felicia tried to communicate with each other by a combination of gestures and lip reading, which was not entirely successful I understand.\n\nThe evening ended when Glenn played his own cadenzas to Beethoven's First Concerto on a chair some inches too high, thereby ruining his left arm and impairing all his concerts from here to eternity.\n\nAlso, Van Cliburn was there in spirit \u2013 all things pianistic dimming in comparison with his coming ticker-tape parade. The pianists, especially the young ones, have a sort of haunted, old look, directly traceable to the affairs in Moscow.18 We shall see just how long young Mr. Cliburn can sustain his glories, but if he can maintain his fees for a couple of years, I predict mass suicide involving quite a few of our friends.\n\nIf the above sounds like sour grapes, it is because Victor signed him over my dead body. But this we can discuss another time, too.\n\n[...]\n\nI saw your Bach program rerun Sunday and it was marvelous.\n\nI hope this finds you as well as the press releases have described you and that you will find time to answer me quickly \u2013 always in the affirmative.\n\nLove,\n\nDavid O.\n\nP.S. Incidentally, I see that you have returned the Duke Ellington scores. What is your impression? Can you use them? I think he is waiting to hear, or at least Irving Townsend in our Pop Department is. My impression is that these could be done in a recording session in addition to all of the others you and I are planning. Anyway, let me know.\n\n417. Leonard Bernstein to William Schuman\n\nGran Hotel Bolivar, Lima, Peru\n\n14 May 1958\n\nDear Bill,\n\nIt['s] so unbelievable to be in the lland of the llamas that I spend most of the day saying PERU over & over to myself, just for sanity. Peru?!? Auckland. Mozambique. And here it is, a nice normal warm capital like any other, beautiful, full of people and a number of things. Felicia has just arrived, making it all perfect.\n\nThe tour so far has been a smash; receptions such as I've never experienced, warmth & love from audience after audience, great reviews \u2013 & the orchestra playing like Gods. Every day they become more and more my orchestra, more than I could ever have expected. Your [Sixth] Symphony was remarked by a Caracas critic to be the most important offering we had, and the main feature of our _three_ concerts there.19 It's been great.\n\nBut now I find myself in a spot with our dear old Sixth [Symphony]: we can't play it except in the capitals, where there's a sophisticated audience of some sort; & the time lapse between capitals is such that the orchestra doesn't retain the piece. Last night, for example, it was really ragged and \"forgotten\" after more than a week; it's not a piece you can tour with; we've had to take it off the La Paz & Quito programs because the enormous altitude prevents the necessary blowing (almost _no_ oxygen at all in La Paz \u2013 13,000 feet up); & I'm scared of the next performance, 8 days away. I think we're going to have to take it off the program for the rest of the tour, heartbreaking as it is; I know you'll understand. It's just plain murder to the players and to the piece. It's just one of those works that has to be hot off the griddle to be played at all well; otherwise it's a haphazard gamble.\n\nWe miss you both, & love you.\n\nWe also love the Sixth!\n\nLenny\n\n418. Leonard Bernstein to David Diamond\n\nGran Hotel Bolivar, Lima, Peru\n\n19 May 1958\n\nDear Dovidl,\n\nImag\u00ednate, here we are in Peru, lland of the llama, lland of llove. Felicia has just joined me (we've been two weeks touring South America already in Panam\u00e1, Venezuela, Colombia, Ecuador) & she'll be with me for the rest of the trip \u2013 Peru, Chile (at last!), Paraguay, Bolivia, Argentina, Uruguay, Brazil, Mexico \u2013 the works. Our reception everywhere has been magnificent \u2013 warm, thrilling, just the opposite of Nixon's, pobrecito. We think of you often, and miss you. I realize I have several letters from you as yet unanswered, & I beg forgiveness. These last months have been wild. But with beautiful, rewarding highlights: Felicia as Joan at the final concerts of the season, brave & inspiring & unforgettable (the whole orchestra in tears, Honegger notwithstanding); the television shows, especially those for the kids; the orchestra itself, which has become _my_ orchestra to an astonishing degree of flexibility, warmth and respect; this exhausting but lovely trip. [...] We'll be home by June 15th. Hope to find a letter from you.\n\nLove from us both.\n\nLenny\n\n419. Rosamond Lehmann20 to Felicia Bernstein\n\nFlat F, 70 Eaton Square, London SW1, England\n\n26 May 1958\n\nMy dear Mrs. Bernstein,\n\nI meant to write this at once, but to my horror found I'd mislaid my address book when I left New York the morning after seeing _West Side Story_ \u2013 and have only just re-discovered it in a folder stuffed with lecture notes etc. etc. I hope for your sake you are still away and will not have been conscious of my discourtesy. I _never_ can thank you properly for your incredible generosity and kindness in enabling me not only to see the opera but to take my two dear far-from-wealthy spinster cousins who had been longing in vain to get tickets ever since the opening night. They are extremely earnest Educationalists, and got me down beforehand by spectacled conjectures about the Puerto Rican Problem being \"too serious a subject for frivolous treatment\" etc. etc. \u2013 but like myself were completely bowled over, breath-taken by the end of Act I. It was easily the most fascinating & exciting dramatic experience I've had for years, and I _still_ feel exhilarated, amazed and heart-wrung by the whole affair. Would give anything to see it again! Words are quite inadequate to thank & congratulate your husband. I have the records \u2013 that's something. It _is_ serious of course, and beautiful; also terrifying, moving, funny & disturbing. It was the high spot of my whole Marathon \u2013 and as I say I can never thank you adequately. I was very sorry indeed to miss you both \u2013 and thrilled to read of your husband's brilliantly successful S. American tour. I hope you are back safe and sound \u2013 & that he is able to have a rest. Or does he never?! I'm only just back, & still semi-prostrate from too many thousands of miles of flying \u2013 but I did enjoy a lot of it & confess I miss being spoilt & made much of! \u2013 as I was, everywhere. Martha [Gellhorn] is still abroad.\n\nAgain, so many thanks.\n\nAffectionately,\n\nRosamond Lehmann\n\n420. Martha Gellhorn21 to Leonard Bernstein\n\nMexico\n\nJuly 4 [1958] \u2013 Independence now from what?\n\nLennypot my dearie one,\n\nI waited for the right time to write about _West Side Story_ but probably the exact right time will never come, so now on a rainy (can you beat it?) Cuernavaca morning, my fourth here, and my first not spent jumping with rage and activity against this house, I shall begin. But I know I am not going to do it well enough.\n\nHow can it be called a \"musical comedy\"? It is a musical tragedy, and were it not for the most beautiful music, and the dancing which is like flying, people would not be able to bear to look and see and understand. Certainly they would not pile into that giant stadium, paying huge sums, in order to be wracked by fear and a pity which is useless because how can help be offered, how can a whole world be changed? Tom and I found it beautiful and terrifying. But then he and Omi22 must speak for themselves. Omi had seen it before, found it more enthralling the second time. Enough about their feelings.\n\nI was literally frozen with fear. Do you realize there is no laughter in it, no gayety that comes from delight, from joy, from being young? You do, of course, and all of you knew what you were writing about. The immensely funny song, \"Please Officer Krupke\" (I will get these titles wrong, but near enough), is not laughter, but the most biting, ironic and contemptuous satire. And I felt it to be absolutely accurate \u2013 not the perfection of the wit, in music and words \u2013 but accurate as describing the state of mind of those young. Again, the Puerto Rican girls' song, when one longs for the beauty of home and the other mocks [\"America\"], is not laughter; but the hardness of life, the rock of life, a dream of something softer (softer inside, where it counts) as against the icy measuring rod of modern big city young. The love songs made me cry (they had before, when I heard the whole show twice in one day, listening to [Irwin] Shaw's record in Switzerland).23 But this time, with the visual picture there, and the murderous city outside, and in America, where _West Side Story_ becomes a sociological document turned into art, they made me cry like a sieve, from heartbroken pity.\n\nBut what stays in my mind, as the very picture of terror, is the scene in the drug store, when the Jets sing a song called \"Keep Cool, Man.\" I think I have never heard or seen anything more frightening. (It goes without saying that I think the music so brilliant I have no words to use for it.) I found that a sort of indicator of madness: the mad obsession with nothing, the nerves insanely and constantly stretched \u2013 with no way to rest, no place to go; the emptiness of the undirected minds, whose only occupation could be violence and a terrible macabre playacting. If a man can be nothing, he can pretend to be a hoodlum and feel like somebody. I couldn't breathe, watching and hearing that; it looks to me like doom, as much as these repeated H-bomb tests, with the atmosphere of the world steadily more and more and irrevocably poisoned. I think that drug store and the H-Bomb tests are of the same family.\n\nWhat now baffles me is that all the reviews, and everyone who has seen the show, has not talked of this and this only: the mirror held up to nature, and what nature. I do not feel anything to be exaggerated or falsified; we accept that art renders beautiful, and refines the shapeless raw material of life. The music and the dancing, the plan, the allegory of the story do that; but nature is there, in strength; and surely this musical tragedy is a warning?\n\nIt shames me to speak of music to anyone, owing to my hopeless ignorance and to the fact that I do not hear it, only feel it. I love your music \u2013 everything you wrote (much more than I like anything you conduct). It may be part of my loving you, but it wouldn't work entirely. I love some people whose writing and painting I deplore. No, it isn't that personal at all. I think I love it because it seems to me real. You'll have to figure that out for yourself.\n\nThank you for giving us perfect tickets, where Tom24 could hear \u2013 you can imagine my anxiety about that, in advance \u2013 thank you very much, darling pie.\n\nI think you must write music, more and more, and I think you will. My theory about this is that what one does and is and how one lives, grows and changes. Americans are fools to fear age. It is needed and proper; all one must certainly do is change with one's age, live one's own age, let one's shape (inside and out) alter as it should. I think that, being you, you had to have the great hectic period of doing everything, being everyone and going everywhere. I think that's raw material; and you had to swallow it all, for you will need it. But I also think you will chuck it, without effort or regret, in time; because that will be the time to work on the raw material yourself, draw your conclusions, make your own private gift out of all you saw, did, heard, felt. I think you will really write music, and be concentrated and used by that, in perhaps eight years from now. All you have to do is not ruin your health before that long slow hard second work-period of your life begins.\n\nYou must try to get a book called _Brighter than a Thousand Suns_ by Dr. Robert Jungk. It was published in Germany, translated and published in England. The first part is slow and tedious going, and one sees how necessary it is, later. It is the story of the atomic scientists; it is the human side of how we have launched ourselves (and how accidental and ignorant and pitiful it all is) into doom. I find it needed reading, and too fascinating to stop. Now I must find some nuclear physicist, who is an honest man, to check with. But it is specially a book for us, who have no part in that world \u2013 and that world this very minute rules us. We must know; we may be ineffective to control our destinies, but we cannot ever be sheep. I do not believe in an atomic-hydrogen war, I don't think it is necessary or will happen. I believe the world is going to be poisoned (literally, physically) without that. It might be that if people realized they were daily and invisibly being led to the slaughter, they would not go in silence. If they knew that right now the entire population of the world is infected, and that growing children are most susceptible to this kind of infection, there would be revolt. Anyhow, you read it. It is certainly the other side of the coin of the mad children, living in the streets and dreaming sick dreams.\n\nOmi and I are here preparing this pleasure dome. We were robbed of course. It is not as bad as the house you and Feli had, but I must say I preferred your wide range of Navajo-Mexican striped rugs to the false pretensions of this house. I have removed every movable object and most of the furniture; the clothes cupboards seem to have been built for 1920 type movie stars and are useful warehouses. The servants are charming and pea-brained. The roof leaks like a faucet and is covered with slender brown boys gently and imperceptibly laying back broken tiles. The view would be perfection, a wide sweep from the black range of the mountains that rise towards the Mexico plateau, the Chinese follies of the Tepoztlan hills, the volcanoes behind them, and to the west the beginnings of the blue Taxco range. The idiot owners have elected to plant mingy palms and other uncertified trees in such a way that the only manner to see the whole view is to lie on one's stomach at certain points in the garden, or climb to the top of the wall. The doom of everyone is to have to walk with fools nor lose the common touch; but how many and heavy the fools, and maybe the touch is not worth keeping.\n\nWhen in N.Y., I seem to have gone mad. Within a week (thinking that I knew what I was doing) I arranged four book contracts, two for Tom, two for me, and five articles for me. People leapt to offer me these contracts, paying more money than I have ever before received. The reason for this is that I really do not want to do any of the work, and I certainly do not want a cent more money than I have. The result is that I have to finish my book of collected war reporting by September 1. Beginning in October, and going through until April, I have to do two articles on England, one each on Poland, Hungary and Czecho. My only hope is that I won't be able to get into the last two countries. On April 15, Tom and I return and drive about this benighted land (not this one, the Estados Unidos) for three months, leaving presumably more dead than alive for some quiet spot where we have agreed to grind out a book on the subject in four months. I am surely mad. The only good I can see in it is that it forces me back to work habits, which I have lost, and will be a long dismal training for my muscles. There isn't a ray of light until a year from this coming Xmas. I also have to deliver a short story to the _Atlantic_. I wrote it years ago but in my usual way, I do not feel I have tinkered enough. (I never believe the thing is ready until I can recite every word of it by heart, and go on changing \"a\" to \"the\" with a maniacal desire for exactitude.) Well. That's what going to America does for one. I have already warned Tom that next spring's three months' jaunt may be the last visit of my life. I know I don't believe in progress. I want to live quietly and harmlessly and perhaps do one or two things right, if possible.\n\nBut on the other hand, I have grown lazy and I need to get back into that awful discipline of three hours a day at the typewriter and nothing at all else happening in the day, so as not to get cluttered in the noggin. So here and now we start. I have seen no one in Cuernavaca yet, and only been busy buying out the grocery stores; but in any case we live so far from the center, and have no car, that I think perhaps I'll only have an occasional loving chat on the Buena Vista terrace. Dread seeing Vera. What can I say to her? It appears that slob Ross [Evans] returned for a visit (sponge on Vera for a change?) last February and again departed \"to look for a job in the north.\" And she loves him. Mr. [Somerset] Maugham is not the only one who knows about human bondage.\n\nI hope little Feli is getting some rest on the Cape. You don't know how to, I think, and perhaps don't really want or need it. But do remember she weighs less than you. I find her always more beautiful, and more miraculous. Alexander has my vote for President right now. I trust you will not ruin Jamie by spoiling (Feli will not help you in that ruin) but I see it will be hard to avoid.\n\nYou know how I love you \u2013\n\nM.\n\n421. Leonard Bernstein to David Diamond\n\nVineyard Haven, MA\n\n16 August 1958\n\nDear Dovidl,\n\nMy 40th birthday approaches, and that makes me sentimental and pseudo-philosophical, and I also think of you when I get that way. I'm going through the usual fortyish motions of stocktaking, examination of life-purposes, re-examination of motives and drives, efforts at great self-knowing \u2013 and that's been my summer. My first free summer in twenty years: and it's been so shocking to have it that I've literally done nothing: not a note, imagine, not a bar, no letters written, only programs planned, sailing, and most important of all, spending huge gobs of time with my children. What splendid companions they are!\n\nBut not a note written: and I wonder, as I study my hairline in the mirror and pray desperately against baldness, whether any composer who is _really_ a composer could go for two months without composing, and doing nothing else either. Where will it all lead? Baldness, I expect.\n\nThe Philharmonic season looms large and exciting and frightening. There will be much more television, more difficult programs, more \"point\" being made, more Handel, more Vivaldi. [Var\u00e8se's] Arcanes [ _Arcana_ ],25 at last, and all kinds of Ruggles and Riegger, & the Sessions Vln. Concerto, and Ives #2 and Aaron Variations & Ned Rorem #3 & Bill Russo & Ken Gaburo and and and. A sort of overall look at the whole picture. Not the _whole_ picture, of course: that's impossible, and I have to leave out all kinds of important fellers like Virgil and you and Marc and [Norman] Dello Joio (important?) and [Paul] Creston (ugh) and Ben Weber and [Andrew] Imbrie and [Leon] Kirchner. The Klee26 arrived, & believe it or not, I haven't had a chance to look at it yet! That's my summer.\n\nI haven't heard your 4th [Symphony] yet on records: I will when I get back to town in the fall.\n\nI hope Goldoni27 is fun, & rewarding on several levels. Felicia's Joan is never to be forgotten. She joins me in\n\nLove,\n\nL\n\nI'll be in Milan for a few days in Nov (10\u201315 or so). Will I see you?\n\n422. Jerome Robbins to Leonard Bernstein\n\n154 East 74th Street, New York, NY\n\n13 October 1958\n\nDear Lenny,\n\nA deep bow of thanks for your wonderful letter. I'm so glad you liked it and I'm so sorry you didn't get to see the Chopin. I think you would have had a ball.28\n\nTomorrow starts rehearsals of _West Side Story._ You've _got_ to come in and take them musically over their material, at least once, so they know what you're about, especially while the European conductor is here. We'll prepare them all and get them ready for you, but you _must_ (IT'S IMPERATIVE) do this for the sake of the success of the show.29\n\nDybbuk Dybbuk Dybbuk.30 I'm sending over an unseen but continually haunting prodder who will creep into your sleep and into your spare moments and will say the words Dybbuk Dybbuk Dybbuk. With this ghost's effort I know that suddenly something will be on paper that will get us all started. I've heard from [Ben] Shahn who is wonderfully enthusiastic and excited about the idea of working with you, so please keep haunted and jot down a few of those scribbles that turn out to be the basis, theme and dramatic motifs for the whole ballet.\n\nLove,\n\nJerry\n\n423. Thornton Wilder31 to Leonard Bernstein\n\n50 Deepwood Drive, Hamden, CT\n\n27 October 1958\n\nDear Lennie,\n\nMrs. Alma Mahler-Werfel has chosen some words of mine as the title of her new volume of memoirs32 and I wish to give a small party for her on the publication of the book. I am asking about twenty friends to meet her at the Algonquin Hotel \u2013 reception room 306 \u2013 on November 11 \u2013 Tuesday \u2013 between 5 and 7.30. She tells me her daughter \u2013 the sculptor \u2013 Gustav Mahler's daughter \u2013 will be there.\n\nDon't trouble to answer this. But it would be a great pleasure if you and Mrs. Bernstein could come.\n\nCordially yours,\n\nThornton (Wilder)\n\n424. Larry Adler33 to Leonard Bernstein\n\n[London, England]\n\n24 December 1958\n\nDear Leonard,\n\nIt being impossible to keep a secret these days, you will no doubt have heard that a certain show whose title contains that part of New York where nobody, but nobody ever goes, opened in London and was not unfavorably received.\n\nI took 2\/3 of my children to see it last night, despite a darkling article in the _Telegraph_ \u2013 \"Should children be _allowed_ to see _W_ [ _est_ ] _S_ [ _ide_ ] _S_ [ _tory_ ]?\" They got it all, loved it all and in their comments were far more perceptive than those of several adults around us during interval.\n\nI think your score is historic. Only in _Porgy and Bess_ have I heard music become both words and plot and character, and it happens again with your music. (I might add a footnote here; in 1954 your score for _Waterfront_ and mine for _Genevieve_ were both nominated for an Oscar and I can tell you that had Dmitri \u2013 tote dat corn, lift dat theme \u2013 Tiomkin not edged us both out, this letter might not have been so easily forthcoming.)34\n\nFurther along in the true confessions hour, I am, or at least was, about to start work on a musical myself. But after that score of yours, where does one go except to say, \"Face facts, Wotan, you ain't ready yet.\"\n\nSo, my heartfelt congratulations. You, as a musician, know how another musician feels when he hears something that says something new, different and honest.\n\nRegards to Arthur Laurents, who got me my tickets. Also, if you see him, from [i.e. to?] the boychick of the fiddle, Isaac Stern.\n\nSincerely,\n\nLarry Adler\n\n425. Louis Armstrong35 to Leonard Bernstein\n\nCBS Television Network, New York, NY\n\n5 January 1959\n\nDear Daddy Bernstein,\n\n_Man_. I sitting in your office rehearsing my lines, and it is _knocking_ me out. You're _My Man_ and that's for sure. From your Swiss Kriss36 Trumpet Player.\n\nRegards,\n\nLouis Armstrong\n\nSatchmo\n\n426. Leonard Bernstein to Martha Gellhorn\n\nArizona Biltmore, Phoenix, AZ\n\n7 January 1959\n\nDearest Marthy,\n\nHappy New Year. At long last, a rest \u2013 although God knows it takes fully as much energy to unwind and force the inactivity as it does to be active. But at least they're not all pushing from all sides: I have only my own sick silly psyche pushing from inside.\n\nI'm not staying at the above \u2013 just using the luxuriousissimo facilities & living with friends. Burtie has been with me, left yesterday, all is calm. We started out last week in Sun Valley. Skied three days on the daisies (and a bit of snow) and then left for the hot glorious desert, horses, tennis, swimming. Lord, if I only had a bit of peace in me \u2013 a bit only, is that too much? \u2013 how I could be enjoying all this! And Poland? And Alaska? And is here next? Did you do the hols in London? Are you as petrified as I of the lunik lunacy?37 What the hell are we fiddling with? When do you arrive in this favorite land of yours for your Okie junket?\n\nI met Ernest Hemingway at Sun Valley last week, and was taken totally by surprise. I had not been prepared by talk, photos, or interviews for a) that charm, and b) that beauty. God, what goes on there under his eyes? What's that lovely adolescent tenderness? And the voice and the memory, & the apparently genuine interest in every living soul: fantastic. We spoke tenderly of you: he said you were brave.38 His present wife seems to be a professional Ja-sayer, though simpatico enough. The question is not How could you have married him, but How could you have done anything else?\n\nDearest love to you, every day, always, dearest potato-pipe. I played tennis today & almost wept with nostalgia for our version of tenny.\n\nWrite me \u2013\n\nL\n\n427. Martha Gellhorn to Leonard Bernstein\n\n20 Chester Square, London, England\n\npostmark 14 January 1959\n\nDearest Lenushka,\n\nI loved your Xmas card, both of you looking so beautiful and so tired and the children so beautiful and benign, like happy little dolls. I am saving my first-in-my-life vote for Alexander who will surely be President unless he decides it's all a silly joke and he'd rather live.\n\nSo much to say but I won't say it, probably. This is my last letter, anyhow, for some time, because now I am going to start on a novel and that means silence, fasting and prayer. A novel about Poland. Most daring. I was there 16 days; and learned more and felt more than I have, probably, since Spain. Terrifying and wonderful nourishing experience. I was also frightened the whole time, and I am not used to being; frightened for everyone because they are too brave. And all my desperate faith in the human spirit was revived and rewarded, because there they _are_. Proof.\n\nShall I say some ominous aunt-like words about peace? I think I will. It is a subject that I have really thought and worked on, you know. So: no one besides yourself will ever help you to get it; everyone, even with the best will in the world, will nibble and shred it. You have to fight for it, yourself, and it is perhaps that most essential fight there is. If you haven't got (and keep clinging to, through every reverse) a hard kernel of your own private peace, maybe no bigger than a pea, you cannot be, do or give any _real_ thing. Practically, I find it works like this: one learns what conditions one needs, for oneself, to bring back or foster one's interior nugget of certainty and calm and happiness. For me, it's absolute solitude and silence, in the country; long walks, no timetable of any kind, no telephone, no mail, no newspapers. Long mooning walks, reading, sleeping a great deal. No booze, simply because booze makes me nervous. And then, after a longer or shorter cure of this (depending on how much my peace has been eaten away) I can start to work: and that sets it firmly. I have no idea what you need, but you must, by now, have learned for yourself. No other person gives it, you know, though anyone can take it away. Sex has nothing to do with it either.\n\nThe Xmas hols, just terminated, ruined me as usual. I cannot bear any season given over to organized official good cheer, and too many people, plans, parties. So, as soon as I'd put little Sandy on his plane for Switzerland, I rushed off to my usual country hotel for three days alone. Whereupon an old friend (known for 30 years, now aged 74) was in the hospital in London, and I had to take over everything by telephone. That fixed the peace allright. I'm hanging on however, and have now got the telephone here turned off all day, will not accept any invitations nor give any, and I mean by God to come back to myself and to where I really live. You see, I get physically sick when the peace all goes. I think you don't do that, though I am not sure. But I think you hardly know who you are, or why you are doing what you are doing.\n\nInterested about Ernest [Hemingway]. Tenderness is a new quality in him; but people do luckily change all their lives and the luckiest ones get better as they grow older. His main appalling lack was tenderness for anyone. I longed for it in him, for myself and for others. I'd almost have settled for others. I do not remember his voice as being anything much, but I always was thrilled by his memory. He was interested in everyone but there was a bad side. It was like flirting. (Like you, in fact, he has the excessive need to be loved by everyone, and specially by all the strange passing people whom he ensnares with that interest, as do you with your charm, though in fact he didn't give a fart for them.) So he would take people into camp; they became his adoring slaves (he likes adoring slaves) and suddenly, without warning, he would turn on them. That was always terrible to see; it made me feel cold and sick and I wanted to warn each new conquest of what lay in wait for him. But one couldn't; they wouldn't believe; they were on the heights of joy \u2013 for he can be a great life-enhancer and great fun, and his attention is very flattering.\n\nBy the time I did marry him (driving home from Sun Valley) I did not want to, but it had gone too far in every way. I wept, secretly, silently, on the night before my wedding and my wedding night; I felt absolutely trapped. When I fell in love with him was in Spain, where for once he did have tenderness for others (not me, he was regularly bloody to me, lustful or possessive, and only nice when he was teaching me, as if I were a young man, the arts of self defense in war. And also he liked being the only man in Spain who took his woman around with him, and I was blonde, very helpful in brunette countries, raises one's value.) I loved him then for his generosity to others and for his selfless concern for the Cause. That was all gone by the time I married him. I think I was afraid of him though I certainly never admitted it to myself or showed it to him. You will also be surprised to hear that I have never been more bored in my life than during the long long months when we lived alone in Cuba. I thought I would die of boredom. But it was very good for me. I wrote more with him than ever before or since in my life, and read more. There were no distractions; I lived beside him and entirely and completely alone, as never before or since.\n\nI am very glad he now speaks pleasantly of me. I never speak of him one way or the other with anyone. The whole thing is a distant dream, not very true and curiously embarrassing. It has almost nothing to do with me. What I write you here is, as you can understand, secret and between us only and forever.\n\nHe ought to be happy and he ought to be gentle; because life has showered gifts and blessings on him; and I hope he is.\n\nConsidering this was to be a quick letter, only saying that I love you and wish you well for 1959 and all years to follow, it has rather swelled, has it not.\n\nMy darling Lenny.\n\nMarthy\n\nP.S. Bertrand Russell uses the word \"impiety\" in relation to luniks and further attempts and he is right.\n\n428. Darius Milhaud to Leonard Bernstein\n\nMills College, Oakland, CA\n\n9 January 1959\n\nMy dear Lenny,\n\nEverybody tells me that you made a magnificent performance of my old _Cr\u00e9ation du monde_. It is too bad I didn't hear it.39 Generally I always listen on the radio to your programs which I love. They are full of miracles. For instance Schumann (Robert!) 4th Symphony sound[ed] like a transparent, light, tender orchestra.\n\nBravo, dear.\n\nBest wishes for the New Year from both of us to your family.\n\nMilhaud\n\n429. Leonard Bernstein to Mary Rodgers40\n\n22 January 1959\n\nMy dear little Miss Rodgers,\n\nI am happy to inform you that you have won the contest for the best word to replace \"classical\". Your magnificent choice of EXACT will ring down through the centuries, and no doubt enter Webster's 567th edition, if only as a footnote.\n\nCongratulations; and please accept the enclosed gift as a token of our esteem and gratitude for your fine thinking.\n\nFaithfully yours,\n\nLeonard Bernstein\n\n430. Darius Milhaud to Leonard Bernstein\n\nMills College, Oakland, CA\n\n29 January 1959\n\nLenny dear,\n\nI was deeply touched by all the nice things you said about _Cr\u00e9ation du monde_ , and you explained everything so _clearly_. You are just marvellous. The performance was _remarkable_ too and you were so exciting in the Gershwin.\n\nLucky Philharmonic!\n\nI hope it will not be years before we see you.\n\nMost affectionately,\n\nMilhaud\n\nI should, I think, tell you that it is preferable not to use all the strings in _Cr\u00e9ation_. It's \"sharper\" with soli.\n\n431. Joe Roddy41 to Leonard Bernstein\n\nLife, Time & Life Building, Rockefeller Center, New York, NY\n\n24 February [1959]\n\nDear Leonard,\n\n1. The _Leonore_ [Overture No. 3] played Thursday night was the work of the best-sounding (that's a loathsome simplification) orchestra I have heard in years.\n\n2. The television program Sunday was \u2013 again \u2013 the best one you have done. The pattern is set and you cannot allow a poor one. That's fine with me. But I defy you to improve on \"How Dry I Am\".42\n\n3. Items 1 and 2 above are not set-ups for a complaint, but I have one. My children are pissed off, which is a concern of yours. They claim that at the last Saturday morning children's concert the TV strong-men blinded them by aiming great flood lights into the audience. Presumably the purpose of this was to make them \u2013 my spawn \u2013 look bright on television screens around the country. They don't give a good damn about being seen around the country because they came there to see you and the concert and they could not see either at times. They claim they cannot hear when they cannot see, but that's the exaggerated howling of the angry young men. I think they have a case.\n\nAnd as for me, I think the picture of the Child Listening Fervently is a wearying clich\u00e9 by now anyway. I strongly suspect you of being Christ (but hell, you know all that) and you of all people know perfectly well that \"suffer little children to come unto me\" is not to be understood this way. Even Kenneth Tynan knows that.\n\nI have urged your appointment as Secretary of State and you will be hearing from the Feds about this suggestion any day now.\n\nHighest regards,\n\nJoe Roddy\n\n432. Jule Styne to Leonard Bernstein\n\n237 West 51st Street, New York, NY\n\n20 March 1959\n\nDear Lenny,\n\nAs you know, I am devastated and shocked... as, no doubt, you were by Mr. Brooks Atkinson's and Mr. Walter Kerr's review of the show.43\n\nI can understand them not liking the songs or a song; or not liking the book or direction; or not liking the performance or a performance... However, for Atkinson, a man of his high intellect, to write in his column his last line \"a mongrel musical drama\" about this show in this day and age, is shocking. This is unfair criticism.\n\nSince you expressed yourself with great joy and thought the show was a beautiful musical and almost felt sure that it was a hit, I would appreciate your writing a letter to the _New York Times_ Mail Bag immediately. I feel a letter coming from you, since they know how honest you are, would be of tremendous assistance to the show. I know you have the courage and honesty to consider writing this letter.44\n\nThanks again for your and Felicia's niceness and God Bless.\n\nLove,\n\nJule\n\n433. Leonard Bernstein to David Diamond\n\n23 May 1959\n\nDavid, David, David,\n\n(That's in answer to _three_ letters of yours.)\n\nThe season is over: we've been to London and back in a week to see my _two_ shows there (imagine that!), & it's as expected: _West Side_ is booming, _Candide_ is limping, & I guess always will. Since our return I've been doing mail mail mail with Helen \u2013 a whole season's worth, Lord \u2013 and spending days on the editing of my book that S[imon] & S[chuster] are bringing out in the fall (nothing new: just a collection of TV scripts & miscellaneous writings. But what time it takes to edit!)45\n\nAnd so, finally, a minute to write you. And say how touched we both are by your wish to present us with the Trittico Dodecafonico! You are sweet. It's awfully hard to tell anything much from the photos: obviously color does a lot of the meaning; but what one can see is _fascinating_. I showed them to Danny S. [Saidenberg] & gave him a spiel, which I hope helps when he visits you this summer.\n\nI've also talked twice to Oliver Daniel46 based on a cooky notion I had that you might do well to just chuck ASCAP at this point & cross the stream. He looked into the story of you & BMI & couldn't learn enough to satisfy him. If you think it's a good idea, write Oliver (he has asked to have you do this: & it is promised to be kept confidential); & tell him the whole sordid tale. I sense an interest there. I wasn't up enough on the facts to give him the whole story. (I am also still trying to get in touch with Nissim.)\n\nI saw David O[ppenheim] yesterday & reminded him about releasing the 4th Symph: he has promised it for the early fall.\n\nAnd Tommy [Schippers] is apparently doing _Klee_ in Russia! Isn't it wonderful? But, as you say, when on earth will he rehearse it? Strange, fancy type fellow, that. I suspect also, frightened to death.\n\nDear David, this time we _must_ meet in Italy! We don't alas, come to Firenze, but we do go to Venice on 26th Sept, & thence to Milan (concerts on the 28th & 29th). Meanwhile, June & July on the delicious old Vineyard, where I hope to recoup some energy & sanity, & maybe even _write_ something, please God!\n\nBless you & love from us both\n\nL\n\n(& Best to Ciro!)\n\n434. Leonard Bernstein to David Diamond\n\nVineyard Haven, MA\n\n[Summer 1959]\n\nDear D,\n\nThe Vineyard is glorious, a blessing every day. I've decided that gratitude \u2013 rather, gratefulness \u2013 is the essence of joy, the basic emotion, what we feel when we hear music we love, or look at our loved ones, or simply breathe on this golden island; and growing old means only losing that emotion. The retention of gratefulness is the guarantee of continued youth, don't you think?\n\nWhen I go into N.Y. to open the Stadium, I shall have a talk with Stanley Adams.47 I've written him already & we have an appointment. I decided to go to the top, after failing with the underlings. We'll see. Meanwhile, write Oliver [Daniel].\n\nAs to the Ford grants, they were part of a special project for _performers_ , who were asked to select composers they wanted concertos from. That's out. But I have a feeling Spivacke48 could be in. Why not write him? I'll put in a blast too.\n\nVery odd to think of you & Tommy S. [Schippers] together, I don't know why.\n\nI know nothing of Marlon [Brando] or his whereabouts. Irene Lee49 will be visiting you shortly \u2013 ask her.\n\nI'll write after Stanley Adams.\n\n [Dosvidaniya]\n\nLenny\n\n(I'm studying Russian out of a little book, & I think of you every time I pronounce a hard L. What a delicious language!)\n\n435. Boris Pasternak50 to Leonard Bernstein\n\n[Peredelkino, near Moscow, Soviet Union]\n\n1\u20133 September 1959\n\nDear Mr. Bernstein,\n\nI am exceedingly touched and most thankfully surprised by your kind friendly wire. If other interjacent notes from me will not anticipate this my uttermost decision I hope to have the happiness to attend your concert on the eleventh. To that end, no daring or intending to trouble you to write anything, I shall by my own care look after four passes (for me and my family) to the performance. Only please indicate my name to the attendant before the door of your artistic room on the evening, that I may be admitted to you after the concert.\n\nBesides that I shall try to get the luck the honour and the right to invite you to dinner at Peredelkino Wednesday the ninth at three o'clock.\n\nI shall confirm it afterwards once more.\n\nObediently yours,\n\nB. Pasternak\n\nSept 2nd 1959\n\nNo it will not go \u2013 I think it better to renounce to that great pleasure and not to meet apart from the concert evening (the 11) when I shall experience myself the delight and ecstasy all the town speaks of, & hereupon I am congratulating you fervently in advance.\n\nExcuse my unexplainable discourtesy. My involuntary ungraciousness is my misfortune, not my fault. But I shall hear and see you.\n\nWith the same devotion,\n\nIdem.\n\n3 September 1959\n\nLast note.\n\nPlease be welcome on the day and hour you dispose the best, except the intervals between 1\u201321\/2 and after 8 in the evening, when I can be about on walks. The best hour remains that of the dinner (3 o'clock). Come _as it were unawaitedly_. Ask the guidance of the concert organisation to provide for the return car. Agree with them upon my being admitted in the evening of the concert in the entr'acte.\n\nI wish you the renewal of your habitual triumphs I know of from hearsay.\n\nRespectfully yours,\n\nB. Pasternak\n\n436. Boris Pasternak to Leonard Bernstein\n\nPeredelkino, near Moscow, Soviet Union\n\n9 September 1959\n\nFor Mr. Leonard Bernstein,\n\nPaste this dedication in your copy of my novel. It was so fine and kind of you to have wired to me from Leningrad, to have got the desire to find and to meet me. After-tomorrow I shall attend to the marvel and triumph of art which are your performances. In grateful presentiment of it.\n\nB. Pasternak\n\n437. Boris Pasternak to Leonard and Felicia Bernstein\n\nMoscow, Soviet Union\n\n12 September 1959\n\nDear friends,\n\nIn the morning of the next day Saturday \u2013 Fatigue, yearning, exhaustedness, like after a sleepless night or a big command event, a great night fire in the town, a conflagration, having devoured [a] lot of houses, or a mighty storm with a powerful inundation.51 So must be art. We must will its produced impression, long, and pine for it. Art must leave us love-stricken and sorrow overcome, like a deep-felt parting or separation. Art is language of greatness, greatness is disclosure, its sight, its tragic and suffering _being exposed to view_.\n\nDon't stand you both so often before my mental eyes, I will be along, don't hinder me to be diligent and working.\n\nB. Pasternak\n\nHearty greetings to all yours; to Mr. St[even] Rosenfeld; to the whole orchestra, to the fiery, dear, expansive Mr. Zimmermann; to Goldstone and Varga.\n\nDear My Felicia, who was the lady you sent after me out of the lobby in the concert-room in the entr'acte? She wore a dark, straight, long dress, not girdled in the waist, that seems a sort of brocade. Her husband was dressed in a light brown suit.\n\nThey figure, they are present in my greetings, I am feelingly asking their names. Having spoken with them, I was so absent minded in the crowd, as to not have demanded to be presented to them. Write me be so kind about them, about her and him.\n\nAnd forget I implore you, my stupid idiotic speeches when in commotion I employed one word instead of another (for instance I used the word \"wife\" for \"woman\" that in reality I intended to say), and gave occasion to think I have been foretelling incidents towards some certain future.\n\nThe Same, in the same state of affectionate devotion.\n\n438. Leonard Bernstein to David Diamond\n\n205 West 57th Street, New York, NY\n\n19 October 1959\n\nDear David,\n\nWhat a shock! The first I heard of Tommy [Schippers] taking off your Klee piece was _your_ letter, which greeted my homecoming. I've thought hard; but there's nothing I can do about it. That he should replace it with yet _another_ Barber piece is too silly, and that I will do something about.\n\nSimultaneously, I heard two fine pieces of news: I ran into Stanley Adams in Washington, who said that all was _set_ & definite for you to get a year's advance! Then Marc told me that the Minna C[urtiss] thing was in the bag for 4 thousand. Wonderful news, & congratulations.\n\nBut meanwhile you say you are broke; I hope the enclosed helps until the big money arrives, which should be soon.\n\nWe finally saw the paintings which are startling & fascinating! More about that later \u2013 I'm dashing off to Springfield & Boston with that goddamn Shosty #5!\n\nLove, & don't despair,\n\nL\n\n439. Karlheinz Stockhausen52 to Leonard Bernstein\n\nMeister-Johann-Strasse 6, K\u00f6ln-Braunsfeld, Germany\n\n8 November 1959\n\nDear Mr. Bernstein,\n\nHow friendly you have written me! It is good that you understood my decision so well. I agree perfectly with your \"postponement\". That gives me an idea:\n\nShould we not give Carnegie Hall the last lubrication before she will die? E.g. with _Kontra-Punkte_ (instead of _Gruppen_ 31st March), and David Tudor as pianist under your direction?\n\nCarnegie Hall would be delighted by such a last honour: she would feel like Carnegie Walhalla, and then she will never be jealous of Lincoln Hall during her long death.\n\nIn the parcel (which I sent to you) you will find a tape 71\/2 inches\/sec. with a copy of an old performance of _Kontra-Punkte_ (1955) and copies of _Gruppen_ performances in K\u00f6ln and Donaueschingen (worse, but with corrections). You will find also a little score of _Kontra-Punkte_ and a record with the electronic composition _Gesang der J\u00fcnglinge_.\n\nA last word on your letter: you should never be sad. You are the only artist since long time ago who gave me the impression to be more than a \"serious\" one, who can fill a room with life, with Unbefangenheit, with Aufger\u00e4umtheit just by his presence, just by speaking and laughing about everything. That's gold of endowment without any merit; and I wish with all my heart that nobody could ever ruin or distort your soul. We live for music \u2013 yes; but we can say as well that music exists for us. There is a secret relationship between your soul and Mozart's soul, perhaps you know it: brilliant seldom stars appearing from time to time at the sky of this earth, light and transparent like angels, making everybody happy for a little moment of this long serious history... homeless like only homeless ones can be.\n\nI know what I say; one half of my soul is like you are, like Mozart's; but I am a strange mixture of heaven and hell.\n\nThanks for having met you, thanks for having heard you and seen you (during your concert I became so angry against you, that I could not stay: I listened too serious, too egoistic); but now, already soon after the concert, I discovered you in my own mind, in my own soul. I even don't think any more that something like this can be destroyed: misdeal yourselves without any limit, spread on this earth as much as possible, reserve nothing.\n\nYours,\n\nKarlheinz Stockhausen\n\n440. Leonard Bernstein to Aaron Copland\n\nPalm Springs, CA\n\n12 November 1959\n\nDearest Aa,\n\nHappy Birthday! I'm all alone, imagine, out in the desert, in a rented house (all Japonee, & ugly) & a wonderful pool, & a Baldwin, and big rocky desert hills, & sun & fine air, all alone thinking of you. How I miss seeing you, & have missed it for years; how much I was looking forward to visiting some Sundays ago, & couldn't because Felicia wasn't well; thinking about how we can splurge it up a year from now at the Philh. for your 60th; thinking about those inside wheels of me that compose music, and are so rusty now (I wrote a bar today!), and how long I can go on being an all-time maestro without writing; & thence to Mahler (I bought lots of albums of Mahler, & I've been listening & crying as I listen \u2013 _Das Lied_ is still one in a million) \u2013 & thence to Bruckner (I bought some of his symphs too, having never heard #6, 8 or 9!!) & find him impossibly boring, without personality, awkward & dull, masked in solemnity.\n\nBut Mahler makes me think of you, hard, and of our music, which I don't think I really understand the direction of any more (or the purpose); & I long to talk to you & have you explain it to me, & reassure me that new music is just as exciting as it was when you showed me all about it 20 years ago.\n\nAnd I long also to kiss you and wish you a very happy birthday \u2013\n\nAlways,\n\nLenny\n\n441. Fritz Reiner to Leonard Bernstein\n\n1320 North State Parkway, Chicago, IL\n\n14 November 1959\n\nDear Lenny,\n\nMany thanks for sending me your book.53 I shall look forward to enjoying it on my Christmas vacation at Rambleside.54\n\nMeanwhile it is gratifying to read in the inscription that my teaching and ideas about music are remembered by my most brilliant and successful student.\n\nThanks again and arrivederci presto.\n\nYour friend,\n\nFritz Reiner\n\n442. Lukas Foss to Leonard Bernstein\n\n11 December 1959\n\nDear L,\n\nHave you answered Stockhausen yet?\n\n\"None of your business Lukas\" \u2013 Right. I am trespassing. But since you showed me his letter,55 forgive my thinking about it. I just thought of the perfect answer. Allow me:\n\nDear Mr. Stockhausen,\n\nI am of course the most gifted man of music in the USA but I have as much in common with W.A.M. as the man in the moon. _And you know it_.\n\nSincerely,\n\nLeonard Bernstein\n\nWho is that man who dares out-flatter the flatterers, who dares manipulate you as if you were vain and childish (like the great S[erge] K[oussevitzky])? He is a man who built an international reputation not on a composition but on promises and promises, on a _platform_. \"I will lead music to...\". Dictators promise and dictators flatter potential enemies into temporary allies, pulling the wool over their eyes. Did not Hitler once call the Italians \"Aryans among Latins\"? Ludicrous flattering which made the recipient quiver with joy. \"Look here, everybody, the man whom I expected to hate me, whom everyone fears, he praises me, he wants to do business with me, he sent me a gift...\"\n\nWhat floors me is: it works. It's so ludicrous that it works. It worked on Stravinsky and it probably works on you.\n\nBut Lenny, suppose for a moment that Mozart were reborn, and that we'd all know: it's Mozart. What kind of a letter would a composer, would Stravinsky, would Stockhausen, would I write to him? Surely not: \"I who am suspended between heaven and hell found you in my soul\" but simply:\n\n\"Dear W.A.M. please allow me to show you my music. I need your advice, your criticism, and I hope you'll like me and the music. Yours...\"\n\nWhen I admire someone I do not \"bestow\" praise. Genuine admiration makes one modest, humble \u2013 in fact \u2013 a beggar.\n\nLukas\n\nP.S. There is no possible apology for this letter of mine. Though I speak the truth it is quite obvious that I wrote it out of jealousy.\n\n443. Francis Poulenc to Leonard Bernstein\n\n13 December [1959]\n\nDear Bernstein,\n\nForgive my late reply to your kind message. I accept very happily your commission for 61\u201362, very flattered to be numbered among the ten eminent composers. Since I will be going to New York in February, we can talk about that. As I am not a man of symphonies _alla Brahms_ or Shosta[kovich], we'll see what I could write, because I want to do a good job. I've just finished my _Gloria_ for Boston (60\u201361), so you don't need to fear a choral work. That said, I thank you and embrace you, hoping to see you soon.\n\nFrancis Poulenc56\n\n444. Joe Roddy to Leonard Bernstein\n\n_Life_ , Time & Life Building, Rockefeller Center, New York, NY\n\n13 February 1960\n\nDear Leonard,\n\nThe Mahler-mania in me is getting out of control and I cannot suppress an urge to sing the Second Symphony this week. I remind you, with good cause, that I am a survivor of City Center performances of the _Symphony of Psalms_ and the _Airborne_ [Symphony] conducted by you, Ninth Symphonys of Toscanini and Koussevitzky, Berlioz Requiem of Munch, and about one thousand ditties ranging from Ives to Byrd led by Robert Shaw when I was a Collegiate Chorale kid.\n\nThe best and worst that can be said of my voice is that it is harmless. It is white. I too am white.\n\nI have crashed many a chorus in recent years, but this time the idea of legality pleases me. This morning I called one F. Austin Walker at Rutgers who said he would be pleased to have me \u2013 the flu and all that having decimated his mankind. If it's all right with you, it's all right with him.\n\nI will call Helen Coates Tuesday for the news of Resurrection or Rejection.57 Because the programs are printed and all that sort of thing, I will not insist on special billing, advance payment or the services of a claque.\n\nBest,\n\nJoe Roddy\n\n445. William Schuman to Leonard Bernstein\n\n130 Claremont Avenue, New York, NY\n\n21 March 1960\n\nDear Lenny,\n\nI was so pleasantly surprised by your charming and thoughtful call that I didn't tell you how touched I was that you should want to celebrate my 50th birthday. And I am delighted that you will celebrate it along with Barber's 50th and the 150th of R[obert] S[chumann], not to mention the Dean's (A[aron] C[opland]) 60th. You set me thinking about the program and I greatly appreciate your having consulted me. Actually, of course, any work of mine that you perform would afford me great pleasure. However, since this is a special occasion, I would like very much to be represented by a work with which I feel a close identification with you and one which will be having its own birthday next season too.\n\nIt will be 20 years next season that you and I first went over my Third Symphony with Kouss. But it is not just for sentimental reasons that this would be my choice for the program. On this occasion I would love to be represented by a piece of some weight and one that has been widely accepted. If you prefer No. IV (a tape of this will be sent to you tomorrow or Wednesday), fine. My idea for the program would be as follows:\n\n1. W[illiam] S[chuman] Symphony No. III\n\n2. Barber Violin Concerto\n\nIntermission\n\n3. Robert Schumann Symphony No. IV\n\nIn my view the order could also be 1, 3, 2, or 3, 2, 1. This is the closest I will probably ever come to making up a Philharmonic program. But I won't be the least bit insulted if you cannot carry out my suggestion.58\n\nOnce again, thank you for your vote of confidence. I told Frankie that Felicia will be calling and we will look forward very much to seeing you \u2013 it has been too long.\n\nAffectionately,\n\nBill\n\n446. Larry Kert59 to Leonard Bernstein\n\n6 April 1960\n\nDear Lenny,\n\nHow excited I am at the thought of you in the pit for the overture. And Carol [Lawrence] coming back next week \u2013 well I am flying.60\n\nSecco Records called me yesterday. I told them to add \"some additions\". Is there anything from _Peter Pan_ I could do? Also what would you say if the album said \"Leonard Bernstein presents Larry Kert\"?61 There would be no extra work involved for you, just your permission I guess. They want me to start on the album around the first of May. I sure hope it's a good one.\n\nCan't wait to see you on April 27th. Nope I'm not bugging you.\n\nSincerely with love,\n\nLarry K.\n\n447. Aaron Copland to Leonard Bernstein\n\nGreen Park Hotel, Half Moon Street, London, England\n\n1 May 1960\n\nDear Lensk,\n\nEveryone's been writing me how wonderfully you did the _2nd H_ [ _urricane_ ].\n\nAlso, was sent the write-ups. Naturally I'm tickled pink. Now I hear you're recording it \u2013 so I can hear it, and maybe CB[S]-TV will screen it for me when I get back end of June. Anyho this is just to say denks and denks again. (Did you get my wire? I really _was_ all agog.)\n\nWhen you get this I'll be in Tokio! (care Amer. Embassy.)\n\nHad a nice concert here with the London Symphony Orch \u2013 big house and lots of enthusiasm. The English \u2013 of all ages \u2013 tend to spoil me anyhow, so I like it here.\n\nI imagine Lukas has given you an earful about the Russkys. (Some nice lady piano teacher at the Leningrad Cons. asked after you most warmly.) It was an experience I wouldn't have wanted to miss.\n\nHave fun with your new house. And love to you and Felicia.\n\nAaron\n\n448. Jerome Robbins to Leonard Bernstein\n\n916 North Foothill Road, Beverly Hills, CA\n\n3 June 1960\n\nDear Lenny,\n\nThank you so much for your letter. It came almost in an ESP way, arriving after I had just had a meeting with [David] Selznick telling him that if he went to New York and enlisted you that then I would be interested in the project, otherwise not. I must have heard you writing a letter a few days before.\n\nThe opera idea about the Chassidic ghetto sounds really exciting. Sure I'd love to work on it.\n\nOur Chino has been set for four weeks now, so unfortunately I won't be able to see Nikiforos. I'll keep him in mind if anything changes or turns up for him.\n\nThe garage idea is a good one, but true to Hollywood standards turns out to be a super garage and I seem to be spending most of my energy in pushing walls closer to each other, washing colors out of the sets, and acting like a sheep dog in trying to keep the script in a nice well directed herd aimed for the success it was in New York. I run from side to side barking warning noises about strayed lines, changed lyrics and cut choruses.62 My they're getting tired of me.\n\nKeep well, rest well and I promise not to pluck you out of your island repose unless I know that only you can do what has to be done.\n\nAll love as always.\n\nJerry\n\n449. Jerome Robbins to Leonard Bernstein\n\n916 North Foothill Road, Beverly Hills, CA\n\n16 June 1960\n\nDear Lenny,\n\nThis just got to me. If this is _your_ version of what happened about \"Somewhere\", it's news to me. Sure, we differed on it \u2013 but nothing was played by any orchestra that you did not know of. Nor would I ever be destructive to your music \u2013 I have too much respect for your taste and talent.63 If it _is_ your version (which I doubt) you owe me an apology I think \u2013 and if it isn't, I think you ought to put it straight. I don't think, Lenny \u2013 if I _am_ to be accused of fang and claw \u2013 that I ever used them on you. I fought _for_ \"Somewhere\" \u2013 over Arthur and occasionally Steve's objections.\n\nMaybe I'm jumping the gun, and if so, forgive me \u2013 I value our collaboration and friendship in spite of difficulties we have when working (I'm sure we both have moments of wanting to brain the other). I find it always the most stimulating and valuable of all I've ever had anywhere. Moreover I always felt from you an admiration for my musicality as I've had for your theatrical ideas \u2013 so the item came as a shock. If that bit came from Arthur or Steve \u2013 I can understand it \u2013 but I'd appreciate hearing from you, and getting me or the paper straight.\n\nSigned \u2013 like in Dear Abby,\n\n'Upset'\n\nJerry\n\n450. Dimitri Mitropoulos to Leonard Bernstein\n\n12 July 1960\n\nDear Lenny, Dearest Friend,\n\nI really ask you to forgive me for my silence, especially after your so generous and wonderful visit to me at the hospital, followed by your nice thoughtful gift of Gauloises cigarettes, which, it goes without saying, I enjoyed to the utmost!\n\nWhat made me especially happy was to think and see that I mean something to you. Besides that, I want to assure you that your wonderful and justified development in the artistic musical world is the best gratification for me since the time I first met you in Boston. I remember in one certain instance when I told you, just like a prophet, that you are the _\u00e9lu_ , and certainly at that time I could not even foresee what happened, today.\n\nSo God bless you, dear friend, and keep on progressing, without paying any attention to criticism \u2013 and certainly you know as well as I that the higher you go, the more you will be criticized.\n\nWith many affectionate regards,\n\nDimitri64\n\n451. Leonard Bernstein to Saul Chaplin65\n\nVineyard Haven, MA\n\n18 July 1960\n\nDear Solly,\n\nThe three scores I've received look lovely (though I'm in no shape this summer to look at them microscopically). I wait in dread for the new version of the Prologue!66\n\nI hear it's all getting VERY expensive. That's life in Hollyburg. If it's not expensive, how can it be good? [...]\n\nMuch love,\n\nLenny\n\n452. Leonard Bernstein to Aaron Copland\n\nVineyard Haven, MA\n\n[July 1960]\n\nDearest A,\n\nA greatly belated welcome home! And the fact that I think of you every day, and often twice a day, does not compensate for not having written you. But then, you haven't written me either (except: loved your cable about the _Hurricane_ ).\n\nBut today I must write, because last night I heard the test-pressing of the _2nd Hurricane_ recording which will be out in time for your birthday. It's badly engineered in places, especially when there is choral complexity (the voices are too distant and unclear, the orch. is too present) \u2013 but in general I think you'll be delighted. Of course, it's similar to the TV version, with me narrating, preserving a line or two of dialogue here & there, & cutting Fat's Song plus 2 other small cuts. But mainly I'm writing because I'm so impressed all over again with the music. It is lovely & endlessly fresh: neither the simplicity nor the grandeur stales. Felicia loves it; Jamie & Alexander sing it marvelously by the yard. I hope you like it; it will be our November release on Columbia, along with _Billy_ & _Rodeo_ , making a delightful, gay (though costly) birthday package!\n\nAnd \u00e0 propos birthday. The Pension Fund concert I had planned for your birthday in Carnegie Hall is off, alas; too complicated to get all the participants I wanted (Ormandy, etc.), & perhaps not the right note for Pension Fund events. _But_ , I have a better idea, which is something nobody else can do for your 60th, and that is to make a whole TV show for the kids (the Shell series, originating in Carnegie) on the subject of the Venerable Giggling Dean.\n\nImagine, Judge-Nose for one hour, coast-to-coast! This will happen on the 12th November (Sat.) at noon in Carnegie Hall, & probably be telecast the following day, Sunday afternoon. I want you to participate, do you hear?! Either to conduct a piece, or play the piano, or maybe narrate the Lincoln piece, or maybe conduct same with me narrating. In any case it should be fun. Please say you'll do it, & that there aren't sixty other conflicting homages on the same day.\n\nA couple of weeks ago I watched & heard you conduct the BSO for 90 minutes on WGBH-TV, and it was a joy. Man, you've improved incredibly! Clarity, meaningfulness of beat, ass not extruding. Only problem: die head too much in die score. You must to know die Musik better (or at least trust yourself more). But the big thrill was hearing the Symph #1 again \u2013 what a scherzo! And I had real pleasure out of your colloquy with Walter P[iston]. Even the Diamond _Rounds_ sounded good! Want to succeed me at the Philh.?\n\nI long to see you, & hear about your phenomenal travels. Any significant liaisons? When will we ever meet again?\n\nI sorely miss T'wood also. My love to it, & its inmates. Me, I haven't written a note.\n\nMuch love,\n\nL\n\n453. Aaron Copland to Leonard Bernstein\n\nBerkshire Music Center, Tanglewood, MA\n\n28 July 1960\n\nDear Lensk,\n\nA big pleasure to get your letter. On my one day in N.Y. before coming here I had Roger Englander show me the _2nd H_ [ _urricane_ ]. That was a big pleasure too \u2013 a revival only you could have made so moving. I hope the recording is as good. (Only one reproach: you didn't mention Edwin [Denby]'s name as collaborator.) Goddard had written me about the birthday package and I'm pleased as punch about that _too_. (He's also bringing out [William] Masselos' performance of the _Piano Fantasy_ which I want you to hear \u2013 performance is _superb_ , I think.)\n\nAbout the TV \u2013 Nov. 12; of course I'll do anything you like. Whatever else happens it will give us a reason to \"confer\", i.e., see each other for a change! The only thing I don't want is to be presented as \"grandpa for the kiddies.\" One item you might consider is a selection of songs from the _Old American Songs_. I did them with The Little Orchestra and W[illia]m Warfield 2 years ago. Warfield does them wonderfully and the orchestral versions are fun-things. (You might show the kids the original versions of the songs I worked with.) If you have a quintet of singers the \"Promise of Living\" from _The Tender Land_ works fine. (Or that and the Square Dance can be performed in the choral version with orch.) Etc. Etc... The hard thing will be to illustrate my \"tougher\" side, no?\n\nThe idea of _you_ watching me conduct for 90 min. struck terror... After 8 performances of the Symph #1 on tour I think I can trust myself to peek outside \"die score\"! Anyway I've been getting lots of conducting practice: in 3 months I've had concerts with 7 different orchestras.\n\nWe need you in Tanglewood \u2013 but badly. That's a whole chapter by itself. Our summer is enlivened by Luciano Berio who is guest composer and has stirred things up considerably. But otherwise, routine reigns. Too bad...\n\nAside from music, I had a lovely time in London and in Tokio. The only thing I didn't do was write music, h\u00e9las!\n\nLove to you always,\n\nAaron\n\n454. Leonard Bernstein to Felicia Bernstein\n\nThe Faculty Club, University of British Columbia, Vancouver, Canada\n\n[14 August 1960]\n\nDarling,\n\nJust a quickie:\n\nWe've arrived67 into a glorious kind of Pacific autumn here, with marvelous light over the sound, snow-capped peaks all around, and really _cool_ air. I'm ensconced in this palatial suite at the University, where it is said Queen Eliz. was impregnated. Probably a canard.68 Anyway I'm sleeping in her bed.\n\nDenver was marvelous.69 Now begins the piano nightmare. (You told me so, I know.)70 And I still can't memorize the Bart\u00f3k or _Daphnis_.71 Paresis.\n\nAll is well, the back is great so far, and the press conferences are enormous & ghastly. I've just written David K[eiser]72 at length, offering myself for a long period on the opening\u2013closing basis of the coming season. OK? Call you from Seattle.\n\nAll my love, my darling,\n\nL\n\n455. Leonard Bernstein to Aaron Copland\n\nMark Hopkins Hotel, San Francisco, CA\n\n26 August 1960\n\nDear Aa,\n\nLoved your letter. First instant to catch up: two weeks gone out of seven, seems like two years \u2013 much work, but also glorious fun. All record-breaking crowds, and screaming ones at that. Like Russia. And Hawaii \u2013 now there's a chapter. Just arrived from there last night, utterly spent. Mon dieu, quelle beaut\u00e9!\n\nSan Fran is all gold and blue, & teeming.\n\nI think your idea of the _American Songs_ is great. But we must be careful not to do too much stuff out of repertoire: there's so little rehearsal time, and you know how it's jammed into the busy week. But I'm sure we can swing some of the songs (& maybe an Emily [Dickinson] one too \u2013 hein?) and mebbe your tougher side through the Variations, which I could show at the piano first. Then perhaps the scherzo of the 3rd Symph, and finally a pop thing like _Rodeo_ or _Lincoln_. How would you like to participate \u2013 that is, in what capacity? Conductor or speaker in _Lincoln_ or pianist or speechifier? Don't bother answering; just mull, & we'll confer when I get home. There's time.\n\nMuch love, I miss you.\n\nLenny\n\n26 Aug 60 (I'm 42!!!)\n\nPS. About Edwin:73 There were supposed to be big credits on the screen about his authorship, which were cut for time reasons. I couldn't really mention him, since we didn't do the play. But I had him invited to the event, in the hopes that I could have him stand up for a bow, but he didn't show. In any case, I had a sweet & very thankful telegram from him after the show. (The recording is poorly engineered \u2013 diction, etc. obscure. Alas.)\n\n456. Marni Nixon74 to Leonard Bernstein\n\nThe Mayfair Hotel, London, England\n\n28 August 1960\n\nDear Mr. Bernstein,\n\nJust saw _West Side Story_ for the first time \u2013 here in London \u2013 and it's a tremendous show! I am \"up\" for the voice dubbing for the picture they are making at Goldwyn \u2013 for Maria's singing voice. Can you help me in any way there? Would certainly appreciate anything you think you can do to help.75\n\nHoping you are well, and I understand we might get together for another fling at Pierre Boulez next March. Hope so!\n\nI'm at present vacationing with my husband [Ernest Gold] \u2013 who just completed the score for _Exodus_ and we've been in London for a while & now we will travel on the continent for a while before going back to Hollywood.\n\nAh \u2013 have you ever been on vacation? I suspect you haven't found the time for years now. It's wonderful!\n\nSincerely,\n\nMarni Nixon\n\n457. Leonard Bernstein to Saul Chaplin\n\n20 September 1960\n\n[Telegram]\n\nDear Solly,\n\nBetween Dixie and Berlin I send urgent pleas to consider rerecording some of the tracks I heard in Hollywood. Cool, Mambo and Jet Song are OK although slowish but America is much too slow and Rumble dies of Adagio. Also rhythms wrong at end of Something's Coming. Prologue of course is impossible and embarrassing. Johnny [Green] know[s] of feelings. Please try to redo and correct as much as possible.\n\nLove,\n\nLenny\n\n458. Leonard Bernstein to Nadia Boulanger\n\nNew York, NY\n\n22 December 1960\n\nMy dearly beloved Nadia,\n\nI am so happy that you have accepted our invitation!76 Not only will your visit give great musical joy to a large public, but enormous personal joy to all of us who for so long have regarded you as the unique and adorable person you are.\n\nDon't you think it would be marvelous to play something of one or two of your former pupils?\n\nI look forward to seeing you with keen anticipation; and I was deeply moved by your beautiful letter.\n\nAlways,\n\nLenny (Bernstein)\n\nJoyeux No\u00ebl!\n\n459. Lukas Foss to Leonard Bernstein\n\nUniversity of California, Los Angeles, CA\n\n[?December 1960 or January 1961]\n\nCarissimo,\n\n_West Side Story_ score arrived and I am having a great time with it. Love it more and more. I am _proud_ to do it!! But _when_ can I get the definitive (final) version? And when is that one and only rehearsal?77\n\nI am anticipating yet another Bernstein feast \u2013 on Saturday eve. May 20th. Ojai Festival.78 Don't know if you know about the Festival. It's famous around here. Stravinsky did _Les Noces_ and other works one spring, Copland conducted it another. This year it's mine. I am doing away with the large orchestra which makes for too skimpy rehearsal time, on the meager budget. I am using never more than 30 players, but the best in Los Angeles. Anyway, on that Saturday, the program \u2013 my pride and joy \u2013 will be:\n\n_Anniversaries_ (Me at the piano)\n\n_Rondo for Lifey_ (Divace + Me)\n\n_West Side Story_ jazzed up (Previn and [Shelly] Manne and... the bass)\n\nMasque from _Age of Anxiety_ (with fade out on pianino)\n\n(Andr\u00e9 plays, I conduct \u2013 or vice versa)\n\n20 minutes of non-jazz improv. (my improv. Chamber ensemble)\n\nMozart C major 4 hand Sonata (Andr\u00e9 & Me)\n\nI think that's a gem of a program. The shift from non-jazz to jazz back to non-jazz is subtle and meaningful thanks to your music and its enormous jazz\u2013non-jazz range. Incidentally Andr\u00e9 and Shelly offered to donate their services and seem as pleased with the whole thing as I am.\n\nLove to Felicia. _Did_ you get the record, notes, charts \u2013 that messy little package? Until February, cher ami\n\nLuke McLuke\n\n460. Leonard Bernstein to Sid Ramin\n\n[January 1961]\n\nSid,\n\nMake it for tpts and tbns (4 each available) and percussion, but if you have time, add optional parts for horns and winds.79\n\nBlessings and luck!\n\nLenny\n\nIf you're rushed, first make it for _brass_.\n\n461. Frank Sinatra80 to Leonard and Felicia Bernstein\n\nWashington, D.C.\n\n12 January 1961\n\n[Telegram]\n\nGreetings,\n\nI thought I'd better send you a rundown of activities along the Potomac. First of all, the workaday side of it, I must ask you not to make any outside appointments for the entire day of the eighteenth which is Wednesday if you remember. This will be a tough day of rehearsal and as it behooves us all to put on a really slick show the next night81 I think we should devote this entire day and night to rehearsals. And you know how much I like rehearsals. The morning of the nineteenth will be final orchestra rehearsals and we will start the dress rehearsal at noon.\n\nNow for the social side of this hoedown. Exhibit A will be a supper party that Ambassador Kennedy is giving in honor of the entire cast immediately after our gala performance. This will be black tie for the fellows and something dazzling for the girls. Exhibit B is the inaugural ceremony itself at noon of the twentieth and the parade which follows. Sections of seats have been allotted for us for both events, for those who want to attend. I must ask you to please tell Miss Lovell in my office at the Statler Hilton Hotel whether or not you want to attend these two functions. Exhibit C is the little wing-ding dinner which I am tossing for all of us at seven thirty on the evening of the twentieth. We will also go _en masse_ from this dinner to the Inaugural Ball at the Mayflower, which is pretty dressy for boys and girls. Black tie or white tie diamond and emeralds and all that jazz. Everything is shaping up for something that we all will be remembering for a long time and believe you me I don't think I have ever been so excited.\n\nLove and kisses and I'll be waiting for you.\n\nFrank Sinatra\n\n462. Saul Chaplin to Leonard Bernstein\n\n11 April 1961\n\nDear Lenny,\n\nAfter much discussion with Bob Wise, it was decided that the opening of the picture would be handled in the following manner: the helicopter shots would be kept the way they are now except for the \"motto\" whistles which will precede the long orchestral note. Also, we _will_ add the finger-snaps during the long note. The general feeling here is that the helicopter shots are very unusual and dramatic, and not at all travelogue-y. It is felt that they do progress the audience to the locale of the picture in a most effective manner. In any event, the opening will be kept this way for the preview. If, at that time, anyone feels that the high shots slow the opening of the picture, or interfere with the dramatic content of the \"Prologue\", the proper changes will be made.\n\nYour recital of Jerry's report to you concerning the picture has been on my mind constantly. He has, conveniently, omitted a significant amount of information in his usual vaguely dishonest manner.82 Isn't it interesting that he didn't think it important to mention various large mistakes in the numbers he shot? Isn't it curious that he didn't mention that the Jets are out of sync during a section of \"The Jet Song\"; or that there's half a bar of music missing during the fugue of \"Cool\"; or how I pleaded with him to just \"try\" the faster version of \"Cool\" and how he refused; or how several sections of \"America\" and \"I Feel Pretty\" wouldn't cut together except through Bob Wise's ingenuity; or how he staged \"One Hand\" and was always aware that we were going to use just one chorus and had no _objection_ whatever? I could further mention how he kept encouraging Natalie Wood to sing her own track so that as late as last Friday I was still having trouble from that quarter. I could go on endlessly reciting blunders, which he neglected to mention to you. But, I assure you, we've corrected, and are correcting, all of his mistakes without talking about them. I hope on the other hand, he mentioned how exciting the \"Quintet\" turned out; or how wonderful the \"Taunting\" is; or how touching \"Somewhere\" is; or many other facets of the picture which, I'm sure, slipped his mind since we managed to muddle through without him.\n\nThe reason for this diatribe is quite simple: the fact that Jerry is going to derogate all of us concerns me not at all; the fact that many people will believe what he tells them, since it's fashionable to regard us all out here as sun-loving, bungling, no-talents, also doesn't concern me. What _you_ believe, however, concerns me deeply. I can only reiterate that never in my experience has so much money and care been expended in the making of a movie. If it is not perfect in every detail, my only answer is \u2013 what is? It isn't because we didn't try like Hell. Jerry, of course, is wildly talented. He is also wildly destructive of people and relationships. For me, one doesn't compensate for the other. He is easily the most reprehensible person I've ever known. And so, when the golden day dawns when I will, at last, be freed from _West Side Story_ , I will make it a life's work never again to mention his name or think of him. That, indeed, will be a time for wild celebration.\n\nI'm sorry to have kept you this long at a time when I know how busy you are. I wish you great success with your tour and I'm sure you'll have much better luck with Alaska than I did. (Remember _Bonanza Bound_?).83\n\nLove and sholom,\n\nSol\n\n463. Leonard Bernstein to Felicia Bernstein\n\nHotel New Nagoya, Nagoya, Japan\n\n30 April 1961\n\nDarling,\n\nHow can I describe the last two days? Paradise. At last, really Japan \u2013 once we got out of mad gay Tokyo. About 16 of us were put up at that famous Japanese inn called Minaguchi-ya, where I think I had the most beautiful day and night anyone has ever had. The gardens \u2013 the beauty \u2013 the sea \u2013 the quiet \u2013 the deep charm of Japanese rooms, the smell of new tatami (the straw mats used for carpeting) \u2013 the elegance of simple flower arrangements \u2013 the marvelous food \u2013 and oh, the girls. We were welcomed in a way that made me feel I'd never been really welcomed before. The girls crowd around, laughing, attending, bubbling, dressing & undressing you, preparing your kimono, your bath (oh that wonderful bath of old scoured wood) \u2013 and with none of the artificial gaiety of the Geisha (who embarrass the wits out of me) but with a natural spontaneous joie de vivre & delight in making you happy. I had the Emperor's suite, mind you, & slept in his bed, & had his breakfast (about 17 courses) and it was coincidentally the Emperor's birthday, so Lennuhtt was the Emperor (which is _Tenno_ in Japanese, so now you can call me Tennuhtt). The morning after we visited the famous nearby temple Shuken-ji, a terribly moving Zen sanctuary with the most overwhelming gardens I have ever seen. The entire side of a mountain covered in every green imaginable, spotted with huge red azalea, & pierced by a long, narrow graceful waterfall from top to bottom. I shall never forget the sound of that silence, or the odors, the color, the peace.\n\nAll this is really to say that in two days you would change your whole mind about things Japanese. I know you would. You would adore the food (most of it anyway), the sense of beauty, the natural grace of people and houses. You'd even change your mind about paper windows & walls & screens & doors & mats & sitting on the floor. It is the way to live. I insist on bringing you here some time, without the orchestra and la Belle and Tourel.\n\nNow today that Paradise is all over, as we've moved into another noisy characterless big city, which doesn't even have the fun of Tokyo, but only the ugliness. But from here on it gets good again \u2013 more Japanese style inns, & Kyoto, which I can't wait to see, & Kobe, & _pearls_. I bought me two black kimonos today \u2013 one silk, one wool: something for you, don't dare really. Bought Axel a great boy-doll in honor of Boys' Day which is the big festival tomorrow. It's a Samurai-boy on a horse, in a glass case, & featuring a huge phallus, which I don't know how I'll explain to him. Then visited Nagoya Castle, a breathtaking piece of architecture, and a museum reminiscent of the Hermitage in terms of costumes, prints, & paintings. Now sleep. Then a koto-player is coming for to play the koto. Then sleep. Day off. Thank God. My big nose is still sick, & needs a big rest. All else is great, concerts et al. are smashes. Minimum of Saudeks [Robert and Elizabeth], etc. All under control. Only I miss you terribly.\n\nMy love, my little maid of Orleans, my swan, I miss you. My dearest love to the littles.\n\nL\n\nO, that inn! I had always thought the idea that Japanese made men happy was a commercial notion from Brando movies: but it's true!\n\n464. Felicia Bernstein to Leonard Bernstein\n\n1 May [1961]\n\nHa-ha Rennuhtt (or is it Chi-Chi),\n\nHip! Hip! Your letter arrived and not so chop-chop either \u2013 but then who can figure it out? I feel I cannot waste a minute if this is to reach you at all.\n\nIt was so wonderful to hear from you \u2013 sort of unbelievable since I confess that I still have the childish notion that the Orient is never-never land \u2013 it's hard to picture you all there. I mean like Helen Coates painted on a kaki-mono or Jack Fishburg84 meditating at a Buddhist Temple, or you for that matter being fed by some dainty geisha \u2013 anyway Tokyo sounds disastrous and shatters all my childhood dreams.\n\nNew York, however, is exactly the same as you left it \u2013 the same set, the same cast of characters. There is _nothing_ new on _any_ front. Alexander is in bed with a cough (not serious). Henry is fine and looks beautiful. He was not allowed by the doctor to go to the country since he's still convalescing \u2013 so I took BB & Ellen instead. I've been with the \"liver pip\" for about a week so I was afraid to go by myself. It was to _die_ there \u2013 no dogwood yet, but forsythia, daffodils, jonquils, pinks, blues \u2013 dreamy! I can't bear that you're missing it \u2013 still, I guess you're getting spring in spades there! But somehow it's not the same when it doesn't belong to you.\n\nI'm off to Rochester in a few days \u2013 I've worked hard and feel secure \u2013 I wonder if they are?\n\nDon't buy me a \"simple\" pearl \u2013 I've decided I don't want rings any more!\n\nKiss you long with all my love,\n\nF\n\n465. Nadia Boulanger to Leonard Bernstein\n\n\u00c9coles d'Art Am\u00e9ricaines, Fontainebleau, France\n\n5 May 1961\n\nDear Lenny,\n\nJust receive[d] the score of _West Side Story_ \"at the request of the composer\". Well \u2013 it sounds rather miraculous as I had just ordered it! Too beautiful \u2013 not to be true!\n\n_Merci_ \u2013 I am enchanted by its dazzling nature \u2013 perhaps facility is a danger, but it is enough to be aware of that and follow it.\n\nUntil soon. I often think of you, of the problems and temptations that your gifts give you \u2013 divergent and convergent.\n\nWith my greatest affection to all of you.\n\nNB85\n\n466. Felicia Bernstein to Leonard Bernstein\n\n8 May [1961]\n\nMy darling,\n\nYour wonderful beautiful letter from Nagoya followed me to Rochester \u2013 it arrived just in time for the concert and inspired me to new heights \u2013 in other words it went tickety-boo to the races! Hollenbach86 turned out to be a jolly good conductor, the chorus was wonderful and your old Maid of Orleans did you proud. And here I am back at the store covered with laurels, my ego in top form for at least six months!\n\nWe are all counting the days to your arrival \u2013 such an enormity of events have taken place since you left that I feel it should all somehow quiet down once you are home safe and sound \u2013 so did it!\n\nLove to all \u2013 kisses to you from the littles \u2013 I love you,\n\nF\n\n467. Leonard Bernstein: Stephen Sondheim Acrostic\n\n5 July 1961\n\nS tephen Sondheim is a maker and solver of puzzles:\n\nT he mind's jig-saw, creativity's crossword, and\n\nE specially the heart's cryptologies.\n\nP uzzler-poet of word and note, now puzzled, now puzzling,\n\nH e may on occasion inch apart\n\nE nough to reveal the delicate cracks between;\n\nN ext moment the pieces are magnetized, spring together with a\n\nJ olt of rightness: himself a puzzle, self devised, self-soluble.\n\nS tephen Sondheim loves Christmas: not\n\nO nly for the riddle of giving the precisely definitive gift;\n\nN ot, surely, for the getting of it; but for the warm\n\nD ecembral restatement of remembrance.\n\nH e is compulsively loyal,\n\nE ven to friends disloyal to each other. Finally,\n\nI f you like his words, wait til you hear his\n\nM usic, _qua solutum acrosticon est_.\n\nLeonard Bernstein\n\nJuly 5, 196187\n\n468. Leonard Bernstein to Arthur Laurents\n\n31 October 1961\n\nDear Arthur,\n\nI've been meaning to tell you, ever since I saw the _W_ [ _est_ ] _S_ [ _ide_ ] _S_ [ _tory_ ] film,88 that I had never realized until that moment how much I admired and, yes, even revered your work on the show, and how much we all owe to you. It becomes all too painfully obvious as one sees the line-by-shot destruction of the book by the H'wood exegists (there's no such word!), and the clearest of all is the line, however fine, between whatever art is, and non-art. I just wanted to say a personal Thank you.\n\nLenny\n\n469. Francis Poulenc to Leonard Bernstein\n\nLe Beau Rivage, Lausanne\u2013Ouchy, Switzerland\n\n1 November 1961\n\nDear Bernstein,\n\nGreat news!! The _R\u00e9pons des T\u00e9n\u00e8bres_ are finished. I hope that you will like them. It's very simple (because of the children) but also, I believe, very moving, with nothing decorative like the _Gloria_ and completely internal. It's penitence, but \"poverty is not a vice\" as Markevitch put it in his _Rebus._ For a long time I've wanted to tell you that I went _twice in five days_ to see _West Side Story_ in Paris. I was _fascinated_ , that's the exact word, by everything that you have expressed and suggested. For someone who loves the interval of a sixth\n\nyou'll know how much I liked the big love duet. Bravo!\n\nThe boys have told me that they played the Double Concerto, and recorded it, with you. What a joy! I cannot wait to hear it.\n\nThank you for remaining a faithful friend.\n\nI embrace you, dear Bernstein\n\nFr. Poulenc89\n\n470. Leonard Bernstein to Igor Stravinsky90\n\n5 January 1962\n\nDear Ma\u00eetre,\n\nIt is our pleasure to hope to make a double celebration of your coming birthday. First we want to devote a special Pension Fund program to your works; this would occur on the 21st of March and be a gala celebration. Then, on the following Saturday, the 24th, we are planning to play another program of your music for a Young People's Concert, and this one would be nationally televised a month or so later.\n\nRemembering your charming cooperation with us on a previous television program, and recalling your kind telegram last season after our _Oedipus Rex_ program, I am encouraged to ask you again if you would not join us on this television show as well. Perhaps you could say a word of greeting to the young people (who, as you have said, understand your music better than anyone!) \u2013 and then, if you wish, conduct the orchestra in some final work, like part of _Petrouchka_ , or whatever you would like.\n\nI don't have to tell you what an honor it would be for us all to have you present (at both occasions, preferably, since they are so close together), besides helping us, reciprocally, to honor you in the way we would like to. To say nothing of my personal delight.\n\nYours always,\n\nLB91\n\n471. Igor Stravinsky to Leonard Bernstein\n\n1260 North Wetherly Drive, Hollywood, CA\n\n11 January 1962\n\nMy dear Leonard,\n\nI was delighted to receive your letter of January 4. I was also pleased to hear of your plans to perform my music during the spring season.\n\nAlas, I cannot appear on television myself before I complete _Noah_92 (when will that be?): on this point my television contract is very strict. And, two times also, I can't be in New York in March. The best I can do is send you my fondest greetings and to hope that all goes well. Perhaps, too, you would be kind enough to greet the children from me.93\n\nCordially,\n\nIgor Stravinsky\n\nP.S. Franz Waxman has urged me to try to persuade you to conduct a concert of my music in his (mid-June) Los Angeles Music Festival. I would be very happy if you would accept, of course, though I know you must be busy at that time (as I am; I can't participate myself).\n\n472. Nadia Boulanger to Leonard Bernstein\n\n1 Sutton Place South, New York, NY\n\n20 February 1962\n\nDear Lenny,\n\nWhat can I say to you? Words seem feeble [...] and I don't know how to tell you how grateful I am, but that must not prevent me from trying. Your spontaneity, your affection touch me deeply \u2013 and after this very moving week, for so many reasons, passed so quickly, alas.94 I drown \u2013 yes, I drown in the memory of _ma Petite_ ,95 hoping for you to receive a little of all that you give \u2013 and a little of the indefinable joy which gives peace to the heart, to life and to the spirit. And at the moment when this mystery is about to bring a new life into your existence I pray to God for [her] and for both of you.96\n\nRead what isn't written, and feel all that goes to you, from the bottom of my heart.\n\nNadia97\n\n473. Fritz Reiner to Leonard Bernstein\n\nRambleside, Weston, CT\n\n5 March 1962\n\nDear Lenny,\n\nCarlotta & I are very happy over the safe arrival of Nina Maria and hope that we shall have the opportunity of seeing her \u2013 as well as the rest of the family.\n\nI felt most unhappy at missing my concerts with the Philharmonic but was forced to give in to the doctor.\n\nWe are being very quiet and getting a good rest before leaving for Chicago on the 16th. Will be back for the rites of spring in Connecticut and hope that your plans are going to include Redding plus a visit to Rambleside.\n\nAffectionate greetings to all and a welcoming kiss to the new Princesa.98\n\nFaithfully,\n\nFritz\n\n474. Rudolf Bing99 to Leonard Bernstein\n\nHotel Sacher, Vienna, Austria\n\n18 June 1962\n\nDear Lennie,\n\nApparently we cannot come to an agreement with your lawyer's requests which I consider totally unreasonable.\n\nI _have_ fulfilled my promise \u2013 I _have_ a signed agreement with Zeffirelli! It is unfair and unreasonable to expect me to release you if Zeffirelli dies or breaches his contract (both of which I hope and trust are only remote possibilities). In such case I gladly agree to _consult_ with you on a substitute but the decision must remain the Management's \u2013 unless you agree to run the Met! Really I feel this has now gone on long enough \u2013 even beyond contract there should remain an ounce of mutual trust and confidence \u2013 so please sign now and I won't worry. I have no intention of killing Zeffirelli and substituting Karajan as Director! You got all you want \u2013 so now please give me what I want: Bernstein!\n\nThanks and regards,\n\nRudolf Bing\n\nNo further \"clause\" is needed \u2013 attach this note to the contract! Bing100\n\n475. Louise Talma101 to Leonard Bernstein\n\nMacDowell Colony, Peterborough, NH\n\n4 July 1962\n\nDear Lenny,\n\nWe miss you just terribly. There's been no real gaiety since you left. We've had a couple of games of anagrams, but without you there's no excitement. You're one of the blessed ones who make everything they encounter come alive. It's a rare and precious gift, and I wish you a long enjoyment of it.\n\nA letter from Thornton [Wilder] contains a message for you: \"Tell Lennie I know all about the Kaddish because it's in _Finnegan's Wake_ , and give him my uproarious regards.\" He also says: \"It sounds as tho' you were a congenial crowd \u2013 the 'Round' gives earnest of that. Anyway, with you and Lenny there the tone is set is How to be civilized though an artist.\" I quote the sentence exactly, punctuation, or rather the lack of it, capitalization and all.\n\nHave a wonderful and refreshing time in Spain.\n\nLove from all of us,\n\nLouise\n\n476. Leonard Bernstein to Adolph Green and Phyllis Newman\n\nBarcelona [written on the headed paper of the ocean liner _Leonardo da Vinci_ of the Italia Line]\n\n3 August 1962\n\nDarling Greens,\n\nDoes this notepaper look familiar? Do you suddenly feel queasy at the sight of it? Does it bring back gorgeous salons filled with square brown chairs, Doman & Pythias, Fancy Hat balls, ping-pong, the Lido Bar? Ah, the beautiful past we have shared!\n\nAnd then, & then, you did Capri, which I trust was heavenly. And I did the highlife of all time \u2013 two weeks of Princesses & Maharanees & phony Barons, parties without end, villas to make you gasp. I was the kid of the moment: it was all insane, ignoble, absurd, & vastly entertaining. But I wouldn't want to live there.\n\nI drove here yesterday, through Provence. God, what charm and beauty! Felicia has been in Paris for a week buying out Chanel: she & Mike join me here any minute now, & then we tootle off as tourists in this wild heat. I'm tired but strangely exhilarated by this feeling of holiday, and I'll be home the 15th to rest and to hug you both. A big kiss to Adam and God bless your new home.\n\nUn abrazo fuerte,\n\nLenny\n\n477. Karl B\u00f6hm102 to Leonard Bernstein\n\nHotel Alden, New York, NY\n\n30 November 1962\n\nMy dear friend Bernstein,\n\nAt the conclusion of my guest appearances at the Philharmonic Hall with your magnificent orchestra,103 I am sending you my heartfelt thanks for your wonderful cooperation and for making it possible for me to be the first \"foreign\" conductor to appear at the new hall with the New York Philharmonic.\n\nEverything was just perfect, and the members of your orchestra are musicians of the highest caliber. I felt at home with them right from the start of the first rehearsal, and I am very grateful to every one of them for their assistance, help and attitude.\n\nMy thanks also go to you personally for your sentiments, and I shall never forget the way in which you behaved when I was very sick in Vienna, after a dangerous eye operation. Then I experienced that you are a real and great human being.\n\nWith best wishes to you, Mrs. Bernstein and your children from Mrs. B\u00f6hm and\n\nYours very sincerely,\n\nKarl B\u00f6hm\n\n478. Morton Feldman104 to Leonard Bernstein\n\n337 Lexington Avenue, New York\n\n7 January 1963\n\nDear Lenny,\n\nThe score you have of _Structures_ underwent surgery last summer. C. F. Peters will send you the new version later in the week.\n\nAt the Stockhausens New Year's Eve. After drinking in the new year he announced \"and now we will have some music.\" Most of the guests' eyes lit up. They thought they were going to twist. He then went to the phono[graph] and played two hours of Stockhausen, Foss and Feldman.\n\nFondest regards from your non analytic\n\nMorton\n\n479. Leonard Bernstein to David Diamond\n\n10 January 1963\n\nDear D,\n\nThis isn't exactly the moment for catching up on ages, literally, of Riverflow, but I have to write you today to say that Lina A[barbanell] died on Monday (or rather Sunday, I think, the 6th). We were all shocked: she had gone into hospital for removal of a tumor, and couldn't take it. Poor great gallant lady. 84 years old! Of course Marc is all but destroyed, and I think he'd deeply appreciate some word from you.105\n\nA is born, B dies, C is in agony, D has some joy, E is humdrum, & we are all of them. I have not yet finished my 3rd Symphony ( _Kaddish_ ) which will be _something_ when and if it gets written. I had hoped by now to have it complete. Alas. Once I have finished it, I can rest in peace: it is my Kaddish for everybody. Last week had joy in it: Mahler #5. Glorious. This week, R[oberto] Gerhard's #1. Next week, East Lynne. \u00c7a continue. Newspaper strike. Jamie has chicken pox. Alexander is a sensitive dream; Nina is funny & bright. Felicia is brave and a little tired. I am exhausted, and off to bed.\n\nLove to you all, and a very happy '63.\n\nL\n\n480. Leonard Bernstein to Olivier Messiaen\n\nNew York, NY\n\n18 March 1963\n\nCher Ma\u00eetre,\n\nIt is curious to be writing to you after some _thirteen_ years \u2013 the year of the first performance of _Turangal\u00eela_. But I think of you very often, and I keep up to date with your music. Last season we had the great joy of presenting the _Trois Petites Liturgies_ of which we have made a record for Columbia to be released next year. I hope that you will be pleased with it.\n\nToday I am writing to you on the subject of your impressive work _Chronochromie_.106 Despite my best intentions, it will not be possible to give this enormous work with the rehearsals that are allowed in a typical week of the season; but I believe that it would be possible to perform just the final part on its own, separately (i.e. the \u00c9p\u00f4de and Coda). Would that be blasphemous and meddlesome? It seems to me that this part, on its own, would be very effective in a concert, and very moving. If you have objections to this, do not hesitate to let me know. My address: 895 Park Avenue, New York, 21.\n\nWith my warmest good wishes,\n\nLeonard Bernstein107\n\n481. Olivier Messiaen to Leonard Bernstein\n\n230 rue Marcadet, Paris, France\n\n26 March 1963\n\nCher Ami,\n\nI was extremely touched by your letter, and thank you with all my heart for it. I, too, think of you often, and \u2013 if you remember \u2013 I have applauded you several times after your marvelous concerts in Paris.\n\nThank you a thousand times for the _Trois Petites Liturgies_ on record for Columbia. Who are the solo Ondiste and pianist? And who is conducting? Is it _you_? (During my trip to Japan, Seiji Ozawa also spoke to me of a performance of the _Trois Petites Liturgies_. Was that the same or another one?)\n\nYour _name_ and the first performance at Boston in 1949 are at the head of the magnificent edition that Durand has given the _Turangal\u00eela-Symphonie_.\n\nNow let me reply on the subject of _Chronochromie_.\n\nThe division of the work into seven sections (Introduction \u2013 Strophe I \u2013 Antistrophe I \u2013 Strophe II \u2013 Antistrophe II \u2013 \u00c9p\u00f4de \u2013 Coda) is a formal division. But there is no break between these sections, and the work forms _a whole_ , _without interruption_. The \u00c9p\u00f4de uses only 18 solo string instruments, in 18 real parts, namely 6 1st violins, 6 2nd violins, 4 violas, 2 cellos. That's not interesting except by contrast with the rest of the work. The rest is written for a very large orchestra, with a solo xylophone and marimba (plus a set of 25 tubular bells for which the part is rhythmically difficult). Finally, all of the work, and above all the two Strophes (where the harmonies of the strings and the woodwind counterpoints of birdsong must underline the _rhythms_ and the _durations_ of the metallic percussion instruments by _coloring_ them) justify the title: _Chronochromie_ , that is to say: _Color of Time_.\n\nWould you like to wait a little while? _Chronochromie_ is entirely engraved, the plates are at the printer at this moment \u2013 and the work will appear in large score and pocket score here in two months, around 15 June, from Leduc, publisher, 175 rue Saint Honor\u00e9, Paris (1er), France.\n\nI will send you a score at that time, and you will see the music for yourself.\n\nThank you again for your letter, and all my best wishes.\n\nOlivier Messiaen108\n\n482. Leonard Bernstein to David Diamond\n\n2 May 1963\n\nDear David,\n\nAs usual, these words are penned in haste. Will there ever be no haste? I'm coming to think that only children, who believe themselves immortal, are blessed with time.\n\nI've been working on Brandeis for you. (California is out.) I suspect that [Arthur] Berger still harbors some ancient grudges; but I approached President Sacher directly, and he informs me that he would like you to come for an interview when you are here. At least it's a step.\n\nDeaths are frequent. Felicia's mother died 2 days ago in Chile, & poor F. has been there for an agonizing week, watching her mother die. A nightmare. And yesterday we lost that angelic Nat Prager, our 2nd trumpet player \u2013 after 34 years of glorious and uncomplaining service.\n\nDeath and spring. I am back with the orchestra again, and love it. The _Kaddish_ is still unfinished, and its premiere is now set for December in Israel. I don't know if it'll ever be ready.\n\nI must fly now to conduct. Bless you, & let me know how spring is in Florence.\n\nLenny\n\n483. Morton Feldman to Leonard Bernstein\n\n337 Lexington Avenue, New York, NY\n\n19 June 1963\n\nDear Lenny,\n\nI had a talk with Jack Gottlieb on the phone this evening, and he gave me some of your thoughts on _Structures_. I was struck by the fact that you felt a lack of \"rhythmic interest\" in this piece, because what I was actually after was an atonal rhythm, or, more precisely, no rhythm.109 It is the juxtaposing of various weights of sound which make for the movement, rather than any rhythmic design. This is equally true of _Out of Last Pieces_ and is in fact one of the basic ideas throughout my work.\n\nThe key to my music is that I want to resolve each piece into one overall color (regardless of how the piece is notated). Because of this, what makes for an \"interesting\" composition for someone else has no place in my thinking.\n\nThat's that \u2013 and what are _you_ doing these days?\n\nMorty\n\n484. Leonard Bernstein to Louis [?]\n\n25 June 1963\n\nDear Louis,\n\nI have finally listened to the recordings of ancient pianists & composers that you so kindly sent me. It has been a ball! Grieg performing his _Papillon_ like a young lady just out of conservatory, Busoni stuttering his octaves, Ravel heavy and rhythmically obscure, [Teresa] Carre\u00f1o running out of gas in the Liszt Rhapsody, [Vladimir] de Pachmann setting an all-time record for ritards at the end of Chopin's C# minor waltz, et al, et al \u2013 and all marvelously authentic, surprising, other-planetary, incredible. It is a thrill to hear these records: we not only extend our knowledge of past pianistic styles, but we gain a fresh view of our own age. And not only pianistically; this glimpse into the past, to the thoughtful observer, becomes nothing less than a revelation of the present!110 I thank you for sending it to me.\n\nAffectionately\n\n[Pencil draft, unsigned]\n\nP.S. Congratulations on the new baby.\n\n485. Leonard Bernstein to David Diamond\n\n24 August 1963\n\nDear DD,\n\nWelcome! I had no idea you were already here: not a word have I had, or Aaron, or anyone. Your birthday card was the first modest sign. You were sweet to send it, to remember. How goes it? Plans? Medical matters?\n\nI am at this second in the hurried grip of time, for a change: my last day in the country. Tomorrow (my birthday!) I rehearse the ork all day for the tour, which starts on Tue[sday]. Gone for 4 weeks to H'wood & back. But I wanted at least to say hello-&-have-a-good-visit-and-a-successful-one before I vanish for a month.\n\nBest news is that I have finished _Kaddish_ this summer. That's all I did. I had virtually no vacation. My text still needs cleaning up \u2013 and a short section or two remains undecided; but actually it's a piece! My first in 6 years \u2013 my first _concert_ piece in 9 years! I can't wait for you to see it. Will you be in town late Sept? (when I return?) Let Helen know: on me \u2013 she can send you my itinerary.\n\nGood health, love to you and Sabina.\n\nL\n\n486. John Cage111 to Leonard Bernstein\n\nStony Point, NY\n\n17 October 1963\n\nDear Lenny,\n\nTwo points. First, I am very grateful to you for having decided to present my work and that of [Morton] Feldman and [Earle] Brown before your audiences. We all admire your courage in doing this at the present time, for actual hostility toward our work is still felt by many people.\n\nSecond, I ask you to reconsider your plan to conduct the orchestra in an improvisation. Improvisation is not related to what the three of us are doing in our works. It gives free play to the exercise of taste and memory, and it is exactly this that we, in differing ways, are not doing in our music.\n\nSince, as far as I know, you are not dedicated in your own work to improvisation, I can only imagine that your plan is a comment on our work. Our music is still little understood and your audiences, for the most part, will be hearing it for the first time. It would seem best if they could do so without being prejudiced. I admired Aaron when he presented my work at Tanglewood, letting the audience know beforehand that, though he didn't share my views, he felt the music, since it was seriously written and had found a following among composers, performers and audiences around the world, had a right to be heard attentively. I feel the opposite way about Smallens who, I am told, after conducting a first performance of Webern for the League here in New York, turned toward the audience and joined them in derisive laughter.\n\nSurely there must be some less provocative way to conclude the program, one which will leave no doubt as to your courage in giving your audiences the music which you have chosen to present.\n\nWith best wishes and friendliest greetings,\n\nJohn Cage\n\n487. Leonard Bernstein to John Cage\n\n[October 1963]\n\nDear John,\n\nYour letter astonishes me. What, for example, makes you think that our orchestral improvisations should in any way constitute a \"comment\" on your work and that of your colleagues? What, again, gives you the idea that everything in this part of the program must be confined to the realm in which you work? The overall idea is _Music of Chance_ and there are chances and chances in your work as well as that of Brown and Feldman and _as well_ in total improvisation. We are trying to have as comprehensive a look at the aleatory world as is possible in half a complete program; and it seems clear to me that improvisation is an essential part of such a look. And, finally, how can you deny that your music enlists \"free play of taste and memory\" when you write for an orchestra that may or may not play at any given time, and if it _does_ play, render approximations?\n\nIf it will make you feel any better, I shall be happy to play the improvisation _before_ your work, thus avoiding the tendentious notion of its being a final comment on the preceding music. I hope that this will alleviate your concern, and prove to you the integrity of my intentions.112 Most cordially.113\n\n488. Claudio Abbado114 to Leonard Bernstein\n\nBerlin\n\n28 October 1963\n\nDear Maestro,\n\nI want to thank you by heart for everything I learnt by you during the weeks that I spent in New York with the Philharmonic. What I learnt from your rehearsing, from the musical and human point of view, I tried now to actuate in the rehearsals of my last concerts. So I succeeded in finding always a human contact with the orchestra, forgetting everything of the dictatorial way that I had in past years. The results have been wonderful, and I could musizieren in a completely new way in the last concerts in Rome, Venezia and Berlin. For all here in Berlin, also because the orchestra is better, I have been particularly happy. With this orchestra, with whom I will conduct also tonight and tomorrow, I have been invited for a European tourn\u00e9e.\n\nComing back to New York, I should be very happy to have the opportunity to speak with you about your music and about your wonderful interpretation of II [Symphony of] Mahler.\n\nGrazie ancora ed arrivederci presto!\n\nClaudio Abbado\n\n489. Mary Rodgers to Leonard Bernstein\n\n24 November 1963\n\nFor many years \u2013 six to be exact \u2013 I've been your highly expendable \"children's expert.\" We all know you don't need a children's expert, and you don't need me for early morning joke telling either. But for me, it's a nice job. I love it. I love you.\n\nIt's occurred to me that I've never bothered to mention this before \u2013 and now seems a good moment.\n\nWhat Kennedy115 did for the affairs of the world, you do for the heart of the world. It seemed to me, tonight, that you two are not (were not) unalike \u2013 in courage, in conscience, in warmth and in purpose.\n\nI'm grateful, _very_ grateful, that there is one of you left.\n\nBless,\n\nMary\n\n490. Leonard Bernstein: Talk given at the \"Night of Stars\" Memorial to President Kennedy at Madison Square Garden\n\nNew York, NY\n\n25 November 1963\n\nMy dear friends,\n\nLast night the New York Philharmonic and I performed Mahler's Second Symphony \u2013 the _Resurrection_ \u2013 in tribute to the memory of our beloved late President. There were those who asked: Why the _Resurrection_ Symphony, with its visionary concept of hope and triumph over worldly pain, instead of a Requiem, or the customary Funeral March from the _Eroica_? Why, indeed. We played the Mahler Symphony not only in terms of resurrection for the soul of one we love, but also for the resurrection of hope in all of us who mourn him. In spite of our shock, our shame, and our despair at the diminution of man that followed from this death, we must somehow gather strength for the increase of man, strength to go on striving for those goals he cherished. In mourning him, we must be worthy of him.\n\nI know of no musician in this country who did not love John F. Kennedy. American artists have for three years looked to the White House with unaccustomed confidence and warmth. We loved him for the honor in which he held art, in which he held every creative impulse of the human mind, whether it was expressed in words, or notes, or paints, or mathematical symbols. This reverence for the life of the mind was apparent even in his last speech, which he was to have made a few hours _after_ his death. He was to have said: \"America's leadership must be guided by learning and reason.\" Learning and reason: precisely the two elements that were necessarily missing from the mind of anyone who could have fired that impossible bullet. _Learning and reason_ : the two basic precepts of all Judaistic tradition, the twin sources from which every Jewish mind from Abraham and Moses to Freud and Einstein has drawn its living power. Learning and reason: the motto we here tonight must continue to uphold with redoubled tenacity, and must continue, at any price, to make the basis of all our actions.\n\nIt is obvious that the grievous nature of our loss is immensely aggravated by the element of violence involved in it. And where does this violence spring from? From _ignorance_ and _hatred_ , the exact antonyms of learning and reason: those two words of John Kennedy's were not uttered in time to save his own life; but every man can pick them up where they fell, and make them part of himself, the seed of that rational intelligence without which our world can no longer survive. This must become the mission of every artist, of every Jew, and of every man of good will: to _insist, unflaggingly_ , at the risk of becoming a repetitive bore, but to _insist_ on the achievement of a world in which the mind will have triumphed over violence.\n\nWe musicians, like everyone else, are numb with sorrow at this murder, and with rage at the senselessness of the crime. But this sorrow and rage will not inflame us to seek retribution; rather it will inflame our art. Our music will never again be quite the same. This will be our reply to violence: to make music more intensely, more beautifully, more devotedly, than ever before. And with each note we will honor the spirit of John Kennedy, commemorate his courage, and reaffirm his faith in the Triumph of the Mind.\n\n491. Walter Hussey116 to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n10 December 1963\n\nDear Mr. Bernstein,\n\nI hope you will forgive me for writing to you and will not think me presumptuous. I did have the pleasure of meeting you briefly in New York when you kindly allowed me to attend one of your rehearsals, at the request of my friend, Dr. Chuck Solomon. But you will not remember this.\n\nThe choirs of Chichester, Salisbury and Winchester Cathedrals combine for a short festival each year which takes place in the three Cathedrals in turn. I enclose copies of the programmes for the last two years to give you some idea of the sort of thing it is. It has proved extraordinarily successful and I think it will be fair to say that it reaches a very good musical standard. Naturally, it is concerned to a great extent with the wealth of music written for such choirs over the centuries, but I am most anxious that this should not be regarded as a tradition which has finished, and that we should be very much concerned with music written today.\n\nThe Chichester Organist and Choirmaster, John Birch, and I, are very anxious to have written some piece of music which the combined choirs could sing at the Festival to be held in Chichester in August, 1965, and we wondered if you would be willing to write something for us. I do realize how enormously busy you are, but if you could manage to do this we should be tremendously honoured and grateful. The sort of thing that we had in mind was perhaps, say, a setting of the Psalm 2, or some part of it, either unaccompanied or accompanied by orchestra or organ, or both. I only mention this to give you some idea as to what was in our minds.\n\nI have always been most eager to do anything I possibly can to foster the ancient links between the church and the arts. Before I came to Chichester when I was in Northampton, I got Henry Moore to carve a Madonna and Child, and Benjamin Britten to write a cantata. I am most eager to carry on this work and it would be a great pleasure and encouragement if you felt you could help us. Please do. We would of course be only too happy to pay a fee to the best of our resources.\n\nI shall be of course delighted to give you any further help I can or information you may require, and again may I express the hope that you will forgive me approaching you.\n\nYours sincerely,\n\nWalter Hussey\n\n492. Iannis Xenakis117 to Leonard Bernstein\n\nBerlin, Germany\n\n7 January 1964\n\nDear Mr. Bernstein,\n\nI want to thank you very much for including me in your concerts in New York. It made me very happy and proud. I think that the piece except its creation by H[ermann] Scherchen had its first real performance on January 64 by you!118\n\nUnder your impulse the series of these concerts brings New York at the head of the cities who care for new music, because of the real popular character you give to them and because of the first quality of the orchestra and of the performers.\n\nI wanted to write you one month ago, but I fell ill and had to support a heavy and painful operation in Paris. Now I can write you, being back in Berlin (I have received a Ford Foundation award and I am an artist in residence of Berlin for one year).\n\nI wish you all the most brilliant success for your effort and that other concert organizations take you as a model.\n\nThanking you again.\n\nYours sincerely,\n\nXenakis\n\nP.S. If there is any time left to you, I would appreciate very much to have your opinion on my piece. I'll send you my book _Musiques formelles_ in French119 (Mr. Karl Haas told me that you speak French) hoping that you'll enjoy it.\n\n493. Harpo Marx120 to Leonard Bernstein\n\n15 January 1964\n\nLeonard,\n\nOn one of your children's concerts I would love to conduct my version of the Haydn Toy Symphony which runs seven minutes. My salary to go to Musicians Aid Soc of N.Y. I have done the symphony on several occasions & most recently with the Philadelphia Symphony using little children to play the toy instruments in the last part. I will be in New York February 14 for two weeks \u2013 do you think you could set it up for that time?\n\nAs I don't read music, in order to stay in good standing with Local 47 I do all my corresponding on score-paper.121\n\n494. Leonard Bernstein to Shirley Bernstein\n\n[New York]\n\n25 January 1964\n\nDarling Hilee,\n\nThis is the first second I've had to write in months. I've been trying to call you for days to \"be\" with you at the death of Marc:122 but no answer, no answer. I called you from Paris (en route home from Israel, having just been told that our plane _would_ stop in London) and then again from the London airport, but no answer, no for an answer. And the wretched luck was that my plane _to_ Israel decided _not_ to stop in London. And so on, for bad communications.\n\nIt's an open season on Kaddish, all right. The President. Marc. I take it you've read substantially what we've read: first an auto accident; then a new story about being beaten by three sailors; then the blessed word _robbery_ reiterated by the Consul in Martinique, & now by the police.\n\nWe pray the story gets no murkier \u2013 or should I say clearer? We are all shocked and miserable, as you must be, & I wish we could be together, at last, at last.\n\nThe _Kaddish_ itself gets its comeuppance in Boston this week, Friday the 31st. I wish you could be there. It will be a regular reunion: B[urton] B[ernstein]'s birthday and all. Felicia is there already, rehearsing. I'm staying out of it 'til the last minute: M\u00fcnch is in love with the piece, but scared witless, & can't beat it (7\/8, etc.) the chorus... Jenny T... oy. Maybe you're wise to miss it.\n\nMarc is dead, & I've lost an arm. Felicia can't stop crying. Come home, darling Shirley \u2013 we want you with us! It's been far too long, too many distances, interruptions, silences. Please come home.\n\nI love you as always \u2013\n\nL\n\nDo you realize we have loved Marc for 25 years?\n\n_Kaddish_ : they loved it in Tel-Aviv, but it may never go in English. I'm nervous.\n\n495. William Schuman to Leonard Bernstein\n\n27 January 1964\n\nDear Lennie,\n\nI couldn't get you by phone to tell you several things.\n\nIn the first place, yesterday's performance of the _Eroica_ was undoubtedly the best, the most exciting and the most moving that I have ever heard. Frankly I had not expected to stay tuned in after the Var\u00e8se. You wouldn't let me get back to work.\n\nI wanted to speak to you about Marc. Actually, you two were so close, I felt the need of making a condolence call. Schuyler Chapin tells me that you and Dave Oppenheim are planning a memorial program this spring. Of course this is wonderful, and I am wondering whether you could consider doing one or two pieces from _Sacco-Vanzetti_.123 And here, I keep wondering whether Marc finished the vocal score. In my last conversation with him \u2013 just one year ago this month \u2013 I gathered that mostly it was done. If this is the case, could not the work be completed and orchestrated by another? If this is a possibility, I am very much afraid that you are the only one who can do it. It's a thought.\n\nAnd mostly, I called to wish you with all affection a stunning success with the new work this week.124\n\nYours,\n\nBill\n\n496. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n10 February 1964\n\nDear Dr. Bernstein,\n\nThis is splendid news! I am indeed delighted to hear that you will write something for our Festival. It will be a great privilege for us all and I can assure you that everyone will do their utmost to do justice to the work.\n\nYes, by all means change from the second Psalm to something more familiar if you wish. We shall be very happy to leave it to you. And as regards the time, of course, we must leave this also to you: but the joint Festival with Winchester and Salisbury goes in rotation and so is only held here once in three years. It is to be held here at the end of July, 1965, and so as you can well imagine we are very much hoping that it may be possible to perform the work then. I shall be sad for it to receive its premier[e] elsewhere! However, I am so proud and pleased that you will write something for us that it would be unfair and presumptuous to try and press further.\n\nI will try to send you within a few days particulars as to the sort of numbers that will be available in the combined choirs, in case this should be of any assistance to you. Please let me know if there is anything further I can do.\n\nAgain, my warmest thanks,\n\nYours sincerely,\n\nWalter Hussey\n\n497. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n14 August 1964\n\nDear Dr. Bernstein,\n\nWe are all tremendously looking forward to the setting of the second Psalm which you kindly said you would write for us for our three Choirs Festival next year.\n\nThis year's festival has just taken place at Salisbury and I thought perhaps you would be interested to see what has been happening, so I am sending you a copy of the programme. A tremendous lot of people turned up and it proved very successful \u2013 I think the performance of Britten's _Cantata Misericordium_ was quite outstanding.\n\nOur Choirmaster and Organist has given me one or two particulars which he thinks might be helpful to you. The string orchestra will probably be the Philomusica of London, a first rate group. In addition, there could be a piano, chamber organ, harpsichord and, if desired, a brass consort (three trumpets, three trombones). It is not really possible to have a full symphony orchestra for reasons of space and expense and the fact that the combined strength of the three cathedral choirs is about 70 to 74 (all boys and men).\n\nThe Festival was founded a good while ago, but then lapsed for about thirty years and has recently been re-started with great success. I think it has throughout its time been of great value to English church music as a means of hearing works of large scale, impossible for any single Cathedral choir, but I am certain it must also provide new works in new idioms to keep the tradition really alive. I hope you will feel quite free to write as you wish and will in no way feel inhibited by circumstances. I think many of us would be very delighted if there was a hint of _West Side Story_ about the music. I hope you will not mind my writing like this, but I talked of it with Chuck Solomon125 when he was here recently, and he said I was certainly to say it to you.\n\nOnce again let me say how enormously grateful I am to you for saying that you will help us in this way. It will be exactly what I should wish for when the Festival is here at Chichester next year.\n\nYours sincerely,\n\nWalter Hussey\n\n498. Leonard Bernstein to Aaron Copland\n\n19 September 1964\n\nSonnet on receiving an honorary doctorate with Aaron Copland126\n\nThis day, my will demurring, I grew old.\n\nI could have written memoirs on that stage\n\nFor the first time. A long wave had unrolled,\n\nAnd beached me, spent with swimming and with age.\n\n_Doctor honoris causa_. First for him,\n\nA craggy cedar planted by the sea\n\nSince Adam. Then they called on me to swim\n\nAshore, and simulate that salty tree.\n\nA poor impostor, I, Not even brave,\n\nA plotter with no plan, and less than bold.\n\nThey fished me, red-eyed flounder, from the wave,\n\nWounded, rigid, open-mouthed, and cold.\n\nWith velvet bait they plucked me from the sea\n\nAnd dropped me, panting, near a cedar tree.\n\nMuch love,\n\nL\n\n499. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n22 December 1964\n\nDear Dr. Bernstein,\n\nI do apologize for burdening you with another letter. I have a horrid fear that you will be regarding me as an arch nuisance but I am most eager that we should have the work which you promised for our combined Choirs in time to learn it and rehearse it properly before the Festival. May I say again that I hope you will feel entirely free to write your setting exactly as you wish to. I hope you would not, at any rate on our behalf, feel any restrictions from the point of view of tradition or convention. The work would not be performed during any sort of religious service and I firmly believe that any work which is sincere can suitably be given in a cathedral and to the glory of God. No one feels more proud and grateful than I do of the tremendous value of the tradition of English Cathedral music, but I am sure that it is a very good thing for something like our Three Cathedrals Festival to have a sharp and vigorous push into the middle of the 20th century, and if you should feel inclined to write something that would do this I am sure nobody could do it better and we should be most happy and grateful.\n\nIt would be a great help if I could know the title and description of the work within the next six weeks as our preliminary announcements and publications will be appearing very soon after that.\n\nAgain, do let me apologize if I seem tiresomely importunate, but Chuck Solomon assures me that we shall get the work and I am most anxious that we should have it in time to prepare it properly. The prospect is a tremendous thrill to us all. I can assure you that the musicians will do their utmost to do [it] full justice, and I do not believe they will fail. I am more grateful to you than I can possibly say.\n\nI do not suppose there is any chance of your being in England at the end of July? We should of course be delighted to welcome you to Chichester and I believe that in the place itself and the Festival Theatre as well as the Music Festival you would find much to enjoy.\n\nWith all good wishes,\n\nYours sincerely,\n\nWalter Hussey\n\n500. Leonard Bernstein to Mary Rodgers\n\n[New York, NY]\n\n[1964]\n\nDear Mary,\n\nI just found a most moving note from you, dated a year ago, apropos the assassination [Letter 489]. In all the grief of those days a lot of mail went unanswered. But this one cannot go unanswered. Thank you, you dear girl, and irreplaceable assistant!\n\nLove,\n\nL\n\n(If you haven't been a \"children's expert\" heretofore, you sure will be after the new one. Have a good one!)\n\n501. Leonard Bernstein to David Diamond\n\n28 January 1965\n\nDear Dd,\n\nFinally the date is settled for your celebration \u2013 the week of 25 April, '66, concerts on 28th, 29th, 30th and 2 May. I had planned for you to conduct the _Rounds_ and the Piano Concerto (for which Schumacher is already engaged); but having since received the 5th Symphony, which I find your absolute _best_ to date, I will take the occasion to open the program with it myself (sorry to do you out of one conducting stint) \u2013 and then you can conduct the concerto with T[homas] S[chumacher]. I'm delighted and hope that you are. (The program will probably close with a staple \u2013 Sibelius #1 or something.127 The whole year is devoted to a survey of the symphony in the 20th century, including all 7 of Sibelius, owing to his centenary. And that's another reason for doing your 5th instead of the _Rounds_. There'll also be some others you'll love \u2013 Vaughan Williams #4, Aaron #3, Nielsen #3, Bart\u00f3k 2-pno Concerto, & Webern Op. 20, Mahler #7, #8 & #9! A big year, a lot of work: I look forward to it.)\n\nWhat are you doing in Florence? What has happened to the Iron Curtain trip? Last I heard, you were in Copenhagen.\n\nYou've probably heard about the collapse of _Skin of Our Teeth_ : a dreadful experience, the wounds still smarting. I am suddenly a composer without a project, with half of that golden sabbatical down the drain. Never mind, I'll survive.128\n\nGood news about Aspen. But the best news is the 5th Symphony. I have the \"waves\" in the first part, and the end particularly. The fugue is a killer. I hereby accept the dedication with due formal ceremony and much affection.\n\nL\n\n502. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n5 February 1965\n\nDear Dr. Bernstein,\n\nI am very sorry to bother you with another letter and I do not want to seem impatient, but I am constantly being pressed by the Organist and Choirmaster for particulars as to title etc. of the work which you most kindly said you would write for us. He is directing the Three Choirs Festival and has got to get his publicity out and tells me that it cannot be held up any longer. I am sure you will understand.\n\nHe also tells me that the music has to be printed and then circulated to the three choirs to practice individually before they come together for common rehearsals. I am sure you are conscious of all this and please do not think me impatient; but your work would of course be the highlight of the Festival and all of us are most anxious that it should be done as well as we can possibly manage.\n\nAgain, my apologies for bothering you with a tiresome letter.\n\nYours sincerely,\n\nWalter Hussey\n\n503. Leonard Bernstein to Walter Hussey\n\nNew York, NY\n\n24 February 1965\n\nMy dear Dean Hussey,\n\nI was on the verge of writing you a sad letter saying that I could not find in me the work for your Festival when suddenly a conception occurred to me that I find exciting. It would be a suite of Psalms, or selected verses from Psalms, and would have a general title like _Psalms of Youth_. The music is all very forthright, songful, rhythmic, youthful. The only hitch is this: I can think of these Psalms only in the original Hebrew. I realize that this may present extra difficulties of preparation; but more important, does it present difficulties of an ecclesiastical nature? That is, are there no objections, in principle, to Hebrew being sung in your Cathedral? If not, do let me know soon, so that I may plunge ahead, and have a working score for you by early April. The orchestration would follow a month or so thereafter.\n\nIf there _are_ objections I should also know soon, for obvious reasons. I would be sad, but I would understand.\n\nFaithfully yours,\n\nLeonard Bernstein\n\nP.S. The Psalms involved would be:\n\nNos. 23, 100 and 131, complete;\n\nNo. 2, vs. 1\u20134; No. 108, vs. 3; No. 133, vs. 1.129\n\n504. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n2 March 1965\n\nDear Mr. Bernstein,\n\nThank you very much for your letter. I was delighted to receive it and am indeed grateful for all the good news it contains.\n\nI do not think that there is any ecclesiastical objection to the use of Hebrew. The meaning of the words could always be supplied by a translation in the programme. It will of course, as you say, present a little problem in the preparation of the work, but no doubt it will be possible for the Hebrew to be printed phonetically as in Bloch's _Sacred Service_.\n\nI will write to you again shortly giving you the numbers in the choir etc. In the meanwhile let me say again how pleased and excited I am at the splendid news of your letter \u2013 and in this the Choirmaster and Organist joins me.\n\nYours sincerely,\n\nWalter Hussey\n\n505. Solomon Braslavsky to Leonard Bernstein\n\nTemple Mishkan Tefila, Boston, MA\n\n16 March 1965\n\nMy dearest Lennie,\n\nLast Friday I received your record of the _Kaddish_ Symphony.\n\nI immediately opened the wrapping and looked for a bill, but instead I found a very \"long letter\" from you consisting of _four_ priceless words. Last Saturday morning I saw your father in the Temple and I told him that I just got a priceless _Purim_ present from Lennie.\n\nWell, needless to say how happy I am to be able to play the Symphony again and again in search for some more jewels of liturgical value which only you are capable of working in such an artistic manner, and for this I am very much indebted to you.\n\nI shall also like to listen again to the recitations so beautifully done by Felicia and evaluate it from its philosophical as well as its artistic point of view. I only regret that I am at present extremely busy with my duties in the Temple and I shall have to wait for a while before I will be able to put aside a few hours for a thorough consideration of this beautiful music.\n\nWith many thanks and all of my best wishes to you and your dear family, for good health and lots of happiness.\n\nYour devoted,\n\nSolomon\n\nP.S. I shall write to you after my first hearing of the records in the near future [...]\n\n506. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n14 April 1965\n\nDear Dr. Bernstein,\n\nI promised to let you have the particulars as regards the choir and the orchestra for the Festival in July. I have now got these from John Birch.\n\nThere are 46 boys, 8 male altos, 9 tenors and 12 basses. All the adults are professional singers and the boys are all members of our three Choir Schools, and I think really very competent.\n\nThe orchestra is the Philomusica of London and the players required for the rest of the programme are strings \u2013 3.2.2.2 (3 players), 1 (1 player) \u2013 18 players in all plus a chamber organ. Other players could be provided if you wished, for example trumpets and trombones, percussion, piano and harpsichord. I dare say you know of the orchestra but I think it is fair to say that they are highly competent players and perhaps the best chamber orchestra in London.\n\nWe are all tremendously excited about _The Psalms of Youth_. I do hope all goes well. I cannot tell you how grateful I am to you.\n\nPlease let me know if there is any further help or information I can give you.\n\nYours sincerely,\n\nWalter Hussey\n\n507. Leonard Bernstein to Walter Hussey\n\n11 May 1965\n\nDear Dr. Hussey,\n\nThe psalms are finished, Laus Deo, are being copied, and should arrive in England next week. They are not yet orchestrated, but should be by June, and you should receive full score and parts in ample time for rehearsal. Meanwhile the choral preparation can start forthwith.\n\nI am pleased with the work, and hope you will be, too; it is quite popular in feeling (even a hint, as you suggested, of _West Side Story_ ),130 and it has an old-fashioned sweetness along with its more violent moments. The title has now been changed to _Chichester Psalms_ (\"Youth\" was a wrong steer; the piece is far too difficult). The work is in three movements lasting about eighteen and a half minutes, and each movement contains one complete psalm plus one or more versions from another complementary psalm, by way of contrast or amplification. Thus:\n\nI. Opens with a chorale (Ps 108, vs. 3) evoking praise; and then swings into Ps. 100, complete, a wild and joyful dance, in the Davidic spirit.\n\nII. Consists mainly of Ps. 23, complete, featuring a boy solo and his harp, but interrupted savagely by the men with threats of war and violence (Ps. 2, vs. 1\u20134). This movement ends in unresolved fashion, with both elements, faith and fear, interlocked.\n\nIII. Begins with an orchestral prelude based on the opening chorale, whose assertive harmonies have now turned to painful ones. There is a crisis; the tension is suddenly relieved, and the choir enters humbly and peacefully singing Ps. 131 complete, in what is almost a popular song (although in 10\/4 time!). It is something like a love-duet between the men and the boys. In this atmosphere of humility, there is a final chorale coda (Ps. 133, vs. 1) \u2013 a prayer for peace.\n\nI hope my score is legible. In order to help with the Hebrew text, I shall enclose a typewritten copy of the words (the Hebrew words of Ps. 2 are a tongue-breaker!). The score contains exact notes on the pronunciation.\n\nAs to the orchestra, I have kept to the prescribed forces, except that there will be a large percussion group necessary (xylophone, glockenspiel, bongos, chimes, etc., in addition to the usual timpani, drums, cymbals, etc.). Also, I am sure more strings will be necessary than the number you list \u2013 especially the low ones. Certainly _one_ bass will not do the trick. One of the three trumpets must be very good indeed, in order to perform several difficult solo passages. There is also an extensive harp part.\n\nOne last matter; I am conducting a program of my own music with the New York Philharmonic in _early_ July, and I have been asked if I could include the _Chichester Psalms_. I realize that this would deprive you of the world premiere by a couple of weeks; do you have any serious objections?\n\nIn any case I wish you well with the piece; and I may even take your performance as an excuse to visit Sussex in late July. I should dearly love to hear this music in your cathedral.\n\nFaithfully yours,\n\nLeonard Bernstein\n\n508. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n11 June 1965\n\nDear Dr. Bernstein,\n\nI have just returned after being away for five weeks as a result of a tiny cerebral thrombosis. In due course the doctors assure me that I shall be quite fit again and need take no notice of it, but just at present I feel slightly old and tired!\n\nThe _Chichester Psalms_ arrived yesterday and I do indeed thank you for them. They are splendid and exactly the sort of thing that I was hoping for. So far as I can judge they seem to be admirable and I thank you most warmly for them.\n\nIf it is at all possible for you to come over and hear them in late July we shall be delighted for you to do so. In this case please come and stay at the Deanery. Would you like to conduct them?\n\nAs to the orchestra, Mr. Birch has got this in hand and will I am sure follow your wishes.\n\nYours sincerely,\n\nWalter Hussey\n\n509. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n8 July 1965\n\nDear Mr. Bernstein,\n\nThank you very much for your letter.\n\nI expect by this time you will have heard from Mr. Robert Lantz giving you a formal invitation to come for the Music Festival, and the final rehearsal and the first performance of the _Chichester Psalms_. It is unfortunately impossible to get lodgings anywhere in this part of the world at the time, they are all booked up eight or nine months ahead for the Goodwood Race Meeting which takes place on the Monday to Saturday of that week. This is most unfortunate for us because it does make it frightfully hard for any people to get accommodation for the Music Festival within a fifteen to twenty mile range. I have had a word with numbers of hotels and they all tell me the same. However, I shall indeed be delighted for you and Mrs. Bernstein to come here and Mr. and Mrs. Robert Elwes, who live just a mile or two from the Cathedral in a very lovely house, will be delighted for the children to go there. They have two children aged 13 and 10 of their own, and a swimming pool, if only the weather will allow them to use it! They are very nice people indeed and I am sure you could feel absolutely happy for them to be there. If this would be satisfactory for you Mrs. Elwes says she would be delighted for them to be there from Thursday until Monday, and of course I shall be only too glad for you to be here.\n\nI am delighted to hear that Chuck Solomon hopes to be in Chichester for the Psalms and I have written to tell him so, but my only fear is that if he had not made the arrangements already he may well find it impossible to get anywhere. It would be great to see him again.\n\nPlease tell me if there is any way in which I can help you further.\n\nYours sincerely,\n\nWalter Hussey\n\n510. Leonard Bernstein to Walter Hussey\n\n17 July 1965\n\nMy dear Dean Hussey,\n\nOur Psalms had their \"creation\" last night, to a standing ovation, and I was overjoyed for you as well as for myself. I enclose this morning's _Times_ review, which pays you just credit for your kindness. Even juster credit is paid you by the program notes. I'm having them sent you: won't you need them for your own use?\n\nI have your last letter, and am most grateful for your solicitude. Mrs. Bernstein and I will be most happy to stay at the Deanery; and my children are terribly excited at the prospect of living with an English family, and making British friends of their own age. Please tell Mr. and Mrs. Elwes how very grateful we are. The only minor hitch is that our visit is planned one day earlier; we should arrive on the 28th and leave on Sunday, 1st August. I hope this does not complicate your and the Elwes' lives.\n\nWe are leaving here the 27th, arriving in London that night, which we shall spend at the Savoy Hotel. We have covering bookings at the Savoy for the whole 10-day period, so don't hesitate to evict us if you would have to; in any emergency there will be a roof over our heads.\n\nLooking forward with immense pleasure \u2013\n\nYours,\n\nLeonard Bernstein131\n\n511. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n22 July 1965132\n\nDear Mr. Bernstein,\n\nI am _delighted_ you & Mrs. Bernstein & the children can come over to England for the first performance of the Psalms. All is ready \u2013 except the weather! Miss Chavez133 of CBS records will meet you at London Air Port and has laid on a car, if you should require it, to bring you to Chichester the next day.\n\nI am very glad that the Psalms have met with such a warm reception \u2013 I'm sure they entirely deserve it.\n\nI wrote to Chuck a while back, but have not heard. Still, he may well turn up when the time comes!\n\nThere was an excellent article about the _Chichester Psalms_ in the _Times_ (of London), but I'm afraid I haven't got a copy now. However, I'll have one when you come.\n\nLooking forward to seeing you with great pleasure \u2013 & some little apprehension!\n\nYours ever,\n\nWalter Hussey\n\n512. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n1 August 1965\n\nDear Lenny,\n\nI hope you arrived safely back at the Savoy last night \u2013 with no breakdowns!\n\nI cannot begin to tell you how grateful I am for the _Chichester Psalms_. This morning the Bishop of Chichester said to the Archdeacon \u2013 they were a new revelation to him & brought home afresh the meaning of them, joyous & ecstatic & calm & poetic \u2013 he said you could imagine \"David dancing before the ark\".\n\nWe were _all_ thrilled with them. I was specially excited that they came into being at all as a statement of praise that is oecumenical. I shall be tremendously proud for them to go around the world bearing the name of Chichester.\n\nI hope the family are enjoying their brief stay in London. They won the hearts of all who met them! But especially Felicia, who has gone straight to the number 1 place of the most charming & attractive of wives.\n\n_Please_ come again any time you can. You will always be welcome & it will be an honour to have you here.\n\nBless you indeed for everything.\n\nWalter\n\nP.S. I wonder if there is _any_ chance of getting the MS \u2013 to go with those of Britten and others who have written things for us? It doesn't matter if it is terribly crossed out & untidy!\n\n513. Felicia Bernstein to Walter Hussey\n\nSavoy Hotel, London\n\n3 August 1965\n\nDear Walter,\n\nThis is the first chance I've had to sit down in relative peace since our drive back from Chichester! As you can imagine it's been non-stop \u2013 finally today I asked for time off to see an old childhood friend and write to you.\n\nWe will all remember Chichester for many reasons but the main reason is you. We talk about you so much and miss you already; so you see, for all our sakes you simply must return the visit. Do come!\n\nBless you for all your kindness and hospitality, and think of us once in a while as you stare at the yellow carpet! A reference to the coffee disaster!\n\nWith best wishes from us all.\n\nFelicia134\n\n514. Leonard Bernstein to Walter Hussey\n\nSavoy Hotel, London\n\n6 August 1965\n\nDear Walter,\n\nWe are all about to leave London, and in this last hour I wanted somehow to talk to you again, to thank you, not only on a social level, but on the deepest personal one, for all the things you are, do, and stand for. I shall carry sweet memories of Chichester for a long time.\n\nThe Psalms are, of course, dedicated to you, and you should receive the very first published copy. Meanwhile I shall arrange to have a photocopy sent you.\n\nAgain, Felicia and I send you our most affectionate thanks.\n\nLenny B.135\n\n515. Leonard Bernstein to Walter Hussey\n\n20 September 1965\n\nMy dear Walter,\n\nChichester seems awfully far away by now, but the memory of our days there is a glowing one, reinforced this week by the test pressing of the recording, which should be out within the month. I'll see that you get one of the first copies.\n\nI have also been busy proofreading the vocal score, which is to appear at the same time as the record, and of course you'll get one of these too.\n\nApropos of the printed score, I must confess that I have yielded to a very human weakness. I could not resist sharing the dedication between you and Chuck Solomon. (You get the first credit, as having commissioned it, and there is then a line dedicating it to Chuck.) I happened to talk to him the other day, and he sounded very _down_ ; and in that instant it occurred to me that this dedication would set him up for at least a year. It's just what he needs \u2013 and I am sure that you, of all people, will understand the nature of this gesture, since you and Chuck are such good friends, and since he was the originator of our relationship, yours and mine. I somehow feel I owe this to him. I hope you agree.\n\nAgain, thanks from us all for your kindness and friendship.\n\nYours,\n\nLenny136\n\n516. Walter Hussey to Leonard Bernstein\n\nThe Deanery, Chichester, England\n\n16 October 1965\n\nMy dear Lenny,\n\nI found your letter here when I returned from a fortnight's holiday, and that is the reason it has gone unanswered so long. I am very sorry.\n\nThank you indeed for writing it. I also carry with me the most happy memories of your visit \u2013 now we are having the most wonderful, cloudless weather that we should have had then!\n\nYes indeed, I entirely understand about Chuck and approve his sharing the dedication. I am sure he will, rightly, be as pleased as a dog with two tails. I sent him, some while ago, a letter thanking him for his part in the _Chichester Psalms_ and telling him it would never have come about apart from him!\n\nOf course I am only to pay the bill for the car-hire. I was only anxious that we should not _both_ pay it.\n\nI am most anxious, and so are many other people, to hear the record. It will be very exciting. It is _most_ good of you to say you will send me a copy of the score and the record. It's far more than I deserve. Have you the original score you wrote? I feel awful asking, but I should love to have it if you can spare it, to put with the others. It doesn't matter how rough it is \u2013 Britten's is in pencil & a mass of scratching out!\n\nGive my love to Felicia and the children. I look back on your visit as a great time and a very happy one.\n\nYours ever,\n\nWalter\n\nP.S. I hear rumours that the BBC want to do the _Chichester Psalms_ again from here, with a different orchestra. I do hope they are true!\n\n517. John Cage to Leonard Bernstein\n\nStony Point, NY\n\n28 October 1965\n\nDear Lenny,\n\nWas very good to be with you, Felicia, and the Fosses. I'm now organizing a show and sale of 20th century music mss. (Stable Gallery April 1966) for the benefit of the Foundation for Contemporary Performance Arts, Inc. Gifts to the Foundation are tax-exempt. Will you give us a page of your work?137 Wd. be very grateful.\n\nAs ever,\n\nJohn (Cage)\n\n518. Leonard Bernstein to Walter Hussey\n\n29 October 1965\n\n[Dear Walter,]\n\nI can't describe to you the surprise and pleasure of receiving your gift. It is at least as extraordinary of you to remember my liking your pen as it is for you to have sent me one.\n\nAnd, of course, the pen's first act-in-office is to write this note, sending thanks and affection.\n\nThe recording and the published score should be arriving any day now. Meanwhile I am sending you my original \"fair copy\" \u2013 though alas, only a photostat of the manuscript, since the true-original is committed to the Library of Congress, as is the case with everything I write. This is the next-best; it is all in my own hand, and I hope it pleases you.\n\nBless you, your health, your work, and your touching generosity in sending me this treasureable pen.\n\nWarmest greetings,\n\nLenny138\n\n519. George Szell139 to Leonard Bernstein\n\nNew York, NY\n\n29 October 1965\n\nDear Lenny,\n\nMany many thanks for your sweet note. Your wishes for pleasure with your orchestra have already come true: I am enjoying myself with them thoroughly and find them in better shape & spirits than I can remember.140 The prospect to spend a little time with you after you return is positively _enticing_. I hope it _will_ happen. Meanwhile, all good wishes for a pleasant & restful vacation.\n\nEver cordially,\n\nGeorge\n\n520. Yo-Yo Ma141 to Leonard Bernstein\n\n138 East 94th Street, New York, NY\n\n21 December 1965\n\nDear Mr. Bernstein,\n\nDo you still remember me? Now I am ten years old. This year I learned with Prof. Leonard Rose three concertos: Saint-Sa\u00ebns', Boccherini's and Lalo's. Last week my sister and I played in a Christmas Concert in Juilliard School. We are invited to give a joint recital in Brearley School on January 19, 1966 at 1:45 p.m.\n\nIf you have time, I would be glad to play for you.\n\nYo-Yo Ma\n\n521. Leonard Bernstein to John Adams142\n\n[27 January 1966]\n\nDear Mr. Adams,\n\nI am touched by your intelligent letter, but hard put to answer it. When you depict me as \"turning my back\" on \"new\" musical trends you do me a disservice, to say nothing of making an irrelevancy. One writes what one hears _within_ one, not without. Lord knows I am sufficiently exposed to the \"influences\" of non-tonal music; but obviously I have not been conditioned by them. Mahler apart, I cannot conceive music (my own music) divorced from tonality. Whether this is good or bad is, again, irrelevant. The only meaningful thing is the truth of the creative act. The rest of the chips will fall where they may.\n\nGood luck to you.143\n\n522. Leonard Bernstein to Felicia Bernstein\n\nHotel Bristol, Vienna, Austria\n\n2 March 1966\n\nFleshy darling,\n\nIt is a week to the hour that I am in Vienna, and this is literally the first moment free. It's been all rehearsals, 10 a.m. to 8 p.m. daily, studying at night, & trying (not too successfully) to get some sleep. The time-change threw me for a loop; I'm just now coming out of it.\n\nBut all goes well.144 The orchestra is cheering, the directors are fawning, the press is voluminous and astonishingly sympathetic. Fischer-Dieskau is a dream-singer \u2013 I've never heard anyone better.145 And the rest of the cast is superb, with a weakness here or there \u2013 but not important. Dr. Caius is a German with a frightful accent in Italian which will sound bad on the record, & Oncina's Fenton is not ideal. But you can't have everything. The Opera people have given me extra rehearsals; the orchestra goes overtime without a word of protest (never before, since Strauss himself, says Dr. Hilbert). In short, I'm a sort of Jewish hero who has replaced Karajan \u2013 and all this at the moment of the general elections which happen tomorrow, & are full of anti-Semitic issues & overtones. It's a strange feeling...\n\nI've just come from a place in Nussberg \u2013 a suburb \u2013 an old palais devoted to vineyard activity, where some members of the orchestra took me to play for me \"Schrammelmusik\" \u2013 old, authentic, \"kitsch\" Viennese tunes arranged for 4 instruments \u2013 the original _Rosenkavalier_ stage-music. Fantastic. Lots of glorious wine, platters of hot roast chickens, hams, Gott weiss was. Very touching, from orchestra men: toasts, good feeling. As I say, a very strange moment for me.\n\nThe Kripsies146 have been here, & invited me daily to his performances & to dinner or lunch \u2013 I funked out ever time, being left with a bad conscience, especially since they sent me (us) huge lilies, & today, as a _farewell_ gift (!) two boxes of Demel chocolates. They're in San Francisco now: and I've written a conscience-letter.\n\nMaazel is also here doing _Carmen_. I heard 10 minutes & left: _bad_. But the musical life is wilder than Milan: everything seems to center around it, & _Falstaff_ is the event everyone is waiting for.\n\nMany of Willie Weissel's147 friends have written or called \u2013 I don't dare to accept or answer: _qu\u00e8 lata_. Luchino [Visconti] is tired & morose, at once doing _Falstaff_ & preparing _Rosenkavalier_ for London, & cutting a movie: he's insane, & surrounded by numerous young Italian assistants, which gives him pleasure. And now I go to be fed & f\u00eated by Regina Resnick [Resnik], who lives across the square, & has been preparing a risotto for days.\n\nAs you can see, it's a fascinating, glorious life so far. The haut-monde is yet to come. Bet you can't wait! But I'll tell you one thing: I can't wait for you to come. You'll love Vienna, somehow I smell it.\n\nHave a great Buffalo, love to Fossies, & come soon. I miss you!\n\nLove,\n\nL\n\nI read a bad Kerr notice of Mendy's148 play in the _Trib_. Are they all bad? Give him & Susan my love. And the kiddies: kiss them. And write me. And make the kiddies write me \u2013 xxx\n\n523. Leonard Bernstein to Sam and Jennie Bernstein\n\nHotel Bristol, Vienna, Austria\n\n19 March 1966\n\nDear Folks,\n\nAt last I have a minute to write you. I am enjoying Vienna enormously \u2013 as much as a Jew can. There are so many sad memories here; one deals with so many ex-Nazis (and maybe still Nazis); and you never know if the public that is screaming _bravo_ for you might contain someone who 25 years ago might have shot me dead. But it's better to forgive, and if possible, forget. The city is so beautiful, and so full of tradition. Everyone here lives for music, especially opera, and I seem to be the new hero. What they call the \"Bernstein wave\" that has swept Vienna has produced some strange results; all of a sudden it's fashionable to be Jewish.\n\nBut I work very hard, practicing & studying, recording (20 sessions!) and rehearsing. So far everything has gone brilliantly, but I'm tired \u2013 too many parties, also. Don't be upset by that bronchitis story: I had it for only two days. And I really feel very well.\n\nThis morning I went to _Shul_ , imagine, with Regina Resnick [Resnik]. The old, famous Wiener Schul, restored as it used to be, on _Judengasse_ (what a name for a street!). But it was warm and heartening. I ran into a Bar Mitzvah _and_ a _Rosh Chodesh_ , got a _Misheberach_ & held the Torah for Rosh Chodesh, got a plug from the Rabbi & even attended the Kiddush afterwards. Very sweet. And tell Prof. Braslavsky that I met his old friend Rothenberg, who sends his greetings. They all remember him here.\n\nNow I have a TV interview, a cocktail at Princess Hohenlohe, and then a dinner party with the recording people from London. So I'm off \u2013 and I send you much love. Soon you should receive some chocolates from Demel \u2013 the best on earth. Be well & take care of yourselves.\n\nYour Wiener Schnitzel,\n\nLenny.\n\nShabbos, 19 March '66 after Havdala[h]\n\nJennie: Every morning I eat Vienna rolls \u2013 what you always used to call _Vianna_ rolls. Remember?\n\n524. Victor de Sabata149 to Leonard Bernstein\n\nSanta Margherita Ligure, Italy\n\n28 June 1966\n\nDear Leonard,\n\nIn my purest and deeply felt joy I read this very morning that _at last_ you plan to write an opera. Very seldom in my life I felt so thrilled and impatient! I wish to have your new score near my heart as soon as possibly! I am sure this will enable me to plunge into _real music_ , a thing that \u2013 let us be sincere! \u2013 I am vainly longing for since centuries. Useless to tell you how often I think of you and your incandescent musical vitality. To know that a Bernstein does exist helps a lot. Ciao!\n\nTuo affezionatissimo,\n\nVictor de Sabata\n\n525. Robert Russell Bennett150 to Leonard Bernstein\n\n150 East 50th Street, New York, NY\n\n26 November 1966\n\nCaro Maestro,\n\nThere is no reason why my opinion should be of especial value to you, but I can't resist sending you this note to voice my enthusiastic approval of your decision to give full time to music composition.\n\nMy opinion is not without substance because I once, when we were both much younger, tuned into a broadcast and hear a Sonata for clarinet and piano written by you, and as I turned away from the loudspeaker I said to Louise, \"This is one of our big composers.\" As years have gone by I felt a certain reluctance to see you pursuing the conducting, and even the composing of Broadway music, as being a waste of that precious commodity, time, when so much is needed for the full realization I had in mind. I never saw Gustav Mahler conduct. If he was as great as I have been told, he is just about the one exception that proves the rule as far as I am concerned.\n\nI leave you to comb through the history of our profound composers and see how they fared at strictly commercial roundelays. Of course, someone will answer this remark by bringing up that innocent era of Mozart and Haydn when \"popular music\" was the reaction of a whole era to the deep expression of Bach, for instance. Be that as it may, you have at least one enthusiastic vote for your career as a real composer.\n\nAll the best as always,\n\nRussell\n\n526. Georg Solti151 to Leonard Bernstein\n\n17 Woronzow Road, London, England\n\n19 May 1967\n\nDear Mr. Bernstein,\n\nI hope that you will not regard this letter as interference, but I felt that I had to write to you about Mr. Wobisch.\n\nI first heard from John Culshaw152 about the reports in the Vienna Press concerning you and Wobisch. Two days later Wobisch telephoned me about another matter, at the end of the conversation he told me more about the reports and how very distressed he was about them. Afterwards, I felt that without wanting to interfere in a matter which does not concern me at all, I had to write to you for purely human reasons.\n\nAs another Jewish conductor, I understand your feelings surely better than anyone else. If somebody, after the Nazi horrors, does not want to work with a German or Austrian orchestra, as is the case with several Jewish artists, I understand only too well. I have been through great soul searching in the past about this, and several times have been on the verge of breaking contact with them. But finally I always had the conviction that one must forgive the past and try to work to help and educate the younger generation in these orchestras.\n\nI am aware of Wobisch's political past, as surely you were before you went to Vienna. However, working with him and knowing him for the past ten years, I have come to the conviction that despite everything he is probably one of the few trustworthy members of that orchestra.\n\nWobisch worked very hard to bring you to Vienna and to prepare your appearances and successes there; I even heard from Mr. Rosengarten of Decca that Wobisch went as far as threatening to change the orchestra's contract from Decca to Deutsche Grammophon unless they were released to make _Rosenkavalier_ with you. As you will know by now this involved the postponement of my own recording of the opera with the orchestra, which should be enough indication of my real neutrality in this issue.\n\nIf Wobisch should have to resign as a result of this controversy with you, I am convinced that not only would this be bad for the orchestra, but that both you and I might well find any replacement totally unacceptable for political and human reasons.\n\nI hope that these few lines may have helped in some way.\n\nWith kindest regards,\n\nYours sincerely,\n\nGeorg Solti\n\n527. Janis Ian153 to Leonard Bernstein\n\n7 June 1967, \"Evening tide\"\n\nHello sir,\n\nExcuse the formality of the address, but if you remember everyone was bopping about calling you \"Lenny\", and since I felt rather strange doing that we decided on sir.\n\nI would have written sooner but... Well, no excuses. I didn't really have much to say except thank you, and I'd said that. But now I want to tell you what's happened.\n\nIf you didn't see, the biggest rock station in California wrote out a public apology for their recent timidity, and thanked you for showing the way. More stations on the West Coast went on \"Society's Child\", and now it's number one in California. I'm waiting for it to hit the top 20, so NY stations will be forced to play it. Except the station manager of WMCA or WABC said he'd _never_ play it because he wanted to keep his children's ears free from the \"objectionable\" lyrics.\n\nAnyway, this is just to say that you're lovely and thanks again for everything.\n\nJanis Ian (me)\n\nP.S. Is it okay if when a reporter asks what I think of you I just say that you're gorgeous and charming?\n\n528. Leonard Bernstein to Lukas Foss\n\nCasa Malone, Orbetello, Grosseto, Italy\n\n8 August 1967\n\nPoor, dear, blessed Luky-Puky!\n\nA grief ago! I pray that period has lengthened to a vague unpleasantness ago. When I read the accounts in the paper I was sure something like what you described had happened: I wanted to rush home and set things right. But I can't rush anywhere: the _dolce far niente_ has taken over. I do nothing. No note written, no score studied, no idea thought out. I'm a fish, living with other fish underwater in my glorious diving gear. I have a rubber motorboat and a divine Maserati (my first & last pure playboy object); my summer romance & constant companion is Alexander; I read the mail & some newspapers; I fret from afar over race riots, Vietnam, tax hikes, bad N.Y. weather, increasing horror in the world from Cairo to Memphis, Tenn. I fret over Myrow's _Salome_ libretto (lousy, pompous, meaningless, imitative in the worst way, and corruptive of youth). I fret over your neoclassicism & Philharmonic tragedy154 (Oh, well; it gave Brigitta a chance for a big triumph, no?)\n\nI don't sleep (it is now 4.30 a.m.). My back has been in agony for a month. Felicia can't take the sun. But the water and sky & air are divine, as is the weather and this house-&-garden; and you have a birthday in a week. Bless you. Time... I am tortured by the passing of time, to the point where I can hardly enjoy the passing of these beautiful days. Each day is a horror because it leads me one day closer to the end of summer; & the guilt of not working is intolerable. But my brain & creative innerds are dormant, or dead. Why? I shriek inside. For what, for whom? Shall I leave music & enter politics? My tune of the summer, obsessive, is the Beatles'\n\nWill you still need me\n\nWill you still feed me\n\nWhen I'm sixty-four?\n\nAt least it's gay and simple and no trouble.\n\nJohn Gruen155 sits with me for hours, a tape-recorder between us, and I talk, talk, talk. I have been photographed to a crisp. Israel was astonishing and semi-sad and like a religious experience. The concerts there were my last conscious acts.\n\nI have a gnawing feeling that David O[ppenheim] is still miffed at me. Ask him. I miss him a lot; if he can, would he write?\n\nRio \u2192 Warsaw \u2192 Buffalo! Only Luky could concoct that itinerary. But at least have fun, feel like Marco Polo, adventurize!\n\nI long to see you in September. I rejoice in your \"fan letter\" and in the good news of the _Phorion_ tape. I pray for you to write beautiful music. I love you.\n\nL\n\nHugs to Corny & Chris-Andrew and L-Baby and the Opps. And the Rivers and rest. How did _Phorion_ phare in Chicago? Phabulously, I hope.\n\nWrite again before you leave.\n\n529. Leonard Bernstein to Stephen Sondheim\n\nCasa Malone, Orbetello, Grosseto, Italy\n\n19 August 1967\n\nDear SS,\n\nThe appearance on my desk, _faute de mieux_ , of a pack of Winstons brings you instantly and clearly into the room. Besides I can't sleep o' nights, nor have I been able to for over a month. And these guilty sleepless hours, drugged yet jumpy, are my only epistolary moments. Reading H. P. Lovecraft this evening has also brought you to mind, as has Nina's incessant playing of the _WSS_ album. In, out, let's get cracking. Neutral territory. One-handed catch. Then the Princes appeared for dinner, reporting you depressed at Merrick's failure to announce your work among his plans, or, indeed, to come up with a theatre. And beside these, I just happen to think of you often, apropos a thousand trivia, all warmly nostalgic.\n\nA strange summer. Glorious weather, sea, boats, diving gear, skis, sun and air \u2013 all the goodies. But a fearsome back (how's your back, Lenny? And now I'm to be 49) prevents aquatic fun, and nameless anxieties (is that the word?) forbid work. Not a note, scores unstudied, books unread. Thoughts and ideas are absent, except for such stuff as _Improvised, eh? Garbled and poor!_ Felicia sleeps badly too: only the children prosper. I teach Alexander Hebrew \u2013 my one real activity. I shudder at the heaps of unanswered mail. I itch.\n\nThere's my report. What's yours?\n\nYou should have come to this Eden-on-the-Sea; we could have moaned together.\n\nIf you see or talk to Jerry, please tell him I'm simply too guilty to write him, owing to the absence of a single idea. Total non-energy.\n\nI hope your musical is ship-shape.156 I read that _Lion in Winter_ is to be cinematized. I hope you are loving somebody, regularly and in bed. I hope the soul-brothers haven't reached Turtle Bay yet. I hope the world can survive its awful weight a bit longer. I hope \u2013\n\nI send you wee-hour love and personalized sentiments. Be dour if you must, but be happy.\n\nLenny\n\n530. Joe Roddy to Leonard Bernstein\n\n25 October 1967\n\nDear Leonard,\n\nBecause you have one of the last grasps left of the human comedy, I am counting on you to see me into, and then out the other side, of this absurd fix I am in. For all the Mahler I never knew before, for Ives, for much Haydn, for an overwhelming _Missa Solemnis_ , for the world's first _Falstaff_ , for the _Chichester Psalms_ , for _Candide_ , for \u2013 well, Christ, for ninety percent of the music that matters to me nowadays, I am in debt to you. Clinking oceans of gold pieces would not repay all. So really, it cannot matter, can it?, that there is an ice cream stand at Expo 67 on which there is writ in chocolate sauce _L. B. OWES J. R. $100_. Debt for debt, mine is hardly worth mentioning.\n\nExcept that, weeks from now when you are savoring a distraction or two instead of settling down to write the next to last song for that Brecht show, into the mind of you will come the picture of me. I will be seen sitting, sitting and sitting at some many Philharmonic rehearsals, day after day after day. Why, you will then ask yourself, did he come round so often? Why, why, why? Then a terrible thought will come over you, a thought so disruptive that it may dislodge forever that shred of melody you were counting on to get you started again, the thought that I was sitting there not watching and listening to you work with the orchestra at all, but instead just waiting there like some mouse creep of a bail bondsman from Baxter Street worrying about his cheesy C-note.\n\nBut, my ever so dear friend, that is not why I was there, nor why I will be there tomorrow maybe. I like it there, but you know all that. I just want to protect you from that blinding light in which I might glow, though dully, like a mouse. I don't want you to lose that shred of tune, and surely you don't want me to sit there at rehearsals with these grotesque introspections.\n\nUnless I know\n\nThat you know\n\nAbout my dough\n\nAt Expo,\n\nI can't show\n\nAt rehearsals any mo'\n\nBo'.\n\nLove,\n\nJoe Roddy\n\n... and I don't even know what present you bought for your sister.\n\n531. Leonard Bernstein to Joe Roddy\n\n27 October 1967\n\nDearest Joe,\n\nWhy, you will ask yourself, $112.49? Precise figures follow: $100, plus interest (at, I believe, the going rate of usury, 4%), making $104, plus interest compounded for compound guilt and shame, making $108.16, plus further compound interest for neglect and discourtesy.\n\nMy only redeeming feature for my life-long inability to remember debts owed is my concomitant inability to remember debts owed to me by others. In short, money is the thing that interests me least of all this world's wonders.\n\nBut:\n\nNow that you know | But be my beau, \n---|--- \nThat I've eaten crow | Dear Joe, \nOver the dough | Fo'- \nYou lent at Expo | ever. And FO'- \nNever go | GIVE! \nAway no mo', |\n\nLove,\n\nLenny\n\n532. Leonard Bernstein to Aaron Copland\n\n[New York, NY]\n\n12 November 1967\n\nDear A,\n\nIt's two days before your birthday, but I'm already thinking hard and tenderly about you; and this note is your birthday present carrying with it such abiding love as I rarely if ever get to express to you in our occasional meetings. I don't know if you're aware of what you mean, have meant for 30 years, to me and my music and so many of my attitudes to life and to people. I suppose if there's one person on earth who is at the centre of my life, it's you; and day after day I recognize in my living your presence, your laugh, your peculiar mixture of intensity and calm... I hope you live forever.\n\nA long strong hug.\n\nLenny\n\n533. Janis Ian to Leonard Bernstein\n\nRichard Armitage Management Corp., 130 East 57th Street, New York, NY\n\n[November 1967]\n\nHello Sir,\n\n(\"Sir\" on account of \"Lenny\" sounds too presumptuous, and \"Mr. Bernstein\" too unpresumptuous)\n\nI guess you know what happened by now, everyone calling up and apologizing for not playing \"Society's Child\", and then playing it and it turned into a top twenty record... and the album too... and the new record looks like it will...\n\nBecause of that, because you drilled me on Spanish, because you're a nice person, I'd like to invite you to my concert.\n\nIt's to be at Philharmonic Hall on December 8 (a Friday night). I'd really like to have you there, and though I can't quite explain why, I'm sure you understand.157\n\nIf you can come, would you please call Jean Powell who's my manager, and let her know how many tickets you'll want. Or ask David [Oppenheim] to call if he's around, as we're inviting him too and it'll be killing two birds with one stone.\n\nI really hope you can come.\n\nYours for sunshine etc.\n\nJanis\n\nP.S. Passed my Spanish Regents with an 86.\n\n534. Leonard Bernstein to Joe Roddy\n\n[New York, NY]\n\n7 January 1968\n\nDear Joe,\n\nYour piece in _Look_ was a fine Xmas present, in that it is always a gift to read something sanely considered and well told.158 I have, naturally, a few objections (oh, two or three hundred) \u2013 nothing sensational, like what's so special about sport-jackets (Dimitri [Mitropoulos] wore them constantly, as do you) and who ever lived like Scott Fitz[gerald] \u2013 anyone \u2013 who, me? \u2013 at 32 W. 10th? 40 W. 55th? The Chelsea? What else? Oh, _On the Town_ is _not_ in any sense a version of _Fancy Free_ : there is not one note in common \u2013 only three sailors.\n\nBut all these I forgive easily; what may take a bit more time is your quoting a quote which is a misquote to begin with, and by Ned Rorem, at that! I never expected that you'd reach that far and that low, just for a kicker. But peace, I'll get over it. And I'll manage to survive not being loved by you, which should patently disprove Ned's quote.\n\nThis note was started as a thank-you, and so it should end, with the addition of a Happy New Year.\n\nLenny\n\n535. Richard Rodney Bennett159 to Leonard Bernstein\n\n[Munich] _as from_ 4 Rheidol Terrace, London, England\n\n25 January 1968\n\nDear Mr. Bernstein,\n\nI wanted to write as soon as I arrived here in Munich, but the time has been so taken up with rehearsals for my opera that this is the first moment I've had.\n\nThe performances of my symphony were, it seemed to me, absolutely perfect.160 They had all the passion and excitement that I hoped I was writing into the work, plus a brilliance & power which I couldn't have imagined. It was a most thrilling time for me, and I am extremely grateful, both for the chance to write the work and for the marvellous performances. I was only sorry not to hear them all. Paul tells me the Saturday one was especially good.\n\nI hated to leave N.Y. but there seem to be all sorts of things in the air & I rather feel I shall be back _quite_ soon. So I hope to see you again before very long.\n\nWith many thanks,\n\nYours,\n\nRichard\n\n536. Leonard Bernstein to Stephen Sondheim\n\nHotel Sacher, Vienna, Austria\n\n19 April 1968\n\nDarling Steve,\n\nLife is good, the gods are kind, _Rosenkavalier_ is sensational, I've never worked so hard, etc. etc. My third act rehearsal was almost ruined by my staying up all night with your enthralling Dodecahedron.161 Things like that. But why I'm _really_ writing is, as they say here, _das volgendes_ :\n\nThe Funke literary effort.162 It was sent me by dozens of people and I never really read it to the end, what with all the hectic goings-on here, until yesterday, when I found myself shocked by the last line.163 Shocked for you, that is \u2013 and I want you to know (as if I needed to tell you!) that, natch, I could never be the source of such a stupid and indelicate remark. But I have talked to Stu[art] O[strow] and told him so, that I will gladly write Funke if you'd like me to, that it's all too silly, life is too short, that I hope you've not been offended, that I love you.\n\nAnd there you are. Tomorrow morning I get up and play a Mozart concerto for thousands of people and I haven't practiced a note. Tonight's _Rosenkavalier_ boasted the presence of Strauss' son who made known that never before... but why go on. Fact is, I miss you and can't wait to get back and dig in.164\n\nLove,\n\nLenny\n\n537. Jacqueline Kennedy165 to Leonard Bernstein\n\nWashington D.C.\n\n9 June 1968\n\nDear Lennie,\n\nIt's 4:00 in the morning \u2013 after this long, long day. We stayed in Washington, at my mother's house.\n\nEveryone has gone to bed but I just want to stay up by myself \u2013 to think about so many things \u2013 and about today.166\n\nIn awful times I think the only thing that comforts you is the goodness in people.\n\nI want to write you that tonight \u2013 or this morning \u2013 whatever it is \u2013 because when I come home I will be so tired \u2013 and I may start thinking about the badness in people.\n\nWhen your Mahler started to fill (but that is the wrong word \u2013 because it was more this sensitive trembling) the Cathedral today \u2013 I thought it the most beautiful music I had ever heard. I am so glad I didn't know it \u2013 it was this strange music of all the gods who were crying. And then \u2013 if only you could have seen it \u2013 it was the time when Ethel had thought of the most touching thing \u2013 having the littlest nephews and nieces, small children, before that terrifying array of Cardinals and gold and Gothic vaults, carry all the little vessels for Communion up to the high altar, so they could have some part in the farewell to the uncle they all loved so much. They were so vulnerable \u2013 and your music was everything in my heart, of peace and pain and such drowning beauty. You could just close your eyes and be lost in it forever. That is what I thought the whole service might be like \u2013 but as you and Monsignor Duffy and the Archbishop have found out, I don't know very much about liturgy and ritual.\n\nSo out of all the confusion \u2013 all your days of conferring, postponing, canceling, adding etc., etc., etc., \u2013 with every one under strain and out of control, I think something emerged that is as beautiful as your Mahler, and that is the way you have been, through all of this.\n\nThe only thing that mattered in the world was that Ethel should have what she wished as music for her husband.\n\nAll the music that meant so much to her \u2013 all the music the church could or couldn't play \u2013 all the intermediaries who cling to their old ways, even through this. Now [we] have the most Ecumenical of Archbishops \u2013 through all that, dear Lennie \u2013 you were so tender and gentle and understanding \u2013 and tactful and self effacing \u2013 so she had _everything_ she wanted, including the last solo that I didn't know of until I heard it \u2013 and that is what I mean by what I said in the beginning of this endless letter \u2013 it is the goodness of people that is the comfort.\n\nI think your goodness and those few soaring moments of Mahler together are more beautiful than if you had played the most beautiful Requiem all the way through.\n\nAnd it was so much more appropriate for this Kennedy \u2013 my Kaleidoscopic brother-in-law \u2013 and his wife who loved him mystically. If there had been anything organized or unified about it, it would not have been Bobby and Ethel!\n\nAnd now we know they were something this world will never see again.\n\nWill you tell your noble orchestra, drowning in heat and cables when I passed them \u2013 that so many people all this day have said: how beautiful you were \u2013 how many people cried \u2013 people who don't know music, and all the ones who were saying the day before, \"You should have so-and-so's trumpets, Fourmier (or a name that sounds like that)167 Requiem, a Beethoven quartet \u2013 a brass quartet?\" etc. etc. all the people who really know music \u2013 which I don't \u2013 today, yours was the time when they wept.\n\nThank you dear Lennie \u2013 I had better not write any more \u2013 I could have spared 6 pages of my mother's writing paper and said all I mean and feel so much more coherently, but I wanted to thank you tonight.\n\nWith my love, and to Felicia\n\nJackie\n\n538. Adolph Green to Leonard Bernstein\n\n[August 1968]168\n\nDear Lenny,\n\nNot being one of the valiant ones (already, is this sentence defiantly grammatical???), I have tasted death on a considerable number of occasions (no, but it's certainly going to be haphazardly, helplessly, convolutedly kind of maybe Jamesian) \u2013 well, anyway, blackest despair \u2013 the dark night of the soul \u2013 the journey into cold sweatland 7 nights a week, the sprawling, crawling, pumping, exploding, expanding and contracting, 4 A.M. \u2013 Mippy-Bosch time \u2013 lost and screaming in the depths \u2013 the pits of hell. (Wow!!)\n\nHowever \u2013 like others of my ilk who have descended whimpering into the maelstrom, I have from time to time been conversely blessed (this is not Jamesian at all, it turns out \u2013 rather Micawber-Dickensian), emerging suddenly from darkness to light, tasting unexpected moments of re-birth.\n\nMost fortunately, in one or two of the instants that I have been re-born, I have known it and recognized it, not by hindsight, in a sad desperate, wandering through the jungles of Recherche \u00e0 temps perdu (wow, wow!!) in search of a landmark that may never have been there \u2013 but known it and gloried in it, at the very moment of borning.\n\nOne such on-the-spot recognized re-birth occurred on a hot July night in 1937, when I, a sallow, bloated (195 lb.) unemployed Hungarian-American Pirate King-to-be, disembarked on the steps of the mess Hall of Camp Onota, to be greeted by my mentor, monitor, R[obert] Weil, counselor of Dramatics, who had loyally and rigorously and self-effacingly plotted and planned my engagement extraordinaire at Uncle Lou's Heavenly Haven for Healthily well-fed young Hebrews.\n\nI was immediately introduced by Robert to you, a handsome lad of possibly 12 to 14, or so you appeared in the well-gathered post-dusk as you came down the steps to greet me and whisked me at once indoors, to the inner sanctum of the outer mess-hall \u2013 with piano, for our first music quiz. About 5 minutes after we had made our first tentative and mutually suspicious hellos and I had triumphantly _not_ identified your Anna Sokolov music (??) as Shostakovitch (was this the future Miss Turnstiles??) and you had sprung up and thrown your arms about me for accomplishing this sensational-non-feat, I felt a sudden, complete exuberance, the fresh air of 1,000,000 windows opening simultaneously and a sense that my life had been building towards a turning point and that it had happened \u2013 now.\n\nMy sense of the \"turning point\" was as sure and conscious, as Judy H[olliday] knew and felt in her own more subterranean way about me when we met the following summer, and she took me up to her apartment and pressed my tattered trousers, on an ironing board with me standing around before the eyes of her horrified mother in my less than provocative jockey-shorts. She told her mother that this unappetizing young stranger and she were going to go places together and very soon (based on no knowledge, information or past association whatever).\n\nBut back to us (or have I left us at all?), since in not too long a time to come from _our_ meeting, we were all 3 and 4 and more bound together with mysterious and continuing consequences that still continue, on and on.\n\nWe trouped the Onota Hills that night, for hours and hours up and down, to the dock and back to the camp gate and up and around the bunks and back and forth, and every moment was a new miracle. I knew I had been listening to music all these years and making my funny and odd full-orchestra phonograph sounds, with soloists thrown in simultaneously, in preparation for this meeting and that I had been carefully rehearsing Sibelius' 5th Symphony to give you its definitive performance that night. All those seemingly hopeless years that I wandered around NY in my sloppy shabbiness, conspicuously sporting rubbers over my shoes in every kind of weather, were not hopeless, were not wandering at all. I had always been on the road, leading straight to these hills, this night.\n\nHave I ever told you that I often used to sing aloud to myself in those days as I stumbled around Times Square or Bryant Park \u2013 Brahms' First, the Sibelius 5, _Petrouchka_ , or whatever, always hoping, some kindred spirit would perk up his or her ears and join me on the next phrase with arms thrown round me? I was looking for _you_ to join me the whole time.\n\nOn and on we hiked that night, and the miracles kept exploding around me. Why had I treasured every word and measure and record scratch of [ _I Wish That I Was Born In_ ] _Borneo_ , all those years from my pre-historic childhood, if not to share them with you with that night? There was no other soul in the universe, besides you and me, to savor and bellow that Crumit masterpiece with every treasured nuance, into the starry midnight sky. Somehow, someway back in Sharon and the Bronx we had stored up _Borneo_ , shored up our memories of it, for this night.\n\nWhat am I wheezily, puffily, floridly trying to tell you??? The simple fact that suddenly there was meaning in my life. I felt _alive_. There was _Borneo_ , Sibelius, T. S. Eliot, _Alice In Wonderland, L'Histoire du soldat, Of Thee I Sing_ , with Auden and Spender, Gilbert and Sullivan, old old movies, Palestrina, Black Pete, eighteen million wires and associations I had been waiting all this time to connect with.\n\nI knew as we walked and sang and talked, that you, the boy L. B., was nothing less than a genius, but this knowledge was only another comfortable fact to me, by now \u2013 part of the magic of our continuing dialogue.\n\nLeonard, my friend, it seems that you are now 50 or about to be, and I am 50 and certainly have been (and never the twain shall meet, and that last phrase certainly means nothing, but my pen jes' wrote it down).\n\nWhatever our ages, and until we stop all walking, we are still taking that walk in the night around the Onota hills. It seems haphazard, and unexplained as ever, but it goes on, and it is all still the same moment of re-birth of me.\n\nHow happy your friendship makes me. It fills me with the simple and complicated joy of knowing there can be a meaning to life \u2013 that our haphazard and rambling walk is filled with endless connections into the past and the future.\n\nHwhatt I'm saying is \u2013 I'm not writing \u2013 I'm only looking for a way to say I love you, my friend. Happy birthday.\n\nAdolph\n\n539. Harold Byrns169 to Leonard Bernstein\n\nKudamm 50A, West Berlin, Germany\n\n28 September 1968\n\nMy dear Leonard Bernstein,\n\nIt was more than a good deed (just here in Germany) to bring Mahler's Fifth. Since I have done this work innumerable times myself (as all other works by our beloved Gustav Mahler), I am in a position to really embrace you as a _Br\u00fcder in Apoll_ , as Mahler called people (la Burlesca of the Ninth). Your insight into the \"cookies\" of Mahler's spiritual secrets touched me profoundly. You see, right after coming back from the USA in 1952, I started to re-introduce Mahler in many a European country. Your Mahler-cycle on records is one of the great achievements of our period, I mean also in a \"missionary\" way.\n\nWhen Mrs. Byrns and I came backstage after your last concert, you asked me \u2013 after Shonah Tovah and before Leihitraoth170 \u2013 if I played still under Mahler. Great Jove, no! I was not even 7 when he passed on.\n\nBut I wanted you to see a fagotto part retouched 3 times by Mahler's own hand. In fact, being a close friend of Alma Mahler and Franz Werfel, I studied most of Mahler's work anew with his _manuscripts_. Quite an exciting experience.\n\nPerhaps, you do remember me, my name was, until 1939 _Bernstein_ and it was our great late friend K[o]ussevitzki who suggested to me to change into Byrns. It was K[o]ussevitzki who gave me the first breaks in the USA.171\n\nAnd more than that: some time ago, many European papers carried headlines in the music columns: \"Two Bernsteins get the Mahler Medal\" from the Bruckner Mahler Society of America, as some people still remember me under my real name (which is your name!).\n\nFrom 1952 I conducted Mahler cycles in many European centers (Vienna, Roma, Torino), where I am for the past 17 years steady guest. Also I am steady guest conductor at the Norddeutsche Rundfunk Hamburg and Hanover (I was actually born in Hanover, where my father had one of Europe's leading concert managements \u2013 Gieseking, Erdmann, Hindemith, etc., etc.) [...]\n\nMahler, my dear Leonard Bernstein, is the very backbone of my spiritual and human existence. I was 15 when I played (4-mains) the _Fifth_ with a schoolmate, and I knew I would be a conductor and my life's main task would be to help Mahler. _His day_ , indeed, _has come_!\n\nI was interested to hear you went to [Walter] Felsenstein's _Traviata_. He is one of my oldest friends and I conducted _Entf\u00fchrung_ and _Zauberfl\u00f6te_ and Mahler's 3rd and 4th at the Komische Oper 9 and 8 years ago! Yes, you are so right (and so said Klemperer): \"the chorus is fabulous\". [...]\n\nAnd now let me sign off. I want to reassure you that this missive is a sign of my \u2013 how shall I say \u2013 inner relation, and _subsequently_ friendship for another Bernstein.\n\nTake good care of yourself and, when I come to New York again, I should very much like to just chatter with you. There are many things about Mahler's life and stories (true ones) that you may not know yet. Have you been to Toblach where I induced the mayor of the town (with the Wiener Mahler Gesellschaft) to have the plaquettes on the tiny cottage and the Bauernhaus (where I stay frequently)?\n\nAgain, lehitraoth, mazal tov,\n\nHarold Byrns\n\n540. Leonard Bernstein to Adolph Green\n\n2 December 1968\n\nTo Adolph, on his 50th(?)172 Birthday\n\nROUND NUMBERS...\n\nAre misery to escape from:\n\nThey're so round.\n\nThe roundest is O,\n\nNext roundest 100,\n\nAnd neither experienceable.\n\nWhich leaves 50,\n\nRound enough, God knows,\n\nA hollow glob\n\nWithin which\n\nWe skitter and slide\n\nLike a doomed bug.\n\nHuis clos.\n\nThere is, however,\n\nComfort, however freezing cold.\n\nTomorrow, yeah, tomorrow,\n\nWe are in our Fifty-First Year,\n\nAnd the perfect N-dimensional circle\n\nIs busted.\n\nSo live for tomorrow (song title).\n\nI did, and am still alive, barely.\n\nSo I clasp you\n\nIn my freezing cold embrace,\n\nAnd comfort you with refrigerated love.\n\nAs the Romans would say,\n\nL\n\n541. Leonard Bernstein to Alan Fluck173\n\nHotel de Paris, Monte Carlo, Monaco\n\n11 September 1968\n\nDear Alan,\n\nFor the puzzle, for your warmth and ingenuity, for all these dear signatures, for the beauty of the idea, for the drawing, for... well for all the love contained in that charming packet, I send my sincerest thanks. It makes me want to sit down immediately and write the Youth Orchestra an overture. Alas, there is hardly even time to write this letter. It was only last night, after a concert and much grisly socializing with princely royalty and TWA officialdom, that I attacked your puzzle, and regained some cheerfulness and faith in what has seemed to me these last weeks a fairly hopeless world. Thank you for that. And today, owing to a sudden bad cold-cum-cough-cum-you name it, I am in the enviable position of being able to cancel lunches, etc, and spend my one free day of the tour in bed, and writing you this letter. I don't mind the work, you understand (although the programs are very heavy) but the \"official\" part of it \u2013 receptions, press conferences, etc. \u2013 have felled me. And so I cough my farewell to you, sneeze a kiss to all the Farnham youth, and thank you rheumily again for the brilliant puzzle. Hope to see you in London.\n\nAffectionately,\n\nLenny B\n\nP.S. Why was the _Candide_ mss. clue (65 across) inverted? Some subtle meaning I've failed to catch?\n\n542. Randall Thompson to Leonard Bernstein\n\n22 Larch Road, Cambridge, MA\n\n27 October 1968\n\nDear Lenny,\n\nI felt grateful to you when I heard that you and the Philharmonic were to play my Second Symphony \u2013 \"our symphony\".174 Now that you have done so and I have heard the way you did it, I have no words worthy to express my gratitude and admiration. At a rough guess this must have been about the six-hundredth performance. I have never heard a more beautiful one, or one that expressed so fully and so lovingly what I wanted to say. My hearty congratulations and thanks to each and everyone concerned. You have given me a joyous experience.\n\nGratefully & devotedly, your old friend\n\nRandall\n\n543. Leonard Bernstein to Alan Fluck\n\n17 September 1969\n\nDear Alan,\n\nI have just listened to young Overbury175 playing the _Anniversaries._\n\nReactions:\n\na) You are an angel to have sent the tape. A perfect birthday present.\n\nb) The boy is marvelous. A natural. One reservation: a feeling of dynamic sameness, lack of contrast. But this may well be due to tape difficulties. The whole tape came over very distantly on my machine, necessitating full volume turn-up.\n\nc) I am moved to write a hundred more _Anniversaries_.\n\nd) I love you for being so good to Helen.\n\nSummation: Thank you, bless you, more power to you.\n\nAlways,\n\nLenny B\n\n544. Elliott Carter176 to Leonard Bernstein\n\nMead Street, Waccabuc, NY\n\n24 October 1969\n\nDear Lennie,\n\nHere is a great deal of the _Concerto for Orchestra_ we spoke about.177 I appreciate your interest and am sending a copy which is still somewhat in the state of a sketch: i.e. the percussion part throughout will probably be revised somewhat in order to scale it up or down in the light of the entire work \u2013 the third movement \u2013 287\u2013419 \u2013 may be slightly revised, and the fourth movement I am still in the state of writing out in score having about 100 measures more to do, which I hope to finish in the next two weeks.\n\n_The score you are receiving is not to be used for conducting_ (if it could with its crazy binding) _because a number of details_ (mistakes and changes) will be different in the final score.\n\nThe work was originally suggested to me by St. John Perse's long poem about America _Vents_ which has a rather large Whitmanesque vision of winds blowing over our continent, changing everything, wiping away the old world, bringing in the new, and of the poet stating:\n\nO vous que rafra\u00eechit l'orage... Fra\u00eecheur et gage de fra\u00eecheur.\n\n... Et vous avez si peu de temps pour na\u00eetre \u00e0 cet instant.\n\nSuch lines as the opening of the poem:\n\nC'\u00e9taient de tr\u00e8s grands vents sur toutes faces de ce monde\n\nDe tr\u00e8s grands vents en liesse par le monde, qui n'avaient ni d'aire ni de g\u00eete,\n\nQui n'avaient garde ni mesure, et nous laissaient, hommes de paille\n\nEn l'an de paille sur leur erre... Ah oui, de tr\u00e8s grands\n\nvents sur toutes faces de vivants!\n\nOr\n\n(Ces grands vents)\n\nSur toutes choses p\u00e9rissables, sur toutes choses saisissables, parmi le monde entier des choses...\n\nEt d'\u00e9venter l'usure et la s\u00e9cheresse au coeur des hommes investis,\n\nCar tout un si\u00e8cle s'\u00e9bruitait dans la s\u00e9cheresse de sa paille, parmi d'\u00e9tranges d\u00e9sinences: \u00e0 bout de cosses, de siliques, \u00e0 bout de choses fr\u00e9missantes.\n\n[...]\n\n(The poem ends:)\n\nQuand la violence eut renouvel\u00e9 le lit des hommes sur la terre,\n\nUn tr\u00e8s vieil arbre, \u00e0 sec de feuilles, reprit le fil de ses maximes...\n\nEt un autre arbre de haut rang montait d\u00e9j\u00e0 des grandes Indes souterraines,\n\nAvec sa feuille magn\u00e9tique et son chargement de fruits nouveaux.178\n\nAs soon as the piece began to take shape, however, I forgot about the poem and find its false epic tone a little too bombastic for my taste. I have quoted it at length because [it] gave me the overall mood of an idea of the work, which was finally reformulated into my own, human and musical terms.\n\nTechnically this is a work built on four main strata of chords, with different interval structures. The normal state of these are five note chords and their complementary seven note chords (of which pairs there are 38, all of which are used).\n\nThe five note chords are associated with the four movements of the work [...] Each of the movements is embedded in the others \u2013 movement I contains elements of II, III, IV etc. as on page 1 of analysis, and emerges and disappears throughout the work.\n\nThe opening and some of the important climaxes (138\u2013140, 285\u20136 and the coda) combine elements of all four movements simultaneously, as indicated at the top of p. 1 of the analysis.\n\nEach of the movements has a characteristic tessitura, as the analysis shows, and a characteristic temporal behavior. Movement II appears in a fast version at the beginning, and during its long statement gradually slows down, a pattern it continues at each secondary appearance. The reverse is true of mvt IV. Movements I and III combine ritardation and acceleration: mvt 1 tends to start each successive pattern of ritardation at a faster point, and mvt III starts each successive pattern of acceleration at a slower point. All of this is not carried out too schematically.\n\nAs for the many \"metrical modulations\" which occur in the work: the tempi of this work should be somewhat flexible since it deals primarily in written out rubati, but the carry-over of note values from one tempo to another should be quite exact so as not to break the continuity. For instance, from 26\u201327 the septuplet of 8th notes of the piano should be exactly the same speeds as the regular 8th notes in 27; in 40 to 41, the piano's triplets in 40 should flow evenly into the septuplet under the triplet, the beats in the piano's left hand in 40 state the tempo of 41, as I have indicated in red pencil [...] The same applies to all other \"metric modulations\" which I think are clearly marked.\n\nThese changes should be fairly accurate as the whole pacing of the work depends on progressive reappearances of materials coming in at different tempi, i.e. mvt IV material comes in bit by bit faster over the entire work, and this will not come out if the tempo changes are not reasonably accurate.\n\nAs you can see, while the work has an underlying structure of chords and tempi, it should be played in a way that gives the impression [of] freedom almost abandon.\n\nIf I can clarify anything please feel free to call on me \u2013 we shall be returning to 31 W 12th St, WA-9-1618, in a week or so, and hope to get in touch with you and see you quite soon after.\n\nWith kindest regards to Felicia,\n\nElliott\n\n1 Bernstein, taped interview with John Gruen, Italy, 1967, transcription online at (accessed 19 March 2013).\n\n2 Jule Styne (1905\u201394), British-born American composer whose Broadway successes included _Gentlemen Prefer Blondes_ , _Bells Are Ringing_ , and _Gypsy_.\n\n3 This telegram was sent the day of Bernstein's Carnegie Hall concert with the New York Philharmonic which included Schumann's _Manfred_ Overture, Strauss' _Don Quixote_ , the American premiere of Shostakovich's Piano Concerto No. 2 (with Bernstein as soloist), and Ravel's _La Valse_.\n\n4 The only New York Philharmonic programme in which Bernstein conducted both Stravinsky's _Rite of Spring_ and a work by Webern (the _Six Pieces for Orchestra_ , Op. 6) was at Carnegie Hall on 16, 17, and 19 January 1958.\n\n5 Robbins' idea for a theater piece based on the Beat Generation was sparked off by the interviews he enclosed with the letter. The first can be identified from Robbins' quotations: it was given by Jack Kerouac to Mike Wallace and published in the _New York Post_ on 21 January 1958, soon after the publication of Kerouac's _On The Road_ (Wallace's first question was: \"What is the Beat Generation?\"); the second was almost certainly another Wallace interview with Beat poet Philip Lamantia. See \"Interview with Jack Kerouac: Lowell Author Gives His Version of the Beat Generation,\" _New York Post_ , 21 January 1958, reprinted in Kevin J. Hayes, ed. (2005), _Interviews with Jack Kerouac_. University Press of Mississippi, pp. 3\u20136.\n\n6 The planned program mentioned in this letter took place a year later, in Carnegie Hall on 9, 10, and 11 April 1959. The concerts included Handel's Harpsichord Concerto in F, Mozart's Piano Concerto in C, K467, the first New York performance of _Symphony of Chorales_ by Lukas Foss, and Wagner's _Tannh\u00e4user_ Overture. Reviewing the concert in _The New York Times_ (11 April 1959), Howard Taubman praised Foss' \"finely balanced\" interpretation of the Handel, in which he played the harpsichord, and enjoyed his stylish Mozart playing that had \"elegance, but not at the sacrifice of virility.\" Foss' _Symphony of Chorales_ was a commission by the Koussevitzky Music Foundation. Taubman was largely unimpressed: \"The first movement has some brilliance, the second some appealing serenity of mood and the third an attractive pastoral quality. But all of it goes on and on with a meandering garrulity. Agreeable ideas are worried and turned ponderous. It is all reminiscent of a diluted Mahler.\"\n\n7 See Letter 416, describing the same evening.\n\n8 _The Firstborn_ by Christopher Fry opened at the Coronet Theatre on 30 April 1958. It included incidental music by Bernstein. The cast was led by Anthony Quayle; it also included Michael Wager, a friend of the Bernsteins.\n\n9 The Aula Magna of Caracas University, built in 1952\u20133. The architect was Carlos Ra\u00fal Villanueva. Alexander Calder's magnificent \"flying saucers\" are memorable visually and useful acoustically.\n\n10 The new President of Venezuela was Wolfgang Larraz\u00e1bal, who was in office for less than a year, from 23 January to 14 November 1958.\n\n11 Vice-President Nixon's trip to South America (27 April\u201315 May) revealed to the US government just how bad relations were with Latin America, and reached its low point on 13 May when he was attacked by an angry mob in Caracas.\n\n12 Felicia had played Joan of Arc in Bernstein's New York Philharmonic performances of Honegger's _Jeanne d'Arc au b\u00fbcher_ on 24, 25, and 27 April 1958.\n\n13 _Time_ (5 May 1958) described the occasion as \"a family triumph\".\n\n14 Brigitta Lieberson, the wife of Goddard Lieberson, who performed as Vera Zorina. She played Joan in the first American performance of the work, given by the New York Philharmonic under Charles Munch on 1 January 1948.\n\n15 La Belle H\u00e9l\u00e8ne was the Bernsteins' nickname for Helen Coates.\n\n16 Between 29 April and 14 June 1958, Bernstein and the New York Philharmonic undertook an extensive tour of Central and South America, giving concerts in Panama, Venezuela, Colombia, Ecuador, Peru, Bolivia, Paraguay, Chile, Argentina, Uruguay, Brazil, and Mexico.\n\n17 Ellen Adler became Oppenheim's second wife in 1957.\n\n18 Van Cliburn had just won the first International Tchaikovsky Competition in Moscow.\n\n19 Schuman's Sixth Symphony (composed in 1948) was played in Caracas on 2 May, then on 13 May in Lima, the day before Bernstein wrote this letter.\n\n20 Rosamond Lehmann (1901\u201390), British novelist. Her friends included many of the Bloomsbury Group, among them Lytton Strachey and his wife Dora Carrington, Vanessa Bell and Duncan Grant, and Leonard and Virginia Woolf. In the 1940s she had a ten-year affair with the poet Cecil Day Lewis. This warmly appreciative letter about _West Side Story_ was written less than a month before the death of Lehmann's daughter Sally \u2013 a tragedy that virtually put an end to Lehmann's writing career and led her into Spiritualism.\n\n21 Martha Gellhorn (1908\u201398), American journalist and author. Gellhorn's work as a war correspondent started with the Spanish Civil War, which she witnessed with Ernest Hemingway (he dedicated _For Whom the Bell Tolls_ to her, and they married in 1940). She covered Hitler's rise to power, and was one of the first journalists to report on the concentration camp at Dachau. She later covered the war in Vietnam, the Six-Day War in 1967, and numerous other international conflicts. She divorced Hemingway in 1945, and in 1954 she married Tom Matthews, editor of _Time_ magazine. The Martha Gellhorn Prize for Journalism was established in her memory. Her friendship with Bernstein was a curious one. Though both were politically liberal, there were few other obvious connections between the two of them aside from being tennis partners when they were both at Cuernavaca in Mexico; Gellhorn was not musical, but she clearly liked Bernstein. For Gellhorn's remarks on Hemingway, see Letter 427.\n\n22 \"Omi\" was Martha Gellhorn's mother, Edna Gellhorn, n\u00e9e Fischel.\n\n23 Gellhorn had already written to Bernstein about hearing the cast recording at Klosters, Switzerland, earlier in the year. In a letter dated 18 March 1958, Gellhorn told Bernstein that she \"wept at parts, and roared with laughter at others. It is almost tangible music.\"\n\n24 Tom Matthews, former editor of _Time_ Magazine and Martha Gellhorn's second husband. They married in 1954.\n\n25 Bernstein's broadcast performance of _Arcana_ was released on CD in _Bernstein Live_ (NYP 2003).\n\n26 Diamond's _The World of Paul Klee_ for orchestra.\n\n27 Diamond was working on a musical comedy based on _Mirandolina_ by Carlo Goldoni.\n\n28 Bernstein had been to see the Broadway run of Robbins' _Ballets USA_ , which included _New York Export: Opus Jazz_ (music by Robert Prince and sets by Ben Shahn), Debussy's _Afternoon of a Faun, 3x3_ (with a score by Georges Auric), and _The Concert_ , the first of Robbins' ballets to music by Chopin. _New York Export: Opus Jazz_ was performed again at President Kennedy's 45th birthday party in the old Madison Square Garden \u2013 the occasion when Marilyn Monroe sang \"Happy Birthday, Mr. President.\"\n\n29 These rehearsals were for the cast that was about to take the show to England (including George Chakiris as Riff, Marlys Watters as Maria, Don McKay as Tony, and Chita Rivera as Anita), and the \"European conductor,\" Lawrence Leonard. It was first seen at Manchester Opera House on 14 November 1958, before heading to London where it opened at Her Majesty's Theatre on 12 December and ran for 1,039 performances.\n\n30 The paragraph about _Dybbuk_ is a reminder of how long Robbins and Bernstein spent contemplating this project. The ballet was completed in 1974, but they had first considered the story soon after collaborating on _Fancy Free_ , three decades earlier, and Robbins was eager to make progress straight after _West Side Story_. The proposed collaboration with the artist Ben Shahn never came about because he died in 1969. (When _Dybbuk_ was presented by New York City Ballet in 1974, the designs were by Rouben Ter-Arutunian.)\n\n31 Thornton Wilder (1897\u20131975), American playwright and novelist, author of two of the most celebrated plays written for the American stage \u2013 _Our Town_ and _The Skin of Our Teeth_ \u2013 and the novel _The Bridge of San Luis Rey_. All three won Pulitzer Prizes for Wilder.\n\n32 Alma Mahler (n\u00e9e Schindler, 1879\u20131964) was married in turn to Gustav Mahler, Walter Gropius, and Franz Werfel. Her memoirs were published in 1958 with the title _And the Bridge is Love_ , a quotation from Thornton Wilder's 1927 novel _The Bridge of San Luis Rey_ (\"There is a land of the living and a land of the dead and the bridge is love, the only survival, the only meaning\"). The sculptor Anna Mahler (1904\u201388) was the second daughter of Mahler and Alma. Like her mother she married several times, including the composer Ernst Krenek, the publisher Paul Zsolnay, and the conductor Anatole Fistoulari.\n\n33 Larry Adler (1914\u20132001), American harmonica player for whom Vaughan Williams and Milhaud composed pieces. Adler was blacklisted in Hollywood and moved to London in 1949.\n\n34 Adler's name was not credited because of his blacklisting; instead the arranger and conductor Muir Mathieson was named as the composer of the score for _Genevieve_ \u2013 an error that was only rectified officially in 1986. The nominations for Best Music Score at the 27th Academy Awards were: _The Caine Mutiny_ (Max Steiner); _Genevieve_ (Larry Adler); _The High and the Mighty_ (Dimitri Tiomkin); _On the Waterfront_ (Leonard Bernstein); and _The Silver Chalice_ (Franz Waxman).\n\n35 Louis Armstrong (1901\u201371), jazz trumpeter and singer. In 1956, he appeared on Bernstein's album _What is Jazz?_\n\n36 Swiss Kriss was a herbal laxative that Armstrong used and promoted.\n\n37 A reference to the early years of the Space Race, starting with the launch of Sputnik 1 in 1957.\n\n38 Gellhorn was married to Hemingway from 1940 to 1945. See Letter 427.\n\n39 Clearly Milhaud had heard the performance by the time he wrote to Bernstein again three weeks later, on 29 January 1959.\n\n40 Mary Rodgers (b. 1931), American composer and author. She is the daughter of Richard Rodgers. A graduate of Wellesley College, where she majored in music, Rodgers had a Broadway hit in 1959 with _Once Upon a Mattress._ She worked as assistant to the producer of Bernstein's Young People's Concerts. In 1972 she published her first children's book, _Freaky Friday_.\n\n41 Joe (Joseph) Roddy (1920\u20132002), American journalist. He worked on the staff of _Look_ magazine and _Life_ magazine as well as writing for _The New Yorker_ , _The New York Times_ , and _Harper's_ magazine. A friend of Bernstein for many years, he was a passionate music-lover who regularly attended rehearsals at the New York Philharmonic.\n\n42 In _The Infinite Variety of Music_ , broadcast on 22 February 1959, Bernstein took the four notes of Irving Berlin's \"How Dry I Am\" (G-C-D-E), showing how they were used by composers from Handel ( _Water Music_ ) to Shostakovich (Fifth Symphony). The script was printed in Bernstein 1966, pp. 29\u201346.\n\n43 The show was _First Impressions_ , a musical based on Jane Austen's _Pride and Prejudice_ , with music and lyrics by Robert Goldman, Glenn Paxton, and George Weiss, and a libretto by Abe Burrows (of _Guys and Dolls_ fame), who also directed. _First Impressions_ opened on 19 March 1959 and closed on 30 May, after just 92 performances. The production was under the overall supervision of the Jule Styne Organization, hence Styne's interest in it.\n\n44 Bernstein didn't write to _The New York Times_ about Atkinson's acidic review. He wrote to Styne that he \"would love to help out and did enjoy the show,\" but that he \"makes it a rule not to do this kind of thing as so many people ask for it\" (from the draft reply in Helen Coates' hand on Styne's letter).\n\n45 _The Joy of Music_.\n\n46 Oliver Daniel (1911\u201390) was an energetic promoter of new music. In 1954 he created the Concert Music Department at Broadcast Music Inc. (BMI), a rival to ASCAP. Daniel also helped to establish Composers' Recordings Inc. (CRI).\n\n47 Stanley Adams (1907\u201394), American songwriter probably best known for writing the English lyrics for _La Cucaracha_. He was President of the American Society, of Composers, Authors, and Publishers (ASCAP) 1953\u20136 and 1959\u201380.\n\n48 Harold Spivacke (1904\u201377), chief of the Music Division at the Library of Congress from 1947 to 1972.\n\n49 Irene Lee Diamond (1910\u20132003), Hollywood script editor and philanthropist. She was the Hollywood story editor who had recommended movie treatments for both _The Maltese Falcon_ and _Everyone Comes to Rick's_ (immortalized as _Casablanca_ ). In later life she became a generous patron of the arts and of AIDS research. She was unrelated to David Diamond.\n\n50 Boris Pasternak (1890\u20131960), Russian poet and novelist most famous for _Doctor Zhivago_ , for which he was awarded the Nobel Prize for literature in 1958. This caused a scandal in the Soviet Union, where the book had been refused for publication owing to its critical stance on Stalin and Socialist Realism (the manuscript was smuggled abroad so that the book could be published). Pasternak at first accepted the Nobel Prize, but after intolerable pressure from the Soviet government (including the KGB surrounding his house in Peredelkino), he was forced to decline it: \"In view of the meaning given the award by the society in which I live, I must renounce this undeserved distinction which has been conferred on me.\"\n\n51 Bernstein's concert on 11 September was Pasternak's first appearance in public after his denunciation following the _Doctor Zhivago_ scandal. Bernstein was overwhelmed by his meeting at Pasternak's _dacha_ , but they never met again as the great writer died a few months later, on 30 May 1960. On 13 July 1960, Rabbi George Lieberman from Long Island, NY, wrote to Bernstein about his admiration for the writer, and to ask whether, during their meeting, Bernstein had touched on Pasternak's Jewish antecedents: \"Was he a formal convert to Christianity? Now that he is no more, we may never know whether he took the step. On the basis of your conversation with him, could you draw any conclusions on this point? Were you in his home in Peredelkino and did you notice any icons in it?\" Bernstein's draft reply is written on the back of Rabbi Lieberman's letter: \"My conversations with P. never touched on this point. (They were in fact virtually monologues by him on aesthetic matters.) But he conveyed the impression of a Tolstoyan Christian, a worshipper of nature and the divine spark in man. I saw no icons at his home. I suspect that he felt little or nothing about _Jewishness_ , though he may have been deeply interested in _Judaism_. This is a guess.\"\n\n52 Karlheinz Stockhausen (1928\u20132007), German composer, and a leading light of the post-war avant-garde.\n\n53 Bernstein's first book, _The Joy of Music_ , was published in 1959.\n\n54 Reiner's home in Connecticut.\n\n55 See Letter 439.\n\n56 Written in French; English translation by the editor.\n\n57 Bernstein's performances of Mahler's _Resurrection_ Symphony in which Roddy sang (as a member of Rutgers University Choir) were given in Carnegie Hall on 18, 19, 20, and 21 February 1960, with Phyllis Curtin and Regina Resnik as the vocal soloists.\n\n58 Bernstein played most of Schuman's suggested program. The only change was to end with Robert Schumann's Third (\"Rhenish\") Symphony instead of the Fourth. The concerts were given on 13, 14, and 16 October 1960 with Aaron Rosand as the soloist in Barber's Violin Concerto.\n\n59 Larry Kert (1930\u201391), American actor and singer who created the role of Tony in _West Side Story._\n\n60 A reference to the only time Bernstein conducted any of _West Side Story_ in the theater: the Overture at the opening night of the Broadway revival on 27 April 1960 (at the Alvin Theatre).\n\n61 Bernstein did not agree to Kert's proposal about the title of his album (a note typed by Helen Coates at the top of the letter says \"cannot use L's name on Album\"). It was released by Secco Records as _Larry Kert Sings Leonard Bernstein_ (CELP 4670). At Bernstein's suggestion (jotted down by Helen Coates at the foot of the letter) the record included \"Build My House\" from _Peter Pan_ as well as songs from _West Side Story_ , _On the Town_ , and _Wonderful Town._\n\n62 Robbins' entertaining description of his time in Hollywood working on the early stages of the film version of _West Side Story_ shows just how tenacious he needed to be (\"like a sheep dog\") and shows the obvious concern he had for the way the show was being treated. However, a sharply contrasting account can be found in the long letter from Saul Chaplin to Bernstein about Robbins' work on the film (see Letter 462).\n\n63 Robbins was angered by an unidentified newspaper report about his treatment of \"Somewhere\" in _West Side Story_. It is likely that he is referring to Bernstein's comments (hitherto kept private) about Robbins changing the orchestration of \"Somewhere\" at the 1957 dress rehearsal of the show in Washington, DC. Amanda Vaill summarized what happened at this rehearsal as follows: \"As the orchestra swelled into the lush refrain of 'Somewhere' after the ballet's end, Jerry sprang out of his seat and ran down to the pit to demand that the conductor, Max Goberman, cut the orchestration and give the first bars to an unaccompanied flute. 'Take that Hollywood shit out!' he cried. Without a sound, Lenny Bernstein got up and went to a bar across the street where Steve Sondheim found him staring at a row of neat Scotches lined up in front of him. Jerry later said he didn't realize Lenny was even in the theatre at the time and thought he'd understand that to make the 'extremely sensitive transition' into the duet work right, 'the song should start simply, purely.' [...] Jerry also admitted that 'my tactics were not of the best'\" (Vaill 2007, p. 287). Evidently, some version of this story had got out and Robbins was irritated. This letter is of more general interest for Robbins' very positive view of their collaborations and their friendship.\n\n64 Mitropoulos died of heart failure on 2 November 1960, a few months after sending this letter.\n\n65 Saul Chaplin (1912\u201397), American composer and film-music supervisor. He first met Bernstein in 1944 and saw him conduct _Fancy Free_ at the Hollywood Bowl in August that year. Chaplin was Music Supervisor and Associate Producer for the film version of _West Side Story_.\n\n66 The Prologue of _West Side Story_ , which was extensively reworked for the film.\n\n67 Bernstein and the New York Philharmonic arrived in Vancouver on 14 August, and gave concerts in the city on 15 and 16 August.\n\n68 Queen Elizabeth II and Prince Philip stayed at the newly opened Faculty Club in July 1959, a year before Bernstein's visit. Prince Andrew was born on 19 February 1960 (so the story is, indeed, a canard).\n\n69 The New York Philharmonic gave a concert in Denver, CO, on 13 August.\n\n70 Bernstein appeared as soloist in Beethoven's Piano Concerto No. 1 in one of the Vancouver concerts.\n\n71 Bart\u00f3k's _Concerto for Orchestra_ and the Second Suite from Ravel's _Daphnis and Chloe_ were on the Vancouver programs.\n\n72 David Keiser was President of the Board of the New York Philharmonic.\n\n73 Edwin Denby, librettist of _The Second Hurricane_.\n\n74 Marni Nixon (b. 1930), American soprano whose concert repertoire includes works by Schoenberg, Webern, Ives, and Boulez, and whose parallel career has been as the dubbed singing voice of several screen legends in film musicals, including Marilyn Monroe (the high notes in \"Diamonds Are A Girl's Best Friend\"), Deborah Kerr ( _The King and I_ ), Audrey Hepburn ( _My Fair Lady_ ), and Natalie Wood ( _West Side Story_ ).\n\n75 On Bernstein's recommendation, Marni Nixon was chosen to dub Natalie Wood's singing voice as Maria, and she was in Hollywood less than a month after sending this letter. She was widely experienced in contemporary classical repertoire (on 31 March 1960 she had given the first American performance of Boulez's _Improvisation sur Mallarm\u00e9 I_ with Bernstein and the New York Philharmonic), and was already familiar to film studios as the singing voice of Deborah Kerr in _The King and I_. She arrived in Hollywood in late September 1960 to work on _West Side Story_ for $300 a day, and had to tread a delicate path, since Natalie Wood was convinced she could do her own singing. Nixon's own account of the recording deserves quoting at length: \"The sessions were set up so that Natalie would first record a song on her own and then I would get up and record the whole song again. I knew that this was going to be embarrassing and traumatic for her, but I was just an employee and went along with it. Musicians, especially ones this good, are noted for their disdain of mediocrity [...] As Natalie sang, they showed their displeasure by playing poorly. They kind of sawed away at the notes instead of playing with the sensitivity of which they were more than capable. Then, when I got up to sing the identical song, the same musicians would sit up in their seats and play with renewed vigor and passion. When I finished a take they would even applaud. I was both very embarrassed and disgusted at their rudeness to poor Natalie who was, after all, doing her best\" (Nixon 2006, pp. 133\u20134).\n\n76 Nadia Boulanger conducted a series of concerts with the New York Philharmonic in February 1962 (see Letter 472).\n\n77 This concerns the first performance of the _Symphonic Dances from West Side Story_ , which Foss conducted with the New York Philharmonic in Carnegie Hall on 13 February 1961 at the _Valentine For Leonard Bernstein_ , a Pension Fund Benefit Concert, concluding a first half that had opened with Aaron Copland conducting the _Candide_ Overture followed by the _Jeremiah_ Symphony conducted by Vladimir Golschmann, with Jennie Tourel as the soloist. The second half of the programme, produced by David Oppenheim and presented by Betty Comden and Adolph Green, was a series of \"Valentine Surprises\" of music from Bernstein's Broadway shows and _Fancy Free_.\n\n78 The Ojai Festival, in California, was directed by Foss in 1961, the occasion on which the programme he outlines was given with Andr\u00e9 Previn, Shelly Manne, and others.\n\n79 Given the scoring (including 4 trumpets and trombones), this note is almost certainly in connection with Ramin's orchestration of the _Fanfare for the Inauguration of John F. Kennedy_ , first performed in January 1961, which Ramin orchestrated for 1 piccolo, 2 flutes (ad lib.), 2 oboes, 2 clarinets, 3 horns, 4 trumpets, 4 trombones, timpani, and percussion.\n\n80 Frank Sinatra (1915\u201398), American singer, actor, and entertainment legend. He had known Bernstein since the early 1940s when they had both worked in New York nightclubs such as the Riobamba (see Letter 115).\n\n81 Sinatra is confirming arrangements for the Inaugural Ball for President Kennedy at which Bernstein conducted his _Fanfare for the Inauguration of John F. Kennedy_.\n\n82 This brutally frank account of Jerome Robbins' work on the movie of _West Side Story_ was intended for Bernstein's eyes only, but it conveys something of the profound frustration felt by the producers \u2013 Robert Wise (also the movie's co-director), Walter Mirisch, and Saul Chaplin \u2013 about what they felt to be Robbins' high-handed and unreasonable behavior on the set, as well as Chaplin's concern that Robbins was giving a very selective account of the movie's problems in his reports back to Bernstein.\n\n83 _Bonanza Bound_ was a 1947 musical set in Alaska in 1898, with a score by Saul Chaplin, book and lyrics by Betty Comden and Adolph Green, sets by Oliver Smith, and musical direction by Lehman Engel. Try-outs opened at the Shubert Theatre in Philadelphia on 26 December 1947, but despite the talent involved in the show, it closed there a week later, never reaching Broadway. Bernstein toured to Alaska with the New York Philharmonic in 1961.\n\n84 Jack [Joachim] Fishberg (1904\u201370) was a violinist in the New York Philharmonic for forty-four years.\n\n85 The first paragraph of this letter is in English. The remainder is in French, translated by the editor.\n\n86 Theodore Hollenbach, conductor of the Rochester Oratorio Society from 1945 to 1986. Felicia was performing Honegger's _Joan of Arc at the Stake_ in Rochester.\n\n87 An earlier version of this acrostic, dated 22 March 1957, is printed in Bernstein 1982, p. 136. It contains several differences from the text printed here.\n\n88 The film version of _West Side Story_ had its premiere in New York on 18 October 1961.\n\n89 Written in French; English translation by the editor.\n\n90 Igor Stravinsky (1882\u20131971), Russian-born composer; a dominant figure in twentieth-century music. His professional relationship with Bernstein was always friendly, but as Charles M. Joseph writes: \"although his correspondence with Bernstein was cordial enough, the [Stravinsky] archives disclose some hostility in Stravinsky and his circle of friends. Bernstein's _The Age of Anxiety_ , Symphony No. 2 of 1949, was inspired by W. H. Auden's book of the same name. Auden was not pleased and quickly distanced himself from any association with the work. He gave Stravinsky a copy of the book in which he wrote on the title page, 'Leonard Bernstein is a shit.' Stravinsky himself, Craft told me, walked out of a performance of the same symphony and wanted to leave _West Side Story_ as well. [...] Perhaps at the root of this reprobation was an envy of Bernstein's public appeal, as well as an aversion to the charismatic conductor's mission to catechize about music, to convince people 'what to feel', as Stravinsky described it.\" (Joseph 2001, pp. 226\u20137). Joseph also points out that \"Commercial record sales were another target of Stravinsky's annoyance. Bernstein was a major competitor for recording royalties, hitting home in a way that Stravinsky could not fail to notice. John McClure of Columbia Masterworks recalled that Stravinsky never accepted the fact that other conductors, especially dynamic ones like Bernstein, could steal the composer's thunder\" (Joseph 2001, p. 229).\n\n91 Pencil draft of the letter Bernstein sent.\n\n92 An earlier title for _The Flood_ , commissioned by CBS Television and first broadcast on 14 June 1962.\n\n93 Stravinsky was eager to see the broadcast of \"Happy Birthday Igor Stravinsky,\" Bernstein's _Young People's Concert_ first relayed on 26 March 1962. He sent a telegram to Bernstein on 6 April: \"Please tell Roger Englander Saturday May 5 is ideal for me. Many thanks. Would you lunch with me before or after the screening? Another great favor I have to ask is could you please record the three snippets John McClure has from _The Flood_. Cordially, Stravinsky.\" A draft reply is written below, partly by Bernstein and partly by Helen Coates: \"Everything set for screening Sat. May 5 at 1 p.m. Delighted to have lunch with you afterwards. Will make every effort to accommodate you on _The Flood_ snippets. Warmest greetings.\" Bernstein did in fact record part of the soundtrack for the CBS television relay of _The Flood_. According to Charles M. Joseph's _Stravinsky Inside Out_ : \"The work was finally recorded at the CBS studios in Hollywood on 31 March. [Robert] Craft led the orchestra for the audio taping (although the televised program led the audience to believe that Stravinsky himself was on the podium), and Leonard Bernstein helped as well. A week later the composer wrote to McClure in New York, providing specific instructions for a passage to be recorded by Leonard Bernstein\" (Joseph 2001, p. 152). Joseph goes on to quote from Stravinsky's 6 April letter to McClure, is which the composer raises the question of how the credits should read for the shared conducting duties: \"It seems to me you invite speculation and call undue attention to a problem by saying two people conducted and not saying who conducted what. Therefore, leave out the word 'conductor' _entirely_. Say 'recording supervised by the composer'\" (Joseph 2001, p. 152). Bernstein himself believed that Stravinsky's coolness towards him near the end of the composer's life was down to Robert Craft: \"I could _kill_ him \u2013 I mean, he spoiled such a lovely relationship between Stravinsky and myself\" (Cott 2013, p. 31).\n\n94 Nadia Boulanger conducted four concerts with the New York Philharmonic on 15, 16, 17, and 18 February 1962. The program comprised Faur\u00e9's Requiem, the premiere of a specially made orchestral version of Virgil Thomson's _A Solemn Music_ , and three Psalms by Lili Boulanger. There was another, more somber reason for this being a \"moving week\": the Faur\u00e9 Requiem on 17 February was dedicated to the memory of Bruno Walter, who had died that day. Bernstein addressed the audience before the performance:\n\nMy dear friends, I bring you the heartbreaking news that Bruno Walter died this morning. It is almost too much to bear. Last year our beloved Mitropoulos \u2013 and now this great genius, who for forty years has been so close to us here at the Philharmonic \u2013 who has guided us so wisely, and so generously brightened and enriched our lives. Like Mitropoulos, he was one of the saints of music \u2013 a man all kindness and warmth, goodness and devotion. We can only mourn, and pay tribute. I would like to add that the Philharmonic and Mlle Boulanger will perform the Requiem of Faur\u00e9 in memory of Bruno Walter.\n\n95 Presumably Boulanger's sister, Lili (1893\u20131918).\n\n96 Nina Maria Felicia Bernstein was born on 28 February 1962.\n\n97 Written in French; English translation by the editor.\n\n98 Nina Bernstein.\n\n99 Rudolf Bing (1902\u201397) was General Manager of the Metropolitan Opera from 1950 to 1972.\n\n100 A draft of Bernstein's reply (in Helen Coates' hand) is attached: \"Whole starting point of this venture was the prospect of collaboration with Z[effirelli] & without that collab. the whole project loses its orig. meaning. It was not simply a case of finding a suitable opera to conduct at the Met, but rather to make my first appear. at the Met in this specific collaborative enterprise. I must therefore insist on collaborating with Z. or have approval of a substitute.\"\n\n101 Louise Talma (1906\u201396), American composer. She was a pupil of Nadia Boulanger and a regular at the MacDowell Colony, along with composer friends of Bernstein such as Irving Fine, Lukas Foss, Harold Shapero, and Arthur Berger. In her will, Talma left a bequest of one million dollars to the MacDowell Colony. Talma's letter refers to Bernstein's first visit to the Colony, in 1962, when he went there to work on the _Kaddish_ Symphony.\n\n102 Karl B\u00f6hm (1894\u20131981), Austrian conductor. He established a warm friendship with Bernstein. The two conductors admired each other greatly.\n\n103 B\u00f6hm conducted a month of concerts with the New York Philharmonic in November 1962, sixteen in all between 1 and 25 November.\n\n104 Morton Feldman (1926\u201387), American composer. A leading figure of the musical avant-garde and a pioneer of Indeterminate music.\n\n105 Lina Abarbanell (1879\u20131963) was an opera singer and casting director. She was Marc Blitzstein's mother-in-law. As a singer, her debut at the Metropolitan Opera on 25 November 1905 was as Gretel in the Met premiere of Humperdinck's _Hansel and Gretel_ , given in the presence of the composer. She later became a successful casting director (including _Street Scene_ and the movie version of _Carmen Jones_ ). Though her daughter Eva died in 1936, she remained close friends with Blitzstein. Bernstein included Lina Abarbanell's name in the opening scat trio of _Trouble in Tahiti_ (\"Who but Abarbanel[l] buys a visa\").\n\n106 _Chronochromie_ was composed in 1959\u201360 and first performed on 16 October 1960 by the Orchestra of South-West German Radio conducted by Hans Rosbaud. The New York Philharmonic performed two movements (\"Strophe\" and \"Antistrophe\") at a concert on 24 July 1965 conducted by Lukas Foss.\n\n107 Written in French; English translation by the editor.\n\n108 Written in French; English translation by the editor.\n\n109 In a marginal note, Bernstein has written: \"Jack [Gottlieb] misrepresented my reaction.\" Bernstein performed (and recorded) Feldman's _Out of Last Pieces_ with David Tudor and the New York Philharmonic in February 1964 as part of an extraordinary programme that included \"Autumn\" from Vivaldi's _Four Seasons_ and Tchaikovsky's _Path\u00e9tique_ Symphony in the first half, and music by Cage ( _Atlas Eclipticalis with Winter Music_ ), Earle Brown ( _Available Forms II_ ) and Feldman's _Out of Last Pieces_. Bernstein was fascinated by Feldman's music even if he wasn't particularly sympathetic towards it.\n\n110 Though the recipient of this letter has not been identified, this pencil draft of Bernstein's letter has been included because of his interesting comments on some of the legendary pianists of the past.\n\n111 John Cage (1912\u201392), American composer and musical pioneer.\n\n112 Bernstein did place the orchestral improvisation at the start of the second half of this programme, which was given on 6, 7, 8, and 9 February 1964. The first half included \"Autumn\" from Vivaldi's _Four Seasons_ and Tchaikovsky's _Path\u00e9tique_ Symphony. According to Harold Schonberg's review in _The New York Times_ (7 February 1964) it was an unusually long evening, ending at around 11.05 p.m. Of the second half, Schonberg wrote: \"these pieces, with their new sounds, apparent chaos and weird textures, shook the audience quite a bit. Not unexpectedly, the most unconventional was Mr. Cage's [ _Atlas Eclipticalis_ ]. He used an orchestra of more than 80 players, and each instrument was equipped with a contact microphone that led into a little preamplifier on the floor. This preamplifier led into an electronic mixer, which fed into six amplifiers, which went to six loudspeakers scattered through the hall. The piano was amplified, and on the podium was, instead of a conductor, a mechanical affair with a spoke that slowly revolved. When eight minutes were up, the piece was over. In the Brown piece, though, two live conductors were needed \u2013Mr. Bernstein and the composer. One might think that Mr. Cage's piece, and the others, would have caused some kind of demonstration. What happened was that during the progress of the work, people walked out. When it was over, there was a more general exodus. There were a few lusty boos, a few counter-cheers, but on the whole the music fell flat. So did the music of Mr. Feldman [ _Out of Last Pieces_ ] and Mr. Brown [ _Available Forms II_ ]. The audience, the part that remained, seemed more amused than anything else. Its amusement had started with a demonstration of IBM music, and an improvisation by full orchestra that lasted a minute and a half.\"\n\n113 Draft reply written in pencil on the verso of Cage's letter.\n\n114 Claudio Abbado (b. 1933), Italian conductor. In 1963 he won the Dimitri Mitropoulos Prize, which enabled him to work for several months with the New York Philharmonic. Abbado has kindly supplied this reminiscence: \"For the 1963\u201364 season, I was assistant conductor at the New York Philharmonic, an opportunity given me by the Mitropoulos Prize, which I won that year. So I had the opportunity to watch and conduct major concerts, working with George Szell, Josef Krips and soloists such as Arthur Rubinstein and David Oistrakh, as well as with Bernstein. Just to mention one episode, I remember very well during the rehearsal of Mahler's Second Symphony that Bernstein went to sit in the hall and asked me to get on the podium. He wanted to hear the very complex part, in the finale, where the principal ideas are played by the small orchestra offstage. Bernstein was surprised and very happy that I was aware of how difficult this passage was, though he didn't know that I had actually already done that symphony in Europe\" (Claudio Abbado, personal communication).\n\n115 President Kennedy was assassinated on 22 November 1963.\n\n116 Walter Hussey (1909\u201385) was Dean of Chichester Cathedral from 1955 to 1977, and before that was vicar of St. Matthew's, Northampton. In both these posts he commissioned an extraordinary range of new music, literature, and works of art for the Church. In Northampton these commissions included Benjamin Britten's _Rejoice in the Lamb_ , Gerald Finzi's _Lo, the full, final sacrifice_ , W. H. Auden's _Litany and Anthem for S. Matthew's Day_ , Graham Sutherland's _Crucifixion_ , and Henry Moore's _Madonna and Child_. At Chichester, he continued his commissions, notably stained-glass windows by Marc Chagall, a magnificent tapestry by John Piper \u2013 and Bernstein's _Chichester Psalms_.\n\n117 Iannis Xenakis (1922\u20132001), Greek composer and architect.\n\n118 On 2 January 1964 (repeated on 3, 4, and 5 January), Bernstein conducted the first US performances of _Pithoprakta_ by Xenakis and _Atmosph\u00e8res_ by Ligeti.\n\n119 _Musiques formelles_ was published in Paris in 1963.\n\n120 Harpo Marx (1888\u20131964; born Adolph Marx), American comedian and actor, the second-oldest of the Marx Brothers (Chico, Harpo, Groucho, Gummo, and Zeppo). He was famous for never talking during performances or on screen (though in fact he had a deep, rich speaking voice). Harpo was also a regular of the Algonquin Round Table (with the likes of Dorothy Parker, Alexander Woolcott, George S. Kaufman, and Robert Benchley). He wrote this letter just before his decision to retire from public life, and he died on 28 September 1964.\n\n121 This letter is written on a sheet of manuscript paper, decorated with treble and bass clefs, rests, a fermata, and a double bar at the end of the last sentence. The titles at the head of the sheet have been filled in by Harpo Marx as follows: Prod.: \"Maybe\", Title: \"Letter to Maestro Bernstein\", Page: \"3,472\", Arranger: \"Harpo Marx\".\n\n122 Marc Blitzstein was murdered in Martinique by three Portuguese sailors on 22 January 1964.\n\n123 The Blitzstein Memorial Concert took place at Philharmonic Hall on 19 April 1964. Bernstein included \"With a Woman to Be\" from _Sacco and Vanzetti_ , extracts from _Regina_ and other songs, and a complete performance of _The Cradle Will Rock_ narrated by Bernstein and directed by him from the piano.\n\n124 A reference to the Boston premiere of _Kaddish_ , conducted by Charles Munch on 31 January 1964, the first performance of the work in the United States.\n\n125 Dr. Cyril [Chuck] Solomon was Bernstein's doctor for many years and a personal friend.\n\n126 Bernstein and Copland were both awarded honorary doctorates of music by the University of Michigan, conferred on 19 September 1964 (the date of Bernstein's sonnet), as recorded in _The Proceedings of the Board of Regents (1963\u20131966)_ , University of Michigan, p. 577 (electronic edition: Ann Arbor, MI:University of Michigan, Digital Library Production Service, 2000).\n\n127 The concerts of Diamond's music took place along the lines discussed here: Bernstein conducting the world premiere of Diamond's Fifth Symphony followed by Diamond conducting his Piano Concerto with Thomas Schumacher as the soloist. In the second half, the eventual choice was Sibelius' Second Symphony.\n\n128 The saga of _The Skin of Our Teeth_ is an unhappy one, though it did eventually provide some of the musical material used in the _Chichester Psalms_. As long ago as 12 September 1962, Sam Zolotow reported in _The New York Times_ that \"the Broadway association of Leonard Bernstein, Jerome Robbins, Adolph Green and Betty Comden will be renewed with the song and dance version of Thornton Wilder's fantastic comedy, _The Skin of Our Teeth_ , winner of the 1943 Pulitzer Prize\" and announced that everything would be \"ready for a Broadway presentation during the 1964\u201365 season. [...] As explained yesterday by Mr. Bernstein, the reason for the long-range project is that 'we haven't started work on it yet.' Miss Comden said: 'We have been meeting on and off whenvever we can. Intensive work, however, will be done during the 1963\u20134 season.'\" A year later, on 29 August 1963, Zolotow wrote of it as \"an early 1965 entry,\" to which Robbins would turn his attention after the upcoming \"Tevye\" \u2013 not yet called _Fiddler on the Roof_. Another year later, on 4 September 1964, Zolotow wrote of the $400,000 investment in the project from CBS (following the success of their backing of _My Fair Lady_ and _Camelot_ ) and promising that Columbia Records would be making the cast recording of _The Skin of Our Teeth_. But by the New Year, the whole project had fallen apart and the persistent Zolotow delivered the bad news on 5 January: \"Leonard Bernstein, Jerome Robbins, Adolph Green and Betty Comden have cancelled their plans to do a musical version of Thornton Wilder's Pulitzer Prize play, _The Skin of Our Teeth_ ,\" adding that Bernstein \"had told a friend that six months of work had gone in the wastebasket due to a dispute with his colleagues.\"\n\nBernstein himself wrote about this project in the poem he composed for _The New York Times_ as a report on his sabbatical in 1964\u20135, published on 24 October 1965:\n\nSince June of nineteen-sixty-four\n\nI've been officially free of chore\n\nAnd duty to the N. Y. Phil. \u2013\n\nFifteen beautiful months to kill!\n\nBut not to waste: there was a plan,\n\nFor as long as my sabbatical ran,\n\nTo write a new theatre piece.\n\n(A theatre composer needs release,\n\nAnd _West Side Story_ is eight years old!)\n\nAnd so a few of us got hold\n\nOf the rights to Wilder's play _The Skin of Our Teeth_.\n\nThis is a play I've often thought was made\n\nFor singing, and for dance. It celebrates\n\nThe wonder of life, of human survival, told\n\nIn pity and terror and mad hilarity.\n\nSix months we labored, June to bleak December.\n\nAnd bleak was our reward, when Christmas came,\n\nTo find ourselves uneasy with our work.\n\nWe gave it up, and went our several ways,\n\nStill loving friends; but still there was the pain\n\nOf seeing six months of work go down the drain.\n\nThe \"loving friends,\" Comden, Green, Robbins and Bernstein \u2013 who remained the closest of friends even after this harrowing project \u2013 had a tough time trying to recapture the success of earlier collaborations such as _On the Town_ (1944) and _Wonderful Town_ (1953). The surviving musical material is fairly desultory: a couple of numbers that were reworked for the _Chichester Psalms_ , and Sabina's opening aria (\"Oh! Oh! Oh!\"), which survives in a pencil sketch.\n\n129 Printed in Hussey 1985, p. 113.\n\n130 Never one to waste a good idea, Bernstein used almost all of \"Mix!,\" a cut number from _West Side Story_ , in the second movement of the _Chichester Psalms_. His recycling of the music originally written for \"Mix!\" had uncanny parallels with the idea of conflict in the original song: \"Lamah rag'shu goyim\" (\"Why do the nations rage\"), a passage from Psalm 2 about the futility of nation fighting nation. Hussey's \"hint\" of _West Side Story_ (see Letter 497) turned out to be a very apt choice.\n\n131 Printed in Hussey 1985, p. 116.\n\n132 Hussey has written \"1963\" but this is an error. The _Times_ article to which Hussey refers appeared on 19 July 1965, three days before he wrote this letter.\n\n133 Quita Chavez was classical promotions manager for CBS Records in London.\n\n134 Printed in Hussey 1985, p. 118.\n\n135 Printed in Hussey 1985, pp. 118\u201319.\n\n136 Printed in Hussey 1985, p. 119.\n\n137 A note in Helen Coates' hand at the top of the letter reads: \"Sketch of Psalm XXIII ( _Chichester Psalms_ ) sent to him Nov. 15th.\" This sketch was requested by Cage as part of his _Notations_ project, started in 1965. Cage donated all the manuscripts he collected to Northwestern University (Evanston, IL) in 1973, where Bernstein's sketch now forms part of the John Cage Notations Project collection.\n\n138 Printed in Hussey 1985, pp. 119\u201320.\n\n139 George Szell (1897\u20131970), Hungarian-born conductor who was Music Director of the Cleveland Orchestra from 1946 until his death.\n\n140 The same day that Szell wrote this letter he conducted the first of four concerts with the New York Philharmonic (28, 29, 30 October, and 1 November 1965), which included Mussorgsky's Prelude to _Khovanshchina_ , Prokofiev's Piano Concerto No. 3 (with Gary Graffman as the soloist), and Tchaikovsky's Fifth Symphony. He remained with the orchestra until 22 November, conducting three further weekly programs in the subscription series.\n\n141 Yo-Yo Ma (b. 1955), American cellist; he had already played for Presidents Eisenhower and Kennedy as a child prodigy by the time he wrote this letter to Bernstein.\n\n142 John Adams (b. 1947), American composer, who was a student at Harvard in 1966. Adams writes about the circumstances of this letter in _Hallelujah Junction_ : \"Despite my hunch that Boulez's was the wrong way to make art, I continued to try embracing the beast. Still in my freshman year, and by way of venting my frustration with the direction contemporary music was heading, I wrote a letter to Leonard Bernstein. I had never met him, but for some reason I felt the need to prick such a famous superstar to see if he might possibly bleed. I thought maybe that sharing my own frustration would perhaps sting him enough to elicit a response. Composed more in the negative spirit of a heckler at a baseball game than in any true seriousness [...] it was prompted by my hearing of his most recent piece, _Chichester Psalms_. [...] In my letter I chided him, asking 'What about Boulez?' A week later there in my mailbox at Wigglesworth Hall was a letter \u2013 from Leonard Bernstein\" (Adams 2008, p. 32).\n\n143 Draft letter written in pencil, unsigned.\n\n144 Bernstein was in Vienna to conduct _Falstaff_ at the Vienna State Opera.\n\n145 Dietrich Fischer-Dieskau sang Falstaff in this production. See Letter 546.\n\n146 The conductor Josef Krips and his wife.\n\n147 William Weissel was the Viennese-born assistant manager of the New York Philharmonic.\n\n148 \"Mendy\" was the nickname of the actor Michael Wager (1925\u20132011). The play mentioned by Bernstein is _Where's Daddy?_ by William Inge, produced by Wager and directed by Harold Clurman. It ran for just 22 performances in March 1966.\n\n149 Victor de Sabata (1892\u20131967), Italian conductor, renowned not only for his operatic conducting (especially at La Scala, Milan), but also for his performances of twentieth-century orchestral repertoire and for his phenomenal musical memory. When Bernstein first saw de Sabata conduct in London in 1946, he described him in a letter to his sister Shirley as \"a wildman\" (see Letter 225), but once Bernstein came to know him at La Scala in the 1950s he developed a warm admiration for de Sabata. In 1977, he wrote a short tribute entitled \"Memories of Maestro de Sabata\":\n\nThe first word that comes to mind as I call up memories of de Sabata is _generosity_. It seemed to me to inform and characterize all his actions: his abundant love for music and for the colleagues with whom he produced it; the abundance of his passions and of his patience; his profound gratitude for his own gifts; his kindness to young performers like me; his devotion to his public, whether in Milan or Pittsburgh \u2013 all of it generous, generous. It was also in the music he composed: the spirit of _abbondanza_.\n\nIt was because of his sudden illness in 1953 that I was called upon to open the Scala season with Cherubini's _Medea_ , Callas and all. There were only six days for me to learn an unknown score, to make cuts and repairs, to meet and cope with Callas (which turned out to be pure joy), to make a difficult debut alla Scala, and all with severe bronchitis. In all this Maestro de Sabata was intensely helpful and encouraging; he gave me the extra measure of courage that I needed. And two years later, when I returned to conduct a new performance of _Sonnambula_ , he virtually saved my life. \"Too slow! Too slow!\" I can still hear him chiding. \"Bellini was Sicilian; and Sicilian blood runs hot! Run with it! Run!\" Who knows what a boring disaster I might have made without that affectionate warning!\n\nThere was much other good counsel besides; and of course he was equally generous with his praise. And this abundance of spirit flowed through all his conducting: one has only to listen to his old recording of _Tosca_ with Callas. I still believe it to be the greatest recording of an Italian opera I have ever heard. I have only to listen to it \u2013 any dozen bars of it \u2013 and the spirit of de Sabata is in the room with me. Leonard Bernstein, May 1977.\n\n150 Robert Russell Bennett (1894\u20131981), American composer and arranger. After studying with Nadia Boulanger in Paris, Bennett began working as an arranger and orchestrator on Broadway, collaborating with Kern ( _Showboat_ ), Gershwin ( _Girl Crazy_ , _Of Thee I Sing_ ), Porter ( _Anything Goes_ , _Kiss Me, Kate_ ), Rodgers ( _Oklahoma!_ , _Carousel_ , _South Pacific_ , _The King and I_ , _The Sound of Music_ ), and many others. He also composed extensively for symphony orchestra, concert band, and chamber ensembles.\n\n151 Georg Solti (1912\u201397), Hungarian conductor. This letter serves as a reminder of the dilemma that faced Jewish musicians working in Vienna during the 1960s. In dealing with anti-Semitism, Solti \u2013 himself a Jew \u2013 had an outlook that was a mixture of humanity and pragmatism: to look forward rather than back, with the aim of rebuilding Europe after the ravages of war. He established a productive working relationship with the Vienna Philharmonic during the 1950s that grew into a warm and enduring association. The same was to happen with Bernstein, but not until he had come to terms with the fact that Helmut Wobisch, the orchestra's manager (and also one of its trumpeters), had been an active Nazi during the Second World War. Solti played a crucial role in this: his advice to Bernstein \u2013 to give Wobisch a chance rather than to condemn him out of hand \u2013 was both wise and realistic: he knew the Vienna Philharmonic very well by 1967, whereas Bernstein had made his debut with the orchestra only the year before; and Solti's letter is also written by someone who had clearly learned to rise above the gossip-mongering that was a constant feature of the city's musical life.\n\nBernstein quickly took his colleague's advice to heart. According to Humphrey Burton (1994, p. 354), Bernstein \"brushed aside [Wobisch's] past: he would refer to him openly as 'his SS man'.\" Wobisch's well-documented past was catalogued in detail in a letter to Joseph Wechsberg from Simon Wiesenthal on 3 February 1967 (a copy of which was sent by Wechsberg to Bernstein) confirming Wobisch's membership not only of the Nazi Party and the SS, but also of the SD (Sicherheitsdienst), the intelligence agency of the SS and the party. Despite these grim associations, both Solti and Bernstein found Wobisch friendly and supportive, and Bernstein was to spend much of the latter part of his career performing and recording the symphonic repertoire with the Vienna Philharmonic (including Mozart's late symphonies, Beethoven, Brahms, Schumann, Mahler, and Sibelius). Solti's letter was a timely and characteristic intervention, encouraging Bernstein to take a conciliatory approach.\n\n152 John Culshaw (1924\u20131980) was a producer for Decca. He is best remembered for producing the Solti _Ring_ and for numerous recordings conducted by Benjamin Britten.\n\n153 Janis Ian (b. 1951), American songwriter, singer, and author. The CBS News Special, _Inside Pop_ \u2013 _the Rock Revolution_ , was broadcast on 27 April 1967, presented by Bernstein and produced by David Oppenheim. In this program, Janis Ian performed \"Society's Child,\" which Bernstein discussed as a social protest song.\n\n154 Foss' \"Philharmonic tragedy\" was the cancellation at the last minute of the first performance of Foss' Variations. According to _The New York Times_ (9 July 1967), this was because \"the materials needed to perform the new work were not ready.\" The other work on the program was Honegger's _Joan of Arc at the Stake_ , conducted by Seiji Ozawa, with Vera Zorina (whose real name, Brigitta, Bernstein uses) playing Joan. _The New York Times_ review suggests that it was indeed the triumph Bernstein suggested: \"Her diction was beautiful, her voice was musical, and her intensity was compelling.\"\n\n155 John Gruen's _The Private World of Leonard Bernstein_ was published in 1968.\n\n156 The musical Bernstein hopes is \"ship-shape\" is _Follies_ (information from Stephen Sondheim). Ted Chapin explains that: \"In June of 1967 _The Girls Upstairs_ [the original title for _Follies_ ] was scheduled for the coming Broadway season, to be produced by David Merrick and Leland Hayward. The plan ultimately fell through\" (Chapin 2005, p. xxii). _Follies_ eventually opened in 1971.\n\n157 Bernstein has written a draft reply at the foot of this letter: \"Dear Janis, I'd love to come but I can't and I'm flattered you asked me & thought of me. I wish you a _howling_ success. LB\"\n\n158 The 9 January 1968 edition of _Look_ magazine (pp. 74\u20137) included an article by Joe Roddy entitled \"How to Think about Leonard Bernstein,\" mostly about his tenure at the New York Philharmonic and his relationship with the orchestra.\n\n159 Richard Rodney Bennett (1936\u20132012), English composer and pianist.\n\n160 Bennett's Symphony No. 2 was commissioned by the New York Philharmonic for its 125th anniversary and first performed on 18 January 1968, conducted by Bernstein.\n\n161 Sondheim's \"Dedicated Dodecahedron\" puzzle was published in _New York Magazine_ on 15 April 1968.\n\n162 \"The Funke literary effort\" refers to an article by Lewis B. Funke published in _The New York Times_ on 8 April 1968, under the headline \" _West Side Story_ Collaborators Plan Musical of Brecht Play\".\n\n163 Derived in part from comments made by Bernstein \"speaking from Vienna,\" Funke ends with what seems like an unduly brusque comment: \"Mr. Sondheim, who will do the lyrics, will have to wait until Mr. Bernstein completes some of the score.\" Understandably, Bernstein was keen to clear up any misunderstanding this remark might have caused.\n\n164 Bernstein's desire \"get back and dig in\" refers to the planned musical based on Brecht's _The Exception and the Rule_ that he was working on with Sondheim and Jerome Robbins. Though several songs were written and there are a number of sketches in the Leonard Bernstein Collection, the project was abandoned.\n\n165 Jacqueline Kennedy (1929\u201394), First Lady of the United States during the presidency of John F. Kennedy, from 1961 until his assassination in 1963. She had come to know and like Bernstein during his frequent visits to the Kennedy White House.\n\n166 Shortly after midnight on the morning of 6 June 1968, Jacqueline Kennedy's brother-in-law, Robert F. Kennedy, was assassinated in Los Angeles. The younger brother of President Kennedy, Robert had served in his brother's administration as Attorney General, then as Senator from New York. He had just won the California Democratic primary in the 1968 presidential campaign, running on a radical platform of social justice and racial equality. Jacqueline Kennedy phoned Bernstein later on the same day (6 June) to ask him to oversee the musical aspects of the funeral Mass at St Patrick's Cathedral, New York. The funeral took place on 8 June; Bernstein conducted the Adagietto from Mahler's Fifth Symphony with thirty members of the New York Philharmonic Orchestra, and part of the last movement of Verdi's Requiem. As reported in _The New York Times_ (9 June 1968), \"Mr. Bernstein's role in the Mass was specifically requested by the Kennedy family, with whom he has been friendly for several years.\" As Jacqueline Kennedy wrote in her letter, the Mahler was played at a particularly touching moment in the Mass, during the Offertory procession. _The New York Times_ described this \"procession by eight Kennedy children who marched in twos up the sanctuary behind two candle bearers to present the hosts and the wine used in the consecration of the Mass.\"\n\n167 Presumably one of the suggestions was Faur\u00e9's Requiem.\n\n168 This undated letter was sent on the occasion of Bernstein's 50th birthday in August 1968. It is particularly valuable for Green's recollections of their first meeting at Camp Onota in 1937. Bernstein and Green were near contemporaries, though not quite as near as Bernstein imagined when he responded a few months later (2 December 1968), with a poem to celebrate \"Adolph, on his 50th(?) Birthday,\" actually his 54th.\n\n169 Harold Byrns (1903\u201377), German-born conductor (born Hans Bernstein) who studied with Erich Kleiber and Walter Gieseking at the Stern Conservatory in Berlin and became Kleiber's assistant. After moving to the United States, Byrns became known as a specialist in contemporary music. In October 1949, he conducted a concert for Schoenberg's 75th birthday in Los Angeles (attended by both Schoenberg and Stravinsky), including the First Chamber Symphony; the same year he made the first recording of Bart\u00f3k's _Music for Strings, Percussion and Celesta_ and gave one of the earliest performances of Stravinsky's Mass. Byrns was particularly devoted to Mahler's music, and was a friend of Alma Mahler.\n\n170 Hebrew greetings, meaning \"a good year\" and \"see you soon.\"\n\n171 Bernstein has added two exclamation marks beside this paragraph: the remarkable coincidence of two conductors called Bernstein, both of whom Koussevitzky attempted to persuade to change their names. Byrns took his advice, whereas Bernstein didn't.\n\n172 In fact, Adolph Green's 54th birthday.\n\n173 Alan Fluck (1928\u201397) was Director of Music at Farnham Grammar School and the moving force of the Farnham Festival with its numerous commissions of pieces for young musicians. Fluck had a warm friendship with Bernstein. He commented on this letter that for Bernstein's 50th birthday he \"made a gigantic crossword puzzle, 50 words across and down. Clues and answers were all based on the life and works of LB. I sent it to him in Brussels and received this [letter] a month later.\"\n\n174 \"Our symphony\" given Bernstein's long history conducting the work, as well as the enduring friendship between the two men. Bernstein conducted Thompson's Second Symphony at Tanglewood in 1940, and later with the New York Philharmonic in 1959 and 1968 (followed by the recording praised in Thompson's letter of 16 January 1970, Letter 545).\n\n175 Michael Overbury (b. 1953), English organist. After his youthful success as a pianist, he was organ scholar at Corpus Christi College, Cambridge, and subsequently held positions at New College, Oxford, St. Alban's Cathedral, and Newark Parish Church, before being appointed Director of Music at Worksop Priory in 1999.\n\n176 Elliott Carter (1908\u20132012), American composer. He studied with Walter Piston at Harvard and later with Nadia Boulanger. Stravinsky called Carter's _Double Concerto_ (1961) \"the first American masterpiece.\"\n\n177 Carter's _Concerto for Orchestra_ was commissioned by the New York Philharmonic for its 125th anniversary. Bernstein conducted the world premiere performances at Lincoln Center on 5, 6, 8, and 9 February 1970, and recorded it on 11 February.\n\n178 The following is an English version of the quoted passages, in the translation by Hugh Chisholm:\n\nThese were very great winds over all the faces of this world, great winds rejoicing over the world, having neither eyrie nor resting-place,\n\nHaving neither care nor caution, and leaving us, in their wake,\n\nMen of straw in the year of straw... Ah, yes, very great winds over all the faces of the living!\n\nOver all things perishable, over all things graspable, throughout the entire world of things....\n\nAnd airing out the attrition and drought in the heart of men in office,\n\nFor a whole century was rustling in the dry sound of its straw, amid strange terminations at the tips of husks of pods, at the tips of trembling things.\n\nWhen violence had remade the bed of men on the earth,\n\nA very old tree, barren of leaves, resumed the thread of its maxims...\n\nAnd another tree of high degree was already rising from the great subterranean Indies,\n\nWith its magnetic leaf and its burden of new fruits.\n7\n\nTriumphs, Controversies, Catastrophe\n\n1970\u201378\n\nBernstein relinquished his post as Music Director of the New York Philharmonic in 1969, but he remained firmly in the public gaze. Two events stirred up controversy \u2013 both of them for questionable reasons. When Felicia hosted a reception (at the behest of the American Civil Liberties Union) to raise funds for the legal costs of thirteen members of the Black Panthers, this was seized upon gleefully by the press and widely misreported. Declassified files reveal that it gave the FBI yet another excuse to take an interest in Bernstein's allegedly suspicious activities. \"Radical chic,\" the phrase coined by Tom Wolfe to describe the event, is a resonant one, and doubtless contributed to sustaining the widely reported but largely mythical version of the story \u2013 that Bernstein gave a party for the Black Panthers. As Bernstein told Jonathan Cott in 1989, \"It's a legend and it dies hard. It _wasn't_ a party and _I_ didn't give it. [...] So what am I to do? You can't beat the legends... except by telling the truth. And ultimately, maybe, legends die.\"\n\nIn 1971, Bernstein caused controversy again, this time at the highest levels of government \u2013 in this case the paranoid and criminal administration of Richard Nixon. The problem was a rumor, investigated by the FBI, that Bernstein's newly finished _Mass_ was intended to embarrass the President by promoting an agenda of peace. Nixon detested Bernstein, and when the FBI passed the investigation back to the White House, what ensued was something close to black comedy. So convinced were Nixon's aides that Bernstein was out to cause trouble that they lost any kind of grip on common sense. Pat Buchanan \u2013 then an advisor to Nixon, later a conservative pundit \u2013 wrote this memorandum on 28 July 1971:\n\nMy view is that we ought to find someone who can definitely translate that Latin Mass Bernstein is working on \u2013 to make sure this is accurate. Then, we might want to sand-bag him; i.e. wait until it is too late for him to change his format \u2013 and then unload on him. Another course would be to have this released to front-page and force him to back down. However, we should be able to get a copy of what he is preparing \u2013 as there will have to be rehearsals \u2013 and once we get that, get us a good Jesuit to translate, maybe Father McLaughlin will do and once translated \u2013 leak the thing. But we ought to move rapidly lest the President be tied into attending and forced to back down.1\n\n_Mass_ was Bernstein's first work to be written in the 1970s, much of it conceived during a stay at the MacDowell Colony. It was followed by three other large-scale pieces. The ballet _Dybbuk_ \u2013 which Jerome Robbins had been urging him to write since the 1940s \u2013 finally saw the light of day at New York City Ballet in 1974: it turned out to be the last of their collaborations. It is a very demanding score, making use of some twelve-tone techniques, about which Oliver Knussen wrote as follows: \"After the militant anti-atonal statements which abounded in his Norton Lectures at Harvard, it is surprising to find Bernstein making use of numerical formulas derived from the Kabbalah [...] and producing his most austerely contemporary-sounding score to date.\"2\n\nThe Norton Lectures, given on six consecutive Tuesday evenings in October and November 1973 and published as _The Unanswered Question_ , were the most fully developed expression of Bernstein's thoughts on music, and his attempt to apply Noam Chomsky's theories of linguistics to it. Bernstein's work was criticized by some academics as unsystematic \u2013 but surely the important point is that his conclusions are so often inherently musical. Virgil Thomson and Bernstein had known each other for thirty years, and Thomson was sometimes a harsh critic of Bernstein's music; but he was impressed by the lectures. He praised Bernstein's \"skill in explaining music\" and went on: \"Myself I find nothing reprehensible about your bringing in linguistics. You needed an authority to support an 'innate musical grammar' and Chomsky's heavy artillery is surely that.\"\n\nThe musical _1600 Pennsylvania Avenue_ turned out to be an unhappy experience. It was overlong when it was tried out in Philadelphia, but the score includes a great deal of music that is beautiful (some of the best numbers were later salvaged by Charles Harmon and Sid Ramin for _A White House Cantata_ ). Written by Bernstein and Alan Jay Lerner as a celebration of the United States Bicentennial, by the time the show opened on Broadway it was doomed \u2013 especially as it had been cut to shreds, against Bernstein's wishes. Friends were well aware of the trials and tribulations, and rallied round: during the try-out in Philadelphia, Robbins did his best to encourage his old friend (\"Take care of your house. You can do it. Come on kid, get the spirit up again. No limp cocks!\"), while Sondheim sent a telegram saying \"you're still the only artist writing musicals with one exception that is.\" The great photographer Richard Avedon was (understandably) overcome by the beauty of the music. But it was to no avail. Despite the wonderful score, and clever lyrics by Lerner, _1600 Pennsylvania Avenue_ was a failure. Another commission intended for the Bicentennial (finished a year too late) was _Songfest_ , an anthology of thirteen poems for solo voices and orchestra. Completed in 1977, it was much admired by Bernstein's old friend and collaborator Oliver Smith, and was praised by others: Oliver Knussen pointed out the lessons Bernstein appeared to have learned from Britten, especially the _Spring Symphony_ (which Bernstein knew well, having conducted it in 1963), but he also valued the originality of the score: \"I can't think of another living composer who could approach Bernstein's complete involvement with and response to such varied texts.\"3 Despite the quality of the music (and the sensitivity of Bernstein's settings), _Songfest_ is hardly ever performed. It deserves better.\n\nFrom a creative point of view, the 1970s must have been rather disheartening for Bernstein: of four major works, only _Mass_ could be counted a success, and even that was the focus of some very hostile criticism. But the musical disappointments were as nothing to the turmoil in Bernstein's personal life. In 1974, Felicia was diagnosed with cancer \u2013 Bernstein's letter to her from New Zealand is full of reminders to see her doctor. But things quickly got even worse. By 1976 their marriage was in tatters: _Newsweek_ announced a \"trial separation\" \u2013 Felicia was increasingly disturbed by what Humphrey Burton has described as \"intimations that her husband was abandoning the discretion that was part of their unspoken covenant\" (Burton 1994, p. 414). Burton's evidence is pretty damning: Bernstein was having affairs with at least two men, one of them Tom Cothran, a young musician Bernstein had met in 1973. Felicia was not prepared to see their family life put in peril and gave him an ultimatum: either he must stop seeing Cothran alone, or he need not come home. Bernstein's daughter Nina \u2013 who was 13 at the time \u2013 recalled some difficult family scenes in an interview with Ginny Dougary in 2010: \"My mother was a fairly conventional lady and so she expected to be treated like one. The deal was that he would be discreet and that she would maintain her dignity. And then he was not discreet, and so that was that.\"4 Bernstein and Cothran set up in a new apartment for a few months, but then Felicia was diagnosed with lung cancer, and Bernstein begged to be allowed back. Nina recalls that \"The whole thing was terribly awkward and painful,\" and, of course, matters were made worse by the intrusive glare of publicity. When Felicia died on 16 June 1978, Bernstein blamed himself. Humphrey Burton writes about this with harrowing honesty: \"The crushing impact on Leonard Bernstein was that he believed himself responsible for his wife's death, and his sense of guilt never left him. Felicia was the greatest love of his life. He never recovered from her loss, and he never forgot the curse she uttered when he told her he was leaving her for Cothran. She had pointed her finger at him in fury and predicted, in a harsh whisper: 'You're going to die a bitter and lonely old man.'\"5\n\n545. Randall Thompson to Leonard Bernstein\n\n22 Larch Road, Cambridge, MA\n\n16 January 1970\n\nDear Leonard,\n\nWhat a glorious recording of my Second Symphony!6 What can I say to express my appreciation and my happiness? The whole interpretation is perfect \u2013 and inspired. The orchestra is superb and seems to be breathing with you all the way through. The engineering is both sensitive and powerful, refined in solo passages and rich in the _tutti_. And throughout, the rhythm is so vital that the whole work throbs with life. I wrote it exactly forty years ago, in this very village, for Koussy. It's yours now, and I see him smiling.\n\nThank you from the bottom of my heart.\n\nTi abbraccio.\n\nRandall\n\n546. Dietrich Fischer-Dieskau7 to Leonard Bernstein\n\nBaur au Lac, Zurich, Switzerland\n\n9 February 1970\n\nDear Lennie,\n\nJust because I don't want to miss a chance of making music with you please give me a hint whether you would like me as Kurwenal in the Bayreuth _Tristan_ production of which I heard.8 You know how opera houses are in their short notice planning.\n\nIn case of \"yes\" it would be a great thrill. Should you already have made an agreement with somebody else, I am still your greatest admirer. Only \u2013 I am dying to sing with you again. So please let me know.\n\nEver,\n\nSir Dieter Falstaff\n\n547. Rabbi Judah Cahn9 to Leonard Bernstein\n\n10 Park Avenue, New York, NY\n\n28 February 1970\n\nDear Lenny,\n\nI received the copy of Rabbi Schindler's letter addressed to you and the note which you added to it.\n\nLenny, since 1934, I have been deeply involved in inter-racial affairs. I believe my credentials are quite adequate to demonstrate that for thirty-six years I have put my physical being, my professional standing, and my financial means on the line any number of times when the struggle for civil rights demanded it, and you know this.\n\nWhen Felicia spoke to me, I suggested that the best way to handle the situation was to drop it.10 Your friends will never question your motives, even if they do not agree with you. To defend yourself is totally unnecessary. Frankly, as your friend for all these years, and one who loves you deeply, I needed no letter explaining why you called a meeting in support of the Black Panthers. I know that you were motivated by the highest ethical imperatives, and you acted in a way that you thought just and right. I do believe, however, that you must give me equal credit for ethics and motivation.\n\nI believe that the need to defend oneself is a necessity, if we are going to defend others. If those I personally defend would, in turn, defend me, then such a defense has a top priority in my scale of values. If, on the other hand, those who seek my assistance would, if they could, destroy me, then logically it would be a mistake to give such a group my special, personal assistance.\n\nHowever, in order not to neglect out concern for all groups who are struggling to achieve civil rights, whether or not we agree with the philosophies of those groups, we should continue to support the American Civil Liberties Union, whose purpose it is to defend the rights of those who espouse unpopular causes.\n\nAt no time, since the Hitler holocaust, has the Jewish people been faced with the complexity of problems which now confront us. Both of us are wholly committed to the future and safety of our people, and I will not, under any circumstances, give special time and energy to strengthening those who are outspoken in their avowed attempt to destroy me. Since Hitler, I take such \"forthrightness\" very seriously.\n\nI will fight for the right of the Panthers to a just trial. I will fight to secure justice for them, as I would for any other group, but I will do so through channels which are already established for such purposes. Frankly, I would not open my home and ask my friends to make special contributions to their cause. This constitutes a difference of opinion. I honor you for your position, and I ask you to honor me for mine.\n\nAs for \"lessons,\" Lenny, as you know, it is not the lessons but the practicing that makes the artist. I have been practicing a long, long time, and I took my lessons when it was not so fashionable.\n\nWith my best wishes and kindest regards to your family.\n\nAs ever,\n\nJudah\n\n548. Leonard Bernstein to Ingmar Bergman11\n\n11 August 1970\n\nDear Ingmar Bergman,\n\nI have just talked with my good friend Humphrey Burton in England who tells me that he spoke to you about the possibility of our working together on a production of _Tristan und Isolde_. Even though Humphrey reported that you had misgivings I want to tell you personally how very much I would love the opportunity of our working together, particularly on a project as fascinating and challenging as _Tristan_.\n\nThe _Tristan_ idea has been growing in my mind ever since Wieland Wagner asked me to do a production at Bayreuth, but there never was time during the years I was with the New York Philharmonic. Now that my time has become more flexible, I return to _Tristan_ as one of the major projects I want to achieve in the near future. The idea would be to produce it at the Bayreuth Festival in 1973 and to record and film it thereafter. I realize that you may not be interested in staging the opera at Bayreuth but I can think of no one on Earth better suited for a free, fresh and \"inner-directed\" film version.\n\nCould we talk about this? I will be going to Japan next month, then back in New York from mid-September through January '71. In early February I go to Paris for concerts and then generally around Europe and Israel until the end of April.\n\nI am your profound admirer, and want to work with you!\n\nIn friendship.\n\nLeonard Bernstein\n\nP.S. Of course, if you were interested in the Bayreuth staging as well, that would be a more than welcome bonus.\n\n549. Leonard Bernstein to Stephen Sondheim\n\n[Christmas 1970]\n\nDear Steve,\n\nThese balls are not meant to symbolize any contribution to your testicular powers. They are simply beautiful carpet-balls ( _bocci_ , I believe), and when I saw them I knew they had to be for you. So Merry Xmas!\n\nAs for your gifts, I bless you for the _Listener_ book, thank you for the _Company_ record, and adore you for the gum-machine.\n\nHappy 1971, and have a glorious _Follies_!\n\nLove,\n\nLenny\n\n550. Leonard Bernstein to Helen Coates\n\nHotel Sacher, Vienna, Austria\n\n18 February 1971\n\nDear H,\n\nThe Paris concerts went gloriously, finally, in spite of all kinds of problems:\n\n1) my usual Parisian diarrhea, really bad this time, 2) a cut thumb, 3) a bad back, 4) the failure of the Berlioz material to arrive until the last day of rehearsal (sent to wrong address!) so that I had to use _their_ material which was full of errors, had no rehearsal numbers or markings or anything. A nightmare, necessitating hours of extra work & fatigue for orchestra & conductor. But finally all was well, with audiences screaming & happy.12\n\nRehearsals here go beautifully: I seem to be playing most of the notes of the Ravel,13 & my stomach, back, thumb, etc. are very well indeed. Tonight I'm even going to the Opera Ball!\n\nI've received both your letters, with enclosures (sad about Ethel L[inder] R[einer],14 horrid and shocking about Debs Myers!).15 Very pleased at your progress report on the dictaphone \u2013 brava!\n\nI'm feeling better than I had expected \u2013 all is well.\n\nLove to all, & to you,\n\nL\n\n551. Leonard Bernstein to Shirley Bernstein\n\nHotel Sacher, Vienna, Austria\n\n25 February 1971\n\nMine Hilee,\n\nStrange, but everything seems to be going well, & better. Or, as they say here (pure Annie M), es geht _g\u00fc-at_. Paris (as always) was a bit of a trial \u2013 the trots in spades, the back, the cut thumb, the usual list. But great concerts, and now Vienna \u2013 why, I even drink the tap water! Back great, thumb prima, stomach glorious (or as glorious as my Tumburger can be). _And_ the concerts are, so far, tops. Sunday I think I conducted one of the really best ones of my life \u2013 including playing the Ravel. It was fleed, my dear, _fleed_.\n\nBut you, mine ape, have you vot to do, vot to eat? I worry about you. How about a letter telling all. Find out my tour schedule from Felicia, & send me a letter. Simple. A.B.C. We leave on tour Sat. a.m. for Munich. Back in Vienna 13th March for a Unicef benefit, then off for another week, then back here for three more weeks. Schreib!\n\nAnd love to Mither when you talk to her.\n\nI hug it upon you.\n\nL\n\nI do a March [?] every day, _plus_ the London Times, _plus_ an occasional Telegraph, etc.16 So you see, my time is well spent, intellectually.\n\n552. Leonard Bernstein to Helen Coates\n\nDan Hotel, Tel Aviv, Israel\n\n16 April 1971\n\nDear Helen,\n\nLife is not too kind to me at the moment: I left Vienna a wreck and went straight to Eilat for some days of sun and sea \u2013 my first four free days in months. Well, there was no sun, no sea \u2013 only stormy weather, nothing to do but watch the mindless hippies on the beach. Very lonely. I came back to Tel Aviv early \u2013 same bad winter weather. We had our first rehearsals yesterday \u2013 the orchestra is in fine form, but I'm not. I'm depressed most of the time, and longing to get home.\n\nI've been worried at not finding any mail from you here. I hope you're all right, and that your brother is not causing you too much anxiety. My mother seems to be enjoying herself, although her activities are necessarily limited.\n\nToday is again a free day, and I'm spending it mostly in bed. It's storming outside. What a non-holiday! People here can't remember such weather at Pesach-time in years.\n\nPlease call Dr Gaylin17 and ask him if he can reserve some time for me the week of May 3rd. I should be back on the 3rd. I have a dentist appointment that day which will have to be cancelled. [...]\n\nEveryone asks for you \u2013 with affection \u2013 Katia, etc.\n\nBe well, and let me hear from you.\n\nMuch love,\n\nL\n\n553. Jamie Bernstein18 to Leonard Bernstein\n\nCambridge, MA\n\nSunday 18 April [1971]\n\nHey, Dadz!\n\nA quiet, pleasant Sunday eve in rainy Cambratch... I stayed here this weekend, cause I've been in New York over Spring break for a week and a half. This was my first week back, in which I had a paper due Monday, a paper due Tuesday, and a paper due Wednesday, for God's sake. So this past week has been an incredible hassle, and this weekend was blissfully uncluttered. Friday night: two girlfriends and I went out and had a deliciously sickenening (whoops!) Italian dinner, after which we played gin rummy and watched C movies on television. Saturday: zero. Slept till twelve (for the first time in months), ate, had a nap, ate, did some leisurely reading. _Then_ Jane Weeks, some other people and I decided to go and see the Russian Easter service, cause we'd always heard how beautiful it was, and Mummy had just given us a big pep talk about it, so we decided to go to the Russian Orthodox Cathedral on the Fenway. Well... it sounded good. We got there, and damned if it wasn't a _modern_ church! Oh, God, were we disappointed! Where were the onion domes, the gold? Just wood and plaster and a gold-plated gaudy chandelier with flickering red electric \"candles\". [...] _No seats_! Hundreds of _ugly_ people, badly dressed, very upper middle, very reformed Jewish in a funny way. So okay, there we all were, waiting for the fireworks, and out comes this little dumpy priest in an ugly ill-fitting robe, and starts mumbling most unmusically.\n\nAnd then the chorus opened their mouths. AAAAAAUUGGHHH!!!! Good _God_! That poor chorus must have been made up of volunteers out of the congregation, I guess, because it really was indescribable. I never did figure out what key they were in.\n\nSo we split. No offense to the Russians, _but_ , as Nina would say, but gee whiz...\n\nWe dropped someone off in Harvard Yard, got back on to Mass. Ave., and _the car stalled_. So we all have to get out and push the car down Massachusetts Avenue at one thirty in the morning like a bunch of Keystone Cops! Oh, God, it was funny. Anyhow, that was Saturday. Today I got up at one, ate, read the paper, went to the library for a while, ate, watched _The Wizard of Oz_ on TV, and now here I sit, tapping away and watching a Diana Ross (of Supremes fame) special. As I said, a quiet, pleasant Sunday eve.\n\nBruce couldn't come up this weekend; he had to go down to Etlontik City to visit his dying grandfather. Oh, dear God, I know what he's going through. Otherwise Bruce is in pretty fine shape. [...] I've been going through some strange changes lately, and I've been a big pain in the ass some of the time, and boy, Bruce was so patient. He has his bad days too, God knows, and then I have to be patient. And sometimes we both run out of patience, but love ain't a bowl of latkes.\n\nBruce leaves on June 3 for his summer job in England. And guess what?! Mummy said I could travel this summer, and stay with Anya, and the Smiths, and stuff like that. OBOY!!! SUMMER!!!!!!! I can't wait. And the bestest part of all is going to be those few weeks when I don't have to do _anything_ but sit in the country and swim and play tennis and read what I _want_ to read, and lie in the sun and see my friends in New York... oboy, is that ever going to be loads of fun. And then when I get to England, I can see Bruce, and I even got invited to West Pakistan by Pinkie Bhutto,19 and if you've been keeping up on your current events, you'll know that Pinkie's father20 is the Alessandri of West Pakistan. Far out! I'd _really_ like to go, and see what the vestiges of the Arabian Nights look like. Hey Daddy... you think you can get yourself invited to Red China to conduct? Boy, that's the place I want to go to more than anywhere else.\n\nWell, it being eleven and all, I think I'll get a head start on this week and go to bed early. Say a big _Hello, Grandma_ to Grandma. I hope she's grooving on her trip, man, and I sure hope you feel better yourself. Anyhows, see you soon. Hurry back; I can't remember what you look like already!\n\nLove and kisses,\n\nJamie.\n\nP.S. I got into Adams House, in case I lived here next year. Eliot House is a most unpopular house these days: all preppies and jocks. Were you one of those? I think if you were going here now, you'd be in Lowell House. Adams House is where all the hippies are. Birds of a feather spoil the broth, I always say... J\n\n554. Oliver Smith to Leonard Bernstein\n\nThe Watergate Hotel, Washington, D.C.\n\n8 September 197121\n\nDearest Lenny,\n\nWhat can I say? Nothing but to hug and embrace you to express my feelings concerning _your_ superb _Mass_. You are such a genius that with you I am speechless with admiration and humility.\n\nTo work on this project was a rare privilege.22 Thank you deeply for making this possible.\n\nAll my love,\n\nOliver\n\n555. Christa Ludwig23 to Leonard Bernstein\n\nParis\n\n13 November 1971\n\nDear Maestro,\n\nI want to say thank you for the recital! I loved so much, no: I _love_ so much to make music with you! I heard the tape and I think we shall do it again. My voice is sometimes unsteady and breathless. And I also think that the placement of the microphone is too close. So, let's do it in Vienna.\n\nBest regards.\n\nYours,\n\nChrista\n\n556. David Charles Abell24 to Leonard Bernstein\n\n466 Poplar Street, Winnetka, IL\n\n21 November 1971\n\nDear Mr. Bernstein,\n\nI would like to thank you very much for the fantastic time I had in Washington D.C. in September. It was thrilling to be in as big and important a production as your _Mass_. I think it is a great piece of music, and my favorite parts are the Epistle, and from the \"Agnus Dei\" to the end. I was an alternate in the Berkshire Boy Choir and I was in over half the performances and all of the recording (the two easiest sessions I have ever seen or been in).\n\nRecently, I heard the piece on radio and was so excited about it that I didn't do my homework so I could listen to it! I can't wait until I can get the record, but the stores around here are slow getting it in. Wasn't the recording changed a little from the performances? (Ron Young singing the first \"I don't know\").\n\nIs _Mass_ going on Broadway or to Los Angeles or any place like that? I'm sure if you took it around to different cities, the crowds would be miles long.\n\nI wish I could re-live the fantastic experience I had in Washington, but anyway it will benefit me in many ways all my life.\n\nSincerely,\n\nYour friend and admirer,\n\nDavid Abell\n\nBerkshire Choirboy\n\n557. Richard Rodney Bennett to Leonard Bernstein\n\n4 Lonsdale Square, London, England\n\n[1971]\n\nDear Mr. Bernstein,\n\nThis is a big thankyou letter. I gather from Sam Spiegel that you put in a very kind word for me over _Nicholas and Alexandra_ , and I am very grateful indeed. It was a very happy and I think successful job altogether, and both Sam and the director, Franklin Schaffner, were splendid to work for, considerate and helpful. So altogether I feel pleased and satisfied and without your kindness it would not have happened. There will be a sound-track album very soon and I will make sure they send you a copy.\n\nI was hoping that Andr\u00e9 Previn would be conducting it, and I know he wanted to do it, but there were contractual problems. The man who always does my films for me25 \u2013 I have a mortal dread of conducting \u2013 did it, and did a very good job.\n\nCongratulations, very late in the day, on the huge success of the Kennedy _Mass_. I have been reading avidly about it, and long to hear an English performance. I'm sure plans for it are well under way.\n\nNow I think of it, I hope you won't be _dismayed_ when you hear the _Nicholas_ score... It's rather a curious mixture stylistically.\n\nExcuse frightful typing.\n\nYours ever,\n\nRichard Bennett\n\n558. Leonard Bernstein to Felicia Bernstein\n\n6 February 1972\n\nF ifty is nothing but twice twenty-five:\n\nL et me take over one half of the weight,\n\nE xtending my years to seventy-eight \u2013\n\nS eventy-eight \u2013 still here, and alive!\n\nH appy birthday, dear Child-Bride,\n\nY outhful always, at my side.\n\nAll my love,\n\nBen\n\n559. Richard Rodgers to Leonard Bernstein\n\n598 Madison Avenue, New York, NY\n\n27 March 1972\n\nDear Leonard,\n\nIs there anything you don't do better than anyone else? Your playing last night was so simple and pure that it gave a grace that little song never had before.26 Your words were kind, too, but taking the trouble to be there was the kindest of all. I appreciate it tremendously.\n\nFondly,\n\nDick\n\n560. Leonard Bernstein to Helen Coates\n\nHotel Sacher, Vienna, Austria\n\n14 April 1972\n\nDear H,\n\nThis is a brief moment grabbed out of a monstrous schedule just to send love and say that all is well with health \u2013 though not with the schedule. There's been some poor planning (too much at once) and the Vienna Philharmonic simply doesn't know Mahler27 \u2013 so it's all from scratch. Besides, I don't feel like performing much these days: I'd rather be quietly composing. But I'll get through it, somehow \u2013 and there is the compensation of beautiful music.\n\nThe sun is just beginning to show itself for the first time since I arrived here.\n\nI hope all is well with you, and thanks for all the envelopes filled with goodies.\n\nLove,\n\nLenny.\n\nFelicia is in Venice for a few days, returning on the weekend. Then, I guess, she'll return to NYC on Monday.\n\n561. Leonard Bernstein to Shirley Bernstein\n\nHotel Sacher, Vienna, Austria\n\n21 April 1972\n\nDearest Hilee,\n\nI think of you with every puzzle (and in Europe that's 2 or 3 a day). But when a _Nation_ puzzle arrived from la Belle with a _whole wrong diagram_ then I had to write. It's not enough to think: one must share. What is happening to our world?\n\nWhat indeed?\n\nI want to write music.\n\nI've had Vienna.\n\nI think I've had performing.\n\nSpring is _not_ here, and all is somewhat sad, foolish, exaggerated. Very tiring indeed.\n\nWhy don't you ever write me, you thriving, prosperous, presidential wonder?\n\nPlease call BB & Ellen & send my love. I can't write any more \u2013 Mahler calls.\n\nAnd special hugs for Ofra.\n\nAnd mostly to mein Schwest \u2013\n\nL\n\n21 Ape, '72\n\n562. Luciano Berio28 to Leonard Bernstein\n\nVilla della Mendola 131, Rome, Italy\n\n28 April 1972\n\nCaro Leonardo,\n\nI want to thank you once more for your precious contribution to _C'\u00e8 musica e musica_.29 The 9th program, where you led the way deeply into yourself and through American music, was certainly one of the best. I hope you will be able to see it soon. In 10 days this incredible TV adventure will be finally over \u2013 and now, because of these 12 TV hours on music I have more \"enemies\" in Italy than I ever had (mainly singers and conservatory professors and one music critic wrote that I have prostituted music).\n\nI will be in NY very soon and I hope to see you.\n\nLove, as ever,\n\nLuciano\n\nP.S. Maybe a correspondent of an Italian newspaper (\"Messaggero\") will call you for a short interview. Please give him a few minutes \u2013 even on the phone. I promised him that you will not push him away.\n\n563. Leonard Bernstein to Shirley Bernstein\n\n3 October 1972\n\n_For Mine Schwest\u00f6ahs_\n\n\"Nothing Could Be Finer Than to Be a Forty-Niner\"\n\nI thought of finding you a gift\n\nOf cashmere, silk or leather;\n\nBut nothing seems so precious as\n\nOctober's Bright Blue Weather.\n\nWhen H. H. Jackson thought it up\n\nShe had her head together,\n\nAnd Hilee in mind, as high she penned\n\n\"October's Bright Blue Weather.\"30\n\nSo may it be for all your life\n\nAnd when your birthdays trouble you\n\nRemember you are shone upon\n\nBy high O. B\u2013B\u2013W.\n\nLove mo\u00f6ahs,\n\nL\n\n564. Jerome Robbins to Leonard Bernstein\n\n7 October 1972\n\nLen,\n\nTried to reach you before departing.\n\nIf the ballet isn't working out (I think it _is_ , and is mostly there) then we'll do it next year when you are less pressured.31 I'll tell the N[ew] Y[ork] C[ity] B[allet] _not_ to count on it for their Gala of June 7th \u2013 that way they don't get fucked up on their money raising \u2013 and if by chance something breaks there for you, and you feel more optimistic \u2013 or one of your miracles occurs \u2013 we can always do it in June with less commotion around it. Anyway, write me in London, care of Royal Ballet, Covent Garden Opera House.\n\nLove to all,\n\nJ.\n\n( _Interplay_32 went very well!)\n\n565. Benjamin Britten to Leonard Bernstein\n\nThe Red House, Aldeburgh, England\n\n[Spring 1973]\n\nMy dear Lenny,\n\nPlease forgive p. c. (I'm not supposed to write!) \u2013 but I was very touched by your wire. The present little medical blow is maddening, boring & v. painful! I am just coming up to one of these miracle modern operations and shall be inactive for several months \u2013 which means I'll miss _Owen Wingrave_ (first time on stage)33 & _Death in Venice_ (first time ever!).34 But they must look after themselves. It was so good of you to think of me with the 1001 things you have to do.\n\nYours ever,\n\nBen\n\n566. Jennie Bernstein to Leonard Bernstein\n\n21 March 1973\n\nBirthday reflections.\n\nThis seventy five year _young_ mother and grandmother is counting her blessings, moments filled with joy and pleasure. I am indeed blessed to have such precious children and grandchildren. All these wonderful things in my life will keep me happy and young. How do I love you all, let me count the ways?\n\nWith all my love,\n\nJennie (Elizabeth Browning)\n\nCirca 1898.\n\n567. Benjamin Britten to Leonard Bernstein\n\nThe Red House, Aldeburgh, England\n\n[?December 1973 or 1974]35\n\nMy dear Lenny,\n\nI am so sorry I haven't answered your last letter \u2013 but writing is still very difficult for me. I do want to write a proper letter to you, & to say how touched both P. & I are that you have come on to the board of the Maltings Friends, how deeply I have been touched personally by your concern about me, & several other things too! Can I have your address to write to, please?\n\nWith much Xmas love from\n\nBen\n\n568. Virgil Thomson36 to Leonard Bernstein\n\n222 West 23rd Street, New York, NY\n\n14 March 1974\n\nDear Lennie,\n\nMy warm congratulations on the success of _Candide_.37 Way back in 1945, when I used to help out the French radio with broadcasts of American music from discs available through U.S. Information Service, by far the most popular with the public were excerpts from _On the Town_. And the French musicians around simply could not get over their astonishment that in New York an _op\u00e9rette_ composed in so advanced a musical style could be successful.\n\nSo still, with _West Side Story_ and _Candide_ you remain world master of the \"musical\". All honor to you!\n\nNow that I have been through the Norton lectures (three by video, all six by reading) it is clear that your skill in explaining music is also tops, as indeed it was when you used to do it at Carnegie Hall.\n\nMyself I find nothing reprehensible about your bringing in linguistics. You needed an authority to support an \"innate musical grammar\" and Chomsky's heavy artillery is surely that. Especially since post-war researches in the physiology of hearing, though they do support a syntax based on the harmonic series as unquestionably built into the human ear, are being treated by both the Germanic twelve-tone world and the French-based solfeggio world as \"controversial\".\n\nSo the linguistic argument, though merely an analogy, as you pointed out, does carry weight. And it enables you to by-pass vested musical interests.\n\nI am sure your conclusions are valid, and I see no reason why you should not have used any material conveniently to hand for expounding them. All the more so since that material is relatively familiar and hence easily acceptable. I enjoyed everything.\n\nMany thanks for the courtesies of your office.\n\nI am returning the borrowed scripts with gratitude.\n\nEver warmly your admirer,\n\nVirgil\n\n569. Alan Jay Lerner38 to Leonard Bernstein\n\n15 May 1974\n\nDear Lenny,\n\nBlessings tonight. I can't wait to see it and hear it.39\n\nIf Jerry wants any more changes after tonight I'll put a contract out on him.\n\nI'm starting the second act40 and waiting for you like Duse for D'Annunzio. The room is ready, the piano has been tuned, the plane is revved up and I have fired my children.\n\nHurry,\n\nAlways,\n\nAlan\n\n570. Maurice Abravanel41 to Leonard Bernstein\n\nUtah Symphony, Salt Lake City, UT\n\n4 June 1974\n\nDear Lennie,\n\nPlease accept my apologies for having taken so long to thank you for your lovely telegram. I passed it on to all the participants who thank you for your kind message.\n\nThe two performances went exceedingly well and were highly successful, and your _Mass_ was exceedingly impressive and moving even in the very large auditorium. We had in excess of 12,000 attending between the two consecutive nights.\n\nIn addition to the fact that I felt it essential to present _Mass_ in our city, I felt very gratified that it was only possible through cooperation among our orchestra, Repertory Dance Theatre, some people from Ballet West, the Music and Theatre Departments of the University of Utah, in addition to three other universities in the state from which we borrowed sound equipment and robes. As far as I know, it is the first time so many organizations worked together for the common goal. 6,195 University of Utah students attended, which is almost one-third of the total enrolment.\n\nI discovered that you and I are part of the establishment and were therefore (especially!) outrageously censured by a small but very vocal group. Also, our top Roman Catholic personage was personally against _Mass_ , but even though quite a few of his people prompted him to do something about it, he refused to take any stand against it.\n\nMy associate conductor, Ardean Watts, did a fantastic job of organization and conducting. He used our entire orchestra plus a dozen extras, 185 singers and 10 dancers. The set was terrific, and I think that you would have been very happy with the whole thing. Of course I knew that you were working on _Dybbuk_ which is a much more important thing.\n\nI was putting in a strong pitch for asking you to serve again on the National Council on the Arts. I realize that right now, as I was told, it would be hopeless to expect President Nixon to appoint you, but I would like to know whether you would be ready to again serve. Even if you could attend meetings but rarely, your name alone would mean a great deal to the Endowment and in particular to its music component.\n\nI am not quite sure that you yourself know how much your name means and how much more it could mean for us, the hewers of wood, who have to confront day in and day out the unrelenting putdown of the symphony orchestra in America.\n\nForgive me for having written such a long letter. Again, many thanks for having written _Mass_ and so much other beautiful music, and for having done so much for music in our nation.\n\nWith best wishes always.\n\nAffectionately yours,\n\nMaurice\n\n571. Leonard Bernstein to Felicia Bernstein\n\nClarendon Hotel, Christchurch, New Zealand\n\n[August 1974]\n\nDearest F,\n\nThis letter has been a-borning for days, ever since arriving in Auckland, but there was always a reason to postpone \u2013 until we'd seen something to write about, until the first concert was over, until the first Mozart Trial by Fire42 was over, etc. Well, they're all over now, and all is more than well \u2013 triumphant concerts, orchestra in top form, even the Mozart last night in Wellington was a joy, with only a moment here and there of Frozen Fingers, and no Heart Attacks to speak of. Mozart again tonight here, and _Steinways_ all the way, which is 50% of the joy. (And I can't tell you the joy of having Axel43 along, he's a glorious companion.)\n\nInterruption: the \"Physiotherapist\" has just arrived for massage purposes. Be back in a flash.\n\nAn hour later:\n\nTerrific treatment \u2013 all those neck and shoulder pains gone. A genius; he brought along his 2-year-old daughter Katie, who \"helped\". The aches aren't serious \u2013 just this sudden spate of thick schedule \u2013 much rehearsing in Auckland, then nightly concerts in a row, all in different cities. Tonight is Christchurch (where I was greeted at the airport this morning by, natch, the Christchurch Hebrew Youth Group or something, with flowers and planted trees in Israel) \u2013 and they have a gorgeous new concert hall where I've just been trying out the piano: I'm really looking forward to this concert.\n\n_Don't forget Dr. Clarke check-up_.44\n\nTomorrow is a free day, and Alexander and I are going skiing in the nearby Alps. Yesterday in Wellington we spent the afternoon sailing in the harbor: glorious it was, with buckets of beer and a jolly crew of real leathery mates. The city is a beauty \u2013 all great green hills around a huge harbor, and the cleanest air in the world. Architecture abysmal, however. Auckland even worse: it's all so new, just a century old, that everything is either the worst Victorian or plastic office buildings. The houses are sort of seaside shacks with a curiously familiar Oaf Gloves look.\n\nBut oh, the country around Auckland \u2013 that whole north tip! You would adore it: all rolling Scottish hills and Scandinavian fjords, massive firs, etc. \u2013 and anomalously, mitten drinnen, _palms_ and other tropicalia. Rain forests. Monster fern-trees, vines. And around it all, the green hills with the millions of little sheep grazing peacefully. I don't know anything quite like it. Also, in that region, we discovered some thermal pools, hot mineral baths, and we became addicted: we went out there twice, and whiled away two mindless afternoons soaking in these pools while it rained on us, shone on us, and rainbows adorned the Hobbema skies.\n\nAside from a bit of a dust-up with the press on first arrival from Hawaii, we've found the people warm and simple and delightful \u2013 sort of provincial British midlands mixed with beautiful Maoris (Tahitians who came over in the 14th century) plus all kinds of other Islanders \u2013 Fijians, Samoans, etc., who keep arriving owing to the population explosion back home. It's all a beautiful mixture.\n\nAnd the milk! The best on earth.\n\nInterruption: concert time \u2013 (to be continued).\n\nIt's now two days later; it's all one can do to find ten minutes to write a letter. So much activity! First the concert itself, Mozart & Mahler, in the great new concert hall here \u2013 and I mean _great_. I've never played the piano so well \u2013 really deep Mozart, and no nerves. You'd have been proud of the Town Hall.\n\nAnd yesterday, a free day. Axel and Paul and I were taken skiing by some charming local ladies & small son \u2013 it was bliss. A perfect day, cool, brisk mountain air and a hot sun \u2013 dream Chilean weather. We came back last night in rapturous exhaustion, Axel and I both with banged-up knees & Paul with torn Pectorals. We're all limping around today, happy as clams.\n\nAxel is actually doing a local crossword (in the authentic British style)45 while I finish this letter, and then we're off to mad gay Brisbane, & the Australian leg commences \u2013 probably all downhill from here on. New Zealand was climactic, and those who say it's a final refuge when the crunch comes are probably right.\n\nWe miss you terribly; it seems incredible that ten days ago I had a last glimpse of you standing in the airport being abandoned by your males. I hope you're not too lonely, that there are all kinds of cheery _visitors_ and chums, and that Nina is also not lonely. I miss her so. And Jamie, of course, would have gone bonkers over all this; it's a pity for her to miss it, and that Axel doesn't have her to share so many things with. But he's having a great time: he even acquired a cute dimply girlfriend of 17 in Auckland.\n\nWell, the siren blows for Australia, and we're off, as Shirley would say, across the straits of Magellan. Take care of yourself, rest, have fun, and I fervently hope to find a letter from you waiting in Brisbane, full of news about the Dakota and Dixie's love life and the great Sublime Kitchen of Life.\n\nHugs and kisses to all, and so much love to you, my angel.\n\nBen\n\nP.S. Don't forget to see Dr. Clarke. Or have I already mentioned that?\n\nxxx\n\n572. Lukas Foss to Leonard Bernstein\n\n12 September 1974\n\nDearest Lenny,\n\nWelcome back. I know you could not care less, but I do: ages ago a Mr. Buketof[f] sent me a form (the 111st of the year) to fill out: the 15 best American orchestral pieces \u2013 for some European library. I sent him my choices. 2 Sundays ago I open the N.Y. _Times_ , find an asinine list of 11 pieces, find myself listed, also among the jurors who \"deliberated\" to arrive at their idiotic choices.46 I called Buketof[f], asked him to fish out _my_ list: sure enough, it has little in common with the published one. Mine starts with _Age of Anxiety_ , proceeds to Aaron's _Variations_ etc. No Harris, no Brown. So I wrote to the _Times_ a small note of protest for publication, calling their list: \"conspicuous for the absence of America's best composers.\" (So far not publ. to my knowledge). You can laugh it off; but I can't write about my frustration to all the colleagues. In fact I won't write to any but you. Basta.\n\nI'm off for 7 long weeks, 3 to Jerusalem, 4 with the Jerusalem Symphony all over Europe. Also trying to write a string quartet. There must be a saner way of life. What is it? How was Kyoto without me and my Buddhist nuns? Congrats!!!\n\nWrite the flute piece.\n\nArigato.\n\nLove to Felicia, Nina, Jamie, Alexander [and] you.\n\nLukas\n\n573. Christa Ludwig to Leonard Bernstein\n\nVienna, Austria\n\n1 February 1975\n\nDear Maestro,\n\nYou are so famous, but in N.Y. the telegram office doesn't know your address! When your wishes for New Year arrived I learned that you have a new home. So I am really very sorry that when I came back from holidays and I saw your cable, my answer couldn't reach you. But I do hope that things will go better for Israel, but: oil! It is a terrible world. But we can hide ourselves a little bit in the music. I am glad that you come back to Salzburg. Why don't we make music together any more?!\n\nThe best to you and your family.\n\nLove,\n\nChrista\n\nI think it is wonderful that our _Lied_ [ _von der Erde_ ] and Brahms Lieder are now on records!\n\n574. Thornton Wilder to Leonard Bernstein\n\nEdgartown, MA\n\n20 July 1975\n\nDear Leonard Bernstein,\n\nAs I told you on the phone:\n\nI did not want an opera to be made of _The Skin of our Teeth._\n\nBut I admired and trusted you, and was persuaded. I trusted you and the fellow-worker you would select.47\n\nWhen your fellow-work fell apart \u2013 who was left to write the book? \u2013 I felt relieved of my commitment to you.\n\nHereafter, while I'm alive no one will write or compose an opera based on that play.\n\nTorn from its context, Sabina's opening aria \"Oh! Oh! Oh!\" sounds awful, unmotivated, synthetic vivacity.\n\nThe nearest thing to it would be Zerbinetta's aria (or rondo). Who cares what her words are, except as are implied somewhat in Ariadne's abandonment? _These_ words bear the weight of a crowded historical story of many facets.\n\nI'm sorry to disappoint you, but my mistake was to have said \"yes\" in the first place; yours, to have not followed through with the original plan offered me.\n\nAlways with much regard.\n\nEver,\n\nThornton\n\n575. Stephen Sondheim to Leonard Bernstein\n\n[New York, NY]\n\n23 July 1975\n\nDear Lenny,\n\nI encountered the enclosed when I was organizing my files (well, it's better than working). I thought you might like to have them for future archivists, as your corrections are on them in your inimitable handwriting.\n\nLove from Kanagawa,48\n\nSteve\n\n576. Stephen Sondheim to Leonard Bernstein\n\n15 November 1975\n\nDear Lennie,\n\nThanks for the good wishes. We got one terrible review (from Kevin Kelly, very bitchy) and one mixed (Eliot Norton, heavy on the good side), the rest raves.49 The show needs an enormous amount of work on details: clarity, making the numbers land (the \"button\" problem again,50 God save us), timing, etc. But the structure is sound and the production startling and terrific. Keep your fingers crossed.\n\nHope things have taken an upturn on your show and your spirits.\n\nBack to rewriting the opening number.\n\nLove to Felicia,\n\nSteve\n\n577. Stephen Sondheim to Leonard and Felicia Bernstein\n\n16 January 1976\n\nDear Lenny and Felicia,\n\nJust a note to thank you for the tree \u2013 I've discovered that it thrives if you keep yelling 'Bonsai!' triumphantly at it. It's easily fooled.\n\nLove,\n\nSteve\n\n578. Stephen Sondheim to Leonard Bernstein\n\n26 January 1976\n\nDear Lenny,\n\nI've responded to the Societi Musiki Shiki as per your request. I sent them my collection of Bernstein birthday and Christmas compositions and advised them to make their own choice. I also offered to conduct for them, since I figure that the placement of downbeats is irrelevant in Japan.\n\nLove,\n\nSteve\n\nP.S. Your letter arrived with no postage and came from the Dead Letter Office. Should I read anything into this? S.S.\n\n579. Jerome Robbins to Leonard Bernstein\n\nThe Barclay, Rittenhouse Square, Philadelphia, PA\n\n[February 1976]\n\nDear Lenny,\n\n_Good Morning!_\n\nI wanted to talk to you last night just for a moment. I understood your black moment of despair \u2013 but for God's sake, you are a big, capable, enormously talented man \u2013 with tons of energy \u2013 and so is Allan [ _recte_ Alan Jay Lerner] and _don't_ sink (like that down moment). It's understandable; but now is the time to muster up all your wonderful optimism and get it still moving, come to the aid etc., and above all \u2013 as in the show, _keep it going!_ You _are_ rehearsing in public, you are in _some_ chaos, but as in our democracy, you must believe your system will work, which I _know_ you do, in order to move it. Your show now is exactly like the one you are writing about.51 Now you _can_ make it work much _much much_ better. Take care of your house. You can do it. Come on kid, get the spirit up again. No limp cocks!\n\nOle Coach Jer[ry]\n\nSee you about 1.\n\nLOVE\n\n580. Stephen Sondheim to Leonard Bernstein\n\nNew York, NY\n\n4 May 197652\n\n[Telegram]\n\nThe point is it's over and you're still the only artist writing musicals with one exception that is.\n\nLove\n\nSteve\n\n581. Sid Ramin to Leonard Bernstein\n\n8 May 1976\n\nDear Lenny,\n\nJust a note to thank you again for your marvelous gift. Not only is it lavish but the inscription on the inside of the beautiful Gucci leather case is something I will always remember and treasure.\n\nAs you may know, Lenny, I'm always on cloud nine when I'm in the same room with you. The show made it possible for us to spend some time together and I savoured every minute.\n\nFrom our pre-orchestration meetings to our post-orchestration meetings to our Fine and Schapiro53 festivities, I look back at the last three months with great affection.54 Especially, one very long and late meeting (in [Apartment] 92 [in the Dakota]) when we talked into the dawn. I'll never forget it.\n\nI'm sorry the show didn't work out for you (for us!) but I will be eternally grateful for the wonderful moments I've had (including that great night at the Variety Club) in just being with you.\n\nGloria joins me in sending you much love.\n\nAlways,\n\nSid\n\n582. Richard Avedon55 to Leonard Bernstein\n\n407 East 75th Street, New York, NY\n\n[May 1976]\n\nDear Lenny,\n\nI know it isn't what you dreamt it would be,56 but you can _not_ be responsible for anything but your music which is _superb!_ I wept during \"Take Care of This House\" as I haven't since \"... and make our garden grow!\"\n\nAnd:\n\nRehearse\n\nSeena\n\nThe dirge during the second act funeral \u2013\n\nThe Red, White, and Blues\n\nand much more. It's just beautiful, Lenny, and everyone near me was moved, and happy, and so was I, and it was because of your music.\n\nYou stand alone. Terrifying, but true.\n\nLove always,\n\nDick\n\n583. Leonard Bernstein to Helen Coates\n\n\"Xmas 1976\" [December 1976]\n\nDearest Helen,\n\nAt this very crucial turning-point57 in both our lives my annual wishes for a happy new year carry very special weight. So: a very Happy '77, with all my love.\n\nAs always,\n\nLenny\n\n584. Leonard Bernstein to Irwin Kostal58\n\n17 June 1977\n\nDear Irwin,\n\nI've just heard the Fiedler\u2013Pops recording of my _Mass_ music, and I am so pleased that I must write you. What a job of sound-making you did! I almost don't miss the voices...59\n\nI hope all is going well with you. My warmest thanks for _Mass \u2013_\n\nand my affection,\n\nLenny B\n\n585. Oliver Smith to Leonard Bernstein\n\n70 Willow Street, Brooklyn, NY\n\n18 December 1977\n\nMy dear friend,\n\nYour _Songfest_60 is a composition of such emotional variety and richness, it is difficult to be articulate about it after one hearing except to express to you the tremendous emotional effect it had on me. It contains ravishing sound, humor, tenderness, strident joy and anger, and sweet melancholy. Whatever I say to you is inadequate in expressing the joy your beautiful music gave me.\n\nYou are such a dear, great artist, and I hug you again with great love and thankfulness.\n\nOliver\n\n586. Leonard Bernstein to Helen Coates\n\nHotel Sacher, Vienna, Austria\n\n28 January 1978\n\nDear Helen,\n\nI haven't written in all these three weeks (this pen is finished!) because I've been spending all my time between rehearsals in bed, sleeping and trying to regain my strength. It's a long, slow haul, and I still don't feel quite up to snuff. The doctor says that I don't need medicine, but a vacation in the sun, and that's just not possible for many weeks to come. So I muddle through, and _Fidelio_ goes surprisingly well (tremendous reaction & critics) considering that I don't feel my full powers, and that I had to cancel so many rehearsals. Tomorrow is the live TV broadcast \u2013 I pray it will go well.61\n\nSo you too have had the flu! Isn't it ghastly? It's as though one's whole body had been attacked, ear-lobes, toe-nails & all. I also have no appetite & have lost weight: I hope yours is restored to normal.\n\nI'm expecting Felicia on the 3rd of February & I hope you can give her a pleasant birthday party. She sounds splendid on the phone.\n\nDo take care of yourself, & write \u2013 I love hearing from you.\n\nAlways,\n\nLenny\n\n587. Betty Comden to Leonard Bernstein\n\n117 East 95th Street, New York, NY\n\n2 July 197862\n\nDearest Lenny,\n\nThe impossible letter must be written. I have seen you, and we have talked, and I have felt close to you, and it is hard to write when you are close, and you know that so much is both expressed and understood without the need for the written word. Yet I think of you and feel for you, and I think of Felicia, and what this last year must have been like for you, and the void that is now, and I wish I could write something to lessen the pain so visible in your eyes.\n\nThat Felicia was an extraordinary person, we all know. She impressed me the moment you brought her to our green living room on 55th Street. Her patrician beauty and her cool sparkle awed me a bit. I later came to enjoy her great earthy streak, crazy sense of humor, and her many sudden radiant bursts of warmth, and I wish I could have gotten closer than I did. I was close enough to feel totally bereft now.\n\nYou must not blame yourself for not coming through this as a kind of patriarchal leader and rock of ages. If I am bereft, what must you be? You are entitled to full grief, and floundering, and, yes, weakness. These are the feelings you expressed to me. This is a deep tragedy you are experiencing. You are so much, Lenny \u2013 so many qualities and gifts and inner voices not given to many human beings. You will find your strength somehow in them \u2013 and in the beautiful elements added to them by the co-mingling of your life and Felicia's.\n\nMuch love, always, from Steve and from me.\n\nBetty\n\n588. Nadia Boulanger to Leonard Bernstein\n\n\u00c9coles d'Art Am\u00e9ricacies, Fontainebleau, France\n\n7 August 1978\n\nDear Lenny,\n\nTo try to be with you in your commensurable distress.\n\nAnd you knew so well, and for so long. Hope she did not suffer too terribly. Life sometimes is so difficult to stand. May your courage be as great as your sorrow.\n\nForgive these poor lines, I feel myself so sick and miserable.\n\nWith love,\n\nNB\n\n589. Andr\u00e9 Previn63 to Leonard Bernstein\n\nThe Watergate Hotel, Washington D.C.\n\n28 August 1978\n\nDear Lennie,\n\nIt bothered me a lot to hear you sounding so depressed when I spoke to you the day after the concert. I thought about it quite a lot. At first I came to the naive conclusion that writing to you about it was none of my business, but then, the more I gave it thought, the more I realized that both as an old friend and as a musician, it _was_. I've been an admirer and a follower and, in a more remote way, a disciple since I first heard you make music in San Francisco in 1950 with the Israel Philharmonic. You've touched, directly or circuitously, a great many musical decisions of mine, but what's more important, the lives and ambitions of every conductor in this country. That's the kind of statement usually found on the scrolls of Doctorates, but for all its grandiloquence it happens to be true. Therefore, if you were to succumb to a depression, however temporary, that would keep you from your usual frighteningly energetic achievements, you'd be letting down an amazing number of musicians. You've kept those of us who grew up in the same years as you feeling young; you've kept those older than you correctly infuriated, and you've been a lighthouse of constancy to all those 20-year-old current phenomena. As a friend, I can see that this is a burden you might not want right now, but as a member of that weird band who feel that a day without music is an irresponsible waste, I have to tell you that you're stuck with it. I'm certainly not entitled to be a spokesman, and all this sounds terrifyingly pompous, but we depend on you and love you and trust you.\n\nWhen I celebrate my 60th birthday, you will be a hell of a lot younger than [...] Karl B\u00f6hm is right now, and I will expect you to play the Triple Concerto while I conduct.\n\nSee you soon.\n\nAndr\u00e9\n\n590. Aaron Copland to Leonard Bernstein\n\nPeekskill, NY\n\n5 September 1978\n\nDear, dear Lenny,\n\nThat was a _beautiful_ note you sent me. I was so pleased that you were pleased with the _Jeremiah_ movement.64 It sure is a beauty!\n\nAnd now the sad part. I'm not going to be present at the \"Remembrance\" for Felicia on the 18th because I am under contract to conduct the same day in Virginia (Norfolk). But my thoughts will be with you and the family that day.\n\nAs ever,\n\nAaron\n\n591. Leonard Bernstein to Burton Bernstein\n\n\"Sharm-el-Skeikh, Tiran Straits, Sinai, Israel (or, by the time you get this, Egypt?)\"\n\n[October 1978]\n\nDearest BB,\n\nI've thought of you all day, driving down the old S[inai] Peninsula (with the McClures and Tommy C.) and hearing your voice from Eilat to Sharm, Bedouin by Bedouin, camel by camel (one dead by the road in rigor mortis) & glorious geological grandeur by g. g. g. But the climax came tonight, walking up the beach from our Tunisian shrimp dinner, as we were confronted by one of the truly great signs of my experience, repeated at intervals along the strand, in the usual 3 languages. The English reads (and this may well merit a _New Yorker_ appearance):\n\nSECURITY WELL DESTINED\n\nFOR ABSORPTION SUSPICIOUS\n\nOBJECTS ONLY.\n\nI cannot get it out of my mind: it defies all intelligibility. \"Only\"!\n\nAside from this I've thought of you often this month (Holidays _very_ late this year) climaxed by _Shmini Hatzsreth_65 (gasp! of childhood total recall) only yesterday. Rosh, Kol, Yom, Succoth, etc. have come and gone in a welter of rehearsals & concerts & much sleeping in between. Very difficult to resume the old schedules, but God knows I've tried, acquitted myself passably, & have these 4 days of desert holiday before attacking the even more rigorous set-up in Vienna. I'll make it...\n\nThis great wonderful Sinai... give it back?!\n\nNot one word have I received from anyone all month. I have phoned & written: not been answered. What is happening to the loving old family? Do call Mamma (if that is indeed her name) & give her my love. Tell Horse she's a cad not to have called or written.\n\nAnd much love to all, & to you, \u2013 mine Brothoass,\n\nBen\n\nBest from Moish [Moshe Pearlman].\n\nI hope the book goes apace.\n\n1 Published by Alex Ross in _The New Yorker_ online: (accessed 26 February 2013). It would be reassuring to think that this bizarre episode was the only time that the words of the Latin Mass came under suspicion from a Presidential aide. Presumably, Buchanan's main cause for alarm might have been the last line of the Agnus Dei: \"Dona nobis pacem\" (\"Grant us peace\"). It's surprising that Buchanan felt the need to \"get us a good Jesuit\" to provide a translation, since he had himself been educated at Jesuit-run institutions: Gonzaga College High School and Georgetown University, and English translations of the Mass were so readily available.\n\n2 Oliver Knussen, \"Bernstein: _Dybbuk_ ,\" _Tempo_ , No. 119 (December 1976), p. 34.\n\n3 Oliver Knussen, \"Bernstein: _Songfest_ ,\" _Tempo_ , No. 128 (March 1979), pp. 21\u20132.\n\n4 Dougary 2010.\n\n5 Burton 1994, pp. 446\u20137.\n\n6 Bernstein and the New York Philharmonic recorded Thompson's Second Symphony for Columbia Records on 22 October 1968, straight after four concert performances. It is difficult to argue with the composer's delighted response to Bernstein's magnificent recording of the work, which has been reissued on CD by Sony Classical (SMK 60594).\n\n7 Dietrich Fischer-Dieskau (1925\u20132012), German baritone who had sung the title role in Bernstein's Vienna production and recording of Verdi's _Falstaff_ in 1966, the reason he signs himself \"Sir Dieter Falstaff\" here.\n\n8 See Letter 548.\n\n9 Rabbi Judah Cahn (1912-84) was the founding Rabbi of the Metropolitan Synagogue of New York, which has always had a reputation as an informal and liberal Reformed synagogue. Cahn was a family friend of the Bernsteins, and he was also passionate about music (he appears as a speaker on Bernstein's recording of Bloch's _Sacred Service_ ).\n\n10 A reference to the fundraiser for the legal defense of the Black Panthers held in Bernstein's apartment. On a number of occasions Bernstein explained that the event had not been intended to endorse the radical (armed) agenda of the Black Panthers. But Tom Wolfe's description of the event as \"radical chic\" (in _New York Magazine_ , 8 June 1970) has refused to go away.\n\n11 Ingmar Bergman (1918\u20132007), Swedish film and theater director, described by Woody Allen (in a TV interview with Mark Kermode) as \"probably the greatest film artist, all things considered, since the invention of the motion picture camera.\" Unfortunately, the plan outlined by Bernstein for a filmed production of Wagner's _Tristan und Isolde_ came to nothing.\n\n12 Bernstein's concerts with the Orchestre de Paris at the Th\u00e9\u00e2tre des Champs-\u00c9lys\u00e9es in February 1971 included Ravel's _Tombeau de Couperin_ and the G major Piano Concerto (directed from the piano), and Berlioz's _Rom\u00e9o et Juliette_.\n\n13 Bernstein's concerts with the Vienna Philharmonic on 20 and 21 Februrary 1971 included Haydn's Symphony No. 102, Ravel's G major Piano Concerto (directed from the piano), and Schumann's Fourth Symphony.\n\n14 Ethel Linder Reiner was a Broadway producer whose credits included _Candide_ in 1956. Her death was announced in _The New York Times_ on 11 February 1971.\n\n15 Debs Myers was described in his _New York Times_ obituary as a \"political press aide.\" He worked on Senator Bobby Kennedy's Senate campaign and on both of Adlai Stevenson's presidential campaigns. He died on 2 February 1971 at the age of 59.\n\n16 Cryptic crossword puzzles.\n\n17 Bernstein consulted the psychiatrist Willard Gaylin regularly in the early 1970s.\n\n18 Jamie Bernstein (b. 1952), the eldest child of Leonard and Felicia Bernstein.\n\n19 Benazir Bhutto (1953\u20132007), the first woman in modern history to lead a Muslim state, attended Harvard (Radcliffe College) from 1969 to 1973. Her nickname at Harvard was \"Pinkie\".\n\n20 Zulfikar Ali Bhutto.\n\n21 8 September was the opening night of _Mass_ at the Kennedy Center in Washington, D.C.\n\n22 Oliver Smith designed the sets for _Mass_.\n\n23 Christa Ludwig (b. 1928), German mezzo-soprano with whom Bernstein collaborated on numerous occasions from the late 1960s onwards, notably in Mahler (Symphonies Nos. 2 and 3, _Das Lied von der Erde, Des Knaben Wunderhorn_ ), Brahms (lieder), and in Bernstein's own work: the _Jeremiah_ and _Kaddish_ symphonies, and, in December 1989, as the Old Lady in _Candide_.\n\n24 David Charles Abell (b. 1958) is now a successful conductor. At the time of writing this letter, he was a member of the Berkshire Boy Choir and had performed in the original production of Bernstein's _Mass_. At the time he conducted Stephen Sondheim's 80th birthday concert at the 2010 BBC Proms, he wrote: \"None of that would probably have happened if I hadn't been in _Mass_ at age 13\" (David Charles Abell, personal communication).\n\n25 Marcus Dods (1918\u201384), British conductor. He was music director for all of Richard Rodney Bennett's major film scores: _Billion Dollar Brain_ (1967), _Far From the Madding Crowd_ (1967), _Nicholas and Alexandra_ (1971), _Lady Caroline Lamb_ (1972), and _Murder on the Orient Express_ (1974).\n\n26 Bernstein played his own arrangement of Rodgers' \"Nobody's Heart Belongs to Me\" from _By Jupiter_ as part of the \"Celebration of Richard Rodgers\" held at the Imperial Theatre on 26 March 1972 in honour of Rodgers' forthcoming 70th birthday (28 June 1972).\n\n27 On 15 and 16 April 1971, Bernstein conducted performances of Mahler's Fifth Symphony with the Vienna Philharmonic. A few weeks later (6 and 7 May) he conducted Mahler's Fourth Symphony.\n\n28 Luciano Berio (1925\u20132003), Italian composer. Bernstein commissioned Berio's _Sinfonia_ , which was first performed by the New York Philharmonic on 10 October 1968.\n\n29 This innovative television series, presented by Berio and broadcast in 1972, included an episode entitled \"Nuovo mondo\" in which Bernstein appeared.\n\n30 \"October's Bright Blue Weather\" by Helen Hunt Jackson (1830\u201385) was first published in her _Verses_ (Boston, 1870). Her poetry was admired by Ralph Waldo Emerson, and she was a lifelong friend of Emily Dickinson. As well as poetry, Jackson also wrote a popular novel ( _Ramona_ ) about the government's mistreatment of Native Americans.\n\n31 _Dybbuk_ was slowly taking shape when Robbins wrote this letter, but there was still much to be done. _The New York Times_ reported (12 July 1972) that Bernstein was planning to take \"nearly a year off from public performing to give his undivided attention to writing music\" and as a result he did get the score finished. The ballet was first performed by New York City Ballet at Lincoln Center, New York, on 16 May 1974.\n\n32 _Interplay_ was Robbins' second ballet (after _Fancy Free_ ). It was set to a score by Morton Gould and first performed at Billy Rose's Concert Varieties on 1 June 1945. It was subsequently taken into the repertoire of the New York City Ballet. In October 1972, the Joffrey Ballet gave a successful revival, the one referred to in this letter.\n\n33 _Owen Wingrave_ was originally composed as an opera for television, first broadcast on 16 May 1971. In this letter Britten is referring to the work's stage premiere, at the Royal Opera House, Covent Garden, on 10 May 1973.\n\n34 The world premiere of _Death in Venice_ was given at Snape Maltings on 16 June 1973.\n\n35 Undated, written inside a Christmas card from Britten and Pears.\n\n36 Virgil Thomson (1896\u20131989), American composer and critic. Thomson had known Bernstein since the 1940s, when he encouraged Bernstein, Paul Bowles, and others to develop as tonal composers. As a critic, he always took Bernstein's music seriously. In John Rockwell's obituary of Thomson in _The New York Times_ (1 October 1989), Bernstein was quoted as follows: \"The death of Virgil T is like the death of an American city: it is intolerable. Virgil was loving and harsh, generous and mordant, simple but cynical, son of the hymnal yet highly sophisticated. He will always remain brightly alive in the history of music, if only for the extraordinary influence his witty and simplistic music had on his colleagues. I know that I am one twig on that tree, and I will always cherish and revere Virgil, the source.\"\n\n37 The 1974 Broadway revival of _Candide_ opened at the Broadway Theatre on 10 March and ran for 740 performances.\n\n38 Alan Jay Lerner (1918\u201386), American lyricist and librettist who was a contemporary of Bernstein's at Harvard. A plan to work together on a show in 1949 came to nothing (see Letter 288), but in 1957 Lerner and Bernstein wrote two choruses for the Harvard Glee Club. In 1976 they collaborated on _1600 Pennsylvania Avenue_ \u2013 a work that has, alas, become more famous for its catastrophic failure on Broadway than for the beauty of Bernstein's score and the brilliance of Lerner's lyrics (both heard to advantage in _A White House Cantata_ , arranged from the show by Charlie Harmon and Sid Ramin after Bernstein's death). Lerner's most productive collaborations were with Frederick Loewe: _Brigadoon, Paint Your Wagon, My Fair Lady, Gigi_ , and _Camelot_ ; he also worked with Kurt Weill ( _Love Life_ ) and Burton Lane ( _On A Clear Day You Can See Forever_ ).\n\n39 The first performance of _Dybbuk_ took place the day after this letter was written.\n\n40 Presumably the second act of _1600 Pennsylvania Avenue_.\n\n41 Maurice Abravanel (1903\u201393), Greek-born American conductor. He was a pupil and friend of Kurt Weill and conducted the original productions of several of Weill's Broadway shows, including _Knickerbocker Holiday, Lady in the Dark, One Touch of Venus, The Firebrand of Florence_ , and _Street Scene_. In 1949 he was conductor for the Broadway run of Blitzstein's _Regina_. Abravanel was Music Director of the Utah Symphony Orchestra from 1947 to 1979, leaving an extensive legacy of recordings. The orchestra's home was renamed Abravanel Hall in 1993.\n\n42 In August\u2013September 1974, Bernstein was on tour with the New York Philharmonic to New Zealand, Australia, and Japan. Bernstein's concerts began in Auckland on 16 August, and ended in Nagoya, Japan, on 6 September. In Wellington (17 August) and Christchurch (18 August) the programme was Mozart's Piano Concerto K503 (with Bernstein as the soloist) and Mahler's Fifth Symphony.\n\n43 Alexander, their son.\n\n44 In June 1974, Felicia had mastectomy surgery.\n\n45 A \"British-style\" cryptic crossword.\n\n46 This article appeared in _The New York Times_ on 25 August 1974. Written by Raymond Ericson and headed \"The Pick of Modern American Music,\" it included no work by Bernstein. According to the report, Igor Buketoff collated the list in _The New York Times_ from the responses of a jury of nine experts. Foss (like Diamond) was one of the American composers who actively supported Bernstein's music from the 1940s onwards. Foss was understandably hurt that his choice of Bernstein's _Age of Anxiety_ had been ignored by _The New York Times_.\n\n47 Thornton Wilder died in December 1975, a few months after writing this letter, but the story of Bernstein's attempt to set _The Skin of Our Teeth_ went back more than a dozen years (see note to letter 500). Why was Bernstein in contact with Wilder about this project as late as 1975? Perhaps it was because the revue _By Bernstein_ that opened on 23 November 1975 at the Chelsea Theater Center, New York, included \"Here Comes the Sun\" and \"Spring Will Come Again,\" both originally written for _The Skin of Our Teeth_ in 1964, before being recycled in the _Chichester Psalms_ the following year. Perhaps Bernstein also contemplated including Sabina's opening aria (\"Oh! Oh! Oh!\"), described by Wilder in unflattering terms in this letter. Sketches for this survive in the Leonard Bernstein Collection.\n\n48 \"Kanagawa\" is a reference to _Pacific Overtures_. Tryouts for this show started at the Shubert Theatre, Boston (8\u201329 November 1975), and then the Kennedy Center Opera House, Washington, D.C. (4\u201327 December 1975), before opening on Broadway at the Winter Garden Theatre on 11 January 1976.\n\n49 The reviews for the Boston tryout of _Pacific Overtures_.\n\n50 The \"button problem\" refers to finding the most effective way to end a song.\n\n51 The tryouts of _1600 Pennsylvania Avenue_ were in Philadelphia: it was proving to be an exceptionally troublesome show. Jerome Robbins was not only Bernstein's most regular theatrical collaborator and a trusted friend, but he was also a brilliant \"show doctor.\" Despite a score that is often very beautiful, _1600 Pennsylvania Avenue_ failed when it moved to Broadway.\n\n52 4 May 1976 was the opening night of the troubled _1600 Pennsylvania Avenue_ on Broadway, where it ran for just seven performances.\n\n53 Fine and Schapiro is a famous Kosher restaurant and delicatessen in New York.\n\n54 Ramin (and Hershy Kay) orchestrated _1600 Pennsylvania Avenue_.\n\n55 Richard Avedon (1923\u20132004), American photographer who helped to \"define America's image of style, beauty and culture for the last half-century,\" according to his obituary in _The New York Times_ (1 October 2004). Avedon and his wife Evelyn were close friends of the Bernsteins, and he loved Bernstein's music (a few years earlier, on 3 January 1972, he had written to Bernstein and Felicia about _Mass_ : \"I play it over and over (not while I'm photographing), and when I'm not listening, I sing 'I Go On' and cry a lot. How can I thank you?\").\n\n56 _1600 Pennsylvania Avenue_.\n\n57 One of the very few allusions in Bernstein's correspondence to his separation from Felicia in 1976\u20137. The story of this traumatic episode is eloquently told by Humphrey Burton (1994, pp. 426\u201341). Felicia and Leonard were reconciled by the summer of 1977, but she was already suffering from the lung cancer that would kill her a year later.\n\n58 Irwin Kostal (1911\u201394), American orchestrator and arranger. One of the original orchestrators of _West Side Story_ with Sid Ramin. His Hollywood credits included _West Side Story_ (again with Sid Ramin) and _The Sound of Music_ (conductor and music supervisor), winning Oscars for both. He also conducted the 1982 digital re-recording of the soundtrack to Disney's _Fantasia_.\n\n59 The album _Music from Mass \u2013 Overture to Candide_ by Arthur Fiedler and the Boston Pops Orchestra was issued by Deutsche Grammophon. It included nine movements from _Mass_ arranged for orchestra by Irwin Kostal.\n\n60 The first complete performance of _Songfest_ took place in Washington, D.C., on 11 October 1977.\n\n61 The live performance of _Fidelio_ from the Vienna State Opera on 29 January 1978 was released on DVD (Deutsche Grammophon 073 4159) in 2006. It's a magnificent performance that has been widely praised, not least by John Steane in _Gramophone_ who said that it constituted \"one of the great artistic experiences of a lifetime.\"\n\n62 Felicia Bernstein died on 16 June 1978, having been ill with cancer for several years.\n\n63 Andr\u00e9 Previn (b. 1930), German-born American conductor, pianist, and composer. Previn was the piano soloist in Beethoven's Triple Concerto (with Yehudi Menuhin and Mstislav Rostropovich) in the gala concert conducted by Bernstein on 25 August 1978. This was a celebration of his 60th birthday with the National Symphony Orchestra, at its summer home, Wolf Trap in Virginia.\n\n64 Copland conducted the Lamentation from Bernstein's _Jeremiah_ Symphony (with Christa Ludwig as the soloist) at the gala concert for Bernstein's 60th birthday at Wolf Trap, on 25 August 1978.\n\n65 Often given as Shemini Atzeret, the \"Eighth Day of Assembly,\" a Jewish holiday sometimes combined with Simchat Torah.\n8\n\nFinal Years\n\n1979\u201390\n\nThere was no lack of glory in the last decade of Bernstein's life \u2013 nor any shortage of love \u2013 but Felicia's furious prediction doesn't feel too wide of the mark either. In terms of composition, the last ten years are difficult to assess: there are some fine pieces, of which _Halil_ for flute and orchestra is certainly among the best. Typically for Bernstein the inveterate self-borrower, the fast section of _Halil_ (starting at p. 15 of the published full score) was derived from an occasional piece: the music he wrote in October 1979 for the 50th Anniversary of CBS in 1978 (the main notes of the theme, C\u2013B flat\u2013E flat, spell out C\u2013B\u2013Es [S] in German). But with the opera _A Quiet Place_ , there's an inevitable sense of declining powers \u2013 made manifest by its integration of _Trouble in Tahiti_ , which emerges as much the strongest part of the work. Bernstein longed to write the Great American Opera, and had done for decades, but while he made some glorious contributions to American musical theater \u2013 _On the Town_ and _West Side Story_ are unquestionably two of the greatest scores ever written for Broadway \u2013 Bernstein felt he should push himself further, and in a more \"serious\" direction. But this was not something he could do by himself. Perhaps if Jerome Robbins had wanted to write an opera, it might have happened, since Robbins was one of the very few people from whom Bernstein took criticism and who had superb theatrical instincts. John McClure, Bernstein's long-time record producer, surely put his finger on the problem of the late years: \"Felicia was vital to his stability as was Jerry Robbins, the only two people who could make Lenny sweat.\"1 It's very hard to escape the feeling that he was right.\n\nThe letters from the 1980s are less substantial \u2013 above all because Bernstein no longer had Felicia to confide in when he was away, but also because of an increasing reliance on phone calls and faxes. Even so, there were still faithful letter-writers, Stephen Sondheim and Jerome Robbins among them, and there are interesting \u2013 often touching \u2013 letters to and from other musicians, especially conductors. Bernstein's profound admiration for Karl B\u00f6hm is apparent from the letter that he sent in 1981 to his ailing colleague on his deathbed. There are delightful letters from Carlos Kleiber (requesting an autograph for his son, but in a way that is full of humor and charm), from Yehudi Menuhin praising Bernstein's controversial performance with the BBC Symphony Orchestra of Elgar's _Enigma Variations_ , and from Marin Alsop early in her conducting career. She wrote to send heartfelt thanks to Bernstein for his inspiration, and for the opportunity to work with him in Japan. It's well known how warmly Alsop admires her mentor, but Jonathan Cott's 1989 interview reveals just how highly Bernstein also thought of her: \"There's a young woman named Marin Alsop. She was a student of mine at Tanglewood \u2013 she did Hindemith's _Mathis der Maler_ and Roy Harris' Third Symphony under me, and she's fabulous, she is simply wonderful\" (Cott 2013, p. 125).\n\nBernstein's 70th birthday brought tributes from friends, celebrities, and even politicians. Ronald Reagan wrote to congratulate him, as did Frank Sinatra (an old friend \u2013 they'd worked together in nightclubs in the early 1940s), Miles Davis, and Claudio Arrau, who recalled fondly the first meeting of Bernstein and Felicia at his party in 1946. But there's a feeling of nostalgia about many of these greetings \u2013 celebrating great times that have, to a large extent, been and gone. In terms of Bernstein's compositions, it certainly seemed to be the case: _A Quiet Place_ (1983, revised in 1984 and 1986) was his last large-scale work.\n\nAnd yet in the concert hall he continued to give triumphant performances with orchestras in New York, Vienna, Amsterdam, Munich, London, and elsewhere: these were not just huge public successes, but inspiring accounts of Schubert and Schumann, or of Copland, Harris, and Ives. There were extraordinary and daring concerts of Tchaikovsky in New York and Sibelius in Vienna, and Mahler performances that it seems too easy to describe as \"revelatory\" \u2013 but Bernstein's understanding of Mahler grew ever deeper, and his interpretations evolved as a consequence: the thrilling drive and drama of his earlier Mahler gave way to a kind of visionary splendor. It's as if Bernstein's frustration about his inability to compose with any consistency over these last few years found a more positive counterpart in his conducting of music by others. After Felicia, he didn't \u2013 indeed, he couldn't \u2013 find any lasting personal relationships. There were some passionate affairs, there was infinite love poured out on his children, but there was a certain loneliness: as a musician, Bernstein's interpretative insights grew deeper and richer, and yet at the same time his extreme celebrity carried with it the inevitable problems of having less time to spend with people he loved and cared about, or even to be alone. He fell out with David Diamond in a viscious exchange of letters, after half a century of friendship \u2013 and the publication of Joan Peyser's tell-all biography hurt him (even though he claimed never to have read it). What disturbed him was probably not so much what it said, but the people who said it: friends and colleagues who Bernstein felt had been disloyal by sharing secrets that should have remained private. But there were still faithful friends, and Bernstein took great pleasure in their company, as they did in his \u2013 for instance, he took Sid and Gloria Ramin to Israel in 1986. His devoted secretary Helen Coates died in 1989, and by the start of 1990 concerns about his own health started to preoccupy him. A malignant tumor was diagnosed and in great secrecy he was given a course of radiation therapy. Still, he managed a trip to Prague to conduct Beethoven's Ninth Symphony (and to spend time talking with President V\u00e1clav Havel). After a few days delayed convalescence, he set off for the first Pacific Music Festival in Sapporo, Japan (where Marin Alsop worked as his assistant), but it quickly became apparent that his condition was worsening. He struggled through the concerts there and in Tokyo. Humphrey Burton quotes an anguished diary entry made by Craig Urquhart2 about Bernstein's dependency on massive doses of painkillers: \"The real question is why he bothers at all. Here is a very sick man who knows he is doing his _danse macabre_ \" (Burton 1994, p. 519). Bernstein had to withdraw at very short notice from his last engagement in Japan \u2013 a big outdoor concert \u2013 earning some criticism in the Japanese press, who were unaware of the seriousness of his condition. He returned to New York and was soon on the road again, for concerts at Tanglewood. The major event was the 50th anniversary concert of Tanglewood and \u2013 on a more personal note \u2013 the 50th anniversary of Bernstein first conducting there. The Bernstein family was out in force for the occasion, including his mother, Jennie. Heavily medicated and fighting for breath, he conducted the \"Four Sea Interludes\" from Britten's _Peter Grimes_ , and the concert ended with Beethoven's Seventh Symphony. Listening to the published recording of this concert, it's immediately apparent that Bernstein was uncomfortable, but in the Scherzo third movement he succumbed to a coughing fit that prevented him conducting for several minutes. The Boston Symphony Orchestra kept playing, and Bernstein was able to resume conducting for the last movement, but only with the greatest difficulty. It was his last concert \u2013 and he knew it. Back in New York he told Craig Urquhart: \"You know it's incredible how I did my first concert at Tanglewood and I did my last concert at Tanglewood. There's a real sense of closure\" (Burton 1994, p. 524). On 5 September his mother Jennie, then in her nineties, wrote: \"You are surrounded by a beautiful family, your children and grandchildren. That in itself should be good medicine for you.\" But by then it was far too late. His apartment in the Dakota began to resemble a hospital ward, and on 9 October a statement was issued announcing his retirement from conducting. Five days later, at 6:15 p.m. on 14 October in his apartment at the Dakota, Leonard Bernstein died.\n\n592. Oliver Smith to Leonard Bernstein\n\n70 Willow Street, Brooklyn, NY\n\n23 July 1979\n\nDearest Lenny,\n\nYesterday I had the enormous pleasure of hearing you conduct at Tanglewood.3 You were simply wonderful. I felt so proud of you. I made a darting trip up to \"welcome\" you back along with the enthusiastic thousands to whom you bring inspiration and joy and so much of your innermost being. Unfortunately I had to rush back and didn't have the opportunity to go back stage, see those eyes open very wide with delight and give you a great hug. I do so now.\n\nYou were simply magnificent. The Haydn Mass in B flat was so joyous I was ready to embrace the church, something I am sure would horrify you. I heard the chorus rehearsing as I came on the grounds, only one half hour before the performance! I thought: just like Lenny.\n\nI was a little apprehensive about the Shostakovich on such a steaming afternoon. You made it absolutely riveting, with all its prolix fascination and strangeness as well as introspective qualities. I was as thrilled as the audience and found a joyous release in joining in the ovation.\n\nIt is wonderful we are finally again doing _West Side Story_.4 It is my favorite theatrical effort. This week I take off to England for a few weeks to rest up for the battles to follow, to which I am looking forward.\n\nMeanwhile I send you my love and a tender embrace of thanks.\n\nOliver\n\nP.S. I am very sorry to miss the Mahler 9th.5 It is something very special to you and your interpretation is the greatest. I shall never forget that great performance in Vienna several years ago at the Musikverein.\n\nLove,\n\nO\n\n593. Leonard Bernstein on Aaron Copland\n\nTribute delivered at the Kennedy Center Honors, Washington, D.C.6\n\n2 December 1979\n\nLast month Aaron Copland celebrated his 79th birthday, out of which evolved a plethora of toasts, lunches, speeches, tributes and honors, of which tonight's honor is certainly the grandest. But if all this happens when he is 79, what volcanoes will erupt when he hits 80 a year from now? I can't begin to imagine; but whatever monster celebrations, fireworks and celebrations may take place, they will never suffice to honor in proper degree this great gentleman of American music. Never have we had a composer of his superb lyric and symphonic quality who has been personally so admired, respected and \u2013 let's say it \u2013 _loved_ by so many people as Aaron. I speak not only of the music but also of the man. Ask anyone who knows him: \"What is Aaron like?\" And they will surely respond by describing the Copland grin, the Copland giggle, the Copland wit and warmth, and width of his embrace.\n\nHe has always had time for everyone \u2013 especially the young (that is the mark of a great man: time for _people_ ); and his unmistakeable sharp \"judge-nose\", as he once described it, has always been sniffing out new talent, in whatever hamlet or continent it might be hiding, to encourage with praise, to nurture with criticism, and to help on its way to public exposure.\n\nAnd yet he is also the most moderate, balanced, objective, sane and non-melodramatic man I have ever known. When he exaggerates, it's to make us laugh; when he understates, it's to point up an irony. Everything else is plain truth \u2013 \"plain\" is one of his favorite words \u2013 and \"truth\" is the very essence of the man.\n\nAll of these qualities \u2013 the generosity, the wit, the quirkiness, the compassion and tenderness and plainness \u2013 all of these inhabit his music with a mirror-like truth. But there are other qualities in the music which reflect aspects of the man he never allows us to see. The music can have an extraordinary grandeur, an exquisite delicacy, a prophetic severity, a ferocious rage, a sharp bite, a prickly snap, a mystical suspension, a wounding stab, an agonizing howl \u2013 none of which corresponds with the Aaron we loving friends know \u2013 but comes from some deep, mysterious place he never reveals to us except in his music.\n\nI have known Aaron intimately for 42 years, and I have only once seen him in a state of anger. Once. And I recall a luncheon date in which he was uncharacteristically quiet, mentioning only that he had a headache. I learned much later that day that his father had died on the previous night. And once \u2013 again only once \u2013 have I seen him weep when, at a Bette Davis movie that caused me to oo and ah and marvel and groan \"NO, NO, NO\" at the unbearable climax (I am always very vocal at the end of Bette Davis movies), he turned to me, his cheeks awash with tears, and sobbed, \"Can't you shut up?\"\n\nNow usually men of such restraint and moderation, who also harbor such tumultuous inner passions and rages, are sick men, psychotics who are prone to unpredictable and irrational explosions. Not so Aaron. The unpredictability is all in the music, which is why that music is so constantly fresh and surprising, as is the music of Beethoven. The man himself is sanity itself \u2013 and that is why the first moment I met him \u2013 on his 37th birthday \u2013 I trust[ed] him instantly and relied completely on his judgment as gospel and have done so ever since. It is my honor to present him to you, my first friend in New York, my master, my idol, my sage, my shrink, the closest thing to a composition teacher I ever had, my guide, my counselor, my elder brother, my beloved friend \u2013 Aaron Copland.\n\nLeonard Bernstein\n\n30 November 1979\n\n594. Aaron Copland to Leonard Bernstein\n\nPeeskill, NY\n\n4 December 1979\n\nDear Lensk,\n\nHow to thank you for that _splendiferous_ talk at the Kennedy Center Sunday night. (Not to mention the special effort needed to be in 2 cities almost simultaneously!)\n\nAnd everyone around me seemed to be enjoying the talk as much as I did (including Mrs. Carter,7 who was seated next to me).\n\nIt was truly a night to be remembered \u2013 thanks a million!\n\nLove,\n\nA\n\n595. Francis Ford Coppola8 to Leonard Bernstein\n\nThe Sentinel Building, 916 Kearney Street, San Francisco, CA\n\n7 March 1980\n\nMy most respected Maestro Bernstein,\n\nCertainly the telephone conversation that my assistant Tess related to me broke my heart. The confusion that comes when four different artists attempt to assemble a collaboration; the complexity of each individual's time, place and temperament \u2013 the more difficult areas of lawyers, agents and deals \u2013 is the reason very often that desirable collaborations never happen at all.\n\nI'm sure you can understand how hard it must be on me, who essentially must be the pivot-point for the project \u2013 and also work as writer and director, when it's quite obvious to me that my ideas are being auditioned by you, and I can plainly see that every so often those with whom I wish to collaborate are obliged to talk to their agents to see whether or not a deal has been struck.\n\nSo that, as I understand it, the collaboration hinged on whether in fact you were committed and approved the project as it had been (however vaguely) outlined by me to you, Betty [Comden] and Adolph [Green].\n\nI didn't know I was expected to call you or that you were waiting to hear from me, as of course I had just assumed that we were all friends enough that if you or anyone wished to talk to me, you would simply call, and not stand on ceremony.\n\nIn short, as I see the entire situation, it is as follows:\n\nMy ideas and concept for this kind of musical\u2013opera\u2013film are too embryonic in form to really be presented to collaborators who must have something defined and specific so that they can do their work. I guess I had hoped that the discovery of this kind of unprecedented film would be made together, on a slow, arduous hit or miss kind of artistic exploration. But I realize now that this process is too difficult to put together between three different groups of artists, who are also all working on other things and have a tight schedule. I also understand, Maestro, that your own composition schedule is heartbreakingly short, and that you can ill afford to waste your time.\n\nI have personal opinions also, about this matter, which I have never mentioned because it would have been rude to you, whom I have always admired. My opinion is that for a composer of your scale and stature to limit his composition, which is the flower of his work, to take a second chair to certain other of the performing arts of which others are perfectly capable, is wrong. But no one can _write_ music like Bernstein. And Bernstein's theatrical music is as good as any that has _ever_ been written.\n\nAnd so I came to say... \"Here, let me give you musical cinema. Take theatrical music to its next step, in a medium that the whole world will see and respond to.\" Well, that is my dream, and I know as sure as I know anything that it is your dream too. Please bear with me \u2013 maybe we'll work more informally, over the next six months or even year.\n\nBut then when we put these sketches together, it would be a pleasure to fully collaborate with you and Adolph and Betty.\n\nExcuse me for the ownership and the beginning of a new movie studio, Zoetrope Studio, which makes you want to say that I have gone from being an artist to a \"mogul\". Zoetrope Studios will create the most modern, electronic studio in the world, and its first work will be TUCKER by Leonard Bernstein, Betty Comden and Adolph Green \u2013 who knows, maybe even Jerome Robbins, and Francis Coppola. I'd sure want to see that film.9\n\nMaybe someday you'll say: \"I didn't know he was going to be Francis Coppola\".\n\nWith sincere love,\n\nFrancis\n\n596. Leonard Bernstein to Stephen Sondheim\n\n16 March 1980\n\nDearest Steve\n\nThis Es\/la10 tritone comes with warmest memories of work and play and friendship. S\u2013L is almost a quarter of a century old, and it seems all wrong that I should not be in attendance on your glorious 50th. But I'll be down in the Caribbean with my children on that date, and will be thinking strongly and affectionately of you. More power to you and much merriment.\n\nL = 50 = Love = Lenny\n\n597. Stephen Sondheim to Leonard Bernstein\n\nNew York, NY\n\n19 March 1980\n\nDear Lenny,\n\nAs usual, clever, appropriate, poetic and touching. Thank you for this one, and the others, and what's behind them.11\n\nI am sorry you won't be here, but I suspect you'll have a better time anyway among the natives.\n\nLove,\n\nSteve\n\n598. Jennie Bernstein to Leonard Bernstein\n\n[August 1980]12\n\nI remember saying, \"Dear son, some day you will have it all.\" Your Dad of blessed memory objected to your career out of his love for you.\n\nYou write and think young. Stay that way my dear, for many more healthy musical years.\n\nWith much love and good wishes,\n\nYour ever loving,\n\nMom\n\n599. Jerome Robbins to Leonard Bernstein\n\n12 November 1980\n\nDear Lenny,\n\nThank you for sending me the tape and score of your latest work.13 I really like it very, very much, and I'm in love with so many of the middle movements. I'd love to do it, but I worry about it because, as you know, music tends to shrink when you add dancing to it, and the pieces themselves are so short to start with that I don't know how it would avoid resulting in broken, tiny pieces. At least, that's my reaction now. Let me know if you have any suggestions, and we'll get together. But I do love the work.\n\nSorry that I've only gotten to this now. I just finished the Mozart Rondo in A minor, and I like it, although like the music, it is a fairly quiet work. If you want to come see it, let me know.\n\nAll my love,\n\nJerry\n\n600. Doriot Anthony Dwyer14 to Leonard Bernstein\n\n3 Cleveland Road, Brookline, MA\n\n25 July 1981\n\nDear Lenny,\n\nJust a note to tell you, in writing, that it was an enormous pleasure you gave me this year, to play your _Halil_ with you. It is rare one flutist plays a solo with a great conductor who is _that_ involved in the same work. I suspected as much and the anticipation nearly tore me apart, and I loved _Halil_ right from the start. Still it was more than that: even if it had _not_ been your composition, I knew once you decided to perform it you were giving it your creative attention & delivery. _That_ is what was added to make this time a magical, great adventure for me.\n\nI _hope_ we can work together again in some similar capacity. I don't expect, exactly. But I will keep on with this stimulus, this gift of your composition & performing with such as you & I _do_ thank you for it \u2013 a lot!\n\nDoriot\n\n601. Leonard Bernstein to Karl B\u00f6hm\n\n13 August 198115\n\nDear Maestro Karl,\n\nThey tell me you are now very ill, worse than our last time in Munich. Some people even imply that you feel your life coming to a close \u2013 for me, an inadmissible thought. The years of physical age have mounted close to 90, and even I, on the eve of my 63rd birthday, can feel their weight and the concomitant panic at time running out before all our works can be finished.\n\nI have always been somewhat amazed at the warmth and musical closeness of our relationship. After all, you were born in the lap of Mozart, Wagner and Strauss, with full title to their domain; whereas I was born in the lap of Gershwin and Copland, and my title in the kingdom of European music was, so to speak, that of an adopted son. That is why I was so surprised to receive your message, some months ago, when you were stopped by illness from completing your _Elektra_ recording, that if you should in time not recover to finish the missing central love scene, I, of all people, must be the one to complete it for you. You can imagine the honor I felt at this request and, also, my sense of inadequacy at the prospect of replacing so great a master.\n\nBut you _must_ recover; I know what your recuperative powers can be. You are resilience itself. I have observed it in brilliant action last January in Munich when I watched and heard your last _Entf\u00fchrung_. It was charming and subtle as ever, but I did notice the difficulty you were having in moving, the extra long time it took you to reach the podium, and the extra effort the singers on stage had to make in order to follow your beat, usually small, but always so clear. Backstage, in the interval, you asked me if you might attend one of my _Tristan_ rehearsals for only 20 minutes \u2013 all you thought your body could bear. Remember that you stayed for all of it \u2013 all 91 minutes of Act I (in a not very well polished first run-through). Remember, if you can, that you came bounding down the aisle to my podium, when orchestra and singers had left, your eyes aflame, and your cheeks ablaze. \"Na Bernstein,\" you said looking up (up!) at me from floor to stage, \"jetztz hab' ich endlich zum ersten Mal im Leben _Tristan_ geh\u00f6rt.\" You looked like a young man, burning, radiant. I was in heaven, not only because of this unbelievable _imprimatur_ from the Wagnerian pope himself, but also because I was watching a mystical, quasi-Faustian rejuvenation. \"Auch das Vorspiel?\", I asked timorously, knowing that there was at least a five-minute difference between your timing of the Prelude and mine. \"\u00dcberhaupt das Vorspiel,\" you answered, and began to give me an extraordinary analysis of what I had just done in terms of phrasing, tempo relationships, etc. You _taught_ me, in wisdom, what I had been performing by intuition. You were a young, strong man.\n\nYou are young. Please stay so, for me, for my colleagues, for the holy art. What you have done in music has already made you immortal; does that not encourage you to remain with us, and teach us forever?\n\nI pray for you, as does the whole world of music.\n\nWith devotion,\n\nBernstein\n\n602. Leonard Bernstein to Richard Horowitz16\n\nAugust 1981\n\nFor Dick H,\n\nDear friend, and cherished colleague, Dick:\n\nBless you for each and every stick,\n\nEspecially these new birthday sticks\n\nWhich are delicate, strong, and _free_ , and six!\n\nLoving thanks,\n\nLenny\n\n603. Lukas Foss to Leonard Bernstein\n\n1 September 1981\n\nHi Lenny,\n\n_Halil_ arrived this morn. What a moving piece. As an x-flutist (I used to play well at 15) I enjoy the flute writing, and I love the Alto Flute's role. The end reminds me of the 1st version of the _Age of Anxiety_ with that reentry for one chord; here it is reentry for 2 notes. It works though and if I can't get Rampal or Galway to play it with me, I'll have my 1st flute in Milwaukee learn it (she is wonderful).\n\nSometime can I hear a tape or will the record be out soon?\n\nLove\n\nEver\n\nLukas\n\n604. David Charles Abell to Leonard Bernstein\n\nHotel Schweizerhof, Berlin, Germany\n\n30 April 1982\n\nDear Lenny,\n\nI want to thank you from my heart for having made possible my debut with _Mass_ in Berlin. When I stepped off the podium last night, I felt so wonderful, I could have conducted the whole piece through right there again. Had I planned it all when I was twelve years old sitting backstage at the Kennedy Center listening to _Mass_ , I could not have done better than the reality. It gives me so much joy to conduct your music. I know it so well and I love it so much and believe in it and understand what it has to say \u2013 it makes me very happy that _Mass_ is the first piece I have conducted professionally. I grew up with _Mass_ , and I hope God will permit me to grow old and die with it too.\n\nThere are some things about this production which would please you very much. I think that the space it is in, an enormous arena, is better for the work than an opera house. The Deutschlandhalle, however, is just _too_ big.17 I would like to see _Mass_ done in a smaller arena in three-quarter round as it is here. When we see each other next, I'd like to discuss this production and what should be done in the future.\n\nEveryone here working on the show \u2013 the cast, the staff, singers, players and dancers \u2013 send their love and gratitude. It is really wonderful working with this Czech orchestra, all very fine musicians, who consider it such an honor to be invited to the West to play an American piece. They understand the music quite well and find a lot of joy in it. As with every production of _Mass_ that I know, the piece has brought the cast together as a family. It is an experience which everyone comes away from a little bit changed. I am always in awe of its power.\n\nI wish you joy with your tours and hope that you are having time to work on your new opera. I have a feeling that \"Tahiti Two\" is a very important work for you and for all of us who know and love you. I am looking forward to hearing it and seeing it and studying it.\n\nI hope to see you in Milano at the end of the month. If not, until July.\n\nShalom,\n\nDavid Abell\n\nP.S. And the flowers! I'm having them framed \u2013 will you sign them? Thanks so much.\n\n605. Yehudi Menuhin to Leonard Bernstein\n\n15 Pond Square, Highgate Village, London, England\n\n[1982]\n\nDear Lennie,\n\nI asked a friend to send me a tape of your _Enigma Variations_ in London recently.\n\nI have just listened to it and it is the most moving performance of the work I have ever heard.\n\nI just wanted you to know how much I admire your interpretation and how I felt it revered Elgar's every indication.\n\nLove,\n\nYehudi\n\n606. Leonard Bernstein to Helen Coates\n\n19 July 1983\n\nDearest Helen,\n\nI want you to have this _in writing_ on your birthday, since flowers can wilt and the telephone-voice is ephemeral. This is only to say, once again and for always, this I cannot imagine my life without you, not one year of it out of the soon-to-be-three decades.18 My admiration, gratitude and deep _respect_ are matched only by the intensity of my prayers for your health and happiness.\n\nLove,\n\nLenny\n\n607. Leonard Bernstein to Stephen Sondheim\n\n28 July 1983\n\nDear SS,\n\nSaw [ _Sunday in the Park with_ ] _George_ Tues eve (but not you). It (and you) is brilliant, deeply conceived, canny, magisterial, and by far the most _personal_ statement I've heard from you. Bravo.\n\nLove,\n\nL\n\n608. Kristin Braly19 to Leonard Bernstein\n\n10 January 1984\n\nDear Man,\n\nSomehow, in the midst of clicking cameras and microphone-bearing reporters, mystified onlookers and frantic volunteers, conducting students, my buzzing musician colleagues [...] the wires, lights, echoes and occasional blurtings of babies, I discovered that Leonard Bernstein was not just an image on a screen, a part of the title on a page of sheet music, a dusty recording on my shelf. What had been a name to me became a heart. His heart was in my heart, and I warmed myself by the fire of his affection. I wanted to hold him in silence. But so many needed him, his touch, his smile, that I became very small.\n\nAt home, after the concert, I found myself alone, surprised and shaken. From all those hands and faces, would he remember mine?\n\nIf only I were a cat needing a fish, how simple life would be! But here I am, traveling the human road, and needing Leonard Bernstein.\n\nKris Braly, viola,\n\nBaltimore Symphony\n\n609. Christa Ludwig to Leonard Bernstein\n\nVienna, Austria\n\n8 March 1984\n\nDear Lenny\u2013Maestro,\n\nIn the last weeks I worked on the _Wunderhorn-Lieder_ and found out that my voice isn't suitable any more to these songs and I don't want to be my own competition to the wonderful records we made \u2013 18 years ago!\n\nSo, I am really very sorry, but wise enough not to sing with you in the coming summer!\n\nI hope I am not making you and Harry [Kraut] too many problems with my cancellation, but _please_ understand my point of view.\n\nLove,\n\nChrista\n\n610. Mary Rodgers to Leonard Bernstein\n\nThe Watergate Hotel, Washington, D.C.\n\nreceived 24 July 1984\n\nDearest Lennie,\n\nThe last time I had as glorious a time in DC was '57 \u2013 _West Side_ \u2013 you asked me if I'd like to work on the Y[oung] P[eople's] C[oncert]s \u2013 and you've been making me happy \u2013 musically, personally, too, ever since.\n\nWith deepest love and praise and delight. _A Quiet Place_ is not uncomplex but boy is it worth it. It's truly wonderful.20\n\nLove you,\n\nMary\n\n611. Oliver Smith to Leonard Bernstein\n\nBox 184, Yellowgate, North Salem, NY\n\n4 August 1984\n\nDearest Lenny,\n\nIt was a wonderful occasion to attend your opera _A Quiet Place_ at the Kennedy Center. The work is very moving, absorbing and complex. All is held together by the wrenching beauty and intensity of the score, which as I told you at the party gave me the good \"old goose pimples\" of emotional involvement. I especially loved the trio at the end of Act I, and all the third act. That does not mean I did not appreciate the second act, which is full of its special delights.\n\nWhile melding the two operas may make for a better balanced evening, I still feel the two operas have very separate emotional values. The performances were superb and could not have been better. It was also beautifully directed and simply, but very effectively designed.\n\nI can think of no modern opera of such intensity or in a sense generosity. By letting it all \"hang out\", you have bared your soul, which few modern composers are prepared to do. Thank you for your beautiful music.\n\nI do love you.\n\nOliver\n\n612. Alan Jay Lerner to Leonard Bernstein\n\n21 Bramerton Street, London, England\n\n16 April 1985\n\nMy dear old chum,\n\nBobby passed on to me your interest in trying to reshape and make something out of _1600_ [ _Pennsylvania Avenue_ ]. I gather that although there is no schedule you would like to get it done some time before the _tri_ -centennial.\n\nTo get to the point: _of course_ a fresh pair of eyes is needed. Even though I've had an eye implant and don't wear glasses any more (can you believe it?) I don't think this brand new one will be sufficient, and I would welcome a writer with ideas on how to reorganise and rewrite.\n\nAs far as the lyrics are concerned, I suddenly seem to have taken on a new lease of life and am scribbling away like fury. I have a musical in rehearsal here in July which John Dexter is directing,21 and am two-thirds through the score of another that Allan Carr plans to put into rehearsal sometime in the autumn. Besides that I am just completing a huge tome for Collins on a history of the musical theatre since Offenbach. So if the right time for you should be impossible for me, and time adjustments cannot be arranged, as much as I would dislike it I would understand if you had to turn to someone else for any additional lyrics. But I truly hope that will be only the last resort. I would love to have another crack at it.\n\nIt is very interesting how _1600_ started because Robby22 remembers its inception one way and I have a clear memory of it in another. Originally I wanted to do five episodes which were critical in the history of the White House. I remember that I thought the entire production would look like a rehearsal, on the theory that democracy is still rehearsing. Robby, on the other hand, is convinced that the original intention was to write a sort of \"Upstairs Downstairs\" history of the White House \u2013 without the upstairs. In other words, it would be told strictly through the eyes of a multi-generational servant family. What I think we got was a mixture of both with moments of the black experience thrown in, all of which added up to a horse with three heads. I still vote for the \"upstairs\" story \u2013 perhaps now even more than ever because the upstairs material is fresher and we have been surfeited with the history of the blacks in America. Another reason is because I don't \u2013 and didn't \u2013 do that sort of thing very well. But, I am open to any and all approaches.\n\nIn any event, I have some two thousand books and crates of pre-play material that are finally about to be shipped over to me and I will be able to examine all those early versions.\n\nI am, at long last, not happily married but ecstatically so to a smashing lady23 \u2013 we've been together for five years and married for four, a track record for me. We bought a house too quickly that was and is too small, but we have finally got around to looking for a place large enough to accommodate all that I left in storage. I adore living in London and I've had the most wonderful five years of my life here. So whenever we meet \u2013 which I hope will be soon, somewhere \u2013 prepare yourself for a bubbling version of your old Virgo friend.\n\nI hear the new _West Side Story_ album is terrific. We have been in Spain for a few days on hol and this is our first day back, but I've already ordered a copy.\n\nI think of you often. And always with love.\n\nAye,\n\nAlan\n\n613. Stephen Sondheim to Leonard Bernstein\n\n[April 1985]\n\nDear Lenny,\n\nThanks for the album \u2013 very impressive \u2013 but I wish you'd have asked me before restoring that first (and not-my-favorite) Jets quatrain.24\n\nLove,\n\nSteve\n\n614. Leonard Bernstein to Stephen Sondheim\n\n25 April 1985\n\nDear Steve,\n\nIt's particularly hard to apologize to today's Pulitzer Winner for a bit of thoughtlessness (or perhaps it's easier: you might understandably be in a euphoric and forgiving mood). In any case, I _was_ thoughtless, so carried away by the fun of presaging the Gym-swing-music that I neglected to consult you for approval. I am sorry, but also (forgive me) singing and leaping about in celebration of your new glory. And does your show deserve it!25\n\nCongrats\n\nBlessings\n\nLove\n\nLenny\n\n615. Leonard Bernstein to Helen Coates\n\nBen-Gurion Airport, Tel Aviv, Israel\n\n27 August 1985\n\nDearest Helen,\n\nA quick note from Ben-Gurion Airport, en route to Munich & Japan and...\n\nThe last two nights have made concert history here (Mahler #9).26 I don't think I've ever heard it played with quite so much passion and tenderness. The orchestra is transformed, the newspapers ecstatic. I'm enclosing the one _non_ -ecstatic review, at least of _Halil_ ; the rest is the usual rave. I personally, on the other hand, have fallen in love all over again with _Halil_ , and Ransom W[ilson] is playing it like a god.\n\nI think of you so often, and send you love from so many friends \u2013 especially the Lishanskys, who gave me a great fish-festival dinner-party after our Haifa concert.\n\nKeep well, and cool. I'm managing to do so, although I frankly don't understand how, given this formidable schedule of rehearsals, travel, concerts, receptions, parties, time-changes... But I've survived my birthday, and feel younger than ever.\n\nA big hug & kiss,\n\nLenny\n\n616. Jerome Robbins to Leonard Bernstein\n\n9 December 1985\n\nLenny,\n\nWhat a _good_ concert!27 Especially the Copland & the last movement of it. Thank you! It was good to see you working again, and so finely. Sorry I couldn't get back stage to say hello & thanks in person.\n\nHave a good holiday season \u2013 & all love.\n\nJerry\n\n617. Leonard Bernstein to Helen Coates\n\n19 July 1986\n\nFor HGC\n\n[Echo from Haydn's _Creation_ ]\n\nThe Heavens are tellin'\n\nThe glory of Helen!\n\nAnd me too, I'm yellin'\n\nHOORAY FOR HELEN!\n\nAnd _I'm_ tellin' Heaven:\n\nShe's now eight-seven;\n\nSo you better keep her well 'n'\n\nHappy! PRAISE HELEN!\n\nAll love, as always,\n\nLenny\n\n618. Yevgeny Yevtushenko28 to Leonard Bernstein\n\nPeredlikino, near Moscow, Russia\n\n[?September 1986]\n\nMy dear Lenny!\n\nI haven't seen you for ages. I send you with a kind help of Sarah Caldwell my script _The End of_ [ _the_ ] _Musketeers_. In my immodest opinion it could be very easily transformed into kind of \"sparkling tragedy\", full of joy and bitterness, strange mixture of French champagne with a womit [vomit]. Probably, only composer who could create music for such kind of theme are you, because you are all of my musketeers and it could be your own confession, like it was mine. Of course it needs a lot of work if you'll like it in collaboration with me and one American poet. I am waiting for your answer. I was hopelessly trying to find you in USA.\n\nMy love and respect,\n\nYevgeny Yevtushenko\n\n619. Leonard Bernstein to Yevgeny Yevtushenko\n\n\"Napoli \u2013 Paris \u2013 Zurich \u2013 Jerusalem\" [sent from Jerusalem]\n\n27 September 1986\n\nMy dearest poet-friend Yevgeny,\n\nI have just tried to telephone you to Peredelkino; I _think_ I got through to someone who understood. In case not, the message was: I finished yesterday reading the _Musketeers_ script. I loved it; I love you.\n\nFurther, I am moved and excited and want to work with you (and, of course, an English or American lyricist, although you are already a great lyricist).\n\n_Further_ : I believe this will be a great musical work \u2013 _not_ an opera in an opera house, but directly for _film_. It is a natural film, \"cast of thousands\" as they used to say in the great old days of Eisenstein and Cecil B. de Mille \u2013 a wildly funny epic that tears out your heart even while you laugh. I don't know exactly how this film should be made, or where; perhaps it is all _animated_ , like Disney cartoons, or played by robots (they make fantastic ones now in California) or eventually by _real_ actors and singers, or all together. Maybe someone like Fellini should direct it; it should be a _major international project_.\n\nWhat is the next step? I don't know now, but I will soon. I will be in Wien, Hotel Bristol, this next week, then back in New York on 8th October. Then I become _composer_ for five months! In this period we can decide many things. Where and when can we meet? Can you call me in Vienna, or if too late, in New York?\n\nBeginning of _big_ song: music already shouting in my head!:\n\nLive before you die!\n\nDon't die until your death!\n\n(in Eb)\n\nCall me, write me, come to me, or all three... we have a lot of work to do, and play...\n\nMuch love,\n\nLenny\n\n620. Sid Ramin to Leonard Bernstein\n\n6 October 1986\n\nDear Lenny,\n\nWhat an experience!29 The Via Dolorosa, The Masada, wading in the Dead Sea, peering into Lebanon, exploring Jerusalem, meeting new friends, hearing and seeing you conduct in Tel Aviv and Jerusalem, and so much more.\n\nThere is no way I can express my feelings towards you and to you when I think of our magnificent trip to Israel. A first visit would always be exciting but the circumstances that permitted me (and Gloria) to share with you, be part of your routine, to be included in your success and adulation, to listen to you, to learn (!) from you, and, most important, to _be_ with you \u2013 is something I will never forget.\n\nIt's incredible for me to realize that we spent so much time together in Israel (and in Fairfield) and the thrill of being with you will never diminish. It seems you are a bit biased. Whatever I do, right or wrong, you always justify my mistakes or ignorance with a kindness and warm admonition that I am grateful for and love.\n\nAnd so, Lenny, I thank you again for a \"dream trip\" and my gratitude for including me in your life knows no bounds.\n\nImagine, from Roxbury to Jerusalem!\n\nWhere next?\n\nWith love, from your devoted and oldest,\n\nSid\n\n621. Sid Ramin to Leonard Bernstein\n\n[New York, NY]\n\n15 October 1986\n\nDear Lenny,\n\nAfter spending so many wonderful hours with you in Israel, I thought you'd like to have something to remind you of the many wonderful hours we spent together over a half-century ago. God, that sounds ancient!\n\nThe first few letters were written in the summer of 1933, when you were in Sharon and I was at Revere.30 Luckily you numbered your pages so that you can now better follow the order in which they were originally written.\n\nIn 1937, when you were at Harvard, I began to \"seriously\" study with you... no more governing chords, finishing chords, pre-finishing chords, etc. Your notes to me show that we really started with the basics... and we're still at it!\n\nNow on to the next fifty!\n\nWith love as always,\n\nSid\n\n622. Leonard Bernstein to Stephen Sondheim\n\n[1986]\n\n_Sorrowful Song_\n\nLast night\n\nI sat down and wrote a poem.\n\nThis morning\n\nI looked at it and didn't like it much.\n\nSo I started\n\nAll over again,\n\nMaking (major and minor but) significant\n\nChanges.\n\nThis evening\n\nI looked again and didn't like it much better.\n\nSo I changed it back\n\nTo Version One\n\nWhich I wrote last night,\n\nAnd this is it.\n\nLove,\n\nL\n\n623. Leonard Bernstein to Harry Kraut31\n\n7 April 1987\n\nDear Harry,\n\nHaving just finished Martin Gardner's brilliant\/hateful wipe-out of occultism (in the _N.Y. Review of Books_ , a propos the collected _oeuvre_ of Shirley MacLaine) I find myself beset by feelings of paradox \u2013 and thinking of _you_. I guess your paradoxical duality is one of the things I most like in you (and, for that matter, in any thinking\u2013feeling person \u2013 including myself, in those ever-decreasing moments when I like myself).\n\nWhich of us worth his salt is not a paradoxnick? There's something in the Bible we all believe, even if not literally; and there's also something in Darwin and Freud that grabs us equally. Wm. Blake vs. Martin Gardner, X vs. Y, and on down the list of all the antitheses that engender free inquiry and democracy.\n\nI like to think of myself, and of you, as primarily rational humanists, but then, there I go inhaling cosmic energies via Aaron Stern.32 And then, there _you_ go, so movingly, pursuing your profound and loving experiment with Patrick Porter.33 I can't tell you how touching I find it.\n\nThis is not a sketch for some future lecture, but a spontaneous love-letter on your birthday. Have a happy one and many more.\n\nLenny\n\n624. Maureen Lipman34 to Leonard Bernstein\n\nLondon\n\n[April 1987]\n\nDear Mr. Bernstein,\n\nThe show is off and the star is off-colour, but the memory is just the grandest. _Wonderful Town_ ended in triumph, with a dynamic show, encores and bravos, a weeping star, the weeping star's children in _Wonderful Town_ T-shirts, coming on stage bearing flowers \u2013 the whole audience on its feet and a mass \"Conga\" round the stage and auditorium.35 The rafters sang with your wonderful music and I hope it continues to do so right through the next production (which should never have been allowed in!)\n\nLooking back, we had just the best notices we could have wished for, but no advertising, no record \u2013 no hype! You can't exist without it now in the West End. We had \"a perfect gem\" as the _Punch_ critic said \u2013 but we needed a master jeweler \u2013 Van Cleef & Arpels even! \u2013 to set us, invisibly, into the bracelet of Shaftesbury Avenue! I wish I had hot latkes [potato pancakes] for every person who came to the dressing room glowing with the pleasure of seeing a _real_ show with a book, lyrics, music and a HEART. After being disappointed by the Phantom of the Chessboard school of musical theatre.\n\nEnough. Onwards. I know you are working on a new project with Stephen Sondheim and I wish you huge success. I'm taking three months off to do a second book and I hope it will be OK to include a wonderful picture of you and I \u2013 me kissing you on the nose, taken by Christina Burton of Watford.\n\nMeanwhile \u2013 this really is the point of this rambling missive \u2013 _Thank you_ , for the privilege of your music in _Wonderful Town_ and for the joy of translating it, through Ruth, for the last year. God bless and take care of you.\n\nLove,\n\nMaureen (Lipman)\n\n625. Leonard Marcus36 to Leonard Bernstein\n\n299 Under Mountain Road, Lenox, MA\n\n20 September 1987\n\nDear Lenny,\n\nMerely a line to tell you that 1) you nearly caused me to have an accident and 2) I heard a performance of the Mahler Second that was more overwhelming than anything I remembered even you ever having done \u2013 and that goes back to my unforgettable first encounter with the work, singing it as a teenager under you at Tanglewood.\n\nEarlier today I drove down to NYC from Lenox. When I got close enough to catch a City station worthwhile fielding, the radio was in the middle of the Mahler. Almost immediately, even with the lo-fi of car radio, I realized I was surrounded by an extraordinary performance. By the middle of the last movement, I concluded that a new champion of the work had succeeded in snatching from you the belt reading \"Most in Tune with Mahler's Soul.\"\n\nI anticipated the penultimate choral chord \u2013 and what genius did it take to bring this epic to its perfect climax simply by expanding a dominant from closed to open position? \u2013 but what I heard so overpowered my expectations that I did something I haven't done in a couple of years.\n\nI cried.\n\nIt only lasted a moment, but in that moment, I nearly swerved into another car as we came to the 59th Street exit of the West Side Highway.\n\nYou needn't worry. By 42nd Street I had already forgiven you. During the interim, Marty Bookspan had announced which Philharmonic concert had just been rebroadcast.\n\nDoes anyone realize how dangerous great music can be? I mean, Plato's dead and all that, but even he could only have had a more formal, Dionysian version of Rock in mind, and everybody knows to complain about _that_. No, I'm referring to the perils of the classics. I hope Surgeon General Koop doesn't find out.\n\nAs always, with love.\n\nLenny\n\n626. Leonard Bernstein to Claudio Arrau37\n\n20 December 1987\n\nDear Claudio,\n\nI am at this moment remembering, with deep emotion, our Brahms D minor in 1946. It was your birthday; and besides playing like a god, you had a post-concert birthday party at your house. It was at this party that I met a ravishing girl called Felicia Montealegre, who was not only a fellow-Chilean of yours and your one-time pupil, but who also shared your birthday, _and_ for three decades thereafter shared my life.\n\nSo you see, my dear Claudio, how closely intertwined our lives have been, with Music playing the r\u00f4le of Destiny. May we long continue this closeness.\n\nA very happy birthday, and may you go from strength to strength.\n\nLove,\n\nLenny38\n\n627. Claudio Arrau to Leonard Bernstein\n\n29 February 1988\n\nDear Lenny,\n\nI cannot thank you enough for your beautiful words of good wishes on the occasion of my 85th birthday. Imagine, 85, hard to believe.\n\nAs I sat holding the Festschrift on my lap, we all remembered our dear Felicia. Your words have special meaning for us and always will.\n\nNow that you yourself are getting to be a grand old man, don't let the thought of age bother you. It really is not so bad. Some of us get better and stronger and I hear that is what is happening to you. So God bless you because nothing is more wonderful than fulfillment in later life.\n\nAll our love.\n\nAs ever,\n\nClaudio\n\n628. Jerome Robbins to Leonard Bernstein\n\n19 April 1988\n\nDear Lenushka,\n\nSorry we didn't meet yesterday. It was _Fancy Free_ 's 44th birthday \u2013 and I was looking forward to giving and getting a big hug!\n\n_But_ , I was very happy to know you were working hard on your new piece \u2013 and I know how much that means to you, _and_ all of us.\n\nWe do have some things to solve, not much, but things we can settle so we can move ahead.39 _W_ [ _est_ ] _S_ [ _ide_ ] _S_ [ _tory_ ] is in pretty good shape as we outlined it with only one spot a bit bumpy. [ _On the_ ] _Town_ needs some talking about. I've a few ideas. So let's get it done & out of the way.\n\nI look forward to Tuesday (and the hug) and any other time you can manage to make for us.\n\nLove,\n\nJerry\n\n629. Jerome Robbins to Leonard Bernstein\n\n117 East 81st Street, New York, NY\n\n10 June 1988\n\n[Note at top:] Lenny: _First, play the tape!_ Then read this.\n\nDear Lenny,\n\nHere's a tape of the _On the Town_ Ballet such as we have put together musically and I have choreographed. It combines the elements we talked about and where I had some problems we made some temporary fill-ins and adjustments. I know there will be places which you think are over-extended musically, such as in the Penny Arcade and in the Dance Hall section which follows it, but they don't seem over-extended when you watch it with dancing. However, I'm certainly looking forward to your reactions and help.40\n\nScott, the pianist, did a wonderful job and I like the total buoyancy of the piece. It still is episodic which curtails a dynamic flow-through feeling. But, I like the little reprise of the fugato leading back into the finale material very much. I know you will love what I have choreographed for the fugato and the music that comes after it. It's all joyous and now that I've completed sketching it, I'll do better by it when I get back to it. But I'm most anxious to get your reaction.\n\nSorry I can't be there to dance it for you. You know how much respect I have for the music and anything we have to add has been an imperative necessity to make the logic and the story work out.\n\nSo how are you? I miss you and hope you're not too tired and that the tour has been wonderful. I cannot come to Chicago but I hope to see you on the 26th of June.\n\nI send you a big hug and await your response.\n\nLove,\n\nJerry\n\nP.S. I've started finding out about the _Dance in America_ tape rights. Of course if I can manage to help, I will.\n\n630. Miles Davis41 to Leonard Bernstein\n\nc\/o Shukat & Hafer, 111 West 57th Street, New York, NY\n\n28 June 1988\n\nDear Leonard,\n\nHaving received the Son[n]ing Award in Copenhagen which only Isaac Stern, Stravinsky, you and I have received42 I am reminded of what an honor it is to be in your and their company.\n\nI also think about the time when my wife,43 who was the lead dancer in _West Side Story_ , said to me: \"Leonard wants you to think about playing this music\", and I replied \"how am I going to play this corny shit\". Needless to say it turned out to be a classic.\n\nYou are one of America's true geniuses along with [Thelonius] Monk, [Dizzy] Gillespie, [Charles] Mingus and [Charlie] Parker. You are a true musician and if you chose to be you could be a great pianist in addition to being a great composer and conductor.\n\nOn this your 70th birthday, I wish you all the best and wish you many more productive years in pleasing the world with your music.\n\nSincerely yours,\n\nMiles Davis\n\n631. Gerald Levinson44 to Leonard Bernstein\n\nSwarthmore College, Swarthmore, PA\n\n9 July 1988\n\nDear Mr. Bernstein,\n\nI send greetings and wishes for much nachas, as well as \"harmony and grace,\" on your birthday. After having made so much great music as composer, conductor, and teacher over the past seven decades, I'm sure you won't be content to rest on your laurels now.\n\nI'll always be grateful for the Leonard Bernstein Fellowship that allowed me to spend an inspiring summer at Tanglewood in '71 \u2013 not least for the unforgettable _Missa Solemnis_ you conducted that season. Hanging around the B.S.O. all summer certainly contributed to my appetite for the big orchestra, of which you've heard some of the fruits. And fellow TMC Fellows of the time are now friends and colleagues. The ties that keep bringing me back to that wonderful place formed then and keep growing.\n\nFinally, here's a coded message for your seventieth. A key is included for the lower stave;45 you're on your own for the upper stave.\n\nMany happy returns \u2013 from Ari, too.\n\nJerry Levinson\n\n632. Ronald Reagan46 to Leonard Bernstein\n\nThe White House, Washington, D.C.\n\n5 August 1988\n\nDear Mr. Bernstein,\n\nNancy and I are delighted to join with your many friends and admirers gathered in Tanglewood to extend warmest congratulations to you on your 70th birthday.\n\nYour remarkable career as a conductor, pianist, and composer has greatly enriched American culture. Your memorable compositions during a long and prolific career have captured the hearts and dreams of generations of your countrymen.\n\nFrom _West Side Story_ to _On the Waterfront_ , your music has cheered us, thrilled us, rallied us, and gladdened us. Today, we salute you for your rare gift for music and your outstanding contribution to the artistic life of our Nation.\n\nHappy Birthday and God bless you.\n\nSincerely,\n\nRonald Reagan\n\n633. Leonard Bernstein to David Diamond\n\n[postmark Stamford, CT]\n\n8 August 1988\n\nDear DD,\n\nThank you for the birthday wishes, but how dare you talk of no communication from me when the last I've seen or heard from you was after I'd lost five years of my life learning & teaching & performing your 9th Symphony and you walked off with your 75,000 bucks and little or no thanks and remained unheard from except via certain people who read your weighty input to a Peyser book which I have promised my children on my honor never to read.\n\n... and I think that after decades of saving you from suicide, mental collapse, poverty, public fantasizing, and generally spoiling other people's lives you may owe me a bit more than a green, posterity-oriented birthday greeting, but never mind...\n\n... (as always, ungenerous to my colleagues)\n\nGoodbye & good luck.\n\nL47\n\n634. Claudio Arrau to Leonard Bernstein\n\n18 August 1988\n\nDear Lenny,\n\nI salute you on the wonderful occasion of your 70th birthday and wish you many more years of success, happiness and fulfillment.\n\nI cannot believe that so many years have gone by since we first met and first performed together and Ruth and I introduced you to Felicia, your beautiful bride to be. Then, I felt like an older brother full of admiration for your God-gifts. Today, I feel more like a loving uncle delighting in your enormous growth and achievements. May the gods continue to carry you to the ultimate portals of your deepest hopes and wishes.\n\nYours ever,\n\nClaudio\n\n635. Frank Sinatra to Leonard Bernstein\n\n[Reno, NV]\n\n25 August 1988\n\nDear Genius,\n\nHappy Birthday!\n\nYou are one of the few who deserves everything warm and wonderful that will be said about you on this marvelous occasion of reaching what Abe Lincoln would have called 3 score and ten.\n\nAnd I think it's sensational this big bash in your honor is being held at your beloved Tanglewood in the shade of the Boston Symphony Orchestra.\n\nBe assured, Lenny, that between songs here in Reno, where I am performing tonight, I raise a toast in your honor, in gratitude for all you have done for the musical world which bows towards you in appreciation this day, and for all you have done for the personal world I alone inhabit and which is a far better place because of your friendship, which I will always cherish.\n\nHappy Birthday, young man. I can hardly wait for your next seventy.\n\nWarmest hugs,\n\nFrancis Albert\n\n636. David Del Tredici48 to Leonard Bernstein\n\n1 September 1988\n\nDear Lenny,\n\nAmong all the wonderful things, I was most moved by _Mass_.49 It is way out there (a place I, too, inhabit), takes every kind of chance, and succeeds wonderfully. The pacing of the whole musical, dramatic unfolding is so skilful!\n\nThoughts of the moment and thanks for the never-to-be-forgotten music.\n\nDavid\n\nP.S. Look forward too, to Fall!50\n\n637. Leonard Bernstein to David Diamond\n\n[postmark New York, NY]\n\n24 October 1988\n\nDear old Dovidl,\n\nI've just returned from Europe and discovered your letter of two months ago. I don't want to discuss details, but I do want to say that I'm sorry I wrote you in the way I did. I should never have sent that letter in such a burst of anger (I can't remember ever having written such a letter to anyone, and besides, the anger was probably related to something else and only triggered by you).\n\nSo, I'm sorry; but I must say I meant every word of it.\n\nShalom uv'rachah [peace and blessings to you]\n\nLenny51\n\n638. David Del Tredici to Leonard Bernstein\n\n28 November [1988]\n\nLenny,\n\nYou were every bit as terrific as I thought you'd be. _Tattoo_52 came to life like I'd dreamed and in 4 different ways, too. I love your Rubens, Van Gogh, Vermeer, Rembrandt versions of the piece!\n\nAnd most of all I love you.\n\nAll my gratitude, love, and envy.\n\nDavid\n\n639. Leonard Bernstein to Charles Harmon53\n\n[New York, NY, with hand-drawn postage stamp \"Namibia State Prison\"]\n\n\"New Year's\", 1989\n\nDear Lito,\n\nOverleaf, behold my refuge from ragweed,54 1941, and the garret where I wrote my Clarinet Sonata, and started a ballet called _Conch Town_ (bits to be found in _Fancy Free_ , all the shows, including the whole tune of \"America\" in _West Side_ ). The house was dark brown then, and all I could afford.55\n\nLove,\n\nLB\n\n640. Stephen Sondheim to Leonard Bernstein\n\n22 March 1989\n\nDear Lenny,\n\nThanks for the telegram, and for the quick glimpses of your face on Sunday night's broadcast when you were listening to the song.56 I empathized with your apprehension at the start of it and was therefore doubly pleased at your relief when you realized that it was going to be affectionate (as well as brilliant, of course).\n\nAs you may have gathered, I called you when you were on vacation \u2013 it was just to give a nostalgic Christmas hello. I was about to do so again two weeks ago but Harry [Kraut] said you were in the slough of despond and it was not a good time. I'll try again. It would be nice to see you. Or at least talk.\n\nLove,\n\nSteve\n\n641. Leonard Bernstein to Marin Alsop57\n\n20 August 1989\n\nMy Marvelous Marin,\n\nThe bronchitis has finally felled me, and I've cancelled everything for this Sunday (a _necessary_ Sabbath!) Forgive me, and do understand that I'm with you, every 16th-note.\n\nBe glorious.\n\nLove\n\nLB\n\n642. Leonard Bernstein to Doriot Anthony Dwyer\n\n20 August 1989\n\nDear Darling Doriot,\n\nForgive & forget! I am in temporary collapse, abed, and have cancelled all activities for the day (including my own kiddies' concert) so that I can do this coming week with full powers. I know you'll understand.\n\nHave a great party \u2013 and I do so look forward to our Shosty!\n\nLove,\n\nLenny\n\n643. Yo-Yo Ma to Leonard Bernstein\n\n[No place]\n\n31 October 1989\n\nDear Mr. Bernstein,\n\nIt was so wonderful to have the chance to see you in New York last week. Thank you for your time and for the lunch. My only regret is that I did not get to see the letters that you received from Boris Pasternak with his comments on the meaning of art.\n\nI am very excited that you are willing to write a trio for Mr. Stern, Manny [Emanuel Ax], and me; I can't tell you what a thrill it is. Although you must hear this from so many people, I would just like to add that those of us who have the privilege of coming in contact with you do feel truly blessed.\n\nWarmest wishes,\n\nYo-Yo Ma\n\n644. Carlos Kleiber58 to Leonard Bernstein\n\nThe Carlyle, New York, NY\n\n[October 1989]\n\nDear Maestro, Dear Lennie!\n\n\"Non per me, ma per altri\", \"che voi dire, per altri\"?59 It's my son: he has become a fan of yours. (This is an understatement.) He is working at Unitel in Munich, translating your Salzau _Romeo und Julia_ tapes\u2013commentaries into German. He \"schw\u00e4rms\" about you on the phone and I hear about Romeo sitting in the garden, the triangle that shouldn't sound like a doorbell \u2013 in short (he is 24) you seem to have revived his interest in music (something I haven't managed to do) and he is pestering me for the following:\n\nHere is a CD of _West Side Story_. Do you by any means think it _possible_ for you to sign the first CD of the set (on the label-side with an indelible thingamajig) with a \"dedication\" to _Marko_ (with a K) _Kleiber_? and leave it (have it left) at the Met or the Carlyle?60\n\nThis would make him the happiest person in the world, renew his respect for me (cause I know you personally) and generally improve the morale all round.\n\nKindest greetings and best wishes from your old\n\nCarlos\n\n645. Stephen Sondheim to Leonard Bernstein\n\n6 November 1989\n\nDear Lenny,\n\nI was playing over my \"Anniversary\"61 and noticed an error, which you might want to correct in future (let's hope) editions: namely: in the third bar of the fifth system, the soprano note on the second beat should be a G, not an A. And why did you change the final cadence from a G major chord to an E major chord? Is there a runic significance I missed?62\n\nLove,\n\nSteve\n\n646. Lukas Foss to Leonard Bernstein\n\n19 November 1989\n\nHi Lenny,\n\nMust write to you because these San Francisco Symphony days you are _with_ me: 2nd half of program: \"Masque\"63 and [ _On the_ ] _Waterfront_. 4 evenings in a row at Davies Hall. It's great. 1st half is [Copland's] _Billy the Kid_ and _Time Cycle_.64 Practicising \"Masque\" again is always a revelation; every note so right, so inventive (what I used to love in Stravinsky). Have to practice a lot to get my fingers to do it and my brain to memorize it all, but it worked. 2 evenings behind me, 2 more to go.\n\nDoing _Waterfront_ without cuts (there is a cut suggested in print[ed score] from 32 to 33 which I really thinks makes for a lot less drama). Enough! Don't want to bore you.\n\nHope you are doing what I _should_ be doing \u2013 composing.\n\nLove,\n\nLukas\n\n647. Stephen Sondheim to Leonard Bernstein\n\n20 December 1989\n\nDear Lenny,\n\nWhat a terrific letter \u2013 thank you! You'll be interested to know that the rhyme you particularly liked (\"He goes..., etc.\")65 was the rhyme that Cole Porter liked when Jule [Styne] and I played a few songs for him. I always detected his influence on your work.\n\nAnd thanks for the advance birthday present. It's indeed tempting to set \u2013 maybe for your 75th. I'm still a slow writer.\n\nIf you have the time and inclination when you get back from tanning yourself, give me a call and let's have our semi-annual evening alone.*\n\nLove,\n\nSteve\n\n*I'll even play you the new score66 SS\n\n648. Marin Alsop to Leonard Bernstein\n\n14 July 1990\n\nDearest Maestro,\n\nI wanted to telephone you, but didn't want to disturb your rest.\n\nThank you for introducing me to Japan \u2013 and vice versa!\n\nThe greatest enticement (if it can be called that) to come on this trip was the opportunity to work with you once again.67\n\nAnd, once again, it was an inspiration. You haven't ever let me down since I first wanted to become a conductor when I was 11 and saw one of your NY Phil concerts! But more than that, you've been a constant source of energy and integrity and leadership and innovation to our world.\n\nI hope only that I can make a small contribution and always make you proud of me.\n\nI love you \u2013 and thank you for helping _all_ of us.\n\nMarin\n\n649. Jennie Bernstein to Leonard Bernstein\n\n5 September 1990\n\nDearest Son,\n\nI have confidence in you, that you are on the right track. I know you are watching out for yourself. No one can do it for you, but you and you alone.\n\nI feel a lot better now because I am looking towards your quick recovery. As you know dear, if you stay well, I will stay well.68 You are surrounded by a beautiful family, your children and grandchildren. That in itself should be good medicine for you.\n\nLooking forward to a Happy (Jewish) New Year for you, and your dear family.\n\nI want to wish you happy composing and so much love.\n\nYour one and only\n\nMother xxx\n\n650. Georg Solti to Leonard Bernstein\n\nChicago Symphony Orchestra, 220 South Michigan Avenue, Chicago, IL\n\n10 October 1990\n\nMy dear Lenny,\n\nI was more than sorry to learn of the announcement you have made yesterday and I would just like to send these few lines, to let you have my warmest thoughts and support, both now and in the future.\n\nIt is wonderful that you will continue to write and teach; do keep in touch and let me know if we can meet when I am in New York next, in April.69\n\nAs ever,\n\nGeorg\n\n1 John McClure, personal communication, 20 February 2013.\n\n2 Craig Urquhart was Bernstein's assistant from January 1986 until 1990. He is a vice-president of the Leonard Bernstein Office, and founded the Bernstein newsletter _Prelude, Fugue & Riffs_.\n\n3 The program on 22 July consisted of Haydn's _Thereisenmesse_ and Shostakovich's Fifth Symphony.\n\n4 This important (and acclaimed) revival of _West Side Story_ opened at Broadway's Minskoff Theatre in February 1980 and ran for 333 performances. It was directed by Robbins \u2013 his first work on Broadway for sixteen years \u2013 with the close involvement of several people involved in the original 1957 production, including sets by Oliver Smith, costumes by Irene Sharaff, and lighting by Jean Rosenthal. The Musical Director was John DeMain, who went on to conduct the world premiere of Bernstein's _A Quiet Place_ at Houston in 1983.\n\n5 Bernstein conducted Mahler's Ninth Symphony at Tanglewood a week later, on 29 July.\n\n6 Bernstein delivered this speech at the second annual Kennedy Center Honors gala on 2 December 1979. Copland, Henry Fonda, Martha Graham, Tennessee Williams, and Ella Fitzgerald were the five artists honored with lifetime achievement awards. The _Washington Post_ (9 December) described Bernstein's tribute to Copland as \"a piece of magic.\"\n\n7 Rosalynn Carter, wife of President Carter and First Lady.\n\n8 Francis Ford Coppola (b. 1939), film director whose screen credits include _The Godfather_ and _Apocalypse Now_.\n\n9 Coppola's _Tucker: The Man and his Dream_ was eventually released in 1988, but as a biographical film rather than the \"musical\u2013opera\u2013film\" outlined to Bernstein in this letter. The genesis and tribulations of this project are described in detail by Gene D. Phillips in _Godfather: The Intimate Francis Ford Coppola_ (Lexington: Kentucky University Press), pp. 261\u201378.\n\n10 \"Es\" (E flat) is normal German spelling for \"S,\" while \"La\" is solf\u00e8ge for A and can be used to spell \"L\"; by a neat coincidence these musical spellings for Stephen and Leonard form a tritone (augmented fourth) \u2013 an interval that is of particular significance in _West Side Story_. Sondheim was born on 22 March 1930 and started working with Bernstein on _West Side Story_ in October 1955.\n\n11 A reference to the piano pieces that Bernstein often sent Sondheim as birthday gifts (Stephen Sondheim, personal communication).\n\n12 An inscription written inside a birthday card.\n\n13 The _Divertimento for Orchestra_ first performed by the Boston Symphony Orchestra under Seiji Ozawa on 25 September 1980. Bernstein has written a note at the top of this letter outlining ideas (perhaps the result of a discussion with Robbins) for a ballet treatment:\n\nFanfare | 1 \u2013 Tutti \n---|--- \nWaltz | 2 \u2013 Diminished corps (girls?) \nMazurka | 3 \u2013 Pas de 6 \nSamba | 4 \u2013 Pas de 4: Kay Thompson & Boys \nTurkey | 5 \u2013 Pas de 2 (Castles) \nSphinxes | 6 \u2013 Solo: joke on [Martha] Graham vs. ballet on cadences \nBlues | 7 \u2013 Solo blues \n[March] | 8 \u2013 a) Adding company gradually during flutes \n| b) Tutti march.\n\n14 Doriot Anthony Dwyer (b. 1922) was principal flute of the Boston Symphony Orchestra from 1952 until 1990. She gave the American premiere of _Halil_ with the Boston Symphony conducted by Bernstein in a Fourth of July concert of Bernstein's music given at Tanglewood. Bernstein inscribed her copy: \"For my beautiful colleague Doriot, with all the old affection and a brand new admiration, Lenny \u2013 4 July '81\".\n\n15 Bernstein received word on 13 August that Karl B\u00f6hm was gravely ill, and immediately wrote to him. It is unlikely B\u00f6hm ever saw this letter as he died in Salzburg the next day, 14 August.\n\n16 Richard Horowitz (b. 1924), American percussionist and baton maker. He joined the Metropolitan Opera Orchestra in 1946 and played with it for 66 years, retiring as principal timpanist in 2012. He made his first baton in 1964 for Karl B\u00f6hm and subsequently made batons for many of the world's leading conductors, including Riccardo Chailly, Carlos Kleiber, James Levine, Charles Mackerras, and Klaus Tennstedt, among others. In 2008 he was interviewed in _The New York Times_ : \"I made Bernstein's batons. He's buried with one of my batons. I think he gave them away more than anything else, gave them to his students. [...] Bernstein's, I made out of corks from Champagne bottles.\"\n\n17 Built in 1935 for the 1936 Berlin Olympic Games, the Deutschlandhalle has a seating capacity for concerts of about 10,000.\n\n18 Bernstein's arithmetic is shaky here: he first met Helen Coates in October 1932, almost fifty-one years before writing this letter. If he was counting back to his New York Philharmonic debut, that had taken place almost forty years earlier, in November 1943. Miss Coates began working as Bernstein's secretary in 1944.\n\n19 Kristin Braly (b. 1948), was a violist in the Baltimore Symphony Orchestra until 2005. She wrote this letter two days after playing in the performance of Mahler's _Resurrection_ Symphony given in aid of Musicians Against Nuclear Arms that Bernstein conducted in the National Cathedral in Washington, DC, with an orchestra drawn from members of the National Symphony and the Baltimore Symphony.\n\n20 The first Washington performance of _A Quiet Place_ was given at the Kennedy Center Opera House on 22 July 1984.\n\n21 The stage version of _Gigi_ , directed by John Dexter, opened at the Lyric Theatre, Shaftesbury Avenue, London, on 17 September 1985.\n\n22 Possibly Robert Whitehead, one of the producers of _1600 Pennsylvania Avenue_.\n\n23 Liz Robertson married Alan Jay Lerner in 1981.\n\n24 The \"Jets Quatrain\" refers to the passage beginning \"Oh, when the Jets fall in at the cornball dance\" in the \"Jet Song,\" included on Bernstein's recording of _West Side Story_ made in September 1984. This passage is not in the first piano-vocal score, nor on the original cast recording. It is printed in the 1994 full score (pp. 43\u20134) and the revised 2000 piano-vocal score, marked in both editions with an optional cut.\n\n25 _Sunday in the Park with George_ won the 1985 Pulitzer Prize for Drama for Sondheim and James Lapine.\n\n26 The performance of Mahler's Ninth Symphony given in Tel Aviv on 25 August 1985 (Bernstein's 67th birthday) has been released on CD by Helicon Classics (HEL029656).\n\n27 On 5, 6, and 10 December 1985, Bernstein conducted an unusual program comprising the Third Symphonies of Roy Harris, William Schuman, and Aaron Copland.\n\n28 Yevgeny Yevtushenko (b. 1933), Russian poet. This was a project that clearly excited Bernstein (see Letter 619), but nothing came of it. Among Yevtushenko's earlier poems, the most famous \u2013 and the one that caused controversy with the Soviet authorities \u2013 was _Babi Yar_ which provided the inspiration for Shostakovich's Symphony No. 13.\n\n29 In Autumn 1986, Bernstein conducted a series of concerts with the Israel Philharmonic to celebrate its fiftieth anniversary. The first performance of _Jubilee Games_ was given in Avery Fisher Hall, New York, on 13 September, and Bernstein then went to Israel to conduct the work in the opening concerts of the 1986\u20137 season. Sid Ramin (who had helped with the orchestration of _Jubilee Games_ ) and his wife Gloria went with Bernstein on this trip \u2013 their first visit to Israel.\n\n30 These are Bernstein's letters to Ramin printed in Chapter I.\n\n31 Harry Kraut (1933\u20132007) was Bernstein's business manager from 1971, and became a valued confidant.\n\n32 Aaron Stern was a close friend of Bernstein's in the 1980s. He believed that the arts could be used as a way to greater self-knowledge and cultural transformation. With Bernstein's support he established an institution, the Academy for the Love of Learning.\n\n33 Probably Patrick K. Porter, who founded Positive Changes in 1987 to help people bring about lifestyle changes through hypnosis and counseling.\n\n34 Maureen Lipman (b. 1946), English actress and writer.\n\n35 The London revival of _Wonderful Town_ opened at the Queen's Theatre in August 1986 and ran until April 1987, with Maureen Lipman as Ruth and Emily Morgan as Eileen. She recalled the occasion of her meeting with Bernstein: \"The memory lingers of our meeting: him in tan leather trousers and a bright turquoise macram\u00e9 pullover, hugging me and telling me I was a wonderful Ruth and my brain saying 'take a mental picture of this moment! It'll never come again!'\" (Maureen Lipman, personal communication).\n\n36 Leonard Marcus, a Harvard music graduate, was appointed editor-in-chief of _High Fidelity and Musical America_ in 1968, a post he held until 1980. Before then he had worked in Minneapolis, had studied conducting with Bernstein and composition with Copland, and was assistant manager of the classical department of London Records from 1959 to 1961. He subsequently worked at Columbia Records and as editor of the Carnegie Hall programs. He became conductor of the Stockbridge Chamber Orchestra (later the Stockbridge Sinfonia) in 1975 and later became the orchestra's conductor emeritus.\n\n37 Claudio Arrau (1903\u201391), Chilean pianist.\n\n38 Written for a published tribute to Arrau on his 85th birthday on 6 February 1988.\n\n39 For the forthcoming _Jerome Robbins' Broadway_.\n\n40 Like Robbins' letter of 19 April, this concerns the preparations for _Jerome Robbins' Broadway_ which opened in February 1989. The show included sequences from _On the Town_ and _West Side Story_ , and early versions of the extracts from _On the Town_ were filmed at piano rehearsals in April and May 1988. This, or something very similar, must be the tape that Robbins describes (copies of these rehearsal tapes are in the New York Public Library of the Performing Arts).\n\n41 Miles Davis (1926\u201391), jazz musician.\n\n42 The L\u00e9onie Sonning Music Prize is Denmark's highest musical honour. The first recipient, in 1959, was Stravinsky, and the second was Bernstein (1965). Other winners included Lutos\u0142awski (1967), Britten (1968), Shostakovich (1973), Messiaen (1977), Stern (1982), Boulez (1985), and Miles Davis himself (1984).\n\n43 Frances Elizabeth Taylor married Miles Davis in 1958; they divorced in 1968. In the original production of _West Side Story_ she played Francisca (credited as Elizabeth Taylor).\n\n44 Gerald Levinson (b. 1951), American composer, a pupil of George Crumb, George Rochberg, and Olivier Messiaen. Levinson met Bernstein on a few occasions, notably at Tanglewood in 1987 when Levinson's first symphony, _An\u0101hata_ , was performed. Levinson recalls: \"In '87 when Oliver Knussen conducted a spectacular orchestra concert which concluded with my _An\u0101hata_ (and included Peter Serkin playing Stravinsky's _Movements_ ), Lenny and Seiji Ozawa followed my piece with the score, and both \u2013 but Lenny in particular \u2013 were very enthusiastic. Backstage afterward he said it brought him to tears ('That bad, eh?' I stupidly responded). I asked if he'd like to keep the score (my publisher, standing beside me, nodded vigorously); he said, 'I don't think I'd ever conduct it, but I'd love to possess it'\" (Gerald Levinson, personal communication, 23 March 2013).\n\n45 The lower stave is written using Messiaen's _langage communicable_ , an alphabetical code for musical spelling (the decoded message is \"Mazel Tov and Gesundheit on your Seventieth\"). The upper stave includes a Bernstein-ish version of \"Happy Birthday,\" followed by references to several Bernstein tunes, described by Levinson as follows: \"Once Happy Birthday is done it keeps on going through a medley of bits of Bernstein tunes: 'Maria,' 'New York New York,' and 'Trouble in Tahiti.' And the durations of the rests, carefully counted out in numbers in parentheses, add up to 70\" (Gerald Levinson, personal communication, 23 March 2013).\n\n46 Ronald Reagan (1911\u20132004), fortieth President of the United States. Reagan was not known for his interest in music, and his conservative political outlook was diametrically opposed to Bernstein's. One of Bernstein's kinder assessments of Reagan's presidency is to be found in his November 1989 interview with Jonathan Cott: \"The last time I went to the White House was during the last days of Jimmy Carter's administration. [...] I love the White House more than any other house in the world \u2013 after all, I'm a musician and a citizen of my country \u2013 but since 1980 I haven't gone back there because it's had such sloppy housekeepers and caretakers. [...] We had eight lovely, passive, on-our-backs, status quo, don't-make-waves years with Ronald Reagan. The _fights_ I had with my mother! 'Don't you dare say a word against our president!' she'd say to me.\" (Cott 2013, pp. 81, 82, and 83).\n\n47 The real source of Bernstein's rage was almost certainly Diamond's contribution to Joan Peyser's book, in which, among other things, he was quoted as saying that Bernstein \"often hurt him very much.\" Diamond replied to Bernstein's furious letter on 26 August 1988 with a long, 12-page diatribe in which he challenged Bernstein's claims, made some harsh comments about the effect of fame on his old friend, defended his own cooperation with Peyser, and ended with a plea: \"Lenny \u2013 don't make our friendship a poisoned one. Make it instead as rich as it once was. Take me out of this horrible depression your letter has caused me. No matter what I will be at your concerts this fall.\" After returning from concerts in Europe, Bernstein replied on 24 October (see Letter 637). But it was too late: after almost fifty years, their friendship was over.\n\n48 David Del Tredici (b. 1937), American composer.\n\n49 _Mass_ was performed at Tanglewood on 27 August 1988 by the Opera Theater of the Indiana School of Music to celebrate Bernstein's 70th birthday.\n\n50 When Bernstein was to conduct Del Tredici's _Tattoo_.\n\n51 This is the last letter Bernstein wrote to Diamond. See note 47 to Bernstein's letter (Letter 633) of 8 August 1988.\n\n52 Bernstein conducted the US premiere of _Tattoo_ with the New York Phiharmonic at concerts on 17, 18, 19, and 22 November 1988.\n\n53 Charles (Charlie) Harmon joined Amberson (Bernstein's publishing company) as his personal assistant in 1982. After Bernstein's death, Harmon contacted many of the people who had been close to Bernstein, requesting photocopies of letters. These are an invaluable addition to the Leonard Bernstein Collection in the Library of Congress. Harmon also edited several Bernstein works for publication, including the full orchestral scores of _West Side Story_ and _Candide_ , and the definitive piano-vocal scores of _On The Town_ and _Wonderful Town_.\n\n54 Bernstein suffered from acute hay-fever all his life.\n\n55 Written on a postcard on the verso of which is a picture of the house in Key West, Florida, where Bernstein stayed in the summer of 1941. An arrow points to the right-hand window on the top floor and reads \"This was my first abode. Cattie room in KW, 1941.\" Another postcard from Bernstein to Charles Harmon (undated, but with the same image of the house in Key West, with an arrow pointing to the same top-floor window) reads: \"This room is where I worked on my Clarinet Sonata, late _August '41_ , when I was fleeing from Ragweed & total hayfever (2 nights & days by train from Boston). LB. P.S. The house was then a drab dark grey-brown.\"\n\n56 Sondheim's reference to \"the song\" concerns his affectionate spoof \"The Saga of Lenny\" (based on the \"Saga of Jenny\" by Kurt Weill and Ira Gershwin) written for Bernstein's 70th birthday gala concert at Tanglewood in August 1988, when it was sung by Lauren Bacall. This was broadcast in the _Great Performances_ series on 19 March 1989, the Sunday night mentioned by Sondheim.\n\n57 Marin Alsop (b. 1956), American conductor, who became Music Director of the Baltimore Symphony Orchestra in 2007. As an undergrautate at Yale, she played with Steve Reich and Philip Glass, developing a passionate interest in new music. A decade earlier, when she was nine years old, Alsop's father had taken her to one of Bernstein's _Young People's Concerts_ : \"I immediately knew that I wanted to become a conductor. Becoming his student at Tanglewood in 1988 was a dream come true. To have a hero exceed my expectations was too much to hope for, but Leonard Bernstein certainly did that and more. I loved being around him, soaking in his way of looking at the world and connecting the dots in life. His encouragement and support were invaluable on every level\" (interview for National Public Radio, 12 October 2012).\n\n58 Carlos Kleiber (1930\u20132004), conductor. Famously elusive, his performances were legendary for their unique combination of blazing intensity and attention to detail.\n\n59 This near-quotation comes \u2013 aptly enough \u2013 from Act II scene 6 of Verdi's _Don Carlos_.\n\n60 This letter was probably written while Carlos Kleiber was in New York to conduct Franco Zeffirelli's production of _La Traviata_ at the Metropolitan Opera.\n\n61 The third of Bernstein's _Thirteen Anniversaries_ , published in 1989, is \"For Stephen Sondheim (b. March 22, 1930)\" dated at the end of the piece \"20 March 1965\".\n\n62 Bernstein's annotations on this letter include \"Right?\" beside the \"G, not an A\" (it is corrected to G in the 1990 revised edition); Bernstein has written \"Beats me\" next to Sondheim's question about \"runic significance\" of the final cadence that ends in D major in the 1990 edition.\n\n63 \"Masque\" is the Scherzo from _The Age of Anxiety_ Symphony, a movement described by Bernstein in 1949 as \"a kind of fantastic piano-jazz [...], by turns nervous, sentimental, self-satisfied, vociferous.\" Foss specialized in playing the solo piano part of _The Age of Anxiety_ : he was the soloist in the first New York performance (26 February 1950; see Letter 295), and he recorded it twice with Bernstein conducting (in 1950 and 1977).\n\n64 Foss composed _Time Cycle_ (for soprano and orchestra) in 1959\u201360. The first performances were given on 20, 21, and 23 October 1960 by Adele Addison with the New York Philharmonic conducted by Bernstein.\n\n65 A reference to lyrics from _Gypsy_ :\n\nWherever I go, I know he goes\n\nWherever I go, I know she goes\n\nNo fits, no fights, no feuds and no egos\n\nAmigos\n\nTogether!\n\n66 The \"new score\" was _Assassins_ , which opened Off-Broadway at Playwrights Horizons on 18 December 1990. It includes a crazed and darkly amusing monologue addressed to Leonard Bernstein by Samuel Byck (1930\u201374), a psychopath who attempted to hijack a plane in order to crash it into the White House and kill President Richard Nixon.\n\n67 In June 1990, Marin Alsop traveled with Bernstein to the Pacific Music Festival in Sapporo, Japan.\n\n68 Jennie Bernstein outlived Leonard. She died in December 1992 at the age of 94.\n\n69 This letter was written a few days before Leonard Bernstein's death on 14 October 1990.\nAppendix One\n\nArthur Laurents (with Leonard Bernstein): Outline for Romeo sent to Jerome Robbins\n\n[New York, undated, shortly before 18 October 1955]\n\n_ROMEO_\n\n_Act One_\n\n_Scene One: Back Alley \u2013 Nightfall._\n\nAgainst music, we see two or three shadowy figures beating up a boy. A lookout signals, the assailants flee. Their clothes are different from the \"cool\" outfits of the boys who stroll on: Mercutio and members of his gang including Romeo and Benvolio. Their exaggerated talk is interrupted by the discovery of a kid who was beaten up: A-rab. Baby, the youngest member of the gang, is shocked by what has been done to A-rab and by the \"Puerto Rican mark\" left on him. The others are enraged, want to have a rumble with the Puerto Rican gang because there have been too many of these raids (by both sides). It is up to Mercutio to decide \u2013 and he does in a \"Let's Have A Rumble\" song with the gang. During this, the formalized ritual of sending scouts to summon the PR gang leader, Bernardo, to a War Council is done. Benvolio, the best fighter, is to be one scout. Romeo \u2013 much to Mercutio's pleasure for M is Romeo's protector and Romeo doesn't partake too much of gang activities \u2013 volunteers to be the second scout. But Romeo's real reason is to see Rosalind who loves dancing and will probably be at the Crystal Cave, the dancehall where Bernardo is and which is neutral territory. The \"Rumble\" song comes back as the scouts start off on their mission.\n\n_Scene Two: Crystal Cave \u2013 Later._\n\nA wild mambo is in progress with the kids doing all the violent improvisation of jitterbugging. Benvolio and Romeo enter, searching for Bernardo. Benny (Benvolio) has to keep Romeo's mind on their mission, for the latter thinks only of Rosalind who snubs R as she whirls by. They find Bernardo and start negotiations. Romeo keeps looking around for Rosalind. (All this in pantomime.) Then Romeo sees a lovely young girl, dressed more simply, more innocently than the others; obviously a newcomer being shown around by an older, more experienced girl: Juliet and Anita. Romeo goes to Juliet and, as they meet, the music goes into half-time, the dancers keep going but as in a dream state, the lights change. Now there is dialogue \u2013 between the two. It is finally interrupted by Bernardo who pulls Juliet away. The music goes back to tempo and the number finishes and the lights go back to normal. Bernardo is Juliet's brother and Romeo's opponent. Thus, the two lovers learn, to their mutual dismay, that they belong to opposing factions. Benny takes Romeo off to report the results of their meeting with Bernardo to Mercutio; and Bernardo, despite the pleading of Anita, sends Juliet home with one of his Lieutenants as escort.\n\n_Scene Three: Gang Hangout \u2013 Later._\n\nA shack of some sort, depending on the designer. Mercutio and the gang are horsing around when Benny and Romeo enter, to report. Romeo, his mind on Juliet, gets the facts wrong so Benny takes over: War Chieftains from the two gangs (Mercutio and Bernardo and aides) are to meet in Doc's Drugstore at midnight. Then Mercutio, in song, proceeds to give Romeo some advice about love: the older bon vivant (probably just old enough to vote if that) to the neophyte. The gang joins in a razzing, possibly they chase Romeo \u2013 who tries to duck them \u2013 in a number which overflows out of the set. And at the end, he does elude them.\n\n_Scene Four: Tenement \u2013 Later._\n\nThis is in the Puerto Rican area and shows the scrabbly building Juliet lives in, with a fire escape. Puerto Rican music from the unseen interior of the flat as Romeo moves in the shadows looking for her house. Then she comes out on the fire escape and the \"balcony\" scene begins. This should go from dialogue to song and back, ending in song. In it, these facts: Romeo works for Doc as drugstore delivery boy and general helper; Juliet sews in the bridal shop and has not been long in this country (let us take the dramatic license of eliminating all accents) and Anita is her confidante and adviser. Plus, of course, R and J's mutual lack of caring about prejudice, gangs, hostility, etc. It might end with \"Good night\" and \"Buenos noches\", the latter repeated lovingly by Romeo.\n\n_Scene Five: Street or outside Crystal Cave._\n\nBernardo taking his girl, Anita, home, before he goes to the Rumble meeting. Various points can come up here: Bernardo's hatred of the \"American\" gang and thus his hate for Romeo as beau for his sister (as opposed to Anita's feeling that love is love and it all ends anyway); note of future disaster, heightened by Anita's plea to B not to get into bloody rumble. She is probably a little older than Bernardo and tho she has been kicked by love before, is still in love with Bernardo \u2013 but expects the worst. Her attitude is explained in a torch version of Mercutio's song after Bernardo has gone off to the rumble.\n\n_Scene Six: Drugstore \u2013 Midnight._\n\nMercutio and his aides impatiently awaiting Bernardo. Doc (possibly a Jew) tries vainly to stop the coming rumble. Here the violence, restlessness, lost feeling of these strange kids should be explored. The bursting inside then which needs a release should build and build until it explodes into a violent cold jitterbug number kicked off by a record on the jukebox. First the boys dance by themselves. But, as the set should show both inside and outside the drugstore, as gang girls come along the street, they grab them as partners, though still maintaining that frozen-faced solo quality such jitterbugging has. At the peak, Bernardo and aides enter; the dance continues to its finish, though the attitude of Mercutio and his gang changes subtly. They are aware of Bernardo but will not, deliberately, stop for him. At the conclusion, silence: the girls go, the boys line up in formalized gang positions. Mercutio, as \"host\", offers cokes; negotiations on the rumble begin. Time: sundown; place: Central Park. But the type of rumble is argued over as Romeo \u2013 who has arrived with a happy \"Buenos noches\" \u2013 argues for the simplest and least bloody kind: a \"fair fight\" between the two best fighters from each side. Romeo, who seems to have grown, to have become stronger as a result of his love, manages to prevail. Partly because, however, of Mercutio's fondness for him and happiness that Romeo is finally interested in the gang. Bernardo wants to fight Romeo at the rumble but Benny is the best fighter for Mercutio's gang. At this point, Shrank the policeman enters. He is suspicious because the boys are so quiet: he is their common enemy and this is the one time all are allied against the same thing. He suspects they are planning a rumble. No one says a word but Shrank, starting quietly, builds himself up to a frustrated frenzy which makes him throw everybody out of the store except Romeo since he works for Doc. Shrank makes a crack against both sides and goes. Romeo and Doc are alone. There must be some sharp note to underline the prejudice that stands between Romeo and Juliet, then Doc goes (his closeness to Romeo must emerge here, too) leaving Romeo to close up, turn out the lights as he sings softly of his love.\n\n_Act Two_\n\n_Scene One: The Neighbourhood._\n\nThis is a musical quintet which covers various parts of the neighbourhood in space and the whole day in time. Its theme is \"Can't Wait for The Night\"; its mood is impatience of different kinds, exemplified by five of the principals: Mercutio (with humor) and Bernardo (with anger) can't wait for the rumble; Romeo can't wait to see Juliet; Juliet, at her bridal shop sewing machine, can't wait to see Romeo. Only Anita strikes a different note: she is afraid of the night because of what the rumble may bring. It should end with Juliet and thus go directly into:\n\n_Scene Two: The Bridal Shop \u2013 Late Afternoon._\n\nEveryone has gone except Juliet who has pretended she has to finish up a wedding veil needed for the next morning. Romeo comes in and they arrange the mannequins as a bridal party, almost like children playing a game, and marry themselves. Here again, dialogue goes in and out of song. The entrance to the shop is at the rear of the stage and as they leave at the end, the curtain closes for:\n\n_Scene Three: Outside the Park._\n\nBernardo and his aides come from one side and, after a moment, Juliet and Romeo from the other. Bernardo is furious that his sister is with a member of the other gang. He wants to provoke a fight but Romeo won't be provoked and Juliet becomes surprisingly strong. Romeo is going to take her home and no one is going to stop that \u2013 and no one does. Alone with his aides, Bernardo says the hell with a fair fight: get ready for a real rumble.\n\n_Scene Four: Central Park \u2013 Sundown._\n\nMercutio and his gang are waiting for Bernardo and his. They, too, are actually prepared in the event that the \"fair fight\" should bust into a bloody rumble. Romeo enters on this and tries to talk them out of it. Bernardo and gang arrive. Romeo tries to prevent any rumble. Bernardo accuses him of stalling and really tries to make Romeo fight, finally spitting at him. Romeo almost lunges, but won't fight. This enrages Mercutio who slams Romeo out of the way, leaps at Bernardo and the fight is on. The scene is probably underscored and, here, breaks out into a stylized gang canon as both gangs take up positions for the fight. It does break out into a fracas when Bernardo, almost beaten, whips out a knife and stabs Mercutio. Romeo, horrified at what has been done to his protector, grabs a broken bottle from A-rab and plunges it into Bernardo. There is a wild moment of mel\u00e9e \u2013 then everybody clears because of the two still bodies on the ground. Both Bernardo and Mercutio are dead. This is horrifying even to the kids. A clock begins to chime as they slowly leave the scene. Romeo stares at the bodies. A police whistle, a siren, the roving light of a police car picks over the ground; the whistles, the sirens louder, music \u2013 the chase is on and Romeo runs as:\n\n_CURTAIN_.\n\n_Act Three_\n\n_Scene One: Juliet's Apartment \u2013 Sundown._\n\nThis is a very crowded place: room made into rooms for all purposes: a curtained corner for Juliet who is dressing up happily as her family sings a gay street song in Spanish (mother, father, uncle). During this, the clock strikes the same hour as in the previous scene. And, after a time, faint police whistles, sirens. But the song goes gaily on until Shrank comes in. Juliet's family's English is too poor to understand what he says, so she must translate the terrible news: their son has been murdered by Romeo. Shrank goes, and the family goes to claim the body. Juliet starts to go with them when Romeo appears on the fire escape which is right outside her little corner. His one drive has been to find her and tell her it was a horrible mistake. But her first reaction is: you killed my brother. He tries to explain but there is a police whistle and shouts: they are after him. Juliet doesn't call to the police. She stands, confused, as Romeo whispers \"meet me at the hangout\" and disappears. As he goes, Anita comes into the flat and sees him. She would call the police but Juliet stops her. Anita's attitude has changed. Bitter, angry over the death of her lover, Bernardo, she tells Juliet to stick to \"your own kind\". This is a duet for both girls. But Juliet's confusion resolves itself during the duet: Romeo is her own kind, for she loves him. And at the end, she starts down the fire escape to meet him. Immediately the apartment moves off (as it did in the balcony scene) and three other fire escapes appear behind it for:\n\n_Scene Two: Love Ballet._\n\nAs Juliet shins down the fire escape, other girls wind down the other fire escapes, all going to meet lovers representing Romeo. The dance goes from forgiveness to love to passion to actual sex. It ends with:\n\n_Scene Three: The Hangout._\n\nRomeo and Juliet are in the positions the dancers were at the end. Romeo sings a happy song to Juliet about what their world will be and, in dialogue, they agree to run away together and be safe with each other. This dream-plan is broken by the arrival of Benny. The police have found out where the hangout is (the gang is constantly moving from one shack to another) and are on their way. Benny is furious with Juliet: a lousy Puerto Rican, in his mind, has prevented Romeo from making a safe getaway. Romeo kicks him out, tells Juliet Doc will know where to find him and runs.\n\n_Scene Four: Streets._\n\nRomeo running from the police who fire at him and wound him. He escapes.\n\n_Scene Five: The Drugstore._\n\nThe same jitterbugging tune is being played in a muted way and the gang is going thru the motions of dancing to avert suspicion from the police. Doc comes up from the cellar to get more bandages and medicine. Romeo is down there: he has been hit badly. Doc goes down again and the gang vents its bitterness against Puerto Ricans, now specifically for murdering their leader and for causing one of their good men to be shot by the police. Juliet comes in, seeking Doc, and all the hatred is turned against her. The kids tell her Romeo is dead and jeer at her extreme reaction. She almost faints and instead of offering her water etc., they hideously offer her all kinds of poison so she can kill herself for love and pay for the evil she has done. This is done with macabre humor. All their prejudice and hate and violence comes out in the taunting until, able to bear no more, she grabs the bottle Benny holds and runs out of the store. Doc returns to see her run out. He doesn't know what the gang has done but realizes they have driven her away. He tells them off for what they really are \u2013 and yet winds himself down because, somehow, what they are is not their fault. He goes to tell Romeo and the kids' reaction is: all that because of a dirty Puerto Rican.\n\n_Scene Six: The Bridal Shop._\n\nThis scene is almost completely in song. Juliet has put on the wedding veil, is arranging the mannequins as she sings. Her strangeness is explained by the empty bottle of poison which she addresses for a moment. She is becoming more and more delirious when Romeo comes in. He is very weak but so happy to see her. She is so happy to see him: in her delirium, she thinks they are at least in their own world which has been transported to heaven. He doesn't realize at first that she has taken poison. But when he does discover the truth, it is too late. She sinks to the floor, he cradles her in his arms, they both start a reprise of their balcony song but they never quite finish. The lights change, the walls disappear, the music soars upward and the audience swoons.\n\n_THE END_\nAppendix Two\n\nBernstein's Letters and Postcards to Mildred Spiegel\n\nIn December 1991, Mildred Zucker (formerly Mildred Spiegel) sent details to the Leonard Bernstein Office of all the letters and postcards she had received from Bernstein. Her annotated list was divided into two sections (\"Contents of Letters\" and \"Contents of L.B. Post Cards\"). The following includes the most significant items, which have been amalgamated into a single chronological sequence.\n\nThe additional indented comments are taken from a long letter of 23 July 1978 that Mildred Zucker wrote to Jack Gottlieb, full of information (\"I have been continuing my homework about Lenny and came across a few more facts\"), and ending as follows: \"I consider it a great privilege to have been a close friend of Lenny's. It was thrilling for me to watch him grow and reach such great heights. Lenny was and still is a great source of inspiration to me. Give him our love.\"\n\n29 December 1935, Boston, MA: Lenny asked me to turn his pages at a lecture.\n\n2 January 1936, New York, NY: Has a lot to tell me. Reminds me to turn his pages.\n\n24 June 1936, Philadelphia, PA: Description of Curtis Institute audition\/judges' reaction. They thought he should be at Curtis.\n\n29 June 1936, Philadelphia, PA: Attended Robin Hood Dell concert \u2013 writes his reaction upon hearing _Romeo and Juliet_ Overture of Tchaikovsky. Appointment with [Jos\u00e9] Iturbi.\n\n\"In June 1936 in Philadephia, Lenny had an audition with Merrs. Simpkin and Lorenz (arranged by Don, his Harvard roommate). They were greatly taken with his playing. Among other things they said: 1) The boy is an artist. 2) He has stage personality. 3) Interpretation a little youthful and immature. 4) He ought to be at Curtis \u2013 not for what they could teach him but for what contacts he could make. 5) Mr. Lorenz confided to a girl, 'You see a great boy \u2013 you'll see a greater man'. They also arranged an interview with Iturbi. While visiting Don at 2008 N. Park Aven., Lenny attended a Robin Hood Dell concert where he heard Spalding play the Mendelssohn Violin Concerto under Alexander Smallens, also the _Romeo and Juliet_ Overture of Tchaikovsky and a rehearsal of Harold Bauer playing the Schumann.\"\n\n3 August 1936, Sharon, MA: Wrote me about an eleven-page letter which he will show me. He has grippe and asked me to visit him.\n\n4 August 1936, Sharon, MA: Invites me to Sharon for lunch.\n\nAugust 1936, Sharon, MA: Eleven-page letter written in the blackest of moods describing a clash with his father who did not want Lenny's friends visiting. He also poured his heart out about his friends.\n\n13 September 1936, Alfred, ME: He is staying at a farm. Will come to visit me at York Harbor, where I am playing with a trio.\n\n\"In September 1936, he wound up at Elm Top Farm in Alfred, Maine for a short vacation. He borrowed the farmer's truck and drove to the Emerson House at York Harbor, Maine, where I was playing with my trio. I was delightfully surprised to see him.\"\n\n9 July 1937, Pittsfield, MA: His activities at Camp Onota as a music counselor \u2013 casting for _The Pirates of Penzance_ , first month, then _Of Thee I Sing_ , second month.\n\n\"In July 1937, on his night off from Camp Onota, he went to Cap Allegro and was in the clutches of a million women, playing all evening \u2013 Lecuona, Ravel and de Falla etc.\"\n\n21 January 1938, Cambridge, MA: Announcement \u2013 he's soloist in Ravel Concerto. He changes the printing and writes Boston Symphony and not State Symphony. [See Letter 18].\n\n6 April 1938, Minneapolis, MN: Describes visit to Mitropoulos \u2013 quiet, interesting week.\n\n2 May 1938, Cambridge, MA: He wrote out two measures of a theme.\n\n7 July 1938, Sharon, MA: Thinking of putting on _Cradle Will Rock_. Rehearsal with Forum Quartet, and off to a publicity tea in Scituate, MA.\n\n\"In July 1938, Lenny was at the Sharon summer home (17 Lake Ave) and entertaining persuasive notions of putting on _Cradle Will Rock_ that summer. He returned my copy of the Bach _Well-Tempered Clavichord_ Vol. I which he had borrowed and marked up for his Harmonic Analysis class at Harvard.\"\n\n25 July 1938, Sharon, MA: Audience reaction to his Newport concerto where he earned $50.00 playing in a home and made many friends. Summer plans.\n\n15 August 1938, Sharon, MA: Describes trip home in the fog \u2013 from the Berkshires, Massachusetts.\n\n25 August 1938, Chicago, IL: En route to the West. Bus driver would not stop at Pittsfield, where Lenny wanted to visit me.\n\n18 September 1938, California: En route home from trip out West. \"Glorious trip. We live in a great country.\"\n\n29 December 1938, New York, NY: Staying in New York for the New [Year] \u2013 work to do \u2013 wishes me a happy New Year.\n\n30 December 1938, Maywood, NJ: Saw _On Your Toes_. Will let me know of his arrival. It may be by plane.\n\n7 July 1939, New York, NY: Having a terrible time with his nerves. No future. Staying with Adolph Green. Looking around for work.\n\n3 October 1939, Woodstock, NY: Went to see [L\u00e9on] Barzin about a conducting class. Out of the question. Lenny's plans depend on Curtis examination.\n\n\"In early October 1939, he went to Woodstock to see Barzin who was doubtful as to whether he'd have a conducting class that year.\"\n\n20 October 1939, Philadelphia, PA: Curtis Institute \u2013 Fierce desire to work hard \u2013 helpful, considerate faculty \u2013 lists subjects and teachers \u2013 majoring in piano and conducting \u2013 practices three hours a day.\n\n9 November 1939, Philadelphia, PA: Studying _Tristan_ \u2013 found a deli that sells Halvah \u2013 more in accompanying letter.\n\n9 November 1939, Philadelphia, PA: Description of piano lesson with Vengerova. Offers to teach me when he returns home. Made friends with Mme Miquelle.\n\n20 January 1940, Philadelphia, PA: Curtis Institute \u2013 Hard work \u2013 one tremendous piece of news \u2013 will tell me when he returns home.\n\n1 February 1940, Philadelphia, PA: Mitropoulos has plans for him to come to Minneapolis and be official assistant conductor and pianist for the orchestra \u2013 be at every rehearsal and ready to take over \u2013 will be presented as soloist with orchestra \u2013 also as a composer.\n\n28 February 1940, Philadelphia, PA: Curtis Institute \u2013 He was the only conducting student to get an A from Reiner. Helen Coates sent him brownies.\n\n29 March 1940, Philadelphia, PA: Writes that my letter was the most wonderful letter he ever had. Heard Cleveland Orchestra and Rodzinski in Wilmington, Delaware. Koussy came to Curtis and remembered him.\n\n10 April 1940, Philadephia, PA: Writing vocal quartets, settings of poems by Kenneath Fearing. No contract from Minneapolis and no confirmation about the Berkshires. Going to Washington with Mme Miquelle for the weekend.\n\n23 April 1940, Philadelphia, PA: Letter explaining why Mitropoulos cannot import him. The union claims that Lenny is not a necessary function that cannot be filled by local people. Manager did not want Lenny because he was a student.\n\n3 May 1940, Philadelphia, PA: Coming home on weekend. Writes me to cultivate my left dimple for the occasion. Sings \"Love in May\".\n\n22 June 1940, New York, NY: Leaving for telecast. Will write from New Hampshire. Writes card in Aaron [Copland]'s studio using his card. I asked him what score he would like. He suggested Schumann's 4th Symphony, _Petrouchka_ , Debussy _Nocturnes_ , _Firebird_ , _Don Juan_ , _Till Eulenspiegel_.\n\n27 June 1940, Hanover, NH: Kenny Erhman looking for him. Leaving for Cranwell School for Boys, Lenox, Massachusetts.\n\n\"In June 1940 he did a television show in Hanover, NH, while staying with Raphael Silverman (Hillyer).\"\n\n15 July 1940, Lenox, MA: Made an auspicious conducting debut. Lenny writes it was terrific, thrilling, awe shedding. Letter to follow.\n\n16 July 1940, Lenox, MA: Conducted his own first concert at Tanglewood \u2013 Randall Thompson's Second Symphony \u2013 marvelous time \u2013 Koussy and orchestra like him. Plans to conduct _Scheherazade_ , Copland's _Music for the Theatre_.\n\n24 October 1940, Philadelphia, PA: Reiner is furious at Koussy for stealing his pupil. Never feels wholly alive in Philadelphia.\n\n13 December, 1940, Philadelphia, PA: Practices four hours at the piano \u2013 Mozart Concerto, restudying Chopin \u00c9tudes in 3rds and 6ths. Teaching \"rich brats\" at the Meadowbrook School on Wednesdays.\n\n18 January 1941, Philadelphia, PA: Life is dull and lonely, but very active. He is broadcasting the Stravinsky two-piano concerto on February 1st, 5:30 p.m. on NBC. He will imagine me at the other piano.\n\n7 February 1941, Philadelphia, PA: Vengerova wants to make a two-piano team with him and another student. She was swept off her feet with their Stravinsky. Feels like he is finishing up a jail sentence and can't wait to get away from there fast enough. _Bored_.\n\n12 March 1941, Philadelphia, PA: Next broadcast April 26th.\n\n21 April 1941, Philadelphia, PA: Plans to be home May 4th. Will be on the air.\n\n21 July 1941, Lenox, MA: Big success conducting William Schuman _American Festival Overture_ on the same program with Koussy conducting the _Faust_ Symphony. Lenny got two more bows than Koussy. Lenny got a screaming ovation. Conducted _Mastersingers_ Prelude at Esplanade \u2013 22,000 people in audience \u2013 $150.00.\n\n2 December 1941, Boston, MA: Theme from Schumann Symphony.\n\n5 December 1941, Boston, MA: Card announcing opening of his studio for the teaching of piano and musical analysis at 295 Huntington Avenue, Boston.\n\n20 December 1941, Boston, MA: Christmas card and invitation to see his new studio.\n\n6 October 1942, New York, NY: No real secure job yet. Doing odd jobs.\n\n23 June 1943, New York, NY: One week late for my birthday. Wishes me success and happiness.\n\n12 July 1943, New York, NY: Will conduct Goldman Band concerts. Leaving for Hollywood in August for a month vacation. Will conduct Boston Symphony concerts in Boston on August 1st and 2nd before leaving.\n\n11 August 1943, Brookline, MA: Rehearsals for Tanglewood \u2013 will be back at end of month for Army induction.\n\n22 October 1943, New York, NY: Written on Philharmonic Symphony of New York stationery. Wonderful and exciting.\n\n15 December 1943, New York, NY: Thanks me for the \"glorious Halvah\". Do I ever get to New York?\n\n26 June 1944, New York, NY: My wonderful cards delight him. Was in the hospital having his septum out. Leaving for Chicago to begin his summer season. Will conduct at Stadium in New York.\n\n20 September 1944, New York, NY: Wants to know how I am. Will conduct _Fancy Free_ in Boston on October 2nd and we will get together. Some day we will play the Saint-Sa\u00ebns 5th Concerto together.\n\n19 February 1945, St. Louis, MO: Conducting was wonderful, responsive orchestra, audiences wild. Middle of a huge tour across the continent through Canada and back.\n\n4 September 1945, New York, NY: Thanks me for my most touching card. He asked if I heard of his acquisition of the New York City Center Orchestra. It was a thrilling birthday gift.\n\n11 December 1945, New York, NY: Came home from a two-week bout in St. Louis and found my heavenly Halvah \u2013 beautiful surprise.\n\n24 December 1945, en route from Minneapolis to New York: He talked of me with mutual friends, and loved me very much.\n\n1 February 1946, New York, NY: Coming in March to Boston and wants to spend some time together. He is redoing the Beethoven 1st Concerto. It is full of me.\n\n9 April 1947, New York, NY: Am I coming to Tanglefoot?\n\n15 April 1948, New York, NY: Off to Europe to conduct in Munich, Budapest, Vienna and Milan.\n\n29 November 1951, Cuernavaca, Mexico: Encouraging me to teach in Israel. He is composing, sunning, swimming. Opera progresses slowly \u2013 throws away more than he keeps.\n\n3 January 1952, Cuernavaca, Mexico: Wishing me happiness for my marriage. He feels lazy.\n\n4 December 1961, New York, NY: He is sick in bed. The Halvah is like Manna from Heaven. It cheered his ailing days.\n\n9 February 1962, New York, NY: He was overcome with nostalgia when he met a mutual friend in Aspen, Colorado. Memories of my Madison Trio, joyful early days.\n\n3 June 1970, Paris, France: Conducted Mahler 3rd in Paris, Verdi Requiem in London, _Fidelio_ in Rome. He can hardly keep up with himself.\n\n10 January 1972, New York, NY: While he was in Vienna, he recalled meeting \"under Beethoven\" when we met for two pianos [under the Beethoven statue] at the New England Conservatory.\nBibliography\n\nArchives\n\nThe vast majority of the letters published in this book are in the Leonard Bernstein Collection, Music Division, Library of Congress, Washington, D.C. Thanks to the work of Charlie Harmon and others in the Leonard Bernstein Office, the Bernstein Collection also includes many photocopies of letters from Bernstein as well as those written to him.\n\nOther collections in the Library of Congress containing letters from Bernstein include those of Aaron Copland, David Diamond, Hans Heinsheimer, Serge Koussevitzky, and Helen Coates, as well as the papers of other members of the Bernstein family, notably Leonard's sister Shirley, his brother Burton, and his wife Felicia.\n\nBernstein's letters to Jerome Robbins about _Fancy Free_ are in the Robbins Papers at the New York Public Library of the Performing Arts, and other letters from Bernstein are drawn from the institutions and private individuals listed below:\n\nLeonard Bernstein Collection, Music Division, Library of Congress, Washington, D.C.\n\nAaron Copland Collection, Music Division, Library of Congress, Washington, D.C.\n\nDavid Diamond Collection, Music Division, Library of Congress, Washington, D.C.\n\nHans Heinsheimer Collection, Music Division, Library of Congress, Washington, D.C.\n\nSerge Koussevitzky Collection, Music Division, Library of Congress, Washington, D.C.\n\nJerome Robbins Papers, New York Public Library for the Performing Arts, New York, NY.\n\nRichard Adams Romney Letters, Beinecke Rare Book and Manuscript Library, Yale University, New Haven, CT.\n\nYevgeny Yevtushenko Papers, Manuscripts Division, Special Collections, Stanford University Library, Stanford, CA.\n\nPat Jaffe, New York, NY.\n\nPhyllis Newman, New York, NY.\n\nShirley Gabis Rhoads Perle, New York, NY.\n\nSid Ramin, New York, NY.\n\nBooks and Articles\n\nAdams, John (2008): _Hallelujah Junction: Composing an American Life_. New York: Farrar, Straus, and Giroux.\n\nBarnouw, Erik (1990): _Tube of Plenty: The Evolution of American Television_. New York and Oxford: Oxford University Press.\n\nBernstein, Burton (1982): _Family Matters: Sam, Jennie, and the Kids_. New York: Summit Books.\n\nBernstein, Leonard (1957): \"Excerpts from a West Side Log,\" _Playbill_ , 30 September, pp. 47\u20138; repr. in Bernstein 1982, pp. 144\u20137.\n\n\u2014\u2014 (1959): _The Joy of Music_. New York: Simon and Schuster.\n\n\u2014\u2014 (1966): _The Infinite Variety of Music_. New York: Simon and Schuster.\n\n\u2014\u2014 (1982): _Findings_. New York: Simon and Schuster.\n\n_Bernstein Live at the New York Philharmonic_ (2000): Disc notes for NYP 2003, New York: Philharmonic-Symphony Society of New York.\n\nBurlingame, Jon (2003): \"Leonard Bernstein and _On the Waterfront_ : Tragic Nobility, a Lyrical Song, and Music of Violence,\" in Joanne E. Rapf, ed.: _On the Waterfront_. Cambridge: Cambridge University Press, pp. 124\u201347.\n\nBurton, Humphrey (1994): _Leonard Bernstein_. London: Faber and Faber.\n\nChapin, Ted (2005): _Everything Was Possible: The Birth of the Musical Follies_. New York: Applause Books.\n\nChaplin, Saul (1994): _The Golden Age of Movie Musicals and Me_. Norman, OK: University of Oklahoma Press.\n\nCooke, Mervyn, ed. (2010): _The Hollywood Film Music Reader_. New York and Oxford: Oxford University Press.\n\nCopland, Aaron, and Vivian Perlis (1984): _Copland: 1900 Through 1942_. London: Faber and Faber.\n\n\u2014\u2014 (1992): _Copland Since 1943_. London: Marion Boyars.\n\nCott, Jonathan (2013): _Dinner with Lenny: The Last Long Interview with Leonard Bernstein_. New York: Oxford University Press.\n\nCrist, Elizabeth B., and Wayne Shirley, ed. (2006): _The Selected Correspondence of Aaron Copland_. New Haven and London: Yale University Press.\n\nDougary, Ginny (2010): \"Leonard Bernstein: Charismatic, Pompous \u2013 and a Great Father,\" _The Times_ (London), 13 March. Online version at www.ginnydougary.co.uk, accessed 19 March 2013.\n\nDunning, John (1998): _On the Air: The Encyclopedia of Old-time Radio_. New York: Oxford University Press.\n\nGordon, Eric A. (1989): _Mark the Music: The Life and Work of Marc Blitzstein_. New York: St Martin's Press.\n\nGottlieb, Jack, ed. (1998): _Leonard Bernstein_ [...] _A Complete Catalog of His Works. Volume 1: Life, Musical Compositions & Writings_. [New York:] Leonard Bernstein Music Publishing Company.\n\nHoopes, Roy (1982): _Cain_. New York: Holt, Reinhart, and Winston.\n\nHouseman, John (1972): _Run-Through: A Memoir_. New York: Simon and Schuster.\n\nHussey, Walter (1985): _Patron of Art_. London: Weidenfeld and Nicolson.\n\nJoseph, Charles M. (2001): _Stravinsky Inside Out_. New Haven and London: Yale University Press.\n\nJowett, Deborah (2004): _Jerome Robbins: His Life, his Theater, his Dance_. New York: Simon and Schuster.\n\nKimberling, Victoria J. (1987): _David Diamond: A Bio-Bibliography_. Lanhau, MD: Scarecrow Press.\n\nLaurents, Arthur (2000): _Original Story By: A Memoir of Broadway and Hollywood_. New York: Alfred A. Knopf.\n\nMangan, Timothy, and Irene Herrmann: _Paul Bowles on Music_. Berkeley, CA: University of California Press.\n\nMassey, Drew (2009): \"Leonard Bernstein and the Harvard Student Union: In Search of Political Origins,\" _Journal of the Society for American Music_ , vol. 3, no. 1, pp. 67\u201384.\n\nMilhaud, Darius (1995): _My Happy Life_ (trans. Donald Evans, George Hall, and Christopher Palmer). London: Marion Boyars.\n\nMoorehead, Caroline, ed. (2006): _The Letters of Martha Gellhorn_. London: Chatto and Windus.\n\nNixon, Marni (2006): _I Could Have Sung All Night: My Story_. New York: Billboard Books.\n\nOja, Carol J., and Kay Kaufman Shelemay (2009): \"Leonard Bernstein's Jewish Boston: Cross-Disciplinary Research in the Classroom,\" _Journal of the Society of American Music_ , vol. 3, no. 1, pp. 3\u201333.\n\nPollack, Howard (1999): _Aaron Copland: The Life of an Uncommon Man_. London: Faber and Faber.\n\nSarna, Jonathan D. (2009): \"Leonard Bernstein and the Boston Jewish Community of His Youth: The Influence of Solomon Braslavsky, Herman Rubenovitz, and Congregation Mishkan Tefila,\" _Journal of the Society of American Music_ , vol. 3, no. 1, pp. 35\u201346.\n\nSeldes, Barry (2009): _Leonard Bernstein: The Political Life of an American Musician_. Berkeley, Los Angeles, and London: University of California Press.\n\nSimeone, Nigel (2009): _Leonard Bernstein: West Side Story_. Farnham: Ashgate.\n\nSwayne, Steve (2011): _Orpheus in Manhattan: William Schuman and the Shaping of American Music_. Oxford and New York: Oxford University Press.\n\nVaill, Amanda (2007): _Somewhere: The Life of Jerome Robbins_. London: Weidenfeld and Nicolson.\n\nZadan, Craig (1974): _Sondheim & Co._ New York: Macmillan.\n\nWebsites (selective list)\n\nLeonard Bernstein Collection, Library of Congress: \n\nOfficial Leonard Bernstein website: www.leonardbernstein.com\n\nLeonard Bernstein's Boston Years: Team Research in a Harvard Classroom [includes interviews with Lukas Foss, Raphael Hillyer, Sid Ramin, and Harold Shapero]: \n\nFBI Records: The Vault (online archive): \n\nThe Harvard Crimson: www.thecrimson.com\n\nInternet Broadway database: www.ibdb.com\n\nNew York Philharmonic Digital Archives: \n\n_The New York Times_ : www.nytimes.com\n\n_Time_ Magazine: www.time.com\nIndex of Compositions by Bernstein\n\nNote: this is an index of works mentioned in the present book. For a complete catalogue of Bernstein's compositions, see Gottlieb 1998.\n\n_Age of Anxiety, The_. _See_ Symphony No. (i)\n\n_Anniversaries_ (unspecified) (i). _See Seven Anniversaries_ and _Thirteen Anniversaries_\n\n_Birds, The_ (1939; incid. music for play by Aristophanes) (i), (ii), (iii), (iv), (v) n73, (vi) n77\n\nFirst performance: 21 April 1939, Cambridge, MA, Sanders Theatre, Harvard Classical Club, Charles T. Murphy and Lawrence B. Leighton (dirs.), Leonard Bernstein (cond.); orchestra members included Raphael Silverman [Hillyer], first violin, J[esse] Ehrlich, cello and David Glazer, clarinet\n\n_By Bernstein_ (revue) (i) n47\n\nFirst performance: 23 November 1975, New York, Chelsea Theater Center\n\n_Candide_ (1954\u20136; book: Lillian Hellman; lyrics: Richard Wilbur, John LaTouche, Dorothy Parker, Lillian Hellman and Leonard Bernstein) (i) n104, (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii) n77, (xxiv), (xxv), (xxvi) n14, (xxvii) n23, (xxviii), (xxix) n59, (xxx) n53\n\nFirst performance: 29 October 1956, Boston, Colonial Theatre, cast incl. Barbara Cook (Cunegonde), Robert Rounseville (Candide), Max Adrian (Dr. Pangloss); Tyrone Guthrie (dir.), Samuel Krachmalnick (cond.)\n\nFirst New York performance: 1 December 1956, Martin Beck Theatre, cast as above\n\nOrchestrations by Bernstein and Hershy Kay\n\n_Chichester Psalms_ (1965) (i), (ii), (iii), (iv), (v), (vi) n128, (vii), (viii), (ix) n142, (x), (xi) n47\n\nFirst performance: 15 July 1965, New York, Philharmonic Hall, John Bogart (boy alto), Camerata Singers, New York PO, Leonard Bernstein (cond.)\n\nFirst UK performance: 31 July 1965, Chichester Cathedral, Choirs of Chichester, Salibury and Winchester Catherdrals, Philomusica of London, John Birch (cond.)\n\nDedication (on printed piano-vocal score): Commissioned by the Very Rev. Walter Hussey, Dean of Chichester Cathedral, Sussex, for its 1965 Festival, and dedicated, with gratitude, to Cyril Solomon\n\nMuch of the score is recycled from earlier music:\n\na) two numbers composed for the abandoned musical _The Skin of Our Teeth_ : \"Here Comes The Sun\", used in the first movement, and \"Spring Will Come Again\", used for the outer sections of the second movement\n\nb) a cut number from _West Side Story_ (\"Mix!\"), used for the central section of the second movement\n\nc) a sketch headed \"Wartime Duet?\" used as the main 10\/4 theme of the third movement\n\n_Conch Town_ (1941\u20132) (i), (ii) n25, (iii)\n\nPlanned ballet, unfinished; manuscript for two pianos and percussion virtually complete. Started during Bernstein's stay at Key West in August 1941, he was \"working very hard\" on it in April 1942 with the plan to \"turn it into a ballet\" ( _see_ Letter 106). Bernstein later used material from _Conch Town_ in _Fancy Free_ and _West Side Story_\n\n_David_ (January 1954) (i)\n\nPlanned \"big three-act opera\" ( _see_ Letter 335), not composed\n\n_Dedication to Aaron Copland_. _See Seven Anniversaries_\n\n_Dybbuk_ (1972\u20134) (i) n145, (ii), (iii), (iv), (v) n39, (vi)\n\nFirst performance: 16 May 1974, New York, New York State Theater, Lincoln Center, New York City Ballet, Jerome Robbins (choreo.), Leonard Bernstein (cond.)\n\n_Extension of a Theme by Adolph Green_ (by 21 February 1943) (i)\n\nFirst broadcast performance: 21 February 1943, New York, WNYC, 6:30\u20137:00 p.m. ( _see_ Letter 125), David Oppenheim (clarinet), Leonard Bernstein (pf)\n\nReworked as the \"Waltz\" variation in _Fancy Free_\n\n_Facsimile_ (1946) (i) n104, (ii) n145, (iii), (iv) n11, (v), (vi), (vii) n70, (viii), (ix), (x)\n\nFirst performance: New York, Broadway Theatre, Ballet Theatre, Jerome Robbins (choreo.), Leonard Bernstein (cond.)\n\nDedication: For Jerome Robbins\n\n_Fancy Free_ (1944, including material from 1941\u20133) (i) n104, (ii), (iii), (iv) n25, (v) n44, (vi) n76, (vii) n141, (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi) n30, (xxii) n65, (xxiii) n77, (xxiv), (xxv) n32, (xxvi), (xxvii), (xxviii)\n\nFirst performance: 18 April 1944, New York, Metropolitan Opera House, Ballet Theatre, Jerome Robbins (choreo.), Leonard Bernstein (cond.)\n\nDedication: For Adolph Green\n\n_Fanfare for the Inauguration of John F. Kennedy_ (by January 1961) (i) n79, (ii) n81\n\nFirst performance: 19 January 1961, Washington, D.C., National Guard Armory, Inaugural Gala, orchestra, Leonard Bernstein (cond.)\n\nA reworking of \"This Turf is Ours\", a cut song from _West Side Story_\n\nOrchestration by Sid Ramin\n\n_Firstborn, The_ , incidental music (1958; incid. music for play by Christopher Fry) (i)\n\nFirst performance: 30 April 1958, New York, Coronet Theatre\n\n_Halil, Nocturne for Solo Flute with Piccolo, Alto Flute, Harp and Strings_ (1981) (i), (ii), (iii), (iv)\n\nFirst performance: 27 May 1981, Jerusalem, Sultan's Pool, Jean-Pierre Rampal (flute), Israel PO, Leonard Bernstein (cond.)\n\nFirst US performance: 4 July 1981, Tanglewood, Doriot Anthony Dwyer (flute), Boston SO, Leonard Bernstein (cond.)\n\nDedication: To the Spirit of Yadin [Tannenbaum], and to his Fallen Brothers\n\nPartly derived from the _CBS 50th Anniversary Music_ composed in October 1977\n\n_I Hate Music: A Cycle of 5 Kid Songs_ (by 14 March 1943; _see_ Letter 133) (i) n57, (ii), (iii), (iv), (v)\n\nFirst performance: 24 August 1943, Lenox, MA, Town Hall, Jennie Tourel (mezzo-sop.), Leonard Bernstein (pf)\n\nDedication: For Edys [Merrill]\n\n_Jeremiah_. _See Symphony No.(i)_\n\n_Kaddish_. _See Symphony No.(i)_\n\n_Lark, The_ (1955; incid. music for play by Jean Anouilh adapted by Lillian Hellman) (i), (ii), (iii) n74\n\nFirst performance: 28 October 1955, Boston, Plymouth Theatre, Russell Oberlin (counter-ten.), members of New York Pro Music Antique, Noah Greenberg (dir.)\n\nFirst New York performance: 17 November 1955, Longacre Theatre, cast as above\n\nMusic from _The Lark_ was later reworked as the _Missa brevis_\n\n_Mass: A Theatre Piece for Singers, Players and Dancers_ (1970\u20131; additional texts by Stephen Schwartz and Leonard Bernstein) 70 n104, 268 n3, 301, 502, 503 (i), (ii), (iii), (iv), (v), (vi), (vii) n55, (viii), (ix), (x)\n\nFirst performance: 8 September 1971, Washington, D.C., Kennedy Center for the Performing Arts, cast incl. Alan Titus (Celebrant), Norman Scribner Choir, Berkshire Boy Choir, Maurice Peress (cond.)\n\nOrchestrations by Jonathan Tunick, Hershy Kay and Bernstein\n\nDedication (added after the first publication of the piano-vocal score): For Roger L. Stevens\n\n_Missa brevis_ (1988) (i) n24\n\nFirst performance: 21 April 1988, Atlanta, GA, Symphony Hall, Derek Lee Ragin (counter-ten.), Atlanta Symphony Chorus, Robert Shaw (cond.)\n\nDedication: For Robert Shaw\n\nDerived from music for _The Lark_ , 1955\n\n_Nicest Time of Year, The_ (by 12 July 1943, _see_ Letter 148) (i)\n\nProbably unperformed at the time. The melody used the following year for \"Lucky To Be Me\" in _On The Town_\n\n_On the Town_ (1944; book and lyrics by Betty Comden and Adolph Green, on an idea by Jerome Robbins) (i), (ii) n43, (iii) n60, (iv) n75, (v) n104, (vi), (vii), (viii), (ix), (x), (xi), (xii) n9, (xiii), (xiv), (xv) n90, (xvi) n173, (xvii), (xviii), (xix), (xx), (xxi), (xxii) n61, (xxiii) n128, (xxiv), (xxv), (xxvi), (xxvii)\n\nFirst performance: 13 December 1944, Boston, Colonial Theatre, cast incl. Betty Comden (Claire DeLoone), Nancy Walker (Hildy Esterhazy), Sono Osato (Ivy Smith), Adolph Green (Ozzie), Cris Alexander (Chip), John Battles (Gabey); Jerome Robbins (choreo.), George Abbott (dir.), Max Goberman (cond.)\n\nFirst New York Performance: 28 December 1944, Adelphi Theatre, cast as above\n\nOrchestrations by Bernstein and Hershy Kay, Don Walker, Elliott Jacoby and Ted Royal\n\nDedication: none, but the _Three Dance Episodes from \"On the Town\"_ are dedicated to Sono Osato (I), Betty Comden (II) and Nancy Walker (III)\n\n_On the Waterfront_ (1954; film score) (i), (ii), (iii), (iv)\n\nSound recording: 25\u20138 April 1954, Hollywood, CA, Columbia Pictures Studio, North Gower Street, orchestra, Morris Stoloff (cond.), supervised by Leonard Bernstein\n\nBernstein also played the piano for part of this recording: he performed the solo in saloon scene, approx. 37 minutes into the film (the manuscript of the music for this scene is headed \"4C - Juke Box\")\n\n_Peace, The_ (1941, incid. music for play by Aristophanes) (i), (ii), (iii)\n\nFirst Performance: 23 May 1941, Cambridge, MA, Sanders Theatre, Harvard Student Union Theatre, Robert Nichols (dir.), Leonard Bernstein (cond.)\n\n_Peter Pan_ (1950, incid. music for play by J.M. Barrie) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii)\n\nFirst performance: 24 April 1950, New York, Imperial Theatre, cast incl. Jean Arthur (Peter Pan) and Boris Karloff (Captain Hook); John Burrell (dir.), Trude Rittmann (music coordinator), Ben Steinberg (cond.)\n\nOrchestrations by Hershy Kay\n\n_Piano Trio_ (1937) (i) n18\n\nFirst performance: 1937, Madison Trio: Mildred Spiegel (piano), Dorothy Rosenberg (violin), Sarah Kruskall (cello)\n\nLater performance: 1939?, Harvard University, Paine Hall, Raphael Silverman [Hillyer] (violin), Jesse Ehrlich (cello), Mildred Spiegel (piano) [according to Raphael Hillyer]\n\nDedication: For the Madison Trio: M.S., D.R., S.K.\n\nOn the last page of the autograph score, Bernstein has written \"Revised Apr. 1937\"; Bernstein's movement listing on the inside front cover of this manuscript calls the work \"Pianoforte Trio, op. 2\", as do the autograph parts\n\n_Quiet Place, A_ (1983, rev. 1984; libretto by Stephen Wadsworth) (i) n46, (ii), (iii), (iv) n4, (v)\n\nFirst performance of original version (in one act; the second work on a double bill with _Trouble in Tahiti_ ): 17 June 1983, Houston Grand Opera, John DeMain (cond.)\n\nFirst performance of the revised version (in three acts, incorporating _Trouble in Tahiti_ ): 19 June 1984, La Scala, Milan, John Mauceri (cond.)\n\nFirst U.S. performance of the revised version: 22 July 1984, Washington, D.C., Kennedy Center for the Performing Arts, John Mauceri (cond.)\n\n_Riobamba, The_ (1942) (i), (ii) n80\n\nFirst performance: 10 December 1942, New York, Riobamba Club, 151 East 57th Street ( _see_ Letter 115), possibly Jane Froman (singer), who headed the bill on the club's opening night\n\nDerived from music in _Conch Town_ and later reworked as the \"Danz\u00f3n\" in _Fancy Free_\n\n_Serenade_ (1947\u20138; 1955) (i); (i), (ii)\n\nPlanned musical setting of James M. Cain's _Serenade_ , not composed\n\n_Serenade after Plato's Symposium_ (1954) (i), (ii) n83, (iii), (iv), (v), (vi), (vii), (viii), (ix)\n\nFirst performance: 9 September 1954, Venice, Teatro La Fenice, Isaac Stern (violin), Israel PO, Bernstein (cond.)\n\nFirst US performance: 15 April 1955, Boston, Symphony Hall, Isaac Stern (violin), Boston SO, Charles Munch (cond.)\n\nDedication: To the Beloved Memory of Serge and Natalie Koussevitzky\n\n_Seven Anniversaries_ (1942\u20133; completed by February 1943) (i) n59, (ii) n110, (iii) n65, (iv) n75, (v), (vi), (vii)\n\nFirst broadcast performance (as _Six Pieces_ and _Dedication to Aaron Copland_ ): 21 February 1943, New York, WNYC, 6:30\u20137:00 p.m. ( _see_ Letter 125), Bernstein (pf)\n\nFirst known public performance: 14 May 1944, Boston, Opera House, Bernstein (pf)\n\nDedications: I. For Aaron Copland; II. For My Sister, Shirley; III. In Memoriam: Alfred Eisner; IV. For Paul Bowles; V. In Memoriam: Natalie Koussevitzky; VI. For Sergei Koussevitzky; VII. For William Schuman\n\n_Six Pieces_. _See Seven Anniversaries_\n\n_1600 Pennsylvania Avenue_ (1976; book and lyrics by Alan Jay Lerner)\n\nFirst performance: 26 February 1976, Philadelphia, PA, Forrest Theatre, Ken Howard (President), Patricia Routledge (President's Wife), Gilbert Price (Lud); Donald McKayle (choreo.), Frank Corsaro (dir.), Roland Gagnon (cond.)\n\nFirst New York performance: 4 May 1976, Mark Hellinger Theatre, cast as above; Gilbert Moses and George Faison (choreo. and dir.), Roland Gagnon (cond.)\n\nOrchestrations by Sid Ramin and Hershy Kay\n\n_Skin of Our Teeth, The_ (1964\u20135) (i), (ii) n31, (iii), (iv), (v)\n\nPlanned musical, not completed. Two numbers (\"Here Comes The Sun\" and \"Spring Will Come Again\") were reworked for _Chichester Psalms_\n\n_Sonata for Clarinet and Piano_ (1941\u20132) (i), (ii), (iii), (iv) n38, (v), (vi), (vii), (viii), (ix) n76, (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi) n144, (xxii), (xxiii) n157, (xxiv) n48, (xxv) n159, (xxvi), (xxvii), (xxviii)\n\nFirst performance: 21 April 1942, Boston, Institute of Modern Art, David Glazer (clarinet), Bernstein (pf)\n\nFirst broadcast performance: New York, WNYC, 21 February 1943, 6:30\u20137:00 p.m. ( _see_ Letter 125), David Oppenheim (clarinet), Bernstein (pf)\n\nFirst New York public performance: 14 March 1943, New York Public Library, League of Composers Concert, David Oppenheim (clarinet), Bernstein (pf)\n\nDedication: For David Oppenheim\n\n_Sonata for Violin and Piano_ (by August 1940; _see_ Letter 58) (i), (ii), (iii), (iv)\n\nFirst performance: 1940, Cambridge, MA, Raphael Silverman [Hillyer] (violin), Leonard Bernstein (piano)\n\nDedication: For Raphael Silverman [Hillyer]\n\n_Songfest_ (1977) (i), (ii)\n\nFirst performance: 11 October 1977, Washington, D.C., Kennedy Center, Clamma Dale (sop.), Rosalind Elias (mezzo-sop.), Nancy Williams (mezzo-sop.), Neil Rosenshein (ten.), John Reardon (bar.), Donald Gramm (bass-bar.), National SO, Bernstein (cond.)\n\nDedication: For my mother\n\n_Symphonic Dances from West Side Story_ (1960) (i) n72, (ii)\n\nFirst performance: 13 February 1961, Carnegie Hall, New York, New York PO, Lukas Foss (cond.)\n\nOrchestrations by Sid Ramin and Irwin Kostal with Bernstein\n\nDedication: For Sid Ramin, in friendship\n\n_Symphonic Suite from \"On the Waterfront\"_ (1955) (i) n79, (ii), (iii)\n\nFirst performance: 11 August 1955, Tanglewood, MA, Boston SO, Leonard Bernstein (cond.)\n\nDedication: For my son, Alexander\n\n_Symphony No. 1, Jeremiah_ (1939\u201343) (i), (ii) n66, (iii), (iv), (v) n52, (vi), (vii) n138, (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n77, (xx) n23, (xxi)\n\nFirst performance: 28 January 1944, Pittsburgh, Syria Mosque, Jennie Tourel (mezzo-sop.), Pittsburgh SO, Leonard Bernstein (cond.)\n\nDedication: For my father\n\n_Symphony No. 2, The Age of Anxiety_ (1945\u20139; rev. 1965) (i) n163, (ii), (iii) n72, (iv), (v), (vi) n111, (vii), (viii), (ix), (x) n2, (xi), (xii), (xiii), (xiv), (xv), (xvi) n90, (xvii), (xviii), (xix) n63\n\nFirst performance of \"Dirge\" (only): 28 November 1948, Tel Aviv, Habima Hall, Gala Soir\u00e9e in aid of the IPO Pension Fund, Leonard Bernstein (pf), Israel PO, George Singer (cond.)\n\nFirst complete performance: 8 April 1949, Boston, Symphony Hall, Leonard Bernstein (pf), Boston SO, Serge Koussevitzky (cond.)\n\nFirst New York performance: 23 February 1950, New York, Carnegie Hall, Lukas Foss (pf), New York PO, Leonard Bernstein (cond.)\n\nFirst performance of revised version: 15 July 1965, New York, Philharmonic Hall, Philippe Entremont (pf), New York PO, Leonard Bernstein (cond.)\n\nDedication: For Serge Koussevitzky, in tribute\n\n_Symphony No. 3, Kaddish_ (1963; rev. 1977) (i) n2, (ii) n108, (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi) n23\n\nFirst performance: 10 December 1963, Tel Aviv, Frederick Mann Auditorium, Hannah Rovina (speaker), Jennie Tourel (mezzo-sop.), Abraham Kaplan (choirs dir.), Israel PO, Leonard Bernstein (cond.)\n\nFirst US performance: 31 January 1964, Boston, Symphony Hall, Felicia Montealegre (speaker), Jennie Tourel (mezzo-sop.), New England Conservatory Chorus, Columbus Boychoir, Boston SO, Charles Munch (cond.)\n\nFirst performance of revised version: 25 August 1977, Mainz, Michael Wager (speaker), Monserrat Caball\u00e9 (soprano), Wiener Jeunesse Chor, Wiener S\u00e4ngerknaben, Israel PO, Leonard Bernstein (cond.)\n\nDedication: To the Beloved Memory of John F. Kennedy\n\n_Thirteen Anniversaries_ (1974\u201389) (i) n61\n\nDedications: I. For Shirley Gabis Rhoads Perle; II. In Memoriam: William Kapell; III. For Stephen Sondheim; IV. For Craig Urquhart; V. For Leo Smit; VI. For My Daughter, Nina; VII. In Memoriam: Helen Coates; VIII. In Memoriam: Goddard Lieberson; IX. For Jessica Fleischmann; X. In Memoriam: Constance Hope; XI. For Felicia, On Our 28th Birthday (& Her 52nd); XII. For Aaron Stern; XIII. In Memoriam: Ellen Goetz\n\n_Trouble in Tahiti_ (1951, libretto by Leonard Bernstein) (i) n46, (ii), (iii) n47, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi) n45\n\nFirst performance: 12 June 1952, Brandeis University Festival of the Creative Arts, cast incl. Nell Tangeman (Dinah), David Atkinson (Sam), Constance Bingham, Robert Kole and Claude Heater (Trio), Elliot Silverstein (dir.), Leonard Bernstein (cond.)\n\nFirst New York performance: 19 April 1955, Playhouse Theatre, cast incl. Alice Ghostley (Dinah), John Tyers (Sam), Constance Bingham, John Taliafero, James Tushar (Trio), David Brooks (dir.), Joseph D. Lewis and Urey Krasnopolsky (pfs), Max Rich (drums), Leonard Gaskin (bass)\n\nDedication: For Marc Blitzstein\n\n_West Side Story_ (1949\u201357; based on an idea by Jerome Robbins, book: Arthur Laurents, lyrics: Stephen Sondheim) (i) n4, (ii) n104, (iii), (iv), (v) n145, (vi) n5, (vii) n132, (viii) n140, (ix) n173, (x) n76, (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix) n58, (l), (li), (lii), (liii), (liv), (lv), (lvi), (lvii), (lviii)\n\nFirst performance: 19 August 1957, Washington, D.C., National Theatre, cast incl. Carol Lawrence (Maria), Chita Rivera (Anita), Larry Kert (Tony), Ken Le Roy (Bernardo), Mickey Calin (Riff), Art Smith (Doc); Jerome Robbins (concept and dir.), Max Goberman (cond.)\n\nFirst New York performance: 26 September 1957, Winter Garden Theatre, cast as above\n\nOrchestrations by Sid Ramin and Irwin Kostal with Bernstein\n\nDedication: To Felicia, with love\n\n_White House Cantata, A_ (scenes from _1600 Pennsylvania Avenue_ ), arr. Charlie Harmon and Sid Ramin (1997) (i), (ii) n38\n\nFirst performance: 8 July 1997, London, Barbican Centre, Nancy Gustafson, Dietrich Henschel, Jacqueline Miura, Thomas Young, London Voices, London SO, Kent Nagano (cond.)\n\n_Wonderful Town_ (1952\u20133; book: Joseph Fields and Jerome Chodorov, lyrics: Betty Comden and Adolph Green) (i), (ii) n43, (iii) n60, (iv) n113, (v) n171, (vi) n58, (vii) n73, (viii), (ix), (x), (xi), (xii), (xiii) n60, (xiv) n128, (xv), (xvi) n53\n\nFirst performance: 19 January 1953, New Haven, CT, Shubert Theater, cast incl. Rosalind Russell (Ruth), Edith Adams (Eileen), George Gaynes (Robert Baker), Jordan Bentley (Wreck), Cris Alexander (Frank); Donald Saddler (choreo.), George Abbott (dir.), Lehman Engel (cond.)\n\nFirst New York performance: 26 February 1926, Winter Garden Theatre, cast as above\n\nOrchestrations by Don Walker assisted by Seymour Ginzler and Sid Ramin\n\nArrangement:\n\n_El Sal\u00f3n M\u00e9xico_ (Aaron Copland), transcribed for solo piano (by 10 December 1940; _see_ Letter 74) (i), (ii), (iii), (iv), (v) n107, (vi), (vii) n160\n\nFirst known public performance: 18 November 1941, Boston, Copley Plaza Hotel, as part of recital for the Chromatic Club, Leonard Bernstein (pf). The program for this concert makes no mention of the Copland transcription being a first performance. Since it had been finished a year earlier, it is likely that Bernstein had played it before, possibly at private concerts\nGeneral Index\n\nAbarbanell, Lina (i)\n\nAbbado, Claudio (i)\n\nAbbott, George (i), (ii), (iii), (iv), (v) n4, (vi), (vii) n6, (viii), (ix) n173, (x)\n\nAbell, David Charles (i), (ii)\n\nAbravanel, Maurice (i) n104, (ii)\n\nAction Committee to Free Spain Now (i)\n\nAdams, Franklin P. (i)\n\nAdams, John (i)\n\nAdams, Sherman (i)\n\nAdams, Stanley (i), (ii)\n\nAdler, Ellen (i), (ii), (iii) n146, (iv), (v)\n\nAdler, Larry (i)\n\nAlbert, Eddie (i)\n\nAlexander, Holmes (i)\n\nAlger, Horatio (i)\n\n_Allegro_ (musical) (i), (ii) n60\n\nAllen, Elizabeth (i), (ii)\n\nAlphand, Claude (i), (ii)\n\nAl Smith Memorial Hospital Fund (i)\n\nAlsop, Marin (i), (ii), (iii), (iv)\n\nAlswang, Ralph (i)\n\nAmber, Lenny (pseudonym for Leonard Bernstein) (i)\n\nAmerican Arts Committee for Palestine (i)\n\nAmerican Christian Palestine Committee (i)\n\nAmerican Civil Liberties Union (i), (ii), (iii)\n\nAmerican Committee for Spanish Freedom (i)\n\nAmerican Committee of Yugoslav Relief (i)\n\nAmerican Council for a Democratic Greece (i)\n\nAmerican Friends of Hebrew University (i)\n\nAmerican Friends Services Committee (i)\n\nAmerican Fund for Israel Institutions (i)\n\nAmerican Fund for Palestinian Institutions (i)\n\nAmerican Legion (i) n64\n\nAmerican Red Cross (i)\n\nAmerican Red Magen David for Palestine (i)\n\nAmerican Youth for Democracy (i)\n\nAndrews, Dana (i) n62, (ii)\n\nAntheil, George (i), (ii), (iii)\n\nArlen, Harold (i)\n\nArmstrong, Louis (i)\n\nArrau, Claudio (i), (ii) n59, (iii), (iv), (v), (vi), (vii)\n\nAskew, Constance (i), (ii)\n\nAtkinson, Brooks (i) n61, (ii), (iii) n77, (iv)\n\nAuber, Daniel Fran\u00e7ois Esprit (i)\n\nAuden, Wystan Hugh (i), (ii) n163, (iii), (iv), (v) n116, (vi)\n\n_Age of Anxiety, The_ (i), (ii) n110, (iii) n16, (iv) n90, (v) n116\n\nAugustine, Stephanie (i)\n\nAvedon, Richard (i), (ii)\n\nAvshalomoff, Jacob (Jack) (i) n49, (ii), (iii), (iv)\n\nAx, Emanuel (i)\n\nBacall, Lauren (Betty) (i), (ii), (iii) n56\n\nBach, Johann Sebastian (i), (ii), (iii) n29, (iv), (v), (vi), (vii), (viii), (ix)\n\nBarber, Samuel (i) n58, (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix) n94, (x) n133, (xi), (xii)\n\n_Adagio for Strings_ (i) n10\n\n_Second Essay_ (i) n10\n\nViolin Concerto (i) n10, (ii), (iii) n58\n\nBarrymore, Ethel (i)\n\nBart\u00f3k, B\u00e9la (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii) n133, (xiv), (xv), (xvi), (xvii), (xviii)\n\n_Concerto for Orchestra_ (i) n71\n\n_Music for Strings, Percussion and Celesta_ (i), (ii), (iii) n169\n\n_Portrait_ Op. 5 No. 1 (i), (ii), (iii)\n\n_Rhapsody_ No. 1 (i), (ii)\n\nBaudelaire, Charles (i)\n\nBauer, Harold (i)\n\nBauer, Marion (i), (ii)\n\nBaxter, Anne (i) n62\n\nBearson, Lawrence (i)\n\nBeat Generation (i)\n\n_Beckoning Fair One, The_ (unrealized film project) (i) n59, (ii) n74\n\nBeery, Wallace (i), (ii)\n\nBeethoven, Ludwig van (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x) n27, (xi), (xii) n151, (xiii)\n\n_Fidelio_ (i), (ii)\n\n_Leonore_ No. 3 Overture (i), (ii) n27, (iii)\n\n_Missa solemnis_ (i), (ii), (iii)\n\nPiano Concerto No. 1 (i), (ii) n32, (iii), (iv) n70, (v)\n\nString Quartets (i), (ii)\n\nString Quartet Op. 131 arr. Mitropoulos (i) n40\n\nSymphony No. 1 (i) n27\n\nSymphony No. 3 (\"Eroica\") (i) n69, (ii) n69\n\nSymphony No. 4 (i), (ii)\n\nSymphony No. 5 (i), (ii) n99\n\nSymphony No. 7 (i), (ii), (iii)\n\nSymphony No. 9 (i), (ii), (iii)\n\nTriple Concerto (i), (ii) n63\n\nBeinum, Eduard van (i)\n\nBellafonte, Harry (i)\n\nBellini, Vincenzo (i) n149\n\n_Sonnambula, La_ (i), (ii), (iii), (iv), (v)\n\n_Bells are Ringing_ (musical) (i) n21, (ii) n5, (iii) n2\n\nBen-Gurion, David (i), (ii), (iii)\n\nBennett, Richard Rodney (i), (ii), (iii) n25\n\n_Nicholas and Alexandra_ (i)\n\nSymphony No. 2 (i)\n\nBennett, Robert Russell (i)\n\nBerezowsky, Alice (i)\n\nBerezowsky, Nicolai (i), (ii) n86\n\nBerger, Arthur (i) n27, (ii) n28, (iii), (iv), (v) n101, (vi)\n\nBergman, Ingmar (i)\n\nBerio, Luciano (i), (ii)\n\nBerkowitz, Ralph (i)\n\nBerlin, Irving (i), (ii), (iii) n42\n\n_How Dry I Am_ (i)\n\nBernstein, Alexander (son of LB, sometimes \"Axel\" or \"Fink\") (i), (ii) n4, (iii) n79, (iv), (v), (vi), (vii), (viii), (ix), (x) n30, (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv)\n\nBernstein, Burton (brother of LB) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii)\n\nBernstein, Felicia (wife of LB) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii), (liv), (lv), (lvi), (lvii), (lviii), (lix), (lx), (lxi), (lxii), (lxiii), (lxiv), (lxv), (lxvi), (lxvii), (lxviii), (lxix), (lxx), (lxxi), (lxxii), (lxxiii), (lxxiv), (lxxv), (lxxvi), (lxxvii), (lxxviii), (lxxix), (lxxx), (lxxxi), (lxxxii), (lxxxiii), (lxxxiv), (lxxxv), (lxxxvi), (lxxxvii), (lxxxviii), (lxxxix), (xc), (xci), (xcii), (xciii) n44, (xciv), (xcv), (xcvi) n55, (xcvii) n57, (xcviii), (xcix), (c), (ci), (cii), (ciii). _See also_ Montealegre Cohn, Felicia\n\nBernstein, Jamie (daugher of LB) (i), (ii) n4, (iii) n57, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv)\n\nBernstein, Jennie (mother of LB) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx)\n\nBernstein, Nina (daughter of LB) (i) n4, (ii) n30, (iii), (iv) n96, (v), (vi), (vii), (viii), (ix), (x), (xi)\n\nBernstein, Samuel (father of LB) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix) n148, (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n47, (xx), (xxi), (xxii)\n\nBernstein, Shirley (sister of LB) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv) n130, (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix) n149, (l), (li), (lii), (liii)\n\nBerrigan, Daniel (i)\n\nBerrigan, Philip (i)\n\nBigelow, Otis (i), (ii)\n\n_Billion Dollar Baby_ (musical) (i) n5, (ii) n31, (iii)\n\nBing, Rudolf (i)\n\nBirch, John (i), (ii), (iii)\n\nBizet, Georges (i) n53\n\n_Carmen_ (i), (ii) n11, (iii), (iv), (v), (vi)\n\nBlack Panthers (i), (ii), (iii), (iv), (v)\n\nBlitzstein, Marc (i), (ii), (iii), (iv), (v), (vi), (vii) n66, (viii), (ix), (x) n104, (xi), (xii), (xiii), (xiv), (xv) n11, (xvi), (xvii), (xviii), (xix) n65, (xx), (xxi), (xxii), (xxiii) n29, (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix) n122, (xxx)\n\n_Cradle Will Rock, The_ (i), (ii) n29, (iii), (iv) n56, (v) n4, (vi), (vii) n123\n\n_Regina_ (i), (ii) n4, (iii), (iv), (v), (vi), (vii) n104, (viii) n123, (ix) n41\n\n_Reuben Reuben_ (i) n4, (ii), (iii), (iv), (v) n30\n\n_Sacco and Vanzetti_ (i)\n\nBloch, Ernest (i) n71, (ii), (iii), (iv), (v), (vi) n9\n\nBlue Angel, The (nightclub) (i)\n\nB'nai B'rith (i), (ii), (iii) n8\n\nBohm, Jerome (i)\n\nB\u00f6hm, Karl (i), (ii), (iii), (iv), (v) n16\n\nBok, Mary Louise Curtis (i), (ii), (iii), (iv), (v)\n\n_Bonanza Bound_ (musical) (i)\n\nBonney, William H. (Billy the Kid) (i)\n\nBoston Symphony Orchestra (i), (ii) n26, (iii) nn72\u20133, (iv) n75, (v), (vi) n94, (vii) n107, (viii) n2, (ix), (x) n163, (xi), (xii), (xiii), (xiv), (xv), (xvi) n75, (xvii), (xviii), (xix) n82, (xx) n112, (xxi) n153, (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii) n73, (xxviii), (xxix), (xxx) n13, (xxxi) n14, (xxxii), (xxxiii)\n\nBoulanger, Nadia (i) n66, (ii), (iii) n88, (iv) n71, (v), (vi) n123, (vii), (viii), (ix), (x) n146, (xi), (xii), (xiii) n76, (xiv), (xv), (xvi) n101, (xvii) n150, (xviii) n176, (xix)\n\nBoulez, Pierre (i), (ii), (iii) n74, (iv), (v) n142, (vi) n42\n\n_Polyphonie X_ (i)\n\nBoult, Adrian (i), (ii)\n\nBourke-White, Margaret (i), (ii), (iii)\n\nBowles, Paul (often \"Pfb\") (i), (ii) n74, (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii) n85, (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv) n11, (xxvi) n33, (xxvii) n102, (xxviii) n36\n\n_Wind Remains, The_ (i), (ii) n88, (iii) n101\n\nBoy Scouts of America (i)\n\nBraggiotti, Mario (i)\n\nBrahms, Johannes (i), (ii), (iii), (iv), (v) n28, (vi), (vii), (viii), (ix) n151, (x) n23, (xi)\n\nDouble Concerto (i), (ii)\n\n_Haydn Variations_ (i)\n\nPiano Concerto No. 1 (i), (ii), (iii), (iv) n59, (v)\n\nPiano Concerto No. 2 (i), (ii), (iii) n46\n\nSymphony No. 1 (i), (ii) n27, (iii) n69, (iv)\n\nSymphony No. 2 (i)\n\nSymphony No. 3 (i)\n\nBrain, Dennis (\"Brahan\") (i)\n\nBraly, Kristin (i)\n\nBrandeis University (i) n83, (ii) n11, (iii), (iv), (v) n62, (vi), (vii), (viii), (ix)\n\nBrando, Marlon (i) n142, (ii) n80, (iii) n105, (iv), (v)\n\nBraslavsky, Solomon (i), (ii) n66, (iii), (iv), (v), (vi)\n\nBrennan, Charles D. (i)\n\nBrennan, Walter (i) n62, (ii)\n\nBritten, Benjamin (i), (ii), (iii), (iv), (v), (vi), (vii) n55, (viii) n68, (ix), (x) n133, (xi), (xii), (xiii), (xiv) n152, (xv), (xvi), (xvii), (xviii) n42\n\n_Cantata misericordium_ (i)\n\n_Paul Bunyan_ (i) n107, (ii) n108\n\n_Peter Grimes_ (i) n107, (ii), (iii), (iv), (v)\n\n_Rape of Lucretia, The_ (i), (ii) n55\n\n_Rejoice in the Lamb_ (i) n116\n\n_Sinfonia da Requiem_ (i)\n\nViolin Concerto (i) n53, (ii)\n\nBroder Jane (i)\n\nBroder, Nathan (i)\n\nBrooklyn Philanthropic League (i)\n\nBrooks, David (i)\n\nBrown, Earle (i) n109, (ii)\n\nBrown, Kay (i)\n\nBruckner, Anton (i), (ii)\n\nBurgin, Richard (i)\n\nBurk, John N. (i)\n\nBurke, Johnny (i)\n\nBurns, John Horne (i) n109\n\n_Gallery, The_ (i)\n\nBurr, Aaron (i)\n\nBurrell, John (i), (ii)\n\nBurrows, Abe (i) n43\n\nBurton, Humphrey (i), (ii) n130, (iii) n151, (iv), (v), (vi) n57, (vii)\n\nBusoni, Ferruccio (i) n10, (ii)\n\nByrns, Harold (i), (ii)\n\nCadman, Charles Wakefield (i)\n\nCaesar, Irving (i)\n\nCage, John (i) n79, (ii), (iii), (iv), (v), (vi) n112, (vii)\n\nCahn, Judah (i)\n\nCain, James M. (i), (ii), (iii) n114, (iv), (v)\n\n_Double Indemnity_ (i) n114\n\n_Mildred Pierce_ (i) n114\n\n_Postman Always Rings Twice, The_ (i) n114\n\n_Serenade_ (i), (ii), (iii), (iv) n119, (v), (vi)\n\nCalder, Alexander (i), (ii)\n\nCalin, Mickey (i), (ii)\n\nCallas, Maria (i), (ii), (iii), (iv), (v), (vi) n149\n\nCamp Onota (i), (ii), (iii) n43, (iv) n168, (v), (vi), (vii)\n\nCampbell-Watson, Frank (i), (ii) n8\n\n_Can-Can_ (musical) (i), (ii) n54\n\n_Carnival in Flanders_ (musical) (i)\n\n_Carousel_ (musical) (i), (ii), (iii) n150\n\nCarre\u00f1o, Teresa (i)\n\nCarter, Elliott (i) n79, (ii), (iii)\n\n_Concerto for Orchestra_ (i)\n\nCarter, Jimmy (i)\n\nCarter, Rosalynn (i)\n\nCasafuerte, Marquise Yvonne de la (i), (ii), (iii), (iv)\n\nChadwick, George Whitfield (i)\n\nChagall, Marc (i) n116\n\nChapin, Schuyler (i)\n\nChaplin, Charlie (i)\n\nChaplin, Ethel (n\u00e9e Ethel Schwartz) (i)\n\nChaplin, Saul (i), (ii), (iii), (iv), (v), (vi) n82, (vii) n83\n\nCharnin, Martin (i)\n\nChasins, Abram (i)\n\nChausson, Ernest (i)\n\nChavchavadze, Prince George (i)\n\nCh\u00e1vez, Carlos (i), (ii), (iii), (iv), (v), (vi)\n\n_Sinfonia India_ (i), (ii)\n\nChavez, Quita (i)\n\nCherubini, Luigi (i) n149\n\n_Medea_ (i), (ii), (iii), (iv), (v) n149\n\nCincinnati Symphony Orchestra (i), (ii), (iii), (iv)\n\nCivil Rights Congress (i)\n\nClaman, Julian (i)\n\nClark, Anne Gibson (i)\n\nCliburn, Van (i)\n\nClooney, Rosemary (i)\n\nClurman, Harold (i), (ii) n143, (iii), (iv), (v), (vi) n148\n\nCoates, Helen (i), (ii), (iii) n3, (iv), (v), (vi), (vii), (viii), (ix), (x) n45, (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx) n96, (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii) n173, (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii), (liv), (lv), (lvi), (lvii), (lviii), (lix) n15, (lx) n44, (lxi), (lxii), (lxiii) n61, (lxiv), (lxv) n93, (lxvi) n100, (lxvii), (lxviii) n137, (lxix), (lxx), (lxxi), (lxxii), (lxxiii), (lxxiv), (lxxv), (lxxvi), (lxxvii) n18, (lxxviii), (lxxix), (lxxx)\n\nCohen, Mrs. Frank (i)\n\nCohen, Yardena (i)\n\nColeman, Francis A. (i) n10\n\nColombo, Ted (i)\n\nColumbia Records (i), (ii) n38, (iii) n120, (iv) n137, (v), (vi) n52, (vii) n62, (viii), (ix), (x) n58, (xi), (xii) n66, (xiii), (xiv), (xv), (xvi), (xvii) n128, (xviii) n6, (xix) n36\n\nComden, Betty (i), (ii) n43, (iii) n54, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii) n21, (xiv) n131, (xv), (xvi), (xvii), (xviii) n2, (xix), (xx), (xxi) n6, (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii) n29, (xxxiv) n58, (xxxv), (xxxvi), (xxxvii), (xxxviii) n103, (xxxix), (xl), (xli), (xlii) n37, (xliii), (xliv), (xlv) n77, (xlvi) n83, (xlvii) n128, (xlviii), (xlix), (l)\n\nCommunist Party (i), (ii) n98, (iii) n49, (iv) n106, (v) n42, (vi), (vii), (viii), (ix), (x)\n\nCone, Edward T. (i) n28, (ii)\n\nConverse, Frederick (i)\n\nCopland, Aaron (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n66, (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii), (liv), (lv), (lvi), (lvii), (lviii), (lix), (lx), (lxi), (lxii), (lxiii), (lxiv), (lxv), (lxvi), (lxvii), (lxviii), (lxix), (lxx), (lxxi), (lxxii), (lxxiii), (lxxiv), (lxxv), (lxxvi), (lxxvii), (lxxviii), (lxxix) n11, (lxxx), (lxxxi), (lxxxii) n28, (lxxxiii), (lxxxiv) n37, (lxxxv), (lxxxvi) n78, (lxxxvii), (lxxxviii), (lxxxix), (xc) n104, (xci), (xcii), (xciii), (xciv) n133, (xcv), (xcvi), (xcvii), (xcviii), (xcix), (c) nn35\u20136, (ci), (cii), (ciii), (civ), (cv) n64, (cvi) n65, (cvii), (cviii), (cix), (cx), (cxi), (cxii), (cxiii), (cxiv), (cxv), (cxvi), (cxvii), (cxviii), (cix), (cxx), (cxxi), (cxxii), (cxxiii), (cxxiv), (cxxv), (cxxvi), (cxxvii) n36, (cxxviii), (cxxiv), (cxxx)\n\n_Appalachian Spring_ (i) n118, (ii) n151, (iii) n170, (iv) n35, (v), (vi), (vii), (viii)\n\n_Billy the Kid_ (i) n30, (ii) n82, (iii), (iv), (v), (vi), (vii), (viii), (ix) n1, (x), (xi)\n\n_Canticle of Freedom_ (i), (ii)\n\n_City, The_ (i)\n\nClarinet Concerto (i), (ii), (iii), (iv)\n\n_Dance Symphony_ (i)\n\n_Danz\u00f3n cubano_ (i) n46, (ii), (iii)\n\n_El Sal\u00f3n M\u00e9xico_ (sometimes \"El Saloon\") (i), (ii), (iii) nn26\u20137, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii) n107, (xiv), (xv), (xvi), (xvii) n160, (xviii), (xix), (xx)\n\n_Hear Ye, Hear Ye_ (i)\n\n_In the Beginning_ (i) n101\n\n_Lincoln Portrait_ (i), (ii), (iii), (iv), (v), (vi), (vii), (viii)\n\n_Music for the Movies_ (i) n63, (ii)\n\n_Music for the Theatre_ (i), (ii), (iii) n63, (iv), (v), (vi)\n\n_North Star, The_ (i) n62, (ii), (iii) n133, (iv), (v) n78\n\n_Ode_ (i)\n\n_Old American Songs_ (i) n52, (ii), (iii)\n\n_Our Town_ (i) n50, (ii), (iii), (iv), (v)\n\n_Outdoor Overture_ (i), (ii), (iii), (iv), (v), (vi), (vii), (viii)\n\nPiano Concerto (i)\n\nPiano Quartet (i), (ii), (iii), (iv)\n\nPiano Sonata (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii)\n\nPiano Variations (i), (ii), (iii), (iv), (v), (vi), (vii) (orchestral version)\n\n_Quiet City_ (i), (ii), (iii), (iv), (v)\n\n_Rodeo_ (i), (ii), (iii), (iv) n150, (v), (vi)\n\n_Second Hurricane, The_ (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x) n73\n\n_Short Symphony_ (i), (ii)\n\n_Statements_ (i)\n\n_Tender Land, The_ (i) n21, (ii), (iii), (iv)\n\n_Third Symphony_ (i), (ii) n37, (iii), (iv), (v), (vi), (vii)\n\nViolin Sonata (i)\n\nCoppola, Francis Ford (i)\n\nCorwin, Norman (i) n62\n\nCott, Jonathan (i), (ii), (iii) n46\n\nCouncil on African Affairs (i)\n\nCourtnay, Alex (i)\n\nCowan, Lester (i) n59, (ii), (iii)\n\nCowell, Henry (i) n74, (ii), (iii), (iv), (v) n28\n\nCraft, Robert (i) n90, (ii) n93\n\nCrawford, Cheryl (i), (ii), (iii), (iv) n65\n\nCreston, Paul (i) n74, (ii)\n\nCulshaw, John (i)\n\nCunningham, Merce (i) n88\n\nCuomo, Ciro (i), (ii), (iii), (iv), (v), (vi)\n\nCurtin, Phyllis (i) n57\n\nCurtis Institute of Music (i), (ii), (iii) n21, (iv) n32, (v), (vi), (vii), (viii), (ix), (x) n66, (xi) n72, (xii), (xiii), (xiv), (xv), (xvi), (xvii) n3, (xviii) n37, (xix), (xx), (xxi) n137, (xxii), (xxiii) n160, (xxiv)\n\nCurtiss, Mina Kirstein (i), (ii)\n\nCzech Phiharmonic Orchestra (i) n98\n\nDale, Grover (i)\n\nDalgliesh, James (i)\n\nDaniel, Oliver (i), (ii)\n\nDavies, Joseph P. (i)\n\nDavis, Bette (i), (ii), (iii), (iv), (v) n63\n\nDavis, George (i)\n\nDavis, Miles (i), (ii) n41, (iii) n43\n\nDebussy, Claude (i), (ii), (iii) n28, (iv)\n\n_La Mer_ (i)\n\nDeLange, Eddie (i)\n\nDel Tredici, David (i), (ii)\n\n_Tattoo_ (i) n50, (ii)\n\nDello Joio, Norman (i)\n\nDeMain, John (i) n4\n\nDe Mille, Agnes (i), (ii), (iii)\n\nDemuth, Charles (i)\n\nDenby, Edwin (i), (ii), (iii), (iv) n73\n\nDe Sio, Al (i)\n\nDe Spirito, Romolo (i), (ii) n102, (iii) n103\n\nDessau, Paul (i)\n\nDetroit Symphony Orchestra (i), (ii) n129, (iii), (iv)\n\nDiamond, David (i), (ii), (iii), (iv), (v) n74, (vi), (vii), (viii), (ix), (x), (xi) n28, (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix) n102, (xxx), (xxxi), (xxxii), (xxxiii) n64, (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv) n27, (xlvi), (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii), (liv), (lv), (lvi) n47, (lvii)\n\nPiano Concerto (i)\n\nPrelude and Fugue No. 3 (i), (ii), (iii)\n\n_Rounds_ (i), (ii)\n\nSymphony No. 1 (i) n66, (ii) n40\n\nSymphony No. 2 (i)\n\nSymphony No. 4 (i) n66, (ii), (iii), (iv)\n\nSymphony No. 5 (i)\n\nSymphony No. 9 (i)\n\n_World of Paul Klee, The_ (i), (ii), (iii)\n\nDiamond, Irene Lee (i)\n\nDimitryk, Edward (i)\n\nDonen, Stanley (i), (ii) n28\n\nDownes, Olin (i) n28, (ii), (iii) n28, (iv) n33, (v) n69, (vi) n94, (vii) n159, (viii) n65\n\nDu Pont, Paul (i)\n\nDuffy, Monsignor (i)\n\n_Duffy's Tavern_ (radio show) (i)\n\nDufresne, Gaston (i)\n\nDuke, Vernon (i)\n\nDulles, John Foster (i)\n\nDunham, Katherine (i)\n\nDvo\u0159\u00e1k, Anton\u00edn (i) n26, (ii), (iii), (iv), (v), (vi)\n\nDwyer, Doriot Anthony (i), (ii)\n\n_Easter Holiday_ (provisional title for _Easter Parade_ ) (i)\n\n_Easter Parade_ (film) (i) n89\n\nEastman School of Music (i), (ii) n37, (iii) n49, (iv), (v) n71, (vi), (vii) n51, (viii) n58\n\nEddy, Nelson (i)\n\nEdens, Roger (i)\n\nEhrlich, Jesse (i) n18, (ii), (iii), (iv), (v)\n\nEhrman, Kenneth (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii)\n\nEisenhower, Dwight D. (i), (ii) n53, (iii) n141\n\nEisner, Alfred (i), (ii), (iii), (iv), (v), (vi), (vii) n97, (viii), (ix), (x)\n\nElgar, Edward (i), (ii), (iii)\n\nEliot, T.S. (i)\n\nEllington, Duke (i), (ii)\n\nEllis, Havelock (i) n104, (ii)\n\nElman, Mischa (i)\n\nElwes, Mr. and Mrs. Robert (i)\n\nEnglander, Lester (i)\n\nEnglander, Roger (i), (ii) n93\n\nEssen, Viola (i)\n\nEuropean Friends of ORT (i)\n\nEvans, Ross (i)\n\nFarbman, Harry (i)\n\nFassett, James (i)\n\nFaur\u00e9, Gabriel (i)\n\n_Requiem_ (i) n94, (ii) n167\n\nFaye, Alice (i)\n\nFederal Bureau of Investigation (FBI) (i) n85, (ii) n105, (iii) n106, (iv), (v) n68, (vi)\n\nFeigay, Paul (i), (ii), (iii)\n\nFeldman, Charlie (i)\n\nFeldman, Morton (i), (ii), (iii), (iv), (v), (vi), (vii) n112\n\nFennell, Frederick (i)\n\nFiedler, Arthur (i), (ii), (iii), (iv) n59\n\nFine, Irving (i), (ii), (iii) n101\n\nFine, Verna (i)\n\n_First Impressions_ (musical) (i) n43\n\nFischer-Dieskau, Dietrich (i), (ii)\n\nFishburg, Jack (Joachin) (i)\n\nFizdale, Robert (i) n62\n\nFlagg, James Montgomery (i)\n\nFluck, Alan (i), (ii)\n\nFonteyn (de Arias), Margot (i)\n\nFord Foundation (i)\n\nFoss, Lukas (i) n58, (ii), (iii), (iv), (v) n89, (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii) n101, (xix), (xx) n106, (xxi), (xxii), (xxiii), (xxiv) n154, (xxv), (xxvi) n46, (xxvii), (xxviii)\n\n_Phorion_ (i)\n\n_Psalms_ (i)\n\n_Song of Anguish_ (i)\n\n_Symphony of Chorales_ (i)\n\n_Time Cycle_ (i) n64\n\nVariations (i) n154\n\nFran\u00e7aix, Jean (i)\n\nFray, Jacques (i)\n\nFreed, Arthur (i)\n\nFrings, Kurt (i)\n\nFry, Christopher (i)\n\nFrye, Peter (i)\n\nFryer, Robert (i)\n\nFuleihan, Anis (i) n28\n\nFuller, Donald (i)\n\nFurtw\u00e4ngler, Wilhelm (i), (ii), (iii)\n\nGabis Rhoads Perle], Shirley [(i), (ii), (iii), (iv), (v), (vi), (vii)\n\nGaburo, Kenneth (i), (ii)\n\nGalway, James (i)\n\nGarbo, Greta (i) n15, (ii) n74\n\nGarfield, John (i) n59, (ii)\n\nGarland, Judy (i), (ii) n89, (iii) n93, (iv)\n\nGebhard, Heinrich (i), (ii), (iii) n18, (iv)\n\nGeer, Will (i), (ii)\n\nGellhorn, Martha (i), (ii), (iii), (iv) n22, (v) n24, (vi), (vii), (viii)\n\nGershwin, George (i) n6, (ii) n51, (iii) n8, (iv) n60, (v), (vi) n150, (vii)\n\n_Porgy and Bess_ (i) n40, (ii), (iii), (iv), (v)\n\n_Rhapsody in Blue_ (i), (ii)\n\nGershwin, Ira (i), (ii), (iii) n63, (iv) n56\n\nGhiringhelli, Antonio (i), (ii), (iii), (iv), (v)\n\nGhostley, Alice (i), (ii), (iii)\n\nGide, Andr\u00e9 (i), (ii)\n\nGieseking, Walter (i), (ii)\n\nGilbert, Helen (i)\n\nGinzler, Robert (i) n4\n\n_Girl with the Two Left Feet, The_ (Revuers sketch with improvised score by LB) (i), (ii), (iii) n65\n\nGlazer, David (i), (ii), (iii)\n\nGlickstein, Izso (Isadore) (i), (ii)\n\nGluck, Christoph Willibald von (i)\n\nGoberman, Max (i), (ii), (iii), (iv) n5, (v) n173, (vi) n63\n\nGold, Arthur (i) n62\n\nGoldman Band (i), (ii), (iii), (iv), (v), (vi), (vii), (viii)\n\nGoldman, Edwin Franko (i)\n\nGoldman, Milton (i)\n\nGoldstein, Leonore (i), (ii)\n\nGoldwyn, Samuel (i), (ii), (iii), (iv), (v), (vi)\n\nGolschmann, Vladimir (i), (ii), (iii), (iv) n77\n\nGoodman, Benny (i) n153, (ii), (iii) n62\n\nGoodman, Zelda (i)\n\nGordon, Beatrice (i), (ii), (iii), (iv), (v), (vi), (vii)\n\nGottlieb, Jack (i), (ii) n24, (iii), (iv)\n\nGould, Glenn (i), (ii)\n\nGould, Morton (i), (ii) n6, (iii), (iv), (v) n32\n\nGrace Congregational Church (i)\n\nGradenwitz, Peter (i), (ii) n152\n\n_The Music of Israel: Its Rise and Growth Through 5000 Years_ (i)\n\nGraham, Martha (i), (ii), (iii) n6, (iv) n13\n\nGranger, Farley (i) n62, (ii), (iii), (iv), (v), (vi)\n\nGrant, Cary (i)\n\nGrayson, Kathryn (i)\n\nGreen, Adolph (i) n17, (ii), (iii) n60, (iv), (v), (vi), (vii) n21, (viii), (ix), (x) n23, (xi), (xii) n76, (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n47, (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii) n58, (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv) n77, (xlv) n83, (xlvi), (xlvii) n128, (xlviii), (xlix) n172, (l), (li), (lii)\n\nGreen, Frank (i)\n\nGreen, Johnny (i)\n\nGreene, Milton (i)\n\nGreenwich House, The (i)\n\n_Greenwich Village_ (film) (i) nn130\u20131\n\nGretchaninoff, Alexander (i)\n\nGrieg, Edvard (i), (ii)\n\nGruen, John (i), (ii)\n\nGuthrie, Tyrone (i), (ii)\n\n_Guys and Dolls_ (musical) (i), (ii) n43,\n\nHaftel, Zvi (i)\n\nHaieff, Alexei (i) n28, (ii)\n\nHalasz, Laszlo (i), (ii)\n\nHall, Radclyffe (i) n20\n\n_Well of Loneliness, The_ (i)\n\nHammer, Alvin (i) n131, (ii)\n\nHammerstein, Oscar (i), (ii) n115, (iii), (iv), (v), (vi) n37, (vii) n60, (viii) n74\n\nHandel, George Frideric (i) n6, (ii)\n\n_Water_ Music (i), (ii)\n\nHanson, Howard (i), (ii), (iii)\n\nHarburg, E.Y. (Yip) (i)\n\nHargail Music Co. (Hargail Records) (i), (ii), (iii), (iv), (v), (vi), (vii)\n\nHarmon, Charles (Charlie) (i), (ii) n38, (iii)\n\nHarris, Roy (i), (ii) n74, (iii), (iv), (v), (vi), (vii)\n\n_Ode to Truth_ (i)\n\nSymphony No. 3 (i), (ii), (iii), (iv), (v)\n\nSymphony No. 5 (i)\n\n_Harvard Advocate_ (i), (ii)\n\nHarvard Scholarship Fund (i)\n\nHarvard University (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv) n41, (xv), (xvi) n59, (xvii) n73, (xviii), (xix), (xx) n77, (xxi) n149, (xxii), (xxiii) n3, (xxiv), (xxv), (xxvi), (xxvii) n39, (xxviii), (xxix), (xxx)\n\nHaydn, Joseph (i), (ii), (iii), (iv), (v), (vi), (vii) n13\n\n_Creation, The_ (i)\n\nMass in B flat (i)\n\nHayes, Helen (i)\n\nHayton, Lenny (i), (ii)\n\nHayward, Leland (i), (ii) n156\n\n_Hazel Flagg_ (musical) (i)\n\nHebrew Sheltering and Immigrant Aid Society (i)\n\nHebrew Union College (i)\n\nHeifetz, Jascha (i)\n\nHeinsheimer, Hans (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi)\n\nHellman, Lillian (i), (ii), (iii) n65, (iv), (v), (vi) n104, (vii), (viii) n23, (ix) n31, (x) n74\n\nHemingway, Ernest (i), (ii) n87, (iii) n21, (iv), (v)\n\n_For Whom the Bell Tolls_ (i) n77\n\nHendl, Walter (i), (ii) n89, (iii)\n\nHenried, Paul (i) n18\n\nHerbert, F. Hugh (i) n4\n\n_Kiss and Tell_ (i)\n\nHerbert, June (i)\n\nH\u00e9rold, Ferdinand (i)\n\nHeusen, Jimmy Van (i) n71\n\nHiller, Wendy (i)\n\nHillyer, Raphael (i) n18, (ii) n73, (iii). _See also_ Silverman, Raphael\n\nHindemith, Paul (i), (ii), (iii) n9, (iv), (v), (vi), (vii), (viii), (ix) n68, (x), (xi), (xii), (xiii)\n\n_Mathis der Maler_ Symphony (i)\n\nViolin Concerto (i)\n\nHitchcock, Alfred (i) n73, (ii) n78\n\n_Rope_ (i) n78\n\n_Shadow of a Doubt_ (i)\n\n_Strangers on a Train_ (i) n78, (ii) n154\n\nHocker, David (i)\n\nHolliday, Judy (i), (ii) n54, (iii), (iv), (v), (vi) n38, (vii), (viii) n131, (ix), (x), (xi) n104, (xii), (xiii), (xiv) n65\n\nHolm, Hanya (i)\n\nHonegger, Arthur (i)\n\n_Jeanne d'Arc au b\u00fbcher_ (i), (ii) n57, (iii) n12\n\nHoover, J. Edgar (i)\n\nHorne, Lena (i), (ii) n65, (iii), (iv)\n\nHorney, Karen (i), (ii), (iii)\n\nHorowitz, Richard (i)\n\nHouse Un-American Activities Committee (HUAC) (i) n70, (ii) n98, (iii) n105, (iv), (v), (vi) n60, (vii), (viii), (ix) n79, (x) n27, (xi) n28, (xii) n31\n\nHoyt, Howard (i), (ii) n131\n\nHughes, Langston (i) n104\n\nHunt, Marsha (Mrs. Robert Presnell) (i) n46\n\nHurok, Sol (i), (ii)\n\nHussey, Walter (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix)\n\nHuston, Walter (i) n62\n\nHutcheson, Ernest (i)\n\nHuxley, Aldous (i)\n\nHyatt, Dick (i)\n\nIan, Janis (i) n38, (ii), (iii)\n\n\"Society's Child\" (i) n38, (ii)\n\n_If the Shoe Fits_ (musical) (i)\n\nImbrie, Andrew (i)\n\nIrvington House, The (i)\n\nIsaacs, Stanley M. (i)\n\nIsherwood, Christopher (i) n107\n\nIsrael, Bob (i)\n\nIsrael Philharmonic Orchestra (i), (ii), (iii), (iv) n150, (v), (vi), (vii), (viii), (ix), (x)\n\nIstomin, Eugene (i)\n\n_It's Always Fair Weather_ (film) (i) n81, (ii) n28, (iii)\n\nIturbi, Jose (i), (ii)\n\nIves, Burl (i) n98, (ii) n104\n\nIves, Charles (i) n74, (ii), (iii), (iv), (v), (vi)\n\nJackson, Helen Hunt (i) n30\n\nJacobi, Frederick (i) n28\n\nJacoby, Herbert (i) n111\n\nJames, Henry (i)\n\nJefferson School of Social Science (i)\n\nJohns, Erik (i), (ii), (iii)\n\nJohnson, Lyndon B. (i)\n\nJohnson, Thor 215, 216m (i)\n\nJohnson, Van (i)\n\nJoint Anti-Fascist Refugee Committee (i)\n\nJones, Sandy (i)\n\nJoyce, Eileen (i)\n\nJoyce, James (i) n87\n\nJudson, Arthur (i), (ii), (iii), (iv)\n\nJudd, George (i), (ii), (iii), (iv)\n\nJuilliard School (i), (ii), (iii) n51, (iv) n73, (v) n90, (vi), (vii), (viii), (ix)\n\nKahn, Louis (i) n162\n\nKaltenborn, Rolf (i), (ii)\n\nKarajan, Herbert von (i), (ii), (iii), (iv)\n\nKarp, Frieda (i)\n\nKates, Helen (i.e. Helen Coates?) (i)\n\nKay, Hershy (i) n4, (ii), (iii) n54\n\nKaye, Nora (i)\n\nKazan, Elia (\"Gadg\") (i), (ii), (iii), (iv) n80, (v)\n\nKeiser, David (i)\n\nKell, Reginald (i)\n\nKelly, Ellsworth (i) n107\n\nKelly, Gene (i) n59, (ii), (iii), (iv), (v) n28, (vi)\n\nKelly, Kevin (i)\n\nKennedy, Jacqueline (i), (ii), (iii)\n\nKennedy, John F. (i), (ii), (iii) n28, (iv) n81, (v), (vi) n141\n\nKennedy, Robert F. (i), (ii), (iii) n15\n\nKennedy Center for the Performing Arts (i), (ii), (iii) n21, (iv) n48, (v), (vi), (vii), (viii)\n\nKent, Rockwell (i)\n\nKerouac, Jack (i) n5\n\nKert, Larry (i), (ii) n67, (iii), (iv)\n\nKidd, Michael (i)\n\nKirchner, Leon (i)\n\nKirkpatrick, John (i)\n\nKirstein, Lincoln (i)\n\nKleiber, Carlos (i), (ii) n16, (iii)\n\nKleiber, Erich (i) n169\n\nKlein, Kenneth (i), (ii)\n\nKletzki, Paul (i), (ii)\n\nKnussen, Oliver (i), (ii), (iii) n44\n\nKober, Arthur (i)\n\nKod\u00e1ly, Zolt\u00e1n (i)\n\n_Psalmus Hungaricus_ (i) n36\n\nKorngold, Erich Wolfgang (i), (ii) n18, (iii) n68\n\nKostal, Irwin (i) n4, (ii) n71, (iii)\n\nKostelanetz, Andr\u00e9 (i), (ii)\n\nKoussevitzky, Moshe (i), (ii)\n\nKoussevitzky, Olga (i), (ii), (iii), (iv), (v), (vi), (vii),\n\nKoussevitzky, Serge (often \"Kouss\" or \"Koussie\") (i), (ii), (iii) n21, (iv) n26, (v) n66, (vi), (vii), (viii), (ix), (x), (xi) n76, (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii) n37, (xxviii), (xxix), (xxx), (xxxi) n89, (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi) n115, (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii), (liv), (lv), (lvi), (lvii), (lviii), (lix), (lx), (lxi), (lxii), (lxiii), (lxiv) n86, (lxv), (lxvi), (lxvii), (lxviii), (lxix) n112, (lxx), (lxxi), (lxxii), (lxxiii) n153, (lxxiv), (lxxv) n160, (lxxvi), (lxxvii), (lxxviii), (lxxix), (lxxx), (lxxxi) n48, (lxxxii), (lxxxiii), (lxxxiv), (lxxxv), (lxxxvi), (lxxxvii) n82, (lxxxviii), (lxxxix), (xc) n73, (xci), (xcii), (xciii) n171, (xciv), (xcv), (xcvi)\n\nKoussevitzky Music Foundation (i), (ii), (iii), (iv), (v), (vi) n6\n\nKraft, Victor (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii)\n\nKraut, Harry (i), (ii), (iii)\n\nKrips, Josef (i) n114, (ii)\n\nKrueger, Karl (i) n129\n\nKubel\u00edk, Rafael (i)\n\nKubik, Gail (i) n74, (ii)\n\nKurnitz, Harry (i)\n\nKyle, Steve (i), (ii)\n\nLa Scala, Milan. _See_ Teatro alla Scala, Milan\n\nLadd, Alan (i)\n\nLamantia, Philip (i) n5\n\nLambert, Constant (i), (ii)\n\nLandowska, Wanda (i)\n\nLaTouche, John (i), (ii) n65, (iii), (iv), (v)\n\nLaufer, Beatrice (i) n79\n\nLaurents, Arthur (i) n42, (ii) n117, (iii) n132, (iv), (v), (vi) n65, (vii) n107, (viii), (ix), (x), (xi) n4, (xii), (xiii) n9, (xiv) n28, (xv), (xvi) n37, (xvii), (xviii), (xix)\n\nLawrence, Carol (i), (ii), (iii), (iv)\n\nLawrence, Marjorie (i), (ii)\n\nLawrence, Peter (i), (ii), (iii)\n\nLeBaron, William (i)\n\nLederman, Minna (i), (ii)\n\nLee, Gypsy Rose (i), (ii)\n\nLehmann, Rosamond (i)\n\nLeichentritt, Hugo (i)\n\nLeinsdorf, Erich (i), (ii)\n\nLenya, Lotte (i) n11, (ii)\n\nLerner, Alan Jay (i), (ii), (iii), (iv), (v), (vi), (vii) n23\n\nLessard, John (i)\n\nLester, Edwin (i)\n\nLevant, Oscar (i)\n\nLevinson, Gerald (i)\n\nLevit, Herschel, Janice and Lois (i)\n\nLewandowsky, Louis (i)\n\nLewis, Frank (i), (ii) n69\n\nLieberson, Brigitta (i), (ii) n14, (iii) n154\n\nLieberson, Goddard (i) n38, (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix) n14\n\nLipman, Maureen (i)\n\nLissfelt, J. Fred (i)\n\nLittle Red Schoolhouse (i), (ii), (iii)\n\nLloyd, Norman (i) n25\n\nLoesser, Frank (i), (ii), (iii) n156\n\nLoewe, Frederick (i), (ii) n38\n\nLogan, Joshua (i)\n\nLollobrigida, Gina (i), (ii), (iii)\n\nLowe, Jack (i)\n\nLongy, Ren\u00e9e. _See_ Miquelle, Ren\u00e9e Longy\n\nLoriod, Yvonne (i), (ii), (iii)\n\nLubell, Bob (i)\n\nLubitsch, Ernst (i)\n\nLudwig, Christa (i), (ii), (iii) n64, (iv)\n\nLynn, Vera (i)\n\nLyons, Leonard (i)\n\nMcCarthy, Joseph (i), (ii), (iii)\n\nMcClure, John (i) n90, (ii) n93, (iii), (iv)\n\nMcDonald, Harl (i)\n\nMcDonald, James (i), (ii)\n\nMacDowell Colony (i), (ii), (iii)\n\nMacDowell, Edward (i) n1\n\nMacDowell, Marian (i)\n\nMcHose, Allen (i)\n\nMcInerney, James (i), (ii)\n\nMcLerie, Allyn Ann (second wife of Adolph Green) (i), (ii), (iii)\n\nMcPhee, Colin (i), (ii)\n\nMa, Yo-Yo (i), (ii)\n\nMaazel, Lorin (i)\n\nMahler, Alma (i), (ii) n169\n\nMahler, Gustav (i), (ii), (iii), (iv), (v), (vi), (vii) n32, (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii)\n\n_Lied von der Erde, Das_ (i), (ii) n23\n\nSymphony No. 2 (\"Resurrection\") (i) n122, (ii), (iii), (iv), (v) n114, (vi), (vii) n23, (viii), (ix)\n\nSymphony No. 3 (i) n23, (ii)\n\nSymphony No. 4 (i) n27\n\nSymphony No. 5 (i), (ii) n166, (iii), (iv), (v), (vi) n27, (vii) n42\n\nSymphony No. 7 (i), (ii) n39, (iii)\n\nSymphony No. 8 (i)\n\nSymphony No. 9 (i), (ii), (iii)\n\nMalipiero, Gian Francesco (i)\n\nMaltz, Albert (i), (ii)\n\nMann, Thomas (i)\n\nManne, Shelly (i) n78, (ii)\n\nMarcus, Leonard (i), (ii) n36\n\nMarcuse, Barbara (i) n172, (ii), (iii), (iv), (v)\n\nMarcuse, Philip (i), (ii), (iii), (iv), (v), (vi), (vii)\n\nMargolies, Henry (i), (ii)\n\nMarkevitch, Igor (i), (ii), (iii)\n\n_Icare_ (i)\n\n_Rebus_ (i)\n\nMartenot, Ginette (i), (ii)\n\nMartin, Mary (i)\n\nMarx, Harpo (i)\n\nMatthews, Tom (i) n21, (ii) n24\n\n_Me and Juliet_ (musical) (i) n59, (ii) n60\n\nMead, Margaret (i) n12\n\n_Male and Female_ (i)\n\nMeck, Nadezhda von (i), (ii), (iii)\n\nMendelssohn, Felix (i), (ii), (iii), (iv), (v), (vi)\n\nMenotti, Gian-Carlo (i), (ii) n94, (iii) n139, (iv), (v)\n\n_Saint of Bleecker Street, The_ (i), (ii)\n\nMenuhin, Yehudi (i) n63, (ii), (iii)\n\nMeredith, Burgess (i) n62\n\nMerrick, David (i)\n\nMerrill, Edys (i), (ii), (iii), (iv)\n\nMessiaen, Olivier (i), (ii), (iii), (iv), (v), (vi) n45\n\n_Chronochromie_ (i)\n\n_Trois Petites Liturgies de la Pr\u00e9sence Divine_ (i)\n\n_Turangal\u00eela-Symphonie_ (i), (ii), (iii)\n\nMetropolitan Opera House (i), (ii) n124, (iii) n150, (iv), (v) n105, (vi) n16, (vii) n60\n\nMetropolitan Synagogue of New York (i)\n\nMeyerson, Seymour (i), (ii), (iii), (iv)\n\nMiddleton, Jean (i), (ii), (iii), (iv)\n\nMignone, Francisco (i)\n\nMilhaud, Darius (i), (ii), (iii), (iv), (v), (vi) n33, (vii), (viii)\n\n_Cr\u00e9ation du Monde, La_ (i), (ii), (iii) n84, (iv), (v)\n\n_David_ (i), (ii)\n\nMiller, Abe (i)\n\nMiller, Arthur (i), (ii)\n\n_Crucible, The_ (i)\n\nMills, Charles (i)\n\nMiquelle, Georges (i) n129\n\nMiquelle, Ren\u00e9e Longy (sometimes \"Re La Mi\") (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii) n45, (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n160, (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi) n129, (xxvii), (xxviii)\n\nMiranda, Carmen (i) n131\n\nMirisch, Walter (i) n82\n\nMitropoulos, Dimitri (i), (ii) n16, (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi) n40, (xvii) n140, (xviii), (xix), (xx), (xxi), (xxii) n40, (xxiii), (xxiv), (xxv), (xxvi), (xxvii) n36, (xxviii) n39, (xxix), (xxx) n94, (xxxi) n114, (xxxii), (xxxiii), (xxxiv)\n\nMishkan Tefila (Synagogue) (i), (ii), (iii)\n\nMontealegre Cohn, Felicia (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi). _See also_ Bernstein, Felicia\n\nMonteux, Pierre (i) n94, (ii), (iii)\n\nMontgomery, George (i)\n\nMontreal Symphony Orchestra (i)\n\nMoore, Henry (i)\n\nMorel, Jean (i), (ii)\n\nMorgan, Brewster (i) n62\n\nMorley, Robert (i)\n\nMoross, Jerome (i), (ii)\n\nMorris, Harold (i) n28\n\nMorris, Marketa (\"The Frau\") (i), (ii), (iii) n35, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii) n130\n\nMorrison, Paul (i)\n\nMoseley, Carlos (i) n8, (ii)\n\nMostel, Zero (i) n104\n\nMozart, Wolfgang Amadeus (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix)\n\n_Magic Flute_ Overture (i) n112\n\nPiano Concerto in C, K467 (i)\n\nRondo in A minor (i)\n\nSymphony No. 36 (\"Linz\") (i), (ii)\n\nSymphony No. 39 (i)\n\nMunch, Charles (i), (ii), (iii), (iv), (v) n14, (vi), (vii) n124\n\nMusicraft (i), (ii)\n\n_My Fair Lady_ (musical) (i), (ii) n74, (iii) n128, (iv) n38\n\nMyers, Debs (i)\n\nNational Association for the Advancement of Colored People (i) n104, (ii)\n\nNational Council of American-Soviet Friendship (i)\n\nNational Federation of Infantile Paralysis (i)\n\nNational Negro Congress (i)\n\nNational Urban League (i)\n\nNational War Fund (i)\n\nNational Youth Administration (NYA) (i), (ii), (iii)\n\nNell, Ren\u00e9e (i), (ii), (iii)\n\nNeveu, Ginette (i)\n\nNew York City Symphony (sometimes \"City Center\") (i) n28, (ii) n69, (iii) n70, (iv) n71, (v), (vi), (vii), (viii), (ix), (x), (xi)\n\nNew York Guild for the Jewish Blind (i)\n\nNew York Philharmonic Orchestra (formerly New York Philarmonic-Symphony Orchestra; often just called \"the Philharmonic\") (i) n62, (ii) n66, (iii) n72, (iv), (v) n93, (vi) n107, (vii), (viii), (ix), (x) n24, (xi) n40, (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix) n125, (xx), (xxi), (xxii), (xxiii) n140, (xxiv), (xxv), (xxvi) n159, (xxvii), (xxviii) n33, (xxix) n41, (xxx) n41, (xxxi) n46, (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi) n51, (xxxvii) n62, (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii) n39, (xliv), (xlv), (xlvi) n44, (xlvii), (xlviii), (xlix), (l), (li), (lii), (liii) n12, (liv) n16, (lv), (lvi), (lvii), (lviii), (lix), (lx) n75, (lxi) nn76\u20137, (lxii) n84, (lxiii) n94, (lxiv), (lxv), (lxvi) n106, (lxvii) n109, (lxviii), (lxix), (lxx), (lxxi) n147, (lxxii), (lxxiii), (lxxiv) n160, (lxxv) n166, (lxxvi), (lxxvii) n177, (lxxviii), (lxxix) n6, (lxxx), (lxxxi) n28, (lxxxii) n42, (lxxxiii) n18, (lxxxiv), (lxxxv) n64, (lxxxvi)\n\nNewman, Harold (i), (ii)\n\nNewman, Phyllis (third wife of Adolph Green) (i)\n\nNielsen, Carl (i)\n\nNielsen, Leslie (i) n49\n\nNixon, Marni (i)\n\nNixon, Richard M. (i), (ii), (iii), (iv), (v), (vi) n66\n\nNorth, Alex (i)\n\nNorman, Edward A. (i)\n\nNorman, Ruth (i)\n\nNorton, Eliot (i)\n\nO'Donnell, Kenneth (i)\n\nOdets, Clifford (i)\n\nOffenbach, Jacques (i)\n\n_Omnibus_ (television series) (i) n47, (ii), (iii), (iv), (v) n33, (vi), (vii)\n\nOppenheim, David (i), (ii), (iii) n21, (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi) n85, (xvii), (xviii), (xix), (xx), (xxi), (xxii) n122, (xxiii), (xxiv), (xxv), (xxvi), (xxvii) n157, (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii) n130, (xlviii) n142, (xlix), (l), (li), (lii), (liii), (liv) n77, (lv), (lvi) n153, (lvii)\n\nOppenheim, Ellen. _See_ Adler, Ellen\n\nOppenheim, Judy. _See_ Holliday, Judy\n\nOrder of the Purple Heart (i)\n\nOrmandy, Eugene (i) n159, (ii), (iii), (iv)\n\nOtterloo, Willem van (i)\n\nOverbury, Michael (i)\n\nOzawa, Seiji (i), (ii) n154, (iii) n13, (iv) n44\n\nPachmann, Vladimir de (i)\n\nParay, Paul (i)\n\nParker, Charlie (i)\n\nParker, Dorothy (i), (ii) n104, (iii) n45, (iv) n120\n\nPasternak, Boris (i), (ii), (iii), (iv)\n\n_Doctor Zhivago_ (i), (ii) n51\n\nPeck, Gregory (i)\n\nPerlberg, William (i)\n\nPerle, George (i) n111\n\nPerle, Shirley Gabis Rhoads. _See_ Gabis, Shirley\n\nPer\u00f3n, Eva Mar\u00eda Duarte de (i), (ii)\n\nPersichetti, Vincent (i) n79\n\nPetrillo, James (i), (ii)\n\nPhiladelphia Orchestra (i) n111, (ii), (iii) n159, (iv)\n\nPiatigorsky, Gregor (i)\n\nPiston, Walter (i), (ii), (iii) n66, (iv), (v) n123, (vi) n152, (vii), (viii), (ix), (x) n176\n\nSymphony No. 4 (i)\n\nPlanned Parenthood campaign (i)\n\nPolignac, Marie-Blanche de (i), (ii), (iii) n36\n\nPollock, Jackson (i) n107\n\nPorter, Cole (i) n74, (ii), (iii) n60, (iv) n150, (v)\n\nPorter, Quincy (i) n74\n\nPosselt, Ruth (i)\n\nPoulenc, Francis (i) n147, (ii) n36, (iii), (iv), (v), (vi)\n\n_Biches, Les_ (i)\n\nConcerto for Two Pianos (i) n62\n\n_Dialogues des Carm\u00e9lites_ (i)\n\n_Gloria_ (i) n62, (ii)\n\n_Mamelles de Tir\u00e9sias_ (i)\n\n_Sept R\u00e9pons de T\u00e9n\u00e8bres_ (i) n62, (ii)\n\nSonata for Clarinet and Piano (i) n62\n\nPower, Tyrone (i) n46\n\nPrall, David (i), (ii), (iii), (iv)\n\nPrall, Margaret (i), (ii)\n\nPresnell, Robert (i)\n\nPressler, Menahem (i), (ii)\n\nPrevin, Andr\u00e9 (i), (ii), (iii) n78, (iv), (v), (vi)\n\nPrince, Harold (i) n65, (ii), (iii)\n\nProgressive Citizens of America (PCA) (i)\n\nProkofiev, Sergei (i), (ii) n68, (iii) n111, (iv) n140\n\nPurcell, Henry (i)\n\n_Pursuit of Happiness, The_ (radio show) (i), (ii)\n\nQuashen, Ben (i)\n\nRains, Claude (i) n18\n\nRaksin, David (i) n73\n\nRamin, Gloria (n\u00e9e Gloria Breit) (i) n95, (ii), (iii), (iv)\n\nRamin, Sid (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi) n71, (xii), (xiii), (xiv) n38, (xv), (xvi) n58, (xvii), (xviii), (xix)\n\nRampal, Jean-Pierre (i)\n\nRapper, Irving (i), (ii), (iii) n18, (iv)\n\n_Deception_ (i) n18, (ii)\n\nRavel, Maurice (i)\n\n_Alborada del gracioso_ (i) n40\n\n_Bolero_ (i), (ii), (iii)\n\n_Daphnis et Chlo\u00e9_ (i), (ii) n71\n\nPiano Concerto for the Left Hand (i) n66, (ii)\n\nPiano Concerto in G major (i), (ii), (iii) n159, (iv), (v), (vi) n53, (vii), (viii), (ix) n121, (x) n123, (xi), (xii) n111, (xiii), (xiv), (xv), (xvi)\n\n_Sh\u00e9h\u00e9razade_ (i) n40\n\n_Tombeau de Couperin, Le_ (i), (ii) n69, (iii) n12\n\n_Valse, La_ (i) n3\n\nRCA Records (Victor) (i), (ii), (iii), (iv)\n\nRead, Gardner (i)\n\nReagan, Nancy (i)\n\nReagan, Ronald (i), (ii)\n\n_Red Channels: The Report of Communist Influence in Radio and Television_ (i) n104, (ii), (iii), (iv)\n\nReichold, Henry (i) n129\n\nReiner, Ethel Lindner (i), (ii), (iii)\n\nReiner, Fritz (i), (ii) n21, (iii), (iv), (v), (vi) n66, (vii) n76, (viii), (ix), (x), (xi) n31, (xii) n37, (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii) n94, (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii)\n\nReitell, Elizabeth (first wife of Adolph Green) (i), (ii), (iii)\n\nResnik, Regina (i) n57, (ii), (iii)\n\nRevueltas, Silvestre (i)\n\nReveuers, The (i), (ii), (iii) n43, (iv) n46, (v) n54, (vi) n55, (vii), (viii) n65, (ix), (x) n71, (xi), (xii) n16, (xiii) n21, (xiv) n38, (xv), (xvi), (xvii), (xviii)\n\nReynolds, Quentin (i), (ii)\n\nRiegger, Wallingford (i)\n\nRiley, Peggy (i)\n\nRimsky-Korsakov\n\n_Scheherazade_ (i), (ii), (iii)\n\nRiobamba Club (i), (ii) n80\n\nRittmann, Trude (i)\n\nRiver, Walter Leslie (i)\n\nRiverdale Children's Camp (i)\n\nRiverside Children's Association (i)\n\nRobbins, Jerome (i), (ii) n21, (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii) n11, (xiv), (xv), (xvi), (xvii) n111, (xviii), (xix), (xx), (xxi) n173, (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii) n4, (xxviii), (xxix), (xxx) n28, (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii) n30, (xxxviii), (xxxix), (xl) n128, (xli) n164, (xlii), (xliii), (xliv), (xlv), (xlvi) n4, (xlvii), (xlviii), (xlix), (l), (li) n40, (lii)\n\nRobeson, Paul (i)\n\nRobinson, Earl (i)\n\nRoddy, Joe (Joseph) (i), (ii), (iii), (iv), (v)\n\nRodgers, Mary (i), (ii), (iii), (iv), (v)\n\nRodgers, Richard (i), (ii), (iii), (iv), (v) n60, (vi) n74, (vii) n40, (viii) n150, (ix)\n\n\"Nobody's Heart Belongs to Me\" (from _By Jupiter_ ) (i), n26\n\nRodzinski (sometimes \"Rodzinsky\"), Artur (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii) n41, (xviii), (xix)\n\nRogers, Bernard (i) n66, (ii) n49, (iii), (iv)\n\nRogers, Buddy (Charles) (i)\n\nRogers, Leslie (i)\n\nRoll, Eddie (i)\n\nRomberg, Sigmund (i) n115, (ii)\n\nRome, Harold (i)\n\nRomney, Richard Adams (\"Twig\") (i), (ii), (iii), (iv), (v), (vi), (vii) n16\n\nRorem, Ned (i) n142, (ii), (iii)\n\nRosand, Aaron (i) n58\n\nRosenthal, Jean (i) n4\n\nRosenthal, Manuel (i)\n\nRoss, Alex (i) n1\n\nRoss, Alvin (i)\n\nRoss, Lynn (i)\n\nRossen, Robert (i)\n\nRoth, Charles (the \"Black Fairy\") (i), (ii), (iii), (iv), (v)\n\nRothko, Mark (i) n107\n\nRouault, Georges (i)\n\nR\u00f3zsa, Mikl\u00f3s\n\n_Spellbound_ (i)\n\nRubinstein, Arthur (i), (ii), (iii), (iv) n114\n\nRudel, Julius (i)\n\nRuggles, Carl (i)\n\nRussell, Anna (i)\n\nRussell, Bertrand (i)\n\nRussell, Rosalind (i) n75, (ii)\n\nRusso, William (i)\n\nSt. Louis Jazz Society (i)\n\nSt. Louis Symphony Orchestra (i) n90, (ii), (iii)\n\nSabata, Victor de (i), (ii), (iii), (iv), (v), (vi)\n\nSaidenberg, Daniel (i), (ii), (iii), (iv), (v)\n\nSaletan, Rhoda (i) n95, (ii), (iii)\n\nSalvation Army, The (i)\n\nSanrom\u00e1, Jes\u00fas Maria (i), (ii)\n\nSapiro, Aaron (i)\n\nSaputelli, Bill (i)\n\nShermerhorn, Kenneth (i)\n\nSchippers, Thomas (i), (ii), (iii)\n\nSchlick, Frederick (i)\n\nSchlick, Robert (i)\n\nSchoenberg, Arnold (i), (ii), (iii), (iv), (v), (vi), (vii) n74, (viii) n169\n\n_Verkl\u00e4rte Nacht_ (i), (ii)\n\nSchreiber, Lew (i)\n\nSchubert, Franz (i), (ii), (iii)\n\nSymphony No. 7 (i.e. \"Great\" C major Symphony) (i) n41, (ii), (iii) n112\n\nSchuller, Gunther (i), (ii)\n\nSchumacher, Thomas (i)\n\nSchuman, Frankie (i), (ii), (iii), (iv), (v), (vi), (vii)\n\nSchuman, William (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi), (xvii), (xviii), (xix), (xx), (xxi), (xxii), (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii)\n\n_American Festival Overture_ (i), (ii), (iii)\n\n_Newsreel_ (i), (ii)\n\nSymphony No. 3 (i), (ii), (iii)\n\nSymphony No. 6 (i)\n\n_Undertow_ (i)\n\nSchumann, Robert (i), (ii), (iii), (iv), (v), (vi), (vii)\n\n_Manfred_ Overture (i) n3\n\nSymphony No. 2 (i), (ii) n123, (iii) n69\n\nSymphony No. 3 (\"Rhenish\") (i)\n\nSymphony No. 4 (i), (ii), (iii) n13\n\nSchwartz, Arthur (i), (ii) n173, (iii) n20\n\nSchwartz, Ethel (i)\n\nSchwieger, Hans (i), (ii)\n\nScriabin, Alexander (i), (ii)\n\nSeeger, Pete (i) n104\n\nSeldes, Marian (i) n46\n\nSelznick, David (i)\n\nSerkin, Peter (i) n44\n\nSerkin, Rudolf (i), (ii), (iii)\n\nSessions, Roger (i), (ii) n66, (iii) n68, (iv), (v), (vi)\n\nShamalter, Gene (i)\n\nShapero, Harold (i), (ii), (iii), (iv), (v) n101\n\nShapirra, Elyakum (i)\n\nSharaff, Irene (i), (ii) n4\n\nShaw, Artie (i) n104\n\nShaw, Robert (i), (ii), (iii) n24, (iv), (v)\n\nShepherd, Arthur (i)\n\nShore, Dinah (i)\n\nShostakovich, Dmitri (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix)\n\nPiano Concerto No. 2 (i) n4\n\nSymphony No. 1 (i)\n\nSymphony No. 5 (i), (ii), (iii), (iv) n17, (v), (vi) n42, (vii), (viii)\n\nSymphony No. 7 (i)\n\nSymphony No. 13 (i) n28\n\nSibelius, Jean (i), (ii), (iii) n151, (iv), (v)\n\nSilverman, Raphael (i), (ii) n73. _See also_ Hillyer, Raphael\n\nSimon, Henry (i), (ii)\n\nSinatra, Frank (i), (ii), (iii) n59, (iv), (v) n155, (vi), (vii) n81, (viii), (ix)\n\nSinger, George (i), (ii)\n\n_Singin' in the Rain_ (film) (i) n43, (ii) n81, (iii) n93, (iv) n29\n\nSirmay, Albert (i), (ii), (iii)\n\nSitwell, Edith (i)\n\nSitwell, Osbert (i) n107\n\nSmallens, Alexander (i), (ii), (iii), (iv)\n\nSmith, Betty (i)\n\nSmith, Oliver (i), (ii) n88, (iii), (iv) n5, (v) n47, (vi), (vii), (viii), (ix) n71, (x), (xi) n83, (xii), (xiii), (xiv), (xv), (xvi), (xvii)\n\nSokolov, Anna (i), (ii)\n\nSolomon, Cyril (Chuck) (i), (ii), (iii), (iv), (v), (vi), (vii)\n\nSolomon, Izler (i)\n\nSolti, Georg (i), (ii), (iii), (iv) n152, (v)\n\nSondheim, Stephen (i) n117, (ii), (iii), (iv) n6, (v), (vi) n28, (vii), (viii), (ix), (x) n63, (xi), (xii), (xiii) n69, (xiv) n63, (xv), (xvi), (xvii), (xviii) nn161, (xix), (xx), (xxi), (xxii) n24, (xxiii), (xxiv), (xxv), (xxvi), (xxvii), (xxviii), (xxix), (xxx), (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi)\n\n_Assassins_ (i) n65\n\n_Follies_ (i) n156, (ii)\n\n_Gypsy_ (lyrics) (i) n65\n\n_Pacific Overtures_ (i) n48, (ii)n49\n\n_Sunday in the Park with George_ (i), (ii) n25\n\n_South Pacific_ (musical) (i) n60, (ii), (iii) n74, (iv) n150\n\nSpeyer, Kiki Jacqueline] [(i), (ii), (iii), (iv), (v), (vi), (vii)\n\nSpeyer, Louis (i) n94\n\nSpiegel, Mildred (i), (ii), (iii), (iv) n18, (v), (vi)\n\nSpivacke, Harold (i), (ii)\n\nSpivakovsky, Tossy (i)\n\nState Department. _See_ US Department of State\n\nStegall, Mildred (i)\n\nStein, Gertrude (i)\n\nSteinberg, Ben (i), (ii)\n\nSteinberg, William (i) n82, (ii)\n\nSteiner, Max (i)\n\nStern, Isaac (i) n10, (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix)\n\nStern, Yossi (i), (ii), (iii)\n\nStevens, Roger L. (i)\n\nStockhausen, Karlheinz (i), (ii), (iii), (iv), (v)\n\nSt\u00f6hr, Richard (i), (ii)\n\nStokowski, Leopold (i) n68, (ii), (iii), (iv), (v), (vi)\n\nStoltzman, Richard (i)\n\nStothart, Herbert (i)\n\nStrauss, Noel (i)\n\nStrauss, Richard (i), (ii) n68, (iii), (iv), (v)\n\n_Don Juan_ (i), (ii)\n\n_Don Quixote_ (i), (ii)\n\n_Rosenkavalier, Der_ (i), (ii), (iii), (iv), (v), (vi), (vii), (viii)\n\n_Till Eulenspiegel_ (i)\n\nStravinsky, Igor (i), (ii), (iii), (iv)\n\n_Agon_ (i)\n\n_Dumbarton Oaks_ (i)\n\n_Firebird_ (i), (ii), (iii), (iv), (v)\n\n_Flood, The_ (i) nn92\u20133\n\n_Histoire du Soldat_ (i), (ii)\n\n_Noah_ (early title for _The Flood_ ) (i)\n\n_Noces, Les_ (i)\n\n_Petrouchka_ (i), (ii), (iii)\n\n_Sacre du Printemps, Le_ ( _The Rite of Spring_ ) (i), (ii), (iii) n112, (iv), (v)\n\n_Symphony of Psalms_ (i), (ii)\n\n_Symphony in C_ (i) n68, (ii)\n\nStresemann, Wolfgang?] [(i)\n\nStroheim, Erich von (i) n62\n\nStyne, Jule (i) n21, (ii) n156, (iii) n72, (iv), (v), (vi), (vii)\n\nSulzer, Salomon (i)\n\n_Swan Song_ (Ben Hecht and Charles MacArthur) (i), (ii), (iii) n60\n\nSzathmary, Arthur (i) n44, (ii) n56\n\nSzell, George (i) n38, (ii), (iii) n114, (iv)\n\nSzigeti, Joseph (i), (ii), (iii) n155, (iv) n17, (v), (vi), (vii), (viii)\n\n_Take Me Out to the Ball Game_ (film) (i) n155\n\nTalma, Louise (i)\n\nTanglewood (Berkshire Music Center) (i), (ii) n21, (iii), (iv) n75, (v), (vi), (vii), (viii) n107, (ix), (x), (xi), (xii), (xiii), (xiv), (xv) n37, (xvi) nn38\u20139, (xvii), (xviii) n77, (xix) n89, (xx), (xxi), (xxii), (xxiii) n45, (xxiv), (xxv), (xxvi), (xxvii) n59, (xxviii), (xxix), (xxx) n72, (xxxi), (xxxii), (xxxiii), (xxxiv), (xxxv), (xxxvi), (xxxvii), (xxxviii), (xxxix), (xl), (xli), (xlii), (xliii), (xliv), (xlv), (xlvi), (xlvii), (xlviii), (xlix) n79, (l) n93, (li) (\"Tanglefoot\"), (lii), (liii), (liv) n63, (lv), (lvi), (lvii) n174, (lviii), (lix), (lx), (lxi) n14, (lxii), (lxiii), (lxiv), (lxv), (lxvi) n49, (lxvii) n56, (lxviii) n57, (lxix), (lxx)\n\nTartini, Giuseppe (i)\n\nTaschner, Gerhard (i) n123\n\nTaylor, Davidson (i)\n\nTaylor, Elizabeth (i)\n\nTaylor, Frances Elizabeth (i)\n\nTchaikovsky, Peter Ilyich (sometimes spelt \"Chaikovsky\" by Bernstein) (i), (ii) n15, (iii), (iv) n74, (v), (vi) n69, (vii) n109, (viii) n112, (ix) n140, (x), (xi)\n\nTchelitchew, Pavel (i) n93, (ii) n107\n\nTeatro alla Scala, Milan (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x), (xi), (xii), (xiii), (xiv), (xv), (xvi)\n\nTheatre Guild (i)\n\nThiede, Alexander (i), (ii)\n\nTompkins, William F. (i)\n\nThompson, Randall (i), (ii), (iii) n21, (iv), (v), (vi) n66, (vii), (viii), (ix), (x), (xi), (xii) n94, (xiii), (xiv), (xv), (xvi)\n\nSymphony No. 2 (i), (ii) n137, (iii), (iv), (v), (vi) n174, (vii) n6\n\nThomson, Virgil (i), (ii), (iii), (iv), (v), (vi) n87, (vii), (viii), (ix) n93, (x), (xi) n107, (xii), (xiii), (xiv), (xv) n102, (xvi), (xvii) n94, (xviii), (xix)\n\n_Four Saints in Three Acts_ (i)\n\n_Mother of Us All, The_ (i), (ii) n1\n\nTortelier, Paul (i)\n\nToscanini, Arturo (i), (ii), (iii), (iv) n64, (v)\n\nTourel, Jennie (i), (ii) n40, (iii), (iv), (v), (vi), (vii) n62, (viii) n77, (ix)\n\n_Tree Grows in Brooklyn, A_ (musical) (i), (ii) n173, (iii) n20\n\nTulin, Mischa (i)\n\nTureck, Rosalyn (i)\n\nTurner, Lana (i) n69, (ii)\n\nTuttle, Frank (i)\n\nTuvim, Judith. _See_ Holliday, Judy\n\nUnited Jewish Appeal (i)\n\nUnited Negro and Allied Veterans of America (i)\n\nUnited Unitarian Appeal (i)\n\nUnited World Federalists (i)\n\nUrquhart, Craig (i)\n\nUS Department of State (State Department) (i) n46, (ii), (iii) n64, (iv), (v), (vi), (vii), (viii)\n\nUS Department of Veterans Affairs (VA) (i), (ii)\n\nVal\u00e9ry, Fran\u00e7ois (i), (ii), (iii), (iv) n36\n\nVan Druten, John (i) n107\n\nVar\u00e8se, Edgard (i), (ii)\n\n_Arcana_ (i)\n\nVaughan Williams, Ralph (i) n33\n\nSymphony No. 4 (i)\n\nVengerova, Isabella (i), (ii), (iii), (iv), (v)\n\nVerson, Cara (i), (ii)\n\nVeterans of the Abraham Lincoln Brigade (i)\n\nVickers, Sylvia (i) n36\n\nVienna Philharmonic Orchestra (i) n33, (ii) n122, (iii), (iv), (v)\n\nVienna Symphony Orchestra (i) n123, (ii)\n\nVilla-Lobos, Heitor (i)\n\nVisconti, Luchino (i), (ii)\n\nVivaldi, Antonio (i), (ii) n109, (iii) n112\n\nWager, Michael (i) n8, (ii) n148\n\nWagner, Richard (i), (ii), (iii), (iv)\n\n_G\u00f6tterd\u00e4mmerung_ (i)\n\n_Meistersinger_ Prelude (i)\n\n_Tannh\u00e4user_ Overture (i) n6\n\n_Tristan und Isolde_ (i), (ii), (iii), (iv)\n\nWallenstein, Alfred (i), (ii)\n\nWalter, Bruno (i) n66, (ii), (iii) n38, (iv), (v), (vi) n94\n\nWalton, William (i)\n\n_Belshazzar's Feast_ (i) n36\n\n_Portsmouth Point_ (i)\n\nWang, Cilli (i)\n\nWarfield, William (i)\n\nWarner Bros. (music publisher) (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix), (x)\n\nWarren, Harry (i)\n\nWasterlain, Viola (i)\n\nWatts, Ardean (i)\n\nWaxman, Franz (i) n34, (ii)\n\nWeatherly, Robert (i), (ii) n90\n\nWeber, Ben (i)\n\nWebern, Anton (i), (ii) n39, (iii), (iv), (v) n4, (vi) n74, (vii)\n\nSymphony Op. 20 (i)\n\nWeil, Robert (i), (ii)\n\nWeill, Kurt (i), (ii) n90, (iii), (iv), (v) n60, (vi) n63, (vii) n38, (viii) n41, (ix) n56\n\n_Lady in the Dark_ (i), (ii) n60, (iii) n41\n\n_One Touch of Venus_ (i), (ii) n90, (iii) n41\n\n_Threepenny Opera, The_ (i) n11, (ii), (iii)\n\nWeissel, William (Willie) (i)\n\nWeizmann, Chaim (i), (ii)\n\nWelch, Carrington (i), (ii) n103\n\nWesselhoeft, Dr. (i)\n\nWhite, Katherine (i), (ii)\n\nWhiteman, Paul (i)\n\nWhittemore, Arthur (i)\n\nWilder, Alec (i)\n\nWilder, Billy (i)\n\nWilder, Thornton (i) n73, (ii) n1, (iii), (iv), (v), (vi) n128, (vii)\n\nWilliams, Susan (i)\n\nWise, Robert (Bob) (i), (ii)\n\nWittgenstein, Paul (i)\n\nWobisch, Helmut (i), (ii)\n\nWolf, Kathryn (i), (ii)\n\nWorks Progress Administration (WPA) (i), (ii), (iii)\n\nWyler, William (i) n62\n\nXenakis, Iannis (i), (ii)\n\nYaddo (artists' community), Saratoga Springs, NY (i), (ii), (iii), (iv), (v), (vi)\n\nYevtushenko, Yevgeny (i)\n\nYoung Men's Christian Association (YMCA) (i)\n\nYoung People's Concerts (i), (ii) n39, (iii), (iv) n40, (v), (vi) n57\n\nZanuck, Darryl (i)\n\nZeffirelli, Franco (i), (ii) n60\n\nZirato, Bruno (i), (ii), (iii), (iv)\n\nZorina, Vera. _See_ Lieberson, Brigitta\n\nZucker, Mildred. _See_ Spiegel, Mildred\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nCopyright \u00a9 2017 by Josh Barkan\n\nAll rights reserved.\n\nPublished in the United States by Hogarth, an imprint of the Crown Publishing Group, a division of Penguin Random House LLC, New York.\n\ncrownpublishing.com\n\nHOGARTH is a trademark of the Random House Group Limited, and the H colophon is a trademark of Penguin Random House LLC.\n\n\"The Kidnapping\" was first published as the winner of the Lightship International Short Story Prize 2013, in _Lightship Anthology 3,_ Alma Books, U.K., 2014.\n\nLibrary of Congress Cataloging-in-Publication Data is available upon request.\n\nISBN 9781101906293\n\nEbook ISBN 9781101906309\n\nFrontispiece photograph by Frederick Bertrand\n\nCover design by Elena Giavaldi\n\nCover illustration by Alexander Henry Fabrics\n\nv4.1\n\nep\nFOR MARIANA\n\n# Contents\n\nCover\n\nOther Titles\n\nTitle Page\n\nCopyright\n\nDedication\n\nThe Chef and El Chapo\n\nThe God of Common Names\n\nI Want to Live\n\nAcapulco\n\nThe Kidnapping\n\nThe Plastic Surgeon\n\nThe Sharpshooter\n\nThe Painting Professor\n\nThe American Journalist\n\nEverything Else Is Going to Be Fine\n\nThe Prison Breakout\n\nThe Escape from Mexico\n\nAcknowledgments\n\nAbout the Author\n\n# THE CHEF AND EL CHAPO\n\nHow the hell \"El Chapo\" Guzm\u00e1n chose my restaurant to come into, I'll never know. It was just like the stunt he's done in a few other cities\u2014Nuevo Le\u00f3n and Culiac\u00e1n. Guzm\u00e1n\u2014\"Shorty\"\u2014it was him, with all his narco clothing. He had on a baseball cap with some of that digitalized camouflage the U.S. Army invented for Iraq, and a beige down parka. It was one of those cold days in June, after the rainy season has started, and the most badass narco in the country must have felt just a touch of a chill. Crazy! In my restaurant. With fifteen bodyguards swarming around him. The guards came in first. They all had AK-47s swinging in their arms. They came in fast and polite, rushing past the ma\u00eetre d'. The leader of the guards, a tall guy with a neatly trimmed thin mustache and a diamond earring, swooped into the center of the dining room and yelled out, \"The Boss will be coming soon. Everyone give us your purses and cell phones and continue with your meal. Nobody leaves before The Boss is done. If you cooperate, everything will be fine. You'll get your purses and phones returned when The Boss is done. Leave your check. The Boss will pay for your meal.\"\n\nI knew Shorty was short, of course, but when he came in, it was surprising to see just how small the biggest drug kingpin was. He walked in quickly, like he knew where he was going. He turned to the first table, to the left, and introduced himself. He removed his cap and said in polite Spanish, \"Hello, my name is El Chapo Guzm\u00e1n. Nice to meet you.\" He smiled and extended his hand to shake with one of the customers, an old man in a blue blazer who, fortunately, had the presence of mind to shake back. The customer looked like he'd just seen a ghost.\n\nGuzm\u00e1n went from table to table shaking hands like a politician asking for votes of approval. But the way he smiled, with a permanent grin and his eyes a little too focused on the clients, he seemed to be saying: You will like me! I'm not so fucking bad, right? After he reached the last table, he chuckled, cracking himself up. He was the most badass jokester in the world. He was the biggest gentleman, extending his hand of courtesy to every diner, after he'd killed hundreds.\n\nEvery day, I drive from the neighborhood of Santa Fe, in Mexico City, to the restaurant, and I pass guys selling tabloid newspapers in the morning. They run up and down the street, at the stoplights, trying to find clients, waving papers in the air, and the covers always have a narco like Guzm\u00e1n with some gory detail like the bodies El Chapo dissolved in acid on a farm after he got pissed off at some other narcos, or like the photos of headless and handless bodies dumped in the middle of the streets of Veracruz. El Chapo killed the son of his brother-in-law. He's fifty-five and head of the Sinaloa Cartel, and he's one smart narco, because he not only escaped from prison\u2014with the help of dozens of people he bought off, pushed out of a maximum-security prison in a laundry cart\u2014he's managed to live to the age of fifty-five when most narcos never make it even close to being a grandpa.\n\nEveryone in Mexico knows about him: how he married yet another young woman, some beauty queen, and how she had twins in a hospital in Los Angeles. How the guy controls all the cocaine, pot, and most of the meth and heroin that's going into the U.S. I've only been in Mexico two years, building the restaurant up, but anyone who's spent time down here knows the names of all these narcos like they're the heroes and devils of the soap operas that are on all day in every housewife's home and in every cantina.\n\nSo it didn't take a genius to know the guy who'd just walked into my restaurant was capable of killing me and every one of my clients, and I was the head chef.\n\n\u2014\n\nEl Chapo asked to be escorted to a private room, in back, where we sometimes have lunches for important business people. My restaurant is in the neighborhood of Polanco, on the border with the most expensive neighborhood of Las Lomas, where all the international banks are located. The food in my restaurant is a mix of French with new American cuisine\u2014meaning anything is OK, fusion with Asian touches, wasabi with bourbon crab, pork with chanterelle mushrooms in a ginger cream sauce with Beluga caviar sprinkled on top, arugula salad with truffle shavings and Cointreau sauce.\n\nI wake up early in the morning and go to the San Juan market, in the center of Mexico City, to buy the freshest produce I can find. The market looks typical, at first, in a wide concrete warehouse, but the stalls are full of the latest vegetables trucked in pickups from small farmers, and there are even a few Korean stands where you can find Asian vegetables that are less common in Mexico City. Fusion cooking has been the rage in the U.S. for thirty years, but in Mexico it's a new thing, so I've received more attention than a comparable chef would get in the U.S. That's one of the reasons I came to Mexico. A friend of mine, who was living in Mexico, came into a restaurant where I was the head chef in Pittsburgh, he tasted some cured duck breast I was preserving in the cellar of the restaurant, he slurped up the homemade vinegars we were using in the salad dressings and to pickle baby carrots and peas, and he told me I could be an instant hit in Mexico City.\n\nMy body is covered with tattoos, with bright oranges and blues swirling in sci-fi flames up my legs and arms, and the thought of going to somewhere new, out of the U.S., appealed to me. I had already done the \"successful chef\" thing in the U.S. I found myself staying after work with adoring clients who had watched too many episodes of _Iron Chef,_ who thought I could throw knives in the air and prepare delicious meals in half an hour or less. The reality is, it takes time to make good food. Those shows are bunk. It takes hours of planning and experimenting. It was nice to ride the wave of the food obsession in the U.S., but I wanted to see if I could go somewhere out of my comfort zone, somewhere where food was not synonymous with pornography, where people still loved the food for how it tasted and not for what it said about them. So I jumped at the chance to open the new restaurant in Mexico City.\n\nI needed clients with money to do the kind of food I wanted. But I was looking for clients who needed their palettes awakened, who hadn't already read about everything in some glossy magazine. I was looking for new markets in which to hunt down ingredients, for new adventures.\n\nEl Chapo was led into the back room, and he sent one of his guards to summon me. If it was adventure I'd been looking for, ironically I was going to get more of it than I'd bargained for.\n\nThere's a fine leather bench along the walls of the back room, and El Chapo was perched on the bench, leaning back, his legs so short I had the sense he was swinging his feet beneath the table when I came in to meet him. A guard blocked the door, to the left, to the main dining room. Another stood pointing an AK-47 my way. El Chapo sat, alone, waiting to speak to me.\n\n\"Sit down,\" he said. He spoke in Spanish.\n\nI sat in front of him. He looked me over and then spat on the polished wood top of the table.\n\n\"What kind of chef's costume is that?\" he said. \"Don't you have any self-respect? I thought you were supposed to be the latest hot chef in this town. Fucking Mexico City. Everyone thinks they're so fucking important in this city. They don't know anything.\"\n\nI never wear a white chef's hat. That seems ridiculously pretentious to me. I usually have on some T-shirt with a retro rock band that I like. I tend to wear bicycle riding pants and clogs in the kitchen, under my apron. I like to ride, whenever I get a little bit of time, and I tend to wear a bicycle cap or a baseball cap with some heavy metal logo. I had on a baseball cap, with the letters AC\/DC on the black front. The truth is, El Chapo and I looked a little similar, each in our baseball cap.\n\n\"What kind of outfit do you want me to wear?\" I asked, politely.\n\n\"Oh, have some self-respect, man. If this is the getup you have to make yourself feel important then don't try to change what you do for me. But you look like some kind of wannabe athlete, not a chef. Everyone wants to be something other than what they are. Politicians pretend to be saints. Crooks act like they love their wives. I would have thought a chef was something different...But I see you're a phony, just like all the rest.\"\n\n\"I'll try to do my best,\" I said.\n\nThis was going to be harder than I thought. I knew that if El Chapo Guzm\u00e1n was coming into my restaurant, he was probably doing it for PR, to let everyone in the city know this was his territory, that he could come and go at will, that they could have a five-million-dollar bounty on his head and he could still make a mockery of every counter-narco cop in the country. If he could come into a fancy restaurant and pay for everyone in broad daylight, then it would be enough to strike terror into every last person in the city. I had heard this was the effect of his stunts like this in other cities. He seemed like a god who could come and go as he pleased, invulnerable to any human boundaries. But if he was coming into my restaurant, in particular, I guessed it wasn't just to make a statement. If he was coming into my specific joint, it was to see if the food was any good. My job then, like any great chef, was to be a magician. A chef who's truly great sits a person down at a table, makes them wait longer than they want until they're beginning to salivate\u2014to be a little cranky, to doubt the abilities of the kitchen\u2014and then comes out with plate after plate of unexpected wonders, with flavor combinations that pop and surprise in perfect ecstasy, until the patron willingly pulls out their wallet, pays much more than they think they should, but without any regret, with a clamoring, in fact, for the next opportunity to eat more of the food. And all the while, the chef has to come up with just a few dishes on the menu that will please everyone. Each person eating thinks the magic has been made just for them, but it's been made for the eighty to a hundred clients of the day.\n\nReluctance, I was used to. Someone who was already saying I looked like a pansy was another thing. It was going to take more than the usual tricks to win El Chapo over. It didn't seem like he was the kind who would want to kill me if I failed to charm him, but it was always an option.\n\n\"Would you like something from the menu, today?\" I asked El Chapo. I used reverse psychology. I knew if I asked him this way, he would say he wanted something made just for him.\n\n\"Do I look like the kind of guy who eats what all the other pigs out there are eating?\" El Chapo said. He pointed to the door to the main dining room. Normally, there's a hum of patrons talking, eating, ordering fine wines, and privately licking their knives, even though that's gauche. You can tell how much your patrons like the food by how much sauce they leave on their plates. I always inspect the plates as they come into the kitchen. Where there are marks where people have been sopping up the sauce with bread, I take note and try to make those sauces more frequently, though you always have to try something new.\n\nThere was barely a sound from the other side of the door to the dining room.\n\n\"Something special, just for you, then?\" I said.\n\n\"Do you know how I became the jefe?\" El Chapo said. \"It wasn't just killing people. Anyone can kill people. Anyone can be the baddest badass around. That will get you about twenty percent of what you need to be the capo.\" He leaned in closer to me, as if he were about to give me the secret key to the universe. \"What made me the jefe was coming up with a better plan. What made me the jefe was someone else telling me what I had to do, and me coming back to them with something better than what they asked for. You want fifteen tons of the product in Chicago, by Monday? OK, I'll get it there. But I'm also going to build a tunnel under the border so we can ship over three hundred tons next week. And I'm going to ship two kinds of product on the same planes into LA. Those idiots. They were just shipping pot on the planes, when they could have been shipping coke.\" He looked under the table, as if he wanted to be sure there were no bugs in the restaurant recording what he was saying.\n\n\"It's making do with less to make more,\" he said. \"So here's what I want you to do. I want you to give me something that tastes so good I almost cum in my pants, that makes me and my compadres slap you on the back and give you an extra million-dollar tip for cooking so well. And I want you to do that without salt, without pepper, and with no more than two ingredients. And if you can do that, then I'm serious. You're going to get some unexpected tip. And if you can't...\" He gave his chuckle again, the same jokester laugh he'd given to the patrons, outside, when he'd introduced himself to them, the great El Chapo shaking their hands. \"If you can't,\" he said as he put his right hand up to his head in the shape of a pistol, cocked his thumb, pointed into his ear, and released. \"Pow!\" he said, and he started laughing.\n\n\u2014\n\nI came down to Mexico City with my wife. We have a four-year-old son, and one of the pleasures of Mexico City has been seeing how kid-friendly, how kid-obsessed, the place is. Mexicans love their kids more than just about any place I've ever been. There's a big park in Polanco, not too far from my restaurant, called Chapultepec, and on Sundays I walk with my wife and son, Jimmy, through the park. I leave the cooking up to the sous chefs on Sunday. I close the restaurant on Monday. It was because of Jimmy that I opened a gourmet hot dog restaurant in Pittsburgh, before moving down to Mexico City. He loves hot dogs. He takes a dog in both fists and shoves it in. There's something primal about the way kids eat food, shoving the good things into their mouths, tossing what they don't like onto the floor. Behind the hidden-flavor sophistication in a fine restaurant I want the patrons to feel that primal energy, to feel like they're pushing food in and tossing it out on the floor, all as they constrain themselves to holding a fork delicately in front of their mouth and then popping it in.\n\nI like to watch the patrons eat like animals. I like to see them roll their eyes upward in delight, just as Jimmy does when he eats a hot dog with mustard and all the fixings.\n\nA couple days before El Chapo came into the restaurant, Jimmy was in Chapultepec Park, next to one of the big artificial lakes where the fountains spout water into the air, and he dashed away from my wife and clambered over the lip of the stone border around the water before I could grab him. He went straight into the water. I thought he could drown, even though the water is no more than a couple feet deep. I had my riding pants on, since I'd gone into the restaurant earlier in the day to check up on how everything was going. (Sundays are a day off, but sometimes I pop in, unexpectedly, to make sure everything is keeping up to standards.) With all my clothes on and shoes, I jumped straight over the lip of the lake into the water and grabbed Jimmy, and threw him up into the air, out of the water. \"Never do that again, Jimmy,\" I said. He'd scared the living daylights out of me and my wife. Jimmy thought the whole thing was hilarious. He laughed and spit water out of his mouth, into my eyes. He had no sense of the danger he was in.\n\nSo this wasn't about me anymore, I realized, as I went back into the kitchen to cook for El Chapo. This was about me and my wife and Jimmy and all the patrons out in the restaurant. Going to a restaurant is putting trust in the chef. A chef is meant to do more than delight. He's meant to block out the pressures of the world, for a few minutes. For a couple hours, the patron sits at the table, with a nice white tablecloth, and they're allowed to forget about the outside world, to forget about their business deal going sour or their dying grandmother, or the problem they're having with their spouse. Food, at its best, can be like an amulet, something that wards off evil like a magic shaman. I could feel the weight of the magic that I needed to perform as I went into the kitchen.\n\nIt may sound crazy, but people like to eat what they are. If they have voracious habits they can't change, they like sweet foods. If they are tight with their money, they prefer to eat bread and mashed potatoes. If they are flamboyant they like elaborately thin vegetables, fried and piled up high like a fancy hat. We are all cannibals, eating ourselves, eating the secrets we have within. There's a reason the pedophile has poor dental work, teeth that have eaten too many sweets.\n\nI went to the main meat fridge in the kitchen, and I started to pull out things that I thought El Chapo might like. El Chapo was a short bull. He was an animal. He was dark and earthy. He snorted when he spoke. Beef, alone, wouldn't be enough to get his attention. I needed something darker. I thought of some wild boar I had in the fridge, some elk or venison. The boar meat might be close, but the elk and venison were too delicate. If I could have found a slab of water buffalo, that would have been about right. I ate a piece of cured water buffalo meat once, in a restaurant in Sri Lanka, and it tasted as black as the thick skin on the animal.\n\nAnd then I realized, only one thing would make El Chapo absolutely happy. Human flesh. I knew human flesh would do it. But I couldn't give him what he wanted unless I could cut that flesh off of myself, to save the life of me and the other patrons outside, waiting to see if they would make it to another day.\n\nI found a block of Wagyu beef and took it out of the fridge. I sliced it in thin delicate shavings and then piled the shavings up on six different plates, for El Chapo and some of his henchmen. The first ingredient would be the beef. The second would be human blood. I took one of the sharp knives off the wall of cutting utensils and cut my thumb. I squeezed drops of blood onto the beef, letting the crimson color sink into its thin wafers. The plates looked colorful and there were only two ingredients. I made the plate for El Chapo bigger than the rest, to fit his big ego. I was about to tell the waiters to bring the dishes out, but then I stopped, for a second, and tasted the meat. The ingredients were wrong. The dish was close but not quite right. There was a bitterness, a tough saltiness to my blood that wasn't quite perfect, as if the totality of my experiences in life came out in the flavor of the blood, and I realized normal blood wouldn't be enough.\n\nEl Chapo had eaten everything. He had eaten, I guessed, in the finest restaurants of Vegas and in London, Tokyo, and Paris. His empire was global. I knew he was a billionaire. This dish had to be unlike anything he had ever eaten in any of those fine restaurants. But normal blood wouldn't do. I thought of the baby piglets, lambs, and veal that were the staples of every great restaurant in Europe. It was precisely the tenderness and innocence of those animals, the very thing that made vegetarians cringe at such dishes, which made meat-eaters relish such food. Like old people rubbing on creams to remove their wrinkles, the restaurant patron craves baby carrots, the youngest peas, and other new-grown shoots and lettuces to pass across their tongue before they are chewed and ingested. The desire of someone like El Chapo\u2014who had devoured young women like his wife, and who dressed like a teenage hip-hop star\u2014to avoid his true age, was strong. He needed to dominate all the people around him, like a dirty old man who preys on young girls. So I knew, suddenly, that I needed the blood of a kid.\n\nI went out the door and into the main dining room. There were a dozen of El Chapo's guards there, watching the patrons eat. Most of the diners' plates were empty, and they were waiting for El Chapo to finish his meal so they could go. When I came into the dining room, I heard a few gasps from some of the clients and saw an older woman start to cry. I had no idea how bloody my apron looked. I saw a young girl with two long pigtails running down the back of her hand-knit blue sweater. Her mother was holding her tight by her side, telling her, gently, not to be worried. I came up to the mother, well done up with makeup, who looked like she went to the spa once a week. I had to convince her to give up her daughter.\n\n\"My son's name is Jimmy,\" I said. \"He looks about the age of your daughter. Please trust me if I tell you I need your daughter to come into the kitchen for a second.\"\n\n\"Why?\" the mother asked. She looked tortured, like she was in the middle of a bank robbery and I'd just become one of the bank robbers.\n\n\"There will be time to explain later,\" I said, in as calm a voice as I could manage. I tried to be commanding. A command would be the only way to get the mother to relinquish her daughter. I said in a soft, but firm and commanding, voice, \"I need your daughter to come back to the kitchen with me.\"\n\n\"Only if I come, too,\" the mother said.\n\nSo I told her all right, she could come, too. The guards of El Chapo told me to hurry up. \"What are you doing wasting time?\" the head guard, with the diamond in his ear, said. \"Get in the kitchen and give The Boss his meal. He's hungry. He's been waiting for you.\"\n\nI went back into the kitchen with the daughter and her mother. I stood in front of the plates of Wagyu beef growing moister and plumper as it absorbed the blood.\n\nI put my thumb up to the face of the mother, where I had the slice from the knife I had made before. The wound was pink and the blood was sticky. \"You see this?\" I said to the mother. \"I need to do the same to your daughter.\"\n\nShe looked at me like I was a crazy terrorist. She tried to pull the knife out of my hand. I had a choice: Give in, let the girl go, admit that I might be crazy, a Dr. Frankenstein in the kitchen, admit that the whole thing was an overreaction on my part. But I have learned, in life, that you have to go with your gut. I didn't think, I just took action. I put my hand up against the mother's mouth and made her stay quiet. I grabbed one of the dish towels and stuck it into her mouth. Her eyes pleaded with me not to do anything to her daughter. I picked up the knife again, I pulled up the hand of the child and cut her thumb, deeper than I would have liked in the heat of the moment, and blood came pulsing out. I lifted her high into the air, with her pigtails flailing behind as she cried out for her mother, who was trying to yell, and I squeezed the blood of her hand onto El Chapo's plate.\n\nThe waiters looked at me like I was insane, but the rule in a fine restaurant is to never question what the head chef demands. They brought the food out to El Chapo, and I made sure they brought the plate with the young girl's blood specifically to him.\n\nI followed out a few minutes later, and I could see El Chapo licking his thumb, picking up the plate to get more of the sauce on his lips and into his mouth.\n\n\"There are only two ingredients in the dish,\" I told El Chapo.\n\n\"What is it? It's fucking good. It's fucking delicious,\" he told me. \"The others agree. I've eaten all over the world, and I've never eaten anything this tasty.\"\n\n\"It's Wagyu beef with human blood,\" I told him. \"The blood is mine.\" I held up my thumb to prove I wasn't lying. There was the cut, irritated and sticky with drying blood. I didn't mention the girl. If El Chapo knew, he might develop a fetish for young blood. He might look for her in the main dining room and take her away. He might rape her. There was no telling what kind of habits he could develop. I've read, from survivors of airplane crashes, that once the taste of human flesh is tried, once the taboo is broken, it's hard to go back.\n\n\"Son of a bitch,\" El Chapo said. \"You think I'm an animal. Some kind of cannibal.\" He spat on the table with venom, like a rattlesnake. I couldn't tell if he was going to pull out a gun and shoot me, or not.\n\n\"I don't think anything,\" I said submissively.\n\n\"You people in the 'normal world,' you know nothing about who I really am,\" El Chapo said. \"A cannibal! Nothing could be more ridiculous.\" He picked up his plate and held it in front of his face, looking into a mirror. He threw the plate against the wall and the white china, with blood, broke into pieces.\n\n\"But I never lie,\" he said. \"Even if someone attempts to trick me. I'm the most honest person you'll find in all of Mexico. Come here!\" I approached, as commanded. He took my thumb and squeezed my wound until I cried out, hoping my finger wouldn't burst. New blood came forth from the thumb. \"You should be happy just to keep your life,\" El Chapo said, looking me straight in the eyes, letting me know he could kill me every bit as easily as he'd tossed the plate. \"But I challenged you, and you gave me something with two ingredients, and it's the best thing I have ever tasted.\"\n\nHe ordered one of his guards to bring me my tip. He pulled out a couple thick piles of hundred-dollar bills from a brown briefcase, and left the money on the table. He put my thumb on the money, until some of the blood seeped into the face of Ben Franklin on the bills. \"Now you are guilty of entering the world of blood, too,\" he said.\n\nHe stood up quickly and walked into the main dining room. The girl was back with her mother, sitting at their table, the mother still in tears. I prayed El Chapo wouldn't discover she'd been involved with the meal. I hoped he wouldn't see the girl's thumb was just like mine.\n\nEl Chapo looked around the room and saw the girl, next to her crying mother. He stood in the middle of the patrons. \"What's wrong, little girl?\" he said to all the customers. He walked up to the girl. \"Didn't you like your meal? Please finish eating, everyone. I hope you've seen the great El Chapo isn't such a bastard after all. Please enjoy your meal. It's compliments of me and my associates.\" He waved his hands wide in appreciation to his guards and bowed to them.\n\nThe mother held her daughter, hiding the girl's face from the \"great\" El Chapo, and he left with his hands raised in the air like a boxer accepting applause after winning another fight, though the room was silent. The guards returned everyone's purses and cell phones, and then disappeared.\n\n\u2014\n\nTwo months after El Chapo came into my restaurant, I went with Jimmy to the Museum of Anthropology in Polanco, not far from where I work. The museum is enormous, with a courtyard where Jimmy can run around, so I like to take him there when I can. At the far end of the museum, like a magnet drawing all the visitors forward, are the rooms with the remains of the Aztec empire\u2014high stone temples covered in alabaster, which had reached up to the broad sky at the center of the ancient city of Tenochtitlan in the middle of what is today Mexico City. The halls with Aztec art are filled with clay statues, and one is of a god that looks like a robot, square with big round eyes. This is Tlaloc, the god of rain and thunder. He controlled the water, necessary for the corn to grow. The Aztecs prayed to dozens of gods, and at the top of their pyramids they sacrificed thousands of people to keep the gods satisfied.\n\nJimmy ran ahead of me in the Aztec galleries, past stone knives with small eyes made of turquoise that the Aztecs used to cut out the hearts of the sacrificed people. Young victims were always considered the most worthwhile for placating the gods. They had more potential energy for the rest of their lives stored within, so they were believed to be the most valuable offerings.\n\nFor two weeks, I hadn't been sleeping well. I didn't know what to do with the money El Chapo had left me as my tip. I kept thinking about what he'd said to me, that now I was guilty, too. My wife had told me I should be practical and use the money to keep building up the restaurant, or to put it into the college fund for Jimmy. She told me I would be turning bad money into good, taking it from the drug dealers and making it into something positive. She told me I should be proud I'd saved everyone in the restaurant that day. But whenever I looked at the money, and thought about the possessed way I'd lifted the girl to put her blood on the beef to give to El Chapo, I wasn't so sure I was a hero. It wasn't just that I'd tortured her mother and cut the girl's finger; I was playing with black chef magic, thinking I could outsmart El Chapo. What if he'd reacted differently to my dish? What if he'd reacted in anger and killed me, or more people? I was just as guilty of giving in to my ego, of thinking I could control him, as El Chapo thought he could control the world.\n\n\"Come over here,\" I told Jimmy. He was lost, wandering among the Aztec gods, and I had to call him a couple more times before he complied. I'd told his mother I wanted to go with Jimmy alone to the museum. Jimmy and I came to a stone pyramid altar, twenty feet high, which used to sit on top of one of the pyramids of Tenochtitlan. From my backpack, filled with snacks and a juice bottle for Jimmy, I pulled out a big envelope with the twenty thousand dollars El Chapo had given me. The envelope was new, but you could still see, faintly, the blood on some of the bills within. \"Take this envelope up to the top of the pyramid,\" I told Jimmy. \"Put it on the round stone.\" I told him to go to the place where the Aztecs used to leave the sacrificed hearts. I looked around to make sure there were no guards coming. \"Keep going! Put it up there,\" I said to Jimmy. He placed the envelope just where I told him, on the stone altar. He made a face at me. He told me he didn't want to come down. \"It's fun up here,\" he said. I could hear the guard coming. I told Jimmy to hurry. I told him we had to go home.\n\n# THE GOD OF COMMON NAMES\n\nThis is a Romeo and Juliet story. But it's set in Mexico, where I live and work as a teacher.\n\nOne day, about a month ago, this black SUV pulls up in front of the private New Hampshire School where I teach in San Jer\u00f3nimo. The New Hampshire School is one of those precious schools where rich Mexicans send their kids to give them an exclusively foreign-feeling education. English is spoken in half of the classes. The students take ski trips to Vail, in Colorado, over Christmas break with the other students. They dream of getting into places like Harvard and Yale, but most end up at places like Bucknell University or, if they're less brilliant, the University of Miami, Florida.\n\nThree blocks up the street from the school, there's a fantastic bakery where you can get croissants stuffed with walnut paste and a dog-grooming store where they import weight-loss dog food for breeds like yellow Labs. There's a wine store with two-hundred-dollar bottles of imported French wine and an electrolysis beauty salon that keeps humming with one laser plucking after another. I'm told permanent Brazilian waxes there are the thing.\n\nBlack SUVs aren't all that uncommon at our school. The kids come with their drivers and bodyguards, who bring them in the morning. But what made this SUV unusual is that it showed up in the middle of the day, when the kids had already eaten lunch and were all in their classes. The other thing that made the SUV unusual was that no kid came out of it. I could see the Cadillac come into the parking lot. I was teaching my usual tenth-grade class in American History and Culture. We had just done a unit on American outlaws like Jesse James and the Chicago gangsters of the '20s, and the kids were into that.\n\nTwo of my students, who almost never pay attention, had perked up during the unit. One was Jos\u00e9 Cachez Jim\u00e9nez and the other was Sandra Fern\u00e1ndez de Guanajuato. I make up a seating chart at the beginning of each semester, where I place the students randomly in new desks. I do this to avoid having cliques form, and I do it because one of the big social differences between American students and Mexican students is how everything in Mexico revolves around the group. American students, of course, like to hang out together, but they tend to break off in groups of two or three after school. But in Mexico it's all about big groups of eight or so buddies or female pals. They do everything together. They play soccer together, go to the movies together, have parties together, and then work together or swap business deals together as they get older.\n\nIn any case, Sandra and Jos\u00e9 ended up randomly in my class sitting next to one another, in the back. There's some strange law about teaching that even when you seat students randomly, the ones in the back tend not to pay too much attention. I would hear them tittering, a few weeks into the class, when my back was turned to the students, while I was writing on the board. I'd see Sandra and Jos\u00e9 casting glances back and forth, even while taking exams.\n\nSandra's face is long and thin; she wears her hair with a lot of wavy curls that she's spent hours combing so they cascade one below the other in bountiful thick semicircles, frozen perfectly with lots of hairspray. If Frank Gehry were a hairstylist instead of an architect, this would be his kind of head of hair. She wears red lipstick that she regularly freshens. Excessive makeup is illegal in the school, but the definition of what's excessive is different in Mexico. Unless you have false eyelashes, you're OK in the New Hampshire School. Outside of school, I saw Sandra in an upscale mall, once, with tight jeans with sequins stitched on the back pockets, in the shape of crosses\u2014but coquettish, round crosses that let you know she's religious and she believes in God but if some boy wants to come along and find his way into her pants, that would be OK.\n\nIn class, the students all have to wear uniforms with green and blue plaid skirts for the girls and blue blazers with the New Hampshire crest for the boys.\n\nJos\u00e9 is the kind of guy who likes to have his blazer always unbuttoned. He sneaks in a soccer shirt when he can, under the blazer, of the national soccer team or of one of his favorite teams in Italy.\n\nI never turn the kids in for clothing violations, because I figure the whole pretension that the kids are future Ivy Leaguers is ridiculous. And besides, they're kids. All I'm interested in is that they learn to read a bit better, that they start to understand just a little of the wide history out there that most of them are oblivious to, and that they consider the possibility there's more than only one right answer. If they can get away from the idea that history is a right or wrong answer, with only dates to memorize, then I feel I've done my part.\n\nThe man who came out of the SUV was wearing a white cowboy hat. He was tan in a way upper-class Mexicans around Mexico City don't like to get, meaning his skin was rough and leathery. He had on a crisp, Western-style shirt with fake cowboy stitching on the pockets, and he was wearing a pair of mirrored aviator glasses. Needless to say, this isn't the typical kind of parent that comes to pick up their kids at the New Hampshire School. Two men in black suits, with white shirts and pink ties, who looked like private security guards, trailed behind the man with the cowboy hat as he came into the school.\n\nFor another couple minutes, I forgot about the man I'd just seen in the parking lot. I continued with the latest course section about American Prohibition and the women's suffragist movement, how women finally fought to get their full right to vote.\n\nThen a knock came at the door to the classroom. I turned away from the blackboard, where I was writing, and went to the door. There was the guy with the white cowboy hat. I looked through the glass window of the door, and for a second I had the certain feeling I should leave the man outside and call security. But I also had the feeling I was going to be in deep trouble if I didn't open the door. So I did. Or, more like, after we made eye contact the man with the hat let himself in.\n\nHe turned to the class as a whole, and there was suddenly complete silence. He walked toward the back slowly, letting the sound of his white crocodile-skin boots click precisely until he got behind Jos\u00e9 and stood next to Sandra. \"Your romance,\" he said in Spanish, \"is over. You hear me?\" He turned to everyone in the class and said, \"It's over.\"\n\nHe took his son Jos\u00e9 by the collar of his green Mexican national soccer team jersey, holding the shirt beneath Jos\u00e9's blazer. The boy stood up obediently and bent over, like a little child hoping not to be caned by his father, even though he was about five feet ten. The father pulled Jos\u00e9 up through the middle of all of the desks until he got to me. I watched the whole rebuke in silence, holding my textbook, trying to think how best to respond to let the father know I was the boss in this room. But get real. He was the boss. So I just stood there silently, obediently, with his two guards at the doorway. \"And you,\" the father said when he got up to me. He spoke in English to let me know he could tell me what to do in English, too. He had a thick Mexican accent to his English. It made \"you\" sound more like Jew. \"Jew,\" he said. \"I am going to hold Jew personally responsible for these two. If they ever\u2014ever\u2014do anything together again, I'm going to come kill you. Is that clear?\" He paused for a second, as if he realized the way he had just spoken to me in public was too much, a bit outrageous for a school, and then he added, \"Look, these two people, they're from different families. That's all you need to know. I'm asking you to keep them apart.\"\n\nHe took Jos\u00e9 away, and Sandra sat in the back of the room crying.\n\n\u2014\n\nApart from the tremendous fear I felt from the man threatening me, it was funny that he called me a Jew. Not funny in the way of _ha ha,_ but funny that he made that mistake of pronunciation, because I am a Jew. I'm a secular Jew, maybe even an atheist Jew, meaning that I grew up as a Jew in Chicago, and my parents sent me to temple, but I don't believe in God. But I do feel an extreme sense of ease in a temple, so it was with great surprise, when I came down to Mexico eight years ago, that I discovered there was a large group of Jews living in Mexico City. The thought that there could be Jews in Mexico had simply never occurred to me. Mexico was a place of tamales and tacos, mariachi bands and guacamole. What would a bunch of Jews be doing in a place like that?\n\nWhen I first arrived in Mexico City, with a backpack, with no real plan other than to find some eventual work, a friend of mine who worked for the U.S. embassy let me stay in her apartment in the neighborhood of Polanco. The first day I arrived in the city, I walked outside to get some groceries. I wandered around the upscale neighborhood. It was a Friday, and around evening I saw all these men walking with black fedora hats and long black jackets toward who knows where. At first I just saw one or two, who stood out oddly against the other people walking to a nearby mall, where there are stores like El Palacio de Hierro, which is like Saks Fifth Avenue. The men looked like they'd been Photoshopped into the scene from Brooklyn or Jerusalem. The few Orthodox men I saw grew into somewhat larger groups as they walked into the distance, to the end of the sidewalk.\n\nWhen I got home to the apartment I was staying in, I asked my friend what these Jews were doing in the neighborhood. \"Oh, there are lots of Jews here,\" she said. \"Maybe forty thousand of them.\" My friend was from El Paso, Texas, and she'd definitely been raised Catholic: hence she used the \"them,\" which is the way even I thought of the Jews, as \"them.\" We were a \"them.\" Always had been, since the days when the Jews first went to Egypt. Even as a nonreligious Jew, it was part of my identity that I was a \"them.\"\n\nThe next week, when Friday came around, I followed the men to temple. I was clearly not dressed appropriately to enter. I stood at the door, watching the men go in, feeling like I should stay outside. But curiosity got the better of me, and I asked one of the men, in broken Spanish, if he had a yarmulke I could borrow. He looked at me suspiciously, at first. He eyed my corduroy pants and white button-down shirt. It wasn't a suit, but it was the best I had in my backpack, and I'd ironed my clothes neatly, and he must have seen the good intentions in the crisp creases because he nodded slightly toward the door and let me in. He gave me a tallit, he found me a yarmulke, and he took me inside the main temple, and in the center of the room I saw the scrolls of the Torah eventually revealed, with handles gleaming so brightly that even in the dim light of the temple they shined like silvery moons.\n\nI returned to the temple the next week, and while the women were separated from the men, and while I still didn't believe in God, and while I fumbled through some Hebrew as the other men recited together in unison, I was dragged to a Shabbat dinner afterward, in the basement of the temple, where candles were lit, and where I first set my eyes on Sara.\n\nShe had soft freckles and her cheeks were a bit pink and upwardly round, her eyes were a turquoise that pulled me into her gaze. She wore a long black skirt and plain black shoes with stockings, as was required for an Orthodox woman to be considered modest.\n\nWhat can I tell you about the walks we took throughout the city the next six weeks? They were the kind of ambles where the sun feels warm and perfect on the skin and on the brow. They were the kind of walks where birds seem to constantly sing. We wandered through the Parque Lincoln, where all the trendy restaurants are located. We saw the people eating their meals happily, laughing, raising glasses of wine and seeing people like us as passersby. We watched the dogs play in the park, leaping for balls thrown by their masters. We walked by the beautiful fountains in the Bosque de Chapultepec. We walked to the ice cream store called La Michoacana, where they served us mango and guava sorbets.\n\nBut after six weeks, one afternoon after I'd taken Sara home, as I stood outside of her apartment under her second-story balcony\u2014not going upstairs because I knew her father was very strictly Orthodox and didn't approve of me\u2014she stood on the balcony waving at me below, and then the screen door opened and her father came out and he stood tall with his black pants and black jacket and white shirt, and with the fringes of his tallit just poking from under his shirt, and he looked down at me with the beard on his face looking heavy, with the weight of gray crowding out the rest of his once-brown beard, and he raised his hand like a hammer, banging it in the air, and he said in Spanish, \"You. You sir. You! Go away. You are not good for my daughter. You are not a real Jew. You! That's enough. I don't want to see you around here anymore.\"\n\n\u2014\n\nAfter school, I coach one of the girls' basketball teams at the New Hampshire School. It's part of my contract. I have to coach a sport. It's part of the motto of the New Hampshire School that students will excel not only in the mind and in the heart but also on the field. Growing up, I used to love to watch the Chicago Bulls play basketball. I marveled at Michael Jordan flying through the air and reverse-dunking the ball. It was like he had escaped the bounds of this earth and the bounds of his background in the 'hood.\n\nI was never a natural-born basketball player, myself. I'm too short, only five feet eight. But I'm pretty quick with my hands, so I could always play as a guard.\n\nI remember on one of those early dates with Sara, as we walked through the Parque Lincoln, I saw some kids shooting hoops in a made-up basketball court. A ball came rolling over to me, and I picked it up and dribbled for a few seconds. I thought of throwing the ball from the distance to show Sara I could make a long shot, but then I thought that would look like showing off, so I decided to throw the ball to her, instead, and when the ball came into her chest, where I expected her to grab it, she didn't know how to fully raise her hands properly, to spread them wide to catch the ball, so it went straight into her chest, hitting against her plain white shirt, smashing her breasts.\n\nShe gave me this look of betrayal, like I had just intentionally hurt her. I went running up to her as she sucked in air, looking at the smudge of dirt on her shirt. \"Why did you do that?\" she said.\n\n\"I'm so sorry,\" I said. But I was more than sorry. I was afraid we wouldn't be able to go out anymore. I was afraid the distance between us might be too great. While I'd been growing up watching TV, and Jordan hit the hoops, she'd been wearing long, modest clothes, and occasionally playing jump rope. Attraction is one thing: turquoise eyes, a beautiful smile. But maybe there was too much to overcome.\n\nI thought about that moment with Sara\u2014it was years ago, before we'd gotten married\u2014as I watched Sandra come out to play on the basketball team. She wasn't very good as a student in class, and she may have spent hours combing her hair in perfect curls, but on the court she was an aggressive tiger, wearing her hair back in a long ponytail.\n\nYou might think that after the incident with Jos\u00e9's father, earlier in the day, Sandra would have gone straight home, but I think, like me, she must have been terrified of deviating in any way from the normal routine.\n\nSo, there she was in her basketball shorts and white court shoes trying to go through the motions of shooting hoops from the three-point range. Usually, she was a good shooter. Usually, the ball went in with a nice swish, her hands following through in a delicate downward dip after the shot. But she was missing them all, today, and I saw her shoot even an outright air ball.\n\nI called Sandra over while the other girls continued throwing hoops.\n\n\"Are you OK?\" I asked.\n\n\"You have no idea what it feels like to have him take Jos\u00e9 away like that. It's none of his business,\" she said.\n\n\"Excuse me,\" I said, trying to be genuinely polite, \"I know it's not my place to ask. But what did you and Jos\u00e9 do together that got him so mad?\"\n\n\"We've just been hanging out,\" she said. She looked at her shoes. She chewed her gum a little, dejectedly. She knew gum was forbidden during practice.\n\n\"Just hanging out?\"\n\n\"His father and my father\u2014 They don't get along. They're from the same business. They compete against each other, and whenever one gets the upper hand they get furious at each other.\"\n\n\"And what business is that?\" I asked. But I already had more than an inkling of the answer. I'd heard the same rumors all the kids in the school had heard. The rumor was the fathers were both in drug cartels, high up.\n\n\"Just a business,\" she said. \"A very lucrative business. So sometimes they get pissed off at each other.\"\n\nShe looked at me stone-faced, as if wanting to let me know she didn't appreciate me asking that kind of question. \"So if Jos\u00e9 comes back to class,\" she said, \"what are you going to do? Are you going to make him sit apart from me? Are you going to tell the principal and make us stay apart?\"\n\n\"I don't know what I'm going to do,\" I said, and I meant it. I didn't have a clue what to do. There was the very real possibility I could get involved in something violent if I didn't do what Jos\u00e9's father said. There was the very real possibility that Jos\u00e9 and Sandra had been making love in some motel, somewhere in the city, and that I shouldn't be encouraging their sex at that young an age. There was the very real possibility I should just shut up and go about my business teaching and pretend this day had never happened. But beneath all those clear, rational ideas was the vision of Sara's dad yelling down at me from the balcony telling me I wasn't a real Jew, that I had no right to be with Sara, even though we knew we wanted to be together.\n\n\u2014\n\nLater that night, I asked Sara if she remembered our wedding day.\n\n\"Of course I do,\" she said. \"Why must you bring up something so bitter?\"\n\n\"Well, do you think your father was right?\" I said. I genuinely wanted to know. I truly needed to know.\n\nHer father had refused to attend the wedding until the end of the ceremony, and just after I broke the glass beneath my foot, indicating that I hoped for a life of happiness together with Sara even as we remembered that joy must always be tempered, her father came into the temple and yelled it was an abomination that we had just married. He stood like a prophet foretelling doom, shaking his hands in the air, yelling against us with so much force that drops of his spit seemed to fly out like bullets, caught in the shaft of light coming down from the highest window of the temple. He was a well-respected man in the Orthodox community, so no one tried to shut him up, but there were a couple of men who walked up to him, put their arms around his shoulders, and tried to give him comfort. The question I wanted to know was whether Sara felt I had been an impostor that day, dressing up for a Jewish wedding when she knew I did not believe in God? And I wanted to know if she thought her father had been right, that the divide between us was too big to overcome? Sara had certainly paid the price in marrying me. We had married in a temple different from the one where she grew up. We had married in a Conservative, but not an Orthodox, temple. There is no such thing as Reformed Judaism down in Mexico City. You are either a Jew or not a Jew. The shades of gray, as found in the U.S., don't exist.\n\nHer father had never spoken to Sara again, after our wedding. In the end, during the last words he said to her, he stood frozen like a statue as he yelled in the temple, shouting out to his daughter, \"Sara, how could you do this to me? You, who were always my favorite daughter.\"\n\nThere was probably nothing crueler I could have asked Sara after she'd made such a sacrifice of her father in order to marry me. But I felt, following the events of the day with Sandra and Jos\u00e9, I wanted to know after eight years whether she was happy with her decision, or with _our_ decision.\n\nWe had just eaten some tacos from around the corner\u2014I had ordered some pork tinga tacos and she had her usual beef. She rarely went to temple now, and we didn't keep kosher at home, but she still never ate pork.\n\n\"I will always feel, in some ways, that a piece of me was cut out that day,\" Sara said, looking at me with the same intensity she'd looked at me the day the basketball flew into her chest. \"I will always feel sad that my father couldn't understand that a good man is a good man whether he is Muslim, Asian, or whatever. You're a good man. That's why I married you. But if I could have married you and kept him happy at the same time, I would have done that. My father used to tell me to never burn any bridges. He used to tell me you never know when you will have to walk back over a bridge you have left behind. I miss the community sometimes. Of course I do. But I don't miss the irrational exclusivity that keeps people in and out. It felt like a prison. That's why I left. And you already know all this, so why are you bringing the subject up?\"\n\nI felt, after dragging her into the muck of our marriage, that I owed her an explanation for what had brought the thoughts of our marriage day back to me. The night before our wedding day, her father had come to me where I lived in the south of the city, far away from his neighborhood of Polanco. I was living alone then. Sara stayed residing with her family until our wedding day.\n\nHer father came to my apartment building. Like Sara, I lived on the second floor, and like Sara, I had a balcony, though not nearly as nice, or as big, of one as the apartment where she lived. Her father had come all the way to the south of the city, but he didn't want to get too close to me, so he didn't knock on the door\u2014at least that's why I think he didn't knock. Maybe he thought somehow I could infect him if we came too close\u2014me an atheistic, secular Jew, him a pure man who studied the Torah into the wee hours of the night. So instead I heard a small rock tap against the window of my balcony door. I went out onto the balcony, and I looked around. I couldn't see anyone, at first. I only heard the sound of a forlorn whistle, the haunting wail like a steam train of a _camote_ wagon, which roasts bananas and sweet potatoes. I thought maybe I had heard a tap that wasn't really a tap, from the stone. But just as I was about to go back in, the form of a black fedora hat and a black jacket, and the silent, bluish-white glow of his shirt became clearer, below me.\n\n\"I have tried with warnings,\" Sara's father said. \"I have tried with reason. Reason is what the Torah teaches us. Two things that are different should not be joined together. But if you will not listen to reason, then I will ask you as a humble man who loves his daughter. Please. Please! I'm begging you. Don't do this to an old man. Don't make me die before my time. Let my Sara marry someone worthy of her. Don't let me lose her.\"\n\nHe was not such an old man. He was no older than sixty-five. His pleading was haunting, touching, and certainly I felt for him. I asked him if he wanted to come upstairs to talk to me. I asked him, couldn't he come upstairs so I could convince him I was worthy of his daughter? But he just shook his head in the moonlight, bending down on one knee, praying as if he could no longer hear me. I went downstairs and opened the door and walked up to speak to him, and when he saw me coming he told me the only way I could redeem myself was to not show up at the wedding tomorrow. \"Better not to come. Better to break her heart than to ruin her life,\" he said.\n\n\"Me, ruin her life?\" I said. \"Haven't you already ruined it? Look at you. Just look at your pale, wrinkled face. You quiver and worry, wringing your hands like some kind of medieval caricature of a Jew in the shtetl. You claim you're the only person who knows right, that there is only one way to live, and that I'm not a real Jew, that I'm a dirty atheist. You negate life itself, wrapping yourself in a small bubble of virtue, blind to everything outside your community, fearful of it, cursing it, and demanding your daughter do the same.\"\n\nI could no longer hold my tongue. I no longer felt the impulse to try to placate him. Who was he to tell me I shouldn't show up at my wedding tomorrow? It was Sara's and my decision to make alone. I told him so. He put his hands up, against his ears, to block the sound of my voice, and then he ran off.\n\nHe disappeared within the shadows of the curvy streets of the night like a chimney sweep, darker than coal. \"It's not my fault, it's yours,\" I whispered after him, as he ran away.\n\nBut now, years later, I wondered if I had done wrong to separate his daughter from him forever.\n\n\"There are these two kids in my tenth-grade class,\" I told Sara, \"and today the father of one of them came to school and said they should break up, they shouldn't be together.\" I told her the rest of what had happened that day.\n\n\"How can you equate the kids of two rival narcos to us?\" she said, at the end of what I told her.\n\n\"I'm not,\" I said. \"I know it's different. But is there any real reason why the two of them should be kept apart, if they love each other, other than that their fathers hate each other?\"\n\n\"You complicate your life,\" Sara said. \"Stay out of it. Let the fathers take care of it. Or, if you really feel like doing something\u2014before you meddle with the two kids in your class, go visit my father and make things right.\"\n\n\"Make things right? You know the fault was never mine. It was him.\"\n\n\"It takes two to build a wall,\" she said.\n\n\u2014\n\nIt was a windy, sunny day three weeks after Jos\u00e9's father had come into the classroom. The days were getting longer. Spring was in the air, even though the seasons don't change all that much in Mexico City. But it felt like a spring day, with a gentle breeze and the sky bluer than usual, without the frequent pollution of Mexico City, and I stood in front of one of the small motels that litter the city, made for romantic affairs, usually between bosses and their secretaries, or between married men and their lovers. This motel was called Amor del Para\u00edso and was located near the Central de Abasto, a market where every kind of food is sold, and where all the restaurants of Mexico City come to buy their fish in the football field's length of stalls.\n\nI had parked my car a block away, after following Sandra and Jos\u00e9 riding in a small, black BMW that belonged to Jos\u00e9. As they drove through town, I saw them lean into each other at the traffic stoplights, Jos\u00e9 kissing Sandra as she turned into him, and I could see him twist his head back, occasionally, to see if any of the security guards of his father were following, or the guards of Sandra's family. He must have sensed someone was on his tail, but the mind sees what it is inclined to see, so he couldn't imagine his teacher was following him.\n\nI had listened to Sara, and also to Jos\u00e9's father when he'd spoken to the students in class. I separated Jos\u00e9 from Sandra, as far away as possible, placing him in the front right corner of the classroom, near the door where his father had peered through the window before telling me what to do with my class. It was embarrassing to move Jos\u00e9. It meant admitting to everyone in the class that I wasn't their leader: that I was a coward, that I would teach them the rights and wrongs of history and about great leaders like FDR but that in this class I wouldn't practice what I would preach. It's hard to talk about the women's suffragist movement when you won't even let a tenth-grade boy and girl, who love each other, sit next to one another.\n\nBeyond separating them, I went and told the principal what had happened in the classroom. This was the ultimate cover-your-ass move. I hated reporting on Sandra and Jos\u00e9. I hated reporting on Jos\u00e9's father, too. It went against every fiber in my body. But Sara had asked me to do this, and she was right that it was common sense. What if Jos\u00e9's father came back with guns? What if he became more violent? The administration had to know what was happening inside the classroom and to decide what to do next. The principal shook her head as I told her of the incident. She wiped her palms on her tight wool skirt and said she agreed Jos\u00e9 and Sandra should be separated in the classroom. We discussed the possibility of expelling both of them, because of the threat of violence in school, but we laughed at the very moment we said this. That may have been the right punishment, but you don't play God with the cartels, you accommodate yourselves to them. For the same reason, no extra security was placed around the school. Everything was left as before, to imply nothing had happened, except the separation of Jos\u00e9 and Sandra in the classroom to comply with the father's will.\n\nThat was the official response. The unofficial response was that I wanted to see what Jos\u00e9 and Sandra were up to. You could certainly call this voyeurism. It's what it was. But I liked to think of it as chaperoning the two of them. I know it might sound insane\u2014me, a defenseless teacher without any weapons, and the people who wanted to hurt Sandra and Jos\u00e9, narcos with guns. Certainly, in class, I hadn't found the strength to stand up and protect them. But it was precisely because I hadn't been able to stand up to Jos\u00e9's father, so far, and because I felt such cowardice was wrong, that I fought to find the strength to try to take care of Sandra and Jos\u00e9 in private. Any teacher worth their salt feels the students in their class are there more than to learn; they're there to grow safely into adulthood. A teacher is as much psychologist, protector, and nurturer as the teacher of a subject. This is why they have parent-teacher conferences. This is why they take care of the students at prom, to make sure they get home safe. This is why they spend their short lunch break trying to help the poor student who doesn't have anyone else to speak to in class. In whatever limited way I could, I wanted to try to protect Sandra and Jos\u00e9. If I could find a way to keep them safe from their parents, I became determined to do that. So, every day for three weeks, I followed them.\n\nThis was the fourth time in three weeks Sandra and Jos\u00e9 had come to this motel. The first time they'd come, I'd stood far in the distance, not approaching closer, waiting a couple of hours until I saw them exit the hidden curve of the motel's in-and-out entrance. They were safe, still together, still unharmed, still undiscovered by their fathers, and I breathed a sigh of relief as they came out, Jos\u00e9 looking carefully left and right to make sure none of his father's security guards were watching.\n\nThe second time, the man working at the check-in desk asked me if I needed anything, and I said I just wanted to know the rates of the place for future use. I was too scared to go any closer, and I left early, back to my home and to Sara.\n\nThe third time, I grew so bold as to walk into the area of parked cars until I found Jos\u00e9's car in front of one of the doors. I put my ear against the door, and I could hear them laughing with two friends, the four of them smoking pot. I heard the high laughs and rambunctious jumping on a bed of four teenagers goofing around. That was enough for me. I didn't need to know more. But I kept following them, feeling like I was their guardian angel. If I had to castigate them, or separate them in public, at least I could protect them in private. I looked for ways they might be able to escape from the motel. I checked out where there were exits and what roads anyone who wanted to harm them would come from. If I knew the exits, maybe I could lead them to safety when the time came, alert them to get the hell out of the place before it was too late.\n\nThis fourth time, as I stood across the street waiting for Jos\u00e9 and Sandra to come out, I saw two black Cadillac SUVs approach from the right, down a one-way street that led to an on-ramp of the highway that passed in front of the innocuous motel. I couldn't tell who the bodyguards inside the SUVs belonged to\u2014Sandra's or Jos\u00e9's family\u2014but I knew they must have something to do with them. By now, I'd discovered there was a small rear entrance to the motel. It was made for the cleaning crews, so they could take laundry in and out. I'd had far too much time to explore the place\u2014the bland beige paint that covered everywhere except the blue palm trees stenciled on the interior courtyard, where the cars were parked, and a red heart painted on the door to the check-in desk.\n\nWhen I saw the SUVs approach, I ran across the street behind them as they pulled in front of the motel, and I hurried along the dying cacti and volcanic rocks littered with juice boxes, condoms, and other trash thrown along the side of the motel. A furniture construction warehouse, small and old, with sawdust flying to the ground toward the motel, ran along the path to the back, and I squeezed between the two buildings hearing the sound of a whirring table saw. It was the sound of wood being ripped in two for cheap pine furniture in poor homes. I reached the back door. It was jammed for a second. I picked up a rock and smashed at the lock. The door wasn't closed with a key, it was just stuck from loose hinges, causing it to lean sideways. When I came into the inner courtyard, I looked fast to see if Jos\u00e9 and Sandra were in their usual room, parked on the left, and they were. Across the thirty-yard courtyard, I could see the two black SUVs. The men had gotten out from the SUVs, and they were entering the check-in area to ask, no doubt, for the check-in book. I'd looked before, the second time I'd visited the place, to see if Jos\u00e9 was smart enough to use a fake name, and he was. At least he had no illusions that no one would come for them. It might take a couple minutes for the men to pry out of the man at the front desk where the two of them were staying. But it wouldn't take longer, so I knew I had no time to waste.\n\nFor a moment, as I stood in front of their door\u2014the cheap wood dry with the dust of the city, the unmistakable sounds of the two of them making love\u2014I thought about turning and fleeing. What the hell was I doing here? What if I was wrong about the guys in the SUVs? What if the men dressed in black that came out of the SUVs wanted nothing to do with them? What if they were looking for someone else? There's an edge, at the point of a knife, when you can fall to the side that gets cut or to the side left safely behind. But I didn't waste time knocking when I found the door unlocked. I opened the door and saw Jos\u00e9 on top of Sandra.\n\nHer hair was sweaty, spread to all sides. Jos\u00e9 looked at me and said, \"Son of a bitch! What are you doing here? I'll kill you.\" He reached for a gun, beneath the bed. He was ready for what he knew was coming. I put my hands in the air and held them there. Jos\u00e9 stood up, his chest hairless, thighs firm from playing soccer, hair still held back from gel and from Sandra's fingers combing through his hair. He was a naked man with a gun. Sandra tried to cover herself quickly with the blanket, but one of her legs strayed out from beneath the covers, and she could barely hide her breasts.\n\n\"What the fuck?\" Jos\u00e9 said. \"Mr. Jacobs. What the fuck? What are you doing here?\" He was still holding the gun pointed at me, but he lowered the 9mm a little.\n\n\"You have no more than two minutes to get out of here,\" I said. \"There are security guards. Coming. Fast. At the check-in desk. You have two minutes to get out of here, at best, or they're going to find you.\"\n\n\"How do you know this?\" Sandra said. \"What are you talking about?\"\n\n\"Get out the window, quick,\" I said, pointing at the window that led to the pathway between the motel and the building where furniture was made. \"Open it and get out now. Or come with me to the back entrance.\"\n\n\"You're crazy,\" Sandra said. \"If they've come this far they're going to cover every way out. They're not going to let us escape.\"\n\nJos\u00e9 put his pistol down, threw on his jeans, and grabbed his pistol again. He ran barefoot, without a shirt, to the window and tried it, but it was locked. I told them we had to go. I told them we should run out the way I'd come. I waved at them to follow me, but Jos\u00e9 and Sandra wouldn't listen. I ran back into the motel parking lot and hoped that if I led by example, they'd follow. I'd done as much as I could. I'd stuck my neck out way too far to stay around to see what would happen next. I ran to the back of the motel, farthest from the check-in desk, heading to the door I'd come in before. When I got to the back, I turned around and saw the men opening the glass door at the check-in desk, and I knew I didn't have time to fully escape, so I hid behind a Coke vending machine.\n\nI can only tell you what I heard next: the sound of men running with polished black shoes over the pavement. The sound of the men running up to the room where Sandra and Jos\u00e9 had been mating. The sound of glass breaking. Perhaps from a chair? Jos\u00e9 told Sandra he'd broken the window. He told her to go out before him. He must have had his arms behind Sandra, trying to push her out the window. The sound of a small gun firing, which must have been Jos\u00e9 shooting at the men running after them. The sound of the guards letting out round after round of their large pistols. The guards weren't from Jos\u00e9's father. They were from Sandra's. They shot Jos\u00e9. \"Come with us, you slut,\" they yelled at Sandra. But she told them she'd rather die than leave Jos\u00e9. \"If you come any closer I'm going to kill myself,\" she said, and she must have found a pistol, because there was one more gunshot and then she didn't make any more noises.\n\n\u2014\n\nA week after the death of Sandra and Jos\u00e9, I was in Polanco, in the evening. It was Friday night and I saw the Jewish men dressed in their black fedora hats, heading to temple. I had walked in the Parque Lincoln earlier in the evening, with Sara, after school. I felt I needed to be back in the park where we used to stroll. I needed the calm order of the tall palm trees that spread over the park, protecting the walkers below like umbrellas. After the shooting of Sandra and Jos\u00e9, I had told Sara immediately what had happened and where I was at the time of the shooting. I had even confessed to her I had been following Sandra and Jos\u00e9 for three weeks.\n\n\"You crazy, irrational man,\" she said, but then she held me tight, stroked my head, and let me know she loved me as much as ever. By her fingertips, I could tell it was precisely the fact I cared so much about Sandra and Jos\u00e9 that had moved her to give me comfort, and to marry me in the first place.\n\nAs we walked in the Parque Lincoln, and I saw some of the Jewish men gathering to go to the synagogue, I said to Sara, \"Why don't we go to temple? Why don't we find your father and tell him we're sorry he's so far apart from us?\"\n\nSara looked at me sternly. She gave me a look that said it was better not to stir up old wounds. What she actually said was, \"But you don't believe in God.\"\n\n\"Well, I don't,\" I said. \"If ever I needed proof of the absence of God, look at what happened to Sandra and Jos\u00e9.\"\n\n\"That's not proof of anything,\" Sara said. \"That's proof bad things can happen to good people, or to people who don't deserve what they get. But that's proof of nothing. Besides, you can't prove whether God exists or not. That's why they call it faith.\"\n\n\"Why should your father care whether I believe in God or not?\" I said. \"Isn't it enough I'm proud to be Jewish? Isn't it enough I try to live my life in a good way?\"\n\n\"For me, it doesn't matter,\" Sara said. \"For me, I can see your intentions are good, and that's close enough to God for me. But for him, God is God and you've betrayed his faith.\"\n\n\"I'd like to go to temple and find him,\" I said. \"I don't want us to live the rest of our lives feeling he has disowned you because of me. I'm not comfortable with that burden.\" Following Sandra and Jos\u00e9's death, I felt a need to go to the temple to find him.\n\nSara relented, after we had a few more rounds discussing the futility of seeking her father out. I saw a man with a black fedora, black pants, and shiny shoes, walking down the Calle Goldsmith toward Avenida Horacio, and I decided to follow him. Sara followed with me, though I could feel her lagging half a step behind, the weight of her resistance wanting to pull me back to the park. There was no need to follow the man; we knew, of course, exactly where we were going. There was no doubt where the temple was where Sara's father would be walking to this very moment, and who knew if this other gentleman was going to the same temple or to another? We walked together in silence for ten minutes, and then came to the imposing door of the temple with Hebrew letters written above the door.\n\n\"I'll wait outside,\" Sara said. \"I'm not dressed properly to go in.\"\n\n\"Please,\" I begged her. \"Do this for me. For us. With a gesture we can repair things. You were the one who told me a wall is built by two people, not by one.\"\n\n\"You go in first,\" she said.\n\nI paused in front of the long handle, feeling the weight of the door, the weight of the thousands of years of tradition the temple represented. But I am a man of faith, too, I thought, even if I don't believe in God. I am a man of faith in the brotherhood of all men, and I belong to my tradition as much as Sara's father, I thought.\n\nWhen I came into the temple the men were already grouped below on the bottom floor, davening, listening to the rabbi lead the prayers and reciting to him their responses to his calls. The women were upstairs in the temple. I saw Sara walk behind me, and then upstairs to the section for women, as I went inside the main temple area, below, to find her father.\n\nThere he was on the right-hand side, in back, as I remembered his place had been eight years before. It was as if he had stood in the exact same spot, praying, the whole time. His beard was now fully gray and his yarmulke sat sternly on his head like an anvil. I didn't want to surprise him too much, by coming up from behind, so I chose to enter his bench row from the center left, where he could see me for a while as I came closer to him. The wood of the long bench, as I moved sideways toward him, was worn smooth from generations of congregants coming every day to pray, especially every Sabbath. When I came close to him, I could see his eyes wander left and he seemed to see me. But he showed no immediate expression. He continued praying, looking at the book with Hebrew lettering in his hand, and calling out in response to the rabbi. It was as if I weren't there, as if an iceberg had come close to his ship, the _Titanic,_ and he refused to acknowledge any ice. His ship was impenetrable, a fortress, impregnable. I came closer and closer, and was close enough that I could easily reach out to his hand to touch it. But still he made no motion to acknowledge me. It was now or never. I could leave this second with things as they were. I could leave with the same wall of silence between him and me and his daughter that had lasted eight years. For a moment, I considered moving back to the left, sidling into the center of the temple, and leaving. But the thought of Sandra and Jos\u00e9 came to me, the sound of Sandra shooting herself, and I whispered into the old man's ears, \"Your daughter loves you and would like to talk to you. And I love you, too.\"\n\nAt the mention of the word \"daughter\" his eyes widened for a second, then they resumed their half-open shape of deep meditative prayer. If this plea wouldn't work, then I thought I would try one more. I turned to him and whispered, \"I'll do anything. I will do anything to make you happy.\"\n\nHe turned up to the balcony to see if Sara was there. He must have found she was. He turned back down, not acknowledging my presence. Then he turned to me directly and said, \"Will you accept there is only one God, that he is the God of Abraham and Isaac, and that he is your God? Will you denounce that you don't believe in God?\"\n\nWhat he wanted me to do was to denounce who I was. What he wanted me to do was to be only who he wanted me to be. What he wanted me to do was to believe in rote faith for no more reason than that I had been born into a faith. And even if I did that, I knew he would find other problems with me: that I wasn't Orthodox enough, that I didn't keep kosher, that I didn't follow my daily life as he wanted me to. He wanted to find divisions; and wanting to find divisions, he would always find them. Who was he to try to tell me what to do, when I had done nothing but try to treat him with respect, and to give love to his daughter? Sara wasn't asking me to make this choice. She had asked me, in fact, not to come see her father. She had had that wisdom.\n\nAnd yet, looking up at the front of the temple at the scrolls of the Torah, it came to me that what the man was asking me to do was to give a name to a feeling that was mutual. We both believed there were mysteries to the universe. We both believed the ways and reasons why men acted were often unknowable, and that they were often worthy of punishment. We both believed a life should be lived as morally as possible. He wanted faith, and faith I couldn't give him. But there is wisdom in seeking common ground, in breaking down walls that don't need to exist. If his faith meant I would honor him and his daughter and that I would seek to live a moral and just life, then I would say what he wanted me to hear. In naming, each hears the name they want to hear, and if it's the same name it will be heard as the same, even if it is different. \"Yes, I have come to tell you I believe in God,\" I said. \"I will no longer call myself an atheist.\"\n\nHe turned to me and hugged me, coldly, but with a pat on the head that let me know he was giving me forgiveness. He looked up at Sara and nodded for her to come down so he could talk to her. He went out behind the main room of the temple to the entryway to speak to her. And as he went up to Sara and hugged her, and whispered something into her ear, speaking to her for the first time in eight years, I thought to myself, I believe in the God of family. That will be my God, however we have to name it. It was not that I would believe in his God. I couldn't. But I would believe in the God of common names.\n\n# I WANT TO LIVE\n\nI was waiting in the waiting room of the Spanish Hospital in Mexico City. I'm a nurse. Or at least I used to be, until I retired down to Mexico. I came down, in part, because the cost of living had become more than I expected in the U.S. and mainly because, at a relatively young age, I decided I wanted to live my life to the fullest. I didn't want to die with the regret I never did all the things I would like to do. So, at the age of fifty, after working as a nurse for twenty-five years in Cleveland, I came down to Mexico City. For five years I integrated myself into the city and the culture. I learned Spanish, taking intense language classes and generally speaking to everyone I could.\n\nThings were going swimmingly until three weeks ago, when I came in to get a checkup at the hospital for the first time. I hadn't exactly come in with no idea of what they would tell me. My mother died of breast cancer and my aunt died of breast cancer and my grandmother died of breast cancer. It runs extra-prevalently in certain families. The same way some families have heart disease or mental health diseases, others have a risk for breast cancer. I usually avoid sharing such information with doctors, precisely because I am afraid of what they will tell me. My whole adult life I was a nurse giving out logical, rational information to patients, but the minute the tables were turned on me and I was a patient, it felt the other way. As a new patient at the Spanish Hospital, three weeks ago, they gave me a sheet with a checklist of all the health problems that might be troubling me. I could honestly check _no_ for almost everything on the list. When it came to a history of breast cancer in the family, I checked _no,_ at first. Then I scratched out my check mark. Then I left the yes box blank. And then, after tapping the pen they had given me on the clipboard, I finally checked _yes_.\n\nAfter a brief talk with Dr. Rodriguez, the general practitioner who examined me, they brought in another doctor, Jim\u00e9nez, a specialist in breast cancer. \"I see you have mentioned there is a history of breast cancer in your family,\" Dr. Jim\u00e9nez said. I explained to him who had died in my family. He insisted we do a standard genetic test to see if I had inherited the gene for having a high probability of breast cancer. The test was arranged that very day; they took a sample. And a week later, two weeks ago, they told me I had the gene BRCA1, which gave me an 87-percent chance I would develop breast cancer, unless I had a double mastectomy.\n\nIt was under these circumstances that I was waiting in the waiting room when I saw a woman across from me waiting, too. She was quite young, no more than thirty. She had long, brown, perfectly combed hair and a stunningly beautiful face. Her eyes were wide and round in a way that reminded me of Japanese anime dolls. Her cheeks were fleshy and healthy. Her ears were petite, and she wore a couple of big fake diamonds that seemed an effort to show off her whole face. It was hot in the waiting room. We were in the waiting room for reconstructive surgery. I was very torn by the idea of having my breasts removed. I wasn't sure, at all, that this was something I was willing to let happen. At the same time, I was deeply worried about the consequences\u2014a near certainty of getting breast cancer, sometime in the next five years\u2014if I didn't.\n\nLooking across at the other woman, I couldn't help but say, aloud, without thinking, \"It must be nice to still be so pretty and perfect.\"\n\nThe woman stared at me, then turned sharply away. I saw her fold one hand under the other in her lap. The suddenness with which she moved her hands struck me as odd. She played at the hemline of her skirt. The skirt was long and floral and accentuated the fact that she had a surprisingly fit body. She was in all ways, it seemed, a near perfect specimen of beauty, and it certainly didn't escape my notice that _her_ breasts were perfectly intact. She had on a red silk chemise that attracted attention to her breast area, and when she bent forward and down to fix the hemline of her skirt, smoothing it, I couldn't help but see her young, well-formed breasts. Her body was evenly tan and her breasts curved firmly upward, so that even with her bending forward, the firmness of the curve, and the health of her breasts, was more than evident.\n\n\"It's so unfair,\" I found myself saying, involuntarily aloud. \"Look at you. Just look at you.\" There were only the two of us in the waiting room, so there was no way for her to avoid me or to misconstrue what I was saying. She looked simply gorgeous in the most stereotypical of ways. She looked like she could be on the red carpet of an Oscars awards ceremony, with cameras taking photos of her. \"You could be a movie star,\" I said. \"And me, now they want to take away my breasts. Option A, they say, is to do a double mastectomy. Option B is to die. I used to tell patients about these kinds of crazy dilemmas and I would tell them all rationally, just like they have told me, but it's different when you're suddenly on the side of being a patient...But you, the gods seem to have made you perfect.\"\n\nIn the quickness with which the woman in front of me kept moving her hands, I could only notice her perfection, even her perfume smelled sweet. The woman tried not to listen to me; she looked away and laughed, a kind of shocked laugh when I continued insisting on her beauty, but when I declared the gods had made her perfect she shot up her right hand, holding the palm in my direction, and on her palm I saw, carved in a pink hideous X, a large scar that took up the whole of her hand. She raised her other hand, and the palm had the same hideous scar. She was marked with two Xs, puffy and thick, fleshy and with a rawness that indicated they would most likely never disappear, even with the best of reconstructive surgery.\n\nShe turned away from me and lifted her perfectly combed hair, held with hairspray that kept it perfectly in place, light brown with streaks of blonde, thick and rich. She bent to show me her neck even more clearly. It was a thin, pale brown, delicate long neck made for caressing; only, in the center of the back of her neck there was another X, marked as crudely as the first two.\n\n\"I'm so sorry,\" I said. \"How did it happen?\"\n\n\"Some things are not meant to be shared,\" she said. She spoke in Spanish. \"You're asking something too private.\"\n\n\"Yes,\" I said. \"I'm listening.\" Over the years, as a nurse, I have learned that what everyone really wants to do, what they really need to do, is to tell their story, locked inside. If they can just get the story out, it can do more good than even the surgery they are facing. I have learned that\u2014more than running around changing IVs and helping doctors prepare for surgery and changing bedsheets for patients\u2014the biggest role a nurse plays is psychologist. \"Yes, tell me. Tell me,\" I said. \"Tell me and maybe your story will help me feel better, too.\"\n\n\"You want me to tell you such a personal story, but you want me to tell it for you rather than for me,\" the woman said. \"Do you think you can stomach such a story?\" She held her palms in the air for me to clearly see the two Xs. It wasn't like me to blurt things out the way I'd done. I must have looked tense, an animal lost at sea. I had no family in Mexico. I was, completely, all alone. She was right, I wanted the story for me and not for her.\n\n\"Yes, I'm sure I want to hear about the scars,\" I said. \"I'm all alone, here in Mexico. I'm waiting to see if they're going to remove my breasts. I never had kids, and the man I was once married to left me. I'm divorced. I could use hearing your story. I'm in deep trouble. I don't know _what_ I will do if they get rid of my breasts. I think I'll feel like I'm just going to die, like they're getting rid of my body and soul.\"\n\nShe sucked in deeply and examined me. I must have looked well on my way to becoming a grandmother. She must have felt pity for a woman my age having to confront breast cancer alone.\n\n\"Forgive me,\" she said, \"but I can see you are a foreigner. Only a foreigner would insist so much I tell a personal story for them. Yet, since you insist, I'll tell you, because it seems it might help you\u2014on the condition you don't judge me before I finish telling you the entire story.\"\n\n\"Oh, I won't judge. I can see someone did something wrong to you.\" Since she spoke to me in Spanish, I did the same.\n\nShe leaned back and closed her eyes, as if retreating to a time far away. She kept her eyes closed as she began to speak, opening them only as she got into her story, so rapt in her tale she barely saw me as she talked.\n\n\"I have been a very foolish person. I have used beauty like a drug. But I became the person who did so much wrong, I think, or who participated in so much wrong because I grew up with nothing, first. When I was a kid, I was completely an orphan. I grew up in Sinaloa, and I grew up in an orphanage. My father ran off, away from my mother before I was even born. I have no idea where he went. He was a man who just wanted to plant his seed in every woman he could find. I hear he was from Brazil, but I really don't know. He came and went, arriving on a boat in a storm and leaving only a few weeks later, after he had wooed my mother. My mother was very beautiful, they tell me, and she died when I was two. She had been raised as an orphan also, so when you say the gods have made me perfect and that everything has been given to me, I laugh, because the only thing I know is that life is like a wheel that circles and circles. What happened to my mother was passed on to me. She chose a bad man to live with and to have a baby with, and I guess I did the same.\n\n\"But I loved my bad man. That's the thing\u2014I suppose I loved my bad man just like my mother loved hers. Bad men are like addictions. They lure you in with sex and fulfilling whatever dream you have, and they make your heart rush with their masculinity, and that's what Enrique did to me.\n\n\"But every addiction starts somewhere, and my addiction to my beauty began when I lived in that orphanage. I didn't know that I was beautiful, and what the power of being beautiful is, until I was six years old. Yes, of course, I knew that people always looked at me longer than they looked at other kids. When you are beautiful people envy you, they look at you and they want to get close to you. But they also want to punish you and harm you because you are everything they want to have and what they are jealous of.\n\n\"When I was six, a wealthy woman came to the orphanage where I was growing up. The orphanage was run by nuns. There were children crying all over the place. Each group of children had to look after the others. One room had the kids who were infants, on up to three-year-olds. Another had the kids who were from three to eight, which is where I lived. And then there were the oldest kids, who were, by then, considered untouchables who would never be adopted by anyone, destined for failure. The three- to eight-year-olds were expected to change the diapers of the babies. I was changing the diapers of one of the infants when the wealthy woman, La Se\u00f1ora Elvira de Castilla, came into the orphanage. She had a long face, full of wrinkles that had been stretched tight with plastic surgery. Her gray hair was as brittle and dry as dusty straw. She walked with her body rigid, like she was afraid of bending over because it might hurt her bony figure. She was someone, I would discover, who always looked in mirrors. The mother superior took La Se\u00f1ora around the orphanage, bringing her into each room, and when La Se\u00f1ora saw me she said, 'Stop. Stop. That one is beautiful. She is exactly what I am looking for.' She grinned at me like a cat looking at its prey. She came up to me and stared deep into my eyes, seeming to see a reflection of beauty too alluring and powerful for her, which reminded her of whom she had once been, or wanted to be. She pinched my cheek until it left a sting I could still feel the next day, when I was delivered by a nun to her dark, front house door.\n\n\"La Se\u00f1ora didn't treat me like an adopted daughter. She treated me more like her servant. She made me sleep in the servants' quarters with the other old muchacha she had. She made me wash the floors and clean food in the kitchen and dust the banisters of her house, where she lived, otherwise, alone. She made me wind up the grandfather clock in the old study where her husband had once worked, before he'd died. She made me stand behind her as she sat at the boudoir for hours putting on makeup, having me comb her hair, one long brushstroke after another, the gray, long hair, like pieces from a skeleton, coming off in my brush.\n\n\"She would barely let me go out, since she barely went out. She had caged birds in her house, in large cages I had to regularly clean. The birds stank, even though they were beautiful, exotic tropical birds from South America, and I would watch them gnawing at the bars of the equally exotic birdcages.\n\n\"So it shouldn't be surprising that, like those birds, I searched for any way I could get out of the house. I looked for excuses to go to the market to buy fruit and vegetables. I never had any money of my own to go shopping. La Se\u00f1ora would give me exactly the small amount she wanted me to spend on the food. She never gave quite enough, so she knew I would have nothing to spend on myself. It was always embarrassing to come to the end of the shopping and to find, with one of the last vendors, that I would have to tell them to take some of the food back because I didn't have enough money to pay for everything. I would walk home eating a mango or plum, and when I got home La Se\u00f1ora would tell me she knew I was eating some of the food and that I shouldn't be such a little thief.\n\n\"At the age of sixteen, then, it shouldn't come as any surprise I ran away, one day. I left without any extra clothes. I was afraid if I left with anything that seemed unusual, La Se\u00f1ora would notice and slam the door before I could leave, or she would immediately call the police and have them search for me. So I ran away with only the black dress I had on, with a white lace top like a maid's bib attached to the dress. For six days, I wandered the streets. I slept in alleyways so no one would find me, in the corners of stairwells, and once even behind a garbage dump, because I had looked in the garbage for food and then had fallen asleep, so tired. It was the day after that that I found Enrique, or, I should say, he found me.\n\n\"I am not going to lie and say he did not look like a drug dealer, or a pimp. He looked like a criminal, straight-up. He had on a white suit with a red handkerchief. He had on a pair of shiny, gray alligator-skin boots that were so polished they looked like leather mirrors. His black hair was gelled back with tons of oil, and his eyes were hidden by dark sunglasses. His smile was almost nonexistent. If he smiled, it was despite himself. But when he took off his dark glasses his eyes looked through me, and through everyone he saw. His eyes came to black points that shined like glass, which seemed to calculate and take everything in all at once, figuring out the value of anything before him, what kind of gold he could turn something into that looked to others like nothing. He had the eyes of someone who won at cards, who could sit down at a table, assess the situation, and walk away with all the money on the table.\n\n\"He took one look at me and said, 'Allow me, please, to take you out to breakfast.' He spoke to me like a gentleman, even though, by that time, my clothes were covered with dirt, and the lace bib attached to the dress was gray with filth.\n\n\"He took me to one of the fanciest places for breakfast, on the main plaza of the city. When he walked into the restaurant, the waiters all stood at attention. The headwaiter came up to him and said, 'Se\u00f1or Enriquez, your usual seat?' and they took him to a table that looked directly out onto the plaza. He ordered me a large breakfast of huevos divorciados, a plate of fruit, coffee with fresh cream, sweet rolls, and an extra plate of strawberries. La Se\u00f1ora Elvira had never allowed me to eat with her, only to serve her. I truly didn't know how to eat in the company of a man who was well-dressed and who wanted me to sit at his side. I ate, sometimes, with my fingers, because I didn't know any better. I picked up some of the pieces of strawberry with my hands, without thinking, though I tried to eat with a fork and knife because I knew that was what a lady was supposed to do.\n\n\"Enrique watched me eat like a wolf, and at one point he began to laugh, after I told him a little about working for La Se\u00f1ora, and as he watched me eat the fruit with my fingers. 'She tormented you,' he said. 'She turned you into an animal. That's what I like, a beautiful animal. The regular women out there, they are either too soft or\u2014if they are tough\u2014they have no physical beauty. But you have both, because of the way you were raised.' He told me he was going to buy me all the clothes I wanted. He told me he was going to let me see just how beautiful I was. He told me I was the most beautiful young woman he had ever seen. He was no older than twenty-nine, but I liked that he didn't look like a boy. I wasn't looking for a teenager then, like the boys I had sometimes stared at with longing in the marketplace. I wanted someone who could keep me safe, and he had everything I needed then.\"\n\nShe stopped, suddenly, and came out of her dream for a second. She looked over at the secretary in the waiting room, but it was clear there was no movement forward from the doctor to speak to either of us. We were two patients who were going to have to wait to see the doctor. It was a Tuesday around lunchtime, and who knew if the doctor was even with a patient, or if it was just going to be one of those waits, in Mexico, that can take forever?\n\n\"And the scars?\" I said. \"What happened with the scars?\" The lady's story was engaging, but I had to admit I was getting impatient to hear about the scars. I was still trying to figure out what any of this had to do with my potential double mastectomy. The lady had suggested there was some connection. I was glad she was getting her story out, and I wanted to hear more, but I was feeling anxious about what the doctor would tell me, and this story wasn't necessarily comforting me.\n\n\"The scars are always there. Even before they give them to you. But I told you, you will have to let me finish the story. You want everything to be revealed quickly, you want this story to be immediately about you. Just listen. This story is not instantly about you. It's about me. And you will have to not judge me.\"\n\n\"I promise,\" I said. \"I guess I promise.\" I was beginning to lose my patience, a bit.\n\n\"Do you want to hear, or not? Just listen to what someone else has to say.\" I heard her say the word \"selfish\" under her breath. She shook her head and pressed her hands, anxiously, one against the other. She looked at the palms of her hands, where the scars were. She looked at them, seemingly in disbelief, and continued.\n\n\"Over the next two years, once Enrique picked me up off the streets, I began to shop and to buy clothes, voraciously. I bought long, bright gowns, purple and gold, with sequins sewn all around the necks of the dresses. I was only sixteen when I met Enrique, and I had never had a quincea\u00f1era party, so Enrique gave me a sort of party like that, even if it was a year too late. He invited the various other small-time leaders of the drug cartel he was in. It was a way of trying to move up the ranks. At the same time as he was selling more drugs and killing more people and proving he could be tough enough to become one of the bigger leaders of the cartel, he wanted to show me off as his girlfriend and as the woman he intended to make his bride. A couple hundred people were going to come to the party, and he wanted me not only to look good but to be able to perform something for the guests. 'I think you could be a good entertainer,' he told me. 'Most of those singers who get all the attention on TV, they start out like us\u2014people who the rich try to keep down\u2014but then they show their pure soul and their talent and they prove the rest wrong. I don't see any reason why you couldn't be a famous actress or an entertainer. Just look at you. You're more beautiful than Ninel'\u2014a popular singer, at the time. 'We're going to turn you into something, too. You and I, we're going to rise, together.'\n\n\" 'Really?' I said. 'You really think I can be something? You really think I could be someone people would admire, with talent?' When I said 'talent,' I was thinking of the soap opera stars. It never occurred to me to think of a serious stage actress or a serious opera singer, or a simple singer in a church choir. I wanted to be like the pop stars and the soap opera stars and the stars of the movies in the theaters, which La Se\u00f1ora Elvira had never let me go to, and that I went to with Enrique religiously.\n\n\" 'With your beauty, anything is possible,' Enrique said. 'Life is for the taking. You have to take what you want. The people who want to stop you from becoming what we want to become are everywhere. But I'll be your prince. I'll make you into what you want.'\n\n\"So I started to take dancing and singing lessons. I started to pose in front of the mirror as I sang, and to practice my moves. I learned to sway dramatically in those most sentimental of moments, when you want to hold the audience enraptured with your song. I started out learning traditional songs, because tapping into the traditional songs was all I knew, at first, and they were easier to learn in voice lessons. I put on more and more jewels. Fake ones, at first, much bigger than these fake diamonds you see me wearing now. I covered myself in big hoop earrings and rubies. I watched videos of the pop stars, and I did the same moves with them on TV, walking back and forth on my imaginary stage. Jumping and shaking and twirling. Enrique told me I had always had the talent within me to be a star, it's just that others were preventing me from being what I could be.\n\n\"You see, we were in this together. The idea was we were going to both rise up until people could admire us the way Enrique said we should be admired. That's why Enrique wanted me to compete to be a beauty queen, and that's why he was strategizing and working so hard to move up the ranks of his cartel. And I want to tell you, I could see how much we needed each other. At the party for me that he threw with his couple hundred guests, he would pull my hand, bringing me forward from one group of invited men to another, showing me off, telling everyone I was his novia\u2014his girlfriend. He told me to wear a bright red dress, because he knew no other woman would have the guts to be dressed so brightly, so young, and at a party where I was meeting all these men for the first time. As each of the men would come up to me and Enrique, to shake his hand and to show their loyalty to him, or for him to show his loyalty to them, the men would whistle under their breath, as if to say, 'Oh my god, how did you get such a beauty?' They gave Enrique the look that said, 'I wish I could be in bed with her. Caray!' They shook their hands rapidly in front of their faces up and down when speaking to the others, just next to us, after they had met me with Enrique. It was the waving of hands that said: 'She's too hot! Fuck, man. That Enrique is one lucky cabr\u00f3n.'\n\n\"You might think a young woman would feel like a cow on display, like a sex object these crude men wanted to rape, if they could. And you might not be far off, if you did think that. But that's not how I felt, at the time, at all. What I felt was that I was radiating. What I felt was that, for the first time in my life, I was the object of attention, not because I was the servant being called to attention by La Se\u00f1ora Elvira but because I was envied and admired.\n\n\"When I sang at that first party, up on stage with a group of sixteen mariachi players Enrique had brought in just to make me look good, I sang so-so. I looked beautiful on stage, but my voice wasn't as good as how I looked. Enrique was furious with me, afterward, that my voice wasn't good enough. 'You are going to have to do better,' he would tell me. 'You are simply going to have to do better. I am not going to let you embarrass me in front of the people who I invite to my parties.' And then he hit me. It was the first of many times he would hit me on the cheek. But out in public, after I had sung that first time, he came up confidently on stage, holding the microphone so close it gave a reverberation feedback, a wince-inducing high-pitched echo as everyone watched Enrique on stage, and he said, 'Wasn't that just beautiful? Please, everyone, give a round of applause for Esmeralda Sanchez. She is going to be the next movie star and pop star of Mexico.'\n\n\"People clapped politely, but without much enthusiasm. Enrique went backstage to tell me to do better, and he hit me. I felt the pain so sharply, after having tried my best, after feeling I was finally beginning to be someone more than what La Se\u00f1ora Elvira had told me I was, after having been presented to all the other men at the party. But I didn't take this as a sign Enrique was at heart what he, of course, was\u2014a violent man. I took it as a sign I had failed and that I needed not to fail the next time. I would need to work harder.\n\n\"And so, I did work harder, one dancing and singing class after another. Pushing myself harder and harder. Singing with records and tapes in the dance studio Enrique constructed for me. The dance studio began to feel like one of those birdcages La Se\u00f1ora Elvira had in her house. At times, I felt I wanted to stop practicing, but Enrique would come and check in on me. He would even lock me in there, occasionally, if I told him I was tired of practicing so much.\n\n\" 'I need a break,' I told him, once.\n\n\" 'You can take breaks another day,' he told me. 'Everyone out there in this city, they think the only way to make it in this world is to be born with connections, to be born rich. But you and I know otherwise. When you have nothing, you have to make it yourself. I'll be your connection for you, but you have to have the talent, too.'\n\n\" 'And how do you show _your_ talent?' I asked him.\n\n\" 'Excuse me?' he said, with indignation. 'Are you saying it doesn't take talent to lead my business? To make my business decisions every day?' He was building up his section of the cartel, more and more. He was rising, quickly, in the cartel. This is what he told me regularly, and it was something I could see, as well. The men around him were becoming more solicitous. They started to walk with him, his old friends, less as equals and more as men showing respect to their leader.\n\n\"A year later, I finally fulfilled Enrique's dreams. Not his final dreams for me at all, to become a big star, but his dreams for me in Sinaloa, as a first step to the big time. I competed to be Miss Sinaloa. My makeup was caked-on thick, and I had extensions on my lashes. I posed in evening gowns and in a green bikini with bows. I crossed one leg alluringly in front of the other. The men whooped, in the audience. They hooted and hollered with lust. The announcer had to tell them to keep their shouts down and to show some respect. I felt the heat of the lights and the sweat on my forehead, and I worried the heat might somehow ruin my makeup, that something out of my control might cause me to be just a little off and to lose the contest, which would cause Enrique to lose faith in me and to hit me, as he had done, by now, a number of times. I began to associate my beauty with the pleasure of Enrique's adoration, but also with the pain when I disappointed him.\n\n\"But on this day, when I completed my dance and singing routine, I was relieved to see I had not failed Enrique. The announcer on stage, dressed in a tuxedo with big white ruffles, placed the crown on my head, and I felt like a true princess wearing that tiara. I felt the love and adoration and appreciation of the crowd. I began to cry on stage, as all beauty queens cry, but I was crying not only because all of my hard work had paid off, and not only because I knew Enrique would be proud of me and would love me more, and because he was right, he had pushed me hard and proven to me that with hard work I could be something more than I thought; I was crying because I felt for the first time I had escaped the clutches of La Se\u00f1ora Elvira and the clutches of the orphaned life, the wheel of life that had first made my mother an orphan and then me. I was escaping from that past. The wheel of life could no longer control me. And so, I cried on that stage, letting my mascara run a bit beneath my eyes, a mark of imperfection that was perfectly meant for the moment.\n\n\"Enrique didn't wait to meet me after the contest, behind stage. He ran up the side of the stage and took the microphone from the announcer, who looked at him as if this was a bit odd, but by then everyone knew who Enrique was and that they shouldn't disappoint him, or their personal safety might be on the line. Enrique lifted my hand in the air, as if I was the champion boxer of the country, as if I was not the beauty queen of Sinaloa but the real queen of the city and nation. 'Ladies and gentlemen, this is my queen,' he said. 'And I wish to announce we are going to get married.' I was as surprised as everyone in the audience. Enrique hadn't said anything definitive about marriage, up to this point. He had kept me close, as if I was the only one who mattered to him, but I knew well that a man like Enrique could lose interest and suddenly want to be with another woman. He was tough and full of intensity, full of whatever he needed at any moment. Announcing the marriage, on stage, I was both filled with elation and relief, a sense that now he would be mine and I would be his, that we would truly fulfill, together, our dreams and his dreams. As I said, I felt he knew more than me at that point. But I also wondered why he hadn't asked me, in private, as he should have. The side of me that had been caged for so long could not help but ask him, behind stage, why he had made the announcement before asking me what my intentions were.\n\n\" 'Are you disobeying me, mi amor?' he said. 'Do you think I don't have good reason to tell everyone out there?' I could see the well of anger within him bubbling to the surface at the idea I had challenged him, for a second. And then he switched moods, entirely, pretending I had said nothing and that he had said nothing that was acrimonious. He swept me off my feet, with my tiara on my head, and told me he loved me and adored me and that he wanted to make me the happiest wife I could ever be. He pulled out a big diamond ring, which was a real diamond, not like the other fake costume jewelry I so often wore. He slipped the ring on my finger and pulled me out to the car, a long, white limousine he had rented just for the occasion. There was a sunroof. We drove around in the limousine, with a jeep of his cartel compadres carrying weapons in front, and a black SUV with support in back. The three vehicles drove around the city, honking, letting everyone know I had just won the Miss Sinaloa contest and that we were going to get married. Once in a while, Enrique would tell me I should stand in the back of the limousine and peek out the rooftop and wave at my public. 'Go ahead, wave! Wave to them. They are all your people, now. They all adore you.' I did as he commanded. I was with my prince, the man I was going to marry. I stood with my head through the roof, waving, with a crowd in the central plaza waving back at me, and Enrique standing next to me. He held my hand, fingering the diamond he had placed on my ring finger. He gave me a big kiss on the lips, in public, in the central plaza. Shouts, egging him on to be manly, made him kiss me again. I wrapped my arms around him, and for a fleeting second, despite the fact I knew I was more and more his possession, despite the fact I knew he was the leader of a cartel that others feared, despite the fact I knew his cartel must be killing hundreds of people, despite all those things I grabbed him closer to me and I felt, 'This is my man.' \"\n\nListening to the woman, who I now knew was Esmeralda Sanchez, who I thought for a second I had seen on TV, once, in a cantina\u2014now that she'd mentioned she'd been elevated to Miss Sinaloa\u2014I could see she was reliving each experience as she told it to me. She sat in the waiting room waving her arm as she'd once waved at the public. Only now her hand had a scar on it.\n\n\"And the scar?\" I said. \"The scars...\" I tried to remind her she was supposed to be telling me about her scars.\n\n\"The story of the scars will come soon enough. But I cannot just tell you the story about them right away, or you won't understand, and you won't understand why this story is important to you and why I'm telling it to you.\" She stood and straightened her long flower skirt and walked to the far end of the waiting room and filled a cup of water from a watercooler. She went to the bathroom to freshen up, it seemed. I asked the secretary if the doctor was back yet, and she said he was still busy with another patient, or maybe out to lunch. I'd wanted the doctor to come quickly, before, so I could talk with him about whether I should have the mastectomy or not, and to have the certainty of a decision made crystal clear. But now, before I met the doctor, I felt I needed to hear the rest of the beauty queen Esmeralda Sanchez's story. I hoped the doctor wouldn't come before she could tell me what I needed to hear. It seemed to me all of a sudden that she must have the answers, like a fortuneteller, as to what I should do, she was telling her story so passionately.\n\nShe came back from the restroom, sat down, and began to tell her story, again.\n\n\"Three years went by, with Enrique's business in the cartel growing and growing, with other narcos feeling more and more threatened by him as he took over more of their territory in the drug business, when one night, on June second, a date I will never forget, a group of men broke into the house where we were living, who worked for another cartel. By this time, we lived in a house on the beach that looked out over the water, with high ceilings, crammed full of the art and objects that Enrique felt proved he had made it in his cartel work. He had been collecting large jade and malachite Buddhas and scenes of ancient fantasy landscapes, in the clouds, from China. He bought them not only in Mexico, from dealers who brought to him the art they thought he would like, but also from places like Hong Kong, where he occasionally went for his work. He had a collection of French crystal statues of sports cars, over fifty objects in a case at the base of the stairs leading up to our bedroom. The objects were truly rare, and at his command a number of lights lit the sports cars when he showed off his crystal to guests. At the front door, when visitors came in, there were porcelain cheetahs; and, in an effort to please him, I'd selected red fur coats and suede leather couches of purples and aquamarines, which made the whole place feel like a very upscale home, with rococo designer furniture. A Porsche 911 and two Porsche SUVs sat in the garage. He'd attained more and more of what he wanted, but more was never enough.\n\n\"I saw him less and less because of his work, and because I tried to appear in movies and TV shows, just as he desired. I had a bit of success but not nearly as much as his. I had the looks but not, necessarily, the full talent. And soon, I had even less of the looks. Much less.\n\n\"The night when they came and hurt me so badly, a group of three men, the rival narcos, came into the bedroom. Two held me down on the sprawling, custom-made bed. They did not rape me. I will never understand why, though I think they understood it would torment Enrique more if he could never know, for certain, whether they had raped me. Instead, they carved one of the Xs on one of the palms and the scar on the back of my neck, and an extra scar which I haven't shown you, an X that is the same as the others, on my stomach, just over my belly button. It is important to know they did not make the scar on my other hand. That would come by me, later.\n\n\"I could go into gory details of how they carved the scars into my body, but why turn this into a horror movie? The facts are bad enough, aren't they? The fact they took their time as they cut me. The fact they knew what they were going to do, before they came. It wasn't a crime of animalism, in the heat of the moment as they fought with my husband\u2014who, in fact, was not there. It was a crime of meditation, a well-thought-out act to send my husband a message that his most precious object, the one he had shown off for years as the beauty queen of Sinaloa, was not untouchable, was scarred and belonged to them, and was deflowered.\n\n\"Only, my husband had already, even before the incident, begun to care less and less about me. He was already beginning to lose interest in me. He was already beginning to look at other women, and I suspect to sleep with them, and once I was scarred by these men\u2014in the middle of the night, with blood soaking into the sheets, and especially with the mark over my womb, which had failed to produce a child for us\u2014he lost all interest, completely. He simply almost never came home.\n\n\"The absence of his attention, even though he had hit me over the years, even though he had always treated me more as his possession than a mutual love built by the two of us, made me feel more and more insecure, more and more desperate to win his attention back. I bought brighter and richer clothes, if that was possible. I went shopping for Louis Vuitton and Prada handbags, dresses from Versace, not the most expensive haute couture dresses because those were out of Enrique's league, but I bought whatever he could afford and whatever he would allow me to afford. I bought gold and platinum baubles and sat in front of the mirror and combed and recombed my hair, feeling the brush scrape against my neck, sitting in front of the boudoir off the room where they had scarred me, combing my hair just as La Se\u00f1ora Elvira once had me comb her hair for hours. Once free of her, I was now acting as if she still possessed me, as I tried to repossess Enrique.\n\n\"But Enrique was with other women now, barely hiding them anymore. Once, when I confronted him about an especially young woman, who looked as young as I'd been when Enrique had first found me on the street, but whom, looking at her at a party, I felt had none of the charm I had originally had, none of my toughness, I wondered what he saw in the woman he was nudging up to so closely at the large party, a party now with five hundred. I had sung at the party, and people had politely clapped. I sang better than that first party Enrique had me sing at. I had the moves down now, after years in the profession. But I was still the mediocre performer I had started out as, a beauty queen who was no longer all that beautiful, with the scars on my hand and neck and on my womb, which I now hid. 'What is it about her that makes her so perfect for you?' I asked him at the party. If it seems strange I asked him then and there, directly to his face, it was no stranger than when you, earlier, asked me why the gods had made me so perfect. What I am saying is, the question was involuntary, a question I should not have asked, unless I wanted to be nearly destroyed.\n\n\" 'What makes her so perfect,' Enrique said, 'is that she is what you once were but no longer are. She has potential, and what kind of potential do you have now, with the scars they did to you? But I want you to know I am not rejecting you because of your scars alone, though they are hideous. I am rejecting you because you have lost your edge, your talent, your drive, your animalism. Look at you now, so soft and draped in all of these clothes. I can barely see the puma inside of you that I once wanted. You have failed to give me a child. You have failed in the most basic task of a real beauty queen. Infertility is hardly becoming. Every king needs his heir. Every king must have his concubines. And I have to say, when I look at the scars on your body, I can't help but wonder if they raped you, too, but you just won't admit it.'\n\n\"They were words, in short, of pure bile. Pure hate. They were words to hurt me like a boy who holds a magnifying glass in front of an ant, keeping the focus of light that is too hot on the ant until it burns and dies. And here is the part where you will surely want to, and have to, judge me. Because after that evening, when he said those things to me and touched his new, young female thing, I went up, into the room in the house where we lived, and I took a sharp knife from the kitchen, a small one usually reserved for paring apples and potatoes, and by my own will and with my own self-loathing and with loathing and anger for Enrique, I took my left hand out, a hand that had never been scarred by the three men who had tied me down, and I carved slowly and methodically, and then faster and deeper, with punctures of pain, with pain more that I had let myself become Enrique's thing and his object, and then that I could no longer be that object of desire for him. I wanted him to know I was in torment. I wanted him to wake up from his smug certainty that he knew everything and controlled everything. I wanted him to feel for a second, when he would hear about this carving of my flesh, and when he would be forced to see it, that I loathed him now and myself now so much that I had had the strength, the animalistic strength, to do this self-mutilation.\n\n\"After sleeping, most likely, with the other young woman at the party, he came in the door at four a.m., smelling of sex, with the smell of his cologne mixing with the perfume of another. He came up to his bedroom. By then, we didn't sleep in the same room. Like a phantom, I walked into his bedroom, once he was undressed and getting into his bed. I held my palm up in the air, with blood dripping on the carpet, on a tiger skin rug he had brought from Hong Kong, which stood at the base of his bed. I held the palm up in the air until he could tell something was wrong, something was desperately wrong.\n\n\" 'I did this,' I said. 'I did this because of you!'\n\n\"He rushed up to me and looked at the hand, and the wound was so fresh even he could not deny what he had done. 'Oh, Esmeralda, Esmeralda, what have you done? What have you done? You should never have done that. Never.' He looked at me with a look of pity, and yes, some concern, and maybe even, oddly, a bit of pride that I could be, momentarily, an animal. But I was no longer the queen he wanted. There was a new queen. She would be his new pretty face. The concubine, in those old Chinese tales I would read later, replaces the queen, and even my most desperate cry out could not win him back.\n\n\"The rest of the details of my story are hardly worth noting, how the police and military finally set up an operation to take him out. Usually they took their cut and left Enrique alone, but one day they came in force to kill Enrique, and he got word of their plan first. He met them in a full gun battle in the center of town. The one surprise for this story, perhaps, is that I fought beside him. I, too, carried an AK-47 and fired in the direction of the police who were firing at my husband. We hadn't had sex for a year, when he was shot and killed. The bullets, they truly riddled his body and he fell in the street. He had taken me with him to flee to the next compound. He had rejected me in almost all ways, but he still seemed to want to keep me in the cage of his house, and he still insisted I continue to try to make it in the music business, though my recordings were fewer and fewer. But when the cops came, I was standing next to him, a loyal follower to the end, admiring his drive and talent, loathing my own body, wanting to do what I could to please him. He was my man, and I foolishly still loved him.\n\n\"But it was not then that I learned what I needed to from the experience. It was two years later, long after all the furs and clothes and houses and boats had been confiscated by the government, and I was living in a very small house that I could barely afford, renting only a room in the top floor which had an attic that was too hot. The police had determined I was not a real narco, just the wife of a criminal, and they had let me free. I was already twenty-six. I was walking through the town, one day, and I saw a young girl, and I could see she was an orphan from the same orphanage where I had grown up. I followed her, as if involuntarily, back to the orphanage. There, despite all the changes to my body, was the structure of the orphanage, almost identical to when I had left. Time had flown by, my body was now permanently marked, and I had attempted to run away from this orphanage for so long, and from the orphanage somewhere else, unknown, where my mother had grown up. And standing in front of that door I realized I had spent my whole life running away from that door and that building instead of running to something. I had been running with the subconscious fear that if I did not become a beauty queen, if I did not please Enrique, if I did not do what everyone else wanted from me, I would end up in the orphanage forever, not just the orphanage in front of me but the orphanage of a trapped spirit, a trapped soul. And in my fear I had ended up scarring my body more than anyone else could have ever scarred me, trying to live for others, instead of ever truly living for me, or ever even truly living. 'I want to live,' I told myself. 'I want to live for me and for me alone and not for anyone else and not out of any more fear.' I opened the gate of the orphanage. I walked into that door, the porthole that had marked the gatekeeper of my greatest fear, and I walked inside. I looked around the rooms of the orphanage\u2014first in the room for the babies, where I had once changed the diapers of the smallest infants, and then in the room where La Se\u00f1ora Elvira had come and snatched me away, when for a fleeting moment I had felt some hope, until the once beautiful witch had dashed my hopes. 'I have paid for my beauty,' I told myself. 'I have paid more than enough.' I walked up to the head nun. I raised my palm to her. 'Mother Superior, I have sinned against myself for not loving myself.' I walked around the old orphanage, looking into the corners of the place that had once been my deepest fear. I was not going to let the place dominate me anymore, if I could. I was no longer going to run away from myself, couching my fears in my body. I was going to find some other orphanage to work in, a place where I could help some new young girls.\"\n\nShe stopped her story abruptly, opened her eyes, and looked straight at me and told me she had gone to work in an orphanage. She had completed her promise to herself. She was exhausted from telling her story, and she quickly walked to a sink in the corner of the room to wash away tears that had come to her.\n\nWhile she was at the sink, I thought about her story. It seemed foolish to try to reduce what she had said to one idea or two. Was she saying beauty is only skin-deep and that I should worry less about any scars to my body, because she had suffered much worse? Was it that I should worry less about my fears, and that I should confront them head-on? Was it that I should escape the cycles of life that she had been caught up in, just as my own family had cycles and cycles of breast cancer? Or was it what I think she was really getting at\u2014that to give means more than to take? It was all of these things, perhaps, but when I walked into the doctor's office, when he finally came, after I'd thanked the woman for telling her story to me, a telling that I believe and hope was of some cathartic use for her, as it was for me, I walked into the doctor's office and I told him I needed a double mastectomy; I told him I had made up my mind firmly and said to him, \"You see, doctor, even with all the pain and cruelty and torture out there in the world, I want to live.\" There was nothing more magnificent, more surprising, more awe-inspiring, more mystical than being alive in the world. I thought of the woman with the scars on her body, who had told me the story. I thought of her in her new orphanage; and even with her scars, because of all of her suffering, because of her honesty and her fortitude, her spirit unbowed by the pain, and because she had been there for me when I needed someone to comfort me when I was alone, she was the most beautiful woman I had ever met. She had given me something when I was feeling wholly selfish and afraid. She was much more than her mother could have ever known.\n\n# ACAPULCO\n\nAt 3:42 a.m., according to the police report, we left the nightclub. But don't take the reports too seriously down here in Mexico. They're a joke. That time seems about right to me, though. I have a watch, an expensive Piaget my father once gave me, and after the whole shooting was through, I heard the faint Swiss, precise ticking and the time said 3:50.\n\nI ended up in the nightclub because the client wanted to show me a good time. Never mind that I have a girlfriend\/partner who I've been living with for two years. She and I are going to get married in a few months. But the way business gets done in Mexico, the older men\u2014say about fifty-five\u2014they don't care about being faithful to their wives. They have \"lovers\" on the side. They go to motels on the edge of town to have sex with their secretaries or with their hidden \"gem.\" Me, I'm not into that kind of tradition. It smacks of a macho culture that I left behind when my parents sent me to study architecture at Harvard School of Design. I grew up in Mexico City. My whole family is from Mexico. But I'm halfway in, halfway out. I consider myself proudly from Mexico, but I'm a citizen of the world. I'm as happy eating sushi in Tokyo as having a quesadilla on the streets of Mexico City.\n\nWe went to the nightclub to close the deal. Gonzalo was the client. Sixty years old. The usual paunch that most older, rich Mexican men have. It's the paunch of the _felicidad_ \u2014the happiness. Gonzalo wears a gold Rolex. It's the kind of flashy watch I would never wear because I come from old money, and his money is new. Who knows from where? There are some questions it's best not to ask, with all the cash floating around Mexico these days. If you're an architect, as I am, you don't ask where the dough is coming from, you just ask, \"What kind of pool do you want?\" You give them an option\u2014large or extra-large. I've designed pools that extend off the house and into the ocean, with a long jetty that slices into the Pacific. To me, it's never made sense why a client needs a pool right next to the warm water of the ocean, but for the clients the pool is where the action is at, the trophy, like the trophy bride, that is always immaculately clean, radiating turquoise up to the sky, the center of their personal temple. And believe me, for the clients, their house is their personal temple.\n\nMy own taste is toward the modern. I'm a good architect but not a great architect. I can do nice, sweeping open-floor plans. I can get you that view you want that makes you feel like your house is worth a few million. I do my best. I struggle to get the details. I love the clean, white look of _pilotis_ from an architect like Le Corbusier. But no one is ever going to remember me. Fortunately, you don't have to be amazing to make a living down in Mexico. You just need to be willing to give the client what they ask for. There is a long tradition of great architects in Mexico, like Luis Barrag\u00e1n, and, as is usually the case, most people don't buy their homes or know about them. Bad taste reigns supreme, everywhere. I try to find the clients who have less bad taste so they don't give me too many problems. I build them what they want, and then, in a few smaller places, I try to design what I really care about. It's the life of an architect. So it goes. And it goes easier, since my father is a fairly well-known architect in Mexico.\n\n\u2014\n\nWhen we came into the nightclub, the first thing I noticed was the pole dancing. This club was on the edge of Acapulco, out of town about fifteen minutes, up on one of the side hills and at a sharp pullover. The club stands alone, with no other stores nearby, and it has a sign with two cartoon-cutout women dancing with martini drinks in their hands, all lit up with fifty flashing incandescent bulbs. Some of the lightbulbs are gone, so the sign looks like some teeth are missing.\n\nGonzalo's driver parked his Mercedes-Benz. Gonzalo got out and I followed him in. It was the third club we'd been to this evening. In Mexico, everything's about excess. If you have a party, the music volume has to be at eleven. If you invite one of your cousins over for lunch, you have to invite their parents and their sisters and brothers. It's the same in business. There's no such thing as going to one club to close the deal.\n\nWhen we came into the club, the driver waited by the door, acting as a security guard. Most of the drivers do that for someone like Gonzalo. He was traveling light this evening. He only had his one driver for security. Often, a guy like him drives with two extra cars around as his security detail\u2014one in front, one in back, hugging close to him through traffic with a couple bodyguards in each car to make sure no one kidnaps him. This isn't just for nefarious people. This is for anyone with money. They never know when some group of professional kidnappers or hit men will try to get them. The guards look like Secret Service details. They dress in dark suits and have earpieces to communicate with each other. The main difference is they don't have to hide their weapons as much in Mexico. They let people know they're nearby. They try to be a presence, without bothering anyone at the party or at the business meetings.\n\nI think Gonzalo was traveling with only his driver as his guard because he wanted to be able to take me to some of the smaller, seedier bars, and with four guards it would have been too intrusive. He told me at the beginning of the night he wanted to take me to some of the traditional nightclubs because the women were better there\u2014not so jaded, he said, not so phony. \"I like them with fat butts, you know what I mean?\" he said. He put his hands up in the air and squeezed, as if he were squeezing a big, round ass. \"For me, if they don't have a pretty face, it's a plus. It means they like what they're doing. It means they probably have a sweet boy back home that they take care of. They're mothers. I like the mothers...\"\n\nThat chain of logic made absolutely no sense to me. If they're mothers they were working in the clubs because they had to, not because they wanted to. But I think it's fair to say Gonzalo had a Mommy complex. He needed some kind of mother he maybe never got as a child, and he wanted me to share his fetish. Me, I wasn't interested, but the guy wanted me to design a four-million-dollar home off one of the nearby golf courses, so I was willing to go along for the ride.\n\nIt was just like what you might expect for a cheapo porno bar. Up on stage, to the left as you came in, the lighting was red and dark, a single white spotlight on the woman at the pole. If she was a mother, she was a young mother, no older than twenty-five. Her nipples were big and brown, so she probably was a mother, and her ass was wide, just like Gonzalo wanted, but otherwise she was slim. She shimmied around the pole. She wore a pair of black-velvet high heels. She had on a red G-string. Otherwise, she was naked, cinnamon brown. Gonzalo walked straight over to her. He pulled out a five-hundred-peso note. The lady turned, with her ass in front of his face. She jiggled it up and down, back and forth, like she was playing the bongo drums, or like some Disney porno film version with her ass as some bongo drums. The standard rule with the pole dancers\u2014as I'd learned earlier that evening, and with other clients closing deals\u2014is that you don't touch the dancer. You can push money in her G-string, but you don't touch the merchandise. For that there are lap dances, and there are always plenty of women circling around the nightclubs to offer lap dances and to fill your drink. Gonzalo came about as close as you can to violating the rule. He put his hands up behind the dancer's butt, he came within a millimeter of touching her, putting his hands around each cheek and pretending to squeeze in the air. This was the third time of the evening he'd done this, as if he thought it was hilarious each time. One of the advantages, from his point of view, seemed to be that for a moment his face and hands were caught in the spotlight, so everyone could see he was the big man, the guy putting in the five hundred pesos. After he stuck the money into the strap, he turned around to me and gave me this look like, Not bad, eh?\n\nI gave him the look of approval he wanted.\n\nTo the right of the door, as you came in, there was a long bar-table with a view of the stage to the left. I took a seat at one of the stools and Gonzalo wended his way back to me. He chuckled as he sat down. His hair was combed straight back like some kind of Latin American dictator from the '80s. He had on a pink button-down shirt, the collar wide open showing some black hairs mixed with gray. He had a number of gold teeth fillings that were more noticeable whenever he put his hand with his Rolex up by his mouth, while laughing. There was an old scar over his left eye. His face was smooth from eating a lot, full of fleshiness, and he had a constant slight sweat from eating and drinking. He seemed to feel a bit hot and sweaty wherever he was.\n\nThe whole evening he'd said normal things to me, talking about girls, asking me about my girlfriend, asking me about my father. There wasn't much to say, he kept the conversation light, but now he leaned in close to me and put his heavy arm around my shoulder and whispered in my ear, \"You're a pussy, but even though you're a cunt they say you're a good architect. I want a palace. You give me a palace and I'll be happy and you'll make some good money. You fuck up, you give me a house that other people don't like, and I'll make your life miserable.\"\n\nThere was no mistaking his words. It was some kind of strange threat, from out of nowhere. How did I get this client? My brother Rodrigo had a friend who played golf in Acapulco. That friend had said he knew Gonzalo on the golf course. Three degrees of separation. In Mexico there's usually a personal link, or nothing happens. Rodrigo's friend was the link. I was going to kill him.\n\n\u2014\n\nThe whole time we were in the club it was pretty empty, other than the women working there, the bartender, and Gonzalo's guard, but there were a few bunches of clients randomly scattered at the tables, some in pairs, chattering, some alone staring in wonderment or, like depressed junkies, up at the pole dancer. There were just enough that I didn't notice two guys who slipped in after we had been in the club for an hour. When they came in, I didn't know they were the reason, but Gonzalo excused himself and said he had to go to the bathroom. That left me alone for a while, sitting in the club without having to pretend to be interested in Gonzalo's conversation. I had a chance to take in the soft Latin beat of the music. There were a few trumpets. It was supposed to be happy music, but in the sad red light falling on the pole dancer on stage the trumpets sounded forlorn, and I wondered what I was doing in this club, so far from my girlfriend, Julieta, in Mexico City. She was under the covers now, sleeping like a babe, with our dog, no doubt, on the floor by her side. We live in a loft I designed in the center of the city, with modern lines that were clean enough to get _Wallpaper_ * magazine interested. They did a photo shoot of the loft\u2014a mix of the modern within a traditional Colonial building, downtown. The truth is, I ripped off the style of another architect whom I admire who isn't that well known. The loft looks beautiful, but it never would have gotten into _Wallpaper_ * without the connections of a friend of my father. My father is one of those great architects of Mexico, who people in the outside world know. I got to thinking about my father, and I asked myself why I was in this dump with Gonzalo, trying to get him as a client, and I realized the only reason I was sucking up to Gonzalo, or tolerating him, was because I wanted to prove to my father I could make it without him, that I could get my own big clients by myself.\n\nMy father's clients are serious. They're famous writers, like Carlos Fuentes, or big industrialists who own steel and tuna factories, or telephone and TV companies. You know where their money comes from. They're old, established families that have owned Mexico from the time of the haciendas or who came over from Spain to make their fortunes. I'm not saying they're all nice, or all have good taste, but in as many deals as my father has told me about, I've never heard him say he went to three crappy strip joints, on the edge of town, to keep the customer satisfied.\n\nWhile I was having these thoughts, Gonzalo's driver came up to me and asked me to please follow him to a back room where Gonzalo had gone, earlier, following the two men. The back room had floor-to-ceiling mirrors on the walls, and a mirror on the ceiling. There were cheap chandeliers, barely casting an orange tiger glow that perfectly imitated the clothing of the three women who were on the gold vinyl couches next to Gonzalo and by the two men who had come in. The women wore leopard-skin bodysuits. They had long nails and they put their arms around the three men. One stuck her hand on Gonzalo's hairy chest and stroked him. There was a low coffee table in front of them\u2014a rectangular cube made of mirrors\u2014and on top of it the two other men had made a bunch of lines of white powder cocaine. A lady in a leopard suit grabbed my arm and made me sit in front of the cocaine. One of the two men slapped me on the back and said, \"Take some. It's good. It's good quality.\" I told him, politely, thanks but no thanks. That's one rule I had for myself. No coke. Ever. I didn't do that, and I didn't need Gonzalo so badly that I was going to get trashed that way. It was one thing to drink shots of booze and to go from cheap club to club to get my 10 percent of a four-million-dollar home that many future clients would walk around along the beach of Acapulco. It was another to be told, like a dog, that I should snort coke with a fat client like Gonzalo.\n\n\"Hey boss,\" one of the two guys said. His face was thick and his ears were missing some of their flesh, like a wrestler who's had them pulled at too many times. \"This guy\u2014what's his name?\u2014he says he's not interested.\" He pointed at the coke. \"What kind of faggot is he? Where did you find this guy?\"\n\nI couldn't tell if Gonzalo had snorted some of the coke, or not. He seemed way too subdued to have taken any. He seemed the opposite, not up but down. He had his head leaning back on the couch, and his eyes half shut, and he was taking in the rubbing of the woman in the leopard suit assigned to him. He was staring up at the mirror on the ceiling.\n\n\"I don't really like him, either,\" Gonzalo said in a fairly muted voice, like he was underwater. \"He's been a party pooper all night. He does look like a faggot. But he's not. His girlfriend lives with him at Corregidora 38, in the Center. They have a dog and they live together in a loft. He's an architect who's going to design my next house for me. And if he screws up, if he makes any mistake on the house, his girlfriend is going to be in trouble.\"\n\nI had never given him my home address. I had never told him I had a dog. I had never told him I lived with my girlfriend.\n\n\"Hey faggot,\" Gonzalo said. \"You've been avoiding the lap dances all night. I've offered you women in each of the clubs and in each place you look at your Piaget watch like you have better things to do and like you don't like my Rolex. I don't care if you think I'm a cheap, fat guy with nouveau riche money. Every penny I have, I made it myself. I started my business myself. I don't sell drugs. I didn't make money the cheap, fast way. These guys are not drug dealers. They're just snorting a little coke. Big deal. Every penny I made, I earned it myself. I didn't come from some family where my father was sending me off to Harvard to go to architecture school. I would have liked that. Anyone would have liked that. But that's not the way it was for me. So, here's what you're going to do. You're going to take off your pants right here, right now, in front of all of us, and then you're going to get a lap dance from one of these women.\"\n\nI was more and more convinced he didn't earn his money, pulling himself up by his bootstraps. If he'd made his money the honest way, people would have told me where his money had come from\u2014but before we got together, no one knew. They said he just had it. I was more convinced than ever he was a small-time drug dealer, one of those mini-jefes down the totem pole who thinks he's a big shot. This kind of scum was getting bigger and bigger in Acapulco. It was infecting the whole country like a cancer.\n\nIn Mexico, earthquakes are common. Every architect has to learn how to build around them. You put big buildings on springs that can move with the earth when it ruptures. You reinforce the beams with thick steel rods. You put on extra concrete so the elevated freeways don't fall. If you do it right, the structure survives with only minor cracks.\n\nThe smart thing to do would be to listen to Gonzalo and just do what he said, to go with the flow, to bend. Who cares if it was humiliating? I could take off my pants. I could take off my underwear. I could let the women rub me until I got hard, with the other men watching like animals. I could be the circus animal and design his house, and take the money that I knew was from illegal sources, and get the house printed up in one of the big Mexican architecture magazines that would lead to other clients. It would be a small price of humiliation to keep climbing up the ladder. My father would see me doing better. He would have to admit I could make it on my own. I stood up with one hand on my belt, trying to decide what to do. Should I open my belt or not? If I opened my belt it would be capitulation.\n\nI opened my belt, and I justified it to myself that it was better to live than to be a dead example of someone who had stood up for his principles against these thugs.\n\n\"That's it,\" Gonzalo said. \"That's it. Now dance, dance with one of these lovely ladies.\" One of the older women with long, red nails opened my pants. I stood in front of the other men, and she felt me up while the men laughed at me like I was a circus monkey.\n\n\u2014\n\nSometimes, I think about my watch. There are a million things happening around the world at any second. There are an infinite number of split seconds in a minute. If Zeno's paradox is right\u2014that between any two points there are an infinite number of halfway points, so we should never be able to get anywhere\u2014then the same must be true about choices in life. What if I hadn't opened my belt? What if I hadn't let myself stay in that room for a stupid lap dance that I feigned interest in? I didn't care about the woman who was, supposedly, turning me on, and I cared even less about her twenty minutes later when Gonzalo finally came out of his stupor enough to say he wanted to leave. The two goons who'd been cutting up the coke on the mirror asked Gonzalo, suddenly, when they were going to get paid for some work they had done and for the delivery of a load. I couldn't hear well, since by now I was standing fairly cold while the woman assigned to me gyrated around my body.\n\n\"Just shut up,\" Gonzalo said. \"You'll get paid when I tell you.\"\n\nThe goon with the ears partially missing said, \"Nothing is free, jefe. You pay up soon or things could get nasty.\"\n\nGonzalo's driver came in, and he wasn't having any of this discussion. He told the two goons to calm down. He was sober. He told Gonzalo he had to go to the car now, that it wasn't safe to stay anymore. He said he would deliver me back to my hotel.\n\n\"We're not going to his hotel, yet,\" Gonzalo said. \"He's coming with me and these two women.\" He grabbed ahold of the woman who had been dancing around me, who'd tried to give me a blowjob in front of the other men\u2014she wasn't fully successful because I willed myself not to get it up for her.\n\nIt seemed like the evening was falling apart, completely. Whether I would get the contract to build the house or not, suddenly seemed up in the air. In the morning, when everyone is recovering from their fiesta, there are times when people pretend whatever happened the night before never happened. And that could mean I would have a contract for the house or not, the next day. Oddly enough, though, and one of the reasons I had opened my pants\u2014other than wanting to live\u2014is that I knew these jefes lived according to their own, perverted code. If they said they wanted me to build the house, and even more so if they threatened me, it meant they had fixated on me for some reason, and they wanted it bad. And once they wanted it bad, they wouldn't let go.\n\nGonzalo's driver hurried me into the back of the Mercedes-Benz with the two women\u2014one for Gonzalo and one, supposedly, for me. I had hopes the driver was just humoring his boss, that Gonzalo would finally calm down and let me end the \"fun\" of the evening and go home to my hotel.\n\n\"You'll get your money next Monday,\" Gonzalo said to the two goons, and this calmed them as we got into the car.\n\nFor a moment, sitting in the back of the car, I thought there would be a happy ending. I was sandwiched between the two women. Gonzalo was in front, telling his driver to go faster. The engine of the Mercedes accelerated hard and we swooped around a curve at what must have been ninety miles an hour. On the straightaways the driver was going a hundred. The precision of the machine, the precision of European engineering, was comforting, and I leaned back into the plush leather thinking this was it, a minor moment of humiliation survived, a day I could live with. My head was foggy from the late night, and the bad perfume of the women next to me pressed into my nose, but I began to think of the house I would design for Gonzalo on the beach. I would build it four stories high, one more than usual to give it the prominence he wanted. There would be circles and triangles throughout, primary shapes, just as Louis I. Kahn did in his Center for British Art at Yale, but I would come up with my own style for this house. I vowed I would never let myself rip off the work of any other architect again. This house would be a turning point in my work, a moment of maturation where I would go from cribbing others. And I would do it precisely because Gonzalo had humiliated me. I would give him a house that was better than he deserved, because it would be for me and not for him, and I would use his filthy money to do it.\n\nThe car slid sideways as the driver slammed on the brakes when the headlights pointed out a tree across the road. From a hundred to zero in six seconds. The women next to me came out of their soft leaning-in, tired from their work, ready for sleep, and screamed. I angled forward once the car came to a stop, with the bright lights against the green branches of the tree in front of us. It was a large tree, masking what was beyond. Gonzalo yelled at the driver, \"Get back. Turn this around. It's a trap.\"\n\nThe driver punched the car in reverse, and I was thrown forward and then back hard into the leather seat. Bullets came from the right as the wheels whirled against the pavement. _TaTat! TaTat! Tattattattattat_. When the projectiles hit the front, they didn't break the glass. There must have been bulletproofing. But the bullets kept coming, the glass bending and molding inward, until there was no more resistance, Gonzalo yelling at his driver to go faster, the front window pocked with flying metal, the women grabbing at me and trying to get down, Gonzalo's head flying back and then his body slumping forward, the driver's foot coming off the accelerator as he slumped, too. I hid beneath the bodies of the women on the floor of the backseat. I couldn't see anything, but I heard some men running up to the car and then away from it.\n\n\"They're dead,\" I heard one shout. \"Come on, let's go, they're all finished. We got them.\"\n\nThen another opened up his machine gun against the side door, in the back. He unleashed a wave of bullets. They hit more and more into the door, just behind me. I felt a sting and a piece of metal go into my body somewhere. I heard someone open the door and look in.\n\n\"Yes, they're all dead,\" he said, and he left the door open and ran away.\n\nHow I'd become twisted, with my back against the floor, I'll never know. My hand was splayed above me and I heard a faint _tick, tick, tick,_ with the precision of a Swiss watch, and saw the time: 3:50. One second earlier, and I might have had a different outcome\u2014I might never have been in that car.\n\n\u2014\n\nA couple days ago, I went to a funeral. It was of the woman who gave me the lap dance. Before the funeral, my girlfriend asked why I had to go. I'd told her everything that had happened that night. I didn't want any secrets between us before we got married. She came and saw me in the hospital, from Mexico City to Acapulco, and while it wasn't clear, at first, whether I would live, I told her, through the breathing mask stuck to my face, what had happened. I didn't want to leave any detail out, so I told her about the strange way I'd ended up in the back room with the mirrors and how Gonzalo had insisted I have a lap dance. I told her about the house I wanted to design for him.\n\n\"Why would you want to design a nice house for him?\" she said.\n\n\"Because I wanted to prove him wrong.\"\n\n\"But you don't have to prove anything to anyone,\" she said. \"Not even to your father.\"\n\nThat's why I like Julieta. She tells me what I want to hear. She didn't understand why I'd want to go to the funeral of the women who worked in the club, but she didn't try to stop me.\n\nThe funeral was in a small church in a small town outside Acapulco. The priest started his sermon as a hundred people listened, crammed into the pews. In the front of the church I could see the two children of Gloria, the woman who'd danced for me. I knew her name, now. The children were ages three and five. Two boys. There was no father present. They stood with their grandmother.\n\nThe priest told everyone gathered that as hard as it was to accept, there was always a reason for everything that happened. Everything was God's will. The ways of the Lord were unknowable and mysterious.\n\nI could think of no reason, whatsoever, for Gloria's death. So a man like Gonzalo could get his sexual kicks? So a man like Gonzalo could make enough money to build a four-million-dollar home? I was just as complicit as her, just the same, trying to make a buck off a guy like him. I had let my ambition get the better of me. I was revolted that I'd let myself stay that night in the room with the mirrors, and that I'd let myself \"dance\" with Gloria\u2014not because there's any shame to sex, but because I had no interest in that sex and because Gonzalo had made me do it.\n\nI walked into the parking lot, where there was dirt around the old cars. I picked up a stick. And in the dirt I wrote, \"For the memory of Gloria, who deserved none of what happened.\" I held on to the stick. I drew circles and squares. I was going to make a memorial to her, in a small place by the beach. I wouldn't make the memorial too big. It wasn't for me, wasn't for me to try to get some recognition in the magazines, it was for her. I drew a tower into the air, which represented me and my hopes. I was going to do better. I was going to try to make something of myself while I still had life.\n\n# THE KIDNAPPING\n\nTen seconds before they kidnapped me, son of a bitch, I was walking my dog, Azteca. There are little stalls all over my neighborhood in downtown Mexico City. They sell everything\u2014paper, plastic beach balls, bolts of cloth, straw hats, tight jeans with sequins, bright plastic Tupperware, blow-up superheroes, chilies, tacos, fried pig skin, speakers for local bands, tools, and wedding cakes. You name it, it's being sold down here in the center of town. I like to watch the hot women walking around my neighborhood. I like to even check out the tranny prostitutes\u2014not that I'd ever want to sleep with one of them, but man, they do it up all the way, you know what I mean? They go whole hog.\n\nSo I was walking my dog late at night. Another dog was barking ferociously from a rooftop. I don't usually go the way I was walking after 10 p.m., 'cause walking down that way gets a little hairy. During the day, it's fine. But at night, you gotta be careful. Normally, no one messes with me 'cause of the dog. I have a Xoloitzcuintle, which is some kind of pure-race dog from the time of the Aztecs. It's one of only three kinds of 100-percent pure, authentic, Mexican dogs. People come up to me all the time to try to get close to the dog. They're a little afraid, but they're curious, and they're especially curious 'cause I'm an American, and they know something ain't right with a white-looking guy, a g\u00fcero, with one of these pure Mexican dogs. So it gets their head spinning and they come up and ask me what my dog's name is, and I say Azteca, and that gets them laughing, 'cause it's almost like calling your dog George Washington, or something unduly serious and patriotic like that, and since my dog's big, they keep their distance, and they like that my dog is big.\n\nBut man, those motherfuckers, when I was walking late at night at 11 p.m., trying just to get to my studio where I do some of my bigger paintings, or more conceptual type of work that takes up some larger space\u2014those dudes just came up to my dog and shot him, point-blank, one shot right through the rib cage, and Azteca twisted his ears first toward them, even before they got the shot off, but he didn't have a chance. It was cruel to do that shot. Azteca got blown down. He cried while his rib cage was jerked around. And then my dog, which had always guarded me so well, was kaput.\n\nI was standing on one of those empty streets downtown, in the center of the old Colonial city, where there's always garbage littered around in the middle of the night after all the people coming and going have bought and sold whatever they're going to take home for the day from the stalls. So I was standing naked in that street, other than my dog, the harsh mercury of some streetlights in the distance the only thing making the street safe. Usually, there's at least someone on the street watching, hanging out, talking with their buddies, but these dudes who did the kidnapping, they knew what they were doing. They must have been watching for just the right moment. There was no one, and I mean no one on the street when I finally saw them coming and when they shot my dog.\n\nOne shot. I looked up the alleyway. A car comes screeching up from the shadows. These two dudes, who smell like a combination of too much piss, alcohol, and bad perfume, push me into the car. One of them puts a gun to my head. He tells me to shut up, in really fast, crude Spanish. He tells me to bend over. They put a blindfold on me. They tie my hands behind my back with the leash from Azteca. Azteca was gone, of course. They left him back on the street, stone-cold dead. I tried to tell myself to stay calm. I told myself just to do whatever they wanted me to, 'cause I usually have this attitude that if you stay calm everything will work out. But it wasn't doing me any good. This one jerk, and I mean that with the greatest of all understatements, he spit in my hair, he told me I was a filthy pig and that my mother would never see me again unless I begged for mercy. Beg, he said. Beg, now. Beg for your life. So man, I begged. Oh yeah, I begged. I tried to do it calmly. I tried to do it with all the sincerity I could muster. I'm gonna beg until they get whatever they feel comfortable with, I thought. One of them hit me on the back of the head, and when I came to I had a large welt on my scalp.\n\n\u2014\n\nI'll tell you what's terrifying. It's not being alone in the place where they take you after they kidnap you. It's the fact that you're _not_ alone, that there are other people with you who you know are going through the same hell they're putting you through.\n\nHere's what they did every day for five straight days. They woke me up in the middle of the night, they slapped me on the side of the head, they did all this after I listened to them cut the finger off some young woman in the other room, and then they took me in the back of the car to an ATM, where they forced me to take out all the money I could, the daily limit, before they slapped me on the head again, beat me in the chest, and stuffed me back in the car. You're like a gerbil on a spinning wheel, while they do this. You're in the ninth ring of hell. I mean, you don't know what day it is\u2014whether it's the same day as the day before. Whether they're just gonna fuckin' waste you right there. Whether they've already found a way to get in touch with your family, and whether your family is gonna pay. It's just one big, black hole of no information. Maybe you'll live, maybe you'll die, and that finger they cut off the woman in the next room\u2014that's real. That's her begging for them not to do it. That's them coming in, in a posse of three, and you hear them scuffling with her, and you hear them shouting at her in Spanish, \"They aren't paying, se\u00f1orita. They aren't holding up their end of the bargain. So they're gonna have to be convinced.\"\n\nThe cry from that woman, there was no faking there. They take a pair of kitchen shears. They run it up along the skin. They scrape your knuckles with the edge of the blade of the scissors, until they bleed. I know how they do it, 'cause later they took one of my fingers off and sent it to my family. This is what they do. They tease you. They make you shit your pants. They have you saying, Please no, please don't do it. Please. No. For Christ's sake, I'm a painter. I'll do anything. I'll get you more money. I'll get you any money you want.\n\nThere were two guys when they did it to me. One was the man I saw shoot Azteca. He had a couple Fu Manchu hairs growing out of his jaw, his face was trapezoidal, his eyes sunk in, his skin leathery like a burnt turtle. He had all the look of a Neanderthal you could imagine. He had a tattoo on his right upper bicep that said MAM\u00c1, with a heart below, and a dagger through the heart. The other guy was skinny, like he was addicted to heroin. He never said anything. He would just hold my arm down against the wood table. He was a lot stronger than you'd expect for a junkie. It was like all his junkieness, all his neediness, came out in his grip. He'd hold my bicep, while the Neanderthal threatened to cut off my finger, and then one day the Neanderthal did it. Blood was gushing all over the fucking place. They cut my pinkie. They said they were gonna start with the small fingers first and then go up the scale, with each request to my family that didn't come through for money. One thing about Mexico, it's impossible to send any mail. You can't get any letter anywhere in less than a week. It takes a full eight days just to get a letter sent from one side of Mexico City to the other. So these guys aren't dumb. They use FedEx. They work for the cartels. They aren't some loners. They're pros. They FedEx'd my finger to my family in Pittsburgh.\n\n\u2014\n\nMy father, he's as straitlaced as they come, as straight a shooter as possible. He never lies. Or, if he does, he does it for a good reason, to help someone. He was the football coach at Pitt for twenty-five years before he retired. I'm forty-four. He's seventy-eight. I'm one of seven kids. I used to joke my parents had kids until they realized the Pope wasn't watching anymore.\n\nMy father, his idea of cruelty was making his players run fifteen miles. He'd have them woken up by his assistant at 5 a.m. on a Sunday, and make them run to our house out of town and press the doorbell. Then he'd have them run back into town before breakfast. If he was torturing someone, it was for their own good. He'd tutor his students personally. When the army called him up, he went. There was no question in his mind ever, what you did; you did the \"right thing.\" The guy has a heart of gold, but he has this sense of duty that sometimes blocks out everything else. So if the FBI told him he shouldn't cave in to any extortion, that it was just going to make things worse, that giving in to kidnappers usually results in having someone killed, that he should just listen to the cops and be firm and not negotiate, but only say a deal could be cut\u2014if they gave me back, no charges would be pressed\u2014then that's what he did. It pained him. This kind of decision pained him. It's not the kind of decision he'd like to make for himself, but the FBI had more experience in these matters, and if that's what they told him then that's what he should do.\n\nMy dad coached a lot of winning teams. He's probably the smart one in the family, but if it hadn't been for my mother, I'd probably be dead now. She sent the first amount of ransom. She sent ten thousand dollars behind my dad's back. She sent the money, so they stopped cutting off my fingers for a while. But it was never clear if they were stopping the finger-cutting exercises because they were just gonna kill me.\n\n\u2014\n\nI'm not sure what the purpose of life is, but I can tell you this, every person has a reason to be on earth, and that reason isn't to be kidnapped, isn't to die in some war, isn't to end up some puffy body left in a trench or to be the residue of some crazy drug war, like what they've got down in Mexico now. I kind of think the purpose of life is to sing. I don't mean, literally, always to sing, but to sing metaphorically, to sing in some way of beauty, to raise the spirits of our voices in hope.\n\nSo here's something I did one day, while they had me kidnapped, sitting on the floor with my hands tied behind my back, my head leaning against a hard concrete wall. I started to sing \"Mary Had a Little Lamb.\" I've got a bit of a gravelly voice. I like to smoke my share of stub cigars. I like to drink mezcal, if it's offered to me. Or, more like, I like to give it to friends if they come around. I like to have spontaneous parties. I like to get a bunch of people together and buy too much food. There's a tranny who works in a restaurant below me, and she makes the best plantain empanadas this side of the world, and just looking at the pride on her face when she brings up the food for a party, that's what I like. I want people to be happy, you know what I mean? So I started singing \"Mary Had a Little Lamb.\" Amazingly, they let me sing, at first. I started singing louder and louder. I wanted the woman in the next room to hear. She'd been going downhill. She'd lost two fingers already, and I thought, if I can sing this song loud enough she'll come back to life. So I just started singing. I don't know why in the hell I chose that song. It's just the one that came to me. It's easy, you know. And after I'd been singing for a bit, I swear I heard that other woman singing, too. She wasn't singing the same song. She started singing some Mexican song. But there it was, some other nice song in Spanish that somehow fit together, perfectly, with \"Mary Had a Little Lamb,\" and even though her voice was weak, she was singing with some kind of inner strength, in perfect harmony with the melody of my own song, and the two of us sang to the concrete ceiling, feeling like we'd stolen a moment, going round and round, repeating the songs a few times, like some kind of chorus of monks but with more joy. They shut us up, of course. They came in and hit me on the mouth. But it didn't matter. It's those moments of joy that make life worth living. It's that harmony, when all around you there's dead silence.\n\n\u2014\n\nHere's what I was deadly afraid of when I was kidnapped in that room. It wasn't that I was going to die. After a while, you get exhausted, you start to lose your energy, but what you're really afraid of isn't that you're going to leave this world, it's all the things you never did outside that room.\n\nI mean here's the thing that was getting to me. I'd never had a relationship for more than two years with any single woman. Like I said, I'm a free spirit. I like to drink, I like to have sex, I like to roll around the world like a nomad. I've been to Cuba six times, and when I'm there I'm happy to find some women, to have a good time, to do whatever comes to me in the moment. But while I was kidnapped, I got to thinking about this one moment with my last girlfriend in Mexico, who I went out with for two years. I'd been shuttling back and forth from New York City to Mexico City for thirteen years, mainly living in Mexico. The rawness of the place intrigues me. It's cheap, and I was able to get started as a painter late in life, at thirty-three, after studying to be an architect.\n\nI wanted to live life large. I didn't want to be stuck just in NYC, like all the other wannabe artists. I wanted to paint with all my heart and soul, with everything I've got, and Mexico was a raw place where I could work, a place that inspired me with the Wild West spirit.\n\nMy girlfriend, she'd been in a band. She was the lead singer, but she was really a good girl at heart, from a pretty wealthy family. She said to me, \"Jakob, you can go around the world wherever you want, you can paint the best painting in the world, but until you find that stone within that makes you distrust everything, that makes you fail to put down roots, that makes you think everything is just of this moment and never of another, just a bang, you're never going to be happy.\" She told me this when I said I wanted to break up with her. I wasn't 100-percent certain why I wanted to break up, but like I say, I'm a nomad, I need change, and at some point, anything\u2014even something good\u2014gets too familiar to me, and I cut things off.\n\nSo I'd sit there in all the dead time during my kidnapping and think about that. Being kidnapped isn't like in the movies. Most of the time, they're not beating you up. Most of the time, they're not torturing you. Most of the time, you're just sitting slumped on the floor, with complete dead time on your hands, with time to think over who you are and what you want and why normal life is so hard.\n\nLike a rolling stone. That's what she meant. I'd been a rolling stone, chasing after the bottom of the hill. Maybe I needed to take the time to grow some moss. I sat on the hard concrete floor and cried.\n\nThe reason I was crying was because I'd been an SOB cheating on her just before we broke up. I mean, sure, things had already fallen apart between us. She was using coke. She was taking too much of the stuff, and I'm just not into that. That shit will mess you up. I stay pretty clean with just mezcal and tequila, beers and some occasional ecstasy or something like that. But coke, that'll mess you up. And she was hiding her use of all that from me. She was pretending she wasn't using that shit, and that's not OK with me. I like to be up front, you know what I mean? If there's a problem, I like to be up front about it.\n\nSo I was in Cuba taking photographs for some new art project. I'm mainly a painter, but I like to go wherever my interests lie, and the changes and things happening in Cuba were just totally interesting to me. Fidel was sick and it seemed like the place was already changing a bit, even though his brother Ra\u00fal still had the reins. I went over to take pictures of the things happening. We went into every nook and cranny of Havana. We got great shots of people living in their poor homes, of people waiting in lines, of guys wearing Nike sneakers\u2014when Nikes had been forbidden, forever. One day, we started taking photos of these two young girls, they must have been seventeen. They thought it was just amazing, us taking photos. I went with this photographer partner of mine. And the two of us, we started charming the girls. We went out to a nice bar with them. We took photos that made them feel like they were superstars that were going to be on the cover of _Vogue_. We never promised them anything. We never said we were shooting for any magazine like that, but you know the hopes of young girls. They saw these white dudes with expensive cameras, and they got excited.\n\nI like to wear a straw trilby hat and a Hawaiian floral shirt, with a good pair of jeans. I've usually got some gray stubble on my face, and these girls, looking at the photos you can see how wide-open their eyes are. We each had our own room, and I took one of those girls for a romp, while my partner did the same with the shorter, brown-haired girl. And, oh my god\u2014the one who I had in my room. She was so frickin' hot. She did all the moves on top of me. She took it all off, I didn't have to even do much of anything. And that's just the way it is, you know, for me. I've got to be in the moment. I've got to be taking it all in\u2014everything, from the blade of grass to that mural of Che Guevara. I want it all.\n\nWhile I was on the floor with my hands tied behind my back, I got to thinking what a dick I'd been. How it didn't matter how hot those girls were. There was nothing exploitative about it. They wanted it just as bad as we did. But what kind of power relationship is that? I mean, doing it with some seventeen-year-old girl while my girlfriend was back in Mexico with a drug problem. That just wasn't cool. That was pathetic. So I cried.\n\nAfter I cried, I got to thinking that a person can justify anything. If I could justify sleeping with that girl in Havana, then why would one of those kidnappers feel anything bad about cutting off one of my fingers? They looked like assholes. They looked like trapezoidal thugs, but each one of them had a family\u2014some mom and sister somewhere who they were probably sending money to. Don't make the mistake of thinking that a kidnapper working for a cartel is some kind of heartless, psychotic goon. They may not have gotten enough love from their mom when they were young, but in their mind they're doing the right thing. Christ, Idi Amin thought he was doing the right thing!\n\n\u2014\n\nI'm of two minds as to whether there's a heaven. My logical mind says no way. I like the art in churches. I love the old wood cherubs painted with such bright cheeks in all the beautiful churches of Mexico. Those statues are incredible. But it's some kind of make-believe Disneyland that's just fun to look at. Whether there's a plan for anything, I doubt it.\n\nBut something let me stay alive. And I don't just mean the ransom money my mom sent, which, obviously, was essential. There are stories all the time, down here, of people who pay everything the kidnappers want, who follow every last instruction, and in the end they kill the person they kidnap anyway.\n\nSo the question is, what makes one live and another die? It could just be the luck of the draw, it could just be the way the dice tumble and fall. But I don't quite see it that way. Or I see it that way, and a little more.\n\nI think I have more to do. I think each person is like a plant\u2014they have a length of time they're supposed to live, like a tomato until it gets ripe. And my ripeness just hadn't happened yet.\n\nThose son of a bitches. They kicked the shit out of me. They came at 2 a.m. It was the same guys who'd cut off my finger, and two more guys. Not that they really needed four to kick the crap out of someone already sitting on the floor, tied up. But they came and took turns kicking me. I was half dazed when they did it, at first. They turned on the one bare bulb that lit up my room. There was gray peeling paint on the wall. The bulb shocked me, so suddenly bright, and I looked up at it and a boot came smack into my face. I swiveled my legs around trying to get one of them to fall, but he jumped out of the way. When you try to fight back with your hands tied behind your back, you mainly pull at yourself, tying yourself into knots. I tried in any case, and I was thrown on the floor by the force of the kicks. Some blood bubbled up from my lip and they carried me out, still trying to struggle with almost no force left. I think I got a good jab with one of my legs at one of them, though. Then it was pretty dark, my head blindfolded, but a little light seeped through as lights from the streets penetrated the cloth. You notice the smells at a time like that. The smell of old vinyl in an old car. The smell of upholstery in what's probably a stolen car.\n\nThey dumped me in the middle of a street, Donceles, in the center of town, with my hands tied behind my back, my legs tied up with Azteca's leash. It was like they wanted to remind me they'd killed my dog. They left the blindfold sack over my head. When they threw me out of the car, one of them said, \"Hijo de puta\u2014chinga tu madre!\"\n\nAfter fifteen minutes, I heard a street dog come up to me. There was a whole pack of them. They do that, sometimes, the homeless dogs of Mexico City running around at night. They come in all stripes, almost all mutts, but every once in a while a purebred gets stuck amongst the others, lost from its fancy neighborhood. I felt the dogs sniff at me. They put their noses up close against my crotch. They sniffed my shoes. They pulled a bit at my pockets. One of the bigger dogs tugged at the sack over my head and then I could see up, the face of a black and brown tiger-striped dog holding the bag in its mouth, the burning lights of a mercury street lamp shining like in a concentration camp, a small star in the sky in the infinite beauty and depth of the universe almost hidden behind.\n\n\u2014\n\nThe party was over. Everyone was gone. It was a month after the kidnapping, and I'd had all my friends over to celebrate the fact I was still alive.\n\nThere was the residue of the party all over the place, glasses and bottles. I've fixed up an apartment in the center of Mexico City. I bought it cheap, for thirty thousand dollars\u2014in an old building from the seventeenth century\u2014and even though the building, as a whole, looks like a dump, I've done a ton of work to fix up my apartment. I have high ceilings. You can see the old beams from the time when the Dominican monks and the Inquisition ruled the day.\n\nThe tranny brought up all the food, earlier. As always, I'd ordered too much. That tranny has been through everything. She's been through kicks and shouts. Mexico's a macho country, and it's not easy being a transvestite. They cackle at you as you walk down the street. But I've never seen a trace of anger on her face. She's got the height and slim hips of a man, but she plays up her freckles.\n\nI ate one of the plantain empanadas that was left. The leftover tostadas looked like a mess, with the lettuce wilted and the tomatoes looking like they had sweated and over-ripened into the sauce. But the plantain was still just right, the filling inside good. The air smelled of smoke. There was still the smell of cigars, and we'd smoked more than a few short Cohibas I'd brought back from Cuba. I caught sight of my pinkie as I took a bite out of the empanada, or of where my finger should have been, where the stub ended with a raw skin wound.\n\nI could view it as a wound or as a badge of promise. I could be tough like the trannies I admired. There was a lot more for me to paint. I went into my studio. I have two studios, one the one I was trying to walk to the night they kidnapped me, the other just off the space I've renovated in my old building. I pulled out a canvas. I put it up on the wall. I took out some spray-paint cans, which I sometimes use for background, and I put on yellows and bright greens. I shook those cans as hard as I could. I sprayed on thick the happiest colors I could think of. I wasn't going anywhere. I was going to stay right here and plant roots. I spray-painted an abstract form of a tree with roots. I added a stone in the middle that wasn't rolling anymore. I might call my ex-girlfriend in the morning. I had no intention of getting back together. We were through, but I wanted to call to check up on how she was doing, to make sure she was hanging in there, to be certain she was more or less OK.\n\n# THE PLASTIC SURGEON\n\nI am a plastic surgeon, and down here in Mexico it helps that I'm tall and blond. I'm six feet two, I like to play basketball, and during the summer when I take a vacation up in Seattle, where I have a house and come from, I like to row on the rough waters of the ocean. I don't spend that much time looking in the mirror. I spend more time looking at my hands, my tools, to make sure they're smooth and in working order. But I care about my body. I like to feel sculpted. Like most of my clients, I think it's important how we look. I'd rather my body look good via hard work. My nose is a bit long, and according to most standards of beauty it's not beautiful, but that doesn't bother me because I think people looking at my body can tell I work out, can tell I do the hard work necessary to make myself look good and healthy. Others don't have that kind of discipline. They might want an easy way to look fit, and while that's not my way, I'm more than happy to do the work for them in surgery, to suck out those extra pounds of fat with liposuction, around the hips and abdomen, or to design what they think is a more perfect face. It makes them happy, which is reward enough for me, and every patient is a challenge, a raw tableau to shape, a goal to work at hard, like paddling from one end of a large bay to the other along the most efficient route possible.\n\nSome patients are more challenging than others, however. Not everything always goes smoothly. Sometimes the best calculations go awry, and that's what's been preoccupying me lately. I had a patient die a week ago. I've been trying to calculate why and how he died, and whether his death is going to mean I have less time left to live.\n\nThe patient was Paco. That's a made-up name, because I'm used to keeping the privacy of patients, and he has a long drug-cartel name in Spanish that means Lord of the Heavens. They called him this, in the Ju\u00e1rez Cartel where he came from, because he hired a fleet of Boeing 737 jets to transport cocaine from Colombia up to Mexico, flying in to municipal airstrips around the country, before sending the coke on to the U.S. through an extensive network he'd built up. Paco was one of the most wanted narcos in the country. It's always hard to tell if the Mexican government really wants to catch its fugitives or not. After all, how else do you land your own 737s around the country, with drugs hidden in the payload, day after day, year after year? After Paco died, I heard estimates he'd acquired a fortune of twenty-five billion dollars. That's a lot of smackers. But I can tell you, for sure, after the DEA came to visit my clinic, the U.S. authorities had been hell-bent on capturing him.\n\n\u2014\n\nI'm not sure how Paco found my clinic. My office is one of the smaller ones in Mexico City, in the neighborhood of Polanco. It's on a side street with shady trees, and the usual mix of private homes and guards and upscale apartments you might find on any street in Polanco. The U.S. embassy owns apartments at the end of the street, and there's an OfficeMax supply store around the corner and a Carl's Jr. hamburger place. I'm not one of the top plastic surgeons in the country, and as you can tell by my description, while the neighborhood is nice, and close to clients who can pay, the area is less chic than Las Lomas, where many high-end plastic surgeons have their offices.\n\nPaco, I'm told by the DEA, had been running from the law a couple years by the time he came into my clinic. I don't mean that's when he began his life of crime. From what I understand, he was born into a cartel family in Sinaloa. Sinaloa is the Sicily of Mexico for drug mafias. It's a bit in the middle of nowhere, on the Pacific coast two-thirds of the way up the map of Mexico toward the U.S. If you're like me, these are just names\u2014places of intrigue where Mexican narcos come from. But something changes, crystallizes, when one of the names that flits across the pages of the newspapers comes into your clinic and you're expected to operate on him.\n\nPaco didn't announce who he was when he came, but he didn't seem like a usual patient. My clinic is called Verde, or Green, and it's meant to give off a clean, fresh spa feeling. Most of my patients come for routine Botox and collagen treatments, to add firmness to their lips and cheeks. I can do it all, however: blepharoplasty surgery for women who feel their eyelids are too dominant; rhinoplasty, reducing the thickness of a nose, adding an angular point, or reducing overprominence of the nose in profile; breasts can be augmented and shaped with silicone and saline implants or by reshaping the areola. Many women need breast reduction to reduce the pain and heaviness of large breasts and to create a firmer, more youthful look. Depending on the surgery, my two assistants can do most of the preparatory work; or, if the case is more traumatic, as with a burn victim, I take over from the beginning. Burn victims are complicated. The more surface area, the greater the risk without a full surgical hospital on hand, and unless the skin area is small enough, I send that kind of patient to the big hospitals, where they can get proper treatment.\n\nPaco's hair was long and uncombed when he came into the clinic. He had a six-inch beard and a thin face, and he looked like he had a brown lion's mane around a face that was too skinny for all that hair. He had a long-sleeve, thick cotton shirt, his sleeves were rolled up, his hands dirty, and his jeans dirty, too, which made him look like a car mechanic or one of the homeless guys that sometimes live in the park along Avenida Horacio, nearby. I thought he might be lost and asking for directions, or wanting some change, when he walked into the clinic. He gave a fake name to the secretary and told her he wanted to speak to Dr. John Franklin.\n\n\"Right here,\" I said to him in Spanish.\n\n\"We need to go into your office,\" he said. He didn't ask if I wanted to do this, he told me what was going to happen. He looked alone, when he first came in, but two others with mirrored sunglasses, dressed in plain clothes with similar jeans and shirts like him, holding AK-47s, came in. They told the secretary to clear out any waiting patients. Veronica, my secretary, is probably smarter than me. She has a way of sizing up every patient as they come in. She often guesses what a client is looking for, before they fill out the forms or have their consultation with me. She wasted no time telling two women, waiting in another room, that she was very sorry but could they come back tomorrow?\n\nFortunately, we'd only been open half an hour. It was 10:30 a.m., so no one was in the surgery room.\n\nPaco refused to speak in the reception area. His guards surveyed the front door while he took me into my office. I have a photo of my girlfriend there\u2014things haven't been going well with her. I have a photo of my girlfriend's golden retriever. I've been in limbo, never quite rooted anywhere enough to feel I can have my own pet. I take off, suddenly, to go sea-kayaking on the coast of the Yucat\u00e1n. So I've never made the commitment to having an animal.\n\n\"I want a full makeover,\" Paco said, when we were sitting in my office.\n\n\"I'll be happy to do whatever I can,\" I said. \"But what, exactly, do you mean by a 'full makeover'? That can mean a lot of different things to different people.\" Some men ask for a tummy tuck or complete removal of the wrinkles around their eyes. Others want things plastic surgeons have nothing to do with\u2014penis enlargement or cures for erectile dysfunction. Ninety percent of my patients are women, but I have an increasing number of men who want to enhance their body.\n\n\"What do you think a 'full makeover' means?\" Paco said, sarcastically. \"I want to change how I look so completely, even a master of disguises won't know who I am. I want this beard gone. I want my nose changed. I want my face altered. My teeth. My stomach. I want to walk out of here a ghost no one can recognize.\"\n\n\"That kind of surgery can take weeks of planning,\" I said, \"to do safely, to assemble the right team, to make sure you approve of the plan.\"\n\n\"Do I look like I have fucking weeks or even days?\" he said. \"I've come to you because I heard you're good\u2014good enough, but not famous. They're unlikely to look for me here. You're a somebody but a nobody, and I'm telling you you'll give me a complete makeover\u2014today\u2014or I'm going to kill you.\"\n\nBy the look of the guards, I knew he wasn't bluffing. If it hadn't been for them, I might have thought he was a schizophrenic with fantasies. In Mexico, the surreal can suddenly become real. One minute you're in your practice with Botox patients, the next a narco is threatening to kill you.\n\n\u2014\n\nIn the popular imagination, a plastic surgeon is a greedy guy who makes millions, who cares only about sucking money out of the vanities of his patients. The patients drive BMWs, they can't accept they're getting older, and they ask the plastic surgeon to give them a \"boob job\" so they can keep their rich husband, or the male clients want to get rid of the wrinkles so they can keep their CEO job and their sporty looks. I'm not going to say there's no truth to this. But in my experience, there are far more women who scrape together all the money they have to make one significant change to their body. They're women working as teachers, secretaries, police officers, and in bakeries. They're not hoping to look like the centerfold in _Playboy,_ and they aren't ditzy women or rich bitches, they're women with souls, and hopes, and dreams, just like anyone, and often they had the misfortune to be born with something less than what society says is attractive. They didn't create the norms of beauty. Society did, and judges them. And often they had fathers who told them, when they were growing up, that they weren't beautiful, or, even worse, mothers who belittled them. That's one of the reasons I choose to have my clinic in Polanco. It's a place where I don't have to cater only to the super-rich. It's a place that feels more accessible to the full variety of patients. I charge different rates for different kinds of patients. I don't tell them this, so they don't feel like charity cases. I ask Veronica to use her judgment, as she watches the patients come in, as to what they can afford.\n\nFor me, plastic surgery is like any other medicine. It's a form of healing. And often I'm healing psychological wounds as I reshape the body. I consider it vital to adhere to the Hippocratic oath\u2014to do no harm. Sometimes that oath can be hard to follow, like when a patient is begging you to make them thinner, and they're already thin. Sometimes, no doubt, some of the patients seem to have a frivolous sense of beauty or are unwilling to accept they're growing older. But I try to guide them gently, following the oath, avoiding any violation of the promise I swore to uphold, passed down from the time of the ancient Greeks.\n\nIt wasn't so easy to know what to do with Paco's request. I could tell he was a criminal, given the look of the guards and his threat to kill me. I could tell\u2014since he was trying to change his appearance, completely\u2014he was trying to run away from the law. I could tell that if he was a narco, which was the most likely explanation for his behavior, he'd probably killed dozens of people climbing to his position as a boss. I had no idea he was the Lord of the Heavens. I would find that out only later. But he had a presence, a way of commanding, that let me know he was a big shot who would fully be willing to kill me, and others.\n\nBy the Hippocratic oath, if I helped him change his appearance to run away from the law, then wasn't I becoming complicit in his crimes and in his harm to others? Performing the surgery would break the rule and put the blood of his hands indirectly on mine.\n\nAnd if I didn't stand up to him, wouldn't I just be placing my life above others? Wouldn't I be saying, in effect, that my life was worth more than the many lives of the people he'd undoubtedly killed, and would kill in the future?\n\nAt the same time, it was hard to see how performing surgery on him was really any different from doing so on any other patient coming into my clinic and asking for a change of appearance. What he chose to do with his life\u2014whether to kill, or not\u2014wasn't an act I, myself, was committing. And, while it could be wrong to aid a criminal, the law of self-preservation is valid, too: sometimes it's necessary to do harm for the greater good. My life was my life. It wasn't worth nothing. And the more I lived, the more I might help others. It may seem impossible, but as I went through preparation for the surgery on Paco, I went through all these ideas as I scrubbed my hands and put on my surgical gloves, as I put on my clean medical scrubs, and as the assistants prepped the surgery room. Nothing was cast in stone. At any moment, I could pull out of the situation and say no.\n\nPaco lay on the surgery table with the LED lights focused on his face, where I would begin. I told one of my two assistants, Jaime, to administer the sedative Dormicum intravenously to begin the process of anesthesia. I told him to be careful not to give too much of the sedative. The risk was too great, otherwise, that something could happen to Paco, that his heart could stop during surgery, and the consequences of the death of a narco were obvious.\n\nPaco overheard the conversation, and he said to Jaime, \"Give me double whatever the doctor said. I'm not going to let you torture me.\"\n\n\"The Dormicum is powerful,\" I said to Paco. My other assistant, Marina, cleaned surgical instruments and placed them by my side. \"If too much is administered, you could die.\"\n\n\"I don't give a shit,\" Paco said. \"If I die, nobody killed me. The only person who can kill the great Paco is Paco,\" he said, using his real name.\n\nI signaled to Jaime not to listen to him, but I can't be sure Jaime understood my signal, or whether he refused to disobey Paco, or whether he followed my instructions. Paco went under, and I began the surgery.\n\nWe worked for eight hours. The two guards with the AK-47s came in, and they stood in corners of the room, one by the door, the other by the respirator. They'd dismissed the secretary, Veronica, and told her if she contacted anyone in the police, or told anyone, at all, what was happening, they'd go to her house and kill her.\n\nThe shape of Paco's nose was wide, like a fistful of lead. I narrowed his nose and shortened it, raising the tip. I opened his eyes wider and lifted his eyebrows. I reduced the widow's peak at his hairline and pulled his skin tight around his ears, so he looked ten years younger.\n\nThe respirator hummed, pumping into his body, but after four hours of surgery the sound paused, and I could tell the machine was somehow cut off.\n\n\"What's going on with the ventilator?\" I asked Jaime, calmly, but extremely firmly.\n\n\"I'm not sure,\" Jaime said, and he checked the machine. An electrical outlet was near one of the guards, and Jaime found the machine unplugged. It was extremely strange that the machine could have become unplugged. I couldn't help but think one of the guards had removed the plug, or had kicked it with his feet. It was bad enough I was operating on a killer, but I could no longer be certain the two guards present weren't trying to kill Paco. What if they'd received orders to make him die during the surgery? What if they'd been bought off, by some rival narcos, to kill Paco and make it look like an accident?\n\nI told one of the guards to put the plug of the respirator in, and he didn't respond. I couldn't tell if this was because he didn't want to hear me, or if he'd simply begun to doze off during the surgery.\n\nA patient can last no more than three or four minutes without the respirator before permanent brain damage sets in. I told Jaime, sharply, to plug in the respirator. He bent beneath the AK-47 of one of the guards and reached to plug it in.\n\n\"What are you doing?\" the guard said. \"Are you trying to kill the jefe?\" The guard seemed to snap out of his stupor. He pulled the plug out of Jaime's hand and plugged it in himself.\n\nBeyond Paco asking for extra anesthetic, this was the second strange thing that had happened since the beginning of the surgery, but I decided to focus on the task at hand. I shifted the shape of the skin around Paco's abdomen and thighs. In the eighth hour, as we neared the end of surgery, I could hear the vital sign of his heart slowing. I asked Marina to check if the intravenous tube with the Dormicum was dripping at the appropriate flow rate. She said the drip chamber was open a little wide, and I told her to cut off administration of the fluid. The heartbeat slowed further. It flatlined. With his face in bandages, Paco closed his mouth and eyes, pushing against the clamps that held them open in their proper place. He looked like a phantom, up at me. I ordered Jaime to administer Flumazenil as an antagonist drug to counteract the effect of the Dormicum. I considered defibrillation to revitalize normal heart functioning.\n\n\"What's happening?\" the guard near the door said.\n\n\"His heart has stopped, temporarily,\" I said. \"We'll get it going again soon.\" I shouted this last bit between teeth as an order to Jaime to prepare the defibrillator.\n\n\"No one kills the jefe,\" the guard by the door said. \"Only the jefe can kill himself.\" His heart wouldn't revive, and the guard by the door took a nearby pillow and placed it over the face of Paco and pressed the pillow into his face. The guard seemed to act to kill Paco, to suffocate him, though the patient was already dead.\n\n\"The jefe told me that if he was suffering I had to put the pillow on him to keep the pain out,\" the guard said.\n\nI've been a surgeon for thirteen years, and I couldn't make any sense of all these strange actions. Had Paco been trying to kill himself when he asked for the extra Dormicum, and when he ordered the guards to put a pillow on his face? Was he constructing an elaborate suicide, to end his life but without disgrace? Had the guards tried to kill Paco in the pay of another, rival group of narcos? Had I failed, completely, to oversee the operation correctly? And was I responsible for his death? The only thing I knew was that Paco was definitely dead, and what it meant for me, I wasn't sure.\n\n\u2014\n\nThe disposal of the body was fast. I didn't have any say in the matter. The two guards took out a camera and took photos of Paco lying on the surgery table, his body already losing whatever pallor he'd had. It seemed an odd trophy shot, almost like the photos of Muammar Gaddafi or Saddam Hussein minutes after they were killed. There's something about the final photo of major tyrants\u2014grainy, fuzzy, and stark\u2014which reminds people, when it comes out in the tabloids, that they committed horrible crimes when they were alive. The guards' photography might imply they were truly on the payroll of other narcos, but there was no smiling, no cheering, as they took these shots. Nor was there any crying, nor wrenching of hair, which would show deep loyalty to the Lord of the Heavens. They stayed as neutral as the mirrors on their sunglasses, as professional as hired hit men, pulling out a body bag that they wrapped Paco up in, after they documented he was dead.\n\nAs quickly as they'd come, more than ten hours ago, they disappeared.\n\nIn some crazy way, the ease with which his body was disposed of, the fact all traces of him were gone\u2014minus pieces of tiny flesh and blood that stuck to the scalpels and other surgical instruments\u2014relieved me. It was like a nightmare that was over, and now I could awake.\n\nBut just like a nightmare, which never quite leaves the sleeper alone as they stay up, walking around in the middle of the night, Paco's sudden appearance and disappearance wasn't so easily sewn up.\n\nI decided there was no way I could sleep in my apartment the night of the surgery. I debated whether I should call the police, but for the same reason I'd performed the surgery to begin with\u2014a desire to live\u2014I decided this avenue was impossible. I decided to keep everything hushed up, to say nothing, and I made Jaime and Marina swear they, too, would say nothing.\n\nI went to my girlfriend's apartment. She's forty, and I'm forty-two. She's a Mexican banker, and she often works late hours. She has a reputation, in her office, for being a tough boss. She directs a group of twenty beneath her, in the bond department. My girlfriend is shapely, her breasts are large, she wears high heels, and she has a high voice, all stereotypically feminine, but these outwardly soft signs mask that she's a perfectionist who needs things done her way. When I came into her apartment, she'd just returned from work, and it was ten at night.\n\nWe often barely said hello to each other, when I came over to her place. If she came in from work first, she was likely to give me a peck on the cheek. If I came in first, I was likely to say a big hello, but then to disappear to take a shower before saying more. I couldn't tell her right away, therefore, what had happened during my day. I thought if I told her she might get scared and insist I call the cops. Or, I thought, she might suddenly worry about her own safety. I wanted everything to be normal, for everything that had happened that day to disappear.\n\nWe'd been going out for three years, sometimes breaking apart then getting back together. It was hard for either of us to know what we wanted. I'd take off for Seattle for a couple months, and she wouldn't come with me, or she'd go off to Turkey and not even tell me she was on vacation. We revolved like two atoms in the depths of outer space, attached but with barely enough energy to stay together, without the necessary warmth from the sun.\n\nA week ago, I'd told her I wanted to have a baby. I admit the idea must have seemed like it came from out of nowhere. I told her this because my father had died a couple months before; my sister was in the process of divorcing; I felt everything in my life pulling apart, and I wanted something to come together. I wanted love\u2014a love I seemed incapable of sparking in my girlfriend. She told me, immediately, she didn't want a child. She told me she didn't want to break up, but that she didn't know what she wanted next.\n\nSo I told her nothing about my day, as I came in. I slept in the bed beside her, feeling the space between us, wanting to come closer, but feeling her repelling me.\n\n\u2014\n\nThe next day, the police came to my clinic. They asked if Jaime and Marina worked for me. In an odd way, I was glad the police came, because neither of my assistants had come into work, and I was worried about what had happened to them.\n\n\"Do you know these people?\" a police officer said, showing a photo of Jaime and Marina, their heads sticking out of two steel drums filled with concrete. Their faces were puffy with lacerations on their cheeks, as if they'd been tortured. \"We found their bodies in the main pond of Chapultepec Park,\" the police officer said. \"Do you have any idea why they might have been killed?\"\n\nI decided I couldn't screw around anymore, hiding what had happened the day before. I told the police exactly what had happened. They thanked me for my time and told me they might need to take me in for further questioning, but at the moment I wasn't under suspicion for the murder of my two assistants. They told me a full investigation would be made of the surgery I'd performed on Paco, that any evidence of negligence during the operation would be looked into, and that I'd remain under surveillance. They told me I wasn't permitted to leave the country.\n\nThe DEA came an hour later and did further questioning, informing me who Paco was and that they'd been on his trail, coming in closer and closer, the last two weeks. \"Don't worry,\" one of the agents said. \"We know you don't have any ties to Paco. We'll do what we can to get the local police off your case. The embassy has been informed of your situation.\"\n\nThe embassy may have been informed of my situation, but the question was, what was my situation? Was there someone out there who would be trying to hunt me down, as they had so quickly hunted Jaime and Marina? I told the DEA I wanted immediate protection. I told them I needed someone to be with me all the time, protecting me. I told them I wanted out of the country, fast.\n\n\"No can do about getting you out of the country fast,\" one of the DEA officers said, \"but we promise to put someone on duty to watch what happens to you. We'll make sure nothing crazy happens. It has all the signs of an inside job\u2014the guards wanted to get him. I don't think they'll have any interest in you.\"\n\n\"Then why'd they kill Jaime and Marina?\" I said.\n\n\"Perhaps because they tried to talk about what happened. You did the right thing keeping silent. As long as you stay quiet, you should be fine.\"\n\n\u2014\n\nThree days later, the news was in all the tabloids. I went with my girlfriend to the Parque Lincoln, near her apartment, to take her golden retriever for a walk. I'd canceled all my appointments. I'd closed the clinic. It was seven at night, late July, in the rainy season, and the sky was overcast. We walked through the park, and as I threw the ball, and as her dog, Maya, returned the ball faithfully, dropping it at my feet, I thought this was all I'd ever really wanted, the simple order of throwing a ball and having a dog bring it back, wagging its tail, showing the love and affection between a pet and its master. I'd never been able to have that kind of clarity in my personal life. I'd been married, briefly, to a woman in Seattle, and then divorced. I had few friends in Mexico City, but I'd been in the city ten years. Life was nothing like paddling hard from one end of a long bay, efficiently, to the other. It was nothing like throwing a ball and having a dog bring it back.\n\nAfter a while of me throwing the ball, my girlfriend went up to Maya and took the ball from her. She told Maya to bring it to her, instead of to me. \"He's not your daddy,\" she said. \"I'm your mother, bring the ball to me.\" It seemed a cruel dig at my request, the other day, to have a child with her. She was saying I wasn't a parent; that she was the only parent.\n\nI knew I was being sensitive, and potentially reading too much into everything, but I didn't want to play with Maya anymore.\n\nAcross the park, in front of a large birdcage a couple stories high, I was positive I could see a man who looked just like Jaime, even though he was dead. I told my girlfriend I'd be right back. I told her to please be patient but that I had to run across the park. I still hadn't told her what had happened to me, the other day.\n\nI rushed across the park to where the man stood in front of the birdcage, wearing the same shirt as Jaime. I went up close and said, \"Hey, Jaime!\"\n\nThe man turned around, completely different from Jaime. It was a father holding a bag of peanuts, and his son came running around the base of the cage, laughing and shouting at the birds inside.\n\nThe boy's screams attracted a peacock inside the cage, resplendent in all the turquoise and blue that nature can conjure up to impress. The bird came toward the boy, the father, and me and opened its wide feathers, forming a big fan, so wondrous it stunned us all into silence. The pattern on the tail of the peacock looked like a thousand eyes staring at me, judging me. The bird shook its tail, and the eyes waved back and forth, evaluating me.\n\nAnd it was then that I knew what I had to do. I ran back across the park toward my girlfriend. She was still holding the ball in her hand. She'd stopped throwing the ball at Maya. She seemed to watch me run back from the cage.\n\n\"I know I've been cold and silent the last few days,\" I said, when I got to her, panting a bit from the run, though feeling invigorated from the dash. \"A few days ago, an unknown man came into my clinic, a narco, and he asked me to completely change his body. I did my best to do what he wanted. He died during surgery, and now I feel I'm being hunted. I don't know if I'm safe.\"\n\n\"Really? What a horrible, crazy thing,\" she said. \"You should have told me,\" she said, softly. \"Come here. Let me give you a hug.\" She came close and pressed her large breasts against my chest. It was the first warmth I'd felt in days, maybe even in months and years. I wasn't sure if things would be better between us, for the long run. I wasn't sure if everything would work out with the police. I could see a DEA agent in the distance, watching us, keeping us safe. I could feel her warm chest pressing into mine, feeling wider than the menacing tail of the peacock, and though Jaime and Marina were dead, filling me with confusion and mourning, for a second, wrapped in the arms of my girlfriend, I was the happiest I'd been in ages.\n\n# THE SHARPSHOOTER\n\nI had been down in Mexico City for two years, working with the DEA and CIA on covert activities to break up drug production and smuggling, when my buddy Charlie was shot. Charlie looked as young as I did, twenty-four, hair buzzed short, much taller than me, and a crack sharpshooter, just like me. Growing up, you would not have expected me to be some kind of crack sharpshooting hero-to-be, because I had dyslexia. I struggled to read at a fast enough pace to pass basic tests. But, through willpower, I learned you can do just about anything you put your mind to\u2014including graduating from college with ROTC.\n\nBefore I came down to Mexico, I remember signing up for a class at Wichita State University. It was a class I really wanted to take, on contemporary American literature, but two weeks into the course I got orders from my superior officer that I was going to have weekly training at the same time as the class, and that I had to give it up. I was pissed off, for a second, that the commander on base hadn't thought enough to give us the schedule for training before the classes at the U. began, but orders are orders. I wanted to do what I was told\u2014I had grown up without much money\u2014and I wanted to get a college education, to progress with my life. If they had told me to walk backward to training, I would have done it.\n\n\"Jeremiah,\" my father told me, after I had learned about the scheduling conflict, \"if they want you to sit, sit. If they want you to shit on command, shit on command. If they want you to roll over, roll over.\"\n\nMy father had been in the military for four years in Vietnam. Before he got his cancer, perhaps from all that Agent Orange over there, he told me, \"The proudest thing I ever did was serve in the military. It made me into a man, out of a lazy sack of potatoes. It taught me order, discipline, and how to serve my country.\" He said he had absolutely no problem shooting gooks as he was told. He said there were people who knew better than those of us at the bottom of the heap, and sometimes you should just listen to them. He said all that second-guessing about the wars in Vietnam, Iraq, and Afghanistan was done by a bunch of crybabies who needed to toughen up and take one for the team.\n\n\"All those liberals,\" he said, \"are always saying President Bush was fucking up, not showing the caskets of soldiers when they came home to Dover Air Force Base. Well what the hell do they expect him to do? Put those bodies on TV?\" He was a big fan of Rush Limbaugh. If liberals said they wanted Hillary Clinton for prez, then they must want some kind of lesbian sex, he said, 'cause lord knows Bill wasn't giving her any. He'd rant about Obama coming from some Muslim country like Indonesia. \"And now he wants us to believe, after he's tampered with his frickin' birth certificate, that he's not an alien.\"\n\nUnder those circumstances, I was first a Cub Scout and then a Boy Scout. I worked my way up to Eagle Scout. In high school, I occasionally walked around the halls in my Boy Scout uniform with merit badges pinned to my sash. When my father would say some of those things, from Rush, it would disappoint me. I agreed with some of the things my father would say, but I've always been a stickler for facts. Facts are what win battles, I thought. Facts are what win wars.\n\nI used to stay up at night, as a kid, reading Time-Life books my father had about the battles of Iwo Jima and the landing of our troops on D-Day. I was impressed by the way Eisenhower had planned for that invasion, not letting any of the enemy know about the buildup of Marine amphibious ships, and keeping everyone on a need-to-know basis. I memorized those signs from the books that said LOOSE LIPS MIGHT SINK SHIPS and CARELESS TALK COSTS LIVES. And one of the phrases I learned is that when you confront the enemy, and when you do something wrong in battle, you have to admit it to _regain the mission_.\n\nWhat I'm getting at is that even though the army exonerated me for what happened six days before I finished my tour in Mexico, the record needs to be set straight. The chips have to fall where they may.\n\nCharlie and I had done everything together for four years when the incident happened, April 26. Stateside, before going down to Mexico, we went through our ROTC training together. Some Marines say the training we get in army ROTC is cushy, but beyond the extra studying with books, and strategy sessions to be leaders\u2014able to explain to the troops, clearly, what our mission is\u2014we go through the same rope courses and rubbing of our faces in dirt. I remember, once, after carrying sixty-pound packs of rocks, running in the hot Kansas sun, with the atrociously humid heat of 114 degrees in the month of August, I was lagging behind Charlie pretty bad, a hundred feet back. We were supposed to be jogging together, as a group. There were thirteen in our training platoon. I was extra-dehydrated. I had failed to obey orders and to drink two quarts before we'd started running. We were out on the WSU golf course, next to the main university campus, and I saw the golfers hiding from the heat under their golf carts, getting out of the carts only to barely lob balls down the fairway. I was tempted to lie down on the patchy grass of the fairway. It was so hot, even the grass was dying. I wanted to sit and say, \"This is it. I can't go on. I can't take it anymore.\"\n\nI had struggled through all the test-taking for years, with my dyslexia. The office of student services, on campus, had told me I had a right to take exams with twice the amount of time, and at first I, regrettably, requested that amount of time. But I told myself it was OK if it was necessary to complete the mission, and I slowly weaned myself off that extra time. I wanted to be a full soldier, complete and proud, just like everyone else, and I learned I could be. Usually I carried my pack, not only keeping up with the group but leading it, yet my legs cramped up bad this day. Charlie could have just left me there to sit on the grass, but he risked his standing in the eyes of the other soldiers\u2014being seen as someone who had a friend that might be going soft\u2014and came back and lifted me off the ground. He carried my pack in front, his in back, and let me hop next to him. We got back a full half hour late to the gym, where all the others were waiting. But Charlie insisted on carrying me to the end of training.\n\nThe soldier, Charlie, watched his buddy, Jeremiah, through binoculars at the Central de Abasto market. For five weeks they had been ordered to direct a sting operation at the market. The place was full of fish stalls, small vendors selling slabs of tuna, shrimp by the kilo, lobsters, oysters, cheap sardines, and tiny fish to cook up and fry in the inexpensive restaurants of Mexico City. The air stank of fish. The stalls were elevated on concrete platforms, where trucks came and unloaded their catch, trucked in from the Gulf Coast on ice. Hundreds of trucks arrived in the first light of dawn, an army that drove from the coast during the night, to bring fresh fish to the whole of the nation's capital, Mexico City. It was precisely all the trucks that made it perfect for smuggling cocaine. Smuggling boats on the coast mingled with other fishing boats near Veracruz, and the trucks at the fish market were so full of blocks of ice and fish, there were a million places to hide kilo bricks of coke.\n\nThe Mexican authorities were corrupt, and Charlie had been bought off, too, but Jeremiah didn't know anything about their corruption. The Mexican authorities pretended they hadn't authorized the operation. They'd said, in front of Jeremiah, gringos should stick to the north where the action was. They insisted all the big cartels were operating and fighting in the north, along the border, or in Sinaloa along the Pacific Coast. It was well known that many of the Mexican authorities didn't want to see what they didn't want to see, so the cocaine, pot, and meth ended up passing through the border into the U.S. and coming by hidden rail cars and in small boats into Texas, or attached to tourists as they flew into New York. The ingenuity of the smugglers was never-ending. The drugs didn't always come on small planes into the deserts of California or into Florida. They came in the bowels of mules, in latex bags in their intestines. They came in boxes stuffed into beehives, they came inside panels of old cars and in trucks as they went over the Mexican border. Everyone was a suspect, and Jeremiah had been trained, during his ROTC classes in Wichita, and then in his Special Forces training, to understand the enemy could be anywhere. So when the U.S. government gave the go-ahead for the sting operation, Jeremiah felt the mission must be getting close to some real drug-busting if the Mexican authorities wouldn't authorize the mission. They must be cutting close to the bone.\n\nCharlie watched as his longtime partner, his buddy Jeremiah, whom he'd gone to Special Operations Group training with, appeared in the distance with a briefcase of cash, to meet his counterpart, the drug dealer Francisco Sosa. Through the binoculars, seeing Sosa live for the first time, Charlie was impressed by Sosa's formality. Often the drug dealers dressed in Hawaiian shirts that draped over their fat bellies, over slacks that fell to white alligator-skin shoes. The look was clich\u00e9d and gave them away. But Mr. Sosa wore a dark, worn-out suit and a frumpy panama hat, which made him look like he was the owner of a large area of fish stalls and truly from the coast, just as he was supposed to be. Through the binoculars, Charlie saw Jeremiah shake hands with Mr. Sosa. Sosa looked left and right, as if making sure no one was watching. He looked convinced the man with the briefcase meant business. He shook hands with Jeremiah again, and took him inside a fish stall.\n\nCharlie watched the whole event from the inside of a fish truck with some dark windows and listening equipment, across the wide parking lot of the market. The outside of the truck looked like a run-down Isuzu delivery truck, one of the miniature vehicles that plied the streets of Mexico City. He switched to an image inside the fish market where Jeremiah talked with Sosa, transmitted by a camera they had put in the market a week ago.\n\nSosa said, \"I promise you the product is fresh and of the highest quality.\" He never referred to the cocaine as coke, but always as \"the product.\" It was known this was the best way to avoid any recordings being used in court, though the judges were mostly bought down in Mexico.\n\n\"Can you give me a taste of it, now?\" Jeremiah said.\n\n\"We can give you a taste of whatever you need. I assure you the product is fresh.\"\n\nSosa told his assistants to bring some of the product. He looked around, to see if he was being followed into the back room of the fish-market stall. The image was clear. The head of a giant swordfish poked a long snout from behind Sosa, so the sword seemed to come out of his hat. Charlie zoomed the camera in on Sosa's hands. There was a brick of the product, a kilo of white cocaine, wrapped in layers and layers of plastic with tape. Sosa cut open the block his assistants had brought him. The outside of the package was brown with dark red blood on top, from the guts of fish the package had been sent in. Sosa stuck in the tip of his penknife, with white mother-of-pearl on the handle, and cut at the dense white brick until a few pieces came off in his hands. He gave the brick to Jeremiah to hold, to feel the weight. He lifted the blade of his penknife up to Jeremiah's face and gave the knife to Jeremiah. Jeremiah picked at the area Sosa had cut. He stuck his finger inside the brick and pulled it out with powder still clinging. He put the finger to his lip and tasted the powder and it numbed and tasted rich and real.\n\n\"My people will expect the shipment in three weeks,\" Jeremiah said. \"Two hundred bricks. No more, no less. On time. To Houston. You know the place. Everything you asked for is here.\" He handed over the briefcase.\n\nSosa took the briefcase and handed it to his assistants. They took the briefcase to a different back room to count. They came back and told their jefe it was all clear. Sosa shook Jeremiah's hand. \"It's a pleasure doing business with you.\" Jeremiah left and kept walking. He took a taxi, as predetermined by the operation, to a safe house.\n\nCharlie sat in the truck and gave a thumbs-up to three DEA and CIA officials in the vehicle. They slapped each other on the back. This was the first test to see if the shipment would work, to be sure they had the real guys. The next time, they would arrest Sosa. With the recordings, they could begin to nail Sosa in the States, or they'd take him out, if necessary, in Mexico.\n\nLater that night, Charlie called his corrupt Mexican counterparts to tell them Jeremiah had delivered the money and that the product would be shipped to their contacts in Houston. It was a perfect scam. The money had come from the U.S. government. The U.S. government thought it was paying for the first phase of a real sting operation. Only next time, when they went in to make the final bust, Sosa mysteriously would not appear. He would be tipped off. Charlie would pocket a million and a half of the current six-million-dollar shipment to Houston, and then retire. Jeremiah had no clue he was being duped by his best friend.\n\nThis is how it happened. This is how you end up shooting your own buddy.\n\nThree weeks before what they call the \"incident\" in the military report\u2014which wasn't an incident but a shootout clusterfuck\u2014I went for drinks with Charlie. It was the day after the first Sosa connection. Charlie wanted to go drinking and whoring, and while I tended to avoid that kind of crap, to remain as upright a soldier as I could, after posing as the decoy with Sosa, which I had been selected for by picking straws with the other guys in the group, I felt I deserved some booze and pussy.\n\nWe started out having a martini at the Condesa DF hotel, a super-fancy place that always seemed out of my budget, where the pool on the roof shines aquamarine, in cool hues, up to the pulsating sky of el DF at night. There were fancy women who went to the hotel, women who wore designer dresses and who walked with imported Italian leather shoes, and me and Charlie sat in one of the white couches with a view of the skyline and of the hemlines of those women as they went by, and it felt like life was good, downing the vodka in shot after shot with a whole bottle Charlie insisted on buying.\n\n\"Ain't this the life?\" Charlie said. He was chewing on some ice, and he popped in a cocktail shrimp from a plate of twenty he had ordered. I thought of the simple bars of Wichita, where the women all look like aging sorority girls by the time they're twenty-five, where the only plan is to get married, find a solid job, and raise the kids, if you don't get caught up in a divorce too soon and they make you pay alimony.\n\n\"I've learned a thing or two over here,\" Charlie said.\n\n\"Like what?\" I said.\n\n\"I've learned black and white can be the same color, if you look at them long enough.\"\n\n\"How's that?\" I said. I was into reading martial arts books and strategy by Sun Tzu about the art of war, and I understood how you could take the power of your enemy and absorb it to fight back at them, but I didn't understand what he was getting at.\n\n\"Well, you know how here\u2014if some Mexican says they're coming to dinner, they may or may not come?\"\n\n\"Yeah, I hate that,\" I said.\n\n\"Well, at some point, I came to realize maybe I was the one with too rigid an expectation the person _should_ come.\"\n\n\"But they should,\" I said. \"Don't give me that crap you're going all culturally relative on me. I hate that relativism. Some things are right. Some things shouldn't be done.\"\n\n\"Okay, okay. Different example. Here, if I call some woman up on a date, and I go out with her and she doesn't like the evening, then she tells me she had a good time. She smiles at me and says I'll look forward to your call. She's giving me the turn-on but she really means the turn-off.\"\n\n\"What's so good about that?\" I said.\n\n\"Because no one has to do what they don't want to. They just do what feels right and worry about everyone else's feelings later.\"\n\n\"It sounds like you haven't thought that one fully through yet,\" I said.\n\nCharlie went over to the pool, where a woman in a long, silk dress with flowers printed on the fabric was standing alone. I watched him from my seat in the leather couch. \"Hey, se\u00f1orita,\" Charlie said. \"How about grabbing one of your woman friends and going out with me and my buddy?\"\n\nThe woman smiled politely at Charlie, and looked down at her shoes.\n\n\"Aren't you interested in going out with a handsome stud like me?\"\n\nCharlie looked as out of place in the hotel as if you had just dropped him out of an army helicopter onto the roof. He was handsome and tall, but his whole body language was stiff. He gestured too much. He waved his hands in the air, trying to persuade the woman she should go with him. He looked like a soldier trying to buy a camel in a foreign land, and these women weren't camels.\n\n\"Excuse me,\" the woman said to him. She was speaking in English, because Charlie had been speaking in English to her. None of us spoke much Spanish. \"I would really like to be able to go out with you, sir, but I'm afraid I'm busy tonight.\"\n\n\"Aren't I good enough?\" Charlie said. \"Don't I look good enough? Don't I look like the kind of guy you would wanna bring back to your mamacita to show off?\"\n\nThe woman winced. She was standing by the edge of the pool, next to a ladder that went in and out of the water, and she started to grip the ladder tightly like she needed something to help her out. She started to step away, trying to walk slowly so she wouldn't seem impolite.\n\n\"Oh, now, don't go away, little honey...\" Charlie said. \"Don't you do that.\"\n\nOne of the men a couple booths away, the manager\u2014a tall Mexican wearing a tailored suit, with a black shirt made of silk and expensive leather shoes, a five o'clock shadow neatly cut in the way of the fashion of the day, and bald in a manly way\u2014came over and said to Charlie, \"I'm sorry, sir, but it seems the woman isn't interested.\"\n\n\" 'Cause she thinks she's too rich or too fancy,\" Charlie said, shaking his head. \"That's what they think, around here.\" He said this turning back to me, but then, speaking so loud he was almost shouting it to the entire rooftop, continued, \"It's all about class, class, class in this fuckin' country. Everyone thinks they were born from some rich Spanish king or queen. Like they're all fucking _g\u00fceros_.\" He was using the word for \"whitey\"\u2014a g\u00fcero. He was saying everyone wanted to be a whitey.\n\n\"I think it's time to hold your liquor better, soldier,\" I said to Charlie. \"It's time for you to get in line for roll call.\"\n\n\"That's your problem, Jeremiah. You're always thinking the only path to take is the one of a soldier. There are other paths, too,\" he said.\n\nIn the heat of the moment, I let the comment slide. I didn't think all that much about what he was saying. I wanted to stop my buddy from getting into a fistfight in a fancy hotel with the other Mexican guy, before he called the bouncers.\n\n\"Come on, soldier,\" I said. I pulled Charlie's arm.\n\n\"She likes me. I'm telling you she likes me. Black is white. I'm telling you.\"\n\nBy that time, the woman had moved quickly to the far side of the rooftop patio. The water from the pool cast strange shadows against everyone's faces. I saw a darkness, and sneer, on Charlie's face I had never seen quite the same way, though he often liked to get into bar fights. I knew the first rule we were taught about being in another country: _Always learn the local traditions and culture_. _Get the other culture on your side_. _Use the strength of the enemy village and government to engage in counterinsurgency_.\n\nThey pulled Charlie off the rooftop\u2014two bouncers\u2014and held his arms behind his back. The bouncers were dressed in crisp white suits, similar to the one of the rooftop manager. They escorted us toward the door. I spoke to the two bouncers in as much broken Spanish as I could. \"Please tell your manager and the se\u00f1orita we are sorry,\" I said. \"We are sorry and we didn't mean any disrespect.\"\n\n\"The fuck I didn't,\" Charlie said. \"I meant every word I said in there. These fancy Mexicans think they're better than everyone.\" When they got us to the front door he said, \"Come on, let's go find ourselves some place that's got some real women to fuck.\"\n\nOutside, he insisted on grabbing his car. He wouldn't give me the keys. He was so worked up, he was going to drive. He was drunk and on his way to getting drunker.\n\nThe rest of the night is more or less a blur. We drove to the Zona Rosa, where the prostitutes hang out. We chose two women. They wore tight miniskirts, the way none of the local women ever did. One had long black hair that cascaded down to her knees. She had on a red skirt, and she let most of her breasts show out of a loose, leopard-patterned camisole. She oohed and aahed at the car as we drove by. \"I want that one,\" Charlie said. \"I want her to give me a fuckin' blowjob.\"\n\nI had no interest in really getting it on with any of these women, but I realized my mission for the evening was going to be to keep Charlie out of any serious trouble and that he'd need me along for the ride. In the military, when you go swimming, when you are out in the bush, you know you always need to have one buddy who's going to keep your ass safe. That's the person you can count on, no matter what. It doesn't matter what they say or what they do, you stick to them like a fly on shit. You are their eyes and ears. You hump them home, if necessary.\n\nI chose the prostitute I thought was the prettiest, but who wouldn't get in the way. I chose a petite thing that looked like she was a good daughter, just trying to make a buck. We went to a bar called the Cuatro Aces, and we sat around with these women getting one drink after another.\n\n\"You see,\" Charlie said to me later in the night, leaning in close to hug his gal, \"these women know a good man when they see one.\" The woman he was with pulled a cherry out of her drink and put it in Charlie's mouth.\n\nWe went to some hotel and grabbed a pair of rooms, and the walls were so thin I could hear Charlie in the next room yelling, \"Come on lady, ride 'em cowboy! Ride it!\"\n\n\"Don't you want to make love?\" the woman I was with asked me. \"Come on, let me make you happy. Let me give you a party.\" It was two in the morning, and I couldn't get out of my mind that we had a ride down to Cuernavaca we were going to have to make the next day, with a couple of Mexican military officers, at 1500 hours, but the booze was pulling at my mind. I'd had a bunch of vodka and tequila. I couldn't help but hear Charlie in the other room saying, \"Yes, yes, more, more. Give it to me. Give it to me. Give it to me, little lady. Give it to me.\"\n\nI may be a soldier, but I'm not a saint. I didn't like paying for my loving, or for whatever sleeping with a prostitute meant. Call it the relief of manly urges. I had no illusions that the women giving us the blowjobs and sex liked it. I knew they were just trying to make some money. _Know the real face of the enemy_. _Don't be fooled by the smiles locals will give you,_ I remembered from my training. I let the woman get on top of me and ride for a bit. My head was swimming. I remembered the tension of going in to see Sosa with the briefcase. It was just \"the mission,\" but it had been a tense one. I tried to let out some of the tension. I got on top, and with the booze flowing through me, I felt more like an animal, spurred by the sounds of Charlie in the other room, and I thrust hard until I came.\n\nThat night was the beginning of something new with Charlie, something that seemed odd to me, after knowing him and living with him in the next-door bunk from base to base and operation site to site for a few years. A couple days later, I saw him with a gold Rolex. He didn't wear the Rolex when we were with the other soldiers, but he put it on when we went out for a drink.\n\n\"What's up with the new watch?\" I said. \"You get it at one of those pawn shops in el Centro?\" I had been walking around the center of Mexico City, where there are plenty of shops where they buy and sell gold by weight. I figured he must have gotten a deal at one of those places.\n\n\"Nah. Not at a pawn shop,\" Charlie said. \"My whole life my dad used to buy things at those pawn shops. This one's for real.\" He pulled it off his wrist and held it in front of his face like he was holding the Promised Land.\n\n\"Must have been expensive,\" I said. \"How'd you get it?\" Our salaries were shit in the army. People always thought we made good money, because we were Special Forces, but we didn't.\n\nCharlie looked at me and said, \"How the hell do you think I got it? Saving up money. I earned it.\"\n\nBut I knew a new watch like that cost thousands, and he was suddenly spending more on a lot of things. He kept going back night after night to that same prostitute we met in the Zona Rosa. I went back with him once, since he was my buddy and I wanted to let him know I was with him, but I couldn't keep up his pace\u2014either with money or with desire\u2014and I let him just tell me stories of conquering love when he came back to the small base where we lived.\n\nHe started buying fancier clothes, too. He told me he was sending money back to his mother, in Wichita, and the day we were going to meet Sosa for the second time, he had a bag all packed, with some of those nice clothes, next to the foot of our bunk, and he said he was going to go on a nice vacation to Canc\u00fan once the operation was over, when we were going to be given a couple days of R&R. The whole thing seemed odd to me, extravagant, and I couldn't figure out where he was getting the money.\n\nThree weeks after the first contact with Sosa in the Central de Abasto, Jeremiah lay on top of a roof looking over the parking lot and the area where the delivery trucks came with fish, where a football field's length of stalls spread one after the other, and where men carried fish in and out of trucks. It was early morning, and the first light of sunrise was hitting the clouds. Looking through the scope of his M86 sniper rifle, trained on the same storefront across the way where he had met Sosa before with the briefcase, to give him the money, he was so focused on his targets he didn't notice the fiery struggle of dawn. Above his shoulder, crouched on the roof behind him, the head of the Special Operations Group, Colonel John P. Saunders, had come to direct the final sting operation. Saunders had been brought in, unexpectedly, three days ago, and the rumor was he was brought in because the operation might be compromised. He was a veteran of foreign battles, head of a counterterrorism group that had operated in Iraq for three years and then in Afghanistan. He was known as a powerful son of a bitch who didn't take whining from his soldiers. With gray hair cut close, patches for valor and leadership pasted across his battle fatigues, he looked like a puma ready to spring off the roof, watching in the same direction as his soldier below him, Jeremiah. His right hand formed a fist, and his knuckles pressed into the roof tar.\n\nAcross the way, and walking up a staircase to meet Sosa, Charlie carried a briefcase identical to the one brought by Jeremiah before. They had all picked straws, again, to see who would carry the case this time. It was dangerous being the one who delivered, if anything went wrong. Professional buyers used different mules to bring the cash all the time. It was a game of musical chairs. Everyone had to have trust, but everyone had to know that at any moment both sides could disappear, vanishing in the night. The world of drug deals was like a mirage, water that came in the desert like an oasis and then went.\n\nIt had fallen to Charlie to deliver the cash this time. Below Jeremiah, where he looked through the scope, the same Isuzu truck with the listening equipment was in the parking lot, with four men. Six other men were also stationed, with short-range Uzi machine guns, dressed in plain clothes, pretending to be workers on the fish docks. Charlie was protected and surrounded. The plan was to have Sosa taken out with one shot when he came to meet Charlie. Jeremiah would aim with precision and shoot. One shot, to the temple, and Sosa would fall. Sosa would likely be outside for only a second before he would want to retreat into the back room, where fighting would be more difficult. So they wouldn't wait to shoot. In the immediate mayhem, Charlie would jump down in front of the concrete platform where the delivery trucks unloaded, to protect himself. He was wearing a bulletproof vest under his loose plaid shirt. While Charlie crouched beneath the concrete barrier, the other six men with guns on the platform would deal with any return fire. It was the most likely way to get Sosa, with a sniper bullet. It was quick, simple, and efficient and sent the right message: this cartel was going to be eliminated. Colonel Saunders had announced the plan to his group three days ago, when he took over direct command. \"In the second raid, once we know they have delivered the packages to Houston, we will eliminate Sosa. We will let them know operations in this area are forbidden and will cease to continue. Are there any questions?\" Colonel Saunders had asked the group.\n\n\"Why not just bring in Sosa alive?\" Jeremiah had said, during the briefing. \"Get more intelligence. Make an example of him, publicly.\"\n\n\"And just how, Second Lieutenant Young,\" Colonel Saunders said, \"do you think we are going to grab a guy like Sosa alive without more manpower down there to do the job, putting them at risk? And what the hell do you think is the probability a Mexican court will not be bought off, or that our own courts in the U.S., if they ever get jurisdiction for the case, and if we can get him extradited to the U.S., won't give some crap that the evidence we've provided is tainted, that we have somehow failed to protect Sosa's rights, while thousands of young Americans are poisoned in their bodies, through the scourge of drugs, because of this man, destroying the manhood of our country and increasing drug use in the ghetto? This operation is going to be clean and efficient, and legal within the authority vested in this group. It is my final decision, and it will be done this way.\"\n\nJeremiah had looked forward, standing at attention, chin flat and eyes looking into the distance, seeking not to see anything other than the command he had just heard, after receiving the dress-down and final order.\n\nOn the rooftop now, he felt the presence of Colonel Saunders next to him, tense, chewing gum, surveying the target area. Jeremiah squinted into the scope of the gun. He followed the crosshairs just in front of Charlie as he walked up the concrete stairs to the delivery platform. The scope made everything clearer than in normal reality, Jeremiah felt. It showed a man as he was, as a physical body, as a plane in space rather than a mind. Through the scope he always felt he was finally in control, that he was master of his defects, that he knew why God had put him on this planet, making it harder for him to read in order to give him another gift instead, the gift of being a supreme shooter, slow and steady, one of the few who could breathe calmly in and out, so calmly the gun barely moved, maintaining only the will of his targeting, until he followed orders and took out the necessary target. In shooting he felt the certainty he was completing his mission. He had taken out thirty-nine targets during his two years, so far, in Mexico. He had shot from rooftops in Ju\u00e1rez and rooftops in Guadalajara, and out of hotel rooms in Acapulco. He had watched men with briefcases scatter onto the pavement. He had missed only twice. All other times, it was one bullet, one carefully placed projectile, preferably in the base of the skull area, where the flesh was softer, or higher up, between the eyes, if necessary, or in the chest as a last resort, where there were more bones to protect the heart. The bleeding was longer there before death.\n\nThrough the scope, he could see Charlie up on the platform, mouth pursed, eyes looking around for Sosa. Six men came out to meet him. Two approached to give him a handshake. They looked oddly relaxed as they gave him the shake. Not tense and distant the way they usually interacted with the client. Charlie stood on the platform, and he seemed to ask for Sosa. He pointed to the interior, from where Sosa had come out the last time, and raised his hand in the air, questioning. He waited and waited. The other men with him milled about, looking left and right to see if anyone was coming. The pause was getting longer and longer. It was far too long to be normal. Looking through the scope, Jeremiah ran his finger up against the trigger. Soon, Sosa would appear. Soon he would shoot and it would be over.\n\nThe men on the platform suddenly crouched down, their knees pointing forward in the direction of the Isuzu, and they began firing with all their intensity at the vehicle. The truck was no more than forty yards away.\n\nThe bullets hit the truck and the windshield crashed in. The driver inside, a Mexican in on the surveillance group, who was used to blending in with the locals, was instantly killed. The bullets came more and more, against the rear of the truck where the four DEA and CIA operatives listened in.\n\nThe men in plain clothes, in the team on the platform, came running from the fish stalls. They were a mix of Mexican military. The Mexican military had been brought in on the raid at the last second. If things got violent, it was going to be necessary to have their approval and participation, and they had finally chosen to be in on the mission.\n\nJeremiah shot at the men on the platform who were shooting at the Isuzu truck. He aimed, with the precision of a mathematician, and killed one man, with long hair and a black leather jacket, who was firing at the truck. He looked to the left and took out a man wearing baggy jeans and a black T-shirt with a skull. He heard the sound of bullets going back and forth, at the Isuzu truck, pounding harder and harder into the truck and at the gas tank, until the tank gave way and the truck exploded.\n\nThrough the scope he could see Charlie had dropped down to the front of the concrete platform where he was supposed to jump once Sosa was shot. But Sosa had never come.\n\n\"Take him out. He's a mole,\" Saunders said. He put his hand on Jeremiah's shoulders.\n\n\"Who?\" Jeremiah said.\n\n\"You know who. You can sense it, solider. He has betrayed this entire group. He has betrayed his country. I received intelligence last evening that he tipped them off and that Sosa wouldn't come. Sosa hasn't come. It's confirmed true. And now you'll get rid of the cancer within the unit before it spreads. Take him out.\"\n\n\"Sir,\" Jeremiah said. \"Sir, this is not how we deal with such situations, sir.\" He couldn't believe his buddy had betrayed him. He couldn't believe he had betrayed the unit. He could not believe he had betrayed his country. He could not believe he would do such a thing. \"Sir, is that a direct order, sir?\"\n\n\"Yes, it is.\"\n\nAnd though in his conscious mind, if he had had time, he would have reflected harder on the fact that he had the right to disobey an order he knew directly to be against the law of his country, he felt the hand of the colonel press hard between his shoulder blades. He saw, out of the corner of his eye, the truck in flames. He had heard, on the group radio, the cries of the men in the Isuzu truck. He saw the men firing back and forth on the platform. He saw Charlie run along the base of the platform, running away from the scene with the briefcase, toward the taxis. He knew, looking through the scope, that if he wanted to, he could shoot the target between his head and shoulder, at the back of his brain, completing the order just as he was told. But he chose not to. He knew he could not do that. He could not kill someone in his group, even if he had betrayed the group and led to the deaths of others. He moved the scope ahead of Charlie, running. Yet he could not do nothing at all, with the colonel there, ordering him, even if his order was unacceptable. He would shoot in front of Charlie to scare him, to let him know he had been found out, to complete, in some form, the order of the colonel, while disobeying him. He had to do so. This was his dilemma. It was a direct order. He aimed wide, to the left, in front of Charlie's feet as they twisted and hurried across the pavement. The bullet sped forward. The gun recoiled. And then it all happened in what felt like a long moment. There in front of him, through the scope, to his surprise, the bullet caught the tips of Charlie's toes. It hadn't totally missed. It had grazed one of his tennis shoes. Charlie bent forward and down, suddenly arrested, as if suddenly realizing he had feet that he had taken for granted while fleeing. He dropped the briefcase and looked for who was shooting, up at the rooftops. He looked in the direction of Jeremiah. He turned back down and reached first for his foot and then for the briefcase on the ground. He hobbled forward with his left foot slightly wounded. And in that moment of arrested running forward, a pause in his running toward the taxis, one of the men on the platform, who had been fighting for Sosa's cartel, found Charlie was running away and turned to Charlie and fired at him and shot Charlie in the head and in the neck. It was that moment of pause, surely, which led to his death, Jeremiah thought. Charlie lay on the pavement of the Central de Abasto, his head caught in a pothole with the sloppy residue of the fish market, bleeding into a mix of water used to hose off the fish stalls, fish oil from scattered fish bones, and the oil of the idling trucks. His body lay twisted, his hand reaching toward the briefcase, his eyes looking up at the sky, blank now, no longer quivering as they had just done, dead, as Jeremiah looked at him through the scope.\n\nHad it all been in my mind? Was Charlie really gone? My buddy gone? Certainly his suitcase, which was at the bottom of our bunk, was gone when I got back to base. I sat on the bunk bed, thinking of Charlie, shocked and crying. Could Charlie have really been guilty, as the colonel said? There was the watch and the sudden money. But I needed proof, much more proof than that.\n\nWhat I knew is that the colonel had gone crazy. Such a direct order to shoot Charlie could only have come from battle fatigue, from too many tours in Afghanistan and Iraq before being sent to the relative calm of the \"pasture\" of Mexico. But six months later, stateside, out of the heat of the moment in the field, I wasn't so sure the order had come from battle fatigue. I had seen headlines in the newspapers that confirmed what I had seen on my own: drone strikes in Pakistan and Afghanistan, taking out people who had never been judged in a court of law. American citizens shot by the drones. Those were true terrorists, true enemies, and before the incident with Charlie, I had never thought about them too much. Yet now I could see, placed within a command environment of choosing who lives and who dies, and deciding that even American citizens could be shot from the sky, and placed in situation after situation where millions of dollars of cash are given out to buy the loyalty of men who are thieves, like the president of Afghanistan, Hamid Karzai\u2014how far of a step was it from those actions, approved every day by my government, to an order that crossed the line, like the one from Colonel Saunders asking me to kill our own man, simply because he had double-crossed us? Yes, in Saunders's heart of hearts, he must have known somehow that what he had ordered me to do was too much, a crossing into the land of some character like Mr. Kurtz in that novel by Joseph Conrad that I read once called _Heart of Darkness_. Coming home, I began to read up more and more, even at the slower pace of my dyslexia, to make sense of what had happened to me down in Mexico.\n\nThis morning it is Memorial Day, and a full year has passed since I left Mexico. I told the other men in my unit, after the battle where Charlie died, what the colonel had ordered me to do. None of the men would believe me, or they chose not to hear. The army wrote up a report about the \"incident,\" as they called it, and they said that after a case of accidental \"friendly fire\"\u2014as they call such shots from us when we hit our own\u2014the soldier, Charlie Reynolds, was lightly wounded and then taken out by the members of the Sosa Cartel. The description is completely accurate as to what happened, physically, and yet it is completely false. The colonel had me removed from the unit. He said I had done great service to my country but that I was getting tired and needed a rest. The army gave me a Silver Star Medal for the many kills I had made as a sharpshooter, and sent me home.\n\nBack home in Wichita, I got to thinking about truth and lies. I thought about My Country 'Tis of Thee Sweet Land of Liberty and shooting and taking out those thirty-nine people in Mexico, by my hand, by my own will, by my own following of commands and orders, by my own volition, by my own feelings of honor and duty, and so on and so forth, and what all that meant. How, exactly, did killing some drug dealers help to raise the American flag high? Where was one drug user who was using drugs less because of any one of those men I had killed? The drugs would come as long as the people wanted them. Remove one drug dealer and another will mushroom up. And another. And another. Every person smoking a joint, every person snorting coke or injecting or snorting heroin, knows what they are doing, knows they are tied into the drug trade, and yet they are not going to stop doing what they are doing. The feeling of getting high is too powerful, too good. The liquid train of drugs will forever come into the country as long as people want to party, as long as people are sad, as long as people feel they have nowhere to turn in their hour of need, when God and Jesus and nothing above will seem to quiet the pain within.\n\nI began to deeply question what I had been a part of. I began to ride down the wide avenues of my city of Wichita, so neat and clean, each road wide enough to have a military parade, the roads paved and graded, and the box stores of Walmart and Best Buy and one soulless mall after another spreading out in an infinite plain of vapid consumption, while there was no money, I had been told, to send me to college without me joining the ROTC. I was not complaining. I was not whining. I was not saying the government should give me a handout. I was not saying that I had never wanted to be an Eagle Scout or a Boy Scout. But what kind of country, I began to wonder, makes it easier for you to get an education if you are willing to kill than if you want to talk about peace?\n\nAfter Mexico, I couldn't stand living with my father. That was no longer an option. I took an apartment for myself. I lay in bed reading, thinking, and recalling the faces of all thirty-nine of the men that I took out with single bullets. I began to drink hard. I drank Jack Daniel's first, and then moved on to cheaper schnapps. For the better part of six months I drank. And then I saw a homeless vet walking down the street, carrying all his possessions in a shopping cart, wandering from here to nowhere, as I'm sure you've seen so many times before. Before I had gone down to Mexico, I had looked at those homeless vets as failures, as aberrations, guys without discipline who just couldn't hold it together. Not like me. I was a real solider. A kick-ass soldier. I was America's finest. I was bred to be the best. Yada, yada. Just like on the army recruitment billboard that I saw every day in my neighborhood that towered above the street, and over the vets who wandered to the vet thrift store to get some clothes. I saw that homeless vet, and I was tempted to give him my alcohol. I no longer judged him. I went home and poured out all the booze into the toilet. Who dares to judge a homeless vet like that on the street, or to even feel pity for him, until he has been in battle?\n\nThis morning, I saw the front page of the newspaper. There was a glowing, patriotic photo of a soldier at Dover Air Force Base prepping an empty dress-uniform to be placed inside the casket of a fallen soldier. The message of the photo was that America takes care of its own. We make sure we treat our soldiers right.\n\nI hopped in the truck my father had said he didn't need anymore. It was an old, beat-up Chevy pickup, with dents on the back from being used on a farm outside of Wichita, moving equipment. I didn't know where I was going to go, at first. I drove around the endless desert of strip malls, along the flat, wide avenues of the city, passing an International House of Pancakes then a Staples, a McDonald's, a Target, a Hog Wild BBQ, a pawn shop and then another pawn shop and then another. I got out of the car at the A-OK Pawn Shop holding my medal for my tour of duty in Mexico in my hand, and walked into the shop and put it on the counter and asked, \"How much for this?\"\n\n\"This? You sure you want to sell this?\"\n\n\"Yes, I'm sure. How much?\"\n\n\"Truth be told,\" the man at the counter said, \"the value is more in what it means than in what you can sell it for. I can't really get all that much money for a medal. You'd be better off going to a specialty collector than trying to sell it in a pawn shop.\"\n\nI looked around at the interior of the store, old electric guitars from dreams of making it big in a band, old tools from mechanics who couldn't find jobs anymore in construction during the recession, or who had been laid off from the airplane factories because of Chinese competition. I saw weed whackers that looked like they may or may not be able to run anymore, things discarded from garages stuffed with too much junk. I saw merchandise that looked like it had been sold by small-time drug dealers\u2014high-end speakers way too scuffed-up to have ever been owned by someone who had real money from a legal job. I saw videos and DVDs, row upon row of movies no longer wanted, that had been seen, digested, and found not worthy of holding on to. I saw people overweight at the checkout counter buying things on credit cards. I saw a man with a dark tan, with a long mustache falling from side to side and a part down the middle, who looked like the peak of his life had been serving in the military and now he was smoking two packs a day and trying to hold down a job to feed his family. I saw all this as I looked around the A-OK Pawn Shop, with wedding rings that were being sold after marriages had gone sour, diamonds that no longer shined.\n\nA young, disillusioned man could look at this forever, bitter, seeing things he wanted to see reflected back at him, seeing his own reflection in objects. And then it was that I saw my own true reflection, in a mirror between the section of the store devoted to tools and the section devoted to jewelry. I looked up behind the man working at the jewelry section, who had told me my medal might not be worth as much as everyone thought, and I caught the reflection of myself. There was a young man looking too tired, too dejected for his age, his short hair no longer looking buzzed, no longer neat and orderly, no longer standing on the drill field at attention. I saw the bags of sleeplessness around the lower edges of his eyes, and the brown pupils that seemed so dark they swallowed into the iris. What was I doing in this pawn shop? How did I get here? I picked up the medal from the counter where I had left it while I wandered around the store. It was time for me to find that core of humanity inside me, again. It was time to erase the image I had seen of myself in the mirror. I picked up an old acoustic guitar, hidden beneath the electric guitars. It was old and used, but there was still life in it. I strummed on the guitar. I didn't know how to strum well, but I kept picking at the guitar for a while, until the man in the store came over and said to me, \"You seem to play pretty well.\"\n\nI went up to the counter and bought the guitar and went outside and put it carefully in the back of the pickup. I drove out to Charlie's grave, on the edge of town, at the small plot where he had been buried a little over a year ago. There was his tombstone, with plastic flowers still fresh. The military wouldn't give him a burial. They had simply sent the body home.\n\n\"Charlie, you fucked up,\" I said, \"you wanted too much, you got greedy. But they did you wrong.\" I looked down at the grave. \"I did you wrong.\" I left the guitar on his grave, and I went home knowing I was going to go back to school. There were other ways to \"Be All That You Can Be.\" If I had made it as far as winning a medal as a soldier, I was going to be far more as a civilian.\n\n# THE PAINTING PROFESSOR\n\nIn Puebla, the professor was losing his mind. After the bullets of the night before had quieted down, he went outside of his compound to inspect the walls of the fortress of his studio. The studio had been a factory, once. Fifteen feet high, thick walls bordered the compound, and there were small watchtowers on each corner where the guards who had once protected the factory could take aim and fire, if necessary. More likely, they hadn't been protecting the factory from theft, but rather keeping out unionized workers in the old days, the professor had thought, when he'd first bought the place.\n\nHe was a professor of painting in Mexico City, and he had bought the studio in the city of Puebla, an hour and a half away, in the late '70s, when traffic was less bad and it didn't take so long to get out of town to his private painting retreat. The factory had been a sock production facility. The interior of the compound had three buildings. One still had the old, rusted metal molds, like upside-down Christmas socks, which threads were wrapped around. It was a bit of a mystery what the metal figures were for, but they looked like a garden of statues to the professor, like pinwheels floating in the air in the first building. He had left the space alone as a monument to what the compound had originally been. In the second building, he had stored his paintings for thirty-five years and used the wide, industrial open space as a studio. Old fiberglass patched the tin of the roof here and there, letting in a tired light. It was like his brain, he thought, fragments of light coming in and bending, diffused. His thoughts were this way, no longer clear and focused the way they once had been. He was in his mid-sixties. For a man who had been so alert only three years ago, the increasing dullness of his mind, the feeling he was no longer who he had only just been, came to him regularly\u2014and more than usual now, as he looked up at the bullet holes in the front wall of his compound. He would have to do something about the holes. He would have to climb up a ladder and patch them up and paint them over with the creamy, pink adobe he had selected for the outside of his home. He had chosen the color because it was innocuous, keeping the world at bay, he'd hoped, shutting out the world enough so he could be left alone inside his artistic compound, but now, it seemed, the gang warfare in the neighborhood wasn't going to leave him alone.\n\nWhere had his mind gone? he wondered, looking up at the wall where the chunk of plaster was missing, high up, just where a stray bullet had wandered, divotting a piece of the old adobe. Stephen Hawking, the physicist, said there were infinite black holes in the universe. But the holes were supposed to be out there, somewhere up in the heavens. He pointed up at the sky, moving his finger up from where he had picked at the wall, cursing the empty space of plaster. So many holes. They were supposed to be affecting the minds of others, and now he couldn't deny the accumulation of the last three years, the feeling in the morning that he couldn't grasp the newspaper firmly, the feeling he would begin an article and then his mind would wander and he would put the paper down and find he was just staring into space, lost in reverie, but with no beauty to the reverie, just a focus of the mind on things that were too small, like dust motes floating in the air.\n\nHis second wife was inside. His first wife had been more than dutiful to him. His first wife had given him two beautiful daughters, even if the younger one found it difficult to find work, after being laid off as a flight attendant. His first wife had watched him rise in the painting world, evolving from a young, modern painter who painted with bright, bold colors, with patterns that had made him a figure in the Mexican art scene of the early '70s, into a painter that had moved on to scenes with abstract backgrounds with shades and folds so subtle they felt as detailed in their brushstrokes as the fabric in a painting by Caravaggio. Most abstract painters relied on quick, violent brushstrokes, but he'd moved the compositions forward with the elaborate shadings of light and dark, integrating the techniques of Old World masters he had found in Italy, into modern fantasy spaces as wide and open as the skies of the universe. That technique wasn't the part his first wife had to be tolerant of. Those were the things that had made him somewhat famous\u2014a painter's painter. It was his drinking. He was a brooder. He looked for the shadow under every rock. He delighted in pointing out the darkness of the sky to the audiences of Mexico, rendered in such precision and with so much powerful energy that even those who thought he should be painting political art\u2014which he despised\u2014had to admit his brilliance. The students had loved him and clamored to be in his small painting studio, where he let in only six students every year. But there were times when he wouldn't show up in class for weeks, when the demons in his mind would take over, when he would simply sit in his room and drink. And then there had been the multiple affairs he had had with his students.\n\nWhich is how he had arrived at his second wife, a very mediocre painter, one he would normally never have taken into his own studio, but whom he was so attracted to he couldn't say no. He had slept with at least a dozen of his former students by the time he met her. His second wife was good to him, she took care of him, and especially now that his mind was less reliable.\n\nA shopkeeper from next door came out to look at the professor looking up at the bullet holes in the wall. \"It's getting worse,\" the shopkeeper said. \"Last night there were eight of them. I watched them from the window of the store. I had the lights off and I could see them, spraying gunfire one at the other. Didn't you see them?\"\n\n\"Oh yes, I saw them,\" Professor Mauricio Sanchez said. When the gunshots had started, at first he had thought it was more of the nightmares, which seemed to echo through his mind too frequently, these days. But then he'd realized it wasn't the case. He'd told his wife, Ana, to stay in bed, to close her eyes and pretend not to hear. Then he'd climbed up the strange, twisting staircase of the third building of the compound, which he had renovated, where he'd created a house of lofts and spaces that felt like the winding layout of a medieval Spanish town.\n\nFrom the rooftop patio he'd peeked over the rim of the wall, the mercury lights in the distance barely giving much sense of the shooters below. The gang members crouched behind cars on both sides of the road that went out of town. He had bought the factory because it was cheap, and it was far from the beautiful, Colonial center of Puebla. The men shot in rapid bursts. He could barely see the car just below him, in front of the compound wall of his studio, but he had a clear view of the cars across the way, where bullets blew through the glass of an old white Ford station wagon. The glass popped and fell within. At least one man was killed. The bullets went past the car into the front of a local hardware store, blowing holes in the glass behind the security bars that were meant to keep out criminals. After the spraying of bullets back and forth, like the roar of infected lions screaming at each other, one of the cars suddenly drove off, tires tearing against the wet pavement, where water had collected from the summer rains. The cancer was gone, for the moment. It was like one of the increasingly frequent headaches that came and went inside Mauricio's brain. They came and rattled. They banged at his temple. And just like the psychiatrists that couldn't give him any medicine that would completely control the headaches, there was no law and order, no police that came\u2014or that would come\u2014to solve the growing violence in his neighborhood.\n\nLast night, in front of that fucking professor's house, that was one fuck of a gun battle I was in, man. That old man, he's been in the neighborhood forever. He's been here since before I was born. This neighborhood is a nothing place, man. This neighborhood is just on the road out of town. I don't know why the fuck he would ever have wanted to come here. I saw the professor last night, looking down at us while we was shooting, and I thought, Why does that guy get to sit up there so nice and comfortable-looking? He was pretending we couldn't see him. He kept peeking over the edge of the roof of his building. I mean what the fuck does he think, that we're just a bunch of cattle down here that can't see nothing? I could have shot that motherfucker to kingdom come last night. I thought about it, too, just to put him in his place. That's why I shot a few bullets up high on the wall, to let him know we can take him out anytime. But I got so much more important things going on than that old cabr\u00f3n, you know what I mean?\n\nMe and my compadres, we never asked for the shit to go down last night, you know what I mean? Yeah, we started selling some drugs in a new area. I admit it. So what? A guy's got to make a living.\n\nI've been selling small packets for four years now, next Saturday. Four years, and I'm still alive. 'Cause I've got what it takes to move up the ladder. Not like a bunch of these sad-asses around the neighborhood. Chucho is the one who hooked me up. He came around four years ago to the little store, around the corner, where my father used to work before they fired him. I was sixteen, then. My father had told me to come help him move some gas tanks onto one of them delivery trucks. My papi never asked me for nothin' unless he needed it, so I went and gave him a hand, and he said he was goin' to give me twenty pesos if I helped him move the gas canisters. Those things are big. Too big for him. And I was trying to pick up change however I could.\n\nSo when I was loading up the canisters, that's when Chucho, who was driving the truck, asked me if I wanted some more regular work, if I wanted to make some real money. He gave me a nod to let me know he was talking about more than fucking twenty pesos. And I was interested, so I went in closer to him and he said, \"Sometimes we've got some things we need distributed around the neighborhood, you know what I mean? I know you. I know all about you. I've heard from your father and others you're a real man. They tell me no one can scare you.\"\n\nSo that's how I started. Delivering small packages, in the gas truck. That's how we moved around town. The old men, like Chucho, they'd sit in the truck while I'd unload the big gas canisters from the back. Those forty-liter tanks hurt like a motherfucker when you take them off the truck. But I'd take them off. I acted like a regular gas man, and then, every once in a while, I'd drop off a package. You know what I mean? A secret package. No one had to tell me what was in them. No one did, at first, but I knew they were drugs. That's how we moved some of the product around the city. It was small shit at first, like that, and then I started moving up.\n\nAnd one time I delivered gas to the professor's house. We had to do some real gas deliveries, too, and he opened them big-ass doors that keep his compound closed. I walked in there, rolling one of those 40L gas tanks on the base, along the ground, the way you have to after you've pulled the thing off the truck. And I noticed he didn't even notice me. It's like that professor, he was just lost in his thinking. He was lost in all those weird pieces of metal that looked like socks hangin' upside down.\n\nIt was strange in there. I couldn't understand why there was so much space for one guy just making some paintings. We had heard about him. A painting professor from el DF. I like painting and all. I had my car airbrushed. Had my friend do it. But I couldn't see no paintings around when I went inside. He had some cool shit in his house. There was some old ceramics and he had a big paper-mache doll of La Catrina\u2014she was all bones and she was a big old skeleton rattling around with a sun hat on her head. I kinda dug that. But I noticed the professor thought I was just some boy. One of the people who just worked for him. Like his muchacha, Carmen. Everyone knows Carmen in the neighborhood. She says nice things about the professor. She says he's a bit weird but that he treats her right. But I didn't feel that one bit. He didn't even notice me. Gave me a two-peso tip, which was the normal tip everyone gave, and I kept thinking\u2014with all that big compound, with all that space, couldn't he have found another peso or two? What's he so hard up about?\n\nThe day he had planned was unraveling. In the morning, after he'd pulled out the ladder and climbed up to the divots from the bullets, and had filled them with some of the adobe he'd found in bags stored in the wide factory space where he also stored his paintings\u2014so many paintings, many under cloth, work from his earlier days that he held on to out of sentimentality, or because they were experiments which he hadn't sold and would be valuable to collectors someday\u2014his muchacha, Carmen, had called to tell him she was feeling sick and she wouldn't be able to make it to clean the house. It wasn't like Carmen to call in sick. She had a mother who had cancer, but even though she needed to help her mother, if she could, she was almost always present. It was one of the things that amazed Mauricio, the way people with the least made the most effort to help others. Some might say it was because they needed money the most and couldn't afford to lose their job, and there was some truth in that, but he felt, without wanting to overly romanticize the poor, that in the case of Carmen, she was simply more giving. He had turned to Buddhism the last few years, because of his older daughter, and because of a half-Japanese, half-Mexican student he'd once taught, who had died tragically in an accident in Japan. He was drawn to the asceticism of the Buddhism he'd learned from the parents of the student. The discipline, the long sitting on a tatami mat focusing and training the mind, had appealed to him. He would sit and think about decoupling from the material world into the spiritual world. And he felt Carmen did this without having to do all of those elaborate meditations. She wanted less, and expected less, and so she seemed happier than those who wanted more, like Mauricio, who, even though he claimed he painted only because he wanted to paint, and even though that was his main reason for painting, his desire to have some fame also was something he couldn't do without.\n\nHe wasn't sure why he needed the attention. He realized, even though he was now an adult man with a gray, rough-cut beard, the need for attention from his second wife and from his former student lovers, and from his children and even from his first wife, made him a little childish. But he liked to give, if he could, so when Carmen called in sick and said she might have to go to the cl\u00ednica, he didn't hesitate for a second and he took her to the hospital.\n\nThe cl\u00ednica was ten minutes away by car\u2014not too far, but Carmen didn't have a car. He took Carmen up to the third floor and waited with her in the lobby thirty minutes until a doctor could see her. The waiting room was crowded, and once she had gone off with a nurse to have a number of tests to see why she was feeling so low in energy, he felt his bladder full, pressing and demanding he take care of it. He felt his body more and more. He had given up drinking four years ago, when he'd understood it was taking over his life. It had seemed better, at first, even though he struggled with the temptation all the time. The only way he could give up the booze was going cold turkey. And, although stopping drinking brought him back on the right track, he found he had this restless energy, a need to constantly move, and with his bladder full, he couldn't hold himself in the lobby anymore and he went off wandering. He found the bathroom on the third floor and it was locked. A sign indicated it was being renovated and was indefinitely closed. He needed to go badly, and he tried to remember just where the hell there was another bathroom in the building. He entered an elevator and wandered down to the first floor where he had once donated blood. In Mexico, you had to donate blood anytime someone you knew needed surgery. He had given blood six months ago, and he remembered there was a bathroom down there. His memory sluiced in and out, sometimes knowing exactly where a bathroom was but then incapable, anymore, of remembering just what he'd eaten for breakfast the day before. Down a hallway he traipsed. He felt his feet shuffle, without the full control of his mind. He felt the weight of his belly, which he hadn't had until the last two years, a byproduct of giving up drinking. He had gone from skinny to fat, a potbelly that must have come as a side effect of the pills the psychiatrist gave him to try to calm his nerves. The blood-donation clinic was just ahead, the place where people waited, and then behind the waiting area the rooms where they took you to give blood. The donation area was closed off, forbidden for people who weren't giving blood, but that's where he had last gone to the bathroom in the hospital, so he opened a door with a sign that said for nurses and patients only. He walked back into an assault of bright fluorescent lights. The halls were suddenly tight like the innards of an endless spaceship, all lit up, far too incandescent. He wondered where the toilet had been. Where had he gone to pee, before? He saw some nurses and patients at the end of the halls, disappearing as he approached wanting to ask for help to direct him, though he didn't want to be found out to have left Carmen.\n\nBefore he could find the bathroom, he opened a door to where he thought he should go, and he entered a room full of complex machines, whirring, with IV needles plugged into the arms of three people donating. This is where he had found himself the last time he'd donated. The machines clicked in powerful ways, letting all the patients know the plasma separators were in control of their bodies, and as he looked at the three patients who seemed like blobs feeding the machines, he suddenly made out the face of one patient and realized it was the man who had been shooting at his studio building the night before. It was one of the men who had been firing wildly. The man's forehead was wide and his black hair greased back. His jaw was tight and triangular. He had the peach fuzz of a mustache he seemed to choose not to shave. He was scowling, it seemed to Mauricio, at Mauricio and at the world. Mauricio didn't mean to stare but his eyes\u2014which were used to taking in all visual details\u2014looked at the mole on the left side of the young man's face, at the premature wrinkles on his forehead, at the wannabe rich gangster fake Gucci glasses perched on his head. This was one of the people taking over his neighborhood and threatening him. He had seen the kid crouched behind the white Ford station wagon the night before, looking up at him as the kid shot in his direction. It had felt like he was intentionally shooting in his direction. And now he stood face to face with this thug.\n\n\"What are you staring at, old man?\" the gangster said. \"Haven't you ever seen someone give blood?\"\n\n\"Yes, of course,\" Mauricio said. \"I was just looking for the bathroom.\"\n\n\"Well, I'm not the bathroom. Even though you might think I'm the toilet. You don't remember me delivering gas to you, once, four years ago, do you?\"\n\nHe tried to remember all the people who had brought him gas, and he could only remember the round face of the man with thick grime on his fingers who had delivered gas the last year, or two. The gas-men came like men from the engine room of a ship, essential for you to move your boat forward in the ocean, and then they disappeared. Sometimes a new engineer popped up on deck, and then was replaced by someone else. The number of gas-men who had come and gone in his adult life melded. He knew he always smiled at them. He knew he always gave them a tip and thanked them; he knew he called out in frustration, sometimes, when his gas ran out and he couldn't find a gas deliveryman. It was strange that\u2014for someone so important, without whom he wouldn't be able to cook or heat his tea in the morning\u2014he couldn't remember any of the gas-men except the one from the present.\n\n\"I know about your two daughters,\" the gangster said. \"I know one of them used to be an airplane flight attendant. I know the other takes yoga classes in the center of town when she comes to visit you. I know you rarely go outside your compound. I know a lot about you, se\u00f1or, and you don't know a thing about me, do you? You think you own the world in there, inside your castle, looking down at us shooting below. You think your daughters are more beautiful than us. But I've got news for you. I sold a small package of coke to your younger daughter. Oh yes, she's a lot more wild than you think. That one who goes biking all the time. That one who you think is fucking up not working anymore. She's lonelier than you, inside your castle.\"\n\nWas it possible this punk could really know so much about him? \"How do you know these things? Who are you?\"\n\n\"I'm like the ghost of the barrio. I'm everywhere and nowhere. I lurk in the shadows. You've seen me all the time in the neighborhood. You just haven't wanted to admit I'm there. But I'm there. Just like that paper-mache statue of La Catrina you have in your kitchen\u2014that bag of bones you've forgotten about. And if you say anything to the cops, I'm going to come and get you.\"\n\nHe was lying, Mauricio thought. He was saying crazy things about his daughters just to torment him. Nothing he said was real. It was all made up. How could he possibly know his daughter had been a flight attendant? How could he possibly know his other daughter liked to do yoga? How could he possibly know he had a statue of La Catrina that he had looked at so many times he'd almost forgotten it was present in his kitchen? And then he realized the boy was no longer a boy. It was the boy who had been around the corner, the one who worked for the small grocery store a block away, but now all grown up. It was the same boy, only he was older now and shooting bullets.\n\n\"Get out of here, old man, and forget you ever saw anything last night, or I'll come and kill you.\"\n\nIt was a threat, like so many other threats to him now: his heart, his lungs, his mind. His mind hadn't been behaving. His mind had been making up dark nightmares and secrets. His mind must be making up this one, too, he thought. This whole conversation was happening in his mind. It wasn't a real conversation. He would shut out the hallucination. He would go. He would find the bathroom. He would go back up to Carmen. He looked away from the bad dream and it merged back into the machines, and the clicking and whirring sounds of blood being sucked in and out, donations being made to save lives of people hidden in back surgery rooms. He felt the texture of the fabric of the earth fold like a drapery around him. The demons were running and running through his brain.\n\nThat old man, when he came into where I was giving blood so they could fix up my compadre, Angelito, he looked like he had just seen some guy trying to rape his daughter. He looked at me like I was the scum of this earth. There he is, thinking he's some kind of clean-cut, perfect professor, and what's he think he's so fancy for? That house, it's just a sock factory. My papi said my grandfather used to work there, fourteen-hour days, sometimes seven days a week. My grandfather, I never knew him. He died before my time, bless his soul, but my papi told me he was one hard motherfucking man. He busted his balls, working in that factory. And then, one day, this guy\u2014the professor\u2014he come into the town and he bought the whole place up like he thinks he's some kind of king, and he locks himself in there, all hidden away. And it's like it's his harem, or something, but they say he never keeps no women in there long-term except his wife and two daughters. He's just some hidden king working on his paintings. And I asked my papi, once, how can I see those paintings? I like paintings. I wanna see 'em, and my papi told me there's no way to see the paintings there, you got to go to some museums in Mexico City. And I'm like, what's he keepin' them hidden for so far away, you know what I mean?\n\nSo one day I was in Mexico City, and I'm in some bus full of people, holding on to the roof with my hands pressed against the ceiling, trying not to fall onto all the other people, 'cause there was a woman with a baby behind me, and we was all swaying like animals stuck together in that bus, with no air to breathe, but my face it gets pressed against a window where the doors are in the back and I look out the window and I see a photo with a poster of some painting by the professor and it says there's a show by Mauricio Sanchez. So I thinks to myself, let me off the bus. Let me off this bus.\n\nI pop out at the next stop, when some guys jump on and off. There's no official stops for the peseros. I squeeze by the baby cryin' and I think to myself I'm going to go see what the old man's art is all about, 'cause I never forgot the time I went into his house to give him that gas, and I thought, that's what someone should do, you know. That's what someone should do to move up the ladder. That's what all of them rich people do. They look at art. And I thought, I want to be someone, you know. So I go up to that museum on the big Avenida Reforma. And it's not like any other building I've ever seen. There's strange statues in front of the building that look like chopped-up metal from a construction site, and the glass on the museum is black and reflecting, keeping the world out, like it's some kind of rich man's glass tomb, and I walk up to the front door and when I get to the door I tug at it, and it's locked, and I think, man I just missed the bus because I was so stupid I thought I should go see this museum, but I know the place ain't for me. Them museums, they say they're for free, but then the guards look at you once you get in there. That's what happened to me. I realized the door wasn't locked, it was just I was opening the wrong side, which made me feel like, why don't they tell you which side to use, and when I gets in there one of the guards he looks me up and down and he tells me it's gonna cost me forty pesos just to look around that building and I say, I just wanna look at the paintings of that professor, Mauricio Sanchez, 'cause he owns this strange fortress-like building in my neighborhood, and they say well, that's fine but it's gonna cost you forty pesos. And I tell them I don't have forty pesos to give them to go into the museum. I just spent three pesos taking the bus, and if I had forty pesos I woulda taken a small taxi. And the guard he tells me I can come back on Sunday when it's free if I want to. And I tell him I'm not going to be in el DF on Sunday. I was just doing a quick in-and-out from Puebla. So he wouldn't let me in without the money. So the only thing I saw was that poster of the professor's paintings.\n\nI slunk outa there and had to wait a bit more to get the bus and when the bus came I felt like a fool for having gotten off to go to the museum, before, so I just walked all the way to where I had to meet these guys to deliver a package. That wasn't a package of drugs, but I was delivering something important\u2014I don't know what that was\u2014in an envelope someone had given me to deliver.\n\nAnd the whole incident, it got me thinking about the professor more and those museums they build that are kinda like the professor's studio, with big doors on them that shut tight. So I thought it was kinda funny when one day I was doing my rounds, making sure all the clients had their stash, making sure the guys below me was delivering, 'cause that's what I spend most of my time doing now, checking up that others aren't fucking up, that they're getting the product to the customers. Lemme tell you, I'm just as good a fucking businessman as them bankers in those big towers. The only difference is they've got the connections from their papis to get there.\n\nAnd so, one day, I'm like checking up on my crew and I see the professor's daughter, and she's buying a little bag of coke from one of my guys. And I'm like, ain't life funny\u2014me I was trying to get into that museum and here's the professor's daughter coming to _me_ to get what _she_ needs. That's the daughter who's the airline flight attendant. She was telling my man that she didn't quite have all the money with her, that she was going to have it for him next time, and asking him if he could just hold her over. And sometimes we'll do that just to make sure the customer don't stop using, know what I mean? So we'll let 'em get away with not paying for a little bag as long as they pay the next time. But I told my compadre, when I seen that daughter, I said to him and to her, right in front of her face, \"If you want some of this, you're gonna have to pay interest. You know what I mean? We could go somewhere and we could have some nice romantic dinner, but that's the only way you're gonna get some of this for free.\"\n\nShe gave me this nasty look like she didn't need me. Her black hair was cut kinda messy and she looked pale like she hadn't found her makeup, and her mascara was put on too thick, like maybe she'd been crying. And I thought, man, I'd like to do her. I would like a piece of the professor's daughter. 'Cause even though she had this messy look, she was as white and pale as they come, and kinda sexy wearing a tight skirt, the way no one does in my barrio\u2014whiter than a g\u00fcera I ain't never tasted in bed before.\n\nAnd she says to me, \"What kind of woman do you think I am? You think I'm some kind of woman who sleeps around for drugs? You think I need your drugs that bad?\"\n\nAnd so, I says to her, \"Well, I didn't see me coming up to you. Looks like you crawled up here to this place all on your own.\"\n\nAnd she just spit on the ground in front of me like she was some kind of boss-queen. She bent her head down to spit, and while she did it I could see her shirt open up a little and I looked down into her chest where her breasts were, and I ain't never seen no breasts that up-close so white and clean. I could almost see her nipples. So I said, \"Sooner or later you're gonna find out everything cost somethin'. Your daddy's been spoilin' you. He's been making you think mangoes fall from the sky. But to get the green mango and turn it orange, without spoiling, you've gotta pick it up off the ground when it falls.\"\n\nShe still gave me some disgusted look like I was just some parasite telling her she'd been sucking at the teat too much of her daddy. I knew she'd go around and find nobody else was gonna give her that coke for free for long. Maybe once. Maybe twice there'd be someone that would give her a packet for free to get her as their customer. But not for long. And I was right. One day, 'bout a month later, I saw her come up to my compadre and she was asking for me and where I am, and I showed up from behind, a back room where I'd been listening to my compay, and when she saw me she said she was still laid off from being a flight attendant and she didn't have any money and could I give her some for free for a couple of days, and I said, \"I told you that's gonna cost something.\" So she said she'd meet me for dinner at this restaurant I told her where to meet me at. And that night I had the first white pussy I ever had in my life. And I can tell you, she didn't seem like she was minding it so much. She looked like she was a nymphomaniac. Some of them rich girls, they'll pretend they don't have a penny and then they'll fuck your brains out. I know she didn't have the money, but she looked like she was OK with it, rocking on top of me in the hotel.\n\nThe professor stood in his studio space in front of a large canvas mounted on its easel\u2014where he had planned on painting earlier in the day before he had to take Carmen to the hospital\u2014and no new painting would come to him. He had been stuck on this painting and series for over six months now. He would put paint on the canvases, and he wouldn't like what he saw, so he would paint over them. Or he would just stand in front of the canvases and nothing new would come to him. Six months ago, he had finally been able to return to his painting, after giving up directorship of the painting department at the university. The stress of running the program, the numerous faculty he had to direct, the meetings with the administration, the setting of standards, the managing of students, had all worn on him for six years and he had had almost no time to do what he was meant to do\u2014paint. It was like having his purpose removed from him, turning him into a body, a vessel instead of a spirit. Halfway through his time chairing the department, he had felt little edges of his mind bend from the stress. He had attributed it, at first, to giving up the drinking, to the effects of trying to go on the straight path. Maybe he needed the juice to keep himself whole. But that was preposterous, he knew, and so he'd continued forward with the Buddhism and with the meditation, trying to heal himself more, which had helped for a while, but the stress built and built. He wasn't meant to be an administrator, even though he was good at it. The university had wanted him to continue directing the program, until they realized his behavior was becoming more erratic. One day, he walked into the university and students heard a howling in his office and they didn't know what was going on. He had locked himself in his office chanting Buddhist prayers and the chanting had risen in sound until it became a full-blown guttural howl, like a sheep trying to find its lamb.\n\nHe resigned quickly. His older daughter took over his painting classes and he moved permanently to Puebla where he thought he could return to being the great painter everyone expected him to be. He had a show lined up in a year. People were expecting great things from him at his gallery in Monterrey. He would have a show at the main museum up there. But six months had passed and nothing but crap would come to the paintings.\n\nStaring at the canvas, in the descending light of the evening as it bent through the fiberglass patches in the roof and slapped against the canvas in harsh dots of light, the light reflected off the white canvas like emergency lights in bright oranges and reds. He watched the light move across the canvas, lost in the sundial. He saw a flash of Carmen coming out of the hospital. He saw a flash of the insolent smile of the gangster in the blood donation room, hooked up to the plasma separator. He saw blood everywhere, sluicing in and out of the machine, and blood on the ground across the street that he had touched earlier in the day with his fingers. The coagulated blood had stuck to his index finger when he'd dragged it through the mud. The images swirled, flashed. He heard a high-pitched buzzing in his ear and then a _gong, gong, gong,_ and he threw the canvas onto the ground and was surprised by the sound of the wood frame clattering against the concrete floor.\n\nHe walked into his younger daughter's room, which she had been using more and more since she was laid off, and since he had moved to Puebla full-time after finishing up directing the painting department. The words of the gangster in the hospital kept coming to him, that his daughter was snorting coke. Had he failed as a parent, too? Was he not there for his daughters? He had wanted to be there for them, always. Often, the painting had taken precedence. Often, his first wife had to do much of the real, day-to-day raising of the children, while he had worked hard in the studio painting and showing his work.\n\nBut he had felt certain he was there for his daughters when they needed him. He would meditate on the floor with his elder daughter. The two daughters looked nothing alike. The elder daughter was plump and happy in her roundness, radiating light off her smile. The other daughter had black hair and pale skin and had inherited his fervent energy, his need to be absorbed in a project at all times. She had finished school and tried to be a chef, but that didn't work out, so she had become a flight attendant. Her eyes often looked sad to him, especially since she had lost her job.\n\nBut perhaps, he now remembered, the look in her eyes had begun earlier, ten years ago, on her twenty-first birthday. That was when he was still married to his first wife, Cristina. A party had been organized at the studio compound in Puebla for his younger daughter. It was the custom of his family to get together, every year, in Puebla for her birthday. The party had started at three, but he had arrived at nine at night, after sleeping in the apartment with one of his female students, whom he had been seeing for six months. His hair was messy from the encounter in bed. He had completely forgotten about the birthday of his daughter. When he arrived at the back of the studio house, at the long patio decorated with his good taste in Mexican pottery that looked like Greek amphorae, traditional Talavera tiles from Puebla, and the trickling fountain, he expected to simply find his good taste reflected back at him, but there, running along the patio, was a table with the tablecloth bent wildly upward at the corners onto the table, as if the process of beginning to clean a big mess had begun. The table was covered with plates dirty with the remains of discarded food, the knives and forks filthy, wine stains covering the tablecloth, and the wax of burnt candles spreading all the way into the fabric.\n\nEveryone had left the party except his two daughters and his first wife, Cristina.\n\nHis wife looked at him quickly, then continued cleaning. His elder daughter said, \"Pap\u00e1, where have you been? You missed the party.\" She looked at him, genuinely baffled he could have forgotten such an important thing.\n\n\"Maybe he was sleeping with some other woman,\" his younger daughter said. \"Maybe he was sleeping with her back in the house in el DF.\"\n\nHer mother, Cristina, told her to shush. Her mother said, in a high voice, she would have no such wild talk in front of everyone. \"Quiet, Alicia. What kind of horrible thing are you making up?\"\n\n\"But it's right in front of your eyes, Mam\u00e1. It's right in front of all our eyes. He's been fucking these women for years. He's been fucking them right in front of our nose, and thinking we can't even see. And there he stands, after forgetting my birthday, and he pretends he doesn't even know what we're talking about. Look at him, with his shoulders hunched, like I'm making things up. Look at him running away, into his studio, he's too cowardly to even admit what he's done.\"\n\nThat was the end of his first marriage. That was the beginning of the sad look on his younger daughter's face. She had found out about the affairs, he had learned later, from the muchacha who cleaned the house in el DF and from former students who eventually spoke to his elder daughter.\n\nInside his younger daughter's room, he didn't want to pry, but he began to look at the surface of her desk and then, after he had made the determination to look inside, her desk drawers. He pushed away pencils and pastels and oils. She had decided to paint since she couldn't find any work. She was painting flowers, lately, and they were three-dimensionally well rendered and with bright colors, but they looked as cold as plastic. He couldn't lie to his daughter about her work, so he told her nothing about them, and since he said nothing she must have known what he was really thinking. He searched inside her closet, inside her shoes. He had no reason to believe anything would be inside them, but he couldn't help but pick up the leather shoes, sniffing the smell of the dry sweat inside like a crazy animal looking for water in the desert. He looked under the bed mattress, and there he found two small plastic baggies, the size of an inch, no more, one with the residue of white powder, and the image of the gangster came back to him telling him his daughter was lonely and unhappy and that she had been buying coke from him, and he went back into the closet looking for the upper reaches of the shelf where he had an old box from his father with a gun his father had received when he was in the Mexican Army. The professor kept the weapon in good condition. He'd had it oiled and maintained regularly, not only to honor his father but because the neighborhood was changing.\n\nHe pulled the gun down with the box and put the box on his daughter's desk and he wanted to swallow the plastic bags and the gun and make them all disappear and he picked up the gun and waved it in the air and hopped on one leg like a warrior and the snaps in his mind were snapping away and he wanted a drink, but he wasn't going to go for a drink, and he bleated and moaned, but no one was in the house with him. His wife had left, back up to Mexico City, after the shootings the night before. She was scared and he had told her she should go.\n\nHe was alone in his castle. He heard the _gong, gong, gong,_ in his brain and he was being called to action. His head felt hollow, and then suddenly full again, like a pile of mercury liquid had been poured into his brain and was sloshing around. But the _gong, gong, gong_ was a knocking at the door. And how had it become so dark? Only fifteen minutes ago, it seemed, the orange light had been moving across the canvas. He pretended the banging and knocking at the door hadn't happened. It might be the police. Maybe it was the police coming to speak to him. When he'd come home from the hospital, when his wife went back up to Mexico City, he had called the cops and told them they needed to do something about the increasing violence in his neighborhood. He had used all of his connections to speak to an assistant and then to the boss of an assistant, on up the chain of command until he'd spoken to the local chief of police of Puebla. Mauricio was known in the city as \"the painter.\" People who knew nothing about art still knew his name in the city, even though he was reclusive, and he had told the chief something had to be done. Things could not continue like this in his neighborhood. The bullets were climbing up the wall. Next time he would peek over the wall and one of the bullets would hit him. There were young children, young men, dying in the neighborhood and no one cared. The chief had said yes, yes, something would be done, police would be sent, but no one had come. But maybe now the police were at the door. The knocking was louder and more insistent and he went through his painting studio and continued through the building with the metal molds of socks and out again to the large front doors that kept the world out.\n\nWhen he got to the doors he tried to gather his thoughts, to shut the creeping mercury off, and he said in a feeble voice that felt to him like a loud voice, \"Who is there? Are you the police? Have you finally come? What took you so long? Who is there?\"\n\nThe knocking didn't stop. It was a persistent bang against the door, but the person banging wouldn't announce who he was. He could open the peephole, but he didn't want to see who it was, he wanted to turn back and run into the house. He felt his father's gun in his hand. The gun felt heavy and pulled down to the ground, the barrel of the handgun pointed at his feet. He needed to look more like a warrior, he thought, so he struggled and lifted the gun into the air and waved it over his head in silent pantomime, wanting to scream, and then he opened the rusted metal box which covered the primitive hole through the thick wood door. On the other side he saw nothing. Not a thing. It must all have been in his head. He must be having hallucinations again. He had to see if it was all just in his head, so he opened the door and looked outside, and he saw emptiness. Absolutely nothing, at all. He left the door open in his shock and scurried back into his compound, forgetting he had left the door ajar, and he went into the kitchen and sat at the kitchen table, an old wood table, and for the first time in years he noticed the paper-mache statue of La Catrina. She was supposed to be a symbol of life after death, but the whiteness of her paper-mache bones came to him now, and he could feel his own mortality coming, and he sat staring at the folds of her rib cage and at her scrawny body, which felt like the withering interior of his body, and in the doorway to the kitchen he felt a presence and he looked over and saw the silhouette of the boy who had turned into a man whom he had seen earlier in the hospital.\n\n\"I told you not to call the cops,\" the man-boy said. \"I told you that if you did I was going to come here.\"\n\n\"Was that you banging at the door?\"\n\n\"What are you talking about, old man? You even left the door open. I've never seen you do that, but you left it open.\"\n\n\"What are you going to do to me?\"\n\n\"You're the one who made the choice to call the cops.\"\n\n\"But they haven't come.\"\n\n\"It doesn't matter. And it won't matter. They won't come. We own the cops. Don't you see? My boss's boss. He owns the cops. That's how I know you called them. But we can't have you making noise. It gets messy.\"\n\n\"Are you going to shoot me?\"\n\n\"You need to show me one of your paintings, first. I tried to see them, once, in the museum in Mexico City, but they wouldn't let me in. So you're going to give me a private showing.\"\n\n\"I'll show you one of the paintings now.\"\n\nHe took the gangster into the studio. He took him in front of the white canvas that he wasn't able to progress on. He picked the canvas up from the floor and put it on the easel. The light switches were far away, in the corner, and he told the gangster to stand in front of the easel with the canvas while he would turn the lights on. The gangster did as he was told. He stood in front of the white canvas. The room was still dark; the professor hadn't turned on the lights.\n\n\"Do you see the canvas? Do you see the whiteness there? That is my troubled mind. No one will want to buy this painting.\"\n\n\"Then that's not the painting I want to see,\" the gangster said. \"I want to see the painting you have that is worth the most money. I want to see the painting that is your most valuable and beautiful one. I want to see the painting that you have been keeping from me and my compays all of this time, hidden in your fortress.\"\n\nThe professor threw on the switch and the bright mercury lights started out dull at first and then they screamed with whiteness so hard he could barely see. He pulled the gun out of his pocket, which he had been hiding. He walked up to the gangster, with the gun pointed at the gangster's head.\n\n\"Go ahead and shoot,\" the gangster said. \"I know you don't have the guts to shoot me or anyone else. That's why you're a professor. You don't have the raw power and beauty and evil inside necessary to shoot another person. It takes a lot of power to do that, to just shoot a man at his face. That's why the new guys they just shoot the AK-47s, because they're too scared to shoot a handgun into another man's face.\"\n\nThe professor could feel the gonging again, the knocking at his temple.\n\n\"Why did you sell the coke to my daughter?\" he said.\n\n\"Because she was a client. Because she wanted to buy, and we don't ask questions. But she didn't always pay, you know. One time she slept with me and I banged her white pussy and I gave her some coke for free.\"\n\nThe gangster reached into his pocket, and he threw some keys in the direction of the professor, hard, making the professor want to protect himself. The gangster fell on the ground and rolled and pulled out a gun, and the professor shot once, twice, a third time into the chest of the gangster while a bullet came out of the gangster's gun, missing the professor, hitting the tin of the roof, breaking a hole through to the dark night outside. The gangster's body lay, spread wide on the floor, the peach fuzz on his face covered with sweat, the hair gel perfectly combed back like he had just looked into the mirror. How could he have shot this man? How? The gun dropped from the hand of the professor and he lay over the body of the gangster and cried. He held the body in his arms and rocked back and forth weeping. He took blood from the chest of the body and went up to the white canvas staring at him like the blank walls of his mind and he smeared the blood on the canvas and rubbed it into the white surface and scratched the blood deeper and deeper into the white paint until the white and the blood dug into his nails. He wanted no part of the blood, but the blood was now stuck hard into his skin, and even after he cleaned his hands he would never be able to clean the blood.\n\n\u2014\n\nFor a year, in the neighboring town over from Puebla of Cholula, the professor climbed the remains of one of the highest pyramids in the world, the Tlachihualtepetl. The pyramid was built by the native Indians and then the church built a sanctuary to the Virgen de los Remedios on top. He did not believe in Christianity, or no more so than he believed that all the gods were the same, but people went to the church to find remedies and he supposed he did the same. From the top of the pyramid, where there is a balcony that gives a view out over the plain that runs until it reaches the base of the giant volcano of El Popo, and in the other direction to the peak of Orizaba, he looked out at the vastness of nature and at the people who dotted the ground. What difference did it make that he had killed one person out of the multitudes, when that person was a gangster? What difference did it make when so many\u2014over seventy thousand\u2014were dying in the drug war? That is what some of his friends told him. The police said he had acted in self-defense and they refused to judge him. There was an investigation. The case was clear, the gangster's weapon was found in the studio, and the shopkeeper had seen the shootout with the gangster the night before. No one blamed the professor. But he blamed himself.\n\nHe had practiced meditation before with his elder daughter, but never with his younger. It was exactly one year, to the day, since the shooting\u2014June 15. He asked his younger daughter to come with him up to the top of the pyramid. He thought it would help both of them to heal on this one-year anniversary. The two daughters came with him. All three of the family walked with yoga mats, and they lay the mats on the ground at the top of the pyramid, on the balcony, in a place they could look out at the width of the valley, with green spreading into the distance until it slowly curved into a darker hue up the majestic mountains. A year ago, El Popo had exploded, but the volcano was calmer now, the lava contained, though there were recent signs of abnormal activity. The professor had been painting for a year, and he would have a show in Monterrey that his gallery owner said would be seen as brilliant. The professor sat on the ground and felt the rays of the sun falling peacefully on his chubby cheeks. He was plumping out more. He was returning to a state of near contentment. His younger daughter sat next to him and she imitated the movements of the elder daughter, folding her legs one on top of the other. All three of them faced the sun, closed their eyes, and held their palms gently in their laps, imitating the pose of the Buddha. The professor felt the presence of his younger daughter beside him. \"I have been blind,\" he said. \"I have been so blind. I was blind to the man I shot. And I was blind to you. I could never see the suffering my affairs with the students had on you, growing up.\" He reached over, with his eyes closed, and touched his younger daughter gently, and she held his hand for a second, her hand warm, then it fell to the side, cold.\n\nHe opened and closed his eyes, and he could not get out of his sight the color red. It was the color of blood and his personal weight to bear. He asked for forgiveness from the heavens. He asked for his daughters to always be beside him. He asked for their love and understanding, and for the power to pay enough attention to them. He asked for his home to no longer be a fortress, to open his art up to the world. He had understood this from the gangster before he was killed. He asked for his mind to be stable. He asked for all of this, silently, and then he turned up to the sky radiating hard and pure light on him, light of judgment and forgiveness, and though he would have to ask for this forgiveness every day for the rest of his life, and though there was much for him to work on, he felt a bubbling lightness inside, a lightness perhaps too strong, too giddy, perhaps unstable, again, perhaps a delusion, his mind sluicing in and out, and he knew the universe was good, and with medication coursing through him his mouth opened partway and formed a vacant, distant smile, and it was not easy to see the trees in the distance.\n\n# THE AMERICAN JOURNALIST\n\nFor twenty years I have been a bounty hunter. Not a guy tracking down bodies, but a journalist based in Mexico City tracking down stories. I've seen it all, drug kingpins strafing bodies with machine guns, firing bullets out of a Hummer while the narcos they're firing at shoot back, until both cars go up in flames. I've seen the nastiest prostitutes working the streets of La Zona Rosa at night, giving blowjobs for a simple sniff of glue. I've seen mothers claiming the Virgin of Guadalupe came to them in a dream, delivering their child immaculately, without sex with their husbands. I've seen swimsuit contestants dive off the cliffs of Cabo San Lucas in Mr. Universe\u2013type contests. I've seen child acrobats beg for money on the streets of Mexico City, at the streetlights, while their fathers swallow flames and show off how resistant their skin can be to glass as they press their backs into broken beer bottles on the hot pavement. I've seen mothers with babies strapped to their shoulders walk a hundred miles to protest the lack of food back home, to politicians who wouldn't give them the time of day. You name it, I've seen it down in Mexico during my time as the main correspondent for the _Houston Chronicle_. And while I was a Marine once, who huffed and puffed carrying a hundred pounds of rocks on my back, in my youth, running through whatever foreign lands they sent me to, I've seen shit down here in Mexico that would make you weep.\n\nThe editors of the paper wouldn't let me file the story about Ernesto. I think because it didn't fit into any of the neat categories I've just described. The papers are fine with stories of poverty or violence, as long as it all fits into a neat bow. A story that fits what we already think is golden. Ever notice that all the roadside bomb stories in Afghanistan are A-OK? Or the stories about drones that have gone astray, killing a few collateral-damage villagers? Stories that make you feel safe, glad you're secure and cozy in the U.S., rather than in some godforsaken place like rural Afghanistan or caught up in the drug war in Mexico, those stories are all right, as long as everyone plays their role correctly. But Ernesto was a bit of an odd tale.\n\nFirst, there was how he looked. Tall, with a big, hairy beard, a white guy with forehead wrinkles who looked like a nightclub bouncer, but he was an insurance agent who worked to save his company a lot of dough in big cases, and he was also a member of a punk rock band. The punk rock part is what ruined it for my editors. What's an insurance company guy doing in a punk rock band? Ernesto collected fine art. He would hang out with some of the top visual artists in Mexico City. He had some of the best oil paintings in his apartment, and he liked conceptual art, like a thousand gold necklaces grouped together, hanging from the ceiling in the shape of a phallus.\n\nBottom line: Ernesto didn't fit the image of a brown Mexican caught up in the violence of Mexico. Which is why they killed the story I sent about Ernesto when he was murdered two days ago. It didn't help that I was with Ernesto when he was shot. The editors told me I was his friend. They said I was writing a biased story from the gut. They told me to take a few days off and then get back to writing \"real stories.\"\n\n\u2014\n\nI met Ernesto a couple of years ago in the port of Veracruz. Some big tanker ship had banged into the pier at one of the enormous port facilities. The case seemed open and shut: a storm had pushed the boat sideways until it smashed into the pier, causing a few million dollars of damage to the pier and sinking a fifty-million-dollar boat. I was already in Veracruz, covering a story about dead bodies showing up from drug violence in the city, and editors always like it if you can get two stories in one location, and they love it even more if there's a story about a natural disaster. Natural disasters fill readers with awe that after all the science we've developed, after all the robots and moon landings, we're still at the mercy of the heavens. Any immense natural disaster story is good, but one in which a complex, expensive machine gets destroyed is even better. It hits home at our vulnerability. So I went over to the port to get the story, as extra gravy, during my time in Veracruz.\n\nErnesto had on a button-down shirt, to give off a look of professionalism, but he was a professional in action, too, his collar wide open, his sleeves rolled up showing off his hairy arms, a clipboard in his hand, and wearing a hardhat. He had half a dozen divers in the water beneath the pier, checking exactly how the hull of the boat hit the dock. He gathered photos. He collected evidence. He proved there was doubt the ship had actually smashed and drowned because of the storm. It could have been due to a drunk captain. The angle of the impact against the dock wasn't congruent with the angle of the prevailing winds and the damage from the storm. In the end, he saved his insurance company fifty million dollars, and only the dock was paid for under the policy.\n\nIf Ernesto had to hire undercover cops to go through garbage in the dumpster behind a bank to prove his case, he would happily do so. He always wore sunglasses, aviator-shaped and large, with a mirrored cover to make it impossible to see his eyes and what he was thinking or calculating.\n\nI'm not sure he ever planted evidence to suit his cause, but I wouldn't put it past him. I kept in touch with him because he was the kind of guy who could give you the kind of dirt, the kind of background information you sometimes needed to finish a story, to give it the color of the panties found from a lover inside the room of a politician, or whatever else the editors in Houston thought could \"sex up the story.\"\n\nBut over time, what I liked about him most was that he would always invite me to the performances of his band, and I'd go.\n\nI wouldn't say his music was punk, per se. It was rock, but with the kind of hammering repetition and rough voice that came out of punk. The drummer of his band had stripes carved on the side of his hair, giving him the look of an attacking tiger. The bass guitarist was round in shape, cutting hard on the rhythm, a fairly well known visual artist. And Ernesto would stand ramrod straight, in a dark nightclub with his mirrored shades on, in the seedy joints he and his pals chose to perform in, where there were sometimes prostitutes. The crowd was upper-middle-class friends from their high school days. The music was a release, a reminder of my own old high school days, a way out of the pressure of the \"real\" stories of one more child caught in the crossfire in a place like Ciudad Ju\u00e1rez.\n\n\u2014\n\nI was writing a story about collectors in Mexico. Carlos Slim is widely known as the richest man in the world, worth personally seventy-five billion dollars. He owns the main telephone company in Mexico, Telcel, a monopoly he was given. He owns bakeries and malls and a chain of fast-food diners. He owns so many things, you can go to malls down here and every single type of shop in the place belongs to him. He's also a collector of art, and he just built a large museum, called the Soumaya, that's so tacky, in many respects\u2014the typography for the name of the museum is the same as the typography for the name of his fast-food restaurant chain, and he has one of the restaurants in the basement of the museum, beneath multimillion-dollar paintings by Van Dyck and Rubens. Slim let his son-in-law design the building, so it ended up a complete disaster of nepotism, a structure with no natural light inside, which makes it almost impossible to see the otherwise fairly good collection. In one of the galleries, there's a silver model of the architectural design of the museum, which rests in a leather case made by Bulgari\u2014part of the whole self-congratulatory monument.\n\nErnesto told me he knew another collector in Cuernavaca who he was willing to introduce me to. It was a politician, a local governor, who collected life-size dolls of _Star Wars_ characters and Superman, all made by a Japanese company called Hot Toys. I had vaguely heard about these kinds of collectibles. I knew there were people who paid a lot for these toys on eBay. But I had no idea just how rare and valuable these kinds of dolls could be. Ernesto told me the governor was paying fifteen to twenty thousand dollars a month on these toys. Some of the individual dolls, with details as precise as the acne scars on a character like Han Solo, were going for as much as ten grand. It was the kind of story that seemed like it could go over well with the editors back home: goofy collections with pop culture dolls that everyone in the U.S. would know about, with the added twist this was all happening in Mexico. My hope was to try to get in the angle about corruption in Mexican politics, the idea that who the hell knew where all the money for this kind of collection was coming from?\n\nI asked Ernesto if the politician was independently wealthy, if he'd had money before he became a politician.\n\n\"His money's all new,\" Ernesto said.\n\nSo I asked Ernesto to introduce this politician to me the next time he went down to Cuernavaca.\n\nA month later, Ernesto called out of the blue. He and his buddies were going to be doing a concert in Cuernavaca, so he invited me down. He set up an appointment with the politician for us to see the doll collection, in the afternoon, the day of the concert.\n\nSome people like to hide their wealth when they know they've acquired it illegally, but others can't help themselves, and they need an audience. That's how a bunch of corrupt guys ultimately get caught. In Mexico there are guys, sometimes, driving three-hundred-thousand-dollar Ferraris around Mexico City, and when you see one you know the guy stole his money, somehow.\n\nI went to the house of the governor with Ernesto, and the governor's maid opened a large wood door, from the sixteenth century. The house was just up the block from the Robert Brady Museum, the home of an American collector who established himself in Mexico in the 1960s, and who was a major friend of the collector Peggy Guggenheim. Brady did a painting of Guggenheim once, with her sunglasses on, holding her three white Maltese poodles. Brady had Frida Kahlo and Diego Rivera paintings mixed in with sculptures of clay penises he'd found around the world. My point is, even the good collectors have some weird taste, a mix of good taste with the bad. It's the rare collector who doesn't have some secret fetish lurking in the background.\n\nErnesto and I went into the home of the governor, and after we made it past the Spanish Colonial door, we entered a stone courtyard where monks used to roam, when they were living behind the main cathedral, and the courtyard was filled with life-size statues of Princess Leia, robots like R2-D2 and C-3PO, and a perfect replica of Sigourney Weaver panting with her sweaty shirt in _Alien_ as she tried to escape from the deathly spaceship. There were figures from _Blade Runner_ and of the Transformers. Each doll was handcrafted in Japan in limited edition, sent by airplane to Mexico City, and then brought to Cuernavaca.\n\n\"Nice dolls,\" I said to the governor. \"How long have you been collecting them?\"\n\n\"They're not dolls,\" he told me. \"They're life-size action figures.\" He slapped the back of a figure shaped like Darth Vader, minutes before Vader fights Luke Skywalker almost to the death.\n\nThe governor looked a bit like a life-size character himself, his hair combed up and back Elvis Presley\u2013style. His cheeks were scarred from what looked like bad skin as a child. He had on a kelly green Polo shirt, with the large figure of a polo rider on a horse. This kind of shirt, with the Polo insignia blown up to be three inches high, is well known as the kind of clothing a certain type of narco likes to wear.\n\nThe violence in Cuernavaca had been getting much worse, lately. There were military checkpoints throughout the city. Rival narco groups were fighting in the streets. Not all the time, like up in the north near Ju\u00e1rez, but more and more. The son of a famous poet was killed, but in his case by the police, after some police working for some of the local drug gangs saw the son catching a glimpse of them. Rather than leave potential witnesses alone, the police just knocked off the son of the poet and his friends. They were found dead in a car.\n\nIt was strange to see these action figures from dramatic, violent scenes when all around Cuernavaca you didn't need action dolls to feel the tense moments of Hollywood; you could just go out your front door and into the main streets of the city.\n\n\"Come here,\" the governor said. \"I'll show you one of my favorite parts of the house.\" He took us into the basement, down ancient stone steps with Roman arches overhead, where cobwebs were growing in the corners. I expected him to want to show us his wine collection. Some of these politicians love wine, and they like to show off how\u2014supposedly\u2014sophisticated they are.\n\nHe came to a large wood door with metal plating, with spikes coming out the plating and with iron bars in the center of the door. The door looked like the entrance to an old debtors' prison. The governor pulled out a metal key about a half-foot long, and he twisted an ancient lock open. He opened the door and brought us into the dank space. There were old wheel-racks of the kind they used to torture the early Christian saints; there were metal pincers hanging on the walls, which he explained were used to pinch the genitalia of bandits and free thinkers during the days of the Inquisition. There were hoists to lower prisoners onto hot, burning embers until they cried out in confession. There were handcuffs with chains attached to the walls where prisoners were whipped, until they fell with the weight of their body in exhaustion to the ground, to the point of suffocation. Some cheap flickering lights wavered back and forth, giving the whole place the look of a haunted house. There were some life-size action torture dolls of prisoners attached to the walls, suffering as a branding iron was placed onto their naked flesh. The doll figures had ripped burlap and old cotton clothing, their hair splayed wildly back, their mouths open in terror, as they were abused.\n\n\"Pretty realistic collection, eh?\" the governor said.\n\n\"It's cool,\" Ernesto said.\n\n\"Cool\" was definitely not the word I would have used. The place was giving me the creeps. The governor walked in front of us, and left us to inspect the room a little closer, and suddenly I heard the main door close, and the governor locked the door.\n\n\"You get a better feel for the whole terror of the place if you're forced to stay in there, for a while,\" he said, and laughed. \"I'll be back at some point,\" he said, and he left.\n\nI tried the door, and it was definitely locked. I tried to see Ernesto's eyes, beneath his mirrored sunglasses, but all I got was a blank, mirrored expression. Only his pursed lips, between his beard, gave a sense he didn't think this was in any way funny.\n\n\u2014\n\nAfter an hour of being locked up, the governor came down to the dungeon and opened up the big wood door. For an hour, I'd gone through all the possible reasons the governor could have locked us in there. One option was that he was just one of those incorrigible gamers, one of those guys who loved to make the world of his magic and make-believe real. This is, more or less, what Renaissance fairs are all about, where people dress up like knights in shining armor to fight other knights, to get the love of a thin woman dressed like a damsel in distress. I did a story, once, about Mexican game-addicts, and the governor was showing some of those tendencies\u2014life-size dolls, the grandiose way he showed us his basement with instruments of torture. It had all the signs of someone in way too deep with his hobbies.\n\nAnother possibility I'd run through was that the governor was sadistic and any minute he was going to come down, personally, dressed in black leather to whip and torture us. This was more the Quentin Tarantino scenario, like in his movie _Pulp Fiction,_ where the Gimp is kept in the basement for ages until turning into some kind of animal.\n\nA final option was that Ernesto had something to do with this treatment. I asked him point-blank: \"Do you owe him any money? Is he usually like this?\"\n\n\"It's the opposite,\" Ernesto said. \"If anyone owes any money here, he owes me a few thousand. I sold him a painting a month ago, by a pop artist in Mexico City, and he hasn't paid me yet. But the painting is only five thousand dollars. He should be able to come up with that amount, easily.\"\n\n\"What's the painting like?\" I asked.\n\n\"It's a pink, airbrush version of Darth Vader in a gay pride parade.\"\n\n\"Is the governor gay?\"\n\n\"He might be. But I don't think so. I don't even think he knows what a gay pride parade is. I think he just saw the painting and loved the fact it had Darth Vader in it. As you can see, he's a bit obsessed.\"\n\nSo, while Ernesto and I stood in the haunted house of horrors, I couldn't come up with the reason we were being kept in such a \"nice\" place.\n\nThe governor came down the steps, his feet dragging against the flagstones to the dungeon, and I could hear a few other men behind him, some yelling from the top of the steps to others at the bottom to be sure to protect the jefe, to stand in front of him.\n\nThe door opened and two guards came and took ahold of Ernesto and me. The governor came in.\n\n\"So, have you figured out why you're here, yet?\" he asked.\n\nI felt like giving a sarcastic answer, that the reason I was here was to write a story about his goofy, life-size action doll collection. Oh, the places writing fun stories for the editors back home will get you. But I've learned, over the years, to keep my sarcastic side in check and to play dumb.\n\n\"No idea,\" I said.\n\n\"None at all?\" the governor asked.\n\nGiven his mocking tone, I tried to think further through the potential options for the fun treatment we were being given.\n\n\"How about if I refresh your memory,\" the governor said. \"A year ago, you wrote an investigative piece about allegations of new drug trafficking in Cuernavaca. You wrote about the marriage of my niece to an important businessman, who had just opened a large apartment complex with a golf course, and you raised suspicions that the owners of the golf course\u2014El Para\u00edso\u2014were involved in laundering drug money. Does this ring any bells?\"\n\nIt did, of course.\n\n\"That article was completely false, and you will write a retraction. You are going to turn in your resignation to your newspaper, and they are going to write an article saying you have been dismissed, completely, for improper conduct. The article will appear on the front page of your newspaper.\"\n\nI was used to hearing about these kinds of requests, for local journalists, on a daily basis. There were dozens and dozens of journalists killed trying to cover the drug war. There were more than dozens told to shut up what they were writing, or they would be killed. The situation for local journalists was horrible, and getting even more so all the time. It was something that we foreign journalists talked about regularly when we got together, informally, at a bar in Mexico City every Friday. The noose of the violence was closing on anyone who wanted to expose the truth. But, so far, almost all foreign journalists had been left alone. And there was a fine line that we all knew, usually, not to cross. Don't give any specific details of the hideouts of the drug kingpins. Don't go into too many details about the people being pulled off buses and shot at gunpoint. Keep the articles from a mid-level distance, about the general trends in the drug wars, about the general territorial battle. As long as the focus was kept a little further back, you could get the broad story out while avoiding putting your own life at risk. But it seemed, in naming the governor's niece in passing, I had put the focus in too close. And\u2014who knew?\u2014maybe he had some financial interest in the El Para\u00edso golf club.\n\n\"I'm sorry if you were offended, in any way, by the article,\" I said. \"I had no idea your niece was mentioned.\" I looked at the guards holding Uzi submachine guns.\n\n\"The question isn't whether I was offended but that you're spreading lies,\" the governor said. \"Do you stand by your story?\"\n\nI have learned that if you are going to be in this business, you can be polite, you can weasel around a bit to get the bad guys you report on off your back, you can tell a few white lies to be able to keep reporting the truth; but there are some moments when you have to stand up to the bullies. If you don't, they'll push you right over.\n\n\"Yes, I do,\" I said.\n\n\"Yes, you 'do' what?\" the governor said. \"You stand by your lies?\"\n\n\"Yes, I do stand by the story.\"\n\nOne of the guards twisted my arm behind my back.\n\n\"You see this guy\u2014Ernesto,\" the governor said. \"He never should have brought you here. I don't know what he was thinking. Maybe he was thinking he's some kind of special smartass. Maybe he was thinking he knows more than he knows about everything. Here's the deal. You are going to leave this place, and you are going to retract the story, and you're going to quit being a journalist, and if I see you write one more story, then I'm going to send some people to your apartment in La Condesa and I'm going to have you taken out. You have one week to leave the country.\" He had his men take us upstairs, and they pushed us out the door. I'd never seen Ernesto, such a big guy, look scared before, but leaving the governor's house in Cuernavaca, he took off his shades and he didn't look so big or tough anymore.\n\n\u2014\n\nBack in Mexico City, I got to thinking about my twenty years of writing stories in Mexico. What had it all been for? After all the stories about every kind of human interest I could find, had I made any difference to the world? Had I done anything to help the people around me, or was I just documenting the freak show and the river of human misery? I'm fifty-two years old. I look younger than my years, with big muscles, with the build, still, of a Marine, but my head is bald and no one can deny the passage of time. I could call the editors in Houston and tell them it was time for the ship to come in. I could ask for an editing job that would be cushier than writing the day-to-day ins and outs of a country on the brink of descending into chaos. If you live in Mexico City, you can have a fine, perfectly nice life. I have a fairly comfortable apartment with modern furniture. If you stay away from the riskier stories, you can do all right as a foreign correspondent. The big papers like the _New York Times_ rotate their staff in and out of countries, so you don't get too comfortable in any one place, so you don't lose your edge and hunger as a reporter looking for stories. They do it to eliminate too much bias, as you get too close to the \"natives.\" Maybe it was time to pack it all in. If I left, I didn't have to call it caving in to a piss-ant, small-time politician in Cuernavaca. I could call it coming to a \"lifetime decision\" after some \"reflection.\" But that was a bunch of bullshit. If I left, it would prove I was becoming soft and that I hadn't given a damn about real journalism, about really standing up for the truth, since the first day I'd arrived in Mexico City, twenty years before.\n\nI decided to ignore the threat from the governor. It's not that he was all bluff, but a lot of these evil guys, lurking in the shadows, just try to scare you and then\u2014like a dog barking\u2014they move on. It didn't make much sense for them to try to take out a foreign journalist, especially an American one. It could lead to too many investigations by the CIA and the State Department.\n\nSeven days had passed and I went to a new, fancy bar in Mexico City called La Romita. There's an old, art deco staircase that sweeps up a couple of floors to the main area. The bar is grandiose, with black and white checkered tiles, bartenders whipping up exotic drinks of mezcal mixed with guava juice that they flamb\u00e9. The customers are trendy, dressed in the latest foreign jeans that go for $250 a pair in the Meatpacking District of NYC. Women come and go in strapless concoctions that show off their Acapulco tans. Men wear leather motorcycle jackets, or the latest glasses and shoes imported from Italy. This is the crowd of young hipsters that live off the bank accounts of their mothers and fathers, the elite-in-waiting of Mexico who, while they're young, party hard, proving how cool they are by how much money they can spend, while beggars lurk on the streets outside.\n\nErnesto's band has an underground, decadent touch, and they were invited to play in the bar, so I chose to go see what was up in the club. I wanted Ernesto to know I wasn't going to leave Mexico. I wanted to let him know he shouldn't worry\u2014any more than me, pretending I wasn't visibly worried\u2014about what had happened in the dungeon the week before.\n\nThe ceiling of La Romita is two stories high, with a glass roof that opens to the sky. A wide balcony faces the street, hovering above the uneven sidewalks below. A couple hundred clients were packed into the space, some standing on the staircase that floats to the upper floor, dressed in their finest clothes, one woman with a Gucci cream-colored dress.\n\nAs the band got louder and louder, Ernesto standing ramrod straight, the crowd was into the decadence of seeing a punk-sounding rock band in such an elegant space. Ernesto spewed out his somewhat filthy lyrics, and people ordered fine martinis with Bombay Sapphire gin.\n\nThe gunshots came from out of nowhere. Maybe the guy who shot Ernesto was up on the top balcony of the bar, shooting down at him. Maybe he was mixed into the crowd. Three shots came fast, and by the sound\u2014so powerful it cut into the amplified guitars\u2014it must have been a 9mm weapon, something as big as a Glock 9.\n\nThe crowd recoiled. People shouted and fell to the floor. The bartender, next to me, couldn't see what was happening, at first. He seemed to think it was all part of the show, of the craziness of the packed bar. He lit another flamb\u00e9ed cocktail, and then, when he figured out what was happening, he doused the flame, dropping the drink on the floor.\n\n\u2014\n\nI filed the story, that night, to the editors. I told them, at last, about the threat I'd received from the governor a week before. I hadn't wanted to tell them, because I was afraid they would pull me back home. I didn't want to leave where I'd been living the last twenty years.\n\nI called up the editors and told them, \"You've got to publish this story. The significance is that the violence is now coming to Mexico City. It's finally infiltrating the capital. This is going to be the next wave of the violence.\"\n\n\"Then get out of there,\" they said.\n\n\"But there's no reason for me to leave, yet,\" I said. \"It's not that dangerous for me, yet.\"\n\n\"Well, either it's dangerous or it's not. And if it's not, then this story isn't something new. Look, Shawn, you've been down there too long. You've been down there twenty years. Maybe it's time to come back to Houston. We could find you a position at the foreign desk, or you could take time off to write a book, for a while. You could take the time you need to get back to some real stories.\"\n\nIt was two in the morning. Normally, this was the time when I would finish up going over a last-minute major piece with the editors, before they put the story to bed. But for the first time I could remember, when I said the story should run, they said no.\n\nI went outside my apartment, taking my dog to the Parque M\u00e9xico, which runs in a big oval shape, with tall palm trees, where there was once a racetrack, decades ago. I walked around the park. It was so late, none of the usual people walking their dogs were in the park. I took out a cigarette, and I looked up at the palm fronds waving in dark silhouettes against the lit-up sky, over the giant valley of Mexico City. It was time to pack it in. It was time to move on. I was no longer strong enough, anymore. I was no longer a bounty hunter.\n\n# EVERYTHING ELSE IS GOING TO BE FINE\n\nTo live in Mexico City you have to pretend there aren't many dangers. There are the occasional bullets, of course, which most of you \"gringos\" read about in the papers. I put the word gringos in quotation marks because I know better than to make that kind of slur, but the honest truth is that's the way we think about you guys to the north. We have all sorts of guesses as to why we call you gringos. One of the reasons, supposedly, is that you guys wore green uniforms when you came down and took what are now Texas, Arizona, and New Mexico during the Mexican-American War. So we called you \"green-gos.\" But that doesn't make much sense to anyone who speaks English, which I do. Take the origin as you like it, when we call you a gringo we mean it as a slur.\n\nI know what it's like to speak to you about these things because I work for an American company, run by the U.S. gov't., called U.S. Wheat. I've been working at the company for four years now. It's not ideal. It's a place I found some work, with a degree in biology, when the economic crisis started in 2008. The goal of U.S. Wheat is to make sure everyone in the world\u2014from the Philippines to the tip of Tierra del Fuego\u2014eats as much wheat grown in America as they can. And by \"America\" I'm using that all-encompassing word by which you guys call the U.S. (even though there are other Americas, like South America, and even though we, in Mexico, think about Mexico as being in North America, which you guys think is in Central America. Don't even get the Canadians going on whether they think they're Americans).\n\nIn the office, I never tell my boss I'm gay.\n\nHe's two hundred and fifty pounds and from Kansas. He wears a crisp, white cotton shirt, ironed by his _muchacha_ \u2014the maid that most upper-middle-class Mexicans have\u2014and he pits out, leaving big sweat stains under his arms. He moves as an imposing presence through the office, speaking to us only when he has something he wants us to do. \"You'll have to go to Colombia at the end of April for the Wheat Conference,\" he'll say. And, even though I already have ten other trips planned around then for work, I'll have to jiggle them around to fit in the trip to Colombia.\n\nNo one else at the office knows I'm gay, either. There are four of us. It's a tight space. One of the four is our secretary, and if I told even one of them, they'd all know right away.\n\n\u2014\n\nIf you looked at me, the first thing you would think isn't: he's gay; you'd think: he's a runner. I'm skinny. I'm five feet seven. My chin is fairly triangular at the bottom. I guess I've been told my face is a bit boyish. I have soft brown eyes, with somewhat long eyelashes. My hair is straight and dirty blond, combed with a part on the side, and a bit long like a boy's, though I'm thirty-seven. It's hard to know what we look like to others, but I would say I look like a runner. Someone else might notice my thin legs. They might notice I always rush around, never really stopping, always in some kind of motion. Even when I'm sitting, my hands are moving, or my mouth is moving. I don't like long pauses between sentences. I like to speak fast, even in English.\n\nSo I'm always doing something: sending off a link to a YouTube video, like of a thirteen-year-old girl singing about what it feels like to suck at the tit of her mother. That song is called \"La Tetita.\" It's campy, with this Peruvian girl dressed up in traditional mountain clothing and a tight blouse, swaying back and forth like a little girl, singing out in this high voice, \"De d\u00eda y de noche, la tetita\"\u2014 _All night and all day, the titty_. I just think that's hilarious. So I send that around to a few of my friends.\n\nOr I go to el Cabaret. I like to go to the Cabaret. It's a theater in the neighborhood of Coyoac\u00e1n where they perform all sorts of outlandish, raucous theater. Men dress up as women. People sing in drag. They do political spoofs, making fun of the president, or whatever. I go with a small group of my gay friends who, like me, are out to each other at night but not to our coworkers or to our parents or to most people during the day.\n\n\u2014\n\nSix months ago, I was out running on one of the elevated freeways that stretch across Mexico City. When I'm heading into a marathon, I'll do a twenty-mile run. I run six miles every day, and then I'll do a twenty-miler at the end of training. It's almost impossible to run in normal places in the city. There are a few running tracks and a couple big parks, like Chapultepec, but those get pretty confining when you want to do twenty miles. There are oncoming buses and cars everywhere in the city, and no one follows the traffic rules in any case, so I run up on the elevated freeways. It's illegal to be up there. But, unlike in the U.S., no one ever comes to force you out of the way. There's a Wild West attitude in Mexico, so if you do things like you know what you're doing, most people just let you do what you want. So I run along the side of the freeway, elevated sometimes more than ten stories in the air. Up there I feel free, like no one knows who I am. I look out over the wide valley of Mexico City, with hundreds of thousands of concrete houses below, stretching for miles into the distance until they roll up the ring of mountains far away. The sky is dramatic, with big gray clouds that come and go, indifferent to the activities of the human beings below. There are more and more skyscrapers in Mexico City. In the center there are few, because of the prevalence of earthquakes. Most are up on a hillside called Santa Fe, and only one big tower, of Ixe Banco, stands in the center. The building is fifty-five stories tall, but it looks minuscule and alone beneath the clouds, standing ostracized, smaller than the power of the choices nature makes.\n\nI wonder, sometimes, looking at the clouds, why God made me gay. It's a question that comes to me every time I run, looking at the immensity of the force of nature and at the Catholic churches that dot the floor of the jammed urban valley below. Call it what you will, a gift, a curse, I'll tell you what they call it here: _maric\u00f3n_. Faggot.\n\nSix months ago, on the segundo piso\u2014the elevated highway\u2014a van pulled up beside me and drove slowly, neck and neck, as I ran forward as steadily as I could in my mental running zone.\n\nThe van was a Ford Aerostar\u2014thirty years old\u2014with the paint peeling, white tinged yellow from the Mexican sun. One of the guys on the passenger side rolled down his window. He was drunk and he held a bottle of tequila.\n\n\"Where you running to, faggot?\" he yelled out the window.\n\nI ignored him. I've discovered silence is the best way to confront that kind of comment.\n\n\"Eh, _maric\u00f3n_! I'm speaking to you! Where are you running to in your faggoty clothing?\" I had on my usual electric-blue nylon running jacket with fast-looking white stripes running down the back. I was wearing my black running tights, which hug me like a second layer of skin. My shoes had neon yellow soles and silver reflective mesh. It was standard gear for any runner.\n\n\"He must be a faggot,\" the man yelled, \"because he doesn't answer me.\" The driver, on the other side, seemed to look into his side mirror to see if any cops were behind him.\n\nHe sped up, pulled over in front of me in the lane I was running in, and halted. Four men jumped out of the van. They were all drunk. They spoke with slurred words to each other.\n\n\"Grab the motherfucker,\" the guy who'd been shouting out the window said. \"Hijo de puta,\" he said, which means son of a whore. \"Just tell me you're a faggot. Just tell it to me. Just admit it. Look at your fagotty-ass haircut and the way you run.\"\n\nI didn't break my normal run as I approached them. I thought I could just ignore them and pass through. In Mexico, a lot of times, you have to bluff. You have to pretend you don't hear or see things, because if you do the whole world will come crashing in on you like a tidal wave. It was only when I got closer to them, and I could see they weren't going to let me go, that I started to sprint to the left to get away from them, but the drunk guy who'd yelled at me out of the van lunged and caught the back of my right leg. He pulled my foot out, suddenly, into the air like a ballerina in an arabesque. Then he pulled my other leg out from under me, and I fell with my chin onto the pavement. I struggled as much as I could. I'm small, and a bit scrawny, but I'm tough.\n\n\"Just let me go,\" I said to the guys, firmly, with my face against the ground. \"This is going to be a problem for you if you keep holding on to me.\"\n\n\"You've got to be kidding me. A problem for me, if I keep holding on to _you_?\" the guy who'd had the tequila bottle said. The others laughed in unison, with me squirming like a worm on the ground.\n\nThey kicked me in the ribs. They kicked my face. They kicked me until I went unconscious, and flopped me like a U over the guardrail as a warning sign to any other faggot who might come along. When I finally came to, I felt a wave of nausea, the shock of vertigo, and I looked down from the second floor of the highway at the pavement far below, cars rushing into the remote, aloof distance of the city.\n\n\u2014\n\nThe next time I saw my family, I did not tell my father what had happened on the highway. I did not tell my sister. I didn't tell the one of my two brothers who was present. I didn't tell my grandmother who was there. I believe I am close to my sister and to my father and my brother and my grandmother, but I have never told them I am gay.\n\nThe reason for us all getting together was to celebrate the upcoming wedding of my sister, which would take place in a few months. She had told me over the phone she was getting married, and I was really pleased for her, because I like the guy she'd been going out with. When my father heard the news, he immediately organized a brunch at one of our favorite restaurants in the center of the city.\n\nMy sister hadn't arrived yet for the brunch. My father and I waited at the crowded restaurant to get a seat. It's an old Colonial building, like many in the center of Mexico City, which reminds you of the conquest of Mexico by the Spanish, when Ferdinand and Isabella, the Catholic king and queen, took Spain back from the Muslims and sent Columbus forth to the New World to bring \"civilization.\" The staircases in the restaurant, of ancient cedar wood, have yellow Sevillian tiles between each step that commemorate the taking of the Mexican people under the \"protective wing\" of the Church.\n\nMy father didn't ask further about my black eye when I told him it was \"nothing.\" It's not that he wasn't concerned, it's that he knows when he's approaching too close into my private life. Sometimes, I imagine he must have known I was gay. It's simply not possible he couldn't know. But we kept complete silence about the matter. I never volunteered if I was going out with someone. I'd simply keep the conversation light, or about something\u2014anything\u2014other than what my sexuality was. As we waited for my sister to arrive with her new fianc\u00e9, my father said we would have to go looking for a bouquet of hideous plastic flowers, that afternoon, to give my sister. It was a running joke in the family to give a kitsch gift to celebrate marriages. It was our way of showing deep love. And I knew I would never get one of those kitsch gifts.\n\nMy grandmother was the only one who didn't follow this protocol of dancing around who I was. Halfway through the meal, later, she turned to me, just after we'd toasted my sister and her fianc\u00e9, and she said, \"And when are you going to get married, Miguelito? You've been a bachelor for too long. Tell us who you're going out with these days?\"\n\nI shrugged my shoulders and said, \"I guess I just haven't met the right one.\"\n\nShe gave me a look like I needed to hurry up. She took my hands in hers, which were old and soft with age, her wedding ring still on her hand, though she had been a widow for twenty-seven years. \"A man without a wife is nothing,\" she said. \"Without a wife a man is like a lost sock fallen to the ground.\"\n\n\u2014\n\nI have been living with a roommate for three years now, and he doesn't know I'm gay. He's home most of the time, and I'm rarely home, traveling the world\u2014especially the Caribbean and South America\u2014promoting U.S. Wheat. He likes to bring friends back to our apartment and to open one beer after another, speaking with his buddies into the night. He's as hetero as they come. He's the kind of guy who plays poker with his friends late into the morning, and wears his jeans a bit baggy, hanging down so his underwear shows. It lets you know he has a lot of hair down there. It lets you know he's a bit stinky.\n\nI take showers every day. I have almost no hair on my body. I like to make sure the bottles in my part of the medicine cabinet are in order. I like to make my bed. I like to cook, when I can, though I'm usually on the road. I don't bring friends over. I don't think they'd like the space. It feels too dominated by my roommate. But I stick with my roommate because he never asks me anything. He just says, \"Hey,\" when I come in the door. He respects my space. I respect his. Everything is properly compartmentalized, and I save on rent.\n\n\u2014\n\n\"If I could have love without sex, that would be my ideal,\" I tell Ra\u00fal. We are on our second date, three months ago. Ra\u00fal is smart. He's a stock trader for Banco Santander. He's older than me, by a few years, and his hair sits like a gray mop on his head. His leather jacket is a size and a half too large for him, and he slouches in his chair, outside at an Italian caf\u00e9 in Polanco, a rich neighborhood of Mexico City, yet in a no-man's-land part of the neighborhood. This caf\u00e9 is known to almost no one. No one fashionable would ever come here or know the place exists. The sign of the caf\u00e9 burns with metallic, fluorescent light. We're the only ones sitting outside, and the caf\u00e9 is about to close.\n\n\"Why would you want love without sex?\" Ra\u00fal says. He seems to ask the question more to the evening air than to me.\n\n\"Because I've never been that interested in penetration, you know? I just want to be hugged. I just want someone to hold me.\"\n\n\"You mean you don't want to be on bottom?\"\n\n\"I don't want to be on top _or_ on bottom. I don't want sex. I want love.\" Like with most people, Ra\u00fal and I lasted no more than a few weeks. We never had sex. Eventually, he got tired of waiting and he moved on.\n\n\u2014\n\nWhen I was thirteen, it happened in back of the cathedral in Cuernavaca where my grandmother used to take me. My parents didn't care about going to church. They went once in a while, and they've been divorced for fifteen years now. But my grandmother took me, regularly, to church, and I became involved. It was the thirteenth of December, and there was a humid chill in the air, though it was a bright, cloudless day.\n\nThe cathedral of Cuernavaca is one of the biggest in Mexico. There's a large patio connected to the main worshipping area. Beneath the portico, around the patio, there are paintings of old frescoes, worn out, with row upon row of the hierarchy of the Church. The priests are scrunched so close together in the paintings, it's hard to tell one from the other. The priests stand over the nuns on the map of the hierarchy. The paintings let you know there's a code of obedience, who is master and who is not.\n\nPadre Francisco took me into his office, after he'd finished the 8 a.m. mass. I was dressed in the long frock of a choirboy. He had never touched me before, but I had heard he'd touched others, and I had remained silent about the rumors I'd heard. He was a major figure in the church, at the time. He told me to put down the large candle he had asked me to carry into his office. He closed the door behind us, and the room was dark except for the big candle, which I had placed on his mahogany desk, where he'd told me to. His cheeks were white with powder, which he seemed to have put on to cover some of the popped capillaries on his face. His glasses were thick, with bifocals and a sharp metallic frame.\n\nWhen I put the candle down, I felt his body suddenly behind me. I felt the weight of his black vestment, a flowing robe as thick as it looked, the heavy cotton brushing against me. He put his claws on my shoulders and squeezed. \"Come now, Miguelito,\" he said. He ran a hand down the side of my ribs. \"The Heavenly Father calls you into his arms,\" he said. He put his lips, old with the juice of wine, against my ear and whispered, \"In the name of the Father I pronounce you my son.\" He lifted my robe and stuck his finger in my asshole. He stuck his penis in me and pushed back and forth, rapidly, for a few minutes, as I lay over the desk with the candle next to my head nearly burning my hair. I didn't say anything. I knew not to say anything. It happened another dozen times, and then he moved on to another.\n\n\u2014\n\nThree days ago, a friend of mine from the Cabaret, Arturo, asked me if I wanted to go to the Vazquez Hermanos Circus with him. The Vazquez Hermanos Circus is one of the five biggest circuses in Mexico. There are more than four hundred smaller circuses that tour around the country. Arturo regularly plays the role of a clown at the Cabaret, and he used to work for the Vazquez Hermanos Circus. After fifteen years of touring with the circus, he gave it up. He loved making the children laugh. He loved handing out balloons to the kids, or play-fighting with the other clowns and watching the whole audience fall back in the aisles with laughter. But the rigors of picking up and traveling after every performance\u2014the circus took six hours to put up, with its three rings, and two hours to take down\u2014wore him out, and he eventually gave up the life on the road.\n\n\"The other thing that got to me,\" he told me once, \"is that when you're a clown no one knows who you really are. They expect you to always be funny. I got tired of people looking at me and assuming the person I was, inside, was the mask.\"\n\nArturo has been a longtime theater partner with me. We go to shows at the Teatro Auditorio Nacional together. Sometimes I get him free tickets, since he doesn't have much money. When the Vazquez Circus came to town, he had two comp tickets, so he asked me to come with him. Arturo can make those sad clown faces, in pantomime, if you don't do what he wants, and I can't resist that look, so I decided to drop what I was doing\u2014preparing for my next U.S. Wheat trip to Nicaragua\u2014to go with him.\n\nDuring the performance, I watched the tightrope walkers sway back and forth as they passed above the arena. I watched the trapeze artists barely grab onto the hands of the receiving man on the other side. I saw a Bengal tiger put his jaws around the neck of the lion tamer. The audience oohed and aahed, on the edge of their seat, wondering if each performer would survive. I was on the edge of my seat, too, but nothing in the show could come close to the fear I experienced after the regular performance.\n\nOn our way out of the ring, I saw a bunch of schoolchildren, tightly grouped together. I saw some priests accompanying them, and it reminded me of the way Padre Francisco used to shepherd us to events around the city of Cuernavaca. Arturo must have seen my reaction because he said to me, \"What's wrong? You look pale as a ghost.\"\n\nI found myself unable to stop looking at one priest in particular. He looked so much like Padre Francisco, though he was much younger than Padre Francisco had been. His metallic glasses were the same, now thirty years out of date. He must have been from the countryside, or he wouldn't have been wearing that kind of glasses anymore.\n\nI pulled my shoulder away from Arturo, who had patted me on the arm to try to comfort me. \"It's nothing,\" I said.\n\nArturo looked at me intently, as if trying to make sure I was truly OK. He tried to distract me. \"Let's go to the back of the circus. I want to introduce you to my friends.\"\n\nAs the audience rushed out of the main tent, Arturo took me back toward the trucks, waiting behind the three-poled, big, movable arena. Dozens of men were already rushing about, putting the hippopotamuses, tigers, elephants, and camels into their cages, giving them straw and snapping the faces of the animals away from the bars if they tried to poke their noses out too far.\n\n\"Come back here, further,\" Arturo said. \"I want you to meet the man who taught me everything I know about acting. He's the one who convinced me to be a clown.\"\n\nWe approached an old, wood trailer, with its red paint beginning to peel off. On the outside center of the trailer a large poster of a clown laughing uproariously was plastered to the wood. The mouth of the clown was open with melodramatic joy, and no sound came out of his mouth, but the promise of laughter was implied.\n\nArturo opened the trailer. I looked around, seeing no one at first, and Arturo pointed toward the floor at a dwarf still dressed in the costume of a clown, with a bowler hat with polka dots, and clown-red cheeks, his lips drawn with makeup in an upward swoop on both sides, a tear penciled in black and white coming down off his left eye. His Rudolph-the-red-nosed-reindeer nose had already come off. But in all other ways he still looked like he had just come from the stage.\n\nWhen the dwarf saw Arturo he began to pull a long handkerchief out of his short pants. He pulled and pulled, and one handkerchief led to another, all connected in a long line; he pantomimed struggling to get to the end of the handkerchief, and the cloth came further and further, coiling into his hands and down to the floor. He pulled, and he pulled, and it seemed the handkerchief would never end. Arturo pantomimed back, in silent laughter, slapping his knees, throwing his head and chest back uproariously, laughing so the whole audience could see. It was a sign of mutual respect, a sign Arturo had learned well from his master, the dwarf. When it seemed the string of handkerchiefs could come out no longer, the dwarf gave one last harsh tug and pulled out a stuffed rabbit that seemed as big as the pant space around his legs.\n\nJust then, a cold draft of air came from behind me, and I turned to see who was at the door. A man with a baseball cap with the logo of Ed Hardy\u2014with a skull on the cap and a bloody dagger through the skull, in the style of a tattoo, with shiny rhinestones like white diamonds studded all over the cap\u2014had punched open the door. He strode into the trailer, moving slowly with his hands patting the sides of his jeans like he was itching to pull out a gun. He wore a red leather jacket, unzipped most of the way to show a white undershirt below. His hair beneath the cap, to the sides, was gelled back, combed too neatly. The cap sat large on his head and cocked to the side.\n\n\"Hey, you! Midget,\" he said. \"Where's your boss.\"\n\nThe dwarf dropped the rabbit to the floor, gently. He seemed calm. He seemed like this wasn't the first tense moment he had experienced in his life.\n\n\"The manager is out now,\" the dwarf said.\n\n\"No, he's not,\" the young man said. \"Who the fuck is that behind you?\" And, for the first time, I noticed a man counting money in the back of the trailer. The man at the end wore a suit. I realized, now, he had been the ringmaster during the performance.\n\nThe young gangster grabbed the hand of the dwarf and lifted him in the air, so he dangled with his legs kicking. The gangster went to the back of the trailer to the ringmaster. \"You're late on your fucking payments,\" he said.\n\n\"Business has been slow,\" the ringmaster said.\n\n\"Liar. The tents are full. I saw it myself, today. And the jefe doesn't care if your business has been slow. I'm taking the midget as payment, for now. The jefe says he wants a midget. He says he wants someone to dance for him. He says he's bored.\" The gangster pulled out a gun. He dropped the dwarf to the floor. \"Get up,\" he told the dwarf. \"Stand up. Or are you too short? Dance for me. Dance for me, you ugly piece of shit. Dance for me and maybe you'll grow.\"\n\nThe dwarf looked at the gun. He stood up from the floor and began to dance.\n\nThe gangster looked over at Arturo and me. \"And you, faggot!\" he shouted, pointing his gun at me. \"What the hell are you looking at, you fucking pansy? Does this make you sick? Does all of this dancing make you sick? Take off your pants. Take your pants off and dance with the midget.\"\n\nI took my pants off and moved next to the dwarf. The ringmaster made no sounds of protest. Arturo said, \"No,\" but the gangster shot at the floor in front of him and the sound of the gun burst through my eardrums.\n\n\"Get in front of the midget,\" the gangster said. I walked in front of the dwarf. \"Now, you! Midget. Pretend to fuck the faggot. Show him who's on top. Get up on a stool and show us you can make him your whore.\"\n\nThe dwarf got up on a stool and pretended to press his body against me like he was making love. And maybe it was the sight of the priest, earlier, coming out of the circus tent, but I felt, once more, the presence of Padre Francisco, and I bent forward crying.\n\nThe gangster leaned back, roaring. \"Oh, I knew it. I knew you were a maric\u00f3n! This is too much.\" He wiped tears of laughter from his eyes. He took the dwarf and went to the door. \"If any of you says a word about this, if any of you says anything, you're dead.\"\n\nHe pointed his hand in the air like a pistol and fired; then he left with the dwarf.\n\n\u2014\n\nThis morning, I went up to the elevated highway to run. I needed time to think. I needed to make sense of what had happened to me at the circus. Arturo was looking frantically for his friend, the dwarf. The ringmaster had told him to keep quiet about the whole affair. He'd told Arturo he would make the payment soon, and that the dwarf would be returned then. He begged Arturo not to go to the police. \"Just shut up about the whole thing,\" he said. \"Shut up about it and everything will turn out best. If you get the police involved they'll come after me. They'll say I did something wrong. They always screw up everything.\"\n\nThe ringmaster was asking for silence in a country that is always asking for silence. This morning, before I went running, I picked up the newspaper, and I saw a headline about the new president, Enrique Pe\u00f1a Nieto. The president told the press they were writing too many negative things about the country. They were writing about too much crime instead of all the wonderful, positive things taking place in the country. \"Write about the real news,\" the president said. \"Don't write about the junk.\"\n\nArturo told me he wouldn't stay silent about his friend. After the circus, I had accompanied him to the police, at the nearest station. We went in to file an urgent report of a kidnapping. We told the police officer exactly what had happened\u2014minus me standing without my pants and the dwarf pretending to have sex with me.\n\n\"When do you think you will have some information?\" Arturo said. \"When can we expect to hear something? I don't want this case to become just like one of the others. I want you to do something about this man. He's like a father to me.\"\n\n\"You can rest assured we will look into the matter fully,\" the police officer said. He took the printout of the case and put it on a stack of others. There must have been twenty-five cases sitting in the tray above his desk. He seemed in no rush to touch any of them. He asked his secretary for some tea. \"Give me some extra sugar,\" he said, and he squeezed at her bottom.\n\nArturo and I had spent the last two days driving around the city looking for any sign of the man with the baseball cap and the red leather jacket. If we could find him, Arturo said, we would tell some other police officers where he was located. Or we would free his friend the dwarf and not even bother with the police. \"That's the only way to get anything done in this country now,\" Arturo said. \"If you see your own bicycle has been stolen, you have to steal it back.\"\n\nI was petrified, of course, as to what would happen to Arturo's friend. The more hours after a crime begins in Mexico, the less chance it will ever be solved. The case slips into a collective amnesia. The whole country is weighed down with the guilt of disappeared people who were loved but forgotten.\n\nBut as I ran, what concerned me most wasn't, I have to admit, the dwarf. What concerned me most were my own efforts at amnesia. I had never told anyone of Padre Francisco. I had never shared my inner secret. And the wound that was festering within me wasn't only the wound of being abused by him. In my mind, the questions that I had turned over and over, and that I turned over and over again as I ran, were: Was I responsible for my abuse? Had I somehow led Padre Francisco on? Had I enticed him? Or, was I gay because he had somehow made me that way?\n\nAnd yet, as I ran along the upper floor of the highway, it came to me that for far too long I had been conflating the two\u2014Padre Francisco and me. I had been assuming there was some relationship between the two, when there was none. I realized this as I thought about the dwarf being forced to entertain the gangster, and as I thought over the whole incident at the circus. The gangster had forced the dwarf to perform for him. The gangster had put a gun in front of him. What choice did the dwarf have? If he didn't dance, he would have been dead meat. The gangster had ridiculed the dwarf for being a dwarf, but he didn't _become_ a dwarf because of the cruelty of being put on exhibition. He was a dwarf and had always been a dwarf. And he would die a dwarf. The cruelty of the gangster had nothing to do with who the dwarf, fundamentally, was.\n\nLikewise, I had intertwined the two. I had confused my sexuality with the abuse of Padre Francisco. I had assumed I was somehow responsible for what he had done to me, and though I knew it shouldn't be the case, I felt guilt for being gay. I felt that being gay was a sin. I felt that my sexuality\u2014who I was at my core\u2014was wrong. But if the dwarf was no more responsible for himself than being born who he was, then why should I torment myself for who I am?\n\nSlowly, methodically, I worked through these ideas as I ran along the pavement. It might seem I should have come to these conclusions years ago. It might seem these ideas were so simple that surely I would have understood them before. But until the violence of the moment of being forced to dance with the dwarf, of being forced to be his victim, playing out the role, in some way, that I had once played out with Padre Francisco\u2014yet as a full grown adult who could now think about what he had just experienced\u2014I could not come to these conclusions.\n\nI ran along the upper roof of the city, and I could see layers of clouds upon clouds, just as the mind is layered in all of its confusion. I saw the twinning of lampposts as they came out of the sides of the pavement, climbing up with hard metal, splitting forcefully into two lights that lead against and away from each other. I saw a crow attack a smaller bird and dive at it and try to humiliate the other bird, forcing it away from its hidden nest beneath the highway. I looked down at the traffic on the lower floor, rushing back and forth, cars coming in and out, and I realized the cars stop for no one, you have to break your own path forward through the traffic.\n\n\u2014\n\nThis morning, after running, after taking a shower to clean every inch of my body, I put on my crispest white cotton shirt. I took out an old iron and unwrinkled the fabric. I put on a tie, although I never wear a tie to the office\u2014only when I go out on the road to sell and promote U.S. Wheat. I put on a blue blazer, a pair of smooth khaki pants, and I polished my shoes. I didn't have to go into the office, today. Technically, I was off because I was headed out on the road tomorrow.\n\nI put my running shoes in the closet. I had been running away from myself for years.\n\nWhen I arrived at the office, I asked my boss, the secretary, and my other coworker for a meeting at eleven. They looked at me, surprised to see me in the office and so dressed up. My boss looked me up and down. He said, \"What's up with the spiffy clothes? You going to communion today?\" He said the last words in a sarcastic tone. He knew I almost never went to church. \"How about at eleven-fifteen?\" he said. He always had to have the last word.\n\n\"I can't do it then,\" I said. \"It has to be at eleven.\" I was tired of him always pushing us around in the office. It was all the same to him\u201411:00 or 11:15\u2014so I told him it had to be at eleven.\n\nAt eleven we all gathered around the conference room table. The boss had a box of Cracker Jack with him. He didn't care for local, Mexican snacks, and he bought junk food at the U.S. embassy commissary.\n\nWhen they were all gathered, I closed my eyes and I saw an image of Arturo's friend, the dwarf. I saw him encouraging me, pulling his handkerchief longer and longer until I laughed. I saw an image, in my mind, of Padre Francisco behind me, and I took the handkerchief and I tied up Padre Francisco with the dwarf's cloth, binding his hands and feet, wrapping the cloth, of bright yellow and blue handkerchiefs, around Padre Francisco's eyes. I did not open my eyes, at first. I kept them closed as I spoke. \"I have something I want to tell you. For thirty-seven years I have denied who I am. I have been in the closet my whole life.\" I paused. My voice had started out quavering, a bit, but now it was strong. \"I am gay,\" I said. I opened my eyes and looked firmly at my boss. His mouth was paused, in midair, with bits of snack floating in mid-arc. And then he continued chewing and said, \"You need to get me that presentation to go to Colombia. You're late on that. Everything else is going to be fine.\"\n\n# THE PRISON BREAKOUT\n\nThis morning, I took the first steps to help a prisoner in a maximum-security prison in Mexico make a breakout. I brought in a metal file in the base of my briefcase. I have a black, simple nylon bag, which I have worn for years, as I speak to death row inmates. Usually I carry the bag in the U.S., bringing files to support and represent death row cases in the U.S. I have been living in Mexico City for twenty-two years now. I am a U.S. national. I am also a writer. I was down in Mexico City for five years, writing away, trying to have an intellectual life, reading books seriously, and living the cheaper life that is possible in Mexico, writing some fiction but then basically getting more and more involved in tracking down nonfiction stories for magazines and putting together a book of essays about all of the humanity that I found swirling about me in Mexico. That wasn't going to pay the bills alone, so I took a job working for the U.S. government, looking into the background of death row inmates in the U.S. who are Mexican. Before any of the cases is finally exhausted someone has to, by law, check to see if there are any mitigating circumstances that might help in favor of the death row inmate. Someone has to go into the town of the inmate, or wherever they come from, and speak to the family members and to the people who grew up with the inmate, to see what kind of circumstances the prisoner grew up in, and to see if there is any background information which the court should know about that might help to explain how the prisoner became such a deadly individual.\n\nThis is strange work, in that I am constantly surrounded by people who know they raised a killer, a true killer; and I myself, while I have a lot of sympathy for these killers, do not deny that in 97 percent of the cases the man in prison is someone who genuinely did murder someone. But the more you get into this business, tracking down information about the people behind bars, you can't help but notice many of the cases do not stack up. Many times, it's clear someone behind bars simply could not have been the person who committed the crime he's been convicted of.\n\nFor years and years, when I came across those cases, I simply presented my findings to the court. I did my research as best I could. My job has not been to reopen cases, and I don't generally see my job that way. As I said, in most cases the man is guilty. But guilty or not, you can say I am hired to see the humanity of the prisoners as they work their way through the U.S. justice system, until most of them, after they exhaust all possibilities, are finally put to death.\n\nMy work, which is paid for, is for the criminal justice system in the U.S. I am not paid to help or to work with prisoners in Mexico. Down in Mexico, they don't have the death penalty. But they do happily put prisoners behind bars.\n\nThe case of J\u00e9sus Martinez, in Mexico, is a case I would normally never have been caught up in, precisely because there is no one to investigate cases like Martinez's in Mexico. There are no court-appointed researchers in Mexico to find mitigating circumstances about someone put in the slammer for life. Finding an \"impartial judge\" is even harder in Mexico. There is not a jury system in Mexico the way we have in the U.S. There are not the checks and balances that can force a judge to hold back, to momentarily pause and consider which ways he may be wrong in his assumptions, or which ways the police may have taken evidence improperly, or which ways witnesses may be lying. Instead, the judge is like a god down here, who processes case after case, slowly, with witnesses often rounded up by the police and told they'd better write down on paper what the police want or they're going to be beaten up. I don't want to crap on the system of justice in Mexico too much. No system is perfect, anywhere. Some systems don't even exist, in some countries\u2014that is, in some countries, where there is, for example, shariah, the victim is lined up at a stake and stoned to death to enforce God's will. Mexico is nothing like that. And yet, even with a judge, a robe, and the occasional Latin word thrown in, it is certainly not a place where _I_ would want to be charged with a crime.\n\nIn the case of Martinez, the main piece of evidence against him is that he is deaf and he knows sign language. Someone went into a bank and pointed to the teller, and without spoken words, using only his hands, directed the teller to give him all the money. He had an accomplice in the case, a man he used sign language with during the robbery. Everyone clearly saw two men using sign language. The teller behind the counter put money into the bag she was given, at a Bancomer bank, as slowly as possible. She followed the procedures she was given when she'd trained to become a teller. She put an ink bomb in the bag with the 500-peso notes the thief was taking. The ink bomb was supposed to explode, later, to mark the bills, so they could be tracked down by the cops. But as she was following the procedures, perfectly, and after she'd pressed the button beneath her counter to indicate a robbery was in progress, and just as the co-conspirator in the robbery was holding an elderly woman hostage in the back of the bank to let the tellers know they should hand over the money quickly and without fuss, the teller handed over the money in the bag the robber had given her, the bag was just in front of the face of the crook at the counter, and the ink bomb exploded in his face. In pandemonium, the second bank robber, holding his gun to the temple of the elderly woman, shot her dead. Both of the men committing the robbery signed back and forth it was time to flee. Apparently, according to someone watching the whole thing\u2014there were fifty at the crime scene in the bank\u2014the first robber signed to the other in sign language, \"What the hell have you done? How could you have killed that woman?\"\n\nThe second signed back, \"I didn't mean to, but you scared me when you jumped back from the bag.\"\n\n\"Run, run,\" the first crook signed.\n\nThey ran out of the bank, to bicycles they had waiting. They were both dressed in black, wearing ski masks. The crime occurred in the part of Mexico City where the Avenida Isabel la Cat\u00f3lica enters an area of printers' rows, where hundreds of little businesses print T-shirts, plastic mugs, and business cards, etc. The two rode off on small BMX bicycles, riding as fast as they could, the first crook holding the bag of money, a burlap sack that normally holds coffee beans and that was covered with ink stains and filled with worthless bills. They swerved in and out of the hundreds of small buyers who had come from around the city to order their little entrepreneurial projects in the printers' rows. The crooks disappeared behind the printing machines, into the cracks of Mexico City. The few, fat cops\u2014otherwise known as _panzones_ \u2014ran after the crooks, one of the cops trying to shoot at the criminals as they escaped, but failing to shoot either of them. Back in the bank, people fell to the ground in fear. The old woman, a grandmotherly type, in a nicely ironed brown dress with a sash carefully tied around her waist, her gray hair neatly combed back in place, had crumpled to the ground, dead. Her family would never see the lovely abuelita again.\n\nNeedless to say, the case was a mess, as all cases are a mess, in that there was a genuine victim, a genuine tragedy at hand. A week later, Martinez was hauled in to the local police station by a local cop. He was a deaf man. He knew sign language. He had a BMX bicycle, which he liked to do tricks on. Never mind he didn't fit any of the other descriptions of either of the thieves. He was short when the men who'd committed the crime\u2014everyone agreed, when I eventually interviewed them\u2014were tall. Martinez didn't have any money when they found him. There was no evidence of any coffee bag and none of the 500-peso bills were discovered with him. His whole family agreed, independently, that he had been at a birthday party at the time of the crime\u2014a party for him. He worked as a mime in a troupe, making money downtown doing street theater, when he wasn't working in the mechanics' garage where he worked the rest of his time. The man who ran the garage was questioned independently, by me, about the birthday party. Everyone agreed on the color of clothes Martinez had been wearing at the birthday party; where the party had been held; that friends of his, who he mimed with, had been at the party; what food they had eaten at the party; and the exact time of the party, because the performance troupe of a friend\u2014a group of clowns\u2014had been brought in to perform.\n\nAfter all my time looking at death row cases in the U.S., and meeting families down in Mexico of the perpetrators of crimes, it was clear to me the case against Martinez was so flimsy, based solely on the fact he was deaf, that I couldn't believe he could ever have been convicted. But there he was, behind bars, in a maximum-security prison in Mexico City, placed in prison for the rest of his life, with no real chance of parole, for the killing of a lovely grandmother who I was certain he'd never met before in his life.\n\n\u2014\n\nI have been thinking about what made me bring the metal file in to Martinez this morning. What made me act? What made me cross the line? I am normally an observer, not an actor. I sit on the edge of the world, like most writers, watching and taking in every detail, seeing more, perhaps, than any person should see.\n\nGrowing up in New York, I began to see lies at an early age. I remember, once, as a boy, watching my brother steal some candy from the store around the corner. He ran and ran with the few pieces he'd stolen from Mr. Horowitz's local grocery on Fourteenth Street and Eighth Avenue. He did it for the sport, more than anything. We didn't have much money, but we weren't so poor he needed to rob that candy. I watched from across the street as my brother came dashing out of the grocery, Mr. Horowitz running onto the street after him twisting and huffing and puffing through the cars, until he hung his head, hangdog on the sidewalk, with my brother running away through the alleyways of the West Village. My brother had taken that man's bread and butter, the fruits of his hard work, and I asked my brother, when he got home and he ate the candy slowly in his bedroom, why he'd done it.\n\n\"What? Done what? I didn't do nothing.\"\n\n\"But you stole Mr. Horowitz's candy. I saw you do it,\" I said. I was no older than seven. It was my first experience with a bald-faced lie, something I knew was wrong staring me in the face. I could have told on my brother. I could have gone to tell my parents, or a teacher, or even Mr. Horowitz.\n\nI was shocked when my brother said, \"Georgie did it.\" Georgie was a black boy who lived down the street. \"Georgie took 'em. I didn't do nothin'.\"\n\nAnd I didn't do nothing, either. I watched, I observed. I saw him make his racist accusation, pinning the blame on another boy, simply because he knew he was an easy target.\n\nAs I grew up, I saw the same thing, over and over. I saw the lies pile up. I'm fifty-two. I was born in 1961. I remember, as a kid, watching President Lyndon Johnson on our black and white TV telling us North Vietnam was a country of Communist invaders that wanted to destroy my country, and I watched the older brothers of my friends go off to Vietnam to protect my country from Communism, from the evil red carpet that was going to take over the world, but I never saw that red carpet come close to my shores, all I saw was the silent, faraway look of those boys as they came home, looking older in their faces, like men, when they were only twenty-two. And that was the big lie, the lie that let me know my country wasn't going to tell me the truth, and as a kid I watched the protests up at Columbia University, and I took the subway up there when I was just twelve, and I saw students screaming through megaphones, I heard them shouting, \"Hey, Hey, Ho, Ho, President Nixon has got to go!\" By then, they were protesting against Watergate and the bombing of Hanoi, and Nixon and Kissinger's plan to end the war with \"Peace with Honor,\" words which even then I could tell had nothing to do with the photos of bombed-out bodies I saw on the front page of the _New York Times_.\n\nMy parents were hardworking Jews who'd immigrated to the U.S. as kids, just before the end of World War II, ferried through England from Germany. I knew that lie, too, when Hitler said he was going to cleanse Europe and make a Reich that would last a thousand years. I grew up with history books in the house, which told of President Wilson and the gassing of World War I, and of the League of Nations, and of the \"War to End All Wars,\" and of Neville Chamberlain's \"Peace in Our Time.\" And I saw the worn books of Yiddish writers that my father would read and finger over and over, which told of the pogroms my ancestors had fled from in Russia before they'd landed in Germany, before they fled to the U.S. The history was everywhere, the history of lies, and power and abuse, and I read those books in bed, sometimes long after my mother said I had to be asleep, with a flashlight under the sheets.\n\nWhen I was older, I knew of other lies, the lies of another president, Ronald Reagan, as he sent money from the secret sale of weapons to the government of Iran, which he'd labeled the devils of Islamic tyranny, but he sold weapons to them anyway, to funnel money off to his secret war in Nicaragua against the Sandinistas. Those were the images I grew up with in college, photos of people dying in mass graves in Nicaragua and El Salvador, all funded by my government.\n\nI am not telling all these things because it is any surprise to anyone hearing about this, I am telling this to give the context to why I took a metal file in to Martinez this morning. Because when I look back at all these pieces of news, at all these incidents, from the stealing of the candy by my brother to the wars and the lies of my politicians\u2014the leaders of the country whose speeches and pronouncements flooded around me like syrup in the air, inescapable, everywhere\u2014I realize that for years and years, my reaction was always one of quiet outrage, of words of denouncement, of curiosity, of anger, of telling others in caf\u00e9s and at my schools that I was upset, but doing nothing. I did nothing. I watched, and knew, and became more and more knowledgeable, and could see the wrong, and I told myself that what mattered was that at least I _knew,_ at least I was aware, at least I was conscious unlike the others who didn't see, who weren't conscious, who didn't even take the first step to responsibility. Because at least I _knew_. I was _informed,_ unlike those other Joes.\n\nAnd when I came down to Mexico City, at the age of thirty, twenty-two years ago\u2014deciding to leave the world of my country behind in the U.S.\u2014wasn't that a political statement, even though I said it was just because I liked Mexico City better than New York? Wasn't I really saying that I wasn't going to participate in the lies of my country anymore? Didn't that make me somehow cleaner and better than my fellow citizens, because I wasn't going to be tainted with the blood of America?\n\nNo, not me. I was a serious thinker, someone who could see things as they were. I read and I read, and I rented a cheap apartment in La Roma, a neighborhood that is now chic in Mexico City, but that wasn't chic at all when I came to Mexico City in 1991. There was a big earthquake in Mexico City in 1985, and the ground of La Roma is on the part of the city that is much looser, since the city was constructed on an old lake in the center of a valley, and the buildings shook and many fell apart in the quake, so the fear of living in such a neighborhood after so many had died meant that the neighborhood was undesirable. So I found a cheap apartment, and I began to live off the savings I had from working as a young journalist in the U.S. I wrote and I wrote, and I conjured up characters that I thought were pregnant with meaning. None of what I was writing was very good, but I was writing so earnestly, with so much belief that what I was writing mattered. Fiction is nothing like writing journalism, and I wasn't making much progress, but all the time I thought I was doing something that might show what I really felt, that might really change the world. Because isn't that what all those great writers, whom I admired so much, like Dostoyevsky, were trying to do, to change the world? Weren't they trying to reflect the pain and torture of our existence back at us so we might take pause and choose to change our ways and reorganize who we were and how we decided to act?\n\nBut my fiction wasn't any good. My characters had no life in them. They were puppets, extensions of my ideology. And after a while I realized that wasn't why Dostoyevsky and the others wrote, in any case. If they brought us to action, it wasn't because they were trying to get us to change our ways, it was a by-product of them simply creating life, and when we looked at life it made us see the need to make changes. If we saw the world as it truly is then it couldn't help but make us pause and reflect that we had to make personal changes, just as the characters always went in these nice arcs from ignorance to enlightenment, to epiphanies that caused them to change their ways.\n\nAnd yet, as I spent days in my small apartment in La Roma, reading and reading, and writing and failing to write well, what I realized is that I had epiphanies up the wazoo. I was one walking, giant epiphany. I had seen the sickness of the world, right in front of me\u2014how far did you really have to go?\u2014looking out at the poverty in my neighborhood, at the poor men sweeping up the streets so the rich men could walk by and make it dirty, again. Did I really need to know more? Did any of us really need to know more? I began to realize the hard thing was to act. The hard thing was to take action, not to know, because knowing was the relatively easy part. I had _known_ that my brother had stolen the candy, I had _known_ the politicians were lying to us all as they waltzed into other countries and stole from those countries and killed their citizens. I had _known_ I was an earnest young man wanting to be pure when I left my country, looking for adventure. It wasn't the knowing that was hard, it was the doing.\n\nSo I took a job doing more of what I thought needed to be done. I took the job to pay the bills, working for the U.S. government with the death row inmates. I took the job meeting the family members of the death row inmates, seeking out any \"mitigating circumstances\" that might influence their sentences. For, Christ!, wasn't the possibility of saving a life the biggest act of doing you could do? The words \"mitigating circumstances\" were the most Orwellian I could think of. Who doesn't have \"mitigating circumstances\" in life? I found the way most of these men grew up poor, and the small shacks where they grew up, and the places where chickens ran around on the street, and where there often isn't any running water, and hell, yes, there are mitigating circumstances, and I presented my findings to the court as ordered, in neat folders, neatly printed out on paper. And all of this was a form of doing, and yet I felt I was still an observer. I wouldn't have broken the law if the law had come looking for me, because I thought the best way to _do_ is from within the system. You can't change the world unless you work within the world the way it is. I didn't want to be one of those guys throwing himself against the barricades in the French Revolution or in the revolts of 1968. I was a thinking man's doer. That's how I would do things. Let others throw the stones; I would do by working within \"the system\" and making it change.\n\nUntil Martinez. Martinez was different. When I saw Martinez, something clicked in me, and I knew I didn't want to be working within the system anymore. I was, for the first time in my life, about to become, as they say, \"radicalized.\"\n\n\u2014\n\nThe first time I saw Martinez, two months ago, he was on the floor in the corner of his cell, smelling of feces that clung to his body. He was in the back, far right corner of the small area that pertained to him, and I thought he wasn't even present. \"Watch out for the prisoner. Keep going forward,\" the warden of the prison told me.\n\n\"Where is he?\" I asked.\n\n\"There. In front of you. To the right, on the ground.\"\n\nI had been in so many Mexican jails before, nothing should have surprised me. They are nothing like an American jail. The prisoners often have to get their food from their family in the outside world. They sleep so many to a bunk, there is frequently someone sleeping on the concrete floor beneath two others in mattresses above. The prisoners do not always have time out in an exercise yard each day. They often have to bribe the guards who run the prison for basic necessities like toilet paper and chances to ask for their cases to be reviewed.\n\nGiven that Martinez was all alone in his cell, I knew it was almost inevitable his cell had been cleared out, just before I saw him, to give the appearance he had the whole place to himself. There was no real chance the whole place was just for him, since I had never seen such a solitary cell for a prisoner, and this most likely explained his crouching on the floor as if he had just been liberated of the presence of other prisoners. I had been told of the existence of Martinez by a friend who worked, independently, for the release of prisoners, funded on his own dime and based on money he raised himself. He had approached me because he knew about my work with death row inmates in the U.S., and I went to the prison in the south of Mexico City because I thought there might be an article here worth writing, not for the American press, but for a slick glossy magazine called _Gatopardo,_ a bit like _Vanity Fair_ but for Mexicans. The thought I might be exploiting Martinez if we put him in the magazine only vaguely crossed my mind. What I thought of, instead, was that this would be a good chance to get my foot in the door with a magazine that paid well, which reached a sophisticated audience, and that I might be able to do something to further the cause of prison reform with the elites of the country.\n\nWhat I didn't expect with Martinez is that he would act out what he had experienced and what he felt, with such strength of facial expressions with so many movements, like a great mime, every bit as good as the French Marcel Marceau. I had little experience with mimes, and little experience with deaf people, and I expected that I would spend most of my time writing questions on a piece of paper, with Martinez writing his answers to my questions back on the same piece of paper. This definitely happened sometimes, but the first thing I didn't fully expect is that Martinez would be a perfectly good lip reader. Once the warden left me inside the cell and the guards closed the doors, and I was warned he was \"very dangerous,\" I approached a table in the middle of the cell, which took up a good chunk of the space, and I sat down, leaving Martinez in the corner to decide how he wanted to respond to me. I had learned from other prisoners, in other jails, that it was often best not to approach them with too much haste or with too much directness, to let them get adjusted to me and to decide whether or not they wanted to communicate with me. Giving the prisoner the liberty of choosing how to react to me placed them in a position out of their normal routine, and allowed them to enter into a different dynamic with me than with the other prisoners and guards.\n\nMartinez took his time in the corner as I sat at the table, Martinez deciding, it seemed to me, whether he even wanted to approach me. I had a piece of paper and a pencil out, ready to write to him. He looked me up and down, moving his head in a smooth, sweeping gesture. He wore a gabardine trilby hat, which tilted back rakishly on his head, which had been shaved until he was bald. He had on a white undershirt that left his shoulders bare, a pair of worn-out jeans, and a pair of even more worn-out flip-flops. His shirt was clean, which led me to believe the prison had washed it for him just for the interview. He walked back and forth in the prison cell, touching each wall in a repetitive movement, ignoring me, back and forth and back and forth as if a man in a play by Beckett, losing himself in the space, which seemed foreign to him now that the cell was empty of the belongings that were usually there, tapping his head, with the brim of his hat, against each wall before turning to take five paces to the other side.\n\nThis went on for five minutes, and I watched him, and then he abruptly sat down across from me in the other chair provided and he put all of the attention that had previously been focused blankly on each of the walls and stared straight into my eyes. It was a challenge to me, I felt. He sat perfectly motionless, with his back straight, his gaze level into my eyes, still not looking down at the paper. The challenge seemed to be: \"Don't come here just to exploit me, to rob from me. Look at me for who I am.\" And then, after he had my full attention, he wrote as much on the paper. He picked up the pencil and wrote in Spanish, \"They tell me you want my story so you can write an article. They tell me this is for an important magazine and that I should behave. What makes you think I want to speak to you? What makes you think you have the right to just walk in here and take my story to put it in a magazine? I am not some plaything, you know, some object of curiosity.\"\n\nI began to write my answer, but he stopped me from writing and pointed at my lips and showed me I should talk. He used his hands to show me I should speak slowly enough so he could read my lips.\n\n\"OK. You've got me,\" I said. \"I admit there is some element of exploitation in this. But how many other people have come, recently, to try to bring your story to the outside world? Without me, you will stay in this prison for the rest of your life.\"\n\n\"That's a threat,\" he wrote on the paper. \"I don't need threats. I am threatened in here every day. Sometimes, when they don't want to do anything, they threaten to never let us get food again, or that we will have to sleep with that bucket of our shit, over in the corner, forever. There is nothing you can threaten me with. I am prepared to stay in here forever, if I have to.\"\n\n\"So what do you want?\" I said. \"To be a martyr?\"\n\n\"I want you to take me seriously. I want you to promise you will keep coming to this prison after you have written your story. I want you to show me the article you write, before you publish it. If there is something that strikes me as false, as a lie, I will cross that part of your story out and I will ask you not to publish it.\"\n\n\"I can't let you tell me what to do,\" I said. \"It's characteristic of a lot of prisoners that they tell me a bunch of lies, and if I let you edit my article then it might end up with a bunch of your fantasies.\" It was, perhaps, bold of me to tell him this, but this was the truth. Prisoners loved to make up stories. I had almost never met a prisoner on death row who didn't claim he was innocent.\n\n\"You think you're smarter than me. You think you're better than me. How much do you really understand of the other prisoners you have worked with?\"\n\nThis simple question was not really new to me. It was a question I had asked myself, before, but the question had never come to me directly from a prisoner, and the force of the question, the immediacy of it, within only minutes of meeting Martinez, made me stand up and come out of my pose of knowingness. He was right in his question, I could see. He had figured me out in only a matter of minutes. I did feel I knew more than most of the prisoners I investigated. I even felt I knew more than most of the people I interacted with. Arrogance had always been my Achilles' heel. I did not mean to look arrogant, but the way I lifted my chin when I spoke, the way I often paused between my sentences, listening to myself speak, as if I had words that were particularly important to say, made me look arrogant. It was not the image I intended to give off, but it was true about me. My head, which was bald, made me look, in some ways, like a mirror image of Martinez across from me. My head was bald from older age and an effort to keep my skull from looking like that of an old man; it was a way of me trying to remain appearing like someone who was cool. It gave me a suave, debonair look that I liked to use to my advantage at parties, to pick up younger women, sometimes feigning interest in what they told me, even if I wasn't interested, to woo them into a bedroom. I had always thought of myself as fair and righteous and just, even though I often doubted that about myself, and here was Martinez who had called me out on my worst inner feelings and habits in a matter of minutes. It was unnerving to see him across from me, reflecting me back to myself so powerfully and so easily.\n\n\"If you want me to leave,\" I said, \"I will go now. I'm sorry if I have disturbed you today.\"\n\n\"That's better,\" he wrote. \"I can see you are speaking to me sincerely now. No, go ahead, now that you have made the effort to come all the way to this prison, we can talk.\"\n\n\"How did you end up here?\" I asked him, even though my friend in the agency had given me the basic blow-by-blow.\n\nHe stood up from the chair and began to mime what had happened to him. He showed me a party, where he was with his friends, laughing, as he blew out a birthday cake. He drew the candles on the cake in the air in front of me. He showed me a clock with the hands at three o'clock at the party, as he ate a piece of cake. Then he drew a line and shifted his arms back and forth, on a flat plane, to show me he was going to another scene. In that other scene there was a thief asking for money at a window with bars. He pulled out his pockets to show that the person was asking for money. He frowned and pouted, to let me know this wasn't him. He pretended to be a nice old lady. He pretended to be another man holding on to this nice old lady, twisting back and forth to do the face of the man with a gun and the face of the old lady. Then he pointed a gun with his fingers at the temple of the old lady and pulled the trigger and she fell to the ground. He lay on the ground, taking in the full pain of the lady, not moving for a good long while to let me know she was really dead. When he finally got up from the ground he mimed tears coming from his eyes, to let me know the whole story was very, very sad.\n\nI was mesmerized by the story he told with his body, by the way he swayed to and fro, arching his back to make me feel the story he was telling; and by the way he made his face extra expressive.\n\n\"You're an impressive mime,\" I told him when he came back to sit in front of me.\n\n\"Thank you,\" he wrote. \"But I want you to understand I'm not being melodramatic. And I'm not doing this to entertain you or to impress you. I mime because that is how I express my feelings and because you are deaf impaired.\"\n\n\"Deaf impaired?\" I said.\n\n\"Yes, you not only can't sign but you have no idea what it is like to be in my world. So by miming maybe I can hint at some of the things you miss in usual body language.\"\n\n\"Are you saying all deaf people can mime, or that they feel like mimes?\"\n\n\"No, of course not,\" he wrote. \"I am just saying that there are many things you might think you know that you might miss. If you come again, I will teach you some more.\"\n\nHe was, according to my friend, no older than thirty-three, but he knew he had things to teach me, someone nineteen years his senior. It was the first of five meetings over the next two months. I came as often as the warden of the prison would let me. They told me I could visit no more frequently than once every two weeks. The reason they gave was so as not to disturb the prisoners, but I sensed it was because every time I saw Martinez they would have to clear the others out of his cell, and Martinez wrote me as much on his paper the next time I saw him.\n\nOn the last visit, two weeks ago, he stopped writing on the paper and he looked me straight in the eyes, and he mouthed at me, insisting I watch his lips and try to read them as he always watched mine. There was something he clearly did not want to write on the paper, which I kept after each interview but that he must have believed the guards could see. \"I want you to help me escape from here,\" he mouthed to me. \"I do not belong here. You know this now, completely. You know every part of my story.\"\n\nHe had told me, by then, how he had grown up, who the members of his family were, what the hardscrabble ways of his growing up had been. He had poured out every part of his youth, all mimed in front of me as I sat transfixed, watching him act. He had given me all his story, and now what he was asking for was something in return.\n\nI stopped and told him, \"That is something I can't do. If I did that, then I would instantly end up in prison myself, and I wouldn't be able to help any of you on the inside.\" It was a perfectly logical answer. He wasn't the first prisoner to ask me to help him try to find a way to escape. Other prisoners had found enough confidence in me to ask the same question. But he was the first prisoner to make me think twice about my reply.\n\nGoing home, as I drove from the prison through the endless concrete small houses of Mexico City, and then as I paced in my small apartment back in the neighborhood of La Roma, I thought about a flame thrower I had seen the night before in the central plaza of Tlalpan in the south of the city. There, the evening before, I had seen a street performer light on fire a large circle with multiple torch endings and throw the circle around and around, larger than a hula hoop around his body, then on the ground in front of him, moving his body in and out of the flames. The risk of what the performer did, only to make a few pesos, trying to wow the crowd, had entranced me. The performer put his life on the line every night, and like many performers in the city, he probably performed at streetlights as well, doing his show, risking burning his hair, all to make a dime.\n\nDid I have the same guts, not to make a dime, but to help a man who had been condemned to prison for life on completely phony charges? By now, I was more than convinced Martinez had never committed his crime. I had, in between prison visits, interviewed all the other people at his birthday party and the few people I could find who had been at the actual bank crime. It was clear he had never been at the scene of the crime. It was also clear the particular judge who had sentenced him had a particularly bad reputation for sending innocent people to prison.\n\nHow much was I willing to get inside the hoop of flame? I thought about my life in Mexico, how comfortable I felt in the country, how every day that I woke up I felt like there was a new surprise with something to teach me and to fill me with curiosity and awe. I thought about the risk I might lose all of that, the sense of place I had become attached to. I thought about the feeling that in this country, I was always alive. It was not that this country was completely better than mine. I wasn't interested, after a number of years, in comparing one country to the other. But what I felt was that here, people cared about me, and I cared about them. There was time to care. We were not separated by all of the barriers of private spaces and private homes, which carved out so much personal space in the United States that it often felt like we had bubbles keeping us from one another. In Mexico, those bubbles felt smaller, and often nonexistent. This had become my home. I had become a permanent resident of the place. To risk all of that, my right to live in the country, in order to bring some means of escape to Martinez, was risking it all. There was the very real risk that if caught, I could be brought in front of the same kind of despicable judge that had thrown Martinez in prison for life.\n\nBut for the first time in my life I knew I had to let more than my rational mind make decisions for me. I had to step into the ring of fire twirling around and around above my head and around my body. I had to risk it all, to take the action to see if I could make one big wrong right. I packed a metal file in the bottom of my briefcase on the next visit. I opened the stuffing of my briefcase carefully, and made a little sleeve on the bottom, and brought the briefcase to a shoe-repair man to have the bag sewn up properly so it would look right. I could slide the metal file in and out easily. It was a small gesture. It was much less than bringing in a gun, something that seemed far beyond my capacity and that might get Martinez caught, as well as me. It was the most efficient act of rebellion I could think of, and I brought the metal file in to Martinez on my visit in the morning.\n\nFor two months, Martinez took that file which I had brought to him and he scraped and pulled and concentrated the edge of the metal blade against the bars of his cell. On the night of his breakout, he waited until the guards came by on their final round of the evening, before the prisoners were supposed to go to sleep, when the guards made their final roll call. The guards called out the names of each prisoner in each cell. They tapped their batons against the metal bars of each cell, echoing the names of the prisoners, and then it was lights out. A guard came and said Martinez's name, specifically. Martinez made the loudest sound he could, a squeal that was high-pitched, approximating a \"yes.\" He knew when to say his name when he felt the vibration of the other prisoners in his cell calling out their names.\n\nThe night was unusual because it was September 15, the night everyone in the prison had just completed celebrating the independence day of the country, the equivalent of the U.S.'s July 4. On this night, all the prisoners were brought into the main courtyard at the same time to watch a concert. It was one of the few times the prisoners were brought out all together, and one of the few times they were entertained together. The daily life of the interior of the prison was run by the warden, in outward appearance. But a high-level narco, from the north of the country in Monterrey, who ran a large cartel on the outside of the prison, was the real head of the day-to-day life. He had been convicted, but his links to the outside world were still strong. He brought in information and got out his plans for his cartel, and he told the prison warden who he wanted for the D\u00eda de la Independencia music. He chose Los Tigres del Norte, a popular band that toured the country. The inmates had all hooted and hollered listening to Los Tigres. The Tigres knew better than to bring in any women with the band, or a riot might ensue. Hundreds and hundreds of tacos and cans of Coca-Cola were downed. The garbage piled up in bag after bag, loaded into extra trash cans in the corner of the central courtyard.\n\nWhen the last of the guards went by, and Martinez had called out his name, and the other prisoners in his cell went to bed, he pulled at the two cuts he had filed on a single bar, pulled out the bar, and shimmied himself through.\n\nI had discussed the breakout plan with him. Martinez had initially thought he could dress up like a member of Los Tigres and waltz out of the concert with the rest of the band in the heat of the mayhem of the prison-mate fans. He was good at acting, and a perfect mime, but the chance he could fool the other band members and the guards as they went out was far too risky, I had told him, so he had scrapped that idea. Instead we had agreed the large quantity of garbage from the get-together was the key to his escape.\n\nFrom his place of freedom outside of his cell, he ran in the dark, unchecked in his small space for the first time in five years. He ran like a cat freed from his enclosure. His body moved with energy, with the training of a professional mime, keeping to the edges of the shadows and out of any few splotches of light. He ran and felt the ground, listening with his hands for the vibrations of any of the guards as they made their rounds. I knew nothing about the layout of the prison, other than what I had seen on my five previous visits, but Martinez was able to tell me where the garbage was gathered and thrown out every night from the high-walled building. He ran to the garbage containers, which were more than usual, from the party, as he had expected, lifted the lid of one of the large containers, climbed inside the gray plastic container, took one last big breath of unfetid air, smelled the residues of sauces and of soda cans, the sticky sweet residue of Coca-Cola mixing with the warm decomposing smell of corn and chilies, and he buried his whole body beneath the mess, breathing through a plastic bottle that he had cut the bottom off of.\n\nIt was similar to the trick the famous narco El Chapo Guzm\u00e1n had used, making it out of prison in a laundry basket. Martinez had thought of that possibility, too, but after Guzm\u00e1n had used the method, all laundry baskets were inspected entering and leaving prison. But the garbage wasn't suspected. Martinez had figured that out. It was considered too disgusting. He sat in the container for four hours, until four in the morning, when the garbage was always brought outside, each container wheeled out and left for the garbage trucks to come an hour or two later, in the early dawn.\n\nAlong the north edge of the prison, when the garbage canisters were wheeled out and left by a single man whose job it was to do so, I waited thirty-five yards across the street, in the shadows, with a motorcycle, watching with binoculars. When the last of the large plastic canisters was wheeled out, and the man whose duty it was to remove the garbage had gone, I watched intently to see if Martinez had made his break. I saw one of the lids of the canisters push up. I saw Martinez leap out. I flashed a flashlight, twice, at the man covered with garbage. We had agreed his chances would be almost zero if someone didn't come pick him up. The most dangerous moment for a prison breakout was just after the prisoner had found a way outside the prison walls. That was when they stood out, looked unusual in prison clothes, and were so dazed from the adrenaline rush of their escape that other people ran into them, saw something suspicious, and reported them. Or the prisoner went off, running cockeyed.\n\nThis was the moment when Martinez had asked me to come get him in a motorcycle. A car was too big. A car might be spotted by one of the watchtowers.\n\nAnd this was the moment that I had hesitated, for a moment, in the planning. Was I really willing to risk my whole life in Mexico and to fully commit myself to this man? Yet here I was, fully committed, fully taking action for the first time in my life. I wore a black ski mask, like the crooks who had genuinely committed the crime Martinez was accused of. The mask hid the white baldness of my head. I had given Martinez the one signal with the two quick flashes, and he came to me, where the motorcycle was hidden in the long underbrush on the far side of the road. Martinez jumped on the back of the motorbike. I barely looked back at him in the rush of the moment. I focused on balancing him on the back and rushing away as quietly as I could, then racing down the big hill from the prison toward the north of the city. I felt his arms grip my waist. I smelled his body, the residue of the garbage, and the smell of his fright. There was a smell in the night unlike any I had ever smelled before, the smell of a man rushing with all of his fright and elation to freedom. We rode the back roads at first, and then hit the main _eje_ boulevards that break through the city. The red lights blinked, from stop to green, and I cut north through the night like a man bringing a slave in the United States north on the Underground Railroad. And like the people who took a risk bringing those slaves north, I brought him into my home first, briefly, to wash and change his clothes, so he wouldn't stand out as he made his journey to freedom. He stuffed his clothes into the kitchen garbage container in my apartment in La Roma. He scrubbed his body in my shower. He cleansed the last of the residue of the garbage, and of the prison, off his body. He put on some clothes, which I had bought for him. It was too risky to give him my own clothes, or anything that might give me away as helping him if he were caught, and too risky to ask for any clothes from his family. His family would be kept in the dark about his escape until he was settled in a new city, months down the line.\n\nHe washed, and he ate the food I gave him, he wrote down on a piece of paper explaining exactly how the escape had gone, and accentuated the words on the paper by miming some of the scarier moments. Then he jumped on the back of the motorcycle, not waiting longer than an hour, since the prison would find out in the dawn he was gone and they might signal a search to begin. But by then, as the first light of the morning came out strong, as the chickens cockled in the morning, as they do in the streets of Mexico City even in the biggest, most densely inhabited neighborhoods, I had brought him to the bus station and placed him on a bus bound for Oaxaca. The price of his freedom would be exile from his home city, where he had grown up. It was too dangerous, even in a city of twenty million, for him to pretend to melt into the fabric of Mexico City. Someone would eventually see him. Someone would eventually recognize his face. He would have to leave the city, where he had spent almost every day of his life for thirty-three years.\n\nStanding by the bus, after I had bought him his ticket, I gave Martinez a hug. He clasped me like a wiry feline. He rubbed the back of my bald head in appreciation, leaned back, looked into my eyes, and I could see him hold back tears. I couldn't hold back my own. I wept as he took one last look at me and then as he turned with only a small bag, with some peanuts and one extra pair of clothes I had left for him and a few hundred dollars in pesos. He would need every peso to start his new life. I had no idea, exactly, where he would go in Oaxaca. We had agreed it was best for me to know nothing about his final destination. Someday, far in the future, when he was safe, he would write to me, but we had agreed there would be no specific day when.\n\nI felt lighter as I rode home on my motorcycle. It wasn't just the lack of an extra passenger on the back of the bike. It was the lack of the fear I realized I had always carried with me. How much does fear weigh? How much does the fear of righting the wrongs of our daily life weigh? More than I could ever have known. When I got home, I made myself a cup of coffee. I brought it out to the balcony of my small apartment. I looked at the men and women preparing for the rest of their day, sweeping up the ground, cleaning their food stalls, and carrying their heavy loads as they made their journey. I saw the hum of the city, and I felt like I was flying above it all, floating over the metropolis. I would not be able to save each prisoner, each person who had been wronged, but at least I had saved one.\n\n# THE ESCAPE FROM MEXICO\n\nLooking back, the summer when I was twelve\u2014I am in my early forties now\u2014a young, horrible man, the head of a gang, tried to kill me. The day it all began, if you can really pinpoint a single moment when an avalanche begins, was on the soccer field, or what we called in Mexico the f\u00fatbol field, behind my school. Now I am an American, an immigrant of thirty years in the U.S. with my mother, but at that time I was a relatively poor Mexican. My mother worked as an elementary school teacher. My father was more or less a good-for-nothing, even then, someone who tacked from one side to another of quick jobs he made up and small entrepreneurial projects. He believed in making the fast buck, a deal for a small plot of land or to buy a bunch of tires, from a scrap heap, and turn them around to sell to another fool who might think lead had been turned into gold.\n\nThe soccer field was so used, it was nothing more than dust. Grass was never planted on the field in any case; that was the kind of school I went to, a dusty building without grass, and as the players ran down the middle of the field, a cloud of dirt would slowly gather around the players as they duked it out and would follow the players, cycloning around them, until the curtain ebbed and flowed to and fro, moving toward me, the goalie, where a breakaway player would suddenly run out of the cloud and shoot with full force at me, a round, chubby boy, a bit taller than the others, and certainly wider, with thicker arms that might stop the ball from reaching its goal.\n\nThe other boys sometimes called me Fatty, or Gordo, though I wasn't really fat in the way of a boy who couldn't move fast, and certainly not the way Americans, who are fat now, look. You see, at that time, to be even a bit chubby meant to be fat, because there really was no extra food and because everyone had a nickname. I was Gordi the Goalkeeper. The _portero_. My hands were very wide for twelve. I lived for soccer. I stood on the line and lunged with all the energy I could muster, when they kicked the ball. I pretended to be someone much older, one of the players in Europe who played in the World Cup or in the leagues of Spain. My favorite goalie was Toni Schumacher, a monster from Germany who led the German team in 1982, leaping in the air and snatching the ball from the other players, accidentally hitting a couple of them in the head as he waded through his opponents, saving the German team, almost winning them the Cup. You could say Schumacher was rough, and some said he was an animal, but I knew he had simply done what he had to, which is the way the game of life is actually played.\n\nThe day my life began to turn, all the players had left their watches at the edge of the field, next to me, just in the back of the tattered net, with me in the center. That was what every player did, so they wouldn't scrape the faces, arms, or legs of the other players as they jumped in the air, or as they practiced their slide tackles into the other players, like sliding in to first base, trying to gain the approval and praise of all the others. So, all the watches were left by one of the goalies, and everyone trusted me, El Gordi, the most. My nemesis in this story, a thin, tall player for his age, who at fourteen was two years older than me, with a bare chest that was already as big as a sixteen-year-old's and with black hair always gelled back, neatly combed over and over, who could be pretty like a model with his body, if he hadn't been so ugly in his face and if he didn't have a tendency to sneer, was a guy named El Farito\u2014The Lighthouse. He was called El Farito not only because he was taller than the rest but because he swiveled his head further than most could, to hit in a header goal. El Farito was also the head of a gang, whose territory included all the area that surrounded my immediate house, to the east side of the school. The west side of the school belonged to another gang.\n\nIf this sounds strange, that one side of the school could belong to one gang and the other side to another, it is really no different than many other artificial lines in the world. As I tell you this story, I am now a bass player, who regularly plays in the back section of some of the best orchestras, and the line between the first violin section and the second violin section, or between the cellos and violas, is as clear as night and day. The audience sees only \"the orchestra,\" but the players know your whole life is different if you are one of the violinists slogging along the underlying melody in the second violins, or if you get to be the first violinist who plays the bright solo moments.\n\nThe gangs in my neighborhood fought over who controlled each inch of the immediate city. El Farito was one of the leaders of a gang called the Nacos. _Naco_ is generally a pejorative term for someone who is a complete good-for-nothing, an uneducated, trashy person who has no respect for others; but they wore the badge proudly. El Farito's watch was plated with gold. He said he had bought the watch himself, and then another time he said his grandfather had given it to him, but everyone knew he had stolen it from one of the watch stands at an outdoor market that went up on Sunday mornings and down on Sunday afternoons. He walked around with that watch like he was a king. The king of Los Nacos. The rest of his outfit was completely incongruous, a sweatshirt with the skull-and-bones of his gang spray-painted by hand on the back, and some wide bell-bottom pants, even though bell-bottoms had gone out of style in the late '70s and this was 1983.\n\nIf only I had watched the watches as much as I concentrated on the soccer ball. I will never know how the watch of El Farito disappeared. The watches of all the players were left next to me, at the very back of the net. I jumped and lunged, trying to reach the crossbar and the far corners. And in one of those lunges my whole life must have changed, as someone came and stole the watch of El Farito from behind me. Some son-of-a-bitch took it, claiming it for his own, just as El Farito had claimed the watch, initially, for his own, but you can see that in the eyes of a gang leader, what he himself has done to another isn't what he sees as the same being done to himself. I imagine one of the sixth graders\u2014or even younger, a boy I wouldn't suspect as I was playing, looking forward\u2014sneaking his hand into the back of the net, seeing the gold plating of El Farito's watch glisten in the sun that barely peeked through the dusty clouds that day when my life changed, and I imagine him thinking the watch would bring thousands of pieces of candy if it were grabbed and sold, or maybe the child was thinking more practically of how to get more food for his grandparents and family.\n\nHowever it happened, when the players all came back to get their watches at the end of the game, El Farito said, \"Where the fuck is my watch, Gordi? Where did you put it? Show me your pockets.\" He came up to me, standing a full eight inches above me, sweating, his hair momentarily out of place until he would get his big blue plastic comb, and he put his wide hands around my chubby neck and told me that if I didn't find his watch he was going to kill me.\n\nThis wasn't a fake threat, coming from El Farito. It was well known, in those days, that to be a member of the Nacos each gang member had to beat someone up until they nearly died, and occasionally, when the battles between the rival gangs of the neighborhood got particularly bad, someone pulled out a knife and stabbed another person, or if things turned really ugly out came a gun and a couple of the opposing gang members were shot. You see, death was a very real thing in my neighborhood. Not the death of old age and natural dying, but death chosen like savage animals marking their territory, peeing on it, like dogs. The goal of all the marking wasn't just to be the bad boys who were the sexiest, but to make money, to survive, to find a millimeter of space and class and sexiness in a world that didn't give a shit about anyone in my neighborhood. We lived far away from the nice neighborhoods of Polanco and Las Lomas. We lived in Iztapalapa, on the edge of the city, where people from the wealthy neighborhoods were afraid to even say the name of my neighborhood. For the people in the rich neighborhoods, ours was forbidden territory, a place known about only in the newspapers where crime happened and the animals fought amongst themselves, and where a \"civilized\" person, someone much whiter than I was in skin tone, would simply never enter.\n\nFor the life of me, I could not figure out how that watch was stolen. As I played, when I wasn't looking at the dusty cloud of players as they moved down the field, I would occasionally look back to make sure everything was there. But without eyes in the back of my head, I had missed the\u2014probably\u2014young thief that took the watch, and El Farito didn't blame the thief, he blamed me. In the neighborhood where I lived, \"don't shoot the messenger\" had a literal connotation. It was assumed that if you had bad news to tell, you were the bad news, or you were hiding some kind of bad news you had caused. I was in the goal. Therefore, I was the thief. El Farito said, \"I don't care how you get the watch back, but you get my grandfather's watch back by the end of the day, or I'll kill you.\"\n\nAnd, as ridiculous as that might sound, that's how the bullying of me\u2014the following of me by El Farito and his gang\u2014started. Naturally, I couldn't find the watch. I looked everywhere around the goal. I lifted the net and kicked the dirt, and asked everyone else around if they had seen who'd stolen the watch, but everyone that had been watching the game looked down at their feet or said \"no\" in that lying way we all knew we lied to each other. You didn't snitch in my neighborhood. The only thing worse than snitching was failing to defend your mother's honor if someone said something nasty about her.\n\n\u2014\n\nBefore the last class of the day is done, a math class that I need to concentrate in harder or I might fail the course, I run out the back door of the classroom, looking left and right down the hallway, hoping none of the members of the gang have noticed me. Three weeks have passed since the day when the watch disappeared, three weeks during which I have begged for anyone who knows what happened to the watch to tell me, but no one tells me. I don't take the books out of my desk. It's too dangerous for me to pick up my books to study, later. I take an unexpected exit from the building, not through the front door but through the gymnasium in the basement and out through an exit where they bring in food to the gymnasium to sell us snacks. I squeeze through the crates where milk is kept for the students, and crouch behind the crates, waiting like a frog hoping not to be eaten by an eagle swooping down from the sky. And like a little mouse, I finally run from my hiding place, outside, zigging and zagging, hugging the walls of buildings and houses, not stopping in the little stores where I would normally stop to buy a bit of candy, because those days are over for good, and I feel my chest filling with air, my lungs punching in and out, trying to support my legs running, as I stop and start, springing down the back streets convinced that around the next corner I will find El Farito. Faster, faster, I think. Cut left. Stop. Look around the corner, run more. They will see me. The walls of the city are full of people who will tell them. Is that man selling tacos on the street in cahoots with them? Is he informing them? Who isn't informing them? Everyone in the neighborhood has some relationship with the two gangs. They cooperate or they are punished. The smallest punishment is spray-paint on their stores. People pay up, extortion money, or their windows are broken. Or, the bigger guys, the ones above the gangs, the real crime syndicates might come in, tipped off by the gangs. The city is a web of punishment and obeying. I am only twelve, but even at that age I already know you obey or they do with you what they want.\n\nWhen I am not running, when I lie in my bed at night, trying to sleep but failing, I dream of buying a gun. A gun is what I need to make El Farito and his guys realize they shouldn't fuck with me. A gun is not easy to find, a gun is illegal, is almost impossible to purchase, especially for a boy like me who has no more than a few savings stored in a clay pig that was given to me by an uncle from Guadalajara. There are coins in there, stuffed until they have filled up the pig, and a jar, but the coins are of small denomination and there are no bills; the coins are from my parents, coins they have given to me or coins I have taken out of their bedroom at times when they are not looking.\n\nThe coins I usually use to buy some candy, but I have been saving them up for a long time to buy an instrument, an electric guitar that I have seen in a store in the center of town, far away from my neighborhood where they don't sell items of luxury like electric guitars. I listen to heavy metal music in my room when I am alone, closing the door that my parents tell me to leave open, not to hide, and I have posters on the walls of AC\/DC and KISS and Judas Priest, and I play records because this is the era before CDs and we wouldn't have the money to get a CD player until much later, in any case, long after most kids have CD players, and I throw my hands around in the air, standing with my chubby body in front of the mirror of my room, imagining I am a rock star. The music pulses through me as I run\u2014electric screeches and jams and scales\u2014and this is the first taste I get, as I run and run and run, hoping to come closer to home, that music can be the thread that saves us, that literally comes down from the sky into the brain like a gift from God, even though I am not a believer in God, but at that time the music came in a pulse of a gift to keep my legs running to give me faith that there might be some way through the slim streets and alleyways. I stop. There in front of me is El Farito. He tells me, \"Why do you bother to go running so hard? You know we can find you anytime we want. There's no escape.\"\n\n\"I don't have your watch,\" I say. \"I confess I don't have it. I wish I had it. I didn't steal it. Someone else has stolen it. I told you that. It's the truth.\"\n\n\"But everyone knows the watch was mine. And who would dare to do something so stupid and brave and foolish, but you?\"\n\nHe has called me brave. Maybe there's a chance. Maybe he wants me to join his gang. Maybe he will not try to find me in the hallways anymore.\n\n\"And brave is stupid,\" he says. \"Do you know that? Brave is what gets a little punk beaten up.\"\n\nMaybe I can reverse, run back from where I have come from, but the time it will take me to turn will give him the time to come on my heels and tackle me. So I run forward, thinking that with the music in my ears like a crescendo of harsh, scraping, rising notes I can break through him to the other side, behind him, floating in and around and above and over, back to my home, and I run as fast as I can, straight at him, straight into his stomach with my head down, like a torpedo trying to ram into a ship that I hope will disappear, a move so unexpected I hope he will let me pass through. I am only twelve and foolish and full of hopes, and he steps aside, and with my eyes closed as I run it seems he will let me go because I have not hit him yet, but then a foot comes out and he trips me and I fall flat on my face with my chubby chin hitting the ground.\n\nEl Farito towers above me and I can't see him because I am splayed forth on the ground, lying like a dead man on the pavement. \"Tomorrow you will bring me all of your money. Every peso you have. You will come to me, personally, and give me the money. I'm sick of watching you run, Gordi. This time, I'm letting you lie on the street like a baby. I could run over you and stomp on you. But I prefer to watch you squirm. Bring it all, tomorrow, and don't say anything to your mother or your father about this, or I will truly kill you.\"\n\nAll this he says with his voice much deeper than you might expect for a guy so tall. His words come from the depths of his body like from the depths of a sewer, bubbling slowly up and out. He comes from behind me, where I lay on the pavement feeling the wet of the rain of earlier in the day mixing with the dirt of the street, and he kicks me in the groin until the sight in front of my eyes switches from the grains of sand and dirt and the black of the pavement to a momentary yellow, a flash of pain. One kick and he is gone. I hobble home, dragging my legs, bent forward, feeling the pain in my crotch. When I come into the house, my mother is there. She looks at me and her eyes well up in tears, a glossiness I hate to see, which comes only in the worst of times I have seen on the face of my mother before, and she says, \"Gordi. Tell me now. Tell me, finally. You haven't said a thing for three weeks. You have barely eaten for three weeks. What is going on?\"\n\nBut I won't tell her. I don't want her to worry. There is nothing she can do. There is nothing my parents can do. My father is barely present, in any case. He comes home, frequently drunk. My mother is the one who meets me at home with snacks and food made just for me. She teaches kindergarten, and she is home, early, before me. She has a plate of tortillas with cheese and green chili sauce waiting for me, and a freshly blended glass of carrot juice, but I can't begin to imagine having any of this now. I go to my room and shut the door.\n\nMy mother comes to the door and opens it, and I have already thrown myself on the bed with my face down, crying into the bed. There is no escape. None. There is no way out of this trap except to get a gun or to tell El Farito that I will join his gang. Join his gang. Get a gun. Join his gang. Get a gun. I go from one to the other, considering the only two options.\n\nMy mother peers into the doorway and she says, \"Gordi. I am going to come each day to school and pick you up. I will take you home each day. What are you running from? Whatever it is, you can't run forever. Whoever it is, they'll get you, if I don't pick you up. I will come for you every day.\"\n\nI don't respond. They are the sweetest words my mother could say. But nothing can save me, unless I can find a watch that has disappeared. No, I must get a gun.\n\n\u2014\n\nWhen I think of what happened to my mother, later in this story, I wonder what would have happened if I had never made the choice to look for a gun. What if I hadn't been so obsessed with the idea of finding the piece of deadly metal? What if I hadn't thought that a gun could solve all my problems? Certainly, I never would have ended up in the U.S., and certainly I would not have had to see the initial indignity my mother was brought to. But when you are twelve and a life-or-death situation seems to be knocking at your door, you knock back, seeking to protect yourself in any way you can. So I took all my money out of my piggy bank and jar, and, since that wasn't enough, I took money from a can where I knew my mother was storing up money for a rainy day, and I went to try to buy a gun.\n\nMy mother had been picking me up at school for two weeks when I went. She came like clockwork, waiting at the front door a few minutes before the final bell of the day rang. Despite the protestations of my math teacher and his stern warnings, and despite the fact I was failing his class more and more, I would leave early and hurry out to the car, where my mother would meet me where she was waiting with the motor running, and she would receive me like a mother back into the womb, protecting me as if she could be a large umbrella keeping me from the rain. I would tell her, \"Mom, this cannot go on forever. You know that.\"\n\nAnd she would say to me, \"Forever is made up of individual points, one by one, and if you are good today, then things will work out in the end.\"\n\nThose two weeks were, in some ways, the happiest of my life, in that I could see how much my mother truly loved me. Every child wants to believe their mother really loves them, wants to believe their mother loves them even more than their mother loves their father, that they are, deep down inside, the most important thing in the world to their mother, but few get the chance to know so clearly the complete devotion of a mother as I did in those days my mother picked me up from school.\n\nIt was like walking among a graveyard of the dead. El Farito was outside sometimes, or other members of his gang, and once they threw a rotten egg at the windshield after my mother had taken me safe into the car. Another time, one of the tires was punched flat, mysteriously, while my mother waited for me in the car. It would have been too dangerous for us to get out of the car to change the tire, so my mother drove home with only three of the wheels working, the other flapping against the ground, the tire air that was missing letting me know this couldn't continue much longer.\n\nSo I went to el Se\u00f1or L\u00f3pez's house. Se\u00f1or L\u00f3pez had been a police officer in his younger days. It might seem like surely I should have known I should try to buy a gun from someone much more nefarious, and I did know that, but I also knew I didn't really have enough money to buy a gun on the real black market. The cost of a gun was not the price itself, it was the price of the risk the seller was taking in selling a gun to someone. If someone was a professional crook, no one thought twice selling the weapon to them, because they knew that person would never lead the authorities back to them. But who wants to sell a gun to a chubby twelve-year-old who stands out like an innocent baby in a field of violence? The seller knows he will be traced to the boy who buys the gun in no time at all, when the foolish, innocent-looking child fails to use the gun properly. So when I had tried to buy a gun on the black market, at first, speaking to people on the street corners who I knew could sell me a weapon, everyone told me, \"It will never happen, Gordi. You're just a kid. You're just a punk.\"\n\nIn my desperation, I went to el Se\u00f1or L\u00f3pez. He was eighty years old, and I told him El Farito was chasing me. The old man lived above a small bodega where they sold rice and eggs and other staples, and he had nothing to do, now that he was old, except to watch the television with the volume cranked up too loud.\n\n\"Why don't you go to the police?\" he said. \"They can help you.\"\n\n\"Excuse me, Se\u00f1or L\u00f3pez, but you know the police will not believe me if I tell them my life is in danger because someone took El Farito's watch. To them this will seem like nothing. To them, I am just a flea, a tiny story while there are big crimes going on. But for me, I know that if I don't get this gun, El Farito and his gang are going to kill me.\"\n\n\"How much money do you have?\"\n\nI laid out all of the money from the piggy bank, jar, and my mother's can on his grimy kitchen table. He was turning blind in his old age. He fingered the bags of coins, and then he fingered the bills I had taken from my mother. He seemed to retreat into the cloud of sound coming out of the TV, some cop show, a rerun of _Starsky & Hutch_. I thought of taking all my money back and running to find someone else. This old man was useless to me, and even as a twelve-year-old I felt he was wasting my time, but then he moved around his kitchen, feeling his way in the dim fluorescent light that harshly lit up his house. He went to his back bedroom, and I watched him crouch beneath his bed. He rummaged around, pulling out stacks of old, musty girlie magazines that he kept and must have once masturbated to, and from behind the magazines, wrapped in a piece of worn red velvet, pulled out a gun.\n\n\"This isn't a toy,\" he told me as he unwrapped it. He put the gun on the bed and I wanted to grab it. The gun looked like a miracle machine that would save me. \"Do you know how to use this?\"\n\n\"Yes, of course I do,\" I lied.\n\n\"Of course you don't,\" he said. He took out six bullets and put them into the police revolver. It had a thick, black plastic handle, with harsh texture; it felt like a handle that had been held firmly to lord it over criminals and to make sure Mr. L\u00f3pez received his small bribes when he was a police officer, as almost all the cops took bribes to survive.\n\nHe put the bullets in the gun and spun the barrel of the revolver. He took the bullets out again. Then he had me put them in one by one. I tried to spin the revolver as he had, and the metal in the barrel seemed like lead in my hands compared to in his. The barrel barely moved.\n\n\"Don't get cocky with this gun. You're not a cowboy. Use this only in the worst cases, in the very last instance, if you have to. A gun is only as strong and smart as the person carrying it. If you try to be a show-off with this, I guarantee they will get you in two seconds flat. Hide the gun in your pant leg. And only if you absolutely need it, take it out.\"\n\nHe was telling me much more wisdom than I could know. The only thing I saw was what felt to me like my salvation. I held the gun, and the weight of the machine seemed to me to offer strength and power and definite protection. I would have kissed Se\u00f1or L\u00f3pez if it weren't so inappropriate. So I simply told him I would bring him more money every week for the next four weeks, as we'd agreed, since the money I had was insufficient.\n\n\u2014\n\nThere is a long gash that runs down my right arm. The gash is hidden under my tuxedo when I play the bass in orchestra concerts, but the gash is always there. It is something music cannot hide. Playing music, as I hold my bow, I can only represent the scar of my body in the slow, sad sections when the plaintive sound of the bass overwhelms the audience, in those long, sighing underlying interludes which a great composer like Beethoven gives a bass player to remind the audience that sorrow is the underlying note of a life as it seeks higher ground.\n\nThree days before I escaped from Mexico with my mother, I met El Farito and four other members of his gang out on the soccer field where the watch had first disappeared. One of the shortcuts home from school was to run out back to the soccer field and jump over a fence at the end of the field, behind the goal where I had defended my team when the watch had disappeared. Jumping down the fence, hanging in the air and then jumping down a concrete wall, saved a couple of minutes on the way hurrying home. This was the first day my mother had failed to come get me. She failed because she had to be in her school late that day to meet the parents of her students, and if she didn't stay at work she would lose her job.\n\nSo I was left to run home, on my own, that day. I brought the gun with me. I had the gun with me every day since I had bought it. The gun was heavy and must have made my pant leg look unusual to anyone who saw me, but I tried to hide it completely. I was out on the soccer field hurrying, at first, toward the back chain-link fence to climb up it, when the rush of the game of soccer that I loved, so much, overtook me. For a second, rather than hurrying on my way, I put myself in the goal to feel the thrill of the space that I wanted to defend. I put my hands in the air, pretending I was stopping balls. I felt the urge to jump from side to side, leaping toward the corners, as I had once jumped every afternoon and as I hadn't done since the day El Farito's watch had disappeared. It was impossible to jump with the weight of the gun in my pants. So I took out the gun, after looking around to see if anyone was coming, and I saw no one. I put the gun in the back of the net, where the watches had been the day they had taken the watch from El Farito. The net was still torn. It was a pitiful net, with holes as large as grapefruits, and the orange netting loose in its fibers like an old net that has been used too many years on worn-out ships at sea. But to me, in that moment, I felt like Schumacher, who had captained the German soccer team in the World Cup. I jumped high to the upper left and felt my body fall to the ground, pretending I had snatched the ball. I got up, dusted the dirt off my thighs, slapping my legs, crouching in the pose of a great f\u00fatbol player ready to block a penalty kick in the finals of the World Cup. The crowd chanted my name, \"Gordi, Gordi, Gordi!\" and I looked out at the wide plain of dirt and imagined the ball coming high and straight at me. And it was in that moment of pure fantasy that I lost sense of time, until in front of me, instead of my fantasy kicker, El Farito and four of his gang members were out on the field. I will never know where they came from, because it seems I should have seen them before they could get so close. By the time I saw them, they were only twenty yards away. I turned and reached for my gun. I didn't have time to think. I turned toward them and fired the gun, but I had no idea what I was doing and the bullet went nowhere near any of them. The loud clap of the gun surprised me. I shot again, and the next bullet went further into nowhere. I saw a ping of dust in the distance. The bullet must have hit the ground. The young men ran up to me. I shot again and hit the arm of one of the boys and he grabbed his arm, but he was in no mortal danger. And the next thing I knew I was on the ground and one of the boys had a machete, and they held the hand I had held the gun in, the gun now in El Farito's hand. And he said to me, \"Where the fuck did you get this gun, you idiot? You don't even know how to fire this weapon. You are going to be in our gang now. But not before we teach you a lesson...Cut his arm,\" he told one of the gang members. \"Let him know he will always be a fucking idiot. From now on, if you can live, you are going to do whatever I tell you. That's your payment for the watch. You are going to be one of the Nacos and you are going to tell your mother you don't belong to her anymore. You're a man now. Tell your mother you're not a baby anymore. Watching her come to pick you up in her car, it's such a pathetic sight to see.\" He raised his hand and held it in the air. The boy with the machete raised his blade in unison with El Farito. El Farito threw his hand down at the ground, where I was pinned by two of the boys. The one who had been shot was barely bleeding, barely grazed. And I saw the blade, as I looked up at the blue sky, which made it seem like the day was as pure as it could be, but was blue in the way of someone who feels they are going to die and be sucked up into the heavens, and out of that blue sky I will never forget seeing, as if in slow motion the black metal blade, thick and wide like a blunt club yet sharp at the edge coming down into my arm, a harsh swack, a smack and slice that caused blood to leap out of my arm, a gash halfway into my arm until it was stopped at the bone. They left me lying on the field, bleeding to death, a future Naco if I could survive, a dead man if I could not.\n\nI still cry when I see that scar, the thick crude stitching up my arm, the long gash that never leaves me. I passed out soon after they cut me, after they looked down at me, laughing, the last image I can remember before I lost consciousness. The dust of the field floated around them, it crept into my wound as blood pulsed out onto the dirt and as I tried to close my hand, reaching for the revolver I no longer had, a phantom reaction, reaching for the weapon I had thought would protect me.\n\nThe door to the embassy loomed like a gaping maw before me, Marisa\u2014Gordi is my son\u2014in downtown Mexico City. I had traveled first by one of the small peseros, which look like small buses, and then by a real bus for two hours to get to the building along the Avenida Reforma. The gang in the neighborhood had spray-painted the car with Gordi's name and had slashed a line over his name. They had punched out all the tires. They had spray-painted the house. Gordi's body had been delivered to me by a nurse from his school, with a lone police officer. I had shouted at them that they should have brought him to a hospital first. What were they thinking? But they told me that if they had taken the time to take him to the hospital he might have died on the way, and the local hospital did not like to get involved with the gang fights. Sometimes they let patients die, intentionally, to avoid having the fighting infiltrate into the building. It was crazy having a hospital that could deny service to its own people. It was certainly illegal to do so. But who was there to check anyone was following the law? The law was what any hospital administrator decided. The choice of whether a gang member who was hurt lived or died could be made by a resident at the hospital in the middle of the night, on a whim. So the nurse had put a tourniquet on his arm and brought my Gordi to me.\n\nThe nurse was a young woman, no older than twenty-five, and she stayed in the house beside Gordi, checking his wound, putting on fresh strips of bedsheets that she wrapped around his arm, as she removed all but the tourniquet. I knew Gordi didn't believe in God\u2014he had told me so, once, such a strange thing for a young boy to claim to know at such a young age\u2014so I went for my own candles with the image of the Virgen de Guadalupe to burn around his bed inside the house. In the first hours he was home, he didn't know who I was. He barely could identify anyone, the nurse, or my husband.\n\n\"We can't keep him here,\" my husband said. \"He'll die and he'll bring the gang into this house.\"\n\nThe nurse said, \"Se\u00f1or, if he goes outside now, he will definitely die. He will not be able to take being transported to a hospital. Let him rest.\"\n\n\"Who knows what kind of criminal activity he has become involved in?\" my husband said. \"He has brought near death for all of us into the house.\"\n\n\"Oh, shut up, Jos\u00e9 Manuel,\" I told him. \"He's just a boy, for crying out loud. He's a boy. They have hurt my boy!\" I kneeled down by the side of Gordi and stroked his head. How could they do this to my boy? My boy. My boy!\n\n\"Then act like his mother and find a way out of this trap. Do something,\" Jos\u00e9 Manuel said. He was nervous and said he was going to look for a gun to protect us.\n\n\"A gun?\" I said. \"The last thing we need is more guns. Can't you see Gordi's gun got him into this trouble to begin with?\"\n\n\"I'm going to get a gun,\" he said, again. And he left the house with me imploring him not to do so.\n\nThe nurse told me she would look after Gordi, she would protect him, and I went upstairs and put on my best clothes and lipstick and looked for all my money, which was less than there had been in the jar where I normally kept it. I realized Gordi must have taken some of the money to buy the gun. There were no reserves to fall back on. There was nothing except my ability to persuade, I knew, and I made the lipstick shine brighter on my lips and ironed my dress before I went to catch the pesero and bus. It was 4 a.m. and we had been up all night with the nurse and in a vigil with Gordi, and Jos\u00e9 Manuel had just left. I looked into the mirror and asked God to help me and made the sign of the cross. There was only one solution I could think of\u2014to fly away, to go to America, to get the hell out of this neighborhood, to leave El Farito and the gangs and the cancer of Iztapalapa. We should have gone before, I blamed myself. I should have known to get Gordi out of his school, before. My sweet, my love. I went into the room and ran my hand across his face, touching his round cheeks and nose and the soft skin of his forehead and felt his slow breathing in and out, so shallow, I could have no certainty he would still be alive when I came back, but the only chance he would live not only for another few hours, but for days, and weeks and months and years into the future, was to go and flee like any other refugee I had ever seen, or the women of World War II holding their arms up in the air crying over the spilled blood of their husbands, their soldiers, and knowing they had to uproot themselves from their farms and go, go, go.\n\nSo I went in the dark of the early morning, catching a pesero crammed with the people of the neighborhood, the muchachas that had to travel far across the city to get to the private homes where they worked, far away across the metropolis. They said little on the pesero in the morning, too tired to chat in the way they would chat coming home. A thin, tiny wisp of a light burned from the ceiling of the bus, the bodies crowded together, even at that early hour, and I made my way until I stood in front of the security gate of the American embassy, where the guards put me in a line, with others who waited, each like a begging ghost in the dawn as it lifted, against a snaking wall, each with hopes, until after a very long wait, it was finally 9 a.m. and the line began to move toward the large metal detector at the front, and white doors that reflected like a mirror back at all of us the hopes we felt and the barrier of the glass.\n\n\"Do you have an appointment?\" the guard asked.\n\n\"No,\" I had to admit. I was sent to another line and to another waiting room. For four hours I sat in a room with faint blue paint on the walls and fluorescent lights, with no windows to give me any sense of the passage of time and no hope there was any way out of the room. It was a trap, a box with one way in and it felt like no way out. The numbers in red, digital, primitive shapes\u2014yet which looked so new to me then, like another world, outside of the world of analog where I lived\u2014changed slowly. They were a sign of hope that this country, while cold and with barriers, might be something different, completely, from the neighborhood back home where Gordi, I hoped, was healing. This was before the days of cell phones. There was no way to check up on Gordi. I waited with all the papers I could think of that I might need: his birth certificate, my birth certificate and the certificate of Jos\u00e9 Manuel, a deed to the house we owned, which was worth almost nothing, a letter which indicated I was employed as a schoolteacher. I took what I had heard from others was necessary, a list of our single bank account and whatever else I could find that might indicate some money, which would show we were wealthy enough to be trusted to go to America and to come back.\n\nWhen my turn finally came, I could feel my lipstick had worn off. Why didn't I freshen it up before I was called up to the window? From the window I was directed to an office. I followed down the dull lighting of a corridor, with white linoleum tiles that shined in a way I had never seen before, so shiny and yet so cold. I knocked on the door, which was already open.\n\nThe man behind the desk did not look up at me, at first. He looked at a folder, among stacks of other manila folders on his desk.\n\nI sat in front of the man until he finally looked up. He looked bored, and tired, and he had a pair of black horn-rimmed glasses and a blue pinpoint shirt, and he had the look of a man who had never heard of my Gordi and would never care. I placed the envelope with all my documents on his desk.\n\n\"You need to wait until I tell you you can put something on the desk,\" he said.\n\nI pulled the envelope back. How could I change this man's thinking? Should I cry? Should I show him I was a mother? Should I stand as straight and still as a fragile, obedient bird? Should I go straight to the point?\n\nI waited until he was ready. \"You may now place your envelope on the table,\" he said. And I did.\n\nHe spoke in Spanish, but with a strange American accent that made his words twist into unrecognizable forms. I knew very little English, only a few words from when I had studied, years before, to be a teacher. But I thought it would impress him if I could speak some of his language, so I said, \"Mr....Mr., I have come here today because I have a son.\"\n\n\"You should speak in Spanish,\" he told me.\n\nI switched to Spanish, as he commanded. Long ago, I had learned you do whatever a bureaucrat wants, or nothing happens. It was the first rule to facing a bureaucrat in Mexico. And giving him a bribe.\n\n\"Se\u00f1or,\" I said. \"I am requesting a visa for tourism for me and my husband and my precious son, Gordi. He has always wanted to go to Disneyland. He wants to go to Disneyland, and my husband and I, we want to give him an opportunity to see the beautiful place of your country.\"\n\n\"And the reason for why you don't have an appointment?\" He opened the envelope I had placed on the table and looked at the papers, as I replied. He scowled at the papers. I could see he was not going to buy my story about Disneyland.\n\n\"You know,\" he said, \"it is a punishable offense to lie about why you want to go to the United States. I see people like you coming in here, every day, saying they want to go see the Empire State Building or the Statue of Liberty. The whole family just wants to up and go for a few days to see the Statue of Liberty. And, strangely enough, they come with papers trying to show me how much money they have, when they clearly have very little. You, ma'am, have a house in Iztapalapa. Does that sound like a place where many people, suddenly, have the money to go for tourism to Disneyland?\"\n\n\"Please. I beg you,\" I said. \"I am begging you. My son Gordi has been attacked by a gang. He is lying at home bleeding now, and I must be a bad mother to have left him, and the only reason I have done so is because I believe in your government. I believe that your government stands for the land of the free, as it says in your national anthem. I believe that you will give a chance to me and my husband and Gordi, and so I am requesting a tourist visa for us to go to Disneyland and to Miami.\"\n\n\"I would be fired, in a moment, if I gave you the kind of visa you are asking for because your son is trying to get out of his neighborhood,\" the man said. \"But I will tell you what. If you will agree to meet me for a drink, then we can resolve this matter, and you and your husband and your son can go to Disneyland. A drink. One time, and at this place.\" He wrote an address on a piece of paper and gave it to me. \"Will you do that?\" He paused. He looked at me with the officiousness of a man who had no scruples. Yet what did I think? Why did I expect him to be different from any other animal? Why did I really think that just because the floors shined more brightly that this bureaucratic dog from the United States would be any different from the Mexican bureaucratic dogs who reached out for money every time we needed something\u2014a permit to buy our house, a permit to register the car.\n\nI knew he meant more than a drink. I knew the words, _a drink,_ were a euphemism for sex. I knew he simply wanted to make me bend to his will to see if he could. I knew he was a pig of a man, who could ask to take advantage of a mother, dressed in her finest, obviously just trying to find a way to save her son and her husband and to get out of a hellhole of a circumstance. I felt like spitting in his face, but, as any mother would, I told him I would be there. I told him I would have the drink with him. I went to the address he gave me later that evening, staying in town until he was done with his day of work. I called Gordi a couple of times to see how he was doing, and the nurse said he was recovering, but nothing was certain. I thought of the gangs, like a cancer coming closer and closer around the house, circling like wolves. I thought of my husband doing who-knows-what to get another gun. And I slept with that officer in Polanco, in the hotel at the address he gave me. His breath tasted like cheap spearmint gum as he kissed me, later. His fingers moved like cold keys of a typewriter across my nipples and my chest. I had never done anything so abhorrent as to let a stranger touch my body, but I did it to get the tourist visas, to be able to take Gordi to a new land.\n\nMy father refused to come with us when my mother came home with the news of the tourist visas. He reacted like a caged cat, pacing through the house, faced with what my mother had done. I was only, finally, coming out of the immediate shock of the blow El Farito had given me. I had missed my mother's gift, her gift of her body to another man, which she had never wanted anyone to find out, but which I found out years later, when my father wrote to me in the United States saying he had reason to believe my mother had done such a \"horrible thing,\" as he put it. It was his reason, his excuse, for not making the trip with us. I heard him from my bed yelling at my mother, \"You smell of another man's cologne.\" I had no idea what he was referring to, as I lay in bed, but his letter years later let me know what he was stating. How he had guessed what had happened to my mother, I will never fully know. He used this as the main pretext for not coming with us. At the time, when my mother had given me more than any son could ever hope or expect from his mother, her own sense of self-dignity, he turned on her and accused her of being unfaithful. For that I can never forgive him, but I know that what he was also afraid of was leaving the home of his birth, the country which, for better and worse, had raised him. There is a smell to the country that one comes from that is either necessary, a smell that cannot be left because it gives you your whole identity, the culture which cradles you and gives you your sense of purpose and norm and a sense of daily routine; or a smell which, for me, after El Farito struck me, would always be the smell of blood and fear.\n\nMy mother snatched me out of that fatherland, like the stork which had originally given me life and which now wanted to ensure my continued survival. With no more than a small suitcase, because my mother insisted we look like tourists and not immigrants to a new land, she made the further difficult choice of not only leaving her home country but also of leaving my father, whom she had been married to, at that time, for twenty years. She had met my father at a wedding party of a cousin, with soft mariachi music accompanying them on their first dance, but she gave up even her husband\u2014who certainly had many faults, but for all of his faults had still been her love, once\u2014so that she could bring me to safety.\n\nWe landed in the middle of the night, at the darkest hour just before the dawn in Miami, me with a teddy bear and my small suitcase, and my mother with an old, beat-up red nylon suitcase that was so old the back zipper was coming undone and the red had turned to a bruised reddish-black. And with no more than that in our hands we arrived in America, two tourists supposedly going to Disneyland, two members of a family now broken, my arm wrapped in gauze which my mother kept as fresh and as clean as possible.\n\n\"What's wrong with him?\" the guard at the border said in the airport in Miami, pointing to my arm, looking at our passports, yet barely showing any real interest in us. Others were waiting. We had the proper visas. My mother merely said, \"His name is Gordi. He is twelve years old, and he has always wanted to come play in this country.\"\n\nThat was enough to have the gentleman stamp our passports and, though he was clearly unconvinced, he was on to the next persons to stamp their passports.\n\nSoon after we arrived in Miami, my mother enrolled herself in night classes so she could say she was a student, worthy of a student visa. She enrolled me in school for the fall, so I could get a student visa as well. Our long climb with the immigration service of the government began, and we stayed in the United States, and I grew up in the country, first as a rebellious kid who, once my arm healed, began to play American football, and then, oddly, I received a minor football scholarship to Southern Methodist University, but the year I arrived at the university their football program was shut down and all of their money was put, temporarily, for a few years, into promoting music instead. In high school, because I had wanted to play the bass guitar in a band, I had started taking stand-up bass lessons in the public school orchestra. I was late to begin the instrument, but the bass is not like the violin, which you have to begin as a child to play well. I played the anchoring notes, and traveled with the orchestra, and became interested in the deep, underlying sounds that seemed to resonate with some inner pain I had kept from my days in Mexico, even though I had grown into a boy who talked a lot and who joked a lot with others in the hallways of my American school.\n\n\u2014\n\nToday, thirty years have passed since my mother took me, bravely, to the United States. Thirty years to the day, and the United States is finally giving my mother her U.S. citizenship. Had we come completely illegally, we might have been able to become U.S. citizens much earlier, but ironically, by coming with official visas, by working through the legal system as you are supposed to, my mother had signed papers with fine print that said when we came as tourists we never intended to stay as immigrants. It has taken this long to work her through the legal system.\n\nI live most of the time in New York, where I eventually attended Julliard for a master's degree, but I have bought a house with my wife down in Florida. She is a composer, and she finds the escape from New York a source of calming, a place where it is easier for her to compose. I follow my wife, who is far more famous than me, around the world. She is the star. I am her supporter, her biggest fan, just as my mother was once my biggest fan, saving me. I pull my wife's suitcases through the airports. I check her in to hotels, as she is brought to compose music for some of the biggest orchestras and quartets in the world. I am a nobody, but I know my wife loves that I am so supportive of her. I live for my wife, as my mother has lived her life for me.\n\nTo celebrate my mother's thirtieth year in the United States, and the envelope that came in the mail this morning, announcing that she is now, formally, a citizen of this country, I take her to the mineral springs that I love so much, near our house just outside the town of Englewood, Florida. The place has the feeling of a spa. When you walk in the door there are lawn chairs spread out on the wide-open, manicured grass. Palm trees sway in the air.\n\nAs is my habit when I am down in Florida with my wife who is composing, she works in the house, locked away from me in her room, while I go to the mineral springs. I like to wear a thick, white, plush bathrobe, which I stole from a hotel in Germany when we were on tour for one of her musical compositions. I have a second, equally plush robe that I give to my mother. We have lunch first, and I order plate after plate of sandwiches, fresh orange juice, beer, borscht soup, since the local Russians like to come to this artesian spring, and then I order a plate of caviar and black bread, which is the most expensive item on the menu. I like to celebrate everything in life. I like to take in every opportunity to have fun, to taste, to play music with friends, to feel just how good life can be.\n\nMy mother, looking at all of the food I have ordered for us, as we sit under a large sun umbrella at the caf\u00e9 of the mineral springs, says, \"You eat like you are always running away from something, Gordi. You eat too much.\"\n\nShe is right, of course, in some ways. But I tell her, \"It is not running away. It is taking in everything, sucking the marrow of the bone.\" I feel my scar under the plush cotton cloth of my robe and pull the sleeve of the arm up, involuntarily, until I can see the scar. My eyes get watery looking at the scar, the deep gash where El Farito almost killed me. I think of my father, who only came once to the United States to see me, six years ago, when I was married to my wife. It is the only time he has ever come to see me. He came on the day of the wedding, though he hid from standing in most of the pictures because he was uncomfortable that my wife is Jewish. There is odd, lingering anti-Semitism there, another story for another time.\n\nMe, I have never been back to Mexico. I can never return. I have no desire to go back to the place where I was almost killed, and to those times of running away from the gangs of Iztapalapa. I order another orange juice and down it in three gulps. My belly is round, but it is a happy belly, not a slothful belly. I take in life. I play in the back of the orchestra, letting my arm sway widely to and fro. \"You look wonderful today, Mam\u00e1,\" I tell her. And she does look happy. \"I can never thank you enough.\"\n\nShe smiles at me, gets up from the table for a second, leaning forward with her relatively small body next to me, her little bear cub, who is now not so little, with some gray around my temples, and gives me a kiss around the back of my ear. When we are done, and she is sitting in the chair with her eyes closed, leaning back, looking calm, I leave her and I walk out into the deep pond of the mineral spring. I walk into the water that others might think is full of muck, burbling up from the origins of a swamp, which is so deep the bottom cannot be seen. It is said the hole of the mineral spring goes down hundreds of feet to another place, to other origins, to a place so deep the bones of dinosaurs can be found there. The explorer Ponce de Le\u00f3n passed near the place in 1513 when he came looking for seven fountains of youth and gold, which he never found, but which I lay floating in comfortably in the deepest part of the water, looking up at the sky as it breezes by. Floating in this water is my favorite activity when I am down in Florida. I like it even more than floating in the ocean. I feel the power of the healing minerals as they swish and move around my body, into my ears and over my scar, and the waters soften the scar, as time does\u2014I should never have bought the gun, my mom and I will always be refugees, I think\u2014and I feel my mother nearby, right next to me even though she is fifty yards away, the calm of another beautiful day in Florida, the calm of a mother's love for her child.\n\n# ACKNOWLEDGMENTS\n\nTHANKS TO Tessa Hadley, Jaime Manrique, Edie Meidav, Susan Burmeister-Brown, Linda Swanson-Davies, Albert Goldbarth, Tom Perrotta, Nathan Roberson, Philip Spitzer, Lukas Ortiz, Christopher Merrill, Jennifer Clement, David Lida, Peter Nazareth, Chris Walsh, Mary Sullivan Walsh, John Matthias, the National Endowment for the Arts, the Hermitage Artist Retreat, Patricia Caswell, Martin J. Sherwin, Marc Nieson, Sherrie Flick, Peter Trachtenberg, James Alan McPherson, Frank Conroy, David Hamilton, Steve Almond, Thorpe Moeckel, Margaret Dawe, Levente Sulyok, Bradley Narduzzi, Roberto Espinosa, Jessica Poore, Rafael Moreno Arn\u00e1iz, Du\u0161an Sekulovi\u0107, Dena Wetzel, Maria Elena Barron, Sandy and Bronwyn Barkan. In memory of my father, Joel Barkan.\nABOUT THE AUTHOR\n\nJOSH BARKAN has won the Lightship International Short Story Prize and has been a finalist for the Grace Paley Prize for Short Fiction, the Paterson Fiction Prize, and the Juniper Prize for Fiction. He is the recipient of a fellowship from the National Endowment for the Arts, and his writing has appeared in _Esquire._ He earned his MFA from the Iowa Writers' Workshop and has taught writing at Harvard, Boston University, and New York University. With his wife, a painter from Mexico, he divides his time between Mexico City and Roanoke, Virginia.\n\n# _What's next on \nyour reading list?_\n\n[Discover your next \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9781101906309\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n 1. Cover\n 2. Other Titles\n 3. Title Page\n 4. Copyright\n 5. Dedication\n 6. Contents\n 7. The Chef and El Chapo\n 8. The God of Common Names\n 9. I Want to Live\n 10. Acapulco\n 11. The Kidnapping\n 12. The Plastic Surgeon\n 13. The Sharpshooter\n 14. The Painting Professor\n 15. The American Journalist\n 16. Everything Else Is Going to Be Fine\n 17. The Prison Breakout\n 18. The Escape from Mexico\n 19. Acknowledgments\n 20. About the Author\n\n 1. Cover\n 2. Cover\n 3. Title Page\n 4. Contents\n 5. Start\n\n 1. iv\n 2. v\n 3. vi\n 4. vii\n 5. \n 6. \n 7. \n 8. \n 9. \n 10. \n 11. \n 12. \n 13. \n 14. \n 15. \n 16. \n 17. \n 18. \n 19. \n 20. \n 21. \n 22. \n 23. \n 24. \n 25. \n 26. \n 27. \n 28. \n 29. \n 30. \n 31. \n 32. \n 33. \n 34. \n 35. \n 36. \n 37. \n 38. \n 39. \n 40. \n 41. \n 42. \n 43. \n 44. \n 45. \n 46. \n 47. \n 48. \n 49. \n 50. \n 51. \n 52. \n 53. \n 54. \n 55. \n 56. \n 57. \n 58. \n 59. \n 60. \n 61. \n 62. \n 63. \n 64. \n 65. \n 66. \n 67. \n 68. \n 69. \n 70. \n 71. \n 72. \n 73. \n 74. \n 75. \n 76. \n 77. \n 78. \n 79. \n 80. \n 81. \n 82. \n 83. \n 84. \n 85. \n 86. \n 87. \n 88. \n 89. \n 90. \n 91. \n 92. \n 93. \n 94. \n 95. \n 96. \n 97. \n 98. \n 99. \n 100. \n 101. \n 102. \n 103. \n 104. \n 105. \n 106. \n 107. \n 108. \n 109. \n 110. \n 111. \n 112. \n 113. \n 114. \n 115. \n 116. \n 117. \n 118. \n 119. \n 120. \n 121. \n 122. \n 123. \n 124. \n 125. \n 126. \n 127. \n 128. \n 129. \n 130. \n 131. \n 132. \n 133. \n 134. \n 135. \n 136. \n 137. \n 138. \n 139. \n 140. \n 141. \n 142. \n 143. \n 144. \n 145. \n 146. \n 147. \n 148. \n 149. \n 150. \n 151. \n 152. \n 153. \n 154. \n 155. \n 156. \n 157. \n 158. \n 159. \n 160. \n 161. \n 162. \n 163. \n 164. \n 165. \n 166. \n 167. \n 168. \n 169. \n 170. \n 171. \n 172. \n 173. \n 174. \n 175. \n 176. \n 177. \n 178. \n 179. \n 180. \n 181. \n 182. \n 183. \n 184. \n 185. \n 186. \n 187. \n 188. \n 189. \n 190. \n 191. \n 192. \n 193. \n 194. \n 195. \n 196. \n 197. \n 198. \n 199. \n 200. \n 201. \n 202. \n 203. \n 204. \n 205. \n 206. \n 207. \n 208. \n 209. \n 210. \n 211. \n 212. \n 213. \n 214. \n 215. \n 216. \n 217. \n 218. \n 219. \n 220. \n 221. \n 222. \n 223. \n 224. \n 225. \n 226. \n 227. \n 228. \n 229. \n 230. \n 231. \n 232. \n 233. \n 234. \n 235. \n 236. \n 237. \n 238. \n 239. \n 240. \n 241. \n 242. \n 243. \n 244. \n 245. \n 246.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzsjfx b/data_all_eng_slimpj/shuffled/split2/finalzzsjfx new file mode 100644 index 0000000000000000000000000000000000000000..478ed3ab361b0d2807ed63792493037c09f60081 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzsjfx @@ -0,0 +1,5 @@ +{"text":" \n# THE WEIGHT OF THINGS\nOriginally published in German as Die Schwerkraft der Verh\u00e4ltnisse\n\nby S. Fischer Verlag, 1978\n\nCopyright \u00a9 1978 by Marianne Fritz\n\nTranslation and Afterword copyright \u00a9 2015 by Adrian Nathan West\n\nThe Afterword was originally published in different form in Asymptote.\n\nThe publisher gratefully acknowledges the financial support received for this translation from the Austrian Ministry of Education, Arts, and Culture.\n\nThe publisher would also like to thank Jeremy M. Davies.\n\nAll rights reserved. No part of this book may be transmitted or reproduced in any manner or in any form without prior permission from the author or publisher.\n\nISBN: 978-0-9897607-7-5\n\nEbook ISBN: 978-0-9897607-8-2\n\nArt on cover \u00a9 Hella van 't Hof\n\nAnonymous, 2014\n\n13 \u00d7 18 cm\n\nEmbroidery on photo transfer\n\nDesign and composition by Danielle Dutton\n\nPrinted on permanent, durable, acid-free recycled paper in the United States of America\n\nDorothy, a publishing project\n\nPO Box 300433\n\nSt. Louis, MO 63130\n\nDOROTHYPROJECT.COM\n\n# THE WEIGHT OF THINGS\n\nMARIANNE FRITZ\n\nTRANSLATED AND WITH AN AFTERWORD\n\nBY ADRIAN NATHAN WEST\n\n# A NOTE ON THE TRANSLATION\n\nMany of the names in The Weight of Things have secondary meanings: Faust, Berta's maiden name, means fist, as well as recollecting Goethe's famous play; Schrei, Wilhelm's surname, means scream or cry; Reichmann is analogous to Richman in English, and so on. The place-names are particularly significant: the town of Donaublau or Danube-blue invokes the waltz by Johann Strauss, and Allerseelengasse, where Berta was born, is named for All Souls' Day, when the dead are mourned. We have elected to leave these proper names in the original to avoid giving the text a parodic tone that would stray from the sense of the original, where they are less obtrusive than they would be in English. There are also scattered allusions to literature and politics from the time of the book's writing: when Berta and her children sing \"I know not what it means,\" this is the first line of Heinrich Heine's Lorelei, and the Isle of the Blessed, a notion Wilhelm hits upon in describing his home country, was a favorite term of Chancellor Bruno Kreisky in his praises of Austria's alleged social harmony.\n\n# WILHELMINE IS NOT BERTA\n\nOf all the events of 1945, there was one Wilhelmine recalled with particularly painful clarity. Wilhelm had hung the necklace with the tiny Madonna around Berta's neck, not hers. This although Berta's belly clearly demonstrated that she, unlike Wilhelmine, was no longer a chaste young woman. For her own sake\u2014and for Berta's!\u2014she ought to have spoken her mind. Maybe Berta had been good enough for Rudolf, but she certainly was never right for Wilhelm!\n\n\"Rudolf is so gentle and indulgent. A born brooder. He needs a firm hand, someone with both feet on the ground. He's a dreamer! His head's in the clouds! He needs a woman who knows what she wants! Berta, he's just not right for you.\"\n\nIf, when Berta had come to her, Wilhelmine had spoken words to this effect, would Berta have ever fallen for Rudolf?\n\n\"Berta, you poor little fool! Couldn't you have found a better time for making babies? Rudolf might not even make it back from the front! And he couldn't slaughter a chicken, let alone raise a child! Too sensitive, too timid, never stuck his neck out for anything. Now he's gone off who-knows-where and the only thing you can think to do is have a kid! Berta! This is crazy!\"\n\nIf, when Berta had come to her, Wilhelmine had spoken words to this effect, would Rudolf's last furlough have produced such a tragedy?\n\nIf so, then Wilhelmine's restraint was her own undoing\u2014for Rudolf had indeed failed to return, and in his place came a stranger, Wilhelm Schrei, who handed Berta a daisy and a letter. She accepted both like precious jewels. \"So. So,\" Berta said. Not even a \"Thank you, sir.\" This stranger had trudged to Donaublau from Frankfurt on the Oder, in such treacherous times, no less, and all Berta had to say was: \"So. So.\"\n\nHeaven knows, that was going too far! Wilhelm, the poor sap, just stood there without a clue what to say. And what did Berta Faust do? She just forgot him; sat down at the table, ran her hand over the tablecloth, trying to smooth it out; said again, \"So. So,\" and didn't even look up.\n\nThat's how Berta Faust was. Never in the present moment; always with her mind somewhere else.\n\nIn her misfortune Berta was at least lucky enough to have her\u2014Wilhelmine\u2014around. It was she\u2014Wilhelmine, and not Berta (!)\u2014who offered Wilhelm schnapps, ham, bread; luxuries that sometimes Berta's father and sometimes she\u2014Wilhelmine and not Berta (!)\u2014used to buy on the black market or from farmers. Wilhelmine let the plainly quite hungry young man eat in peace and asked questions\u2014questions that really Berta herself should have been asking\u2014only once Wilhelm had stopped his frantic pawing at the ham and bread.\n\n\"All right. Rudolf's left us. That must have happened to a lot of people. At least tell us: how did he go? Was it over quick, or did it drag on?\"\n\nThe poor fool was naturally so daunted and perplexed by Berta's thoroughly inhospitable silence that he hemmed and hawed, refusing to speak of such things, though this was the perfect time to sort them out once and for all.\n\nNo, Berta showed not the least consideration for that weary soldier. In the end, Wilhelmine even had to do Berta's grieving for her.\n\n\"The poor, poor dears! They were made for each other! And that's not all: a baby on the way, no husband in the house. Such a mess! What hope is there for a pregnant woman stuck living with her father\u2014an old gravedigger to boot! The women in your family have always been blessed with useless men, Berta, but this Rudolf of yours has topped them all! It's enough to make you jump out of your skin. Berta, say something!\"\n\nBerta giggled and said: \"So. So.\"\n\n\"What's so funny?\" Wilhelmine hissed at Berta. Then she turned back to Wilhelm: \"Karl was undoubtedly the most respectable of the three Faust boys. But there's been no word from him since this May. We did at least find out he'd been laid up in the military hospital in Castelfranco, which is somewhere near Modena. Richard, his twin brother, wasn't much of a letter writer either. After March of 1944, not a single word. Upper Silesia isn't even that far. You'd think he could at least have gotten away for a few days' leave! And then Wastl, Wastl clean forgot he even had a mother and sister back in Donaublau waiting for word from him. Wastl in Grajewo, Karl somewhere near Modena, Richard in Upper Silesia\u2014that's all we ever knew. You call that staying in touch? They could at least have sent a letter home with someone, don't you think? In case they didn't make it? Shouldn't they have thought of that? Berta, say something!\"\n\nBerta kept silent.\n\n\"No. This can't go on. Someone needs to do something. What's your story, sir? Are you planning on staying in Donaublau, then? Nowadays all the cities look more or less the same. A heap of rubble is a heap of rubble no matter where you go. Nowadays everyone has to start from scratch. The Faust boys used to sleep in the next room. That means we've got three empty beds, two dressers, a table, and three chairs\u2014will that do for you?\" And Wilhelmine stared at Wilhelm expectantly.\n\nWilhelm stayed. Yet it was not until the 13th of January, 1960 that the ring\u2014the obligatory wedding ring Wilhelm had properly owed to Wilhelmine from that very first day forward\u2014that Wilhelm at last placed this ring on her finger. Still, she at least got it in the end.\n\nBut the necklace with the tiny Madonna\u2014that remained with Berta.\n\nThe degree to which the necklace with the tiny Madonna had become an object of Wilhelmine's ambitions was something Wilhelm had never understood. Over time, he even forgot there ever was a necklace with a tiny Madonna.\n\nFor Wilhelmine, doubting and brooding were like alcoholism or drug or nicotine addiction, problems of the sort Wilhelm tended to evade with an \"if\" and a \"but,\" an \"on-the-one-hand\" and \"on-the-other.\" Wilhelmine assumed that it would always be her judgment\u2014accompanied by some of Wilhelm's \"on-the-one-hands\" and \"on-the-others,\" his \"ifs,\" his \"ands,\" and his \"buts\"\u2014that would guide them through their various marital disputes. Yet even understanding full well how things were between them, there remained in Wilhelmine a vague anxiety that she had never been able to shake and that she could express only by occasionally exclaiming: \"Wilhelm, something has to be done!\"\n\nTheir household was consummately organized. Her position as cleaning woman for Dr. Ulrich Reichmann's clinic and the Donaublau branch offices of the Mueller-Rickenbergs fulfilled both her financial and her creative needs. And what remained of Wilhelmine's energy was burned up in her and Wilhelm's ceaseless marital combat.\n\nIn the long run, however, Wilhelmine couldn't ignore it any longer: her ambition craved a new object, and she resolved at last to take the matter of Berta in hand. Intuition called to her as she was preparing breakfast with quick and nimble gestures: \"Today the matter shall be settled for good and all.\"\n\nWilhelm had always hoped to dodge this particular turn of fate, indeed had steadfastly fought\u2014successfully for the three years they'd been married\u2014to prevent its ever coming to the fore. But Wilhelm was getting older, and the older he got, the more inevitable it became that, like it or not, he would one day be forced to face up to a certain fact, which he had known in his heart all along: Wilhelmine was no Berta!\n\n\"Let's leave things be\u2014what's done is done! I can't undo it!\" Quite often during the past three years, this phrase would serve as the second-to-last word in each new installment of their perennial discord. His wife, the resolute Wilhelmine, saved the last word for herself, having always at hand an irrefutable riposte to whatever objection Wilhelm might offer. Convinced that every marital dispute inevitably required a winner and a loser, Wilhelm stuck to the latter role, in which he felt perfectly comfortable. Far more upsetting to him was the realization, clearer and clearer as time went on, that Wilhelmine was not satisfied simply to cling to her various judgments, but was also inclined, under certain circumstances, to actually act upon them.\n\n\"On the 13th of January, 1960, we were married.\" These were the words with which Wilhelmine, determined to realize her new objective regardless of the cost, opened their breakfast conversation on the 13th of January, 1963. And with a sinister single-mindedness, she smacked the dome of her egg until it was pulp.\n\n\"And the 13th of January was Berta's birthday,\" Wilhelm countered, taking pains to give an impression of composure, of even temper, and in particular to radiate docile humility, though gloomy thoughts were now wandering like ghosts through the cells of his brain. Yet his efforts were in vain, and his gloom could not be hidden from Wilhelmine, for it had a marked effect on Wilhelm's smile.\n\n# WILHELM, THE SMILER\n\nWilhelm's smile won him extraordinary benefits. In particular, it brought in generous tips, a most welcome supplement to his salary as chauffeur and Come-hither-boy to Johannes Mueller-Rickenberg, which the Schrei household had come to rely upon as a regular addition to its income. These tips, in turn, became further incentive to Wilhelm's talent for smiling; the smiler Wilhelm knew how to work up the proper smile-combination much as a chef, when cooking, strives for the right mix of spices. All that was needed, according to Wilhem, the chauffeur and Come-hither-boy, was a proper assessment of his rider.\n\nWilhelm's smile told Johannes Mueller-Rickenberg that Wilhelm was meek and a little dumb, but also discreet, and that in his work as a chauffeur and Come-hither-boy he was quick, agile, and dependable. It reminded Johannes Mueller-Rickenberg that he could be sure of his Wilhelm seeing both everything and nothing, hearing both everything and nothing, understanding everything and grasping nothing. In a nutshell: Wilhelm was a storybook chauffeur and ideal Come-hither-boy. It was to reinforce this image of himself, in the eyes of Johannes, that Wilhelm's smile was cultivated. He knew just when to sneak a pinch of acquiescence into the recipe and when to leaven his stupidity with a dose of wit, knew when and in what place and under which circumstances his smile should suggest presence of mind, how his smile should corroborate for this or that passenger what the passenger already knew. He had a skeptical smile, a brooding smile, a sly smile, a moronic smile and a shrewd smile, a clear-eyed and a purblind, a dutiful smile, the smile of a deferential and devoted spirit, as well as the smile of an obstreperous spirit prone to criticizing everything, and then yet another that was never indelicate or frank, always \"on-the-one-hand... on-the-other,\" always ready to let one or another nuance recede, dwindle, or simply vanish. Nor was the pursuit of his bonus the sole motive for Wilhelm's really quite cunning talent for smiling. His thoughts were already inclined by nature to \"ifs\" and \"buts,\" to \"it merits consideration\" and \"in view of the circumstances,\" to \"on the one hand, when you consider\" and \"on the other hand, you have to bear in mind,\" and this reticence enriched the vast range of his smiles. He believed all and nothing, doubted all and nothing, was a born dreamer who never dreamed. In a nutshell: he was a worthy representative of his nation.\n\n# THE SMILER WILHELM AND WILHELMINE'S AMBITION\n\nThe smiler Wilhelm could clearly sense, even while exercising his smile, that somehow he had already made a wrong move. There was something too deliberate, too intricate in the way Wilhelmine spooned out the remainder of her egg. After long consideration, he couldn't decide which was more worrisome: the sudden failure of his smile to have its accustomed effect, or the ultimatum issued by his Wilhelmine, which, though given three years ago, now came back to him violently and unexpectedly.\n\n\"If you want to marry me,\" she'd said, \"then it must be on January 13th.\"\n\nReliving again the terrible anxiety of that day, Wilhelm suspected his third anniversary might begin with yet another ultimatum. Even with his \"if\" and \"but,\" his \"on-the-one-hand\" and his \"on-the-other,\" he would have to tread carefully, very carefully, given the circumstances. Of course, when you considered Wilhelmine's irritable disposition, caution itself might here be in some sense misguided, since Wilhelmine was as likely to interpret it as weakness. Yet as he thought back further, to the days leading up to the 13th of January, 1960, he decided on caution after all. A bit more caution back then and things might have gone better than they did.\n\n\"Wilhelmine! What's gotten into you! January 13th is Berta's birthday! No. We'd best avoid that day.\"\n\n\"So what? Berta's birthday? Are we supposed to walk around her on tiptoes for the rest of our lives? It's January 13th or it's never!\"\n\n\"Wilma, dear, you're acting like a child! And it's not only Berta's birthday, you know, it's also an unlucky number.\"\n\nWilhelmine shook her head and broke out laughing, uproariously, maybe also a little perversely. \"So now you're superstitious? Just like our poor Berta\u2014she's superstitious, too! No, Wilhelm, no: if you're too big a coward to take me to the registrar's office on January 13th, then don't bother. What I need is a man, one who's there for me in spring, summer, fall, and winter\u2014who's there every single day! I'm not interested in half-measures.\"\n\nUndoubtedly he had been right not to make Berta's birthday a point of contention before, but that did not necessarily mean\u2014did it?\u2014that hinting at his old objections now, three years later, was a good idea. In the end, Wilhelm forced a decision: he would wait. He would see what precisely his Wilhelmine was going to propose. It could be she was somehow in the right and he somehow in the wrong. The only compelling certainty here was that Wilhelmine was on the attack: everything else was open to doubt.\n\nBut now it came to him how last night in bed he had, in fact, gotten carried away and let slip a certain vow: \"My golden girl. My little spitfire,\" he'd started, sliding up beside Wilhelmine. \"Tomorrow is our third anniversary. You didn't forget that, did you? Shall I make an oath to you, my turtledove? You have my word of honor as a chauffeur\u2014but what am I saying? You have my word as an upstanding man, who always lives up to what he says\u2014that from this day on, even more than before, I'll beckon to every wish that falls from your lips,\" and here he planted, with great relish, a gentle kiss on the face of his golden girl. \"I will fulfill your every desire, even before you've come out with it. Dear little Wilmerl, you busy bee, my exalted, my one and only beloved! Mark my words! Mark them well! This is what our third anniversary means to me! By God, let there be no doubt! This is what it means to me!\"\n\nThere was one positive thing to be said for this vow, at least from a certain perspective: it provoked an uncommon stirring in Wilhelmine's body, in a manner that was much to his benefit.\n\nBut now he worried: what if she took his vow too seriously? In view of Wilhelmine's inclination to demand fulfillment of every last verbal concession she'd wrung out of him, usually without warning, he might have done better to vow henceforth never to make his vows so hastily, and to leave himself more ways out.\n\n\"It just so happens that the 13th of January, 1963, is not only our third wedding anniversary, but is also Berta's fortieth birthday,\" said Wilhelmine. \"This is what I've meant all this time I've been saying to you: something has to be done! I've meant, of course, that we should pay Berta a visit and cheer her up. We'll have to do it sooner or later anyhow, if only for decency's sake. And if not now, then when?\"\n\n\"I knew it,\" groaned Wilhelm, and then a breadcrumb slipped down his throat and he absolutely could not cough it out, and tears welled up in his eyes, and his extravagantly robust circulation colored his face red, right up to his thinning hair.\n\n\"What's that, Wilhelm? You knew what? You mean that you yourself had already taken Berta into consideration? Well, I'm not at all surprised! When Wilhelm promises something, he comes through. It's no coincidence that you and I have the same name\u2014no coincidence! We are simply one heart and one soul, right to the very end.\"\n\nAnd Wilhelmine, who so relished stripping every hint of subtlety, every ambiguity from a conversation, felt her pulse race, and her eyes, her mouth, and the wings of her nose quiver, so that her well-nourished face suddenly flushed with an autumnal beauty. And it was only after she'd given Wilhelm a full dose of this Indian-summer glow that she decided at last to take note of his struggle with the breadcrumb.\n\n\"Wilhelm! Were you choking on something?\" she exclaimed thoughtfully, once he had coughed it out.\n\n\"No. No,\" Wilhelm said, then hacked again, dried his eyes with a napkin, ran his right hand over his prematurely gray hair, hacked once more into this same hand, which he was now holding out in front of him, trying to bring his perturbation to an end as gracefully as possible and to thereby recover the self-restraint proper to a self-possessed man of the sort who faces life's many ups and downs without fear.\n\n\"It was nothing,\" he said, and with a gracious, singularly discreet nod of the head, he acknowledged the last-but-one interjection of his Wilhelmine, whose morning mood, cheery though it already seemed to be, no doubt held plenty of room for improvement: \"I agree with you entirely!\"\n\nAfter a long pause in the conversation, he proceeded: \"You know, Wilhelmine, at first I did have the impression I was choking on something. Now, however, it appears I must have been mistaken.\"\n\nAnd Wilhelm went out and picked the most beautiful rose from the rosebush to underscore the solemnity of this 13th of January, 1963, and he passed it with a smile to his bride, who could do nothing but giggle virginally, coquettishly, blushing: this was how touching it was to her.\n\n\"You shouldn't have, Wilhelm,\" Wilhelmine gasped. \"Oh, you shouldn't have!\" She interpreted Wilhelm's smile, and the gesture that accompanied it, as absolute submission to her plan, albeit with an \"if\" and \"but,\" an \"on-the-one-hand\" and an \"on-the-other.\" Her sordid ulterior motive, the slightly indecent exhilaration she felt in connection with the idea of at last paying Berta a visit, all this she kept prudently to herself.\n\nAnd so Wilhelm's failure to speak up had at least been successful in staving off Wilhelmine's ill humor. Over the course of that afternoon, and even right there at the table eating lunch, he concocted many new ideas, some remarkably good ones, for managing the coming visit as painlessly as possible. Yet somehow none of these ideas made the prospect seem better or the pain less inevitable: there was no getting around the menace in Wilhelmine's words.\n\n\"You're perfectly right, Wilhelm. Your suggestion is an excellent addition to our plans for our third anniversary. Really, a lovely thought. A remarkable thought. After all, something has to be done! It has to! It has to! We can't just think of ourselves, Wilhelm. We can't act as though Berta's already dead and buried!\"\n\n# A MAN, A WORD, AND THEN YOU'RE LOST\n\nWard 66 was equipped with a cage. This cage divided the left-hand row of beds visible from the door into two sections.\n\nBefore Berta Schrei lost her voice, she often used to say, \"A man, a word, and then you're lost. But still, Little Mother, if I may be allowed to speak,\" and she would reach out with her twitching crone-hands toward the cage bordering her bed, looking at the old woman with bulging eyes\u2014eyes in which fear, indeed raw dread, was flickering\u2014and would speak further only when the old woman granted her permission with a nod.\n\n\"Don't I have pretty wallpaper? Little Mother, I know. I know. A man, a word, and then you're lost. I know I shouldn't speak. But, I mean. If I might be permitted to make such a remark, does the Little Mother agree?\"\n\nEach time, the old woman would nod graciously.\n\n\"Yes. Yes. Pretty wallpaper,\" Berta would say, staring enraptured at the bars of the cage around her. At that, the old woman would praise Berta's observance, and express her pleasure that Berta was beginning to gain control of the dreadful specter of her morbid imagination. And this was yet one more reason for the Little Mother to give Berta her blessing, and so she did give her blessing to her dear Berta.\n\nBasking in the glow of this blessing, Berta would murmur to herself at short intervals the lesson she had drawn from life: \"A man, a word, and then you're lost.\"\n\n# WILHELMINE, THE MODEL BOOKKEEPER\n\nIt was in one such moment, as a-man-a-word-and-then-you're-lost Berta was striving to conjure up some more beautiful truth for her cage, that Wilhelm and Wilhelmine, entering the courtyard of the fortress where she was held, crossed paths with a distinguished man of around fifty with a half-bald head, hamster cheeks, and little pig's eyes.\n\nWilhelm, the chauffeur, removed his hat with the utmost reverence, and blood rose into his cheeks.\n\nConfounded, Wilhelmine cried, \"What are you...?\" She couldn't resist gawking a bit at the little half-bald orb approaching.\n\n\"This is the Distinguished Dr. Primarius Gottfried Trimm,\" Wilhelm whispered, looking imploringly, if not quite with outright horror, at his wife, who in general had little understanding of social hierarchies.\n\n\"Aha,\" Wilhelmine said, adding: \"He's a bit of a porker.\"\n\n\"Wilhelmine!\"\n\n\"Or does he have a glandular condition?\"\n\nWilhelm squeezed his eyes shut, opened them, and stared at Wilhelmine, feeling a prisoner to his destiny.\n\n\"Wilhelmine,\" he said, and then, as if not even he believed that his words would have any effect: \"He is a highly distinguished man.\"\n\n\"And what's that to me?\" Wilhelmine hissed.\n\nThey stood perplexed in the courtyard of the fortress, with Wilhelmine pointing first to its northern wing and then to its eastern one.\n\n\"Berta is either there or there. She has to be around here somewhere. Wait! I'll ask the good Dr. Primarius.\" And Wilhelmine turned in the direction of the parking lot. Suddenly, Wilhelm became a man; his doubting and brooding compulsion vanished entirely; he grasped Wilhelmine by the arm and said, \"That is out of the question. That I will not accept.\"\n\nSo much resolve puzzled Wilhelmine and gave her pause.\n\n\"Don't drag me around like that.\"\n\n\"Leave him in peace. He doesn't like to be spoken to while he's thinking. And he's almost always thinking.\"\n\n\"So he's a busy sort, then?\"\n\n\"He visits the countess even more than her own daughter does. And listen up, Queen Penny-Pincher... Even when he's busy all month long making his rounds among the rich and famous, he still meets one-on-one with his brother-in-law, and his brother-in-law is my estimable employer, whose tips are as bountiful as my salary is meager. And that's one plus you're in danger of minusing, dear Queen Penny-Pincher!\"\n\n\"Fine. Fine,\" Wilhelmine said morosely, and: \"Don't make a speech about it! How am I supposed to know what goes on in some bigshot's head!\"\n\nWilhelmine had a guilty conscience.\n\nAnd, having just had her distinction as Queen Penny-Pincher called into question, she changed the subject: \"We need a plan, Wilhelm! We can't just both barge in there. You have to break the news about us to Berta as considerately as possible. Berta could be capable of anything, we don't know what state she's in. Give me a sign as soon as you've cleared everything up.\"\n\nBut then: Wilhelmine was filled with anxiety. What if Wilhelm couldn't find the right words and his ineptitude made Berta more distraught? What if Berta was already so far gone that she would no longer even understand what Wilhelm said to her? Indeed, it was a thoroughly heroic decision, her letting Wilhelm go in there alone, and hopefully Wilhelm would know how to appreciate the gravity of the responsibility she had conferred on him. Hopefully! Staying here, chained like Prometheus, waiting and hoping that Wilhelm would act reasonably despite the absence of her prudent guidance was a burden that could hardly be borne. Still, better that than having to hear another word about this Trimm and his tips! This \"distinguished\" personage who wandered about so deeply involved in his \"important\" problems that Your Lowliness, Trimm's brother-in-law's chauffeur and Come-hither-boy, dare not so much as utter a word to him, even here on the public grounds of Berta's fortress!\n\nIn summary: Distinguished Dr. Primarius Gottfried Trimm headed for the parking lot; Wilhelmine headed for the courtyard, yanked first one way and then another by all sorts of misgivings; and Wilhelm headed into the fortress, still full of the hope that something unforeseen might occur and save him from having to drain this cup of sorrow.\n\n# I WAS COMPELLED TO COME UP HERE\n\nAnd thus Wilhelm Schrei came skulking in, decked out in his Sunday best, with a bouquet of light pink roses, following on the heels of Head Nurse Gotaharda, who felt bound by duty to commemorate this highly unusual event: Ward 66 had a visitor. \"Berta, my dearest child, do you know what the Head Nurse has for you today? You don't? But it's visiting hours, isn't in Berta? And still you don't know?\" These were the words of this angel of angels, her baroque face beaming with those glorious feelings of triumph that sometimes fill people in the know when they have a chance to lord their secrets over their faintly hopeful, warily ignorant victims.\n\nAs Wilhelm Schrei sought out that forgotten face, he smiled soft and meek, and thought to add a pinch of embarrassment to his smile: \"Forgive my coming here!\" he was attempting to say, but without weighing down his declaration with words. \"I just couldn't do otherwise. I was compelled to come up here!\" The smile he actually produced, however, when he finally found Berta Schrei, was a trifle softer, a trifle more embarrassed, as if to beg her pardon for being there at all.\n\nSitting on her bed, twirling her thumbs, Berta dared to look up briefly, only to drop her head in shame and twirl her thumbs more frantically while a very faint pink colored her cheeks.\n\nAnd Wilhelm had a feeling he had often felt before: as if a hot and hazy pall of smog had gathered above the city of Donaublau, then sunk slowly down into the ravine between the houses of Allerseelengasse, as if wanting to settle in there for good.\n\n# WILHELM AND THE PALL OF SMOG OVER THE CITY OF DONAUBLAU\n\nWilhelm had long compared this pall of smog with a hippopotamus: \"A spiteful beast that just has to come to rest right where I'm standing, lying, sitting, or walking! It's trying to crush me with its tonnage!\" he would yammer, and his imagination deluded him into believing this hulking hippopotamus really was stampeding toward him. Sleepless, he would toss and turn, sit up in bed, shout in terror to his Wilhelmine, who would be snoring away rhythmically. It usually took a number of formidable shoves to wake her, and then she would dismiss Wilhelm's fears as more or less unmitigated gibberish.\n\n\"Wilmerl! Call the doctor! My little bride! I need help! Do you hear me! I need help! Wake up now!\"\n\nHe'd gasp for breath, feel the heavy beat of his heart, which struck him as strangely irregular. It galloped like a startled racehorse, and he thought for a moment it would leap from his ribcage unless he kept his hand firmly pressed over the obstinate beast; if not, he would lose his best racehorse\u2014his only racehorse, as it happened, so that its loss would be existentially catastrophic.\n\nAnd yet no sooner had he managed to calm himself through this downward pressure of his hand than his heart would begin decelerating, beating slower and slower until it was barely detectable, and then, it seemed, vanishing completely.\n\nAt this stage of Wilhelm's valiant struggles to stave off death, his wife would generally rouse herself and express her opinion on the matter, a bit impatiently, though not without a certain kindness: \"Look, it's not complicated. It's the humidity, that's all. Wait for the rain to come, then you'll feel all better. You're not going to die every time.\"\n\nDie: that was the operative word.\n\nWilhelm's teeth began to chatter, his body to tremble; even the springs in their marriage bed started to quiver, and hefty Wilhelmine was rocked back and forth, just like a boat over choppy waves\u2014as she noted to herself, with less bewilderment than irritation. She snatched at the small lamp on her nightstand, determined once and for all to snuff the life out of Wilhelm's ridiculous fears. And indeed, as soon as the room was filled with the lamp's soft half-light, Wilhelm left off with his chattering and trembling.\n\nOnce, Wilhelmine had dared to take a boat trip with Wilhelm on the lake, and on that occasion she'd become ill, as she soon alerted him, shouting, \"Wilhelm dear! I'm seasick! I can't take this constant rocking back and forth! The boat, Wilhelm dear! The boat is tipping over! I can't swim! This is just horrible! Are you trying to do me in? This rocking has got to stop! Right now! Turn back right now!\" It seemed she had a strong aversion to back-and-forth rocking of any sort\u2014and thus she felt great relief in these moments as well when the rocking of her bed came to an end.\n\nAnd yet no sooner had he gotten used to their bedroom, and grown calm again in the veil of half-light, than Wilhelm began to worry about the shadows cast by the dresser, the mirrored vanity, and the rubber plant, which filled the room with their ghostly apparitions.\n\nThen Wilhelm's chattering and trembling started up even more insistently than before.\n\n# WILHELM, THE SMILER, DISCOVERS TO HIS RELIEF THAT HE IS AN AVERAGE CITIZEN\n\nBut he knew what was and was not proper behavior in public, and so the brave Wilhelm who struggled valiantly against death in the nighttime remained a secret known only to his Wilmerl, while in Ward 66 there now stood a shell of a man wiping dollops of sweat from his forehead with a large white handkerchief before pushing it laboriously back into his pocket, then taking it back out immediately to extravagantly blow his nose. Once this nose-blowing procedure had been brought to an end, he turned his attention to his head of gray hairs, which he used his newly freed hand to smooth over, at which point there arose the possibility of tugging at his ear, at his nose, and then at his ear once more.\n\nHe had the feeling there was something about this place that restricted his musculoskeletal options. And so he gave up on his fidgeting and settled on a single, unassailable declaration: \"I'm here, Berta. Yes. I'm here.\"\n\nHe laid his hand on his heart, though he didn't dare press down firmly in public. Berta giggled, embarrassed, and glanced over at him, and he thought that if it weren't this particular place\u2014and on January 13, 1963\u2014he might be justified in supposing that a heavy, hazy pall of smog really was bent on choking the life out of him. In view of the place and time in which he'd found himself, however, it seemed unlikely some sudden rain would blow in to save him, to clear the air and fill him with renewed cheer and well-being. On the other hand, who knows? There was surely much more between heaven and earth than an average citizen like Wilhelm could ever imagine.\n\nEver since stepping into Ward 66, Wilhelm's brain cells seemed veiled in a thick waft of fog, so that he could make out his thoughts only vaguely, and he had to proceed slowly and carefully, feeling them out, to be able to tell one from another at all. He resolved to restrict his thoughts to a level appropriate to the circumstances, to concentrate his energies, like a chauffeur driving in the fog who has to focus his attention on the oncoming cars: on seeing them for what they are, on not drifting too near them, on recognizing trees in the roadside shadows, concrete dividers in the spectral darkness, on knowing the median isn't just a harmless fringe, to grasping, above all, that what surrounds him is real space, not some sort of vacuum, as the fog would prefer him to think\u2014to the extent that a fog prefers anything\u2014and to understanding that this material world is more resilient than he, so that failing to respect it, approaching it with arrogant recklessness, incautious stubbornness, or dogmatic inflexibility, would be extremely dangerous.\n\nWilhelm, the smiler, discovered to his relief that he had been, was, and would remain an average citizen. He looked at Berta and thought to himself, \"Am I a murderer? Am I a suicide? Both, even? No, I'm a chauffeur, a chauffeur above all else. My task is clear. I steer the car from one place to the other without putting myself, my passenger, or anyone else in unnecessary danger.\"\n\n# WILHELM, THE CHAUFFEUR\n\nWilhelm, who had always compensated easily for the errors of other drivers and the behavioral abnormalities of pedestrians, animals, and suchlike, to the benefit of all parties concerned, was praised at times by one or the other of the gentlemen he steered through the city's maze of streets\u2014or even hundreds of kilometers into the countryside\u2014and this praise would plunge him into deep uncertainty.\n\nThe question being, ought he to heartily accept or somehow disparage this praise of his automotive prowess? \"Yes, it's true, I've still never once had so much as a fender bender, nor have I caused anyone else to have a fender bender, or anything of the sort. You see? I am neither a suicide nor a murderer. I am a chauffeur. My duty is to preserve and protect the lives of distinguished gentlemen by means of my humble talents.\" Then would come an unassumingly bashful smile, and thus would he parry the praise of the powerful business attorney Dr. Ulrich Reichmann, whose importance Wilhelm would often emphasize with the words: \"For every five pages he writes, he's earned 100,000 schillings!\"\n\nThere had been a time when Wilhelm labored under the erroneous assumption that a good attorney must go to court as often as possible if he wants to make any money. Under his current employer, the great landowner and industrialist Mueller-Rickenberg, Wilhelm had also come to concern himself for the spiritual condition of his passengers\u2014but back when he worked for Dr. Reichmann, he had once tried to express his admiration for that man's professional accomplishment, only to be curtly interrupted with the observation: \"A good attorney never stands before the court.\" Wilhelm had not failed to notice the soft reprimand in Dr. Reichmann's voice.\n\nIt wasn't until later, sometime after Dr. Reichmann had come to rely upon Wilhelm and his pleasant, dependable smile, that Wilhelm understood the favor the attorney had done him in highlighting a decisive gap in his education. He discovered that Dr. Ulrich Reichmann belonged to a particular group of people, well-versed in the law, whose profession consists of merging business enterprises, or organizing corporations into their ideal legal form\u2014the ideal legal form being, it goes without saying, the one in which the least taxes are paid and the company is least vulnerable to any official prying. In brief: Wilhelm came to understand it was pure nonsense to think a good attorney must go to court as often as possible if he wanted to make any money.\n\nAnd so Wilhelm the chauffeur could now confidently sum up Dr. Reichmann's character with the same words that had formerly caused him such confusion: \"A good attorney never stands before the court.\" Whenever he uttered these words, he would nod knowingly and not without satisfaction. Indeed, he took great pleasure in springing this sentence on the uninitiated.\n\nRegarding his own profession, however: one passenger, the poet Fonderstrassn, had once waxed so lyrical in his praises of Wilhelm that the chauffeur felt duty-bound to put a damper on his enthusiasm.\n\n\"Maestro! Maestro! Don't offend a chauffeur! I'll admit that there may indeed be some artistic skill to driving through the fog, to recognizing a stag for a stag and not some trick of the mind, some phantom seen through windshield wipers that a driver might ignore or not, as he likes, without suffering any damage... That much I'll give you wholeheartedly! But sir, don't scatter your genius at the feet of such triviality. Your eloquence is, if I may venture a most deferential criticism, too lofty for a chauffeur's quite inconsequential duties. Only great men, men of significance, men struggling with complicated, contradictory, nay, tragic intrigues\u2014that is what should inspire you, that is what makes you a maestro. Not a little, undistinguished man, a nobody, a faceless, colorless Come-hither-boy... a simple chauffeur.\"\n\n# BERTA SCHREI HAD THOUGHT ABOUT WILHELM A GREAT DEAL\n\nSince 1958 Berta's doubting and brooding compulsion had centered solely on Wilhelm. The administration at the fortress found nothing objectionable in this; in fact they encouraged Berta's reflections on Wilhelm with utterly unbounded broadmindedness. As it happens, everything Wilhelm had ever disclosed to her bore some relation or other to his m\u00e9tier of chauffeur and Come-hither-boy. Even his child-rearing maxims would eventually wend their way back to his experiences as a chauffeur and Come-hither-boy.\n\n\"Berta. You need to teach your children to be more flexible. The world is one obstacle after another\u2014and it's in man's nature to put himself in harm's way. You have to pay close attention, like a good chauffeur, and take care to avoid collisions. What do you think would happen if I said to myself sometime, Hey, look at that fellow driving right down the middle of the street, does he think he owns the road? Well, I'll show him! What would be the outcome then, Berta? No, for better or for worse, I have to swerve out of his way. And if the other children continue to beat our little Berta black and blue, well, then our little Berta simply has to learn. If I go out looking for a collision, I too will end up black and blue; if, however, I know how to avoid a collision, then I will not end up black and blue. It's a question of wits, Berta. And we're all responsible for having that kind of wit.\"\n\nWilhelm dispensed generic philosophical advice of this character in hopes of raising their children, Little Rudolf and Little Berta, as competent citizens with both feet on the ground. Berta, his then-wife, might complain as often and in as many ways as she pleased: Wilhelm would just smile and philosophize away all her questions about upbringing, and Berta would twirl off back into her mind, where she would wrestle with the endless \"ifs\" and \"buts,\" the \"on-the-one-hands\" and \"on-the-others,\" today forbidding what tomorrow she would naturally allow (constraining this naturally with an \"if\" and a \"but,\" an \"on-the-one-hand\" and \"on-the-other\"), and the next day permitting what she would saddle with a stern prohibition one day later.\n\nWilhelmine, her friend, was repulsed by Berta's incapacity to keep her doubting and brooding compulsion from getting in the way of her children's upbringing.\n\n\"Berta! My little catastrophe! Your children have no use for your doubts and brooding! They need a steady hand, Berta! Don't fuss for so long. Just say yes or say no and then stick to it! One time this way, one time that way, it makes people lose perspective, if not their minds to boot! You've got no backbone!\"\n\nAnd yet if Berta would go on insisting her children be cautious and flexible, before later switching gears with Little Rudolf or Little Berta, or both at the same time, this was because, irrespective of Wilhemine's resolute example, Berta\u2014so fickle, so soft and acquiescent\u2014did indeed lack a backbone.\n\nBerta Schrei was thankful for every hour allowed to her for raising her children. She brought them up with heartfelt devotion, and considered it a lost opportunity, something to be mourned, if at any hour she failed to share an important life lesson with her two little sprouts; she would bewail herself, bemoan herself, bethink herself a bad mother, swear to improve until the next unpardonable oversight, which she would aspire to correct at the first available opportunity. That is how she brought her children up\u2014with great ardor\u2014until the year 1958.\n\n# BERTA GREETS WILHELM; DOES BERTA SCHREI HAVE A VISITOR?\n\nBerta said: \"So. So,\" and a drawn-out, seemingly endless sigh rose from her ribcage. Whenever Berta used to sigh that way, Wilhelm always smothered her fears, misgivings, doubts, and broodings about her children with his smile and his bits of chauffeur-philosophy.\n\nBy now Head Nurse Gotaharda had left Ward 66 to the visitor and his secrets, and the Wise Little Mother watched Wilhelm and Berta closely from the corner of her eye. Scarcely ten minutes had gone by since the Wound of Life had made its treacherous entrance into the Wise Little Mother's realm. It was that whore of a Head Nurse, who threw herself into the arms of anyone who came her way, who had befouled their corner of eternity, and it weighed upon the Wise Little Mother now, a heavy burden. The old woman resolved to alert Berta to the evil forces around her.\n\n\"You must reflect on what is happening here, Berta dear. Reflect and be careful. Life is a wound, and this wound... this wound is slow to heal.\"\n\nThrice she made the sign of the cross with her right hand, staring straight at the grating over the window, thrice more made the sign of the cross staring straight into the bars of the cage, and then turned back to Berta: \"Life, so say the wise, appears incurable.\" From her bed the old woman peered up at Wilhelm. \"Bring word of life, and you bring pain. Bring word of death, and you bring deliverance. So it is. Indeed. Are you bringing deliverance then?\" she purred at him.\n\nBerta giggled, and the Wise Little Mother laid her hand on Berta's shoulder, whispering serenely, \"Yes indeed, our Berta has a visitor. Berta Schrei, is it? Berta Schrei has a visitor? Is that right? Does Berta Schrei have a visitor?\" A murmur coursed through the rest of the ward. Berta Schrei has a visitor. Does Berta Schrei have a visitor?\n\nTears streamed over the face of Eulalia Mondschein, and she proclaimed, with mounting intensity: \"All my fault! My own irreparable fault!\"\n\nViktoria L\u00f6ffelholz shouted \"Oh!\" clapped a hand over her impertinent mouth, and then looked over at Wilhelm, begging his pardon, before vanishing beneath her bed, where she cowered, ashamed, until the visitor took his leave.\n\n\"Without a doubt. Life is visiting Berta. Berta Schrei. And life... We know this, don't we, my dear Berta? We know this. Life is hope and hope is a wound. Yes indeed. It's come full circle. Isn't that so?\"\n\nAnd turning to Wilhelm, she brought her observations to a close: \"Did you not know? Did you not know that hope is a wound? What do you know then?\"\n\nShe scuttled back to her bed, twiddled her thumbs, and prayed her \"Hail Mary, full of grace...\"\n\n# BERTA SCHREI HAS A VISTOR\n\nIn Ward 66, on the 13th of January 1963, it was apparent that the Little Mother must once have sung in the church choir, for only a trained singer could sing as well as the Wise Little Mother did.\n\n\"Hail, O star of the ocean, God's own mother blessed!\" The newness of the sound remained new, and it raised gooseflesh up and down Viktoria's spine. Eulalia's body began to rock back and forth, faster and faster, and her guilty conscience overwhelmed her: to temper her tears, she kept glancing at the cage, beseechingly, and this helped her to banish her sickness back inside her body. Eulalia felt the old need, so long in abeyance, but now vital, even urgent, for self-recrimination, to still the gods' wrath and arrive at some insight capable of striking a decisive blow against her illness:\n\n\"Eulalia, Eulalia, behave yourself. Eulalia. Eulalia. Don't take more blame on your shoulders. Eulalia. Eulalia. It's just your disease. You can't give in. Eulalia. Eulalia. Speak the words that will banish your sickness. Speak them yourself, or else let the gods speak them. Eulalia. Eulalia. Everything is fine. What's trying to come out of you isn't you, Eulalia. No. It's the sickness. Eulalia. Eulalia.\"\n\nIt seemed the Wise Little Mother had successfully initiated a general rebellion against life\u2014which just today had wormed its way with such shameful presumption into Ward 66. She took note of her victory with equanimity, and with equanimity she struck up her song one more time, from the beginning.\n\n\"Hail, O star of the ocean, God's own mother blessed!\"\n\nBerta giggled. She said: \"So. So.\"\n\nWilhelm looked down at his right hand and concluded he had five fingers on it, and the same for the other one too: not three, seven, or twelve. This somehow impressed and emboldened him a great deal; he almost felt himself the sort of man who could face life's ups and downs without fear.\n\n\"Berta, my dear. I've brought you something,\" he said, and laid the bouquet of roses gently on her lap. Berta looked up, lowered her eyes again, then her hands twitched slowly upward in the direction of her neck, feeling for the necklace with the tiny Madonna inside her institution uniform.\n\n# THE NECKLACE WITH THE TINY MADONNA\n\nBerta Schrei always spent a long time observing her Madonna trinket carefully in the washroom mirror; in the mornings, before the fortress day began, and in the evenings, before the fortress night, and during the morning and evening ablutions, she was always the second-to-last\u2014and the Wise Little Mother the last\u2014in the line of miserable souls into and out of the washroom... and for good reason.\n\nFor Berta Schrei was the only one in the entire fortress with a necklace with a tiny Madonna.\n\nFor years, the struggle to win back this prize had been waged by the shy and brooding and not at all shrewd Berta with surprising ingenuity and tremendous shrewdness, day in and day out. Yet in the end she had only to win the support of the Wise Little Mother, who entered into tireless negotiations with Nurse Franziska Querbalkner\u2014remarkably, as the old woman's general preference was to have nothing to do with Querbalkner whatsoever.\n\nTo have recruited the old woman, this tireless servant of the fortress regime, as her own sworn advocate in a battle against that very same institution\u2014this was a great triumph for Berta, and had made all the difference in her struggle. From then on, her hope of getting the necklace back from the fortress's administration soared, onward and upward. Until finally, on the very day when Wilhelmine had married Wilhelm, Berta got her Madonna back at last.\n\nIt was on that day\u2014Berta's birthday\u2014that Nurse Franziska smuggled the necklace out of the fortress depository and passed it to Berta, adding emphatically that she should hide this diadem of hers\u2014that was the nurse's word\u2014as safely as possible inside her institution uniform. \"If you don't keep an eye on that diadem of yours, it'll get lost again\u2014forever, Berta. Understand? So be careful. Don't forget. Keeping it safe will be an almost impossible task! To accomplish it will require constant attention and watchfulness and great cleverness too\u2014for the man who makes light of what he's won is never able to defend it, and the man who doesn't know how to defend what he's won will lose far more than just his treasure!\"\n\nBerta nodded, and as the nurse laid the chain hastily around her neck, Berta let it vanish inside her institution uniform and stared at nurse Franzi with a look that seemed desperate for approval. Querbalkner nodded her head several times, impressed, and then squeezed Berta's hand approvingly before stepping away from Ward 66. For the rest of that day Nurse Franziska felt like a person who's accomplished the impossible, while Berta Schrei rushed to have herself blessed by the Wise Little Mother, not just once, but several times.\n\nWilhelm smiled when Berta held out the tiny Madonna trinket to him; her face looked almost shrewd. It struck him then that sometimes a certain unfamiliar pressure in the stomach region could be more dangerous than the most excruciating cramps. Stomach cancer, for example, did not necessarily announce its treachery in the body with great fanfare; in fact it was inclined to do quite the opposite. He had read that or else heard it somewhere, sometime.\n\n\"I'll visit you more often from now on. You know that it's your birthday today, right? I have a feeling this could be a new beginning.\"\n\nWilhelm believed what he was saying, and Berta stopped twirling the Madonna back and forth between her fingers. She strained to hear his voice, which reached her ears like the song of the Lorelei calling out to the fishermen. Berta thought she could smell the beguiling odor of the roses creeping up into her nostrils, and Wilhelm's voice and Wilhelm's proximity turned time backward.\n\n# THE BOAT TRIP\n\nBerta was walking to the lake behind Wilhelm; the children were up ahead; there were summer blossoms for the eyes and rich aromas for the nose. Overcome by the opulence of the vegetation, Berta's doubting and brooding compulsion seemed to be swept away, and she thought to herself:\n\n\"I am a blank slate.\"\n\nAll around, the vibrant green was like a broad canvas upon which an exuberant painter had daubed generous swaths of color, as though his feelings and his passion for these many shades of green were not the least bit limited by the price of paint. To the children's delight, when they'd made it to the shore, Wilhelm coughed up the fare for the boat trip without any \"ifs\" or \"buts,\" any \"on-the-one-hands\" or \"on-the-others\"! To Berta's delight, he even launched the boat without any \"ifs\" or \"buts,\" \"on-the-one-hands\" or \"on-the-others.\" The children were happy! Berta was happy! Wilhelm looked at Berta again with the eyes of Private First Class Rudolf, with the eyes of Wilhelm the returnee.\n\nBut soon they began to cry, Little Rudolf first and then Little Berta, and the children tendered their own \"ifs\" and \"buts,\" their own \"on-the-one-hands\" and \"on-the-others.\" They began by calling the boat trip into question, and ended by declaring it a torment. Rudolf, the little scamp, let it be tearfully known that he wished only to lie flat on the bottom of the boat, then stood up from his seat to get ready to do so, and stumbled. Nor was there time to consider the cause of his stumbling, for he instantly plunged into the water\u2014a dire situation indeed, since, like his mother, Rudolph could not swim. He would likely have drowned had Berta not dove into the water to snatch the scamp by a tuft of hair, thus startling\u2014at last\u2014the clueless Wilhelm into a demonstration of his swimming prowess.\n\nWith characteristically laggard urgency, Wilhelm saw what was happening and dove to rescue the two of them, neither of whom would have been there to rescue even a moment later. Eventually the Schrei family made it back into the boat, hurt, coughing, disconcerted, and one lesson the wiser. All Berta's anxieties about herself, which had been growing over the course of the year, were confirmed once again. Their lovely outings inevitably ended with her and the children embroiled in some catastrophe that Wilhelm somehow always knew how to fix.\n\nThey had christened their new car with this excursion to the lake. Whereas on the drive there Berta had been proud to sit beside Wilhelm, on the way home she cowered in the back, her right arm around Little Berta, her left arm around Little Rudolf, both of them cuddled up to their maker, as if they wanted to crawl back inside her and resume their places there.\n\nBerta contemplated the boy's profile. Rudolf drowsed, the corner of his mouth tending softly downward. She didn't need to lean over to know that the other corner of his mouth was likewise tending downward and that two steep wrinkles of displeasure were etched between his brows, and this knowledge triggered the same sensation that always beset her in late autumn, when she would lose herself staring at the barren branches of the trees that writhed so strangely toward the heavens, giving an impression of such extraordinary muteness that even the din of the city seemed to die away, and she would strain her ears, then, listening only for a scream. A scream always bored into Berta like a bark beetle through wood before finally, freed from the clutches of decay and death, it would echo off into the sky. Berta was afraid of late autumn, which she used to call the leafless season, and felt relieved, in a certain way heartened, when the first snowfall came, bringing with it the perennial hope that the bare branches, writhing strangely heavenward, would soon be covered by a blanket of ice.\n\nLittle Rudolf slept more fitfully than Little Berta. Eight-year-old Berta slid into sleep like the Madonna herself\u2014who occupied her proper place over the marital bed in the Schrei bedroom, in a painting with a ponderous gilded frame: the Madonna with the Christ child. The Madonna's face had long troubled Berta. In all the time she'd looked at it, not a single wrinkle had ever crossed the Madonna's face. Why should it be free from even the slightest trace of life's hands, their pounding and molding? The Madonna seemed untouched by those thick fingers, their rolling, pressing, flattening\u2014untouched, that is, by daily life, by the weight of the earth itself, or by the weight of those circumstances that Berta referred to, simply, as life as such.\n\n\"Inwardness. Inwardness is what eludes me. I'm too caught up in the world, too concerned with surfaces...\" Thus Berta often rebuked herself, and frequently and sincerely tormented herself, trying to get at something, the thing she could never quite reach, the truly inward gaze.\n\n# PRIVATE FIRST CLASS RUDOLF\n\nIn February of 1945, Berta experienced a moment of freedom from the weight of things, in particular from that weighty circumstance historians call the Second World War. The thirty-one-year-old Private First Class Rudolf was standing at the window, staring out, his elbows propped on the window frame, his head buried in his hands, which were frozen from the cold, and saying to her, \"You'd be better off out in the countryside.\"\n\nBerta, standing at a prim distance from Rudolf, shrugged her shoulders and replied, \"I can't leave Mama alone, and Mama won't budge from this street. 'I've grown old here, I'm staying here,' she says.\"\n\n\"And Wastl? What about Wastl?\"\n\n\"His last deployment was supposed to be in mid-January, in Grajewo.\"\n\n\"And the twins?\"\n\n\"Nothing. Not from Richard, anyway.\"\n\nPrivate First Class Rudolf stood up straight, tore his eyes from the wreckage of the houses on the opposite side of the street, and turned to face Berta. \"So. Nothing new.\"\n\nBerta shook her head in confirmation.\n\n\"But Karl?\"\n\n\"In the field hospital in Castelfranco, somewhere near Modena. And you?\"\n\n\"Back to Denmark. For the time being.\"\n\n\"So. So.\"\n\n\"And the old gravedigger?\"\n\nBerta shrugged. Rudolf laid his hands on her shoulders. She turned away and her body grew stiff. Rudolf let his hands fall, stood slouching for a moment in front of her, then turned and walked toward the dresser, in the top drawer of which lay his fiddle, which he'd left in Berta's hands before he set off for duty with the words: \"Until the war is over!\"\n\nNow the former teacher turned back to Berta, smiled like a child, rolled up his secret marching orders, and tried to leaven the glum moment with a harmless clich\u00e9, emphatically delivered: \"All this will be over before long!\"\n\nBerta clapped her hands, spun in a circle several times, snapped to attention, and marched back and forth across the small room, a female soldier who didn't take her soldiering especially seriously, whose manner was rather carefree and amused. She hummed the \"Radetzky March\" and let herself fall laughing on the bed, sat up, breathing heavily, giggled again at Rudolf, full of mirth, and then, abruptly, fell to thinking.\n\n\"Any last requests?\" asked Rudolf, courage and discouragement struggling in his eyes. Berta laid an index finger on her lips, thought for a moment, and then called out in a victor's pose: \"I do! I want to hear the 'Aquarelles' by Strauss! Yes. Yes. That's what I want.\"\n\nRudolf's brow furrowed, then he smiled and said a bit curtly, \"If you wish.\"\n\nBerta forgot about the weight of things, the cold in the unheated room; forgot her nineteen-year-old brother, from whom they'd had not one word since March of 1944. All she knew was he'd been stationed to post number 34421-A and had departed from Upper Silesia. At least she could still write to Karl: Wastl was off at a training camp, some fourteen kilometers from Grajewo; Rudolf unexpectedly here on leave, Mother doing well, she herself doing well too; but how would she fill in the blank space that corresponded to Richard?\n\nThe giddy thaw of youth was still on Berta's face, the belief that the present is always the best time to spread your arms wide and embrace the whole world, all at once, as best you can.\n\nThere was a trembling around Rudolf's mouth; the tacked-on second skin of the Private First Class was flaking away, the man inside was coming to life, was urging him toward the woman, and in proximity to her sensuousness and warmth, the waltz with all its frivolity\u2014which he had always dismissed, thinking it sordid and juvenile\u2014took on a new appeal. And when Berta then requested that Rudolf the fiddler play \"The Blue Danube,\" even he began to feel that the Danube was truly so blue and eternal, and that despite everything, a person had a right to be happy, that there was no time for sorrow and gloom, that it was possible to abandon oneself to the soft butterfly's flight of a melody. \"And when the people start to complain \/ Shove a few more in the oven's flame...\"\n\nWith his eternal weakness for muses, this music teacher at a girl's secondary school in the city of Donaublau now discovered, with almost ferocious voluptuousness, the Johann Strauss inside him, and defended this silly Johann Strauss heroically against his own rather nettlesome character, ordinarily so inclined to stubbornness: \"So I'm a Johann Strauss. Fine then. Who cares about those Beethovens or Alban Bergs? They find a hair in every soup. Live and let live, that's what I say. Live and let live!\"\n\nIn Berta's presence, Rudolf had felt strengthened in his struggle against his own irascible character. With a look of solemn inwardness, Berta defended her \"Blue Danube\"; her mouth flapped open and closed, diligently and eagerly, like a duckling waddling in that perfectly blue river. At the same time, her eyes turned naughtily to Rudolf, as if inviting him over, and her singing faltered before she slipped back with renewed intensity into that inwardness betrayed by her child-like voice, and she managed to convince Rudolf that he had never really been a soldier in the German army, that it was only a nightmare he shouldn't take too seriously. He was the music teacher at the girl's secondary school in the city of Donaublau. And he was more than that. In him Johann Strauss had found the most ardent admirer and ally!\n\nWhen Rudolf laid the fiddle back in the dresser and walked over to Berta, who was now sitting on her bed, she no longer turned her head away, and her body had slackened; it felt soft and pliant.\n\nThe next morning, the thirty-three-year-old bachelor with an irascible character, who had so passionately deflowered the twenty-two-year-old neighbor girl Berta Faust, promised himself he would make sure that\u2014as soon as this nightmarish war was over\u2014the broodingly shy Berta with the vast astonishment in her eyes would get a man with Alban Bergian faithfulness and Beethovenesque rigor.\n\n\"Why should I take Strauss from her? To ignore the 'Aquarelles' and the 'Blue Danube' would mean to ignore what brought us together. And what, after all, does the 'Blue Danube' have to do with 'Deutschland \u00dcber Alles'? They're water and fire\u2014and I prefer to stick with water.\"\n\nThese were his thoughts while he sullenly observed the sleeping creature beside him.\n\n# DUTY IS DUTY, SCHNAPPS IS SCHNAPPS\n\nBerta Schrei, with her immense affection for Johann Strauss, could only wonder why it was that this man Wilhelm was so difficult to fit into her life. Looking up at the rearview mirror, she saw Wilhelm's eyes: fixed rigidly, or so it seemed, to the road. She shut her mouth, and when for a moment Wilhelm's eyes met hers, more by chance than design, it wasn't the stare of Wilhelm the returnee she saw\u2014the Wilhelm who looked at her with Rudolf's eyes\u2014but instead of Wilhelm the chauffeur, humming some vague tune as he adjusted the rearview mirror. What Berta refused to say aloud began to teem in her mind with dreadful intensity.\n\n\"When our little girl sleeps, she looks like the Madonna,\" Berta had wanted to say, but did not.\n\nWilhelm's mouth was as tightly sealed as Berta's throughout the drive home. This left time for Berta to indulge her thoughts about Little Berta's resemblance to the Madonna. During dinner, when Berta was finally on the verge of revealing this discovery, which to her felt quite essential, the telephone rang, and Wilhelm, chauffeur and Come-hither-boy, said, \"Very well.\" And, \"Yes, naturally.\" And, \"In a quarter-hour it'll be taken care of.\" And, \"But that goes without saying. It's what I am paid for!\"\n\nIn truth, Wilhelm Schrei had, like Berta, felt something needed to be said about the events of earlier that day; yet the Come-hither-boy and chauffeur to Mueller-Rickenberg\u2014a vocation by which he assured the Schrei family's continued existence, and not too shabbily at that\u2014knew he must set aside his private life as soon as duty called. Berta understood this, too; she no longer hemmed and hawed about it, as she'd done in the old days. She stood up without a word, left for the bedroom, and took Wilhelm's uniform down from its hanger. She was standing with her back to the Madonna. But then she happened to look at it. Forgetting the chauffeur's uniform draped over her arm, she stood rooted to the spot and stared, as though the real reason she had entered the bedroom was to make an especially attentive, conscientious study of the Madonna's face. The likeness really was astounding. When Little Berta slept, she was the perfect image of the Madonna.\n\n\"When she's asleep, you know, she's not so caught up in the world, so concerned with the surface of things. The stamping and molding hands of life, the rolling, pressing, and flattening fingers\u2014the weight of things, life as such, it can't hurt her so long as she's asleep. It's that simple. Sleep startles everything away. Everything and everyone.\"\n\n\"Berta! What's keeping you so long?\"\n\nWilhelm's shout from the bathroom only confirmed what Berta was feeling. And after his bellowing, Little Berta came in to her mother with a wrinkled nose, the corners of her mouth turned contemptuously downward, to ask, harsh and a bit imperious, \"You're not right in the head, are you Mama? Papa needs his suit!\"\n\n\"So. So,\" Berta said, and after a long pause: \"Papa needs his suit? Yes. That may well be.\" She let the suit be dragged off her arm and didn't even wince when Little Berta stuck out her tongue, in all its splendor, as far as possible, before running off to Papa with the suit.\n\nWilhelm, now in his work uniform, tapped Berta on the shoulder, all the while smiling his Wilhelm-smile, ever at the ready for whatever occasion may arise, and the affectionate knowingness that lay behind it brought back to Berta her worries about the weight of things; she nodded in affirmation, saying, \"Well then. You have to hurry if you want to arrive on time. Go now, Wilhelm. Go now. I have a lot to do here.\"\n\n\"When I've returned,\" he said, \"perhaps we'll have the chance to talk...?\"\n\nBerta nodded, cast a thoughtful glance at Wilhelm's chauffeur cap: \"It really is a handsome uniform. It suits you so well,\" she said, and turned toward the dresser. \"Let's not hold each other up,\" she added, and before he left apartment 12 in building 13 on Allerseelengasse, he went into the kitchen to say good-bye to Rudolf and Little Berta with a kiss on the head for each. Berta looked back thoughtfully at the painting of the Madonna, sighed, and turned to the kitchen, where Rudolf and Little Berta were jabbing listlessly at their plates and observed her entrance with sour looks on their faces.\n\n\"The nightgown really suits you,\" said Berta to Little Berta, and thought of the Madonna's blue, undulating cloak in the painting. Little Berta said nothing. Rudolf said nothing. The dinner dragged on; Berta's attempts to brighten Rudolf and Little Berta's faces only clouded them over further.\n\n\"All in all, it was still quite a nice trip, in some ways?\"\n\nNothing.\n\n\"Isn't there anything nice we can say about it? About the new car, for instance? Or the pretty roses? Did you see the pretty roses?\"\n\nNothing.\n\nBerta leaned her head toward Rudolf as though she was hard of hearing. \"What did you say?\" She strained to listen for a moment, then turned to Little Berta and asked her, \"Or was it you who said something?\"\n\nNothing.\n\nAfter turning over in her mind what else there might be for her to say, Berta shrugged and stood up: \"Well then. It's getting late.\"\n\nRudolf nodded in agreement, tilted his head to one side, and appraised his mother with a critical-contemplative gaze.\n\n# I KNOW WHAT'S WHAT\n\nAfter a while, the boy asserted firmly: \"Anyhow, I'm thirteen.\"\n\n\"Thirteen years old, Rudolf. We say: I'm thirteen years old.\"\n\n\"And the 13th is when you tumbled into the world.\"\n\n\"I don't believe it can be said that I tumbled into the world. I arrived in the world. That is proper. That much certainly can be said. And not just on the 13th, but on the 13th of January, 1923.\"\n\n\"You can't swim.\"\n\n\"That can also be said.\"\n\n\"It can't not be said,\" Rudolf replied sardonically and scratched the back of his head. \"And I can't swim! And that one over there\u2014she can't swim either!\" The boy pointed at Little Berta, who was wriggling back and forth in her chair.\n\n\"We don't raise our voices, Rudolf. One doesn't raise one's voice. People who raise their voices often have something to hide. People who raise their voices are often in the wrong and that's why they yell so loudly. And that one there, that one there has a name. Her name is Berta. Do you not know that?\"\n\n\"The chauffeur can swim though.\"\n\n\"You mean your papa?\"\n\n\"No. The chauffeur.\"\n\n\"The chauffeur didn't come into the world on the 13th!\" Little Berta shouted triumphantly, scratched her chin, went on thinking for a moment, and then reached a conclusion that catapulted her from her seat: \"And you neither! And me neither! Only that one there! She's the only 'thirteen' person!\"\n\nBerta shrugged and said: \"So. So.\"\n\nWith contempt, Rudolf eyed his sister, some five years younger, and said, \"Anyway, I'm a Friday child, I tumbled into the world two months early, and my father is the music teacher without a head and yours is the chauffeur who still has his.\"\n\n\"Rudolf! Where did you get all that from?\"\n\n\"I know what's what, and anyway Aunt Wilhelmine says it too: he's Rudolf through and through, that's what she says about me\u2014maybe not as handsome or as clever, but still: Rudolf through and through.\" And Rudolf stared straight ahead. \"Can that be said?\" he asked trenchantly.\n\n\"Well now. That's all very much a matter of perspective.\"\n\n\"But I won't become any music teacher at a girl's secondary school. At best, if I'm lucky, I'll be a street sweeper or a coal shoveler.\"\n\n\"Rudolf!\" Berta sat down again and stared at her thirteen-year-old son as if he were indeed a bad omen. \"There are many reasons you shouldn't say things like that,\" Berta countered. \"And bear in mind you have weak arms...\"\n\n\"And two left hands.\"\n\n\"I wouldn't say that,\" Berta objected.\n\n\"But it's true. And also I have lousy luck\u2014so I won't even be a street sweeper or a coal shoveler.\"\n\n\"Maybe... I mean, it's conceivable... maybe you'll become a chauffeur?\" Berta asked, looking at her son full of expectation. Rudolf waved her off and said, \"You're not right in the head.\"\n\n\"All right then. Your objection stands. Driving school is still school, I suppose. And school doesn't quite suit you...\"\n\n\"Right! And I can't play any instrument the way a music teacher could, and I also can't sing, and I can't catch a ball.\"\n\n\"Well then.\"\n\n\"As soon as I fall down, my nose starts bleeding.\"\n\n\"Perhaps.\"\n\n\"And I fall down all the time.\"\n\n\"That may well be.\"\n\n\"My nose bleeds a lot, and then I cry.\"\n\n\"That much is evident!\"\n\n\"And I go to a special school.\"\n\n\"Well, yes.\"\n\n\"I'm dumb.\"\n\n\"No! That's not true! I mean, there are nicer ways of putting it. No, no, dumb you are not.\"\n\n\"I'm to be taken out of special school after the fifth year. Now, am I dumb or am I not?\"\n\n\"Well, if you want to take such a superficial perspective. I mean... One shouldn't do that\u2014you have to look at the situation more deeply, more carefully. After that, you may well see things in a more reasonable light. Don't you think?\"\n\n\"I was born too early.\"\n\n\"Yes, that's true. Yes.\"\n\n\"I'm ugly.\"\n\n\"Rudolf!\"\n\n\"No one likes me. I'm annoying.\"\n\n\"No. No. It's not like that. No. No. That is completely, completely wrong. I, for example, like you, and Papa likes you too.\"\n\n\"Papa is dead. He can't like me. I have no papa. I don't want another one.\"\n\nLittle Berta patted her brother's hand and said, \"I'm not dead. I like you too.\"\n\nRudolf looked scornfully from Little Berta to their mother and back: \"And who likes you? What do you think, Mama? Who likes her!\" And Rudolf pointed at Little Berta, then at his mother, who slapped his index finger away and declared, \"We do not point at people. That's something we do not do\u2014no, that is something we absolutely do not do.\"\n\n\"Who then, Mama? Tell me. And who likes you? Aunt Wilhelmine? The chauffeur? I think they're happy when they can get a little peace from your constant blabbering.\"\n\nRudolf stood and paced moodily back and forth several times in the kitchen, then turned finally toward the door and marched off sulking to bed.\n\nLittle Berta looked at her mother, full of reproach, threw back her head, stomped one foot, and said, \"You're not right in the head!\" And she added, her features twisting into a grimace: \"Everyone knows it. Aunt Wilhelmine knows it, and Papa knows it, and me, I know it too!\" Little Berta stuck her tongue out at her mother, and the hate from her eyes collided with Berta's despair. Little Berta threw the kitchen door open and slammed it closed over and over. And Berta winced each time, as though someone were lashing her with a whip. This loud, portentous exit served its purpose: Berta did not dare follow her children into the bedroom.\n\n# THEY WAIT FOR WILHELM\n\nThat evening Berta decided, without a single \"on-the-one-hand\" or \"on-the-other,\" no longer to dole out further lessons to her children about the weight of things. Instead, she would try to introduce them to life as such through stories she would invent more or less from scratch and tell them night after night, and Berta Schrei was a good storyteller, at least at these moments. She would tell these stories of her own invention\u2014they hardly ever had happy endings\u2014and then she would wait an hour, two hours, three. After the third she felt secure in her hope that the weight of things had given up its pursuit of her children, and that Rudolf and Little Berta, ferried over now to the realm of dreams, were safe and sound, far from the grasp of the wrenching and molding hands of life.\n\n\"That's all it is. The three of us are just too caught up in the world, too superficial, somehow. What we lack is inwardness, a belief in ourselves.\"\n\nBerta could hear the wrathful voice of Wilhelmine: \"Berta! What's to become of your children? They'll end up losing their heads just like your doomed Rudolf!\"\n\nBerta looked thoughtfully at the clothesline that was stretched from the window frame to a hook that Wilhelm had nailed in over the stove, where their wet clothes were now hanging.\n\n\"Wilhelmine. Yes. Yes,\" Berta said, and wiped her eyes with the back of her hand. For Berta, Wilhelmine was the embodiment of life as such, the way it spoke to her, the way it prophesied a dismal end for her, for her and her children.\n\n\"Somehow we just do everything wrong. Somehow we just don't fit,\" Berta had thought, crouched down with one arm around Little Berta, the other around Rudolf, sitting between the children in the new car, while Wilhelm, the chauffeur, said nothing, his eyes fixed rigidly on the road.\n\n\"He knows it. Yes. Of course. Everyone knows it.\"\n\nLater, Berta felt the need to verify the intuitions that had struck her on the drive home.\n\nShe'd crept into the bedroom, knelt down by Little Berta's bed, and in the soft half-light of the lamp compared Little Berta to the Madonna from the painting. She hadn't been wrong; even scrutinizing the girl so closely, even with such careful consideration, the resemblance was still there.\n\n\"When Berta sleeps, she looks just like our Madonna. Not just outwardly, not just superficially. It is the inward gaze of sleep. It's what silence does for her,\" she murmured to herself.\n\nAround midnight Berta was sitting on her bed; upright; a rigid pillar; with a strange gaze fixed rigidly on her daughter's face.\n\n\"If only Little Berta could always sleep this way, she would look like our Madonna forever. If only she could always sleep this way\u2014if.\" In her mind, Berta worked and reworked this thought, which sprang from the basic resemblance she had noticed in her child, but she neither spoke of it aloud nor even murmured it to herself. It was just a thought; nothing more and nothing less.\n\nShe waited up for the chauffeur Wilhelm until four in the morning; sitting upright in bed; rigid and with that same strange gaze fixed on her daughter's face.\n\nAt around four-thirty in the morning, Berta finally fell asleep, only to be jolted awake by Wilhelm: \"Stop that screaming! You'll wake the children! Berta! Stop that screaming!\"\n\nIt was too late. Rudolf and Little Berta sat straight up in their beds, terrified, then shot sinister glances at their mother, who said to them, finally shaken out of her sleep, \"It's nothing. Everything's fine. I was just dreaming.\"\n\nWilhelm shook his head, more dumbfounded than annoyed, lay back down, said, \"What were you dreaming about, then?\" and immediately turned away from her onto his side: \"I'm tired. Wake me around eleven. We're driving to Felsenstein.\"\n\n\"To Felsenstein?\"\n\nWilhelm mumbled something not even he could understand, and soon afterward the regular course of his breathing made clear that he had fallen asleep once more.\n\n\"So. So,\" Berta Schrei said, \"to Felsenstein,\" then looked perplexed from Wilhelm to Little Berta, from Little Berta to Rudolf, and from Rudolf to Little Berta once more.\n\n\"The resemblance really is astonishing. I need to think about it more,\" Berta said to herself, then bent over Wilhelm, and with a push of the button on the lamp on the night table, draped the room in darkness.\n\n\"This is no coincidence. And the dream I had is also no coincidence.\"\n\n# A CHARMINGLY UNASSUMING CREATURE INDEED\n\nBerta Schrei, who saw in her dreams allusions to life as such, and therefore found them worth taking to heart, lay on her back in bed, her eyes turned toward the window, and decided it would be better to stay awake. She looked at the blinds, remembered how Wilhelm had carved the wooden slats himself in the workshop of his colleague Ferdinand Wolf, and how Wilhelmine had sewn the fabric holding them together.\n\nWhen the work was over, Berta had offered a shot of schnapps to her laborers, and they all stood in the bedroom to marvel at this masterpiece: Ferdinand Wolf, Wilhelmine and Wilhelm, Rudolf and Little Berta.\n\n\"Really. It's beyond belief. How did you ever do it?\" Berta asked, and stared at Wilhelmine, astonished. Wilhelmine responded to her astonishment with pity, saying, \"Well, it's nothing. You take some old fabric out of your dresser, you reuse it, andsoonandsoforth. Anyone can do it.\"\n\nAt that Wilhelm nodded, stroked Berta's cheek, and said, \"Don't rack your brains over it. The important thing is we have our blinds now. What do you say, Ferdinand?\"\n\nFerdinand held out his shot glass to Berta a fourth time, let it be filled with good schnapps, drained it in one go, and declared with admiration, \"I never knew a chauffeur to understand anything other than motors, speedometers, and the joy of steering passengers around the countryside\u2014but I can't say I'm surprised. Yes, I have long suspected there was more to you than that! And now I know for certain.\"\n\nAfter delivering this little speech, Ferdinand felt a most urgent need for a fifth shot, and Berta was amazed how quickly Wilhelmine noticed this. With a jerk, the schnapps bottle was wrenched from Berta's hands, then passed back, and the glass was emptied straightaway once more.\n\n\"Berta. Keep your eyes open,\" said Wilhelmine in a slightly reproachful tone, shaking her head. \"You're always off somewhere lost in your thoughts.\"\n\nBerta clapped her hand over her mouth; she cried out, \"Oh, forgive me, Mr. Wolf! I was just stunned at the sight of so much made from so little.\"\n\nFerdinand Wolf nodded knowingly and said, \"So it is, Mrs. Schrei my dear, so it is. If you keep your eyes open and have a bit of common sense, the improbable becomes probable.\" Rudolf yanked at his mother's skirt, she bent down to him, and he whispered in her ear, \"Help me, Mama, help me!\" He pointed to the floor where a small puddle was forming. Berta took the boy by the hand, excused herself, and tried to cover up Rudolf's shameful mishap insofar as was possible. She said, \"Our able craftsmen will have to excuse us for a moment. I imagine they may enjoy a bit of time to themselves anyhow, and now they don't have to worry about our meddling.\"\n\nThe able craftsmen Ferdinand and Wilhelm broke out into hearty laughter. Wilhelm clapped his Berta good-naturedly on the shoulder and Ferdinand Wolf said, \"What a charmingly unassuming creature.\"\n\nBut Wilhelmine threw her hands up in horror, pointed toward the puddle, and cried out, \"Berta! What an absolute mess! When will you finally teach your boy how to use the toilet?\"\n\nLittle Berta wrinkled her nose and said, \"It wasn't me, Uncle Wolf, that doesn't happen to me anymore!\" Uncle Wolf cleared his throat, stroked Little Berta's cheek, and Wilhelmine said, \"Rudolf! Take your little sister as an example!\"\n\nBerta fled to the bedroom with Rudolf, and Wilhelmine waddled into the kitchen, returning with a rag to wipe up the calamity. \"It's repulsive! Repulsive! He really is a bit old for this sort of thing!\"\n\nFerdinand Wolf nodded knowingly and declared, \"Yes. Yes. Raising children. Hard work. It's not so easy. I know. You have to be resourceful. You really do. It's hard work indeed.\"\n\nWilhelm looked at his colleague Ferdinand Wolf, bursting with gratitude, and Wilhelmine said irately, \"That doesn't change the fact that it is something she has to deal with!\" And with a stern, sideways glance at Ferdinand Wolf: \"The hard work, that is.\"\n\n# \"I WAS JUST DREAMING...\"\n\nRudolf hung on the cross. Around him stood scattered groups of people, all of whom pinned him in their withering gaze. \"Where's my mother?\" Rudolf cried from the cross.\n\n\"Your mother is in her grave,\" a faceless voice answered him from the multitude, and Berta became conscious of herself lying under the earth, a few meters from the cross. She tried to shift her coffin lid and to cry out with all the force of her love:\n\n\"Rudolf! I'm still alive! I'm coming! Wait for me! Be patient! I'll get you down from there! Rudolf!\" The dreaming Berta observed this other Berta, as powerless and voiceless as a corpse in her fruitless struggle.\n\n\"Where's my mother?\" Rudolf cried a second time and looked down onto the nearby hillock where there was no cross and no flowers, only shoveled up dirt, as on a molehill. A figure without a face, a torso on two legs, pulled away from the group and said, \"There she lies. Let her rest. It'll be over soon. You'll understand when the sun has reached its zenith.\"\n\nAfter the headless figure had spoken, the scattered groups merged into a single human mass. All of them had their heads at their sides, holding them in their hands and resting them on their hips. Each head was the same as the others. And all the heads resembled helmets.\n\n\"What did I ever do to you?\" Rudolf cried. And since no one answered him: \"Why am I hanging here on this cross? Why?\"\n\nOne headless creature after another stepped forward out of the mass of people. The voices of women, girls, men, and boys drowned out one another in turn:\n\n\"You can't catch a ball.\"\n\n\"You can't play an instrument.\"\n\n\"You can't even sing.\"\n\n\"You always fall down.\"\n\n\"Your nose bleeds.\"\n\n\"You have two left hands.\"\n\n\"You can't do your sums.\"\n\n\"You can't even remember the Ten Commandments.\"\n\n\"You can't write on your own.\"\n\n\"You can't even copy things down.\"\n\n\"All you can draw are animals, and houses, just barely\u2014you can't draw people with two hands, ten fingers, two feet, and a head. Your people have five eyes and monstrous mouths. Your people have seven heads or no head, twenty-three fingers or none at all.\"\n\n\"You can't catch frogs.\"\n\n\"You can't even make it to the bathroom when you need to go.\"\n\n\"You're a bed-wetter.\"\n\n\"You can't throw a punch.\"\n\n\"You don't know how to fight.\"\n\n\"You're a weakling.\"\n\n\"You always have diarrhea.\"\n\n\"You bite your nails.\"\n\n\"You stutter when the teacher asks where you're from.\"\n\n\"You're an idiot.\"\n\n\"You can't swim.\"\n\n\"You're a crybaby.\"\n\n\"You grind your teeth at night.\"\n\nRudolf yelled down from the cross, \"But I'm not a bad person!\"\n\nAnd the headless ones answered him in a chorus, \"You are good for nothing.\"\n\nThe people put on their helmets, now they had heads again, only one still stood there headless, then stepped forward and pointed at the sun, saying, \"It is done. The sun is at its zenith,\" and then it threw its helmet on Berta's grave, and a tremor bore through Rudolf's body like a bark beetle through wood. But the scream that Berta always waited for in the leafless season, that one definitive scream, never came.\n\nAnd Berta Schrei shouted for her son, and a ghastly thousandfold echo rang out of the earth, and the mass of people, already dissolving, scattered in all directions, with each person running for dear life. Voiceless Berta shouted down the hurricane-like storm that seemed to have driven, in a matter of seconds, grayish black banks of clouds from all four corners of the earth to gather above the cross where Rudolf was hanging.\n\nWhen the clouds burst and the rain began to lash down, everyone had already fled into their homes and hovels, and Berta's cry told them that Rudolf's life had slipped away. His head hung down, and the hard rain battered shut the lid of Berta's coffin, which had finally, momentarily, come open. Berta knew she had died before she'd been able to shelter Rudolf from the terrible weight of things.\n\n# HUNTING SEASON IS HUNTING SEASON\n\nThe chauffeur and Come-hither-boy Wilhelm Schrei telephoned to Donaublau from Felsenstein. In one of the apartments behind the filth-gray fa\u00e7ades, which gave the narrow alley the aspect of a ravine, Berta Schrei, mother of two children, housewife, and helpmeet, grabbed the receiver.\n\n\"So. So. As you say. Hunting season is hunting season. I understand. Of course, Wilhelm. I've always understood that. Wilhelm, are you unhappy with me?\"\n\nAnd.\n\n\"Yes. As you say. Everything's in order here. That's right.\"\n\nAnd.\n\n\"If four weeks have already passed, then two weeks will certainly pass too.\"\n\nAnd.\n\n\"So. So. But there's no need for that. Everything here is exactly the same. Why should you worry? All that's changed is Berta and Rudolf are back in school. As you say. That's right.\"\n\nAnd.\n\n\"No. Nothing new, it's just that I've gotten a letter asking me to go to the school. On account of our little girl.\"\n\nAnd.\n\n\"Why shouldn't the letter mean something good? Isn't Berta a quick learner?\"\n\nAnd.\n\n\"But of course I'm going. Why should I get worked up about it?\"\n\nAnd.\n\n\"I wanted to tell you something else. The resemblance, Wilhelm, the resemblance. What's that? You have to go? So. So. No. It's nothing special. Of course. You shouldn't keep him waiting. If they have orders for you, then they have orders for you. Of course. As you say.\"\n\nAnd.\n\n\"I know. Yes. Yes.\"\n\nAnd Berta laid the receiver back down in the cradle as tears ran down her face.\n\nWilhelm's three days in Felsenstein had turned into five days, five days into seven, and one week into two, then three, then four, and now six.\n\nFive hundred kilometers away, the chauffeur and Come-hither-boy Wilhelm Schrei was being kept exceedingly busy. Naturally, hunting season began much earlier for his employer than it did for all the other dignitaries summoned to the Mueller-Rickenberg hunting lodge, many of whom Wilhelm transported there himself. Moreover, hunting season imposed a number of supplementary duties. And, of course, Wilhelm Schrei was thoroughly devoted to his profession.\n\n# THE DRAGGING OF THE SCHOOLBAGS\n\nBerta, who slept badly whenever Wilhelm was off being a chauffeur and Come-hither-boy, had grown used to Rudolf shuffling off glumly each day on his walk to school, and then returning home disheartened. She saw no point in sharing this information with Wilhelm, in treating it like news, since the truth was that every year, as soon as the summer holidays were over, Rudolf started back up with his skulking.\n\nWhat did seem worthy of attention, at least at first, was Little Berta's dragging her schoolbag behind her by its strap. She seemed more and more like Rudolf every day. The children's mother always knew exactly when they were headed home from school: she could hear their bags being dragged along the street, a peculiar noise that bore into her ears like some sort of grim portent. Whenever she heard it, she would have to clutch the table tight and hold her breath to keep from losing her composure. Little Berta would then open the apartment door, skulk past her mother into the bedroom, and sit down in a corner to stare off into space.\n\nBerta Schrei would wipe away the trail the schoolbag had left behind on the floor.\n\nOnce it had been only Rudolf who dragged his schoolbag, now it was Little Berta too, and so inevitably the floor would be soiled anew each day, the rug pushed out of place. No sooner had Berta wiped away her daughter's mess than Rudolf would drag himself in, skulk past, just as glum and silent, and sit in a corner of the bedroom staring into space, unruffled by his sister's presence there.\n\nBut now even Little Berta's retreat into the bedroom was no longer news.\n\nOnce Berta had wiped away the traces left behind by Rudolf's schoolbag, she went into the bedroom herself.\n\n\"So. So,\" she said. \"That rain just won't let up.\"\n\nNothing.\n\n\"Papa called.\"\n\nNothing.\n\n\"He's doing well.\"\n\nNothing.\n\n\"He's very busy.\"\n\nNothing.\n\n\"Would you like to come to the table for lunch?\"\n\nNothing.\n\n\"Surely you're hungry?\"\n\nNothing.\n\n\"Well now. I just mean. It's time to eat.\"\n\nLittle Berta raised her head, glared past her mother toward Rudolf, and said, \"She just means. She always just means. I mean, I mean. She really isn't right in the head.\"\n\nRudolf shrugged and kept quiet.\n\n\"But a person has to eat, no? I mean, isn't that so?\"\n\nRudolf glared past his mother to Little Berta and said, \"You're right.\"\n\nLittle Berta shrugged and kept quiet.\n\nBerta said, \"So. So.\" And after a long pause for thought: \"Even if I'm not right in the head,\" and she tapped her left temple, \"am I not still your mother? Am I not always your mother?\"\n\nLittle Berta replied drily, without changing her expression, \"Unfortunately.\"\n\nRudolf shrugged.\n\n\"I got a letter,\" Berta said.\n\nRudolf looked up at his mother for a moment and then went back to staring blankly ahead; he droned, \"You can save yourself the trip. The teacher'll just tell you she's dumb,\" and he pointed at Little Berta. \"That's the whole story.\"\n\n\"What?\" asked Berta. \"Who's supposed to be dumb?\"\n\n\"That one there! Who else?\" And Rudolf pointed at Little Berta, who, like him, was glowering straight ahead.\n\n\"Rudolf, what are you talking about? Our Berta is a quick learner. Isn't that right, Berta? Aren't you a quick learner?\"\n\nLittle Berta pursed her lips and said, \"What can you expect, with a mother like that.\"\n\n\"Where did you learn to speak this way?\"\n\n\"From me,\" Rudolf answered and scratched the back of his head. \"And I heard it from Aunt Wilhelmine.\"\n\n\"So. So,\" Berta said, and considered. \"But I was always a quick learner. Straight As. Yes. Yes. Look now! All As on my transcript. You don't believe me? Ask Aunt Wilhelmine then. You're being unfair. I was never dumb. Never!\"\n\n\"So. So,\" said Little Berta, and grinned maliciously; but then she hunched over and shivered with an overwhelming sense of defeat. Rudolf rose up from his corner, walked stiffly over to Little Berta, patted her shoulders, awkwardly caressed her hair, looked down at his sister with the expert eye of a specialist\u2014for he was a specialist in the matter of defeat\u2014and said, \"Don't take it like that. You get used to it.\" A moment later, the little boy, the specialist in matters of defeat, crashed past his mother and out of the bedroom. He ran through the living room and off down the hallway.\n\nHe was truly frantic. At that time of year, diarrhea permeated his day-to-day life as breathing permeates prayer.\n\n# THE CONVERSATION WITH THE TEACHER\n\nThe conversation with the teacher ended the same as the one with the principal. But Berta still couldn't understand. She stood gawking at the teacher as if she'd been clubbed over the head, staring at her in the hope that in the next moment, or the moment after the next, at least, the teacher would offer some consolatory words about her and the little girl she had brought into the world.\n\n\"My dear Mrs. Schrei. Maybe Berta really is a quick learner. However, in view of the circumstances. I have forty children in my class. I can't just concentrate on one individually until, sooner or later, most likely later, she's ready to show me she has at least some notion of what one plus one might equal...\"\n\n\"But Little Berta knows that!\" her mother countered, and the teacher shrugged ruefully.\n\n\"But she doesn't know it when I ask her. And I can't know what's going on in her head. She doesn't even know the name of the city where she was born, the city where she lives. I can't worry myself constantly over what's going on in her mind. Just try and imagine, worrying constantly over forty little minds. It's impossible! Utterly impossible! Now, I know this isn't exactly an ideal solution. But how can I get through my material if I have to worry what's going on in each of forty little minds? It simply can't be done.\"\n\n\"Give her another year. I know Rudolf was a bit of a bad student from the beginning. But Berta. Isn't Berta a quick learner? Hasn't it always been that way? Why should things be different now?\"\n\n\"My dear Mrs. Schrei. You're giving this little transfer too much significance. What's really happened, after all? Berta's changing classes. That's it. Try to look at the whole thing as a kind of organizational matter. It really is nothing. It says nothing about Berta's future development. She may yet grow up to be a Madame Curie!\"\n\n\"No. I just can't accept it. You can't turn my daughter into a special student. That's something you just cannot do! I always got straight As, Miss, and my daughter will also get nothing but straight As. That I promise you! On my life: I will make sure of it. Just one more year. A year!\"\n\n\"Calm down, Mrs. Schrei. Please, don't get so worked up...\"\n\n\"My Berta is going to be placed in a special school! My Berta is supposed to be an idiot! And I should calm down?\"\n\nThe teacher tried to console Berta Schrei with sympathetic words as Berta walked out of the classroom, waving her hands.\n\nIn a trance, she walked in the direction of Allerseelengasse 13, bravely choking back her tears. But hardly had she taken the first step onto the sidewalk of her street than the shame of it all shook her and the defeat streamed from her eyes, and it was more blind than seeing that she stumbled along Allerseelengasse, till she made it to building 13.\n\n# THE USUAL, WILHELM, THE USUAL\n\nAfter her conversations with the principal and the teacher, Berta stopped upbraiding her daughter. She no longer bothered to remind Little Berta to cinch her schoolbag tight or to carry it up on her back.\n\nWhenever Wilhelm called from Felsenstein to find out what was new, Berta told him that everything was the same.\n\nShe didn't mention how Little Berta was skulking around the house like Rudolf, dragging her schoolbag like some heavy, nearly immovable burden, and she likewise neglected to mention their daughter's transfer to a special school.\n\nWhen Wilhelm asked whether she had gone to see the teacher and, if so, what the latter had wished to tell her, Berta replied, \"The usual, Wilhelm, the usual.\" She also failed to mention Little Berta's transformation from conscientious and motivated to slovenly and utterly indifferent.\n\nThere was a time when, if Little Berta's fountain pen had slipped from her hand, or if she'd dripped a small blot of ink in her notebook, she would rush to rewrite its entire contents in a fresh one, and very neatly at that. Her once precise, rounded, and actually quite beautifully formed letters seemed now to jut up and down every which way across the page. One would lean to the left, another to the right; there wasn't a single example of a letter keeping within its prescribed lines. Little Berta's writing trailed upward, downward, ran aground on ink blot after ink blot, and she would say laconically, without fail, about each assignment she worked on: \"I can't do it.\"\n\nNot once did she manage to write out a text correctly, let alone form a complete sentence on her own. Little Berta and Rudolf vied for the title of most forgetful, recollected neither the Ten Commandments nor the name of their hometown; they even swore they didn't know the name of the street where they played. If Berta lost patience, refused to believe her children's confusion, and shouted out in desperation, \"All right, out with it! How much is one plus one? You must at least still know that!\" still she would get no more of an answer than the teacher had. Shrugged shoulders. Silence. Perhaps a meager \"I don't know.\"\n\nThe result was that Berta herself came to doubt her children's comprehension, and in spite of Wilhelm's constant questions, was reduced to simply repeating: \"Everything's the same as always. What do you want me to say, Wilhelm? Everything's as it was. That's right. Yes. As you say.\"\n\n# ANOTHER DREAM, WHICH GAVE HER MUCH TO THINK ABOUT\n\nThe night after the conversation with the teacher, Berta Schrei had another dream, which gave her much to think about. Little Berta was sitting at home with Rudolf; they were doing their homework. When the doorbell rang, Little Berta ran to the door, opened it, and a corpse was flung in at her feet, \"The corpse! The corpse!\" Little Berta screamed, \"Rudolf, the corpse is here!\" She walked in a circle around the corpse, and pointed at it as Rudolf ran over. The two children stared at each other, slapping their thighs with laughter, then dragged the corpse by its legs into the living room with their combined strength. They knelt down by it, stared at it, winked at one another, stood up, danced a \"Ring Around the Rosie,\" then knelt back down beside the corpse again. Hours passed, days, until hunger drove them out of the apartment; they walked from one house door to the other. Beggar children, imbeciles, with mindless hunger shining from their eyes.\n\nIn the dream Berta saw herself lying there, the corpse in the living room, saw her children on their journey, saw an edict tacked up to the notice board of the building: \"Beggars, imbeciles, and maniacs are to be turned away from the door.\" She saw her children at the trashcans, how they struggled with stray dogs over a bone. Once they'd rifled through the trashcans and waste heaps of Donaublau and crammed themselves full of whatever they found there, sometimes confusing an old scrap of fabric with a bit of food, they turned back to Allerseelengasse, saw a new edict tacked up on the notice board of building 13, collided with one another, giggled, danced a \"Ring Around the Rosie,\" and assured each other that they couldn't read the edict's words: \"The corpse in question is not to be buried.\"\n\nAgain and again, they danced their \"Ring Around the Rosie\" in front of the notice board, then with great hooting and giggling, madness in their eyes, the trashcan and waste-heap odors and scraps clinging to their bodies, they stormed up the stairs, once more to wait for hours, for days beside the corpse, until hunger drove them back out. They wandered like ghosts through the city of Donaublau, decrepit, dingy, feral creatures that everyone ignored, and in the face of whom everyone fled.\n\nHand in hand they skulked, stumbled, ran from door to door, from trashcan to trashcan, from waste heap to waste heap, later returning to their mother's corpse, to slap their thighs with laughter, to yowl out dirges, to cry, to dance in a strangely disjointed manner, pausing once in a while to kneel down by the corpse, then staring off into space. And the dreaming Berta found it odd that no one came to bury her, to pull the children away from their deathwatch, or at least to beat them to death.\n\n\"Life as such. Life as such. Did you deny it?\"\n\n\"Life as such. Life as such. Did you steal it?\"\n\n\"The weight of things. The weight of things. Did you forget it?\"\n\n\"The weight of things. The weight of things. Did you dream it?\"\n\nThe dreaming Berta wanted to come to her corpse's aid, to close Rudolf's and Little Berta's mouths, but scream as her body might, it was voiceless. The strain only made something crack in her head. The children could see clearly how the corpse had begun bleeding from its eyes, nose, and ears. They pointed at it with their spindly fingers, giggled, and chanted their song more ghoulishly and boisterously than ever.\n\n\"Life as such. Life as such. Did you deny it?\"\n\n\"Life as such. Life as such. Did you steal it?\"\n\n\"The weight of things. The weight of things. Did you forget it?\"\n\n\"The weight of things. The weight of things. Did you dream it?\"\n\nAnd at that they returned to dancing in their strangely disjointed manner.\n\n\"Soon they'll come to lock away the corpse! Soon they'll come and seal up the apartment! Rudolf! Berta! This living room will become a coffin! Rudolf! Berta! This apartment will become a grave!\"\n\nWilhelm came, along with the work crew who'd been employed to seal off apartment 12 in building 13 on Allerseelengasse, to keep the stench of rot inside.\n\nWilhelm signaled for Berta's brood to go with him; they shook their heads no; Wilhelm thought a while and decided: \"When I return from Felsenstein, I'll pick you up. In the meantime, Berta will look after you.\"\n\nThe voiceless Berta screamed, \"I'm not Berta! Wilhelm! I'm only the corpse! The corpse in question is me! Take the children with you!\"\n\nThe children were nailed inside the apartment with the corpse in question. As their tomb was quite generously proportioned, with numerous burial chambers, it didn't occur to the children at first that they'd been buried alive. With time, though, the madness of hunger began to ravage the children's brains; they began to circle the corpse; the madness of hunger tore their jaws open wide. For long days their hunger encouraged them, before they finally wedged their spindly fingers into their mother's rotting flesh and gnawed down to her bones.\n\nThe children's skin slowly dried out until they looked more like prunes than people, walked crablike on all fours, howled indecipherable sounds and pounced on one another. When Wilhelm finally knocked at the door, they could no longer speak or stand upright. That noise, the uproar from inside the tomb, the scraping and the scratching, couldn't possibly come from his children. Dumbfounded, Wilhelm asked himself whether a corpse could become restless in its grave, but he was probably just exhausted, he thought, and so it was best to assume he hadn't actually heard anything.\n\n\"Odd. I could have sworn they were in there,\" he mused, shaking his head, consoled by the thought that there was, after all, far more to the world than an average citizen like himself could ever imagine.\n\nHe walked downstairs, exited the building, more bemused than unsettled, more soothed than perturbed, more concerned with his duties as chauffeur and Come-hither-boy than with the strange goings-on in the thirteenth building on a small street in the city of Donaublau.\n\n# WILHELM MAY SOON COME BACK\n\nNightmares of this sort hacked away at Berta Schrei's evenings, poisoning her sleep and weakening her defenses against the weight of things, making it harder and harder to make peace somehow with life as such and survive for the time that remained until Wilhelm's return from Felsenstein. She chose not to tell Wilhelm about her nightmares over the phone; in view of their regularity, they were hardly news. And what Wilhelm wanted to know was news, not her day-to-day.\n\nAnd though she had been yearning for weeks now for Wilhelm's return, with ever-mounting intensity, yet she had also thought it better not to mention her yearning.\n\nInstead, she set down the receiver each time hoping quietly that Wilhelm might come home soon, and that he would be able to explain to Little Berta how her transfer at school was of hardly any significance in her life to come, and to explain to her\u2014to Berta\u2014how to help her children put up with their benighted existence, and how she herself might maneuver through the leafless season that was ineluctably approaching.\n\nIf only the leafless season were already past, and the children, especially Rudolf, were cheerful, looking forward to the Christmas festivities, and the boy could take the winter break to get used to the diarrhea that tormented him each autumn when school started up, to get used to his stuttering, to his invariable I don't know when standing in front of the teacher, and to his bedwetting, and especially to his headaches.\n\nThen, perhaps, Rudolf's restless nights would grow gradually more serene, his tooth gnashing would abate, and he wouldn't cry so often in his sleep.\n\n# THERE ONCE WERE A PRINCE AND A PRINCESS\n\nAn hour came when Berta gave in, when she stopped sharing important life-lessons with her offspring, when she seized on an opportunity she'd long ignored. One morning, noticing the children were loath to get out of bed and put on their clothes, she said nothing, but called the teacher, said the children were sick, and sat down by them on the bed. After three days of this, Berta's new life with her children was settled.\n\nThe children would stay in bed in the mornings, Berta would bring them breakfast, and once their appetite had been sated, all three would begin to sing. If a false note rang out, Berta didn't reproach them, not even once; and, anyway, because they were her children, and she herself had a knack for music, they had a special flair for microtones, semitones, and other obscure intervals. They sang with all their might, and Rudolf and Little Berta managed to read their song lyrics from the songbook without difficulty. They began their singing hour with \"Danube so blue, so blue...\" continued with \"There once were a prince and a princess...\" and finished with \"I know not, what does it mean...\" then started over with \"Danube so blue, so blue.\"\n\nAfter the singing hour they joked and played around. There was no homework, no Fs, no letters home, no black-and-blue marks on their bodies from fighting or gymnastics, no jabs and no jeers, no \"ifs\" and \"buts,\" no \"on-the-one-hands\" or \"on-the-others,\" and no warnings about behaving better and minding their manners.\n\nBerta told her stories to a rapt audience while she cooked, ironed, and put things in order. Rudolf and Little Berta never left their mother's side; they circled around her, every single day, caressed her, comforted her when cleaning up seemed to drag on, when she broke a glass while washing up, if she burned the potatoes till they were black rock-hard clumps, if there were scorch marks on the hand towels or nightshirts after she ironed. She might interrupt the Sisyphean task of putting things in order to look for the Faust family photo album or her collection of letters from the front: from her father, her brothers, or Big Rudolf. At last she would come to the solemn decision to put everything in order once and for all.\n\nDespite, or perhaps precisely because of her conscientiousness, she never made much progress in this final task. The children told her it was just too hard to pick out one fixed place for every single object in their apartment, but they also tried to help their mother and not shirk their own responsibility. Dreadful to think how many possible places there were for each and every object in the apartment! Berta could order things this way or that, or some other way altogether. In the end, nothing came of it but pointless discussion, a kind of defeated meditation on the weight of things, useless both to her and her offspring.\n\n# JIBBER-JABBER! WAR!\n\n\"Yes. Yes,\" Berta recounted to her offspring for the umpteenth time. \"That's what it was like then. My mother always used to say, 'Not for the life of me, Berta. If there's a direct hit on building 13, then that is the way God wants it. I'm too old to start over again somewhere else anyhow. And suppose building 13 isn't hit, suppose I was to move away to exactly the place where I would get hit? It's not logical. That can't be what God wants. What am I getting at? I'm saying you shouldn't uproot an old tree and plant it somewhere else. That's no good for it. No. I'm staying here.' Me, I said to my mother, Mama, maybe you're right. But this is war.\"\n\nBerta smiled. \"And do you know what my mother answered then?\"\n\nRudolf waved her off and said, impatiently, \"Jibber-jabber! War! If I hear that one more time... True as the Lord God hung up there on the cross. I'm here and I'm staying here!\"\n\nBerta nodded and stared straight ahead, reflecting.\n\nLittle Berta asked, breathless, \"And did building number 7 get hit?\"\n\nBerta nodded, and Rudolf said, \"Of course it did. What else?\"\n\n\"And Grandma was right there in building number 7? 'I'm just going over for a little chat.' Didn't she say that?\" asked Little Berta, who always savored the same questions whenever this story was told. Her mother shrugged and kept silent, as she did every time after this question was asked.\n\n\"Christ Almighty! Christ Almighty!\" Little Berta shouted and scratched her chin.\n\n\"She was very strict but very proper. My mother. A wise woman.\"\n\n\"I like you better,\" Rudolf purred, and laid his head in Berta's lap. Little Berta growled, \"Make a little room for me too! You always have to have everything just for yourself,\" and Rudolf scooted reluctantly to the side, so Little Berta too could have the chance to put her head on her mother's lap.\n\nBerta laid her left arm around her daughter and her right arm around her son. She was kneeling in the bedroom on the marriage bed; Berta counted the chimes from the wall clock: it was eight in the evening, two days, or so she thought, before Wilhelm's return from Felsenstein.\n\n# HURRY HOME TO YOUR WIFE\n\nBerta Schrei did not know that the chauffeur and Come-hither-boy had lied to his employer for reasons that even he himself did not understand: \"My wife is bedridden at home right now with a fever of 104. I hoped it would subside with the passing of time. But now I've just been informed that the fever refuses to go down.\"\n\n\"But Wilhelm, why didn't you say so earlier? I was already thinking to myself,\" Johannes Mueller-Rickenberg said, \"something's not right with Wilhelm. So: how many days do you think your Dulcinea's recovery will take?\" Wilhelm declared it would be three days at most, then stressed his regret that they had fallen into this predicament. Two thin creases of displeasure emerged between the eyebrows of Johannes Mueller-Rickenberg's forehead. The down-turned corners of his mouth, with the lower lip drawn over the upper, signaled to Wilhelm his employer's inclination to put the chauffeur to a test.\n\n\"If you must rush off to your Dulcinea, you'll also have to come up with some excuse to give the countess, because I'm staying put\u2014don't forget to report to her that my own health is in an extremely precarious state right now. That'll make her stop fretting so much about my absence. And let's not forget the most important thing of all: the crime novels. Tell the countess I've brought them over with me personally from France. Just pick up a few French and English ones on the way home. She needs about two or three more weeks' worth of crime-fodder. Keep that in mind when you do your shopping! And once your Dulcinea recovers, don't neglect to tell her we'll be busy for a little while longer in Felsenstein this year. That's all.\"\n\nJohannes Mueller-Rickenberg nodded, turned around, and crossed the yard, with its shading of fall colors, his hands clasped behind his back and his shoulders bowed slightly forward. Wilhelm knew his master would be heading back to the house; he was in a hurry. His new conquest, Francesca, a fresh import from Brittany, needed attending to.\n\n# GRAVEDIGGER, THEN INFANTRYMAN, FOR A TIME\n\nBerta answered Rudolf's question for the umpteenth time.\n\n\"What was Grandpa again?\"\n\n\"A gravedigger. An infantryman. A gravedigger. When he was an infantryman, he deserted,\" Berta chuckled. \"He hid out in the hayloft of a farmer named Zweifel in Gnom. Until the war was over. Zweifel's daughter smuggled him bread and speck. The farmer could never know about it. And he never did find out.\" Berta chuckled again. \"Yes, yes. Old farmer Zweifel was a distinguished gentleman in those days.\"\n\n\"That's not what I mean! In October! What happened with Grandpa in October?\" Rudolf asked. It was only this one story, the story about October, that managed to hold his attention and calm him down somewhat.\n\n\"Of course. It was in the newspaper. Someone found a body. In the Mueller-Rickenberg forest, in the vicinity of Gnom near St. Neiz am Gr\u00fcnbach. Under a mound of earth, neatly covered with spruce branches. A grave. Shot in the head. Buried three to four weeks before. The corpse was gagged and its hands were bound behind its back.\"\n\n\"And the corpse was the gravedigger?\"\n\nBerta shrugged. \"He went into the forest, and he never came back out.\"\n\n\"But how do you know that that was Grandpa's body?\" Rudolf insisted.\n\n\"Well, you know, they notified me.\"\n\n\"Why didn't you go looking for Grandpa? Maybe that unidentified corpse wasn't actually him?\"\n\nBerta said: \"That's just how it happened.\"\n\n\"Did Grandma see the corpse? What did Uncle Wastl think, and Uncle Karl and Uncle Richard? Did they really all think the same thing, that the corpse was definitely the gravedigger's?\"\n\n\"It happened in October!\" Little Berta said, shoving her brother with her elbow. \"Are you stupid? They were all dead by then.\"\n\n\"Aha,\" said Rudolf at length, and then once again, \"Aha.\"\n\n\"Why was my grandpa a gravedigger?\" Little Berta asked, and Berta shrugged:\n\n\"He just was. He got his job back without any difficulty as soon as the war was done. He had no reason to disappear. The war was already well over.\"\n\nRudolf patted his sister several times on the back and said, \"Nothing can kill the gravedigger\u2014he always keeps his word. When he says he's coming home from the war, then he'll come. And you know why?\"\n\nHis sister, bored, answered, \"Of course!\"\n\n\"Then say it! Say it, if you know why!\"\n\nLittle Berta, insulted, remained silent. Berta answered in her stead: \"'Because I'm a gravedigger. And we must have gravediggers if order is to be kept. That means I have no choice but to survive. It may be I'm the last one who still knows this craft! What happens to a craft when none of its masters return from the front?'\"\n\nBerta chuckled and said, \"He was just a devoted gravedigger. He just knew what his work meant to people. He loved what he did.\"\n\nBerta's gaze fell on Rudolf's profile, and her thoughts turned vaguely to her father, the gravedigger, who had been an infantryman for a time, and she said, \"Yes. Yes. You really were a good gravedigger.\"\n\nRudolf's eyes closed, but he went on speaking: \"Maybe a gravedigger is something I can be, too,\" and Berta thought how very close she and her children had become over those past three days. She didn't know exactly why, but somehow it seemed right to stay up through the rest of the night. Either to avoid the nightmares, or to sift through her own life and the lives of her children.\n\nMaybe it was all of those things, maybe none of them.\n\n# PRECISE, CLEAR, WELL-ROUNDED LETTERS\n\nBerta Schrei had given birth to a boy in the year 1945 and to a girl in the year 1950 and in the year 1958, before the beginning of the leafless season, she saw fit to impart a certain piece of information to Wilhelm, a thing it seemed essential to impart, precisely and without distortions, in very legible letters: precise, clear, and well-rounded. She had always gotten As in handwriting. Her last report card attested to the fact: a certain Berta Faust, resident of building 13, apartment 12 in Allerseelengasse, Donaublau, had graduated with a cumulative grade of A.\n\nSolely for reasons of thrift, Berta conveyed her disclosure in the same blue envelope that had come from the school, an envelope that served to hide\u2014with extraordinary tact\u2014the substance of the letter inside it, just as when the teacher, from pure sympathy and the warmest sense of compassion, had taken pains\u2014again, with great tact\u2014to soften the substance of the school's decision with regards to Little Bertha, a decision that now loomed all the more ominously before Berta's eyes, implying that her daughter, who had always learned everything so easily, might be an even worse student than Rudolf. \"So much compassion,\" as Berta knew already on her way home, \"means that her case is hopeless.\"\n\nAnd on that long night, Berta Schrei came to a decision: \"Two hopeless cases are a doubly heavy burden.\"\n\n# AS SHADOWY AND GHOSTLY AS THE WAR YEARS\n\nWilhelm, still in his chauffeur's uniform, had imagined that on returning from Felsenstein, at around eleven in the morning, he would be able to march in triumphantly, right before Berta's eyes, and announce, \"What do you say, Berta? Two days earlier than expected! Can you believe it?\"\n\nHe crept in like a thief, laid his ear against the kitchen door's keyhole, giggled to himself, rubbed his hands together cheerfully, then threw the door wide open: \"Berta, it's me!\" he shouted, his arms stretched outward.\n\nThen he dropped into the chair disappointed, took off his chauffeur's cap, thought to himself that his Berta must be out shopping, and cheered up once more as he imagined the look on her face when she came back home and found him sitting there in the kitchen, a sight as perfectly natural as it would be utterly unexpected. He laughed, pleased with both himself and the circumstances, and just then, his eyes fell on the blue envelope on the table.\n\n\"I have brought my cursed creations to an end. Your Berta, who loves you.\"\n\nHe read and smiled, read again and smiled again. At first he didn't understand, nor would he manage to understand later, when every trace of Berta Schrei had been struck from his life, and his children were lying in the ground.\n\nIt wasn't until the second year following these events that the notion gradually dawned on him that the Earth's shadow wasn't simply passing him by, like a dream; that the Earth was a place he could make his home. He arrived somehow at the sense that he was a fully fledged resident of the Earth, and beyond that, the son of a nation, and not just of any nation, but the Isle of the Blessed.\n\nIt was with the aid of this feeling, as vague as it was pervasive, that he managed eventually to accept that, at a given moment in his life, a person existed who was capable of anesthetizing her children with sleeping pills and strangling them with her bare hands, only to fall on a butcher knife afterward with all her weight. It came to light that what she'd supposed to lie behind her left breast was instead concealed behind her right: a medical rarity. This circumstance assisted the doctors, who did still want to do their best to save this person's life.\n\nAnd in fact Wilhelm could just manage to accept it, so long as he allowed the events before the leafless season in 1958 to remain for him as shadowy and ghostly as the war years.\n\nMuch later, when he heard Wilhelmine's decisive proviso to the proposal he'd never actually made\u2014\"If you want to marry me, then it must be on January 13th\"\u2014he understood, then, it was time to ask for Wilhelmine's hand.\n\nSomehow, it was Wilhelmine's insistence on that date that restored to him the reality of that person, aforementioned, and she recovered her proper name, Berta.\n\n# THE BEST MOTHER OF ALL\n\nBerta's creations, Rudolf and Little Berta, were buried. Only that other creation, Berta Schrei, remained alive. And when the elder Berta saw that Little Berta, as she lay there still, bore no resemblance whatsoever to the Madonna from the painting, even though she'd finally been salvaged from the molding hands of life, she understood then that her delusion, founded on the casual resemblance of an image and a face, had been dispelled the moment it was carried out, and that reality was now spiraling into absurdity. Rudolf looked nothing like the Christ child either.\n\nBerta Schrei roared at the Madonna and the Christ child. Her yearning for an ideal, her wish to shelter Little Berta and Rudolf from the weight of things, had ended in a madwoman's double murder and her own failed suicide.\n\n\"How does it feel to be forty?\" Wilhelm asked, and stroked Berta's stringy mane, running the index finger of his right hand over the arrow-straight part in her hair.\n\nBerta giggled and said, \"I really need to go back to the hairdresser,\" and tried with a jittery movement of her hand to make her girlish haircut look more womanly.\n\n\"Curls always look good on you. On the other hand, a hairstyle without curls also has a certain appeal. Curls all the time, perhaps that's a bit much? What do you think?\"\n\nBerta felt the time had come to smile bashfully like a girl and turn her eyes away from Wilhelm's gaze. Meanwhile, the Wise Little Mother had been waving her hands protectively, each gesture a bit more emphatic than the one before. But no one in Ward 66 paid much attention to these efforts to alert them to what was happening, and this indifference was yet another sign, to the old woman, that it was time to teach Life, the intruder, a definitive lesson. She stood up from her bed, shuffled over to Wilhelm, and stood there for a while in silence, folding her hands slowly and deliberately and staring down at Berta's head; with the meekest look in her repertoire of meek looks, she purred at Wilhelm, \"No doubt, no doubt,\" but with a gently threatening undertone. Wilhelm at last deigned to look at her, disconcerted.\n\n\"Our Mother Fortress, the best of all mothers, has brushed Berta's hair straight. Her hair no longer knows the winding, wending, crimping, curving folds or furrows of the wound, Life. It has achieved a state of peace, it has come to rest. Isn't that right, dear Berta? Your hair has come to rest. Is that not so? Has it not achieved peace? Has it not come to rest?\" Berta looked at the Wise Little Mother, and her body swelled with guilt. Wilhelm waited in vain for the long sigh that he knew so well. But it died away before it could escape her.\n\nWilhelm grasped Berta's upper arm, gave her a soft but determined shake, and looked at her through the eyes of Wilhelm the returnee, a man who had tried, at the behest of his comrade, Private First Class Rudolf, his first and only friend, to explain the war to a certain Berta Faust. But when he stood there before her then, on that first day of June, in 1945, he got the feeling somehow that there was nothing to explain, and Wilhelmine's prying inquiries had burdened more than just his eardrums.\n\n# THEN I WILL LOOK AFTER YOUR BERTA\n\n\"Wilhelm!\" Several times on the journey from Denmark to Frankfurt on the Oder, Private First Class Rudolf shouted his friend's name, with unexpected vehemence, taking him firmly by the upper arm.\n\n\"Wilhelm! Do you remember Berta's address? Do you?\"\n\nWilhelm nodded placidly; Rudolf shook him and demanded he recite it.\n\n\"Where does she live? Tell me right now! Where does Berta live?\"\n\nOnly when Wilhelm had laid his calming hand on Rudolf's thigh and answered his question in a tranquil voice did Rudolf begin to breathe easily again.\n\n\"Berta Faust. Allerseelengasse, building 13, fourth floor, apartment 12, Donaublau.\"\n\n\"Exactly. You've got it. That's very good. You have to pay close attention to everything I tell you. You have to look at Berta with my eyes, otherwise you won't understand her. It's important. Whoever sees Berta with my eyes will know that he has to protect her from all the filthy people who want to abuse her and disgrace her. She's a child, Wilhelm. Even if her body is soft and warm, not awkward and bony like it used to be. She had no idea what men and women did together, I had to teach her that, the only men she knew were her father and brothers. The morning after, when she woke up, she looked at me like a baby. Curious. Trusting. Like a newborn suckling, imagining things would stay this peaceful forever. So curious, so trusting\u2014she was an easy mark! Not even a child is so na\u00efve. And yet I did it\u2014you see? I did it with a baby!\" And Rudolf's face clouded over, and he had to force away his thoughts. He reached for Wilhelm's upper arm again, then, and exclaimed, like a man doubting his own sanity: \"And just because I played her the 'Aquarelles' and the 'Blue Danube'!\" He shook his head and laughed with rage.\n\n\"Do you get it? 'The Blue Danube': Strauss, of all things. Do you understand me?\"\n\nWilhelm laid his calming hand again on the Private First Class's leg and said, \"It can't go on much longer.\"\n\nAnd again Rudolf turned, again he took his friend's arm, again he shook him and made his plea, as desperate as it was unnecessary: \"If they get me, Wilhelm! I mean if they get me! You're sure you're listening to me?\"\n\nWilhelm nodded and said, \"There's no need to worry.\"\n\n\"Wilhelm. Swear to me. Cross your heart. Look me in the eyes. If they get me, do you know what you have to do then?\" Wilhelm acceded for the umpteenth time. He knew that Rudolf would never rest until he'd taken his hand and sworn and promised something that to Wilhelm went without saying.\n\n\"Fine. I swear it. If they get you, then I'll take care of your Berta,\" and he smiled at his friend good-naturedly. \"But honestly, Rudolf, it's an unnecessary and ridiculous thing to swear. I think it's more likely they'll get me. That would make more sense. If I'm gone, no one will notice.\"\n\nRudolf brushed off his objection. The gesture was curt, but brimming with contempt. \"Go tell Goebbels that. Maybe he'll see the contradiction between making babies and waging war... You'll never be a proper soldier, you know. You can't use logic to win a game of chance. Not your kind of logic, in any case!\" Private First Class Rudolf began to pontificate as if he were once again Music Teacher Rudolf, in the days when he'd pronounced upon the arcana of harmony before his students' baffled eyes: \"The front has its own logic, and it's as simple as can be. Them or us. Period. If a single one of them is left, that's one too many. Whoever grasps that has grasped the essence of the matter. Do you see, Wilhelm? This is the secret of politics by other means.\" And Rudolf chucked his friend on the chin, snickered, and concluded, with something that nearly resembled mirth: \"No, you'll never grasp that. You could never understand a thing like that.\"\n\nWilhelm answered gently, \"If I understood it, I would stop it.\"\n\nAt this new instance of Wilhelmish logic, Rudolf cackled and threw back his head.\n\n\"That is what I would do. If it had a logic, I mean. Because in that case, a person could stop it by being logical. That seems reasonable,\" Wilhelm continued.\n\nNow it was Rudolf who patted Wilhelm's thigh: \"You clearly think madness has no logic. But that's a decisive misapprehension! Everything has its logic. Everything, understand? But let's leave that aside for now...\"\n\nWilhelm was well aware of his friend's disdain for his inferior intelligence, and knew Rudolf considered any attempt to think about the war to be far too difficult a task for Wilhelm's brain. Because he was so proud of his friendship with this clever man\u2014a music teacher in a secondary school, not just a common trade school, but one where Latin and Greek, French and English were taught\u2014he was able to forget the insults Rudolf was wont to lob at him each time Wilhelm made the least attempt at independent thought.\n\n\"Maybe we'll both make it out alive. Then you can come with me to Donaublau. I'll get you a job at the secondary school. As a janitor or something. Let me take care of it. I get along well with the principal. Let me take care of it.\"\n\nWilhelm revered Rudolf, and he liked the idea of going to Donaublau with him, it sounded nice, peaceful. He was familiar with Rudolf's tendency to swing from irate rebellion at the interminable madness of those years to soft, tranquil daydreams\u2014and then to the naked fear he felt for Berta, the alpha and omega of his sentimental daydreams.\n\n\"Don't forget the 'Aquarelles'! Or the 'Blue Danube' either! You have to learn to play the fiddle. Swear to me. Cross your heart. I can feel it\u2014I'm not going to make it back. There's no use saying otherwise. I know.\" As he said this, the Private First Class pointed at his temple. So firm was his belief that the assembly lines of war had already manufactured the decisive bomb or bullet that would bring his life to a close, that at times even Wilhelm came to think his friend's survival unlikely.\n\nWilhelm never promised Rudolf he'd learn to play the fiddle; he was too old for that, he felt. But he did repeatedly swear, whenever Rudolf demanded it, that he would look at Berta through Rudolf's eyes. It wasn't hard to do: Rudolf could paint an aquarelle of Berta with words, so convincingly that Wilhelm could never see her except as Rudolf did, even if he tried.\n\nAnd he knew he was thinking with Rudolf's mind when he concealed from Berta the particulars of his friend's death. But Wilhelmine insisted on wheedling it out of him.\n\nIt hadn't been easy, but he'd managed to make his way to Allerseelengasse. Building 13 was the only one remaining that was halfway habitable. On the fourth floor, he'd knocked at door 12 and seen Berta with Rudolf's eyes, a thing that wasn't hard for him to do.\n\nAnd Wilhelm kept the oath he had sworn to his friend. He stayed.\n\nHe even tried to learn the fiddle, but gave it up after a time. Berta started crying, utterly bewildered, when she heard the violent moans and groans he coaxed from the instrument; at last she took the fiddle away from him, saying, \"Leave it, Wilhelm. Put it in the dresser. Leave it alone. It's out of tune. I can't tune it and you don't know a music teacher who can. Just forget about it.\"\n\n\"Be patient with me. I'm learning. Believe me, I really am,\" he objected, and Berta simply smiled.\n\nThis smile he attributed to whimsy rather than contempt. And with time, Berta managed to convince him that she wouldn't trade him for any man in the world. Not even for the best fiddle player.\n\n# BERTA'S ANSWER\n\nWilhelm shook Berta more and more violently, and his breath came faster and his face twisted.\n\n\"Why? Berta? Why?\"\n\nBerta giggled. \"So. So,\" she said, and her eyelids began to flutter open and closed, open and closed.\n\n# WILHELMINE ON HER WAY\n\nWilhelmine was restless, pacing back and forth in the courtyard. Her eyes scanned the north and west sides of the fortress. There was no sign of Wilhelm anywhere.\n\nWilhelmine was indignant, highly indignant. So great was her indignation that she waddled past the porter to see for herself whether everything in the fortress was in order.\n\nShe wheezed up and down the staircases, one hand balled into a fist, the other clutching her handbag.\n\nHer indignation propelled her forward, and it was not until she became aware of the conflict between her movements and her indignation that sensible thoughts were able to once more work their way into her head. All the doors were closed, she saw; she was merely marching upstairs and down. So she waddled back to the porter, though it was difficult to track him down in this labyrinth. From him she learned that in order to get to Berta she had to take one specific staircase. She found this one staircase, wheezed her way up, sucked in several deep breaths while standing outside the door to the ward, then pressed the button, and the door opened.\n\nShe zipped past Sister Franziska Querbalkener, who called after her, \"Hello! You! Madame! Where are you trying to go! That is against the rules! Hello!\"\n\nOnce more, Wilhelmine stopped for a moment to catch her breath, and she used this moment to give Sister Franziska Querbalkener an annihilating stare: \"I must be admitted to Ward 66, to see a certain Berta Schrei,\" she said, in a tone that said quite clearly: \"Don't hold me up with your needling. I'm in a hurry!\"\n\nSister Franzi shook her head. \"Ma'am! You're going the wrong way! Over here! Over here!\" Wilhelmine again welled up with indignation. Yet after an arduous struggle with herself, she thought it best to let it pass, even if letting things pass was far from her strong suit. She nodded solemnly and said, \"Of course. I know that. I'm not an idiot, you know.\" And she marched past Sister Franziska Querbalkener with her head held high.\n\nShe stopped one more time, to gather herself in front of the door to Ward 66. She caught her breath before opening the door. Now she was in her element. She would create order, set matters aright, and save that catastrophe called Berta from Wilhelm's doubting and brooding once and for all.\n\n\"So!\" she said, and, \"Here I am!\" pinning Wilhelm in a devastating stare that clearly indicated what awaited him as soon as they had left the fortress behind. \"Wait and see! Just you wait!\"\n\n# INSEPARABLE FRIENDS\n\nWhen Wilhelmine caught sight of Berta, she went up to her with arms wide: \"Berta! You poor thing! How are you though? Just look at you! This hair! Who gave you this outrageous haircut?\" She accosted Berta's left cheek with a kiss, her right with another, examined her from all sides, took a look around the room, and declared with satisfaction, \"The bed linens are clean and everything seems to be in its place.\"\n\nShe ran a finger lightly over Berta's nightstand and announced, thoroughly contented, \"Indeed. No dust. That's what I call order. See, Berta. Things have actually turned out quite well for you. They take care of you here. Here everything is in order, everything has its place.\"\n\nShe pawed Berta's cheek and then offered good-naturedly, \"Isn't that right, you poor little thing? It's not so bad as all that. What do you say, my little dear?\"\n\nBerta giggled and dropped her head, ashamed. Hardly had the door opened when Wilhelm jumped up, jerking back his hand and sending the roses flying from Berta's lap onto the floor. Desperately, he racked his brains for a clever turn of phrase, something to make light of the situation. He soon found it, or so he thought, for as soon as he spoke it aloud, he immediately regretted opening his mouth:\n\n\"You know our Wilhelmine, Berta! She sweeps in, tosses everything into disarray, and then, miracle of miracles (!), God looks down and all is squeaky clean.\"\n\nWilhelmine's eyes were on the Madonna. She sat down by Berta on the bed, ran her hand over Berta's hair, reached nonchalantly for Berta's necklace, and in a tone so gentle it made the sweat bead up on Wilhelm's forehead, she cooed, \"Berta! My child! My little disaster! You're still holding onto that Madonna trinket!\" She smiled, stared into Berta's eyes, and Berta giggled, more ashamed than ever, before lowering her eyes to the floor.\n\nWilhelmine rested her right hand on Berta's lap and left it there a long while, till Berta's fluttering eyelids showed she had seen what there was to see on Wilhelmine's ring finger. Berta reached up, let her arm drop, said, \"So. So,\" and set to twiddling her thumbs. Her own ring was locked up in the fortress depository.\n\nWilhelmine rolled the Madonna trinket back and forth between her fingers. When Wilhelm nudged her, she responded with an obstinate shrug. He wiped his forehead with the large white kerchief Wilhelmine had embroidered with a W and an S, then tugged at his necktie, opened the top button of his white shirt, and pushed the cloth down into his collar, mopping up the sweat on his neck.\n\n\"Wilhelmine. We have to go.\"\n\nBerta looked up, her eyes met Wilhelm's, and in that moment he longed for the floor to open up and swallow him and Wilhelmine whole.\n\n\"So. So,\" Berta said, and turned toward Wilhelmine: \"He's a good chauffeur. Does he take good care of you?\"\n\nWilhelmine tried to look solemn, but was incapable of hiding the triumphant gleam in her eyes. Berta had understood!\n\n\"Yes. What can I say. There's the occasional 'if' and 'but,' the occasional 'on-the-one-hand' and 'on-the-other.' But I don't suppose he's a bad man. He's managed to make a little something of himself. No, he's not bad. If you keep your ears open, if you look around the neighborhood, you can see he's better than most.\" Wilhelmine sighed. \"If only I could break this habit of his, this doubting and brooding. That would be progress. But he just can't seem to give it up.\"\n\n\"Well then,\" Berta said and twiddled her thumbs busily.\n\nWilhelmine felt the clock ticking and reached for Berta's treasure one more time. \"Believe me, my dear, believe me\u2014life on the outside is no cakewalk. I would be happy to change places with you. You have everything here, there's nothing for you to worry about, you're always taken care of. No noise from the city, no bills, no rushing here and there. Just quiet, blessed quiet. No one curses at you, no one tells you off, no one makes you toe the line. You always were a bit of a disaster, weren't you? Even as a child, no? But now all that's finally over. You deserve a rest. You shouldn't feel the least pang of conscience. I understand you, you know. Once someone's been through what you have, they deserve a bit of peace and quiet. If they hadn't sent you here, you'd be made out to be a black widow, a vulture, killing her children out of pure selfishness, to satisfy her own wanton urges, who knows, to have more time for her lover. Oh! People! Sometimes they're so horrible, and from pure ignorance, too! People! I... I have to tell you that! But that's no reproach, my little catastrophe. How could I reproach you? It was your disease that dragged you into this story, wasn't it? Isn't that the case?\"\n\nBerta giggled while Wilhelmine pawed her hand: \"It's true, it's true. That damned illness. And anyhow, things are better for your children now, up there with the Lord God. They really were such miserable things, such sad little waifs. Most likely they'd never have come to any good. You shouldn't reproach yourself. Everything does turn out for the best. And I mean everything. The Lord God makes sure of that. He knows there was nothing you could do about it. He knows what he's doing. You poor, poor little thing, poor Berta. How can one person be so unlucky!\"\n\nAnd Wilhelmine pulled Berta's head into her bosom, and as she looked down, pity and sympathy filled her shifty eyes; it was a commendable display of fellow feeling for the fate of Berta Schrei. The mercy she bestowed on this poor disaster made her eyes grow damp, and tears rolled down her fat cheeks as she turned Berta's face to hers and gazed deep into her eyes. Moved by Wilhelmine's compassion and understanding, Berta felt for the necklace with the Madonna trinket.\n\nPlacid and comprehending, Wilhelmine smiled, caressed Berta once more on the cheek, and said, \"We're the closest of friends, isn't that so? We women understand one another. If only I could show you, in spite of everything, just how close I feel to you. But there's nothing so dear to me that it would make a fitting gift for you. Really, I have nothing!\" And Wilhelmine was aggrieved.\n\n# BERTA HANDS WILHELMINE THE CHAIN WITH THE MADONNA TRINKET\n\nA few tears rolled down Berta's cheek, a tremor rose up inside her and tugged at the corners of her mouth. Berta fidgeted with the clasp of the necklace for what seemed to Wilhelmine an eternity before she finally figured out how to open it. She cupped her left hand, laid the necklace inside it with her right, and stared at it, contemplative, almost brooding.\n\nWilhelmine was worried Berta might have second thoughts. Wilhelm held his breath. Just then he swore to himself that if Wilhelmine tried to snatch it, he would disregard her virtues and end their marriage right then and forever.\n\nBut she didn't try to snatch it. Berta raised her cupped hand to Wilhelmine and said, \"I have this,\" looking at her as if she wished to say, \"This is everything.\"\n\nWilhelmine gave an extravagant sob, then moaned, \"Berta! My poor dear! You can't give that away! Berta! I simply can't accept it! You have no idea what you're doing!\" She turned away, indignant.\n\nWilhelm sighed with relief. His fears had been needlessly bleak.\n\nBerta came down from her bed, stood like a sparrow before Wilhelmine the dove, got up on her tiptoes, teetered, then, clumsily, but with unswerving resoluteness, placed the necklace with the Madonna trinket around Wilhelmine's neck. When she saw what she'd done, she sat back on the bed, said, \"So. So,\" and began twiddling her thumbs frantically.\n\n\"Wilhelmine!\" Wilhelm shouted. He had forgotten he was in a public place. \"Wilhelmine!\"\n\nWilhelmine looked over at him, helpless and contrite: \"What should I do then, Wilhelm? What should I do? Wilhelm? You give it back to her. I can't bring myself to spoil her happiness. You do it,\" she whispered to him, and shrugged feebly.\n\nWilhelm knelt down in front of Berta, gathered up the fallen roses, and laid them back on her lap. \"You have given Wilhelmine great joy. That was very generous of you,\" he said, and cursed himself ten, twenty, a hundred times for every word he spoke.\n\nBerta looked down at the roses' slightly blemished buds, plucked off the petals and the leaves, giggled, and fell silent.\n\nThe inwardness she had struggled for, tirelessly and to no purpose, now suffused her face, and it would never leave her thereafter.\n\nThe Wise Little Mother saw this, and was pleased.\n\n# WILHELM CRIED AND LEFT\n\nAnd Wilhelm cried.\n\nThe last time Wilhelm cried, he had been crawling after Rudolf's head. Finally he grabbed hold of a shock of hair and pulled up on that trunkless head, thinking to seal away in his memory the features of his only friend, until he realized the thing he held was no longer Rudolf at all, it was little more than a scalp, that there was nothing left of Rudolf to hold onto; so he gave up and kissed what remained of Rudolf's head, blown off by the machinery of war; he kissed the eyes, nose, mouth, cheeks, and forehead over and over, he daubed himself with Rudolf's blood, longing to reattach the severed head to its torso, to breathe life back into his friend amid the torrent of bullets and grenades. His comrades in the trenches looked on with disgust as this bizarre spectacle unfolded, and went on dealing death from their various emplacements.\n\nThis time, Wilhelm didn't wail or bellow as he had then, and his features weren't quite so contorted. Instead, his sobs were soft, silent in fact; he cried without making a sound.\n\nBerta's hands edged warily toward his face, and she wiped away his tears, helpless and a little awkward. She looked past him as she did so. There was nothing left to say.\n\nWilhelm stood, turned around, and walked out. Wilhelmine watched him, perplexed, lingered a while longer in Ward 66, and finally said, \"Well, Berta, until next time! Of course we'll come visit you again!\" Berta got up and pressed the roses into Wilhelmine's hands, so forcefully that to refuse her would be uncalled-for. Berta turned, looked over at the Wise Little Mother, said, \"So. So,\" shrugged, and giggled.\n\nWilhelmine left.\n\n# WAS THE LONGING STILL THERE, AND THE BURNING SILENCE?\n\nHaving turned all the \"on-the-one-hands\" and \"on-the-others,\" all the \"ifs\" and \"buts\" over and over in her mind without arriving at any conclusion, the a-man-a-word-and-then-you're-lost Berta turned to the Wise Little Mother for advice. \"My Berta was looking inward, and so was my Rudolf. They really were, weren't they? So why is it, Wise Little Mother, if I may ask you a question, why is it my Berta looked so little like the Madonna from the painting, and my Rudolf so little like the Christ Child?\"\n\nThe Wise Little Mother twiddled her thumbs and said nothing.\n\n\"If I may ask the Wise Little Mother's opinion. Could it be that the molding hands of life disfigured them once it was over? I'm only wondering. Could it be that the molding hands of life outwitted me? Is that something we should consider?\"\n\nThe Wise Little Mother folded her hands and ardently prayed her \"Hail Mary, full of grace...\"\n\nThe a-man-a-word-and-then-you're-lost Berta had really believed that the only thing she needed was to get the Madonna necklace back from the fortress depository, after which she'd be able to figure out whether or not the molding hands of life had played a filthy trick on her. Or perhaps she simply needed the Madonna trinket as a barrier to shield her from what she'd seen as she carried out her deed.\n\nBut even once she had it, a-man-a-word-and-then-you're-lost Berta had still never mastered the inwardly turned gaze she had believed was her only means of finally triumphing over the molding hands of life. Only when she hung her Madonna around Wilhelmine's neck did her doubting and brooding compulsion fall away from her like withered autumn leaves, and then she understood that she had lost, and that life, with its molding hands, the weight of things, had won. Was the longing still there, and the burning silence? Giggling Berta's inward gaze gave no indication one way or the other. Only on occasion did the thing the old woman called the wound of life still flare up in Berta's eyes.\n\n# A NEW RESPONSIBILITY FOR THE WISE LITTLE MOTHER\n\nThe old woman, well schooled in all that pertained to the wound of life, had seen straight through the events of 13 January, 1963.\n\n\"There's no doubt about it. The wound of life is challenging me, trying to step in and hinder Berta's recovery.\"\n\nPutting a stop to this was her most sacred duty, and to that end, she philosophized away that particular day and its lamentable events, until finally they seemed like a mere trifle. But trifles of this kind still damaged the fortress, creating the impression it was nothing more than the inevitable result of an upbringing that had failed out there in that Sodom and Gomorrah world, where the wound of life holds court and Babylon founders. And this was a terrible affront to the fortress's edifying work, its mission of scrupulously cultivating sainthood, which it carried out in the most exemplary manner.\n\nIn other words: the old woman would not stand for a challenge to her merit with respect to Berta, for whom she felt great responsibility, as if she herself had borne her. No, she would not be defied by the wound of life, which had crept its way so shamelessly into Ward 66.\n\nOn January 14, 1963, the Wise Little Mother stood witness to Berta's final deliverance, a responsibility the old woman accepted with great solemnity.\n\nOn January 14, 1963, Berta washed the man, the word, and all her lostness out of her life. This is according to the words of the old woman who watched her do it.\n\nBerta stood at the sink and washed herself slowly and deliberately. Then a-man-a-word-and-then-you're-lost Berta swam down the drain with the sweat from yesterday's nightmare and into the fortress's sewers.\n\n\"This is your awakening, Berta dear, a blessing from our fortress. Here is where you sweat out life, that awful dream. Isn't that true, Berta dear? That life is no more than an awful dream?\"\n\nBerta giggled.\n\n\"Yes. Indeed. Guilt cries out from this dream. Indeed. The dream never leaves you in peace, it forces visitations on us, it never frees us from guilt. Everywhere we turn it confronts us with Sodom and Gomorrah. Say it now, Berta dear: the visitation has passed, the terrible dream is washed away, the colossus of life has been toppled. There is no more Sodom and Gomorrah, no more guilt, and no more dream.\n\n\"It is true, Berta dear. Yes. Indeed. The fortress has heeded your wishes. It is speaking to you now. It speaks to you before God and the saints. They hold the keys to your redemption. Yes. Indeed. Hear us now, Berta Schrei. You have been chosen for eternal peace. I bless you: Ego te absolvo.\"\n\nBerta giggled.\n\n\"Dear Berta, you owe this to the fortress. The fortress alone listens to you. It alone loves you. It understands you. It feels what you feel. It speaks for you before God and the saints.\"\n\nBerta giggled.\n\n# AFTERWORD\n\nOn Marianne Fritz\n\nOf all the crooked arrows in the critic's quiver, the one labeled \"genius\" seems to fly most erratically. Rather than a substantive gesture of praise, in its normal application it tends to serve as an aggrandizement of the critic himself, who thereby claims the capacity to appreciate this ethereal property. Yet beyond able craftsmen of the Richard Ford stamp or crowd-pleasers whose affability calls their integrity into question, there is a class of artists whose work is so strange and extraordinary that it eschews all gradations of the good and the mediocre: genius and madness are the only descriptors adequate to its scale.\n\nSuch is the case of the Austrian novelist Marianne Fritz (1948\u20132007), whose work is little known outside a small but fervent circle of admirers. Praise, though scant, is neither tepid nor inconsiderable: from 1978, when she received the inaugural Robert Walser Prize for the unpublished manuscript of The Weight of Things, her first novel, to her winning the highly prestigious Franz Kafka Prize in 2001, her writing was repeatedly honored with awards and stipends. Regarding Naturgem\u00e4\u00df1, Fritz's unfinished magnum opus, Elfriede Jelinek commented, \"It is a singular work, before which one can do nothing but stand, like a devout Muslim before the Kaaba.\" W. G. Sebald also dedicated to her a section of the late poem \"In Alferm\u00e9e.\" Here the image of Fritz working through her exhaustion, \"one hand on the keys of her machine,\" recalls the passage in The Rings of Saturn on the melancholy of scholars and weavers, \"harnessed to the machines we have invented.\"\n\nA contrasting view was held by Thomas Bernhard, who addressed his esteemed publisher, Siegfried Unseld, with characteristic charm in 1986:\n\nBefore my departure I have had another glance at your recent publishing catastrophe: the 3,000 pages you have had printed and allowed to appear are the greatest embarrassment I have been acquainted with to this day. To print and bind over 3,000 pages of mindless proletarian trash with all the bombast of a centenary event belongs, quite frankly, in the record books: as a world record of stupidity. I am not speaking so much of the begetter of this idiocy, but of the fact that the publisher has handicapped himself by releasing this fatuous vulgarity.\n\nThe project that inspired these extremes of reverence and ridicule began with The Weight of Things, which was followed two years later by The Child of Violence and the Stars of the Romani. Whereas The Weight of Things covered the years 1945\u20131963, in her second novel Fritz would approach the period she saw as the key to understanding the disaster of Western civilization: the years surrounding the First World War. The action of the book centers around Kaspar Zweifel, a sensitive young man who dreams of moving away to America. In 1914 he loses his beloved and is sent away to fight for the Austro-Hungarian army in the suicidal battles on the Isonzo front in present-day Slovenia. When he returns to the village of Gnom, he takes over his father's farm, makes a conventional marriage, and begins to complain of the \"mongrelization\" of the Austrian countryside. One night, drunk, he rapes a gypsy woman, and she and her community flee the area in fear of further violence. The woman returns in June of 1923 to leave Zweifel's child in the village parish house.\n\nThe novel went nearly unnoticed. While its rancorous subject matter and its convoluted structure were unsuited to a broader readership, its failure to engage with the more self-consciously avant-garde aims of the famous Grazer Gruppe writers left it adrift with respect to Vienna's literary establishment. Yet it proved a turning point for Fritz, for its stylistic departures and the broadening of the author's vehemence and scope.\n\nOnly five years later, she would publish the provocatively titled Whose Language You Don't Understand, the 3,392 page prodigy that was the object of Bernhard's derision. It was natural that critical discussion surrounding a book of such proportions would not be confined to its literary import. Fritz's neologisms, intentional misspellings, and readiness to violate the rules of grammar in favor of the construction of her own eccentric poetic idiom left the text unfit to be fed through the computer her publisher used to weed out errors, and her proofreader gave up after a thousand pages, saying it was impossible to distinguish mistakes from the distortions characteristic of the author's style. Critics began to talk of classifying books by their weight, and spoke openly of the point at which they had abandoned their reading.\n\nWhose Language You Don't Understand is the chronicle of the Nulls, a poor family residing in Nullweg, number 0, in the village of Nirgendwo2, in June and August of 1914. Thomas K. Falk, in World Literature Today, points out that the date is significant not only for its proximity to the Great War, but also as a marker for a period in which the traditional agrarian economy gave way to industrialization, when those who had previously worked the land became a despised and neglected appendage to the modern capitalist state. It is in this book that Fritz's partiality toward the insulted and injured becomes explicit: the patriarch, Josef Null, is killed in a worker's demonstration, as is his third son, Josef II; one of his brothers, the Dostoyevskian August, is a farmhand and an anarchist and murders the landowning parents of his girlfriend Wilhelmine; another becomes a deserter and is chased down by the military and shot. Their mother is confined to the fortress first mentioned in The Weight of Things, and their home is destroyed, lest it serve as a remembrance of the possibility of resistance.\n\nEleven years later, Naturgem\u00e4\u00df began to appear: five volumes in 1996, another five in 1998, just shy of 7,000 pages reproduced directly from Marianne Fritz's typescript. Set largely in Przemy\u015bl, an \"eternal death-territory\" in southeastern Poland, it examines the lives of many of the characters in her previous books over the war years of 1914\u201315. Her publisher found setting the book's first part impossible; the printed version consists of a bound facsimile of the typescript. At first, Fritz limited herself to employing a variety of fonts, spacings, margins, and unusual typographical markers, but when she learned how the book was to be released, she began to incorporate drawings, maps, coded marks meant to establish links between various characters and situations, and copies of the innumerable notecards\u2014her second memory, in the words of her partner Otto D\u00fcnser\u2014that she used to keep track of the hundreds of characters and place names and thousands of events that made up her novel. (One critic complained the reader was forced to turn the book back and forth like a steering wheel.) In her refusal of unilinear narrative, Fritz had largely dispensed with the traditional paragraph; text was often inverted or written at an angle, or a central letter would form the axis for three words that would be based on it, their letters curved to fit inside a drawing that appears to represent an oblique phase of the waning moon.\n\nThe comparison with Joyce is both obvious and inapposite, and has been made by critics within and beyond the German-speaking world. Against the clear dissimilarities in tone, subject matter, and working method stands the male genius as the archetype against which the female author is to be compared. If she passes muster, she may be crowned a \"female Joyce\"; if she falls short, her work is disqualified as an extravagance. A further injustice to a female writer of such staggering ambition is the lack of a feminine analogue to the stereotype of the \"mad genius\": women's madness cannot invoke the dignity of the late Nietzsche or Thomas Chatterton, but is classed with the aberrations of hysterics and cat ladies.\n\nMarianne Fritz may represent a limit-case of the blurring of life into literature. The small apartment she shared with her partner in Vienna's 7th District, now preserved for visitors, is lined floor-to-ceiling with bookshelves, map cases, and card files; there are no accommodations for guests or relaxation, virtually no concessions to the demands of ordinary existence. When her writing went well, she would not leave her work room for weeks at a time; it was only when she had elected to take a \"philosophical pause for thought\" that she would discover what season it was. In fact her labor transcended the possibilities of a lone author, and she engaged her partner to toil in the archives, bringing home to her mimeographs of war correspondence, newspaper reports, ministerial records, and battle plans. \"Going out was my job,\" D\u00fcnser recollects. \"At first she would research with me in the War Archives in the Stiftgasse. We were to look through 30,000 photos. Then she said to me: Otto, you do this.\"\n\nIt is possible that a fictional topos as dense and expansive as Fritz's is inconceivable outside of such isolating circumstances; that like a phantom limb, such imagined worlds may arise only after conventional reality has been hewn away. To say this is not to reduce Fritz's work to aleatory hallucinations or to question her artistic integrity. On the contrary, just as Rimbaud argued for the systematic derangement of the senses, Fritz's subjection of herself to the conditions necessary to reconceive the situation of war in all its staggering complexity must be considered a conscious, programmatic gesture.\n\nIn Naturgem\u00e4\u03b2 III, her work begins to revolve around Der Giftpilz, Ernst Heimer's notorious anti-Semitic children's book published by Nazi propagandist Julius Streicher. Reversing the text's original schema, it is Nazism rather than Judaism that becomes the poison mushroom growing in Austria's soil. For the writer and critic Klaus Kastberger, this episode leads to the essential question for an understanding of Fritz's oeuvre: in what does the subjective experience of such a disaster as National Socialism consist? Paradoxically, the first person or free indirect approach is inadequate, for inner experience obscures the historical and social conditions by which it is determined. To Fritz, fidelity was to be found instead in the minute recreation of a society down to its very fundaments, and to this end, nothing was irrelevant. The purpose of imagination became less the fashioning of persons and events from whole cloth than the rectification of those gaps in official documentation through which the lives of so many at society's bottom ranks had filtered:\n\nwhat moves me are the \"blank spaces,\" the \"not established,\" the \"crossed out,\" the \"unmentioned,\" the \"irrelevant,\" the \"superfluous,\" the \"redundant,\" the fact that so much \"information\" is actually lived through [...]3\n\nFritz's writing has been described as a counter-history or a writing against history, but it is a counter-historiography as well, a supersession of the Great Man Theory (and the notion of free will underlying it) through a vision of historical events overwhelming their radically passive subjects. It is not, for all that, lacking in moral force. Fritz, who was from a humble background and only came to pursue literature after completing vocational studies to prepare her for secretarial work, shows an all-consuming contempt for the structures of class oppression. Indeed, her entire oeuvre works toward a vindication of the lives of the poor, men and especially women, who were expelled from the dignified arenas of Austrian society in the first half of the twentieth century and crushed like roaches under the millstone of history. Freedom is not absent amid the duress of time past, but remains in the artwork as an imaginative horizon: the space of the dreaming mind that cannot but turn away from its own degradation, and the compulsion to defiance that imbues it like a calling.\n\n\u2014Adrian Nathan West\n\n1 Naturally, or, more precisely, in accordance with nature.\n\n2 That is, the Null family, residing in Number 0, Null Street, in the town of Nowhere.\n\n3 From a letter from Marianne Fritz to her editor included in the volume Was soll man da machen.\n\n# ABOUT THE AUTHOR\n\nMarianne Fritz (1948\u20132007) was an Austrian novelist. She lived most of her adult life in a small apartment in Vienna's 7th District, devoted entirely to her writing and dependent on small subsidies and occasional prize money. Her first book, The Weight of Things, marked the beginning of an ambitious cycle of novels with the overarching title of Festung, or \"The Fortress,\" comprising Das Kind der Gewalt und die Sterne der Romani, Dessen Sprache du nicht verstehst, and the gargantuan Naturgem\u00e4\u00df, the third volume of which she was preparing at the time of her death.\n\n# ABOUT THE TRANSLATOR\n\nAdrian Nathan West is the author of The Aesthetics of Degradation as well as the translator of numerous works of contemporary European literature. He lives between Spain and the United States with the cinema critic Beatriz Leal Riesco.\n\n# DOROTHY, A PUBLISHING PROJECT\n\n1. Renee Gladman Event Factory\n\n2. Barbara Comyns Who Was Changed and Who Was Dead\n\n3. Renee Gladman The Ravickians\n\n4. Manuela Draeger In the Time of the Blue Ball\n\n5. Azareen Van der Vliet Oloomi Fra Keeler\n\n6. Suzanne Scanlon Promising Young Women\n\n7. Renee Gladman Ana Patova Crosses a Bridge\n\n8. Amina Cain Creature\n\n9. Joanna Ruocco Dan\n\n10. Nell Zink The Wallcreeper\n\n11. Marianne Fritz The Weight of Things\n\n12. Joanna Walsh Vertigo\n\nDOROTHYPROJECT.COM\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n# The Triumph of Modernism\n\n# THE Triumph OF Modernism\n\n* * *\n\nTHE ART WORLD , 1987 \u2013 2005\n\nHilton Kramer\n\nROWMAN & LITTLEFIELD\n\nLanham \u2022 Boulder \u2022 New York \u2022 Toronto \u2022 Plymouth, UK\n\nPublished by Rowman & Littlefield\n\n4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706\n\nwww.rowman.com\n\n10 Thornbury Road, Plymouth PL6 7PP, United Kingdom\n\nDistributed by NATIONAL BOOK NETWORK\n\nCopyright \u00a9 2006 by Hilton Kramer\n\nFirst Rowman & Littlefield edition published in 2013\n\n_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 written permission from the publisher, except by a reviewer who may quote passages in a review.\n\nBritish Library Cataloguing in Publication Information Available\n\nThe hardback edition of this book was previously cataloged by the Library of Congress as follows:\n\nKramer, Hilton.\n\nThe triumph of modernism : the art world, 1987\u20132005 \/ Hilton Kramer.\n\np. cm.\n\nIncludes bibliographical references and index.\n\n1. Modernism (Art). 2. Art, Modern\u201420th century. 3. Art, Modern\u201421st century.\n\nI. Title.\n\nN6494. M64. K73 2006\n\n709.04\u2014dc22\n\n2006017584\n\nISBN 978-1-56663-708-4 (cloth : alk. paper)\n\nISBN 978-1-4422-2321-9 (pbk. : alk. paper)\n\nISBN 978-1-4422-2322-6 (electronic)\n\n The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences\u2014Permanence of Paper for Printed Library Materials, ANSI\/NISO Z39.48-1992.\n\nPrinted in the United States of America\n\nTo Roger Kimball\n\n# Preface\n\n* * *\n\nIn _The Triumph of Modernism_ I have collected a selection of my writings on art from 1987 through 2005. Doubtless many readers, reflecting on the curious vicissitudes of cultural life over the last few decades, will regard the title of this book as a provocation. Modernism? Triumph? If there have been any cultural triumphs in the last twenty years, have they not been primarily at the _expense_ of modernism? Have we not lately witnessed the triumph of postmodernism\u2014a knotty, perhaps an incoherently knotty, term that nonetheless bespeaks the failure or disintegration of the entire modernist enterprise?\n\nPerhaps. But it is worth stressing that the dynamics of what we have come to call postmodernism are neither as novel nor as revolutionary as its champions seem to believe. More than thirty years ago, in an essay called \"The Age of the Avant-Garde,\" I wrote what was in effect an obituary for the avant-garde. Among other things, that essay was an attempt to distinguish the career of the avant-garde from the larger, more vital imperatives of modernism. The central issue, the central distinction, has to do with seriousness, with the allegiance art maintains to the pulse of lived experience. To a large extent, I argued, the avant-garde had repudiated that fundamental seriousness. Its idols were figures like Duchamp, its governing institution was the museum\u2014conceived less as a repository of cultural achievement than as a terminus in which the monuments of the past could be subjected to the dissolving bath of irony and historical dislocation. \"Duchamp's legendary assault on the work of art as traditionally conceived,\" I wrote, \"demonstrates that there is no such thing as an object or gesture that, within the magical museum context, cannot be experienced as art, and this demonstration has the effect of consigning both the idea of tradition and the museum itself to a limbo of arbitrary choices and gratuitous assertions. Which is exactly what our culture has now become.\"\n\nI believe that the experience of the past thirty-odd years has amply confirmed that judgment\u2014though I would add that our culture has become something else as well. The aftermath of the avant-garde has not been the sole province of \"arbitrary choices and gratuitous assertions\"\u2014the sole province, that is to say, of forces dedicated to disintegration. There have been countervailing movements and oppositional figures, though the specific gravity of our cultural life today dictates that genuinely oppositional figures appear chiefly as proponents of recuperation. This is one of the signal ironies of this age of irony and institutionalized subversion: that the pressure of human experience should finally work to subvert the ethic of subversion and erase the grinning rictus of Duchampian frivolousness. We have not, perhaps, traveled very far down that road of recuperation. But I remain convinced that the chief critical task facing us is the task I identified at the end of \"The Age of the Avant-Garde\": the task of \"renegotiating our pact with the past and reexamining all those restrictive clauses that have so often rendered our commerce with 'tradition' simply foolish and parochial.\"\n\nProperly understood, modernism is not only the ally, it is also the spearhead of that renegotiation. For modernism denominates not a particular \"stance\" or style\u2014it is by disposition neither figurative nor abstract, for example\u2014but rather a discipline: the discipline of truthfulness, the rigor of honesty. _The Triumph of Modernism_ is intended as a contribution to that open-ended process. The book records my assessment of a wide range of artistic endeavor and critical intervention. To be sure, it minutes a variety of aesthetic failures, wrong turns, and outright depredations. But its guiding ambition is to discriminate between the genuine and the fraudulent in cultural life. Many of the artists, critics, and institutions I discuss in these pages have been more or less witting collaborators in this effort. This book is a testament to their story\u2014the story of the triumph of modernism.\n\nEarlier versions of most of this material appeared in _The New Criterion_ , _Commentary_ , and the _New York Observer_. I am grateful to the editors for permission to include that work here. I am also grateful to my colleagues at _The New Criterion_ for their help in the preparation of this manuscript.\n\nH. K.\n\nDamariscotta, Maine\n\nJuly 2006\n\n# The Triumph of Modernism\n\n# I \nModernism and Abstraction\n\n## Kandinsky and the Birth of Abstraction\n\n* * *\n\n> _For me, the province of art and the province of nature thus became more and more widely separated, until I was able to experience both as completely independent realms. This occurred to the full extent only this year_.\u2014Vasily Kandinsky, in 1913\n\nThe birth of abstract art\u2014art that makes no direct, immediately discernible reference to recognizable objects\u2014has long been recognized as a fateful event in the history of art. Yet the intellectual origins of this event, which promptly established abstraction as one of the central traditions of twentieth-century painting and sculpture, have remained a vague and little-understood subject for the vast public that now takes abstract art for granted as part of the familiar scenery of modern cultural life. That the emergence of abstraction early in the second decade of this century represented for its pioneer creators a solution to a spiritual crisis; that the conception of this momentous artistic innovation entailed a categorical rejection of the materialism of modern life; and that abstraction was meant by its visionary inventors to play a role in redefining our relationship to the universe\u2014all of this, were its implications even dimly grasped, would no doubt come as a shock to many people who now happily embrace the history of modern art as little more than a succession of styles, or art fashions, which may have something to do with the history of taste but do not have much to tell us about life.\n\nAs for the academic study of modernism, Professor Rosalind Krauss no doubt spoke for many of her colleagues in the universities when, in the 1980s, she declared that she found it \"indescribably embarrassing to mention _art_ and _spirit_ in the same sentence.\" Yet the artists who first created abstract painting\u2014Vasily Kandinsky and Piet Mondrian preeminently among them\u2014did not share that embarrassment. They really did conceive of their artistic endeavors as serving a spiritual mission, and if, some eighty-plus years after the birth of abstraction, we still take an interest in the art of its progenitors, we are obliged, I believe, to examine the ideas that shaped it, however odd or alien those ideas may look to us today.\n\nThis is certainly the case with Kandinsky, whose work is the subject of a current exhibition at the Museum of Modern Art that takes the birth of abstraction as its principal focus. Organized by Magdalena Dabrowski, _Kandinsky: Compositions_ is a small exhibition about a very large subject. Yet it has the virtue of concentrating our attention on one of the pivotal moments in modern cultural history: the years 1910 to 1913, when Kandinsky was deeply engaged in painting the pictures that led him into the realm of abstraction and, at the same time, formulating the theories that supported this audacious endeavor. The exhibition includes some of the most beautiful paintings that have ever been created in the name of abstraction, and the greatest of them\u2014 _Composition V_ (1911), from an unidentified private collection; _Composition VI_ (1913), from the Hermitage Museum in St. Petersburg; and _Composition VII_ (1913), from the Tretyakov Gallery in Moscow\u2014have the additional interest of being relatively unfamiliar to the American public. The many studies that are also included in the exhibition give us a vivid account of the inspired, painstaking pictorial thought that Kandinsky lavished on this crucial turn in his artistic development. If only because of the masterpieces of lyric abstraction the show contains, it would have to be considered an important occasion. What may not be so apparent is that it also marks an event in modern intellectual history.\n\nKandinsky was very much a man of ideas\u2014a cosmopolitan artist and intellectual straight out of the emancipated bourgeois intelligentsia that dominated the life of art and social thought in Russia in the last decades of the nineteenth century and the early years of the twentieth. His keen interest in music, theater, poetry, philosophy, ethnology, and myth was an essential element in his thinking about art. So was his deep interest in the occult\u2014an interest that he shared with Mondrian and certain other creators of early abstract art. The character of Kandinsky's mysticism was different from Mondrian's, however\u2014more socially and emotionally engaged, and less inclined to take refuge in strategies of withdrawal from the trials of life. Kandinsky relished intellectual combat and critical controversy. Even his celebrated treatise _On the Spiritual in Art_ , which he published in the period covered by the current exhibition, bristles with argument, assertion, and polemical heat.\n\nThe paintings that Kandinsky first produced in his pursuit of abstraction, culminating in the _Compositions_ he created in 1913, are likewise very different in character from Mondrian's. Whereas Mondrian's aesthetic of abstraction derived from the \"logic\" of Cubism, Kandinsky's had its origin in the more disorderly, emotion-charged conventions of Expressionism. While Mondrian's mode of abstraction found its ideal in a pictorial style of straight lines, primary colors, and rectangular structures, Kandinsky's abounded in graphic improvisation, coloristic invention, \"hidden\" images, and a tumult of painterly dynamism\u2014in other words, lyric inspiration. Whereas everything in Mondrian's abstraction seeks fixity and order, Kandinsky's embraces the fluidity and flux and thus the sensuality of the mind's impressions as an essential ingredient of his art.\n\nFor both Mondrian and Kandinsky, the artistic base from which they made their fateful leap into abstraction was landscape painting, but their respective approaches to landscape were, again, very different. Whereas Mondrian's was that of an ascetic determined to strip nature of its mutable attributes, Kandinsky's was that of a mystical lyricist for whom nature is an enchanted realm of poetry and symbolism. Yet for both, the leap into abstraction was at once guided and sanctioned by their faith in the metaphysics of the occult, which in the end emancipated them from the mundanity of the observable world.\n\nKandinsky enjoyed a distinct advantage, however, in coming from a vastly richer intellectual and cultural milieu\u2014one in which all artistic issues were subjected to intense analysis and informed argument by artists and critics who, in many fields, were energetically engaged in advancing new ideas. The Symbolist movement in Russian literature, music, and philosophy in the 1890s, itself deeply immersed in mystical systems of belief, had already disposed Kandinsky to look beyond the boundaries of naturalism and materialism for the foundations of his art. Munich, where he established himself at the turn of the century, was also the scene of a good deal of avant-garde activity that devalued any abject dependency on the imitation of nature in favor of stylized and symbolic representations of it. Compared to the pinched and provincial cultural atmosphere that Mondrian experienced in Amsterdam in the nineties, Munich was a veritable hotbed of modernist thought, much of it addressed to imaginative departures from academic convention and the canons of realistic depiction in the arts.\n\nIn his pursuit of abstraction, moreover, Kandinsky was, if anything, even more directly indebted to the doctrines of theosophy than Mondrian, himself a confirmed acolyte of the occult. There are many pages of _On the Spiritual in Art_ , as well as other writings by Kandinsky during the period of his turn to abstraction, that are barely intelligible without recourse to the ideas of Madame Blavatsky and Rudolf Steiner. Both are briefly mentioned in _On the Spiritual in Art_ , but their influence\u2014and that of two other theosophical writers, Annie Besant and C. W. Leadbeater\u2014on Kandinsky's artistic thought was far greater at this crucial juncture in his development than the artist could bring himself to disclose at the time. His entire _Weltanschauung_ , his worldview of life itself, was shaped by their philosophy of the occult.\n\n_On the Spiritual in Art_ takes as its point of departure a religious crisis that Kandinsky describes in terms virtually paraphrased from these theosophical writers. \"Our souls, which are only now beginning to awaken after the long reign of materialism, harbor seeds of desperation, unbelief, lack of purpose,\" he writes in the Introduction, echoing the theosophists' diagnosis of the spiritual malaise that had overtaken mankind in the modern world. He continues:\n\n> The whole nightmare of the materialistic attitude, which has turned the life of the universe into an evil, purposeless game, is not yet over. The awakening soul is still deeply under the influence of this nightmare.\n\nOwing to the nature of this spiritual nightmare, Kandinsky writes, the artist's whole world of feeling has been radically altered. In an apparent reference of the \"decadent\" art movement of the 1890s, Kandinsky declares that \"coarser emotions such as terror, joy, sorrow, etc., which served as the content of art during this period of trial, will now hold little attraction for the artist.\"\n\nWhat, then, was to take the place of this failed artistic content? Kandinsky speaks of the artist's yearning \"to awaken as yet nameless feelings of a finer nature.\" He invokes the possibility of an art that \"also has an awakening prophetic power,\" and links this to \"the spiritual life, to which art also belongs and in which it is one of the powerful agents.\" Yet because it is an essential part of Kandinsky's purpose in _On the Spiritual_ to discredit alternatives to the \"spiritual\" program he envisions for the art of the future, he inveighs against a number of the reigning artistic orthodoxies of the period before outlining the terms of his agenda.\n\nThe primary target of his criticism is, of course, naturalism in all its materialist manifestations. Scorn is lavished upon the social radicals in the art world who favor naturalism\u2014the \"atheists\" and \"socialists of various shades,\" as Kandinsky calls them, given to quoting from Marx's _Capital_ and announcing that \"God is dead.\" Yet the aestheticism which, with its credo of _l'art pour l'art_ \u2014\"art for art's sake,\" had itself rebelled against materialist orthodoxy is also denounced as a \"dissipation of the artist's powers.\" For Kandinsky, this soulless aestheticism, which attempted to make a religion of art itself, was as much a threat to the \"spiritual in art\" as the most orthodox forms of the materialism it opposed.\n\nIt is at this point in _On the Spiritual in Art_ that Kandinsky invokes the name of Madame Blavatsky. He cites theosophy as \"one of the greatest spiritual movements\" of the new century, and commends its mystical \"path of inner consciousness\" as the proper approach to \"problems of the spirit.\" He then reminds his readers of Madame Blavatsky's prophecy that \"in the twenty-first century this earth will be a paradise by comparison with what it is now\"\u2014a prophecy that, in Kandinsky's view, the new art of abstraction was destined to play a crucial role in fulfilling. For in Kandinsky's conception of the artist's new vocation, he was destined to become a spiritual leader of mankind\u2014still another idea he derived from theosophical doctrine. Other than a brief footnote reference to Rudolf Steiner, no further direct mention of theosophy occurs in _On the Spiritual_ , yet the entire work is based on theosophical ideas.\n\nIt wasn't only in his conception of the artist as a spiritual leader, moreover, that Kandinsky drew upon occult doctrine. _On the Spiritual in Art_ is, after all, a treatise on the aesthetics of painting, and a good many of what appear to be purely aesthetic observations\u2014about form, about color, about spatial relationships\u2014are similarly derived from theosophical pronouncements. In this respect, _Thought-Forms_ \u2014the theosophical classic by Annie Besant and C. W. Leadbeater that was translated into German in 1908\u2014seems to have been of capital importance. When, for example, in the chapter of _On the Spiritual_ devoted to \"The Language of Forms and Colors,\" Kandinsky sets about the task of assigning specific meanings to specific colors, he is clearly appropriating an occult practice.\n\nThus about the color blue Kandinsky writes that \"the deeper the blue becomes, the more strongly it calls man toward the infinite, awakening in him a desire for the pure and, finally, the supernatural\"\u2014an obvious echo of the assertion in _Thought-Forms_ that \"the different shades of blue all indicate religious feeling.\" When he writes about green that \"pure green is to the realm of color what the so-called bourgeoisie is to human society: it is an immobile, complacent element, limited in every respect,\" this description, too, bears a distinct resemblance to the statement in _Thought-Forms_ that \"green seems always to denote adaptability; in the lowest case, when mingled with selfishness, this adaptability becomes deceit,\" etc. And while it is true that Kandinsky takes certain liberties with this theosophical practice, deviating from some of the meanings assigned to colors in _Thought-Forms_ and other occult writings, he nonetheless follows the practice of conferring upon each color a significance that is finally metaphysical.\n\nSo, too, with Kandinsky's conception of pictorial form. In _The Sounding Cosmos_ , the definitive study of the place occupied by occult doctrine in Kandinsky's theory and practice of abstraction, the Finnish scholar Sixten Ringbom reminds us that \"thoughts and feeling, theosophy taught, have direct formative powers, and in _Thought-Forms_ there is an important passage which only required an artistic application in order to serve as a manifesto for non-objective art.\" In this passage there are said to be \"three types of form-producing thought, of which the first two take the image of persons or material objects, and the third takes a form entirely its own, expressing its inherent qualities in the matter which it draws round it.\" In this third category, wrote the authors of _Thought-Forms_ , \"we have a glimpse of forms natural to the astral or mental planes\"\u2014in other words, abstract form. To which Ringbom adds:\"If, as theosophy maintained, there was a world of form and color independent of material objects, then such forms could be exploited artistically and provide the content replacing the object.\"\n\nIn his pursuit of abstraction, Kandinsky similarly divided his paintings into three categories\u2014Impressions, Improvisations, and Compositions\u2014and while they are not in every respect congruent with the division of form outlined in _Thought-Forms_ , they nonetheless trace a similar course. As Magdalena Dabrowski writes in the catalogue accompanying the current exhibition at MOMA:\n\n> Impressions were the pictures stimulated by \"direct impressions of external nature,\" expressed in \"linear-painterly form.\" Works that were intended to convey \"impressions of internal nature\" and that represented \"chiefly unconscious, for the most part, suddenly arising expressions of events of an inner character\" were called Improvisations. [Kandinsky] described the most ambitious of these categories, Compositions, as \"the expressions of feelings that have been forming within me in a similar way (but over a very long period of time), which, after the preliminary sketches, I have slowly and almost pedantically examined and worked out. Here, reason, the conscious, the deliberate, and the purposeful play a preponderant role. Except that I always decide in favor of feeling rather than calculation.\"\n\nIn a long career that was copious in production, Kandinsky reserved the term Composition for only ten paintings between the years 1910 and 1939. The first three Compositions were, alas, destroyed during the Second World War, and so we know them today only from preliminary studies and photographs. The last three, though they are included in the current exhibition, fall outside the period that is its principal focus\u2014the period that saw the birth of abstract painting. It is therefore in _Compositions IV_ through _VII_ and their related studies that we can follow Kandinsky's development as he struggled to formulate his artistic ideas and create the pictorial language that enabled him to produce his first abstract paintings.\n\nBut did Kandinsky really succeed in these works in creating an art of pure abstraction? I believe he did; but early on in the history of abstract painting, as its champions divided into factions and each upheld an orthodoxy that precluded the admission of any rival version of abstraction, questions were raised as to whether Kandinsky's _Compositions_ could be accepted as authentic examples of abstract art. No less a critic than Mondrian himself was one of the first to express some doubts on the matter. In the September 1918 number of the magazine _De Stijl_ , five years after Kandinsky painted his first abstract paintings, Mondrian published an essay called \"The New Plastic in Painting.\" With that essay he reproduced two pictures, Picasso's Cubist masterpiece _Le Violin_ ( _Jolie Ev_ a) (1912) and Kandinsky's _Composition VI_ (1913), and insisted that the Cubist painting, even though it still \"represents particular things,\" came closer than Kandinsky's in expressing \"the human spirit\" because it \"introduces the straight line where it is not directly seen in the object.\" Then, confining Kandinsky to a footnote, Mondrian wrote as follows:\n\n> Kandinsky too broke the closed line that describes the broad contour of objects, but as he did not sufficiently tense the natural contour, his work remained predominantly an expression of natural feeling. Comparing the works of Picasso and Kandinsky . . . one sees clearly how important are the tension of curved line and use of the straight. Kandinsky's generalized expression, like Picasso's, came about through the abstraction of naturalistic form and color: but in Kandinsky's line still remains a vestige of the contour of objects, whereas Picasso introduces the free straight line. Although Picasso still uses fragments of the contour of things, he carries them to determination, whereas Kandinsky leaves somewhat intact the confluence of line and color found in nature.\n\nMondrian was certainly right in claiming to find in Kandinsky's _Composition VI_ some trace of a \"confluence of line and color found in nature,\" but his argument with Kandinsky cannot be understood entirely in terms of the aesthetics of abstraction. It was because Mondrian had come to believe that the spiritual mission of abstract art could not be fully realized except by means of straight lines that he condemned Kandinsky's lyrical abstraction as \"predominantly an expression of natural feeling.\" Among theosophists, which Mondrian and Kandinsky both were, this was a pronouncement of spiritual failure. Needless to say, this was not the way Kandinsky saw the matter. By 1913 he believed he had arrived at a stage in his development when he was able, as he wrote, \"to experience both [the province of art\u2014meaning spirit\u2014and the province of nature] as completely separate realms.\" Within a decade after 1913 he would, to be sure, abandon the Expressionist component in abstraction and make use of straight lines. But the straight line never became for him, as it did for Mondrian, either a sign of his spiritual mission or an exclusive component of his aesthetic. Yet for both of these masters of abstraction, theirs was an art born of this strange alliance of aesthetics and mysticism.\n\n[1995]\n\n## Mondrian and Mysticism: \"My Long Search Is Over\"\n\n* * *\n\n> _The irrational bears the same relation to the rational that the unknown bears to the known. In an age as harsh as it is intelligent, phrases about the unknown are quickly dismissed. I do not for a moment mean to indulge in mystical rhetoric, since, for my part, I have no patience with that sort of thing. That the unknown as the source of knowledge, as the object of thought, is part of the dynamics of the known does not permit of denial. . . .We accept the unknown even when we are most skeptical. We may resent the consideration of it by any except the most lucid mind; but when so considered, it has seductions more powerful and more profound than those of the known_.\u2014Wallace Stevens, circa 1937\n\nWhat was the nature of the quest that moved the Dutch painter Piet Mondrian (1872\u20131944) to abandon the representation of nature in favor of an art of pure abstraction? What, exactly, did Mondrian _believe_ he had achieved? In any attempt to address this question, we are obliged to deal with the fact that abstract art\u2014and not only Mondrian's\u2014was born of an alliance of aesthetics and mysticism. We are obliged to examine the ideas that shaped the artist's search for the absolute in art. Ideas, of course, are no substitute for the experience of art, and in Mondrian's case are certainly not to be taken to be the \"subject matter\" of his painting. Yet without some grasp of the intellectual history that led the artist to make his fateful leap into abstraction, the spiritual imperative that governed Mondrian's art is unlikely to be understood. This is a subject that takes us into the mind of modernism at one of the key turning points of its history. It therefore seems an appropriate subject to explore on the occasion of the great Mondrian exhibition that opens next month at the Museum of Modern Art in New York\u2014and all the more so since the exhibition itself, as it was presented at the National Gallery in Washington this summer, gave the subject scant attention. In Professor Yve-Alain Bois's essay for the catalogue accompanying the exhibition, for example, the author seems vaguely embarrassed that an artist of Mondrian's stature might be seen to be seriously implicated in a system of ideas and beliefs which for most of us cannot claim a very persuasive intellectual provenance.\n\nIt won't do to dismiss the issue with condescending references to Madame Blavatsky or attempts to substitute the ideas of more respectable thinkers like Goethe and Hegel in examining the intellectual origins of abstraction. For the problem posed by the relation that obtains between mystical belief and artistic innovation in the creation of abstraction is one that goes beyond Mondrian himself. In the period that saw the genesis of abstract art\u2014roughly the decade beginning in 1910\u2014what is particularly striking about the outlook of the artists primarily responsible for creating abstraction is their espousal of occult doctrine. So prevalent was a steadfast belief in the occult among the pioneers of abstract art\u2014not only Mondrian but also Kandinsky, Malevich, and a good many of their disciples\u2014that we have no alternative but to regard such doctrine as a basic component of their artistic vision.\n\nIt should be understood, however, that the particular school of mystical thought that exerted the greatest influence on the creation of abstract art\u2014the system of occult belief generally known as theosophy\u2014was in this period a widely established component of Western cultural life. While deliberately esoteric and by no means a popular creed, theosophy was not an interest limited to coteries of avant-garde painters and sculptors. Still less was it an interest confined to a circle of intellectual cranks and charlatans, though it did indeed attract a large number of crackpots and crazies. (It still does.) As Wallace Stevens was to observe some years later, \"There are, naturally, charlatans of the irrational. That, however, does not require us to identify the irrational with the charlatans.\"\n\nStill, it was an odd group of savants and mystagogues that shaped the course of modern theosophical thought, thereby exerting a decisive influence not only on the origins of abstract art but on other key developments in twentieth-century art and culture. The high priestess of the movement was indeed the Ukrainian-born adventuress Helena Petrovna Blavatsky (1831\u20131891), known to her followers the world over as H. P. B. She was said to have commanded occult powers from an early age. She certainly possessed a magnetic personality, making her way in the world\u2014which in Blavatsky's case meant Europe, America, and Asia\u2014as an emancipated woman without apparent means, yet always attracting (it seems) a circle of admiring men and women who believed in her spiritual gifts, supported her occult endeavors, propagated her theosophical doctrines, and remained undaunted by her scandalous reputation.\n\nIn New York, in 1875, H. P. B. founded the Theosophical Society and promptly produced two large tomes\u2014 _Isis Unveiled_ (1877) and _The Secret Doctrine_ (1888)\u2014that became the sacred scriptures of the movement. The claim advanced by this movement, which came to be called the Universal Brotherhood of Humanity, was that theosophy had triumphantly resolved the conflict between science and religion caused by the Darwinian theory of evolution. \"It was the genius of H. P. B.,\" writes the historian James Webb in _The Occult Underground_ , \"to apply Darwin's theory to produce a _hopeful_ resolution to the human condition.\" Webb continues:\n\n> Whereas others saw only the destruction of the sustaining myth of man's divine origins, H. P. B. discovered that evolution could apply also to the \"spiritual\" aspects of existence. Man had evolved from apes\u2014perhaps; but he had a noble destiny. Just as _homo sapiens_ had evolved from a lower form of animal life, and that form in its turn from a lower\u2014a vegetable, protoplastic, or unicellular existence\u2014man as at present constituted was on his way to higher and better things. Evolution continued on a cosmic scale, with each individual born and reborn thousands of times until he had achieved earthly perfection.\n\nTo achieve the exalted goals that theosophy now invoked as a spiritual imperative for art, it was nonetheless necessary for art itself to remain firmly within the realm of aesthetic experience. An art born of an alliance of aesthetics and mysticism was still obliged to traffic in material objects, and thus remain accessible to visual perception, if its spiritual function were to be realized. It could not\u2014or not yet, anyway\u2014so completely dematerialize itself as to become invisible if it were to continue to function as art. When the Epoch of the Great Spiritual was finally attained in some faraway future existence, no doubt this last concession to material contingency would be dispensed with; and art, too\u2014to the extent that it continued to be needed\u2014would become pure spirit.\n\nBut meanwhile, in the spiritually imperfect present, an art that aspired to make itself the instrument of pure spirit\u2014to align itself with what Kandinsky described in _On the Spiritual in Art_ as \"the struggle toward the non-naturalistic, the abstract, toward inner nature\"\u2014had no alternative but to make its case as an earthly cultural enterprise. It thus remained subject, in this respect, to the \"laws\" and conventions and material conditions governing the medium in which the artist worked. It was for this reason that even the most mystical of the pioneer creators of abstract art found themselves engaged in a dialectic that embraced two very different aspirations\u2014on the one hand, that of advanced artistic thought, with its headlong drive to discover new modalities of pictorial expression, and, on the other, that of a spiritual ideal that looked upon art as a means to an end that transcended art itself.\n\nFor observers who mistakenly tend to identify mysticism with a retreat from the practical affairs of the world, it must sometimes seem a paradox, if not an outright contradiction, that artists of a mystical temperament are perfectly capable of pursuing their worldly ambitions with the kind of uncompromising zeal that we observe in the careers of the early abstractionists. But the truth is that in every realm of high endeavor in which we find mysticism serving as a spur and an ideal\u2014in movements of political revolution and campaigns for sexual emancipation, for example, no less than in avant-garde art movements\u2014this loyalty to occult doctrine has proven to be a powerful incentive to practical achievement. Within the imaginative faculties there is nothing that so stirs the will to accomplish extraordinary things as the conviction that one is possessed of absolute knowledge about the ultimate destiny of human affairs. This was precisely the kind of mystical conviction that the pioneer creators of abstraction brought not only to their own innovations as artists but to the formation of avant-garde movements that changed the direction of modernist art and greatly expanded the place it would come to occupy in modern cultural life.\n\nIt is this belief in the absolute that accounts for the speed as well as the confidence which the pioneers of abstraction brought to their break with what they saw as the last vestiges of representation in art, and for the success they achieved in winning converts to their radical view of artistic expression. These artist-acolytes of the occult really did believe they were fulfilling a unique spiritual mission in this fateful act of artistic innovation, and they persuaded a great many others to join them in that belief. Theirs was, in this sense, a religious movement as well as an artistic one, and they brought to it the kind of intensely felt faith that we associate with spiritual yearning and religious conversion.\n\nYet it was as artists first of all, and only secondarily as exponents of the occult, that they exerted their deepest influence on art itself. In the latest developments of the art of their time, which in the case of Cubism had already brought painting to the very verge of abandoning representation, they shrewdly discerned a \"logic\" that cried out for completion. This gave to abstraction a powerful sense of purpose\u2014a kind of onto-logical imperative\u2014that was all the more compelling when combined with the promise of a spiritual vocation. As artists, the pioneers of abstraction had largely mastered in their earlier paintings the rapid succession of styles that had moved from Realism and Impressionism to the Post-Impressionist, Fauvist, Expressionist, and Cubist art which dominated the European avant-garde in the last decade of the nineteenth century and the first decade of the twentieth. Because it was from inside this fast-paced arena of pictorial revision that Mondrian, Kandinsky, and Malevich made their historic assault on what remained of traditional representation in art, their views as well as their example carried a special authority\u2014particularly for the younger members of the avant-garde who harbored an intense appetite for the absolute and a headlong impatience to see it realized. For them, everything that had occurred in the history of painting from Courbet to Picasso was suddenly relegated to the past by this unexpected leap into abstraction. In their very different ways\u2014for Mondrian, Kandinsky, and Malevich, despite similar convictions about abstraction and the occult, were distinct individualists with quite separate programs for their art\u2014the pioneers of abstraction opened up a new artistic world to their contemporaries. It was a world vastly different in its metaphysical and artistic outlook from that of the avant-garde movements which immediately preceded the emergence of abstraction.\n\nIndeed, from the perspective of the modernist movement in Paris, where the prehistory of pictorial abstraction had largely taken place and which continued to command an unrivaled authority and influence as the capital of the international avant-garde, the new abstractionists signified something alien and suspect. The modernism of the School of Paris had been anything but monolithic, of course. Its cosmopolitanism was legendary, and it was justly famous for its receptivity to a wide range of advanced ideas and personalities. Theosophy and other theories of the occult were well known there, too, though they did not lead to a movement advocating abstraction. In fact, none of the ideas contending for attention in the world of the Paris avant-garde openly advocated as _a matter of principle_ the total rupture with representation then beginning to be championed by the pioneers of abstraction. Parisian modernism was, by and large, far too attached to the things of this world to forgo their pictorial representation as a permanent imperative. The Paris avant-garde was, in this respect, invincibly materialist in its outlook on life\u2014still closely tied to observable sensation, and thus ineluctably earthbound in its basic character. However hermetic its allusions to the things of this world might at times become, Parisian modernism was nonetheless steadfast in its allegiance to them as an indispensable point of departure.\n\nIt was this allegiance to the things of this world that the mystagogic convictions of the new abstractionists now put into question for the first time, and it was for this reason, above all, that they were either ignored, rejected, or consigned to a marginal status by the Parisian avant-garde. (Even the quintessentially French color abstractions of Robert Delaunay, which the poet Apollinaire dubbed \"Orphic Cubism\" and which derived from the color theories of the Neo-Impressionists, were more sympathetically received in Germany than in Paris.) The doctrinaire abstraction that was rooted in theosophical ideas was therefore seen to constitute, in the early years of its development, a counter-avant-garde opposed to the mainstream artistic interests of the School of Paris. It was, as far as Paris was concerned, virtually an underground movement that did not enjoy the critical acclaim, or even the kind of critical scandal, that had marked the emergence of Fauvism and Cubism before the First World War and would again mark the emergence of Dada and Surrealism in the years following the war.\n\nIt was in advanced artistic circles outside of Paris\u2014in Russia, Germany, and the Netherlands\u2014that the new vision of the pioneer abstractionists met with the most combustible and consequential response. Of the vanguard figures who set out to create an abstract art based on a visionary synthesis of aesthetics and mysticism, only Mondrian lived and worked in Paris for extended periods of his development, first in the years between 1912 and 1914 and then again from 1919 to 1938. It was in the first of these periods, when he remained virtually unknown to the Paris avant-garde, that Mondrian created his earliest abstract paintings.\n\nMondrian had made the decision to dedicate his life to art long before he turned to the occult, but it was as a dedicated theosophist that he created his first abstractions. After a strict Dutch Calvinist upbringing, he found in theosophy a refuge from the religious turmoil of his youth and early manhood. For a time he even kept a portrait of Madame Blavatsky on the wall of his studio\u2014an icon of his spiritualist endeavors. (As Yeats later recalled, \"To [Madame Blavatsky's] devout followers she was more than a human being.\") In later life Mondrian's friends often spoke of his \"priestly air,\" and there was no question that for him the artistic vocation would never be entirely separable from his intense religious feelings.\n\nBorn in Amersfoort, Utrecht, the Netherlands, in 1872, Mondrian was educated at the Calvinist Primary School in Winterswijk, where his father\u2014a stern disciplinarian\u2014was headmaster. This was the only formal schooling Mondrian received before enrolling in the Academy of Art, in Amsterdam, in 1892. In his adolescence he was given some informal art instruction by his father, an amateur draftsman, and by an uncle, who was a professional painter in The Hague. His own development as a draftsman was sufficient to enable him to qualify, at the age of twenty, for the modest post of teacher of secondary-school drawing. This was his situation when he entered the Academy.\n\nNothing in Mondrian's early life as an art student or as a young exhibiting artist gives us any hint of the gifts that would one day make him a leader of the international avant-garde. He was anything but precocious. He attended the Academy off and on until 1896, and began exhibiting in Amsterdam the following year. But the pictures he then produced\u2014at a time, it should be remembered, when Van Gogh and Seurat had already completed their work, when C\u00e9zanne was painting some of the greatest pictures of his late period, and Art Nouveau was the fashion\u2014still consisted of naturalistic landscapes and still lifes in the Dutch academic tradition. Even by provincial academic standards, moreover, Mondrian's early career was anything but promising. In 1901, at the age of twenty-nine, he failed to qualify for the Dutch _Prix de Rome_ , owing to a lack of proficiency in rendering the figure. Henceforth, until his turn to abstraction, Mondrian's principal subject matter was landscape. He was also devoted to painting flowers, a practice that provided him with a living of sorts even after he had turned to abstraction.\n\nFollowing the _Prix de Rome_ fiasco, nearly a decade elapsed before any significant trace of modernist influence made itself felt in Mondrian's landscape painting. This was slow in coming for several reasons. Holland in the nineteenth century had remained something of an artistic back-water as far as modern painting was concerned. News of avant-garde developments in Paris and other cosmopolitan capitals did not easily penetrate the provincial art life of Amsterdam in Mondrian's student days. (\"Van Gogh was dead,\" wrote Michel Seuphor of this period in Amsterdam, \"but still unknown.\") Shut off from such developments in a very conservative cultural milieu, Mondrian himself did not awaken to the implications of modernism until he was well into his thirties. This awakening occurred, as it happened, coincidentally with his formal commitment to theosophy.\n\nIt was thus around 1909\u2014the year that Mondrian officially joined the Theosophical Society of Amsterdam, with a certificate of membership signed by Annie Besant\u2014that he belatedly developed the rudiments of a modernist pictorial style. It began with a modified version of Seurat's Pointillist method, combined with a Symbolist attitude toward color, that was applied to the rendering of Dutch landscape motifs\u2014haystacks, windmills, tree forms, and seaside dunes. This initial approach to an independent style was already a step in the direction of abstraction, for nature was not so much described in these pictures of haystacks and dunes as made the point of departure for more emphatically articulated pictorial structures that relegated their subjects to a secondary status. Such pictures were, in any case, quickly succeeded by the artist's determination to submit these same landscape motifs to the even more radical Cubist innovations of Braque and Picasso, a few examples of which Mondrian saw for the first time in an Amsterdam exhibition in the fall of 1911. Those few proved to be enough to change everything.\n\nMondrian's encounter with Cubism was clearly the critical turning point of his artistic development. While it would be another year or two before he finally expunged the last traces of traditional representation from his painting, what can only be called his conversion to Cubism\u2014now abetted by the anti-materialist imperatives of his theosophical faith\u2014put Mondrian's life as an artist on an entirely new footing. Approaching his fortieth birthday, released at last from the tethers of academic tradition, philistine taste, and the provincial culture of his youth, Mondrian was ready to enter the ranks of the avant-garde. Artistically and spiritually, the obscure Dutch artist who finally quit Amsterdam for Paris in the last days of December 1911 was on his way to a new life\u2014a life that in the coming decade would change the face of Western art.\n\n\"We may say,\" wrote Michel Seuphor in his study of Mondrian, \"that it was in Paris in 1912 . . . that the life of the great painter began.\" It was, from the outset, a life of personal isolation, artistic dedication, and mystical reflection. The evidence suggests that it was also a life devoid of erotic engagement. Upon his arrival in Paris, where he lived and worked in a studio borrowed from a Dutch acquaintance, Mondrian had few friends, knew little French, and allowed himself almost no distractions. His first\u2014indeed, his sole\u2014priority was to come to terms with the Cubist ideas he had only just discovered.\n\nTo this task he now brought a sense of purpose that was entirely new to him. Quickly mastering the ambiguities that were central to the Cubist aesthetic\u2014ambiguities that set the need for a recognizable motif dialectically at odds with a method of pictorial composition designed to eliminate its distinctive features\u2014Mondrian painted with a confidence and drive he had never before enjoyed. His mission was at once exalted and implacable. Cubism became the battleground on which the artist who had been nurtured on the specifications of Dutch landscape painting fought his last protracted conflict with nature. In this struggle, Mondrian produced his first masterpieces\u2014the series of _Compositions_ based on tree forms in which the last remnants of naturalistic detail are no sooner evoked than all but extinguished in favor of pure painting.\n\nTechnically, Mondrian's tree _Compositions_ of 1912\u20131913 may not qualify as pure abstractions, for the presence of the fractured motif, while diminished almost to the point of extinction, is still minimally discernible. (This tends to be true of nearly all the earliest abstractions.) Yet aesthetically, in the way such pictures are experienced as integrated structures of pictorial form, they have unquestionably crossed the boundary separating abstraction from representation. And during the remainder of this first period of residence in Paris, Mondrian eliminated virtually all association with nature from his pictorial vocabulary. By imposing upon each successive painterly composition of soft-edged Cubist forms an increasingly insistent network of hard vertical and horizontal lines, he left no trace of\u2014and finally no space for\u2014an observed object or remembered motif.\n\nThe result was a series of paintings dominated by an irregular, improvised, and as yet unsystematic version of the abstract \"grid\" that, a few years hence, would become a compelling principle in Mondrian's classic geometrical abstractions. These first abstractions were not yet governed by any kind of absolute rule. They were not painted according to a theory. Mondrian was following the dictates of his sensibility\u2014a sensibility that was then more confident, perhaps, about what it wished to eliminate than about what it wished to retain for painting\u2014as he worked his way toward what he took to be \"the logical consequences\" of Cubism. Still, the artist who reluctantly returned to the Netherlands in the summer of 1914, on the eve of the First World War, had nonetheless succeeded in transforming the \"logic\" of Cubism into an art of abstraction. If he was not yet a recognized leader of the avant-garde\u2014for abstraction was not itself a widely recognized artistic \"cause\"\u2014he was already producing the kind of art that would serve as a foundation for a new movement. As Theo van Doesburg, the ideological leader of De Stijl, afterward acknowledged: \"Mondrian was the first to realize, in 1913, by a consistent elaboration of Cubism, this new plasticism as a painting.\"\n\nWhat role did Mondrian's belief in the occult play in this headlong pursuit of abstraction? The notebooks in which he began, in 1914, to keep a record of his ideas about \"Art and Reality\" clearly establish the centrality of his quest for \"the spiritual\" in all of his artistic endeavors at the time. This is the key passage:\n\n> _Art and Reality_. Art is higher than reality, and has no direct relation to reality. Between the physical sphere and the ethereal sphere there is a frontier where our senses stop functioning. Nevertheless, the ether penetrates the physical sphere and acts upon it. Thus the spiritual penetrates the real. But for our senses these are two different things\u2014the spiritual and the material. To approach the spiritual in art, one will make as little use as possible of reality, because reality is opposed to the spiritual. Thus the use of elementary forms is logically accounted for. These forms being abstract, we find ourselves in the presence of an abstract art.\n\nCareful attention must be paid to the way Mondrian invokes the concept of \"reality\" in his writings, for in a very short time it would come to signify something very different\u2014the exact opposite, in fact\u2014from his identification of reality, in this passage, with the material world of \"our senses.\" This is, of course, what most of us mean by reality. As he refined his philosophy of art, however, bringing it into stricter alignment with occult doctrine, \"reality\" became a term wholly reserved in Mondrian's later writings for the realm of \"the spiritual.\" But this terminological revision does not reflect any change in the essential direction of his thought about abstract art. Fundamental to its creation, in his case, was a steadfast metaphysical belief in the division that separated \"the spiritual and the material,\" and, as an artistic corollary to this, a further belief\u2014which he was not yet ready to act upon to the fullest degree\u2014in \"the use of elementary forms\" as the most expeditious means of achieving \"the spiritual\" in art.\n\nWe cannot know what immediate course Mondrian's painting\u2014as distinct from his ideas about painting\u2014would have followed if the war had not disrupted his artistic pursuits in the summer of 1914. Summoned to his father's bedside in July of that fateful year, he was not expecting this duty visit to his family to amount to anything more than a brief interlude. As it happened, of course, he soon found himself trapped in the Netherlands for the duration of the war. As the country was neutral in the war, Mondrian was never in any danger from the military operations that were devastating other parts of Europe. Yet he felt painfully cut off from the important work he had started. Separated from his paintings, which had been left behind in Paris, and from the milieu in which they had been produced, Mondrian lost some of his artistic momentum. Nevertheless, in this period of enforced repatriation he devoted a great deal of thought to formulating his future artistic program.\n\nIn this pursuit he soon discovered that he was by no means as isolated in his new artistic and philosophical outlook as he had believed himself to be. \"The war,\" writes the Dutch historian H. L. C. Jaff\u00e9, \"had brought many Dutch artists who had been working abroad, back to their native country, and these came back to the Netherlands charged with the results of their studies and full of new and promising ideas. In the Netherlands they found an atmosphere of spiritual tension, generated by the fact of the country's being a neutral enclave, an oasis in a continent at war.\" It was thus in the war years that Mondrian came to know, among others, the painters Theo van Doesburg and Bart van der Leck, the architects J. J. P. Oud and Robert van't Hoff, the designers Vilmos Huszar and Gerrit Rietveld, and the Belgian sculptor Georges Vantongerloo\u2014the artists who, with Mondrian, founded in 1917 the avant-garde movement called De Stijl (the Style). This was to have an immense influence not only on the future of abstract art throughout the Western world but also on modern architecture and design and on the aesthetic, social, and pedagogical theories that supported them.\n\nDe Stijl was, from the outset, something more than an art movement. It was a social and cultural program, based on a visionary amalgam of idealist aesthetics, industrial mechanics, and utopian politics. Its ambition was to redesign the world by imposing straight lines, primary colors, and geometric form\u2014and thus an ideal of impersonal order and rationality\u2014upon the production of every man-made object essential to the modern human environment. Rejecting tradition, it envisioned the rebirth of the world as a kind of technological Eden from which all trace of individualism and the conflicts it generates would be permanently banished. Its goal was to liberate European culture from the \"archaistic confusion\" of its past in order to create an entirely new and completely harmonious civilization.\n\nIn this ambition, notwithstanding its rationalist claims, a belief in the occult was to play an important role. In 1915 the Dutch writer M. H. J. Schoenmaekers published a treatise called _The New Image of the World_ , described by its author as a work of \"positive mysticism.\" This and a subsequent volume, _Plastic Mathematics_ (1916), exerted a crucial influence on Mondrian and the other founders of De Stijl. According to Jaff\u00e9, \"Both Mondrian and Schoenmaekers lived, at the time, in Laren and . . . saw each other frequently and had long and animated discussions.\" Schoenmaekers's ideas were directly incorporated into the De Stijl program from 1917 onward, as well as into Mondrian's later theoretical writings. The very term that Mondrian adopted for the geometrical abstractions of his later years\u2014Neo-Plasticism\u2014was borrowed from Schoenmaekers's wartime writings.\n\nSchoenmaekers was a Neo-Platonist in quest of a reality more absolute than any that could be discerned in the natural world without the aid of mystical illumination. Nature, from the perspective of this \"positive mysticism,\" was looked upon as a mystery to be scrutinized and penetrated\u2014a view that was more or less in accord with theosophical doctrine, and that also bore a distinct resemblance to the Symbolist philosophy of Mallarm\u00e9. Truth was defined as the reduction of \"the relativity of natural facts to the absolute, in order to recover the absolute in natural facts.\" What was especially important in this theory was that art, if sufficiently awakened to its mystical role, was accorded a central place in the metaphysical process. \"In art,\" Schoenmaekers wrote, mysticism \"creates what we call, in the strictest sense, 'style.'\" Style in art is characterized as \"the general in spite of the particular\" and the means by which \"art is integrated in general cultural life.\" Thus the joint task of the artist and the mystic is \"to penetrate nature in such a way that the inner construction of reality is revealed to us.\"\n\nIt was not only for its philosophical affirmation of the mystical view of \"reality,\" moreover, nor for the role it envisioned for art in divining its \"inner construction,\" that Schoenmaekers's theories were important for Mondrian in his thinking about abstraction. Schoenmaekers also specified the nature of the forms (rectilinear structures of the horizontal and the vertical) and the colors (the primaries: red, yellow, and blue) to be used in this artistic quest for the absolute, and then elaborated upon their cosmic significance. \"This new plastic expression,\" Schoenmaekers wrote in _The New Image of the World_ , \"is of this world. . . . But it is a new earthiness, an earthly heaven.\" And an \"earthly heaven,\" or some artistic analogue thereof, was indeed what the artists of the De Stijl group, Mondrian most conspicuously, were determined to create in their abstract art.\n\nIt must not be supposed, however, that Schoenmaekers's ideas triggered an abrupt or immediate break in Mondrian's pictorial practice. The purely geometrical, or Neo-Plasticist, abstractions, with which Mondrian came ultimately to be identified, were never painted according to a theoretical formula, and there was certainly no sudden conversion to the purely geometrical as the result of his response to Schoenmaekers's philosophy. On the contrary, Mondrian seemed at first somewhat daunted by the vision of an art that might jettison all the variations in tone and contour that had remained a staple of his painting even when he abandoned conventional representation in the most highly developed of his \"Cubist\" abstractions. In later years, with his painting confidently entrenched in Neo-Plasticist principles, he might dismiss such residual signs of painterly tradition as a throwback to \"Impressionism\" or even naturalism; but during this difficult period of transition to a more absolute mode of abstraction, when every mark on the canvas was an exploration of the unknown, Mondrian nonetheless held fast to such vestiges of tradition even as he was inching his way toward a conception of painting that would finally do without them.\n\nThus the pictorial means traditionally associated with the depiction of observable objects and the space they occupy\u2014shifts of tone and color value, and the variations in drawing that serve to hold them in place\u2014survived a while longer in Mondrian's painting even when their representational function had already been forfeited. Clearly it was not an easy matter for a painter of Mondrian's background and experience, even at this relatively advanced stage of his development, to bid an irrevocable farewell to such a fundamental pictorial practice. The depth of the perturbation caused by this radical option may be inferred from the fact that only one painting by the artist survives from the year 1916. This was the crucial period in which Mondrian was struggling to reconcile the demands of his art with the appeal of Schoenmaekers's ideas. The abstraction that he produced that year\u2014 _Composition: 1916_ , now in the collection of the Guggenheim Museum\u2014is at once a summation of the \"old\" painting and a partial preview of the new. Its forms are predominantly rectilinear but still soft-edged, \"open,\" and irregular. Its black lines hold firmly to right-angled verticals and horizontals, but at the side edges of the painting they shade off into veiled greys. Its colors\u2014blues and blueish greys, pinks and greyish pinks, yellowish ochres that in places turn brown\u2014are cousin to the primaries but remain equivocal, impure, and atmospheric. There is no discernible subject in _Composition: 1916_ , but for anyone who knows the _Pier and Ocean_ drawings by Mondrian that preceded it there is an unmistakable visual echo, or memory, in its fragmented grid structure. This is painting with a pronounced appetite for an absolute it does not yet command the means of realizing.\n\nThe breakthrough came the very next year. \"For the moment at least,\" Mondrian wrote to a friend in September 1917, \"my long search is over.\" While he had produced little new painting during the previous year, he had devoted much time and thought to the writing of a treatise of his own\u2014\"The New Plastic in Painting\"\u2014which van Doesburg began serializing in the inaugural issues of _De Stijl_. At the same time, Mondrian painted his first geometrical abstractions\u2014the series of pictures entitled _Compositions in Color_ or _Compositions with Color Planes_. These \"flat\" pictorial compositions of rectangles in primary colors on a grey ground closed the door on the kind of illusionistic space\u2014the space of representation\u2014that the artist had been struggling to transcend since his first encounter with Cubism six years earlier. In 1917, Mondrian had at last brought both the theory and the practice of pure abstraction into a satisfactory alignment.\n\n\"The New Plastic in Painting\" owed a good deal to Schoenmaekers's mystical philosophy, of course, but it was more directly addressed to the problems of painting. It was at once a manifesto for a new pictorial aesthetic\u2014an aesthetic of pure abstraction\u2014and a restatement of the occult, anti-materialist doctrines that supported its creation. \"The new plastic,\" Mondrian wrote in the introduction to this historic treatise, \"cannot be cloaked in what is characteristic of the particular, natural form and color, but must be expressed by the abstraction of form and color\u2014by means of the straight line and determinate primary color.\" What is remarkable about \"The New Plastic in Painting\" is not only its triumphal affirmation of pure abstraction as an aesthetic \"absolute\"\u2014a word that was repeatedly sounded throughout the manifesto\u2014but the anxious attention that is lavished on the need to emancipate painting from the thralldom of what is variously called \"nature's appearance,\" \"naturalistic representation,\" \"naturalistic form,\" \"naturalistic color,\" and so on. Nature haunts the pages of \"The New Plastic in Painting\" like a specter that can be neither exorcised nor appeased. It can only be resisted by an act of transcendent artistic conscience that makes the pursuit of the \"universal\" a quest so absolute that all the contingencies of nature are stripped of their importance.\n\nImplicit in this view of what Mondrian also called \"the style of the future\" was an idea of evolution\u2014the evolution of art in the service of the evolution of spirit\u2014akin to the theosophical doctrines of Madame Blavatsky and Rudolf Steiner. According to this conception of evolution, mankind had once \"lived in harmony with nature's rhythm.\" With the development of human consciousness, however, \"there automatically ensued a disharmony between man and nature.\" It was thus to be the function of the new abstraction, or what Mondrian sometimes called \"abstract-real painting,\" to give us \"the image of this regained harmony.\" This was to be achieved, Mondrian avowed, by the creation of a pictorial composition \"more mathematical than naturalistic,\" an art of \"pure relationships\" that eschewed \"all that was capricious\" in nature in order to achieve \" _the most constant, the most determinate plastic expression of equilibrated relationship_ \u2014composition in rectangular planes.\" (The emphasis is Mondrian's.) Hence the _Compositions with Color Planes_ that set him on his new course in 1917.\n\n\"The New Plastic in Painting\" concludes by celebrating the emergence of \"a purer and purer mode of expression in art\" as a triumph of the modern era, and Mondrian was himself to live by this artistic faith for the rest of his life and make it a source of inspiration for others. All the same, an existential uncertainty about the role of nature\u2014in art and in life\u2014remained a vexing issue for Mondrian, and his philosophy of abstraction cannot be fully comprehended in isolation from it. Toward the end of \"The New Plastic in Painting,\" his ruminations on nature suddenly take the form of an arcane discussion of the male and female principles transposed to the realm of mystical archetypes. Identifying nature with the female principle and spirit with the male, he concludes that \"The female and male elements, nature and spirit, then find their _pure expression, true unity, only in the abstract_.\" (Again, the emphasis is Mondrian's.) This is clearly the sheerest mysticism, and more the expression of a lingering anxiety about nature than a resolution of the problem it posed for a sensibility nurtured on the occult. Yet it was upon such an uneasy mystical foundation that Mondrian's absolutist aesthetic of abstraction finally came to rest.\n\nIn this amalgam of mysticism, anxiety, abstraction, and a search for the absolute, we are reminded, as we often are with Mondrian, of Mallarm\u00e9\u2014especially of that passage in Mallarm\u00e9's prose in which the poet speaks so poignantly of \"isolating oneself, disdainfully, in the serenity of abstraction.\" \"Serenity appears as a quality both of the effect of the finished work,\" wrote one of Mallarm\u00e9's commentators, D. J. Mossop, \"and of the artist's feelings during the act of composition. It is, in turn, inseparable from the idea of harmony. The poet achieves harmony and serenity in himself and in his work by withdrawing from too active a participation in the passions of his theme so as to abstract a perfect order from the disorder common to the impressions made by normal experience and the feelings of lyrical inspiration.\"\n\nSuch was Mondrian's outlook on art and life, too, as he finally achieved \"the serenity of abstraction\" and embarked upon the work that would shape the course of abstract art for decades to come.\n\n[1995]\n\n## Art, Revolution, and Kazimir Malevich\n\n* * *\n\nThere are careers in the arts that epitomize the spirit of their time in so many salient respects that they attain a kind of mythological status in the eyes of posterity. The life and work of the Russian painter Kazimir Malevich (1878\u20131935), whose art is currently the subject of an extraordinary exhibition, constitute a career of this sort. In every aspect of his endeavors\u2014in his downfall as much as in his triumphs\u2014Malevich was the very archetype of the avant-garde artist in the age of revolution. Embracing without qualm or limit every extreme of belief and aspiration offered up by the tumultuous era in which he pursued his radical ideals, Malevich looked upon art as at once an absolute and an instrument\u2014as pure spirit, on the one hand, and a means of imposing a new social order, on the other. In this view of art and its functions, the work of art was thought to encompass and affect every sphere of life and thought\u2014and to do so, moreover, through the suppression of all visual reference to earthly existence. So total was the mission of art to be in Malevich's conception of it that it was only through the most complete abstraction that both its transcendent and its earthly goals were believed to be fully realizable. Upon abstract art there was thus conferred a burden and a status that in more traditional societies were considered the province of religion, politics, education, philosophy, and culture itself. No wonder Malevich called the movement that he founded Suprematism, for the role that he envisioned for art was nothing less than a universal supremacy in all fields of human endeavor.\n\nThat such a conception of art might harbor some profound contradiction, not to say a fundamental delusion; that it might deceive the artist himself as to what was central to his art and what merely circumstantial and peripheral; that in the end it was founded on a misconception not only of what art was capable of achieving but what the realities of life itself consisted of\u2014about none of this does Malevich appear to have been even vaguely concerned while he was riding high as a leader of the Russian avant-garde in the years immediately preceding and following the Bolshevik Revolution of 1917. Though more of a mystic than a political thinker, more indebted to the doctrines of P. D. Ouspensky than to those of Marx or Lenin, Malevich seems nonetheless to have truly believed that the ideas governing his conception of art\u2014ideas that are inseparable from the spiritualist, anti-materialist tenets of theosophy\u2014were in perfect agreement with the ideological imperatives of the new Soviet state.\n\nMalevich was by no means alone, of course, in his conviction that some ideal harmony united a belief in the occult with a commitment to Communism. Theosophy commanded a significant following among artists in many fields at the turn of the century, and not only in Russia; and it was often the case that an allegiance to theosophical doctrine inclined these believers in the occult to seek earthly salvation in utopian or otherwise extreme political positions. Figures as different as the Irish poet William Butler Yeats and the Dutch painter Piet Mondrian followed quite similar courses at the time, Yeats combining his belief in the occult with an attachment to fascism and Mondrian seeking to unite his theosophical faith with some vague notion of an ideal utopian future. What made Malevich's situation so special was the opportunity that the Revolution of 1917 gave to the entire Russian avant-garde to make their dreams of a new world come true. In the vast upheaval of the Revolution, it hardly mattered for the moment that the romance of the occult was not finally reconcilable with the imperatives of a Communist society. In the short term, at least, the occultists and the Communists were agreed in wishing to overturn every tradition and destroy every remnant of the culture and the society of pre-Revolutionary Russia. The result was one of the most bizarre intellectual alliances in the annals of the modern era: an art movement that owed much to the irrationalist, anti-materialist doctrines of the occult was empowered by the Leninist leaders of the Revolution to create a new culture in the name of dialectical materialism.\n\nWe have long known about the tragic fate that overtook this misbegotten alliance of art and politics early on in its history. Even before Lenin's death in 1924, many of the stellar talents of the avant-garde had fled into exile\u2014mainly to Germany and France\u2014in order to avoid the catastrophe that was clearly on the way. By 1922 the process of systematically stripping the avant-garde of its powers, which in the first years of the Revolution had been enormous, had begun; and under Stalin modernism was categorically condemned as \"bourgeois\"\u2014a charge likely to lead directly to the Gulag, if not to the firing squad. The dreams of the avant-garde were quickly supplanted by the long dark night of the Stalinist terror.\n\nIt was not the least of Malevich's distinctions that he survived this ordeal, even managing to die of natural causes in his own bed\u2014itself something of an achievement for a figure of his eminence in the Soviet Union in the year 1935. That the price of survival was a fundamental betrayal of his own artistic ideals is made plain for us to see in the current Malevich retrospective\u2014the most complete account of the artist's work we have seen in this country. When we come to the last room in this exhibition and find the painting called _Reapers_ (circa 1929\u20131932), which depicts in a more or less realist manner three robust peasant women harvesting the grain, the only appropriate response is a feeling of utter revulsion. Not only does the painting represent an abject surrender to Stalin's newly proclaimed doctrine of Socialist Realism, but its very subject matter\u2014those well-fed peasant women harvesting the grain in what looks like a pastoral idyll\u2014is the most cynical propaganda. For it was precisely at this moment in history that Stalin was carrying out his program of collectivization, which caused the deaths of millions of peasants and resulted in the destruction of Russian agriculture.\n\nEven if we knew nothing of Stalin's murderous policies, however, this painting would be difficult to take in purely artistic terms, for it marks a return to everything Malevich had categorically rejected in his art. In the development that is clearly and beautifully traced in this retrospective exhibition, we follow an immensely gifted and adventurous painter as he makes his way through the various styles\u2014from Impressionism and Symbolism to Fauvism and Cubism and Futurism\u2014that have defined the course of modernist art, until the moment arrives when he takes off into a realm entirely his own with the invention of a special form of abstraction. There is something very moving about this exhibition, for Malevich was in many respects a far better painter\u2014once he hit his stride, anyway\u2014than he has generally been thought to be in the past. Certain of his Cubist and Cubo-Futurist pictures are undeniable masterpieces of their kind. They owe much to the precedents of the School of Paris, but it was in the nature of the Russian avant-garde to seize upon the most radical of these precedents and then carry them a step further\u2014and to do so, moreover, with a speed and confidence and inventiveness that were often unrivaled elsewhere. Malevich was particularly audacious in this respect. In the Suprematist abstractions he produced in the decade beginning around 1915, he created not only some of the most original abstract paintings of the age but certain pictorial conventions that have continued to serve as guidelines for the more austere varieties of abstract art down to the present time. Compared with the works Malevich produced between 1915 and 1926, much that was acclaimed for its originality and audacity in the Minimalist art of the 1960s, for example, seems distinctly secondhand. Malevich created the aesthetic territory that the Minimalists later colonized and expanded.\n\nI doubt if there has ever been a more beautiful presentation of Malevich's work than the one that Angelica Rudenstine has given us at the National Gallery in Washington, where this retrospective opened its current American tour. The emphasis in the installation and layout of the exhibition in Washington is decidedly\u2014and properly, in my view\u2014on the _art_ , rather than on the ideas governing the art; and this emphasis has the salutary effect of establishing Malevich as an aesthetic sensibility, as the successful exponent of certain aesthetic strategies and devices, and thus tends to minimize the politics, the mysticism, and the rest of the ideological baggage that occupied such an important place in the artist's life and thought. This seems to me exactly the right emphasis, for it is _only_ as an artist of considerable distinction and originality that Malevich continues to make a major claim on our attention. As a thinker he was quite hopeless. His copious writings, maddeningly repetitious and replete with every kind of mystification about art, politics, history, and the life of the spirit, now can have only a documentary and historical interest for us. There was also in Malevich what can only be described as a certain nuttiness. In no other member of the Russian avant-garde was the confusion of religion and politics quite so complete. He was given to comparing Lenin to Christ, and in what was, perhaps, the most extreme avowal of his irrational conflation of art, religion, and politics, he recommended to his Soviet comrades that Lenin's body be placed \"in a cube, as if in eternity,\" and urged that \"Every working Leninist should have a cube at home, as a reminder of the eternal, constant lesson of Leninism,\" etc. \"The cube,\" writes Robert C. Williams in _Artists in Revolution_ (1977), \"would symbolize Lenin's immortality. . . . In painting, the cube moving through two-dimensional space at an angle had created the patterns of Suprematism; in life, it would now help to build a new Soviet culture.\" One hardly knows whether to laugh or cry at the intensity of Malevich's conviction in such matters. Yet it was this same kind of conviction that led to the creation of such an extraordinary body of paintings.\n\nWe shall be a long time separating the art of the Russian avant-garde from the political tragedy that encloses it, but it is now more important than ever before that we do so, and this exhibition marks an important step in that direction. Because politics plays such an important role in the life of art today\u2014such a destructive role\u2014there is a tendency to sentimentalize and idealize the political circumstances in which Malevich and his Russian contemporaries produced their art; a tendency, in other words, to make a romance of revolution in its relation to the art of the avant-garde. To do so, however, requires a large measure of historical amnesia. When we come to that last room in the Malevich retrospective and encounter _Reapers_ , we find that Malevich, too, was finally devoured by the Revolution he had so ardently supported. He survived by dishonoring the art that won him his place in history. We should not contribute to that dishonor by making a romance of the politics that was finally so murderously at odds with the art often produced in its name.\n\n[1990]\n\n## Abstraction and Utopia\n\n* * *\n\nI. _From Theosophy to Utopia_\n\n> _[Suprematism] will liberate all those engaged in creative activity and make the work into a true model of perfection_.\u2014El Lissitzky, 1920\n> \n> _Abstract art, I thought, creates new types of spatial relationships, new inventions of forms, new visual laws\u2014basic and simple\u2014as the visual counterpart to a more purposeful, cooperative human society.\u2014_ L\u00e1szl\u00f3 Moholy-Nagy, 1928\n> \n> _In the future, the realization of pure plastic expression in palpable reality will replace the work of art. But in order to achieve this, orientation toward a universal conception and detachment from the oppression of nature is necessary. Then we will no longer have the need of pictures and sculpture, for we will live in realized art_.\u2014Piet Mondrian, 1942\n> \n> _To construct a utopia is always an act of negation toward an existing reality, a desire to transform it_.\u2014Leszek Kolakowski, 1968\n\nAbstract art\u2014painting and sculpture that makes no direct, immediately discernible reference to recognizable objects\u2014was born of an alliance of modernist aesthetics and the occult doctrines of theosophy in the second decade of the twentieth century. Its first masterworks were produced by Vasily Kandinsky in Germany, Piet Mondrian in the Netherlands, and Kazimir Malevich in Russia. Yet no sooner was this new artistic convention established as an influence on the European avant-garde than it was quickly appropriated by still another mode of thought\u2014utopianism\u2014which similarly rejected commonly held conceptions of \"reality\" in the name of a visionary ideal. This utopian ideal, while no less mystical in some respects than that of theosophy, nonetheless differed from it in locating the realm of future perfection in the material world. Toward the material world utopian ideology adopted an attitude as radical\u2014and, in fact, as otherworldly\u2014as any to be found in occult doctrine, but it did not question the existence of the material world. Its goal was to change it\u2014to eliminate its imperfections in order to establish an ideal harmony, at once social and spiritual, in every sphere of earthly human endeavor.\n\nWhereas theosophical belief is essentially religious in nature, utopian thought is fundamentally political, for it seeks to realize its model of human perfection by bringing about a radical transformation of society and its institutions. While both envision an ideal future, it is in the nature of utopian thought to imagine that the particular \"heaven\" it hypothesizes can be created on earth.\n\nIt should not be supposed, however, that an allegiance to mysticism precludes a commitment to utopian ideology. On the contrary, these apparently contradictory doctrines have proved to be highly compatible systems of belief. Both refuse to acknowledge either the limits of human nature or the legitimate conflicts of interest that characterize virtually every form of human society. Both are attempts to transcend the contingencies of history for the purpose of achieving an eternal order so perfect as to lie beyond the need of dissension or modification. Yet utopian ideology departs from occult belief in a fierce yearning to effect a sweeping transformation of material life by means of political action.\n\nThe political models to which the pioneers of abstract art attached their utopian dreams, though generally of a socialist character, were by no means uniform, however much they may have been thought to resemble each other in the eyes of their adversaries. The leading figures of the Russian avant-garde\u2014Malevich, Tatlin, Rodchenko, Lissitzky, Popova, et al.\u2014were fervent supporters of revolution, and welcomed the Bolshevik seizure of power in 1917 as the dawn of a new millennium. And it was in Russia, in the immediate aftermath of the Revolution of 1917, that these champions of abstraction came to enjoy an unprecedented authority over the conduct of cultural life.\n\nThe utopian impulses of the De Stijl movement in the Netherlands, which also dates from 1917, were more restricted, and certainly less strident. While no less militant than the Russian avant-garde in its artistic objectives\u2014and initially sympathetic to the Bolshevik cause\u2014the De Stijl group remained aloof from any organized involvement in revolutionary politics. De Stijl was a movement confined to avant-garde artists, architects, designers, and art theorists. While in some respects an anti-war movement that took the horrors of the First World War to represent everything that was moribund in the European past, it never came under the jurisdiction of political commissars or party functionaries. Its primary influence\u2014its primary achievement, too\u2014would always be aesthetic. Its concept of utopia had more to do with ideas about painting and a redesign of the man-made environment than with programs of political action. It helped, of course, that the Netherlands itself remained neutral in the war.\n\nIn Germany, and in Central and Eastern Europe, where both Russian modernism and the De Stijl movement wielded immense influence in avant-garde circles, the relation that obtained between the aesthetics of abstraction and the utopian impulse varied from group to group\u2014and sometimes from year to year\u2014in the wake of the Bolshevik Revolution and the termination of the First World War.\n\nThe most celebrated citadel of avant-garde design to be established in Europe in the period between the two world wars\u2014the Bauhaus, founded by the German architect Walter Gropius in Weimar in 1919\u2014initially billed itself as the \"Cathedral of Socialism,\" and from the outset was imbued with a radical fervor that was believed, especially by its enemies on the political right, to reflect revolutionary goals. Yet the radicalism of the Bauhaus, while generally socialist in its announced ideals, was in practice a more volatile and bizarre amalgam of artistic, mystical, and collectivist ideas than its socialist reputation suggests. Its ranks were rife with the followers of occult sects, bohemian lifestyles, food cults, and sundry other manifestations of a freewheeling counterculture.\n\nAt the same time, practical considerations required the Bauhaus to collaborate with industry and commerce, for its principal endeavors were addressed to the theory and practice of architectural, industrial, and graphic design, not to the production of fine art. In its early years its contribution to abstraction was more theoretical than practical. It wasn't until the late 1920s, when the Bauhaus had entered upon the crisis that preceded its shutdown in 1933, that the most distinguished artists on its teaching staff\u2014Vasily Kandinsky and Paul Klee\u2014were first permitted to offer studio instruction in abstract painting.\n\nIt was in Russia, far more decisively than elsewhere, that abstraction was first drawn into the orbit of utopian thought in ways that determined its fate. In no other country did its peculiar combination of aesthetic radicalism and metaphysical yearning so quickly impel abstraction in the direction of political revolution. And in no other country did a triumphant revolution go so far in embracing abstraction as an instrument of cultural and political change. It was therefore in Russia, in the decade following the Revolution of 1917, that a new alliance of abstract art and utopian ideology was put to its severest historical test.\n\nThe artistic momentum that set the Russian avant-garde on this radical course was already well established before the Revolution itself. The roots of Russian modernism\u2014abstract art included\u2014are to be found in the cosmopolitan culture of the pre-Soviet Russian liberal intelligentsia that was in close touch with the latest artistic and intellectual developments in France, Germany, and Italy. The ideas of the Expressionist movement in Germany, including Kandinsky's _Blue Rider_ group in Munich, and those of the Futurist movement in Italy were well known in Moscow and St. Petersburg, but it was the art of the Paris avant-garde that dominated the thinking of the Russian modernists in their accelerating quest for the absolute.\n\nCrucial to this development were the great collections of the modernist School of Paris that had been amassed by two remarkable Russian businessmen, Sergei Shchukin and Ivan Morozov. With their immense holdings in the most advanced painting of the Paris school\u2014especially the works of Matisse and Picasso\u2014these collections played a central role in acquainting the artists of the burgeoning Russian avant-garde with splendid examples of the new Fauvist and Cubist art long before such objects were exhibited in significant numbers in the museums of the Western world. Then, too, a number of the artists who would shortly be counted among the early creators of abstract art in Russia made their personal pilgrimages to the French capital. Mikhail Larionov, who in 1913 issued a manifesto on behalf of Rayonist painting\u2014a more or less abstract pictorial style claiming to be based on \"forms chosen by the will of the artist\"\u2014had visited Paris as early as 1906 in connection with the Russian exhibition organized by Sergei Diaghilev for that year's Salon d'automne. (It was in the Salon d'automne a year before that the Fauvist paintings of Matisse were first exhibited.) Liubov Popova, one of the stars of the early Soviet avant-garde, served an apprenticeship to Cubism in Paris in 1912\u20131913.\n\nAround that time, too, Vladimir Tatlin, who would win lasting fame for his _Monument to the Third International_ (1919), wangled his way into Picasso's Paris studio, where he had his first glimpse of the artist's constructed Cubist reliefs\u2014a new mode of sculpture not yet known to the public. This led, in 1915\u20131916, to the creation of Tatlin's even more radical Corner Counter-Reliefs, abstract constructions made of metal and other materials that, unlike Picasso's constructed sculpture, no longer made any discernible reference to recognizable objects. Similar pilgrimages to Paris were made by Naum Gabo, Ivan Puni, and others, and key documents of the Paris avant-garde, like the writings of Albert Gleizes and Jean Metzinger on Cubism, were promptly translated into Russian.\n\nRussian modernists were by no means alone, of course, in submit-ting to the influence of the Paris avant-garde in this period, but what they made of that influence was nonetheless remarkable even in an era known for its artistic audacities. For it was in the nature of Russian modernism to seize upon every innovation imported from abroad and propel its development in an even more radical direction than its originators had envisioned. As a result, the pre-Soviet Russian avant-garde promptly created more extreme varieties of pure abstraction than could be found in the advanced art of any Western European or North American art capital.\n\nIn this headlong rush into pure abstraction, a volcanic mixture of occult doctrine, political fervor, futuristic fantasy, and incendiary dreams of cultural revolution acted as a powerful stimulus. However pure or simplified its forms, abstract art in Russia was seldom looked upon as art-for-art's-sake even by pronounced aesthetes. Whatever its roots in mystical doctrine, abstraction was conceived to serve social, historical, and other purposes larger than itself. In the years leading up to the Revolution of 1917, cultural life in Russia acquired an increasingly apocalyptic character that favored extreme solutions to all the problems of art, life, and thought. In this circumstance, abstract art\u2014precisely to the extent that it eliminated all trace of or reverence for the world as it was\u2014had the advantage of addressing its public as, in Leszek Kolakowki's words, \"an act of negation toward an existing reality.\" This in itself gave to the aesthetics of abstraction a utopian imperative.\n\nII. _Theo van Doesburg and De Stijl's Utopian Aesthetic_\n\n> _The creation of a new world has commenced_.\u2014Theo van Doesburg, _De Stijl_ , August 1921\n\nTo succeed in elevating its acts of negation to the level of a comprehensive and influential artistic program, every avant-garde movement requires, in addition to the achievement of a master talent to set its aesthetic course, the services of a robust _animateur_ \u2014an artist-activist capable of commanding attention, forging alliances, winning converts, and otherwise advancing the cause. Mondrian, while undoubtedly De Stijl's master talent and the only major artist to be closely associated with the group, was temperamentally ill-equipped to perform such worldly tasks. The role of the movement's _animateur_ therefore fell to Theo van Doesburg (1883\u20131931), a determined avant-gardist painter, writer, and intellectual whose artistic ambitions often exceeded the scope of his talents but whose gifts as a cultural impresario and propagandist were irrepressible.\n\nIt was van Doesburg who organized De Stijl as a movement and established its presence in the European avant-garde in the 1920s. He wrote some of its key documents, he was the editor of its journal\u2014called _De Stijl_ \u2014and he was its principal liaison with other avant-garde groups, particularly Dada and the Bauhaus. He became, in fact, something like the movement's Apollinaire or Marinetti\u2014a tireless proselytizer of its radical aesthetic who was himself \"the prototype of the anti-bourgeois,\" as the Dutch writer Joost Baljeu called him. As early as 1914, though he had yet to produce a significant work of art or anything like an artistic manifesto, van Doesburg imagined himself making, as he said, \"a daring, spiritual crusade throughout artistic and intellectual Europe.\" His opportunity came with the founding of De Stijl in 1917.\n\nThe ideas and aspirations that Mondrian and van Doesburg brought to De Stijl were anything but identical, however. For Mondrian, painting was a spiritual vocation. Once established, the principles of his Neo-Plasticism\u2014the reduction of pictorial form to the square and the rectangle, with a palette limited to black, white, and the primary colors\u2014remained inviolate. Anything that suggested a residual dependency on subjective expression was regarded as a failure to conform to the universal \"reality\" Mondrian believed he had achieved in his Neo-Plastic compositions. Hence the strict observance of a pictorial syntax based on the right angle. As Mondrian once observed to the American painter Carl Holty: \"Curved lines are too emotional.\" \"Curved lines,\" of course, was a reference to Nature. Even shades of grey had to be eliminated from the painter's palette as an impermissible compromise.\n\nWhatever may have been his interest in some possible equivalent of Neo-Plasticism in the fields of architecture and design\u2014he seems to have been content to consign such endeavors to some hypothetical future\u2014Mondrian himself remained exclusively occupied with the \"reality\" of his own painting and the synthesis of aesthetic and occult ideas governing its creation. Yet under van Doesburg's leadership, De Stijl embraced a cultural agenda that went beyond painting to include not only architecture and design but literature, music, and indeed\u2014at least in theory\u2014all of cultural life. As a consequence, Mondrian's relation to De Stijl was bound to be more problematic than anyone, least of all van Doesburg, was willing to acknowledge at the outset of their collaboration. Mondrian was nothing if not candid in underscoring for van Doesburg the exact nature of his own artistic interests. \"You must remember,\" he wrote to van Doesburg in February 1917, \"that my things are still intended to be paintings, that is to say, not part of a building. Furthermore, they have been made in a small room.\" Which, at the very least, placed his painting at a certain distance from the architectural ambitions van Doesburg envisioned for De Stijl.\n\nYet the originality of Mondrian's achievement as a pure abstractionist was fundamental to the founding of De Stijl. The aesthetic that Mondrian was perfecting during the early years of the war was understood to be essential to the movement's success\u2014so much so that van Doesburg agreed to postpone the publication of the journal that became _De Stijl_ until Mondrian felt ready to participate. This proved to be a shrewd decision, for the treatise that Mondrian wrote for _De Stijl_ at van Doesburg's instigation on \"Neo-Plasticism in Pictorial Art,\" which was serialized in the issues of 1917\u20131918, did much to define the radical aesthetic character of the entire movement. For the moment, moreover, van Doesburg's own work as an abstract painter, though it still derived from the kind of representational subject matter that Mondrian had now permanently abandoned, showed every outward sign of conforming to the master's prohibitions. So certain fundamental differences that separated their respective views not only of art but of the movement itself were left in abeyance.\n\nYet painting was never more than one of van Doesburg's many avant-garde interests, and not always the dominant one. He certainly did not share Mondrian's view of painting as a spiritual vocation. For the whole mystical element in Neo-Plasticism he seems to have felt little affinity. He was more interested in the social functions of abstract art than in its religious significance or its occult sources. He was thus just the _animateur_ that was needed to bring De Stijl's mystical conception of abstract art into alignment with the aspirations of utopian politics. It is no easy task, however, to establish the exact character of van Doesburg's own political beliefs. He was anything but a stickler for ideological clarity or linguistic precision.\n\n\"Capitalists are deceivers,\" he wrote in 1921, \"and so are the Socialists.\" And further: \"The first, second and third Socialist Internationals constituted ridiculous nonsense,\" and so on. What he then called for as an alternative to this \"nonsense\" was an \"International of the Mind,\" which he seems to have conceived of as something resembling De Stijl's ambition to transcend what was stigmatized as Europe's tradition of \"intellectual and material individualism.\" In the utopian \"International of the Mind\" that van Doesburg envisioned, individualism in society was\u2014like lyricism in art\u2014rejected in favor of something more \"universal.\" In practice, this might mean nothing more revolutionary than the effort of abstract artists like himself, who made an absolute of straight lines, geometric form, and unmixed color, to ally themselves with like-minded modernist architects in creating immaculate urban designs devoid of traditional decorative embellishments. Or it might signify something more political\u2014the dream of a utopian collectivism in which the De Stijl aesthetic would inevitably be established as the final arbiter not only of art and design but of all cultural life. Van Doesburg's writings on the subject, while unmistakably collectivist in spirit, are consistently vague about particulars.\n\nLecturing on \"The Will to Style\" in Germany in 1922, for example, he insisted that \"Only collective solutions can be decisive in the realms of both politics and art,\" which seemed to place his program squarely in the camp of the Bauhaus radicals and the Bolshevik loyalists in the Soviet avant-garde. Yet van Doesburg himself turned out to be too much of an unreconstructed, ego-driven modern individualist to conform even to the very limited \"collective solutions\" of his comrades in the De Stijl circle\u2014Mondrian in particular\u2014never mind the more sweeping solutions advocated by his counterparts in the Soviet avant-garde.\n\nBy 1926 van Doesburg had openly broken with the principles of Mondrian's Neo-Plasticism in order to advance his own mode of abstraction. This was grandly dubbed Counter-Plastic Elementarism, which amounted to little more than the introduction of the diagonal into Neo-Plasticism's orthogonal orthodoxy. Then he compounded the heresy by ridiculing the concept of \"spirit,\" which he now ascribed to \"witches, fortune-tellers and Theosophists\"\u2014in other words, charlatans. This was, of course, a calculated affront. Theosophy was the philosophical foundation on which Mondrian had based his entire abstract _oeuvre_. It was one of the core doctrines on which De Stijl had been founded. The very terms \"De Stijl\" and \"Neo-Plasticism\" were derived from the mystical writings of Mondrian's philosophical mentor, M. H. J. Schoenmaekers. This was heresy indeed. It was also a sign that van Doesburg, having established relations with Dada and the Bauhaus, now saw himself as the _animateur_ of a much larger project\u2014a Europe-wide, if not worldwide, avant-garde in the service of a utopian aesthetic.\n\nIn retrospect, van Doesburg can be seen to have been preparing himself for this larger role for some years. A man much given to adopting pseudonyms\u2014even \"Theo van Doesburg\" was a pseudonym; his real name was Christiaan Emil Marie K\u00fcpper\u2014van Doesburg had established a separate identity as a Dutch Dadaist poet, calling himself I. K. Bonset, as early as 1918 when he began a correspondence under that name with Tristan Tzara, one of the original instigators of Dada. (Mondrian, who was at first unaware of the deception, actually cautioned van Doesburg not to publish I. K. Bonset in _De Stijl_.) \"There is little doubt,\" writes Joost Baljeu, \"that van Doesburg saw Dada's revolutionary character and its engagement in the destruction of an old culture as a necessary preparation for the realization of De Stijl's utopian aims.\" Be that as it may, van Doesburg was clearly attracted to the anarchist-nihilist element in Dada even as he was promoting the principles of utopian order as the leader of De Stijl. Under still another pseudonym\u2014Aldo Camini\u2014van Doesburg also adopted the stance of a Dada anti-philosopher, drawing his inspiration from Tzara's _Manifesto of Mr. Aa the Anti-philosopher_ (1918), to attack Kant, Hegel, and the whole school of German idealist philosophy (for which other members of the De Stijl circle felt a close affinity) as well as Schoenmaekers, De Stijl's resident philosopher.\n\nMeanwhile, van Doesburg was also busily at work establishing his bona fides with the leadership of the Bauhaus. He later claimed that Gropius had invited him to join the Bauhaus faculty, but this, like so many of van Doesburg's boasts, turned out to be something of a fiction. According to Gropius,\n\n> I have never invited van Doesburg to the Bauhaus. He came there of his own initiative because he was attracted by our courses. He hoped to become a professor here at the Bauhaus, but I did not give him a position, since I judged him to be aggressive and fanatic and considered that he possessed such a narrow, theoretical view that he could not tolerate any diversity of opinion.\n\nThe latter point isn't quite correct, however, for van Doesburg wasn't at all averse to a \"diversity of opinion\" so long as he was free to claim every one of the diverse ideas he espoused as wholly his own.\n\nIn the end, it remains difficult to know what, if anything, van Doesburg really believed in beyond a vague, idealized utopian future from which the \"old culture\" of Europe had at last been expunged\u2014a project in which the aesthetics of abstraction was seen to play a key role, and in which van Doesburg himself would be the intellectual leader. It was not to be, of course. By the end of the 1920s the modernist avant-garde was already in retreat in the Soviet Union. In Western Europe, Dada was succumbing to the blandishments of Surrealism, which rejected the aesthetics of abstraction, and the Bauhaus was heading toward political disaster. As it was, van Doesburg was probably fortunate to die when he did\u2014in 1931, at the age of fifty-eight\u2014with his utopian dreams intact and his megalomania undiminished. Had he lived another decade, he would probably have ended up as a university professor in capitalist America, which, while not a fate worse than death, would not have borne much resemblance to that \"daring, spiritual crusade throughout artistic and intellectual Europe\" that he had imagined for himself in 1914. But then, the entire character of \"artistic and intellectual Europe\" was altered when Hitler came to power in 1933 in ways that van Doesburg had never imagined, and, in that altered Europe, De Stijl's vision of a utopian future was one of the many avant-garde dreams that died of inanition.\n\nMondrian, however, proved to be a hardy survivor of De Stijl's demise and illusions. Working in Paris in the heyday of Surrealism's dominion in the 1930s, he went his own way, secure in his convictions, confident of his goals, never wavering in the faith that supported an art untouched by adverse circumstance; and in the end, in New York City of all places, found his final inspiration in the closest thing to the utopian future he had ever encountered.\n\nIII. _Abstraction and Revolution_\n\n> _Utopias are an expression of aspirations that cannot be realized, of efforts that are not equal to the resistance they encounter.\u2014_ Alexander Bogdanov, _Red Star_ (1908)\n> \n> _A revolution strengthens the impulse of invention. That is why there is a flourishing of art following a revolution. . . . Invention is always the working out of impulses and desires of the collective and not of the individual_.\u2014Vladimir Tatlin, _International Iskusstv_ (1919)\n> \n> _The reality principle\u2014_ la force des choses _\u2014will, in the end, always prevail over utopian passions_.\u2014Irving Kristol, \"The Adversary Culture of Intellectuals\" (1979)\n\nThe fate of abstract art in Russia in the decade following the Revolution of 1917 proved to be one of the most tragic chapters in the annals of twentieth-century modernism. Its only near rival among the political calamities that beset the modernist movement in the early decades of the century occurred in Germany in the 1930s with the unleashing of Hitler's war on \"degenerate\" art. There was this important difference, however, between the Nazi campaign to abolish modernist art in Germany and that of the Soviet Union under Lenin. Hitler never pretended to set up as a patron of modernist art. However grudgingly, Lenin\u2014or the regime over which he presided as dictator, anyway\u2014did so pretend for a very short but crucial period, thus inducing the illusion among members of the Russian avant-garde that modernism in the arts would be embraced by the Bolsheviks as a privileged coefficient of their political revolution.\n\nThe fragility of this ill-fated misalliance between modernist art and political revolution was signaled by the fact that the Bolshevik loyalist to whom Lenin entrusted authority over the arts in the early years of the Soviet regime\u2014Anatoly Lunacharsky, named People's Commissar for Public Enlightenment in 1917\u2014was not himself exactly an enthusiast for avant-garde art. He was, to be sure, far more closely acquainted with the modernist movement than Lenin and far more tolerant of its innovations, having come to know some of the \u00e9migr\u00e9 Russian modernist painters in Paris before the First World War. Yet, while not as paranoid about the avant-garde as Lenin, who regarded it\u2014not wholly incorrectly, alas\u2014as a refuge for \"escapees from the bourgeois intelligentsia,\" Lunacharsky clearly harbored serious doubts about the modernists he officially supported. To direct the Visual Arts Section of Narkkompros, the huge bureaucracy in his charge, Lunacharsky appointed not one of the militant advocates of Russian abstraction but David Shterenberg, a modernist painter he had met in Paris who was more amenable to welcoming a broader range of artistic loyalties than was common among the doctrinaire abstractionists. (Shterenberg, though a supporter of the Revolution, was not himself a Bolshevik.) Lunacharsky also named Marc Chagall, another of the modernist painters he got to know in Paris, as Commissar of Art in the artist's home town of Vitebsk. Abstract art, which in retrospect remains the most significant achievement of the Soviet avant-garde in this period, was never a branch of cultural life to which either Lunacharsky or his comrades in the Bolshevik hierarchy felt any personal or ideological commitment. On the contrary, abstraction was often under attack in the official Soviet press as a development inimical to the goals of proletarian culture even while the abstractionists enjoyed the official support of Lunacharsky's bureaucracy.\n\nFrom the outset, then, the relation that obtained between an avant-garde that made an orthodoxy of abstraction and the firebrands of what, in the period from 1917 to 1921, was called War Communism, was highly problematic. Within the ranks of the dedicated abstractionists, moreover, loyalty to the Bolshevik regime was similarly riddled with doubt and suspicion. Vasily Kandinsky, the most accomplished of the abstract painters to enjoy official support, was never a true believer in the Revolution. Forced by the outbreak of the world war to leave Germany for his native Russia, where he was living when the Revolution erupted, Kandinsky married the daughter of a Russian general, and in 1922, having correctly perceived that the avant-garde would have no future in the Soviet Union, he decamped for the Bauhaus in Weimar.\n\nArtistically, however, Kandinsky's period of repatriation in Russia brought a major change in his work. Under the influence of Malevich and his Suprematist circle, Kandinsky abandoned the painterly expressionist manner that had marked his first abstract paintings for the geometric and other hardedge forms now favored by the Russian abstractionists. Indeed, he became enough of a militant in his pursuit of this newly adopted pictorial orthodoxy to join Malevich in ousting Chagall from his official post in Vitebsk on the grounds that the latter's work was insufficiently abstract\u2014and thus, in the minds of the abstractionist militants, insufficiently revolutionary. In 1921 a disillusioned Chagall departed once again for Paris, where he had made his first reputation as a member of the European avant-garde before the Revolution. He thus preceded Kandinsky into permanent exile in Western Europe by a year. Whatever differences divided Kandinsky and Chagall in the realm of pictorial aesthetics, they turned out to be very much alike in rejecting the political control of art that was already in progress in the Soviet Union.\n\nAnother prominent defector from the Soviet avant-garde was the sculptor Naum Gabo (1890 \u20131978), who left Russia for Germany the same year as Kandinsky. Gabo was one of the pioneers of the kind of open-form abstract sculpture that, somewhat confusingly, has come to be called Constructivist. What lends confusion to the term in relation to sculpture like Gabo's is that while the latter is literally a construction of thin, sheetlike materials\u2014first painted cardboard, then wood and metal and transparent plastics\u2014that are cut and joined to shape three-dimensional \"volumes\" consisting of space rather than mass, Constructivism as a movement in the Soviet avant-garde represented a philosophy of art diametrically opposed to Gabo's. Gabo regarded his constructions as works of art conceived in a spirit that reflected new scientific thinking about space and matter. This was why he called his constructed sculptures \"Realist\" rather than Constructivist. For, in the Soviet context, Constructivism was one of the names given to a movement that repudiated fine art\u2014art that was now stigmatized as mere \"speculative activity\"\u2014in favor of the kind of functionalist art that was more explicitly designed to serve the practical and propaganda interests of the Revolution: posters, photography, book illustration, even the design of clothing, tableware, and furniture. It was mainly in the West that sculptures like Gabo's came to be called Constructivist, and Gabo himself eventually accepted the name when he came to live first in England and then in the United States.\n\nWhat adds still another layer of confusion to the issue is the fact that many of the leading advocates of Constructivism in the Soviet Union\u2014Tatlin, Rodchenko, and Lissitzky, among them\u2014first distinguished themselves as votaries of abstractionist fine art. Then, to advance the cause of Constructivist applied art, they adapted the formal vocabularies they had themselves invented for their fine-art abstraction to the design of useful objects, while at the same time declaring in the absolutist political spirit of the time that the fine arts, as traditionally conceived, must now be permanently rejected as a useless and harmful relic of a reactionary bourgeois culture. In a manifesto written in May 1921, Alexander Rodchenko (1891\u20131956), one of the most gifted abstract artists of his generation, thus repudiated all of his past artistic accomplishments when he announced that \"up to now we did not see this simple thing called life, we did not know that it was so simple, clear, which has to be organized and purged of all kinds of adornment.\"\n\nHaving succeeded in purging painting and sculpture of everything but their most minimal abstract forms, Rodchenko and his Constructivist cohorts then undertook the much larger task of redesigning the objects of daily life for the Soviet masses along similarly minimalist lines. This was yet another utopian endeavor bound to fail\u2014and fail it largely did in a very short time. Whatever small success this Constructivist initiative could make claim to was largely confined to the VKhUTEMAS, the government-sponsored design workshops where revolutionary theory enjoyed a distinct priority over the actual production of makable objects. This is one of the reasons, incidentally, why theory-obsessed academics in our own day feel such a strong affinity for the failed experiments of the Soviet Constructivists: they exist mainly in the realm of theory.\n\nGiven the conditions of Soviet society in the immediate aftermath of the Revolution of 1917, it was the very nature of the Constructivist enterprise that doomed it to failure. For one thing, the Bolshevik commissars on whose support these experiments in modernist design depended were, if anything, even more philistine in their aesthetic tastes than the pre-Revolutionary bourgeoisie the Constructivists scorned. Then, too, the faltering Soviet economy was simply not equipped to produce most of the projects that were proposed. As a consequence, most of these projects remained on the drawing board. And needless to say, the Russian masses weren't exactly clamoring for modernist design, and it certainly wasn't in order to drink out of Suprematist teacups or buy their daily copies of _Izvestia_ at a Constructivist kiosk that the Bolsheviks had fought a bloody civil war.\n\nThe ultimate historical irony of the whole Constructivist misadventure is that what survives of its endeavors\u2014Rodchenko's brilliant photographs, Popova's extraordinary textile designs, and Alexandra Exter's sets and costumes for certain theatrical productions\u2014are now admired for precisely the aesthetic, which is to say, the formal, ideas they embody rather than for the political motives that prompted their creation. What also needs to be noted about this failed utopian venture is the role it played in terminating one of the most remarkable chapters in the entire history of abstract art. Years before Stalin was in a position to demonize abstraction as \"bourgeois formalism,\" the campaign to condemn the creation of abstract painting and sculpture as a cultural activity politically at odds with the goals of the Revolution and thereby to render abstraction officially insupportable was initiated from within the ranks of the Soviet avant-garde itself. Stalin only codified and criminalized what Constructivist militants like Tatlin and Rodchenko had first proposed in rejecting abstract art, and then promptly condemned Constructivism as well.\n\nIV. _Abstraction and the Bauhaus_\n\n> _\"Art and Technology, a New Unity\" is the slogan on our Bauhaus poster at the railroad station. Oh dear, will there really be no more art than the technology necessary to become profitable from now on? Since times are bad, art has to give in, and authority is granted it by associating with usable things\u2014a hateful conception_.\u2014Lyonel Feininger to Julia Feininger, 1923\n> \n> _Our aims are becoming more and more clear and have been expressed very precisely in an article by Moholy. They go against my grain in every respect, distress me terribly. What has been \"art\" for ages is to be discarded\u2014to be replaced by new ideals. There is talk only about optics, mechanics and moving pictures. . . . Klee was very depressed yesterday when we talked about Moholy. He called it the \"prefabricated spirit of the time.\"_ \u2014Lyonel Feininger to Julia Feininger, 1925\n> \n> _The concepts of mechanics, dynamics, statics and kinetics, the problem of stability, of balance, were examined in terms of three-dimensional forms and the relationship between materials was investigated as were methods of construction and montage. . . .Today's sculptor knows scarcely anything about the new fields in which the engineer works_.\u2014L\u00e1szl\u00f3 Moholy-Nagy, 1928\n> \n> _It took quite a while to get under way the kind of work which later made the Bauhaus a leader in designing for the lighting fixture industry_.\u2014L\u00e1szl\u00f3 Moholy-Nagy, 1938\n\nIt is one of the paradoxes of the legend of the Bauhaus, which the German architect Walter Gropius founded in Weimar in 1919 to reform and revitalize modern architectural practice, that it is often thought to have been one of the principal citadels of abstract art in Europe in the 1920s. The fact is, however, that no instruction in abstract painting\u2014or, indeed, painting itself\u2014was permitted at the Bauhaus until the last five years of its fourteen-year existence, and even then it was accorded a very low priority. Gropius disliked easel drawing, and his aversion to it was shared by Mies van der Rohe, who presided over the school in its final years. Traditional instruction in drawing and painting was never part of the Bauhaus curriculum.\n\nThe legend of the Bauhaus as a citadel of abstraction nonetheless persisted because a number of distinguished artists who have passed into history as masters of abstract art\u2014Kandinsky, Klee, Albers, and Moholy-Nagy\u2014were among the best-known members of the Bauhaus faculty. Then, too, an even larger number of lesser painters who had been students at the Bauhaus in the 1920s went on to acquire reputations as abstract artists working in what came to be recognized as a Bauhaus style. Most important of all was the fact that the basic course of instruction at the Bauhaus was designed to comprehend the entire range of visual experience in ways that eliminated from consideration all but its most universal and axiomatic attributes. As a consequence, abstraction of a certain kind\u2014geometrical, Constructivist, and depersonalized\u2014came to look as if it had been conceived to illustrate the theoretical principles of Bauhaus pedagogy.\n\nThe _Vorkurs_ , or required preliminary course at the Bauhaus, was concentrated on theories of form and perception, on the one hand, and on the nature of materials, on the other, as a preparation for training in crafts and design. \"What made the Bauhaus preliminary course . . . unique,\" writes Frank Whitford, \"was the amount and quality of its theoretical teaching, the intellectual rigor with which it examined the essentials of visual experience and artistic creativity.\" What this meant in practice may be seen in a passage from an essay written in 1916 by the Austrian artist Johannes Itten, who three years later was brought to Weimar to create the original _Vorkurs_ for the Bauhaus:\n\n> The clear geometric form is the one most easily comprehended and its basic elements are the circle, the square and the triangle. Every possible form lies dormant in these formal elements. They are visible to him who sees, invisible to him who does not.\n> \n> Form is also color. Without color there is no form. Form and color are one. The colors of the spectrum are those most easily comprehended. Every possible color lies dormant in them. Visible to him who sees\u2014invisible to him who does not. . . .\n> \n> Geometric forms and the colors of the spectrum are the simplest, most sensitive forms and colors and therefore the most precise means of expression in a work of art.\n\nBy the time Itten arrived at the Bauhaus in 1919 to set up its _Vorkurs_ , such ideas were already well established in the De Stijl group in the Netherlands and the Suprematist circle in Russia, and in fact both De Stijl and Suprematism were to exert a profound influence on Bauhaus theory and practice as well as its disposition to think of both in utopian terms. It was certainly not the purpose of the _Vorkurs_ to prepare its students for careers as abstract artists. Its goal was to prepare students for a much more ambitious task: the application of such theories of abstraction to a redesign of the entire man-made universe. In this respect, Bauhaus pedagogy had much in common with that of the VKhUTEMAS in the Soviet Union, especially after 1923 when the Hungarian artist L\u00e1szl\u00f3 Moholy-Nagy (1895\u20131946) was appointed to replace Itten as director of the preliminary course.\n\nItten, who was more of a mystic than a political ideologue, had derived some of his ideas about color and form from Kandinsky's _On the Spiritual in Art_ , published in 1912. (It was to teach in Itten's version of the _Vorkurs_ , to which his own ideas had already contributed a good deal, that Kandinsky was invited to join the Bauhaus in 1922.) Spiritual matters were not of compelling interest to Moholy, however. In politics he was a radical who had supported Bela Kun's short-lived Soviet in his native Hungary in 1919, and was still identifying himself as a Communist in the early 1920s. As a radical he was inevitably drawn both to the _r\u00e9volt\u00e9_ art and politics of the Dadaists and to the authoritarian absolutism of the Russian Constructivists. (Whatever differences may have divided Dada from Constructivism, both had formed an alliance against the despised bourgeoisie.) In some respects, Moholy took the Bauhaus's commitment to socialism more seriously than Gropius himself\u2014though Gropius had been a supporter of the ill-fated German revolution in 1919 and designed a _Monument to the Fallen of the March Insurrection_ in 1920\u20131921.\n\nWhat Moholy's arrival at the Bauhaus signified was a change in the atmosphere as well as the ideas that would now govern the Bauhaus. \"Even Moholy's appearance,\" writes Frank Whitford, \"proclaimed his artistic sympathies\"\u2014and his political sympathies, too, as Whitford's further account makes clear.\n\nItten had worn something like a monk's habit and had kept his head immaculately shaved with the intention of creating an aura of spirituality and communion with the transcendental. Moholy sported the kind of overall worn by workers in modern industry. His nickel-rimmed spectacles contributed further to an image of sobriety and calculation belonging to a man mistrustful of the emotions, more at home among machines than human beings. His dress stamped him as a Constructivist, as a follower to Vladimir Tatlin and El Lissitzky, who rejected all subjective definitions of art and were scornful of the idea of the artist as the inspired maker of unique objects stamped by his personality.\n\nLike Rodchenko, to whose example he was also strongly drawn, Moholy was a man of remarkably versatile talents, excelling as a painter, photographer, graphic designer, product designer, sculptor, and visual theorist. Unlike Rodchenko and the Soviet militants, however, he was not prepared to reject the art of painting in order to concentrate exclusively on the applied arts. Nor, for that matter, did his radical sympathies prevent him from allying himself with capitalist industry in the realm of the applied arts. In this respect, Moholy's career\u2014first in Germany, and then in America, where he later founded the so-called New Bauhaus in 1937\u2014marked the demise of the utopianism which had prompted the creation of the Bauhaus as the \"Cathedral of Socialism\" in the immediate aftermath of Germany's defeat in the First World War. The hope of redesigning the man-made universe did not entirely fail, to be sure, but whatever successes were realized proved to be peculiarly dependent upon the least utopian of all economic and social systems: capitalism. It was as a handmaiden to capitalist enterprise and corporate competition in the marketplace that many of the Bauhaus's modernist design ideas enjoyed their greatest prosperity and influence. It was a happier fate than that suffered by modernism in the Soviet Union under Stalin, but it was no less a refutation of the utopianism to which the founders of the Bauhaus had initially aspired.\n\nAll utopian endeavors derive their moral force from a firm belief in the perfectibility of man. It was one of the purposes of the original _Vorkurs_ at the Bauhaus to achieve this radical transformation of the human species by, in effect, creating the conditions for a radical revision of human consciousness. The late Reyner Banham put it very well when, in a discussion of the _Vorkurs_ in _Theory and Design in the First Machine Age_ (1960), he cited \"the determination to cleanse every incoming student's mind of all preconceptions and to put him, so to speak, back into Kindergarten to start again from scratch\" as one of the defining innovations of the Bauhaus preliminary course. This, as Banham further observed, was clearly designed\n\n> to return incoming students to the noble savagery of childhood. To have gone so far against established precedent without moving forward into a mechanized culture, meant that Itten had to go right outside the general body of Western, rational thought and under his influence Bauhaus students involved themselves in the study of mediaeval mystics like Eckhart, and Eastern spiritual discipline such as Mazdaznan, Tao and Zen.\n\nThus the socialist utopianism to which the Bauhaus dedicated its fortunes at the outset contained a large admixture of occult belief that was not to be exorcised until the arrival of Moholy, a man \"more at home among machines than human beings.\" Under the guidance of this radical materialist, it was no longer the perfectibility of man that occupied the Bauhaus agenda but the perfectibility of machines, and that effectively closed the book on the utopian aspiration.\n\n[1997]\n\n## Abstraction in America: The First Generation\n\n* * *\n\n> _A work of art is an abstract or epitome of the world_.\u2014Ralph Waldo Emerson, in \"Nature,\" 1849\n> \n> _Let us look at this American artist first. How did he ever get to America, to start with? Why isn't he a European still, like his father before him?_ \u2014D. H. Lawrence, 1923\n\nAbstract art made its debut in America at the same time as it initially emerged in Europe\u2014which is to say, in the two or three years preceding the outbreak of the First World War in 1914. Yet the earliest achievements of American abstract art were not destined to have an impact comparable to that of the European masters of abstraction. No artists of the stature of Kandinsky, Mondrian, or Malevich appeared on the American art scene to galvanize a movement in favor of abstraction, and nothing was produced in the realm of theory that came even close to the urgent appeal that was sounded in Kandinsky's _On the Spiritual in Art_ or the Suprematist manifestos of Malevich. American abstraction in this period was essentially the creation of isolated talents operating in a lonely and leaderless intellectual environment. What the American critic Sadakichi Hartmann had observed in 1898 still described the generation of Arthur Dove, Marsden Hartley, and John Marin a dozen years later. \"Many of our American painters, particularly the better ones,\" wrote Hartmann, \"have, in strange contrast to French artists who move in the midst of life and society, a peculiar trait in common: they love solitude and shun life.\"\n\nUnlike its more robust counterparts in France, Germany, and Russia, moreover, the American avant-garde in the second decade of the century was deeply afflicted with a sense of its own provincialism and powerless-ness. As a consequence, its efforts to achieve a modernist art that was authentically American tended to be more a matter of individual aspiration than of group endeavor or shared ideals. Added to this was the nature of the relation that obtained between the modernist artist in America and the dynamism of the society in which his work was created. Nowhere else in the world was the material progress of society so far advanced as it was in the United States. Yet the innovative spirit that animated technological change and capitalist enterprise in America met with firm resistance in the realm of culture and the arts, which, unlike the world of business and politics, remained a citadel of the genteel tradition.\n\nIn Europe the avant-garde mind saw itself as the coefficient of a modernity which society at large had not yet achieved. In America, however, modernity of the sort that the European avant-garde envisioned for its future\u2014skyscraper architecture, high-speed locomotion, assembly-line production, electrified advertising, and similar feats of technological innovation\u2014was already commonplace. What was missing\u2014and thus left to the individual talent to invent for itself\u2014was an artistic milieu that was open to a comparable level of radical innovation. Such attempts as there were to create a milieu of that kind were greeted with a philistine response so adamant and so uninformed that it inevitably had the effect of confirming the modernist's profound sense of alienation from the mainstream interests of American life.\n\nThe solitude that Hartmann characterized as the condition of the American artist was a direct consequence of this alienation. What also contributed to it was the influence of well-established American tradition\u2014the Emersonian tradition of free-thinking mysticism, which elevated solitude to the status of a moral vocation. Compounded of French utopian thought, German metaphysics, and Unitarian liberalism, this was a tradition that made of individual consciousness a sovereign of all it surveyed in the life of the times. It set the soul in opposition to the institutions of society while allying itself with the beauty and beneficence of nature. It was thus a tradition that conferred an exalted spiritual sanction on any expression of individualism or isolation.\n\nFor the American modernists who came of age in the decade preceding the First World War, this Emersonian tradition, which embraced the kindred radicalisms of Whitman and Thoreau, defined the very atmosphere in which they committed themselves to the life of art, and the ideas it fostered\u2014especially its peculiar combination of mysticism, independence, and rebellion\u2014remained the substratum of their artistic thought even as they looked to artistic developments abroad for the means of implementing them.\n\nIt certainly did much to shape the ethos of the first generation of avant-garde art in America\u2014the embattled coterie of painters, photographers, and writers who sought refuge from the regnant philistinism of their day in the Photo-Secession Gallery, which Alfred Stieglitz had founded in New York in 1902. One of their number\u2014the painter Marsden Hartley (1877\u20131943)\u2014was given Emerson's _Essays_ to read by his drawing teacher in art school in Cleveland, and it became for him, as he later wrote, the text \"I was to carry in my pocket for at least the five ensuing years\u2014reading it on all occasions as a priest reads his Latin breviary on all occasions, it seemed so made just for me.\" Emerson provided, he said, \"the religious element in my experience.\" Soon thereafter, Whitman's _Leaves of Grass_ acquired a similarly high place for him as a sacred text, and in the volume of his own essays, _Adventures in the Arts_ , which Hartley published in 1921, Whitman was coupled with C\u00e9zanne\u2014the ultimate standard of excellence for an American modernist of Hartley's generation\u2014as \"the two most notable innovators\" of their times.\n\nIt was hardly surprising, then, that when Hartley wrote a letter to Stieglitz in 1913 from Munich, where Vasily Kandinsky and his circle of _Blaue Reiter_ painters had come to talk to him about his abstract paintings at the Goltz Gallery, Hartley hastened to compare this momentous encounter to Emerson's historic meeting with Whitman to discuss _Leaves of Grass_. In Hartley's greatest moment of triumph as an abstract painter, it was from the annals of the Emersonian tradition that he still derived his touchstone of distinction.\n\nAbstraction and the Stieglitz Circle\n\n> _In the little gallery of the Photo-Secession, where Mr. Stieglitz and his disciples hold forth for months together, there is never so much as a lead pencil sketch in the little exhibitions which may be properly said to have so much as a shred of a subject, and the word of all other words that may be constantly overheard in the discussions there is the word \"pure.\"_ \u2014Henry McBride (1913)\n\nIt has to be understood that it was never the specific purpose of the Photo-Secession Gallery\u2014or 291, as it was commonly called (after its address on Fifth Avenue)\u2014to foster the development of abstract art. 291 was founded by Alfred Stieglitz, who was himself a photographer of immense distinction, to promote an interest in photography as a fine art. Beginning with a Matisse exhibition in 1908\u2014the artist's first in the United States\u2014Stieglitz also took up the cause of the European modernist masters. By 1913, when the Armory Show in New York gave the American public its first comprehensive account of the modernist movement, Stieglitz had already exhibited C\u00e9zanne, Picasso, Brancusi, Nadelman, and Picabia, and was thus well established as the country's unrivaled champion of modernism. In that role he published an excerpt from Kandinsky's _On the Spiritual in Art_ in the gallery's magazine, _Camera Work_ , in 1912. Yet abstraction was not the aesthetic absolute for the painters of the Stieglitz circle that it was for Kandinsky himself.\n\nIf the word \"pure\" was often heard in that quarter to describe the modernist ideal toward which the painters were striving, its precise meaning nonetheless remained ambiguous\u2014as ambiguous, indeed, as the paintings themselves. It might be as easily applied to John Marin's landscapes and cityscapes, in which recognizable motifs remained distinctly discernible within a pictorial structure that was highly abstract, as to the more hermetic abstract paintings from nature by Arthur Dove (1880\u20131946), which the artist actually called _Abstractions_ as early as 1910\u20131911. At times the word \"pure\" was merely a synonym for \"modern,\" and at others it denoted the complete absence of a recognizable subject. At all times it was a way of distinguishing the modernist idea of pictorial form\u2014form that was no longer tethered to a strict description of the observable facts\u2014from what were regarded as the \"impure\" varieties of academic realism that were in thrall to such facts and still dominated the American art scene.\n\nThis whole notion of pure painting recalls the similarly ambiguous Symbolist concept of pure poetry, and there is a parallel to be observed between Mallarm\u00e9's call for artists \"to paint, not the object, but the effect which it produces,\" and certain strains of Emersonian thought. In writing about Thoreau in 1862, Emerson claimed that \"His [Thoreau's] power of observation seemed to indicate additional senses. . . . And yet none knew better than he that it is not the fact that imports, but the impression or effect of the fact on your mind. Every fact lay in glory in his mind, a type of the order and beauty of the whole.\"\n\nSomething akin to this attitude toward the observable facts of nature, which are taken to represent the \"order and beauty\" of something larger than the facts themselves, is what governed the early abstractions of Arthur Dove. The exact status to be assigned to Dove as a pioneer of abstract painting remains an unsettled question. We know for a certainty that he created some of the earliest abstract paintings, yet we also know that abstraction was never for Dove the single-minded pursuit of the absolute that it was for the acknowledged pioneers of abstract painting. Dove's abstractions are not the kinds of paintings that categorically separate themselves from the experience of nature. Their imagery is more like an Emersonian \"epitome\" of nature than a repudiation of it. Their light is an earthly light\u2014the light of the sun and the moon as it is experienced in daily life. Their forms and their color, too, evoke the organic forms and colors of the natural world.\n\nDove's was an essentially pastoral sensibility. He was not a man of wide cosmopolitan culture. Farming was an occupation that sustained him in hard times, and boats were one of his passions. As a young man he had gone to Paris and even showed in the Salon d'automne in 1908 and again in 1909, and it was on his return to the United States that he earned a precarious living as an illustrator in New York. But cities held no romance for him. He belonged to that class of abstractionists who are totally devoted to nature, totally dependent on it for their visual inspiration and eager to wrest from it something permanent and quintessential. What the critic Paul Rosenfeld said of him in 1924\u2014\"There is not a pastel or drawing or painting of Dove's that does not communicate some love or some sensuous feeling of the earth\"\u2014defined his quality. So did Georgia O'Keeffe's more specific observation that \"Dove comes from the Finger Lakes region. He was up there painting, doing abstractions that looked just like that country, which could not have been done anywhere else.\"\n\nThe modernist conventions Dove learned in Paris and helped to domesticate in America were straightaway placed at the service of this communion with nature. His primary interest was in a certain abstract visual rhythm\u2014elusive, organic, and irregular, constantly changing and constantly enchanting\u2014to be discerned in the diurnal life of the landscape and the seasons and in those objects (both natural and man-made) that nature transforms by its vivid orchestration of light and shadow. It was no doubt for this reason that he moved without qualm or compunction between abstraction and representation in his painting, allowing his inward response to nature to determine the terms of his pictorial style. He was, in this respect, anything but an ideological acolyte of abstraction, which remained for him a contingent option in art rather than an absolute commitment. It is for this reason, perhaps, that his influence on the history of abstract art rarely made itself felt beyond the boundaries of the Stieglitz circle itself.\n\nMarsden Hartley's \"Cosmic Cubism\"\n\n> It is not like Picasso\u2014it is not like Kandinsky not like any _\"cubism\"\u2014it is what I call for want of a better name subliminal or cosmic cubism\u2014It will surprise you\u2014 \n> _ \u2014Marsden Hartley to Alfred Stieglitz (1912)\n\nMarsden Hartley's brilliant, short-lived career as an abstract painter traced a very different course. Although he was reared in provincial and impoverished circumstances in northern New England, Hartley became a man of wide cosmopolitan culture\u2014not only a brilliant painter but a self-educated poet and critic who contributed to _The New Republic_ , _Seven Arts_ , _Poetry_ , and other leading literary journals of his day. Despite his penchant for solitude and the personal problems caused by his homosexuality, especially in America, his friendships and connections in the years immediately preceding the First World War extended beyond the Stieglitz circle in New York to include Gertrude Stein's salon in Paris and the _Blaue Reiter_ painters in Munich. It was, in fact, in Europe that Hartley made his most significant contribution to abstract art. Yet the misfortunes of history\u2014in this case, the eruption of the war in Europe in 1914\u2014had the effect of limiting that contribution to a very few years, 1912\u20131915, in a career that was otherwise to be devoted to representational painting of an equally high order. In those few years, however, Hartley produced some of the masterpieces of early abstraction.\n\nThey were not the work of a young or untried artist. Hartley was in his mid-thirties when he went to Europe for the first time in 1912. He had already had two solo exhibitions of his New England landscape paintings at 291, and that was where, before his departure for Paris, he also had his first opportunity to study the work of C\u00e9zanne, Matisse, and Picasso. All were to have an immense and immediate impact on his pictorial development. What ignited the spark that accelerated Hartley's headlong plunge into abstraction, however, was his unexpected discovery in Paris that his deepest affinities\u2014both in art and in life\u2014might not be there, despite the support he received from Gertrude Stein, but with the German avant-garde and Germany itself. For while the Picassos and Matisses in the Stein collection confirmed for Hartley his initial turn to an art more radical than anything he had attempted in the past, it was the German connections he made in Paris, and which then developed into the great romance of his life in Berlin, that provided the existential impetus for the painting that was to be his most distinctive contribution to abstraction.\n\nIn Paris, where the French had made Hartley feel like an outsider, his German friends offered encouragement and were eager to acquaint him with artistic developments in Munich and Berlin. One of those friends was a young sculptor, Arnold R\u00f6nnebeck, who was accompanied by his American fianc\u00e9e, a singer. He promptly made a bust of Hartley that was exhibited at the Salon d'automne. (Some years later, R\u00f6nnebeck emigrated to America where he became the director of the Denver Art Museum.) Another was R\u00f6nnebeck's cousin, Karl von Freyburg, whose early death in the war inspired one of Hartley's finest abstract paintings, _Portrait of a German Officer_ (1914). The third was a handsome Swiss poet, Siegfried Lang, with whom Hartley seems to have established an immediate romantic attachment\u2014a devotion subsequently transferred to von Freyburg when Lang returned to Switzerland and Hartley settled in Berlin.\n\nWith these German comrades acting as translators, Hartley was introduced to the _Almanach_ of the _Blaue Reiter_ group, an anthology of pictures and essays launched by Kandinsky and Franz Marc in Munich in 1912, and was also able to make his way through Kandinsky's _On the Spiritual in Art_ for the first time. Being of a mystical temperament himself, Hartley proved to be an eager convert to Kandinsky's call for an art of spiritual transcendence that would manifest itself in abstract forms, and he straightaway set about the task of creating paintings of this persuasion himself.\n\nAlthough Kandinsky was already deeply committed to this spiritual mode of abstraction in his own work, the _Blaue Reiter_ program embraced a far wider range of aesthetic affinities and possibilities. In the pages of its _Almanach_ , illustrations of Kandinsky's abstractions shared attention with the modernism of the School of Paris\u2014the Cubism of Picasso, the Fauvism of Matisse, and the so-called Orphism of Robert Delaunay, in addition to the less classifiable paintings of Henri Rousseau\u2014as well as a miscellany of European folk art, medieval woodcuts, and primitive tribal art. (The latter no doubt reflected the ideas of Wilhelm Worringer, whose _Abstraction and Empathy_ [1908] exerted a considerable influence on the _Blaue Reiter_ circle.) Yet all of these disparate interests were seen to serve the same overarching mystical purpose. C\u00e9zanne, too, was drafted into service as a precursor of the _Blaue Reiter_ 's spiritual mission, and so was El Greco, a recent discovery of the German critic Julius Meier-Graefe, who was also a champion of C\u00e9zanne. \"In their views of life,\" wrote Franz Marc in an _Almanach_ essay called \"Spiritual Treasures,\" \"both [El Greco and C\u00e9zanne] felt the _mystical inner construction_ , which is the great problem of our generation.\"\n\nNothing could have been better calculated to win Hartley's allegiance at that juncture in his career. The _Blaue Reiter_ 's alliance of modernism and mysticism was irresistible. He saw it as the invitation to join the European avant-garde that he believed had been denied him in Paris. And his German friends offered Hartley something else as well\u2014a comradeship in which his homosexuality proved not to be an obstacle. That was something he seems never to have felt confident of in America, even in the relatively permissive atmosphere in the Stieglitz circle. The fever of inspiration in which Hartley hastened to join the German avant-garde was thus fueled by a fusion of sexual passion, aesthetic daring, and intellectual ambition unlike anything he had felt in the past.\n\nYet it was in fact in Paris, before his departure for Germany in January 1913, that Hartley produced the initial attempts at the \"cosmic cubism\" that constituted his first significant series of abstract paintings. Always a quick study, Hartley seems to have grasped the essential point about abstraction upon his initial contact with its sources. As to exactly what those pictorial sources were in his case, Hartley himself was not inclined to be entirely forthcoming. If the \"cosmic\" element in these abstractions owed much to his encounter, in Paris, with _Blaue Reiter_ mysticism, their formal language still owed even more to Picasso's Cubism, and, in all probability, to Delaunay's Orphic variations on Cubism, which were more adventurous in their use of color than Picasso's. About his debt to Delaunay, Hartley may have had personal reasons for remaining silent. He clearly felt an intense dislike for the man himself, whom he had met in Paris, but he seems nonetheless to have taken some cues from his work. Certainly the disklike forms in Hartley's early abstractions and their color-driven dynamism are traceable to Delaunay, who in this period, not incidentally, was more highly esteemed in Germany than in Paris.\n\nHartley's response to Kandinsky was far more complicated. His prior immersion in Emersonian mysticism had naturally disposed him to respond with enthusiasm to Kandinsky's advocacy of the occult as the basis for an abstractionist aesthetic. About Kandinsky's paintings, however, he expressed some serious doubt. While hard at work on his own abstract paintings in Paris in December 1912, Hartley wrote to Stieglitz\u2014in the very letter in which he announced his venture into \"cosmic cubism\"\u2014that \"In Kandinsky's own work I do not find the same convincing beauty that his theories hold\u2014He seems to be a fine theorist first and a good painter after.\" And writing to his niece at the same time, Hartley was even more negative: \"he is a pure theorist and his work is quite without feeling\u2014whereas I work wholly from the intuition and the subliminal.\"\n\nWhatever Hartley may have meant by this criticism, it is to a large extent belied by Kandinsky's discernible influence on his own early abstractions. Like many of Kandinsky's _Improvisations_ and _Compositions_ of 1911, Hartley's 1912 \"cosmic\" abstractions have the look of imploded landscapes reconstituted on a celestial scale, and they are similarly dependent upon heavily marked black outlines for their basic pictorial structure. It may be, however, that Kandinsky's paintings were thought by Hartley to be too impersonal to satisfy the kind of emotion that Hartley himself was endeavoring to bring to his own abstractions. For what Hartley seems to have found missing in Kandinsky was precisely the kind of erotic fervor that was then impelling his own creative life.\n\nIt wasn't until Hartley himself went to live in the German capital, however, that he was fully able to give expression to this erotic impulse in his work. Berlin was clearly the great romantic idyll of Hartley's life. The sexual freedom he experienced in Berlin, which centered upon a young German officer he had met in Paris, was never entirely distinguishable in his own mind from either his mystical beliefs or his aesthetic aspirations, all of which disposed him to find in the colorful pageantry of German military life both an imagery ideally suited to the artistic imperatives of abstraction and a pictorial correlative for its erotic subtext. It was in this sense that Hartley's \"cosmic cubism\" made of abstract painting the vehicle of a sexual romance.\n\nIn that endeavor, Kandinsky was indeed less important to Hartley than Franz Marc, who, though not himself either an abstractionist or a homosexual, proved to be a devoted artistic ally. In Marc's highly colored, folkloristic Expressionism, which encompassed a pictorial romance of another kind, Hartley found a Northern, mystical, modernist sensibility akin to his own, and Marc responded to this aesthetic kinship with the kind of encouragement, criticism, and useful contacts that Kandinsky could never bring himself to offer. It was largely owing to Franz Marc that Hartley was invited to exhibit with the _Blaue Reiter_ group in Munich and become, in effect, a bona fide member of the pre\u2013World War I German avant-garde.\n\nOne important consequence of this was an invitation to exhibit his work in the 1913 Berlin Herbstsalon, in the company of Kandinsky, Klee, Arp, L\u00e9ger, Chagall, Boccioni, Delaunay, and other luminaries of the European modern movement. Five paintings by Hartley were hung between pictures by Kandinsky and Henri Rousseau. As he was the only American artist in the exhibition, Hartley was quick to entertain the illusion that he had at last \"found [his] place in the art circles of Europe.\"\n\nThis short-lived idyll was decisively shattered by the war, which had the paradoxical effect of inspiring Hartley's most important contribution to abstraction while denying him the possibility of its continued development. Writing to Stieglitz from Berlin in 1913, Hartley had already indicated a \"desire\u2014to make a decorative harmony of color and form,\" which was his way of talking about abstraction, in his paintings of German military subjects. The outbreak of the war, though it brought him immense personal grief, initially accelerated Hartley's determination to pursue this pictorial project, and with the death of Karl von Freyburg in October 1914 it was transformed into a series of memorials for the great love of his life. For Hartley, this encompassed not only the fallen von Freyburg but also everything he had come to cherish about his life in prewar Germany.\n\nAbout the political ramifications of the war, Hartley remained invincibly unconcerned. He seems to have had no faculty whatever for understanding the political situation in which he found himself. While he acknowledged to Stieglitz that he was dealing with \"teutonic\" themes in these pictures, he never gave a thought to the way the war might affect their reception in the United States, where anti-German sentiment was already running high. (The United States did not officially enter the war against Germany until 1917.) Hartley saw the war entirely in terms of his personal loss, and it was as an expression of love and loss that he created the abstract elegies of 1914\u20131915.\n\nThe problem was, of course, that by constructing his \"decorative harmony of color and form\" in the elegies around specific, highly legible symbols of German military glory\u2014most conspicuously, the Iron Cross that had been posthumously awarded to von Freyburg\u2014Hartley made it inevitable that his paintings would be taken to be pro-German political statements. It may have been for their perceived political character that they were favorably received when they were exhibited in Germany, first in Frankfurt and then in Berlin, in 1915. But such a reading of his work seems never to have crossed Hartley's mind. While nothing was sold in these shows, he nonetheless believed, again mistakenly, that he had achieved a considerable success with them solely on the basis of their artistic merits. The exhibition in Frankfurt had even caught the attention of a correspondent for the _New York Times_ , who, though he may not have known much about modern painting, certainly understood that Hartley's pictures had been inspired by the war. He described them to _Times_ readers as \"a snarl of triangles, squares, rectangles, flags of all nations, in glaring, solid, primitive colors, shuffled together, [which] produces a picture puzzle that absolutely defies you to say it isn't a battle.\" In New York itself, to which Hartley had reluctantly returned at the end of 1915, the political consequences of the war could no longer be avoided.\n\nStieglitz had been warning Hartley about the difficulty of finding buyers for abstract paintings of any persuasion. Even among the small number of patrons who had an interest in modernist art, abstraction still met with firm resistance. Hartley's new abstractions would therefore have been a hard sell at 291 even if they had been free of Teutonic motifs. But abstractions that were seen to honor Germany's role in the war placed the whole matter even farther beyond the pale of acceptability.\n\nWhen, despite his misgivings, Stieglitz gamely mounted an exhibition of Hartley's German abstractions at 291 in the spring of 1916, it was, of course, a fiasco. Hartley's belated attempt to forestall censure by falsely denying the symbolic character of his abstractions\u2014\"There is no symbolism whatsoever in them; there is no slight intention of that anywhere,\" he wrote in the catalogue of the show\u2014persuaded nobody. The symbolism was simply too blatant to be missed. And as the exhibition itself was not so much decried as snubbed, Hartley was denied even the satisfaction of causing a scandal. The net effect of the exhibition was to bury Hartley's hope of making his mark as an abstract painter whose work would rank with that of the leaders of the European avant-garde.\n\nIn the immediate aftermath of this debacle, Hartley made one last attempt to salvage his venture into abstraction. In a series of low-keyed, highly formalized compositions based on Provincetown motifs drawn from boats, houses, and still-life objects in the summer of 1916, he abruptly lowered the emotional temperature of his painting, eradicating all trace of Expressionist symbolism and high color in favor of muted tones and crisp, flat, neutered forms. Under more auspicious circumstances this radical change in style might well have served as the basis of a new development in abstract painting. Whatever they still owed to Picasso's Cubism, these cool abstractions were something new\u2014an anticipation, in some respects, of the kind of close-valued, clean-edged abstract styles that would later be developed on both sides of the Atlantic. But Hartley's was not a sensibility that could sustain an impersonal style devoid of existential imperatives. By temperament an Expressionist, he would never find an art of pure form sufficiently compelling to satisfy his inflammable personality. So he abandoned abstraction, which had yielded him his first masterpieces, and entered upon his long quest for a representational style in which the demands of both life and art might be given their due.\n\n[1999]\n\n## Was Rothko an Abstract Painter?\n\n* * *\n\n> _You might as well get one thing straight . . . I'm_ not _an abstractionist_.\u2014Mark Rothko to Selden Rodman, 1957\n\nIt has long been one of the curiosities of abstract art that so many of its practitioners have denied that they were in fact abstract artists. To have created works of art that were seen to be \"merely\" a mode of abstraction has been, for these artists, a considerable vexation, and some have gone to great lengths to\u2014in their view\u2014set the record straight on this question. But whether they actually succeeded in this project of denial or only added further impediments to our understanding of abstract art\u2014including their own\u2014remains a matter of debate.\n\nIt is a debate, moreover, that is certain to acquire a renewed momentum now that abstract art has lost its radical status and has passed into the hands of the art historians as a subject for research and interpretation. It is when an art no longer commands the distinction of radical novelty, and therefore no longer serves as a model for a new generation of \"advanced\" artists, that the task of explaining its \"meaning\" becomes especially problematical. A change in taste occurs, and the old aesthetic loyalties no longer obtain with the force they once did. There is a need to understand what it was about this art that held so many gifted artists in its thrall and persuaded so many connoisseurs of its importance. At the same time, however, there is likely to be a revisionist impulse at work in this search for \"meaning\"\u2014an impulse that seeks to align the old art under study with the interests of the new art that has displaced it. If, as happens to be the case at the present moment, these interests are more closely focused on the \"content\" of art than they are 6 5 son its form, then we can expect the new interpretations of abstract art to follow a similar course.\n\nThis is what I believe has now happened with the study of abstract art in general and Abstract Expressionism in particular. Abstraction continues to be copiously produced, of course\u2014and sometimes at a significant level of aesthetic achievement. But it can no longer be said to set the agenda for new artistic developments. (This doesn't mean that it might not do so again at some future date, but that is a different question.) Hence the proliferation of attempts to explain what abstract art was really about in its heyday.\n\nWhat makes the whole issue especially perplexing is the even more fundamental question of whether this debate about the \"meaning\" of abstraction, though initially prompted by the effort of certain abstract artists to deny that they _were_ abstractionists, makes any discernible difference in our experience of the works themselves. When, for example, we find ourselves in the presence of Mark Rothko's _Number 22_ (1949) in the permanent collection of the Museum of Modern Art, what is there in what we see in the abstract forms of this painting that can be taken to refer to, or to embody, the metaphysical \"subject\" which the artist himself claimed for the work? Is that claim to be taken literally? If not, then in what sense is it to be taken? If we cannot actually _see_ the artist's metaphysical \"subject\" in the physical object he has created for the avowed purpose of giving it expression, how can that subject be said to be present in the work?\n\nQuestions of this sort have often been raised about Mark Rothko's abstract paintings\u2014the paintings, that is, that he produced from 1949 until his death in 1970\u2014and with good reason. For it is well known that Rothko himself spurned the very idea that he was in any sense an abstract painter, yet the paintings themselves have long been taken\u2014both by those who love them and by those who do not\u2014to represent a particularly pure and even extreme form of abstraction. In his lifetime, Rothko made this issue of denial a crucial test for his admirers, and as a result they tended, by and large, to support his claim without significantly clarifying it. On at least one occasion, however, the claim was upheld so zealously that even Rothko appeared to shrink from endorsing it. I refer to the catalogue written by Peter Selz for the Rothko exhibition he organized at the Museum of Modern Art in 1961. In the short essay that Mr. Selz wrote for that catalogue, he began by declaring that \"[Rothko's] paintings can be likened to annunciations,\" and invoked the example of Fra Angelico to support such a reading. He ended with a flight of interpretative fancy that, though it was obviously intended as lavish praise, the artist himself recoiled from. The relevant paragraphs of Mr. Selz's text are too long to quote here in their entirety, but the following excerpts will convey something of their flavor:\n\n> Eventually as other images occur to the viewer the metaphor of the creation of some universe becomes paramount. And increasingly\u2014in the mind of this writer\u2014these \"shivering bars of light\" assume a function similar to that loaded area between God's and Adam's fingers on the Sistine ceiling. . . . Rothko has given us the first, not the sixth, day of creation.\n> \n> The open rectangles suggest the rims of flame in containing fires, or the entrances to tombs, like the doors to the dwellings of the dead in Egyptian pyramids, behind which the sculptors kept the kings \"alive\" for eternity in the _ka. . ._ . Indeed, the whole series of these murals brings to mind an Orphic cycle; their subject might be death and resurrection in classical, not Christian, mythology; the artist descending to Hades to find the Eurydice of his vision. The door to the tomb opens for the artist in search of his muse.\n\nThis was surely less a case of an \"artist in search of his muse\" than of an art historian in search of something\u2014anything\u2014to say about an art he couldn't comprehend. Never mind that Rothko himself, with all his talk about myth and tragedy and timeless subjects, had been responsible for leading Mr. Selz astray. Artists are given license in these matters that art historians appropriate at their peril. What Mr. Selz wrote on that occasion was all the sheerest nonsense, and was perceived to be nonsense at the time. Sidney Geist wrote a hilarious and devastating analysis of the essay in a short-lived publication called _Scrap_ , and Rothko himself then asked the museum to withdraw the essay when it sent his exhibition abroad. I followed these developments with a keen interest while they were occurring, and not only because Mr. Geist and his collaborator on _Scrap_ , Anita Ventura, were friends and colleagues but because the essay that I had commissioned the late Robert Goldwater to write about the Rothko show for _Arts Magazine_ , which I was then editing, was the one that Rothko selected for the catalogue accompanying the European tour of the exhibition.\n\nWell, all of this happened a long time ago, and would scarcely be worth recalling now if not for the publication of a new book on Rothko that makes an even more elaborate attempt to illuminate the artist's \"subject\" than the one Mr. Selz undertook nearly thirty years ago. Given the changes that have occurred in the intellectual climate of the art world, it is my impression that this new book\u2014Anna C. Chave's _Mark Rothko: Subjects in Abstraction\u2014_ is likely to meet with a more sympathetic response than Mr. Selz's essay did. And, of course, the artist is no longer around to register his objections and thereby discredit a fanciful interpretation of his work. We are indeed in a new period, and I have an awful feeling that Professor Chave's book may very well represent its spirit\u2014at least as far as the study of abstract art is concerned.\n\nThus, where Mr. Selz spoke only of \"images [that] occur to the viewer,\" which obliquely acknowledged that they were, after all, _his_ images rather than Rothko's, Professor Chave seems to believe that the images\u2014or \"traces,\" as she sometimes calls them\u2014that occur to _her_ in studying Rothko's paintings are actually present (in some sense) in the work itself. At least she seems to believe this some of the time, and her professors at Yale, where the original version of this book was accepted as a doctoral dissertation in art history, were apparently persuaded that this represented a perfectly legitimate reading of the artist's classic abstract paintings. Yet Professor Chave also appears, at other times, to believe that Rothko was indeed an abstract painter, and that her principal \"discovery\" in this book\u2014that the abstract paintings are filled with discernible references to entombment and nativity motifs (among much else)\u2014in no way contradicts that belief. Which, at the very least, raises the whole question of what an abstract painting is. Is it an art in which we can find \"traces\" of whatever our hearts and minds and the fashionable methodologies of academic study may wish us to find, or does what can be seen\u2014and not seen\u2014in the paintings place a limit on what can be said to be present in the painting?\n\n\"No doubt Rothko,\" Professor Chave writes, \". . . did not consciously make all the quotations that may be discerned in his art, but that does not mean that they are not there.\" She then continues:\n\n> The influence and the perpetuation of existing cultural forms is not something artists can fully escape, either by design or by ignorance. Tradition, as Harold Bloom wrote (following Freud), is \"equivalent to repressed material in the mental life of the individual.\"\n\nWhat, then, does the \"repressed material\" of Rothko's \"tradition\" consist of, according to Professor Chave?\n\nThe general answer she gives to this question is\u2014well, almost everything that has been done in Western painting since the Renaissance.(Unlike Mr. Selz, she doesn't go back to the Egyptian pyramids, though in truth there is nothing in her method to preclude such an option.)\"Landscapes\" and \"portraits\" are discovered to abound in Rothko's abstract paintings\u2014\"Any of Rothko's classic paintings could be described as having a portraitlike aspect or as bearing a trace of portraiture,\" etc.\u2014but it is in what Professor Chave calls their \"vestiges or traces\" of the figurative religious painting of the Renaissance that these paintings are claimed to be especially rich. Painful as it is for me to reproduce such nonsense in these pages, I am obliged to quote a representative passage from Professor Chave's book, lest the reader think that I am exaggerating the extreme\u2014and extremely foolish\u2014degree to which she has carried this bizarre project of interpretation:\n\n> The pose of the sleeping infant Christ in Bellini's _Madonna and Sleeping Child_ prefigures the martyred Christ in the \"sleep of death,\" as found in Bellini's _Piet\u00e0_ , for example. Those images of Rothko's that parallel the pictorial structure of a piet\u00e0, such as _White Band (Number 27)_ and _Number 20_ , 1950, might be said at the same time to parallel the structure of a conventional mother and child image. Further, and by the same logic, such tripartite pictures as the untitled yellow and black painting of 1953, which bear a relation to the pictorial structure of an entombment, also bear a relation to certain conventional adoration or nativity images; in the first instance the deceased is laid on the ground before its attendants, and in the second, the infant lies sleeping on the ground at its mother's feet. But because Rothko's abstract figures are far removed from human form, in any case, the specific scale of the supine or horizontal figure relative to that of the erect or vertical figure\u2014the factor that explicitly tells viewers of a mimetic image whether they are looking at an entombment or a nativity\u2014has no bearing in Rothko's work. This basic schema, with a narrow band sandwiched between or below two larger masses, could yield an icon for the entire \"human drama,\" then, from the \"tomb of the womb\" to the \"womb of the tomb.\" His abstract figures could do (at the least) a double symbolic duty, that is, even where the symbols involved were opposing ones, such as birth and death. Rothko's image-sign enabled him to elide or dismantle such conventional binaries and present them not as polar opposites but as interconnected, two sides of the same process or phenomenon.\n\nNo doubt it was on the basis of interpretative feats of this kind that Professor Chave received her present appointment to the Harvard faculty. One hardly knows whether to laugh or cry.\n\n\"Viewers [of abstract painting],\" Professor Chave writes, \"are left in the tense and tantalizing position of feeling that they can almost but never quite grasp a message that they may sense as inhering in the pictures.\" And earlier on she observes that \"The message of the abstract painting veers decisively away from the textual or narrative, the explicitly recognizable and specifiable, into a more indeterminate realm.\" Part of the problem with this kind of pictorial analysis, of course, is its crass assumption that an abstract painting is something like a telegram containing a \"message\" in a code we are called upon to crack. Professor Chave doesn't call Rothko's paintings telegrams, to be sure\u2014she calls them \"a palimpsest of traces,\" as any academic semiologist in good standing would, I suppose. \"The larger purpose of this book,\" she writes, \"is to show how Rothko's pictures function polysemically,\" which is as close as Professor Chave comes to acknowledging that the aesthetic issues in Rothko's paintings\u2014and, indeed, the aesthetic experience they afford, or fail to afford\u2014have little or no interest for her. Her method in this book obliges her to deal with Rothko's paintings as if they really are _texts_ to be decoded. It is therefore no wonder that it is their alleged \"message\" that interests her, for there is nothing in her method that can account for anything else. And since the \"message\" itself is an invention of the author, the whole enterprise turns out to have remarkably little to do with Rothko or his art.\n\nWhat it does have a lot to do with, however, is the current disfavor with which academic opinion views the aesthetics of abstraction and the priority that academic research now accords to the study of \"content\" in art even in cases, like this one, where the content has to be invented in order to serve as a subject suitable for study. I have to say that Professor Chave is quite brilliant at this kind of analytic invention; I am in as much awe of her powers of imagination as I am of her sheer intellectual effrontery. But the sad truth is, this book has nothing to tell us about Rothko or about abstract painting. It is a book\u2014one among many, alas\u2014about the crisis that has overtaken the study of art history in the universities.\n\nMark Rothko was hardly alone in rejecting the idea that he was in any sense an abstract artist. Many of the most illustrious names in the history of abstract art are known to have made similar denials. That many abstractionists were uneasy about venturing into this new aesthetic terrain, that they harbored doubts about the \"meaning\" of what they produced as abstract artists, that they sometimes made claims for their work, especially for its \"content,\" that defy easy understanding and sometimes even credulity\u2014these and many related questions have long been familiar to anyone who has studied the subject. But the answers to such questions do not lie in the impulse to \"interpret\" abstraction or to allegorize it or decode it as if it were a hermetic text. A painting isn't a text, and to pretend that it is one is to denigrate the very idea of abstract art, and to add yet another chapter\u2014given its academic provenance, a particularly awful one\u2014to the history of philistine response to its existence.\n\n[1989]\n\n## Jackson Pollock and the New York School\n\n* * *\n\n> _The American artist with any pretensions to total seriousness suffers still from his dependency upon what the School of Paris, Klee, Kandinsky, and Mondrian accumulated before 1935. . . . All excellence seems to flow still from that vivacious, unbelievable near past which lasted from 1905 until 1930 and which not even the First World War, but only Hitler, could definitely terminate_.\u2014Clement Greenberg, 1947\n> \n> _Every intelligent painter carries the whole culture of modern painting in his head. It is his real subject, of which anything he paints is both a homage and a_ critique, _and everything he says a gloss_.\u2014Robert Motherwell, 1951\n> \n> _We Americans have the technique to bring something to performance so well that the subject is left out. There is nothing we throw away so quickly as our_ donn\u00e9es; _for we would make always an independent and evangelical, rather than a contingent, creation. . . . We throw away so much and make so much of the meager remainder. We make a great beauty, which is devastated of everything but form and gait_.\u2014R. P. Blackmur, 1958\n\nIs it possible that the significance of the New York School has been misconstrued? Is it possible that the much vaunted \"triumph\" of Abstract Expressionism in the 1950s\u2014when, for the first time in our history, an entire \"school\" of advanced American painting commanded international attention and acclaim\u2014might not, after all, have been the unqualified artistic success it is nowadays taken to be by all the organs of established cultural opinion? That the emergence of the New York School in the decade that followed upon the end of the Second World War brought an immense change in the fortunes of American art is certainly beyond dispute. That this momentous change was closely linked to a decline in the fortunes of modernist art in Europe\u2014the art from which Abstract Expressionism derived its principal impetus and ideas\u2014remains a far more contentious issue. That the implications of that decline did much to determine the scope of what it was possible for the New York School to achieve is a question that few chroniclers of the Abstract Expressionist movement have been willing to confront. From a critical perspective of this persuasion, however, the emergence of the New York School is best understood as at once an epilogue to and a quintessentialization and apotheosis of the great epoch of early-twentieth-century modernist painting in Europe.\n\nSuch a view is, of course, very much at odds with current critical orthodoxies. On the one hand, there is the citadel of institutional opinion that insists on celebrating the New York School as a grand departure from European tradition, quite as if Abstract Expressionist painting was to be seen as some sort of analogue to Whitman's poetry or jazz. On the other hand, there is a counterorthodoxy prevalent in the academy that holds to a belief that the American avant-garde somehow \"stole\" its artistic ideas from their legitimate European custodians under the malign auspices of the cold war. The first attempts to aggrandize the New York School by exaggerating its originality and uniqueness, while the second attempts to demonize it for purely political purposes. (To this demonizing political agenda, moreover, the academic left has lately added equally irrelevant charges of racism and sexism.) Yet neither of these critical orthodoxies seems true either to our experience of the art itself or to the historical circumstances in which it was created.\n\nWhat is essential to understand about the historical origins of the New York School is that it emerged from a world in which the political geography of modernist art had suffered a severe contraction. Even before the Second World War brought the public life of art in Europe to a virtual standstill, modernism had been outlawed in two of its vital centers\u2014first in Russia under the Soviet regime of Stalin, and then in Germany under the Nazi regime of Hitler. This malevolent fate had particularly grievous consequences for the votaries of abstraction, which was everywhere seen\u2014by its champions no less than by its enemies\u2014to represent the most extreme manifestation of the modernist spirit in art, and was thus even more vulnerable to hostile opinion than its figurative counterpart.\n\nWhat is also important to understand about the fate of modernist art in the decade prior to the Second World War is that even in those countries where it was more or less accepted as a legitimate, though still controversial, component of modern cultural life\u2014which was largely the case in France, Britain, and the United States\u2014abstraction continued to meet with fierce institutional resistance. In France, except for L\u00e9ger and Delaunay, the masters of the School of Paris were hostile to pure abstraction. In the 1930s, moreover, the dominance of Surrealism on the Paris art scene had the effect of marginalizing even abstract artists of the stature of Mondrian and Kandinsky. (That the abstract art of Kandinsky had exerted a crucial influence on Mir\u00f3, the greatest painter to receive the imprimatur of Andr\u00e9 Breton, hardly mattered. Kandinsky remained excluded from the Surrealist canon.) In Britain, too, despite a spirited campaign mounted in the critical writings of Herbert Read on behalf of Henry Moore, Barbara Hepworth, Ben Nicholson, and Naum Gabo, abstraction was generally regarded by both the cognoscenti and the public as something alien and absurd. Even a modernist critic as enlightened as Roger Fry had harbored serious doubts about its aesthetic efficacy.\n\nThe situation in America was somewhat different, however. While mainstream critical opinion in the United States was similarly hostile or indifferent to the achievements of abstract art in the 1930s, there was nonetheless some significant institutional support for its European masters. The Museum of Non-Objective Art\u2014later to be better known as the Solomon R. Guggenheim Museum\u2014was founded in New York in 1937 as a kind of shrine to Kandinsky and his followers. (Jackson Pollock worked there, doing odd jobs, in 1943.) Even earlier, in 1927, A. E. Gallatin had opened his Gallery of Modern Art, a collection that favored abstraction, at New York University, and earlier than that, in 1920, Katherine Dreier had founded the Soci\u00e9t\u00e9 Anonyme, which was also sympathetic to abstraction. Most important of all was the founding in 1929 of the Museum of Modern Art, where in 1936 Alfred H. Barr, Jr., organized a major exhibition of _Cubism and Abstract Art_ , which gave the public of its day its first comprehensive account of the history of abstract art in Europe.\n\nYet not a single American painting was included in the _Cubism and Abstract Art_ exhibition, though Alexander Calder was represented by his mobile sculpture. This implicit downgrading of American abstraction deeply embittered our embattled native modernists, who in the 1930s were fighting a lonely struggle for recognition against the popularity of reactionary Regionalist and Social Realist painting. This led to a public protest in 1939 when a group called the American Abstract Artists, which had been organized in 1936 to advance the cause of abstract art in the United States, issued a broadside that asked the question, \"How Modern Is the Museum of Modern Art?\" It was signed by fifty-two artists and caused the museum a good deal of embarrassment. It did not, however, prompt any immediate change in policy.\n\nThe reason, though never publicly avowed, was nonetheless clear. American abstract art in the 1930s was not deemed sufficiently powerful or original to merit more vigorous support. As one of the museum's curators, James Thrall Soby, confided on another occasion: \"You cannot possibly present twentieth-century American painting as we have presented School of Paris painting. The revolutionary impact is not there.\" This is a judgment that posterity has somewhat modified\u2014for there was more going on in American art in the 1930s than the museum quite appreciated\u2014but not essentially altered. Even the most advanced American art of the time was still largely tethered to European initiatives.\n\nThis was a situation that the eruption of the Second World War radically changed. The fall of Paris to the Nazi army in June 1940 marked the end of an era in European modernism. If the modernist movement in art was to have any immediate future, it would be left to artists in the United States\u2014both native modernists and European \u00e9migr\u00e9s\u2014to create that future. Almost overnight, then, New York became the sole remaining outpost of the modernist movement, and thus the de facto capital of the international avant-garde. This was as much of a shock to the Americans as it was to the Europeans, for modernist art was essentially a European creation and American artists were used to regarding themselves as the underdogs of the movement. That its future might now be in American hands was a daunting prospect for which few American talents had prepared themselves. Yet it was in this atmosphere of shock, dislocation, and worldwide crisis\u2014a crisis in the very civilization that had produced the modernist movement\u2014that the New York School emerged in the early years of the war.\n\nLike so many of the other American painters who constituted that first generation of New York School painters, Jackson Pollock (1912\u20131956) was abysmally ill-prepared, by training or temperament or intellect, to respond to this daunting challenge. That he succeeded to the extent that he did\u2014which was for a very brief period between 1947 and 1950\u2014was itself a remarkable feat, a triumph of ambition and short-lived inspiration over a severely handicapped and unruly personality. The figure he most resembles in American cultural history isn't Walt Whitman but Hart Crane, whose horrific career traced a similar course of self-destructive combat with extreme circumstance to wrest from the conventions of European modernism an authentic statement of American experience.\n\nIn September 1939, when the war in Europe commenced, Pollock was a twenty-seven-year-old painter whose accomplishment was negligible and whose personality was already showing signs of fracture and disarray. For two years he had been under psychiatric treatment for alcoholism, and would remain under treatment\u2014mostly of a Jungian persuasion\u2014for some years to come. As an artist he was still very much under the influence of his teacher and mentor, Thomas Hart Benton, one of the leaders of the Regionalist school. He had also taken a keen interest in the Mexican muralists, especially Jos\u00e9 Clemente Orozco and David Alfaro Siqueiros, who in the 1930s were very much in vogue in the United States. (They indeed enjoyed more favor at the Museum of Modern Art than most American modernists.) It was from Siqueiros in the 1930s that Pollock acquired an interest in the use of enamel paint and some of the techniques for dripping and pouring it, though it would be another decade before he put them to significant use in his own painting. It was not until 1940 that, as his brother Sande wrote at the time, Pollock had \"finally dropped the Benton nonsense.\"\n\nWhat seems to have changed everything for Pollock was seeing Picasso's _Guernica_ at the Valentine Gallery in New York in the spring of 1939, and then, later that year, the mammoth exhibition _Picasso: Forty Years of His Art_ at the Museum of Modern Art. It was his intense encounter with Picasso that marked Pollock's entry into the arena of modernist painting, which Benton had famously repudiated\u2014an arena in which Picasso would presently be submerged, in Pollock's case, by the impact of Kandinsky, Mir\u00f3, Masson, and Matta. It was to a large extent the therapeutic ethos of Jungian psychoanalysis, however\u2014an ethos that conferred immense metaphysical authority on so-called archetypes, which were believed to lie buried in the recesses of unconscious memory but were waiting to be magically summoned for conscious purpose\u2014that determined the way Pollock performed in that arena.\n\nWhat ignited this heady amalgam of modernist pictorial aesthetics and Jungian psychology in Pollock's painting was the Surrealist doctrine of automatism, which was itself derived from the Freudian concept of the unconscious. This isn't the place to explore the central role played by psychoanalytic theory and practice in the art and culture of the 1940s in this country. Suffice it to say that it was enormous. It effectively supplanted the role played by Marxist thought in the 1930s, and exerted an even greater influence on the conduct of life. Neither the art of the New York School nor the poetry of the same period\u2014the poetry of the generation of Delmore Schwartz, John Berryman, Randall Jarrell, and Robert Lowell\u2014can be fully understood or seriously assessed in isolation from the culture of psychoanalysis. Nor, for that matter, can any other area of American cultural life in the years between the Second World War in the 1940s and the Vietnam War in the 1960s, a subject that still awaits a social historian to do it justice.\n\nBe all this as it may, it was the doctrine of automatism that emancipated Pollock from the morass of modernist eclecticism and pastiche in which his painting remained mired during the war years when his art was just beginning to attract serious attention. His conversion to automatism also liberated him from the bogus mythification of those tacky Jungian \"archetypes\" that identify paintings like _The Moon Woman_ , _Male and Female_ , _Guardians of the Secret_ , and _The She-Wolf_ (all from 1942\u20131943) as the work of an artist whose raw command of the painterly medium, while still pretty crude, was more impressive than his muddled attempts to deal with symbols ostensibly dredged up from the unconscious\u2014but actually acquired from the intellectual hearsay of the period\u2014as his principal subjects. It wasn't until Pollock got rid of his subjects\u2014or what R. P. Blackmur called \" _donn\u00e9es_ \" in the passage I have quoted above\u2014that he was able to achieve an art that, as Blackmur wrote, \"is devastated of everything but form and gait.\" It was his total embrace of automatism in the \"drip\" paintings of 1947\u20131950 that made the breakthrough possible.\n\nIt is one of the many problematic aspects of the retrospective _Jackson Pollock_ , which Kirk Varnedoe has organized at the Museum of Modern Art this winter, that it treats the artist's oeuvre as if it were a consistently sublime achievement rather than what it is: a highly uneven body of work which, even in its rare moments of original accomplishment, is scarcely comparable to the greatest painting of the modernist era. This approach to the Pollock problem seems to have been dictated by a mistaken belief that Pollock's life constitutes one of the master myths of twentieth-century American life. This belief is baldly stated in the opening pages of Mr. Varnedoe's essay for the catalogue of the exhibition, an essay whose very title\u2014\"Comet: Jackson Pollock's Life and Work\"\u2014puts us on notice that we are invited on this occasion not merely to examine an artist's work but to participate in an historical romance. \"The span from Elvis Presley's first record in 1954 through Jasper Johns's first show in 1958 formed a key divide in American life,\" Mr. Varnedoe writes, \"and Pollock's car crash (like the actor James Dean's the year before) was one of its benchmarks. In retrospect, that accident says 'end' as surely as Sputnik (launched the next year) says 'beginning.'\"And further:\n\n> Pollock now looms as a central hinge between the century's two halves, a key to how we got from one to the other in modern art. As the pivot on which prologue and coda balance, he has become in history, still more than he was in life, a legitimator: validation accrues to the lineage fertile enough to have spawned him, or to followers clever enough to have properly read his message; and competing claims abound. Even more broadly, any theory of cultural modernity has to claim the summit he occupies before it can assert dominion on the territory; and in this way accounts of Pollock also become litmus tests for broader philosophical and political positions about the meanings of his epoch. Trying to write his history inevitably broaches larger queries about our own.\n\nIn the presence of such claims to world-historical importance, then, Mr. Varnedoe embraces the notion that \"normal critical criteria\" are still beside the point in discussing Pollock's art, for how can mere art criticism hope to account for a phenomenon like \"a central hinge\" of an epoch that goes back a hundred years? Even in the vast literature devoted to Picasso, I cannot recall anything to equal this bizarre exercise in museological hyperbole. It makes even William Rubin's misguided attempt to place the paintings of Frank Stella in the company of Shakespeare and Dante seem almost modest by comparison. In the end it is a testimony not to Pollock's artistic achievement but to something far more mundane\u2014a curator's anxiety that a claim to anything less than the colossal and world-shaking will fail to bring in the crowds needed to justify an exhibition mounted on a blockbuster scale.\n\nAlas, it is in the interest of museum marketing and mythmaking that war has been declared on so-called normal critical criteria in the organization of this retrospective and its accompanying catalogue, and not because such criteria are no longer relevant to what the artist accomplished or failed to accomplish in the course of a very troubled life.\n\nWhat \"normal critical criteria\" amount to on this occasion turn out, for the most part, to consist of the critical observations made by the late Clement Greenberg in the course of his close involvement with Pollock's painting. While paying a kind of pro forma lip service to Greenberg's accomplishments as a critic, Mr. Varnedoe mounts a major assault on just about every aspect of his critical thought. Not only are we treated to the now familiar reference to Greenberg \"as a tyrant of taste in the early 1960s\" but to a summary attack on Greenberg's pessimistic views of American society in the immediate aftermath of the Second World War and the sense of \"alienation\" with which he identified both the New York School and the vicissitudes of high cultural aspiration in that period.\n\nTaking what can only be called a _Saturday Evening Post_ cover view of this postwar period, Mr. Varnedoe evokes a jolly time \"when a world war against totalitarian powers had just been won, when prosperity had returned after the bitter years of the Depression, and when an enormous boom in new births apparently signaled a groundswell of confidence and relief.\" Apparently he is unacquainted with W. H. Auden's _Age of Anxiety_ (1947), or the ballet later based on it, or with the pervasive influence in these years of such unjolly writers as Kafka and Kierkegaard and Sartre and Camus and Maritain\u2014but why go on? About the spiritual temper of that postwar period Mr. Varnedoe remains as innocent\u2014or ignorant\u2014as one of those newborn babes he cites as evidence of a \"groundswell of confidence.\" Can he really have no idea of the spiritual disarray and psychological breakdown that the war and the Holocaust and the atomic bomb brought in their wake?\n\nThere is certainly plenty to take issue with in Clement Greenberg's criticism, and with Greenberg himself, but on this subject\u2014the moral and cultural temper of the period in which the New York School emerged\u2014his pessimism was both accurate and prophetic. He wrote as follows in 1947:\n\n> The morale of that section of New York's Bohemia which is inhabited by striving young artists has declined in the last twenty years, but the level of its intelligence has risen, and it is still downtown, below 34th Street, that the fate of American art is being decided\u2014by young people, few of them over forty, who live in cold-water flats and exist from hand to mouth. Now they all paint in the abstract vein, show rarely on 57th Street, and have no reputations that extend beyond a small circle of fanatics, art-fixated misfits who are as isolated in the United States as if they were living in Paleolithic Europe.\n\nAlthough Pollock's principal critical champion, Greenberg didn't write as a cheerleader\u2014and it is for that reason, I suppose, that a curatorial cheerleader like Kirk Varnedoe is concerned to discredit him. While Greenberg did not hesitate to describe Pollock in 1947 as \"the most powerful painter in contemporary America and the only one who promises to be a major one,\" he also said of Pollock's painting that \"its paranoia and resentment narrow it; large though it may be in ambition\u2014large enough to contain inconsistencies, ugliness, blind spots, and monotonous passages\u2014it nevertheless lacks breadth.\" It is for judgments like this that Greenberg has had to be demonized once again on the occasion of this retrospective.\n\nFor the entire premise on which this retrospective has been organized is the belief that Pollock ushered in a new golden age in American art\u2014a golden age represented by the likes of Cy Twombly, Jim Dine, Roy Lichtenstein, Claes Oldenburg, Carl Andre, Frank Stella, Lynda Benglis, Richard Serra, and Brice Marden, among many others. One can only say that golden ages aren't what they used to be. On this matter, too, what Greenberg wrote about the Museum of Modern Art in 1947 remains amazingly relevant, for some things haven't changed in the last fifty years. \"Pusillanimity makes the museum follow the lead of the powerful art dealers. . . . But it cannot be blamed too much, since it reflects rather accurately the prevailing taste in American art circles.\" If there really was \"a tyrant of taste in the early 1960s\" on the New York art scene, it was Leo Castelli, not Clement Greenberg\u2014but to acknowledge that would capsize the premise on which this Pollock retrospective has been organized.\n\nA more reliable key to our understanding of Pollock's painting is, in any case, to be found in Greenberg's observation that it \"lacks breadth.\" But to reflect on that realm of aesthetic actuality is to remove Jackson Pollock from the romance of history and return our attention to what the artist accomplished and failed to accomplish in the only important pictures he ever created.\n\n> _Somewhere deep in every American heart lies a rebellion against the old parenthood of Europe_.\u2014D. H. Lawrence, 1923\n\nUntil well into the 1940s, Jackson Pollock's painting remained locked in a struggle to master and overcome the influences of the European modernist painters he took as his models\u2014primarily Picasso, Kandinsky, and Mir\u00f3. That he was not himself an artist in their class was more or less taken for granted even by his most ardent admirers at the time. It is doubtful that Pollock himself believed he was an artist in their class. Yet what he brought to his encounter with these European masters was a fierce determination to produce an art that would somehow carry him beyond the styles and conventions he was wrestling with. Among much else, this meant that traditional easel painting had to be abandoned in favor of something the European masters of modernism had not yet attempted to place at the center of their art\u2014a mode of mural-scale wall painting of a radically subjective character.\n\nPollock seems to have understood that as an easel painter he would never be able to trump his European masters. Not only did they bring to the tradition of the easel picture a richer and more complex command of experience than any that was within his own reach, but they had effectively transformed the content of the easel picture to conform to the imperatives of that experience. Mural-scale wall painting of a certain persuasion\u2014painting that would be tethered not to the service of some social cause but to the only subject that governed the artist's will: his own troubled psyche\u2014offered a way to circumvent the tradition to which Pollock could not finally make a significant contribution.\n\nIt was in the interest of circumventing that tradition that the influence of the Mexican muralists, the public art of Thomas Hart Benton, and Pollock's own response to Picasso's _Guernica_ all played a crucial role in determining his leap into mural-scale wall painting in the late 1940s. It was in the scale and ambition of such painting, not in its ethos or imagery, that Pollock saw the possibility of a radical artistic opportunity. For the social content of 1930s mural painting Pollock felt no affinity whatever. Indifferent to politics and fundamentally anti-social in his personal behavior, Pollock had only one subject that commanded his loyalty: his own appetites, ambitions, and compulsions, which years of Jungian psychoanalysis had elevated in his own mind to the status of a cosmological imperative. (Hence his megalomaniac assertion that \"I _am_ nature!\") It wasn't until he was able to bring those appetites and ambitions into alignment with a pictorial technique allowing them unfettered expression\u2014unfettered, that is, by the traditional tools and constraints of easel painting\u2014that Pollock was able to achieve an art uniquely his own.\n\nIt is for this reason that the \"drip\" abstractions of the late 1940s and early 1950s are inevitably the central focus in any comprehensive account of Pollock's oeuvre. It is one of the distinctions of the _Jackson Pollock_ retrospective which Kirk Varnedoe has organized at the Museum of Modern Art that it assembles a larger number of these abstract paintings than has ever before been seen in a single exhibition. This, in my view, is also one of the principal liabilities of the exhibition, for what has come to be regarded as \"classic\" Pollock is not, after all, an art of infinite variety. It is maddeningly repetitious in its formal rhythms. It is paltry in its command of color, for classic Pollock is largely based on light-dark contrasts rather than chromatic structure. Classic Pollock also lacks breadth, as even Clement Greenberg acknowledged, in its range of feeling and invention.\n\nGiven these limitations, such paintings are best seen in isolation from each other. They actually gain something from being viewed in the company of abstract paintings unlike themselves in method and imagery\u2014which is generally how we do see these Pollocks in museum collections\u2014for the contrasts to be observed in such contexts have the effect of underscoring the element of furious, headlong energy in Pollock. When a large number of Pollock's abstract paintings are seen in close succession, however\u2014as they are in the MOMA retrospective\u2014their much-vaunted energy tends to deflate and flatten into decorative tedium. The more we look, the less we find. What may strike us initially as random and reckless in the labyrinthine traceries of dripped and poured pigment soon reveals itself to be involuntarily governed by an ineluctable and compulsive monotony. Pollock's unconscious, to the extent that it was called upon by his automatist methods to supply him with images, forms, and ideas, turned out to be a very limited resource. But then, so too was the element of conscious control in Pollock. Which may be why classic Pollock turned out to have a lifespan of less than five years.\n\nEven so, the paintings produced in the early stages of this short-lived period are more interesting than those that came at the end. If we take _Cathedral_ (1947) as marking its beginning, then it was pretty much over by 1950, the year in which Pollock produced _Lavender Mist_ , _Autumn Rhythm_ , and a number of similar paintings in which he seems to have been able to effect a more workable equation between uncertainty and control than either before or after. That there remained a large element of uncertainty in Pollock's own view of what he was doing in the drip abstractions even in his best years is evident in his introduction of Mir\u00f3esque cut-out figures in 1948\u20131949. In the eight-foot-wide _Out of the Web_ (1949), for example, the figures actually dominate the weblike traceries of pigment. By the time we get to _Blue Poles_ (1952), uncertainty has been supplanted by predictability. _Blue Poles_ is the Abstract Expressionist version of Salon painting. Classic Pollock had become its own clich\u00e9. And in the few years that remained, Pollock is seen to be attempting something like a return to easel painting.\n\nNo one has given us a better account of the artistic implications of the shift from easel painting to wall painting in Pollock than the late William C. Seitz in his early study of _Abstract Expressionist Painting in America_.1 Until 1946 [wrote Seitz] Pollock's paint surfaces were passionately molded with brush and knife in what Thomas Hess characterized as a \"heavy, dark Expressionist idiom.\" The irreducible unit of his style, despite rectilinear structure, was the individual stroke, though its identity was apt to be lost in the total textural maelstrom and the optical pulsation effected by variegated color. Out of this \"total textural maelstrom,\" according to Seitz, came \"Pollock's identification of passion with nonobjective brush tracks,\" and this in turn led him to a pictorial method in which \"tools seldom touched the painting surface\"\u2014the method Seitz described as \"the flow or drip of enamel from brush, stick, or can.\"\n\nSeitz then addressed the crucial question of \"the degree and nature of the control that the artist intentionally or unconsciously exerts over his medium.\" While reminding us that \"The idea of accident is deeply entrenched in modern tradition,\" Seitz went on to observe that Pollock had\n\n> demonstrated that direction can be given to even so fluid and willful a medium as poured enamel. Control here applies not to the exact track and shape of each brushstroke [sic] to be sure, but to the types of relationship inherent in the process. Individual passages . . . at no point evidence the direct touch of the painter. Rather, it is his entire bodily activity that from a distance influences, but by no means determines, his configuration. Accident, gravity, and the fluid response of the paint combine with human gesture to form a structure that is the result of their interaction.\n\nAbout the pictorial structure that results from this interaction, Seitz also offered some cogent observations.\n\n> It is easy to see three-dimensional structure in Pollock's pictures; and as one's consciousness moves, in exploring his endless space of cellular division, time is involved as well. Finally, following the perceptual jolt by which one's impression of a visual field shifts, hollow space becomes flat surface. What was open structure is seen as a network of lines that weave above and below each other across the canvas, and the spectator is excluded. The picture, still vital, becomes a wall decoration.\n\nNow the possibility that it might be the fate of mural-scale abstract painting to be experienced as wall decoration\u2014\"wallpaper,\" as Harold Rosenberg derisively characterized it in \"The American Action Painters\"\u2014was an issue that was deeply troubling to virtually all of the painters of the New York School. That is why there was so much heated discussion lavished on the importance of the \"subject\" from the earliest days of the Abstract Expressionist movement in the 1940s. It was believed\u2014some of the time, anyway, by some of the artists\u2014that a compelling subject would forestall abstraction's descent into wall decoration. There were New York School painters\u2014Mark Rothko prominently among them\u2014who insisted that the only legitimate subject for abstract painting was some mythic or mystical vision of human tragedy, yet few observers of his art could find any hint or suggestion of such a subject in the paintings he actually created. Willem de Kooning, on the other hand, downgraded the very notion that \"content\" might play an important role in painting\u2014\"It's very tiny\u2014very tiny, content,\" he said\u2014but his own argument was undermined by the series of _Women_ paintings that placed him in a direct line of descent from Picasso, for whom a subject had always been crucial. There was never any consensus on the subject of a \"subject\" in the New York School, nor could there be\u2014for it was in the very nature of Abstract Expressionist painting for each of its practitioners to conduct a radically subjective dialogue with the art of the European modernists they aspired to supplant.\n\nIn this respect, certainly, Robert Motherwell was far more candid than most of his colleagues in the New York School when he observed that \"Every intelligent painter carries the whole culture of modern painting in his head. It is his real subject, of which anything he paints is both a homage and a _critique_ , and everything he says a gloss.\" Yet the idea that \"the whole culture of modern painting\" might in fact be the unacknowledged subject of Abstract Expressionist painting\u2014which I firmly believe to be the case\u2014was deeply distasteful to many painters in the movement, who regarded it as an affront to their high-risk, existential ambitions as members of an embattled avant-garde. It was an idea that for many people in the art world conjured up discredited notions of art-for-art's-sake and \"significant form\"\u2014theories that were believed to give priority to art over life, and thus inevitably put in question the ambition of the New York School painting to produce an art so radical that it would at some level challenge the life of the time.\n\nIt was for this reason that Harold Rosenberg's theory of \"action painting\" scored such an immense success when it was first promulgated in 1952. In a single bold stroke, Rosenberg's essay \"The American Action Painters\" appeared to remove the entire Abstract Expressionist movement from both the history of modern painting and the pettifogging distinctions of aesthetic discourse by situating it instead in some exalted realm of existential \"action,\" where, as the author claimed, \"what matters always is the revelation contained in the act,\" and such mundane matters as \"Form, color, composition, drawing . . . can be dispensed with.\" This was the key passage in what amounted to a declaration of independence from what D. H. Lawrence had called \"the old parenthood of Europe\":\n\n> At a certain moment the canvas began to appear to one American painter after another as an arena in which to act\u2014rather than as a space in which to reproduce, re-design, analyze or \"express\" an object, actual or imagined. What was to go on the canvas was not a picture but an event.\n\nIt was further claimed by Rosenberg that \"The new American painting is not 'pure' art, since the extrusion of the object was not for the sake of the aesthetic.\" On the contrary, Rosenberg wrote, \"The new painting has broken down every distinction between art and life.\" From this perspective, then, \"The critic who goes on judging in terms of schools, styles, form\u2014as if the painter were still concerned with producing a certain kind of object (the work of art), instead of living on the canvas\u2014is bound to seem a stranger.\"\n\nIf \"every distinction between art and life\" had been eliminated in this painting, what it called for wasn't art criticism or aesthetic judgment but some existentialist version of psychoanalysis. Since the painter has become an actor, the spectator has to think in a vocabulary of action: its inception, duration, direction\u2014psychic state, concentration and relaxation of the will, passivity, alert waiting. He must become a connoisseur of the gradations between the automatic, the spontaneous, the evoked.\n\nThe notion that \"What was to go on the canvas was not a picture but an event\" was the sheerest nonsense, of course. But it proved to be very seductive nonsense. For its effect was to provide the Abstract Expressionist movement with an exciting new dramaturgy in which the artist now emerged as an existential hero and his painting was to be seen not as an aesthetic endeavor but as the cynosure of a heroic private action that was not to be judged by aesthetic standards. It was, alas, a very \"European\" theory, which derived from ideas to be found in Breton's surrealism, Freud's psychoanalysis, and Sartre's existentialism, yet it proved to be so appealing that it was somehow exempted from any negative association with \"the old parenthood of Europe.\"\n\nThere was, however, a dirty little secret involved in the promulgation of Rosenberg's theory. For it was well known at the time that Jackson Pollock was the artist upon whose recent work\u2014the mural-scale \"drip\" abstractions of the late 1940s and early 1950s\u2014Rosenberg had based his theory of \"action painting.\" And it was also well known in New York art circles that Rosenberg actually despised Pollock's paintings and had a very low opinion of the man himself. Yet because Pollock was the only painter then at work whose pictorial practice\u2014especially as it was recorded in the photographs by Hans Namuth\u2014seemed to lend itself to Rosenberg's existential \"action\" scenario, he shamelessly exploited Pollock's notoriety without according him appropriate recognition or even mentioning his name. There was thus at the very core of this famous essay on \"The American Action Painters\" an act\u2014one might even call it an existential act\u2014of unconscionable bad faith. Yet because Rosenberg was so enamored of his own theory and saw that its publication would bring him a celebrity he had never before enjoyed\u2014which, of course, it promptly did\u2014he turned his private mockery of Pollock's painting into a bogus manifesto for the entire Abstract Expressionist movement.\n\nThe unhappy history of this episode in critical legerdemain is now worth recalling not only because it became so inextricably involved in public perceptions of the art of the New York School for so many years but also because something akin to Harold Rosenberg's bad faith in promoting his \"action\" theory seems to have governed the very conception of the _Jackson Pollock_ retrospective at MOMA. Unlike Rosenberg, Mr. Varnedoe cannot be accused of despising Pollock's painting, yet implicit in the atrocious way he has presented the artist's drip abstractions to the public\u2014and without the drip paintings there would be no Pollock retrospective\u2014is an assumption that these are not works of art that can be expected to sustain our attention without recourse to the mythology surrounding their creation.\n\nHence the unforgivable decision to install a life-size replica of the interior of the Long Island barn that served as Pollock's studio in the exhibition itself. That studio interior is, of course, the _mise en sc\u00e8ne_ of the media-generated legend of \"Jack the Dripper\" and all the other nonsense written about Pollock's antics as an artist. Owing to the photographs of Hans Namuth that documented Pollock's \"performance\" in that studio, it has become a space almost as famous as the paintings that were created in it. At MOMA, moreover, the replica of this studio space is adorned with a copious selection of the Namuth photographs of Pollock at work, and on a video monitor at the entrance there is a film version of the same subject. This replica of the Pollock studio turns out to provide the climactic moment in what is a large and at times a very wearying exhibition, and its function in the retrospective is clearly to give the public something other than Pollock's paintings to look at. To an understanding of Pollock's painting this replica of his studio space contributes nothing. Its purpose is purely dramaturgical. By evoking the legend of the painter performing in the studio, it succeeds in shifting attention from the paintings to the artist himself as he has passed into the mythology of modern cultural life. It certainly succeeds as show biz but is finally very damaging to the public's understanding of the art.\n\nWhat the conception of the Pollock retrospective owes to Rosenberg's \"action\" theory, and what both owe to the folklore generated by Namuth's studio photographs and films, is made explicit by Pepe Karmel, Mr. Varnedoe's collaborator on this exhibition, in his essay for the catalogue accompanying the retrospective. This essay on \"Pollock at Work: The Films and Photographs of Hans Namuth,\" which runs to some 45 large pages plus 102 footnotes\u2014roughly two-thirds as long as Mr. Varnedoe's own essay on the life and work of the artist\u2014is something of a museological phenomenon in itself. I, at least, cannot recall another museum text of this length in which the most mind-numbing pedantry is so seamlessly combined with such shameless legend-mongering to produce such a negligible critical result. For what momentous discovery does Mr. Karmel come up with after his protracted study of Namuth's negatives and the outtakes from the Pollock film project? \"Pollock's achievement, in his pictures of 1947\u201350,\" writes Mr. Karmel in conclusion, \"was to transform graphic flatness into optical flatness\u2014to show that by piling layer upon layer, sign upon sign, you could generate a pictorial sensation equivalent to that of the primordial vision field.\" He acknowledges that \"The impact of this discovery is evident in Pollock's painting,\" and then adds: \"But only through the films and photographs of Hans Namuth can we understand the technique that made it possible.\"\n\nYet in this pedantic endeavor Mr. Karmel is guilty of engaging in the very same critical strategy he takes Harold Rosenberg's \"action\" theory to task for. \"Rosenberg's rhetoric encouraged artists and critics to focus on Pollock's actions rather than on the images resulting from them,\" Karmel correctly observes, and then at much greater length than Rosenberg attempted does exactly the same thing. But since the whole conception of this retrospective is to exploit the Pollock myth rather than to give the public a disinterested account of the artist's accomplishment\u2014which really doesn't require an exhibition on this scale\u2014it was essential for MOMA, too, to concentrate on \"Pollock's actions\" rather than his art. So the paint-splattered floor of the artist's studio is given a full-color reproduction in the catalogue, and the studio itself has been replicated as a shrine where not the paintings of Jackson Pollock but the photographs of Hans Namuth documenting their production may be pondered as the sacred relics of the artist's legend. All of which has less to do with the life of art than with the current business practices of the art-museum industry.\n\n[1999]\n\nFootnote\n\n1. Originally written as a doctoral dissertation at Princeton University in 1955, Seitz's _Abstract Expressionist Painting in America_ was not published as a book until 1983, when Harvard University Press brought it out with a forward by Robert Motherwell and an introduction by Dore Ashton.\n\n# II \nModernism and Its Critics\n\n## Clement Greenberg in the Forties\n\n* * *\n\nWith the publication of the first two volumes of Clement Greenberg's _Collected Essays and Criticism_ , we are at last on our way to having a comprehensive edition of the most important body of art criticism produced by an American writer in this century. The two volumes now available\u2014 _Perceptions and Judgments, 1939\u20131944_ , and _Arrogant Purpose, 1945\u20131949\u2014_ bring together for the first time Mr. Greenberg's critical writings from the decade in which he emerged as the most informed and articulate champion of the New York School as well as one of our most trenchant analysts of the modern cultural scene. The two additional volumes promised for the future will presumably bring his remarkable critical oeuvre up to date. John O'Brian, the editor of the series, has so far carried out his duties with an exemplary scholarly tact\u2014both of the new volumes contain useful notes, interesting biographical chronologies, and appropriate bibliographies\u2014and the books have been produced by the University of Chicago Press in a very readable format. Under any circumstances the publication of Mr. Greenberg's _Collected Essays and Criticism_ would have to be considered a capital intellectual event; but in the present climate, when so much that passes for serious criticism is in reality some form of academic twaddle, commercial hype, or political mystification, the appearance of these volumes also has a wonderfully bracing effect. For this is criticism that does not require us to surmount some impenetrable rhetorical barrier in order to discover what it is actually up to. It is plainly written, cogently argued, and admirably precise in the many discriminations and formulations it is concerned to make. For anything remotely comparable in intellectual quality or aesthetic intelligence, one would have to turn to the most illustrious of Mr. Greenberg's predecessors\u2014to Julius Meier-Graefe in Germany and Roger Fry in England. In his own generation he stands alone.\n\nIt is one of the many virtues of the present edition of Mr. Greenberg's writings that it places them firmly in the context of the history which prompted them in the first place. When, some twenty-five years ago, Mr. Greenberg published a selection of his essays under the title of _Art and Culture_ , he was frank to say that that book was \"not intended as a completely faithful record of my activity as a critic.\" Although drawn from the first twenty years of Mr. Greenberg's production, many of the essays in _Art and Culture_ had been substantially revised in order to give the reader a clear account of the author's current views. \"Not only has much been altered,\" as Mr. Greenberg noted in his preface to _Art and Culture_ , \"but much more has been left out than put in.\" _Art and Culture_ was certainly a very distinguished book. Indeed, for a quarter of a century it has remained without rival as the premiere work of art criticism in our time. Yet the emphasis that _Art and Culture_ placed on the present inevitably had the effect of foreshortening our understanding of the past\u2014both the critic's past, with all of the shifts in emphasis and perspective that any full-scale critical career is bound to encompass, and that of the art he was attempting to come to grips with in a succession of discrete encounters. It is only with the publication of _The Collected Essays and Criticism_ that we have now begun to have placed in our possession a complete account, month by month and year by year, of this crucial critical history.\n\nIt is another of the virtues of this edition that it illuminates\u2014and with a fullness and candor that was no doubt inappropriate to the purposes of _Art and Culture\u2014_ the broad range of literary, artistic, and political interests that did so much to shape Mr. Greenberg's outlook as a writer. The first entry in Volume 1 is a review of Bertolt Brecht's _A Penny for the Poor_. The second is the author's now classic essay \"Avant-Garde and Kitsch,\" written at the age of thirty and as much concerned with literature as with art. The third is an equally important but not so well-known essay, \"Towards a Newer Laocoon,\" which attempted to explain in historical terms how it happened that modern art, in its \"revolt against the dominance of literature\"\u2014the dominance, that is, of \"subject matter at its most oppressive\"\u2014came to place \"a new and greater emphasis upon form\" and upon the physical medium. \"Towards a Newer Laocoon\" was, as Mr. Greenberg acknowledged, \"an historical apology for abstract art,\" and may thus be considered something of a manifesto for the \"formalist\" view of art which he has steadfastly maintained throughout his subsequent career as a critic.\n\nInterestingly, he was writing in these early years as a Marxist\u2014an anti-Stalinist Marxist of the Trotskyist persuasion, to be sure, but a Marxist all the same. In this phase of his career Mr. Greenberg was _very_ much a fan of the _Partisan Review_ circle that looked to Trotsky as a political guide in attempting to \"save\" the Revolution from what was looked upon as Stalin's \"betrayal\" of it. In the fourth entry in Volume 1 of _The Collected Essays\u2014_ an essay called \"An American View,\" written for Cyril Connolly's magazine _Horizon_ in 1940 and published in London during the Blitz\u2014we are given a clear statement of the political folly that this ideological outlook entailed. \"Hitler realizes this much, that in order to keep capitalism there must be fascism,\" Mr. Greenberg wrote; \"Trotsky realizes that in order to keep democracy there must be a socialist revolution. . . . We must choose: either capitalism or democracy. One or the other must go. If we insist on keeping capitalism then we cannot fight Hitler. He must win.\"\n\nCyril Connolly promptly and correctly responded to this mistaken analysis by writing an article in the same issue of _Horizon_ which firmly rejected Mr. Greenberg's claims. \"It is obvious that this view,\" Connolly wrote, \". . . rests on an over-simplification of the facts, and if put into practice would lead to disaster.\" (Mr. O'Brian quotes Connolly's response in a footnote.) Before the end of the forties Mr. Greenberg had more or less come to agree with Connolly, describing himself in a _Partisan Review_ symposium in 1948 as an \"ex- or disabused Marxist.\" And as an editor of _Commentary_ from the late forties to the end of the fifties, he was, of course, in the forefront of those ex-radicals who played a role in articulating the case for what came to be characterized as anti-Communist, or cold war, liberalism\u2014a position from which, I believe, he has never since deviated.\n\nWhat needs to be borne in mind about the role of Marxism in Mr. Greenberg's early writings is that 1) it did much to shape his view of history, and 2) it was never allowed to distort his judgment of particular artists. The extent to which a Marxist-influenced teleology came to be subsumed in his reading of art history is another matter. It can be argued\u2014I have argued it myself\u2014that some residue of the Marxian dialectic disposed him to find in art history, and most particularly the history of modern art, something in the realm of aesthetics (\"subject matter at its most oppressive\") that is more or less akin to the Marxist notion of class struggle as an iron law of historical development.1 Something of the sort is certainly to be found in such essays as \"Avant-Garde and Kitsch\" and \"Towards a Newer Laocoon,\" and it persists as a kind of leitmotif in a great many other essays, too. Yet it must be said that Mr. Greenberg's response to particular works of art is, more often than not, remarkably free of the ideological strictures to be found in his more theoretical pronouncements. And even in \"Towards a Newer Laocoon,\" he was at some pains to underscore the limit that he placed on his own theoretical imperative:\n\n> My own experience of art has forced me to accept most of the standards of taste from which abstract art has derived, but I do not maintain that they are the only valid standards through eternity. I find them simply the most valid ones at this given moment. I have no doubt that they will be replaced in the future by other standards, which will be perhaps more inclusive than any possible now. And even now they do not exclude all other possible criteria. I am still able to enjoy a Rembrandt more for its expressive qualities than for its achievement of abstract values\u2014as rich as it may be in them.\n\nThis is not the observation of a closed mind, and the application of \"other possible criteria\" occurs more frequently in Mr. Greenberg's criticism than one expects from a writer so given to theoretical absolutes.\n\nThere is much discussion of poetry as well as of politics in these first two volumes of _The Collected Essays;_ and fiction, philosophy, criticism, and even on one occasion the ballet\u2014Anthony Tudor's _Dim Luster\u2014_ are likewise given some interesting attention. The forties was also a period in which Mr. Greenberg was much occupied with his translations of Kafka, and Volume 2 contains the first of his essays on Kafka, written in 1946. There is little question, I think, that Mr. Greenberg had both the resources and the sensibility to become a first-rate literary critic. But it was the art rather than the literature of his time that claimed his primary allegiance, and it is, of course, as a chronicle of his art criticism that this new edition of his writings makes its principal claim on us.\n\nDuring the period under review Mr. Greenberg was writing most regularly for _Partisan Review_ and for _The Nation_. As Mr. O'Brian points out in his introduction to Volume 1, \"His longer, justly celebrated essays were written primarily for _Partisan Preview_ , while his shorter, until now mostly unrecovered articles, appeared mostly in _The Nation_.\" Read now in sequence, these articles give us an amazingly detailed, close-up view of the art life of the time\u2014not only in their account of the new art that was being shown in New York for the first time, but also in their response to the work of the established masters.\n\nIn regard to the former\u2014Mr. Greenberg's account of the new art of the forties\u2014this new edition of his writings amply confirms his reputation as the preeminent critic of the New York School. In this first decade of its emergence he seems to have missed very little. As early as 1943 he wrote of David Smith: \"Smith is thirty-six. If he is able to maintain the level set in the work he has already done . . . he has a chance of becoming one of the greatest of all American artists.\" About Jackson Pollock that same year he wrote: \"Pollock has gone through the influences of Mir\u00f3, Picasso, Mexican paintings, and what not, and has come out on the other side at the age of thirty-one, painting mostly with his own brush.\" About Robert Motherwell's first one-man show in 1944: \". . . he has already done enough to make it no exaggeration to say that the future of American painting depends on what he, Baziotes, Pollock, and only a comparatively few others do from now on.\" In 1945, on the occasion of Pollock's second one-man show: \"the strongest painter of his generation and perhaps the greatest one to appear since Mir\u00f3.\" And so on. It isn't at all the case that Mr. Greenberg was especially easy on the artists he admired, either. He could be severely critical\u2014as he was, for example, with Arshile Gorky. Yet when Gorky's _The Calendars_ was shown in the Whitney Annual in 1948, he did not hesitate to pronounce it \"the best painting in the exhibition and one of the best pictures ever done by an American.\" That same year\u2014\"a remarkably good one for American art,\" as he noted\u2014he spoke of Willem de Kooning as \"one of the four or five most important painters in the country\" on the basis of the artist's first one-man show. One could go on quoting from other reviews as well, and we are talking only about the period that ended in 1949\u2014the terminal date of Volume 2.\n\nThe figure who looms behind a good many of these judgments\u2014not in the sense of dictating them, for it was never a case of that, but by way of providing a reading of modern art that gave one the essential clues to its further development\u2014is the painter and teacher Hans Hofmann, about whom there are fewer references in Mr. Greenberg's writing in this period than we might have expected. Hofmann is first mentioned in a footnote to \"Avant-Garde and Kitsch\" in 1939, but his own painting is not discussed until 1945, the year of his second one-man show in New York. When, on that occasion, Mr. Greenberg spoke of Hofmann as \"a force to be reckoned with in the practice as well as in the interpretation of modern art,\" it was an acknowledgment of an influence that had already played an important role in the critic's thinking\u2014an influence duly celebrated two years later in an essay written for _Horizon_ on \"The Present Prospects of American Painting and Sculpture\":\n\n> Hofmann will in the future, when the accomplishment of American painting in the last five and next twenty years is properly evaluated, be considered the most important figure in American art of the period since 1935 and one of the most influential forces in its entire history, not for his own work, but for the influence, enlightening and uncompromising, he exerts. . . . Hofmann's presence in New York has served to raise up a climate of taste among at least fifty people in America that cannot be matched for rigor and correctness in Paris or London.\n\nClearly the critic included himself among the beneficiaries of this influence, and so must we. It may even be that much that we still admire in Mr. Greenberg's writings about art in this period owes something to the fact that as Trotsky's influence\u2014and the influence of Marxism generally\u2014steadily waned, that of Hofmann's \"formalist\" reading of modern painting continued to cast its powerful spell.\n\nIt would be a mistake, however, to suggest that it is primarily as a sort of retrospective tip-sheet on the emerging eminences of the New York School that this critical chronicle is of interest now. What has kept Mr. Greenberg's criticism alive is the quality of the thought that he lavished not only on individual artists as he encountered their work for the first time but on the larger artistic and historical issues that their work was found to raise. In this collected edition of the criticism one can see more clearly than before how the famous longer essays\u2014\"The Decline of Cubism,\" \"The Crisis of the Easel Picture,\" \"The New Sculpture,\" and \"Our Period Style,\" all written in the years 1948\u20131949\u2014grew out of the individual encounters with specific objects. However generalized, Mr. Greenberg's observations on individual artists, on art history, and on history itself may sometimes appear to be, they were always firmly anchored in concrete aesthetic experience. It is another of the strengths of _The Collected Essays_ that it makes this internal connection between apodictic judgment and aesthetic actuality far more explicit than it has been hitherto. One may still want to quarrel with the judgment, but one knows better now on exactly what basis it has been arrived at.\n\nAnother large interest of these volumes is to be found in Mr. Greenberg's writings on European art, and specifically on the School of Paris. The forties was, of course, the decade in which the decline of Paris as the art capital of the West and the emergence of New York as its successor became an undeniable fact of international cultural life. In Mr. Greenberg's critical chronicle of this fateful decade we do not find a celebration of the decline of Paris\u2014and still less, any gloating over it\u2014but there is no denial of it, either. If anything, his discussion of it\u2014which, characteristically, takes the form of discussing individual artists\u2014is marked by a tone of deep regret. For the art produced in the heyday of the School of Paris remained for him, even in the forties, a touchstone of aesthetic quality. Again and again in these essays he makes it clear that for him Matisse still loomed as \"the greatest master of the twentieth century\"\u2014a judgment that was by no means commonplace even in Paris in the forties\u2014and Cubism remained, as he wrote in 1948, \"the only vital style of our time, the one best able to convey contemporary feeling, and the only one capable of supporting a tradition which will survive into the future and form new artists.\" The intensely serious attention given as a matter of course to Picasso, Bonnard, Chagall, Leger, Braque, Mir\u00f3, and other luminaries of the School of Paris is itself a reflection of the central place Mr. Greenberg continued to accord these artists. The welcome he gave to Dubuffet in the late forties, moreover, makes nonsense of any notion that his was a criticism in any way governed by chauvinistic bias.\n\nIn this respect, it should be said that the campaign lately mounted by certain Marxist historians to suggest that in the 1940s New York somehow contrived to \"steal\" modern art from Paris by means of a government-sponsored conspiracy will find nothing in _The Collected Essays_ to support what is essentially an exercise in political paranoia (if not indeed an attempt to repeal history itself). To Mr. Greenberg, as to many others at the time, it came \"much to our own surprise,\" as he wrote in 1948, that \"the main premises of Western art have at last migrated to the United States, along with the center of gravity of industrial production and political power.\" His complaint was that Americans were still so piously attached to the prestige of the School of Paris that they could not bring themselves to recognize what was going on in their own country\u2014but this was an aesthetic judgment, not a political one. And the fact that Paris still loomed so large in the eyes of the New York art world, that art produced in Paris still held such unbroken sway over what dealers showed, critics wrote, collectors bought, and museums acquired\u2014all this, too, makes nonsense of the charge that some secret conspiracy was at work to elevate New York at the expense of Paris. The plain truth is, Paris _did_ suffer a decline so irreversible that it has not recovered to this day. It is, in any case, against the background of this abiding piety about art in Paris and its corollary\u2014a general disinclination to believe in the importance of American art\u2014that Mr. Greenberg's writings in the forties about both French and American art need to be read.\n\nFinally, it is worth pointing out that there is much in these volumes on a variety of other subjects that remains of great interest. About the paintings of Arnold Friedman, an American artist only now\u2014forty years after his death\u2014being accorded anything like the recognition he deserves, Mr. Greenberg wrote beautifully and repeatedly. His obituary of Mondrian is still a very moving document:\n\n> [Mondrian's] pictures, with their white grounds, straight black lines, and opposed rectangles of pure color, are no longer windows in the wall but islands radiating clarity, harmony, and grandeur\u2014passion mastered and cooled, a difficult struggle resolved, unity imposed on diversity. Space outside them is transformed by their presence.\n\nAbout Edward Hopper, too, an artist one hardly expected Mr. Greenberg to have much interest in, he had some _very_ sharp things to say. Also about Max Beckmann. Reviewing an exhibition of the paintings Beckmann produced in Holland during the Second World War, Mr. Greenberg wrote in 1946 that there were five or six pictures in the show that \"warrant calling Beckmann a great artist, even though he may not be a great painter.\" He then went to say of Beckmann:\n\n> He is certainly one of the last to handle the human figure and the portrait on the level of ambitious, original art. . . . The power of Beckmann's emotion, the tenacity with which he insists on the distortions that correspond most exactly to that emotion, the flattened, painterly vision he has of the world, and the unity this vision imposes\u2014so realizing decorative design in spite of Beckmann's inability to think it through consciously\u2014all this suffices to overcome his lack of technical \"feel\" and to translate his art to the heights.\n\nOn visual art other than painting and sculpture there are likewise some brilliant perceptions. Here is Mr. Greenberg on the photographs of Henri Cartier-Bresson:\n\n> The unusual photographs of the French artist Henri Cartier-Bresson . . . provide an object lesson . . . in how photography can assimilate the discoveries of modern painting to itself without sacrificing its own essential virtues. One thing that painting since Manet has emphasized is that a picture has to have a \"back.\" It cannot simply fade off in depth into nothingness; every square millimeter of picture space, even if it represents only the empty sky, must play a positive role. This, Cartier-Bresson, like his fellow-photographer Walker Evans, has learned preeminently. At the same time, unlike Edward Weston and the later Stieglitz, he has not forgotten that photography's great asset is its capacity to represent depth and volume, and that this capacity's primary function is to describe, convey, and make vivid the emotional \"use-value\" of beings and objects. It is to anecdotal content that Cartier-Bresson, rightly, subordinates design and technical finish.\n\nHe writes well on the cartoons of William Steig and David Low, too.\n\nNotwithstanding the great variety of subjects that is dealt with in this critical chronicle, we are seldom in any doubt that it is one subject, above all\u2014the emergence of abstract art, which Mr. Greenberg described in 1944 as \"one of the most epochal transformations in the history of art\"\u2014that lies at the center of _The Collected Essays_. In these first two volumes Mr. Greenberg's response to this \"epochal\" event, while generally positive and welcoming, remains fluid and analytical. He is fully aware of the aesthetic costs entailed in the triumph of abstraction, and not reluctant to talk about them. If he can already be said to be writing here as a \"formalist\" critic, he also shows himself to be open to, and even enthusiastic about, a good deal that the formalist criticism of a later generation has shut its eyes to. What _The Collected Essays_ of the forties makes abundantly evident, moreover, is that he is a bigger and more various writer than he has generally been thought to be. This being the case, the future volumes in this series will be keenly awaited.\n\n[1987]\n\nFootnote\n\n1. See \"A Critic on the Side of History: Notes on Clement Greenberg,\" in my book _The Age of the Avant-Garde_ (New York, 1973).\n\n## Clement Greenberg and the Cold War\n\n* * *\n\nNo art critic of our time has been the subject of more discussion than Clement Greenberg, who was born in 1909 and published the bulk of his critical writings between 1939 and 1969. Yet the nature of that discussion has at times been so contentious, not to say acrimonious, that the effect has been to obscure the virtues that made this criticism loom so large\u2014and for so long a time\u2014in the minds of both his admirers and his adversaries. It seems to me unlikely that the publication now of two further volumes of Mr. Greenberg's _Collected Essays and Criticism_ will do much to alter this situation. The academy, the museums, the media, the art journals, and a good deal of the intellectual press, not to mention foundations, corporate sponsors, and the cultural agencies of government, are now in the hands of apparatchiks who have a vested interest in defending both the kind of art and the kind of writing about art that Mr. Greenberg has famously deplored, and it is not to be expected that they will surrender their animus on the present occasion.\n\nOn the contrary, opposition is likely to be intensified, for the discussion of the issues raised in Mr. Greenberg's criticism is even more adamantly politicized today than it was in the days when he was still a regular contributor to critical opinion. In a culture now so largely dominated by ideologies of race, class, and gender, where the doctrines of multiculturalism and political correctness have consigned the concept of quality in art to the netherworld of invidious discrimination and all criticism tends to be judged according to its conformity to current political orthodoxies, even to suggest\u2014as Mr. Greenberg's writings invariably do\u2014that aesthetic considerations be given priority in the evaluation of art is to invite the most categorical disapprobation.\n\nSo rapidly has the radicalization of critical opinion accelerated in the past decade that in the seven years that have elapsed since the publication of the first two volumes of _The Collected Essays and Criticism_ , the editor in charge of this otherwise exemplary edition of Mr. Greenberg's writings\u2014John O'Brian, now professor of art history at the University of British Columbia\u2014has clearly felt obliged to abandon the politically neutral tone he brought to the presentation of the earlier volumes and to adopt a more belligerent voice for the later volumes. No doubt this is due, in part, to the shift that occurred in Mr. Greenberg's own political views over the course of his critical career, and in even larger part to Mr. O'Brian's disapproval of that shift.\n\nIn his early years as a critic, Mr. Greenberg was a Trotskyist\u2014which is to say, an anti-Stalinist Marxist\u2014yet by the end of the forties he had already described himself as an \"ex- or disabused Marxist,\" and by the fifties he had joined the ranks of the anti-Communist liberals. (Mr. O'Brian prefers to call the position of the latter \"Kantian anti-Communism.\") From the perspective of the academic Marxists who came out of the sixties, this put Mr. Greenberg on the wrong side of the cold war, making him politically suspect if not actually retrograde, and it is more or less from that perspective that Mr. O'Brian has written his introduction to the last two volumes of _The Collected Essays and Criticism_.\n\nHence the tremendous emphasis that Mr. O'Brian places on the cold war as the principal influence on Mr. Greenberg's criticism in the fifties and sixties. The main charge is that an \"acquiescence to the _Pax Americana_ and its policies was accompanied by a corresponding shift in [Mr. Greenberg's] stance as a cultural critic.\" To support this charge, Mr. O'Brian dwells at some length on the long essay that Mr. Greenberg published in _Commentary_ in 1953 under the title \"The Plight of Our Culture,\" which is now reprinted in Volume 3. This is indeed an important essay that ought to be better known. It is not only the best response to T. S. Eliot's _Notes Towards the Definition of Culture_ that I know of, but also one of the most cogent analyses of the problem of democratic culture any critic has given us in the last forty years. Unfortunately, it is one of the odd features of this new edition of Mr. Greenberg's writings that Mr. O'Brian has allowed his own political animus to distort its meaning. My guess is that he was so concerned to bring his reading of Mr. Greenberg's later criticism into line with the political views of his friend and colleague Serge Guilbaut, the author of _How New York Stole the Idea of Modern Art_ , that he scarcely noticed the degree of misrepresentation that such an effort required.\n\n\"In 'The Plight of Our Culture,'\" Mr. O'Brian writes, Mr. Greenberg \"revoked his earlier criticism of mass-circulation magazines and their blurring of distinctions between high and low culture.\" It is also claimed that \"Greenberg deduced that the newly dominant culture of the middle classes had the capacity to resist dilution and adulteration by mass culture as well as produce what he still most desired: . . .'formal culture with its infinity of aspects, its luxuriance, its large comprehension.'\" For Mr. O'Brian, then, \"the transformation in Greenberg's thinking was an about-face,\" and \"in the space of a couple of years, pessimism about the culture of modernity had given way to optimism\" for purely political reasons. \"Thus Greenberg's Cold War politics and cultural optimism merged,\" he writes, and this \"ex- or disabused Marxist\" is now said to have joined other New York intellectuals in believing that \"democracy and capitalism . . . already were demonstrating what might be accomplished in the realm of middlebrow culture.\"\n\nAll of this suggests, of course, that \"The Plight of Our Culture\" is a rousing, politically inspired apology for middlebrow culture, and hence a retreat from Mr. Greenberg's vigorous defense of high culture in his classic essay \"Avant-Garde and Kitsch,\" first published in 1939 and reprinted in Volume 1 of _The Collected Essays and Criticism_. Yet, in fact, \"The Plight of Our Culture\" is one of the most thoughtful and categorical indictments of middlebrow culture any American has ever given us. Far from representing an \"about-face\" on the imperatives of high culture, this very dour analysis of its fate under the pressures of democracy and capitalism puts a good deal of the blame for its problematic condition on the corrupting force of middlebrow culture. What does make \"The Plight of Our Culture\" different from \"Avant-Garde and Kitsch\" is its refusal to formulate its subject in orthodox Marxist terms (though Marxist thought still exerts a considerable influence on its analysis of culture) and its acknowledgment of the changes that had lately occurred in the relation of middlebrow culture to high art.1 Yet to characterize this examination of middlebrow culture as \"optimistic\" requires a suspension of attention to its most salient points, if not indeed a flight of ideological fancy.\n\n\"The Plight of Our Culture\" occupies some thirty pages in the new edition of Mr. Greenberg's writings, and cannot therefore be summarized in its totality. But here, anyway, are some of the relevant passages from Mr. Greenberg's analysis of middlebrow culture:\n\n> The liberal and fine arts of tradition, as well as its scholarship, have been \"democratized\"\u2014simplified, streamlined, purged of whatever cannot be made easily accessible, and this in large measure by the same rationalizing, \"processing,\" and \"packaging\" methods by which industrialism has already made lowbrow culture a distinctive product of itself. Almost all types of knowledge and almost all forms of art are stripped, digested, synopsized, \"surveyed,\" or abridged. The result achieved in those who patronize this kind of capsulated culture is, perhaps, a respect for culture as such, and a kind of knowingness, but it has very little to do with higher culture as something lived.\n> \n> The middlebrow in us wants the treasures of civilization for himself, but the desire is without appetite. . . . A sense of continuity with the past, a continuity at least of truth, of enduring relevance, belongs to a genuine culture almost by definition, but this is precisely what the middlebrow does not acquire. . . . He might be able to do so, eventually, by exerting humility and patience, but these he is somehow never able to muster in the face of culture. In his reading, no matter how much he wants to edify himself, he will balk at anything that sends him to the dictionary or a reference book more than once. (Curiosity without energy or tenacity is a middlebrow trait wherever and in whomever it appears.) Towards his entertainment, no matter how much he wants it to be \"significant\" and \"worthwhile,\" he will become recalcitrant if the \"significance\" is not labeled immediately and obviously, and if too many conditioned reflexes are left without appropriate stimuli. What the middlebrow, even more conspicuously than the lowbrow, wants most is to have his expectations filled exactly as he expects to have them filled.\n> \n> Middlebrow culture, because of the way in which it is produced, consumed, and transmitted, reinforces everything else in our present civilization that promotes standardization and inhibits idiosyncrasy, temperament, and strong-mindedness; it functions as order and organization but without ordering or organizing. In principle, it cannot master and preserve fresh experience or express and form that which has not already been expressed and formed. Thus it fails, like lowbrow culture, to accomplish what is, perhaps, the most important task of culture for people who live in a changing, _historical_ society: it cannot maintain continuity in the face of novelty, but must always forget and replace its own products.\n\nAs for the relation that obtains between high culture and middlebrow \"standardization,\" Mr. Greenberg can hardly be said to take an optimistic view. \"High culture, however\u2014authentic, disinterested culture\u2014has so far suffered more than it has gained in the process,\" he writes. And his characterization of popular culture doesn't offer much of a basis for optimism, either:\n\n> At the same time lowbrow, \"machine,\" commercial culture is there everywhere to offer its relief to all those who find any sort of higher culture too much of an effort\u2014lowbrow culture being powerful not only because it is \"easy\" and still suits the majority, but also because it has replaced folk culture as the culture of _all_ childhood, and thereby become our \"natural,\" \"autochthonous\" culture. (And, unlike folk culture, lowbrow culture neither contributes\u2014at least not fundamentally\u2014to high culture nor effaces itself in its social presence.)\n\nYou may, if you like, call this optimism or an \"about-face\" or\u2014what is really implied by the charge\u2014cold war propaganda, but it sounds both pretty grim and pretty accurate to me, and not exactly a cheerleader's view of either capitalism or democracy.\n\nIt is also true that Mr. Greenberg acknowledges that \"like lowbrow culture, middlebrow culture is not all of a piece,\" that \"the good and the bad are mixed,\" and that middlebrow art \"is not wholly adulteration and dilution.\" This, too, strikes me as true, at least for the time in which it was written, but not to offer much solace. Really, the only thing that Mr. Greenberg found to admire in \"the middlebrow's respect for culture\" was, as he wrote, that \"it has worked to save the traditional facilities of culture\u2014the printed word, the concert, lecture, museum, etc.\u2014from that complete debauching which the movies, radio, and television have suffered under lowbrow and advertising culture.\" Yet anyone reading this passage in the 1990s must be aware of how much more has been surrendered by these \"traditional facilities of culture\" to the demands of lowbrow, popular culture over the last forty years. But even forty years ago, when our institutions of high culture had not yet sunk to their present levels, Mr. Greenberg's view of even this development was anything but rosy:\n\n> But doesn't the damage still outweigh the gains [he asked], and can any amount of improvement at the lower levels compensate for deterioration at the highest, where the most authentic manifestations still have their being, where the forms and values of every other level originate\u2014no matter how perverted subsequently\u2014and where our experience is still most significantly and enduringly preserved?\n\nAs the passages I have quoted from \"The Plight of Our Culture\" attest, this is an essay that raises fundamental questions about the fate of high art in our society. They are the kinds of questions, moreover, that have acquired an even greater urgency today when the middlebrow culture described by Mr. Greenberg in 1953 has been largely gutted of whatever virtues that could once be claimed for it and, for both political and commercial reasons, is now effectively supplanted by something much worse. (See, for an egregious example, Tina Brown's _New Yorker.)_ For Mr. O'Brian to reduce these questions to a highly simplified scenario of cold war politics not only misrepresents the content of \"The Plight of Our Culture\" but renders it irrelevant to our current cultural concerns\u2014which is, to say the least, a curious policy for the editor of these books to pursue.\n\nWhat Mr. O'Brian clearly cannot forgive is that Mr. Greenberg had taken the anti-Communist side in the early stages of the cold war, and, as he writes, \"remained committed to the Cold War agenda of the U. S. government.\" This is the \"original sin\" that for Mr. O'Brian has left an ineradicable taint upon everything that has been gathered in Volumes 3 and 4 of this _Collected Essays and Criticism_ , right down to its last pages. For \"in 1969,\" he writes, in what turns out to be the closing item in Volume 4, Mr. Greenberg \"conducted a lengthy interview with Lily Leino for dissemination by the U. S. Information Agency, the umbrella organization for the Voice of America.\" I frankly rejoice that someone in the USIA bureaucracy had the brains to ask a writer of Mr. Greenberg's distinction to speak on the Voice of America broadcasts, but for Mr. O'Brian it remains a mark of the writer's contract with the Devil.\n\nAnd what sort of thing was Mr. Greenberg broadcasting for the Voice of America in 1969? Recalling a period twenty-five years before, \"when everybody was so sure that Americans couldn't produce art of any consequence\u2014and that included Americans themselves,\" Mr. Greenberg was, among other things, complaining that\n\n> a lot of inferior art is taken seriously all over the world. It's a paradoxical situation when someone like Rauschenberg\u2014who's nowhere nearly as good as Eakins, Homer, Ryder, or the early John Sloan, or Milton Avery, not to mention Marin\u2014is viewed a major figure because of the credit American art in general now enjoys in the world.\n\nHe also had some praise for Andrew Wyeth! But it was probably this sort of thing that Mr. O'Brian found particularly offensive:\n\n> I've seen some contemporary Soviet art, and as far as I can tell, art in the Soviet Union is controlled by Philistines. The same appears to be true in China and in every other place where Bolsheviks are in power. I call them Bolsheviks instead of Communists because I feel that that's more accurate, more specific. \"Socialism\" in backward countries means Bolshevism\u2014Stalinism, if you want\u2014and that means something barbaric, because \"socialism\" in a backward environment becomes, among other things, an aggressive expression of backwardness.\n\nSome of us knew that all this was perfectly true when Mr. Greenberg made his broadcast in 1969, and now all the world knows it was true. Yet because he spoke the truth under U. S. government auspices, Mr. O'Brian apparently finds it tainted. That is indeed the triumph of ideology over veracity.\n\nThere is no question that the cold war played a significant role in shaping American cultural life in the decade and a half that followed the end of World War II, and that ex-radical intellectuals of Mr. Greenberg's generation made an important contribution to the formulation of that role. I said as much more than thirty years ago when Mr. Greenberg's first book of essays, _Art and Culture_ , was published, and Mr. O'Brian accurately quotes me to that effect. \"To understand _Art and Culture_ is to understand a great deal about the artistic values that came out of the war and the Cold War years,\" I wrote in _Arts Magazine_ in October 1962; \"to question it is to question some of the salient achievements and aesthetic beliefs of those years.\"\n\nWhat is so curious and distorting about Mr. O'Brian's characterization of Mr. Greenberg's role in this intellectual history is that he leaves the \"salient achievements\" of the cold war period\u2014which means, in this context, the Abstract Expressionist movement\u2014unquestioned while at the same time endeavoring to reduce the \"aesthetic beliefs of those years\" to a purely political scenario. It is in this respect as well as others that he reveals his loyalty to the radical politics of the sixties, with its anti-American paranoia, its sentimentalization of Marxist ideology, and its adamant refusal to acknowledge the moral superiority of American democracy over Soviet tyranny.\n\nFor what, after all, was the cold war about if not the conflict between American (and Western) democracy and Soviet tyranny? Mr. O'Brian is not, to be sure, one of those nut cases who attempt to demonstrate that the fame and influence of Jackson Pollock was the creation of Nelson Rockefeller and John Foster Dulles and that even Harry Truman had somehow contrived to promote the interests of American abstract painting even while going through the motions of condemning it. He leaves the ideological dirty work to Serge Guilbaut and the cadres of radical art historians who have now succeeded in portraying the achievements of the New York School as nothing but a tainted product of America's role in the cold war. Yet what he has given us in a large part of his introduction to Volumes 3 and 4 of Mr. Greenberg's _Collected Essays and Criticism_ is a somewhat more respectable and respectful version of the same ideological narrative.\n\nThere is no way to understand what in 1962 I spoke of as \"the salient achievements and aesthetic beliefs\" of the war and the cold war period without some acknowledgment of the lethal effects that Stalinist influence had on American art and culture in the 1930s. The Stalinist-inspired Popular Front culture of the thirties was, in all its essentials, an irredeemably philistine and middlebrow culture, and it laid upon the arts in this country a curse of mediocrity and sentimentality from which it did not begin to recover until the 1940s. (That is the reason why William Faulkner and Wallace Stevens were dismissed as reactionary eccentrics in the thirties while Grant Wood and Thomas Hart Benton were acclaimed as geniuses.) If there is a serious criticism to be made of Mr. Greenberg's essay \"The Plight of Our Culture,\" it would have to do with his failure to acknowledge the extent to which the middlebrow culture of the forties and early fifties had been fashioned in the Popular Front culture of the thirties.\n\nIn its artistic and cultural interests, this was one of the things that Trotskyism set out to combat in the thirties, and it was out of that conflict that Mr. Greenberg wrote his essay \"Avant-Garde and Kitsch\" in 1939. But even stripped of its Stalinist distortions, the Marxism of the thirties proved to be a poor guide to what was actually happening in American art in the forties, and it is much to Mr. Greenberg's credit that he recognized the change for what it was. (By the same token, it is much to Mr. O'Brian's discredit that he still doesn't.) This was the political drama that was made manifest in virtually every area of postwar American cultural life. Whereas in the early plays of Arthur Miller, for example, you see the middlebrow sentimentalism of the Popular Front mind reenacted with a vengeance, in the criticism of Lionel Trilling you see the attempt to liberate literary and social thought from its corrupting influence. It is in that context that Clement Greenberg's criticism of the fifties and sixties needs to be understood, and it is a sad commentary on the intellectual life of the 1990s that the editor of this fine edition of _The Collected Essays and Criticism_ still doesn't get it. Hence his determination to reduce every aesthetic idea, if not aesthetics itself, to a suspect political datum.\n\n[1993]\n\nFootnote\n\n1. Still, there was much in \"Avant-Garde and Kitsch\" that did not belong to Marxism, and much, too, that is perfectly consistent with the view of high art to be found in \"The Plight of Our Culture\" and all the later criticism. For example:\n\n> . . . it is true that once the avant-garde had succeeded in \"detaching\" itself from society, it proceeded to turn around and repudiate revolutionary as well as bourgeois politics. The revolution was left inside society, a part of that welter of ideological struggle which art and poetry find so unpropitious as soon as it begins to involve those \"precious\" axiomatic beliefs upon which culture thus far has had to rest. Hence it developed that the true and most important function of the avant-garde was not to \"experiment,\" but to find a path along which it would be possible to keep culture _moving_ in the midst of ideological confusion and violence. Retiring from public life altogether, the avant-garde poet or artist sought to maintain the high level of his art by both narrowing and raising it to the expression of an absolute in which all relativities and contradictions would be either resolved or beside the point. \"Art for art's sake\" and \"pure poetry\" appear, and subject matter or content becomes something to be avoided like a plague.\n> \n> It has been in search of the absolute that the avant-garde has arrived at \"abstract\" or \"nonobjective\" art\u2014and poetry, too. The avant-garde poet or artist tries in effect to imitate God by creating something valid solely in its own terms, in the way nature itself is valid, in the way a landscape\u2014not its picture\u2014is aesthetically valid; something given, increate, independent of meanings, similars or originals. Content is to be dissolved so completely into form that the work of art or literature cannot be reduced in whole or in part to anything not itself.\n\n## T. J. Clark and the Marxist Critique of Modern Painting\n\n* * *\n\n> _The question is, finally: how could there be an effective political art? Is not the whole thing a chimera, a dream, incompatible with the basic conditions of artistic production in the nineteenth century\u2014easel painting, privacy, isolation, the art market, the ideology of individualism? Could there be any such thing as revolutionary art until the means existed\u2014briefly, abortively\u2014to change those basic conditions: till 1919, when El Lissitzky puts up his propaganda poster outside a factory in Vitebsk; or 1918, in Berlin, when Richard Huelsenbeck has the opportunity, at last, to \"make poetry with a gun in his hand\"? \u2014_ T. J. Clark, in _The Absolute Bourgeois: Artists and Politics in France, 1848\u20131851_\n> \n> _\"It turns out we are part of the superstructure,\" M. said to me in 1922, after our return from Georgia. Not long before, M. had written about the separation of culture from the State, but the Civil War had ended, and the young builders of the new State had begun\u2014for the time being in theory only\u2014to put everything in its proper place. It was then that culture was assigned to the superstructure_.\u2014Nadezhda Mandelstam, in _Hope Against Hope: A Memoir_\n\nThere are times when, owing to its larger ramifications, a single academic appointment at a leading university very quickly acquires the status of an historical event for those alert to its meaning. The appointment of T. J. Clark as professor of art history at Harvard University in 1980 was for some observers an event of this kind. For what it signified was a decisive shift in the way the study of art history\u2014and most particularly, of course, the history of modern painting, which is Professor Clark's academic specialty\u2014would henceforth be pursued as an intellectual discipline at this venerable seat of learning.\n\nIn an earlier generation the Harvard art history department, together with its illustrious museum-training program at the Fogg Art Museum, had won a high reputation among the _cognoscenti_ of the art world as a citadel of humanistic scholarship and aesthetic connoisseurship. There had indeed developed a distinctive Fogg tradition at Harvard. It was a tradition based, above all, on the close, comparative study of individual art objects, and one of its goals was to produce a certain kind of expert intelligence\u2014aesthetic intelligence. In the perspective of this tradition, it was assumed that for the mastery of such intelligence the cultivation of sensibility\u2014a sensibility for objects\u2014was quite as essential as the acquisition of historical knowledge; that, in other words, aesthetic intelligence was virtually meaningless if separated from the disciplines of connoisseurship.\n\nThis, in any event, appears to have been the ideal that once governed the Fogg tradition at Harvard, and from the influence it exerted on the museums and the universities, on art collecting and on art scholarship, there can be no question that the art public in America had over the course of many years reaped some significant benefits. There are certainly more than a few masterpieces to be seen in our public collections today which owe their presence there\u2014and thus their place in our experience\u2014directly or indirectly to precisely the kind of connoisseurship that concentrates its principal interest on aesthetic quality. For this reason, the Fogg tradition may be accurately described, I think, as an example of cultural elitism that serves the public interest.\n\nIf there was also something to be said against the Fogg tradition\u2014that it tended to be snobbish and clubby; that it was often condescending to outsiders, operating in the world of money, jobs, appointments, and preferments, as something of an old-boy network; and that it produced its full share of duffers who had all the appurtenances and pretensions of the connoisseur without any discernible trace of the sensibility or knowledge essential to the connoisseur's task\u2014well, that too may be taken as a measure of how far the ideal often fell short of being realized in an imperfect world. The ideal itself, however, was not only a noble one, but one that had the merit of addressing itself to something at once central and indispensable to our aesthetic experience.\n\nBy the time Professor Clark came to Harvard in 1980, this Fogg tradition was already on the wane. If its demise was not quite complete, its days were nonetheless clearly numbered. The cultural insurgency of the sixties had done its work at Harvard, no less than elsewhere in the academic world, and a new administration at the university was obviously prepared to welcome not only radical change but a change in the direction of an avowedly left-wing radicalism. Art history was by no means the only field in which this new direction made itself felt at Harvard. Its successful penetration of the law school faculty was sufficiently dramatic in its consequences to cause articles to be written about them in _The New Yorker_ , the _New York Times_ , and other publications. For those who understood what was happening to the study of art history at Harvard, the change represented by Professor Clark's appointment was seen straightaway to be no less fateful.\n\nWhat this change entailed, moreover, could never have been in doubt. Harvard certainly knew what it was getting. Professor Clark, a British scholar trained at Cambridge University and the Courtauld Institute, had in 1973 published two works\u2014 _The Absolute Bourgeois: Artists and Politics in France, 1848\u20131851_ , and _Image of the People: Gustave Courbet and the 1848 Revolution\u2014_ which spelled out his philosophy of art history in unmistakable terms. They had instantly established him as one of the most programmatic and plainspoken Marxist writers then concentrating on the field of nineteenth-century French art. The author of _The Absolute Bourgeois_ and _Image of the People_ was not one of your new-style, hermetic Marxists, given to employing an esoteric jargon and an impenetrable syntax. There was nothing arcane in Professor Clark's vocabulary and nothing obscure or ambiguous in his ideology. The opening chapter of _Image of the People_ had been devoted to a disquisition \"On the Social History of Art,\" which made the flagrantly political character of the inquiry altogether explicit; and lest there remain any lingering doubts on this score, Professor Clark wrote a new preface for inclusion in both volumes when these were reissued in paperback editions by the Princeton University Press in 1982.\n\nIn this preface he underscored the fact that both _Image of the People_ and _The Absolute Bourgeois_ \"were written for the most part in the winter of 1969\u201370, in what seemed then (and still seems) ignominious but unavoidable retreat from the political events of the previous six years. Those events haunt the books' best and worst pages, and provide throughout the main frame of reference for the history of a distant revolution and its cultural dimension.\" The main purpose of the preface was thus to provide a summary of what Professor Clark called \"the books' chief problem\"\u2014which is to say, their principal political objective.\n\n> So I owe it to the reader, confronting these relics of a slightly better time, [he wrote] to state again what I took to be the books' chief problem. They wished to establish what happened to art when it became involved, however tangentially, in a process of revolution and counter-revolution: in other words, when it lost its normal place in the machinery of social control and was obliged for a while to seek out other spaces for representation\u2014other publics, other subjects, other idioms, other means of production. As my Jamesian title _[The Absolute Bourgeois]_ was meant to imply, one could hardly have expected the story, in an order already irredeemably capitalist, to be other than one of failure to find such space. Bourgeois society is efficient at making all art its own. The conditions for a rendezvous between artistic practice and the only class which henceforth could produce and sustain alternative meanings to those of capital did not exist in 1850, and have only existed in botched and fragmentary form in the century since\u2014against the grain of the Bolshevik Revolution, around the edges of the Workers' Councils in Berlin, Barcelona or Turin, and nowadays [1981] perhaps in the struggle of the Polish working class to reimpose soviet rule.\n\nThere is much that could be said about the outlook that is revealed in these observations. One notes, for example, the easy, unargued assumption that the primary \"meanings\" to be gleaned in a work of art are those which derive from the economic and political situation in which it is produced. Then, too, there is the implacable hatred of capitalism and its corollary detestation of the institutions of bourgeois society (except, perhaps, where appointments to the Harvard University faculty are concerned). There is the inevitable\u2014for a Marxist\u2014glorification of revolution, and the cynical\u2014indeed sickening\u2014reference to \"the struggle of the Polish working class to reimpose soviet rule.\" From these and other attitudes so emphatically avowed in the preface to the new editions of _Image of the People_ and _The Absolute Bourgeois_ , as well as in the books themselves, it will be evident that Professor Clark not only approaches the study of art history from the perspective of a doctrinaire Marxist, with everything this implies about the primacy given to politics in his entire intellectual enterprise, but that he wishes to be clearly understood to be doing just that. Somewhat less evident, perhaps, at least to the uninitiated, is what follows from this: that in the perspective of this politicized history of art, art itself is definitively consigned to what in classic Marxist parlance is called the \"superstructure\"\u2014a subject to which we shall return in due course.\n\nNow Professor Clark has given us a new book, his first since joining the Harvard faculty. It is called _The Painting of Modern Life_ , and it bears the subtitle, \"Paris in the Art of Manet and His Followers.\" Once again the ostensible subject is nineteenth-century French painting. The names of Manet, Monet, Pissarro, Degas, Seurat, and other artists of the Impressionist era appear frequently, and reproductions of their works, both in color and in black and white, adorn its pages. Yet it will come as no surprise to any reader acquainted with _Image of the People_ and _The Absolute Bourgeois_ to discover that the real focus of _The Painting of Modern Life_ is not on the history of _art_. What interests Professor Clark is something else.\n\nIn this new book it is mainly, but not exclusively, the issue of _class_. In its fundamental concerns, _The Painting of Modern Life_ is yet another contribution to the propagation of that mythic phenomenon which lies at the heart of the Marxist conception of history: class conflict. This is Professor Clark's real subject in this book. Everything else\u2014certainly everything involving aesthetics\u2014is placed in a subordinate relation to that subject. And while Professor Clark is shrewd enough not to insist on the fact that this is his principal subject in this book, he doesn't deny that it is, either. He even offers us some ruminations on the concept of class at the outset of the discussion, but at the same time cautioning the reader that these may appear to be \"very general, not to say banal, propositions on the nature of society as such.\" He is anything but a fool. He knows exactly what he is doing. But he is also somewhat fearful\u2014as he has every reason to be\u2014of being seen to be an exponent of what he himself dismisses as \"the worst pitfalls of vulgar Marxism.\" Rather than dwell on the \"vulgar\" concept of class conflict, then, he avails himself of the more sanitized and eminently more fashionable nomenclature of semiotics. Thus the new revised standard version is: \"Society is a battlefield of representations.\" Behind this semiotic smokescreen, however, the same old Marxist scenarios remain firmly in place. Nothing has changed but the words.\n\nThe truth is, there are no instruments fine enough to measure the difference between the vulgar Marxism that Professor Clark professes to scorn and the particular variety of Marxist dialectics that governs every page of _The Painting of Modern Life_. In the one variety of Marxism, as in the other, neither the creation nor the experience of art is believed to enjoy the slightest degree of aesthetic independence from the iron laws of history. Everything in art is seen to be determined by the economic \"base\" and what are now called its \"representations.\" As a result, art is simply\u2014and sometimes very simply indeed\u2014treated as a kind of epiphenomenon of the historical process, of interest mainly as a reflection of the depredations of bourgeois society and the strategies that have been devised to conceal or compensate for their outrageous moral consequences.\n\nIn keeping with this view, Professor Clark writes of Seurat's _La Grande Jatte_ that \"it attempts to find form for the appearance of class in capitalist society\" and \"that the forms it discovers are in some sense more truthful than most others produced at the time; and that it suggests ways in which class might still be painted.\" (There is thus an assumption here that \"ways in which class might still be painted\" is, or ought to be, an important item on the agenda of contemporary art today.) And just as \"the intermingling of classes\" is said to be \"Seurat's subject,\" and hence the basis of his appeal for us, the real interest of Manet's _Olympia_ is said to be \"that class was the essence of Olympia's modernity and lay behind the great scandal she provoked.\" Of Manet's _La Musique aux Tuileries_ , moreover, Professor Clark writes that it is \"hardly a picture of modernity at all, as it is sometimes supposed to be, but, rather, a description of society's resilience in the face of empire.\" Class is thus assumed to be the key to every aspect of painting by Manet and his followers, and Professor Clark is sometimes quite cross with those painters whose work fails to support him on this point.\n\nMuch of _The Painting of Modern Life_ is devoted not so much to painting itself as it is to the work of Baron Haussmann in redesigning and reconstructing the city of Paris in the nineteenth century and to the ways in which painters depicted the life of the city as the result of this work. Professor Clark is very concerned about those artists who are deemed to have failed to grasp the real\u2014which is to say, the class\u2014meaning of this phenomenon. Monet and Renoir, for example, are among those who are given failing grades on this score.\n\n> It should go without saying that this situation\u2014Haussmann's work and its aftermath\u2014[Professor Clark writes] presented painting with as many problems as opportunities. Naturally it offered occasions for a meretricious delight in the modern, or proposals in paint that the street henceforth would be a fine and dandy place. (I cannot see, for example, that Monet's two pictures of _Le Boulevard des Capucines_ in 1873 do more than provide that kind of touristic entertainment, fleshed out with some low-level demonstrations of painterliness. Where Monet went, Renoir inevitably followed; his image _of the grands boulevards_ in 1875 is untroubled by its subject's meanings, and not helped by this innocence.)\n\nNow it needs to be pointed out, I think, that when Professor Clark writes the words _I cannot see_ in this passage, he isn't kidding. He sees only what he wants to see, and he clearly cannot find anything worth looking at in a painting if it fails to denote signs of class distinction, or other evidence of social conflict, with the requisite degree of documentary explicitness. To the extent that he can be said to have an \"eye\" for anything in painting, it is an eye for \"its subject's meanings.\" But these \"meanings\" are rarely, if ever, those determined by the painter. They are entirely predetermined by Professor Clark himself and his \"method.\" Which is to say that he has eyes only for the meanings he has gone in search of, and no other meanings\u2014least of all those which are known to have been of primary interest to the painter in question\u2014have any reality for him.\n\nAs for the matter of how painters see their subjects, Professor Clark turns out to be a curious (and sometimes comical) guide. Consider, for example, the famous question of _flatness_ in Manet's painting. Professor Clark does not deny, of course, that it is an issue in Manet, but he insists that it is not so much an aesthetic issue as an ideological and social one. Thus the very term is rejected in the discussion of _Olympia_ , for it is deemed insufficient to account for the \"circuit of signs\" that is held to be responsible for the radical nature of the painting's subject. Yet in speaking of Manet's _Argenteuil, les cantiers_ , flatness is accepted because it is found to serve a documentary function in the painting. Taking note of \"the unlikely construction of the black straw hat\" in this picture, Professor Clark writes: \"it has to be said . . . that Manet _found_ flatness more than invented it; he saw it around him in the world he knew.\"\n\nOn the other hand, what Monet's eye\u2014the most celebrated \"eye\" in Impressionist painting\u2014found in the world _he_ knew is judged to have little or nothing to do with visual observation and everything to do with the presuppositions of political ideology.\n\n> There is a rule to these paintings [by Monet], and it might be stated roughly as follows [Professor Clark writes]: Industry can be recognized and represented, but not labour; the factories have to be kept still, as if that were the guarantee in an art which pretended to relish the fugitive and ephemeral above all else. Industry must not mean _work;_ as long as that fictitious distinction was in evidence, a painting could include as much of the nineteenth century as it liked. The railway, for instance, was an ideal subject because its artifacts could so easily be imagined as self-propelled or self-sufficient. The train went discreetly through the snow. . . .\n\nAnd so on. Nor was Monet's eye the only one to be governed primarily by social considerations of this sort. Why did Pissarro, at a crucial phase in his development as a painter, turn to the Neo-Impressionist methods of Seurat? Not, according to Professor Clark, for aesthetic reasons\u2014which are never mentioned in this book. What prompted Pissarro's interest in Seurat's pictorial innovations was, it turns out, the burning issue of \"leisure in class conflict.\" (Here the \"battlefield of representations\" is abandoned in favor of the more familiar Marxist terminology.) Of Neo-Impressionism Professor Clark writes:\n\n> I do not believe that its vehemence (or its appeal to Pissarro) can be understood unless it is seen as deriving from an altered view of leisure, and of art as part of that leisure\u2014which in turn derived from a new set of class allegiances.\n\nI await with interest the monograph which either Professor Clark or one of his students must surely now be preparing for the purpose of tracing the relationship of every one of Pissarro's changes of style to the artist's shifting perception of \"class allegiances\" in the society of his day. Who knows? In the magical world of Marxist dialectics it might even be disclosed to us that it is in this issue of _class_ that the key to Pissarro's influence on C\u00e9zanne\u2014and perhaps on Matisse, too\u2014is finally to be found. No doubt all will be revealed to us in the fullness of time.\n\nIt would be unrewarding to dwell further on Professor Clark's treatment of particular artists or particular works of art in _The Painting of Modern Life_. The field of inquiry which it is the purpose of this book to explore and enlarge upon contributes nothing to our understanding of painting _as art_ , and is not meant to. On the contrary, its purpose is to destroy\u2014or, as we say nowadays, deconstruct\u2014the very idea that art is in any sense an autonomous enterprise or that its highest achievements often add up to a good deal more than the sum of the social and material circumstances of its creation. (Also deconstructed in the process\u2014and thrown, as it were, onto the ash heap of history\u2014is the concept of aesthetic intelligence that once governed the Fogg tradition at Harvard.) There is a striking irony in this, of course. For in denying to art the spiritual freedom that it has sought for itself and very often achieved in the modern era\u2014most notably, of course, in precisely those bourgeois capitalist societies which Marxist criticism is most eager to condemn\u2014writers of this persuasion inevitably exhibit a kinship with the kind of reactionary and philistine taste which led so many benighted critics to condemn modernist art in the first place.\n\nI wonder if it isn't, after all, a sense of this kinship that has prompted Professor Clark to devote so much of his intellectual energy, in this book as in his others, to disinterring a great many of the articles and reviews that attacked Manet and his contemporaries in their own day. Professor Clark's devotion to this \"lost\" critical literature, which he has culled from the annals of nineteenth-century French journalism, amounts at times to an obsession. He can hardly set down three consecutive paragraphs of his own prose without inserting into them some lengthy passage from an obscure review of the exhibitions in which Manet, Seurat, and others showed their work. Excerpts from this vast reservoir of critical writing, usually given in English in the main body of Professor Clark's text and in the original French in the voluminous notes, actually make up a sizable part of _The Painting of Modern Life\u2014_ by far the more valuable part, in my opinion. Professor Clark's real gifts are those of a researcher, not a critic. Documents, not art objects, are his forte\u2014which is yet another reason, I suppose, why he is inclined to deal with art objects as if they, too, were little more than social documents. Given his interest in nineteenth-century French art criticism, one rather regrets that Professor Clark hadn't settled for producing an anthology of critical writing along the lines of the collections given us in recent years by Elizabeth Gilmore Holt.1 But of course it is not in the interest of his political scenario to do so. His quotations from French art criticism, like his \"readings\" of French paintings as sociological reports, are made to serve a purpose. Both are part of the indictment he has drawn up for the purpose of unmasking modernism itself as a failed revolutionary impulse. What is intolerable to a Marxist like Professor Clark is that modernist art, despite its many and famous conflicts with bourgeois society, has proved in the end to be an essential component of bourgeois culture, and accepted as such. For that reason alone modernism must now stand condemned as a coefficient of capitalism\u2014than which, from a Marxist perspective, no worse crime in the realm of cultural life is conceivable. It must therefore now be discredited and deconstructed to make way for that future \"rendezvous,\" as Professor Clark has called it, between artistic practice and \"the only class\"\u2014the proletariat, of course\u2014capable of ushering in a new golden age.\n\nIt is no wonder, really, that Professor Clark is haunted by the specter of \"vulgar\" Marxism. Some forty-one years ago, in the aftermath of the radical movement of the thirties, Arthur Koestler wrote an essay called \"The Intelligentsia\"\u2014it was published in _Partisan Review_ and then reprinted in _The Yogi and the Commissar\u2014_ in which he discussed the falsity of this Marxist mode of historical analysis. He spoke of \"the fatal short cut [taken by Marx and Engels] from Economy to 'Superstructure': that is, culture, art, mass-psychology\"\u2014a shortcut that has been endlessly retraced by Marx's intellectual heirs ever since. \"Marxian society,\" Koestler wrote, \"has a basement\u2014production, and an attic\u2014intellectual production; the staircase and the lifts are missing.\" Koestler went on to point out that \"it is not as simple as that: the triumphant class creating its own philosophic superstructure to fit its mode of production like a tailored suit.\" Now despite everything that has happened in the last four decades, and despite the new terminology that is invoked to give this famous \"short cut\" a new currency, this image of a basement and an attic with the staircase and the lifts missing is, as it happens, a very accurate description of the historical method employed by Professor Clark in the writing of _The Painting of Modern Life_. For much of this book, he rum-mages around in the basement of nineteenth-century French society, retracing and reinterpreting events in the economic life of the period. To make his ascent from this basement of \"production\" to the attic, where the form of \"intellectual production\" called painting is to be found, he is obliged to avail himself of that magic carpet he calls historical materialism. No wonder, then, that when he arrives at his destination he has some difficulty discovering\u2014to use Koestler's other image\u2014a \"tailored suit\" that entirely pleases him. History is not, after all, a fertile field for the discovery of miracles.\n\nThe question to be asked about _The Painting of Modern Life_ , and indeed about Professor Clark's entire intellectual enterprise, is: Why? Why now? What accounts for its governing animus, which is a profound and unappeasable hostility to modernist art, and why has it won such a favorable response both inside the academy and out? In attempting to answer such questions, it is no longer sufficient to say that a phenomenon like Professor Clark is part of the price we continue to pay for the political upheavals of the sixties. This is true as far as it goes\u2014it is indeed the basic truth of the historical situation in which we find ourselves today\u2014but it does not entirely account for the place which this Marxist critique of modern painting now occupies in contemporary cultural life or for its reception among people who believe themselves to be interested in art, people who believe art to be vitally important to their lives.\n\nThe answer, I think, lies in the sense of keen disappointment\u2014a disappointment that amounts at times to a feeling of rage\u2014among people of a generally liberal and leftist outlook over recent changes in our relation to the achievements of modern art. Brought up to believe that the essence of the modern movement was to be found in its _r\u00e9volt\u00e9_ impulses, that in some sense modernism _was_ revolutionary in its ultimate goals as well as in its technical and expressive innovations, these people feel betrayed by their discovery\u2014which history has now made all but unavoidable\u2014that they were mistaken, or at least misinformed, about the thing they loved so much and placed so high a value on.\n\nModernism not only did _not_ turn out to be, as they were for so long led to assume it was, coextensive with the impulses of political revolutionism, but was in some respects deeply hostile to those impulses and incompatible with them. Modernism really did turn out to be an essential component of bourgeois culture. And to this fact of history, which they now recognize _as_ a fact of history, they cannot reconcile themselves. It leaves them bereft of a sense of cultural purpose. Never mind that these disappointed acolytes are the same people, for the most part, whose embrace of modernism has confirmed its central place in bourgeois cultural life. Such paradoxes of history do little to assuage the bitterness of their disappointment. They were brought up, after all, to believe that they could have the cake of revolution and eat it too, and now that they have discovered otherwise there is no consoling them. Their grief is great, and it has disposed them to welcome any and all attempts to discredit the achievements of the very thing in which they once placed their fondest hopes.\n\nThere remain, of course, for the people I speak of, certain sacred moments in the history of modern art that are exempt from the general condemnation. The Russian avant-garde in the early years of the Bolshevik Revolution, the German Dadaists in the upheaval of the First World War and the German revolution\u2014these, above all, have not lost their luster. For true believers in the doctrine that modernism must remain synonymous with revolutionism if it is not to be despised and rejected, these remain the exemplary episodes in the history of artistic production (as they call it). Hence the loving references to these episodes in Professor Clark's writing. (Hence, too, the whole program of a critical journal like _October_ , by the way, which draws its name and indeed its _Weltanschauung_ from a similar reading of history.) Hence, above all, the need to discredit the idea of _aesthetic_ achievement in art and replace it with scenarios claiming to represent an authentic _social_ history of art.\n\nProfessor Clark is only one exemplar of this movement which has made such headway in the academy and the art journals (including such eminently bourgeois journals as _Art in America_ and _Artforum)_ in recent years, and Harvard's is by no means the only department of art history to have felt its influence. Yet precisely because Harvard for so long represented a philosophy of art that has now been made a special object of ideological deconstruction, T. J. Clark's appointment to that faculty has inevitably acquired a significance that reverberates far beyond the groves of academe.\n\n[1985]\n\nFootnote\n\n1. Holt's two major works on the history of art in nineteenth-century Europe are _The Triumph of Art for the Public_ (New York, 1979) and _The Art of All Nations_ (New York, 1981).\n\n## Rembrandt as Warhol: Svetlana Alpers's \"Enterprise\"\n\n* * *\n\nFrom time to time we are given a new book that so vividly illuminates the intellectual ground we stand on that it instantly acquires the status of an emblematic event. As far as the study of art history is concerned\u2014and more particularly, what has gone wrong with it\u2014I believe we have now been given such a book in Professor Svetlana Alpers's _Rembrandt's Enterprise: The Studio and the Market_. Professor Alpers is one of the leading representatives of the kind of art history that now dominates the profession, and is widely recognized as an academic eminence of considerable power and influence. Occupying a senior position on the art history faculty at the University of California at Berkeley, she is the author of an earlier and much-praised study of seventeenth-century Dutch painting, _The Art of Describing_ (1983), and serves as an editor of two academic journals, _Representations_ and _The Raritan Review_. She is also, I believe, an adviser to the Getty operations that now play such a conspicuous role in the funding of art historical research. She has been a fellow of the Center for Advanced Study in the Behavioral Sciences at Stanford, a fellow at the Institute for Advanced Study in Princeton, and the recipient of numerous other fellowships and awards. _Rembrandt's Enterprise_ is itself based on the Mary Flexner Lectures delivered at Bryn Mawr College in 1985\u2014though, as we shall see, it was the more academic rather than the political parts of the book that were mainly vouchsafed to the audience at Bryn Mawr.\n\nClearly, we have in Professor Alpers one of the brightest stars now traversing the academic firmament. This makes it all the more imperative that we understand the nature of _her_ enterprise, which I believe to be a wholly destructive one. To understand what she has set out to accomplish in this book, it is essential, however, that we have some sense of the intellectual situation of which _Rembrandt's Enterprise_ is so important an expression. Only then will it be possible to see why her book is indeed an emblematic event\u2014emblematic, that is, of the catastrophe that has overtaken not only the study of art history but much else that we prize in the life of the mind.\n\nIs it an exaggeration to speak of a catastrophe in these matters? I think not. For it is one of the truly horrifying characteristics of the present age that so many of its most gifted scholars and intellectuals\u2014especially in the fields of art history and literary study\u2014are nowadays almost exclusively engaged in systematic and politically inspired demolition projects. In this respect they differ so greatly from even the most critical and contentious of their predecessors as to constitute a separate profession. To say that they no longer belong to the tradition of humanistic study is hardly sufficient to describe what it is they are actually doing, for it is precisely this humanistic tradition and its accomplishments that all their intellectual energies are mobilized to \"deconstruct\"\u2014which is to say, destroy. They may, perforce, still feel themselves obliged\u2014as Professor Alpers does in her study of Rembrandt\u2014to lavish attention upon the achievements which this tradition has elevated to a position of the highest esteem; but if so, it is only for the purpose of nullifying that esteem and placing those accomplishments in question.\n\nIt needs to be emphasized that there is much more involved in this phenomenon than a mere revision of received opinion. The greatest scholarship, like the greatest criticism, has always had a revisionist effect upon our thinking, for new research and new ideas in any field of study are bound sooner or later to dislodge established beliefs, topple entrenched intellectual authority, and introduce new interpretations and judgments. It is one of the purposes of new knowledge, after all, to alter our understanding, and it is all the more to be welcomed when it puts us more deeply in touch with the spirit as well as the substance of the objects of our study.\n\nThe academic projects I speak of, however, are different in purpose and in spirit from the scholarship and criticism of the past. It is the unmistakable goal of this new learning (as I suppose it must be called) not to bring us a deeper appreciation of the achievements of the past but, on the contrary, to discredit them in some fundamental way and thereby render the whole idea of achievement\u2014in art or in anything else having to do with aesthetic or moral distinction\u2014highly suspect if not indeed completely fraudulent.\n\nProjects of this kind may take a variety of forms\u2014or, as they say in the academy, may employ a variety of methodologies\u2014but fundamental to all of them is the attempt to make the study of art a branch of the social sciences. By effecting this change of intellectual venue, these scholars hope (as they see it) to demystify their subject, strip it of its claims to any sort of aesthetic or spiritual achievement, and thus reduce it to a level where it is no longer to be differentiated from any other form of material culture. The very idea that a work of art may be thought to transcend the material conditions of its creation is an anathema to this new learning, which is rigidly deterministic in its entire outlook on art and life. The particular mode of determinism deemed most appropriate to the subject under study will _vary_ , of course, according to the special ideological goals to which the study is tethered. The feminists and \"gender\" specialists will naturally favor a purely biological determinism; the Marxists (of whatever stripe) will similarly favor a purely economic or \"class\" determinism; and so on, as the emergence of sundry hybrid or crossover ideologies prompts the creation of ever new methodological instruments. But the fundamental goal remains the same: art must be categorically removed from the realm of aesthetics and placed firmly in a realm where the only legitimate questions are those that can be asked about the material\u2014which is to say, the political and economic\u2014conditions of its production.\n\nIn the field of art history, this movement\u2014in which Professor Alpers plays a major role\u2014now commands the greatest academic prestige and the greatest academic patronage. Books by its leading practitioners and their many acolytes flow from the presses\u2014especially the university presses\u2014in ever greater numbers, and academic appointments are governed by its followers with a degree of control that old-time union leaders would have every reason to envy. The movement's hegemony\u2014to borrow the atrocious term that is so often on the lips of its representatives\u2014is becoming so nearly complete that newcomers to the field can no longer hope to have much of a career in art history if they are so rash as to register their dissent from its fundamental premises. Young art historians\u2014and many who are no longer young\u2014now face the choice of joining the herd or seeking employment elsewhere (meaning, of course, outside the academy). Under the circumstances, it is no exaggeration at all to speak of a catastrophe.\n\nIt is one of Professor Alpers's distinctions as a writer to make this fateful turn in the outlook of art history\u2014and indeed, in the whole notion of what art is\u2014sound at times as if it were a weighty and even disinterested enterprise. Less obviously strident than many of her political colleagues and often more diligent in her research, she is remarkably adept at mimicking the tone of the old-style scholar when it serves her purpose, and thus at giving us the impression\u2014at first glance, anyway\u2014that there is nothing of the vulgar ideologue to be found in her arduous historical labors. The learned references accumulate with the requisite density, and there is even a show of civility (of a sort) toward one or another of her illustrious predecessors. Writing, for example, about Julius Held's assertion that Rembrandt's is \"the work of a man who never compromised, who never permitted himself to be burdened with a chain of honor, and fiercely maintained both the integrity of his art and his freedom as a man,\" Professor Alpers observes (characteristically) that \"this is not wrong, but the terms which are brought together here are unexamined.\" What she means is that they are not sufficiently political, and she then hastens to apply her own, more explicitly political terms to the question at issue here, which is Rembrandt's \"individualism\" and which is said in her account, not surprisingly, to be \"a function of the economic system in which he lived.\"\n\n_Rembrandt's Enterprise_ is not a long book. The text consists of only 122 pages divided into four chapters, and there are 32 pages of notes. But it sometimes reads like a long book while one is trying\u2014especially in the first three chapters\u2014to figure out exactly where it is going and to what purpose. The observation I have quoted about Rembrandt's individualism, for instance, does not turn up until page 114, in Chapter IV, which is entitled \"Freedom, Art, and Money\"; and by that time, of course, we know where we stand\u2014or rather, where Professor Alpers has been standing from the beginning. (The reference to Rembrandt as a man who \"loved only three things: his freedom, art, and money,\" comes from J. B. Descamps's _La Vie des Peintres Flamandes, Allemands et Hollandois_ , published in Paris in the eighteenth century.) But in the first three chapters of _Rembrandt's Enterprise_ , though they are not without some interesting observations on the life of art in Rembrandt's time and on his own working methods, the gravamen of the argument is often elusive. As one makes one's way through the twists and turns of these first three chapters, one more and more has the sense of an indictment being drawn up, and while there is never any question in one's mind that the culprit\u2014namely Rembrandt\u2014will be found guilty, the crime with which he is to be charged nonetheless remains obscure for pages at a time. It is only in Chapter IV that it is openly stated. Interestingly, it was this Chapter IV, which contains the substance of Professor Alpers's political charge, that was omitted from the Mary Flexner Lectures at Bryn Mawr. Which leads one to wonder what it was that the students in the audience at Bryn Mawr understood Professor Alpers to be talking about.\n\nIt must be said, too, that as a writer Professor Alpers has an extremely irritating habit of setting out to make a point, or at least of inducing the illusion that a point is about to be established by means of all the learned references that are set before us with such a showy display of erudition, and then, without the point ever really being made, of looking back on what she has been saying as if an irrefutable conclusion had been reached. This rhetorical strategy, if it can be called that, has the effect of rendering a good deal of the scholarship in the book either completely redundant or simply pedantic. Certainly much of it, though not without interest, proves to be irrelevant to her principal theme. But as we don't know for certain what the principal theme is until we get to that crucial fourth chapter, we are left to savor the erudition for its own sake\u2014which is harmless, I suppose, even though it suggests that Professor Alpers is not primarily a thinker at all but a pedant who is dependent upon ideology for the substance of whatever thought she wishes to convey. It is, in any case, only when she comes to make an ideological point that her writing acquires an unmistakable clarity.\n\nWhat, then, is the charge that Professor Alpers is so concerned to make in this book? What, in fact, is Rembrandt's \"enterprise\" claimed to have been?\n\nThe very word _Enterprise_ in the book's title alerts us, of course, to the fact that we are to regard Rembrandt as first and foremost an entrepreneur, and the book's subtitle, _The Studio and the Market_ , underscores the extent to which we are being invited on this occasion to consider Rembrandt's art primarily as a business. But there is more to Professor Alpers's charge than a mere description of the artist's role in marketing his pictures. She would certainly not want to be seen as one of those \"vulgar\" Marxists who see nothing in an artist's work but its economic function\u2014although, in fact, when one has penetrated Professor Alpers's new-style academic jargon (Rembrandt's studio is described as \"enabling,\" etc.) and waded through the 258 footnotes, it is quite apparent that her argument turns out to be far more dependent upon such crude Marxist formulations than she could ever bring herself to acknowledge.\n\nIt is in the very last paragraph of Chapter III that Professor Alpers gives us a first, brief, tantalizing glimpse of the case she means to bring against the artist she is writing about. (How frustrating it must have been for the Bryn Mawr audience to have the Mary Flexner Lectures conclude on a note that, in truth, marks their real beginning!) Writing in this chapter about _The Jewish Bride_ , long considered one of Rembrandt's greatest works, she takes up the matter of the \"universality\" that has been traditionally attributed to so many of the artist's greatest works, and sets about the task of deconstructing the term. It isn't a true universality that can be claimed for _The Jewish Bride_ , according to Professor Alpers, but only an _\"effect_ of universality\" (her emphasis), and she then explains how this artificial \"effect\" is achieved:\n\n> He makes a portrait more than a portrait _by_ elevating the genre. [This, by the way, is not said in admiration.] But this _effect_ of universality is achieved by masking or hiding the economic and social basis of the transaction between the painter and his sitters.\n\nThe suggestion of \"individuality\" in Rembrandt's subjects is likewise said to be something \"imposed on them\" by the \"master's domination\" of them. \"One might say,\" she writes, \"that rather than serving [his patrons], Rembrandt exercises his authority over them,\" and since we all know that for writers like Professor Alpers authority is always to be considered a very bad thing, this too figures in the indictment. Thus she concludes Chapter III with the following analysis:\n\n> The nineteenth century credited Rembrandt with being uniquely in touch with something true about the individual human state. I would put it differently. It was something stranger and more unsettling. Rembrandt was not the discoverer, but one of the inventors of that individual state. And so his late works became a touchstone for what western culture, from his day until our own, has taken as the irreducible uniqueness of the individual.1\n\nWhen we turn to Chapter IV of _Rembrandt's Enterprise_ , we are bluntly informed that Rembrandt was \"a man of the market,\" and this is intended to mean something more than the mere fact that he sold his paintings for money. And it is not meant to mean anything good, of course. The fact that Rembrandt achieved both a personal and artistic freedom by means of the market is taken by Professor Alpers to be crucial to the indictment she draws up against him.\n\n> He had a \"propensity to truck, barter and exchange,\" in Adam Smith's famous phrase [writes Professor Alpers], and to make works suitable to such transactions. As a master in the studio he made himself a free individual, not beholden to patrons. But he was beholden instead to the market\u2014or more specifically to the identification that he made between two representations of value, art and money.\n\nIn Professor Alpers's view, the market economy provided Rembrandt with his \"models or definitions of self and art,\" and the market itself is said to have marked \"the establishment of a system of representation.\" (It is at this point in the argument that Professor Alpers effects a marriage of sorts between the crude Marxist formulations that lie at the heart of her analysis and the familiar twaddle of poststructuralist academic jargon.) Since it was from the market that Rembrandt is alleged to have derived his \"models or definitions of self and art,\" and the market was itself \"a system of representation,\" it is no longer a mystery as to where Professor Alpers's argument is going. \"Rembrandt was also a maker of representations,\" she writes, \"hence the appropriateness of his making art into a commodity to be exchanged in a marketplace so perceived.\" And then, just in case we are in danger of missing the point, she writes: \"One can say of this artist what Adam Smith said of mankind, 'Every man thus lives by exchanging or becomes in some measure a merchant.'\" The next paragraph begins: \"Let us consider Rembrandt's paintings as commodities . . .\"\n\nBut what was it that Rembrandt was actually selling in the market\u2014besides his paintings, that is? He was selling what Professor Alpers is fond of calling \"the individuality 'effect,'\" and this is said to be \"a function of the economic system in which [Rembrandt] lived and in which he played such an active part.\" There then follows this remarkable passage:\n\n> It was in Rembrandt's time that the individual came to be defined in what were then new, economic terms. Those familiar words from the American declaration of human rights, \"life, liberty, and the pursuit of happiness,\" are a reworking of Locke's \"life, liberty and estate\" which constituted his definition of property: \"By Property I must be understood . . . to mean that Property which Men have in their Persons as well as Goods.\" On this view, to be an individual is defined by the right to property. And the most essential property right of each man, and hence the grounding of this notion of the individual, curious though it might sound put in this form, is the right to property in one's own person. Freedom, then, is defined as proprietorship in and of one's own person and capacities. It is the proprietorial quality of this notion of the individual to which I wish to call attention. It was Rembrandt who made this the center of his art\u2014the center, even, of Art.\n\nIn reading this passage, it is important to understand\u2014it does, admittedly, take some effort\u2014that Professor Alpers is not lavishing any praise on either Rembrandt or liberty or the concept of the individual. Far from it. She is concerned to unmask the true nature of Rembrandt's enterprise. And what was that? Well, he stands charged with having commodified _himself_ by virtue of having painted and marketed his own self-portraits. On page 118 we are finally given Professor Alpers's definitive judgment on the great self-portraits: \"Rembrandt was an entrepreneur of the self.\"\n\nAll through the reading and rereading of this last, ghastly chapter of _Rembrandt's Enterprise_ , I was overcome with a sense of having read it, or something very like it, before. It is a feeling one often has in reading the \"new\" art historians, and the new literary scholars, too. No matter what their subject, and no matter how \"original\" their treatment of it, there is a political scenario that tends to repeat itself over and over again, endlessly striking the same notes and leading to the same conclusions. In the case of _Rembrandt's Enterprise_ , however, it wasn't of another art historian that (I came to realize) I was being reminded. It was of that eminent Marxist literary theorist and academic, Professor Fredric Jameson. At the risk of depressing the readers of this article even more than I already have, I want to quote the relevant passage from Professor Jameson's essay on \"Postmodernism and Consumer Society\"\u2014which, as a matter of fact, I have quoted once before in _The New Criterion_2 but which I must return to once again because it explains so much about the nature of Professor Alpers's enterprise.\n\n> The great modernisms [Professor Jameson wrote] were . . . predicated on the invention of a personal, private style, as unmistakable as your fingerprint, as incomparable as your own body. But this means that the modernist aesthetic is in some way organically linked to the conception of a unique self and private identity, a unique personality and individuality, which can be expected to generate its own unique vision of the world and to forge its own unique, unmistakable style.\n> \n> Yet today [we] . . . are all exploring the notion that that kind of individualism and personal identity is a thing of the past; that the old individual or individualist subject is \"dead\"; and that one might even describe the concept of the unique individual and the theoretical basis of individualism as ideological. There are in fact two positions on all this, one of which is more radical than the other. The first one is content to say: yes, once upon a time, in the classic age of competitive capitalism, in the heyday of the nuclear family and the emergence of the bourgeoisie as the hegemonic social class, there was such a thing as individualism, as individual subjects. But today, in the age of corporate capitalism, of the so-called organization man, of bureaucracies in business as well as in the state, of demographic explosion\u2014today, that older bourgeois individual subject no longer exists.\n> \n> Then there is a second position, the more radical of the two, what one might call the poststructuralist position. It adds: not only is the bourgeois individual subject a thing of the past, it is also a myth; it _never_ really existed in the first place; there have never been autonomous subjects of that type. Rather, this construct is merely a philosophical and cultural mystification which sought to persuade people that they \"had\" individual subjects and possessed this unique personal identity.\n\nProfessor Alpers does not tell us which of these \"positions\" she is espousing in _Rembrandt's Enterprise_. I suspect it is the first\u2014but then it hardly matters. The point is, it is this political interpretation of the concept of the individual and of Rembrandt's role in fostering it that _Rembrandt's Enterprise_ is really about. This, anyway, is its manifest subject. But I think the book is also to be taken as a political allegory about the relation of art to life in our time as well. Indeed, the more one thinks about _Rembrandt's Enterprise_ , the more Professor Alpers's \"Rembrandt\" comes to resemble an artist like Andy Warhol, the most successful \"entrepreneur of the self \" in the art of our own era. Rembrandt as Warhol: such is the fate of even the greatest art when it is so categorically removed from the realm of aesthetics and made to do duty as just another counter in the dialectic of material culture. Such, too, is the dismal fate of art history when the study of art is no longer its primary concern.\n\n[1988]\n\nFootnotes\n\n1. One notes with interest the lower-case \"w\" in this reference to \"western\" culture. Are we to take this as Professor Alpers's contribution to the current debate over the place to be accorded to courses in Western civilization in the liberal arts curriculum? It would seem so.\n\n2. See \"Modernism and Its Enemies\" in _The New Criterion_ , March 1986.\n\n# III \nModernism and After\n\n## ESSAYS\n\n### Richard Serra at MOMA\n\n* * *\n\nThe Richard Serra exhibition which has come to the Museum of Modern Art this spring is an event bound to disturb\u2014perhaps even infuriate\u2014a great many more people than it pleases. In this respect, as in others, the exhibition is completely faithful to the spirit of Mr. Serra's work, which from the outset of his career\u2014some twenty years ago\u2014has been conceived of as a series of provocations. Yet it would be a mistake, I think, to allow this posture of provocation to set the terms of our response to the sculpture which we encounter in the MOMA exhibition. It would also, alas, be a mistake to discount it entirely. For the element of provocation is an integral part of Mr. Serra's artistic program. It would therefore be absurd to ignore it.\n\nThe problem, in Mr. Serra's case, is to determine exactly what _artistic_ role the artist's posture of provocation plays in the creation of his sculpture. This is not an easy task, however, for the artist himself has done much to obscure the issue, and his many \"activist\" champions\u2014no doubt taking their warrant from the artist's own attitudes and then raising the ante, so to speak, in their own political interests\u2014have done even more. We need to be reminded, then, that it is as an _artist\u2014_ and not as something else\u2014that Mr. Serra's primary claim on our attention rests. To see what that claim really amounts to, it is therefore essential that we separate the art from the artist's reputation. And this, in turn, means that on the occasion of the present exhibition at MOMA, we must separate the art from the texts describing it in the catalogue accompanying the show\u2014texts which MOMA itself, in an extraordinary disclaimer, has all but disavowed but which it has obviously deemed politic to publish without interference.\n\nThere is no mystery, of course, as to how this unusual situation, in which the museum finds itself publicly at odds with its own publication, has come about. The museum had apparently intended for some time to devote an exhibition to Mr. Serra's work. Before much headway could be made in that endeavor, however, Mr. Serra suddenly emerged as a figure of considerable controversy. The large outdoor sculpture called _Tilted Arc_ , which he created for the plaza of the Jacob K. Javits Federal Building in lower Manhattan on a commission from the General Services Administration, became the focus of a fierce debate. A great many people who work in the Javits building\u2014people who have to live with the sculpture every working day\u2014strongly objected to it. (The sculpture consists of a Cor-Ten steel wall, measuring 120 feet in length and 12 feet in height, which bisects the building's plaza.) This led to a hearing on the suitability of the work. As a result of the hearing, a decision was made to order the removal of _Tilted Arc_ to another site, and this prompted further debate. (It did not cause the work to be removed, by the way. At this writing, it remains in place.) Naturally, given the current climate of opinion, the whole episode made of Mr. Serra something of an artist-hero and, owing to the fact that _Tilted Arc_ was a government commission, something of a political martyr as well.\n\nAs I have elsewhere given my opinion on this work and the controversy surrounding it, I do not propose to go into the matter again in any detail here.1 Suffice to say that I was not one of those who succumbed to the charms of _Tilted Arc_ , nor was I persuaded that the decision to order its removal from the Javits plaza constituted a crime against art, society, or anything else. On the contrary, it seems to me that there are legitimate reasons for objecting to the work and that the entire controversy has done much to illuminate the deeply problematic nature of the project that has come to be called public art\u2014a vast and expensive program upon which the government embarked with a great deal of naivet\u00e9 and a woeful paucity of serious thought about its implications or indeed its essential purposes.\n\nWhat is of more direct concern here, however, is the role that the _Tilted Arc_ affair has played in the presentation of the MOMA exhibition. By its very nature, of course, _Tilted Arc_ could not be included in the exhibition. There is, to be sure, a photograph of the work in the show, as there are of the artist's many other immense public sculptures both here and abroad. But photographs are not sculpture, and the photographs in question are in many cases extremely misleading. This is certainly the case with the photograph of _Tilted Arc_. It does not in any sense suggest what it is actually like to experience the sculpture _in situ_ , which is what the controversy is really about. _Tilted Arc_ is by no means the only work photographically misrepresented in this way in the MOMA show. A sculpture I very much admire\u2014the construction called _Carnegie_ , which was erected last fall outside the entrance of the Museum of Art at the Carnegie Institute in Pittsburgh\u2014is similarly misrepresented by an aerial view of a kind which is vouchsafed to almost no one but the birds (and professional photographers seeking bird's-eye views). We can hardly be surprised, then, that it is one of these aesthetically sanitized photographs of _Tilted Arc_ that adorns the cover of the MOMA publication, which is not really a catalogue of the exhibition but a propaganda offer-ing wholly devoted to exalting Mr. Serra in his role as artist-hero and political martyr.\n\nThe question which therefore most urgently needs to be asked about the sculpture of Richard Serra at this juncture is this: What is it we are left with when all the controversy and role-playing is put aside? With this question in mind, the MOMA exhibition turns out to be an event of exceptional artistic interest. By no means is everything in the exhibition an unqualified artistic success, in my opinion. But there is much that is. And there can be no question, I think, that Mr. Serra has taken sculpture in a direction that is new, however much his work may owe to the ideas of his contemporaries. What is original in his work may not always be to our liking, perhaps\u2014but it nonetheless leaves the art of sculpture significantly altered.\n\nEvery artist is shaped to one degree or another by his moment of entry into the art scene, and Mr. Serra has proved to be no exception to the rule. His moment of entry came in the turmoil of the sixties, and in his work there are still to be discerned some of the characteristic forms of that period and some of its characteristic attitudes, too (not all of which found their way directly into art at the time). Much of the new art of that period\u2014especially the new abstract art\u2014was not only highly formalistic in spirit but cool and impersonal in feeling. Elementary shapes\u2014geometric or at least straight-edged\u2014predominated, and in sculpture there was a distinct preference for outsize scale and for an industrial or fabricated look in the finish of the work. Such were the established characteristics of Minimal sculpture when Mr. Serra arrived on the scene in the sixties, and he very quickly made them his own. Minimalism became the \"tradition\" to which he attached himself.\n\nTo this tradition, however, was added something else: the notion that both the process by means of which the sculpture is executed and the specific space in which it is created must henceforth be regarded not merely as the circumstantial conditions of the work's realization but as essential, defining aspects of its artistic existence. Hence the emergence of so-called process art and site-specific sculpture, in both of which Mr. Serra played a much publicized role.\n\nNot all of the influences that shaped Mr. Serra's outlook in the sixties were strictly aesthetic, of course. There was also the _r\u00e9volt\u00e9_ ethos of the counterculture, which called for a programmatic resistance to whatever could then be regarded as established social practice. The innovations inherent in both process art and site-specific sculpture were born of this spirit of resistance, but in themselves they proved to be feeble as well as temporary challenges to the artistic status quo\u2014and no challenge whatever to the established social order. With few exceptions, they were easily assimilated into the patronage system they were ostensibly designed to challenge, and have remained a permanent feature of an expanded art world ever since. What began as a radical attack on established artistic conventions was very quickly embraced as a new artistic orthodoxy. Which is why a \"radical\" artist like Mr. Serra could so swiftly become the darling of cultural bureaucrats, bourgeois collectors, and highly placed academics throughout the Western world. It was not that he had failed to produce something new, but that within a decade of his debut he was clearly seen to be taking sculpture in a direction which, although new and in some respects startling, was nonetheless consistent with what prior modernist practice had prepared the way for.\n\nI have already observed that Minimalism was the tradition to which Mr. Serra attached himself at the outset of his career, and he remains very much a Minimalist sculptor today\u2014but a Minimalist with a difference. To the purely formalistic attributes of Minimalism he has added something odd and unexpected\u2014an element of dread, or what might even be called fear, which I do not recall ever before encountering in a sculptural work in quite the same way. In much of Minimalist sculpture, no matter how elephantine the scale of the work may be, the highly simplified forms together with their impersonal surfaces and rigorously logical structures tend to have a rather benign, if not indeed a sedative, effect on any reasonably knowledgeable spectator. Just about the last thing in the world we expect to experience in the presence of these works is a sensation of dread. Far from posing either a physical or theoretical threat to our existence, Minimalist sculpture tends for the most part to relax the spirit and offer us the visual equivalent of nirvana\u2014which is to say, deliverance from all sense of pain, anxiety, or trauma.\n\nThis is distinctly not the case with many of the steel constructions we encounter in the Serra exhibition at MOMA. For one thing, these sculptures occupy the museum's spaces with an authority that is all but indistinguishable from a certain mode of aggression. Their outsize forms tend either to impede or direct our movements in or around them; to take an interest in these works is, to some extent, to surrender to their scale, which is not so much a human scale as a scale designed to diminish our sense of autonomy and volition. I speak particularly of the room-size constructions which, by means of immense steel slabs, divide space in such a way as to hold us captive, direct our vision, and control our movements as a condition of seeing the work at all. I cannot speak for others, of course, but I doubt if I am the only spectator to find the experience of such works vaguely threatening. Enclosed in the vaultlike spaces that the sculptor has created, one naturally wonders\u2014not continuously, but from time to time\u2014what would happen if one of these weighty steel members were suddenly to topple. Or rather, one doesn't wonder\u2014one knows for certain that in such an event one would be crushed to death. Of course nothing does topple. Everything remains securely in place, and one is free to explore the work in perfect safety. And yet the implied threat is there\u2014and it persists in remaining a constituent element in the experience of seeing Mr. Serra's work for as long as one lingers in its presence. Even with the constructions that one cannot enter\u2014which are roped off from the viewer and which we therefore see from a space we do not share with the work itself\u2014the delicate balances sustained by the huge steel plates give the impression of a precariousness that reinforces this sense of an implied threat and thus surrounds us with an atmosphere of dread.\n\nI at least know of nothing quite like this in the history of sculpture. As Dr. Johnson famously said of the prospect of hanging, it is an experience that concentrates the mind. We leave this exhibition with a _very_ vivid sense of having seen something and felt something we have never seen or felt before, and the memory of it is not soon dissipated. It remains fixed and unforgettable. Whether we like it or not\u2014whether, indeed, we are meant to like it\u2014is another question, and I _very_ much doubt if this question of our \"liking\" the work is of compelling interest to Mr. Serra. I rather think he would believe he had failed in his mission if we were found to like it too quickly. (Which, if true, must make his current popularity among cultural bureaucrats something of a problem for him.) His _primary_ aim (I gather) is to affect and manipulate our response, to impose a certain control on our perceptions, and thus to shatter our complacency in the face of the art experience, and in this he certainly succeeds. The marriage of Minimalist form and existential dread turns out to be a potent combination.\n\nI note with interest, by the way, that neither of the two sculptures that MOMA has acquired for its permanent collection\u2014 _Cutting Device: Base Plate Measure_ (1969), which is the first work we see on entering the exhibition, and _Modern Garden Arc_ (1986), which reposes in the museum's elegant sculpture garden\u2014can be said to generate this sense of dread. They belong to Mr. Serra's more purely formalistic works. Which is entirely understandable. Vastly more space would be required for one of Mr. Serra's room-size constructions than MOMA has, or will ever have, available in its permanent collection. And there are probably other reasons why the museum has settled for what is, in truth, only a token representation of the artist's work in its permanent collection. Mr. Serra is not, after all, primarily a museum artist. As that wall of photographs documenting Mr. Serra's many public commissions reminds us, his major works have mainly been created for large outdoor sites, and I very much doubt if MOMA is ever going to turn over its beautiful garden to something akin to the _Tilted Arc_. Were it ever to do so, the uproar likely to ensue would make the whole controversy that has raged over the commission for the Javits plaza look like a pastoral idyll.\n\nInstead of this unpleasant and indeed unthinkable prospect, the museum has turned over its book on Richard Serra to two authors\u2014Rosalind E. Krauss and Douglas Crimp, co-editor and managing editor, respectively, of _October_ magazine\u2014who, for the most part, completely ignore the exhibition that MOMA has mounted on this occasion and offer the museum's public absolutely nothing in the way of an intelligible guide to the artist's work. In the customary _October_ manner, the one gives us a poststructuralist disquisition of insurmountable opacity while the other rides his political hobbyhorse for a replay of the _Tilted Arc_ melodrama, grandly announcing that we have now, as a result of this controversy, witnessed \"the redefinition of the site of the work of art as the site of political struggle.\" No wonder, then, that William Rubin, the director of the museum's Department of Painting and Sculpture, felt obliged in his preface to this publication to declare that \"The Museum of Modern Art disagrees with the rhetorical tone and historical polemic of much that has been written about _Tilted Arc_ here as elsewhere.\" We can assume that Mr. Serra got the publication he wanted and that getting it was one of the conditions he laid down for agreeing to this exhibition. That is the way it usually goes nowadays with artist-heroes, especially those honored few who have also been elevated to the status of political martyrs. They get everything they ask for. But the fact remains that this MOMA exhibition is itself in no sense a \"site of political struggle,\" nor is Mr. Serra himself of any outstanding interest as a political figure. He _is_ of great interest as a sculptor, however, and it is a pity that an institution of MOMA's stature could not bring itself to offer its public even a hint about what the real nature of that interest might be. Its abject surrender in this matter has surely set a precedent it will live to regret, for the next time around the ante will be raised to a degree guaranteed to be even more unpleasant than anything the museum might have faced on this occasion.\n\n[1986]\n\nFootnote\n\n1. See \"Art Since the War: Who Will Write Its History?\" in _The New Criterion_ , Summer 1985, pp. 7\u20138.\n\n### The Death of Andy Warhol\n\n* * *\n\nThe death of Andy Warhol was bound to be a media event, and so it was. For the media, after all, it was like a death in the family. Here was a figure who was famous for being famous, for knowing the famous, and for serving as an avatar of fame, and nothing so pleases the media as an opportunity to celebrate one of their own creations. The front-page obituary in the _New York Times_ , the special segment on the MacNeil-Lehrer News Hour, the cover story in _New York_ magazine, not to mention the many pictures and news stories in the daily tabloids and on the network news programs\u2014the coverage could hardly have been better (or worse) if Warhol himself had orchestrated it. Which, in a sense, he had. The most distinguishing characteristic of this prodigious outpouring of commentary, homage, and celebrity worship was the way it confined itself to the terms that Warhol himself had set for the discussion of his life and work. Even writers who, on other occasions, find it appropriate to apply more elevated standards to art (and life) proved ready and eager to suspend them in discussing Warhol and his significance. It was as if no language but Warhol's own\u2014the language of hype\u2014could be expected to have any meaning when it came to explaining just what it was that made him important.\n\nAll the same, it is my impression that Warhol's death caused some of these volunteer laureates of hype and celebrity a good deal of uneasiness. Suddenly they were on the spot. They couldn't in good conscience bring themselves to say that Warhol had been a great artist, even though they had often written about him as if he were. They couldn't, in many cases, even bring themselves to explain why he should be considered an important artist, even though they had long taken it for granted that he was. No one is under oath, of course, in writing obituary notices. Even so, there was a general tendency on this occasion to take refuge in the subject's fame, in his personality, in his business affairs and his entourage, even his wig, and leave the art more or less unexamined. Amid the expected encomiums, there was in fact a discernible hedge and wariness to be observed in the claims advanced for Warhol's artistic achievement. It turned out that almost no one could bring any conviction to the task of specifying what that achievement had consisted of.\n\nThe truth is, even among his friends and admirers it was widely recognized\u2014even if seldom admitted\u2014that, although being an artist was essential to his social position, Warhol was far more important as a social phenomenon than he ever was or could be _as_ an artist. That was what gave him his special aura, after all\u2014and his influence. _As_ an artist he ended his career exactly as he began it\u2014as a gifted _commercial_ artist with a flair for the arresting graphic image and skillful layout. He was never much of a painter, and as a sculptor he didn't exist. (As for his movies, it is probably enough to observe that, having served their purpose\u2014which was to propagate the Warhol myth\u2014they had long ago predeceased their creator. Of aesthetic merit they had none whatever.) Warhol's \"genius\" (if it can be called that) consisted of his shrewdness in parlaying this essentially commercial talent into a career in an art world that no longer had the moral stamina to resist it: a career that would have been unthinkable, for example, ten years earlier.\n\nTo an extent that was unrivaled among his contemporaries in the Pop Art movement, if only because they knew so much less than he did about the dynamics of the fashion world, Warhol understood that something momentous had happened to the art scene in the late fifties\u2014that the relation which had formerly obtained between art and fashion (and hence between art and publicity, and between art and money) had undergone a decisive change. Which is to say that he understood the success of Robert Rauschenberg and Jasper Johns better than they did. (Not that they were slow to catch on\u2014but that is another story.) Their success owed much to the international acclaim that Abstract Expressionism, having passed through its avant-garde phase, was receiving for the first time and to their putative role as both the heirs and the rebels of that movement. Warhol understood that something else was involved; that with the success of Rauschenberg and Johns, the age of the avant-garde had drawn to a close. Their success had been, initially, an art-world success, but it quickly developed into a much larger phenomenon. A _boundary_ separating art and fashion had been breached, and never again would the really big successes of the art world be confined to that world. Henceforth success in that art world would be played on a much larger stage.\n\nRauschenberg and Johns came out of the school of Marcel Duchamp. They stood in relation to Abstract Expressionism very much as Duchamp had stood to Cubism\u2014as plenipotentiaries to the world of publicity and chic, agents of \"advanced\" styles that offered just the right dash of scandal (the stuffed goat, the paint-splattered bed, the targets and flags, etc.) to excite a taste that had wearied of the rigors of abstract art. Warhol, on the other hand, belonged to the school of Cond\u00e9 Nast, where such \"scandals\" are recognized as marketable commodities and routinely processed as the materials of fashion. In moving from a career in fashion illustration to a career in this swiftly expanding art world, Warhol saw that the product needed only the most superficial modification to win attention and garner rewards\u2014rewards far greater than any to be found in the realm of fashion illustration. Art, no matter how debased, still offered a kind of status that was denied to advertising, but otherwise there were now fewer and fewer differences separating the art world from the advertising world. The ethos was getting to be essentially the same. Success was the goal, the media would provide the means of achieving it, and what the media loved more than anything else\u2014certainly more than art\u2014was a product based on their own stock-in-trade. And that, from his \"Marilyns\" to his \"Maos,\" is what Warhol mainly specialized in giving them. Considering the magnitude of his accomplishment in this respect, it would be churlish to deny that he did indeed have genius of a sort. No ordinary talent could have done it\u2014but the talent, alas, was not primarily an artistic talent.\n\nAs everyone knows, the art world never really recovered from this fateful incursion. As a movement Pop Art came and went in a flash, but it was the kind of flash that left everything changed. The art public was now a different public\u2014larger, to be sure, but less serious, less introspective, less willing or able to distinguish between achievement and its trashy simulacrum. Moreover, everything connected with the life of art\u2014everything, anyway, that might have been expected to offer some resistance to this wholesale vulgarization and demoralization\u2014was now cheapened and corrupted. The museums began their rapid descent into show biz and the retail trade. Their exhibitions were now mounted like Broadway shows, complete with set designers and lighting consultants, and their directors pressed into service as hucksters, promoting their wares in radio and television spots and selling their facilities for cocktail parties and other entertainments, while their so-called education programs likewise degenerated into sundry forms of entertainment and promotion. The critics were co-opted, the art magazines commercialized, and the academy, which had once taken a certain pride in remaining aloof from the blandishments of the cultural marketplace, now proved eager to join the crowd\u2014for there was no longer any standard in the name of which a sellout could be rejected. When the boundary separating art and fashion was breached, so was the dividing line between high art and popular culture, and upon all those institutions and professions which had been painstakingly created to preserve high art from the corruptions of popular culture. The effect was devastating. Some surrendered their standards with greater alacrity than others, but the drift was unmistakable and all in the same direction\u2014and the momentum has only accelerated with the passage of time.\n\nThere was no more telling symbol of this surrender of standards than Warhol himself, the cheerful nihilist who became the unlikely culture hero of a new era. Anyone who was around at the time will remember exactly when it was that we were given our first taste of its lethal character. It came, of course, in the early sixties with the false dawn and specious glamour of the Kennedy administration\u2014the media-fabricated glamour that in retrospect has turned out to be the perfect political analogue for what was occurring on the cultural scene. There is an important book to be written about the cultural fallout of the Kennedy years, and Warhol's would by no means be the only significant career that would require a place in the story. So, among many others, would Norman Mailer's and Arthur Schlesinger, Jr.'s and George Plimpton's, and so too, in their own spheres, would the history of _New York, The New Yorker_ , and the _New York Review of Books_ , not to mention less exalted branches of the media. Still, in any such study a special place would have to be reserved for the Warhol phenomenon, which gave to the art world\u2014but not to the art world alone\u2014a model that has proved to be so irresistible that it is now a permanent, and permanently disabling, component of cultural life. It was really that ghastly model, rather than the man or his art, that the media celebrated so copiously in their obituary notices of Andy Warhol\u2014a model that continues to exhibit a potency his art could never achieve.\n\n[1987]\n\n### How Good Was Gauguin?\n\n* * *\n\nI. _The Gauguin Myth in the Age of Media Hype_\n\nThe process by means of which certain artists\u2014certain modern artists particularly\u2014attain a mythical status that transcends all aesthetic considerations is one that is so familiar to us that we hardly any longer pay it much attention. It is now taken for granted as one of the facts of modern cultural life. This does not mean that the myths themselves have lost their power to influence our thinking about art, but precisely the contrary. Both the myths and the process of mythmaking are now so firmly established in our consciousness that they continue to exert an influence even when the effort is made to discount their effect and rid our minds of their disfiguring fictions.\n\nYet it is only on a certain kind of artist that such a mythic status can be conferred. Great achievement in art is not itself a sufficient condition; at times, indeed, it is not even a necessary condition. It is life, not art, that confers the status I speak of, and not just any life, of course, but one in which an extreme degree of nonconformity or suffering or violence or saintliness or promiscuity or sheer human oddity is seen to be inseparable from the artistic vocation itself. Matisse, who was probably the greatest painter of the twentieth century, was never a suitable candidate for the role\u2014too bourgeois in appearance, for one thing\u2014whereas it fit Picasso to such a degree that he came finally to see his own life entirely in its terms. C\u00e9zanne, with his banker father and uneventful love life, was never a candidate either, though he was a greater painter than either Van Gogh or Gauguin, who now stand for us as the very archetypes of the mythic artist.\n\nIt was the writers of popular fiction who transformed the lives of Van Gogh and Gauguin into these archetypes, thereby changing forever\u2014even for those who resist the fictionalization\u2014the way their art would enter our lives. As the protagonists first of widely read novels\u2014 _Lust for Life_ and _The Moon and Sixpence\u2014_ and then of the trashy movies based on them, these two painters acquired an aura that has lent itself to an ever increasing vulgarization with each passing decade. Moreover, it is this vulgarization and not their intrinsic artistic merits that has so greatly transformed the market value of their pictures. Now not even the art museums can any longer be counted upon to act as a bulwark against such a sweeping distortion of artistic values.\n\nIt is certainly owing to this mythic aura and the vulgarization it has brought in its wake that it has lately proved to be so easy to divide the short, tragic career of Van Gogh into a series of isolated agons, each of which could be packaged as a separate, highly promotable exhibition melodrama. And since it is now the tendency of the museums to market the materials of high art, wherever possible, as if they belonged to the realm of popular culture, we can probably expect to see even worse things in the future. Thanks, for example, to Mrs. Huffington's ghastly biography of Picasso, who can doubt that we shall soon see this artist accorded the ultimate degradation\u2014a television docudrama mini-series certain to be the purest and most lurid soap opera.\n\nBecause of all this, it was with a certain feeling of dread that some of us greeted the announcement of a major Gauguin exhibition\u2014a show destined to be mounted on the blockbuster scale\u2014and there was nothing about the initial promotional atmosphere surrounding the announcement to allay our worst expectations. At a posh press luncheon in New York months in advance of the actual opening of the exhibition in Washington, the featured speaker\u2014the one who caused, and was expected to cause, the biggest stir\u2014wasn't a scholar or connoisseur who could shed new light on an artist already so familiar to us, though such experts were briefly heard from, but a movie actor, Donald Sutherland, who had been invited (hired?) to explain to the company assembled at Le Cirque\u2014writers, publicists, museum officials, and AT&T executives (the sponsors)\u2014what it had felt like to _play_ Gauguin.\n\nIn the event, Mr. Sutherland conducted himself with exemplary dignity and tact, but the quality of the actor's presentation, however admirable, is not the point, of course. It was his presence on such a program that struck entirely the wrong note\u2014wrong because it was the responsibility of the art museums organizing the Gauguin retrospective not only to place their primary emphasis on the artist's aesthetic achievement but to make every effort to separate his name and his work from the kind of vulgarization that had already done so much to distort the nature of that achievement. By choosing instead to exploit the Gauguin myth at the very outset of their campaign to win maximum media attention for this exhibition, the museums made yet another contribution to undermining the standards that it is their primary function to uphold.\n\nWas it even necessary for the museums to make this obeisance to crass commercial interests? Certainly not as far as the public was concerned. It is my impression, anyway, that nothing in the world\u2014nothing short of a total news blackout, that is\u2014could nowadays have kept a major Gauguin retrospective from being one of the most popular exhibitions of our time. The likelihood is that it was not the public but the sponsor of the exhibition\u2014in this case, AT&T\u2014who needed, or at least wanted, this approach in order to be certain that _its_ costly role in the event would receive the requisite visibility. None of us is so naive as to believe that corporate sponsorship of art exhibitions\u2014or any other cultural programs, for that matter\u2014is ever entirely cost-free. But too little attention, I think, is given to what the actual cost might be both to the cultural institutions which receive the funding and to the public which is supposed to be the primary beneficiary of the sponsor's largess. In the case of public television, for example, we have seen the whole idea of a \"noncommercial\" medium give way to one in which it is now taken for granted that commercial messages\u2014and sometimes quite elaborate ones\u2014will now be the norm. We seem, alas, to be moving in that direction in the museum world, too. Hence the featured presence of Donald Sutherland at a press lunch for a Gauguin exhibition, for in the advertising business making use of show-business celebrities to plug a company name has long been a standard practice. For museums to lend themselves to this practice is not a happy augury.\n\nII. _Separating the Art from the Legend of the Artist_\n\nWhen we turn to the actual exhibition\u2014the largest, apparently, ever to be devoted to the artist's work\u2014we are reminded, alas, that it was Gauguin himself who was the first to set this process of mythification in motion, and this makes the task of separating the art from the legend of the artist at once more difficult and more imperative. As Fran\u00e7oise Cachin writes in her splendid essay for the catalogue of the exhibition: \"The hero of his own history, [Gauguin] also saw himself as the hero of painting of his age, which seemed to him decadent.\" It is important to keep this notion of the decadence of the age firmly in mind in thinking about Gauguin, for it is one of the keys to the artist's desperation as well as to his achievement.\n\nIt can be said, of course, that every great modern painter has seen himself as \"the hero of painting in his age, which [seems] to him decadent,\" and to the extent that this is true, Gauguin was certainly one of the principal figures who created this role for the modern artist. It is a role, moreover, that manifests itself _in the work_ as well as in the legend. For the pressure to make every painting\u2014every drawing and print and sculpture, too\u2014count as a radical revelation; the horror of backsliding into something merely familiar and already accomplished; the sense of engaging in a fierce competition not only with one's contemporaries but also with one's predecessors and one's posterity; above all, the determination to make of art something absolute and ideal, something from which all the chaff of common experience and conventional feeling has been once and for all time expunged, leaving only the purest aesthetic essence, or what Gauguin himself often called \"the abstract,\" to be experienced in the created object: in all this we are made to feel not only the harrowing and all but impossible nature of the task that the artist set for himself but the role that each fresh assault on the creative enterprise was obligated to play in bringing that task to a triumphant realization.\n\nNot every attempt at the task succeeded, to be sure\u2014not in Gauguin's case nor in anyone else's; but the unremitting pressure is there, all the same, even in the paintings that aren't exactly masterpieces, the paintings that remind us, over and over again, that with Gauguin the will to _be_ a certain kind of artist was at times even more important than either talent or vision in enabling him to accomplish what he did. In the work of almost no other modern painter of comparable stature, indeed, are we made to feel so emphatically or so often the presence of the artist's will in determining the momentum of his development, the audacity of his ideas, and the quality of even his greatest pictures. This, too, was a matter in which Gauguin served as a model for the future.\n\nIt is this sense we so often have in Gauguin of the creation of art as an act of will that leaves us wondering, even in the face of his greatest achievements, if there wasn't, after all, something rootless and self-determined and artificial in what this renegade from bourgeois life set out to accomplish. Of course it was in the nature of the Symbolist aesthetic that Gauguin so passionately espoused to embrace artifice and indirection as the instruments of revelation, and he was in this respect completely faithful to the spirit of Rimbaud and Mallarm\u00e9. Yet this impulse to accord to the artist's will a kind of sacred priority cost Gauguin something as a painter. It is only in an exhibition on this scale that we can see how much remained _un_ developed and _un_ resolved in his work\u2014that we can see the profusion of ideas that were never pursued and the array of possibilities never explored. There are paintings here\u2014and they tend to come earlier rather than later, from the _Quarry in the Vicinity of Pontoise_ (1882) to _Fatata te mou\u00e0_ (1892)\u2014out of which other painters, and I mean other great painters, might have made entire careers (and in some cases actually did). That fear of lapsing into the kind of decadence he most despised won for Gauguin his many artistic victories, but did he entirely escape the shadow of the decadence he so greatly dreaded? As a painter, I don't think he did. Which is why, I suppose, that in the end painting alone could not satisfy his aspiration. The last paintings of Gauguin are by no means his best or most original work: that is another lesson of this vast exhibition. In the end he had to turn to words\u2014and to the kind of printed images that are much akin to words\u2014to make his final avowals. The sheer exertion of will could no longer make the art of painting its most direct or satisfactory instrument.\n\nStill, Gauguin emerges from this exhibition as a greater painter than some of us, at least, had thought him to be in the past. He was not what I would call a natural painter. He never commanded the kind of fluency we find in Pissarro and Monet and Degas and Picasso and Matisse. He could take nothing for granted, and there was never anything relaxed or easy in his work\u2014not even in the Impressionist pictures, which sometimes have the character of a lesson to be mastered, a test to be met before he could be released to pursue a vision of his own. He is also an artist who tends to look better and stronger when seen, as he is in this retrospective, in isolation and on a great scale than he usually does in the company of his contemporaries. While his ideas were sometimes profound and even prophetic, his art\u2014dare one say it?\u2014was sometimes shallow, aesthetically shallow in a way that C\u00e9zanne's, for example, never was. All of which makes Gauguin an odd kind of classic\u2014an artist who can never quite be relied upon to be great, but who was great often enough to make his place among the masters a persuasive proposition if not always an absolute conviction.\n\nIs it finally possible to separate the art from the legend of the artist? I am not sure that it is. For Gauguin will always remain the painter laureate of all those members of the middle class, wherever they may be, who harbor the illusion that what is \"primitive\" or distant or alien\u2014whether it is found in faraway islands and what is now called the Third World or in the underclass of our own society or in our own most cherished fantasies\u2014is somehow more real and certainly less decadent than the common realities of bourgeois life. That this standard of the real is, in the end, itself an illusion does nothing to diminish its appeal, and while that appeal persists Gauguin will remain its poet.\n\n[1988]\n\n### John Szarkowski's \"History of Photographic Pictures\"\n\n* * *\n\n> _A picture has been said to be something between a thing and a thought.\u2014_ Samuel Palmer\n\nOf the many things to be noted about the book and the exhibition that John Szarkowski has produced for the Museum of Modern Art under the title _Photography Until Now_ , the first is this: the book is without doubt one of the best ever written on its subject, which is the history of photography, and the exhibition is one of the most beautiful that the museum has devoted to the medium. Neither is without significant flaws, to be sure, but given the scale of Mr. Szarkowski's project\u2014which encompasses the prehistory of photographic thought, going back to Leonardo, as well as a century and a half of actual photographic production\u2014such flaws as it has are not, for the most part, unduly disfiguring.\n\nAt least this is true as far as the historical material is concerned. When it comes to the contemporary pictures, however, the exhibition goes into something of a nosedive. While certain reasons for this conspicuous drop in aesthetic quality can be adduced from the way Mr. Szarkowski treats this period in his book\u2014he speaks, for example, of a sense of \"a diminished role for photography, even a kind of disenfranchisement\"\u2014what it comes down to, I think, is the same pattern of decline that has lately been discernible elsewhere in the visual arts.\n\nThat we are now in a fallow period in the visual arts, if not indeed a period of decadence, is not something that the curatorial mind can generally be expected to acknowledge. Yet Mr. Szarkowski comes remarkably close to doing just this when, in the concluding chapter of his book, he says of contemporary photographers that \"As with poets and composers, the relationship between their work and their lives has become casual and improvisatory\"; and further, that the time has now passed when photographers could really believe that \"they had a privileged access to truth and thus to power.\" Beside such doleful pronouncements, the claim that \"a score or more of photographers are now doing work of vitality and original beauty, under conditions that do not seem propitious\" sounds a little hollow, and the exhibition doesn't do much to support the claim, either.\n\nIt is, in any case, for its splendid account of what Mr. Szarkowski calls the \"history of photographic pictures,\" rather than for its inventory or assessment of current accomplishment, that _Photography Until Now_ commands our admiration. Yet it is important to make a distinction here between the book and the exhibition. About the book, Mr. Szarkowski writes that he has \"attempted . . . to sketch out a history of photographic pictures, organized according to patterns of technological change.\" This interpretation of technological change is broad enough to encompass, as he notes, \"not only . . . the chemical and optical issues often thought of as constituting photographic craft, but . . . methods of distribution of photographic imagery, economic constraints, and professional structures.\"\n\nWhat this means, in effect, is that Mr. Szarkowski has written a social history of photography, and\u2014in my view\u2014a very good one. The exhibition, on the other hand, although divided into sections that conform to the organization of this social history, is more the work of a connoisseur of photographic art than of a social historian. This accounts for the remarkably high level of quality that is sustained throughout most of the historical sections of the exhibition. Where this high level collapses into something else, we can usually feel the intervention of the social historian who cannot resist the temptation to document a development that, whatever its other claims to our attention, is devoid of aesthetic interest. It is only in the interest of social history, for instance, that items such as a population chart from a New York newspaper in the year 1900 or a layout devoted to Mussolini in a 1934 issue of the London _Weekly Illustrated_ have found their way into the company of so many photographic masterpieces.\n\nIt is, moreover, the connoisseur rather than the social historian who is responsible for the brilliant installation of the exhibition. Mr. Szarkowski has always had a keen sense of what might be called the dramaturgy of a large photographic exhibition, which in the wrong hands\u2014and with the wrong eyes\u2014can so rapidly degenerate into either an exercise in tedium or a riot of showy effects. In this exhibition he is very adroit at striking a balance between familiar and unfamiliar pictures in a way that illuminates both. He is very good at rhyming pictures that belong together and, where needed, coming up with striking juxtapositions that refresh the eye, foil expectation, and offer instruction as well as pleasure. Mercifully, there is no tendency here to treat the installation as, primarily, a problem in layout design, and thus to treat pictures as mere visual counters in some larger design strategy. In most cases the exact pictorial weight of a photograph is carefully calculated and given the space and the company that are appropriate to its character and its scale.\n\nAgain, this connoisseur's delicacy (if one may call it that) tends to break down in the contemporary section of the exhibition, but there it is often a case of trying to make bricks without the requisite straw. Pictures such as Robert Rauschenberg's, Andy Warhol's, and Cindy Sherman's are so fraudulent and crass in the use they make of the photographic medium that their sheer vulgarity is bound to capsize whatever company they are made to keep. The historical part of the show is not without its own episodes of misplaced emphasis\u2014Mr. Szarkowski seems, for example, to have acquired a sweet tooth for anonymous pictures of early industrial plants that cannot always be justified by the photographic results\u2014but by and large, it is his connoisseur's eye that remains in control. It was certainly the connoisseur's eye, by the way, that elected to omit Robert Mapplethorpe from this exhibition\u2014a decision that renders a more salutary and unanswerable judgment on the whole Mapplethorpe controversy than any other I know about.\n\nWhen we turn to the book Mr. Szarkowski has written to accompany this exhibition, we find it is the social historian who is mainly in command, with the connoisseur usually playing a more ancillary role. About the role of early photography in politics and in the lives of the lower classes, for example, Mr. Szarkowski has some interesting things to say. Writing about Mathew Brady's 1860 portrait of Abraham Lincoln, he tells us that Lincoln is thought to have considered the popularity of this picture, which was printed in huge numbers, one of the reasons for his election. \"The frontier lawyer,\" Mr. Szarkowski writes, \"who was said to look 'half-alligator and half-horse' looks in Brady's portrait rather more like Gregory Peck. It is not known how many of Brady's portraits of Lincoln were sold in carte-de-visite form, but the one made by Alexander Hesler the following June was said to have sold more than 100,000 copies.\"\n\nHe reminds us that the daguerreotype, on the other hand, \"found its special function in making pictures for the smallest and most private of audiences\u2014the families of uncelebrated men and women. Daguerreotypists made portraits of millions of people who were unknown beyond their own village, people whose forebears had never before been portrayed as individuals. . . . It is poetically just that the daguerreotype belongs to the century that saw in Western countries a radical expansion of suffrage. After Daguerre the yeoman, and then the peasant and the proletarian, would also have visible ancestors and family histories.\"\n\nIn Mr. Szarkowski's account of the nineteenth-century interest in photographing ruins, however, the social historian is joined by the connoisseur of pictorial aesthetics in defining a major genre of early photography. \"Of all the surprises that might persuade one to postpone one's journey,\" he writes, \"a ruin was the delight that came most naturally to Talbot's mind.\" William Henry Fox Talbot (1800\u20131877) was, of course, one of the inventors of photography. Mr. Szarkowski continues:\n\n> In this taste [Talbot] was altogether representative of the educated classes of his time and place. Ruins were revered both for their meaning and their aspect: both as documents to be read for what they told of the medieval or ancient past, and as embodiments of the picturesque, a word that resists precise definition, but that expressed a taste for the used above the new, the irregular over the geometric, the rough rather than the smooth. The ruin was a central icon to the Romantic imagination, and if, by 1850, the idea had been well worked over by painting and poetry, photography brought to it a new specificity and conviction. And the calotype, which itself tended toward the rough and irregular, and did not look too insistently new, was an excellent tool for the issue at hand.\n\nAbout many subjects, indeed, Mr. Szarkowski writes with a critical clarity that is admirably unclouded by nostalgia. His judgment of the way photography was used\u2014and misused\u2014in _Life_ magazine properly explodes one of the abiding myths of photographic history:\n\n> In 1972 _Life_ closed its doors; many of the other magazines that had supported photo-journalists had already done so, or would shortly. The reason usually given for their failure is the rise of television. . . . But one might say that the magazines had failed on creative grounds before television became a competitor, and that if they had succeeded on those grounds they might also have survived. It is difficult to identify a photo essay from the best days of the experiment in which consistently superior photographs and rigorous writing augment and transform each other, to achieve that new means of expression to which editors, photographers, writers, and art directors paid continual lip service.\n\nWhat is also a major part of this story, which Mr. Szarkowski does not go into, is the politics of photojournalism in its heyday. The most celebrated photojournalists of the thirties and forties belonged to the left, whereas the proprietors of the mass-circulation magazines they worked for were generally on the right; and this meant that an ideological civil war was endemic to the entire enterprise. I frankly doubt that the history of photojournalism can be entirely understood in isolation from this political conflict.\n\nThere is much more to _Photography Until Now_ than can be described here, of course. And, it is also important to take note, it is a major event in John Szarkowski's own career. Mr. Szarkowski has been the director of the Department of Photography at the Museum of Modern Art since 1962, and in _Photography Until Now\u2014_ both the book and the exhibition\u2014he has given us both a summary of his thought in this field and a quite specific sense of what he believes the canon (dare one use the word?) of great photography to consist of. There is no one better equipped to essay these tasks than he, and his achievement in _Photography Until Now_ will itself stand as a landmark in the intellectual history of photography.\n\nThat it is an achievement attended by a certain air of melancholy is itself a reflection of the situation in which photography now finds itself. With his customary candor Mr. Szarkowski takes note of the degree to which photography has lately taken refuge in the academy and in the art world in the interest of survival, and while he doesn't offer any assessment of what losses this shift has entailed for photography\u2014losses in quality and in expressive power as well as fundamental purpose\u2014he shows himself to be keenly aware of them. It was certainly bad luck for photography that it attached itself to the academy at the very moment when the academy was itself suffering a calamitous crisis of confidence in its own purposes, and it was a similar misfortune for photography to become so dependent on the art world at a moment when its standards, too, were in a tailspin. But perhaps this is just another way of saying that photography is now, for better or for worse\u2014usually worse, I'm afraid\u2014so much a part of the crisis that the institutions of our cultural life are passing through that it has virtually ceased to have an identity or a destiny of its own. If this is indeed the case, then _Photography Until Now_ may serve to remind us of what this field of achievement consisted of in the halcyon years of its development.\n\n[1990]\n\n### Reflections on Matisse\n\n* * *\n\n> _Sometimes it has been conceded that I have a certain technical ability but that all the same my ambition is limited, and does not go beyond the purely visual satisfaction such as can be obtained from looking at a picture. But the thought of a painter must not be considered as separate from his pictorial means, for the thought is worth no more than its expression by the means, which must be more complete (and by complete I do not mean complicated) the deeper is his thought. I am unable to distinguish between the feeling I have about life and my way of translating it_.\u2014Henri Matisse, in \"Notes of a Painter\" (1908)\n\nMatisse first emerged as a leader of the School of Paris in the fall of 1905. That was the year the critic Louis Vauxcelles, who admired the thirty-six-year-old painter even if he did not understand him, coined the term _fauves_ \u2014wild beasts\u2014to describe the work of Matisse and his followers in the Salon d'automne. However ironical or absurd this coinage may now seem to us as a description of Matisse's pictorial aesthetic, at the time it had the effect of conferring on his work the status of an avant-garde provocation.\n\nThe evidence suggests that this had never been Matisse's intention. The role of _peintre maudit_ was not something he coveted. He had submitted his art to the lessons of the masters. He was forging a style based on the discoveries of Pissarro, Seurat, and C\u00e9zanne. Nothing about his experience or his temperament placed him in the line of those _r\u00e9volt\u00e9_ personalities who specialized in uproar and anarchy. Yet in the fast-paced world of the Paris avant-garde, where radical innovation was a cynosure of distinction and reputations were made overnight by coterie opinion while the public remained largely oblivious to what was really happening in art, Matisse was thrust into a position wholly alien to his sensibility. For a couple of years thereafter, until Picasso, twelve years his junior, emerged with _Les Desmoiselles d'Avignon_ in 1907 to challenge his rival's leadership of the avant-garde, every negative opinion, whether academic or \"advanced,\" only added to Matisse's preeminence.\n\nAfter the impact of _Les Desmoiselles_ , however, Matisse was never again mistaken for an avant-garde incendiary. With the bizarre painting that appalled and electrified the cognoscenti, which understood that _Les Desmoiselles_ was at once a response to Matisse's _Le Bonheur de vivre_ (1905\u20131906) and an assault upon the tradition from which it derived, Picasso effectively appropriated the role of avant-garde \"wild beast\"\u2014a role that, as far as public opinion was concerned, he was never to relinquish.\n\n_Le Bonheur de vivre_ , owing to its long sequestration in the collection of the Barnes Foundation, which never permitted its reproduction in color, is the least familiar of modern masterpieces. Yet this painting was Matisse's own response to the hostility his work had met with in the Salon d'automne of 1905, a response that entrenched his art even more deeply in the aesthetic principles that had governed the Fauvist paintings which had caused such a furor and which did so on a far grander scale, too. Had it been better known, its role both in Matisse's development and in Picasso's, too, would have been more readily understood, and it is a great pity that it could not be secured for the marvelous Matisse retrospective that John Elderfield has now organized at the Museum of Modern Art. For it is a painting that stands at the crossroads of modern art. No sooner was this masterwork completed, elevating Matisse's art to a new level of achievement, than it was eclipsed\u2014in avant-garde opinion, anyway\u2014by Picasso's bombshell, and this, in turn, did much to determine the way modern art was comprehended from that day until ours.\n\nWhereas Matisse had drawn upon a long tradition of European painting\u2014from Giorgione, Poussin, and Watteau to Ingres, C\u00e9zanne, and Gauguin\u2014to create a modern version of a pastoral paradise in _Le Bonheur de vivre_ , Picasso had turned to the alien traditions of primitive art to create in _Les Desmoiselles_ a netherworld of strange gods and violent emotion. As between the mythological nymphs of _Le Bonheur de vivre_ and the grotesque effigies of _Les Desmoiselles_ , there was no question as to which was the more shocking or more intended to be shocking. Picasso had unleashed a vein of feeling that was to have immense consequences for the art and culture of the modern era while Matisse's ambition came to seem, as he said in his \"Notes of a Painter,\" more \"limited\"\u2014limited, that is, to the realm of aesthetic pleasure. There was thus opened up, in the very first decade of the century and in the work of its two greatest artists, the chasm that has continued to divide the art of the modern era down to our own time.\n\nThere can be little doubt, I think, that what might be called Matisse's \"tradition,\" by which I mean not only Matisse's own accomplishment but his influence on other artists and the whole spirit of his aesthetic, has been less widely understood than that of Picasso, whose art and life have dominated so much of our thinking about art in this century. The fact that we have had to wait twelve years, since the great Picasso retrospective that was mounted at MOMA in 1980, for a comparable retrospective devoted to Matisse is only another reminder of this basic difference in our apprehension and valuation of their respective accomplishments.\n\nMatisse seems to have understood the implications of the chasm that separated him from Picasso straightaway, and in the immediate aftermath of _Les Desmoiselles_ , as the juggernaut of Cubism was launched upon the Paris art scene, he wrote his great defense of his artistic philosophy in the \"Notes of a Painter,\" which was first published in _La Grande Revue_ on December 25, 1908. Owing to a single, long sentence in this manifesto, however, Matisse seemed to incriminate himself as some kind of escapist, at least in the minds of the more _r\u00e9volt\u00e9_ elements of the avant-garde, and as a result this beautiful avowal of his faith in pure painting made him the target of further hostility for many years to come. Certainly nothing could have made his distance from the spirit of _Les Desmoiselles_ more vivid than the famous passage in which Matisse wrote:\n\n> What I dream of is an art of balance, of purity and serenity, devoid of troubling or depressing subject matter, an art which could be for every mental worker [ _travailleur c\u00e9r\u00e9bral_ ], for the businessman [ _l'homme d'affaires_ ] as well as the man of letters [ _l'artiste des lettres_ ], for example, a soothing, calming influence on the mind, something like a good armchair which provides relaxation from physical fatigue.\n\nYet what immediately precedes this passage in the \"Notes of a Painter\" is an observation about Giotto that is as essential to Matisse's aesthetic credo as his talk of balance, purity, and serenity.\n\n> A work of art must carry within itself [Matisse wrote] its complete significance and impose that upon the beholder before he recognizes the subject matter. When I see the Giotto frescoes at Padua I do not trouble myself to recognize which scene of the life of Christ I have before me, but I immediately understand the sentiment which emerges from it, for it is in the lines, the composition, the color. The title will only serve to confirm my impression.\n\nWhich is why, of course, so large a part of \"Notes of a Painter\" is devoted to lines, composition, and especially color\u2014to the \"means\" by which an artist expresses his \"thought\"\u2014not only in Matisse's own work but in that of a great many other painters as well.\n\nIn his reaffirmation of this philosophy of art, Matisse made a definitive statement about his own artistic goals\u2014goals that it must have seemed to him were imperiled by _Les Desmoiselles_ and the response it instantly ignited. But he also enunciated one of the categorical statements of faith in modernist aesthetics\u2014that belief in the power of \"the purely visual satisfaction\" of a painting to convey its meaning and spiritual value. This is the belief, whether consciously acknowledged or not, that draws us to the great works of the past even when the stories they retell are no longer of compelling interest to us, and it is what still draws us to many modern works\u2014even, I daresay, _Les Desmoiselles d'Avignon_ \u2014long after their subjects have lost their power to excite our attention.\n\nMatisse never abandoned this faith in what he called the painter's \"thought.\" He restated it nearly thirty years later in the remarks he published in _The Studio_ magazine under the title \"On Modernism and Tradition\" (1935). \"A great modern attainment is to have found the secret of expression by color, to which has been added, with what is called Fauvism and the movements which have followed it, expression by design; contour, lines and their direction,\" he wrote, adding that \"if I am not mistaken, only plastic form has a true value, and I have always believed that a large part of the beauty of a picture arises from the struggle which an artist wages with his limited medium.\"\n\nAmong \"the movements\" that Matisse had in mind in this statement, Cubism was undoubtedly the most important, and despite his initial recoil from _Les Desmoiselles_ , he monitored the progress of Cubism with close attention to its great contribution to the \"plastic form\" he valued above all else in painting. He made ample use of it, too, which the current retrospective documents in greater detail than any other Matisse exhibition we have seen. Matisse was never a dogmatist in aesthetic matters. He was open to every possibility except the abandonment of the painter's \"thought\" to something outside of the \"means\" used to express it.\n\nIt is one of the many strengths of the retrospective that John Elderfield has organized at MOMA that it has been conceived and executed with a very clear and subtle understanding of the arguments and counterarguments that have dogged Matisse's reputation since that fateful day in 1907 or thereabouts when he was effectively supplanted by Picasso as the leader of the Paris avant-garde. These, as Mr. Elderfield knows very well, are arguments and counterarguments about the nature of art itself, and not only about Matisse's and Picasso's. Yet the fate of Matisse's reputation, and thus of our understanding of what Matisse achieved as an artist, remains ineluctably joined to the ways in which Picasso has come to define the idea of modern art in this century, and it is Mr. Elderfield's distinction in this exhibition and in the great catalogue he has written to accompany it to have explored this relationship more thoughtfully than any of his many predecessors. The result of this critical endeavor is the best account of Matisse's art and thought that any writer has given us.1\n\nThat the \"Evaluation of Matisse's importance is still tied to evaluation of Picasso's\" is indeed the _donn\u00e9e_ of Mr. Elderfield's approach to his subject, and in the series of oppositions he cites\u2014between the hedonist and the radical, between the artist \"on holiday\" and the artist \"at war,\" between the champion of \"harmony\" and the exponent of \"dissonance,\" etc., which in the end come down to an opposition between the artist as \"French bourgeois\" and the artist as \"international bohemian\"\u2014he places Matisse's art in the historical framework that we can all instantly recognize as having dominated all discussion of these artists for as long as we can remember. No other writer has explored the implications of these facile oppositions so tellingly or related them so directly to the fact that, as Mr. Elderfield writes, \"the very idea we have of modernism contains, if not indeed comprises, the idea of a protracted struggle for dominance between opposing systems, represented by Matisse and Picasso, each claiming for itself proprietary rights on an essential modernism.\"\n\nMr. Elderfield also gives us a very exact account of the reason why Matisse later emerged as the winner, so to speak, in the contest that he seemed for so long to have forfeited.\n\n> Matisse's reputation as a radical artist was not to achieve its full height . . . until after his death. It came as the result not only of broadening critical acknowledgment of his work . . . it also, and more importantly, was the result of Matisse's influence on new art. His paintings of the 1930s and 1940s had been perceived as maintaining the vitality and innovative possibilities of that art in a period of great stylistic confusion, when the tradition of Picasso's Cubism decayed into mannerism and mediocrity. His example had certainly been critical to the development of American Abstract Expressionist painting in that way. But it was not until the 1960s, when so-called Color Field painters, Minimalists, and Pop artists could all, in their different ways, find inspiration in Matisse's work, that he fully recaptured the interest of the avant-garde, was acknowledged as a true radical, and again began to be described as the most important twentieth-century painter.\n\nTo which Mr. Elderfield promptly adds the questions that continue to haunt both the debate about Matisse and Picasso and the larger debate about art itself:\n\n> But still, was not Picasso, perhaps, a more \"serious\" artist? And is he not a more influential one? For some, the old questions remain, especially now that the magnificent _Alterstil_ of modernism itself\u2014the American painting of the 1950s and 1960s that came so to value the purely visual, and therefore Matisse\u2014is past.\n\nThis retrospective is undoubtedly the finest exhibition of its kind ever mounted, and insofar as it is possible to do justice to an artist of Matisse's stature in a single exhibition, this one certainly succeeds beyond cavil. And quite apart from what it does for Matisse, it also has the effect of making one feel a lot better about the century in which we live\u2014a terrible century in so many ways, yet one in which we can nonetheless feel an immense sense of pride if, beside its unremitting record of suffering, bloodshed, and tragedy, it can also boast of an achievement as sublime as Matisse's. Yet there is something about this wonderful event that induces a feeling of sadness and loss as well\u2014a feeling that one of the great possibilities of the human mind may now have spent itself. When we exit this exhibition and return to the sordid cultural landscape of this last decade of the century, it is hard to believe that we shall ever again witness anything like it, now or in the foreseeable future.\n\n[1992]\n\nFootnote\n\n1. John Elderfield, _Henri Matisse: A Retrospective_ , with contributions by Beatrice Kernan and Judith Cousins (New York, 1992).\n\n### Philip Johnson's Brilliant Career\n\n* * *\n\n> His wit all see-saw, between that and this, \n> Now high, now low, now master up, now miss, \n> And he himself one vile antithesis.\n> \n> \u2014Alexander Pope, \"Epistle to Dr. Arbuthnot\"\n\nThere are certain figures in the arts who, although minor in accomplishment and equivocal in their aesthetic influence, are so completely representative of the spirit of their age that they come to occupy a historical position far greater than the intrinsic merits of their work could ever justify. Their function in cultural life is not so much to create as to incite and impersonate.\n\nSuch figures tend therefore to commence their careers as epigones of talents more radical than their own. Often they are drawn to extreme positions that promise deliverance from the conventionality of their origins, but they are not equipped by sensibility or conviction to distinguish between the appearance and the reality of the idols and ideals they choose to emulate. When they feel called upon at opportune moments to attempt originality or innovation\u2014as they inevitably do\u2014they lapse into parody and pastiche, sometimes deliberately, sometimes not.\n\nThe m\u00e9tier of such figures is not, in any case, either ideas or artistic creation. It is publicity, showmanship, and the exercise of power. Once they have captured the limelight, they become addicted to it, for the withdrawal of attention is tantamount to oblivion. Their careers thus become a succession of feints and assaults designed to command the public stage and add luster to their visibility\u2014even at the price of obloquy and scandal. In our time this usually means pleasing important people while pretending to offend established taste.\n\nThe American architect Philip Johnson is a particularly vexing example of this type. For more than sixty years\u2014he was born in 1906\u2014he has been a formidable presence on the American cultural scene. No other figure of his time\u2014no other architect, certainly\u2014has so successfully combined so many roles in the arts and their institutions. In addition to his work as an architect and architectural publicist, Johnson has been an eminent museum curator (at the Museum of Modern Art), an influential university professor (at Yale), a leading collector and promoter of contemporary art, and a tastemaker and reputation broker on a huge scale. He has also been an immensely successful businessman, and equally successful, too, as an adviser and mentor to people who are even richer and more powerful than he.\n\nYet Johnson's career, though undeniably brilliant, encompasses no single accomplishment that can be cited as a quintessential example of his ideas or his vision. In lieu of a personal aesthetic or a recognizable style or a definitive philosophy\u2014or even, alas, a coherent development\u2014what characterizes his work is a series of brilliantly performed charades in which other people's ideas, other people's tastes, and other people's styles have been appropriated, exploited, deconstructed, and repackaged to advance the prosperity of his own reputation and influence.\n\nOne consequence of that influence and the enormous power of patronage and preferment that it commands has been a notable reluctance among architectural critics and others to undertake the kind of frank and comprehensive assessment that is normally devoted to the work of famous figures on the cultural scene. Compared, for example, with the attacks that have been mounted in recent years on the accomplishments of Alfred H. Barr, Jr., the founding director of the Museum of Modern Art who launched Johnson's career as an architectural expert and art collector, or on the work of Mies van der Rohe, whose ideas exerted the single greatest influence on Johnson's early architectural thought, Johnson himself has remained remarkably exempt from searching criticism. It is indeed a measure of Johnson's power that despite the many really bad buildings he has created over the last thirty years or more, and the many ideological hostages to fortune he has given his enemies over the course of his career, he remains in nearly total control of his own reputation. Although he is widely recognized as the bad boy of contemporary architecture\u2014he once proudly proclaimed himself to be an architectural \"whore\"\u2014an aura of artistic distinction and superior taste has nonetheless insulated Johnson from serious attack.\n\nHow is one to explain this odd situation? It is certainly not a question of his being a beloved figure; he is not. No doubt, fear of reprisal accounts for a good deal of the reluctance to challenge the basis of Johnson's artistic celebrity, but there are other reasons, too. One of them is that he has played an important role in launching the careers of many of those who would now be in a position to write the requisite critiques of his work, and who may well be hesitant to seem lacking in gratitude. Another is that Johnson has proved to be very adept at anticipating such critiques in his own frequent utterances about his work, making effective use of a combination of campy self-mockery and swaggering self-aggrandizement as a means of trivializing the case against him before others have had the opportunity to state it more cogently.\n\nBut a more fundamental reason than any of these has been the sweeping destruction of architectural standards which Johnson himself\u2014both as an architect and as an architectural thinker\u2014has made fashionable both within the profession and in the public perception of current architectural practice. This destruction of standards has advanced for a couple of decades now under the rubric of \"postmodernism,\" which is less a movement than a collective agreement to reduce architecture to the level of kitsch and elevate kitsch to the status of art.\n\nJohnson did not himself create the postmodernist phenomenon\u2014creation, even of this sort, has never been his forte\u2014but he was quick to identify himself with its fortunes and soon became its most prestigious representative. What made the move especially piquant for a sensibility as perverse as Johnson's was the contradiction it embraced. For in the first phase\u2014the Miesian phase\u2014of his architectural career, Johnson was an ardent champion of the very modernism that the postmodernists were busily maligning. But it is entirely characteristic of Johnson that the second half of his career should have been devoted to destroying the standards that served as a touchstone of quality in the first. Infidelity has long been a distinguishing feature of Johnson's character, and in this respect the promiscuous shifts of attachment and betrayal in his artistic tastes are an accurate reflection of the way he has lived his life in both the private and public realms.\n\nOwing to the checkered character of that life, it was not to be expected that Johnson would ever authorize a frank, full-scale biography. Indeed, the original agreement that obtained between him and Franz Schulze, the author of _Philip Johnson: Life and Work_ , stipulated that the book would not be published in its subject's lifetime. It was on this assumption that Johnson enthusiastically cooperated in the preparation of the book and encouraged intimate friends and professional associates to do likewise. Yet as the book approached completion and Johnson, despite the frailties of age, continued to command the limelight, he changed his mind about delaying publication. His appetite for publicity undiminished, Johnson gave Schulze the go-ahead, thereby assuring himself of what was likely to be his last sensational fling in the public eye.\n\nAlways a shrewd connoisseur of the _Zeitgeist_ , Johnson understood very well that in the debased cultural climate of the 1990s, even the most scandalous revelations in Schulze's biography would not only have lost their power to cause him public embarrassment but would, on the contrary, contribute to the glamour of an invincibly chic reputation. And in this hardheaded assessment Johnson has been proved largely correct. For what emerges from Schulze's biography is, among much else, a portrait of the architect as an immoralist\u2014precisely the kind of immoralist in whose mind a deep-rooted aestheticism combines with a remorseless nihilism to render everything but personal gratification and public approbation utterly meaningless. And that, rather than the admired or despised architect, is the figure who is now being celebrated as he approaches his ninetieth birthday.\n\nPhilip Johnson was born to wealth, privilege, and propriety in Cleveland, Ohio, the son of a successful, Harvard-educated lawyer and a reclusive, Wellesley-educated mother whose primary interests, according to Schulze, were \"good manners and lofty ideas appropriate to her concept of [the Johnson family's] station and mission in life\"\u2014\"a mother of majesty rather than intimacy.\" After the death of his older brother when Philip was two years old, he was subjected to what Schulze describes as \"an uncommonly protected upbringing.\" Neither of his parents was young. When they married in 1901, his father was thirty-nine and already twice widowed; his mother, then thirty-two, had been, in keeping with the conventions of the day, well on her way to spinsterhood. Philip and his sister Theodate, the only member of the family to whom he was ever close, later believed their mother had had a lesbian attachment before her marriage\u2014but this was pure speculation and may have been, on Philip's part at least, a self-justifying belief.\n\nHomer Johnson was remarkably generous to all of his children, and in the case of his single surviving son he lavished so much wealth upon him that by the time Philip graduated from Harvard he was richer than his father. Still, the father seems never to have known what to make of his brilliant, troubled son and was fairly obtuse about what he could not avoid knowing. When, for example, he discovered that Philip, while at Harvard, had consulted a Boston neurologist about his homosexuality, he simply declared that \"Boys don't fall in love with boys\" and advised his son to forget about it. Many years after Homer Johnson died, in his ninety-eighth year, the son's judgment on him was characteristically cold and dismissive: \"He wasn't any use in the world.\"\n\nHis mother, however, was more interested in things of the mind, and although no more adept than her husband at understanding their son, she exerted a far greater influence on his development. \"Since she understood more of the mind than of the heart,\" Schulze writes, \"she looked upon Philip's special place in the family as an excuse, indeed an inspiration, to design his intellect first and worry over his psyche later, if at all.\" It was a recipe for disaster, and the disasters promptly manifested themselves in the form of violent tantrums, abysmal loneliness, and mental breakdown.\n\nYet however much he may have disliked his mother, she remained Johnson's principal intellectual confidante until well after his student years. \"Their mutuality endured,\" Schulze writes, \"and she became his favorite correspondent throughout his school years away from home and his early professional life.\" At Harvard one of his major concentrations was in Greek, which his mother had taught him as a boy, and her notion of the Johnson family's \"station\" in life, which placed them well beyond the mundane standards of ordinary folk, was one that the adult Johnson adopted as a personal credo.\n\nThis sense of a special station was greatly abetted by Johnson's enthusiastic discovery of Nietzsche while he was studying philosophy\u2014his other academic interest\u2014at Harvard. According to Schulze, it was Nietzsche who turned Johnson in the direction of art and politics, and whose ideas can be seen in retrospect to have shaped the immoralist impulses that governed the rest of his life and work.\n\nDuring most of his undergraduate career at Harvard (protracted to seven years because of episodes of mental breakdown), Johnson seems not to have been much interested in either art or architecture, and indeed he had no very specific notion of what he wanted to do. Except for study at Harvard and travel abroad, his life remained closely tethered to that of his family. (On one of his trips he took to wearing Arab dress and enjoyed what Schulze describes as \"his first fullfledged, 'consummated' sexual experience\" with a guard in the Cairo Museum.) The fortune that Johnson's father had already settled upon him did not make the choice of a vocation a pressing one, and his father's predictable suggestion that he study law was easily rejected. So was the offer to teach Greek at Oberlin, where his father was a trustee.\n\nWhat chanced to give Johnson his first sense of a commanding purpose to which he could harness his considerable intellectual energies was a meeting in 1929 with the twenty-seven-year-old Alfred Barr, who later that year would be named director of the Museum of Modern Art in New York. Barr had given a series of five widely noticed lectures at Wellesley on the entire history of modernism, from Post-Impressionism to the Bauhaus and Russian Constructivism, and he was on the lookout for recruits to the cause. Johnson had only lately acquired a curiosity about modernist buildings in Europe, but at their first encounter at Wellesley, where Johnson's sister was a student, Barr appears to have converted him on the spot. Schulze helpfully illuminates the psychological process at work. Johnson, he writes,\n\n> had a habit of seeking out authority figures . . . who, as if reenacting [his mother's] role in his younger life, could take command of his emotional loyalties at the same time they nurtured his intellectual ambitions\u2014could dominate him, that is, as they aggrandized him.\n\nAs discussions for the new museum in New York were already in progress, Barr apparently hinted that there might be a place for Johnson on its staff, and gave his eager recruit detailed instructions on what he should see on the European trip he was planning for the summer of 1929. Thus began Johnson's initiation into the modernist movement that was to dominate the whole first phase of his professional career.\n\nWhile Johnson was abroad that summer, the announcement of Barr's appointment was duly made. Johnson could not immediately join the museum staff, however, because he had not yet completed the course requirements for his Harvard degree. This he did in the spring of 1930, meanwhile commuting to New York where he quickly became a member of Barr's inner circle. By that time he had also teamed up with the architectural historian Henry-Russell Hitchcock, Jr., and together they planned a book that in 1932 was published under the title _The International Style: Architecture Since 1922_. This, as Schulze correctly observes, \"went on to exert a tremendous impact on architecture worldwide.\"\n\nJohnson was distinctly the junior partner in this collaboration, but it nonetheless established him as an authority on a subject that, three years earlier, he had barely been acquainted with. Even more important to his new career was the show \"Modern Architecture: International Exhibition,\" which he organized at the Museum of Modern Art in 1932. This was a major event in the campaign to establish modernist architecture on the American scene, and it launched Johnson as a tastemaker in a field that was still highly controversial and in which he was himself still something of an amateur, neither a trained scholar nor a professional practitioner. He proved, however, to be a great showman, and this was an immense advantage in the new museology which Barr introduced to the public with the Museum of Modern Art\u2014a museology in which installation, presentation, and propaganda were to be as important in shaping public response as the objects on display.\n\nJohnson was clearly one of the rising stars in the constellation of talents that Barr was counting on to run the museum in the first decade of its activities. In Barr and his wife Marga, moreover, Johnson found a friendship unlike any he had ever known. They were more than mentors to him; they were \"family\"\u2014the kind of cosmopolitan, uncensorious family Johnson preferred to his own. In Marga Barr, especially, he found a worldly confidante with whom he could frankly discuss anything, including the vagaries of his now very active sex life. Although more or less contemporaries\u2014Barr was one year older than Johnson\u2014Johnson became, in effect, the Barrs' adopted \"son.\" They traveled together, they rented Manhattan apartments in the same building, and they were closely bound to the same mission\u2014the museum and the modernism it espoused. Or so it seemed, anyway, to the Barrs and their circle.\n\nYet the museum, as it turned out, did not really satisfy Johnson's in-choate and unappeased Nietzschean ambitions for himself and the world. To fulfill those ambitions, art was deemed to be insufficient. Only politics would do\u2014and politics, as it happened, of a particularly loathsome character.\n\nThe turning point came in 1933 with Adolf Hitler's assumption of power in Germany. Johnson adored Germany. He was fluent in the German language and well versed in the avant-garde cultural life that marked the last years of the Weimar Republic. He had taken full advantage, too, of the sexual opportunities it offered to a man of his tastes. That Hitler was the sworn enemy of everything Johnson seemed most to admire in Weimar Germany hardly seemed to matter. There is no other way to put it: Johnson fell in love with the Nazi regime.\n\nThe Barrs, who were in Germany when Hitler came to power, were horrified. When they met with Johnson in Europe shortly thereafter, their disagreement was total and intense.\n\n> Alfred deplored the takeover [Schulze writes]; Philip was exhilarated by it. Alfred foresaw a brutal repression of freedoms in all walks of German life, leading to an atrophy of national culture as a whole. Philip, remembering the Potsdam rally at which he found himself transfixed by the Nazi spectacle and transported by the charisma of Hitler, saw a _\"nationale Erhebung\"_ (national resurgence), an amazing restoration of confidence among the German people, who only shortly before had seemed defeated by the Depression.\n\nFor Johnson, as Schulze shrewdly observes, \"Hitler and the Nazis had color. . . . There was dash to these Nazis: the way they dressed and sang and marched and fought; their impudence, their bravado, the sexuality Philip could not help but project on them.\" And the Nazis also met the requisite Nietzschean standard. \"He concentrated his attention on the Nietzschean text in _Der Wille zur Macht_ and its thesis that 'the will to power' constitutes man's fundamental motivating force\"\u2014a doctrine, as Schulze also points out, that \"must have appealed to the elitist view in which Philip had been nurtured since birth.\"\n\nIn the face of these attractions, which were as much aesthetic and erotic as they were political, the arts were no longer a priority for Johnson. \"If in the arts [Germany] sets the clock back now, it will run all the faster in the future,\" he wrote at the time. Even the Bauhaus, which Johnson had acclaimed four years earlier as a new architectural ideal, he now condemned for bearing \"irretrievably the stamp of Communism and Marxism.\" As for Hitler's racial doctrines, Johnson never allowed them to interfere with his sex life. In that respect, at least, he remained a follower of the Weimar ethos rather than the Nazi. Thus, writes Schulze, \"he took his first serious lover [in New York] in 1934\"\u2014at the very moment when he was committing himself to the fascist cause, to which racial theories were central. \"Jimmy Daniels was a black caf\u00e9 singer whom Philip later called the first Mrs. Johnson.\"\n\nStill, as a consequence of Johnson's political conversion, when he returned to New York from Germany he resigned his post at the Museum of Modern Art to devote himself to politics. Then he did something really bizarre for a white homosexual with a black male lover. With a school friend who was working at the museum, Johnson attempted to join up with the Southern political machine led by the demagogic governor of Louisiana, Huey Long.\n\nIt was a move widely reported in the New York press at the time. The headline of a story written by Joseph Alsop in the _Herald Tribune_ read: \"Two Quit Modern Art Museum for Sur-Realist Political Venture.\" The _New York Times_ reported of Johnson and his colleague that \"Recently they became convinced that, after all, abstract art left some major political and economic problems unsolved. Consequently both have turned in their resignations [from the Museum of Modern Art] and will leave as soon as practicable for Louisiana to study the methods of Huey Long.\" Once again Alfred Barr was horrified, tried to change Johnson's mind, and failed.\n\nNeedless to say, Huey Long was not interested in Johnson's services. But Johnson's next attempt to attach himself to a homegrown version of a Nazi-type movement met with a more enthusiastic response. He discovered the Reverend Charles E. Coughlin, the Roman Catholic priest whose Sunday afternoon radio broadcasts from the Shrine of the Little Flower in Royal Oak, Michigan, commanded an audience of thirty million listeners. By February 1936, Johnson was in direct contact with Coughlin, and though he found the man himself a \"crashing bore,\" he nonetheless enlisted in his political cause, which at that moment aimed to challenge President Franklin D. Roosevelt in the 1936 election. Johnson went to work for Coughlin's weekly paper, _Social Justice_ , and was much involved in the single biggest rally of the priest's political campaign. Schulze writes:\n\n> At Riverview Park [in Chicago], an enormous amusement complex on the city's North Side, a throng estimated by the _Chicago Tribune_ at between 80,000 and 100,000 heard Coughlin speak from a huge platform that Philip had taken special pains to design. It was modeled after the one he had seen used so effectively at the 1932 Nazi rally: \"A special stand, bordering on the moderne,\" the _Tribune_ reported, \"had been created at one end of the field. It provided a glaring white background 50-feet wide and 20-feet high for the solitary figure of the priest.\"\n\nEven though the campaign of Coughlin's Union party flopped, Johnson labored on, attempting to stimulate some kind of political youth movement along the same lines. That, too, flopped.\n\nIt was in the aftermath of the Coughlin debacle that Johnson returned to Germany. The highlight of his 1938 journey was the Nazi _Parteitag_ in Nuremberg, marking the fifth year of Hitler's ascension to power. Johnson characteristically described the Nuremberg rally as \"even more staggering\" than Wagner's _Ring_. He remained unbothered, moreover, by the fact that Hitler had officially declared war on _Entartete Kunst\u2014_ \"degenerate art\" of precisely the kind he had so recently championed as a member of Alfred Barr's inner circle at the Museum of Modern Art. As Schulze observes, \"the romance of the thing overpowered him.\"\n\nWhen he returned to New York in the winter of 1938\u20131939, Johnson tried to buy the _American Mercury_ , the monthly magazine founded by H. L. Mencken, complaining that \"the Jews\" had ruined it, and when that, too, fell through, he traveled once again to Europe\u2014this time venturing as far as Poland. \"The Polish tour,\" writes Schulze, \"only reinforced Philip's preconceptions of the backwardness among the Poles and the Jews, while reminding him of the superiority of German society and the German military force.\" When the war started with Hitler's invasion of Poland, he therefore headed straight for Berlin in order \"to prepare for the most exciting episode of his summer: the German Propaganda Ministry had formally invited him to follow the Wehrmacht to the front.\"\n\nHe owed this opportunity to Father Coughlin's _Social Justice_ , for which he wrote five articles in the summer and fall of 1939, condemning England, praising the Germans, and dismissing the United States as \"the best misinformed nation in the world.\" From Johnson's FBI file, which the Bureau began assembling during the war, Schulze has retrieved a letter, believed to have been written to a friend in December 1939 when Johnson was back in the United States. This is the key passage:\n\n> I was lucky enough to get to be a correspondent so that I could go to the front when I wanted to and so it was that I came again to the country that we had motored through, the towns north of Warsaw. . . . The German green uniforms made the place look gay and happy. There were not many Jews to be seen. We saw Warsaw burn and Modlin being bombed. It was a stirring spectacle.\n\nThis is almost too much even for Schulze's studied detachment. While he makes a feeble attempt at a psychological analysis of Johnson's behavior\u2014based on some ill-digested ideas out of William James\u2014he is clearly both appalled and bewildered by his own account of this crucial period in Johnson's life, which is the most complete we have been given so far.\n\nIn 1939, Philip Johnson's romance with Hitler and the Nazis had begun to attract the interest of a journalist named Dale Kramer\u2014no relation to the present writer\u2014who in the October 1940 issue of _Harper's_ gave an account of Johnson's political affiliations in an article on \"The American Fascists.\" The magazine was on the newsstands in September when, according to Johnson's FBI file, he had arranged a meeting at the German embassy in Washington. It may have been pure coincidence, but, writes Schulze, \"three days after the alleged appointment at the embassy in Washington Philip was back in Harvard, enrolled as a student in the school of architecture.\" His affair with Nazism was over.\n\nOn his return to Harvard, Johnson, at age thirty-three, was no ordinary student. He was older, richer, and more knowledgeable than other students. Moreover, in the decade since he had received his undergraduate degree, the university\u2014or its Graduate School of Design, anyway\u2014had abandoned its traditional curriculum to embrace the modernism which Johnson had championed in the early 1930s. The book that Johnson and Henry-Russell Hitchcock had published in 1932, _The International Style_ , was now a required text.\n\nThis gave him an immense advantage, to say the least. And so did his wealth, which enabled him to design and build for himself a Miesian house in Cambridge while he was still enrolled as a student. To be sure, Johnson could not entirely escape the consequences of his political past, but in the end his fascist involvements proved to be more of a temporary inconvenience and occasional embarrassment than a permanent bar to advancement. This was all the more remarkable in that the new Harvard design faculty was dominated by three refugees from Nazi Germany\u2014Walter Gropius, the founder of the Bauhaus; his Bauhaus colleague Marcel Breuer; and Martin Wagner, the former director of planning for the city of Berlin. But the bad blood that existed between Gropius and Johnson, for example, had much more to do with the latter's loyalty to Mies van der Rohe than with the former's love affair with the Nazis. (Years later, when he was in a position to do so with impunity, Johnson dismissively characterized Gropius as \"the Warren G. Harding of architecture.\") Breuer, on the other hand, treated Johnson with cordiality and respect.\n\nMeanwhile, Johnson was making some pro forma attempts at his own political rehabilitation. When he joined a Harvard Defense Group (a civil defense organization) in 1941, it fell to the young Arthur Schlesinger, Jr., to dismiss him from the organization on political grounds. An attempt to take a wartime job at the Office of Facts and Figures in Washington likewise failed as soon as the agency got a look at his FBI file. Still, Johnson weathered these and other travails\u2014including a brief stint in the U. S. Army, into which he was drafted in 1943\u2014with remarkable ease, given the strong anti-Nazi sentiments at the time.\n\nWhen he got out of the army in 1945, with the country still waging war against Japan, he promptly set about mapping the course of his new career. Toward this end he opened his own architectural office in Manhattan (though he was unlicensed to practice in New York State); and, with the help of Alfred Barr, he insinuated himself into his old position in the architectural department of the Museum of Modern Art. The renewed MOMA connection proved indispensable to his new career as a practicing architect, restoring Johnson to a position of authority in the field even before he had built enough to earn it.\n\nThen, as always, Johnson was not a man\u2014or artist\u2014to allow principle to interfere with opportunity. He straightaway designed a prefabricated house for a _Ladies' Home Journal_ competition, and arranged for the house to be exhibited at the Modern. \"Remarkably,\" writes Schulze, \"it was a deep bow to the functionalism Philip had so long abhorred.\" Even his renewed campaign on behalf of Mies concealed a growing private doubt about the value of the Miesian aesthetic\u2014which did not prevent Johnson from mounting an exhibition devoted to Mies at the Modern or from designing for himself the famous Glass House in New Canaan, Connecticut, a house that is itself little more than a pastiche of the Miesian style.\n\nThat house promptly proved to be so egregiously uninhabitable even for an aesthete like Johnson that he at once undertook to build an alternative, _anti-Miesian_ retreat on the same property that would afford him the privacy and amenity which the doctrinaire transparency of the Glass House precluded. In this regard I recall a lecture Johnson delivered at Harvard in the spring of 1951 (when I was a student there). Asked by a member of the audience if the Glass House was not fundamentally incompatible with the needs of family life, Johnson declared with his customary hauteur that the family should be abolished.\n\nFor Johnson, the Glass House of 1949 served a purpose far more important than domestic amenity. More than anything else he had done or would ever do, it established Philip Johnson the architect as a reputed master of the same modernist style upon which Philip Johnson the museum curator and author had conferred a renewed legitimacy and glamour. Whatever his doubts about the failings of the Miesian ideal, and despite the inevitable jokes about \"Mies van der Johnson,\" the Glass House was a brilliant gambit that completely succeeded in its purpose. To this day it continues to be regarded as a classic of American modernist architecture, despite the fact that architects as eminent and as different as Frank Lloyd Wright and Mies himself dismissed it with contempt.\n\nThe payoffs came right on schedule. In 1950, Johnson was commissioned to design the MOMA annex on West 53rd Street\u2014another exercise in Miesian pastiche\u2014and in 1953 the Abby Aldrich Rockefeller Sculpture Garden. (It was the elder Mrs. Rockefeller who, when told something of Johnson's Nazi connections, reportedly said that \"every young man should be allowed to make one large mistake.\" But it is doubtful that Mrs. Rockefeller ever knew the full extent of Johnson's involvement.) The sculpture garden may be the single most beautiful thing Johnson ever designed, but it is an essay in museum installation, not building design, and serves a strictly aesthetic function. In any case, these MOMA commissions set the stage from which Johnson's subsequent architectural career was launched.\n\nNot everyone could be expected to be as forgiving as Mrs. Rockefeller, however. The experience of World War II and the impact of the Holocaust remained powerful moral and emotional issues, especially in the intellectual, cultural, and business worlds of New York. This obliged Johnson to ponder, as Schulze writes, \"the need . . . to be free of public perception of his kind of political taint.\" Success, in other words, required some public gesture of atonement if the stain of anti-Semitism were to be effectively neutralized.\n\nWhen, therefore, Johnson learned that a Jewish congregation in the suburb of Port Chester, New York, was looking for an architect to design a new synagogue, he seized the opportunity with his characteristic bravado. \"His proposal to design [the building] without fee,\" writes Schulze, \"was hard for the potential clients to resist, coming as it did from an architect, and a New Canaan neighbor, whose professional reputation was growing at about the same rate his political past seemed to be receding in most people's minds.\" The building that Johnson designed was an architectural hodgepodge devoid of aesthetic distinction, but for the architect it served its purpose well enough. It was another brilliantly cynical move, and there was no way for his clients to know that the sculpture he selected for the interior\u2014an abstract metal relief by Ibram Lassaw\u2014was virtually identical with the one that already adorned his secluded New Canaan bedroom, the scene of rituals of a very different sort. Whether or not Johnson intended this as a private joke cannot be known; but it is not the kind of aesthetic irony that would have been lost on him.\n\nIt is worth reflecting, in this connection, upon the remarkably large number of prominent Jews who, in the course of Johnson's architectural career, have made major contributions to his reputation and success. Conspicuous on this roster of art patrons, intellectuals, businessmen, museum trustees, critics, and architects are such luminaries as Lincoln Kirstein, Edward M. M. Warburg, Ronald Lauder, Samuel Bronfman, I. S. Brochstein, Rosamond Bernier, Ada Louise Huxtable, Paul Goldberger, Robert A. M. Stern, and Peter Eisenman. To them should also be added Shimon Peres, who in 1960 arranged for Johnson to design the nuclear reactor at Rehovot in Israel, and Meyer Lansky\u2014\"one of the panjandrums of organized crime in America,\" as Schulze writes\u2014for whom Johnson was planning to design a huge gambling casino\u2013hotel in Havana when Fidel Castro's revolution altered their respective schedules.\n\nThis is by no means an exhaustive list, but it is sufficient to underline the extraordinary success Johnson achieved in his programmatic effort to induce a kind of collective amnesia or suspension of curiosity about his Nazi past, even among people who had every reason to be alert to such matters. It was not until 1988, when Johnson was eighty-two and, as Schulze says, \"probably the most famous figure in the American architectural world,\" that the critic Michael Sorkin attempted to awaken a new generation with a scathing article in _Spy_ magazine. But the venue of Sorkin's attack was itself too disreputable for the article to have much of an impact. At the altitude of eminence which Johnson now occupied, with the press in his pocket, the _beau monde_ at his feet, and the profession lavishing hosannas upon him, he was for all practical purposes beyond the reach of criticism or exposure.\n\nWhat in retrospect is especially remarkable about these adroit historical maneuvers is that the man who effected them remained at best a very mediocre architect and at times a disgracefully bad one. The art museums he designed for Lincoln, Nebraska; Utica, New York; and both Corpus Christi and Fort Worth, Texas, are among the worst such buildings erected in this country in his lifetime\u2014and all the less to be forgiven as coming from a connoisseur canonized by the Museum of Modern Art. None of these buildings is in the same league with a contemporary masterpiece like Louis Kahn's Kimbell Art Museum in Fort Worth or the building Edward Larrabee Barnes designed for the Walker Art Center in Minneapolis. The New York State Theater, which Johnson designed for New York City Ballet at Lincoln Center, is so undistinguished that many people find it hard to believe it was done by a reputable architect at all; it looks like pure developer's boilerplate.\n\nThese failed buildings, moreover, date from before the time when Johnson developed a conscious policy of socking it to his corporate clients with aggressive displays of imperious bad taste\u2014before, that is, he perpetrated postmodern monstrosities like the AT&T Building and the so-called Lipstick tower in New York. For some two decades now, Johnson has been functioning\u2014with what degree of deliberateness, one can only surmise\u2014as the Andy Warhol of American architecture, \"quoting\" a bit of Gothic here, a bit of Chippendale there, and sporting a display of self-aggrandizing kitsch everywhere, in more or less the same manner in which Warhol famously \"quoted\" the Campbell Soup label.\n\nSome of this has proved to be too much even for Johnson's most loyal admirers. The critic Ada Louise Huxtable spoke for many when, in an address to the American Academy of Arts and Sciences in 1980, she condemned buildings like AT&T and the Pittsburgh Plate Glass headquarters in Pittsburgh as \"shallow, cerebral design,\" \"bad pieces of architecture,\" and \"clever cannibalism.\" But the most devastating judgment came in 1976 from an even more surprising quarter, the Museum of Modern Art, which rejected Johnson's bid to design the most ambitious expansion in its history. Johnson had been, in effect, MOMA's house architect for a quarter-century and its principal architectural authority for longer than that; he was now passed over for the commission that would give the museum its architectural identity for many decades to come. (The job went to Cesar Pelli.)\n\nIt was tantamount to a divorce, and Johnson suffered what may have been the single biggest disappointment of his career. \"Hence his reaction,\" Schulze writes, \"which was to reverse the intentions he had earlier had for the disposal of his estate and not to leave the museum his collection of painting and sculpture or even his country property in New Canaan.\" Still, even this has not quite ended Johnson's association with the museum, for MOMA is planning to mark his ninetieth birthday in 1996 with a publication and exhibition devoted to the paintings, sculptures, and design objects he has already given the museum since its founding.\n\nWhat audacities Johnson may be preparing for that occasion is anyone's guess, but it is a mercy that neither Alfred nor Marga Barr will be around to witness them. On the last pages of _Philip Johnson: Life and Work_ , Schulze brings us up to date on his subject's current views with an account of an address Johnson delivered last year (1994) at the Austrian Museum of Applied Arts in Vienna:\n\n> The subject was Stalinist architecture, and Philip used the occasion not only to demonstrate again that his basic view of the world was as consistent as the styles of his architecture were not, but to affirm an opinion he had seldom had the opportunity to express: his admiration for a boulevard in the former East Berlin that had been the object of almost unanimous scorn among Western critics. It was the notorious Stalinallee, later renamed the Karl-Marx-Allee, still later, after the fall of the Berlin Wall, the Frankfurter Allee. Philip had long found a grandeur in it that others had dismissed as totalitarian pomposity. Now he called it \"the new Champs-Elys\u00e9es,\" a product of \"romantic daring\" and \"the dream of the East for monumentality.\"\n\nSchulze may well be correct in his observation that \"At eighty-eight, [Johnson] was where he was always most comfortable, at the head of the parade of contemporary taste.\" But given his history, there are other ways of reading his remarks in Vienna. Coming upon them in his biography, I was reminded of a conversation I had with Marga Barr in the last year of her life. I was then working with her on the preparation of a \"Chronicle\" of Alfred Barr's career for publication in _The New Criterion_. (It was published under the title, \"Our Campaigns,\" in a special issue of the magazine in the summer of 1987.)\n\nOn one of the mornings we had set for a meeting in her apartment, the _New York Times_ published Johnson's proposed designs for the rehabilitation of the Times Square\u201342nd Street area. I found them even more wretched than some of the awful things he had already built, and I was eager to know what Marga thought of them. In recounting to me the story of Alfred's career, she had had frequent occasion to speak of Johnson, and she always did so with fond affection\u2014for the record, so to speak. That morning I asked if she had seen the paper, and she rather glumly acknowledged that she had. I then asked what she thought of the kind of buildings Johnson had lately been designing\u2014and hastened to add that she was under no obligation to discuss the subject if she preferred not to.\n\nIn responding to difficult questions, Marga had a way of turning away for a few moments while she composed her thoughts and then facing her interlocutor with a very determined look. This is what she did that morning as she said to me: \"I feel about Philip today the way I would feel about a beloved son who had gone into a life of crime.\"\n\nIt was a hard thing for her to say, but Marga Barr was not in the habit of shirking the truth. After reading Franz Schulze's life of Philip Johnson, I would say that she had gotten the matter exactly right.\n\n[1995]\n\n### Duchamp and His Legacy\n\n* * *\n\n> _For Marcel Duchamp the question of art and life, as well as any other question capable of dividing us at the present moment, does not arise_.\u2014Andr\u00e9 Breton, in _Litt\u00e9rature_ , 1922\n> \n> _He approaches life as he does the chessboard: the gambits fascinate him without leading him to imagine that there is a meaning behind it all which might make it necessary for him to believe in something. . . . Duchamp's attitude is that life is a melancholy joke, an indecipherable nonsense, not worth the trouble of investigating_.\u2014Hans Richter, _Dada: Art and Anti-Art_ , 1964\n\nWhat accounts for the immense intellectual prestige which the mystique of Marcel Duchamp has enjoyed in this country\u2014if only in certain circles, to be sure\u2014for more decades than most of us can now remember? I speak at the outset of the artist's mystique rather than of his art, for it is not so much Duchamp's art as the penumbra of ideas and mystifications associated with it that has cast a spell over the minds of so many otherwise intelligent people, both in the art world and beyond it. As an artist, after all, Duchamp produced one of the smallest _oeuvres_ in the modernist canon. He was always a lazy revolutionary. His insouciance was often a mask of indolence, his indolence a reflection of the accidie or ennui he perfected as a way of life and somehow managed to endow with an aura of sanctity. That it was a sanctity of negation only enhanced its prestige.\n\nHow much that aura of sanctity owed to the effortless celebrity that Duchamp achieved in America while he was still in his twenties is a matter about which, of course, we can only speculate. What is a certainty, however, is that America was essential to the kind of career he elected to pursue once that celebrity had been thrust upon him in the Armory Show of 1913. The painting that made Duchamp an overnight sensation in that show\u2014the _Nude Descending a Staircase_ , painted in 1912\u2014wasn't the masterpiece of audacity it was taken to be. As William S. Rubin observed in his \"Reflexions on Marcel Duchamp\" in 1960,\"the _Nude_ hardly compares in adventurousness with the great Analytic Cubist paintings of the same year\"\u2014the paintings of Braque and Picasso. Indeed, to place the artist who painted the _Nude Descending a Staircase_ in the company of Braque and Picasso, never mind Matisse and L\u00e9ger, is to define him as a minor figure. In the Paris avant-garde of the period before the First World War, Duchamp did not rank at all.\n\nOnly in America was he mistaken to be a major representative of the modernism that had been created by talents more robust than his. The truth is, the sensation that the _Nude_ caused in New York in 1913 was more a reflection of _our_ provincialism than of Duchamp's originality. Yet who can doubt that the entire career of the Duchamp who is now so admired, so solemnly studied and so widely emulated\u2014the Duchamp of _The Large Glass_ , the \"Readymades,\" and the enigmatic notes and clues that accompany them\u2014was determined by this early episode in mistaken identity? For that episode and the publicity it generated placed upon Duchamp's every subsequent artistic effort an obligation to come up with something that would prove to be equally provocative and controversial.\n\nIt is in this respect that Duchamp belongs to the history of the avant-garde in America rather than to its greater counterpart in Paris. It wasn't until the emergence of the Surrealist movement in the early 1920s that Duchamp secured an anchor to which he could attach himself in the Paris avant-garde. Even then, of course, it suited his temperament to use the Surrealist camp as a convenience, as it were, rather than be used by it. Mark Polizzotti, in his new biography of Andr\u00e9 Breton, gets the matter exactly right when he describes Duchamp as an \"amiable fellow traveler\" of both Dada and Surrealism who viewed the younger Breton \"with a mixture of respect and sarcastic amusement.\" But then, of course, he viewed everything in life with sarcastic amusement.\n\nWith the Dada movement Duchamp had closer affinities, and many people still persist in portraying him as the very archetype of the Dada spirit\u2014a practice that Duchamp did nothing to discourage. He certainly shared the Dada taste for creating scandal, yet with neither of the two principal activities that occupied the Dada movement\u2014art and politics\u2014did Duchamp have much sympathy. He professed to scorn the creation of the first (though in private he continued to pursue certain projects), while about the second he refused to pay it even the compliment of rejection. Alongside the copious _oeuvres_ of the leading artists of the Dada movement\u2014Jean Arp, Kurt Schwitters, Max Ernst, and Man Ray\u2014Duchamp's is little more than a provocative patchwork; and with the politics of a Dada militant like Richard Huelsenbeck, Duchamp had no affinity whatever. One can well imagine with what \"sarcastic amusement\" he might have observed Breton's tortuous attempt to establish an _entente cordiale_ with the French Communist party in the 1920s, but it is doubtful that he paid the matter the slightest attention. The sheer solipsism that circumscribed Duchamp's mental universe precluded such interests.\n\nIt was this inviolate solipsism, to which his works, his utterances, and his public career made so many arch allusions, that combined with the glamour of Duchamp's peculiar fame to create his special mystique. By suggesting that he believed in nothing, least of all in art, while at the same time enjoying the status of an admired and controversial artist, Duchamp left it to others to discover and codify what, if any, might be the private pieties and proclivities that actually governed his life and thought. Has any other modern artist ever given so many interviews about such a small number of objects without revealing anything very definite about their meaning or intention?\n\nThose interviews were often brilliant performances. They introduced just enough variation into Duchamp's account\u2014or rather, nonaccount\u2014of his thought to sustain the interest of his acolytes while disclosing little, if anything, that wasn't already established as a coefficient of his personal myth. It was in this sense that Duchamp may have been the greatest \"performance artist\" of his time\u2014an artist, in other words, whose \"works\" are seen to be mere props for the ritual reenactment of a private destiny.\n\nAll through the later decades of his life, when it pleased Duchamp to beguile successive generations of artists and art-world functionaries as an eminence who had achieved a special status precisely because he had given up the mundane task of creating art, it was his historic act of negation that was assumed to be the basis of his moral authority\u2014a moral authority that depended, paradoxically, on the good opinion of the many other artists who treated him as a guru and hailed him as a master while they themselves refused, of course, to follow his example in abandoning the creation of art. These artists understood very well that giving up art _\u00e0 la Duchamp_ would get them nowhere. Duchamp had already done that to great acclaim, and it was not a gesture that could be repeated with anything like the same effect. Besides, these artists were eager to get on with their careers and achieve the kind of success that had come to Duchamp himself at an early age. They hadn't yet done enough in their art or achieved enough public notice to endow any decision they might make to abandon the creation of art with the requisite shock and significance.\n\nTheir problem\u2014the problem of the Duchampian epigones\u2014was indeed paradoxical: how to honor and participate in the spirit of the master's negation of art while continuing with the creation of art. The solution, of course, was the sham compromise that goes by the name of \"anti-art,\" which may best be defined as a species of art that claims, if only provisionally, to be something other than what it is\u2014namely, art\u2014in order to be judged by criteria other than those currently in effect or, better still, not to be judged at all but simply embraced as a form of expression more imperative than mere \"art.\"\n\nMuch of the art of the last fifty years\u2014much, certainly, that belongs to what the art establishment believes to be \"avant-garde\"\u2014is anti-art of this persuasion, and Duchamp is indeed its patron saint. He set this train on its course, after all, with the \"Readymades\"\u2014the snow shovel, the urinal, the bicycle wheel, etc.\u2014that mocked the very concept of artistic creation and were yet quickly accepted, albeit in quotation marks so to speak, as authentic and even profound works of art. Today, nearly eighty years after Duchamp perpetrated this bluff on the New York art world, his anti-art legacy fills our museums and in some other quarters, too\u2014the academy and the media\u2014commands an esteem that is often greater than any enjoyed by works of art created by more traditional means. All of which is a reminder, if we still need one, that the conventions of this anti-art legacy have themselves now come to constitute an academy of sorts.\n\nIt is a measure of this academic status that the study of Duchamp is now something of an academic industry\u2014so much so, indeed, that in this last decade of the twentieth century it looks more and more as if the university will be the final resting place of Duchamp's reputation. This may tell us more about the fate of the university at the end of the twentieth century than it tells us about Duchamp\u2014for much of the academic literature on Duchamp amounts to little more than pedantry in the service of uncritical adulation\u2014but it also suggests that the appropriation of Duchamp by the academy may be the ultimate Duchampian irony: a revolution in sensibility buried in a cemetery of scholarly footnotes.\n\nTo this vast literature on Duchamp there has now been added something new\u2014an avowedly \"humanistic\" and \"old-fashioned\" historical approach to the subject called _The Private Worlds of Marcel Duchamp_. Its author\u2014Jerrold Seigel\u2014is not an art historian or art-world groupie but the William J. Kenan Professor of History at New York University. His aim in writing about Duchamp is, as he notes in the preface to this study, \"to clarify certain features of the avant-garde and of modern cultural history more generally.\" In this respect, at least, his study of Duchamp may be considered a sequel to his earlier history of _Bohemian Paris: Culture, Politics, and the Boundaries of Bourgeois Life, 1830 \u20131930_ , published in 1986. And while in much of the book Professor Seigel doesn't write as a critic of Duchamp\u2014on the contrary, he is clearly captivated by the audacity of Duchamp's performance and determined to give it the benefit of many doubts\u2014in the end _The Private Worlds of Marcel Duchamp_ is really a contribution to the demystification of his subject.\n\nIt is for this reason that one wishes the book had begun where it ends, with the chapter on \"Art and Its Freedoms.\" For it isn't until the last page of _Private Worlds_ that Professor Seigel explicitly raises the question that is now the principal reason for us to take an imperative interest in Duchamp and his legacy. This is the key passage:\n\n> As the claim that dissolving the separate sphere of art would liberate energies capable of creating new forms of life loses what persuasive force it once bore, we realize more and more that art owes what power it has to enrich the rest of life to the very separation and independence against which many avant-garde projects were directed; what we gain by dissolving the boundaries between art and life turns out to be much less than what we risk losing.\n\nThis is immediately preceded by an observation that is even more devastating in its assessment of Duchamp's influence:\n\n> As some recent critics have begun to argue, the consequences of merging art into life have not been what Breton and others hoped, leaving life untransformed but art much weakened by the absence of criteria to decide whether any given object belongs within its sphere and, of those objects that do, which are good. The sovereignty of the artist who claims the right to declare that art is whatever he or she designates has clashed with the equal authority of audiences to accept or reject what is offered them; the result is only a higher level of mutual suspicion and confusion.\n\nThis is indeed where a study of Duchamp and his legacy ought now to begin, and it is one of the disappointments of Professor Seigel's book that the subject comes too late to be fully explored. There are some excellent things in this last chapter on \"Art and Its Freedoms\"\u2014most notably, the comparisons that are made with Picasso, John Cage, Paul Val\u00e9ry, Jean-Fran\u00e7ois Lyotard, Michel Foucault, and Nietzsche\u2014but even these are somewhat blurred in the final equivocations about what in Duchamp's legacy remains \"vital and worth preserving.\"\n\nAs for what Professor Seigel means by those \"private\" worlds he al-ludes to in his title\u2014in the end they amount to little more than the defeated desires of a Freudian family romance. Professor Seigel describes his study of Duchamp as \"only mildly and partially psychobiographical,\" but in fact the book is regrettably \"old-fashioned\" in the way it accords intellectual priority to a quite conventional Freudian analysis of Duchamp and his _oeuvre_. Did Duchamp feel \"neglected as a boy\"? Did he suffer from \"his mother's indifference\"? Did he entertain fantasies of incest about his sister Suzanne? Did all this leave him sexually \"dead-ened\" in later life? Was he impotent or merely sexually indifferent to the many women with whom he was thought to have had affairs of a sort in his maturity? Or was he a secret homosexual? And was much of his art, in any case, some sort of masturbation fantasy?\n\nAbout most of these questions we have no definitive answers. Professor Seigel's speculations are better informed than most others I have read, but they leave us wondering if he quite understands the extent to which they, too, contribute to the mystification of his subject's reputation. For what they add up to is a protracted case of voyeurism haunted by impotence, and they leave us with a distinct impression of Duchamp as a man whose relation to art was remarkably parallel to his relation to sex\u2014an impression of total freedom won at the price of a bemused and self-absorbed sterility. And if Professor Seigel believes what he says about Duchamp's damaged psyche, then what does it mean for him to invite us to \"celebrate the courage and originality with which [Duchamp] explored those inner spaces\" on the last page of this book?\n\nThe question is an important one, for what I have called Duchamp's mystique derives, above all, from the conception of radical freedom he brought to the artistic vocation\u2014a conception of freedom so fundamentally inimical to the artist's vocation that it renders it impotent. For in art, as in life, there are only choices to be made, obstacles to overcome, contingencies to engage and accommodate. To \"drop out\" is to abandon effort and inspiration in favor of a radical passivity\u2014which is what the Duchampian idea of total freedom finally comes down to.\n\nWhy, then, should we speak of \"courage\" when what we are dealing with is failure\u2014a failure of spirit, certainly, and very likely a failure of talent as well? Failure in art, failure in life: isn't that the unacknowledged subtext of the story that is told in _The Private Worlds of Marcel Duchamp_?That Duchamp had the \"genius\" to turn that failure into a myth of success and superiority is an important part of the story, too, of course\u2014maybe even the more important part, for it returns us to the American setting of his career and the role played by American provincialism and gullibility in the creation of the Duchamp cult. And to the study of that subject \"psychobiography\"\u2014Freudian or otherwise\u2014doesn't have much to contribute.\n\nIn the end, alas\u2014and despite the demurrals of its last chapter\u2014 _The Private Worlds of Marcel Duchamp_ is a study remarkably reluctant to face up to the implications of its own findings. Inside this book there is a better book waiting to be written, but the one we have been given leaves the subject of Duchamp and his legacy safely enclosed in the equivocations of the academy.\n\n[1995]\n\n### L\u00e9ger's Modernism\n\n* * *\n\n> _If pictorial expression has changed, it is because modern life has necessitated it. The existence of modern creative people is much more intense and more complex than that of people in earlier centuries. The thing that is imagined is less fixed, the object exposes itself less than it did formerly. . . . The view, through the door of the railroad car or the automobile windshield, in combination with the speed, has altered the habitual look of things. . . . The compression of the modern picture, its variety, its breaking up of forms, are the result of all this_.\u2014Fernand L\u00e9ger, 1914\n\nI once had a student who wrote a term paper in which he compared the forms in Fernand L\u00e9ger's _Three Women_ (1921), in the collection of the Museum of Modern Art, with the forms of the engine room of the navy submarine in which he had served during the Korean War. While older and with more experience of the world than the other undergraduates I was then teaching, this student was new to the mysteries of modern painting. Mine was the first art history course he had ever signed up for, and he had never before entered an art museum before I assigned the class to study certain pictures at the Museum of Modern Art\u2014not slides or other reproductions but the actual paintings\u2014as a preliminary to reading anything about them. His reaction to L\u00e9ger was instant and electric. He seemed to understand everything about the painting, including its humor.\n\nOne of the things that were interesting about the immediacy of this response to L\u00e9ger was its relation to the student's wartime experience.. For there was no way, in that early stage of his study of modern painting, that he could have known of the role that L\u00e9ger's own military experience in the First World War had played in the development of his art. About the effect of that experience on his artistic thought, L\u00e9ger was nothing if not explicit and unequivocal. \"During those four war years,\" he later recalled,\n\n> I was abruptly thrust into a reality which was both blinding and new. When I left Paris my style was thoroughly abstract: period of pictorial liberation. Suddenly, and without any break, I found myself on a level with the whole of the French people; my new companions in the Engineer Corps were miners, navvies, workers in metal and wood. Among them I discovered the French people. At the same time I was dazzled by the breech of a 75-millimetre gun which was standing uncovered in the sunlight: the magic of light on white metal. This was enough to make me forget the abstract art of 1912\u201313. A complete revelation to me, both as a man and as a painter. . . . Once I got my teeth into that sort of reality I never let go of objects again.\n\nIt was not that L\u00e9ger was ever inclined to glorify the violence of the war. Whatever his affinities with the Futurists in other respects, he did not share their cult of violence. Nor, for that matter, was his conversion to a modernism based on machine forms quite as sudden, perhaps, as his later account of it implies. He had, after all, reflected on the way machine technology had \"altered the look of things\" in 1914, prior to his wartime encounter with \"the magic of light on white metal,\" and the abstract paintings he produced in the _Contrast of Forms_ series in 1913 had already infused the Cubist aesthetic with a machinelike dynamism quite unlike anything to be found in Picasso or Braque.\n\nYet what John Golding has written about the effect of the war on L\u00e9ger's artistic thought\u2014that \"no other artist of his generation was to extract such positive conclusions from its squalor and horror\"\u2014is certainly true. L\u00e9ger came out of the war with a clear conviction that modern technology, both as an experience to be encompassed in art and as a determinant of its pictorial form, had something important to contribute to modern painting, and it was upon that conviction that all of his subsequent achievements were based.\n\nIt wasn't quite the case, however, that in the pursuit of those achievements in the decade following the war L\u00e9ger had altogether forgotten his interest in abstraction. Almost alone among the luminaries of the School of Paris in the early decades of the century, L\u00e9ger understood the importance of Mondrian and the De Stijl group. In an essay on \"Abstract Art\" for _Cahiers d'Art_ , in 1931, L\u00e9ger said of these Neoplasticist painters that \"they are right to say that Cubists' works cannot go as far as their own inquiries do,\" and hailed them as \"the true purists.\" Abstract art, he wrote, \"is the most important, the most interesting of the different plastic trends that have developed during the last twenty-five years,\" and predicted it would have staying power. Yet he readily acknowledged that as a painter \"I have stayed at the 'edge' without ever involving myself totally in their radical concept.\"\n\nDid he nonetheless derive something from their example? I think it is undeniable that he did. There would scarcely be such a preponderant use of black, white, and brightly colored geometric forms in L\u00e9ger's paintings of the late teens and early twenties if not for the innovations of Mondrian and his circle. \"They have daringly used as the 'main character' the _colored plane_ which has obsessed painters since 1912,\" L\u00e9ger wrote, adding that \"the geometric shape rigorously limits it. . . . Color is locked in and must remain fixed and immobile.\" Even a painting as crowded with curvilinear forms as the _Three Women_ contains much that answers to this description.\n\nIt is often said of L\u00e9ger that, alone among the early masters of the School of Paris, it was he who took Cubism out of the studio and into the street\u2014which is to say, into a fateful dialogue with the iconography of modernity. The nod to Futurism, just before the First World War, and the bow just after to its polar opposite\u2014De Stijl\u2014certainly bears this out. What he wanted for his painting was the dynamism of the former, minus its violence, and the order of the latter, minus its mysticism and metaphysics. Once L\u00e9ger adopted Cubism as the key to his pictorial ambitions, moreover, he seems to have felt impelled to endow it with a more impersonal profile than would be possible in an art that drew its subjects from the intimate objects of the studio or the familiar haunts of an artists' bohemian milieu. Before the war he solved this problem most successfully by turning away from objects altogether in favor of Cubist abstraction in the _Contrast of Forms_ pictures. During the war that was a solution that no longer commended itself to his chastened view of the place he wanted his painting to occupy in the social order to which he had been awakened by the war itself. Hence what looks like a retreat into the quasi-Futurist figurative Cubism of _The Card Game_ (1916). This is not to say that the latter isn't a terrific picture, but it has the period look of the prewar avant-garde.\n\nThis is why, in the \"Fernand L\u00e9ger\" exhibition that Carolyn Lanchner has organized this spring at the Museum of Modern Art, our encounter with one of the first of the big postwar paintings\u2014 _The City (_ 1919)\u2014has the effect of a clash of cymbals announcing a new era. This is a far more original picture than is commonly acknowledged, and not the least amazing thing about it is its buoyant affirmation of modernity. The modern city that is celebrated in this painting is, however, an entirely visual rather than an existential or social phenomenon. There is no throng, no traffic, no momentum that derives from the depiction of an urban mass enacting the grueling rituals of workaday life. There is no suggestion of urban anxiety, either. In L\u00e9ger's _City_ , \"the _colored plane_ ,\" here given an insistently vertical accent, is indeed the \"main character,\" while what traces remain of the city's shadowy inhabitants are reduced to the status of robots or silhouettes\u2014in other words, abstractions. For that matter, L\u00e9ger's entire conception of the modern urban scene in this painting is a variety of visionary abstraction\u2014a syncopated gloss of impressions in which the painterly dabs and touches of, say, Pissarro's city pictures of the 1890s, with their grey weather and distant vistas, are cheerfully transmuted into the close-up, immaculate, highly colored, hard-edged, fragmentary forms of a twentieth-century metropolis. It was also, of course, a largely imaginary metropolis, for the actual city of Paris in 1919 was anything but the model of modernist geometric construction that is depicted in _The City_. . . .\n\nIn this respect it hasn't been sufficiently noted that _The City_ was a remarkably upbeat painting for an artist in L\u00e9ger's position to produce at that moment in modern history, scarcely a year after the armistice that ended the First World War. The war had been an immense trauma for the men of L\u00e9ger's generation and the cause of widespread cynicism and demoralization in its aftermath. Compare the effect of the war on minds as different as Bertolt Brecht and Max Beckmann in Germany, T. S. Eliot in England, and Ernest Hemingway in America, and you might suppose, if you didn't know better, that the war had scarcely touched L\u00e9ger in any adverse way. In neither _The City_ nor in the pictures that L\u00e9ger produced in the decade that followed\u2014the decade in which he painted his greatest pictures\u2014is there a hint of anything like the anguished disillusionment and cynical detachment that we find in Brecht's _In the Jungle of Cities_ or Beckmann's paintings of the 1920s or Eliot's _The Waste Land_ or Hemingway's _In Our Time_. Nor is there the least suggestion of the kind of angst that permeated the work of the New Objectivity movement in German painting that is exactly contemporary with L\u00e9ger's paintings of the 1920s. There is instead an almost utopian vision of a technological order in which models of beauty and perfection are drawn from the world of machine-made objects.\n\nUnderlying L\u00e9ger's modernism in the 1920s, then, is a certain dissociation of sensibility\u2014for his was a modernism accompanied by a rejection, which we must suppose to be deliberate, of everything that was dark and desperate and unresolved in the experience of modernity. It was not at all the case that L\u00e9ger was himself indifferent to the failings of modern society, but he seems to have reserved his feelings about all that for his politics\u2014the politics of a sentimental leftism that eventually brought him into the orbit of the French Communist party. In his art, however, he specialized in a kind of comic modernism, a modernism of happy endings. There is certainly a deadpan comic undercurrent in paintings like _The Mechanic_ (1920), _Woman and Still Life_ (1921), _Reading_ (1924), and the delightful _Composition with Hand and Hats_ (1927), which obviously owes a lot to the movie comedies of the 1920s. The _Three Women_ is itself\u2014among much else, of course\u2014a comic masterpiece.\n\nBy the 1930s and 1940s, this comic vein in L\u00e9ger's paintings lost some of its deadpan humor as the pictures got bigger and their subjects more traditionally satirical. From his version of _Adam and Eve_ (1935\u20131939) to his not very amusing _Leisure, Homage to David_ (1948\u20131949), L\u00e9ger becomes a more problematic artist. The element of visionary abstraction that served as a kind of pictorial conscience in L\u00e9ger's masterpieces of the decade following the First World War is largely gone, supplanted by a larky pictorial vaudeville that often degenerates into mural-scale cartoons. In this connection I think it is fair to say that L\u00e9ger's interest in the movies contributed far more to his painting than his interest in modern architecture did. It was always a fallacy that modern painting would somehow achieve some sort of higher social redemption by placing itself at the service of modern architecture, and L\u00e9ger\u2014owing, no doubt to his political sentiments\u2014was in his later years much tempted by that false prospect. In a lecture called \"The Wall, the Architect, the Painter\" (1933), L\u00e9ger welcomed such collaboration with modern architecture. \"The simplified and rational architecture that is going to conquer the world must serve as a possibility for reviving this collective art that created immortal masterpieces before the Renaissance,\" he said. \"The evolution of society may be tending toward this new order. Easel painting survives\u2014and will always survive\u2014but it can be broadened by the Renaissance mural.\"\n\nBut the \"simplified and rational architecture\" that did conquer the world proved to be deeply inimical to a modernist revival of the Renaissance mural. As late as 1950, L\u00e9ger was still harboring the illusion that mural painting might still be revived, and now he saw it as a vehicle for abstract painting. \"I believe and I maintain,\" he wrote, \"that abstract art is in trouble when it tries to do easel painting. But _for the mural the possibilities are unlimited_. In the coming years we will find ourselves in the presence of its achievements.\" He was wrong on both counts, about abstract easel painting no less than about the possibilities of abstract mural painting. The net result for his own painting, however, was to render it more and more superficial, not only in its imagery but in its command of feeling. And it wasn't as if the result succeeded with the public, either. George Heard Hamilton was surely right when he wrote that \"there is little evidence that L\u00e9ger's efforts to create an art with wide popular appeal has met with much response from the people for whom he intended it.\" L\u00e9ger's modernism remained an art of the museums.\n\nIt is interesting, in this regard, to see L\u00e9ger's modernism in the 1920s in relation to the turn that Picasso's modernism took in the same period. On this subject, John Golding's essay on \"L\u00e9ger and the Heroism of Modern Life\" must be regarded as the classic text.1 Writing about what he calls L\u00e9ger's \"Neo-Classicism\"\u2014a term I should myself hesitate to apply to the artist's machine-form pictures of the twenties\u2014Mr. Golding observes that \"he avoided all the overtly classicizing attributes in which Picasso rejoiced; there are no white draperies; no garlands of leaves and flowers, and his figures wear a garb that is strictly contemporary and yet timeless in its simplicity.\" And further:\n\n> The pitchers and the urns, which can transform a Picasso figure into a caryatid or a river goddess, become, in L\u00e9ger's work, the implements of domestic, workaday life. The latent sensuality of Picasso's nudes, and the languor that characterizes the contemporary work of Neo-Classicizing artists such as Derain or La Fresnaye, these were qualities completely alien to L\u00e9ger's art. He treats his figures, not without an underlying compassion, but with a detachment that was fundamental to his vision of the modern world; his figures are geared to contemporary life, and it is as part of the mechanism of life in general that they achieve significance.\n\nYet the similarities are found to be just as striking as the differences:\n\n> Both adopt for their figures static, monolithic poses, and both make use of similar, heavy, generalized forms to render their limbs; the heads are composed of simple, almost geometrical shapes and the features are rendered in the same incisive, sculptural way. At times L\u00e9ger's figures echo those of Picasso so closely that one cannot help suspecting him of consciously trying to inject a quality of contemporary slang into Picasso's Neo-Classical vocabulary.\n\nMr. Golding misses the comic element in L\u00e9ger's parodies\u2014if that is what they are\u2014of Picasso's Neoclassicism, but he is otherwise correct about the parallels, and this raises the question of whether L\u00e9ger's upbeat 1920s modernism was any less of an escape from the traumatic effects of the First World War than Picasso's fantasias on a classical dream world. In retrospect, both now seem to have been in flight from precisely the kind of ill-fated modernity that the war itself represented for an entire generation.\n\nThe exhibition that Ms. Lanchner has organized in \"Fernand L\u00e9ger\" is not, alas, the full retrospective one had hoped it would be, but it is a splendid exhibition all the same. It is certainly odd that in an era that has witnessed the rise and fall of Pop Art and of so much machine-made Minimalist art, no New York museum thought to mount a retrospective of L\u00e9ger's work as a relevant point of reference. But it may be that L\u00e9ger himself, while still an undoubted classic of modern painting, no longer looms as the master artist he was once thought to be. In the absence of a full retrospective, the question remains open, but in this abridgment of L\u00e9ger's _oeuvre_ it looks more and more as if the L\u00e9ger destined to survive is the L\u00e9ger of the years 1910\u20131930.\n\n[1998]\n\nFootnote\n\n1. See John Golding, \"L\u00e9ger and the Heroism of Modern Life,\" in _Visions of the Modern_ (Berkeley, 1994).\n\n### Bonnard and \"the Stupidities\"\n\n* * *\n\n> _I agree with you that the painter's only solid ground is the palette and colors, but as soon as the colors achieve an illusion, they are no longer judged and the stupidities begin_.\u2014Bonnard to Matisse, 1935\n> \n> _In short, there is something abject and sinister about Bonnard's late bathers. . . . The metamorphosis may be gorgeous, but it is also a kind of elegant_ pourriture, _exquisite rot, canvases shimmering with the iridescence of putrefaction, glowing with the ooze of the_ informe. _It is significant that Bonnard's work is at its best when he kills off or mutilates his subject: Marthe dismembered or floating in deathlike passivity is the heroine of his most exciting canvases_.\u2014Linda Nochlin, _Art in America_ , July 1998\n> \n> There is decadence that excites and decadence that enervates. _Bonnard's is the second sort: edgeless, nerveless, weird, fussy. He isn't mindless, exactly, but he withholds his mind from his transactions with painting. For thought, he substitutes maundering on autopilot. His compass is a vague tastefulness.This comforts people who dislike thinking. I submit that such people are already comfortable enough, on their own lookout, and should not be indulged_.\u2014Peter Schjeldahl, _The Village Voice_ , July 28, 1998\n\nWho could have imagined that in 1998, more than half a century after the artist's death, the paintings of Pierre Bonnard (1867\u20131947) would still be a subject of controversy? Yet in certain influential quarters of the art world this season's Bonnard exhibition at the Museum of Modern Art has elicited a response at once so intemperate and so bizarre that at times it went beyond controversy to become a moral indictment. This may strike us as a very odd thing to happen to an artist as reclusive as Bonnard was in his life and as exclusively concentrated on the gravity of the painter's vocation as he remained throughout all the later decades of his long career. But it would probably not have surprised Bonnard himself. Responses of this kind were precisely what he had in mind when he spoke of \"the stupidities\" that painting of his persuasion\u2014so radically entrenched in the language of \"the palette and colors\"\u2014was likely to inspire in critics who regard such a pictorial practice as a wicked indulgence.\n\nWhat is it, then, about Bonnard's painting that excites such a heated condemnatory response? Make no mistake: Bonnard really incites these critics to paroxysms of denunciation and ridicule. Linda Nochlin, the Lila Acheson Wallace Professor of Modern Art at New York University's Institute of Fine Arts, went so far as to say of Bonnard's late _Bather_ pictures that \"I am so repelled by the melting of flesh-and-blood model into the molten object of desire of the male painter that I want to plunge a knife into the delectable body-surface.\" (She is speaking of Bonnard's paintings of his beloved wife, Marthe.) Peter Schjeldahl, still in mourn-ing over the lost illusions of the 1960s counterculture, adds to the indictment by speaking of these same paintings as \"eye-candy,\" \"trivial,\" the residue of \"a masturbatory trace,\" and \"an art that is on its last legs as a culture-changing enterprise.\" These writers clearly disapprove of Bonnard's marriage as well as the paintings that derived from his domestic situation, and are made wildly angry by the high reputation he has achieved as a modern master\u2014a reputation that Mr. Schjeldahl dismisses as having \"one foot in the museum and the other in among glossy-magazine perfume ads.\"\n\nWe are given an essential clue to the nature of this critical distemper in Mr. Schjeldahl's reference to an art that is seen to fail in its duty as \"a culture-changing enterprise.\" Professor Nochlin's version of this charge is that Bonnard's later paintings mark \"a retreat from the public world.\" In other words, Bonnard's paintings of his wife at her bath fail to meet what might be described as a politically correct, social activist quotient in art. And if this is indeed the charge, then it must be said that it is at once correct and utterly ludicrous. These paintings most certainly are insufficiently _engag\u00e9_ in the political sense to qualify as social activist art. They are definitely not what we used to call \"avant-garde\" either. They have not been conceived to give us a _nouveau frisson_. Yet because both Professor Nochlin and Mr. Schjeldahl are savvy enough to know that the age of the avant-garde is long past, they are obliged to recast their indictments to conform to a roster of socio-sexual offenses more appropriate to the postmodernist culture wars than to the old arguments over \"advanced\" art.\n\nProfessor Nochlin thus speaks of dismemberment and mutilation\u2014terms more aptly applied, perhaps, to a discussion of _Les Demoiselles d'Avignon_ than to an elucidation of such masterpieces of memory and desire as _The Bath_ (1925), _The Bathroom_ (1932), and _Nude in the Bath_ (1936). Mr. Schjeldahl, too, writes as if Bonnard has violated some sort of social compact in failing to depict his wife as an aging senior citizen. \"He painted her at least 384 times,\" Mr. Schjeldahl writes, \"never with real attention to her face and invariably, even when she was 72 years old and dying, as a nubile young woman.\"\n\nAlas, it isn't only in regard to the art of painting that such criticism may be relegated to what Bonnard called \"the stupidities.\" Is it really necessary to explain to grownups like Linda Nochlin and Peter Schjeldahl that when a man and a woman have lived together in a conjugal relation over a long period of time they inevitably accumulate a reservoir of erotic memory that profoundly alters the ways in which they see and experience each other's physical presence? A gesture glimpsed in old age may evoke emotions tethered to a distant past that are far more vivid than the actualities of present experience. Couples who have lived together a long time do not meet each other as utter strangers upon awakening to the light of day. They tend to see each other through an involuntary scrim of memory, desire, familiarity, and regret. In such lives, the past lies close to the surface of all waking experience, ready at any moment to illuminate a gesture, an utterance, even a silence, with some freshly remembered simulacrum. A sudden look of delight, a cry of pain, or something as ordinary as the sound of a footstep on the stair\u2014or the ablutions of the bath\u2014may unexpectedly trigger feelings that, however brief in duration, wipe away the years to reveal a continuum of emotion, attachment, and association that is all the more precious for being unanticipated and unwilled. Does all this really have to be explained to writers claiming to be educated, sentient adults? Apparently it does, to judge from \"the stupidities\" that have been addressed to Bonnard's _Bather_ paintings on the occasion of the MOMA exhibition.\n\nSomething like this realm of experience was, in any case, central to Bonnard's project as a painter in all the later decades of his life. His word for it was \"poetry\"\u2014a poetry of feeling that for Bonnard was the poetry of life itself\u2014and he was indeed its unrivaled pictorial master in the art of the modern era. This \"poetry\" is not to be mistaken for a romantic idyll, however. There was little about the anxiety, melancholy, and despair of conjugal experience that Bonnard did not comprehend\u2014and make the subject of his paintings\u2014long before he embarked upon the late _Bather_ pictures. In the early bedroom paintings\u2014 _Man and Woman in an Interior_ (1898) and _Man and Woman_ (1900), among others\u2014we are given glimpses of a world of sexual melancholy more often associated with the paintings of Edvard Munch and the drama of Henrik Ibsen, both potent influences in the circle Bonnard frequented in this period, than with the _luxe, calme et volupt\u00e9_ of the Impressionist masters. When he does come to paint a version of the latter in _The Earthly Paradise_ (1916\u20131920), he sets the autobiographical scene in Eden just before the fateful Fall\u2014a fate Bonnard well understood to be his own in this central attachment of his long life.\n\nNor was there anything about the ravages that time inflicts on the frailty of human flesh that Bonnard did not understand and experience at firsthand. But this difficult subject was something he reserved for his own remarkable self-portraits, which are sometimes terrifying in their pitiless candor. To his obsessive depictions of the aging Marthe, however, Bonnard brought an unfailing moral delicacy that gives priority to an intensity of emotion that precludes the very possibility of the kind of pitiless candor the painter lavished on his self-portraits\u2014a moral delicacy in which time is dissolved in favor of memory and feeling. Why this moral delicacy\u2014and the exquisite painterly invention that is its exact pictorial correlative\u2014should now be regarded as a personal and moral failure, if not something worse, rather than as a triumph of aesthetic sensibility, is a matter that has more to do with the vagaries of postmodern sexual politics than with the achievements of art.\n\nReading the criticism I have been quoting here\u2014\"decadence that enervates,\" \"a masturbatory trace,\" \"exquisite rot,\" and so forth\u2014I am reminded of the fatuous way in which Marxist critics used to write about Proust. Comparisons between Proust and Bonnard have been made many times, of course, but nowhere has the comparison done more to illuminate Bonnard's painting than in Timothy Hyman's new book _Bonnard_ , which in my judgment is by far the best thing ever written about the painter. In calling our attention to what \"aligns Bonnard's vision with that of his near contemporary Marcel Proust (1871\u20131922) in his vast novel _A la recherche du temps perdu_ ,\" Mr. Hyman writes:\n\n> We know that the artist read _A la recherche_ probably before 1925, and reread it after 1940. Proust's artist-character Elstir may contain a component of Bonnard (he may be Monsieur Biche, \"Mr. Bitchy,\" at Madame Verdurin's _petit c\u00e9nacle_ , or little group), but the real identity is with the narrator, \"Marcel\"; just as Proust writes the book of himself, so Bonnard will paint the picture of himself. Bonnard's self, like Marcel's, is not that fixed and continuous character of our usual social being. What he has to record is a sudden stabbing involuntary vision of things, and of his place among them. Jean Clair writes of Bonnard being \"giddy in the astonishment of the relived moment.\" He is a myopic spectator, moving through a \"floating world,\" until arrested by a sudden focus. (The transition from blur to focus is essential to his painterly language.)\n\nAnd further:\n\n> Freud downgraded ecstasy, writing it off, along with daydreaming, as an aberration\u2014as atavistic forms of mental life, and regressions to \"primary process\" thinking. But Bonnard's first-person art could be seen as a sustained argument for the validity of such modes of being, often in association with Marthe. Her bathroom at \"Le Bosquet\" was for Bonnard what the famous \"cork-lined room\" was to Proust: a place of incarceration, and the alchemical chamber in which the base metal of everyday was transformed to gold.\n\nIt is in passages like these that Mr. Hyman's _Bonnard_ seems to give us not only a glimpse into the painter's creative imagination but also an authentic account of our own experience of the paintings themselves. It goes far in establishing Bonnard's greatness as a painter and the depth of feeling and thought upon which the paintings are based.\n\nYet the sad fact is that attacks on Bonnard are hardly a recent development, though I know of nothing in the past that quite matches the virulence of the criticisms I have mentioned here. The only thing approaching it is Picasso's now legendary hostility to Bonnard\u2014\"That's not painting, what he does\"\u2014and the attack on Bonnard in _Cahiers d'Art_ in 1947, the year of the painter's death, an attack that Picasso is believed to have inspired. This article asked the question, \"Pierre Bonnard\u2014est-il un grand peintre?\" and answered with a resounding negative judgment: \"It is evident that this reverence [for Bonnard] is shared only by people who know nothing about the grave difficulties of art and cling above all to what is facile and agreeable.\" As I wrote in 1984 on the occasion of the Bonnard exhibition at Beaubourg in Paris: \"Some version of this judgment will always, I suppose, be the view of professional avant-gardists in every generation, and it is for this reason that Bonnard has survived, at least until now, as a kind of displaced person in the art of our time.\"1\n\nMoreover, this attack in _Cahiers d'Art_ was echoed, or at least paralleled, in avant-garde circles in New York at the time. Clement Greenberg, for example, wrote about Bonnard in 1947, on the occasion of an exhibition at the Bignou Gallery, and then again in 1948 when MOMA devoted its first exhibition to the artist. These were crucial years in the development of the New York School, and it was from the perspective of this American avant-garde that Greenberg made his curiously divided assessment of Bonnard's accomplishments. For while he clearly admired Bonnard and gave his readers a close account of his painterly qualities\u2014qualities that, as he wrote in 1947, \"led Bonnard to paint more and more abstractly; the greater the attention to pigment and brushstroke the less becomes the concern with the original idea of the subject in nature\"\u2014he couldn't quite forgive Bonnard for retaining his attachment to the object.\n\n> Here . . . is a way of approaching abstract painting that makes a detour around Cubism and yet arrives at the same place in the end [he wrote in 1947]. But Bonnard never abandons the object, and never will\u2014nor does he violate it as Picasso has done, while still retaining it. He holds on to the third dimension more tenaciously. He may simplify nature but he does not reorganize it with respect to anything except color; and so the world he shows us disorients no one familiar with that of Monet or Renoir.\n\nGreenberg understood that what Bonnard \"seems to want is a big _flat_ picture with the massiveness and weight of Tintoretto or Veronese,\" and admired what he called Bonnard's \"audacity\" in this respect. Yet because the artist was still found wanting by the avant-garde standards of that particular moment in American painting, the final judgment is highly equivocal.\n\n> But the intimacy of Bonnard's art, its concentration on gentle pleasures, and the fact that it smells permanently of the fashions of 1900\u20131914, expressing as it does the desire of the French middle classes to make history stop and stand still at 1912, and leave them undisturbed in the enjoyment of the modest but refined amenities that the Third Republic had permitted them to accumulate\u2014all this should not mislead us into thinking that [Bonnard] lacks ambition as a painter.\n\nWhen Greenberg returned to the subject in 1948, his assessment was equally equivocal:\n\n> The question why an artist who painted as consummately as Bonnard should have failed quite to attain major quality is perhaps best explained by a certain aspect of French tradition. . . . He experimented within the limits set for him, but did not try very hard to break through them.\n\nAll of this is on a much higher level of critical discourse than what we have been getting in response to the current Bonnard exhibition at MOMA, but it is also a reminder of the degree to which judgments of Bonnard's achievements have been\u2014and continue to be\u2014hostage to whatever mix of avant-garde sentiment and radical political ideology happens to be rampant. The aesthete in Greenberg clearly responded with enthusiasm to Bonnard's painterly qualities, yet a certain residue of Marxist sentiment\u2014by the late 1940s, Greenberg was speaking of himself as a \"disabused Marxist\"\u2014combined with his vocation as a champion of the American avant-garde to render Bonnard finally unacceptable.\n\nOutside the circles of that avant-garde, however, Bonnard became something of a hero and a model for a younger generation of painters\u2014for Nell Blaine and Fairfield Porter and the painters upon whose work they were to exert an influence. And even in the circle of the Abstract Expressionist painters, at least one painter\u2014Mark Rothko\u2014found in that 1947 exhibition at the Bignou Gallery the foundation of the mode of color abstraction that became his signature style. In the current Rothko exhibition at the Whitney Museum, you can see the results of his encounter with Bonnard in 1946 \u20131947, and it would be a brave critic who pronounced Rothko a greater painter than Bonnard. For notwithstanding the \"stupidities\" that the current Bonnard exhibition has met with in New York, I believe that the artist's greatness is now beyond dispute, or ought to be. Still, Bonnard's own pessimism on this question cannot be ignored, either. \"Speaking, when you have something to say, is like looking,\" he said. \"But who looks? If people could see, and see properly, and see whole, they would all be painters. And it's because people have no idea how to look that they hardly ever understand.\"\n\n[1998]\n\nFootnote\n\n1. See \"Bonnard in Paris\" in _The New Criterion_ , May 1984.\n\n## A GALLERY CHRONICLE\n\n### Mapplethorpe at the Whitney: Big, Glossy, Offensive\n\n* * *\n\nIn the Robert Mapplethorpe retrospective that is currently installed at the Whitney Museum of American Art, there is so much that is highly problematical and so much that is simply offensive\u2014so much, indeed, that many people will still want to consider pornographic\u2014that we have no choice but to conclude that it was precisely its air of scandal and provocation that commended the work to the museum's staff and made it seem not only a suitable but an irresistible subject for a large, glossy, sensational exhibition.\n\nThis impression\u2014that it is Mr. Mapplethorpe's subject matter that is primarily, though not solely, responsible for this large-scale show\u2014is reinforced, moreover, by the fact that he is in many other respects a photographer who contributes little that is new to the language of photography. The formal conventions within which he works clearly owe much to the style that Edward Weston and his many followers perfected years before Mr. Mapplethorpe was born, and this is not a style that in its original form could be expected to elicit much interest nowadays at the Whitney.\n\nWhat Mr. Mapplethorpe has made of this style is certainly remarkable. While adhering to and intensifying the formalist conventions established by Weston, he has also transformed them into a vehicle for a highly specialized sexual vision. (Even this transformation was anticipated, of course, in Weston's many photographs of female nudes and in the famous picture of his son's naked torso.) Not every one of Mr. Mapplethorpe's pictures has a sexual subject, to be sure, but a great many of them do, and the exhibition as a whole has a very distinct erotic flavor. It encloses us in a realm where a certain kind of homoerotic taste is allied with and aggrandized by a particular kind of purist aestheticism. If the aestheticism remains tethered to the Weston style, the taste derives its imperatives from something quite different\u2014a homosexual subculture in which extremes both in physical endowment and in the trappings and rituals of sexual practices generally considered perverse are greatly idealized.\n\nIt is this taste for the idealization of the physical and erotic extreme that gives Mr. Mapplethorpe's work its characteristic emotional charge. What his admirers most enthusiastically respond to is the exceptional candor as well as the exceptional craft he brings to this project of idealization. Especially where he concentrates his attention on subjects heretofore considered \"forbidden\"\u2014subjects, that is, usually confined to porno publications\u2014Mr. Mapplethorpe provides his adoring public with the kind of gratifying _frisson_ that, happily for the rest of us, it seldom encounters elsewhere on the art scene.\n\nRichard Marshall, the curator of the Whitney show, correctly identifies one of the most important of these subjects as the \"sadomasochistic theme,\" and duly notes\u2014quite as if he were listing the components of a landscape or a still life\u2014\"the elements of threat, danger, pleasure and pain that often underlie [Mr. Mapplethorpe's] imagery.\" I shall spare the reader any attempt to describe the really repulsive pictures on this sadomasochistic theme, but there are several things to be said about them. One is that they generally don't amount to much as examples of the art of photography, and it is really not as outstanding works of art that they are included in the Whitney show. No doubt the subject so absorbed the photographer that he had little time to embellish the pictures on this theme with his usual repertory of artistic effects. The most respectable of them\u2014the double portrait of \"Brian Ridley and Lyle Heeter\"\u2014is nothing but a second-rate imitation of Diane Arbus.\n\nThen there is the problematic nature of the way these pictures are discussed in the exhibition catalogue\u2014one of the most elegant publications, by the way, that the Whitney has produced in some years. For Mr. Marshall, who acknowledges that Mr. Mapplethorpe made these pictures \"not as a voyeur but as an advocate\" and \"sympathetic participant,\" and claims that they \"confirmed his position as an artist of strength, confidence and talent,\" the SM photographs simply represent a happy confluence of the \"prevalent aesthetic issues of the 1970's\" and \"the dominant social and sexual issues of the decade.\"\n\nOn the other hand, Ingrid Sischy, another contributor to the catalogue, writes as if the votaries of sadomasochistic sex constituted an oppressed ethnic minority in need of a UN resolution guaranteeing their total and absolute freedom to do whatever they desire. Hers is surely one of the oddest documents ever published in an art museum catalogue. It is a cry of pain over the fate of sexual freedom in the age of AIDS\u2014a subject eminently worthy of examination but not one that belongs to the province of an art museum.\n\nThe best of Mr. Mapplethorpe's pictures are those that are concentrated on the bodies of nude figures\u2014mostly male figures. Some of these\u2014the frontal head and shoulders picture of \"Ken Moody,\" for example, and the \"Lisa Lyon\" pictures\u2014bear a remarkable resemblance to certain sculptures of Gaston Lachaise. By and large, however, these photographic celebrations of exemplary physiques, with their theatrical lighting and arty poses, combine the familiar conventions of fashion photography\u2014especially that of Irving Penn and Richard Avedon\u2014with the formalist conventions of the Weston style.\n\nWhat Mr. Mapplethorpe adds to this expertly crafted pastiche\u2014it constitutes, I suppose, his artistic signature\u2014is an impulse that Richard Howard, still another contributor to the catalogue, describes as the wish \"to restore aesthetic dignity to the [male] genitals.\" Mr. Howard is probably right in observing that in Mr. Mapplethorpe's photographs \"the male genitals are often presented . . . as surrogates for the face.\" Less admiring observers might conclude that this, in effect, reduces the bodies of Mr. Mapplethorpe's subjects to the status of impersonal sexual objects, but this isn't the way Mr. Howard sees the matter. He concludes his homage to Mr. Mapplethorpe by comparing his work to \"the sonatas of Scarlatti [and] the last paintings of Mondrian\"\u2014a comparison that would certainly have horrified Mondrian, and that I find utterly ridiculous.\n\nBut it is not inappropriate for Mr. Howard's extravagant praise to adorn the catalogue of the Mapplethorpe exhibition, for it is just this sort of hype and special pleading that has made it possible for the Whitney to give us this bizarre exhibition. Alas, the work itself does not, in my opinion, even begin to justify an exhibition on such a major scale. Mr. Mapplethorpe is unquestionably an accomplished photographer, but there remains much in his work that is not only derivative but second-rate. And it doesn't lack, either, the kind of vulgarity and ostentation\u2014I am speaking of aesthetic vulgarity\u2014that is only the other side of a pervasive sentimentality.\n\nAs for Mr. Mapplethorpe's famous framing devices\u2014the use he makes of fabric, mirrors, etc., in order to turn his pictures into one-of-a-kind art objects\u2014all one can say is that they represent a tacit recognition that the photographic images are often not in themselves of sufficient power or interest to stand on their own. If you have a taste for campy minimalism, you will probably love these devices. I find them pretty trashy.\n\nThe truth is, there are about a dozen pictures in this show that have some real artistic quality\u2014but you couldn't mount a major retrospective on an accomplishment of that size. And so it has been left to the \"forbidden\" subject matter to carry the burden of the show. There is no denying that the result is a sensational exhibition\u2014but the sensation in question doesn't add much to the experience of art.\n\n[1988]\n\n### Kiefer at MOMA\n\n* * *\n\nNo exhibition of an artist of the 1980s had been more eagerly awaited than the Anselm Kiefer retrospective that opened at the Art Institute of Chicago at the end of 1987. And on that occasion the Kiefer retrospective was, as expected, loudly acclaimed as an epochal event. This remarkable German painter, born in 1945, was heralded as the saint and savior of art in our time. Indeed, he was celebrated as something more than an artist. _Time_ magazine spoke for many when its art critic announced that Mr. Kiefer was not only \"the best painter of his generation\" but that his work constituted \"a victory for the moral imagination.\"\n\nYet this excessive judgment has very quickly receded into history, and is unlikely to be reaffirmed with the same unequivocal enthusiasm now that the traveling Kiefer retrospective has come to its last stop at the Museum of Modern Art. It is my impression, anyway, that feeling is no longer running quite so high in Mr. Kiefer's favor\u2014that it has now become somewhat compromised by elements of doubt and disappointment. Last year the question seemed to be: What makes Anselm Kiefer so great? Now, less than a year later, the question one hears is: What was it that made Anselm Kiefer seem so important? It would be a mistake to conclude that the artist's spectacular reputation is already pass\u00e9. It isn't. Mr. Kiefer continues to command immense respect and even adulation. Yet something has undeniably changed. A significant shift has occurred in the way his work is perceived, and it has occurred in a remarkably short time. How are we to account for it?\n\nSpeaking for myself, I can say that I, too, was immensely impressed and moved the first time I saw a show of Anselm Kiefer's paintings at the Mary Boone Gallery. The large, overscale images of landscape and architecture, with their melancholy allusions to Germany history\u2014and most particularly German history in the Nazi period\u2014were like nothing else to be seen in contemporary painting. The dour palette\u2014all greys, blacks, and earth colors\u2014seemed in itself a sober reminder of the death and destruction we associate with the Hitler era, yet there was also in the work a very clear attempt to evoke the kind of mythic grandeur we had come to associate with the classics of German romanticism. The artist was clearly attempting to deal with the tragedy and horror of the cultural tradition into which he had been born, and born in the very year that Germany itself had been reduced to the kind of rubble that was so movingly evoked in the physical quality of the paintings on view.\n\nI returned to that exhibition several times, and on each of those visits I found what appeared to be a permanent company of young artists in attendance. They were clearly in a state of enchantment and awe over the artist's work, and not only, I believe, because of the unusual element of pictorial drama to be found in it but also because of the sense of moral gravity that it conveyed. I have to say that I was never again able to experience anything like that initial response to Mr. Kiefer's art. Yet I continued to admire it, and probably for some of the same reasons those awestruck artists did. In an art world that was rapidly becoming ever more vulgar, commercialized, and cynical, Anselm Kiefer seemed to be in possession of a vision that was infinitely more serious and thoughtful than anything to be found among his immediate contemporaries. When, a year or two later, I found myself serving on an advisory committee for the 1985 Carnegie International Exhibition, I did not hesitate to award Mr. Kiefer one of the top prizes, and in the context of that exhibition I still believe that he had earned it.\n\nWhen I went to Chicago in the fall of 1987 to see the huge retrospective that has now come to MOMA, it was inevitably with mixed feelings about what I would find. I still counted myself among the Kiefer admirers, but I wasn't quite prepared for the way so many other members of the press\u2014and I mean _in the exhibition_ and not only in what they wrote\u2014responded to the event. To describe that response as excessive and irrational hero-worship would scarcely begin to give an account of the kind of adulation that was so openly displayed on that occasion. There was an almost religious quality to the response. It was as if the press, whose members are normally so skeptical and impious in their general outlook on art and life, had found in Mr. Kiefer a redeemer in whose works it could at last truly believe.\n\nThis response was to some extent abetted by the highly dramatic installation that the show was accorded in Chicago. At MOMA, with its rather prosaic gallery spaces and bright lights, there is nothing akin to the cathedral-like heights and twilight-of-the-gods atmosphere that made the Art Institute a perfect setting for Mr. Kiefer's dark, oversize paintings. Under MOMA's bright lights, some of the magic has evaporated. The work simply doesn't cast the same spell. But for myself, the spell was already receding, if not completely gone, in Chicago. What struck me most forcefully on that encounter with Mr. Kiefer's work\u2014one that lasted several hours\u2014was the problematical nature of the artistic enterprise the artist was engaged in. And it is this question\u2014the question of whether Mr. Kiefer has actually succeeded in his attempt to make the tragedy of history an integral and indispensable element in the art of painting\u2014that must now be faced in any serious discussion of his achievement.\n\nOur answer to this question is likely to depend on the degree to which we are inclined to accord to any artist's subject matter a radical priority over the aesthetic means employed in its expression. The question is rendered all the more moot in Mr. Kiefer's case because of the extreme difficulty many of us are likely to have in identifying his subject matter or motifs with any exactitude. I hasten to add that in the book-length catalogue that Mark Rosenthal, the curator of twentieth-century art at the Philadelphia Museum of Art, has written to accompany the exhibition we are provided with an expert guide to virtually every nook and cranny of Mr. Kiefer's complex iconography. Yet it is one thing to read about the artist's symbols and allusions in a book and quite another to experience those symbols and allusions _as art_. Purely as a pictorial performer Anselm Kiefer remains a very impressive talent, but the longer I look at his paintings the more I come to feel that his subject matter has a way of dissolving\u2014for the spectator, if not for the artist himself\u2014into a generalized gloom that is not altogether untouched by something that can only be described as sentimentality.\n\nMr. Kiefer's sentimentality is itself a complicated affair, of course. It is a heavy, portentous sentimentality, a very Germanic and romantic sentimentality\u2014but it is sentimentality, all the same. You can feel it with a special acuteness in one of the big, recent pictures that wasn't included in the Chicago version of the retrospective\u2014a huge grey painting of many panels called _Women of the Revolution_ (1986 \u20131987). There are no female figures explicitly depicted in this painting. But its surface is covered with a great many small picture frames inside of which are dried flowers or leaves or other materials intended to evoke feelings of loss, memory, and death. The names of the departed women are inscribed on the picture surface to commemorate their spirits. I have to say that the sheer pathos of this picture came as something of a shock to me, for I hadn't expected Mr. Kiefer to go quite this far in surrendering to the sentimentality that elsewhere in his work remains under a tighter rein.\n\nAnselm Kiefer's best paintings remain his visionary landscapes, and some of them\u2014 _The Mastersingers_ (1981\u20131982), _Nuremberg_ (1982) and the earlier _Cockchafer Fly_ (1974)\u2014are very fine indeed. But anyone who now comes to this retrospective\u2014which was jointly organized by the Art Institute of Chicago and the Philadelphia Museum of Art\u2014in search of an artist-redeemer is in for a disappointment.\n\n[1988]\n\n### Frankenthaler at the Whitney\n\n* * *\n\nIt is no secret in the art world that abstract painting has fallen on hard times. I do not mean economic hard times. The runaway prices commanded by Jackson Pollock, among the dead, and Frank Stella, among the living, show no sign of declining, as far as I can tell, and many other big names in the field still enjoy immense prestige. But economic prosperity should not be mistaken for aesthetic vitality. There is a crisis in abstract painting all the same\u2014an artistic crisis, which stems from a loss of faith in the aesthetic basis of art. You cannot have a vital movement in abstract painting without a strong belief in the efficacy of the aesthetic impulse, and that is what we seem for the moment to have lost. Not all of the troubles that beset the current art scene can be blamed on money matters. Ideas still count for something, too\u2014and it is the very idea of abstraction in painting that is now in trouble.\n\nThe 1980s have witnessed a widespread revolt against abstraction, which many younger artists\u2014not to mention critics, curators, and collectors\u2014have come to identify with a modernism they believe to be moribund. I think they are wrong about this, woefully wrong, but a lot of people have nonetheless come to believe it. And in art, belief matters a lot.\n\nBelieving modernism to be moribund, a new generation of artists has put its faith in that curious amalgam of fantasy and philistinism that goes by the name of postmodernism. Whether this proves to be an enduring tragedy for our culture, or only a sterile and sometimes comic interlude, no one can yet say. What we can be certain about, however, is that there is no place for abstraction in this postmodernist scenario. Abstraction requires an absolute conviction in the power of pictorial form to speak directly to the mind and the emotions without recourse to adventitious \"subjects\" or easily identified symbols. Toward such a conviction the postmodernist attitude is one of mockery and derision. Rejecting the aesthetic impulse as an illusion and originality in art as a myth, the postmodernist camp\u2014or should one say, Camp postmodernism?\u2014relies instead on parody, allegory, and what is called \"appropriation.\" It relies, in other words, on the recycling and repackaging of received images. Whereas it had been the abiding ambition of modernist art to \"make it new,\" in postmodernism\u2014as Calvin Tomkins wrote a few years ago\u2014\"nothing here is new, only 'neo.'\"\n\nIn vivid contrast to the aesthetically degraded atmosphere that this postmodernist imperative has imposed upon the current art scene, the Helen Frankenthaler exhibition that has now come to the Museum of Modern Art recalls us to a very different attitude toward art\u2014a point of view quintessentially modernist in its aesthetic ambition. Ms. Frankenthaler is not only one of our best abstract painters; she has been one of our best and purest painters of any persuasion for a remarkably long time. She entered upon her professional career as a painter at an early age, having her first solo exhibition at the Tibor de Nagy Gallery in New York in November 1944, just before her twenty-third birthday. This made her the youngest member of the so-called second generation of the New York School. At twenty-four, in her second solo show, she exhibited _Mountains and Sea_ , a painting that was to become a landmark in American art\u2014a classic of abstract painting that did much to determine the course of her own artistic development and, owing to its influence, that of abstract painting itself.\n\n_Mountains and Sea_ (1952) is the first of the forty paintings we see in the current MOMA exhibition, which has been organized by E. A. Carmean, Jr., the director of the Modern Art Museum of Fort Worth, Texas. The show is called a \"paintings retrospective,\" which means that it is not a comprehensive survey of everything the artist has done in the period under review (1952\u20131988). As Ms. Frankenthaler has been a prolific as well as a highly original artist, a true retrospective would require a much larger exhibition than the present one. Yet the decision to limit this show to a carefully selected succession of very individual pictures seems right.\n\nI use the word \"individual\" advisedly, for Ms. Frankenthaler is not the kind of abstract painter who replicates a carefully designed trademark image from picture to picture. She is anything but an abstract serialist, remaking the same picture over and over again in cold blood, as it were. Each of her paintings is a separate conception, and it requires a certain space\u2014a mental as well as a physical space\u2014to be properly seen and felt. It is, moreover, painting in which the emotions are directly engaged. Ms. Frankenthaler works in what is essentially a lyric mode of abstraction, where the quality of feeling governing virtually every aspect of the painting's execution is crucial to the pictorial outcome. Paintings of this kind are not ideally seen in exhausting numbers, for each of them obliges the spectator to begin, so to speak, at the beginning with each encounter. Which is why I think this exhibition is the right size for the kind of aesthetic experience it offers.\n\nWriting about her work some years ago, I said that Ms. Frankenthaler was the painter who had put Abstract Expressionism on a diet. The wonder is that she was able to effect this radical change in the Abstract Expressionist aesthetic so early on, and at a moment when challenges to its authority were not exactly welcome. By thinning out her paint to the consistency of a wash and then pouring it directly onto the canvas, instead of applying it with a brush, the artist eliminated some of the heaviness and opacity\u2014those weighty encrustations of pigment\u2014that had already come to characterize certain aspects of the Abstract Expressionist school.\n\nThis was an innovation that allowed her to bring to large-scale abstract painting something of the fluency and transparency of watercolor. What resulted\u2014first in _Mountains and Sea_ and thereafter in such pictures as _Eden_ (1956), _Round Trip_ (1957), and _Winter Hunt_ (1958)\u2014still had something of the look of Abstract Expressionist painting. Its forms remained basically gestural and improvised, and there was a good deal of drip and splatter. Yet the whole quality of feeling was different\u2014lighter, more lyrical, less earthbound. Something genuinely new had been added to the history of abstract painting, and it straightaway had an effect on that history, leading directly to what came to be known as Color-field painting.\n\nI once half-jokingly described _Mountains and Sea_ as \"a sort of _Desmoiselles d'Avignon_ of the Color-field school,\" and I see that Mr. Carmean quotes this remark in his catalogue for the present exhibition. What I had in mind, as I expected my colleagues to recognize, was An-dr\u00e9 Breton's characterization of Joan Mir\u00f3's great painting, _The Birth of the World_ (1925), as \"the _Desmoiselles d'Avignon_ of the _informel_ ,\" the _informel_ school of abstract painting being a sort of French counterpart to the Abstract Expressionist school in New York. Thinking abut it now, I see that there is even more of a connection between Mir\u00f3's painting and Ms. Frankenthaler's than I realized at the time. Mir\u00f3, after all, had similarly put the rhetoric of Surrealist painting on a diet, and by remarkably similar means\u2014though there is no way that Ms. Frankenthaler could have seen _The Birth of the World_ at the time she painted _Mountains and Sea_. Although Mir\u00f3's masterpiece now hangs in the permanent collection of MOMA, it had never been shown in public before 1968.\n\nIf, in the 1950s, Ms. Frankenthaler succeeded in putting Abstract Expressionism on this radical diet, she went on in the 1960s to produce a remarkable series of masterpieces in the Color-field style she had invented. The current exhibition has an especially rich representation of pictures from this period, a total of thirteen paintings, beginning with _Swan Lake I_ (1961), that will be a revelation to those who have not had an opportunity to see them before. Neither then, nor since, has Ms. Frankenthaler succumbed to the temptation to repeat her successes. While remaining loyal to certain ideas, she has never allowed herself to become hostage to a method. I cannot honestly say that the latest painting in the exhibition\u2014the very large _Casanova_ (1988)\u2014is one of my favorites, but I nonetheless admire the unflagging invention that she continues to bring to every picture. And the seven pictures selected to represent the 1980s in this exhibition give us only a cursory account of what she has been up to in this very productive decade.\n\nOne leaves this exhibition feeling more convinced than ever that Ms. Frankenthaler is a major artist, not only in the history of American painting or abstract painting but in the very rich history of modernist art itself. But one also leaves this show with a sad sense of how isolated painting of this kind and quality has become. The ideals that nourished it\u2014the ideals of a modernism that gave an unquestioned priority to the authenticity of aesthetic experience over every other consideration in art\u2014no longer occupy the place they once did. The fashions that have taken the place of those ideals have reduced whole areas of the art world\u2014and this includes large parts of our museums\u2014to a vulgarity and meretriciousness without parallel since the worst days of the official nineteenth-century salons. It is no wonder that abstract painting has suffered a crisis in an artistic environment that is so debased.\n\nOn a happier note, it should be observed that Ms. Frankenthaler has been unusually lucky in her critics and commentators. On this occasion, in addition to the fine catalogue written by Mr. Carmean, we have also been given a splendid new monograph on the artist's work by John Elderfield, the curator at MOMA responsible for the very beautiful installation of this show. Mr. Elderfield's has the additional interest of providing us with a fairly detailed account of the New York scene in which Ms. Frankenthaler came of age as an artist\u2014one of the best accounts of New York in the fifties I have read.\n\n[1989]\n\n### The Citadel of Modernism Falls to Deconstructivists\n\n* * *\n\nPity the poor curators at the Museum of Modern Art! They preside over one of the greatest collections of modernist art in the world\u2014perhaps _the_ greatest\u2014yet nowadays it seems to give them nothing but nightmares. Indeed, a kind of collective nightmare seems to have descended upon all the curatorial departments at MOMA as one after another the curators struggle and compete to free themselves of the onus of\u2014well, of modernism.\n\nAt modernism's once-proud citadel, modernism is now the big bogyman\u2014or should one say, in keeping with the new mind-set, bogyper-son? To be suspected of a secret allegiance to modernism is now, for every ambitious curator at the museum, to be deeply compromised. The aesthetically correct thing to be associated with is, of course, postmodernism. That is the new orthodoxy to which all curatorial assent is now directed. That is the new party line at MOMA, and meeting it with the requisite dedication and enthusiasm inevitably entails a mighty effort of deconstruction. Hence the nightmares.\n\nWe were given a glimpse of what this effort means in practice when, as part of last fall's _Dislocations_ show at MOMA, we were offered in lieu of certain paintings in the permanent collection\u2014Seurat, Modigliani, et al.\u2014some idiotic words _about_ these pictures written on the walls where the pictures themselves normally hang. Earlier on, an even more ambitious example of this deconstructivist program was given to the public when the late Scott Burton was allowed to dismantle certain Brancusi sculptures in order to exhibit their pedestals as autonomous works of art\u2014an idea that would have certainly given Brancusi some nightmares.\n\nThe truth is, wherever one turns at MOMA these days, there is evidence of this impulse\u2014this itch to deconstruct and desecrate and mock the legacy of the modernist art that is the museum's principal reason for existence. Sometimes this impulse takes the form of a stupid little joke\u2014e.g., hanging a self-portrait by Frida Kahlo next to a framed mirror so that visitors to the permanent collection can be made to feel that their reflected images in the mirror are just as important as the artist's image of herself. This is the kind of thing that now passes for curatorial initiative at MOMA.\n\nMore and more often, however, this deconstructivist impulse takes the form of big stupid exhibitions and their catalogues. The latest of these shows now occupies a good deal of space on three floors of the museum under the title _Allegories of Modernism_. Organized by Bernice Rose, senior curator of MOMA's Department of Drawing, it is a show that cheerfully bids farewell to the whole idea of drawing and substitutes in its place the usual inventory of postmodernist inanity. There are paintings here that aren't drawings. There are objects here that aren't drawings. There are photographs, comic strips, and other printed materials that aren't drawings. There are even some drawings here that aren't really drawings but badly failed attempts to make something that may or may not resemble a drawing. As a drawing exhibition, _Allegories of Modernism_ is a fraud and a shambles. As a guide to the new orthodoxy at MOMA, however, it is highly instructive, and there is this, too, to be said for the show: It doesn't assault the ears with the kind of noise that had to be endured in the _Dislocations_ exhibition. We are allowed to suffer through this inventory of failed art in silence.\n\nFor this overscale massacre of the art of drawing Ms. Rose has rounded up all the usual suspects. Jean-Michel Basquiat, Jonathan Borofsky, Mike Kelley, Jenny Holzer, Julian Schnabel, David Salle, Nancy Spero, Francesco Clemente, Sherrie Levine, Doug and Mike Starn are here along with sundry fashionable Germans and the ubiquitous Bruce Nauman, who seems to be every curator's favorite these days at MOMA. At times, it seems, Ms. Rose has gone to great effort to single out precisely those artists who are manifestly devoid of any real talent for drawing\u2014Basquiat and Schnabel, for example\u2014or to bring us items that are wholly removed from drawing itself\u2014Mr. Nauman's _Model for Animal Pyramid II_ , which is a rather silly collage made of cut-and-taped photographs. The more we look at the work that Ms. Rose has assembled for this _Allegories of Modernism_ show, the more we come to realize that it should really have been called _Allegories of Drawing_ , for the show as a whole does tell a sad tale about what has happened to the very idea of drawing under the authority of postmodernist ideology.\n\nTo understand the real meaning of this show and what it signifies for the museum, however, it is necessary to consult the catalogue, and that is itself a very melancholy task. Ms. Rose is no newcomer to MOMA. She has been around for some years, and it is clear that the mental labor involved in the need to effect her conversion to the new postmodernist program has not been easy for her. For one thing, this postmodernist orthodoxy requires of a curator the mastery of a new language. The old lingo, you see, is tainted\u2014just like modernism itself. Its clarity is now thought to be sort of repressive, so you are obliged to muddle up the language with a lot of jargon and other obstacles to clear thinking in order to qualify as a fully emancipated postmodernist.\n\nFor this purpose Ms. Rose has gone to a good source for her postmodernist models\u2014to the contributors to _October_ , a journal that can always be counted upon to be reader-proof, so to speak. Some of us can remember a time when Ms. Rose wrote perfectly intelligible prose, but now under the imperatives of the postmodernist program she writes things like this:\n\n\"The 'iconography,' or lexicon, of postmodernism is the abstraction and reworking\u2014the 'personification'\u2014of modernist style itself, so that style is read through style, with the body of modernism serving as the original text. Remembrance of the past is iconographically integral to the new language age of art. Within this general area of agreement, there is an enormous range of play inherent in the new mode, and drawing, with its enormous potential for overwriting, has become a primary vehicle for this postmodern allegory.\"\n\nSomehow as we read along in this catalogue, though we may doubt that \"overwriting\" has much to do with the art of drawing, it does seem to have something to do with Ms. Rose's new prose style. Where it will all end\u2014Ms. Rose's text, I mean\u2014we can more or less divine, too, and she does not disappoint us. In the last paragraphs she brings on the big guns\u2014Terry Eagleton, the English Marxist literary theorist, and Theodor Adorno, the German Marxist philosopher. This, I suspect, is foreign intellectual territory to Ms. Rose, but she has tried hard to learn her lessons well. And so on page 113 of this (postmodernist?) parody\u2014or allegory?\u2014of an exhibition catalogue, she leaves the last word to Mr. Adorno, believing that \"out of his negativity,\" as she writes, \"comes affirmation,\" a neat postmodernist formation. \"Perspectives must be fashioned that displace and estrange the world, reveal it to be, with its rifts and crevices, as indigent and distorted as it will appear one day in the messianic light\": this is the message that Mr. Adorno, a Marxist lapsing into the language of Christian theology, has for connoisseurs of postmodernist drawing.\n\nIt would all be very comical if it did not portend a ghastly future for the Museum of Modern Art. We were given a warning of things to come a couple of years ago when Kirk Varnedoe, the then newly installed chairman of MOMA's Department of Painting and Sculpture, published _A Fine Disregard_ , his deconstructivist assault on the thought of Alfred H. Barr Jr., the founding director of MOMA, and Clement Greenberg, our leading modernist critic. And since then we have seen one department after another fall into line with the new orthodoxy. Peter Galassi, the new director of photography at MOMA, has already announced a postmodernist extravaganza to come\u2014one that will revise the definition of photography to conform to postmodernist doctrine. And Ms. Rose has meanwhile rushed in to dismantle the definition of drawing along the same lines. She has done her job well in _Allegories of Modernism_ , for as the result of this show just about everything in the world can now be considered a \"drawing\"\u2014or maybe a \"text,\" since according to postmodernist doctrine there is no real difference between a drawing and a text.\n\nI could not help but wonder what Paul Sachs, the great Harvard connoisseur whose name used to adorn the Department of Drawings galleries at MOMA, would have thought of this debacle. Appropriately, however, his name was removed from the Drawings galleries in the 1980s\u2014a more symbolic act, it now seems, than we could have appreciated at the time.\n\nIn the early decades of the Museum of Modern Art, the enemies of modernism were all on the outside, shaking their fists and lavishing their anathemas. Nowadays they occupy the curatorial offices, quietly deconstructing from within. This is the real allegory of modernism in our time.\n\n[1992]\n\n### Fairfield Porter, a Master Long Ignored\n\n* * *\n\nAt the time of his death in 1975 at the age of sixty-eight, Fairfield Porter had been a conspicuous and well-connected figure on the New York art scene for many years. Admired and influential as a painter, he also commanded great respect as an art critic, first as a writer for _Art News_ and then as the critic for _The Nation_. Yet, though Porter's paintings were often praised by painters, poets, and other critics, there was a distinct reluctance in New York art circles to grant him the status of a master, which was what in the end he certainly was.\n\nThe museums paid him no attention, and his work remained unknown to the public that knew\u2014or thought it knew\u2014all about Jackson Pollock, Jasper Johns, and Andy Warhol. He was never a big name. Among the curators who were then responsible for the making and breaking of contemporary reputations, Porter's paintings were often regarded\u2014to the extent that they were regarded at all\u2014as too agreeable to be important. There was the story, which I know to be true, of the Whitney curator who had to be dragged to see a private collection of Porter's paintings. He had put off the dreaded visit for months even though the collection was hanging in an apartment only a block or two away from the museum. When at last he finally had a look at them, he pompously announced to the lucky collector of those paintings that \"This is too tame for us.\" That's the way it often was for Porter.\n\nI am reminded of this curious history by the excellent exhibition, called simply _Porter Paintings_ , that has now been organized at Hunter College, and reminded, too, of some of my own experiences of that history. When I first wrote about Porter's work in the _Times_ , in the late sixties, one of my colleagues on the paper\u2014a critic in another field who lived in a state of acute anxiety lest he be caught missing out on the latest trend\u2014took me aside to warn me that I would ruin my reputation if I continued to write about such boring, conventional stuff.\n\nThen there was the day, a few years later, when my wife and I took Henri Cartier-Bresson to see an exhibition of Porter's paintings at the Hirschl & Adler Galleries. We were having lunch, and Henri was returning to Paris at the end of the day. He had time to see only one exhibition that afternoon, and he asked us what he should see. Simultaneously, we both instantly said: \"Fairfield Porter!\" The name, of course, was unknown to him. As soon as he began to look at the paintings, however, Henri was beside himself with excitement\u2014and Henri Cartier-Bresson, I should add, is a man difficult to please in artistic matters. As it happened, an old friend of Henri's\u2014the late Monroe Wheeler, for decades one of the senior members of the Museum of Modern Art hierarchy\u2014walked in while we were looking at the Porters. In his characteristically abrupt manner, Henri rushed over to Wheeler and, without any preliminary greeting, said to him: \"Monroe, what kind of a friend are you? All these years I have been coming to New York, and you never told me about Fairfield Porter!\" But then, I cannot recall ever seeing a Porter painting hanging in the MOMA galleries.\n\nWhen the time came in the early eighties to mount a Porter retrospective, it was well known that no New York museum wanted it. The show was organized by Kenworth Moffett at the Museum of Fine Arts in Boston, and took even that museum by surprise by turning out to be one of the most popular shows in its history. When the Whitney was finally persuaded\u2014or coerced, really\u2014into taking a severely abridged version of the Boston retrospective, which was enormous, the staff hung the paintings against brightly colored walls that nearly killed them\u2014an act of revenge, I suppose, for the humiliation of having to handle such boring, conventional stuff. Or did they feel that the paintings needed a little pepping up? Who knows? To this day New York has never seen a comprehensive Porter retrospective.\n\nThe show that William C. Agee has organized at Hunter College with three of his students\u2014Malama Maron-Bersin, Sara Rosenfeld, and Michele White\u2014is not, of course, intended to be a retrospective. The 23 paintings in the show are all drawn from the magnificent gift of 250 works that Porter's estate made in 1979 to the Parrish Art Museum in Southampton, where the artist had lived for many years. Still, as the organizers of the show point out in the splendid catalogue, the result gives us \"an overview of [Porter's] art. There is at least one painting from each decade of his working career, with the first dating from 1938, and the last from 1963, two years before his death.\" It was from around 1951 that Porter really got going as a painter, however, and the bulk of the show is happily concentrated on the period from 1952 to 1973.\n\nOne of the things that made it difficult for Porter's contemporaries to place him, or to place themselves in relation to Porter's painting, was the fact that while he painted in a very accessible representational style, he belonged in fact to the Abstract Expressionist generation. He readily acknowledged the influence of Willem de Kooning, whom he met in the late thirties and whose work he bought long before the art world even knew the name. Yet Porter himself always looked a little askance at the idea of abstraction. The other great influence on his painting was Vuillard, whose paintings he saw at the Art Institute of Chicago in 1938.\"I looked at the Vuillards,\" Porter said, \"and thought that maybe they were just a revelation of the obvious, and why did one think of doing anything else when it was so natural to do that.\"\n\nYet more than a decade elapsed before Porter could fully realize his own version of this \"revelation of the obvious,\" and it was de Kooning's handling of the medium in his paintings of the late forties and early fifties that made it possible, oddly enough, for Porter to follow through on what he had seen in Vuillard years before. The influence of Vuillard is easily seen\u2014from the wonderful painting of _Laurence Typing_ (1952) to the more problematic pair of paintings called _Lizzie, Guitar, and Christmas Tree_ (1973)\u2014but throughout Porter's later painting de Kooning nonetheless remained a presence. Or, as the authors of the catalogue essay put it, \"Porter's procedure was to fuse Vuillard's world of direct observation with the broad and physical paint handling of de Kooning and Abstract Expressionism.\"\n\nThere are some outstanding paintings in this show\u2014the two portraits of the artist's wife Anne (1965 and 1967) are both amazing pictures, as indeed are most of the portraits here. I would like to have seen some of the small flower paintings included in the show, for among them are some of the best things Porter painted. The landscape called _Penobscot Bay with Peak Island_ (1966) is fine, too, but it would have been nice to see more of the Maine landscapes, which are such an important part of Porter's oeuvre. No matter. This is still one of the best painting exhibitions in town, and Mr. Agee and his students are to be congratulated on a job well done\u2014and a job all the more welcome because of the fate Porter's work has suffered at the hands of the New York museums. Their catalogue, too, is first-rate, with a fine introductory essay and good selections from Porter's own writings and from the critics of his work. What a pleasure it is to read a publication coming from the academy that is completely untouched by the ghastly jargon and odious politics that nowadays tend to dominate academic discourse on art. Bravo!\n\n[1992]\n\n### Hoving's Biteless Barking\n\n* * *\n\nEveryone who was around at the time (1967\u20131977) has a favorite memory of Tom Hoving's years as director of the Metropolitan Museum of Art. Mine is the story of his performance at an off-the-record luncheon with the top brass of the _New York Times_ toward the end of his tempestuous tenure at the Met. For this story to be fully understood, however, something must be said about the special relationship he had with the _Times_ while he was the director of our greatest art museum. For late-comers to this saga, it should also be mentioned that in the first years of Mr. Hoving's directorship of the Met, I was the _Times_ 's art news editor, and in the later years its chief art critic.\n\nThe _Times_ loomed very large in Mr. Hoving's career at the Met. It couldn't have been otherwise, for as a self-confessed \"publicity hound,\" he was peculiarly dependent upon the _Times_ to gratify his single most ardent desire, which was to see his name continuously and conspicuously in the news. From the perspective of a publicity hound like Tom Hoving, it hardly mattered whether the news was good or bad. He well understood that in the magic world of publicity, even a knock can sometimes be a boost, and in the early days at least, there were many more boosts than knocks. In respect to the attention that was lavished on his every word and deed and misdeed, the _Times_ rarely let him down. It was the _Times_ , more than any other branch of the media, that made Tom Hoving a New York celebrity\u2014which is to say, a person who is famous for being famous.\n\nOn the other hand, it was also the _Times_ that published the most damaging criticism he ever received as director of the Met, and in the end the _Times_ played a significant role in causing Mr. Hoving to resign his position at the museum. Hence his intensely conflicted, love-hate relationship with the paper he so tirelessly courted and so shamelessly attempted to manipulate by fair means and foul. Hence, too, the space that Mr. Hoving now devotes to the _Times_ in the book he has written about his troubled tenure at the Met\u2014 _Making the Mummies Dance_. The bulk of the _Times_ 's coverage of Mr. Hoving's activities at the Met does not, to be sure, receive any mention at all in _Making the Mummies Dance_. In keeping with the self-aggrandizing tenor of the book, most of the critical stuff the _Times_ published about the Hoving era at the Met has been dropped down the memory hole. But even so, some eighteen lines of the book's index list references to the _Times_ , more lines than are devoted, for example, to the Met's acquisitions during Mr. Hoving's ten years as director of the museum\u2014itself an interesting index of what still looms largest in his retrospective assessment of his achievements as the Met's director. It is for this reason, as well as others, that a more accurate title for the book would have been: _Confessions of a Publicity Hound_. But accuracy, alas, has never been one of Mr. Hoving's cardinal concerns.\n\nIt certainly wasn't his principal concern on that day in the late 1970s when he came to the _Times_ as the guest of honor at the \"publisher's lunch.\" In those days, the publisher's lunch at the _Times_ was a highly formalized \"informal\" ritual to which well-known figures in public life\u2014the secretary of state, say, or the president of Lincoln Center, or some visiting foreign statesman\u2014were invited for an off-the-record discussion of current problems with the paper's senior editors and executives. Staff writers like myself were invited to attend these lunches, which were held in the publisher's private dining room, only when their journalistic responsibilities were directly related to the professional field of the guest of honor.\n\nWhile it was understood that the purpose of these lunch discussions was to provide \"background,\" it was also expected that they would generate exclusive news stories for the _Times_. Either way, the rule prohibit-ing direct attribution was strictly enforced. Thus it was in the nature of this journalistic ritual for the advantage to lie with the guest of honor, who was in a position to disclose the kind of confidential information, real or imaginary, that would reflect favorably on himself and unfavorably on his adversaries, and to do so without risk to himself. The whole process was, in effect, a high-level form of leaking, and like most journalistic leaks, it gave the leaker a marvelous opportunity to inflict damage on his rivals while enjoying complete anonymity.\n\nI attended a number of these lunches during my tenure at the _Times_. The most hilarious was the one given for Michel Guy when he was the French cultural minister. The editors were expecting some grave reflections on the French political situation, but all Mr. Guy wanted to talk about was the sculpture of Mark di Suvero, which he had just installed in the Tuileries in Paris, and the choreography of Merce Cunningham, which he also adored. The meeting was not considered a success. The most lugubrious of the lunches I attended was the one given for the loathsome Armand Hammer, who was then making use of his position in the art world\u2014as a collector, dealer, and museum benefactor\u2014to advance the interests of his company, Occidental Petroleum, in Mexico. As a result of that lunch, I was assigned to review a tenth-rate exhibition of Mexican art that Hammer had brought to Washington and New York. I panned it, of course, which (I was told) made Hammer very angry, but there were no other repercussions. Neither the publisher nor the editors of the _Times_ ever interfered with anything I wrote about art events\u2014or, indeed, anything else that I wrote.\n\nThis was true, by the way, even in the case of my criticisms of Mr. Hoving and the Met. Although the publisher of the _Times_ was a member of the museum's board, I was never asked to change a single word that I wrote about the Met. On the rare occasions when I saw the publisher, who always treated me with great courtesy, he would simply say something like, \"Boy, am I going to catch hell at the next board meeting for what you guys are writing about the Met!\"\n\nIt seemed to me at the time that it was an odd moment for the publisher of the _Times_ to invite Mr. Hoving to speak at one of these lunches. Everyone in the museum world and in the media knew that Mr. Hoving's days at the Met were already numbered, and as a member of the Met's board, the publisher was in an even better position than the rest of us to know the extent to which the board's confidence in its current director had been shattered beyond repair. My guess at the time was that Mr. Hoving, either directly or (more likely) through an intermediary, had requested the invitation\u2014perhaps in the hope of generating some sort of _Times_ story that might save his neck. As it turned out, I had quite underestimated how far Tom Hoving was prepared to go in the misinformation game.\n\nConversation at lunch that day got off to a slow start, partly because Mr. Hoving seemed to be in a particularly dispirited state of mind and partly because Charlotte Curtis, who had risen from her position as chatelaine of the paper's women's pages to become editor of the Op-Ed page, derailed the discussion by trying to get Mr. Hoving to agree that New York was losing its preeminent position as the art capital of the nation. (In her invincible ignorance, Charlotte cited Georgia O'Keeffe's fame as a resident of New Mexico, and it was crushing news for her to learn that O'Keeffe's reputation had actually been made as a member of the Stieglitz circle in New York.) So it wasn't until the coffee was served that Mr. Hoving dropped his bombshell.\n\nSuddenly sporting that smarmy smile that always signified some noxious mischief in the making, Mr. Hoving let drop the \"news\" about an agreement that was being negotiated between the Museum of Modern Art in New York and the National Gallery of Art in Washington. Surely we must all know about it, he said. In an instant, all eyes at the table were turned to me. I was the only person present who was supposed to know about such things, and I had to admit that I knew nothing about this alleged negotiation or agreement. It was a nice moment for Tom Hoving and a bad one for me, for I had been the most vocal of his critics and it was perfectly clear that the assembled editors were furious at the thought that I might be responsible for the _Times_ missing out on a major art story.\n\nThen, with all the sangfroid that Mr. Hoving had learned to muster in tight situations, he allowed himself to be persuaded to tell the following tale. The Museum of Modern Art, he said, was in grave financial peril\u2014graver than was known, and so grave, indeed, that it might have to be permanently shut down. The National Gallery, however, was then in the process of promoting its new East Building, which was planned to be the site of a new program in modern and contemporary art. The problem, according to Mr. Hoving, was that the National Gallery had \"nothing\" to put into its new facility. What could be simpler, he suggested, than for MOMA to close up shop and transfer its great collections to the National Gallery? This, we were told, was the nature of the agreement then being negotiated by Paul Mellon and the Rockefellers.\n\nIt was all the sheerest baloney, of course\u2014and instantly recognizable as such by anyone who had Mr. Hoving's number. And yet, despite the Niagara of misinformation that had flowed from the Met in the Hoving era, the editors inevitably found the story electrifying\u2014as, of course, Mr. Hoving knew they would. The publisher's lunch was hurriedly concluded, and within minutes a major investigation was set in motion in the newsroom of the _Times_ to confirm the story from other sources. Needless to say, it could not be confirmed because it was a complete fabrication, and so the story never ran. Like so many of Tom Hoving's ill-judged initiatives, it caused a lot of commotion and produced nothing but hot air.\n\nWhy, it will be asked, would a man in Mr. Hoving's position pursue such a foolhardy course? The main reason, of course, is that he is basically an extremely foolish man\u2014at heart, despite his age and experience, still an unruly adolescent out to shock his elders and embarrass his benefactors. But there was also the immediate hope of avenging himself on the two institutions\u2014MOMA and the National Gallery\u2014that, in his view at the time, he had ample reason to resent. MOMA had stood in the way of realizing his ambition to make the Met's twentieth-century department a major player in the field of modern and contemporary art, and the National Gallery had often stolen his thunder in the field of blockbuster exhibitions. (See, in this connection, his unlovely remarks about Carter Brown, then the director of the National Gallery, in _Making the Mummies Dance_.) Then, too, if he could also have caused the _Times_ a major embarrassment by getting this phony story into the paper, that would have added to the fun. And there was always the outside chance that the story, if published, would have caused so much fuss and confusion elsewhere that the Met's board might not have thought it a propitious moment to dump its own wayward director.\n\nWe shall be a long time assessing the full impact of Tom Hoving's directorship of the Met on our art museums, and on their place in American cultural life. Yet in a way that its author hardly intended, the publication of _Making the Mummies Dance_ makes that assessment a lot easier. For this is an egregiously shallow and mean-spirited book, and a book, moreover, that has absolutely nothing to say about the life of art in our time. For serious thought it substitutes malicious gossip and the tiresome rituals of self-promotion. It thus reminds us of the intellectual void that was always at the center of Tom Hoving's tenure as director of the Met. It spells out in gruesome detail what happens to a great art institution when it passes into the hands of a spoiled smarty-pants whose principal success was to gull a lot of important people (who should have known better) into believing that money, politics, publicity, and what Tom Hoving liked to call \"flash\" and \"fun\" would serve as an adequate substitute for vision and leadership. In _Making the Mummies Dance_ , the momentous affairs of one of our greatest cultural institutions are reduced to the level of talk-show scuttlebutt, which is the only level on which Tom Hoving has ever excelled.\n\n[1993]\n\n### A Wretched Nauman Show\n\n* * *\n\nMasochists of the art world! Your hour has come at last. No longer are you to be denied the stupefying levels of tedium, torture, disgust, humiliation, intimidation, and victimhood that constitute your greatest artistic pleasures. If a programmed mental regression accompanied by an unrelenting assault on the senses is what you need in order to experience high pleasure in art, then your fondest desires are about to be gratified: the Bruce Nauman show has opened at the Museum of Modern Art, and it is indeed a masochist's delight.\n\nWhereas other contemporary exhibitions merely pretend to offer a radical quantum of pain, insult, and degradation sufficient to the needs of the most jaded art-world tastes, this one really delivers. Your aural faculties are horribly assaulted. (Bring earplugs if you are unduly sensitive to high levels of incessant, repulsive noise.) Your sense of delicacy is repeatedly and pointlessly violated. (Leave the kids home if you don't want to have to explain the effigies of dead animals, the sexual mayhem on the videos, or the masturbation joke in _Manipulating a Fluorescent Tube_.) After the Nauman show, the next _Oprah_ will look like a rerun of _Rebecca of Sunnybrook Farm_ , and the most nerve-jarring police sirens will sound to your beleaguered ears like the anodyne strains of Vivaldi.\n\nBruce Nauman's language\u2014for he uses a lot of words in his work\u2014is coarse or stupid or unintelligible where it is not actually obscene, which it often is. His admirers, however, regard Mr. Nauman's use of language as quite profound, though they appear to be in some disagreement about the exact character of this alleged profundity. Robert Storr, the curator of painting and sculpture at MOMA, believes that \"Nauman's method . . . has much in common with that of Ludwig Wittgenstein,\" whereas Kathy Halbreich, the director of the Walker Art Center in Minneapolis, finds a more appropriate comparison in the writings of \"the great 17th-century metaphysical poet John Donne.\"\n\nBut preposterous claims of this sort represent nothing more than a kind of _deformation professionnelle_ that nowadays is a common affliction among curators of contemporary art, who are routinely obliged to defend the indefensible and confer meaning where none exists. The fact is, Mr. Nauman's amateur wordplay bears no resemblance whatever to Wittgenstein's work, and neither does the adolescent fixation on body parts bear any resemblance to Donne's poetry. You'd have to be either a fool or a curator of contemporary art to believe otherwise.\n\nI haven't myself seen, or have any wish to see, the Nauman video called _Black Balls_ , which Ms. Halbreich cites as a Donne-like work. But from the still reproduced in the catalogue of the current exhibition and Ms. Halbreich's own account of the work (\"Eroticism is frustrated, frozen, even when the artist applies theatrical makeup to his own testicles,\" etc.), it can reasonably be assumed by anyone familiar with Donne's poetry that her memory of it is less than perfect.\n\nWhat, then, is really going on in this huge exhibition, which alternates between extreme tedium and extreme nastiness? The show is divided into two parts, with the early work\u2014beginning in the 1960s\u2014confined to the museum's basement galleries, and the later work occupying an immense amount of space on the third floor. The course traced by the artist's development is from a kind of graduate-school version of Dada to what must now be called bureaucratic Dada\u2014the kind of overscale video and\/or installation art that requires large art bureaucracies like the Walker Art Center and the Museum of Modern Art to serve as its accomplices and an agency like the National Endowment for the Arts to provide a certificate of respectability.\n\nGraduate-school Dada is little more than a familiar rehearsal of Duchampian jokes and stunts that, owing to the success of Jasper Johns, have been commodified as \"avant-garde\" bourgeois entertainments. Bureaucratic Dada, however, is art-world show business, requiring expensive production facilities and an audience of determined masochists. What unites the smaller-scale graduate-school Dada and overscale bureaucratic Dada is an expanding atmosphere of boundless, adolescent narcissism, which the hapless art public is invited to scrutinize for signs of deep cultural significance.\n\nI shall have to leave it to the psychiatrists to explain exactly why, in Mr. Nauman's case, this unrelieved narcissism is, as Mr. Storr writes in the catalogue accompanying the exhibition, \"increasingly prone to brutal fun.\" Certainly there is a great deal in the exhibition, especially in the third-floor galleries, that is exceedingly brutal. Whether it is also fun depends, I suppose, on whether you delight in the imagery of dead animals, as in the installation called _Carousel_ , or in the spectacle of slapstick sexual violence in the video sequence called _Violent Incident_ , or in the disgusting details of another sequence called _Clown Torture_.\n\nFar from finding it fun myself, I thought this exhibition the single most repulsive show I have seen in the nearly fifty years I have been going to MOMA. It is also one of the biggest. At a press conference the other day, Mr. Storr proudly pointed out that the Nauman retrospective is, after the great Picasso and Matisse retrospectives, the third largest in MOMA's history. Why a great institution like the Museum of Modern Art should take such great pride in showing such nihilistic rubbish on such a grand scale is, in the end, a far more interesting question than any raised in the exhibition itself. We know, of course, that Bruce Nauman has won a great many prizes, that in Europe he is considered one of our greatest living masters, and so on. Yet the work remains nihilistic rubbish all the same. Why taxpayers should be called upon to support an event of this kind\u2014and why indeed a supporter of conservative causes like Ronald S. Lauder should help finance it\u2014needs to be explained. These are more compelling questions than any to be found in the mind of Bruce Nauman.\n\n[1995]\n\n### How Long Will Curators Ignore the Great Nerdrum?\n\n* * *\n\nThe strangest painter exhibiting new pictures in New York at the moment is undoubtedly a fifty-one-year-old Norwegian artist by the name of Odd Nerdrum. He isn't exactly new to New York. He has been exhibiting here since the 1980s, and in certain circles\u2014not the most fashionable, to be sure\u2014he may even be said to have become something of a cult figure. He has remained unknown, however, to the public that takes its cues from the museums specializing in contemporary art. Indeed, he seems to represent everything that our curators of contemporary art most despise\u2014a highly accomplished reactionary technique combined with an attitude of extreme moral gravity.\n\nAll the same, my guess is that before the end of the decade, we shall be seeing Mr. Nerdrum's paintings in our museums. We may even get to see a retrospective. For he is a force to be reckoned with. Which is why his current exhibition at the Forum Gallery is not to be missed. When you see the work, you may very well dislike it intensely. But you will not soon forget it, and you will certainly not remain indifferent to it. Afterward, you may even find that a lot of contemporary painting looks perfectly trivial by comparison.\n\nMr. Nerdrum is that rare thing\u2014a deeply disturbing artist. I mean genuinely disturbing, not merely irksome or cleverly provocative. He is disturbing in ways that pose more questions about his art\u2014and about art itself\u2014than can be easily answered. His pictures do not seem to belong to the contemporary world at all. Yet they don't quite belong to the past, either\u2014despite the resemblances they obviously bear to the work of certain Old Masters. They reject the present and exploit the past in favor of a realm of pictorial fable, allegory, and myth that offers the viewer a grim symbolic account of the human condition in extremis.\n\nThis is not the kind of task that we look to the art of painting to perform for us nowadays, and a large part of Mr. Nerdrum's appeal is that his art is so resolutely indifferent to contemporary conventions. In everything but his view of human nature, which is as bleak and unforgiving as anything to be found in Nietzsche, Freud, or Heidegger, Mr. Nerdrum is a radical anti-modernist. His pictorial style, with its swaggering obeisances to Rembrandt, Brueghel, and Caravaggio, consigns the whole realm of modernist practice to oblivion. This is an art deeply rooted in the twilight chiaroscuro of the northern latitudes\u2014an art that is premodern rather than postmodern in its fundamental loyalties.\n\nAll of this\u2014the contrived fables, the emotional violence, the atmosphere of primeval menace\u2014would be a matter of indifference or mere curiosity if Mr. Nerdrum did not bring to his art such a formidable command of his medium. Painting at this level of ambition requires exceptional technical powers, for it invites comparison with the great painting of the past at every turn, and it is one of the astonishments of Mr. Nerdrum's art that it so consistently meets this challenge with such unflagging virtuosity.\n\nIn addition to his mastery of the figure and of landscape, moreover, Mr. Nerdrum brings to his painting a first-rate dramatic imagination. With paintings like _Man with a Horse's Head_ or _Girl with a Fish_ or the sensational _Woman Killing an Injured Man_ \u2014all dating from the last two years\u2014the only appropriate comparisons that come to mind are not with other contemporary painters, but with artists in other fields. It is in relation to certain films by Ingmar Bergman or with the poetry of the English writer Geoffrey Hill that Mr. Nerdrum's accomplishment is best understood. In that company, the spiritual turmoil that lies at the center of Mr. Nerdrum's pictorial vision may be more clearly perceived. And in that company, Mr. Nerdrum's visionary ambitions for art no longer look quite as anomalous as they would if we limited our perspective to the spiritually destitute conventions of contemporary painting. Seen in relation to Bergman and Hill, moreover, Mr. Nerdrum no longer appears quite so isolated or orphaned as a late-twentieth-century artist. He turns out to be our contemporary, after all.\n\nSomething that another fine poet, William Logan, recently wrote about Geoffrey Hill comes to mind in attempting to place Mr. Nerdrum's art in the cultural life of our time. \"Critics who would pigeonhole Hill all too conveniently as a conservative,\" wrote Mr. Logan, \"fail to understand the radical nature of his mistrusts, from which no complacent sentence is safe.\" In Mr. Nerdrum's case, the \"radical nature of his mistrusts\" encompasses just about all of modernist art, which makes it very convenient indeed to pigeonhole him, too, as a \"conservative.\" And insofar as it gives us a key to a keener understanding of Mr. Nerdrum's painting, I see nothing wrong with regarding him as a conservative\u2014though I think myself he is more of a radical reactionary in his attempt to reconstitute a lost tradition in his art.\n\nIt would be a mistake, however, to view Odd Nerdrum as just another offbeat novelty that has been tossed up by the carnival of the contemporary art scene. He is more important than that and, I am sure, he will be seen more and more as important. Frankly, I don't think the future of painting lies in his direction. He is too much of an individual and his art is too closely bound to the eccentricities of his own sensibility for him to become the leader of a \"school.\" (Not to mention the fact that few artists today are technically or spiritually equal to the challenge of following his lead.) But I do think that the future of painting will be affected by his example, if only because of the profound reexamination of modernist practice it demands. And from the perspective of Mr. Nerdrum's art, this includes postmodernist practice as well.\n\nIt is going to take a while for the implications of Mr. Nerdrum's audacious innovations to be fully grasped. Meanwhile, the ten paintings on view at the Forum Gallery constitute an excellent introduction to his work.\n\n[1995]\n\n### Best-kept Secret in Art? Stettheimer\n\n* * *\n\nThe paintings of the American artist Florine Stettheimer (1871\u20131944) have long been one of the best-kept secrets in the art of this century. Those who were in on the secret during her lifetime\u2014Marcel Duchamp, Virgil Thomson, Henry McBride, and Carl Van Vechten, among other eminences of the period\u2014held her work in high esteem, and they were right to do so. For Stettheimer is a very remarkable artist. Yet except for her participation in a single avant-garde event, she remained unknown to the public. That event was the 1934 premiere of the Virgil Thomson\u2013Gertrude Stein comic opera, _Four Saints in Three Acts_ , for which Stettheimer designed the sets and costumes. But even that event, though it garnered a good deal of public attention, still left her paintings unknown.\n\nIt was Stettheimer's wish, moreover, that they remain unknown. In her will, she stipulated that all of her pictures were to be destroyed upon her death. Fortunately for us, this harsh sentence\u2014born, I think, of her belief that her paintings were too private to be understood by anyone outside her immediate circle of family and friends\u2014was never carried out. Instead, her sister Ettie set about the task of placing as many of the pictures as she could in important museum collections.\n\nThat task was made somewhat easier than it otherwise might have been by the fact that immediately following Stettheimer's death, her well-connected friends went to work organizing a memorial exhibition of her paintings. It took place at the Museum of Modern Art in 1946. Marcel Duchamp selected the pictures, and Henry McBride wrote the catalogue. If the show didn't exactly put Stettheimer on the map\u20141946 wasn't a good year for an art as esoteric as hers was thought to be at the time\u2014it at least saved the work from oblivion. It couldn't save it from obscurity, however. Notwithstanding the imprimatur of MOMA, Stettheimer's pictures remained in private hands or in the storerooms of the museums to which they had been donated. Occasional attempts to revive an interest in them were unavailing.\n\nThe exhibition called _Florine Stetheimer: Manhattan Fantastica_ , which Elisabeth Sussman and Barbara J. Bloemink have organized at the Whitney Museum of American Art, is the first New York museum show devoted to the artist in nearly half a century. I am happy to say that it is a very good show. It concentrates on the work of Stettheimer's artistic maturity, which dates from around 1917, when the artist was in her forties. It was only then that she broke loose from her academic training\u2014she had studied at the Art Students League in New York and the academies of Berlin, Munich, and Stuttgart\u2014to follow a course of her own.\n\nThat course consisted of the development of a style that compounded a very personal wit and an aestheticism that concentrated its attention on the _mise en sc\u00e8ne_ of the artist's life\u2014a life bounded by art, money, and the rituals of a privileged bohemia. It was a style decisively shaped by Stettheimer's extensive knowledge of art in Europe, where she lived with her mother and sisters for a part of each year from 1898 to 1914, but it was wholly devoted to New York subjects.\n\nStettheimer's New York was indeed as personal as her style, and in her art she treated it as comic exotica. Its center was the elegant salon she established for her artist and writer friends, almost all of whom were better known than she was, and the country entertainments she organized for their amusement. This was a society of sharp wits, intellectual dandies and worldly aesthetes, and Stettheimer's depictions of it were, as the critic Paul Rosenfeld once wrote, \"tinged with irony and merriment.\"\n\nYet if there is a lot of comedy in these pictures\u2014as there certainly is\u2014there is also an element of inventive aesthetic artifice that places Stettheimer's painting at a great distance from the art of her American contemporaries. The elongated stylized figures in her paintings clearly owe a good deal to the art of Aubrey Beardsley and to Jugendstil illustration, though the affectionate mockery they convey is often closer in spirit, if not in form, to the drawings of Max Beerbohm. There is, in any case, a lot of nineties \"decadence\" and aestheticism in Stettheimer's art.\n\nIt is also much influenced by the Russian ballet and the American circus. Nijinsky's dancing and L\u00e9on Bakst's sets for Diaghilev's Ballets Russes, which Stettheimer saw in Paris in 1912, constituted one of the central aesthetic experiences of her life, and one of her paintings\u2014 _Beauty Contest_ (1924)\u2014is dedicated \"To the Memory of P. T. Barnum.\" The gestures traced by the figures in her group scenes and portraits are indeed \"choreographed\" to resemble those of ballet dancers, mimes, and circus acrobats. Often her canvases are divided into separate, simultaneous scenes like those of a pantomime or the rings in which circus artists perform. But every figure in these paintings is a specific personality drawn from Stettheimer's circle of family and friends. It is autobiography turned into harlequinade.\n\nThe most distinctive painterly feature of Stettheimer's art, however, is its extraordinary coloristic invention\u2014that palette of hot reds and purples, magenta pinks and theatrical blacks, often set in an icy white light and embellished with yellow and gold, is like nothing else in the American art of her time. This she owed principally to Bakst and the tradition of Byzantine-Russian design that Bakst drew upon in his sets for the Ballets Russes. It is the kind of color you often find in the early paintings of Marc Chagall, who studied with Bakst in Russia, and in some of the Russian Symbolist painters. In appropriating this chromatic convention for her own pictorial purposes, Stettheimer adapted it to her own interest in cosmetics and theatrical _maquillage_ , especially in the portraits of the 1920s, thus making it an instrument of contemporary feeling.\n\nBeginning in the late twenties, with her _Cathedrals of Broadway_ (1929), and continuing into the early forties with the _Cathedrals of Art_ (1942), Stettheimer expanded her subject matter to include satires of the New York social and cultural scene. These comic tableaux are also delightful pictures, and are in some respects, I suppose, Stettheimer's aesthetic \"answer\" to the heavy-handed and badly painted social realist painting of the period. But upon revisiting them on the occasion of this exhibition, I found them a lot thinner in substance and style than the more private group scenes and portraits of the twenties. Even among the paintings of the thirties, they aren't in the same class with the wonderful _Family Portrait No. II_ (1933).\n\nStettheimer was a true original, and the current exhibition at the Whitney at last does justice to her achievement. Since we live in an imperfect world, however, it wasn't to be expected that this event would escape the kind of ideological packaging that is now the fashion at the Whitney and elsewhere on the museum scene. So in the catalogue accompanying the current show, Stettheimer is made out to be some sort of radical feminist with a discernible political agenda. This turns Stettheimer's career into a comic opera of a very different sort, and is tone-deaf to the entire spirit of her work. But it's a small enough price to pay\u2014if pay it we must\u2014to have Stettheimer's paintings given their due.\n\n[1995]\n\n### Reintroducing Nadelman\n\n* * *\n\nAlthough he was one of the most accomplished sculptors of his generation\u2014a great generation\u2014and some of his drawings are among the best of their time, Elie Nadelman (1882\u20131946) remains an elusive figure for many people on the New York art scene. Part of this has to do with the fact that with every passing season the New York art scene becomes more and more amnesiac, if not simply ignorant, about its own past. And part of it has to do with the fact that Nadelman's art requires a certain sensibility\u2014a certain cultural depth\u2014to be fully appreciated.\n\nYou have to know something about the art of the past\u2014sources as diverse as Greek sculpture and American folk art\u2014to understand the impulses and traditions that shaped Nadelman's vision. It helps, too, to know something about his \"take\" on American culture and the ways in which this mandarin European talent transformed himself into an authentic American artist during the last three decades of his life, which he spent in New York.\n\nThese are all things that were better known in the past than they are today. In 1973, Lincoln Kirstein published a big, brilliant, beautifully written study of the artist, and in 1975 the Whitney Museum of American Art\u2014hard as it may now be to believe such a thing\u2014devoted a marvelous exhibition to his work. Long before that, in 1948, the Museum of Modern Art had organized a show of his sculpture, and even longer ago than that, in 1915, Nadelman was one of the few \"new\" sculptors whom Alfred Stieglitz favored with an exhibition in his gallery. Yet despite this impressive history\u2014and the fact that some wonderful examples of his work can be seen in the museums\u2014Nadelman still, or again, needs an introduction to an art public that seems at times to have the attention span of a gnat.\n\nWe are fortunate, therefore, that in one quarter at least\u2014the Zabriskie Gallery\u2014we have been given some salutary reminders of Nadelman's artistic importance in exhibitions that are always intelligent introductions to the artist's oeuvre and yet can be counted upon to come up with some surprises even for aficionados. The current Nadelman exhibition, which Virginia Zabriskie has organized to open her new space in the Fuller Building, is the sixth show she has devoted to the artist. It is the kind of small, perfectly judged, utterly delightful exhibition that is nowadays one of the rarest pleasures on the New York art scene, and it has the additional interest of calling our attention to the fact that the Fuller Building, on the northeast corner of 57th Street and Madison Avenue, is itself the site of a Nadelman architectural commission. The clock flanked by a sculptural relief of _Construction Workers_ that adorns the 57th Street fa\u00e7ade of the building above the main entrance was created by Nadelman in the early 1930s. Ms. Zabriskie's new gallery space is, appropriately, on the forth floor directly behind this Nadelman work, which the many art lovers who frequent the galleries in the Fuller Building are nowadays scarcely aware of.\n\nThe show itself consists of seven sculptures, seven drawings, and\u2014the real surprise\u2014eleven early photographs of Nadelman sculptures, many of which have been lost and presumed to have been destroyed. These photographs, all but one taken by Mattie Edwards Hewitt around 1922, are themselves very beautiful. The earliest\u2014Eug\u00e8ne Druet's photograph of a lost _Plaster Head_ (circa 1909)\u2014is a real knockout. Druet was the art dealer who gave Nadelman his first one-man show in Paris in 1909\u2014a show that caused a considerable stir at the time. Druet was also an accomplished photographer whose pictures of Rodin sculptures are, like the Nadelman photograph in the current exhibition, works of art in themselves.\n\nThe earliest of the sculptures in the show\u2014the _Semi-Seated Tubular Nude_ (circa 1909\u20131910), in carved wood, and a marble _Head_ (circa 1910)\u2014date from the period when the exhibition at the Galerie Druet had made Nadelman something of a sensation on the Paris art scene. Even before that exhibition, in 1908, Leo Stein took Picasso to Nadelman's studio, and scholars of the period have long believed that the bronze Cubist _Woman's Head_ that Picasso created in 1909\u2014the first of his Cubist sculptures\u2014owed a good deal to what he saw on that visit.(Druet's photograph of Nadelman's _Plaster Head_ in the current show seems to confirm that link.) It will be interesting to see what, if anything, John Richardson will have to say about the Nadelman-Picasso connection in the forthcoming second volume of his Picasso biography.\n\nSmall as the Zabriskie exhibition is, it nonetheless touches upon the whole range of Nadelman's accomplishment. His art is at once very classical in its formal loyalties, and very modern in its range of feeling. It is an art of great elegance, wit, and character\u2014a character that owes as much to the style of the American musicals that Nadelman adored as to the exalted European traditions he revered. Nadelman was himself a connoisseur and collector of considerable distinction. His most important discovery, when he settled in the United States around the time of World War I, was American folk art, which was then very little understood in \"high art\" circles in this country, and he promptly set about amassing a major collection of it. When the Wall Street crash of 1929 left him and his family in need of money, he attempted to interest New York City in acquiring the collection for a folk-art museum, but nothing ever came of the proposal\u2014and the collection was dispersed. Today you couldn't buy a single major work from that collection for what the whole collection would have cost the city in the 1930s.\n\nBorn in Warsaw when it was still a province of the Russian Empire, Nadelman studied at the Warsaw Art Academy, then continued his studies for six months in Munich before settling for a time in Paris. He seems never to have been an avant-gardist, however. In 1912 he quarreled with Marinetti, the leader of the Futurists, whose _Manifesto_ famously called for the destruction of the museums. For Nadelman, a speeding motorcar was never thought to be more beautiful than the _Victory of Samothrace_ , and his art reflected that prejudice in favor of tradition. Yet he found in America\u2014the kind of civilization the Futurists clearly envied\u2014a whole new impetus for his art.\n\nElie Nadelman is still a bigger and better artist than he is now commonly thought to be\u2014when he is thought about at all\u2014and in this exquisite exhibition at the Zabriskie Gallery, we are once again reminded of his quality.\n\n[1995]\n\n### Alex Katz Blossoms\n\n* * *\n\nThis year the American painter Alex Katz observes his sixty-ninth birthday. And this season, as it happens, he is having the best exhibition he has had in his long career\u2014the best, anyway, that I have seen in the four decades or so that I have been familiar with his work. The show on view at the Marlborough Gallery is large, ambitious and beautiful. Not all of it is of uniform quality, to be sure. I have often had my doubts about Mr. Katz's large-scale figure paintings, and the examples there\u2014though they are not without a good deal of interest\u2014do not persuade me to allay those doubts. But the big new landscape and cityscape pictures achieve a level of quality that seems to me to be well beyond even the finest of Mr. Katz's earlier paintings. For that reason alone, this show is something of an event.\n\nEven very good painters do not necessarily get better as they get older. Some do, some don't. Some lose their nerve, some suffer losses of energy and momentum, some merely repeat themselves with diminishing effect. It is given to others, however, to achieve an almost preternatural clarity and concentration as they reassess their goals and their means of achieving them. Mr. Katz appears to belong to that lucky class of painters to whom age brings not only an increase in fluency, confidence, and ambition but something else as well: a certain gravitas.\n\nIn this exhibition you see the evidence of it in the landscapes and cityscapes\u2014paintings in which the bright, voluble, insouciant spirit of the artist's earlier work has been supplanted by something less sociable and more austere, something timeless and impersonal. The weight of the artist's interests\u2014which is to say, the emotions that inspire his gravest efforts\u2014seems to have shifted from society to nature. No longer does the social milieu of the young, the hip, and the immaculately dressed command the kind of priority in Mr. Katz's painting that it once did. It is still vividly represented in the figure paintings, of course\u2014but these cool young mannequins of the downtown Manhattan scene are no longer drawn from the artist's own generation. Nowadays they tend to be drawn from his son's generation, which inevitably makes them actors in someone else's play.\n\nIs it for this reason, too, that the backgrounds in the figure paintings tend to be more interesting, _as_ painting, than the figures themselves?\n\nNature has, in any case, overtaken society as Mr. Katz's principal subject, and is now the locus of his finest endeavors. Landscape has always been one of the artist's interests, of course, but it hasn't in the past assumed the kind of priority or claimed the kind of scale that it enjoys today. Does this mean that Maine, where Mr. Katz has spent his summers for some decades now, has supplanted Manhattan as his primary inspiration? To some extent, it obviously has. When you walk into the Marlborough Gallery and see the mural-size canvas of light filtering through the spring foliage in the distance\u2014the painting called _May_ (1996) is twenty feet wide and ten feet tall\u2014you are certainly made to feel that you are a long way from the streets of downtown Manhattan.\n\nThis painting may be the single finest thing Mr. Katz has ever done, and it ought to go straight into some museum collection, though given the taste of our museums today, the chances of this happening are not terrific. _May_ and certain other paintings in the current show\u2014 _Coleman Pond III_ (1995), _10 A. M_. (1994), _Dark Reflection_ (1995), _Poplars_ (1994), and _Yellow Morning_ (1996)\u2014suggest, moreover, that in the Maine landscape and nightscape Mr. Katz has found his Giverny, the kind of \"late\" subject that allows for the maximum expansion of imaginative energy.\n\nIndeed, it is of late Claude Monet that one often thinks in looking at these overscale paintings of nature\u2014not as a direct influence, perhaps, but as a premise that has lodged itself in the minds of several generations of American painters, both realists and abstractionists, since the rediscovery of the Giverny murals in the 1950s found them to be among the most radical paintings produced in the early decades of the twentieth century. It is worth recalling, too, that prior to his total absorption in his water garden at Giverny, Monet had also been a painter of the modern urban scene.\n\nSo decisive has been this shift to nature in Mr. Katz's work that it now does much to determine the way even his urban subjects are pictorially conceived. Another of Mr. Katz's major efforts in the current show is the mural-size _City Landscape_ (1995), which, if not for the city lights in the night sky shining like scattered moons, could easily be mistaken for a depiction of the Maine woods after dark. Without their titles, certain other paintings in the show\u2014 _Manhattan Evening_ (1995), for example\u2014might also be mistaken for Maine subjects. Which suggests that Maine is now less a discrete subject for Mr. Katz's painting than it is a realm of pictorial fantasy to which he has ready access wherever he finds himself.\n\nMany years ago\u2014in 1960, in fact\u2014the late Fairfield Porter, an excellent critic as well as a fine painter himself, said of Mr. Katz, \"He is not overwhelmed by nature but stands outside it; it is outside him and includes his subjectivity.\" Like almost everything else Porter wrote about painting, it was a sharp observation. Yet as Mr. Katz has developed over time, I think he has tended to remain less \"outside\" the subjects he has drawn from nature, and that nature itself has come to occupy a larger and larger place in what Porter called \"his subjectivity.\" Again, one thinks of Monet at Giverny.\n\n[1996]\n\n### Welcome, Professor Forge\n\n* * *\n\nPainters who are wonderfully articulate about painting\u2014their own as well as others'\u2014are often regarded with a certain suspicion. What they are suspected of is being better at talking a painting than at painting one. The suspicion is quickly elevated to a certainty, moreover, if the painter in question is also known to be a first-rate writer on art. By and large, the art world tends to be more comfortable with talents that are nicely segregated, and trespassers are sometimes punished.\n\nYet despite the fact that the art world is full of clever people who are better at talking than at painting, the suspicions I speak of cannot invariably be relied upon to place an artist's work. Chore though it may be for some overburdened souls, we are still obliged to examine pictures before we pronounce judgment on them\u2014to look at them, as I heard someone say the other night, \"one at a time.\"This is a phrase that made a deep impression on me as soon as it passed the speaker's lips, for it had never before occurred to me that painting could be looked at any other way. Well, we learn something every day, especially at exhibition openings.\n\nI was thinking of all this when, a few days ago, I went up to New Haven\u2014not, I must admit, one of my favorite cities\u2014to see the Andrew Forge retrospective at the Yale Center for British Art. Everyone in the art world either knows Mr. Forge, or knows of him. But not as a painter. He is known as a teacher, a lecturer, and a writer. He taught first in London, then in New York, then for twenty years at the Yale University School of Art, where he also served as dean. He has been one of our best teachers of painting\u2014and a much beloved figure in that capacity. For almost as long as I can remember, he has also been one of our best writers on art\u2014author of, among much else, two marvelous books on Pierre Auguste Renoir and Claude Monet.\n\nYet it is no exaggeration to say that _as a painter_ Andrew Forge has remained a largely unknown talent on both sides of the Atlantic. Unless you had the good fortune over the last two decades to be closely involved with the Yale School of Art or the New York Studio School, where Mr. Forge has also been a teacher and lecturer, you are unlikely to have seen his pictures. It is for this reason that _Andrew Forge: A Retrospective_ at the Yale Center for British Art is something of an event\u2014a bigger event, indeed, than the artist's small circle of admirers may have expected it to be.\n\nWith this exhibition, Mr. Forge, who was born in 1923 and first studied painting with William Coldstream, Kenneth Martin, and Victor Pasmore at the Camberwell School of Art in London, is now seen to be something more than a very good painter. He must now be regarded\u2014in my view, anyway\u2014as one of the most accomplished painters of his generation anywhere. This is the kind of exhibition that requires us to make a revision in the way we think about the achievements of painting since the 1970s, for in any serious account of the subject, Mr. Forge will henceforth have to be afforded a significant place.\n\nBe warned, however, that Mr. Forge's paintings are not the sort that can be taken in by a hasty glance. They really do have to be looked at \"one at a time.\" Or, as Mr. Forge has said of his own pictures, \"They are a long time in the making; they also take a long time to look at.\" This is painting that seems to begin in a realm of unstructured painterly abstraction and only gradually, by way of small poetic increments, discovers through a visual memory and association its ostensible \"subject\" or motif. What begins as a lyrical improvisation of luminous touches (or \"dots\") of color reveals a wealth of perceived experience\u2014the fugitive experience of nature as it reposes and recomposes itself in a pictorial imagination alert to its every visual nuance.\n\nCertain paintings seem to address the subject of seasonal change, and are on that account named after months of the year. Others may or may not be phantom cityscapes or evocations of a nude torso, but what is never in doubt is the nearly flawless pictorial command. This is painting of tremendous virtuosity and tact\u2014the kind of painting that requires years of experience with the medium, and with life, to achieve.\n\nThe paintings I speak of\u2014which, for want of a better term, may be described as postabstract neo-impressionist\u2014all date from 1973 to the present. Several were completed this year, and it is painting of this persuasion that dominates the current retrospective. The exhibition commences, however, with a very northerly _Winter Landscape_ from 1951 and a series of cold-climate nudes and landscapes from the later 1950s and '60s. What is remarkable about these earlier paintings is their command of and sheer affection for the medium, and they leave us in no doubt that it was in this period that Mr. Forge discovered his true vocation.\n\nThen there is a hiatus of a decade, 1964 to 1973, that is not represented in the exhibition. It was in 1973 that Mr. Forge came to New York to teach, first at Cooper Union, and this change of venue seems to have provided the artist with the impetus\u2014and, I suspect, the time\u2014that his talents needed for their full realization. It is, in any case, for the paintings he has produced since the 1970s that he has now won a place for himself in the history books. A few years ago, Mr. Forge wrote something about Monet that I believe offers us an important clue to his own pictorial endeavors, for at the time of writing he was also hard at work on many of the paintings in this exhibition. \"To paint,\" he observed, \"is as necessarily sequential as to play chess. Meanwhile, somewhere between time 'out there' in the changing light and time 'here' in the sequence of changes on the canvas, there is another unfolding in time, the gradual movement toward understanding in which the evolving painting reveals its nature.\"\n\nFor the viewer, too, a good deal of the pleasure to be derived from this exhibition comes from that \"gradual movement toward understanding\" the nature of Andrew Forge's achievement.\n\n[1996]\n\n### The Somber Brilliance of Braque's Late Work\n\n* * *\n\nIt may sound odd to suggest that an artist as famous as Georges Braque (1882\u20131963) is nowadays underappreciated\u2014or even, perhaps, unduly neglected. His celebrated collaboration with Picasso in creating Cubism is, after all, universally recognized as one of the pivotal events of the modern movement. But of Braque's artistic achievements after that historic collaboration, the public\u2014on this side of the Atlantic, anyway\u2014has rarely taken an excited interest.\n\nBraque was never the kind of artist who was supported in the public mind by a compelling personal myth. There were no sexual scandals or political outrages associated with his name. He had but a single wife, to whom he remained devoted, and he played no role in public life. To be interested in Braque\u2014in Braque after Picasso, so to speak\u2014you had to be interested, really interested, in painting rather than the things that sometimes surround it, and that is often a rarer interest for the art public than is commonly acknowledged.\n\nFor people of so-called avant-garde taste, moreover, post-Picasso Braque represented another kind of problem\u2014the problem of Late Cubism, which in the period immediately following World War II was often thought to represent the academicization of a once revolutionary style. This, like a good many other avant-garde illusions of the period, is an aesthetic prejudice that is nowadays somewhat abated. But in its day, it no doubt contributed something to a disinclination to comprehend the magnitude of Braque's pictorial achievement in the later decades of his life. As a result, Braque came to be regarded as a master who could be taken for granted, an artist assured of his place in history but without a compelling appeal for contemporary tastes.\n\nFor all of these reasons, it is especially regrettable that the exhibition called _Braque: The Late Works_ , which has just opened at the Royal Academy of Arts in London, will not be coming to New York. For myself, anyway, this exhibition has proven to be something of a revelation. It is at once deeply serious and deeply moving. For this is a very somber exhibition in which we are made witness to a great painter's attempt to come to terms with the gravity of feeling generated by war, conquest, and their aftermath\u2014meaning, of course, the condition of France in the 1940s and '50s\u2014and yet remain faithful to the purely aesthetic demands of his art.\n\nWhat these late works of Braque constitute, in effect, is a series of elegies or meditations on the experience of enduring that catastrophe. At the same time, these paintings are also a series of meditations on the nature of painting itself, about its capacity to encompass a world that resists fixity and settled definition. They are not what in any conventional sense could be called history paintings. Yet, they must now be seen to have a place among the relatively few works of genius\u2014beside, that is, the sculptures of Alberto Giacometti and the plays of Samuel Beckett, especially _Waiting for Godot_ \u2014that still have more to tell us about the spiritual history of the war and its aftermath than almost anything else produced at the time.\n\n\"People don't take sufficiently into account the dark forces that impel us,\" Braque remarked in 1939, and that observation could easily have served as an epigraph to the _Late Works_ exhibition. What makes Braque's artistic success so extraordinary in the difficult endeavor he set for himself in this period is the self-imposed restrictions he rigorously observed in pursuing it. For this is largely an exhibition limited to still lifes and studio interiors, which are not genres traditionally associated with the elegiac mode\u2014not in modern painting, anyway. Unlike Picasso and Matisse, Braque was never primarily a painter of the figure, and he didn't become one in the late works. He had begun as a landscape painter, and, as this exhibition reminds us, he returned to landscape in his very last years. But it was in the paintings of objects and the space they occupy that Braque concentrated his elegiac vision of \"those dark forces that impel us,\" thus transforming the still life and the studio interior into vehicles of tragedy.\n\nTo encompass the \"dark forces\" that haunted him in this work, Braque's palette grew progressively darker as he explored his restricted themes. Though there are moments of light and occasional forays into vivid color, the late paintings are largely executed in greys, blacks, and earth colors. And what an amazing range of greys and blacks Braque was able to summon to his purpose in these paintings! He had always, of course, been one of the consummate painterly craftsmen of his time, a painter with a total command of his medium who was also deeply suspicious of his own facility and determined to resist it. The discipline he had acquired in this regard was put to magnificent purpose in the late works, where every temptation to an easy appeal is resisted and the reigning temper of his art is governed by an elevated and deeply pondered austerity.\n\nFrom the first masterpiece we encounter in the Royal Academy show\u2014the _Large Interior with Palette_ (1942)\u2014to the marvelous room in which we are surrounded by the black _Atelier_ paintings of the late 1940s and early '50s, and the _Billiard Table_ pictures that immediately precede them, to the last room that contains not only the last, very melancholy landscapes but an utterly astonishing painting, from 1960, of a rusted-out plow reposing in a field of bright sunshine, we are shocked into discovering, or rediscovering, what a great painter Braque was in these years. Even where color is allowed, on occasion, to assert its prerogatives\u2014as in two of the most ambitious of the paintings of the 1950s, _Atelier VIII_ (1954\u20131955) and _The Echo_ (1953\u20131956)\u2014the mood remains somber, and the elegiac spirit unbroken. As for all the old arguments of Late Cubism, if we remember them at all after seeing this exhibition, they are bound to seem faintly ridiculous.\n\n_Braque: The Late Works_ is not, to be sure, one of those overblown blockbuster exhibitions of the type that is nowadays devoted to Picasso\u2014and, for that matter, to the likes of Jasper Johns and Andy Warhol. It numbers a mere forty-five pictures that are hung in four ample galleries: one could wish that it were somewhat larger than it is, that the walls of the galleries containing it hadn't been painted a hideous green for the occasion, and so on. But the show itself has been splendidly organized by John Golding, who has also contributed an excellent essay to its catalogue. And it would not have been wise, in any case, to make the exhibition very much larger than it is, given its emotional tenor. If it succeeds in making us wish to see a good deal more of late Braque in the future, that too is all to the good. My principal regret is that the show will not be seen in New York, where I think serious painters are more and more in need of such an exhibition.\n\n[1997]\n\n### Richard Diebenkorn at the Whitney\n\n* * *\n\nFew painters of his generation in America are more highly esteemed today\u2014esteemed, that is, as painters\u2014than the late Richard Diebenkorn (1922\u20131993). Yet visitors to the Diebenkorn retrospective that Jane Livingston has now organized at the Whitney Museum of American Art will quickly discover much that is new to them. Few of us have seen as many examples of the artist's Abstract Expressionist paintings from the 1950s as Ms. Livingston has brought together in this show. There are also some rarely seen landscapes, figure paintings, and still lifes, shown in the company of Diebenkorn's best-known representational paintings. And if only for its stunning selection of paintings and drawings from the artist's Ocean Park series\u2014surely Diebenkorn's crowning achievement\u2014this exhibition would be a capital event. So there is a lot to look at in this retrospective, at least for that surviving remnant of the art public that still takes a serious and knowledgeable interest in the fine art of painting and in the kind of drawing that sustains it. There is also a lot of art history packed into this exhibition. I don't mean the history of reputations and publicity and shocks to the solar plexus, but the history in which we can trace an artist's thinking as he attempts to comprehend both the artistic past and the art of his own contemporaries for whatever may be usable in extending the range of his creative endeavors. Every artist carries in his head\u2014which is to say, his imagination\u2014certain touchstones of quality and achievement that are felt to have a direct bearing on his own developing vision of what at any given moment may be possible in art, and Diebenkorn wasn't shy about acknowledging and acting upon such models. He clearly considered himself to be part of a tradition\u2014a modernist tradition, in his case\u2014that he hoped to advance at something like the same level of quality that he admired in the masters he favored.\n\nHe thus set a very high standard for himself, and for many of his contemporaries as well\u2014a standard that invites us, too, to see his work in relation to the tradition he aspired to uphold. In large part, it was an American standard, and in an even larger part, perhaps, it was a School of Paris standard. But this conjunction of American experience and Parisian modernism is itself the tradition that has so often produced the best American modernist art in this century.\n\nIt is for this reason that on the basis of this retrospective alone one could almost write a comprehensive chronicle of the aesthetic relationships that have tethered the fate of American modernist painting in the last half of the twentieth century to the precedents and standards of modernist painting in Paris in the early decades of the century. In what other American pictorial oeuvre do we find the ideas of such diverse talents as Edward Hopper and Willem de Kooning, Piet Mondrian and Henri Matisse, so deeply pondered, so openly acknowledged, yet so delicately assimilated and so triumphantly transcended?\n\nAt the same time, there is something else in Diebenkorn's art that is important for us to take account of\u2014its West Coast, Californian character, which gives to even his most abstract paintings the look and feel of something pastoral. This is true even of Diebenkorn's California cityscapes, in which the man-made urban environment dissolves into a pastoral patchwork of California light and shadow. In this exhibition the Californian accent announces itself straightaway in the Abstract Expressionist paintings of the 1950s. De Kooning is clearly a central influence in these paintings, and there are hints of Franz Kline and even Mark Rothko. Yet the light in these paintings is not the light of the New York School, and in these paintings, too, the forms are more suggestive of pastoral pleasures than of urban anxieties.\n\nThis may be one of the reasons why it took a while for Diebenkorn's painting to be accepted by critical opinion in New York. In his Abstract Expressionist period, he didn't seem \"tough\" enough to meet the New York\u2014or perhaps I should say, the Tenth Street\u2014macho standard. And then when he started painting figures in the California landscape, the partisans of the New York School hastened to nail him as a provincial, after all. It was all a part of conventional art-world wisdom of the period. (And it wasn't only to Californian figurative painting that it proved to be prejudicial; Fairfield Porter's paintings of Maine and Long Island were treated with the same condescension.) Forty years later, many of those overrated Tenth Street reputations are forgotten, and Diebenkorn has emerged as an underrated master.\n\nThere are certain moments in this retrospective in which we are given such a clear glimpse of the artist rethinking the very basis of his art that they have the quality of high drama. One of these moments comes in the group of small painterly still-life pictures from the early 1960s that occupy an entire wall of the exhibition. Suddenly we are transported back in time to Manet, or some memory of Manet, in paintings that seem to sever all connection with the abstractions and even the landscapes and figure paintings that occupy adjacent galleries. One is reminded of the way Matisse often shifted his ground backward, so to speak, before undertaking his next radical departure.\n\nBut it is in the paintings that register Diebenkorn's most fateful encounter with Matisse's art that the most dramatic moment in this exhibition occurs. When Diebenkorn finally got to see the Matisses in the Hermitage collection in Leningrad (as it was then called) in the 1960s, it was clearly the most decisive moment of his career. From that moment, everything shifts onto a higher plateau of painterly aspiration. You see it first in such pictures as _Recollections of a Visit to Leningrad_ (1965), which should really have been called _Recollections of Seeing the Matisses in Leningrad_ , and the magisterial _Large Still Life_ of 1966. Yet the greatest artistic consequence of that encounter with Matisse in Leningrad was the heroic Ocean Park series, which Diebenkorn commenced to create shortly thereafter.\n\nGoing through this extraordinary retrospective, what came to mind more than once was that remarkable passage in the text that Matisse wrote for _Jazz_ in which he warned that an artist should never become a prisoner of his own style, a prisoner of his own reputation, or a prisoner of his own success. Then he recalls that the Goncourt brothers wrote of certain Japanese artists that they even changed their names in the course of their careers in order to pursue new departures. \"I like that,\" Matisse wrote, \"they wished to safeguard their freedom.\"\n\nFew artists of our time have made more of that kind of \"freedom\" than Richard Diebenkorn\u2014which is why this retrospective is such an absorbing event. It needs to be remembered, however, that you do have to bring a certain interest in the art of painting to fathom what, in Diebenkorn's case, that kind of freedom made possible.\n\n[1997]\n\n### Arthur Dove at the Phillips Collection\n\n* * *\n\n\"Americans are supposed to paint as if they had never seen another picture.\" That disheartening observation was made in a moment of exasperation and despair by the American painter Arthur Dove (1880\u20131946) sometime in the 1930s\u2014a decade that was not an easy period for an artist of his persuasion. By the mid-1930s, Dove had for a quarter of a century been creating pictures that are both remarkably original and yet very much akin to the paintings of certain other modern masters. Yet pictures of the kind he favored, which were either completely abstract or tending toward abstraction even when openly addressed to subjects drawn from nature, were still regarded as an alien presence on the American art scene. There was not yet much of an awareness in the public mind that modernism, while still castigated by academic realists, social realists, regional muralists, and sundry other champions of \"tradition,\" was now itself a vital tradition that an artist like Dove could continue to draw upon for inspiration and support. For many benighted souls, modern art was still thought to be a hoax perpetrated by people who \"had never seen another picture.\"\n\nWith what ardor, invention, and lyric grace Dove did indeed continue to work in this modernist tradition in his later years is made wonderfully vivid for us in the superb retrospective of his work that has now been organized at the Phillips Collection in Washington. _Arthur Dove: A Retrospective_ is quite the best exhibition of this artist's work I have ever seen\u2014and I think I've seen all the major Dove shows of the last forty years. Beginning with a _Still Life Against Flowered Wallpaper_ (1909) and ending with _Flat Surfaces_ (1946), his last major painting, this retrospective does full justice to every phase of Dove's development. There are things in this exhibition that I, at least, do not remember seeing before, and even the most familiar paintings and assemblages are seen in a new light in a survey that encompasses some seventy-eight objects.\n\nIt has become the custom in discussions of Dove's work, owing to the small oils he called _Abstractions_ in 1910\u20131911, to make much of his status as a pioneer of abstract painting\u2014even, perhaps, the pioneer, superseding the claims made for Vasily Kandinsky, as Debra Bricker Balken puts it in the catalogue of this show, \"by maybe a year.\" But as Ms. Balken also points out, the question of what Dove knew or did not know about Kandinsky at that early date remains unresolved and is probably unresolvable. It is exacerbated, moreover, by the attempt made by Dove's dealer and patron, Alfred Stieglitz\u2014who really was familiar with Kandinsky's ideas\u2014to capitalize on Dove's early abstractions in order, as Ms. Balken writes, \"to buttress his [Stieglitz's] growing claims for America's artistic parity with Europe.\"\n\nIn my view, however, \"America's artistic parity with Europe\" did not then exist, and to claim otherwise places an aesthetic and historical burden on those early abstractions of Dove's which the work itself cannot support. To my eyes, anyway, those pictures read more as symbolist paintings based on motifs drawn from nature than as abstract paintings that abjure all reference to recognizable objects. It is no insult to Dove, either, to be reminded that in 1910\u20131911 he was not yet an artist in Kandinsky's league. Alfred Stieglitz had many admirable qualities, as we see illustrated in his early support of Dove, but it is worth remembering that as a dealer he was no stranger to what later came to be called hype. In the New York art world, he was indeed one of its pioneers.\n\nIt isn't, in any case, in the early work in this Dove retrospective that the artist's finest achievements are to be found. Dove doesn't really hit his stride as a modernist painter of remarkable originality until the 1920s when, for the first time, almost every object he produces\u2014collage and assemblage as well as well as paintings\u2014is wholly individual in conception and beautifully realized in the execution. Still largely based on nature, the paintings move in and out of the realm of abstraction so consistently that the distinction often made between abstraction and representation in his work hardly matters.\n\nThe most audacious of his pictures in the 1920s are undoubtedly the assemblages that make use of unconventional materials\u2014which have, alas, become all too conventional in our own day\u2014and the paintings that are similarly executed in mixed media. _Rain_ (1924), for example, is an assemblage composed of twigs and rubber cement on metal and glass. _Starry Heavens_ (also 1924) is a painting executed in oil and metallic paint on the reverse side of a glass plate backed with black paper. The most famous of the assemblages\u2014 _Goin' Fishin'_ (1925)\u2014uses bamboo and the sleeves and buttons of a blue denim shirt, while the extraordinary painting called _Hand Sewing Machine_ (1927) applies oil pigment, cut and pasted linen, and graphite to a sheet of aluminum.\n\nOddly enough, it is in this period, too, that Dove produced his most Kandinskyish abstract painting\u2014 _George Gershwin\u2014\"Rhapsody in Blue,\" Part II_ (1927), which has the look of the kind of \"Improvisation\" that Kandinsky himself had abandoned more than a decade earlier. The connection probably had more to do with the parallels both painters believed united abstract painting and classical music than with any conscious effort on Dove's part to imitate Kandinsky\u2014but the resemblances are nonetheless striking.\n\nBy the 1930s the assemblages and the unconventional materials are largely abandoned as Dove, whose health had begun to deteriorate, concentrates exclusively on the art of painting. In this retrospective, \"late\" Dove emerges as the primary Dove, for it was in his last years that he created his greatest abstract paintings\u2014 _Naples Yellow Morning_ (1935), the three _Sunrise_ paintings (1936), _Flour Mill II_ (1938), _Rain or Snow_ (1943), _Sand and Sea_ (1943), _Roof Tops_ (1943), _That Red One_ (1944), and the final _Flat Surfaces_ (1946).\n\nWhat William Agee says about _Flat Surfaces_ in the catalogue of this retrospective\u2014\"It is a singularly personal variant of abstract art of the time, while closely paralleling the work of Robert Motherwell, and within a few years, of David Smith and Ellsworth Kelly\"\u2014might also be said of certain other late abstractions in the exhibition. My own favorite is _That Red One_ , which I think is a finer, deeper painting than anything I know of in the oeuvres of either Motherwell or Kelly.\n\nThe extent to which Dove was consciously working within the tradition of modernist abstraction during this late period is well documented for us by Mr. Agee in the catalogue. \"In April 1936, while visiting New York,\" he writes, \"Dove had seen and commented favorably on the landmark exhibition Cubism and Abstract Art at the Museum of Modern Art. A week later, on April 23, he visited A. E. Gallatin's Gallery of Living Art at New York University, where he would have seen a broad group of important abstract and nonobjective paintings, including work by Mondrian. . . . Two months later, Dove went to the trouble of translating an article by Mondrian, most likely Mondrian's response to an inquiry into the state of modern art published in a 1935 issue of _Cahiers d'art_.\" And so on. Dove, who spent time in Paris before the First World War, knew what the standards of achievement were for the kind of art he aspired to create, and in the late work he met those standards more consistently than at any other time in his life. It was a considerable achievement, and it is our good luck that this retrospective does it such justice.\n\n[1997]\n\n### Richard Pousette-Dart at the Met\n\n* * *\n\nAmong admirers of the late Richard Pousette-Dart, the exhibition of his paintings that has now been organized by Lowery Stokes Sims at the Metropolitan Museum of Art is likely to elicit some mixed emotions\u2014on the one hand, gratitude and delight that this still underrated and often misunderstood modern master has at last been accorded official recognition by the greatest of our museums; yet, on the other hand, disappointment that the exhibition has come so late and is so woefully unequal in scale to the actual dimensions of the artist's achievement. Richard Pousette-Dart was one of the youngest and most precocious painters of the New York School. He was also one of the most productive, enjoying a career that was far longer than almost any of his contemporaries in the Abstract Expressionist movement. He came to his vocation as an abstract painter earlier than most, and in the painting called _Symphony No. 1,The Transcendental_ (1941\u20131942), he produced at the age of twenty-six what may very well be the first large-scale masterwork of the New York School. And he continued to create paintings of remarkable quality, variety, and size for another fifty years.\n\nYet it is a fact of cultural life in the 1990s that this artist's work still remains more or less unknown to the many people in the New York art world who fancy they know everything there is to know about the art of the New York School. All of this might have definitively changed had the marvelous retrospective organized by Joanne Kuebler in 1990 at the Indianapolis Museum of Art come to New York. But it didn't, and so an entire younger generation in the New York art world\u2014including some of its critics\u2014has been left in ignorance about a body of work never actually seen in all of its amazing variety and depth.\n\nIt is no criticism of Ms. Sims, who has devoted many years to the study of the artist's work, to report that the exhibition she has now organized in _Richard Pousette-Dart, 1916\u20131992_ , is not a retrospective on the scale of the Indianapolis exhibition. Consisting, as it does, of thirty-three paintings from the years 1939\u20131990, plus some early drawings, a few sketchbooks, and a display of brass objects, the current exhibition at the Met is more in the nature of an introduction to the artist's oeuvre than a full-scale account of it. As such, it serves its purpose well, and it has the additional virtue of bringing us certain pictures that few of us are likely to have seen before. The scandal is that an artist of Pousette-Dart's accomplishment still needs an \"introduction\" to our increasingly amnesiac art public.\n\nIt needs also to be said, however, that Pousette-Dart's paintings do not easily conform to certain ideas that have come to be identified with the art of the New York School. There is little in his painting, for example, that resembles the kind of athletic improvisational gesture that characterizes certain works of Jackson Pollock, Willem de Kooning, and Franz Kline, and there is even less that conforms to the hard-edge geometrical forms we find in Ad Reinhardt or the fields of unbroken color in Barnett Newman. From the outset, Pousette-Dart was a painter who brought a more deliberate, painstaking concentration to the physical realization of the painted surface. While many of his contemporaries in the New York School worked toward simplifying their forms and thinning their facture to a watercolorlike transparency, Pousette-Dart remained committed to working by means of a slow accretion of dense painterly touches that brought to his pictures a visual richness and complexity that could not be achieved by any other means.\n\nThe miracle is that a pictorial method so dependent upon the working and reworking of the painting's material density should result in an imagery of such extraordinary spiritual radiance. But this is the miracle we encounter in all of the artist's most successful pictures. It is all the more remarkable when we see, as we are able to do in the current exhibition, that this essential impulse in Pousette-Dart's work has its origins in an art immersed in nocturnal shadows\u2014not only in _Symphony No. 1_ but in _Figure_ (1944\u20131945), _Fugue No. 4_ (1947), and certain other early pictures\u2014and only slowly found its way toward celebrating a kind of celestial light in _Path of the Hero_ (1950), _Chavade_ (1951), and in so many of the paintings that follow in the remaining decades of the artist's life.\n\nIt was in the pursuit of that celestial light, which gave expression to the artist's mystical beliefs, that Pousette-Dart's pictorial imagination was most deeply engaged throughout his long career. Lest it be thought peculiar that painting of this abstract persuasion should owe so much to mystical belief, it is worth recalling that the principal pioneers of abstract painting\u2014Piet Mondrian, Vasily Kandinsky, and Kazimir Malevich\u2014were all similarly disposed to identify their pictorial innovations with a variety of mystical ideas. Far from being an eccentric in this respect, Pousette-Dart may be said to have carried forward one of the central traditions of modernist painting in allying his art with the realm of mystical thought.\n\nFor nonmystics like myself, however, this inevitably raises the question of exactly how we are to respond to the \"content\" of an art so deeply immersed in a mode of mystical belief. The best answer to that question that I know of is to be found in one of the essays of Wallace Stevens, written around 1937, in which this great poet offered some reflections on the problem of \"the irrational\" and \"the unknown.\"\n\n\"The irrational bears the same relation to the rational that the unknown bears to the known,\" wrote Stevens. \"In an age as harsh as it is intelligent, phrases about the unknown are quickly dismissed. I do not for a moment mean to indulge in mystical rhetoric, since, for my part, I have no patience with that sort of thing.\" But then, Stevens reverses his course by observing: \"That the unknown as the source of knowledge, as the object of thought, is part of the dynamics of the known does not permit of denial.\" And further: \"We accept the unknown even when we are most skeptical. We may resent the consideration of it by any except the most lucid mind; but when so considered, it has seductions more powerful and more profound than those of the known.\"\n\nThis may be a useful thought for visitors to the Pousette-Dart exhibition at the Met to bear in mind as they encounter some of the artist's statements that are inscribed on the walls of the show and quoted at greater length in Ms. Sims's contribution to its catalogue. As Stephen Polcari also makes clear in his essay for the catalogue, to ignore such questions in relation to the art of the New York School is to ignore something essential about the ideas that shaped the early course of the Abstract Expressionist movement. It is certainly to ignore something essential about the ideas that governed the art of Richard Pousette-Dart.\n\nBut in regard to our understanding of that art, the current exhibition at the Met must be regarded more as a beginning than a culmination. There is far more to be discovered in Pousette-Dart's oeuvre than the public has ever been shown or the critics have ever written about.\n\n[1997]\n\n### Gustave Courbet at Salander-O'Reilly\n\n* * *\n\nLate Courbet? It's my impression that, unlike late Monet or late C\u00e9zanne, late Courbet is not a subject commonly given special attention, even by the artist's admirers. Gustave Courbet (1819\u20131877) has long been recognized as one of the greatest masters of nineteenth-century French painting, yet much of the last decade of his often turbulent career was\u2014in everything but his painting\u2014a debacle that ended in humiliation and disgrace. It began, to be sure, with the clamor of acclaim. With characteristic bravado, Courbet had mounted a mammoth retrospective of his own work\u2014it consisted of more than one hundred pictures exhibited in a private pavilion\u2014at the time of the Paris World's Fair in 1867, and it proved to be a triumph. The painter who had earlier on been mocked as an oaf and an upstart was now embraced as a living master. As Jean-Jacques Fernier, curator of the Mus\u00e9e Courbet in Ornans, writes in the catalogue accompanying the exhibition of _Gustave Courbet: Later Paintings_ , which has now been organized at the Salander-O'Reilly Galleries, \"During the late 1860s, Courbet was as famous in Frankfurt, Montpellier, Munich, and Deauville as Picasso was on the C\u00f4te d'Azur, or Andy Warhol was in New York. Courbet was a 'star.'\" In 1870 he was even offered the Legion of Honor, which he grandly refused\u2014though a year earlier he had readily accepted a decoration from the King of Bavaria, where his work had won the esteem of the leading German painters of the day.\n\nThen came the Franco-Prussian War, in which France suffered a devastating defeat. Always something of a political radical, Courbet joined the revolutionaries of the Commune that temporarily attempted to govern the city of Paris. When the Commune fell, he was arrested and given a short prison term. Upon his release in early 1872, he found the official art world closed to him. Then a vindictive French government imposed an enormous fine for Courbet's alleged role in the destruction of a public monument while a member of the Commune. Never a teetotaler, he was also drinking too much, and his health was failing. He fled to Switzerland, where he set up a new studio in a small town on the Lake of Geneva. It was there that he died an invalid on the last day of 1877.\n\nYet, being Courbet\u2014which is to say, a painter of irrepressible energy, ambition, and genius\u2014he continued in those last years to work like the inspired monomaniac that he was. According to Mr. Fernier, \"Courbet's production after 1870 is vast,\" and it includes some of his loveliest and most original paintings. Most of these are devoted to subjects drawn from nature\u2014landscape, water, and sky\u2014which are painted with a mastery that is all the more breathtaking when we come to understand the ordeal that Courbet was enduring at the time of their execution. It is only in certain of the \"darker\" paintings of the late period, in which storm clouds may be seen to portend a menacing fate and mountain peaks loom as symbols of an insurmountable destiny, that we may be tempted to see evidence of Courbet's troubled state of mind. On the other hand, it is more likely that he was drawn to these subjects because they offered a fresh challenge to his pictorial powers.\n\nIn the exhibition at the Salander-O'Reilly Galleries, which brings together thirty-four paintings from museum and private collections here and abroad, the earliest of these \"later\" pictures is the spectacular _Fox in the Snow_ (circa 1860), from the Mus\u00e9e Courbet. Considering some of the nonsense that has recently been written about Courbet's animal pictures, it is a salutary experience to be reminded that Courbet was, above all, a virtuosic painter of realistic subjects drawn from nature and not at all a harbinger of the kind of late-twentieth-century postmodern art theory that has now made the study of painting such a farce in our universities.\n\nIt is still another virtue of the current exhibition that it includes two splendid portraits, both painted in 1862, to remind us of Courbet's debt to a pictorial tradition that he continued to venerate even in the course of the many radical changes he brought to it. There is also the wonderful _Portrait of Jo, the Beautiful Irish Girl_ (1866), from the Nelson-Atkins Museum in Kansas City, which was last seen here in the _Courbet Rediscovered_ exhibition at the Brooklyn Museum in 1988. And lest we forget that Courbet was also one of the great painters of the female nude, there is the terrific _Bather_ (circa 1865).\n\nYet the primary focus of this _Later Paintings_ exhibition is inevitably on the landscapes and seascapes, and among them are three extraordinary examples of the so-called wave paintings that must be counted as some of the most moving and original of Courbet's late masterpieces: _The Wave, Stormy Weather_ (1869), _Waves_ (1869), and _The Black Rocks_ (1872). Seeing this _Stormy Weather_ painting, I could not resist wondering if Marsden Hartley had ever had a glimpse of this remarkable picture, for between it and some of the stormy-sea pictures that Hartley painted in Maine in the early 1940s there are many striking resemblances. Even more haunting, in some respects, is _The Black Rocks_ , with its clouds, which seem to be illuminated by an unearthly fire that is reflected in delicate touches on the black rocks in the foreground. That Courbet was able to create so fine a painting at a moment when his worldly fortunes had sunk so low is astounding. But then, of course, Courbet went in for astounding his public. He was the very archetype of the impudent genius\u2014a terrible showoff, in fact, who had the irritating habit of always producing something worth showing.\n\nOne of the latest and gloomiest of the paintings in the show\u2014one of the toughest as well\u2014is a small canvas called _Landscape in the Alps_ (1874), from the Art Institute of Chicago. If only for the brute force with which the mountains in this picture have been laid in with the palette knife, this painting, too, is an unforgettable experience. I don't ever recall seeing this picture on public view at the Art Institute, but there are a number of other pictures in this exhibition that will be equally unfamiliar to most of us.\n\nNeed I add that _Gustave Courbet: Later Paintings_ is an exhibition not to be missed?\n\n[1998]\n\n### Jinny and Bagley Wright Collection at the Seattle Art Museum\n\n* * *\n\nThere are times when one has to travel a certain distance to acquire some perspective on the recent history of the New York art scene. It's not that travel is likely to alter our judgment of artists whose work is already well known to us. That\u2014in my experience, anyway\u2014rarely occurs. The true believers remain secure in their faith; the skeptics remain skeptical. Yet a change of venue\u2014and the kinds of changes in the conversation about art that is its inevitable accompaniment\u2014often does give us a vivid sense of how such work is perceived when it comes to rest in collections far from the milieu of its initial reception. In other words, when the work ceases to be contemporary and becomes historical. Or, as we say now, when it enters the canon of established taste. I have been thinking about this experience since returning earlier this month from Seattle\u2014a city I had not visited since the 1980s\u2014where I had gone to attend the opening of the Virginia and Bagley Wright Collection of Modern Art exhibition at the Seattle Art Museum. While I was delighted to see Seattle again, it was mainly to artistic developments and shifts of taste in New York over the last fifty years that the exhibition of the Wright collection turned my attention.\n\nAmong the earliest works in the show are Willem de Kooning's _Woman_ (1943), Arshile Gorky's _How My Mother's Embroidered Apron Unfolds in My Life_ (1944), David Smith's _Royal Incubator_ (1949), and Mark Rothko's _Number 10_ (1952). Among the most recent are Anselm Kiefer's _The Wave_ (1990) and Katharina Fritsch's _Man and Mouse_ (1991\u20131992), which are also among the largest works in the show. Thus the course that has been traversed on the New York scene from the heyday of the New York School in the 1940s and '50s to the German incursions of the '80s and '90s is well documented in this exhibition, and so are all the best-known intervening developments\u2014hard-edge abstraction, Color-field painting, Pop Art, Minimalism, Neo-Expressionism, and sundry escapades claiming to be postmodern.\n\nOpening at the same time as the show at the Seattle Art Museum, which numbers about one hundred works, was a new public exhibition space in the city devoted to other works in the Wright Collection. Although smaller in scale than the museum show\u2014a mere twenty-two items\u2014the artistic and historical mix was similar, ranging from de Kooning's _Warehouse Mannequins_ (1949) to Tony Cragg's _Eroded Landscape_ (1998). Taken together, then, both shows had the effect of confronting the visitor with the problems that are inherent in current thinking about the canon of modernist art.\n\nOnly the other day, it seems, many of the artists represented in the Wright collection exhibitions were still looked upon as contemporary\u2014meaning, among much else, that their artistic importance was still in contention. Yet in these exhibitions they are presented to the public as exemplars of modern art\u2014which, if it means anything in a period more and more eager to be seen as postmodern, means that the artists in question are believed to occupy a place in a tradition deriving from C\u00e9zanne, Matisse, Picasso, Braque, Mondrian, Mir\u00f3, Kandinsky, Brancusi, and certain other modern masters. That, however, is a very stiff standard for most of yesterday's contemporaries to meet, and in my view very few of them do in fact even come close to meeting it.\n\nYet I believe the Wrights are correct in calling theirs a collection of modern art, for every work in these two exhibitions derives its aesthetic sanction from one or another aspect of the modernist movement. If inspiration is drawn from Duchamp rather than from Mondrian, that is certainly no reason to regard the result as postmodern, for it is only within the boundaries of a modernist sensibility that Duchamp's readymades have any meaning at all. The whole notion of postmodernism is little more than a critical fiction, a strategy designed to escape the standards and strictures of modernism itself when it is not simply a polemical ploy designed to discredit the ideas of Clement Greenberg.\n\nWhat is nowadays advertised as postmodernism would more accurately be described, I think, as late modernism or even low modernism\u2014modernism engaged in a fierce polemic with itself. Polemics of that kind have been a constituent strain of the modernist movement from the outset, however, and it doesn't advance our understanding of anything that has occurred in the art of the 1980s and '90s to call it something else. As we are all, perforce, the creatures of a modernity we cannot escape, so is our art of every persuasion a modernist art of one variety or another.\n\nIt is indeed the primary interest of the Wright collection that it embraces that variety without attempting to resolve its many conflicts and contradictions. The story of how the collection was assembled is recounted by Bagley Wright in a wonderful essay for the catalogue of the museum show called \"Our Lives as Collectors.\" In the early 1950s, Jinny Wright, fresh from studying art history at Barnard, was working at the Sidney Janis Gallery. Bagley Wright, just out of Princeton, where he had majored in English, was working on the night shift of the _New York Mirror_ , a long defunct tabloid he describes as \"the last of the spit-on-the-floor dailies.\"\n\n\"What a marvelous time it was to be a collector,\" he writes, \"when only a few collected and the best paintings being done anywhere in the world were available for a few hundred or at the most a few thousand dollars. From this period came the de Kooning, Guston, Rothko, and Pollock that formed the foundation of our collection.\" About acquiring the Rothko, he tells the following story: \"The first really important large painting Jinny acquired was Mark Rothko's _Number 10_ , but she got it only after a protracted negotiation. The problem wasn't the price ($1,200), it was Jinny. Was she worthy, in Rothko's judgment, to own one of his masterpieces? His dealer at that time, Betty Parsons, arranged a meeting. Rothko was charmed by Jinny. With certain restrictions\u2014no exhibitions, no photos\u2014she was allowed to have it. I doubt that Rothko had sold more than a dozen paintings in his life, but like all the Abstract Expressionists, he had a high opinion of himself and his calling.\"\n\nIn this memoir, Mr. Wright also gives us a shrewd and affectionate account of Clement Greenberg and pays tribute, too, to the dealer Dick Bellamy, both of whom he acknowledges as important influences on the collection. Especially in his evocation of the New York art world in the 1950s and early '60s, he has a very engaging story to tell, and he tells it well. He doesn't hesitate to recount the kind of resistance the Wrights' efforts on behalf of some of their favorite artists have sometimes met with, either. When the Virginia Wright Fund commissioned a Mark di Suvero sculpture for the campus of Western Washington University, it was assumed it would be a great hit. Instead, however, \"They denounced his unpainted steel construction as the detritus of a military-industrial society. Only after it was painted red did they grudgingly accept it as art.\"\n\nIt was even worse with a Richard Serra commission. Failing to find an admiring public, Mr. Wright writes, \"It has become a maintenance problem, the target of graffiti and scurrilous pamphlets plastered on its surface.\" All of which is a reminder that not all the criticism that is directed at the varieties of modernist art in the Wright Collection is what is written in the papers.\n\n[1999]\n\n### Honor\u00e9 Daumier at the Phillips Collection\n\n* * *\n\nThere are retrospective exhibitions of well-known artists that ought to be accompanied by a warning label that would read: Do not understand this artist too quickly! In other words, what you know, or think you know, may not be all there is to know. The French painter, sculptor, and caricaturist Honor\u00e9 Daumier (1808\u20131879), whose work is currently the subject of a large and wonderful exhibition at the Phillips Collection in Washington, D. C., is an artist who would certainly benefit from such a cau-tionary injunction, especially on this occasion. For this is, astonishingly, the first ever retrospective to give the public a comprehensive account of the entire range of Daumier's achievements. Famous he is, of course, and rightly so, for Daumier is an immensely appealing and accomplished artist. In every aspect of his work, he brings us closer to the life of his time at every level of society and culture than any other visual artist of his period, which was one of the greatest periods in the history of French culture. It is for this reason that his oeuvre has often been compared to Balzac's. Yet Daumier's fame mainly derives from but one aspect of his copious and many-sided endeavors: the pungent and abundant caricatures of contemporary social and political life that made him something of a culture hero in the middle and later decades of nineteenth-century France. That he might also have been a painter of considerable consequence, a sculptor of distinction, and a draftsman of genius are not matters that, until now, anyway, the public has been in much of a position to appreciate.\n\nIt is one of the many virtues of this Daumier retrospective that it accords to all of the artist's accomplishments the kind of concentrated attention they merit. The caricatures that made him famous are by no means neglected or diminished. They did indeed occupy a very large part of Daumier's life. He produced nearly five thousand of these satirical prints over the course of a fifty-year period, and for one of them\u2014his horrific and hilarious _Gargantua_ (1831), a scatological caricature of the French monarch Louis-Philippe\u2014he was arrested and even served a brief prison term. The labor of study and selection that enabled the curators of the exhibition to reduce this enormous corpus of prints to a representative sampling of seventy-four lithographs for this show is impressive, to say the least, and does much to explain why such a retrospective had not been attempted before.\n\nEven in this aspect of Daumier's oeuvre, which is the most familiar to most of us, there are many revelations to be savored. For they contain a range of subjects and sympathies, of irony and indignation, comedy and pathos, that is not customarily associated with the art of caricature. There is also a technical virtuosity that confers on each of its subjects, from the most repulsive to the most sympathetic, a finely judged emotion that is totally devoid of cant or sentimentality. No profession or social class is exempted from Daumier's undeceived comprehension of the human comedy, and every comic touch is accompanied by a mordancy that can make us shudder even before we have stopped laughing. It is no wonder that Baudelaire compared Daumier's caricatures to the plays of Moli\u00e8re.\n\nIn this exhibition, however, this classic corpus of caricatures shares attention with no less than seventy-five paintings, fifty-seven drawings and watercolors, and thirty-nine sculptures. It is the paintings that are likely to be the greatest revelation in this retrospective, for we have never before seen so many of them assembled in a single exhibition. Painting was anything but a sideline for Daumier. It was in fact the medium in which he made his most profound contribution to the art of his time. That he appears to have been entirely self-taught as a painter makes his achievement in this medium all the more remarkable.\n\nAs a painter he is usually classified as a Realist, but with a contemporary Realist like Courbet he had little in common. Daumier's is a much darker sensibility than Courbet's, a sensibility that is closer to that of the \"black\" paintings of Goya than to the sunlit world evoked in Courbet's landscapes. Daumier's is also a more urban, more cosmopolitan vision than Courbet's. The solitude that is Daumier's most powerful subject in his paintings is an urban solitude. The virtuosic attention to detail that is one of the glories of the artist's lithographs is often abandoned in the paintings in favor of a pictorial drama of light and shadow, with shadow the dominant element. Not always, to be sure. In a painting like _A Third-Class Carriage_ (circa 1862\u20131864), in which a subject explored in the artist's lithographs is carried over into painting, certain details remain. But even there they are enclosed in a pictorial structure of shadow and silhouette. And in the later masterpieces like _Two Sculptors_ (circa 1872\u20131875) and the first version of _The Painter at His Easel_ (circa 1870\u20131875), detail is radically occluded as we are plunged into a maelstrom of painterly dusk.\n\nWith these late paintings, which were produced when Daumier had abandoned caricature and was living in impoverished circumstances, it hardly makes any sense to consider him a Realist. He had in fact become a kind of Expressionist _avant la lettre_ , and this was what he was seen to be by the very few connoisseurs of modern painting in the early decades of the twentieth century who were drawn to his work in this medium. One of the most perceptive of these connoisseurs was indeed Duncan Phillips, who, in founding the Phillips Collection in 1921, created the first museum of modern art in the United States. Phillips understood better than most that Daumier had created something distinctly modern in his painting. Which is why some of the greatest of Daumier's paintings, including the two late paintings already mentioned, were acquired early on for this distinguished collection of modern art.\n\nNo doubt this is also why the Phillips Collection is the only American venue for this marvelous Daumier retrospective, which has been organized by the Phillipses in collaboration with the National Gallery of Canada and the R\u00e9union des Mus\u00e9es Nationaux in France.\n\n[2000]\n\n### Thomas Eakins at the Philadelphia Museum\n\n* * *\n\nFor a large part of the American art public, the Philadelphia painter Thomas Eakins (1844\u20131916) stands alone and unrivaled as the classic American representative of the Realist style. With subjects ranging from water sports and baseball to affectionate scenes of domestic life and portraits that are penetrating character studies of his Philadelphia contemporaries, Eakins created a corpus of American types and pastimes that, for sheer quality, gravity, and intelligence, no subsequent champion of the Realist style has been able to match.\n\nThat his career was also marked by scandal, rejection, and outright character assassination only adds to the luster of a fame that has elevated Eakins to the status of a national icon. We like our artist-heroes to be fiercely independent and disruptive, and Eakins leaves little to be desired in this respect, too. He made it a habit to offend established taste, not only in painting but in the realm of public morals as well, and this has also endeared him to posterity. Not that offending established taste was a difficult thing to accomplish in the Philadelphia of his day, which was nothing if not staid in its standards of respectability. Eakins even dressed for the role, affecting the kind of shabby attire that polite society looked upon as impermissibly \"bohemian,\" especially for a teacher at the remorselessly respectable Pennsylvania Academy of the Fine Arts.\n\nThus, while Winslow Homer was undoubtedly more beloved, and both William Merritt Chase and John Singer Sargent were far too eager to please to be cast in a heroic role, Eakins remains a singular figure in the American art of his time\u2014and indeed, in ours as well. Which is one reason why the comprehensive retrospective that has now been organized at the Philadelphia Museum of Art\u2014 _Thomas Eakins: American Realist_ \u2014is bound to be a huge success. Another reason, of course, is that Eakins brought a powerful, if narrow, talent to an oeuvre that is wonderfully accessible to every level of public perception. This is pictorial Realism un-burdened by the modern appetite for ambiguity, irony, or obscurity. It is an art in which easily recognized sentiments and actions are clearly expressed in a manly, robust style that commands attention without challenging the boundaries of the mainstream imagination.\n\nThat the aesthetic elements in Eakins's art were deeply conservative, with affinities closer to academic tradition than to avant-garde innovation, has seldom been held against him. In fact, it is seldom even acknowledged. Owing to his reputation as a rebel and martyr at the Pennsylvania Academy, Eakins has been transformed in our literature into a kind of honorary modernist r\u00e9volt\u00e9, yet this is a classification that is denied in every square inch of painted canvas Eakins ever put his hand to. In every respect but one\u2014his unremitting and incorruptible candor in registering the impact of his personal response to the American life of his time\u2014Eakins remained a votary of academic tradition.\n\nYet this radical candor\u2014which was abetted in Eakins's case by his keen interest in scientific standards of objectivity, and which was itself a spur to clarity of expression\u2014was sufficiently rare and sufficiently affecting to set him apart from his contemporaries. Yet it was a candor expressed in an academic mode. No one understood the academic roots of Eakins's art better than his widow, Susan Eakins, who was herself a gifted painter, very much influenced by her husband's pictorial practice. Thus, when Lloyd Goodrich's pioneering study of the artist, _Thomas Eakins, His Life and Work_ , was published in 1933, she was appalled to find that Goodrich had \"disregarded Eakins' roots in the French academic tradition,\" as Carol Troyen writes in the catalogue accompanying the current retrospective.\n\nSusan Eakins understood that, far from being a painter \"little influenced by others,\" as Goodrich had insisted, Eakins had been deeply influenced by his period of study in Paris under Jean-L\u00e9on G\u00e9r\u00f4me, a _retardataire_ Academician whose work, as Goodrich surely understood, commanded nothing but contempt in advanced art circles in the 1930s. Susan Eakins thought otherwise, of course, observing that \"when a man cannot understand the greatness of G\u00e9rome I cannot think he understands Eakins.\" To what extent Susan Eakins came to understand that her husband had been a far greater painter than G\u00e9rome\u2014and greater by virtue of his radical candor in rendering the look and feel of contemporary life\u2014must remain a matter of speculation.\n\nIn the current exhibition we are given a glimpse of the kind of academic painter Eakins might have remained absent that candor in an ambitious painting of a conventional religious subject, _The Crucifixion_ (1880), which\u2014in my view, anyway\u2014is the least persuasive of all of the artist's major canvases. Compare it to the intensity of concentrated attention in _The Gross Clinic_ (1875) or the pathos of _The Concert Singer_ (1890\u20131892) or the controlled passion of the rowing pictures, and you see the difference between an artist attempting to conform to convention and an artist inspired by the depth and complexity of his own response to life.\n\nIn no other Eakins exhibition that I've seen have we been given such a detailed account of the intellectual and aesthetic labor that Eakins invested in perfecting his Realist masterpieces. In addition to the expected drawings and oil sketches in the Philadelphia retrospective, we are also given a plethora of his sculptural studies and the photographs that were used in the preparation of his major paintings. Indeed, there may be a few too many photographs in the current show for most viewers, for they are not all invariably interesting, and the sculpture, too, is mainly of interest as preparatory studies for the paintings. The sculpture, certainly, can leave no one in doubt about Eakins's academic loyalties.\n\nIt is in the paintings\u2014and in the paintings alone\u2014that Eakins's genius was fully developed. And it is in the paintings, too, that he remains an isolated figure in the history of American art. No painter before Eakins was as audacious as he was in his account of American experience, and none that came after was in a position to resist with impunity the innovations of the modern movement. In his lifetime, Eakins paid a high price for his singularity of purpose\u2014the singularity of a radical conservative\u2014yet it's hard to believe that he could have achieved what he did without the social resistance he met with at almost every stage of his career. It is in this respect that he can be said to resemble one of his idols, Walt Whitman, who was the subject of one of his finest portraits.\n\n[2001]\n\n### The Thaw Collection at the Met and the Morgan Gallery\n\n* * *\n\nOf the many intellectual faults that have plagued the study of art in recent decades, one of the least forgivable has been the campaign to discredit the idea of connoisseurship. At the outset, about a quarter-century ago, it was a campaign largely confined to university art history departments, where training in connoisseurship\u2014which concentrates on aesthetic distinctions\u2014came to be more and more disparaged, at times even vilified, in favor of sociological and political standards in assessing artistic achievement. Inevitably, this campaign spread to the museums and the media as our academic institutions turned out cadres of art history graduates to serve as curators and critics. Since their training disposes many of these graduates to regard the study of art as a branch of the social sciences\u2014if not, indeed, as a political science\u2014this is the primary intellectual perspective they bring to their curatorial and critical endeavors. As a consequence of this folly, it's not uncommon to find curators and critics in high places who are simply incapable of making an independent assessment of the aesthetic quality of the works of art they address. Hence their abject reliance on the reputations and fashions of the moment.\n\nThe good news, however, is that connoisseurship is alive and flourishing elsewhere\u2014most conspicuously among independent collectors, who, through patient study and analysis of the specific objects that interest them, have proven again and again to be fully capable of distinguishing aesthetic quality from its absence. These independent collectors are the unsung heroes and heroines of the art world, and not only because of their exemplary judgment but also because they remain the principal donors of great works of art to our museums.\n\nI have been vividly reminded of all this once again by two current exhibitions: _The Thaw Collection: Master Drawings and Oil Sketches, Acquisitions Since 1994_ at the Morgan Library, and _Nomadic Art of the Eastern Eurasian Steppes_ at the Metropolitan Museum of Art. From the collections of drawings that have been assembled by Eugene and Clare Thaw since the 1950s and donated in stages to the Morgan Library, we have come to expect both top quality and remarkable scope, and the latest survey of their acquisitions adds something new: some forty-nine oil sketches on paper and canvas dating from the eighteenth and nineteenth centuries.\n\nOutstanding among them is a superb landscape by the English master John Constable, _A View of Hampstead Heath_ (circa 1821\u20131822), with its breathtaking masses of clouds in a brilliantly illuminated sky. The biggest surprises, however, are likely to be found among the many drawings and oil sketches by German and Scandinavian artists. Museum-goers who have lately discovered the work of Danish artist Vilhelm Hammershoi (1864\u20131916) will be pleased to find an exquisite pencil drawing by him, _Study for Interior of San Stefano Rotondo, Rome_ (1902); and admirers of the German painter Adolph von Menzel (1815\u20131905), who now enjoys something akin to a cult status, will be thrilled by the undated drawing, _Sleeping Youth, His Head and Arm Resting on the Back of a Sofa_.\n\nNor have Italian, French, and Dutch masters been neglected in this latest selection of the Thaws' drawings, for strongly represented, too, are works by Vittore Carpaccio, Ingres, Delacroix, Degas, Pissarro, Redon, Mondrian, Morandi, Seurat, Corot, and Rembrandt. It all adds up to a tremendous feast for the eye and the mind\u2014and a feat of connoisseurship that is without equal in recent times.\n\nThe sheer scope of the Thaws' artistic interests has long been recognized in the art world; Mr. Thaw is also a recognized authority on the work of Jackson Pollock. But nothing could have prepared us for his latest collecting project: the more than two hundred objects, largely drawn from his collection, in the exhibition of _Nomadic Art of the Eastern Eurasian Steppes_ at the Met. Dating from the tenth century B. C. to the second century A. D., these small, sometimes miniature sculptural objects in bronze, gold, silver, and jade were produced to serve as horse harnesses and chariot fittings, belt ornaments, garment plaques, weapons, and vessels. Dominated by animal imagery, which at times is highly realistic but also given to fantastic stylization, they are certain to be a revelation to visitors of the exhibition.\n\nAccording to James C. Y. Watt, chairman of the Met's Department of Asian Art, \"The present-day knowledge concerning the art of the nomadic world is insufficient for a systemic art-historical exposition.\" Undaunted by this lack of documentary literature, Mr. Thaw has assembled a dazzling collection of these bizarre and often enchanting objects, and the Met has responded to still another of his feats of connoisseurship by mounting this exhibition in an installation that is exemplary both for its design and its discreetly informative wall texts.\n\nOne of these days, when training in connoisseurship is restored to its rightful position in the academic study of art history, collectors like Eugene and Clare Thaw and institutions like the Morgan Library and the Metropolitan Museum will be remembered for upholding appropriate standards while the tenured radicals in the universities betrayed their academic responsibilities. Meanwhile, for anyone who wants lessons in connoisseurship, _The Thaw Collection: Master Drawings and Oil Sketches_ is on view at the Morgan Library, and _Nomadic Art of the Eastern Eurasian Steppes: The Eugene V. Thaw and Other New York Collections_ remains on view at the Met.\n\n[2002]\n\n### Lois Dodd at Alexandre Gallery\n\n* * *\n\nLet's face it: There's a class of highly accomplished American painters whose work has been consistently rejected by the New York museum establishment when it comes to mounting full-scale retrospective exhibitions. Although otherwise diverse in style and spirit, these painters can generally be characterized as representational but not confrontational. They deal with recognizable subjects but do not set out to shock or dismay; on the contrary, their work offers us aesthetic delight and intellectual probity. They shun the so-called cutting edge and give us instead a deeply pondered personal vision of a world we can recognize as our own. These painters are not discriminated against according to generation. I'm old enough to remember a time when Milton Avery (1885\u20131965) and Fairfield Porter (1907\u20131975) were both relegated to the class of talents denied museum retrospectives. Why? They were apparently considered insufficiently avant-garde by doctrinaire modernists and yet too modern to be embraced by doctrinaire traditionalists. So it was left to posterity: The museums granted them posthumous recognition.\n\nThe exhibition of paintings by Lois Dodd at the Alexandre Gallery is a reminder that at the age of seventy-five, she too now belongs to this class of neglected talents. For close to half a century, Ms. Dodd's work has been seen and admired in more than fifty solo exhibitions. One of her recent shows ( _Women at Work_ , at the Caldbeck Gallery in Rockland, Maine), a rare (for Ms. Dodd) and often hilarious survey of naked female figures performing common household tasks, was a runaway sensation. It was the kind of show that visitors returned to again and again\u2014as I did. The show was great fun as well as an experience of high-octane painterly virtuosity.\n\nIt's unlikely, however, that a single New York curator ever bothered to see it, even though Ms. Dodd has been a presence on the New York art scene for decades. She studied at Cooper Union in the 1940s; she was one of the founders of the Tanager Gallery in the '50s; and she taught at Brooklyn College for many years. Like a great many other New York painters, past and present, she has found some of her best subjects in rural Maine.\n\nAs is often the case in New York these days, the galleries provide what the museums overlook or deny. While the exhibition at the Alexandre Gallery is hardly the Dodd retrospective we needed, it does have the great virtue of giving us a concentrated account of one of the artist's most inspired inventions: the complex, highly poetic pictorial compositions based on the structures and settings and shifting light to be seen in and around the windows and doorways of old Maine houses.\n\nTo these pictorial inventions, Ms. Dodd brings a commanding mix of realism and abstraction. Fidelity to exact observation is tempered and elevated by the discipline of formal rigor. If there are identifiable affinities or sources for such a style, they are likely to be found in the precisionism of Charles Sheeler and certain other varieties of American Cubism, as well as the geometric abstraction of Piet Mondrian. Yet nature, too, is given its due, especially in the depiction of foliage and seasonal change. And it's in her brushy, fragmented renderings of nature\u2014often seen reflected in rectangular window panes\u2014that Ms. Dodd also reminds us that she came of age as an artist in the era of Abstract Expressionism.\n\nIt's interesting to observe how this range of abstractionist conventions and styles is adapted to the spatial ambiguities of her window and doorway subjects. Is there an allusion, perhaps, to the all-black abstract paintings of Ad Reinhardt or even the early abstractions of Malevich in the series of black rectangles to be seen in the painting called _Barn Window with White Square_ (1991)? Probably not, but the comparison nonetheless leaps to mind.\n\nOn the other hand, there is certainly something Hopperesque in the haunting _Night House_ (1975), even though Ms. Dodd's use of realism differs in many respects from Edward Hopper's vein of anecdotal melancholy. (In an essay for the catalogue of the current show, John Yau makes a strong case for Ms. Dodd as \"Hopper's heir,\" but I remain unpersuaded; the complexities in every development of her oeuvre are too distant from Hopper's dour, unforgiving realism. No heir of Hopper's could have possibly conceived of a show like _Women at Work_.) It is, in any case, in a painting like _Falling Window Sash_ (1992) that the artist's stunning combination of realism and abstraction is most elaborately developed.\n\nWell, it's clearly going to take a while for the New York museums to catch up with Lois Dodd's fast-paced development\u2014and it may not happen in her lifetime, or mine. But when such a retrospective does come to New York, it will be a smash.\n\n[2003]\n\n### Christopher Wilmarth at the Fogg Museum\n\n* * *\n\nThe American sculptor Christopher Wilmarth (1943\u20131987), whose drawings are the subject of a splendid exhibition at Harvard University's Fogg Art Museum, was one of the most remarkable artists of his generation. He was remarkable not only for extending the aesthetic frontiers of sculpture and drawing but for the subtlety and depth of thought he brought to their creation. The materials he favored\u2014steel and glass\u2014obliged him to master the crafts of a seasoned factory worker. Yet from the outset of his career as an artist, Wilmarth's closest affinities were with certain aspects of Matisse and then, in his later work, with the sensibility of the nineteenth-century French poet St\u00e9phane Mallarm\u00e9. Wilmarth's description of his own work as \"physical representations of states of mind, reverie, interiors, spirits\" is itself reminiscent of Mallarm\u00e9's most celebrated dictum: \"Describe not the object itself, but the effect it produces.\" Until now, Wilmarth's sculpture\u2014large, highly poetic constructions of steel, etched glass and steel cable\u2014have been better known to the art public than the drawings that were central to everything he aspired to achieve as an artist. That perception is now likely to be altered, however, by the exhibition called _Christopher Wilmarth: Drawing into Sculpture_ , which Edward Saywell has organized at the Fogg. This enchanting exhibition, together with Mr. Saywell's illuminating catalogue, establishes beyond doubt the crucial role that drawing occupies in the Wilmarth oeuvre. Indeed, we are reminded that certain works that we might have regarded as sculpture\u2014the small, exquisite wall constructions made of etched glass and steel cable\u2014were actually entitled _Drawings_ by the artist.\n\nAbout these particular works, Mr. Saywell writes: \"Most extraordinary is the group of drawings that [Wilmarth] made in the early 1970s, not with pencil and paper but with etched glass, which acts as the drawing layer, and steel cable, the calligraphic surrogate for the graphite mark.\" Hence the title of this exhibition: _Drawing into Sculpture_. Yet the bulk of the drawings in the exhibition are indeed on paper, usually graphite on gesso-prepared off-white wove paper; and the graphite is sometimes combined with graphite wash, as it is in _Invitation #1_ (1976). The drawings on paper encompass a broad range of function, style, and expression. Some are diagrammatic drawings for future sculptures while others are highly finished drawings of sculpture made earlier. As Mr. Saywell also writes: \"For the most part drawing was a retrospective activity for Wilmarth, a means for him to return to and assess specific aspects of existing sculptures, occasionally years after their completion. Acting almost like an appendix or index to the sculptures, his drawings helped mediate the transition from one sculpture, or group of sculptures, to the next.\"\n\nStanding quite apart from the many abstract drawings in the show are a knockout figurative drawing of a female nude, _Shifrah_ (1964), in which Wilmarth made the most explicit acknowledgment of his debt to Matisse; and a related drawing, _Yolande_ (1965), in which the Matissean forms, though still explicitly legible, are in the process of being transmuted into something more identifiable as Wilmarthian.\n\nIn some respects, however, the most haunting works in the show are the abstract drawings of heads, and the _Breath_ series that recorded Wilmarth's aesthetic and spiritual response to Mallarm\u00e9's poetry. In the late 1970s he was asked by the American poet Frederick Morgan to recommend an artist to illustrate his translations of seven poems by Mallarm\u00e9. As soon as Wilmarth read these translations, he felt he had found something like a soul mate in Mallarm\u00e9 and undertook to create the illustrations himself. Immersing himself in a study of Mallarm\u00e9, Wilmarth came to the realization that \"this is a guy who just lives in his head,\" and this proved to be the key to his Mallarm\u00e9 drawings and the blown-glass sculptures of heads that followed. As Mr. Saywell observes of this encounter: \"Using that metaphor as the key, Wilmarth settled on the image of an oval sphere to translate the poetry of Mallarm\u00e9. For each of the seven poems, he took advantage of the many metaphorical connotations of the oval as heart, soul, or mouth, to make a series of drawings first in charcoal, then in pastel.\" These and the watercolor-and-graphite drawings in the _Breath_ series constituted Wilmarth's own homage to the poet's vision.\n\nNear the entrance to the Wilmarth exhibition at the Fogg, a wall text quoting a passage from the artist's writings underscores the extent to which he had embraced this Mallarm\u00e9an vision in his own art. It reads: \"Light gains character as it touches the world; from what is lighted and who is there to see. I associate the significant moments of my life with the character of the light at the time. . . . My sculptures are places to generate this experience compressed into light and shadow and return them to the world as a physical poem.\" Wilmarth's drawings are also an integral part of that poetry.\n\n[2003]\n\n### Max Beckmann at MOMA\n\n* * *\n\nIn a rare collaboration between two elite art dealerships, Richard L. Feigen and Co. and the Jan Krugier Gallery have joined in organizing an exhibition devoted to a pair of major artists\u2014Pablo Picasso (1881\u20131973) and Max Beckmann (1884\u20131950)\u2014whose works, though they belong to the same generation of European modernists, are rarely seen in close, comparative proximity. The result, on view at the Feigen gallery, is Beckmann-Picasso\/Picasso-Beckmann, a show that is not to be missed. It's not only of compelling interest in itself, but it explores a relationship in the history of modern painting that, as far as I know, has not heretofore been closely studied.\n\nPicasso needs no introduction; he has long enjoyed a fame that's universal in scope. This doesn't mean that his work is universally understood, however: in Picasso's case, it has often meant the opposite. What many people think they know about Picasso has far more to do with his public persona\u2014his sex life, his politics, his longevity, his fecundity in creating a variety of styles in a variety of media, and his unembarrassed love of the limelight\u2014than with his artistic achievements. Such has been his celebrity that his name is now synonymous with the very idea of the artist-genius, even among people who remain baffled by his work.\n\nMax Beckmann, on the other hand, has never enjoyed a celebrity on this scale, even in America, where he lived and worked as an \u00e9migr\u00e9 in the last years of his life and as a teacher, first in St. Louis and then in New York, influencing an entire generation of American painters. Moreover, Beckmann's private life, which was centered on a long and happy marriage, remained private, and his politics\u2014insofar as he can be said to have had any\u2014were limited to his opposition to the Nazi regime in his native Germany, which famously declared art of his persuasion degenerate.\n\nYet, despite Beckmann's opposition to the Nazi regime, his German background has been a problem for many art lovers. This has less to do with politics than with style and aesthetics. The fact is, for a majority of the art public in this country, the school of Paris\u2014Picasso, Matisse, Leger, Bonnard, et al.\u2014is greatly favored over that of the modern German school, which for tastes nurtured on Parisian aesthetics is often found to be too harsh, too strong, too naked in feeling.\n\nEven if you reject this criticism of modern German art as a caricature, as I do, it has to be acknowledged that it exists, and it has sometimes proved damaging to Beckmann's reputation. In the presence of a powerhouse talent like Picasso's, however, Beckmann's strengths as a draftsman and painter show to great advantage, and he emerges in the current exhibition as a powerhouse himself\u2014and does so without the presence of a single example of his finest work, the great triptychs of the climactic years of his maturity.\n\nIn this connection, it has to be said that it's one of the downsides of this Picasso-Beckmann standoff that it's mostly limited to smallish paintings and works on paper. (The few sculptures hardly seem to be part of the show.) By the same token, it's one of the upsides of the exhibition that it makes us all the more eager to see a Picasso-Beckmann show on a far grander scale. Not until we have an opportunity to see _Guernica_ (1937) and other Picasso paintings on a similar scale in the same space as some Beckmann triptychs will we be in a position to make a definitive judgment of their respective pictorial achievements.\n\nMeanwhile, in the show that we've been given at Feigen, Beckmann more than holds his own. In the very first room, Beckmann's _Oyster Eaters_ (1943) commands a painterly sensuousness that triumphs over Picasso's schematic _Bouquet_ (1969), just as his radical _Crouching Woman_ (1930) renders Picasso's more delicate _Femme Dormeuse_ (1937) almost mute. For this viewer, anyway, even Picasso's more powerful _Femme Assise en Robe Grise_ (1943), with its emphatic black structure, has too formulaic a look to be persuasive.\n\nThe really big surprise in this exhibition is its revelation of Picasso's keen interest in Beckmann's paintings, which seems to have had a longer history than any of us suspected. Once we've been alerted to this connection and its implications, both Beckmann and Picasso emerge somewhat changed for us\u2014changed, that is, from powerful individualists into comrades in pursuit of very similar goals.\n\n[2004]\n\n### Manet at the Art Institute of Chicago\n\n* * *\n\nThe French artist \u00c9douard Manet (1832\u20131883) is often said to have been the first modernist painter\u2014the father, as it were, of Impressionism and the great succession of avant-garde movements that followed in its wake. In fact, there's just enough truth in the claim to render it plausible, even at this distance in time. But it was never the whole truth about Manet\u2014who could scarcely touch a brush to a canvas without recalling the Old Masters\u2014and it has frequently led to a misunder-standing of the artist's relation to the tradition whose lofty standards the nineteenth-century reactionaries in the French Academy accused him of violating. By their lights\u2014not always the brightest\u2014the champions of the academy undoubtedly had a point: though Manet clearly made a close and sympathetic study of certain Italian and Spanish masters, it was a study that resulted in a determination to modernize his medium and thus bring it into close alignment with contemporary experience. The benighted exponents of \"tradition\" could not forgive him for this audacity, which made him a hero to the newly emergent avant-garde.\n\nTwo provocative figure paintings of 1863\u2014 _Le Dejeuner sur l'Herbe_ , based on Giorgione's _F\u00eate Champ\u00eatre_ , and _Olympia_ , based on Titian's _Venus of Urbino_ \u2014provoked famous scandals. And it is mainly for his figure compositions\u2014of which, in my view, the greatest is the _Bar at the Folies-Berg\u00e8re_ (circa 1882)\u2014that Manet is widely admired today.\n\nBut in the exhibition called _Manet and the Sea_ , currently at the Art Institute of Chicago, the focus is on a very different aspect of the artist's achievement. For most of us, Manet's interest in marine painting has remained a more or less marginal subject, but it turns out to have been a major one for the artist himself. Although we were recently given a glimpse of Manet's initial foray into marine pictorial subjects (a battle scene at sea in a show called _Manet and the American Civil War_ at the Metropolitan Museum), the exhibition in Chicago is a far more ambitious production, with nearly one hundred marine works, including paintings, watercolors, and drawings by Manet and his contemporaries\u2014among them Delacroix, Courbet, Whistler, Monet, Renoir, Jongkind, Boudin, and Morisot.\n\nWhile it would be going too far to suggest that with _Manet and the Sea_ the artist emerges as a radically changed painter, energetic engagement with marine subjects does seem to have unleashed an impulse in Manet that we do not find in his earlier work\u2014an impulse to confront the untamed forces of nature. In this respect, too, Manet was once again in the avant-garde, for as John Zarobell reminds us in his catalogue essay for _Manet and the Sea_ : \"There was, in fact, something like an explosion of marine painting in the 1860s among artists not connected with the Academy or bound by official commissions. . . . As the behemoths of official marine painting declined in artistic importance, the field was left open to experimental artists who did not much care about their relationship to the artistic establishment in Paris and who sought, in the sea, a means of furthering their inquiries into the relationship between the self and the natural world.\"\n\nManet was in one respect better qualified to deal with the subject than many of his contemporaries: At the age of sixteen, he had gone to sea as a sailor, spending three months on a ship bound for Brazil in order to qualify for a career as a naval officer. But he wasn't much good at passing the requisite exams for such a career, and it's our good fortune that he fell back on pursuing an artistic career as an alternative.\n\nAs a consequence of his experience as a sailor, Manet was on more intimate terms with the sea and its manifold mysteries than most of the other painters of the time\u2014and far less daunted by it. Courbet, especially in his wave paintings, often responded to the challenge of sea painting as exercise in physical combat, in which he was determined to subdue an unruly antagonist; Renoir remained utterly undaunted, painting his ocean waves with the same brushy delicacy he brought to the depiction of bourgeois ladies' frilly dresses; and Monet appears to have been steadfastly faithful to his direct observation of the sea and its seaside visitors. But Manet seems to have made no distinction between observation and invention in rendering a subject he knew by heart, so to speak.\n\nWhat lends still another layer of interest to _Manet and the Sea_ are the many glimpses it gives us of those visitors to the seaside who, either as swimmers or leisurely observers, inaugurated a pastime so familiar to us today. Between paintings like Manet's _On the Beach at Boulogne_ (1868) or the _Departure of the Folkestone Boat_ (circa 1868\u20131872) and the steady production in our time of similar seaside scenes at Provincetown, the Hamptons, and Cape Ann, there's an obvious connection\u2014despite vast differences in aesthetic quality. The sea may be eternal, but the social uses that are made of it bear the stamp of period sensibilities, and that's another part of our history that's illuminated in this dazzling exhibition.\n\n[2003]\n\n### The Matisse Collection at the Met\n\n* * *\n\nThe late Pierre Matisse (1900\u20131989), the younger son of the French master Henri Matisse, was one of the most illustrious art dealers of his day. For the art public of my generation, his gallery in the Fuller Building at 41 East 57th Street was as much a fixture on the modern art scene in New York as the Modern, Guggenheim, and Whitney museums, and its opening in 1931 actually preceded that of the Guggenheim by six years. For some sixty years, the Matisse Gallery remained one of the centers of contemporary art life in New York. This is one reason why the current exhibition of the Pierre and Maria-Gaetana Matisse Collection at the Metropolitan Museum of Art is a significant event. Another is that the Met's acquisition of the Matisse Collection is itself an important chapter in the history of the museum's ongoing effort to accord the masterworks of modern art a status akin to that of its celebrated collections of the Old Masters. It should be remembered that in the early years of the Matisse Gallery, it was the policy of the Met\u2014and, indeed, most of the older museums\u2014not to acquire or exhibit the work of \"untested\" living artists.\n\nIf this policy now strikes us as bizarre, it should also be revealed that as late as 1950, when E. H. Gombrich published the first edition of his classic history, _The Story of Art_ , the chapter devoted to the great flower-ing of modernist art in the first half of the twentieth century was called \"Experimental Art\"\u2014in other words, art that had not yet proven to be of permanent interest. That's not the way we think about art today, of course, but I wonder if I'm alone in feeling a certain nostalgia for this discredited policy. It would certainly have saved us from having to wade through a lot of the fashionable trash that has glutted museums in recent years.\n\nFashionable trash was never of much interest to Pierre Matisse. His principal mission as a dealer was to bring the controversial achievements of the Paris avant-garde of his own generation to the attention of the American art public. His own favorites in that generation were Balthus, Dubuffet, Giacometti, and Mir\u00f3\u2014especially Mir\u00f3, to whose work the Matisse Gallery devoted an astounding thirty-seven exhibitions.\n\nThe works of these artists are now blue-chip investments on the international art bourse, but some of them were a hard sell when they were first exhibited in New York\u2014Balthus because of his concentration on erotic subjects, and Mir\u00f3 because his work was often found to be incomprehensible. (Often, to fully understand Mir\u00f3, you have to have a sophisticated sense of humor.) In one of Mir\u00f3's letters to Pierre Matisse\u2014I quote from memory\u2014the painter thanked him for his support, adding that he sometimes thought it must require as much courage for the dealer to exhibit such paintings as it took him, the artist, to create them.\n\nIn some quarters even the beloved Giacometti was controversial\u2014but for an opposite reason. Hard-core Surrealists could never forgive Giacometti for abandoning the orthodoxies of their movement in favor of what they regarded as the more traditional style of his sculptural figures and still-life paintings. Oddly enough, it was the so-called Art Brut of Jean Dubuffet that was an instant hit with the Upper East Side collectors, who still favored the work of the School of Paris over anything produced by the burgeoning New York School of Abstract Expressionism.\n\nAnd yet, the exhibitions that the Matisse Gallery devoted to Mir\u00f3 and Giacometti in the late 1940s proved to be landmark events\u2014but again, for opposite reasons. Mir\u00f3 exerted an enormous influence on the Abstract Expressionists while Giacometti's figurative sculpture, drawings, and paintings were a source of inspiration and moral support for a younger generation of American painters determined to resist the temptations of abstractionism. For both of these opposing camps, the Matisse Gallery provided a standard of achievement.\n\nThe exhibition of the Matisse Collection at the Met is the first in a series of three consecutive shows over a yearlong period designed to bring us highlights of the collection. In addition to examples of the artists already mentioned, it also includes some excellent works by Matisse p\u00e8re that Pierre Matisse inherited from his father's estate. Of lesser interest are paintings by Ren\u00e9 Magritte and Paul Delvaux and sculpture by two British artists, Raymond Mason and Reg Butler. Of great interest, however, are two powerful paintings by Andr\u00e9 Derain\u2014a still life, _The Table_ (1911), and a flamboyant full-length painting of a woman called _The Black Feather Boa_ (1935). These two monumental pictures leave one yearning to see a full-scale Derain retrospective, but that's probably too much to hope for\u2014Derain is so utterly unfashionable today. Both for artistic and biographical reasons, Balthus's portrait of Pierre Matisse (1938) is a must-see painting.\n\n[2004]\n\n### John Walker at the Center for Maine Contemporary Art\n\n* * *\n\nFor the many people, whether tourists or natives, whose favorite memories of paintings of Maine are largely defined by the work of Winslow Homer in the nineteenth century and the Wyeth clan in the twentieth, the art of John Walker is bound to come as something of a shock. Everything traditionally associated with the beloved imagery of the Maine coast and its weather-beaten landscape\u2014the illustrational clarity, the crystalline light, and the abundant detail of a down-home naturalism\u2014is totally absent from Mr. Walker's paintings. Inducements to nostalgia are nil.\n\nWhat one encounters instead in the artist's latest exhibition\u2014 _John Walker: A Winter in Maine, 2003\u20132004_ , at the Center for Maine Contemporary Art in Rockport\u2014are huge, sprawling expressionist canvases and smaller oil sketches on paper that give the observer what's best described as the clamdigger's view of the Maine landscape. In this view the terrain tends to be muddy, the atmosphere overcast, the sky a distant band of mottled light, and the boundaries separating land from sea all but overwhelmed by a painterly virtuosity that's easily mistaken for outright abstraction. Yet as the eye habituates itself to these bold, highly charged depictions, what comes into focus are some of the most extraordinary landscape paintings of the modern era. Not since John Marin burst upon the American art scene in the 1920s and '30s have paintings of Maine succeeded to a comparable degree in setting a new standard for pictorial innovation in the art world at large.\n\nLike many Maine painters, Mr. Walker is, as Mainers say, \"from away\"\u2014in his case, originally from Britain; he was born in Birmingham in 1939 and studied at the Birmingham College of Art in the 1950s. Then came Paris, where he studied at the Acad\u00e9mie de la Grande Chaumi\u00e8re in the 1960s, and New York, where he came under the sway of the regnant Abstract Expressionists.\n\nNowadays Mr. Walker divides his working life between a coastal property in South Bristol, Maine, the _mise en sc\u00e8ne_ of his current work, and Boston University, where he's a member of the art faculty. (He often brings his students to Maine as part of their course of instruction.) In New York his work can often be seen at Knoedler & Company.\n\nIt's sometimes said of the Abstract Expressionist painters that they could be divided into two classes: those who put everything\u2014which is to say, more than merely enough\u2014into their pictures, and those who left out as much as possible while still giving us something to look at in what remained.\n\nMr. Walker unquestionably belongs to the first category, for his appetite for overloading his canvases is unstinting, and he has found in the dour attractions of a muddy bay in South Bristol a correlative in nature that allows him to create a landscape art in a medium that is not only reminiscent of the viscous facture often seen in the work of the Abstract Expressionists but at times actually incorporates mud itself\u2014or what's sometimes called \"sea cake\" in the titles of his paintings\u2014into the painted surface. What Mr. Walker's \"sea cake\" paintings recall for me are the lines from the \"Little Gidding\" section of T. S. Eliot's _Four Quartets_ : \"Dead water and dead sand \/ Contending for the upper hand.\"\n\nFor Mr. Walker, mud has clearly acquired an aesthetic, if not indeed a mystical significance. In a recent interview, Bruce Brown, one of the curators of the current exhibition, asked him, \"Technically, how do you get the mud to stick to the canvas and why do it?\" This was Mr. Walker's response: \"I've experimented with mixing various mediums with the mud. Basically dirt turns into cement, really. The fact that I take in these beautiful surroundings\u2014the muddiest, smelliest, dirtiest cove to paint in\u2014allows me to get beyond the beauty of the tourist sort of Maine. Mud has been a reoccurring theme in my paintings for years. . . . I have certainly always thought of paint as being colored mud. As you know, while I was involved with the first group of landscape paintings, I was concurrently painting my father's recollections of the First World War where mud was the theme\u2014not only his recollection, but almost everyone's from that war. I like the fact that mud is dirty. If I'm painting and a clammer comes along and digs those big, dirty holes right in front of me, I truly believe that what I'm doing on canvas is just a pastiche. I really am moved when I see that his is the artwork and mine is just an impression. It always shocks me that these people come along and dig great holes and walk away from it and it looks just wonderful.\"\n\nWell, as I say, this is no longer the Maine of Winslow Homer and the Wyeths.\n\n[2004]\n\n### R. B. Kitaj at Marlborough Gallery\n\n* * *\n\nThe career of R. B. Kitaj, whose work is the subject of a very large exhibition at the Marlborough Gallery, is like no other in the annals of American art. For one thing, no other American artist in recent memory has accorded to literary and historical subjects so large a role in shaping his pictorial ideas. For another, none has devoted so much attention\u2014and so much writing\u2014to serving as a mythographer of his own life and work. Then, too, there's Mr. Kitaj's \"obsession\" (his word) with the \"Jewish Question,\" which he insistently identifies as the principal focus of his work. It's important to understand that Mr. Kitaj's interest in Jewishness has everything to do with history and almost nothing to do with religion. Neither prayer nor piety\u2014never mind religious orthodoxy\u2014comes to mind in studying his work and his many commentaries on it, but rather a kind of intellectual combat addressed to enemies real and imagined. He is the most autobiographical of living artists (hence the title of the show at Marlborough: _R. B. Kitaj: How to Reach 72 in a Jewish Art_ ) and the most unforgiving in responding to the adverse judgments of his critics.\n\nOwing to his appetite for polemic and public controversy, moreover, Mr. Kitaj has often cast himself as a kind of art-world adversary or outlaw. And yet his career has been a resounding success since he first caused a stir as a student at the Royal College of Art in London in the early 1960s. The very scale of the show at Marlborough\u2014fifty-four paintings and thirty-four works on paper, plus a selection of prints and a large-format catalogue running to nearly ninety pages\u2014is further testimony to that success.\n\nThis is not to say that Mr. Kitaj has been spared harsh attacks on his work. Far from it. No sooner had he been elevated to stardom by the London critics\u2014who compared him with T. S. Eliot and Ezra Pound\u2014than those same critics turned against him when his subsequent work failed to meet their extravagant expectations. Never one to settle for half-measures, Mr. Kitaj responded to this censure by denouncing the London art scene\u2014in terms guaranteed to offend\u2014and promptly left town. He now makes his home in Los Angeles.\n\nIn his fury at the London critics, Mr. Kitaj has been guilty of some remarkably intemperate outbursts. Not only did he accuse them of anti-Semitism but he also suggested that their negative response to his work in London had somehow been the principal cause of his wife's unexpected demise. Grief over the untimely death of a spouse is certainly understandable, but an unfounded charge of murder is nonetheless an outrage\u2014and only one example of Mr. Kitaj's many rhetorical excesses.\n\nGiven all this uproar\u2014so much of it provoked by the artist himself\u2014it's a mercy that Mr. Kitaj draws as well as he does, for a highly accomplished draftsmanship remains the fulcrum of his art. Indeed, one sometimes has the impression that for him\u2014especially in dealing with literary subjects (Kafka is a special favorite)\u2014drawing is a form of writing by other means, a medium in which the literary and the graphic are in perfect accord.\n\nAt the risk of inspiring yet another wave of invective from Mr. Kitaj's pen, I am obliged to observe that in many of the paintings in the current exhibition I do not find a comparable accord. Drawing remains the strongest component in his paintings, but it's unmatched by any comparable command of chromatic invention. More often than not, Mr. Kitaj's draftsmanship tends to be coarsened by its intercourse with color; and color in his work, where it's not simply illustrational, has the character of something added to the armature of the artist's draftsmanship. Though Mr. Kitaj's palette can be said in some respects to be Matissean, its application remains devoid of Matissean subtlety and control. Here color is at once hectic, nervy, breathless, and undisciplined, often suggesting that it's struggling to keep ahead of a draftsmanship that has already outpaced it.\n\n_R. B. Kitaj: How to Reach 72 in a Jewish Art_ remains on view at the Marlborough Gallery. The press release reminds us that Mr. Kitaj's work is \"in the permanent collections of fifty-five museums throughout the world\"\u2014not bad for an artist who continues to regard himself as a rebel and an underdog.\n\n[2005]\n\n### Marvin Bileck and Emily Nelligan at Alexandre Gallery\n\n* * *\n\nSome years ago I began hearing about the drawings of the American artist Emily Nelligan, and it was mostly other artists who were talking up her work. I soon discovered that it was these other artists who were also buying Ms. Nelligan's drawings. The mainstream galleries hadn't yet heard of Emily Nelligan, and neither had most of the critics and collectors. When I finally did get to see a rather modest gallery show of her drawings, they were a revelation of a remarkable talent. This was soon followed by a show in which Ms. Nelligan's drawings were exhibited with the prints of her husband, Marvin Bileck, a renowned printmaker and illustrator. It was in 2000 that the first full-scale exhibition of Ms. Nelligan's drawings was organized at the Bowdoin College Museum of Art in Maine. Now the Alexandre Gallery in New York has mounted a splendid survey of the drawings and prints devoted to the subject that inspired both artists over a period of fifty years: the landscape of Great Cranberry Island, off the cost of Maine near Mount Desert.\n\nEmily Nelligan and Marvin Bileck met in the 1940s, when both were art students at the Cooper Union in New York. For many years they divided their time between their home in Connecticut, where they spent winter and spring, and Great Cranberry Island in summer and fall. Sadly, Bileck died in April at the age of eighty-five while preparations were in progress for the current exhibition.\n\nWhile both artists devoted their principal endeavors to works on paper depicting the visual enchantment of this isolated landscape, they brought very different sensibilities to their graphic accounts of it. Ms. Nelligan's medium is unfixed charcoal on white paper, which she transforms into virtuosic varieties of shadow and light. The sea, the sky, and the fog-misted earth are devoid of sharp contours in these drawings, and seem instead to have an apparitional character. From every smudge and erasure of charcoal dust she has been able to wrest subtleties of coastal light unlike anything that has heretofore been seen in pictures of Maine or any other coastline. In contrast to those images we carry in our memories from paintings of Maine by Winslow Homer and John Marin, Ms. Nelligan's drawings are nocturnes in which light is more fugitive than shadow and a velvety darkness dominates every prospect.\n\nBileck's forte was close-up, linear observation of nature in decay. Fallen trees, tangled underbrush, and lonely rock formations were frequent subjects for both his graphite drawings on paper and his etchings. The sky all but disappears as the artist concentrates his attention at ground level, where a once-vital jumble of roots and stumps and broken branches has weathered into a baroque phantasmagoria of seasonal decomposition. Bileck's command of detail in depicting such profusion, especially in the etchings, is worthy of comparison with the finest tradition of Northern Europe graphic art.\n\nFor readers\u2014and aspiring printmakers\u2014who are curious to know more about Bileck's working methods, the essay for the catalogue of the current show, written by Alison Ferris, provides some useful information. \"Bileck,\" Ms. Ferris writes, \"works on his plates over the course of a number of years. As a result, one print might be very different than a previously made print even though they were made from the same plate. Additionally, each print varies and is determined by how wet the paper is, how much pressure Bileck decides to put on it, and if and how he might burnish the back side of the paper before it's dry. Sometimes he even considers one of his counter prints the finished product. All this indicates that Bileck is perfectly correct when he says that no one else could print his work. The copious amount of time that Bileck spends on his prints instills in them a complexity that distinguishes them from the spontaneity of his drawings, some of which are completed in a day or so.\"\n\nFor connoisseurs of drawing and prints, the Marvin Bileck and Emily Nelligan exhibition is clearly an event not to be missed.\n\n[2005]\n\n### Oscar Bluemner at the Whitney\n\n* * *\n\nThere are artists who, despite their abundant gifts, seem destined to endure a melancholy fate, and one of them was Oscar Bluemner (1867\u20131938), the subject of a fine exhibition at the Whitney Museum of American Art. Bluemner was too \"advanced\" for the traditionalists at a time when modernism was still a contentious issue, and he was too tactless and outspoken in his relations with the modernists of the Alfred Stieglitz circle to benefit from their support. He remained an archetypal outsider in whatever milieu he frequented.\n\nHe was born in Germany, where he was trained as an architect, and it was as an architect that he initially established himself in New York. In 1904 he designed the Bronx Borough Courthouse, but this remained his only major completed work in architecture, which he then abandoned in favor of painting.\n\nIn 1910, Bluemner met Stieglitz, whose 291 Gallery was already an established haven for the American avant-garde. Bluemner's first solo exhibition at 291 came in 1915\u2014a series of eight landscape paintings\u2014and this further confirmed his commitment to painting. So, too, did his contributions to the prestigious Forum Exhibition of Modern American Painters.\n\nYet a variety of vexations continued to make both art and life extremely difficult for Bluemner. Money had always been a problem for him, and it was a problem that became more acute over the course of his career. When he could no longer afford to live in New York, he settled in the New Jersey countryside; and when he could no longer afford the materials he needed for his oil paintings, he turned to watercolor as an alternative. Yet his slide into penury proved irreversible.\n\nThen, too, there was the problem of Bluemner's German identity. This became especially acute in 1917, when the United States declared war against Germany, which inevitably provoked a terrific wave of anti-German sentiment. This added considerably to Bluemner's woes. In every respect but one\u2014his painting\u2014he was in the wrong place at the wrong time. And even as a painter, he was in some respects ahead of his time. Had he come of age as an artist in the era of color-oriented abstraction, he would very likely have been acclaimed an avant-garde master. The series of paintings called _Suns and Moons_ , which Bluemner created in 1926\u20131927, are virtual prototypes for a kind of color abstraction that later became commonplace in American abstract painting.\n\nThe show that Barbara Haskell has organized in _Oscar Bluemner: A Passion for Color_ thus has an interest for us that extends well beyond Bluemner's personal misfortunes. It's an exhibition that anyone curious about American modernism will want to see. And Ms. Haskell is to be congratulated, also, for the accompanying catalogue. She has triumphantly succeeded in rescuing an ill-fated painter from the tragedies of his life in a way that will allow posterity to appreciate his accomplishments.\n\n[2005]\n\n# IV \nModernism and Its Institutions\n\n## Modernism and Its Institutions\n\n* * *\n\n> _I couldn't portray a woman in all her natural loveliness. I haven't the skill. No one has. I must, therefore, create a new sort of beauty, the beauty that appears to me in terms of volume, of line, of mass, of weight, and through that beauty interpret my subjective impression. Nature is a mere pretext for a decorative composition, plus sentiment. It suggests emotion, and I translate that emotion into art. I want to expose the Absolute, and not merely the factitious woman_.\u2014Georges Braque, circa 1908\n> \n> _Subject, with her, is often incidental_.\u2014Wallace Stevens, on the poetry of Marianne Moore, 1935\n> \n> _Content is a glimpse of something, an encounter like a flash. It's very tiny\u2014very tiny, content_.\u2014Willem de Kooning, 1963\n\nAlthough we have lately been advised that \"the days when one could sit down with an easy mind to write an account of something called modernism are over,\"1 there nonetheless remains very little in our experience of the arts even in this first decade of the twenty-first century that can be separated from the traditions that were established by what used to be called the modern movement but that nowadays tend to be known collectively as modernism. As I shall be using the term here\u2014that is, modernism as a movement in literature as well as the visual arts\u2014it was never monolithic in style, ideas, or impact. In its heyday, which by my reckoning dates from the 1880s to the 1950s, it encompassed a broad range of styles, from realism and symbolism to pure abstraction, and a variety of anti-styles we associate with the legacy of Marcel Duchamp and Dadaism.\n\nIn many respects, modernism is identified as much by the traditions it rejects as by the innovations it embraces. In his _Histoire de la litt\u00e9rature fran\u00e7aise de 1789 \u00e0 nos jours_ (1936), the French historian Albert Thibaudet highlighted the radical nature of the modernist enterprise when he spoke of the commitment to the idea of \"indefinite revolution,\" of\n\n> a right and duty of youth to overturn the preceding generation, to run after an absolute. If the poets were divided into \"normal,\" or \"regular,\" and free-verse, literature was divided into normal literature, and literature of the \"avant-garde.\" The chronic avant-gardism of poetry, the \"What's new?\" of the \"informed\" public, the official part given to the young, the proliferation of schools and manifestos with which these young hastened to occupy that extreme point, to attain for an hour that crest of the wave in a tossing sea\u2014all this was not only a new development in 1885 but a new climate in French literature. The Symbolist revolution, the last thus far, might perhaps have been definitively the last, because it incorporated the theme of chronic revolution into the normal condition of literature.\n\nThibaudet's announced subject was Symbolist literature; but his diagnosis of the way a commitment to \"chronic revolution\" had become \"the normal condition of literature\" pertains equally to the whole project of modernism. The ultimate aesthetic and spiritual fruitfulness of this chronic revolution differed sharply among genres. In the pictorial arts, revolution first of all entailed rejection of the moribund conventions of nineteenth-century academic instruction, which had elevated a narrowly conceived mode of depicting the observable world to the status of an aesthetic and cultural absolute.\n\nWhat modernism rejected in architecture was ornament, decorative embellishments, and explicit references to historical precedent. The Viennese architect Adolf Loos put the case against ornament with histrionic astringency in his 1908 essay \"Ornament and Crime.\" \"The evolution of culture,\" Loos wrote, \"is synonymous with the removal of ornament from utilitarian objects.\" According to Loos, \"freedom from ornament is a sign of spiritual strength.\" In the modern world, \"art has taken the place of ornament,\" and a hankering after ornament is, for Loos, a sign of spiritual backwardness or criminality. \"Anyone who goes to the _Ninth Symphony_ and sits down and designs a wallpaper pattern is either a confidence trickster or a degenerate.\"\n\nLoos may have written with tongue at least partly in cheek, but he nonetheless summed up a cardinal element in the brief of classic architectural modernism. We are now perhaps in a position to question how fruitful the rejection of ornament really was as an aesthetic desideratum: what seemed terribly brash and exciting in 1908 looks rather different when it succeeded in transforming whole cities into monotonous rows of rectangular glass and steel plinths.\n\nAs far as painting and sculpture are concerned, modernism introduced a radical revision in the very concept of representation, the implications of which are admirably summarized in George Heard Hamilton's introduction to his classic history, _Painting and Sculpture in Europe, 1880\u20131940_ (1967):\n\n> In the half-century between 1886, the date of the last Impressionist exhibition, and the beginning of the Second World War, a change took place in the theory and practice of art which was as radical and momentous as any that had occurred in human history. It was based on the belief that works of art need not imitate or represent natural objects and events.\n> \n> Therefore artistic activity is not essentially concerned with representation but instead with the invention of objects variously expressive of human experience, objects whose structures as independent artistic entities cannot be evaluated in terms of their likeness, nor devalued because of their lack of likeness, to natural things.\n\nBut revolution and rejection do not tell the whole story of modernism. As I argued long ago in my essay \"The Age of the Avant-Garde\" (1972), the impulse to wage war on the past had a constructive as well as a rebellious side. What was most conspicuously embraced by modernism in the literary arts were so-called free verse ( _vers libre_ ) in poetry, which entailed an abandonment of traditional rhyme and meter, and the \"stream of consciousness\" technique in fiction, which was introduced to literature in English by James Joyce but was made more accessible to public comprehension by the popularization of Freudian psychoanalytical therapy. These innovations entailed a rejection of nineteenth-century narrative conventions in favor of more hermetic literary structures based on myth, symbolism, and other devices more commonly found in poetry, especially modern poetry, than in prose fiction.\n\nMoreover, owing to the resolute and often vindictive resistance that such innovations met with in the arena of public taste, it was probably inevitable that ways would be sought to circumvent that resistance. It was, in any case, in direct response to such prohibitions that the modernist impulse was driven to create institutions of its own in order to safeguard the survival and prosperity of its aesthetic initiatives. It was probably inevitable, too, that in the early history of these institutions, an attempt would be made to minimize the sometimes controversial content of modernism\u2014not only its sexual explicitness but its political provocations as well. After all, in a period when even as blameless a book of short stories as Joyce's _Dubliners_ met with refusal by its first printers on the grounds that certain passages were deemed to employ improper language, and a masterpiece like _Ulysses_ was legally banned in the United States, there was ample reason to be cautious about publicizing the content of certain modernist works.\n\nOn the large and thorny question of modernism's content and its relation to modernist form, however, it must also be said that certain modernists have themselves been complicit in minimizing its content even under conditions where prudence was no longer required, as the epigraphs quoted above from Georges Braque, Wallace Stevens, and Willem de Kooning suggest. For anyone who was present on the New York art scene in 1953, for example, when de Kooning exhibited the first of his sensational _Women_ paintings at the Sidney Janis Gallery, the much-quoted claim made ten years earlier, obviously in reference to abstract painting, that the content in the creation of a work of art is \"very tiny,\" would have sounded absurd. The ferocity with which de Kooning attacked his subject in the _Women_ series left no one in doubt about the importance of its content\u2014both for the artist and viewer\u2014and the un-raveling of de Kooning's talent in the aftermath of the _Women_ paintings, when he returned to a mode of abstraction that was sadly depleted of both form and content, only underscored the point.\n\nSimilarly, Wallace Stevens's observation that in the poetry of Marianne Moore, \"Subject . . . is often incidental,\" may indeed apply to some of the poet's minor later work, but elsewhere Moore's poetry positively bristles with difficult subjects. In the early masterpiece from the 1920s called _Marriage_ \u2014a work that, in my view, occupies a place in Moore's literary oeuvre akin to that of _The Waste Land_ in Eliot's\u2014the intensity and indignation prompted by the subject of matrimony is anything but \"incidental.\" Its treatment is ferocious, as the poem's opening lines attest:\n\n> This institution, \n> perhaps one should say enterprise \n> out of respect for which \n> one says one need \n> not change one's mind \n> about a thing one has believed in, \n> requiring public promises \n> of one's intention \n> to fulfill a private obligation: \n> I wonder what Adam and Eve \n> think of it by this time, \n> this fire-gilt steel \n> alive with goldenness; \n> how bright it shows\u2014 \n> \"of circular traditions and impostures, \n> committing many spoils,\" \n> requiring all one's criminal ingenuity \n> to avoid!\n\nIn a new edition of _The Poems of Marianne Moore_ , edited by Grace Schulman, _Marriage_ runs to eight and a half pages, and remains to its very last lines one of the most caustic poems in the language\u2014one of the scariest, too.\n\nGeorges Braque was a far gentler soul than either Willem de Kooning or Marianne Moore, and it was characteristic of his moral delicacy to wrap his statement about the artist's subject in a mantle of modesty, only to disclose in the end that, like so many other modernists, his too had been a pursuit of the Absolute. Braque's version of the Absolute no doubt differed from that of the true firebrands of modernism\u2014among them, Piet Mondrian, Mies van der Rohe, Ad Reinhardt, and Donald Judd\u2014yet where would any of them have been without the prior existence of Cubism, the creation of which was owed to Braque's collaboration with Picasso? It might even be said that Braque created the foundation upon which these firebrands were able to take their stand.\n\nThe great irony, as I pointed out in \"The Age of the Avant-Garde,\" is that this effort to place tradition under the pressure of a constant revaluation had an unexpected effect. It resulted in the virtual dissolution of any really viable concept of tradition. And this sense of dissolution\u2014of the muteness or sterility of tradition\u2014is at the heart of the situation in which we find ourselves today. Without the bulwark of a fixed tradition, modernism found itself deprived of its historic antagonist. Much to its own embarrassment, it found that it had itself become a tradition\u2014albeit one without obvious progeny. With its victory over the authority of the past complete, its own raison d'\u00eatre had disappeared, and it had, in fact, ceased to exist except as an imaginary enterprise engaged in combat against imaginary adversaries.\n\nIn fact, in its most fruitful manifestations, the modernist attitude toward tradition was as much a distillation or transformation as a rejection of tradition. Indeed, a revaluation of the past is precisely what the modernist masters were forcing upon the official guardians of taste. One thinks in this context of T. S. Eliot, himself a modernist pioneer, who outlined in \"Tradition and the Individual Talent\" (1919) a vision of originality in which \"not only the best, but the most individual parts of [a poet's] work may be those in which the dead poets, his ancestors, assert their immortality most vigorously.\" \"No poet, no artist of any art,\" Eliot wrote in one of the most famous passages of that famous essay,\n\n> has his complete meaning alone. His significance, his appreciation is the appreciation of his relation to the dead poets and artists. You cannot value him alone; you must set him, for contrast and comparison, among the dead. I mean this as a principle of esthetic, not merely historical, criticism . . . [for] what happens when a new work of art is created is something that happens simultaneously to all the works of art which preceded it. The existing monuments form an ideal order among themselves, which is modified by the introduction of the new (the really new) work of art among them.\n\nEliot's effort was not to subvert tradition but, on the contrary, to salvage it from the sclerotic imperatives of an exhausted antiquarianism or impotent gentility. To what extent Eliot succeeded may be open to question\u2014his conception of an \"ideal order\" of \"existing monuments\" may in the end have been too purely aesthetic to bear the existential burden of vitality with which he invested it. Nevertheless, Eliot's example illustrated the apparently unavoidable paradox that the advent of modernism brought with it the seeds of its own perpetual renovation: the anti-institutional impulse of modernism would ultimately be brought to bear on the seemingly unassailable institution that was modernism itself. The question that confronts us today is to what extent the institutions of modernism can survive the consuming effects of their own self-immolation\u2014survive, I mean, as vital sources of cultural renewal, not simply as vacant bureaucratic placeholders.\n\nWhat are the institutions that either created or were commandeered to accommodate modernism's battle of the Absolutes? Some of them have become so well established as fixtures of our cultural landscape that we can hardly imagine a time when they didn't exist\u2014among them, the art galleries, with their one-man shows of new art; the art museums that vie with each other for the privilege of being the first to embrace what is certain to be controversial; the \"little magazines,\" literary quarterlies, and small presses without which modernist literature would never have prospered; the more problematic writers' \"workshops,\" which seem now to have degenerated into an academic racket (one recalls Kingsley Amis's remark that much of what was wrong with the twentieth century could be summed up in the word \"workshop\"); and the Armory Show type of large international exhibitions that grew out of the various \"independent\" and \"secessionist\" modernist movements of a century or more ago.\n\nVirtually all of these institutions were created by artists working in collaboration with amateur, noninstitutional collectors, just as the little magazines, small presses, and literary quarterlies were created by poets and critics who understood that mainstream publishers were not in a position to respond to the challenges of modernist literature without the kind of spadework that only noninstitutional amateurs could provide. As Lionel Trilling wrote in his essay \"The Function of the Little Magazine\":\n\n> To the general lowering of the status of literature and of the interest in it, the innumerable \"little magazines\" have been a natural and heroic response. Since the beginning of the [twentieth] century, meeting difficulties of which only their editors can truly conceive, they have tried to keep the roads open. From the elegant and brilliant _Dial_ to the latest little scrub from the provinces, they have done their work, they have kept our culture from being cautious and settled, or merely sociological, or merely pious. They are snickered at and snubbed, sometimes deservedly, and no one would venture to say in a precise way just what effect they have\u2014except that they keep the new talents warm until the commercial publisher with his customary air of noble resolution is ready to take his chance, except that they make the official representatives of literature a little uneasy, except that they keep a countercurrent moving which perhaps no one will be fully aware of until it ceases to move.\n\nNow, nearly sixty years after Trilling wrote \"The Function of the Little Magazine\" to mark the tenth anniversary of _Partisan Review_ , the kind of \"little magazine\" he described in that essay is virtually extinct\u2014as, of course, is _Partisan Review_. Some literary quarterlies have survived but have diminished in number and\u2014with the shining exception of _The Hudson Review_ and (if I may say it) _The New Criterion_ \u2014in quality, too, as the blight of deconstructionist, poststructuralist, and other varieties of anti-literary \"theory\" has triumphed over literary intelligence. When we look back today on the much-maligned New Criticism, which was largely the creation of modernist poets\u2014among them, T. S. Eliot and Ezra Pound\u2014it looks like a Golden Age compared to the kind of academic obscurantism and political shadow-boxing that have supplanted it.\n\nAs for the fate of literature itself in the hands of the mainstream book publishers, we find that virtually all of the major houses in this country are now wholly owned subsidiaries of foreign conglomerates whose standard of achievement has less to do with literary quality or innovation than with access to media promotion, movie and television tie-ins, prize-winning, and other coefficients of high profitability. Mercifully, we can still count on a number of smaller presses and some of the better university presses to save the situation, but the downside of this benefit usually entails significantly reduced royalties for the worthy writers who do get published.\n\nExactly how it came to pass that a nation as prosperous as ours could not summon the resources to resist the takeover of its book publishing industry by an ailing Europe remains to be explained, but that takeover is now a _fait accompli_ , and we shall be obliged to suffer its consequences for a long time to come.\n\nIn some respects the institutions that serve the visual arts\u2014especially the museums and the galleries\u2014might seem to present a much rosier prospect, for even in periods of low economic growth they have continued to prosper. Indeed, headlong and often heedless expansion of both collections and exhibition space and the funds required to support them has been the rule in the art museums for some years now. Modernist art of various persuasions has been the driving force as well as the principal beneficiary of this very expensive expansiveness as museums have hastened to respond to new artistic developments while at the same time attempting to catch up on the earlier innovations they missed out on. This museological scenario is now so familiar to us that we sometimes forget that the compulsion on the part of museums to keep abreast of radical innovations in contemporary art\u2014and even, when possible, to anticipate and assist in creating a demand for them\u2014was itself a momentous innovation in the way museums come to identify their interests and responsibilities. For this compulsion obligated the museums to become, in effect, not only collectors but also promoters of the art in which they were seen to have a vested interest\u2014a vested interest, that is, in both the objects acquired for their collections and in the careers and celebrity of the artists who created them.\n\nHence the elements of hucksterism and entrepreneurial cynicism that have coarsened the character and spirit of so much museum activity today. Consider the advent of the museum called Tate Modern in London. Born in a blitz of publicity the like of which had formerly been reserved for pop stars and consumer gadgets, Tate Modern was the huge spin-off from the venerable Tate Gallery (now rebaptized Tate Britain). Housed in a gigantic renovated power plant on Bankside across the Thames from the original Tate Gallery, Tate Modern is ostensibly devoted to modern art. In fact, as the Tate's chief commissar Sir Nicholas Serota put it when announcing the bifurcation of the original Tate Gallery, Tate Modern is really devoted to \"new narratives\" of art\u2014\"new narratives,\" alas, in which the energy and seriousness of modernism is trivialized and distorted in order to accommodate its repackaging as a postmodernist exercise in chic cultural shallowness.\n\nWhen we enter a monstrosity like Tate Modern in London, we are straightaway put on notice by the noise, the crowds, the theatrical lighting, and the general atmosphere of vulgarity and tumult that art has been used as bait to attract a segment of the public\u2014free-spending youth\u2014for which aesthetic achievement is, if not a matter of indifference, certainly not a compelling priority. And to assure a steady supply of the only kind of new art that is guaranteed to be a turn-on for this public, there are the proliferating productions of Charles Saatchi's gang of YBAs\u2014Young British Artists\u2014and the Tate's own atrocious Turner Prize\u2013winners, who can be counted upon to maintain the requisite standard of titillation.\n\nUnlike the old Tate Gallery, our own Solomon R. Guggenheim Museum has not had to change its name but only its character to adjust to the new entrepreneurial standard. I wonder how many of the people who went to ogle Matthew Barney's freak show at the Guggenheim have any idea that this museum, with its once incomparable collection of paintings by Vasily Kandinsky, was founded in 1930 as an institution devoted to the achievements of abstract painting? (Its original name was the Museum of Non-Objective Painting.) Yet, just as its influence in that respect was contributing something important to the emergence of Abstract Expressionist painting in New York\u2014the young Jackson Pollock, among other artists, worked there as a guard in its early days\u2014the Guggenheim initiated the first of its ongoing efforts to reinvent itself. This project of reinvention has left the museum stripped of anything that can be called an identity and has required, among other depredations, the sale of a great many of its Kandinsky holdings and works by other modernist masters. Today it is an institution better known for its exhibitions of Norman Rockwell and Harley-Davidson motorcycles and its branch museums abroad than for anything that advances an understanding of modernist art.\n\nTwo of the other New York institutions that were founded to serve the interests of modernist art\u2014the Museum of Modern Art and the Whitney Museum of American Art\u2014are also at a crossroads that will determine their future course, but for very different reasons.\n\nMOMA is now in the throes of yet another of its periodic expansion plans, one of the biggest in its history. When the expansion is completed in 2005, it is expected to provide the museum with far more space for showing its permanent collection as well as for its temporary exhibitions program. Meanwhile, the museum and its public are making do with an abridged, unappealing facility in Long Island City, MOMAQNS, for a reduced exhibition schedule, and a theater on East Twenty-third Street in Manhattan for its popular film program.\n\nAt this point, we can only speculate about what this expansion will bring in the way MOMA's permanent collection\u2014and thus modernism itself\u2014is to be presented to the public for the remainder of the twenty-first century. If there is good reason to be hopeful about this outcome, it is mainly because John Elderfield\u2014no doubt one of the most qualified senior curators in the field today\u2014has been called upon to head the curatorial committee that will oversee the installation of MOMA's permanent collection in its new building. If there is also good reason to be anxious about the outcome, it is owing to the debacle of the museum's _MOMA2000_ exhibitions, which radically recast the history of MOMA's permanent collection to conform to a \"new narrative\" emphasizing social content at the expense of aesthetic innovation. This was a shift in perspective that, among other losses, had the effect of consigning the history of abstract art\u2014one of the central developments of modernist art\u2014to the sidelines. Given the theme-park character that governed the organization of _MOMA2000_ , there was no way that the aesthetics of abstraction could be given its due. It thus remains to be seen whether Mr. Elderfield will have sufficient authority to rectify such disastrous errors of judgment in the newly expanded MOMA.\n\nAbout the future of the Whitney Museum, too, we can only speculate. Its recent history, marked by a succession of incompetent directors and a board of trustees that seemed at times to have lost its mind, has been so dismal that almost any change is likely to be a change for the better. The good news is that the Whitney's plan for a harebrained expansion of its own has been canceled for financial reasons. The appointment of Adam D. Weinberg, a former curator at the Whitney, as the museum's new director also gives us reason to expect significant improvement. It will not be easy, however, for the Whitney to win back the respect it has lost among artists as well as the critics and the public. A good place to start would be either the overhaul or the outright abandonment of the Whitney's Biennial exhibitions, which in recent years have gone from being merely ludicrous to wholly contemptible.\n\nAs for the international exhibitions like Documenta in Germany and the monster Biennials in Venice and S\u00e3o Paulo, they have now become cultural dinosaurs with no useful functions to perform and therefore no reason to exist. There was a period, of course, when exhibitions like the 1910 Post-Impressionism exhibition in London, the 1913 Armory Show in New York, and the 1938 International Exposition of Surrealism in Paris really did bring the public news of important avant-garde developments in modernist art. But the age of the avant-garde is long gone. Its celebrated scandals and audacities have passed into the possession of the academic curriculum, to be catalogued, codified, and otherwise processed for doctoral dissertations and classroom instruction. The pathetic attempts at artistic insolence, mostly having to do with sexual imagery and political ideology, that turn up in the Whitney's Biennial exhibitions and the art departments of colleges and universities are better understood as efforts to attract publicity and what in the business world is called \"market share\" than as anything that can be regarded as avant-garde. In a culture like ours, in which, alas, everything is now permitted and nothing resisted, the conditions necessary for the emergence of a genuine avant-garde no longer exist. It doesn't change anything, either, to adopt the term \"trangressive\" as a substitute for \"avant-garde,\" for where boundaries no longer exist it is impossible to violate them. \"Transgressive\" is a term that belongs to the history of publicity rather than the history of art. Today there is no avant-garde, and the big international shows are mainly devoted to marketing and politics. Modern systems of communication have, in any case, rendered the big international shows irrelevant.\n\nFar more important to sustaining the aesthetic vitality of modernist art, however, has been the institution that we do not usually even think of as an institution: I mean the commercial art gallery. The art gallery as we know it today is, after all, a modern creation, barely a century old, and it performs a service for art unlike that of any other institution. It keeps the public in constant touch not only with current developments on the art scene but also with revivals of the work of earlier artists that the museums and the critics may have overlooked or underrated, and it does so at no financial cost to the viewer. More often than not, it is the gallery dealer, not the museum curator, who discovers significant new talent, for nowadays most curators make their \"discoveries\" in the dealers' galleries.\n\nThis is a cultural service more often enjoyed than acknowledged, but for many of us the art galleries have been a fundamental part of our aesthetic education. In this connection it is worth recalling Clement Greenberg's tribute to the late Betty Parsons, whose gallery introduced Jackson Pollock, Hans Hofmann, Mark Rothko, Barnett Newman, and Richard Pousette-Dart, among other artists, to gallerygoers of my generation. In 1955, on the tenth anniversary of the Betty Parsons Gallery, Greenberg wrote:\n\n> Mrs. Parsons has never lacked for courage. It is not a virtue signally associated with art dealers (or, for that matter, with art critics or museum directors either), but then she is not, at least for me, primarily a dealer. I have seldom been able to bring her gallery into focus as part of the commercial apparatus of art (I am not sneering at that apparatus); rather, I think of it as belonging more to the studio and production side of art. In a sense like that in which a painter is referred to as a painter's painter or a poet as a poet's poet, Mrs. Parsons is an artist's\u2014and critic's\u2014gallery: a place where art goes on and is not just shown and sold.\n\nJust as modernist art in America was, initially, an extension and appropriation of European modernism, so was a gallery like Betty Parsons's in a tradition that grew out of the precedents set by Vollard and Kahnweiler in Paris and Alfred Stieglitz's \"291\" gallery, which introduced C\u00e9zanne and Matisse as well as Marsden Hartley and John Marin to the New York public even before the Armory Show and long before the museums awakened to the achievements of modernism. The same could be said of the Weyhe Gallery's efforts on behalf of Gaston Lachaise and Alfred Maurer and the exhibitions devoted to Stuart Davis at Edith Halpert's Downtown Gallery. At every stage in the history of modernism in America, it was the galleries that set the pace in recognizing artistic achievement. This is not to suggest that all of our art dealers are sainted figures, but merely to point out that in New York, anyway, we are blessed with an extraordinary number of galleries that are places \"where art goes on,\" and they should be given their due in any account of modernism and its institutions.\n\nFinally, it is inevitable\u2014or at least expected\u2014in any discussion of modernism that the question of postmodernism will rear its ugly head. Or should I say, its wrongheadedness? For the entire concept of postmodernism is based on a fundamental misconception\u2014a belief that the modernist era in art and culture is over and has been supplanted by something radically different. What we find this usually means when we get down to specific cases is a mode of art or thought in which some element of modernist sensibility has been corrupted by kitsch, politics, social theory, gender theory, or some other academic, pop-oriented, anti-aesthetic intervention. Modernism has aged, to be sure, as modernity itself has aged, and in the process modernism has undeniably lost its capacity to shock or otherwise disturb us. But except for the short-lived antics of Dada and Surrealism, shock was never the essence of modernism. It was, rather, an inspired and highly successful attempt to bring art and culture into an affective and philosophical alignment with the mind-set of modernity as we know it in our daily lives. Modernism endures, and does so, in part, anyway, by virtue of the institutions it has created to serve the needs of a public that is today more enlightened intellectually and aesthetically than at any other time in our history. Postmodernism, in contrast, has created no institutions of its own, largely because postmodernism is nothing but a mind-set of deconstructive attitudes in search of a mission.\n\nAs I have noted elsewhere, we are far more aware than earlier generations of the many divisions, contradictions, and countervailing tendencies within modernism. We are more aware, too, that what goes under the name \"postmodern\" is really only modernism gone rancid\u2014modernism with a sneer, a giggle, modernism without any animating faith in the nobility and pertinence of its cultural mandate. At any rate, it is a crowning irony that much that modernism in its original formulations took as essential to its identity should now appear just as dispensable as representation, narrative, ornament, and other traditional aesthetic accoutrements once did to the early modernists. It turns out that no style or genre has a monopoly on the modernist spirit. Its core is not synonymous with abstract art or plotless novels or Miesian glass boxes. On the contrary, the core of modernism lies in an attitude of honesty to the imperatives of lived experience, which means also an attitude of critical openness to the aesthetic and moral traditions that have defined our culture. The problem with postmodernism is not that it embraces architectural ornamentation or representational painting or self-referential plot lines. The problem is postmodernism's sentimental rejection of the realities of modern life for the sake of an ideologically informed fantasy world. In this sense, modernism is not only still vital: it remains the only really vital tradition for the arts. It is, in many ways, a broken or fragmented tradition, a tradition bequeathed to us not in a perfectly legible script that guarantees our continuity with the past but in the hieroglyphic syllables of a past that must be continuously redeciphered in order to shed intermittent light on our half-plotted itinerary. We see, then, that what modernism requires is not a commitment to \"chronic revolution\" but rather permanent restlessness. It yields insights but not finality. The traditions it recovers are partial, fragmentary, but also forthright and nourishing: \"these fragments I have shored against my ruin.\"\n\nLooking back on the history of modernism in the twentieth century, what is especially striking is the violence that was directed against its achievements by the most horrific totalitarian regimes in recorded history: the Nazis in Hitler's Germany and the Communists in Stalin's Russia. And if we ask the question of what it was about modernist art that prompted such a massively destructive response, I believe the answer is clear: modernist art was seen to provide a spiritual and emotional haven from the coercive and conformist pressures of the societies in which it flourished. Modernism represented a freedom of mind that totalitarian regimes could not abide. It is in this sense, perhaps, that the infamous _Degenerate Art_ exhibition that Hitler devoted to modernist art in Munich in 1937 may now be seen to have marked the beginning of the \"postmodernist\" impulse. And just as modernism survived the determined efforts of Hitler and Stalin to impugn and destroy its artistic achievements, so, I believe, will modernism and its institutions continue to prosper in the face of the nihilist imperatives of the postmodernist scam.\n\n[2004]\n\nFootnote\n\n1. Paul Mattick, _Art in Its Time: Theories and Practice of Modern Aesthetics_ (New York, 2003), p. 9.\n\n## Has Success Spoiled the Art Museum?\n\n* * *\n\nOf all the institutions of high culture that have undergone significant change in recent decades, none has been more radically transformed than the art museum. In every aspect of its function, its atmosphere, and its scale of operations, in the character and number of the events that it encompasses, in the nature and size of the public it attracts, and in the role it plays in codifying\u2014and at times deconstructing\u2014our ideas about what art is, the museum has been so dramatically altered in our lifetime that in many important respects it can no longer be said to be the same institution we came to in our youth. And of all the changes that have overtaken the art museum in our time, the most crucial has been the elevation of change itself to the status of a governing principle.\n\nIn the past we looked to the art museum for our touchstones of artistic quality and achievement. It was in the museums that we learned to become connoisseurs of artistic accomplishment\u2014to become intimate with and knowledgeable about works of art over the long term; to acquire a sense of judgment and discrimination about their special attributes, about their differences and resemblances, and their complex relation to one another and to ourselves. It was through this process that our pleasure in art was deepened and our understanding of it enlarged, and this was as true for the amateur public, which we are all a part of at the outset, as for the professional. \"The Louvre,\" wrote C\u00e9zanne, \"is the book in which we learn to read,\" and it was understood that learning of this sort could not be hurried by crash courses, superficial entertainments, or overnight conversions. Like all learning that is serious, that becomes a personal and permanent acquisition and thus a part of our lives, it required time and application, and one of its essential preconditions was a certain stability in the museum itself\u2014the kind of stability that precludes precipitous change and a fickle, continual tampering with the objects of our scrutiny. We did not look to the art museum for news but, on the contrary, for what remained vital and enduring after it had ceased to be news.\n\nNowadays, however, we expect of our museums that they will be dynamic rather than stable, that they will no longer be guided by fixed standards or revered traditions, but just the reverse\u2014that they will shed convention, defy precedent, and shatter established values as often and as eagerly as the most incendiary avant-gardist of yesteryear. We demand of our museums that they bring us news as regularly as the media itself, and this inevitably entails a shift of attention from the permanent to the temporary\u2014a shift now everywhere in evidence in the museum world and officially codified in the name of the museum facility in Los Angeles that calls itself the Temporary Museum of Contemporary Art, or the Temporary Contemporary, as it is better known. It is thus one of the paradoxes of the art museum as a cultural force that as the reign of the avant-garde has drawn to a close as far as the creation of new art is concerned, its program of incessant and omnivorous change has been enthusiastically embraced\u2014some would say promiscuously embraced\u2014by the institution that was long thought to be the principal counterweight to the restless avant-garde spirit.\n\nThe art museum thus finds itself at the end of the twentieth century in the curious position of having supplanted art itself as the leading advocate and agency of innovation in our artistic affairs. What was formerly a highly individualistic impulse has now become an established bureaucratic practice, and there can be no question that the museum's adoption of this radical role has been accompanied and indeed accelerated by widespread approbation and applause. There are dissenting voices, of course, but if the command of money, resources, publicity, prestige, and sheer numbers of people is a measure of success in the life of an institution, the transformation of the art museum into a vehicle of headlong cultural change must be pronounced a resounding success.\n\nYet, just as Braque once observed that in the making of art every acquisition involves an equivalent loss, it may be appropriate to ask what this huge success has cost us. While there is little likelihood that the momentum that has propelled our museums on their present course will soon be reversed, it nonetheless behooves us to see as clearly as we can where this course is taking us and what further sacrifices are to be exacted as the price of developments still to come. For certain sacrifices have already been made, as almost anyone whose experience of museums goes back a few decades will readily attest, and we may be sure that an institution that has so irreversibly tethered its fate to the principle of dynamic change will be called upon to make a good many more.\n\nOne thing is beyond dispute. As a result of the immense growth of museums and their embrace of the principle of dynamic change, more people than ever before have been induced to take some sort of interest in art, or induced at least to take an interest in matters that have the appearance of being somehow related to art. It must also be said that this new multitudinous public has access to a program of events, exhibitions, entertainments, publications, etc., far more various and far more numerous than any that was offered by museums in the past. Many of these activities are not, to be sure, what I have heard described by museum officials as \"object-oriented.\"\n\nThat is, they are not concentrated on specific art objects. But then, museums have long ceased to be places that people visited solely or primarily for the purpose of looking at works of art. They are now the venue of all sorts of other activities that can be described as museum activities only in the sense that they take place under the roof, or at least under the auspices, of the museum. Yet with serious artistic matters, with learning or intellectual improvement or aesthetic enlightenment of any sort, these activities often have little or nothing to do. They come under the general heading of social diversions.\n\nWe now take it for granted that the art museum is an appropriate place in which to order lunch or dinner, buy something to wear, do our Christmas shopping, see a movie, listen to a concert, attend a lecture on anything under the sun, possibly even art, and also on occasion participate in a wedding party, a cocktail party, a charity benefit, a business reception, a fashion show, or some other lavish social event for which the museum is deemed a suitably prestigious facility. In the museums that now offer such services and distractions, it is still possible, of course, for people to look at works of art, and many do. Yet except in the highly publicized special exhibitions that draw the crowds, the galleries containing works of art are likely to be emptier of visitors, even on week-ends, than they were a generation ago. This is no doubt some consolation to the beleaguered connoisseur looking for a quiet corner in which to have an unimpeded view of a favorite masterwork, but it is not exactly a happy index of the museum's interest in cultivating a loyal public for its greatest treasures. The overall atmosphere in our big museums, anyway, does not really lend itself to the cultivation of such a public, which cannot be expected to prosper in an environment of hubbub and hucksterism. In lieu of this public the museum now has members, who avail themselves of the social diversions and add to the hubbub without in any way constituting a serious public for art.\n\nThis development is no longer to be regarded as a peculiarly American phenomenon, by the way. We may have pioneered the move toward a broad diversification of the art museum's interests and functions, but the phenomenon itself is now widespread. When we enter the Louvre nowadays through the famous glass pyramid, we descend by escalator into a vast, bright, sprawling space that resembles, more than anything else, the interior of a huge international airport in the busy holiday season. There are lines everywhere\u2014a line to enter the pyramid itself, and then more lines at the cloakroom, the washrooms, the ticket booth, and the cafeteria. (Except in the immediate vicinity of the famous tourist attractions\u2014the Victory of Samothrace, the _Mona Lisa_ , et al.\u2014the crowd is considerably thinner in the galleries, where most of the art is.) And at this \"new\" Louvre or Grand Louvre, as it is now to be called, there is worse yet to come. According to an interview with Michel Laclotte, the director, \"in the future, a shopping mall and a substantial underground parking lot will complete the new underground areas of the Louvre.\" We are assured that despite this vast expansion of the Louvre's functions and facilities, \"in Michel Laclotte's mind, no concessions have been made to mere trends or fashions\"\u2014a statement which, if it means anything, can only mean that the director of the Grand Louvre is oblivious to what is going on under his own authority.1 For at every turn in the labyrinthine sprawl of the new Louvre we are in the grip of every trend and fashion that has beset the art museum for several decades now.\n\nAdding a huge underground shopping mall to the airportlike entrance to the new Louvre will have the virtue, I suppose, of carrying the logic of the new museology to its ultimate conclusion. Martin Filler, writing in the London _Times Literary Supplement_ , gave us an excellent account of where this logic was heading even before the Grand Louvre was on the drawing boards. \"I. M. Pei's East Building for the National Gallery of Art in Washington, completed in 1978,\" wrote Mr. Filler, \"is dominated by a vast, glass-roofed atrium\u2014replete with escalators and potted ficus trees that give it the aspect of an upmarket shopping mall\u2014while the oddly shaped galleries relegated to three towers on the periphery of the central courtyard and the subterranean changing-exhibition spaces seem like grudging afterthoughts. Intentional or not, the architect's adaption of the imagery of the contemporary market place reinforces the notion of museum-going as yet another consumer activity.\" In this respect, at least, it must be acknowledged that the French, albeit with some help from us\u2014Mr. Pei is also, of course, the architect in charge of the Grand Louvre\u2014have outdone even American efforts in this direction by unit-ing the shopping mall and the art museum in a single gargantuan facility. Alas, one is reminded that Andy Warhol's joke about the many things that department stores and museums have in common is turning out, as so many of Warhol's jokes have, to be a deadly prophecy.\n\nIt would be difficult to say whether this rampant commercialization and consumerization is only a symptom or a primary cause of the altered relationship that now obtains between the art museum and its public, but there can be no question that this relationship is now the main issue facing the art museum as an institution. It determines much of its budget, its program, its choice of personnel, and its general ethos. To a larger extent than is commonly supposed, the museum has become a captive of its own success, and what is so striking now about its relation to its public is the extreme degree of anxiety and uncertainty that has come to characterize every aspect of it. Often the problem comes down to catering\u2014in every sense\u2014to a public that does not know exactly what it has come to the museum to do or see. This public wants, at one level or another, to be amused, instructed, gratified, edified, or otherwise diverted. Nowadays, too, with the toxin of multiculturalism threatening the intellectual integrity of arts institutions of every kind, the public may also come to the museum in search of political satisfaction. The experience of art is likely, in all too many cases, to be entirely peripheral to what is looked for in the museum.\n\nWith museumgoing now separated in this way from a genuine interest in art\u2014and by this I do not mean only a professional interest, for it is the amateurs of art who have traditionally constituted the true museum public\u2014it was inevitable that the art museum would be obliged to devise new stratagems for attracting its public and retaining its loyalty. That these stratagems have, for the most part, less to do with education than with marketing is a reflection of the museum's low estimate of the public it has won for itself with such great effort. Hence the sometimes extreme measures that even great museums are driven to in their quest for the high attendance figures that are now needed to justify the expense of major events. When, a few years ago, I heard the radio commercials for the Caravaggio exhibition at the Metropolitan Museum of Art\u2014commercials that attempted to make the show sound as much as possible like an X-rated movie\u2014I knew that we had entered a new period in the relation of the museum to its public. This was, after all, advertising aimed at the \"educated\" segments of the public, and it was not only the appeal to a prurient interest that was obnoxious. It was also the cynicism implicit in the whole endeavor, for the museum officials responsible for this advertising campaign surely knew that the exhibition they were touting in this unseemly manner was incapable of satisfying the kind of sexual curiosity the ads were designed to arouse. As a result, the art was demeaned, the public misled, and the institution degraded. This was not an isolated case, either. The same advertising strategy was used by the same museum to promote a showing of Balthus's paintings, too. In other words, a policy had been established to cope with a pressing need\u2014the need to entice a public in whose artistic judgment the museum has no confidence whatever.\n\nThis low estimate of the public's interests and appetites, with its corollary reliance on marketing rather than education as a means of recruiting new constituencies, is even more vividly in evidence in the kinds of new museum architecture that have been created at such vast expense in recent decades. The field of museum architecture has in fact emerged in this period as a cultural battleground of considerable significance, for in the debates that the new museum architecture has prompted, as well as in the particular buildings it has produced, virtually all the most contentious issues concerning the future of high art in our democratic society have been broached, and in them we have been given a glimpse of the way these issues are likely to be resolved, or left even more conflicted and unresolved, for many years to come.\n\nThe first thing we notice about a great deal of the new museum architecture\u2014and this is as true of the refurbished Los Angeles County Museum of Art as it is of the new Louvre or the East Building of the National Gallery in Washington\u2014is that the palace-of-art, or temple-of-art, ambience has been dramatically supplanted by the atmosphere of an emporium, a recreational facility, or a transportation center. This isn't the whole story of the new museum architecture, to be sure\u2014the Kimbell Art Museum in Fort Worth and the Picasso Museum in Paris afford a very different order of experience\u2014but it is the largest part of the story, and the part most likely to be emulated in the future. The grandest of the new museum spaces tend to be allotted not to the contemplation of art objects but to the activity of the crowd\u2014 _as_ a crowd. The debate that has attended the creation of the new museum architecture is, moreover, almost wholly shaped by this shift from an environment of reverence, as it may be called, to an atmosphere of activity and consumption. And while it is less a debate about the philosophy of art than about the philosophy of leisure, it nonetheless has grave implications for the place to be accorded high art in our culture.\n\nLike all cultural questions these days, this debate about museum architecture also has an important political dimension. In the policy disputes governing the new museum architecture, no less than in other branches of the arts today, we have been witnessing the struggle\u2014sometimes it seems as if it were the final struggle\u2014between those who, on the one hand, believe that the future of our civilization depends on safeguarding the integrity of high art from the leveling imperatives of popular culture and populist politics, and those who, on the other hand, are dedicated to destroying\u2014in the name of democracy and equality, of course\u2014all sense of hierarchy and all distinctions of quality in the life of culture itself. The latter have been especially concerned to discredit the idea of the museum as a \"temple\" or \"palace\" of art, which in their view is not only an outmoded museological paradigm but a politically vicious one, and they argue in favor of the kind of design, the kind of program, and the kind of technology that will shatter forever the entire range of museum practices traditionally used to reinforce the sense of hierarchy and the distinctions of quality they so intensely despise. Needless to say, connoisseurship at any level is dismissed as an interest identified with the odious elitism of the past. In the brave new world of the new museum\u2014the populist museum, as it may be called\u2014every interest, including a lack of interest in art, is to be on an equal footing with every other interest. The art object is thus to be denied its traditionally \"privileged\" place in the institution that was originally created to preserve and celebrate its existence.\n\nThe most respectable exponents of this populist museum model\u2014Mr. Pei is, I suppose, the outstanding example among the architects\u2014bear no grudge against art objects, of course, but they tend, all the same, to favor the kind of consumerist museum plan described by Martin Filler and so enthusiastically embraced by Michel Laclotte. But there is also a more radical faction in this debate that takes as its primary goal the downgrading\u2014or, as they say, the desanctifying\u2014of the art object in favor of technological simulacra that can now make images of the art object instantly accessible to ever larger numbers of people. Not the experience of art but the consumption of \"information,\" not the ideal of high culture but the easy, ephemeral pastimes of entertainment, are what now, in this view, should be made the museum's priority.\n\nIt is from this perspective\u2014best described, perhaps, as that of a technocratic populism\u2014that we have lately been given a glossy book called _The Museum Transformed_. Its author is Douglas Davis, who was for many years a writer for _Newsweek_ ; its subtitle is \"Design and Culture in the Post-Pompidou Age\"; and its foreword is by Jack Lang, the French cultural minister, who is himself a champion of the technocratic populism Mr. Davis espouses. _The Museum Transformed_ is, as an intellectual artifact, a perfect example itself of the way the appeal to fashionable consumption is nowadays combined with the denigration of the art object and the advocacy of radical cultural policy in discussions of museum matters. The format of the book is that of the upscale coffee-table picture book, with its many color plates and its lavish use of decorative typography, yet in the nearly two hundred illustrations to be found in _The Museum Transformed_ , which surveys a good many of the art museums that have been recently built or significantly altered, we are afforded very little sense of what it is like to look at works of art in these structures. You can turn many pages of this big book without having a glimpse of an art object, and this, I think, tells us most of what we need to know about the author's view of the museum's real function today. It is scarcely a surprise, then, that Mr. Davis's allegiance turns out to be clearly committed to what he calls the \"post-object\" role of the museum, which, owing to what he also calls the \"post-contemporary condition\" of our culture, is\u2014or at least ought to be\u2014in the happy position of dispensing with the traditional obligation to provide, as he says, \"first-hand access to the sacred object.\"\n\nWhile it may still be necessary for the museum to provide such access to the art object, it is Mr. Davis's view that \"the evolving museum must see itself as a medium of information and pleasure.\" This, in any case, is the direction in which \"the audience and the program of the museum\" are said to be moving, and ought to be moving, and an institution that remains \"committed to posing as the cathedral of the irreplaceable object\" is merely pursuing a lost cause. Instead of a concern for what Mr. Davis calls \"unique totems sanctified by scholars and historians\"\u2014one of the author's many sneering epithets for the accumulated aesthetic achievements of an entire civilization\u2014museums are advised to look to technology and the theories of Walter Benjamin for proper guidance in these matters. Why should the actual art object matter, anyway, when technology has already rendered its physical presence obsolete? He thus invokes \"the presentation of information about the world's authentic treasures, now easily recalled for our eyes and minds by electronic media or by holography, in media varying from the television set to the home computer . . . or to monitor terminals installed in the museum itself,\" and this development is acclaimed as the happy realization of Benjamin's theory of the desanctified art object.\n\n> Long ago [Mr. Davis writes] the German critic Walter Benjamin prophesied that the mechanical reproduction of works of art through photography and film would diminish our insistence on confronting the \"aura\" of art only in its original state. . . . Now we have the means to \"move\" any treasure in a matter of seconds from the vault of the Louvre to a terminal in Columbus, Ohio, before a viewer conditioned by now to intuit the lost \"aura\" of a work of art in its duplicate state.\n\nBut is it really the \"aura\" of the work of art that we go to a museum to have an experience of, or is it something else, something far more concrete that cannot be experienced in \"its duplicate state\"? This refusal to make a distinction between aesthetic experience and what is now called \"information\" is central, of course, to this utopian\u2014or should I say dystopian?\u2014vision of the museum of the future. In his chapter on \"The Museum in the Next Century,\" Mr. Davis writes:\n\n> Because the possibilities for instantaneous transmission of visual and textual information are now almost limitless thanks to computer systems of instantaneous retrieval . . . each museum is now potentially every museum.\n\nThe awful thing is that Mr. Davis is by no means alone in believing in this chimera\u2014nor alone, either, in his eagerness to see this museum of the future quickly and fully realized. Almost nothing that the new museum architects have so far produced for us comes close enough to the ideal he envisions. The Pompidou Center in Paris is the closest approximation, but its futuristic mission has been subverted, in Mr. Davis's view, by old-fashioned curators who persist in their benighted attachment to the art object and to devising sympathetic ways of exhibiting it\u2014admittedly, an uphill task at the Pompidou Center. Certainly the art object is nowhere mentioned in Mr. Davis's evocation of what the \"next\" museum will ideally offer us:\n\n> The grand staircase, the broad walkable corridors, the honeycombed asymmetrical galleries, and of course the elegant theaters and restaurants are the obvious interior elements that define the potential of the \"next\" museum.\n\nThe curious thing is, except for the remaining presence of those \"unique totems sanctified by scholars and historians,\" this sounds to me a lot like the museum buildings we have lately seen erected, but Mr. Davis is apparently the kind of purist who will not be satisfied until the museum is entirely object-free.\n\nIs that day close at hand? I frankly doubt it. The art object is not going to be eliminated from the museum, but what is already upon us in the museum is an altered attitude toward the work of art. It is no longer trusted to speak for itself. It needs, so to say, to be brainwashed. It must now be seen to speak for, or about, something else. The way in which the art object is presented in the museum, the way in which it is interpreted, deconstructed, \"contextualized,\" politicized, and otherwise de-aestheticized, reflects a loss of confidence on the part of museum professionals in art itself.\n\nIt is also a reflection of the shift that has taken place in the academic study of art\u2014the shift from connoisseurship to theories and methodologies derived from the social sciences. The immense influence of structuralist and poststructuralist theory has also played a crucial role in this trend among museum professionals to de-aestheticize the interpretation of art and treat the art object as if it were just another artifact in what is now called material culture. Aesthetic categories are now shunned in the museum in favor of sociological and political categories. The charge of \"formalism\" is now almost as dreaded in the museum world as it used to be in the Communist world. While militant Marxists remain a small minority among museum professionals, the Marxist notion of relegating art to the so-called superstructure, where it is seen to be an embodiment of social and economic forces more profound than itself, has now become part of the conventional intellectual wisdom of a great many curators who have never read a line of Marx himself.\n\nOften entire exhibitions are now organized on the basis of such ideas. The show called \"The West as Art,\" which caused such a furor at the National Museum of American Art in Washington last season, was but one of many recent examples, and most of them cause no furor at all. And this changed attitude toward art is also beginning to be reflected in the way museums present their permanent collections to the public. This summer at the Whitney Museum of American Art in New York, for example, the seasonal \"Selections from the Permanent Collection\" exhibition was mounted under the title \"American Life in American Art, 1950\u20131990,\" and it was billed as a survey of \"postwar paintings and sculpture exploring themes from prosperity to critiques of the American Dream.\" When one ascended to the third floor of the museum to see what had been selected to represent this documentation of \"prosperity\" and \"critiques of the American Dream,\" there were familiar paintings by Milton Avery, Richard Diebenkorn, and Lee Krasner, and sculptures by David Smith, among many other objects, that had nothing whatever to do with these subjects.\n\nI don't suppose that most visitors to the Whitney were even aware that these and the other works of art in the exhibition were intended by the museum staff to be \"read\" as social documentation and criticism. But that will change. With the announcement in July that Benjamin H. D. Buchloh, a left-wing stalwart of the _October_ magazine circle, has been appointed to a position that makes him, in effect, the Whitney's chief political ideologist, you may be sure that we shall be seeing a good many more \"critiques\" of this sort far more aggressively pursued. According to David A. Ross, the new director of the Whitney, Mr. Buchloh \"will have a key role in providing an environment that encourages critical study and theoretical inquiry into the practices, institutions and discourses that constitute the field of culture.\" This, you may be sure, sounds the death knell to whatever remained of a remnant of a serious and disinterested attitude toward art at the Whitney. From now on it will be more and more politics and less and less art, though the politics will be presented, of course, in the name of art.\n\nAnd this political, anti-art course at the Whitney will be pursued in tandem with a revivified emphasis on consumerism at the museum. Mr. Ross's other big initiative this summer was the refurbishment of the Whitney's restaurant and the installation of a new chef. If you think that consumerism and political radicalism represent conflicting polarities in the new museum world, you are mistaken. Nowadays they live on very easy terms with each other, for each conforms to a fashion of the moment, and they are alike in diverting the museum from its real function. Neither contributes anything whatever to our understanding of art. That is what makes them so attractive to the new breed of museum director. You will see the same program of consumerism and left-wing ideology in command at the expanded Guggenheim Museum whenever its director gets around to reopening it. In the new museum there is no need to eliminate the art object. You may instead render it socially and intellectually invisible.\n\nOf the art museum at the end of the twentieth century it may truly be said that nothing has failed like success.\n\n[1991]\n\nFootnote\n\n1. See Raoul Ergmann's interview with Michel Laclotte in _The New Louvre: Complete Guide_ (Paris, 1989).\n\n## Tate Modern Inside and Out: The Museum as Culture Mall\n\n* * *\n\n> Absence of heart\u2014as in public buildings\u2014 \n> Absence of mind\u2014as in public speeches\u2014 \n> _Absence of worth\u2014as in goods intended for the public_ ,\n> \n> _Are telltale signs that a chimera has just dined_ \n> _On someone else; of him, poor foolish fellow, \n> Not a scrap is left, not even his name_.\n> \n> \u2014W. H. Auden, in \"The Chimeras\"\n\nOf the late, lamented Tate Gallery in London it can be said that only a name is left, but now clipped of its definite article and divided into dubious duplicate, Tate Modern and Tate Britain\u2014a reminder, if we need one, that two negatives cannot be expected to produce a positive result. Yet this ill-conceived project clearly represents the spirit of the age, which in art and in life is besotted with an appetite for destroying what is good by enlarging it to a scale of extinction. It puts us on notice that in the twenty-first century we shall need no wars to devastate our monuments to the past. Our cultural bureaucrats have shown themselves to be fully capable of performing the task for us.\n\nAt the outset of these melancholy reflections on Tate Modern, I must acknowledge that I put off visiting this bizarre creation\u2014or should I say bazaar creation?\u2014for a longer time than was, perhaps, appropriate for a professional critic of art and its institutions. Yet I had my reasons. The publicity campaign preceding the breakup of the old Tate Gallery into two new museums\u2014Tate Britain, which is now to be devoted to British art, and Tate Modern, which ostensibly provides London with its first museum of modern art\u2014was not of a sort that inspired confidence in the entire enterprise. Whether we call it hype or spin or (as we used to say) false advertising, it was clearly a campaign to prepare the public for an art museum that was to be something other than a traditional art museum. What this other thing might be remained a little vague, but one somehow had the impression that the advancement of an aesthetic understanding of works of art would not be its top priority.\n\nThe announcement of the Tate's breakup coincided, moreover, with the Blair government's promotion of a policy called Cool Britannia, which cheerfully promised to extirpate all signs of Britishness from the UK's institutions and their symbols\u2014a campaign of self-annihilation that, in my view, is a death warrant for much that I have cherished in Britain during my frequent visits over the course of some four decades. Suddenly the very words \"British\" and \"English\" were being stigmatized as \"racist\" by wacko members of the Blair government\u2014a sure sign that \"New Labour,\" as the Blair government bills itself, was launching a far-reaching _Kulturkampf_ bound to affect the art museums, which in Britain (as in most of Europe) are state institutions.\n\nMy mounting feeling of dread was further advanced when, shortly before the opening of the new Tates, I attended a lecture at our own Museum of Modern Art in New York by the mastermind of this project, Sir Nicholas Serota, who outlined his program of \"new narratives\" for these institutions. Sir Nicholas spoke like a man who had lately emerged from a conversion experience somewhere in the vicinity of Disneyland or some other American theme park. Themes, themes, and more themes were now to determine every aspect of the museums' \"new narratives,\" while historical chronology and the presentation of art according to periods, styles, and movements were to be consigned to oblivion as outmoded, if not indeed\u2014perish the thought\u2014elitist.\n\nThus distinctions between past and present, never mind the very idea of tradition in art, were to be abandoned in favor of something like a boundless eternal present in which an anaesthetic populist ideology would trump every other intellectual imperative. Thinking back now about Sir Nicholas's talk at MOMA that evening, it occurs to me that his only remaining attachment to tradition of any sort is to be found in his acceptance of a knighthood from Her Majesty's government. Was this a sign of hypocrisy? Perhaps. But as it was also a sign of power, it was deemed to be acceptable. All of which I took to be a dour augury, and I was in any case fully occupied elsewhere. I was in no hurry to visit the ruins of what I had for so long regarded as a beloved embodiment of the civilization that had successfully survived the punishing wars and revolutions of the twentieth century with its democracy, its civility, and many of its cultural traditions intact.\n\nBy March, however, I felt I could no longer defer this professional obligation, and I set off for London with a heavy heart. Arriving at night, I had my first encounter with the new, dumbed-down BBC\u2014another sign of the times\u2014when I attempted to find a news broadcast on the television set in my hotel room. What I found was a dispiriting variety of tabloid journalism, delivered in a distinctly downmarket accent, that was determined to whip up hysteria over the spread of foot-and-mouth dis-ease in the countryside while exploiting the already rampant cynicism over the sleaze-scandals of the Blair government. The obvious correlatives that obtained between the Blair government's hype and the Tates' spin strategy were ominous indeed.\n\nStill, London looked beautiful the next morning as I set off on foot from my hotel in Knightsbridge to see \"The Genius of Rome\" exhibition at the Royal Academy in Piccadilly. Spring comes earlier to London than to New York, and the parks were already ablaze with daffodils and primroses in bloom. The sun was shining, the skies were blue, and \"The Genius of Rome\" show proved to be every bit as marvelous as I had expected it to be. It was a pleasure to be able to revisit this exhibition twice again during my week in London. Marvelous, too, was the big exhibition of Rembrandt's etchings at the British Museum.\n\nI was less thrilled\u2014appalled, in fact\u2014by the new, horribly overscale entrance to the British Museum, which now encloses the museum's legendary Reading Room in a faceless structure of monstrous height and girth. No matter how well acquainted we may be with the many examples of execrable museum architecture in our time, atrocities of this magnitude have not lost their power to shock when we first encounter them. It therefore seemed prudent to put off my initial visit to Tate Modern for another day.\n\nThat day, too, was bright with sunshine and blue skies, which made the taxi ride to Bankside, across the Thames, a very pleasant journey. Despair descended, however, immediately upon entering the vast ascending ramp, fully wide enough (it seemed) for two-way truck traffic, that serves as the principal pedestrian entrance to Tate Modern. This unwelcoming space in what was once a power station seems deliberately to embrace the featureless brutality of an underground parking garage. Until then I had thought that the lugubrious Hayward Gallery, designed in the 1960s in what its creators were happy to call the Brutalist style, was the worst museum building in London, but that distinction now has been won by Tate Modern. It was almost enough to make one lapse into nostalgia for the kinder and gentler horrors of the Muse\u00e9 d'Orsay in Paris, which was originally built to serve as a major railway terminal and still feels like one.\n\nI had only just entered this behemoth, but it was already enough to make one experience a profound sense of loss\u2014a loss not only of beauty and amenity but of civilization itself. This was an experience that was to be repeated many times as I began to make my way through the organized chaos of the place, for upon ascending to the top of the unlovely entrance ramp, the visitor to Tate Modern suddenly encounters a somewhat different prospect: the gloom of an underground parking garage dissolves into something resembling a gigantic airport arrivals terminal, where crowds, confusion, and commercialism reign and the only work of art to be seen is what I suppose must be called a sculpture constructed of discarded office furniture. This turned out to be one of Damien Hirst's nonzoological inspirations, and it prompted me to wonder if this mini-monument to Neo-Dada was meant to represent some sort of farewell to the Tate Gallery of old\u2014a memento mori, as it were. Or was it, perhaps, to be construed as a symbolic portrait of Sir Nicholas himself\u2014a portrait of the museum director as deranged bureaucrat? When I put this question to a harried clerk at the Information counter, he was unable (or unwilling) to shed any light in the matter. Given the unrelieved din that made even a brief exchange of words something of a feat, I could hardly blame him.\n\nThen it became a question of what to attempt to see first: the reinstallation of the Tate's modern art collection or its mammoth inaugural theme exhibition\u2014\"Century City: Art and Culture in the Modern Metropolis.\" Given the museum's chaotic layout, there was no consecutive way of seeing either without skipping floors and doubling back. For the sections of \"Century City\" devoted to London and Bombay were right there on the first floor, behind the Information counter, while the sections devoted to other cities in this mad m\u00e9lange of what it pleased its sizable cadre of curators to call art and culture but was often neither\u2014Paris, Rio, Vienna, Moscow, New York, and Lagos\u2014were on the fourth floor, sandwiched in between the two floors, three and five, devoted to a reinstallation of the museum's collection.\n\nIn the end, the order in which the visitor to Tate Modern looks at anything\u2014whether walls festooned with words or rooms in which magazines and newspapers vie for attention with works of art\u2014hardly matters. For every object and every wall text is manacled to its assigned themes and classified according to its subject and sub-subject. In the \"Century City\" show, each metropolis was also strictly confined to a specific time frame. This was lucky for Paris, which was assigned the years 1905\u20131915, and for Vienna, which was given 1908\u20131918, and so some art worth looking at was actually on the walls. The way it was presented, however, was another story\u2014a \"new narrative.\"\n\nYou might think it impossible to devote even a thematic show to the art and culture in Paris in the years 1905\u20131915 that could reduce its high artistic achievement to a tedious, incoherent jumble. Yet Serge Fauchereau, the curator of the Paris section of \"Century City,\" somehow managed to pull off this unimaginable feat without a hitch. A few of Picasso's and Braque's finest Cubist collages, for example, were hung in a narrow corridor in which they could hardly be seen if three or four people were present at the same time. And remember, \"Century City\" was designed to draw huge crowds\u2014and did. The sound track\u2014Stravinsky, I think\u2014didn't help matters, either. It only added to the din.\n\nNew York, to be blunt about it, was given the shaft in \"Century City.\" To provide even a foreshortened glimpse of art in New York in the twentieth century, what period would commend itself? The years before and during World War I when Alfred Stieglitz was showing John Marin, Marsden Hartley, Arthur Dove, Paul Strand, et al., and publishing _Camera Work_? Or the period in the 1940s and '50s when the Abstract Expressionists captured the attention of artists and critics on both sides of the Atlantic, and New York itself became the art capital of the Western world?\n\nForget it. The geniuses who organized \"Century City\" were clearly determined to cut New York down to size\u2014the smallest possible size. So the New York section of the show was limited to a five-year slot, 1969\u20131974\u2014compared, say, to the fifteen-year period, 1955\u20131970, assigned to Lagos, and the fourteen years, 1950\u20131964, given to Rio de Janeiro. The point, apparently, was to find a time when New York in particular and the United States as a whole could be depicted as a society on the skids.\n\nThe iconic object here was the silk-screen print of a demonic Richard Nixon which Andy Warhol produced for the George McGovern campaign in 1972. And to underscore an alleged absence of significant art in New York in this half-decade period, the curator of the New York section\u2014Donna de Salvo\u2014featured the work of Lynda Benglis, Hannah Wilke, Vito Acconci, and Gordon Matta-Clark. The creepiest thing of all, perhaps, was the first wall text that greeted visitors to the New York section of \"Century City\": a quotation from that celebrated connoisseur of urban amenity, the late Nikita Khrushchev, whose statement deplored the absence of \"greenery\" in Manhattan. Apparently nobody took the trouble to show him Central Park when he made his famous foray to New York to announce that the Soviets would \"bury\" us.\n\nDoes all this suggest that \"Century City\" was, in its principal focus and energy, more of an exercise in left-wing political pieties than a serious inquiry into the worldwide achievements of art and culture in the twentieth century? You bet. Hence the exuberant attention lavished upon the Moscow section of \"Century City,\" which focused on the years 1916\u20131930. Did anything even vaguely unpleasant happen to the modernist movement in Russia in this period? Well, sort of. Lutz Becker, the curator of the Moscow section of the show, does very briefly acknowledge that \"Mayakovsky's suicide and Malevich's three-month imprisonment as a suspected spy in 1930 were for many a signal of the beginning of Stalin's terror.\" But that's the last we hear of such unpleasantness, and even this less than satisfactory reference to the Terror that vanquished the Soviet avant-garde is quickly followed by a statement that is patently false. \"Throughout the 1920s,\" writes Mr. Becker, \"artists had contributed to the new society voluntarily and with enthusiasm, with a minimum of Party interference.\" The fact is, you either had to give up your artistic freedom in favor of party hack work or leave the country to survive. Which is why Rodchenko, Stepanova, Lissitzky, and other true believers in the Revolution became, in their graphic art, apologists for Stalin's Terror, and also why, by 1923, Kandinsky, Chagall, Gabo, Pevsner, and many other talents had already fled to the West.\n\nWhat was finally even worse than the political slant of \"Century City\" was its pervasive contempt for art itself\u2014a contempt that refused to acknowledge that works of art do not, in the character and quality of the experience they afford, differ from any other variety of material culture. As a consequence, it was only as a variety of material culture that works of art were presented to the public in this \"new narrative\" of social ideology.\n\nAlas, it cannot be said that the reinstallation of the Tate's collection of modern art fared much better. In this \"new narrative,\" an elaborate system of classifying objects according to themes reduces all of the art to four basic categories: \"History\/Memory\/Society\"; \"Nude\/Action\/Body\"; \"Landscape\/Matter\/Environment\"; and \"Still Life\/Object\/Real Life.\" Except for the artists who are given the privilege of having rooms largely devoted to their work\u2014Bruce Nauman and Louise Bourgeois among the Americans, and Richard Deacon and Ben Nicholson among the Brits\u2014the idea seems to be to pit unlike talents against each other in a game of survival-of-the-strongest. Thus a Richard Long mud wall is pitted against a water lilies painting by Claude Monet in the \"Landscape\/Matter\/Environment\" section. The most horrific of these confrontational installations is the room in the \"Nude\/Action\/Body\" section in which several really grim paintings by Lucian Freud and Christian Schad are pitted against a single exquisite work by Bonnard\u2014 _The Bath_ (1925), the first of the many paintings Bonnard devoted to the subject of his wife lying full length in a bathtub, and a picture that was much beloved by an earlier generation of English painters. This grotesque abuse of a masterpiece ought really to be called museum rape, and it tells us everything we need to know about Sir Nicholas's \"new narratives.\" In one of the installations at Tate Britain, by the way, Lucian Freud is similarly pitted against a Whistler portrait of a girl in a white dress.\n\nWhy, one is finally left wondering, this persistent, pernicious imperative to degrade art of superior subtlety and delicacy\u2014Monet, Bonnard, et al.\u2014by means of incongruous juxtaposition with the kinds of art that are blatantly inimical to its spirit? We can only guess at the motives governing this malign impulse, which is as pervasive in the installation of the Tate's permanent collection of modern art as it is in so much of the art selected for the \"Century City\" installations. My own guess is that it may be an attempt by a younger generation of curators to reduce the masterworks of modern art to the debased levels of feeling which dominate the so-called Young British Artists\u2014think \"Sensation,\" which came to the Brooklyn Museum by way of the Royal Academy and the Charles Saatchi collection. The current London art establishment has so much invested (in every sense) in the international prosperity of these YBAs that it was probably inevitable that their supporters would make an attempt to bring everything else down to their level. From this perspective, it makes perfect sense, then, that the first work which the visitor to Tate Modern encounters is that construction of discarded office furniture by Damien Hirst.\n\nIt makes the same kind of sense for the New York section of \"Century City\" to be dominated by some overscale urban debris\u2014no doubt rescued from the demolition of a New York building\u2014by Gordon Matta-Clark and by Lynda Benglis's photograph of herself in the nude sporting a sizable dildo\u2014a picture that once served as an advertisment in _Artforum_. If items like these were meant to signify that New York art in the early 1970s is only of interest to London now as a kind of prelude to the Brits' own YBAs, the point is well made. But for most of us in New York, Gordon Matta-Clark and Lynda Benglis were entirely marginal to what art in New York was up to in that period. At every turn at Tate Modern, a line remembered from William Hazlitt seemed to sum up the whole spirit of the place: \"People who lack delicacy hold us in their power.\" Is this really the way to run a museum?\n\nThe truth is, Tate Modern isn't an art museum at all. It's something else\u2014a culture mall in which bits and pieces of twentieth-century art, culture, and politics are deboned and prepackaged for quick and effortless consumption. It is the function of this culture mall to reduce the achievements of high art to the sensation levels of tabloid journalism and pop entertainment. It addresses itself not only to a pop sensibility but to a pop attention span, and with a similar need for interminable hubbub. Is it any wonder, then, that it has proved to be a tremendous success, especially appealing to the young? But this success has been purchased at what is also a tremendous price. _Absence of heart, Absence of mind, Absence of worth_ :Tate Modern has them all.\n\n[2001]\n\n## The Man Who Created MOMA\n\n* * *\n\n> _A work of art is an infinitely complex focus of human experience. The mystery of its creation, its history, and the rise and fall of its esthetic, documentary, sentimental, and commercial values, the endless variety of its relationships to the other works of art, its physical condition, the meaning of its subject, the technique of its production, the purpose of the man who made it\u2014all these factors lie behind a work of art, converge upon it, and challenge our powers of analysis and publication. And they should be made accessible to other scholars and intelligible to the man off the street_.\u2014Alfred H. Barr, Jr., 1946\n\nOf the many museums that were founded in the United States in the course of the last century, none has exerted a greater influence on public taste, on American intellectual life, and on the life of art itself than the Museum of Modern Art, which made its debut in New York in 1929 under the directorship of Alfred H. Barr, Jr. With one exception, moreover\u2014the Phillips Collection in Washington, D. C., founded by Duncan Phillips in 1921\u2014none was so clearly the creation of a single governing intelligence.1 The ideas and the aesthetic judgments that shaped the formation of MOMA in the early decades of its existence were primarily Barr's. They were the ideas and judgments of a man who was not yet thirty years old when the museum opened its doors to a wary, uninformed public in the first year of the Wall Street crash.\n\nIn his own generation the young Alfred Barr was uncommonly well prepared to undertake the daunting task that had been offered to him. His intellectual history actually began in high school when, as Sybil Gordon Kantor writes in her forthcoming biography, _Alfred H. Barr, Jr.and the Intellectual Origins of the Museum of Modern Art_ , \"Barr's interest in history was already piqued . . . when his Latin teacher, William Serer Rusk, awarded him Henry Adams's _Mont-Saint-Michel and Chartres_ as a prize.\"2 It was in his sophomore year at Princeton, however, in Charles Rufus Morey's course in medieval art, that Barr decided on a career in art scholarship.\n\nMorey's teaching methods prompted Barr's interest in the sources, patterns, chronology, and spread of a style\u2014an approach that he would apply to modern art, first in his teaching modernism at Wellesley in 1926\u20131927, and then in the very structure of the Museum of Modern Art and its multidepartmental organization. Barr later attributed his famous 1929 plan for the establishment of the various curatorial departments at the Museum of Modern Art to Morey's course.\n\nAfter Princeton, in his graduate studies at Harvard with Paul J. Sachs\u2014himself a collector and connoisseur\u2014the now legendary Museum Course gave Barr his first practical training in the aesthetic and administrative aspects of museum work. It was a mark of Sachs's esteem for Barr that he also provided him with the funds he needed for further study in Europe.\n\nModern art was not yet, of course, an accepted subject for study in the universities. The pioneering course in modern art that Barr initiated at Wellesley is said to be the first of its kind in the American academy. It became a subject of lively public discussion when Barr's remarkably comprehensive \"Modern Art Questionnaire\"\u2014an entrance exam he devised to screen students for his Wellesley course\u2014was published in its entirety as a feature in _Vanity Fair_ , a magazine then widely read in literary and art circles.\n\nBy the time he came to occupy the first directorship of the Museum of Modern Art in 1929, Barr had succeeded in establishing himself as the country's leading authority on the modern movements in both Europe and the United States. From the outset, his approach to the study and presentation of modernism was wide-ranging and ecumenical. It went beyond painting, sculpture, and the graphic arts to embrace architecture, industrial design, theater, movies, and, at least in principle, literature and music. Thus, among the subjects to be identified in his Modern Art Questionnaire, which was an inquiry into what we should now call cultural literacy, were James Joyce and Arnold Sch\u00f6nberg as well as Henri Matisse and Frank Lloyd Wright\u2014though not, oddly, either T. S. Eliot or Pablo Picasso. But then, the fame already enjoyed by Eliot and Picasso may have disqualified them for the purposes of the questionnaire as too easy.\n\nBarr's ecumenical view of modernism was similarly to be observed in the concentration he lavished upon painting, sculpture, and the graphic arts, which inevitably, then as now, have been MOMA's central focus. This was a more unorthodox approach to modernist art in the early decades of the twentieth century than it may appear to be today, for we are all now, albeit in varying degrees, ecumenical in our understanding of what constitutes modernism in the visual arts. In the United States no less than in Europe, however, every avant-garde movement tended to be absolute in its dismissal of aesthetic alternatives. In Paris the Surrealists regarded Matisse as a bourgeois lightweight while the votaries of pure abstraction looked with horror upon the literary and erotic subject matter of the Surrealists. And in New York in the 1930s, the first decade of MOMA's existence, the art scene was deeply riven by a political as well as an aesthetic factionalism. The political left favored social realism, the populists promoted regionalism, aesthetic conservatives condemned every deviation from the conventions of realism while the underdog abstractionists struggled against all of these currents to establish their aesthetic validity.\n\nIt was Barr's distinction that as MOMA's first director he refused to be drawn into this factional warfare, preferring instead to search out aesthetic merit and historical significance in whichever variety of modernist endeavor he encountered. From his study trips abroad in the 1920s he was already well versed in the politics as well as the aesthetics of the principal avant-garde movements in France, Germany, the Netherlands, and Russia. Indeed, he witnessed at firsthand the political suppression of modernist art first in the Soviet Union in the 1920s and then in Nazi Germany in the early 1930s, and wrote eloquently about these baleful developments at the time. In politics he remained an old-fashioned liberal, as immune to every totalitarian temptation as he was to the absolutism that virtually every faction of the avant-garde made claim to.\n\nIn 1949, in a contribution to a symposium on \"The State of American Art,\" Barr made a clear statement of his liberal distaste for what he dubbed the \"battle of styles.\" Observing that \"Abstraction and realism are crude and ambiguous terms but they have a present usefulness as indications of polarity,\" he issued a salutary warning against the politicization of art which had again become an issue in the late 1940s, owing to some stupid congressional denunciations of modernism as \"Bolshevik.\"\n\n> An actual \"battle of styles,\" as for instance between realism and abstraction, is desirable only to those who thrive on a feeling of partisanship. Both directions are valid and useful\u2014and freedom to produce them and enjoy them should be protected as an essential liberty. There are, however, serious reasons for taking sides when one kind of art or another is dogmatically asserted to be the only funicular up Parnassus or, worse, when it is maliciously attacked by the ignorant, the frightened, the priggish, the opportunistic, the bigoted, the backward, the vulgar or the venal. Then those who love art or spiritual freedom cannot remain neutral.\n\nFor his rejection of political attacks on modernism, Barr won much praise, of course, in the liberal press. But for his refusal to be drawn into the aesthetic factionalism of the art world, Barr was roundly attacked by all sides. Lincoln Kirstein denounced him and MOMA itself for promoting the work of Matisse, of all people, and Clement Greenberg criticized him for his allegedly laggard support of the Abstract Expressionists, while Meyer Schapiro condemned what he characterized as Barr's \"essentially unhistorical\" account of the origins of abstract art.3 Yet in retrospect Barr's disinterested and unhurried concentration on aesthetic achievement in a wide range of modernist art proved to be a key to MOMA's success in creating the single greatest collection of modern art in the world. This was itself a considerable intellectual feat.\n\nYet it is one of the curiosities of Barr's pivotal career\u2014a career largely responsible for establishing modern art as a subject for serious critical and institutional attention in this country\u2014that he has rarely, if ever, been acknowledged to have made a major contribution to American intellectual life. This is owing, in part, to the fact that for much of the twentieth century the principal chroniclers of American intellectual life did not take much of an interest in the visual arts. But it also has something to do with Barr's lifelong refusal to become a public personality. Both in his voluminous writings on art and in his public statements about the place of art in our democratic culture, he was seen\u2014by his own design\u2014to be speaking for the museum rather than for himself. And as he was neither an academic nor a journalist nor a charismatic public speaker, it rarely occurred to the chroniclers of twentieth-century intellectual life to accord Barr any attention in their historical surveys of its achievements. More often than not, it was Barr's critics in the academy and in the press who were assumed to occupy the intellectual high ground\u2014and thus worthy of historical analysis\u2014while Barr himself was relegated to the faceless ranks of the museum bureaucrats. It didn't help matters, either, that no collection of his writings on art was published in his lifetime. The splendid volume called _Defining Modern Art: Selected Writings of Alfred H. Barr, Jr_. , edited by Irving Sandler and Amy Newman, was published posthumously in 1986, five years after Barr's death.\n\nGiven this curious history, in which Barr's achievements have somehow been allowed to eclipse our understanding of the man himself, the publication of Ms. Kantor's intellectual biography must be considered something of an event. For this is the first extended account we have been given of Alfred Barr as an intellectual force not only in the art world but also in the larger cultural life of his time. Its focus is emphatically upon ideas and their consequences rather than on the private life of its subject. It examines the conflicts, both personal and institutional, which strongly held beliefs are bound to generate in the arena of cultural debate, but this is not the kind of biography that goes on the prowl for scandal or psychoanalytic revelation. This is not to say that Barr fails to emerge as a distinct personality in this book. Far from it. Ms. Kantor concentrates on character, intellectual development, and the acquisition of skills\u2014in Barr's case, not only the critical skills needed for making a wide range of difficult and often controversial aesthetic judgments but also the political and diplomatic skills essential to professional survival in a milieu as highly charged with oversize egos, big money, conflicting tastes, contending ideas, and the fierce competition for power, influence, and preferment, as MOMA was in the first decade of its existence.\n\nThat Barr succeeded for as long as he did in imposing his vision of modernism, which was essentially but not exclusively the vision of an aesthetic formalist, is an amazing story, and Ms. Kantor tells it well within the limits of the scope of her study. For this intellectual biography comes to an end in 1943 when Barr was precipitously fired by the chairman of MOMA's board of trustees, Stephen Clark. As Ms. Kantor writes:\n\n> Barr had been having difficulties for the preceding six years on a variety of fronts: his tastes were running ahead of the trustees, his writings were always behind schedule, and, in the area of administration (which he disliked), competing forces were undercutting his efforts. . . . His enemies ranged over a broad front of staff members, hostile critics of modern art in general, or adherents of one mode of art or another.\n\nBarr had never been physically robust, and he was increasingly subject to nervous collapses, insomnia, and other ills of the body and spirit. Yet he faced down his adversaries in the museum and outside the museum with an extraordinary display of tenacity and conviction. Although he had been fired from the directorship of the museum, he refused to acknowledge that he had been booted out of the museum itself. He continued to turn up at his office and pursue his curatorial tasks.\n\n> Without fanfare or argument\u2014never mind such a thing as a lawsuit\u2014he simply stayed on and on, continuing to meet his writing commitments, advising on acquisitions and exhibitions until, amazingly enough, in 1944 he was appointed chairman of MOMA's Modern Painting and Sculpture department, which had always been his principal interest anyway.\n\nThere are, of course, many ways to write the history of modern art, and one of the most important, certainly, is to write its intellectual history. It is to the intellectual history of modern art that _Alfred H. Barr, Jr. and the Intellectual Origins of the Museum of Modern Art_ makes an indispensable contribution. It is also timely, for with MOMA currently undertaking the largest expansion in its history, Ms. Kantor's disabused account of its past gives us a needed perspective from which to judge the museum's future.\n\nAbout the future of MOMA we can only speculate, of course, but certain auguries have already alerted us to what might be expected of the greatly expanded MOMA that awaits us. One, certainly, was the break with MOMA's past\u2014in effect, a decisive rejection of Alfred Barr's carefully formulated conception of modernism\u2014that was mounted in the series of exhibitions and catalogues called _MOMA2000_ (October 1999\u2013January 2001).4 An even earlier augury was the publication of _A Fine Disregard: What Makes Modern Art Modern_ (1990) by Kirk Varnedoe, the current director of MOMA's Department of Painting and Sculpture, which grandly promised to bring us \"an entirely new vision of modern art's origins and its subsequent meanings.\" Still further portents may be found in the kinds of changes in programs and exhibitions that have lately been attempted in a number of other museums ostensibly devoted to modern and contemporary art\u2014at Tate Modern in London, for example, and at the Solomon R. Guggenheim Museum and the Whitney Museum of American Art in New York.\n\nNotwithstanding certain differences in emphasis and in sheer intellectual competence, what all of these developments have in common are some sweepingly revisionist attitudes toward art and history: 1) a distaste for the fundamentals of historical chronology; 2) a refusal to accord the aesthetics of style a priority over the subject matter or \"content\" of works of art; 3) a bias that favors tendentious sociology at the expense of connoisseurship; and 4) an appetite for populist appeal. It is this nexus of ideas and attitudes that united such otherwise disparate exhibitions as \"High and Low: Modern Art and Popular Culture,\" which Kirk Varnedoe and Adam Gopnik organized at MOMA in 1990, as well as large tracts of _MOMA2000_ , \"The American Century\" at the Whitney Museum, \"Century City: Art and Culture in the Modern Metropolis\" at Tate Modern, and the Guggenheim's ill-fated turn to exhibiting motorcycles, haute couture, and the pictures of Norman Rockwell.\n\nAll but the Rockwell exhibition have been large and ambitious projects, encompassing hundreds of objects and occupying vast quantities of exhibition space as well as many hundreds of pages of catalogue text. They have been hugely expensive, too. Yet their net effect has been to render the aesthetic history of modern art incomprehensible to anyone coming to them without a firm command of the subject. Intellectually they represent the same corruption of scholarship that has led to the teaching of art history in the universities as, in effect, a branch of the social sciences. For a broad public\u2014Barr's \"man off the street\"\u2014they virtually guarantee historical confusion and a corruption of taste. That some of these monster exhibitions have been box-office hits is not surprising. To package art for the public on the basis of its alleged subject matter is always an easier \"sell\" than an attempt to instruct the public in aesthetic distinctions, for it turns every work of art into an easily apprehended illustration of something other than itself. Yet if the term \"modernism\" in art is to mean anything other than a strictly period classification of objects, an aesthetic comprehension of the art object itself and its relations to other objects of a similar class must be seen to be central to our experience of the art. When such objects are presented to us as illustration, however, it is no wonder that museums are tempted to embrace authentic illustration\u2014e.g., Norman Rockwell\u2014as a substitute for the real thing. As H. L. Mencken famously said, no one has ever lost money underestimating the intelligence of the American people.\n\nAlfred Barr's conception of modernism, in his installations of MOMA's great collections and in many of the museum's major exhibitions during his tenure, provided a coherent account of both the master currents of modern art and its minor tributaries. It was a conception that was bound to be modified, augmented, or otherwise revised in the course of time, especially in relation to subsequent developments in modern art itself and in the literature of criticism and scholarship that they have generated. Yet none of the attempts to overturn the basic tenets of Barr's conception of the aesthetic and intellectual morphology of modernism have proved to be persuasive. Kirk Varnedoe's attempt to provide \"an entirely new vision of modern art's origins\" foundered on the debacle of \"High and Low,\" while Robert Storr's attempt to rewrite the history of modernism in the section of _MOMA2000_ called \"Modern Art Despite Modernism\" was obliged to ignore the fact that much of what he was concerned to reclaim for modernism had long before been encompassed by Barr's ecumenical embrace of modernism's diversity of style. Even worse, the decision to deny the history of abstract art a section of its own in _MOMA2000_ and then consign actual examples of abstraction to various categories of subject matter\u2014thus once again demoting abstract art to the realm of illustration\u2014meant that the public for Barr's 1936 exhibition of \"Cubism and Abstract Art\" was in a better position to understand the aesthetics of abstraction than the much larger public for _MOMA2000_ two-thirds of a century later.\n\nThe truth is, the greatest monographic exhibitions at MOMA in recent decades\u2014among them the 1980 Picasso retrospective organized by William Rubin, the 1992 Matisse retrospective organized by John Elderfield, and the current retrospective devoted to Alberto Giacometti\u2014have all had a close kinship to Barr's pioneering exhibitions. Yet at a press briefing in November for MOMA's forthcoming _Matisse Picasso_ exhibition in 2003, neither Glenn Lowry, MOMA's current director, nor either of the MOMA curators involved in the organization of this big show\u2014Kirk Varnedoe and John Elderfield\u2014thought it appropriate to make a single reference to the definitive exhibitions and scholarly monographs that Barr devoted to Matisse and Picasso during his long association with the museum. They spoke of MOMA's long-standing commitment to the achievements of Matisse and Picasso, but of the man who initiated, nurtured, and institutionalized that commitment they said nothing. For some of us in the audience at that press briefing, this startling omission said more about MOMA's current attitude toward its own history than anything that was actually said about the forthcoming exhibition.\n\nAbout what the new mammoth MOMA now under construction will bring\u2014it is expected to open in 2005\u2014we hear many things, and we can hope that some of them are true. One is that _MOMA2000_ will _not_ be a model for the reinstallation of the museum's permanent collection. That would be good news, indeed. But we also hear that the entire first floor of MOMA's expanded quarters will be devoted to contemporary art, and if true this is not such good news. For in practice\u2014if the present and the recent past are any guide to the future\u2014this is likely to mean a wholesale transfer of the contents of the more trendy galleries in Chelsea to MOMA's midtown facilities. And the present is not, after all, a period that has produced artists who can be claimed to rank with the masters of modernism. Even the few exceptions to this judgment that come to mind are not, for the most part, the artists in whom MOMA's curators take much of an interest. The very thought of a mammoth new exhibition space devoted to the overscale detritus of a post-Duchampian folly and its allegedly \"postmodern\" explorations of \"transgressive\" subjects is chilling.\n\nMy own guess is that what MOMA will do best in the future is what it has done best in the past: pay homage to the masters of modernist achievement, and thus face up to the fact that its principal mission lies in sustaining the tradition that it did so much to establish. And in the execution of that mission, the challenge will still be to meet the standards of the man who created MOMA.\n\n[2001]\n\nFootnotes\n\n1. The same might be said of the Barnes Foundation, founded by Albert C. Barnes in Merion, Pennsylvania, in 1925, but for one crucial difference: the Barnes Foundation was specifically created to function as a school, not a museum. Its subsequent transformation into a public art museum, resulting from dubious court decisions, openly contravened its founder's purposes and program.\n\n2. Sybil Gordon Kantor, _Alfred H. Barr, Jr. and the Intellectual Origins of the Museum of Modern Art_ (Cambridge, Mass.).\n\n3. Meyer Schapiro's criticism of Barr was occasioned by the latter's catalogue essay for one of the most important exhibitions on MOMA's early history, _Cubism and Abstract Art_ (1936). Writing as a Marxist in those days, Schapiro decried Barr's refusal to discuss the social and political aspects of abstract art. Yet Barr himself had already addressed those issues in his early writings: see, for example, \"The LEF and Society Art,\" published in _Transition_ in 1928, and reprinted in _Defining Modern Art: Selected Writings_. For my own analysis of Schapiro's political perspective on modern painting, see \"The Apples of Meyer Schapiro,\" in my book _The Twilight of the Intellectuals_ (Chicago, 1999).\n\n4. For two views of _MOMA2000_ , see Karen Wilkin's \"Rethinking Modernism\" in _The New Criterion_ , January 2000, and my own essay \"Telling Stories, Denying Style: Reflections on 'MOMA2000'\" in _The New Criterion_ , January 2001.\n\n## Does Abstract Art Have a Future?\n\n* * *\n\n> _It is hard to tell if abstract painting actually got worse [after the 1960s], if it merely stagnated, or if it simply looked bad in comparison to the hopes its own accomplishments had raised_.\n> \n> \u2014Frank Stella, _Working Space_ , 1986\n\nIt must be acknowledged at the outset of these observations that the question of whether abstract art has a future is anything but new. The question of abstraction's future has been raised many times in the past. Historically, the question of abstraction's future is as old as abstraction itself, for the birth of abstract art some ninety years or so ago immediately prompted many doubts about its artistic viability. No sooner did abstract art\u2014particularly abstract painting\u2014make its initial appearance on the international art scene in the second decade of the twentieth century than the doubts about its future course began to be heard.\n\nMany good minds have raised such doubts, and many benighted minds have done so as well. There are highly accomplished artists and critics who have taken sides on the question, as well as respected museum curators, art collectors, and art dealers. Over the course of time, in fact, debate about the future of abstract art has been an equal-opportunity enterprise to which the smart and the dumb, the advanced and the reactionary, the informed and the misinformed have all been eager to make a contribution.\n\nI have a particularly vivid memory of an evening in 1954 at the Artists Club in New York when no less an eminence than the late Alfred H. Barr, Jr., the founding director of the Museum of Modern Art, announced that the age of abstraction was drawing to a close and would now be succeeded by, of all things, a revival of history painting. The occasion was a panel discussion on what was then called the \"New Realism.\" It was organized by John Bernard Myers, the irrepressible director of the Tibor de Nagy Gallery, which represented a number of the painters under discussion\u2014among them, Larry Rivers, Fairfield Porter, and Grace Hartigan.\n\nThe Artists Club had been founded, of course, as a forum dedicated to the advancement of Abstract Expressionism. It was therefore inevitable that Barr's provocative pronouncement\u2014and indeed, the very subject of the panel\u2014would be greeted with a vociferous mixture of skepticism and hostility. It had clearly been Johnny Myers's intention to create such a stir on that occasion, for he knew very well that public controversy would have the effect of making some of the figurative painters he represented better known to the art world. He also knew that Barr had just acquired Rivers's _Washington Crossing the Delaware_ , a modernist version of Emanuel Gottlieb Leutze's well-known nineteenth-century history painting in the Metropolitan Museum, for MOMA's permanent collection, and could therefore be counted upon to acclaim Rivers's picture as a significant development on the contemporary art scene.\n\nTo discuss this controversial development, however, Myers had deliberately convened a panel that could be relied upon to be evenly divided about the significance of Rivers's painting. Joining with Barr in extolling the virtues of _Washington Crossing the Delaware_ and the shift in direction it was said to represent was Frank O'Hara, already well established as a poet and art critic. Opposing this view, not surprisingly, was Clement Greenberg. Although he had praised Rivers's earlier work, Greenberg clearly had a low opinion of both _Washington Crossing the Delaware_ and Barr's claims on its behalf. Relations between Greenberg and Barr had never been anything but icy, and Greenberg remained, in any case, firm in his often-stated belief that abstraction represented what he characterized as the \"master current\" of the modernist era.\n\nI was the fourth member of the panel, a newcomer to the art scene whose sole claim to attention on that occasion was an essay I had recently published in _Partisan Review_ that was highly critical of Harold Rosenberg's theory of \"Action\" painting, then a hot topic in the art world. This criticism of Rosenberg, then an arch rival of Greenberg's for the critical leadership of the New York School, had the effect of allying me on the panel with Greenberg even though some exhibition reviews I had lately contributed to _The Art Digest_ were unmistakably sympathetic to a wide range of contemporary figurative painting. I didn't think much of Rivers's _Washington Crossing the Delaware_ , however, and much as I admired Alfred Barr, I didn't believe that a revival of history painting was either imminent or even possible. Like many first-rate art historians, critics, and connoisseurs, Barr was better at codifying the past than at predicting the future.\n\nI mention all this as a reminder that some of the harshest criticisms of abstract art, and some of the direst predictions of its demise, have come from within the ranks of the modernists themselves. In the 1950s, few living persons could rival Alfred Barr in his knowledge and understanding of abstract art, yet he seems genuinely to have come to believe that its end was nigh\u2014and indeed inevitable. In the 1960s, Philip Guston, after winning considerable acclaim as a convert to Abstract Expressionism, won even greater acclaim for loudly denouncing abstraction in favor of a pictorial style derived from comic-strip and other populist images. Then in the 1980s came Frank Stella's Charles Eliot Norton Lectures at Harvard, _Working Space_ , with its litany of contemporary abstraction's repeated failures to produce an art that we had any reason to admire. Stella, too, looked to populist imagery for inspiration\u2014in his case, urban graffiti\u2014with results that are depressingly familiar.\n\nGiven this checkered history, in which so many gifted people have said so many foolish things about abstract art\u2014and we haven't even mentioned the openly declared enemies of abstraction\u2014why does one feel compelled to raise the question of its future yet again? I think there are several good reasons for doing so. One is that by any objective measure, the place occupied by new developments in abstract art on the contemporary art scene\u2014what we see in the galleries and museums and read about in the mainstream press and in the art journals\u2014by this measure, the place occupied by new developments in abstract art is now greatly diminished from what it once was.\n\nIn this country, certainly, you have to go back to the 1950s and 1960s to recall a time when new developments in abstract art had shown themselves to have the effect of transforming our thinking about art itself. This is what Kandinsky, Mondrian, Malevich, and others accomplished in the early years of abstract art. It was what Pollock, Rothko, de Kooning, and others in the New York School accomplished in the 1940s and 1950s. And, for better or for worse, it was what Frank Stella, Donald Judd, and certain other Minimalists accomplished in the 1960s.\n\nI don't myself see anything comparable to this group impact on aesthetic thought happening at the present moment. I haven't observed anything of comparable magnitude occurring in the realm of abstract art since the 1960s. Let me underscore here that I am speaking about group impact on aesthetic thought, about art movements and not about individual talents. I am speaking about the impact of abstract art\u2014or more specifically, abstract painting\u2014on cultural life. That kind of impact on cultural life can rarely be achieved by an individual talent working in isolation from a group of like-minded, or similarly minded, talents.\n\nI think there is a reason why the place occupied by abstract art is now so radically diminished not only on the contemporary art scene but in cultural life generally. At least I have an hypothesis as to the cause or causes of the diminished power and influence that abstraction has suffered since the acclaim it met with and the spell it cast in the 1960s. And let me be clear once again about what I mean in speaking of abstraction's diminished powers. I mean its power to set the kind of agenda that commands the attention of new and ambitious talents\u2014and at times, indeed, even the emulation of established talents.\n\nBefore speculating about the causes of this diminution in what may be called, for want of a better name, movement abstraction, I think it important to recall something that has been central to the aesthetic history of abstraction itself\u2014specifically, its inevitably symbiotic relation to representational art. As all of us know (but sometimes forget), abstract art\u2014especially abstract painting\u2014derives, aesthetically, from representational painting. Whatever the degree of purity abstraction can be said to attain, it cannot make claim to a virgin birth. If abstract painting could be said to have a genetic history, its DNA would instantly reveal its debt to some representational forebear. Whether abstraction derives from Cubism or Impressionism or Fauvism or Neo-Impressionism or Expressionism or some combination of these developments, its antecedents are traceable to the aesthetic vitality of representational painting. This is true of abstract sculpture, too\u2014for abstract sculpture comes out of abstract painting\u2014specifically, Cubism and Cubist collage.\n\nIn my own speculations about the fate that movement abstraction has suffered in our own day, two specific developments seem to nominate themselves as the cause or causes of our current impasse. The larger and more general cause is the fate of painting itself\u2014its fate as a factor in cultural life generally as well as in the life of art. If we look back on two recent developments\u2014the series of exhibitions at the Museum of Modern Art called _MOMA 2000_ and the transformation of the Tate Gallery in London into two really bizarre institutions\u2014Tate Britain and Tate Modern\u2014we are obliged to recognize two things: 1) that on both sides of the Atlantic, abstract art has been marginalized by the institutions that were formerly in the vanguard of its public support and presentation, and 2) that painting itself is well on its way to being similarly marginalized.\n\nIt was certainly striking that in the vast logistical planning that went into the organization of the _MOMA 2000_ exhibitions, no place was accorded to the birth and development of abstract art. There were plenty of examples of abstract art in the separate little shows that constituted _MOMA 2000_ , but those examples were in every case presented to the public on the basis of their subject matter, their so-called content, and not on the basis of their abstract aesthetic. John Elderfield, one of the curators involved in the organization of _MOMA 2000_ , at least had the decency to acknowledge this perverse treatment of abstraction as a failure of thought, but elsewhere in the voluminous texts that accompanied _MOMA 2000_ the aesthetic history of abstraction was consigned to oblivion.\n\nAt Tate Modern in London there was also a discernible hostility to all aesthetic considerations, for painting and sculpture of every persuasion were similarly presented to the public on the basis of their thematic \"content.\" And in the Tate Modern's initial blockbuster exhibition\u2014a real horror called _Century City:Art and Culture in the Modern Metropolis_ \u2014painting could scarcely be said to have been given even a marginal status. In the section devoted to New York, for example, the twentieth century was represented by work from the period 1969\u20131974 with the principal focus on Andy Warhol, Lynda Benglis, Vito Acconci, Adrian Piper, Gordon Matta-Clark, and Mary Miss. There were no paintings, abstract or otherwise. Even in the sections devoted to Vienna 1908\u20131918 and Paris 1905\u20131918, where paintings could not be avoided, they were presented as a kind of tribal art of interest for its social content and political imagery.\n\nIn my own thinking about this fateful shift of priorities away from the aesthetics of painting, both abstract and representational, in favor of a political, sexual, and sociological interest in art-making activities, two historical developments\u2014one within the realm of art itself, the other in the larger arena of intellectual and cultural life\u2014appear to have shaped the situation in which we find ourselves. In the art world, the emergence of the Minimalist movement, which has been so central in determining the fate of abstract art since the 1960s, went so far in diminishing the aesthetic scope and resources of abstraction that it may in some respects be said to have marked a terminal point in its aesthetic development. At the same time, in the larger arena of cultural life, the fallout from the 1960s counterculture left all prior distinctions between high art and pop culture more or less stripped of their authority. It was hardly a coincidence that Minimalism and Pop Art made their respective debuts on the American art scene at the very same moment. However they may have differed in other respects, they were alike insofar as each constituted a programmatic assault not only on the Abstract Expressionism of the New York School\u2014their initial target\u2014but also on the entire pictorial tradition of which the New York School was seen to be a culmination.\n\nIt was left to Donald Judd, the most militant of the Minimalists, to spell out exactly what was at stake in this project to sever all ties to the cultural past. About what Judd contemptuously called \"the salient and most objectionable relics of European art,\" he was nothing if not explicit: \"It suits me fine,\" he said in a radio interview in 1964, \"if that's all down the drain.\" He clearly meant it, too, for what was needed, in his view, was an art that would radically occlude all connection not only with the great traditions of the distant past but also with the kind of latter-day modernism that he had come to regard as the depleted remnants of a moribund culture. For Judd, art itself had become a utopian project.\n\nThere is a passage in an essay by Lionel Trilling\u2014\"Aggression and Utopia,\" published in 1973\u2014that neatly defines the spirit that came to govern this utopian project and so much else in the Minimalist movement. Never mind that Trilling was writing about a Victorian utopian romance\u2014William Morris's _News from Nowhere_. His admonitory analysis of its utopian vision applies with uncanny accuracy to the Minimalist project. This is the key passage:\n\n> . . . the world is an aesthetic object, to be delighted in and not speculated about or investigated; the nature and destiny of man raise no questions, being now wholly and finally manifest. . . . [I]n Morris's vision of the future, the judgment having once been made that grandiosity in art is not conformable with happiness and that Sir Christopher Wren had exemplified radical error in designing St. Paul's, the race has settled upon a style for all its artifacts that is simple and modestly elegant, and no one undertakes to surprise or shock or impress by stylistic invention.\n\nThis, in my view, is the Ground Zero from which the aesthetics of abstraction has not yet recovered.\n\n[2002]\n\nA NOTE ON THE AUTHOR\n\nHilton Kramer was born in Gloucester, Massachusetts, and attended the public schools there. He received a bachelor's degree from Syracuse University and studied at Columbia, Harvard, and Indiana universities before teaching at the University of Colorado, Bennington College, and the Yale School of Drama. Mr. Kramer became editor of _Arts Magazine_ in 1955 and art critic of _The Nation_ seven years later. In 1965 he was appointed art news editor of the _New York Times_ , then its chief art critic, a position he held until 1982, when he left to found _The New Criterion_ , of which he is now co-editor and co-publisher. He was also art critic of the _New York Observer_ from 1987 to 2005. Mr. Kramer's other books include _The Twilight of the Intellectuals_ , _The Age of the Avant-Garde_ , and _The Revenge of the Philistines_. He lives in Damariscotta, Maine.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n**TABLE OF CONTENTS**\n\nTitle Page\n\nDedication\n\nBook One\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-one\n\nBook Two\n\nChapter Twenty-two\n\nChapter Twenty-three\n\nChapter Twenty-four\n\nChapter Twenty-five\n\nChapter Twenty-six\n\nChapter Twenty-seven\n\nChapter Twenty-eight\n\nChapter Twenty-nine\n\nChapter Thirty\n\nChapter Thirty-one\n\nChapter Thirty-two\n\nChapter Thirty-three\n\nChapter Thirty-four\n\nChapter Thirty-five\n\nChapter Thirty-six\n\nChapter Thirty-seven\n\nChapter Thirty-eight\n\nAcknowledgments\n\nAbout the Author\n\nAlso by William Bernhardt\n\nCopyright Page\n**FOR HARRY\n\n** I am so proud of you.\n**BOOK ONE**\n\n> **THE GROTESQUE**\n> \n> And all my days are trances; \n> And all my nightly dreams \n> Are where thy dark eye glances . . .\n> \n> \u2014\"To One in Paradise\" \n> EDGAR ALLAN POE\n**CHAPTER ONE**\n\nThree times I've fired my weapon. Three times. Twice because I had to. The third time was optional. But I never plugged anyone for making a pass at me, no matter how tempting it might be. It was a rule. Until that night in early October. When the whole damn mess began.\n\nI really don't know how it happened. For starters, I looked like hell and I knew it, despite what the guy was saying. It was all bullshit.\n\n\"Has anyone ever mentioned that you have a gorgeous pair of eyes?\"\n\n\"Only my ophthalmologist,\" I told the kid in the Polo.\n\n\"No, seriously, you do. My mom says I've always been an eye man.\" He leaned closer. I could smell the whiskey on his breath. \"Are they different?\"\n\n\"Different from . . . your mom's?\"\n\n\"From each other. It's like . . . your right eye is darker than the left.\"\n\nI nodded. \"Cat scratch. When I was five.\"\n\n\"Well, it works for you. Gives you an exotic aura.\"\n\n\"If you like that, wait till you see my athlete's foot.\"\n\nHe smiled, which wasn't his best look. \"You know what? You're funny.\"\n\n\"Not another reference to my appearance, I hope.\"\n\nHe scooted his chair closer to mine. \"Look,\" he said, his voice suddenly low and tremulous. \"I think it's obvious what's happening here. Why don't we cut through the baloney, go back to my place, and give each other what we both know we want?\"\n\n\"At the moment, there's only two things I want.\"\n\n\"And they would be?\"\n\n\"Another bourbon. Neat.\"\n\n\"I can arrange that. What else do you want?\"\n\n\"You to leave.\"\n\nThe bar, Gordy's, was a hellhole I'd discovered when I was working on a case. Mind you, Vegas has some beautiful neighborhoods. This just wasn't one of them. Cops get called to some of the seediest parts of the city\u2014actually, I think I've been to all of them. My specialty is the psychological profiling of deviant personalities. They call me a detective, but what I really do is provide detailed descriptions of creeps they haven't been able to catch, which can be plenty challenging. I love it. Anyway, I tracked some low-life child molester here. Hated him but loved his bar. I bonded with it; I don't know why. It wasn't at all a _Cheers_ thing. Barely anyone there knew my name, and I liked it that way.\n\nThe d\u00e9cor was deadly. Tacky like the worst small-town plywood watering hole, except this was buried in Vegas's old downtown. Noise thundered relentlessly, assaulting your eardrums, not just music but an endless stream of chatter\u2014sports, politics, and lame come-on lines. The place stank, maybe because drunks kept leaving the men's room door open, maybe because a wino on one of the bar stools kept vomiting on himself. Even the tables reeked, moldering wood soaked in way too much spilled hooch. There was a staleness to the air that made your head throb the second you stepped inside, that made cigarette smoke seem like a welcome alternative. And Gordy's teemed with men of the worst sort\u2014not the bikers, pimps, prostitutes, mobsters, gamblers, and bookies that gave Vegas its colorful reputation, although they were there in force, but preppy types from UNLV in starched golf shirts who knew they could treat anything with breasts like dirt and still get laid because they were so damned hot and hunky.\n\nBe it ever so humble.\n\nI wasn't even thinking about work, so it came as a surprise when I saw Hikuru Mikimoto enter this two-bit saloon. He was a big-time drug dealer. And I hate drug dealers. I'd been consulting with some of the boys in Narc, trying to draft a profile that might help them find him. I really wanted to help, to prove that I could still do the job, but we'd been looking for more than three weeks without results. And then I just look up and there he is.\n\nI wasn't entirely sure I was up to an arrest, but I couldn't let a godsend like this slip through my fingers. I pushed to my feet, bumping the table over, and fumbled for my badge.\n\n\"LVPD. Freeze, Mikimoto!\"\n\nHe was a middle-aged Asian man, his paunch masked by a black T-shirt and what looked to be an Armani sport coat. As soon as I spoke, he took a decisive step backward. And two men behind him surged forward.\n\nPersonal goons. This was going to be more complicated than I had realized.\n\nThey came on strong and quick. My only chance was to take them out before they could gang up on me. I pulled my gun and fired, but the shot went wide. It hit the mirror behind the bar and shattered it. The lounge lizards sitting at the bar scrambled. A second later, one of the goons knocked the gun out of my hand. I did a quick spin behind the table and a swing kick with my left leg, catching him full in the face. He dropped like a sandbag and didn't get up. The other one lunged from behind and grabbed me around the throat. I bit down on his arm, and when he released his grip, I gave him an elbow to the solar plexus. He doubled over. I grabbed him by the ears and propelled him into the hardwood bar.\n\nStupid fool didn't know when to quit. He pulled himself together and came at me again. I whirled around at the last moment and used a move they'd taught me at the academy, a little Judo 101, to flip him over my shoulder. He flew forward and crashed into that splintered mirror. Big chunks of glass sprayed the room. All the patrons ducked for cover.\n\nMikimoto tried to run away. Not likely. I dove for him, brought him down hard. By this time, the rest of the customers were racing for the doors, desperately trying to get out of my way. None of them offered to help.\n\nI straddled Mikimoto, pinning him facedown against the filthy glass-strewn floor. He was raging, babbling incoherently in some language I didn't understand.\n\n\"You're under arrest,\" I said, wishing to God I had a pair of cuffs. \"You have the right to remain silent. If you choose to waive that right\u2014\"\n\nMikimoto swung around with a speed that caught me by surprise. He had a small switchblade in his hand.\n\nNow that pissed me off.\n\nI twisted his arm at the socket, breaking it. The knife clattered to the floor. I wrenched his hand back, pinching it in the soft fleshy part between the thumb and forefinger. He screamed. With his slicked-back hair in my fist, I pounded his head against the floor.\n\n\"Goddamn drug dealer,\" I muttered. \"Preying on kids. Pulling a knife on me.\" I shoved his face down again, hard, and then repeated it, again and again and again.\n\nI felt someone pulling on my shoulders, trying to interfere. Another accomplice?\n\nNo. It was Harry, the old guy who worked behind the bar.\n\n\"Susan!\" He'd been shouting, but for some reason it hadn't registered until now. \"Stop it! Stop it!\"\n\n\"Keep cool,\" I said as I let Mikimoto's limp head flop to the floor. \"This creep's the worst scum in Vegas. Pushes hard drugs to schoolchildren.\"\n\n\"Who the hell are you trying to kid?\"\n\nI didn't understand him, didn't get it at all. But as I stared at Mikimoto's face, it seemed to, I don't know, sort of shimmer. Like a shape-shifter in a science fiction movie.\n\n\"This is police work, Harry,\" I growled, still staring at the face on the floor. \"I'm doing my job.\"\n\n\"You're drunk off your ass is what you are. Did you bloody that kid up just 'cause he was trying to make time with you?\"\n\nI kept watching as the face changed, the whole body changed, and instead of a slick black T there was a pink Polo. How had the drug scum pulled this off? I wondered. Disguising himself as some preppy creep!\n\nI pushed up to my feet. All at once, I realized how wobbly I was. The room began to spin, so I sat down again. The problem with that was, my eyes went back to the face, that kid's face, and I saw all the splattered blood and swollen flesh surrounding it. That finely chiseled face was like a pound of ground round.\n\nStrong hands rummaged under my coat, taking my flask, and I didn't resist. \"I told you to lay off the sauce an hour ago,\" Harry said. \"Didn't know you had a private stash, damn you. How the hell am I going to explain this?\"\n\nThe room was still spinning, even though I was sitting. I felt like I might rip my stomach out with a dull knife if I could. Then I noticed that I was bleeding, too, that I was sitting in a pool of glass, and that there was an especially large shard right in front of me, and I recall thinking someone should do something about that because it could hurt someone, and then I grabbed it and jabbed it into my left wrist. Blood spewed everywhere.\n\nI fell over onto the floor, head first, and the rest of the world went away. After that, I don't remember anything. I assumed I was dead.\n\n\"Am I dead?\" the young girl asked.\n\nHe stared down at her, stretched out on the table before him, a luminescent tableau so full of innocence and youthful curiosity. Her lengthy stay in the basement, so far from the bright lights of the city, had caused her skin to etiolate, but rather than detracting from her natural splendor, it seemed to enhance it. The primordial was strong with her, he sensed. He had chosen well.\n\n\"Of course you're not dead, my darling. You can see, can't you? Hear, smell, taste, and touch?\"\n\n\"I can't move. Not at all. Nothing below my neck.\"\n\n\"I know.\"\n\n\"I think I've wet myself, but I'm not sure.\"\n\n\"You have.\"\n\n\"Even talking is hard.\"\n\nHe brushed a hand gently across her forehead, straightening her bangs. \"I'm so sorry.\"\n\n\"And I'm scared. Really scared. You're not going to hurt me, are you, mister?\"\n\nHe was short of stature, but he liked to think he had a certain presence just the same. Did his accent thicken as he spoke to the offering? He suspected that it did. The genteel Southern gentleman rose to the surface.\n\nHe turned and gazed out the window, just above ground level. The sky was clear as glass; the air was pungently sweet. And oh, the stars\u2014! The stars seemed to go on forever, traveling from his private retreat all the way to Dream-Land. Heaven was real here, far removed from the decay of the city, the fiberglass fa\u00e7ades and organic stench. He did not look down but across, outward, into the desert, the vast untouched expanse, the low-lying Spring Mountains, feeling the arid warmth as it bathed and reassured him.\n\n\"Mister?\" Her voice was slow and stuporous.\n\n\"Yes?\"\n\n\"Am I\u2014am I\u2014\" Her hair was caught in her mouth. She tried to blow it away, but it was sticky and wet and wouldn't go and there was nothing she could do. She was like a rag doll, unable to help herself.\n\nHe reached down and brushed the hair out of her mouth. \"Is that better, my dear?\"\n\n\"Yes. Thank you.\"\n\n\"And your question?\"\n\nHer eyes were swollen and red from the anger phase. Screaming, shouting, threatening. Testing the waters, learning the abject futility of it all. Now she was more subdued, acquiescent. \"Am I naked?\"\n\n\"Yes, love. You're just as God made you.\"\n\n\"W-Why?\"\n\n\"Because I wanted to see you as you truly are.\"\n\n\"Did you . . . do anything to me? While I was out?\"\n\nHe pressed a hand against his black cotton vest. \"What manner of monster do you take me for, madam?\"\n\n\"Well . . . I didn't know.\"\n\n\"There has been no physical impropriety, I can assure you of that.\"\n\n\"Well . . . that's . . . good, I guess. So . . . could I have my clothes back?\"\n\n\"I'm afraid not.\" He reached down and brushed another strand of hair out of her mouth. He held it for a moment, staring at the root. \" 'The life upon her yellow hair but not within her eyes . . .' \" He looked at her with opprobrium. \"You're not a natural blonde.\"\n\n\"No.\"\n\n\"But your\u2014your\u2014\" His face flushed.\n\n\"Dyed that, too.\"\n\n\"Oh, my. Oh, my.\" He assumed a stern expression. \"My dear girl, this will never do. I mean, it simply isn't done.\"\n\n\"All the girls at my high school were doing it.\"\n\n\"Then I shall see that you never return to that pubescent whorehouse.\" He cleared his throat, fanning himself. \"I couldn't help but notice when I undressed you. You were wearing\"\u2014he bore a pained expression, as if the very words hurt him\u2014\"thong underwear. Do your parents know about this?\"\n\n\"No. A girlfriend bought them for me. Amber.\"\n\n\"I thought as much. Well, I destroyed the offending article.\" He leaned in close and whispered, \"There's only one kind of girl who needs thong underwear, Helen. And you aren't that kind.\"\n\nShe spoke hesitantly, her words still slurring occasionally. \"You can't know what kind of girl I am. You don't know me.\"\n\n\"You're wrong, my lovely. I've been watching you. When you slipped out.\"\n\n\"What?\"\n\n\"I've even been to your Web site. I know you've been unhappy. I know your mother doesn't understand you. I know you were contemplating leaving home for good. You want something better than what you have been given, something richer. A Dream-Land.\"\n\nThe fear in her eyes was so intense he felt it in his heart as if it were his own. He had always been like that, sensitive to a fault, so in tune with the feelings of others that it sometimes became unbearable. He wished there were some way to turn it off, to flick the switch, to distance himself. But he had learned long ago that distance was not an option for him. He was a part of this world, and so he would remain. And if he could not escape the world, then his only recourse was to make it a better world.\n\nStaring down into her fearful eyes, golden locks encircling her face, it was impossible not to think of another girl, another innocent, from that lost time so long ago. Following him through the forest, splashing him at the beach, she was the best of him, too pure to be tainted and forever young.\n\n\"Mister? Do you think if maybe I promised not to wear that underwear anymore, you could, um, let me go?\"\n\n\"But my sweet, we have so much work yet to do.\" He returned to the basin and placed a washcloth in damp water. With great vigor, he began scrubbing her face.\n\n\"Mister? You're . . . you're hurting me.\"\n\n\"It's got to come off. All of it. A good girl doesn't need paint to make herself attractive.\"\n\n\"But\u2014you're tearing\u2014\"\n\n\"A little elbow grease. That's what's wanted here.\" He made a small gasping noise. \"Are those eyelashes false? Pity.\" He ripped them off.\n\n\"Oww! Mister, please\u2014\"\n\n\"And the same goes for these earrings, I'm afraid. Imagine piercing your flesh so you can adorn yourself with colored glass. They've got to go.\" He yanked them off through the lobe.\n\nThe girl shrieked. \"Please! Oh, my God, stop, please!\"\n\n\"Don't fret. I'll get something to stanch the bleeding.\"\n\nThe girl began to tremble helplessly.\n\n\"And that leaves us with the problem of the hair. What to do about the hair?\"\n\n\"Maybe\u2014I could just wear a wig?\"\n\nHe considered. \"I fear that would only intensify the artificiality. No, there's only one thing to be done.\" In the cabinet beneath the basin, he found a battery-operated electric shear.\n\n\"Please, don't. Please.\" She breathed heavily, twisting her head back and forth.\n\n\"I don't want to. But I have no choice.\" He switched the clippers on. \"Please don't move.\"\n\nHe applied the shear to the crown of her head and moved over the crest in a long straight line, like a Marine barber buzz-cutting a new inductee. The girl let loose hopeless streams of tears that she couldn't wipe away.\n\nIn a matter of minutes, the cutting was done and the hair was gone. He took a brush and dustpan and cleaned up the girl and the table she rested upon. He retrieved his damp warm cloth from the basin and used it to gently, tenderly rinse her face and scalp.\n\nHe ran the palm of his hand across the top of her head. \"Smooth as a baby's bottom. There now. That wasn't so bad, was it?\"\n\n\"You\u2014you cut off my hair,\" she said, her voice loose and broken.\n\n\"It was a necessary unpleasantness, but there's an end to it. Now we can relax and wait until the moment of\u2014\" He stopped short. His mouth twitched. \"You've painted your fingernails.\"\n\nHer eyes were wide, pleading. \"Everyone does it!\"\n\n\"No, not everyone.\"\n\n\"Please don't hurt me anymore. I'm begging you.\"\n\nHe smiled reassuringly. \"Fear not. You'll barely feel it.\"\n\nIt was a simple procedure; they were artificial, press-on augmentations that left little dots of glue on the true nails after he tore them off. Then he pulled a chair to her side and rested. This process was more difficult than he had anticipated. He gazed out the window into the crepuscular sky, contemplating the outlines of the pretend palaces of the Strip, the headlights rushing from one nowhere to the next, hustling people about like the miserable ants they were. He was so fortunate to be here\u2014sanctified, removed, anointed. So lucky.\n\nHis eyes turned upward, tracing the rectilinear line where the horizon melted into the sky. This was his favorite kind of dusk, with no moon and just enough light to turn the sky a rich roseate blush. Gazing at this masterpiece painting, he thought: who could doubt that there was a plan for us?\n\n\"Look at the stars,\" he said after he wiped the tears and blood from her. \"You can see the heavens so clearly. There must be a million of them. They're beckoning to us, leading us to the truth, telling us how we can live among them. But so few listen. So few can.\"\n\n\"Mister,\" she said. Her voice was dry and coarse, a staccato grating. \"Are you going to kill me?\"\n\n\"Why would you ask that?\"\n\n\"Because you took me and you brought me here and I can't move and, and\u2014\" Her voice broke down. \"And I think maybe you're going to kill me.\"\n\n\"Well, I'm not. Not precisely.\"\n\n\"Then why? Why have you kept me here so long? What are you going to do to me?\"\n\nHe pressed his head close against hers, and his eyes shone with reflected starlight. \"Something wonderful.\"\n**CHAPTER TWO**\n\nAm I dead? I wondered.\n\nMy last night in detox, I woke around five A.M. and saw David standing at the foot of my bed.\n\n\"Sugar bear?\" I whispered, only marginally awake. My eyes were filmy and I knew I was mumbling, but I didn't think it would matter.\n\n\"I'm here, Susan,\" he answered. \"How are you?\"\n\n\"Not so good.\"\n\n\"Sorry to hear that.\" My God, but he was handsome. Made my whole body go warm and liquid. \"I've been keeping an eye on you.\"\n\n\"Have you? I haven't felt it.\"\n\n\"Wasn't sure you'd really want to.\"\n\n\"Don't be dumb.\" I tried to move but my body wasn't responding. Just as well. I'd have been crawling all over him, probably violating several hospital regulations. \"Did you see what happened?\"\n\n\"Yeah. You screwed up big-time.\"\n\n\"Didn't mean to. I was . . . confused.\"\n\n\"It happens. So you're leaving today?\"\n\n\"Thank God. I threatened to bust some doctors' heads if they didn't let me out.\"\n\nHe tucked his head, letting all that jet-black tousled hair cascade over his eyes. \"I'm not sure that was the right thing to do.\"\n\n\"I had no choice. They're killing me.\"\n\n\"You need help.\"\n\n\"You've got some nerve, saying that to me.\"\n\nHe was so strong, even when he was silent. Muscled arms. Adorable chin dimple. \"I miss you.\"\n\nI reached out to him, but it was like touching a bubble: the instant you do, the filmy surface wraps around your hand and evaporates. I wanted to feel David so much. But my fingers fluttered like butterfly wings in the empty air.\n\nDid I mention that I hated Dr. Coutant? Detested the man. I was only in that Popsicle joint six days, but it felt like a month in hell, thanks to Dr. Coutant.\n\n\"Let me state again that I oppose your early release. I think you need more time.\"\n\n\"Especially when you're billing by the hour, right, Doc?\" I said it only because I knew it would infuriate him. The guy had been trying to get my goat all week\u2014how could I resist the chance to give back a little of the same? I've been around doctors enough to know that they love to trash lawyers and other professional clock-watchers while ignoring the fact that their own bills are higher than anyone else's.\n\nHe had me at a card table in the main lobby of the detox ward, by the nurses' station, down the hall from the private rooms. The wing was all done up in calming shades of beige, with padded sofas and soft carpets. Like an airport lounge. \"As long as the city's health insurance is footing the bill, what do you care how much I make?\"\n\n\"I was just saying\u2014\"\n\n\"The fact is, Ms. Pulaski, you have a serious chemical addiction, and six days in detox isn't going to cure it. You need some time in a professional rehab facility.\"\n\n\"Do tell.\"\n\n\"I don't get the sense that you're taking this seriously.\"\n\n\"Not as seriously as you, certainly. . . .\"\n\n\"Your addiction, I might add, was fueled by severe emotional problems, which you also are not dealing with.\" He was a stout, short man. When he went into his sapient counselor mode, he leaned back, his arms folded across a belly not even his white coat could disguise, and used an orotund, patronizing voice that affected me like teeth on tinfoil.\n\n\"Hey, I listened to your lectures. I took notes, even.\"\n\n\"That's not going to help when you get the urge to drink.\"\n\n\"Look, Doctor, I was never really addicted to it. I just let it get out of control. I'm not going to do that anymore.\"\n\nHe fingered the rim of his glasses. \"That is, quite literally, what they all say.\"\n\n\"But in my case, it's true. I won't\u2014\"\n\n\"Ms. Pulaski, you do yourself no favors by minimizing your actions. You went on what apparently was a three-day bender that culminated in serious\u2014\"\n\n\"I made a mistake\u2014\"\n\n\"You had an alcoholic delirium, turned violent, and nearly killed a man!\"\n\nI clammed up. It was obvious he wouldn't let me go until he felt I had been sufficiently punished, so I just let it ride. He could inveigh against me till his beard turned gray.\n\n\"We need a plan,\" Coutant said, frowning. \"I'm never comfortable releasing a patient unless he or she has a road map for overcoming the addiction. I want you to attend classes.\"\n\n\"Classes? As in school?\"\n\n\"IOP. Intensive outpatient therapy. I'll put the information in with your release papers. Our group leaders are very gifted. You'll be surprised how much you'll benefit from it. And you should supplement that by attending a registered AA group.\"\n\n\"So, we're talking, amateur shrinks trying to get inside my head?\"\n\nHe stared down at the table. I could tell he was choosing his words carefully, being oh so tolerant, which really drove me bananas. \"You have a lot of anger, Ms. Pulaski. You may not be aware of it, but you do. That's what drives your self-destructive behavior.\"\n\nI can't stand this business where some guy with a beard and a Ph.D. spends an hour with you and thinks he knows your whole life story. \"Look\u2014am I getting out or not?\"\n\n\"I don't have the power to keep you against your will. I wish I did.\"\n\n\"Then give me my Get Out of Jail card and let me go.\"\n\n\"But we need a plan.\"\n\n\"Why? So you can magnify the paper jam in your file?\"\n\n\"Because if we don't, you'll start drinking again. Too often alcoholics have to hit rock bottom before they stop abusing alcohol and start putting their life back together.\"\n\n\"That's bullshit. If I say I won't drink, I won't drink.\" I got up and walked to the door\u2014locked, of course\u2014making it clear I was ready to leave. Hadn't I played his games long enough? Why was he so determined to prove that I had dark, insuperable problems that only he could solve? I had a job and a house and a niece who must be bouncing off the walls by now. Why should I let these people drain the insurance companies dry making referrals to one another? I had a life to live.\n\nGod bless Lisa. She was waiting for me when I finally got out of the treatment facility.\n\n\"Hail fellow well met,\" I said, wiggling my fingers in the air. \"Thanks for the lift.\"\n\n\"My pleasure. Missed you, sweetie.\" She gave me a peck on the cheek.\n\n\"Ditto. Now get me the hell out of here.\"\n\nLisa is my oldest friend. We've known each other since kindergarten. I stole her cookie the first day and we've been girlfriends ever since, through school and college and her marriage to a bodybuilder with a leather fetish and her divorce and subsequent endless string of dates to guys remembered only for their kissing ability or lack thereof. Not to mention all my little difficulties.\n\nShe put her convertible into first and screeched out of the parking lot. She couldn't begin to afford this car; it cost more per month than her apartment. But she loved it. Lisa was an inveterate speed freak, always had been. In kindergarten, it was the swings. Now it was a Porsche. I think she's in one of those Oprah categories: Women Who Love Their Cars Too Much. And why not? We both pretended there was nothing orgasmic about the smile on her face when she shifted into high gear.\n\n\"Jesus Christ, I can't believe I spent six days in there.\" It felt good, letting the wind rush through my hair\u2014like being in a cool shower, that tingly sense of something cascading over your entire body. She was taking me down the Strip\u2014what outside Vegas is called U.S. 9. Hotels and casinos. Volcanoes and pirate ships. Fabulous multicolored neon view. Bright lights, Sin City.\n\n\"They told me they wanted you to stay longer,\" Lisa said.\n\n\"There's no money in miracle cures.\"\n\n\"That's a bit cynical, even by your standards.\"\n\n\"Did you ever wonder why it's always a twelve-step program, Lisa? Three steps would be insufficiently profitable.\"\n\nShe looked gorgeous in this car, with the wind whipping her hair back like a model in a shampoo commercial, which was probably another reason it was worth the money to her. She had long hair, perfectly blond, not a trace of dishwater. Or black roots. She worked out and never ate anything and looked great. She had those wonderful slender arms with firm muscles, the telltale identifier of a gym girl. I worked out, too, but on me it looked bulky and formidable, not sleek and sexy. If Lisa weren't my best friend, I'd hate her.\n\n\"Anything interesting happen in there?\" she asked.\n\n\"Not much. It was evil.\"\n\n\"Evil?\"\n\nI nodded. \"Doctors with lots of questions. Nurses taking your vital signs for no apparent reason. Some old gal who gave me a daily physical and enjoyed it way too much, if you know what I mean.\" Lisa giggled, which made me smile for the first time today. She was a tough audience. If I could make her laugh, I was doing something right. \"Twice a day some guy would come in and lead a group session on the evils of substance abuse, and the whole time he's drinking coffee by the gallon.\" I gave her a grim look. \"And they made us play games.\"\n\n\"You're kidding me.\"\n\n\"Scout's honor. Monopoly. Clue. Even Scrabble. It was compulsory.\"\n\n\"Now that's evil.\"\n\nI hesitated a moment. \"I saw David.\"\n\nHer reaction was just a beat delayed, though she tried to act as if I had said nothing unusual at all. \"You did?\"\n\n\"Yeah. In my room.\"\n\n\"Were you having, um, dreams?\"\n\nA nice way of putting it. \"I don't know. I guess. Didn't seem like it.\"\n\n\"But Susan, you know . . .\"\n\n\"Yeah. I'm not that far gone yet.\"\n\n\"Well.\" Lisa focused her attention on her driving.\n\n\"He said he missed me. He said he was watching me.\"\n\n\"That's nice.\"\n\n\"Do you believe that?\"\n\n\"Believe what?\"\n\n\"That dead people watch us after they're gone. That they're up in the clouds, keeping tabs on the people they knew.\"\n\n\"Sure.\"\n\n\"I don't.\" I extended my arm out the passenger-side window, letting the wind ripple through my fingers. \"I'm not even sure I believe in an afterlife. But if there is one, I can't imagine why anyone there would waste it watching the folks back on earth. I mean, if that was what you wanted to do, why leave in the first place?\"\n\nShe didn't have an answer, so she concentrated on her driving, which was just as well.\n\nHe managed to get Helen into the back of his pickup, but just barely. Even though she was wrapped securely and he was using the flatbed dolly, it was hard work. But no matter, he consoled himself. At the end of days, one does not dwell on the mundane.\n\nAfterward, he washed his hands, then dried them with a daisy-pattern dish towel, one of the few possessions of Nana's he had saved. She had taken such pride in those towels, he recalled, back when they were new. A small token of simple beauty in a life of utter squalor, he supposed. Where had she gotten them, anyway? Had the bank been giving them away? The gas station? A free gift in a box of detergent? He couldn't remember.\n\nHe walked back out the front door, whistling. Whistle while you work\u2014that was what those dwarves said. He chuckled. And people called him short.\n\nGinny had loved that movie. Nana had the tape and they'd watched it together, over and over. He preferred livelier fare, truth be told, but his sweet Virginia loved it, and that was good enough for him.\n\nIt was a radiant night, almost a cerulean blue, and teeming with shadows. Perfect for his appointed task. Of course, he had planned it that way. Every detail in place, every jot and tittle. Just as it should be. As it was destined to be.\n\nHe had almost returned to his truck when he spotted the redheaded woman from the house next door. Divorc\u00e9e, mid-thirties. Camille, she was called, like the victim in \"Rue Morgue.\" Happily, she did not have her boyfriend with her today.\n\n\"Ernie?\"\n\nHe stopped and waited as her crunched-gravel footsteps caught up to him.\n\n\"Hey, I'm sorry about last night.\"\n\nHe pulled himself upright. \"My dear, there's nothing to be sorry about.\"\n\nShe grinned. \"I love that accent. I wish Ty could do it. Gives me shivers.\"\n\nHe felt his face flushing, his stomach roiling.\n\n\"I tried to stop him, but he had to stir things up. You know how men are. Well, maybe you don't.\" She leaned in close. \"They're all assholes. Except for you, Ernie.\"\n\nHe had been playing Mozart very loud last night. It was necessary to ensure that no one heard the screaming. In any case, it hardly merited her boyfriend's overreaction.\n\n\"He isn't a bad guy,\" Camille said, \"not compared to some of the others I've had. But sometimes he can get . . . out of control.\"\n\nYes, he thought, especially when Ty was bored and hungry for a fight. He wanted an easy knock-down-drag-out he knew he could win, and since he was a big, gym-muscled black man, he felt no compunction against taking on a somewhat smaller neighbor. People always picked on short men, always had and always would. It had taken all his comity and bonhomie to get rid of the thug without an incident, but he'd managed.\n\nCamille stood awkwardly for a moment, her fingers fidgeting, her breasts all but spilling out of the flimsy halter top. He felt her discomfort, her longing to say something effervescent or witty, some pointed observation that would elicit his approbation. The woman liked him, strange as that seemed. Apparently Mandingo wasn't keeping her satisfied. She yearned for something different, someone smart, someone who could elevate her life from the drudgery and banality that presently characterized it. She was vulnerable. He had once considered her for an offering, but she was too old, too large. She could never fit the specifications.\n\n\"Say, what you got in the truck?\"\n\nHe stepped forward, blocking her approach. \"Just some trash.\"\n\n\"Trash? I saw how you strained to scoot that load off the dolly. What are you throwing away, barbells?\"\n\n\"Books. Nothing in the world heavier than books, you know.\"\n\n\"And you don't want them? Seems like I see you reading all the time.\"\n\n\"My interests have . . . evolved.\"\n\n\"Oh, yeah? What do you\u2014\"\n\n\"Camille . . . I must beg your pardon.\" He edged away. \"I need to take care of this. Immediately.\" He slid into the cab of the truck and started the engine.\n\n\"You know, Ernie,\" she said through the open window, \"you could come over sometime. Ty isn't around that much these days. We could have some fun, I think.\" She reached out and touched him on his cheek.\n\n\"Must dash, Camille. Please give my best to your beau.\" He rolled up the window and sped away. He drove quickly, but not too quickly, making sure she wasn't following him.\n\nIt should take no more than twenty minutes to get to the hotel, which was providential, because he had a limited window of opportunity during which he could get this bundle into his room without being spotted. From there, delivery to the ultimate destination would be a simple matter.\n\nHe tried to whistle again, but he seemed to have lost the tune. The encounter with Camille had unnerved him. It had been a parlous moment, he realized, when she'd reached toward the bundle in the back of the truck. Not that he had done anything to betray himself, nothing she was ever likely to comprehend, this woman who couldn't master subject-verb agreement and didn't have the sense to dispense with the psycho stud who was servicing her. But he didn't like the unexpected. He had planned everything with meticulous care. Any deviation from the designated path could only delay the Golden Age.\n\nI am the perfect passenger. Lisa has told me so on numerous occasions. The absolute antithesis of the bossy backseat driver. I figure if you're behind the wheel, then you're calling the shots. I don't mess with your radio, I don't tell you when to change lanes, and I don't plot the course. So I sat quietly as Lisa took me all the way down the Strip, even though this was tourist season (every season is tourist season in Vegas) and the traffic was atrocious.\n\nI love this town. Lived here all my life; never had the desire to go anywhere else. There's so much more to Vegas than what the tourists see. But the truth is, I love what the tourists see, too. From the austere Nevada mountaintops to the concrete palaces to the sex clubs and the gluttonous buffets, I love it all. Even the Liberace museum. Honest.\n\nMy house was in a gorgeous neighborhood called Summerlin, near one of the area's many man-made lakes. Okay, ponds, depending on what you're used to. It was just down the road from the largest of Vegas's many pet cemeteries, where all three of my German shepherds are interred. I sat quietly while Lisa cruised through the south side. But when we were a good twenty minutes or so away from where we should be, I felt I had to speak.\n\n\"Are you taking me home?\"\n\n\"Um, no.\"\n\nI sensed immediately that this was a subject she had been deliberately avoiding. \"I've been gone a long time, Lisa. I've got a million things to do.\"\n\n\"Yes, but . . . not there.\"\n\nI laid my hand on her shoulder. \"What gives? Tell me the truth.\"\n\n\"The truth is . . .\" I could tell this was agonizing for her, which didn't make me any less insistent. \"You don't have a house anymore.\"\n\n\"The goddamn bank.\"\n\n\"Yeah. 'Fraid so.\"\n\n\"I'm in the hospital for one lousy week and they take the house?\"\n\n\"They told me you hadn't made a payment for four months.\"\n\n\"I've had a lot on my mind, as you well know.\"\n\n\"They said they've called and written and left messages but never heard anything from you.\"\n\n\"I don't recall any messages.\"\n\n\"And then when they found out you were in detox and no one knew when you might get out, that was it. They foreclosed.\"\n\n\"Can they do that? So fast?\"\n\n\"Evidently.\"\n\n\"Greedy bastards. What do I have to do to get it back?\"\n\nThe beautiful blond speed demon became a timid little girl. \"I . . . I don't think you do, Susan. It's gone.\"\n\n\"Goddamn it.\" I slapped my hand down on the dash. \"God _damn_ it.\"\n\n\"I boxed up all your stuff and put it\u2014\"\n\n\"I don't care about that,\" I snapped. \"Where's Rachel?\"\n\nLisa's respiration accelerated. Her grip on the steering wheel tightened. All bad signs. \"Oh, Susan, I'm so sorry.\"\n\n\"Where is she?\"\n\nHer forehead creased. \"They've put her in a foster home, sweetie.\"\n\n\"Who did?\"\n\n\"The state. NDHS. Human Services.\"\n\n\"I'm sick for one week and they confiscate my niece?\"\n\n\"Susan, think. You never formally adopted her. She was just living with you, and that was okay for a while. But after . . . you know . . . what happened . . .\"\n\n\"They had no right. None.\"\n\n\"They say you can visit. I mean, you can't take her away or anything, but\u2014\"\n\n\"I'm her family! I'm the only family she has!\"\n\n\"I know. I told them. I offered to let her stay with me till you were released. I tried everything I could think of\u2014\"\n\n\"Apparently you didn't try hard enough.\"\n\n\"Hey, don't kill the messenger, okay?\" For the first time, Lisa's voice rose. \"You're not the only one who's had a shitty week, you know? I've been under continuous fire, trying to straighten out your\u2014\"\n\nShe stopped just short, for which I will always love her.\n\n\"They had no right to take my niece away.\"\n\nStill driving, Lisa reached across and squeezed my hand. \"Honey, you're an alcoholic\u2014\"\n\n\"I am not.\"\n\n\"\u2014and there's no way the state is going to let a fifteen-year-old girl stay with a noncustodial nonparent alcoholic with proven violent tendencies.\"\n\n\"I've never hurt Rachel. I would never hurt her.\"\n\n\"I know, Susan. But you practically killed that chump at the bar, and endangered everyone there, and that's all they're seeing.\"\n\n\"Goddamn it!\" I pounded the dash over and over again, which I'm sure Lisa did not appreciate, but she didn't say anything. \"Goddamn them all to hell.\"\n\nShowing her usual perspicacity, she let me stew for a while and didn't speak again until it was necessary. \"We're almost to my place, Susan. Come in with me. I'll start a fire. You can put on some woolly pajamas and I'll brew some tea and you can just chill for a while, okay?\"\n\n\"No. Take me to the office.\"\n\n\"Susan\u2014\"\n\n\"It's no good, Lisa. You know I can't tolerate just sitting around, and I would hate being coddled even worse. The best thing for me to do is get back to my job and forget\u2014\"\n\nShe started to cry. This really bothered me because, for starters, Lisa is my friend, and furthermore, it seemed like if anyone should be crying it should be me\u2014and I wasn't, so what right did she have?\n\nAnd I wondered what she could possibly be holding back that was worse than what she had already divulged.\n\n\"Susan . . . you don't have a job anymore.\"\n\nNo.\n\n\"Don't blame Chief O'Bannon. It's not his fault. IA was all over what happened, and that boy's family is threatening to sue the department. O'Bannon had to do something.\"\n\n\"So\u2014so\u2014\" I was having trouble forming the words. \"So\u2014they suspended me?\"\n\nShe shook her head. \"No.\"\n\n\"What? _What?_ \"\n\nShe pulled up to a light, then turned to face me. \"They fired you, Susan.\"\n\n\"No way. I'll just talk to O'Bannon.\"\n\n\"He didn't want to do it. But he had no choice.\"\n\n\"This can't be right. It can't be.\"\n\n\"But there are lots of things a trained psychologist can do, Susan. It might be good for you to get away from police work, where there are so many . . . reminders. This could be a golden opportunity. Look on the bright side.\"\n\nSure. Other than that, how did you like the parade, Mrs. Kennedy? \"This isn't right. It can't be. I'm the best profiler O'Bannon has, and he knows it. I'm the one who solved the Wyndham killings. I'm the one who\u2014\"\n\n\"That was before,\" Lisa said forcefully.\n\n\"Before this one stupid little incident?\"\n\n\"Before you started drinking.\" The word hit me like a brick. She continued to talk, but it rolled off me like water on a slick surface. There was a liquor store on the corner, and another on the corner after that. Liquor was everywhere. It was pervasive, and not just here in Sin City, either. I spotted an ad for some tarted-up booze, Chivas Regal or some other stuff I couldn't afford. I remembered the smoky scent of a good scotch, the warm assurance as it glided down my throat.\n\n\"Lisa . . . could you stop the car?\"\n\n\"You can't drink, Susan. Not at all. Not even once.\"\n\n\"I need . . . something. I can't . . . everything . . . it's all . . .\"\n\n\"I'll stay with you tonight.\"\n\n\"You don't have to.\"\n\n\"I'm not asking\u2014I'm telling. I'll stay with you.\"\n\n\"I'm not going to drink.\"\n\n\"Then you won't mind my being there.\"\n\n\"I don't need a babysitter.\"\n\n\"I've heard the first night is the hardest. For people in your situation.\"\n\nSo that's what I'd become. A situation.\n\nI closed my eyes and tried to conjure up the memory of David just as I'd seen him that morning, but it wouldn't come. At best, I got a turbid glimmer, a toothy smile, a dimpled chin. Pieces of the whole.\n\nIt seemed I had nowhere to go and no one to see. Nothing to do. Nothing to live for.\n\nThe throbbing in my left wrist intensified. Beneath the bandage, it was sending me a message.\n\nIf ever there was a girl who deserved a drink, it was me.\n\nHe lifted his spade and began to dig. The soil was soft and loose, as he had known it would be. It was only about two feet deep, but that would be sufficient. It didn't really need to be buried. It was the suggestion that was important. The re-creation of the sacred image.\n\nDespite the simplicity of the task, he found himself tiring and perspiring. But this entire area was deserted and he knew it would remain so until six in the morning, so it didn't matter how long he took. Just so the job was done right. According to plan.\n\nHe slowly lowered the long box off the dolly and into the freshly dug cavity. He lifted a spadeful of dirt and tossed it onto the box. The resultant clamor caught him by surprise.\n\nMerciful Zeus. How could I be so forgetful? He leaned over the edge of the pit and lifted the half lid from the top section of the box.\n\nHelen screamed.\n\nHe clamped his hand over her mouth. \"My dear, I can't allow you to make a commotion.\"\n\nShe struggled to get free of his hand. She tried to bite him. She spit on him. Nothing worked.\n\n\"I'm going to release you in a moment. And when I do, I don't want to hear any more screaming. You know, I could've deadened your entire body. And I still can, if need be. Do you comprehend what I'm saying?\"\n\nSlowly, he removed the hand from her mouth. She did not scream.\n\n\"Now that's more like it.\"\n\nWith the half lid open, she was visible from her bare shoulders up. \"I couldn't breathe in there, mister. I thought I was going to die.\"\n\nHe made no comment.\n\n\"I could tell you were moving me, but I didn't know where we were going. And then I heard that thumping on the lid and I didn't know what was happening and I hate confined spaces and I panicked.\"\n\n\"Of course you did. Entirely understandable.\"\n\nShe craned her neck, trying to see something other than the walls of the box surrounding her, gazing straight up at the cobwebs and skeletons and white sheet ghosts. \"What is this place?\"\n\n\"A gallery. A tableau, if you will. To honor the prophet.\"\n\n\"I don't know what that means,\" she said, choking. \"I want to go home.\"\n\n\"And you will, in a sense,\" he said reassuringly. \"To a far, far better home than any you have known before. Better than the one you kept sneaking away from.\"\n\n\"How do you know that? Did Amber tell you?\"\n\n\"You must pardon me, but I really don't have time to continue this conversation.\"\n\nHer eyes were red and watery. \"You're going to kill me, aren't you?\"\n\n\"Do you see a gun in my hand? Am I wielding a knife?\"\n\n\"Answer me!\"\n\n\"You have the potential to be reborn. To become something greater than you ever dreamed possible. To usher in a milestone in the evolution of mankind.\"\n\n\"Please don't kill me. I'm begging you! I'll do anything. You want sex? I can give you sex. You want to put it in my mouth? I can do that. Hey, I'd like to do it. I'd enjoy it. Just give me a chance, mister.\" She was babbling, rambling, desperate. \"You've already hurt me so bad,\" she cried, her voice breaking. \"Please don't kill me. Please let me go home.\"\n\n\"I can't,\" he said gently. \"I know this is difficult for you, but it is for your own good. I'm helping you.\"\n\nShe stared at him, breathing in short, quick gasps. \"Would you come closer, please? I'd like to whisper something to you. It's a secret.\"\n\nHe almost did it. But at the last moment, he stopped. \"You were going to bite me.\"\n\n\"I\u2014I don't know what you mean.\"\n\n\"And you thought if I came close enough you could bite my nose, maybe even my eyeball?\" He made a tsking sound. \"I suppose anything is possible for a girl who would wear thong underwear.\"\n\nHer eyes closed. Her last hope was lost. There was nothing left now.\n\n\"This is a coffin, isn't it?\"\n\n\" 'I could no longer doubt that I reposed in a coffin at last.' \" He caressed her smooth bald head. \"Don't focus on the here and now, my dear. The all-too-present present. Give your mind to the ineluctable truths of the cosmos. Cast your eyes to the stars!\" He gestured heavenward, or ceilingward, like a master showman unveiling his main attraction. \"There is so much more out there, so much more that we can become.\"\n\nA tiny light blazed one last time. \"I think you're a crazy man. I think you're an impotent little\u2014\"\n\nWith a single smooth gesture, he flung the lid of the wooden coffin closed and locked it. \"I daresay that's enough of that,\" he murmured quietly. \"I detest vulgarity.\"\n\nHe picked up the spade and continued shoveling dirt. It didn't take long. As he worked, he could hear her. She could move a little now, but alas, there was nowhere to go. She beat against the lid and scratched and clawed and screamed, but to no avail. In just under an hour the air would be exhausted, but the screaming and clawing would end well before that. She would be at peace.\n\nAnd so would he. \" 'But out of Evil proceeded Good,' \" he chanted softly to himself, \" 'for their very excess wrought in my spirit an inevitable revulsion.' \" Cheerfully, he continued the spadework until the pit was full and smooth with only the merest tip of the coffin exposed to bring about its eventual discovery. And this time, the tune returned to him, and he whistled while he worked.\n**CHAPTER THREE**\n\nI awoke bathed in sweat. My cammies clung to me like glue, my whole body was cold, and there was an empty aching in my chest that felt as if someone had ripped my heart out while I slept. My wrist was throbbing, talking to me. I tried to stand but found I couldn't. Finally settled for rolling onto my back. I forced myself to inhale and exhale, hoping it would steady my nerves, which it never did.\n\nI'd had some bad mornings before, but never anything like this. I felt as if I'd broken something, some part of my natural living apparatus. Am I dead? I wondered, and some part of me hoped I was.\n\nAfter about two minutes that seemed like ten hours, Lisa came into the room. She cradled my head in her lap and squeezed my hand tightly, stroking the side of my face.\n\n\"You can do it, sweetie,\" she said in soft, dulcet tones. I knew she was trying to be comforting, but I couldn't stop thinking that she sounded like a cross between a kindergarten teacher and a hooker. \"I'm here for you. They told me you might have mornings like this.\"\n\n\"I\u2014don't know what's happening,\" I said, breathing deeply. \"I slept okay in the hospital.\"\n\n\"They were giving you drugs. Librium. It's a benzodiazepine. Eases the withdrawal symptoms. This is your first morning off.\"\n\n\"God.\" My lips were dry and cracked. \"Lisa, I think I'm going to need a little something, just so I can function. It'll be the last\u2014\"\n\n\"No, Susan. No booze.\"\n\n\"I don't want the alcohol. I just need to kill this pain\u2014\"\n\n\"No booze.\"\n\n\"It's not like I want to get drunk. Once I get this anxiety under control, I won't touch another\u2014\"\n\n\"That's exactly what they told me you'd say, Susan. The answer is no. You've got to ride this out.\"\n\n\"Fine.\" I knew she was right, but at the same time I was cursing her under my breath, thinking what a stupid woman she was, how she didn't understand at all. I began plotting how I could get rid of her so I could live my own life. It wasn't as if one stupid drink was going to put me back on the skids and start me hallucinating. I could control it, and this time I would.\n\nTook almost an hour to get to the point where I could walk. Showering made me feel a lot better. I cleaned up, put on some jeans and a clean shirt. Used some damn bubble-gum-flavored toothpaste I found in Lisa's medicine cabinet. Considered myself lucky I didn't have to use a Barbie toothbrush.\n\nShe'd already found me an apartment, not far from hers, and had my stuff moved into it. At the same time, she made it clear that I was welcome to stay with her as long as I wanted or needed. She hadn't lived with anyone for over a year, so she had a second bedroom that did nothing but collect dust. It was a generous offer. But I couldn't help noticing that she didn't suggest that I move into the spare bedroom on a permanent basis. We'd tried being roommates once, when we were both in college. It was a disaster. The fact that we are still friends is a testament to the healing power of forgiveness, or perhaps the evanescence of human memory.\n\n\"I want to see Rachel,\" I told Lisa as I finished dressing.\n\n\"I knew you would,\" she replied. \"You've got an appointment at noon.\"\n\n\"So now I need an appointment to see my own niece?\"\n\nShe didn't say anything.\n\nI peered into Lisa's big lighted makeup mirror. At thirty-four years of age, I was still subject to pimples, and sure enough, there was a big one right at the top of the bridge of my nose. Makeup or straight pin? \"Well, that'll be okay. We can get some lunch.\"\n\n\"You . . . aren't permitted to take her away from her foster home.\"\n\n\"What?\"\n\n\"I'm sure they'll change that in time, but for now, your visit will be restricted to the Shepherds' home.\"\n\n\"Who the hell are they to tell me when and how I can see Rachel?\"\n\n\"They're just trying\u2014\"\n\n\"I'm her only living relative.\"\n\n\"They know that, and\u2014\"\n\n\"They've got no business giving me orders.\"\n\nI could sense that Lisa was getting irritated. \"As far as NDHS is concerned, you're the sozzled psycho who attacked a kid in a bar and then\u2014\" She paused. \"They're not going to let you take Rachel anywhere.\"\n\n\"Have they been spreading these slanderous remarks all over town? I will fight this\u2014\"\n\n\"Susan.\" I could tell she was checking herself, biting back words. \"Give yourself a little time. You're not going to put everything back together in a day. Your main focus should be on getting healthy. And staying healthy.\"\n\n\"You're right, of course,\" I said, not meaning it at all. I finished brushing my hair and adjusted the lay of my bra. Nothing worse than poorly aligned mammaries. \"So what have you planned to kill time till noon?\"\n\n\"I thought we might go to the mall at the Venetian. Do a little window-shopping for purposeless trinkets we can never afford. Maybe bump into Michael Jackson. Get a coffee. Ride a gondola.\"\n\nI shook my head. \"Thanks, Lisa, but I'd go nuts. Will you take me downtown?\"\n\n\"You mean downtown as in police headquarters?\"\n\nI touched my nose. \"Got it in one.\"\n\n\"I told you already. You're no longer employed there.\"\n\n\"Then I need to clean out my office, don't I?\"\n\nLisa looked very dubious. \"I suppose.\"\n\n\"So let's roll.\"\n\n\"And you won't cause any trouble?\"\n\n\" _Moi?_ Wouldn't dream of it.\" I tucked in my shirt, wishing I still had a holster to strap on. \"I'm finished with that sort of thing. From now on, I'm going to be amiable and even-tempered at all times. Positively serene.\"\n\n\"W _hat the hell did you think you were doing?\"_\n\nLieutenant Barry Granger stared at me with a placid expression that in no way disguised the contempt I knew he felt for me. It was quite a change. I still remembered a time when his stares signaled thinly disguised lust, or later, stares of envy, and, more recently, pity. I probably liked this one best.\n\n\"Clearing out the trash,\" he explained, his feet propped up on my desk. \"Making myself at home.\"\n\n\"In my office?\"\n\n\"Not anymore.\"\n\nIncredible. In the space of one week O'Bannon had reassigned my office space. Actually, I could live with that, even understand it. But what I couldn't understand was giving it to this pig Granger.\n\n\"This is unacceptable,\" I said. My emotions were also pretty undisguised.\n\n\"O'Bannon needed a new detective.\"\n\n\"I was only gone a week. I've taken longer vacations.\"\n\n\"Not in the drunk tank.\"\n\nI could feel my rage rising, and I really wanted to ream him in the worst way. But that would be playing into his hands. \"Why would he promote you?\"\n\n\"I was next in line.\"\n\n\"I was still around.\"\n\n\"You were drunk on your ass.\"\n\n\"And even drunk on my ass I would be a better detective than you.\"\n\nTo his credit, Granger remained calm. \"I don't think Internal Affairs would agree.\"\n\n\"O'Bannon can't afford to replace a seasoned\u2014\"\n\n\"You don't get it, do you?\" Granger was a handsome man, which made him all the more difficult to bear. He had sandy hair and a sun-baked complexion. He wore a light stubble, which even I had to admit made him look damn sexy. How did men maintain a light stubble, anyway? At some point, don't you have to shave or it turns into a beard? Do they make special razors with dullish blades for guys who look good with a little growth? \"You're a behaviorist. I'm a homicide specialist. It's not that I'm replacing you. O'Bannon just needed someone he could depend on.\"\n\n\"Go to hell.\"\n\n\"And he can't count on you any more than\u2014\" He stopped himself, the bastard. Apparently this line was too low even for him. \"Than anyone could.\"\n\n\"How the hell would you know?\"\n\n\"I've seen what you've done to yourself. I've seen what you've turned into.\"\n\n\"There have been some extenuating circumstances, you son of a bitch. I lost my husband!\"\n\n\"And I lost my partner!\" He lashed out, letting loose something I knew he'd been holding back a long time. \"Did you forget that?\"\n\nGranger had been David's last partner. Granger wasn't around when he died, but in a way, I think that made it worse. Sometimes cops carry this \"my partner\u2014my life\" routine too far\u2014the influence of excessive episodes of _T. J. Hooker._ But David and Granger had been close. Granger had genuinely loved David. And admired him. I knew that. But it didn't make me like him. How dare he fling the loss of my own husband in my face to score points, as if somehow he had more right to grieve than I did?\n\n\"I didn't forget,\" I said, pushing past him. \"Where are my files?\"\n\n\"You don't have any. They've all been reassigned.\"\n\nI was reaching my limit. One more remark like that, and I knew I'd hit him. So much for my attempt to prove that I don't have any more violent tendencies. \"I'm talking to O'Bannon.\"\n\n\"He's at the crime scene. Likely to be there all morning.\"\n\n\"What crime scene?\"\n\nGranger squinted slightly. \"You _have_ been out of touch, haven't you?\"\n\n\"Don't play games with me, Granger. Tell me where he is!\"\n\n\"No, that wouldn't be prudent.\"\n\nI swore under my breath. Then I swore over my breath, several times. \"You're still sulking because I wouldn't suck your dick that time at Gordy's, aren't you?\"\n\n\"Jesus Christ, that was what? Ten years ago? Before you hooked up with David? I'd just met you.\"\n\n\"Yeah, but you haven't forgotten. Who was drunk _that_ night?\"\n\n\"I think you should go now, Susan.\"\n\n\"I think you're right. The smell in here is getting intense.\" I bolted for the door. \"Have fun playing detective, Granger. If you need any help\u2014go fuck yourself.\"\n\nI slammed the door\u2014 _my_ door\u2014just for dramatic effect.\n\n\"How'd it go?\" Lisa asked as I slid into the passenger seat.\n\n\"Fine. Swell.\"\n\n\"No problems?\"\n\n\"Nah. They were all very kind. Glad to see me. Granger and I embraced.\"\n\n\"Granger?\"\n\n\"Yeah. Who'da thought?\" Well, if I couldn't live the fantasy, I could at least make Lisa think I did. While she drove to the house where Rachel was staying, I used her cell phone to call Eleanor, one of the young girls in Dispatch. She was nice, I thought she liked me, and I also thought she was gullible enough to believe anything.\n\n\"Eleanor? Susan Pulaski. I've got some lab reports for O'Bannon. Is he still at the crime scene?\"\n\n\"Yeah,\" she answered. I heard a dozen other lines working in the background. \"But I thought you\u2014\"\n\n\"He brought me back in. Said he needed my expertise on this one. So he's still out there?\"\n\n\"Far as I know.\"\n\nI noticed a light drizzle on the dashboard. \"You think I should bring him a raincoat?\"\n\n\"Nah. He'll be inside the hotel.\"\n\nGetting warm. \"And still working in that . . . that\u2014what do they call it?\"\n\n\"The Edgar Allan Poe Ballroom?\"\n\n\"Right, right.\" That would have to be the Transylvania. One of the newer \"family\" resorts on the Strip. \"Dumb of me. I never paid much attention in English class. I was thinking of that other guy. Hawthorne.\"\n\n\"No, it's Poe.\"\n\n\"Okay, thanks.\" For more than you know.\n\nAnnabel Spencer gazed nervously at the mirror-paneled walls, the smoky glass ceiling, the translucent corbels. Cameras behind every one of them, she thought. She had read that the camera positions were changed regularly to prevent anyone from knowing for certain what the security people could and could not see. Men on metal catwalks hidden behind the ceiling peered down 24\/7. There was no telling how many people were watching her right now. Did they know she wasn't supposed to be here? How long would it take them to figure it out? And once they knew, what would they do to her?\n\nAs soon as she passed through the front doors of the casino, she knew her face was scanned by a computer running facial recognition software\u2014developed at MIT, of course\u2014which converted a fluid digital image of her face into mathematical data, translating facial landmarks into algorithms, then comparing the results to the millions of faces in a shared database. She didn't know if anyone had her cheekbones on file yet. In this day and age, it was impossible to be certain. The latest hot cyber-rumor held that Big Brother was constantly scooping up people's faces, taking them off Web sites and newspapers and airport security scanners. Today, almost anyone could be identified by an entity with the financial resources to pay for the data. Privacy was an illusion, or perhaps more accurately, a luxury that many big businesses could no longer afford. So the casino hard drives must be whirring away, she mused, trying to come up with a match, determining if she was a player or a patsy, a tourist or that most dreaded of all evils: a card counter.\n\nGiven the famous hatred of casinos for card counters, the minor detail that she was underage might seem negligible\u2014although it would certainly give them a no-questions-asked excuse to bar her from the premises. Not that they needed one. They could expel anyone, and would, if they thought the player was too successful at the blackjack table. It hardly seemed fair\u2014they only allowed people to play the game if they weren't very good at it. Forget the signs saying the dealer rests on seventeen; the signs should read: SUCKERS ONLY.\n\nLike many of her fellow students, Annabel considered herself a freedom fighter, not a card cheat. But she knew the casino would take a different view. Dark rumors circulated throughout the MIT campus about what happened to card counters in the back rooms of Vegas gambling emporiums. Since they were private property, they could ban anyone they wanted, and if you violated their ban, you could be charged with trespassing. But this was the town the mob built, after all, and casino bosses were known to employ means not sanctioned by law to convey the message that card counting was disfavored. She'd seen more than one MIT hotshot return to class on Monday morning looking as if he'd lost a fight with a tractor. And a few had disappeared altogether\u2014never to return.\n\nTrying to remain calm, Annabel strolled into the blackjack pit and scanned for an open seat. She was wearing her best DKNY _Sex and the City_ black dress, and for an MIT math major, she looked pretty damn hot, if she did say so herself. Not that she wanted to attract attention\u2014she just wanted to look as if she belonged there. Because everything depended on her having a successful night. Everything in the world.\n\nShe saw her mother's face on an overhead TV in the bar and froze. If her mother found out what she had done . . .\n\nGod. Annabel didn't even want to think about it.\n\nA zombie dressed in tattered rags bumped into her backside. Clumsy, or copping a feel? Hard to be sure, but she was not inclined to give a guy in tatters and black eyeliner the benefit of the doubt.\n\n\"Sorry about that, ma'am.\"\n\nWell, at least he was a polite zombie. \"No problem.\"\n\n\"Can I get you something? Drink? Change?\"\n\nShe suspected he really wanted to give her something altogether different. Damn those horny zombies. \"I'm fine, thanks.\"\n\nHe moved on down the aisle, dragging his leprous skin and putrescent sores with him, brushing aside a cobweb and blending into the crowd of ghosts, witches, and assorted ghouls.\n\nWelcome to Transylvania.\n\nNot that she was an expert, but in her mind a casino should be sleek and elegant, like the ones in James Bond movies, where the men wore tuxes and the women wore beaded floor-length gowns with d\u00e9collet\u00e9 bodices. None of that here\u2014this joint was pure Disneyland. Dolorous Victorian d\u00e9cor and bar stools that creaked when you sat on them. Your \"ghost host\" checked you in; the bell captain wore a long black hooded robe and carried a scythe. The gambling pit was decked out like a haunted house, with shutters blowing in the wind and tombstones with corny epitaphs. A Vegas casino with a Halloween party motif\u2014it was enough to make you barf. And there were cops swarming everywhere, which also made her nervous. Apparently some horrible crime had taken place in one of the ballrooms; an entire wing of the hotel was roped off. But like it or not, this was the only house in town that still dealt a six-deck shoe all the way to the sixth deck. So here she was. Anyone want to suck her blood?\n\nShe found a stool at a hundred-dollar blackjack table and slid onto it. She liked sitting at the far left\u2014Position Three, they called it at MIT. It gave her the most time to count cards before she had to play. Most amateurs thought the game of blackjack favored the dealer. It didn't. The dealer was stuck hitting sixteens and standing on seventeens whether it was smart or not. He didn't have the luxury of sitting pat on a twelve and waiting to see what happened. The only advantage the dealer had was that he played last. If you went bust, he collected your bet and kept it\u2014even if he later went bust himself. A small point, but one that made a huge financial difference to the casinos. The only way to overcome that advantage was to count cards.\n\nShe ran the rules through her head. They weren't complex, but the cards moved so fast it was hard to keep up. Plus one for twos, threes, fours, fives, and sixes; minus one for tens and face cards. Take that running count and divide it by the number of decks in the shoe not yet dealt to come up with the true count. When the true count was plus eight or better, start increasing your bet. And most importantly, look casual, distracted, unconcerned, lusty, possibly intoxicated\u2014but never give the slightest hint that you're counting.\n\nShe pulled out five hundreds and placed them on the table, waiting for chips. A moment later, a small man in a black vest slid onto the stool beside her. He had thinning hair and a little Hitler-style black mustache.\n\n\"Having any luck?\" he asked.\n\n\"Just got here,\" Annabel answered succinctly. She didn't want to be unfriendly\u2014but she didn't want to be friendly, either.\n\n\"I haven't caught a break all night,\" he said.\n\n\"Sorry to hear that.\"\n\n\"Guess the stars just aren't right.\"\n\n\"Maybe a zombie priestess put the hoodoo on you.\"\n\nThe corner of his lips turned up. \"Anything's possible.\"\n\nA waitress came by the table. She was dressed in a witch getup, but Vegas-style, with a plunging neckline, an obscenely short skirt, and fishnet hose. She did at least wear the traditional black pointed hat. Annabel ordered a mineral water. The man beside her asked for a Bloody Mary.\n\nThe first several decks were average. That was okay. She knew counting required patience. Favorable decks were the exception, not the rule. For the first hour and a half, Annabel remained almost exactly even. The count never rose above a plus five. She'd managed to rack up a few extra bucks by doubling down and splitting at strategic moments, but she'd had no justification for seriously increasing her wager. And the man sitting next to her talked almost continuously the whole time. He seemed uncommonly interested in her\u2014where she was from, what she did for a living, whether she was here alone. Another hand or two and he'd probably be asking what kind of underwear she had on.\n\nAbout two-thirds of the way through the next shoe, the count rose to plus twelve. With all those ten-spots in the deck, it favored the players, and coming this late in the shoe, the count was extremely reliable. It was time to increase her bet. Her hand trembled a bit as she pushed three hundred bucks' worth of chips into the circle. She was nervous, putting so much money on the line. But she had to do it. Everything depended on her game. Including Warren.\n\nShe felt a tapping on her bare shoulder. \"Excuse me, miss.\"\n\nShe turned and saw a silver security badge. Damnation. Busted.\n\n\"Could I speak to you for a moment? In private.\"\n\nHer first instinct was to run, but she got a grip on herself and followed the man to a private corner at the edge of the pit, just beneath a cartoonish polystyrene Satan. Had he caught her counting? Did he know she was underage? And what would he do about it? Her knees knocked. She was beginning to wish she'd had something stronger than mineral water.\n\n\"I've been watching you at the blackjack table, ma'am.\"\n\nGreat. So he knew she was counting. She should've realized that\u2014\n\n\"The man sitting next to you has been watching, too.\"\n\nHer brain jerked back front and center. \"He has?\"\n\nThe guard nodded. \"I've had my eye on him for a while. I think he's a thief\u2014pickpocket or purse snatcher or some such.\"\n\n\"How can you tell?\"\n\n\"I've been at this job awhile, ma'am.\"\n\n\"Of course. I'm sorry\u2014\"\n\n\"I'd keep my distance from him. And definitely keep your eyes on your chips.\"\n\n\"Sure, I'll\u2014\"\n\n\"When you sat down at the table and pulled out your money, you flashed a huge wad. Looked to me like several hundred bucks, maybe more. And if I saw it, every crook in the joint saw it as well.\"\n\n\"I guess I should've been more discreet.\"\n\n\"We do our best to keep the casino safe, but the reality is, we have robberies every day. These creeps prey on nice people like you.\"\n\n\"I'll try to be more careful.\"\n\n\"That would be wise.\" He smiled. \"Now go back there and get lucky, okay?\"\n\n\"Sure. Thanks.\" She returned to the table. The dealer resumed the game. She won the hand. The count was still high, so she let it ride.\n\nNot a minute passed before her neighbor spoke again. \"Car parked next to a hydrant?\"\n\nMaybe if she was cooler to him, he'd go away. \"No. Just a private matter.\"\n\n\"Trouble back home?\"\n\n\"Nothing important.\"\n\nMore cards were dealt. Annabel got a natural blackjack, which paid off one and a half times the wager. She'd just made over two thousand dollars.\n\nAt this rate, she'd be married by sunrise.\n\nThe Shepherds' house was in one of the middle-range suburbs, not nearly as nice as the neighborhood where my house is\u2014was\u2014but decent. The vast preponderance of white picket fences and close-set houses told me the designers were striving for that old-time community feel. That these foster parents would live in such a neighborhood told me a lot about them from the get-go.\n\nIt was a brick two-story with green shutters and beige trim. A white trellis up the left side, with ivy snaking through the latticework. A long, curving walk led to the front door. The paint was fresh and the garden had been tended. It was all so _Leave It to Beaver_ it momentarily engaged my gag reflex.\n\n\"They put Rachel in this Stepford house?\"\n\nLisa ignored the commentary. \"She was lucky to get a good placement as quickly as she did, Susan. I'm told the Shepherds are very nice people. She has her own room.\"\n\n\"She had her own room at my place.\"\n\n\"Which is now the bank's place.\" She popped open the car door. \"Come on.\"\n\nI didn't have to walk all the way up those steps. Rachel saw me coming. She burst out the front door and ran toward me, leaping into my arms.\n\nI winced. I'd forgotten about my wrist, still wrapped and bandaged. The impact opened the wound and it began to bleed. I yelped.\n\n\"Oh, my God!\" Rachel shrieked. \"I'm sorry.\"\n\n\"It's nothing. It was already bleeding before I got here.\"\n\n\"Is that where you\u2014you\u2014\"\n\n\"Cut myself. Accidentally.\" I took a step back and gave her the once-over. \"Well, suburbia hasn't ruined you. Yet.\"\n\nShe laughed a little, but I could feel her edginess. \"So . . . you're all right?\"\n\n\"Of course. I was a little sick, but I'm better now.\"\n\n\"Can I go home with you, then?\" She threw her arms around me again and squeezed tightly. God, but this girl loved me. I could feel the affection rushing through her fingertips. And I loved her back. I would do anything for her.\n\nBehind her, standing in the doorway, I saw what must be the Shepherds. An elderly couple, trim and tidy, with smiles plastered on their faces that had nothing to do with happiness. They were watching every move I made.\n\n\"Rachel, honey, I guess the man says you have to stay here for a while. But I'm going to hire an attorney and fight. I'll get you back.\"\n\n\"Please hurry. I can't stand it here.\"\n\n\"Why?\" I put a finger under her chin and raised her eyes to mine. \"Have they hurt you?\"\n\n\"No, nothing like that.\"\n\n\"Made you do a lot of chores?\"\n\n\"They haven't made me do anything. They're just so . . . you know. Boring. I want to go home.\"\n\n\"About that.\" I figured the best approach was to come right out with it. \"I lost the house.\"\n\n\"What?\" Almost instantly, water welled up in her eyes.\n\n\"Yeah. Bank took it.\"\n\n\"But\u2014all my stuff\u2014\"\n\n\"Lisa saved it. And I've got a new place.\"\n\nShe wiped her eyes. \"Where's the new house?\"\n\n\"It's . . . an apartment. But it's nice-sized and there's a room for you. So all your stuff will be waiting for you once I bust you out of Stalag 13.\"\n\nRachel was a lovely girl\u2014she took after my brother, not me. She had auburn hair, getting redder by the day. Clear-skinned and pretty. I knew for a fact she was popular with the boys at school. So far she'd shown good sense about that, for which I was grateful. \"Can I see it?\"\n\n\"Well . . . let me talk to Ozzie and Harriet.\"\n\nTheir expressions changed when they realized I was coming for them. They made no approach but braced themselves, as if I were some malevolent force they couldn't escape. I knew it would be awkward\u2014a meeting between the guardian who'd lost custody and the guardians who'd been given custody. But I wanted to minimize the discomfort. What was the point in hassling them? Better to try to charm them, at least until such time as a court put custody back where it belonged.\n\n\"Hello,\" I said, hand extended. \"I'm Lieutenant Susan Pulaski. I want to thank you for taking care of my niece while I was in the hospital.\"\n\nThe man took my hand and shook it feebly. The woman just stared.\n\n\"Looks as if you have a fine home.\"\n\n\"We like it,\" he said. \"Been here twenty-seven years.\"\n\n\"That's wonderful. I've just moved into a new place. Rachel is anxious to see it.\" I laughed and acted very casual. \"Probably what she really wants to see is her stuff. Pick up her Discman and some eyeliner. Mind if I show her the new spread?\"\n\nThe Shepherds exchanged a silent look.\n\n\"What do you say? I'll have her back in an hour.\"\n\nThe man took forever, like he had to crank up his motor before he could speak. \"We were told not to let you remove her from the premises.\"\n\n\"Oh, I wouldn't leave town. I just\u2014\"\n\n\"We were told not to let you take her anywhere.\"\n\nStay calm, I told myself. A portrait in tranquility. This old man can't get to you.\n\n\"We only want what's best for the child,\" the woman interjected.\n\n\"Well, ma'am, I've kept Rachel for the last three years, so I feel as if I can be trusted to\u2014\"\n\n\"We were told that Rachel has had a most unstable home environment,\" the man said. \"That you worked odd hours, were gone for extended periods of time. That she was habitually late for school and missed extracurricular activities, the few she was involved in. That her grades have dropped dramatically.\"\n\n\"We've both had a difficult year. I'm sure you know why. But Rachel is tough. All us Pulaski girls are.\"\n\n\"Nonetheless\u2014\"\n\n\"She'll bounce back. As long as her spirit isn't smothered under your two-car-garage, sex-every-Thursday mentality.\" Damn. Shouldn't have said that . . .\n\n\"I don't appreciate that kind of talk.\"\n\nI couldn't stop myself. \"And what is it you've got to give her that you think is so hot? A riding lawn mower and a color TV? Now I remember where I've seen you before. In the dictionary, under _mundane._ \"\n\n\"You can stop that abusive talk right now.\" He was nervous, twitchy. My God, what had they told him about me? \"You aren't going to take her anywhere.\"\n\n\"Is that right.\"\n\n\"That's right.\"\n\n\"And what makes you think you could stop me?\"\n\nHe leaned close to me and sniffed the air. \"Have you been drinking?\"\n\n\"Are you out of\u2014\"\n\n\"I know all about your substance abuse. The NDHS people told me. And Rachel told me.\"\n\n\"Look\u2014I don't do that now.\"\n\nHis silence clearly communicated how little weight that statement carried with him.\n\n\"I'm serious. I'm not drinking anymore.\"\n\n\"Good.\"\n\n\"I mean it.\"\n\n\"Good.\" He paused. \"But you're still a drinker. And drinkers are liars.\"\n\nIt took all my restraint\u2014never my best quality\u2014to keep from decking him. \"Look, Mr. High and Mighty, I don't need any crap about\u2014\"\n\n\"This interview is over.\" He raised his voice. \"Rachel. Come in now.\" She reluctantly obeyed.\n\n\"Wait just a goddamn minute. I'm not\u2014\"\n\n\"If you don't leave my property immediately, I'll call the authorities.\"\n\n\"I didn't do anything. I just\u2014\"\n\n\"Goodbye.\" He tucked Rachel behind the door, but she rushed forward and hugged me, hard. He eventually pried her away. Then he shut the door and left me standing like a goon on his well-swept porch. Alone.\n\nAnnabel raked in the dough, all the while giggling and acting as if she barely knew what she was doing, as if her big wagers were made out of boredom rather than expertise. She'd ask the dealer for advice\u2014they do that now, ever since Vegas decided to become friendly. Not that they could tell her anything she didn't already know. It was all for show.\n\nBack around 1994, a team of six MIT students began flying to Vegas to prove what many had speculated for years\u2014that a keen understanding of advanced mathematics could pay off at the blackjack table. Half the team\u2014the Spotters\u2014would spread through the casino and sit at various blackjack tables making normal bets, never varying the amount\u2014but counting cards. Some of these guys were absolute geniuses. They could bring off a slick trick called shuffle tracking, which took advantage of probability-distribution mathematics and the fact that most dealers do a light shuffle so as to resume the game as quickly as possible. Shuffle tracking allowed the Spotters to follow a pocket of favorable cards from one shoe to the next, calculating the amount of low-card infiltration caused by the shuffle. Some Spotters would even intentionally blow hands to control the flow of cards\u2014that is, to make specific cards they were tracking come up when and where they wanted them.\n\nWhen the deck favored the players, the Spotters would signal one of the Gorillas, who wandered the floor, usually pretending to be drunk or inexperienced or both. When they got the signal, they sat at the Spotter's table and made the big bets, cleaning up until the Spotter signaled them to leave because the deck was no longer favorable. The Gorillas couldn't be accused of counting cards; they barely looked at the cards. And it was nearly impossible to nail the Spotters, since they never made any money and often lost. At the end of the day, all members of the team shared the loot. In this manner, the MIT invasion managed to make big bucks and not be detected.\n\nFor a few years. Then the casinos caught on and came down on the students\u2014hard. Not only were they all banned, but some were beaten, apartments were robbed and raided, and everyone was terrorized. House rules were changed to make counting less reliable. But new students kept coming. They became the scourge of Vegas; rumor had it the big security firms were willing to take drastic steps to stop the students. Annabel thought it seemed like a crazy risk and had never expected to do it herself.\n\nUntil she found out she was pregnant.\n\nWarren had saved her life. When she first came to MIT, she was all alone, had no friends. She was awkward and isolated and tended to stutter in class. Even with a famous mother, she was a standout nerd\u2014and at MIT, that took some doing. She knew some of the boys made fun of her behind her back. But not Warren. He adopted her, took care of her, showed her the ropes, invited her to parties. They'd been going out for more than a month before he even tried to have sex with her. And by then, she was so in love that she melted like an ice cube.\n\nShe loved Warren, but he was in no position to marry her, not now, when they were both in school and had no money. He told her it would be a mistake and she knew that he was right. Once again, he was looking out for her. But she couldn't bear the thought of having an abortion. And even less could she bear asking her mother for help. The mother who could spend hours with the network suits she belittled but couldn't find time to see her daughter win the Academic Bowl state finals. She had never given Annabel any direction or advice or help other than financial. Her mother had chosen to make her job Priority One, to never be seriously involved in Annabel's life. Well, fine. She wasn't about to go running to the woman now.\n\nWhich meant she needed to come up with some cash, fast. So she'd flown west, slapped on this blond wig, and come to the Transylvania. And on the eighth hand, the last dealt from the favorable shoe, she split two tens\u2014not normally a smart move, but the deck favored her, and the dealer had a six showing, his worst possible card. It paid off. She doubled a two-thousand-dollar bet twice, took all her winnings, and quit. Now she had the stake she needed. She could marry Warren, give this baby a name, and continue their education without involving her mother.\n\nShe pushed away from the table. \"That's enough for me. It can't get any better than that.\"\n\nThe little man sitting beside her grabbed her hand. \"You're not leaving, are you?\"\n\nShe yanked her hand away. Something about the man's touch made her skin crawl. \"I'm afraid I am. B-B-Best of luck to you.\"\n\nShe cashed out, then quickly made her way to the elevators that descended to the parking garage. She didn't want to risk being stopped by pit bosses or mashers or thieves. All she wanted was to get back to Warren and start their life together. This development wouldn't advance her plan to be the youngest woman ever to win the Fields Medal, but she didn't mind. She liked the idea of settling down, being a married woman. And she loved the thought of being a mother. Annabel would take a different approach than her own mother had done. Better. She would turn her heartache into empowerment.\n\nThe elevator doors dinged and she stepped out into the cold, barren garage. It was quiet and dark, shadowy. This was one area where the haunted house motif did not need to extend, she thought.\n\nHer pace quickened; she heard each step echo in her wake. Chill bumps rose on her arms and legs. She moved even faster.\n\nThat was when she heard the footsteps.\n\nNo one had come off that elevator since she arrived, she was certain of that. Nonetheless, someone else was here. She fumbled in her purse for her keys. She was practically running now, her heart thumping in her chest. She was alone, dressed provocatively, carrying a big wad of cash\u2014an obvious target, an easy one. Please, God, just get me back to the rental. . . .\n\nShe rounded a lane of cars and sprinted. Just a few more steps and she'd be safe. Just a few more steps . . .\n\nHe jumped right in front of her. She screamed.\n\nIt was the man from the blackjack table. The little man with the mustache.\n\n\"Where you going in such a hurry?\" he said, his vulpine eyes dancing.\n\n\"Leave me alone.\" She held her purse up, brandishing it like a club.\n\nHe was quick, smooth, as if he'd had martial arts training. He knocked the purse out of her hand, then grabbed her by the hair. The wig came off in his hands.\n\n\"Not a natural blonde? That's disappointing.\" He tossed the wig away and grabbed her brunette locks, jerking her head back. He pressed his face close to hers. \"If you take that pretty dress off yourself, I won't have to rip it.\"\n\n\"Please leave me alone. Please.\"\n\n\"After, bitch.\"\n\n\"You don't want me. I\u2014I'm pregnant.\"\n\n\"Sure you are,\" he said, flinging her back against the hood of the car. He grabbed the front of her dress with both hands, one over each breast. \"Don't forget that I gave you a chance to do this the easy way.\"\n\n\"Freeze!\"\n\nHer assailant's head jerked around.\n\nIt was the security guard from the casino. He stood about ten feet away and had his gun pointed right at them. \"Good thing I kept my eye on you. I thought you were a thief, not a rapist.\" He stepped closer, keeping his weapon level. \"Now let go of the little lady.\"\n\nInstead the assailant thrust Annabel forward, locking his arm around her neck. \"She's my shield, man.\"\n\nThe guard continued his steady approach. \"Do you think I can't hit you without hitting her? Think again.\" He adjusted the aim of the gun, obviously training it on the man's head. \"She won't even get dirty.\"\n\n\"All right, all right!\" The man released Annabel and pressed himself face-first against the car. \"I give up.\"\n\n\"Very wise.\" Lowering his gun, he grabbed the assailant's right arm and swung it behind his back.\n\nIn a sudden flurry of movement, the man whipped around and knocked the gun out of the guard's hand. He pushed the guard backward and tore off running.\n\n\"Stop!\"\n\nHe bolted down the parking garage, back toward the stairs.\n\nThe guard scrambled under a car to retrieve his gun, then gave chase. Before he was halfway across the expanse, the man had disappeared through a stairwell door.\n\n\"Damnation!\" The guard pulled out his radio and called for help. Then he returned to Annabel's side. \"Are you okay, ma'am?\"\n\n\"I'll live,\" she said, steadying herself with one hand on the car roof. \"You think they'll catch him?\"\n\n\"The way he was running? He'll be at the Luxor before the men upstairs are on their feet. I'm sorry.\"\n\n\"Hey, I'm just glad you showed up when you did. You're my hero.\"\n\nHis eyes twinkled. \"Told you that man was dangerous, didn't I?\"\n\n\"Yeah, you did.\"\n\n\"You'll need to fill out a report.\"\n\n\"Ohhh . . .\"\n\n\"You can do it tomorrow if you'd rather.\"\n\n\"I would, thanks.\" She smiled. He was kind of cute, really.\n\n\"Got your purse?\"\n\n\"Sure.\" She bent down and picked it up. \"Well, thanks again.\"\n\n\"Of course.\" He started to go, then stopped. \"One other thing.\"\n\n\"Yes?\"\n\n\"That dress you're wearing? I don't like it.\"\n\n\"Excuse me?\"\n\n\"Look at yourself in the mirror. Your breasts are on display, as if you were a southern plantation slave girl.\"\n\n\"I\u2014I didn't realize\u2014\"\n\n\"Your teats are for nursing children, my dear. Not for attracting men.\" His voice seemed to slow, to acquire a more pronounced drawl. \"Not for producing unholy thoughts. Luring men to their doom. Throwing your sex at them like some kind of harlot.\"\n\nShe turned toward her car. \"I\u2014I think I should go.\"\n\n\"Too late for that, petunia.\" One hand on the back of her head, he jabbed a syringe into her neck. Her legs wobbled.\n\nHe scooped her up into his arms and carried her to his pickup. \"I am sorry about the pain, my dear. But you won't feel it for long.\"\n\nAnnabel was dazed, limp. \"What . . . what are you going to do . . . ?\"\n\n\"I'm going to help you. Help you be something better than you are. Something wonderful.\"\n**CHAPTER FOUR**\n\nChin up, Susan, chest out, I told myself as I made a beeline toward the yellow crime scene tape. Walk like you know exactly what you're doing and you're in a hurry to get there. That's how I've managed to bluster my way past guards, thugs, reluctant witnesses, and on one occasion, Secret Service agents.\n\nI nodded at the patrolman posted by the entrance to the ballroom\u2014and kept on walking. I could read his confusion, his uncertainty. No doubt he'd heard that I'd been relieved of duty and didn't know what I was doing here. But he didn't stop me. There was also a hotel security guard standing by the door, a little guy with a tangled mess of black hair and a big bad gun. He was watching me carefully, too. But I kept on walking.\n\nThis ballroom was something else. I had been out to this hotel before, not for work but for pleasure. I liked the joint. It appealed to my sense of the macabre. Of all the themed casinos that had sprung up over the last couple of decades, this was my favorite. It was built back in the early Nineties, when Steve Wynn and some of the other high rollers were doing their Vegas Is for Families initiative. Disneyland of the Desert, that's what they wanted. That's when the new improved Strip got Treasure Island (pirates) and the Excalibur (Camelot) and the Luxor (fantasy Egypt). Then we got the geographical reconstructions\u2014the New York, complete with a fake Statue of Liberty, and the Paris, complete with a fake Eiffel Tower (like the original, only better-lit). By the end of the decade, the pendulum had swung back again and Vegas was refocusing on its old reliable: vice. This has always been a city of addictions\u2014booze, drugs, sex, money, risk\u2014and now they were back in fashion. Most of the new resorts focused on providing premier shopping or replicating high-dollar vacation spots. Truth was, most of the chumps who came to Vegas had never been to Europe and would be bored stiff at the real Bellagio. But they loved the chance to pretend to be cosmopolitan sophisticates\u2014with girlie shows and free drinks, of course.\n\nThe Transylvania had come in about the same time as Treasure Island and the Excalibur. With a choice spot just up the Strip from Circus Circus (now there was an interesting concept\u2014Gambling for the Whole Family!), it had done a brisk business, specializing in those with a taste for the outr\u00e9. Most of the joint was more tongue-in-cheek than terrifying. Scary more in the sense of, say, Disneyland's Haunted Mansion than, say, a Jason movie. That showed in the galleries, too. That was another hot Vegas trend\u2014everyone wanted a exhibition. The Bellagio had originally sported Steve Wynn's art collection. The Venetian had a Guggenheim museum. Mandalay Bay had art treasures collected from around the world. And the Transylvania had a series of ballroom galleries exhibiting re-creations of various fright classics such as _Frankenstein, Dracula,_ and _Dr. Jekyll and Mr. Hyde._\n\nThis gallery was dedicated to Edgar Allan Poe. I didn't know enough about his work to identify most of the references. But there was a spooky Victorian house, crumbling and decayed, with a fa\u00e7ade that looked like a human face\u2014windows for eyes, et cetera. There were cobwebs and skeletons and, of course, the requisite graveyard, which was where I found Chief O'Bannon, crouched on the floor examining something in a tiny evidence baggie. He was surrounded by a swirl of activity, at least a dozen forensic technicians carefully combing the site with dusters and infrared lights. Chemical swabs. Tweezers. I wondered if they knew about me. For a brief moment, I thought about turning tail and running before I was spotted. But that's not my style. That would be too sensible.\n\nOne of the techs approached O'Bannon. Tony Crenshaw. His specialty was dactylograms\u2014that's _fingerprints_ to the rest of the world\u2014but he was so good O'Bannon let him mess around in hair and fiber and pretty much anything else he wanted to do. I decided to keep my mouth shut for a moment\u2014a novel idea, for me\u2014and just listen.\n\n\"We've gone over the box pretty carefully, sir. Lots of good trace evidence.\"\n\n\"From the victim?\" O'Bannon asked in his usual gruff manner. \"Or the killer?\"\n\n\"Certainly from the victim,\" Crenshaw said, wincing slightly. \"But we're hoping we'll get something from the perp.\"\n\n\"What have you found on the girl?\"\n\n\"Hair. A few latent prints. Blood.\"\n\n\"How much?\"\n\n\"Not a lot. She does not appear to have been wounded in any significant way.\"\n\n\"Anything else?\"\n\nCrenshaw hesitated a moment. \"Sir . . . have you looked inside the box?\"\n\n\"Briefly.\"\n\n\"The inner side of the lid?\"\n\n\"No. Why would I?\"\n\n\"Claw marks.\"\n\nO'Bannon squinted. \"Like a wild animal?\"\n\n\"Like she was desperate to get out. The marks match the victim's fingers, which you may have noticed were raw and bloody.\"\n\nHis eyes narrowed. \"You mean\u2014she was still alive when\u2014\"\n\nCrenshaw nodded grimly. They both looked as if they were about to be sick. I crept forward to get a look at the box they were talking about.\n\nJust below O'Bannon, sticking out of the mock graveyard adjoining the haunted house, was an open coffin. With a very scratched inner lid.\n\n\"Christ,\" O'Bannon said, wiping his brow. \"What have we got now?\"\n\nCrenshaw shook his head and went back to work. O'Bannon did the same, but I could tell he was shaken.\n\nA moment later, O'Bannon spotted me out of the corner of his eye. \"What the hell are you doing here?\"\n\nI moved closer, hoping to avoid a scene. \"Looking for you.\"\n\n\"You should've been stopped on the other side of the tape.\"\n\n\"C'mon. You know no one can stop me.\"\n\nHe grunted unhappily.\n\n\"I wanted a few words with you, Chief. About my job.\"\n\n\"Here's your few words: you don't have a job anymore.\"\n\n\"Chief, I know I kind of screwed up.\"\n\n\"That's like saying Rush Limbaugh is kind of conservative.\"\n\n\"Let me make it up to you. Reinstate me.\"\n\n\"No can do.\"\n\n\"Please.\"\n\nHe started to speak, then stopped, glancing at all the people surrounding us. He grabbed my elbow and dragged me off to where we would be less conspicuous, then looked me straight in the eyes, glowering. \"Do you have any idea what I've had to deal with this past week, while you were off taking your rest cure?\"\n\n\"It was hardly\u2014\"\n\n\"I've been dealing with a family\u2014a very rich and influential family\u2014that doesn't understand why one of Las Vegas's finest beat their oldest son to a pulp. It's amazing how unreasonable people can be about things like that.\"\n\n\"The kid was a jerk.\"\n\n\"Oh, well, in that case, you should've just killed him.\"\n\n\"Chief\u2014\"\n\n\"They've been threatening to sue the department, something that would decimate our already strained budget, not to mention create some incredibly bad press.\"\n\n\"If there's anything I can do\u2014\"\n\n\"But there isn't. You created a big shitpile and left me to clean it up.\"\n\nI had to take a step back. O'Bannon was way angrier than I had anticipated. I'd never seen him like this, and I'd known him even longer than I'd been on the force. His left eye was twitching, for God's sake, and the little purple veins on the bulb of his nose were throbbing.\n\n\"Of course, I've had to deal with the IA boys, who were all over us, calling for an investigation, policy changes. The usual bull. I tried to point out that you were not exactly acting ex cathedra when the incident occurred\u2014although apparently in your drunken stupor you thought you were\u2014but that didn't placate them. They're demanding prompt action, which is just IA code for 'throw us a scapegoat.' \"\n\n\"And I'm the scapegoat.\"\n\n\"What'd you want me to do, pin it on the kid you creamed?\"\n\n\"So you just fired me. While I was seeking medical attention.\"\n\n\"Technically, I didn't fire you. Internal Affairs did.\"\n\n\"Bastards.\"\n\n\"IA wanted to go the whole dishonorable-discharge route\u2014put a permanent stain on your record that would guarantee you couldn't get hired as a security guard at Piggly Wiggly. But I told them you were dealing with personal problems and had a chemical dependency and a lot of other crap, so they just fired you and left it at that.\"\n\n\"Thanks so much.\"\n\n\"Don't thank me. I didn't do it for you. I did it for your dad. _He_ earned my loyalty.\"\n\nI can't pretend that one didn't kick me in the teeth pretty hard. My father was a cop. He and O'Bannon had been peers. I think they were even partners for a while.\n\n\"Chief\u2014please. Listen to me.\"\n\nHe pushed my hands away. \"Pulaski, we've got nothing to say.\"\n\n\"I need to work right now.\"\n\n\"They're hiring at McDonald's.\"\n\nI gestured toward the coffin. \"Looks like you've got a weirdie on your hands. Some kind of psycho?\"\n\n\"God, I hope so. Maybe if we have a real case to work, IA will ease up on my former behaviorist's drunken brawls.\"\n\n\"You're going to need someone with expertise.\"\n\n\"I've promoted Granger to homicide detective.\"\n\n\"Granger couldn't detect his ass with both hands.\"\n\nO'Bannon leaned right into my face. \"But you know what? When I get back to the office tonight, he'll be sober.\"\n\nThey found Helen. Just as he'd planned.\n\nHe was experiencing so many emotions at once, it took a moment to sort through them, to catalog them in his brain. There was a calm, yes, definitely a peace that came from knowing it had begun. His actions had begun to concatenate. He had taken the prophet's nebulous meliorisms and converted them into a concrete plan of action. There was no chance of turning back, losing faith, rethinking the agenda. It was done, and because it was done, nothing would ever be the same.\n\nBut he couldn't deny that he also felt a certain disappointment. Did anyone appreciate what he had conceived? Of course not. Who could appreciate how dramatically their lives would soon be changed, irrevocably? What had begun with Helen would soon affect every soul on this earthly plane. _Helen, thy beauty is to me \/ Like those Nicean banks of yore . . ._\n\nHe continued observing the busy proceedings in the ballroom; his security uniform gave him an entrance pass to anywhere he wanted to go. So much energy, but so little of any importance being accomplished. They would not be among those who understood, these poor functionaries charged with an impossible task. Not a one of them.\n\nExcept perhaps . . .\n\nHe watched the woman, the tall one, the one most of the others studiously avoided. What had she done to make herself such a pariah? he wondered. He sensed that he himself might be more welcome than she was at the moment. And yet there was something special about her, something he sensed, something he felt in his heart.\n\nHe watched as she approached the remains of dear Helen. The others had obsessed over her shaved head\u2014still did, in fact. He could well imagine what they were thinking. Obsessive-compulsive, perhaps. Organized nonsocial, the shrinks would proclaim. Woman-hater, the chief was probably grumping. Fools. Children.\n\nBut not the tall one. Her eyes had already moved to the torn earlobes. Then the damaged fingernails. She knew instinctively what was important. What was telling.\n\nShe was strikingly attractive\u2014tall, slender, athletic. Her face seemed drawn, strained, but her features were sound. And she had hair the color of his totem.\n\nBut there was more than that. Her eyes.\n\nThey were not the same. One was darker than the other. The left was a smoky gray, but the right was pure ebon.\n\nHe closed his eyes. _Now all my hours are trances; \/ And all my nightly dreams \/ Are where thy dark eye glances . . ._\n\nHer presence here was no accident. She had been sent to him.\n\nShe was part of the plan.\n\nCould it be she who would share his dream? That he might work in solitude no longer?\n\nNo, Ginny, I am not being unfaithful. But it's hard to be alone always. So hard . . .\n\nHe opened his eyes and peered all the more intently at the woman. She was older than his usual; she could not be an offering. But why not a colleague? Poe had taken a young girl for his bride, but he had chosen an older woman for his companion, his true soul mate. Maria Clemm had been with him longer than his child wife, and in the end, she had been far more important to him.\n\nHe gazed longingly at her eyes again, wondering at their importance. And this time, he saw more than just their color. There was pain in those eyes. She had been hurt, this one. Scarred. She was still reeling. Trying to find her place in a world that had turned against her.\n\nShe needed him.\n\nHis shift was over. One quick stop at the dentist's office, and then he could return his attentions to the new offering. Darling Annabel. But all the while he walked and later drove, he thought about the woman he had seen in the gallery, the one with the hair of the raven. How could he reach out to her? How could he make contact? A blessing such as this could not be ignored.\n\nI tried to give the IOP classes a chance. I really did. I played Dr. Coutant's game. But it didn't take long to realize that this wasn't going to be helpful.\n\nAs I predicted, it was mostly a big group therapy session run by two former users, neither of whom was remotely qualified to lead a big group therapy session. They'd probe with their little questions, trying to get people to talk. I was willfully noncooperative. I couldn't relate to anything anyone was saying, most of which was incredibly stupid. As a trained psychologist, I resented seeing these nudniks turn therapy into Ted Mack's _Original Amateur Hour._\n\nOkay, so I was the only one in the group with a college degree. I wasn't going to be snotty about it. I've never had any trouble mixing with people from all walks of life. And I remembered Dr. Coutant saying that intellectuals rarely did well in these programs. Does that mean they're only successful with dunderheads? People who buy into anything anyone tells them? At any rate, I had a hard time relating to the travails of guys working on the loading dock making eight bucks an hour who got hooked on street drugs mostly out of boredom because it wasn't football season and there was nothing on television. And I detested hearing people whine about their personal problems, most of which didn't amount to a hill of beans.\n\nOf course, they had the AA twelve steps up on the wall. We all recited them in unison, then the group leader talked about each of them, even though the first eight or so all seemed to me to say pretty much the same thing over and over again. What is all this \"admit that I am powerless\" crap, anyway? Wouldn't it be a better technique to admit that I am powerful, that I have the strength to overcome my troubles? I had a real problem with this sniveling approach to better health. I couldn't help wondering if that was why AA and other similar programs didn't have a better recovery rate.\n\nWe also had this guy, Herb, a little salt-and-pepper-haired man who fancied himself a motivational speaker. He had lots of standard routines, gimmicks, anecdotes, acronyms. I thought it was just a matter of time before he tried to sell us his three-hundred-dollar award-winning series of inspirational cassette tapes. He asked us how we were, starting with me.\n\n\"I'm fine,\" I said succinctly.\n\n\"That's not an answer.\"\n\n\"I'm fine,\" I repeated, a little louder.\n\n\"That's not an answer. That's a blow-off.\" He pointed to a poster on the wall next to the twelve steps. It was basically a long list of adjectives. \"Pick three that describe how you are.\"\n\nOkay. His class, we'll play it his way. I chose three at random. \"Optimistic. Determined. Reverent.\"\n\nHerb arched an eyebrow.\n\nFor the following hour, we were treated to this blustering rodomontade about Herb's successful battle against demon rum. We were supposed to be inspired, but I couldn't help thinking that if the guy had ever had one day like most of mine, he'd be back in the gutter with a dollar bottle of muscatel.\n\nThen, for his next act, Herb wrote PEOPLE PLACES EVENTS on the chalkboard. \"You abuse substances,\" he announced, \"because of one of these three things. Something a person in your past did. A place that hurt you. An event that traumatized you.\" He used examples from his own life, so we got to hear about how his mother threw plates at him when he was six and how his drunken daddy left when he was nine and how he got busted up in 'Nam. And oh yeah\u2014his daughter is a sex addict, so he won't speak to her anymore. Thanks for sharing.\n\nMental note: next time I develop an addiction, sex addiction sounds a lot more fashionable, not to mention pleasurable, than substance abuse. All the major movie stars are sex addicts, right? But no one treats them like they would a wino. Alcoholism gives a girl ruddy skin and liver damage. Sex addiction adds luster.\n\nAnyway, this guy's sermon opened the floodgates on what all the women in the group wanted to do anyway\u2014blame it on their spouses. This was not remotely helpful to me, but I have to admit listening to it had a certain addictive quality, like tuning into a poorly written soap opera\u2014just one damn thing after another. I listened to hours of the running battles between Jill and Buddy, every last mean thing he ever supposedly said. Oddly enough, she never did anything to provoke his invective. At least as she told it. Jacqueline was slightly more honest. She admitted that she argued back when her husband came after her for no reason. But she was blameless. Those bad men made those nice girls drink.\n\nYeah, right.\n\nSo at the end of this interminable three-hour session, one of the leaders, a heavyset gal named Margie, decided to pick on me. I hadn't said much, so I guess she felt obliged to try to get me into the whining bee.\n\n\"Why do you think you drink, Susan?\"\n\n\"Drank. I don't do that anymore.\"\n\nThe smile only flashed for a second, but I didn't miss it. \"Why do you think you drank?\"\n\n\"I liked it.\"\n\nShe batted her long false eyelashes. \"That's all?\"\n\n\"Yup. Tastes good. Makes me feel good. What more do you want?\"\n\n\"You don't think it . . . could have anything to do with your work?\"\n\n\"Of course not. I love my work.\" I left out the detail that I didn't actually have work at the moment.\n\n\"You don't think it could relate to . . . what happened with your husband?\"\n\n\"No,\" I said, giving her a stony glare. \"I don't.\"\n\n\"You took a piece of glass\u2014\"\n\n\"I don't want to talk about that.\"\n\n\"You seem to have a lot of anger. Why is that?\"\n\n\"Maybe I don't like having people prying into my private life.\"\n\n\"We're only trying to help you.\"\n\n\"You're talking about things you know nothing about.\"\n\n\"Then tell us. We want to learn.\"\n\n\"There's nothing to learn.\"\n\n\"No problems?\"\n\n\"No. Everything is fine. Perfectly pleasant.\" I even forced myself to smile.\n\nMargie leaned forward. \"Susan . . . what's your secret?\"\n\n\"Excuse me?\"\n\n\"What's your secret?\"\n\n\"I don't have one.\"\n\nShe paused a moment. \"I remember a patient I had about three years ago. He said everything was fine, too. Then one day I asked him, 'What's your secret?' And he looked at me for a long time. Then he finally told me he was sexually abused when he was six years old. And he started crying. All that fear and anger tumbled out of him. He's been well ever since.\"\n\nSo that was what she wanted. Some big fake Hollywoodesque _Prince of Tides_ \u2013like single-event explanation for every problem I've ever had. Talk about trite. I considered making one up: _I was stolen by Gypsies and forced to work on a coffee plantation in Kenya._ But I knew I would be the only one laughing.\n\n\"There's no secret. Your whole approach is psychologically wrongheaded. You can't boil all of a person's problems, all of their life, down to one person or incident.\"\n\nShe remained implacable and insistent. \"Susan . . . what's your secret?\"\n\nWhat was this\u2014some kind of hypnagogic brainwashing technique? \"How many times do I have to say this? There is no secret.\"\n\nI could see the other patients in the room shaking their heads sadly. Margie sighed. \"I hope that's true, Susan. For your sake. You have a lovely smile, and I'm glad to see you using it occasionally. Beautiful eyes, too.\" She paused, staring at me. \"But when I look into those eyes\u2014I see pain.\"\n\nAt the next break, I left, never to return.\n\nHe heard Annabel even before he opened the door. That was unfortunate. He had not expected her to regain consciousness so quickly.\n\n\"Who is it?\" she shouted as he came through the door. She twisted her head around; it was all she could move. \"Who's there?\"\n\n\"Just me again, my lovely. Back from the trenches.\"\n\nShe was lying flat on the table, as pure and unsullied as the day she was born. Her wide limpid eyes stared up at him. No restraints were necessary. She would not be able to move her limbs for some time.\n\n\"Why am I here?\" she asked. \"Where are my clothes?\"\n\n\"You're in my laboratory,\" he said, maintaining calm, genteel composure. \"And it was necessary to remove your garments.\"\n\n\"I don't\u2014don't know what you\u2014what you\u2014\" She was trying to fight the drug, to gather her strength. It was hopeless, but he couldn't help but admire her for trying. \"I don't know what kind of sick bastard you are\u2014\"\n\n\"I assure you my parents were married at the time of my birth, dear.\"\n\n\"My boyfriend knows where I went! And when I don't return, Warren will do whatever it takes to find me.\"\n\nA feisty one, this offering. A pleasant contrast to her predecessor. Exhilarating. \"Dear me. Do you suppose he might call the law enforcement authorities? My heart's atwitter.\"\n\n\"Worse than that, you asshole. He'll call my mom.\"\n\n\"Do tell.\"\n\n\"That's right. She may be shit as a mother, but she's got power and money and she won't let you get away with this.\"\n\n\"You should not speak ill of your parents, Annabel. It's most unbecoming.\"\n\n\"They'll show you unbecoming, you sicko.\"\n\n\"Annabel, I don't care for your tone.\"\n\n\"You won't care for anything when my mother gets her hands on you. Now let me out of here!\"\n\nHe looked down at her with genuine sadness. \"That, I am afraid, I cannot do. My plans for you are not complete. But when they are, you will be released.\"\n\n\"And I'll be free to go?\"\n\nHe hesitated only a moment. \"You'll be free.\"\n\n\"What are these plans?\"\n\n\"You'll see.\"\n\n\"What are you going to do to me?\"\n\n\"Nothing improper, I assure you.\"\n\n\"Then why did you take my clothes off?\"\n\n\"Did you think I had no legitimate purpose? That I did all this for my personal delectation?\" He touched her sternum experimentally with his finger, then drew a long, slow line down her breast bone. He used his most mellifluous voice, hoping to calm her. \"You can't appreciate what I'm doing, but it is for your own benefit, to help you fulfill your destiny, to complete your spiritual efflorescence. It was necessary.\"\n\n\"Necessary for what? For you to get your rocks off?\"\n\nHe twitched. \"You know, my dear, my plan does not require me to advance to the next step until tomorrow. But under the circumstances, I think mayhap it's best that we proceed immediately.\"\n\n\"What do you mean? What are you talking about?\"\n\nHe set down the brown bag he had brought in with him. \"Allow me to prepare my tools. Then all will be made clear.\"\n\n\"What's that? What is it?\" Panic set in. Her voice quavered. She was no longer shouting, no longer demanding. She was scared.\n\nHe opened the bag wide and held it before her so she could see the scalpel, clamp, wedge, prong, and drill. \"These are my dentistry tools.\"\n\n\"W-W-What are you going to do with those?\"\n\n\"I'm afraid you are due for an oral examination, darling.\"\n\nHer eyes grew wide and watery. \"Are you\u2014are you going to hurt me very much?\"\n\n\"Yes, I'm afraid I am. I can't deaden this pain\u2014not without making you unconscious. And you need to be awake as long as possible. So you can appreciate what is happening to you.\"\n\nHer face caved, as the horror of her situation became clear. \"P-P-Please don't do this. I'm going to be married. I'm going to have a baby.\"\n\n\"I don't think so.\" He leaned forward, the metal clamp and wedge reflecting light into her eyes. \"Open wide.\"\n**CHAPTER FIVE**\n\nThe lawyer sat on the other side of his desk, wearing a three-piece suit with a watch chain dangling from his vest pocket. His expression was so earnest it made me want to barf.\n\n\"You must understand, Susan. There are many competing factors involved here.\"\n\n\"What's so complicated? I'm her only living relative.\"\n\n\"Granted. But there are complications.\"\n\n\"I've been raising Rachel for three years without any problems.\"\n\nHis head swayed. \"Well . . .\"\n\n\"Certainly nothing major. And then I get sick for a week and they steal her away and stick her in a foster home.\"\n\nHe took a deep breath, then slowly released it. \"I don't want to make you angry, Susan. But if we're going to get anywhere, we have to be realistic. You did not get sick. You were committed to a detox clinic. Because you are an alcoholic.\"\n\n\"That's bullshit.\"\n\n\"That's the position of the state, at any rate. And NDHS is not going to allow an alcoholic to retain sole custody or guardianship of a minor without a fight.\"\n\n\"NDHS needs to mind its own business.\"\n\n\"This _is_ its business.\"\n\nJerk. Of course, like any good cop, I'd been trained to despise lawyers, so having to go to one for help was excruciating. I'd used this guy, Quentin Delacourt, a few times after David died, for wills and estates stuff that I never really understood. But he didn't know me. And I didn't much care for having him make these blanket proclamations about who and what I was.\n\n\"I'm not going to sit here defending myself to my own attorney,\" I said. \"Will you take the case or not?\"\n\n\"That depends on what you mean.\"\n\n\"I mean getting my niece back.\"\n\n\"Are you sure that's what you want?\" He leaned back in his burgundy leather chair, adopting what he undoubtedly thought was a deep, contemplative pose. \"There's no rush. Maybe you should give yourself some time.\"\n\n\"I gave myself a week. Look what happened.\"\n\n\"Give yourself a month. Just to relax. No stress, no work. And no alcohol, of course. Give your body a chance to recover. I don't think you've taken any time to get your head together since\u2014\"\n\n\"I'm not wealthy, Mr. Delacourt. I don't have the luxury of indulging myself in some spiritual walkabout. I want my niece back\u2014now. I want you to file a motion or whatever it takes to get her yanked out of that hideous foster home.\"\n\nHe opened a file on his desk and thumbed through it. \"From what I can see, the Shepherds' home is far from hideous. Apparently they have taken in many minors from\u2014um\u2014difficult situations.\"\n\n\"Rachel was perfectly happy with me. And she hates it where she is.\"\n\n\"That's to be expected, at least at first.\"\n\n\"So will you bring the motion or not? I want a hearing.\"\n\nHe gave me the contemplative look again, this time even steepling his fingers for added effect. \"If you're sure that's what you want, then I'm honor-bound to get it for you. But I can't say that I'm optimistic about your chances. NDHS wouldn't have intervened unless they thought they had due cause. They've really papered this file.\"\n\nHe was starting to piss me off, big time. \"How much mileage can they get out of one stupid mistake?\"\n\n\"It goes way beyond that.\" He continued thumbing through the file, not making eye contact with me. \"You're currently unemployed.\"\n\n\"I'm going to fix that.\"\n\n\"I won't lie to you, Susan. As long as you're unemployed, there's no way you're going to get custody.\"\n\n\"I told you, I'm going to fix that.\"\n\n\"Rachel's grades have been falling for the past year.\"\n\n\"It's hard losing a father figure. For the second time. Hasn't been a real picnic for me, either.\"\n\n\"Her school counselor says she's been depressed.\"\n\n\"I think that's somewhat understandable, given the circumstances.\"\n\n\"NDHS says you often leave her alone at night, while you're ostensibly working.\"\n\n\"Ostensibly? What is that supposed to mean?\"\n\nHe closed the file. \"One has to wonder, given the frequency . . .\"\n\nI stood up. \"What are you implying?\"\n\n\"Have you been drinking around Rachel?\"\n\n\"I never drink at home.\"\n\n\"I ask again, have you been drinking around Rachel?\"\n\n\"Absolutely not. Look, I screwed up once. I wasn't drinking that often.\"\n\n\"If you make that claim at the hearing, NDHS will bring out your blood work and prove you a liar.\"\n\n\"What the hell do they know about it?\" I screamed. And immediately felt embarrassed. I was proving myself just as unstable as NDHS said I was. Playing into their hands. \"Let those sons of bitches lose a husband! Let's see how well they handle it.\"\n\n\"We all know you've had some difficult trials. But the focus at the hearing will be on what's best for Rachel.\"\n\n\"I'm best for Rachel. She needs me!\"\n\n\"Or is it more that you need her?\"\n\nI felt as if the top of my head was going to blow off. Literally. I gripped the edge of the man's desk, consumed with fury. \"Are you going to get me the damn hearing or not?\"\n\n\"I'll get the hearing,\" he said quietly. \"But I can guarantee the attorney for NDHS will be much rougher on you than I have been. And if you behave in court the way you're behaving now, you haven't got a snowball's chance.\"\n\nIt took him more than three hours to remove all of Annabel's teeth. He used no anesthetic and nothing to stanch the bleeding. She bled profusely, down her chin, onto her bare neck and shoulders, mixing with saliva and coagulating to create a nasty bubbling paste. He had no means of measuring the quantity of blood lost, but it seemed enormous, an endless flow from those torn and ravaged gums. He had no idea if, as in the story, she would bleed to death, but it certainly seemed possible. And if she did not do so as a natural consequence, he knew how to see that she did.\n\nAfter the first tooth was removed, she began to scream. He turned the Mozart up high. Fortunately, Camille's boyfriend was not nearby; it would have been an inopportune time for a visit. After the third tooth, the screaming ceased, as did the threats about her mother's revenge, the name-calling, the inappropriate remarks about his parentage. By the tenth tooth, she was entirely broken, shattered, incoherent. She began to hallucinate. She lost all sense of who and where she was. She called him Warren, made sexual advances. She alternated between pleading for mercy and babbling about her schoolwork. He was moved by her pain, truly. It took great force of will to remind himself that there was a higher purpose behind all this. The path to godhood was strewn with sacrifice.\n\n\"Am I dead?\" she asked at one point just before unconsciousness descended, her pulpy gums mashing together. \"I feel dead.\"\n\n\"The sleep will come, my sweet Berenice.\"\n\n\"Thasss good,\" she said, and her last thought was expressed as a single word. \"Why?\"\n\n\" 'And still the phantasma of the teeth maintained its terrible ascendancy,' \" he murmured, and with gentle fingers closed her eyelids for the last time. \"Make a place for me in the firmament, my darling. Tell my love I will soon be reunited with her. That we will all be together once again.\"\n\nHe walked upstairs and gazed out his window at the striking sunset. It was a glorious evening, bright and clear, a vivid orange and blue curtain draped across the skyline. He could almost feel the warmth of the light emanating from the Luxor pyramid, a forty-billion-candlepower beacon said to be the most powerful spotlight in the world. It shone for ten miles into space, carving a swath through the heavens. Was it the light that illumined sweet Virginia's face? Was it a trail of bread crumbs Annabel might follow? There was no basis for these beliefs in the prophecies. But it seemed so in his dreams, and the prophet did teach that dreams were a portal into the land of the ideal. If he could dream there, he could be there. And so he would. And so would they all.\n\nAfter dark, I told Lisa I was going to the grocery store for some Vanilla Heath Bar Crunch. Hated to lie, but it seemed simpler than having an argument about whether I should do what I already knew I was going to do. I had to see O'Bannon. And I'd decided to confront him in an environment in which he couldn't so easily blow me off.\n\nShort drive with the top lowered down the streets of neon did me a world of good. Saw the transparent dome of the Fremont Street Exposition, my new favorite tourist joint. Basically, they tarted up the old Strip so it could compete with the new, and did a darn good job of it, in my opinion. Hey, beats seeing David Cassidy lip-sync or the ten billionth magic show, right?\n\nOnce I arrived at his house, I pounded on his front door, literally pounded. Guess my rage was still boiling. When I thought about what he'd done to me, after all I'd done for him, the cases I'd solved, only to have him shaft me the first time I'm vulnerable\u2014it just infuriated me. I pounded and pounded, and when he didn't come to the door immediately, I started shouting.\n\nSome kid opened the door.\n\nOkay, _kid_ might be pushing it, but he seemed like a kid to me. He was probably in his early-twenties. He had peach fuzz on his upper lip and cheeks, and long brown hair that wasn't very well groomed. He was dressed in a green T-shirt and was tall and lanky. Actually, my first thought was of Shaggy from those Scooby-Doo cartoons.\n\n\"Do I know you?\" he asked.\n\nI blinked. \"I . . . don't think so.\"\n\nWe stared at each other.\n\n\"Is this the O'Bannon residence?\"\n\n\"Uh-huh.\"\n\nMore staring.\n\nFinally he spoke: \"Did they give you ice cream?\"\n\n\"Umm . . . excuse me?\"\n\n\"I wondered if maybe they gave you ice cream. They gave me ice cream. I like ice cream, don't you?\"\n\n\"I never had much of a sweet tooth. Who is they?\"\n\n\"The hospital.\"\n\n\"The . . . hospital?\"\n\n\"I had to be in the hospital once and I didn't like it. But they gave me ice cream and I liked that.\"\n\nI checked my reflection in the glass in the door. Did I look sick? Was I still wearing the ID bracelet? I had on a short-sleeved blouse, but I had replaced the gauze bandage on my wrist with a plain wide Band-Aid. \"How do you know I've been in the hospital?\"\n\nHe gave me a sheepish look. \"Did you have to wear one of those gowns with your butt hanging out?\"\n\n\"Um, I don't\u2014\"\n\n\"Do you mind if I say _butt_? I think _butt_ is a funny word, but my dad says I shouldn't say it.\"\n\n\"How did you know\u2014\"\n\n\"This.\" He grabbed my arm\u2014a little too roughly, but of course he didn't know I'd recently punctured myself. At least, I don't think he knew. \"See?\" He turned my left arm around so the crook of the elbow faced up. There was a red pinprick where they had drawn blood and now that I noticed it, a couple of faint bruises above and below. \"I bet you have small veins. I have small veins, and it took them three times to get the needle into me.\"\n\n\"Could I possibly see your father?\" I assumed O'Bannon was the previously mentioned dad and I was speaking to his twenty-something boomerang boy.\n\n\"Do you have a dog?\" he asked.\n\n\"Not currently. Why?\"\n\n\"I don't like dogs. Do you like dogs?\" With each new remark, I became increasingly aware of the oddness of his voice. He talked too loudly, for starters, given that we were only about a foot apart. And the inflections were all wrong. It was almost like one of those computer-generated voices that are assembled by syllables, so the intonation goes up and down with no relation to what is being said.\n\n\"Well . . . some dogs. I used to have German shepherds\u2014\"\n\n\"If you have a dog, I can't let you in the house.\"\n\n\"That's cool. No dog.\"\n\n\"Did you know that dogs were first domesticated around fourteen thousand years ago near Israel?\"\n\n\"No . . .\"\n\n\"I think it was a mistake.\"\n\nI stared at him, but not as intently as he was staring at me. He never actually made eye contact. Every time my eyes came near his, he averted them. I tried to get a fix on who I was talking to, without success. There was something odd about his expression, a certain vacancy behind the eyes. Almost as if he wasn't really there. Like his body was on the front patio with me but he wasn't. Like he was gazing out through an invisible acrylic barrier\u2014he could see through it, but he couldn't make contact. It disturbed me.\n\nOne of my chief assets has always been my hyper-empathy. I've had it since I was a child. I don't know why. And I can't really explain what it is. But I've had this ability to tune into what other people are feeling. It isn't a mind-reading trick. But it's real. Of course, it's always been a great asset to me in my work as a behaviorist. One of the reasons I've been able to create useful profiles is that I've had this talent for understanding what motivates people, what impels them to take action. But it wasn't doing me any good with this guy. When I put out my feelers toward him, I got nothing.\n\n\"So . . . is your dad home?\"\n\n\"He's in his study.\" No movement.\n\n\"Can I talk to him?\"\n\nAnd the kid disappeared. Didn't say yes, no, or go to hell\u2014just turned around and dived back into the house.\n\nSo what was I supposed to do? I decided that I'd been invited in, so I stepped into the foyer and closed the door behind me.\n\nAnd nothing happened. After a minute or so of waiting, I escorted myself down the front hallway in the direction I'd seen the boy go.\n\nI found O'Bannon's study. It was a gorgeous room, with walls of dark wood paneling and hundreds if not thousands of books. Two of the walls were completely covered with bookshelves stuffed to capacity. Beautiful books, leather-bound editions with gilt-embossed lettering on the spines. Some of them were police-related\u2014criminology texts and such. But most were fiction, literature I was supposed to read in college but never did: _Wuthering Heights, In Search of Lost Time, Ivanhoe, Bleak House, One Hundred Years of Solitude._\n\nChief O'Bannon sat in a plush recliner, obviously designed to be a reading chair. The spine of his book faced me. He was perusing _Jane Eyre._ Yes, you heard me correctly. My tough-guy supercop boss was reading _Jane Eyre._\n\nAnd he was drinking. There was a crystal snifter on the table beside him, and an open decanter of brandy beside that.\n\nI could smell it, even across the room. It smelled good.\n\nThe kid who had opened the door was sprawled across the carpet. He was reading something, too.\n\nI guess this room was far enough back that O'Bannon was insulated from the noise at the front of the house. Neither of them appeared to have heard me approach. Neither realized that I was standing in the doorway staring at them.\n\n\"Chief?\" I said quietly.\n\nHe jumped, actually jumped, out of his chair, slamming his book shut. His elbow knocked over the brandy snifter. It spilled onto the table and soaked some papers. One of them caught my eye. It was a photocopy of a page torn from a spiral-bound notebook, filled with block lettering and symbols.\n\n\"Damn it.\" O'Bannon tried to wipe the mess up with his sleeve. Fortunately, there had not been much left in the snifter. Once he had the mess contained, he focused his attention on me. \"What are you doing here?\"\n\nI pointed. \"Your son let me in.\"\n\nHe looked down at the kid, frowned, nodded. \"Well, what do you want?\"\n\n\"You know what I want.\"\n\n\"It isn't going to happen.\"\n\n\"Just listen to me for a moment.\"\n\n\"No.\"\n\n\"Come on. We can help each other.\"\n\n\"I don't think so.\"\n\n\"I need work.\"\n\n\"And that's why you came here?\"\n\n\"I want to help.\"\n\n\"I don't need help. I need peace and quiet.\"\n\n\"Chief, listen to me!\"\n\n\"Go home.\"\n\n\"A-A-Are you guys fighting?\"\n\nIt was the kid, still lying on the floor, but now twisted around and watching us like a spectator at a tennis match.\n\n\"No,\" O'Bannon snapped. \"We are not fighting.\"\n\n\" 'Cause it sounds like you're fighting. Do you need to go to time out?\"\n\n\"Darcy . . .\"\n\nThe boy looked at me. \"Do you know that if you make him mad, you will have to go to time out? Or maybe military school.\"\n\n\"Darcy!\" O'Bannon barked. \"We are not fighting!\"\n\nThe kid's eyes widened. He ran the tips of his fingers through his hair, as if he were washing it with invisible shampoo. He made a strange, excited noise, over and over again, something between a snort and hysterical giggling.\n\n\"Hey, it's okay.\" I don't know why, but I walked over to him and tried to lower his hands. \"Your dad and I work together. We always talk like this. It doesn't mean anything.\"\n\nHis arms were stiff and resistant. \"S-S-S-Sometimes my dad talks like that, and it means he's mad.\"\n\n\"Well, it doesn't now. Your daddy adores me and I know it. Even when he tries to hide it.\"\n\nApparently O'Bannon'd had enough of this fun. \"Susan, you're wasting your breath. The only reason I'm tolerating this intrusion is out of respect for your father. But that respect can only go so far.\"\n\n\"Look,\" I said, \"I know you've got a weird homicide on your hands. I know the victim was buried alive. Obviously, you've got a seriously twisted killer. My specialty. You need me.\"\n\n\"Like hell. I assigned the case to Granger.\"\n\n\"Right, Granger\u2014hawkshaw extraordinaire. Give me a break, Chief. Granger doesn't know squat about aberrant psychology. Except maybe what he picks up when he looks in the mirror.\"\n\n\"We'll catch the killer, Susan. Without your invaluable assistance.\"\n\nI pointed to the photocopy on the table, the one with all the gibberish. \"Is that part of the case?\"\n\nHe shrugged. \"We found that inside the girl's coffin. But we can't make heads or tails of\u2014\" He stopped short. \"You are not getting involved, Susan. Give it up. Go home.\"\n\nWhy was everything so difficult? Everyone kept babbling about how they wanted to help me, but every time I needed help, no one could deliver. \"Chief, I need my job back. If I don't get work, I'm going to lose Rachel.\"\n\n\"That ship has already sailed.\"\n\nI was so frustrated, so furious, my hands tightened into fists.\n\n\"Are you going to spank my dad?\" the kid\u2014Darcy, I guess\u2014asked.\n\nIf I thought it would help . . . \"No. I told you, we talk like this\u2014\"\n\n\"Because whenever Unca Donald gets mad like that, he gets a switch and chases after his nephews and spanks them.\"\n\nThere was something so strange about this young man, something so childlike and yet not, something unnerving because it was so ineffable. But I kept my attention focused on his father.\n\n\"Is this because of that jerk's rich family?\" I said. \"Because if you're doing this because you're afraid of a lawsuit\u2014\"\n\n\"There isn't going to be a lawsuit.\"\n\nThis about-face caught me off guard. \"Excuse me? This afternoon, you said\u2014\"\n\n\"No lawsuit.\"\n\n\"Someone pulled some strings?\"\n\n\"For you? Hell, no. You just got lucky.\"\n\n\"What happened?\"\n\n\"At the hospital today, some helpful RN knocked over a table where the frat boy had laid the pants he was wearing when you worked him over. And guess what fell out of his pocket? A little baggie filled with crystal meth.\"\n\n\"You're kidding me.\"\n\n\"Like I said, you got lucky. Of course, there's no evidence that he was using on the night in question. But it would certainly explain his aggressive and violent behavior.\"\n\n\"Did he have aggressive and violent behavior?\"\n\n\"Sure. That's how the fight started, right? You had to defend yourself.\"\n\n\"To tell the truth, I'm a little fuzzy on the details. . . .\"\n\n\"At any rate, that's going to be our story, and it persuaded the guy's family to entertain second thoughts. Even if they managed to overcome this brilliant defense at trial, the whole world would know their son was a drug user. Possible dealer. They didn't think it was worth it.\"\n\n\"That's fabulous.\"\n\n\"That's dumb luck, which does not in any way excuse or justify what you did. I can't use a cop I can't trust, Susan. I can't use a cop who might lose it at any moment. I can't use a cop who might be sneaking drinks on the side.\"\n\n\"Chief, that's all behind me.\"\n\n\"So you're all cured now, huh?\"\n\n\"Well . . . yeah, I'm over it.\"\n\n\"Fit as a fiddle. One hundred percent. Ready to carry a gun again. Ready to be some other cop's partner. Ready to have someone count on you, depend on you. Ready to have someone put their life in your hands.\"\n\nI didn't bother answering.\n\n\"I care about you, Susan. I _do._ And that's why I'm saying this. I will not reinstate you. You need to get help\u2014professional help. This may be your last chance to save yourself before you've lost everything.\"\n\n\"I've already lost everything.\"\n\n\"No, you haven't. But you will if you start drinking again. Now get out of here.\"\n\nI wanted to scream. \"Fine! Have it your way. You won't see me back here begging. Not for a blue moon. Not if\u2014\"\n\n\"And try not to be so angry all the time. Relax. Read a book. Get healthy.\"\n\nI couldn't leave without giving it one more try. I dropped a scrap of paper on his table. \"I'm leaving you my new address and phone, just in case you change your mind.\"\n\n\"I won't. Go.\"\n\nDarcy walked me to the door. He still talked in that too loud voice with the odd inflections, but he seemed to have lost the stutter. \"Will you be coming again in two and a half years?\"\n\n\"Why\u2014?\"\n\n\"That's what you said.\"\n\n\"I said I'm never coming back here.\"\n\n\"Uh-huh.\" He gave me that sheepish grin again. \"What you said was: 'You won't see me back here begging. Not for a blue moon.' \" The amazing thing was, he not only repeated what I had said verbatim, he mimicked my inflection. Then he shifted into a colder, almost singsong tone, as if he were reciting in front of a schoolroom. \" 'A blue moon is the second full moon in a given calendar month, which in North America occurs approximately once every forty-one months, or approximately every two and a half years.' \"\n\nI gave him a long look. \"I'm not coming back.\"\n\n\"Oh. Is the reason that you will not come back because of me?\"\n\n\"Of course not.\"\n\n\" 'Cause if it is, it's okay. I know sometimes people don't like me. My dad says it's not my fault, but he still yells at me sometimes. Everyone yells at me sometimes.\"\n\n\"It wasn't you. Really.\"\n\n\"Then\u2014\" His hands began to flap up and down. \"Do you think that maybe you could come again sometime? My dad gets real lonely. I think he misses having a girl around.\"\n\n\"I think your father made it clear. . . . Never mind.\" I wiggled my fingers, smiled faintly, and left. I probably should've said something more, but I couldn't manage it. I was so depressed, so frustrated, so . . . empty. I didn't have anything to spare for anyone else.\n\nAt first I thought the tall girl had a dog and I was worried because I don't like dogs and dogs are scary mean and smelly, but it wasn't a dog. The tall girl was sweaty and the hospital smells were still there a little and maybe one of those perfumes that she put on way too much of. I liked the girl and I think Dad likes her even though he was mean to her and she said he wasn't mean to her but he was and I know that look like the coach at the YMCA when I was in the eighth grade. He smelled too and Cleanliness is next to Godliness and get your hands out of your hair and what is that smell and I liked her she was nice to me. I don't know why she was nice to me but she was she touched me and I don't like it when people touch me but sometimes I wish they would more. _My love for Heathcliff resembles the eternal rocks beneath\u2014a source of little visible delight, but necessary._ I was right when I said I hoped she would come back because Dad is lonely but sometimes I'm lonely too and I like the tall girl with the eyes that don't match and the bump on her nose and the yellow Post-it stuck to the sole of her shoe. The girls at the day care act nervous around me but this girl didn't I think she liked me at least some maybe a little but when Dad talks to me I can't talk any more and he gives me that look like he's so disappointed in me. He spent hours on that simple puzzle and he never asked me never once did he ask me even though I like puzzles and he doesn't. The girl was keeping secrets and not just that she's been in the hospital but I like the girl and I hope she comes back.\n\nI wonder if she has babies because my dad said you need a girl to have babies and I like babies. Babies are nice to me. They don't mind when I'm around.\n\nHe rolled her remains into an old carpet, which made it easier to transport her to the pickup. He was learning, wasn't he? He was an innovator, never content with the status quo, always searching for ways to improve himself.\n\nHe drove the short distance to his chosen disposal place. This would be much easier than the last. Almost no chance of being spotted here, not this time of night. It was perfect\u2014thematically appropriate (though the police were unlikely to get the joke), risk-free, and certain to be discovered.\n\nOnly an occasional plane passing overhead reminded him where he was. The glittering silver created a reflected brightness, but if he stayed on the far end of the lot, he would be safe. He opened a curved and unlocked silver door and laid her body to rest.\n\nHe hesitated. A ceremony before he departed seemed appropriate. Perhaps\u2014some sort of prayer.\n\nHe lowered his head and spoke in susurrous tones.\n\n\" 'And so, all the night-tide, I lie down by the side of my darling\u2014my darling\u2014my life and my bride.' \"\n\nHe took a handful of dirt and, just to give this a semblance of burial, sprinkled it over the body. \" 'In her tomb by the sounding sea.' \"\n\n\"You've got a date? A true-to-life, honest-to-God date?\"\n\nLisa batted her eyelashes. \"Control your envy.\"\n\n\"And this isn't with . . . what was his name?\"\n\n\"The Human Plunger? Not him.\" Lisa had an endearing habit of identifying all her boyfriends by reference to their kissing technique.\n\n\"No, I mean the advertising guy.\"\n\n\"The Tongueless Titan. Ditched him.\"\n\n\"Dare I ask why?\"\n\n\"Kissed hard with his mouth open, but never any tongue.\"\n\n\"Was that bad?\"\n\n\"It was weird. Like he was administering CPR. And I asked myself, if he's so reserved with his tongue now, is he ever going to\u2014\"\n\n\"Lisa!\"\n\n\"Well, a girl has to wonder.\"\n\nI leaned through the open car window. \"Have a good time. I'll expect a full report.\"\n\nLisa took my hand and squeezed. \"Are you sure you'll be okay, honey?\"\n\n\"Of course I will.\" Lisa was so sweet. She would do anything for me, I knew that. It frustrated her, not knowing what to do, what she could do.\n\n\"I don't feel right, leaving you by yourself.\"\n\nI patted her reassuringly on the shoulder. \"Don't be silly. You can't babysit me forever.\"\n\n\"But it's only\u2014\"\n\n\"We'd drive each other crazy and we both know it. You're my best friend, Lisa, but even you can only tolerate me in limited doses.\" We both laughed. \"You've gotten me a great place to stay. I need some time to get it in order.\"\n\n\"I could help.\"\n\n\"Hey, you did all the packing. It's only fair that I unpack.\"\n\n\"There's no rush.\"\n\n\"I want the place looking nice before Rachel comes back.\"\n\nLisa fell silent.\n\n\"You run along,\" I said quickly, papering over the gap. \"I'll call you later.\"\n\nShe smiled a little. \"Okay. But if you need something, call me on my cell.\"\n\n\"I will.\"\n\n\"I won't be far. I could be at your place in a heartbeat.\"\n\n\"You shouldn't say that. I might call. Interrupt some CPR.\"\n\nHer smile faded a bit. \"I wish you would. But I know you won't.\"\n\nLisa hugged me again, then drove off in a sports car cloud of smoke and I made my way to the new apartment.\n\nShe'd done a terrific job, especially considering how little time she'd had to work on it. Wasn't too expensive, either. The monthly rent was considerably less than my mortgage had been, when I paid it, so I might end up with a little extra spending money\u2014a pleasant thought.\n\nWhat little furniture I had was in storage, so I was stuck with the rudimentary apartment-provided stuff. The mattress was lumpy and hard, but I had a hunch I'd be sleeping soundly anyway.\n\nI started by doing what little had to be done to make the joint habitable\u2014sheets on the bed, Mr. Coffee in the kitchen. Most everything else could wait. I really needed to relax. What an ordeal this day had been. I was exhausted.\n\nI should've just watched television, but I couldn't resist going through the packing boxes, making sure everything was safe and still in one piece. Lisa had taken great care with my belongings. But she couldn't know everything. She couldn't know that the scruffy, torn T-shirt that looked as if it must be a dust rag was actually my favorite pajama top. She couldn't know that I folded my sweaters along the vertical bias, not the horizontal. And she couldn't know that I had left a full bottle of bourbon in my gym bag.\n\nBut I knew.\n**CHAPTER SIX**\n\nHe had just unfolded the morning paper when Harv Bradford entered the canteen.\n\n\"Can you believe those cops are still running around outside, Ernie? Took me twenty minutes just to get into the hotel.\"\n\nHe shrugged. \"They have a job to do. They must keep the crime scene secure.\"\n\n\"Yeah, right.\" Harv poured himself a cup of coffee, took a sip, then winced. \"Are we reusing yesterday's grounds?\" He tossed the drink into the sink. \"You had any contact with the cops?\"\n\n\"No.\"\n\n\"Neither had any of the boys on the night shift. Kind of a snub, if you ask me.\"\n\n\"They're LVPD. Why would they consult with us?\"\n\n\"We're the hotel security force. We work every damn day right here where they found the body. Seems like we might be able to tell them a thing or two.\"\n\n\"Such as what?\"\n\n\"Well . . . I don't know exactly. But something. At any rate, they could ask. To treat us like we don't exist . . .\" Harv shook his head. \"Just seems disrespectful.\"\n\n\"City cops never have any respect for private security,\" he replied. \"They call us rent-a-cops.\"\n\n\"I think that sucks.\" Harv was a little over six feet, but he carried a spare tire that made the gray uniform bulge in all the wrong places. He looked ridiculous, out of shape, stupid. And he wondered why the police didn't want to consult with him. \"I could tell them a thing or two.\"\n\nHe lowered his paper. \"You know something about the body they found?\"\n\n\"Well, no. Not exactly. But it's possible I might've seen something without knowing I saw it, you know what I mean?\"\n\nI know you're a fool, he thought.\n\n\"Hey! Is there something in the news about it?\"\n\nHarv snatched the paper out of his hands without even asking. It was because of his height, of course. Because Harv towered above him, that gave him free rein to disregard common courtesy.\n\n\"What do you know?\" Harv said, slapping the paper. \"The Transylvania made the front page. Did you read it?\"\n\n\"I _was_ reading it,\" he answered sharply.\n\n\"This is pretty cool. Look at the size of the headline. I bet this is getting national play.\"\n\n\"We can but hope.\"\n\n\"Kinda exciting, ain't it? Being a part of a big story like this.\"\n\n\"I don't believe that either of our names is mentioned.\"\n\n\"Maybe not. But it happened right where we work. And I know the guy who found the body.\"\n\n\"You're a celebrity, Harv.\"\n\n\"If you ask me, this is what the money boys get for choosing such a creepy theme for this place.\"\n\n\"The children like it.\"\n\n\"Yeah, and since when did Vegas care about children? This new crowd\u2014they got more bucks than brains. I liked the town better when the mob ran it.\"\n\n\"Those were the good old days.\"\n\n\"You really don't think those cops would want to talk to me?\"\n\n\"I really don't, Harv.\"\n\nThe paper crumpled in his hands. \"Know what? I always wanted to be a cop. A real one, I mean. When I was a kid. But I couldn't afford the school and I couldn't pass the test. So I went into private security.\"\n\n\"And isn't that satisfying? You wear a uniform. You have the occasional opportunity to hustle prostitutes. Strong-arm card cheats.\"\n\n\"It ain't the same. People look up to cops.\"\n\n\"Do they?\"\n\n\"Cops are like heroes. They make TV shows about cops. When was there ever a series about a private security guard?\"\n\nAnd that of course proves, a posteriori, that security guards are without merit. \"I suppose you have a point.\"\n\n\"I mean, here I am, right on the premises, with a badge and a gun and everything, but those guys outside would never dream of asking for my help. Wouldn't even cross their minds.\" He released a slow sigh. \"Here.\" He tossed the paper back. \"All that little print makes my head hurt.\"\n\nThere were smudges on the main story, big black remnants of Harv's Frankensteinian thumbs. He hated that. He didn't want to read a paper that had been pawed over by illiterates. And this one was important; he needed this story for his History. He would have to pick up another copy on his drive home.\n\nHis eyes returned to the main story under the banner headline:\n\n **MURDER VICTIM \"BURIED\" \nIN CASINO GRAVEYARD**\n\nBY JONATHAN WOOLEY\n\nAn unidentified nude female corpse completely shaved of body hair was discovered early Tuesday morning in a mock graveyard located at the multimillion-dollar Transylvania resort hotel, authorities revealed yesterday afternoon. The body was placed in a wooden coffin and buried under a thin layer of dirt. The graveyard is part of the hotel's Edgar Allan Poe gallery, one of several horror-themed tableaus on the ballroom floor.\n\n\"We're just glad the body was discovered before the doors were opened,\" said Transylvania owner Katherine Wentworth. \"We wouldn't want any of our guests to be disturbed.\"\n\nPolice officials remained tight-lipped about the investigation, but LVPD Chief of Police Robert O'Bannon indicated that they were pursuing several leads.\n\n\"Obviously, the irony of depositing a body in a fake graveyard was more than someone could resist. We've taken evidence, which should allow us to identify the victim in time. The large number of tourists coming in and out of Vegas makes a quick ID difficult. Nonetheless, we have all our top officers working on it and have every reason to believe we will identify the victim\u2014and her assailant\u2014in short order.\"\n\nAt a press conference later in the day, a representative of the LVPD Homicide Department, Lieutenant Barry Granger, stated that preliminary tests indicated that the victim had died of suffocation. Several unanswered questions still remained about . . .\n\nIn other words, they knew nothing. He allowed himself a tiny smile. They didn't know who Helen was, they didn't know who he was, and they had no glimmer of the magnitude of what they had stumbled across. At least not most of them . . .\n\nHis eyes scanned the page and then the continuation on page three, searching for the information he wanted. Yes, yes, he knew O'Bannon, that blowhard was on television all the time. There were repeated references to Lieutenant Granger, who during his initial crime scene appearance seemed almost deliberately slow-witted. But what of the raven-haired beauty? Who was she? What was she doing there? Given the way she was treated by most of the other police officers, it was tempting to conclude that she was an unauthorized visitor, that she had no connection to them. But he knew that was wrong. He had seen the way she moved, the way she carried herself. She was on familiar ground. She had done this before. Had she been brought in from another jurisdiction? He had to find out.\n\n\"So I'm thinking maybe I'll just march right up and introduce myself to 'em. What'd'ya think about that?\"\n\n\"What?\" He looked up. Had Harv been babbling the entire time he was reading? \"Who?\"\n\n\"The cops, Ernie. I'm thinking maybe I'll tell them I'm available. Who knows? They might like the chance to work with someone who knows the lay of the land.\"\n\n\"They won't give you the time of day,\" he replied. \"If you had something to tell them, then maybe\u2014\"\n\nHe stopped short. That was it. If he had something to tell, something they really wanted, he could command anyone's attention. Even hers.\n\n\"I'll be out this afternoon,\" he said, tossing down the paper. \"Cover for me in the casino.\"\n\n\"Sure, but\u2014\"\n\n\"Keep your eyes open and your lips sealed,\" he added as he slid into his coat. \"And most importantly, Harv\u2014don't hassle the police and don't go near the Poe room. You never know what might happen there.\"\n**CHAPTER SEVEN**\n\nThe phone must've rung twenty times before it finally registered in my brain. Exerting all my available strength, I managed to pull my head out from under the pillow. It throbbed. More than throbbed\u2014it felt as if someone were running an electric mixer inside my cranial case, scrambling my brains.\n\nI grabbed the receiver, knocking over the end table in the process. The bottle fell to the floor with a bang but thank God didn't quite break.\n\n\" 'Lo,\" I managed. My tongue felt like Velcro.\n\n\"Pulaski? Is that you?\"\n\nI stiffened. It was Chief O'Bannon. \"Yes, sir.\"\n\n\"Are you all right?\"\n\n\" 'M fine.\"\n\n\"Took you forever to answer the phone.\"\n\n\"Sorry. I was in the shower.\"\n\nHe was silent for a moment. \"Are you able to come out to a crime scene? As soon as possible.\"\n\nTruth was, I felt like shit. But I wasn't going to tell him that. \"You want me at a crime scene? After last night\u2014\"\n\n\"I've changed my mind. Decided to give you a second chance.\"\n\n\"Out of the goodness of your heart?\"\n\n\"Out of respect for your father. And David.\"\n\n\"Bullshit.\" I shifted the phone from one ear to the other and pushed my aching self up to a sitting position. \"There's been another murder, hasn't there?\"\n\n\"Yes, but\u2014\"\n\n\"Buried alive?\"\n\nAgain the silence. \"No. This one's worse.\"\n\n\"You've got a psycho on your hands.\"\n\n\"Looks that way.\"\n\n\"And that's the real reason for this call. Not any charity toward me. You need my expertise.\"\n\n\"Look, I don't have time for this. Are you coming or not?\"\n\nGet a grip, I told myself. Take the job before he changes his mind. \"All right, I'm in. I'll stop by headquarters and get my badge and\u2014\"\n\n\"No badge. No gun.\"\n\n\"But you said\u2014\"\n\n\"I'm willing to hire you on a part-time consulting basis with respect to this one case. That's it.\"\n\n\"No way.\"\n\n\"Those are my terms. Take it or leave it.\"\n\n\"You\u2014\" I pounded my fist into the pillow, biting back what I really wanted to say. \"Why are you doing this to me?\"\n\n\"You're not ready to be reinstated.\"\n\n\"How the hell would you\u2014\"\n\n\"You have a problem, Susan. A major problem. And until you've overcome it, you're not going to play on my team. But I still need a behaviorist, at least until the Feds move in.\"\n\n\"Feds?\" I whistled.\n\n\"So I'm asking one last time. Will you take the consulting position or not?\"\n\n\"I'll take it.\" Even though I found the whole situation offensive, I needed work if I wanted to get Rachel back.\n\nHe told me where to meet him. \"I'll be there in about an hour.\"\n\n\"Twenty minutes,\" he answered. \"Or you're off the case.\"\n\n\"But I just got up\u2014\"\n\n\"I thought you just got out of the shower.\"\n\nI pressed my hand against my forehead. Dumb, dumb, dumb.\n\n\"See you in twenty.\" He made a grunting noise, then disconnected.\n\nMy entire body ached. The head was the worst, but it was just part of the overall miserable package. I felt broken, shattered, both in body and in spirit. I looked down at the bottle rolling around on the floor\u2014empty\u2014and it made me sick in so many ways I couldn't count them all.\n\nMy first night alone since the big breakdown, and I'd found a hell of a way to celebrate. I couldn't be trusted for one night. I'd done exactly what everyone said I would do, and I was lying about it afterward, just as everyone knew I would.\n\nHell. I had work to do. That would be my cure\u2014bury myself in a case. Get too busy to indulge in bad habits. I'd made one mistake, but I was determined not to let it happen again.\n\nThe hollow anxious aching in my chest reasserted itself. My wrist tingled.\n\nI found the crime scene on the back forty of McCarran International. Out here in the desert, you could barely tell you were anywhere near a major city. The airport terminal was to the north; other than that, it was big-sky country. Hard to imagine what could've inspired those Mormons to settle down here all those years ago. John Fremont, now mostly remembered for the tourist trap street that bears his name, first wrote about the area he discovered in 1844 while he was out harassing Indians. But Mormon cattle ranchers set up the first settlement in these fertile plains, around 1855. By 1905, we had a train station\u2014and casinos\u2014and they held the first Las Vegas land auction, the event that put this city on the map.\n\nBugsy Siegel always gets the credit for founding Vegas\u2014especially after they made that movie with the far-too-handsome Warren Beatty\u2014but he was only one of several people who established Vegas as a fantasy pleasure destination. He was a gangster, for God's sake, not a visionary. There were already a couple of hotels out here when he made the scene. If he'd been that insightful, he'd have bought all the land in the area, not just one lot, right? Meyer Lansky and a host of investors\u2014one of whom probably had Siegel offed\u2014were also major players. But everyone remembers Bugsy. There's even a memorial garden shrine to him, out at the modern-day Flamingo. Lisa and I went there once, just for laughs. It was a hoot. Of course, I was snockered at the time.\n\nAfter I parked my car, I stumbled down a sharp paved declivity to the recessed tarmac where the body had been found. The crime scene was in the midst of dozens of disabled aircraft. Apparently this was where the big birds came to die. One of the patrolmen on duty filled me in. The body had been stashed inside one of the retired jets. Judging from appearances, this young naked woman had already been dead before she was brought here. Why would the killer stash the corpse in an abandoned airplane? How was this connected to the woman who had been buried alive?\n\nI wasn't surprised to see Granger lurking about. Wasn't surprised, but wasn't happy, either. His face expressed his feelings about me pretty clearly, too.\n\n\"Took you long enough,\" he grunted.\n\n\"You're in a jocund mood this morning. Someone put castor oil in your coffee?\"\n\n\"Just for the record,\" he said, \"your involvement in this case\u2014or any other case, for that matter\u2014is against my strong objection.\"\n\n\"I assumed any smart idea wouldn't come from you,\" I replied, walking past him without stopping.\n\nI found O'Bannon inside the plane, running down a checklist with Crenshaw. When he saw me, he glanced at his watch. \"Twenty-nine minutes,\" he said. \"And you look like hell.\"\n\n\"Top of the morning to you, too,\" I answered, smiling.\n\nHe walked up to me and widened my eyes with his fingers, like I was a damn cow he was thinking of buying. \"Your eyes are bloodshot.\"\n\n\"I had trouble sleeping. You know how it is. First night in a new place.\"\n\nHe frowned, then sniffed.\n\n\"Anything on my breath?\"\n\n\"About half a tin of Altoids, unless I'm mistaken.\" He gave me a long look, and believe me, I didn't need hyper-empathy to know what he was thinking.\n\n\"Who's the victim?\" I asked, hoping to redirect his scrutiny from me to, well, anything. I stepped closer to the body, which the coroner's assistants were in the process of transporting. She was young, probably sixteen or seventeen. Something odd about the way her face was set, but she had been a pretty thing, that was obvious, and she still had her hair, unlike the last one. Fingernails, too. Her skin was an icy white, so drained of color her lips were almost invisible. There were no apparent wounds or injuries.\n\n\"Don't know. She was found with no identification. We'll run her picture in the paper and with luck someone will recognize her.\"\n\n\"What makes you think this is the same killer?\" I asked, although I was certain it was. \"Seems like an entirely different MO.\"\n\n\"He left another note.\"\n\nO'Bannon handed me the evidence, already encased in a transparent sheath. It was similar to the one I'd spotted last night\u2014letters and symbols and general nonsense on a sheet of lined notebook paper. \"Any idea what this is?\"\n\n\"No. I've got some of our biggest eggheads working on the first one, trying to see if it's some kind of code. So far, no luck. It may just be psychotic rambling.\"\n\n\"Can I get a copy? I know someone who might be able to help.\" I surveyed the crime scene. \"Who was the first responder?\"\n\n\"Harrelson. Lucky choice, really. He did a solid, clean job.\"\n\nThe first responder has the job of securing the crime scene. This is critical, not only to obtaining pure and useful information, but to being able to use that information later at trial. He or she must protect the evidence, then initiate safety procedures\u2014in this case, make sure the killer wasn't still around\u2014and then contact the proper criminalists and finalize the relevant documentation, which with a homicide, was enormous. Once the crime scene experts and homicide investigators arrived, supervisory authority passed to them.\n\nContrary to what everyone thinks from watching _C.S.I.,_ the Vegas Metropolitan Police Department has no department called a C.S.I. Level III. Or for that matter, a Level II or Level I or Level 427.5. Those TV creations are a blanket fiction that allows characters to do the work of a wide range of criminalists: forensic lab techs, photo techs, latent print examiners, firearms experts, medical examiners, document experts, hair and fiber teams, and evidence custodians, just to name a few. The only thing the TV show doesn't exaggerate is the importance of this work. Most cases are solved\u2014and proven in court\u2014thanks to the work of these technicians.\n\n\"How was she killed?\" I asked O'Bannon.\n\n\"Naturally, Dr. Patterson won't offer an opinion this soon. But judging from her skin tone, she bled to death.\"\n\nI was puzzled. \"You mean, she had internal bleeding?\"\n\n\"No.\"\n\nI glanced again at the body. \"I don't see any injuries.\"\n\n\"Right. That's the mystery.\"\n\nI stared down at the corpse, hoping to get some kind of fix on who she was or what she had been doing. What happened to you? I wondered. Who did this? And why?\n\nI scrutinized the whole picture, the neck, the chest, the legs. Not only were there no signs of a wound, there were no signs of any kind of struggle. No signs of restraint, except perhaps some faint redness across her upper arms. Were you too scared to fight? I wondered. She looked healthy enough. Why didn't she claw his eyes out?\n\nI put the coroner techs on hold and, against their heated protest, took a closer look at the body. I found signs of a body piercing on her navel. But the stud was gone. Ripped out.\n\nA pattern was forming in my mind. Far from a complete picture\u2014a hint at best. But something. Taking two tongue depressors from one of the coroner's boys, I pried open her mouth. And gasped.\n\nNow I knew why there was something odd about the set of her jaw. Her teeth had been removed. Every single one. The tearing of her mouth, her gums, was enormous; the extraction had not been executed by a trained professional. This was how she had bled to death. Not from any bodily wound. From the mouth.\n\nThank God I'd had no time for breakfast. Throwing up would not only be unprofessional, it would convince O'Bannon I'd been drinking. I'd seen some seriously twisted, weird, ugly stuff in my time, and it took a great deal to get a gasp out of me, even on a day like this when I was well off my game. But I was sickened by the thought of the pain she must have endured, both mental and physical. This was not the work of any ordinary killer. Not even an ordinary psychopath. This was something\u2014someone\u2014altogether different.\n\n\"I'll want to see the coroner's preliminary report,\" I said, letting her mouth relax. \"As soon as it comes in.\"\n\n\"Natch.\"\n\n\"I'll want the files on the first victim, too. Everything you've got.\"\n\n\"I've already had them sent to your office.\" O'Bannon coughed. \"Your temporary office. Downtown.\"\n\n\"Criminalists got anything useful yet?\"\n\nO'Bannon shrugged. \"Not that I've heard.\"\n\n\"What about blood splatters?\"\n\n\"Do you see any?\"\n\n\"No.\"\n\n\"Neither do we. Even after we went over the area with leucocrystal violet.\"\n\nWhich confirmed my feeling that the young lady was killed somewhere else. And cleaned up afterward. \"Firearms?\"\n\n\"No indication.\"\n\n\"Forensic entomology? Botany? Zoology?\"\n\n\"Possible they'll turn up something. But so far, no.\"\n\n\"Hair and fiber evidence?\"\n\n\"Nope.\"\n\n\"How could the guy bring a corpse all the way out here without leaving something behind?\"\n\n\"By being very careful.\"\n\nAnd that in itself was telling.\n\nI searched for, spotted, then approached Crenshaw. He was crouched on the ground, going over the metal floor of the plane with a small brush. Beside him was his fingerprint examiner's field kit, a five-level tool chest filled with everything he might possibly need\u2014powders, lifting tape, ink, flashlights, petri dishes, baggies, tweezers, distilled water, and a lot of other stuff I couldn't identify. \"How's it hanging, Tony? Seen any exciting friction ridges lately?\"\n\nHe smiled a little. \"Are you working this case?\"\n\n\"Strange but true. Got any identifiable prints?\"\n\n\"Not yet, but I'm still working. I'll have to take some of this stuff back to the lab before I can be sure.\"\n\n\"I would've thought the killer would get his paws all over the place, dragging a heavy corpse into the plane.\"\n\n\"I would, too, but he didn't. We found nothing inside the plane\u2014except for one little smudge. On the body.\" He pointed down at the corpse with which I was now altogether too familiar. \"Probably touched her before he transported her. Possibly even before he killed her. Maybe when he undressed her.\"\n\n\"Could the print belong to someone other than the killer?\"\n\n\"Anything's possible, but I got it off her back, so it isn't her own. If she's been captive for a while, it almost has to be the killer's.\"\n\n\"What is it? Index finger?\"\n\n\"Unfortunately, it doesn't appear to be a finger at all. I can get a print off any section of volar skin\u2014fingers, soles, lips, ears. This is a palm. It could be worse\u2014some courts won't admit non-hand or -foot prints. But it could be better, too. Although palms are just as unique in pattern as fingerprints, no one is databasing them.\"\n\n\"So even if your print pays out, we won't be able to run it through VICAP.\"\n\n\"Right. We might use it to verify a suspect\u2014once you have one. But that's it.\"\n\nI nodded. \"Keep looking.\"\n\n\"Will do.\"\n\nI wandered around a bit longer till I found the impression examiner, a woman about my age named Amelia Escavez. She'd joined the force maybe six months before.\n\n\"Whattaya got?\" I asked, crouching at her side. God, this felt good. Back in the swing of things. Doing what I did well.\n\n\"Tire print.\" She tended to be succinct when she spoke to me. Perhaps if I'd ever asked her out to dinner, made a friend of her, she'd be more forthcoming. But of course I hadn't.\n\n\"The killer?\"\n\n\"Possible. He must've used some kind of vehicle to get the body to that plane. Since he couldn't get through the locked gate, he presumably needed something sturdy enough to make it down that steep off-road slope. And the airport officials tell us none of their personnel has had any reason to be out here recently. So . . .\"\n\nShe reached into her field kit, took out a fixative, and began stabilizing the impression. She'd use dental-stone casting or some similar material to transfer the print. I noted that her kit was even bigger than Fielder's. She seemed ready for anything we might throw at her\u2014evidence vacuum, envelopes, bottles, boxes, cutting implements, disposable filters, glass slides, measuring tools, bindle paper, lifters, acetate covers, lifting tape, even an infrared spectrophotometer. Left the electron microscope in the car, I supposed. Looked cool, though, I had to admit. Maybe I should get a kit. What would I put in mine? Rorschach ink blots, multiphasic personality tests, a copy of _The Silence of the Lambs . . ._\n\n\"It's a small print,\" she explained. \"There was a spot of oil, still somewhat damp, on the pavement. That's what caught it.\"\n\n\"Just the one?\"\n\n\" 'Fraid so. I looked for a matching opposite-side impression but didn't get one. This concrete isn't a very good surface for that sort of thing, absent the oil.\"\n\n\"Can you identify the tread?\"\n\n\"I don't have enough to do it by sight, but once I get it into my computer, I may be able to give you a brand or even make. The FBI has a huge tire tread database.\"\n\n\"I need anything you can give me now. Can you at least put me in the neighborhood?\"\n\nShe hesitated. I could see she was reluctant to make an unverified guess that might come back to haunt her if it turned out to be wrong, especially since she didn't have any reason to trust me. But she did it anyway. Good woman. \"Looks like a pickup to me.\"\n\nI nodded. Yes, that seemed right. Would make it easier to transport the body, and you could get it down that sharp slope.\n\n\"Any footprints?\"\n\n\"I wish. Sorry, no.\"\n\n\"You'll get me a copy of that print?\"\n\n\"Sure.\"\n\n\"Lifting material?\"\n\n\"Thought I'd use overlapping tape affixed to white card stock. Soon as it dries a little more.\"\n\n\"Sounds like a winner. Thanks.\"\n\nI stayed another hour or so, chatting up the techs, the ones who would talk to me, and trying to learn whatever I could. For the most part, I just absorbed. The place, the victim, the whole scenario. Tried to get inside the killer's head. What was he playing at? What made him do the things he did? I don't like to admit it, but I was more than a little creeped out. Maybe it was just the effect of being hung over on a body that was already in poor condition, but I couldn't shake this ominous feeling. I mean, I've worked some horrible crimes in my time, but that business with the teeth\u2014who would be capable of that?\n\nAt the edge of the crime scene, I saw O'Bannon motioning to me.\n\n\"I'm on my way back to HQ,\" he explained. \"Will I see you there when you finish up here?\"\n\n\"Sure.\"\n\n\"I'd like it if you could drop by my house tonight. Maybe around nine-thirty.\"\n\nMy eyes narrowed a bit. \"May I ask why?\"\n\n\"Well, I'm not coming on to you, if that's what you're thinking. Bring a chaperone, if you like.\"\n\n\"What's the occasion?\"\n\n\"I want to review the case. Get your preliminary thoughts. You know the press is going to be all over this case. I want to be ready to tell them something.\"\n\n\"And it couldn't wait until tomorrow morning?\" I looked him straight in the eyes. \"You don't want to hear my thoughts on the case. You want to make sure I'm not drunk.\"\n\n\"I'm trying to help you, Susan.\"\n\n\"You think if I'm alone tonight, I'll drink. Do you know how offensive, how utterly\u2014\"\n\nHe cut me off with a harsh glare. \"Don't be stupid, Susan. When people offer you help, take it.\"\n\n\"I don't need help!\"\n\n\"See you at nine-thirty,\" he said curtly. He waved to Granger, and together the two of them left the scene.\n\nWhen I got back to headquarters, I was in for a few surprises. The temporary office set up for me was a desk, an old crappy one, wedged into an alcove just a few feet from the men's room. This not only guaranteed that I would be constantly bombarded with manly odors, but also ensured that every guy in the building would trip over me at least twice a day.\n\nFine. Let O'Bannon play his little games. I was going to solve his case.\n\nFirst thing I did was call that pettifogger of mine and tell him I was gainfully employed.\n\n\"So how much are you going to make as an LVPD consultant?\" Delacourt asked.\n\n\"I'm a little fuzzy on that.\"\n\n\"Not as much as your previous salary.\"\n\n\"Probably not.\"\n\n\"Well, it's better than nothing.\" Man was damn hard to please. \"Look, I got you a hearing. Two weeks from yesterday.\"\n\n\"Two weeks? Why so long?\"\n\n\"In case you haven't heard, Susan, the Las Vegas family courts are swamped, and there was no emergency.\"\n\n\"I think there's an emergency!\"\n\n\"The child is not in danger.\"\n\n\"I think she is. The longer Rachel is forced to stay with Darby and Joan, the more they're going to warp her mind.\"\n\n\"Susan, I had to pull a lot of strings to get two weeks. Would you just ride with it?\"\n\nI closed my trap. \"Anything I can do?\"\n\n\"Yes, since you ask. You could become a model parent. Keep regular hours. Stay sober. I got a report from NDHS that says you're not going to the IOP classes.\"\n\nGoddamn spies. \"I can't very well earn a living and support my niece if I go to classes all day long.\"\n\n\"A sound point, but not one they're likely to be sympathetic to. I'll let you know if anything else turns up. In the meantime\u2014keep your nose clean. For your sake, and Rachel's.\"\n\nAfter that entertaining badinage, I got to work. My desk was already buried in paper. The reports on Murder Victim Two were just trickling in, but O'Bannon had sent over the voluminous reportage on Murder Victim One. I started at the top and tried to become familiar with the case, all the while pretending that I didn't feel as if I'd been buried alive myself, as if my head weren't throbbing, as if I didn't desperately want a drink. Not to get drunk\u2014that had been a stupid mistake and I wouldn't do it again. I just needed a little pick-me-up, something to take away the pain so I could focus on my business. I'd stop after one.\n\nI started with the autopsy protocols for the first murder and what little preliminary information they had provided regarding the second. I'd done this often enough to know I could safely skip the pages of minutiae on the body organs and glands, which would probably not be helpful and which I wouldn't understand even if it were. In both cases, the coroner reported an increase in serotonin and histamine levels. In the first murder that was to be expected. She had a long, painful time to be terrified before she finally suffocated. But it was present in the second victim, too. What's more, the second victim's gum wounds showed much higher histamine than serotonin. So she lived a good while after the teeth were removed. O'Bannon was right\u2014she'd bled to death, aspirating blood making it increasingly difficult to breathe. A slow, painful passing.\n\nWith considerable reluctance, I opened the envelope marked PHOTOS. Death had caught Victim One's face in a hideous rictus, eyes and mouth wide, terrified, like that Edvard Munch painting. She died screaming, with no one to hear her.\n\nThe forensic analysis reports on Murder One were not helpful. The criminalists had vacuumed and grid-searched thoroughly. Even taken the trap from the vents and examined them. Found a few soil deposits, but nothing useful. Organic stains were unavailing. The preliminary victimology reports were even more of a joke. We didn't know who either victim was, making it almost impossible to speculate as to motives or why the victims were chosen.\n\nNow here was an interesting tidbit\u2014Victim Two was pregnant, just barely. Did the killer know? Was that why she had been chosen? Or why she had been killed?\n\nI read the handwriting, ink, and paper analysis that had been performed on the two messages left with the corpses. Although the notes were handwritten\u2014probably a necessity given that many of the characters used don't appear on a standard keyboard\u2014the writer was probably not using his usual hand. A rightie using his left, or vice versa. It was the most common way of concealing handwriting. It would account for the shakiness of the lines, the inconsistency in character size. But that made it impossible for the expert to draw any conclusions regarding the personality being masked.\n\nThe handwriting expert did provide one interesting bit of information: the writer was using a fountain pen, gold-tipped broad nib. In this day and age. When you could find a fifty-cent Bic in any drugstore. He was using a fountain pen and a blotter.\n\nI spent the rest of the workday messing about in the database for psychological profiling of serial killers maintained by the FBI's Behavioral Science experts. I am a huge admirer of the work John Douglas did, interviewing serial killers and cataloging the patterns and similarities in their backgrounds, as well as their modus operandi. But I didn't find much that pertained to the case at hand. With each new piece of information, however small, I got a growing sense that we were dealing with something entirely out of the ordinary, something I had never seen before.\n\nMaybe something no one had seen before.\n\nLas Vegas Metropolitan Police Department\u2014Central Division\u2014was more hectic than M&M World on a Saturday night. Chaos reigned supreme. You'd think they had slots in there, the way people wandered in and out, back and forth. Twenty conversations going at once, not to mention a few scuffles and one flat-out fistfight between a plainclothes vice cop and a young lady of the evening he had in custody. The room was humid and noisome, reeking of sweat and stale coffee and fetid breath.\n\nHow did people work under these conditions? he wondered. What utter banalities these officers were, with their rolled-up sleeves and underarm stains. All they needed were Irish accents and a box of crullers to complete the picture. It seemed unlikely that this crowd could apprehend a purse snatcher. And these were his adversaries?\n\nHe was astonished. And in all honesty, a bit disappointed.\n\nWhere was she, the raven woman? The one who tried to understand.\n\nHe'd chosen his disguise with exquisite care. It had to be subtle; she would detect any major attempt at subterfuge. And he did not want to bury himself so deeply that she could not perceive his true self. Just enough that any formal description she might give at a later time would be useless. He wore a false mustache, a simple bit of misdirection, but one that seemed to alter the entire character of his face. He'd forgone his contact lenses and was wearing the wire-rimmed glasses of his early youth. And in his boldest stroke, he'd darkened his hair. He had considered going blond\u2014he'd always fancied the effete, sensitive poet look\u2014but he sensed that she would be more comfortable with dark hair. Black, like her own. Black like the raven.\n\nHe would be forced to prevaricate, for his own safety, and he wasn't happy about that. A southern gentlemen does not tell tarradiddles. Except, he felt it fair to add, in self-preservation. His appearance itself was a lie, come to that, so what additional damage to his integrity could a few words do?\n\nHe waited for what seemed an interminable time but saw no traces of her. In fact, he saw no female at all, discounting the ones wearing handcuffs. How was he to learn anything about her when he didn't even know her name?\n\nHe approached the front desk clerk. \"Beg pardon, sir. I need to speak to someone.\"\n\nThe clerk looked up. \"Wanna give me a hint?\"\n\n\"It's about the young woman found at the Transylvania.\"\n\n\"Okay,\" he said wearily. \"What about her?\"\n\n\"If you don't mind, I need to talk to someone working on the case.\"\n\n\"Granger isn't here.\"\n\n\"Actually, I need to speak to the woman. . . .\"\n\nThe clerk seemed lost. \"The dead woman?\"\n\n\"No, the one working the homicide.\"\n\n\"Granger hasn't brought in any female detectives.\"\n\nPatience, he told himself. Patience. \"I wonder if perhaps you might be mistaken. I'm quite certain I saw her yesterday at the crime scene.\"\n\nNo reaction from the clerk.\n\n\"I don't recall her name, but she was quite tall. Slender.\" He paused. \"And hair the color of the raven.\"\n\n\"You talkin' about Pulaski?\"\n\n\"Perhaps I am.\"\n\n\"She isn't on the case. Not officially, anyway. She isn't even on the force anymore. She's just been brought in to give advice or something. Weirdos are her specialty.\"\n\n\"Do tell.\"\n\n\"Yeah.\" He lowered his voice. \"Big mistake, if you ask me.\"\n\n\"You don't care for . . . Miss Pulaski?\"\n\n\"Not that I'm one to talk out of school.\" He lowered his voice to a whisper. \"Bitch and a half.\"\n\n\"I'm sorry to hear that.\"\n\n\"She was never exactly Shirley Temple. But since her old man died . . .\" He whistled.\n\n\"Is she . . . coping with her loss?\"\n\n\"Oh, yeah. She's coping.\" He made a drinking motion with his hand. \"In the worst possible way. Look, if you want to talk to her, she's coming up the stairs right now.\" His voice dropped again. \"But if I were you, I'd stick with Granger.\"\n\nHe pivoted and cast his eyes down the staircase that bisected the room. There she was. Hair as black as the night. And those eyes\u2014that magnificent dark eye!\n\n _It was the Eye that transfixed, that vexed me . . ._\n\n\"Are you Lieutenant Pulaski?\"\n\nThe woman peered at him, a quizzical expression on her face. \"Sort of.\"\n\n\"My name is Ethan Jenkins. I need to talk to you.\"\n\nShe gave him a quick once-over. \"I'm sorry, but I'm very busy right now.\" She started to pass.\n\n\"I know. I heard about the second victim on the news. That's why I'm here.\"\n\nShe stopped. \"You know something about the murders?\"\n\n\"I think so, yes. I saw the most recent casualty, just two days ago. At the Tropicana. She was gambling.\"\n\nPulaski's eyes narrowed. \"You're sure about this?\"\n\n\"I am. I saw her picture. It's the same woman. And someone was following her.\" He looked around, frowning at the distracting clamor all around them.\n\nShe pointed her head toward the front door. \"Walk with me.\"\n\n\"So he escaped?\"\n\n\"I'm afraid so,\" he explained. \"I just happened to be in the parking lot at the time and was able to intervene on her behalf. I'm no fighter\u2014I'm an accountant, actually. I think he had some martial arts training.\"\n\n\"We haven't received any reports from the Tropicana.\"\n\n\"I don't think they know it happened. As far as I could tell, no one at the hotel was aware of the incident.\"\n\n\"With all those cameras they have? I'm surprised anyone could escape notice.\"\n\n\"I believe the cameras are in the casinos. Not the parking garage.\"\n\n\"But muggers love to lurk in parking garages!\"\n\n\"Lieutenant, I've been in Vegas long enough to understand that casinos install those cameras to protect the casinos, not the patrons.\"\n\n\"Right, right.\" She pulled a notepad out of her back pocket. \"Do you know if she filed a report?\"\n\n\"I told her she should, and I assumed she would.\"\n\n\"But she didn't. The creep must've gotten her before she had a chance.\"\n\n\"It would appear that way. If only I'd realized. I thought he was gone for good. I never imagined he might come back and try for her again. He must've really wanted the woman's purse.\"\n\nSusan shook her head slowly. \"I don't know what this guy is, but he's not a purse snatcher.\"\n\nIndeed? He remained silent.\n\n\"I don't think he was after her money. And I don't think it was a random selection. It was . . . considered. I think he'd been watching her, investigating her, and she met . . . some kind of criteria.\" She paused. \"I think he _chose_ her.\"\n\nIt was all he could do to keep from kissing her. She was brilliant. Instinctively brilliant. \"But how can you know?\"\n\n\"I can't _know._ Not exactly. I just . . . get stuff. About people.\" She frowned. \"It's hard to explain. It's more a feeling than anything else. A sense of what fits and what doesn't.\"\n\nHis first impression, even as he had observed her from afar, had been correct. She was a very special woman, this Lieutenant Pulaski. Her intuitive powers were great, possibly even equal to his own. She would be a worthy opponent.\n\nOr better yet, a partner.\n\n\"And you're sure this was the same girl we found at the airport?\"\n\n\"Positive. I talked with her for some time.\"\n\n\"But she didn't give you her name?\"\n\n\"No.\"\n\n\"Did she tell you where she was staying?\"\n\n\"I'm sorry, no.\"\n\n\"Well, we'll check the records at the Tropicana. Mention any friends, family?\"\n\nShould he tell her about Warren? Of perhaps even the girl's mother? That would be a bold move, wouldn't it? But too fraught with risk. Lieutenant Pulaski might not have much data to work with at the moment, but she was not stupid. Far from it. He couldn't know too much.\n\n\"I'm afraid we never talked about anything other than the incident.\"\n\n\"Sure. I was just hoping.\" She scrutinized the notes she had scribbled onto her pad in a hand far too tiny and slovenly for him to read. \"Can you give me a description of the assailant?\"\n\n\"I can try. He was a big man\u2014strong, obviously.\"\n\n\"Height?\"\n\n\"Almost six feet.\"\n\n\"Hair?\"\n\n\"Blond. With lots of curls. And he had a beard.\"\n\nHer head tilted to one side. She was considering, he realized, running the description through her mental database. And it didn't fit. Because it wasn't true. But she couldn't know that for a fact. Not yet.\n\n\"I wish we could be sure the man you saw was the killer. But . . .\" Her voice drifted.\n\nThey had rounded a complete city block and returned to the street on which they had started. \"I suppose it's possible this guy had nothing to do with her murder,\" Susan said, closing her notebook. \"But it's still worth checking out.\" She extended her hand. \"Thank you for coming forward, Ethan.\" She paused, not in front of headquarters, but across the street from a place called The Golden Bear. A bar and grill. \"Now, if you'll excuse me\u2014\"\n\n\"Lieutenant?\" He took her hand.\n\n\"Yes?\"\n\nHe glanced across the street. \"That is not what you need.\"\n\n\"Excuse me?\"\n\n\"A little early, don't you think?\" He smiled. \"If you start drinking now, you'll be useless for the rest of the day. Then you'll become angry at yourself for drinking and being useless. Wouldn't it be smarter not to start?\"\n\n\"You're pretty damned impertinent, Ethan.\" He thought she would be mad, but the steam never rose. The corner of her lips turned up. \"But you're right. I guess I needed to hear someone say it.\" She checked her watch. \"Anyway, I've got an appointment with an old friend. Thanks again.\"\n\nHe hesitated. \"I . . . wouldn't mind seeing you again sometime, Lieutenant.\"\n\nShe grinned. \"Call me Susan.\"\n\nHe watched as she made her way up the steps and away from him. He would see her again. He knew it. But there was much work to be done in the meantime. Another offering to be secured\u2014according to the prophet, a trinity was necessary to bring about the holy objective.\n\nAnd he had to find an axe.\n\nHis plan would move forward. And Susan Pulaski would be a part of it. Onward unto glory, now and forever.\n\nI was still thinking about that distinctly decent-looking witness when I arrived\u2014perhaps I should say descended\u2014into Colin's office.\n\nHe peered across his cluttered desk at me, one hand still on the computer keyboard. Was he umbilically attached to the thing?\n\n\"You want to consult with me? On a murder case?\"\n\n\"If you're willing.\"\n\n\"Well, sure, yeah. I mean, I guess.\" He rubbed his hand through tousled hair that looked as if it hadn't been combed for days. \"But why me?\"\n\n\"Because of all David's friends, you were the . . .\" Nerdiest? \"The best at solving puzzles.\"\n\n\"And you think that's what these are?\" He took the photocopies of the two messages left at each of the crime scenes. \"Anyone else working on this?\"\n\n\"Chief O'Bannon has assigned his best and brightest, but I don't think they have any real expertise in this field. He also faxed copies to the professional code breakers at the FBI and even cryptanalysts at the CIA\u2014but so far, they got nothing. He's talking about running the messages in the newspaper, a prospect I really dread.\"\n\n\"Why?\"\n\n\"Because the press will eat this up with a spoon. They've already gone big with the photos of the victims. These coded messages will take the story right into comic book land. The press will glamorize them and hype them and make it all the more difficult to conduct a serious investigation.\"\n\n\"It may just be delusional psychotic ranting.\"\n\n\"I'm hoping not.\"\n\n\"Why?\"\n\n\"If it's gibberish, as my superior suspects, it's not going to be any help to me. But if it's a message, something the killer so desperately wanted to say that he left behind a potentially incriminating note . . . well, that could tell me a great deal.\"\n\nColin laid the copies flat on his desk and stared at them. He was wearing a T-shirt and sweatpants, mismatched, but he worked at home and I suppose if you know you're not going anywhere you don't have to dress for it. His posture recalled a vulture, his neck craned over the desk, his glasses so thick he could probably perform microsurgery without additional instrumentation. David had met Colin in college and they'd stayed in touch after, right up until David's death. They weren't best friends\u2014their tastes were worlds apart, Colin being more cerebral\u2014but they were close enough that I knew him, and I knew what he did for a living, too.\n\nHe created puzzles. Crosswords, mostly, but also acrostics and word searches and this godawful impossible wordplay-infused variant of traditional crosswords called cryptics, clearly the products of demented brains. Best of all, I knew he considered himself an expert on cryptograms. An entire shelf on the wall behind him was dedicated to codes and ciphers.\n\n\"Well, if it is a code, it isn't a simple substitution code, I can promise you that,\" he said about a minute later.\n\n\"Simple substitution code?\"\n\n\"Yeah. One letter representing another. Like the cryptograms in the newspaper. Easily solved by reference to letter frequency and patterns of orthography.\"\n\n\"Orthography?\"\n\nHe grinned. \"That's what the nonpuzzling world calls _spelling._ \"\n\n\"And how do you know it isn't one of those . . . simple substitution codes?\"\n\n\"Because if it was, I would've solved it already.\" He continued staring at the pages. \"There are more than twenty-six characters in use here, which also rules out a simple substitution. Some of these symbols aren't letters at all. Decoding is also complicated by the fact that there do not appear to be any breaks for sentences or words. The symbols are grouped in large blocks, and I rather suspect that the blocks may not be in the proper order. Looks like it may contain some decoy characters, too.\"\n\n\"Decoy characters?\"\n\n\"Right. Blanks. I see that _Q_ appears in here twelve times, far more than any other letter, which would suggest that it represents _E._ But it doesn't.\"\n\n\"How do you know?\"\n\n\"Because if it did, I would've solved it already. No, it's more than just a cipher.\"\n\nI wondered if I should subpoena the subscription list to his puzzle magazine. There had to be something wrong with people who spent their spare time busting their brains over stuff like this. \"If it isn't a substitution code, what is it?\"\n\n\"Well, I'd say there's a remote possibility it's a translated anagram.\"\n\n\"Huh?\"\n\n\"Letter scrambles. Like those Jumbles in the paper. He takes the message, then rearranges the letters at random. In a message this long, it would come out looking like gibberish.\"\n\n\"How would I ever solve it?\"\n\n\"You wouldn't. Or me, for that matter. A computer might, once it figures out what to do with the nonalphabetic entries. Might at least be able to generate a menu of workable solutions.\"\n\n\"Any other possibilities?\"\n\n\"Oh, there are lots. Codes are literally as old as language, and over time people have devised a lot of devious ones. It's possible that breaking the code requires reference to some external text. Like you have to know what page of the King James Bible to use as a reference key. Those were popular during World War I. Or it's possible the solution requires a code-breaking machine, like Enigma in World War II.\"\n\n\"I doubt if that's the case.\"\n\n\"Why?\"\n\n\"Because if this code requires either of those two external devices, we have no realistic chance of solving it. And I have to think at least some small part of this guy wants us to solve it. Otherwise, why would he leave it? He wants it to be hard. He wants us to appreciate his brilliance. But eventually, he wants us to read the message.\"\n\nHe nodded. \"You know, there is precedent for this sort of thing.\"\n\n\"Codes?\"\n\n\"Left behind by psychopathic killers, yeah. You heard of Zodiac?\"\n\n\"Of course. Studied him in school.\" I snapped my fingers. \"He left messages, too, didn't he?\"\n\n\"Yup. Coded. His crypts have appeared in some of the puzzle magazines. They were insidious. Stumped all the experts, including the government. Three of the four were never solved. Only one was. As I recall, it was a schoolteacher who finally cracked it, some regular Joe who saw the codes in the paper and worked on them in his spare time. Took months.\"\n\n\"I don't have months, Colin. This guy's on a killing spree.\"\n\n\"Understood. I'll do my best.\" He shrugged. \"Maybe I just think this because I want to think it, because it would be more fun, but I don't think this is gibberish. I don't think it's the work of an amateur, either. I suspect your killer knows something about codes. A little, anyway. My hunch is there's a message hidden in there, but it's incredibly complex. Different. And it's going to take a different kind of brain to figure it out.\"\n\n\"Then I've come to the right place.\"\n\nHe sat up, stretching. \"I'll give this top priority. I'll even put off doing today's _New York Times._ \"\n\n\"I'm surprised those puzzles are any challenge for you.\"\n\n\"I do them without the grid.\"\n\nI blinked. \"You mean, without the little white and black boxes?\"\n\n\"Right. I work out the grid on my own, from the clues and their numbering.\"\n\n\"If you keep talking like this, Colin, I'm going to move you to the head of my suspect list.\"\n\nHe grinned. \"Hey, you holding up okay? I heard you were having some problems.\"\n\nMy chin rose. \"None to speak of.\"\n\n\"If you need anything\u2014\"\n\n\"I don't need any help.\" I paused. \"But thanks.\"\n\n\"Okay.\" He reached down and pulled up his socks, which didn't match. God, but this man needed a wife. \"You know, when David died like that\u2014it hit us all pretty hard. But I have to think it hurt you most of all.\"\n\n\"Nothing I can't handle.\" I pushed myself out of his chair, bringing this line of conversation to a dead stop. \"If you get anything on those codes, let me know, okay? The sooner, the better. No telling how many young women's lives may be at stake until we catch this whack job.\" I left as I came in, all business, no crack in the exterior.\n\nI couldn't afford a crack, and I damn well resented his trying to pry one open. We live in such a Jerry Springer world\u2014everyone wants to go public with all their problems. Whatever happened to the virtue of circumspectness? When did we become such a nation of whiners?\n\nBesides, I had work to do.\n**CHAPTER EIGHT**\n\nI was pissed as hell about being forced to report to O'Bannon's house just so he could make sure I'd been behaving myself. At the same time, I knew if I didn't appear, he'd jerk my tenuous little consulting position like the handle on a one-armed bandit. For now, I had to play it his way.\n\nI left the top down and stoked myself on the night air. Did I mention that I love this city? People talk about New York and its nightlife, but for my money, Vegas has it beat. People crawled down Fremont till the wee hours of the morning, and for the most part, they enjoyed themselves, acting like kids, blowing money they don't need, being royally entertained. Granted, our shows may not have the sophistication of Broadway, but people came to Vegas to have fun, not to get clubbed over the head with Pulitzer Prize\u2013winning angst. And for the most part, the tourists were nice folks. Writers always portray them as seekers of sin, but what I see is mostly plain, decent folks who want to get away, play, gamble a little, gorge themselves at a buffet, and sleep sweet dreams.\n\nWhile I drove, I called Lisa on my cell. To my surprise, she was home.\n\n\"How's it hanging?\"\n\n\"Oh, fine. I'm washing my nylons.\" Which was code for no date. \"I checked by your place but you were in absentia.\"\n\n\"On my way to O'Bannon's. He's got me working a new case.\"\n\nShe gasped a little. \"Not those girls who\u2014\"\n\n\"That's the one.\"\n\n\"Oh, geez, Susan. Do you think you're ready for this?\"\n\nI tried not to take offense. Anything she said arose from her concern about me. \"Best thing. Keep me off the streets. How was your big date last night?\"\n\n\"Ohh.\" I didn't have to see her to see her face falling. \"Disappointing.\"\n\n\"Not a tiger?\"\n\n\"More like a lap dog.\"\n\nThat was a new one. \"I'm not sure I\u2014\"\n\n\"Visible tongue. Before my mouth was even open. I think I'm going to become a nun. Stop by on your way back?\"\n\n\"I . . . it'll probably be too late. Definite date for tomorrow?\"\n\n\"All right. You're sure you're all right?\"\n\n\"Couldn't be better. Couldn't be better.\"\n\nBy the time I got to O'Bannon's, it was almost ten, but I hoped he'd cut me some slack since I'd been working like a busy beaver. I was surprised to find that kid of his on the front porch\u2014sort of. He stood just off the edge of the concrete, about three feet from the door. His entire body was stiff, shoulders hunched, like he'd just been injected with a paralytic drug.\n\n\"Hey, Darcy,\" I said, flashing my best smile.\n\n\"Do I know you?\" he said, but his expression almost immediately brightened. He remained stiff. \"I like your voice. You're Dad's friend from work.\"\n\n\"Yup. He inside?\"\n\n\"Yes . . .\"\n\n\"Shall we go see him?\"\n\nIt was difficult for him to speak, but when he finally started, the words burbled out in that strange voice of his, too loud, the inflection all askew. \"Would you be afraid of spiders? Because some girls are afraid of spiders. Did you know some, a lot, a lot of girls are afraid of spiders?\"\n\n\"I've heard that.\"\n\n\"So, if, then, are you afraid of spiders?\"\n\n\"Nah. I'm not afraid of anything.\"\n\n\"Do you think that would be poisonous?\"\n\nHe pointed downward. I didn't see anything. \"Can you be more specific?\" I moved in closer and eventually realized there was a small gray spider on the front porch, barely noticeable. \"I don't think it can hurt us, Darcy.\"\n\n\"Did you know there are four kinds of poisonous spiders indigenous to North America?\" He was lecturing me, but he never made eye contact. \"The widow spiders, the recluse spiders, the hobo spiders, and the yellow sac spider.\"\n\n\"I'm fairly certain this little fella is none of the above.\"\n\n\"You can never be sure. Sometimes they can trick you. Did you know that sometimes spiders can try to trick you?\"\n\n\"I didn't know that.\"\n\n\"I think that maybe we should go around to the back door.\"\n\n\"Not necessary. I'll take care of it.\" I raised my foot, hovering over the offending obstacle.\n\n _\"No!\"_ Darcy fairly screamed. He raised his hands, flapping them in the air. \"Don't kill it!\"\n\n\"I thought you were afraid . . .\"\n\n\"I don't like to hurt things. I don't think anyone should hurt anyone, do you?\"\n\n\"But . . . it's a spider.\"\n\n\"It's still alive. Isn't it still alive?\"\n\n\"But eventually we need to get inside. Don't we?\"\n\n\"Maybe you could just . . . just . . .\"\n\n\"Capture and release?\"\n\n\"That would be good.\"\n\nI picked up a fallen leaf, slid one edge under the spider, and tossed it into the hedges.\n\n\"Will he be all right there?\" Darcy asked.\n\n\"That's its natural environment,\" I assured him. \"It'll be like a kid in a candy store.\" Judging from Darcy's nonresponsive expression, I needed to modify my clich\u00e9. Abstract language was lost on him. \"It'll be like\u2014what's your favorite room in the house, Darcy?\"\n\n\"The library.\"\n\n\"It'll be like you with a good book in the library.\"\n\n\"Oh.\" His head twitched a couple of times. \"Good.\"\n\n\"Can we go in now?\" I looked back at the front door and found O'Bannon standing there, watching. \"Chief.\"\n\nHe pushed open the screen door. \"Come on in. Darcy, why don't you go finish your book?\"\n\n\"Oh.\" He seemed reluctant to leave. \"Okay.\"\n\nOnce we were alone, O'Bannon showed me into his den.\n\n\"Sweet boy you've got.\"\n\n\"That's one word for it.\"\n\n\"Wouldn't even hurt a spider. That's pretty endearing, in this day and age. Most guys his age spend the evening pretending to blow people to smithereens in video games.\"\n\n\"He's always been like that. Entirely nonaggressive. Doesn't want to hurt anyone. Kids at school used to rough him up pretty bad. But he never struck back. Never did anything to anyone. Practically a saint.\" He stopped by his desk and gave me a searching look.\n\n\"You going to sniff my breath?\"\n\n\"Don't have to.\"\n\n\"Because you trust me?\"\n\n\"Because I've been on the police force thirty-four years, which unfortunately gives a man a lot of experience with alcoholism. I can spot a drunk at ten paces.\"\n\n\"Ooh. Scary.\" I jerked my thumb toward the other room. \"So, that boy of yours. Autistic?\"\n\nHe raised an eyebrow. \"Not bad, Pulaski. Most people think he's retarded. Or just weird.\"\n\n\"Well, I am a trained professional. I've worked with autistic children.\"\n\n\"I guess that gives you an advantage.\" I could tell this wasn't something he talked about often. At the same time, I got the impression that while he was accustomed to keeping it private, it might be a comfort to talk about it with someone who had some understanding of the condition. \"Most people, all they know about autism is what they got from that Dustin Hoffman movie, where he acted all weird and twitchy\u2014but not particularly autistic. People looking for a hyperactive toothpick counter aren't going to peg Darcy.\"\n\n\"He must be very high-functioning. He's obviously smart, and he speaks clearly enough. He interacts. He's in our world, at least to some extent.\"\n\nO'Bannon nodded, dropping onto the large sofa. \"It wasn't always that way. At the age of three he was profoundly autistic. Didn't answer questions, showed no emotions, was totally withdrawn. Took years of aggressive behavioral intervention programs to get him where he is today. Worked with a guy named Dr. Lovaas out in L.A. for years. Made a huge difference.\"\n\n\"That's controversial therapy, isn't it?\"\n\n\"It shouldn't be. Got too many damned romantics trying to pretend that there's something special or magical about an autistic kid's private world. He's in his own little paradise, they say. Why do we have to force him to conform, to live in our world? What bullshit. Anybody who's ever spent time with a seriously autistic kid knows better. They're confused, frustrated, unhappy, isolated. Bringing them into our world, trying to make it possible for them to interact, to be productive, is a gift.\"\n\n\"I'm sure you're right. He seems very knowledgeable.\"\n\n\"Well, he remembers everything he ever read. Wish I could say the same for myself. Autistics are known for their prodigious memories. When Darcy's focused, he's incredible. The trouble is getting him to focus, getting him interested, giving him the proper incentive. He's flooded with stimuli. His head is like a radio receiver getting messages from fifty different places at once. It takes a lot of concentration to sift through that, to focus on one particular line of thought.\"\n\n\"Aren't there drugs that could help him?\"\n\n\"No. Everything his docs ever tried just made him listless. Dreary. I'd rather have an autistic boy than a zombie.\" He drew in his breath. \"I've had to face facts. He'll never be normal, no matter what I do.\"\n\n\"Well,\" I said, depositing myself on the sofa, \"normal is overrated.\"\n\nO'Bannon's gaze turned inward. \"You can't imagine how hard it was on Connie and me. Our first child. By the time he was two, he was reading. Honest to God. Knew the alphabet, could count as high as you want, remembered everything you told him. Sang, even played the piano a little. Had perfect pitch. We thought we had a little genius on our hands. Then somewhere around his third birthday, it all started to go south. He didn't come when called. He disappeared, hid. All his incredible speech disappeared.\"\n\n\"That's frequently the way neurological disorders work. At first they manifest as prodigious intellectual abilities. Then the other shoe drops.\"\n\n\"Yeah.\" O'Bannon kept his face stoic; everything I got, I got from the eyes. He did a good job of keeping it in. But I suppose he'd had a lot of practice. \"He does have some special abilities. Memory. Math. Reads fast. Plays the piano\u2014and he never had lessons. Does the whole puzzle page in the _Courier_ in less than five minutes.\"\n\n\"Okay, now I'm starting to hate him.\"\n\nO'Bannon grinned. \"He really tries to get along in the world, to understand what's going on around him, hard as it is for him. He's done a hell of a lot with the hand he's been dealt.\"\n\n\"But I notice he's still at home. What is he, twenty-one, twenty-two?\"\n\n\"Twenty-six. It's the angelic countenance that throws you off. Autistics are renowned for their sweet good looks. As for moving out, I don't know if he'll ever be able to live independently. He sure isn't now. Most autistics end up in some kind of home, but . . . I don't know. I couldn't bear it. He's my only family.\"\n\n\"I get that.\"\n\n\"I had hoped that one day, with enough therapy, he might be able to do some kind of work. But that's probably not realistic. He helps out at a day care center. He loves to work with small children. I think he's comfortable around them in a way he'll never be with adults. But . . .\" His eyes wandered. He was feeling uncomfortable. I could tell he wanted to change the subject. \"So what've you got on this case? Solved it yet?\"\n\n\"Give me another ten minutes.\"\n\n\"Have you at least got a working theory?\"\n\n\"Guy's hard to get a grip on. A lot of the information I've received is contradictory. And it doesn't fit the standard profile of a psychopathic sexually motivated serial killer.\"\n\n\"Are you sure that's what this perp is?\"\n\n\"I can't imagine anyone burying a woman alive, or tearing out all her teeth, for any logical reason. Can you?\"\n\n\"But does he have to be sexually motivated? Neither of the girls was molested.\"\n\n\"He's probably impotent. That's not uncommon. They can't get off the normal way. They get gratification from killing and torturing the helpless. Any progress on identifying the victims?\"\n\nHe shook his head. \"Nothing so far. Eventually someone will miss them. But it might take a while.\"\n\n\"If we knew how the killer was selecting his victims\u2014or why\u2014we'd be better able to protect the populace. This has already gotten a lot of press\u2014\"\n\n\"And it's going to get a lot more.\"\n\n\"Right. So we need to be able to tell them something about the man.\"\n\n\"Are we sure it's a man?\"\n\n\"Serial killers almost always are. Comes with the testosterone.\"\n\n\"But you said this killer doesn't fit the standard patterns.\"\n\n\"True enough.\" Leaning back, I noticed that Darcy was crouched in the entryway. How long had he been there? \"Need something, Darcy?\"\n\n\"W-W-Would it be okay if I sat with you guys? I think that would be fun. Wouldn't that be fun?\" He walked over to the sofa and sat between us, brushing shoulders.\n\nHis father grunted. \"Darcy, we're talking about a case. You should go to your room.\"\n\nDarcy was crestfallen.\n\n\"I don't see any reason he can't stay,\" I cut in. \"Maybe he'll inspire us.\" I don't know why I did it, except that I could see that he really wanted to be there with us. And I've always had a soft spot for Shaggy.\n\nO'Bannon spoke as if he weren't there. \"He isn't usually this social. Especially with strangers.\"\n\n\"But we are not strangers,\" Darcy said, looking at me. \"Are we strangers? Don't I know you?\"\n\n\"Yes, of course you do.\"\n\n\"That's what I thought. Then we are not strangers. What are we talking about?\"\n\nI suppressed a smile, then leaned forward so I could see the chief. \"I don't suppose any of your experts have had any luck on the coded messages.\"\n\n\"No,\" O'Bannon grunted, reaching back for one of the two photocopies. \"They don't think they are messages. Which of course is the obvious response when you aren't able to decode them.\"\n\n\"I took them to a geek friend of mine. I'm hoping he'll figure something out. I'd really like to know what they say, but\u2014\"\n\nDarcy jumped in. \"W-W-Would it be okay if, do you think, could it be okay if I took a look at them?\"\n\n\"Darcy,\" his father said, \"this is police business. It might be better if you watched one of your videos.\"\n\n\"Please,\" he said. \"I want to read them.\"\n\nRead them, I noticed he said. Not solve them. Read them. \"Why don't we give him a shot?\" I took the copies and handed them to Darcy.\n\nO'Bannon's face was pained. \"Susan . . .\"\n\n\"You said he was good at puzzles. Maybe he'll see something the experts missed. Maybe he'll spot some clue that will make it possible for the experts to\u2014\"\n\nDarcy cut me off. \" 'Deep, deep, and for ever, into some ordinary and nameless grave.' \"\n\nO'Bannon and I both looked at him. \"What?\"\n\nHe repeated it.\n\n _\"What?\"_ his father said, irritated. And then his voice softened: \"Is that a poem or something?\"\n\nDarcy looked up nervously. \"I\u2014I\u2014I\u2014I think that's, what you were, you were, can't you read it?\"\n\nI took the photocopy he was holding, the one that had been stuffed inside the coffin with the girl who was buried alive. \"You're saying that's the translation of this message?\"\n\n\"That's what it says.\"\n\n\"You decoded it?\"\n\nHe gazed at me with that deceptively vacant expression. \"That's what it says.\"\n\n\"Darcy . . .\" His father sighed. \"We could all make up something that could be the message. Sometimes it's fun to pretend\u2014\"\n\n\"I\u2014I\u2014I\u2014\" I noticed his stuttering became much more pronounced when he was speaking to his father. \"I read it.\"\n\n\"That's impossible. You looked at it for, what? Twenty seconds?\"\n\n\"Does Darcy normally make things up?\" I asked.\n\n\"Well . . . no.\"\n\n\"Does he tell lies?\"\n\n\"I don't think he knows how.\"\n\n\"Is he given to flights of fancy?\"\n\n\"He wouldn't understand what a flight of fancy was.\"\n\n\"Probably never engaged in imaginative play as a child, right? No make-believe. No cowboys and Indians. That would be typical for kids with his . . . situation.\" I turned back to Darcy. \"Say it again. Read it.\"\n\nHe glanced down at the paper. \" 'Deep, deep, and for ever, into some ordinary and nameless grave.' \"\n\n\"Sounds like something that belongs in a coffin, doesn't it? Darcy, is it a substitution code?\"\n\n\"What?\"\n\n\"Does one letter\u2014or symbol\u2014stand for another?\"\n\n\"I guess\u2014I mean, no, not really. Well, sort of.\"\n\nI resisted the urge to pound my head against the wall. \"Darcy, can you explain to me how you can read this?\"\n\n\"Can you, do you\u2014I mean\u2014I just read it.\"\n\nNo cheese down that tunnel. \"I wonder what it means. Maybe if I went on the Internet\u2014\"\n\n\"It's from a book.\"\n\nI gaped. I mean, it was incredible enough that this kid might solve the puzzle the experts couldn't. But now he was going to identify the quotation?\n\nHe jumped off the sofa and raced to the wall behind the desk, one that was covered with books. It was as if someone had flipped the ON switch; suddenly, he was alive, more than alive. His eyes darted back and forth, his hands flapped frenetically. His fingers brushed against the spines of the books as if they were his friends.\n\nIt didn't take him long. He pulled down a large black volume, crouched on the floor, and began rifling through the pages. O'Bannon stood to the side, watching.\n\n\"Here it is,\" Darcy said, barely a minute later.\n\nI knelt down beside him and read the passage he indicated. There it was. Verbatim.\n\n\"Darcy . . . how did you find it so quickly?\"\n\n\"I\u2014I read it once.\"\n\n\"Recently?\"\n\n\"No. But I remembered.\"\n\nI flipped the book around and looked at the spine. _Tales of the Grotesque and Arabesque,_ it said. The subtitle was: _The Poems and Stories of Edgar Allan Poe._ The quote he had found was from one of the stories: \"The Premature Burial.\"\n\nOf course.\n\nBut what was the connection to the second story? That awful business with the teeth . . .\n\n\"Darcy, I want you to look at the second message.\"\n\n\"I already did.\" He held it in his hands. \" 'In the multiplied objects of the external world I had no thoughts but for the teeth.' \"\n\n\"Do you recognize that quote? Is it from\u2014\"\n\nHe was already turning the pages. Less than thirty seconds later he found it. In a short story I'd never heard of, titled \"Berenice.\" I flipped to the end. _. . . there rolled out some instruments of dental surgery, intermingled with thirty-two small, white, and ivory-looking substances that were scattered to and fro about the floor._\n\nThe story was about some maniac pulling out a woman's teeth. And letting her bleed to death.\n\n\"I told you he had a prodigious memory,\" O'Bannon said. \"Eidetic. Not only can he recite back word for word something he read years ago, he can tell you what page it was on.\"\n\nI'd heard of such things in school, but never actually encountered it. \"And he reads a lot?\"\n\n\"Constantly.\"\n\n\"That's . . . an amazing gift.\"\n\n\"Yeah. With virtually no practical application. Or so I thought.\" O'Bannon frowned.\n\nI held the book up. \"I read a little Poe in college. Kind of liked it, as I recall. But I didn't see the pattern.\"\n\n\"And now you do?\"\n\n\"I'm beginning to. Thanks to your son.\" I patted Darcy on the shoulder. He immediately reacted to the touch of my hand. It was sort of like he was wriggling away and sort of like he was snuggling against it. \"I hope you're not one of those collector types who won't lend reading materials.\"\n\nO'Bannon shook his head. \"It's just a book.\"\n\nI hope he agreed that this interview was over. Because I had some real work to do now and I wanted to get to it. The book was more than a thousand pages long, for God's sake. It would take me hours to get through it, to see what other clues might be in there. I needed to reexamine everything in light of this new lead. And then . . .\n\nAnd then I got an idea. \"Is the first crime scene still restricted?\"\n\n\"Sure. We've got men posted.\"\n\n\"Good. I want to go back out there tomorrow morning.\"\n\n\"Okay.\"\n\n\"Would it be all right if I bring Darcy?\"\n\nDarcy's eyes lit.\n\nO'Bannon's didn't. \"Are you out of your mind? What for?\"\n\n\"He might be useful. He sure as hell was tonight. There might be more Poe clues lying about. Stuff only he would spot.\"\n\n\"Susan.\" He sidled closer to me and lowered his voice. \"Darcy may look like a man, but inside, he's a little boy. You saw him with that spider. What's he going to do at a crime scene?\"\n\n\"The body has been removed.\"\n\n\"But still\u2014\"\n\nI hated this business of talking about Darcy as if he weren't present, when he was standing barely two feet away. But I had a strong feeling that I was right about this. I couldn't even explain why, not coherently. But when you're Empathy Girl, you learn to trust your instincts. \"You said you wished he could learn to do something productive. Hold a real job. So humor me here. Maybe you've got a budding detective on your hands.\"\n\n\"Susan . . .\"\n\n\"As I recall, Sherlock Holmes was pretty odd himself.\"\n\n\"Susan!\" I was not prepared for his anger. \"You've known Darcy for what? Ten minutes? I've been living with him for twenty-six years.\" His voice dropped. \"There's no way in hell he could cut it as a cop.\"\n\n\"I'm not asking you to give him a badge. Just let him tag along. Humor me.\"\n\nDarcy jumped in. \"C-C-C-Can I go, Dad? I would like, would you, I could, I could be good. Can I go with that one?\"\n\nHe gave me a long look. \"Don't make me regret bringing you in on this case, Pulaski.\"\n\n\"I won't. Darcy\u2014pick you up at nine in the morning.\" I winked. \"Don't be late.\"\n\nSee I knew I could help I knew I could help if only he would let me but he wouldn't but the girl did and the girl's name is Susan I heard that was her name and she was nice to me just like she was before. There was a girl at the clinic who was nice to me like that and she was pretty too but not as pretty as Susan and she told me to read one of her music books so whenever she forgot the words I could tell her and she was nice but I think maybe Susan is nicer and smells better. She has something funny about the way she smells but it's not so bad I remember people's smells and everyone has a smell if you smell hard enough. I'm glad I could remember those stories like I did and my dad didn't he read the stories too but he forgot and I remembered. _You shouldn't be reading those horrible stories. You should be reading a nice book like_ The Hardy Boys _or_ Two Years Before the Mast. Susan says I could go see where they found this lady's body and I don't like killing I think it's mean to hurt things but if I go with her it would almost be like I'm a policeman and I know that would make my dad happy even though he pretends like he doesn't care whether I'm a policeman but he does. I hope it's not too gross. I don't like gross. I would never hurt anyone, no matter what.\n\nMostly I want to go because I'll go with Susan. I like Susan. Susan is pretty even though she has a cigarette burn on the back of her right pant leg and she bites her fingernails so much two of them have been bleeding. Susan is babies and sugary and I like the way she flips her hair back when she's being funny even though I don't understand the joke I can tell she's being funny and she smiles at me and lets me sit next to her and I think she must like me and that's good because I know I like her.\n**CHAPTER NINE**\n\nHe felt such intense revulsion that it became difficult for him to breathe. He was physically ill. He knew his face was ashen, and he feared he might soon relinquish custody over his lunch. It was so disturbing, so depraved.\n\nCertainly he had expected to be offended. But he had no idea how bad it could be. Sexual relations were a gift given us for the perpetuation of the species, not, he thought, a commodity to be bought and sold. But here, in this Haunted Palace of a sort Poe never imagined in his most fevered dream, it was all garishly on display, everywhere he turned. He had never seen so much unclothed flesh in his entire life\u2014and it sickened him. Everything here sickened him.\n\nFrom the start he had understood that his visit to Nighthawks was one of duty, not pleasure. Vegas sex clubs were notorious, and this one had a reputation worse than most. Its dark ambience, the decorative whips and chains, bespoke a debased sensibility with a strong sadomasochistic bent, inimical to all standards of decency. Not a place for the avatar of the prophet. To begin with, there was the music\u2014which was not at all musical. How could this electronic rap dissonance be music, which by definition is a melody played in rhythm and in counterpoint to a harmony? Where was the melody in this hip-hop mishmash? It was just sound, mindless decibels, played blaringly, unbearably loudly. And the light was blinding\u2014silver shards glittering all about him, reflecting off the mirrored walls and the discoth\u00e8que balls on the ceiling. It was a grotesque de Sade bacchanalia, all justified by suggestions that indulgence and degenerate fantasy fulfillment were salubrious for the psyche. Well, he did gainsay it, as would any decent soul with an eye on Dream-Land.\n\n\"Good evening,\" said the woman at the door, who was wearing a skintight black leather bodice exposing extraordinary mammarial engineering. \"I am the mistress of pain.\"\n\n\"Good evening to you, madam,\" he replied in his most elegant southern accent.\n\nShe rammed a riding crop under his chin. \"Your heart's desire can be yours. All you need do is ask.\"\n\n\"Most obliging.\" He removed the crop and stepped inside. The d\u00e9cor reminded him of those hideous films of the 1960s purportedly based upon the prophet's stories. Victorian furnishings, faux marble pillars, red curtains, padded sofas and love seats. A throne at each table, such as it was. Waiters dressed in silk Italianate tunics. He almost expected to see Vincent Price emerge from behind the drapes. But in fact, the most noteworthy figures inside were women, naked or all but. He didn't object to nudity in and of itself. But it was never meant to be distorted and turned into a weapon, much less an industry. He was surrounded by unclad women, more than a hundred of them in bikinis, G-strings, negligees, all manner of exiguous attire. Some wearing nothing more than a few carefully cantilevered scarves. None of them much older than their teens. Undulating and thrusting and rubbing and pressing. Trying to excite the worst of passions. Parading their sex for the entertainment of the unworthy.\n\nHe staggered through the narrow corridors, his mouth dry, searching for a spot with an open seat and a modicum of oxygen. The music, the smoke, the cachinnation, and the Caligulan revelry all assaulted and oppressed him. Most of the rooms had stages upon which young women removed their clothes in time to the rhythm of that relentless music. He saw one stage\u2014he couldn't help but look\u2014with an uncommonly limber woman spread across the floor, twisting and writhing like a snake, hands flat, breasts pressed against the stage, her thighs locked around the head of a middle-aged man in a blue leisure suit. In some of the smaller, more private alcoves, women performed one-on-one, straddling the men's laps, rubbing themselves against their patron's personal areas for his despoiled gratification.\n\nHe was tempted to run outside, retrieve the axe from his truck, and bring them all to account for their crimes against decency.\n\nBut that was not the plan. He pressed his hand against his forehead, forcing himself to maintain focus. He had a destiny to fulfill, and he would not shirk it.\n\nHe found an empty chair wedged between two young men in matching shirts, both in the throes of lap dances. He tried to make himself comfortable, but the girls on either side constantly poked him with their stiletto heels or other protuberances. They giggled, smiled, then returned to their business. Their _business._\n\nA woman wearing a red lace teddy appeared before him. She had no concept of personal space\u2014or perhaps she did\u2014and stood so close to him that the tips of her fairly enormous and probably artificial breasts touched his face.\n\n\"You look as if you could use a friend.\"\n\nHe tried not to stammer as he spoke. \"We could all use a friend.\"\n\n\"I'd like to be yours.\" She had vivid red hair\u2014not natural, he felt certain\u2014parted in the center, and a mole strategically positioned just below her lower lip. He rather suspected that wasn't natural, either. She appeared to be about twenty, which in this place made her a senior citizen. \"Can we do business?\"\n\n\"I'm looking for a girl.\"\n\n\"That's why I'm here.\"\n\n\"No, you don't understand. I'm looking for a particular girl.\"\n\nHer smile faded a few notches. \"Don't be put off by the laugh lines, Skippy. I'll rock your boat like it's never been rocked before.\"\n\n\"I'm sure, my dear, but\u2014\"\n\n\"Just give me a chance.\" She pressed a knee into his lap and leaned closer. \"I know what you want.\"\n\n\"I don't believe you do.\"\n\n\"Trust me.\" She squeezed.\n\n\"Stop that!\" His voice came out much louder than he intended as he slapped her hand away. Fortunately, the music was so thumpingly loud that even his immediate neighbors did not notice. He took several deep, cleansing breaths, trying to regain his genteel demeanor. \"Listen to me. I am looking for a specific girl who works here. Her name is Lenore.\"\n\nThe redhead arched an eyebrow. \"You like them young, don't you?\" She pulled away. \"What else is new? Give me a minute, slick.\"\n\nHe waited. While he did, the young man to his left apparently reached climax, shouting and bellowing and putting a very satisfied expression on the face of the purposeful titian-haired teenager who climbed off his lap. Money changed hands, a lot of it.\n\nAnd then he saw Lenore. She was an Asian girl, as he'd known, but her hair was dyed blond. Or perhaps it was a wig? She was much smaller than her predecessor, and younger. Almost a child. Poe would've loved her. He thought he perhaps loved her himself, in his way.\n\n\"April said you wanted me?\" she said with a ruby-red pout.\n\n\"She was correct.\"\n\n\"Okay, so a table dance is two hundred, all right? You want anything more, we negotiate.\"\n\nHe gazed at her, the impossibly rouged cheeks, excessive bee-sting lipstick, breasts like pomegranates. She was wearing a tight red bustier with dragons embroidered on each side. She was a lovely thing, delicate as a rose blossom.\n\nHe had been right. She was the offering. And the third would fulfill the prophecy.\n\n\"This may seem odd to you, dear,\" he said, oozing gentility, \"but all I want to do is talk.\"\n\n\"You like to watch. That's okay, I get it.\"\n\n\"No, ma'am. Listen to me carefully. I want to talk. With you.\"\n\n\"Believe it or not, mister, that's about the only thing we're not allowed to do here. They don't want us wasting time with conversation. And they don't want patrons getting hung up on a particular girl and starting some kind of trouble.\"\n\n\"I can pay you. Well.\"\n\nShe pursed her oh-so-red lips together. \"I don't know.\"\n\n\"Please. I'll make it worth your time.\"\n\nShe considered a few more moments. \"I wouldn't do this if it hadn't been such a shit of a night.\" Her eyes scanned the room, checking for supervisors, then scrutinizing the numbered lights on a neon sign by the door that told her where vacancies existed. \"Okay, look. I can get us a couch in a semiprivate room. But it'll be three hundred to me. And you'll have to tip the bouncer.\"\n\n\"And we can talk?\"\n\n\"You can do anything you want. I'll be working. Come on.\"\n\nShe led him through the madding crowd to an alcove farther down the main hallway. After he took care of the bouncer, Lenore gave him a gentle push onto a black upholstered couch. A moment later, a woman wearing a black dominatrix outfit appeared bearing a tray with two glasses of champagne.\n\n\"I'm sorry, ma'am,\" he said, charming her with his smile, \"but I do not partake of strong spirits.\"\n\nThe waitress stared at him. \"It's just champagne.\"\n\nHe wagged a finger. \"Nonetheless. Spirits destroyed the prophet, you know.\"\n\nThe waitress and Lenore exchanged a look, then a shrug. The waitress disappeared.\n\nLenore reached behind herself and snapped open the bustier.\n\n\"Just a moment,\" he said, holding out a hand. \"You don't need to do that. I want to talk.\"\n\n\"No doubt.\" She pushed his hand away and crawled onto his lap. Her bare breasts tickled his nose.\n\n\"I mean it!\" he said, holding her back. \"This is not\u2014\"\n\n\"Do you want me to lose my job?\"\n\nHe relaxed. Even in a semiprivate room, the night must have a thousand eyes. \"At least give a man a chance to breathe, would you?\"\n\nLenore giggled. \"Whatever.\" Her hips began to sway.\n\n\"That's not necessary, either.\"\n\n\"Got to please the client.\"\n\n\"Rest assured you will receive my highest encomiums.\"\n\n\"Just relax,\" she said, stroking the back of his neck. \"We have to look as if we're doing proper business. Even if we're not. Believe me, girls who don't follow the rules don't last long here. And I've got a living to make.\"\n\n\"Some living. A girl your age. Performing lap dances for strangers.\"\n\n\"I don't do lap dances,\" she replied. She squeezed her thighs together, tightening her grip on his groin. \"I do friction dances. It's my specialty.\"\n\nHe felt his internal temperature rising.\n\n\"Now what is it you wanted to talk about, you stud?\" she growled in his ear, her hips grinding. She was eager and energetic but not that practiced. \"Don't I interest you even a little?\"\n\n\"This isn't\u2014isn't\u2014\"\n\n\"I know what to do.\" Her hand found the zippered fly of his trousers.\n\nHe knocked her hand away. \"Stop!\" This was becoming too intense, too potentially awkward. \"I want to go somewhere private.\"\n\n\"We are somewhere private.\"\n\n\"Someplace else. Away from here. Someplace we can do . . . more than this. You know what I mean.\"\n\n\"I don't know if I think that's a good idea. . . .\"\n\n\"Please. Vouchsafe me this one cherished boon.\"\n\nShe peered at him with a harsh eye.\"If we leave the club, I'll be out for the entire evening.\"\n\n\"Yes.\"\n\n\"You'd have to compensate me for the loss. Me and the management.\"\n\n\"I understand.\"\n\n\"Do you? We're talking, like, three thousand dollars here.\"\n\n\"I can do that.\"\n\nShe gave him a long look. \"I can't pretend it wouldn't be good for me. Bring my average up. You're sure?\" She hesitated only the merest of moments. \"Okay, I'll do it.\"\n\n\"Most munificent of you.\"\n\n\"Let me clock out and get my coat.\" She stopped just before she left the room. \"You're sure? You're serious about this?\"\n\nHe nodded, smiling pleasantly. \"Dead serious.\"\n\nI was pumped. For the first time since I got out of the hospital I was actually feeling somewhat good. I might not have the case solved, but I'd had some breakthroughs\u2014the eyewitness, and now the decoded messages. My first steps in the psychologically right direction.\n\nI decided to treat myself. Dinner at Elmer's. Not a million-course buffet, not fancy French cuisine. No elaborate d\u00e9cor. No d\u00e9cor at all, really. Just good old American down-home comfort food, ribs and chicken-fried steak served straight, at a very affordable price. Once upon a time, Vegas was famous for places like this, for their ninety-nine-cent all-you-can-eat shrimp and buck-ninety-nine filet mignon. Nowadays, the big resorts hired Michelin-quality chefs to entice people to pay for the prestige of a ridiculously overpriced meal in a room with minor French impressionist paintings. Wolfgang Puck had four restaurants here, for Pete's sake.\n\nElmer's was much more to my liking. It had a lot of sentimental value. David and I used to come here on our anniversary. I hadn't been back since he died, and hadn't wanted to. But I had a sense now that I was ready.\n\nI felt like Dolly Levi after a long absence. I smiled at the ma\u00eetre d', a freckle-faced kid engaged in a losing battle with acne. \"Party of one, please. I'd like a table\u2014\"\n\n\"By the window. With a view of the skyline.\" He grinned. \"Good to see you again, Miss Pulaski.\"\n\nI was floored. \"What a memory you must have.\"\n\n\"Not at all. You're one of our most regular customers.\"\n\n\"I\u2014you mean, before\u2014\"\n\n\"On the seventh, every month. It's your anniversary date, right?\"\n\n\"Well, yes, it was, but\u2014\"\n\n\"And you never miss a month. Very admirable.\"\n\nMy neck stiffened. \"But\u2014I haven't been here for more than a year.\"\n\nHe blinked, still smiling. \"You were just in last month.\"\n\n\"I\u2014was?\"\n\n\"And the month before that. And the month before that.\"\n\n\"But\u2014I don't . . .\"\n\n\"Shall I have the bartender bring you your favorite?\" He winked. \"Or maybe we should save time and have him bring you a pitcher.\"\n\nMy stomach felt like lead.\n\n\"When you're finished, let me know. I'll call you a cab.\" He winked again. \"I think that's best, don't you?\"\n\nThe elation I'd felt before deflated like a collapsed artery. The gnawing in my gut, the panicky, breathless, acidic sensation reasserted itself. \"All of a sudden, I\u2014I'm not feeling well. Maybe I'll skip dinner.\"\n\nI stumbled out of the restaurant, knowing damn well that I was not skipping dinner. Only mutating its form.\n\n\"Where are we going?\" she asked as they made their way to his truck. She had pulled a white embroidered wrap\u2014a kimono, perhaps?\u2014around herself to cover her virtual nudity.\n\nThey arrived at the pickup. He opened the passenger side door for her. \"Huh. I didn't figure you for a pickup man.\"\n\n\"What did you have in mind?\"\n\n\"I don't know. Thought you were more the Lexus type.\"\n\nHe closed her door securely, then walked around to the other side. \"I find my truck very practical. And reliable.\"\n\n\"Yeah, I guess that's right. Mind if I turn on the radio?\"\n\nHe winced, ever so slightly. \"Would it be more music such as what they play in your workplace?\"\n\n\"Guess you're pretty tired of that, huh?\"\n\nHe smiled. \"I don't mean to be bilious with you. But one must have standards.\" He looked over his shoulder, making sure the path behind him was clear. \"That noise does not even qualify to be called music. It is an assault on the eardrums.\"\n\n\"It wouldn't be so bad if they didn't play it so loud. But it helps with the dancing. And it creates a party atmosphere. Most of the girls are so stoned they don't hear it anyway.\" She glanced down at the seat. \"What's this?\"\n\nShe had found the axe which he had left lying on the floor beneath the glove compartment.\n\n\"That's . . . just what it appears to be. I have some stumps on my property that require removal.\"\n\n\"Oh.\" She handed it to him. \"Creepy.\" She slid into the seat. \"Do you have one of those new places out in Grover Mills? I've heard those are\u2014\" She stopped again. \"Now what's this?\"\n\nEdgar turned. His eyeballs bulged as he realized what she had found. \"Don't\u2014\"\n\n\"There's something rattling around in there.\" She picked up a shoe box he had left under the seat. \"I think it may be broken.\"\n\nShe opened the box.\n\nAnd screamed.\n\nThe box fell out of her hands and all of sweet Annabel's teeth, all thirty-two of them, caked with blood, flew across the cab of the pickup.\n\n\"Oh, my God,\" Lenore said, pressing her hands against her mouth.\n\n\"I can explain,\" he said rapidly. \"I'm a dentist and\u2014\"\n\n\"I heard about that girl\u2014\" She pushed open the passenger side door. \"I'm getting out of here.\"\n\nHe grabbed her arm. \"Please don't.\"\n\n\"Look, keep your money. I'm leaving.\"\n\n\"But I can explain.\"\n\n\"Let me go!\" She brought her fist down on his arm, as hard as she could. It was all he could do to hang on.\n\n\"Give me one more chance.\"\n\n\"I'm not giving you anything, you pervert.\" She sank her teeth into his wrist.\n\n\"Oww!\" She'd hurt him, broken skin.\n\nShe scooted toward the door, but he managed to grab her shoulders and yank her back. The kimono slipped, exposing her. \"Stop this immediately!\"\n\n\"You stop, asshole.\" She rolled back, bringing her legs around and kicking him in the face. He slammed back against the driver's-side door.\n\nShe was already upright and moving toward the open door. With a sudden lunge, he sprang forward and grabbed her by the neck, then flung her head against the dash.\n\nShe was slowed, but not unconscious. \"I'll scream . . . ,\" she mumbled.\n\n\"No, you won't.\" He snatched a loaded syringe out of the glove compartment and jabbed it into her neck. \"Sweet dreams, Lenore.\"\n\nIt took more than a moment for him to regain his composure. Perhaps the insufferable music was a blessing after all, he noted, since it ensured that no one inside could possibly hear anything that happened out here.\n\nHe had to remember not to blame the girl. What could she know? She only acted out of fear, ignorance. He bent over and lightly brushed his lips across her forehead. _Thus I pacified Psyche and kissed her, \/ And tempted her out of her gloom . . ._\n\nHe threw a tarp over her, hid the axe behind the seat, and pulled out onto the street.\n\nThat was sloppy, he scolded himself, driving away. He'd been clumsy, foolhardy, as if he wanted to be caught. And it had almost cost him everything.\n\nHe could take no more risks. His work was too important. She was the third, the final component in the sacred trinity. First Helen, then Annabel, and now Lenore. The chosen offerings. After this, there would be only rejoicing. He was the Instrument who would usher in the Golden Age. As it was meant to be. As it was foretold.\n**CHAPTER TEN**\n\nIt was a miracle I woke at all, much less before nine o'clock. I didn't know which was dragging me down more\u2014the reading all night or the drinking all night. I'm not sure when I finally gave it up. I was in the middle of \"A Tale of the Ragged Mountains\"\u2014a jolly Poe yarn about someone being killed via a poisonous leech\u2014when my eyelids finally gave in.\n\nA quick glance at the watch told me I had less than thirty minutes until I was supposed to pick up Darcy, who was about ten minutes away. My first instinct was to just grab my keys and go\u2014what would he care if I was groomed or not? But O'Bannon might be lurking about, and he'd know something was up if I came in looking disheveled, distraught, or drunk. He'd fire my sweet ass in a heartbeat, and then I'd have no chance of reclaiming Rachel. So I showered quickly, steaming the smell out of my skin, and I brushed my teeth relentlessly.\n\nI stared at the smoky brown liquid resting at the bottom of the bottle on my nightstand. If I polished that off, I could ditch the bottle. That would be smart. Get rid of the evidence, just in case Lisa or O'Bannon dropped by.\n\nBut if you start drinking first thing in the morning, I reminded myself . . .\n\nDon't be idiotic. It was barely a swallow. I raised the bottle to my lips and downed it. It burned going down, but it burned good.\n\nAnd then I brushed my teeth some more.\n\nI may have violated a few traffic regs making my way to O'Bannon's, but they were minor ones, I'm sure. After I'd been in the car a few minutes, my cell beeped out the theme from _Dragnet._\n\n\"Pulaski.\"\n\n\"Susan, it's Colin. I've got something for you.\"\n\n\"Talk to me, Einstein.\"\n\n\"Those messages you left behind\u2014they really are messages. Coded messages. I can confirm it.\"\n\nSince he couldn't see my face, I figured it was safe to smile.\n\n\"It's a code, but an insanely complex one. A normal substitution cipher has twenty-six characters, for obvious reasons. This one has three hundred and forty.\"\n\nI raised an eyebrow.\n\n\"And given the relative brevity of the messages, that leaves a lot of characters to decode on not much information. Over half of the symbols appear only one or two times. See, he starts by substituting each letter for another symbol, then transposes these symbols, creating a hybrid substitution-transposition cipher. The cipher alphabet changes after every seven letters. So each letter of the plaintext is represented by several different symbols in the cipher-text. It's called polyalphabetic cryptology.\"\n\n\"So,\" I ventured, \"have you cracked it?\"\n\n\"No way. It's a major accomplishment just to have figured out what it is. I suspect a full decoding will require one of those mainframe code-breaking computers at the CIA. All I have is a few words. That first cipher says something about a grave. A deep one, I think.\"\n\n\" 'Deep, deep, and for ever, into some ordinary and nameless grave.' \"\n\nThere was a pause. \"Could be. Where'd you get that?\"\n\n\"And the other one reads, 'In the multiplied objects of the external world I had no thoughts but for the teeth.' \"\n\n\"Teeth?\" He fairly squealed into the receiver. \"No wonder I didn't\u2014 What kind of message is that?\"\n\nA damn good question. \"You ever read Edgar Allan Poe?\"\n\n\" 'The Gold Bug.' \"\n\n\" 'The Gold Bug'?\" I hadn't gotten that far yet. \"What's that?\"\n\n\"Short story. Big hunt for pirate treasure. Which they find by solving a substitution cipher.\"\n\n\"Poe wrote about ciphers?\"\n\n\"He was the first writer to ever use one in a story, if I'm not mistaken. He's probably the granddaddy of American codes and code breaking. He was really into it. The story gives a mini-lecture on how to solve cryptograms. As I recall, he ran ciphers in whatever magazine he was editing at the time and challenged people to send him one he couldn't crack. I don't think he was ever stumped.\"\n\nCodes were important to Poe. And so of course they were important to anyone to whom Poe was important.\n\n\"You know, Susan, if your killer's really into Poe, he may be up to some seriously bizarro business.\"\n\nI let Colin sign off, with that sobering thought ringing in my head.\n\nAfter a brief stop, I pulled into the O'Bannon driveway at a quarter after nine. Darcy was waiting for me on the front porch.\n\n\"You came!\" he said, plainly delighted. Was he worried that I wouldn't? I approached bearing tall ones from the corner Starbucks clone. I needed a cobweb clearer, and I thought he might, too.\n\n\"How 'bout a cup of jamoke? Your choice\u2014regular, or the more exotic white chocolate mocha.\"\n\nHe stared at them, not taking either. \"On Wednesdays I have two eggs and bacon and the eggs sunny-side up and not touching the bacon and a half glass of orange juice at eight o'clock. Then I take Bus 17 to the day care center so I can be there by nine, except today I called and told them I wasn't coming so it's okay that I'm not there yet.\"\n\nAnd they say autistics are inflexible. \"So you want coffee or not?\"\n\nHe was still staring at the cups. \"Did you know that most coffee comes from the west coast of Africa?\"\n\n\"I thought South America . . .\"\n\n\"Less than ten percent comes from South America. Most comes from Africa, where acid rain is constant and bathes the coffee beans all year round.\"\n\nThat explains the rich aroma. \"Come on, Darcy. Choose your poison.\"\n\n\"Caffeine has been used as a highly effective poison in many agricultural arenas.\"\n\n\"Well, I'm not a plant. Take one.\"\n\nHe hesitated, looking at the cups the same way he had looked at the spider. \"Did you know that caffeine is more addictive than cocaine?\"\n\nMy arms were getting tired. \"No . . .\"\n\n\"It's also a diuretic. It dries you up and creates an addiction. Causes headaches and diarrhea and other physiological ailments.\"\n\nI sighed. \"You don't want any coffee, do you?\"\n\n\"Scientists say that it's best to avoid addictive substances. Do you think that it's best to avoid addictive substances?\"\n\nThe cups were starting to burn my fingers. I poured the mocha into the grass. \"Absolutely. Horrible things, those addictive substances. Avoid at all costs.\"\n\nThe Poe Gallery. Of course. The significance of this burial site was increasingly apparent, thanks to Darcy.\n\nThere were only a few cops left at the Transylvania. A couple of techs and one uniform standing guard. Another day or two and the room would likely be released back to the hotel. What they would do with it, I had no idea. I suspected the memory of a real corpse turning up in this Disney-Meets-Death joint might spoil the fun for some of the patrons.\n\nTony Crenshaw was on the scene. I decided to walk over and give him a good chin-wag. \"Anything new?\"\n\n\"No. Not here, anyway. There's a rumor the coroner might release something later today.\"\n\n\"Be still my heart.\"\n\nI noticed that Darcy was hanging back. Probably shy around people he didn't know. I thought about introducing him around, then decided it was probably best to let him absorb the crime scene in his own way at his own speed.\n\n\"You see my e-mail on the Poe connection?\" I asked.\n\n\"Jesus Christ, yes. The whole office was buzzing about it this morning.\"\n\n\"What's the general opinion at the watercooler?\"\n\n\"Either the killer is crazy or you are. Possibly both.\"\n\nI smiled a little. \"What do you think?\"\n\n\"I think you should've kept your theories to yourself. Something that warped is bound to leak out.\"\n\n\"What if it does?\"\n\n\"This is nutty stuff, Susan. The press is already leading with this story. If they get wind of this development . . .\" He made a slashing gesture across his throat.\n\n\"I just wanted to get my theory on the table so I could get input from the other geniuses in the department.\"\n\n\"Other?\"\n\nAnd of course I wanted to show everyone that I wasn't too inebriated to do my job. That O'Bannon hadn't committed a spectacular error in judgment by involving me. I wanted to make sure I got credit. Even if my brilliant breakthrough was mostly attributable to a twenty-six-year-old autistic guy. \"You read much Poe, Tony?\"\n\nHe shrugged. \"I'm more of an Anne Tyler man myself.\"\n\n\"Just as well. Better for you.\" I saw that Darcy was hovering over the grave site. The body was gone, of course, but the coffin was still where it had been found, the half lid open. I walked over to him. \"Kind of spooky, isn't it?\"\n\nHis hands were flapping in tiny circles. \"I think it's strange that that one died. Do you think it's strange that that one died?\"\n\nI didn't quite grasp his line of reasoning. \"I . . . think the whole scenario is strange.\"\n\n\"In 'The Premature Burial,' the man gets away.\"\n\nI thought back to my reading the night before. It was all a bit hazy, but I could recall a few details. \"Yeah, that's right.\"\n\n\"In 'The Fall of the House of Usher,' Madeline is buried alive.\"\n\n\"Oh, yeah?\"\n\n\"She gets out, too.\"\n\n\"Huh.\" Hadn't gotten to that one yet.\n\n\"In 'Ligeia,' a woman is buried alive.\"\n\nJeez Louise. Poe was really hung up on this plot device, wasn't he? \"Does she get out?\"\n\n\"Sorta.\" He glanced down at the coffin. \"But that one did not get out, did she?\"\n\n\"No, Darcy. She didn't.\" Which raised an interesting point, at least in my head. If my killer was trying to re-create scenes from Poe, and the characters in the stories don't die, why had he let this victim die? What purpose did he think the deaths would serve? \"She must've been awake, though. Look how she clawed at the lid, trying to get out.\"\n\n\"At the top. Not at the bottom.\"\n\n\"I suppose it's everyone's first instinct to use your fists to try to pound your way out of something.\"\n\nDarcy's head tilted to one side, as if that computerized brain of his was momentarily processing information. \"Are your arms stronger than your legs?\"\n\nI thought for a moment. \"I suppose not.\"\n\nHe pointed to the coffin. \"Were that one's?\"\n\n\"I don't know for sure, but\u2014\"\n\n\"Did you know that men usually have more upper-body strength than women?\"\n\n\"Well, I've won a few arm-wrestling matches in my time.\"\n\n\"But girls sometimes have strong legs, don't they? Especially young ones. How old was this one?\"\n\n\"We're not sure. Sixteen or so.\"\n\nHis head tilted again. \"Why do you think she did not use her legs?\"\n\n\"I . . . guess she didn't think of it.\" Which sounded lame even as I said it. \"Or she couldn't.\"\n\n\"Were her legs tied up?\"\n\n\"No.\"\n\n\"Then\u2014\"\n\n\"He's using a drug.\" Damn. It was obvious\u2014once the kid fed it to you. \"Some kind of paralytic agent. That's how he controls his victims.\" I pondered a moment. \"But nothing showed up on the preliminary tox screen.\"\n\nDarcy jumped up and down eagerly. His voice sounded as if he were reciting at high speed. \"Some paralytic drugs, like poisons, will not be detected by general screening procedures. Progressive laboratory screening from the general to the more specific is required. For example, an anion gap on the electrolyte panel combined with metabolic acidosis on arterial blood gases would prompt an inquiry into ASA, methanol, or ethylene glycol as potential etiologic agents.\"\n\nI stared at him, doing my best not to gape at this astounding feat of mimesis. Even if he didn't understand everything he was saying, his memory was preternatural. \"Did you go to medical school while your father wasn't looking? How on earth do you know this?\"\n\n\"My dad has books about criminology. Lots of them. When books at his office were replaced by newer editions, he always brought the old ones home.\"\n\n\"For you to read. And memorize.\" I began to get the picture. \"Darcy, you're a one-man forensic lab. I'm going to call the coroner. Patterson hates interference from detectives, but if I gently pass the word that a singular screen for paralytic agents before he files his report might be a good idea, he can do it and act as if it were his idea.\"\n\nI punched my cell and delivered the message. When I finished, I found Darcy staring at the double doors that led to the main hallway outside.\n\n\"That's how the killer got in,\" I explained. \"Looks like he fired up an acetylene torch to weaken the chain, then used his spade to break it.\"\n\nDarcy didn't respond. He continued staring at the door, the jamb, the place where the chain had been.\n\n\"We think he came in around two or three in the morning,\" I continued. \"The coroner should be able to nail the time for us. There's a service driveway on the other side of the hallway. No one would've been out there early in the morning. He'd have the time and opportunity to drive up from wherever, break in, and do whatever he needed to do.\"\n\nDarcy was still staring at the door, doing that weird head-tilting thing.\n\n\"We've photographed the door, of course, and the chain is back in Evidence. I can take you to see it, if you like. I can show you the enlargements of the doorway\u2014\"\n\n\"That's an ugly black spot.\"\n\nHe was pointing to a longish horizontal mark, slightly triangular, that scarred the green paint at about the level of the doorknob. It was widest on our side, then narrowed toward the hallway.\n\n\"Yes, very unattractive. Hotel employees can be such slobs. Probably some workman carting in furniture in a big hurry.\"\n\nDarcy scraped the black place with the tip of his fingernail. It flaked off a bit. \"It's burnt.\"\n\n\"Burnt?\"\n\nHe nodded. \"My dad works with a torch, in the garage. For his pottery.\"\n\nOkay, this time I had to do a triple take. \"Your dad makes pottery?\"\n\n\"Yeah. I like pottery. Do you like pottery? He's good at it. He has a kiln for the big stuff, but he uses a torch on the ashtrays.\"\n\n\"Let me make sure I'm getting this. Your dad makes ashtrays?\"\n\n\"And bud vases.\"\n\n\"Chief of Police Robert O'Bannon makes bud vases?\"\n\n\"I think he is very good at it. He whistles while he does it, so I guess it makes him happy.\"\n\nI had no reason to doubt the guy. But I had a hard time conjuring a mental image of this big gruff cop straddling a potter's wheel, much less baking dainty knickknacks.\n\nDarcy redirected my attention to the black scar. \"This is from the torch.\" He tucked his chin, made that irresistibly silly face. \"One time my dad made a mark like that on his workbench. He said some very bad words. I could tell you about them but he told me I was not to repeat them.\"\n\n\"That's quite all\u2014\"\n\n\"The only other time I heard my father talk like that was when I dropped his first-edition copy of _Brideshead Revisited_ in the bathtub and\u2014\"\n\n\"Is there some significance to this, Darcy?\"\n\nHe had no answer. I could tell something was bothering him. But the cranial computer hadn't quite lined up all the data yet.\n\n\"Don't sweat it, Darcy. You've been brilliant enough for one day. I can't believe you know so much about toxicology.\"\n\n\"Do you think that I know a lot of things? My dad says I know more stuff than an encyclopedia, but it doesn't matter because I don't have the sense to know what to do with it.\"\n\n\"That's okay, my friend,\" I said, linking my arm around his. \"Because I do. Let's go to the next crime scene.\"\n\nDid I make her happy I think maybe I made her happy and she looked happy but I remember sometimes Mommy looked happy and it turned out she wasn't. She made me go to those places I didn't like and Susan brought me to this place I didn't like but it's okay if it makes Susan happy. I found out that Susan is not married anymore and if she's not married then she could marry me and we could have babies. She doesn't have any babies but she could I know she could and that would be even better than the day care center. _An unhappy alternative is before you, Elizabeth. From this day you must be a stranger to one of your parents._ I don't know why it made her so happy all I told her was what was obvious but her voice was faster and higher and she touched my arm so I think that means she was happy so I'm happy too. I wonder if my dad will be happy. I wonder if he will let me go out more. Maybe he would even let me be a policeman. He thinks I don't want to be a policeman, but I do. I want to make him happy. I just don't understand why people do all these things they do. People always think I act crazy but I think the things other people do are crazier and I don't understand them.\n\nMaybe Susan can explain it to me. I would like that.\n\nEven as I drove him out to the second crime scene at McCarran, I knew I had no right whatsoever to expect Darcy to come up with more breakthroughs, or to spark new ideas in me, or whatever it was he was doing. But I couldn't help hoping, just the same. I wonder if that's why God stopped with the miracles after the New Testament days. People are never satisfied. They always want more.\n\nI asked one of the patrolmen to give Darcy access to the crime scene, then let the boy do as he pleased. Far be it for me to direct him. He spent almost an hour just wandering in and around the plane. He crouched down where the body had lain, looking at it from all angles, his head tilted with that uncanny android expression on his face. He paced the perimeter of the entire pavilion where the retired planes were stored. But he always came back to the place where the body had been found. He crouched down on all fours and pressed his face to the floor, his butt sticking up in the air. He looked like a bird dog sniffing for a scent. I could see one of the techs watching him out of the corner of his eye, laughing. I gave him the finger and told him to get to work. Asshole.\n\n\"The criminalists have been all over this site,\" I told Darcy, just to hear myself talk. \"They didn't find much. No useful stains or residue. No blood. A few bits of carpet lint on the body.\"\n\nDarcy's head rose, as if he had detected a new scent. \"Carpet? Or rug?\"\n\n\"Whatever. Something from her home or his that\u2014\"\n\n\"I think maybe it was a rug. Do you think maybe it was a rug?\"\n\nI paused. \"I don't know. I'm not sure I see the\u2014\"\n\n\"Did you know that murderers sometimes wrap bodies in rugs to make them easier to carry? During the 1934 torso murders in Cleveland, the killer\u2014\"\n\n\"But if the killer brought the corpse in a rug, where is it?\"\n\n\"I think maybe he must've taken it home with him.\"\n\nI considered. \"No. The body couldn't have been wrapped, and he must've dragged it part of the way. There was dirt on the body.\"\n\nDarcy crouched down low again. \"There's a lot of dirt on the floor of the plane. Why would there be dirt on the floor of the plane?\"\n\n\"I . . . guess the door blew open and\u2014\" Right. We were in a sea of concrete. \"He brought the body from somewhere else. Probably got tired at some point and dragged it. It got dirty.\"\n\n\"I didn't see the body,\" Darcy said. \"Was it all scraped up?\"\n\n\"Well, no. Actually, it seemed well cared for, even washed. Almost like he really cared about her.\"\n\nDarcy brushed his hands off and stood. \"I think that maybe he brought this one here in a rug.\"\n\nAnd damn it all, I knew he was right. \"But where did the dirt come from?\"\n\n\"I think that maybe he brought that, too.\"\n\n\"But why? For what reason?\" I asked, but I realized that brilliant as he was, Darcy could never deduce the answers to these teleological questions. Because that required someone who understood the wild and wooly ways of people. Perhaps someone blessed with hyper-empathy. Yours truly. I thought a moment. . . .\n\nBecause this was a burial, that's why. At least in his deranged mind, this was a burial, so he'd brought dirt. Why hadn't I seen it before?\n\nI knew the answer to that question, too.\n\nBecause I hadn't had Darcy before.\n\nI put my arm around his neck and gave him a squeeze. He pulled away but not too hard. \"Darcy, my friend, I think this could be the beginning of a beautiful friendship.\"\n\nHe smiled cautiously, eyes up but chin tucked. \"Did I\u2014did I do something good?\"\n\n\"You hit the ball out of the park, slugger.\" Bizarre as it sounds, being with Darcy, working with him\u2014if you can call it that\u2014made me feel . . . better. Stronger. Like maybe I really could still do this.\n\n\"Does this mean that we could go for custard later? I like to eat frozen custard. Custard is my favorite thing because I know whenever I have custard it's going to be a Very Excellent Day. On Wednesdays, my favorite is vanilla toffee, but my dad likes the bubble gum flavor\u2014\"\n\n\"Your dad likes bubble gum custard? Get out of here.\"\n\n\"Sometimes he gets Oreo Cookie Mash. If we went now I think my dad wouldn't mind and it would be okay and I don't think it would spoil our dinner, do you?\" He grinned that goofy grin again. Irresistible.\n\nFrom a landing bay outside the Arrivals deck, he watched the crime technicians work through high-powered binoculars. His hand still hurt from the bite, but he had cleaned and dressed it and all his fingers were functional, if a bit weak. He was wearing his uniform, which was not the same as those of airport security but was more than enough to keep people from interfering with him. When a man in a uniform went to the airport, no one messed with him, especially these days.\n\nWatching the crime techs at work was fascinating\u2014and rather discouraging, too. It was of considerable interest to him to see where they focused their attention. The careful steps they took to ensure that no tracks or footsteps were obliterated\u2014not that he had left any. All right\u2014one tire track. He was human. What could they possibly do with that? He would change the tires on the truck tonight, just to be safe.\n\nThese people spent hours coating the area with a fine white dust, hoping to find a fingerprint. Absurd. To leave behind clues of that caliber, one would have to be a fool. No, he wanted to scream at them, I didn't leave behind any hair\u2014I wore a net the entire time. Did they think they were dealing with a beer-sozzled redneck? A tripped-out tourist? Had he not made it abundantly clear that he was a serious man doing serious work?\n\nThank heaven Susan finally arrived. His sweet, beautiful Cassandra. Yes, his totem had been right, had been prophetic. It was still early days, but he felt certain that she would be the one to appreciate what he had done. She'd barely been on the scene a minute yesterday before she had Annabel's mouth open, something none of her predecessors, not even the representatives from the coroner's office, had the sense to do. _Her understanding encompasseth mountains. . . ._\n\nBut Susan's return to the scene today was fraught with disappointment. For starters, she did not come alone. Who was this new companion? He hadn't been at the police station. He was dressed casually and didn't appear to have any official status. And yet, Susan talked to him constantly. It was as if she was feeding him information, soliciting his opinion about every aspect of the case. Was he her friend, her partner? Or something more? He seemed younger than she, surely too young to be . . .\n\nAnd yet there was something between them, something real, important. He tightened the focus on his binoculars, zooming in for a closer look. He stared at the young man's eyes, watching the way he moved, the way he talked, for a considerable period of time. And yet, he couldn't get a reading on him. The young man's eyes were like reflecting mirrors. There was something . . . elusive about him. Something inscrutable. As if there was an emotional lacuna where a soul should be. As if . . . as if he wasn't entirely a part of this world.\n\nDream-Land? No. He might not be able to discern what drove Susan's young cohort, but he was certain it was not enlightenment. There was more confusion about him than determination.\n\nAnd yet he couldn't help but wonder. . . .\n\nDid he pose a threat? The others, with their by-the-book approach and mundane sensibilities had little hope of ever discovering the truth. But this new interloper . . .\n\nHe would have to keep a close eye on this young man.\n\nOn his way out of the airport, he bought a newspaper from a self-serve kiosk. He would want the day's story for his History. The article about the discovery of Annabel's body was rather disappointingly small, even though there were no other stories of great import. Was this town so jaded that murder no longer captured its imagination? What was this distorted mentality that bestowed more attention on Siegfried and Roy than a messiah?\n\n **SECOND BODY FOUND IN \nABANDONED AIRCRAFT**\n\nBY JONATHAN WOOLEY\n\nAnother naked corpse was discovered late Friday afternoon by FAA investigators in one of the many abandoned aircrafts housed on the rear field at McCarran International, authorities confirmed at a noon press conference. Although officials did not offer an opinion as to whether this death was linked to the body found two days before at the Transylvania Resort Hotel, they did acknowledge that both bodies were entirely naked, and that both were found with a mysterious written message.\n\n\"We're doing everything we can to solve this case,\" said Chief of Police Robert O'Bannon. \"Even calling in federal experts. But it would be a mistake to prematurely conclude that the killings are linked without more information.\" When pressed, Chief O'Bannon acknowledged that there was a strong possibility that both crimes were committed by the same assailant or assailants. He offered no opinion as to who that might be or what motive might lie behind the crimes. He did note that the two deaths were very different in nature. The woman found at the Transylvania died of asphyxiation, while preliminary indications were that this latest victim had died from exsanguination.\n\nEven if the crimes are linked, O'Bannon noted, it is important not to create a panic. \"Las Vegas is a large city, and there is no reason to believe that anyone, especially tourists, are in danger.\" Lieutenant Barry Granger, the homicide detective assigned to the case, also urged that . . .\n\nHe closed the paper, disgusted. Why did they always interpellate O'Bannon and Granger? They didn't know anything. O'Bannon was a supervisor and Granger was a fool. Why didn't they talk to Susan? Susan understood that this was not just another murder case; he was sure of it. She could give them a story worthy of publication. He would be interested to hear what she had to say. What she thought of this case, these crimes.\n\nWhat she thought about him.\n\nHe had always considered himself a paragon of decorum and chivalrous behavior. He had modeled himself after the prophet in word and deed, sensibilities, even adopting the euphuism of his era. But there were so many unanswered questions. He would follow Susan constantly now, as much as he was able. To keep an eye on her. To ensure that her interest didn't flag. And as to this new man\u2014perhaps he would have to arrange a meeting. If there was danger afoot, he had to know. So he could take the appropriate action to eliminate it.\n**CHAPTER ELEVEN**\n\nThere were some advantages, I was beginning to learn, to having an office adjoining the men's room. True, it meant I was on display like a museum exhibit, or perhaps more accurately, like a zoo animal, caged all day long. But it also meant that no one could avoid me, certainly no one male, no matter how much they might like to do so. Even Granger did not possess a cast-iron bladder. Earlier, he had tried to slip past by pointedly looking the other way, so I complimented him on something, I think on the way the coffee stain on his tie matched his underarm stains. For his afternoon visit, he decided to take the offensive. And he was pretty darned offensive about his offensive, too.\n\n\"Pulaski,\" he growled, even before he got to my desk. \"What the hell were you doing with O'Bannon's kid?\"\n\n\"I was investigating, sir. It's what investigators do. Perhaps you were absent the day they covered that at the academy. Darcy was quite useful.\"\n\n\"I know the kid's story, Pulaski. And I saw how he acted at the crime scene. There's no way\u2014\"\n\n\"You don't know anything about it.\"\n\n\"I know he's been nothing but a burden to his father. I know O'Bannon never mentions him. Never.\"\n\nI wondered if that was true. \"There's no reason for anyone to be embarrassed about a neurological disorder. It's no one's fault. It just happens. We don't know why.\"\n\n\"None of which explains why you were dragging this poor boy all over two grisly crime scenes. Are you trying to suck up to O'Bannon?\"\n\n\"Don't be imbecilic.\"\n\n\"You really think if you play out this charade with O'Bannon's son you'll get your old job back?\"\n\n\"The thought never entered my mind,\" I said quite honestly.\n\n\"Well, leave the kid alone. Let him work in that day care or whatever. No telling what a crime scene might do to a mental case like that. Might give him ideas, even.\"\n\nGod grant me the serenity to accept the things I cannot change and not kill Granger in the process. \"I'm just trying to catch the murderer. Just like you.\"\n\n\"Which reminds me. You were not hired to sniff around the forensic evidence or to repeat the work of others more qualified to do it. You were hired to prepare a psychological profile. Have you done that?\"\n\n\"I'm working on it. It's difficult when we don't know who the victims were.\"\n\n\"I can help you with that.\" He slapped a folder down on my desk. \"If I were you, I'd get something down on paper. The FBI profiler will arrive tomorrow.\"\n\nMy head jerked up. \"What?\"\n\n\"I'm told he's one of the best. Most of the boys think O'Bannon hired you so he could tell the press he was doing everything possible to catch the killer. But after the Feebie arrives, I can't imagine what the point of paying for an outside consultant would be.\" And on that note, he galumphed toward the men's room.\n\n\"Thanks for stopping by, Granger,\" I couldn't resist saying. \"You honor David's memory.\"\n\nHe stopped short, his shoulders rising. \"Better than you do,\" he muttered under his breath. Then left.\n\nI didn't allow myself to focus on what the son of a bitch had said or what he meant by it. I had work to do. If Granger remained in charge of this investigation, the killer could work his way through Poe's collected works three times over without getting caught.\n\nA federal profiler? Why? Had O'Bannon requested it? Because he had no faith in me anymore?\n\nI put it out of my mind. If the man was coming, I would have something ready to show him.\n\nThe file Granger dumped on me was more helpful than I could've imagined. The two victims had been identified, thanks to the pictures run in the daily paper. The first was named Helen Collier, and she lived here in the Vegas 'burbs with her mother. Mom had been visiting friends and didn't know her daughter was missing\u2014till she saw her pic in the paper at the airport on her way home. Hell of a shock that must've been. Helen had been a petite girl and cute, judging from her school photo. Too cute to end up suffocating, clawing for air in a buried coffin.\n\nThe second girl was named Annabel Spencer. She was originally from New York but was going to school at MIT. No one had any idea what she was doing in Vegas. The only person who even knew she was gone was her boyfriend, and so far he wasn't providing any details. But here's the kicker: Annabel Spencer was the only daughter of Dr. Fara Spencer, acclaimed TV shrink. Her afternoon TV show was huge; Lisa swore by it. Sort of a cross between Dr. Phil and Judge Judy, Fara alternated between administering homespun wisdom to the worthy and tongue-lashing the unworthy. Her hour was the hottest thing in syndication.\n\nAnd her daughter was the second victim. As if this case could get any weirder. Or more complicated.\n\nI jotted down the address of the Collier residence. That had to be my next stop.\n\nI was almost out the door when an intern brought me a message.\n\n\"Lieutenant?\"\n\nShe was a Hispanic woman who I knew worked in toxicology. She'd been here more than two years, but if she hadn't been wearing a badge I wouldn't have known her name was Jennifer. \"Yes?\"\n\n\"I have a report for you.\"\n\n\"I'm kind of in a hurry, Jen. Can you give me the highlights?\"\n\nShe spoke quietly, almost timidly. \"Despite Patterson's protestations, we ran the tests you requested, the progressive tox screen.\"\n\n\"And?\"\n\n\"A lot of time had passed on both corpses, but we found distinct traces of tetraodontoxin.\"\n\n\"And that is?\"\n\n\"A neurotoxin, basically. It causes paralysis, speech difficulty, shallow breathing, and slowed pulse till it gradually wears off, usually permitting brain functions and speech before it allows movement of the rest of the body.\"\n\n\"Is it dangerous?\"\n\n\"Given in large doses, it can be fatal.\" She paused, fidgeting with her fingers. \"Do you like sushi?\"\n\n\"I'm more a cheeseburger and fries girl. Why?\"\n\n\"Heard of fugu?\"\n\n\"The blowfish that's a Japanese delicacy, but if it's not prepared right it can kill you?\"\n\n\"That's the one. The toxin is found in the ovaries of the blowfish, and it isn't destroyed by cooking. Prepared properly, just enough remains to give diners a pleasantly flushed and tingly feeling. Prepared improperly, it's fatal.\"\n\n\"I think I'll stick with Quarter Pounders. Any idea where the killer might've gotten this stuff? I assume it's restricted.\"\n\n\"Theft from a lab or clinic or drugstore or hospital is always a possibility. Or he might've gone to Haiti.\"\n\n\"Haiti?\"\n\nShe nodded. \"I understand you can buy it on the street there. It's used in voodoo religious rituals. That's how they zombify plantation workers.\"\n\n\"And that's how our man keeps his victims under control. Wow. Thanks.\" I grabbed my coat.\n\n\"Lieutenant?\" Jennifer hadn't budged. \"We never would've found that if you hadn't asked for the additional tests. How did you know?\"\n\nI smiled. \"Years of experience. And training. Proper training is so important.\" I grabbed the report and headed out the door. Darcy, you little genius. I'm keeping you in this investigation. Whether Granger likes it or not.\n\nI've lived in this city all my life, so there was no excuse for it. If you live by only one rule in Vegas, it should be this: avoid the Spaghetti Bowl at all costs. But it would be the quickest way to get to the Colliers' place if traffic was down . . .\n\nBut traffic was never down, not in the loop formed by I-35 and U.S. 93-95 around downtown Vegas, joining just to the north and west of the Old Strip. The long strands of highway gave it its culinary name, and all the locals knew traffic there could be more congested than L.A. Fortunately, I had Darcy in the car, and he entertained me by reciting the crime statistics for every violent homicide in Vegas for the last fifty years.\n\nI left Darcy outside while I went in to meet the first victim's mother. Mrs. Collier was remarkably informative, all things considered. It had to be unbearable, losing a daughter. I still stung from being separated from Rachel; imagine how it must feel to know your little girl was gone forever, and worse, that the psychopathic killer who snatched her did it while you were off on a holiday. The guilt would be enormous.\n\n\"She was the sweetest girl,\" Mrs. Collier kept saying. She had not wanted to talk to me. She had already been quizzed extensively by Granger's investigators. \"Sweet, sweet, sweet. All my friends told me that girls were the worst. That once they were teenagers they became monsters. But not my Helen. She was always so sweet.\"\n\n\"I'm sure she was,\" I said, wishing the woman would come up with a more useful adjective.\n\n\"She never went in for those naughty activities some of the other girls did. She didn't chase after boys. She didn't stay out late. She didn't like to party. She preferred to tap-dance.\"\n\n\"Did you say tap-dance?\"\n\n\"That was her grand passion. Her idol was Shirley Temple, ever since she was a baby. She could've been great, given her chance.\"\n\n\"Was she taking lessons?\"\n\n\"Yes. She and her best friend, Amber, went to Miss Claire's School of Dance on South Fremont. Almost every Friday, they went to the dance recitals held in the basement at Trinity Episcopal.\"\n\n\"Any nights other than Friday?\" I asked, remembering that she was last seen on a Thursday.\n\n\"No.\"\n\n\"Never?\"\n\n\"Absolutely not. I had a strict policy on bedtime by ten o'clock on all the other nights of the week, but on Fridays I let her stay up till midnight. I checked to make sure she was in bed, safe, every night. And I locked the doors.\"\n\n\"Could I see her room?\"\n\nAfter a moment of hesitation, she escorted me upstairs.\n\nIf I hadn't known the girl was sixteen, I certainly wouldn't have gotten it from her living quarters. It was all decked out in pink frillies and lace and flowery dust ruffles and stuff I don't even know the names for. My parents never went in for this junk. Even at an early age it was clear I wasn't a Barbie girl. But evidently Helen was.\n\nOr someone thought she was.\n\n\"Had this furniture some time?\" I asked the mother, who was hovering awkwardly by the door. I figured she'd probably bought it when the girl was five and couldn't afford to replace it later.\n\nI was wrong. \"Actually, we picked this set up last year. They were having a sale at Conway Brothers.\"\n\nHmmm. \"You know, I'm probably going to be a while. I like to soak up the atmosphere, get a feel for who your daughter was. You don't have to wait for me. I'm sure you have many other things to do.\"\n\nCouldn't be any less subtle than that. \"Very well.\" She was still reluctant, but she retreated. \"I'll be in the kitchen. If you need anything.\"\n\nThank heaven. Behaviorists don't work well with an audience. I couldn't get inside Helen's head with her mother monitoring and censoring me the whole time. Helen's desk, her closet, and for that matter her entire room, were uncommonly clean. I realize I was not the prototypical teenager, but my room had never looked like this. Had her mother cleaned up before the cops arrived? Or did Mom always keep the joint like this? Was she one of those hausfraus who scurried around telling people to take off their shoes and not touch anything, making it more like a museum than a living environment? How would young Helen react to being raised by a single parent who had that obsessive-compulsive approach to life?\n\nI combed through Helen's closet, finding nothing of interest. She had a lot of clothes and tons of shoes, but I supposed that wasn't unusual for girly-girls of her age. All her outfits appeared to be of the sort her mother would approve. Tasteful knee-length tea dresses that kept the body well covered. Pep club uniform. One-piece swimsuits. No Britney Spearsish midriff-revealing outfits. No hip-hugging blue jeans. No cleavage-boosting brassieres or clinging sweaters. No Victoria's Secret lingerie.\n\nMaybe her idol really was Shirley Temple.\n\nNah.\n\nI checked the bathroom, too, but I felt certain that if there had ever been anything of interest, Mom would've removed it. I not only found nothing useful, I found nothing that suggested this girl had ever hit puberty, unless you counted the box of tampons shoved to the back of the cabinet beneath the sink. No pills, no diaphragm. Not even Clearasil.\n\nI was hoping for a diary, but no such luck. In the bottom desk drawer, however, I found a stack of collage books. Helen was a scrapbooker. But this was not your garden-variety scrapbook. There were no pictures of actors or pop stars, no Eminem or Brad Pitt. Most of the pictures came out of magazines, and all were of people in authority, people in helping professions. Police officers, doctors in white coats, firemen. Wholesome role models.\n\nWas this girl really the Pollyanna her mom thought she was?\n\nPossible. But I still didn't think so.\n\nOn the back page of one of the scrapbooks, I found a Web URL. I jotted it down in my notepad. I was putting them away when something spilled out of one of them, something that had been wedged between the pages.\n\nA torn bus ticket. Now that was interesting. She wouldn't need to ride the bus to get to Trinity Episcopal.\n\nI didn't expect to find anything useful under the bed. Wasn't that the first place parents always looked? That was where I'd kept my pot when I was her age. And God knows I'd gotten caught often enough.\n\nBut under Helen's bed, hidden in a small box wedged between the bottom of the mattress and the wooden slats, I found an outfit of clothes. It was all black. A sheer, tight lacy bodice. An equally tight, short leather skirt. Matching bra and shoes. Fishnet hose. A pair of black Ray-Bans with purple lenses. Something that looked like a white shoe polish brush but which I knew (thanks to Rachel) was actually used to put a temporary streak of color in your hair that washed right out once you were home from your revels. All told, a very exotic, erotic, interesting little outfit.\n\nGranger's investigators would've seen this, too, of course, but they wouldn't grasp the significance. They'd laugh embarrassedly, or maybe make some off-color joke about the little girl getting some action. Then they'd close the box and put it away and proceed to look for bloodstains or something else they could understand. But to me, this box spoke volumes.\n\nHelen was a closet Goth girl.\n\nDownstairs, I found that one of Helen's friends had arrived. I knew from a picture wedged into the side of the mirror above Helen's dresser that this was Amber. She was more distraught than the mother, her cheeks still red, her eyes watery. When I asked if I could have a few words with her, I thought she might faint. But she agreed. That only left the more difficult chore of getting rid of Mom.\n\n\"I don't see what you could possibly have to ask that I couldn't hear. This is about my daughter, after all.\"\n\n\"That's just it,\" I tried to explain. \"Your presence could . . . inhibit the discussion.\"\n\n\"This is still my home, and if you're going to talk to my daughter's best friend, whom I've known since she was six, you're going to have\u2014\"\n\n\"If you won't cooperate with me, ma'am, I'll be forced to call some uniforms and take her downtown. Is that what you want?\"\n\nShe stared at me stonily, lips tightly pursed.\n\n\"They'll come with the siren blazing. They'll put cuffs on her. She'll ride in the back of the cop car and be processed and printed and strip-searched before being interrogated.\" All of which was total bullshit, but I figured this lady wouldn't know.\n\nShe relented. \"Very well. But Amber, dear, listen to me.\" She took the girl's hand, and I got the immediate impression the girl wished she wouldn't. \"You don't have to say anything. You don't have to tell this woman anything. If at any time you want the questioning to stop, you just call for me. Understand?\"\n\n\"Yes, Mrs. Collier.\"\n\nThe woman disappeared herself, leaving us alone. Amber was taller and beefier than Helen had been, with lighter hair and a way of talking that seemed both lazy and smart.\n\n\"Kind of controlling, isn't she?\" I said, hoping to break the ice.\n\nAmber shrugged. \"I'm used to it.\"\n\n\"I guess you must be, if you've known her since you were six. Were you over here a lot?\"\n\n\"Most times we hung at my house. It's closer to school, and my dad keeps the pantry well stocked. Over here I was always worried that I might drop a cookie crumb on the carpet and give Mrs. Collier a heart attack.\"\n\nI grinned. Mordant Humor R Us. \"But you and Helen were tight?\"\n\n\"Yeah. Best buds.\"\n\n\"And when the two of you took off on Friday nights, you weren't going to a church and you weren't going to any Shirley Temple show either, right?\"\n\nNow she became wary. Which I could understand. Why should she trust me? \"What makes you think that?\"\n\n\"My psychic powers. Am I right?\"\n\nShe didn't answer.\n\n\"I found one of Helen's bus tickets.\"\n\nStill nothing.\n\n\"Found Helen's party suit, too, and I feel certain she wasn't wearing that getup to any church.\"\n\nAmber smiled a little.\n\n\"Where's the Goth scene these days, Amber? Was there a bar you two liked? Maybe something on campus?\"\n\n\"Nothing like that,\" she said quietly.\n\n\"Did you go down to the Strip? Pretend to be hookers just to amuse yourselves?\"\n\nI was getting warm, but I hadn't arrived. \"We did go to the Strip sometimes.\"\n\n\"To do what?\"\n\n\"Whatever. Just hang. Went to shows sometimes.\"\n\n\"And not tap dancing.\"\n\n\"Helen was more into heavy metal.\"\n\n\"But there wasn't always a concert.\"\n\n\"Sometimes we'd just walk. Go to the mall at Caesar's or the Aladdin. See what was happening at the hotels.\"\n\nOf course. \"The Transylvania. She liked the Transylvania, didn't she? Where else would a Goth girl go?\"\n\nAmber nodded. \"She got off on all that creepy stuff. Haunted houses. Horror movies.\"\n\nSure she did. Anything that was the antithesis of her mother. That was her quiet rebellion. \"Anyplace else?\"\n\n\"There was this club near the Transylvania. An Army grunt hangout. Helen was kinda sweet on military types.\"\n\n\"Do you know where she went the night she disappeared?\"\n\n\"No. I had to go to Los Angeles with my parents. So I guess she went out without me.\"\n\n\"Maybe with another friend?\"\n\n\"Maybe. But I don't know who it would be.\" Her eyes lowered. \"I bet she went alone.\"\n\nI bet she did, too, damn it. That's why she'd been so easy to snatch. \"Do you have any idea what happened to her?\" I asked, but I knew Amber didn't and I was right.\n\nI left the house excited. I still had a long way to go, but I was definitely making progress. And the bizarre thing was, I wasn't anxious to get back to HQ and wow O'Bannon. I wasn't aching to spill the beans to Lisa.\n\nI couldn't wait to tell Darcy.\n\nI found him more or less where I'd left him, out in front of the house. He was crouched down in the rather bosky garden that lined the north side of the house.\n\n\"Do you think Helen wore a size six?\" he asked as soon as he saw me. \"Because I think maybe she was a size six.\"\n\n\"No,\" I replied. \"She was too busty.\"\n\nHe looked at me, puzzled. I made an explanatory gesture. He blushed, then averted his eyes and ran his fingers through his hair.\n\n\"Did you know that I was asking about her shoe size?\" he muttered, staring at the ground. \"I was asking about her shoe size.\"\n\n\"Oh, geez, sorry.\" Pretty adorable really, watching him flush up like a radish over nothing. \"Size six, huh?\" I remembered the shoes I'd seen in the girl's closet. \"That sounds about right.\"\n\n\"I think she was a size six,\" Darcy repeated, still flapping his hands nervously. \"At first I thought maybe her mother was a size six. But I saw her feet when she came to the door and they were like boats.\"\n\nI giggled. I thought I was allowed, since my feet were also of the boatish variety. \"Why were you wondering about Helen's shoe size?\"\n\nHe pulled me into the garden, behind a row of hedges, then crouched down and pointed. Behind the hedge, close to the house itself, there was a faint but discernible impression in the soil. A footprint. The tread looked like some kind of spiked-heel number.\n\nI looked up. We were directly beneath Helen's bedroom window. There was a drainpipe attached to the wooden siding that could provide some support. Not that much was really required. Her window wasn't that high off the ground.\n\nThanks to Darcy, I had a pretty good idea how Helen could walk on the wild side on nights other than Friday. Even if her mother did make sure she was in bed at ten and locked the doors.\n\n\"You've got a good eye, Darcy. That looks like it could be a size six. Maybe seven.\"\n\n\"Six.\"\n\n\"Well, to be sure, we should\u2014\"\n\n\"It's six and five-twelfths inches long. That's a size six.\"\n\nI'd been around this wunderkind long enough to know not to argue. \"Let's get some plaster out of my car and make a cast.\"\n\n\"And after that?\"\n\nI grinned. Something about this guy brightened my spirits, just being around him. \"I think you've earned a custard. Don't you?\"\n\nHe grinned excitedly. \"Very Excellent Day! Very Excellent Day! Are you going to try the Strawberry Mash?\"\n\n\"Maybe. What about you? Vanilla Toffee again?\"\n\n\"I usually have Vanilla Toffee on Wednesdays and Strawberry Mash on Thursdays, unless there's a new flavor, and then I substitute the new flavor for whichever flavor on my list has the most letters in its name. If there's a tie, I cross out whichever one comes last in the alphabet, unless the Thursday falls on the last day of the month, 'cause then I reverse the alphabetical order and . . .\"\n\nHe was so close. She was the third and final offering, and once he was done with her his work would be complete. He had crossed the Rubicon. The Golden Age would soon be upon them.\n\n\"You hurt me,\" he said as soon as Lenore opened her eyes.\n\nIt was a long while before she could reply. Her eyelids fluttered as she slowly shook off the soporific. She parted her lips, then worked them slowly, soundlessly, as if taking them for a test drive. She tried moving other parts of her body and soon found that she could not.\n\nHe watched it all, reading her emotions as they raced through her head. Her first instinct was panic, but she stifled it. Even in this dazed state, she was smart enough to realize a cool head would be required if she was going to save herself. Her next emotion was anger, but that too she managed to sublimate. She thought that he was probably some kind of sexual deviant\u2014how could she know?\u2014and that she was more likely to survive by acting submissive and helpless. And waiting for her opportunity.\n\nIt was more than a minute before she actually spoke. \"I\u2014I'm sorry. I can see your hand is sore.\"\n\n\"I don't mean there,\" he said. He placed his injured hand over his heart. \"I mean here.\"\n\n\"I\u2014I\u2014I'm sorry,\" she said. She must be tired, lethargic from the drug. But he still sensed that she was playing him, exuding vulnerability until she had enough strength to make a break for it. Poor little offering.\n\n\"There was no justification for that sort of behavior,\" he said firmly. \"You forced me to retaliate in kind. I was not pleased.\" He lowered his head. \"I abhor violence.\"\n\n\"I\u2014I guess I just panicked.\"\n\n\"So you did.\"\n\n\"Why can't I move my arms or legs?\"\n\n\"I've given you a little something.\"\n\n\"Is it . . . permanent?\"\n\n\"It will wear off altogether soon, if I don't give you another dose.\"\n\n\"I\u2014I'd rather you didn't.\" She was laying it on a bit thick now, he thought, with the stuttering and plangent baby-girl vocalization.\n\n\"Then I won't.\"\n\n\"Really?\"\n\n\"I give you my word. No more injections.\" He paused. \"It won't be necessary.\"\n\n\"That's good. I'm glad you feel that way. Um . . .\" She batted the lashes over those lovely Asian eyes. \"Sir? Am I naked?\"\n\n\"You are. Cap-a-pie. And let me just say\u2014never have I had an easier time removing someone's clothing.\"\n\n\"So you've . . . you've done this before?\"\n\n\"Once or twice.\" She was testing, exploring. To his surprise he saw that she was already able to move the fingers of her right hand, just a bit. A strong girl, this one was.\n\n\"Are you a dentist?\"\n\nHe cleared his throat. \"I don't have a degree. But I am not without skill.\"\n\n\"Are\u2014are you going to remove my teeth?\"\n\n\"No, dear.\"\n\n\"Are you going to remove . . . anything?\"\n\nHe sighed. \"Yes, I'm afraid I am. I do regret it. But it's essential.\"\n\n\"What . . . are you going to take?\"\n\n\"Your head.\" He revealed the axe she had discovered in the truck. \"I should never have left this lying about. That was inexcusable.\"\n\n\"Please don't,\" she said. Her voice was tiny, almost invisible.\n\n\"I have no choice, my darling.\"\n\n\"I'll do anything. Anything you want.\"\n\n\"That's most generous of you. But I can't accept your offer.\" He closed his eyes. \" 'Vainly had I sought to borrow, from my books surcease of sorrow\u2014sorrow for the lost Lenore.' \"\n\n\"Don't, sir. Please don't hurt me.\"\n\n\" 'For the rare and radiant maiden whom the angels name Lenore.' \"\n\n\"Please. _Please!_ \"\n\nHe smiled at her. \" 'Nameless here forevermore.' \" And then he raised the axe over his head.\n**CHAPTER TWELVE**\n\nGot to work without incident, thank God. Took a little more than I should've before I brushed my teeth, but it was right there in the open bottle, and I had to be sure I could work without distraction today, without that stifling, panicked feeling, without my temper getting out of control. I mean, it was one thing to be drinking last night. You needed something to get you through all those bizarre Poe stories. But in the morning? I probably shouldn't have. . . .\n\nDamn them all. I can handle it. I can handle it. It'll wear off in an hour or so, and I am not going to make a habit of it. It was just this one last time. . . .\n\nI slid behind my desk, bound and determined to avoid the obvious stereotype. Sure, I know the clich\u00e9. The FBI comes to town and the local cops get bent out of joint. They're coarse and resentful. The Feebs are all cool, steely-eyed authority. There's a lot of chatter about jurisdiction\u2014wait, no\u2014\"turf.\" That's the way it's supposed to happen, in TV shows and movies. And, unfortunately, in real life.\n\nBut I wasn't getting sucked into that trap. I didn't need any more problems and I certainly didn't need anyone filing negative reports on me. I had to keep my job and to stay on my best behavior, at least until that custody hearing. So I was prepared to suck it in and be deferential. Why not? We were both trained professionals. Psychology was a fluid science. Two professionals could hold differing opinions and neither necessarily be wrong. There was nothing threatening about it, no harm in having a partner.\n\nJust so I was the one who caught the killer.\n\nMaybe half an hour later, Granger strode superciliously to my desk, avoiding eye contact, white shirt in tow. I braced myself for the inevitable fatuous remark.\n\n\"And this is the former Lieutenant Pulaski whom you've heard so much about,\" he said. \"She has been working on a temporary basis as a consulting profiler. Up until now, anyway.\"\n\nSubtle, Granger. Very subtle. I stood and held out my hand.\n\nThen my eyebrows rose, of their own accord. I was prepared for the Fed to be cool and authoritative. I was not prepared for him to be hunky.\n\n\"Patrick Chaffee, Behavioral Science Unit. Good to meet you, Lieutenant.\" His grip was firm but not oppressive. He was a couple of inches taller than me, which is saying something. He had a kind face, a friendly one. He seemed relaxed, at ease. Not like he was planning some macho squeeze play.\n\n\"Call me Susan. And just for the record, I've been working as a behaviorist for\u2014\"\n\n\"Oh, I know, I know,\" he said, still shaking my hand. \"I'm familiar with your work on the Wyndham case. I read your report in the American Academy Journal.\"\n\n\"You did?\" I said, totally nonplussed.\n\n\"Absolutely. It made the rounds at Quantico. First-rate work. Thorough and innovative.\"\n\nDid I say he looked like a nice guy? Obviously, he was the spawn of Satan. \"We got lucky on that one.\"\n\nHe blew air through his lips. \"There's no such thing.\"\n\n\"Look,\" I said, \"I've got all the files you'll want to see. I'll clear out and let you dig in.\"\n\n\"I'd rather you walked me through it.\"\n\nYet another surprise. \"You would?\"\n\n\"Absolutely.\"\n\nMy eyes narrowed. \"So . . . when you say you're from the FBI, would that be the one in D.C.? In the J. Edgar Hoover Building? Or is this perhaps some kinder, gentler FBI?\"\n\nHe laughed. \"Let me clarify, okay? This is still a Vegas PD case. Two killings, weird as they are, aren't enough to put it on our threshold. I was just asked to help. Although with you on retainer, I'm not sure why they bothered.\" He flashed his smile, the sort of smile that turned George Clooney into a twenty-million-a-flick property. \"Think we can work together?\"\n\nMy chin rose slightly. \"Possible.\"\n\nGranger looked disgusted. \"I'll leave you two psychos alone,\" he said, chuckling quietly at his own nonjoke. Nebbish.\n\nPatrick clapped his hands together. Did I mention that his eyes were blue? Oh, man, his eyes were blue. Vivid, liquid blue. \"Shall we get started?\"\n\nAs it turned out, he wasn't reading anything until he had a shot of java in him. A man after my own heart. Literally, I hoped. I took him down to the kitchen. Despite his initial generosity to me and my favorable first impression, I thought it was important to set a few ground rules.\n\n\"Let's just get this straight up front,\" I said, passing him a Styrofoam cup filled with the brackish stuff that passed for coffee around here. \"You may be the big-shot FBI behavioral specialist. That's okay, I can respect that.\"\n\nHe took it straight\u2014no cream, no sugar. Brilliant. \"I sense a _but_ coming.\"\n\n\"But I know my stuff, too, even if I didn't train at Quantico. I've been working this beat for nine years and I've earned my propers.\"\n\n\"Understood.\"\n\n\"So let's skip the usual business of lording it over me because you're fed and I'm not. I've studied John Douglas's work on sexual killers, all the interviews, all the compare-and-contrast. Hell, I've read every word the man wrote.\"\n\n\"I was trained by John Douglas.\"\n\n\"And I'm not inexperienced. My work has led to the capture of twenty-seven sexual or habitual offenders.\"\n\n\"Excellent. I've caught forty-two, myself.\"\n\n\"And I am up-to-date on the new research in my field. I read the Behavioral Science Unit's annual report from cover to cover. I read last year's twice.\"\n\nHe smiled. \"I wrote it.\"\n\nI leaned back in my chair. \"You're doing this to me on purpose, aren't you?\"\n\n\"And loving every minute of it.\" The twinkle in his eye was irresistible, even though every instinct in my body told me I should resist.\n\n\"They gave you the scoop on me, didn't they?\"\n\n\"I've seen your resum\u00e9, yes.\"\n\n\"That's not what I mean.\" I watched his eyes carefully.\n\n\"What do you mean?\"\n\n\"I think you know.\"\n\n\"That your father was a cop\u2014till he was murdered? And the case remains unsolved.\"\n\n\"That was a long time ago. What I'm concerned about . . .\"\n\n\"The drinking?\"\n\nI nodded.\n\n\"I'm okay with that.\"\n\n\"Not a problem?\"\n\n\"Long as you're sober when we're working, I don't figure it's any of my business.\"\n\n\"You aren't afraid I'll relapse and destroy the case or something? Everyone else is treating me like the Creature from the Black Lagoon.\"\n\nHe shrugged. \"I might have a little more perspective on this than they do. I used to be hooked on heroin.\"\n\n\"Heroin? You?\"\n\nHe spread his hands. \"See? Least booze is legal.\"\n\n _\"Heroin?\"_\n\nHe nodded. \"But I kicked it. You will, too.\" He crumpled his empty cup in his fist. \"Tell you what. Let's hold off on the files. Show me the crime scenes. Take me to the house where the first victim lived. We can read papers later. Make it a late night. Maybe an all-nighter.\"\n\n\"Sounds great.\" I headed toward the door.\n\n\"So,\" he said, stopping me. \"You think we can work together?\"\n\nWhat could I say? I gave him my best squinty-eyed, tell-me-no-lies look. \"Did you really have a heroin habit?\"\n\nHe grinned a little as he led the way out. \"You'll never know.\"\n\nAfter two burials, a hanging was almost exhilarating. He had allowed himself to have fun with this one\u2014why not? If the eyes of the world were going to be focused on his work, as it now seemed evident they were, he should make the most of it. He should see that the word was given to those with the perspicacity to understand it. And for the rest\u2014well, at the very least, he could entertain them.\n\nHe'd almost fallen asleep waiting for the owners to shut off the power. It had been a sea of lights, a blazing neon panorama, garish and lovely all at once. And so much to choose from! What sort of message should he seek? The profound? The prophetic? The risible? Most of the older casinos had deposited signs here, as well as restaurants, hotels. Even theme parks. There were so many possibilities. . . .\n\nIn the end, the choice was obvious. Puns might be the bailiwick of the insipid humorist, but this was irresistible. A huge, towering, crane-held sign. The front proclaimed FIRE SALE. While the back read HALF OFF.\n\nThat was where he left Lenore, the biggest part of her, anyway. Hanging beneath the HALF OFF.\n\nSuch a beau geste! How would dear Susan react? he wondered. It was amazing how completely the woman had come to dominate his thoughts in so short a time. It was impossible to resist thinking about her. Somehow, knowing that there was someone out there with the potential to appreciate his work made what he did so much more thrilling. The thought of carrying on without her was intolerable. But what if she was replaced? He had read in the paper that the FBI was sending the LVPD a federal expert in the same field. What if O'Bannon decided Susan's services were no longer needed?\n\nHe couldn't let that happen. He would have to do something to prevent it.\n**CHAPTER THIRTEEN**\n\nOddly enough, Granger had not arranged high-speed Internet access for me, possibly because he hadn't arranged a computer terminal for me, so I had to sneak into O'Bannon's office while he convoked his top detectives in the conference room. Once I was in, I checked out Helen Collier's Web site. She had obviously done it all herself. The signs of amateur webmastering were everywhere. The layout was functional but unadorned. A lot of hyperlinks led to nothing. But what was there was interesting.\n\nHelen had scanned some of her own artwork, the same kind of drawings and collages I'd found stuffed in the bottom drawers of her desk. I wasn't surprised. One look at the meticulous living room was sufficient to tell me that Mrs. Collier was never going to put her daughter's art on display in the house. Not even on the refrigerator. So Helen had found another way to exhibit it. She'd kept a blog, too\u2014a Web diary. It hadn't been updated for two months, but what she had written was fascinating. Darcy was right\u2014she'd been creeping out at night through the bedroom window for at least a year and a half.\n\nAll the photos of Helen were distorted, maybe for security reasons, maybe just because she thought it was cool. But using my imagination, I could get a pretty good idea what she had looked like when she hit the street in that outfit. False eyelashes, black fingernails, big hoop earrings. She would definitely attract attention. Even more clearly, she had a taste for the dark side. I could see this dodgy girl talking to a stranger, particularly if he gave her some reason to trust him. I could even see her getting into his car. Making the biggest mistake of her too short life.\n\nI should've stopped reading the blog right then and there, but of course I didn't. I kept moving backward in time until I got to an entry describing a family trip to Carlsbad Caverns. As soon as she was down in the cave, she'd freaked. Totally lost her head. Turns out that prissy mother of hers used to exact punishment by locking her in a small, dark closet and she'd been claustrophobic ever since.\n\nSo just imagine what happened when Helen found herself locked up in that coffin. No light, no air. Barely able to move. No one to hear her screams.\n\nSmall wonder her fingers were shredded, the lid of the coffin was so scarred.\n\nI had to catch this killer. Soon.\n\n\"Seen this?\"\n\nPatrick tossed the morning paper on my desk. The double-sized headline was easy to read: KILLER INSPIRED BY POE!\n\nI scanned the story by Jonathan Wooley, the reporter who had been covering the case. He knew about the quotes and he knew the murder methods re-created scenes from Poe's fiction. \"I thought we were keeping this to ourselves.\"\n\n\"So did O'Bannon,\" Patrick informed me. \"He's furious. Who do you think leaked it?\"\n\n\"I have no idea. For his sake, I hope O'Bannon doesn't find out.\"\n\nPatrick propped his feet up on the edge of my desk, leaning his chair back against the men's room door. It was generous of him to stay out here with me. I knew perfectly well Granger had given him a nice private office.\n\n\"I read your preliminary profile. Good, solid work.\"\n\n\"Thank you. I appreciate your\u2014\"\n\n\"So you won't mind, I hope, if I say we should tear it into pieces and start from scratch.\"\n\nSlow burn. \"You think I'm on the wrong track.\"\n\n\"Not at all. I just prefer to build from the ground up. I've had previous cases where I came in late and tried to operate within the parameters of preexisting profiles. It doesn't work. Even when I have full and free license to edit.\"\n\n\"Okay.\" I was not going to throw a fit. I was not going to act defensive. There would be no turf war, damn it. \"Why don't you work up your own profile, then we'll compare\u2014\"\n\n\"No, no,\" he said, looking at me with those baby-blue eyes that could probably persuade a chimney to give up smoking. \"I want us to do it together.\"\n\n\"Look, you don't have to humor me\u2014\"\n\n\"Not at all. You've got the experience with this case, not me. And you've got a solid background in behavioral sciences. I might be able to contribute some of the latest thoughts and theories. We'll work together.\"\n\nLike I said before, almost too perfect. \"Okay, where do we start? What do we know?\"\n\n\"Statistically speaking,\" Patrick began, \"our killer is most likely a white male between the ages of twenty and forty-five. Over ninety percent of all American serial killers are.\"\n\n\"The cops already know that. What else can we give them?\"\n\n\"Let's start with preliminary classifications.\"\n\n\"Organized and disorganized?\"\n\n\"Essentially. But that terminology has fallen out of favor. Roy Hazelwood has modified Douglas's work somewhat in this regard. He prefers to start by distinguishing between the impulsive offender and the ritualistic offender.\"\n\n\"I'd say our guy is ritualistic.\"\n\n\"Definitely. A thinking killer. Someone who has spent an enormous amount of time working out his fantasy and bringing it to life. He's not taking the easy way, or the approach that would be most likely to avoid detection. He's planning everything in accordance with some loony scheme.\"\n\n\"The Poe fetish.\"\n\n\"So it seems. Bringing those weird stories to life has become an id\u00e9e fixe for our man. But what does he hope to accomplish?\"\n\n\"Good question. Wish I had an equally good answer.\"\n\nHe sat up to let one of the sergeants pass into the bathroom. \"Hazelwood has delineated the five components of the ritualistic killer: relational, paraphilic, situational, victim demographics, and selfperceptional.\"\n\n\"You're going to have to explain.\"\n\n\"Relational has to do with the relationship between the victim and the offender\u2014or more accurately, what he fantasizes the relationship to be. Girlfriend? Wife? Slave?\"\n\n\"And the answer is?\"\n\n\"We don't know. We need more information. Your coroner says the victims haven't been sexually molested, at least not in the sense of penetration. Our man may be a kidnapper, but he's no lothario.\"\n\n\"Probably impotent.\"\n\n\"A distinct possibility, but we both know there are still ways for a crazed man to inflict sexual damage and humiliation on a helpless woman. If we knew more about what he does with them before he kills them, that might yield some answers. Or if we knew how he selects them. How he lures them in.\"\n\n\"Next component?\"\n\n\" _Paraphilia_ is the currently vogue term for sexual deviation. Voyeurism, pedophilia, necrophilia, transvestitism\u2014you name it.\"\n\n\"You think this guy can't get it off the normal way, so he's grabbing little girls off the street.\"\n\n\"I'm not saying that. This could be a twisted form of sexual sadism. A way of asserting his power over them. He renders them powerless with the drug, then subjects them to some Poe-inspired horror. A form of slavery, I suppose.\"\n\n\"But there's no indication that he's trying to break their will. Play with their minds. Turn them into true slaves.\"\n\n\"Not yet, maybe. But this guy is just getting started.\" A grim expression crossed his face. \"Let's hope we catch him before it gets to that.\"\n\n\"Situational?\"\n\n\"That's key to understanding what our boy is up to. What's the situation he's trying to create? What setting is he trying to realize?\"\n\n\"I'm not sure I follow.\"\n\n\"For instance, when I'm giving lectures back at Quantico, the setting I'm trying to create is a classroom. The relationship is teacher-student.\"\n\n\"I got you.\"\n\n\"Or here, for instance, with us, the setting is master-servant.\" His eyes sparkled. \"The young prot\u00e9g\u00e9 learns at the feet of the seasoned master.\"\n\n\"Is that what this is? I thought it was more like the hopeful acolyte worships at the temple of the earth goddess.\" Okay, maybe that was a little obvious, but he'd started it.\n\nHe cast his eyes about. \"Not much of a temple.\"\n\n\"I'm a rose-colored-glasses girl.\"\n\nHe dragged the conversation back on track, darn it. \"So I'm thinking the setting this guy wants to create must be a sort of torture chamber.\"\n\n\"Like Robert Leroy Anderson?\"\n\nHe arched an eyebrow. \"Very good. You are up on the literature.\"\n\n\"I do my best.\"\n\n\"So he's using Poe for inspiration but is basically serving his own sadomasochistic need to inflict pain on helpless victims.\"\n\nMy face scrunched. \"I don't know.\"\n\n\"You have a different theory?\"\n\n\"No. I don't know. Maybe you're right. I just sense there's something more going on here. He's had so many opportunities for cruelty, but actually there's been little evidence of it. Kidnapping and murder, yes, but\u2014I don't know. Sadomasochistic lust just doesn't explain everything.\"\n\n\"Which leads us to our fourth component. Victim demographics.\"\n\n\"Well, they were both young girls. Teens.\"\n\n\"Both girls look young for their age.\"\n\n\"That's true. A baby-doll fetish?\" I shrugged. \"They came from very different backgrounds. One was solidly lower-middle-class. The other came from a super-wealthy background, daughter of a celebrity. Both appear to have been raised by their mothers.\"\n\n\"But did the killer know that?\"\n\n\"Seems unlikely.\"\n\n\"So he was just going by appearance?\"\n\nI have to admit, I hadn't thought I'd like working with a partner, but I did. Bouncing ideas off someone who had the same grasp of the field was exciting, almost electric. I felt a tingling run through my body that wasn't all about serial killers, either. Good thing Patrick wasn't in any position to make advances. I would've melted like a custard. \"I don't think so.\"\n\n\"Why not? It's the logical conclusion.\"\n\n\"Despite the age and gender similarity, both girls looked quite different.\"\n\n\"He can only choose from what's available.\"\n\n\"This is Vegas, Patrick. You can find anything you want, and plenty of it. Take a short walk down the Strip and you'll find a dozen girls who fit any possible physical description. No, he chose these victims because they fit some specific parameter\u2014we just haven't figured out what it is yet.\"\n\n\"And as to the killer's self-perception?\"\n\nI pondered a moment. \"That's more difficult.\"\n\n\"He obviously likes being in control. Exerting power over others.\"\n\n\"Ye-es . . .\"\n\n\"He enjoys inflicting pain on his victims.\"\n\nMy neck twisted. It would be easier just to agree than to try to explain my reluctance. But as always, I had to go with my feelings. \"We don't know that.\"\n\n\"Susan, think about what he did to these two girls.\"\n\n\"I know. But that doesn't mean he enjoyed it.\"\n\n\"What other possible reason could he have for burying a woman alive? For making someone bleed to death?\"\n\nI shrugged. \"This may be a rather heterodox theory, but I don't believe this guy perceives himself as an evildoer. Or even a punisher. He's communicated with us twice, but there've been no jeremiads about whores and harlots. No suggestion of guilt on the part of the victims. I get the sense that he somehow thinks what he's doing is . . . honorable. That he's acting purposefully to accomplish . . . something.\"\n\n\"Like what?\"\n\n\"I can't imagine. You think our guy has a personality disorder?\"\n\n\"Duh.\"\n\n\"Psychopath?\"\n\n\"Actually, we don't use that term anymore.\"\n\n\"Oh, spare me.\"\n\n\"The currently preferred mental health term is antisocial personality disorder. APD for short.\"\n\n\"Whatever. You think that's our guy?\"\n\n\"Not if you're right that he thinks he's doing a good thing. That would be more like . . . I don't know. Schizoid personality disorder.\"\n\n\"Or a narcissist.\"\n\nPatrick batted a finger against his lips. \"That's not bad. Delusions of grandeur. Belief that he's special and his actions can't be comprehended by ordinary people. Feeling of divine entitlement.\"\n\n\"If I'm right, what does it tell us?\"\n\n\"That he needs constant admiration. That he won't hesitate to take advantage of others in order to achieve his plan, whatever it is. That he will be indifferent to or unaware of the needs or feelings of others. Basically, the world is his stage, and the rest of us are just props at his disposal.\"\n\n\"How does that help us catch him?\"\n\n\"Well, he'll be seeking attention. Praise, even.\"\n\n\"He's going to try to contact us, isn't he?\"\n\n\"Almost certainly. He already has, with those coded notes that were bound to lead us to the Poe connection. But he'll do more. He'll talk to us.\"\n\n\"Good. That would help me understand him, what he wants. Empathize.\"\n\n\"With a serial killer? Is that a good idea?\"\n\nI stood up and stretched, wondering whom I'd have to sleep with to get a fresh cup of coffee. \" 'Is that a good idea?' hasn't really been the touchstone question of my life.\"\n\nWe were well on our way to a solid and surprisingly useful profile when we were interrupted by O'Bannon's secretary, Madeline. She looked put out, probably because she'd had to walk across the station to deliver the message herself\u2014Granger hadn't given me a phone.\n\n\"O'Bannon wants to see you. Right now.\"\n\nAs if she thought I might keep the man waiting for an hour or so. Maybe stroll down to the pub and have a few drinks first. Whatever\u2014I didn't have to look at the woman to see that she had it in for me. So I obediently pushed myself off the Naugahyde and headed for O'Bannon's office. I was almost there when I was accosted by an attractive middle-aged woman with large red spectacles and a neatly tailored suit. I didn't have to look at the label to know it hadn't come off the rack. And I didn't need an introduction to know who she was.\n\n\"Are you Susan Pulaski?\" she asked.\n\nMoments like this, one has to wonder about the wisdom of the saying \"Honesty is the best policy.\" \"I am.\"\n\n\"I'm Fara Spencer. The mother of Annabel Spencer. She was\u2014\"\n\n\"I know who she was,\" I said, sparing her the explanation. \"I'm sorry for your loss.\"\n\n\"Thanks, but that really isn't good enough. I want to catch the bastard who did this.\"\n\n\"We all do, ma'am.\"\n\n\"That's what I keep hearing, but as far as I can tell, no one is doing anything.\"\n\n\"I can assure you\u2014\"\n\n\"I have some serious complaints about the way this investigation is being handled.\"\n\nI tried to edge past her, but she wasn't budging. \"Any complaints should be directed to Lieutenant Granger. It's his case.\"\n\n\"My understanding is that he's essentially a supervisor. All my sources tell me that in a case such as this, a proper psychological analysis is critical to catching the killer. And that's your department, right?\"\n\n\"I'm also working with\u2014\"\n\n\"So let me be blunt, Ms. Pulaski. Do you think you're up to this?\"\n\nEvery joint in my body stiffened. \"I've been working as a behaviorist for over\u2014\"\n\n\"I'm aware of that. I'm also aware that you were recently fired and have not been reinstated. I know about your personal problems. As well as the . . . addiction that led to your hospitalization.\"\n\n\"You're terribly well informed, aren't you?\"\n\n\"In my business, it's essential.\"\n\n\"Well, in my business, it's essential to know what the hell you're talking about. We're running a first-rate investi\u2014\"\n\n\"I've had recovering alcoholics on my show, Ms. Pulaski, as well as experts in the field of substance abuse. And I know you can't just square your shoulders and be cured a week after you go into rehab.\"\n\n\"It was detox, not\u2014\"\n\n\"Frankly, I'm appalled to think that a critical role in the apprehension of my daughter's killer has been relegated to someone who only weeks before was suffering paranoid alcohol-infused delusions and behaving in a violent and psychopathic manner.\"\n\n\"We don't say _psychopathic_ anymore,\" I told her through clenched teeth. \"I'm surprised all those experts on your show haven't told you that.\"\n\n\"My point, Lieutenant Pulaski, if indeed it is still appropriate to refer to you as a lieutenant, is that you have no business working on this case. I want you to resign so that someone better qualified can take your place.\"\n\nAll right then, the gloves were off. \"Ms. Spencer, this isn't some daytime TV show dispensing feel-good bullshit to bored housewives. This is reality. And the reality is, I'm good at what I do. You're not going to find a replacement who does any better.\"\n\n\"I find that very difficult to believe. Your judgment is clouded.\"\n\n\"Ma'am, don't talk to me about clouded judgment. With all due respect, I'm not the one who just lost her only daughter. In every case of this nature we have grieving parents, and they are almost always obstacles, not assistants. We put up with it because we are service-oriented professionals and we realize that dealing with the death of a loved one is difficult.\"\n\n\"Certainly you're a testament to that,\" she said dryly.\n\nI sucked it in, showing a degree of restraint that surprised even me. \"My point is, we're doing everything we can to catch the killer and we will continue to do so. If you're not going to help, get the hell out of the way!\" I pushed her aside and marched on toward O'Bannon's office.\n\n\"You haven't heard the last of this,\" she shouted after me, confirming what I already knew all too well. \"I'm not that easily brushed off.\"\n\nI entered O'Bannon's office and slammed the door behind me. He was sitting at his desk, pretending to rifle through some papers, but really just marking time till I arrived.\n\nAnd to my surprise, Darcy was there, too, sitting in a chair just behind him.\n\n\"Hey, Darce,\" I said, wiggling my fingers. \"How's my main man?\"\n\nHe blinked. \"Did you know that coffee is the second largest trade commodity in the world market? Americans consume more coffee than the inhabitants of any other nation on earth.\"\n\n\"It's my fault. I skew the average.\" I turned my attention to the boss. \"What's up, Chief?\"\n\n\"How are you getting along with the Feeb?\"\n\n\"Swimmingly.\"\n\n\"No complaints? On either side?\"\n\n\"Not that I'm aware of.\" I looked at him. \"You seem somewhat incredulous.\"\n\n\"I'm glad to hear you've finally learned to play well with other children.\"\n\n\"So what's Darcy doing here?\"\n\nO'Bannon squirmed slightly. Not physically; he was too savvy for that. But I saw it, just the same. \"He . . . asked to come. He wanted to see you.\"\n\nI smiled. \"No new developments today, Darce. No more crime scenes.\"\n\n\"Nonetheless, he . . . wanted to be involved.\"\n\n\"Okay by me. But Granger apparently has a problem\u2014\"\n\n\"I'll speak to Granger. He's pissed at me already because of the press conference.\"\n\n\"We're doing a press conference? I thought you hated\u2014\"\n\n\"We don't have any choice. Do you realize how much attention this case has been getting?\"\n\n\"I've seen the local papers.\"\n\n\"It's not just local. It's everywhere. Only thing worse than a serial killer is a serial killer during a slow news week. I guess it's to be expected\u2014this case gets weirder and weirder every day. The press has a million questions about the Poe connection. And now we've got a beloved television celebrity involved. We're the lead story on CNN Headline News, for God's sake.\"\n\n\"So if the press is all over it already, why hold a conference?\"\n\n\"We've got to do something. Tourism is down dramatically. People are canceling their vacations. I guess no one wants to gamble badly enough to risk being buried alive. Or bled to death.\"\n\n\"But unless they're young women\u2014\"\n\n\"Which is one thing we need to explain. But the main chore will be to convince everyone that we're working hard and we have substantial leads.\"\n\n\"We do?\"\n\nHe ignored me. \"I've got the Chamber of Commerce breathing down my neck, Susan. The mayor. The hotel commission. We need to put on a dog-and-pony show.\"\n\n\"Okay. So Granger is traumatized because he has to go before the press?\"\n\nO'Bannon's eyes drifted down to his desk. \"Granger isn't doing it. You are.\"\n\n\"Me? But he's\u2014\"\n\n\"You know as well as I do that once those reporters get going with Granger he'll come off looking like a doofus. He's a good cop, but quick wit isn't his specialty. Besides, what does he have to tell them? The only person who's come up with anything useful is you.\"\n\n\"And Darcy. But what if the press hassles me about my recent demotion to consultant status?\"\n\n\"Tell them it's none of their damn business. Stick to the case.\"\n\n\"But\u2014\"\n\n\"Don't bother arguing. You're doing it. Unless you'd like me to terminate that consulting agreement.\"\n\nI fell silent.\n\n\"Good. The press will assemble at five. So go change your clothes and do something with your hair and get ready. If you want to bring your FBI guy, that's fine, but I want you to do the talking. We have to impress upon them the fact that the LVPD is in charge and has the investigation under control.\"\n\n\"Great. Just great.\" I checked my watch. \"I'd better\u2014\"\n\nHe held up a finger. \"One more thing.\" He turned. \"Darcy, can you get me some more coffee?\" O'Bannon held out his mug. \"I'd really appreciate it.\"\n\nDarcy tilted his head. \"D-D-Did you know that Americans drink over forty million cups of coffee a year?\"\n\n\"Fascinating. So refill my mug, will ya?\"\n\nDarcy left the office. And O'Bannon gave me the harshest look I'd had since I got out of detox. \"Don't hurt him.\"\n\nI was totally flummoxed. \"What?\"\n\n\"You heard me. I don't want my boy hurt.\"\n\n\"I wouldn't dream of hurting him. He's adorable. And I think he loves working on this case.\"\n\n\"What he loves is\u2014\" He stopped, shook his head.\n\n\"I'd think you'd be pleased. You know, in his own weird little way, he has a real aptitude for detective work.\"\n\n\"There is no way in hell Darcy could cut it as a detective. He can't even carry on a coherent conversation.\"\n\n\"He might need help in certain areas, but who doesn't? Lots of people can't carry on conversations. I don't know anyone who can do what Darcy does.\"\n\n\"How could he interview a suspect? How could he organize his thoughts and come up with a theory? Write a report? It's ridiculous.\"\n\n\"I think you're being too hard on him.\"\n\n\"I didn't just stroll into his life last week. I think I know a little something about what he can and cannot do.\" He muttered something under his breath. \"Look, you can take him around to the crime scenes. Let him talk to the techies. Fine. But I don't want him hurt. Are we clear?\"\n\nI stood quietly at attention. \"Yes, sir.\"\n\nThe door opened again and Darcy entered. \"Here's your coffee, Dad.\"\n\nO'Bannon took a deep drag, then winced. \"What the hell is this? This isn't\u2014\"\n\n\"I got you decaf. Because caffeine is not good for you.\"\n\n\"Decaf?\"\n\n\"In controlled studies conducted at Stanford University, caffeine and caffeine withdrawal were linked to headaches, nosebleeds, stomach disorders, irritability, impotence . . .\"\n\nO'Bannon pressed a hand against his forehead. \"One more thing to remember about this press conference, Susan. Your psycho killer may be watching.\"\n\n\"Almost a certainty,\" I said. \"So I'll be careful not to make him feel challenged, offended, maligned. No telling what he might do if that happened.\"\n\nHe was disturbed.\n\nHe had done everything according to plan. He had sacrificed the offerings. He had followed the directions in the prophet's work. But the Golden Age had not come. Ginny had not been returned to him.\n\nWas it possible he was wrong?\n\nIt must simply be delayed. A transformation of this magnitude cannot come about overnight. This would give him more time to get the word out. The media coverage had exceeded his most fevered imaginings. It seemed he was everywhere, or his work was\u2014on newspapers and magazines, on the television, on street corners and newsstands and even the giant electronic billboard on the MGM Grand. A condign response to actions of this boldness, of this import. His great commission had been to spread the good news, to tell those who would hear of the coming of Dream-Land. He'd become a sensation.\n\nSuch success could only presage greatness. Such acceptance could only validate the rightness of his path. With this degree of exposure, he could be assured that any receptive ears would hear the message. Not everyone would understand it, of course. Some would write it off as just another news story. Another pathetic wretch trying to get his fifteen minutes. But the enlightened would see more. The prophet had known his message would not be heard by everyone: _To the few who love me and whom I love\u2014To those who feel rather than to those who think\u2014To the dreamers and those who put their faith in dreams as the only realities\u2014I offer this Book of Truths. . . ._\n\nOn the television, a popular talk show host was interviewing an English professor about how Poe's dark and nihilistic visions might inspire an unbalanced personality. The professor appeared delighted to be consulted. Not surprising, in this age in which colleges push professors to become media consultants as much as they push them to publish. An expert in American literature probably receives few calls from the six o'clock news.\n\n\"Tell me the truth,\" the host said, leaning forward in her swiveling chair. \"This Poe stuff is mostly for kids, isn't it?\"\n\n\"Not at all,\" the man replied, straightening the cuff on his tweed jacket. \"Poe was an important figure in American literature\u2014indeed, in world literature. He invented, or at least defined, the modern short story as a literary form. He invented detective fiction, wrote the first true science fiction story. He invented symbolist poetry and the New Criticism, which would be fully realized only half a century later, during the Modernist era. He may not be America's greatest writer, but I would be hard pressed to identify one whose contributions were more widespread. Poe had a huge impact on many great writers. Oscar Wilde. Jules Verne. Thomas Pynchon. Nabokov. Poe has fallen out of favor with my academic colleagues at this time, who tend to favor Faulkner as the chief figure in American literature. But Poe's influence has been vastly greater.\"\n\n\"But Dr. Watson . . . Poe may have been a good writer, but wasn't he kind of a freak?\"\n\n\"Much of the Poe persona as we know it today was the creation of his literary executor, Rufus Griswold, who was jealous of Poe's work and sought to destroy his reputation. He began this stereotype of Poe as a dark, cruel, nasty, abusive alcoholic. In fact, most accounts from contemporaries who knew Poe describe him as charming, witty, intelligent, generous, courteous, even chivalric. Women adored him; toward the end of his life he was seeing several wealthy socialites at once. He had lived in the North, the South, and even England, but always considered himself a southern gentlemen and behaved accordingly. Except when he was drinking, of course.\"\n\n\"But those stories he wrote\u2014that's really twisted stuff. Burying people alive\u2014\"\n\n\"Premature burial was a widespread and much discussed phobia in the nineteenth century. Not just with Poe. There had purportedly been a true incident that got great play in the papers. People began buying coffins with escape hatches that could be activated from the interior or that had a bell the interred could ring if consciousness returned.\"\n\n\"Okay, and how about that teeth-pulling business?\"\n\nThe professor held up his hands. \"Make no mistake about it\u2014Poe wrote some strange tales. His imagination was given to the macabre and sensationalistic. But he was trailblazing\u2014writing a kind of story that had never been attempted before. And to some extent, all of his work is united by his strange belief system. Poe believed that there was another world, a better one, that we saw glimpses of when we dreamed. Poe even believed it was possible . . .\"\n\nHe switched channels to the press conference, which was late in starting. No doubt they had woodshedded Susan, drilling her on what could and could not be revealed. With so many eyes watching, they must be concerned about her somewhat mercurial temperament. Either that or they were pouring coffee down her gullet, sobering her up, poor thing.\n\nHow interesting that she should share the prophet's infirmity. Remarkable\u2014or a sign of shared destiny?\n\nAt last, Susan approached the podium, looking elegant in a sleek blue jacket and white slacks. In the background, he spotted Lieutenant Granger and that new companion, the one who looked as if he barely knew where he was. Susan adjusted the microphone and began.\n\n\"My name is Susan Pulaski. I'm a behavioral expert and consultant to Lieutenant Barry Granger and the Vegas PD. I have a prepared statement, and then I'll be able to answer some questions.\n\n\"Let me make one point up front. Our investigation into these murders is ongoing. All available resources have been assigned to this case, and we are also working in consultation with federal authorities and various experts in the fields of serial crime and sexual deviancy.\"\n\nHe raised an eyebrow. Sexual deviancy? He had never touched any of these girls in that way. Surely she must know that.\n\n\"All our available manpower and then some is working around the clock to bring this killer to justice. While we are not prepared to make an arrest at this time, we have several very promising leads. We are working closely with a top FBI profiler and we hope to be in a position to make an arrest soon.\"\n\nA bit of bravado? he wondered. Probably fed to her by O'Bannon, necessary to keep the press at bay. But unwise. He had no intention of being apprehended.\n\n\"This is what we know for certain. The first victim, as I believe most of you already are aware, has been identified as Helen Collier. She died of asphyxiation after being buried in a mock coffin at the Transylvania hotel. The second victim was Annabel Spencer, a mathematics student from MIT, apparently a weekend visitor who had been gambling at some of the local casinos. She died from acute blood loss. Her body was hidden in an abandoned aircraft stowed on a back lot at McCarran International. We have one possible eyewitness and a host of physical evidence.\"\n\nHands shot up, but Susan ignored them, continuing with her statement.\n\n\"It is true, as has been speculated in the press, that the killer obtains inspiration from the works of nineteenth-century American writer Edgar Allan Poe. Both bodies were left with cryptic messages drawn from the works of Poe. The murders appear to be reenactments of scenes from Poe. While this is colorful and good for headlines, it is not particularly helpful in tracking the killer, nor is it unprecedented. Past serial offenders have been known to derive inspiration from the zodiac, old movies, _The Catcher in the Rye,_ even Beatles songs. What is important is that we focus not on the superficial trappings but on the personality that lies beneath it. Understanding that personality, we believe, is the key to preventing future crimes.\"\n\nShe folded her notes and looked out into the crowd. \"I can take a few questions now.\"\n\nAgain, the hands flew upward. Susan scanned the gallery. She's looking for something, he realized, but what? A familiar face? A friend somewhere in the valley of the vultures?\n\nShe pointed.\n\n\"Lieutenant Pulaski, do you have a description of the killer?\"\n\n\"I'm not prepared to provide that information at this time.\"\n\n\"Does that mean you don't? Because if you do\u2014\"\n\n\"I'm not prepared to provide that information at this time.\"\n\nHe supposed she was hesitant to reveal the description he gave her when he was disguised as Ethan. Very smart, Susan. He hadn't given her enough detail to make it possible for anyone to find the killer (even were it accurate). Revealing it now only risked potential embarrassment if (when) it turned out to be wrong.\n\n\"Do you have a psychological profile of the killer?\"\n\n\"We are developing and refining our profile every hour of every day. I am not authorized to provide details of the current profile to you.\"\n\nAn anorexic reporter in the front row cut in. \"You know, Lieutenant Pulaski, that will lead some to speculate that you have no valid profile.\"\n\nSuch thinly disguised calumny. Susan was doing a good job of masking her reaction, but he felt it, all the same. \"I can assure you, ma'am, that my colleagues and I are more than able to perform our jobs, and as I mentioned, we have been working in consultation with federal specialists. But we do not feel it would be advantageous to reveal everything we know about the killer at this time.\" She apparently couldn't resist adding, \"Use your brain. It isn't hard to figure out why.\"\n\n\"What's his motive?\" someone else asked.\n\n\"We can only speculate. Certainly we do not believe there is any rational motive, such as greed or jealousy. Given the vagaries and inconstancies of the psychotic mind, determining what delusion motivates him can be extremely difficult.\"\n\nHe felt a flash of anger\u2014 _psychotic?_ \u2014then checked it. No, he told himself, don't blame her. She doesn't understand, not yet. She couldn't possibly know.\n\nA handsome middle-aged man with a salt-and-pepper goatee spoke up. \"Jonathan Wooley, _Vegas Courier._ \"\n\nHe inched forward. Wooley had written all the best pieces he had pasted into his History.\n\n\"Ms. Pulaski, I respect that you don't want to reveal everything you know\u2014you don't want to force the killer to change his MO, and you probably need some undisclosed details to distinguish crank informants from people with actual knowledge. But surely you can understand our position. Basically we're asking: are we safe?\"\n\n\"The LVPD is taking all possible steps to ensure\u2014\"\n\n\"Pardon me, but you weren't able to stop the previous killings and you've given us no reason to believe you can prevent any future ones. Are the citizens of Las Vegas safe?\"\n\nHe noticed that Susan's fingers tightened, almost imperceptibly, on the lectern. He knew what she was thinking. She wanted a drink.\n\n\"The LVPD is advising citizens to take extreme caution until the assailant is apprehended, particularly young women who are\u2014or look\u2014ages fourteen to twenty.\"\n\n\"Should we close the casinos? Are the discos safe? How can people protect themselves?\"\n\nThe hesitation in her voice\u2014first time yet\u2014told him this was something that had been discussed before the conference. He suspected that she favored telling people to lock themselves in their homes and shutting down the whole city. But of course the powers that be would never permit that.\n\n\"We have to keep our heads on straight and not let fear get the best of us. For the most part, people should proceed with their normal lives. But they should exercise extreme caution, particularly those in the target gender and age group. Don't travel alone. Have someone walk you to your car. Don't speak to strangers. And most especially, don't get in a car\u2014or truck\u2014with someone you don't know.\"\n\n\"But how can we know what situations create jeopardy, when we know so little about the killer?\" This Wooley was relentless. He supposed that was why the man was a successful reporter, but it was beginning to wear a bit thin. \"Are his victims chosen at random?\"\n\n\"I've said all I have to say on the matter,\" Susan replied firmly. \"Now if there's nothing more, I\u2014\"\n\n\"Perhaps you wouldn't mind answering this question, Lieutenant Pulaski. That is, _former_ Lieutenant Pulaski.\" A camera pivot revealed a woman barreling her way through the heart of the assembled press corps. \"Are you qualified to work on this investigation?\"\n\nHe edged toward the television. It was that woman, the TV host. Annabel's mother.\n\nSusan cleared her throat. \"Mrs. Spencer, as I think you know, Lieutenant Granger is in charge of the investigation. I'm only consulting on\u2014\"\n\n\"You're the behavioral expert. The only one on the force.\"\n\n\"Actually, I'm working with a representative from the FBI who\u2014\"\n\n\"But so far, you haven't come up with anything. My daughter's killer is still at large. And I am outraged.\"\n\nThe reporters drank up this unexpected bit of conflict. Pens scribbled madly. The minicams shifted their gaze from the podium to the aggrieved mother.\n\n\"Mrs. Spencer, everyone at the LVPD is sorry for your loss\u2014\"\n\n\"But not sorry enough to do anything about it, apparently. Why hasn't this man been caught? Why aren't you doing more?\"\n\nSolipsistic firebrand. Quarrelsome quidnunc. She had no right to embarrass Susan in public. She was just trying to steal the limelight, the glory hound. Everything Annabel had said about her was true. She was unworthy, as was this behavior. She'd be better served to consider how her daughter could be with child and in Las Vegas without her knowing about it.\n\n\"Again, let me make it clear that everyone is doing their best\u2014\"\n\n\"But that's not good enough, is it?\" The mother stepped up on the raised platform. \"I am publicly calling here and now for a clean sweep of the LVPD team and full federal assumption of this investigation.\"\n\n\"That isn't even constitutional and it wouldn't\u2014\"\n\n\"Talk is meaningless. We're looking for a deranged killer, a psychopath. We're not going to catch him with these Deputy Fife officers.\"\n\nSusan's cheeks burned. \"I can assure you, ma'am\u2014\"\n\n\"In the meantime, I'm demanding the immediate removal of several LVPD personnel who have done nothing but obstruct reasoned efforts to bring this fiend into custody. Beginning with Robert O'Bannon.\"\n\n\"That's uncalled for. Chief O'Bannon has years of\u2014\"\n\n\"His conduct has been grossly negligent.\"\n\n\"He knows more about police work than you could learn in\u2014\"\n\n\"He hired a drunk to work on my daughter's case!\"\n\nThe harsh words sliced through the press conference like a laser. Susan took a step back, almost staggered.\n\n\"I guess you didn't cover that in your opening statement, did you, Ms. Pulaski? Perhaps you'd like to explain why you are no longer a member of the LVPD. Why your employment was terminated. Perhaps you'd like to explain where you were residing only ten days ago or why you have a bandage on your wrist.\"\n\nSusan's teeth were tightly clenched. \"This conference is over.\"\n\n\"I'm calling a press conference of my own!\" the mother all but shouted. \"I want some action! And I'm willing to put up my own money to see that it happens.\"\n\n\"I don't need this,\" Susan said, folding up her materials.\n\nThe mother grabbed her arm. \"I want my daughter's killer found!\"\n\nSusan pushed her away. \"We all do, ma'am.\" She left the stage, and the live feed gave way to background commentators rattling about the \"surprising and dramatic turn of events\" at the press conference. Although they didn't come right out and say they agreed with the mother, they were quick to note that she was not the only one who had criticized the way this investigation was being handled. They couldn't resist adding that there were many unanswered questions about Susan Pulaski's involvement, suggesting that they had known all along about her alcohol addiction and time in detox which, of course, none of the lazy mouthpieces had.\n\nHe shut off the television in disgust. His anger at that hideous woman and the ignorant press was intense, but not so intense as the sorrow he felt for his poor damaged Susan. She must be devastated. She would try to act tough, as if it didn't bother her. But it would. It would eat away at her like an earwig burrowing through her brain. And before the day was done, she would drink.\n\nThe department would be all over her, pressuring her, just as they would now be pressured to terminate her. His resolve was redoubled: he had to give her something. Something to make her indispensable to the investigation.\n\nThat contemptible Spencer woman\u2014how he loathed her. She was projecting, of course. Annabel had said she was a wretched, inattentive mother. No doubt she was trying to sublimate her guilt by lashing out at others. She had slandered him and, more importantly, his Susan. If it happened again, he would be forced to take action. For Susan's sake. And that of the world to come.\n\nI don't like that one with the frozen hair and the painted face and the big glasses and the guitar calluses on her left-hand fingers. I didn't like it when that one was mean to Susan there was no reason she should not be mean and that's what they taught me in Sunday school didn't she ever go to Sunday school that one was mean I'm sorry she lost her little girl but maybe if she'd been watching she wouldn't have lost her and it isn't Susan's fault. She was mean to Susan and said lots of loud things I didn't understand and she smelled bad too like something out of one of those bottles at J. C. Penney's where the lady is always asking if you want a sample and I don't I don't I hate those smells get those smells away from me and I screamed and knocked over her tray and my dad was so mad at me and she smelled just like that. I didn't think my dad would let me come to his office today but he did and he said I could come again if I wanted to if I didn't get in Susan's way because Susan has problems and he doesn't want me to make them worse like I would. _The world is full of obvious things which nobody by any chance ever observes._ I got to see Susan and I know she likes coffee even though it's bad for you so I read up everything about coffee so I could tell her and I think maybe it made her happy but I saw her talk to those camera people and saw the glasses lady be so mean and for no reason. Bad lady! I bet she likes spiders and I bet she has a dog.\n\nShe shouldn't have been mean to Susan. No one should ever be mean to Susan. I want her to be happy. I want to have her babies.\n\nI turned on the water, stepped into the shower fully clothed, and screamed. Screamed like a banshee, an elemental force of nature. Everything that had been pent up inside me I tried to release in one piercing blast. All my anger at the lawyers, at the people who took Rachel from me. At Dr. Coutant. At O'Bannon. And most of all at that obscene television gorgon Fara Spencer.\n\nAnd David. David, David, David. I shouted and shouted until my throat hurt.\n\nIt wasn't enough. It was still with me. It would always be with me.\n\nWhat had that hideous bitch thought she was doing? As if nothing mattered but her own self-centered quest for vengeance? Normally, I was scrupulously considerate of the feelings of the bereaved, however their actions might complicate my work. But this woman had crossed the boundaries. She was using her celebrity status to interfere with the detective work. Worse, she was threatening my job. I knew that O'Bannon would be under all kinds of pressure to cut me loose. It would be the easiest thing to do, especially now that he had a federal behaviorist working on the case. And if I lost this job, my chances of getting Rachel back were less than zero.\n\nI stumbled out of the shower, dripping water all over the floor. I didn't know what I should do. I didn't know what I could do. I dreaded going back to work tomorrow, knowing what everyone would be saying, or at least whispering. Knowing it was just a matter of time before the word was given. Before I was pink-slipped into oblivion.\n\nI tore through the sacks in the kitchen, all the shopping I had done on my way home from the office. I thought about calling Patrick\u2014after the conference, he had more than hinted that he would be available if I wanted to get together tonight. But I couldn't, not now, not on these terms. Not when I was so goddamned vulnerable. There was only one thing I could do under these circumstances.\n\nI had swallowed half the bottle before I came up for air. After that, I don't remember much of anything.\n**CHAPTER FOURTEEN**\n\nI skulked into the office feeling like death on a plate, wondering if they could tell. If they would suspect. Everyone always suspects the worst, don't they? At least about me they do.\n\nI had not gone by the O'Bannons' to pick up Darcy this morning\u2014for a reason\u2014so I was surprised to see him sitting beside my desk.\n\n\"Morning, Darce,\" I said.\n\nHe scrunched his face up. \"You smell funny.\"\n\n\"I do?\" I had showered, groomed, perfumed, brushed my teeth about six times, and consumed Altoids as if they were a breakfast food. \"Must be my perfume.\"\n\n\"Chanel No. 5?\"\n\nI stared at him. \"That's right. How did you\u2014\"\n\n\"One of the ladies at my day care wears Chanel No. 5. Till she stopped 'cause they use animals to test stuff.\"\n\n\"I see,\" I said, amazed. We were a good five feet apart. \"You like it?\"\n\nHe shook his head. \"Stinky.\"\n\nI laughed.\n\n\"I like the normal way you smell better.\"\n\n\"Okay. I'll lay off the Chanel.\"\n\nHe didn't say anything.\n\n\"Because that explains why you thought I smelled bad.\"\n\nHe fidgeted with a lock of hair curled behind his ear. \"Did you know that a bottle of vodka contains more sugar than a Giant-Size Snickers?\"\n\nI slid behind my desk.\n\n\"That's fascinating, Darcy. Did you get that from one of your almanacs?\"\n\nHe shook his head. _\"Hollywood Squares.\"_\n\nI was relieved to see a kid from the mail room loping up to my desk with a small package. \"We found this on the front counter. It's addressed to you. Someone must've dropped it off when no one was looking.\"\n\nI took one of the end flaps and started to tear it open.\n\n\"I don't think that's wise.\" Patrick appeared at the top of the stairs. \"It could be dangerous. I'll call for a fluoroscope.\"\n\n\"What, like it might be a bomb? Nah.\"\n\n\"You can't be sure.\"\n\n\"Well, I've read two-thirds of the way through the complete stories and poems of Edgar Allan Poe, and I've yet to encounter anyone being blown to smithereens.\"\n\n\"Susan, you know the proper procedure as well as I do. Let's call the bomb squad and\u2014\"\n\n\"He doesn't want to kill me. He wants to impress me.\" Once I had the wrapping off, I closed my eyes and opened the box.\n\nNothing happened.\n\nI peeked into the box. My lips parted.\n\nI guess my shock registered, because Patrick immediately said, \"You know, we can't be sure this came from the killer. A case that garners as much publicity as this one is bound to generate copycats. This could be from some crank or would-be martyr or\u2014\"\n\nI pursed my lips. \"No. This is from the guy.\"\n\n\"How can you be sure?\"\n\nI tilted the box toward him. And thirty-two blood-caked teeth slid into view.\n\nIt took us a few minutes to notice, but there was a note in the box, too, taped inside the lid. I put on some plastic gloves and carefully unfolded it.\n\nThe same old code, or another one like it. But it appeared to be longer than the previous messages. \"Darcy?\"\n\nI showed him the note. It was fascinating to watch him go into action, his head tilting, his eyes slightly contracting. \"Is it the same cipher?\"\n\n\"No,\" he said quietly. He continued staring at the paper. \"This is a toughie.\"\n\nI suppose the killer realized, since we'd made the Poe connection, that we had broken his code. So he provided a new and even more insidious one. \"Look, Darcy, it isn't fair to put you on the spot like this. I'll call my friend Colin\u2014\"\n\n\"No.\" He seemed bothered. I don't know if he was offended or just determined. As always, he was hard to read. But I let him work.\n\nIt was a full five minutes before he spoke again. \" 'Misery is manifold. The wretchedness of earth is multiform.' \"\n\n\"I remember that,\" I said. \"It's from one of the stories.\"\n\nDarcy nodded. \" 'Berenice,' \" he said, pronouncing the _c_ as if it were English, not like the Italian _ch_ sound. How would he know? \"The first line.\"\n\nThe first line. Something we would be sure to identify. He was making it easy for us. \"That's the story where the psycho yanks the girl's teeth. We already knew about that. I was hoping for something\u2014\"\n\nDarcy interrupted. \"There's more.\"\n\nPatrick and I huddled around him, as if we might be the slightest help.\n\n\"It's hard,\" Darcy said quietly. \"This one has five different symbols for each letter, but some of the letters don't appear five times in the entire message.\" As far as I was concerned, that made the whole thing impossible, but not a minute later Darcy read us the remainder: \" 'I hope you like my present, Susan.' \"\n\n\"Susan?\" I said, eyes wide. \"It says that? He calls me by name?\"\n\nHe nodded, then continued: \" 'There will be more messages. But only for you.' \" He looked up. \"And I guess the rest of it is numbers.\"\n\n\"Numbers?\"\n\n\"Seven of them.\"\n\n\"A phone number. Can you read it? Did he give us a number where we could contact him?\"\n\nDarcy's head twitched. \"Do you think the bad man lives with you?\"\n\n\"What? What are you talking about?\"\n\nDarcy looked at me with sad eyes. \"It's your phone number.\"\n\nThank God I'd hidden the booze before I left the apartment that morning, because when I returned, it was with an entourage of twelve cops of various types, not to mention Darcy. While the trace team set up some extensions and all their recording equipment, I scurried around picking my underwear up off the floor and other such essential housecleaning chores.\n\nPatrick had come, too, and I have to admit it gave me a bit of a charge, having him there, in the very place where I sleep and all. I made a resolution\u2014as soon as all these interlopers cleared out of my apartment, I was going to insist that he go out with me. Shameless, I know, but let's face it\u2014I'm a squeaky wheel.\n\nAnd I didn't want to spend another night alone. Not if I could help it.\n\n\"We've added two extensions,\" Tony Crenshaw explained, \"so we can listen in and make a recording. We've also added an open line to Bell's electronic switching center\u2014ESS\u2014and to humor Agent Chaffee, a hotline to the FBI's communications room.\"\n\n\"Sounds good,\" I said. \"How long does it take to make a trace these days?\"\n\n\"Not as long as you might think. Depends on how the call arrives, but if it comes on all-electronic switches\u2014and if it's a local call from a residence, it will\u2014we should have it in a minute.\"\n\n\"So I have to keep him rambling for sixty seconds.\"\n\n\"Not even. We've got a tone generator connected to your line. As soon as he calls, we'll pick up and start tracing\u2014but the tone generator will make a false ringing noise and fake him into believing you haven't answered. We'll cut it after four rings\u2014otherwise he might hang up.\"\n\n\"So that leaves me with, what? Fifty seconds of talk?\"\n\n\"We could have someone else answer. Try to put him on hold.\"\n\nI shook my head. \"He won't buy it. He may be crazy, but he's also smart.\"\n\n\"Fine. After the fourth ring, click the interrupt button quickly to simulate the sound of the phone picking up. Then start talking.\"\n\n\"Don't seem too eager to chat,\" Patrick said. \"He'll get suspicious.\"\n\nI agreed. \"I think I can keep him talking. He's concocted this brilliant scheme\u2014in his eyes\u2014but has no one to appreciate it. He wants me to be his audience. Who better to appreciate what you've done than the police officer who's trying to catch you? He wants my admiration.\" I gave Patrick a sly smile. \"But I'll make him work for it.\"\n\nDarcy sidled up beside me. \"I think sometimes people's voices on the phone sound scary. Do you think sometimes people's voices on the phone sound scary?\"\n\nI patted his shoulder reassuringly. \"I can handle it.\"\n\nHis father cut in. \"Darcy, why don't you catch the bus home? Susan is very busy right now.\"\n\nDarcy frowned, then started stuttering again. \"I\u2014I\u2014I would rather stay with Susan, I think. If\u2014if\u2014if\u2014\" He swallowed. \"If you get scared, Susan, I'll talk to him for you.\"\n\nMy eyes got strangely itchy. \"Thanks, Darce. Appreciate it.\" Which I did, especially given how scared he was of the voice on the other end of the line.\n\nAfter that, we sat around and stared at the phone. We had no idea when he might call. I wished I'd bought the latest _Cosmo_ or something. My new apartment was distinctly short on reading materials, other than that Collected Poe. All my books were still packed up in boxes. It occurred to me that Darcy could probably recite any number of books to me from memory, but I declined to ask.\n\n\"Remember this,\" Patrick said, \"next time you're asked to talk about the glamorous and exciting world of law enforcement.\"\n\nI laughed. \"When I finally lose the rest of these chumps,\" I said, \"wanna go out and get\u2014\" I checked myself. \"A sandwich?\"\n\nHe beamed back a smile filled with potential. \"I'd like that.\"\n\n\"It's a date,\" I said, just in case there was any doubt about the direction I was heading. Before I could elaborate, the phone rang.\n\nWe all stared at each other. This was it. This was really it.\n\nThe machine picked up the line and the tone generator kicked in. I patiently waited for four rings. Crenshaw gave me the signal. Then I clicked the interrupt, took a deep breath, and spoke.\n\n\"Hello. Susan speaking.\"\n\n\"Hey, Suze. Whatcha been up to?\"\n\nBreath poured out of me like a deflated balloon. It was Lisa.\n\n\"Haven't seen you today.\"\n\n\"Well . . . I've been busy. O'Bannon is working me like a plantation owner.\"\n\n\"Want me to come over?\"\n\n\"I do, but the problem is\u2014\" I glanced at O'Bannon. He gave me the okay. \"I'm currently surrounded by about a dozen police officers.\"\n\n\"What? What's going on?\"\n\n\"I can't go into it now. How about I meet you for breakfast tomorrow? Krispy Kremes sound good?\"\n\n\"I could live with that. Eight too early?\"\n\n\"See you then.\" I hung up the phone and stared out into a sea of irritated faces. \"Well, forgive me for having friends.\"\n\nAnd then we all sat down to wait some more.\n\nAfter another hour or so, Patrick excused himself so he could review the security detail. I felt like I was probably safe at the moment, since I was surrounded by half the Vegas police force, but whatever. Maybe he just wanted to stretch his legs.\n\nI killed time by showing Darcy the two card tricks I had learned in Brownies about twenty-five years ago. The problem was, he could always see how it was done. In fact, after watching it once, he could duplicate the trick himself.\n\n\"Hey, Chief,\" I said, \"did you know you have a potential cardsharp on your hands? I could see him running a three-card monte operation on the Strip.\"\n\nO'Bannon grunted.\n\nWe were all having a perfectly merry time\u2014when the phone rang. Tone generator, four rings, and then I was on.\n\n\"Hello. Susan speaking.\"\n\n\"Did you enjoy my gift?\"\n\nI closed my eyes and tried to focus. Friendly, but not too eager. Don't challenge him. \"It got my attention. I think the forensic lab will probably have more fun with it than I did.\"\n\n\"They won't learn anything.\"\n\n\"You never know.\"\n\n\"What are you doing, Susan? You're not falling back into bad habits, are you?\"\n\nWhat the hell was that supposed to mean? \"Just spending a quiet evening at home.\"\n\nHe chuckled. Actually chuckled. \"I rather doubt that.\"\n\n\"So what are you up to, anyway? Why are you doing this?\"\n\n\"I can't tell you. You're not ready.\"\n\n\"Aw, please. Fill me in. So I can appreciate what you're doing.\"\n\n\"You're very clever, Susan. If you open your mind, you will find the truth. I only have time to tell you this: you would look beautiful in neon.\" And then the phone went dead.\n\nI checked my watch. Thirty seconds, tops.\n\nI looked at Crenshaw. He was on one of the extensions, talking to the switching center. \"Did we get a fix?\"\n\nAfter a few moments, he put down the phone. \"No.\"\n\nGranger pounded a fist into his palm. \"Nothing at all?\"\n\n\"It's local. We're sure of that. But we don't know where he is.\" He looked at me sadly. \"I think we should leave the recorder on and the equipment in place. In case he calls again.\"\n\n\"Sure,\" I said. \"But he won't. He's already accomplished what he wanted.\"\n\n\"We'll get that tape recording to the sound lab. See what they can tell us.\"\n\n\"And then?\" Patrick asked.\n\n\"Then,\" I said, \"we have to figure out what he meant by that gibberish about the truth. And that crack about me looking good in neon.\"\n\n\"It was just bullshit,\" Granger said. \"He was flirting with you.\"\n\n\"Maybe, but I don't think so.\"\n\n\"Then what?\" O'Bannon asked.\n\nGood question. I fell back against my sofa and thought a good long while before answering. \"It was a clue. To whatever he wants us to find next.\"\n\nThis was a new experience for me. Being at a bar\u2014The White Feather\u2014but not ordering a drink. It was almost as seedy as Gordy's, but it was near my apartment, and they served sandwiches as well as libations.\n\nThe three of us found a table\u2014Darcy tagged along\u2014and I ordered a club soda. Patrick relaxed as soon as I placed my nonalcoholic drink order. Maybe Darcy did, too. I felt proud of myself. I'd passed the test.\n\nAnd periodically, as needed, I excused myself to powder my nose. And safe within the confines of a stall with the door closed, I took a deep swig from the flask tucked inside my jacket. Crunched a few Altoids and I was back in action.\n\nAs I passed by the bar on my way back to our table, this guy leaned backward from his bar stool and blocked my path. He was big and black and had a shaved head. He couldn't have exuded more testosterone if he'd poured a bottle of it over his head.\n\n\"What you doin' in a dive like this, Susan?\"\n\nI gave him a look. I knew him from somewhere . . . the courtroom, that was it. He was a bail bondsman. He'd made a few passes at me through the years; I'd never given him the time of day. I thought his name was Jake, but I wasn't sure enough to give it a try. \"Just drinking in the atmosphere.\"\n\n\"I think you came lookin' for me.\"\n\n\"Do you now?\"\n\n\"Yeah. Me, or somebody like me.\" He had a laconic, deliberate way of speaking that set my teeth on edge.\n\n\"Well, you're wrong. I'm with someone. Two someones, actually.\"\n\n\"You sure about that?\"\n\n\"Yeah, positive.\"\n\nHe gave me a look I didn't like at all. \"You change your mind, you come back and see me, okay? I'll give you what you want. I'll make you feel alive.\"\n\nJesus Christ, some guys. I pushed past him and returned to my seat.\n\n\"You've changed your perfume,\" Darcy said upon my return.\n\nWas he talking about Chanel No. 5 again? Or did he know what I'd been doing in the ladies' room? The kid was a damn bloodhound. \"Yeah. I prefer something smokier when I'm out at night.\"\n\n\"What do you think he meant?\" Patrick asked. He was nursing black coffee, but oddly enough Darcy hadn't given him the lecture on the evils of caffeine. \"That crack about neon.\"\n\nI waved my hand, smiling, my eyebrows arched provocatively, leaning forward. I didn't want to be obvious, but I was feeling pretty bold. Maybe it was the Jack Daniel's coursing through my system. \"I don't want to talk about business tonight.\"\n\n\"Neon is an inert gas,\" Darcy informed us. \"It's odorless, colorless, tasteless, nontoxic, and monatomic. In a vacuum tube, neon glows reddish orange. Its chemical symbol is Ne. Its atomic number is ten.\"\n\n\"Fascinating,\" I said, fluttering my eyelashes, still gazing at Patrick. \"But like I said, I don't want to talk about work.\"\n\nDarcy grew quieter. \"I was trying to be useful. It's the most important thing in life. To be useful.\"\n\n\"That's a lovely sentiment.\"\n\n\"John Adams, the second president of the United States, said that in 1814 to his\u2014\"\n\n\"Darcy . . .\" I sighed. \"It's late. I appreciate you seeing me here, but this might be a good time for you to head home.\"\n\n\"Are you also going home? I'm sure the police officers have left.\"\n\nI ignored him. \"I'll be along. But you should go now. I'll pick you up early tomorrow morning. We'll see what we can get out of that phone call. Till then\u2014get some rest.\"\n\n\"I never sleep more than four hours a night. Some nights my dad turns out all the lights and I just stare at the clock for hours.\"\n\n\"Darcy\u2014\" I tried to smile. \"There's a bus stop at the corner. Just turn left outside the door. You can ride it all the way home without changing.\"\n\nHe stood, looking uneasy, fidgeting with his hands. \"Do you think that you will be all right?\"\n\nAdorable. I was amazed at how much affection I felt for him, after knowing him only a short while. But I wasn't going to get laid with him around. \"I can take care of myself, Darcy. Good night.\"\n\n\" 'Night.\" He stumbled out of the bar, glancing back at me over his shoulder.\n\n\"Cute kid, huh?\" Patrick said, sipping his coffee. \"And what a memory.\"\n\n\"Yeah. Incredible.\" I leaned forward. Had I remembered to unbutton the top button of my shirt? \"I don't want to talk about him, either.\"\n\n\"Indeed. What do you want to talk about?\"\n\nI leaned in even closer. \"Come to think of it, I don't really want to talk at all.\"\n\n\"I'm astonished.\" My God, but he was sexy up close. Or far away. Or with a paper bag over his head. \"I was told you were very aggressive.\"\n\n\"Me? I'm a pushover. Try me.\" Just another inch, and my lips were planted firmly on his. They tasted sweet. I'd almost forgotten. He put his hand at the base of my neck and sent tingles radiating up and down my spine. This was going to be good. I knew this was going to be good.\n\n\"What do you want?\" he whispered to me.\n\n\"I want to feel . . . something different,\" I said, peering into his beautiful blues. \"I want to feel like I'm really alive. Not just going through the motions.\" Enough with the damn talking. I pressed my lips against his and we didn't come up for air for a good long while. I didn't care who saw or what they thought. I needed this. I needed this.\n\n\"Come home with me,\" I said finally. \"Or I'll go with you. Whatever's closer. What'd'ya say?\"\n\nHe looked at me a long moment. His expression alone was sufficient to convey everything I needed to know. Everything I didn't want to hear. \"No, I can't.\"\n\n\"But\u2014I thought you felt like\u2014\"\n\n\"I do.\"\n\n\"Then come on.\" I snaked my hand between his legs and gave him a squeeze in a strategically chosen area. \"I'll make it worth your while. It'll be great. Promise.\"\n\n\"I know it would,\" he said, pushing my hand away. \"But you'd hate me in the morning.\"\n\n\"Don't be dumb. I\u2014\"\n\n\"You would. You'd be embarrassed and ashamed and regretful. We wouldn't be able to work together anymore.\"\n\n\"Are you saying\u2014we can't ever\u2014\"\n\n\"No. I'm just saying it's too soon.\" He stood. \"I think it's best if I go.\"\n\n\"Please\u2014\"\n\nHe took my hand. \"You'll be glad. Later.\"\n\n\"I doubt it.\"\n\n\"You will.\" He let my hand fall and left.\n\nWay to go, Susan, I told myself bitterly. Chased him right out of the bar. Came on too damn strong, just like always, and bulldozed him right out of the ballpark.\n\nI needed a drink, and I wasn't talking about club-fucking-soda, either. I made my way to the little girls' room again and raised the flask. It burned good, going down. After a swallow of that, I felt much better about myself. What the hell\u2014I knew I was going to drink it all eventually. I pressed the bottle to my lips\u2014almost as tasty as Patrick was\u2014and emptied the flask.\n\nI left the restroom feeling stronger, complimenting myself on handling it and not losing the knack and walking and talking as if I really were only drinking club soda. I had about decided to give up on the place and go home when I saw a man's leg blocking my path.\n\n\"Can I buy you a drink now, Susan?\"\n\nIt was that same guy, Jake or whatever, the shaved head. \"I don't drink anymore.\"\n\n\"Fine. You figure out what you want yet?\"\n\n\"I . . .\" I looked at him through blurred eyes. \"I want to feel something different. Alive.\"\n\n\"I can handle that.\"\n\nHe threw some bills on the bar, took me by the hand.\n\n\"We can't go to my place,\" I said. \"Is your place near?\"\n\n\"Not near enough.\" He led me down an alleyway behind the bar. It was dark, but not so dark I couldn't see it was filthy. Nothing but dirt and slime and upended trash cans. \"This'll do.\"\n\nHe whipped me around and grabbed my hands. \"You ready to feel somethin' different?\" He didn't wait for me to answer. He ripped my jacket open, sending the empty flask smashing onto the pavement. With unrestrained brutality, he pulled down my pants and panties, then sat me down on one of the trash cans.\n\n\"I'm thinkin' a little back door action might do the trick,\" he said while he unzipped. \"That work for you?\"\n\nI didn't say anything.\n\nWith one powerful move, he pushed me to my knees, then swung me around until I was spread stomach down across the trash can. No talk, no warning, no foreplay.\n\n _\"Ahhh\u2014!\"_\n\nIt was my first time and the pain was searing. I wondered if he had done this before, used the same lines, gotten what he wanted the same way. I wondered where he came from, what he really liked, what he saw when he looked into my eyes. I didn't think he was heartless or even particularly selfish. He just had needs, like we all did. When it was over and he woke up the next day, he might feel a little guilty about it.\n\nBut I wouldn't, and I wouldn't care if he did. I needed to feel something. Pain. Humiliation. Rage. It was all the same. I just wanted to feel alive again.\n**CHAPTER FIFTEEN**\n\nThe next morning, I hurt like hell. My head throbbed, sure, always, but that wasn't the worst of it. I could barely walk. Somehow I managed to stumble out of bed and make it to the front door before the bell had rung more than, oh, fifty times or so.\n\n\"Susan. My God, what happened to you? We were supposed to have breakfast, remember?\"\n\nLisa. \"I, uh, slept in. I had kind of a rough night.\"\n\nShe rushed in, putting her arms around me. \"You look like someone beat the hell out of you.\"\n\n\"No, no, just trouble sleeping.\"\n\nShe stiffened slightly. \"Susan, I'm sorry. I know I'm not your mother, but the counselors told me that the best way to be your friend was to try to help you keep your promises. Have you been drinking?\"\n\n\"What? Are you kidding? No.\"\n\n\"Really?\"\n\n\" 'Course. You called last night.\" Thank God I could remember that much. \"I was home, remember?\"\n\n\"But\u2014\"\n\n\"Wanna smell my breath?\" What a bluffer I am.\n\n\"Frankly, no.\" She guided us both to the sofa in the living room. It was green and faded and showed traces of all the cop butts that had been on it the night before. \"I got enough of that last night.\"\n\n\"The new guy?\"\n\n\"Oh, yeah.\"\n\n\"Memorable?\"\n\n\"Human Dental Pik. Tongued every incisor in my mouth. Thought he was attempting a root canal. Do you still want to see Rachel?\"\n\n\"Right. Damn.\" I brushed my straggly, stinky hair out of my eyes. \"I forgot.\"\n\n\"How long has it been since you've seen her?\"\n\n\"Well, uh . . .\"\n\n\"When I talked to her last night, she said you hadn't been in for several days.\"\n\n\"Well, she should know.\"\n\n\"Susan, this is exactly the kind of behavior that caused you to lose her in the first place.\"\n\n\"Look, Lisa, I had a horrible night. That killer, the Poe guy, he called me. Here.\"\n\nHer eyes ballooned. \"He called _you_? Why?\"\n\n\"Hard to say. I think he was threatening me. Or trying to help me. Or none of the above.\"\n\n\"Oh, my God.\" She cradled my stinky head in her hands. \"No wonder you're a wreck.\"\n\nSo here I was, using a serial killer to excuse my erratic behavior. I felt pathetic.\n\n\"That explains why you were so weird when I called. What did he say?\"\n\n\"Well, I think he may have given us some clues.\"\n\n\"My God, Susan. Should you be working on this case? He knows who you are.\"\n\n\"True.\"\n\n\"I don't think this is good for you at all. Especially not now. You need out.\"\n\n\"I need work.\"\n\n\"I'm talking to your doctor.\"\n\n\"Don't you dare.\"\n\n\"Susan, it's for your own good. I'm just trying to help.\"\n\nMy blood burned. \"I don't need help!\"\n\n\"You do.\" Despite my nastiness, she hugged me all the tighter. \"I love you, Susan. I'm not going to let you kill yourself. I'm not.\"\n\n\"May I come in?\"\n\nWe had left the front door ajar, and Darcy was standing just outside, peeking through. Lisa still had her arms around me, and I could see he was confused. \"Is that one your girlfriend?\" Darcy asked.\n\nI cleared my throat. \"Well, yes . . .\"\n\n\"Did you know that four percent of all women prefer other women to do sex with?\"\n\n\"Darcy . . .\"\n\n\"I don't think there's anything wrong with that, do you? But my dad says it's bad.\"\n\n\"Darcy . . .\"\n\n\"But the weird thing is he loves his videos where women are with other women and\u2014\"\n\n\"Darcy!\" He fell silent. I had to remind myself\u2014he doesn't see faces; he can't read facial expressions. He's not going to get any nonverbal cues. You want him to be quiet, you gotta say so. \"What are you doing here?\"\n\nHe looked nervously at Lisa, fidgeted with his hands, then started again. \"I think maybe Mr. Granger figured out what that 'neon' remark meant. He called my dad. The killer wanted us to go to this place where they store old neon signs.\"\n\n\"Why would he want us to go there?\" But I was certain of the answer before I had finished asking the question. \"Let's go.\"\n\nThe usual rant against Vegas is that it isn't really a city, just an oversized vacation destination. After all, there's no urban blight, like cooler-than-cool New York. There are no high-rises, other than the hotels. How can that be a city? What some of these geniuses don't get\u2014Hunter S. Thompson, for one\u2014is that this is a _Western_ city. It's out in the fertile desert plains. It's meant to be flat. The houses were designed to go out, not up, in the traditional hacienda style. It was built for the automobile, not the pedestrian. No one wants to walk in Vegas. It's too hot. You don't walk home from work. You don't walk to the store for a loaf of bread. That doesn't make it any less of a city.\n\nI got to see a lot of that flat landscape on my way to the Vegas Neon Graveyard. We parked and walked down a lovely pine-bordered alameda till we arrived at the main lot. A lot of memories were stored there, especially for someone like me who'd lived in Vegas all my life. Remember Sassy Sally's, on Fremont? That huge neon marquee was here, in all its glory. Or the gigantic guy in the leisure suit who used to shoot pool on the marquee for Binion's? Also here. Some of this stuff went back to the very earliest days of postwar Vegas. Remember the crown that used to sit atop the Royal Nevada? Well, neither did I. I'm not that old, for God's sake. But one of the lab techs pointed it out to me.\n\nApparently the guy who owned this place was a collector who just couldn't bear to see these Vegas icons destroyed. So whenever one of the casinos or hotels replaced their signage, he bought it, usually dirt cheap. He'd had to move his collection three times, on each occasion to a larger tract of land. He was outside the city now\u2014probably too far from the Strip to attract much tourism, not that the average Vegas visitor was all that interested in historical memorabilia. He sold new signs, too, but I got the impression that was mostly a front to finance the acquisition and upkeep of this gaudy but sentimental collection.\n\nIt must've seemed like a unique and harmless specialty field. Until the headless corpse turned up.\n\nThe other half, the head, was hanging separately.\n\n\"Half off,\" I muttered, wishing to God I had something to drink. \"I guess that's a joke.\"\n\nDarcy looked at me, puzzled. \"Is it funny?\"\n\n\"No, but\u2014\" How was I going to explain this? His father had warned me that most humor passed Darcy by, and given all the language oddities associated with autism, he was less likely to get wordplay than anything. \"Never mind.\"\n\nDarcy turned away, staring at what looked like the old frontispiece from the Horseshoe. Poor kid. I remembered what a gentle spirit he had\u2014how he wouldn't even hurt a spider. And here I was dragging him around to see decapitated corpses. Well, technically, he'd dragged me, but still. Maybe Granger was right.\n\nSpeaking of whom: \"Glad you could make it, Lieutenant Pulaski. Hope you don't mind us taking your clue and running with it.\"\n\nJust had to rub it in, didn't he? \"Not at all. Give my congratulations to your team. How'd you figure out what it meant?\"\n\nGranger gave me the sort of smile that makes you desperate to erase it with the flat of your hand. \"That's why we're called detectives.\"\n\n\"Of course.\"\n\n\"What were you doing last night after we left your place?\"\n\nWas there something about my face? Clothes? Smell? Had that brute from the bar gone to the press? \"I was a little shaken up, actually. After that phone call.\"\n\nI brushed him off and scouted around for Darcy. He'd been here a good two minutes. More than enough time for the Boy Wonder to start making his bizarre and brilliant deductions.\n\n\"So what Poe story are we in now?\" I asked. Darcy was staring at the ground. \"I've read several that involved decapitations.\"\n\n\" 'The Black Cat,' \" he said flatly.\n\n\"I kind of remember that one. Similar to 'The Tell-tale Heart,' right? Guy tries to kill a cat, and when this woman rushes in to saveit, he kills the woman instead. What makes you think that's the one?\"\n\nI could tell this poor gentle soul didn't want to talk about it. But he did. For me. \"He used an axe.\"\n\nEww. \"Did you get that from the coroner?\"\n\nHe shook his head.\n\n\"Then how?\"\n\n\"I looked.\"\n\nDouble eww. 'Course, I looked, too, but apparently I lacked Darcy's eye for detail. Or maybe it was that I hadn't read Carston's twelve-volume _History of Criminology_ cover to cover and memorized each page. \"Did you have a chance to look at the note?\"\n\nHe nodded, then recited it for me\u2014without looking. \" 'From the one Particle, as a center, let us suppose to be irradiated spherically\u2014in all directions\u2014to immeasurable but still definite distances in the previously vacant space\u2014a certain inexpressibly great yet limited number of unimaginably yet not infinitely minute atoms.' \"\n\n\"That's kind of different. Is that Poe? I don't recognize it.\" Not that my brain came equipped with photographic memory. \"Do you?\"\n\nDarcy shook his head. It bothered him. I could see that.\n\n\"Maybe our killer has begun composing original works? In the style of the master?\"\n\nAgain he didn't answer and I didn't blame him. Even as I suggested it, it didn't sound right. The psycho was still using Poe for his blueprint. There was no indication that he had broken free of that particular part of his psychosis. So where did this bizarre and cryptic message come from? Evidently I needed to know more about Poe than was contained in his complete works. I resolved to stop by the city library on my way home from work and see what I could learn about the man himself.\n\nI saw Tony Crenshaw near the spot where the torso still dangled. His body was stiff, almost rigid. Only his mouth moved. He reminded me of one of those talking statues at Caesar's Palace.\n\n\"Got anything for me, Tony? More of those rug fibers?\"\n\n\"Not a one.\"\n\n\"Anything at all?\"\n\n\"Not much. No prints, that's for damn sure. Nothing we can get DNA from. A few almost microscopic traces of clothing.\"\n\n\"Can you tell what it is?\"\n\nHe shrugged uncomfortably. \"I'd like to get it under a microscope before I make any definite pronouncements. But it looks as if it might be lace. Red lace.\"\n\n\"Lace.\" I turned to Darcy. \"You remember any references to lace in the Poe stories?\" He didn't. \"Well, he couldn't have carried her here in anything made of lace. Wonder why he didn't use the rug?\"\n\n\"He may have destroyed it after the last time,\" Tony suggested.\n\n\"That one is small,\" Darcy said, edging forward. It took me a moment to realize he was talking about the victim.\n\n\"Yeah, it's a shame.\"\n\nLines crossed his forehead. Obviously, I hadn't taken his meaning. \"She's light. Especially in . . . two pieces.\"\n\n\"Are you saying . . . he didn't need the rug?\" And as soon as I said it, I realized Darcy was right. That Asian girl couldn't've weighed a hundred pounds\u2014even before she was subdivided.\n\n\"We found a robe covered with blood,\" Tony explained. \"We figure he used it like butcher paper\u2014put it under her when he killed her to soak up the blood. But it's possible . . .\"\n\nI followed his meaning. Wrapped her in her own robe and tossed her into the back of the truck. The robe was already wrapped in plastic for protection, but I gave it a good once-over. It was silken, if not actually silk, and rather exotic. Long, sinuous Chinese dragons were stitched all over it. Victim was very involved with her Asian heritage? No, that wasn't it. . . .\n\n\"Hey, I've got something.\"\n\nIt was Jodie Nida, a tech from the coroner's office. Given Patterson's normal reticence to say anything in advance of the official report, this was nothing short of amazing. I hurried to her side. Darcy followed close behind.\n\nWouldn't you know it\u2014she was examining the disembodied head. I felt my gorge rising, and this time it wasn't due to Granger's belligerence or drinking my dinner last night.\n\n\"Let me guess\u2014she died of lack of oxygen to the brain.\"\n\nNida didn't smile. \"Technically, that's correct. It appears to have taken two blows to sever her head. Not the easiest task, even with a sharp blade. And there are other body wounds, probably slipups. One is about two centimeters long between the fifth and fourth intercostal spaces at the medial border of the right breast. Another wound at the anteromedial right deltoid.\"\n\n\"Administered before he cut off her head? Or after?\"\n\n\"I can't be sure. But that isn't why I called you over here. You see it?\"\n\nI had no idea what she was talking about.\n\n\"The teeth,\" she said helpfully.\n\nI leaned in closer, even though it was the last thing on earth I wanted to do. She used a dentist's tool to pull back the edge of the mouth.\n\n\"See?\" There was a small but discernible red dot on one of the victim's incisors.\n\n\"Blood?\"\n\n\"Looks that way. We'll confirm it in the lab.\"\n\n\"Could this mean . . . she tried to defend herself?\" I hoped for a yes. It would be nice to think she hurt this bastard before he chopped her in two.\n\n\"Either that or he treated her to a T-bone steak, rare. But I'm seeing no signs of a recent meal, and I've found skin flakes on the lining of her mouth. I think she got him.\"\n\n\"Good for her.\" I wondered what had allowed her to get in even that tiny blow of resistance that none of the others had managed. \"Looks like she was in good shape.\"\n\n\"Agreed,\" Nida said. \"Excellent muscle tone, even after rigor. I'm guessing she was an athlete.\"\n\n\"Dancer,\" Darcy said quietly, looking away, hands flipping.\n\nWe both turned and stared at him.\n\n\"How can you possibly\u2014\"\n\n\"Did you see the calluses on her feet?\" he asked.\n\n\"Darcy, anything could've caused that. She might have a job that required her to stand on her feet for long periods. Waitressing, or\u2014\"\n\n\"Dancer,\" Darcy gently insisted. \"Did you know that dancers get special calluses in special places? Look at the calluses on her feet.\"\n\nI did. I saw how they formed a semicircular arc at the base of her heel. A firm ridge down the sides and across the center.\n\nNida and I stared at each other, utterly wordless.\n\n\"Maria Tallchief had the same calluses, according to this biography my dad brought home from the library once. Did you know that she was the first truly great prima ballerina in America? She was born in Fairfax, Oklahoma, and\u2014\"\n\n\"No, I didn't,\" I said, not informing him that I didn't even know who Maria Tallchief was. My mind had already wandered to another place. Red lace. Exotic wraps. Dancing. I punched my cell phone. \"Madeline? Check the missing-persons reports for the last few days. See if anyone is missing an exotic dancer. Showgirl. Anything like that.\" If I could figure out where the victim came from\u2014sooner, this time around\u2014I might finally figure out this creep's pattern and anticipate the next one. \"I wish I understood how he chooses his victims,\" I said, not really realizing I was speaking aloud. \"Or how he chooses where to leave his victims.\"\n\n\"They are all graveyards,\" Darcy said.\n\n\"Huh?\" I snapped out of my reverie.\n\n\"Did you notice that they are all graveyards? The make-believe graveyard at the Transylvania Hotel. The old airplane graveyard at McCarran. The neon sign graveyard.\"\n\nI slapped myself for being such a stooge. It was obvious\u2014after Darcy explained it to you. \"What could be more natural for a Poe freak? Fake graveyards. He wouldn't use a real one. That would be too ordinary. Too nonpsychotic. So he put a twist on it. His idea of a joke.\"\n\n\"Is this funny?\" Darcy asked, and this time, I thought it was a valid question. It could be humorous, in a twisted sort of way. Dragging the police all over town, from one faux grave site to the next. What fun it would be for him to\u2014\n\nI froze up. He wouldn't want to miss all the action. He must be watching. Just so he could savor the fun a little longer.\n\nI looked all around. We were on the very outskirts of town, surrounded by high, dusty hills. Lots of potential vantage points.\n\nI waved for Granger. I wanted him to send his boys into the hills to see what they could find. But even as I told him, I knew they wouldn't catch our killer. He was much too careful for that. He was already leaving.\n\nBecause he'd been watching me all along. I was certain of it.\n\nA cold shudder coursed down my spine. He was way ahead of us. He had been way ahead of us all along. We were just playthings to him.\n\nI grabbed Darcy and tugged him back toward my car. I didn't want to be here any longer. I didn't want to be anywhere near here.\n\nHow could I possibly catch this man when he was so much smarter than I was?\n\nI don't like going to these dead places. I don't like seeing dead people or touching them or thinking about them. I don't like killing things. No one should ever kill someone. I would never kill someone.\n\nSusan was still stinky today. She smells like that yucky bottle with the brown stuff that looks like apple juice but isn't that my dad opens at night when he's reading or writing or thinking about Mom. It makes him stinky, too, but he never drinks enough of it to last long. I know Susan has brushed her teeth and crunched on mints and she has a pretty smile doesn't she but she's still stinky I can smell it even when she thinks it isn't there anymore. I wish Susan wasn't stinky because I like Susan but I don't like stinky. People should not be stinky.\n\nCoffee is also stinky, but not as much stinky. I hope I never find the axe I hope no one does because I don't want to see it like when Margaret Hayes bumped Alice Tucker at school and she cut her lip on the water fountain and the blood streamed everywhere and I thought it was funny and everyone told me it wasn't funny but I couldn't help but laugh because she looked so silly with her lip swollen and that red stuff all over her face. I got in trouble but I don't know why because I didn't push her and she wasn't really hurt but I wish the bad man would not cut up his people I don't like it when he cuts up people. I don't like him at all but I like Susan and if Susan wants to take me to these places I guess I'll go.\n\nWhy doesn't she see it? It is so obvious. I would tell her but I'm afraid then I'd get in trouble like I did with the water fountain because I don't understand something like they always tell me I just don't understand and I don't want to get in trouble anymore ever again. Especially not with Susan.\n\nI wasn't particularly upset when I returned to headquarters and found Patrick sitting at my desk. I'm not that easy to set off. But I was surprised\u2014and enraged\u2014to find daytime TV's favorite non-doctorate-possessing doctor sitting on the opposite side.\n\n\"May I ask what you're doing here?\"\n\nDr. Spencer barely even looked up. \"I'm trying to catch the man who killed my daughter. Not to mention another\u2014make that two other girls.\"\n\n\"Don't you have some important housewives' crisis you could be working on? The heartbreak of psoriasis or something.\"\n\nThat got her. \"I've begun my own investigation. I'm setting up shop at the Transylvania, where this case began. I came here to inform the local authorities. Even though we are conducting separate inquiries, I hope we can still share information.\"\n\n\"I'll bet you do. Since we have some and you don't.\"\n\nPatrick evidently thought this would be a good time to intervene, with the hope of possibly avoiding bloodshed. \"Under instructions from Washington, I've given Dr. Spencer selected portions of our draft profile, Susan.\"\n\n\"What?\"\n\n\"Particularly those parts dealing with what we know about the killer's preferred victims.\"\n\n\"Are you crazy? That could compromise the whole investigation! What if she reveals everything we've got to her television audience? She could force the killer to change his MO.\"\n\n\"That is a possibility, but\u2014\"\n\n\"This stinks to high heaven, Patrick. Did she threaten the department?\"\n\nSpencer rolled her eyes. \"Don't be so melodramatic, Lieutenant. I simply informed your superiors that I will be giving a prominent prime-time interview tonight. They thought it best to arm me with information so that I could protect potential victims. Frankly, I don't understand why you haven't already done this yourself. I can only assume that it is another reflection of your . . . currently unstable condition.\"\n\nThe problem was, she said it so convincingly I almost believed it myself. \"You're just ticked off because you couldn't get me fired.\"\n\n\"That was never my goal.\"\n\n\"Like hell. You gave it your best shot at that press conference. Turned out you didn't have as much clout as you thought.\"\n\nShe rose out of the chair and looked me square in the face. \"The only thing that saved you, Lieutenant Pulaski, is that timely care package that you say came from the killer. Since he has chosen to communicate with you directly, your superiors thought it would be unwise to dismiss you at this time. But that won't last forever.\"\n\nShe grabbed her coat and purse but couldn't resist a final addendum. \"Not if I have anything to say about it. And believe me, I do.\"\n\nThey didn't want to let her see me, damn it. And I had made a point of stopping at a gas station, checking my looks. My breath. Putting on makeup. Hell, I even tweezed! And Ozzie and Harriet still didn't want to let me see her.\n\n\"I called NDHS. They say I'm allowed.\"\n\n\"At designated times,\" Ozzie said. He was standing tall, but his nervousness showed. We both knew I could knock him down like a bowling pin. But that probably wouldn't be in my long-term best interests. \"You've missed the last two.\"\n\n\"I've been very busy at work. Big murder case. Maybe you've read about it.\"\n\n\"We saw you on the TV,\" Harriet said. She was barely visible under the crook of her husband's arm. \"I thought that Spencer woman was very rude.\"\n\n\"And honest,\" her spouse groused.\n\n\"Look, could I just see my niece? I'm not planning to take her away. I don't understand why this is such a big deal.\"\n\nI stared at the man. We were practically nose to nose. \"You're trying to sniff my breath.\"\n\nHe gave me a \"Who, me?\" look.\n\n\"Here, let me make it easier for you.\" I leaned forward and breathed on his nose. He winced. \"Okay?\"\n\n\"You come back at your designated visitation time. You've got one on Monday after school.\"\n\nI wanted to scream. \"Why are you being like this?\"\n\n\"If we are to establish any order in young Rachel's life, we have to maintain a schedule that she can depend upon and\u2014\"\n\n\"Susan!\" Without warning, Rachel surged past him. He reached for her, but she was too fast for him. She threw her arms around me and hugged tightly. I buried myself in her lovely auburn hair. \"Susan! God, I've missed you!\"\n\n\"I've missed you, too, honey.\" I stared at her, long and hard. \"You look great. Is that a new dress?\"\n\n\"Yeah.\" She whispered in my ear. \"I think they're trying to buy my good graces. Of course, it isn't working.\"\n\nThat was my Rachel.\n\n\"Where have you been?\"\n\n\"I've been chasing this killer, sweetie. Have you read about it?\"\n\n\"Are you kidding? They don't talk about anything else on television. Have you seen the guy?\"\n\n\"No. But I've talked to him.\"\n\n\"Really!\"\n\n\"Yup. Called me on the phone last night.\"\n\n\"Get out of here!\" She was so pretty, so pure. God, but I loved this girl. \"All my friends are jealous that I know the famous Susan Pulaski.\"\n\n\"Famous?\"\n\n\"Don't you know? Everyone watched that press conference.\"\n\n\"Swell. All of it?\"\n\n\"Yes. Even when that cow attacked you. My friends are boycotting her show now.\"\n\n\"Well . . . don't be too hard on her. She's lost her only daughter.\"\n\nShe gave me another squeeze. \"Susan, how long till I can come home with you?\"\n\nAbove us, I saw Ozzie's frown intensify. \"My lawyer is working on it. We're supposed to have a hearing in a few days.\" I pulled her away a bit and addressed her captors. \"Mind if I take her to my car? I'd like to talk to her privately for a moment.\"\n\nOzzie was succinct. \"No.\"\n\n\"We're not going anywhere.\"\n\n\"You step off the front porch, I call the police.\"\n\nI sighed heavily. He was probably bluffing. But given the current delicate circumstances, I couldn't take the risk. \"Rache, I'm sorry I haven't been by. But it's important that I work on this case.\"\n\n\"I know. He's so sick.\"\n\n\"It's more than that, honey. I have to be able to tell the court that . . . that I'm working. That I have a steady income. That I'm gainfully employed. I need good references.\" I could see she didn't really understand. But that was okay. Just so she knew I hadn't forgotten about her. \"But as soon as I get you back home, we're going to spend some major time together.\"\n\nShe looked at me carefully. \"Just the two of us?\"\n\nI didn't know what she meant. Well, I couldn't be sure. \"Just the two of us.\"\n\n\"You'll come by again soon?\"\n\n\"Sure. How about tomorrow night?\" Which of course was not my next scheduled visitation.\n\nI heard the tiniest hesitation in her voice. \"Oh\u2014geez. I have church tomorrow night.\"\n\n\"Church?\" I gave Ozzie and Harriet the long look. \"Trying to bring her to Jesus?\"\n\n\"They've got a big youth group,\" Rachel explained. \"It's kind of cool, actually.\"\n\n\"It is?\"\n\n\"Yeah. I mean, it's real queer banana,\" she added hastily. \"Corny like you wouldn't believe. But I'm tolerating it.\"\n\nUh-huh. I gave her foster parents another once-over. They were more dangerous than I realized.\n\n\"You won't let that crazy man hurt you, will you, Susan?\"\n\nI stood and smiled. \"Are you kidding? I'm going to put him behind bars where he can't hurt anyone. Just a matter of time.\"\n\nIt was well past dark, but he continued reading, reading and rereading, poring over the prose-poem that for him held all the keys to understanding. The answers were there, buried beneath its cryptic passages. They had to be.\n\nHad he erred? Had he somehow misinterpreted the prophecies? Why had the Golden Age not begun?\n\nAll along, he had been buoyed by an innate confidence, an ineffable sense of rightness. He had always been an edacious reader, but for years now he had perused nothing but the texts, reading them over and over, subjecting them to the most intense lucubration. He had discovered the truth and he would use it to work miracles. But the offerings had been made, the triumvirate had been sacrificed. Each of them\u2014Helen, Annabel, and the lost Lenore\u2014in turn had been translated in a manner prescribed by the texts. But there had been no passage to Dream-Land. No Golden Age.\n\nNo Virginia.\n\nWhy had it not happened as prophesied? _The final globe of globes will instantaneously disappear, and God will remain in all._\n\nThere must be something he was missing, something he had yet to do. But what was it? What could it be?\n\nHe staggered away from his reading table, his hand pressed against his brow, his heart filled with sorrow. Why did it have to be so hard? Was this despair that the prophet had felt? Was this why he had ultimately failed? Why he had drunk himself to a crapulous demise on the streets of Baltimore?\n\nHe threw his arms up toward the heavens. Why must the road to redemption be strewn with thorns? Would he never find peace?\n\n _Don't go near the ocean, Ernie. Nana told you not to go near the ocean._\n\nHe closed the door to his bedroom and entered the living area, then turned on the television and began scanning channels. It was a little early for the news broadcasts, but perhaps there would be something to transport his mind for a brief time. . . .\n\nIn only a few moments, he had found a program of interest. That woman. The mother. And another man, prematurely gray hair, bulge around the center.\n\nThey were talking about him. His work.\n\nThat part didn't interest him much. The media attention had ballooned to such an extent that he had almost tired of hearing about himself. Idle speculation, repetition, sidebars on other cases not remotely similar to his own. It was all the same uninformed claptrap, over and over again, signifying nothing.\n\nBut this woman had something very different to say.\n\n\"Dr. Spencer,\" the host said, \"you've been caught in the eye of the hurricane. After building a career based upon helping others, including those bereaved by violent crime, you find yourself crime's victim. To the rest of the country, perhaps even the world, this is a fascinating, gruesome murder mystery. But to you\u2014it's personal. How are you dealing with the loss of your daughter?\"\n\nShe was wearing a red dress, he noticed, red like the blood of the offering, with a neckline more suitable for a prostitute than a mother. No doubt she had used that costume to get on the air, to excite impure thoughts in unsuspecting men. She cared nothing for sweet Annabel. She craved attention for herself.\n\n\"It's a struggle, Chet. I won't lie to you. Pulling myself out of bed. Facing a new day. Confronting the horror of . . . of what happened to Annabel. A loss like this\u2014it's just devastating.\"\n\n\"I can only imagine,\" he said, his eyes watery. \"And yet you've managed to keep going.\"\n\n\"It's been hard. But I\u2014I have an obligation to Annabel. And her child\u2014my unborn granddaughter. Annabel was a fighter, right to the very end.\" As if she would know. \"So I have to be, too. I have to be strong for her.\"\n\n\"I think you're doing an impressive job of that, wouldn't you agree?\" The live audience unleashed a supportive round of applause.\n\nSpencer smiled slightly. \"I've always thanked the Lord for my blessings\u2014especially Annabel. No matter how busy things were in the world of television, I never forgot that Annabel was my top priority.\"\n\nHe choked. Not what she told me, you two-faced harridan. You were an absentee mother who didn't even call on the weekends. She told me you couldn't identify her boyfriend, didn't even know she had one. Why don't you explain to the man why you didn't know your own daughter was pregnant? Why she flew to Vegas rather than to you for help?\n\n\"Now Dr. Spencer, you've been quite active in the investigation into your daughter's murder.\"\n\n\"Chet, I have no choice. She was the dearest thing in my life.\"\n\n\"And you haven't been afraid to criticize the law enforcement officers investigating the case, either. You've been quite vocal about your objections.\"\n\nShe paused thoughtfully. \"I never thought of myself as a tub-thumper. But how can I remain silent? This killer tortured and murdered my daughter! Most crimes are solved shortly after the crime is discovered or they aren't solved at all. I first talked to the LVPD officers twenty-four hours after my daughter's remains were discovered, and they knew nothing. That hasn't changed\u2014even now, when a third victim has been discovered. All my suggestions, all my offers to help fell on deaf ears. And they've made the most inexplicable, unforgivable personnel assignments.\"\n\n\"You're talking about the behavioral expert, aren't you? Susan Pulaski.\"\n\n\"Among others. God knows I hate to single out the only woman working on the case. But she's an alcoholic. Barely out of rehab. It's inexcusable.\"\n\nShe was doing it again. Making her ad hominem attacks on Susan for her own petty reasons. Spreading Susan's secrets to every moron with a television. Had she no sense of decency? Of propriety? How would she like it if her secrets were bared on the open airwaves?\n\n\"Now, to be fair, Dr. Spencer, my sources tell me that Lieutenant Pulaski does have some solid experience with aberrant criminal psychology. And she's not exactly in charge of the case, is she?\"\n\nThe woman squinted slightly, but there was no wrinkle. Plastic surgery, he surmised. Probably lots of it. And she was barely middle-aged. How telling. \"Any contact is too much, Chet. I want my daughter's case to have the best. I insist upon it. That's why I've taken steps.\"\n\nHe reached for his teacup, but his hand was shaking. Steps? What . . . steps?\n\n\"Please tell the audience what you've done, Dr. Spencer.\"\n\n\"I can't tell you everything. But I've hired private detectives, several of the best. They're looking into this case, and they've already made several interesting discoveries. Things the police totally overlooked.\"\n\n\"Can you give us an example?\"\n\n\"Sorry, Chet, no. You never know who might be listening.\"\n\n\"I understand.\"\n\n\"I would like to say this, though.\" She turned slightly, adjusting her seat so that she was looking not at her host but directly into the camera. \"My first instinct was to make an appeal to the killer. But everything I've learned about this case, everything my detectives have discovered, suggests that it would be useless. This man is sick. A sexual deviant. Someone who likes to torture little girls. My experts tell me he probably started when he was young, maiming animals, deriving pleasure from it. Setting fires. They tell me he enjoys torturing his victims before killing them, that it makes him feel powerful, sexually gratified, taking off their clothes, doing\u2014\" Her voice choked. \"Doing hideous things to them.\"\n\nHis lips parted as he stared wordlessly at the television screen. No. _No!_\n\n\"The experts tell me it's even likely that . . . that . . .\" She turned away, wiping her eyes. \"That he probably . . . did things to Annabel and the others . . . after they were gone. That he would seek sexual gratification from the dead.\"\n\nHe stumbled backward, knocking over the chair. Calumny!\n\n\"We are dealing with the worst scum who ever walked the face of the earth. A human worm. So I won't bother appealing to his better nature. But I will say this to all the other people out there, the good people, the ones who want to catch this man as badly as I do. He has struck three times. Common sense tells me someone must know something. Someone must work with him. Someone must live next door to him. Someone must've sold him a cup of coffee. Someone must've seen or heard something that made them suspicious.\"\n\nShe leaned into the camera. \"Please come forward. Call me at the command center I've set up at Las Vegas's Transylvania Hotel. I will personally reward anyone who brings us useful information with a no-questions-asked award of a hundred thousand dollars. All you have to do is call.\"\n\n\"Talk about putting your money where your mouth is.\" The host gazed at her with adulatory eyes. \"But Dr. Spencer, shouldn't any potential witnesses call the police?\"\n\nShe drew in her breath. \"Of course, I can't suggest that any informant should not contact the police. All I can ask is that you call me, too. Give me a fighting chance to find this monster.\"\n\n\"Given that kind of incentive, Doctor, I think anyone out there with information will be calling you first.\"\n\n\"That filthy murderer had better hope they don't.\" Her eyes lowered, then darkened. \"Because if I get to you first, mister, it won't be so I can read you your rights.\"\n\nHe clutched the remote, punching the power button, then flinging it at the set.\n\nHe was breathing rapidly, perspiring. His entire body was shaking.\n\nShe was coming after him. That woman was coming after him. That damnable whited sepulcher\u2014pretending to be so noble, when in fact she was as base and vile as the serpent. Destroying his reputation, tainting his good work with her relentless animadversions.\n\nShe was threatening him, threatening him with her money and her detectives and her sick sick words. She had called him a sexual deviant. A torturer of young women. She had sat there in front of thousands of people, perhaps millions, and told them he was a demented necrophiliac!\n\nAnd she had sent them hunting for him, enticing them with her petty little cumshaw.\n\nHe paced around the living room, trying to calm himself, to get a grip on his thoughts. This could not be permitted. He was working at a sacred cause. He sought the truth and the light, the Golden Age. And he wasn't just doing it for himself; he was doing it for all of them. Even Annabel. Even that hideous woman, so determined to repugn him at every step!\n\nHe had tried to maintain some degree of gentility throughout this process, but if more direct means were required, then he had no choice but to provide them. Even if it wasn't in the plan, even if she could never be an offering. She must be stopped. And so she would be. And so would be all those who stood against him at the dawn of the Golden Age.\n**CHAPTER SIXTEEN**\n\n\"Her name was Lenore Johnson,\" Granger said, not bothering with any niceties such as \"Good morning\" or \"Hello\" or even \"How's tricks?\"\n\n\"Lenore? The lost love in 'The Raven' is named Lenore.\"\n\n\"Must be a different chick,\" Granger brilliantly opined. \"This one worked at an S&M club a few blocks off U.S. 69.\"\n\nI stared at the photo he slid onto my desk. It was her, all right. I'd recognize that head anywhere. \"Lenore Johnson, huh? Not very Asian-sounding.\"\n\n\"Mixed-race. The Asian is all on her mother's side.\"\n\n\"Positive ID not twelve hours after we found the decapitated body. Nice work. How'd you manage it?\"\n\n\"That's why\u2014\"\n\n\"\u2014they call you detectives. Right, I remember. Know what, Granger? You're full of it.\"\n\n\"Least I still have a job.\"\n\nWhy, why, why? Why did he have to be such an asshole? \"It wasn't my fault O'Bannon yanked my badge. He was being pressured\u2014\"\n\n\"Don't make excuses. By all rights, you should be out on your\u2014\"\n\n\"Can't you see that I'm trying!\" I screamed at him, so loud that I attracted attention all across the office. \"Can't you see I'm trying to do better? I need this job!\" My eyes began to water up. I hate that. It's so . . . girlish. I felt humiliated. \"Why do you have to hate me so much?\"\n\n\"You know the answer to that question,\" Granger said quietly.\n\n\"Do you think you're the only one who loved him?\" I cried. \"You goddamn, self-righteous\u2014\"\n\n\"Hey, whoa, _qu\u00e9 pasa,_ man?\" It was Patrick, riding in on his white horse. \"What's going down? Big case discussion?\"\n\nI threw myself into my not very comfortable desk chair and wiped the water from my face. \"Granger was telling me about his big breakthrough in the case. He's identified the last victim.\"\n\n\"That's his big breakthrough?\" Patrick turned toward Granger. \"I thought one of your detectives recognized her from the head shot. If you'll pardon the expression.\"\n\nI raised an eyebrow.\n\nGranger stammered. \"W-Well . . . we were assisted by a certain degree of facial recognition pattern in\u2014\"\n\nI gave him a look that would turn flesh to stone. \"You mean your great moment of deduction came because one of your detectives is into S&M?\"\n\nHe cleared his throat. \"It's important that detectives stay hip to the mean streets of the city. You never know when\u2014\"\n\n\"Oh, give me a break.\" I did my best imitation of his whiny voice. \" 'That's why they call us detectives.' Jesus. More like 'That's why they call us porn addicts.' \"\n\n\"Hey,\" Granger said, \"I didn't see you finding the corpse. Figuring out what that 'neon' reference meant.\"\n\nPatrick spoke again. \"Didn't O'Bannon tell me that the owner of that sign graveyard found the corpse? And called you.\"\n\nI couldn't believe it. \"And this was how you obtained your other big insight?\" I stood up and did my best impression of a macho stud walk\u2014even groped myself. \"We be studs. We be detectives.\"\n\n\"You're disgusting,\" Granger said, walking away.\n\n\"Sticks and stones,\" I muttered, watching with pleasure as he departed. Probably stupid to piss him off so badly. But if he'd had the clout to get me fired, it would've happened a long time ago.\n\nDarcy made his way up the stairs. I grabbed his hand. His face lit up like a lightbulb. \"Darcy, guess what the third victim's name is.\"\n\nHe thought for, like, a nanosecond. \"Lenore.\"\n\n\"What? Someone already told you.\"\n\n\"I do not think anyone told me. I just got off the bus. But I thought that maybe all of the girl's names were from Poe poems. And Lenore is the most popular\u2014\"\n\n\"Why didn't you tell me?\"\n\n\"I do not think you asked me.\"\n\n\"You don't have to be asked, Darcy. If you know something, you should just . . . volunteer it.\"\n\n\"My dad does not like it when I do that. He says I tell people a lot of boring things they don't want to hear, and sometimes I say things that get me into trouble because I don't understand. He says I should be quiet unless\u2014\"\n\n\"Listen to me, Darcy. New rules. If you know something\u2014anything\u2014you tell me. Immediately.\"\n\nPatrick stepped in. \"So the victims' names are all found in these poems?\"\n\n\"Right. I was suspicious when we had an Annabel\u2014'Annabel Lee' is one of Poe's most famous verses. But having a Lenore clinches it. That's the name of the girl in 'The Raven,' and he used the name in other poems as well. Always to represent some lost love. Actually, all of these names represent some unrequited or lost love.\" I tossed Darcy my library copy of _Poe: His Life and Legacy_ by Jeffrey Meyers. \"Would you mind reading this tonight?\"\n\n\"Okay,\" he said with alacrity. \"Why?\"\n\n\"Because you'll remember it.\"\n\nA line crossed Patrick's forehead. \"You really think this biographical material will be important?\"\n\n\"It may be critical,\" I answered. \"It may hold the key to the whole puzzle. Even that last message.\"\n\n\"So in your view, our killer thinks he's Poe?\"\n\nI squirmed. \"I don't know that he literally thinks he's Poe. It's more that . . . that . . . he takes inspiration from him, his work. Not just when he's selecting his murder methods, but\u2014everything.\"\n\n\"Edgar Allan Poe is his role model?\"\n\n\"Kind of, yeah. Which explains a lot. According to this book, the public image of Poe as this ghoulish creepazoid is inaccurate. His work was creepy, but he wasn't.\" Except when he was on a drinking binge, of course. \"He thought of himself as a proper southern gentleman. He was offended by vulgarity, impropriety.\"\n\n\"And what does that tell us about the killer?\"\n\n\"Well, for starters, it might explain why he removed the painted nails. Piercings.\"\n\nPatrick nodded slowly. \"Shaved Helen's hair.\"\n\n\"Because it was dyed an unnatural color.\"\n\n\"This is beginning to make sense. I mean, a twisted, narcissistic, antisocial, delusional kind of sense.\" He thought for another moment. \"But if the women in these poems represented some sort of Poe ideal\u2014\"\n\n\"They all died,\" I said, thinking off the top of my head. \"That's the key. Helen was a woman he admired when he was an adolescent. Annabel Lee and Lenore were versions of his wife, Virginia, who died of tuberculosis.\"\n\nDarcy spoke. \" 'And so, all the night-tide, I lay down by the side \/ Of my darling\u2014my darling\u2014my life and my bride . . .' \"\n\n\"Exactly. Annabel Lee, Lenore, and a dozen other characters in Poe's poems and stories. They're all his dead wife.\"\n\n\"But,\" Patrick said, \"what's the point of it all?\"\n\n\"I don't know. But that quote\u2014the last one. I think that's the key. We have to figure out what it is. What it means.\"\n\n\"And,\" Patrick added, \"we need to make a list of all the female names used by Poe in his stories and poems.\"\n\n\"It's going to be a long list,\" I said, \"but I agree. It might be useful. Maybe we can put out some kind of warning. Darcy, are you up for it?\"\n\n\"Did you know that Poe wrote fifty-three poems and seventy-three short stories?\"\n\n\"No, but you do, which is why you're the best man for this job.\"\n\n\"So the guy has been choosing women with these Poe names,\" Patrick said, his mind still racing. \"But why so young? Are they easier to control? Is the killer a repressed pedophile?\"\n\nI shook my head. \"Don't you know?\"\n\n\"What?\"\n\nEven when the agent was a decent guy, knowing something the FBI didn't was not an altogether unpleasant sensation. \"That bride of Poe's? Virginia Clemm? He married her when she was thirteen.\"\n\nOvernight, the Van Helsing Ballroom had been converted into the nerve center for Dr. Fara Spencer's Wanted Dead or Alive operation. The room was a beehive of noise and activity, a bombinating assault on the senses. And yet, he observed, it was not chaotic. There was an almost serene order as all concerned careered from one area to the next going about their designated tasks. A dozen operatives milled through the room in straight ties, white shirts, and rolled sleeves, some of them private detectives, some retired police officers, some specialists hired to lend expertise or to screen potential informants. Security officers were posted on all doors. Interviews were conducted in private alcoves. Two rows of phone banks, with over two dozen phones, filled the length of the ballroom, and they were constantly ringing, ringing, ringing . . . _to the tintinnabulation that so musically wells, \/ From the bells, bells, bells, bells, bells\u2014_\n\nHe soaked up the view, smiling. The hive was running smoothly. But where was the queen bee?\n\nDr. Spencer did not so much walk as march into the ballroom, two men on either side of her, several behind, all talking constantly. At least one of the hangers-on was a reporter; he was not sure about the others. He knew she had a fleet of so-called behavioral experts advising her on the case, suggesting potential avenues to explore. Earlier, while posted at the front door, he had managed to overhear most of an absurd exchange between the queen and two of her minions.\n\n\"Fundamentally,\" a pedant in horn-rimmed glasses had explained, \"serial killers can be divided into two categories. Social and nonsocial. Organized and disorganized.\"\n\nHe had to bite his lip. Even given the vagaries of modern psychiatry, it was absurd. The whole world divided into four lame labels. And these people called themselves experts.\n\n\"So which is this pervert?\" Spencer had asked.\n\n\"Keep in mind that we're working with precious little information,\" the partner said, an obese man in an unseemly green tie. He was making excuses for himself in advance, as they always did. \"But all indications are that he is very organized. Three crimes so far\u2014that we know of\u2014and he still hasn't left behind any determinative trace evidence.\"\n\n\"And we can safely assume that he has some social skills,\" Horn-rims intoned. \"Since he appears to have been able to capture his victims without the use of force. So far as we know.\"\n\n\"All right,\" Spencer said, holding up her hands. \"So he's an organized social. What does this get us?\"\n\nDespite his profound dislike of this contemptible woman, she did have a knack for cutting through the balderdash.\n\n\"Well,\" Horn-rims said, \"once we've made our diagnosis, we can get a fix on who the killer is.\"\n\n\"I don't care who he is,\" Spencer shot back. \"I don't want to know his inner child. I just want the bastard to fry.\"\n\n\"Right, right. So we must create a profile\u2014\"\n\n\"Forgive me for saying so, but this is starting to sound like the same bullshit I got from the police department's so-called expert. I'm not laying out all this money to get more of the same.\"\n\n\"Of course not. I'm probably not explaining myself clearly.\" He licked his lips and tried again. \"Once we know who the killer is\u2014what kind of person he is\u2014we can begin to anticipate his moves. Perhaps even trace him to his lair.\"\n\nHis _lair._ He stifled his laughter. It was like something out of a comic book.\n\n\"Now we're talking.\" Spencer's interest level markedly increased. \"So how do we do that?\"\n\n\"Well, Doctor, one salient fact we have observed is that all the victims have been women.\"\n\nBravo, Auguste Dupin. How much was she paying these fools?\n\n\"According to the group consensus, our killer is driven by a psychosexual hatred of all women.\"\n\nStill pretending not to listen, he stifled a yawn. Wrong.\n\n\"He is physically unattractive, or at the least not handsome. Because he is unable to attract women, he came to hate them.\"\n\nWrong, wrong, wrong. Every bit of it.\n\nThe partner cut in. \"Perhaps it would be more accurate to say that he focuses the sublimated anger he bore toward his mother figure on randomly chosen women.\"\n\nRandomly chosen? Now that was truly offensive.\n\n\"This speculation is all well and good, I suppose,\" Spencer said, \"but where does it get me? How do I catch him?\"\n\n\"By keeping a close watch on all the places it would be easiest for an organized social to find women alone,\" the partner answered. \"The streets, obviously, especially downtown. Strip clubs, casinos. We've prepared a list.\"\n\nSpencer took the proffered paper. \"I don't see this hotel listed.\"\n\nHorn-rims nodded. \"Our feeling is that he's unlikely to strike at a place already associated with him. Plus, since organized socials tend to follow the media coverage of their crimes obsessively, he must be aware that you've set up shop here. He won't come anywhere near this hotel.\"\n\nIt was all he could do to keep from laughing.\n\nUntil the woman stood beside him.\n\n\"So . . . your name is Ernie?\" Spencer said, reading his badge.\n\nHe stiffened. \"Yes, ma'am.\"\n\n\"Keeping an eye on things?\"\n\nHe was on loan-lease from the hotel to Dr. Spencer and was currently assigned to watch one of the side doors. \"Doing my best. Lot going on here.\"\n\n\"Yes, there is. Try not to be distracted by it all.\" She leaned in closer. \"I'm probably safe here, but, still my experts say there is . . . some reason to believe the killer might lash out against me.\"\n\n\"Surely not, ma'am.\"\n\n\"I know, it seems incredible. But I'm being extra careful, just in case. There's more than one way to skin a rabbit.\"\n\nShe was so close to him. Mere inches away. In his coat pocket, he had a hypodermic, loaded and ready. In the wink of an eye he could have it in her throat.\n\nBut then there would be the difficult matter of her escorts, getting her to his truck on the other side of the casino, removing her from a ballroom teeming with people. . . .\n\n\"I'll make sure nothing happens here, ma'am.\"\n\n\"I'd appreciate that.\"\n\n\"And if I may say so . . .\" His southern accent came trippingly off his tongue. \"Speaking on behalf of all the security officers here at the Transylvania, we very much appreciate and admire what you're doing. We're behind you one hundred and ten percent.\"\n\nHer eyes went slightly out of focus, glistening. \"Thank you, Ernie. That . . . that means a great deal to me. I hope we can talk again sometime.\"\n\n\"I feel certain we will, ma'am,\" he said, smiling sweetly.\n\n\"Seriously, Darcy, I think it would be better if you stayed in the car.\"\n\n\"But I do not want to stay in the car. I cannot help you if I stay in the car.\"\n\n\"You can. I'll tell you all about it.\"\n\n\"I want to go in with you.\"\n\nWe were parked outside Nighthawks, still in my car. Truth was, I would rather he came in, too\u2014you never knew what he might notice. But I had a feeling it might not be a good idea. \"You see, Darcy . . . this place . . . this is a grown-up place.\"\n\n\"All those places where girls died were grown-up places.\"\n\n\"Yeah, but this is . . . this is . . .\" I took a deep breath. \"Darcy, has your dad ever . . . had that talk with you?\"\n\n\"My dad talks to me all the time. Sometimes I wish he wouldn't talk to me so much.\"\n\n\"Yeah, but has he ever talked to you about . . . the birds and the bees?\"\n\nHis eyes widened. \"I've read lots and lots of books about birds and bees. Did you know that the hummingbird is the fastest\u2014\"\n\n\"That's not what I mean, Darcy.\" I took a big breath. \"See, Darcy . . . this is a sex club.\"\n\nHe looked at me, then at the building, then back at me. \"Do you mean\u2014they do sex in there?\"\n\n\"Yes. I mean, probably not. Certainly not all the\u2014\"\n\n\"I have never seen anyone do sex. Can I go in with you and see it?\"\n\nI pressed the palm of my hand against my forehead. \"Are you sure you'll be okay with this, Darcy?\"\n\n\"Of course. This is not like gong to all those places where people got killed. Killing is bad. But I think doing sex is a good thing. Do you think doing sex is a good thing?\"\n\nI popped open the car door. \"C'mon, Casanova.\"\n\n\"Good afternoon,\" the woman in black said as I stepped through the door. \"I am the mistress of pain.\"\n\n\"Stow it,\" I replied, flashing my ID. \"I'm the mistress of pain in the ass.\"\n\nShe blocked my path, pressing up against me. \"You prefer to be dominant?\"\n\n\"I prefer to get what I need without any hassles.\"\n\nHer fingers toyed with my collar. \"You should open yourself to new experiences. I could\u2014\"\n\nI slapped her hand away. \"Let's get one thing straight from the start. If you think you're going to intimidate me with your lesbian chic bullshit, forget it.\"\n\nDarcy was behind me, staring with a total absence of subtlety. But of course subtlety was a personality quality he didn't have. The mistress was wearing brown riding pants, very tight, and a black leather bustier top. Not hard to guess where Darcy was staring.\n\n\"Do you have trouble breathing?\" he asked. \"Because it looks like you might have trouble breathing.\"\n\nThe mistress pointed her riding crop at him and winked. \"Breathing is overrated.\"\n\n\"Do they make you wear that?\" he continued. \"Are you being punished because you misbehaved?\"\n\n\"No, dear.\"\n\n\"I've never seen a shirt that had to be tied up in the front like that. Do you like to be all tied up like that?\"\n\nShe turned her withering expression my way. \"One of your crack detectives?\"\n\n\"As a matter of fact, yes. I have a few questions for you.\"\n\n\"I've already spoken to several\u2014\"\n\n\"You haven't spoken to me. I have some additional queries.\"\n\n\"Well, I have a business to run.\"\n\n\"I can change that.\"\n\nShe jammed that crop under my chin, forcing it up. \"You like to play rough. Is that what you're into?\"\n\n\"You don't have any idea what I'm into.\"\n\n\"I've been at this a long time. I can tell what a client wants in about thirty seconds.\"\n\nWe were practically nose to nose. \"I'm not a client, and if you don't cooperate, I'll send Vice over to shut you down for good.\"\n\n\"Bitch.\"\n\n\"Lady, you don't know what bitch is till I get started. So are you going to talk to me, or what?\"\n\n\"Well . . .\" She glanced beyond the red curtain, down a corridor. \"These are business hours.\"\n\n\"And what business would that be? Nothing illegal, I hope.\"\n\n\"Of course not.\"\n\n\"I would certainly hate to find out there was\"\u2014I made a little gasp\u2014\"prostitution on the premises, because that's still not legal in Vegas.\"\n\n\"Contact dancing is permitted. At least for now.\"\n\nShe was right. Despite Vegas's rep as Sin City, prostitution had never been legal here. Customers had to leave town and go to joints like the famed Chicken Ranch for that. For it to be legal, anyway. In reality, prostitution was not uncommon. A lot of it passed under the guise of \"outcalls\" or \"room dancing.\" Escort services with girls who met you at your hotel room fronted a lot of it, too. After a 2002 law change, lap dancing became technically legal in Clark County, but dancers were not permitted to touch or sit on the customer's genital area, which some would have said was the definition of a lap dance. What many people didn't realize was that the Las Vegas Strip\u2014and most of the clubs on it, including this one\u2014were outside the city limits. Municipal officials had no jurisdiction.\n\n\"Is that what you give your customers?\"\n\n\"In part. We're about fantasy fulfillment.\"\n\nDarcy probably only understood a tenth of what was being said, but he was still red in the face, and I knew it was only going to get worse. \"Darce, why don't you take a stroll around the premises while we talk? See if you can spot anything the detectives missed.\"\n\nI could tell he didn't want to leave me, but he did as I asked. If only my previous partners had been so compliant.\n\nThis chick was way over the top, but by Vegas standards, she was a perfectly average, ordinary working girl. After all, Vegas was the one city where a girl with no training, no education, and not incredibly bright could still make a good living, own a house, raise kids. Thanks to the Culinary Union, even nongaming cocktail waitresses got nine bucks an hour, plus tips, which was where the real action was. Gaming waitresses got fourteen. Where else could a cocktail waitress afford a mortgage and car payments? Where else could a high school dropout park cars and make enough to send his kids to college? Call girls\u2014even run-of-the-mill ones\u2014took home anywhere from five hundred to three thousand bucks a night, depending upon what exactly they were willing to do. Anywhere else in the world, this woman would be sleeping under a bridge in a cardboard box. In Vegas, she was the mistress of pain.\n\nHey, it wasn't called Sin City because of Wayne Newton.\n\n\"What can you tell me about Lenore Johnson?\"\n\n\"Nice girl,\" she answered. \"Did whatever she was asked. I like that in an employee.\"\n\n\"How long had she been here?\"\n\n\"About three months.\"\n\n\"Know anything about her background?\"\n\n\"She came from Kansas, poor girl. Father was a police officer. She didn't do drugs\u2014something of a rarity in this field. She was well mannered, respectful. Didn't have the attitude a lot of my girls get. She was trusting.\"\n\nWhich was probably what killed her. \"Did she do outcalls? For sex?\"\n\n\"Not to my knowledge. That would be illegal, you know.\"\n\n\"Yeah, but did she do them?\"\n\n\"I don't think so. She was a good girl.\"\n\n\"But she was working here. The night she disappeared.\"\n\n\"Yes, and two of my girls saw her leave with a customer. One of them actually referred her to him. She blames herself.\"\n\n\"She shouldn't. He picked his victim based on her name and her appearance, not any referral.\"\n\n\"Really?\" For the first time, her mistress-of-pain fa\u00e7ade cracked a bit. \"I'll tell her that. I hope she takes some comfort from it.\"\n\n\"I'll want to talk to all the, um, employees who saw this guy.\"\n\n\"I'll assemble them. But you won't get much. They can't describe his face.\"\n\n\"Surely if they saw him right here\u2014\"\n\n\"But my girls are trained never to look a customer in the eyes. The entire face is to be avoided, as much as possible. We don't want attachments forming. It clouds the judgment. In this line of work, it's important to retain a certain professional detachment.\"\n\nJust my damn luck. My only eyewitnesses are sex merchants who've been trained not to look at people's faces. \"I'll still want to talk to them just as soon\u2014\"\n\nI was cut off by a piercing scream from down the corridor. I raced past the mistress, fumbling for my weapon, remembering that I wasn't allowed to have one anymore. Damn!\n\nAnother cry, this one even more terrified than the first. I tracked it to a closed door, grabbed the knob, flung it open.\n\nThere were two women in the bed, both stark naked. The one on top, the one with the surgically enhanced knockers who was holding a huge dildo in her right hand, appeared to be the trained professional. The skinny girl who had pulled the covers up to her neck was undoubtedly the customer.\n\nDarcy was on the floor at the side of the bed, hunched over in a fetal position, rocking back and forth. He was making strange nonsense noises, babbling, whimpering.\n\nThe mistress came in behind me. \"What happened, Kimberly?\"\n\nThe silicone princess dropped her equipment. \"We were just\u2014\"\n\nThe mistress shot her a harsh warning look.\n\n\"\u2014having a conversation,\" she continued. \"And this simp comes rushing into the room.\"\n\nDarcy looked as if he were having a total meltdown. I'd never seen him like this. He began pounding his head against the floor. I ran to his side and wrapped my arms around him. \"Darcy\u2014what happened?\"\n\nHe flapped his hands, rubbed the sides of his head. \"Did you think that one was in trouble? Because I thought she was in trouble.\"\n\n\"But why\u2014\"\n\nHe couldn't stop rocking. \"She was screaming. Screaming real loud. I thought the big one was hurting her.\"\n\nI closed my eyes. \"So you rushed in to help?\"\n\n\"And I saw she wasn't wearing any clothes, and I remember the bad man took all the girls' clothes away, and I tried to help, and she hit me with\u2014with\u2014that thing.\" He was hand-flapping with a frenzy. His voice was never well modulated, but now it sounded as if he was shouting. \"Why did she do that? Why did she hurt me? I don't think people should hurt each other!\"\n\nI took his wrists and tried to get him under control. \"It was a misunderstanding, Darcy. She wasn't hurting the other woman. She was just\u2014\"\n\nOkay, where did I go from there? Even if O'Bannon had had that little talk with his son, would it have covered activities such as the one he'd just stumbled upon?\n\n\"Let's go back to the car, Darcy,\" I said. \"We'll get a custard or something.\"\n\n\"Why would she scream if she wasn't hurting? I screamed last night when I stubbed my toe because it hurt and she was screaming and she didn't have any clothes on and\u2014\"\n\n\"Come on,\" I said firmly. \"We're leaving.\"\n\nI made a few excuses to the mistress and got him the hell out of there. Damn it all. I should have seen that coming. Maybe Granger had been right. Maybe I didn't have any business dragging Darcy to these horrible places. All kinds of traumatic things might be going on inside his head that I knew nothing about. I had enough problems without playing with fire of this magnitude. O'Bannon's autistic son. Christ, what was I thinking?\n\nThose are bad girls and I know they are and they were doing bad things. Bad people go to hell and I don't want to go to hell. Mr. Strickland said that we have to behave ourselves and if we didn't we'd go to hell and he took me by the hand away from the others and told me he knew what I was thinking that I had these ideas and all the boys like me did and we couldn't control them but I had to or I would be a dirty boy and I would go to hell. Bad girls! And the smell was so yucky on the big girl with the mole under her right knee and the holes all up her arm. Like the smell of Mommy's dishwashing gloves when Mommy was still alive.\n\nI hope Susan doesn't stop taking me places even though I had a fit and Dad told me to control myself but I couldn't help it and I wanted to rip my hair out but I didn't and I hope Susan doesn't stop taking me because I was bad but I'm afraid she will because she has been smelling really funny bad and it isn't funny and Dad wouldn't let me read the D. H. Lawrence books because he said they would be bad for me and I think this is all scary and I wish people wouldn't do those things to other people. Bad girls! Bad girls!\n\nMidnight. Most of the operatives had gone home, but Dr. Spencer and several others were still in the hotel ballroom. The phone rang incessantly. He had an hour to go before his shift ended.\n\n\"This is really something, isn't it?\" Harv said with his usual conversational panache.\n\n\"Did you have a specific _this_ in mind,\" he replied, \"or just a general _this_?\"\n\n\"This. Everything.\" He waved his arm about. \"The whole works. Can you believe this operation was pieced together by one woman? What a pistol.\"\n\n\"A . . . pistol?\"\n\n\"Yeah, you know. A hot tamale. Proactive. Ballsy.\"\n\n\"I don't see that her efforts have produced much in the way of results.\"\n\n\"Give her time, Ernie. They will. Everyone knows it. She must have that sick son of a bitch quaking in his boots.\"\n\nThat might be something of an exaggeration, all things considered. \"You seem to be enjoying this assignment.\" Which might explain why Herb was still hanging around, even though his shift had ended half an hour ago.\n\n\" 'Course I am. Didn't I tell you I always wanted to be a cop?\"\n\n\"Yes, but you weren't and you still aren't. You're a security officer temporarily assigned to a private room. No one here is a cop.\"\n\n\"I'm a lot closer than I was bagging pickpockets in the blackjack pit. I mean, you can feel the excitement in here. You can breathe it. Makes my whole body tingle. Hell, I'm having a moment as we speak.\"\n\nThat was really more information than I required, he thought ruefully.\n\n\"On this detail, we're a part of something that matters. The whole world is watching this investigation.\"\n\n\"The whole world is watching the police investigation. This gang is little better than a well-financed vigilante squad.\"\n\nHarv ran a hand through his russet curls. \"You're pretty damn down on this operation. But I know you volunteered for it. Why? If you dislike it so much, why don't you go back to policing the slot machines?\"\n\nWell, there was a very good answer to that question, but he wouldn't be sharing it with Harv. \"I need the money.\"\n\n\"That bonus in the pay envelope was pretty good, wasn't it? I may be able to take Elaine on that Halloween holiday she wanted.\" He took a handkerchief out of his back pocket and wiped it across his brow. \"No disrespect to my Elaine, but Dr. Spencer looks pretty damn good for her age, doesn't she?\"\n\nHe made a slight clicking noise with his tongue. \"Too much hair. And the plastic surgery was a mistake.\"\n\n\"Yeah, like you'd kick her out of bed for eating crackers.\" He laughed. \"The ladies are tough on you little guys, aren't they?\" He gave his co-worker a gentle jab that was not returned.\n\nA few minutes later, a trim black man with a cell phone in each hand approached them. \"Which one of you two officers is in charge?\"\n\n\"I am,\" Harv chirped.\n\nHe burned. Was that because you're so incredibly tall? \"We're both of equal rank and stature on this security detail,\" he replied.\n\n\"Well, the doctor needs someone to drive her to the airport.\"\n\nBehind them, Dr. Spencer approached with her usual no-nonsense deliberateness. \"Hello again, Ernie. Car's parked out back, same lot you people use. These bodyguards can get me to the parking lot. But I need a driver. So which of you lucky boys is going to do the honors?\"\n\nHe cut Harv off before he could speak. \"I would be pleased to escort you, ma'am.\"\n\n\"Sure you wouldn't mind? It isn't an official part of your job description.\"\n\n\"That doesn't matter. I would be honored.\"\n\n\"Well, if you're sure you don't mind, let's be off.\"\n\n\"Maybe I should be the one to go with her,\" Harv cut in.\n\nHe felt his jaw clenching. Steady, old boy. Steady . . .\n\n\"Well, you're still on duty, aren't you?\" Harv added.\n\n\"My shift ends at one.\"\n\n\"And we don't want to leave the door unguarded.\"\n\n\"But if the good doctor is leaving now\u2014\"\n\n\"I got off half an hour ago. I'll take her.\"\n\n\"I could still\u2014\"\n\n\"No, he's right,\" Spencer said. She was looking at him, staring with an intensity he had not felt before. Did she . . . see something? Was he too anxious? Was it possible she suspected? \"It makes more sense for him to go.\"\n\n\"Yes, of course,\" he said, acquiescing as gracefully as possible. \"Take care of the good doctor, Harv. You never know what might happen out there.\"\n\nInfernal imp, he thought, swearing to himself as he pushed his way into the break room. He'd almost had the woman exactly where he wanted her, where he needed her. A moment of vulnerability handed to him like a gift. Until Harv, with his inveterate imbecility, pursuing his infinite dreams with his infinite ego, intervened.\n\nHe poured himself a cup of the hotel's mediocre coffee. In truth, he did not feel overly disappointed. Was it possible his heart wasn't in this work? It had seemed important, but there was no denying that it was not in the plan. And for that matter, the plan had failed to produce its desired results. Perhaps this was a signal that it was time to rest and reconsider. Perhaps he should take a leave of absence, at least until the TV doctor lost her zeal. He could travel, read. And then perhaps\u2014\n\n ** _\"Why have you betrayed me?\"_**\n\nThe voice was deep and reverberating, shaking him to his core.\n\n ** _\"Why have you betrayed me?\"_**\n\n\"I\u2014I don't understand.\"\n\n ** _\"Why have you strayed?\"_**\n\n\"I don't know what you mean.\"\n\n ** _\"What did I tell you to do?\"_** The voice came from nowhere and everywhere at once, from no one place. **_\"What did I tell you to do?\"_**\n\n\"I made three offerings. Just as the texts prescribed.\"\n\n ** _\"And were you successful?\"_**\n\n\"No. Nothing happened.\"\n\n ** _\"What have you learned from this?\"_**\n\n\"That\u2014that perhaps I was wrong. Perhaps this is not the way.\"\n\n ** _\"No!\"_** The dark voice split his skull, knocking him to his knees. **_\"You failed because your faith was weak.\"_**\n\n\"That's not true. I believed. I did everything I could to\u2014\"\n\n ** _\"You failed because your offerings were unwilling. They withheld their spirit. You did not ascend.\"_**\n\n\"I've done everything I know to do.\"\n\n ** _\"Why did you not take the doctor?\"_**\n\n\"I didn't get a chance. Another man\u2014\"\n\n ** _\"That is the answer of the weak! The dissembler. The betrayer.\"_**\n\n\"No! I\u2014I\u2014\"\n\n ** _\"Your faith is not strong. That is why Dream-Land did not accept your offerings. To succeed you must be strong.\"_**\n\n\"Please\u2014\"\n\n ** _\"From this moment forward, everything will be different. You will act as prescribed by your destiny. You will be my Instrument.\"_**\n\n\"Yes. Yes!\"\n\n ** _\"Would you know the secret?\"_**\n\nHe braced himself against the wall. The voice was so loud, so overpowering. \"What\u2014secret?\"\n\n ** _\"Would you know why you have failed?\"_**\n\nTears came to his eyes. \"Yes. Please!\"\n\n ** _\"Then I will tell you. But not until you have done what you know you must.\"_**\n\n\"But I\u2014I\u2014\"\n\n ** _\"It is your Destiny. Do this and I will give you the knowledge that you seek.\"_**\n\n\"I will. I swear to you. I will!\"\n\nIt had taken almost half an hour to finish all the checks, sign the papers, and put the bodyguards at ease, but Harv and Dr. Spencer eventually entered the employee parking lot. It was ridiculously dark, Harv thought. Of course, he came out here every night and it had never bothered him before. But it was different when you were on the job. In this blackness, how well could he protect her? How could he know what lurked in the hedge beyond the north perimeter? How could he know who might be watching from one of the balconies? Maybe he should've asked those bodyguards to come all the way to the car. . . .\n\n\"Best to get right to it, ma'am,\" he said, trying to sound very official. Spencer pointed toward the far end of the third row of cars and they headed that way. \"I've really enjoyed working with your team,\" he added.\n\n\"I'm glad,\" she replied, walking with determination and deliberate speed. \"I'm grateful for your assistance.\"\n\n\"This is a great thing you're doing. A great thing. If you don't mind my saying so . . . you do your daughter proud.\"\n\n\"Well . . . thank you.\" They rounded the end of the third row and started down it.\n\n\"I just wanted to tell you, while I had the chance . . . and I hope you don't mind . . .\"\n\n\"What is it?\"\n\n\"Well, this security work, it's a fine job and all, but it isn't what I really want to do, you know? I mean, I'd hate to think of me ending up like one of those guys you see at the mall, gray hair and a paunch that stretches out a mile in front of the uniform. So old they don't let them carry a gun.\"\n\n\"We all get old, Harvey.\"\n\n\"Yeah, but you're doing work that's important. Not just this but that . . . that stuff back in New York, helpin' people find fulfillment and all. Anyway, I just wanted to say\u2014after this is all over and you've caught the dirty bastard who took your daughter's life, if there's anything I can do for you\u2014\"\n\n\"If I ever need a security officer, you'll be the first one I call.\"\n\n\"Well . . . yeah. That'd be nice. Or anything. There's a lot I can do. I'm good with a wood lathe and I make these pillboxes that my wife gives all her friends at Christmas. They really love 'em. I even play a little banjo. . . .\"\n\nThey reached her car. \"You get in the passenger seat, ma'am. I'll drive.\"\n\n\"If you don't mind.\" She slid inside.\n\nHarv walked around the back. Just as he came by the trunk, the headlights lit on a car at the far end of the row.\n\n\"Who the hell is that?\" Harv muttered. He kept walking.\n\nThe car peeled out. Tires squealing, accelerating.\n\nHeading straight toward him.\n\n\"Son of a bitch,\" Harv muttered. \"Stop! You're under arrest!\"\n\nThe car kept coming. Even faster.\n\nHarv drew his weapon. \"Stop!\"\n\nDespite the haze of the headlights, Harv realized\u2014it was his own car.\n\nAnd it was about to kill him.\n\nHe fired twice. Both shots hit the windshield. But the car kept coming.\n\nHarv turned, trying to get out of the way. But he wasn't nearly fast enough. The car surged forward, engine roaring, till it smashed into the back of Dr. Spencer's car.\n\nBisecting Harv.\n\nHe unbuckled his seat belt and stepped out of the car. Then he paused, gazing down at what was left of his colleague. \"Guess you're not so tall now, huh?\"\n\nHe opened the passenger-side door. Spencer was conscious, but her eyelids were fluttering. The twisted position of her body told him she had not been wearing a seat belt. A streak of blood ran down the right side of her head. She was breathing rapidly, panting, cowering, her hands clutched to her chest.\n\n\"Come,\" he said simply. \"We don't have much time.\" He held out his hand.\n\nSpencer stabbed him with a pocketknife, ripping the flesh of his palm. Then she pushed him away and tried to scramble out of the bucket seat.\n\nHe grabbed her throat with his uninjured hand and squeezed. He knocked the knife out of her hand and shoved her forcefully back into the car, falling forward, halting his descent with a hand on the floor mat. She reached up and scratched the side of his face.\n\n\"Oww!\" His face flushed red with rage. He pulled the syringe out of his coat pocket and jabbed it brutally into her neck. A few seconds later, the struggle was over.\n\nHis hand was bleeding. He had to be careful not to let it drip; the police would have a field day with a blood or skin sample and he didn't have time to perform a thorough cleanup. Even this time of night, someone must've heard the crash. He pulled a tissue out of his shirt pocket and wrapped the wound, wincing at the hideous torn mess she had made of his hand.\n\nHe put the pain out of his mind. It had been a brilliant stroke, using Harv's own car. Since the man had never been able to keep his mouth shut longer than it took to breathe, he'd known which car was Harv's and where he left his keys.\n\nHe could not fault Dr. Spencer for attempting to defend herself. But he could fault her for what she had done before, how she had publicly and brutally maligned him. He could not let this offense pass. He had been told he must be strong, and so, like it or not, he had to act. For his honor. And for . . . for . . .\n\n _For the love of God, Montresor!_\n\nYes, he thought, as he lifted her body and carried it back to his truck. For the love of God.\n**CHAPTER SEVENTEEN**\n\nI saw him again. Standing at the foot of my bed, just like before. He was wearing pajama bottoms and no shirt. I always thought he was sexiest when he wore pajama bottoms and no shirt.\n\n\"Are you attracted to him?\" David asked.\n\n\"I dunno,\" I mumbled, still barely awake, if at all. \"Maybe.\"\n\n\"It's okay, you know.\"\n\n\"Well, yeah, we're both consenting adults, even if he is\u2014\"\n\n\"I mean with me. It's okay with me.\"\n\n\"It is?\" I tried to pull my head off the pillow, but my body wasn't responding to commands.\n\n\"Of course.\" He looked so strong, so manly, like when we were first married. Like a man who was capable of doing anything. \"I don't expect you to become a nun.\"\n\n\"Well, sure, but still\u2014\"\n\n\"You have my blessing.\"\n\n\"That's damn straight of you.\"\n\nHis face dissolved into that smile, that silly, toothy smile that used to turn my insides into goulash. \"Well, I love you, you know.\"\n\n\"You do? Even\u2014\"\n\n\"Even.\"\n\nIt felt good talking to him, so warm and comforting. I felt as if I could sleep forever after that, like I could smell him, right there next to me in bed. So you can imagine the shock I experienced when I blinked and saw Lisa peering down into my face.\n\n\"What the hell have you done to yourself?\"\n\nI tried to bring myself around, but my head was screaming and there are some levels of pain that are impossible to mask.\n\n\"You're hungover.\"\n\n\"Don't be stupid.\" Did I slur my speech? \"I had a tough night.\"\n\n\"I can see that.\" And then the most horrible thing happened. Horrible and horrifying. She started to cry. \"Damn it, Susan, you said you weren't going to do this anymore.\"\n\n\"I\u2014I wasn't\u2014\"\n\n\"I'm supposed to be looking after you. God, what a shitty job I've done. I mean, you've only been out of the hospital for\u2014\"\n\n\"Lisa,\" I said, concentrating on proper pronunciation, \"it's not your fault. I was just feeling a little stressed.\"\n\n\"Oh, spare me your rationalizations.\" She pressed her hand against her forehead. Tears streamed around the palm of her hand. \"I talked to Dr. Coutant yesterday. He told me you missed your appointment with him. That you haven't been going to IOP.\"\n\n\"I've been working.\"\n\n\"They have night sessions.\"\n\n\"I've been working nights, too. You don't want me to lose this job, do you?\"\n\n\"Frankly, I don't care about your job, Susan. I don't even think this is a good job for you, not now. What I care about is you!\"\n\nAnd she did, too. I could see that in her watery eyes, not that I needed additional proof. I felt like something someone might scrape off the heel of their shoe. \"Look, Lisa, I'm sorry. It won't happen again. Honest.\"\n\n\"That's such bullshit.\"\n\n\"No, I mean it this time.\"\n\nShe walked away from the bed and stared out the window. \"It's just like they told me. You're an alcoholic. And alcoholics are liars. They'll say and do anything. Because deep down, no lover on earth can take the place of a chemical addiction. Addicts will lie, cheat, and steal to get their fix.\"\n\n\"Honey, please\u2014\"\n\n\"I'm moving in with you, Susan.\"\n\n\"Now wait just a\u2014\"\n\n\"Don't waste it. I should've never let you stay by yourself. That was stupid. I'll pick you up at work and we'll spend the evenings together.\"\n\n\"I don't need a damn babysitter.\"\n\n\"You don't know what you need. And you'll start going to IOP at night, too, no matter how many crazed killers are on the prowl.\"\n\n\"And supposing I don't agree?\"\n\nLisa sucked in her breath. \"Dr. Coutant has been asked by Chief O'Bannon to report on your progress.\"\n\n _\"What?\"_\n\n\"So that O'Bannon can evaluate whether you're ready for reinstatement. And I talk to Coutant. Often. So you see, Susan, if I give you a bad report\u2014\"\n\n\"You wouldn't.\"\n\n\"I sure as hell would.\"\n\nAt this point in time, I think it would be fair to say I hated Lisa. She was my best friend and always had been\u2014and I hated her. I knew that she was trying to help me, that everything she said was right. I knew how hard it was for her to play tough with me. But I still hated her.\n\n\"That sounds fun, Lisa. A sleepover. Maybe you can paint my toenails. We can braid each other's hair.\"\n\nMy sarcasm was wasted on her. \"Clean yourself up and get dressed. I'll drive you to work.\"\n\n\"I have a car.\"\n\n\"I'll drive you to work.\"\n\n\"Lisa, it's eight o'clock in the morning. I wouldn't\u2014\"\n\n\"I'll drive you to work.\"\n\nI rolled out of bed, trying to pretend like I wasn't hurting, thinking maybe I wouldn't go into the office at all.\n\nThen the phone rang.\n\nI'd never expected this investigation to bring me back to the Transylvania. Thus far, the killer had been careful not to repeat himself. But I suppose he didn't have much choice. She wasn't coming to him. He had to take whatever chance he could get.\n\nIt wasn't hard to figure out what had happened here. Both cars were still attached to one another. The parked car was particularly damaged. Anyone who had been in there must've been injured, maybe seriously. Blood was everywhere. Maybe it was just the fact that I'd drunk all my meals for several days running, but it was getting to me. I was sick in so many ways I couldn't enumerate them all.\n\nThe press was there in force. Not surprising, given who the abductee was. The news was already blanketing the airwaves. This was now the biggest story anywhere, everywhere, all across the nation. Maybe even internationally. The reporters were demanding answers that we didn't have, acting as if it was our fault that we didn't. They shouted questions at me as I passed, which I ignored. O'Bannon was dealing with them, or trying. He looked like a drowning man facing a tsunami.\n\nLisa dropped me off at the outer perimeter of the crime scene, where Patrick was waiting to brief me.\n\n\"How did he find her?\" I asked.\n\nPatrick shrugged. \"Everyone on earth knew she was at the Transylvania. It had been on the news. She even mentioned it on her show. She was doing live remotes from the ballroom. He may have known what car she was using. Possible that he scouted all the parking lots\u2014including this private one for employees and VIPs\u2014found her car, waited for his chance.\"\n\n\"Why go after the guard? He's never killed a man before.\"\n\n\"It was necessary to get to Spencer.\"\n\n\"And why did he want to do that so badly?\"\n\n\"She practically put out a hit on him.\"\n\n\"Yeah, but so what? He likes attention. He's been craving a challenging opponent.\" I shook my head. \"It wasn't the reward money. It was the character assassination.\"\n\nPatrick didn't get it. \"Is it possible to assassinate the character of a serial killer?\"\n\n\"It is in his mind. Remember\u2014he's a proper southern gentleman doing some kind of respectable, perhaps even sacred work. And then this woman goes on television and suggests that he's a pervert. It's an affront to his honor.\"\n\nChief O'Bannon walked between us, a grim expression on his face. \"They're demanding a press conference, people. What can we tell them?\"\n\n\"Stall,\" I advised. \"We need more time.\"\n\n\"They're very insistent.\"\n\n\"Tell them to cool their heels.\"\n\n\"That won't cut it. In their view, we've let a serial killer snatch one of America's most beloved television personalities. One who recently suffered a great personal loss\u2014thanks to the same killer. All the news reports have been critical of us.\"\n\n\"Okay, then promise them a conference\u2014later. After we've had more time to sift through the evidence.\"\n\nHe grunted. \"Can I at least tell them whether we think Spencer is still alive?\"\n\n\"There's no corpse,\" Patrick said. \"Why bother taking her if he's going to kill her?\"\n\n\"He's killed all the others.\"\n\n\"But not right away,\" I insisted. \"There's no reason to believe this will be different.\"\n\n\"So there's still a chance. If we move quickly.\"\n\n\"But I wouldn't say that to the press,\" Patrick quickly interjected. And we all knew why. Because we didn't want to be blamed if we didn't find the killer in time to save her. As things stood, we weren't even close.\n\n\"We've got to do something,\" O'Bannon growled. \"You bring Darcy out today?\"\n\n\"No . . . I thought it would be best not to.\"\n\nO'Bannon nodded curtly. \"He seemed pretty upset last night. Had to give him something to get him to sleep. Haven't seen him that anxious in a while. You take him to an espresso bar?\"\n\n\"No. He . . . saw something that bothered him.\"\n\nTo my surprise, O'Bannon didn't ask any more questions. Sighing heavily, he headed back toward the press corral.\n\nOver by the smashed Chevy, I heard a cry. Either pain or exultation, I couldn't tell. But a few moments later, I saw Tony Crenshaw running toward me.\n\n\"What?\" I wasn't going to get my hopes up, but his eyes were like Christmas lights. \"What have you got?\"\n\n\"You are going to love me so much,\" Tony said, obviously pleased with himself. \"You are going to fall down on your knees and kiss my feet. Perhaps even some more sensitive spots.\"\n\nI saw Patrick giving me a sidewise glance. \"Cut the fantasy and tell us what you've got already.\"\n\nBut he wasn't giving it up that easily. \"Most forensics would've missed it, of course. You do the outside of the car, sure. The upholstery. But how many would've bothered to check the floor mats? Especially with a guy who has been so scrupulous in the past?\"\n\nI grabbed him by the lapels. \"Prints? Are we talking about prints?\"\n\n\"It's possible. . . .\"\n\n\"I'm not your joytoy, Tony. Don't play with me. Have you got a print?\"\n\n\"There's definitely something there. I'll take the mat back to the lab and try vacuum metal deposition. It's great for lifting prints off plastic. I'll get something for you.\"\n\nI couldn't restrain myself. I pulled him closer and smacked him a big one, right on the lips. \"But that's as far as it goes,\" I cautioned.\n\n\"Looks like there was a tussle,\" Tony explained. \"We found a few drops of blood in there that didn't come from Spencer. I think she hurt him. He fell forward, his hand went down on the mat, he forgot to clean up. Maybe didn't have time. The rental company tells me those mats were washed just before Spencer rented it, so . . .\"\n\nI couldn't help myself. I kissed him again.\n\nHe arched an eyebrow. \"Could we take this somewhere?\"\n\nI released his lapels. \"Yeah, back to the lab. Now.\"\n\nShe must've heard him coming. The steps leading to the basement were wooden and creaked when he stepped on them.\n\n\"Who is that? Let me out of here!\"\n\nThe basement was dark, but he liked it that way. When he first moved in, he had been down almost every day, oiling hinges, wiping the walls with mildew remover, but at some point he had realized that was futile and foolish. He was denying the basement its true nature, its essential basementness, so to speak. Certainly the prophet would not have approved. So these days, he let things be what they were. As a result, the basement had acquired a distinctive odor, not noisome, but a lovely evocative mustiness. The air seemed thick and earthy; the walls bore a thin filmy layer of green growth.\n\nHe found Dr. Spencer in the alcove on the far side, strapped to the table, just as he had left her.\n\n\"You can't get away with this,\" she shouted. Her words echoed through the basement, reverberating off the stone walls. \"I've had people following me everywhere I go.\"\n\n\"Then where are they, madam?\" he replied, smiling sweetly. \"I'm afraid I find that statement lacking a certain credibility. You see, I am intimately familiar with all your security precautions. I struck at a time when your protection entrusted you to the custody of the rather poorly chosen and recently bifurcated Harv Bradford.\"\n\nBetween the two of them, forming a partition between the alcove in which she lay and the rest of the basement, was a four-foot-tall brick wall.\n\n\"You're the security man, aren't you?\" she said, staring at him. \"Back at the hotel. The other one.\"\n\n\"What a memory you have. Spectacular.\"\n\n\"Let me tell you something, mister. You didn't know half the precautions I was taking. I had eye-in-the-sky copters watching me. There's a homing device in my shoe.\"\n\n\"I don't think so.\"\n\n\"As soon as they zero in on my position, your ass\u2014\"\n\n\"Madam, you have been here more than twenty-four hours.\"\n\nShe paused, her mouth sucking air. \"But\u2014I just\u2014\"\n\n\"You've had a nice long nap, courtesy of my favorite pharmaceutical. But I am forced to conclude that if anyone had the slightest notion where you were, they would have long since arrived.\"\n\nShe was silent for a while. \"What do you want? Are you going to strip me naked and . . . do whatever it is you do?\"\n\nHe struggled to maintain control. \"I have not removed your clothing because you are not now, nor could you ever be, an offering. And may I add that I have never assaulted or in any way behaved inappropriately with any of my offerings. Your public accusations were offensive and ungrounded.\"\n\n\"You killed three girls!\"\n\n\"That is correct, in a technical sense. But there was no sexual misconduct, my dear pseudo-doctor. I'm sure the police have conducted tests establishing that for a fact. They in all likelihood have shared that information with you. Nonetheless, you appeared in a public forum and made your vile accusations.\" He paused. \"A gentleman's reputation is his stock in trade. You have impugned my personal integrity. For that, you must be punished.\"\n\nHe had prepared the mortar earlier. He added some water from the sink, loosening it. He stirred it with the trowel. It was ready.\n\n\"I can't believe this,\" Spencer said. Her voice was hoarse from shouting and it had acquired an edge, but one born more of fear than of menace. \"I've got a serial killer complaining that I tarnished his reputation.\"\n\nHe slapped mortar down atop the partial wall, then pressed a brick into place. It held.\n\n\"I mean\u2014don't you see a certain irony in that?\"\n\n\"I see that you understand nothing,\" he said, applying another brick, then another. \"I see that you categorize me with the insane, or those who kill for pleasure or sexual gratification. Insult upon insult.\" Another brick. Then another. Then another.\n\n\"What are you doing?\" she asked.\n\n\"Just finishing a little something I started. I'm a great devotee of home improvement.\" Another row finished. Then another.\n\nTime passed. She watched him work. As each new brick fell into place, her voice became more strained. \"Look . . . I never meant to offend you. I just\u2014I loved my daughter.\"\n\n\"You did not, madam.\"\n\n\"How dare you\u2014\"\n\n\"Annabel talked quite a lot about you before she was offered. She told me that you never paid any attention to her, hadn't for years. That you were always absent, obsessed with your career. Your work, that was what you loved. Not her.\"\n\n\"How can you presume to\u2014\"\n\n\"I have it from the best authority, wouldn't you say?\"\n\n\"Annabel was a child. She couldn't understand what a working single parent\u2014\"\n\n\"She knew whether her mother paid attention to her or not.\"\n\n\"Paid attention to her? I lavished every possible attention on her. Did she tell you she was going to MIT, for Christ's sake? Did she tell you what she was driving?\"\n\n\"The checkbook, my dear woman, is no substitute for parenting.\" He slapped another layer of brick on the wall. It now reached two-thirds of the way to the low basement ceiling. He had to stand on a ladder to continue his work. \"Could you perhaps name one of her friends? No guessing, now.\"\n\n\"What difference does it make? Friends come and go.\"\n\n\"Did you know she was pregnant?\"\n\nHer voice came back a little quieter. \"The police told me.\"\n\n\"Did you know the name of her beau, before the police found him?\" He smiled, stirring the mortar. \"His name was Warren. He and Annabel were in love and wanted to marry. She came to Las Vegas to make some money playing blackjack so they could start their life together.\"\n\n\"She could've just asked. . . .\"\n\n\"She felt she couldn't. She knew you would be importunate and disapproving and would not give her anything, so she didn't bother. Because you were too wrapped up in your own world to be a part of hers.\"\n\nDr. Spencer was quiet for several layers of brick. When she spoke again it was with a sullen defiance. \"So you're doing this to punish me for being a bad mother?\"\n\n\"Not at all. You're the one who brought up your relationship with your daughter.\"\n\n\"Then why?\"\n\n\"I cannot let you continue to interfere with my work. My plan.\"\n\n\"What are you talking about? What do you think you're doing?\"\n\nHe slapped down another slab of mortar. \"At the moment, I'm building a wall.\"\n\n\"Around me? Is that your sick idea? You're going to entomb me in your basement?\"\n\n\"More or less.\"\n\n\"And I'm supposed to lie here till the end of time?\"\n\n\"Goodness, no. That would be cruel and inhuman.\" He continued laying the bricks. \"This is a small alcove, and the way you're breathing, you'll use up the air quite quickly. I doubt if you'll last more than an hour or two.\"\n\nHer arms and legs stiffened, straining against the leather straps. It seemed her muscular control had returned. \"Sadist!\"\n\nHe made a _tsk_ ing sound, then continued working. Barely half a foot remained between the top of the wall and the ceiling.\n\n\"Please don't do this.\" Her voice finally cracked. \"I'll give youanything you want. Just let me go. I won't hunt you anymore, I promise.\"\n\nHe sighed a little as he slapped down the bricks. It was so disappointing. In the end, they all gave way to weakness.\n\n\"What is it you want from me?\"\n\n\"Only your death. A slow and painful one. A terrifying demise.\" The wall reached the ceiling. There was only one opening remaining, one space he had left vacant at eye level. \"I'm afraid this is where I must bid you adieu,\" he said, peering through the gap.\n\n\"Please don't leave me in here!\"\n\n\"Dr. Spencer, you are wasting precious air. Instead of this useless caterwauling, may I recommend that you spend your remaining time coming to terms with your Maker? Use these last precious hours to commit yourself to your faith. If you have one.\"\n\n\"You'll pay for this!\"\n\n\"And if not,\" he said wearily, \"this might be an opportune moment to adopt one.\"\n\n\"For God's sake\u2014\"\n\n\"Yes. _For the love of God._ \"\n\nWith a splash of mortar he wedged the final brick into place, then plastered over the wall to ensure that it remained airtight. Not a bad bit of masonry, if he said so himself.\n\nShe did scream, of course, even though it was the stupidest thing she could possibly do. She threatened and pleaded and repeated the vile insults that had made this action so necessary in the first place. After a while, he realized he did not need to subject himself to this. He went upstairs, closing the basement door behind him, and waited for the screaming to end.\n**CHAPTER EIGHTEEN**\n\n\"Explain this to me again,\" I asked Tony.\n\n\"My pleasure.\" His nose was pressed against the glass of the vacuum chamber as he repeated the entire exegesis. \"We put the floor mat in there with a milligram of gold in the heating element, then sealed it. The pumps suck out the air and create a vacuum. The gold boils, almost into a steam. A thin invisible layer coats the plastic. The gold will sink into the oil from the print, leaving only the ridges uncoated. Then we do it again, this time with zinc in the heating element. The zinc vaporizes, then recondenses only on metal\u2014in other words, the gold from the previous treatment. And the result?\" He directed our attention to his nearby computer monitor. \"A great big beautiful high-contrast reverse-image print.\"\n\n\"Nice little gizmo you've got here,\" I murmured softly.\n\n\"Glad you think so, Susan,\" he replied. \"Because vacuum metal deposition costs a fortune, what with the gold and all. I'm telling Granger you authorized it.\"\n\nI hunched over his shoulder, peering at his computer screen, but no matter how much I squinted, no matter which way I turned my head, no matter how long I let my eyes go fuzzy, I couldn't make out the print. \"The lines all look the same to me,\" I said, admitting defeat.\n\n\"Don't sweat it,\" he replied. \"Psychos all look the same to me.\"\n\nWhat we were looking at was a computer enlargement of the print he had found on the floor mat in the car from which Fara Spencer was taken. It wasn't all there\u2014a chunk from the upper left never came clear\u2014but Tony assured me that was enough to make a match. And this time it was a forefinger, not a palm print. I was trying not to get my hopes up, but we were all hoping this would allow us to identify the killer. With Patrick's assistance, he'd already fed the print to FINDER, the FBI's automatic fingerprint reader and processor. If this print or anything like it had been recorded by any computerized law enforcement agency in this country or several foreign nations, they could give us the identification we so desperately needed.\n\n\"We've got mail,\" Tony said, pointing at his screen. \"Three partial matches.\"\n\nI watched as three more prints appeared on the screen in a vertical column opposite the original. Tony scrutinized each whorl and swirl.\n\n\"Well?\"\n\n\"Give me a minute.\"\n\nI saw that each of the match prints had a name beneath it with a hyperlink to a full FBI bio. If we could get a name, maybe even an address, this killer could be behind bars by midnight.\n\n\"No,\" Tony said, after dragging the suspense out for what I thought was an ungodly length of time. \"None of these work.\"\n\n\"What do you mean?\"\n\n\"They aren't him. There are similarities, sure. Enough to pass the computer software match threshold. But they aren't the same.\"\n\n\"You're sure?\"\n\nHe was still staring at the screen. \"Much as I wish I weren't. Besides, none of these guys comes close to matching your description. This one's a woman. The next is a guy in his seventies.\"\n\n\"But we were sure that print came from the man who abducted Fara Spencer.\"\n\nHe pushed back away from the computer, rubbing his eyes. \"So now we know that our guy has never been arrested. Never run for political office. Never taken the bar exam. He's managed to get through life without being fingerprinted. He's never done anything like this before.\" He slid out of his chair and switched the power off his monitor. \"Or if he has, he's never been caught.\"\n\nHe ambled up the sidewalk outside Central Division headquarters trying to concoct a suitable conversation starter. As it happened, the young man sitting on the front steps eliminated the need.\n\n\"Are you a grown-up person?\"\n\n\"Ye-es . . .\"\n\n\"You must be kind of a short person. Are you kind of a short person?\"\n\n\"I am as God made me.\"\n\n\"I'm six foot one. Do you know how tall the Sears Tower is?\"\n\nHe tugged at his collar. All his initial impressions were correct. There was something strange and more than a little disconcerting about this man's demeanor. The way he struck up a conversation, albeit a nonsensical one, with a total stranger on a Vegas street. His voice was simple, almost childlike. And yet he was an adult, somewhere in his mid-twenties by appearances.\n\n\"No, I'm afraid I don't.\"\n\n\"It's one thousand four hundred fifty-four feet tall. One hundred and three floors. It used to be the tallest building in the world. Not anymore.\"\n\n\"Fascinating.\"\n\n\"Do you know how tall the Empire State Building is?\"\n\n\"Not exactly.\"\n\n\"It's one thousand four hundred fifty-three feet. One foot shorter than the Sears Tower. One hundred and two floors. Have you ever talked to a midget?\"\n\nHe stiffened. \"I'm not sure what\u2014\"\n\n\"I saw a midget once and I talked to her. I got in trouble for talking to her but I don't know why because I didn't do anything to hurt her.\"\n\nThere was something wrong with this man, a discernible . . . vacancy. He didn't lack intelligence or language. His syntax was skewed, but there was a distinct legerity to his responses. At the same time, there was a profound oddness about him: the way he held his head when he talked, the curious inflection, the unvaryingly excessive volume.\n\n\"I'm Darcy O'Bannon the second. My dad named me for my uncle, he's dead. My uncle, not my dad.\"\n\n\"Please to meet you, Darcy.\" He extended his hand, but Darcy did not take it. Instead he stared at it, as if hesitant to make contact. \"My name is Ethan.\"\n\n\"Are you a jockey?\"\n\n\"Uh . . . no . . .\"\n\n\"Because I read that jockeys have to be short and they like jockeys to be short so you should be a jockey.\"\n\n\"No, I'm . . . I'm an accountant.\"\n\n\"How tall do you have to be to be an accountant?\"\n\n\"I'm not aware of a height requirement.\"\n\n\"I think I'd like to be a jockey. I rode a horse once and I liked that. It went really fast and I like to go really fast. Do you think I could be a jockey?\"\n\n\"Uh . . . probably not, given your height. But I'm no expert.\"\n\n\"Willie Shoemaker won eight thousand eight hundred thirty-three races, did you know that? He was four foot eleven. But he got rich. I think my dad would like me better if I were rich.\"\n\n\"Darcy . . . I'm looking for Lieutenant Pulaski. Do you know where she might be?\"\n\nDarcy cocked his head to one side. \"Do you know Susan?\"\n\n\"I've had the pleasure of meeting her, yes.\"\n\n\"You're not going to take her away from me, are you?\"\n\n\"I'm . . . not sure what you mean.\"\n\n\"Whenever I really like someone somebody else takes them away or tells me I can't play with them anymore. I'll be sad if Susan goes away. I like her a lot. Do you think she's pretty?\"\n\n\"Most striking.\"\n\n\"I think so, too. But she's not the prettiest woman ever. Some people say Cleopatra was the prettiest woman ever but did you know archeologists dug up a coin with her face on it and she wasn't pretty atall?\"\n\n\"I didn't know that.\" He suppressed a smile. And he had worried that this harmless meshuggener might be a threat, a rival, that he might come between himself and Susan. Obviously, that was not going to happen. What was she doing with this boy? Was he some sort of charity work, a Good Samaritan exercise? Was this Susan's plan for worming her way back onto the force? Earning Chief O'Bannon's favor by babysitting his brain-addled son?\n\n\"Do you know what the tallest building in the world is?\"\n\n\"Uh . . . the Sears Tower?\"\n\n\"Wrong!\" He made a honking noise and pointed. \"Faked you out. It used to be the Sears Tower, but now it's the Petronas Tower in Malaysia. It's one hundred and ten stories tall. That would be two hundred and sixty-four of me stacked on top of each other.\"\n\n\"Imagine.\"\n\n\"Would you like to see the Sears Tower and the Empire State Building stacked on top of the Petronas Tower? I would. Do you know how many stories that would be?\"\n\n\"Rather a lot.\"\n\n\"One hundred and two plus one hundred and three plus one hundred and ten. Know what that is?\"\n\n\"Sorry, I've never been good with numbers.\"\n\nDarcy's head tilted. \"But I thought you said you were an accountant.\"\n\n\"I . . . I rely heavily on my calculator.\"\n\n\"Accountants are good adders. I read that in a book. My dad took me to an accountant once and he could add five-digit numbers in his head. So can I but he was the only other person I ever saw who could. Why can't you add three-digit numbers?\"\n\n\"Well . . . of course . . . I wasn't really listening.\"\n\n\"Are short accountants not as good at adding as tall ones?\"\n\nHe stepped onto the sidewalk. \"I really must be going.\"\n\n\"Goodbye,\" Darcy said. \"You might think about seeing if you could become a jockey. 'Cause I'm not sure how good you're going to be as an accountant.\"\n\nHe hurried back to his car, wrapping his jacket tightly around himself. That had been an unforgivably stupid mistake. He'd relaxed his guard, thinking this mental deficient could pose no danger to him, and as a result, he'd made a foolish error. If the boy were not so pitifully without guile, he would've become suspicious, perhaps conveyed his suspicions to Susan. And that could be disastrous.\n\nAt least he'd ascertained that there was no romantic affiliation between the two. Now that he knew he had a clear field, he would contact Susan again. Soon.\n\nWhat bothered him was his inability to read the young man. The connections this Darcy's brain made were unpredictable. Illogical. There was no way of anticipating him.\n\nIf it became necessary, the young man would have to be removed. For Susan's sake. And his own. And that of the world to come.\n\nBy the end of the day, several more FBI agents had made the scene. In addition to our rent-a-behaviorist, we now had agents from CIU\u2014the Critical Incident Unit. From the shadowed basement of the J. Edgar Hoover Building to the sunny Vegas strip. We also had some liaisons with VICAP\u2014the Violent Criminal Apprehension Program\u2014some of them pretty famous, names I recognized from the _Law Enforcement Bulletin._ They were cataloging and analyzing data, comparing these crimes with others the feds had encountered. Patrick had them all gathered in a back room and was bringing them up to date on the case.\n\nI saw a thick stack of paper on the corner of my desk\u2014summary reports on all the confessions that had arrived in the last few days. We'd been getting them since murder number one, but they were skyrocketing now that a TV celeb was involved. If I worked this job till doomsday, I would never understand what it was about high-profile media cases that made perfectly harmless people crawl down to the station to give false confessions. Unless they were obviously bogus, they had to be checked out, at least a little, which diverted our already strained time and manpower.\n\nI scooped up a stapled document that someone had dropped on my chair. Looked like the Feebs had been busy.\n\n **SECONDARY VICTIMOLOGY REPORT, BSS04-67 \nSUBJECT: EDGAR**\n\nI had to grin a little. I knew the feds always gave their serial killers a pet name. So this one was Edgar. Cute. I scanned the report.\n\n **VICT1\u2014** Helen Collier, Clark County Police Dept, homicide\n\nWF, DOB 1-9-87, DOD 10-4-04\n\n **DEATH:** Asphyxiation\n\n **WEAPON:** Coffin\n\n **POD:** Transylvania Hotel ballroom\n\n **BODY FOUND:** Transylvania Hotel\n\nNo witnesses. Forced entry. No trace evidence attributable to assailant.\n\n **VICT2\u2014** Annabel Spencer, Clark County Police Dept, homicide\n\nWF, DOB 8-15-86, DOD 10-8-04\n\n **DEATH:** Exsanguination\n\n **WEAPON:** Dental tools\n\n **POD:** Unknown\n\n **BODY FOUND:** McCarran Airport, retired aircraft field\n\nNo witnesses. No forced entry necessary. Only trace evidence attributable to assailant is a tire track and a partial palm print.\n\n **VICT3\u2014** Lenore Johnson, Clark County Police Dept, homicide\n\nWF, DOB 7-13-85, DOD 10-13-04\n\n **DEATH:** Decapitation\n\n **WEAPON:** Axe\n\n **POD:** Unknown\n\n **BODY FOUND:** Neon sign graveyard\n\nNo witnesses. No forced entry. No evidence attributable to assailant.\n\n **VICT4\u2014** Harvey Bradford\n\nWM, DOB: 01-04-60 DOD: 10-17-04\n\n **DEATH:** Massive bodily trauma\n\n **WEAPON:** Automobile\n\n **POD:** Hotel parking lot\n\n **BODY FOUND:** Hotel parking lot\n\nSecurity guard at Transylvania Hotel. Killed by impact of his own car. Died instantly.\n\n **VICT5\u2014** Fara Spencer\n\nWF, DOB: 10-16-61 DOA: 10-17-04\n\n **STATUS:** Unknown\n\nMother of Vict2. Abducted after car incident with Vict4. No witnesses. Unidentified partial forefinger print possibly attributable to assailant.\n\nPretty damn thin, when you got right down to it. I thumbed through the additional material attached to the cover report. Not much there. Was it any wonder we hadn't caught this clown yet? About the only thing this report didn't cover was the phone call I'd gotten from Edgar\u2014great, now they had me calling him that\u2014and his present. And I received a detailed analysis on those barely an hour later.\n\n\"The lab found nothing useful on the box, the wrapper, or the teeth,\" Tony Crenshaw explained. \"Neither did Latent Prints.\"\n\n\"Voiceprints?\"\n\n\"Nothing. We have a pattern now we can compare against any future communications. But even maximum volume magnification failed to turn up any useful background noises. There were few clues in his language other than the obvious one\u2014 _neon._ No distinctive patterns. Southern accent, to be sure, but the experts think he was affecting that.\"\n\n\"Because he wants to be like Poe. But he isn't really from Virginia.\"\n\n\"That's our guess. Other than that, no real clues. He chose his words carefully. Probably rehearsed what he was going to say before he called.\"\n\nThat would be consistent with the organized sociopath I had in my head. \"I'd like a copy of the tape of the phone call.\"\n\n\"Sure.\" Tony seemed hesitant. \"You know he was trying to rattle your cage, right? Talking about your bad habits and all. Doesn't mean he knows anything. It was an easy shot. Everyone has some secret. He wants to scare you. Chase you off the case.\"\n\nDid he? Maybe. But to me, it seemed more like he was trying to impress me. To win me over.\n\n\"But you're not going to let him get to you, right?\"\n\nMy eyes lifted. \"What's this? A trace of concern? For _moi_?\"\n\nHis eyes darted to the carpet. \"Granger doesn't speak for the whole department, Susan. You still have friends around here. Lots of them.\"\n\nWell, that was cheering. \"Thanks, Tony. I appreciate it.\"\n\n\"All the guys loved David. You know that. And we think what happened . . . wasn't right. The way you were treated.\"\n\n\"Tony, I really don't care to\u2014\"\n\n\"Just wanted you to know.\" He gave me a mock salute. \"Back to the trenches.\"\n\nI thrust my nose into my paperwork, trying not to be resentful. Of course, that was what it would be about. David, not me. Everyone had loved David. Everyone.\n\nI'd loved David, too, damn it. But that wouldn't bring him back.\n\nI was on the phone with my lawyer, which is about as unpleasant as life gets.\n\n\"If the judge isn't going to decide anything, why do I have to be there? I'm in the middle of a major investigation.\"\n\n\"Do you want custody of Rachel or not?\"\n\n\"That's why I got this job! On your advice.\"\n\n\"Then you must be there. Dressed conservatively. Sparing makeup. You need to sit next to me and be untempermental, cool, and well mannered.\"\n\n\"In other words, exactly not like myself.\"\n\n\"Whatever.\"\n\n\"I can't believe this judge is so shallow he's going to make a decision based upon whether I wear red.\"\n\n\"He might not do it consciously. But judges are influenced by their subconscious impressions, just like everyone else in the world. The most important part is that you remain calm. NDHS will try to convince the judge that you're unreliable. You have to show him that you can be restrained and responsible, fit to raise your niece. You have to tell him you've sworn off booze and you're working to stay clean.\"\n\nI felt an itching in my chest that wouldn't go away. \"I can do that.\"\n\n\"Good. See you there.\"\n\nDamn everything. As if I didn't already have enough on my plate. Now I was going to have to deal with the American legal system. The tenth circle of hell.\n\n\"Eureka!\"\n\nI glanced up. Darcy was standing at the edge of my desk, grinning like a sheepdog.\n\n\"Eureka!\" he repeated.\n\n\"Okay, I'll bite. What are you talking about?\"\n\n\"Eureka means 'I found it.' Did you know that historians say that's what Archimedes said when he discovered about the displacement of liquid? He jumped out of the bathtub yelling, 'Eureka! Eureka!' \" Darcy giggled. \"He was strange.\"\n\n\"Thanks for the ancient Greek perspective, Darce. Why are you telling me this?\"\n\n\"It's the name of a thing by Edgar Allan Poe.\"\n\n\"What? I read his complete works\u2014\"\n\nDarcy corrected me as gently as possible. \"I think maybe you read his _Complete Stories and Poems._ That's what everyone reads. No one reads _Eureka._ Even my dad doesn't have a copy.\"\n\n\"Then how do you know\u2014\"\n\n\"I ran that code message that the bad man left with the teeth through the Internet.\" I swore silently. I should have thought of that myself. \"Even Dad says I'm good with computers. Do you Google? I love to Google. You never know what you'll get. This one time\u2014\"\n\nI cut in. \"So what is it?\"\n\n\"Poe called it a prose-poem.\"\n\n\"What the hell does that mean?\"\n\nDarcy's face reddened. \"I don't know. It's like about stars and stuff.\"\n\n\"Stars?\"\n\n\"Science. Where the stars come from. Planets. Heaven.\"\n\nI was thoroughly confused, and Darcy wasn't making it any better. \"Can you get me a copy?\"\n\n\"I could not find the whole thing on the Internet, only some parts. Then I went to the library, but they don't have it.\"\n\n\"Keep trying. And Darce\u2014I owe you a custard.\"\n\n\"When do you think that will be? Sooner than a blue moon, I hope, because a blue moon\u2014\"\n\n\"I remember. We'll do it, Darcy, promise. But first I think I need to read that book.\"\n\nHe left, and a few minutes later my phone rang, as it always seems to do at the most annoying and least convenient times. \"Yeah?\"\n\n\"You look lovely in a black turtleneck. You should wear that more often.\"\n\nHim.\n\nI stood up, waved at the boys on the lower floor. I pointed to the receiver, trying to get the message across. Men can be so slow-witted. Eventually enough ribs were jabbed and the tracing began.\n\n\"Uh, sorry. I was . . . distracted.\"\n\nHe laughed. \"Have they instigated the trace now? Can we talk?\"\n\nI tried to concentrate. I had to learn as much as possible. And I had to keep him talking. \"I don't suppose you plan to stay on long enough for us to trace the call, so why bother?\"\n\n\"Oh, you never know, dear. I might give you a sporting chance.\"\n\nDamn straight of you. \"Look, I don't know what to call you.\"\n\n\"The FBI agents call me Edgar, don't they?\"\n\nAnd how the hell did he know that? \"Is that okay?\"\n\n\"It's as good a name as any.\"\n\n\"Is that who you are?\"\n\nAnother pause. \"I'm an acolyte. Not a prophet.\"\n\nOooo-kayyy. \"Can you explain to me what that means?\"\n\n\"I'd like to. Because you don't have to remain behind, mired in this miserable life you've made for yourself. I can help you. I have only your best interests at heart.\"\n\nI knew my goal\u2014not only to keep him talking, but to keep him off whatever prepared script he had in his head. The more he extemporized, the more likely he was to tell us something useful. \"Have you hurt Fara Spencer?\"\n\n\"I'm afraid I can't answer that question.\"\n\nI tried something else. \"I got your note. Tracked down the quote, too.\"\n\n\"Did you really?\"\n\n\"Yeah. _Eureka,_ right? I'm trying to find a copy.\"\n\n\"Don't bother, dear. I've brought you one.\"\n\nAt that point, I looked up and saw three Feebs standing in front of me, waving their arms. Meddlers. Couldn't they see I was trying to focus?\n\nI covered the talk end of the receiver and mouthed: \"What?\"\n\nOne of them held up a note he'd scribbled furiously on a legal pad. WE'VE TRACED THE CALL.\n\nWhy the hell tell me? Just go already.\n\nThen came the follow-up message. HE'S IN YOUR APARTMENT.\n\nI felt my heart stop. I uncovered the receiver, breathless. \"Edgar?\"\n\nThe line was dead.\n\nOf course, he was long gone by the time the police arrived. True to form, he had gotten in and out without leaving a trace of himself behind. All he'd left was a paperback Dover Press edition of _Eureka: A Prose-Poem_ by Edgar Allan Poe, which the techs were treating to a microscopic scrutiny. I knew they wouldn't find anything\u2014nothing useful, anyway. Although he'd left no evidence, we could tell where he had been. Footprints in the carpet. My underwear drawer left open. An indentation on my bedspread.\n\nNone of my neighbors had seen him. One reported spotting a nondescript meter reader, so we were guessing Edgar had used that disguise. But he'd only seen the man from a distance, so he had no useful information.\n\n\"I don't get it,\" Patrick said. \"He's been so careful before. So calculated. Why would he come here? Why take the risk?\"\n\nI thought about that. In the early days, all his actions had seemed well planned. Careful. But he was becoming increasingly impulsive, or at least more varied in his approaches. Acting on emotion. Kidnapping Fara Spencer, essentially for spite, even though she didn't fit his profile. And now this. How could burglarizing my apartment fit into his fabulous master plan?\n\nOf course, there had been other cases of serial killers who became involved, even obsessed, with one or more of the officers trying to catch them. But just as there was something very different about this killer, there was something unusual about the attention he was paying to me. Like I'd told Tony before, I didn't get the sense that he was perpetuating a cat-and-mouse game for his own amusement. It was more like he was trying to . . . win me over. Seduce me. Even this in-your-face power play had an element of seduction about it. _I have only your best interests at heart._\n\n\"You're getting round-the-clock security,\" Patrick said. \"Don't bother arguing. Should've done it after those damn teeth arrived with your name on the package.\"\n\n\"Does this mean you think\u2014\"\n\n\"You already know what I think. He'll only be content with presents and phone messages for so long. He's working up his nerve. Till he comes after you.\"\n\nHe could almost pity her as she lay on the table, her eyes closed. If only he could forget all that she had said and done. Forgive. But he could not. That power was no longer his.\n\nAt last she awoke, blinking, a dumbfounded expression creasing her brow.\n\n\"Am I dead?\"\n\n\"Of course,\" he said, leaning into her face. \"Welcome to Hell.\"\n\nShe gasped. \"You.\"\n\n\"Did you enjoy your nap, Dr. Spencer?\"\n\n\"But I\u2014I thought\u2014\"\n\n\"I know. You thought you were dead.\"\n\n\"I remember the wall. And . . .\" Her words came slowly, as she retrieved them through a dense fog. \"It was hard to breathe. And then\u2014hard to think.\" Lines formed around her eyes. \"Then I don't remember anything.\"\n\n\"You passed out,\" he explained. \"All but asphyxiated. Yes, you were a goner, as the moderns say.\" He opened his black bag and began laying out the instruments. \"But I rescued you. Am I your hero?\"\n\nShe tried to struggle but soon realized it was useless. She had regained control of her body, but she was firmly affixed to the table. \"Why?\"\n\n\"What fun would it be simply to kill you? A mere two hours of torment. When you deserve ever so much more.\" He held the instruments up before her face. They glistened in the light from a large overhead lamp. \"Do you recognize these?\"\n\nShe squinted. \"Are those\u2014surgical instruments?\"\n\n\"Indeed. Have you ever seen a straight razor?\" he asked, swishing a scalpel in the air. \"One sometimes reads about them in books\u2014an orangutan did great damage with one in 'The Murders in the Rue Morgue.' \"\n\n\"What\u2014what are you going to do with that?\"\n\n\"Don't you know? You're the doctor.\" He smiled. \"Ah, but you're one of those odd television doctors who aren't actually doctors.\"\n\n\"I want to know what you're planning to do!\"\n\n\"Apologies in advance. I don't have a hospital gown for you.\"\n\nShe craned her neck, realizing for the first time that she was naked. \"Why have you taken my clothes?\"\n\n\"Standard pre-op procedure,\" he replied.\n\nShe closed her eyes tightly. She was trembling, but despite her fear, she kept her voice remarkably strong. \"Is this another pathetic attempt to scare me?\"\n\n\"Alas, no. I have to move on to other responsibilities. This time you're going to die.\"\n\n\"Of course you have to say that. To terrorize me.\"\n\n\"Believe whatever makes you happy, Doctor.\"\n\n\"Listen to me\u2014you're a sick man. Ill. You're\u2014\"\n\nHe pressed the razor against her throat. \" _Stop it!_ \"\n\nShe quieted. But her eyes continued to peer at him, refusing to look away.\n\n\"I've always been fascinated by the idea of open heart surgery, as I'm sure was that orangutan,\" he said jauntily.\n\n\"Please don't do this.\"\n\n\"I must admit I have butterflies in my tummy. I've done a lot of reading on the subject, mind you. But this is my first attempt.\"\n\n\"Please.\"\n\n\"Tell me if I have this correct, _Doctor._ Are you knowledgeable about cardiopulmonary bypass? Percutaneous transluminal coronary angioplasty? Can you say that three times fast?\"\n\nThe doctor did not respond.\n\n\"I gather the typical heart operation begins with the all-important opening of the chest to expose the heart. Is that right?\"\n\n\"God, no. Please.\"\n\n\"And the most common way to do that is to slice down the middle of the chest, dividing the breastbone. Am I going to need a bigger knife?\"\n\n\"I'm begging you.\"\n\n\"Then I expose the heart by dividing the protective covering\u2014the pericardium.\" He clapped his hands together. \"This is going to be delightful.\"\n\n\"At least put me out. Show me that mercy. I know you have the drugs for it.\"\n\n\"Ah, but that would spoil the fun. Tell me, Doctor\u2014do you use a pump oxygenator? I'm fascinated by those little machines.\"\n\n\"This is wrong. You're not thinking rationally or you wouldn't want to do this.\"\n\n\"I'm not sure where to make the first incision. My books don't show. Do I go through the rib cage? I think I'll try. I'll start here. Then here.\" He stair-stepped the scalpel down her rib cage, stopping to press in at the valley between each rib. \"Then here, then here, then here. And when we're done, your lungs will be thoroughly punctured. Will your heart stop? Because I know it's important that the heart stop before we take it out.\"\n\n _\"Take it out?\"_\n\n\"Come now, Doctor\u2014you didn't think I'd go to all this trouble and not come away with a souvenir, did you?\"\n\n\"Please listen to me. You need help.\"\n\n\"Don't presume to psychoanalyze me. You're the one on the couch.\"\n\n\"I'm not qualified to psychoanalyze you, and I don't really believe in all that bullshit anyway. But you should see a professional.\"\n\n\"Doctors. Always making referrals.\"\n\n\"I can't believe anyone wants to live as you do. I know you must be tormented. Do you have hallucinations? Do you hear voices?\"\n\n\"I've had about enough of\u2014\"\n\n\"We can block out those voices. We can suppress the irresistible impulses. We can help you.\" She strained against her bonds. \"I will personally ensure that the finest doctors are\u2014\"\n\n _\"Stop it!\"_\n\nHe poured a drink down her throat and then, when the convulsions ceased, he raised the blade of the scalpel and thrust it downward, cutting between her top and second ribs. A terrible hissing sound followed as air escaped from her lungs. Blood rushed up her throat and out her mouth.\n\n\"Where is that heart? _Where is it?_ \"\n\nThe razor plunged again, this time between the second rib and the third.\n\nHer body rocked. Gases seeped out of the gaping wounds. Despite the restraints, she jerked and spasmed as if she were in seizure. Blood gushed from the openings in her chest, her mouth, even her ears.\n\n\"Where is it?\" he cried. \"Where is it?\" He stuck the blade into her body again and again, until blood streamed from more than a dozen places. \" 'Dissemble no more!' \" He slashed wildly with the knife, cutting her arms, her legs, her torso, slicing open her chest, drenching himself, staining everything in sight. _\" 'I admit the deed! It is the beating of the hideous heart!' \"_\n**CHAPTER NINETEEN**\n\nI didn't get free of the cops till midnight, and even then, since my apartment had turned into a crime scene, sort of, I was going to have to stay at a hotel for a few days, with security detail in tow. Which was all right with me. As long as it wasn't the Transylvania.\n\nThank God, Patrick agreed to meet me at The White Feather. I made some excuse about why we should go separately\u2014the real reason, of course, was so I could leave the security guys posted outside the front door, dash to the bar in the back, grab a fifth, then down it in the ladies' room. I knew I'd promised Lisa I wouldn't, but these were pretty damn extenuating circumstances. I mean, the man had been on my bed, for God's sake. He'd gone through my drawers. When I thought about this psycho pawing my underwear, I got physically ill. I felt like a rape victim, even though he hadn't laid a finger on me.\n\nPatrick was very good about it, very sweet. I knew he was tired and probably wanted to go home, but he stayed with me just the same. God, I wanted his arms around me. I just wanted to feel safe. I just wanted to feel.\n\nThe waitress came by. \"Another whiskey?\"\n\nI stiffened. \"I'm drinking club soda.\"\n\n\"Okay. Want another one?\"\n\nI shook my head. Patrick got another beer. Half an hour later, we were still talking.\n\n\"You were pretty tough on the boys,\" I said. \"Back at my place.\"\n\n\"With good reason. Someone does a home invasion on a member of the team, that's serious business. We have to take care of our own.\"\n\n\"Thanks for sticking up for me.\"\n\nHe shrugged. \"It's what partners do.\"\n\nGod, he was handsome. I felt an itching I couldn't scratch and suddenly I didn't want to be in this bar anymore. \"Patrick? You did tell me you were unmarried, right?\"\n\n\"Ye-es.\"\n\n\"Do you think you could get rid of those security guys for a while?\"\n\n\"I could, but why on earth\u2014\"\n\n\"Tell them you're going to stay with me.\"\n\nHe raised an eyebrow. \"I am?\"\n\nI gave him my best big long lusty. \"Unless you still think it would damage our working relationship.\"\n\n\"Probably would. But at the moment, I'm not sure I care.\" He pushed himself out of his chair. \"I'll go talk to the uniforms.\"\n\nI'm sure he thought we were going back to his motel room, but we never made it that far, at least not at first. I led him by the hand to his car out back, slid onto the hood, and reeled him in.\n\n\"Now, wait a minute,\" Patrick protested.\n\n\"What?\" I said, grinning in what I hoped would seem a lascivious expression. \"Don't you wanna make a girl happy?\"\n\n\"We're professionals. We shouldn't\u2014\"\n\n\"Don't be a spoilsport.\"\n\n\"But someone might come. . . .\"\n\n\"Who cares?\" I bit him on the side of the neck.\n\nHe pulled away slightly. \"It would be wrong of me to\u2014\"\n\n\"Oh, don't be so damn good. Just this once.\" I unbuckled his belt and reached inside. After that, I knew I had him. He didn't care who was watching. I pulled him inside me and felt the warmth, felt the glow, felt good, felt safe. For a little while.\n\nOh, Susan. Oh, my dear, sweet Susan. I wanted so much for you. I wanted to elevate you, to cherish you, to escort you through the gates of Dream-Land. I tried to win you over, to help you see the light, to seduce you with the truth.\n\nBut now I see that you have been seduced by another master altogether.\n\nI followed you because I wanted to help you, to learn more about you. Because I cared for you. And I was concerned, genuinely desperately concerned when I saw you enter that bar, knowing your weakness as I do. But that was nothing compared to the abject horror I experienced when you emerged. How could I know you had an addiction worse than alcohol, an addiction to decadence, to evil? I could never have believed it\u2014until I saw you roll onto your car like the most debased jezebel, like the village harlot, an impure woman less worthy than the dust. Yes, I know you've been drinking again. But that is no excuse. There is no excuse.\n\nI know now what I must do.\n\nI must show you the error of your wanton ways. I must show you the result of indulging your passions, let you see your ultimate destiny if you continue on this wicked path. I must crush the spiritual depravity, modify your behavior with an experience so ghastly those old instincts will be dissipated, now and for all time. You must hit rock bottom before you can be cured.\n\nI know you will not come of your own accord. You are willful, stubborn, eternally contumacious. But I can break you. And I will. Not because I want to. Not because I will enjoy it. But because it must be done.\n\nI have only your best interests at heart.\n**CHAPTER TWENTY**\n\nI was used to waking up groggy, disoriented, not knowing where I was. I was used to a throbbing head, pulsing temples, dry cottony mouth. I was thoroughly familiar with finding I had forgotten to put on my jammies. And I was not altogether unaccustomed to finding myself in a strange bed.\n\nBut being handcuffed to it? That was different.\n\n\"Wha . . . tha . . .\" My eyes felt as if they had been pasted shut, and I couldn't wipe them clear since both wrists were cuffed to the headboard. What the hell had happened to me? Could I have been abducted by\u2014\n\nThat's when I felt it\u2014the cold clutching at my heart. The paralyzing, stabbing pain in the chest. Shortness of breath. Panic.\n\nEdgar. Had he found me? Had he given me his drug and chained me here, waiting for the right moment to begin whatever sick Poe-derived deprivation he had in store for me? I pulled at my bonds, but they were secure. I was chained down like a dog on a leash, utterly at his mercy, powerless to help myself. Any moment, he would return with his axe, his dental implements, his\u2014\n\n\"Ready for some coffee?\"\n\nPatrick appeared in the doorway, carrying a small tray with two cups. \"Maybe I'm wrong, but you seemed less the tall-glass-of-OJ type and more the stiff-cuppa-joe type.\"\n\n\"Why the hell am I chained up?\"\n\nHe looked up absently. \"Oh. Right. Sorry about that.\" He put the tray down and fumbled in his pockets for a key.\n\n\"You forgot? Is this some twisted power trip for you? You get your jollies by chaining up women with your big stud FBI toys?\"\n\n\"I didn't want to do it.\"\n\nI quieted. \"You didn't?\"\n\n\"Do you remember anything about last night?\"\n\nThinking hurt, but I made myself do it anyway. I recalled the phone call from Edgar, of course, the bar, the thing on the hood of the car. After that, it got a little hazy. Well, actually, it was a void.\n\n\"Maybe you're used to, um, this sort of activity, Susan, but I have to tell you\u2014I'm not.\"\n\n\"Look, just undo the cuffs, okay?\" He reached over and freed me. He smelled good. He was already scrubbed and dressed and aftershaved and ready to tackle the day.\n\nI didn't realize how stiff my arms were until I could move them again. They ached. I managed to work them back down to my side. They tingled as if they had been asleep for a thousand years. \"Where are my clothes?\"\n\nHe pointed. I crawled out, clutching the sheet to me, and started dressing. \"I hope I didn't\u2014\"\n\n\"No. You were great.\"\n\n\"I . . . was?\"\n\n\"Unpredictable. Intense. But great. Really.\" He grinned. \"Something like that is good for you every now and then. Shakes things up a little.\"\n\n\"Yeah. I feel the same way,\" I said, wondering what the hell had happened.\n\n\"I got some food from the diner downstairs if you want it. But my hunch is\u2014\" I made a gagging face. \"Yeah. That was my hunch.\" He smiled. \"I'll be in the next room. When you're ready, I'll drive you to work. You don't have a car, remember?\"\n\n\"Okay.\" It went against the grain, but damn it, I had to say it. \"Patrick?\"\n\n\"Yeah?\"\n\nI tried to smile. \"Thanks.\"\n\nI actually ventured a slice of toast as we drove to the office. And even after we arrived, I wasn't quite ready to say goodbye. We walked together to my desk.\n\n\"What's this? Another package?\" I picked it up. It was about the size of a bowling ball, wrapped in brown paper. \"It has my name on it.\"\n\n\"Susan! Get rid of it!\" Patrick cried. He shouted for assistance, but I was already unwrapping it. \"Susan! We need to have it\u2014\"\n\n\"It's not a booby trap. He wants me alive.\"\n\n\"Not again! For God's sake\u2014\"\n\nToo late. I lifted the lid.\n\nThe stench emanating from that box was unlike anything I had ever smelled in my life. And I've been around corpses, sickness, all kinds of filth.\n\n\"My God!\" Patrick cried, covering his nose and mouth, staring at the wet, viscous, blackish red lump in the box. \"What is it?\"\n\nThere was a card, hand-lettered in block print. He hadn't bothered to encode it.\n\nDR. FARA AND I HAD A NICE HEART-TO-HEART. SEE?\n\n**BEHAVIORAL PROFILE\u2014EDGAR**\n\nBY SUSAN PULASKI, M.A., LVPD, \nAND PATRICK CHAFFEE, BSS, FBI\n\nBased upon what is generally accepted about serial killers and their crimes, Edgar is probably a white man, between twenty and forty. He is more likely a book-reader than an athlete. He may have some physical deformity. He is literate, perhaps highly so. He is intelligent, as evidenced by his familiarity with and adoption of the works of Poe and his proficiency with ciphers. Various witness statements have described him as both tall and short, thin and wide. Although this could indicate that Edgar is in fact two people, it is more likely that some of the descriptions are inaccurate. At this time, we have no way of knowing which reports are erroneous.\n\nAlthough he has used a southern accent in his telephone communications, that is probably an affectation associated with his idolization of Poe. If he has any natural accent at all, it is more likely that his origins are in the western United States, Nevada or the surrounding states. Although he has used many Vegas-area locations for his crimes, we cannot assume that he is a native or even that he currently resides here, especially given the propensity of serial killers to move from one place to another. It is possible that the Sin City reputation attracted him. Many of his actions\u2014punishing strippers, removing body adornments or nail paint, dyed hair, etc.\u2014evidence a desire to enforce old-fashioned values.\n\nThere are no indications of great wealth, but he must have some income flow. Several of his crimes have required unusual props or equipment. All have involved a drug that cannot be obtained legally in this country without a prescription. Tire tracks suggest that he drives a truck.\n\nSerial killers commonly bear great hostility toward women, often triggered by a negative early female influence. In this case, however, despite the fact that he has murdered at least four women, there are some indications that he holds women in an almost Victorian-era reverence. His background may reflect the conflicted influence of both females he adored and females who abused him. In any case, he likely had a violent and chaotic childhood with little stability. Broken marriages, domestic violence, and early exposure to death are all likely. It is also likely that the male head of his household was absent for a protracted period during his childhood.\n\nGiven this rather bleak upbringing, detectives should look for an adult who as an adolescent, or even earlier, was lonely, isolated, withdrawn, angry, and violent. He likely had an active fantasy life in which he imagined himself an important or powerful personage. His fantasies may have involved domination and retribution against those who he felt wronged him. He was probably preoccupied with sex, even more so than most adolescent boys, and had no close friends, much less a girlfriend or sexual partner. During these years, his psychological disorder would have become progressively more apparent, making interpersonal relationships more unlikely. Sports, extracurricular activities, and hobbies would not have appealed to him. He may well have developed an interest in pornography, possibly involving young girls.\n\nThe psychological portrait of Edgar that emerges from all the information we have gathered is that of a narcissistic, self-absorbed, antisocial individual. He has an insatiable desire for attention. Despite his antisocial tendencies, deep down he wants to impress, wants people to be appreciative of his work. The crimes Edgar has committed evidence an ability to compartmentalize and rationalize extreme behavior. Thus far, he has acted in conformity with a preestablished pattern, but his recent variation from his previous victimology model\u2014in order to wreak revenge on Fara Spencer\u2014suggests that his innate psychological controls may be slipping.\n\nEdgar's demands for attention\u2014the coded messages, the gifts, the phone conversations, depositing corpses in clever \"theme\" locations where they are certain to be found\u2014are all classically infantile. Presumably his basic needs were not met early in life and he is psychologically overcompensating for that deficiency now. He has not progressed beyond the self-absorption that characterizes the infantile stage. Although he justifies his acts with some purpose we do not as yet understand, fundamentally he is trying to give himself the psychological nutrients he did not receive in youth.\n\nA sense of superiority and a desire for control characterize the antisocial personality disorder and paranoid personality disorder. The _DSM-IV_ does not require us to choose between the two diagnoses, and indeed, Edgar shows traits of both. Because Edgar is afraid of being controlled by outside forces, he will increasingly attempt to control others and subjugate them to his will.\n\nAnother disturbing possibility is dissociative personality disorder\u2014what is commonly called multiple personality disorder. Although this has been used in the past as a legal defense by multiple murderers who showed no true pathology of it (Bianchi, Gacy), there are instances of it being a bona fide aspect of the psychological makeup of a serial killer. Of course, even normal personalities can develop imaginary friends and playmates, talk to themselves, etc. But for those suffering from the disorder, one or more alter personalities acquires a specific sense of self. The alter personality can become an outlet for the individual's worst instincts and desires. Should such a personality emerge, the scale of Edgar's activities could escalate to a horrifying degree. . . .\n\n **O** 'Bannon looked up from the thirty-page document he held in both hands. \"Good report, Susan. Damn good.\"\n\nI fluttered my eyelashes. \"Well, I try. Do you mean you've actually read it?\"\n\n\"Twice. And I plan to read it again tonight. I get something new out of it each time. I'm giving you a special commendation for it.\"\n\n\"Well, be sure to give Patrick credit, too. He came in late, but he's been a huge help.\"\n\n\"Whoever did it, it's brilliant. I think you've nailed him.\"\n\n\"I think so, too.\"\n\n\"Has the lab said whether that heart belonged to Fara Spencer or not?\"\n\nI sat on the opposite corner of his desk, facing him. \"At this point all they can say for certain is that it came from an adult female of approximately her age. But they're planning a DNA analysis\u2014compared against a sample taken from her daughter's corpse\u2014as soon as possible. And of course it was pretty sliced up. In addition to all the other exclusionary factors in Edgar's profile, I think we can rule out the possibility that he's a trained surgeon.\"\n\n\"And the coded message? The one in the bottom of the box.\"\n\n\"To my astonishment, Darcy wasn't able to solve it instantly. But he's working on it. Apparently Edgar made this one even more devious than the previous ones.\"\n\nO'Bannon folded his hands in his lap. \"The boys tell me you're seeing Patrick Chaffee. Socially. True?\"\n\nI squirmed. \"Sorta.\"\n\nHe nodded. \"Fine man. Solid. Far as I can see. He'd be good for you.\" He nodded again, then turned his eyes toward the window. \"Don't hurt my boy.\"\n\nI rose to my feet. \"I'll take care of him, Chief. Promise.\"\n\nWe were in a classroom setting\u2014a private conference mandated by O'Bannon between me and all the detectives on Granger's team. I didn't know what the point was, with Granger so openly hostile to my work. When he was with his boys, at any rate. O'Bannon told me he'd seen Granger after hours marking up a copy of my report with a yellow highlighter.\n\n\"So we're looking for some freak who's talking to himself?\"\n\n\"Well, perhaps,\" I said, with a degree of tolerance that startled even me. \"And there are other markers, too. The fake accent. The assumed Victorian sensibility. The obsession with the works of Poe.\"\n\n\"How could we search for any of that?\"\n\n\"Well,\" said one of Granger's new lieutenants, \"we could check the libraries. See who's been reading Poe.\"\n\n\"Or the video stores,\" suggested another. \"Aren't there a lot of Poe movies?\"\n\n\"Both good ideas,\" I said, giving them the verbal pat on the back I knew they'd never get from Granger. \"And don't forget this little prank he's playing with the bodies, depositing them in faux graveyards. How many of those can there be?\"\n\n\"A good Internet search could tell us,\" said the first lieutenant. \"I'll get on that immediately.\"\n\n\"Even if you don't find him, any action that disrupts his usual patterns could be valuable. Remember\u2014we've gotten more information from his last little escapade with Fara Spencer than all his previous murders put together. Why? Because that crime wasn't part of his plan. For once, he acted impulsively. He saw an opportunity and he took it. If we can get him to do that again, he may make another sloppy mistake. He may be smart, but he's still working under some handicaps.\"\n\n\"Like what?\"\n\nI gave the class a little smile. \"He's nuts.\"\n**CHAPTER TWENTY-ONE**\n\nIf I wasn't mistaken, my lawyer's office was in one of the high-rises Howard Hughes used to live in while he was hiding out in Vegas, before he took up semipermanent residence at the Sands in 1966. Everyone seemed to think he'd lived here forever, maybe even died here, but in fact he was only in Vegas about four years, made some bad investments, split. From publicity hound to Vegas recluse\u2014although I suppose the hermit routine was another way of generating attention. But while he was here, he influenced world politics\u2014and football games\u2014consorted with (and according to some, bribed) both LBJ and Nixon, got progressively balmier, and tried to buy ABC to prevent miscegenation on _The Dating Game_ (true!), all the while protected from the world by his bubble of loyal attendants. Until he died. But hey\u2014he got a parkway named after him, one you just about had to use to get to the airport. Could be worse.\n\nWhen we arrived at the courtroom, I saw that the Shepherds were looking their usual saintly selves. He was wearing a plain-vanilla suit and tie; she was wearing a cotton print maxi. Did they always dress like that, I wondered, or was this a courtroom ploy to show how different they were from me? Particularly in dullness. I would've preferred to avoid them altogether, but I didn't want to be rude. The judge's bailiff or someone might be watching.\n\n\"Nice to see you again,\" I lied. \"Rachel couldn't come?\"\n\n\"The judges prefer not to have the minors present at these hearings,\" Mrs. Shepherd explained. \"If the judge wants to talk to her, he'll call her to his chambers. Besides, she has basketball practice.\"\n\n\"Rachel?\"\n\n\"Yes. She's joined the school team.\"\n\n\"Rachel is playing a team sport?\"\n\nThe woman was so tiny she seemed to bob when she spoke, like one of those mechanical storks you saw at truck stops poised over the rim of a glass. \"She's enjoying it. Making new friends.\"\n\n\"Is she any good?\"\n\n\"Well,\" Mr. Shepherd explained, \"she's inexperienced. She hasn't played as long as most of the other girls. But she has the height, and she's not without talent. I think she has some natural athletic gifts. I'm surprised you didn't encourage her to play.\"\n\n\"Well, I . . . didn't . . . I . . . thought it best to focus on academics.\"\n\n\"Her first game is Monday night. You should come. I think she'd like that.\"\n\n\"I'll try. I've been busy with this investigation.\"\n\n\"Of course.\"\n\nDelacourt shot me a look, and I amended, \"But I'm always ready, willing, and able to spend time with Rachel.\"\n\nGoddamn those Shepherds, anyway. Did they do it on purpose\u2014always making me feel inferior to their pedestrian middle-class blather? I threw myself into my chair. At least they hadn't turned her into a cheerleader. Yet.\n\n\"You know the judge will be watching you,\" Delacourt said to me quietly.\n\n\"Is it my hair or this new Wonderbra?\"\n\n\"He'll be watching your demeanor. Trying to judge whether you're capable of raising Rachel. I told you this already, remember?\"\n\n\"You also said nothing would be decided today.\"\n\n\"That doesn't mean he can't start thinking about it.\"\n\n\"And he can tell what kind of parent I'd be from looking at me?\"\n\n\"He can tell a lot. He's been doing this for thirty years. He can tell if you're drunk, which thank God you don't appear to be. He can tell if you're able to control your temper.\"\n\n\"So I will.\"\n\n\"Then we have nothing to worry about.\"\n\nThe first fifteen minutes of the hearing were boring beyond belief. Lawyers talking lawyerspeak to other lawyers. Occasionally I'd hear my name and my interest level would increase. But after another ten seconds or so of _parens patriae_ and _guardian ad litem_ my head would be in another place.\n\n\"Ms. Pulaski?\"\n\nI was pretty stunned to realize the judge was talking to me. I rose to my feet. \"Yes, Your Honor?\"\n\n\"Do you agree with what the counsel for NDHS said?\"\n\nI hated these memory tests. Especially when I hadn't been paying attention. But I figured if the lawyer who wanted to give Rachel to the Shepherds had said it, it couldn't be good. \"No, Your Honor, I certainly don't.\"\n\n\"Good. Neither do I.\" I sat back down. Judge Gaynor was in his late fifties, but his hair was still jet black and his face relatively unlined. He had a clipped tone to his voice but seemed to make a point of avoiding rudeness to anyone. \"In fact, I'm rather disappointed to hear the state make the argument.\"\n\n\"Your Honor,\" the other lawyer protested, but the judge waved it away.\n\n\"We need public servants. Now more than ever. We are perhaps only beginning to appreciate the enormous benefits provided to us on a daily basis by the law enforcement community. Their job is difficult and the hours are long. We should honor their dedication, not use it as a weapon against them. I've never heard of anyone losing custody because police work was inherently demanding. Or dangerous. And we're not going to set a precedent in my courtroom.\"\n\nI only half understood what was happening. But I had the sense to know it was good. \"Thank you, Your Honor.\"\n\n\"I am aware that Ms. Pulaski is a behaviorist and that she is working on the current spate of killings that have plagued our community. I commend her for taking on this challenge.\"\n\nMy God\u2014was it possible? The judge actually liked me?\n\n\"The state's concerns about her income and employment status seem to me totally without merit. I also note that you have found a new place to live.\"\n\n\"Yes, sir. Although my job recently forced me to make yet another move. It's small, but\u2014\"\n\nHe nodded. \"We like our parents to have homes, but we're certainly not going to evaluate their worthiness based upon square footage. Especially not for a dedicated public servant.\"\n\nFor the first time, my spirits swelled. Maybe, just maybe, I had a chance to win this thing.\n\nOf course, it couldn't last. \"I am, however, concerned about the state's allegations regarding the personal problems that arose after the loss of your husband. According to the Human Services' brief, you're an alcoholic. Is this true, Ms. Pulaski?\"\n\nWhat to say? I didn't like being labeled, and I didn't think it was fair or accurate, but if I quibbled with him, they would say I was in denial. As I peered into the judge's eyes, I realized that he already had all the factual information he needed. He had asked the question to see how I would respond.\n\n\"I have problems with alcohol, Your Honor. That is absolutely true. But I'm dealing with it. I've given up drinking. Totally.\"\n\nHe looked at me intently but didn't say anything.\n\n\"I've completed a detoxification clinic. I'm attending Intensive Outpatient classes downtown.\"\n\nHe shuffled some papers. \"I have a report from a . . . Dr. Coutant, who treated you at the detox center.\"\n\nI felt as if my heart had been stabbed with a dull pizza knife.\n\n\"According to him, you haven't been attending the IOP classes. Is that true?\"\n\n\"I . . . uh . . . I have been absent for a while. This case takes up so much of my time.\"\n\n\"Surely your recovery comes first.\"\n\n\"I plan to start up again, just as soon as\u2014\"\n\nJudge Gaynor cut me off. \"Alcoholics always plan to do something in the future. Just as they always say they're not going to drink anymore. It's part of the disease.\"\n\n\"But\u2014\"\n\n\"I admire your work, Ms. Pulaski. Truly I do. But unless and until you've dealt with this problem, no court on earth is going to grant you custody. It would be irresponsible. And certainly not in the best interests of the child. Quentin?\"\n\nMy lawyer rose. \"Yes, Your Honor?\"\n\n\"At our court date, I'm going to want full and complete medical records on your client.\"\n\n\"Yes, sir.\"\n\n\"I'll want attendance records from the IOP classes. And evidence of enrollment in a certified AA program. And I want periodic blood tests.\"\n\nHe breathed heavily. \"Yes, sir.\"\n\nThe judge gave me one last look. \"If those tests show alcohol in your bloodstream, Ms. Pulaski, you might as well not bother coming to court.\"\n\nThe rest of the hearing was a blur. I don't remember what happened. I know Delacourt got me to a car, made some inquiries about my state of mind, and entrusted me to the care of my security detail. Meanwhile, my brain stayed on one topic. What the judge had said. How he'd built my hopes up. Then cut them out from under me with a single swath of his vicious scythe.\n\nThe hardest part was not the embarrassment, not being called a drunk in open court. Nor was it the judge's demands and requirements. Not even the blood test. Damned intrusive, but in my heart I knew it was a perfectly reasonable request under the circumstances. No, my problem was not the fact that the judge was trying to force me to give up drinking.\n\nMy problem was that I knew I couldn't do it.\n\nWhy won't the letters do what I want them to do? Normally they dance around and they tell me what they mean and I get it and I can tell Susan but this time I can't tell her anything. I know the Bad Man made it harder because he knew I was good at this just like that teacher in the tenth grade who was mad at me because I showed him that he was working the quadratic equation wrong and so he made me stand at the front of the class and gave me calculus problems and he was trying to make me sad because I hadn't read any calculus books and he made me stand there the whole time and work on it and I did get it eventually I did but by then everyone had gone home and when I showed the teacher the next day he said that I had cheated and I got in trouble. I don't know why I always get in trouble and I'll probably get in trouble if I solve this puzzle but I want to do it because Susan wants it and if Susan wants it then I have to do it.\n\n _Talk to me!_\n\nHe's trying to fool me he's using the same letters over and over again but changing what they stand for like before but more often it's like the letter only appears once and then it's something else and that's why the letters can't talk to me he won't let them. But I can sorta see the words even when he keeps changing everything and he won't tell me where one word ends and the next begins and I think he messed it up in a few places but there's still a pattern and I can trace that letter that keeps coming over and over again I thought it must be _E_ but it isn't _E_ and if it isn't _E_ maybe it's _S_ and if it's _S_ why does it keep coming up again and again like one word with two _S_ 's close together like he's talking to someone and\u2014\n\nSusan.\n\nThis was a letter to Susan like maybe they were all letters to Susan but this one really is because her name is in it. _Talk to me!_\n\nSusan oh no no no no no Susan I have to call Dad or someone or that Patrick and get them to get to Susan oh no quickly Susan oh no oh no oh no no no no no no . . .\n\nDAM YOU IM ACELERATING YOUR EDUCATION YOURE NEXT SUSAN\n\nI felt like death on a soda cracker, to use one of David's favorite phrases. I felt as if I had wrapped Rachel up in a box and gift-wrapped her for those narrow-minded prigs she'd been living with. I'd been kidding myself, pretending that I might be smart enough to catch this killer when I knew damn well I wasn't. Not now. Probably not ever. I was as depressed as I ever remembered being in my entire life. And the worst part of it\u2014I wasn't even free to drink. Not in a public place, anyway, not when I knew spies might be watching me at any time.\n\nWhy did they have to make everything so hard? Why couldn't they understand? I'm trying, but I'm not Wonder Woman. I'm not perfect. But Rachel wants to live with me, and I want to live with her. Why isn't that enough?\n\nJust to make matters worse, Lisa was out with some new kisser of the week, and Patrick was too busy babysitting the new feds to step out with me. Not that I really wanted to wake up handcuffed to the headboard again. But I didn't want to be alone, either. So instead of chatting with a handsome FBI agent, I was in The White Feather wondering if one of the security guys posted outside could be lured in for a club soda.\n\nWas there anything I hadn't screwed up? Would there ever be?\n\nDavid could answer that question. Yes, he certainly could. But I might not like the answer.\n\n\"Susan? Is this chair taken?\"\n\nYou can imagine my relief when I saw that accountant-informant guy, Ethan, standing at the other end of the table. He was a small guy, but still reasonably attractive. He wasn't Patrick, but now that I looked at him, I realized they did have a similar look about them. And Patrick had given me a pass. So a girl could hardly be faulted for exploring alternatives.\n\nI invited him to have a seat. He ordered a Coke. I continued nursing my club soda. We talked about nothing in particular for a while.\n\n\"I know you've been buried in that case,\" he commented after a while. \"But you haven't said anything about it.\"\n\n\"And I'd like to keep it that way, if you don't mind.\"\n\n\"What do you think that guy wants?\"\n\n\"Who? The killer?\"\n\n\"I can't help being curious. After all, I saw the man.\"\n\n\"He doesn't know what he wants. He's delusional. Psychos of his caliber work up these grandiose schemes that only make sense in their own minds. If there.\"\n\n\"Are you sure? I'm no expert, but it seems to me as if he might be . . . working toward something.\"\n\n\"And this is based on seeing him in the parking lot?\"\n\n\"And everything I've read. You don't think he has a greater plan?\"\n\n\"Not one that makes any rational sense.\"\n\n\"Do you think you're close to catching him?\" He seemed awfully interested in this killer. Didn't he get that I didn't want to talk about it? \"I mean, I'm wondering if I might have to come in and pick him out of a lineup or something.\"\n\n\"It's possible. I can't talk about how close we are.\"\n\n\"Sure. I understand.\" Finally we moved on to other topics. A few minutes later he got around to asking if I was married.\n\n\"Widowed. You?\"\n\n\"No. Wanted to. But never met the right girl.\"\n\n\"You wanted to be married? Why? Just for the pleasure of sharing a bathroom?\"\n\nHe smiled shyly\u2014almost boyishly. \"A stable lifestyle is the key to harmony. A family. You have any children?\"\n\nI felt my buzz fading. \"No.\"\n\n\"Pity. You'd make a great mom.\"\n\n\"Tell it to the judge.\" He couldn't possibly know how sensitive a nerve he'd just struck. \"I'm sorry, long story. Stupid.\"\n\n\"No, not at all.\" To my surprise, he put his arm around me and held me. Warmly. There was nothing sexual about it. He was just . . . comforting. \"Best to let it out.\"\n\n\"No. I just . . . oh, God.\"\n\nThe bartender came by, obviously wondering if something was wrong, if he was a masher trying to overpower me or something. \"She's just not feeling well,\" he explained, and now that he mentioned it, I realized I wasn't feeling well. My eyesight was getting fuzzy. My limbs were stiffening.\n\n\"I\u2014I think I may need to go to the bathroom.\"\n\n\"That's okay,\" Ethan said. \"Let me help you.\"\n\n\"I\u2014don't\u2014need\u2014\" All of a sudden, I could barely move. Or speak. He helped me off the stool and all but carried me back to the ladies' room. To my surprise, he went in with me.\n\nAnd locked the door behind us.\n\n\"Is your head feeling heavy, dear? You're starting to slur, and for once, it isn't the copious quantities of booze you've imbibed. I put a little treat in your drink.\"\n\nI tried to do something, but it was impossible. I was alert and fully cognizant of what was happening. But I couldn't move.\n\n\"I didn't want to do this, Susan. I wanted you to come willingly, eagerly. You made this necessary.\"\n\nHe opened the window in the back of the bathroom and stuffed me through it. He was stronger than he looked.\n\n\"We're going back to my place now. I've got my truck parked in the alleyway. You're familiar with that alleyway, aren't you? Your security detail is taking a nap, but we'll use the back way just to be careful.\"\n\nAs promised, his truck was right there. He lifted me into the cab, pulled a blanket over me, and started the engine.\n\n\"I'm going to give you something to help you sleep now, Susan. You'll wake later. We'll talk. You won't believe all the preparations I've made.\"\n\n _Noooo!_ I wanted to scream. _Someone help me!_ But I couldn't do it. Couldn't do anything. I was worse than putty in his hands.\n\nThe syringe descended toward my neck, a silver drop dangling from the tip. I didn't feel it penetrate, but I knew it had. And a few seconds later, I was gone.\n\nGranger didn't even wait for the car to stop. He leaped out the passenger side door, stumbling, recovering, racing. Darcy followed close behind. Despite repeated efforts, they'd been unable to reach Susan or any of her security detail on their cells.\n\nGranger raced into the bar and flashed his badge. \"Susan Pulaski. Tall, dark hair. Tough. You know her?\"\n\nThe barkeep shrugged. \"Sure. Seen her on TV.\"\n\n\"Where is she?\"\n\n\"She's over\u2014\" He turned and almost pointed before he realized the chairs were vacant. \"That's funny. She was sitting right there.\"\n\n\"Was she by herself?\"\n\n\"At first. Then some other guy joined her.\"\n\n\"Who?\"\n\n\"I didn't know him.\"\n\n\"And this didn't concern you?\"\n\nThe barman's eyes crinkled. \"I thought he was trying to pick her up. My only concern was that she might wipe the floor with him.\"\n\n\"What did he look like?\"\n\nHe pondered. \"Kind of nondescript. I didn't really notice.\"\n\nDarcy stood by the empty chairs. \"Did you know that she left her purse? Do you think Susan would go away but leave her purse?\"\n\nGranger gripped the bartender by the shoulders. \"Did you see where she went?\"\n\nThe barkeep shook his head. \"No.\"\n\n\"I did.\" It was one of the waitresses. \"She went to the ladies' room. Her friend was helping her walk. She looked kinda sick.\"\n\n\"Sick? As in ill?\"\n\n\"Well, I've seen that look on her before.\" She winked.\n\nGranger plowed through the crowd, carving a path to the ladies' room. Darcy followed. \"Clear out!\" Granger shouted. And then he kicked the flimsy door to pieces.\n\nThe ladies' room was empty. But the rear window offered a view of the alley behind. And it was open.\n\n\"Goddamn it. Goddamn it!\" Granger shouted. He pushed his head through the window. Nothing there.\n\n\"Do you know where she is?\" Darcy asked. He was pacing in a small circle, running his fingers through his hair. \"Did she go through the window? I think she could fit. Do you think Susan would crawl through the window?\"\n\n\"No,\" Granger said bitterly. \"Not on her own.\"\n\n\"Then . . . then . . . what . . . what do you think happened to her?\"\n\nGranger didn't respond. Because as strange and stupid as Darcy seemed at times, he already knew the answer to that question.\n\nHe sat patiently by her bedside throughout the night, wiping her brow, stroking her face, brushing her hair. He could not have been more tender, more loving. Lying quiet like this, freed from the stress of the world, from the demons that plagued her, there was something almost angelic about her countenance. She wore no jewelry, no finger paint, no hair dye, not even makeup. Pure as the day she was born.\n\nHe felt an aching yearning for her. _With a love that the winged seraphs of Heaven coveted her and me . . ._\n\nWhen at last she woke, the face she saw first was smiling.\n\n\"Good morning, sweet Susan.\"\n\n\"W-W-Www . . .\" She licked her lips, stretched her jaw as if breaking it in for the first time. \"Where . . .\"\n\n\"You're with me, my darling. In my safekeeping.\"\n\n\"I'm alive?\" She looked around as much as possible, given her restraints.\n\n\"I wanted to talk with you. Before . . . I'm forced to proceed.\"\n\nHer eyes darted about the room, obviously trying to learn as much as possible about her location. \"What's that noise? Sounds like a waterfall. Where am I?\"\n\n\"I can't tell you that.\"\n\n\"Why am I strapped down?\"\n\n\"I couldn't very well let you run free. I did leave you your clothes, and I left the restraints as loose as possible\u2014\"\n\n\"Let me out of here!\" Her voice was finding its strength. \"Now!\"\n\n\"I'm afraid I can't do that, Susan.\"\n\nShe bucked and heaved, straining against the belts that held her to the table. \"Let me . . . go!\"\n\n\"Susan . . . please remain calm. We don't have that much time. I need to tell you about my work.\"\n\n\"You don't have any work. You're a psychotic killer.\"\n\n\"Just let me talk to you a few moments.\"\n\n\"Let me go. Then I'll listen.\"\n\n\"Be reasonable. I know what I have to say may seem . . . unconventional at first, but I'm confident that you're capable of assimilating it.\"\n\n\"I want out of here!\"\n\n\"We could be such a team, Susan, you and I. There is another world, Susan, a better world. And we can go there together. I can make it happen. Give up this mortal plane that has so ill-used you and join my quest! We'll meet Virginia\u2014\"\n\n\"Who?\"\n\n\"My dear late sister. We could all live together in peace and\u2014\"\n\n\"Listen to me!\" Susan took several deep, invigorating breaths. \"I know this will be hard for you to understand, but it's the truth. You are not well. You don't do these hideous things because you have been chosen or because a prophet is speaking to you. You do them because you are ill.\"\n\n\"No, Susan, no.\" His eyes became clouded. \"Please don't.\"\n\n\"This is something I know about. I also know that people like you can be helped. With invasive drug therapy.\"\n\n\"Chemical emasculation.\"\n\n\"Or intensive behavior modification. Or a combination of the two. If you let me, I will try to help you. But if you go on killing as you have been, you're going to end up dead. Do you hear me? They will catch you in the end. And they will kill you.\"\n\nWas it possible she was right? She seemed so earnest, so good, and her dark eye shone as clear as the\u2014\n\n ** _\"You know what you have to do.\"_**\n\n _\"No!\"_ he shrieked, hands pressed against his ears. \"Go away! I'm not ready.\"\n\nSusan's brow creased. \"What? Are you talking to me?\"\n\n ** _\"You know what you have to do.\"_**\n\n\"There's still time. I think she's beginning to see, just a little. . . .\"\n\nSusan strained futilely against her restraints. \"I don't know what you're talking about.\"\n\n ** _\"You have strayed. You have allowed this witch of Endor to seduce you.\"_**\n\n\"No, you don't understand. I just\u2014\"\n\n ** _\"I am the Raven!\"_**\n\n\"I . . . know . . . I\u2014\"\n\n ** _\"Then you know what must be done.\"_**\n\n\"I don't want to hurt her. If there's any chance\u2014\"\n\n ** _\"No more chances! She must be broken! She must be shown the grievous error of the path she has chosen.\"_** The voice thundered in his head, knocking him back against the table. Susan stared at him, brow creased, lips parted.\n\nHe stumbled to his feet, wiping the water from his eyes. He staggered until he found the black bag, then withdrew a syringe. \"I'm so sorry, Susan.\"\n\n\"You don't have to do this.\"\n\n\"I do. I don't want to hurt you. But I have no choice.\"\n\nSusan twisted back and forth. \"You always have a choice. It's never too late.\"\n\n\"It is,\" he said, his voice cracked with pain. He clenched the syringe in his fist and thrust it into her neck. Almost immediately, her eyelids fluttered closed.\n\n\"Goodbye, Susan,\" he choked. \"Goodbye and farewell. Please remember that I loved you.\"\n\nAfter that, there was only blackness. For both of them.\n**BOOK TWO**\n\n> **THE ARABESQUE**\n> \n> I was a child and she was a child, \n> In this kingdom by the sea: \n> But we loved with a love that was more than love\u2014\n> \n> \u2014\"Annabel Lee\" \n> EDGAR ALLAN POE\n**CHAPTER TWENTY-TWO**\n\n\"Damn it, Boss, it's been five days.\"\n\n\"I know how long it's been, Granger.\"\n\n\"The press is asking questions. A lot of questions.\"\n\n\"The press can go screw itself.\"\n\n\"Did you see those pictures?\"\n\nO'Bannon stiffened, like a metal rod had replaced his spine. \"Yes.\"\n\n\"I have to assume we're not the only recipients. I'll bet they went to all the papers, all the TV stations. They're going to want to know what happened to our behaviorist.\"\n\n\"Who the hell doesn't?\"\n\nGranger's face twisted up. \"Boss\u2014those pictures. They're\u2014we have to tell them something! Try to explain\u2014\"\n\n\"Why?\" O'Bannon pushed away from his desk, matching Granger bellow for bellow. \"It would be different if we knew something. But let's face it\u2014we don't. We don't have the slightest idea where she is or whether\u2014\" He stopped short of saying it.\n\n\"Face facts, Boss. Five days. None of this guy's previous victims lasted five days. All of them were killed within a day or two of their capture.\"\n\n\"Shut up.\"\n\n\"I'm telling you, we've got to face up to it.\"\n\n\"And I told you\u2014\"\n\n\"Susan is dead.\"\n\nO'Bannon froze.\n\nDarcy sat behind him, in a chair in the corner of his office, not saying a word. Listening to everything.\n\n\"Darcy,\" O'Bannon said, lowering his voice, \"would you leave us alone for a moment?\" He fumbled in his pocket for change. \"Run across the street and get yourself an ice cream. Get us both an ice cream. Okay?\"\n\n\"No.\"\n\nThe response was so startling that both O'Bannon and Granger were thrown. \"Excuse me?\"\n\n\"I think I would rather stay here.\" As if a sudden chill had come over him, Darcy's hands rose and flapped against themselves. \"Did you know there was a woman kidnapped in Vancouver in 1979 who was held for sixty-seven days till she escaped? She overpowered her kidnappers and ran away. She hadn't been harmed.\"\n\nO'Bannon and Granger looked at one another. \"Darcy . . .\"\n\n\"In Omaha, in 1984, a teenage girl was kept in a basement for over six months until she was found by the police. She wasn't hurt, too.\"\n\nO'Bannon felt as if he had gained a hundred pounds in the space of a second. \"Darcy, we have to be realistic about this.\"\n\n\"Elizabeth Smart was gone for eight months, but she came back. Patty Hearst was held for a hundred and twenty-four days until she was found by the FBI, but she had been brainwashed into joining their cause. Do you think Susan could be brainwashed?\"\n\n\"No,\" O'Bannon grunted. \"I think anyone who tried to play mind games with Susan would probably end up in the loony ward himself.\"\n\n\"Actual brainwashing in the post\u2013World War II era has been quite rare and mostly ineffective,\" Darcy recited. \"Several instances of the Stockholm syndrome have been documented, but it is still uncommon.\"\n\n\"What have you been reading?\"\n\n\"Unless there's torture involved. Do you think Susan might be tortured?\" His voice remained as loud and expressionless as ever, but the flapping of his hands accelerated.\n\n\"No, son.\" He took the boy's hands and brought them to rest. \"I don't think so. But we have to accept the fact that Susan is prob\u2014is\u2014may not be coming home.\"\n\n\"I think that I should have figured out that message sooner. Do you think that I should have figured out that message sooner? Should have should have should have. That makes it my fault that the Bad Man got Susan.\"\n\n\"Of course it isn't.\"\n\n\"For that matter,\" Granger said, \"I wish I'd reacted faster.\"\n\n\"There was no way anyone could have predicted what this bastard would do. He took out three security officers, for God's sake, with that magic drug of his.\" O'Bannon pressed the heels of his palms against his forehead. \"Darcy, please go get some ice cream.\"\n\n\"No. I will stay here.\"\n\n\"Darcy, we have our best officers working on this case. The best thing you\u2014\"\n\n\"I am not going for ice cream. I am going to help.\"\n\n\"Darcy, you're not a police officer.\"\n\n\"I could be!\" He turned, and although he did not quite make eye contact, he looked in the direction of his father. \"You\u2014you\u2014you always w-w-wanted me to be a policeman. So I will be a policeman.\"\n\n\"Darcy . . .\"\n\n\"You\u2014you d-d-don't think I can do it.\"\n\n\"These men have been through years of training and\u2014\"\n\n\"I can do it. I can do training.\"\n\nO'Bannon turned for help to Granger, who was pointedly looking away as if he were not paying attention. \"You have some . . . special challenges, Darcy.\"\n\n\"Everyone has special challenges.\"\n\n\"But we have to be realistic and\u2014\"\n\n\"Do you think that any of your officers could decode the Bad Man's messages? Because I do not think any of your officers could decode the Bad Man's messages.\"\n\nO'Bannon craned his neck. \"That was a special situation. You need to go home, Darcy. When I get back tonight, we'll talk more. Maybe we can make popcorn.\"\n\n\"No!\" Darcy threw down his hands. \"I do not need popcorn or ice cream or going home. I will help on this case, even if you don't think I can. Susan thought I could help and I did help. I did!\" He marched to the door, his eyes watering. \"I m-m-may be an idiot, Dad, but Susan needs me. And I am going to help her.\"\n\nAm I dead? I wondered as the light streamed into my eyes. It wasn't a warm sensation\u2014more like the grinding of gears against metal, brakes after the brake pads have worn. I should've been surrounded by darkness, I remember thinking, but instead I was immersed in light, too much light, white hot and blinding. I wanted to turn away from it but found that I couldn't move.\n\n\"Hello, sugar bear.\"\n\nDavid again. I wasn't surprised. Wasn't exactly relieved, either. But he was something to look at. As always.\n\n\"What are you doing here? Do you know what happened to me?\"\n\nHe didn't have to answer. He gave me that soft, knowing look, the one he always used to disarm my wrath.\n\n\"I feel so . . . stupid. So ashamed.\"\n\n\"You shouldn't. It wasn't your fault.\"\n\n\"Wasn't it?\" A thought occurred\u2014I should try to open my eyes. I needed to orient myself, separate the real from the imagined. \"I think I'm . . . broken.\"\n\n\"You did before.\"\n\n\"But this is . . . different. This seems . . . I don't know. Final. Unrecoverable.\"\n\n\"I've felt like that.\"\n\nI looked at him unflinchingly, peering into the depths of those overwhelming blue eyes. \"I guess you must've.\"\n\n\"But here's what you need to remember, Susan.\"\n\n\"You're giving me advice?\"\n\n\"This is what you already know. What you've always known. What I could never figure out. When you forgive others, you're not doing them a big favor. You're doing yourself a big favor.\"\n\n\"David.\" All at once I wanted to reach out, wanted to hold him, wanted to bring him back to me and never let him go. \"Am I dead?\"\n\n\"What do you think?\"\n\n\"I think . . . I saw someone. Something. Before you. It was beckoning to me.\"\n\n\"And you think it was . . . ?\"\n\n\"You know what I think it was.\"\n\n\"You think it was Death.\"\n\n\"Can you tell me what to do?\"\n\nHe smiled that goddamned irresistible grin. \"Well, if he wants to play chess with you, don't.\"\n\n\"How long has he been in there?\" Lisa asked.\n\n\"Days,\" O'Bannon replied. They both peered through the window of the door to the small police library where Darcy had taken up residence. He had his books, an evidence file, and an Internet connection. \"I've been sending in food\u2014pizza and stuff\u2014and sometimes he eats it. I don't know when was the last time he slept.\"\n\n\"What does he hope to find?\"\n\n\"I don't know. I've asked, but he can't really explain it. He just starts stuttering and flapping his hands. I think maybe he has some crazy idea that if he punishes himself enough, it will bring Susan back.\"\n\n\"What do you think?\"\n\nO'Bannon looked away. \"You know what I think. What everyone thinks.\"\n\nLisa's face crumpled. She pressed a hand against the wall to prop herself up. \"I should've been with her that night. She called me, but I was off on a date with some loser I didn't even know. If I'd been with her\u2014\"\n\n\"He might've taken you both.\"\n\nShe shook her head, brushing away her tears. \"No. I could've stopped it. It's my fault.\"\n\nO'Bannon walked to the water fountain and splashed cold water on his face. \"You sound like Darcy. He says the same thing.\" He used his shirtsleeve to dry himself off. \"Susan had been so depressed, felt so isolated. I think she thought she had lost everything, that no one loved her anymore.\" His teeth clenched. \"My God. She had no idea.\"\n\n\"Is there anything more you could be doing?\" Lisa asked, her voice cracking. \"Anything I could be doing?\"\n\n\"Believe me, we're trying everything possible and then some. We've reassigned all available manpower and borrowed more from neighboring jurisdictions. We're tracking down every lead we get, every sighting. So far, they've all been bogus.\"\n\nLisa covered her face. \"I've\u2014I've seen the pictures.\"\n\n\"Damn! What irresponsible\u2014\"\n\n\"I made him. This reporter I know at the _Courier._ He showed me.\" She pushed herself into the corner, eyes wet and wide. \"My God, do you think it's even possible? Could she still be alive?\"\n\n\"I don't know,\" O'Bannon said, swearing under his breath. \"And at this point, I don't know if she'd want to be.\"\n\nThe first sense to return was my sense of smell. There was an acrid bitterness in the air, kind of like coffee left too long on the burner. And there was something else, something fouler. Sour milk. No, that wasn't it, but it was like that. Stale, stinky. Something I didn't want to be near.\n\nThen I heard the sounds. Wind whistled in my ears, and I felt cold. And a pounding, crashing sound, an auditory sense of motion. It was that rushing water noise, same thing I'd heard before. A forceful sound, the kind that could sweep me up and wash me away.\n\nI wished it would.\n\nMy eyes finally opened on a bleak field of gray. Was I in the desert? I wondered. On a mountaintop? Took me more than a few moments to realize what I was staring at had to be an overcast sky, since I was lying flat on my ass. Literally. Because I was naked.\n\nI suppose that shouldn't have been a big surprise, all things considered. I was on something hard and flat and grainy. Dirty. It was in my skin and under my fingernails. I wondered how long I had been there. A long time, I thought.\n\nI tried moving and was amazed to find I actually could. My joints were stiff, stiff to the point of near immobility. My skin hurt. But I forced myself. I sat up, and it couldn't have taken more than half an hour or so.\n\nI surveyed my surroundings. I was in some kind of gravel pit, white and chalky, no one else around as far as I could see. The crashing sound I'd heard was water, huge tumbling quantities of water, tumbling down not far from where I lay.\n\nEventually I had to turn my attention to myself, in all my glory. My skin was red and scorched, except for the bruises, which were many. I was exposed, floppy, veined, dirty, about as unattractive as it was possible for a woman to be. I repulsed myself.\n\nWhat the hell had happened? I tried to recall, but the effort made my head pound. I remembered being captured. I remembered lying on his table. I remembered being scared, so scared, like I haven't ever been before, not even when I found out about David. I remembered hating myself because I was helpless. I should've figured a way out. I should've pulled some clever last-minute trick that saved the day. Instead, I became a victim. Another pawn in the hands of a psycho who had proved himself a thousand times smarter than I was.\n\nI had no sense of time and no timepiece. I'd forgotten what little I ever knew about telling time with the sun, so I can't possibly say how long it was before I noticed the boxes. Seemed like an eternity that I sat there thinking, crying, cursing, not able to move, not wanting to move. But eventually that passed, or at least subsided. And I turned my attention to the shoe boxes he had left at my feet.\n\nThere were two of them, each with a message scribbled across the top in indelible black marker. The first box read: TO HELP YOU REMEMBER.\n\nLike Alice in Wonderland, I slowly opened the envelope inside, not wanting to know what it contained, but unable not to look.\n\nThe envelope contained pictures, lots of them. Polaroids, amateur stuff, obviously taken by Edgar himself. They all had the same subject. Me.\n\nThey must have been taken while I was under the influence of his drugs. My eyes were open, but there was no one home. I could tell. There was no me in there. Only my body. My naked body.\n\nI had been posed, over and over again, different for each picture. He had . . . made me do stuff. He had me playing with myself. Touching myself. Sexual poses, me on all fours, me with my legs spread, me dry-humping the furniture. One nasty pose after another. In some of them, he'd given me props. A broomstick. A Coke bottle. A dildo stuck in my mouth. A dazed, zoned expression on my face, like I liked it. Like I was drunk and I liked it.\n\nI fell forward on my hands, heaving. He must not have fed me, because nothing came up, much as I tried. I hurled so hard I expected the lining of my stomach to spew out. I felt sick. Betrayed. Abused. Raped.\n\nI wanted to throw the pictures away, to lose them, to forget they ever existed. And then I saw the sheet of paper in the bottom of the box. It was a mailing list. All the places he had sent copies of the pictures. All the local television stations. National news agencies. Local radio shows. Police headquarters. The FBI. Chief O'Bannon.\n\nHe'd sent the pictures to Chief O'Bannon's home.\n\nDarcy.\n\nI fell forward, scraping my breasts against the gravel, wanting to hurt myself, wanting to die, wanting this to all be over, just please, please let it be over. I pounded the box with all the force my fists could muster, which wasn't enough to dent cardboard. Look what he's done to me, David. Look what I let him do to me.\n\nOf course it was just a matter of time before I opened the second box. It had been labeled, too: TO HELP YOU FORGET.\n\nOnly one thing inside that one. A quart bottle of scotch whiskey.\n\nI ripped the lid off the bottle and pressed it to my lips. I was hungry, starved, thirsty, desperate to forget. I opened my mouth and let the liquor course down my throat.\n\nI gagged. The booze spilled everywhere, all over me. I bathed in it. As soon as I'd stopped choking, I tried again. I would use more restraint this time, I told myself. Just take a sip. A little sip, then another. Sip myself into oblivion. I raised the liquid salvation to my lips.\n\n _This is what he wants you to do._\n\nI stopped. Where had that come from?\n\n _This is what he wants you to do. Why do you think he gave it to you?_\n\nI pulled the bottle away and stared at it, as if I had never seen such a thing in my life. He was manipulating me, just as he had done from the start. As I had allowed him to do. This is why he let me live. This is why he gave me the bottle. Because he knows I won't be able to resist.\n\nAnd he was so right. So bloody goddamn right.\n\nHe was trying to break me, to destroy what little was left so he could scoop up the pieces and reshape me into whatever he wanted me to be.\n\nI pushed up to my feet, amazed that I could do it, and walked toward the noise. I stood naked before the god of the waters, staring down from the precipice. It had to be a hundred feet down to the basin, maybe more. I didn't even have to climb over a barrier. Just one simple step. That's all it would take to end it, to find peace. Hell of a lot simpler than slashing my wrists with a shard of glass. No one would care. Not after they saw those pictures. And everyone would see those pictures.\n\nThe thundering crash of the water crescendoed in my ears.\n\n _That's what he wants you to do._\n\nI stared down into the maelstrom. And saw something I had never seen before.\n\nThat's when I made up my mind.\n\nFirst, I got the goddamn pictures and tossed them in. The next bit was harder, a lot harder. But I did it. I turned the bottle upside down and let it pour out into the abyss. It would've been simpler to just toss in the bottle, but this was more satisfying. It occurred to me that I might be spiking the Vegas water supply. Well, tough.\n\nThe booze was gone. The photos were gone. The need to destroy myself was gone, at least for the moment. I was naked, and I didn't know where I was, and I had no idea how to get back to civilization. Or even if I should.\n\nI fell back onto the gravel as if I were a bag of boneless meat. And stayed there. In time, I fell asleep. Not unconsciousness, not druginduced stupor, but the real thing.\n\nAnd I even dreamed. Or something like it.\n**CHAPTER TWENTY-THREE**\n\nPatrick marched into headquarters, his face taut and lined. He threw his coat at the nearest hook on the rack. It missed, fell in a crumpled heap on the floor. He didn't notice. He slid behind Susan's desk and started reviewing all the reports on Susan's disappearance. Just as he had every day this week. Over and over again.\n\n\"You've got messages,\" Madeline shouted from the lower floor, waving pink slips in the air.\n\n\"Give them to someone else,\" he said, his face buried.\n\n\"They want you.\"\n\n\"I don't have time for crackpots and false confessions.\"\n\n\"Some of them say\u2014\" She paused, lowered her voice. \"They've seen Susan.\"\n\n\"I've followed up on twenty-two Susan sightings. Granger has done more than that. Not a damn one has led to anything.\"\n\n\"You got a problem?\"\n\nPatrick whipped his head up. Somehow, O'Bannon was right in front of him. \"Sorry. I've been . . . immersed.\"\n\n\"So I hear. Obsessed, some say.\"\n\nPatrick craned his neck. \"Sir, when an officer is down\u2014\"\n\nHe waved it away. \"You don't have to tell me, Chaffee. I've known Susan all my life. I used to diaper the girl's bare bottom.\" O'Bannon's eyes briefly closed. He looked tired, aged. \"Madeline says you were making a stink about the files.\"\n\n\"I was trying to find out everything I could about Susan. Her background, personnel file, police record.\"\n\n\"You think the key to finding her is in her past?\"\n\n\"I don't know. But profilers are supposed to absorb all the data, collect every scrap of evidence, then come up with some brilliant conclusion. And I've read everything else.\" He paused. \"Except one file. It was logged into the computer index. But I couldn't find it. Madeline thought maybe you had it.\"\n\n\"She was right. It's restricted.\"\n\n\"I don't know why you pulled it, but if there's any possibility that it could help us find her\u2014\"\n\n\"It's not about Susan.\"\n\nPatrick stopped, thought a moment. \"I found it listed in her directory.\"\n\n\"A cross-reference. It's about her husband. He was a cop, too.\"\n\n\"David.\"\n\nO'Bannon frowned. It was obvious that this was a subject he preferred to leave alone. \"How much do you know about him?\"\n\n\"Not much. Except that he's dead. And his loss seems to have really hit Susan hard.\"\n\n\"It did.\"\n\n\"Started her alcoholism.\"\n\nHe shrugged. \"Certainly a contributing factor.\"\n\n\"I know he was a detective. Worked with Granger.\"\n\n\"Know anything about his death?\"\n\nHe shrugged. \"Police work is dangerous. I assumed he was killed in the line of duty.\"\n\nO'Bannon drew in his breath, then slowly released it. \"The first part is right. The last part is wrong.\"\n\n\"He didn't\u2014?\"\n\n\"They'd been married eight years. Susan probably wasn't the easiest person on earth to live with, but then David had a temper on him, too. They fought, but no one thought much about it. In a lot of ways, they were perfect for each other.\"\n\n\"Chief, are you saying\u2014\"\n\n\"They had a big fight that day. Right here at headquarters. Everyone watching. He stormed out. And that was the last time we saw him.\"\n\nPatrick's lips parted. \"No.\"\n\n\"Yeah.\" He handed Patrick a thin file. \"Put his weapon in his mouth and blew his brains out.\" O'Bannon shook his head. \"I guess he won that argument.\"\n\nI was sitting at a dinner table lit with candles. The soft, rosy glow cast a warm aura across the sumptuous spread. I felt all warm and snuggly.\n\n\"What happened?\" I asked.\n\nRachel answered. \"We found you, remember? In the desert. The police brought you back here.\"\n\nI turned my mind back, a mental process that produced physical pain. I remembered wandering around the desert, or trying. But my legs still didn't work well and could only move a few feet at a time. Something had happened to my right leg, or maybe it was the lingering effect of the drugs. I couldn't seem to remain conscious long enough to focus my thoughts. I was hungry. And thirsty. I had thought myself very noble when I poured the booze into the brink, but I later came to regret it. I needed to drink. I didn't need a drink. There was a difference. I kept telling myself.\n\n\"I told you I was going to prepare a very special dinner,\" Rachel said. \"Don't you remember any of this?\"\n\n\"I\u2014I\u2014\"\n\nAnother voice from down the table. \"Surely you remember the kiss I gave you when you got home.\"\n\nIt was Lisa. Lisa!\n\n\"I mean,\" she continued, \"I don't normally go in for kissing chicks. But when my homegirl has been lost in the desert for damn near a week, that's different. I kissed with wild abandon.\"\n\n\"I hope this doesn't mean you're going to give me a nickname.\"\n\n\"Well . . .\"\n\n\"Or if you do, I want it to be Goddess.\"\n\nLisa laughed. \"Goddess it is.\" She held up a platter. \"I made my artichoke dip.\"\n\nI gasped. Lisa made awesome artichoke dip. Three different cheeses, mayonnaise, and oh yes I think there's some artichoke in there somewhere. It is to die for. She hadn't made it in months. I'd almost forgotten how good it was.\n\nRachel held out yet another platter. \"Don't forget the buffalo wings.\"\n\nI gazed down in ecstasy. Another favorite. With bleu cheese on the side, not ranch dressing like some lame-o joints served. \"Did you make this?\"\n\nRachel squirmed. \"In a sense.\"\n\n\"In what sense?\"\n\n\"In the sense that I drove to Chili's and picked it up.\"\n\nI laughed and pushed six of them onto my plate.\n\nLisa chirped up again. \"Don't forget the potato skins.\"\n\n\"Potato skins? What kind of a meal is this, anyway?\"\n\n\"All your favorite junk,\" Rachel explained. \"You deserve it.\"\n\nIt was too good to be true. \"What about the Shepherds? Will they be joining us?\"\n\nShe shook her head. \"They've given up the battle. NDHS, too. It's sad that it took your being kidnapped for them to realize what a wonderful guardian\u2014parent, really\u2014you've been to me. They've all agreed that you should have custody.\" She giggled. \"I'm so glad, Susan. It's what I've wanted all along. I'm coming home.\"\n\n\"And . . . and the basketball? And that church group?\"\n\n\"Oh, the Shepherds made me do that stuff. All I want is to be home with you.\"\n\nThe flickering glow of the table filled me. I felt a warmth inside, a contentment. Something I couldn't remember having felt for a long time.\n\nGranger cleared his throat. \"Susan, would this be a good time to tell you something I've wanted to say for a good long while?\"\n\n\"Well, that depends . . .\"\n\n\"I'm sorry. About the way I've behaved.\"\n\n\"Oh, you haven't\u2014\"\n\n\"Sure I have. I've been a regular bastard and I know it. I knew it when I was doing it. But I just\u2014I just\u2014\"\n\n\"I know. David.\"\n\n\"It's not that. Not just that.\" He sighed. \"It's because you're such a good cop. And I know it. Hell of a lot better than I am. Smarter. I feel inferior around you.\"\n\n\"You shouldn't. Let it go.\"\n\nAt the end of the table, one chair was empty. The place had been set. \"Why isn't someone there?\" I asked.\n\n\"Because you haven't decided,\" Rachel said.\n\nI gazed about the table. \"Surely all this food isn't for me.\"\n\n\"Of course not.\" Rachel laughed, then pointed.\n\nDavid was sitting at the other end, facing me.\n\n\"You're back,\" I whispered.\n\n\"Surely you knew I couldn't stay away for long.\"\n\nI nodded. \"Thank you.\"\n\nHe seemed puzzled. \"For what?\"\n\n\"For helping me. Back in the desert. Helping me survive.\"\n\n\"Oh. Well, there's a problem with that.\"\n\n\"Problem.\" My heart raced. All of a sudden I couldn't catch my breath. \"What problem?\"\n\n\"You see, Susan . . .\" I knew he didn't want to go on, but he did. He was always braver than I was. \"You're still in the desert.\"\n\n\"I\u2014I am?\"\n\n\"Yes, honey.\"\n\n\"But I can see you. And Rachel and Lisa. And all this food.\"\n\n\"Because you're starving.\" He laid his hand gently on mine. \"You're dying, sweetheart.\"\n\nI opened my eyes. The sun blistered. I didn't know where I was, found it painful to move. My naked skin was burned and scratched and bleeding. I had chills and sweated and shook.\n\nI had never left the desert. I could hear the crashing of water, but it was more distant than it had been before. Why weren't there any people around? Shouldn't there be people? How long had I been wandering? Weak, exposed, lost. Broken.\n\nHad David said I was dying? But I was already dead. Surely I was already dead.\n**CHAPTER TWENTY-FOUR**\n\nShe isn't dead she isn't dead she isn't dead I don't believe that she is dead Mom Mommy is dead and they wouldn't tell me and everyone looked at me so sad and I didn't know why and Uncle Braden smelled like rosewater but she couldn't be dead because if she's dead then it's my fault and we won't have babies and I won't get to be a policeman and most of all I won't get to see her and I like her I really really really like her she's nice to me.\n\nWhy couldn't I have read that message sooner?\n\nDAM YOU IM ACELERATING YOUR EDUCATION YOURE NEXT SUSAN\n\nDad says I shouldn't use words like that and I should forget I ever heard them and so I did forget and it took me longer for the letters to talk to me and the Bad Man took Susan. It's my fault because I'm so stupid stupid stupid I'm a retard just like they say at Dad's office I'm a stupid stupid retard. _Your mother can't be with you anymore, Bambi._ This Bad Man is playing with us he likes to give us clues but we don't know what they mean and he took Susan and we don't know where and please let her still be alive please please please Mr. Strickland said that Jesus saves and Jesus protects so please take care of Susan and keep her alive.\n\nDAM YOU IM ACELERATING YOUR EDUCATION YOURE NEXT SUSAN\n\nAlso if he spelled better it wouldn't have taken so long and he made mistakes translating the words into code, too. He left out the apostrophes and he didn't put a period at the end of the sentence and Mrs. Calloway in first grade said I should always put a period at the end of the sentence but I don't think that way and she smelled moldy like she didn't brush her teeth enough and he put in the wrong kind of _dam_ I would've gotten it sooner if he checked his spelling.\n\nUnless that's the trick.\n\nDad! I need my dad or Patrick or someone I hate it when they make games with words I never get those stupid jokes because words just say what they say and he used the wrong one unless he meant to use the wrong one because he thought it was funny and maybe we can still help Susan maybe it's not too late for Susan please don't let it be too late for Susan please please please please please.\n\nWhy does everyone who's nice to me have to go away?\n\n\"Paaaa-trick!\" Before he could look up, a body fell across the desk, almost head-butting him in the process. It was Chief O'Bannon's son, Darcy.\n\nMadeline came running up behind him. \"I'm sorry, sir. I told him you didn't want to be disturbed.\"\n\n\"It's okay.\" He helped the young man off his desk. \"Haven't seen you in days.\"\n\nFor once, Darcy's conversation was remarkably direct. \"I know where she is.\"\n\nPatrick felt a deep sadness in his heart. He knew the boy had a crush on Susan. Even if he wasn't physically demonstrative, his devotion couldn't have been more evident. Given his preexisting emotional fragility, her disappearance must be tearing him up. \"Did you have a dream about her?\"\n\n\"I know where she is! I figured it out.\"\n\n\"I'm sure you did.\"\n\nDarcy grabbed his shoulders and shook him. \"Did you know that in World War II the Allies invented a code that could only be solved if you had a special machine because they used hidden cells and multiple substitutions and\u2014and\u2014\" His voice began to break up, like it was tumbling into a funnel. \"Did you know that these codes are impossible to solve and that must be why the Bad Man used it when he wanted to say something that we wouldn't get until it was too late?\"\n\nPatrick didn't know what to do. If this were a normal man, he'd think he was on the verge of a mental breakdown. With an autistic, he didn't know what it meant. Except that he needed help. \"I know you're good with codes. You figured out what Edgar's warning said\u2014\"\n\n\"But not what it meant!\" Darcy flapped his hands, rocking back and forth. He broke away from Patrick and circled around the desk, again and again, with increasing speed. \"My dad says I don't get jokes.\"\n\n\"Well, sometimes I'm not the quickest\u2014\"\n\n\"But I do! I do get jokes. Maybe I don't think they're funny, but I know when people are kidding, some of the time. I knew when the other kids were making fun of me.\"\n\n\"Darcy\u2014\"\n\n\"But I hate puns. Why should one word have more than one meaning? It's confusing and it doesn't make any sense.\"\n\nHis agitation was intensifying. Half the office was watching now. Out of the corner of his eye, Patrick saw Madeline go for Chief O'Bannon, probably thinking he was the only one who could get the boy under control. \"Darcy, I think the best thing would be for you to go home now. Get some rest. If anything happens\u2014\"\n\n\"You thought he spelled it wrong!\" Darcy shouted. \"But he didn't spell it wrong. He's too smart for spelling wrong. It was a clue.\"\n\n\"What?\"\n\n\"About _dam._ 'Damn you.' But he left off the _n._ Because he didn't mean that kind of dam.\"\n\n\"As I recall, he misspelled a couple of words.\"\n\n\"To fool us. He's smart and tricky. You said so in your paper. 'The subject is possessed of extraordinary intelligence and imagination.' So why would he misspell words?\"\n\nDespite Darcy's convoluted, histrionic manner, Patrick was beginning to grasp his point. \"Are you saying\u2014?\"\n\nDarcy climbed up on the desk chair and began jumping up and down. \"She's at a dam! He took her to a dam!\"\n\nO'Bannon appeared at the top of the stairs. \"Come on, Darcy. Let's go home.\"\n\n\"She's at a dam! You have to go and find her!\"\n\nSomehow, O'Bannon managed to drag his son away, apologizing to everyone as they passed. Darcy kept on screaming. He looked back, his hands stretched toward Patrick.\n\n\"Go to the dam! Go to the dam!\"\n\nPatrick fell back into a desk chair. Was it possible?\n\nHoover was the closest and most famous dam, but hardly the only one. How could they know? It was probably nonsense. He couldn't take tips from a hysterical autistic boy. Surely they weren't that desperate. If there was any chance of finding her, it would only come from good solid detective work. Surely.\n\n\"You aren't real,\" I said as I ran my fingertips down David's perfectly sculpted chest.\n\n\"Does it matter?\" he replied. \"I'm the only game in town.\"\n\nI laughed and pulled him closer. \"Do that thing.\"\n\n\"What thing?\"\n\n\"You know. That thing you do.\"\n\n\"With . . . what part of my body?\"\n\n\"Your nose, silly.\" I laughed. I loved it when he was like this, all tender and attentive. Happy. No moods, no complaints, just him and me. \"The way you crinkle it.\"\n\n\"I do not crinkle my nose. That's a girl thing.\"\n\n\"You do. Sometimes.\"\n\n\"I do not.\" He pressed against me, letting me feel the ripple of his rib cage, his strong thighs pressing between mine. Letting me remember how it was before . . .\n\nAnd then he took me. All in a rush, the way I liked it best, the way that always gave me goose pimples. Orgasm was nothing compared to the creeping, dizzying head rush that hit when he came after me like that. It was all I could do to breathe, to prevent myself from perishing from a surfeit of pleasure.\n\n\"Susan?\"\n\nI blinked rapidly. That wasn't David.\n\nA shadow fell across my face. \"I'm sorry it took us so long.\"\n\n\"Patrick . . .\" I grinned a little. \"Could you wait a minute? My husband and I were having a thing. . . .\"\n\nHe put his hand on my forehead. \"Why didn't you stay where Edgar left you?\"\n\n\"I wanted to stay. David told me I should go.\"\n\n\"David?\"\n\n\"That's what Edgar wanted, you know. He wanted me to stay at that place and he wanted\u2014well, he didn't get it. I wouldn't give it to him.\"\n\n\"You're delusional, Susan. Which is no surprise, given your condition.\" He shouted over his shoulder. \"Can I get some help?\"\n\nI laughed. \"Whatever. You're not real, anyway.\"\n\nHe kept on shouting. \"Get a stretcher! Start the IV! She's dehydrated and starved, with a serious case of exposure.\"\n\n\"You shouldn't be looking,\" I said, giggling a little. \"I'm naked.\"\n\n\"Not anymore.\" He spread his coat over me. It felt warm and scratchy. \"We'll get you to a hospital.\"\n\n\"No, you won't. I'm just dreaming you. But it's a good dream. Nice of you to come.\"\n\n\"I didn't come alone.\"\n\nA moment later, I saw Darcy rush forward, hovering behind Patrick. \"Susan!\"\n\nGood grief, who let _him_ into this hallucination? \"Darce.\" I tried to wiggle my fingers but couldn't. \"Nice to see you too.\"\n\nHis face was weird, eyes wide and kinetic, as if he were being pulled a thousand directions at once. Just as well that he was a stoic sort who wasn't comfortable with human contact, because\u2014\n\n\"Ooof!\"\n\nDarcy threw himself on top of me, squeezing me in his arms. \"Susan! Susan!\"\n\nWell, I couldn't dream this, could I? \"Darcy?\" I said weakly.\n\nHe was screaming and crying all at once. The volume made me wince, another element I couldn't be dreaming. \"I told them it was a dam. I told them it was a dam.\"\n\n\"All right, son, please move aside. Let us get her into the ambulance.\"\n\nI could tell he didn't want to let go. Funny thing was, I didn't really want him to let go. But they pried him away and hoisted me onto a stretcher. Darcy insisted on riding in the ambulance with me. I could tell Patrick didn't like the idea, but he didn't want to take the time to argue about it.\n\nIt was a pleasant little ride to the hospital, my friends all around me. I slept a little, listened a little, maybe both at the same time. It was nice. Darcy held my hand the whole way.\n**CHAPTER TWENTY-FIVE**\n\nShe's alive! She's really alive and I knew she would be except I didn't know but I hoped and she is she's alive alive and we found her and I guess Jesus does save because I prayed and I prayed and there she was she's all beat up and she lost her clothes but she's alive and I'm so happy I was so sad and scared but she's alive and she let me hold her hand in the big car on the way to the hospital.\n\nHer hand felt nice.\n\nThe next time I opened my eyes\u2014at least, the next time I opened my eyes and remembered it\u2014I felt much better. Which was not to say I felt good\u2014I was feeble and tired and had trouble speaking. I felt like hell, like I still had one foot in the grave. But definitely better than before.\n\n\"It's because they've been force-feeding you,\" Lisa explained. \"Through the tubes.\"\n\nI made a purring sound. \"Must be yummy stuff. Could I get some to take home with me?\"\n\n\"You don't want it.\" She was sitting beside my hospital bed, her arm snaking through the steel bars and resting on mine. \"May be good for you, but it makes your skin cold as ice. You'll never get a date.\"\n\n\"Might be better for everyone.\" Rachel was also there\u2014really, truly there. I don't know how she'd managed to slip away from her keepers, but I was grateful they'd allowed it. \"Help me out, Rache\u2014have I given you a hug yet?\"\n\n\"Yes, but here's another.\" She leaned across the bed and gave me a squeeze I could feel through the sheets. God, but that was good. What had I done to merit this attention, this warming affection? It reminded me how pathetically I'd abused their friendship, lying, hiding, them trying their best to help me while I acted as if my only friend was that revolting bottle. I didn't deserve them. \"I love you, Susan.\"\n\nDamn it if I didn't start to cry. Must be the medication. The tears just welled up in my eyes and there wasn't a thing I could do to stop them. I choked, couldn't speak. I was going to start earning this friendship, this affection. I had to do better, for them if not for me.\n\n\"Love you, too,\" I snuffled, wiping water from my eyes. \"So this is real, right?\"\n\nRachel and Lisa exchanged a look. \"As far as we know.\"\n\n\"And you're real. This is actually happening.\"\n\n\"Are we delving into existentialism here?\" Lisa asked. \"Because if so, I need to go home and reread my Kierkegaard. Then I'm sure we could have a very deep and profound metaphysical conversation.\"\n\n\"Please don't.\" I squeezed Rachel's arms. \"Did I mention that you look great?\" Very fresh-faced and healthy\u2014the picture of an all-American teenage girl. She was wearing a little makeup, which she'd never done before. But I had to admit it looked good on her. She was dressed well, too, in a skirt and some kind of fancy pastel T-shirt. The jeans I was accustomed to seeing her in didn't show off her legs, which were truly excellent legs. And it appeared that Mrs. Shepherd, unlike myself, knew the current location of her ironing board.\n\nRachel laughed, obviously pleased. \"I think it's the basketball. I've been getting lots of exercise.\"\n\n\"Heard you were quite the defensive player.\"\n\n\"Yeah, I do the pick. I mean, they give me that because I can't shoot. But maybe next year . . .\"\n\n\"I have no doubt.\" I gave her another squeeze. \"You can do anything you want to do. You know that. Why don't we get together, soon as the warden releases me. Maybe even this weekend.\"\n\nShe winced. \"This weekend?\"\n\n\"Uh-oh. Big game?\"\n\n\"Actually . . . it's the science fair.\"\n\n\"You're in the science fair?\"\n\n\"Yes!\" she said, bubbling with enthusiasm. \"I made this cool automated mini-cyclotron thing that demonstrates the principle of torque. Do you know what torque is?\"\n\n\"Will you think less of me if I say no?\"\n\n\" 'Course not.\"\n\n\"No.\"\n\n\"Well, then. You need to come see my exhibit.\"\n\n\"Engineering . . . that's Mr. Shepherd's field, isn't it?\"\n\n\"Yeah, before he retired. He's been a big help. I mean, it's my work. But he was able to, you know, give me some assistance. . . .\"\n\nThat I couldn't have given you in a million years.\n\nThere was a knock on the door. Patrick poked his head in.\n\n\"Come on,\" I said. \"We're having a party.\"\n\nHe did, and Chief O'Bannon came in behind him\u2014with Darcy. Darcy hung back and didn't say a word. But he seemed pleased to see me. And I was pleased to see them.\n\nAnd then I remembered the pictures, which Patrick and the Chief had seen\u2014I could tell just by looking at them. And I wished I could crawl into a coffin and close the lid behind me.\n\nPatrick gently laid his hand on my shoulder.\n\n\"Are you feeling any stronger?\" he asked.\n\nThere were times when I wished I didn't have this empathic gift. \"Oh, God, you want to talk business, don't you?\"\n\nHe squirmed slightly. \"If you're up to it.\"\n\nLisa jumped to her feet. \"If you guys are going to start talking serial killer, I'm out of here.\" She looked at Rachel. \"Give you a ride home?\"\n\nRachel nodded. I waved at Lisa on her way out. \"Call me tonight.\"\n\n\"Okay. It'll be late. I've got a date.\"\n\nOf course she did. \"Don't kiss anything I wouldn't.\"\n\n\"That\u2014\" She stopped, made an erasing gesture with her hands. \"Too easy.\"\n\nAfter they were gone, Patrick began his subtle probing. Apparently he had been chosen as point man. Darcy and the chief sat quietly behind him.\n\n\"I know no one has specifically asked you about what happened yet,\" he said.\n\n\"True.\"\n\n\"And I'm sure it's the last thing in the world you want to talk about.\"\n\n\"True.\"\n\n\"But the doctors say you may be here for weeks, and we just can't wait that long. What can you tell us about him?\"\n\n\"He was very weird, Patrick. Babbling. I think our theories about multiple personalities must be right. He was talking to a voice I couldn't hear, acting as if it were controlling him.\"\n\nHe pondered a moment. \"We've neither seen nor heard any trace of him since you were kidnapped. There've been no more killings or abductions. No letters or phone calls or packages.\" He smiled a little. \"We were wondering if maybe you'd killed the bastard.\"\n\n\"I wish.\"\n\nPatrick grew quiet. I knew what was on his mind. \"Susan . . . we've seen the pictures.\"\n\n\"I know,\" I said quietly.\n\n\"The media got them, too. But to their credit, they haven't run them, not in the papers or TV. But they have . . . talked about them.\"\n\nI closed my eyes. I felt even more naked, more exposed. More raped. But what did I expect? Deference from the media? Right, and a Corvette for Christmas.\n\n\"Does Rachel know?\"\n\n\"She hasn't seen the pictures, but . . . she must've heard or read something.\"\n\nAnd the custody judge who already thinks I'm unfit, too, no doubt.\n\nI looked across the room at Darcy. Had he seen those lovely glamour shots? Would he understand them if he did? Impossible to know. But when I peered into those expressionless eyes, I was sure I saw something. If not a total comprehension of how I had been compromised, then at least a knowledge that I had been hurt. And a sorrow. For me.\n\n\"I want to assure you,\" O'Bannon said firmly, \"that this will not in any way affect your consulting relationship with the LVPD. As soon as you're released, if you want to continue working, we want you.\"\n\n\"Thanks, Chief.\"\n\n\"Although I think it might be best if we took you off this case.\"\n\n\"No way in hell.\"\n\nSilence.\n\n\"Look, I wasn't drinking. I don't care what it looked like in those pictures. I wasn't drinking.\" At that time.\n\n\"Susan\u2014\"\n\n\"I'm telling you, I didn't drink!\"\n\nPatrick grinned, damn him. \"I know.\"\n\n\"You\u2014do?\"\n\n\"Blood test. Your blood alcohol was a big fat zero. If you'd been drinking, we'd have found a trace, even after all the time you spent out in the desert. He used his drug on you.\"\n\nThank God I managed to resist Edgar's little bottle of temptation. \"So I can stay on the case, right?\"\n\nI could see O'Bannon wasn't happy, but he wasn't going to argue with me. While I was stretched out in a hospital bed with an IV in my arm, I had the upper hand. Momentarily.\n\nDarcy was the one who broke the silence. \"Why didn't they bring you ice cream? When you're in the hospital, they're supposed to bring you ice cream.\"\n\nI couldn't help but grin. \"As soon as I get loose of this joint, Darcy, you and I are going on a custard binge. We need to make up for all those potential Very Excellent Days I missed.\"\n\n\"Is there anything more you can tell us?\" Patrick asked. \"Did you see his face?\"\n\n\"Yes. And as it turns out, I've seen him before. But he was disguised. He's smart, Patrick. Smarter than we ever realized.\"\n\n\"I think that has become abundantly clear. It's just unfortunate that you were drugged. I wish to God we knew where he took you.\"\n\nI drew in my breath, wriggled up against my pillow. I had to seem strong for this. \"I know where he took me.\"\n\n _\"What?\"_\n\nI let the memories trickle in, unwanted as they were. The rushing of water, even when I was flat on my back on his table. We were near the dam, even then. And I saw enough of the interior to make a pretty good guess about the exterior. \"Approximately. I can find it, anyway. But I want to go with you.\"\n\n\"Susan\u2014he could still be there.\"\n\n\"No. He's smart, remember? He's gone somewhere else.\"\n\n\"But you can't be sure\u2014\"\n\n\"Oh, yes,\" I said, hoping my resolve was evident. \"I can be sure. But we still might find something of interest. Now how fast can you get me out of here?\"\n\n\"Susan, the doctors say\u2014\"\n\n\"I don't care. We have to act fast. And no, I can't give you directions. I have to go.\"\n\n\"Susan, you've been through a horrible ordeal.\"\n\nMy eyes narrowed. \"And I want to make sure Edgar never has a chance to do that to anyone else. Ever.\"\n\nThe doctors pitched a fit, but I lied through my teeth and told them I felt fine, and eventually the need to track down this maniac won out over medical prudence. They gave me some pills to help with the pain and a few hours later I was in a car with Patrick trolling around the dam, searching for something I recognized. I knew I could find it. And I did.\n\n\"This is it,\" I said.\n\nI was certain I was right, even though I'd never seen it from the outside. It was a small cabin, a shack, really, stuck in some of the scrubbiest country you could imagine, not far from the Hoover Dam. The spindly trees and faint vegetation weren't enough to make anyone forget we were in the desert. The joint was probably intended as a weekend retreat for boat or fish fans. \"Let's go.\"\n\n\"We need a warrant,\" Patrick cautioned.\n\n\"You're a fed. Don't you carry them around in your back pocket?\"\n\n\"No. But I can send a fax via my cell phone. And my address book has the numbers of a lot of judges.\"\n\nWell, that wasn't too shabby. Granger put a phalanx of officers around the perimeter, and we waited.\n\nAn hour later we were inside.\n\nThe ground level was perfectly ordinary. Tacky furniture, no food, a dinky television. But I knew there had to be more. It didn't take me long to find the basement door. It was locked, not that it mattered.\n\nThe light switch didn't appear to work, so we had to resort to those cool pencil-thin flashlights like you see cops use on television. It was dark and dank, stereotypically basement-like. There was no wind, but I felt a chill just the same. I usually got my impressions from people, not places, but this little dungeon had a palpable ambiance. It was terrifying, threatening, oppressive. Insane.\n\n\"Maybe you should stay upstairs,\" Patrick whispered to me.\n\nNo. In truth, I still felt weak, nauseous, barely able to stand, but I wasn't going to let them shut me out. I inched forward, shining my light ahead. The more I saw of this room, the more I recognized. The warped wooden walls. The high window, probably the only source of exterior light\u2014and the passageway for the sound waves that brought me back here. The table. His goddamn table with the restraining straps. And there was a stench. A putrid, almost unbearable stench.\n\nI heard a sound, sprang around. The beam of my flashlight crisscrossed the room. Just Granger, creeping up behind me. This was a big basement, I saw now. Maybe it was just an illusion, but it seemed as if it was bigger than the house. Like it stretched on forever.\n\nThen I jumped. Way up in the air, like a human bottle rocket. Dropped the flashlight and everything. And I had practically been expecting what I found. But that isn't the same as seeing it.\n\nThere was a body hunched behind the table. A corpse. My God\u2014had she been there the whole time I'd been held down here? The whole time he'd been playing with me?\n\nIt was Fara Spencer. Her eyes were wide open, her face frozen in an expression of fear or panic or whatever her intense final horrific emotion had been. Her skin was gray and seemed stretched, barely covering the prominent bones of her chin and cheek. She was naked, with a huge blood-caked cavity in her chest. She'd been decomposing for more than a week, but you could still tell who it was. Even if you wished you couldn't.\n\nI clamped a handkerchief over my mouth. The nausea was almost overwhelming. \"Call the coroner,\" I muttered.\n\n\"Already on it,\" Granger said, and I guessed he stopped to dial his cell, and Patrick was getting a close-up look at Fara, which explains why I was the first lucky devil to see the really big surprise Edgar left for us.\n\nFor me.\n\nAt first I thought it was plowed soil. Had he been gardening down here? I wondered. Potatoes, maybe? But my first impression was wrong. There was dirt, and evidence of digging. Two spades were propped up against the far wall.\n\nNot a garden. A graveyard. A real one, this time.\n\nThe mounds went wide across the floor and deep into the background. But they weren't buried. Not entirely. As if to create a memorable tableau, he'd left parts sticking up out of the ground. A decayed arm. A rotting leg. Sometimes a face. And at this stage of decomposition, they all seemed sadly the same. Small. Young. Female. Dead. Long dead.\n\nMy God, I thought, as the aching in my gut, in my heart, intensified to unbearable proportions. Must be more than a dozen of them.\n\nWe thought Edgar had five victims. We thought he began with Helen Collier.\n\nWe were wrong.\n\nMy head became unbearably heavy. My legs began to ache, pinpricks running up and down them. I remember thinking, I ought to get to a chair. But there was no chair, and I sure as hell wasn't going back to that table. I heard Patrick scream out my name. I saw the dirty ground, the corpse-strewn soil rushing toward me.\n\nAnd then I was out.\n\nSo they finally found it, he observed, smiling to himself. The audi-tion. The warm-up act. It seemed more impressive, viewed from this height. Almost disturbing for its . . . wastefulness. But this had been the work of his previous incarnation. Not him. Another person altogether.\n\nHe had expected them two days ago, and was startled to see not only that they finally arrived, but that they had brought Susan with them. She must've insisted.\n\nShe had not been drinking. Had she resisted the temptation he'd laid before her? Had he broken her, or had she somehow managed to reassemble herself? He would have to wait quietly and watch. Proceed with the new plan, with the implementation of the secret he had been given. And when the time came, hope that Susan was ready for him.\n\nNo wonder he had found himself attracted to Susan. It was all so clear now, now that he knew everything.\n\nShe was the Vessel.\n\nHe put the binoculars back in their leather case. It would be so easy to pick them all off, one by one, leaving nothing but a few more corpses littering this potter's field. And why not? He could do anything now, anything at all.\n\nHe'd had to isolate himself these past few days, return to the texts, meditate. Commune with his totem. He eventually realized that his flaw was not so much in his actions, nor in his plan\u2014but in himself. He could not force the offerings\u2014they had to come willingly. These paltry reincarnations were woefully insufficient; something far greater was necessary to merit the meed he desired. And he had to secure a Vessel worthy of the soul with which he sought reunion.\n\nHe was a new person now, a new man with a new plan.\n\nHe was ready to Ascend.\n\nHis long days in the xeric wasteland had been fraught with temptation, but he had resisted them. His passage had been filled with torments, but he had weathered them. His last night in the Spring Mountains had been the time of his translation. He had offered his very essence, everything he had. He had gone without food, without water, re-creating the vision quest that first revealed his true destiny. He'd stripped and pounded himself with sand, abused himself with the cactus flower. He'd bled and he'd wept. And when at last he'd fallen down on the rocky crag, exhausted beyond reckoning, he believed he had failed.\n\nThe truth had come to him in a dream, as did all knowledge of that other blessed world.\n\n ** _\"You have done well,\"_** said the voice, the one that could not be ignored. **_\"You have pleased us, and so you shall become one with us. The time of your Ascension is at hand.\"_** And so the voice which he had once heard in his head became his own. He became the totem. And the totem was he.\n\nHe looked down now upon those pitiful fools, scurrying about like infantile ants. He laughed, and even his laughter was filled with power. He was the mountain now, and they but grains of sand, a part of him, but not of him, in his world, but not of his world. He was invulnerable, indomitable. He had slipped beyond the boundaries of time and space\u2014 _By a route obscure and lonely, \/ Haunted by ill angels only, \/ Where an Eidolon, named Night, \/ On a black throne reigns upright, \/ I have reached these lands but newly . . . \/ From a wild weird clime that lieth, sublime . . . \/ Out of Space, out of Time . . ._\n\nHe stretched his arms toward the sky, letting the stardust settle all around him, feeling at home and at one with the cosmos. I am larger than death, he knew, and greater.\n\nThe man I once was is no more.\n\n **I am the Raven.**\n**CHAPTER TWENTY-SIX**\n\nAfter I came to, I stayed outside the basement while the crime techs did their work. There was nothing I could contribute at this stage. Better to let the experts work unimpeded. I took another pill, rested in the backseat of Patrick's car. I still felt drawn, unsteady. And I'd be lying if I didn't admit I was embarrassed. Fainting was so amateurish, and worse, so . . . girlish. Even if I did just get out of the hospital. It was exactly what the old-guard grunts expected someone like me to do.\n\nAfter another hour or so, I got bored and slowly made my way inside. I saw Amelia Escavez outside the house. She had a football-sized metal frame on the ground and was pouring plaster inside it.\n\n\"Another tire track?\"\n\n\"Footprint. Wanna see?\"\n\nI did. \"Think it's him?\"\n\n\"A definite possibility. There are several of them around the place.\"\n\n\"Anything that might help us find him? A distinctive tread, maybe?\"\n\n\"I don't think so. But it'll be good for confirmation if you do catch him.\" She quickly corrected herself. \"When. I mean\u2014after\u2014\"\n\n\"I know what you mean.\"\n\nI entered the house. Crime lab guys in coveralls were working over the shack, upstairs and down, hoping against hope for any trace of a clue that might tell us where this man was now. Using something called gentian violet, which stained skin cells left behind on adhesive surfaces, they'd managed to lift prints off a piece of masking tape. But since Edgar didn't appear to have a record, that wouldn't get us far. Ditto for the hair and fiber traces. As always, Edgar hadn't given us anything that would help us find him. Didn't even have the courtesy to leave a forwarding address.\n\nWhen I got to the basement, I found Darcy hunched over the remains of Fara Spencer. Now that was a bizarre sight. Here was a kid so innocent, so gentle, he literally wouldn't step on a spider. Terrified of puppy dogs. But he had no difficulty working around a corpse. Of course, the corpse could do him no harm now, but that wouldn't comfort most people. Only Darcy's brain was free from those irrational emotional associations we have about the dead.\n\n\"Did you ever eat bugs?\" he said when he saw me approach.\n\nI don't know. Maybe it was the combination of that horrible corpse\u2014which was crawling with bugs\u2014and the suggestion of eating them that made me certain I was going to heave, the only thing that would be even more embarrassing than fainting. I placed a hand against the wall to steady myself and willed my stomach to behave. Most importantly, I kept my eyes locked on Darcy\u2014not the corpse he was scrutinizing, and not the field of corpses that lay beyond.\n\n\"No, my tastes run more to meat and potatoes.\"\n\n\"Did you know that some people in some countries eat bugs all the time? I don't like eating bugs. When I was in grade school, some of the other kids told me to eat bugs and I did. But then they laughed at me.\"\n\nI'd wager he got a lot of that in school. \"I don't blame you for not liking bugs.\"\n\n\"But I do. Bugs are cool. Except for spiders. Just not for eating.\" He grinned guilelessly.\n\nWell, of course\u2014what little boy didn't like bugs?\n\n\"I used to collect them, but my dad made me throw them all out.\"\n\n\"Spoilsport.\"\n\n\"Then I started reading about them in the library. Did you know that blowflies love dead bodies?\"\n\n\"I, um, think I heard that mentioned at the academy.\"\n\n\"They love to lay their eggs in dead people. This body has bugs all over it.\"\n\n\"Yup.\" I didn't look.\n\n\"Do you know how long it takes a blowfly to go from egg to larva to pupa to adult?\"\n\n\"Oh, the answer's on the tip of my tongue. . . .\"\n\n\"When it goes from egg to first instar larva, it's about this big.\" He held his fingers only a few millimeters apart. \"Then it sheds its skin and goes to second instar larva\u2014this big.\" The fingers widened. \"Then it sheds its skin again and does the third instar.\" His fingers were more than ten millimeters apart now. \"After that, it becomes an adult.\"\n\n\"That is so cool. You know, I should go to the library more often.\"\n\nDarcy pointed at the squirmy, whitish bits crawling out of Fara's nose. \"Those are third instar. She died ten days ago.\"\n\nI did a double take. \"But how could you possibly\u2014\"\n\n\"It takes eight days to get to the third instar.\"\n\n\"But you can't know when it was born.\"\n\n\"Blowflies like to lay their eggs in dead people two days after they die. Eight days later, these bugs are third instar. So she died ten days ago. Right?\"\n\nThe things that emerged from that wild gallimaufry brain of his. I wondered if maybe he was just talking through his hat, trying to impress me. But I really didn't think he possessed the slightest instinct for deceit.\n\n\"He's right.\"\n\nI turned and found Jodie Nida, the coroner's assistant, standing behind me. \"You know the five stages of decomposition?\"\n\n\"Um, gross, grosser, more grosser . . .\"\n\n\"No. Initial decay, putrefaction, black putrefaction, butyric putrefaction, and dry decay. Each is associated with a different type of insect infestation. Your young associate was describing the putrefaction stage\u2014that's when blowflies like to join the party. The gas formed by bodily organisms causes the body to swell. And smell. Blowflies groove on that. Which is why the body has those larvae crawling in and out of every available orifice.\"\n\n\"Except her mouth,\" Darcy said, not looking up.\n\nJodie examined the body. Bugs all over the nose, the eyes, the ears, the cavity in the middle of her chest. But not the mouth. \"You're right, kid. Guess they didn't like her lipstick.\"\n\nDarcy tilted his head to one side. \"Which do you think would be grosser\u2014her lipstick or that big hole in the middle of her chest?\"\n\nJodie chuckled. \"Blowflies have different tastes than you and me.\"\n\n\"I had this teacher once, Miss Overton, who tried to kiss me on the forehead every day. She wore lots of lipstick. I thought it was yucky.\"\n\nWe all laughed, which was pretty amazing, given the circumstances and surroundings. Here we were, standing in a field of corpses. And what does Darcy think is gross? Lipstick smooches.\n\nAnd you know what? He's right.\n\n\"Twenty-two corpses, total,\" Patrick announced.\n\nMy eyes closed. \"Jesus God. All young girls?\"\n\nHe nodded. \"Brace yourself. You're about to be inundated with white shirts.\" So he knew how respectfully we locals spoke about his brethren. \"No question now but that this case is federal. And major.\"\n\nJust as well, much as I hated to admit it. It wasn't as if we were closing in on him. \"Anything new from forensics?\"\n\n\"No. But they have instructions to copy us on all reports. And there will be a lot of them.\" He looked at me, and his eyes seemed to soften. \"I'm thinking we need to spend the night together.\"\n\nCould he possibly mean what he was saying? After what happened to me? \"Are you saying\u2014\"\n\n\"Pull an all-nighter. Like we were in college. Rework the profile from top to bottom, incorporating all this new information.\"\n\nI stared at him. \"That won't take all night.\"\n\nHe looked back at me. \"Then we'll have to think of something else to do.\"\n\nMy cell phone rang. I opened it up, not even thinking.\n\n\"Hello, my dear. Glad you made it home all right.\"\n\nI tensed up, my back rigid. Bugs were crawling all over my skin, sending prickles of fear coursing through me. I could never mistake, never forget, that voice.\n\nI mouthed to Patrick, \"It's him.\"\n\n\"I made it, all right. No thanks to you.\"\n\n\"Susan, you wound me. Did I hurt you?\"\n\n\"Damn straight.\" I realized this probably wasn't the psychologically soundest technique for extracting information, but I couldn't help myself. I hated the bastard. I hated him. I could never forgive what he did to me.\n\n\"I caused you no physical injury. And I gave them a fair clue so they could find you.\"\n\n\"So what do you want? A medal?\"\n\n\"You.\"\n\nMy heart stopped. I almost couldn't speak. \"You\u2014had me.\"\n\n\"Not your body. Your mind. I want you to see what I see. I want you to join the enlightened. The name Pulaski is etched on the rolls of Dream-Land. You are to be the Vessel.\"\n\n\"What about the other girls? The twenty-two we found in your basement. Were their names etched, too?\"\n\n\"I haven't given up on you, Susan. You're very special to me.\" He paused. \"I slept with you, Susan. Did you know that?\"\n\n\"I\u2014I\u2014\" The gorge rose in my throat. I felt physically ill.\n\n\"While you were sleeping. I held you close, pressed up tight, your naked body against mine.\"\n\n\"You\u2014\"\n\n\"It was meant to be, Susan. I knew that. Every time I touched you. Our destinies are intertwined.\"\n\nI screamed into the phone. \"You had no right to\u2014\"\n\n\"I did what had to be done. Just as I do now. You will come to understand this, Susan. Soon. The Day of Ascension is almost upon us.\"\n\n\"What's going to happen? What are you planning?\"\n\nHe sounded rapturous. \"Something immense. Something spectacular.\"\n\n\"Could you be a little more specific?\"\n\n\"It starts tonight. The new offerings will be chosen. And it ends as the prophet would've wanted it.\"\n\n\"What? What are you talking about?\"\n\n\"The end of the world, Susan. The end of this world. The start of the world to come.\"\n\nAnd on that note, he hung up.\n\n\"We have to put our plans on hold,\" I told Patrick. \"I mean, for the\u2014the\u2014you know.\" I stuttered like a stupid schoolgirl. \"We should go on patrol. Even if it's hopeless and futile. We should try. Tell Granger to get every man he has out on the street.\"\n\n\"You think Edgar is going to do something tonight.\"\n\nI felt a chill spread through my entire body. \"I'm certain of it. I don't know what. But he's starting something. Something new, something horrible. Something worse than anything he's done before.\"\n\nJudy and JJ and Tiffany waited until they were sure no one was watching, then tiptoed past the PARDON THE INCONVENIENCE cones and slipped behind the door.\n\n\"I can't believe we're doing this!\" JJ squealed.\n\n\"Me neither,\" Tiffany said breathlessly. \"Wait till we tell the rest of the squad. They'll die!\"\n\nThe three girls clung to the shadows draping the walls, then crept through the cobwebs and pumpkins and papier-m\u00e2ch\u00e9 ghosts.\n\n\"Do you think he's been here?\" Judy whispered.\n\n\"You know he has,\" JJ replied. \"I mean, how could he not? That's where they found the first one. It's probably why they've shut it down.\"\n\n\"Shhhh,\" Tiffany said. \"We can't stay long. Mrs. Cross will miss us. I just wanted a little souvenir. Maybe get my picture taken in the graveyard.\"\n\nThey were wearing matching uniforms, V-neck sweaters and short pleated skirts, both in orange and black. They were all three teenagers, all three blond, all decked out in makeup and sports bras.\n\n\"All right,\" Tiffany said, passing a palm-size metallic object to JJ. \"Here's my Advantix. Take me on the porch with the graveyard in the background. I'm going to try to look scared. Does this look scared?\"\n\n\"More like you're having an orgasm.\"\n\nTiffany knocked her on the shoulder. \"You whoredog.\"\n\n\"I'm not a whoredog.\"\n\n\"Are.\"\n\nNot much of the haunted house fa\u00e7ade was left. The hotel appeared to be in the process of creating a new decorating scheme.\n\n\"Stop!\" she hissed. \"Did you hear something?\"\n\nThey sneaked back into the shadows, stopped, listened intently. JJ wished she had thought to bring a flashlight as well as a camera. The atmospheric darkness, while useful for avoiding detection, became oppressive when she suspected someone else was in the room.\n\nSuddenly, just at the edge of her peripheral vision, Tiffany saw a flicker of movement.\n\n\"Over there!\" she hissed.\n\nThe others turned just in time to see the approach of a slight dark figure, hair wild, arms waving, running toward them. \"Nevermore!\" he bellowed.\n\nThe girls screamed. \"It's him! It's him!\"\n\nOne after the other they ran toward the front door\u2014only to find it securely locked. They pounded with their fists, but it would not budge, and no one came to their aid.\n\nThey could see him more clearly now. He was wearing a black waistcoat with a ribbon tie. He had a small mustache and a furrowed brow and eyes that peered at them like daggers. He spoke again, an evil smile playing on his lips. \" 'Once upon a midnight dreary, while I pondered weak and weary . . .' \"\n\n\"Help! Please! Someone!\"\n\nThe dark figure held his arms over his head like a monster and shouted: _\"Boo!\"_\n\nAnd then he began to laugh.\n\nA moment or two later, the girls stopped pounding. They stared at the formerly menacing figure, now prostrate on the floor, giggling at them.\n\n\"S-S-So\u2014you\u2014thought\u2014\" It was difficult for him to speak, he was laughing so hard. \"You thought I was the guy?\"\n\nJJ felt foolish and annoyed. \"Well, you look like him.\"\n\n\"And how would you know what he looks like?\"\n\n\"You've got that whole Poe thing going on.\"\n\n\"Ah. That would be a clever disguise.\" He pushed himself up off the floor, brushing the seat of his pants. \"My name is Elliot Barnes. I'm an actor. I used to work here till they shut down the Poe display. And you are . . . ?\"\n\nWith some reluctance, JJ introduced herself and her two friends. \"We're here for a cheerleading competition. We had some spare time, and we're not old enough to gamble, so we thought . . .\"\n\n\"That you'd come see where the first body was found? The parking lot where Dr. Spencer was kidnapped?\" He shook his head. \"Rather morbid bit of spectating.\"\n\n\"But this story is huge! And it started right here,\" Tiffany said. \"It's all they talk about on the news!\"\n\n\"And isn't that a sorry statement. On a slow news week a celebrity hangnail could command national attention.\"\n\nJJ cleared her throat. \"Can you tell us why they're shutting down this room? I think it's kinda cool.\"\n\n\"Well, after the body was found here, it seemed in bad taste, even by Vegas standards. There was some thought of simply eliminating the graveyard, but now, with all the attention this case has been getting, Poe has lost his fun factor. We're going to remodel the room. This haunted house will become the cathedral Notre Dame de Paris. Should be ready by Halloween.\"\n\n\"Bitchin'.\"\n\nThe man stopped, tilted his head, looked at her strangely. \"Would you mind saying that again?\"\n\nJJ gave him a look. \"Why?\"\n\n\"Humor me.\"\n\n\"Oooo-kayyy.\" She glanced at her friends and shrugged. \"Bitchin'.\"\n\n\"Perfect delivery.\" He snapped his fingers. \"Has anyone ever told you that you resemble Britney Spears?\"\n\n\"Me?\" She pressed her hand against her chest. \"No.\"\n\n\"I'm surprised. Are you familiar with our Legends show? It's an impersonator gig. They're all the rage right now\u2014hotter than the buffets. Everybody's got one, but ours is the best. We've been looking for a Britney Spears.\"\n\n\"Serious?\"\n\n\"Dead. If you're interested, I could set up an audition.\"\n\nShe hesitated. \"I can't sing all that well.\"\n\n\"You don't have to. We play records\u2014all you do is move your lips. Well, for a Britney Spears show, I suppose you move everything. But you don't sing.\"\n\n\"I can dance,\" JJ said, bubbling. \"I'm a cheerleader. I know how to move.\"\n\n\"I'll bet you do.\"\n\n\"What do you think, girlfriends?\" She turned to her cohorts, then back to him. \"You got any parts for my friends?\"\n\n\"I can't guarantee anything. But Britney usually performs with backup dancers, doesn't she?\"\n\n\"Cool! What would I have to do?\"\n\n\"Just audition.\" He handed her a card. \"Here's my address. If you could come by tonight around midnight\u2014\"\n\nJJ's brow furrowed. \"Is this your place?\"\n\n\"Yes. I just moved in. Why?\"\n\nJJ stared at the card pensively. \"I don't know. . . . I'm not sure I'd feel right going to a stranger's house alone. I mean, I'm sure you're a nice guy and all, but what with . . . you know. All that's been going on.\"\n\n\"I can assure you I'm perfectly safe.\"\n\n\"Oh, yeah, sure, but\u2014\"\n\n\"I'm not really Edgar Allan Poe, you know. I just play one on TV.\"\n\nShe giggled. \"Well . . . could we all come? Together?\"\n\nHe clasped his hands together. \"I think that would be lovely. Strength in numbers, right?\" He beamed. \"I have a very good feeling about this audition. I have something marvelous to show you.\"\n\n _\"Freeze!\"_\n\nThe girls jumped. From behind the crumbling haunted house fa\u00e7ade, a man in a uniform came running toward them.\n\n\"Elliot, you're under arrest.\"\n\nThe man in the Poe getup threw up his arms. \"Damn! What are you doing here?\"\n\n\"Protecting these foolish young ladies from you.\" He grabbed the man's wrists and handcuffed him. \"Hotel security, ladies.\"\n\nJudy took two steps backward. \"But\u2014doesn't he work here?\"\n\n\"No, he doesn't, and if you don't mind my saying so, miss, you were foolish to think he did just because he's in a Poe getup. I've been listening, waiting until he solicited an illicit rendezvous with minors. This man is a known exhibitionist.\"\n\n\"A\u2014what?\"\n\n\"He had something marvelous to show you, all right. But it wouldn't have gotten you a part in a show.\"\n\nJJ gulped air. \"Oh, geez, yuck. I feel so stupid.\"\n\n\"You should. Do you girls understand that there is a sadistic killer on the loose in this town? This isn't the time to go sneaking away from your teachers to see if you can get a souvenir from a crime scene. Young women are being tortured and killed. And the way you were acting, you could well have been next on the list.\"\n\n\"I'm sorry, sir,\" Tiffany jumped in. \"We weren't thinking.\"\n\n\"Yeah,\" Judy said. \"We'll go back where we're supposed to be. We promise.\"\n\n\"As if that was worth anything.\" He pushed his arrestee toward the door. \"You just stick with me, young ladies. I'll drop him off at our holding facility. And then I will personally escort you back to your sponsor.\"\n\n\"You don't have to do that. We can just\u2014\"\n\n\"With all due respect, miss, I'm not offering you a choice. I'm in charge of security here, and the hotel doesn't need any more young corpses turning up. You're sticking with me.\"\n\nTiffany gazed at him with admiration. \"He's right. He's got a job to do. Let's go with him, girls.\"\n\nThey followed the guard out of the ballroom.\n\nThe irony, he realized, a few minutes later as he led them out to his truck, was that all the publicity about the mad Poe killer had made it not tougher but easier to obtain offerings. When people are afraid, they put their trust in authority figures. A little too much so, as it turned out.\n\nHe could hardly contain his delight. This was so perfect, and so much simpler. Why not do all three at once? Now that he knew the truth, knew all the secrets, everything was so clearer. But those cheerleader outfits would have to go. Garish colors, preposterously provocative short skirts, even matching colored underwear. They looked more like streetwalkers than schoolchildren. Shocking. And the makeup would have to come off, all of it. The jewelry. And God knows, the studs. He just hoped none of them had tattoos. That could be time-consuming. And painful.\n\nHe smiled with the sweet contentment of a man who enjoys his work, who knows that his endeavors are worthwhile. There was so much to be done. So much wonderful work to be done.\n\nThe Raven never rests.\n\n\"We're not going to find him, are we?\"\n\nIt was two in the morning, and I suppose Patrick was tired of humoring me. \"Did you think we would?\"\n\nNo. Of course not. Catch a killer before he strikes when you haven't got the who or what or where, only a deadly certainty that it will happen? Not likely. But I needed to try. If there was any chance of preventing a girl from experiencing what that man did to me, I had to try.\n\n\"We can pack it in,\" I said, trying to be charitable. \"If you want.\"\n\n\"I can take it as long as you can,\" he replied. \"I'm a fed, you know. We're invincible.\"\n\n\"I've heard that. But I've never had a chance to prove it.\"\n\n\"You came damn close the other night.\"\n\nOuch. Me and my smart mouth.\n\nI looked out the window again, searching for some basis\u2014any basis\u2014to change the subject. Barry Friedman, my favorite comic, was playing at the Excalibur. What a treat that would be. Put all this misery aside and just laugh for a while. But I knew that wasn't an option. Didn't matter where I sat\u2014I wouldn't be thinking about the jokes.\n\nPatrick's face was a study in chiaroscuro as the car oozed down the street, segueing from one bright light to the next. A handsome, strong face. One I'd never taken the time to sort out my feelings for. Oh, sure, I'd had sex with him. I think. I'd yelled at him, bossed him around, been rude as hell to him. But how did I actually feel about him? How did I feel about anything? Why didn't I know? Had the booze deadened me? Or was I just dead and using the booze to hide the ugly truth from myself?\n\nHadn't had a drink all day. Hadn't had a drink since Edgar grabbed me. Another good thing about wasting the night trolling the streets of Vegas with Patrick. No opportunity. Of course, I felt hellish, but there were extenuating circumstances. Patrick had wanted me to check myself back into the hospital, and I have to admit that I was tempted. But I couldn't let this manhunt go on without me. I was needed here.\n\nOn the left, just past Circus Circus, I saw the Transylvania, where the killer had dumped his first victim. Where he'd taken Fara Spencer. I was tempted to pull over and check the joint out. But why? Edgar was much too smart to go there again.\n\n\"Maybe it is time to call it quits,\" I said.\n\n\"Want me to come back to your place?\"\n\n\"Nah. I'm pooped.\"\n\n\"I can sleep on the couch.\"\n\n\"Not necessary. But thanks.\" A nice guy. Which no doubt explained why I still felt ambiguous toward him. God forbid I should get hooked up with someone nice.\n\nExcept that David had been nice, hadn't he? Once upon a time. Before the troubles started.\n\nMy God, David. Was that the real reason I was ditching Patrick tonight? Because I was still hung up on my dead husband? Or more accurately, because I still hadn't forgiven my dead husband?\n\nFunny how much clearer you can see things when you're sober.\n\n\"Just drop me out front,\" I told him. \"They've got so many people watching my place now, Houdini couldn't get in.\" I leaned over and kissed him on the cheek. \"Night, Patrick.\"\n\n\"Night.\"\n\nAnd I headed back to my boozeless, snoozeless, antiseptic hotel room, a yearning in my chest, my body complaining because I wouldn't give it what it wanted, my heart aching because even if I didn't know her name, I knew there was a girl out there dying tonight. One more person I had failed to save.\n\nI pressed up against the door, eyes clenched shut. So this is what life is like sober? Wonnnnnnnnderful.\n**CHAPTER TWENTY-SEVEN**\n\nYou'd think nothing on earth could be more innocent and stress-free than a stroll through the forensic lab. You don't expect screaming and shouting\u2014that happens upstairs, where we high-IQ detectives hang out. And you certainly don't expect to see your toxicology expert getting into it with the boss's son.\n\n\"Please please please please please please please please please please please,\" Darcy said, over and over. He wasn't exactly shouting. His voice was always loud. Near as I could tell, his theory was that if he didn't give his opponent a chance to argue with him, then he won the argument. An approach I have to admit I've used once or twice myself.\n\n\"Listen to me!\" Jennifer Fuentes (yes, now I knew her last name) was trying her best not to lose it. \"There's no poison!\"\n\n\"Please please please please please please please please please please please.\"\n\nJennifer was totally losing that cool detached scientist thing.\n\n\"What's going on?\" I asked.\n\n\"This guy is making me crazy!\" Jennifer said. \"The chief asked me to humor him. He didn't say I had to take orders from him. Especially not stupid ones.\"\n\nDarcy looked at me, his face brightening. \"Did you sleep well?\"\n\n\"As a matter of fact, I did. Why do you ask?\"\n\n\"Your breath.\" And then he started right back up again. \"Please please please please please please please please please please please.\"\n\n\"Would you make him stop that?\" Jennifer begged.\n\n\"Sorry. I work with him, but I don't control him.\"\n\n\"Try!\"\n\nI shrugged. \"Darcy, lay off already. Before you get carpal tongue syndrome.\"\n\nHe did. Instantly.\n\nWow. Feeling more powerful than a locomotive, I asked Jennifer, \"What does he want?\"\n\nShe rolled her eyes. \"He's got this crazy theory that Fara Spencer was poisoned.\"\n\n\"Poisoned?\" I winced. \"Darcy, I think we all know how she died. You may have noticed that big hole in her chest?\"\n\nDarcy flapped his hands. \"Did you know that one in five domestic murders are committed with poisons you can obtain without a prescription?\"\n\nNo, and I was happier not knowing. \"Any chance he's right?\" I asked Jennifer. \"I mean about the poison.\"\n\n\"None.\"\n\n\"You did a tox screen?\"\n\n\"Of course. Came up dry.\"\n\n\"But as I recall, your previous tox screens didn't detect the drug Edgar was using to paralyze his victims.\"\n\n\"That was a totally different situation. We couldn't miss the cause of death.\"\n\nYou wouldn't think. Still, Darcy had been right before. . . .\n\n\"You know, Jen,\" I said, slow and cautious, careful not to bruise any egos, \"Fara Spencer was killed a good ten days before we found her. Any chance the poison might've broken down in the body? So it wouldn't show through normal toxicology tests?\"\n\n\"Yes, it's possible, but we have no reason to believe that happened. Anyone can see how the woman died.\"\n\n\"Would you mind testing a tissue sample?\"\n\n\"For what reason?\"\n\n\"To make me happy.\" Seemed like a better answer than _Because I said so._\n\n\"This is very irregular.\"\n\n\"Story of my life.\"\n\nShe fidgeted with her rubber gloves. \"I suppose I could cut away a little something near the exposed chest. . . .\"\n\n\"Mouth,\" Darcy said.\n\n\"Huh?\" we replied in unison.\n\n\"Do you think that maybe you could take the tissue from her mouth? Because I think you should take tissue from her mouth.\"\n\n\"Why?\"\n\n\"Did you notice that there were no blowflies in her mouth? I bet blowflies don't like poison. I don't think I would like poison. Do you?\"\n\nThe toxicologist and I exchanged a look.\n\n\"Jen, do the test. I want the report on my desk ASAP.\"\n\nHe held the tip of the pendulum delicately between two fingers. He had honed the blade until it was razor-sharp, and he did not want to cut himself. He pulled it back to the height of its arc, then released it.\n\nJJ screamed.\n\n\"I suppose you know how this works,\" he said, reclining in a chair near her table. \"Everyone does. Even those who have never read the story. Have you read the story, JJ?\"\n\n\"N-N-No.\"\n\n\"Seen the film, perhaps?\"\n\nHer voice was choked and broken. Her eyes were fixed on the steel blade swinging back and forth only a few inches above her chest.\n\n\"Maybe. I\u2014I'm not sure.\"\n\n\"No matter. I just didn't want you to have any erroneous misconceptions. You see, in the original text, the narrator escapes. Oh sure, he's sliced once or twice across the breast\u2014\"\n\nJJ's face turned ashen.\n\n\"But he survives. My dear JJ\u2014\" He took her hand and squeezed it. \"You will not.\"\n\n\"W-W-Where are my clothes?\"\n\n\"Burned. Nasty provocative little things. I'm astonished any reputable high school would allow you and your raffish companions to wear them\u2014much less make them an official uniform.\"\n\n\"Did you . . . do stuff to me? While I was out?\"\n\n\"Like what?\"\n\nTears welled up in her eyes. \"Like . . . sex stuff?\"\n\n\"Would it bother you if I did?\"\n\n\"I'm only seventeen, and I'm still a virgin and\u2014\"\n\n\"Liar.\"\n\n\"I am!\"\n\n\"My dear, I can assure you I conducted a most thorough examination while you were unconscious.\" He looked at her sternly. \"You are no virgin.\"\n\nHer eyes were trained on the pendulum. \"How\u2014how high up is that thing?\"\n\n\"At present, it swings about four inches above your lovely chest, but it is descending at a rate of an inch every minute. So you see, you still have a little time to enjoy the lovely mesmeric swinging\u2014before you feel its cold blade slicing your flesh.\"\n\nHer head whipped back and forth, her face contorted with fear. \"Why are you doing this to me? I haven't done anything to you.\"\n\n\"Oh, my sweet thing. Please listen.\" He scooted his chair closer to the table\u2014though careful to stay out of the arc of the pendulum. \"I know this will be difficult for you to comprehend. So much of what we are told in life is simply . . . wrong. The emphases are put in all the wrong places. Look at you. Devoting yourself to cheering for the accomplishments of others instead making accomplishments of your own. Dressing up in that blatantly objectified costume that can serve no possible purpose other than the titillation of the dominant male hierarchy. Painting your lovely face.\"\n\nHe reached forward and stroked her cheek. \"You have fine features, my precious. Why would you smear paint all over them? Because society has taught you that your God-given looks are insufficient. In order to be attractive to men\u2014and of course that is your principal function in life\u2014you must add artificial color. It's a shame.\"\n\nJJ licked her lips. \"If\u2014if I promise not to wear makeup, would you please stop that\u2014swinging thing?\"\n\n\"I merely use this as an example of what society has done to you. Just as it has taught you that because I take lives I must be some kind of monster. Just as it has taught you that your ephemeral life here on earth is so precious you must cling to it even when it is perfectly evident that your time is coming to an end.\"\n\n\"I\u2014I don't want to die!\"\n\n\"Darling,\" he said, leaning close and whispering, \"your life on earth is over. But because of my work, because of your sacrifice, we will all be translated to a better world, a happier one. We will leave behind this earthly plane of disappointment, discontent, and disillusionment. We will usher in a Golden Age.\"\n\nShe trembled so much it was difficult for her to speak clearly. \"Is\u2014that\u2014why I'm strapped to this table?\"\n\n\"I would like to believe you have the strength to remain in position when the pendulum begins its final descent. That you would not run or attempt to save yourself. But the flesh is weak, even when the spirit is willing. And so much is at stake. I felt a few precautions were in order.\"\n\n\"Where are my friends?\" Her eyes followed the blade, back and forth, back and forth. It was so close now it never escaped her line of sight.\n\n\"They are in other rooms. Enjoying similar experiences I've devised for their delectation.\"\n\nShe stared at the blade, barely an inch away now. \"Is it going to hurt very much?\"\n\n\"Yes,\" he said, stroking her brow, \"I'm afraid it is.\" He pushed to his feet. \"It's almost time. I'll leave you alone now.\"\n\nShe quivered, then rocked hysterically, crying, wailing. And the pendulum kept swinging. She screamed hysterically. \"Stop it! Please help me! Please!\"\n\nThe pendulum swung again and this time she felt it crease her exposed flesh. She cried out. But it did not stop. Again it swung and again it cut her. A thin line of blood trickled to the surface. She cried out uncontrollably, insanely, crazed, her eyes wild with frenzy. The next pass would be the one, she knew. The next swing of the pendulum would kill her.\n\n\"Please, God! Someone! _Help me!_ \"\n\nThe pendulum descended even lower, sweeping toward her chest\u2014\n\nThen stopped.\n\nShe was so hysterical she couldn't hold still. She arched her back and twisted, flinging herself from one side to the other, straining against her bonds, as if she'd lost all sense of time or place, all reason, all sanity.\n\nAbove her, holding the pendulum barely an inch from her breast, the Raven smiled.\n**CHAPTER TWENTY-EIGHT**\n\nThe only thing more frustrating than knowing a killer is on the loose and not being able to do anything about it is knowing a killer is on the loose and not having anything to do. I was totally stymied. Waiting for reports. Waiting for lab results. Waiting for someone to give me the magic piece of information that would allow me to catch the miserable table-strapping picture-taking bastard once and for all. But that magic bullet was not forthcoming.\n\nI thumbed through the stack of information that had trickled across my desk. They still hadn't gotten a fix on who owned or had built the cabin out by the dam. Speculation was that hunters or fishers had slapped it together, maybe dug the basement to store or cure fresh kills. Edgar found it and took it over. Maybe killed the original occupants, who knows? There were few other dwellings in the area, and they had found no one who had any knowledge of who lived there. Some of the new FBI personnel working the case had managed to track down the identity of two of the girls found in Edgar's basement\u2014two out of twenty-two\u2014by comparing the physical remains against old missing-persons reports in the FBI database. They were both runaways, both last seen in small towns in northern California about six months before. Although it was difficult to make reliable determinations about bodies so decomposed, the coroner believed they had been killed first, then brought to the shack sometime afterward. The logical conclusion was that our Edgar had a previous life\u2014one in which he buzzed up and down the coast killing helpless girls, then dumped their corpses back here. All before the Poe motif fully developed.\n\nI was feeling better. Not 100 percent, not even close, but given what that bastard put me through, I was pretty damn solid. I called Rachel, but she was out. Basketball game. Seems the team was still undefeated and if they won another game would be guaranteed a spot in the play-offs. Bully for them.\n\nCalled Lisa, too, but she was not at home.\n\nFound a book on the corner of my desk, one I'd forgotten about in all the turmoil following my abduction. Edgar Allan Poe's _Eureka: A Prose-Poem._ The only Poe I hadn't read yet, as far as I knew. And weird as all get-out.\n\nI opened the book and started to read. It was hard going. Strange. Poe as writer qua astrophysicist. Lots of cosmological theorizing, but couched in unscientific, poetic language that made it extremely difficult to follow. I'd read Poe's bio\u2014he was no scientist. Why had he written this? It was like Carl Sagan on an acid trip.\n\nI had to reread a passage three times\u2014some babble about irresistibly attractive forces\u2014before I got any sense of what he was talking about. Then it occurred to me that what he was describing, an enormously powerful force in space sucking everything toward it, sounded a lot like a black hole. Did we know about those in Poe's day?\n\nThen there was the passage in the coded message Edgar sent us: _From that one Particle, as a center, let us suppose to be irradiated spherically\u2014in all directions\u2014to immeasurable but still to definite distances in the previously vacant space\u2014a certain inexpressibly great yet limited number of unimaginably yet not infinitely minute atoms._\n\nWhich, placed in context, sounded for all the world like the big-bang theory, once I read it over about six times and decoded some of the nonscientific terminology. My history of science was sketchy, but I thought that idea came later, that in Poe's era people were still mostly buying into the Adam and Eve bit. How could Poe know this?\n\nNormally, I tried to empathize with a living, breathing person, but this time, I let my mind wander into the psyche of this writer, long dead, who had penned this bizarre work. What was he trying to accomplish? And what did Edgar\u2014our Edgar\u2014get out of it? Why was this book so significant to him that he led us to it? It was baffling.\n\nUntil I started to see a weird sort of pattern emerging, a secret latticework woven between the sentences. And some disturbingly familiar terminology. Dream-Land. Ascension. Golden Age.\n\nThat was when I started to get it.\n\nI was so absorbed in my reading I didn't even notice the woman standing at the other end of my desk. She had to clear her throat, then drum her fingers.\n\n\"Thallium.\"\n\nI looked up. It was Jennifer Fuentes, the toxicologist.\n\nI squinted. \"You're saying I need Valium? Do I seem stressed?\"\n\n\"Not Valium. Thallium. A deadly poison.\"\n\nI pulled my head out of the book. \"And the reason you're saying this is . . .\"\n\n\"I found it in Fara Spencer's mouth, just like the O'Bannon kid predicted. I used a wide range of reagents for different hard-to-detect poisons. Thallium clicked. The spectrophotometer confirmed it. It had broken down, as any poison would over that period of time. So to double-check, I put the sample in a graphite tube and heated it to vaporize the poison. Put it under the blue light. Voil\u00e0. Thallium. Judging from what was left more than a week after her death, I'd say it was a significant dosage.\"\n\n\"Enough to kill her?\"\n\n\"Oh, yeah. Instantly.\"\n\n\"So he took the heart out after she was dead.\"\n\n\"I think so. Immediately thereafter, before the blood had a chance to coagulate.\"\n\n\"But she wouldn't have felt the pain.\"\n\n\"Not if she was dead.\"\n\nOf course not. He'd captured her, sure. Probably terrorized her, just like he did me. And he'd taken the heart, because that's what Poe wanted him to do, and that's what he wanted to mail to me. But he couldn't do it while she was alive. She wasn't an offering, and outside of his twisted plan for redemption, he lacked the requisite cruelty. Or at least one of his personalities did.\n\nAnd Darcy had known it all along.\n\n\"Tell me about thallium, Doc. Is it hard to get, like that voodoo zombie stuff?\"\n\nShe shook her head. \"Rat poison, most likely. Contains thallium sulphate. Half the people in Vegas probably have it in their garage, never suspecting how deadly it can be.\"\n\n\"But Edgar would know. Edgar knows everything.\"\n\n\"I feel like an idiot. I would've missed it altogether if it hadn't been for that kid.\"\n\n\"Don't feel bad, Doc. You were meant to miss it.\"\n\nJennifer walked humbly back to her office and I returned to my reading.\n\nI'm getting close to you, Edgar, I thought. I was still missing a few key pieces of information, but it was starting to fall into place, like snowflakes on a Colorado mountaintop. And the few things I didn't know yet, I knew where to find.\n\nIn this book.\n\nThere was a reason why Edgar was who he was, why he did what he did. And I was going to discover it. Before his damned Day of Ascension. Before it was too late.\n\nWho are you? Where did you come from?\n**CHAPTER TWENTY-NINE**\n\n\"Please, Nana, please. We'll be careful.\"\n\n\"No. You're too young. You children should stay near the house.\"\n\n\"We're old enough. Honest.\"\n\n\"Ernie, I've given you my answer.\"\n\n\"But Nana!\"\n\nThe twins had lived with their grandmother for almost a month before she relented, on a bright summer morning when the California air was so cool and the sun so warm even she must've found the temptation irresistible. Her small rural house backed up against a forest full of brush and hidden dangers, maple and oak and tall pines, redwoods and relicts of redwoods. Even better, not a quarter mile beyond the forest was a vast expanse of beach, a private access road to the Pacific Ocean. Nana's family had bought this choice land not far from Salinas at the turn of the century and never let it go, even though they were poor as dirt and it came to be worth a substantial sum. But what good did it do the twins if they weren't allowed to leave the house?\n\n\"I want you children where I can see you. No telling what kinds of mischief you might get into.\" She had a cat in her arms, a huge peach-colored Maine coon that stared at Ernie with eyes that never blinked. \"I know what you were up to when your parents weren't watching and I won't have any of it. You just stay put.\"\n\n\"But there's nothing to do in the yard!\"\n\n\"And what is it you think you're going to do in the forest? Don't you know that place is full of ticks? Poison ivy? Do you know what poison ivy looks like? I'll bet you don't. You get a dose of that and you'll be miserable for days. Snakes out there, too.\"\n\nErnie was unconvinced.\n\n\"Did I tell you about the wild boars? My mother saw one, just before the Lord took her home. Teeth like razors. Could eat little things like you two in a single gulp.\"\n\nBut Ernie did not relent. To him, the forest was an unexplored wonderland teeming with adventure. Even from a distance he could see its dark and foreboding corners, and the mystery made it all the more alluring.\n\n\"We could play lots of games in there. We could build forts and play hide-and-seek.\"\n\n\"You can do all that in the yard.\"\n\n\"Not good. Not like we could there.\"\n\nNana's eventual surrender was inevitable, given how both Ernie and Ginny pounded her with the relentlessness only eight-year-olds can muster. \"But just for one hour. And I want you to carry this tin whistle. You get in any trouble, meet up with a wild boar or something, you blow on it, hard. And when you get back, I'll be inspecting you for ticks.\"\n\n\"All right, Nana. We will, Nana,\" he said, throwing himself up against her and hugging her tightly.\n\n\"That's enough of that,\" she said, pushing him away.\n\n\"And what about the beach? Can we go there, too?\"\n\n\"Under no circumstances are you to go near the beach. I couldn't hear you out there even if you did blow that whistle. Now get along, before I change my mind.\"\n\n\"Yes, Nana,\" he said, already running.\n\nAnd so the revels began. They were like pagans, Ginny and Ernie, romping through the forest, worshiping each other and secret gods known to no one but themselves. They would pretend to be astronauts on another planet searching for new life-forms and alien civilizations. They would play endless games of chase. There were no other children living anywhere near them, and they never minded in the least. They were a world unto themselves.\n\nOn some afternoons, Ernie's favorite ones, after the running and chasing were done, they would sit together on their makeshift table, a huge stump of a tree that had been logged a generation before. Hidden amidst the maple and second-growth redwoods, they would tell each other everything. They had no secrets. Why would they? Each was an extension of the other. As far as they were concerned, they were two parts of the same person.\n\n\"Do you ever miss Mom and Dad?\" Ginny asked, one such afternoon.\n\n\"I dunno.\" He stretched out, sunning himself. \"Kinda sorta. You?\"\n\n\"Maybe. Sometimes.\"\n\n\"I don't miss Mom yelling all the time.\"\n\n\"No.\"\n\n\"And I don't miss Daddy's spankings. Which really weren't spankings because they weren't on my butt.\"\n\n\"He never spanked me.\"\n\n\"That's 'cause he liked you. He spanked me all the time. He just liked you.\"\n\n\"Yeah,\" she said, drawing her arms inward as she spoke, staring at the leaves. \"He sure did like me.\"\n\n\"I hated it when they acted all weird and crazy and couldn't hardly walk.\"\n\n\"Me too. But that was when Daddy liked me the most.\"\n\n\"And Nana's pretty nice. Even if she is old and kinda strange.\"\n\n\"Yeah.\"\n\nThe breeze blew a trace of honeysuckle between them, rustling the leaves and giving them both a slight chill.\n\n\"But I still miss Mom and Daddy. Sometimes,\" Ginny said quietly.\n\n\"Yeah. Me too, I guess.\"\n\nTrue to her word, when they returned from the forest each day, their grandmother performed full and thorough inspections.\n\n\"Those ticks are insidious. They dig down deep and they never let go. Strip!\"\n\nAnd she meant it. The inspection did not begin until both children were standing before her starkers. \"No underwear. Nothing.\" Now they really looked like wild animals, primordial wood nymphs, hair tangled and full of leaves, even bugs, dirt sweat-stained to their skin. Their grandmother checked each nook and cranny, every fold and orifice. The children didn't much enjoy this assessment. But it did not deter them. Next day, they were back in the forest.\n\n\"Do you want to touch it?\"\n\nHe had seen her looking at him while he went to the bathroom behind a tree. It was not the first time.\n\n\"No way. Gross.\"\n\n\"Isn't gross. It's just me.\"\n\n\"Well, I don't have anything like that.\"\n\n\"That's 'cause you're a girl.\"\n\nThey sat on the stump in silence for a few moments. He knew what she was thinking. It was always like that, and not just with her. He could tell what anyone was thinking, sometimes before they knew themselves. And he knew Ginny's mind as well as he knew his own.\n\n\"Okay,\" she said, something like fifteen minutes later, out of the blue.\n\n\"Okay what?\"\n\n\"Okay I want to touch it.\"\n\nHe considered. \"You'll have to let me touch yours.\"\n\n\"I don't have one.\"\n\n\"Well, whatever you got, I want to touch it.\"\n\n\"Okay.\"\n\nErnie dropped his shorts. And his Sears-bought Underoos, which drooped over his shoes.\n\nAnd she touched it.\n\n\"That wasn't so much,\" she remarked, then giggled. \"It's all sticking out now.\"\n\n\"That's not funny.\"\n\n\"Why's it doing that? Just 'cause I touched it?\"\n\n\"You didn't do it right,\" he said defensively. \"You're s'posed to kiss it.\"\n\n\"How would you know?\"\n\n\"I saw it in one of Dad's books. You kiss it and it makes people happy.\"\n\nShe obliged. Afterward, he did the same to her. It was weird and gooshy.\n\nLater that day, when the inspection was done, their grandmother kept Ernie after she let Ginny get dressed and go up to her room.\n\n\"I saw what you did,\" Nana said. \"You nasty little boys with your nasty little things, you're all alike. Well, I won't have it. You understand me? I won't have it.\"\n\n\"Y-Y-Yes, Nana,\" he managed to squeeze out.\n\nThe cat curled around his grandmother's ankles, staring at him with large accusing eyes.\n\n\"There will be no nastiness on my property, hear me? Your father may have gone in for that sort of thing, but I most certainly will not. The Good Lord doesn't like it. And neither do I.\"\n\n\"Okay, Nana.\"\n\n\"Nasty. Nasty, ugly boys. Here. Put this on.\" She held out a sky-blue pinafore, one of Ginny's.\n\n\"But\u2014that's a dress\u2014\"\n\n\"Put it on!\" she commanded. He obeyed. She did up the buttons in the back, tied the bow, and spun him around a few times, admiring him. \"There, that's better. Not so nasty now, are we?\"\n\n\"But Nana\u2014\"\n\n\"Since you're so interested in girls, we'll let you be one from now on.\"\n\n\"Nana\u2014\"\n\n\"Go to bed now. And leave the dress on. You are not to remove it until I give you permission. Understand?\"\n\nHe did as he was told, but the dress was scratchy and uncomfortable and he had a hard time sleeping. She made him wear it forever, days, weeks, including the night one of Nana's lady friends from town came over. Nana thought the children didn't know because the lady came after bedtime, as she had come many times before. But the dress bothered Ernie so much he couldn't sleep, and when he heard screaming, it scared him. Scared him so much he ran into Nana's bedroom.\n\n\"What the hell is this?\" the lady said, sitting up in the bed.\n\nNana leaped out of her bed, wrapping a sheet around herself. \"Nasty little boy! Nasty little boy!\"\n\n\"Nana, I was scared\u2014\"\n\n\"Did you think you might see something? Is that what you were hoping for? That you might see something nasty?\"\n\n\"No\u2014I\u2014I\u2014\"\n\n\"I tried to help you. I tried to make you not be such a dirty little boy. But I can see this dress wasn't enough. Not nearly enough.\" Enraged, she ripped it off him, her nails like claws, then ran to a sewing kit resting on her dresser.\n\n\"Nana, please\u2014\"\n\n\"My last sewing needle,\" she said, holding the silver shaft between thumb and finger. \"Shame to have to use it like this. But what's got to be done's got to be done.\"\n\nShe lowered the needle and jabbed it into his little penis.\n\nErnie screamed. He tried to squirm away, but the old woman's hand held him firmly in place. Only after an eternity had passed did she yank the needle out, leaving a trickle of clear fluid in its wake.\n\nErnie clutched himself. He fell to the floor, curled up, hoping if he squeezed hard enough it wouldn't hurt anymore.\n\n\"I don't want you playing with Ginny. Not in the forest. Not anywhere.\"\n\nHis eyes widened with horror. This was even worse, the most horrible punishment possible. \"B-B-B-But\u2014\"\n\n\"You heard me, boy.\" She held the dripping needle between her fingers. \"Do I need to tell you what will happen if you disobey me?\"\n\n\"N-N-N-No, ma'am.\"\n\nShe would not help him or allow the lady in her bed to help him. Ernie had to crawl to his bedroom, holding himself, biting back the pain. He lay on his bed for hours, not sleeping, empty, devastated, in every kind of agony. A swelling began and he didn't know what to do about it. He started to cry, and once he started, he found he could not stop. His whole body shook with the force of his tears. He did not cry so much for the aching, although that was great. He cried for Ginny. Because she was his life. And now his grandmother had taken his life away.\n\nThe next few weeks were torturous. The swelling eventually subsided, but in its wake the shaft of his penis turned an odd mottled color and bent to the left as if it were permanently broken. He was not allowed to play with Ginny or even to talk to her. He sat at the opposite end of the breakfast table and if he so much as glanced her way his grandmother made a motion toward her sewing kit. He learned to stare at his cereal bowl. At night he would lie awake in his bed, sleepless, thinking of nothing but her.\n\n\"Ernie?\"\n\nHe sat bolt upright, on that memorable night so long into their forced separation.\n\nIt was her.\n\n\"Come outside.\"\n\nIt was dark as a cave in his room. What time was it, midnight? Later?\n\n\"Hurry!\" He heard her tiny footsteps scampering down the stairs.\n\nHe followed, feeling his still broken member as he climbed out of bed, knowing all too well what he risked. But Ginny was calling, and he could not resist following her.\n\nThe floorboards creaked as he entered the hallway. He could hear his grandmother snoring; she was sound asleep. But he also knew how quickly she could rouse herself, given sufficient motivation.\n\nDown the stairs and out the door. He found Ginny by the clearing at the edge of the forest, waiting for him.\n\n\"I'm not sure we should be in the forest at night,\" he said, even though he desperately wanted to go. \"It could be dangerous. The ticks and boars and stuff. We might not see them.\"\n\n\"We're not going to the forest,\" she said, her eyes sparkling like a gift. \"We're going through it. To the beach.\"\n\nThe beach! He felt a clutching at his heart. Their grandmother had expressly forbidden it. \"Nana won't like this. If she finds out\u2014\"\n\n\"Nana won't find out.\" She took his hand. \"If she does, we'll run away together.\"\n\nShe led the way through the edge of the forest on a worn path that seemed safe even when they were traveling with only the moon for a flashlight. Perhaps fifteen minutes later, they arrived at the beach.\n\nErnie was stunned. He had seen it from the road, of course, but never like this, never this close. Even from the edge of the forest, he could feel the spray on his cheeks. He could hear the thunderous roar. It was so loud\u2014how could they not hear it back at the house? The waves crashed against the surf with a shuddering violence. The ocean seemed to go on forever and forever, receding into the horizon.\n\n\"Better build a fort fast,\" Ginny said, racing barefoot across the sand. \"You're about to be under attack.\"\n\nThey played together like they had never played before. In retrospect, they were probably not there more than an hour or two, but it seemed a glorious eternity, a Golden Age. They built forts and lobbed sandballs at one another. They fashioned castles and dug tunnels between them. They played chase and ran like the sea breeze, squeezing the sand between their toes. Ernie caught her and they both tripped; he fell down on top of her.\n\n\"Do you think I'm pretty?\" she asked as he lay against her, sand skimming her hair.\n\n\"Of course you are.\"\n\n\"I think my nose is too big.\"\n\n\"Is not. You're pretty.\"\n\n\"Then how come you never kiss me?\"\n\n\"I\u2014I didn't know if\u2014if I\u2014\"\n\n\"It's okay.\"\n\nIf she said it was okay, then so it must be. He pressed his lips against hers, hard, like he'd seen his father do it to that woman in the drugstore. It didn't feel very warm, but it made him go all tingly and stiff just the same.\n\n\"That wasn't very good, was it?\"\n\n\"Don't worry,\" she said, laughing. She pushed him away and ran free. \"You'll get better.\"\n\nAnd she was right. They came out to the beach every night that week, running and playing, hugging and kissing, happier than they had ever been before, in their kingdom by the sea.\n\nThey were always quiet when they left, always careful to brush the sand off themselves before they returned home. They gave their grandmother no cause for suspicion or alarm. Ernie was deliriously happy, and each night seemed more intense, more momentous than the one before. But he could never entirely shake his sense of foreboding. He knew that what they were doing was wrong, or at least that their grandmother would think so. Didn't that mean they would be punished? Would she return to her sewing basket? Would she hurt him again?\n\nWhat he did not realize was that his punishment, when it came, would be ever so much worse.\n\n\"Ginny, look out! It's a big one!\"\n\nThe wave crashed down on them, huge and impenetrable as a wall, knocking Ernie off his feet. \"Ginny!\"\n\nHe scrambled up, fighting the pounding of the water, but his feet sank into the sand. \"Ginny!\"\n\nShe had been digging a tunnel when last he'd seen her, burrowing through the sand, connecting his castle to hers. He called and called for her, but she did not answer him.\n\n\"Ginny!\" Another wave crashed down on him. Ernie choked on the stale salt water, coughing and spewing up something gray and bitter. He had been daydreaming, not paying attention, still tasting his sister's salty kisses on his lips. Now he was soaked, mired down in the sand. And she wasn't answering.\n\n\"Ginny!\"\n\nHe struggled to make his way to the two castles and found a half-dug tunnel between them. Ginny's feet were sticking out of the sand. Her head was buried somewhere beneath.\n\nThe tide had come in while she was burrowing. The tunnel had collapsed and she'd been buried, unable to escape.\n\nHe began digging as fast as he could, pulling out great clumps of wet sand, trying to find her head, but it was slow work for little hands. He knew every second counted. He called out her name again and again, crying into the night wind, but there was never any response.\n\nHe didn't know how long it took, a minute, twenty, he couldn't tell. He excavated her head and finally managed to roll her out of the muck. He brushed sand from her mouth, her nose, her eyes, all the while screaming out her name.\n\nHer eyes remained closed.\n\nHe opened her mouth and blew air into it like he'd seen people do on television, but she did not respond. He was scared and alone and he didn't know what to do. He raced back and forth on the beach, the death clock ticking away in his brain.\n\nHe had to get help. Grown-up help. He started running toward the house, racing recklessly through the forest. His legs and feet were cut in a dozen places but he never stopped, never for a second. What would he do when he got there? If he told Nana what he had done, what they had been doing\u2014he knew what would happen. But Ginny wasn't moving. He had to do something. He had to do something.\n\nAs soon as he arrived at the house, he called the police. They were the ones best able to help Ginny\u2014and perhaps to protect him. Nana heard him talking and came downstairs, but the police arrived before she could do anything to him.\n\nErnie found it difficult to function, to perform even the simplest tasks. He was weary and heartsore and scared and confused. He answered the policemen's questions as best he could and took them out to the beach.\n\n\"Dear God,\" the cop said when they finally arrived. \"Why didn't you call us sooner, son? We might've saved her.\"\n\nThe rest of the night was a hideous blur. There were questions and questions and questions. He was still wet and cold and miserable. And all the while, his grandmother stared at him, her eyes dark as coals and cold as night. He knew what she was thinking.\n\nAround three A.M., they took Ginny's body away. They would not let him kiss her goodbye. He would never see her again.\n\nErnie didn't know who all these people were. He'd never known Nana had any friends; only her nighttime lady friends came to the house. But at the reception after the funeral, the place was packed with strangers.\n\nNo one would talk to him, not at the funeral and not now. He knew why. Some of them thought maybe he'd done it on purpose. They thought he was a bad seed, a chip off his father's block. They blamed him not only for Ginny's death but for his grandmother's sudden decline. Everything.\n\nSomeone had brought food, a couple of casseroles and some bean salad, but he didn't eat much, even though he'd taken nothing all day. The first bite died in his throat; it seemed tasteless.\n\nThe minister was the only person there who didn't have wrinkles. He was new, maybe thirty. Ernie knew his grandmother didn't like him. Ernie didn't like him much, either. But he was the only one in the house who would talk to him.\n\n\"You mustn't blame yourself,\" Reverend Barton said. \"God called her home, that's all. She's in heaven now, with the angels. We should be so fortunate.\"\n\n\"It doesn't seem right,\" Ernie mumbled.\n\n\"It never does. We think, why did it have to be her? But remember\u2014she went to be with Jesus. That's a good thing, not bad. The Lord God moves in strange and mysterious ways. There is a plan, even if we have not yet discovered it. Evidently, God needed her more than we did.\"\n\nErnie looked up at the minister, his eyes pleading. \"What I don't understand is, why didn't God take me, too? We belong together.\"\n\n\"You will be together again one day, God willing.\"\n\n\"But why not now? I feel so\u2014awful. I never should've gone out there with her.\"\n\nReverend Barton knelt down and took the boy by the shoulders. \"It's not your fault, son. You were God's instrument. You helped Him fulfill His plan.\"\n\nThat night, she came for him.\n\n\"Ernie,\" she said, standing in the dark at his bedroom door. The cat was curled between her ankles. \"Wake up.\"\n\n\"Can't . . . ,\" he moaned, pretending he had been asleep.\n\nShe grabbed his head by the hair and jerked him upright. Ernie sputtered, wild-eyed, drool spilling from his lips.\n\nHe looked down at her other hand. She was holding the needle.\n\n\"Thought those damn fools would never leave.\"\n\n\"Please don't hurt me, Nana. It was so bad last time. Please don't do it again.\"\n\n\"You're all alike,\" she said, her eyes glistening like the silver dagger she held in her hand. \"Your father took my first little girl away from me. And you took my next one.\" She grabbed him by the collar of his pajamas, shaking him. \"Did you do anything to her before you killed her?\"\n\n\"No!\"\n\n\"You disgusting men with your disgusting little things.\" She shook him even harder. \"Tell me the truth. What did you do to her? So help me, I'll\u2014\"\n\n\"No!\" He broke away, scrambling across the bed. He dove through the doorway but miscalculated in the darkness, banging his head on the wall. He leaped to his feet, stubbing his toe in the process. She reached out just in time to grab his leg.\n\n\"Gotcha!\" She jabbed the needle into the soft underside of his foot.\n\nErnie shrieked, then tore himself away from her. He pushed ahead, but the cat raced in front of him and made him stumble. He collided with the banister, headfirst. She came after him, her teeth bared, her needle shining in the reflected moonlight.\n\n\"Please don't hurt me,\" he whimpered. \"Please don't.\"\n\n\"You'll take your punishment, Ernie. If I have to chase you to the ends of the earth. God punishes sinners. God and me.\"\n\nShe reared up before him, her needle poised like a dagger.\n\nErnie kicked her in the stomach.\n\nFor a long moment, she seemed suspended in air. He could have grabbed her hand and pulled her back onto the landing. But the needle was in that hand.\n\nHe let her fall.\n\nWhen at last her body stopped, the tumbled heap at the foot of the stairs did not stir.\n\nErnie moved quickly. He gathered together everything he wanted to save, threw it in a bag, and hid it in the forest. Then he killed the cat. He took his time about that, releasing much that had been pent up for so long, in a slow, protracted, highly gratifying dissection. Then he burned the house down.\n\nThe books he'd read had given him a good idea how to do it in such a way that it would not be obvious that he had done it, at least not to the rural cops he'd met a few days before. He left the gas burners on for a long time. He tossed a match. Then he went outside and watched it burn.\n\nThere was nothing they could do to him that wasn't going to happen already. This way, no one would ever know for certain what happened. No one but him. And Virginia.\n\nHe was the Instrument, he murmured to himself as he watched the shutters and shingles turn to ash. Now he needed to find his God.\n\nIn and out of foster homes and reform schools all his teen years, Ernie never stayed in one place for long. Soon after he left his grandmother's house he developed a stutter that plagued him until high school. The permanent deformity to his private member made gym class a nightmare and sports an impossibility. But he continued to search for some explanation, some meaning in his life, in the tragedy that had visited him. Which was what led him, during his sophomore year of college, to journey out into the desert to attempt a vision quest.\n\nHe had never seen such a desolate environment\u2014flat, barren, bleak. Heat rose from the pavement creating miniature mirages, smoothing the road ahead. A man named Ralph Studi acted as his spiritual guide and instructor. The first three days, he learned, would be spent in preparation. The last four days he would spend in the desert, alone. \"All this training will be geared toward one central objective\u2014your spiritual growth. At all times, the emphasis will be on grounding you in the Spirit. Not just absorbing but owning the lessons learned. Because the true work of the vision quest begins when we return to our people.\"\n\nThe first day on his own, out in the wilderness, he was bored to tears. The second day, he was starving\u2014and bored to tears.\n\nThe third day, he saw the Raven.\n\nHe had fallen asleep, or thought he had. His legs were aching from the stiff sedentary drain of remaining in the sacred circle for so long. He kept the fire burning, even though the air was hot and oppressive, even at night. He longed to stretch his legs, to partake of the tiny ration of water he had been permitted. Sweat dripped down the sides of his face. His eyelids closed and he drank in the heady smell of smoke and whatever was in that wood they gave him to burn. He thought he was asleep. But when the Raven spoke to him, he was wide awake.\n\nHe couldn't move. Somehow, the Raven had imparted a paralysis that he couldn't shake.\n\n ** _\"Why have you strayed from the Path?\"_**\n\nErnie didn't know what to say. He couldn't believe he was talking to a raven, but there it was, perched on his shoulder. It wasn't like other birds. It was larger, its face more expressive, more human. Its eyes terrified him.\n\n\"I\u2014I didn't know\u2014\"\n\n ** _\"The Path was shown to you. But you have not followed it.\"_**\n\n\"Well . . . I've been busy with classes and\u2014\"\n\n ** _\"There are no excuses.\"_**\n\n\"Look, just\u2014just tell me what to do. I'll do it. Really I will.\"\n\n ** _\"You know what to do.\"_**\n\n\"I don't. But if you could give me a little hint\u2014\"\n\n ** _\"Nevermore!\"_** The coal-black eyes flared, angry and intense. Ernie tried to inch away, but he was still unable to move. **_\"I have been with Virginia.\"_**\n\nErnie's hunger was supplanted by an aching in his chest, a new emptiness. \"You've seen her?\"\n\n ** _\"I have been with Virginia. And so could you.\"_**\n\n\"But how\u2014\"\n\n ** _\"In my realm, all are reunited. All are one.\"_**\n\n\"I don't know what that\u2014\"\n\n ** _\"You have the potential for greatness. You could be what I am.\"_**\n\n\"I\u2014I'll do whatever I must. Whatever you want.\"\n\nThe Raven unfolded its wings and the span was endless, a dark umbra that spread from one perimeter of his vision to the other, swallowing him. He screamed, and somehow, the act of screaming ended the visitation. He was wide awake, able to move, fully conscious that he was alone in the middle of the desert.\n\nBut he was certain he had been visited by his totem, and that the visit had meant something. What was he was being called to do?\n\nThe sweat on his brow had vanished, replaced by a fevered chill. He rubbed his hands up and down his arms, trying to warm himself. Something had changed, something inside him. He didn't know how or what exactly. But he knew he would never be the same.\n\nIt was hard to avoid the sorrowful look in the eyes of all who shook Ernie's hand as he left the questing headquarters. Word had gotten out. They all were aware that he had been visited by the Raven, the totem of death, and they all believed that meant he would soon be dead.\n\nBut Ernie knew differently. The Raven might be the totem of death, but not his own. This visitation had a different meaning.\n\nHe returned to school, ostensibly focusing on his studies, but obsessed with the Raven's words, trying to uncover the mysteries of his path. He graduated with honors and became a teaching assistant while pursuing his Master's in American Literature. But although he performed his appointed tasks with excellence, his heart was no longer in them. And his soul was in another place altogether.\n\nHe had not wanted to revisit the works of Edgar Allan Poe. He remembered those stories from his childhood as dark and gruesome, obviously the product of an unstable, demented mind. But he was TA-ing an American Lit survey course, and of course he had to grade the exams, and he couldn't do that unless he refamiliarized himself with the texts. So he sat down in his room late one evening, alone as always, with a thick volume of Poe.\n\nHe had not intended to read the entire book. A few of the major works would do, surely. He started with the poems, lovely things, sonically immaculate, if rather syrupy. But so much of it reverberated in strange and unforeseen ways.\n\n _She was a child and I was a child in that kingdom by the sea. . . ._\n\n _Such love that the winged seraphs of heaven coveted her and me. . . ._\n\nHad Poe really written that about his lost child bride? Could anyone but Ernie himself have written that?\n\nIt was only a short while before he reread Poe's great masterpiece, \"The Raven.\" Eighteen immaculately rhymed quatrains, with the Raven as the harbinger of death. Could this possibly be a coincidence?\n\n _And the raven, never flitting, still is sitting, still is sitting . . . just above my chamber door. . . ._\n\nErnie felt as if his brain had been opened wide. As if the sun had dawned for the first time. After the poems, he pored through the stories, over and over again. It was only after he had read them many times that he began to see beyond the superficial entertainments and realize that there was something important buried within them. The similarities, the points of correspondence, were too great to be coincidental. Just as the Raven had spoken to him, so it also must have spoken to Poe. He found a tantalizing clue in one of the worst of the tales, \"Ms. Found in a Bottle\": _It is evident that we are hurrying onward to some exciting knowledge\u2014some never-to-be-imparted secret, whose attainment is destruction._ Yes! And another story\u2014\"The Premature Burial\": _To conceive the horror of my sensations is, I presume, utterly impossible; yet a curiosity to penetrate the mysteries of these awful regions, predominates even over my despair, and will reconcile me to the most hideous aspect of death. The boundaries which divide Life from Death are at best shadowy and vague. Who shall say where the one ends, and where the other begins?_\n\nThat was what the Raven was trying to tell him. Death was not an ending but a translation, a passage from one borderland to another. But this wasn't a Christian fantasy, a heaven up in the clouds such as they spoke about in Sunday school. This was something real. The Dream-Land Poe described in his poems existed, and his sweet Virginia must be there. The narrator in \"Ligeia\" brought back his love. Could Ernie not do the same? But what was the mystic formula that the prophet hinted at but never described? How was this magnificent end to be accomplished? How could he enter Dream-Land? How could he make Poe's Golden Age a reality?\n\nThe answer came to him in the December of that year, not from his intensive studies, not from his work, but from a purely adventitious discovery in a small coastal California town. In a used-book store, he found an obscure Poe work, something that hadn't been in any of the anthologies. It was titled _Eureka._\n\nHe had seen references to this work in some of the biographies he had consumed, but they were all brief and dismissive. A failed effort, they called it. A hopeless mishmash. It seemed useless and irrelevant, and for that reason, and because it wasn't in any of his books anyway, he had never bothered to read it.\n\nWhen he did, it gave him the answers he had so long sought. The path.\n\nPoe believed in dreams, not just sleeping dreams but waking ones, believed they were glimpses into another world, a better world, one to which we could all be translated. He limned a memorable, if enticingly vague, portrait of this world in his poem \"Dream-Land\": _All that we see or seem is but a dream within a dream._\n\nMuch of _Eureka_ was concerned with Poe's efforts to create a new cosmology, which was rejected by contemporaries because of his lack of scientific training, and yet in retrospect, some of what Poe wrote was positively prescient. Poe solved Olbers's paradox\u2014why the sky is dark at night\u2014envisioned black holes, and was the first to describe the universe as expanding, then contracting. He proposed the big bang theory, which would not be formally discovered until seventy years later, by Alexander Friedmann, a Russian mathematician who was very fond of the works of Edgar Allan Poe.\n\nHow had Poe known, long before the scientific observations had been made that could prove it?\n\nThe Raven told him, of course.\n\nPoe believed that man was a mere extension of the Deity. He believed that as man shrinks into spatial nothingness he will regain his lost harmony and become absorbed into a perfect, mystical unity. He wrote: _The pain of the consideration that we shall lose our individual identity [in death] ceases at once when we further reflect that the process, as above described, is neither more nor less than the absorption, by each individual intelligence, of all other intelligences (that is, of the Universe) into its own. That God may be all in all, each must become God._\n\nAfter he read that passage, Ernie wept. He threw his hands up in the air, ecstatic, euphoric. He had wondered for so long, had needed to know. And all the while, the prophet had been trying to tell him.\n\nThese were not mere stories, mere poems. They were blueprints.\n\n\"I'm sorry, but I just don't get this Poe stuff. I mean, I know he was great and brilliant and all, but to me, it just seems gross.\"\n\nErnie was in his tiny TA's office, opening his mail, trying to appear interested.\n\n\"I guess, I grew up in the suburbs, you know what I'm saying? We didn't have guys sealing each other up in the basement or swinging big pendulums over their chests. And what was the deal with that story where the guy yanked out that woman's teeth? I mean, this Poe guy had issues, if you ask me.\"\n\nErnie tried to smile. She was a tiny thing, presumably eighteen, but she looked younger. She had a round face and large eyes and long, straight blond hair. He knew she had a reputation as a partyer. She wore far too much makeup. \"So what, pray tell, can I do for you, Miss Swanson?\"\n\nShe shifted from one side of the chair to the other, crossing her legs. \"See, I know the final is supposed to be like, final, but I think I didn't do so hot on it.\"\n\n\"Why do you believe that?\"\n\n\"Well . . . I never finished the reading. I mean, I'm sorry, but those stories were just so _wrong._ \"\n\n\"That's a pity.\"\n\n\"Yeah, but my sorority is counting on me to keep up the academic average, and I wondered if there wasn't some way I could . . . make it up.\"\n\n\"I'm afraid Professor Levy doesn't give second chances.\"\n\n\"Would he have to know?\"\n\n\"I could hardly offer a makeup exam without his authorization.\"\n\n\"I wasn't really thinking about another test.\" She slithered off the chair and onto her knees, just before him. \"I was hoping I could make it up . . . some other way.\"\n\n\"Miss Swanson, I'm sure I don't know. . . .\"\n\n\"Come on,\" she said, rubbing her hands up and down his pant legs. \"I know you pretend to be above it all with your big words and your old-fashioned suits. But I'll bet there's a real man in there somewhere.\"\n\n\"Miss Swanson, this\u2014this is most inappropriate.\"\n\n\"Sure?\" She unzipped the fly of his pants.\n\n\"Miss Swanson!\"\n\n\"Come on. I'll do you a favor, you do me one.\" Her hand reached inside his pants. \"And I'll bet\u2014\" She stopped, choked. \"Oh, my God! What's wrong with you?\"\n\nErnie hurriedly tucked himself back inside. \"It's nothing.\"\n\n\"Nothing! It's gross!\"\n\n\"I had an accident. When I was a child.\"\n\nShe stepped away from him, her face stricken. \"I'm sorry. I don't mean to be cruel. I was just\u2014startled.\" She waved her hand back and forth, as if fanning the air. \"Look, let's just forget it.\"\n\n\"But\u2014you said\u2014\"\n\n\"I can do a lot, but I can't do that. I'll just take the D.\"\n\n\"Nevermore?\" he whispered.\n\nHe wasn't sure how it happened. But he heard the Raven speaking to him, loud, insistent, and he saw the girl, so like Poe's own, and he felt his shame and embarrassment, and he was desperate to find the path, to know what it was he was supposed to do. And a moment later, the letter opener was jutting out from her left temple. She was dead in seconds.\n\n ** _\"You have done well,\"_** the Raven intoned. **_\"You have begun your journey. But there is still much to be done. Much to be discovered.\"_**\n\nAnd so Ernie quit his position at the college and trolled up and down the coast of California, through Montana, then Nevada, refining his prowess and technique as he traveled, finally making his way to Las Vegas, where the final secrets were revealed to him, and the countdown to Ascension could begin at last.\n**CHAPTER THIRTY**\n\n\"So what you're basically telling us is, Edgar is Jesus Christ?\" Granger wheezed.\n\n\"In his mind, yes.\" I was back in the classroom again, except this time it was packed beyond capacity, not only with Granger's increasingly sizable team, but with all the new FBI agents on the case, most of whom I hadn't formally met. And here I was lecturing these feds, debriefing them as if I were some kind of behavioral genius. Patrick had gracefully allowed me to take the lead, thereby ensuring that I would be kept in the loop and given a decent modicum of respect. But this case was federal now. We were still allowed to play. But they owned the sandbox.\n\n\"This stuff is all fine for the college professor crowd,\" Granger said, \"but how is it going to help us catch the guy?\"\n\n\"If you don't understand who he is, you'll never get him. You spent valuable man-hours last week having your men blanket all the S&M clubs and similar places Edgar would never dream of visiting.\"\n\n\"One of his victims worked in an S&M club!\"\n\n\"He went there because his victim of choice was there. That's no indication that he liked it. I'll bet he hated it and left as soon as possible.\"\n\n\"Excuse me.\" This came from one of the agents in the front row. \"In your opinion, will he continue to abduct only girls with given names found in the works of Poe?\"\n\n\"Frankly, no.\" I saw their looks of disappointment\u2014one of the few useful leads lost. But I had to give them the straight scoop. \"Too restrictive, now that everyone knows. He won't be able to find an Annabel this side of the Rocky Mountains. And let's not forget about his last victim\u2014there are no Faras in Poe. I think we have to assume he's over that, or that he's taking different instructions from whatever voice is talking to him now.\"\n\n\"I read your report on the _Eureka_ book,\" he commented. \"Fascinating. Do you have any idea what he might be planning to do? To bring about this Golden Age?\"\n\n\"I'm afraid I don't. Except that it will have something to do with Poe. The connection may be mostly in his mind. But there will be one.\"\n\n\"And do you have any theory about when this might happen?\"\n\n\"According to that last phone call, he's already started.\"\n\nHe scribbled something into his notepad. \"I assume someone has reviewed the missing-persons reports?\"\n\n\"Yes, but remember, this is Vegas. There were eighteen missing-persons reports filed last night. Four of them concerned teenage girls.\"\n\n\"Any likely suspects?\"\n\n\"A group of three. Wandered away from a cheerleader clinic. No one has seen them since.\"\n\n\"Three? At one time?\"\n\nI nodded grimly. \"As I said, Edgar's actions will escalate. Until his plan is completed. During the phone call, he spoke of a day of ascension\u2014when something big was going to happen, something that would change everything. I called some of the local Christian churches. They say Easter is generally considered the day of ascension. But since this is October, I doubt if that's Edgar's target date. He's planning his own ascension, on his own timetable. Like any other self-respecting savior.\"\n\nThe Feeb almost smiled. \"Does this put us in the role of Judas Iscariot?\"\n\nI returned the expression. \"I'll be happy to kiss the man on the cheek. Next time I see him.\"\n\nDuring the drive to Carson City, Darcy read police reports to me. It was funny listening to him, and not just because of that uninflected voice. His vocabulary was incredible; we never hit a word he didn't know. But his pronunciation was often far from the mark. I got the impression he had done a good deal more reading than he'd done talking. I suppose he wasn't the first person to find books more comfortable than other people. But I still liked being with him, and I know he liked being with me. And that felt good.\n\n\"Did you know there are over nine hundred missing-persons reports filed in Clark County each year?\" he asked as he shuffled between files.\n\n\"Your point being?\"\n\nHe was staring at a group of photographs. The cheerleaders. \"They seem like nice girls, don't you think? I hope the Bad Man doesn't do anything mean to them.\"\n\nPoor sweet Darcy. \"I could be wrong. But how else do you explain their disappearance?\"\n\n\"Spontaneous combustion?\"\n\n\"Seems unlikely.\"\n\n\"White slavery ring? Did you know that white slavery rings are still active in Kuwait and many Middle Eastern nations? But I don't know about Las Vegas.\"\n\n\"Let's hope that isn't it.\" What kind of books did O'Bannon have in that library? \"I can't be certain, Darcy. But my instincts tell me Edgar grabbed these girls. And I've learned to trust my instincts.\"\n\n\"Me too,\" Darcy said, surprising the hell out of me. \"You're usually right.\"\n\n\"Well, I don't know if I'd go that far.\"\n\n\"Ninety-three point six six percent of the time so far.\"\n\n\"Thanks, Spock.\" I wasn't going to ask how he'd calculated that. Or what he considered to be my mistakes.\n\nHis head tilted to one side. \"You smell good today.\"\n\n\"I do? Oh\u2014you mean no coffee breath.\"\n\n\"Uh-uh. Something else.\"\n\nAnd I guess I knew what that was, too. I'd made it through the night again without taking a drink. And I could do it again. I knew I could. I had the strength now. And the really strange thing was that I knew I was getting that strength\u2014at least in part\u2014from Darcy.\n\n\"I didn't like that funny smell. I like Susan smell better.\"\n\nGood thing I knew he was autistic. Otherwise I might have him arrested.\n\nWe didn't know that Tiffany was dead. But I still found my voice choking, my eyes tingling, throughout the interview. Was this the first time I'd done something like this since David? Or perhaps, was this the first time I'd done something like this sober since David?\n\n\"It seemed such an innocent, harmless activity,\" Mrs. Glancy said. \"Cheerleading camp. What could happen?\"\n\nIt's not your fault, I wanted to tell the woman. But it wasn't my place. \"The team sponsor says she and the others disappeared after dinner. Around nine.\"\n\n\"And that is so unlike Tiffany.\" She dabbed her eyes. She was medium-sized and of medium weight, with a pleasant face that had probably aged twenty years in the last twenty hours. She did all the talking. Her husband was a physician, and like most doctors I had encountered, words were not his best thing. He sat beside her, not speaking, barely moving. Stunned.\n\nTiffany was their only child.\n\n\"She's never run off before?\"\n\n\"Of course not,\" the mother said. \"Tiffany is a good girl. Responsible. She's on the honor roll, you know.\"\n\n\"I, um, didn't.\"\n\n\"She's not just some dumb blond cheerleader,\" her father said, speaking for the first time. Apparently this was a point he felt compelled to make. \"She had a real head on her shoulders.\"\n\n\"And so kind,\" the woman continued. \"So considerate of others.\" Her face flattened for a moment. \"I'll bet it was that JJ's idea. I never cared much for her.\"\n\n\"That was one of her friends, right?\" I checked my files. \"One of the other cheerleaders who disappeared.\"\n\n\"I often told Tiffany she should be more careful. People judge you by your friends. But you know girls that age. They don't listen. Do you have a girl that age?\"\n\n\"Niece,\" I offered.\n\n\"Oh, well then, you know. They don't listen. Not a bit. Even the smart ones.\"\n\nDarcy sat in an overstuffed chair, picking at the armrest. I knew he was uncomfortable. All this misery\u2014he absorbed it like a sponge. He might not understand emotion, but in a way, that could increase the discomfort of being around it. I just hoped he wouldn't have another breakdown. I couldn't deal with that now.\n\n\"Does she have any friends in Las Vegas?\"\n\n\"Not that I know of.\"\n\n\"Does she have an interest in gambling?\"\n\n\"Of course not.\"\n\n\"Rock and roll? There are several rock stars playing the Strip this week.\"\n\n\"She's more interested in Broadway. Show tunes, you know.\"\n\n\"Does she like to . . . dress up?\" Had to tread carefully here. But I hadn't forgotten that Helen had a secret life her mother knew nothing about. It was possible this one did, too. Possible they both made the same mistake that put them in Edgar's clutches.\n\n\"Dress up? How?\"\n\n\"Oh . . . provocatively. Sexy.\"\n\n\"My Tiffany would never do that.\"\n\nI had to push. \"Those cheerleading skirts are usually pretty short.\"\n\n\"That's entirely different. That's a sign of school spirit.\"\n\nMmmm. \"You think she'd have any interest in sex clubs?\"\n\nMrs. Glancy clutched her bosom. \"I\u2014I\u2014never\u2014!\" She looked at her husband, who was no use at all. \"Are you planning to help our Tiffany or destroy her reputation?\"\n\n\"I'm planning to find her, ma'am. And that means I need to know as much about her as possible.\"\n\nDad cleared his throat. \"I think maybe I should call Dick Conners.\"\n\nThe family lawyer, no doubt. That would gum up the works. And frankly, I didn't have the time. Not with the clock on these girls already ticking. \"Does she have any hobbies? Interests?\"\n\nThe woman was still glaring at me, but she eventually answered. \"She likes to collect Dumbo figurines. You know, the flying elephant. She must have a hundred of them. And she wants to be a policewoman.\"\n\nThat's one you didn't hear every day. \"Tiffany wants to be a cop?\"\n\n\"Yes. Especially after nine-eleven. She always has been very respectful, even worshipful, around our public servants. Heroes, she calls them. Police, firefighters. All that.\"\n\nA cheerleading policewoman. Couldn't hurt. \"Is she friendly? Outgoing? Would she talk to strangers?\"\n\n\"Very friendly. But I would hope she has the sense not to talk to someone she doesn't know. Especially in Las Vegas.\"\n\nI folded up my notepad. This was getting nowhere. Time to search the girls' room. They would protest\u2014I might even have to sit through a phone call to Dick Conners\u2014but eventually they would relent. Because whatever their faults or foibles, they wanted their little girl back. And they knew the longer she was gone, the less likely that became. As did I.\n\nIt was nice being with Susan again. She's almost like she used to be before the Bad Man took her but sometimes her hands shake and I can tell her stomach hurts and she looks like she's going to cry but her smell is better and she's back and she let me read to her while she drove the car. I wish I could drive the car but they wouldn't let me get a license and I know I could do it but not unless they let me try and maybe Susan would let me after we catch the Bad Man and her hands don't shake so much anymore. I know she likes me. I know she likes me.\n\nNext time I'm going to ask her about babies.\n\nTiffany was the strongest, as it turned out. Who would have guessed that the spoiled rich girl would be the most resilient of the threesome? Hidden depths, he supposed. Dark secrets such as the prophet often saw lurking just beneath the surface. But those depths held dangers. They had to be eliminated.\n\nHe'd taken her through the entire \"Pit and the Pendulum\" scenario, just as he had the other two. But whereas Judy and JJ had disintegrated into hysteria, Tiffany had kept hold of her senses, even after the blade gave her a few rather significant slices. She had remained defiant, even through her pain.\n\nAdditional measures were required.\n\nWhen Tiffany awoke, she found herself strapped to the table, still naked. It was amazing, he noted, how nakedness and physical discomfort increased their vulnerability. He had made the room positively frigorific, so much so that her body was covered with goose pimples.\n\n\"Good morning, Tiffany.\"\n\n\"You can't hurt me,\" she said through dry and cracked lips.\n\n\"I'm glad to hear that, my dear. It will make what I have to do next so much less trying. Are my hands cold?\"\n\nHe pressed the palms of both hands down on her abdomen. She flinched.\n\n\"I feared they might be. Hard to keep warm this time of year. The temperature is having a rather remarkable effect on your body.\"\n\n\"I don't know why you're doing this, you sick fuck, but it isn't going to work! I'll never do what you want. If you're going to rape me, then go ahead and do it. Get your filthy rocks off so I can get out of here!\"\n\n\"My dear Tiffany, you mistake my intentions altogether. And you have a mouth like a sewer.\"\n\n\"You're a disgusting little creep. I bet your thing is just as short as you are. That's probably why you have to get your thrills hurting teenage girls.\"\n\n\"My darling\u2014\"\n\n\"Where're Judy and JJ? What have you done to them?\"\n\n\"They're in another room.\"\n\n\"Are you torturing them, too?\"\n\n\"Not at all. They're being quite compliant. Only you are\u2014\"\n\n\"Then let me see them!\"\n\n\"I'm afraid that isn't possible yet. But in time\u2014\"\n\nShe twisted and strained against the straps, trying with all her might to get free.\n\nHe laid a hand on her shoulder, gently pushing her back to the table. \"You've cut yourself.\" He pointed to an abrasion across her left breast, just above the nipple.\n\n\"It was your damned pendulum, you\u2014\"\n\n\"Looks nasty. Could be infected. Needs attention.\"\n\n\"Leave me alone!\"\n\n\"Fear not, I know just the thing.\" He lifted a bucket and placed it on the edge of the table where she could just see it. \"Heavy. Needs to be stirred.\" He took a large wooden ladle and swirled it through what appeared to be a thick gray muck. \"There. That's better.\"\n\n\"What is that? What are you going to do with it? Are you going to put that on me?\"\n\n\"Of course not. This is not the salve. This is but the living environment.\" He dipped the ladle into the bucket, this time just skimming the surface.\n\nIt came back with something.\n\nHe brought the ladle around so that she could see it, letting a splotch of gray goo splash down on her neck. It was small, thin, and writhing, greenish black in color. As he held it close to her face, the putrid smell made her turn away.\n\n\"What the hell is that?\"\n\n\"Don't you know, Tiffany? It's the best thing for an infection. In the prophet's time, all the best physicians used them, a practice that has sadly fallen out of favor.\" He leaned in closer, pressing the lip of the ladle against her cheek. \"It's a leech.\"\n\n\"Get it away from me!\"\n\n\"No, no, you don't understand. It's a good little creature. It'll clean your wounds. Suck out the poison.\"\n\n\"I said, get it away!\"\n\n\"Don't be silly. You'll hurt his feelings. Now where was that wound? Oh, yes.\" He tilted the ladle until the leech slowly oozed out and plopped onto her left breast.\n\n\"Get it off me! Get it off!\"\n\n\"Don't fuss so. Let it do its work.\"\n\n\"Get it off!\" Her voice screeched, panic rising. She squirmed as the slimy creature oozed its way across her. \"Get\u2014it\u2014off!\"\n\n\"Many hardy souls such as yourself are quite resilient when it comes to physical torment or fear, yet still have a weakness. Spiders, perhaps. Loud noises. But I suspected that you might have a touch of tactile defensiveness. We all do, of course, to varying degrees. But your case might be more extreme.\" He smiled. \"Oh, look. The little beastie has found the wound. Engaging suckers.\"\n\n\"Please make it stop.\" She was sobbing, her voice bubbling, tears streaking. \"Please make it stop.\"\n\n\"Just leave it to him. He knows what's best for what ails you. Oh\u2014look! Another wound.\" With his fingertip, he traced a line up the inside of her upper thigh. \"Fortunately, I have more of these salutary animalcules.\"\n\n\"No! Not there!\"\n\nHe plopped another leech onto her leg. Tiffany writhed and shivered, thrusting herself forward and backward, right and left, trying to shake it off, to no avail.\n\n\"Is that a pimple on your face? Oh, I hate those.\"\n\n\"Please don't. Please don't.\"\n\n\"Here comes another helper.\"\n\n\"Not on my face! Please! Not on my\u2014\"\n\nHe dropped it just above her upper lip. It immediately began to slither toward her mouth.\n\nHer eyes widened with fear and helplessness. She didn't dare speak, but she bucked against the table, thrusting her hips, squirming, trying to relieve herself of the slimy creatures.\n\n\"Honestly, Tiffany, you will carry on, won't you? About three puny leeches? The way you're behaving, I might as well\u2014you know what? I think I will.\"\n\nHe tilted the bucket and dumped it, gray ooze and leeches alike, on top of her. Tiffany was deluged with the muck, in her eyes, in her mouth, every crevice and orifice.\n\nShe sputtered and spit, trying to keep it out of her mouth, but it was useless. She spat out a leech, gagging. She shuddered, unable to hold still. Her eyes were wide as balloons, her breathing a rapid-fire succession of jagged intakes, her chest heaving. She couldn't speak, but was reduced to making incoherent guttural noises, vacant and horrifying.\n\n\"How does it feel?\" he asked, truly curious. \"Having that sucking sensation all over your body, on your hands, your face, even your most private parts? Is it too awful? Or is it, as you young girls say, a turn-on?\"\n\n\"Please make it stop, please make it stop, please make it stop . . .\"\n\n\"Oh, Tiffany, my dear, you don't need me to do that. It will stop, sooner than you might imagine. You see, those leeches are more than just disgusting. They are poisonous. Instead of ridding your body of toxins, they are actually infecting you, tainting your bloodstream with a potent cyanide derivative.\" He looked at her levelly. \"You will be dead soon, Tiffany. And then you won't be able to feel the little creatures at all.\"\n\n\"Pleeeeeeeeeeeease!\" At least, that's what he thought she was saying, but her shrieks were so piercing at this point he couldn't make it out with any degree of fidelity.\n\n\"Goodbye, Tiffany. I'll let you spend your final moments in peace.\" Even as she cried and pleaded with him, he laid down the bucket, wiped his hands on a towel, and left the room.\n\nShe screamed for more than an hour before it was over.\n\n\"Who the hell is the leak?\" Patrick bellowed, slamming the front door behind him.\n\nI looked up, as did virtually everyone in the building. Outside, I could see reporters' faces jammed up against the glass like trick-or-treaters with their own perverse way of celebrating the forthcoming holiday. We were under siege. Had been all day.\n\nPatrick stomped through the aisles, pushing aside locals and feds alike. I had never seen him in such a state. \"It's one thing to leak our theories. New developments. But we don't even know for sure that Edgar took these girls!\" He pounded his fists against the staircase banister. \"For all we really know, they could be holed up at the Flamingo with their quarterback boyfriends!\"\n\nI knew what he was talking about. The morning papers had leaked the names of the three potential abductees\u2014Tiffany and Judy and JJ. While cautiously reporting that the LVPD was investigating the possibility, they strongly suggested that it was a fact\u2014that the girls were now dead and that it was all our fault.\n\n\"Is this tirade supposed to accomplish something?\" I asked when he made it to my desk.\n\n\"Excuse me very much,\" he spat back. \"You may be used to this kind of amateurism, but at the Bureau, we don't countenance leaks. They compromise the investigation!\" He headed down the stairs. \"I'm going to talk to our criminalists.\"\n\nBecause those FBI guys are so much smarter than I am? Even though they don't have the sense to take off their sunglasses when they come inside? Even though they dress like extras from _Men in Black_?\n\nI gave the desk a shove, kicked back in my chair. I shouldn't let it get to me. He was just frustrated, like everyone else connected to this case. But he doesn't have to take it out on me. Especially after all I've been through. After all that we've . . . shared.\n\nDamn. Amazing how much less sexy guys are when they're acting like assholes.\n\nTiffany did not die. The leeches were infected with a mild paralytic, enough that she might well think she was dying (and did). But not nearly enough to kill her. He had planned it that way.\n\nShe had screamed and begged for mercy till her voice was shredded and her tears were dry and there was nothing she could do but wait to die. But she did not.\n\n\"Tiffany! Still with us? Lovely.\" She was shivering, making quiet sobbing noises, her naked body covered with dried slime, leeches, and the remnants of leeches. They had sucked all over her body; her flesh was variegated with bruises and discoloration. Her eyes were cloudy and her expression was vacant, but he could see she was still there. Part of her, anyway.\n\n\"I'm sure you must be thinking horrible thoughts about me right now, thinking I've been terrible to you. If only you could understand that it is not so. Per contra, this has all been for your benefit.\"\n\nShe shuddered, trembled, but did not attempt to answer.\n\n\"I know you're probably not able to speak, so I won't expect you to hold up your end of the conversation. I'll do it for you. Look what I brought.\"\n\nHe loosened the straps just enough that she could peer up and see what he had brought with him.\n\nTwo bizarre hairy orange costumes. Orangutans.\n\n\"I knew you were anxious to see your two friends, so I brought them with me. They're quite agreeable these days. They put on their costumes without any hesitation. I hope you will, too.\" He leaned closer. \"I'm afraid I exaggerated a bit beforehand, my sweet. The leeches drugged you, but they didn't poison you. This time.\"\n\nShe stared at him, shivering, her eyes wide and lost, like a broken doll.\n\n\"No need to feel left out, my dear. I've got a costume for you, too.\"\n\nIt was like the others, except that a huge section was cut out at the bosom, with a similar cutaway at the groin.\n\nShe trembled as she spoke. \"If\u2014if\u2014if\u2014if if if if if I put it on, will you get this . . . slime off of me?\"\n\n\"My dear, you'll be able to do it yourself. I'll provide the soap and water. You can pass the day away in indulgent lavation.\"\n\nHe unstrapped her. She took the suit like an automaton, barely thinking. He handed her a towel and she wiped herself off, wiped and wiped and wiped, leaving red abrasions and in some places bleeding. She picked the leeches away, crying as she did it, from her waist, her breasts, her pubis. Some left round sucking circles, others left blood oozing from her skin. But she did not scream or misbehave. She just sobbed quietly, desperately.\n\nThen she took the suit. \"Seems . . . wet.\"\n\n\"Coated with paraffin wax. I'll tell you why in a moment. Go on now. You must be anxious to get dressed.\"\n\nShe pulled the suit over herself. Her breasts hung out through the opening at the upper torso, while the midriff cutaway exposed her genitalia.\n\nShe put it on without complaining.\n\n\"What you have to understand,\" he said gently, almost paternally, \"what it took me so long to grasp, is that it isn't enough to simply make an offering. The offerings must be true participants. Willing.\"\n\nHe took a book of matches from his pocket, struck one, then lit a small torch\u2014a club with oily rags wrapped and secured at the top.\n\n\"Hard to imagine, isn't it? That all this might have some wondrous purpose? But I can assure you that it does. One day, all humanity will give you its thanks. Your suffering will lead this troubled world to eternal bliss.\"\n\nHe smiled genially, then passed her the torch. \"Set your friends on fire, Tiffany.\"\n\nShe looked at him, peering through the eye holes in the orangutan mask.\n\n\"Don't worry. They won't resist. They only exist to please me. Set them on fire.\"\n\nHer arm twitched.\n\n\"If you refuse, of course, I'll have no choice but to put you back on the table. Perhaps it will be time to restart the pendulum. Instead of putting it across your chest, we'll let it take off an arm. Or a leg.\" His eyes narrowed. \"Or perhaps I should bring out more leeches. Would you like that, Tiffany? Shall we bring them back?\" He paused. \"Or perhaps I should give one of your friends the opportunity I now offer you. Remember\u2014you're wearing a suit, too, highly flammable. Perhaps I should ask JJ if she will light _your_ fire.\"\n\nTiffany hesitated. Her arm moved indecisively.\n\n\"What shall it be, Tiffany? Them, or you?\"\n\nShe took the torch.\n\n\"That's a dear. Finish it up now.\"\n\nShe did not cry or wail. It was almost as if sensation had fled from her, as if the idea of resistance was beyond imagining.\n\n\"Do it, Tiffany. Do it now.\"\n\nThe wax caught fire immediately. She stepped back from the flame, dropping the torch. One suit caught the other and in only a few seconds both were consumed in a blistering inferno.\n\n\"They won't suffer long. The suits will fuse to their skin and the smoke will choke them and they'll die before they experience . . . too much of the burning.\"\n\nTiffany crumpled to the floor, the orangutan suit bunched around her, her head buried beneath her hands. Her limbs were limp, as if all strength, indeed the bones themselves, had disappeared. He sensed that she wanted to cry, but no tears would come. There was almost no feeling at all, just a deadness, and a felt horror, not at him, but at herself.\n\nBlessed be the prophet. The time was at hand.\n**CHAPTER THIRTY-ONE**\n\nThe phone rang. Lisa.\n\n\"Calling to brighten my day with a report on last night's lip-lock?\"\n\n\"No. You wanna shack up together?\"\n\nI drew in my breath. \"But Lisa\u2014this is so sudden. We haven't even kissed. And I know how important that is to you.\"\n\n\"I'm serious, Suze. I've found a place. Fabulous. Big. Guy needs a housesitter. He's going to be gone for years. We could have it for next to nothing.\"\n\nI sat up. Living in hotel rooms and barren apartments was getting way stale. \"Sounds too good to be true.\"\n\n\"I haven't even told you the best part yet. It's in L.A.\"\n\nEverything seemed to go into slow motion. I drew the receiver from my face and stared at it. I noticed I was breathing deeply.\n\n\"You mean . . . leave Vegas?\"\n\n\"Well, it would be a hell of a long commute.\"\n\n\"But\u2014I'd lose my job.\"\n\n\"Technically, you already did that, sweetie. And this consulting thing can't last forever.\"\n\n\"What would I do?\"\n\n\"I don't know. But you are a trained psychologist, remember? There are a lot of things you could do that don't involve poking around corpses or getting nabbed by serial killers. Personally, I think it would do you good. To get away from all the . . . reminders.\"\n\nMy eyes went into deep focus. There was a certain truth to what she said, of course. She wasn't the first to suggest that I should leave. And if I won the custody hearing, I could take Rachel anywhere I wanted. Maybe I could go back into clinical work. Maybe get a cushy job as a corporate trainer.\n\nI wondered if this house in L.A. had palm trees. I always wanted a house with palm trees.\n\n\"I don't know, Lisa. I'll have to think about it.\"\n\n\"Okay. There's no rush. Let's talk tonight, okay, girlfriend?\"\n\nAfter she hung up, I had a hard time getting my head back where it belonged. Police work had been a part of my life for so long. It was the one thing my father and I'd had in common, and then after he was gone, something David and I had always shared. To some extent it had _been_ my life, especially after David was gone. But what had it ever done for me? Made my life a misery. Gotten me kidnapped, abused. Reviled by my colleagues. Driven me to drink.\n\nAnd there would always be room for another shrink in La-La Land, right?\n\nI might still be thinking about it if Granger hadn't burst into what I laughingly called my office. \"Are you as sick and tired of these goddamn Feebs as I am?\"\n\n\"Probably not,\" I said honestly. \"Why?\"\n\n\"They've taken over the whole damn investigation!\"\n\n\"And this surprises you? Granger\u2014it's what they _do._ \"\n\n\"Hey, wouldn't it be great if you or I could crack this case? Show up those Junior J. Edgars?\"\n\n\"Yeah. Especially if it were me.\"\n\nDarcy shuffled in behind him. He'd been acting like the department errand boy all day, couriering things from one department to the other, delivering messages, even going for coffee. Anyone else his age might've found it demeaning, but Darcy wanted to be on the premises, even when he and I weren't doing anything. And not an hour passed that he didn't find some reason to come by my desk.\n\n\"Have you been thinking about that cheerleader one, Tiffany? 'Cause I've been thinking about that one.\"\n\n\"Really? Why?\"\n\n\"Did you know that when most children run away from home, they have some kind of place in mind where they're going?\"\n\nI slowly rose out of my chair. \"Are you saying you think Tiffany was running away from home?\"\n\nMore shrugging. Staring at the carpet. \"Not exactly.\" His hands began to pump the air. \"But I don't think she'd leave unless she had someplace she wanted to go.\"\n\nHe was giving me all the bread crumbs, but I wasn't following the trail. \"We've quizzed everyone. They say she had no friends or family around here.\"\n\n\"She was interested in police work. She wanted to be a policeman.\" His chin rose. \"Lots of people wish they could be a policeman.\"\n\n\"So what are you saying? Maybe she decided to visit headquarters?\"\n\n\"Did you watch the news last night? 'Cause I watched the news last night. It was all about Edgar.\"\n\n\"Yeah, it has been for\u2014\" Wait a minute. Praise God, I was starting to see the glue. \"Tiffany would've known about the Edgar murders. And she was interested in police work.\"\n\nGranger jumped in. \"So she might've decided to do a little investigating?\"\n\n\"Come on, Granger, we see it all the time. Whenever a case gets a lot of attention. The rubberneckers turn out at the scene of the crime, at the courthouse, whatever. Some people thrive on this kind of stuff.\"\n\n\"But where would she go? That shack out by the dam? That strip joint?\"\n\nI knew the answer before he'd finished speaking, knew it with a clarity that startled even myself. \"Where the first body was found. Where Fara Spencer was killed. The Whitechapel of this whole case.\" I paused. \"The Transylvania. That's where he grabbed them. Because that's where he is.\"\n\nThe girls were in their respective stalls, performing their nine-thirty poses.\n\nThey sat on the cold, bare floor in a confined area with nothing to do, nothing to look at, cold, dirty, naked. Every waking hour he would bring them a picture, usually something torn out of a porn magazine, always a woman in some demeaning pose. He would give them whatever they needed to re-create the scene. And then he would wait.\n\nNot a word need be spoken. They knew what he wanted. And he rarely had to wait long to get it, certainly not after the first day. They knew what disobedience would bring. No food, for starters. No water. Not even a clean pan for their excrement. And quite possibly a return visit to the pendulum. Or the leeches. Or whatever else was required.\n\nThere had been no disobedience for a long time.\n\nAfter they assumed the pose, whatever it was, he snapped their picture with his Polaroid, then posted it on the wall next to them. A little something to remind them who and what they were now. What they had become.\n\nHis. They belonged to the Raven, heart and mind and soul.\n\nJudy and JJ had not been in those orangutan suits, of course, although it would've been magnificent, their ashes rising in an incandescent blaze, a magnificent incarnation of the prophet's tale of little Hop-frog's revenge. But the shock of thinking she had killed them\u2014willingly\u2014had been more than enough to break Tiffany. She had ended up even more deeply subservient than the two who had crumbled first. She was a sock puppet with his hand inside her.\n\nNow the three of them were so compliant, so eager to please him, that a picture was not even necessary. As soon as he entered the room, Tiffany began to assume a variety of poses, running through her repertoire, reenacting the photos on the wall. Anything to please him.\n\nPerfect.\n\nEverything at the hotel was proceeding apace. The Poe room was gone, The Hunchback of Notre Dame tableau was all but complete. It was not one of the prophet's works, but it would serve his purposes just the same.\n\nAll he lacked was the Vessel. Susan. Perhaps he had given her too much time, hoping that the time bomb he'd left ticking in her head would bring her to him of her own accord. One way or the other, once he had secured the Vessel, all his preparations would be complete.\n\nTiffany slithered up to him and wrapped herself around his feet. She pointed to her mouth, begging for food. Pathetic thing. He shook his head; he had not even brought a cube of sugar. Didn't matter.\n\nShe pulled up his pant leg and began to lick his ankle, purring.\n**CHAPTER THIRTY-TWO**\n\nI ripped the information out of the printer just as quickly as it emerged. For once, even my inner Luddite was glad we lived in the computer age. Once I convinced the management of the Transylvania that we should be permitted access to their records\u2014by giving them no choice whatsoever about it\u2014getting what we wanted was a relative snap. Compiling a list of all the guests who had stayed at the Transylvania since the body of Helen Collier was found was a cinch. Then we winnowed it down to a shorter list of all male guests who fit the current profile. I started with the names of men who had been staying at the hotel for a while and were still on the premises. As soon as I had the names and addresses, I transmitted them by fax to headquarters, where Madeline and Patrick ran Internet and FBI checks on them. She could also tap the DMV records and see if they owned a pickup.\n\nWe'd been at it for hours, and so far, we didn't have any suspects who fit all the parameters. But I wasn't worried. Names were still flying out of the computer. I knew we were on to something. I felt it in my heart, my bones. We were on the right track, finally.\n\nMost importantly, I felt good. Even though I had no right to, not yet. But I did. I felt strong. I felt sober. I hadn't had a drink for days, and I was dealing with it. The shakiness was fading. I didn't think about it all the time. I had something more important to occupy my brain.\n\n\"Still no match?\" Granger asked. He pushed away from the computer terminal and stretched. He'd been at it for hours. Turned out he was pretty good with those evil little machines.\n\nI didn't look up. I was scanning names, faxing, periodically talking into my cell, and chatting with Granger, all at once. I don't need a computer to multitask. \"Not yet. But he's in there somewhere. Give me some more names.\"\n\n\"That's going to take a while. We've covered Tower One, but they keep Tower Two in a separate database. It'll take a while to load.\"\n\n\"All right. No point in me standing around while you work.\" I grabbed my coat. \"I'm going to slip out for a minute. I'll be back soon.\"\n\n\"You're leaving the hotel?\"\n\n\"Right.\"\n\n\"There are about a zillion bars out there.\"\n\nMy buoyancy submerged, but only for a moment. \"I'm going to visit my niece. I am not sneaking out to get a drink, Granger.\"\n\n\"I know,\" he said.\n\nThat caught me by surprise. He did?\n\n\"But\u2014why make life difficult for yourself?\"\n\n\"You want me to wear a chastity muzzle?\"\n\nHe smirked. \"You're still a potential target. Take one of the uniforms with you. Take Berman.\"\n\n\"I outweigh Berman by fifty pounds. How's he going to stop me from doing anything?\"\n\n\"He's Church of Christ. He sees you order a drink, you'll get a lecture so harsh it might save even _your_ soul.\"\n\nAgainst my will, I found myself smiling. Why did Granger have to display these occasional flashes of human-beingness? It made it so much harder to hate him.\n\nHe'd been more than a bit worried when he saw Susan at the hotel. He had followed her discreetly, just to make sure she wasn't getting too close. Happily, she never came near the ballroom. But after she left the hotel\u2014\n\nHe had no idea what an astounding discovery he would make.\n\nHow had she managed to keep this from him so long? He had researched everything he could find about her. He'd hacked into her police file, searched the newspaper morgue, performed repeated Internet sweeps, quizzed her when she was barely conscious and unable to resist. But somehow, through it all, she had managed to withhold one detail.\n\nThere was another Pulaski. A little girl.\n\nJust the age he liked them.\n\nHe'd run a computer search through the city database and come up with a name: Rachel Pulaski. A daughter? No. If she and her deceased husband had procreated, it would have appeared in the public records. Same for any adoption. A cousin?\n\nA niece, as it turned out.\n\nHer brother, the one who died in the traffic accident. That must be the answer.\n\nBut why was the girl living with strangers? Why wasn't she with Susan? She must've lost custody, or been unable to obtain it. So she was reduced to occasional visitation.\n\nA rapid-fire synaptic flurry crackled in his head. New ideas flooded to the surface. Was this why he'd been unable to break Susan, why she had not become his willing partner like Tiffany and her friends? His quest for Susan was always marred by the fact that she was not suited to be an offering, much less the Vessel.\n\nBut Rachel was. She so perfectly, delicately, wonderfully was.\n\nHis premonition had been right. The name Pulaski would be writ in the roll call of Dream-Land. But not Susan Pulaski. _Rachel._\n\nHe must have her.\n\nOriginally, he'd been trying to reincarnate Virginia as she once was. Of course that was impossible; her flesh was dust. But like the prophet's Ligeia, her spirit could be recaptured, brought back to the earthly plane. If only he had the proper Vessel.\n\nHe would have to remove her, to condition her, and he had little time. But he was sure it could be done. And then Virginia would return to him. And together, they would leave this horrid world behind. And create a far better one.\n\nThe only thing worse than Granger acting like a human being was Granger trying to be consoling.\n\n\"It's not your fault, Susan. It was a good theory. I thought we were on to something, too. But even the best theories don't pan out sometimes.\"\n\n\"He's here. I know he's here.\"\n\nHe actually laid a hand on my shoulder. And the worst of it was, I let him. \"We went through all the records, Susan. Twice. And we didn't come up with anything.\"\n\nI pressed my palms against my forehead, running every scrap of Edgar-data through my brain for the millionth time or so. \"We must have something wrong. In the profile. The description. Something.\"\n\n\"Susan, you've looked at everyone who has stayed here in the last month who even remotely fits your profile. You came up with zip.\"\n\nTruth hurts. He was right. I'd played my best hand and come up short. The review of the Transylvania's guests had yielded nothing.\n\nWhere was all my buoyancy now? All that blinding self-confidence? The girl who was going to catch the bad guy and never drink again? Where had she gone? Now when I looked in the mirror, I just saw a big placard reading LOSER. LOSER\u2014AND DRUNKARD.\n\nMy wrist throbbed.\n\nGranger was shuffling papers, obviously making moves to get the hell out of this tiny hotel office. \"You look beat. Why don't you let me drive you home?\"\n\nSo I won't stop somewhere and drown my sorrows in alcohol? \"Sure. Thanks. Darce?\"\n\nAfter he'd done everything he could back at headquarters, Darcy had joined us here at the hotel. I don't know why. But I made sure he went over every name, every bio, every scrap of information I had, just in case I missed something. When your PC fails you, put a human computer on the case, right? He'd stared at those lists till his eyes watered.\n\n\"Did you know Einstein wrote his Special Paper on Relativity three times before he realized that space was curved? That was what made the whole thing make sense.\"\n\nI gave him a tired grin. \"I'm much too feeble to grasp Einsteinian physics at the moment. Or any other moment, actually.\"\n\n\"Can I go over the lists again?\"\n\n\"No, Darcy,\" I said, clapping him on the back. \"We're all going home now.\"\n\nThe fax machine pinged.\n\n\"Did Madeline have anything else to send us?\" I asked.\n\nGranger shook his head. \"Madeline has gone home.\"\n\nThat was intriguing enough to keep me by the machine a few seconds longer. And halfway through the cover sheet, I had an even better reason.\n\n\"It's from Edgar.\"\n\nHow did he know we were here? I took the sheet and stared at it. Another coded message. But this time it was all ones and zeros.\n\n\"He's really taken this multiple-substitution code gimmick to the outer limit.\"\n\nDarcy snatched it from me. \"I think that this must be binary code. Do you think that this is binary code?\"\n\n\"What, like computer talk?\"\n\n\"Can I use this please?\" He was already scooting in front of the hotel's PC.\n\n\"Sure. Ain't mine.\"\n\nHe tapped information into a black screen on the computer, fingers flying faster than I could follow. It all looked like gibberish to me.\n\n\"Hey,\" I said, \"this probably won't be important. He wants to brag, impress me, maybe scare us a little. But he won't give us anything we can use to stop him. And I'm not going to stay up all night so I can read a quote from Edgar Allan Poe's grocery list.\"\n\n\"I think that maybe this is a hyperlink,\" Darcy said, not that it meant anything to me. \"The code is much easier than the last one, if you know COBOL.\"\n\n\"Well, that's nice, but it could still take hours and\u2014\"\n\n\"Got it.\" Darcy clicked the Enter button a few more times. A Web browser came up, and a few moments later I saw the hourglass symbol that told me it was traveling to a new destination.\n\n\"You think Edgar has his own Web site?\" Granger asked.\n\n\"I wouldn't be surprised. There are plenty of public-access servers that allow anonymous uploads. But what would he post? Poe's Greatest Hits? MP3s for the Golden Age? Photos of his vic\u2014\"\n\nMy tongue froze in my mouth. The graphic image in the Web browser had begun to resolve.\n\nIt was a photograph, presumably taken from a distance with a digital camera. I had no problem recognizing one of the people in the photo. It was me. Didn't have to work much to identify the other person, either. The one I was talking to.\n\nIt was Rachel. He knew about Rachel.\n\n\"Oh, God,\" I said as a cold sweat broke out all over my body. \"Oh, my God.\"\n\n\"I'm calling for backup,\" Granger said, already dialing his cell. \"What's her address?\"\n\n\"Oh, God,\" I repeated uselessly. Rachel. All I had left\u2014\n\n\"Susan! Give me her address!\"\n\nAnd I did. Pulling myself together as best I could, I grabbed my car keys and raced for the front door. Please, God, don't let us be too late. Please, don't let us be too late.\n\nBut my own words haunted me, even as I raced out the door and into the stifling desert night air.\n\nHe wouldn't give us anything we could use to stop him.\n\nRachel raised the window and leaned out into the cool night air.\n\nNot a creature was stirring, as the poem went. Excellent.\n\nIt was drizzling outside. The white trellis attached to the front of the house was slick. She would have to be careful.\n\nShe hoisted herself through the window, flipped her feet around, then slowly descended onto the trellis. The Shepherds were nice folks, but they had to be crazy not to see how easy it was to get out of this room. Just too innocent to consider the possibilities, she supposed. Or perhaps it was some sort of test.\n\nDidn't matter, she thought, as she dropped like Spider-Man down the side of the house. At first, she'd thought the Shepherds were pathetic, hilarious, ridiculous. But over time, she had sort of gotten used to the regularity of life with them. She had found new friends, new interests. It wasn't all bad, really.\n\nBut her first loyalty was to Susan. Had to be. She was family. And she knew how much Susan loved her. Knew how much Susan counted on her, needed her if she was to stay sane. When she'd visited earlier today, Rachel could see how edgy she was. Then when she got the call from a co-worker telling her how much Susan had fallen apart, how desperate she was\u2014Rachel knew she had to go to her.\n\nShe hadn't packed much, but then, she didn't have much to pack. She'd thrown some clothes in a backpack and scraped together all the money she had\u2014less than twenty dollars. It would be enough. She'd catch a city bus to Susan's neighborhood, then hike the rest of the way to her hotel room. Simple.\n\nShe checked carefully in all directions before she emerged from the concealing shadows of the house. No signs of life, not even a car in the distance. She made a break for the sidewalk.\n\nVegas buses ran all night long. She supposed they had to. Businesses were open all hours and people had to get to work. She crossed the first street, then another, then several more, moving much faster now, making her way toward the bus stop at the corner.\n\nIt was lighted; she wasn't crazy about that. A safety precaution, she supposed, but tonight, she preferred to remain in the darkness. She was pretty sure the Shepherds hadn't heard her leave, but you never knew.\n\nShe checked the posted schedule. Bus should be along in about five minutes. This time of night, of course, it was impossible to be sure. She decided to move a few feet down the street, out of the light, at least until she heard the chug-chug of the diesel engine or the high-pitched squeal of hydraulic brakes. . . .\n\n\"Excuse me, miss. Rather late, isn't it?\"\n\nShe froze. Where the hell had he come from?\n\n\"Mind if I ask where you're going?\"\n\nHe was wearing a uniform. Wasn't the usual uniform, though. She couldn't read the name on his badge.\n\n\"I'm going to visit my aunt. You got a problem with that?\"\n\n\"No. Do your parents know about this?\"\n\n\"My parents are dead.\"\n\nThe man gazed at her with an intense focus. She couldn't recall ever being subjected to such severe scrutiny before. What was he looking for?\n\n\"I'm sorry to hear that, miss. Is your aunt your legal guardian?\"\n\nRachel hesitated barely a second. \"Yes, she is.\"\n\n\"And does she know you're out at this time of night?\"\n\n\"Ye-es. It was . . . unavoidable. There was a mix-up at school and\u2014\"\n\nThe man's eyes twinkled. \"I think you're lying to me, miss.\"\n\n\"How would you know?\"\n\n\"Come with me.\"\n\nShe shrugged his hand away. \"I'm not going anywhere except on a bus.\"\n\n\"I'm afraid I'll have to insist. Consider yourself under arrest.\"\n\n\"You can't arrest me.\"\n\n\"I can. Suspicion of delinquency.\"\n\n\"Let me see your badge.\"\n\n\"If you know what's good for you\u2014\"\n\n\"I know not to get in a car with some weirdo just because he claims to be a cop.\"\n\nThe man reached into his wallet and flashed something quickly. \"Okay? Now, if you'll just come along . . .\"\n\nShe grabbed the wallet and reopened it. \"This isn't a LVPD badge.\" She read the small print. \"You're just a private security cop. You can't arrest me.\"\n\nHis face seemed to transform, harden. His voice acquired an accent. \"Don't be difficult, Rachel, my dear. It's for your own good.\"\n\nShe took a step back. \"How do you know my name?\" Her eyes widened. \"You're him. You're the guy. The one who grabbed Susan.\"\n\n\"It would have been so much easier if you'd simply come when I asked.\" He seized her arm, his grip so tight it hurt. She twisted back and forth, trying to break free.\n\n\"Give up,\" he said, smiling. \"You can't get away.\"\n\n\"Wanna bet, asshole?\" She kicked him hard on the kneecap. He buckled. The next kick went into his groin. He released her arm.\n\nRachel turned to run, but before she could get away he wrapped an arm around her throat, pinching her trachea, yanking her down to the sidewalk. Rachel coughed and sputtered helplessly; she felt the air draining out of her lungs.\n\nStill pinning her to the wet pavement with his left hand, he began fumbling for something in his jacket pocket with his right. Rachel saw her chance\u2014probably her last one\u2014and took it. She brought both fists around and pounded the arm that choked her, knocking it away. Before he could react, she slapped him on both sides of the head, right over his ears.\n\nHe screamed. His face contorted with pain and, even more, astonishment.\n\n\"You hurt me,\" he murmured.\n\n\"About time someone did.\" Rachel fled. Without looking back, she tore across the street and dove between two houses. Now she wouldn't mind so much bumping into a cop, but what were the chances? She raced down the slick pavement, slipping and sliding, gliding down alleyways, creating shortcuts between houses and crisscrossing through residential streets. At least she knew the neighborhood. She had that advantage. Her sneakers slipped on a wet patch and she went flying feetfirst into a chain-link fence. She was dazed and her face was scratched, but she knew she couldn't indulge herself in rest.\n\nShe checked over her shoulder, down the alleyway. Was he still there? She didn't see him, but somewhere in the darkness, she thought she heard the footfalls of someone, someone running. She couldn't keep this up. She'd never make it back to the Shepherds' house on foot. She needed help.\n\nShe raced out into the street. Surely someone would come, even this time of night, anyone, it didn't matter, just so she could get out of here before that guy caught up to her. Please!\n\nIn the distance, she saw the gleam of a pair of headlights. She shouted and waved, but it did not slow. What had Susan taught her? If you're not sure someone will cooperate, don't give them an alternative. She ran out in the center of the road, waving her arms wildly, forcing the oncoming vehicle to stop.\n\nWhen it did, she ran to the passenger side. \"I need a lift. Please! Someone's chasing me. I think it's that guy, the Poe freak.\"\n\nThe young girl poised behind the wheel was about Rachel's age. She seemed confused initially, but after a moment she said, \"Get in.\"\n\nRachel did. The second she closed the door, the truck peeled out. Rachel whipped around, peering out the back window, searching for a trace of the man who had been chasing her. Nice try, you sick pervert, she thought to herself. But you can't have me.\n\n\"Where do you want to go?\" the driver asked.\n\n\"Back to my\u2014no, just take me to a gas station or something. Anything public that has a phone.\"\n\n\"I think there's one on the corner of Maple.\"\n\n\"Great.\" Rachel tried to relax. \"I really appreciate this. You've saved my life, and I'm not exaggerating.\"\n\n\"Glad I could help.\"\n\n\"You and me both.\" Rachel crumbled against the back of the cab. She was still breathing hard; her pulse was racing. But she was safe. \"By the way, my name is Rachel.\"\n\n\"I'm Tiffany.\" All at once, the truck ground to a halt.\n\n\"Wait a minute. What are you doing?\"\n\nThe girl did not respond. Her face was like a mask, expressionless. Her eyes were wide and hazy.\n\nThe passenger side door opened. Rachel screamed.\n\n\"We meet again.\" It was him, the security guard. The killer. He was leaning into the cab. He held a hypodermic needle in his left hand.\n\n\"Help!' Rachel tried to crawl out the other way, but the woman driving would not budge.\n\n\"You've done a good job, Tiffany,\" the man said as he crawled after his prey. \"You will be rewarded.\"\n\n\"Leave me alone!\" Rachel tried desperately to escape, but there was nowhere for her to go. She kicked and clawed at him, without avail. Behind her, the girl called Tiffany grabbed her arms and held her in place.\n\n\"So much spirit. Just like your aunt. And my Ginny. You are the perfect Vessel.\"\n\nRachel tried to resist, but she was helpless, powerless, and even as she thought about somehow trying to get away, the needle jabbed her in the throat.\n\n\"Your devotion to your aunt is admirable. I was quite certain that phone call would bring you out.\"\n\nThe instant the needle left her neck, Rachel felt her strength fading, the lights dimming.\n\n\"S-Susan . . .\" she said as her eyelids fluttered closed.\n\n\"Susan can't help you now, my sweet. You will not see her again.\" He put the syringe back in his pocket, zipped it up, and gently slid his arms under her still, limp body. \"But the happy thing is, after you've been with me for a while, you won't want to.\"\n\nTwo squad cars were already parked out front by the time Granger and I made the scene. Two uniforms were at the front door. I whipped the car around, tires squealing, parked in the middle of the street, and raced across the lawn.\n\nThey didn't answer the doorbell, just as they hadn't been answering the phone. I looked up and saw a window open. The window in what I knew was Rachel's room.\n\nI kicked the damn door open.\n\nWe raced inside, guns raised. Circling in formation, we flooded the downstairs, the living room, the kitchen. No signs of life, good or bad.\n\nWe found the Shepherds huddled upstairs in their bed. They'd thought it best not to open the door this time of night. They'd tried calling the police but couldn't get a dial tone.\n\n\"Damn,\" Granger said under his breath. \"He must've cut the phone line.\"\n\nI bolted into Rachel's room.\n\nThe breeze coming through the open window put a chill in the air. The lace drapes fluttered up and down. The room was silent.\n\nShe was gone. He had her.\n**CHAPTER THIRTY-THREE**\n\nRachel woke screaming.\n\nShe was naked and strapped to a wooden chair in the middle of what looked like a basement. She couldn't get free; she couldn't move. She was barely able to squirm.\n\nOnly a few seconds later, he entered the room carrying a large oaken bucket.\n\n\"Where am I now?\" she shouted. He'd repeated this pattern over and over again, ever since he abducted her, taking her to some new location, bringing her around for a few minutes, then drugging her again. She'd lost all sense of time and place.\n\n\"A little change of scenery while I finish my preparations. There's so much to do. Did you like what I showed you earlier? Did it not seem a wondrous staging ground to ring in the Ascension? You'll be returning later, after we've had a little fun.\"\n\n\"Where are my clothes, you pervert?\" Her voice was hoarse and strained. \"Why did you take them?\"\n\nHe smiled pleasantly. \"I didn't wish to get them wet.\" And then he dumped the bucket on her.\n\nIt was filled with water, ice cold. It hit her like an arctic tidal wave. She thought she was going into shock; for a moment, it felt as if her heart actually stopped beating. She shivered uncontrollably, convulsing. She had never felt so bitterly frozen in her entire life.\n\n\"C-C-C-Could I please have a blanket? Or s-s-s-something. I-I-I\u2014\"\n\n\"I'm sorry. That isn't an option.\"\n\n\"B-B-B-But I'm s-s-s-so c-c-cold.\"\n\n\"Yes. But fear not\u2014later you'll be hot. So terribly hot. Then cold again, then hot. Cold, hot, cold, hot. All the livelong day.\"\n\nShe peered up at him, her eyes cloudy, her flesh a mottled pink covered with chill bumps, her arms clutched as tightly as possible to her exposed chest. \"Why?\"\n\n\"I should think that was obvious. To eliminate that trademark Pulaski stubbornness. Happily, I don't require your total subservience. There simply isn't enough time remaining. I already have my three offerings. You're the Vessel. But fear not\u2014it's a most important role.\"\n\nShe drank in air in deep, convulsive gasps. \"Why . . . are . . . you . . . like this?\"\n\nHe looked at her for a long moment. \"A curious question. Why are any of us the way we are? There's no satisfactory explanation, is there? Would you like me to tell you a sad story? Blame it all on my tragic childhood? Mommy didn't love me. Daddy hit me with a hairbrush. Simplistic balderdash. We are what we are.\" He adjusted the lay of his vest. \"I am the Raven. Everything else was mere prologue.\"\n\nPeople were talking to me, shouting in my ear, demanding answers to their endless questions. I couldn't process it all, couldn't deal with it. Why had he targeted Rachel? How had I known he was coming here? Why hadn't I done something about it sooner? Each time I started to give a coherent answer something else interrupted, a new demand, a false hope, a neural spasm in my brain. This could not be happening. This could not be happening.\n\nRachel!\n\n\"Everybody out of the house!\" Patrick shouted. And when had he shown up? At least someone had the sense to preserve what was now a crime scene. God knows I hadn't.\n\n\"Call the techs. Get Crenshaw. Get O'Bannon!\"\n\nPatrick barked out orders with impressive efficiency and organization. It should be me, I heard the voice inside my head say. It should be me.\n\n _Rachel!_\n\nAll at once, every single living memory, every photograph, every reminder of what that man had done to me flashed through my head.\n\nAnd now he had my niece.\n\nPatrick pulled the questioners off me. I knew the respite would be a brief one, but I was determined to make the most of it. I found a sofa and sat, steadying myself. My hands were shaking again. My stomach was sick, tossing, craving. I knew what I wanted, what I needed. I felt it with an urgency I had not experienced before, not since I woke up by the dam, not even when I saw the pictures.\n\nHe had won, he and the bottle. I knew I would get drunk tonight. I knew I would get drunk and stay drunk and be a drunk for the rest of my life. I had tried so hard. But I wasn't strong enough.\n\nI felt a sharp aching in my left wrist. God, I'd almost forgotten that was there. If only I'd done it right. I might be gone, but Rachel would be safe.\n\nThis was my fault. This was all my fault.\n\nNext thing I remembered was Patrick sitting beside me, Darcy hovering behind him. Patrick was careful to keep his face sympathetic but calm, strong. And Darcy was expressionless. Was that because he wasn't picking up on the nonverbal clues, didn't comprehend my fear and sadness? No, I think he got it, perhaps more than anyone. He just didn't know what to do about it. Who could?\n\n\"We have to ask you some more questions, Susan. Not just about what happened tonight, but about Rachel in general. Anything you can tell us about her that might be helpful.\"\n\nI felt so useless. I couldn't speak. Couldn't even raise my eyes to his.\n\n\"But I don't know why it has to be done here,\" Patrick continued. \"Best if we take it back to headquarters.\"\n\nSo I won't be around if the crime techs start making discoveries? Like Rachel's blood? Evidence of her death or torture?\n\n\"Why don't you ride back with me? I'll get a sergeant to bring your car\u2014\"\n\n\"No, I'll drive,\" I said, snapping out of it with a suddenness that startled both of us.\n\n\"I don't think that's wise.\"\n\n\"I can do it. I'll meet you back\u2014\"\n\nHe grabbed my wrist and held it tight. \"We need you on this, Susan. We need you one hundred percent.\"\n\nI knew what he was saying. He knew why I wanted to drive myself. But it was all so far beyond my control.\n\nO'Bannon crouched down beside me. \"Susan, I'm sorry to have to do this to you, but you're off the case.\"\n\n\"What?\"\n\n\"Your niece is a victim now. You're too close. I could overlook your own involvement, but not hers. Effective now, your consulting contract is canceled. You can keep the desk. I'll try to assign you something else when an appropriate case comes along. But as for now\u2014\"\n\nThat was when the phone rang, cutting him mercifully short. I don't know what I thought I was doing. But I was sitting right next to it. So I picked it up.\n\nNo. I have to be honest. I knew who it was. And I knew it was for me.\n\n\"Yeah.\"\n\n\"I have her.\"\n\n\"Son of a bitch.\" I clenched the receiver so tightly my fingers turned white. \"Why Rachel?\"\n\n\"I needed her. She's the Vessel.\"\n\n\"You said you cared about me, you bastard!\" I shouted, feigning a toughness I did not feel. \"If you do anything to her, anything like what you did to me\u2014\"\n\n\"Please calm yourself, dear. This is pointless.\"\n\n\"I'll make your god Poe look like an unimaginative grandma when you see what I can do. Have you hurt her?\"\n\n\"Of course not.\"\n\n\"What is it you want?\" I cried. \"What is it you want from me?\"\n\n\"Now? Nothing. Nothing at all.\"\n\n\"Then why\u2014\"\n\n\"I just called to tell you that you needn't worry. I have Rachel, and I will take good care of her, after my fashion. There's no chance that you'll catch me or recover her. So relax and enjoy what little time is left.\"\n\nMy head felt thick and unresponsive. There must be something I should do, something I should say. But what was it? \"What do you mean, what little time is left?\"\n\n\"I've told you before, Susan. The end times are upon us. I have everything I need now. Everything.\"\n\n\"Let me talk to her. If you really haven't hurt her, let me talk to her.\"\n\nA long sigh. Followed by: \"Five seconds.\"\n\nThe phone passed. \"Oh, my God, Susan, it's him. It's really him. I haven't been this scared since that day when we rented a video just after my parents\u2014\"\n\n\"Time's up.\"\n\n\"Bastard!\" I wailed, my voice hoarse. \"You could at least let her finish the sentence.\"\n\n\"I'm afraid we must go, just in case you're tracing.\"\n\n\"Can I talk to her again tomorrow?\"\n\n\"I . . . doubt she'll be . . . able to communicate clearly.\" I heard him sigh. \"I wanted so much to save you, Susan. But I couldn't do it. And who else is going to try?\"\n\n\"What are you talking about?\"\n\n\"I think you know. Good night, Susan. Try not to make a mess of it this time.\"\n\nThe line disconnected.\n\nWhile they were all babbling about the trace and the recording and what it meant, I stumbled to my car and drove away, fast, before Patrick got up the strength or numbers to stop me. My heart was pounding and my brain was racing. A thousand thoughts cruised through my head at once. It was like being drunk without being drunk. Was this what they called a dry drunk? I couldn't focus. Couldn't get a grip on myself, on anything.\n\nExcept one thing. I knew where I was going. Gordy's. Back where this all began. It was appropriate, no? Symmetrical.\n\nHad I ever really thought for a minute I could give up drinking? Who was delusional now? The bartender would still serve me, I thought, and if he didn't, there was a liquor store next door. Hell, that might be quicker. What did it matter? No shortage of places to get drunk in Vegas.\n\nSoon as I got there, I parked, popped open the car door, put one leg out\u2014and froze.\n\nNot voluntarily. I wanted to move. I kept telling myself to move. It was as if I'd lost all control, as if some alien being had taken over my body.\n\nI closed my eyes and saw Darcy\u2014Darcy, of all people\u2014in my mind's eye. The autistic savant, the boy who didn't comprehend emotion, but who nonetheless had given me so much emotional support. He was just staring at me. He liked me, I'd have to be blind not to see that, but he wasn't happy to see me. He was sad. So sad.\n\nRachel wasn't sad. Worried, not sad. I saw almost everyone I knew, Lisa, Patrick, Granger, the chief, my parents, my suspects, all of them, all of them, all of them.\n\nDavid.\n\nThey were so sad.\n\n _That's what he wants you to do._\n\nI somehow managed to get my leg back inside the car and close the door, but that was such a strain that I decided to forget about trying to move again for a while.\n\n _Try not to make a mess of it this time._\n\nMy wrist throbbed. Throbbed, like an aching in the hollow of my heart.\n\n\"Don't let him win, sugar bear.\"\n\n\"It's so . . . hard,\" I said, even though I knew I wasn't speaking.\n\n\"Naturally,\" David replied, with his understanding smile. \"It's meant to be.\"\n\n\"I wish you hadn't done it, David. I wish you hadn't.\" I folded over on the seat, hands tucked into my lap, cradling like a fetus. \"I just wished you'd loved me enough to stay.\"\n\nDavid looked at me with heavy eyes. \"I'm sorry, Susan. It's hard to admit, but\u2014there are times when love has nothing to do with it.\"\n\nI lay on the seat like a pathetic baby, which is exactly what I was. \"I don't forgive you, David. Not now, not ever. I will not forgive you.\"\n\nHis eyes only deepened. \"This is my last visit, Susan.\"\n\n\"What? Why?\"\n\n\"Because you need to get on with it. And you won't, as long as I'm around.\"\n\nAnd then he was gone. And I lay across the front seat of my beat-up car, crying into the vinyl, hurting, hurting so much.\n\nBut I was still inside the car.\n**CHAPTER THIRTY-FOUR**\n\nWhen I woke up, I didn't know how much time had passed. Somehow, all the smoke and cobwebs that once had fogged my brain had cleared, like someone had gone in with a mini-vac and sucked it clean. The aching, the craving, was still there. But it was manageable. I could make it. I knew that I could make it.\n\nRachel needs you, the voice in my head insisted. You have no more time to waste.\n\nAnd yet I didn't immediately start the car. I sat up straight and stared into the mirror. All I could see were my eyes, but somehow, that was enough.\n\nI could catch this man, I told myself, looking right into those red, tired, mismatched eyes. I had the means, the gift. If only I could put it all together . . .\n\nI tried to let my mind drift, free-associate. I thought if I opened things up enough, I might spark a connection, discover whatever it was I knew but my conscious mind had not yet seen.\n\nRelax, I told myself. Breathe in, hold it, release. Breathe in, hold it, release.\n\nI had been so sure I had him, back at the Transylvania. I could almost feel him in my grasp. But I'd come up short.\n\nPull back, Susan. Let your mind wander. . . .\n\nHad the three cheerleaders come to the Transylvania? Had the others?\n\n _Helen is a good girl. She would never do something like that. . . ._\n\n _Annabel was brilliant, an honor student even at MIT. I made sure she knew how to apply herself, how to turn heads. . . ._\n\n _The most important facet of the narcissistic personality is the absolute certainty of his own superiority, that he's right and everyone else is wrong. . . ._\n\n _He's smart, phenomenally smart. Deranged, but smart. . . ._\n\n _She made scrapbooks, just like I did as a girl. She even posted some of her art on her personal Web page. . . ._\n\nMy eyes opened.\n\nUniforms.\n\nThat was the key, damn it. Uniforms.\n\nWhat did Helen have on the walls in her bedroom? What did she have pasted into her scrapbook, on her Web page? Not rock stars. Not TV hunks. Cops, firemen, doctors, pilots . . .\n\nAnd what did they have in common? Uniforms. Where did she sneak out to in her black leather bad-girl getup? A biker bar? The teen stud club? No. The Army grunt hangout. Because that's where she would find men in uniforms.\n\nHelen had a thing for uniforms. She liked them.\n\nShe trusted them.\n\nTiffany admired policemen, firemen. She dreamed of one day being a cop herself, because she admired them so.\n\nShe trusted them.\n\nThere's more, I heard a voice within me saying. Keep working it, keep digging. . . .\n\nDarcy had shown me the burn mark where the door had been forced, the door to the ballroom where Helen Collier was found. But why was that significant?\n\nBecause it pointed _away_ from the room, not toward it. Because the chain had been torched from the inside.\n\nEdgar had already been inside when he brought out his acetylene torch. He'd had access to the room. Breaking the chains andforcing the lock had been just another clever trick to throw us off his trail.\n\nMy respiration spiked. I was breathing hard and heavy, my heartbeat racing. I was getting there. I knew I was getting there.\n\nI stormed into headquarters, taking them all by surprise. The feds appeared to be reorganizing our offices into an FBI hostage crisis center. Which wasn't a bad idea, in theory. But I knew that by the time they were finished, it would be too late for Rachel.\n\nPatrick was in the chief's office, conferencing. Darcy sat silently behind O'Bannon's desk.\n\n\"Susan!\" O'Bannon bellowed. \"Where the hell have you been?\" He looked at me suspiciously.\n\n\"Go ahead, sniff my breath. I haven't been drinking.\"\n\n\"Then what? Damn it\u2014this is your own niece.\"\n\n\"I know that,\" I said firmly. \"I also know he won't kill her. Not yet. He might . . . do things to her. But she's strong. She'll survive. I did.\"\n\n\"Susan, our investigators have a thousand questions\u2014\"\n\n\"And I'll answer them. But in exchange, I want five plainclothes answering to me and complete freedom.\"\n\nThey stared at me, all of them, speechless.\n\n\"And I'd like Patrick, if the Feebs can spare him. And Darcy,\" I added. \"Most importantly, Darcy.\"\n\nO'Bannon stared at me uncomprehendingly. \"Have you taken complete leave of your senses?\"\n\n\"Just the opposite. Regained them, finally.\"\n\nHe looked as if he were about to burst a blood vessel. \"Even given the bizarre assumption that I said yes, what do you think you'd do?\"\n\n\"Go back to the Transylvania.\"\n\n\"You already played that hunch! It was a good theory. But it didn't pan out. None of the guests\u2014\"\n\n\"He isn't a guest. He works there.\"\n\nPatrick stepped forward. \"Susan, I looked at the employee rolls. I didn't see anyone who\u2014\"\n\n\"Then we need to line them up and let me look. I'll recognize the rat bastard.\"\n\n\"Are you sure?\"\n\n\"Positive. I don't know why I didn't see it before. It's obvious, once you know.\"\n\n\"Know what? What do you think he does?\"\n\n\"I'm not sure. But I know he wears a uniform.\" I paused. \"I think there's a good chance he's a cop.\"\n\nHe frowned. \"A cop?\"\n\n\"Or something like a cop. Don't they have security at the Transylvania? I thought I remembered seeing some.\"\n\n\"Of course they do. But they might contract the security out, like most of the big houses.\" He snapped his fingers. \"Which would explain why he didn't turn up on the employee rolls.\"\n\n\"I need to get over there immediately.\" I turned to O'Bannon and looked him square in the eyes. \"With your permission.\"\n\nHe barely hesitated a second. \"Consider yourself back on the case.\"\n\n\"Good. I'll stay in touch.\"\n\n\"You won't have to. I'm coming with you.\" He pulled out his desk drawer and tossed something onto his desk. A gun. My gun. \"I think you may need this.\"\n\n\"I don't know. If you're not\u2014\"\n\nHe pressed it into my hand. \"I insist.\"\n\n\"We need to blanket the hotel,\" Patrick said. \"Make sure he doesn't slip out before we identify him. How much time do we have till this Day of Ascension?\"\n\nI checked my watch. \"Only a few hours.\"\n\n\" _Hours?_ Then the Day of Ascension\u2014\"\n\n\"When else?\" I led the way to the door. \"Today. Halloween. At the witching hour.\"\n**CHAPTER THIRTY-FIVE**\n\n\"You think this place will be ready in time, Ernie?\" Martin asked.\n\nHe was calm and confident. \"I don't see why not. The grand opening isn't until midnight.\"\n\n\"But there's so much still to do.\" Both pairs of eyes scanned the ballroom. The fa\u00e7ade of the Notre Dame cathedral was largely in place, but some of the surrounding decorations were in pieces on the floor, waiting to be assembled. Exposed scaffolding occupied a corner of the room. \"I hear the hunchback is still experimenting with his makeup. And what's with these bells?\" He gestured toward the huge six-foot bells that were being hoisted into place at the front of the cathedral. \"Those mothers are huge. And heavy. Why would the hotel lay out so much for bells?\"\n\n\"You can't do _The Hunchback of Notre Dame_ without bells.\"\n\n\"Hey, I been meaning to ask\u2014what were you doing in the ventilation shafts last night?\"\n\nHe stiffened. \"Last night?\"\n\n\"Yeah. I saw you crawling out of that shaft over at the north end of the casino. I didn't even know that was big enough to get into. What were you up to?\"\n\n\"One of the patrons reported smelling smoke. I didn't detect it myself, but I thought it best to be certain.\"\n\n\"Huh. Well, they never covered that when I came on. Maybe you can show me how to get in there later tonight.\"\n\nHe touched the syringe in his pocket. He could take this man out if necessary. Quickly and quietly. \"Tonight would not be a good day, what with all the work going on. Perhaps after the Halloween celebration.\"\n\n\"Good point. Okay.\"\n\nHis hand relaxed. Just as well. Another dead security officer would draw more attention to the hotel\u2014and he had directed too much attention here already. \"If you'll excuse me, I, uh, need to check on something in the storeroom.\"\n\n\"You've been spending a lot of time in there lately.\" Martin chuckled. \"You got a naked girl up there?\"\n\nHe smiled. \"Yeah. Four of them.\"\n\nHe left the ballroom and headed for the elevator bank. There was still much to be done, so many arrangements to finalize. Everything had to be right, just perfect. But soon he would be able to cordon off the ballroom so he could finish his preparations. Rachel would not participate willingly, but the other three would, and they would help him with her. He had put so much time and effort into this, not just with the offerings, but everything. Obtaining the C4 on the Vegas black market. His unbroken brown study of radio signals and electronics and incendiary agents. Everything that was required.\n\nThe hotel had spent thousands advertising this event, generating publicity for the grand reopening of this ballroom. But they would be celebrating ever so much more than those dullards imagined.\n\nThis celebration would be a cataclysmic event. An apocalypse for some, an ascension for others. The end of days.\n\n\"No, it can't wait until tomorrow!\"\n\nI pounded my fists together for emphasis. I wasn't trying to threaten the man\u2014well, actually, I was, wasn't I? If I couldn't convince him of the urgency of the situation one way, I was prepared to try another.\n\n\"But today is a very special day,\" Bloomfeld insisted.\n\n\"You don't know the half of it.\"\n\n\"All our resources are taxed to the limit.\" This guy had annoyed me when I was investigating the first crime scene and he hadn't grown on me any in the interim. He was probably perfect for micromanaging the organizational details of a hotel but he was useless to me. \"Our hotel is booked to capacity. Our Halloween celebration is generally considered the best anywhere. Thousands of people come to the Transylvania from all over the world.\"\n\nPatrick stepped between us. \"I don't give a damn about your tourists getting their ghost and ghoul fix. Four girls have been kidnapped.\"\n\n\"My staff is already being pulled six ways at once,\" Bloomfeld continued.\n\nI shot him the harshest look I could muster. \"I'm chasing a serial killer here, a killer who\u2014unless he's stopped\u2014is going to try something very bad tonight, probably at midnight, which is less than three hours away. I think that's a little\u2014\"\n\nThis was where Bloomfeld did his best to pretend he had a backbone. \"I have a responsibility to my guests. They expect a party that\u2014\"\n\n\"I'll cancel the damn party if you don't cooperate with me! I'll shut the whole hotel down.\"\n\nHe froze, his face more horrific than any of their gargoyles. \"You can't do that.\"\n\n\"I can and I will. I won't let another girl die because you were too busy entertaining to help. Now you can deal with me, and we can go through your security contractor's employment records, or I can shut the whole joint down. What's it going to be?\"\n\nAs if he had a choice. I held all the cards. And I had to admit\u2014it felt good to be effective again, to be back on top of my game. Or getting that way.\n\nBloomfeld started gathering the records.\n\n\"Great technique,\" Patrick said quietly. \"Where'd you study, Nazi Germany?\"\n\nI suppose I should've been more respectful to Bloomfeld, since we were in his office in his hotel. Well, next week, I'd send him a Hallmark. Right now, I had a job to do. And some lives to save.\n\nBy eleven P.M., I had winnowed it down to five names. Five possibles who fit most of the criteria. The women were eliminated, of course. All the men of the wrong age group. Everyone who was physically too large to be Edgar. I classified them by economic group, by educational background, by family relationships. Anyone who listed a parent as a Person To Contact in an Emergency was eliminated. And in the end, I had five names.\n\nOne of them was Edgar. I was certain of it. But which one?\n\nI peered at the pictures, the files, everything that was known about them. Darcy hunched over my shoulder. I had seen this man, damn it. I had talked to him. I should be able to pick him out of a photo lineup. Shouldn't I?\n\nThree of them were private security, where we had focused this search, but I was also considering a part-time tennis instructor and an actor who worked in the evening Spookapalooza show playing Edgar Allan Poe. I phoned Madeline and told her to run Net checks on all of them\u2014to learn as much as she could as quickly as possible. I instructed Bloomfeld to round up all the suspects. And I let my mind do what it did best. From here on out, I knew finding Edgar would not be a matter of logic or analysis. Intuition had to take over. My instincts had to tell me which of these men kidnapped Rachel. And how to get her back.\n\nThe more I dwelt on it, the more I gravitated toward the three security officers. The tennis dude wore a uniform of sorts, but was it a uniform that instilled trust? Would Annabel, the MIT student, have given him the time of day? Judging from his photograph, he was a cute guy, not much older than she was. But somehow, I just didn't believe it.\n\nThe actor who played Poe was an obvious choice\u2014too obvious. If his Poe connection were that apparent, would he have given us the Poe-derived clues, the quotations, the literary death methods? I had seen this guy on television once or twice since the Poe connection was leaked, being interviewed as a local expert on \"the Dark Bard of Baltimore.\" No, he was way too high-profile. I didn't buy it.\n\nAnd then there were three. Damon William Cantrell. Jeffrey Henry DeMouy. Ernest Lee Abbott.\n\n\"What do you think, Darcy?\"\n\nDarcy stared at the pictures. His brain was in motion, I could tell that. But this wasn't what he did best, was it? When he met them, he might notice the telltale smell of perfume or the stain of a certain kind of ash found only in Sumatra or whatever. But what could he do with a photo?\n\n\"I don't think I like this one,\" Darcy said. He pointed to the file photo of Cantrell, but I noticed he wasn't actually looking at it. \"His hair is like John Wayne Gacy's hair.\"\n\n\"Anything else?\"\n\n\"Did you know that John Wayne Gacy is considered the most successful American serial killer? He made even more deaths than Ted Bundy.\"\n\n\"Anything else?\"\n\nHe tapped another picture, the one of Abbott. \"I think that maybe I have seen him before.\"\n\n\"Really? Where? In the hotel?\"\n\nHis face twisted up. \"I don't remember,\" he said\u2014words I never expected to hear coming out of that mouth. But such was the irony of being autistic. He could remember chapter and verse about anything he read. But he was useless with faces. Some researchers thought autistic people didn't really even see faces, their expressions and distinctions. Just a pink blur. Which would explain why they were so poor at picking up on visual clues, facial expressions, and body language.\n\nDarcy was not going to pick the lucky winner.\n\n\"Where is Patrick, anyway?\" I said. I wanted to get his opinion on this, before Bloomfeld arrived with the suspects. \"He should be back by now.\"\n\n\"I'll go look,\" Darcy said. He probably realized he wasn't much help here. So he would be of use another way.\n\nAnd I continued to stare at the pictures. Will the real Edgar please stand up?\n\nI have to find Patrick Susan wants me to find Patrick and she's so worried and scared about her niece Rachel who seems nice but I hope she doesn't like Rachel more than me or she can like us both and is Rachel like her baby because I want her to have real babies and maybe she won't maybe she won't if something happens to Rachel like Mommy never had any more babies after me and Dad tells people that they couldn't but they could I know they could Mommy told me they could but they weren't going to because I was a difficult boy and they didn't want any more difficult boys. We have to stop the Bad Man because he hurt those girls and he hurt Susan and he might try to hurt more people and it's not right to hurt people. I would never hurt anyone. Hurting is bad.\n\nI can't find Patrick there are so many people in this gambling room and so much smoke I hate smoke I don't know why people smoke it's bad for you and it's disgusting and it should be illegal it makes my eyes hurt so I went into the ballroom with all the weird decorations. I couldn't see Patrick but I saw this guard guy and he was in a big hurry and I don't know why I even looked at him except he was carrying an axe and that seemed weird and then I looked at him some more and I wasn't sure if I knew him I never know if I know people but he smelled like someone I knew his smell was familiar and then he said something and I heard his voice and I remembered the guy on the street and all that talk about how tall he wasn't except then he had a mustache and a different color hair and glasses and he looked different but he said something again and I knew it was him.\n\nHe must be the Bad Man.\n\nHe recognized me too and I made a joke about did he have any more good puzzles I could solve and he didn't and I could see he was going to hit me just like the kids at school used to hit me and I should've done something about it. I should've stopped him but then I would have to hit him and it isn't right to hit people it isn't right and I don't want to hurt anyone and I didn't do anything and then he took the other end of the axe and he hit me and I fell down and then there was nothing.\n\n\"Please,\" Rachel gasped. \"I can't stand it anymore. It hurts.\"\n\n\"Only for a little while, my dear. Soon it will all be over.\"\n\nShe'd been hanging upside down for far too long. Blood rushed to her head, making it throb so intensely she could barely think. \"Where's Tiffany? And the others. Where did they go?\"\n\n\"They're such dear girls, so eager to please. Nothing I ask is too much.\"\n\n\"Because you've tortured and brainwashed them.\"\n\n\"Rachel!\" He tightened the ropes around her wrists and ankles, making sure she was secure. \"Don't speak like that. I've told you what is at stake. I've explained to you about the Ascension, about Dream-Land. About my sweet Virginia. The whole majestic plan.\"\n\n\"I don't want any part of your plan!\"\n\nHe took her chin\u2014upside down before him\u2014and held it in his palm. \"Would you prefer to be like the other heathens, those who remain on this plane and melt into nothingness? Or would you be translated into a Golden Age?\"\n\n\"I would rather be at home in clean clothes.\"\n\n\"Don't be petty. Why can't you see what I can see?\"\n\n\"Because I'm not insane.\"\n\nHe clamped the chloroform-soaked cloth over her nose and mouth, his hands shaking with rage. She was unworthy, but that spirit would soon be gone, replaced with that of his lost Virginia, and once he and she were reunited, nothing else would matter.\n\nI was practically out of my mind when I finally heard the doorknob click. It was barely half an hour before midnight. Did they not understand? Midnight was the dreamtime, according to Poe. Later would be too late. Especially for Rachel.\n\nBloomfeld had two men trailing behind him whom I immediately recognized from their file photos. Two suspects. Two Edgar possibles.\n\nBut only two.\n\n\"Apologies,\" Bloomfeld said. He could be quite polite, once you put the fear of death into him. \"Couldn't find the third officer.\"\n\n\"We need him,\" I said.\n\n\"We'll find him in time, I'm sure. He's supposed to be working in the ballroom, but no one could locate him. It's already packed in there\u2014hundreds of Halloween revelers. Ran into your partner, though, that FBI man. Sent him into the crowd to find the guy while I brought you these two.\"\n\nI stared at the photo of the missing security guard, Ernest Lee Abbott. I mentally added a mustache, changed the hair, put dark glasses on him.\n\n\"He's normally very reliable. That's why we asked him to help with the crowd control. Everyone is doing the work of three.\"\n\nI could imagine the man's lips moving, his face. His eyes taking that somewhat menacing, somewhat sorrowful expression that told so much about him.\n\n\"If you want, I'll go back to the ballroom and look some more. He's probably behind the cathedral, helping with some last-minute crisis. Whose idea was it to do the Hunchback, anyway? I always thought it was too literary. Kids today, they don't know anything about French literature. They probably think\u2014\"\n\n\"Hunchback?\" I closed my eyes and let my mind wander again, but this time, it went straight to the source. The key clue. The one that hadn't fallen into place before.\n\n _I haven't been this scared since the day we rented a video just after my parents\u2014_ That was what Rachel had said, during that brief phone call. Everyone thought she was terrified, babbling, me included. But we were wrong. Rachel is a tough girl, a smart one.\n\nShe was trying to give me a clue.\n\nWhat was the movie? What was the damn movie?\n\nOf course.\n\nWe'd rented _The Hunchback of Notre Dame._ The Disney version. The first day I brought her home. After her parents were killed.\n\n\"Take me to this ballroom,\" I said, rising out of my chair. \"Now.\"\n\nBloomfeld stuttered, \"B-B-But I rounded up your suspects\u2014don't you want to interrogate them?\"\n\nI shook my head. \"It's the other one. Abbott. He's Edgar.\"\n\nBy the time I made it to the ballroom, I still hadn't found Patrick, Darcy hadn't returned, and it was barely ten minutes until midnight. Ten minutes.\n\nRachel! I wanted to scream out her name, but I knew that wouldn't help, not in this earsplitting chaos. Please, God, don't let me be too late. Don't let me be too late.\n\nEven though the Halloween party had not officially started, the ballroom was packed. I could see where Bloomfeld might've had difficulty finding one security cop in this swarm. I might have trouble finding myself in here. At least half the partygoers were in costume, many of them masked. If Edgar was one of them, how would I ever find him?\n\nThink, Susan. _Think!_\n\nHe wouldn't be out here mingling, would he? He has some tremendous master plan in the works, something wonderful, something terrible. Something involving Rachel. He couldn't have her out here, whether she was costumed, dead or alive. Could he?\n\nWhile I was trying to crawl into Edgar's brain, I saw Chief O'Bannon enter the ballroom. I showed him the photo of Abbott.\n\n\"You're sure it's him?\"\n\n\"Damn straight.\"\n\nHe smiled a little. \"Good girl. Knew you could do it.\"\n\nHe took the left side of the room and I surged into the right. I saw the great fa\u00e7ade of the cathedral of Notre Dame at the far end of the ballroom, a focal point for all the festivities. I moved toward it. I've never been to Paris, but it looked pretty damn real to me, except that it wasn't quite finished. There was still some scaffolding, several raised platforms on wheels, off to the side. The ballroom was festooned with confetti and orange and black ribbons and banners. And where was the hunchback? He would emerge later, I guessed, probably from the top of the cathedral, ringing those four huge bells, two on each side of the central spire.\n\nI moved toward the cathedral. It seemed like the place Edgar\u2014Abbott\u2014was most likely to be. And I knew Rachel had seen it before, right? That was the whole point of the clue.\n\nSomeone dressed in a jester costume fell into me, tumbling backward. I went for my gun. Jesus, was I on edge. I shoved him out of the way and tried to plow a trail through the dense horde. They were getting increasingly crazed, ebullient, nutty, which I suppose was to be expected as the clock approached midnight. I could smell alcohol breath every which way I turned. It made me sick.\n\nWhich was certainly a good sign.\n\nEventually I forced my way to the back of the room. It was a high-quality cathedral, made of some kind of molded fiberglass, stained to the proper shade of gray. Someone had spent some real money on this. After trying several false apertures, I found a door on the far side that worked.\n\nI stepped into the cathedral, such as it was. It was dark back here, darker than I liked. The cathedral touched the ceiling and, despite the openings for the bells, little light crept through.\n\nThis was his place. I knew it, as sure as I'd ever known anything in my life. I could feel it.\n\nI drew my weapon. I'd let IA argue later about whether I had cause or not. Right now, I wanted a gun between me and him.\n\nI stepped into the darkness, marking a path I thought was parallel to the front of the cathedral. The entire area was small, close, silent. And dark. Did I mention that it was dark?\n\nI took baby steps, inching forward, fighting the desire to rush ahead. I wanted to find Rachel. I had to find her before it was too late. But Edgar had proven how dangerous he could be, how smart. I had to be careful. I couldn't save her if I were dead.\n\nI kept moving forward, one dark step at a time.\n\nTill I saw someone.\n\nAt first, I couldn't make out who it was. His face was masked by shadows. He was sitting on the floor, looking up at me.\n\n\"Patrick!\"\n\nHe was staring with a strange, vacant expression on his face. I holstered my gun and ran toward him. \"Patrick!\" I said, grabbing his arm. \"Patrick! What are you\u2014\"\n\nI gasped.\n\nHis head fell forward into my lap. Just his head.\n\nI screamed like a siren, like a child at a horror movie, like the weakest sister who ever lived. Blood spilled all over my turtleneck, my pants. The head fell to the floor but didn't roll. It just impacted with a sickening splat and lay there, staring up at me. It had been sliced clean\u2014by a pendulum? I wondered\u2014at the base of the neck.\n\nMy God, my God, Abbott killed Patrick, he _killed_ him, and if he killed Patrick\u2014\n\nAn even deeper horror clutched at the base of my spine.\n\nIt was so unlike him to be gone so long. . . .\n\n\"Darcy!\" I turned and ran back the way I came, feeling stupid, feeling powerless, terrified. He'd gotten to Patrick, he'd gotten to Patrick but please not Darcy please please _please_ not Darcy . . .\n\nThat was when the bells began to ring. Did that mean it was midnight? Even with all the noise out front, the ringing of the huge bells was deafening. I was just beneath them, and the unrelenting clanging seemed to crush my skull. It was oppressive and mind-numbing. Why would the hotel want to\u2014\n\nI looked up.\n\nMy heart stopped. I couldn't breathe. My fingers were cold, as if all the life had been sucked out of me.\n\n _Rachel!_\n\nBecause the apertures were recessed, it wouldn't be visible from out front, but back here I had a clear view of four young girls strung up one to a bell, tied to the clappers, dangling head-down. Swinging back and forth. Their heads smashing against the sides of the bells.\n\n\"Rachel!\" I shouted, even though I knew she couldn't possibly hear me. Even if she were alive. If the sound was killing me down here, what must it be doing to them?\n\n\"Rachel!\"\n\nI forced my brain to calm, slow down\u2014 _think_! There must be something I could do. The bells had to be activated by some sort of mechanism. I needed to find the controls. Maybe I could ask someone. If not, I could climb up on that scaffolding out front. . . .\n\nI raced toward the door. And I had almost made it when a hand burst out of the darkness. It grabbed me by the throat and slammed me against the wall. Before I could react, his other hand took my gun.\n\n\"Hello, Susan,\" Edgar said, smiling. \"Good to see you again.\"\n**CHAPTER THIRTY-SIX**\n\n\"Aren't they magnificent?\" His eyes rolled up in his head, even as his hand remained tight around my throat. \" 'The throbbing of the bells! Of the bells, bells, bells, bells, bells!' \"\n\nI tried to speak but couldn't. His grip was too tight.\n\n\"I hope you've enjoyed my little tableau. Your friend Patrick tried to spoil it, you know. But I couldn't allow that. Not when I'm so close.\"\n\nHe loosened his viselike fingers just enough that I could speak. \"Why did you have to kill him?\"\n\n\"I'm afraid I lacked the time for a more subtle response.\"\n\n\"Those bells are killing Rachel. And the other girls.\"\n\n\"Not killing. Translating.\"\n\n\"Have you hurt her?\"\n\n\"I haven't put a mark on her.\"\n\n\"I said, have you hurt her?\"\n\n\"Not as much as you have.\"\n\nI thought I was going to explode. \"What the fuck are you talking about?\"\n\n\"Nothing you don't already know. You've damaged that poor girl with your drinking, your temper, your self-indulgent weltschmerz, your flights of martyrdom. She's felt alone in the world, unwanted. Forgotten by the only family she has.\"\n\n\"Let me take her down.\" The anger had left my voice. I was begging. \"Let me save her before it's too late.\"\n\n\"You won't be able to get to her.\"\n\n\"And why the hell not?\"\n\nHe removed a small radio transmitter from his jacket pocket and pushed the first button on the keypad. \"Because the hotel is on fire.\"\n\nThe building exploded. That's what it seemed like. The sound was deafening, utterly drowning those bells, which had been unbearable only moments before. I couldn't see them, but I could hear the crowd screaming, running, crying. I could imagine the pandemonium that must have descended. And even though I couldn't hear it anymore, I knew Rachel was still getting her brains splintered by that damn bell.\n\n\"This is just the start,\" Edgar said, almost giggly with excitement. \"I've got ten C4 charges set all over the hotel, conveniently close to the gas mains. Disconnected the sprinkler system, too.\" His eyes were wide and manic. I could barely stand to look at him. \"This whole place is crumbling! Isn't it wonderful?\" He was totally consumed by his delusion, far worse than when I had seen him last. All vestiges of sanity, of humanity, were gone. \"It's 'The Fall of the House of Usher,' the greatest of the prophet's stories, coupled with the greatest of his poems. 'By the mystical magical tolling of the bells, bells, bells, bells, bells!' \"\n\nIt was getting hot back here. The fire outside was superheating the ballroom. Smoke inundated this dark, narrow passage behind the cathedral as well, making it difficult to speak or breathe.\n\nHe pushed a button and I heard another explosion. This one was farther away, but I was certain it was still inside the hotel. Maybe the casino. Maybe the spa. No telling how many people would be hurt or killed.\n\n\"It's not too late for you,\" he said breathlessly. \"You could be Madeline to my Roderick. You could join us, Susan, join Rachel and Ginny and me. We'll unite as comrades in the Golden Age.\"\n\nI thought fast. \"I'd like that.\"\n\n\"This can be the Day of Ascension for all of us, a passage from this virulent world to one of\u2014\" In the midst of his rapture, he loosened his grip on my neck and body. And that was all the invitation I needed. Mustering my strength, I bodychecked him against the cathedral. His head slammed back against a wooden beam. While he was momentarily stunned, I knocked the detonator out of his hand.\n\n\"No!\" he screamed, but he was much too late to stop me. I scooped the detonator off the floor and shoved it into my pocket, then raised my fist to deliver a knockout blow to his solar plexus.\n\nAnd he raised a gun. My gun.\n\n\"You don't deserve to ascend,\" he said bitterly, blinking from the pain, blood dripping down the side of his head. \"You will die right here in this miserable world where you belong.\"\n\nHe fired.\n\nO'Bannon swore, but it wasn't productive, because no one could possibly hear him. What the hell had happened? There'd been an explosion, and seconds later the whole ballroom was on fire. Within moments the front doors were congested and blocked. Bedlam ensued. Rabid partygoers were punching, screaming, crying, reeling, desperate to get away from the flames. Smoke billowed through the enclosed area, making it difficult to see or breathe. The air was thinning. Without an alternative exit, they'd all suffocate, maybe even before they burned.\n\nFighting his way through the mass hysteria, O'Bannon got to a side door\u2014led to the kitchen, if he wasn't mistaken. There was a crowd around, trying unsuccessfully to open it. Seemed to be locked from the other side. Well, he had the cure for that.\n\n\"Stand back,\" he bellowed. He pulled out his weapon and fired three times at the lock mechanism.\n\nGood thing it wasn't chained, or even that might not have worked. As it was, the bullets weakened it enough that he could kick the door open. As soon as he did, the crowd surged through it, coughing, crying, gasping for air. But they had a way out. At least until that one was blocked.\n\nA second explosion rocked the room. My God, he wondered, where did that one go off? How the hell did he stop this?\n\nAnd what happened to Susan?\n\nLast he'd seen of her, she'd headed behind that fake cathedral. She hadn't emerged, at least not that he'd seen.\n\nHe scanned the still-packed room. No sign of her, and he couldn't believe she'd just leave, not in the midst of all this chaos.\n\nThere was only one reason she would be back there while this turmoil was raging.\n\nCautiously, gun still in hand, he made his way toward the cathedral.\n\nThe bullet missed, at least in the sense that it didn't kill me on the spot. It seared my right arm, creating a fierce burning pain that brought sudden tears to my eyes and gave me a bad case of the shakes.\n\n\"I hate this,\" he said, and to my astonishment I saw that he had tears in his eyes as well. \"The brutality of it. Firearms. This is not the way it should be. Why are you making me do this, Susan? Why?\"\n\nHe lifted his arm and I could see that he was going to shoot again, going to shoot to kill this time, from a distance so short he couldn't possibly miss.\n\nI'm sorry, Rachel, I thought. I failed you. Just as I've always failed you.\n\n\" _Susan! Duck!_ \"\n\nI recognized O'Bannon's voice, but even if I hadn't I would've obeyed. A bullet whizzed over my head and struck near Edgar\u2014but not near enough. Edgar gritted his teeth, shifted his aim and fired, not once but three times. I heard a grunt that told me one of the slugs had made contact, followed by the sickening thud of a body hitting the floor.\n\n\"Nooo!\" I screamed. I rushed forward while Edgar's attention was focused on his new victim, tackling him under his gun arm. He fell back against the fa\u00e7ade. The gun went flying. In this darkness, there was no way of knowing where it had gone. I punched Edgar again and again and again and he didn't resist. I didn't give him a chance. I wanted to hurt him. I wanted to beat him senseless.\n\nWhen I was sure he wasn't going to get up again, I ran to the chief.\n\nThe bullet had caught him in the lower stomach, below his vest. It didn't look as if it would be fatal if he got help in time. But I knew that stomach wounds were the most painful an officer could suffer.\n\nHe was shaking, too stricken to speak. I told him not to try, then called for immediate medical assistance. I assumed the ambulances were already converging, given the conflagration outside. I gave them O'Bannon's location.\n\n\"You're going to be okay,\" I told him, and hoped he believed it. \"Just stay put. Don't try to move.\"\n\nI wanted to remain with him, but I couldn't. Rachel was still up there, and the other girls. Every additional second they spent upside down in those things could be fatal.\n\nI realized Edgar must've used the scaffolding to get the bodies up into those bells. So I would do the same to get them down.\n\nBy the time I ran through the cathedral door again, the ballroom was perhaps half empty, which was a damn good thing, because the flames were spreading fast. At least a third of the room was already ablaze. The air was thick with dense black smoke. Everyone was coughing and choking, black stains under their noses and mouths. I was finding it hard to breathe myself. But I put that out of my head. I had to get to Rachel.\n\nThe scaffolding levels were maybe seven feet apart. Edgar had no doubt used a ladder, but he hadn't left that behind for me, so I just vaulted it. Up on the first level, I found round steel pylons, buckets of mortar, tools, signs of a barely completed construction process. I leaped up, grabbing the edge of the next level, my bullet-creased arm aching, and swung myself around. On the next riser, I was level with the bells.\n\nUp close, I saw that Rachel was tied tightly across her entire body, ensuring that she couldn't move or escape. But the other three girls were only bound at the feet, just enough to keep them on the clappers. Why hadn't they escaped? Was it possible they'd let Abbott put them up there?\n\nRachel's eyes were open, but I couldn't gauge how conscious she was, hanging upside down for so long, her head thumping against the bell, that incredible noise shattering her eardrums. The side of her head closest to me was bleeding\u2014not a good sign.\n\n\"Rachel!\" I shouted. No reaction of any kind.\n\nThere was a narrow catwalk on the front of the cathedral, probably to give the workmen access to the bell chambers. With a cautious, tentative step, I edged off the riser onto the catwalk. From there, I was able to reach out and grab the edge of the bell.\n\nIt fought me. Nearly knocked me off the cathedral. I wobbled and teetered, noticing for the first time just how damn high up I was. But I held on to that damn bell.\n\nIt stopped. No more swinging. Rachel hung motionless in the center.\n\n\"Rachel!\" Still no response.\n\nI didn't know whether I should untie Rachel first or stop the swinging of the other girls' bells. And while I was deciding, the cathedral suddenly shot out from under my feet and I tumbled down into the smoky abyss.\n\nCorrection: my feet were knocked off the platform. By Edgar.\n**CHAPTER THIRTY-SEVEN**\n\nSomehow I managed to grab the edge of one of the risers and swing myself onto the second level. Edgar jumped down on the other side.\n\nHis face was bruised, bloodied. He was wheezing with each breath, coughing. And he had my gun.\n\n\"You've ruined everything.\" His voice was harsh and gravelly from the smoke, and perhaps my beating. \"My ascension. Virginia's return. Dream-Land.\"\n\n\"You need help,\" I managed. \"I told you that before.\"\n\nHe held out his hand. \"Give me the detonator.\"\n\n\"So you can set off the other eight bombs? And kill even more people?\"\n\n\"It's for the greater good.\"\n\nWhile we talked I mentally measured the distance to the next riser, the chances of me making it before he could shoot. Where were the cops, the firemen? They were bound to appear soon. If I could just stall, just keep him talking. . . .\n\n\"You're starting to sound like me,\" I said.\n\n\"I'm nothing like you, Susan.\"\n\n\"You are. You're rationalizing. Trying to justify the horrible things you do.\"\n\n\"I'm trying to give us a new world. A better world! One that isn't so . . . hard. You of all people should appreciate the value of that.\"\n\n\"You have a good heart,\" I said, and I truly believed it. \"Maybe we all do. But it went wrong somehow. You haven't done anything wonderful. You've killed innocent people.\"\n\n\"No!\" He fired. It missed me, but not by so much that I didn't feel my heart skip several beats. \"I've studied the prophet's words. I was given the secret.\"\n\nDown below, the flames were everywhere. I knew it would not be long before the entire room, infrastructure and all, came crashing down. Where was my backup?\n\n\"Give it up, Abbott. Let me get you some help.\"\n\nHe inched forward. \"I want the detonator! Now!\"\n\n\"Not gonna happen. Not now. Not ever. No matter what you do.\"\n\nHe rushed me. I was caught off guard by the sudden change, not to mention the fact that I was on the edge of a riser some fourteen feet off the floor. I wrapped myself around his body and wrestled him down. He couldn't get me in his sights, so he clubbed me over the head with the gun butt. That hurt. I fought to block out the pain, keep myself conscious.\n\nHe seemed possessed, as if one of Poe's worst monsters had taken over his body. He kicked me repeatedly. I pushed up to my hands and knees and he kicked me again, flattening me. I felt something in my chest snap.\n\n\"You don't have to do this,\" I said, spitting blood. He kicked me in the mouth, loosening a tooth. \"Like Poe said in _Eureka,_ we're free spirits. We can chart our own destiny.\"\n\n\"In Dream-Land.\"\n\n\"No. That's where you got it wrong. He was talking about finding your own Dream-Land. He was talking about making _this_ world a Dream-Land.\"\n\n\"Liar. False prophet.\"\n\nI hauled myself up, hoping to make one final run at him, but he saw me coming. With both hands, he grabbed me by the neck and flung me backward. Off the riser. Into the flames.\n\nI woke up and there were noises everywhere there were noises everywhere and they were so loud and it was just like the inside of my head except it was outside and the room was on fire everything was on fire and I don't like fire I'm afraid of fire. Everyone was running and fighting to get out and I was going to get out too except I heard my dad calling and he sounded just like he did that time when he told me my mommy was dead and I wouldn't get to see my mommy anymore. I ran to him and I saw him and he was hurt and he had red all over him. He was crawling and barely pulling himself along and his gun was in his hand and I don't like guns I don't like them at all but he was too weak to lift it up. I shouted for someone to help but no one was listening until someone told me he'd talked to the hospital and ambulances were coming and I tried to get my dad out of there before the entire room burned down and there was a scream! There was a scream I knew it was a scream and I knew who it was even before I saw her it was Susan and she was falling falling and she fell so far please God don't let her be dead please don't please don't please don't please don't let the Bad Man take away our babies. Susan is my friend doesn't he know Susan is my friend just like Brian in the fifth grade was my friend until they took him away to another school but Susan is still my friend and I want her to stay to be my friend. Please don't hurt her. People should not hurt each other! People should not hurt each other!\n\nI hit the floor back first, head tucked, then rolled, like they teach us at the academy, but I was certain I'd broken my right arm. Maybe a leg. It sure hurt enough. Possibly had a concussion. I didn't have time to do a personal inventory. As soon as I opened my eyes, he was standing there, hovering over me, his foot between my legs.\n\n\"You know I loved you. You know that.\" He looked like Satan, with belching smoke and billowing flames in the background.\n\nBlood trickled out of my mouth when I tried to speak. I knew I couldn't escape. He could pump three rounds into my skull before I could blink. \"You had a damn funny way of showing it.\"\n\n\"I let you live.\"\n\nA silence fell, blocking out the fiery chaos surrounding us.\n\n\"I did that for no other,\" he continued. \"It was not the will of the Raven, but I did it, because I so desperately wanted you to see the path.\" His lips trembled. \"You must hate me.\"\n\n\"No.\" And as I spoke, staring up at his twisted, pathetic face, I realized it was true. Whatever anger I'd had, whatever enmity I'd borne, was gone now. \"I don't hate you. I did. But not anymore. I told you already. We're a lot alike.\"\n\n\"We are?\"\n\nI nodded. \"Both haunted. Both screwed to the max.\" I wiped the blood from my mouth. \"I used to think you were evil. Like if I demonized your psychosis, that somehow made it easier to deal with.\" I laughed. \"Hell, you're not evil.\"\n\n\"I'm . . . not?\"\n\n\"No. You're just a poor schmuck who misses his sister. Like I miss my husband.\"\n\nHe hovered over me, gun still pointed, listening.\n\n\"I couldn't forgive my husband for what he did. And I channeled all that anger against you. But that's no way to live. I'm not going to spend my whole life angry, tearing myself apart. I forgive you.\"\n\n\"You\u2014what?\"\n\n\"You heard me. I forgive you.\"\n\nHe hesitated, gun wavering, sweat and blood trickling down the sides of his face. \"You know I can't let you live.\"\n\nI spat more blood out of my mouth. Something inside me was broken. I couldn't remain conscious much longer. \"So if you're going to kill me, do it already. What do I get, the axe? I don't think you have time for dental surgery.\"\n\nHis face knotted up. \"You are so . . . _hard_ on me.\" He pressed the gun against the side of my head. \"I'm sorry, Susan. Goodbye.\"\n\nThe gun fired. I winced. And waited, expecting to feel the intense pain\u2014and release\u2014that did not come. Always I think I've made it, but it never, never comes.\n\nI opened my eyes. Abbott had crumpled to the floor. And behind him stood Darcy, shaking from head to foot, his normally inexpressive face contorted with pain, his eyes streaming tears.\n\nHe was holding his father's gun.\n**CHAPTER THIRTY-EIGHT**\n\nAfter that, everything got kind of fuzzy. I know police and fire teams invaded the ballroom, and I know they got me out of there. Darcy hovered by my side the whole time. He was horribly torn up about what he had done. I knew it would haunt him for a long time, maybe forever. But at that moment, all he seemed to care about was me.\n\nNext couple of days were pretty much a haze, too, but I eventually got the lowdown on what had happened in the aftermath. There were surprisingly few casualties from the fire. Many injuries, lots of smoke-inhalation-related respiratory problems, but only a few fatalities, mostly because O'Bannon had blasted open an exit to speed up the evacuation. That ballroom and the one adjoining it were wrecked, but most of the rest of the hotel was still sound. I'd prevented Abbott from detonating the remaining incendiary bombs, which were found and removed.\n\nO'Bannon was seriously wounded and would be in the hospital for months. He'd already had his phone rerouted to his room in the recovery ward and had all his open files sent over. He might be laid up, but he was definitely not out of commission.\n\nPatrick had been killed with an axe, which Abbott had apparently brought along to cut the rope he used to string up the girls. We assumed he'd come upon Abbott at work and Abbott killed him. Hid the body where he thought no one would find it, at least not before the explosions started. What a waste. He was a good man, a kind man. A rarity, in our field. I miss him.\n\nAbbott died, almost instantly. Darcy's gunshot got him in the brain. Although I could empathize with the pain his life had brought him, I had no regrets about his execution. He had crossed the threshold into utter psychosis. No drug therapy ever would have brought him back. It was better this way.\n\nRachel was alive. The bells hadn't been as hard as they looked\u2014not real iron. She still had a concussion and had suffered some hearing loss, at least temporarily. But she was alive. And the docs told me that if I hadn't gotten to her when I did and stopped that bell, she might not have made it.\n\nThat was something, anyway.\n\nThe other three girls, Tiffany and Judy and JJ, were also alive, but seriously messed up, far worse than Rachel. They hadn't been in their bells as long\u2014apparently they had helped him secure Rachel\u2014but they were suffering severe psychological trauma from their time in captivity. It would be a long while before they were normal again, if indeed they ever were. But they were alive, and where there's life, there's hope. Right?\n\nThis time, I let the docs keep me in the hospital just as long as they wanted. I was in no hurry, and it gave me time to do some thinking. Which for me, was long overdue.\n\nAfter six days, I was released. My arm was in a cast, my leg bore a brace, and I had a cracked rib, but I was out of there. Lisa picked me up at the hospital.\n\n\"You're sure you want to do this?\"\n\n\"Positive.\" I wondered if I needed a friend who was a better driver, because each little bump of her Porsche radiated through my tethered arm and leg. \"Thanks for being my chauffeur.\"\n\n\"Hell, honey, you're unsafe at any speed when you're well. No way I'm letting you drive.\" She paused. \"But this could wait.\"\n\n\"No, I want to do it now.\" I reached out and lightly ran my fingers across her cheek. \"I love you.\"\n\nShe kept her eyes fixed on the road. \"Tell me the trauma of your near-death experience hasn't made you realize that you are at heart a lesbian.\"\n\nI smiled. \"No.\"\n\n\"Not that there's anything wrong with that. God knows you couldn't be a worse kisser than some of the male lovers I've had.\"\n\n\"No. Just wanted to tell you. I know you've been taking care of me. Not just the big stuff, like driving and finding places for me to live, moving me and taking care of my life while I drank myself into oblivion. I know you're the one who put Sugar Babies in my empty holster. Who quietly replaced my ratty old black turtleneck with a much nicer new one. Who kept taping Dr. Phil and leaving it on my VCR.\"\n\nLisa's eyes crinkled. \"That's what friends are for.\"\n\nI laid my head on her shoulder. \"Alcoholics don't usually have friends. They don't deserve them. But you stuck with me through it all. I won't forget it.\"\n\nShe blushed, actually blushed. \"Have you given any more thought to L.A.? It's a great house in a great town. Swimming pools, movie stars. It would be good for you.\"\n\n\"I know it would. You're right, as always.\"\n\n\"You flatterer. So . . . chick night tonight? TNT is running a _MacGyver_ retrospective.\"\n\n\"You're on, girl.\"\n\nMy esteemed lawyer, Quentin Delacourt, stared uncomprehendingly across his desk. I knew I should be taking his mystification more seriously. But he was wearing a red bow tie, and how can you take anyone seriously when they're wearing a red bow tie?\n\n\"I don't understand,\" he said. \"You want to give it up?\"\n\n\"Right. Throw in the towel. Call it quits.\"\n\n\"But\u2014\"\n\nThe Shepherds were also in his office, at my invitation. \"I just wanted you all to know. The battle is over.\"\n\n\"But Susan\u2014\" The lawyer leaned forward. \"Do you understand what will be the consequences of this action?\"\n\n\"Yeah. I get that.\" There was a sudden thickness in my chest that I tried to ignore. At any rate, I wasn't going to let it show. \"Just get me some visitation rights, okay? So I can see her every now and again.\"\n\n\"That won't be any problem,\" Mr. Shepherd said. \"Whatever you want.\"\n\nI turned toward him, shutting the lawyer out. \"You've been pretty hostile to me in the past.\"\n\nHe cleared his throat. \"It's possible that . . . my opinion has changed.\"\n\nHis wife cut in. \"I don't want to intrude, but . . . may I ask why you're doing this?\"\n\nI sucked in my breath. \"Because you're better for her than I am. I know that now. I guess I always did, really, but I didn't want to admit it. I'm not saying this is forever\u2014I'm going to try like hell to pull myself together, and if I do, I'll want to talk about custody again. But for now\u2014this is best for Rachel.\"\n\nMr. Shepherd held out his hand. \"You're doing the right thing, Lieutenant.\"\n\nI took the hand with my good arm and shook it firmly. \"I know I am. But that doesn't make it hurt any less. Take good care of my girl, okay?\"\n\n\"We will. And Lieutenant?\"\n\n\"Yeah?\"\n\n\"You take good care of yourself.\"\n\nOne last stop before the hearing. I could tell Lisa didn't think this was a good idea, but she took me anyway. Let me stop by the florist, then onward.\n\nNot many people at the cemetery this time of day. A groundskeeper, a few scattered mourners. Found David's grave in no time at all. It looked pretty scruffy, barren, unkempt. 'Course, I hadn't been here since the day he was interred.\n\nI stood there just staring at the grave for the longest time before I finally spoke. \"Look, it's not like it was a gigantic surprise or anything. I knew that you were . . . confused. I knew your interest in me . . . sexually . . . was declining. I'd seen the way you turned away whenever a hot-looking guy passed us in the mall. How unconvincing you were, laughing much too loudly whenever Granger made crude remarks about a Super Bowl cheerleader's anatomy. And I know, intellectually, as a psychologist, that it was no reflection on me. Not that that's stopped me from engaging in humiliating, degrading affairs, desperately trying to prove to myself that I might actually be desirable to someone.\"\n\nI drew in my breath, then slowly released it. \"My point is, I had my suspicions for a long time. I just didn't expect to have them confirmed the way I did. To come home and find you . . . you . . .\"\n\nI pressed the heel of my hand against my forehead, trying to stop the mental movie from replaying. \"I'm sorry I threw that huge fit in the office. I had no right to do that, not in front of your friends, co-workers, even if they didn't know what the hell I was talking about. They still don't know. I never told anyone and I never will. But what you did, David\u2014\" I felt myself tearing up, something I promised myself I would not do. I steeled myself, then started again.\n\n\"I mean, bottom line, I didn't care about any of that. You were what you were. But whatever you were . . . I needed you. Rachel needed you. And to just . . . leave us like that, leave me feeling guilty and betrayed and . . . alone. That was what hurt, David. That was what screwed me up the most. That was what I couldn't forget\u2014or forgive. We didn't get a chance to work it out. I didn't even get a chance to say goodbye.\"\n\nThe wind whistled through the barren oaks that dotted the yellow field. A few crows circled overhead, singing their sad songs. \"But you were right. It's time to move on, sugar bear.\"\n\nI crouched down and laid a single red rose across his resting place. \"Consider yourself forgiven.\"\n\nThe Bad Man still comes for me, but he comes in my dreams. Daddy says that it isn't real but it is real I know it is just like I dreamed that Mommy would leave and she did and she never came back and now the Bad Man is dead but he keeps coming for me and I don't know when he will ever stop.\n\nI saw Susan in the hospital and she looked broken but better and I asked one of the doctors who looked at me like I was a weirdo but he told me she could still have babies and that made me happy.\n\nI don't miss the Bad Man but I miss being a policeman. I'm glad Susan is getting out of the hospital so I can be a policeman again. Susan is my friend. Everything has been better since she came to see my dad that night and I don't know if she knows that she makes me happy but she makes me get tingly when she winks at me and has sort of a happiness beam that she shoots out and I feel like I could do more things when she's around I feel like I could do anything I could focus like my dad tells me to focus focus and I could be of use to people. If Susan wanted me to.\n\nIt's lonely here without my dad. I used to dream about being alone and not having my dad scowling and being disappointed in me all the time but now that he is gone it isn't nearly as nice as I thought it was going to be.\n\nI was feeling fairly buoyant when I hobbled into the hearing. And devastated when I left. Like what little I had left had been ripped away from me. As if I had nothing, nothing at all.\n\nNever being one to display much decorum, much less sense, I confronted him in his hospital room.\n\n\"You did this to me, didn't you?\"\n\nO'Bannon sat up. \"What are you talking about?\"\n\n\"I had my hearing today. With IA. For reinstatement.\"\n\n\"How did it go?\"\n\n\"I thought it went brilliantly. They complimented me on my work on the Edgar case. Talked about the pleasure they got from the fact that all those FBI dudes went home empty-handed while one of theirs made the collar. Talked about my impressive courage and resilience. How I seemed to be conquering my personal demons. I thought I had it made in the shade.\"\n\n\"And?\"\n\nI punched his pillow. \"And then they pulled out the report you filed. You blackballed me, you son of a bitch.\"\n\n\"Hardly that. I just said\u2014\"\n\n\"You knew they wouldn't reinstate me against your wishes. Your recommendation was critical!\"\n\n\"Susan, listen to me.\"\n\n\"Why should I, you bastard? I did your dirty work for you! I caught your killer. I even\u2014I even\u2014what he did to me\u2014\" I broke down. Just lost it.\n\nO'Bannon intervened. \"Susan, stop.\"\n\n\"Why should I?\" I screamed. \"I wanted my job back! Don't you understand\u2014it's all I have left!\"\n\nHe looked at me with tired, cheerless eyes. \"You're not ready, Susan.\"\n\n\"Who the hell are you to judge?\"\n\n\"You know it as well as I do. If I reinstate you, that means you carry a gun. That means maybe a partner depends on you for their life. Are you ready for that kind of responsibility?\"\n\n\"I caught Edgar!\"\n\n\"You're an alcoholic, Susan. We both know it. I think you're trying to pull yourself out of that gutter, but how can I know whether you'll make it? You're a brilliant behaviorist, but until I'm certain you're one hundred percent, I will not put another officer's life in your hands.\"\n\nI fell back in my chair, feeling all the pain, the hurt, the futility wash over me. \"What can I do?\"\n\n\"You can go back to those IOP meetings, for starters. Join AA. Get a sponsor. Read the Big Book. Work the steps.\"\n\n\"I'm not the talky-feely type.\"\n\n\"You'll force yourself. You'll get better. And when your doctor tells me you're solid, I'll put you back on the team. In the meantime, your consultation contract continues. Believe me, I can find plenty for you to do. You won't be bored.\"\n\nHe fiddled with the controls on his hospital bed, raising himself. \"And now that we've got that out of the way, would you mind dropping by the house to check on Darcy? He called the front desk\u2014he's having some kind of problem. He's been all by himself since I went into the hospital. He's a good kid, but\u2014you know how he is. He needs someone looking out for him. And God knows there's no one he likes better than you.\"\n\n\"Oh, that's not\u2014\"\n\n\"Don't kid a kidder, Susan. He adores you. I'm his old man, sure, but I know the score. I love him, but he's wary of me. Too much discipline\u2014or attempted discipline, anyway. Too many mistakes. Too many unresolved issues. And I'm laid up. So would you run by and see what's going on? He probably just needs someone to hold his hand for a minute. Would you do that?\"\n\n\"If I say yes, will you reinstate me?\"\n\n\"Hell, no. But I'd consider it a personal favor. I think your daddy would, too.\"\n\nBastard would play any card in his deck, wouldn't he? \"Fine, I'll go. But you can stuff your damn consulting contract.\"\n\n\"Are you sure? Why?\"\n\n\"After I see Darcy, I'm blowing town.\"\n\nI rang the bell and Darcy came to the door almost immediately. His eyes were like balloons. His hands were flapping. He ran around in circles, screaming, barely coherent, even worse than when I'd taken him to that sex club. \"Fire! Fire!\"\n\nI raced inside. The kitchen was indeed on fire, flames shooting out from the microwave oven. Looked like he'd been reheating some Pizza Hut chicken wings, but he'd left the food in the box with the foil wrapping. Darcy ran circles around the kitchen table, screaming, running his fingers through his hair. He collided into the wall. He fell backward against the table and hit his head.\n\nI grabbed him and held him in place. \"Darcy, where is the fire extinguisher?\"\n\nHe was so messed up he couldn't talk, could only point. I opened the pantry door and grabbed the extinguisher. A minute or so later, the fire was out. But the kitchen was a mess. As was Darcy.\n\nHe crumpled on the floor, hunched over the linoleum, rocking back and forth, babbling incoherently, hitting himself in the face.\n\n\"I called and asked Dad about dinner but Dad couldn't fix dinner so I thought that's fine I'll fix my own dinner and I did but the oven was mad at me and it started a fire and I didn't know what to do and . . .\"\n\nOn and on and on. He hit himself so hard he made bruises.\n\nI had to do something. I reached around him with my good arm and grabbed both hands, restraining him. Becoming his human straitjacket.\n\n\"All I wanted was something to eat but there was no one here and there's never anyone here anymore and I was all alone and I didn't know what to do and did you know that sixty-seven percent of all domestic fires begin in the kitchen but I opened the microwave and the flames just leaped out they just leaped out like they were trying to get me they wanted to punish me because I did a bad thing a really really bad thing . . .\"\n\nI hugged him tighter and tried to speak in a soft, soothing voice. I figured it didn't really matter what I said. He just needed to hear someone. It was hard, because I had one arm in a cast and the other ached at the wrist, but I held on to him.\n\n\"It hurt so much and I was all alone and I didn't know why the Bad Man came why the Bad Man always comes when I'm asleep I didn't want to hurt him I didn't want to hurt anyone I didn't hurt Mommy I really didn't but he was going to hurt Susan because I wanted to ask you about babies and I couldn't let him hurt Susan . . .\"\n\nGod, my heart ached for him. He couldn't be left on his own like this.\n\nI whispered into his ear. \"It's all right, Darcy. Susan is here. Susan is right here.\"\n\n\"And sometimes it's dark and I hear these noises and I don't know what the noises are and I don't like it when people touch me why do people always want to touch me I want to be touched but when they touch me it makes me want to run away and I don't want to be here by myself anymore I don't I don't I don't . . .\"\n\nI felt myself choking, feeling his pain, wondering what it must have been like for Chief O'Bannon, raising this boy by himself all those years, dealing with this kind of panic attack not just once when you happen to drop by but every day, every day of your life.\n\nThe words tumbled out of me. I didn't even think before I spoke. This boy had done so much for me, had supported me throughout this whole horrific case. Maybe it was time I returned the favor. \"It's all right, Darcy. I'm here. And I'm not going anywhere.\"\n\nI held him like that for more than an hour before he calmed down. I didn't mind. Even though it hurt, I didn't mind. Once he was calm again, I fixed a proper dinner, then cleaned up the kitchen mess and made myself a place to sleep on the couch.\n\nI took a shower, and when I stepped out of the bathroom with\u2014thank God\u2014a towel wrapped around myself, I found Darcy standing outside the door.\n\nHe was gasping for air and dripping with perspiration. And he was holding a frozen custard in each hand.\n\n\"I hope that you are in the mood for custard. I thought that you might be in the mood but I wasn't sure so I ran all the way to Third Street. And back.\"\n\n\"Just because you wanted me to have a bedtime snack?\"\n\nHis face was like a shimmering sheet of tinfoil. \"Because any day you have a custard is a Very Excellent Day. And I thought that maybe you could use a Very Excellent Day.\"\n\nThat night, before I fell asleep, I cried. Streams of tears, endless flows of salt water, cascading down my face. But it was a good cry. One I'd been saving up for a long time.\n\nGuess I won't be going to L.A. after all.\n**ACKNOWLEDGMENTS**\n\nNo writer can ever tackle anything so large and daunting as a novel without getting a lot of help, and I'm certainly no exception. I want to thank everyone who helped me during the time I spent creating Susan and Darcy's world and who assisted with the enormous research required to bring the characters to life.\n\nThe United States is currently in the midst of an autism epidemic\u2014and no one knows why. Autism Spectrum Disorders have increased over 500 percent in the last decade; the Department of Education reported an 18 percent increase in those seeking special services for autism from 2003 to 2004. In 2004, the Department of Health and Human Services issued an Autism Alert to the nation's pediatricians in an effort to improve data collection to try to determine the cause of this epidemic and to aid in earlier diagnosis. I want to give special thanks to perhaps the leading pioneers in autism research, Ivar Lovaas and Bernard Rimland. Lovaas pioneered the use of behavioral intervention, which has been incredibly useful to many parents trying to recover a child who seemed lost to this neurological disorder. Rimland pursued biomedical research and, as a result, has produced a therapeutic protocol that many parents believe significantly assisted, or even cured, their children. Both approaches are most effective when instituted in the child's life as early as possible. (In case you're wondering, the only reason Rimland is not mentioned in the book is because his protocol would not have existed when Darcy was a child.) Those wanting to know more about Lovaas and his Institute for Early Intervention should visit www.lovaas.com. Those seeking more information about Rimland and the Autism Research Institute should visit www.AutismResearchInstitute.com or consider attending one of his periodic DAN (Defeat Autism Now) conferences (www.DANconference.com). Parents reeling from the shock of this diagnosis and wondering where to begin would do well to read _Let Me Hear Your Voice_ by Catherine Maurice, the inspiring story of one parent's successful battle against this strange and terrifying disorder.\n\nIt would be impossible to write a book about a criminal behaviorist without becoming familiar with the work of the two best-known names in the field: John Douglas and Roy Hazelwood. Douglas developed criminal profiling techniques during his twenty-five years with the FBI and subsequently wrote fascinating books based on his experiences, such as _Mindhunter_ and _The Anatomy of Motive._ Hazelwood built on and expanded his work; his psychological insights are perhaps the best recorded in his book _Dark Dreams._ I also must thank my friend Dave Johnson for his insight and information about the inner life of a police station and those who work there. If the characters in this book do not always behave as model police officers, however, it's not Dave's fault; it's because these characters, like most people I know, are not perfect.\n\nAnd those who want to know more about card counting and other blackjack techniques developed at MIT and elsewhere may wish to read _Bringing Down the House_ by Ben Mezrich, and the classic _Beat the Dealer_ by Edward Thorp. If you're thinking card counting will allow you to go to Vegas and get rich quick, though, please think again.\n\nI also want to thank the many fellow writers who agreed to read an early draft of this book, indulging my friendship, not to mention my insecurity about a book I knew was a departure from my previous work. Many thanks to Jodie Nida, John Wooley, K. D. Wentworth, James Vance, and my wife, Kirsten. The worst of it is, the only compensation they received was having characters in the book named after them. But at least none of their namesakes were tortured by poisonous leeches.\n\nVery special thanks to my agent, Dan Strone, who believed in this book from the start, and my long-standing friend and editor, Joe Blades, who supported this effort just as he has everything else I've tried for the past fifteen years. No writer could possibly be more fortunate than I've been.\n\nI invite readers to send me their thoughts via email at WB@williambernhardt.com. You can also visit my website at www.williambernhardt.com.\n\nAs the academician in the book says, Edgar Allen Poe's contributions to American literature are enormous, and if you haven't read any of his work, this might be a good time to start. Just don't take it too seriously, okay?\n\n\u2014WILLIAM BERNHARDT\n**ABOUT THE AUTHOR**\n\nWILLIAM BERNHARDT is the author of many novels, including _Primary Justice, Murder One, Criminal Intent, Death Row,_ and _Hate Crime._ He has twice won the Oklahoma Book Award for Best Fiction, and in 2000 he was presented with the H. Louise Cobb Distinguised Author Award \"in recognition of an outstanding body of work in which we understand ourselves and American society at large.\" A former trial attorney, Bernhardt has received several awards for public service. He lives in Tulsa with his wife, Kirsten, and their children, Harry, Alice, and Ralph. Readers can e-mail him at WB@williambernhardt.com or visit his website at www.williambernhardt.com.\n**Also by William Bernhardt**\n\n _The Ben Kincaid Novels_\n\nPRIMARY JUSTICE\n\nBLIND JUSTICE\n\nDEADLY JUSTICE\n\nPERFECT JUSTICE\n\nCRUEL JUSTICE\n\nNAKED JUSTICE\n\nEXTREME JUSTICE\n\nDARK JUSTICE\n\nSILENT JUSTICE\n\nMURDER ONE\n\nCRIMINAL INTENT\n\nDEATH ROW\n\nHATE CRIME\n\n _Other Novels by William Bernhardt_\n\nTHE CODE OF BUDDYHOOD\n\nDOUBLE JEOPARDY\n\nTHE MIDNIGHT BEFORE CHRISTMAS\n\nFINAL ROUND\n\n _Edited by William Bernhardt_\n\nLEGAL BRIEFS\n\nNATURAL SUSPECT\n_Dark Eye_ is a work of fiction. Names, characters, places, and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is entirely coincidental.\n\nCopyright \u00a9 2005 by William Bernhardt\n\nAll rights reserved.\n\nPublished in the United States by Ballantine Books, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.\n\nBallantine and colophon are registered trademarks of Random House, Inc.\n\nCataloging-in-Publication Data can be obtained from the publisher upon request.\n\nBallantine Books website address: www.ballantinebooks.com\n\neISBN: 978-0-345-48179-5\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n# City Living\n\nWill McIntosh\n\nwww.orbitbooks.net\n\nwww.orbitshortfiction.com\n\nBegin Reading\n\nMeet the Author\n\nBonus Material\n\nAbout Orbit Short Fiction\n\nNewsletter\n\nTable of Contents\n\nCopyright Page\n\nIn accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\nCity Living\n\n\"Hey.\" Willard grabbed my shirtsleeve. \"Would you look at that.\" He pointed at a white truck parked down the street. It had a big, square back end with _Good Humor Ice Cream_ on it, along with a picture of an ice cream on a stick, dipped in chocolate. The truck was surrounded by kids with cute little dirty knees, their eyes round. \"Ice cream on a stick! I seen that in _Life_ magazine. I want to get me one.\"\n\nWillard pulled me through the crowd toward the truck. \"Ice cream on a stick. I guess you gotta eat it fast before it melts off.\"\n\nWe each bought one for ten cents. In Siloam you can get a cup of hand-scooped ice cream for a nickel, but I reckoned it was worth an extra nickel to tell folks we had ice cream on a stick in New York City. We stood on the sidewalk, happily gnawing the chocolate off to get at the ice cream. Willard kept holding his up, showing it to others who'd bought one, like we were part of a special club or something.\n\nI took the opportunity to soak the people in\u2014what they were wearing, what they did with their hands while they stood, how they talked. My neck had been bent back looking at the giant buildings when we first came through the gates, but now I was fine-tuned on the littler things.\n\nI was eyeing a perfect New York City woman (small and stylish; her hair short so her earlobes poked out underneath dark curls) when a loud honk startled me out of my boots. It sounded like a giant goose. A second one followed, then a third, calling back and forth to each other, coming from the tops of the buildings.\n\nAll around us people sat down in the street. Every one of them, right down on the pavement.\n\n\"Sit down,\" an old geezer near me said. He reached up and tugged my shirtsleeve so hard he partway untucked my shirt.\n\n\"What for?\" I asked.\n\n\"The city's about to move.\"\n\nI nearly dropped my ice cream. \"What do you mean it's about to move? It ain't supposed to leave for two days!\" Willard said, taking the words right out of my wide-open mouth.\n\n\"It's the emergency signal,\" a yellow-haired woman said, sounding a mite impatient with having to fill us yokels in. \"It was installed during the war.\"\n\nI could just picture it: soldiers pouring out of the buildings carrying rifles, Berlin barreling toward New York with its big guns booming. Hard to believe that was only four years ago. Seemed like a lifetime.\n\nI took a seat on the sidewalk, shifted when I came down on some man's ankle and he bellyached about it. \"How long until we start moving?\" I was excited as all get-out.\n\nThere was a commotion down the street, all the people talking at once to the people around them. A man poking his head out of a phone booth seemed to be at the center of it.\n\n\"Chicago _attacked Boston_ ,\" I heard a man in a brown fedora say to the woman sitting next to him.\n\n\"No way,\" I said to Willard. \"Not possible.\" Someone down the line must have gotten it wrong, like happens in a game of telephone. \"An American city attacking another American city?\" Some of these cities were rumbling that they weren't quite part of the United States no more, being as they could mosey down to Mexico whenever they felt like it, but no way Americans would spill the blood of other Americans. The war was over, Hitler and Tojo was dead, but, no way.\n\nThe honking stopped. The last honk echoed across the skyscrapers, then that got drowned out by a downright deafening rumble. It reminded me of boulders coming down in a rockslide.\n\nThe street jerked underneath me. It jerked again, and I was thrown backward, like I was on a train that was pulling out of the station full-steam. I almost fell on the people behind me\u2014it didn't help that Willard grabbed hold of my collar to keep from falling back himself. The pulling went on for a few seconds, then everything seemed to get still again.\n\nA stiff wind was blowing down Madison, but besides that and the rumbling, you'd never know you were on something that was moving. People were standing, brushing off their behinds, and going back to heading wherever they were heading. The only difference was that instead of wearing their hats everyone was carrying them, on account of the stiff breeze. New York City was on the move, and I was on it. I let out a whoop of pure pleasure.\n\nWillard looked at me like I was nuts. \"How are we gonna get back home if New York is leaving?\" he asked. He looked downright scared.\n\n\"Relax. We'll hitch back. Give us a chance to see more of the country.\" I started walking, not sure where I was going but liking the feel of city pavement under my country shoes. Even if Chicago _had_ attacked Boston (which they hadn't) there was nothing for me to do about it, so why not do a little sightseeing?\n\nWillard huffed. \"That's easy for you to say. All you'll miss is a few medical classes.\"\n\n\"If I end up being your doctor you might wish I hadn't missed those classes,\" I joked.\n\n\"I don't show up for work, I don't get paid.\" Willard stopped. \"Where are we headed, anyway? I thought we was going to see the Yankee game.\"\n\nWhat I really wanted to see was the engine under the city. They didn't give tours, but I figured there must be some way down there.\n\nA fetching young woman hurried by. \"Excuse me.\" I trotted to catch up with her.\n\n\" _Now_ where are you going?\" Willard shouted at my back.\n\nThe woman glanced at the steel bracelet on my wrist, which they put on you at the entry gate when you pay your two dollars. She kept walking.\n\n\"Pardon me, but can you tell me how a curious country boy might get down to see the glorious engine that runs this fair city?\"\n\n\"He can't,\" she said. She was walking darned fast. Her high heels clicking on the sidewalk made it sound like a tiny horse was running alongside me.\n\n\"Not for just a minute or two?\"\n\n\" _I've_ never seen the navigation center. No one gets down there.\"\n\n\"Oh.\" I tried not to sound disappointed.\n\n\"If they let people see it, the design secrets could get out. The designers don't want that.\"\n\n\"The designers,\" I said.\n\n\"Yes. The dream team.\"\n\n\"Tesla, Crowley, Gurdjieff, Bohr, and Jung.\" I ticked them off on my fingers.\n\nShe gave me a puzzled look, like she'd just heard a steer sing opera.\n\n\"So is it true that y'all give blood to keep the city moving? It said so in _Life_ magazine.\"\n\nThe woman smiled her city smile. \"Did it?\"\n\n\"Yes, ma'am.\" When the war ended, everyone thought they'd just put the cities back where they'd been, the residents who'd been temporarily kicked off would climb back on, and that would be that. Then the mayor of Chicago got his city moving, and of course everyone else had to follow.\n\nWillard's cheap old pocket watch was suddenly six inches in front of my nose, blocking my view of the street. It was quarter till one. \"Joe DiMaggio is taking batting practice right now. If I'm gonna get stranded in Texas or California or wherever we're headed, I want to see the New York Yankees while I'm taken there.\"\n\nI sighed through my nose. \"Fine. We'll see the Yankees.\"\n\nThe subway was lit bright, but smelled cool and damp like a cave. We waited by the tracks until we heard a whistling sound, then wind rushed through the tunnel, bringing a whiff of something dead with it.\n\nI couldn't see anything through the train's windows, which was disappointing. I once read that there were all kinds of tunnels under New York, dug for sewers and electricity and trains, and that people lived in them and ate rats and never came out of them. I would've liked to see one of those people.\n\nWhen we got aboveground again, Willard rushed us along toward the gates of the stadium. \"Come on, I don't want to miss Joe Di's first at-bat!\"\n\n\"Game's been canceled!\" a newspaper seller shouted at us. He was a fat man, with hair poking out at the neck of his collar.\n\n\"Oh, no,\" Willard moaned. He snapped his fingers. \"Dang it.\"\n\n\"Due to the emergency.\" The newspaper seller pointed at the stadium. \"They set up a sixteen-point event instead.\"\n\nWillard and me looked at each other and shrugged.\n\n\"What's a sixteen-point event?\" Willard asked.\n\nThe newspaperman frowned at us, then noticed the steel bracelets we were wearing. \"Oh. Never mind. City business.\"\n\n\"I wouldn't mind seeing a sixteen-point event,\" I said.\n\nThe newspaperman laughed. \"Trust me, you don't. Anyway, they won't admit you.\" He gestured at the bracelet.\n\nI looked down at the danged bracelet, then at the gate into Yankee Stadium. \"I'll take one of them newspapers, if you don't mind.\" I pulled a nickel out of my pocket. Then I moved Willard out of range of the newspaper seller's ears. \"I'm gonna see what's going on in there.\"\n\n\"Now why would you want to go and do that? The man just said you'd be sorry if you did.\"\n\n\"Never mind.\" I clapped him on the shoulder. \"I'll be back in no time.\"\n\nI put the newspaper over my hand to hide my wrist and stepped into line. There was no admission, we just waltzed right in, no tickets or nothing. I wanted a seat up high so I could see everything. Even if there wasn't gonna be a ballgame, I was still at Yankee Stadium.\n\nIt sure seemed like a game was about to get started\u2014boys were climbing up and down the rows selling hot dogs and sodas from steel boxes strapped around their necks, and music was piping from loudspeakers set all around. \"You Oughta Be in Pictures\" was giving way to \"Moonglow\" as I grabbed the railing and headed up.\n\nI picked a row, scooted sideways past people already sitting, my feet crunching peanut shells on the floor. The whole place smelled like old hot dogs. I hiked up my trousers and took a seat, wondering what this was all about.\n\nThere was a commotion twenty rows below. People were standing and pointing, grabbing their friends by the shoulder. I bent left and right, trying to get a look, and then I saw what the commotion was about, and I plopped down in my seat, my mouth hanging wide open.\n\nThere was no mistaking the man making his way up the stairs. It was the old Bambino himself, now the Yankees' manager. Babe Ruth was dressed in his pinstripes, smiling and waving, a bigger man than I'd've guessed from the pictures I'd seen, with skinny legs but a big middle. Two men in dark suits were with him, and as they made their way up to my level one of them gestured for the Babe to sit, just one row in front of me and a mite to my left.\n\n\"I guess this spot is as good as any,\" Babe said. \"Seems like I shouldn't have to come to one of these things. I do enough for this city.\" The Babe looked uneasy; there was sweat in his eyebrows, and that bulldog face of his was twitching. Now that I thought about it, a good helping of the crowd looked nervous.\n\nOne of his companions leaned his head close to the Babe's. \"If it wasn't at the stadium I could've gotten you out of it, but for something held here, in place of a game, it would've looked bad.\"\n\nThe music went out and everyone got real quiet. Babe looked at the man beside him and whispered, \"I sure hope your hunch about picking a random seat is right.\"\n\nNow I spotted the New York Yankees ballplayers. They were sitting in the front row along the first-base line, all in a neat line in their crisp white uniforms.\n\n\"Here it comes,\" someone said.\n\n\"What?\" I asked, looking around. Everyone was looking at the field. I squinted; the dirt between the pitcher's mound and second base was swirling like there was a little tornado on the field. A dark hole opened up, right there in the field. I stood, wanting to see better.\n\n\"Sit down, you idiot!\" someone shouted. \"You want to draw its attention this way?\"\n\nI sat. Draw its attention? I shook my head. I was confused as could be, and didn't want to draw too much attention to my own self. I didn't know if I could get into trouble for crashing a city folk party, and I didn't want to find out. I was mighty curious about what was going on, though. Mighty curious.\n\nSomething floated up out of the hole.\n\n\"Jesus Lord,\" I whispered.\n\nI could hear it more than see it. The sound went straight to my guts, not touching my ears at all. It was a scraping sound, only too deep and low to sound like anything anybody might scrape. There was a musical quality to it\u2014the worst, scariest music I ever heard.\n\nIt rose out of the dark hole. I couldn't quite focus my eyes on it. It was shifting, sliding\u2014an oily mess of tangled black tubes that flapped here and there, and pointed silver teeth that cranked open and snapped closed. As it moved, it left a trail of black bubbles that drifted toward the stands. When the bubbles popped they sprayed the crowd underneath with black goo that made them shout and twist.\n\nI wasn't so curious no more. I wanted out. I looked around: No one was moving from their seats. I had a hunch it would be a bad idea to leave mine and draw \"its\" attention.\n\nThe thing floated around the stadium, getting closer to the seats until it floated right over the low wall and stopped in front of a woman in a flowery dress with a bow on the collar. She covered her face with both arms and screamed. The thing hissed at her; she screamed louder. I leaned forward, trying to see. There were spiny things stuck in the woman's face, like porcupine quills, only black. She was making no move to pluck them out. Maybe she didn't want to make the thing angry.\n\nIt moved on. The crowd leaned from side to side, ducking, avoiding, praying. Every so often it stopped in front of someone and hissed, or wagged black tubes at them, scaring the bejesus out of them.\n\nI kept walking back through how this all began, trying to remember if I'd seen anyone with guns forcing people inside. There hadn't been no guns, everyone had strolled in like this was some sort of garden party.\n\nThe thing rose into the upper deck.\n\n\"Oh, no. Get outta here, go on,\" the Babe said. He shooed at it with four fingers without lifting his hand off the armrest.\n\nIt kept on drifting around the horseshoe of the upper deck until it was in my section. I didn't look at it; I thought it might be less likely to notice me if I pretended I didn't notice it.\n\n\"Please, not me. Don't let it choose me,\" a man nearby said. I thought it was the same man who'd told me to sit down.\n\nWhat did he mean, choose him? It was gonna choose someone? I was so confused, and I was so scared my hands and knees were trembling. I turned toward the woman sitting a seat away on my left and whispered, \"What's going on?\"\n\nHer eyes opened real wide; she gave me a sidewise glance and she shook her head ever so slightly.\n\nThe thing shifted its direction and moved right toward me. I'd never been so scared. This was no game; I was sure of that now. It changed directions again, and my knotted shoulders relaxed.\n\nThen it reversed itself and moved right toward me again.\n\nI thought maybe I should pray, but I was too scared to think of any words. The thing just kept coming until it was right in my face, then it stopped. It was so close I could have touched it. I looked at my shoes as its music shook my insides. When I glanced up for just a second I could see that its skin was moving\u2014swirling and crawling and twitching. Tubes snaked out to my left and right. They wrapped around me, curling tighter until I couldn't exhale or I would touch them. It hissed at me furiously; the stink was horrible, like rotten fish and sewer water. I squeezed my eyes shut, trying hard not to scream because I didn't want to make it any angrier than it already was.\n\nThen it was gone. By the time I peeled up one eyelid it was a dozen feet away. My shirt was soaked in black oil, but besides that I was fine. It hadn't even quilled me.\n\n\"Lucky,\" the woman to my left mouthed.\n\nThe thing hovered in front of Babe Ruth. The Babe looked away from it; I guess that was everyone's instinct, to look away. The thing hissed; the Babe scrunched up his face, balled his fists, and jerked back in his chair, but then it floated past him, to the man in the suit sitting next to him. It wrapped itself around the man the way it did to me. It flattened itself out so the tubes looked more like barbed whips. Or maybe it was a trick of the light. One of the tubes lifted high in the air, real slow, then came down hard. The man screamed. The Babe moaned a curse. Another tube lashed the man. I closed my eyes and stuck my fingers in my ears, but I could still hear him screaming. Around me I heard people gasping and Oh-Lording and that-poor-manning. From the relief in their voices I figured it must only choose one person.\n\nThe thing didn't kill the man, only whipped him so bad they had to carry him out on a stretcher. Still, I wouldn't have wanted to be him. As I headed down the steps I got to wondering if maybe these weren't my kind of folks. I got along with all kinds of folks, and it never occurred to me that there could be somewhere where I felt out of place, but suddenly I wasn't so sure I belonged in a city.\n\nThe lines to get out were longer and slower than the lines to get in. I wanted out of there as quick as I could. I couldn't figure out what that thing _was_. It wasn't any creature I ever seen or read about. It didn't even seem like it had been all there, even when it was there. I kept peering toward the front, trying to figure out what was taking so long, and finally I spotted the holdup. People were going behind a curtain, one by one. After a moment they would come out the other end. I waited my turn, wondering what was behind the curtain, hopping from one foot to the other. Willard had been waiting a long time.\n\nWhen I got closer I could tell I wasn't going to like whatever was going on behind that curtain. People were groaning, gasping, some downright shouting in pain. As a medical student I had some idea of how much pain it took to squeeze those sounds out of grown men and women, and I was sweating as the line inched along.\n\nWhen it was finally my turn, I strode on back as if I did it all the time. I didn't want to raise any suspicion. There was a doctor and a nurse on the other side, both dressed in white. The doctor held a long, long, long needle. He held it pointed at the ceiling. There was nothing in it.\n\n\"Name and number?\" the nurse asked, holding a pen to a clipboard.\n\n\"John...Smith,\" I said. She looked up when I didn't immediately give her my number. Problem was, I could make up a name, but I didn't know how long the darned number should be. I opened my mouth. \"Eight. Six. Four.\" I put a lot of space between them, stopping after each one and watching the nurse. \"Seven. Six. Seven. Two.\"\n\nShe nodded. I shut my mouth. The doctor yanked up the back of my shirt and stuck the needle into my lower back, shoving it in, moving it around like he was fishing for something in there. I ground my teeth against the pain, thinking at the same time that I knew just what they was after. The adrenal glands are right on top of the kidneys. That's where he was poking the needle. I leaned and watched the doctor draw the plunger back, filling the syringe with adrenaline.\n\nIt all fit together. Being scared stimulated the adrenal glands to make more adrenaline. Scare sixty-two thousand people, you got yourself a whole lot of adrenaline. What I didn't get was what you do with all that adrenaline.\n\n\"What happened to _you_?\" Willard said when I met him outside the drugstore. I looked down at my shirt. It looked like I'd had a bucket of black paint thrown on me.\n\nI put an arm around Willard's shoulder, as much to steady myself as to be friendly. I was mighty shaken. \"Come on, I'll tell you about it while we walk.\"\n\nWe passed that same newspaper seller. \"Here,\" I said, handing him the paper. \"I'm done with it.\"\n\n\"Thanks.\" He straightened it up, put it back on the stack.\n\nWe had crossed Eleventh Avenue, and for the first time I could see it: The tops of trees passed by like we were in a river of leaves and branches. I hurried over to the edge.\n\nIt was an awesome sight, standing on the edge, gawking down through the branches at the ground thirty feet below. We passed a brave fellow standing not fifty feet from the city, staring up, his hand shielding his eyes to improve his view. Straight down, the city's stone foundation tore into the Georgia clay, spraying out a steady stream of red dirt and stone and whatnot. We were traveling in a previously traveled track, so there was no risk of flattening anybody's house or town. Not that cities didn't do that on occasion, if they needed to get somewhere and there was no empty way to get there.\n\n\"Good golly,\" I whispered. The locals were just walking on by, not even looking at the passing countryside. I guess they were used to it by now. Me, I could've stood there all day.\n\nIt took two days for New York to reach Chicago, even with the extra adrenaline squeezed out of the residents. I still couldn't figure what to make of that.\n\nI used the time to see the sights of the big city. I got to see way more of New York than I had expected, though mostly we were just marking time, waiting for New York to catch up to Chicago. Rumors flew all over the streets, that Boston was in flames, that Moscow had crossed into the US and attacked Chicago, instead of Chicago attacking Boston. All we knew for sure was that something bad was happening.\n\nWhen news spread that we was getting close, Willard and me followed a crowd to Inwood Park, which was the front fender of this big ole truck.\n\nThe park was brimming with curiosity-seekers; I found space on the low stone edge of the city. We sat and waited, one of us occasionally using the binoculars I'd bought to watch for Chicago as forest and livestock and farmhouses rattled by.\n\nA city woman was standing just a few feet away, her nails painted red, her high-heeled shoes shiny black. She was watching the horizon just like everyone else.\n\n\"Afternoon, ma'am,\" I said, tipping my hat.\n\nShe glanced at me sideways. \"Afternoon.\"\n\nI offered her the binoculars. \"Care to take a look with these?\"\n\nShe thanked me, and took the binoculars, and that got us to talking about this and that as we waited and watched, watched and waited. Her name was Lois\u2014a big-city name if there ever was one. Eventually I got around to asking Lois about the points and all.\n\n\"So what happens if you don't make your points?\" I asked. Now that I understood how you got points it opened up a whole slew of mysteries.\n\nShe set the binoculars on the wall. \"First you lose your electricity, then your water. Then one day you come home and someone else is living in your apartment.\" Willard pulled the binoculars over to him by the strap. \"It doesn't happen often. Most people want to do their part to keep the city strong.\"\n\nI rubbed the spot above my kidney where the needle had sunk in. \"It's quite a price.\"\n\nLois gave me one of those looks that said she felt sorry for me because I was an ignorant hick. \"When your country was being threatened, weren't you willing to fight and die for it?\"\n\n\"My daddy did fight and die for it. Died on Atlanta, in France.\"\n\n\"That's how we feel about New York. I gladly take the needles. Sharing our life fluids joins us to the city.\" She waved a hand in the air, looking all dreamy. \"If you became a citizen, you'd understand.\"\n\nShouts lit up the park. \"Lemme see them specs,\" I said to Willard, wiggling my fingers.\n\nWillard went right on looking. \"I don't see nothing. What are they shouting about? Wait. I see something.\"\n\nWe waited, looking from Willard to the open fields and back to Willard. \"Well?\" I said.\n\n\"I see the city, but I can't see nothing but buildings.\"\n\nI pulled the binoculars out of his hand and took a look for myself. I couldn't see nothing but buildings, either, but Chicago kept getting closer, and soon I could focus on one street. I could see people walking in the street\u2014but not nearly as many as you'd expect in a big city. There were plenty of cars, parked ass to bumper down both sides of the street, but I couldn't see who'd be driving all those cars. I could see a shoeshine stand with the rags draped on a hook, waiting, but there was no one around it.\n\nAs we got even closer, I could take a look at the people up close, and they didn't look right at all. They were walking with their heads down, taking slow, careful steps like they was afraid they might fall down, or step in something they didn't want on their shoes.\n\nNew York jerked to a stop. I'd gotten used to the whistling sound the air made when the city was moving, and now it felt deadly still. I offered the binoculars to Lois.\n\nThere was an awesome rumbling, like the engine of the world's biggest motorcar coming to life. Lois nearly dropped the binoculars over the wall as Chicago leapt forward, jerking and bouncing. It headed straight for New York.\n\n\"Looks like they want to talk,\" Willard said.\n\nLois and me didn't answer. There was something wrong about Chicago. I wanted to get another look at those people walking the streets, but it seemed rude to ask for the binoculars back.\n\nChicago kept on building steam, adjusting itself a bit to the left or right to make sure it was coming right at us. It reminded me of something. I guess a city can't look angry or crazy, but that's what came into my head\u2014that it was like an angry bull.\n\n\"It's gonna run right into us!\" Willard said, his hand knifed over his eyes to shade them from the glare.\n\nThe honking alarm went on, and this time I didn't need nobody to tell me what to do. I hopped off the wall, helped Lois down, and then planted my behind on solid ground.\n\nNew York jerked to life. We waited a nickel's worth, then got back on our feet to watch. New York cut sideways, out of Chicago's path, picking up speed in a hurry. Chicago swerved toward us; it was going to be a close thing.\n\n\"Have they lost their minds?\" Lois asked, holding her hat so it didn't fly off.\n\nI gritted my teeth; if Chicago hit, it would hit right about where we were standing. It was too late to run\u2014if it hit it would chew up this whole park and more.\n\nPeople were running anyhow, as Chicago got bigger and bigger, its skyscrapers rising up over our heads. Willard took two steps. He looked at me, his face a big question mark. I shook my head. \"Too late.\"\n\nNew York shifted direction, digging a big curve and kicking up dirt as it wove forward, in the opposite direction from Chicago. The walls of the two cities matched up, sliding by each other like two pirate ships passing on the water. I watched the Chicago street closest to the wall, watched the people on it peering between their fingers like they was watching _Frankenstein_ at the movies. They didn't want the cities to collide neither.\n\n\"There's someone on the wall,\" Lois shouted, pointing.\n\nSure enough, there was a Chicago man balancing on the wall, his arms spread, his knees bent.\n\nHe jumped toward us, feet pedaling the air. It didn't look like he had enough giddyup to make it. I reached out; the man just made the wall and I grabbed hold of one of his hands as he clawed at the bricks. Lois leaned out and grabbed him under one armpit. We pulled him up and over.\n\n\"Thank you, thank you,\" the man babbled before he even had a chance to brush himself off. \"Oh Lord, thank you.\" He had a cut on his head, just over his eye, and his hands and arms were scraped bad, but he didn't seem to notice. He turned and leaned his hands on the wall, watching Chicago retreat. \"Thank you, Lord.\"\n\n\"What's going on over there, friend?\" I asked, putting a hand on the man's back, trying to calm him. His face was shaking. His hands, too, but I never seen someone so scared that his face shook.\n\nHe fixed me with a crazy stare. \"Your worst nightmare, that's what.\"\n\n\"Look out! What are they doing?\" Willard said, still watching over the wall. The rest of us joined him. Chicago had turned left, right into the path of a small town.\n\nIt plowed right over the town.\n\n\"Jesus, Lord,\" Willard wailed.\n\nWe could hear screams mixed with crunching wood and brick as Chicago mowed down house after house. People ran, but where was there to go? It was a whole city; no one was going to outrun a city.\n\nThere was a commotion behind us; we turned to see soldiers rushing toward us, some carrying rifles, some pulling cannons.\n\nWe took the fella from Chicago, whose name was Perry, to a diner on Eighth Avenue. We offered him scotch, but he wanted tea. He was a wispy sort, skinny and small with nervous hands and a mustache that had grown in too thin, leaving spaces where the skin showed through. He didn't want to talk about Chicago\u2014he held the cup close to his face like he could hide behind it and kept shaking his head and saying, \"You don't want to know.\"\n\nI did want to know, so finally I reminded him that he'd be nothing but roadkill if it wasn't for us, and that maybe he owed us an explanation.\n\n\"All right.\" Perry put the cup down, leaned back on the couch. \"I'll tell you. But I don't want to.\" He closed his eyes, took a deep breath, and let it out with a wheeze. \"I don't want to think about it. I want to pretend it was a bad dream.\" He looked like he was going to cry.\n\nHe collected himself, and finally started talking.\n\n\"One of the five lived in Chicago\u2014Crowley, the conjuror. He didn't hold any office, but he may as well have been king the way people did what he wanted. What he wanted was more life fluids for the city. He pressured Mayor Cermak into raising the requirements again and again, until we were all being siphoned on a weekly basis.\"\n\n\"Oh, my,\" Lois said, shaking her head.\n\nPerry shook his head in a 'You ain't heard nothing yet' way. \"Then people started dying. In their bathtubs, elevators, the subway. At first it seemed like a rash of freak accidents, but then word spread that there was a detail being covered up by the coroner: the victims were dry\u2014they'd been drained.\"\n\nWillard opened his mouth to speak, but Perry waved him silent.\n\n\"There was panic. A lot of residents would have fled, but Chicago was moving, and it kept moving. And people kept dying. More 'accidents.'\" He stopped, fixed me with a stare that dared me to hear what he was about to say. \"Then the city began killing openly. Gouts of steam might come up through a grate in the sidewalk as you walked over it, boiling you alive. The corner mailbox might close on your arm and squeeze until it cut your arm right off\u2014\"\n\n\"Stop it!\" Lois leapt to her feet. \"You're lying. Obviously you're lying. If you're not going to tell the truth, why\u2014\"\n\nPerry laughed. \"What a wonderful thought. God, how I wish I was lying. How I wish all the things I've seen...\" He balled his fists and rubbed his eyes, like he was trying to erase all the things his eyes had seen.\n\n\"Go on, Perry,\" I said, nice and soothing.\n\nPerry dragged his hands down his face, making pink pockets under his eyes. \"Those of us who managed to stay alive have been sleeping in the parks, keeping away from the places where the city gets you.\"\n\nLois snorted, like she thought it was nothing but a fish story, but the rest of us ignored her. I believed the man, crazy as the story was. Maybe I believed him _because_ the story was so crazy\u2014no one would make up such a yarn and expect anyone to believe it.\n\n\"There aren't many people left alive in the city to provide fluids. But it keeps going. I'm not sure how.\"\n\n\"You talk like the city is alive,\" I said. \"You don't mean that, do you? That Crowley fella's got to be down there pulling the strings.\"\n\n\"Oh, the city's alive,\" Perry said.\n\nNobody said nothing for a while. I sipped my scotch, glad to have something stronger than tea to soothe my nerves. Now I wasn't sure if I believed Perry or not. The whole durn city was alive? That was a big acorn to swallow.\n\nI'm not the sort of man who likes to hear things secondhand\u2014I like to dig right into the incision and see what's going on under the skin. But there was no way I was going to get into Chicago to see for myself, even if I was crazy enough to want to.\n\n\"For the longest time I didn't understand what Crowley needed all that extra fluid for,\" Perry said, breaking the long silence. \"The city didn't move _that_ much faster than before. Then it came to me: That's how Crowley brought the city to life. Or more to life, anyway. I think they're all a little bit alive. I think that's how the dream team got them to move.\"\n\nThat got me thinking: If I couldn't see what was going on under Chicago, seeing the engine that drove New York City was the next best thing. I needed to figure out how to get down there.\n\nAs the others jawed about Chicago, I thought it through. There's usually a way to get where you want to go if you're persistent enough. And I was a persistent son of a gun. My momma said that to me all the time.\n\n\"Lois?\" I said, interrupting the conversation. \"You've honestly never seen the engine that drives New York?\"\n\nLois shook her head. \"No one is allowed down there except the people who run it. I once tried to talk my boss into getting me a tour, and he claimed that even _he'd_ never seen it.\n\n\"Who's your boss?\" I asked.\n\n\"I'm a secretary for the commissioner of the Department of Sanitation.\"\n\nShe looked like she wanted me to act impressed, but I was too busy thinking about that engine. Now I really wanted to see it. How does a person get under a city? The trains ran under the city. I thought about the story I read about tunnels, about people living down there and not coming up for years. Maybe some of them tunnels led to the engine room? It was worth a shot; sitting in a diner sure wasn't getting me anywhere.\n\nThere was a bowl on our table filled with matchbooks. I dug out a handful and headed for the door.\n\n\"Where you going?\" Willard said, standing.\n\n\"You stay here.\" I put a hand on his shoulder. \"I'll be back soon.\" Willard protested, probably because he didn't like the idea of being left with a couple of city folk he didn't know, but I insisted.\n\nOn my way to the subway station, I thought about Chicago. Most people looking at a city that sucked up more and more of its residents' juices would think of it like a drug addict or some such, but it made me think of a virus, because of my doctoring background and all. A virus sucks the juices out of its host, but knows if it sucks too hard the host will die and it'll starve. Chicago reminded me of a virus that had developed too big an appetite, draining off the hosts. Of course that was only if Chicago really was alive, and Crowley or some other fella wasn't down there steering. I still wasn't convinced of that.\n\nI took the train to the Garment District. The article I read in _Life_ magazine about New York City said the engine was underneath the Garment District. When I got to my stop I walked to the far end of the train platform, where the tunnel opened up. I hopped off the platform when nobody was looking and ducked into the tunnel, easy as pie.\n\nAbout a hundred feet in, I lit a match. There wasn't much to see besides the tracks, concrete walls, and a couple of Coke bottles probably dropped by track workers. I shook the match out and moved on another hundred feet.\n\nThe fifth time I lit a match I spied a ladder disappearing into a hole in the floor near the wall. I climbed down.\n\nI spent the next two hours winding through a maze of sewer tunnels, maintenance tunnels, stairways, ramps, and such, every so often hitting dead ends that forced me to back up and try another way, all the while lighting matches so I didn't break my leg. I didn't see no people living down there, only rats and bugs. Finally, I heard a faraway thumping sound that was so deep I felt it in my guts more than my ears. I followed the sound. It kept on getting louder until I felt like I was standing next to someone beating on one of them big parade drums.\n\nI saw beams of light squeezing through a crack in the side of the sewer tunnel I was walking in. The crack was big enough for me to squeeze through as well. I ended up in a little room with a couple of tables, some cabinets, and an icebox. I thought maybe it was a break room for workers. I went to the door, opened it just a little, and looked out.\n\nLooking through that door was like being tossed off a thousand-foot cliff. My legs lost all their standing power and I nearly fell on my backside.\n\nThe electrical currents made the most sense to my eyes. They were jumping through the air like lightning bolts, crackling and dancing, but at least I knew what they were.\n\nBelow them were people lying on tables with their eyes taped shut. Their lips moved like they was trying to form words, or from the looks on their faces, maybe screams. There were tubes feeding out of them\u2014fleshy, like intestine, but you could see partly through them so the color of what was in them came through. Some were dark red with blood, some milky white with I don't know what. There was black, and rust, and green.\n\nBig pools with geysers of colored liquid spurted from the floor toward big colored balls that spun in the air like little planets in a solar system. As far as I could see they weren't hanging from strings, they were just spinning in thin air.\n\nThere were long tubes of skin wriggling like angry babies, glass chambers filled with bubbles, a cube made of water spinning in the air. I couldn't figure out what the walls were made of. It was soft and shiny, wrinkly in places. Faces climbed the walls, then disappeared at the top, replaced by more faces down below.\n\nOne of the faces winked at me; or at least, I thought it did.\n\nThe sounds were just terrible. Moans and groans, burbling and boinging. Burps and farts so loud they nearly busted my eardrums. Everything I'd read about cities talked about the _machines_ that drove the cities, the _engines_. These weren't machines, that was for damned sure.\n\nPeople ran around in this mess, crawling underneath things, climbing up things, shouting back and forth. They looked exhausted. The one closest to me\u2014a gray-haired man\u2014was tugging on fat nipples all hanging in a row, like he was playing an instrument. Different color liquids squirted from each one he tugged and fell into a hole that looked like a big mouth, toothless like my grandpa's.\n\nHe looked my way, did a double take, and stopped what he was doing. \"Hey!\" he shouted. \"Who are you?\"\n\nI closed the door and ran. I'd seen enough anyway. Enough to give me nightmares for the rest of my life. Quivering like a newborn piglet, I traced my steps back to the train station and headed back to the diner.\n\nI had no trouble believing that Chicago was alive. These people were playing with things they didn't understand\u2014living things.\n\nI burst through the door of the diner, found my friends where I'd left them. \"We got to take Perry here to see your boss.\" If her boss was the commissioner of something, then he was important, and someone important needed to hear what Perry had to say, and what I had to say too.\n\nLike I suspected, Lois's boss led us right in when we explained who Perry was. Perry told him what he knew about Chicago, and lickety split, Lois's boss was leading us to _his_ boss, who was the mayor of the whole danged city.\n\nThe mayor wasn't in his office. He was on the roof, watching Chicago. Lois's boss took us up.\n\n\"Oh Lord,\" Willard said when we reached the roof and looked beyond the city walls.\n\nChicago was chasing us. New York was high-tailing it, but Chicago was closing in.\n\n\"Jost,\" the mayor said to Lois's boss, \"set up extraction stations all over the city. Send out a directive to the citizenry: We need fluids, all sorts, as much as we can get.\"\n\nJost looked big-eyed scared, like he didn't think he knew how to do what the mayor was telling him to do. \"Should I set up events to stimulate adrenal flow?\"\n\nThe mayor jerked a thumb toward Chicago. \"There's a mad city chasing us. I think everyone's scared enough.\" He noticed us for the first time, looked us up and down. \"Who are you?\"\n\nJost ran off, leaving us to explain ourselves to the mayor. We watched as Chicago closed in. Every so often it changed course so it could run over a farmhouse or a pasture full of cattle. I wondered about that\u2014why would it waste time wrecking things if it was trying to catch us? Off in the distance, I heard people scream as Chicago bore down.\n\nThen it came to me. \"It's eating!\" I said out loud.\n\nThe mayor grunted, the way people do when they aren't really listening to you but pretend they are.\n\n\"I'm telling you, it don't have any nourishment left inside its own walls, so it's eating.\"\n\nChicago was closing in. It was so close I could see people on its streets. Looked like there were even less than there'd been before. New York started a wide turn, trying to shake Chicago, but it wasn't working this time.\n\n\"It's going to hit us!\" Lois cried, covering her mouth.\n\nWe all took a step back, away from the edge of the roof as Chicago plowed right into the tail end of New York. Stone and concrete flew as the underside of Chicago lifted up, sliding into New York. Two New York skyscrapers toppled back into Chicago's own buildings as Chicago jerked forward like it was gonna mount New York. It crushed buildings and people, trying to climb right on. It was an awesome, terrible sight.\n\nNew York jolted forward, a burst of speed that pulled it free of Chicago and dropped Chicago back to the ground with a _boom_. Chicago sat there a minute, like it was catching its breath.\n\nI turned to Lois. \"Do you know where we're headed?\"\n\n\"No,\" she said as Chicago started moving, coming after us again.\n\n\"Well, where are we now? Do you know?\"\n\nShe looked up toward the blue sky, trying to recall. \"Mississippi, I think.\"\n\nI checked the sun. We were heading northeast. \"If we go this way, there's gonna be more and more people for that thing to feed on. Hey!\" I grabbed the mayor's shoulder. He looked at me for real now, startled. \"Where are we going?\"\n\n\"New Jersey. Detroit and Baltimore are there.\"\n\n\"No, no,\" I said, shaking him like I could shake sense into him. \"Don't you see? You got to starve it.\" I pointed southwest. \"If we head that way we'll hit Kansas, Texas. Nothing but dirt and the occasional possum.\"\n\nThe mayor squinted, like he was thinking hard. \"What is this again?\"\n\nI took a deep breath, trying to keep my patience. \"Chicago is sucking up the people it runs over in them houses. That's where it's getting the nourishment to keep moving. You need to go where there ain't many people.\n\nHe looked back at Chicago, watched it veer to run over a little hamlet set along a lake. People ran from their houses, screaming.\n\nThe mayor put a hand on his forehead, then nodded real slow. \"My God, I think you're right.\"\n\nHe shouted to a man in a black suit and hat who was hanging back by the building's water tower. The man ran right over. The mayor told him to send word to change course.\n\n\"Watch it, watch it!\" Willard cried.\n\nChicago was closing again, moving fast.\n\nCannons flashed and boomed from atop some of the buildings along the edge of New York, hitting the front end of Chicago. A few shots hit directly on the wall in places where it hadn't been crushed in the collision. Bricks and mortar sailed into the air.\n\nChicago kept on coming though, like a shark smelling blood in the water. In the spaces between buildings I caught glimpses of people in downtown New York fleeing uptown. They were running, riding bicycles; cars and buses were jammed in the intersections, honking and bumping into each other.\n\nBelow us the street was one long line of people waiting to donate fluids. Mommas with their babies, soldiers, old folks, everybody was out there. I watched a kid who couldn't have been fourteen running with something in his hand. It must have been some of the fluids they were collecting, because he was shouting something, and people got out of his way like he had smallpox or something.\n\nThe mayor shouted orders; New York left the wide trail we'd been following, plowed into virgin forest. Chicago clipped us as we turned, taking out a half dozen tenement buildings and a green rectangle of park.\n\n\"Come on!\" the mayor shouted. \"Where's the extra juice?\"\n\nWe watched Chicago close, close.\n\nA jolt came that nearly knocked me off my feet. A couple of people on the roof did fall down. Everyone cheered. Chicago faded behind us as our extra juice kicked in.\n\nOur little gang left the roof, went to the closest extraction station to do our part. There weren't no monsters to scare us, but the needles and tubes scared me pretty good by themselves. They stuck me in all sorts of places. It was terrible, but I gritted my teeth and took it.\n\nBy the time we got back to the roof, there were a dozen people up there with the mayor. He gave me a big friendly hello and slapped me on the back as we turned to watch Chicago. It was half a mile behind.\n\n\"My navigators are plotting a course through the most sparsely populated areas,\" he told me, his words a mite hard to understand because he was chomping on a big cigar.\n\nWaiters brought us dinner on the roof\u2014beef wrapped up in pastry dough, and champagne. I never had champagne before. I never had beef wrapped up in dough neither, come to think of it. By the time we got to dessert (a sort of cake filled with a chocolate pond), Chicago was a good mile behind. The mayor told the man in the black suit to get them to slow New York down, so Chicago wouldn't give up following. We watched Chicago gain on us for a while. It was moving a good deal slower. It was getting hungry.\n\nIt kept on slowing, and so did we. The landscape got scrawnier, and by the time the sun set there was nothing out there but scrub pines and jackrabbits reflecting in the moonlight.\n\nWe stayed on the roof, and it was like a party. They brought up three musicians with fiddles and I danced with Lois, who was surely paying me more attention than she had before. I looked into her eyes as we danced, enjoyed the feel of her waist. On other rooftops other people were having parties of their own. None of them was as fancy as ours, but they looked like they was having fun.\n\nJust before sunrise, Chicago jerked to a stop. Cheers rose up from the rooftops. New York swung around and pulled near Chicago, though not too close. We watched soldiers trot across the open ground, shoot ropes over Chicago's walls and climb up and over.\n\nIn no time, we got the word: Chicago was dead.\n\nThe mayor turned and offered me his hand. \"I'm very grateful, citizen.\"\n\n\"I ain't a citizen,\" I said, holding up my arm to show him the bracelet I'd all but forgot. \"I'm just a visitor.\"\n\nThe mayor looked left and right, found the fella in the black suit. \"Get someone to take that bracelet off him.\" He turned back to me, held up his hand like he was a priest set to benedict me. \"By the power vested in me, I proclaim you a citizen of New York, with all of the benefits afforded by said citizenship.\"\n\nI thanked him. It was a right friendly gesture on his part. I wasn't so sure I wanted to stay and be a citizen, though. I motioned at Chicago. \"So, what are you gonna do about the thing living under your own streets?\"\n\nHe gave me a puzzled look.\n\n\"You know, the one with all those faces and tubes and such.\"\n\nNow the mayor looked stunned, but he didn't ask me how I knew what it looked like. Maybe he figured it was a lucky guess. \"What do you mean, what am I going to do about it?\"\n\nI ran a hand through my hair, trying to pick out a way to say something that wouldn't be polite if I didn't say it just right. \"I guess what I'm trying to say is, if you keep on feeding those mouths down under the streets, what's to keep the same thing from happening to New York?\"\n\nThe mayor chuckled, polite-like, the way you do when someone says something ignorant.\n\n\"From what your friend Perry tells us, Chicago got greedy. We're not going to make that mistake. We'll take it slowly, maintain control. And one day...\" He held up his hand, flat, with the palm down, and swept it around like it was a bird. \"One day we're going to fly.\" He opened his eyes wide and smiled at me.\n\nI smiled back. \"Well, good luck with that. But I think I'll sit that one out on good old Mother Earth.\" I clapped him on the shoulder and headed for the stairwell. Lois, Willard, and Perry followed.\n\n\"Do you realize what the mayor just did for you?\" Lois said, catching up to me.\n\n\"I ain't staying in this city if y'all intend to keep on feeding that thing in your basement.\"\n\n\"We can't just _stop_. This is the future. You heard what he said, we're being careful.\" She stopped walking. \"Charles, slow down!\"\n\nI stopped, went back and put my hands on Lois's shoulders, and looked her right in the eye. \"You need to stop. This ain't safe. We were desperate when they created these things. I ain't sure anyone really knows what could come of this.\"\n\n\"So we should just go back to being a lump?\" She jerked her thumb over her shoulder. \"Like those towns back there that Chicago ate?\"\n\nThere was something in her eyes, a look I'd seen somewhere before. Maybe in my uncle Ed, when he was looking for a dollar to buy his next bottle. \"Didn't you hear the mayor? He wants to _fly_ this thing!\"\n\n\"The mayor knows what he's doing,\" Lois said. She didn't sound all that convinced.\n\nIt suddenly struck me that city living had pried Lois's senses right from the hinges. The same with everyone else in New York. Did the president know what had happened to Chicago, what might happen to New York and the others? Surely he knew. I tried to imagine the US Army's bitty little howitzers, its planes like gnats, taking on New York. What was easier to imagine was another of these cities gone insane, plowing over little old Siloam, eating me and Willard, Momma and Daddy, everyone, like it was nothing, then moving on to find the next town.\n\n\"I guess I'm just not a city boy, Lois.\"\n\nShe looked at me with her big, brown, crazy eyes. \"Maybe you could be, if you tried.\"\n\nI shook my head. \"No. I'm just a country doctor.\" I gave her shoulders a squeeze, then headed for the city gates with Willard huffing to keep up.\n\n\"How we gonna get home, Charles? We ain't got but a few dollars left.\"\n\n\"We'll figure something out, Willard. Don't you worry.\" It was the least of our worries. We needed to leave the country, move to some island that needed doctors and didn't have any living cities. I needed to convince as many of my friends and kin as I could to come along.\n\nThere was a crowd gathered in the middle of the street on Eleventh Avenue, looking down at something in the road. Police officers were detouring traffic down side streets while a couple of other officers kept the crowd back. We went over and eased ourselves to the front to see what was going on.\n\nA fellow had fallen into an open sewer while crossing the street, and workers were down there trying to bring him up. I eyed the manhole cover, lying in the street a foot from the open hole. Probably some yahoo had pulled it off as a prank. I couldn't help thinking, though, that the mayor had fed the city a heap of fluids so they could outrun Chicago. A heap of fluids.\n\n\"Let's get going,\" I said to Willard, tugging on his shirt.\n\nThrough the gates I drank in the sight of solid, unpaved ground. Planting my feet on grass and dirt would make me feel at home enough for now. I turned to get one final eye-level look at New York City. I squinted toward the skyscrapers on the far end, where Chicago had taken its bite. I'd swear some of the ones Chicago knocked down were looking a little less knocked down. It was a long way off, though, and it might have been a trick of my eyes.\n\n# Meet the Author\n\n _Paul Harrison_\n\nWill McIntosh is a Hugo award winner and Nebula finalist whose latest novel, _Defenders_ , has been optioned by Warner Bros. for a feature film. His previous novel, _Love Minus Eighty_ , was named the best science fiction book of 2013 by the American Library Association, and was on both Io9.com and NPR.org's lists of the best SF novels of 2013. His debut novel, _Soft Apocalypse_ , was a finalist for a Locus Award, the John W. Campbell Memorial Award, and the Compton Crook Award. Along with four novels, he has published fifty short stories in venues such as _Asimov's_ , _Lightspeed_ , and _Science Fiction and Fantasy: Best of the Year_. Will was a psychology professor before turning to writing full-time. He lives in Williamsburg, Virginia, with his wife and their six-year-old twins. You can follow him on Twitter @willmcintoshSF, or on his website, www.willmcintosh.net. \n\n# Also by Will McIntosh\n\n _Soft Apocalypse_\n\n_Hitchers_\n\n_Love Minus Eighty_\n\n_Defenders_\n\n### WILL MCINTOSH SHORT FICTION\n\n\"The Perimeter\"\n\n\"The Heist\"\n\n\"Watching Over Us\"\n\n\"City Living\"\n\n## If you enjoyed \nCITY LIVING, \nlook out for\n\n## DEFENDERS\n\n### _by Will McIntosh_\n\n_Our Darkest Hour._\n\n_Our Only Hope._\n\n_The invaders came to claim Earth as their own, overwhelming us with superior weapons and the ability to read our minds like open books._\n\n_Our only chance for survival was to engineer a new race of perfect soldiers to combat them. Seventeen feet tall, knowing and loving nothing but war, their minds closed to the aliens._\n\n_But these saviors could never be our servants. And what is done cannot be undone._\n\n# Prologue\n\n# Lieutenant Enrique Quinto\n\n _June 26, 2029. Morris Run, Pennsylvania._\n\nIt was a quaint Pennsylvania town, many of the buildings well over fifty years old, with green canopies shading narrow doorways. Even the town's name was quaint: Morris Run. If not for the abandoned vehicles, filthy and faded by two years of exposure to the elements, and the trash stacked along the sidewalk, Quinto might have expected someone to step out of the Bullfrog Brewhouse and wave hello.\n\n\"Lieutenant Lucky?\" Quinto turned to see Macalena, his platoon sergeant, making his way to the front of the carrier. Quinto wished he'd said something the first time someone called him Lucky, but it was far too late now. Most of the troops he was leading today probably didn't know his real name.\n\n\"One of the new guys shit his pants,\" Macalena said when he drew close, his voice low, giving Quinto a whiff of his sour breath.\n\nQuinto sighed heavily. \"Oh, hell.\"\n\n\"The kid's scared to death. He hasn't been out of Philadelphia since this started.\"\n\n\"No, I don't blame him.\" Quinto looked over Macalena's shoulder, saw the kid perched on the side of the carrier, head down. He was about fourteen. The poor kid didn't belong out here. Not that Quinto couldn't use him; they called raw recruits \"fish food,\" but sometimes they were surprisingly effective in a firefight, because they were too scared to think. The starfish could get less of a read on what they were going to do, which way they were going to point their rifles. Usually the newbies didn't shit their pants until the shooting started, though. \"Does he have a spare pair?\"\n\nMacalena shook his head. \"That's the only pair he owns.\"\n\nQuinto reached into his pack, pulled out a pair of fatigue pants, and handed them to Macalena. \"I hope he's got a belt.\"\n\nMacalena laughed, stuck the pants under his armpit, and headed toward the kid.\n\nWhat an awful thing, to be out here at fourteen, fifteen. When Quinto was fourteen, he'd spent his days playing video games, shooting bad guys in his room while Mom fetched fruit juice and chocolate chip cookies and told him when to go to bed.\n\nThey reached the end of the little downtown, which was composed of that single road, and the landscape opened up, revealing pine forest, the occasional house, mountains rising up on all horizons. There was little reason for any Luyten to be within eight miles of this abandoned backwater town, but they were all out there somewhere, so there was always a chance they'd be detected.\n\nQuinto tried to access his helmet's topographical maps, but the signal still wasn't coming through. He pulled the old hard copy from his pack, unfolded it.\n\nThe carrier slowed; Quinto looked up from the map to see what was going on. There was a visual-recognition drone stuck in a drainage ditch along the side of the road. As they approached, the VRA drone\u2014little more than a machine gun on treads\u2014spun and trained its gun on each of the soldiers in turn. When it got to Quinto, it paused.\n\n\"Human. _Human!_ \" Quinto shouted, engaging the thing's vocal-recognition failsafe. It went on to the next soldier.\n\nIt was always an uncomfortable moment, having a VRA drone point a weapon at you. You'd think it would be hard to mistake a human for a Luyten.\n\nFailing to identify anything that resembled a starfish, the gun spun away.\n\n\"Get a few guys to pull it out of the ditch,\" Quinto said. Four troops hopped out of the transport and wrestled the thing back onto the road. It headed off down the road, continuing on its randomly determined route.\n\nPleasant Street dead-ended close to the mouth of the mine, about half a mile past an old hotel that should be coming up on their left. When they got to the mine they'd have to unseal it using the critical blast points indicated on the topo map, then a 2.5-mile ride on the maglev flats into the mine, to the storage facility.\n\nIf someone had told Quinto two years ago that he'd be going into an abandoned mine to retrieve seventy-year-old weapons and ammo, he would have laughed out loud.\n\nIt wasn't funny now.\n\nThe locomotive and five boxcars were parked right where they were supposed to be\u2014as close to the mouth of the mine as the track would allow. They were late-twentieth-century vintage, the locomotive orange and shaped like a stretched Mack truck. Quinto called Macalena and his squad leaders, instructed them to set the big recognition-targeting gun they'd brought along in the weeds on the far side of the road, and place two gunners near the entrance with interlocking fire. When that was done, they got the rest of the squads moving down the tunnel. The quicker they moved, the sooner they'd be out of hostile territory and back in Philly.\n\nQuinto took up the rear of the last carrier for the ride down into the mine. He was not a fan of deep holes with black walls, and when his CO had first laid out the mission Quinto had nearly crapped his own pants.\n\nMacalena climbed in and took the seat beside him.\n\n\"So what are we looking for? I cannot for the life of me guess what we're doing in here.\"\n\nQuinto smiled. It must seem an odd destination to the rest of the men, but they were used to being kept in the dark about missions. The fewer people who knew, the less likely the starfish were to get the information. Or so the logic went.\n\n\"The feds have been sealing huge caches of weapons in old mines for the past two centuries, waiting for the day when Argentina or India or whoever took out our more visible weapons depots. They coat them in Cosmoline and pretty much forget about them.\"\n\nMacalena frowned, sticking out his big lower lip. \"You mean, old hand grenades and machine guns and shit?\"\n\n\"More or less. Flamethrowers with a pathetically limited effectiveness range, eighty-one-millimeter mortars, LAW rockets, fifty-cal MGs.\" Most were outdated weapons, but simple, easy to operate.\n\nMacalena shook his head. \"So we're that desperate.\"\n\nIn the seat in front of them a private who was at least seventy was clinging to the bar in front of her seat. She was tall\u2014at least six feet. The slight jostling of the carrier was clearly causing her old body discomfort. It was true what they said: There were no civilians anymore, only soldiers and children.\n\n\"Yup. We're that desperate,\" Quinto said. \"They've destroyed or seized so much of our hardware that we have more soldiers than guns.\"\n\n\"What's Cosmoline?\" Macalena asked.\n\n\"I didn't know, either; I had to look it up. It's a grease they used back in the day to preserve weapons. Once you chip away the hardened Cosmoline, the weapons are supposed to be like new.\"\n\nMacalena grunted, spit off the side. \"Dusty as hell in here. And cold.\"\n\n\"Let's be glad we're not staying.\"\n\nMacalena's comm erupted, a panicked voice calling his name.\n\n\"What have we got?\" Macalena asked.\n\n\"Vance is dead. Lightning shot, from the trees to the left of the mine.\"\n\n\" _All stop!_ \" Macalena shouted. The carrier slowed as Quinto dropped his head, covered his mouth as the implications sunk in.\n\nLucky no more.\n\n\"Where are you now?\" Macalena asked the private.\n\n\"Inside the mine, about a hundred yards.\"\n\n\"Stay there.\"\n\nQuinto looked up at Macalena, who raised his eyebrows. \"What do you want to do?\"\n\nHe wanted to get as deep in the mine as he could, and stay there, their backs against the wall, weapons raised until the starfish came to get them. Of course the Luyten would never come down, because they were reading his thoughts right now. Plus it was far easier to blow the mouth of the mine and leave them to suffocate.\n\nQuinto ordered the small caravan to turn around and head toward the mouth.\n\nThey barely got moving before they heard the flash-boom of a Luyten explosive. The cave shook; bits of dirt and debris spewed at them, then everything settled into silence, the cave now truly pitch-black, save for the carriers' headlights.\n\nThey climbed out of the carriers. Some of the troops cried, and there was no shame in that. One woman went off to the side of the tunnel and knelt in the rubble to pray. Quinto didn't know their names, because he hadn't served with them long. Troops came, and died, and new troops came. Only Lieutenant Lucky went on, mission after mission. Quinto realized he'd begun to believe he really was lucky, or special. Destined to see the war to its end.\n\nIt killed him, to think he wouldn't get to see how things turned out, whether the bad guys won, or the good guys pulled something out of their asses at the eleventh hour.\n\nQuinto used the walkie to apprise HQ of their situation, so HQ wouldn't wonder when Quinto's platoon never returned.\n\n\"Lieutenant?\" Macalena said. He was studying the topo map he'd borrowed from Quinto. \"Did you see these?\" A few of the enlisted came over to look at the map over Macalena's shoulder as he ran his finger along black lines set perpendicular to the mine. \"There are five vertical shafts sunk along the length of the mine. I'm guessing they were escape routes in case of collapse, or ventilation, or both.\"\n\nQuinto looked up from the map, impotent rage rising in him. \"Jesus, Mac, couldn't you have waited a half hour to notice this?\"\n\nIt took Macalena a second to understand. When he did, he grimaced, curled his hand into a fist, crumpling a section of the map. He turned and walked a dozen paces down the shaft, cursing quietly, viciously.\n\nEven Macalena was too green for this war. He'd been in the infantry for only four months; before that he'd been writing military technical manuals. The army needed fighters more than writers these days.\n\nIf Macalena had waited even fifteen, twenty minutes before examining the old map, chances were the Luyten would have been out of range, and they could have climbed out of this hole and gone home.\n\n\"We need to move,\" Quinto said. \"The fish are going to find those exits and seal them up. Spread out, find the exits. When I get to the surface I'm going to set off a Tasmanian devil, give us some breathing room. As soon as it's spent, get out there. Understood? Let's move.\"\n\n\"Couldn't we just stay down here? Dig our way out when they're gone?\" It was the kid who'd crapped himself, looking absurd in Quinto's big pants. \"If we go up there now, they'll kill us. I mean, maybe they'll get distracted by something and leave...\" He trailed off.\n\nEveryone stared at the ground, except for the soldier who was praying.\n\n\"Let's go,\" Quinto said.\n\nQuinto grasped the cold rung of the ladder that had dropped down when they unsealed the iron hatch.\n\n\"Good luck to you, Lieutenant,\" one of the troops waiting to follow him called. It was Benneton, the old woman. The kid who'd crapped his pants was there as well, along with four others.\n\nQuinto looked up into darkness. \"Here we go.\" He headed up the ladder. A lot of people who'd been as lucky as Quinto might have been tempted to believe the streak would hold, but Quinto knew his past held no hint of his future. More to the point, he knew he had no future.\n\nIt was a forty-foot climb according to the map, but adrenaline made it effortless. When he reached the top, he twisted the seal on the hatch, then pushed with his back and shoulders to force the hatch open. Daylight flooded into the dusty shaft as dirt and moldy leaves rained down on him.\n\nThe kid, who was just below him, passed up the Tasmanian devil. Reaching among the big spines jutting from the central carbon-fiber sphere, Quinto activated it, tossed it outside, and pulled the hatch closed.\n\nThe buzzing of razor-sharp shrapnel hitting, and then burrowing around inside everything within five hundred yards, would have been reassuring if Quinto weren't absolutely certain the starfish had retreated outside the Tasmanian devil's range as soon as Quinto thought about using it. At least it would back the fish up so they wouldn't be able to pick off Quinto and his troops as they climbed out of their holes.\n\n\"Here we go,\" Quinto said to the boy. \"Have your weapon out. Run as fast as you can. Try to take one with you.\" His guess was that Benneton would stay behind, shoot from the cover of the shaft until the Luyten cooked her. That's what Quinto would do in her situation; it would probably afford her a few more minutes of life. He took a deep breath, trying to grasp that this was the end, this was the moment of his death, but he couldn't.\n\nAs soon as the Tasmanian devil went silent, Quinto threw open the hatch, his heart thudding wildly, and ran.\n\nTheir carriers were trapped in the mine, so his best chance would be to make it to the locomotive. Of course the Luyten would have fried the locomotive, so really there was nothing to do but run, and when the fish closed in, turn and fight.\n\nTwo hundred yards ahead, he spotted four of his troops running north, into the woods, toward the nearest cover. That probably made more sense than what Quinto was doing, but all of the moves open to them were losers. It was always the same: The fish knew their exact location, but they had no idea where the fish were. If you could catch a fish out in the open, it couldn't dodge automatic weapons fire, but you almost never caught them out in the open.\n\nQuinto glanced back, saw the kid was two steps behind, his dirty cheeks tracked with tearstains.\n\nThe locomotive had been melted to a lump. He kept running. Everyone but he and the kid had headed north. Since Quinto wasn't dead yet, it was safe to assume the fish had gone after the larger group first. If he could get outside their range, which meant seven or eight miles, he and the kid might have a chance. Quinto pushed himself to pick up the pace, but when he did the kid started to fall behind, looking panicked. Quinto slowed.\n\nIn the distance, Quinto heard the worst sound in the world: the sizzle-crackle of a Luyten lightning stick, a sound as much felt in your body as heard by your ears. Then another. He was spared the pungent, unearthly sweat smell of the weapon. He was too far away.\n\nWhen he'd made it through the town, Quinto took another glance back. The kid was a hundred yards behind, one hand clutching his side. No way this kid was going to run another four or five miles. Panting, his throat coated in phlegm, Quinto considered leaving him behind. No. No matter how fast he ran, he wasn't going to outrun Luyten on foot. He could try calling HQ and beg for a carrier to come get him, but they'd only tell him what he already knew: They weren't going to feed the fish any more than they had to.\n\nSo he stopped, pulled out his comm, and waited for the kid to catch up. The kid stopped beside him, put his hands on his knees.\n\n\"You want to call anyone? Your mom or dad alive?\"\n\nThe kid eyed the comm. \"Just my little sister.\" He swallowed, looked at Quinto. \"We're going to die, aren't we?\"\n\n\"Yeah. We are.\"\n\n\"Maybe they got distracted by something. Maybe the others killed them.\"\n\n\"Maybe,\" Quinto said. He thought he heard the snap-crackle of something moving through the woods to the north. \"Come on.\" He tugged the kid's jacket and headed into the woods on the opposite side of the road.\n\nShould he call his own mother to say goodbye? He would like that, but he didn't want to risk having her on the line when he died. He didn't want that to be her last memory of him.\n\nBranches whipped his face as he tore through the brush. It was pointless, but he couldn't relinquish that last millimeter of hope that he might get lucky, just one last time. He barreled down a slope as the landscape opened, then splashed through a stream and raced up the bank.\n\nHe spotted a flash of crimson ahead, behind a thick cover of green leaves, and stopped short. The kid stopped short beside him, looked at him, questioning, just as a bolt of lightning burst through the foliage.\n\n# Oliver Bowen\n\n _March 9, 2030 (nine months later). The South Pacific._\n\nThe door was locked. The room was comfortable, replete with a well-stocked kitchen and an entertainment system that was so up-to-date it contained movies yet to be released. But the door was locked.\n\n _You're considered a risk. They don't know the extent of my power to influence you._\n\nOliver turned in his rotating chair to face Five, whose accommodations were less plush. Behind the carbon alloy mesh that separated them, Five's room was empty except for a water dispensation device that resembled a giant hamster lick. Five was lying flat, his appendages splayed like the spokes of an elephant-sized wheel. His skin had a stony, mottled texture, and there were bristles protruding at evenly spaced intervals across it. The cilia protruding from the tips were as thick as nautical rope, and transparent.\n\n\"Because you were able to win over a thirteen-year-old boy, they think you might be able to convince me that I'm fighting on the wrong side? That's absurd.\"\n\n _But they don't know that_ , Five said. _They think you've become too familiar with me. Too friendly._\n\nThe CIA yanks him out of his position at NYU three days after the invasion begins, shifts him from Research to Interrogation as their field agents die off, tells him to figure out how to communicate with Luyten, and when he succeeds, he becomes a suspected sympathizer? Beautiful.\n\n _The next time someone comes, ask them when you'll be informed where we're going._\n\nOliver couldn't help laughing. \"You mean _you_ don't know?\" He waved in what he guessed was the direction of the submarine's bridge. \"Pluck it out of someone's mind.\"\n\n_I don't have to pluck. Your minds are all laid out in front of me. No one on this vessel knows._\n\n\" _No one_ knows where we're going?\" It seemed an absurd notion, though it also made sense. If no one on board knew where they were going, or why, a Luyten who happened to be flying nearby\u2014within their eight-or-so-mile telepathic zone\u2014wouldn't be able to find out, either. The mission must be important. \"How are they navigating if they don't know where we're going?\"\n\n_They're given a set of coordinates corresponding to a point in the ocean, and when they reach it, they're given another._\n\n\"So where are we?\"\n\nOliver jolted back in his chair as one of Five's mouths opened, revealing a bobbing, twitching hole ringed with teeth that resembled the spines on cacti. Smacking, hissing air and background sounds like water draining came from the hole, the sounds so unearthly and repulsive that at first Oliver didn't register that they were approximating words.\n\n\"Find out where we're going,\" Five said aloud.\n\nThe ubiquitous hum of the sub's engine was the only sound in the room as Oliver composed himself. Ultimately it didn't matter whether the Luyten communicated telepathically or using spoken words, but it was still profoundly disturbing to hear the thing speak.\n\n\"You're just full of surprises, aren't you?\" Oliver said.\n\n\"Unlike you.\" Somehow the creature managed to inject a note of irony, and perhaps contempt, into the awkwardly formed words.\n\nOliver slid out of the chair, went right up to the nearly invisible net of carbon fiber that separated them. \"Don't assume you know my mind just because you can read my thoughts. We may not be as simple as you think.\"\n\n\"Yes, humanity is the pinnacle of evolution. The chosen ones, the purpose for the existence of the entire universe. How could I forget?\" Aware that Oliver was having trouble understanding his strangely formed words, Five simultaneously broadcast his words directly into Oliver's mind, giving him the uneasy sensation of hearing the words with an indescribable overlap. \"I know your mind better than you.\"\n\nOliver grunted, folded his arms across his chest. \"Right.\"\n\n\"You're uneasy. You're afraid I might try to prove my claim.\"\n\nIt was pointless to disagree. Oliver had quickly learned how absurd it was to deny what you were thinking or feeling to something who knew precisely what you were thinking and feeling.\n\n\"You love your wife now\u2014\"\n\n\" _Shut up._ I don't want to hear about Vanessa. _Just leave it._ \"\n\nFive waited patiently through Oliver's outburst, then continued. \"After her affair, her denials, the angry divorce... now you love her. Before, when you claimed to love her, you also despised her.\"\n\nOliver turned, went to the door, and thumped on it with the flat of his palm. \"Hey, come on. Unlock this door. _I'm_ not the POW.\"\n\n\"There's an irony you're not aware of, in your newfound feelings for your wife. Should I share it with you?\"\n\nOliver turned to face Five, who was running the fine cilia that served Luyten as fingers across the stump of the limb he'd lost. \"No. Thanks for the offer, but, no.\"\n\n\"It's something you'd be interested to hear.\"\n\nWhen Oliver didn't answer, Five continued. \"All right, then why don't I move on? What else can I tell you, to demonstrate you're as simple to read as I think you are? How about your deepest sexual cravings? Some of these you would never admit to yourself. For example, you'd like to be tied up, gagged with your own dirty sock, and spanked by a woman twenty years older than you.\"\n\nOliver couldn't care less about his repressed sexual desires. They were what they were; he couldn't control them, only whether he acted on them. But Oliver knew Five was only playing with him now. It had already dropped the bait it knew Oliver couldn't resist.\n\nFive grew quiet, waiting for the question it already knew was coming.\n\n\"Fine. What's the irony I'm not aware of?\"\n\nAll of Five's eyes fixed on Oliver. \"The irony is, your instinct to love her is right, because she never had sex with Dr. Paul.\"\n\nAs the words registered, Oliver's vision darkened around the edges, as if he were going to pass out. In some ways, he wished he would. \"You told me she had. You gave me specific details.\"\n\n\"I lied.\"\n\nAn icy numbness crept through him. He'd destroyed his marriage on the word of an alien bent on wiping out the human race. He'd taken Five's word as unassailable proof, because Five could reach right in and pluck the truth out of Vanessa's thoughts. Only he'd forgotten Five had abilities beyond reading minds. The ability to lie, for instance.\n\nHe'd told Vanessa he knew she was lying, said her unwillingness to admit the affair bothered him more than the infidelity itself. The floor, which was nothing but steel under a thin layer of beige carpeting, lurched beneath him, either because the sub was adjusting course or his knees were wobbling.\n\n\"Why would you lie? I didn't even ask you about Vanessa\u2014you volunteered the information.\"\n\n\"I did it to serve as a reminder.\"\n\n _\"A reminder of what?\"_\n\n\"That I might be lying to you at any time.\"\n\nIt dawned on Oliver that he had no way to contact Vanessa, and had no idea when he would, because he didn't know where he was going, or why. When he did finally contact Vanessa, would an apology make any difference? He'd trusted the word of a Luyten over hers.\n\nThis was going to torture him. In all probability that was Five's intention in telling him now. Or maybe he was lying now, simply to distract Oliver at a crucial juncture.\n\n\"Maybe,\" Five said.\n\n# About Orbit Short Fiction\n\nOrbit Short Fiction presents digital editions of new stories from some of the most critically acclaimed and popular authors writing science fiction and fantasy today.\n\nVisit www.orbitshortfiction.com to learn more about our publishing program\u2014and to join the conversation. We look forward to hearing from you. \n\n### Thank you for buying this ebook, published by Orbit.\n\nTo get news about the latest Science Fiction and Fantasy titles from Orbit, along with special offers and exclusive content, sign up for the Orbit newsletter.\n\nSign Up\n\nOr visit us at www.orbitbooks.net\/booklink\/\n\n## Table of Contents\n\n 1. Cover\n 2. Title Page\n 3. Welcome\n 4. City Living\n 5. Meet the Author\n 6. Also by Will McIntosh\n 7. Bonus Material\n 8. About Orbit Short Fiction\n 9. Orbit Newsletter\n 10. Copyright\n\n## Navigation\n\n 1. Begin Reading \n 2. Table of Contents\n\nThis book is a work of fiction. Names, characters, places, and incidents are the product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is coincidental.\n\nCopyright \u00a9 2015 by Will McIntosh \nExcerpt from _Defenders_ copyright \u00a9 2014 by Will McIntosh \nCover design by Kirk Benshoff \nCover illustration by Kirk Benshoff \nCover copyright \u00a9 2015 by Hachette Book Group, Inc.\n\nAll rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher is unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nOrbit \nHachette Book Group \n1290 Avenue of the Americas \nNew York, NY 10104 \nwww.orbitbooks.net \nwww.orbitshortfiction.com\n\nOrbit is an imprint of Hachette Book Group. \nThe Orbit name and logo are trademarks of Little, Brown Book Group Limited.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nFirst eBook edition: February 2015\n\nISBN: 978-0-316-34130-1\n\nE3\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n# Copyright\n\nCopyright \u00a9 2018 by Mark W. Moffett\n\nCover design by Fort\n\nCover image copyright \u00a9 Amitus\/Getty Images\n\nCover copyright \u00a9 2019 Hachette Book Group, Inc.\n\nHachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.\n\nThe scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nBasic Books\n\nHachette Book Group\n\n1290 Avenue of the Americas, New York, NY 10104\n\nwww.basicbooks.com\n\nFirst Edition: April 2019\n\nPublished by Basic Books, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc. The Basic Books name and logo is a trademark of the Hachette Book Group.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nThe Library of Congress has cataloged the hardcover edition as follows:\n\nNames: Moffett, Mark W., author.\n\nTitle: The human swarm : how our societies arise, thrive, and fall \/ Mark W. Moffett.\n\nDescription: New York : Basic Books, [2019] | Includes bibliographical references and index.\n\nIdentifiers: LCCN 2018038761 (print) | LCCN 2019006348 (ebook) | ISBN 9781541617292 (ebook) | ISBN 9780465055685 (hardcover)\n\nSubjects: LCSH: Sociology. | Communities.\n\nClassification: LCC HM585 (ebook) | LCC HM585 .M64 2019 (print) | DDC 301\u2013\u2013dc23\n\nLC record available at https:\/\/lccn.loc.gov\/2018038761\n\nISBNs: 978-0-465-05568-5 (hardcover); 978-1-5416-1729-2 (ebook)\n\nE3-20190305-JV-NF-ORI\n\n# Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nDedication\n\nEpigraph\n\nIntroduction\n\nSECTION I: AFFILIATION AND RECOGNITION\n\nChapter 1: What a Society Isn't (and What It Is)\n\nChapter 2: What Vertebrates Get out of Being in a Society\n\nChapter 3: On the Move\n\nChapter 4: Individual Recognition\n\nSECTION II: ANONYMOUS SOCIETIES\n\nChapter 5: Ants and Humans, Apples and Oranges\n\nChapter 6: The Ultimate Nationalists\n\nChapter 7: Anonymous Humans\n\nSECTION III: HUNTER-GATHERERS UNTIL RECENT TIMES\n\nChapter 8: Band Societies\n\nChapter 9: The Nomadic Life\n\nChapter 10: Settling Down\n\nSECTION IV: THE DEEP HISTORY OF HUMAN ANONYMOUS SOCIETIES\n\nChapter 11: Pant-Hoots and Passwords\n\nSECTION V: FUNCTIONING (OR NOT) IN SOCIETIES\n\nChapter 12: Sensing Others\n\nChapter 13: Stereotypes and Stories\n\nChapter 14: The Great Chain\n\nChapter 15: Grand Unions\n\nChapter 16: Putting Kin in Their Place\n\nSECTION VI: PEACE AND CONFLICT\n\nChapter 17: Is Conflict Necessary?\n\nChapter 18: Playing Well with Others\n\nSECTION VII: THE LIFE AND DEATH OF SOCIETIES\n\nChapter 19: The Lifecycle of Societies\n\nChapter 20: The Dynamic \"Us\"\n\nChapter 21: Inventing Foreigners and the Death of Societies\n\nSECTION VIII: TRIBES TO NATIONS\n\nChapter 22: Turning a Village into a Conquering Society\n\nChapter 23: Building and Breaking a Nation\n\nSECTION IX: FROM CAPTIVE TO NEIGHBOR... TO GLOBAL CITIZEN?\n\nChapter 24: The Rise of Ethnicities\n\nChapter 25: Divided We Stand\n\nChapter 26: The Inevitability of Societies\n\nConclusion: Identities Shift and Societies Shatter\n\nAcknowledgments\n\nAbout the Author\n\nAlso by Mark W. Moffett:\n\nPraise for The Human Swarm\n\nReferences\n\nNotes\n\nIndex\n_I dedicate this book to three remarkable individuals._\n\n_Firstly, my mentor Edward O. Wilson, out of awe for his poetic spirit, his decades of building connections across the sciences, and his indefatigable support of so many careers,_\n\n_my off-kilter one included;_\n\n_To the late and great Irv DeVore, who fostered critical thinking across generations of anthropologists and put up with hours of conversation with this biologist;_\n\n_And to Melissa Wells, my splendid wife and partner,_\n\n_who believes in me._\nAnd what, I said, will be the best limit for our rulers to fix when they are considering the size of the State and the amount of territory which they are to include, and beyond which they will not go?\n\nWhat limit would you propose?\n\nI would allow the State to increase so far as is consistent with unity; that, I think, is the proper limit.\n\nVery good, he said.\n\n\u2014PLATO, _The Republic_\n\n# Introduction\n\nFor as long as human societies have existed, people have found themselves changed, the members of those societies transformed in their mind's eye into exalted Human Beings. Yet as powerful as belonging to a society can be in raising the citizens' collective self-image, it's not their fellow members that they see most differently\u2014the foreign undergo the more radical, at times dreadful, transformation. In each person's mind, entire groups of outsiders can turn into something less than human, a kind of vermin, even.\n\nThat foreigners might be considered contemptible enough to crush underfoot, like insects, is the stuff of history. Think back to the year 1854, Washington Territory. Seattle, chief of the Suquamish Tribe and namesake of the newly founded city, had just heard Isaac Stevens, the territory's recently appointed governor, speak before the tribal elders. Stevens had explained that the Suquamish were to be relocated to a reservation. Standing to respond, Seattle towered over the slight governor. Speaking in his native Duwamish, he bemoaned the abyss between their societies and recognized that the days of the Suquamish were numbered. Yet he was stoical about the news: \"Tribe follows tribe, and nation follows nation, like the waves of the sea. It is the order of nature, and regret is useless.\"\n\nAs a field biologist I make a living thinking about the order of nature. I've spent years contemplating the concept we call \"society\" while exploring human tribes and nations. I find myself endlessly captivated by the phenomenon of foreignness: the way it turns what objectively are minor differences into gulfs between people that have ramifications reaching into every corner of life, from ecology to politics. The goal of _The Human Swarm_ is to take in as much of this broad sweep as possible by investigating the nature of the societies of _Homo sapiens_ as well as those of other animals. A principle thesis of the book is that, uncomfortable as it may sound, human societies and the societies of insects are more similar than we might like to believe.\n\nFor humans, any little thing can signal foreignness, as I have experienced many times. Mortified souls looked on in India when I picked up my food with the incorrect hand. In Iran I tried to nod yes when to the locals a nod meant no. Sitting on moss in the New Guinea highlands, I watched _The Muppet Show_ with an entire village on an ancient television run off a car battery. Knowing I came from America, and that _The Muppet Show_ was American, every man and woman looked at me quizzically when a pig, a species they revere, waltzed on the screen in a dress and high heels. I talked myself past machine guns during the Tamil ethnic uprising in Sri Lanka and sweated as wary Bolivian bureaucrats sorted out who this very strange person was and what I was doing\u2014or should be allowed to do\u2014in their country. At home I have seen my fellow Americans behave with equal discomfort, bewilderment, and, at times, anger toward outsiders. In a primordial reaction, both sides think _how strange that person is_ despite their profound similarities as human beings with two arms, two legs, and a desire for love, home, and family.\n\nIn _The Human Swarm_ I examine society membership as a particular component of our sense of self, which should be considered (as I do in the final chapters especially) in tandem with race and ethnicity\u2014identifications that can exert the same primacy and emotional draw. The elevated significance of our societies\u2014and ethnicities and races\u2014compared with other aspects of our identity can seem preposterous. Nobel-winning economist and philosopher Amartya Sen, for one, struggles to make sense of why people collapse their identities into groups that override everything else. Using the deadly conflicts in Rwanda as a case in point, Sen rues the fact that \"a Hutu laborer from [the capital city of] Kigali may be pressured to see himself only as a Hutu and incited to kill Tutsis, and yet he is not only a Hutu, but also a Kigalian, a Rwandan, an African, a laborer and a human being.\" These and other sorts of breakdowns are one theme of the chapters ahead. When convictions about what a society stands for or who belongs come into conflict, suspicion mounts and bonding fails.\n\nThe word \"tribalism\" comes to mind, denoting people drawn together by anything from a love of car racing to the denial of global warming. The idea of tribe, used loosely in this way, is a common subject for bestseller lists. However, when we speak of the tribe of a New Guinea highlander, or of tribalism with respect to our own connections to a society, we have in mind how a lifelong sense of belonging elicits love and loyalty\u2014yet also, expressed in relation to outsiders, how it can promote hatred, devastation, and despair.\n\nBefore we get to these subjects we will attend to the most basic of questions, namely: what is a society? As we shall see, there's a major difference between being social\u2014connecting positively with others\u2014and the situation, far less common in nature, wherein a species supports the separate groups we call societies that endure over generations. Being part of a society is not open to choice; the individuals who count among its members are normally clear to all. Outsiders, with their foreignness unmistakable from their appearance and accents, gestures, and attitudes about everything from pigs to whether tipping is seen as an insult, are admitted with difficulty. And then, in many cases, they are fully embraced only after the passage of time\u2014decades or centuries, even.\n\nAside from our families, our societies are the affiliations we most often pledge allegiance to, fight for, and die for. But in day-to-day life the primacy of societies is seldom obvious, forming only a portion of our sense of self and our recognition of how others are different. As part of our daily experience, we join up with political parties, book clubs, poker groups, teen cliques. Those of us on the same tour bus may even bond together, thinking more highly of our travel companions than those on other buses for a time, and as a result perhaps working fruitfully as a group through the problems of the day. A predisposition to join groups shapes us as individuals and has been a subject of extensive research. Meanwhile our society hums along, as easily overlooked as our heartbeat and breath. It comes to the fore, of course, in times of group hardship or pride. A war, terrorist attack, or the death of a leader can shape a generation. Yet even in uneventful times our society sets the tone of our days, influences our beliefs, and informs our experiences.\n\nPondering the sometimes insurmountable differences between societies\u2014be they the swarming populations of continent-wide nations like the United States or local tribes in New Guinea\u2014raises questions of utmost importance. Are societies, and the labeling of others as _foreign_ , part of the \"order of nature\" and therefore unavoidable? Bound by a sense of superiority and vulnerable to the enmity of other groups, is each society doomed to flounder and fall as Seattle supposed, as a consequence of either skirmishes with other societies or a sense of alienation that percolates up among the members of the society itself?\n\n_The Human Swarm_ is my attempt to answer these questions. The argument moves from natural history to prehistory and the fickle course of civilizations\u2014from the mud walls of Sumer to the electronic vastness of Facebook. Behavioral scientists tease apart human interactions in narrow contextual frames, for example by using games of strategy to illuminate how we treat each other. But I endeavor to take a broader approach. Understanding the origin, maintenance, and dissolution of societies\u2014how necessary they are, how they come about, and why they matter\u2014will take us through recent findings in biology, anthropology, and psychology, with some philosophy thrown in for good measure.\n\nHistory plays into the narrative too, albeit more for the patterns it reveals than for its specifics. Each society has its own saga, yet I propose that there can be common underlying forces that both keep societies together and cause them to crash and burn. The fact is, whether through conquest, transmutation, assimilation, division, or death, all societies\u2014animal or human, low-key hunter-gatherer or industrial powerhouse\u2014reach an end. This impermanence is easily overlooked given their longevity is measured in human lifespans. Obsolescence is assured not by hostile neighbors or environmental ruin (though such factors have played a memorable role in the decline of some societies), nor by the fleeting lives of the people themselves, but rather by the transience of the identities the members present to each other and the world. Differences between people carry great weight, with changes slowly turning what was once the familiar into the foreign.\n\nThe human connection with societies has deep origins that extend into our animal past. However, the idea of describing societies in terms of membership, with its implications of ingroup and outgroup, which I adopt from psychology, is unconventional in biology. My colleagues routinely have an aversion, rarely mentioned explicitly, to talking about societies. As an example, even though the English vernacular offers words for the societies of many species (take, for instance, \"troop\" for monkeys and gorillas; \"pack\" for wolves and African wild, or painted, dogs; \"clan\" for spotted hyenas and meerkats; or \"band\" for horses), researchers often shun those terms and simply assign the term \"group,\" with a consequent loss in clarity and understanding. Imagine sitting through a lecture, as I once did, in which an ecologist spoke about a group of monkeys that \"separated into two groups\" and later \"one of the groups clashed with yet another group.\" It took feverish concentration to decipher those sentences: what he meant was that the members of a single monkey troop had headed in two directions, with half that troop coming upon and vigorously defending themselves against another troop. While a troop is undeniably a group, it is a group of a very special sort, being set off from all other monkeys by a closed and stable membership that makes it not just worth fighting over, but worthy of being labeled by a term of its own.\n\nOnce a group\u2014a pack, clan, troop, pride, and so on\u2014forges this sort of singular identity that extends beyond the workaday ties of parents rearing offspring, being part of that society can offer a lot. What characteristics do we share with those animals? How do we differ\u2014and more importantly, does it matter?\n\nWhile animal models are helpful in illuminating the value of societies, they are insufficient to explain how humans got to where we are today. As natural as our great nations seem to most of the world's population, they are not necessary. Before the flowering of civilizations (by which I mean societies with cities and monumental architecture), we humans ranged across the habitable surface of Earth in societies as well, though far smaller ones: in tribes reliant on simple gardens and domesticated animals, or as hunter-gatherers taking every meal from the wild. Those societies were the nations of the day. The forebears of every living person were once affiliated with them, going back eons to a time when humans were all hunter-gatherers. Indeed, many of the peoples of New Guinea, Borneo, South American rainforests, sub-Saharan Africa, and other parts of the world retain their primary connections to the few hundred or thousand individuals in a tribe that carries on for the most part independently of their national government.\n\nTo characterize the earliest societies we can draw on evidence from hunter-gatherers of recent centuries and the archaeological record. The vast countries that now cause hearts to swell with pride would have been unfathomable to our hunter-gatherer ancestors. We will explore what made this transformation possible, leading to societies that continue to discriminate against outsiders even though they have become so numerous that most members are unknown to each other. The casual anonymity that characterizes contemporary human societies may seem unremarkable, but it is a big deal. The seemingly trivial act of entering a caf\u00e9 full of strangers without a care in the world is one of our species' most underappreciated accomplishments, and it separates humankind from most other vertebrates with societies. The fact that the animals of those species must be able to recognize each individual in their society is a constraint most scientists have overlooked, but it explains why no lions or prairie dogs will ever erect cross-continental kingdoms. Being comfortable around unfamiliar members of our society gave humans advantages from the get-go and made nations possible.\n\nThe united multitudes that compose modern human societies are unique in the chronicle of life for species larger than a fingernail. However, my training has been with creatures smaller than that: the social insects, exemplified (from a personal bias) by the ants. The idea behind this book entered my head when I encountered a battlefield kilometers long in a town near San Diego, where two supercolonies of the Argentine ant, each billions strong, defended their turfs. These Lilliputians initially led me, back in 2007, to the question of how a vast number of individuals, ant _or_ human, can truly be a society. This book will address how, much like humans, ants respond to each other in such a way that their societies can be anonymous: we\u2014and they\u2014have no need to be familiar with one another as individuals to keep our societies distinct. This ability opened up the possibility for humans to transcend the size limitations of most other mammal societies, as was first seen in hunter-gatherer societies that grew into the many hundreds and in the end paved the way to history's grand republics.\n\nHow are anonymous societies realized? Our ant-like approach to identifying each other depends on shared characteristics that mark individuals as fellow members. But these markers\u2014which in ants are simple chemicals and in humans can range from clothing to gestures and language\u2014are insufficient to completely explain what holds civilizations together. The conditions favorable to expanding human societies, compared to those of small-brained ants, have been stringent and fragile. People draw from other epoch-tested skills in their mental toolset to make life tolerable in response to growing numbers of fellow members. The enhancement of differences between individuals by way of jobs and other distinctions (our groupiness) is part of the package. Perhaps more surprisingly, the emergence of inequalities, which I shall discuss especially as manifested in the rise of leaders, is likewise important to building a society's population. We take such phenomena for granted, but they varied remarkably among hunter-gatherers, some of whom lived in roving societies of equals.\n\nThe coexistence of various races and ethnicities within a society has largely come about since agriculture, an extension of the willingness, previously mentioned, to accept distinctions between individuals, including the authority of others. Such a union of formerly independent groups was unheard of among hunter-gatherer bands, and indeed is found in no other species. Nations would not have taken hold without people repurposing their cognitive survival tools to take in, and adjust to, different ethnic groups. This allowance for diversity comes with stressors that can ultimately strengthen a society, but may also tear it apart. So while the success of the melting pot is good news, only so much melting goes on, with the issue of we-ness at the root of riots, ethnic cleansings, and holocausts.\n\nThroughout, my aim is to intrigue you with mysteries, some of consequence, others odd but enlightening. For a preview, elephants of the African savannas have societies while Asian elephants lack them. We will tackle the curious issue of why, given that humans are so closely related to two other species of ape, the chimpanzee and the bonobo, ants should do all manner of \"human\" things like build roads, create traffic rules, have public hygiene workers, and work on assembly lines while those apes don't. We will consider whether a primal-sounding scream called the pant-hoot could have been our remote ancestors' first small step toward patriotic flag waving, and in a sense the foundation of our continent-wide kingdoms. How can I, a foreigner, manage to look past human differences and carry on in other societies when most animals, ants decidedly among them, are unable to do so? Or, a question for the history buff: might the outcome of the US Civil War have been influenced by the fact that most Southerners of the time still thought of themselves as American?\n\nGeorge Bernard Shaw wrote, \"Patriotism is, fundamentally, a conviction that a particular country is the best in the world because you were born in it.\" So what does it mean if an inherent component of the human condition is to hold tight to a society, idolizing it while so often slighting, distrusting, demeaning, or even hating foreigners? That fact is a wonder of our species, and one of my reasons for writing this book. While we have gone from small societies to vast ones, we have retained an uncanny awareness of who fits in and who doesn't. Yes, we build friendships with foreigners, yet they remain foreign. For good or bad, the distinctions remain\u2014along with equally pronounced and often disruptive distinctions mounting within the societies themselves, for reasons I hope to make clear. How we approach the similarities and differences determines the nature and future of societies.\n\n# THE JOURNEY AHEAD\n\nOur inquiries will lead us down not one long road but along many interconnected paths. From time to time we will circle back to look at subjects like biology and psychology from new angles. Our course won't always be chronological as we draw not just from human history but from our own evolution in order to come to grips with what we do and how we think. Given what can seem like a journey that meanders through multiple and varied disembarkation points, a preview of what lies ahead is in order.\n\nI have divided the book into nine sections. Section I, entitled \"Affiliation and Recognition,\" takes in the wide range of vertebrate societies. Its opening chapter considers the role of cooperation in societies, which I endeavor to show is less essential than the matter of identity; societies consist of a distinct set of members in a rich tapestry of relationships, not all of which are harmonious. Chapter 2 covers other vertebrate species, especially the mammals, to illuminate how societies, despite whatever imperfections in the system of partnership that exists within them, benefit the members by providing for their needs and protecting them. The third chapter probes into how the movements of animals within and between societies are important to the success of the various groups. One versatile pattern of activity, fission-fusion, creates a dynamic that helps explain the evolution of intelligence in certain species, humans most obviously among them, and the subject will come up repeatedly in this book. Chapter 4 investigates how much the members of most mammal societies must know about each other for their societies to stay together. Here, I reveal a limiting factor in the societies of many species: all their members are obliged to know each other as individuals, whether they like each other or not, restricting the societies to, at most, a few dozen individuals. This sets up a puzzle about how the human species broke free of such a constraint.\n\nThe second section, \"Anonymous Societies,\" addresses a group of organisms that readily crash through this population limit: social insects. One of my objectives is to break down any aversions you, the reader, may have about likening insects to \"higher species,\" especially humans, by making clear the value of these comparisons. Chapter 5 reports on how social complexity generally climbs with an increase in the size of insect societies with features like infrastructure and division of labor becoming more complex, a trend paralleled in humans. Chapter 6 looks at how most social insects, and a few vertebrates such as the sperm whale, demonstrate affiliation with a society by using something that marks their identity: chemistry (a scent) in ants, and a sound in whales. These simple techniques are not constrained by the limitations of memory, and thus permit the societies of certain species to reach immense sizes, in a few cases without an upper bound. The chapter after that, \"Anonymous Humans,\" spells out how humans employ the same approach: our species is attuned to markers that reflect what each society finds acceptable, including behaviors so subtle they may only be noticed subliminally. By this means people can connect with strangers in what I call an anonymous society, thereby breaking the glass ceiling in the size societies can achieve.\n\nThe three chapters included in Section III, \"Hunter-Gatherers Until Recent Times,\" ask what the societies of our species were like before the advent of agriculture. I cover people who existed as hunter-gatherers up to recent times, ranging from those who lived nomadically in small, spread-out groups, called bands, and others who settled down for much or all of the year. Although the nomads have gotten most of the attention and are treated as the gold standard for our ancestral condition, a readily defensible conclusion is that both options have been within the reach of human beings likely going back to the origins of our species. We can also conclude that hunter-gatherers were not archaic people living an archaic mode of existence. Their people must be recognized as essentially no different from us: humans, as it were, \"in the present tense.\" Despite traces of ongoing, even rapid human evolution in the past 10,000 years, the human brain clearly hasn't been restructured in any fundamental way since the appearance of the first _Homo sapiens_ . This implies that notwithstanding any human adjustments to modern life, we can look to the lifestyles of hunter-gatherers in recorded history and consider the nature of early human societies as the bedrock that underlies our own.\n\nWhat concerns us most are the extraordinary differences between the nomadic hunter-gatherers\u2014equality-minded jacks-of-all-trades, who solved issues by discussion\u2014and settled hunter-gatherers, whose societies often became open to leaders, division of labor, and disparities in wealth. The former social structure points to a psychological versatility we still possess, even if most people today behave more like settled hunter-gatherers. Two conclusions of Section III are that hunter-gatherers had distinct societies and that those societies were distinguished, just as societies are now, by markers of identity.\n\nWhat that means is that at some point in the distant past, our ancestors must have taken the crucial but heretofore overlooked evolutionary step of making use of badges of membership that would, in time, permit our societies to grow large. For clues about how this happened, Section IV, with one chapter, transports us into the past and also scrutinizes the behavior of modern chimps and bonobos. I put forward the hypothesis that a simple shift in how the apes use one of their vocalizations, the pant-hoot, could make that sound essential for identifying each other as society members. Such a transformation, or something like it, could have easily occurred in our distant ancestors. Ever more markers would have been added to this initial \"password,\" many of them connected to our bodies, transforming them into flesh-and-blood bulletin boards for displaying human identity.\n\nHaving looked at how markers of identity originated, we are in a position to explore the psychology underlying those markers and society membership. The five chapters of Section V, \"Functioning (or Not) in Societies,\" review a fascinating range of recent findings about the human mind. Most of the research has focused on ethnicity and race, but should apply to societies as well. Among the topics are the following: how people see others as possessing an underlying essence that make societies (and ethnicities and races) so fundamental that they think of these groups as if they were separate biological species; how infants learn to recognize such groups; the role stereotypes play in streamlining our interactions with others, and how those stereotypes can become tied to prejudices; and how the prejudices are expressed automatically, and unavoidably, often leading us to perceive an outsider more as a member of his or her ethnicity or society than as a unique individual.\n\nOur psychological assessments of others are many and varied, including our penchant for ranking outsiders as \"below\" our own people or in some cases as subhuman altogether. The fourth chapter of Section V elucidates how we apply these assessments of others to societies as a whole. People believe that the members of foreign groups (and their own people as well) can act as a united entity, with emotional responses and goals of its own. The final chapter steps back to draw from what we have discovered about the psychology of societies and the underlying biology to pose more sweeping questions about how family life fits in the picture\u2014whether, for example, societies can be understood as a kind of extended family.\n\nSection VI, entitled \"Peace and Conflict,\" takes on the issue of the relationships among societies. In its first chapter I document the evidence from nature, which shows that while animal societies need not be in conflict, peace between them is relatively rare, present in just a few species and supported by situations of minimal competition. The second chapter then highlights hunter-gatherers to examine how not merely peace but active collaborations between societies became options for our species.\n\nSection VII, \"The Life and Death of Societies,\" examines how societies come together and fall apart. Before writing about people, I survey the animal kingdom, concluding that all societies go through some sort of lifecycle. Although, as we shall see, other mechanisms for starting new societies exist, the pivotal event in most species is the division of an existing society. The evidence from chimpanzees and bonobos, bolstered by data on other primates, is that a division is preceded by the emergence, over months or years, of factions in the society, which increases discord and ultimately causes a split. The same formation of factions, usually over the passage of centuries, takes place with humans also, except for a key difference: the primary pressure that severed human factions was when the original uniting markers keeping a society together were no longer shared, leading people to see themselves as incompatible. This section lays plain how people's perceptions of their own identities change over time in a way that could not be stopped in prehistory, mainly the result of poor communication across hunter-gatherer bands. For this reason, hunter-gatherer societies split apart when they were tiny by today's standards.\n\nThe expansion of societies into states (nations) was made possible by the social changes I lay out in section VIII, \"Tribes to Nations.\" Some hunter-gatherer settlements and tribal villages with simple agriculture took the first tentative steps in this direction as leaders extended their power to take control of neighboring societies. I begin by describing how tribes were organized into multiple villages, each of which acted independently much of the time. What leaders these loosely connected villages had were not very proficient at sustaining social unity and curtailing social breakdowns, in part because they lacked the means of keeping their people on the same page with regards to identifying with the society\u2014things such as roadways and ships that connected people with what their compatriots were doing elsewhere. Growth also required societies to expand their dominion over the territories of their neighbors. This didn't occur peacefully: across the animal kingdom I find little evidence of societies freely merging. Human societies came to conquer each other, thereby bringing outsiders into their fold. Occasional transfers of membership take place in other species too, but in humans such exchange was taken to a new level with the advent of slavery, and finally, the subjugation of entire groups.\n\nNow that we understand the forces that can cause small societies to scale up to large ones, including the nations of today, the final chapter of Section VIII evaluates how these societies tend to meet their end. What's typical of societies put together by conquest isn't division between factions, as we saw earlier for hunter-gatherers, nor utter collapse, though it can happen, but rather a fracturing that almost always occurs roughly along the ancient territorial lines of the peoples that have come to make up the society. Large societies may be no more durable than small ones, fragmenting on average once every few centuries.\n\nThe final section carries us along the circuitous route that led to the rise of ethnicities and races and the at times muddy waters of current national identities. To become an interlocking whole, a conquering society had to make the shift from controlling what had been independent groups to accepting them as members. This requires an adjustment in people's identities, in which ethnic minority groups adjust to the majority people\u2014the dominant group that most often founded the society and controls not only its identity, but also most of the resources and power. This assimilation would be accomplished only to a degree, for this reason: ethnicities and races\u2014as demonstrated earlier in the book for individual persons and for societies as well\u2014will be most comfortable together if they share some commonalities and yet differ enough to feel distinct. Status differences emerge among the various minorities too, and may change over the course of generations\u2014though the majority almost always stays firmly in control. Bringing the minorities into the fold as society members entails allowing them to intermix with the majority people, a geographical integration of populations that not all past societies have permitted.\n\nThe second chapter of Section IX tackles how modern societies have made the friendlier incorporation of large numbers of outsiders possible through immigration. Such movements have seldom occurred easily, and, as in the past, have assigned lower power and status to the immigrants, who may face the least resistance when they take on social roles that minimize competition with other members while giving them a sense of value and esteem. The identity immigrants had once treasured in their ethnic homeland is often recast into broader racial groups. The shift in perception may initially be pushed on the newcomers, but they can accept the changes because of the advantages of having a more extensive base of social support in the adopted society. The chapter closes by describing how criteria for citizenship have come to deviate from the psychology of how people register who has a rightful place in a society. The latter is heavily influenced by people's attitudes about how important a society should be in providing for different individuals or groups versus protecting themselves\u2014attitudes relating to patriotism and nationalism, respectively. Variation among the members in these points of view may well be required for a healthy society, even though it also compounds the social conflicts that make headlines today. Given these stresses, a final chapter, \"The Inevitability of Societies,\" raises the issue of whether societies are necessary.\n\nIn making what inferences I can in this book, I admit up front that a unified field of study of societies is a distant dream. All too often, academic disciplines foster a habitual concentration on certain modes of thought and a disdain for the unfamiliar by dividing the intellectual world into mutually alien societies known as biology, philosophy, sociology, anthropology, and history, thus leaving much room in the nooks and crannies between for debate. For instance, \"modernist\" scholars of history view nations as a purely recent phenomenon. My contention will be that the national pedigree has ancient roots. Some anthropologists and sociologists go a step further and see societies as entirely optional, with people forming such unions when it serves their interests. My goal is to show that membership in a society is as essential for our well-being as finding a mate or loving a child. I will frustrate some in my own discipline of biology, too. I have listened to biologists adamantly oppose the idea that societies ought to be examined as groups of distinct identity and membership when their study species don't quite match with this criterion\u2014a passionate reaction that more than anything makes plain the cachet of the word \"society.\"\n\nDisputes among the specialists aside, readers of every political persuasion will find both good and bad news in the current science. Whatever your social views, I urge you to consider insights from fields beyond your usual interests to become aware of how your own, often subliminal, biases and those of people around you\u2014writ large, across multitudes\u2014might affect both the actions of your country and your daily conduct with others.\n\n# SECTION I\n\n# Affiliation and Recognition\n\n# CHAPTER 1\n\n# What a Society Isn't (and What It Is)\n\nSeen from the top of the stairs in the main concourse of New York's Grand Central Terminal, swarms of people whirl and eddy under the famous four-faced clock. The staccato clatter of shoes on Tennessee marble and the clamor of voices, swelling and subsiding like an ocean in a seashell, reverberate across the cavernous acoustical marvel. The vaulted ceiling, depicting 2,500 stars frozen in place along their regimented paths one October New York night, makes a perfect counterpoint to the tumult of humanity below.\n\nThe sheer number and diversity of people rushing past each other, or clustered here and there in conversation, make this scene a microcosm for human society in its entirety: not a society as a voluntary association of people, but a society as an enduring group, the kind that occupies a territory and inspires patriotism. When we think of such societies, we may think of the United States, ancient Egypt, the Aztecs, the Hopi Indians\u2014groups that are central to human existence and are the cornerstones of our collective history.\n\nWhat are the characteristics of a population that make it a society? Whether you have in mind Canada, the ancient Han dynasty, an Amazon tribe, or even a lion pride, a society is a discrete group of individuals amounting to more than a simple family\u2014more than one or both parents with a single brood of helpless offspring\u2014whose shared identity sets them apart from other such groups and is sustained continuously across the generations. Indeed it may eventually come to produce other such societies, as when the United States split off from Great Britain or one lion pride becomes two. Most importantly, the membership of a society changes rarely and arduously; such a group is closed, or \"bounded.\" Though its constituents can vary in the intensity of the passion they attach to their national consciousness, most prize it well above any other sort of membership\u2014their ties to those nuclear families aside. This significance is conveyed among humans by commitments to struggle, even die, for the sake of the society, should the situation demand it.\n\nSome social scientists see societies primarily as constructs of political convenience, an arrangement that emerged in recent centuries. One scholar with such a view, the late historian and political scientist Benedict Anderson, conceived of nations as \"imagined communities,\" since their populations are too numerous to allow for members to meet face to face. Indeed, I concur with his basic idea. By serving to distinguish _us_ , those who belong, from _them_ , the outsiders, shared imaginings are all we need to create societies that are true and tidy entities. Anderson also proposed that these made-up identities are artificial products of modernity and mass media, and this is where he and I diverge. Our shared imaginings bind people with a mental force no less valid and real than the physical force that binds atoms to molecules, turning them both into concrete realities. This has been the case for all time. In fact, the concept of imagined communities holds true not just for modern societies, but for all the societies of our ancestors, likely from their remote, prehuman origins. Hunter-gatherer societies, held together by a sense of common identity, did not depend on their members establishing one-to-one relationships\u2014or knowing each other at all, as we shall examine; among the other animals, too, societies are represented firmly in the minds of their members, and in that regard are also imagined. This is not to diminish the societies of human beings. Societies are rooted in nature and yet have come to flower in elaborate and meaningful ways that obviously are peculiar to our species, a subject this book will pursue.\n\nI think the point of view I lay out captures what most people have in mind when we speak of a \"society.\" Of course, any word must encompass some variation, and no animal society is equivalent to a human society just as no two human societies are equivalent. I offer this to those concerned about where to draw a line: The utility of a definition is best shown by how much we learn from anomalous situations where the word doesn't quite work. Pushed sufficiently hard, any definition outside those for mathematical terms and other abstractions will break down. Show me a car and I may show you a pile of junk that once functioned as a car (and maybe in a mechanic's mind still is one). Show someone a star, and an astronomer points to a mass of convergent superheated dust. The hallmark of a good definition is not just that it tidily delimits a set of _x_ 's, but that it also breaks down when things get conceptually intriguing about _x_. Hence countries exist that strain my view of a society as a discrete group with a shared identity in ways that are informative. Iran, for example, counts many ethnic Kurds among its citizens despite the government suppressing their identity as a group, while at the same time those Kurds conceive of themselves as a separate nation and claim rights to their own territory. Situations where groups such as the Kurds have identities that clash with the society throw light on the factors that can serve, over time, to further empower and expand societies or rend them and start new societies. Conflicts over identity can arise in other animal societies, too.\n\nMany experts in my field of biology, and anthropologists also, present a different definition of a society, describing it not in terms of identity but rather as a group organized in a cooperative fashion. Though sociologists recognize collaboration as vital to a society's success, it has become rare in that field to equate a society with a system of cooperation. Still, thinking of a society this way comes easy, and for obvious reasons: we humans have evolved such that cooperation is central to our survival. Humans out-cooperate other animals, having sharpened our skills of communicating our own intentions and deducing those of others in the pursuit of shared goals.\n\n# WHAT HOLDS US TOGETHER\n\nIn considering cooperation, as contrasted to social identity, as an essential feature of societies and a basis for distinguishing one society from another, one place to start is with a hypothesis developed by anthropologists to explain the origin of intelligence. It postulates that as our brains have gotten bigger, our social relationships have likewise grown, each driving the greater size and complexity of the other. Oxford anthropologist Robin Dunbar has described a correlation between the size of a species' brain\u2014to be precise, the volume of its neocortex\u2014and the number of social relationships individuals of that species can sustain, on average. Our close relation the chimpanzee, per Dunbar's data, manages about 50 coalition partners or allies: call these 50, with whom an individual cooperates most generously, its friends.\n\nBy Dunbar's calculation, when it comes to humans, the average Joe can sustain about 150 close relationships, his specific bosom buddies changing over time as friends are made or lost. Dunbar characterizes this figure as \"the number of people you would not feel embarrassed about joining uninvited for a drink if you happened to bump into them in a bar.\" This has become known as Dunbar's number.\n\nThe \"social brain hypothesis\" leaves plenty of room for argument. For one thing, it's reductive: no doubt you find advantages to having lots of gray matter besides keeping track of the Toms, Dicks, and Janes you meet\u2014finding food, making tools, and other skills also require cognitive effort. Context matters, too. At a professional conference, for example, an academic is likely to share common interests with a high proportion of the attendees and might willingly join any number of folks uninvited at that bar. Also, friendship isn't a binary, yes\/no category. If Dunbar's number proved to be 50 or 400, it would simply have indicated a greater or lesser degree of basic intimacy and rapport.\n\nNo matter how much human brainpower is devoted to relationships, though, our social circles don't come remotely close to matching the size of nation-states. The disparity between your capacity to make room in your life for 150 chums and a chimpanzee's competence at handling 50 is too small to explain human societies today, with their staggering sizes\u2014or the smaller societies of our past, either. In fact there has never been a human society, from the Stone Age to the Internet era, that consisted solely, for more than a passing minute, of a band of brothers: a clique of mutually shared friends and family living in requited admiration. To think otherwise would be to misapprehend the nature of friendship, and hence our private networks of friends. Whether in overpopulated India, or on the Polynesian island nation of Tuvalu with its 12,000 citizens, or among the miniscule El Molo tribe on the shores of Lake Turkana in Kenya, no one befriends or cooperates with everyone in a society: they pick and choose. When Jesus said to love your neighbor as yourself, he didn't mean that you must be a friend of everyone. The El Molo aside, our societies encompass at least some folks\u2014more often a great many\u2014whom we will never meet, let alone befriend. And never mind those we don't select as buddies, or those who reject us\u2014our worst enemy almost certainly carries the passport of our country.\n\nData on how individuals interact reveal the same discrepancy between Dunbar's number for a species and the size of its societies. A chimpanzee society, called a community, often has well over a hundred members, but even a community of 50, which by Dunbar's calculation could consist entirely of bosom buddies, in truth never does.\n\nThe \"cognitive constraints on group size\" (Dunbar's phase) that puzzle some devotees of the social brain hypothesis do so because of the confusion of social networks (for which social ties vary in strength and depend on each person's perspective\u2014described, for example, by Dunbar's number) and distinct groups (most notably the societies themselves). Both play a part in the lives of humans and other animals. Societies, with their unambiguous boundaries, provide the richest soil within which long-term, though perhaps ever-changing, cooperative networks can grow. While networks may at times encompass everyone, they flourish best among those members who get along with each other well, built on the smarts and cooperative skills at each person's disposal.\n\nSocieties, then, staked out as different from each other by the identities of the members, rest on more than personal networks of allies. And unlike other species, humans keep social life functioning and social networks strong with rules administered in a host of formulations from one society to the next. We thrash out practices\u2014and penalties\u2014to enforce fair exchanges and ethical behavior that can operate across the many for the mutual good. The trash collector does his part by picking up the garbage of strangers in return for a paycheck. He buys coffee from the corner shopkeeper he doesn't know and may speak to unfamiliar faces by the hundred at a church or union meeting. But the governance of such interactions has limits. Despite the shared economic and defensive benefits a society confers, disagreements between factions, especially about what counts as doing one's part and mutual good, can be painful. Such melees are the least of it, however. No society is without crime or violence (the antithesis of cooperation) committed by one full-fledged member or group of members against another. Yet societies can endure for centuries even if dysfunction speeds their dissolution\u2014the Roman Empire immediately comes to mind, though there have been countless others.\n\nIn general, however, societies will favor cooperation. How much selfishness or divisiveness it takes to rupture a society is likely much greater than intuition would suggest. In _The Mountain People_ , the British-born anthropologist Colin Turnbull chronicled moral decay among Uganda's Ik people during a disastrous famine in the 1960s that brought a disregard for social bonds and led to the deaths of children and the elderly. Turnbull's account showed just how far a society could unravel under stress; nevertheless, the Ik carried on. Likewise Venezuela remains intact despite repeated economic collapses and a murder rate in its capital city of Caracas that in some years exceeds that of a war zone. Whenever I visit an intrepid friend there, he drives us at high speed over harrowing backstreets to avoid shootings by _motorizados_ along the highway. Despite it all, he loves the place. The surprising fact is that Venezuelans show every bit as much attachment to and pride in their nation as Americans show in theirs. Societies have survived worse. For example, during the California gold rush the homicide rate was dramatically higher than that of modern-day Venezuela.\n\nWhile disagreements and disagreeableness can fray the fabric of societies, their positive counterpart, cooperation, does not necessarily knit societies together or separate them from other societies. This is true even when cooperation contributes to the social capital that builds up among the members and enhances the productivity of the whole. The biggest problem with predicating life in a society on cooperation is it simply ignores much of what makes existence in a society a challenge. The nineteenth-century social theorist Georg Simmel interpreted cooperation and conflict as inseparable \"forms of sociation,\" each unimaginable without the other. To focus overmuch on cooperation is to cherry pick.\n\nIn the societies of our ape relatives, kindness and cooperation are likewise just a fraction of the picture. Chimpanzees intimidate each other or fight outright over status, with those on the losing end occasionally ostracized or killed. The paltry assistance the apes render one another, outside of mother-infant bonds, usually materializes when several animals work together to unseat competitors, fighting so that one of them can attain the alpha male status. Chimps also join up to hunt red colobus monkeys, by some accounts making the kill by acting in parallel more than by collaborating. Whichever chimp ends up with the meat may give others a taste, but only after being begged. The bonobos, which look like chimps with small heads and pink lips, are more beneficent yet will steal food from society-mates when they can, and are no more inclined to teamwork.\n\nEven social insects, the very symbol of mindless cooperation, can face domestic strife and act selfishly. Though across most social insect species the queen is normally the only individual to reproduce, in honeybees and some ants a few workers subversively lay eggs. Their nests can be veritable police states, with workers on a vigilant quest to destroy any egg that isn't the queen's. Whatever the species, individuals that don't pull their weight also drag others down; how societies, ranging from arthropod to human, deal with cheaters and enforce fairness is a field of study in itself.\n\nGiven a society with its clearly defined membership, how much cooperation is required to hold it together, for animals generally? In theory, not much. Expelling foreigners may be the minimum collaboration needed. Imagine a loner creature that controls an exclusive space, or territory, by throwing rocks at anyone drawing near. Now imagine a few such creatures settle a territory together. Each throws stones at outsiders exactly as before but with one difference: they leave each other alone. This tacit agreement to \"do no harm,\" so to speak\u2014to coexist in peace\u2014amounts to cooperation of a rudimentary sort.\n\nOf course, societies could not have evolved if they did not offer competitive advantages to either the group (evolutionary biologists call this group selection), its individual members, or both. What might be the attraction for such an uncharitable bunch? Such a society would make sense if, say, ten animals throwing rocks this way could hold on to more than ten times the territorial space that they could have alone, or seize a higher quality territory with less effort or risk to each member. It may also be that purely by keeping others out, they exclude those whose steadfastness at rock throwing is uncertain while sharing exclusive opportunities to mate with each other (even if a lot of infighting remains over who has sex with whom).\n\nSurveys of the animal kingdom show societies can exist with just a little \"prosocial\" behavior between the members: call it proto-cooperation, doing good turns by accident, perhaps. Madagascar's ringtailed lemurs come close to displaying such minimal expectations, in that troop members help each other little except to join forces in attacking outsiders. One expert opines that marmots, a kind of alpine squirrel, don't even _like_ each other, yet still find such behavior as huddling for warmth sufficient motivation to stay together, while another authority has described a social badger clan as \"a tight community of solitary animals.\" Even people retain commitments to groups whose members don't get along, with our helpfulness often dependent on what our society demands of us.\n\n# SOCIETIES THAT GET ALONG\n\nWhen the Spanish ships the _Santa Mar\u00eda_ , the _Ni\u00f1a_ , and the _Pinta_ first arrived in the New World, one society greeted, and the other enslaved. The local group of Ta\u00edno Indians, an Arawak tribe depicted by Christopher Columbus as \"stark naked as they were born, men and women,\" swam or paddled canoes out to greet the newcomers. Without understanding a word the Spaniards said, the Indians showered them with fresh water, food, and gifts. Columbus recorded a more cynical, European reaction to the Indians: \"They would make fine servants.... With fifty men we could subjugate them all and make them do whatever we want.... As soon as I arrived in the Indies, on the first island that I found, I took some of the natives by force in order that they might learn and might give me information of whatever there is in these parts.\"\n\nThe stark contrast between these alternative outlooks, one of open trust and the other of cunning and exploitation, is disturbing, but hardly shocks us. We humans have a flair for identifying who's in our society and who isn't, then drawing a sharp line between what psychologists call the ingroup and the outgroup, even when we are friendly with the latter. We learn from childhood to regard foreigners as a possible threat or\u2014as the Arawak and Columbus both did in differing ways\u2014as an opportunity.\n\nThus we are presented with another reason why cooperation doesn't always indicate where one society ends and the next begins. Some among the horde in Grand Central concourse are no doubt foreign nationals navigating productive relationships with US citizens. Hence just as it's possible to have enemies as well as allies in one's own society, it's possible as well for members of one society to communicate with members of another toward the goal of friendship and cooperation. Camaraderie occurs between nonhuman societies, too\u2014in rare cases. The bonobo has been called the hippie ape for preferring peace to provocation. Yet I would wager that individual bonobos find the occasional adversary in other communities. Even a peacenik doesn't get along with everyone.\n\nThe ease with which people today jet off to other countries has brought our contacts with foreigners to a new level, one that, as we will come to grips with later, is without parallel in nature. Indeed, modern life challenges our tolerance for others by twisting and stretching our identities in novel ways. But societies have remained with us throughout it all.\n\n# COOPERATION WITHOUT SOCIETIES\n\nFollowing Terry Erwin into a Peruvian rainforest means getting up at dawn. While the wind is still low, this entomologist at the Smithsonian's natural history museum loads a machine called a fogger with a biodegradable insecticide and aims its spout upward so a pale gray mist climbs into the trees. There's the light sound of rain, but no water is falling\u2014instead, the patter is the sound of tiny bodies hitting sheets stretched over the ground. Over the years, Erwin has learned just how rich the tropics are. By his estimation, 30 billion individuals belonging to 100,000 species live in a single hectare of the rainforest.\n\nWherever I go, I am awestruck by the diversity of living things. The data of Erwin and others push us to look at societies from the broadest possible perspective of global biodiversity. What is striking is that the vast majority of organisms get by just fine as lone individuals. That's true of more than 99 percent of the species perched in the trees, in Peru or elsewhere. Putting aside the obligation to mate and possibly raise young, it's not always clear why one should stay close to others at all. As beings capable of taking pleasure in the company of others, we humans seldom think about this question. But birds of a feather, whether people or actual birds, are potential competitors for the same resources: food and drink, opportunities for sex, and a place to call home and rear offspring. In many species, individuals cluster together only incidentally, to fight or scramble over food like so many squirrels grabbing nuts. Solitary existence is a safe approach for keeping what one has worked hard to get. For life in a crowd to pay off\u2014any crowd, let alone a whole society\u2014something must be gained in dealing with needy and greedy others.\n\nOne option is to cooperate with those others when the situation is right, and this possibility suggests a final difficulty with associating cooperation with societies: while cooperating individuals can be said to be social, that doesn't mean they compose a society. In his important book _The Social Conquest of Earth_ , my hero and mentor, the ecologist Edward O. Wilson, observes that animals that are social\u2014that come together at some point in their lives to pull off something mutually advantageous\u2014are ubiquitous.\n\nEven so, few species have taken the step of evolving societies. Contemplate the two most basic social units: a mated pair, and a mother with offspring. Not all animals display even these sorts of social pairings. Salmon cast off eggs to be fertilized in the water column, and turtles abandon their clutches after hiding their eggs in the sand. Yet newborns and hatchlings are fragile, so it can be strategic to support them while they are helpless. In all birds and mammals, and in some species of other animal classes, mothers are the caretakers during this critical time. In a few cases, as in the American robin, dads lend a hand. Still, that's usually as extensive, and as lasting, as the group gets: most such little families operate solo, not as part of an enduring society.\n\nNor do networks of allies or intimate friendships need a society to blossom. For example, orangutans don't have societies and are loners much of the time, but primatologist Cheryl Knott tells me females who meet in adolescence will hang out now and then later in life. Or consider that two or more cheetahs, often though not always brothers, will collaborate to hold a territory. Yet in my estimation, friendships, as distinct from sexual partnerships, prosper most often in societies\u2014a pattern that suggests societies' dependable memberships provide a stable foundation for the sort of close relationships that intrigue adherents of the social brain hypothesis.\n\nNevertheless, being with others\u2014even if in fleeting relationships not specifically established for childrearing or friendship\u2014may be beneficial. Think of chorusing birds that come and go like rowdy teens at a soiree. Such flocks are social affairs that can attract whoever's nearby, protecting the birds from predators, connecting them to mates, or stirring up bugs to eat. Some birds burn less energy migrating in a V formation than flying alone. Schools of sardines and herds of antelope offer similar payoffs even when the participants make no commitment to a particular ensemble. In addition to such joint benefits, there can also be cases of altruism where an animal helps others at some cost to itself. A few fish in a school of minnows will swim forward to examine a predator, with the school apparently learning how dangerous the situation is from how aggressively the predator reacts. When relatives are involved, such generosity has a special evolutionary logic because individuals can promote their own genes by helping those kin, as shown in an elementary way by a pair of robins rearing their progeny: that's kin selection.\n\nProximity can offer a safety in numbers that's outright self-serving, as I observed in my first tropical expedition while an undergrad. In Costa Rica I joined lepidopterist Allen Young, who asked me to record the behavior of the caterpillars of the orange-spotted tiger clearwing butterfly. The lumpy larvae ate the leaves of a weedy plant, rested, and moved in tight groups. Spiders and wasps were the bane of their existence; the caterpillars positioned on the outside of a cluster were the ones these predators picked off and ate first. I concluded that their survival instinct was to pack together, each pushing between the others, leaving the weak ones to their demise. Writing up the results I discovered that W. D. Hamilton, a famed biologist, had already proposed this centripetal behavior for fish schools, mammal herds, and the like, and given it a name: they are selfish herds. Despite their selfishness, my caterpillars helped each other, if only by accident. Alone they had a hard time cutting past the tough hairy leaf cuticles to begin feasting; as a group they fared better since the first caterpillar to succeed opened up the leaf for everyone to feed.\n\nThe important point is that my caterpillars\u2014like schools of minnows, nesting robins, and flocking geese\u2014cooperate but don't have societies. As long as their companions were of similar size, the larvae seemed to get along fine with each other when I combined the broods of their species. The same is true whenever individuals can come together willy-nilly, for example when another species, the tent caterpillars, join up to weave a silk tent that's bigger and better sheltered from cool weather. Similarly, the African bird known as the sociable weaver inserts its nest among many others to produce a large communal structure that provides air-conditioning for all residents. The birds move into and leave from these colonies as they choose, albeit over months rather than constantly like birds in many flocks. While some birds get to know each other, a colony, like a flock, isn't closed to outsiders. Any new arrivals are tolerated so long as they find a spot to nest.\n\nIn short, those who equate societies with cooperation have it backward. A typical society encompasses all manner of relationship, positive and negative, amicable and embattled. Given that cooperation can flourish both within and between societies and where there are no societies at all, a society is better conceived not as an assembly of cooperators, but as a certain kind of group in which everyone has a clear sense of membership brought about by a lasting shared identity. Membership in societies of humans and other species is a yes\/no matter, with ambiguity rare. The prospects for alliances, whether from friendship, family ties, or social obligations, may rank among the paramount adaptive gains of having societies in many species, yet aren't necessary to the equation. A misanthrope with no family, filled with contempt for all humanity, can still claim your nationality. That's true whether he lives as a hermit outside the system or as a parasite of others within it. The members of a society are united by their identity, whether or not they are in regular contact or willing to help each other out\u2014though the membership they have in common can be a solid first step in making such relationships a reality.\n\nSo what came first, chicken or egg\u2014membership or cooperation? Whether more than a bare minimum of collaboration had to exist for societies to evolve, or memberships had to be worked out before long-term cooperation was at all likely, remains an open question. Whichever was originally the case, the next chapter introduces us to the many advantages that societies offer our vertebrate cousins in nature.\n\n# CHAPTER 2\n\n# What Vertebrates Get out of Being in a Society\n\nAnimals in a society struggle just as solitary species do. They face conflicts over who gets what, as well as the right to mate, make a home, and raise young. Not all succeed. What membership provides is a certain security in facing the broader world. This is true even in societies whose members do little for each other beyond driving off outsiders. The gamble is that being part of a successful, or dominant, society will allow each member to ultimately get a more substantial slice of the overall pie than the portion it would have earned alone or in a weaker society. While loose and temporary groups have advantages, once animals adapt to life in permanent societies, it can be problematic for them to go back to surviving on their own. Anyone not in a society, or in a failing society, is at risk.\n\nVertebrates, and to be more specific the mammals, provide a good starting point as we consider the benefits of societies, especially since we are mammals ourselves and our evolution as such is a core theme of this book. That's not to say no other vertebrates have societies. In some bird species, such as the Florida scrub jay, fledglings help their parents raise younger siblings; given this \"overlap of generations,\" as biologists describe it, their groups represent a simple sort of society. Or consider the shell-dwelling cichlid, _Neolamprologus multifasciatus_ , a fish native to Lake Tanganyika, Africa. Societies of up to 20 cichlids guard a cluster of shells they dig out of the sediments. Each fish has a personal shell abode in what one biologist described as a \"complex of apartments that would do modern public housing proud.\" An alpha male does the breeding, with outsiders of either sex inserting themselves into the colony membership only at rare intervals.\n\nThe mammals that live in societies are far more widely known, and talked about, than society-dwelling fish or birds. Even so, revisiting them with the issues of membership and identity in mind yields fresh perspectives. Consider two beloved examples, the prairie dogs of the North American plains and the African savanna elephants. Both have societies, but those turn out not to be the groups of either that garner the most attention. People traditionally think of prairie dogs as living in colonies or towns, and elephants in herds; yet a colony or herd is rarely one society but rather a multiplicity of societies, antagonistic to each other in prairie dogs and often convivial in the elephant.\n\nNo individual prairie dog identifies with, or fights for, a colony; rather, its loyalty is to one of the small land-holding groups within it, sometimes called a coterie (the word is appropriate, meaning an exclusive group). In perhaps the best-studied of the five prairie dog species, the Gunnison's prairie dog, each coterie contains up to 15 breeding adults, including one or more of each sex, occupying a vigorously defended area up to a hectare in extent.\n\nSavanna elephants are, in contrast, widely sociable across their populations, yet one grouping merits being called a society. The core groups, or simply cores, have a membership of up to 20 adult females, accompanied by their calves. The societies are female affairs. Each male goes his own way when he matures and never becomes part of a core. Cores often can be picked out by eye from their responses to each other, even at times when hundreds of elephants, and many cores, socialize. To retain their distinct memberships, the cores ordinarily keep outsiders, even animals the members are fond of, from lingering with them in a long-lasting way. The relationships between cores are complex. The cores form connections, called bond groups, but these social networks are inconsistent, with differences in opinion as to who belongs\u2014core A can bond to B and C at the same time that C avoids B. Only the elephant cores themselves retain a distinct membership over the long haul.\n\nThe savanna elephant's life in a core differs from the two other elephant species, the forest elephant of Africa and the Asian elephant, which, though social, lack distinct societies. Whether the absence of societies makes those species less sophisticated depends on what is meant by \"sophisticated.\" The answer may well be that they are more so, since Asian elephants have heavier brains relative to their body weight than the savanna elephant; maybe the savanna elephant's focused dependence on its core simplifies its life by reducing its everyday social obligations to just those few core-group companions. Loner species such as weasels and bears need to be totally, and constantly, self-reliant, which may explain why they can be smarter than many society-dwelling species, as measured by their adroitness at solving puzzles. Asian elephants, socially gregarious as they are, could well face constant cognitive challenges because they live with few clear social boundaries compared to Africa's savanna species, with its cores.\n\n# THE ADVANTAGES OF SOCIETIES FOR MAMMALS\n\nLooked at in sweeping terms, societies give mammals\u2014from savanna elephants and prairie dogs to lions and baboons\u2014multiple avenues for providing their members with both security and opportunity, shielding them from the perils of the external world while offering them entr\u00e9es to the resources they jointly hold. Loosely speaking, these safety nets fall into two, loosely overlapping categories: societies provide, and they protect.\n\nAmong their functions in providing resources is access to dependable long-term helpers, which can be an asset for mothers feeding and housing the young. Gray wolves and birds such as the Florida scrub jay, among others, are cooperative breeders, with their societies based around extended families in which offspring assist the parents or close relatives in raising siblings. Babysitting while mother goes off to feed herself is a common task across many species, but helper meerkats also tidy the burrow and supply the young with insect prey. In some monkeys a female helper contributes little of value; however, if she has never had young of her own before, she profits from having the practice of handling a baby\u2014with the nervous mother keeping a close eye on her.\n\nOther benefits of societies evince how lifelong members who know each other well can obtain meals by efficient group efforts. Big game predators dispatch prey that amount to a windfall for everyone. However, cooperation is more obligatory in some species, such as painted dogs, than in others\u2014lions can be lazy about joining a group hunt and often don't obtain more meat for themselves than they would by stalking prey alone. Some behaviors common across mammal societies are important even though little improved over what we see for temporary aggregations like bird flocks. Meerkats and ringtailed lemurs, for instance, rely on their numbers to locate patches of food, and by staying clustered together stir up lots of insects for easy catching. Baboons are known to hang tight with the most successful forager in their bunch, occasionally even stealing his or her morsels.\n\nAmong the bottlenose dolphins of western Florida, societies perhaps serve primarily for another purpose: enabling the animals to adapt to local conditions. Raising a dolphin is a communal responsibility, with the pups learning traditions passed down through the generations. The elders of some communities, for example, teach the young to team up to circle fish until their prey clump up and can be driven onto the beaches. There the dolphins snatch their flopping prey before wriggling back down into the water again. Such social learning likewise plays a role in species such as the chimpanzee.\n\nThe protection a society affords its members can be just as crucial as the resources they can access. Of course the two are linked. Because chimpanzee females can defect to another society if they don't get what they need for their offspring, the males commit to the brunt of the toil and the danger of guarding the members and keeping their resources from bellicose outsiders. There's no better immediate motivation for the males, it would seem, than sex. Beyond food and drink, many species secure a den for the young or hold on to valued topographic features: for gray wolves, a lookout; for horses, a windbreak; and for many primates, safe spots to sleep. Spaced out across their \"towns\" like suburban homes, the mounds controlled by prairie dogs, much like the shells of the cichlid fish, are important as living spaces for individual members\u2014or two or three members per mound in the case of the prairie dogs. All must be defended from outsiders.\n\nWith regard to the control of resources, the costs incurred by coping with rivals among the members themselves can be compensated for by the effectiveness of having more eyes and ears to detect competitors and other threats from outside, more voices to give warning should enemies appear, and more teeth and claws to fight back. Safeguarding young is critical, as when elephants in a core group shield everyone's calves against lions or when horses in a band encircle the foals to kick at wolves. At times all concerned take part in defense. An entire baboon troop, right down to mothers carrying their infants, will commit to mobbing a leopard together, with some of the monkeys receiving massive slashing wounds in their attempt to corner and, occasionally, kill it.\n\nA society's most intense competition is typically with other societies of its own species. The best defense can be a good offense. A common way to monopolize commodities from conspecifics isn't to protect the assets directly but to lay claim to the space where they exist. Territoriality is an option if an area can be exclusively controlled or at least heavily dominated. So it is that species take a wall-building approach to international relations: in the shell-dwelling cichlid fish, there are literal walls of sand thrown up between the apartment complexes of different groups, whereas mammals like gray wolves and painted dogs mark their spaces with a scent. Prairie dogs take advantage of visual landmarks such as rocks or brush; however, even borders cutting across open ground remain stable from parent to offspring. The rodents know exactly where they are and protect their space by driving out or killing trespassers. When a species is territorial its societies can be identified simply by mapping out the movements of the animals to show that separate groups occupy different regions.\n\nThe societies of a few species, such as the horse, the savanna elephant, and the savanna baboons, aren't territorial. Instead these animals cohabit the same terrain as other societies of their kind. Even so, they rarely wander haphazardly, but stay in a general region each group knows best. These societies don't quarrel over access to the land itself, but rather over the resources it contains, which are generally too scattered to make defending a whole stretch of ground practical. A strong society can seize either the territory or its resources from a less populous society or lone animal. Not all mammals struggle with their closest neighbors, however. Both bonobo and coastal Florida bottlenose dolphin societies stick to what are essentially private plots of ground but fight little when outsiders visit, a border-without-walls accommodation that suggests they face less competition with the neighbors.\n\nSometimes being in a society serves in part to cut down on the harassment animals experience with their own species merely by reducing their interactions to a merciful few. A band of horses usually contains adults of both sexes, for example, but some bands do fine with no stallion. \"Better the devil you know than the devil you don't\" would be a spot-on slogan for such all-female bands when they acquiesce to a male joining them. No matter how pushy that stallion may be toward the mares, he can repel the endless stream of other pestering bachelors that come along. In other species, members of a society will mate with outsiders, liaisons that the animals of the opposite sex try to stop. A passerby ringtailed lemur male finds eager sex partners if he can slip into a troop undetected by its resident males. Similarly, a female prairie dog can cross into another territory for a dalliance, although she will be attacked if discovered in the act.\n\nA final bonus of societies is their internal diversity. Numbers are good for more than the mere repetition of eyes and ears, teeth and claws: the members' varied strengths make up for their individual deficits. A monkey with bad eyesight or an injured leg, or which is simply poor at finding food, can tag along and benefit from those with eagle eyes and healthy limbs, even when the strong don't intentionally set out to aid the weak. And it may be the weak can also fill a social role, perhaps by tending the babies.\n\n# RELATIONSHIPS WITHIN A SOCIETY\n\nIn all these matters\u2014from dodging or fighting predators or enemies to avoiding harassment, finding resources or mates, harvesting food, doing chores, and teaching and learning\u2014opportunities arise for cooperation or altruism between animals. Though identifying as members, rather than cooperation, is the feature that unites a society, cooperation can obviously be a benefit of society life, even if the members have conflicting stakes in what goes on or when helpfulness doesn't extend uniformly to everyone. Selfishness, as seen in the caterpillars that hide behind each other to escape being someone's lunch, is seldom a prime motivator for associating with others. But that members commit such perilous acts as baboons ganging up on a leopard points to an alignment of interests\u2014though such alliances are better developed, or expressed more widely, in some species than in others.\n\nMany instances of affiliation _within_ a society, such as those personal social networks designated by Dunbar's number, are customized to the individual. Often one's closest ally and best friend is a family member or potential sex partner, but not always. Both gray wolves and horses turn to special companions for solace and support. Similarly lionesses that happen to rear cubs at the same time form a tight partnership called a cr\u00e8che. Bonds between male bottlenose dolphins, often childhood buddies of different moms, last a lifetime and serve the pair as they court females together and drive away adversarial males.\n\nTight friendships and networks of allies don't guarantee that life in a society is a cakewalk, as the previous chapter made clear. Not every person can afford a penthouse suite, and not every wolf is top dog. Societies can become veritable battlegrounds of a social, and at times physical, sort, with each individual's portion of group resources at stake. Animals often stick to a society in spite of the power plays and upsets, pain and persecution, holding out for the opportunities that come along, each struggling for its version of the American dream. In this, some have it harder than others. Power differences can exist not only between societies, but among the individuals within them. Control by dominant individuals, often at the top of a social hierarchy, presents a tough bottleneck to advancement in many primate societies, resulting in intense physiological stress. The same is true of spotted hyenas\u2014the only hyena species to form societies. In particular, the most alpha male spotted hyena still has a smaller penis and less testosterone than his savage female counterparts (which have a sexual appendage called a pseudopenis) and a status so low that any cub can drive him away from food. By comparison, savanna elephants, bottlenose dolphins, and bonobos lead idyllic lives, yet they still face discord: unpopular elephants are mistreated by their sisters, dolphins spar over mates, and a mother bonobo intervenes to frighten off any males that pester her son while he's trying to have sex.\n\nDominance has its advantages, even for those who fail to achieve it. Certainly once a hierarchy is set in place, influenced by each individual's physical and mental endowments\u2014and in some species, based on the status of its mother\u2014conflict may decline, a bonus for all. With its position worked out, a monkey of low standing can stop wasting time confronting much higher-ranking individuals and focus on improving its station among those at its own, perhaps miserable, level. Without reconciling in this way to their situation, members would wear themselves out. For people as well as other animals, societies would fall to pieces under the general, unrelenting struggle to get ahead.\n\nIt seems, then, that both the affable bonobo dependent on his mother's protection and a baboon stressed over its social standing live by the rule Michael Corleone pronounced in _The Godfather, Part II_ : \"Keep your friends close, and your enemies closer.\" Within the confines of a society's predictable membership, animals can keep tabs on both friends and adversaries, refining how they handle their relationships, positive and negative, and sometimes even working with competitors to their advantage.\n\nPerhaps this explains the bonobo, a species for which the advantages of societies are not obvious given that these apes, in striking contrast to the chimpanzee, compete little, cultivate foreign friends and sex partners, face few predators, and seldom need help finding meals or catching large prey. Indeed, bonobos can be outgoing, even generous to outsiders. It seems they could behave the same way if they lived with no distinct communities to speak of, yet the community borders remain in place. Why, then, have societies? Possibly their communities exist for the most minimal of reasons: so bonobos can structure their days around a clear and manageable set of individuals. This hypothesis is reasonable, albeit not altogether satisfying. I am inclined to think the societies offer them more than is clear to scientists at the moment. Be that as it may, one thing is certain: bonobos, like the chimps, and like savanna elephants and prairie dogs, spotted hyenas, and bottlenose dolphins\u2014and other vertebrates such as Florida scrub jays and shell-dwelling cichlids\u2014have lives firmly rooted in their society.\n\nOne psychologist has declared of humans that, as a consequence of our evolutionary past, our \"sense of personal security and certainty\" is strongest when we belong to a group perceived as different, and distinct, from outsiders. This statement could well be true of any number of animals that depend on being provided for and protected by their fellow society-mates. In fact, much of society life bears on how those members interact, yielding dynamic movements within and among societies that have been critical in shaping the social evolution of species, humans among them, a subject that will be addressed forthwith.\n\n# CHAPTER 3\n\n# On the Move\n\nOur Land Rover lurched to a stop in view of something wonderful\u2014a painted dog, ears held up to form satellite dishes. I couldn't contain my excitement\u2014these group hunters are rare here in Botswana and throughout sub-Saharan Africa. Painted dogs live in packs, and so others had to be nearby. Yet this one was alone and nervous about it. She paced, then paused, made a loud hoot, listened, hooted again. Within a few seconds of her third try we heard the response of her pack companions and instantly the female bounded away in their direction. One bumpy, banging minute later our vehicle landed among a crowd of the wild dogs, dozing, leaping, growling, sniffing, and playing in a tight bunch all around us.\n\nSocieties offer benefits that a solo existence cannot\u2014that much is clear. But the shape those benefits take has everything to do with movement. The patterns by which the individuals of a society move and occupy space establish the interplay of individuals and groups. For a painted dog or monkey, being separated from society-mates is an emergency situation on the order of a three-alarm fire. Should a meerkat, for example, be too engrossed in eating a scorpion to notice the departure of its clan, the laggard gives a lost-call until it hears a response and reunites with the others, its distress reflecting the risk that a predator or enemy takes advantage of its plight to go on the attack. For horses, too, there can be a panic for a stallion on a walkabout or a mare that falls behind with her foal. The wanderer may climb a hill to catch sight of its band and promptly run to catch up.\n\nThat is not the case in all animal societies: in other species, being spread out is the normal state of affairs. Prairie dogs are an obvious case, distributed as they are among the semiprivate mounds from which each of them forages every day. Their burrows stay put for generations. However, the most intriguing of these habits of dispersal is a dynamic kind of nomadic life. In these fission-fusion species, society members temporarily cluster here and there in social groups that form, dissolve, and reform elsewhere. Since most animals seldom have to be in immediate proximity, a bit of fissioning and fusioning can be detected in the societies of just about any species. Still, fission-fusion is the workaday behavior of a handful of mammals. Spotted hyena, lion, bottlenose dolphin, bonobo, and chimpanzee societies rarely converge on one place. For other animals, fission-fusion is more context-specific: packs of gray wolves and cores of savanna elephants split up for times when it's beneficial for finding food. Fission-fusion sounds esoteric, but there's an abundance of reasons to look at this lifestyle. The fission-fusion species include virtually all the brainy mammals that anthropologists studying the social brain hypothesis salivate over\u2014most significantly, us, _Homo sapiens_.\n\nThe animals of a fission-fusion society travel as best serves their social success. Put simply, this is where social smarts come into play. Moving with few constraints, these animals have the luxury of picking their companions, spending quality time with a friend or mate while finding relief from adversaries. For spotted hyenas such relief takes effort\u2014the species is competitive from birth, at times to lethal effect. Observing the animals with hyena expert Kay Holekamp in Kenya, I delighted in the sight of cubs frolicking at the den until I realized the object of their tug-of-war was a playmate's corpse. As soon as the young are big enough to leave the den, they scatter across the community, seeking out allies while approaching any other clan member with caution.\n\nFreedom of movement allows social interactions to grow more complex, and it also pays off whenever personal relationships become hard to manage. Options open up that are impossible for animals living \"face to face\"\u2014monkeys, for example, that have no choice but to stay constantly together in a troop. Object to the local scene? A cunning chimpanzee of low social rank may find an opportunity to slip off to a quiet spot in his community's territory for a private liaison with a female. Even more surreptitiously, a gray wolf or male lion can sneak away to visit another pack or pride as a first step to defecting\u2014transitioning to become a member of another society can require this sort of painstaking duplicity. No surprise that many fission-fusion species are so bright.\n\nFission-fusion offers other advantages. Spreading out allows for a territory to sustain a greater population, as everyone in the society isn't clambering on top of each other to gain access to the same resources. Imagine what would occur if a whole community of 100 chimpanzees stayed in one tight troop: the land they passed through would be picked clean, forcing them to move relentlessly and scramble for each tidbit like shoppers on Black Friday. Instead, the chimpanzees keep apart, meeting in temporary groups, called parties, of a large size only where they hit a resource jackpot, such as a tree weighed down with fruit.\n\nFission-fusion has its downsides. When members are spread thinly across their territory, enemies can enter and strike small groups or singletons with little danger to themselves. The attackers are also less likely to become the target of a massive counterattack before they make an escape. Such an assault on at most a few individuals would be suicidal if directed at 100 or so chimpanzees in one party.\n\nPerils of assault aside, the scattering of individuals means fission-fusion animals can better monitor a big space for thieving trespassers, since the society has eyes and ears almost everywhere. Maybe that's why chimpanzees never attempt incursions to feed from the neighbor's fruit trees. By comparison, animal societies that stick together rarely know when an intruder enters the far stretches of their home range. For instance, baboons and painted dogs, which spend their days very near others of their troop or pack, can't do much about a rogue foreign animal visiting the far corner of their territory. In this way, spreading out can help the animal society defend more ground.\n\n# TOGETHER OR APART\n\nIt can be easy to claim that a species lives by fission-fusion based on our human perceptions. However, it's important to consider that animals perceive space differently depending on their sensory acuity and how they stay in touch. In short, whether the members of a society are close together or far apart isn't for us to decide. Writing of the hardships baboons face, the South African naturalist Eug\u00e8ne Marais dubbed their lives \"a continual nightmare of anxiety\"\u2014but it's not hard to imagine that anxiety as being aggravated by their constant presence together. GPS data from a troop of baboons confirms that, at one particular area where they stay quite packed, the animals never venture more than a few meters apart. In reality, though, while a baboon is almost always aware of the presence of others, it is unlikely to register the dozens in its troop from moment to moment. Most of its fellows will often be out of sight, vanishing behind its back or looping around a bush. Since baboons' sight and hearing are not much keener than those of a human, the best they can do is keep tabs on whichever comrades are nearest and catch up when they move. When practiced by all, these behaviors should be enough to keep a troop in formation.\n\nSpecies in which society members stray farther apart are often associated with superb sensory abilities. Elephants are aware of the sounds of their companions across several kilometers. While a person might presume a couple of elephants resting under a tree are isolated from those few eating leaves out of view, the animals, emitting a continual low-frequency call known as rumbling, are connected. This ability to communicate over long ranges permits a core's scattered members to coordinate their activities. Spread out they may be, but until they wander very far indeed, the elephants can be better informed about each other's movements than the baboons in a compact troop.\n\nTogetherness, then, is in the eye (or ear or nose) of the beholder, and even fission-fusion animals can panic when truly isolated. All of which is to say that for society members to act in concert, or at least respond to each other effectually, what makes a difference isn't how close to each other they are in our eyes, but their knowledge of the location of others. This knowledge can be limited to a local area, as in baboons, or extend far and wide, as in elephants. The members of a society are _together_ as long as they are aware of each other.\n\nContrast elephants (which scatter, yet can still often act as one since they remain connected by their acute senses) with similarly spread-out chimpanzees. With vision and hearing like our own, a chimp may ordinarily just perceive those near it. In fact, the sensory limitations of chimps, for all intents and purposes, render their communities much more fragmented than the cores of the savanna elephant. If a danger presents itself\u2014a hostile outsider comes by\u2014only those chimps that happen to be together at that place have any chance in organizing a defense. So while a community is always dispersed, the chimps spend most of their time among others, in one of a community's scattered parties.\n\nYet in even the most dispersed fission-fusion societies, animals usually have some means to check in at intervals with those who aren't too remote. Spotted hyenas whoop and holler at any provocation; chimpanzees stay in touch by loud pant-hoots; lions roar; wolves howl. As long as lines of communication exist, individuals aren't completely cut off.\n\nWe are still learning what information the cries convey. The animals may use them simply to update others on their whereabouts and space themselves accordingly, though some vocalizations can be calls to action. Lions can summon assistance to battle or to chase down dinner. As many as 60 hyenas have converged in an attack on an enemy clan, a nasty and deafening event, while fewer come together to stage a hunt. Communiqu\u00e9s can be hacked. Hyenas make so much ruckus they often draw in enemies and end up in a brawl.\n\nAmong the chimpanzee calls, the pant-hoot is the noisiest, reverberating for a couple of kilometers. The name is onomatopoeic\u2014no human makes any noise like it in polite society (unless you are Jane Goodall impressing an audience). The cry expresses excitement among apes on an outing together, a _yay team_ cheer that strengthens the bonds between males and helps parties within earshot keep tabs on each other. Discovering fruit sends chimps into a pant-hoot frenzy, attracting others to any spot where there's plenty to eat. But it's the pant-hoots from a foreign community that really agitate the apes. They embrace, stare toward the uproar, and sometimes pant-hoot back in a hollering match.\n\nMost chimpanzee territories are expansive enough, though, that the parties separated by the greatest distances are likely to be out of earshot; hence in practice the entire community is never alerted to a situation, and certainly never acts as one. By comparison, bonobos form similar parties, but they are larger, longer lasting, and are often all in listening distance. There can be an active exchange of information across an entire bonobo community. That skill is on display at sunset, when the bonobos voice a special \"nest-hoot\" that guides everyone to the same general area for bedtime. Primatologist Zanna Clay tells me that should a bonobo party wander beyond the hearing range of the others, they forgo giving the nest-hoot, apparently aware they are on their own\u2014and fine with it.\n\nSomeday we may learn that animals impart more information than we have given them credit for. Gunnison's prairie dogs communicate in a complex range of tones; it seems they are able to formulate a slightly different squeak of alarm when, say, a tall fellow in a red shirt runs into view compared to a short, slow-moving guy wearing yellow, or when a coyote comes along versus a dog. Whether, and how, the little fellows use this information isn't known. Other species may similarly communicate more than we know now. Could a howling wolf be giving its pack-mates instructions over a great distance? _We over here smell the blood of the enemy\u2014come prepared for battle!_\n\nWhat of humans? I will have more to say later on the topic of dispersal in our species, and its significance to the origins of our intelligence and our societies\u2014but here's a preview. Before agriculture tied people down, hunter-gatherers had the option to spread out. As it does for other species, this fission-fusion reduced competition within a human society and enlarged the population the land could support. At the same time it permitted each individual to sort out how, and how much, to interact with special others. Chimpanzees might be incapable of taking full advantage of the fusion aspect of fission-fusion to draw together faraway members, whereas for humans, keeping abreast of each other at a distance has been a key to success. Most people in early societies were usually too far off to yell, making bits of ingenuity such as smoke signals and drums indispensable. Such techniques had limitations, as did all methods of long distance communication before Morse code transmitted news in real time. Many prehistoric signals might have conveyed about as much information as a text message reading \"Hello.\"\n\nNevertheless, early records are full of clues that hunter-gatherers communicated quite well, especially in an emergency. Messengers making the rounds perhaps represented the pony express of the day\u2014running was no problem; the human body is built for endurance. By such means a thirsty populace gathered at the last puddle, and anyone stumbling on game or an enemy drew in others for feast or fight. In one of the first European contacts with native Australians, the crew of a Dutch ship kidnapped a man on April 18, 1623. The next day they were faced with 200 Aborigines brandishing spears: apparently word had spread fast.\n\n# SHIFTING ALLEGIANCES\n\nSocieties are closed, yet not impregnable. Movement between societies is essential to the health of each group. Without opportunities for individuals to transfer from society to society, the rare sneaky infiltrator aside, populations would become inbred, especially when the societies are small. Savanna elephants circumvent this predicament with males that range free. Bulls spend much of their time with a mix of other males, sticking closest to a male chum or two. Unaffiliated with a core, they mate with whomever they choose (but rarely without a fight from another bull). In most cases, though, societies comprise both adult males and females. In such species, transfer between societies can be a mandatory step in reaching adulthood. Youngsters may strike out from their birth society after its members, who knew them growing up, avoid them as mates. Primates represent a diversity of sexual dispersal patterns. In the highland gorilla and a few other primates, either sex can make the move. In other species, just one sex relocates. In the hyena and many monkeys it's the male that leaves; in the chimp and bonobo the female does\u2014although a few female chimps stay where they were born. An animal may relocate again later in life if its status is so miserable that it's pushed out of a society or escapes voluntarily. Not only are hyena males beaten up by the females, but the females quickly lose interest in newcomers as sex partners, forcing many males to try their luck in another clan.\n\nSometimes an animal is on its own for a while before it finds a new home. Actually, acceptance by a new society can be hard to come by\u2014to keep their boundaries distinct, barriers to entry are high. A forceful individual may initially bully its way into the group by fighting those of its own sex, who see it as a contender for the society's private supply of mates. But attaining permanent group membership can require being found desirable by the opposite sex, too. Hence a freshly arrived male baboon insinuates himself into a new troop by beating off the local males while befriending the ladies. A wannabe female member of a chimpanzee community typically makes her first appearance when she is sexually receptive. This strategy attracts an entourage of male enthusiasts that shield her from females who would rather not have her stay.\n\nEven so, the female chimp must contend with those females after she is taken in, just as a newcomer baboon is stuck dealing with the current male hierarchy; with chutzpah and female support he rises to its top. This aggravation doesn't exist in species where newbies force the previous male occupants out of a society. Male lions and horses hang with other bachelors until they earn a place in a society, typically by driving off its current male residents. They may not do it alone: two or more male painted dogs, lions, and at times horses can team up to stage a hostile takeover and are likely to remain longtime allies thereafter. Male ringtailed lemurs and Ethiopia's baboon-like geladas cultivate similar bromances. While the geladas maintain their ties, lemurs often forget their partnership once they have inserted themselves among the other males of a troop.\n\nIn other cases, joining a new society is a matter of perseverance rather than force. From time to time a female elephant will defect to another core, sometimes after its members repeatedly nudge her away. A male hyena or female monkey hovers on the fringes of a clan or troop, putting up with an onslaught of abuse until its presence becomes routine and tolerated. The ease of acceptance can depend on the straits of the destination society. A wolf pack will normally admit a lone male only after days or weeks of attempts, if ever; but under the right circumstances a lucky dog can be welcomed almost immediately. That was demonstrated when, in 1997, a Yellowstone pack, whose alpha male had been killed by humans, was observed taking in a roaming wolf. Keeping their distance for hours, the loner and the pack howled back and forth until they came face to face. Eventually a juvenile broke the ice, running forward to greet the male. Only six hours later, the whole pack mobbed him enthusiastically. Any dog owner would recognize their behavior: tail-wagging, sniffing, and play. Even though his adopted pack had killed two of his siblings the year before, he instantly became the new alpha animal.\n\nIn species where only one sex leaves, it's the less adventuresome sex that has the easier life. Like a town resident who inherits the family farm, such an animal retains childhood friends and relatives as well as familiarity with its home turf, an area it knows like the back of its hand. The grown male chimpanzee, a lifelong homebody, goes back to his favorite childhood haunts, whereas a sister who moves to a new society must start from scratch, severing not just her personal relationships but also all bonds to her birthplace.\n\nIn all of the foregoing, we have seen that the ways animals travel and keep track of each other inside their societies can influence the benefits societies have to offer; we have also looked at how movement between societies, accomplished at rare intervals and with difficulty, allows for changes of allegiance that turn a former foreigner into an established part of the social scene. Next we consider what, and how much, mammals must actually know about each other to reduce the flow of individuals in and out of their packs, prides, troops, and coteries, so each society can function as a clear, independent, and enduring unit.\n\n# CHAPTER 4\n\n# Individual Recognition\n\nSocieties, as we have seen, are not impregnable. An elephant can seek her fortune in a new core group, male hyenas and female chimpanzees can jump ship to other clans and troops, and a lone male wolf can be crowned leader of a foreign pack. This movement of mammals between societies brings to the fore the conundrum of how membership is achieved and how it stays crystal clear.\n\nIt's possible for groups other than societies to stay cleanly apart indefinitely, even when the animals involved know nothing about each other. For instance, certain social spiders come together by the hundreds to weave communal webs to catch prey, the average person's bad dream. Because the webs are spaced and the spiders stay put, the colonies normally never mix. Yet put one colony next to another, and the spiders merge without making any distinction between them. You wouldn't know it without doing this experiment, but the colonies prove to be completely permeable\u2014open to outsiders coming and going. Thus it would overstate the matter to say the spiders are part of a _society_ : they display no affiliation with the colony that one could construe as membership. Too much permeability can sometimes be an issue even for species with clear societies. Where domestic honeybees evolved in western Asia and Africa there was little opportunity for confusion between hives. These days, beekeepers put the nest boxes so close together that some workers drift\u2014fly astray and go about their business in blissful ignorance, working for the wrong colony, resulting in a loss of beepower for the home hive.\n\nAmong most vertebrates, permeability is restricted to rare and arduous transfers between societies of individuals seeking mates. These species keep outsiders out by forming individual recognition societies: each animal must recognize every other member as an individual, regardless of whether that peer was born in the group or admitted from outside. In the mind of a savanna elephant or bottlenose dolphin, then, every Tom, Dick, and Jane in the society must be identified as Tom, Dick, or Jane. Not that they use personal names\u2014except perhaps the dolphins. These cetaceans can catch the attention of a friend with custom \"signature whistles\" that some researchers take to mean \"over here, Jane!\"\n\nKnow everyone by heart and of course you know who belongs\u2014piece of cake. Animals can also learn to identify individuals who are not part of their society. And those individuals are often, but need not be, enemies. Primatologist Isabel Behncke tells me that certain bonobos groom each other when communities come together. While not exactly international diplomacy, grooming appears to be a sign of extrasocietal friendships. This openness doesn't lead to misunderstandings as to who goes where. After some frivolity, the apes go home, memberships intact. Chimpanzees fail to have such foreign friends except in the Ta\u00ef Forest of Africa, where they have to be circumspect about it: females from different communities, who presumably knew each other in their youth before they emigrated elsewhere, will get together. The two meet surreptitiously, as if aware they would be killed if anyone else found them grooming each other. A society, then, seldom includes _everyone_ animals know. Rather, an animal sees its society as a particular set of individuals: an \"us\" versus \"them\" distinction.\n\nSuch capacities in an animal may seem surprising, but many vertebrates can store information about others of their kind, tagging each with a category that we (a language-using species) might call \"citizen,\" and then further classifying their peers within that group. Consider baboons: they recognize rank, family, and coalitions within their troop and use these categories to predict how others will behave. Because, for example, the females largely inherit their social ranks from their mothers, forming support networks known as matrilines, baboons expect a low-status female, no matter how assertive her personality, to back down in a face-off with a female of high rank, no matter how infirm or shy. Indeed, \"in the mind of a baboon... social categories exist independent of their members,\" biologists Robert Seyfarth and Dorothy Cheney have said. Doubtless that applies not only to their matrilines but to their societies\u2014their troops\u2014as well.\n\nAfrica's vervet monkeys don't just recognize which monkeys are foreign, they often know which troops those others are linked to. Cheney and Seyfarth found that vervets recognize each other by voice and paid more heed when the cries of a member of a neighboring troop came from the wrong place\u2014from the territory of yet another troop. The vervets bounced around manically like spectators anticipating the blows in a boxing match. Their expectations fit what one would guess to happen in this circumstance. The monkeys' reaction indicates that they realized the neighbor whose voice they recognized must have entered another troop's land, behavior that generally occurs during a brawl between troops. So beyond differentiating their own society from the catchall category of \"everyone else,\" vervets grasp that foreign monkeys are split up among multiple troops as well.\n\nA membership system based on individual recognition works because the members agree on who belongs. At times there can be differences of opinion, but across all society-forming species these are rare and temporary, limited to the transitional moments when an individual is being expelled from, or joining, a society. A stallion, excited at the prospect of another mate, may encourage a mare to join his band while the band's mares, still seeing her as an outsider, simultaneously try to drive her off. The endgame for the mare, as she works to wear down this female opposition, is not merely to be recognized, but to be recognized as one of the group.\n\n# RECOGNIZABLY DIFFERENT\n\nRecognizing one's group-mates, of course, requires that each member be distinguishable in some way. Across the animal kingdom, recognition can come in many sensory modalities. The vocalizations of most social mammals, from the calls of vervets to the roars of lions, vary from one individual to the next. Vision can also be important: capuchin monkeys can quickly distinguish between photos of their own troop members and those of outsiders. Depending on context, the distinctive spots of spotted hyenas serve variably as camouflage and as means of identification at a distance on the open savanna. Lacking markings as distinct as a hyena's spots, chimpanzees are drawn, as people are, to faces, notably the eyes, as a key distinguishing feature. Their voices, too, are as distinctive as those of humans, and they can also tell each other apart perfectly well by their rear ends, a gift of discernment yet to be tested in humans. Horses give way at a water hole if they see members of a dominant band approach within fifty meters. Monkeys grow so familiar with one another that in Uganda the red colobus and other species\u2014blues, mangabeys, and redtails\u2014play together with everybody zeroing in on certain interspecies friends. In studying how well lions know each other, George Schaller, a senior conservationist at the Wildlife Conservation Society, has thoughtfully observed their behavior both within and across prides:\n\nNo matter how widely females are scattered or how frequently they meet each of the other members, they still constitute a closed social unit which strange lionesses are not permitted to join.... A pride member joins others unhesitatingly, often running toward them, whereas a stranger typically crouches, advances a few steps, then turns as if to flee, and in general behaves as if uncertain of its reception.\n\nThe cats pounce on a lion that seems strange until it is identified as one of their pride. Mind you, no one has set out to methodically prove that lions or any other animals recognize each and every member of their society, but the assumption seems safe based on this sort of observation.\n\nRecollecting others must have been a necessary precursor to the evolution of societies in vertebrates. Indeed, I wouldn't be surprised if individual recognition is universal among mammals and birds, given that fish, frogs, lizards, crabs, lobsters, and shrimp all have the capacity. We should expect this. Even for animals that don't live in societies, telling one individual from another is still important, whether in fighting over a territory, dominating others, finding a mate, or distinguishing one's infant from somebody else's. And so hamsters, asocial as they are, integrate the smells from different parts of the fuzzy bodies of others of their species into a representation of each individual, much as our minds process facial lineaments to create a comprehensive representation of a person. Emperor penguins and their chicks experience long separations when the parents leave for days to bring home a fish. How do they find each other among the thousands in a colony? By listening. In the same way that we filter the din of a cocktail party, the birds selectively hear the calls of their kin: _Did I hear Tom on the other side of the iceberg?_\n\nHowever crowded their conditions, neither hamsters nor penguins have societies. And however widespread such feats of memory, life in a society presents a whole different ball of wax when it requires a blanket knowledge of _everyone_ \u2014or at least a knowledge of all other members to some minimal degree. But how minimal?\n\nBiologists studying the issue of recognition and relationships have focused on the strongest connections in a society: the individuals that know each other best, and how they interact. Yet as a consequence of this seemingly sensible choice, an equally compelling area of investigation has been neglected\u2014the question of how much the individuals that interact _least_ in a society actually know about each other. It is conceivable that two members who happen never to approach each other may be ignorant of the other's existence. Their lack of contact could also be a sign of indifference, disdain, or an inability to start a relationship because they move in different social circles. The two may overlook or avoid one another as a strategic choice, or it could be that, as with the guy you have seen sitting in the coffee shop a hundred times, they never get around to introducing themselves because there are just not enough hours in a day.\n\nIt may be even simpler. You might be completely unaware of the fellow drinking his coffee, but still register him in the back of your mind. Does the following experience sound familiar? One day I felt that something was different about my favorite caf\u00e9. After a moment I realized a regular patron was gone, a person who, for the life of me, I couldn't recount in detail even if I had to. When we take a moment to perceive someone, like a coffee shop patron, as a person, we are said to individuate him\u2014render him as a Tom, Dick, or Jane. People, and presumably many animals, don't have the endurance to individuate everyone. We store our knowledge of many others at a schematic, often subliminal, level.\n\nHere's a thought experiment: Suppose humans related to everyone entirely in this schematic way. A scientist would pronounce that we are _habituating_ to the distinctive quirks of each individual, which is to say that our subconscious registers his or her personal attributes at the same time we tune out those attributes from our everyday awareness, much as we do with ambient noise we don't notice until it goes quiet. A species might still form societies by individual recognition at this subliminal level, without being mindful of a single individual around them\u2014what a weird and impersonal world that would make!\n\nDespite that possibility, I suspect most vertebrate societies are small enough that animals are cognizant of the other members, not just in the minimalist way in which I was vaguely aware of that caf\u00e9 patron, but very well indeed, whether they choose to take notice of them constantly or interact with them often.\n\n# DEMANDS OF MEMORY\n\nThe members of a troop of baboons or vervet monkeys stay close enough together to encounter each other daily, if not every few minutes. Constant exposure to the same familiar faces should make individual recognition a breeze. However, because out of sight can mean out of mind, and some animals in a fission-fusion society can be absent from their companions for a while, recollection can sometimes be a challenge. For example, biologists spend months between sightings of certain shy chimpanzees, often beaten-down females who are members of the community but keep to themselves in a private corner of its territory. Almost certainly other chimps see such loners only occasionally and after long intervals, too. Clearly chimps need good memories.\n\nAnthropologists have designated our species as \"released from proximity\" because we not only remember others but also keep track of relationships with people we don't see for lengthy periods of time (even maintaining them through intermediaries\u2014friends of friends and the like). For us, trust is easily regained or suspicions rekindled. This \"release\" pertains to other mammals as well, with society-mates reuniting peaceably after a protracted absence. There are many records of animals that, recognizing others despite changes wrought by age, take up their relationship without skipping a beat. Biologist Bob Ingersoll recalled visiting a chimp that hadn't seen him in more than 30 years. \"At first she didn't really recognize me, and I didn't recognize her. Then I said, 'Mona, is that you?' And she immediately signed to me, 'Bob. Hug hug hug.'\"\n\nMemories aren't always so long. Horses don't recollect their own offspring after separations of more than 18 months. By that time the colt or filly will have bonded to a society\u2014a band\u2014of its own. Perhaps, in species where animals permanently depart, good recall would be a waste of mental energy and a burden.\n\nThe demands of memory include inserting oneself into the memory banks of others. In an individual recognition society, being correctly identified, let alone being identified as one of the group, ultimately takes becoming known to all. An outsider joining a society faces risks due to its initial status as a stranger: anyone who doesn't recognize the newcomer may attack. To sidestep this difficulty, as primatologist Richard Wrangham pointed out to me, the new arrival will likely stick near those who are acquainted with it already, so that those animals it still hasn't met will see it as belonging there\u2014 _a friend of our friend must be a friend_ \u2014or at minimum, a compatriot.\n\nYoungsters born in a society are not exempt from this quandary. The process of becoming a familiar presence begins early, with a mother picking out her infant. Birds distinguish their young by an age when there are likely to be mix-ups, for example when the chicks leave the nest and might intermingle with other chicks: species that pack into colonies, for example, can tell one chick from another almost from birth. For a youngster to become part of a society, though, not just mom but everybody else has to learn to identify it eventually. Fortunately, a baby is harmless at first, reasonably ignored by everyone other than the mother. Problems come when it reaches an age at which others may mistake it as a possible foreign threat. All the adults in a troop of dwarf mongoose jointly anoint the juvenile members with an excretion from their anal glands, perhaps a sign of acceptance. Until then, like the immigrant, youngsters of every species stick by those who know them best, safely presenting themselves as _the friend of a friend who must be a friend_.\n\n# INDIVIDUAL RECOGNITION AND THE SIZE OF SOCIETIES\n\nAll the societies of the mammals I've mentioned share one conspicuous feature in common: they are small, with often a dozen or two and seldom over 50 members. Lions don't sweep over the Serengeti in prides of a thousand to fell a herd of wildebeest. Prairie dogs never take dominion over a territory in the style of a human nation, their burrows spreading across the landscape with a population at ready to repel all foreign prairie dogs. And other apes don't rise up in armies like those in _Planet of the Apes_.\n\nIn some instances the ultimate explanation is ecological. One thousand lions would have trouble feeding themselves day after day, and such an enormous pride would starve. But it's safe to predict that the societies of most of our vertebrate cousins are consistently small for a more prosaic reason: as Dunbar recognized for friendships, keeping track of many individuals, in this case sometimes even very minimally, is hard.\n\nWith the exception of humans, and our astounding capacity to live together, ape societies peak at about 200 members in the chimpanzee. Most vertebrate groups much larger than this are aggregations\u2014schools, flocks, colonies, herds, and the like, from which individuals come and go with impunity\u2014not societies, which have rigid memberships. A school of herring located not far from Manhattan (and rivaling the size of that island) once held 20 million fish. The sky's the limit for the red-billed quelea, which turn and wheel over Africa in flocks of a million, while equal-sized herds of wildebeest thunder below. As for nesting birds, the largest colonies may be the 4 million sooty shearwaters that breed on Isla Guafo, Chile.\n\nSuch groups consist of opportunists. Penguins huddle yet, bonds between parents and chicks aside, are indifferent about whom they huddle with. During migrations, \"the assembled wildebeest are a crowd of strangers,\" as one wildlife biologist put it. The same can be said for caribou and American bison, which may stick by their calves and a couple of friends but join herds made up of varying collections of unknown others.\n\nA mammal society reliant on individual recognition faces a limit to the number of others, whether those members be friend or foe, that each animal will be able to track and recall. As societies approach that maximum, memories fail, putting a hard cap on their size.\n\nThe grass-browsing gelada, looking like a baboon in a Halloween mask, is a telling example of the constraints of memory. Since many hundreds of the animals forage in a herd, we might imagine a gelada's recall of others would be legendary. Yet when primatologist Thore Bergman watched how the animals responded to recordings of other geladas, he concluded that they are almost as ignorant of those in the crowd around them as are the bison. Almost, but not quite: their recognition of others was capped at twenty to thirty individuals. Bergman deduced that groups within the herds, each composed of one to a few males and several adult females, are the true social entity of the gelada: in short, its societies. Herds come to be when dozens of these troops, technically called units, forage across the same ground. An alpha male gelada doesn't seem to know anyone outside of his troop. This incomprehension leads him to treat the many outside males moving around him all day in the herd as interchangeable potential philanderers of his females and usurpers of his position.\n\nThis may make geladas seem dim-witted, yet an important lesson emerges. Animals reliant on mutual recognition are compelled to be at least minimally aware of everyone in their society, and that renders sociality at a massive scope prohibitive. Our own kind, _Homo sapiens_ , has obliterated the glass ceiling of those other species. No matter what our intelligence, humans would never have been as successful as we are today without going large.\n\nBefore turning to humans, however, we continue our exploration of nature with another high point in social evolution, the insect societies. Not only do these arthropods include the vast majority of the society-dwelling organisms, but among their colonies are some of a scale and complexity that are indeed immense, and which I believe shed light our own societies. Here's a cryptic preview of the conclusions ahead:\n\n_Chimpanzees need to know everybody._\n\n_Ants need to know nobody._\n\n_Humans only need to know somebody._\n\nAnd that has made all the difference.\n\n# SECTION II\n\n# Anonymous Societies\n\n# CHAPTER 5\n\n# Ants and Humans, Apples and Oranges\n\nWhen I first made my way into a tropical rainforest as a young man, what enthralled me most wasn't the monkeys, parrots, or orchids, alluring as they were, but sliced leaves the size of coins, hoisted in a parade of ants a foot wide and the length of a football field.\n\nYou don't have to like entomology to be a fan of ants. We modern humans may be genetically close to the chimp and bonobo, yet ants are the animals we resemble most. The similarities between our species and theirs tell us a lot about complex societies and how they emerge. In fact there are many modes of life among the over 14,000 or so ant species, and even more for social insects in general, a category that includes termites, social bees, and certain wasps.\n\nThe leafcutter ants of the American tropics exemplify the ant's potential for social complexity. Within their nest, the green leaf banners are broken down to a substrate on which the ants raise their food crop, a domesticated fungus they tend in globular gardens ranging from baseball to soccer-ball size. There are ants with a nomadic, even fission-fusion lifestyle, but species that settle down, such as the leafcutters, can do so in spades. Basing foraging operations out of a nest or central lair is uncommon in vertebrate societies except for species that stash their young in a den or primate troops that keep returning to a tree to sleep, yet is commonplace in ants. The size of leafcutter settlements can be gargantuan: in the French Guiana jungles I came upon a nest the square footage of a tennis court. A drawback of such a metropolis is the same faced by a human city: pulling in enough resources means a lot of commuting. From the far corners of that large nest spring a half-dozen speedways along which the workers doubtless hauled hundreds of pounds of fresh foliage over the course of each year. To unearth just part of another colony I once hired six men with pickaxes and shovels near S\u00e3o Paulo. The bloody bites I sustained that week didn't stop me from feeling like an archaeologist exhuming a citadel. Hundreds of gardens grew in chambers arrayed along meters upon meters of superbly arranged tunnels, some at least six meters below the surface. Scaled to human dimensions, their subway system would be kilometers deep.\n\nThe superabundance of activity in the nest of any ant species makes plain to the most casual observer that, among the social insects, the payoffs of group living are many and varied. Ant workers stake out territories, boldly gathering food even from our dinner plates, and rear their progeny in elaborate safe havens. Communicative, persistent, hard-working, battle-ready, risk-taking, and highly organized, whether they are agronomists, herders, or hunter-gatherers, ants form elaborate labor forces of superb military operatives and diligent homemakers, masters at both protecting and providing for their colonies. The leafcutters, for one, have societies decidedly more complicated than any other nonhuman animal, and carry out mass-scale agriculture to boot.\n\nLikening people to ants can raise hackles. Comparing ourselves to other mammals comes easier, because we are mammals ourselves, as our hair, warm-bloodedness, and ability to lactate confirm. For all that, while watching a nature documentary about mammal societies, you probably won't find yourself exclaiming, _Eureka, they are so like us!_ What similarities exist can be subtle. More often we are struck by the differences: quirks such as the fact that male elephants are outcasts\u2014not members of any society at all, properly speaking. As for those relatives of ours, the chimpanzee and bonobo: how like them are we? Physically we resemble both apes, owing to the genetic closeness of our species. But what of our mode of social life? Most of the similarities that have been brought up, often in the context of evolutionary psychology or anthropology, bear more on broad facets of cognition than on the details of the apes' social organization we would otherwise think of as specific to people.\n\nThose similarities are seldom as great or exclusive as they might seem. Insomuch as chimpanzees and bonobos think as we do, the parallels often extend to other animals, too. Both are like us in recognizing themselves in a mirror; then again so do dolphins, elephants, and magpies, and there's a claim, widely doubted, that ants do, too. At one time chimps were thought to be unique among nonhumans in making tools\u2014employing twigs to snare termites is one example. Yet now we know of other toolmakers, such as woodpecker finches that poke twigs to nab bugs.\n\nChimpanzees do resemble us in how they handle conflicts. For example, some individuals achieve influence through brawn, while others rely on brains. To mention one situation: a female chimp can stop irate males from using a rock in a fight by walking over to take it away\u2014repeatedly, if necessary. Such political maneuverings, at least, might seem unique to chimps, bonobos, and us, but in fact they likely aren't. The discovery of such habits is a direct result of our opinion that these apes, which we recognize as human relatives, are worth close study\u2014as one scientist put it, the experts have been \"chimpocentric.\" The assassination of \"40F,\" the alpha female of the Druid Peak wolf pack at Yellowstone, is a case in point. The evidence suggests the pack rose up to kill her after she ferociously attacked two of them. The researchers who observed these acts wrote, \"Her life and death could be summarized with the old phrase that often applies to human tyrants: if you live by the sword, you may die by the sword.\" The politics of wolfdom can be complicated indeed. Once this kind of scrutiny is applied elsewhere, doubtless more species will be identified that rival apes and wolves in their mastery of social machinations.\n\nThe fact is that though we share 98.7 percent of our genes with chimpanzees and bonobos, it's our differences that are most striking. Indeed we are as different from them as apples are from oranges. In both apes, relationships are dictated by strict hierarchies of power, which are tyrannical in the chimp\u2014especially in the males. On maturing, females of both species abandon childhood kith and kin for another community, never to return. The females are sexually receptive only on occasion, this condition made obvious by their swollen rear ends. The female chimp can pretty much either be beaten up or ignored by males except on the rare days she's in heat, at which point she often has sex forced on her. No wonder neither chimps nor bonobos have pair bonds or an extended family life, and mothers get little child support from dad\u2014or anyone. Females aren't especially skilled at befriending each other, either; in fact a harried chimp mom must give birth at a private site to avoid having her baby killed.\n\nSo while I will point out engrossing similarities between the societies of humans and other vertebrates in the pages ahead\u2014oftentimes in regard to the benefits reaped from being in a society, and how those societies interact\u2014for the most part the social lives of other mammals, those of our ape cousins included, can seem downright weird, if not altogether inhuman. And next to their weirdness is ours as mammals: No chimp has to grapple with the rules of a speedway or the upkeep of a homestead. Nor does it contend with traffic congestion, public health issues, assembly lines, complex teamwork, labor allocation, market economies, resource management, mass warfare, or slavery. As alien as insects seem to us in appearance and intelligence, only certain ant and human societies do such things, along with a few other social insects such as honeybees and some termites.\n\n# PUTTING TOGETHER A LARGE SOCIETY: LESSONS FROM THE ANT\n\nThe parallels between certain social insects and modern humans are largely an outcome of one underlying commonality between us: our societies are populous. Scientists investigating animal behavior have too often been narrowly devoted to the evolutionary relationships between species, when many features of societies have more to do with scaling\u2014sheer numbers\u2014rather than pedigree. The societies we have looked at among the nonhuman vertebrates, including the nonhuman apes, contain at most a few dozen individuals. A large leafcutter nest harbors a workforce in the low millions.\n\nOnce a population grows that large, all types of complexity can emerge\u2014in fact often have to emerge to get jobs done. The coordination seen among hunting groups in a chimp or painted dog society is mercurial enough to amount to wishful thinking compared to the elaborate way some predatory ants organize hunts, with some workers stopping the quarry in its tracks, others applying the death blow, and others dismantling the corpse into slabs of flesh carried off in coordinated teams. Most vertebrates lack the labor force to take on such specific roles; nor must they operate in this way for the members to get the food they need.\n\nThe same is true for housing and infrastructure. The burrows of prairie dogs can be intricate; they connect underground with hibernation chambers and dead ends that thwart predators. Yet, contrasted with the monumental architecture of a leafcutter nest or a honeybee hive, the dwellings of these rodents seem a relic of the Animal Stone Age.\n\nBut\u2014I imagine your protest\u2014no human hunters have ever caught prey in the way ants do, and no human residence resembles one of their nests. Any two things, apples and oranges included, have innumerable characteristics in common and an equally immeasurable number of dissimilarities. What interests someone, similarities or differences, depends on that person's point of view. Identical twins aren't identical in the eyes of their mother; and, as will become important when we look at psychology, the members of one race, though outsiders may fail to differentiate among them, don't look alike to each other. If nothing else, remember this: _comparing identical things is deadly boring._ Making comparisons is most fruitful when parallels are noticed between ideas or things or actions ordinarily treated as distinct. Hence slavery in ants, where individuals work against their interests immersed in another society, is different from how Americans carried out slavery, and which in turn differed from the treatment of those defeated in war by ancient Greeks.\n\nPeople and ants reach different solutions to the same general problems, sometimes by using completely different approaches; but then again so can different human societies or different ant societies. In some parts of the world, we drive on the left side of the road, in others on the right. On busy routes in colonies of Asia's marauder ants, incoming traffic streams down the highway center while outbound ants take to its flanks, a three-lane approach no human district has tried. Both patterns bespeak the importance of getting goods and services to the right places safely and efficiently when the populace depending on them is enormous and doesn't, and possibly can't, all go out foraging.\n\nThink about the allocation of goods and services. Humans show a lot of variety here: Marxist societies don't handle problems the way capitalist ones do. Ants have solutions of their own. In the red imported fire ant, for example, the flow of commodities is regulated according to what is available and what's needed, a supply-and-demand market strategy. The workers monitor the nutritional cravings of other adult ants as well as the brood, and change their activities as the situation requires. In a nest laden with food, scouts and their recruits hawk merchandise by regurgitating samples to \"buyers\" in the nest chambers, who in turn roam through the nest to distribute meals to whoever wants them. If these middlemen find their customers sated on meat (dead insects, perhaps), they peruse the marketplace for other commodities, until they find, possibly, a seller offering something sugary. When a market becomes glutted and sellers can no longer peddle their wares, both buyers and sellers engage in other jobs, or take a nap.\n\nHow does division of labor play out in the ants? Certain jobs can be farmed out to specialists that execute them exclusively, or with greater frequency and precision than others. Age has a role in determining that frequency: because young ants happen to still be near the brood in the nest chambers where they themselves were reared, they start as nurses, grooming and feeding larvae on behalf of their mother (reversing the chronology for humans, in which the elderly help tend the grandkids). But things can get complicated. Just as aspects of people's appearance offer clues about their career\u2014the person with a suit and briefcase may well be a lawyer; those with hardhats and lunch pails are probably construction workers\u2014division of labor in ants can also be associated with appearance. The fairness of stereotypes of spindly human office workers aside, in many ant species different workers do come in body sizes and proportions to suit their role.\n\nFor ants and most other social insects, the most fundamental specialization is unlike anything known in people, and in fact is closer to the situation for gray wolves or meerkats: usually only one female (the queen in social insects) breeds. This aspect of the life history of ants, whereby several generations of young raise siblings, makes the group to which they have membership more than an everyday family and hence worthy of the word \"society.\"\n\nFurthermore, ant societies are sisterhoods (which is why I often describe workers using the pronoun _she_ ). That's not unheard of in vertebrates. For the savanna elephant, too, all the adult members of a society, or core, while not always sisters, are female. However, for many ants, sex matters little in practice because workers can't reproduce; the ovaries of leafcutter workers don't function. Meanwhile ant males, like honeybee drones, are socially useless: their sole contribution is to have sex and die. Termites, on the other hand, show sexual parity: there's a king in the nest, as well as a queen, and the workers are of both sexes.\n\n# DIVIDING UP THE WORK IN AN ANT EMPIRE\n\nDivision of labor in the most extreme leafcutter species is like nothing else. The growth spurt of some larvae yields a soldier caste that guards the other nest denizens. The large soldiers additionally function as a crew for heavy-duty roadwork, clearing a highway so that food, materiel, and personnel flow smoothly. The hefty leafcutters that carved slices into my skin as I dug up the colony in Sa\u00f5 Paulo belonged to this soldier caste.\n\nTo tend their gardens, ants form assembly lines involving all but the soldiers. Medium-sized workers cut the foliage and pass it to slightly smaller ants to be conveyed to the nest in the long parades. Once the greenery is dropped off in a garden chamber, yet smaller ants mince it into pieces that still littler ones crush into a pulp. Then even more diminutive workers integrate the resultant mulch into the garden with their forelegs, where tinier ants \"plant\" tufts of fungus and then prune them over time. The tiniest ants of all meticulously weed the gardens of inedible fungi and contagions. The ants also apply their version of pesticides, produced by their bodies, to the gardens to keep the yields high. The necessity of all this labor, from planting to tending and harvesting the crop, should resonate with any farmhand. No vertebrate other than humans, regardless of how smart or populous, has taken even a rudimentary step toward domesticating its food, something leafcutters, and a few other insects, have done many times.\n\nOne problem with mass production is handling the waste. The matter wouldn't occur to a chimpanzee, I assure you. Nor should it: as in the sparse human communities of Tibet, where toilets have never been required and going off in the woods remains the fashion, chimp excrement vanishes in the soil before it can build to a health catastrophe. But leafcutter nests need full-time disposal squads. Moreover, their nests are built to keep fresh air circulating. After over 150 million years of colonial evolution, ants have come to invest much more of their GDP in public safety and recycling efforts than we do.\n\n# COMPLEXITY IN SOCIETIES, LARGE AND SMALL\n\nA wonderful thing about ants is that it's possible to compare complicated societies like those of leafcutters with the colonies of species that live very simply. Some colonies are small indeed, like those of _Acanthognathus teledectus_ , or trap-jaw ants, containing a couple dozen workers. Each colony takes over the hollow core of a twig on the floor of a tropical American rainforest\u2014the ant analog of a cave. Such a colony doesn't need a highway or assembly line or complicated teams any more than a similar-sized hyena clan, gray wolf pack, or small human tribe, would, beyond perhaps coordinating a few individuals to take down prey. The trap-jaw ants face no crisis distributing supplies or managing waste, and when there is danger, they just run away. Other than the queen, they are equal in size and do the same things. To get all the colony's tasks done, each has what amounts to a Swiss army knife built into her face. The \"trap jaws\" are lengthy mandibles that end in spikes; every ant is, in effect, a bear trap, configured to carry out the killing and retrieval of prey entirely on her own. Specialization is minimal: not only are there too few individuals to fill different roles, but overspecializing is risky when a group is small. A military squadron that loses its sole radio operator may be doomed. A trap-jaw ant colony is simply so small that its workers have to be generalists. Populous societies offer a redundancy that allows a more tailored labor force, as a comparison of the range of job listings in New York City to those in a village will show.\n\nWhat makes the leafcutter story all the more remarkable is that the species evolved agricultural practices by taking much the same path as humankind. The leafcutter's progenitors made their first move to gardening 60 million years ago, as analysis of the genetics of the ants and their fungi reveals. Ants that live as those ancestors did still exist, and their social complexity is nearly as modest as that of the trap-jaw ant. Such societies consist of a few dozen to a few hundred workers raising tiny gardens of wild fungi; in many ways, in fact, they resemble the small human tribes that often cultivated wild species they took from nature, growing only what they needed in small plots of ground near simple abodes. Twenty million years ago, long before _Homo sapiens_ appeared, some of these easily overlooked ants domesticated their fungus\u2014it came to depend on the ants' care. Thus modified, the fungus no longer propagated in the wild yet now could be farmed at a mind-boggling scale; the colonies exploded in population much the way human societies did after agriculture got its start in places like the Nile Valley, supported by domesticated grain.\n\nMany commonplace features of such large societies, such as infrastructure and systems of waste management, are virtual requirements when many settle in one place. Yet complexity and numbers don't invariably go together, as an invasive species, the Argentine ant, confirms. Most Argentine ant \"supercolonies\" encompass far greater multitudes than any leafcutter nest, but the workers are one size, show no specialization, lack aptitudes for assembly lines or elaborate teamwork, and erect no monumental central abode. Relying on roaming habits akin to those of the spotted hyena or bonobo, Argentines ramp up fission-fusion to the _n_ th degree. Spreading every which way, they turn any suitable corner of the territory at once into part of the nest and a place to forage. The simplicity of a supercolony reminds us that while we are sure not to see an Empire State Building in a tribal village\u2014there's only so much complexity a few souls can create and manage\u2014there won't be skyscrapers or good plumbing in every major metropolis.\n\nDespite their societal simplicity, however, the effectiveness with which each Argentine ant supercolony remains identified as one unit can help us answer the very big question of how humans put together truly immense societies, as we shall discover next.\n\n# CHAPTER 6\n\n# The Ultimate Nationalists\n\nThough it's debatable whether insects erected the first society, without question today's social insects are the master society builders. Insects have had this success despite their tiny nervous systems. Many aspects of cognition don't require much of a brain, however. Chances are good that insects have the mental capacity for subjective experiences\u2014a unified perspective on the world that yields a sense of \"self,\" even if that self is simple from our point of view. Unable to get inside their heads, we can't say for sure. Yet a part of their triumph can be chalked up to something straightforward indeed: an efficient approach to distinguishing societies. No species illustrates this better than the Argentine ant.\n\nBirders casually call any nondescript bird a \"little brown job.\" If there ever were a little brown job among the ants, it would be this species, originally from northern Argentina but globally invasive. Argentine ants the color of weak tea routinely overran my pantry when I lived near San Francisco, as they do in millions of homes in the Bay Area. Feeble of jaw, those nuisances didn't look like they amount to anything, lacking even stingers. Yet the species represents a pinnacle of social evolution. I could have plucked an Argentine ant from my house in Berkeley, driven 800 kilometers to the Mexico border, dropped her off, and she'd have been just fine. In a very real sense she would still be home. Improbable as it might seem, the ants swarming around the feet of the customs officers inspecting my passport would have been members of the same ant nation that ran roughshod through my kitchen up north.\n\nIf, on the other hand, I had taken the same ant about 60 kilometers north of Mexico, to the outskirts of San Diego, and deposited her a centimeter or two beyond a border that means nothing to us but that ants mark off with their lives, things would have gone very differently for her. There, too minute and hidden in the grasses to be noticed by human suburbanites, she would have confronted the ants' own border patrol and likely joined the tiny corpses heaped along a narrow line extending block after block beneath the grass of manicured lawns, where over a million ants die each month in what is arguably the largest battlefield of all time.\n\nTo the west of that border lies the holdings of the Lake Hodges Colony, a kingdom of the same Argentine ant species spread over 50 square kilometers. The dominion to the east is claimed by what the experts dub the Large Colony, a single social entity whose territory stretches from the Mexican border to California's Central Valley past San Francisco. Given that a backyard in Southern California can sustain a million Argentine ants\u2014a footstep on the tidiest of lawns stirs up a host of them\u2014Large Colony clearly contains billions of the workers. Little wonder, then, that entomologists call these ant republics supercolonies.\n\nFour supercolonies are known in California: the abovementioned two, and two others. Given the right level of moisture, nothing seems to stop their continued growth\u2014except combat, along contested contact zones that extend for kilometers, bringing to mind the trenches of the Western Front in World War I. _Homo sapiens_ , it turns out, isn't the only imperialist organism. What stupefies is the length of time the supercolonies have been at war. The species, immediately recorded in the newspapers, arrived in California in 1907. Each supercolony started off as a few ants, possibly lodged in the potting soil in separate shipments of houseplants. Their domains expanded incrementally over the ensuing decades, wiping out other ant species until they came to abut each other. Then the fights began. The front lines shift glacially month after month, a few meters one way, then the other.\n\nInternally, however, the supercolonies run so smoothly and at such an immense scale that people in nations\u2014with their meddling, sharp differences of opinion, cheating, selfishness, outright aggression, and homicides\u2014look positively dysfunctional by comparison.\n\nBefore they stumbled on the war zones near San Diego, everywhere scientists went they found Argentine ants living in bliss, an observation that led them to conclude the ants all belonged to one happy family. That is, until 2004, when researchers collected samples of ants from different neighborhoods that, by pure happenstance, fell within the territories of two different supercolonies. To the scientists' shock, a fight raged the instant those ants were combined, killing many of them. The abrupt about-face this caused in the experts' understanding of Argentine ants hints at how toilsome it can be to interpret societies in nature.\n\n# THE IGNORANT ANT\n\nIf big-brained vertebrates can typically manage only societies of a few dozen individuals, what allows the ant, with its minimal nob of neural tissue, to do infinitely better? As hard as it is to fathom a leafcutter ant nest of a million, the Argentine empires boggle the mind.\n\nClearly an ant can't be acquainted with each fellow member of its society as wolves and chimps are with theirs. This isn't because insects inevitably fail to recognize each other as individuals. For instance, the North American paper wasp _Polistes fuscatus_ excels at face recognition, much as humans do. Recognizing each other is important when they first join up to construct a nest because of the wrestling matches that determine who breeds. Later in the season, a colony can grow to 200\u2014a lot of faces to recall. However, by then there's little to fight over and in all probability the wasps no longer keep track of everyone.\n\nNevertheless, the paper wasp is an exception: most social insects don't have individual recognition. Ant and honeybee workers don't know anyone as an individual. Associations between ants\u2014the team pinning down an enemy combatant, for example\u2014are impersonal. The best a worker can do is discriminate among types of individuals\u2014tell soldiers from worker ants or a larva from a pupa, for instance, and most essentially a queen from everybody else. The ants do manifest personality differences, with certain workers putting in more effort than others, for example, but those do-gooders go unrecognized for their trouble. That means ants avoid ever having to confront a rival or build an alliance within their societies the way we vertebrates do\u2014the queen aside, no ant picks favorites, and every other worker in the colony is equally on the same side. To the ant, only the society, not the individual, matters.\n\nThe fact that ants don't need to know each other explains why you can move an Argentine ant anywhere in its supercolony and she will go right into interacting as she had before, but now with whatever random colony-mates happen to be there. In fact, as far as experts know, an Argentine ant, lacking a central nest, wanders indiscriminately across her supercolony's territory until the day she dies. If ants are forever strangers, moving through a crowd of unknown others, how can they tell where their society ends and the next one begins?\n\n# ANONYMITY\n\nIn 1997 two chemists were rearing Argentine ants and cockroaches in a lab for pest control studies. An assistant decided the roach supply would make good Argentine ant chow. Pure scientific serendipity ensued. One day the ants began killing each other rather than carving up the roaches. The cause was quickly discovered: That morning a technician had tried feeding the ants a different roach, a pest from Africa. Any ant that so much as touched one of these brown-banded cockroaches was summarily slaughtered by her fellows.\n\nThe problem was the smell. Most insect communication is based on chemicals known as pheromones, with specialized glands releasing compounds to signal emergencies or signpost a route to food. Colony membership is marked by chemistry, as well. Although ants don't tell individuals apart by their personal aromas the way hamsters do, they do recognize each other as nest-mates\u2014or as foreign\u2014using an odor as a shared sign of identity. As long as an ant displays the correct emblem\u2014as long as she smells right, which requires that she have the right combination of molecules known as hydrocarbons on her body\u2014her colony-mates admit her as one of their own. The scent (or taste, if you wish to call it that, since ants detect the emblem by touch) is like a flag pin, one that every ant must wear. An ant that shouldn't be there is quickly detected by her alien scent. Since ants have no white flag of surrender, more often than not the outsider is killed, like the unfortunate Argentine ant that snoops across a territorial border. Brown-banded roaches, perhaps by coincidence, have critical components of the scent the ants use to cue in on colony mates, and on those from foreign colonies. When an ant touched the roach provided as a food source, she transferred these hydrocarbon molecules to herself, in effect putting on the uniform of the opponent, and was then misidentified as an enemy.\n\nThose identifying markers are what enable social insects to transcend the widespread vertebrate requirement of having firsthand acquaintance with each other. Whether we are talking about several trap-jaw ants intimately clustered inside a twig or a billion Argentine ants spread far and wide, the members need never have met, nor even come close to each other, let alone recall one another. Species that mark their aggregate identity have what I call anonymous societies.\n\nMarkers can have a few origins. The roach example is proof that the environment can influence colony identity, but all the members of a supercolony are not likely to share a single environment. That the abutting borders of supercolonies are measured to the centimeter for territories that extend across kilometers of varied habitat indicates that genetics must play the overriding role. Indeed, the hydrocarbon scent is coded in the genes, with diet generally having little effect.\n\nIt's easy to assume that the actions of ants, including their response to a colony marker, will be genetically prescribed in a fixed, simplistic way. Actually, most aspects of behavior have built-in, or innate, components regardless of species\u2014even human activities like learning a language, which our infants do as a matter of course. Social psychologist Jon Haidt has described such innate features as organized in advance of experience. When a living thing is likely to encounter a range of situations, the blueprint of its nervous system\u2014its organization in advance of experience\u2014must allow for flexibility. For social insects, just as in other animals, that means less flexibility than in humans. Yet don't give them short shrift. Tiny they may be, but insects are flexible nevertheless.\n\nIn ants this flexibility can extend to how they come to know their national scent. This is demonstrated by the slavemaking ants, which exploit the way ants learn their identity to take slaves that do all the domestic chores of the slavemaking colonies. The slavemakers aggressively raid other nests, most often those of another ant species. Mature individuals are no good to them: Nationalists to the core, adult ants will die before accepting a foreign flag. Instead the slavemakers seize the colony's young, favoring the pupae, the quiescent stage before the insects turn into what we recognize as an ant. Pupae don't have the colony scent yet. Normally an ant emerges from the pupal state in the nest of her mother queen and quickly learns the scent of her own society, finding it agreeable for the rest of her life. But a kidnapped ant is duped, imprinting on her captors in much the way a hatchling chicken will imprint on you if you are the first thing it sees rather than its mother. Emerging in a slavemaker nest\u2014and ignorant that anything is amiss\u2014a stolen ant adopts the local colony scent as her \"nationality\" and dutifully sets to work unaffected by differences between slave and slavemaker in size or color that seem more important in our eyes. Even so, occasionally the slaves figure out something's wrong and make a run for it, but their captors usually force them back.\n\nThe favorable reception of slavemaker by slave is the least of it. Here's where an ant's adaptable brain comes in: To steer clear of a social meltdown, each slave and each slavemaker must welcome all the other slaves in the nest as well, no matter how many colonies they have been pilfered from. Yet despite the differences in scent produced by each individual, neither slavemaker nor slave has a problem identifying all the others as members of \"their\" society. Underlying this adaptability is grooming. Among primates grooming serves to bond friendly individuals, but the experts suspect that grooming in ants hardens attachments at the society level by blending the odor of the nest-mates to ensure they achieve a standard scent\u2014hence some of the slavemakers' scent rubs off on young slaves, marking them as part of the colony, and the slaves modify everyone's odor as well. Commingling the aromas has unexpected results. Should a slave blunder onto the colony of her birth\u2014the one to which she truly belongs, with her actual colony-mates and sisters\u2014she is attacked as the enemy. It's a story with the trappings of a Greek tragedy.\n\nAcquiring a national marker can be like receiving the key to the city: all is possible. In an orchard in Australia, I tore apart a weaver ant nest to find an orange arachnid five millimeters long. Many painful ant bites later I identified the species, _Myrmaplata plataleoides_ , a spider that doesn't spin webs but joins the weaver ants as if it were a citizen of the colony itself. This identity theft is achieved when the spider absorbs the colony's aroma by stealing its brood. Cloaked in the stench of nationalism it enters the nest unimpeded to eat more larvae snatched from the nursing workers. Once it takes on the identity of the colony, the eight-legged interloper has it made. Still, there's also a risk: should it amble away from that nest to another weaver ant colony, it will be attacked, not as a spider, but as an invading ant.\n\n# ANONYMITY WITH A BACKBONE\n\nAt this point, I must admit I've been holding back. There's at least one vertebrate that lives in anonymous societies, that in fact marks its societies with a scent much as ants do: the naked mole rat. Neither mole nor rat, these wrinkly, furless, pink rodents, residents of African savannas, \"violate even the most liberal standards of animal beauty,\" admit two leading experts on the species. Their markers may explain how the colonies edge past the population maximum of roughly 200 that is characteristic of most other mammal societies, with the largest ever recorded having 295 members.\n\nUnlike worker ants, naked mole rats recognize each other as individuals, as well. Whether the average mole rat gets much out of this\u2014whether it has a best buddy, maybe\u2014isn't clear. Also unknown is whether a mole rat in a large colony can recollect each of its fellow members or, as I predict, must rely on scent when it comes across those wrinkly blobs less familiar to it.\n\nThe decidedly ant-like nature of the naked mole rat astonishes. The only cold-blooded mammals, they shiver together on cold nights, as honeybees do. Their snuggle sessions may give rise to the colony-wide scent, similar to the function of grooming in ants. Naked mole rats, like Damaraland mole rats, another African species, employ a division of labor with a bulky reproductive queen. In this they resemble termite more than ant: The queen selects two or three males to serve as kings, her exclusive mates.\n\nThese mole rats are termite-like in their subterranean proclivities, too. By necessity they are all about infrastructure, digging tunnels that meander beneath thousands of square meters to gain access to bulbs and tubers, their sole nutrition. Instead of the colony moving about in a troop-like unit, workers come and go from a central, insect-like nest, a collection of chambers packed in an area a half meter to a meter in diameter. The rodents live nomadically, shifting their nest location in the burrow labyrinth every few weeks. Among the workers, the biggest individuals tend to act as soldiers, defending the colony from snakes and any foreign naked mole rats detected by their alien aroma.\n\nI wouldn't be surprised if markers signaling society membership are to be found among other mammals, especially if they involve smells or sounds people can't detect. Spotted hyenas, for instance, rub their rears on clumps of plants, an activity called pasting. Each hyena has a scent of its own, and the exchange of smells at posts is thought to be a way for the animals to be in tune with other individuals of their clan. However, the blending of the pastes of the different clan members generates an odor unique to the group. The hyenas could, in theory, tell clans apart by the smell. Even if they did, though, each clan has only a few dozen individuals, few enough that the animals are already well versed about who belongs and who doesn't. The clan scent would serve at best as a kind of backup, while during the course of a hyena's daily life individual recognition would be key.\n\nMore persuasive evidence of anonymous societies exists in birds. The pinyon jay of the southwestern United States flocks in the hundreds, a normal enough behavior for birds. Yet here's a clue that something fabulous is going on: when one flock meets another, the jays combine into a cloud that later cleanly separates back into the original flocks without the least uncertainty. This reminds me of how savanna elephants peacefully intermix, at certain times in great numbers, but always go back to their original cores. What's remarkable is that while a core consists of just a few elephants, as many as 500 jays make up a flock. So what superficially amounts to just a modest throng by bird standards is on closer inspection actually a populous and well-formed society, each \"flock society\" retaining a membership of multiple extended families throughout the year. A flock concentrates on a tract of land averaging 23 square kilometers, though it isn't territorial: the jays don't defend the area or the food in it, and they often enter the airspace of neighbors. For much of the year the flock flies as a tight mass to hunt seeds and insects. In the breeding season, pairs of birds, devoted to their mates for life (the pinyon jay is more monogamous than humans), spread out over one stretch of the flock's land to rear their young in nests\u2014and yet even then they identify with the same group of birds.\n\nNo one can say for sure how the flock societies retain their clean separation, how the members identify their society-mates both when the birds are in tight flocks and when they disperse among nests. Certain of the jay's vocalizations\u2014including the \"near\" call, a soft nasal sound\u2014are distinct for each bird. Even so, it's hard to credit the jays with employing individual recognition, because that would seem to require them to keep track of hundreds of others while on the wing. Indeed, studies have shown that no jay can recognize every flock member, just its mate and young and a select few others in its personal support network that it can draw on as a respite from hassles with dominant flock members over food and nest material. Hypothetically, knowledge of some, but not all, of the members might seem sufficient to keep the full contingent of 500 threaded together into a flock, perhaps if each bird were comfortable so long as it heard the near call of at least one familiar jay. In practice, however, such alliances within the society wouldn't keep a flock intact. Over time it would fragment or combine permanently with other flocks.\n\nWith so many birds involved, something more than individual recognition must be operating to keep birds of a feather flocking together. Almost certainly the pinyon jay forms anonymous societies, and indeed its vocal repertoire includes a couple of likely markers for flock membership. The jays repeat a _rack_ call in response to predators, a sound that varies enough to identify both the specific caller and the flock to which it belongs. Even more significant is the _kaw_ the jays utter when in flight. This sound, which birders listen for to identify the species, likewise differs between flocks, as we would expect for a marker. I have no doubt the _kaw_ , the _rack_ , or both express the bird's identification to a flock, explaining how the jays keep their memberships straight.\n\nPinyon jays receive definite inducements to live in permanent societies rather than in temporary flocks. First, the safety bonus: Each jay buries a private stash of seeds for hard times (though much like human societies, pinyon jays have their share of thievery). Sentinels stand watch in the trees overhead on the lookout for predators like foxes that can kill the digging birds. The members also coordinate their activities. When the babies are guarded in the nest by mom, all the fathers go off to hunt together, scaring up more bugs as a group than they could as individuals, and returning all at once every hour to bring meals to their respective nests. After the fledglings leave those nests, they socialize in a nursery group under the watchful eye of a couple of adults who pull guard duty for the day while the other parents are off finding meals for their offspring. A year later the youngsters separate from the fledglings, joining what amounts to a boisterous teenage gang, until they are ready to commit to a mate, often after being admitted to a new flock. Pinyon jay lives are organized to a tee.\n\nFinally, evidence is accumulating that yet another group of vertebrates, the cetaceans, create anonymous societies, and that these can sometimes reach stupendous populations. The case for such societies is especially compelling for the sperm whales, the squid-chomping toothed species of _Moby Dick_ fame. Indeed, this behemoth has what I'd grant are societies operating independently at two separate levels.\n\nAt first blush, the sperm whales appear to have rather middling societies for a vertebrate, with units consisting of 6 to 24 adult females plus their offspring, all of which stick together. (Adult male sperm whales, like the male elephants, roam as they wish, mate with whoever they can, and don't participate in the female-only societies.) Unit societies persist for decades; in fact, most females stay with theirs for life, though a few transfer for unknown reasons, so that many units come to contain unrelated individuals.\n\nYoung sperm whales learn short sets of clicks called codas; think of them as a letter or two extracted from a Morse Code message. Certain codas differ slightly from one unit society to the next. The whales emit these when their unit draws near another, apparently permitting the units to recognize each other and coordinate their movements.\n\nRemarkably, the units are part of clans, an arrangement that neatly defines the whales in a dual way. Five clans range across the Pacific, each marked by a specific set of codas. The clans are made up of hundreds of units spread over thousands of square kilometers. While their daily lives are spent in their units, the whales also value clan membership. Only units of the same clan come close to each other, and indeed will hunt together for a time\u2014think of it as a sort of fission-fusion. It's unlikely whales from different clans fight, given how easily such large animals could hurt themselves. Moreover they aren't territorial (albeit in the Atlantic some clans live far apart). They simply avoid each other.\n\nThe units provide sperm whales many of the payoffs of societies in other animals: protection against predators, joint care of the young, and opportunities for sharing their accumulated knowledge; whereas the advantage of belonging to a clan appears to reside in how the whales forage\u2014whether the units of a clan dive or travel together, or stay near islands or in open water. These details matter, as they result in each clan catching different species of squid. One clan does really well during warm, El Ni\u00f1o years, when squid are hard to come by. The hypothesis is that the whales capture prey proficiently only when they buddy up with other units of their own clan, making use of the same hunting technique.\n\nThese foraging differences are not genetically determined. With males open to mating with females in any clan, all the clans have the same genetic makeup. The strategies have to be cultural: the whales learn the clan ways from their elders, much as dolphins learn fishing tactics. The simplicity of the codas must make errors about clan membership exceedingly rare.\n\n# BREEDING AN ANT KINGDOM\n\nThe pinyon jay, the insect-like naked mole rats, and the sperm whale represent evolutionary sidebars among the societies of the vertebrates, most of which operate by individual recognition. Yet the anonymous societies of ants, and most definitely the tiny minority of ant species with vast supercolonies, still stand out for their intricacy, their efficiency, and their size. How does the identity of an ant colony initially fall into place, and how is it that Argentine ants scale those identities up to supercolony status?\n\nAn ant society typically germinates from a preexisting one. The process begins when a mature colony rears winged queens that flutter skyward to mate in midair, often with a few winged males from other colonies, before each queen drops to the earth and digs a little \"starter\" nest by herself to raise the first of her many broods of workers. The workers become the sinew of a society. Genetic and environmental factors cause them to produce a scent and identity independent of all others, including the queen's birth colony. The entourage expands in population through the generations until the colony approaches a mature size, specific to its species, at which point part of its brood production shifts over to queens and males that depart to birth yet another generation of societies, one year after the next. The original mother queen stays with her colony, which survives as long as she does. That can be long indeed: a quarter century in leafcutters. With her demise the workers go into a funk and expire soon after. Give a colony all the food and space in the world and it won't last longer than its queen.\n\nArgentine ant supercolonies owe their ever-burgeoning populations to a twist in the story: a big supercolony is home to not one queen, but millions, because the queens never fly away. Moving on foot between nest chambers spread throughout the territory, they stay in their native society to lay more eggs, which hatch into ants that stay as well. Year after year the supercolony expands its range to occupy any suitable nook or cranny.\n\nSo long as the same scent is produced throughout the supercolony's often vast domain, the society remains intact. This consistency may seem impossible to achieve, yet we can imagine a method for self-correction built into the system. Suppose a gene affecting the colony marker mutated in one queen out of the many. Any other genetic change in her behavior or morphology wouldn't matter to her acceptance. However, if her smell no longer matched the identity of the ants around her, the workers would kill her before she laid eggs. The mutation would vanish with no trace. The outcome of this incessant purging is the ants' adherence to a common identity not just in one nest, as in most ant species, but over distances of hundreds of kilometers. With a uniform identity from one end of their range to the other, supercolonies achieve a kind of immortality. Think of California's four colonies as the very same societies that invaded the state a century ago. Occasional rumors to the contrary, they show no sign of slowing down.\n\nThis reality is hard to grasp when you stand in California with the ants of Large Colony swarming to every horizon. Some biologists question whether a supercolony can really be one society. A few go through contortions to propose that, because a supercolony's population is never a continuous throng of ants across the ground, it isn't really a society at all, but rather a constellation of many societies. This patchy distribution of ants, however, has more to do with the suitability of the habitat than with the inhabitants' social behavior or identity. The ants avoid overly dry areas, for example. Yet turn on lawn sprinklers on a parched day and what had been two patches of ants expand and combine, trouble free, into one.\n\nWhen pressed on the question of whether these ants in their billions can represent a society, the same experts are likely to respond cautiously that they _act_ as if they are one society, even with their patchiness and the existence of genetic variation from place to place within a supercolony. My response is to say, but of course! What criterion makes sense in determining what qualifies as a society other than the choices of its own members as to who should be there and who shouldn't? As long as the ants accept one another and reject outsiders, the span of earth they occupy, and how varied the members are, doesn't matter any more than it matters in a nation such as the United States, with all its ethnicities and political squabbling.\n\nFor the ants, simple markers are up to the task. When entomologist Jerome Howard took a leafcutter worker from one side of her colony to the other\u2014a distance of several meters\u2014the ants at the new spot sometimes paused to inspect the newcomer. Maybe the population of such a metropolis, with its many highways and byways, doesn't completely intermingle, so that subtle distinctions in scent accumulate from place to place\u2014a trifling difference in the national flag. Nevertheless, after a second of mild interest the new arrival was permitted to continue about her business without a hitch\u2014she is still treated as a member of the colony.\n\nGiven the scale of their colonies, Argentine ants seem staggeringly united. The species highlights how individuals can remain members of a society regardless of how little they cooperate or interact. The species does well on ocean voyages; that's how the four supercolonies arrived from Argentina to the United States in the first place. Leapfrogging about as well on our planes, trains, and automobiles, the supercolonies have made their way across the globe while still holding on to their identity, much the way Hawaiians retain a common nationality with denizens of the US mainland. Large Colony has traveled to, and seized control of, 3,000 kilometers of European coastline and other faraway parts of the planet, Hawaii included. Meanwhile, other supercolonies have toeholds in places like South Africa, Japan, and New Zealand.\n\nThe combined weight of the ants in invasive supercolonies can reach and exceed, literally, the mass of a sperm whale, which raises the question of how they got that way. Perhaps the most remarkable thing about the transcontinental societies is how much larger they are than those of the Argentine ants in their original homeland. The societies of this species in Argentina are positively petite: at most a kilometer wide, certainly jaw-dropping for an ant but not much to look at by California standards. This difference appears so radical that one might conjecture that a major evolutionary change must underlie it. I imagine that aliens might have assumed much the same for humans if they had landed on Earth 20,000 years ago to find societies of a few hunter-gatherers, then returned centuries later to discover China with its billion souls. A far simpler explanation for the superlarge societies of modern humans and Argentine ants alike is that no dramatic transformation was required: in both species the expansion of societies became certain when the conditions were right. This extravagant capacity for indefinite growth, not size as such, is what sets supercolonies apart from the societies of other species. Even several dozen Argentine ants in a shipment of plants are, then, a supercolony (or at least a fragment of one). The capacity for societies to grow indefinitely is a true rarity, characteristic of just a few ant species, possibly the clans of sperm whales, and human beings.\n\nAside from their rapacity for expansion, Argentine ant societies are not much different from those of other ant species. They direct their aggression at outsiders and show little animosity toward colony-mates, like all ants do. Furthermore, the societies in Argentina operate no differently from the immense colonies overseas, except that their growth is choked by an overabundance of nasty neighbor ant colonies. The condition that triggers the explosive expansion of a supercolony abroad is the lack of competition. The supercolonies that got to California had nothing to stop them from conquering the state\u2014until their growth came to a halt when they met and started fighting each other.\n\nIn the upcoming chapters I will argue that humans likewise did not need to change in any essential way for the small unions of prehistory to grow, given the opportunity. All the elements needed for the success of empires were already built into the Paleolithic mind, right down to a very human fixation with markers of identity.\n\n# CHAPTER 7\n\n# Anonymous Humans\n\nThere is little in the history of life more remarkable than a human strolling through a coffee shop. The patrons can be utterly unknown to us, and\u2014nothing happens. We do fine, stay calm encountering those we have never met. This tells us something unique about our species\u2014opposable thumbs, upright stance, and smarts aside\u2014because most other society-dwelling vertebrates cannot do this. A chimp bumping into an unknown individual, never mind a caf\u00e9 full of them, would fight or flee like the damned. Only a young female stands a chance of surviving the confrontation without risking combat\u2014but she had darn well better be receptive to sex. Not even the bonobo would walk by an individual it doesn't know with indifference. Yet humans have a talent for dealing with strangers and moving seamlessly in their company. We enjoy being surrounded by a sea of others at a concert, theater, public park, or fair. We grow accustomed to each other's presence and befriend the few we want to, in a kindergarten, summer camp, or job.\n\nWe allow such anonymity by recognizing certain signs in others that fit our expectations\u2014characteristics that act as markers of identity. Among our markers are those indicating every imaginable aspect of identity. A six-carat diamond ring signals wealth and status, and some markers, like a quirky way of making an arrowhead, can be specific to a person. However, in this book the word \"marker\" and its synonyms will generally specify the attributes people associate with their society. The recognition of markers is a human aptitude that most animals lack, with the exception of a few vertebrates such as the naked mole rats and sperm whales and most social insects. Few social scientists will have anything to do with such comparisons, however; indeed, they may be puzzled by (even unreceptive to) the idea that ants, presumably lacking conscious behavior, have an identity at all. Yet though they lack the capacity for self-reflection, ants and other social insects resemble us in an elementary but striking manner by having anonymous societies.\n\nMost mammals, indeed most vertebrates, lack anything they could reliably use to mark their society. The horses in a band never have the same gait or style of whinnying, for example. In most situations the lack of markers keeps vertebrates focused on managing individual relationships, in contrast to ants, for which such familiarity is nonexistent. People fall in the middle, with a selective focus on cultivating key social connections without taking on the obligation of keeping track of everyone in the society. We form variable relationships based on our social histories with others, treating only some people as individuals. Distilling the differences to fewer words, we return, at last, to the formula given at the end of chapter four: to function as a society, chimpanzees need to know everybody; ants need to know nobody; humans only need to know somebody.\n\nThe \"somebodies\" to which each of us are connected fall into expanding circles of social relationship, from most intimate to most abstract: our spouse, our nuclear family, our extended family, our 150 or so friends, and the many hundreds we know less closely. Beyond that is everyone who identifies with our society as a whole, whether it's a tribe or nation, which in all but the most diminished societies like Kenya's El Molo encompasses many who are strangers to us. Most of these connections, aside from our overarching allegiance to our society, are social networks that differ from person to person. Overlying this list are the people with whom we share special affiliations; some proudly display markers of their own, like a Chicago Bears fan wearing a team cap. And of course our knowledge of others isn't restricted to our society. We resemble bonobos and savanna elephants in not just knowing members of other societies, but even befriending them.\n\n# MARKING OUR SOCIETIES\n\nFlags, patriotic songs, and similar in-your-face signs of nationality are only the most obvious of the varied ways people indicate and perceive their connections to a society. Any characteristic of a society can serve as a marker, consciously created or not, so long as when aberrations in it become great enough, the members detect that something is amiss. Whereas ants and naked mole rats rely on an odor to identify society-mates, and sperm whales exclusively on vocalizations, for _Homo sapiens_ just about anything will do.\n\nCertain markers are displayed continuously, and often conspicuously, such as socially prescribed forms of dress. Others come into play on occasion and have to do with values, customs, and ideas. Some markers require conscious intent\u2014carrying a passport, for instance\u2014while others are cues over which the members have no control, such as, in certain societies, skin color. A marker need not be connected with a person at all; the identity of a group does not just permeate the people but may extend to anything the society claims, perhaps a location such as Bunker Hill or an object like the Liberty Bell. Historical events enter the national consciousness, too. A few may be as old as the societies themselves, yet novel ingredients come into being all the time. In America, the September 11 attacks of 2001 were a defining moment. The site of the Twin Towers in New York, the numbers 9\/11, and the date they signify all became part of how Americans view themselves and how others perceive them, which demonstrates how quickly new aspects of our identities can gain prominence.\n\nThe markers of a society can appear trivial or just plain weird to outsiders. For example, the custom of eating with the hands in India; or, in Thailand, primarily with a spoon, never chopsticks. There are capricious matters of taste, though an open-minded outsider can still appreciate their beauty: think of the tonal system of Indian music or the black-on-black pottery designs of Pueblo American Indians. Silly and arbitrary or not, some of the differences can have life-and-death consequences, such as the practice of driving on either the left or right side of the road. Still other markers are icons that bear a relationship with something out there in the world, such as many Egyptian hieroglyphs that are instantly recognizable for what they stand for. Even the oddest markers can seem logical to those who use them. The bald eagle and the bear, mighty predators, respectively represent the strength of the United States and Russia, and ties to such markers can be a powerful social glue.\n\nMarkers can be reinforced, or outright determined, by biology. Certain mutations spread among people with the advent of animal domestication several thousand years ago that allowed adults to digest the lactose found in milk. In Tanzania, the cattle-herding Barabaig, who find milk delicious, live near Hadza hunter-gatherers, who are nauseated by dairy products. Doubtless this gulf in diet further amplifies the physical and cultural separation of their peoples.\n\nWhile sight and hearing are the primary senses involved in humans perceiving markers, the importance of taste is evident. \"What is patriotism but the love of the food one ate as a child?\" goes a Chinese proverb, and there's wide latitude in what people salivate over. I've eaten fried centipedes in China, pickled wasps in Japan, pig embryos in Thailand, toasted ants in Columbia, dried mopane worms in South Africa, raw termites in Namibia, beetle grubs in New Guinea, and diced rat in Gabon. To those who crave such delicacies, the disgust of outsiders is a shock. And while we never accord scent the primacy ants do, people sometimes comment, unfavorably, on the breath or body odor of ethnic groups.\n\nMany of the attributes I call markers fall under the rubric of culture. Though this word is often associated with the pantheon of a society's intellectual and artistic achievements, it refers broadly to the full complement of characteristics passed between generations, mainly by active teaching. Among the most studied of these expectations are norms. Those are the understandings citizens share regarding their values and moral codes, including predilections to be generous or helpful and beliefs about what's fair and proper.\n\nStriking cultural norms and attributes like food taboos or national banners get the most attention, but a miscellanea of more subtle markers are both profoundly important and easy to overlook. A scene from the 2009 Quentin Tarantino film _Inglourious Basterds_ comes to mind. A British spy masquerading as a Nazi orders three beers at a German bar by raising the three middle fingers of one hand rather than extending his index finger, middle finger and thumb as Germans do\u2014the ensuing firefight is classic, heart-stopping Tarantino.\n\nA study of culturally specific gestures alone could fill volumes. In Italy everyone's hands seem to be constantly in motion. Waving one hand horizontally in front of the body with the first finger and thumb touching to produce an O means perfect; chopping with one hand is a warning to be careful; waving both hands up toward the face says this is boring. Isabella Poggi, a psychologist, has classified more than 250 such gestures, many unique to Italy, centuries old, and often more reliable in their meanings than spoken vocabulary. Gestures are plausibly more primeval than words, coming so automatically that blind people wave their hands when addressing a blind audience.\n\nEasier still to miss than gestures are nonverbal accents that are absorbed subliminally rather than taught. These often fall below our radar even though they vary from one society to the next. In the final decade of his life, Charles Darwin published _The Expression of the Emotions in Man and Animals_ , in which he found that the most basic human emotions are universal across our species. Nevertheless, the facial muscles involved in conveying those emotions are brought into play ever so slightly differently across societies. It has been shown, for example, that Americans who fail to distinguish Japanese nationals from Japanese Americans in photographs when both are expressionless can pick out Japanese Americans, their fellow citizens, when the faces in the photographs display anger, disgust, sadness, fear, or surprise. In another experiment, without being able to explain how, Americans often guessed whether a person was American or Australian by watching how he or she waved a hand or walked, which suggests that the entire body can be turned into a nonverbal accent. Such differences, unlike gestures but like many smells, are impossible to express in words.\n\nThe ability to register such details comes from repeated exposure. Their comprehension is akin to that of a few individuals who could distinguish the incoming Nazi fighters and Allied airplanes during World War II. The British were desperate for more plane spotters. The trouble was that no one with the skill could explain how they did it. The only successful training method was for students, not having any idea what to look for, to guess. The spotter corrected them until they got it right. Societal markers may often be acquired and recognized this way, without comprehension or calculated thought.\n\n# RECOGNIZING MARKERS\n\nOur societal markers are so clear that some can even be noticed by animals that don't use markers for their own species. Elephants detect different human tribes and have expectations about their behavior. In Kenya they fear the Maasai, who spear pachyderms as a rite of passage, but are indifferent to the Kamba, who don't hunt elephants. Elephants hide in high grass when Maasai approach, possibly because they can distinguish the body odor of those tribal members, whose diet is based heavily on cattle, from that of the vegetable-loving Kamba. The elephants seem to be able to distinguish the tribes by attire, as well, and will attack cloth of the red color favored by Maasai.\n\nHumans evolved markers of identity for the same reason: our safety and well-being can depend on them. The psychologist Gordon Allport, who founded the study of personality, has explained that \"the human mind must think with the aid of categories... orderly living depends on it.\" The ability is clearly rooted in our animal legacy: pigeons, for example, not only put things in categories like bird and tree; they can be trained to distinguish Picasso's paintings from Monet's. All animals can categorize. No elephant or human, baboon or hyena is forced to recognize each and every elephant that passes by to know one when it sees one, picking out traits like the trunk and its trumpeting call. Still, an elephant _society_ , having no physical or behavioral traits specific to its group, isn't as obvious to the eye as elephants themselves are as a species.\n\nWhile most vertebrates don't wear their group identity on their sleeve (or in their voice or odor), their knowledge of who belongs yields societies no less valid and concrete than those based on markers. In fact, humans also have groups that are defined purely by individual recognition. Take the sports teams in my neighborhood as a kid. We couldn't afford team jerseys, so no outsider could have picked out who was on which team if we were all mixed together in a lineup. Yet we were never confused. What set our team apart was in our heads. We knew each other as individuals, after all. Much as observers can discern a wolf pack or lion pride by watching the animals interact, onlookers could have figured out our teams by watching us play.\n\nEven when humans do know each other, markers are advantageous. My childhood team would have been jazzed to wear team jerseys if we'd had them. Not only might they have bolstered team pride, but they would also have deemphasized our differences and perhaps have given us a more unified and intimidating appearance (all subjects for later in this book). Jerseys would have helped us play, too: wearing them, we could have identified our team members by color even out of the corners of our eyes when the action unfolded fast. Speedy and accurate recognition is a real plus when mistakes are costly; without it, you can lose the ball to the other team or, in the case of societies, your resources or life to a hostile foreigner. Jerseys might additionally have sped the process of bringing on board another member. Sure, we might not have completely trusted any new kids at first, but given satisfactory appearance (the jersey), and behavior (including how they got that shirt), they would likely have been quickly identified as part of the group even by the players who didn't know them.\n\nAs for societies, markers add up to an indelible awareness of who _we_ are, yoking people who haven't met to a shared worldview even when those traits do not require our attention. During the average moment, our markers are so familiar and expected that we notice them no more than we see the sky's precise shade of blue. Yet we hunger for them when they are absent. This is why, when starved for others \"like us\" on a trip abroad, we seek out a bar, restaurant, or hangout of people from our nation, and greet the expatriates there, somehow familiar though unknown to us, as old friends.\n\nDespite their usefulness, adding markers to distinguish our society from outsiders brings no fundamental change: we can be a society with them; we can continue to be a society if we lack them\u2014up to a point. Having markers becomes increasingly valuable, and eventually essential, for keeping societies straight as their populations grow\u2014and, in humans, for distinguishing all sorts of groups within each society, as well. The fez of a Shriner or a Chicago Bears cap are not like anything in nature: the same fellow can identify himself as a Bostonian, a fire fighter, a conservative, with none of the categories invalidating any other\u2014or his identity as an American. Many aspects of human identity come with markers that we may display proudly, but most can be taken on and off as needed. Beyond discriminating worker from queen (and in some species, from soldier), no ant breaks down her identity into such groups. Nor, as it turns out, do vertebrates, which never separate cleanly into lasting cliques within societies, and certainly not aided by markers: even though the wolves in a pack can vary in coat color, the blacks never set themselves apart from the grays. Hence the billions in an Argentine ant supercolony appear to be a homogeneous continuum, while you and I differ in a hundred ways in how we fit into our society (for instance, I suspect you don't study ants).\n\n# THE ROLE OF LANGUAGE, AND WHAT REALLY MATTERS\n\nLanguages, dialects, and accents are the most studied, and perhaps the most potent, human markers. Most societies, and many ethnicities such as Jews or the Basques, have their own language, or version of a language. The evolutionary linguist Mark Pagel writes of the biblical story of the Tower of Babel, in which God gives people different languages to prevent them from uniting to build a tower tall enough to reach heaven: the irony of this story, Pagel points out, is that \"language exists to stop us from communicating.\" Above and beyond the vast differences between languages, every language offers words that describe its own speakers and how they see themselves\u2014and words that portray those who belong to different societies. Labels matter, including the names of societies themselves, which possess \"phenomenal power,\" as one African scholar argues; even six-year-olds prefer kids whom they are told are from their country. One of the greatest threats a human society can face is a competing claim from another society for the same name, illustrated today by the impassioned dispute over the word \"Macedonia,\" which the country of Macedonia has taken for itself but which millions of Greeks apply to their ethnic group.\n\nNevertheless, language differences are neither necessary nor sufficient to keep societies distinct. \"Once a group has lost its language, it will generally lose its separate identity,\" one linguist claims. Certainly this can be true when one society overpowers another and turns its people to serfs or slaves, circumstances under which many markers, language included, are lost or recast. Yet groups can lose their native tongue and keep their identity and independence, disproving the absolute sway of language over other aspects of identity. Africa's Pygmies, part-time hunter-gatherers, abandoned their own languages over the past 3,000 years in favor of the speech of their agricultural neighbors, whom they live with part of the year. Pygmy societies remain culturally intact even though mere traces of their original tongues exist\u2014a few words, mostly describing forest animals and plants.\n\nThe crucial point is that the rich combination of markers enables people to unmask those who don't belong without hearing them speak a word. That said, languages stand out in their importance for human identity. A language or dialect is almost impossible to replicate precisely unless you learned it growing up, giving languages a priority in unmasking the foreigner among us. The book of Judges (12:6) in the Bible recounts how the Gilead soldiers rooted out the people of an Israeli tribe that spoke with a faint accent:\n\n[A Gileadite soldier] would tell him, \"Say the word shibboleth.\" If the fugitive would say sibboleth, because he could not pronounce the word correctly, they would grab him and kill him at the shallow crossings of the Jordan River. At that time, 42,000 men from Ephraim died.\n\nThe value of language as a \"spot check\" of identity is heightened by the level of nuance it conveys: people glean far more from it than is incumbent on them to understand what's being said. Even a child registers a speaker's native tongue before she or he finishes uttering one word. The same is true of other markers, of course: we are more aware of how our comrades walk or smile than we need to be to walk or smile ourselves. But what makes language especially worthy of attention is that speaking the local tongue is a matter of pride for humans in a way that carrying oneself like a local is not. Perhaps only traditions like fire walking, practiced by some Polynesians, among others, signal a commitment to a group no language can convey. However, while people rarely perform such extreme rites, they talk all the time in voices that carry around corners and through crowds, their accents immediately signaling their native or foreign status. Few other aspects of identity elicit such an instantaneous gut response.\n\n# PERMISSIBLE VARIATION, ECCENTRICS, AND OUTLIERS\n\nPeople treat their differences in markedly varied ways. For example, it's not always the case that \"a language is a dialect that has an army and navy,\" as the Yiddish linguist Max Weinreich once quipped. Societies don't need to be marked off by their way of speaking, as the Pygmy example shows. The ways societies use languages reveal that a facility with foreign practices can be not only tolerated but encouraged. Where countries pack into a small space, such as Europe, people commonly speak languages in addition to their native tongue, oftentimes those of neighbors, trading partners, or erstwhile colonial powers. Multilingualism isn't a recent novelty. In pre-contact Australia, many people spoke multiple tongues, often because they, or a parent, had married into the society. Some nations, such as Switzerland, may have several native languages. Others may share a single language with other societies, as do the United Kingdom, Australia, and the United States, with each having multiple dialects of its own.\n\nThe key of ethnography, the science of documenting cultures, isn't so much to figure out the similarities and differences among societies, but to find out what their people consider important, in language or in anything else. What a linguist might determine to be two languages could be interpreted by the speakers as one, or vice versa. What I pointed out for Argentine supercolonies is true as well of the societies of all species: it's the choices the individuals in a society make concerning what signals identity that determines who is a member, and not what you, as an outsider, might assume is important. Tweak the scent of an ant colony, and you will find that some variations make a difference and some don't. As for people, variations, for example in language, similarly don't necessarily jeopardize our identities\u2014it depends what the variations are. I once met a Peruvian who is often mistaken for a foreigner in his homeland because of his failure to produce the rolling Spanish _r_. Whether he's less a Peruvian for this awkwardness is up to other Peruvians. Societies make room for differences of equal significance to speech. Modern-day religions, to give one example, extend across, and coexist within, territorial borders, yet they generally (but admittedly not always) do so without confusing people about who belongs where.\n\nSociety members aren't stamped from a cookie cutter\u2014societies allow for similarity _and_ diversity. For a person to be welcomed, his or her behavior has to lie within permissible social boundaries, as was true even for hunter-gatherers: \"What he does, and how he does this is his, not his neighbor's affair, as long as he remains within the realms of accepted norms of behavior,\" wrote Bushman expert Hans-Joachim Heinz. Social guidelines, often unspoken, inform these evaluations and constrain our choices; as the Japanese saying goes, the nail that sticks up gets hammered down. Naturally there can be issues of interpretation and taste. Ultraconservatives, for example, have little tolerance for the archliberal and vice versa, even though each grudgingly recognizes that the other belongs to their society. By expecting an appropriate level of conformity\u2014such as the tacit agreement and childhood training not to readily show anger in some Asian cultures\u2014the members of a society detect renegades and aliens.\n\nSocieties differ in the degree to which they value conformity. Some promote eccentricities as a sign of individuality and enterprise, and proclaim as a core value the right to be different. Still, all societies, including those taking pride in their commitment to personal freedom, exist by exacting a loss of choice on members for the sake of safety and predictability. The precision with which we must learn from and imitate each other\u2014the narrowness of the boundaries\u2014is more stringent for some behaviors and in some situations than others. The fact is the repulsion felt toward a person who ignores important norms of behavior can be so profound that the social deviant can be treated more harshly for the same offense than a foreigner\u2014an overreaction known to psychologists as the black sheep effect. Outliers poorly matching expectations are ostracized, stigmatized, pressured to change, or treated as foreign, depending on the kind and extent of their aberrance. Such censure reins in what goes on in a society.\n\nAnimals also have limited tolerance for odd behavior. Even species with individual recognition societies endure minimal deviation. As a rule, elephants support the sick, but they, and chimpanzees, are also known to ill-treat gimpy or unhealthy individuals.\n\nSocial insects are the supreme conformists. Ants allow for very little individuality when it comes to identifying with their colonies. This is reflected in the ants' absolute commitment to their anonymous societies based on a scent alone. Their rigid identities give us good reason to call an ant colony a superorganism. Individual ants are bound to a colony as cells are to an organism in the following manner: the ants identify each other by detecting the hydrocarbon markers on their body surfaces, and in a healthy society invariably evade or destroy foreign ants with different markers. Likewise, the cells in a body identify each other by detecting chemicals on their surfaces, with the immune system killing foreign cells bearing the wrong signals. On this basis your body, with its trillions of member cells, represents a society of a microbial sort. While disgruntled people can leave their society, even defect to another, an ant worker, linked for all time to nest-mates sharing her scent, can be stressed to death and still not abandon her nestmates. Admire the devotion of the cell to its body or the social insect to its colony, but from the human point of view their total immersion into a greater whole feels dystopian. Human societies aren't superorganisms, nor would we want them to be\u2014we cherish our choices as individuals too much.\n\nAs in other animal societies, the young in human societies are outliers as well. Not unlike ant larvae, which don't yet have a colony scent, children have at best a nascent identity. No child earns a respected place in a society merely by being alive, however. In contrast to the young in individual recognition species, which simply must come to know and be known by everyone, human youngsters have to detect and learn the markers that permit them to fit in. Until our progeny master that task, they are weak and safely ignored. At that young age, adults recognize that they belong from their proximity to parents or friends.\n\n# STRAIN ON THE BRAIN\n\nAs much as humans can still rely on individual recognition for memberships in some groups, might species with societies that depend on individual recognition have at least the potential to use markers? While we know that primates can learn to associate a plastic token with a nonsocial object (such as food), we understand far less about their ability to represent other social agents with markers. So while monkeys don't ordinarily use signs to differentiate themselves from foreigners, I'd love to introduce a marker into a troop to see if it catches on. Any able-bodied primate will tear off a red jersey, but what about paint: can it learn to expect troop-mates to have a dab of red on their foreheads? If so, would it welcome a foreigner wearing the mark of its \"team\"? Could a supersized troop stay together if its young acquired identical red splotches, enabling them to keep track of each other without knowing one monkey from the next? And would tensions mount if half the troop were dabbed with blue? I anticipate the answer to these questions is no. Monkey minds aren't wired for markers, and would fail to recognize them even should such differences exist.\n\nNow, before we get all superior about humankind's use of markers to recognize and respond to one another, let us consider the cognition required for marker recognition. To suggest that monkeys aren't wired to readily grasp markers of identity isn't to imply that the ability requires a great intellect. The social brain hypothesis posits that handling more social relationships requires a large brain. But for those who point to this theory to predict that populous societies require formidable mental power, the lowly ant presents a serious challenge. Given their social complexity and flexibility, ants do a lot with a mere 250,000 or so neurons. As Charles Darwin proclaimed, \"The brain of an ant is one of the most marvellous atoms of matter in the world, perhaps more marvellous than the brain of man.\" Of course, human markers are vastly more numerous and diverse than those of ants, which don't wear ethnic hats or speak with an accent. Yet the aroma that serves as the ant badge of identity could be more elaborate than one might suppose. Each scent is composed of a cocktail of hydrocarbon molecules that differ in both kind and concentration from one colony to the next. In practice, then, the scent represents a whole collection of markers, not just one, with some molecules probably influencing the ants' interpretation of the scent more than others. Ants are also nimble in responding to foreign colony odors, for example distinguishing members of neighbor societies they know well from members of a colony they have never encountered. The latter, being an unknown threat, are likely to be attacked much more violently.\n\nMany academics may protest that human markers can be supremely complicated, and indeed some are, such as the memorization of an arcane religious text. Humans turn labeling into an art by imbuing many of our markers with significance, often at multiple levels, to create symbols\u2014an impulse that arguably distinguishes us from other animals. For the Irish the shamrock is at once a plant claimed to predict the weather; a Celtic emblem of good luck; and a tool Saint Patrick used to teach the Druids about the holy trinity.\n\nEach person develops beliefs regarding \"his personal affiliation with certain symbols, or, more accurately, with what certain symbols stand for,\" Edward Spicer, an anthropologist who studied American Indian tribes, pointed out. Sociologists balk at the notion that any animal, lacking symbolic capabilities, could have anything resembling a nationality, while anthropologists consider the employment of symbols as critical to the emergence of humanity. Tellingly, however, most people can't tell you much about the meanings of the symbols they prize. Americans belt out _The Star-Spangled Banner_ without knowing what it means to spangle\u2014or remembering the words. \"It is likely that even people who are expert in the use of symbols\u2014shamans, priests, or sorcerers\u2014cannot state precisely what a particular symbol is all about,\" social anthropologist Mari Womack reminds us.\n\nIn truth, it isn't necessary to understand how or why something is profound, or for that matter to possess a deep meaning at all, for us to be sensitive to it\u2014or to its absence, or appropriateness. People don't have to burden their overloaded cranial wetware with what a symbol signifies. What meaning it has, if any, may reside in the eye of the beholder.\n\nProducing and perceiving markers therefore need not take much effort, and, once learned, the markers can be applied to indefinite numbers of individuals with no additional mental demands and no obligation to maintain a relationship. Among ants, brain size actually _declines_ in species that have large societies. Workers in a small colony are mentally challenged, as much as ants ever are, because they are jacks-of-all-trades. In a big colony, a soldier ant may attack enemies but seldom, if ever, nurses the brood, a job for smaller workers. Not only do ants save mental exertion by being ignorant of each other as individuals, but in large colonies this reduction of skills further diminishes the demands on their brainpower.\n\nFor people, too, as long as the strangers around us look and act reasonably, ignorance can be bliss. Imagine if you felt obliged to introduce yourself to, and get to know, everyone you came across. The mental demand would be overwhelming. By comparison, markers are simplicity itself. While managing the markers of our society could have contributed to the enlargement of the forebrain of humans compared to other animals, it isn't likely to be the primary cause. Sitting in a public place, you register the physical, cultural, and other traits of those around you without much thought, a minimal investment of effort. A psychologist would say that markers reduce the cognitive load of social surveillance, freeing you up to read this book or converse with friends at the caf\u00e9. Even small tribes like Kenya's El Molo, with its handful of people, take advantage of the low cost of social surveillance made possible by their shared dress, language, and so on\u2014they may know every other El Molo intimately, but the El Molo are still an anonymous society. In fact, since the advent of farming, humans have undergone a reduction in overall brain volume the size of a child's fist, attributable, perhaps, to the extent to which we have grown more dependent on others for tasks from cooking to construction.\n\nOf course, humans adapt their behavior to circumstances, and that can include adjusting to foreign cultures. After months in rural India I subconsciously took on the accent and side-to-side head wobble while speaking. Subsequently in Singapore I shifted to their accent, adding the word \" _lah_ \" to the end of sentences for emphasis. The tweaks in my speech patterns seemed to help the locals understand me, and I once proudly helped an Indian tourist communicate with a Singapore shopkeeper by \"translating\" between the two English dialects. Even so, I'm sure my foreign manner of speaking was transparent\u2014if my ineptitude at putting on the Indian men's sarong, called a lungi, or a hundred other things didn't give me away.\n\nAs for your sizable brain, that has less to do with the population of your society than with your ability to immerse yourself in the lives of those who matter to _you._ Markers not only take the cap off the size of societies; they make social life less complex. Why is it then that so few other vertebrates employ them? Possibly their societies function best with populations so small that individual recognition suffices. Yet even in a small society keeping track of everyone individually requires more mental bandwidth than the simplicity of noticing a reliable marker. Think of the difference between anonymous and individual recognition societies as another take on the apples and oranges idiom: animals know who fits in which category by registering either their similarities (picking out a marker they share, much as one picks out oranges by their orange color) or their dissimilarities (picking out each member from its personal features). The former demands less concentration. Animals reliant on individual recognition do what they can to circumvent this cognitive burden. Primatologist Laurie Santos tells me that when macaque troops face off, both sides bunch up. Clumping serves as a defense shield, but also minimizes confusion. So long as each monkey sees at least one foreign individual on the other side, it can take for granted the whole lot of them is foreign. In the same way, it can assume a monkey grooming its friend is a member even when the groomer's back is turned.\n\nComplicated personal interactions, mostly with the few dozen souls in our intimate social networks, are at once a hallmark of humanity and a holdover from individual recognition societies in other primates. At the same time, humans know not just friends and relations, but many others to varying degrees of familiarity. So while most mammals treat each society member as an individual and from that personal knowledge build a collective identity, people enjoy the ant-like option of ignoring and even being ignorant of each other. Like the ants, we relate to strangers based on whether they share our identity.\n\nThis section laid out how the societies of ants show the same trend of increased complexity with population size that human societies do. Yet adding citizens to a nation, like adding ants to a nest, doesn't need to entail any additional strain on the brain. By employing markers of identity, we, as members of anonymous societies, are gifted with the ability to think of a stranger as one of us. Modern human societies, in all their sometimes continent-swallowing grandeur, are underlain by this feat of the imagination, as was true, as well, for the small societies of our ancestors. That includes the societies of the agriculture-lacking humans of the past millennia, who were really no different from you and me. To understand people now, we must understand people then.\n\n# SECTION III\n\n# Hunter-Gatherers Until Recent Times\n\n# CHAPTER 8\n\n# Band Societies\n\nThe sun had turned a faint red by the time South African anthropologist Louis Liebenberg brought our jeep to a halt near the !Kung Bushmen camp at Gautcha Pan in the Kalahari of Namibia (the ! represents a click sound). Louis spoke to a young man named !Nani about a noxious beetle grub his people use for poison arrows. Yes, !Nani said, he knew of some in walking distance; he would take us.\n\nThe next morning we took !Nani in our car over a flat expanse broken by scrubby thorn trees and the occasional bloated baobab tree. Clearly a Bushman's idea of walking distance differed from ours. After guiding us by what seemed like quite a few kilometers, !Nani stopped us at a patch of glossy-leafed shrubs. There he used a traditional Bushman digging stick to excavate the deadly pale larvae and showed us how to squeeze their toxic juices onto an arrowhead.\n\n!Nani and his fellow !Kung, along with other Bushmen and hunter-gatherers elsewhere, don't practice agriculture or raise livestock. They depend entirely on the food available from nature, hunting game with poison arrows and other simple implements and gathering vegetable foods across a vast \"walking distance.\"\n\nFor more than a hundred years, anyone seeking to understand early humans has turned to the recorded evidence of how hunter-gatherers have lived in recent centuries. Of special interest has been a pattern wherein hunter-gatherers moved in little groups, known as bands. Each band traveled around to put up camps from which they sought food and water. I call such nomadic hunter-gatherers band societies, which, as this and the next chapter will address, typically consisted of a few bands. I distinguish band societies from tribes, a word typically used, as I do in this book, to describe simple settled societies, most of which were dependent on horticulture, where plants are cultivated in gardens rather than plowed fields; as well as more mobile pastoralists who tend domesticated herds. (To confuse matters, \"tribe\" remains the word of choice for North American Indians, many of whom lived in band societies.) From the perspective of the study of human origins, horticulturalists and pastoralists are latecomers, less pivotal when it comes to grappling with the root characteristics of humanity. Agriculture is such a recent innovation that even the nations of the modern world need to be interpreted in light of how hunter-gatherer societies functioned.\n\nArchaeologist Lewis Binford asked a native Alaskan man to encapsulate his itinerant existence in a band society. \"He thought for a moment and said, 'Willow smoke and dogs' tails: when we camp it's all willow smoke, and when we move all you see is dogs' tails wagging in front of you. Eskimo life is half of each.'\"\n\nAs poetic as this old man's words were, and as important as hunter-gatherers have been to anthropologists, there hangs over the enterprise the question of whether peoples who have recently lived by hunting and gathering are accurate mirrors to our past. For centuries hunter-gatherers have had to either adapt to the presence of farmers and herders or be pushed by them onto harsh, unproductive land. We know they could have undergone profound changes before the first explorers documented their lifestyles. Even shortly before: when the pilgrims arrived in America to found Plymouth Colony, the first Indian to greet them already spoke English (taught, apparently, by British fishermen). Nearly two centuries later, on their cross-country expedition, Lewis and Clark came upon tribes that already rode horses, animals that had been extinct in North America for thousands of years but were reintroduced by Europeans. Clearly the Indians of early photographs had already reinvented themselves many times over.\n\nSimilar stories can be told about any hunter-gatherers. The Bushmen, a race of short, slim, reddish-tan, baby-faced people distinct from other Africans, are especially valued in studies of human evolution. They lived across a swath of southern Africa\u2014in the same general area and a similar sort of desert and savanna habitat where humans evolved\u2014with genetic evidence suggesting they split from other human populations in our deep past. Nevertheless they were influenced by centuries of interactions with Bantu herdsmen who came from the north long before Europeans arrived.\n\nBefore Europeans made contact, Australia was one of the few major locations on Earth where local hunter-gatherers had few encounters with agriculturalists. Aborigines occupied the continent for 50,000 years following an early diaspora out of Africa. Arguably that makes Aborigines a reliable source of insight into the past, yet the northernmost Aborigines traded and intermarried with tribes from the Torres Strait Islands, who raised crops of taro and banana. Additionally, for a period starting in 1720, a fleet of Indonesian fishermen came to northern Australia to hunt sea cucumbers. The Indonesians took some Aborigines to visit their home city, Makassar. From them the Aborigines learned to make dugout canoes and shell fishhooks. They also garnered a taste for new songs and ceremonies, goatees, smoking pipes, sculpting wood, and painting skulls. And of course like all peoples the Aborigines were not culturally stagnant; they invented the boomerang and conceived of spiritual cycles called the dreamtime, items and ideas found nowhere else yet shared widely across the continent.\n\nWhen Europeans arrived, so did colonial diseases and military actions that decimated populations before anything but crude descriptions of their societies had been recorded; as one prominent anthropologist concluded, \"The traditional local groupings of most Aboriginal tribes changed quickly and radically under the impact of European settlement.\" For those studying how societies maintain their identities and separation from each other, that is a serious problem.\n\nEuropean contact with the hunter-gatherers represented a culture shock that is hard for us to conceive. It would have changed everything. Before Europeans appeared, hunter-gatherers likely had their own version of the geocentric view held by many civilizations until Copernicus proved Earth revolves around the sun. As the nineteenth-century explorer Edward Micklethwaite Curr told it, \"Tribes living on the sea-coast probably think the world more extensive than those which dwell inland, and it is likely that those which occupy large tracts of desert country have more enlarged ideas on the subject than tribes in a productive, thickly tenanted neighbourhood.\" Of the inland hunter-gatherer societies, Curr wrote, \"Their idea was that the world was a plane extending about 200 miles on every side, and that their country was its centre.\" Ponder then what an Aborigine would have felt when European boats anchored off their beaches. It would have been as if Martians landed on the White House lawn. Their worldview, right down to many of their perceptions of their own societies and each other, would have been shattered, not gradually over years or even days but in one mind-blowing flash. Any once-divisive differences between Aboriginal groups would be exposed as slight and unremarkable. Although the societies of hunter-gatherers remained standing, their original sense of social identity and being distinctive from their neighbors became almost impossible to accurately reconstruct.\n\nFor all these reasons, I will speak of hunter-gatherers, and later of tribal societies, in the past tense, not to suggest the people themselves are gone, but that their original ways of life largely are. The few hunter-gatherers I have had the good fortune to spend a little time with like !Nani had already been required to stay in one place and stop hunting game. Nevertheless, we have some assurance that the hunter-gatherers studied in recent centuries can reveal something about how our ancestors conducted their business, at least in broad strokes. Band societies as diverse as the Arctic Inuit and African Hadza are alike in ways that suggest human nomadic existence has a blueprint\u2014foraging widely for wild food was part of a package that, as a rule, included other social building blocks, to be discussed in the pages that follow. Along the way we shall investigate what societies represented for these hunter-gatherers, and why so many anthropologists have downplayed or overlooked the societies. What should become apparent is that hunter-gatherers lived in anonymous societies just as we do today.\n\n# FISSION-FUSION AND THE HUMAN CONDITION\n\nThe feature that distinguished band societies from other modes of life wasn't specifically hunting or gathering, which people still enjoy, if only when they bag deer or excavate truffles. Nor were these hunter-gatherers totally distinct for their nomadism. Pastoralist tribes like the Huns could disperse to camps part of the year to secure pasturage for their domestic animals. What most set them apart was the pattern of movement of their members: nomadic hunter-gatherers spread out by fission-fusion, with people roaming with considerable freedom.\n\nFission-fusion nevertheless generally took a regimented form for the nomadic hunter-gatherers of recent times, as it presumably did for those living prior to the invention of agriculture as well. People mostly clumped here and there in bands. Each band consisted of on average 25 to 35 individuals comprising several, usually unrelated, nuclear families, often spanning three generations. A person could visit other bands, yet tended to keep a long-term connection with one. Shifts between bands usually came about with little effort but not often, a far cry from the eternally fluid movements of chimpanzees and other fission-fusion species characterized by ever-changing parties.\n\nAlso uniquely, the members of a human band would separate each day into work groups\u2014confusingly called parties\u2014to find food. Each night everyone would return to the location that the band had chosen to put up camp, typically for a few days or weeks. This protracted commitment to a band, with each band having a home base that changed frequently, is unique to our species. The closest analog among other fission-fusion animals occurs in the gray wolf and spotted hyena during the weeks that the pack or clan brings back meat to pups in a den. In all three species, the base shifted occasionally both to keep enemies off their scent and to put the residents in reach of fresh hunting grounds.\n\nThe ecologist Edward O. Wilson has argued that having a well-protected home base is at the heart of our humanity. In this view, early people were like ants foraging from a central place. I propose an adjustment to this view. Certainly, a band organized its camps more than a wolf pack does a den, and staying put for a while provided relief for the elderly, infirm, and young. Yet the camps were usually short-lived and not especially secure. Indeed, bands moved around in part to protect themselves. Some Ache and Andaman Islander hunter-gatherers, from Paraguay and in the Bay of Bengal, respectively, lived under constant threat, in this case from outsiders, and changed locations often\u2014sometimes daily. A band's members could jointly fight off a leopard, although a baboon troop ganging up on the same cat would be pretty much as efficient; and one could argue that the multiple family fires set up around an encampment kept predators at bay more than would the single fire of a family on its own. But these were the extent of defenses at a band level: fortifications protecting everyone became prominent only for people settled in villages.\n\nOur pedigree of living in nomadic bands that were part of a society-encompassing fission-fusion webwork of movements and relationships suggests spreading out was (and is) as consequential to human success as staying put. The time spent in camps allowed humans to hone their social and cultural lives based on extended face-to-face interactions in ways no other fission-fusion species can, but at the same time their mobility allowed an expansion of a society's realm beyond anything achieved previously by the societies of any other vertebrate land animal. This required an unsurpassed intelligence to monitor daily relationships with individuals in the same band, and to keep up, as well as people could, with the goings-on of others far afield.\n\nFission-fusion in other animals looks remarkably disorganized compared to that in band societies, but a chimpanzee (or bonobo) party and a human band aren't irreconcilably different. Certain chimps actually come close to settling into temporary home bases. Although most chimps are forest dwellers, some live in savannas, the habitat in which we evolved. Savanna chimps get little notice, though their social patterns more closely approximate those of hunter-gatherer bands. For one thing, they have to hoof it almost as much as nomadic hunter-gatherers. Primatologist Fiona Stewart estimates a community of 67 in Tanzania ranges across an area totaling somewhere between 270 and 480 square kilometers, the animals about as thinly spread as hunter-gatherers were in such environments.\n\nParties of savanna chimpanzees also have more stable memberships and locations compared to those of forest communities. Several factors are in play. The animals can't easily meander over to the next party, which could be quite far off. Their favored tree clumps are also few and far between, and the distance often encourages several chimps to stay around one spot for a few days before trying another site. Recent hunter-gatherers also liked clumps of savanna trees for their camps, not only because of the shade, but for the water and other supplies associated with them. In addition, savanna trees are seldom sturdy enough for the apes to bend leafy branches into a canopy bed the way they do in forests. Instead the chimps often bed on the ground, bundling grasses or folding saplings into structures that crudely resemble the temporary shelters that Bushmen construct of grass and sticks. They spend time in caves, as well, to dodge the midday sun. Foraging from these rest sites, savanna chimps kill small primates with spears made from branches and shovel out edible tubers with sticks in something of the manner of hunter-gatherers.\n\nIt is no great stretch to imagine that the common ancestor we share with chimps was capable of similar maneuvers, not to mention the long line of hominids that followed. The evolution of the social skills that individuals needed to stay with the same society members each night; the taming of fire by our predecessor _Homo erectus_ 400,000 years ago; and cooperation in sharing food before bedding down would have given humans a haven worth returning to in regular fashion. Proper camps would have gotten their start.\n\n# THE REALITY OF HUNTER-GATHERER SOCIETIES\n\nThere are shelves of articles and books by anthropologists who write of hunter-gatherers based solely on what happened in bands, while ignoring how hunter-gatherer bands identified with a broader society. Sometimes, when they acknowledge the connections between bands at all, their solution is to claim that the nomads didn't have societies to speak of. This view often intimates that hunter-gatherer culture varied from one band to the next, with no wider human affiliations and nary a sharp boundary to be seen.\n\nI find this doubtful. For one thing, all other human populations live in groups we call societies. Additionally, the hunter-gatherer bands documented over the last few centuries were affiliated with societies. Indeed, there is ample evidence that membership in societies was a crucial aspect of hunter-gatherer existence\u2014one that is indispensable to understanding how the societies of today came to be. My term \"band society\" could suggest that one band constituted a society, and that was true at times, but societies generally extended across a specific set of bands. I am not surprised these societies have been missed or misinterpreted. Until primatologists like Goodall deduced that chimps had societies from the animals' spacing and confrontations with outsiders, the consensus was that fission-fusion animals were \"open-group species,\" without social borders. So it made sense that these human bands, whose movements were marked by fission-fusion as well, wouldn't have had them either.\n\nBecause the members of human band societies, as well as chimpanzee societies, rarely come together, finding out where a society stops and the next starts can be tough, but sharp separation in the memberships of societies are in place just the same. Numerous accounts tell how hunter-gatherers of recent centuries felt secure in the presence of their \"own kind.\" Asked who they were, typically a hunter-gatherer would give you the name for a community encompassing several, often a dozen or more, bands spread over a wide geographic range. These band societies had populations ranging from a few dozen to perhaps a couple of thousand individuals.\n\nTo support the notion that hunter-gatherers do not sort out into distinct societies, some anthropologists put forward the example of the peoples of Australia's Western Desert, a desolate region where resources are scarce and people sparse. The Aborigines across that region have been reported to lack any clear boundaries with their neighbors. This idea is dubious. The anthropologist Mervyn Meggitt recalled a conversation with one of the Aboriginal groups of bands from the Central and Western Desert that lays out the Us-versus-Them view of the people of that region in no uncertain terms.\n\n\"There are two kinds of blackfellows,\" they say, \"we who are the Walbiri and those unfortunate people who are not. Our laws are the true laws; other blackfellows have inferior laws which they continuously break. Consequently, anything may be expected of these outsiders.\"\n\nCertainly the Western Desert peoples intermarried, but that's true for societies generally. It's also true that to survive, they might have had to be open to border crossings. Robert Tonkinson, an anthropologist, has written, \"In areas as harsh as the Western Desert the need to assert a particular identity has to be balanced against the need to remain on good terms with neighbors.\" Amen to that. Yet the peoples of the Western Desert fell into groups as distinct as those of Aborigines elsewhere. Even if, by necessity, they exhibited an unusually generous cooperation with outsiders, societies don't dissolve when they are kind to each other. The border between France and Italy isn't as fraught as the one between North and South Korea, but it is present nonetheless.\n\nWe think of nations, which academics call states, as having governments and laws, and band societies have neither, formally speaking. Still, hunter-gatherers expressed the same comfort and trust around the people of their own society that we do at present, and in many ways their societies were analogous to our nations. The nineteenth-century historian Ernest Renan believed nations to be a modern phenomenon. Yet his definition of a nation as a strongly bonded people with a shared heritage of memories and a desire to identify collectively is an apt description of band societies, too. Hence with some justification, \"nation\" has replaced \"tribe\" as the word of choice for many Native Americans. For them, the bonds are all important.\n\nWhile the institutions of modern nations are efficient at inspiring passions toward our society on grandiose scales, the affections we discern as a national consciousness have a deep history. As linguist Robert Dixon informs us, again about the Aborigines:\n\nA [band society] appears in fact to be a political unit, rather similar to a \"nation\" in Europe or elsewhere\u2014whose members are very aware of their \"national unity,\" consider themselves to have a \"national language,\" and take a patronizing and critical attitude towards customs, beliefs and languages that differ from their own.\n\nLike nations, and in fact all other human societies, band societies identified with an expanse of ground that they exclusively occupied. They were territorial\u2014wary of, and often hostile to, outsiders entering this area. Nomadic they may have been, but the overall movements of band-dwellers could be just as confined as those of the people who came to depend on agriculture. There were exceptions. Territoriality went away when nothing was gained from it. In the American West the tribes of the Great Basin wandered with relative impunity where pine nuts were so abundant that defending a supply was pointless. Yet generally each society took over an area ranging from hundreds to thousands of square kilometers, with the partitions tacitly determined in a time before living memory.\n\nUnlike chimpanzees and bonobos, who can move anywhere within the space claimed by their community (with some biases to favorite spots by individual chimps), each cluster of people\u2014each band\u2014usually had primary use of only a portion of the society's territory, a space that its residents knew like the backs of their hands, and sometimes inherited. A fondness for home didn't originate with people moving into a physical structure but rather with their connection to the land. The Australian anthropologist William Stanner brought this idea to life:\n\nNo English words are good enough to give a sense of the links between an Aboriginal group and its homeland. Our word \"home,\" warm and suggestive though it be, does not match the Aboriginal word that may mean \"camp,\" \"hearth,\" \"country,\" \"everlasting home,\" \"totem place,\" \"life source,\" \"spirit centre\" and much else all in one.\n\nSome anthropologists call the land crisscrossed by each band a \"territory,\" but use of the landscape was more flexible than that word suggests. Like the neighbor casually knocking at the door to ask for a cup of sugar, any socially upright person could usually traipse onto another band's space. As long as permission was sought, the local band might share water or space to hunt if supplies were short elsewhere, or allow the visitor to stay around and chat with friends and kin. Just as individuals of the same band freely shared and borrowed from each other, reciprocity across the members of a society was routine.\n\nThere was more goodwill between bands than typically existed between the societies that contained them, whose claims were rigidly enforced when necessary. Among the Bushmen, bands of the same society occupied contiguous spaces, whereas a no-man's-land lay between different societies. Similar unoccupied gaps separate the societies of other species, too\u2014communities of chimpanzees, packs of wolves, and colonies of fire ants.\n\n# ANCIENT RACES\n\nThat an Aborigine could speak of \"other blackfellows\" reflected a clear awareness among native Australians of who was who and at the same time calls to mind the issue of race. Physical expressions of genetic traits that vary widely across populations have become valued in humans in a manner without parallel in other animals. We associate many nations, and certainly the groups within them, with a race. At the same time we have been so distracted by the distinctive physical appearances of hunter-gatherers from different parts of the world, such as the Aborigines or the Pygmies, that we have often ignored their societies. We need to give some thought, then, to the racial identity of hunter-gatherers, and how much it mattered to them.\n\nMany sociologists and others claim races are socially constructed\u2014arbitrary products of the imagination. This is sensible given that what we often describe as races would have evolved as gradual shifts in physical traits across continent-wide expanses, with all sorts of intermediates between. Yet in the context of societies composed of people of distinct lineages that have come in contact, usually as a result of movements of entire populations with different physical looks, race can have a robust meaning. Societies of other species don't migrate enough for race-like distinctions to crop up between neighbors in this way.\n\nAll Aborigines descended from the same migration of people, so the members of neighbor societies on that continent have been similar in appearance, skin pigment included. Aside from some contact with Indonesians along the coast, categorizing people by well defined, genetically inherited features wouldn't have come up for pre-contact Aborigines, who would have been attuned mostly to social differences from one society to the next. Native Australians became Aborigines (\"blackfellows\") and developed a sense of otherness and self-awareness as a race only after Europeans populated Australia.\n\nIn Africa, the crucible of evolution for the human mind and body, the situation was different. Though the dissimilarities among the peoples of Africa may not seem as black and white (literally) as many of the distinctions we take for granted today, some of Africa's neighboring hunter-gatherers were \"as different genetically from one another as are the major ancestral groups of the world,\" as two experts put it. This proximity of unlike groups could reflect the immensity of time during which human societies have diverged and moved around that continent to establish societies that ended up living near groups sharply distinct from their own. And so it was that for the past several millennia that the Pygmies have formed alliances with racially distinct farmers, while Bushmen have lived among the Khoikhoi pastoralists for perhaps 2,000 years and among the darker-skinned Bantu, who entered their lands over the past 1,500 years.\n\nYet despite the long contact with very dissimilar peoples, and certainly not blind to their resemblance to each other, the Bushmen never felt themselves to be a single unit, worthy of a name and acknowledgment of affinity, any more than America's original inhabitants thought of themselves as Indians before Europeans coined the term. Bushmen was a category that the Bushmen, who thought of themselves in terms of their societies such as the !X\u00f5 and the !Kung, didn't recognize. Even now they consider themselves Bushmen\u2014or San, a pejorative name given to them by their Khoikhoi neighbors, meaning rascal\u2014only when they leave the bush to take jobs elsewhere. As with other human differences, a feature becomes a marker of identity only if the people themselves choose to see it that way\u2014and that includes race.\n\nClear physical dissimilarities between broad groups living in proximity weren't exclusive to the hunter-gatherers of Africa. For instance, the Ache of eastern Paraguay had (and have) pale complexions compared to other native peoples of the region. Of course, slight differences existed within these races, too. People sometimes connected a certain appearance with a single group, such as the Ypety, a group of Ache that tended toward darker skin and thicker beards than other Ache. But for the most part the broad groups we think of today as races don't seem to be central to hunter-gatherer thinking. Only complete outsiders, usually European (to whom all hunter-gatherers across a region looked the same), lumped them together as one group\u2014ignoring the fact that hunter-gatherers actually belonged to multiple societies.\n\nOtherwise, sociologists are correct in identifying race as a recent invention. In this regard, race can be thought of as a rough appraisal of the overall appearance of vast numbers of individuals who have no clear common background. The modern fetish for races is an artifact of confounding people from a mishmash of sources into broad classes that hadn't existed before. The hunter-gatherers of a region often shared a far tighter genealogical heritage, but ironically those people seldom could have cared less about their racial commonalities. Some of these genealogical races were so scarce or limited in distribution that they were represented by a handful of societies: the Hadza are all of one group; those we call the Ache had four societies; Tierra del Fuego's Yamana, now culturally extinct, had five; and the Andaman Islanders were originally separated into thirteen societies. Meanwhile, \"the\" Aborigines, spread across the Australian continent, had societies numbering five or six hundred.\n\n# ANONYMOUS NOMADS\n\nCommonalities with chimpanzees and other fission-fusion mammals notwithstanding, the societies of hunter-gatherer bands were anonymous\u2014they were dependent on markers of identity rather than the members' personal knowledge of each other. Individuals were regularly spread out to such a degree that not all the strangers they came across belonged to other societies. As one anthropologist who lived among the Bushmen nearly a century ago put it, \"The more widely separated bands of a tribe have no personal knowledge of or direct contact with each other.\" Even today the Hadza, with a thousand members, think of themselves as a people despite their affiliations with small bands. Many individuals never come into contact with, or know, Hadza from the far side of their society's territory.\n\nThis would suggest that hunter-gatherers conceived of their societies as being united around a common identity\u2014language, culture, and other markers. The anthropologist George Silberbauer wrote in 1965 that calling the G\/wi Bushmen a \"tribe\"\u2014that is, a society\u2014was merited \"by the fact that there are also other features of their culture, besides language, which are held in common.\" Regrettably he never spelled out what these features were. Finding myself struggling with the scarcity of information about hunter-gatherer identities, in the summer of 2014 I dropped by the home of retired Harvard professor Irven DeVore, a foremost authority on Bushmen. At the age of 80, sporting a thin white beard and flanked by tribal totems, Irv had come to look like a wizard. He pointed out that many relevant details about the interconnections between bands\u2014the markers they shared\u2014might have gone unasked or unrecorded, even if they hadn't already faded or changed since the people's exposure to Europeans.\n\nOne reason for this oversight may have been that many features of hunter-gatherer societies appear muted from the point of view of contemporary observers. Although we are still finely tuned to very small dissimilarities between people, and often exaggerate their importance, our world now revolves around supernormal stimuli, pushing our senses to intensities not experienced in the past. Many of our societal markers, from Big Ben to Times Square, are inflated for maximum effect. For hunter-gatherers the signage of daily existence was understated. Each nuance about the natural world and the people in it filled their senses. Distinctions between their neighbors that to us would be slight would have stood out as plainly as grass blades bent by a passing antelope.\n\nThe dialects among the Bushmen have been easiest to notice and are well studied. Bushmen today speak one of two dozen or so basic languages (or dialects thereof). Unfortunately, numerous other languages, and Bushmen groups that spoke them, have gone extinct. However, there are many markers outside of language that distinguish Bushmen societies. Anthropologist Polly Wiessner has described certain goods\u2014such as oracle discs used to divine the future, wood forks for puberty ceremonies, and aprons worn by women\u2014as identifying them as part of a group of bands regardless of how little their people interacted. Depending on their \"band cluster\"\u2014their society\u2014the !X\u00f5 Bushmen spoke different dialects and made arrows that differed in shape. People from one !X\u00f5 band cluster recognized the arrows from another such group of bands \"as coming from !X\u00f5 'who are not our people,'\" according to Wiessner. Meanwhile she found that another group of Bushmen, the !Kung, numbering 1,500 to 2,000, had an arrow style of their own.\n\nDeVore pointed out to me that some distinctions among Bushmen may have evaporated when they stopped painting and engraving on rocks, the reasons for this artwork long forgotten. The Bushmen also ceased making pots once they began trading goods with their Bantu neighbors centuries ago. Archaeologist Garth Sampson collected pottery shards from a thousand ancient Bushmen campsites in South Africa. He found unique pottery motifs in certain areas, and concluded that each area had been used by a different group of Bushmen, all of them now extinct. Concentrated in one tract of real estate, for example, were pots with comb-stamps, a technique of pressing the edge of a mussel shell into the still-soft clay that must have been invented by the local society.\n\nOther hunter-gatherers had more distinctions between their societies. The classic scene in Western films where the cowboy sees Indians in war paint and tribal regalia and proclaims, \"Here come the Apache,\" has a rough basis in reality. \"A knowledgeable person can look at a pair of Plains moccasins from a hundred feet away and say, those are Ojibwa; those are Crow; and those are Cheyenne,\" Michael O'Brien, an archaeologist, told me. \"The leatherwork is distinctive but the beadwork is what gives them away.\" Decorations on pots and tepee designs are also conclusive markers. This goes for war bonnets, too: feathers could be splayed flat or erected into a tube. Some tribes replaced the feathers with buffalo or antelope horns. The Comanche did away with the bonnets entirely, and instead wore the whole buffalo skin with horns attached.\n\nAs for the Aborigines, \"our laws are the true laws,\" the Walbiri people would say, a clear statement of how important customs and beliefs were in marking off one group from another. As for more subtle matters, historian Richard Broome wrote of the Aborigines that \"even gestures can be misinterpreted, as winks and handshakes in one group are mere twitches or touches to the other.\" Still, from what I can gather, sharp distinctions between Aboriginal societies (or ethnolinguistic groups, as anthropologists awkwardly call them) were few, aside from language. Hairstyles could vary; the Urabunna, for example, encased their coifs in a \"net-like structure,\" while their neighbors inserted emu feathers. But artistic expressions, such as in body scars (their version of tattooing), are thought to have changed only at a wider, regional level.\n\nThere is intriguing information on group identity for hunter-gatherers elsewhere. Among the Andaman Islanders, who for the most part fiercely avoided outsiders until early in the last century, were the Onges who painted their bodies, while the Jarawas tattooed themselves by puncturing their skin with slivers of quartz. African Pygmies differed in the quality and cadence of their music, the dances accompanying their performances, and the instruments they played (and mostly still play). The four Ache societies had different myths and traditions, as well as their own musical instruments and singing styles. What made one of those groups, the Ypety Ache or Ache Gatu, so extraordinary (and feared by the other three Ache societies) isn't so much that they ate human flesh, but that a people with only about 40 living individuals kept up an elaborate ceremony and lore around eating the deceased. As one observer described it more than 60 years ago, \"The Ache [Gatu] consume their dead to keep their souls from invading the bodies of the living... carried away after the meal by the smoke that rises from the ashes of the skull.... The soul climbs into the sky to be lost in the upper world, the Invisible Forest, the Great Savanna, the land of the dead.\"\n\nCannibalism aside, most of the differences I have pinpointed among hunter-gatherer societies might seem trifling, yet to them they were momentous indeed. The oddest beings a hunter-gatherer came across in his or her life would have been neighbors who were actually quite similar, and had very similar things. Still, their differences would have evoked a visceral reaction of anxiety or dread far in excess of what you or I feel during encounters with most foreigners in this day and age. After all, given the low density of most hunter-gatherers in antiquity, people would have encountered an infinitesimal fraction of the outsiders most of us are familiar with today.\n\nThe sparseness of nomadic hunter-gatherers affected relationships not just between societies, but also within them. While their societies were a step larger than those of other vertebrates, on a daily basis people carried on their lives among just the few in their band. That made each band equivalent to a close-knit neighborhood, even if a mobile one. Socially, then, nomadic life was something we would recognize. Economically and politically, not so much. All of these aspects, from enjoying each other's company to issues of workflow and group decision making, were fine-tuned and performed in combination to facilitate the success of these peoples in challenging landscapes.\n\n# CHAPTER 9\n\n# The Nomadic Life\n\nTo get some idea of how a band society functioned in the day-to-day, think of each band, with its two- or three-dozen residents, not merely as a neighborhood in the social sense. Having to operate independently, it also had to serve as a local manufacturing center. Picture not a steel works, but a minimal organized unit of production, something no city quarter can claim. The factory wasn't elaborate. The people had no need for complex or permanent infrastructure. As with animals in a small society, notably the simplest ant colonies, they put together simple dwellings, and anything else they wanted, from materials gathered on the spot. Forget pharmacies. In Australia's Northern Territory, I once watched an Aborigine treat a case of congestion like his hunter-gatherer grandparents did. The man pulled a tentlike nest of weaver ants from a tree, mashed the feisty swarm into a pulp, and held the pulp to his nose for a few raspy inhales. Trying it myself, I smelled the intense Vicks VapoRub scent of eucalyptus oil.\n\nFew tasks in a band factory demanded multiple players. When something had to be made in steps, like a spear or lean-to, usually the same person (or maybe two) carried out the process from start to finish. Teamwork, when required at all, was seldom elaborate, although felling a giraffe or mammoth might have taken the coordinated efforts of several men, and perhaps butchering it required the ant-like participation of everyone available.\n\nDivision of labor by sex and age was the backbone of the factory. In a human band almost invariably the men hunted large game or fished, while the women, often weighed down by breastfeeding children (which made hunting impractical), gathered the bulk of the band's calorie intake in the form of fruits, vegetables, and small prey like lizards and insects, and cooked dinner.\n\nThe rhythm of expeditions orchestrated by sex added a layer of complexity beyond that of the movements of almost all other animals. Outside of foraging gangs of certain ants, the most specialized outings in the animal kingdom may be when a party of chimpanzees hunts for primate meat, or when gray wolves or spotted hyenas gear up for the chase.\n\nAs a factory, each band was assembled redundantly: there could be multiple hunting and gathering parties, and the families performed many chores on their own. Because they had to be mobile to keep their stomachs full, the hunter-gatherers couldn't accumulate stuff, and didn't have ownership in the way we think of it. People could possess just what they could lift\u2014about 25 pounds of supplies is a figure sometimes given, the weight airlines permit for a carry-on bag. (There were a few exceptions: Inuits could carry more on dog-powered sleds, the Plains Indians on travois made of nets between two crossed poles, and other groups loaded canoes or dugout boats.) Left behind for later return were fire pits, heavy stones to grind seed, and hunks of rock used in making tools.\n\nWhat made the band an economic unit was the swapping of goods and information between people. If someone could not make an item when the need arose, then giving and lending were expected, with the understanding that someday the giver would ask the receiver, or someone in the receiver's family, for something in return. The exchange most significant for survival of the factory was of food. The Batek, a group I have visited in Peninsular Malaysia, saw food sharing not as a benevolent act but as a reflection of the fact that all food belongs to the forest, not to the person who found it. A man who made a kill would share the meat with the band. In the Ache and some Pygmies, a hunter never consumed so much as a mouthful of what he caught. This generosity contrasts sharply with the stinginess of chimpanzees and the only slightly improved attitude of bonobos, who still never split food among the many, yet was sensible for humans. Hunters can kill far larger game than the whole band can eat at once, and animal flesh must be consumed immediately. With giving back the long-standing practice, a hunter who passed around meat one day could be sure someone else would fill his family's bellies the next\u2014the original social security package.\n\nAn upside of this factory life was that people weren't occupied with raising crops or struggling to take in excess food. They ended up with leisure time\u2014a commodity that earned these nomads the moniker \"the original affluent society.\" The only thing stockpiled was time to devote to social relationships. The typical population of a band turns out to be ideal. Enough meat, produce, and goods were procured, processed, and exchanged on an average day to keep everyone fed and comfortable. Success became problematic when fewer than 15 or more than about 60 individuals resided in a band.\n\nA band's size was self-regulating. The individuals and families, although interdependent, did as they wished. People could leave an overcrowded band to join another or go it alone for a time. Dispersal was an annual tradition each autumn for Nevada's Western Shoshone, when each family went its own way to collect the same nuts favored by pinyon jays. They split up not because food was scarce, but because it was scattered, which also gave them the excuse to visit friends elsewhere.\n\n# JACK OF ALL TRADES\n\nAt present people rely heavily on experts, big (think Steve Jobs) and small (the watch repairman). By comparison, people in a band society followed the jack-of-all-trades strategy of ants in small colonies\u2014and for the same reason. When a labor force is minimal, a dependence on specialists can spell disaster. This is especially true if one member dies with no trained replacement. That, and the bare-bones nature of ownership, meant everything about their nomadic way of life pretty much had to fit inside each member's head, even if peers, and especially the elderly, were useful at reinforcing what was correct.\n\nThis is hard to grasp now. No one person can make a pencil, let alone an iPhone or a car, from scratch. Expectations of what people should do by age or sex were the only job descriptions. Otherwise the sole specialized line of work in a band was as a medicine man. Even then, Aboriginal healers, whose training in that role could take years, were still expected to carry out all of life's mundane tasks themselves.\n\nThe \"one brain fits all\" approach put a limit on the complexity of a band society and the labors its people performed. The baffling instruction manuals of modernity suggest passing on directions is no easy task even for details that are permanently recorded, and of course hunter-gatherers had no writing. The inventor of the boomerang, clever at carving wood as that Aborigine must have been, also had to find a way for the average Joe to construct one without much fuss, or the discovery would have vanished in a generation.\n\nAlthough all members of a band carried the same survival kit, differences among individuals in creativity and skill would of course have been manifest. As philosopher Gunnar Landtman recognized in the 1930s, among the Bushmen, \"some women are more clever and industrious in making beads than others, some men handier at twisting rope and boring pipes, but all know how these things are done, and none devote their life to them.\" Still, evincing one's talents was a delicate matter. Because of the intimacy of life around camps, band societies rarely tolerated a showoff. There are endless stories of hunter-gatherers teasing the successful or talented person, statements like this one included: \"When I kill anything, it is usual for our fellows, especially my Bushmen, to exclaim: 'how small!' The wounded one that escapes is pronounced wonderfully fat.\"\n\nMockery aside, everyone would have known who the best hunter or gatherer was, no matter how humble others expected him or her to act. Recognizing expertise is in the blood of our species\u2014even a three-year-old realizes that individuals differ in skill and knowledge, and children seek out the appropriate person to solve a problem by the time they are four or five. This inclination likely originated deep in our evolutionary past. The societies of many species benefit from the varied talents of their members, and some animals gravitate to individuals proficient at the task, as chimpanzees do when learning to crack open nuts.\n\nAmong !Kung Bushmen, unproductive people without a noteworthy skill were said to be _tci ma\/oa_ or _tci khoe\/oa_ , meaning \"nothing thing,\" or useless. Men whose talents accrued benefits to others were called _\/\/haiha_ ( _\/\/aihadi_ in the case of women), which crudely translates as \"one who has things\" (don't ask me how to pronounce the words). So without specialists or jobs, band societies offered opportunities for dexterous toolmakers, dazzling storytellers, skilled mediators of social conflicts, and thoughtful decision makers. Gifted people who didn't gloat and took teasing in stride avoided the fate of the characters in Aldous Huxley's _Brave New World_ , who had their creativity squelched. Instead, they were able to use their talents to enhance their quality of life in permissible ways, for instance by attracting a desirable mate.\n\nSome psychologists hold that individuality wasn't prized until the late Middle Ages. Yet as long as their actions weren't anathema to the whole tribe, nomad hunter-gatherers had a wide latitude for making unusual choices\u2014especially given their freedom to escape those who disagreed with them by moving over to another band. Differences in how a task was performed could be condoned from one person to the next. A man might have a special way of fashioning arrows, if only to provide evidence a kill was his and thereby slyly show off his skill. This latitude for individuality must have been a wellspring for innovation, especially since the pool of talent included hundreds of people across several bands. Even if each family were obliged to fabricate an axe of their own, they might try to duplicate the methods of the one person most skilled at axe making.\n\nJust as they had no specialists, hunter-gatherers seldom had special interest groups aside from their families and the hunting and gathering parties that went out each day. There were no political parties or fan clubs, no fashion followers, preppies, hippies, yuppies, or geeks\u2014and certainly none that wore the equivalent of a team jersey. Cliques of the women who especially enjoyed knitting didn't call meetings to order; exclusive fraternities didn't make merry. The group that could be most closely compared to such an affiliation was the band itself, to which people gravitated based on social compatibility, and to some extent, kinship. But a band's members didn't ordinarily believe themselves to be distinct from or superior to the other bands of their society. Individuals could compete, yet while modern towns and cities have rivalries, I've seen no evidence to suggest bands did (bands never, for instance, went head to head at group sports). Other than the personality mix of the residents, and perhaps proximity to a nice waterhole, there was no special advantage of being in one band versus another.\n\nOverall, the lack of groupiness _within_ a band society is another reason to suspect that the society would have been of inordinate consequence to its people. At present we devote tremendous effort to affiliations that shift depending on whether we happen to be attending church or practicing with our bowling team, in a protean aspect of identity referred to as the group self. In a band society all this zeal would have been directed at the only membership most of the nomads had, nuclear family aside: their society. Among other indications that life in the society completely expressed their worldview, the bands didn't distinguish sacred beliefs (usually relating directly to nature) from other aspects of life. Likewise, rituals, entertainment, and education\u2014all matters except family matters were part and parcel of their relationship to the society as a whole.\n\nAs in present-day societies, the fact that individuals interacted with some members more often than with others was of little consequence: the focus of identity and affiliation with their entire society made the ties among all members as clear-cut, if not more so, than those enjoyed by citizens of a modern nation.\n\n# GOVERNMENT BY DISCUSSION\n\nWhen the Canadian anthropologist Richard Lee asked a Bushman whether his people had headmen, the man slyly responded, \"Of course we have headmen! In fact, we are all headmen. Each of us is headman over himself!\"\n\nAmong the jobs absent from bands were leaders. Anyone eager to make decisions for others was a problem for people who spent most of their lives in small groups day after day with little privacy. Ridicule and jokes were weapons deployed against those seeking to sway others, as often they were used to deflate other shows of skill or superiority. To be influential, then, flamboyance wasn't an option; keeping the economy of a band humming along required nuanced social skill, with a goal to, in the words one anthropologist, \"persuade but not command.\"\n\nHumility, again, was key. The socially successful band member was a diplomat and master of the debating arts, delicately guiding the argument without being pushy, and ceding to those with more insight in a given situation. No wonder hunter-gatherers took delight in games less competitive than those of today. With a few exceptions, such as tugs-of-war or a pastime among Bushmen boys of throwing wands at hummocks to see whose bounced the farthest, winning wasn't the point.\n\nA person could of course lead in the limited sense of serving needs of the moment, as we all do from time to time. Such a leader might turn up when decisive action is required and there is no opportunity to discuss the matter. This occurs in animals, for example when a motivated lioness heads the charge on a zebra. Or an individual may have information of immediate value, as when a returning honeybee scout performs a \"waggle dance\" to inform other bees about exactly where to look for flowers.\n\nBands squashed any assertive display of pretensions or attempt to direct others by what's known as a reverse dominance hierarchy. The majority would collude to put a stop to the egotist, the power-hungry, the showoff. Similar tactics are on display in inchoate form when chimpanzees or spotted hyenas gang up on the offending individual, but in a band these actions were effectual enough to subvert whatever rigid primate hierarchy our forebears inherited. Reversals of dominance aren't infallible. All of us learn the hard way that successful tyrants collude, as when tough kids in league with each other wreak havoc on a schoolyard. But such power plays would have had limited success. Anthropologists like to say that hunter-gatherers could vote with their feet. When tribulations were rife, they left for another band. With no way to seize political control across all the bands, bullies could be safely avoided.\n\nThe failure of anyone to dominate, and the resistance of the group to being pushed around, created equality across bands. There are precedents for such egalitarianism among the animals\u2014prairie dogs, bottlenose dolphins, and lions also have no leaders and little dominance. By comparison, life under the most caring alpha chimpanzee can be a rough go as his underlings vie for position.\n\nWhat the lack of differentials in power or influence most brings to mind are the social insects. Their efficient hustle and bustle makes it easy to assume someone is in charge\u2014but no. Think the queen heads a colony? She is at the head of no army, gives no orders to nest builders or nursery attendants. Rate her existence miserable. After one fling with sex, an ant queen is stuck belowground. Cranking out eggs is her only role. Meanwhile each worker operates on her own in the throng, or carries out tasks with those that happen to be near her, and eats, sleeps, and works as she chooses.\n\nAre social insects egalitarian? Workers of the most colony-centric species don't tussle even when perilously short on food. Nonetheless they display a kind of reverse dominance in one instance of social discord. Patrolling workers of honeybees and some ants and wasps smash eggs laid by fellow workers who dare compete with the rightful egg-layer\u2014the queen.\n\nHunter-gatherer egalitarianism didn't mean parity to a tee. It didn't always apply to families\u2014some dads have always ruled with an iron fist. And while material wealth varied little, differing degrees of adroitness in diplomacy and other skills created disparities. In this the bands remind me most of lions, an egalitarian species with no dominance hierarchy, and yet the animals bicker at a kill. Equality means equality in opportunity, not outcome. Among humans it never arises spontaneously. In the words of social anthropologist Donald Tuzin, \"For Americans at least, 'egalitarianism' has a gentle Jeffersonian ring to it, evoking images of ruggedly decent, buckskin-clad pioneers working together in harmony for the good of one and all. The truth is actually quite the opposite: egalitarianism is typically a rather savage doctrine, for it involves constant vigilance and intrigue among society's members as they struggle to stay equal to each other.\" No wonder gossiping is considered such a primordial human skill.\n\nNor was equality expressed perfectly, with expectations consistently varying by sex and age in the areas of childrearing, cooking, and hunting, as we have seen. For the most part, however, each voice was heard, which gave Bushmen, among others, more sexual equality than exists in most societies today. When an issue arose, every affected party spoke until a decision was reached by consensus in what must have been the primary source of entertainment before TV. Theirs indeed was the original \"government by discussion,\" as the British prime minister Clement Attlee once described democracies.\n\nOvercoming discord was a primary concern. While band members had few formal means to regulate behavior, they shared beliefs about how people should be permitted to act. Today, we think of these as _rights_. In a way, identifying with those rules was the measure of citizenship: the duty to behave properly and participate in matters important to the group. \"Our laws are the true laws,\" said the Walbiri man, and by this he meant their ethical code.\n\n# COLLECTIVE DECISIONS\n\nCommunal actions are determined by collective choices in other species too. A meerkat that can't find a meal gives a \"moving call.\" That lone voice won't spur on the clan, but should a second animal echo it, the entourage heads to a new spot. In this situation the dominant meerkat has no more power to sway the group than anyone else. Painted dogs sneeze to chime in about whether to go for a hunt. Honeybees and some ants use a similar technique when they select a new nest by a kind of voter turnout to each location under consideration. Who told you democracy was a human invention?\n\nThe army ant is an exemplar of organization without oversight. In Nigeria I blundered into a carpet of army ants 30 meters wide, a horde setting out to seize quarry with their flesh-slicing jaws in a campaign of shock-and-awe. No ant chief directs the throng. Ant swarms rely on swarm intelligence, with each individual contributing at most a bit of information\u2014perhaps signaling the location of prey or an enemy with a pheromone. Yet the overall pattern that arises at a mass level makes strategic sense in that millions of ants end up taking a productive course without anyone at the lead.\n\nSocial coordination in army ants scales up beyond anything shown by human bands. While pooled decision making can be effective for small human groups, our societies have grown ever more dependent on centers of power to carry out large-scale endeavors, and we are all the more vulnerable for it. A leader can be a society's Achilles heel. Because individuals often step up to leadership roles for personal gain, and indeed can be popular despite an arrogant lack of concern for others, a Hitler or Pol Pot may bring a society to ruin. Then again, a society might be thrown into equal disarray when the death of a well-meaning leader leaves a void. Thus, for a society that seeks to defeat another, regicide\u2014the killing of kings\u2014is a cheap alternative to war. With no commander to take out, you can step on ants till the cows come home and they will continue to invade your pantry.\n\nInformation, as well as the faculty to use it, is distributed across marching ants\u2014and across hunter-gatherers spread out in bands. To some extent social media has brought about a return to this group decision making by enabling likeminded individuals to pursue sweeping collective actions without oversight and at almost no cost. In a historically important early example, a Philippine court was forced to impeach President Joseph Estrada in 2001 after Epifanio de los Santos Avenue in Manila was engulfed by protestors. What brought so many together was a rapid exchange of several million texts reading _Go 2 EDSA, wear Blk_. The very act of treating such a text as legitimate required trust that those forwarding it were coequals. That day saw the reverse dominance hierarchy firmly back in control in the form of what futurist Howard Rheingold calls a \"smart mob.\"\n\n# THE ADVANTAGES OF SOCIETIES FOR HUMANS: LIVING IN BANDS\n\nJust about every boon of membership in an animal society applies to our species also, in how societies both provide for and protect their members. Consider the matter from the perspective of a band society. Membership granted the nomads a balance between the stability of a reliable home base (actually many, shifting at intervals for each band) and flexible mobility, with the band and the society serving different functions. A band was the unit of daily interaction, its intimacy on par with that of monkeys that stay in contact. Even in the band, the connections between members were dynamic, allowing for daily separations into hunting and gathering parties. Groups of people could jointly defeat enemies or predators, obtain food (and widely share it, unlike most primates), and tend the children. In the latter task, which bonobos and chimpanzees do poorly at best, women took chief responsibility but men participated as fathers, as did grandparents: a role for the elderly uncommon among animals, though female bottlenose dolphins too old to reproduce aid in child care. Most remarkably, social learning was pivotal, and intricate, for humans: children garnered lessons not just from mom and dad but from other adults, absorbing the rhythm of life in their society.\n\nIn contrast to the rapport afforded by a band, the societies that hunter-gatherers belonged to were about identities\u2014inclusion and exclusion. Even though each band carried out its quotidian chores alone, the society offered a security blanket, supporting reliable market and marriage relations beyond the local fireplace. Hence while people across all the bands were less intricately and constantly reliant on each other than we have become in nations, their inclusion in a society was of immense survival value, providing the recourse to share goods and information and act together as problems and opportunities came to light.\n\nHunter-gatherers in their band societies minimized petty altercations and stayed egalitarian by dispersing in factory-efficient bands. How then did humans come to take advantage of the social glue afforded by their shared identities and forge civilizations?\n\nThe process began with the hunter-gatherers themselves. As it turns out, my description of their societies, while in line with much of what has been written about them, has been incomplete: I have not conveyed a sense of the possibilities open to them. Human fission-fusion can be reconfigured to permit people to distribute themselves in a diversity of alternative ways, from simple village to urban throng, in the service of feeding and defending themselves. No less than with the farmers that followed them, how hunter-gatherers lived varied by circumstance. Resources permitting, a society could settle down even as its members continued to forage from nature. Their aversion to accepting the authority of leaders could be put aside, and displays of talent at a particular task could become a justifiable point of pride or even a necessity. Both the benefits and strains of participation in a society would have intensified, as the coming chapter will establish.\n\n# CHAPTER 10\n\n# Settling Down\n\nBetween Mount Eccles and the sea in Victoria, Australia, on a lava plain laid down by a volcanic eruption about 30,000 years ago, are the archaeological remains of hundreds of dwellings. The structures cluster in groups of a dozen or so, some so large they are partitioned into apartments. People by the thousands settled across that expanse in these small villages, members of settled tribes that jostled, fought, and forged lasting alliances.\n\nThe region around the villages was transformed into a vast, managed waterscape, with streams and rivers variously dammed and diverted to create a labyrinthine yet integrated drainage system. The waterways, which extend for kilometers, are ancient, many dating back 8,000 years, with the system reaching its full glory 600 to 800 years ago. The canals were used to harvest wild game\u2014a species of eel\u2014with traps reaching a hundred meters long and constructed in some places of stone walls up to a meter high. The people also carved out artificial wetlands in which the young eels could thrive until they were large enough to eat, and caught the fish in such abundance that the excess could be preserved and stored for the off-season.\n\nLike all the Aborigines elsewhere in Australia, the people at Mount Eccles lacked domesticated food. This entire elaborate infrastructure was the brainchild of hunter-gatherers. And yet the homes look to have been permanent, and some may have been occupied year-round\u2014the descendants of the original residents claim this was so. Indeed, the lesson of the Mount Eccles Aborigines is that, even before societies took up farming, people had the option to reside in what I call a settled hunter-gatherer society.\n\nThe social lives of the hunter-gatherers who clustered in bands present us with a riddle. Many aspects of their existence seem antithetical to modern life. Most of us gladly follow a leader we respect and some of us strive to be leaders ourselves. Nomadic hunter-gatherers regarded both choices with disdain. We subscribe to a social hierarchy, not only fostering status distinctions, but also looking up to the powerful and prestigious, admiring those of breeding and affluence, and spending inordinate amounts of time tracking the private lives of stars and presidents. Karl Marx saw the history of societies in terms of class struggle, yet hunter-gatherer bands had no classes to struggle over. From whence, then, came the value that civilizations have accorded to accumulating wealth, if hunter-gatherers owned little and were willing to give it all away? Where did our personal ambitions come from and how far back in time did they exist? Finally, as it is clear that we aren't hardwired to be generalists in the manner of ants in small colonies, from whence our desire to be different from each other, for example by excelling at a narrow career path?\n\nSettling down forced people to draw upon a set of implements from their mental toolkit distinct from the ones hunter-gatherers employed in bands. Although conditions we take for granted\u2014such as social inequality, job specialization, and acquiescence to leadership\u2014were far from ubiquitous among settled hunter-gatherers, each passing generation spent at one place made those conditions more likely to come to the fore. Because of this elaboration, anthropologists generally describe settled hunter-gatherers as complex, in contrast to so-called simple hunter-gatherer band societies. But fission-fusion offered its own complications. These included the laboriousness of tracking down food that could be spread thinly, as well as in finding good campsites and the struggle to maintain social parity. For this reason I will avoid labeling either as simple or complex. Rather, I focus on the constancy and compactness of their living places as the ultimate cause of the complexity (or simplicity) of hunter-gatherer lives.\n\nMany descriptions of human prehistory assume that food domestication made way for the power and role differentials of civilizations to come into play. True, agriculture tipped the balance. However, just as chameleons change color as conditions demand, humans reconfigured their social lives\u2014transitioning from equality and sharing to deference to authority and hording, and from roaming to setting down roots\u2014as suited the situation. This can be hard to believe because we have come to think of hunter-gatherers much as they would have thought of us: as an alien life form. Yet human cognition nonetheless remains adjustable across all the social options that had once been available to hunter-gatherers.\n\n# GATHERING TOGETHER\n\nPeople can be content when right on top of each other, thanks in part to anonymous societies that allow us to tune out others. Most animals have very a narrow tolerance for their spacing from society-mates. Despite their need to stay with the troop, baboons keep a cautious distance from all but their closest allies, whereas army ants constantly stand practically on top of each other. Pinyon jays are more flexible, thriving in dense flocks part of the year and paired up on nests when rearing chicks. Paradoxically, even most fission-fusion animals don't take advantage of the full range of dispersal possibilities, since, in species other than humans, wolves, and elephants, the individuals only put up with a few others nearby at a time. When Goodall stocked her research station with enough bananas to keep her community fed, the chimpanzees didn't all settle at that spot but rather underwent a meltdown that drove them completely apart (a topic for later). For humans, that would be no problem. Even people who have until recently lived in hunter-gatherer bands can take to city life (though as with all people who grew up in a radically different society, the cultural shifts can often be arduous). Meanwhile, the tourist immersed in Grand Central's multitudes can board a train to Blue Mountain for a pleasant walk with barely a human in sight that same day.\n\nSome of the versatility of the human expression of fission-fusion was on display when hunter-gatherer bands converged in aggregations of as many as a few hundred. These occasional social gatherings were staged during a season when resources were bountiful enough to sustain everyone for a few weeks. During this time, people massed together much like monkeys in a troop, except lodged in one place. The chosen site could be a watering hole or other productive location. Each year some otherwise roving Andaman Islander societies rebuilt residences they kept near the ocean, and bands would rendezvous to fish for a couple of months. The pole-and-palm-frond structures reached well over ten meters in diameter and several meters in height, making them larger than my Brooklyn apartment. Inside, each family had a sleeping platform and fire. The communal domiciles were maintained for so many generations that each was associated with piles of refuse in some places thousands of years old and more than 150 meters in circumference and 10 meters high. For another example, every autumn in North America, bands of certain tribes gathered to frighten herds of bison over cliffs. No chimpanzee community enjoys each other's company this way, as one grand assembly; the apes parcel out their social engagements in smaller doses.\n\nIn some fundamentals, life during a band get-together was little changed from normal, with each band often keeping its status as a \"neighborhood\" by camping a bit apart. Yet the throngs were the reunions of the day, enlivened by gossiping, gifts, songs and dances, and, much as occurs when elephants aggregate, with males on the prowl for conquests. While the most common instances of socializing between bands were visits by lone travelers or families to those they knew personally, these assemblages were greased by the people's shared identity. Surely, this must have given gatherings a significance in buttressing group practices and behavior, if just by circumstance. I've seen no evidence decisions were made that affected the collective. Still, the activities must have conveyed a sense of common purpose.\n\nThe gatherings were a step in the direction of permanent settlements. But while they may have reaffirmed people's affiliations, this type of assembly, in which the itinerant hunter-gatherers simply stopped moving, was doomed for several reasons. Nearby sources of food would be cleaned out, offal and wastes piled up (a problem that the nomads managed poorly), and biting insects had a field day. Worst of all, with so many personalities right on top of each other, jealousies and resentments came to the fore. After all, what reunion doesn't reunite a few archenemies?\n\nIndeed, the foremost reason people have never scaled up the egalitarian lifestyle of band societies is that, like most mammals, we squabble a lot. Just as rock concerts can devolve into social entropy, large gatherings could end in brawls. Murders peaked at this time. Bands retired to their homelands, with some musical chairs as people joined a mate or friends living elsewhere. \"Free societies are societies in motion, and with motion comes friction,\" the novelist Salman Rushdie has said. And so it was literally for roving hunter-gatherers.\n\n# THE RAINY-DAY MENTALITY\n\nIn some cases bands developed practices that made their wild foods productive enough to approach the yields of agriculture. Even without sowing seed or domesticating the food, certain modes of extracting from the environment made staying put for a time easier. In 1835 the Australian surveyor-general Thomas Mitchell recorded a vista \"extending for miles\" on fire-cleared plains by the Darling River that had \"the agreeable semblance of a hayfield.\" The Wiradjuri people reaped the millet with stone knives and heaped it to dry in what was clearly a time-honored process. Despite claims about what constituted the Paleo diet, the grain was ground and baked into bread.\n\nIn some instances hunter-gatherers fiddled with nature to improve its productivity. The Ache of South America established plantations of _guchu_ beetle larvae, which mature to a hefty ten centimeters in dying Pindo palms, by felling and sectioning trees for the beetles to find. The meanderings of the bands had to be choreographed so everyone got back in time to harvest the plump insects. The delicacy was so relished that if the Ache had found a way to safely raise enough of them at one spot, my guess is they would have taken up residence right then and there as beetle farmers.\n\nTo drop the itinerant mode of life required bounty on an ongoing basis. That included any excess that could be stored to tide people over times of dearth. Among the insect societies operating this way are harvester ants that jointly build well-protected underground pantries, where they keep seeds fresh for months. However, most vertebrate societies fail to show delayed gratification by means of a group effort. For instance, each pinyon jay caches its own seeds and attacks any member of its flock who steals from it.\n\nDescribing the gathering practices of a Bushman society, the anthropologist Richard Lee explained, \"The !Kung do not amass a surplus, because they conceive of the environment itself as their storehouse.\" Nevertheless, cultural practices that yielded a lasting food supply could support long-term home bases. Both Bushmen and Hadza took a small step in that direction by drying meat (though not enough to keep them at a place long). Inuits did one better by putting seal carcasses on ice\u2014for them the outdoors represented one giant refrigerator. When the Western Shoshone split into families every autumn to enjoy pinyon nuts, they gathered the excess in baskets. During the winter doldrums they came together to consume them, turning a seasonal bust into a social and culinary boom.\n\nFew environments offer the opportunity to accumulate food in bulk from nature year after year. Even when the resources are present, the risk of committing to a site is great since the option to move elsewhere should things go awry can vanish when neighbors also decide to settle down. Examples include the Mount Eccles Aborigine tribes and the J\u014dmon of Japan, who lived most or all of the year in settlements. Some of these villages were small and simple, like those of the few New Guinea hunter-gatherers, who caught fish and ate the pith of wild sago palms on land too swampy to raise the pigs and yams favored by their horticultural neighbors. More elaborate were the settled hunter-gatherers of North America: the Calusa of southwestern Florida, the Chumash of the Southern California coast and Channel Islands, and, by far the best studied, the tribes of the Pacific Northwest, which ranged from the tall forests of Oregon north to the stunted landscapes of coastal Alaska. All these tribes mass-collected and stashed aquatic foods to keep themselves fed through thick or thin.\n\nThe Pacific Northwest was densely populated by hunter-gatherers when Europeans came to North America. Much as the Bushmen and Aborigines had never seen themselves as a group, so too, the peoples of this region had no term in their languages that encompassed everyone living a settled lifestyle and that distinguished them collectively from the roving bands living further inland. The reason for this semantic omission may be that among the settlements there were populations from varied Inuit and Indian backgrounds, each made up of multiple land-holding tribes that had taken to this way of life. Most lived off the ocean, but a few dependent on salmon found places to permanently congregate along rivers.\n\nSome sites in the Pacific Northwest were occupied for centuries by a couple hundred to nearly 2,000 individuals. The most developed of these were impressive indeed. Their longhouses and other residences could be enormous, the biggest on record extending 200 meters long and 15 meters wide. That's as spacious as the homes of many modern celebrities, though several families shared the building; a small settlement might occupy one longhouse, while a large village spanned several structures.\n\nThe societies were neatly set apart by markers of identity, more exuberantly so than for people living in small bands. The differences were striking and well documented in the Pacific Northwest. Most stunning were the labrets, ornamental plugs worn in perforations in the cheek or lower lip, ranging from ivory disks to multicolored beadworks. Labrets made their appearance 3,000 to 4,000 years ago and revealed much about the social and economic standing of the wearer, though their primary and original value was to connect people with a tribe. In the far north, for example, the Aleut tribes were distinguished as well by their tattoos, nose pins, necklaces, and the choice of animal fur for their parkas. Wherever foreigners might be encountered, whether at a friendly meeting or in battle, there was no marker more crucial for the Aleut to wear than a tribal hat that resembled the beak of a bird, painted with brightly colored decorative motifs.\n\n# LEADERSHIP\n\nMuch of the intricacy of settled life bears on handling the personal discord and logistical issues that routinely broke up groups of nomadic hunter-gatherers. That this was a problem for the nomads is supported by the observation that individual hunter-gatherer bands around the world, across habitats as divergent in the resources on offer as tundra and rainforest, kept their size to a few dozen individuals. Social failure, not some inescapable issue around hunting and gathering, would explain this regularity. The bands of some Bushmen societies, for example, grew unwieldy and dysfunctional every two or three generations.\n\nTo circumvent this dysfunction, people in settlements came to tolerate those with a knack for making decisions they could abide by. There are animals that show few differences among members in power and influence and do fine\u2014ants operate through a kind of collective intelligence. People aren't ants, for which markers are sufficient to keep a society together\u2014transcontinentally in the case of Argentine ants. Social conflicts have existed for all humans, band societies included. But where did the human proclivity to either lead or follow leaders come from? Certainly it reflects an upbringing in which we defer to parents who know more than us, and who expect us\u2014indeed, often coerce us\u2014to behave in a manner appropriate to our society and station, and yet it has come to extend throughout our lives. Today we are surrounded by authority figures from teacher and boss to sheriff, the president, and Congress. For societies to function today, people must know their places under different circumstances and behave accordingly, whether to lead or to follow.\n\nIn many vertebrates, a dominant animal can remind us of a leader. Dominance affects who interacts with whom, and how. Still, while expressions of power and control of resources can be important components of high status, they don't necessarily put the one at the top of the heap in charge. Among most species the \"top dog\" is likely to push others around without mobilizing anything significant for the group. Some alpha individuals have a certain influence: naked mole rat queens nudge and nip at the workers, supposedly to prod them to work. Following the most powerful, highest-ranked animal can give others safe cover if something goes wrong\u2014the alpha ringtailed lemur typically sets the path taken by her troop for the day. That's not the case in bands of horses, though, where the mares, and not the lead stallion, set the route.\n\nWhat people more typically think of as a leader, however, is someone who takes a substantial role in directing a society's affairs. The alpha male and female wolf do just that: in addition to setting the pack's direction, they get everyone moving and initiate hunts on prey and attacks on foreign wolves. In an elephant core the oldest animal earns her matriarch position not by pushing anyone around. The others defer to her in apparent recognition of her accumulated wisdom on such matters as which outsiders are friends. She also intervenes when tensions surface between her underlings and comforts the hurt parties after, something dominant chimpanzees with political savvy do as well. Even so, adroit negotiators in these species have limited impact. They cannot lay out long-term courses of action for the whole society the way monarchs and presidents do (albeit admittedly such leaders today rarely carry out such agendas without the approval of others).\n\nLeadership in this strict sense is rare in nature. Yet we have seen that even people don't require leaders. In hunter-gatherer bands everything from daily activities to long-term plans was a matter of discussion. When more than a few dozen people resided alongside each other for the long term, however, this egalitarian approach became untenable. In the first villages the easy nimbleness of moving away from pushy individuals or acting together to knock them down a peg was lost: at best one could walk to the far end of town.\n\nUnder these circumstances many factors played a role in determining who became a leader. Magnetic personalities garnered support but would come along rarely in a band society or small settlement. We don't see many John Kennedys step forward in nations of millions. Still, having even a modestly talented person in charge can offer advantages when acrimony is high. At such times the effort to be an authority figure pays off, as does willingness to bow to such a person. The two must go hand in hand for leadership to work out, with people putting aside disagreements to support an individual rising to the occasion. Then as now, humans would have gravitated to those who made themselves the center of attention and responded to issues quickly. Part of that faculty, first honed in band societies, was public oratory, a skill vital to the incipient stages of leadership. In what's called the babble effect, chatty people have always held a certain sway, though Bushmen youths, in their egalitarian bands, are said to have stifled the urge to follow them.\n\nSo while hunter-gatherer settlements didn't put together anything like a government, they did have influential people, if not clear leaders. As an example, the chiefs of the fisher-people around Mount Eccles were treated as nobility, able to declare war and claim the best spoils. The New World ruler closest to a king in stature was the Calusa chief, who kept the peace seated on a stool, an unassuming throne by current standards, in a building that, according to one historian, was \"said to hold 2,000 people without being crowded.\"\n\nThe Chumash and Pacific Northwest chiefs, although ostentatious, were less overtly assertive of their power. They had to act cautiously compared to the army-backed heads of large agricultural societies, relying less on coercion than persuasion and recompense, such as feasts to encourage people to meet obligations. Leaders have always been virtuosos at political maneuvering and at guarding their own interests. Yet the chiefs often presented themselves as exemplary members while displaying some of the humility, integrity, and steadfastness that were expected of people in a band. These are still admired qualities in leaders today, perhaps a holdover from egalitarian times. Indeed, by convincing people to work together, chiefs ensured that elements of the egalitarian mind-set stayed intact. Even then their sway was limited. In a never-ending push and pull between leader and followers, small settlements supported chiefs upon whom the people exerted control to some extent. Pacific Northwest chiefs sought the support of informal councils that had a say in the pedestrian aspects of village life. This was leadership by committee, with the committee taking up the role handled by the entire band in a nomadic society.\n\n# BACK AND FORTH BETWEEN WAYS OF LIFE\n\nThe troubles that make leaders necessary need not be internal to the society: some threats come not from discord between the members but from outsiders. History's most revered leaders\u2014George Washington, Abraham Lincoln, and Franklin Roosevelt are some American examples\u2014came along when citizens wanted someone they could trust in times rife with conflicts. This could be another path to the emergence of leaders among hunter-gatherers, as well. Early accounts by European settlers tell of hundreds of Aborigines in pitched battles that presumably took organization, and according to some descriptions these had strong headmen. Bushmen embraced leaders when making a united front against enemies, too. That was so for the =Au\/\/ei living in what is now Botswana (the = and \/\/ represent click sounds). The records concerning these Bushmen from the early decades of the nineteenth century recounted the =Au\/\/ei as seasonally occupying villages surrounded by defensive palisades. To obtain sufficient food while so entrenched, the people innovated methods for culling game by driving herds into pits that were a maze of yawning chasms. The =Au\/\/ei of that time were warriors prone to blood vengeance who burned wagons, stole cattle, and collected tribute from other Bushmen.\n\nThat a society adapted to leaders and village life didn't mean the change was permanent. During the final decades of the nineteenth century, =Au\/\/ei reverted to roving, leaderless bands. By 1921, headmen had reemerged with the reduced role of orchestrating armed clashes, this time without the people settling down. The leaders gained influence during periods of encroachment by armed tribes such as the cattle-rearing Oorlam. When the incentives to fight diminished, the leaders fell out of favor and the people reverted back to an egalitarian band life. One might assume it would be their _original_ way of life, except there is no telling how far back in time the requirement for self-defense might have arisen. The dynamic =Au\/\/ei culture suggests that changes in social organization come and go, leaving hardly a trace.\n\nFamously, nomadic American Indians had leaders, too: senior individuals whose roles continue to the present day. It is impossible to say how often those hunter-gatherer tribes, with their portable tepees and wigwams, were led by chiefs prior to the arrival of Europeans, who brought with them horses and guns. What's clear is that when dangers were ongoing, nomadic societies embraced greater social complexity. Among the Plains Indians, warrior associations\u2014the West Points of the day\u2014put select men through rigorous training to prepare for combat.\n\nAt the other end of the gamut from the =Au\/\/ei, who spent most of their time as nomads, hunter-gatherers widely thought of as fully settled were just as malleable. Even the Pacific Northwest Indians, the gold standard for settled peoples who foraged from nature, didn't always stay put. Villages either moved or disbanded as suited them. Some longhouses were seasonal residences, with families traveling by freight canoe to homes elsewhere. There's evidence of temporary camps, too, suggesting people would go out to hunt game, perhaps like sportsmen pitching tents today. Probably few hunter-gatherers lived at the absolute extremes, as perennial homebodies or incessant wanderers.\n\n# LIVING WITH DIFFERENCES\n\nOnce people had settled down for long enough, with property no longer restricted to the items each family could haul around, possibilities opened up for expanding technologies. The revolution in social order that ensued affected the very structure of day-to-day existence.\n\nMany of the technologies generated by settlements improved food production. The Pacific Northwest Indians, for example, built massive seaworthy canoes, equipment to bulk-process fish, and watertight containers for prolonged storage of dried seafood. The range of devices diversified until fish could be caught in a number of styles of nets, or clubbed, speared, harpooned, brought in on a line, or trapped in weirs. The wide assortment of intricate gear permitted a few to collect much of the food for all. That was important given that the concentration of bodies in one place often made it impractical for everyone to fan out and procure food individually. Bringing home the bacon\u2014or salmon\u2014became one of many social duties to benefit from specialized know-how.\n\nA person's interests and talents, say, in knapping arrows or weaving cloth, therefore found a more positive and open reception than they had in bands. In the Pacific Northwest some occupations became hereditary, with parents passing on their specific skills. The growing need for expertise meant the mechanics of every aspect of the society's overall functioning was no longer within anyone's grasp\u2014which also meant that the sum total of the knowledge in the society was expanding. Just as staying in one place assured that social complexity stopped being limited by the physical baggage members could carry on their backs, settling meant social complexity was also set free from the cultural baggage all must carry in their heads.\n\nSince the pioneering work of Emile Durkheim, sociologists have regarded specialization as a force for enhancing the cohesion, or solidarity, of societies. This it certainly is, though the cohesiveness he had in mind wasn't entirely restricted to people in settlements. The jacks-of-all-trades in a band society likewise relied on an exchange of goods that contributed to the people's solidarity. While trading among bands could occur between people blessed with different talents, the clear-cut jobs in many settled societies accentuated the effect such that people grew ever more interdependent, in a trend that continues right up to the present day.\n\nJob differences would also come to simplify interactions with little known others and total strangers as they grew more common in larger settlements. We can figure out how to behave toward someone merely by identifying the person's role\u2014based on the police uniform the person is wearing, for example\u2014without knowing anything else about him or her. Similarly, a worker ant interacts appropriately with a soldier of her colony, whether or not the two have been in contact before.\n\nMammals other than humans seldom take on special jobs other than caring for the young and certain temporary tasks like sentry duty. One could argue that the quality of life for chimpanzees and bonobos might be enhanced if one ape expert at finding nuts turned her nut supply over to another dexterous at cracking them open. Nothing about this is complex: many harvester ants have one type of worker that collects seeds and larger workers that chop them apart for all to eat. Yet the closest any vertebrate comes to this separation of tasks is the naked mole rat, whose relatively large societies support the ant-like division of labor into queen, kings, workers, and soldiers.\n\nIn our species, specialization informs not only how we interact with strangers and others based on their social positions, but also how we identify with other people in general, including their connections with all kinds of special groups. For nomadic hunter-gatherers this social differentiation within the society tended to be less important, beyond sex and age differences. The people of many Australian societies, however, belonged to groups known as skins and moieties. Children were assigned to a skin next in sequence from that of one of their parents, and to a moiety based on their connection to an ancestral being and ties to a plant or animal species, connections that determined how they socialized and whom they could marry.\n\nAmong settled hunter-gatherers, specialist groups would often proliferate over time: secret societies that offered ancient rituals and hidden truths, shaman societies, and others. A likely scenario is that this multifarious identification spun out of the primal affiliation to the society itself to create scores of collectives of lesser urgency, stature, and duration. It's curious how seldom other animals do the same within their societies, beyond the social networks of females that serve especially to raise and protect offspring in mammals from dolphins and hyenas to some primates. Nor does any one of these animals think of itself as being associated with similar-minded companions who, say, really relish fruit. Whether chimpanzees feel solidarity when on a hunt or battle together, as people do, is hard to say.\n\nGroupiness in human settlements would have reduced internal competition and divided social stimulation into chunks that were manageable and gratifying. The notion of optimal distinctiveness from psychology helps explain this. People have the most self-esteem when they achieve a balance in their sense of inclusion and uniqueness. That is, they wish to be similar enough to others to feel a part of their groups, but they also want to be different enough to be special. While membership in a large group may be held dear, it fails to fulfill a desire for specialness; this can motivate people to set themselves apart from the crowd by connecting to more exclusive groups. Nomadic hunter-gatherers had small enough societies that this was seldom an issue. Aside from membership in a few groups such as moieties and skins, everyone's quirks and personal social ties were sufficient for them to feel unique in a society of hundreds. There was no need to express differences by identifying with a job or club. In fact, it might have been discouraged. As settled societies grew larger, though, humans felt a greater urge to distinguish themselves. For the first time, the fact one would want to know about anyone was, _What is it you do?_ 42\n\n# FEELING SUPERIOR\n\nA second, perhaps more dramatic shift was the emergence of status differences between society members, above and beyond submission to a leader. People in bands and in some of the simplest villages considered meeting short-term needs of finding food and shelter a satisfying goal and thought of life as hard only when this minimum wasn't met. Compared to the mostly utilitarian items in a nomad's meager travel pack, largely interchangeable and readily loaned or exchanged, settled hunter-gatherers had a greater number and variety of things. While ownership was a fuzzy concept in a band society, settlements could in time concentrate resources in ways that individuals were able to control. Oftentimes, materialism took off extraordinarily. People not only possessed but inherited property, some of it unavailable to others. The Pacific Northwest Indians could inherit anything, even the right to sing a song or recount a tale.\n\nOwnership became a sign of status in an expanding hierarchy of wealth and influence. In what was the death knell of the ethic of everyday sharing enforced by band societies, chiefs in particular were self-aggrandizers. Most of the time, chiefs would solidify their standing by taking advantage of community productivity to horde part of the surplus for themselves. The cycle of disparity was continued after the chief passed his status and riches to his children, or someone else of his choosing (most chiefs were male). Wandering bands would be nonplussed by both the behavior and the objects, which had little to do with a chief's labor, other than that of managing debts owed him. Pacific Northwest chiefs put their economic clout on conspicuous display at ceremonial feasts called potlatches, calculated investments for political gain. There, the chiefs would dazzle by giving away or even destroying fine food and material goods that had taken years to accumulate, making such persons \"rich for what they dispensed and not for what they hoarded.\" So wrote the American anthropologist Morton Fried.\n\nPeople in settlements came to refocus their goals from maximizing their free time to earning power and esteem. In the Pacific Northwest, admired artisans, sought out for their masks, home decorations, and totem poles, were among the few who plied their trade full time and were compensated well enough to earn a rank just below the aristocrats.\n\nThe Pacific Northwest nobility, lacking the power to force their will on average citizens, had a fallback option to get jobs done: slavery. The slaves were captives or their descendants, part of the plunder taken in ambushes of other tribes. As tribes grew in intricacy, they adapted their markers accordingly, modifying them to display people's position in the hierarchy of prosperity and influence in addition to their tribal membership. Only slaves, who had no rights and were not considered members, wore no labrets.\n\nI am amazed how readily people who lived as equals in bands adjusted to inequality. People accommodated to slaves, aristocrats competed for positions of authority, and overbearing chiefs were deposed, and still there are no records of revolts among the general population of these societies. Perhaps this was because, even as the elite took disproportionate control of resources, everyone was protected and fed. In any case, the uprisings that might have materialized in hunter-gatherer bands to quell the power-hungry grew harder to arrange. It wasn't just that more individuals, with more opinions, were involved: support was also harder to rally. Smart chiefs could draw on the blessings of hangers-on and toadies as well as other elites\u2014a military officer and a priest backed the Calusa chief. A further factor no doubt was that once lives depend on status differences, a known property of human psychology comes into play: the disadvantaged see those on top as deserving their position. This tendency may have evolved from power hierarchies like those in chimpanzees, with our acclimation to differences in stature one of the ancient implements in our species' mental toolkit.\n\n# SETTLEMENTS AND POWER DIFFERENCES IN PREHISTORY\n\nBefore some point in our past, fission-fusion must have been the sole life choice for our ancestors. I say that because chimpanzees, bonobos, and humans have in common fission-fusion into small groups. The simplest explanation for this shared trait is that the ancestor from whom all three species sprung had this mode of life, too. With spreading out the only game in town, the nightly encampments of early proto-humans must have served as incubators for the social dexterity we would need to take up residence at one place. But that split between us and the other two apes occurred 6 million years ago or so\u2014long before our ancestors were anything one could call human. How far back in the human evolutionary lineage did settlements appear?\n\nPutting down roots, along with the concomitant development of the social features often connected with settlements such as leadership and inequality, has been portrayed as a watershed step for humanity. It's true the full potential of the settled life has been achieved since the development of agriculture. I see no reason, however, why these customs couldn't have sprung up, as the situation merited, with all the versatility shown by the =Au\/\/ei, from the dawn of humanity. The constant efforts expended by people in bands to keep on a level playing field suggests that egalitarianism wasn't the original condition of humankind but rather an option recently perfected. After all, even the affable bonobo is only somewhat egalitarian: while bonobos may not abide bullies, let alone leaders, they can be competitive and often at best tolerate many others in the community. Clashes over power and resources are a part of the human heritage we have always expressed. If people have been open to adopting the customs of specialization and status for a very long time, why then did so many hunter-gatherers in recent centuries live in egalitarian bands rather than in settlements? I expect that hunter-gatherers hiked less and settled more before agriculturalists laid claim to the world's choicest, and most fertile, real estate.\n\nThat Aborigines could make bread from wild grasses that looked, to the eyes of outsiders, like fields of cultivated grain shows that differentiating between settled hunter-gatherers and farmers is, essentially, nit-picking. Indeed, while anthropologists traditionally lump tribes like those of the Pacific Northwest together with band societies under the rubric of hunter-gatherer, it was the reliability of the local harvest, rather than whether food was domesticated, that mattered most. Shifts from nomadism to settlement and from hunting and gathering to agriculture were gradual. The hunter-gatherers of the Fertile Crescent stayed put in villages for centuries before gradually domesticating the sheep and wheat that they ate. Once people had agriculture, however, they could increase food production far beyond the yields from nature. Many settled hunter-gatherers were, from the perspective of population growth and the cultural extravagance that can accompany it, a civilizational dead end. Domesticating the aquatic life that nourished most of those societies was impractical, with no way to control its reproduction to feed expanding societies, or to allow those societies to spread far from their sources of wild food. By contrast many domesticated crops and livestock can be taken from their ancestral habitat by locating or creating environments to suit them. Shepherds found pasturage for their sheep and farmers terraced and irrigated the land until their communities lived everywhere. Not that agriculturists were always driven to scale up their production. North America was occupied in part by corn-planting farmers, but even after centuries those tribes were hardly more elaborate than those of Pacific Northwest hunter-gatherers.\n\nReliant as we are on domesticated food and biased toward a way of life in which anonymous societies have been amplified to massive scale, we are prone to think of our ancestors as progressing from simple to like us. Yet transitions from hunting and gathering to farming were not a given. Whether in villages or bands, few hunter-gatherers saw raising food as a move forward. \"You people go to all that trouble, working and planting seeds, but we don't have to do that. All these things are there for us,\" an Aboriginal woman was reported to have remarked to a white settler. \"We just have to go collect the food when it is ripe.\" A traveler to the Andaman Islands in the mid-twentieth century wrote that the natives there reacted similarly to the prospect of planting coconuts, asking why anyone should \"care for a tree for ten years to get its nuts when the island and seas around are teeming with food for the taking?\" So it was that the Chumash hunter-gatherers traded fish for the corn grown by their farming neighbors, not once taking up shovels and hoes themselves. Even the African Pygmies who lived for centuries as seasonal laborers for farmers and therefore understood agricultural techniques intimately never took to sowing seeds full time. As it would turn out, giving up hunting and gathering was no advance in quality of life. After the advent of farming, people grew smaller, weaker, and more sickly as they struggled to nurture and harvest crops\u2014conditions that wouldn't be reversed until the invention of the plow and harnessed oxen.\n\nTaking to cultivation at all but the smallest scale of simple gardening had another drawback that no early farmer could have predicted: it could ensnare a society in a plant trap. A trap, because the option of going back to hunting and gathering full time faded away once an expanding society committed to agriculture. Sure, hunter-gatherers such as the Lakota and Crow and some small South American tribes once farmed, but gave up on the practice. Yet once a society grew to a huge population, or was packed in tight with other agricultural societies, the numbers of people would be too great to be supported by native foods and starvation would be guaranteed.\n\nPre-agricultural settlements at concentrated and dependable food supplies were the original testing grounds for civilizations, their people having taken small steps toward nationhood in the modern political and organizational sense. There are excellent grounds, however, not to lose sight of the full variety of lifestyles that existed among hunter-gatherers, roving bands included. All the hunter-gatherers of recent history were like us\u2014fully evolved humans who exhibited a wide range of social possibilities we have largely abandoned but remain entirely capable of today. This is what makes both nomadic and settled hunter-gatherers relevant in understanding modern people. It is also why I will return to them often in presenting topics extending from psychology to international relations to the ways people structure their societies.\n\nThe continuing biological evolution of the human species notwithstanding, I see little reason to expect there were radical changes in our social potentials over long tracts of the distant past, either. Early peoples must have already evolved the versatility to spread out in small bands, or settle for stretches of time into enclaves, even if these were modest by today's standards. Both modes of life assembled around markers of identity, making for anonymous societies that were simpler than, but in the fundamentals little different from, what we have now. For answers to how our anonymous societies came to be, whether set in place or mobile, we must trace what likely happened much farther back in prehistory, before our days as hunter-gatherers in the mold of Bushmen or Aborigines, back when humans were neophytes on Earth.\n\n# SECTION IV\n\n# The Deep History of Human Anonymous Societies\n\n# CHAPTER 11\n\n# Pant-Hoots and Passwords\n\nThe gated community of Pinnacle Point lies at the edge of the resort town of Mossel Bay along South Africa's coastal garden route, its manicured golf course extending to a precipice above the Indian Ocean. Descending the flank of this bluff are wooden stairs custom-built by a local ostrich farmer, commissioned by archaeologist Curtis Marean of Arizona State University. Part of the way down, hidden by a tarp, is a surreal world. To the right behind the tarp is a shallow cavern where researchers sit at rickety tables, busy at laptops. Blocking a sunny ocean view to the left is a hillock of sediments, illuminated by floodlights and pin-cushioned with little orange flags. Other scientists slowly carve steps into this mound. Three surveyors manning high-tech instruments stand between the computer data experts and the excavators, every few seconds calling out, \"Shooting... taken!\" to hypnotic effect as they map the coordinates of each newfound item. Together, the archaeologists are working to uncover every trace left behind by the people who visited here off and on, spanning an era from 164,000 to 50,000 years ago. Most of the remains are simple artifacts made of minerals, stone, or shell, representing some of the best information we have about how our early ancestors lived. I visited Pinnacle Point curious about the findings. Was there evidence the humans of that remote time were cut from the same cloth as those of the modern era?\n\nBecause both we and our closest extant relatives, the chimpanzee and bonobo, live in societies, we can be reasonably confident the people of Pinnacle Point did, too. The evidence from this site and elsewhere, scattered and incomplete as it is, points further to the conclusion that those humans could have made the transition to anonymous societies deep in our species' past and provides strong clues as to how this may have happened. Ultimately however, the answer may lie not in the archaeological record but rather in what we said to one another, before we even learned to speak.\n\nFive to seven million years ago, our predecessors split off from an ape whose other descendants would evolve into the bonobo and chimpanzee. The successive descendants of these forebears were a varied lot. Human minds automatically simplify any such a profusion by arranging it in a line from primitive to complex. This linear thinking distorts the facts. At most points in the past several million years, multiple humanlike species prospered concurrently to form a bushy family tree. All but one of the branches came to a dead end, leaving us as the sole survivor.\n\nThe earliest of the species, the australopithecines and their progenitors, resembled other apes to the novice eye. Our genus, _Homo_ , originated about 2.8 million years ago. Some of these early humans, such as _Homo erectus_ , left Africa, and later species like the Neanderthal of Europe and Southwest Asia and the \"hobbit\" of Indonesia, _Homo floresiensis_ , evolved elsewhere. But like our earliest ancestors, _Homo sapiens_ is African in origin. Everywhere else in the world we are an invasive species, just as Argentine ants are invasive to California and Europe.\n\n# PUZZLING OUT THE PAST\n\nHow early _Homo sapiens_ lived and\u2014even more indecipherable from the evidence\u2014how they identified themselves and others presents an enigma almost beyond solution. Further obscuring our view is that for a good part of the period of residency of our species on this planet, including much of the timespan when humans camped at Pinnacle Point, _Homo sapiens_ was in trouble. The harsh arid climate over many centuries kept our numbers down. DNA data indicates that at one point just a few hundred people remained, fewer than many of today's endangered species. It is humbling to think how close we came to failure.\n\nThe archaeological evidence is all the more scarce because hunter-gatherers had no reason to overbuild objects that would withstand the ravages of time as we do today, by manufacturing bottles that will last an eternity for sodas gulped down in minutes. The survival of their art in deep caves makes plain that those places held a significance, probably spiritual, to the people. However, we only have those works, some up to 40,000 years old, because the conditions in the caves proved ideal for preservation. Luck can smile on the archaeologist: even chimpanzees create an archaeological record. The hammers they use to crack open seeds, identifiable as tools by their accumulation under trees where their surfaces were worn down from repeated bashing, have been traced back 4,300 years.\n\nArchaeologists must heavily depend on stone tools to understand early humans, too, but judging from recent hunter-gatherers, these would have been only a small part of their traveling kits. Most of the artifacts they left behind have vanished. Among the Australian Aborigines, for example, paintings drawn on the desert floor with colored sands were gone with the next wind. Similarly, the twigs, teeth, bone, thorns, and foliage employed in ceremonies would either have rotted away or have become indistinguishable from other detritus. So while we can surmise the people's favorite spots to bed down at Pinnacle Point, we don't expect relics of their bedding, and we can only guess as to whether they made baskets or wove cloth. Many of the crucial attributes of the societies, notably how people managed their social relations, leave little, if any, trace.\n\nEven settlements in prehistory would be hard to detect if early homesteads were rudimentary, or made of ephemeral materials. The stone walls of the canals and homes near Mount Eccles have been mainly reduced to rubble, in spite of the fact that some were occupied a mere two centuries ago. However, vestiges of huts a few tens of thousands of years old, which look to be suitable for long-term stays, have been unearthed in Europe. Structures possibly dating back hundreds of thousands of years, well before humans evolved to their present form, are reported at sites such as Terra Amata, France. Some claim these are remnants of edifices made of tree branches braced with stones. If this is true, those dwellings would have been big enough to house many people.\n\nThe most recent revolution for hunter-gatherers wouldn't have been a settled life, but the creation of artifacts that persisted long enough for archaeologists to find evidence for such settlements, or at least for places people would often gather. The earliest known enduring piece of architecture, and a monumental one at that, is G\u00f6bekli Tepe, on a ridge in the southeastern Anatolia region of Turkey. Construction began there at least 11,000 years ago, before any plants or livestock had been domesticated. Proclaimed by one archaeologist as \"a cathedral on a hill,\" G\u00f6bekli Tepe is the oldest known religious site. Arranged in circles on a slope are T-shaped limestone monoliths, three meters tall and up to seven tons in mass, engraved with stylized animals: spiders, lions, birds, snakes, and other mostly dangerous species, all made improbably with simple flint tools. The antelope that frequent the area must have drawn hunter-gatherers to this corner of the countryside from midsummer to autumn. Archaeological explorations have uncovered evidence that G\u00f6bekli Tepe was a center for feasts featuring the earliest known bread and beer made from grain reaped from wild grasses. To create such a formidable structure, the site's builders must have lived nearby for all or part of the year. Though similarly ancient dwellings have yet to be found nearby, other researchers working a few hundred kilometers to the south have unearthed substantial homes and elaborate headdresses for hunter-gatherers going back 14,500 years, well before G\u00f6bekli Tepe. The Natufian settlements prove that village life, and the disparities in status associated with it, arose long before people succeeded at domesticating a thing. Other substantiation of differences in wealth indicative of village life includes elaborate burials near Moscow going back 30 millennia; the clothing on the bodies is adorned with thousands of ivory beads, which would have taken years to produce.\n\nThe scarcity of still earlier archaeological finds has been used to assert that Stone Age people never settled down, and, moreover, they had little or no art, music, or rituals, no complex weapons, nets, traps, or boats; perhaps they couldn't even speak. Some claim that humans evolved the brainpower for abstract thought and complex reasoning only recently. That would seem to make the first _Homo sapiens_ little more than bumbling cartoon characters.\n\nThere was more to them than that. The Pinnacle Point team has excavated a considerable quantity of cultural material over the past 15 years. During my two-day visit, they dug up shells that were remnants of a meal, a hearth on which those mollusks were cooked, bits of a red pigment called ocher, and a quartzite blade the size of a pocketknife. Throughout the strata are bladelets made by heat-treating silcrete rock, so toothpick thin that their only function would have been as spearheads or arrow darts.\n\nSimple stuff, perhaps, but we can at least say the coastal visitors had a sense of esthetics. As early as 110,000 years ago they started bringing helmet shells and dog cockles to the cave, pretty objects scrounged from the shoreline by individuals reminiscent of beachcombers today.\n\nThe archaeological record indicates that life improved for _Homo sapiens_ over those early ages, and remarkably so in the past 40,000 or 50,000 years. The number and sophistication of artifacts increased, among them the outstanding cave paintings of Lascaux\u2014of which Picasso reportedly said, \"We have learned nothing.\" Additionally, tools became better designed and more diverse, and also left a greater imprint in the archaeological record.\n\nIndeed, many of the implements carried by modern hunter-gatherers can likely be traced back to these times. In the past decade, an array of artifacts that recent Bushmen would have considered necessities of life have been unearthed by archaeologists in a South African cave where they had been buried for 44,000 years. Among the preserved items are sticks used to dig up beetle larvae and tubers, bone awls, wood segments notched for counting, beads made of shell or ostrich egg, arrowheads (at least one of which was decorated with ocher), resin for attaching arrowheads to the arrow shaft, and an applicator to apply poison to the tips of the arrows. Any Bushman from the past few centuries would know these objects very well.\n\nAt the same time that recent Bushmen would have recognized those items, they doubtlessly would have seen them as somewhat foreign, just as they did the tools made by a different Bushmen society living in their own time. In fact, the ancient material goods so resemble those of the Bushmen of a century ago that it seems logical to conclude that their owners would have likewise made them a touch differently from place to place, and just as quickly come to associate those variations with the particular Bushmen societies of their day.\n\nThe Pinnacle Point artifacts offer plenty of hints that early people might have taken an interest in the stylistic flourishes commonly associated with more recent human societies. Starting 160,000 years ago, ocher (iron oxide) was carried into the caverns to char in fires. Heating is an almost certain sign the intent was adornment: it turns ocher the vibrant red of blood. Hunter-gatherers the world over, including North American Indians like the Chumash, painted their bodies with ocher designs that bespoke their identity. Many Africans, Bushmen included, still do. One hundred kilometers down the coast from Pinnacle Point, researchers have excavated a 100,000-year-old ocher workshop complete with grindstones, hammerstones, and pigment stored in abalone shells. Also in Blombos cave were 71,000-year-old hunks of ocher scratched with geometric patterns and snail shells with holes drilled in them so they could be strung like beads.\n\nNonetheless, the artifacts and living conditions at Pinnacle Point and Blombos might have seemed abysmally primitive even to a Bushman of 44,000 years ago. So stark are the differences between those periods of prehistory that many anthropologists have argued that the clear cultural changes that unfolded for humans starting between 40,000 and 50,000 years ago must have been predicated on a substantial, and rather sudden, evolutionary transmutation. This claim is implausible. It makes little sense to believe that cognitively \"modern humans\" sprang into being during that 10,000-year interval, long after the emergence of our species. This idea is equivalent to thinking that because the people of the early Industrial Revolution led squalid and unsophisticated lives compared to those of us in the modern era, the mental endowments of humans from the eighteenth century were fundamentally inferior to ours. Two scientists put the matter succinctly: \"The archaeological record tells us only what people did in the past, not what they were capable of doing.\"\n\nPointing to scratched bones and shell necklaces as Stone Age versions of the national flag, archaeologist Lyn Wadley has proclaimed that humans became behaviorally modern the moment they committed to storing abstract information outside their brains. The problem is of course that we can never prove what our ancestors thought about ocher designs or styles of beads and arrowheads. Were these informative to them or the equivalent of aimless doodles? Still, if a certain nonutilitarian item turns up repeatedly in the Pinnacle Point caverns, such as a specific kind of shell, it may have been valued as a marker: significant things tend to reappear, the way cats do in Ancient Egyptian art. Even so, for an artifact to designate a society, it wouldn't just reoccur; it would be specific to a group. Too few archaeological sites exist to resolve the pattern for early people. The best we can do is infer that a certain object could have served as a marker because it did for recent hunter-gatherers.\n\nThat the first Bushmen likely had societies differentiated by markers opens up the possibility that anonymous societies could have an older genesis, perhaps extending to the birth of _Homo sapiens_ or even earlier humans. A research team in Kenya's Eastern Rift Valley has uncovered evidence going back 320,000 years of complex technologies and what could be interpreted as symbolic behavior. The archaeologists propose that the obsidian tools and ocher showing signs of grinding to make dye found there might have been used by the people (possibly _Homo sapiens_ ) to decorate each other and mark out group identities. That these materials were precious to their owners is clear. The hunks of the ocher and obsidian had been hauled to the site, the obsidian coming from locations up to 91 kilometers away.\n\nCaves would have been far from the first and only places our forebears expressed their identities. I have no doubt the people at Pinnacle Point embellished trees with carvings and boulders with drawings long weathered away to proclaim _this is ours._ Since the birth of _Homo sapiens,_ Africa may well have been awash with signs of societies hardly less plentiful than the flags that flutter across continents today, whether placed to warn, celebrate, or express a reverence for the land. The reason for my confidence is the ease with which humans could have evolved to depend on markers, as studies of other primates suggest.\n\n# EVOLVING MARKERS\n\nCultures and the symbols attached to them now saturate human societies to such a degree that it's hard to imagine the earliest people could have had societies without them. But symbolic cultures and large populations aren't compulsory for anonymous societies to exist. The identities of ant colonies, no matter what their size, are laid out as chemistry, plain and simple, without symbolic information (that we know of). Their communal odor need only distinguish _us_ from _them_ (with things getting a little more complex for ants that distinguish between enemy colonies). The same goes for pinyon jays with their kaws, sperm whales with their clicks, and naked mole rats with their scent.\n\nThe simple markers of the first human societies needn't have conveyed anything abstract like patriotism or the people's connection to the past. Such qualities could have been added later. Once we stop assuming that markers must carry profound meanings, the dawn of anonymous human societies becomes easy to envision.\n\nWhat the first markers may have afforded was a reduced likelihood of error about who goes where. Every society dweller faces risk from misidentification. Errors can go both ways. They may involve confusing a potentially dangerous foreigner for a member and being attacked for it, or concluding, perhaps in a pinch, that a member doesn't belong and mistakenly going on the attack. Both blunders can be avoided if the individual in question gives a clear indication that he or she is not a threat.\n\nUnderlying the development of such a marker might have been the drive to match the behavior of others in one's group. Our early ancestors, like many animal species, would have excelled at social learning. This talent can generate culture, that is, socially transmitted information taken in aggregate, including traditions such as those of meerkat clans that sleep in later than their neighbors, or dolphins and whales that pass on tactics for fishing to their offspring. People learn a Pledge of Allegiance in some parts of the world and become skilled with chopsticks in other places.\n\nCopying isn't restricted to species that are social or brainy\u2014wood crickets menaced by spiders learn to hide by watching experienced crickets. Still, social learning can be of supreme importance in species with societies. Monkeys raised in a troop trained to prefer corn dyed pink to corn dyed blue will shift to blue if they change membership to a troop that prefers it, one study showed, even when both colors are readily accessible. The fact that youngsters duplicate their elders results in variations among chimpanzee communities in how the apes employ stones to crack nuts, fish for termites with modified twigs, use chewed leaves to sponge up water that's just out of reach, or hold each other while grooming.\n\nCompared with human conventions, however, variations in chimpanzee cultures are simple and few, and as far as anyone can tell, they don't matter for social acceptance. Chimps don't register leaf-sponging techniques to keep track of who belongs where; nor do dolphins monitor differences in fishing strategies. There's nothing to suggest that a chimp deviating from the local tradition of, say, holding hands while grooming is noticed by its companions, let alone avoided, corrected, scolded, or killed. Beyond sometimes ostracizing a disabled individual, chimps don't look at unfamiliar behaviors with alarm, which is to say they haven't taken the step of perceiving their differences as markers. The female who transfers to a new community apparently adjusts to local habits without suffering consequences for any bad manners she unwittingly exhibits. The community comes to adapt to her presence as an individual; this acceptance isn't based on whether she assimilates to their ways.\n\nThat's true except for one possible, and very intriguing, exception: the loud call chimpanzees make to keep in touch with each other, the pant-hoot.\n\nIn certain species even loose and temporary groups adopt the same vocal signal to serve the needs of the moment: birds, for example, can match each other's calls during the time they spend together. Sometimes this vocal matching may approximate the sort of mirroring people do when they synchronize with each other's speech patterns and mannerisms as a sign of sharing common ground\u2014even a monkey prefers humans who adopt its behavior. For chimpanzees, not only do the animals distinguish the voices of individuals\u2014just as monkeys can, and indeed, just as people know whether Tom, Dick, or Jane is speaking\u2014but they also fine-tune how they give their pant-hoot by listening to others, until the exact same pant-hoot sound is shared across a community. Indeed, the pant-hoot \"accent\" turns into an indelible part of each community's repertoire.\n\nEven though the pant-hoot differs from one community to the next, the apes don't seem to use this accent to identify whether individual chimpanzees are part of their community the way we detect a person who isn't from here by his or her dialect. Instead the pant-hoots are thought to serve primarily as what I call a group-coordination signal, to help assemble and mobilize the society members (as vocalizations are used in most bird flocks) and to monitor the locations of chimps from other communities. This is a common function of vocal cries in species where society members claim ownership of a site or coordinate behavior at a distance. As an example, greater spear-nosed bats employ a screech specific to each roosting group, which guides companions away from foreign bats to the fruit-bearing trees of their territory. Once there they continue screeching, one assumes, to assert their property rights.\n\nChimpanzees know the pant-hoots of neighbors well enough to anticipate the consequences of an encounter. Pant-hoots from their own community elicit a pant-hoot in response. Those from a foreign community well known to them provoke an advance\u2014even an attack should the listeners judge from the sound that they have a numerical advantage. Unfamiliar pant-hoots more often initiate cautious retreat: nothing distresses the apes more than a community of chimps previously unknown to them. PET scans show a response by the chimp brain that intrigues, too. In a pattern yet unexplained, pant-hoots differ from other chimp calls in that they fail to activate the posterior temporal lobe. Since this region of the brain is associated with emotions, foreign groups, perhaps, get a chilly reception.\n\nA group-coordination signal is a marker of a sort, one used to coordinate activities and lay claim to space, rather than necessarily to discriminate individual members from outsiders. We have them, too. Nowadays national flags and monuments act as effectively as the vocalizations of apes or scents deposited by wolves or ants to indicate the domain of each society. Converting a group-coordination signal such as a pant-hoot into a marker for recognizing fellow members is a cinch, however, and could have been driven by the simplest of reasons. By a slight customization in how such a call was given and received, the members of a society would be able to confirm whether an individual they had trouble identifying was in fact one of them. The pant-hoot\u2014or anything like it that could be presented as needed to make one's group identity known\u2014would become a proof of membership, a simple marker serving as a password.\n\n# THE PASSWORD\n\nThat early human markers were passwords is a hypothesis, albeit a simple one. Our other close relation, the bonobo, has a vocalization called the high hoot that each community seems to employ much the way chimps use the pant-hoot. That makes it reasonable to think our common ancestor with those apes had a similar call that could be adapted as I described. People seldom depend anymore on passwords to recognize each other, though the soldier approaching his platoon in wartime is wise to give a sign of reassurance he's friendly. Along a similar vein, a Yanomami tribesman returning to their village in South America's Orinoco basin rainforest will cry out the word \"friend.\"\n\nIn fact, the experts may be underestimating the chimpanzee in this regard. Though chimps don't pant-hoot as a regular greeting upon meeting each other, the call could still act as a makeshift indicator of community identity. Primatologist Andrew Marshall told me about a chimp who failed to pronounce the pant-hoot of his zoo community. This social misfit was barred from grooming sessions and wasn't permitted around food until everyone else had had their fill. Eventually the other males drove him into a moat, where he drowned. It's impossible to prove that his odd call was the reason for his ill treatment, but the idea rings true.\n\nThe initial human password could have been marks on the body made by ocher, equivalent to wearing a national flag. Since marks can easily be missed, I think a sound broadcasted like a pant-hoot was more likely, however. In that case dialects would have existed before words. Language, prior to the emergence of writing, is another trait that leaves no trace in the archaeological record. No one can be certain when our species began to speak, yet it may be that a password had to evolve first to support the profound levels of trust needed before productive conversations could be had. Mastering the sound would have become imperative in childhood, more so than copying such behaviors as seed-cracking techniques had been before. It would have been essential for newcomers to the society, as well, although both the young and immigrants would have had to be cut some slack until they got the password right.\n\nMy conjecture is that the first passwords materialized as our ancestors began moving from the forests out onto the African savannas. Judging by their digestive tracts, teeth, and tools, our progenitors ate more meat than chimpanzees and bonobos, and hunting or scavenging savanna game animals would have required roaming vast tracks of land. While chimps might see some reclusive community members only occasionally, for people in the most far-flung band societies, years might have gone by before they happened upon members of the most distant band. Even if those individuals had met before, lapses in recognition as time passed, owing to both changes in physical appearances and fading memories, could have led to precarious moments of uncertainty. A dependable password would not just avoid this dilemma but, in time, would allow for members who had never met and, perhaps, were completely unknown to each other.\n\nFor animals relying on individual recall, we would expect that movements across extremely wide expanses might cause a society to come apart, because long-separated individuals would have forgotten each other, if they weren't complete strangers. Human societies avoided this fate when the members could depend on a password to distinguish both acquaintances and unknown fellow members from foreigners.\n\nBecause markers permit individuals to forget others while retaining their societal connection to them, and even to be comfortable around outright strangers, they allowed societies to expand not only in space but in population. Chimpanzee communities top out at a couple hundred, those of bonobos at somewhat less. There's suggestive evidence that the societies of our ancestors pushed beyond these sizes very early on. Based on estimates of brain volumes, two anthropologists concluded that the social networks of _Homo erectus_ and early _Homo sapiens_ individuals were already bigger than those of chimps. This would indicate their societies were already well in excess of 200. Another study based on archaeological evidence projected that societies had already reached populations of several hundred individuals by the time the genus _Homo_ emerged 2.8 million years ago\u2014when the human diet had first begun to include more flesh. Both results suggest markers are a venerable part of our heritage, having come into play long before the first complex artifacts, like cave paintings, found their way to the fossil record.\n\nThere's no way to know if our ancestors' societies could already include members who were unknown to each other early on, when markers were limited to a simple password, or if being around unfamiliar others became possible only after markers grew in sophistication and diversity. Whichever was the case, from that point on early humans were freed from the requirement that they be acquainted with other members, regardless of how often they came across each other or how far apart they called home. With people at ease for the first time with complete strangers, I would describe our species as \"released from familiarity.\"\n\n# LIVING BILLBOARDS\n\nA password, given as needed, would have been a crucial development in human evolution, one that has been previously overlooked. Societies of a few hundred, though huge by the standards of most mammals, are absurdly tiny compared to those of today. At the time, however, they would have been a breakthrough as our forebears faced ever more imposing rival societies of their own kind\u2014as similarly large band societies have done up to the present era. Nevertheless, to truly secure a society's membership, what had been one signal had to evolve into an entire system.\n\nA problem with having a single marker is that it leaves societies vulnerable to cheating or mistakes that might enable outsiders to tiptoe in\u2014or perhaps even confound the memberships of entire societies. Nature is rife with examples of things going awry when species depend on signals that are too easy or too few; think of the spider that freely enters a nest of hostile ants after dousing its body with the ant's password, the colony odor. Still, I don't imagine a human society has ever taken over a foreign competitor by the stealthy copying of its identity. What would have made such a move impossible is that our initial passwords were eclipsed by more and more varied and inimitable markers\u2014intricate rites, for example. Cracking the elixir of molecules comprising an ant's scent is far simpler than faking the mixture of markers people register about each other. So while no marker needs to be costly, over the passage of evolutionary time, certain markers, and definitely a society's entire package of markers, grew exceedingly hard to duplicate. It came to be, then, that even without shouting the word \"friend,\" people could make their identity quickly obvious and unimpeachable. Even if one individual misses the telltale signs that a person doesn't belong, someone else will catch the error. This reliance on the collective to pick out intruders is found also in insects: a foreign ant that makes it by the first nest entry guard will be detected by one of the sentries it passes next.\n\nUniversal among people, and probably among the first markers to follow passwords other than perhaps simple gestures, are those on the body. Humans evolved to be living billboards for their identities. Our naked skin and head hair, anatomical features that distinguish us from other primates, represented an early canvas for expressing ourselves and learning about others at a glance, both in terms of who they are as persons and as society members.\n\nIn Gabon, I once showed a book of pictures of various African tribes to a crowd that had gathered, voices abuzz. The images drew curious stares and some heated exchanges; fingers landed on jewelry, plumed hats, and any other detail that struck the locals as bizarre. The bits of the photographs that got most smudged, however, were the unfamiliar (and thus patently absurd) hairstyles. Why did humans evolve mops of hair amenable not merely to the grooming we see among other primates, with their short fur, but also to being elaborately trimmed and sculpted, if not because that styling can be accomplished in culturally distinct ways?\n\nHair has had to be styled through every age of humanity if just to keep it out of the eyes. In ancient China, unmanaged hair \"invariably denotes figures outside the human community: barbarians, madmen, ghosts, and immortals,\" according to one historian. The terra-cotta warriors in the tomb of Qin Shi Huang, the founding emperor of the Qin dynasty who died in 210 BC, are precisely rendered with coifs connoting their ethnic origins. Among the Kayapo of Brazil, one anthropologist has observed, \"each people has its own distinctive hairstyle, which stands as the emblem of its own culture and social community (and as such, in its own eyes, for the highest level of sociality to have been attained by humanity).\" Certain North American Indians cut their hair short while others let it grow to the ground; some had a row of bangs across their brow; others shaved the crown (or, notably the Mohawks, the sides) of their heads. Hair could be parted or braided various ways or wrapped in beaver pelts or formed into a hornlike lock. Even people like the Bushmen, whose kinky hair never grows long, take the time to pretty it up.\n\nThe bare skin of our species is no less suitable to flaunting our identities. Darwin thought that bare bodies evolved to make females sexier, an idea a male chimpanzee might refute. What hairlessness did give people was a surface on which they can etch, sketch, paint, perforate, tattoo, or swath to define who they are. Personal customization may have started with ocher crayons prepared on a fire to generate differences in style that anthropologist Sergei Kan aptly describes as transforming the natural skin into a social one. Outsiders so dislike the facial tattoos of women in northern Myanmar that marriage to their own tribe is all but assured.\n\nThe Iceman, a 5,300-year-old mummy discovered in the mountains between Austria and Italy in 1991, bore 14 sets of tattoos made by poking soot into cuts on his back and ankles. Our bodies would have been available as billboards long before the Iceman died. How far into the past our unruly beards and mops of hair go no one can say, but human skin has been all but hairless for 1.2 million years, taking us back to the early days of _Homo erectus_.\n\nThe cartoon caveman (usually a man) of everyday imagination cared not a whit for his appearance. Rating ourselves above nature, we imagine this brute to be crude, unkempt, and unwashed. This persistent stereotype, which can extend to contemporary people of other cultures, expresses belief more than reality. Social primates groom each other for hours\u2014scruffiness is a sign of bad health. Tailoring hair is the human version of their primping and cleaning, with the motive not so much the bonding of groomer to groomed but for the groomed person to consummately broadcast his or her identity. Before vanity mirrors, we would have been completely dependent on others to help us with our appearance, notably to manage \"the bits around the back,\" as primatologist Alison Jolly once put it. A caveman wouldn't have been merely tidy, but elegant. As Jolly notes, \"Figurines of 25,000 years ago show women naked with beautiful braids.\"\n\nIn recent millennia tinkering around with our skin and hair have become just part of our fetish to turn ourselves into walking advertisements. Around the world the signage on our bodies would extend from head to toe with deformations of the skull; lengthening of the neck, earlobes, or lips; chiseling of the teeth; decoration of the nails; and shrinking of the feet. The earliest rationale for clothing wouldn't have been modesty but the further embellishment of the human figure\u2014one more way to own our identity.\n\nWhat may have begun as occasional announcements of identity would have multiplied into an ensemble of markers integral to peoples' moment-by-moment experience of others. Their reliable presence eliminated the constant need to keep track of who was who. This ongoing broadcasting of identity would have become mandatory. As societies grew sufficiently populous and dispersed, humans needed continual reassurance about everyone's attachment to the group. Society-specific routines would no longer have been copied at a whim but enforced: a person who behaved or was clothed unacceptably now would shock even his or her friends. Comparisons with outsiders would have been part of the equation. Though not everything foreign had to be maligned\u2014one society could covet another's goods, for example\u2014a sign of allegiance copied from a foreigner might be enough to get a person banished.\n\nThe ability to conceive of fellow society members in the abstract allowed people to relax around anyone in a crowd who fit their expectations without taking the slightest notice of them as individuals. It could be tough going to keep a marriage partner or significant friend happy if you had to mull over each unfamiliar face that passed by.\n\n# CULTURAL RATCHETING\n\nThe nitty-gritty details about our social lives became a core part of our suite of markers. Humans came to have cultures: rich and complex systems that we teach each other and that differ from one society to another. Cultures went far beyond serving to identify where people belonged; they kept their members safe and fed and in general held deep significance for those members in a way no chimpanzee would recognize.\n\nAll kinds of social traits eventually emerged as an outcome of our ancestors' creation, modification, and diversification of cultural behaviors. By this means people incrementally innovate and improve on everything they do, a unique human trait called cultural ratcheting. Ratcheting ramped up in the last 50,000 years and especially in the past 10,000 until it went on not just from century to century but from year to year. Some refinements came to be shared across society boundaries, like the trendy cell phone models of today. Along the way, people turned anything they like into a marker for their identities to create a nexus of emblematic complexity next to impossible for outsiders to replicate.\n\nThough we can't imagine a world without this perennial novelty, much of it is a product of modern consumer capitalism. Such constant tinkering is not obligatory to human life. The sparse archaeological footprint of early societies suggests that innovations were exceedingly rare for most of the span of human existence, with minor variants yet few steps forward. The experiences of 99 percent of the human generations that have ever lived were pretty much the same as those of their parents and the parents before them, their societies varying by shifts too slight to register in the dust of ages. Case in point, the ratcheting evident at Pinnacle Point was only a hairsbreadth above the zero rate of change for the nut-cracking chimpanzee, which has used similar rocks as hammers for thousands of years.\n\nEarly humans may have innovated little because they were few. Cultural elaboration takes an ample population. Despite my earlier assertions about the self-reliance of hunter-gatherers, recollection of how to do things wasn't _entirely_ stored in each person's head. We constantly remind each other about everything imaginable. The responsibility for remembering is distributed across everyone\u2014call it a collective memory. Lacking books or Internet, our ancestors relied on each other. The more they communicated, the less they forgot, reducing the onus on each person to know how to accomplish each task down to the last detail. Human learning is imperfect and skills can deteriorate over time. With enough people in contact, however, collective memories can extend widely and efficiently, not just from band to band but across societies that learn from their neighbors.\n\nCollective memory wouldn't have gotten us far earlier than 50,000 years ago, when humans were scarce. In the way that job specialization can endanger a small society, overdependence on the knowledge of others has its risks when people are few. Basic survival skills could disappear by sheer bad luck. This is named the Tasmanian effect. Many anthropologists think the Aborigines there forgot such skills as fire-making and fishing after being isolated 8,000 years ago by rising oceanic waters that made Tasmania an island.\n\nNeanderthals suffered from low densities, too. Their brains were heftier than ours, yet their societies simpler. This simplicity has often been taken as stupidity, but because the game they hunted in the harsh north supported a very sparse population, Neanderthals might simply have been stuck in the same sort of rut as early _Homo sapiens_.\n\nAnother impediment to progress when people were few may have been the echo principle, named to convey how the past echoes through the ages. Durable goods can be discarded but never entirely forgotten. Evidence of prior generations is everywhere, extending people's collective memory into the past. A figurine or axe wedged in the soil was almost as likely to have been made millennia ago as yesterday. Never losing sight of their predecessors' designs, early humans could have kept returning to them.\n\nWith the goods of populous modern civilizations both swamping and outshining the castoffs of distant eras, this practice is almost lost on us. Arrowheads were common on the ground in Colorado when I was a kid, and yet even though we snapped them up as souvenirs, we had little motivation to figure out how to make them. But for early Africans, and even later for the first humans to colonize Europe, who got by at low densities too, the echo principle may have been a lifesaver. By studying the scraps of past ages, talents such as crafting tools and figurines didn't indefinitely vanish from their lives. This may explain how many artifacts were replicated with so little variation over immense spans of time.\n\nRather than a fundamental brain upgrade, a more plausible explanation for the proliferation of material goods 40,000 to 50,000 years ago was a surge in populations at that time. The population explosion was attributable to a congenial climate across Africa and the spread of humans throughout the Old World, then or slightly earlier. The resultant boost to the collective memory caused both practical technologies and other aspects of identity to burgeon. More than ever before, humans would also have come into contact with foreigners. Struggles to keep themselves straight from those outsiders would explain why variation in objects likely to express memberships, such as in necklaces and art, took off at this time. Such markers can spread fast. Anthropologist Martin Wobst reasons that tailoring objects to convey the message _we made this_ would start a chain reaction. Once one society added, say, a motif to its pots, the unadorned wares elsewhere instantly signaled that their makers _didn't_ belong. As a result, people responded by innovating alternative styles that put the spotlight on their own identities. Presto: new versions of goods cropped up in territory after territory, those markers fueling the cultural enrichment of societies just as they had originally facilitated the growth in their populations.\n\nBecause our identities emerge in part as a response to contact with alien groups and the things they make or do, I propose a rule: the more societies interact with different competitors, the greater the number, complexity, and conspicuousness of the markers they display. When societies pack together, then, their peoples are likely to grow more unlike each other to avoid confusion and to protect themselves. This explains the varied labrets distinguishing the numerous tribes of the Pacific Northwest. Reflect on the high recognition factor of adornment and ritual across New Guinea, an island densely occupied by more than 1,000 tribes known for their colorful and elaborate costumes, compared with the relative uniformity of their sparsely populated Aboriginal relations next door in Australia.\n\nThe brilliant cacophony of art and decoration, language and activities that distinguish societies the world over would grow increasingly elaborate. The origin of all this diversity traces back to a fundamental shift to anonymous societies that occurred at the inception of our species, or earlier. To recapitulate, the markers our societies use would have evolved by gradual steps from behaviors that perhaps, as I've imagined here, were originally similar to those still seen in chimpanzees and bonobos today. First, there would have been a password. Subsequent markers would have involved the use of the whole body as a canvas for expressing membership in a society, but these would have left little trace in the archaeological record. Several tens of thousands of years ago more complex societies prospered when human populations swelled and interacted sufficiently for people to collectively remember, produce, and improvise on far more complicated social traits, partly to set themselves off from their neighbors.\n\nThe path leading from the individual recognition societies of other primates to the fully human anonymous societies, with all their cultural extravagancies\u2014unknown in the world of ants, with their simple markers and preset social lives\u2014was a long one. It vastly predated civilizations. The evolution of anonymous societies has been part of an enormous rewiring project extending from the cerebral cortex to the lower brainstem. Much of the necessary neural circuitry grew out of what would initially have been a rudimentary interplay of stimulus and response to markers and the groups sharing them. Since then our revamped brains have come to associate our representations of individuals and societies with an unruly republic of emotions and meanings that animates our conduct from one moment to the next, and across the years. The interplay of such behaviors, largely still unaddressed by evolutionists, is being laid bare by the discipline of psychology.\n\n# SECTION V\n\n# Functioning (or Not) in Societies\n\n# CHAPTER 12\n\n# Sensing Others\n\nSpend time among hunter-gatherers and you realize sleeping through the night is a modern idea. On my trip to Namibia, I listened as Bushmen conversed under a chiseled Milky Way in voices elaborate as birds, full of clicks, twang, and twitter. Their huts were barely visible in the flicker of family fires as they shared both traditional tales and events of the day with gusto and moments of dramatic acting. When the sun was up their conversations centered on daily business. Nighttime was story time, with tales conveying the big picture of a proper social life and affirming people's connections with the greater society.\n\nYears later their animated storytelling came vividly to mind as I stood behind Uri Hasson, a professor of psychology and neuroscience at Princeton University, while he hunched over a computer displaying a series of brain scans. Hasson had been observing the cerebral activity of people watching a movie. The snippets he played were open to interpretation: some viewers might suspect the husband in the film was unfaithful while others might conclude the wife was a liar. When Hasson looked at scans of the viewers' brains, he found they differed accordingly. If, however, audience members talked to each other as they watched a video, their cerebral cortexes synchronized: the same portions of their brains lit up when they began to follow the story line with a unified point of view. Hasson calls this mind-to-mind coupling \"a mechanism for creating and sharing a social world.\" The pleasure the Bushmen expressed that starry night must have come from such a melding of minds.\n\nI've asserted that every society is a community that has to be socially constructed in the imaginations of its people. This applies to the details of how the society operates, too. Humans turn everything they do into a kind of story, and interpret their lives in light of what it says. Further, that story expands through people's constant interactions into a society-wide narrative in which everyone plays a part. From the moment of birth we enter the web of expectations this larger story presents, with rules and expectations about work, money, marriage, and on and on. The full narrative brings life to a society's most beloved social markers and imposes structure and meaning to the world by creating an integrated framework in which people function. Passed down through history and slightly reshaped by every generation, the narrative affects how we conceive of our society and the line we draw to divide ourselves from outsiders. It isn't the specifics of how we are governed so much as the psychology that underlies this narrative, and how it affects our identification with others, that sets the line. The formation of human identities, and our reactions to those identities, guide our lives in ways scientists are trying to understand.\n\nThus far in this book, we have approached our investigation of the origins and evolution of societies in phases. We began in the animal kingdom, considering how various species create societies and what they gain from their memberships. Next, we investigated the versatility of the original societies of our species, finding that hunter-gatherers had well-defined societies set off from each other by markers of identity, which have persisted as the organizing principle of societies since the origins of humanity. To ferret out where those identity signals may have come from, we then explored the remote past, where we saw that markers likely began as simple passwords. But the story has grown much more complicated than simply recognizing the identities of other people. As I will describe throughout this section, the human relationship with markers, and the groups they establish, evolved along with a rich underlying psychology.\n\n# DYING FOR A FLAG\n\n\"Humans follow a flag like an experimentally imprinted duckling, a ball,\" says Iren\u00e4us Eibl-Eibesfeldt, a pioneer in studying human action through the lens of how animals behave. Evidence suggests that learning markers and using them to categorize people, places, and things is instinctive\u2014organized in advance of experience.\n\nEven though the inspiring stories we share like the raising of the American flag at Iwo Jima add to the meaningfulness and importance of markers, knowing those meanings isn't imperative for us to be aroused by anything that acts as a potent marker. Nor does such a signal need to be connected with a person to set off an impassioned reaction: either its presence (a stirring anthem) or its absence (imagine a town where Americans shoot bald eagles) engages the limbic system, the brain's emotional center. Those circuits can go off like a firestorm when a powerful marker is evoked\u2014an act of violence becomes all the more horrendous when it also involves the destruction of a national monument.\n\nGiven the right onlooker and context, even the simplest object or word is enough for a strong emotional response. As an example, picture an equilateral cross with arms hooked at ninety-degree angles. Seeing this apparently banal design can put a Holocaust survivor into a faint. Yet pausing to think about the symbolic implications of the swastika isn't required for the horror it may cause. Once an anguished response is set by a kind of Pavlovian conditioning, casting it aside is no more possible than stopping our gag reflex at outlandish ethnic food: take a moment to visualize walking cheese, once popular in Corsica, wriggling under maggot-power. Even so, at the time they were in widespread use, swastikas made Nazi hearts swell with the same rapture we expect to experience when we stand before our national flag at the ball game.\n\nPeople do love a flag. Today, even \"mild-mannered Danes go nuts over their national colors,\" says historian Arnaldo Testi. Indeed, he goes on, \"in democratic, secular republics, the flag acquires an even greater, near-sacred, civic importance, if anything, in the absence of other public binding icons, like king or God.\" This mania ties to our most essential groups: to fight or die for the flag is a personal pleasure and honor.\n\nHow a mere color pattern, shape, or sound can precipitate fervor or fear in the human brain is poorly understood. The response emerges in childhood, and no wonder. In the United States, flags are displayed in classrooms, and the Pledge of Allegiance is often recited in preschools (though children can now opt out). By six years old, a child perceives flag burning as bad; pride in country comes soon after.\n\nThe omnipresence of signs of national identity accords everyone a similarity of experience that primes our feelings even when our minds are elsewhere. Faced with adversity, symbols turn into beacons that galvanize us to action. A recording of the national anthem became a hit single in 1990 after the United States committed to the Gulf War; American flag sales exploded in 2001 the day after the 9\/11 attacks.\n\n# HOW BABIES CATEGORIZE PEOPLE\n\nMuch of the psychology of interactions between human groups concern how we respond to the markers directly associated with individual people. The efforts of Hasson and other psychologists show that our brains are constructed to socialize, and recognizing the identity of others is a part of that process. An essential first step is to notice other beings at all. Suppose you are playing checkers on a computer, thinking your opponent a machine, and midway through the game you find out that a person, not a computer program, is behind the opponent's moves. On cue your mental activity would shift to areas of the brain reserved for interactions with other people, including your medial prefrontal cortex (at the front of the cerebrum) and your superior temporal sulcus (a furrow in the temporal lobe). This would occur even if the information was wrong and no person was there. Same game, same IBM, yet you switch to another mental state\u2014the state that is normal when we attend to people, and infer what is on their mind.\n\nAs social beings, we depend on our awareness of others. A mental acuity we share with the chimpanzee and bonobo, honed by the upbringing of all three species in fission-fusion societies where individuals come and go, is our adeptness at registering slight differences\u2014and commonalities\u2014between persons. Psychologists have come to fascinating conclusions about how people attend to one another, both as individuals and as members of groups. Almost all the research concerns not independent societies, but rather races in urban settings. Hence the examples I give reflect that bias. Nevertheless these attributes would have originally evolved back when hunter-gatherers experienced societies more uniform than those of today, as responses to the markers of their own and other societies. It makes sense then to assume that societal markers such as language or adornment will yield similar results. (Why there should be a near equivalence of human psychological responses to races and societies will be a subject for later.)\n\nRecognizing human groups begins very early in life and cannot be suppressed. From molecules transmitted through the amniotic fluid and later in the breast milk, infants get a taste for what their mother eats, including any strong flavors, such as garlic or anise, preferred by her ethnic group. One-year-olds observing individuals who speak their language anticipate that they will favor similar foods, and that people from other backgrounds will have a different diet. By the age of two, this expectation becomes cemented as a preference: children that age and up are partial to anything eaten by the members of their own group, whether it's a fried scorpion or a tuna melt. This is an expression of a partiality for the familiar, and therefore what's safe, as anyone knows who's watched a baby burst into tears over the slightest strange thing.\n\nEven a three-month-old hones in on faces of its own race. By five months this preference extends to those speaking the parents' language, and with their dialect; a child grows up finding strange accents exhausting to comprehend and is sensitive to differences in speech. Between six and nine months of age, babies become good at categorizing people of other races based on facial cues, but after that they become worse at distinguishing individuals of races other than their own. This precedes the loss of adeptness at learning foreign languages after the age of five. I anticipate a baby's response to equally striking markers of identity such as hairstyle or dress will show a similar decline.\n\nEibl-Eibesfeldt is right in that the response to membership badges could be as much a reflex for a human baby as it is for a chick connecting to its mother (or to a ball, if its misfortune is to think a ball is mom). That imprinting occurs for each species in its own time window. You may wish to be reassured that such instincts are more hardwired in chicks than in humans with all their smarts, but the distinction isn't clear-cut. Flexibility is required for the survival of all creatures, not just humans. A chick must adapt to its mother's appearance after she molts, which gives her a new look. Should mother be eaten for dinner, the chick can imprint on another hen. You need not be a brainy vertebrate to handle such eventualities. Even ants sometimes adjust to the identity of workers of other ant species, and learn to treat them as nest-mates, if the foreigners are introduced into the colony before they mature.\n\nWe may be more flexible than most animals in recognizing the members of our societies as well as the ethnicities and races we are born into within them, yet the difference is one of degree. Some vertebrates can even hone their skills on other species. A human baby exposed to monkeys by the age of six months becomes adroit at distinguishing monkeys as individuals, and monkeys reared by people demonstrate similar lifelong talent at telling humans apart. I wonder if a youthful exposure to the wrong species explains the vervet monkey observed living and mating with baboons in the 1960s, and some other oddball companionships like bears playing with tigers.\n\nIn short, before a baby can read, before it speaks or comprehends language, memberships such as race and ethnicity are distinctions it attends to effortlessly, without any input from elders. But even though we are hardwired to pick out basic group memberships like society and race, how we think about them is far more multilayered than merely identifying features the way ants detect a colony scent. People discern the categories of living things, humans included, as being manifested in their very core, as we shall now consider.\n\n# THE HUMAN ESSENCE AND \"ALIENS\"\n\nStarting at about three months old, infants begin to attribute an _essence_ to every life form, an elemental _thing_ at its core that makes it what it is and not something else. This mental construct is registered in the cosmologies of hunter-gatherers, who, as animists, were convinced that spirits organize the world. They saw animals and plants, and each other, as suffused with essences. Paraguay's Ache, for example, thought a child was given its essence by the man who provided meat to its mother while it was still in the womb.\n\nPeople ascribe essences to species based on how they look and behave. Yet the essence that underlies those features, and not the features themselves, is what matters. That's how the brain handles discrepancies. A child has no problem with the notion of a chair turning into a table when its back is chopped off, but knows a living thing is different: a butterfly remains a butterfly even with its wings torn off; a swan raised by ducks is still a swan.\n\nWe act as if the attributes of swans are embedded in their very atoms. Geneticists speak of the swan's DNA to support this view; however, genes are mutable and allow for transitions, say, when one swan species evolves into another. With their belief that essences are stable, people don't allow for such intermediates. In experiments, children perceive an image morphing from lion to tiger to be one species or the other and not both, even if each child differs in where he or she sees this switch occurring as the cat changes appearance. Essences thoroughly lodge living things in the categories where we think they belong.\n\nHere's where I believe societies, certainly as they existed for early humans, differ from most groups in modern life. Joining a book club or bowling team is optional, and no one would presume anyone would be constrained to choose a spouse from either of these groups. Nor do we expect architects and lawyers to riot or declare war on each other, as anthropologist Francisco Gil-White points out. This is not to deny that some people have stronger allegiances to their faith or the New York Yankees than to their country. Yet a Yankees jersey, goth apparel, or job uniform can be discarded without undermining who we think a person is: the sports-obsessed or counterculture youth is _going through a phase_. By contrast, starting at the age of three, people see their society, and their race or ethnicity, as essential and inalterable features of identity, fixed permanently by an essence, the way species are. More so than many betrothals, a national or ethnicity identity stays with us till death do us part. For us, they are, in a word, natural.\n\nOn this basis the single human species is splintered into many. Outsiders\u2014the people of other societies, and nowadays other ethnicities and races\u2014are treated as if they were distinct creatures. Their markers are passed on to their offspring as dependably as the traits distinguishing a swan from a duck, so that their memberships persist through time. Even so, because societies and ethnicities are perceived to run deep\u2014to be _in the blood_ \u2014we can be open to the possibility that an individual who looks like a member of one group is part of another, the same way we can accept, appearances to the contrary, that whales are mammals, not fish. To think an odd person belongs, we must be convinced he or she was born of members of our group and that his or her kids will be members of that group, as well. We can attempt to conceal our pedigree by immersing ourselves in the lifestyle of another group, but however elaborate the cover-up, anyone who knows better will feel our essence remains. Purging ourselves of our essence is well nigh impossible, an idea as absurd as a baby swan training to be a duckling.\n\nWhat of individuals who marry or are adopted, like a swan by ducks, into another group? While such family members can be treated lovingly, their passage to full acceptance can be long and hard. As reveals of their inner essences, our eagle eyes still detect differences, for instance in the descendants of immigrants. Even people who are products of intermarriages over several generations may not blend in fully. This also explains why, though Americans use the term _biracial_ to label someone born of parents of two races, the child of a white mother and an African American father, no matter how light-skinned, will be understood in the context of white-dominated America as African American\u2014a point of view once known formally as the one-drop rule, where the slightest trace of black ancestry renders the person black.\n\n# ORDER FROM CHAOS\n\nRegardless of modern obsessions with racial groups, actual physical differences aren't the only metric of identity. The man with a drop of \"black blood\" can be as white as any Caucasian. And to the bafflement of outsiders trying to understand the interminable conflict of Israelis and Palestinians, the people involved look much alike: Haredi rabbis and Islamist imams even sport the same beards. Genetics proves they are of a single, narrow stock, a similarity that the people themselves would have a hard time conceding.\n\nYet what physical differences exist are hard to ignore; in fact we tend to magnify them in our heads. Over the past 12,000 years our rules of social engagement have adapted first to narrowly defined groups, such as the homogeneous people of the neighboring hunter-gatherer society, and later to races as we think of them today\u2014races that now apply broadly to people fitting loose physical descriptions that we boldly assert are so clear-cut we code them by color: white, black, brown, yellow, red. That the categories are artificial can be shown by how readily they shift. Early in the twentieth century few Americans considered Italians white, let alone Jews, Greeks, or Poles, the distinctions made in the United States at that time suggesting a probably exaggerated virtuosity at detecting fine-grained differences among humans. Meanwhile the British of the time referred not just to Africans, but to Indians and Pakistanis, as black.\n\nCompare those crudely defined color categories of race to what I call \"genealogical races\" such as the Bushmen or Ache, each of which, while divided into multiple societies, could be traced back in time as one reasonably tight overall population. Though our aptitude at picking out different human morphologies may be ancient, it's unlikely that a sensitivity to human variation originally evolved to respond specifically to racial traits varying across populations of our species such as skin color, most of which arose as fine gradations across vast distances\u2014continents even.\n\nCertainly categories of race or ethnicity are rarely fashioned from one set of human features alone. Just as the child makes sense of an image of a lion that changes into a tiger by seeing it as one or the other but not as both, the mind sees faces intermediate between groups (races, in most studies) as a match to one or the other. Yet we are also sensitive to other nuances of identity that render categories less ambiguous and more sharply defined than objectively they really are. When researchers concocted a face to be intermediate in appearance between two races but gave it the hairstyle typical of one of the races, the person was categorized as of the race associated with that hair. Put a guidepost like an Afro or cornrows on the head, and subjects confidently proclaim the person black.\n\nThat's how Rachel Dolezal, a woman of European descent who had pale skin and straight blond hair as a kid, passed as African American, becoming the president of Spokane's NAACP chapter. She gave special consideration to the style and texture of her coif while keeping her skin tan to look as if she had the requisite drop of \"black blood.\" Few questioned her self-stated identity before her duplicity was exposed in 2015. Once it was disclosed that no drop was there, however, her black essence vanished, rendering her apparent racial markers superficial and irrelevant.\n\nHairstyles label us so strongly that if a hairdo associated with a dark-skinned race crowns the head of someone with racially indeterminate features, we read his or her skin as darker than it is. The skin tone shift happens in much the way we perceive the same line to be longer or shorter if we draw arrowheads pointing inward or outward at its tips.\n\nThe optical illusion works because the brain dislikes confusion\u2014including in social categories. If someone's identity confounds us, our improvisational gray matter pieces together where she fits in from other information we consider useful. Along with hairstyle, the Jewish skull cap, the Sikh turban, and the black garb of a Greek widow are showy examples. More muted signs could be used by hunter-gatherers, for whom physical appearance would have seldom differed by much, and may not have mattered when it did. By this mental procedure people turn artificial (constructed) categories into social realities.\n\nEarlier I brought up how markers speed up the recognition of groups. What's remarkable is how swiftly and unthinkingly the information is absorbed. We register a person's ensemble of markers with no deliberation. Alex Todorov, an exuberant Bulgarian with a dramatic mop of hair and an office near Hasson's in Princeton's psychology department, has shown that a face glimpsed for a tenth of a second, a span of time too short to enter our awareness, is subconsciously assessed in terms of the person's emotional state, sex, race, and (Todorov assures me) ethnicity and society, too. Our nimbleness with markers must reduce our cognitive load\u2014the conscious effort we expend. And as Solomon Asch, a pioneer of social psychology, recognized in the 1940s, we can no more prevent these impressions than refrain from hearing a melody.\n\nShould markers, even in aggregate, come up short, signs of identity have occasionally been forced on people. Jews were compelled to wear a yellow badge in France during the Middle Ages and the Star of David in Nazi-controlled Europe. The rationale given by authorities was that the badge served to indicate shame, but surely a reason to coerce visible marks was that people of an ethnicity were now impossible to misidentify. Proof that words harm as much as any stick or stone, even word of mouth can mark people, with rumors about family backgrounds the downfall of many under anti-Semitic regimes.\n\nWhen push comes to shove, people are liable to be so anxious to avoid mistaking unfamiliar outsiders for members of their society that they slip up, concluding that someone who comes across as odd doesn't belong, even when he or she does. This tendency is so predictable psychologists name it (rather clumsily) the ingroup-overexclusion effect. On occasion the consequences of exclusion from the ingroup are cataclysmic, as a World War II\u2013era news report indicates:\n\nMistaken by Nazi administrators for deported Jews, a trainload of German refugees from Hamburg were exterminated by Gestapo officials in the \"death chambers\" of a Jewish concentration camp near Lwow, the Manchester Evening Chronicle reports today.... When the train arrived, the hungry and exhausted German refugees looked not much different from the hungry and exhausted Jews who have always been brought to this camp to their death in gas chambers. The Gestapo guards lost no time in stripping the newcomers naked and in sending them into \"gas chambers.\"\n\n# SUBCONSCIOUS ALARMS\n\nHow do our minds process the identities of the people we meet? Once we distinguish a person from an animal, a computer, or other object, the brain switches to taking in information about that individual, assessing, among other factors, whether he or she may be a threat. Such judgments are indispensable in an uncertain world, a matter of life and death for hunter-gatherers coming upon someone unexpectedly or for troops in a war. But the evaluations occur even in less fraught situations as part of the background activity of our nervous systems.\n\nThe mental reaction that ensues depends on who the people are and how familiar we may be with them or their group. If the person is unknown to us but is identified as being one of our own or from a trusted other group, our familiarity with his or her markers of identity make the stranger seem less strange. Whether or not we make the effort to individuate the person\u2014treat him or her as an individual\u2014we have some reassurance about the person's beliefs and behavior, though we might still be on guard if someone seems shady or unpleasant, or is simply having a bad day.\n\nForeigners\u2014and definitely those from a disliked or unfamiliar society\u2014can raise subconscious alarms regardless of their behavior. At a minimum we may feel discomfort in their presence. Among Bushmen, even a trace of a foreigner, like an oddly made arrow found embedded in the ground, induced anxiety since the conduct of its maker couldn't be predicted. One researcher studying the G\/wi Bushmen noted the \"reassurance and lessening of tension that is seen when a stranger is recognized as a fellow G\/wi.\"\n\nThe difference between a stranger (an individual we don't know, whether he or she is part of our society) and a foreigner or \"outsider\" (a member of a different society, whether we know him or her or not) is a big one. The two are clearly distinct in our species: we can be friends with the foreign exchange student in our class, while we have never laid eyes on a mysterious next-door neighbor. Yet psychologists often confound strangers and foreigners. Insofar as the English language muddies this distinction, it is unfortunate. The word \"xenophobia,\" for example, is often applied willy-nilly to our negative reactions toward strangers and foreigners alike. The fact is human minds have likely evolved to respond to strangers and foreigners differently, and foreign strangers most strongly of all.\n\nOur instantaneous and unthinking categorization of people has obvious adaptive advantages. As we have seen, it would have alerted us to others whose actions couldn't be predicted and reduced tensions toward those who resembled us. Our evaluations not only occur beneath our conscious radars but also affect us to the bone. For instance, most participants in a study who watched a video of people being stuck with a hypodermic needle perspired less and showed less activity in the bilateral anterior insula section of the brain when the person being injected was of a race different from theirs, a neurobiological response indicating diminished empathy. Even a chimp will display this selective connection to others, yawning only in response to the yawn of another individual that belongs to its own community. This reaction betrays a basic ingredient in human interactions: we act as if the essence of our own people is superior, as if we are more human than other groups. When facing someone who doesn't fit, we disengage. In extreme situations, the brain activity of a person viewing an individual he or she considers foreign looks the same as that of a person viewing an animal. Once people are typecast as outsiders, the mind abandons nuance and may expel them from the human category entirely. Taken together, these reactions scaffold the rickety edifice of human stereotypes.\n\n# CHAPTER 13\n\n# Stereotypes and Stories\n\nStereotypes are mental shorthand, an unavoidable outcome of partitioning our experiences into categories that make the world comprehensible. Without such expectations about regularities we would be chronically surprised, not anticipating the lily's scent or the hot sting of a bee. Walter Lippmann, the journalist who gave \"stereotype\" its modern meaning, wrote about them this way: \"We notice a trait which marks a well-known type, and fill in the rest of the picture by means of the stereotypes we carry about in our heads.\"\n\nSo it is that by categorizing an object as a chair by absorbing its characteristic legs and seat, and thereby anticipating that object will serve our desire to sit, we drop into it without expending a thought (perhaps one day being surprised when, not noticing it's made of paper, we, and the chair, crash to the ground, our chair-stereotyping having failed).\n\nOf course, we don't usually associate stereotypes with furniture. When applied, as the word usually is, to people, stereotypes are socially shared, simplified predictions created by the overloaded brain to evaluate others. Some of our predictions are inoffensive. We give money to a caf\u00e9 barista without thought. Indeed, we treat him or her, most blatantly when distracted or in a rush, more like a machine to get us caffeinated than a person. The barista can be fine with that, treating us in turn as just another in a long line of customers. Yet we also create stereotypes about the people of societies and the ethnic groups within them, for example anticipating their actions, including how those others will act toward us, or prejudging what they will think about the people of our own group. The essences we imagine in others correspond not only with verifiable human markers, but with the baggage of our beliefs and biases about the people who have them. These generalizations can harm by yoking the people of a certain society or background to a package of undesirable and largely outright incorrect assumptions.\n\nOur personal biases are more widespread than we might like to believe. One of the powerful tools for exposing our stereotypes is the Implicit Association Test. The test consists of a series of randomly paired images flashed on a screen, one a word and the other of a person, most commonly with either a dark- or a light-skinned face (though two ethnic groups would work as well). The onlooker is asked to connect a type of face with a certain kind of word. For example, one person might be told to press a button only if the word shown alongside a dark face has a positive connotation, such as _peace_ or _joy_ , and not press it if the word is negative, such as _violence_ or _disease_ , and do the opposite for light-skinned faces.\n\nIt turns out that just about everyone is a racial profiler. Almost all Americans perform the task faster and with less effort (with fewer mistakes) when asked to match dark-skinned faces with unfavorable words and light-skinned faces with positive words. This is true even for those who find stereotypes objectionable and take pride in being unprejudiced. The result comes as a complete shock to most subjects of the experiments, who are seldom aware of their own prejudices. More shocking still, according to the social psychologists who invented the test, is that \"awareness of the hidden biases did not seem to help us to eradicate them.\" Practice doesn't improve the results.\n\nIn the musical _South Pacific_ , the composer-librettist team of Richard Rodgers and Oscar Hammerstein put to song the conventional view of how these prejudices first enter our heads:\n\n_You've got to be taught to be afraid_\n\n_Of people whose eyes are oddly made,_\n\n_And people whose skin is a diff'rent shade,_\n\n_You've got to be carefully taught._\n\n_You've got to be taught before it's too late,_\n\n_Before you are six or seven or eight,_\n\n_To hate all the people your relatives hate,_\n\n_You've got to be carefully taught!_\n\nThis is erroneous: no boy or girl has to be methodically taught such things. Rather, learning stereotypes is an extension of a child's developing mastery of detecting patterns\u2014the primary job of any immature creature striving for independence.\n\nFor the child, distinguishing categories of people is just the start. A child associates those categories not only with patterns of what he or she sees people do, but also with what others say they do, and further, whether those actions are evaluated as good or bad. This discrimination doesn't require a motive. People don't have to feel that the others are adversaries or competing with them somehow. Such biases emerge easily even when the groups are totally arbitrary. For example, kids randomly assigned to one of two groups predict that those in the other group do more bad things. In effect, groups\u2014and especially enduring and significant ones like nations and ethnicities\u2014are represented by mental files that fill up beginning in childhood with rules of thumb ranging from the reasonably inoffensive, such as _Italians eat pasta_ , to such problematic conclusions as _Mexicans do menial work_.\n\n\"Racism isn't something that happens to the child, it is something he does,\" argues the psychologist Lawrence Hirschfeld. By the age of three years, children don't just recognize races; they perceive races as more than skin deep and freight them with stereotypes. This transpires inexorably, as the child assimilates the attitudes of others, not exclusively the child's parents, to assemble a point of view. Actually, mom and dad often have surprisingly little influence. Even the progeny of progressive parents absorb the biases of their society. Young people are little prejudice machines, displaying negative attitudes of the same magnitude as adults but often doing a worse job at covering them up.\n\nFiguring out where they themselves fit in is important to children, too. To do this, children prioritize their time by observing others. Rather than studying their reflection in a mirror, they pay most attention to those of the same race and ethnicity as their parents or caregivers. When adopted by another ethnicity or raised by a foreign nanny, however, children seem to figure out their own background and how they themselves are expected to act from how other people treat them. That a child accepts the identity thrust on him or her by others is a symptom of the intense desire humans express to find their place. No doubt if your infant brain had been transplanted into the head of a child in a Senegalese jungle or a housing complex in Macau you would have grown up exactly as at home in that social setting as you are with your own identity today. Children can even absorb two cultures, albeit as they mature they are likely to see themselves primarily as a member of one or the other.\n\nAlthough no specific prejudice is set by genetics, that our proneness to be biased is innate has a troubling effect: once formed, stereotypes resist revision. This unfortunate fact, a leading group of psychologists has concluded, \"may explain, in part, why conflicts among different language and social groups are pervasive and difficult to eradicate.\"\n\n# SNAP JUDGMENTS\n\nIn all these matters, snap judgments are the norm. Alex Todorov, the Princeton psychologist, finds that in the same tenth of a second or so it takes us to stick a person into a category, we have already come up with opinions on his or her trustworthiness, among other stereotypes. An outsider to our group incurs the quickest and most superficial evaluations. He or she excites the anxiety-triggering amygdalae, the same portion of the brain activated when we slap away a bee, heart racing. Our amped-up, quick-and-dirty response to whatever we deem a potential threat is \"resistant to change and prone to generalization,\" one neuroscientist writes. What amazes me is how we make such verdicts before a person registers in our conscious mind. If we continue to examine a face for more than the blink of an eye, the only result is to justify conclusions already formed unconsciously. In effect, in that critical first moment of perception, we don't see a person. Instead, we register our stereotypes about him or her, erecting a simulacrum in our mind's eye that overrides most qualities of the real individual.\n\nAny departure people show from our stereotypes about them is a measure of their individuality. Valuable information, wouldn't you think? Nevertheless, we give only people from our own society or ethnic group serious consideration. For those we grant such special privileges, we absorb personality details beginning in the fusiform gyrus, the section of the temporal and occipital lobes devoted to face recognition. If we interpret an ambiguous face as black after it's been \"tagged\" with an Afro, for instance, we are more likely to form a more complete impression of the individual if we are black ourselves. This reaction intensifies should we be especially likeminded with the person, say in our political leanings: in response, neural firing shifts to the lower part of the medial prefrontal cortex, the same region that activates when we think about ourselves.\n\nThe price of this automated filtering system is the bottleneck it places on understanding outsiders. Not that we can't decide to put in extra effort, say, by learning such social niceties about an outgroup person as her name. However, given how clumsily we individuate those not of our \"kind,\" Martin Luther King's dream remains an uphill battle: well intentioned people often fail to judge individuals from another group solely by the content of their character.\n\nIn a sense, in the instant of encounter, we automatically judge the book by its cover, taking more than a cursory glance inside only if the cover passes muster. What's more, we recollect people more accurately over the minutes, days, and weeks after meeting them if they are of our group. If they are not, any specifics we may have discerned about them fade from memory. One result is the sad fact that many false imprisonments are based on the eyewitness testimony of accusers who are of a different race than the accused.\n\nWhat would seem to be a positive sign for group relations within and between societies is that sufficiently powerful signals can supplant malignant prejudices. For example, if a man of a different race or ethnicity wears the jersey of our sports team, the jersey can become paramount in how he is evaluated: given our shared allegiance, we may forget his outsider status. Unfortunately, the change will be temporary, and only affects the fellow wearing that jersey. See him or someone else of his ethnicity sporting another shirt and our ingrained biases snap back into place. Attempts to suppress our own negative views can backfire through a \"don't think of an elephant\" effect, where we end up thinking of something\u2014such as an elephant\u2014even more after we try to force our attention onto something else. We end up activating our prejudices even more strongly. Ironically this occurs because the energy we burn in attempting to overcome stereotypes can wear us out, with the result that we can slip up and behave worse.\n\nOne might think the problem could be mitigated if different ethnic groups or societies got to know each other better. However, as can be seen from the racial prejudice against African Americans throughout American history, even where black people live in large numbers, frequent contact isn't sufficient to override the effects of stereotyping. Psychologists Kimberly and Otto MacLin compare the problem to traffic in Manhattan, where pedestrians see thousands of cars but, never having a reason to own a car, fail to distinguish one from another. At best, they may know the markings that label a car as a taxi. To distinguish foreigners as _persons_ rather than in terms of a stereotype, we can't merely be exposed to them as a New Yorker is to cars. To stretch the \"apples and oranges\" idiom in a new direction: fairness to outsiders means going beyond the easy-out of seeing them as the equivalent of apples and ourselves as all sorts of interesting varieties of citrus. Attending to outsiders in such detail doesn't occur effortlessly, or often.\n\nOnly bigots and na\u00efve small children flaunt their biases. As adults, most of us either rationalize any prejudgments we are aware of and the anxieties they elicit, or we submerge them so deeply we fail to notice their existence. Yet the Implicit Association Test tells us that we don't need to be radically xenophobic to show prejudice toward outsiders. Even among well-meaning people perceptions unavailable to conscious inspection still influence responses ranging, in the words of two social psychologists, \"from derogation to degradation to disconnection.\" Just about every store owner, employer, traffic cop, and passerby distinguishes between ingroup and outgroup members, reacts to certain outgroups with heightened vigilance, and is less likely to come to their aid. Because the same people are more apt to feel concern for those they flesh out as individuals\u2014a penchant known as the identifiable victim effect\u2014they may also have trouble treating outsiders fairly because of the strain required to distinguish one from the next. Yet most people haven't the foggiest clue that they misbehaved. Such dynamics play out in every interaction between societies and between the ethnicities within them, and the impact on interpersonal relations is beyond calculation. All these biases stand up to contradictory information. We exclude from our prejudices the friend who is a citizen of a nation we dislike without that person improving our assessment of his or her people\u2014our buddy is an exception to the rule.\n\nDespite the negatives, the human aptitude for dealing with utter strangers serves to simplify positive interactions both within and between societies in a way unachievable for species whose members must know each other to function socially. People can register someone as foreign but still enter into a productive relationship with him or her. A modest adverse reaction can be benign, keeping individuals engaged yet at a formal distance. This is only partially good news. Beyond our everyday biases toward treating our own people subtly better than others is the danger, given a provocation or opportunity, that any disgust or jealousy can be purposed to justify actions against entire classes of outsiders.\n\nSocial pressure to conform can turn stereotypes into a self-fulfilling prophecy. Individuals tend to try hardest to shine at skills that the stereotypes they have internalized about themselves suggest they will be good at, with the prevalent clich\u00e9s in the United States being that black people excel at sports and Asians at mathematics, undermining whatever other potentials they had and thereby further supporting the stereotypes. Whether talent differences generate stereotypes or stereotypes foster differences in talent is an open question, but individuals can face backlash from their own group and peers should they break from expectations about group behavior. This suggests that our biases aren't limited to outside groups: we expect people like us to act a certain way too.\n\nWhat this boils down to is that we act in a biased manner even when we believe we don't. This impulse must have evolved to nudge people to serve their group's best interests even when their relations with a foreign people were good. When ties failed, then as now, undisguised discrimination would rise to plain view.\n\n# REMEMBERING, FORGETTING, MEANING, AND STORIES\n\nThe research makes clear that when it comes to other people, we respond automatically, our positive or negative feelings and biases triggered within milliseconds of an encounter. I expect that our responses to the markers themselves are every bit as automatic, and that only when called on to explain ourselves afterward do we rationalize our reactions by spelling out what a marker symbolizes to us. That's not to claim that its meanings are insignificant. The concepts and stories we grow up with touch upon many aspects of our identities and guide how we interpret our place in a society and the world. In the last chapter I explained that humans turn everything into a story. The vital cultural particulars that people chose to convey\u2014for example, the account of Betsy Ross sewing the stars and stripes on the first American flag\u2014give those details an emotional weight that improves their later recollection and continuity. Such stories resemble stereotypes in that they save us mental effort by cutting through a morass of information to remind us about what really is important in our connections to others.\n\nThe stories of a society may tell of the aspirations of the people and accounts of their past. A story can be recent, perhaps recounting the success of that year's Olympic athlete, but the most important ones endure over the generations: Italians still commemorate the Roman Empire, and the citizens of India can surely recount the contributions of such early dynasties as Maurya. The rendition of a society's birth especially can be a source of inspiration and pleasure whether a person's forebear signed the Declaration of Independence or he or she has just become a naturalized citizen. No origin story is a straightforward recounting of events, however. Creating what amounts to a collective historical consciousness is delicate business. What matters isn't truth but _tale_ , one that conveys a proud past and bravery in a crisis for the sake of the group and its values, where the abiding question that all men and women confront in their lives, \"who am I?,\" turns to \"who are we?\"\n\nAfter Vietnam broke away from the Han dynasty of China in the tenth century, the scholars of the time set down accounts of the country's history, which served that function nicely. For centuries afterward, a founding dynasty known as the Hung Kings was considered part of the Vietnamese heritage. The archaeological record has revealed that those kings were medieval fabrications. In the first half of the nineteenth century, revolutionaries did the same for the Han, an ethnic group that has come to compose the majority of China's population. They invented a streamlined history and ancestor for them all, the mythical Yellow Emperor.\n\nHappy is the nation without a history. So said the seventeenth-century thinker Cesare Beccaria. I interpret his claim to mean that a complete and accurate rendering of the past undermines the bonding function of collective memory. Think of history as the contents of a box of mementos found in an attic, from which we draw what we desire and bury what's best forgotten. Whether the narrative that results from storing the keepsakes together is error-free, a complete fable, or something in between, any challenge to the story is frowned upon or forbidden. A well-crafted history serves the interest of putting the people in the best light and gives shape to their future\u2014although an artful leader can manipulate the contents of the attic box to build a passionate following. At the same time people sear into memory every detail of the tales that they share, as the historian Ernest Renan reflected, \"Forgetfulness, and I would even say historical error, are essential in the creation of a nation.\" So we have Turkey's obstinate refusal to acknowledge the Armenian genocide and the American account of its war of independence, which in the eyes of the British has many faults, such as greatly underplaying the role of French support of the rebel cause.\n\nHunter-gatherers, lacking written accounts and mechanisms for uncovering upsetting facts, were masters of selective recall. They relished stories, but they were more devoted to explaining nature than they were to recognizing their forefathers' achievements. Not all societies treat history prominently, and least of all people living in bands. While the necessities of daily living, such as how to make a fire, were conveyed as assiduously as possible, forgetfulness was absolute when it came to bygone days. For preliterate societies the past was seldom an unfolding saga worth memorizing. Rather, they saw time as at once endless and twisted into cycles similar to the phases of the moon, virtually ensuring that history, as the philosopher George Santayana predicted, would repeat itself.\n\nThis makes for an intriguing contrast with our current fascination with history. By all accounts band societies lived and breathed in the present tense. One of the few instances I've read of hunter-gatherers telling stories of long ago were Aborigines recounting the arrival of Indonesian fishermen three centuries earlier. When I asked about this lack of interest in former times, anthropologist Polly Wiessner proposed that the past gained significance\u2014and I imagine, too, a set chronology associated with themselves and their land\u2014only after people created political systems that had to be justified and conveyed through the generations, such as that set out in the US Constitution.\n\n\"This land was made for you and me,\" sang Woody Guthrie. This sense of joint ownership of a space, as with the things in it, would appear to be vital in a person's connection to the society. I've told you how hunter-gatherers were almost always territorial, as people are generally. More than the founding-father tales central to the ethos of nations, what most resonated with hunter-gatherers was this attachment to the land, and its sacred sites, such as the reverence of Bear Butte mountain in South Dakota to the Cheyenne. This intimacy conveys how people can be willing to die for their homeland as a question of cultural survival. It is _here_ that their knowledge of the world applies, where their sacrosanct ways permeate the soil. In fact, stories and space can be linked. Those with expert recall assign their memories to locations in an imaginary place or landscape. Both memory landscapes and real landscapes are encoded in the hippocampus. Australian dreamtime falls into this memory tradition, its potent stories connecting to places in such detail that Aborigines could reconstruct the lay of their land without maps.\n\nCitizens today retain this conviction of sharing a national territory even as many of them have come to possess, as individuals, just a tiny piece of earth within it. Even if \"a nation is the same people living in the same place,\" as James Joyce's fictional character Leopold Bloom observed, no one has to walk every inch of their society's territory to feel a spiritual connection. Its borders are as fixed in the mind's eye as the boundary in identity that separates Us from the Them who live on the other side. Indeed, the expanse of ground too wide for anyone to ever see in its entirety is no less concrete in the people's imagination than the vast community of compatriots, most of whom they will never meet. And so emotive pronouncements about the homeland grace our national anthems: there are Fuji's \"shores of golden sand and sunshine\"; Bulgaria's beauty and loveliness\u2014\"ah, they are boundless\"; the Chilean countryside as \"a wonderful copy of Eden.\" Meanwhile Guyana is \"set gemlike and fair, between mountains and seas,\" and \"among the lands\" Lesotho \"is the most beautiful.\" People can adore a foreign land, but few merge this feeling with the warm glow of belonging and deep connection to stories that ties them to their country of birth.\n\nNot that people require a territory to tenaciously identify with their group. Perennial wanderers who originated in northern India, the Romani, often called gypsies, are a durable ethnicity that retain a common culture despite being landless since spreading over Europe a thousand years ago. Still, without a home, or at least no claim to a homeland, a group, ethnic or otherwise, can seem impotent. Hence the emotional resonance of the epic stories of dispossessed Jews or Palestinians seeking their own plot of ground.\n\nStories, and the territory they are often anchored to, are the great binders of societies. The minds of the Bushmen, at their camps, must have linked up firmly as they told each other stories of mutual significance, building what one expert proclaimed to be the group mind. Whether taught at school or around a fire, stories handed down through the generations, and the land and people described, frame our common upbringing and destiny: they remind us either that we are in this together or that the others\u2014 _they_ \u2014are in this together in ways that may not serve our interests. Counterintuitive aspects of a traditional tale, whether Homer's Cyclops or Moses parting the Red Sea, are memorable and impossible to confuse with the beliefs of others. A story's themes can be so pointedly improbable that outsiders inevitably find them absurd, yet people accept their idiosyncratic facets without question. The role of such myths isn't to convey logic but rather to elicit emotion and bond us to the place we live, and to each other. How a story is presented is important, too. Spoken aloud in the hushed tones or uproarious chants of the devoted, not only are stories likely to be remembered, but mangling the account can be a travesty or sin. Properly recounting a story becomes a common practice of our upbringing and part of our very souls.\n\nA people's connection to their land adds to the emotional pull of the membership they share, just as their narratives and stereotypes about themselves and others create an identity that gives life meaning. As we shall learn, what makes each of these elements formidable is that they all can be leveraged to maintain our superior self-image. They can also be weaponized against groups that compete with us, or offend us. Alas, this is what makes them so frightening.\n\n# CHAPTER 14\n\n# The Great Chain\n\nAmong hunter-gatherers, the Jahai of Malaysia call themselves the _menra_ , or \"real people.\" This is also the meaning of _Dana-zaa_ , the name the Canadian Beaver Indians give their kind, and _mihhaq_ , the word used by the Kusunda of Nepal. \"Even the gentle San [Bushmen] of the Kalahari,\" E. O. Wilson tells us, \"call themselves the !Kung\u2014 _the_ human beings\"\u2014true, yet even there the designation \"human\" applies not to every Bushman but merely those Bushmen who belong to the !Kung society.\n\nIt should come as no surprise that the people of a society can harbor a sense of their superiority, though rarely do we stop to contemplate how extreme, and at times even outrageous, this feeling can be, as the critic Ernst Gombrich has nicely spelled out.\n\nI know a wise Buddhist monk who, in a speech to his fellow countrymen, once said he'd love to know why someone who boasts that he is the cleverest, the strongest, the bravest or the most gifted man on earth is thought ridiculous and embarrassing, whereas if, instead of \"I,\" he says, \"we are the most intelligent, the strongest, the bravest and the most gifted people on earth,\" his fellow countrymen applaud enthusiastically and call him a patriot.\n\nThat we cultivate and maintain this confidence in our supremacy, and the way we register the relative status of those in other societies, are human universals. Surely Americans didn't invent the idea that their nation is the greatest. \"Somewhere there may be a rationalist country that asks its children to withhold their emotional evaluation of their homeland until they've carefully compared its inventions, heroes, and breathtaking natural wonders with those of its competitors,\" social psychologist Roger Giner-Sorolla says. \"But that country is probably not where you or I grew up.\" This outlook is embodied again in national anthems, which voice almost identical themes. Among them, the adulation of national history and heroes, expressions of pride in the people's work ethic, commitment, and courage, and in the peace, security, liberty, and\u2014the most overused word of all\u2014the freedom a nation offers. For many people the very idea of plausible points of comparison between their society and any other one is absurd, even when the similarities are clear. Social minutia can be enough to arouse a supercilious attitude toward others who in other respects may be all but indistinguishable, a phenomenon that Freud named the narcissism of small differences. A society's members seldom have to be sold on why their ways are best: they absolutely know how things should be, and for them a life arranged that way is worth living.\n\nA sense of superiority is just part of the picture. Compounding the situation is the unnerving ease with which people are capable of devaluing strangers. If extraterrestrials were to land on Earth and witness human behavior, they might rate people socially inept: without considering the matter, we bounce between treating an animal as a person, and a person as if he or she were an individual human being, a coffee dispensing device, or an inferior kind of animal. Given the practical payoff of a skill at _humanizing_ animals (hunters, for example, often imagine what a deer is thinking to predict its next move), it's a shock to discover just how readily people _dehumanize_ other humans. The names many hunter-gatherers give themselves reflect how humans treat outsiders differently from, and usually inferior to, their own people or other groups they know and trust. Even the names many modern nations apply to their own citizens, such as _Deutsch_ for the Germans and _Dutch_ in the Netherlands, derive from the word in their languages for \"human.\"\n\nThe architecture of minds built to perceive human groups as having their own essences doesn't grant equality to them all. We see our society and those of other people as falling into a hierarchy along with other living things, a notion codified in the Middle Ages as the Great Chain of Being. Typically the royal We surmounts the chain (surpassed only by God and the angels). Other humans follow in a descending order, some of them, Aristotle announced, \"as much inferior to their fellows... as beasts are to men.\" This hierarchy continues its plummet through the natural world, with \"some animals,\" as Orwell wryly wrote in _Animal Farm_ , \"more equal than others.\"\n\nThe scale wasn't fashioned in an ancient Greek ivory tower; people intuited the universe this way before words were scratched on parchment. More than likely it's a basic feature of our psychology. Research shows that children think of people as superior to animals, and of outsiders as closer than their own group to animals. Furthermore the proneness of hunter-gatherers and tribal peoples to describe themselves as _human_ suggests this type of thinking is typical even among societies that are small and have much in common with their neighbors (far more in common than you, with your wider knowledge of the world, likely share with a Tibetan yak herder today). Their use of epithets meaning _nonhuman_ or _animal_ also suggests those people felt entitled to treat at least some outsiders categorically as other species, an outlook that would naturally affect their relationships.\n\n# RANKING OTHERS\n\nPsychologists have a lot to say about how people rank outsiders. The status we assign different groups is associated with beliefs we hold about how, and how well, their people express emotions. First, a bit of background. Six emotions are often considered basic\u2014happiness, fear, anger, sadness, disgust, and surprise. These are discrete, hardwired, physiologically distinct mental states expressed first in infancy and recognized worldwide with slight cultural adjustments. We combine these feelings into secondary emotions, culturally influenced expressions of ourselves in relation to others. We use secondary emotions to interpret each other's intentions and respond to what others think about us as persons or as representatives of groups. Many of these complex sentiments are positive, such as hope and honor, but there are negative ones like embarrassment and pity. These shared secondary emotions are important to bonding the members of a society, who are motivated by the satisfaction of pride or patriotism and the avoidance of shame or guilt to act together in costly situations. Secondary emotions require learning and more brainpower than basic emotions and emerge later, while a child is absorbing its identity.\n\nPeople widely intuit that all humans express the basic emotions, and that we share these as well with some animals (picture a happy dog wagging its tail). We think of secondary emotions differently, treating them, along with such qualities as refinement, self-control, and civility, not just as unique to humans but as less developed or questionable in peoples other than our own. We especially doubt the emotional complexity of those near the bottom of the Chain of Being, whom we see as driven almost entirely by the basic emotions of animals. We assume such groups lack self-control as well as the capacity for nuanced feelings, and that they can't be dealt with rationally. We discount evidence that they express regret\u2014a secondary emotion\u2014for any transgressions we attribute to them and doubt their sincerity. Because we connect our august standing with moral caliber, we may associate such pariahs with immorality: they do not (and, we often think, cannot) follow proper ethical codes. Their presumed moral failings thus place them outside the boundary of fair treatment\u2014our scope of justice. To top that off, the dehumanized people elicit few secondary feelings from us in return, a coldness that worsens matters by seeming to confirm the other side's assessment of our own emotional disabilities.\n\nOur unequal treatment of outgroup members is aggravated by miscommunication. Much is lost in translation not only because of possible language differences but because of a lack of adroitness at reading even the basic emotions on a foreign face. While we observe all manner of details about people like ourselves, in interactions with outsiders who make us uncomfortable we detect only the most extreme expressions (notably anger and disgust). Hence research has shown that American Caucasians are faster to register rage on black faces than on white ones and, when presented with a racially indeterminate person, are more likely to conclude the face is of the black race when it expresses anger. Individuals most susceptible to associating an ethnicity with danger\u2014namely, racists\u2014make the quickest, shallowest assessments. The darker the skin, the more dangerous the man. On June 27, 1994, _Time_ magazine published a mug shot of O. J. Simpson, accused of murder, that had been tweaked to blacken his face, as became obvious when _Newsweek_ published the same image, undoctored. The outcry forced _Time_ to pull the issue from newsstands.\n\nEven figuring out when a foreigner is truthful can be hard. In a recent study of Turks and Americans, participants failed to catch each other's lies. Likewise the clich\u00e9 of the inscrutable Chinese, whose emotions, unreadable to Europeans, are assumed to be undeveloped, while Chinese citizens, acting by other rules, find Europeans overreactive. The social implications are real. I have seen tourists rant away while missing the irritation on the face of an \"inscrutable\" native of China or Indonesia, where under ordinary circumstances emotions are understated. In part to counter the passivity Europeans and Americans see in them, some Asians have had the fold in their eyelids removed surgically.\n\nTwo aspects of our perception underlie our emotional reactions to outsiders. One, dubbed warmth, is a measure of their trustworthiness, judged with the blazing speed of a first impression. The other, appraised slowly and determined by our recognition of the group's position in the hierarchy, is competence. Competence is a measure of their strength as a people, and hence their capacity to act on their views about us\u2014that is, whether they will be able to help us or cause us harm. We respond differently depending on these instinctual assessments, pitying members of groups we see as warm but incompetent (Cubans, for instance, in a study of Italian subjects); envying competent others who are perceived as cold toward us (Japanese and Germans, in the same study of Italians); and craving interactions with competent groups that treat us warmly. Finally, disgust tinged with anger\u2014contempt\u2014is an at-times undisguised reaction to those regarded as hostile and inept (a response many Europeans have to the Romani).\n\nThe disgust brought on by the Romani and other denigrated people place them firmly among the vermin at the nadir of the hierarchy. Examples can be dredged up by the thousand. The name given to Ache hunter-gatherers by their farming neighbors is Guayaki, or \"ferocious rats.\" Proclaiming that \"not every being with a human face is human,\" as one political theorist of the Third Reich put it, the Nazis likened Jews to leeches and snakes. During the Rwanda genocide of 1994, the Hutus compared the rival Tutsis, a mildly different racial group of slightly taller people, to roaches. The tactic of equating black people with apes began with the first European contacts with West Africa. We like to think this egregious view is ancient history, yet the Implicit Association Test demonstrates that the characterization remains embedded in American minds. These aren't throwaway comparisons equivalent to disparaging someone for acting like a pig or complimenting the person for being smart as an owl. Registering a people as disgusting and subhuman allows us to deal with them the same way. Such was the fate of the war prisoners at Abu Ghraib, photographed in humiliating ways by American prison personnel in 2003 and 2004. At best their misfortunes are disregarded or treated with schadenfreude\u2014pleasure at their pain.\n\nDisgust is a complicated emotion. It is directed not only toward certain humans and animals, but toward anything contaminated or unhygienic. This includes such outlandish ethnic dishes like walking cheese, as well as anything\u2014or anyone\u2014reminding us of our unclean animal natures. Disgust toward people and disgust toward unsanitary things appear to be essentially interchangeable, a product of activity in two areas of the brain: the insula, a deeply folded portion of the cerebral cortex, and the amygdalae, a pair of almond-sized clusters of neural tissue buried in the temporal lobes, part of the brain's rapid-response system. Meanwhile the medial prefrontal cortex, a lump of brain tissue involved in human interactions, fails to activate, as if we were merely confronting an object. I suspect the Gestapo's misidentification of the trainload of Germans accidentally sent to the gas chamber for \"delousing\" resulted in part from the travelers' repellant appearance after a long journey in crowded, less-than-hygienic conditions. That the American downtrodden have been obliged to use separate drinking fountains and toilets makes sense if you think cleanliness is next to godliness and believe that the badly sullied lie at the filthy and godless end of the scale.\n\nImmigrants, often belittled as unclean or as vermin, are especially vulnerable to extreme xenophobia. Indeed, disgust may have evolved in humans as a way to keep diseases from entering a society by enforcing a bottleneck for contact with outsiders. Such infections as avian flu and Ebola can trigger the same fear today. A territory closed off to outsiders functions as an island to which some parasites have a hard time making a journey. Diseases became unintended weapons in conquest. Colonialist societies often transmitted illnesses to which the native population had no immunity. The Arawak contacted by Columbus were among the many American tribes who succumbed to smallpox.\n\n# SURVIVING AT THE BOTTOM\n\nWhether or not marginalized peoples succeed in building relationships with more powerful neighbors, their low status puts them at a psychological disadvantage as well as at an economic one. Not only do they feel inferior, they can be downright depressed about themselves, as a !X\u00f5 Bushman's explanation of the modern-day beliefs of his people discloses:\n\n[The God] _Gu\/e_ first made all the people alike, later he divided them into different peoples, some to work for others. Nevertheless, when God first made man he started by creating the white man and then the black one. With the pot scrapings he had left he made the Bushmen, that is why the Bushmen are so small and why they have so little sense compared to other people. _Gu\/e_ did the same thing when he created the animals. First he made the large ones, then the smaller ones and with the pot scrapings, the smallest creatures.\n\nYet even people battered by outsiders and with poor self-esteem strive to appreciate their own distinctiveness by finding things specific to their group to value and give their lives meaning. From what I know about the !X\u00f5, for example, I wouldn't be surprised if they take what pride they can today from their superior knowledge of nature.\n\nThat the values and identity of a society can be protected in the face of demonstrable foreign superiority underscores that the very idea of what it means to be human can be open to interpretation. Each society or ethnic group puts a favorable spin on its standards and defining features. Thus, without completely dehumanizing the people of other societies and ethnicities, we can ascribe to them, and to ourselves, differing qualities. The Chinese perceive their nation as admirably communal, while Americans take pride in individualism. We can recognize foreigners as clever, more ambitious, or more expressive than ourselves, in which case we may decide that our straightforwardness, contentment, or reserved nature is appropriate, and excusable, deportment. By calling our shortcomings \"only human,\" we keep ourselves as close to the top of the hierarchy as we can. We rationalize that while those others have a lot going for them, they fall short in categories where we shine. Perhaps they end up working like machines, are hopeless at the things in life our group finds important, or are wanting in moral principles.\n\n# PREJUDICES AMONG ANIMALS AND IN EVOLUTION\n\nA susceptibility toward seeing disliked outsiders as disgusting vermin suggests that prejudices against humans and creature phobias may share an underlying psychology. Once I spent a day watching Barr Taylor, a psychiatry professor at Stanford, treat a patient with arachnophobia. Dr. Taylor introduced her to spiders incrementally, one body part at a time, starting off by drawing a head, adding the abdomen, then each leg. What had been frightful grew routine as the woman absorbed its details. In this reading of the issue familiarity breeds understanding, or at least tolerance. A failure to properly register outsiders as individuals could betray a similar discomfort with people we find unintelligible\u2014a squeamishness about looking them in the eye.\n\nEquating outgroups with vermin may have antecedents among the other primates. Male macaques readily associate troop members with fruit and foreign monkeys with spiders in a simian version of the Implicit Association Test. Conceivably, then, dehumanization predates language. A reasonable proposition is that our ranking of societies and ethnic groups in the Great Chain emerged out of the dominance hierarchy that organizes the relationships of individuals _within_ the societies of many animals. For a macaque or baboon, every moment is dictated by status differences in which dominants pester subordinates. As such, perceiving foreign societies and their members below the hierarchy of its own troop would be cognitively simple, and a plausible precursor to the dehumanization of groups.\n\nStill, among species other than ours, it's hard to conceive of the elaborate prejudices that humans show. One reason for this is biologists don't know whether any other animal has the secondary emotions that seem to be a necessary underpinning for human appraisals of outsiders. But putting that question aside, taking stock of a foreigner's warmth, competence, emotional depth, or status, shallow as those evaluations might be, only makes sense given the chance to build a relationship. That's impossible, for example, in chimpanzees, whose sole recourse on encountering a foreign animal is to flee or bludgeon it to death (some encounters with lone females excepted).\n\nThe emergence of anonymous societies no doubt shaped our perceptions of foreign societies as species. Markers in effect are traits we broadcast to establish ourselves as bona fide human beings. The showcasing of our identities using our bodies as a canvas possibly got its start from our ancestors' reactions to Neanderthals and other now-extinct offshoots of our family tree, whose weak chins, projecting faces, sloping foreheads, and behavioral anomalies marked their own bodies as distinct. If so, our response to foreigners that once were truly distinct species eventually transferred to different societies of our own species.\n\nThat we assess outsiders this way, and see in them what we fear, may have been a smart response in our distant past, especially before 40,000 or 50,000 years ago when contact with outsiders was rare, knowledge about them sketchy, and life spans too short to depend on intimate learning. Generating stereotypes (even if little more than hunches to stave off an overload of uncertain information) gave our ancestors a quick-and-dirty guide to predicting whether foreigners might benefit or hurt them. Reliance on prejudices would be less risky than figuring out the individual personalities of those whose morals, motivations, and modes of expression were strange and whose ties to them were shaky or nonexistent.\n\nThe philosopher Immanuel Kant asserted that our moral concern should take in the human race. It should be clear by now that the mind doesn't readily accommodate such a point of view. Situating ourselves at the summit of the Great Chain of Being may explain how we can hear about deaths from wars overseas every day and still enjoy dinner. One upshot of judging other people's mental and emotional capacities as impoverished, and their ethics as lacking, is that we block ourselves from interacting with them by ordinary means. Committing atrocities, facilitated by psychological assessments of outsiders as inferior, may represent a considerable payoff in pure evolutionary terms. Once the deed is done, seeing those we wronged as less than human turns into a self-preservation mechanism for assuaging guilt, allowing us to move on. Generations can pass before we recognize our actions as heinous, should our selective memories allow it at all. In fact, just as we doubt expressions of regret in others, saying we (as a society) are sorry is a surprisingly recent idea, and doesn't come easy. Buried in the 2009 Defense Appropriations Act was an apology \"for the many instances of violence, maltreatment and neglect inflicted on Native Peoples by citizens of the United States.\" It didn't make the front page.\n\nThus far, I have described social stereotypes as characterizing the individual members of a society or ethnicity. We have yet to tackle how our biases also channel our perceptions of a group considered in its entirety. Moving forward, I will explore how a society can seem to be a single thing, how it can respond with one voice, and how it can carry out actions as a united front.\n\n# CHAPTER 15\n\n# Grand Unions\n\nThe greatest displays of power in prewar Nazi Germany were the rallies at Nuremberg. Each year hundreds of thousands of enraptured people\u2014eventually the number climbed to nearly a million\u2014filled an arena bordered by 130 searchlights bright enough to be seen a hundred kilometers away. During fervent speeches wave after wave of troops marched in goose step under stupendous flags and banners to the thunder of Wagner.\n\nThe spectacle made plain that the Nazis placed the Germans at the summit of the Great Chain of Being. But for people watching the newsreels from elsewhere in the world it also played into a human predilection to conclude that members of outside groups\u2014Germans in this case\u2014are of a kind. The extravaganza was a show of such solidarity that its countless participants, all looking and acting alike and identifying with the same symbols, seemed to blur in the minds of those terrified of the Nazis into a swarm of a dystopian uniformity that people normally associate with insect colonies. Indeed, George Orwell called nationalism \"the habit of assuming that human beings can be classified like insects and that whole blocks of millions or tens of millions of people can be confidently labeled 'good' or 'bad.'\"\n\nEven when groups don't overawe the way the Nuremberg throng did, the members can merge together like so many ants because their shared unfamiliar appearance and ways of doing things make it a struggle to tell them apart. This is in sharp contrast to the people in our own group. Today the problem extends both within and across societies. _They all look alike_ : Caucasians say this about Asians, who in turn say the same thing about Caucasians. It's a view that misses the trees for the forest since both races show the same variability in features. _They can't be trusted_ : this prejudice was once as sweepingly common among hunter-gatherer societies as it is today.\n\nOnce we shoehorn others into categories, the chanting horde of a hostile country or the cheering supporters of a friendly one meld in the mind's eye as if the individual people were so many cells in a Frankenstein-esque body, a flesh-and-blood entity with a personality and ambitions of its own, able to sustain itself for the long haul. Psychologists say the group has a high entitativity, the quality of being perceived as an entity. Overbearing displays of symbols such as the flags and music at Nuremberg add to the effect. They make foreigners look not just more stitched together but more competent (and thus a greater possible threat) and coldly intimidating, and therefore untrustworthy. Our sizing up of a group's strength and prospects for destructive competition with us in turn influences whether we think of it as an enemy, an ally, a needy dependent, and so on.\n\nWhile the creators of the fanfare at Nuremberg undoubtedly took pleasure in impressing the outer world, the core of their game plan was to strengthen the German people's identification with the nation. For the participants, of course, the whole business was intoxicating. You can predict that the Germans underwent much the same reshaping as they did in the minds of foreigners, except with a different outcome. While most of the time people find the recognition they crave by expressing their connections to a society through their contributions as individuals, and seeing one another as persons, the joy of being peas in a pod wins out when they feel part of a unified crowd. The participants' impulse to express their individuality is put aside and their differences forgotten, leaving the sensation coursing through their veins of being both like each other and one with a united ensemble.\n\nConceiving of our society as a concrete entity is a normal experience nurtured by any national celebration, with its parades, fireworks, and flag waving. The perception begins in infancy with the urge to belong, a comfort with like others that transforms into strong bonds with our society as we age. That sense of being part of something greater leads us to inflate how much we have in common with other members, whether we individuate them or not. It also hones the distinctions we perceive between our society and those far more homogeneous outsiders, who appear all the more identical when they frighten or anger us\u2014while our inattention to detail further enhances the impression that each group is united.\n\nThose in power contribute to this entitativity. In extreme cases, as with Hitler during the Third Reich, the leader becomes a symbol and an embodiment of the identity of everyone else, as formidable as any totem pole. People may similarly look upon a foreign leader as representative of his or her society. This view enhances the stereotype bias that prompts us to treat outsiders as carbon copies. The leader becomes the original from which the copies were stamped.\n\nOur belief in this carbon copying requires no proof that the foreigners really are alike. If we assume all persons of a nationality or ethnic group are the same and we come across a hostile one, we are apt to expect them all to be hostile. As a first approximation, people are taken to represent their society, much as their leader does in the international arena. This is especially true for any personality traits we dislike. That's no different from how we generalize about other threats. It is adaptive to reason that all bees sting if one does. Slapping bees is expedient. Our response to societies, treated as species, can be equally simplistic, and sometimes just as deleterious to outsiders. Better safe than stung.\n\n# SOCIETY BECOMES SELF\n\nIn considering the sense in which societies are entities, it may be helpful to step back for a moment and overview the extraordinary range of human connections, from the most personal to the most abstract. Different styles of interaction work well in groups of different sizes, and these styles and group sizes have been important from early times. Most intimate are relationships between pairs such as a wedded couple or a subordinate and boss. Then there are the relationships within the handfuls of people who perform a task together. For hunter-gatherers this meant parties that gathered plants or hunted game, and now it applies to workplace collaborations where decisions have to be made smoothly to achieve a goal. _Where can we dig up these tubers?_ became _how do we market this car?_\n\nNext up were the original hunter-gatherer bands of two- or three-dozen souls that today correspond in size to many classrooms, business departments, and hobbyist clubs. Conflicts can be harder to manage in these larger groups, but people are still tied together by a sturdy knowledge of individual others. A few hundred to a few thousand had an allegiance to a band society, and today we see corresponding numbers in social groupings like church congregations, conferences, and schools, institutions that are pivotal for exchanging information and resources across a broader community. At this tier, collections of people extend well beyond individuals who can directly interact, or even know each other. Most of the anonymous crowd identifies instead in some symbolic manner.\n\nIt would go too far to say that a classroom is equivalent to a hunter-gatherer band, or certainly that a group as ephemeral as a conference is equivalent to a band society. Yet our contentment with the intimacy of the classroom or with the more abstract joy of being at a conference alongside scores of fellow enthusiasts may echo how our psychologies interfaced with others in early bands and societies, respectively. Regardless, for most individuals the societies themselves, no matter how populous, retain their primary hold, their interwoven markers setting up the conditions for people to see each other, and foreign individuals, as part of permanent, unified wholes.\n\nSo it is that people find significance in both their fellow members and their society, which is perceived as an entity amounting to more than the mere sum of its members. We see ourselves as playing a role in our society's continuance, carriers of its celebrated history, traditions, laws, and social mores\u2014all those markers that bind us into the future. It's fair to say that people feel they live on by way of their progeny and also through this union: love of country becomes a vicarious path to immortality. The early ethnographer Elsdon Best reported on one expression of the phenomenon among native New Zealanders:\n\nIn studying the customs of the Maori, it is well to ever bear in mind that a native so thoroughly identifies himself with his tribe that he is ever employing the first personal pronoun. In mentioning a fight that occurred possibly ten generations ago he will say: \"I defeated the enemy there,\" mentioning the name of the tribe.\n\nThus if our nation's soldiers die in war, our response is distress or anger or fear. If our national team triumphs at the Olympics, we are thrilled not purely out of joy for the team, but because of a sense we won the game ourselves. Uniting around other citizens comes with a surge in pride and awareness of strength and glory. This sort of ethnocentric love some claim is a function of oxytocin, a hormone that dampens the amygdala anxiety response while increasing our empathy for similar others\u2014positive feelings also associated with the flag. Moments of such solidarity, and oneness, can represent the highlight of a person's life. Americans felt this affinity at that moment _we_ landed on the moon, or, for the Brits, the _we_ -ness that came with crowning a king or queen. We forget our differences when buoyed by the identity we share, and, in an effect called a group emotion, our intense, and overlapping, feelings draw us closer. Ask a person how she feels, and she might say quite happy; however, if she then hears about an act of terrorism against her country, she will likely report extreme sadness or anger. Intrinsic to human self-esteem, group emotions focus pride, attention, and energy at a national level, so that one's society inherently becomes part of the self.\n\nWhen reactions do spread, at times like a contagion, the response is heightened the closer together individuals are, whether excited monkeys converging on a fruiting tree or thrilled fans cheering across a sports arena. Yet modern-day crowds aren't necessary to cultivate group emotions and bonding. Togetherness on a more modest scale will do. For roving hunter-gatherers, their connection with their society must have peaked when the bands gathered in one place. Members would have confirmed their solidarity not merely by swapping goods and building camaraderie, but more fundamentally by identifying with the assembled community. The get-togethers would have offered a sense of the sacred in the form of collective pride and patriotism. People across bands reaffirmed their unity through feasting, stories, songs, and dances.\n\nThe pleasure of carrying out these activities, and of ceding one's individuality to the group, may spring from an impulse to act as we think we must to be accepted. This is seen in how we mirror not just the vocal tics and gestures but the emotions of those we respect, which we tend to do most with those of our race or ethnicity. These actions are unintentional, induced in part by mirror neurons in the premotor cortex that fire whether we engage in an action or watch another carry that action out. The behavior has genetic underpinnings: newborns imitate the sadness, fear, or surprise of others. Mirroring others, and emotional responses at a group level, spring up in other animals, too, like the excited monkey troop. Chimpanzees match the playfulness or anger of apes in a video.\n\n# ACTING AS ONE\n\nNot long ago I was in Tokyo enjoying the crisp January air at the Teppozu Inari Shinto shrine, recognizable by its century-old torii gates and arched rooftops, when more than a hundred individuals, stripped down to white loincloths and head cloths, gathered in the inner courtyard around a pool filled with ice, and began to chant. About 30 of them climbed in, sunk to their knees until the freezing water reached their waists, and replicated the motions of rowing a boat as their voices alternated from a melodious prayer to a two-note chorus and then to words spoken in a rumble. After a few frigid minutes the devotees climbed out to be replaced by the next set of people.\n\nThe ritual is known as the Kanchu Misogi winter purification festival. I stood watching with Panos Mitkidis, affiliated with the Center for Advanced Hindsight at Duke University, where psychologists investigate human thought processes, including, in Mitkidis's case, how people act in unison under trying conditions. Mitkidis had worked up a sweat despite the chill while his team gathered data as the day's events unfolded.\n\nRituals consist of replicated sequences of actions that have no obvious practical utility in and of themselves. The rallies of some mammal species are the behaviors closest to rituals that we see in nature. Gray wolves leap on each other and yip, while spotted hyenas go through bouts of rubbing one another, tails bristling. The act of rallying animates the participants to take risks they would avoid when alone\u2014attack another pack or clan, perhaps. This boldness may loosely reflect what in our species is called the discontinuity effect: groups interacting with other groups are inclined to be more competitive and less cooperative than the individuals who make up the groups would be when interacting one on one. Of course there's a difference between humans and these other animals: people often have symbols\u2014a flag\u2014to rally around. Slavery ants may be an exception to this rule. Sharing a pheromonal flag, the workers rush about in a swarm at their nest before marching off to seize a fresh batch of servants.\n\nHumans follow ritualized patterns more than we may imagine, going far beyond simply mirroring each other's speech and emotions. As children we become clever at duplicating the complex actions of others, typically to no obvious end other than to demonstrate our commitment to a group. Think of a child gaining the acceptance of a hip clique of schoolmates by abiding whatever play rules they concoct. _We dress just so_ \u2014okay, check Performing the nonessential bits correctly\u2014getting the ritual right\u2014is a unique way humans assert their identities, to their society included.\n\nMitkidis finds that those who participate in intense rituals achieve a heightened solidarity that rises above the identity-affirming power of ordinary social markers by making the participants very cohesive indeed, the sort of behavior augmented into outright hazing by fraternities that forge collective unity through hard knocks. When formal expressions of devotion are repeated regularly, people can be driven to so align their fates that they may commit to perilous concerted actions. Cults and gangs instill a sense of unity in much this way; that's especially true for such criminal organizations as the Mafia, which enforce the members' commitment across the generations.\n\nRituals even more extreme than the winter purification festival cause a radical incarnation of group emotions known as identity fusion. Participants see themselves and their group as truly one and the same, pumping up their commitments to each other while ensuring they stringently uphold group standards. Such \"rites of terror\" are performed seldom and often by a select few. In nations today they are practiced only in the most intense military programs or during actual combat engagements. Hunter-gatherer and tribal groups commonly exhibited such costly and hard-to-fake rites, which extended to producing painful and irreversible markers like body scarring when their people were embroiled with hostile outsiders on an ongoing basis. Think of their acts as simulating the excruciating pain the players expect during the ordeals to come, readying them for taking action together.\n\nAs an ant biologist I'm especially intrigued by the warrior initiation ceremony still performed by the Sater\u00e9-Maw\u00e9 of the northern Amazon. The rite requires young boys to be stung by inch-long, lozenge-shaped horrors some call bullet ants for good reason. The shock of a bullet ant sting once knocked me down even though I had brushed the ant off before it could give me a full injection. That's nothing compared to the experiences of adolescent Sater\u00e9-Maw\u00e9, stabbed by dozens of bullet ants over five minutes in a level of torment surely matching the agony of many battle wounds. The Sater\u00e9-Maw\u00e9, not surprisingly, are a fight-prone people.\n\nA people don't require rituals, of course, to be willing to have each other's backs. In dangerous times they may pull together as resolutely as a band of horses does when confronted by a pack of wolves. A commitment to fight for the common good can be a source of joy and inspiration, but it presents dangers, too. Compatriots respond as one even when some have contrary feelings about the group's course of action. For example, they might hold back their opinions because of fear of rejection or of being seen as cowardly, or simply to go along with the thrill of the moment. As independent-minded as people can be as individuals, their differences can be put aside to shore up a group response. Developmental psychologist Bruce Hood remarked of such situations, \"Whatever self we believe we have is swamped by others.\"\n\nNot that everyone will join the displays of unity. Cooperation and trust may be magnified, yet those attitudes aren't always embraced by all the members. Should some openly dissent, the majority, wrapped in the flag, will find it a problem. Championing a personal view can be wildly unpopular or even considered traitorous, a betrayal of loyalty so heinous that in medieval Europe it was deemed a sin worthy of flaying or disembowelment. _We are in this together_ turns into something more severe: _You are either with us or against us_. Better to cave in, or at least portray that you had no option if you are called to justify yourself later. Nazi criminals pleaded the Nuremberg defense, arguing that they were just obeying orders. Not only, then, do we abandon our better judgment to our blind faith in our society, but we are absolved of fault for its actions or ours on its behalf.\n\nPeople who choose to administer electric shocks to others after being told to do so by someone in authority show a dampening of brain activity, suggesting that those acting under orders emotionally distance themselves from the consequences. When we aren't simply listening to a commander but following the communal will and are drunk on group emotions, our sense of culpability may vanish entirely. Our fervor sanctions a path we wouldn't want to be held accountable for pursuing if we had done so alone. All the better, a feeling of alike-ness and interchangeability renders us secure in our anonymity. Whether in a military formation like those at Nuremberg or in the chaos of a battlefield, common soldiers are trained to act as dependably as clones, impossible to tell apart.\n\nIf a throng comes together without a plan or clear leader, its actions can be as much an emergent property of the collective as the pooled behavior of an army ant swarm. The mass of humans may give the appearance of achieving objectives through a will of its own despite each individual contributing little. However, it's possible for the response of a crowd of humans to be smartly coordinated and yet stupid\u2014maladaptive, or to people's advantage but normally immoral. A leaderless group is likely to yield sound decisions only under carefully constructed conditions in which a population is polled after each person has been given the leeway to assert his or her individual preference uninfluenced by others. Otherwise, what's often called swarm \"intelligence\" may more accurately be described as mob rule, with the participants surrendering their individual will and succumbing to the hysteria of the group. Joining the masses gives the normally powerless their crack at awe-inspiring powers, right up to gang violence and genocide.\n\nAs a successful and envied ethnicity (one considered cold but competent by hostile outsiders), Jews faced a classic reaction that came to a head under the Nazis. Although they were shown grudging respect and their businesses patronized before the Holocaust, the attacks on them started as troubles escalated\u2014blaming the victim, massively amplified. This pattern has been repeated throughout history. Chinese residents were assaulted for similar reasons during upheavals in Java in 1998. Korean Americans in the Los Angeles riots of 1992 were targets of the sort of collective hysteria that has at times escalated to mass murder. Rationality has little bearing on what transpires when tensions are high. The ease with which ordinary people are roused to harm based on the haziest of justifications is dismaying. After the Rwanda genocide most Hutus admitted that the Tutsi had been fine neighbors. Instrumental in changing this point of view wasn't just the spread of comparisons of Tutsis to bugs: treating the murderousness as normal encouraged previously held but hidden prejudices to surface until, in the words of one of the participants, \"when the violence began, it fell upon us like a sudden rain.\"\n\nSmall groups of thoughtful individuals may fare little better. Groupthink can materialize, a desire for fraternity and we-ness eclipsing the goal to achieve sound solutions. People can misrepresent their perceptions of facts to conform to what the group expects.\n\nA tragedy of the human condition is that our group's behaviors can justify the fears and prejudices that outsiders have about us. We can indeed turn out to act like each other, and as a unit. The perceived interchangeability of people most sharply matches reality when tensions are high and we may come to be as much in lockstep as soldiers on parade. Within our society, and equally in theirs, such tension stimulates cooperation, or at least acquiescence to the society's course of action. Subsequently, it reinforces the sensation that our fates are linked\u2014and at odds with those similarly united others.\n\nThis feeling of alikeness with others can extend to everyday behavior, too. The J\u00edvaro of Ecuador were gardeners and game hunters. For the J\u00edvaro, killing those who \"speak differently\" (that is, other tribes), and then using a special kit to shrink their victims' heads, was a must. This viciousness toward foreigners can be contrasted with the relative gentleness of Bushmen societies in recent history. The brutality of the J\u00edvaro and the kindness of Bushmen are stereotypes certainly open to objection, vehemently so when assumed to apply to individual persons. Yet stereotypes can reflect actual group differences in what is called a collective personality, arising at a population level. This has even been shown to be true for animals; for example, honeybee colonies of the same population differ strikingly in aggressiveness. For humans, the collective personality emerges in much the same way as more obvious markers for the society, as a product of repeated social interactions that cause individuals to look and act alike. The British famously are more reserved than Americans, and Americans are more reactive than the French to displays of temper. Differences in national character probably go way back, with the !Kung Bushmen more openly expressing anger than G\/wi Bushmen, and Bushmen being on average more timid than other southern Africans. These \"personalities writ large\" add further to the impression that societies are entities.\n\nIn examining societies from different angles, I've skipped over one incontrovertible fact of psychology: most people's well-being rests heavily on their family ties. Having established how human societies might have functioned from their earliest beginnings, and how the human mind responds to people of the same and different societies, we now take a step sideways to explore the relationship between families and societies.\n\n# CHAPTER 16\n\n# Putting Kin in Their Place\n\nIt makes sense to insist that a society be more than a simple nuclear family\u2014more than merely two parents with a generation of their dependent offspring. But that doesn't preclude the possibility that families and societies might have similar psychological and biological underpinnings. Could a society represent a kind of extended group of kin, either in terms of the actual pedigree of its people, or perhaps psychologically, in the minds of its members? For that matter, what should we consider kin, in the broad sense of an extended family? Do families have the same sort of membership and clear identity that we expect of societies? How integral, extensive, universal, or accurate is our knowledge of our biological kin\u2014and do the logic and emotion of our connections with our families apply to the way we connect with societies?\n\nFamilies, of course, are central to daily human life, and in ways not seen in other species. Take the general expectation that dad will stick around with his offspring. No dolphin, elephant, chimpanzee, or bonobo grows up knowing a father. And parents with kids are just a part of the picture. Human family relations are labyrinthine compared to the kin connections of other animals. Among people, not only do parents participate in their children's lives\u2014and their grandchildren's\u2014for as long as they live, but we keep track of a whole array of relations on both mom's and dad's sides of the family. We know our siblings and both parents' siblings, and the assorted spouses and offspring of them all. Humans don't just mate for life (or at least try to)\u2014their bonds tie them to networks of kin.\n\nInvestigations of those networks by biologists and anthropologists have been an important foundation for the exploration of social behavior, and principally cooperation and altruism. By the 1960s biologists had developed a body of work around the theory of kin selection. The theory holds that species evolve behaviors that favor the propagation of the genes of relatives into the next generation. Ever since, scientists have construed kinship as a core driving force not only in families but also in our connections to a society: If societies are imagined communities, goes one idea, the human mind imagines a society to be an extension of the family. This would in turn lead us to view our society-mates as if they were kin.\n\nIndeed, there is some overlap in how people express their bonds with kin and their bonds with society. Yet evidence from the animal kingdom and human world alike suggests that understanding and keeping track of kin and understanding and keeping track of society members are distinct tasks, generally carried out to navigate different, though at times overlapping, problems and solutions.\n\n# KINSHIP AND SOCIETIES IN NATURE\n\nFor kin selection to work, individuals must be able to pick their kin out of the crowd or at least favor kin by happenstance. For ants and other social insects, this isn't an issue: usually the society _is_ kin. In such species, a colony is one teeming nuclear family, made up of the queen mother and multiple generations of her offspring.\n\nYet beyond the ant colony the idea of a society as kin becomes cloudy. A savanna elephant core, often mislabeled as a family, can allow complete outsiders to join its ranks, and those outsiders, as far as anyone has determined, are treated as equals thereafter. That a core typically ends up consisting of kin is a by-product of its history. Because siblings grow up together, they are apt to stay together. Gray wolf packs can be family affairs, but they, too, have proven capable of letting in outsiders as permanent members.\n\nWhatever bloodlines do exist can be transitory. This is true even for species like the wolf or meerkat, whose societies can resemble a social insect colony in that they sometimes consist of a family with multiple generations of young that may assist in rearing pups and other labor. A pack or clan can go on for generations, unity uninterrupted. But look closely: breeding males and females come and go, sometimes peacefully, at times replaced by outsiders in a fight. Ten, twenty, or fifty years hence the same society, on the same bit of earth, can be stolen out from under one lineage after another until it's composed of animals with no blood relationship to its founders.\n\nOften in gray wolves and savanna elephants, and invariably in many other mammals from bottlenose dolphins to highland gorillas, societies are composed of multiple lineages of kin. In fact there are societies with no kin connections at all. None of the adults in a horse band are likely to be related. This is a result of how a horse society forms. Maturing horses of both sexes bond not with childhood friends, as elephants do, but with individuals they meet after hoofing it from the society of their birth. Adult horses lose all connections with the kin they grew up with. Kin selection theory would predict that an animal would risk its life for kin. However, the unrelated members of these horse societies stick by each other for the long term, even standing as one to keep wolves away from a foal.\n\nAmong the most stable of all mammal societies are roosting communities of the greater spear-nosed bat, which aren't kin at all. Each cave in their habitat harbors multiple societies, roosting at separate sites, consisting of 8 to 40 females that first met as young adults, and one pretty expendable male. Completely unrelated the females may be, yet they treat each other well. The females forage together in a territory they share, and become so committed to the group throughout their 16-plus years of life that when a baby falls from their roost, they protect it from foreign bats until mother comes to the rescue.\n\nClearly societies offer advantages\u2014in protecting the young, for example\u2014that can keep animals together regardless of whether they are related. While kinship connections may add to the motivation to stay in certain societies, the presence of kin isn't obligatory for a society to carry on fruitfully. A jointly beneficial coexistence is the key to its success.\n\nWhat of the nonhuman apes? When adolescent female bonobos and chimpanzees transfer to another community, the females are no longer around any of their kin, yet (most notably in bonobos) they bond readily. Conversely, males stay around their kin yet their closest allies tend not to be brothers or sisters. Rather they befriend individuals with compatible dispositions. Sociable or rambunctious chimps stick together, a nifty advantage of being free to wander in a fission-fusion society.\n\nFurthermore, in genetically diverse societies, both human and nonhuman, kinship tends to be impractical to figure out. Consider the relationship of the baby to its mother. No family member should be simpler for a child to recognize. However, there are stumbling blocks even here, as the baby must learn to differentiate its mother from the nannies, grannies, and others who often help with kids\u2014caretakers that exist in many cooperative breeding species with societies. Humans have a solution: a baby listens to its mother in the womb and bonds to the face associated with that voice within three days of birth. A similar shortcut isn't likely for recognizing any other relative, dad included.\n\nClues may exist, in the form of similarities among family members. For example, a hamster can not only tell other hamsters apart but is able to detect a relative it's never met by comparing the other's scent to its own. In one study, chimpanzees were able to match photographs of unfamiliar baby apes to photos of the mothers of those babies more often than chance would dictate. For all that, resemblances, even among immediate family members, can be spotty. Human fathers are known to overestimate how much they look like their child. While researchers have shown that outsiders can pick out an infant's parents more often than expected by chance, errors abound. Growing up, people come to recognize family members based not on looks or any proof of genetic relatedness, but rather from trusting others to tell them about who is who, and from being together from birth.\n\nPutting aside the difficulties of detecting kin, there is the question of what kinship even means for humans and other animals. People conceptualize categories of kinship relation, like _mother_ or _brother_ , and likely other vertebrates do too, as has been documented for baboons. Yet what goes on in the head of a baboon interacting with its sister isn't likely to correspond to a person's idea of _sister_. Baboons rely on association for support, more than on relation or resemblance. Ties between the monkeys arise from intimate familiarity rather than actual, unknowable, genealogy. Each female keeps track of her matriline\u2014a personal network of females that are typically such relations as her offspring, sisters, mother, and so on. In practice, however, who a female includes in her matriline hinges on her tastes, so kin may be left out and agreeable nonrelatives added in.\n\nWho makes the cut? Shakespeare put it best when he said \"nature teaches beasts to know their friends.\" Animals often build supportive relationships with those who hung around mother in their infancy. A childhood pal needn't be kin, although the likelihood is good that some are. This might be an older sibling who still clings to mother, but it might be an unrelated playmate, too. Life can also come to revolve around kin through happenstance, as when a male baboon sticks by a past mate in anticipation of further dalliances with her. In the process he ends up befriending her infant, which has a fair chance of being his.\n\nFor the most part, then, animals pursue social connections and not biological families as such. Studies of the psychology of relationships suggest people respond to kith and kin much the same, and value them equally. As the saying goes, friends are the family you choose. An interchangeability of the two can be critical for people with small or broken families, or the aged who have lost all the other family members of their generation.\n\nA band society contained many family lines, with friendships driving social choices as much as, if not more than, kinship. A couple customarily raised their kids in the same band that was occupied by one or the other set of grandparents and possibly a sibling or two. Otherwise, their relations\u2014the families of their brothers and sisters, cousins, aunts and uncles\u2014were spread among the society's bands. What kept a band together was compatibility. As with other fission-fusion species, people seek out likeminded others. Among Bushmen, even today insofar as their hunter-gatherer activities continue, \"bands have marked personalities: one may consist of people who are by habit quiet and serious, another may be a gay crowd or perhaps people who enjoy a bit of bite in their humor,\" one anthropologist reported. Not that a band was homogeneous in the personalities of its members; like any social group, people could get stuck with an outcast nobody liked\u2014perhaps an annoying relative of some abashed person in the same camp.\n\nEven so, proximity can be a deciding psychological factor in our evaluation of others as possible close kin. People tend to avoid sex with anyone they were in frequent contact with as they grew up; this seems to act as a heuristic for avoiding incest. Roaming hunter-gathers also tended to find mates outside their childhood social sphere, choosing to marry residents of bands at some distance from their own, where any kin were likely to be far removed.\n\n# THE HUMAN RELATIONSHIP IQ\n\nThe obvious difference between people and other animals is that once a toddler has a vocabulary, it can be taught not only who is in its family, but also about the family tree relationships between those individuals (even for kin seldom seen). Indeed, the most archaic and widespread words may be \"mama\" and \"papa\"\u2014words a baby as young as six months may apply correctly to its parents. Yet the meaning behind the words comes later, starting with the discovery that other kids have different mamas and papas.\n\nWhat surprises is how poorly infants grasp such relations. A child needs to mature enough to employ language in building complex representations of relationships before it understands who connects with whom, and concepts like _uncle_ take years to comprehend. Understanding kinship can be as arduous as memorizing multiplication tables, with the caveat that while multiplication is the same anywhere mathematics is taught, what children learn about kin depends on what their society requires of them. The intricacy of our cognizance of kinship can change over the course of history as well as across cultures. In Modern English, a first cousin may be a father's brother's son or a mother's sister's daughter, a distinction for which Middle English once had separate words. Even within a society, understanding of kinship can vary from person to person depending on the extensiveness and closeness of the family. Those from small or estranged families might have trouble sorting out relationships others take for granted. ( _A second cousin? What exactly is that?_ ) As the biologist David Haig, an expert on family relationships, once quipped to me, \"It's a wise child who knows its father, an even wiser one who knows the father's first half cousin.\"\n\nGiven the significance with which scientists imbue relatedness, one wonders how many humans have an aptitude for keeping track of their family trees. How many would pass an IQ test measuring knowledge of kin? The convoluted classic 1940s song \"I'm My Own Grandpa\" starts with the singer's father marrying the daughter, by another marriage, of the singer's wife; through a Gordian knot of events the crooner arrives at the realization contained in the title. Following its lyrics gives me a headache, and I suspect many other people would fail a relationship IQ test that included this song.\n\nAn intimate knowledge of kin relationships doesn't seem to be essential for human life. Much like tribes that have no words for numbers beyond one and two, some societies are less obsessed than others with sorting out these tangles of kinship, at least as relatedness is conveyed by their vocabularies. Residents of parts of eastern Polynesia have just a few terms expressing kinship. The Amazonian Pirah\u00e3 tribe is even more stripped down, having a single word that applies to both parents, all four grandparents, and all eight great-grandparents. Their one word for \"child\" is used for grandchildren and great-grandchildren, too, while the word for \"siblings\" applies as well to the parents' siblings and the children of siblings. The Pirah\u00e3 tongue apparently lacks recursion\u2014think of \"one's mother's father's mother\"\u2014a kind of looping that is required to represent distant kin.\n\nDespite this linguistic simplicity, the lack of a name for a category might not keep the Pirah\u00e3 from grasping the differences among family members. We can't say from the available evidence, but perhaps those Indians still intuit how someone fits in a genealogy based on her age and who bore her without having a specific term to distinguish sister from cousin. The other possibility is that a category like \"cousin\" is unfathomable to the Pirah\u00e3, much as gravity went unnoticed by most people before Newton gave that word a meaning. This alternative seems plausible given the vexation Pirah\u00e3 experience thinking about, and communicating, other sorts of categories for which they have few terms, such as numbers.\n\nThe point is this: the familial connections that are recognized in a given society take years of training to comprehend, yet a three-month-old, too young to speak, deftly detects members of societies or ethnicities. A practical reason for our struggles with kin is that our relatives aren't a distinct group with a clear cutoff\u2014ever more kin can always be added by going yet one more generation back. The child's relative ease with ethnic and racial groups testifies to the importance of broader groups than kin in our evolution. Societies and the differences that mark them, not families beyond their most immediate members, are an indispensable component of the human mental world.\n\n# FROM FICTIVE KINSHIP TO EXTENDED FAMILIES\n\nEven for individuals who have an impressive relationship IQ, knowledge of their actual family tree is likely to go only so far. People's remembrance of relatives is generally restricted to those alive during their lifetimes, such as the great-grandparents they met as children. Few people recall much about more distant ancestors (except the one who did something to brag about). Even societies that practice ancestor worship seldom require anyone to master genealogies, but rather to honor their ancestors across the board.\n\nPeople in most hunter-gatherer bands put even less emphasis on knowing their exact biological connections. What made retaining such information unlikely was a tenuous relationship to the past and its people. Mentioning ancestors was bad luck and so more or less taboo for Bushmen, which would have made figuring out distant blood relations unlikely. Native Australians never spoke of the dead, who were consequently forgotten within a generation. Indeed, an astonishing reason for language change among some Aborigines was that any vocabulary word that perchance sounded like the name of a person had to be avoided after his or her death. The people had to invent a new word.\n\nAs far as hunter-gatherers were concerned, culture and other markers trumped genetics. Families in some American Indian tribes could adopt children taken in battle, a practice that beefed up the tribal reserve of warriors. What bound adoptees to a tribe wasn't blood but their adherence to tribal ways learned alongside other children. This shared upbringing meant that both the families and their societies were culturally uniform and genetically diverse. Even when genealogies were deemed important, they could be invented, as group histories can be. Members of Central Asian tribes that claim a clear pedigree turn out to be no more related to those in their own tribe than to the population as a whole.\n\nWhat did play a role among hunter-gatherers was the terminology associated with kinship. In band societies, fictive kinship, sometimes called cultural kinship, was commonplace, a method of giving each person a symbolic relationship to others. Thus, people used words such as \"father\" or \"uncle\" to address fellow society members, and all fathers and uncles were equal. This may explain why, even though Bushmen valued close relatives such as their genetic parents, grandparents, and siblings, their languages had no word for _family_. When Bushmen spoke of kin, they didn't necessarily mean a blood relative. What they often had in mind was anyone sharing a given name, which bespoke this fictive connection. A man couldn't marry a woman who was considered his sister no matter how distantly related. The primary value of fictive kinship was to maintain social networks based around rules concerning everything from monumental issues, like marriage, to such details as who should exchange gifts with whom. We retain a whiff of this view of relationships when we label an unrelated individual as if she were family\u2014right, sister?\n\nOne reason not to invest too much significance in distant blood relatives is that, measured genetically, only one's immediate family is a big deal. Who's a relative is, indeed, relative. We are all related at some level. Beyond the very closest family ties, kin merge into the general population in terms of their shared DNA, and quickly. Do the math and you find that first cousins share 12.5 percent of their genes by descent; second cousins drop to a measly 3 percent. Even two people taken from a community at random may have that tiny fraction of genes in common by chance alone. Whatever relationship IQ humans possess is therefore likely concentrated on a few closest kin.\n\nIn practice however, who we consider family is influenced by culture. Many Latinos, to give one example, rely heavily on extended families. Even so, few cultures set absolute rules for family membership beyond the immediate family. In describing our family we don't expect to come up with the same set of names as would one of our cousins, although we can anticipate some overlap. Despite that, connections to a broader web of blood relations can have value for survival and reproduction when those kin are obliged to come to each other's assistance. Of course, helpfulness occurs between nonrelatives, too, and can be equally expected. The phrase \"treat each other like family\" describes how common such cooperation is among members of tight-knit groups such as military units and religious congregations. Neither is it accurate to say that such aid comes simply or dependably even from genetically close family members. Recall how you clashed with a brother or sister, no bombshell given that sibs compete for the attention of harried parents. Dickens, in _Bleak House_ , mentions \"a melancholy truth that even great men have their poor relations.\" However, families can rise up to punish any among them who isn't generous. This makes family obligations hard to escape. For most people, regardless of any unpleasantness, kin are kin for life.\n\nIf bands of hunter-gatherers thought so little about actual genealogies, where did our current obsession with, and dependence on, extended family relationships come from? Family trees became a concern for hunter-gatherers who abandoned the lifestyle of owning no more than they could carry. Settled peoples who stood to inherit a social position and material items had good reason to know their family trees. Likewise, in industrial societies, extended families have been most nurtured when there is wealth to share. The very size of those societies also puts a premium on people reliably building a broad set of relationships and kin networks represent a consistently available avenue for people to do that. The learned skill to conceive of extended families\u2014and to keep track of, and value, those relationships\u2014has been a recent add-on in human evolution. It requires complex communication and learning and is heavily dependent on what each society expects.\n\nMight people's relationship to their society be a clerical error in the mind, a misfiring of their mental representation of kin? That would be as if people mistook their society for an enormous family stretching, as the patriotic song puts it, from sea to shining sea. Some hunter-gatherers did seem to apply an approximation of this representation of the society as kin. For them fictive kin were constituted to take in the whole society. Each individual could identify every single other member with a kin term. Such universal kinship persists in faint echoes in words like \"brotherhood,\" shorthand for the ethos of treating everyone as if they shared blood. As we have seen, though, kinship truly was metaphoric among hunter-gatherer bands, having little to do with blood\u2014and their societies even less so.\n\nFor our species, societies of distantly related others are the norm. No society, even the \"motherland\" of a small tribe, was ever made up of the offspring of one mother, the way a colony can be for insects. Even in social insects\u2014whose success is often explained in terms of kin selection theory and the tight genetic relatedness of the mother queen to daughter workers\u2014a queen will mate with several males, producing progeny with different fathers. More impressively, an Argentine ant society, for one, is no tight family. A big supercolony has genetically varied queens. Yet no ant prefers her immediate kin or even recognizes which queen is her mother or which individuals are her siblings. As in any other ant species, identification with the colony, and not kinship, is the focus of each worker's industry.\n\nI find it doubtful that human societies are simply amplified versions of family relationships. However, this does not rule out the possibility that family bonds played a role in the earliest incarnations of societies, in humans and other species. The first glimmer of a society would have sprung up among our forebears well before the chimp, bonobo, and human species split off from each other, set in motion perhaps when a proto-ape extended its attachment to its infant to others. It may have been that, since primates don't breed as prolifically as ant queens, the full advantages of group life\u2014for example in staving off belligerent outsiders\u2014couldn't be attained without extending the size of the group beyond one family. Some anthropologists argue that the markers that identify societies or ethnicities (dress, hairstyle, and so on) substitute for similarities between family members. I don't find this convincing given that kin are seldom consistently alike. Still, from the perspective of very deep history, a society might represent a kind of motherland.\n\nAll of this said, there is no denying that kinship is a potent force. The obligations to immediate families are as entrenched in the physical operations of the brain as are the commitments to societies, with societies and families involving inherently different aspects of life. Scientists could benefit from a course correction by focusing a bit less on kin and more on the psychological, and biological, basis of societies.\n\nAs has become clear, the human psychology around societies is far ranging. In the past several chapters, we have teased apart how members of a society are registered as having essences, rendered equivalent to biological species. Additionally, we investigated how our inexorable, blazingly fast, and clannish evaluation of other people is based on this perception. Our biases extend right down to how human or animal-like we perceive their capacity for emotion and the overall warmth and competence of their \"kind.\" We have seen, too, how these assessments play out at the level of populations. We are disposed to discount the differences between individuals and to perceive the members of other societies\u2014and to a lesser extent those of our own\u2014as both similar and forming a unified whole. Finally, we have considered the ways in which the psychology of families relates to perceptions of our society, granting the influence of both in human affairs. What we have concluded is that even though biological kinship has a solid basis in genes shared by descent, whereas societies are communities conjured up purely by the imagination, the role of societies in human psyche and thought is foundational, and of vital importance.\n\nThat's because choices regarding who should be treated as members of a society can be critical to survival, whether those humans happen to be blood relations or not. When people are perceived as foreign, all bets are off. The likelihood of rivalry\u2014and collaboration\u2014between societies will be our upcoming topic.\n\n# SECTION VI\n\n# Peace and Conflict\n\n# CHAPTER 17\n\n# Is Conflict Necessary?\n\nOn a trek with the research team of primatologist Richard Wrangham in Uganda's Kibale National Park, I had my first encounter with wild chimpanzees. My heart almost jumped out of my body when I heard the hooting and hollering of a dozen acrobatic chimps rummaging for fruit in a fig tree that loomed over lesser trees above me like a shadowy fist. With their thick bodies, the apes were more intimidating than I had expected. They were endearing nonetheless, holding hands, playing tag, and cuddling. It was part frat party, part Zen gathering. I was surprised to find myself at peace, as if I were among friends.\n\nThe reading material I had brought along, including Wrangham's _Demonic Males: Apes and the Origins of Human Violence_ and writings by Jane Goodall, undermined my euphoria. I could only imagine the shock Goodall had felt in 1974 when, after years of relative calm at Tanzania's Gombe National Park, a bloodbath began. One community of chimps incrementally destroyed another in what became a very lopsided four-year war. The chimpanzee violence evoked the worst facets of human behavior, in which the members of a society, reacting en masse to others whom in many instances they don't even know, cast aside their qualms about violence to attack outsiders.\n\nThe capacity for such violence is a thread that joins humans to chimpanzees and other species. Voltaire wrote, \"It is lamentable that to be a good patriot one must become the enemy of the rest of mankind.\" Chimps, classified in the genus _Pan_ , seem always to be willing to battle the rest of Pankind. Still, though it contains more than a grain of truth, Voltaire's blunt conclusion goes too far\u2014at least for people. Humans are flexible in our range of options for obtaining resources and bringing down oppressors, including aggression, tolerance, and collaboration between groups. How human societies choose between these options is the subject of the current section. In focusing first on murder and mayhem, this chapter, more so than the last few, returns to nature to see what insights can be gleaned about human behavior. In our morbid captivation with the dark side of our past, what we really want to know is whether human societies are fated for violence.\n\nBefore Goodall witnessed Gombe's killing spree, documented conflicts among chimps involved animals _within_ a society. Males were known to fight, at times to the death, over social position; and females, it was subsequently discovered, were capable of killing the infant of a rival. It wasn't simply that no one noticed violence between chimp societies. Most thought societies didn't exist; they weren't aware that chimps lived within strict territorial borders, let alone that they protected their space ruthlessly. You may experience a sense of d\u00e9j\u00e0 vu: the primatologists' initial ignorance of chimpanzee communities recalls the entomologists' original view of the supposedly peaceful Argentine ants, before the ants were found slaughtering each other along territory borders. Because society membership is not always visible in daily life, societies, profoundly important as they are, can be easy to miss.\n\n# ACTS OF EXTRAORDINARY BRUTALITY\n\nAggression between societies is unlike anything normally seen between fellow society members. Within a chimpanzee community, aggression mainly involves fights between individuals, or sometimes several allies ganging up on a victim. Fights between groups within a community, involving many individuals on both sides, haven't been reported. A party of chimps, for example, may be cautious upon drawing near another party of their own community but are never outright hostile. Foreign societies are the targets of group violence.\n\nAmong chimpanzees, such violence is unleashed in raids on the neighbors. Their fission-fusion social structure makes the animals vulnerable to attack. Raids stay away from any parties that, by the sound of them, are large enough to put up resistance and instead single out whatever target, male or female, they happen to find alone. These attacks seem to be the goal of the raiders. The apes don't appear to be hungry and don't stop to feed. To keep the enemy from staging such stealthy assaults and getting out unscathed, patrols keep vigil along the borderlines, at times moving quietly, other times with bravado. Both patrols and raids are almost entirely composed of males, the hypercompetitive sex focused on territoriality.\n\nThe raids come out of the blue. By killing off foreigners, raiders can over time weaken and sometimes eventually eliminate the targeted society, as occurred at Gombe. The long-term consequence is that the aggressors expand the society's realm into the adjacent territory, improving the community's access to food to raise young and attracting more females\u2014even adding a surviving female or two from the losing group.\n\nThere are animals for whom conflicts between societies are tests of strength that as a rule fall short of violence. Ringtailed lemur troops face off with females cuffing and lunging and giving a ruckus of cries, while the males flutter their tails to waft an intimidating scent. Meerkat clans leap face to face in a war dance, their tails erect. Even in these species, though, matters escalate if the two sides are equally matched and neither backs down. Losers are wounded or killed and sometimes forfeit their property. A few other species are more like the chimpanzee in their unrestrained violence. Fights between spotted hyena clans and naked mole rat colonies can be bloodbaths. Closest to the chimp in attack strategy is the New World spider monkey, another fission-fusion species. The males join up to raid the neighbors, taking the unusual measure for a treetop animal of tiptoeing single file on the ground as quietly as chimps on a raid. Yet it's the gray wolf that most closely matches the chimpanzee for pure viciousness. Wolves regularly kill members of foreign packs, often in the course of brazen invasions of another territory in search of game.\n\nWolves don't dispatch members of outside packs with the quick chomp to the neck they employ to take down elk. While visiting the wolf researchers based at Yellowstone, I learned that a pack had just killed an old female and her companion from another pack. Both died from abdomen and chest bites apparently inflicted over many hours. As for the Gombe violence, \"There were gang attacks of extraordinary brutality,\" Goodall would recall. \"They did things to their fellow chimps that they would never do within a community but which they do when trying to kill a prey animal.\" That's an understatement: the viciousness of chimps and wolves toward foreign animals can exceed what they employ to bring down prey or kill rivals within their own community.\n\nViolence is far less common in humans than chimpanzees, which experience aggression every day. That said, conflicts between human societies reach extremes of depravity and probably always have. Evidence of early wholesale slaughter among hunter-gatherers comes from the ancient cemetery site of Jebel Sahaba, northern Sudan, where 58 men, women, and children were buried approximately 13,000 to 14,000 years ago after each was impaled with 15 to 30 spears or arrows, more than necessary to kill a person, suggesting a community was brutally razed. Accounts also exist of Aborigines slaying each other in outright battles, one involving 300 individuals. An early European traveler recounted both sexes \"fighting furiously and indiscriminately, covered with blood... without intermission for two hours.\" In the end, the winners tracked the losers to a camp, where they beat them to death. The account continued: \"The bodies of the dead they mutilated in a shocking manner, cutting the arms and legs off, with flints, and shells, and tomahawks.\" Warriors across history have made trophies of their victims' body parts, from shrunken heads to scalps to genitalia, often to strengthen the essence of their own people by capturing the life force of an alien group. The pincushion-worthy surfeit of arrows and grotesque mutilations bring to mind the wolf killings at Yellowstone and chimps mauling outsiders more hideously than if they were prey animals. An otherwise psychopathic quality of violence is normalized when denigration turns to demonization. What would at any other time be condemned as heinous becomes reason for celebration.\n\nAs it is in other animals, group identity is key to understanding when and why aggression turns, literally, to overkill. Neighboring bands of roaming hunter-gatherers more than likely belonged to the same society and would have shown no animus. Of course, not everyone got along, and conflicts between individuals could turn nasty, but whole bands didn't hold grudges or regard the other bands of their society as hostile. Group aggression was generally directed at other societies. The extent and form of this violence has long been a contentious question in anthropology. What's certain is that the nomadic hunter-gatherers shied from high-risk engagements. Their situation was comparable to that of ants with small colonies, which similarly have no permanent structures and few possessions to protect: when outsiders threatened, it made sense to simply move away. The nomads carried out more dangerous actions only in times of crushing competition and conflict, as we saw when =Au\/\/ei Bushmen turned to battling aggressive neighbors in the nineteenth century. Massacres such as the one at Jebel Sahaba would have been rare for people in bands\u2014there was probably a settlement of hunter-gatherers at that site. For human nomads as for equally fission-fusion chimpanzees and spider monkeys, furtive strikes were the preferred choice.\n\nRaids were often justified as retribution against perceived wrongdoing\u2014maybe witchcraft or territorial incursion by the other side. Even if the raiders could identify the specific person who had carried out the offense that spurred their attack, they usually targeted whomever they could easily lay their hands on. Indifference as to the choice of victim was a convenient outcome of seeing outsiders (in the case of nomads, probably ones the aggressors knew) as identical and interchangeable. It also enabled the interlopers to get in and out quickly and unscathed. Once a people are reduced to a category, they are all equally targets. The biblical prescription of an eye for an eye does not distinguish between eyes; injustices inflicted by just one or a few foreigners are viewed as righted with attacks on anyone of their \"kind.\" Where this social substitution backfires is that the victims are viewed by their compatriots as unique and blameless persons. The suffering inflicted on those victims is felt as harm to all; thus retaliation is inevitable. No animal we know of directs aggressive reprisals at foreign groups in this way.\n\nGiven that hunter-gatherer bands had few material goods, what were the payoffs of these engagements, other than the immediate sating of bloodlust? In human societies, as in chimpanzee societies, the aggressors were predominantly males who wielded control of a private domain that (framed from a biologist's perspective) produced resources essential to raising offspring. One valued resource was the women who birthed the children, and women could be seized during raids. In addition the territory itself could be an asset worth taking. Descriptions of hunter-gatherers annexing territories are scarce, however. A detail provided by the anthropologist Ernest Burch about the I\u00f1upiaq Eskimo gives a clue as to why: \"The great majority of people thought that their own estates were the best places to live, and they could hold forth at length on what was special about them.\" Perhaps a grass-is-always-greener perspective seldom made sense in traditional societies where people grew up wise to every nook and cranny of the home turf. Coveting a neighbor's land would be illogical unless it offered a huge improvement over a region already intimately understood. Nonetheless, raids could continue until a rival group was abolished, and no swath of ground would have gone unclaimed for long. And as tiny as band societies seem to us now, their relative sizes must have been critical to their success. Bigger groups could have displaced smaller ones even when they chose not to make a show of their might\u2014the motto \"peace through strength\" taken to its prehistoric fountainhead.\n\n# VIOLENCE AND IDENTITY\n\nAnother payoff of aggression lies in reinforcing unity. \"Are wars... anything but the means whereby a nation is nourished, whereby it is strengthened, whereby it is buttressed?\" the Marquis de Sade wrote after the French Revolution, bringing to mind the Nazi rallies at Nuremberg. When the members' rancor is directed at foreigners, their identification with their society heightens; their sense of common purpose and shared fate, of the people rising as one, is invigorated. In the Yankee North of the United States, it took the Civil War to forge that sort of oneness. \"Before the war our patriotism was a firework, a salute, a serenade for holidays and summer evenings,\" Ralph Waldo Emerson mused. \"Now the deaths of thousands and the determination of millions of men and women show that it is real.\"\n\nWilliam Graham Sumner, the sociologist who coined the term \"ethnocentrism,\" wrote in a much-discussed passage a century ago: \"The exigencies of war with outsiders are what make peace inside, lest internal discord should weaken the we-group for war.\" To Sumner, external war and internal peace play out a horrid game of interdependence. Any competition and conflict with outsiders redirect people's attention from their competition and conflicts with each other and toward their identity as a group.\n\nWhether or not violence toward outsiders is obligatory to keep a society intact, it is well understood that contrasting ourselves to outsiders, and primarily those we cast as enemies, helps place our society front and center in daily life. We are drawn together by the urge for self-protection. The Israeli psychologist Daniel Bar-Tal tells his readers that every society picks out groups to \"serve as symbols of malice, evil or wickedness,\" whose threat, even if real, tends to be played up. In recent years Russia, North Korea, and Iran have taken on that role in the United States. Should an adversary be lacking, people may flounder until a new one is found\u2014or invented. We rise up as one against terrorists, asylum seekers, illegal aliens taking jobs, or members of our own society thought to have false beliefs, shifting our ire from one to another with ease. When this bad blood is deeply embedded in a group's self-identity, it becomes too precious to relinquish. Such intractable positions are held by many Israelis and Palestinians; each side has come to function with exceptional solidarity and staunch commitment to recognizing their differences.\n\nMaking matters worse, our flawed capacity for assessing risk means that our responses to outgroups are frequently overreactive. Part of the problem is the way people cherry-pick information, a proclivity that doesn't bode well for national or ethnic relations. We are more likely to remember a foreigner doing our society harm than one performing an equal measure of good. News of terrorists triggers our prejudices even though dying from an act of terrorism is much less likely than a fatal slip in a bathtub. Our touchiness makes sense when you consider that the human mind would have evolved to discern anything motivated to hurt the people who came together in the small groups of earlier epochs. Our hypersensitivity to such malicious hazards\u2014a holdover from the distant past\u2014can set humans too hastily on the warpath.\n\nThen again, maybe anxiety about physical harm isn't the sole cause of our trepidation. In understanding our fears, we should not underestimate the role of identity. That includes our grasp of the identities of disliked or dreaded outsiders. Humans are susceptible to latching on to negative stereotypes even when they have no basis in reality (perhaps explaining the belief once commonly expressed among tribes that the feared people of the next tribe over were cannibals). Intense emotions can be bolstered by potent aspects of human markers, especially objects of symbolic power and how they are treated. And so it is that if foreigners show deference to our flag, we register them as trustworthy and are warm in return. Conversely, any perception that a symbol is mistreated enrages: consider the clamor that arises when our leader is beaten in effigy. Our attachment to emblems of identity suggests that the reason we panic about terrorism and not about tubs is not that terrorists are likely to endanger us as persons but that they can harm objects of symbolic importance to our society. Think of the Twin Towers and the Pentagon. Fears of a repeat of 9\/11 have put group emotions on high alert.\n\n# MOVING AWAY FROM VIOLENCE: LESSONS FROM NATURE\n\nThe question of questions remains as to whether Alfred Lord Tennyson's characterization of nature, \"red in tooth and claw,\" applies to societies. Does structuring the world into groups inherently lead to alienation between people? This is indeed the case for ants, which show no alternatives to conflict between the societies of their species, invariably battling or scrambling for resources before neighboring colonies snap them up. One might expect it to hold true for people too, for a simple reason: there's only so much room for the societies of this world, just as there's only so much space for individuals. If societies thrive from the competitive edge they afford their members against outsiders, it may seldom pay to willingly relinquish anything to outsiders unless the act entails some obvious reciprocal gain. Hence the Latin proverb _Homo homini lupus_ : man is wolf to man.\n\nAnimals can let down their guard around outsiders under some circumstances. Something approaching harmony has even been documented in the pugnacious gray wolf. Wolves, like spotted hyenas, can pass through foreign territories to pursue migrating herds. Admittedly it's hard to tell if this is politeness by the owners or sneaky trespass. More convincing is the behavior of wolves at Algonquin Provincial Park in Canada, where 40 years ago the packs initiated a tradition that's been seen nowhere else. Instead of occupying their territories year round, the wolves follow the exodus of deer each winter to a small plot of land near the park called the deer yard. Here, the deer become concentrated enough to feed all the wolves. The packs don't exactly turn convivial, yet at tenfold their usual density, they live in peace. In one instance two packs joined for a time without incident; three packs once ate, with no fuss, from the same carcass over the course of a day. Such cautious indifference can be considered a minimalist form of partnership. It's also a testament to the flexibility of an ordinarily xenophobic species that we assume operates on instinct.\n\nAmong the species celebrated for their friendliness, the savanna elephants and bottlenose dolphins stand out. In the elephants, the affection is typically most intense in cores that share a history, having split off from each other in the past. Members of bonded cores perform greeting ceremonies, rumbling and trumpeting while flapping their ears and spinning in place. The members of one core know those of the other, and certain individuals are tight friends. The dolphins in Florida enter each other's communities and interact in a friendly way. But not all is honey and roses even for these species. The battle scars of male bottlenose dolphins have been attributed to border skirmishes. Savanna elephants steer clear of cores they don't know or don't like by eavesdropping on the others' infrasonic calls. Often a big core, or one with a strong matriarch, will displace a weak one from a tree or water hole. These power plays can affect the health and reproduction of the individual members and the survival of the societies themselves.\n\nMost renowned for peaceful relations between societies is the bonobo, with communities openly intermixing. Each community has a home range, yet rather than being rigidly defensive about these spaces like the chimpanzee, bonobos pass through to make social calls, the kids tagging along. Bonobos even prefer to give food to strangers over community members, a remarkable indicator of how enthusiastically they build foreign relations. That said, bonobos are hardly cavalier about their social visits. While callers don't sneak in like raiding chimps, territorial borders exist and are crossed with due caution. The residents may respond to their appearance with frantic chases and screams, bites, and scratches. Everyone usually calms down, but should a social mixer not be in the cards, the visitors go home. Moreover, certain communities are never seen together. Much the way two individuals can be on bad terms, some societies apparently have their irreconcilable differences. Under these circumstances the bonobos do stick to their side of the territorial lines.\n\nSuch caveats aside, why are bonobo societies generally so permissive? The rarity of violence has been attributed to the general abundance of rich sources of food in their habitats. If that's the case and bonobo communities are only fair-weather friends, good relations could be rescinded just as the truce among Algonquin wolf packs would collapse if the winter deer population didn't fill the stomachs of all of them. During intervals of conflict, the merits of being in a strong bonobo community would become obvious. Luckily hard times seem to be few and far between in the part of the Congo that all bonobos call home. In any case, bonobos are fully capable of putting up a fight. \"Veterinarians sometimes have been called in following altercations to stitch back on a scrotum or penis,\" anthropologist Sarah Hrdy dryly reports of captive bonobos. In addition to the occasional assault, they may kill in the wild. In one case several bonobos ganged up on a male of their own community. Researchers suspected a murder, though no body was found.\n\nEven in some of the least violent species, the most the animals manage in terms of foreign relations is to ignore or avoid outsiders, as the wolves in the deer yard do. Sperm whales live among foreign clans but stay out of their way\u2014they are so massive that any clash could be life threatening. The geladas are masters of disregarding outsiders: their troops intermingle in a show of indifference, aside from the caution the alpha males display around bachelors out for trouble. This may tell us only that among these primates, which eat abundant grasses, there is little or no competition over food.\n\nSpecies usually eager to fight outsiders can have their moments of tolerance, too. When a lion pride splits up and each half takes over part of the original territory, they appear to give the others breathing room to settle in. Even so, within a year or two the former pride mates are just as hostile toward one another as they are to complete strangers. Baboon and mountain gorilla troops intermix and let the youngsters play together when males have no sexually receptive females to come to blows over. Prairie dogs allow for a d\u00e9tente in communal foraging grounds outside their territories on land too poor to dig burrows, and therefore apparently not worth defending.\n\nAnd what about the chimpanzees\u2014can the territorial males reverse the demonic nature that Wrangham attributes to them and coexist with neighbors? When it comes to accommodating foreign communities, this species may be at the bottom of the list. The best that can be said is that some populations raid and kill foreigners less often. Their relative restraint has more to do with lack of opportunity than peacemongering, however: the chimps in those areas stick to big parties and consequently are seldom vulnerable to attack.\n\nGiven the genetic proximity of both chimpanzees and bonobos to humans, it seems reasonable to conclude that our base level of distrust of foreigners, and appetite for doing them harm, is part of our shared heritage with the chimp, while the ability to put aside our qualms and bond, therefore, is a gift we have in common with our bonobo relations. Our two cousin species become contrarian angels standing on our shoulders, whispering their good and bad advice. Fortunately the human species has cut down on the impulsive sort of violence common in chimpanzees, our greater emotional control and toleration of each other an adaptation we share with bonobos. Yet the message from nature is hardly optimistic. Time and again we find that animal societies can be on good terms, or at least not harmful to each other, when conditions suit such behavior: when there's little competition for property or mates, or when fighting over them is not worth the trouble. Such ideal situations are seldom lasting. In Algonquin wolves, for example, violence returns as conditions go downhill, after the deer spread out again for the summer and become more difficult to hunt. Once the packs are compelled to compete over food again, the wolves reestablish their nastier habits.\n\nConsidering the state of affairs we see across species when competition is intense, and the human response to competition and clashes of identity with foreign powers, it seems fair to assume that equanimity with the neighbors will be a perennial struggle. Plato is said to have claimed that only the dead have seen the end of war, and no doubt that's true. War can unite people, as William Sumner described, and as troubling as that seems, a society unable to draw on this selective focus would be defenseless in times of peril.\n\nNevertheless, this much is certain: societies don't need to be antagonistic toward outsiders any more than they require their members to cooperate within their borders. Inimical societies spend most of their days not fighting, even when their peace is uneasy. They may not have a high regard for the other people, how those people live, or their symbols, yet they aren't forever burning each other's flags, either. At times that's all two societies can achieve\u2014the forbearance to set aside disputes and go about their business like the wolves in the deer yard. Such arrangements have worked out during periods of intense conflict, if only briefly, an example being the Christmas truce of 1914, when the German and Allied troops felt free to enter the no-man's-land along the Western Front and sing carols and drink together for the day. Even seeing foreigners as beneath us doesn't need to equate to hostility, as is attested to by the macaque. I have mentioned that these monkeys promptly connect an outsider with vermin ( _horrors, a spider!_ ); the usual outcome of this bias is that the troops give each other space rather than go on the attack. The negative stereotyping and species-like distinctions between societies that come so naturally to humans don't force a violent response out of us, and perhaps this monkey-like lack of intolerance, so to speak, was a steppingstone to the first rudimentary human alliances.\n\nWhile a common adversary can unquestionably motivate people to rally around their society, war is a tactical choice, and we can have enemies we feel no compulsion to kill. Indeed, the fact that international alliances occur at all is proof positive that being loyal to, and favoring, one's own society is a psychologically independent act from thinking poorly of outsiders. One can exist without the other. So long as their mutual antipathy is not deeply set, the cooperation between human societies can surpass anything observed in nature. Up next, the conditions under which such warm relationships thrive, and what that tells us about humanity.\n\n# CHAPTER 18\n\n# Playing Well with Others\n\nLike many explorers, I've had the experience of trading fishhooks for something I needed\u2014in my case, a dugout canoe made by a tribe in the rainforest near the Pacific coast of Colombia. Nothing seemed remarkable about closing the deal, even if it did require downing a fermented beverage that wasn't exactly to my taste. Yet however ordinary that transaction may have seemed, given everything we know about strife between societies in the natural world, this willingness on both sides not only to forgo hostilities, but to see a foreigner as other than toxic, is pretty special.\n\nAs you may have surmised based on the verdict of the previous chapter, any positive contacts between animal societies are problematic, and actual collaborations between them are rare or absent. Oftentimes their positive interactions look more one-directional. Rather than representing one side of a trade or agreement, the bonobo's generosity in giving food to strangers is better thought of as an olive branch. The apes don't appear to expect anything from the foreign animals in return other than, perhaps, their tolerance\u2014let alone to team up with them to get anything done as a group. Savanna elephant and bottlenose dolphin societies similarly show no evidence of moving beyond camaraderie to partnership. The arrangement closest to an alliance exists in sperm whales, which team up with other unit societies to catch squid more effectively than either unit could alone.\n\nHuman societies can accomplish a greater payoff by working together rather than against each other, mitigating any competition over resources, a feat that other animals seldom achieve. The societies of our species can, for example, turn scarcity into plenty by extracting more from the environment with outside help (just as two whale units do when hunting squid). In the face of the very real risks posed by foreigners, it's worth asking how and why humans evolved these enterprises. And once they did, the question becomes how they were able to balance the demands of being in an alliance considering the pressures for their societies to remain separate, and distinctive. Evidence from hunter-gatherers guides us in addressing these issues.\n\n# THE VARIETY OF ALLIANCES\n\nOne of the most remarkable collaborations to be found among hunter-gatherers anywhere was the one between the eel-collecting Aborigines of the Mount Eccles region of Australia. At least five groups speaking different dialects of Gunditjmara, and quite likely other peoples of the region as well, constructed the expansive waterworks that made it possible for all of them to catch the fish. Their relationships weren't without bloodshed. Sometimes wars broke out. But compared to the battle-prone Indians of the Pacific Northwest, similarly dependent on fish runs, the Australian tribes treated each other almost collegially, and with a clear mutual payoff: everyone's success relied on their labor to maintain the waterways. Eel harvesting was an international effort.\n\nSome hunter-gatherer alliances involved the movement of personnel to achieve a mutual goal, usually defense. Although the behavior is unknown in other animals, combining forces against a common enemy has been a justification for binding human societies, much as it can be for binding the members of a single society when outsiders threaten. The American Indians collectively called the Iroquois, hunters with rudimentary farming who occupied what is now western New York State, hammered out a confederacy before the arrival of Europeans, sometime between 1450 and 1600. Member tribes were self-sufficient and independent; however, a grand council met when necessary to direct their relations and coordinate defense against outsiders, Europeans eventually among them. Beyond that, the tribes collaborated in an ever-changing manner to suit their interests, as tribal and hunter-gatherer peoples were able to do elsewhere in North America and around the world.\n\nFor humans, the creation of plenty by way of intergroup relations is often accomplished through trade. The possibility of this arrangement is universal. The Arawak presumed that Columbus, though from a society far outside their experience, would be willing to swap goods. Unfortunately, in seeking mutual gain they found their own ultimate demise, slain by forced labor, musket fire, and diseases until they vanished as an independent entity. Lewis and Clark passed through the territories of Native Americans who were normally hostile to outsiders and might have had no compunction about killing them\u2014except the Europeans presented themselves as prospective trade partners. To sustain these associations over the long term requires a delicate interplay among the parties. Ideally, they must see each other as independent but on a par in strength and importance. Either that, or the stronger group can dictate the terms of the arrangement to the disadvantage of the others.\n\nThe Arawaks' fate reminds us that contact with foreigners is a fraught business. Humans have always turned first to those of their own society, whose shared identity makes social and economic interchanges easier to navigate. There's greater risk of duplicity or misunderstanding in negotiating with those whose ways and values, and, of course, language, are foreign. Judging from research on interactions between races in modern societies, getting past the differences may consume enough mental attention to make mistakes likely. Adding to the anxiety is the possibility of rejection or reprisals should things go sideways. Self-sufficiency is desirable. Everything, then, from finding a mate to procuring meals to keeping enemies at bay, tends to be handled internally whenever possible.\n\nNor do societies have to be open to trading and cultural exchanges. Some Bushmen cooperated with neighbors readily, some did not. Think of the dividing line between societies as a bottleneck to regulate the movement of people, information, raw materials, and prepared goods. All these assets move reasonably unimpeded within a society, but in a regulated, often sharply curtailed flow among societies. Once in place the bottleneck can be widened or narrowed depending, respectively, on the payoff or loss brought about by contact. When a foreign society was disliked, its influence felt as a blight on the local culture, the bottleneck could tighten. Even so, acceptance of desirable goods and innovations, whatever the source, was unstoppable, which could account for the wide spread of newly invented stone tools in the early archaeological record.\n\n# KEEPING THINGS MOVING\n\nIt's hard to say when the first collaborations existed between human societies. Not only are hunter-gatherers known to have traded with farmers for centuries, they have long sought out commodities from each other's societies as well. Unfortunately, alliances, and even trading, don't leave clear evidence. Finding a tool chipped from rock that could only have come from a site many kilometers away may be used as evidence of trade between societies. Yet the implement could have arrived there by other means. Rather than being acquired in a transaction with outsiders, possibly the same intrepid person carried it the whole distance, or it was exchanged between members of one society whose territory covered that area.\n\nOr perhaps its migration occurred over a series of steps from one society to the next, in what's called a transfer chain. Illustrative are ancient pieces of Chinese pottery that switched hands until they reached villages far off in the interior of Borneo. Even then trade doesn't always explain the movements. _Coenobita compressus_ , a type of hermit crab, shifts to a larger shell each time it outgrows its home. Oftentimes the shell it chooses had been previously jettisoned by another crab, much as one person may carry off a bauble dropped by another. The shells move under crab power 2,410 meters a year on average, one fusspot crab to the next. Scaled to human dimensions, that's equivalent to each shell voyaging a kilometer every single day.\n\nNuts buried by rodents are an example of a transfer chain, in this case propagated by theft. Squirrels steal acorns that others have buried for later eating. In this way, a nut bounces from one spot to another for kilometers until it sprouts or one of the thieves eats it. Some ants and honeybees rob other colonies, grabbing food from foreign workers and dashing off. These examples remind us that stealing is a time-honored alternative to trading as a means of moving objects over the landscape, one that predates humankind.\n\nWhile we don't know whether early human societies depended more on commerce or plunder, brute force would seldom get people what they craved on a reliable basis. This reflects that human necessities are more complicated and diverse than those of squirrels or honeybees. Once people started to rely on goods like arrowheads and body paint, locating adequate food and water was no longer their sole concern. Not all the merchandise a society fancied was likely to be available on the real estate they controlled. While people met whatever material needs they could locally, building relationships with foreigners open to long-term commerce became, in many instances, not a luxury or token gesture of friendship, like a bonobo's gift of tidbits to an outsider, but a requirement.\n\nStill, commerce may have begun with something very like the bonobo's easy and amiable contacts with outsiders. Even before that, perhaps, mere mutual tolerance was sufficient to enable the flow of goods in both directions. Band societies often acquiesced to outsiders entering their land to harvest a desired resource themselves. Not that it was ever possible to completely stop trespassers. While a blackbird able to scope out every corner of its small domain from a central perch can see and try to block all incursions into its private space, the dominions of the societies of humans and many other animals are too expansive for that. A meerkat clan will go so far as to brazenly sleep in one of the burrows of another clan when its owners are elsewhere. At best the residents can confront the interlopers they discover, repulsing them most zealously from the interior of the territory, the geographic center that in many other species contains the den.\n\nBut hunter-gatherers were canny at reading the environment. American Indians and Bushmen not only detected days-old footprints, they could identify the sex, age, and often the exact person who made them. Intruders were likely to be found out if only after the fact. That made it prudent to avoid retribution by seeking approval to enter the land of foreign groups. In any case outsiders often couldn't simply creep in and find what they wanted\u2014they depended on the locals' up-to-date advice on when and where to go. Thus guarding the land was usually unnecessary as well as impractical. Permission to access resources could be sought to solve an urgent predicament. Accounts of hunter-gatherers are full of stories of water holes drying out and of game migrating from one territory into another. In any case, the tolerance for visits came with conditions. Both sides were expected to return the favor on a continuing basis, and a reliance on such reciprocity kept people on their best behavior.\n\nDepending on the cost and benefits of defending or granting access to resources, hunter-gatherers differed in their possessiveness toward their realm. Some took the take-no-prisoners approach of chimps, others selectively protected resources as baboons do, while yet others were as open to outsiders as bonobos\u2014with the possibility of negotiating all kinds of concessions in between. But rarely was there ambiguity about who owned what. Precious assets, among them materials venerated for their symbolic importance (a pigment used in a ceremony, perhaps), could escalate rivalries. In general, though, the low cost of negotiating an easement added to the reasons not to seize foreign territories. This has always fostered contact, and familiarity, between human societies.\n\nThose with a superabundance of a valued resource could afford to be generous. Bogong moths make annual flights up Australia's Snowy Mountains in such numbers that when the time was right, foreigners visiting for the season were given spots on the slope where they caught and consumed around a kilogram of insects per person each day. The tradition lasted a thousand years. That weight in moths has the fat content of thirty Big Macs. By season's end the formerly scrawny people headed back to their home territories plump and content\u2014and one imagines with the expectation that they owed the locals something in return.\n\n# THE FIRST MARKETPLACES\n\nRegardless of whether tolerance of outsiders entering one's territory to harvest goods was the forerunner of trade, trade in the usual sense began when parties swapped goods face to face. The first simple markets may have been arranged at territory borders to respect each side's private spaces while allowing spot checks to ensure that exchanges were equitable.\n\nBut what was fair? Dealings among people from the same band society customarily took the form of casual exchanges. There was enough trust that exact parity wasn't expected, much the way Christians exchange holiday gifts. Should one side come up short, they could make it up next time. Trade between societies was different, being less predictable, with more haggling and oversight and risk of relations turning sour.\n\nThe easygoing attitude toward exchanges within societies faded as settlements grew. When traders usually offered very different goods and services and were strangers or barely knew each other, specific values had to be affixed to their offerings. As a result, interactions within societies became more on par with commerce between societies. Settlements of the Chumash hunter-gatherers of California used beads as a form of currency, setting value to goods in the modern sense.\n\nFor recent hunter-gatherers, transfer chains fueled primarily by trade accounted for wide connections between band societies. Such items as medicinal plants, grinding stones, and ocher moved from one Australian group to the next, sometimes across the breadth of the continent. Their value grew with distance, as was true for American Indian goods, too. Pearl shell may have seemed magical stuff when it arrived far inland for use in ornaments. Some items weren't used as originally intended. Boomerangs ceased being made in northern Australia centuries ago. Southerners who continued carving them were eventually able to swap the boomerangs for other goods when a fad swept across the north to employ these missiles not as weapons, but as percussion instruments for musical events.\n\nIn addition to raw materials and worked-up products, band societies traded ideas. Anything from a trendy word to an improved technique for making a tool could be copied across great distances. The circumcision performed during initiation ceremonies on Aboriginal boys was probably learned from Indonesian traders in the 1700s. The procedure gained traction over a wide expanse of Australia, some taking the practice to extremes by slicing the penis its full length. The Aborigines also copied each other's songs and dances. Best documented was an example first reported in 1897. The Molonga ritual of the Workaia people entailed several long nights of fantastical performances, the key characters in elaborate costume. Over the next 25 years, the Molonga spread across a 1,500-kilometer span of central Australia, even though only the Workaia understood its words.\n\nFor societies to interact without a hitch, some awareness of a social connection was a boost. Hunter-gatherers often arranged marriages between their societies for the purpose of courting allies; spouses usually had opportunities to visit home, giving them the equivalent of dual citizenship, something unheard of in other animals. Understanding each other was key. It helped that as a consequence of the history of intergroup relations, many hunter-gatherers spoke the languages of their neighbors. Both Australians and the Great Plains Indians also widely shared sign languages used in diplomacy. Some of the gestures were visible from afar so that negotiators could communicate beyond the distance a spear could travel. The same gestures had a second function: warriors on a raid could signal to each other to soundlessly coordinate an attack.\n\n# TRADING AND CULTURAL DIFFERENCES\n\nJust as individuals interact more comfortably the more they have in common, the same is true of societies: similarities ease the way to friendship. For example, similar languages and compatible cultures simplified the partnership of the Iroquois. Archaeologists speak of the interplay between societies as an \"interaction sphere,\" in which comparable values and other aspects of identity facilitate the traffic of goods. The very act of trading further enhances the resemblance between the societies. This is decidedly true when the transferred goods are more than simply raw materials, such as when the groups trade new ways of making or doing things or the manufactured items themselves.\n\nYet societies have to stay sufficiently distinct to preserve their members' sense of worth and meaning, or so the findings of psychologists would suggest. Herein lies the balancing act that has influenced the course of so much history. Commonalities are a plus\u2014to a point. Too much interchange can be seen as a threat a people's unique identity. Perhaps compounding the problem, being alike can also backfire should societies find themselves desiring, and coming to blows over, the same scarce items.\n\nEarlier I introduced the theory of optimal distinctiveness, in which individuals strive to be enough like other members of their society to earn their respect yet at the same time seek to be different enough to feel special. A reasonable hypothesis is that in building relationships with their neighbors, societies similarly gravitate to this middle ground\u2014an intense bonding that arises from the solace brought on by their similarities, and pride at their distinctiveness. To be a robust society, or a well-adjusted person, is to be the same _and_ different. Even the most look-alike societies must preserve hallmark differences close to people's hearts.\n\nReducing overlap among societies could correspondingly diminish competition; so goes a theory that has been proposed to explain, for example, distinctions in diet between tribes living in close proximity in the Amazon. And a desire for cherished differences could be fulfilled productively by the emergence of distinct economic roles between societies. After all, parties offering identical wares have little reason to trade. A society could exchange a surfeit of tools made by its members for items they found too arduous to produce themselves. Given the versatility necessary in a hunter-gatherer band, whose members honed generalized skills that differed only by sex and age, specialization among people may have originated at a societal level before working its way down to the individuals within societies.\n\nWhile such differences in aptitude were not ubiquitous between hunter-gatherer societies, the evidence suggests they often occurred. \"Each locality tended to make certain objects with a skill or flair which was admired in other localities,\" Geoffrey Blainey, an Australian historian, writes of the Aborigines. Different groups crafted spears, shields, bowls, grinding stones, jewelry, and so on. Indeed, Blainey adds, \"much of the specialization had existed for generations, and its origin was even the theme of tribal myths.\" In the Pacific Northwest, the blankets woven by the Chilkat Tlingit and the adze blades of certain other tribes were exchanged (or stolen) along the length of the coast. Numerous other examples of reciprocal relationships have been documented between small tribal societies dependent on domesticated food. For instance, the Fur agriculturists of the Sudan supplied millet to various cattle-herding tribes in return for milk and beef.\n\nBy achieving an optimal level of distinctiveness, societies would, firstly, become interdependent over long spans of time, their peoples finding all the more reason to avoid acrimony and seek each other out for their mutual benefit. Though to my point here, each society would preserve its boundaries regardless of how much its people were exposed to outsiders or how much they swapped with them or otherwise relied on them. That was true whether\u2014to consider two examples given from Australia\u2014they were barely scraping by in the Western Desert or were well-fed fishermen managing the waterways full of eels around Mount Eccles. No nation was ever ruined by trade, Benjamin Franklin said, and this is so not just economically but socially. The Mandan and Hidatsa of the North American Great Plains maintained clear identities even as their cultural centers transformed into trade hubs, a development that forced other tribes to learn their languages. As for the Iroquois, the confederation had to be loose enough that its tribes didn't cede autonomy or land. In fact, there was little contact between the common people of different Iroquois tribes and their separation persisted, and arguably strengthened, in spite of their interdependence.\n\nDissimilar peoples find ways to profit from each other, too. In fact, the path can be expedited the more radical their distinctions. Australian Aborigines welcomed the Indonesian fishermen who visited the northern coastline in the eighteenth century, while Bushmen traded goods with the Bantu herders who have lived among them for two millennia. Pygmies and their farmer neighbors took things a step further by codifying a relationship that bolstered the survival of both in a forest habitat where filling one's belly can be a struggle, whether by tilling poor soil or hunting scarce jungle prey. Subsisting as hunter-gatherers much of the time, every Pygmy group became associated with a village. There, each Pygmy kept a lifelong relationship with one of its farmers, working his fields for part of the year and providing him with bush meat and honey in return for crops and other goods. Connections between Pygmy and farmer are so ingrained and ancient that some farmers believe the Pygmies first introduced them to the forest.\n\nOne proposed role of markers is that they keep people from imitating outsiders when foreign ways could inflict damage. This I doubt very much. Of course, an outside influence can be detrimental, as when harmful drugs cross a border. But even if neighbors differ sharply, both tend to adopt from the other what suits them, without the interchange bringing about a calamity. Indeed, people can turn the mismatch into a boon. Certainly the problem of communication with a dissimilar other could present an imposing handicap at first. Still, their peoples would likely have vastly different material needs; even if one considered the other its inferior, they might not be rivals for the same things. In fact, their ways of life and skill sets, such as those of the Pygmies and the farmers, could be complementary.\n\nThe similarities and differences between societies, great or small, must factor into people's views about outsiders\u2014for example, their warmth and competence. That would in turn influence the extent to which the outsiders are seen as a formidable, potentially hostile entity or as persons who can be negotiated with in good faith. Once early humans refined their options for playing well with others, what emerged from these assessments were interactions between societies that could be as varied, fine-tuned, and adjustable over time as they are today. The constituent tribes of the Iroquois had been violent toward each other prior to their union; indeed, the peace between them was brought about only through war. As one expert on them grimly noted, \"At times the best way to make somebody stop fighting is to fight him till he does stop.\" After the Iroquois had negotiated an armistice, however, the tribes farther afield took their turn at feeling nervous. Harmony between societies can ironically bring about violence across a region, by establishing a more dangerous opponent for those left out of the entente. One enemy replaces another.\n\nThe neural wiring of the amygdala remains in place to fire its age-old cautionary reflexes of fight or flight. Overcoming those base urges to establish mutual trust between societies prejudiced against each other is a thorny business, a core problem of diplomacy. Even for groups on excellent terms, sublimated biases ensure the playing field is never perfectly level, as each side jockeys for a better deal. Our collective identities embolden us to be self-serving and cutthroat, undermining good relations and creating fertile ground for the birth of enemies when times are hard. Rivalries between groups do not create ethnocentrism; however, the competition draws out its more repugnant facets.\n\nHow do we avoid conflict when resources and opportunities dry up? In recent centuries, even with mass atrocities factored in, the probability of dying from an act of aggression between societies has declined globally. Arguably peace is fostered by the increased contact between countries. Nations also draw ever more on talents and resources from beyond their borders. Ideally, this interconnectedness and interdependence should see nations through periods of shortfall, at times when what tranquility exists between the societies of other animals tends to collapse. Avoiding violence when the potential social and material gains from war are high takes more than good intentions, however: it requires cultivating, and recognizing, the greater payoff, over the long term, of peace over conflict\u2014even among hated adversaries. Whenever that minimum can't be met, every nation must be committed to act against those who refuse to abide by the rules that safeguard the international order. This is a lofty goal. Hopefully it's an achievable objective in light of the perils of modern warfare. In no other species do societies coordinate to preserve the peace.\n\nThe volatility of relationships between societies finds its parallel within a society, where bonds among people are never static. The members' identities undergo trajectories of change over the very long term that can be predicted in broad outline, a fickleness that bears on the rise and fall of societies.\n\n# SECTION VII\n\n# The Life and Death of Societies\n\n# CHAPTER 19\n\n# The Lifecycle of Societies\n\n\"We do not even know how to determine approximately the moment when a society is born and when it dies,\" the eminent French scholar \u00c9mile Durkheim lamented well over a century ago. Despite the obvious practical and academic importance of the major questions regarding societies\u2014how they are founded, how they develop, and how new societies replace them\u2014no definitive answer has revealed itself since Durkheim's assertion in 1895. Durkheim pointed out that even the biologists of his time had resolved little about the life and death of societies. Although those lifecycles have by now been well studied for certain groups of organisms, the topic has indeed been largely ignored by the natural sciences as a subject of general inquiry right up to the present day. As for sociologists and historians, their reflex tends to be to treat the birth or dissolution of any society\u2014be it early Egypt or the former Czechoslovakia\u2014as an affair exclusive to its time and place.\n\nOf course, the devil is in the details. Nevertheless, the rise and fall of societies across the natural world indicates that our social groups have been engineered by evolution to come and go, much as the bodies of individual organisms are. This ebb and flow of societies is linked to how members perceive the identities of others. The behavior of animals and the ways in which human beings formulate our identities in changing social environments are closely tied to societal loss and rebuilding. Related to this deep question of identity is the critical topic of trauma and whether it is a sad necessity in the lifecycles of societies.\n\nThe dynamics of society genesis and transformation play out in a manner unique to each species, constituting the basic historical chronicle for its kind. That narrative depends on the rules by which the members of the species interact and identify each other, and the resources available at a certain time. Yet a theme emerges: Life requires the constant fulfilling of needs\u2014food, sanctuary, mates. When those needs aren't met, escalating physical and social stressors spur the decay of societies. More often than not, the troubles will be most severe when a society has overgrown what its environment can support. Even though a large society can overrun smaller neighbors, its swelling population aggravates competition among its own members, with the added burden on every individual to keep track of who is who\u2014should the societies of a species have opportunities to grow to a size where that is a problem. The members' proficiency at managing relationships and coordinating activities declines. This provokes a shift in allegiance to a subset of the society membership\u2014splinter groups in which everyone fares better.\n\nThe separation of subgroups into independent societies is de rigueur in vertebrates. Several lionesses, for example, will leave a pride that's grown too big to feed all its members. Should an aggressive male join a pride, females rearing cubs fathered by another male may depart, splitting the pride to avoid infanticide by the newcomer. Lions in an oversized pride are forced to make a fresh start with those they know and get along with the best, as is typical of species that rely on individual recognition. Such a severing of relationships differs sharply from the phenomenon I've been calling fission: the temporary and casual separation of a society's members that regularly occurs in fission-fusion animals\u2014species such as lions, chimps, and humans where individuals are free to separate and rejoin each other. When a society divides, there's little chance of piecing it back together.\n\n# A FRESH START IN THE CHIMPANZEE AND BONOBO\n\nHow new societies are born remains the most serious gap in our knowledge of our fellow great apes, the chimpanzee and bonobo. This seminal event is rare: vertebrate societies generally arise once in decades, if not centuries. The infrequency presents a problem. Data is scarce, for one thing. Worse, it makes it easy to shrug off incidents critical to the creation or destruction of societies, to ignore them as anomalies simply because they are uncommon. Such an occurrence might be a new arrival from outside the society or the death of a key animal in it. Either change can threaten the stability of the group.\n\nAn excellent example is the brutal conflicts among the Gombe chimpanzees documented by Jane Goodall in the early 1970s, puzzling at the time but which turned out to shed light on how societies split apart. Primatologists now realize what set off that spate of especially virulent nastiness: what had been one society had, under the watchful eyes of Goodall and her assistants, divided into two. The cleavage was the endpoint of a protracted process. A first indication that something was awry came in 1970, when some chimps were clearly associating more with each other than with the rest of the community to create two subgroups, which I will call factions. There's no evidence the factions had already been present, even loosely, when Goodall first came to Gombe ten years before. In any case by 1971 they had firmed up, one habitually occupying the north and the other the south part of the territory.\n\nInitially, members of those splinter groups mixed amicably when they did get together. The dominant males in the factions would furiously charge each other when they met, but that was nothing unusual, as contenders for alpha status within a community often challenge rivals. However, in 1972 the factions parted ways, establishing independent societies that no longer mingled. Goodall, recognizing that the chimpanzees had separated into two societies with distinct memberships, named the communities Kasakela and Kahama. Violence began after that split, with the Kasakela apes raiding the weaker Kahama to the south, in the end obliterating the Kahama community and taking back much of its territory.\n\nThe two-step process at Gombe\u2014the emergence of internal factions followed by division\u2014seems to be omnipresent among primates that live in societies, having now been documented for the troops of at least a couple dozen monkey species. Why it takes place I can only conjecture. Just as people do, other vertebrates seek out allies and mates, dodge or fight enemies, and ignore others. Among chimpanzees or bonobos on the move by fission-fusion, each individual can pick whichever of the shifting parties fulfills its best interests at the time. Normally an ape cultivates relationships widely, creating social opportunities wherever in the territory it travels. Such behavior helps keep the whole community interlinked. But with stress on the upsurge, perhaps because there are too many chimps, factions must take shape as the members focus their attention toward a more manageable set of generally agreeable individuals. At first those apes will remain part of the original community, the factions mixing without mishaps, given the many social links remaining between them. Because of their greater time apart, however, whatever allies each animal had on the \"other side\" will eventually be phased out of its life with the firmness we would expect of a person dropping a friend who joins a cult. Months or years after the factions initially form, they sever relations\u2014among chimpanzees all remaining ties between them are cut (with the rare exception, brought up in Chapter 4, of friendships between females of different communities, which they keep secret). A single community has generated independent entities, a pair of societies just as incompatible as any two colonies of ants.\n\nPrimatologists admittedly know few specifics about how the story unfolds in the chimpanzee. The only division anyone has watched is the one at Gombe. Likewise, there is a single record of a bonobo community division. The course of events pretty much paralleled that at Gombe. Unlike Gombe, the bonobo factions were already in place when the study began, so we can't say how or why they formed. The two subgroups were stable over the entire nine years of study prior to the breakup, aside from two females switching sides, and one male temporarily doing so. As time passed the factions had noisy quarrels. After they did divide, the communities kept apart for a year, at which point they became amicable, as is commonplace for the separate societies in this species.\n\nDivisions must be triggered when societies approach their uppermost populations. Chimpanzee communities seldom reach much beyond 120 individuals, and those of the bonobo are a bit smaller. At that size a community can be thought of as mature since it is large enough to dominate its neighbors, but such a size comes with difficulties. The relations within the society become strained, as they do among lions in an oversized pride who no longer know each other all that well. This might make it seem that maturity is the point at which societies divide, which must often be true. Yet at least at Gombe that wasn't the case: the community there broke up when it numbered only about 30 adults. Clearly the pressures that sever a society can build at any time. At Gombe, the researchers probably set off the division when they supplied bananas to attract the apes for study\u2014a seemingly good idea with unintended consequences. Chimpanzees mostly avoid competing with others in the community by dispersing. The tactic works out because most of their foods have a patchy distribution. But once all the Gombe animals were scrambling for the same concentrated food source, conflicts escalated when the individuals that would later form Kasakela, and wipe out Kahama, took control of the baits. The antagonism between those factions then went from bad to worse when the fruit was removed and they all faced a shortfall.\n\nA struggle for dominance also could have pushed the division along. Gombe's factions crystallized in the months after Leakey, the dominant male, died, leaving a power vacuum to be filled. The runner up, Humphrey, refused to concede the ascendency of Charlie and Charlie's brother Hugh. One can imagine the dispute between the males compelled everyone else to choose a side, the preferred option being the subgroup offering superior social stability, defense capabilities, food, and mates. Alternatively each of the apes may have simply chosen the portion of the territory that one or the pair of other alpha males happened to favor. Humphrey preferred the northern half of the territorial range where the Kasakela community eventually came to be. Fights over alpha status have fragmented societies in species from mountain gorillas to horses and wolves. A baboon troop can split if the females pick different male favorites or rebel against a tyrannical female.\n\nSocial antagonism isn't always a factor when a society divides. Across the social insects, including the majority of ants, queens leave their birth nests to found new nests on their own, without clashing. Honeybee and army ant societies can't function without big populations, and instead form colonies by division, but the mechanism is different from the procedure in other animals and requires no aggression. The workers separate into two groups, half sticking with the original queen and the others following a new one, her daughter. Everything goes smoothly despite this distinction in royal allegiance. And there are cordial divisions even among the vertebrates. An oversized elephant core often grows uncoordinated with the death of its matriarch, the instability causing the members to cluster around different females nearest to her in seniority; these factions increasingly separate until they become independent\u2014at times, though not always, on friendly terms. Sperm whales, too, form new societies without much strain, as when an outsized unit of more than 15 adults has trouble carrying out activities. The unit diverges into subgroups that wander ever farther away until they stay apart, without social tensions. But otherwise among the vertebrates some acrimony is pretty much guaranteed.\n\nWe have a lot yet to learn, and fingers crossed another division of chimpanzees is on the way that can be subject to a more detailed documentation. One community in Uganda has peaked at 200 members, the largest ever recorded. Factions formed within it at least 18 years back. That the community has persisted suggests the apes can sort out their social preferences over a very long time before achieving independence. I imagine each faction as a recipe on simmer, the individual members adjusting to the overall m\u00e9lange before congealing into a functional society that can stand alone.\n\n# OTHER WAYS TO FOUND A SOCIETY\n\nDivision isn't the only means to generate societies, probably even in our own species. In some other mammals, a solitary individual or mated pair can establish a society in a style paralleling that of most ants and termites, with their dispersing queens and males. Compared to a division, this course puts the individual at risk. The security of sticking tight to a society is a benefit that the members seldom forgo except under conditions of great opportunity (free space opens up that can support a lone animal) or peril (as when an animal or a group of them is aggressively pushed out by competitors). Among the mammals, only naked mole rats go solo as part of a regular cycle of society formation. Naked and defenseless they may be, but mole rats of both sexes, fattened for the trial ahead, make a dangerous sojourn aboveground, wandering from their birth nest to dig a starter chamber for a new colony. There the lone animal waits for a mate, or several mates, to discover it. Occasionally a pair of prairie dogs, and possibly of spotted hyenas, set up shop on a bit of land unoccupied by others of their species, although division is the usual course in both animals. As a last example, a pregnant gray wolf may go off on her own yet has it tough hunting and warding off enemies unassisted. Such a lone wolf is rarely lone for long, though life is hardly less dicey if a male joins her, considering that a pair of wolves is no match for a full pack.\n\nIn a pinch a chimpanzee can squeak by on its own as well. At one site in Guinea, males sometimes abandon their birth community, a behavior typical of females. A male deserter doesn't have the female's option of joining another community, presumably because its resident males would kill him. The apes aren't cramped for space in Guinea, however. Should the male be able to find a safe haven between community territories, he will stay put and try to mate with any female passerby seeking to immigrate. Whether such a female will remain with him to start a community from scratch is unknown, though their chance of success must be slim.\n\nPutting aside for now the topic of the importance to humans of being in a society over the long term, the \"obligatory interdependence\" some claim for our kind is a touch exaggerated. Apart from our childhood reliance on our elders, going it alone, or as a couple or family, is occasionally expedient. I've mentioned the Western Shoshone breaking up seasonally into families, but their bands would get back together again each year. Few have survived in unbroken isolation the way 24-year-old hiker Chris McCandless attempted to do in Alaska in 1992, with the tragic results recounted in Jon Krakauer's book _Into the Wild._ The perilousness of living alone puts the chance of a hermit couple begetting a whole society from scratch near zero. William Peasley's 1983 account _The Last of the Nomads_ tells the gripping tale of Yatungka and Warri, Mandildjara hunter-gatherers who struck out alone in Australia because their relationship was not recognized under tribal law. The pair was rescued several years later following their near-death in a drought. With good weather they may have had grandchildren by now. Even so as a germ of a society their descendants would have been dangerously inbred. Overall, then, going solo is a last resort. This option depends on numbers: ant colonies cast off prospective queens and the males to mate them by the hundreds, so colonies reproduce even as almost all queens die. No vertebrate breeds so prolifically.\n\nWhere a couple fails, a small group might stand a chance, in humans and other species. The exodus of a few animals from a society is called \"budding\" if they succeed in forming an independent group. Ideally they need not go far. Several wolves or lions might carve off a corner of their former society's territory, taking advantage of their access to an area they already know intimately. Should such a little entourage travel farther, its gains can be extraordinary if, by serendipity, the journey takes it to unoccupied lands flowing with milk and honey, or the equivalents thereof for the species. The supreme example of such a coup is an Argentine ant invasion, where what originally had been a handful of colonists mushrooms into supercolonies of billions. Some prehistoric migrations of humans would have been of this ilk. As with all invasive species, early peoples had the most success when they located a spot with few to no competitors. Some North American tribes started this way, as when the Athabaskans of the subarctic moved to what is now Mexico and the American southwest to become the forebears of the Apache and Navajo over a half millennium ago. More dramatic still was the relocation of hardy souls to truly far-off lands; whoever sailed on the first raft from Asia to Australia is a case in point. For such people, cut off entirely from their former society-mates, all of the terrain was theirs to claim. For each fortunate boatload that survived the journey, horrific numbers must have perished.\n\nStill other modes of forming a society exist in nature, but they seem to be largely closed to our species. A meerkat clan or painted dog pack often begins when a few males from one clan or pack join several females from another. This group-dating approach bestows the relative safety of a goodly population on the fledgling society from the get-go. A band of horses often originates when rambling individuals from different sources link up in what might be considered a mini version of a melting pot. The closest approximation of this among humans occurs when people from decimated groups come together to form a community, as happened with some American Indians and with escaped African slaves, known as Maroons, who erected societies scattered throughout the New World.\n\n# BREAKING HUMAN SOCIETIES\n\nDivision, then, seems to be the usual road to birthing societies in humans and most other vertebrates. Dividing entails clear advantages across species\u2014after all, both sides generally start off populous. The division of a human society is not likely to resemble the stress-free automated undertaking typical in honeybees, however. Groups of disgruntled mutineers don't rise up within insect societies, but people, after all, are typical quarrelsome vertebrates. Enough information exists that we can identify factors that would have caused societies made up of bands of hunter-gatherers to break down, and evaluate how those factors may play into the dissolution of settled societies, including the nations of today.\n\nDivisions wouldn't have been pushed along in nomadic hunter-gatherers by local social troubles, such as family disputes or social overstimulation for people who had little privacy. Given how fluidly people could move to bands located elsewhere in the territory, such conflicts could be resolved without the pain of a societal rupture. Tellingly, a dysfunctional band that split up left everyone's sense of who they were\u2014their identity\u2014unaffected. Life went on after people sorted out with whom in their society they were more comfortable. A societal division would more typically be the outcome, instead, of rifts between broader groups of people, across many bands. It's here that the two-step course of action observed in our primate cousins and other mammals would have played out in our species: factions would have emerged to which members became increasingly attached, followed, often years later, by a severing of relationships.\n\nWhat led to such factions is the question, since many of the factors that drive their occurrence in other vertebrates appear to matter little in human societies. Still, they are worth considering in turn. Shortages of food, water, mates, or safe havens, significant in divisions in other species, ultimately helped precipitate the fall of many human societies. Even so, resource shortfalls aren't necessary to the process. Because band societies were leaderless, their fate would not likely have hinged on the actions of specific individuals, in the sense described for the Gombe chimpanzees, which favored different contenders for dominant status. In any case, a division couldn't be coerced; people in bands would rise up against the idea. Difficulties in coordinating activities could have presented occasional problems, but there was usually little need for those living far apart in a band society to cooperate in any numbers anyway. Also, because band-dwelling people put little emphasis on blood kin outside their immediate families, weakening ties of biological kinship across expanding populations would not have mattered either. After all, fictive kin, people one could call father, aunt, and so on, would still be everywhere in the society. However, staying connected with allies might have become more difficult as societies expanded beyond a thousand people, but probably not by much. And because of the use of shared markers large (rituals, language) and small (mannerisms, gestures), the presence of strangers\u2014or leastways individuals who were ever more unfamiliar\u2014would no longer have been an issue the way it could be for chimps or bonobos in a growing community.\n\nIndeed, by sharing markers across their society, people add a twist to the mammalian pattern of separating into factions before dividing\u2014and this shift from individual recognition to anonymous societies would have made a difference in the proximate means by which societies unraveled. The crucial role markers play in societal breakdowns might not be immediately obvious. After all, shared markers have the power to mitigate the tensions between individuals that drive other primates to sever their connections. A recurrent theme of history is that when humans identify with each other strongly, they not just endure but join together and flourish under all but the most brutal conditions. Starve or persecute them, pack them together or spread them apart, yet rest assured: the ties that bind people most tenaciously, beyond those to their immediate families, will be their identification with a society. Where relationships would have permanently eroded in a monkey troop or prairie dog coterie, markers give humans the resilience to stay faithful to other members. In point of fact, given the advantage of a big population in out-competing other societies, we might predict that once our ancestors used markers to distinguish insider from outsider, human societies could swell without bounds. After all, when the markers of a social insect are reliable, no more effort has to be expended to bond to an astronomically huge society than to a tiny one\u2014Argentine ants continue to share an identity with the other ants in their supercolony even after it has spread across continents, obliterating all contenders from the territory.\n\nHowever, there's a distinction to be made between the fixed molecular brew that defines an ant society and the indescribably diverse markers that bind a human one. Despite the social sturdiness markers give humans, over time the stability they confer cannot be trusted. Our markers aren't set in stone, but are subject to alterations, yielding social class distinctions, regional variations, and more. Even though population size isn't an issue for _Homo sapiens_ in the same way it is for chimps, disruptive differences in markers are certain to materialize in our species when many individuals interact seldom\u2014the usual situation when early humans spread out in bands. The more changes in a society's markers to accumulate without the members adjusting to them, the more factious that society is likely to be. Eventually\u2014among bands of people, sooner rather than later, as I will illustrate next\u2014every society reaches a breaking point.\n\n# CHAPTER 20\n\n# The Dynamic \"Us\"\n\nThe Walbiri people, living in the deserts north and west of Alice Springs, Australia, and famed for their tribal dances and art, were recorded by anthropologists in the 1950s as believing they had existed, with the same religious ties to the land. Yet for them as for all of us, societal stability is an illusion. We suffer from a cultural amnesia, a selective memory that lets us imagine our people's undergirding essence as bedrock, etched out by its valued labels of identity. In truth, markers are in flux. The number of stars on the US flag increased from 13 to 50 without undermining the citizens' connection to their nation\u2014indeed, the multiplication of stars has been a point of pride. Even ways of doing things that seem essential to the life of a society, such as keeping slaves, mutate or die out. What matters in the long run isn't the specific markers we prize at the moment, but that whatever markers are in vogue ensure a society's separation from outsiders\u2014and this means societies are open to transformation. Ever since the ratcheting of human cultures accelerated in prehistory, societies have been subject to steady enhancement, reinterpretation, and outright reconstruction. The way things are properly done changes with the times without destabilizing a society or eroding the discontinuities between that society and others. When this social elasticity fails, however, we find ourselves on unstable ground.\n\n# IMPROVEMENTS AND INNOVATIONS\n\nOver the long term, then, the boundaries between societies trump the markers that define them. Still, the members act to minimize changes affecting what they regard as their society's preeminent qualities. For preliterate peoples many ingredients of identity, from spiritual beliefs to dances, were preserved through the ages with surprising accuracy. Repetition and ritualization served to \"comprise almost unbreakable 'codes' for the uninitiated,\" according to anthropologists at the University of Connecticut, all but guaranteeing that minutiae stuck. You don't have to turn to hunter-gatherer ceremonies and storytelling to be convinced of this point: early Greeks orally transmitted the _Iliad_ and the _Odyssey_ before inventing an alphabet. For those aspects of culture that require this kind of perseverance to learn, humans generally rise to the occasion\u2014call it being mature, since the rites of passage central to most societies signify the taking on of the behaviors and responsibilities of adulthood. Despite this continuity of tradition, however, hunter-gatherers had no immutable standards about how to act, and certainly no means of enforcing their markers across the centuries. Early humans did not live in a vacuum, static and unreactive, even if the archaeological evidence indicates they underwent change at a near imperceptible pace.\n\nSkills imperative for survival stayed most firmly in place, the clearest evidence of this being the consistency of stone tool types over long stretches of time. That didn't stop people from going so far as to reconfigure their livelihood when they had to, despite the fact that some stuck obstinately with their ways regardless of the dire consequences. Greenland's Vikings, for example, were apparently tethered to the homeland by sporadic trade and pressured by the church to keep their agricultural customs. A few of their communities may have starved after woeful attempts to raise imported livestock rather than take up the local Inuit practices of hunting whales and seals.\n\nNevertheless, a willingness to pursue opportunities is a trademark of being human. The Pum\u00e9 are a superb exemplar of such adaptability. On the savannas of Venezuela, where farming would be hard, Pum\u00e9 in hunter-gatherer bands feast on lizards, armadillos, and wild plants, while along the rivers, Pum\u00e9 villages raise gardens of manioc and plantains. Among the Pum\u00e9 these differences mean little. All of them carry out the same all-night Toja ritual, share the same language, and think of each other as Pum\u00e9.\n\nThe flexibility of human identities means that differences in subsistence aren't the cornerstone of differentiating our societies the way they are for animal species, with their niches or ecological roles. Yes, societies can opt for distinct approaches to nourish themselves, which can reduce competition. Coastal societies might reasonably rely on fishing, for example, while their neighbors hunt game\u2014and the people could see these choices as part of what defined them. Yet societies occupying the same habitat can eat the same food and make the same tools, the only outward distinctions between them being matters of arbitrary variation in myths or dress.\n\nNot all changes in identity come about on purpose. People enact the traditional ways as best they can, but faulty recall across the generations can alter their behavior inadvertently and at times detrimentally, such as when, as I instanced previously, the once ocean-savvy Tasmanians forgot how to fish. Written records slow but don't stop the loss, even for the things people treasure most. Poor recollection and new frames of mind can affect perceptions of well-documented events when we envision the past in allegorical terms. Members of preliterate societies, dependent on each other's memories, lived out a game of telephone, in which sentences spoken from person to person grew garbled and unrecognizable\u2014except for them the garbling could extend across everything they do.\n\nShifts in speech itself evidence this drift most vividly. Even in our age of globalized interchange, a rich diversity of languages and the dialects within them persist. The speech approximating that of the Midwestern states is often pointed to as standard for American English. After listening to this accent on television and radio for generations and adopting bits of it, however, English-speaking populations the world over have kept their own distinctive speech patterns and trajectories of language change. Midwesterners themselves continue to deviate from \"their\" standard: a shift in vowel sounds around the Great Lakes began in the 1960s, typified by the lengthening of the _a_ sound in certain words so that \"trap,\" for instance, has come to sound like _tryep_. Linguists delight in such variations, recording them within all societies from !Kung Bushmen bands to British royalty.\n\nAnything that can serve as a marker of identity, be it language or cuisine or gesture, is constantly being reshaped in this manner. Some shifts arguably are a consequence of the boredom people feel constantly doing things the same way. Novelties can be introduced bottom up, say with the influx of goods and ideas by trade or theft, or are an outcome of the diffusion of trends among the populace. The essayist Louis Menand summarized the evidence: \"We get attracted to things that we see other people are attracted to, and we like things more the longer we like them.\" Hunter-gatherers did not adopt fads with the regularity that we take for granted, from changing hemlines to the popularity of cell phone apps, nor did their cultures boast the shifting subcultures of modern life. Yet they painted their skin and played music in ways that varied, if only subtly. People no doubt tentatively espoused, and in time came to like more and more, novel social choices.\n\nComplete novelties found their place, too. Imagine a hunter-gatherer coming up with something radical. If the innovation was invaluable it would spread no matter its source. Otherwise the reaction an individual had to it would depend on the innovator. People prefer to follow those who match their values, but they may cut a quirky close companion some slack. While hunter-gatherer bands had no clear leaders, novelties could be introduced top down by anyone with a modicum of influence. Such a role model could nudge everyone's choices in new directions, likely by arousing the subconscious impulse to mirror the behavior of admired others. For example, people could follow the advice of someone with spiritual skills, as this story told by an anthropologist who observed the Andaman Islanders lays bare:\n\nLong-established customs may be altered overnight as the result of a \"revelation\" by some seer, only to have the new customs overthrown themselves in the course of time by the next \"revelation.\" I had an experience of this in the Onges when Enagaghe, a well-known seer, announced one day that he had received an order from the spirits about the way in which hunting trophies were to be displayed. No longer were pig jawbones to be impaled one behind the other on poles hung horizontally along the sloping roof of the hut a little above the hunter's bed.\n\nToday much cultural variation emanates from teens, who lead the way in challenging \"proper\" behavior. Even though their choices can't extend outside a society's permissible wiggle room without a backlash, as time passes, everyone from hippie to skinhead affects the mainstream; counterbalancing the shift are older generations, who hold back the generational changes until their influence wanes. Such battles between young and old seem timeless, but whether they were waged in hunter-gatherer societies isn't clear: most accounts of people living in bands focus on how children learned traditions rather than bucked or invented them. Kids are kids, though, and defiance appears integral to how a human child grows to be an independent being. As boys and girls are generally, the children of hunter-gatherers would have been open to new experiences, from messing about with hairstyles to exploring uncharted terrain. When chic ideas, methods, or products emerged in the distant past, it seems more likely than not a youth introduced them.\n\n# THE BIRTH OF AN OUTGROUP\n\nThat the details of how band societies split up are incomplete is to be expected, given that catching one of them at the moment of a division is almost impossible\u2014in fact has never been done. The frequency with which languages are born might be taken as a rough guideline to how long societies endure. Languages drift apart over time much as genetic sequences drift between species in what's known as a molecular clock, and measurements of this linguistic drift suggest they divide on average about every 500 years. Not all societies develop what linguists judge to be their own language, however\u2014some show only a difference in dialect. Hence a half millennium might overestimate the life span of societies. Still, the few estimates of the longevity of band societies suggest the figure isn't far off. Nor is this longevity unique: chimpanzee communities achieve a similar age.\n\nYet while the final sundering might occur once in approximately five centuries, the lead-up to that moment takes long enough to leave behind plenty of evidence as to what went on. By combining this fragmentary information with what we understand about how human groups split apart generally, I aim for a general picture of what the lifecycle of societies looked like throughout most of human existence.\n\nAmong bands the game-of-telephone effect was compounded by an inaccurate knowledge of what went on elsewhere. Variations crept in most prolifically when society members were seldom in contact. In an anonymous society, individuals living far apart might not need to know each other, yet if their markers were to remain the same, they did need to be aware of what distant members were doing. Certain factors could amplify the spatial variation between widely separated people in a band society. On a territorial border, obviously, exposure to outside ideas and goods was greater. Outlying bands with the most contact with foreign societies and the least exposure to other parts of their own society would come to differ from their compatriots elsewhere. Making things even more complicated, different stretches of the border would abut different neighbors. Hence bands faced sharply divergent problems and opportunities from place to place that compounded their differences in identity with others of their society. As an outcome, people at the territorial margins grew marginalized. Factions were born among these outliers.\n\nA human attribute that helps keep a society together in the face of potentially disruptive diversity is that people can be blind to such differences even when confronted by them. Philosopher Ross Poole articulated this perfectly: \"What is important is not so much that everyone imagines the same nation, but that they imagine that they imagine the same nation.\" Even when discrepancies were noticed\u2014in hunter-gatherers, perhaps during a reunion of bands for a feast\u2014people would have sought to avoid openly expressing dissension around matters of identity if confrontation was likely, much as we do today. Nevertheless, at a certain point, even differences that once might have appeared random and inconsequential could come to be seen as important, and too uncomfortable to ignore. The reunions were times of animated gossip. Among the subjects must have been any oddball behavior, especially when people saw members with whom they had little or no acquaintance do something unexpected. Humans tend to project more negative motivations on strangers, and of course the larger a society is\u2014even a society of hunter-gatherers\u2014the more such poorly known others there are. With minds swayed one way or another and no leader to compel anyone to conform, the stage would be set for ever more distinct and independent factions to emerge. Call them outgroups in the making.\n\nIn the same way that a chimpanzee's fate can be determined by which emerging faction it joins and thereby where it ends up living, and with whom, a person's fate could rest on his or her faction of choice\u2014possibly more than on his or her choice of spouse. In making their decision, however, hunter-gatherers would often have had little to go on, especially since they were unlikely to have any idea what lay ahead. Early humans, like animals experiencing a society breakup, would seldom have gone through a division before. They could not fully comprehend their shifting circumstances or accurately envision an ideal outcome. To make matters worse, studies of decision making tell us that people can be remarkably uncertain about their best interests even when a lot is at stake\u2014for example, often they go along with an idea they think is popular but which few actually believe. As regards to issues of identity, the cart can come before the horse. Whether a choice is bad or good may only be sorted out after people are forced to take a stand on the subject.\n\nDespite all that, it's more than likely most hunter-gatherers' choice of faction would be predictable. Humans find contentment in being with familiar others\u2014for a band society, namely, those of their band and, probably, other bands nearby. People's connection to a certain general area within the territory\u2014their \"home turf\"\u2014might in itself be a binder. Even the factions at Gombe formed around chimpanzees with a fondness for the same bit of earth. What made human factions very likely to correspond to where people spent most of their time was that newfangled markers would spread from their point of origin; because of this diffusion, hunter-gatherer factions, like many cultural distinctions today, would be regional.\n\nDifferent factions didn't have to be adversarial, and certainly not right away. Like the Gombe chimpanzees that at first still socialized, the people remained identified with the original interconnected society. In recent times the Walbiri had four subgroups on good terms, each with its take on dreamings and rituals, while the Comanche fell into three factions with their own minor dialects, dances, and military associations. Given that human minds treat societies as species, I imagine we handle this variety within a society much as we do differences within an animal species, for example dog breeds that we (and, as it turns out, the dogs also) see as variations on a single theme; likewise we register members of other factions within our society as versions of our own \"kind.\"\n\nTroubles mounted when the factions became a source of irritation. \"The master defense against accurate social perception and change is always and in every society the tremendous conviction of rightness about any behavior form which exists,\" psychologist John Dollard averred. Once again it's important to recognize that it was the members themselves who decided what behavior was proper or offensive. Just about any difference could set off this \"rightness\" response and initiate the process of firming up the factions.\n\nOne imagines the division might be precipitated by either an accumulation of many peculiarities or that one especially bothersome difference; in his children's story _The Sneetches_ , Dr. Seuss described a world where those who had stars on their bellies refused to associate with those who did not. Among the little changes that could turn into something of belly-star importance, language would be a chief contender, a truth made plain by the Tower of Babel story. Hunter-gatherer societies could develop several regional dialects as they grew. In the 1970s a linguist reported that the Dyirbalnan of Australia who live in the northern part of their territory not only have their own dialect but call themselves by a different name, suggesting a division wasn't far off.\n\nDoubtless the black sheep effect could be a factor, where people grow hostile toward any member whose outrageous displays seem like an affront to conceptions of what their society is. Psychologists study black sheep as one or a few persons: rebellious teen turned criminal, maybe. But what if, as American sociologist Charles Cooley said, \"the one who seems to be out of step with the procession is really keeping time to another music\"? Such an outlier may not be treated as a black sheep among the company he or she keeps. People weed out some social variants and absorb others into their permissible diversity. What they allow may not match throughout the society, however. Among the likeminded, a person others may see as deviant can thrive, his or her choices emulated by those tuned in to the beat of a different drummer. The eating of the dead by the Ypety Ache, an act steeped in spiritual meaning yet misunderstood and feared by other groups of Ache, may have been a deal-breaker back when those groups parted ways.\n\nBarriers in geography could bring about enough divergences that a society could cleave without any response to deviant behavior. Humans who became utterly isolated, such as the first Aboriginal arrivals to Australia, were cut loose to transform in any direction that suited the place. Elsewhere, topographical features led people to split up despite some contact with their former society-mates. Another group of Ache divided in the 1930s when a highway was constructed down the middle of their territory. Afraid of the foreigners who traveled that route, the hunter-gatherers gave the road a wide berth. With the social ties between them reduced almost to naught, the Yvytyruzu Ache drifted from the Northern Ache until both groups thought of themselves as independent people.\n\n# THE ULTIMATE SEVERING\n\nDescribing the fall of the Roman Empire, the nineteenth-century American senator Edward Everett wrote that the society had broken \"into hostile atoms, whose only movement was that of mutual repulsion.\" We can assume that for hunter-gatherers, too, the division would have been set in motion when each faction came to see the other as a hostile atom, its identity intolerable, its actions falling beyond the society's boundaries of acceptable behavior, if only because of disagreements as to where those boundaries lay. If a society gives the world meaning by conveying a story about how life should work, what had been one story would have split in two.\n\nNo one has specifically modeled how a society might divide, but research by social psychologist Fabio Sani, much of it with colleague Steve Reicher, on schisms within different sorts of groups suggests factors that might come into play for entire societies, too. The Church of England split up after 1994 when the members who saw the ordination of women as contrary to the true nature of the church went their own way by devising other denominations. In a second example from the same time, the Italian communist party entered the mainstream and took a new name, causing a minority faction to splinter off into a new party that stuck to its principles and retained the party's original symbols. In both situations the members who felt that the refinements enhanced their identity seized on the changes as mandatory. For them, those changes strengthened the group. The other members, though, interpreted the refinements as harmful deviations from the intangible essence that defined them, threatening their solidarity. The belief that their identity would be subverted drove a wedge between them and supporters of the change.\n\nWe might expect that most divisions have been prompted, now and in the past, by alterations in identity on both sides. Sani's studies suggest there can be imbalances, however. The more conservative faction, the one espousing the least transformation\u2014in hunter-gatherer times perhaps the people in the core of the territory who were buffered from outside influence and retained most of the society's oldest attributes and original name\u2014might have attracted those individuals we speak of today as nationalists, who abhor change. In the view of such people, dangerous behavior, which might have originated with a single black sheep, would have spread. What had been one estranged person worthy of chastising, or worse, would have metastasized into a faction acting inappropriately, even maliciously, as a unit. Such a faction can elicit a strong commitment from its members because of how likeminded they are. Factions express a narrower set of behavior than present across the whole society, making their solidarity easy to come by. However, the more radical side\u2014occupying the fringes of the territory\u2014would have seen things the same way, and felt just as unified about their own viewpoint. Because they would have seen the changes they promoted as essential to strengthening the same society, for them the dissidents would have been those conservatives refusing the improvement.\n\nWhen factions emerge around differences they find disagreeable, we can expect little effort by either side to see things from the other side's perspective. Psychological studies show that this lessening of attention to the other side manifests even for factions that are just getting their start and so are still trivially different. As a result, communication breaks down and factions strengthen. To individuate the others becomes not merely hard, but destructive. Doing so might give us cause to question our beliefs when we are confident our motives are pure and they are in the wrong\u2014evil, even. From the vantage point of both sides, then, the threat that sundered the society isn't a shortage of food or of places to live, even when the dire straits brought on by such misfortunes may factor into its demise. Rather it is defects in the collective identity that had once tied them together.\n\nWe have seen that people conceive of societies as if they were separate species. In fact the metamorphosis of societies as they splinter off from each other has been labeled \"pseudospeciation,\" in effect turning what had been one species of human into two. In this splitting, markers serve a role similar to genes. As it does for biologists chronicling the origin of species, the pervasiveness of change makes figuring out who divided from whom in prehistory a challenge. The trials of anthropologists are made all the greater by the readiness with which societies trade, borrow, and steal. Darwin's final words about living things in _Origin of Species_ apply equally to societies: \"Endless forms most beautiful and most wonderful have been, and are being, evolved.\" A more detailed consideration of the psychology behind the separation of human societies will confirm that this beauty hasn't come easy.\n\n# CHAPTER 21\n\n# Inventing Foreigners and the Death of Societies\n\nThe disbanding of a society is a time of reinvention. Any reading of history suggests society breakups mirror the breakup of a marriage. When one can't turn back from a split, years of repressed opinions come pouring out that may express the opposite of what had been professed a month, if not the day, before. As pressures to conform to social norms shift, diminish, or vanish entirely, people on both sides gain the latitude to explore ways of interacting that had been out of favor or considered heretical. Previously unacceptable acts can leap to the forefront, helping each group distance itself from those who are now _other_ , reimagined as outsiders, such that they come to appear ever more foreign.\n\nThe evidence indicates that many of the modifications of daughter societies\u2014their character displacement, to borrow again a term from biology\u2014occur in the initial years after they go their separate ways. Their newfound freedom of expression may be a reason why. That's when language\u2014and no doubt many other, less studied, aspects of identity\u2014undergoes the fastest rate of change, before settling into a relative stasis thereafter. Indeed, distinctions between societies, often enough, are an outcome not of their ignorance of each other due to geographical separation, but of their awareness of and interaction with each other. This would be conspicuously true after societies split up. The opportunities for independent thought and invention afforded by a newly minted society, leading to a convergence of perceptions around themes the members can celebrate as their own, can make its formative years a golden age. For example, the Declaration of Independence and US Constitution remain the reference points that Americans turn to for guidance when questions about the nation's governance arise. Based on what is known about modifications in identity, I believe this would have been the case over the course of our evolution as it is now.\n\nYet there would be a deeper psychological impetus for a reworking of identity to bloom right after a division. The sense of being adrift, their fates severed from the meaning and purpose the larger society once provided, would heighten the urgency of the people's search for a strong identity, and essence, that stands apart. Moreover, their identification with each other must actually _matter._ Certain groups, such as people experiencing homelessness or those who are obese, may be marginalized but don't create societies with identities of their own. Neither do sick or disabled chimps or elephants, even when others treat them as outcasts. These outliers fail to bond since they do not see others with their condition in a good light. They lack what psychologists describe as positive distinctiveness.\n\nHence the insights of psychologists suggest that the members of a start-up society will toil to distinguish themselves favorably. To achieve this, they improvise cherished attributes or express old ones in a special way. The process is analogous to the development of traits that biologists studying the divergence of species call isolating mechanisms. Whatever commonalities remain with the other society can be denied or ignored. Like divorcees not on speaking terms, the societies can break off contact, which would mean any shared history would be eschewed or forgotten. Regardless, no matter how alike the newbie societies might seem to outside eyes, reunification would quickly be impossible.\n\n# DIVISIONS AND THE PERCEPTION OF US AND THEM\n\nOne remarkable aspect of a society division is that the relationships between the former comrades have to be recast at the level of individuals\u2014every one of them.\n\nA split must yield an unequivocal grasp of who belongs where, if only so each offshoot maintains order and independence from the start. The traumatic nature of this recasting of identity in chimpanzees is what made the Kasakela raids on the Kahama community all the more horrifying. The slayed animals were not just known to them; many had been among their friends. Best buddies Hugo and Goliath found themselves on opposite sides of the chimpanzee schism, but continued grooming each other as the factions drifted apart (with Goliath on the losing side). Hugo took no part in his killing, yet another male, Figan, did, even though Goliath had been \"one of his childhood heroes,\" Jane Goodall recounts.\n\nWhat doomed Goliath was the shift that occurred in how the chimpanzees discerned their erstwhile compatriots\u2014a change in how they categorized each other. The apes' deftness with categories reminds us that, as one primatologist put it, \"chimpanzees, like humans, separate the world into 'us' versus 'them.'\" Here's Goodall's further take on the subject:\n\n[The chimpanzees'] sense of group identity is strong and they clearly know who \"belongs\" and who does not.... And this is not simple \"fear of strangers\"\u2014members of the Kahama community were familiar to the Kasakela aggressors, yet they were attacked brutally. By separating themselves, it was as though they forfeited their \"right\" to be treated as group members. Moreover, some patterns of attack directed against non-group individuals have never been seen during fights between members of the same community\u2014the twisting of limbs, the tearing off of strips of skin, the drinking of blood.\n\n\"The victims,\" Goodall concludes, \"have thus been, to all intents and purposes, 'dechimpized.'\" As dehumanization likely can be for people, the ability to switch off an awareness of belonging to \"our kind\" becomes a mechanism by which the members of a new society firm up the irrevocable separation from their former society-mates. This change of perception can be a gradual process, preceding the division itself. In the division of a macaque troop, for example, fights shifted from conflicts between individuals in the different factions early on to showdowns between the factions acting en masse as the breakup approached. It was as if the monkeys no longer treated the others as separate beings but as a collective.\n\nWhat happened at Gombe to ultimately cause one group to hive off from the other\u2014the tipping point, after all those years as factions that tolerated each other, when the chimpanzees at last severed any remaining ties? Somehow they all recast their former fellow members as dechimpized others. Did the scientists miss a row that caused each and every ape to change its outlook on the members of the other faction?\n\nThat some sort of deciding incident could play a role in the division seems plausible for monkeys that stay in constant contact with everyone in their troop and therefore seldom miss anything of consequence. This thoroughness of knowledge is impossible for chimpanzees, spread out as they are. Not every ape would have borne witness to any pivotal event, and chimps can discern little from others about what transpired. This draws our attention to a key difference between chimp and bonobo divisions and those of people: in deciding such matters as _Who do I hang with?_ or _Are the others now foreign?_ , our ape relations at best act solely on what little information they are able to glean from community members that perchance are close by. While the dynamics of how divisions unfold might have evolved before speech, humans can ascertain what's taken place elsewhere, and express and bolster opinions as to who must be banished from the society and who belongs.\n\nWhatever caused the final sundering of the Gombe community, one thing seems certain: dechimpization could not have come about by each animal negotiating a change in relationship with every other one, individually. Perhaps by the time of the division, the apes were prepared to revise their treatment of those in the other faction all at once. Voil\u00e0! Past companions became foreigners in a single transference of identity, and a society was born. For aggressive animals like chimps and gray wolves, such a boundary shift distinguishing who is in from who isn't would render past relationships of little consequence. In one fell swoop, the once beloved Goliath became something alien\u2014even dangerous.\n\nOf course, chimpanzees, wolves, and most other mammals do not have markers to which they affix their identity. What our species evolved to do (if we give this crude hypothesis any credence) is connect this mass transference of identity to the unique attributes that the members come to associate with their faction. This shift in perspective creates stress lines along which people clash while they are still part of the same society. We can expect that once humans see a group of compatriots as coming together around behaviors they find truly unforgivable, even appalling, those people become irredeemably _other_. Prejudices now come into sharp relief, as their markers, viewed unfavorably, turn into the chief thing registered about them.\n\nOther species with anonymous societies behave much the same way, though some use a far more formally organized and painless strategy than humans. In a dividing colony of the honeybee, the two offshoot hives will share the same identifying marker at first\u2014the same odor. As far as anyone knows the only reason the offshoots don't recombine is that one of them flies to a location far from the original nest. While the matter hasn't been studied, a reasonable guess is that once the two colonies settle at their respective spots, differences in diet, or more likely in the genetics of each queen's offspring, cause their \"national\" scents to drift apart. As a result, each colony, belatedly, achieves an identity of its own.\n\nA review of civil conflicts shows that shifts in how humans respond to others in their own society can turn, disconcertingly, on a dime, ranging all the way to outrageous dehumanization and outright brutality. The best-documented examples concern modern ethnicities rather than emerging factions, and show the problem at its extreme. Sleuthing by the Polish American historian Jan Gross confirms that everyday citizens can sever relationships as utterly and gruesomely as any chimpanzee. Gross reconstructed the story of Jedwabne, a Polish town that butchered more than 1,500 of its Jewish residents over the course of one day in 1941. While I doubt violence of this intensity was common in hunter-gatherer societies, a breakaway faction could well be looked at as inferior. Depending on a people's response to its members' newly adopted ways, this could even manifest with disgust from its inception. Forged on a sword of social disintegration, this outlook could be tough to alter. Rapidly, the groups would split firmly apart.\n\nThe very sense of being spurned can be a psychological blow. Indeed, rejection even by those we have rejected ourselves can bring on pain or depression. It hurts to be excluded by a group we dislike; among the studies that attest to this is one titled \"The KKK Won't Let Me Play.\" An anticipated outcome is that people who identify strongly with a faction bond all the more tightly the worse they are treated. Those joining separatist movements in Quebec, Wales, Scotland, and Catalonia unify with profound indignation around anything they perceive as unfair, such as overtaxation and repression of civil rights, sharpening the fault lines along which their societies could cleave.\n\nFor the most part, close ties tend to weather a societal division. Having grown up together, family members often think similarly and are likely to choose the same faction. This may explain why hunter-gatherers were on average related to others in their society to a slightly greater degree after it splintered, a pattern also present following divisions in other primates. The trend is sharper still in tribal settlements. People in them depend on extended families to share goods and bequeath property, causing most divisions to fall neatly along family lines. Of course those kin and allies that do take opposing sides in a social schism find their relationships severely tested. Anyone who shifts allegiance to another faction runs the risk of being disowned and replaced with friends whose loyalty is aligned. History is rife with tales of brother killing brother. That was at no time more poignantly documented than during the American Civil War, known as a war of brothers for how family and town loyalties were sundered, damaging relationships for generations afterward.\n\nI don't know what the tipping point might have been to cause early hunter-gatherers to finalize a split, but recorded human history and evidence from other primates suggest that a complete lack of enmity would have been rare. This remains true even after the advent of language, when our ancestors could in theory have annulled the relationship by level-headed negotiation. Fights broke out in the events leading up to the division of the Gombe community; however, these altercations were of a fairly normal sort for chimpanzees. Small consolation, that: the killing started once the community came apart.\n\nYet for people the comparison to divorce, with the common presumption of irreconcilability, may not be the best analogy for picturing the aftermath of a division. As with some other mammals, outright violence need not play into the division or mar the relationship between the societies thereafter. Savannah elephants (and it seems bonobos, too) can face turmoil and uncertainty as divisions play out, and endure a similar global recasting of membership. However, they often (although not always) rebuild ties later. With those species, as with ours, a division might be more usefully likened to conflicts of teens with parents: a fraught growth phase that must be passed through for both parties to achieve independence. In all but the most tumultuous circumstances there may be a fair prospect of later making amends, even if the two sides are now firmly separate.\n\n# THE MAGIC NUMBER\n\nFor hundreds of millennia the volatility of human identities is what made the breakup of societies certain at a small size. Indeed, that size was so predictable that some anthropologists have proclaimed 500 a \"magic number.\" As a ballpark average for all parts of the globe, this was the number of people living in a band society. Much as 120 seems to be the size beyond which a chimpanzee community is likely to become unstable, it's reasonable to think that 500 represented an approximate upper limit on the population of a stable society for most of the prehistory of _Homo sapiens_.\n\nOne can infer a practical reason for a society to contain at least 500: by some calculations, a population of this magnitude gives humans the opportunity to select a spouse who isn't a close relative. This may explain why people, unlike the many mammal species that live in societies of a few dozen, rarely show the restless and risky drive to join a foreign society. Thanks to the abundance of mates, most people throughout history have had the option to stay with their birth society for life. But what set the divisions in motion at this size rather than at some larger one, which would have given our ancestors an even wider choice in spouses, as well as the defensive advantages of a more imposing group? That number doesn't seem to reflect the checks and balances of societies living enmeshed in nature, as ecological factors like predators and food availability differ remarkably between the jungles and tundras where hunter-gatherers lived. The territories hunter-gatherers occupied differed in total area in those different ecosystems, with Arctic people ranging farther, but the populations of the societies were much the same everywhere.\n\nThat band societies hit a low population ceiling may have been a function of the psychology governing the expression of human individuality. Maintaining a balance was essential: members had to feel similar enough to each other to share a sense of community yet different enough to think of themselves as unique. I argued in Chapter 10 that people had little motivation to set themselves apart from others when everyone in the society lived in a few bands, hence the conspicuous paucity of cliques among those hunter-gatherers. But once their population swelled, those same people would hunger for the differences offered by connections with narrower groups. This increased drive toward diversification of identity would promote the emergence of factions that ended up causing discord among the bands and severing their relationships. The situation would be different for societies that settled down, eventually in colossal populations. Unlike people in bands, most settled people found opportunities to connect to social groups that were not factions but were rather widely acceptable and necessary for the society to function\u2014jobs, professional associations, social clubs, and niches in the social hierarchy or among extended kin.\n\nI have left open the question of what is special about the anthropologist's magic 500. My suspicion is that it's at about this population that the human wherewithal to have even the crudest knowledge of everyone in the society begins to be sufficiently strained for individuals to lean heavily on markers as a crutch in their interactions. In groups larger than 500, humans would start to feel truly anonymous, something that has no effect on ants but that undercuts people's desire to matter as individuals. This loss of self-worth would make people eager to enhance their distinctiveness by accepting what novelties come their way. Without the tight organization and oversight of a major settled community, such novelties would spur a division. What I can't say is whether this attribute of social identity evolved in the Paleolithic to keep band societies at a size that (for reasons yet to be fully understood) was ideal for life at that time. Alternatively, it could have been an adaptation to enhance people's individual social interactions, in which case perhaps the 500 number was an accidental outcome of this psychological feature. More likely both were important.\n\n# HOW SOCIETIES DIE\n\nSocieties come and go. Ant and termite colonies ordinarily die with their queen. Every generation of colony starts afresh with a new foundress. Some mammal societies reach almost as decisive an ending. Should the breeding pair in a gray wolf or painted dog pack or a meerkat clan die with no viable successor, their society is doomed. Unlike the insects, however, the other members live on and with luck disperse to join a pack or clan elsewhere.\n\nDivisions ensure that most vertebrate societies don't come to such an absolute dead end. Under ideal conditions a society can propagate like an amoeba that keeps dividing. The biologist Craig Packer has watched several lion prides divide through a dozen generations over five decades of research in Africa.\n\nClosest to an immortal society are the supercolonies of Argentine ants and a few other ant species, whose members keep the colony identification as they spread globally. Even for them, though, new supercolonies presumably come along at some point. For this to transpire, the identity of the ants from the same supercolony must change to become incompatible. Because ants never experiment with the behavioral choices that incite people to part ways, their divergence in identity would be underlaid by genetics. Perhaps a queen with a mutation affecting her \"national\" scent avoids being killed by her birth colony by lucking onto an isolated spot. There she establishes a nest with her unique identity odor, which could\u2014however improbable it might be\u2014grow into a supercolony.\n\nAmong the vertebrates, the bottlenose dolphin communities along the Florida coast may match supercolonies in permanence, albeit certainly not (given their individual recognition societies) in scale. A community occupying the same corner of Sarasota Bay has stayed steady at about 120 animals for over four decades of study, its territory handed down through the generations\u2014inheritance of real estate being a prime advantage of society membership in many species. Genetic evidence suggests that this community has been there for centuries.\n\nYet despite the potentially endless repetition of divisions, the demise of societies must be inevitable at some point, if not for the bottlenose dolphin then for everyone else. Even an amoeba, which divides endlessly, faces an uphill battle when its diet is curtailed. Eking by at the point of starvation, one of the two amoebas dies after each division, leaving its sister to split again. This genetically programmed response keeps the overall number of amoebas near the carrying capacity\u2014the highest number the environment can sustain.\n\nThat sharp death rate brings to mind the tragedy that befell Gombe's chimps when one community wiped out the other, and for good reason. Malthusian realities apply to both, as follows: Populations of even the slowest-growing species reach the carrying capacity in a few generations. Therefore, barring environmental change, about the same number of chimpanzee communities will have existed on any suitable patch of habitat throughout the millennia. Before the chimps (or lions or humans) attain this peak in population, the products of each division can find space to settle without much of a fight. Yet once societies are cheek-to-jowl, conflict will likely loom large not only between neighbor societies but also internally as the members brawl and scramble for resources, straining their relationships until they crumble. Should a large society fissure under these overcrowded conditions, its members will be no better off. The animals will continue to struggle over the resources that had been accessible to all of them when they were part of one society. Division in itself doesn't ease population stress.\n\nStill, dividing a society can provide a savagely practical way around the logjam. When a community of chimpanzees, all competing with one another, is riven in half, the daughter society that knocks out the other\u2014as Kasakela did Kahama\u2014seizes everything for itself while coincidentally reducing the level of conflict among its own members. It's the culling game plan of a starving amoeba implemented by brute force. With societies at or near carrying capacity, as societies generally will be, the way forward may indeed be cruel. If your former society-mates, now on their own, didn't bring you down, other societies could. One doesn't need to be a mathematician to understand that over time most societies die out.\n\nThe merciless calculus of survivorship tells us that, in the ultimate sense of evolutionary causes, competition lies at the root of why societies divide. Nevertheless, given how much their markers shifted from place to place, division of band societies must have been unavoidable at some point, regardless of population pressure. The question becomes what occurred when humans filled the landscape to capacity, so that the offshoot societies, butting up against competitive neighbors, had no place to go or space to grow. People tend to unite around a threat when competition with outsiders is their overriding concern. A hemmed-in society's unified front would impede the emergence of conflicting points of view, presumably including the factions whose conflicts are a prerequisite of division. Possibly a society in such a bind could maintain a single coherent, if slowly changing, identity even if barraged by outside influences. Still, I suspect this situation would delay the society's division though not stop it. Indeed, should social friction impel a division when the daughter societies are packed tight, the loss of the more vulnerable of those daughters may be all but guaranteed. Arguably, this is what took place in the forest patch at Gombe when Kahama chimpanzees had little room to retreat. Such an end must have befallen many human societies in prehistory, often from attrition brought about by raids, their few survivors, usually women, perhaps joining the winning group.\n\n\"The past is a foreign country,\" or so the writer L. P. Hartley claimed. The archetypal rules by which humans alter their identity thus ensure such a built-in obsolescence. Even the societies that prevail transmute with time to forms their original founders would reject as insufferably foreign. Given time, a society alters as inevitably as does a biological species that evolves until the living generations wouldn't recognize their predecessors if they met them, and paleontologists deem it worthy of a new species name.\n\nSo it was in countless repetition before the dawn of history, societies severing from each other, some wiped out yet even the winners in time changing beyond recognition to divide again. Every sundering would have been a source of heartbreak and agony, carried out for reasons that were momentous at that moment but ultimately forgotten. These divisions have been a part of the rhythm of life as primal as love or personal mortality, yet unfolding over a long crawl of generations beyond our ability to grasp. Win or lose, a society, like its individual members, is fleeting.\n\nHow many societies has the cycle birthed? If we can take estimates of the total number of languages to have ever existed as a rough indication, human societies have numbered into the hundreds of thousands. Since not every society has a distinct language, it's conservative to assume a million-plus societies have come and gone, each made up of men and women confident of their society's significance, abiding existence, and success over its predecessors, of which yours is one.\n\nThus the sundering and death of societies is certain. In most vertebrate species, to review, the process has nothing to do with the markers that make people relaxed around each other. Chimpanzees have socially learned traditions, yet don't discriminate against those who act strangely. Furthermore there's no reason to think they come up with customs that some members take on and others spurn, bringing about a severing of a community. For humans, however, being part of a society comes with obligations of acting properly and abiding by its rules and expectations. Still, those rules and expectations are a moving target. Ever since humans first employed markers for societies, those societies have broken down around factions of people who take alternative views about their identity. Our psychology orchestrates the change by turning what once was familiar into the foreign.\n\nIn charting the birth of the societies of our hunter-gatherer forebears, I've said little that is specific to the rise and fall of nations, which will be one of the subjects of the section to follow. Before getting to this question I must address the social pathways that made such immense societies possible. As it turns out, what drove the success of civilizations was anything but dovishness. Societies that turned to conquering each other\u2014in the course of time incorporating ethnicities and races\u2014enter the picture now. The residual effects of their struggles, some ancient, some ongoing, linger in every corner of the earth's surface today.\n\n# SECTION VIII\n\n# Tribes to Nations\n\n# CHAPTER 22\n\n# Turning a Village into a Conquering Society\n\nTen thousand years in the past\u2014a single heartbeat ago in the drawn-out prehistory of our species\u2014the last ice age began to draw to a close. As the climate warmed, some hunter-gatherers turned to farming, a shift known to archaeologists as the Neolithic Revolution. This metamorphosis took place independently in six parts of the world: first and most prominently in Mesopotamia, a region of the Middle East, 11,000 years ago; followed by the area China now occupies, 9,000 years ago; in highland New Guinea, 7,000 years ago, if not earlier; in central Mexico and roughly contemporaneously in the Andes centered around Peru between 5,000 and 4,000 years ago; and in the eastern United States, between 4,000 and 3,000 years ago.\n\nFrom these humble beginnings, enormous kingdoms emerged in four of the areas: China, the Middle East (I include India here, which was sustained by crops brought in from that region), Mexico (starting with the Maya and later the Aztecs), and the Andes (culminating in the Inca). Nothing seems grander and more deserving of detailed treatment than the forging of a civilization, with its cities and elaborate architecture and culture. Yet after the growth of societies beyond the size of hunter-gatherers living in bands, starting with people settling down into tribes, our story accelerates, for the simple reason that the prerequisites for turning those tribes into civilizations are simpler than they may appear.\n\nWe don't see anything that approximates a contemporary nation in scale or intricacy among other mammals; in nature only certain social insects develop what in effect are civilizations of their own. Clearly the evolution of anonymous societies was crucial for civilizations to come into existence and spread. But anonymity is insufficient to explain either why we humans are able to develop large societies, or how we maintain them. A number of extra factors must come into play\u2014some, like a sufficient food supply, are quite obvious. Others, such as having the means to work out social problems and to give the members ample avenues to distinguish themselves, are much less so. Achieving anything close to what we call a civilization took something further still: violence and the machinations of power.\n\nHistory books overflow with the splendor of nations, their clashes and partnerships, the strivings of their colorful characters, of governments that march forward or fall to the wayside. We take the details personally, often because these are the stories of our people, and they matter to us. For all that, though, the differences between nations and most of the societies that preceded them are matters of degree, rather than of kind.\n\nI say _most_ of the societies that preceded them because one transition was a paramount shift as societies grew in size and complexity. Societies began to absorb each other. From there the road to nations as we know them today, opened up by the Neolithic Revolution, was short indeed. For these conquering societies to take root, the several elements that needed to be in place began with the most basic of resources.\n\n# FOOD AND SPACE\n\nHaving more people requires more food. This self-evident truth makes it easy to assume that a sufficient supply of food is enough to spur the growth of societies. It is not. Consider the monkeys clamoring around the marketplaces of New Delhi. Urban macaques live off the fruits (and meat and vegetables) of agriculture, stolen from street vendors. While the relative plenty does indeed support a higher overall population of monkeys, urban troops nonetheless remain much the same size as those of their country bumpkin and deep-forest brethren. There are simply more troops, packed tightly as to leave no unoccupied space. Much the same goes for the Argentine ants still living in Argentina, where colonies, penned in by many hostile neighboring colonies, cannot grow very large, however much they have to eat. Those in California, of course, have escaped that constraint.\n\nPlainly, humankind has not ended up like urban monkeys. The Nile Valley wasn't home to thousands of mini-Egypts, but instead to the grand one that birthed Ramses II. That said, across wide expanses of the globe humans actually did respond to the availability of dependable food, whether from farming or abundant wild resources, by propagating many little societies instead of creating one large society. In the 1930s, for example, when outsiders first hiked into the New Guinea Highlands, people already occupied it in the hundreds of thousands. A walk of just a few kilometers was all it took to reach the territory of another settled tribe. Each tribe committed to the food supply generated in their area, which typically included plants and animals domesticated on that island. Explorers found the same pattern in the Amazon basin and elsewhere. Although most of these horticultural tribes had less flamboyant cultures than the hunter-gatherers of the Pacific Northwest, they also lived in villages, or at least had a central lodge as a retreat (although a few tribes, especially in parts of Africa and Asia, were nomads that herded domesticated animals). In the past, a tiny minority of these tribal groups would prove to be points of transition to the grand societies of today. Understanding how tribes were organized, and the special features that permitted these few to increase in size and complexity, is the next part of our journey.\n\n# VILLAGE SOCIETIES\n\nLife in a tribe could be one big soap opera. As among the settled hunter-gatherers, there was no shortage of occasions for petty squabbling and violence. Conflicts included disputes over matters that might spoil a family reunion, such as what's for supper, but also accusations of sorcery, fights over spouses, and arguments about the distribution of responsibility. These disagreements could precipitate the breakup of a village, the people at times becoming so disgruntled they moved far off to avoid each other as much as possible. Many villagers would experience such a social cataclysm one or more times in the course of their lives. Prehistoric villages in the American Southwest, for example, typically lasted from 15 to 70 years. An example of village fissioning is that practiced by the Hutterites. Arising in the sixteenth century in what is now Germany, this extant communal sect came into being relatively recently by the standards of the groups we normally refer to as tribes. After several centuries of migration, Hutterites emigrated in 1874 from Russia to the American West, where they live in colonies of up to 175, each of which operates a farm. Social stresses rise as a colony grows, until finally the members arrange for the colony to split, an adjustment made on average every 14 years. Although these transitions are handled in a more orderly fashion than the fissioning of preliterate villages, the dynamics are much the same.\n\nFor tribes to stay together at all, solutions were necessary to ameliorate, or at least manage, conflict. In this, most tribes dependent on horticulture devised strategies similar to those of hunter-gatherers who settled down. One approach to emerge repeatedly was to reduce competitiveness between people by adding dimensions to the social distinctions acceptable in their society. Among these were differences in their work and status. Thus, even if some tribes started out as egalitarian as hunter-gatherers in bands, such an outlook seldom lasted. Also contributing to the members' differences were opportunities to be a part of social groups. Across the tribes worldwide the differentiation arguably reached its apogee in New Guinea. The most complicated were (and still are) the Enga, a regional population that manages a social existence that can seem to have a Rube Goldberg\u2013like intricacy. The thousand-plus individuals within each Enga tribe celebrate their history as a people. However, every tribal member is born into a clan and subclan that claims its own garden space, and at times there can be bickering, or outright fighting, between clans. Nevertheless each tribe has remained intact for the long term.\n\nSome sort of centralized management for dealing with social issues is also all but assured in human settlements, even if rudimentary in the simplest of them. As I described earlier for settled hunter-gatherers, people ensconced at one place had more patience for displays of authority than hunter-gatherers did in bands\u2014yet often just a bit more patience; each village tended to have a headman, but his significance came to the fore during conflicts, and even then he spent most of his time convincing people rather than leading them.\n\nStill, even tribes such as the Southeast Asian highlanders described by the anthropologist James Scott in his book _The Art of Not Being Governed_ didn't lack governance. The title alludes to tribal efforts to avoid being devoured by the powerful civilizations spreading like amoebas from the lowlands. The mountain-dwellers had chiefs who could be tyrannical. Elsewhere the topic of who should have power could itself be a subject of contention. Supervising social matters didn't always require a certain person at the helm. The Nyangatom of southern Sudan and Ethiopia are pastoralists distributed across many villages, each of which moves several times a year to spots suitable for the herds (they can almost be portrayed as hunter-gatherers who happen not to hunt since they drive domesticated cattle rather than stalk wild ones). The Nyangatom have a few specialists, such as men skilled at castrating cattle or making symbolic scars on the chests of warriors, and yet they maintain peace without a permanent ruler; instead, every male has a crack at leadership of the tribe, shared with others of his age cohort.\n\nAs for tribal peoples that live in settlements\u2014whether they survive by hunting and gathering, gardening, or gasoline-fueled agriculture\u2014social friction has limited their populations at one place to something on the order of a hundred or several hundred individuals, up to a few thousand for New Guinea highlanders, who spaced their homes out in a way I imagine reduced strife. A Yanomami village in the South American rainforest, where by contrast everyone hammocked almost on top of each other in an oval shelter, was often smaller, with as few as 30 but up to 300 members.\n\nIn some situations, one such village was all there was to a society. Often, however, there was a grouping above the village. The Hutterites, the Yanomami, and the Korowai of New Guinea, renowned for their tree houses, are all examples of tribes comprised of more than one village. The structural and functional equivalent of a nomadic hunter-gatherer society was such a regional cluster of settled people\u2014call it a tribal or village society.\n\nAs anthropologists have often widely underappreciated band societies in favor of studying bands, the individual village has similarly been a centerpiece of anthropological research, rather than all the villages collectively. This bias in attention is due, firstly, to the autonomy of the villages. No outsider, even from another village of the tribe, told the village residents what to do, any more than one band of hunter-gatherers had any say over another. But another reason researchers have focused on the village as the level of study is that the relationships between villages can be dramatic: villages were often famed for their conflicts with each other, including, among the Yanomami, revenge killings.\n\nJust the same, the tribes were important to their people. Yanomami villages had deadly feuds\u2014reminiscent of the Hatfields fighting the McCoys, though in the case of the Yanomami, multiple families were involved. Even so the villages continually renegotiated their relationships. Fights alternated with reconciliations involving marriages, feasts or trade. Everyone had friends in other Yanomami villages, and, as with members of hunter-gatherer bands, people could relocate to another village\u2014although villagers, committed to tending a garden (and in New Guinea, domesticated pigs), accomplished such a move less readily than did the nomads who foraged unencumbered for wild foods. Indeed, much as a villager could change village, whole villages could fuse. These dynamics worked as they did for bands of hunter-gatherers, with the fissioning and fusing of villages driven by social relationships. The biggest distinction between village societies and band societies is simply that villages changed their locations (usually to freshly cleared gardening spots) and split and combined less often.\n\nSeen this way, a village and a band were not much different. As with the people in a band, the residents of a village seldom had to think about the larger society of which they were a part, but it was there for them the rare times they needed it. Their mutual identity came to the fore when a difficulty or opportunity arose at a societal scale. The villages of J\u00edvaro in present-day Ecuador, famed for their shrunken heads, joined forces when raiding foreign tribes. In 1599, in the largest assault of this kind, 20,000 J\u00edvaro from many villages freed their territory from foreign rule in a coordinated massacre of 30,000 Spaniards. Such village societies also had words that encompassed their whole society, just as hunter-gatherers did. Yanomami, for example, is the name the people give themselves, while Yanomami tapa is their way of indicating all their villages. As we saw with many band societies' names for themselves, the monikers Yanomami and J\u00edvaro mean \"human.\"\n\nCollective markers of identity defined these tribal clusters of villages as a society just as they bonded other societies across the globe. The Yanomami are a case in point. The common dress, housing, rituals, and other shared traits of the members enabled the fission and fusion of villages\u2014similarities that the people recognized. When the people of a village came to an impasse and went their own ways, the parting was just like that of a band of hunter-gatherers. Even if those individuals felt animosity toward each other, in the end they nonetheless had the same language and way of living, and remained part of the same society. Differences did accumulate, however, across the villages of a tribe. For settled tribes as much as for hunter-gatherer bands, that's where things went south permanently. The Yanomami today seem to be diverging into several tribes, each with a population of a few thousand, as a result of changing identities. Indeed, some linguists distinguish five, still very similar, Yanomami languages. The Yanomami themselves are aware of the divergences, and they mock the odd Yanomami from distant villages.\n\nTribes like the nomadic Nyangatom and the settled Enga and Yanomami managed to stay intact at overall populations often in the many thousands\u2014larger than was normal for a band society. At the same time these observations about the intolerance of the Yanomami toward differences among their own people point to one reason why very few tribes took the path to building an empire. Being hemmed in by neighboring tribes wasn't the only obstruction. A tribe came up against the same problem hunter-gatherers faced: its own people's identities started to clash.\n\nYet should a tribe have managed to retain a consistent identity, it still would never yield a sprawling civilization solely from the growth of its population. This would have been true even under the most ideal conditions\u2014a favorable birth rate afforded by ample food and space, able leadership, and plenty of social differentiation. That these attributes were insufficient is demonstrated by the fact that all large human societies, on close inspection, contain not the descendants of a homogeneous stock of people, but populations of diverse heritage and identity. The failure of hunter-gatherer and tribal societies to adjust to variation in markers stands in stark contrast to the resounding success of nations in this regard. Indeed, to understand the birth of civilizations we must grasp how they came to be made up of a mixture of citizens, eventually comprising today's ethnicities and races.\n\n# SOCIETIES DON'T FREELY MERGE\n\nOne possible explanation for the heterogeneity of civilizations is that growing a society entails societies voluntarily merging. The evidence indicates this isn't so. Across the animals the merger of societies is so rare as to be nearly nonexistent even in species where societies compete little. Bonobos and chimps exemplify this: the only \"mergers\" of their communities strain the word's meaning. Primatologist Frans de Waal tells me that bonobos who are strangers to each other can forge a community from scratch with little fuss. Bonobos' aptitude for befriending strangers must simplify their adjustment to each other. Such arranged societies are artifacts of zoo confinement, however. In nature, bonobo communities, even if on good terms, keep their memberships apart. Integrating captive chimpanzees into one community is by comparison a nightmare, requiring months of careful introductions, with lots of bloody skirmishes along the way. The only reason the xenophobic chimps adjust to each other is that, with their original communities lost to them, they, like captive bonobos, are refugees, with no choice. Similarly the handful of permanent mergers recorded for wild monkeys occurred after the animals' troops were decimated down to a few. The survivors, like apes in a zoo, were a kind of refugee, forgoing their past societies to join another as a matter of survival. Their link-ups don't come close to the mass unions we see between entire human groups.\n\nUnder ordinary conditions the same holds true among social insects: combining mature colonies isn't a part of their pin-brained ethos. To my knowledge, permanent mergers of healthy societies take place only in African savanna elephants, and then rarely, and just between two cores that once belonged to a single core that divided in the past. By reconstituting their original membership, sometimes years after splitting up, the pachyderms apparently affirm they never forget. Otherwise, once a society has gelled so that its members identify with each other and stay above a sufficient head count to persevere, that society remains distinct from all others.\n\nThe same is true of humans: once society members have their identity set in place, a merger freely made with another society is highly unlikely. I've seen no evidence, for example, that goodly numbers of foreigners were absorbed into hunter-gatherer bands. This is true even in cases of close cultural similarity between the outsiders and the society they might conceivably join. So while bands of the same hunter-gatherer society could fuse, as could villages of a tribe, societies stayed firmly apart. The hurtle of merging societies is compounded in humans by the unlikelihood that each can adjust to the foreign identities of the other. The only examples of mergers again bring to mind a fugitive situation: coalescent societies come about when people are too few to subsist on their own. Coalescence was the fate of uprooted American Indians from the 1540s through the eighteenth century, especially in the Southeast, after years of dying from fighting Europeans and their diseases\u2014among the famous examples are the Seminoles and Creeks. Once combined, these refugee populations often took the name and much of the lifestyle of the dominant tribe in their mix, with a few allowances for the social markers of everyone else.\n\nEven taking to a foreign lifestyle doesn't lead to a merger. For example, the Fur people of Darfur live on arid land that normally supports little livestock. A lucky family with surplus cattle can keep their livestock fed only by pulling up stakes to join what are known as the Baggara. But this is no change in identity. The Baggara aren't a tribe; rather, the term is an Arabic word that applies to the herding lifestyle, which many tribes in Darfur pursue. So although a Fur family might become herders among those other tribes, and even be respected enough by the other herders to gain acceptance as allies, they remain distinct. Even a Fur who marries into another tribe among the Baggara lacks the upbringing to be mistaken for a native-born member of that tribe.\n\nDespite the human capacity for generating foreign partnerships, full mergers of societies are likewise never an outcome of alliances. Indeed, psychologists find that societies heavily dependent on each other are inclined to set themselves apart. The Iroquois confederacy was crucial in fighting mutual enemies\u2014originally other Indians, then Europeans\u2014with the tribes tasked with defending different borders of their combined realm. However, the independence of its six tribes was never in doubt. Coalitions such as theirs could be a source of pride, yet that didn't diminish the importance of their original societies.\n\nOf this, then, we can be reasonably sure: societies, from the conjoined bands of hunter-gatherers to great empires, never freely relinquish their sovereignties to build a still greater society. Aggressive acquisitions of both people and their lands, and not willing mergers, brought different societies into one fold. Greek philosopher Heraclitus was right to proclaim war to be the father of all things. In locations from the Middle East to Japan, China to Peru, the only way a society created a civilization was to combine an explosion in its population with the expansion of its realm by force or domination.\n\n# ACCEPTING AN OUTSIDER\n\nThe absorption of outsiders into human societies would not have begun with aggression. It started\u2014harmlessly enough considering that both sides in the arrangement could benefit\u2014with the acceptance of the occasional outsider as a member, as is necessary for finding sex partners in many species. While band societies tended to be populous enough for members to choose a mate from among themselves, some transfers would have taken place to seal a partnership between groups and to minimize inbreeding over long spans of time.\n\nBringing in such a foreigner would not have been an easy proposition even for early humans. The mate (often a woman) or adopted refugee or outcast would have borne the effort of adapting. A few of the newcomer's exotic behaviors might have been encouraged, as when a society gained from his or her skills\u2014a better option than trading for a tool you can't make was getting ahold of the toolmaker! Nevertheless, given how people nurture and protect their identities in the face of contact with outsiders, a newcomer would have had little effect on the society's conduct. Any new arrival inept at or averse to changing his or her ways faced a hard life and possible rejection. Still, the foreign-born could alter his or her identity to suit the new society only so far. Conversion of membership can never be absolute. Even when people try hard to fit in, their inner essences remain alien, unalterable. After years among the Yanomami, anthropologist Napoleon Chagnon wrote,\n\nMore and more of them began to regard me as less of a foreigner or a sub-human person and I became more and more like a real person to them, part of their society. Eventually they began telling me, almost as though it were an admission on their part: \"You are almost a human being, you are almost a Yanomam\u00f6.\"\n\nGiving credence to the Yanomami perception of Chagnon is the fact that no newcomer to any human society could ever perfect all the markers of a foreign people. Telltale signs would divulge a person's origins even if he or she managed the changes that mattered most to the society members well enough to find a place among them.\n\nEven so, adding an outsider or two in this manner wasn't sufficient for anything remotely close to an entire ethnic _group_ to materialize. An unpleasant truth is that ethnicities made their nascent appearance by way of an industry that didn't make wide headway except among people who had settled down: slavery.\n\n# TAKING IN SLAVES\n\nSlavery is an almost exclusively human activity. Of course there are the slavemaker ants overviewed earlier in the book. Among other vertebrates the only behavior at all like enslavement occurs among langur monkeys. A female that has never birthed an infant can steal one from another troop to raise it (though the odds of a practice baby surviving are dismal). In West Africa, raiding male chimps sometimes goad a foreign female to their territory for sex rather than kill her, but she slips off for home at the first opportunity, the very same day.\n\nSeizing a foreigner and keeping him or her around permanently was seldom an obvious option for people in bands. Escape was too easy. Even so, raiding parties could take any surviving women, who had little alternative but to marry the winners. Slavery was regularly practiced by a few band and small tribal societies, such as among the Indians of the Great Plains, who did not just take captives but traded them as commodities. Although such captives could run away, they may have found their former identity so defiled, they could never return home. We find an example in an 11-year-old Spanish girl captured by the Yanomami in 1937. After 24 years, Helena Valero ran off only to find her half-indigenous children shunned by the Spanish community since, as she bitterly recounted to anthropologist Ettore Biocca, \"she was an Indian and her children were Indians.\" A woman seized by the Comanche in 1785 refused rescue despite being the daughter of the Chihuahua governor. She sent back word that she would be miserable going home with a face inked with tribal tattoos, the sort of indelible marker that all but ensures a lifetime commitment to a society\u2014and turned that Mexican woman into a foreigner among the people of her birth. In both instances the captive was European, yet tribal people caught by another tribe faced the same problem.\n\nBondage grew in significance as settlements became organized to hold captives, although not all settled people kept slaves. Even the Northwest Indians had been living in settlements for centuries before taking to slavery with a vengeance. Often those tribes made sure slaves had little opportunity for escape by kidnapping them on expeditions to villages so remote that getting home was well-nigh impossible.\n\nSlavery took the inequality in the relationship of newcomer to receiving society to the absolute extreme, granting the latter total dominance over a prisoner. Captives retained their status as aliens and were dissuaded, or prohibited, from identifying with the society. Not surprisingly, perhaps, given the importance of markers in human life, some sort of brand could be forced on seized men and women to ensure their identification as slaves. Tattoos as well as actual branding with a hot iron were prevalent in the Americas and in medieval Europe. Shaving heads was widespread, too: with hairstyle a matter of pride as an indicator of identity, the loss was an intended psychological blow. Adding insult to injury, many societies put slaves through fiendish initiations and banned them from using their birth names, practices that severed any hope a slave might have had of reestablishing former connections while making clear to everyone his or her meager status and loss of meaningful identity.\n\nOnce slaves had been rendered damaged goods, permanently unable to go home, slaveholders could take advantage of the slaves' insights about the societies of their birth by bringing them along to capture more slaves. Slaves were valuable, too, for their language skills in negotiating truces or trade deals. One of history's best-known captives was Sacagawea, who was kidnapped from the Shoshone by the Hidatsa at the turn of the nineteenth century and later served as a guide for Lewis and Clark.\n\nThe payoff of having slaves was enormous. A hostage taken during a brief assault could yield a lifetime of labor at a cost to the captors no different from what they expended on beasts of burden\u2014food and shelter\u2014without the time and expense of raising him or her from birth. The fact is, North American Indians lacked work animals, so the slaves of Pacific Northwest tribes were as economically crucial as horses or oxen were for many Old World societies. Indeed, history is rife with explicit comparisons of slave to beast. More than anything else, such comparisons starkly reveal the antiquity of the human penchant to consider only one's own people fully human and to assign varying lesser degrees of humanity to outsiders. Even egalitarian hunter-gatherers could think of foreigners as less than human; slavery put the concept into daily practice, while giving those nonhumans value as goods. Writing of the Pacific Northwest Indians' view of other tribes, one scholar tells us that \"free populations of the coast can be seen as analogous to uncaught salmon and uncut trees. And just as the fisherman turned the salmon into food and the woodworker trees into shelter, the predatory warrior converted freemen into wealth.\"\n\nA slave's status as an animal, extreme as it might be, was a straightforward extension of the imbalances in prestige that often emerged between people in settled societies. Shorn of identity, the lowest of the low, slaves fell neatly at the base of the pecking order generated by human minds. This seemingly preordained hierarchy allowed their abysmal rank to register as natural and just to people from before Aristotle's time and on through the millennia. Indeed, one reason to take slaves from afar was that the novelty of the captives' appearance made it easier to see them as distinct and thus inferior.\n\nWhile members of the elite owned most slaves, their existence was a boon for low-status society members, too, who were freed from the view of themselves as occupying a society's bottom rung and from doing the degrading labor that came with it. This suggests another reason hunter-gatherer bands seldom took captives: bondage rarely made sense when everyone carried out equivalent tasks with leisure time to spare. Supervising slaves would have simply added to their workload. Yet not every slave was treated badly or did menial or hazardous labor such as hauling waste or quarrying stone. Slaves performed best in prime condition, and a leader's slaves had to be worthy of his status. Regardless of their roles, however, slaves were identity-affirming reference points. For example, the historian Theda Perdue explains that among the Cherokee, slaves \"functioned as deviants,\" an invaluable service given that \"the members of a society frequently establish their identity not by proclaiming what they are, or the norm, but by carefully defining what they are not, or deviance.\"\n\nOnce a society was dependent on slaves, it typically had an ongoing commitment to seize more, because slaves rarely bred enough slaves. Males were routinely castrated to make them easy to handle, and stress could depress reproduction for both sexes. Just as slavemaking ants must undertake repeated raids, at times on the same nests, to replenish slaves that have no queens of their own to reproduce, slaveholding humans had to conduct further raids, often on the same \"inferior\" societies, to keep up slave numbers.\n\n# THE CONQUERING SOCIETIES\n\nThe very existence of slaves required a society's social boundaries to expand to allow for their numbers and strangeness, a radical achievement. Yet in its initial form among the hunter-gatherer and most of the tribal societies in which it was practiced, enslavement meant adding only a few souls now and then. Despite the fact that these enslaved were greatly outnumbered and could be treated as mere animals, they were a harbinger of the diversity within societies to come. Indeed, by their mere presence slaves made the inclusion of outsiders in substantial numbers a comprehensible notion. However, the question remains how societies came to engulf entire populations and consider them as fellow members.\n\nWhat started the process along was a change in the motivation for warmongering. When people set down roots around a rich source of food, wild or domesticated, they often had to defend themselves from greedy neighbors. Pacific Northwest tribes had substantial homes and valuable fish runs and stockpiles that could be seized; they also possessed the manpower to protect them or to steal booty from others. Traces of fortifications are ample evidence that ancient villages worldwide needed to defend against the menace of outsiders. The mere existence of a settlement would likely have amplified the fear and distrust, since the concentration of individuals at one place can create an appearance of potentially dangerous unity\u2014a logical reaction for outsiders to have, as individuals in close proximity can act in concert quickly.\n\nThe density of goods, which necessitated a defensive posture, led to the expansion of societies in space as well as population. Although taking someone's land, let alone the people on it, would rarely have occurred to nomadic hunter-gatherers, tribal societies found that the spoils of a fight could be multiplied manyfold when a pugnacious group annexed a productive territory and let its residents live. In fact, once people began to live in anything larger than band societies that could fall apart after repeated raids, completely destroying a foreign group was seldom either the goal or the outcome of clashes in the past. In this light, much of the history of the past century, such as the Nazi vision of a world without Jews or the dedication of ISIS to obliterating Shia Muslims, looks all the more aberrant. The Bible claims every man, woman, and child of the Canaanites of Sodom and Gomorrah were killed, but as genetic studies have shown, they are the ancestors of the modern-day Lebanese.\n\nWiping people out makes no sense given the incentives to profit from a conquered population. Much as enslavement can prove economically more lucrative than killing people, subjugation applied that calculus to taking in entire societies, from which tribute or labor could be extracted on a continuing basis. No other animal shows anything approximating this colonial conquest. It's absent among ants, for which surrender followed by subjugation is impossible. In dispatching the spoils of war, across all the ant species there are only two options: to take slaves or to wipe out the losers, in which case cannibalism is common, as it has been for humans.\n\nHunter-gatherers in bands couldn't pull off conquests, yet villages could. Not that all tribal societies took to appropriating territories. Indians across the Pacific Northwest ranged far to seize slaves but rarely if ever took control of a foreign people and its land. By contrast a tribe with a culture bent on expansion and dominion most often fought with the occupants of territories adjacent to their own, both because a hostile neighbor presented more of a threat than one far off, and because the people on nearby properties were easier to control. The victors who used this tactic are called a chiefdom; their leader, a chief.\n\nChiefdoms were never more than a minority of societies. Still, European explorers came upon chiefdoms by the hundreds, some with many thousands of subjects. Much of eastern North America, for instance, was claimed by chiefdoms renowned for maize agriculture and earthwork mounds. Not every chiefdom relied on agriculture, however. For example, the Calusa of Florida were a chiefdom of settled hunter-gatherers.\n\nChiefdoms were a turning point as pivotal to the advance of societies as was the evolution of markers of identity. No civilizations would exist without the pattern set in motion by chiefdoms after the Neolithic Revolution: the subjugation of foreign societies rather than routing, ruining, enslaving, or murdering them.\n\nTo conquer others, a village required an effective supervisor. While tribal leadership, as noted earlier, was usually weak, the occasional person could gain prominence. Such a figure, coined a Big Man by anthropologists (as with chiefs, most were male), achieved a following usually after showing himself to be a superb warrior. In New Guinea, where tribes were at odds constantly, there were (and still are) many Big Men. Depending on the threat to their people, these headmen could shift in influence\u2014and then disappear, as I related for the =Au\/\/ei Bushmen, who transitioned almost directly from bands to Big Men societies and back within historical memory. When there was a perceived risk of neighbors striking first, Big Men could turn to the brute-force techniques employed by many alpha male chimpanzees, but to much more far-reaching effect. In that situation a Big Man could come to seem indispensable for coordinating the many, who would have acquiesced to being a united hive for the sake of strengthening the \"we-group\" for war, as sociologist William Sumner put it.\n\nA Big Man could become a chief by taking control of other villages. That didn't always happen by seizing enemies. At times he strong-armed a permanent union from what had originally been a fair alliance between friendly autonomous villages, put together to serve the needs of the moment in combating a common adversary. Such a predatory individual could then commandeer the whole region as a base to further expand his domain. Vigorous chiefdoms came to take over what had once been many independent villages, and eventually entire other chiefdoms, to attain populations in the tens of thousands or more.\n\nFew chiefdoms lasted long. For one to persist, its chief had to put a stop to insurrections over the long haul. Like a Big Man, a weak chief had to continue to earn his people's respect, and their faith in him rarely lasted and seldom automatically extended to his children. A chief's best bet was to act on people's dread of being attacked by keeping the battles going. Ultimately, though, a chiefdom had to persist through times of peace, which required that a chief's position, and that of his chosen heirs, be set firmly in place. Inherited status exists among certain animals: a female spotted hyena or baboon assumes the social position of her mother. As for humans, supporting leaders over the long term may be aided by a psychological disposition to see the status quo as legitimate. Displays of power were a basic part of the job, and the extravagant apparel of royalty traces back to headdresses of early chiefs. Today the most downtrodden are prone to believe the high standing of others as warranted, and to assume that important individuals are smart and competent. This perhaps innate belief could have evolved to protect people from the impulse to try to overthrow the powerful and thereby put themselves in danger, which would explain why people concentrated in settlements have always been susceptible to dictators, autocrats, and notions of divine right. Belief that only a deity has power over a leader ensures his ascendency.\n\nMaintaining control of many people, especially when they belong to multiple groups, has always been a laborious business. For an expanding chiefdom to continue functioning, the defeated, though they might be dehumanized to an extent, couldn't be as denigrated as slaves. Their former identity was not entirely lost to them. Many remained on their land with family and community, a situation that allowed their populations\u2014unlike those of most slaves\u2014to increase. Yet life in a chiefdom could be tough. The residents of independent villages, like members of a band, had little reason to exert themselves more than was necessary to get by. Once conquered, however, the vanquished may have been a step up from slaves, but they were still often regarded much like resources to be exploited. Consolidation into a larger society meant that market relations had spread beyond local village campfires, and in theory the goods taken from the subjugated bolstered the economy to the benefit of all. Nevertheless, the spoils disproportionately went to the chief's people or were funneled into further conquests. Compounding the rapacity of chiefdoms was the demand for resources by an expanding part of the society, from priests to artists, disconnected from food production altogether.\n\nIn a band and village society, the constituent villages and bands had been fully able to act on their own and did so most of the time. With chiefdoms, such loose connections between the populations of a society became a thing of the past. Chiefdoms were thus a formative step in a consolidation of societies into a unit\u2014a litmus test for what we think of as a sturdy nation today. To endure for generations, societies starting with chiefdoms needed to achieve what is impossible in other species: a sustained tolerance, if not melding, of groups that had been formerly distinct. Perhaps counterintuitively, then, this formation-of-the-whole was strongest not in societies whose members were most alike, but in those where people of varied origins came to coexist and depend on each other. This was notably true for the state societies that would rise from the most lavishly successful chiefdoms, whose political organization and stability, and influence in welding together people with different backgrounds, we will consider next.\n\n# CHAPTER 23\n\n# Building and Breaking a Nation\n\nFive and a half millennia ago, Uruk, a handful of interconnected towns located east of the Euphrates River, in what is now Iraq, swelled in population and complexity. The largest of them housed thousands of people, supported by a range of goods and services the likes of which had not been seen before. There were streets, temples, and workshops. The numerous tablets with cuneiform writing recovered in the area suggest that many aspects of life were carefully watched over. Uruk is among the first examples of a society that transformed so starkly that what began as a chiefdom had taken on a new means of organization: what academics call a state society (or nation, as I will sometimes refer to modern states, following everyday, informal usage). Although some of the earliest states were no bigger than hamlets by modern standards, they were nevertheless the sort of society we pledge allegiance to today.\n\nStates shared a number of important attributes from their initial appearance. Crucially, in a state society the leader escaped many of the encumbrances that burdened chiefs. A chief had a limited power base, and could be relatively easily overthrown. The fatal flaw in a chiefdom was the chief's inability to delegate authority. When the chiefdom grew large, the former chiefs of its subjugated villages could be allowed to retain their position, but the paramount chief had to oversee each of them personally. This kind of flimsy oversight, based largely on the leader's dominance or powers of persuasion, became impractical once the combined territory took more than a day to traverse.\n\nAll that changed with the emergence of states. Heads of state not only asserted the sole right to enact their will, they could back the claim up with the support of a formal infrastructure. In a state, division of labor and hierarchies of control made their way into institutions dedicated to governance. It was, then, with the proud birth of bureaucracy that societies came to improve their cohesion and rule great expanses. When one state conquered another, each of the former state's territories was typically converted into a province, its capital refashioned into an administrative center. Government agents, each master of a particular job, were distributed as needed. This system of oversight meant that societies could be ruled more coercively than before, even if lag times in communication from the capital to the outer reaches continued to be a handicap for early states. In fact, with a strong enough infrastructure, a state could survive the overthrow of its leaders or reign in their worst impulses.\n\nStates are distinct from chiefdoms in a few other particulars as well. For one, laws are truly laid down: while people carried out vigilante justice in societies with weak leaders, in a state those with authority mete out punishments. In states, too, notions of private property are fully realized, including luxury goods sought after by an upper class. Indeed, while people in chiefdoms could gain prestige and sometimes showed differences in social class, in states inequalities reach extremes. Differential access to power and resources can be either earned or inherited, with some people working for others. Finally, states extract tributes, taxes, or labor from citizens in a more formal way than chiefdoms; in return, they offer infrastructure and services that ensure the members are more dependent on their society than ever before.\n\n# ORGANIZATION AND IDENTITY IN STATE SOCIETIES\n\nStates around the world bear similarities not just in such defining features as their administration of power, but also in the organization of those infrastructures and services. Like any society, a state is a problem-solving organization, and its big problems often require complex solutions. On this point, we discern in states many of the patterns we have already found among the social insects. When a society, human or ant, becomes sizable, the demands on it to provide for and protect its members grow intricate and diverse. Consequently, so must the means by which these obligations are met. Methods must be found for the transport of supplies, troops, and other personnel when and where goods and services are required. Failing to meet basic needs could be cataclysmic. Hence, while there's more than one way to configure a state worthy of the term \"civilization,\" with its impressive urban centers, the range of possibilities is actually quite constrained. When states and their cities expand, land use becomes more structured, institutions from intellectual academies to police forces grow more elaborate, and the range of job opportunities skyrockets.\n\nThere are also improved economies of scale. For instance, it can be easier to feed and house each member, and these lowered costs can lead to a resource surplus that ants invest in warfare and humans in the military, though our species can divert the surplus as well into the sciences, the arts, and such nonessential projects as the Taj Mahal, the pyramids, and the Hubble telescope, which require ant-like levels of coordination and industry.\n\nIn fact, the similarities between world civilizations, even those with completely separate histories, are remarkable, if not downright eerie. Historian and novelist Ronald Wright put it this way:\n\nWhat took place in the early 1500s was truly exceptional, something that had never happened before and never will again. Two cultural experiments, running in isolation for 15,000 years or more, at last came face to face.... When Cort\u00e9s landed in Mexico he found roads, canals, cities, palaces, schools, law courts, markets, irrigation works, kings, priests, temples, peasants, artisans, armies, astronomers, merchants, sports, theatre, art, music, and books. High civilization, differing in detail but alike in essentials, had evolved independently on both sides of the earth.\n\nMany of these innovations not only made it possible to feed and house large populations; they also contributed to keeping societies intact by influencing how the people thought of others. Keeping up an affiliation across a few hunter-gatherer bands was improbable enough; maintaining a common identity grew daunting as societies expanded\u2014at times to the breadth of a continent\u2014and tribes turned into nations, with all their varied subjects. Much of the problem was, and is, a matter of connectivity. To slow disruptive shifts in identity, the population must be interconnected: the more, and more up-to-date, knowledge that citizens have about each other, the better. People can either stop a change or adjust to it, yet only if there is an efficient exchange of information within the society.\n\nOne factor that allowed human interactions to skyrocket was our species' evolved flexibility with regard to personal space. As described earlier, this was important even for the first human settlements. This flexibility has reached its zenith today, however. The populations of Manila and Dhaka work out to roughly one person for every 20 square meters, a million times the density of some hunter-gatherers. Our comfort with the proximity of others depends on upbringing. Yet barring a disability such as agoraphobia or monophobia (a fear of crowds and of being alone, respectively), people existing cheek by jowl exhibit few pathologies.\n\nProximity is just one, very low-tech, way for people to stay in tune with the identity of others. Even still, it is not essential. After all, part of the population of any state must live in rural areas to cultivate the crops. Societies developed other methods of maintaining contact throughout their realms. The domestication of the horse in Eurasia, the invention of writing by Mesopotamians and of ocean-going ships by Phoenicians, the long-distance roads of the Inca and Romans, the printing press in Europe\u2014all such innovations promoted the stability and expansion of societies. In addition to facilitating the transport of goods and the extended control of central authority, such innovations improved the spread of information\u2014and notably, information about identity. This was true not just for states. Once they had horses, tribal nomads such as the Tartars and hunter-gatherers like the Shoshone kept their identities intact over much greater distances than would have been possible when their ancestors moved only on foot, though the complexity became far greater as states took in more, and more dissimilar, groups.\n\nJump ahead to ancient Rome. At its height, all but the most remote parts of the empire were firmly linked by an identity that encompassed vocabulary; fashion from apparel to accessories to hairstyles; crafts such as pottery, floor mosaics, and stucco walls; casual customs, formal traditions, and religious practices; cuisine; and home designs and improvements such as plumbing and central heating. Identification with Rome carried over as well to public works like the layout of towns, roads, and aqueducts. This is not to say the provinces conformed to a uniform standard of Romanization. Rome, like all societies, made allowances for diversity, and across its range people expressed their identification with the empire with geographically local flourishes that reflected their ancestry\u2014and class-based ones, as well, with the affluent opting for the most and most expensive emblems of Roman identity. This sweeping accommodation of markers required efficient communication beyond anything that was possible in a tribal society.\n\nWhatever form leadership took\u2014the rotating arrangement, leadership by committee, or single-person rule\u2014a leader helped shape the social fabric, and part of that mission was shoring up people's identities. Sometimes the influential just set down what was broadly agreed to be acceptable behavior, sometimes their own eccentricities became fashionable, and sometimes leaders could compel behavior of their choosing, setting standards for anything from speech to dress. Given the vulnerability of their positions, the first and most innocuous of these roles, serving as the Voice of the People, was the bread and butter for the leaders of tribal societies and chiefdoms. By setting a firm example, an effective leader could secure his position by contributing to the citizens' sense of shared identity and fate, helping to keep the people's bonds strong even when they were numerous. But once a population was firmly under a leader's thumb, his or her authority tended to amplify: kings seldom felt obliged to show the generosity of a headman at a potlatch. In many historical instances, the influence of leaders reflected a firm control of the roadways, printing presses, and other means by which information flowed across their societies.\n\nBy the time a society manifested the organization of a state, the role of religion had typically changed in ways that further enforced people's identification. Hunter-gatherers admired those imbued with healing or spiritual skills, yet their animist philosophies asked little of their followers. Most tribes and chiefdoms weren't much different in this regard, but the invisibility that a state's large population afforded to its members required they be more strictly supervised. The notion of omnipotent gods provided a mechanism to influence people's behavior even behind closed doors, through fear of divine punishment.\n\nSo long as their governance isn't overly despotic, the benefits states offer can be profound. Not only can the hive-like level of interaction within them strengthen people's collective identity; the rapid-fire exchange of information brings on the opposite of the Tasmanian effect, in which sparse and weakly connected populations forget the innovations of their forebears. Once multitudes interact, fresh perspectives don't just catch on, they are caught in the mill of constant social change. The cultural ratcheting that began in earnest 50,000 years ago has accelerated to the point that no one grows old in a society that hasn't altered markedly since he or she was born. Such enhancements make identifying with a society more of a moving target than it once was. What's more, the mass connections proliferating in states are matched by mass ignorance. In contrast to the near comprehensive cultural smarts of hunter-gatherers living in bands, not even the leader of a state knows more than a fraction of what is required to keep the society functioning. People today often find it necessary to track rapidly changing social trends in deciding what they themselves should do.\n\nThe safeguarding and control of large populations require a heightened level of organization that can be traced back to the first state societies. States assembled armies to put down rebellions and carry out further territorial invasions using offensive strategies that changed as societies grew. Cautious raids made sense for hunter-gatherer and tribal peoples, since they could afford to lose few fighters and the aim was to inflict damage or kill rather than subjugate. Big Men and chiefs often had to personally lead warriors in order to motivate and maintain their following. Even then each combatant was prone to act alone, with plans breaking down in the heat of the moment with little accountability.\n\nBy contrast, the ruler of a state remained safely ensconced in the capital, where he or she could direct attacks and, if the battle was won, oversee the takeover of the territory and its survivors. Military tasks could be delegated to specialists. For a state to sustain a large population and fuel these war efforts, farming, usually of energy-rich grains such as wheat, rice, and corn, had to be carried out at massive scale. A state might overrun its competitors not just because of its greater supply of fighters, but also through its superior tactics, weaponry, and communications. Notable too was a state's tight control over its regiments, composed of citizens drafted for war. The army's strong collective identity, instilled through training and a submersion in patriotic symbols, cemented the soldiers' resolve to \"stiffen the sinews, summon up the blood,\" as Shakespeare put it. Disciplined training ensured much greater reliability, and uniformity, among the troops. This uniformity and the sheer scale of wars guaranteed the impersonal nature of the enterprise. All traces of individuality were stifled. Combat between large states took on ant-like qualities in giant anonymous societies. Social substitutability\u2014the sense that retribution for wrongdoing can be inflicted on anyone in the guilty group\u2014was amplified from the time when raids of hunter-gatherers killed whatever foreigner they came upon, often someone they knew: armies faced soldiers who were interchangeable, and indistinguishable, strangers. Negative stereotypes about the enemy would have clouded any perception of them as individuals, as often happens with foreigners. There was also the fact that the enemy attacked in numbers and dressed in uniforms that made it next to impossible\u2014and completely unnecessary\u2014to tell them apart.\n\n# THE MARCH OF CIVILIZATIONS\n\n\"War made the state, and the state made war,\" sociologist Charles Tilly aptly observed. There have been no truly pacifist states. Whether we speak of a chiefdom or our country, peace masks generations of power plays and almost always swordplay. Any society larger than a handful of villages is composed of once-independent groups. Minoa, a Bronze Age civilization on the island of Crete, was renowned for a tranquil culture of merchants and artisans. Yet even the population of Minoa, serene in its heyday, must have come together, in this case before historical records, by force. The same is true of the modern-day people of such long-peaceful states as Luxembourg and Iceland, when their history is traced back far enough. Just as chiefdoms swallowed tribes and then each other, the pattern of the expansion of the nations and empires that ensued stayed the same. Throughout recorded history conquering was followed by consolidation and control, repeated ad infinitum.\n\nThe birth of states in conflict, and their obligatory inclusion of people from varied sources, have a simple explanation: by the time states emerged, essentially no unoccupied habitable land remained. Some group\u2014whether roaming hunter-gatherers, a tribe, a chiefdom, or a state\u2014was already there and willing to do whatever it could to retain its independence. Any expanding society had to push aside, conquer, or destroy other peoples whose territories would have been part of an unbroken quilt on a map. Not that every battle was directed at the control of land\u2014much of the time the plan was to bring back booty and slaves. All the same, the most avaricious and successful states pushed their territorial borders ever outward. Only a few hunter-gatherers would have been left to eke out an existence in places skipped over because the land was too barren for agriculture.\n\nSuperiority in battle wasn't enough, however, for a society to make the climb from chiefdom to large state. The few titanic civilizations that emerged generally did so in settings where the societies they took over were packed in a tight space. Under this condition, described by the anthropologist Robert Carneiro as circumscribed, conquest paid off royally. \"War appears when mobility is not an option\" is how the anthropologist Robert Kelly framed it. Tribes farming fertile sites encircled by inhospitable regions experienced such circumscription, locking them in a fight from which in time just one would emerge. Think of the Nile Valley hemmed in by deserts, where ancient Egypt took hold; or the Hawaiian and Polynesian island chains, specks in an overwhelming ocean, where giant chiefdoms, some containing 100,000 persons, claimed their domain.\n\nThough far from guaranteed by a circumscribed setting, civilizations were more likely to emerge in such places than elsewhere. Where there was no circumscription, a chiefdom or state could reach a modest size, then be unable to pursue further expansion as surrounding societies shuffled their locations to escape takeover. That was the situation in New Guinea, where an entire tribe, for example of Enga, could move as one unit to keep from being cornered\u2014reminiscent of the flight response a small ant colony can employ to avoid conflicts. To settle in a new place, the fleeing tribe would have to negotiate each shift through alliances with its new neighbors. Because people develop emotional ties to territories, these migrations signaled extreme duress. Yet apparently as a result of this fleet-footedness, the island of New Guinea never supported even a small state society until the country called Papua New Guinea was founded at the eastern end of the island in 1975. Even then it took years for many of PNG's citizens to learn that such a thing as a \"country\" existed. To the majority of the people, their tribes continue to matter most.\n\nWhen an aggressively expanding society couldn't be escaped, often victory was pulled off without bloodshed. Neighbors could be cowed by displays of prowess heightened by uniforms and flags that blurred the marshaled troops into a united \"bristling monster,\" as one battle account described. \"It was an explicit policy of the Incas, in expanding their empire, to try persuasion before resorting to force of arms,\" Garcilaso de la Vega, an early seventeenth-century chronicler of South American history, explained. But showmanship no doubt set the tone of the negotiations, and it may have been safer to acquiesce to subjugation than to be incessantly terrorized by hordes of well-armed outsiders.\n\n# FRAGMENTATION, SIMPLIFICATION, AND CYCLES\n\nIt bears reiterating that nothing about the transition to states was preordained. Nevertheless, the emergence of large, complex states\u2014civilizations\u2014can have the appearance of inevitability, if only because their expansion slashed the number of small societies. No wonder states have been by far the rarest form of society, but are now ubiquitous.\n\nConquering chiefdoms and states sprang up the world over after the advent of agriculture. Most were a flash in the pan, this transience a reminder that no level of complexity secures a society's continued existence. The epigraph of this book quotes Plato, who wrote that a state must only be allowed to increase to a size consistent with unity. Should the learned rulers of such a state heed Plato's advice, however, its unity would still be transient. States undergo a cycle, never an enduring stasis or inexorable ascent.\n\nIn _Collapse_ , Jared Diamond explores how such factors as environmental degradation and competition can hasten a society's demise. But what Diamond calls collapses are a few extreme instances of what is actually the ever-changing nature of societies. Most significantly, such a collapse\u2014when the word is used to indicate the abrupt termination of a society rather than a bleak economic downturn\u2014is more accurately a fracture. Though band societies and tribes divide, the modus operandi for breaking apart chiefdoms and states has been a complex yet often predictable sort of splintering. These breaks were sure to happen, in the end, regardless of resource abundance. More specifically, the manner by which chiefdoms and states absorb outsiders makes them likely to sunder along the demarcations on a map that conform roughly to their past military occupations, even if political and topographic factors and the vagaries of war can have an impact on where the dividing lines ultimately fall. For example, when a Maya civilization shattered, its families didn't just slip into the jungle, and certainly not across a wide area or over the protracted time the word \"collapse\" might suggest. Generally, a king lost control. The top tier of the society vanished. Former sacred and symbolic objects and public works were often desecrated. Outlying districts, subjugated in a distant era, would thereby break free under leaders who no longer had anyone to dictate what they could do. The fracturing of the final Maya kingdom, Mayapan, decades prior to the arrival of the Spanish, left 16 small state societies for the conquistadors to find.\n\nConquering and fractioning alternated through time. Had the Spaniards made it to Mexico a century later, at a different point in this loop, they may well have been met by another Maya empire like Mayapan, reborn, or perhaps\u2014if a recent breakdown had been especially harsh\u2014by isolated farming villages. In the former case the palaces, temples, and art would have stunned them far beyond anything they actually did see. These public works fall to ruin each time control of labor and essential supplies dry up. As a result, the smaller societies left behind after a breakdown experience a cultural simplification. Though often still in possession of their palaces, their more rural populations would no longer have the resources and manpower to maintain them in their previous splendor; in extreme situations people could see-saw between agriculture and hunting and gathering.\n\nThe Roman Empire foundered, in the end, into smaller states as well. These stayed intact by delegating leadership responsibilities to lords in what became a feudal system, a form of social organization in many ways functionally equivalent to a chiefdom. At that point societies, in the sense of people's identification with a greater group, would seem to have lost their sway across much of Europe. Yet careful scholarship demonstrates that the local lords crippled, but didn't erase, the sense of belonging peasants felt with the region beyond the fiefs. That nations quickly sprung up during and after the Middle Ages reveals how people can reconnect to feelings of solidarity that stretch far back in time.\n\nCycles of conquest and dissolution are evident everywhere. Historical records are replete with chiefdoms and states that gained, then lost, members and property after expanding too far to hold their territory. This latter phenomenon, which Yale historian Paul Kennedy calls \"imperial overstretch,\" though it's not at all specific to empires, is precipitated by the way that overambitious societies can be worn down by invasions and civil wars. In such affairs, economics mattered. Control of distant peoples could be an expensive logistical ordeal. There would be a fine line between perceiving an outlying population as a benefit or a nuisance, either because it competed with its conquerors for resources or because it didn't contribute enough to justify the conqueror's investment, or both. Meanwhile the province seldom recouped enough rewards to offset the resources taken from it, and the imbalance fomented discontent. Overburdened families could creep into the hinterlands, living as refugees beyond the reach of the government. Yet countless times through history those people who chose to stick to their land eventually won their independence, whether through the conqueror's willing retreat or in a struggle.\n\nImperial overstretch could have been motivated by the fact that the most exotic and desirable goods often came from societies that were radically different and immensely far away. At some point the practical option shifted from conquest to commerce, a calculus that led to construction of the Silk Road and the trade networks of modernity.\n\nOther patterns emerge at grand scales. Mesopotamia and Mesoamerica were like Europe: collections of states within a geographic region that, due to both trade and their historical connections from past conquests, shared much in common. Both individual societies and these regional expressions of them could go extinct. Sumer, Akkadia, Babylonia, Assyria, and other states around the Tigris-Euphrates river system came and went. Still, they all exhibited common threads we call Mesopotamian, with overlapping artistic styles and polytheistic religions. Then, in the first century AD, after three millennia of brilliance, Mesopotamian cultural traditions, political organizations, and languages all but vanished. They were brought down by the invasion of nomads arriving from the south and west, leaving only trace influences on the modern populations of the Middle East.\n\nAll this was a wild ride played out in super slow motion. Each rise and fall left its imprint on the population of a region, yielding a fresh medley of cultural ingredients. A conquered people re-freed could revert to the ways of their forebears, but not entirely. The greater society of which they had been a part left its mark, as they would have learned the language or taken to the beliefs of those who had once been in power. Across a geographical expanse, then (for example, over the full extent of Mesoamerica occupied by the many and successive Mayan civilizations), people retained evidence of their past unions in the increasing similarities of their identities. Such an overlap would have made their conquering and control of each other easier the next time round.\n\nThe imprint of societies past may explain certain curiosities of world geography. Distantly related peoples of the South American rainforest, for example, have similar crafts, traditions, and languages not easily explained by trade. Their hybrid cultures could be a signature of powerful chiefdoms, now vanished, that overpowered and refashioned the identities of many. Archaeologist Anna Roosevelt, great-granddaughter of Theodore, has argued that the Amazon Basin once had urban centers. The evidence for these cities, such as prehistoric canals I have seen in Suriname, lies buried beneath a green carpet.\n\n# THE ADVANCE AND RETREAT OF NATIONS\n\nFor the last century, at least, most wars have been civil, often brutal attempts to break societies rather than expand their reach. The factors already spelled out for divisions of early human societies continue to play a role. Identity remains pivotal in both building and fragmenting nations, with a key difference. Schisms in a band society emerged from the ground up across a territory because of distinctions brought about by faltering communication between people who had been pretty much the same when the society formed. Factions arose from scratch among the society's members in a way that would have been hard to foresee when the tribe was young. For nations nowadays, however, many factions are symptomatic of groups whose dissimilarities extend into the remote past\u2014indeed, that predated the state. Populations struggling for independence, such as the Scots or Catalonians, aren't concerned only with political and economic interests, but have a substantial psychological driver: they picture themselves as different going back to the mists of time.\n\nIt is fascinating to consider what this means in psychological terms. The response of early humans to a schism in their ranks would likely have emerged very slowly, until a crisis point was reached among members whose forefathers, at the time of that society's birth, would have seen each other as equals. By contrast, the citizens of a state can react to many uprisings by zeroing in immediately on the timeworn social differences of the recalcitrant outliers, a choice that at times can bring centuries-old conflicts to renewed prominence. In all likelihood, derogatory stereotypical beliefs would surface, ideas that could be traced back, in a rough form at least, to a time when those people belonged to separate societies.\n\nSeparatist groups today draw on, and indeed manipulate to intense emotional effect, symbols with a meaningful connection to their past to keep their people united in the face of hardships or to catalyze their secession. The breakup of Yugoslavia yielded a half-dozen republics that recast the anthems, flags, and holidays previously valued by their people; they also added new ones on the spot. These revolutions hark back to how hunter-gatherers swiftly diverged to establish differences after the trauma of a division. But laying claim to time-honored markers is a quick-and-dirty way to revitalize a people's freshly minted independence.\n\nAs happened in the dismemberment of early state societies, such as the Maya, the populations of the resultant countries often must get by on less than they did as part of the more inclusive nation. Unless the original regime had been extremely repressive, this makes partition a step back economically. Today, seceding nations receiving little outside assistance rebound slowly\u2014among the products of former Yugoslavia, Bosnia and Kosovo remain in the doldrums. My sense, though, is that the intense feelings of identification with the new society can offset any loss in the quality of life. After all, forced compliance to a distant ruler will often be a social drain. Fortunately the basic kernels of social existence, such as the social support local communities provide individuals and families, seem resilient to the rise and fall of societies. So, while the division of a band society might affect all the members much the same, the breakup of a state is a blow to the powerful but could make far less difference to everyone else.\n\nIn all these matters, geography truly is destiny. For a secession to succeed, the clashing group typically has to populate a specific portion of the country's territory, which they often claim as an ancient homeland. This means the break ends up falling between stretches of terrain heavily populated by ethnicities that once had societies of their own there. Nevertheless, drifts in identity from place to place still have an impact, much as they once did across hunter-gatherer bands. These changes help generate regional cultures found in every nation and influence all manner of things from cuisine to politics.\n\nThe likelihood that such geographical variation on its own will tear a country apart appears relatively small, however, even when those differences are played up. For example, slavery, not identity, overwhelmingly drove the American Civil War. At the time, most Southerners thought of themselves first and foremost as American. What we think of now as Southern culture became a point of pride only after the war ended. During the fighting, the Southern intelligentsia did promote the superiority of Southern ways to stir up a regional allegiance, including claims that Southern whites were their own ethnic group\u2014descendants of a refined British bloodline that adhered more closely to America's founding principles than did the Northerners. Yet their appeals to Southern commonalities didn't have much effect. Most Southerners, uncertain of the sensibleness of secession, were inspired less by the Confederacy than by the obligation to defend their families. The absence of an intensely shared identity, separate from that in the North, was the key factor underlying the Confederacy's struggle to stay unified, and ultimately its defeat.\n\nWhile many secessions have featured issues of identity that include reassertions of former territorial rights, a good example being the tense but ultimately peaceful separation of Norway from Sweden in 1905, there are, of course, exceptions. Political issues can come into play to create nations that bear no connection to any formerly autonomous group. When Venezuela and Panama withdrew from Colombia (originally Gran Colombia) in 1830 and 1903, respectively, difficulties in travel between their territories sharpened the political disconnect. As a result, they became separate countries without any venerable split in heritage among the people. Alternatively, outside forces can prevail, as in the 1945 partitioning of North and South Korea, which was in no small part an outcome of the rivalry between Russians, Chinese, and Americans; or in the mostly artificial lines separating Pakistan and Bangladesh from India, negotiated by the British with the rulers of the principalities set up during Britain's control of the region until 1947. At other times novelties introduced by colonialism were a factor. That was arguably the case in the separation of Eritrea from culturally more traditional Ethiopia: the otherwise similar Eritreans had been shaped by years of Italian culture and rule by the time their country was annexed to Ethiopia after World War II. They won their war for independence in 1991.\n\nNo society can persist unless being a part of it is important to its people. Despots keep dysfunctional nations intact for a time, but people with little attachment to a group are less stalwart and industrious. The USSR is one example of a nationality that deteriorated after being imposed on its people not even a century prior. Like Yugoslavia, it balkanized into states to which people felt more committed\u2014in the Soviet instance, Latvia, Estonia, Lithuania, Armenia, and Georgia were the offshoots with particularly deep historical continuities.\n\nArchaeologist Joyce Marcus has found that state societies across antiquity had a finite life span, normally ranging from two to five centuries. This duration suggests states are no more persistent than hunter-gatherer band societies, which the evidence indicates endured for a few centuries as well. We must ask why nations aren't more stable given the controls they put in place, the services they provide, and the improvements in information flow that expose the citizens to each other's perceptions of proper behavior. Despite the benefits of living in a state society, one recurring flaw is that, as an archaeologist proclaimed, states are \"ramshackle contraptions, at best half understood by the people who made them.\" That courts of law, markets, irrigation, and so on had to exist in some form didn't mean people always put them together well. Much of the cobbled structure of a state is grounded in a feeble understanding of the origins of people's devotion to their country and each other, and how these can be managed. Nations can be administered without undue force so long as people feel they share an identity and common purpose. Yet as conquests thundered into human history, achieving satisfactory bonds across a society grew increasingly problematic. Markers of identity came to vary radically, leaving members struggling to achieve visions for their society that were at cross-purposes.\n\nI have shown that accommodating outsiders at a mass scale is unheard of in other animals. The most other vertebrate societies do is to bring on board the occasional lone mate or refugee. Adding foreigners as a class began in our species with the taking of captives and slaves but ramped up with the wholesale acquisition of populations not by mergers gladly made but by force. Over time state societies emerged that were better able to administer and control their varied populations\u2014including what we now think of as ethnicities and races. How these populations get along together is the topic of the next section. State societies, as we have seen, are ephemeral, and to keep going at all, they have to function in the face of their ramshackle structure and disparate populace. Success requires that they retain their cachet as the group that their people value above all other memberships. This means a state has to impart to its citizens a consciousness of being cut from the same cloth, even though\u2014with the incorporation of outsiders\u2014those citizens are often clearly different. In this, acceptance of domination and control has played a critical role.\n\n# SECTION IX\n\n# From Captive to Neighbor... to Global Citizen?\n\n# CHAPTER 24\n\n# The Rise of Ethnicities\n\nStepping from my apartment building in Brooklyn, I cross paths with young men and women walking their dogs or running for the subway into Manhattan. I turn the corner onto Atlantic Avenue to see Sahadi's, a century-old grocery store that smells of fresh-ground spices and Mediterranean cuisine. All around are Middle Eastern restaurants and markets where American citizens of Arabic descent shift effortlessly between the languages of their ancestors and American English. I settle down at my favorite caf\u00e9, joining an African American couple, an Arab American family, and a Mexican American man talking to one of several white people at the small round tables near me.\n\nFor a New Yorker, this is an everyday experience. And yet from the point of view of a hunter-gatherer from 10,000 years ago, my splendid spring day would have been beyond comprehension. The incorporation of massive numbers of different peoples has been the single most radical innovation in the history of human societies. It gave societies the option to grow by absorbing formerly separate, often adversarial groups to create ethnicities\u2014that is, groups that once comprised their own societies but have come to occupy the same society (and that, with the passage of time, usually see themselves, often wholeheartedly, as part of that society). While not everyone lives among the sharply different ethnicities around my neighborhood, all societies today, even those of apparently homogeneous population, ranging from small nations like Liechtenstein and Monaco to expansive ones such as Japan and China, are admixtures of people, the distinctions among their citizens merely dimmed with the passing centuries. The other radical changes that took place over the millennia, from innovations in politics to new religious beliefs to scientific achievements, have required minor adjustments by comparison to this acceptance and intermixing of human kinds.\n\nIn chiefdoms and the first small states, the mingled populations wouldn't have been what we consider ethnicities and races today. The differences in identity between the conquerors and subjugated would have been minimal to none. More than likely they were merely adjacent villages of the same tribe, once autonomous but now politically fused. Expand enough, though, and a chiefdom or state society would take in not only peoples of distinct societies but groups defined by distinct languages and traditions. Marked differences presented challenges for communication and control, yet had their advantages: the subjugated were unambiguously _other_ and could be treated as such. There was less moral imperative to be concerned with the newcomers' welfare. Indeed, by some definitions an empire is a state stretching far enough afield to control peoples of markedly different cultures\u2014a strategy that would turn to colonialism once conquests leapfrogged abroad.\n\nAs more peoples were added, encounters with unfamiliar others within one's own society became par for the course. Societies swarmed with people, most of whom were strangers. More bizarrely still, those strangers, no matter how dissimilar, could be recognized and treated as fellow members. How did societies reconcile all this diversity and turn _them_ into _us_?\n\n# CONTROL\n\n\"When peoples were conquered, incorporated into provinces and, in due course of time, became part of an integrated empire,\" one scholarly treatment of the emergence of racism in the Roman Empire tells us, \"this entailed a process of ethnic disintegration and decomposition.\" During the gobbling up of foreigners, populations could be transformed into ethnicities that interacted well together. This didn't happen across all societies, however. Consider the following passage from a study of the Incan Empire:\n\nOne of the reasons for the Incas' success was their use of the existing political and social structures of conquered peoples for ruling them. Instead of trying to change the people's lives, they tried to maintain continuity so the subjects' lives were disrupted as little as possible.... They assigned conquered leaders positions of authority in the government, gave them high-status gifts, and honored their religious beliefs and practices. In return, the Incas expected the conquered people to work hard for them to produce food, cloth, pottery, buildings, and other large and small items and to be obedient and loyal subjects.\n\nThe peoples subjugated by the Inca essentially retained the identities they had had as an independent society. From this you might gather that the Inca were less domineering toward the defeated than the Romans, but I believe this misreads what was a harsh form of indirect rule. Although the conquered peoples received some food and goods from the Inca, the residents of most Incan provinces had no real social status within the empire. They had almost no contact with the regular Incan population, and were prohibited from modeling themselves after their overlords. With their continued foreignness bringing them social exclusion, those unfortunates were for all time wholly and undeniably _other_.\n\nIf the Inca had instead chosen to allow their subjugated peoples into the broader society, as Rome and the Chinese dynasties eventually did with the populations of many conquered provinces, those people may have come to identify not just with the Inca, but as Incans themselves, taking pride in the empire even if their stature in it continued to be less than equal. As it was, the occupants of the outlying provinces were beaten into submission, unrests crushed, and unruly villages transplanted to land other than their own to guarantee their subservience. The only possible good to come to them from this abuse was protection, as the Inca fought back tribes beyond their territories. One supposes the border populations may have preferred to bow down to the enemy they already knew. Nothing matches a common foe in drawing people together, and that's true whether those people are separate societies dependent on each other for security, fully accepted groups of the same society, or\u2014across much of the Incan Empire\u2014residents of districts that are browbeaten by their superiors.\n\nDoubtless the system worked. When the Spanish reached Peru in the sixteenth century, they stumbled on an empire containing perhaps 14 million people. This was remarkable given that the Inca had been a pastoral tribe little more than a century prior (albeit a tribe that had the advantage of being able to build upon the foundations of earlier kingdoms in the region). It's one thing, though, for a country to marginalize small groups within it, as the United States would do later with native tribes; it is quite another to lord over a civilization the bulk of whose population is ill-disposed to its rulers and doesn't identify with them. I question how long the Incan Empire would have stayed intact unless they consolidated their provinces by motivating the people there to get behind the ruling population.\n\nThe comparison of the Incan and Roman methods of administering a society highlights the distinction between controlling people through out-and-out domination and incorporating them into the society. In the first instance, obedient local leaders were often kept in place after the society was subjugated. They would be handsomely paid to oversee the extraction of goods and services from their people, who received few benefits in return. Alternatively, the population could be brought into the fold with the expectation that, even if underdogs, they would come to see themselves as a part of the society. In that case the central government typically took on the administration of their needs in recompense for goods and services. The more thoroughly the people were incorporated, their allegiance and identity recast and former government disbanded, the less able they were to try to break away from their conquerors.\n\nState societies have attempted manifold approaches, from cruel domination to generous incorporation, depending on the tractability of a people and the profitability of giving them greater status with fewer constraints. The Romans put up with the cultural and religious differences of freeborn people from regions across the empire. However, they still kept unruly outliers, such as certain African districts, on a tight leash while absorbing acquiescent peoples like the Britons. A state's strategy could change over time, too. Over the centuries the Japanese maintained an ambivalent relationship with the Ainu hunter-gatherers of Hokkaido island, at some points attempting to bully them into renouncing their ways and conform with the general population, at other times keeping them well apart from everyone else.\n\nThe China subcontinent fascinates in that conquests there began early and to great success, eventually creating a virtual uniformity among what's now regarded as the Chinese, or Han, race, presently amounting to 90 percent of China's people. The scale of this outcome can be attributed to the policy of early dynasties of accepting anyone converting to their culture, writing, and sometimes their language. This tradition traced to Confucius, who promoted the idea that people could become Han simply by committing to a Han mode of life.\n\nBased on the evidence of ancient texts as well as changes in identity expressed in everything from architecture to lacquer work, archaeologists have teased apart how the Qin (221 to 207 BC) and Han (202 BC to 220 AD) dynasties brought together much of the population that would eventually become modern China. Unlike the Romans, who exported to their subjects a miscellany of improvements in plumbing, lighting, and other basics, the Chinese dynasties offered few quality-of-life benefits to the outlying masses, depending more on their military presence to crush repeated uprisings. Some of the tactics used by the Qin and Han were common to many territorial expansions around the world. Both dynasties concentrated on unifying the districts nearest their imperial centers, believed to have been located around the putative birthplace of the original Han in the north. Trusted subjects were encouraged to colonize those districts to ensure the ascendency of the Han culture. Among the provincials, the wealthiest would be first to realize the desirability of teaching their children Han customs. Over the centuries this education filtered down through the social strata to yield a widespread Han identification by the time of the Ming dynasty in the fourteenth century. The focus of the dynasties on the accessible territories could explain why they repeatedly lost control of their most remote compartments, including what would become Korea and Vietnam.\n\nWithin the borders were pockets of indigenous societies the dynasties failed to bring into the mainstream. Often the groups inhabited mountainous regions unsuited for cultivation, offering little in return for the effort of subduing them. Some ethnic areas, notably those of the Tibetans and Uygurs to the west and the Wa near the Burmese border, would in the end come under dynastic control, but even then the authorities held these people at a distance as substandard, mirroring the attitude of the early Japanese, who thought of the Ainu as dogs. An unwritten policy was to leave the languages and practices of such heathens intact. In the sixteenth century, the Ming dynasty even walled the rebellious Miao into their mountain strongholds, repressing these and other peoples as any colonial power would. By retaining their identities, the social outliers fulfilled the role that provinces at the hinterlands perhaps had for the Inca, and that slaves had for societies such as the Cherokee. The titan of Greek poetry Constantine Cavafy was right to ask, \"Now what's going to happen to us without barbarians? \/ They were, those people, a kind of solution.\" Simply by existing, the barbarians cast light on what was proper and right.\n\n# ASSIMILATION\n\nWhat remains to be illuminated is how interactions among such conjoined societies became realigned from enslavement or subjugation into mutually beneficial relationships that no longer required force. In such realignments, societies emerged that, despite any social chasms, inspired the loyalties exhibited by members of small societies in the past. Somehow people had to accept those who had been outsiders, with incompatible identities, even though, as we have seen, societies spurn being combined as intensely as a body rejects a skin graft that doesn't match the immunological identity of the host. I see the success of these mergers as an outcome of a human attribute, coopted for a new purpose: while people cannot simply drop one identity for another, their markers, and how they are deployed, have always had to be malleable for humans to adjust to the perennial social changes _within_ their societies.\n\nIn describing human psychology in earlier chapters, I assumed that research findings about different ethnicities and races within a society could be extrapolated to attitudes about members of foreign societies. But this reasoning implies the converse: that the psychological toolkit our ancestors evolved to interact with foreign societies has been co-opted and reconfigured to allow the ethnicities and races of a society to coexist. Reflect on how this may have occurred. Humans express pride in their society, a cognizance of its specialness and the difference between their identity and those of foreigners. When conquering tribes began to take those same outsiders into their societies, the same mental circuits already employed to distinguish, and respond to, foreign groups would have been pressed into action to make sense of the relationships between the ethnicities within a society. If this description is correct, membership in a society and in an ethnicity or race can in many ways be equated. However, there's one important difference. Ethnic groups have come to invest part of their identity and social obligations into the greater society of which they become a part. These groups act, to some extent, like societies _within_ a society.\n\nIn this final stage of our narrative, humans have taken a path for which there are few parallels in nature. This societies-within-society complexity finds its closest approximation in the ocean-wide clans of the sperm whale, each containing hundreds of unit societies of a few adult females and calves. Yet the similarity is superficial. Units don't differ in their behavior and power relationships as ethnicities do. What membership in the same clan provides is the opportunity for units to team up to catch prey effectively by using a shared approach to hunting, and that is all. By contrast, ethnic distinctions permeate every aspect of human life.\n\nIf ethnicities are to find each other palatable within the greater society, the way forward is for them to align their identities well enough to override the significance of whatever differences remain between them. Since differences between ethnicities, like differences between societies, are written on the body\u2014in the form of markers affecting actions and appearance\u2014then grafting outsiders onto a society requires those people to adjust to the local \"writing.\" That is, to assimilate.\n\nAssimilation becomes more challenging when vast cultural distances need to be bridged and the people have the numbers and resources to fight back, an obstacle that can put the brakes on the expansion of empires. It tends to occur in a particular manner, as I will describe, because the way outsiders are brought into a society guarantees that each society solidifies in great part around one ethnicity. Almost always, this dominant people founded the society, originally occupied its heartland, and came to crown the Great Chain of Being, with other groups suffering negative prejudices by comparison to them.\n\nMuch of the power is in the hands of this core group of people, and especially the leaders and nobility, which are mostly of the dominant group and tend to support its interests. Hence assimilation is asymmetrical. The onus is on other ethnicities to comply with the dominant culture. Indeed, from this point on I will use the word ethnicity as it tends to be used in everyday practice, to describe not the dominant people, but such lower-status groups. The dominant group can compel those ethnicities to fit in, or if an ethnic people find that the changes are to their benefit, they may take them up of their own accord. Often a bit of both goes on, as well as some accommodations to still other ethnicities. Whether the changes are forced or freely made, some tolerance on the part of the dominant people\u2014certainly more than the Inca usually showed\u2014is entailed while the newcomers learn what's expected.\n\nNo surprise here. The same adjustments would have applied to an individual member of a hunter-gatherer band who had married into another society; he or she, too, would be expected to act within the social boundaries permitted by the adopted society. Until then, the locals had to make some allowances for the newcomer's foreign ways. The very possibility of assimilating an entire population must have had its origins with such elective movements of solitary individuals seeking to become members of another society, the sort of transfer widely seen in animals and which would have occurred throughout our evolutionary past.\n\nAn ethnicity's acceptance absolutely requires that it curb anything that could cause its members to be branded black sheep, such as traditions the dominant culture considers immoral or reprehensible. The banning of the potlatches of the Pacific Northwest Indians in 1884 by the Canadian government, which declared them wasteful and uncivilized, is one of countless examples. Dominant populations have perennially taken it upon themselves to \"civilize\" people by enforcing standards of acceptable behavior as well as submission to their proper social place by those \"savages\"\u2014a term settlers in the United States often applied to American Indians, although the sentiment existed before recorded history. Columbus spotted slaves the very day the _Santa Mar\u00eda_ set ashore in the New World, where Native Americans rationalized the taking of captives as \"taming\" them.\n\nWhile this \"civilizing\" carried over to less remarkable behaviors than potlatches, an erasure of the boundary between themselves and other ethnic groups would never have been the goal of a dominant people. The end result of assimilation is a sort of merger, but not one entailing a loss of separate identity. Rather, the identity of what had once been an independent society is refashioned in the image of the dominant group\u2014to a degree.\n\nI say \"to a degree\" because if outlandish behavior on the part of a subordinate group made the dominant people uncomfortable, so too could excessive conformity. Indeed, too many similarities between ethnicities can infringe on people's desire to uphold valued differences, and ultimately make prejudice worse. By the same token, too much assimilation can shatter the self-esteem of an ethnic people. So while dominant groups drive much of what happens, the outlook of the ethnicities plays a role, too. Adjusting to the expectations of the people in power increases an ethnicity's stature, or at least legitimacy, just so long as its population stays distinct enough not to intrude on the unique identity of the dominant group. Hence Jews were targeted by the Nazis not because of their failure to fit into German culture, but rather at least in part because they were recognized as distinct yet couldn't be told apart from other Germans much of the time. Menorahs and kosher diets went unseen behind closed doors. This uncertainty was used to stoke fears that the Jews were using their assimilation to hide wealth and influence, and malevolent intentions. Their fluency with the dominant culture put the Jews in harm's way as much as their differences did.\n\nOf course, every ethnicity has some influence on the dominant culture, nudging the boundaries of what a society permits when that society embraces its cuisine, music, and other cultural highlights\u2014just as the same society could import such elements from other societies. In the process of Romanizing its outlying districts, for example, Rome also took the best those people had to offer\u2014perfumes, dyes, spices, and wine. Even with this fabulous assortment of ethnic traits, we may nevertheless distill a set of shared features that describe the society in toto (as in _the_ Canadian culture). Yet most societies are no longer defined by one culture, in the sense people usually employ this term. Instead the members home in on their commonalities while making allowances for more diversity than people ever did in the past\u2014the ethnic richness itself is a wellspring for national character in places like the United States and Singapore. To describe this tolerance and, indeed, positive attitudes between groups, psychologists speak of a superordinate identity cultivated through society-wide commonalities. Such an identification diminishes Us-versus-Them differences to allow for an inclusive We mentality. It's a point of view necessary for societies-within-society to function.\n\nTo be sure, converging on a perfect match between ethnicities would be impossible even if a loss of self-esteem weren't an issue. Individuals who distance themselves from their ethnic background not only fail to lose their accents and other ethnic tics, but usually they unwittingly hold on to the underlying codes of conduct of their ancestors and instill them in their kids. Differences will persist across generations even when a near fusion of groups occurs over the long march of history\u2014in the case of the Han, two millennia. Just as the Romanization of people varied across the far-flung Roman provinces, the locals across any society put their own spin on their identity, intentionally or not\u2014and other members notice this. While the Han living in the earliest Chinese dynasties might have seen the provincials who adopted Han ways as legitimate fellow Han, they would nevertheless have registered differences. From what evidence exists, they would have felt those Han to be inferior, even if far less inferior than the ethnic groups of China that weren't Han.\n\nEarlier I argued that hunter-gatherers should be said to have had nations despite lacking the governmental infrastructure associated with modern states. But the fact is that nations\u2014in the sense many scholars think about them, as independent groups of people sharing the same cultural identity and history\u2014really existed only in hunter-gatherer days, when societies were far more uniform. Every state society, scrutinized carefully enough, is an ethnic mishmash. Still, I will continue to bow to common parlance and refer to today's societies as nations.\n\n# DOMINANCE\n\nThe dominant group's control of the society's power and identity can reasonably be called the founder advantage. However, what matters most for dominant group membership is sharing the ethnicity of those founders rather than tracing one's actual ancestors back to the birth of the nation. Hence Caucasians have stayed dominant in the United States even though the forefathers of many European Americans came to the US after 1840. Meanwhile, virtually all African Americans are descended from slaves that arrived before that country's independence, which is to say the average black family has been American longer than the average white family. Reportedly Winston Churchill said that history is written by the victors. Indeed, the finely edited histories that nations promulgate put the dominant people in the best light and affirm their claims to power and status. This suggests another reason that hunter-gatherer bands, being egalitarian, had little interest in histories.\n\nWhat keeps the dominant group in power? Its people are often called the majority. As that term implies, they typically make up most of the population, especially in the dominant group's original territory. That other ethnicities, or minorities, sometimes compose the bulk of the populace, however, suggests that being on top can be a matter of circumstance more than numbers. That was true for the white Afrikaners who ruled over native African peoples in apartheid South Africa, and in an even more bluntly repressive form in \"slave societies\" such as those of the early Greeks, where captives outnumbered freemen. It was true as well for societies created by nomadic herders such as the Mongols, who exploited their equine skills to take over much larger agricultural societies.\n\nGenerally the dominant group is less able to cling to control in places where it is outnumbered, as its people tend to be in outlying provinces. This was an issue in the Roman capital, where slaves grew more numerous than citizens. Roman slaves came from many sources and couldn't be distinguished in a clear way from the freeborn population, which was likewise diverse. The government chose to ignore this complication after realizing that branding their servants would have made their numerical advantage transparent and fostered rebellion. Minority groups with low populations but relatively high economic success can be seen as threats, too. The Chinese in countries such as Malaysia have faced this difficulty, as have the Jews at various times in history.\n\nWhile the people of outlying districts, who are likely to be most distinct from the dominant people, have had a high success rate of breaking free, it's been rare for them to seize control over the dominant group and the entire country. \"We confront here almost the political equivalent of a Newtonian law,\" two political scientists submit: \"Bodies in power tend to stay in power unless acted upon by outside forces.\" This indicates how effective the dominant people are at protecting their position, being in control of the police and the military; but it also testifies to how difficult it can be for minorities, often geographically and culturally disparate even if large in combined numbers, to join forces. There have been successful overthrows. One occurred in Rhodesia when the white population lost its exalted position and the country was renamed Zimbabwe in 1980, putting an end to years of guerrilla war. This was an exception that proved the rule: the white \"majority,\" former colonists who had separated from the United Kingdom just fifteen years before, were vastly outnumbered by tribal groups, the disparity in numbers making their hold on power fragile.\n\nThe control of the dominant group extends from everyday matters of identity to the symbols that a society holds most dear. Such majority control of societal markers makes the position of minority groups tenuous. People of Asian descent born in the United States may celebrate their country's flag, yet experimental studies show that they more readily associate that flag with white people than with fellow Asian Americans. Even when a member of a minority group may be part of a family of proud citizens going back generations, at some level there is the sense of being \"a perpetual foreigner in one's own land,\" one study says. Meanwhile \"the white Protestant American is rarely conscious of the fact that he inhabits a group at all. _He_ inhabits America. The _others_ live in groups,\" sociologist Milton Gordon writes. This explains what may be the most significant defining feature of dominant people, and why they are rarely spoken of as an ethnicity. People of the majority enjoy the luxury of expressing themselves as unique individuals more than do minority people, who end up diverting more time and effort into identifying with their own ethnic group.\n\nThere are serious consequences to a majority associating \"their\" country mostly with their own people and less so with other ethnicities. Laboratory studies suggest that the dominant people often express uncertainty about the loyalty of minorities. Because of how people can grow to accept the biased views that others repeatedly express about them, this mistrust can spark the very behavior the majority fears, by marginalizing minorities. The people of especially denigrated groups may come to behave as they are accused, even to think of crime as a reasonable choice when few other options are open to them. This alienation presents another reason, aside from their disconnection from the symbols and wealth of the nation, for minorities to express less attachment to their society.\n\nThere's a sense in which today's minorities serve a role little different from that of the indigenous tribes within the Chinese dynasties, such as the mountain-dwelling Miao: they act as points of comparison for the dominant people, who remain the \"pure\" representatives of the society. Hence the awkwardness Asian Americans feel on being asked where they are from, when, in contrast to people of European ancestry, they aren't expected to say a place like Peoria (a common response to such an answer being, \"No, where are you really from?\"). The most maligned minorities are kept at the greatest social distance, as is true for African Americans. Although still stigmatized, the biracial children of Asian and Caucasian parents, for instance, find it easier to fit in than those with \"black blood.\"\n\nThe link between the society and its dominant people runs deep. Ask a person to imagine a citizen of his or her country. If that person is from the United States, the image that almost always pops immediately into his or her mind, whatever the sex or race of the individual, is a white male. The British political thinker T. H. Marshall wrote of citizenship as \"a claim to be accepted as full members of the society.\" Given what we know about the advantages that the majority people share, and about human psychological responses to ethnicities and races, the trick here is the word \"full.\"\n\n# STATUS\n\nThe status relationships among ethnicities and races are more complex than the mere acceptance of the majority's dominance; the standing of the minority groups within the Great Chain of Being turns out to be fluid when looked at over the decades and centuries. Changes in status are uncommon among the social networks and few groups present in the societies of other animal species. Infrequently, a matriline in a troop of baboons wins a fight with a higher-status matriline and its females end up with access to a better sleeping spot and more food. Human ethnic and racial groups rarely shift rank through outright aggression in this way. Instead their position creeps up or down based on shifts in the society's perceptions of them. Nor are all people of an ethnicity of-a-kind in their station. A family that has moved up\u2014often in part by taking on more of the characteristics of their adopted country\u2014may not associate with the poorest or most recently arrived newcomers among their ethnicity.\n\nOne reason that change comes slowly has to do with the way established social standings are accepted, and not just by those in power. People often see the position of their ethnicity and race as natural, unalterable, and warranted, in much the same way they perceive their social status as individual persons to be what they deserve. A default view is that the world is fundamentally just; thus people's troubles, and those of their group, are justified. As a team of leading psychologists put it, \"Instead of resentment toward the privileged and sympathy for the underdog, on average, people endorse the apparent meritocracy and infer that (for groups) high status invariably represents competence.\" As a result, according to another group of authors, \"people who suffer the most from a given state of affairs are paradoxically the least likely to question, challenge, reject, or change it.\"\n\nThe strength of this conviction can't be denied. It's true today for members of the lowest caste in India, the untouchables, and doubtless was so for any slave resigned to his or her fate throughout history. This acquiescence to social position must have been pivotal to the success of societies going back to the first chiefdoms and states. What might have been expressed in hunter-gatherers as caution, aversion, or disgust at foreigners was redirected toward classes of people within the society in an effect so pervasive that even the downtrodden maintained a poor opinion of themselves. The outcome, as our overview of psychology revealed, is that ethnicities coexist despite the social stigmas they bear.\n\nIn fact, similar power distinctions to those among ethnicities have always existed between societies, too, as in the jostling of the United States with emerging economies like India and China that vie for status on the world stage. When it comes to enduring a low status as an ethnicity or as a society, Pygmies provide a case in point. These Africans occupied, and many still occupy, a position somewhere between independent hunter-gatherer society and minority group in an agricultural community. During the season when they move into villages to serve as field hands, they accomplish the drudge work that the farmers avoid, this longstanding arrangement making them the original migrant workers. Their relationship to the farmers affirms that ethnic interactions within a society retain some of the flavor of alliances between societies. Even though Pygmies treat one another as equals, as bands of hunter-gatherers are wont to do, and even though the farmers have a high regard for the Pygmies as musicians and shamans, their social status is clear. Farmers sometimes say the Pygmies \"belong to\" them, and the Pygmies know and heed their place. \"In the forest, Aka [Pygmies] sing, dance, play and are very active and conversant. In the village their demeanor changes dramatically: they walk slowly, say little, seldom smile and try to avoid eye contact with others,\" says Barry Hewlett, an anthropologist, describing a subservient bearing which primatologists will instantly recognize among rank-conscious baboons, and which the rest of us have come to associate with broken minorities.\n\nThe Pygmies take on the role of servants willingly in exchange for goods they can acquire no other way. Minorities within a society would seem to be similarly motivated to accept their position. Indeed, by preserving their independence and the freedom to go back to their forest territories when they wish, the Pygmies have dodged the unremitting inequalities confronted by ethnicities locked into a society controlled by a majority people. Pygmy bands have been known to abandon ties with one farming village and establish a relationship to a village elsewhere. In effect those Pygmies exchange one dominant group for another that perhaps offers a better relationship.\n\nThat an entire group of former outsiders\u2014an ethnic group\u2014can be accepted in a society, and move up in status within it, may trace back to human responses that evolved to survive enslavement. No slaves could improve a bad situation by switching communities as Pygmies do, but their ability to adapt to captivity helped see them through, and at times allowed them to improve their lives. Among the Comanche, slaves could be accepted into the tribe, to a degree, when they were judged to have achieved proper _personhood_ by taking on the key markers of their society. That meant mastering Comanche customs and language. This took place most successfully for outsiders captured as children and raised by Comanche families. Children have always been ideal war booty, firstly because they are easy to control, and secondly because, with their plastic identities, they assimilate (achieve personhood) more completely than adults. In fact, according to one study, youngsters go through a critical period before the age of 15 when they are most open and adept at absorbing a culture. Adoption by a family was one thing, however; being fully embraced by the society was another\u2014the latter is a hurdle still faced by cross-culturally adopted children. Generally, though, if a slave had the opportunity to assimilate, his children or grandchildren were received as favorably as many second- or third-generation citizens of countries today. The Comanche offered slaves a shortcut, but the bar was high. To be a true Comanche, and marry a Comanche, a slave had to perform an act of heroism on the battlefield.\n\nA slave's prospects for moving up, and possibly becoming a member, depended on the rules of the society. Some upgraded their status if only to become powerful slaves, as was true in the Ottoman Empire; or earned their freedom as individuals, as did the Comanche slaves, or at times as a class. The Greeks were democratic but rarely freed slaves; the Romans on the other hand took many slaves but readily liberated them. Indeed, Roman slaves could achieve citizenship more easily than foreigners who had resided in Rome for generations. Yet former slaves could be stymied in their effort to raise their status. Evidence of a shackled background, such as the widespread association today of dark skin with past slavery, could be a hindrance. As a result, a formal change in standing didn't always translate into social improvement. Roadblocks to advancement went up after the American Civil War when few people hired emancipated slaves, many of whom ended up in conditions more deplorable than those they had suffered when enslaved. The hurdles that societies put in the way of social acceptance and standing for ethnic groups are less imposing, but exist nonetheless.\n\n# INTEGRATION\n\nOne cornerstone of the process by which ethnicities become interwoven into a society as valued members is integration, a word I use to describe how an ethnic people who had originally lived spatially apart now come to intermingle with the dominant group. Even simple chiefdoms, made up of a few villages, didn't always permit this easy mixing, but instead kept people separated out onto their village properties. Letting members of an outlying province leave their own land at all, other than as slaves stripped of their identity, would be a concession in that the dominant people would view it as a risk. This tight control on travel within the society is unusual. In other animals, freedom of movement by all the members across the territory is normal. In a few species the members split the territory among themselves, with individual prairie dogs and female chimpanzees and bottlenose dolphins often favoring certain areas. The use of space by band societies was often a bit more compartmentalized than this; the bands stuck to the \"home\" corner of the overall territory their residents knew best. Yet in all these examples, individuals could choose to go elsewhere in the territory.\n\nHuman territoriality became far more complex as societies expanded to take in foreign groups and turned them into ethnicities. The Inca forced most of their subjugated peoples to stay apart, both socially and geographically. Usually, only the few ethnic individuals assigned a high status to collect tributes from their people for the Inca were permitted on the famous highways. Even beneficent societies weren't always open to giving ethnic groups free passage to the capital. Most or all minorities, and especially the lowborn, stayed put in the region of their birth. To earn the right to move about, the provincials would have to consistently demonstrate loyalty and submissiveness until the dominant group was convinced there was no longer a need for force. The imposed separation would continue until the group had assimilated sufficiently for its people to be regarded as trustworthy. Making the changes required that the dominant group condone the ethnic people learning about, and taking on, the country's culture, even if from a safe distance. Chinese dynasties kept provincial subjects from the imperial center until their Han behavior passed inspection. By that time those people might have already achieved the \"felt closeness\" of being Han. The process primed them to be interwoven into the wider population by cultivating an allegiance that put a higher value on the concerns of their society than on their ethnic interests. For the tolerant and self-confident Romans, integration often came before assimilation, with many distinct groups ensconced in the capital.\n\nUltimately, however, allowing an ethnicity to spread from its place of origin served to firm up the majority's grip on their power. A dispersed people were likely to end up less identified with their group and with a weaker voice than a concentrated one. At the same time, the dominant people can more easily hold sway over an ethnicity's original territory if that ethnicity has scattered beyond its former homeland. Discord in societies is lowest when ethnicities are either well dispersed or well mixed with the general population. In the latter case, with integration encouraged, the path is opened for positive interactions and fast assimilation. Citizens of the majority group learn about, and ideally adjust to, the aliens in their midst, who become a familiar and nonthreatening presence. The minorities, adhering to the \"when in Rome\" dictum, in turn fine-tune what is expected of them, and gain social standing for doing so. Properly accommodating to others is impossible without this firsthand exposure. The absorption of a minority into a society slows, however, if na\u00efve newcomers, still following their old customs, continue to arrive from the home country.\n\nThe difference is huge between staying put in a province among those of one's background and roaming among the dominant group as a free person. It's hard to say how far back in time such integration existed. While the Roman Empire is viewed as an early experiment in multiculturalism, the Greek civilizations preceding it were open only to those of Greek origin while other ethnicities were prevented from traveling beyond city ports. For earlier states, such as Babylonia, there simply is no evidence to say whether freemen from outlying provinces lived with impunity among the dominant people.\n\nOf course, integration has never meant random mixing. The ancient city of Teotihuacan, established around 100 AD near what is now Mexico City, had a distinct barrio occupied by the Zapotec from farther south. In Rome, there's evidence of the Jews and eastern immigrants concentrating in certain outer boroughs, and the ethnic temples of that city must have been colorful gathering places. Almost certainly ethnic neighborhoods existed in other early cities, though little proof exists, most likely because the quarters occupied by different groups shift during the life of cities, confounding the archaeological evidence. Indeed, studies have shown that modern ethnic populations expand and contract both locally and regionally over time, contributing to the dynamic range of \"flavors\" in how national identity can be expressed across the landscape in the same country.\n\nVestiges of spatial segregation remain even when people move away from their original homelands, sometimes by force. American Indian reservations are a kind of holdover of territorial separation\u2014as once vilified groups kept apart from the general population, tribes were shunted off any valuable land and, beyond this, often had restrictions placed on their movements. In practical terms, some inner-city neighborhoods function the same way.\n\nNot that all spatial segregation is bad, especially between groups of similar prosperity and status, as is the case when a neighborhood of working-class Italians abuts one of working-class Hispanics: ethnic neighborhoods can simply reflect people's desire to seek out like others. This self-sorting is an unintended consequence of personal choices that traces back to the more understated sorting-out of hunter-gatherers among the bands of a society. While the reduced contact caused by this sorting contributes to ignorance about those in other neighborhoods, data from modern communities show there's little harm so long as the interactions between enclaves are positive. A backlash can occur if a neighborhood is too self-contained, though, with the residents cocooning themselves exclusively with like others. In that case, people's insularity can contribute to impressions that they are still _foreign_ , stirring up resentment among other society members. Even then, however, ethnic communities can serve as staging points that let the new arrivals sidestep culture shock and yet assimilate, often over the course of generations.\n\nThe absolute comingling of different ethnicities and races takes integration a step further, yet presents hazards of its own. Studies suggest that taking down ethnic separations, both social and geographic, has to be handled delicately. The dominant people must be open to having unorthodox families one door over. Their confidence in their own identity and sense of security must be great enough to override a there-goes-the-neighborhood attitude toward minority families crossing urban borders. For minority families, the departure from an ethnic neighborhood into the community at large can represent a social boost, yet also poses risks. The social distance between groups still needs to be respected. Anyone who pushes too hard to fit in with the majority is in danger of becoming a \"marginal man,\" as sociologist Robert Park perceptively wrote, forsaken as \"one who lives in two worlds, in both of which he is more or less a stranger.\"\n\nNo matter the spatial distribution and degree of interaction of ethnicities and races, they must work together to enable state societies to function across human lifetimes. In the pages ahead, we shall contemplate these interactions in more detail, with close attention to immigration and group relations within the nations of today.\n\n# CHAPTER 25\n\n# Divided We Stand\n\n_G ive me your tired, your poor, your huddled masses yearning to breathe free_: thus goes the sonnet by Emma Lazarus mounted on a plaque inside the pedestal of the Statue of Liberty. Yet in view of their vast differences, what's always most defined the masses who come to unfamiliar shores seeking refuge is less their escape from fatigue, poverty, and past oppression than that the immigrants are admitted as total aliens. They are not _us_.\n\nThe commingling of ethnicities in my Brooklyn neighborhood came about no differently from the diversity found across many communities around the world today. Most black Americans aside, they didn't come to be there because they, or their parents or family members of earlier generations, had been enslaved or subjugated. Immigration, in the sense of the influx of a substantial population from foreign soil, has become the primary means by which foreign peoples enter a society and become members. This latter, more benevolent practice, and how it has contributed to people functioning together as a society, is the theme of this chapter.\n\nImmigration differs from subjugation in that the parties choose to receive one another. The destination society sometimes even encourages the entry of immigrants, in the spirit of the Lazarus poem, with the anticipation that they will stay of their own volition. Such voluntary acceptance over time of whole populations was all but nonexistent in early human history. Of course, absorbing immigrants is rarely indiscriminate: each person usually undergoes a vetting process. And at times the jarring exposure of the host society population to the new arrivals may set off a backlash\u2014a reaction reflective of concern about immigration in many countries today. All the same, what I find extraordinary is that the receiving society often permits the newcomers to immerse themselves in its general population from the start. Even when many of the incoming people move into ethnic neighborhoods, they have the opportunity to rapidly assimilate.\n\nHow did nations come to be open to such a generally cordial influx of so many new members? Hunter-gatherer bands could take in an occasional refugee who had escaped some tragedy with his or her birth society. At its root, immigration is this kind of acceptance of dispossessed peoples, amplified manyfold. I suspect that historically this acceptance occurred in steps. In its earliest form the dispossessed would have originated not from foreign societies, but from within the state itself. After all, when a people were subjugated, there would have followed a period when their status as foreign or native was murky. That's a reason why early states often regulated the movement of their outlying peoples. Until the outliers had assimilated enough to be relied on, their exodus from their birth lands into the rest of the society's territory, if permitted at all, would be treated as a germinal sort of immigration. Maybe the movements seen today between foreign societies got their start from there.\n\nWhile the process of adding members to a society by immigration isn't manifestly hostile, immigration doesn't amount to a merger of equals. Fundamentally little has changed. The issues of dominance and status described in the previous chapter apply to immigrants, too. The ingroup\/outgroup psychology essential to subjugating others remains. This lays bare that the bulk of the huddled masses throughout history have indeed been the beaten people of beaten races that the Lazarus poem suggests. Often the newcomers arrive poor in material wealth and social standing alike, their condition unlikely to change in the foreseeable future. Thus the Chinese who went to California and Indians to Africa in the nineteenth century were cheap labor, paid but detested. The best that newcomers can hope for is to arrive at a time when bigotry is widely discouraged.\n\nThe obstacles immigrants face are many. Their passion for their new society and its treasured symbols, their knowledge of how to act, and their personal connections with other citizens\u2014all take time to build, leading to a deep skepticism from other society members about their steadfastness. Insult to injury, like divorcees tainted by past marriages gone wrong, new immigrants can be further stigmatized by the failure to stick with their birth society. In some ways the plight of immigrants is reminiscent of the difficulty other kinds of animals face in transferring between societies. There too, new arrivals, in this instance usually arriving one at a time, go through the wringer until, with time and luck, they are accepted.\n\nNew arrivals, then, are strangers in a strange land, and no matter how low their social stature or how they try to fit in, the native population can question whether their strangeness will be damaging. Just as people can recoil from too great an influx of traded goods as a threat to their culture, they can regard immigration as eroding their hold on their society. This is true, ironically, even in the United States, a land of immigrants. Thomas Jefferson fretted that the flood of immigrants in his day would \"warp or bias [America's] direction, and render it a heterogeneous, incoherent, distracted mass.\"\n\nThe frequent association people make between those looking for asylum and other foreign arrivals with sickness and disgusting things now takes on layered meanings. An immigrant may carry not just biological but cultural diseases that can corrupt our identity\u2014amoral behaviors prominently among them. Perhaps such nativist fears exist because there are few means to constrain the behavior of immigrants. By comparison, the subjugated people of the past could in effect be quarantined on their own property until assimilation removed any \"black sheep\" social impurities. Despite these anxieties, as we have seen, societies seem amazingly resilient to the arrival of people and their practices, just as they are to anything brought in through trade with the immigrants' country of origin\u2014witness the worldwide popularity of Chinese, Italian, and French cuisines, which haven't reduced the United States or any other society to the incoherent mass Jefferson predicted. This robustness is evinced by languages, which take words from other tongues without blurring the line between them even after prolonged contact. Aside from pidgins cobbled together to promote communication between groups (formalized into creoles if they turn into the speakers' lingua franca), languages never descend from more than one parent tongue.\n\n# ROLE\n\nThe heterogeneity of societies that so concerned Jefferson can contribute to their success. That would have been true of states formed by conquest, but it has been amplified greatly by immigration. The strength of heterogeneity can be understood in terms of the optimal distinctiveness model described earlier\u2014the tendency of individuals, ethnic groups, and societies alike to gravitate to a middle ground of being alike and different. By this model, ethnicities search out, and enhance, their commonalities, even while striving against cultural loss, in the end intermixing with the most ease when everyone perceives resemblances, yet not equivalence. The outcome of this dynamic aspect to people's superordinate identity is a balancing act in which they demonstrate their fidelity to their ethnicity and to the society as a whole. As a result the majority's negative reaction toward those who were once seen as foreign is alleviated while the society is kept front and center in its citizens' lives.\n\nFor groups perceived as contributing something positive, the social advantages can be impressive. Adding ingredients to a society's culture is important. In addition, a people can fortify their ethnic pride by slipping into the system of role differences. This option has been open to ethnicities for a long time: for example, the marble workers from Bithynia, an ancient province in what is now Turkey, renowned across the Roman Empire, or the Rucana of the Incan Empire, assigned to carry the emperor's royal litter. Such distinctions may offer a psychological edge when a small number of talented individuals influence the wider perception of their ethnic group. But friction is reduced if many individuals of an ethnicity move into occupations that few others do. By contrast, minority people can face reprisals when they choose a career favored by the dominant group. For this reason, someone taking a desirable job is evaluated more negatively, as a rival, should he or she happen to be a member of a minority. This sensitivity to people's choice of profession declines in years of little social competition, or when a society needs an injection to its labor force. Economic booms and wars are times when outsiders may be actively brought into a society and some ethnicities can flourish. Even the Comanche hunter-gatherers gave war captives a social footing as warriors when their fighting force needed replenishment. At the other extreme, minorities can be rejected when competition turns severe. The normally tolerant Romans, for example, turned xenophobic in years of famine, when some ethnic groups were banished from Rome.\n\nImmigrants without a notable skill still had a way forward. The baseline expectation would be that ethnicities\u2014like slaves and the subjugated in earlier times\u2014would assume responsibilities for which the locals had no taste. These could be menial jobs that took no special abilities. Or a group might take to employment that required training but was low in status. An example was the black Americans who became barbers for white clients in the nineteenth century. Offering an expertise with a deep foundation, however, could give anyone a boost. In the case of barbers, Italians who happened to have a family history in that trade became popular and had displaced most black barbers by 1910.\n\nIn Chapter 18, describing alliances, we saw the antecedents of this kind of specialization in patterns of commerce between societies. Those with something unique to contribute\u2014boomerangs or blankets among hunter-gatherers; wines, cheeses, and perfumes by the French today\u2014made for favored trade partners. Bringing talented outsiders into the society as members circumvented the need for fickle trade negotiations. In a small society, as mentioned earlier, marrying a foreigner could do the trick. African communities that seasonally employed Pygmies relied on them to get food from the forest as well. A few farmers wedded Pygmies whose proficiency at procuring meat and honey was a boon for the villages, reducing their dependence on Pygmies from outside. Alternatively, skilled labor could be taken by force. The captives most valued by the tribes of the Pacific Northwest were artisans, and slaves brought metalworking, carpentry, painting, and other practical and artistic skills to Islamic regions.\n\nBut with the right inducements, no force is required to bring in individuals with a special gift. A small proportion of immigrants have always been highly qualified persons whose arrival is a sign of brain drain from their birth countries. When the academy Plato had founded in Athens was shut down in 529 AD, its scholars went to the Sasanian Empire in what's currently Iran, hauling the scrolls along. Julius Caesar granted citizenship to doctors and teachers, vocations in short supply in parts of the world today as well. But historically the role most frequently filled by qualified outsiders might have been that of merchant. The laws established by King Hammurabi of the first Babylonian dynasty in 1780 BC included the right of foreign tradesmen to set up shop, and in the course of time many of them became naturalized.\n\nEven for sought-after people, however, anything close to equality comes hard. \"We see the ugly heads of racism and nativism coexisting with our rhetoric of welcome and tolerance,\" write two authorities on immigration. That's not to say immigrant and adopted country can't mutually prosper. In fact, what makes any ill treatment tolerable is the anticipation of an improved quality of life despite the abuses people might suffer. Skilled immigrants, often coming from societies too poor to support them, could take unfilled elite jobs, much as the unskilled could turn to humble work without stepping on any toes. In either case new arrivals might rely on fellow ethnic residents to help them get a start.\n\nOf course, societies also cultivate options for temporarily using foreigners to solve short-term labor needs without bringing them on board as citizens, in effect importing them until their task is complete, as witnessed by the seasonal travel of farm hands between Mexico and the United States. Even when migrants are permitted to stay in the receiving country, they may not be granted legal rights, or may earn them only after the slow crawl of time, their alien standing making them all the more easily disposable.\n\nIn all these matters it seems that stereotypical beliefs about talents or deficiencies have been significant to the coexistence of ethnicities, much as similar beliefs have been key to the coexistence of social classes going back to the earliest settled hunter-gatherers.\n\n# RACE\n\nPeople assimilating into a society face a quandary\u2014a mismatch between how they think of themselves and how everyone else perceives them. Immigrants, more often than not, discover that the identity they have prized their whole life is meaningless in the new country. Thus the immigrant to Europe or America who grew up a proud member of one of the Tsonga tribes of Mozambique is recast as Mozambican at best, and more often as merely black\u2014a designation that has no social relevance across most of Africa.\n\nBy this process of amalgamation, broadly defined races spin out from the groups that originally mattered to people. This simplification of identities reflects that the immigrants' original loyalties tend to be too complex to be understood and appreciated in their adopted country. Immigrants therefore find themselves obliged to undergo the ethnic disintegration and decomposition mentioned earlier. And so it was that the Shoshone, Mohawk, Hopi, Crow, and other tribes \"became Indians, all of us more or less identical in practical terms,\" the Comanche Paul Chaat Smith laments, \"even though until that moment, and for thousands of years before, we were as different from one another as Greeks are from Swedes.\" The overhaul was no easy task. \"The truth is that we didn't know a damn thing about being Indian. This information was missing from our Original Instructions. We had to figure it out as we went along.\"\n\nSuch sweeping categories of racial identity call to mind how Europeans put a name to Bushmen as a class while failing to appreciate the diversity of those hunter-gatherers, who didn't see themselves as Bushmen but as members of many distinct societies. In a similar fashion, 20 centuries ago, in what would become modern-day China, the dominant Han spoke of everyone in the south as Yue, describing their tattoos and unbound hair in an oversimplification of what must have been immense, now largely forgotten, diversity.\n\nA crude lumping into groups like \"black\" echoes the attitude of the conquerors of bygone days toward foreign societies. While as far as we know chimpanzees treat all outsiders with equal dechimpizing venom, humans direct their dehumanization selectively, but the people receiving the brunt of this ill will don't always see themselves connected with the groups they find themselves pigeonholed into. The early Chinese, Greeks, and Romans gathered what intelligence they could against their most threatening foes, but otherwise were uninterested in differentiating outsiders. Ignorance can indeed be bliss. Military might assured their victory, so why make the effort? \"Countries whose very names we did not know precisely in former times we now rule,\" wrote Roman historian Cassius Dio in the second century AD. Human beings don't need to accurately compare themselves to outsiders, or even have the foggiest notion about them, to be confident about their own position in the world and the correctness of their way of life. Even now many Americans and Europeans envision Africa, home to by far the greatest diversity of people on Earth, as the Dark Continent\u2014one monolithic social block.\n\nAs for what transpires within a society, the widening of people's identities isn't entirely a loss for minorities\u2014they can in part drive it. Their transformation resembles what happened during the formation of coalescent societies in which refugees from different tribes joined up to survive. By dropping their primary identity for a broader definition of who they are, minorities gain the base of support essential for their social and political survival within the general population of a nation. This is true whether they decide to work alongside the dominant group or rebel against it. The Haitian uprising at the end of the eighteenth century might never have succeeded, for example, if the slaves had stuck firmly with their allegiances to their original African tribes (some of which must have once been enemies). Nor of course are the broadened groups exactly homogenized: if their community has a strong enough presence, aspects of people's identities can remain connected to an ethnicity that is a subset of a race, such as the Japanese or Korean subcultures of Asian Americans.\n\nAlso, no matter how irrelevant today's ethnicities and races may have been to a people's ancestors, each group has, in its new setting, created its own way of living. The identities of former immigrants undergo revolutionary alterations, which tie them to other immigrants and at the same time set them apart from the populations of their ancestors. Visiting Israel a few years ago I was surprised how uncommon bagels are there: many Jews learn about bagels, Italians about spaghetti and meatballs, and Chinese about chop suey only after emigrating to North America or Europe, as the case may be. Immigrants and their descendants are a people reinvented, and turning back is difficult or impossible. \"I used to worry I would visit China, and people would look at me and not let me go home because I was so Chinese,\" the writer Amy Tan once told me. She met the opposite response. \"The way you walk, the way you look, the way you act\u2014nothing about you is Chinese,\" she was told. While the Chinese word for nationalism derives from their word for the Chinese race, Tan's ancestral background, to truly be _Chinese_ isn't so simple.\n\nThis calls up the oft-expressed American ideal that a blending produces a superior sort of person. \"Here individuals of all nations are melted into a new race of men,\" wrote the French-born New York farmer J. Hector St. John de Crev\u00e8coeur in an essay from 1782. But for societies composed of different human groups, the pur\u00e9e intimated by the motto _e pluribus unum_ is never achieved. Certainly the dictum that \"all men are created equal\" applied to egalitarian, ethnically uniform hunter-gatherer bands more than to any society since. Even when ethnic relations are positive, no nation is the idealist's melting pot any more than its members are ever completely coequal. This reflects in part how people give up a degree of freedom, and to some extent their equality, to gain the security and social and economic payoffs of belonging to a nation, with some ethnicities relinquishing more than others.\n\nMany sociologists speak as if ethnicities could be thoroughly digested into the whole. Insofar as this occurs at all it takes considerable time, as realized most remarkably with China's ancient and almost monolithic Han majority. The urge to be the same but different guarantees the coalescing doesn't carry on to completion across a society, that diffuse categories like black and white are as much melting as the melting pot entails.\n\nThe majority people can likewise widen their identity to absorb other groups, as the Han once did, though only after the newcomers lose some of the trademarks of their past lives. Italian Americans \"made the cut\" as white a century ago after growing a bit less Italian and more American. In the northern United States this transformation appears to have reflected the psychological need for whites to present a strong ingroup. I say this since the status of Italians changed at the same time black communities were growing at such a rate that the difference between the whites and Italians felt less relevant than it had before. The fact is, faced with increasing minority populations, the dominant group may be obliged to broaden its membership to retain a grip on power, in this instance by exchanging one outgroup (Italian) for another (black). Nowadays, only about one in four Americans traces his or her ancestry to the British Protestants who once formed the country's majority and ethnic core. Yet a gradual acceptance of other people as white, Italians among them, has ensured that Caucasians continue to make up a majority (about two-thirds) of Americans.\n\nAt the same time that a society broadens its boundaries to accept outsiders, the ethnicities and races within it remain divided. Perceived distinctions can extend to the very _humanness_ of the groups. There's no reason to expect this to change, even with, for example, the increasing acceptability of interracial marriages. Hypersensitive eyes will pick out different ethnicities and races, especially if a group is blighted by ignoble status, for centuries to come\u2014even if inaccurately. Hence a destitute American of unclear race is prone to be registered as black, just as is a non-African person with an Afro.\n\n# CITIZENSHIP\n\nGauging who belongs in a society, usually an easy task in hunter-gatherers, has become a challenge given the scale of immigration over the last centuries. The difficulty is exacerbated in the United States as an outcome of its population being almost entirely composed of immigrants from many sources. To this extent Thomas Jefferson was right about the potential warping influence of immigrants: a society must keep strong a filigree of common culture and collective belonging to sustain itself. Jefferson had such a core identity in mind when formulating American ideals about rights, religious commitment, and work ethic. Since there was little history to be shared among its citizens at first, awareness of being one nation by necessity relied less on people's ethnic origin and communal stories and more on symbols that were put in place almost de novo, making flags and fanfare a prominent part of American life ever since. With sturdy symbols as rallying points, Jefferson's American creed served to generate a spirit of purpose and solidarity.\n\nStill, America's openness to immigration got off to a slow start. The authors and signers of the Declaration of Independence and the Constitution certainly weren't all that diverse: aside from two of Dutch ancestry, a few were from Great Britain\u2014England, Ireland, Scotland, and Wales\u2014and the others were American-born descendants of British citizens. Nor was citizenship dispensed generously at first, and racism was everywhere. As originally manifested by the Founding Fathers, nationality was proffered largely to Europeans, with immigration encouraged from the northern and western parts of that continent. Even this was open-minded for the time.\n\nIn its current, widely recognized meaning, citizenship is a form of membership that extends beyond a sense of belonging to include basic rights and legal status and a role in politics. The broad application of citizenship in the US, so defined, built up slowly. Women were enfranchised in 1920, but in practice that right primarily applied only to white women. Native Americans became citizens in 1924, but whether they could vote was left up to the states until as late as 1956. People of Chinese descent, including those born in America, were blocked from citizenship until 1943. Those of Asian Indian ancestry waited until 1946 for the vote, while other Asian Americans earned the right in 1952. The path for African Americans was tough. The Fifteenth Amendment, ratified in 1870, granted black men the vote, but the states' adherence was spotty until the passage of the 1965 Voting Rights Act.\n\nThe modern definition of citizenship means that the prerequisites to be a legal citizen for countries worldwide have in practice been reduced to a few conditions, among them minimal requirements, to the extent these can be measured, that immigrants identify with the overall population and follow its basic social mores. All the same, an immigrant's grasp of the adopted land may be quite detailed, if only because of the requirement that he or she pass a civics test to become naturalized. Immigrants are likely to learn more about the principles and symbols of the nation than its long-time citizens, who are liable to have never considered their meanings deeply despite professing\u2014and expecting\u2014a devotion to them. Indeed, most Americans would fail their country's naturalization exam.\n\nOaths of allegiance, like oaths of marriage, are intended to seal the deal. Still, the complexity of human identities makes fitting in, let alone acceptance, onerous even when the devotion of immigrants to the new country is strong from the start. Knowing facts won't cut it. Membership, at the most intimate level of interactions, isn't an intellectual exercise but a way of _being_. Impossible to prescribe in an immigration law is the deep tissue of national identity: those myriad details like learning how to walk or talk \"like an American\" (or a French person). Such minutiae aren't things people easily notice, let alone refine from practice the way they learn to ride a bike. A generation or two often goes by before an immigrant's family members adopt such details. That such markers aren't mandated underpins a basic point that integration demands an allowance for differences.\n\nRegardless of the reasonable probability that fellow members pick each other out by gait, accent, or smile, the fact that minority individuals are asked where they are from tells us that the days of near certitude in distinguishing citizen from alien are over. We have ceded the task to state agencies. What this means is that even while our commitment to societies is undiminished, if not ramped up by the rhetoric of governments, possessing a passport has been largely uncoupled from how our brains register who _belongs_ : citizenship and our psychological assessments of membership don't always mesh.\n\nThat was apparent, and extraordinarily so, way back in the Roman Empire, when in 212 AD virtually all foreign residents were declared citizens by decree. And yet this was done primarily for the practical purpose of taxing them: as a psychologist might predict and historical evidence tells us, the prejudices of the Roman majority remained entrenched. The record is full of disparaging statements about ethnicities by those who felt that Rome was \"aglut with the scum of humanity,\" or so the poet Lucan complained.\n\nGut responses about who truly belongs can go south quickly, notably when anyone of a marginalized ethnicity commits a crime. So when an American citizen with Afghan parents shot up a Florida nightclub in 2016, killing 49, the horrors set off a different and stronger outrage than if a majority person had pulled the trigger: one directed at a whole group of people perceived by many to share responsibility for the offense. Meanwhile a white committing such an atrocity, like Timothy McVeigh, who killed 168 in the 1995 Oklahoma City bombing, is more often seen as a deviant individual, personally accountable for what happened.\n\nThroughout the history of nations, one loathed group has replaced another in endless succession, their trustworthiness, indeed their merit and citizenship, called to question in a roller-coaster of perceptions. A hunger for scapegoats can lead people to frame ethnicities in sinister light. Tolerance has varied precipitously, and in general tracks the rise and fall of economies. By the late nineteenth century in America, Italians and Irish were thought less desirable than Norwegians, Germans, and English. Considered impossible to assimilate and cultural poison, those immigrants were branded with the word \"Irishism,\" used to describe the perceived depravity of the people from that land.\n\nDiscrimination against ethnic citizens often comes to a head during disputes with foreign powers. Those even vaguely associated with the enemy state du jour can face backlashes, if not rampant dehumanization. For the United States, sovereign adversaries have variously included Native Americans, Britain, France, Morocco, Tripoli, Algiers, Mexico, Spain, Japan, Germany, the USSR, Cuba, China, North Korea, and Iran and other Middle Eastern countries. Each time, Americans of those ancestries have suffered, with the contempt for Japanese Americans during World War II being particularly hard to fathom today.\n\nThe New Yorkers I've spoken to attest that after the 9\/11 attacks were carried out by Muslim extremists, flags were on especially prominent display in the shops of Muslims, or those easily confused with Muslims, who found their standing as countrymen on shaky ground. By this public show of a clear identifier, the shopkeepers made sure no one mistook them for the enemy during a time when the cost of being misidentified was high\u2014the so-called ingroup-overexclusion effect. When groups feel threatened by their own society, displays of patriotism while toning down their ethnic origins is par for the course.\n\nSome countries support only the most tenuous hold on their citizenry. Without a strong core identity for their people to celebrate, they are in danger of fragmenting into the \"natural\" units from which their members derive their primary identity and primeval ties\u2014the petty subgroups that built up among the bands of our hunter-gatherer predecessors writ large. The reason so much of the globe consists of artificial nations to which the people have little commitment is that their national boundaries were drawn up after the First World War to register not the homogeneity or solidarity of their people, but the economic interests of Britain, France, and the United States. The outcome of those decisions has been that the populations of these regions often remain more attached to their original tribes and ethnic groups than to their country. That's notably true when those groups retain ancient connections to a territory, and may be hostile to other tribes that are now in the same country. When passions are directed keenly toward local peoples in preference to the government, it's a challenge to act together let alone as a functional part of an interconnected world. A country made up of such a regionally fragmented populace can be more of a delicate confederacy in the service of economic gain than a nation.\n\nThis sense of a fragile alliance holds to some measure between all ethnic groups, insofar as the dominant people are the preeminent owners of the nation's symbols, power, and wealth. Such inequalities ensure that when a society is under pressure, the feeling of unity is brought to bear with most conviction among the majority, who have more to lose, and more feebly among any ethnicities treated as second-class citizens.\n\nThis disparity between majority and minorities in their perception of proprietorship over the society is the Achilles heel of nations. Even in cosmopolitan America, where all men are created equal by decree, respect for citizenship is one thing, for the diversity of those citizens another. Though minorities can have clashing interests and prejudices about each other, support for pro-diversity policies such as affirmative action is greater among them, with diversity reinforcing diversity in ways that bring discord with the monolithic majority. In such matters both the minorities and the majority can be equally self-serving.\n\n# NATIONALISTS AND PATRIOTS\n\nNot only did Paleolithic societies lack ethnicities, but their people would not have abided the emergence of radical groups such as the American Tea Party and Occupy Wall Street movements. Had partisan confrontations as fervent as those of the last decade arisen among hunter-gatherers, the society, with its people's conflicts over identity pushed to a crisis point, would have severed in two; yet even with regional differences building up in the United States and other nations, people with varied social and political leanings are too intermixed for their societies to readily fracture. We are stuck with each other.\n\nSome of the convictions people hold dear bear on ethnicities and races. Here's the rub: the predisposition to dominate other groups, whether those groups are other societies or ethnicities within a society, influences people's views concerning whether the primary role of societies should be to protect as opposed to provide for their members. We are aware that the societies of most animals serve both functions. Protection focuses on hostile outside factors\u2014notably other societies. The provisioning role extends to the care of those in it.\n\nPeople's outlooks on many pressing social issues betray how these roles are valued differently depending on whether individuals subscribe to patriotism or to nationalism. As most psychologists use the words today, these are habits of thought that represent distinct expressions of how people identify with their society. Sometimes lumped together, patriotism and nationalism become plain, and clash with each other, in troubled times. Depending on the person, perspectives shift to more nationalist or more patriotic viewpoints during periods of stress. Yet each individual usually sticks within a narrow range of attitudes over the course of his or her life; the sentiments emerge in childhood under the dual influence of inheritance and upbringing.\n\nThe fundamental difference between nationalism and patriotism is that while individuals with both outlooks are devoted to their society, they relate to it differently. Patriots display pride in their people and a sense of shared identity and particularly of _belonging_ ; such a feeling comes naturally to those born in a country but can be acquired by immigrants. With most of their passion directed at their own group, patriots prioritize the needs of its members: making sure they have food, housing, an education, and so on. Nationalists have similar emotions but couch their identity in glorification. Their pride connects with prejudice. As obsessed as patriots can be with caring for the members, nationalists are absorbed with preserving a superior way of life by keeping the society safe and sound and putting their own people prominently on the world stage.\n\nWhere it gets interesting is that patriots and nationalists have divergent ideas of who constitutes \"their own people.\" Indeed, among the aspects of their identity nationalists admire are those that set the trusted majority apart. It's this position they guard. The extreme nationalist ardently protects each detail of that identity to keep the nation firmly associated with the angels. The priorities of nationalists include staunch demonstrations of loyalty, accepting customary rules of order, obeying leaders whom they see as responsible, and maintaining the established social relationships, most clearly between ethnicities and races. All of these values came to the fore as people settled down and began dominating others. Tradition-driven nationalists believe in their country no matter what. They commit to the status quo, at times at odds with those democratic ideals that allow for transformation: their personalities are less open to new experiences and social change. Compare this _my country right or wrong_ stance to the outlook of patriots, who likewise give their country a high standing yet believe it must be earned rather than fought for, allowing that there are possibilities for improvement.\n\nIn their attention to differences between groups, nationalists treat both people of other nations and minority citizens as outsiders, taking a narrow view of who is, at the heart, truly part of the society. They're more comfortable with the majoritarian idea of democracy in which the dominant people should have the primary say in governance. Their perspectives on moral and legal issues reflects this. I believe it fair to say that to a nationalist, a person of another ethnicity, citizen or not, is relatively more _foreign_.\n\nEarlier I called ants extreme nationalists because they stick tight to their colony marker\u2014its scent\u2014as a stamp of their identity. Indeed, though in our species a patriot can become as teary-eyed as any nationalist in displays of allegiance to a flag or anthem, nationalists are supersensitive to those symbols. For them brief exposure to a flag or an idolized leader incites an intense reaction\u2014as does the absence of such an emblem when one is expected. Thus the uproar about gymnast Gabby Douglas not placing a hand over her heart while the American national anthem played in the 2016 Olympics, a lapse that to a nationalist made her gold-medal win too much about herself and not about the United States. The reaction was a sign of the sentiment that societies are entities: people don't compete in the games, countries do.\n\nBoth the nationalist and the patriot perspectives can be logically consistent, with nationalists being more risk averse and on guard against anything that may contaminate their culture. They prefer to err on the side of separatism, erecting boundaries that might alienate those whose interests could differ from their own, while patriots are more sympathetic to opportunities for trade and cooperation with outsiders.\n\nIn short, the nationalist is suspicious of diversity, while patriots often welcome it. Or at least they tolerate it, because even a patriot, no matter how equality-minded, isn't immune to prejudice: the ardor that patriots reserve for fellow society members of their own race or ethnicity still leads to discrimination as they subtly, and unwittingly, treat those like themselves more fairly.\n\nWhy did these differences in patriotic and nationalistic attitudes evolve? The fact is that a clash in perspectives within societies, although at times so extreme as to verge on the dysfunctional, may have always been integral to human survival. Our varied expression of social viewpoints probably connects back to \"timeless social concerns,\" as one research team put it. Each outlook is beneficial in certain contexts. This dimension of our social identity may be an adaption to balancing the needs for protecting and provisioning the society. Even though people with opposing perspectives might not see eye to eye, a society with too few or too many individuals at either end of the spectrum could be open to catastrophes. This promotion of behavioral diversity has parallels in unlikely animal species. Social spiders are most successful when their colonies contain both individuals that retreat from danger but fastidiously tend the nest, and bold ones that put more effort into defense against social parasites, which steal the colony's food; the colonies of certain ant species function most efficiently when they contain a similarly effective mix of personality types.\n\nFor humans, the hazards of a population overly committed to either the nationalist or patriot extreme are manifest. Nationalists see the patriot's greater openness to weak borders and sharing across ethnicities as promoting social dependence and cheating, fears that reflect the competitive nature of groups present across species. Meanwhile, the prevalence of nationalists across societies, convinced their ways are right and prepared to fight for them, means the dangers that nationalists fear can indeed be realized. Still, by readily espousing oppression and aggression, extreme nationalists bring to mind the historian Henry Adams's description of politics as a systematic organization of hatreds. Their outlook feeds on certain facets of psychology. It's intoxicating to fall in line against an enemy, at times at a whiff of trouble. For those swept up in a nationalist perspective, the swell of group emotions and awareness of common purpose gives life a greater meaning. Not just morale, but mental health improve among civilians when nations face conflict. The fact is, trigger-happy societies have long had an edge, with the impulse for war and the fear of attack critical in driving many social and technical innovations and the expansion of states. What's more, nationalists, adhering to a narrow interpretation about what behaviors are proper, have the advantage of being far more tight-knit and homogeneous than patriots and better able to act together. All this is to say that the patriot's vantage point is and always will be a more onerous path.\n\nBecause of the partiality for their group, displayed by patriots and nationalists in different ways, the troubles our societies face go deep. It's bad enough that a wicked act by one minority person\u2014the Florida nightclub shooting, for example\u2014can set off outrage at an entire minority population. But mistreatment can carry over to ethnicities unconnected to the tragedy. That's an outcome of how stereotypes strip away detailed understanding, making it easy to conflate groups to the point of creating such fuzzy and nonsensical categories as \"brown people.\" Even when no conflation exists, prejudices can be linked, with the denigration of one people associated with the devaluation of others. Persons who fear for their safety, jobs, or way of life indiscriminately lump _them_ together much as ancient societies did with the \"barbarians\" beyond their borders. The impulse is so strong that when a sample of Americans was asked what they thought of Wisians, nearly 40 percent regarded them poorly and did not want them as neighbors, even though they could have known nothing about them since the researcher had made the name up.\n\nSocieties contain ethnicities and races that stick together despite the members' prejudices about each other. The usual view, voiced by William Sumner more than a century ago, is that friction with outsiders draws a society together. Clearly that's not always true. The external forces that promote civil peace primarily galvanize the dominant people while often straining their ties to a society's other ethnicities when those groups are regarded as part of the problem. This tension among the members can cause a kind of social autoimmune disease, turning a society against itself. For all these tribulations, we may reasonably ask whether societies are necessary at all.\n\n# CHAPTER 26\n\n# The Inevitability of Societies\n\nCan we discard our societies, combine them into one, or at least make them secondary to a more universal union of humanity?\n\nHere is a snippet of history that reads like a parable. For centuries, the Pacific island of Futuna, a low chunk of volcanic rock, at 46 square kilometers in size, offered space and resources for just two chiefdoms\u2014Sigave and Alo. These societies, claiming opposite ends of the island, were in almost constant conflict, pausing only briefly now and then for island-wide ceremonies featuring a psychoactive drink made from a shrub native to the western Pacific. One wonders if this enabled their people to tolerate each other for the day. I can only imagine their spear-throwing clashes were a primary motivator in their lives, the Arab-Israeli conflict in a microcosm. One might expect that in such a confined space, and over the course of so much time, one chiefdom would have conquered the other. That this never happened might bear on the human craving for an outgroup, if not an outright opponent _._ Could Alo have continued on without Sigave\u2014a society in a vacuum? Would it, alone in the world, even be what we could call a society?\n\n\"Go to the ant, thou sluggard, consider her ways and be wise,\" advised King Solomon, a keen observer of nature. Indeed, the Argentine ants battling for control of Southern California bring to mind a second hypothetical about what might happen if one society were to extinguish all others. For the ants, the final colony's national emblem (its scent) would no longer represent a marker; rather their odor would become synonymous with being an Argentine ant. By this means, the species would achieve the universal peace that experts believed it had attained before they discovered the boundary wars between the supercolonies.\n\nOne lesson is that while we may indeed learn a few things by considering the ways of the ant, such as the value of investing in roads and sanitation, it would be unadvisable to emulate them. Peace would have depended on the ants' nonpareil knack for slaughter, carnage that, in sheer numbers, would exceed the most nightmarish episodes of human history.\n\nYet a second lesson, to the point here, is that calling a group a society\u2014and recognizing any markers that identify the members as a society\u2014makes sense only when more than one society exists. The implication is that the compulsion to be part of a society must be matched by the imperative to identify an outgroup\u2014Sigave for Alo and vice versa, or at least \"others\" vaguely rendered, as the barbarians were for the Roman Empire and Chinese dynasties\u2014if only as a standard for comparison and a source of gossip, if not denigration. In this sense Futuna's chiefdoms, as simple and similar as they may be by our standards, exemplify human nature stripped to its heart.\n\nOr is that so? In a study entitled \"Us Without Them,\" psychologist Lowell Gaertner and colleagues found that when people need each other they can build a communal identity without contrasting themselves to outsiders. Such mutually dependent people may feel as if they are functioning as a unit. These feelings can foster conviviality and bonding\u2014something we might expect of a ship's crew struggling together in a storm. But it would be a stretch to call such a group, no matter the interdependence of its members or their level of cooperation, a society. For one thing, the shipmates would already identify with a society, whatever nation or nations they came from.\n\nTake this a step further: Let's say the crew becomes shipwrecked and loses all contact with the outer world. Needless to say their former identities wouldn't vanish overnight. Twenty-five years after the 1789 mutiny aboard the HMS _Bounty_ , the crew was found to have fled to Pitcairn Island. Rediscovered by an American ship 18 years later, the mutineers, and the Polynesians and Tahitians who accompanied them, were still recognizably English, Polynesian, and Tahitian. But suppose they had lived and remained undetected. In the passage of time, would they, or their descendants, have reimagined themselves into what we would recognize as one clearly defined and unified society?\n\nIt isn't easy to find examples of a single collection of people utterly cut off from the rest of the world for generations. Some of the Vikings reaching Iceland and North America may have survived in isolation. However, they stuck to their trademark Viking mode of life sufficiently to link back up again without difficulty with Vikings in Europe; but at most their separation lasted decades, their origins never passing out of living memory. Prehistoric peoples reached remote islands, too, but in most places either were in contact with tribes on other islands or had room to split into more than one society. Futuna had its two, and Easter Island harbored 17 adversarial giant-rock-head-erecting tribes, while in Australia the hundreds of aboriginal societies that thrived in pre-colonial times all descended from one group that landed on the continent by way of Asia.\n\nOne possible historical example of a lone society can be found on Henderson. At 37 square kilometers, this Polynesian island is thought to have offered too little space and too few resources to allow its few dozen residents to divide into two societies like Alo and Sigave. Having no more wood to make boats, the sixteenth-century Hendersonites were cut off from their trading partners on Pitcairn and Mangareva, islands 90 and 690 kilometers distant, respectively. By the time Spanish explorers found the island in 1606, its inhabitants had died out. No one can say how those few souls thought of themselves\u2014whether, for instance, they still considered themselves a tribe and gave themselves a name. My guess is that by holding on to dim recollections of others _out there_ , passed down through the generations, those desperate few last islanders retained some shred of identity as a society, with a sense of Us versus Them never expunged from their minds.\n\nThe existence of foreigners would have had to wane even from legend and myth over the passage of many generations for the Hendersonites to see themselves as completely alone in the world. Had the Hendersonites lasted that long, might they have still expressed a need to belong, a yearning for oneness and _we_ -ness? Or would any commitment they once had to a common identity have petered out? Perhaps in that case what we would find on Henderson would be individuals who continued to be social with friends and family but no society to speak of. That seems to be the view of the anthropologist Anya Peterson Royce: \"The hypothetical group on an island with no knowledge of others is not an ethnic group; it does not have an ethnic identity; it does not have strategies based on ethnicity.\"\n\nOne could disagree with Royce, pointing to the human craving to explore commonalities, demonstrated by the ease with which we model ourselves on the quirks of those we admire. Trendsetters drive the popularity of many practices, while the habits of disliked people may be shunned. Copying a leader or venerated individual could result in conventions amounting to a kind of identity, even for the long-suffering Hendersonites. Those islanders' lives must have been stripped to the minimum by the Tasmanian effect, described earlier, whereby people in sparse groups forget aspects of their culture. Even so they no doubt still had a lot in common from growing up with and learning from each other. However, without an _other_ , their alikeness would be reduced to such irrelevance that Royce has it right: it would be hard to regard them as a society (and certainly not as an ethnic group).\n\nTheir similarities might no longer matter in the profound way that shared characteristics do when they serve as societal markers. But perhaps the people's mere knowledge of each other would be enough to make them a society. Do species with individual recognition societies\u2014which lack markers\u2014perceive themselves as a collective regardless of the existence of outsiders, or do their societies collapse without an \"other\"? Field biologist Craig Packer tells me such a dissolution is the fate of isolated lion prides: the members disperse into ever smaller groups. No real surprise given that a primary function of societies is to succeed against competitors. This pressure disappears when there are no others. Yet the fate of lions may not be a reliable indicator for humans. Lions, which can hunt alone, get along by themselves better than people, who ache to be around more than spouses to avert loneliness and give their lives significance, if not to stay safe and fed. That's one reason why failing human societies seldom utterly collapse but rather split into smaller societies in which people retain their support networks, even if in a simpler setting. Escaping a broken society doesn't mean giving up on societies entirely.\n\nOn that basis we can expect that whether or not their relationships merit the name society, isolated humans will stick together more than lions, even if it's merely the Us-without-Them devotion we see among shipmates. For sure that's true of chimpanzees. A lone chimpanzee community stays interconnected in the same way that chimps do anywhere else, or so research on a group on its own in a gorge at Kyambura, Uganda, tells us.\n\nYet such isolated groups would be quick to adjust to changing conditions. For the isolated chimpanzees, the Argentine ant supercolony that kills off all enemies, or for the tribal islanders who forget all others, their identity as a society would leap back to significance the moment they encountered outsiders again. The islanders would accentuate whatever meager social features set them apart, firming up a boundary between themselves and the newcomers (that is, if the invaders didn't overrun and kill or assimilate their little collective first).\n\nAn experiment in 1954 roughly approximated this situation, demonstrating how rapidly some of the features of societies can come into play when conditions are right. Twenty-two 12-year-old boys at Robbers Cave State Park in Oklahoma were randomly assigned to two groups that were originally kept on their own. Once the groups saw each other, from a distance at first, and then came into contact, they were quick to develop separate identities: favorite songs, propensity for cursing, and so on. The boys gave each group a name, drew its totem animal on their shirts, and waved flags emblazoned with the same symbol when they competed in sports. The violence that the Rattlers and Eagles soon displayed toward each other was alleviated after the researchers had them work together, but their differences remained.\n\nEven if no foreigners were to come along from outside to disturb a lone community\u2014if, for example, a hyperaggressive society conquered the entire globe, so that no outsiders existed\u2014their achievement would be short lived. Once humans in isolation grow to more than a very few in number they seem to privilege some individuals over others, creating conditions under which multiple societies are born. The requisite foreigners would be birthed from within. The union would fragment, as have all nations, and all the societies that came before them.\n\n# DREAMS OF A UNIVERSAL SOCIETY\n\nWhat we know about human cognition therefore doesn't bode well for those who think humanity may someday blanket the earth in one borderless population, leaving no person an outsider. Still, even though societies will never dissolve away, perhaps another scenario could play out that would effectively remove them from the picture. With the number of societies generally declining century after century, we might imagine that all the remaining nations of the world will someday drop their boundaries sufficiently to create a cosmopolitan community that is more significant to people than the societies themselves.\n\nSome claim the internationalization of culture (think McDonald's, Mercedes-Benz, _Star Wars_ ) and connections (with Facebook bringing people together from Aa, Estonia, to Zu, Afghanistan) are a harbinger of a Berlin Wall\u2013type collapse of state borders. This is false. Societies have never freely merged, and that won't change. Populations worldwide may devour KFC, Starbucks, and Coca-Cola while screening Hollywood blockbusters\u2014and enjoy sushi, flamenco, French couture, Persian carpets, and Italian cars. They may adopt, and at times be swamped by, cosmopolitan trends. But no matter how many exotic influences nations absorb or how many foreign connections they make, such nations do not slip into disarray, and they retain their people's fierce devotion. After all, from time immemorial societies have taken what they wanted from the outside world to claim as their own, and have been all the stronger for doing so. Even the Statue of Liberty, the quintessential symbol of the United States, was designed and first erected on French soil by the same Mr. Eiffel who gave Paris its Tower.\n\nNotwithstanding this rigid boundary maintenance, humankind can erect umbrella organizations comprised of many nations. Yet such a universal group would also fail to completely supersede our bonds to societies, as demonstrated by the most binding association of societies in the anthropological record. In northwest Amazonia reside 20 or so tribes, or language groups, known collectively as the Tukanoans. Each has its own language or dialect, some similar, some mutually unintelligible. The tribes are so cross-connected as to be tied economically, each specializing in goods it exchanges with the others. Between them are what amount to obligatory trading relationships of an unusual sort: marriages within a tribe are improper. \"Those who speak the same language with us are our brothers, and we do not marry our sisters,\" the people say. Thus a bride marries into another tribe, where she learns the local tongue. In case you imagine the arrangement to be an aberration, similar across-the-board spousal interchanges are recorded for New Guinea.\n\nOne explanation for this arrangement is that it reduces inbreeding in very small societies. We see this in many nonhumans as well, such as in chimpanzees, where the females avoid mating with kin by similarly transferring between communities. When populations are minuscule, as those of the Tukanoans have been on occasion in the past, the only marriage options actually could be between brothers and sisters, an incestuous act to which our species has an innate abhorrence. This is a far more likely problem for the Tukanoans than for the countries of today. The psychological aversion to marrying siblings seems to overpower any trepidation the Tukanoans might have had about firmly linking their societies. Their compulsory spousal exchanges have created what to my mind are the tightest alliances ever recorded, currently totaling about 30,000 Amazonian souls. Yet for all that, the Tukanoan tribes remain clear and separate, each confined to specific areas.\n\nPutting aside the unusual situation of the Tukanoans, the failure of alliances to supersede people's affiliation to a society holds true universally, the United Nations and the European Union included. These intergovernmental organizations don't earn people's emotional commitment because they lack ingredients that make them _real_ for the members. The EU may be the most ambitious attempt at economic integration conceived; yet it will never supplant the nations within it. The members don't see the EU as an entity worthy of their loyalty the way they do their countries, and for several reasons. First off, its borders are not fixed\u2014in fact, are subject to revision as states enter or go. Additionally, its members have a history of conflict dating from the Middle Ages, and a split already exists from east to west between communist and capitalist cultures. To top all that off, the EU offers no grand foundation story, no venerable symbols or traditions, and there's little sense anyone would fight and die for Europe as they might for their nation. That makes the EU a political coalition much like the Iroquois Confederacy was, though with less power. Each member country handles issues relating to its citizens' identity and remains the focus of their self-worth, an outlook that makes membership in the EU secondary and disposable. An analysis of the 2016 Brexit vote shows that those most strongly thinking of themselves as English went against staying with the EU. Those voters saw an economic and peacekeeping tool as a threat to their national identity.\n\nFinancial and security issues hold the EU together. The same can be said for Switzerland, a country subject to perennial scrutiny because, as the four languages and complex territoriality of its people attest, its nationhood rests on a detailed social and political alliance between 26 local communities, or cantons. These self-governing settlements act in many respects as miniature nations nestled in a mountain landscape that enhances each one's autonomy. \"Each Canton has its own history, constitution, and flag, and some even have an anthem,\" the political scientist Antoine Chollet reports, such that Swiss \"citizenship refers to _one who can vote_ and nothing more.\" The formation of the Swiss confederation required rewriting accounts of the past to maintain a sense of equality between the cantons. This challenging step was necessary for the cantons to survive over the centuries when they were forced to negotiate their interests with far larger and more powerful neighboring countries.\n\nThe EU and Switzerland are regional entities kept together by the perceived need to counter hazards from outsiders, which gives both a reasonable chance of success. A global human union would have no such motivation, and would therefore be far more precarious. One possible means of attaining global unity might be to shift people's perception of who is an outsider\u2014a point Ronald Reagan made often: \"I occasionally think how quickly our differences worldwide would vanish,\" he remarked in an address to the United Nations, \"if we were facing an alien threat from outside this world.\" Popular science fiction tales like _The War of the Worlds_ depict all of humankind pulling together against a common enemy. Yet our societies will endure even this. Space aliens wouldn't make nations irrelevant any more than Europeans arriving in Australia caused Aborigines to dispense with their tribes. That would be so regardless of how much the aliens shattered the beliefs people held about their own societies, whose beloved differences would now look trivial by comparison. Moreover, when societies of our species turn to one another, whether for commercial advantage or to defend against aliens, that reliance doesn't diminish the weight they place on their differences. The notion of cosmopolitanism, the idea that the people of the world will come to feel a primary connection to the human race, is a pipe dream.\n\n# SOCIETIES AND BEING HUMAN\n\nLet me put forward a final question. What might happen if people could forgo their markers or somehow put aside the drive to categorize each other with labels? In such a world the only differences people would perceive would be between individuals\u2014not between groups. One supposes that under such circumstances our nations would disintegrate entirely, but it's hard to predict what would rise in their stead. Perhaps our affiliations would coalesce around local neighborhoods, or around the people we know best, with the global population splintering into millions of micro nations. We might foresee a return to the individual recognition societies of our forebears, where everyone knew everyone else.\n\nOr, by discarding our differences, or our penchant for making judgments about the differences, could we achieve the opposite result, doing away with societies entirely? Would the beehive of networks built up through international travel and Facebook friendships interlink us so indiscriminately that we would actually secure that elusive panhuman unity, encompassing every man, woman, and child?\n\nOur reliance on markers may trace back far into the human past, but what comes naturally isn't always desirable, and fortunately our intelligence gives us some prospect of breaking free from our biology, and history. When changes concern the matter of how we mark off our identities, though, any alteration would be extremely difficult and take more than education. While casting off ethnic and societal markers may sound good at first blush, the move would undoubtedly mean the loss of much of what humans cherish. Nationalists or patriots, people care about their memberships and few would willingly give them up. Nor could they, because human responses to groups are involuntary. Markers are two-edged swords, causing us to discount those who differ from us, yet at the same time imparting an esprit de corps with complete strangers who fit our expectations. To abandon human markers would strike against timeless psychological yearnings. No doubt if a mass hypnotist caused us to forget our differences, we would scramble to discover new differences to hold dear. The only way to retool this human attribute would be for a surgeon with near-miraculous understanding of the nervous system to ablate portions of the brain. The result of this science-fictional adjustment would be a creature we wouldn't recognize as ourselves. I'm unsure how one could measure whether such people were any happier than we are today, but surely, _they_ would no longer be _us_.\n\nFor humans in our current condition, the question of whether societies need to exist really boils down to whether people must be in one for their emotional health and viability. I think so. \"A man must have a nationality as he must have a nose and two ears,\" wrote Ernest Gellner, a prominent thinker on nationalism. Gellner\u2014who went on to argue, mistakenly, that the human need to be part of a nation is nothing more than a contrivance of modern times\u2014never fathomed how right his statement was. The mind evolved in an Us-versus-Them universe of our own making. The societies coming out of this psychological firmament have always been points of reference that give people a secure sense of meaning and validation.\n\nTo say a person has no country calls to mind dysfunction, trauma, or tragedy. Without such an identity, humans feel marginalized, rootless, adrift: a dangerous condition. A case in point is the homelessness felt by immigrants who have lost connections to their native land only to face the sting of rejection by their adopted country. Social marginalization has been a motivator stronger than religious fanaticism, explaining why many terrorists originally took to extremism only after being excluded from the cultural mainstream. For the socially dispossessed, radical views fill a void. Organized crime groups likewise commandeer some of the properties that give a society its vitality by providing social pariahs with common goals and a sense of pride and belonging. This we saw in a very embryonic form with the socially well-adjusted young boys at Robbers Cave.\n\nThe evidence presented in this book points to societies being a human universal. Human ancestors lived in fission-fusion groups that evolved, by simple steps, from individual recognition societies to societies set apart by markers. The ingroup-outgroup boundaries of society membership would have made it through this transition unaltered. That means that humans have always had societies. There was no original, \"authentic\" human society, no time when people and families lived in an open social network before deciding to set themselves apart into well-defined groups. Being in a society\u2014indeed, multiple, contrasting societies\u2014is more indispensable and ancient than faith or matrimony, having been the way of things from before we were human.\n\nThe last Henderson Islanders must have been starved not only physically but socially\u2014apathetic, or so I imagine, about the meaning of their lives. Rest assured, few facets of life match a society in striking passion in the human heart, so long as other societies exist to compare with our own. Provided we don't re-engineer ourselves and instead choose to remain fully human, societies, and the markers that bring us together and set us apart, are here to stay, signifying the boundaries between people in our minds, and setting the borders between them physically, across the earth's surface.\n\n# CONCLUSION\n\n# Identities Shift and Societies Shatter\n\nHe who should wish that his fatherland might never be greater, smaller, richer, poorer, would be the citizen of the world.\n\n\u2014VOLTAIRE\n\nMaking their way across the African savannah, the Australian coastline, and the American plains, our ancestors moved in small affiliated bands of lifelong fellow travelers. Month after month they pitched their camps and searched for food and water. Rarely did they encounter other human souls. I find it hard to imagine seeing as few strangers day to day as they did throughout their entire lives. With the passing of ages, societies have swelled to the point that we now move like ants through an anonymous swarm. Many in the throng are far less like us than the people that hunter-gatherers came across, over hundreds of generations, were like them.\n\nSo rarely did our ancestors encounter foreign peoples that outsiders seemed to occupy a realm between reality and myth. Aborigines guessed the first Europeans they met were ghosts. Over time our view of the members of other societies has changed radically; today, foreigners don't seem outlandish or otherworldly, as they once routinely did. As a consequence of global exploration starting in the fifteenth century, and more so today thanks to tourism and social media, contact between people from far-flung parts of the globe is now commonplace. Outright incomprehension of outsiders is no longer the excuse it often was in prehistory. In those times so little was known about foreigners that barbarian hordes could be treated as if they were monsters under the beds of children.\n\nStill, humankind continues to express its relationships to societies in ways that people seldom recognize or avow. Yet our minds, configured to interact with fewer individuals and groups, are overloaded. And so this book has drawn from many fields of study to apprehend the nature of societies and understand how we cope with the overload, with many revelations along the way. Here I distill these down to a few essential conclusions.\n\nThe most fundamental of these is that societies are not solely a human invention. Most organisms lack the closed groups we call societies, but in the species that do have them, societies serve in diverse ways to provide for and protect the members. In all such animals, individuals must recognize each other as belonging to the same group. This membership can offer advantages regardless of whether or not those members cooperate or have any other social or biological relationship.\n\nWhile societies are not unique to humans, they are necessary to the human condition, having existed since ancient prehumans diverged on the evolutionary tree from other apes. A million human societies may well have come and gone. Every one has been a group closed to outsiders, a group that its members were willing to fight for and that at times they died for. Each commanded intense commitments from its members extending from birth to death and through the generations. Until recent millennia, all of those societies were small communities of hunter-gatherers, but that didn't mean their attachments to their societies were any weaker than ours today.\n\nIn the societies of our prehuman ancestors, as in those of most other mammals, the members had to recognize each other individually to function as a group. The resulting constraint on memory put an upper limit of roughly 200 on the size of societies in most animals. At some point in our evolution, probably before the origins of _Homo sapiens_ , humans broke this glass ceiling by forming anonymous societies. Such societies, found in humans and a few other animals\u2014notably ants and most other social insects\u2014can potentially attain vast sizes because the members no longer must remember each other personally. Instead, they rely on identifying markers to accept both individuals they know and strangers who fit their expectations. Scents serve as markers in the insects, but humans go broader. For us, markers range from accents and gestures to styles of dress, rituals, and flags.\n\nMarkers are essential components in all societies of more than a few hundred individuals, whether mighty human or mere insect. At a certain point of growth, however, markers alone are insufficient to hold a society of humans together. Large human populations depend on an interplay between the markers and an acceptance of social control and leadership, along with increasing commitments to specializations, such as jobs and social groups.\n\nIn early humans, new societies were born in a two-step process that has its parallels in other vertebrates. The process begins, usually very gradually, with the formation of subgroups within a society, brought on by divergences in identity. Years later, those identities diverge to such an extent that they become incompatible. The factions then separate to permanently form distinct societies.\n\nThe ability to accept strangers born in the same society as fellow members does not on its own account for the enormous growth of human societies. What made such vast expansions possible was the acquisition of people from other societies. Outsiders had to make accommodations to the expected identity to be accepted as part of the society. The addition of foreigners in numbers, initially by force through slavery and subjugation and more recently by immigration, generated the mixtures of ethnicities and races found within societies today. The relationships between these groups retain the imprint of differences in power and control that in some cases extend back to before recorded history.\n\nThe differentiation of identities among society members continues to be a source of disruption. However, rather than dividing in the manner of hunter-gatherer groups, today's societies more often fragment along geographical fault lines that roughly reflect the claimed ancestral homelands of the ethnic groups that have come to live within them.\n\nThe need for societies, by its very ancientness, has shaped all aspects of human experience. Most notably, relationships between societies have profoundly influenced the evolution of the human mind, which would in turn affect the interactions among the ethnic and racial groups that emerged later in our history. While we may not be operating out of pure ignorance of outsiders the way early humans often did, automatic responses reflect our proneness to harbor stereotypes about different groups and about the superiority of our own group. The psychology underlying our identification with societies and ethnicities is registered in our every action. Our reaction to each person we see, how we vote, and whether we approve of our country's decision to go to war have all been shaped by processes deeply embedded in our biology. The buzzing confusion of modernity may simply exacerbate our reactions.\n\nEven as we face this deluge of social interactions as individuals, our nations grow ever more interdependent. Yet we are who we are, and so our societies still put inordinate effort into jockeying for territory, resources, and power, just as societies of animals have always done. We attack. We cajole. We blame. We abuse. We insulate ourselves from foreign powers we don't trust by partnering with those we do. Such alliances, unique to humankind, may save us. Nevertheless, they can bring further uncertainty and destruction, angering or striking fear into those who have been left out.\n\nWe can hope the past decades have brought change: aware of their interdependence and leery of the costs of modern conflicts, most nations have ceased trying to conquer each other outright. Our global knowledge of humankind has turned the extraordinary ordinary, creating an everyday reality that would have been unachievable when intergroup contact was rare and limited. Luckily not only can we serenely enter a caf\u00e9 full of strangers, but we aren't alarmed if the latte sippers are different from us, whether they are members of ethnic pockets of our society or come from abroad. Given the chance, we may shake those people's hands with barely an increase in heart rate. Yes, among the others that we are compelled to live with there are still people whose identities irritate, repel, offend, or scare us. Yet for all the fraught social navigation and the painful run-ins, seen in an evolutionary and even a historical context the casualness of this mixing is a big deal.\n\nThrough it all people's identification with the societies that shelter them from an unpredictable world has remained steadfast. A sense of belonging inoculates us against outside influences. And our commitments have been invigorated by perceptions that our nations and tribes are venerable and timeless. An accurate reading of the past, and of the human social condition, requires us to squarely face the fact that such beliefs about social stability are a comforting illusion. New groups will assuredly gain traction. Tensions over national or ethnic differences won't go away. Human markers have evolved to facilitate not just the forces that bind society members together, but also those that tear them apart. They do the former by enabling strangers to view each other as fellow members; they do the latter because, over the passing of generations and across geographical distances, identities shift and societies shatter. All societies are ephemeral to the core\u2014fleeting, as Chief Seattle foretold. Italy, Malaysia, the United States, Amazonian tribe, or Bushman band society: all act organically, their continuation requiring a dynamic responsiveness never free of conflict and anguish. Even so, the hard reality is that the social warp and woof can support only so many alterations. At some point the social fabric can no longer be mended, and ultimately the territorial integrity of every nation will be challenged, and fail.\n\nAs we look to the future, it may be illuminating to reframe the discussion around an idea cherished by people everywhere, and which I have brought up only in passing thus far: their freedom. Americans have declared pride in their freedom ever since winning their independence, and yet at the time of the Revolutionary War the British thought of themselves as free compared to more repressive European societies of the day. Indeed, much of human activity can be understood in terms of our pursuit of the choices a society opens to us. But this freedom for personal expression is never simple. Permissiveness is never without limits. Any society defines itself by what it won't tolerate, and the behavior required from its members; thus, by its nature, membership in a society reduces people's options and entails a loss of freedom. For most species the constraint is simply that individuals are required to interact with other members and consort little if at all with outsiders. Human societies come with further obligations. We must act appropriately, adhering to whatever markers set us apart from outsiders. We are free insofar as our actions fall within the most essential of these rules, and adhere to our commitments to the society and our station and status within it. Generally speaking, the more privations a society has undergone, the more rigid the expectations placed on its people. Extremist regimes aside, by and large citizens everywhere gladly accept such restrictions. They believe in the _rightness_ of their society and find comfort within the limitations it imposes on them. In return societies give a great deal, including a feeling of ease, even camaraderie, around likeminded others; the security and social support that comes with membership; and access to resources, opportunities for employment, suitable marriage partners, the arts, and much more.\n\nWhile people value their freedom, in truth, socially imposed limits on freedom are as indispensable to happiness as freedom itself. If people are overwhelmed or unsettled by the options open to them and other members, and by the acts of those around them, what they feel is chaos, not freedom. What we think of as freedom, then, is invariably quite restricted. Yet only an outsider would suggest that the restrictions imposed by a culture are oppressive. For this reason, societies that promote individualism, like the United States, and those that nurture a collectivist identity, such as Japan or China\u2014where people put greater emphasis on their identification with the collective and the support this gives them\u2014can equally celebrate the opportunities and happiness their society offers. Regardless of a society's permissiveness, unity falters if its citizens have the freedom (or feel they should have the freedom) to act outside the comfort zone of others, whether that means women voting, protestors burning the flag, or nontraditional couples claiming the right to marry.\n\nThese are weaknesses of the social fabric many societies struggle with today. But ethnic diversity presents an even greater complication in the pursuit of unity and freedom. The difficulty is balancing one group's pursuit of freedom with another group's comfort. All too often, inequalities in personal freedom emerge between groups. Minorities must fit in with what the society finds acceptable\u2014especially with what the dominant group expects. Yet they are also obliged not to overly emulate the dominant people, leaving the majority to its separate and privileged status. Minorities are thereby put in the position of having to invest in identifying not only with the society where they are citizens, but with their own ethnicity, too. Hispanic Americans, for example, are constantly registered by their fellow Americans, and almost constantly see themselves, as Hispanic. By contrast, the dominant members, in primary control of the symbols and resources of nations, much more rarely need to think about their own ethnicity or race, other than as they need to when pulling together during times of economic trouble. This gives them greater freedom: majority individuals enjoy the luxury of considering themselves as one-of-a-kind, idiosyncratic persons.\n\nThe United States, which overran the native peoples of the subcontinent to cobble together a society composed\u2014slaves and their descendants aside\u2014of people who for the most part came voluntarily, is an experiment with few parallels in history. Its population derives from many sources, with no ethnic groups numerically swamping broad geographical regions of the country where they can lay claim to ancient territorial roots. As a result it lacks the fracturing points that riddle many Old World societies brought together by forceful incorporation followed by an interminable history of resentment. A lack of ready-made fault lines may give America a measure of durability despite its political turmoil. Still, what will become of that country, as it approaches the two-and-a-half-century mark, is uncertain. A single question overrides all others: namely, whether the United States can persist as one nation, indivisible, in a productive relationship with the rest of the world as its remaining superpower. With latitude for diversity reducing requirements for citizenship to a few conditions, social understanding and an openness to adapt will be essential, as it is for many countries.\n\nWhat's the best-case scenario for the future of our societies? What will create social health and longevity? The current trend has been for societies to move away from a support of diversity to focus their national identity more narrowly on their dominant group. Even so, minorities aren't going away. We may slow immigration, but it's no longer tenable to expel ethnic people as the Romans sometimes did when times got bad. Fortunately the United States is among a growing number of nations today that are exceptional not just in their ethnic plurality but in their richness of all kinds. Such nations pride themselves on their wealth of job opportunities, religious choices, sports fandoms, and other interest groups. This cornucopia can amplify a society's strength by giving its citizens many options that add layers to their personal identities and affinities with others. Those able to reach outside their own ethnicity and race, or find commonalities with people like themselves yet with different outlooks, have the chance to bond over other shared enthusiasms; think of the study that shows that a person's race may be overlooked should he or she be wearing the jacket of a beloved team. Such cross-connections can be individually fragile but strong in bulk, keeping a society whole in the face of upheaval. Governance matters, too. Nations comprised of distinct ethnicities function well when their institutions support diversity. As long as interactions stay productive, prejudice declines. This phenomenon is true no matter how insular people may be in their choice of friends or how little face time they spend with other ethnicities.\n\nDiversity presents social challenges at the same time that it brings creative exchange, innovation, and problem solving, fueled by varied talents and perspectives. How long any society can remain strong in the face of the shifting relationships among its people remains a troubling question. For a society to stay intact other than by force, all of its communities must be motivated to rally with equal passion around a core identity\u2014easier said than done given the majority's greater freedoms and power to manipulate the rules of the game in its favor, often through institutions largely under the control of its social upper crust. A nation supporting strong ties among its people and similarly masterful in its dealings with other societies would serve the greater well-being of its citizens, extending its time on Earth and making its legacy a high point in the story of humankind.\n\nIt would be the height of foolishness to think such an outcome could be achieved through cheerful goodwill or careful social engineering. However optimistic you feel about our adroitness as problem solvers, human minds\u2014and the societies we fabricate when those minds interact\u2014are malleable only to a degree. Our willingness to enjoy an advantageous social position, even hurt each other in the service of preserving our sense of dominance and superiority, is an abiding human trait.\n\nOur misfortune has been, and will be always, that societies don't eliminate discontent; they simply redirect it toward outsiders\u2014which paradoxically can include the ethnic groups within them. Our improved knowledge of others hasn't always been enough to improve how we treat them. We will need to better understand the urge to view other peoples as less human, even bug-like, if our species is to break from the history of dissonance between groups that traces back beyond antiquity. We also must know more about how people reformulate their identities, and respond to each sea change with the least damage possible. _Homo sapiens_ is the only creature on Earth that can do this. If our dispositions toward outsiders vary, some of us inclined to caution, others to trust, nevertheless we share an aptitude for harnessing connections with seemingly incompatible others. Our salvation lies in reinforcing this gift, guided by the increasingly refined discoveries of the sciences covered by this book. The good news is that humans have some capacity to counter our inherited propensities for conflict through deliberate self-correction. Divided we will be, and divided we must stand.\n\n# Acknowledgments\n\nOne underappreciated aspect of roughing it in remote parts of the world is that one is forced to slow down. From long days spent as a biologist waiting out tropical downpours under a tarp or lumbering along on a camel over bleached bone and sand, I've come to learn that true creative time is the time _between_ things, empty intervals a packed calendar rarely affords. In _The Journey_ the poet Mary Oliver writes of following one's personal calling. By taking a \"road full of fallen branches and stones\" one finds that little by little a new world opens up. The poem resonates with me because a lifetime on such roads has given me an abundance of time to contemplate what I have seen and motivated me to write this book.\n\nBecause making connections between fields with divergent vocabularies and approaches sometimes required me to simplify arguments to make them accessible for a general audience, I've included a more thorough bibliography than usually found in non-academic writing to give readers with different backgrounds the option to pursue particular threads; even so, I've had to be selective. This book and its technical predecessor (Moffett 2013) would have also been incomplete without going beyond the literature by seeking out the advice of experts who proved generous, from reading drafts to tolerating na\u00efve questions. A gratifying response was \"I never thought of that.\" I freely admit that someone far smarter than me should have written this book; any errors of interpretation are mine.\n\nThe people below have kindly helped me on issues ranging from why hamsters get excited about body odor and how radio programs can accelerate a genocide to the devotion of immigrants to their country. Among the names that follow, those in _italics_ reviewed parts of the manuscript: Dominic Abrams, Stephen Abrams, Eldridge Adams, Rachelle Adams, _Lynn Addison_ , Willem Adelaar, Alexandra Aikhenvald, Richard Alba, Susan Alberts, _John Alcock_ , Graham Allan, Francis Allard, Bryant Allen, Warren Allmon, Kenneth Ames, David Anderson, Valerie Andrushko, Gizelle Anzures, Coren Apicella, Peter Apps, Eduardo Araral Jr., _Elizabeth Archie_ , Dan Ariely, Ken Armitage, Jeanne Arnold, _Alyssa Arre_ , Frank Asbrock, Filippo Aureli, Robert Axelrod, Leticia Avil\u00e9s, Serge Bahuchet, Russell Paul Balda, _Mahzarin Banaji_ , Thomas Barfield, Alan Barnard, _Deirdre Barrett_ , Omer Bartov, Yaneer Bar-Yam, Brock Bastian, Andrew Bateman, _Roy Baumeister_ , James Bayman, Isabel Behncke-Izquierdo, _Dan Bennett_ , Elika Bergelson, Joel Berger, Lu\u00eds Bettencourt, Rezarta Bilali, Micha\u0142 Bilewicz, Andrew Billings, Brian Billman, Thomas Blackburn, Paul Bloom, Daniel Blumstein, Nick Blurton-Jones, Galen Bodenhausen, Barry Bogin, Milica Bookman, Rapha\u00ebl Boulay, Sam Bowles, Reed Bowman, Robert L. Boyd, Liam Brady, Jack Bradbury, Benjamin Braude, Stan Braude, Anna Braun, Lauren Brent, _Marilynn Brewer_ , Charles Brewer-Carias, Charles Brown, Rupert Brown, Allen Buchanan, Christina Buesching, Heather Builth, Gordon Burghardt, and David Butz.\n\nFrancesc Calafell, Catherine Cameron, Daniela Campobello, Mauricio Cantor, Elizabeth Cashdan, _Kira Cassidy_ , Deby Cassill, Emanuele Castano, Frank Castelli, Luigi Luca Cavalli-Sforza, Richard Chacon, Napoleon Chagnon, Colin Chapman, Russ Charif, Ivan Chase, Andy Chebanne, Jae Choe, Patrick Chiyo, Zanna Clay, Eric Cline, Richmond Clow, Brian Codding, Emma Cohen, Lenard Cohen, _Anthony Collins_ , Richard Connor, Richard Cosgrove, Jim Costa, Iain Couzin, Scott Creel, Lee Cronk, Adam Cronin, Christine Dahlin, Anne Dagg, Graeme Davis, Alain Dejean, Irven DeVore, Marianna Di Paolo, Shermin de Silva, Phil deVries, _Frans de Waal_ , Oliver Dietrich, Leonard Dinnerstein, Arif Dirlik, Robert Dixon, Norman Doidge, Anna Dornhaus, Ann Downer-Hazell, Michael Dove, Don Doyle, Kevin Drakulich, Carsten De Dreu, Christine Drea, Daniel Druckman, Robert Dudley, _Lee Dugatkin_ , _Yarrow Dunham_ , Rob Dunn, _Emily Duval_ , David Dye, _Timothy Earle_ , _Adar Eisenbruch_ , Geoff Emberling, Paul Escott, Patience Epps, Robbie Ethridge, Simon Evans, Peter Fashing, Joseph Feldblum, Stuart Firestein, _Vicki Fishlock_ , Susan Fiske, Alan Fix, Kent Flannery, Joshua Foer, John Ford, AnnCorinne Freter-Abrams, Doug Fry, and Takeshi Furuichi.\n\nLowell Gaertner, Helen Gallagher, Lynn Gamble, Jane Gardner, _Raven Garvey_ , Peter Garnsy, Azar Gat, Sergey Gavrilets, Daniel Gelo, Shane Gero, Owen Gilbert, Ian Gilby, _Luke Glowacki_ , Simon Goldhill, Nancy Golin, Gale Goodwin G\u00f3mez, Alison Gopnik, Lisa Gould, Mark Granovetter, Donald Green, Gillian Greville-Harris, _Jon Grinnell_ , Matt Grove, Markus Gusset, Mathias Guenther, Micaela Gunther, Gunner Haaland, Judith Habicht-Mauche, Joseph Hackman, _David Haig_ , Jonathan Hall, Raymond Hames, Christopher Hamner, Marcus Hamilton, Sue Hamilton, Bad Hand, John Harles, Stevan Harrell, Fred Harrington, John Hartigan, Nicholas Haslam, Ran Hassin, Uri Hasson, Mark Hauber, Kristen Hawkes, John Hawks, _Brian Hayden_ , Mike Hearn, Larisa Heiphetz, Bernd Heinrich, Joe Henrich, Peter Henzi, Patricia Herrmann, Barry Hewlett, Libra Hilde, Jonathan Hill, Kim Hill, Lawrence Hirschfeld, Tony Hiss, Robert Hitchcock, Robert Hitlan, Michael Hogg, _Anne Horowitz_ , Kay Holekamp, Leonie Huddy, Mark Hudson, Kurt Hugenberg, Stephen Hugh-Jones, Marco Iacoboni, Yasuo Ihara, Benjamin Isaac, Tiffany Ito, Matthew Frye Jacobson, Vincent Janik, _Ronnie Janoff-Bulman_ , Julie Jarvey, Robert Jeanne, Jolanda Jetten, Allen Johnson, Kyle Joly, Adam Jones, _Douglas Jones,_ and John Jost.\n\nAlan Kamil, _Ken Kamler_ , _Robert Kelly_ , Eric Keverne, Katherine Kinzler, Simon Kirby, John Kloppenborg, Nick Knight, Ian Kuijt, S\u00f6ren Krach, Karen Kramer, Jens Krause, Benedek Kurdi, Rob Kurzban, Mark Laidre, _Robert Layton_ , Kang Lee, James Leibold, Julia Lehmann, Jacques-Philippe Leyens, Zoe Liberman, Ivan Light, Wayne Linklater, Elizabeth Losin, Bradley Love, _Margaret Lowman_ , Audax Mabulla, Zarin Machanda, _Richard Machalek_ , Cara MacInnis, Otto MacLin, Anne Magurran, Michael Malpass, Gary Marcus, _Joyce Marcus_ , _Curtis Marean_ , Frank Marlowe, _Andrew Marshall_ , William Marquardt, Jos\u00e9 Marques, Anthony Marrian, Abigail Marsh, Ben Marwick, John Marzluff, Marilyn Masson, Roger Matthews, David Mattingly, John (Jack) Mayer, Sally McBrearty, Brian McCabe, John McCardell, Craig McGarty, William McGrew, Ian McNiven, David Mech, Doug Medin, Anne Mertl-Millhollen, Katy Meyers, Lev Michael, Taciano Milfont, Bojka Milicic, Monica Minnegal, John Mitani, Peter Mitchell, Panos Mitkidis, Jim Moore, Corrie Moreau, Cynthia Moss, Ulrich Mueller, _Paul Nail_ , Michio Nakamura, Jacob Negrey, Douglas Nelson, Eduardo G\u00f3es Neves, David Noy, and Lynn Nygaard.\n\nMichael O'Brien, Caitlin O'Connell-Rodwell, Molly Odell, Julian Oldmeadow, Susan Olzak, Jane Packard, Craig Packer, Robert Page, Elizabeth Paluck, Stefania Paolini, David Pappano, Colin Pardoe, William Parkinson, Olivier Pascalis, Shanna Pearson-Merkowitz, Christian Peeters, Irene Pepperberg, Sergio Pellis, Peter Peregrine, _Dale Peterson_ , Thomas Pettigrew, David Pietraszewski, Nicholas Postgate, Tom Postmes, Jonathan Potts, Adam Powell, Luke Premo, Deborah Prentice, Anna Prentiss, Barry Pritzker, Jill Pruetz, _Jonathan Pruitt_ , Sindhu Radhakrishna, Alessia Ranciaro, Francis Ratnieks, Linda Rayor, Dwight Read, Elsa Redmond, Diana Reiss, Ger Reesink, Michael Reisch, Andres Resendez, _Peter Richerson_ , Joaqu\u00edn Rivaya-Mart\u00ednez, Gareth Roberts, Scott Robinson, David Romano, Alan Rogers, Paul Roscoe, Stacy Rosenbaum, Alexander Rosenberg, Michael Rosenberg, Daniel Rubenstein, Mark Rubin, Richard Russell, Allen Rutberg, Tetsuya Sakamaki, Patrick Saltonstall, Bonny Sands, _Fabio Sani_ , Stephen Sanderson, _Laurie Santos_ , Fernando Santos-Granero, Robert Sapolsky, Kenneth Sassaman, Jr., Chris Scarre, Colleen Schaffner, _Mark Schaller_ , Walter Scheidel, Orville Schell, Carsten Schradin, J\u00fcrgen Schweizer, James Scott, Lisa Scott, Tom Seeley, and Robert Seyfarth.\n\nTimothy Shannon, Paul Sherman, Adrian Shrader, Christopher Sibley, James Sidanius, Nichole Simmons, _Peter Slater_ , Con Slobodchikoff, David Small, Anthony Smith, _David Livingstone Smith_ , Eliot Smith, Michael Smith, Noah Snyder-Mackler, Magdalena Sorger, Lee Spector, Elizabeth Spelke, Paul Spickard, G\u00f6ran Spong, _Daniel Stahler_ , Charles Stanish, Ervin Staub, Lyle Steadman, _Amy Steffian_ , Fiona Stewart, Mary Stiner, Ariana Strandburg-Peshkin, Thomas Struhsaker, _Andy Suarez_ , Yukimaru Sugiyama, Frank Sulloway, Martin Surbeck, Peter Sutton, Maya Tamir, Jared Taglialatela, John Terborgh, G\u00fcnes Tez\u00fcr, John and Mary Theberge, Kevin Theis, Elizabeth Thomas, Barbara Thorne, Elizabeth Tibbetts, _Alexander Todorov_ , Nahoko Tokuyama, Jill Trainer, _Neil Tsutsui_ , Peter Turchin, Johannes Ullrich, Sean Ulm, Jay Van Bavel, Jojanneke van der Toorn, Jeroen Vaes, Rene van Dijk, Vivek Venkataraman, Jennifer Verdolin, Kathleen Vohs, Chris von Rueden, Marieke Voorpostel, Athena Vouloumanos, Lyn Wadley, Robert Walker, Peter Wallensteen, Fiona Walsh, David Lee Webster, _Randall Wells_ , Tim White, Hal Whitehead, _Harvey Whitehouse_ , Polly Wiessner, Gerald Wilkinson, Harold David Williams, Edward O. Wilson, John Paul Wilson, _Michael Wilson_ , Mark Winston, George Wittemyer, Brian Wood, _Richard Wrangham_ , _Patricia Wright_ , Tim Wright, Frank Wu, _Karen Wynn_ , Anne Yoder, Norman Yoffee, Andrew Young, Anna Young, Vincent Yzerbyt, and Jo\u00e3o Zilh\u00e3o.\n\nFor my research on animal societies in the field, Melissa and I were generously hosted by Elizabeth Archie for baboons, Filippo Aureli for spider monkeys, Anthony Collins for chimpanzees, Kay Holekamp for hyenas, Cynthia Moss for savanna elephants, Daniel Stahler for wolves, and Randall Wells for dolphins. Special thanks go to the National Geographic Society for a long history of funding my research; Gerry Ohrstrom for assistance when I needed it early in my writing; Richard Wrangham for supporting my visiting scholar position at Harvard's Department of Human Evolutionary Biology; Allen Rodrigo and team for a sabbatical at the former National Evolutionary Synthesis Center (the now sadly defunct NESCent); Lynn Addison for much wise counsel; and Ted Schultz and the Department of Entomology for my ongoing status as a research associate at the National Museum of Natural History (Smithsonian Institution). I am fortunate to have so many true friends.\n\nI am also grateful to my agent and adviser, Andrew Stuart, for guiding me through the complexities of modern book publishing, and to my editor, Thomas \"T.J.\" Kelleher, as well as Roger Labrie, Bill Warhop, and editor-in-chief Lara Heimert at Basic, for envisioning _The Human Swarm_ as a book of major scope.\n\nFinally, I have been thankful every day for Melissa and her extraordinary tolerance of my single-mindedness with this book when we could have been going about our usual business of tracking down astonishing creatures in wondrous places.\n\n#\n\n**Mark W. Moffett** is a tropical biologist and research associate at the Smithsonian and a visiting scholar in the Department of Human Evolutionary Biology at Harvard University. Called a \"daring eco-adventurer\" by Margaret Atwood, and recipient of the Lowell Thomas Medal for exploration, Dr. Moffett's writing has appeared in _The Best American Science and Nature Writing_ and he has been a guest on _The Colbert Report_ , _Conan_ , NPR's _Fresh Air_ , and CBS _Sunday Morning_. This is his fourth book.\n\n# ALSO BY MARK W. MOFFETT:\n\n_Adventures Among Ants:_\n\n_A Global Safari with a Cast of Trillions_\n\n_Face to Face with Frogs_\n\n_The High Frontier:_\n\n_Exploring the Tropical Rainforest Canopy_\n\n# Praise for _The Human Swarm_\n\n\"This fine work should have broad appeal to anyone curious about human societies, which is basically everyone.\"\n\n\u2014 _Publishers Weekly_ , Starred Review\n\n\"Our times are filled with garage start-ups that become Silicon Valley behemoths overnight. Such scaling-up pales in comparison to humans going from hunter-gatherer bands to our globalized world in the blink of an evolutionary eye\u2014and thus now, a stranger a continent away can be killed when we press a button operating a drone, or rescued when we press a button marked 'Donate now.' In _The Human Swarm_ , Mark Moffett charts the science of this scaling up of human societies, and its unlikely evolutionary consequences. This highly readable book is ambitious in its interdisciplinary breadth, rigorous in its science, and deeply thought-provoking in its implications.\"\n\n\u2014Robert Sapolsky, author of _Behave_\n\n\"A tour-de-force.\"\n\n\u2014Donald Johanson, discoverer of the famous missing-link fossil \"Lucy\" and founder of the Institute of Human Origins\n\n\" _The Human Swarm_ is a book by a biologist that should fascinate any thoughtful reader and deserves to be taken seriously by psychologists and social scientists alike.\"\n\n\u2014Roy Baumeister, author of _Willpower_\n\n\"A truer statement hasn't been uttered: 'Our _groupiness_ shapes human history.' Moffett's book is a wide-ranging, deeply interesting analysis of how large numbers of individual agents become a society. His travels in the world and through vast intellectual landscapes give him a unique view of why we are the way we are, both in our similarity to other living beings and our differences from them\u2014our ability to include once alien groups into our own, for example. There is no other book I've read recently that made my neurons pop at the rate this book did.\"\n\n\u2014Mahzarin Banaji, author of _Blindspot_\n\n\"This is a book of amazing ideas, many of them counterintuitive. Mark Moffett's astounding stories of animal societies persuaded me that the future of human cities have been foretold by the ants. Read this manifesto if you like to have your mind changed.\"\n\n\u2014Kevin Kelly, founder of _Wired Magazine_ and author of _The Inevitable_\n\n\"In the past quarter century, there has emerged a genre of Big History that includes such epic books as Jared Diamond's _Guns, Germs, and Steel_ , Steven Pinker's _The Better Angels of Our Nature_ , and Yuval Noah Harari's _Sapiens_. Mark Moffett's _The Human Swarm_ is destined to be included in future lists of such books that not only add to our understanding of who we are, how we got here, and where we're going, but change our perspective of how we fit in the larger picture of life on Earth. A magisterial work of monumental importance.\"\n\n\u2014Michael Shermer, publisher of _Skeptic_ magazine and author of _The Moral Arc_\n\n\"Homo sapiens is a small-group social animal that physically seems to be limited to personal relationships with a few individuals. Nonetheless humanity is struggling to deal with societies of billions as human technologies now pose existential threats tied to those numbers. In _The Human Swarm_ , Mark Moffett presents an intriguing overview of the biological roots and cultural evolution of this now-critical situation.\"\n\n\u2014Paul R. 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The syndrome of group-focused enmity. _J Soc Issues_ 64:363\u2013383.\n\nZinn H. 2005. _A People's History of the United States_. New York: HarperCollins.\n\nZvelebil M, M Lillie. 2000. The transition to agriculture in eastern Europe. In TD Price, ed. _Europe's First Farmers._ Cambridge: Cambridge University Press. pp. 57\u201392.\n\n# Notes\n\nIntroduction\n\n Breidlid et al. (1996), 14. Different accounts exist of what Seattle said (Gifford 2015).\n\n Sen (2006), 4.\n\n For perspectives on \"tribe\" in its looser meanings, I recommend Greene (2013).\n\n Of course humans can have these impulses diverted to develop powerful bonds to other groups such as cults (see Chapter 15 and Bar-Tal & Staub 1997).\n\n Dunham (2018). Taken further, people divided into groups with the flip of a coin almost immediately value their group-mates more than those assigned to the other group (Robinson & Tajfel 1996).\n\n Quoted in Dukore (1966), 142.\n\n For evidence of ongoing human evolution, see Cochran & Harpending (2009).\n\nChapter 1\n\n The self-sacrifice in humans often demands cultural indoctrination (Alexander 1985).\n\n Anderson (1982).\n\n My thanks to Emilio Bruna, the editor of _Biotropica_ , for permitting me to adapt my thoughts in this paragraph from Moffett (2000), 570\u2013571.\n\n Thanks to David Romano and G\u00fcnes Tezc\u00fcr for advice on the citizenship and individual rights of the Kurds across the Middle East.\n\n For example, Wilson (1975, 595) defines a society as \"a group of individuals belonging to the same species and organized in a cooperative manner,\" adding the diagnostic criterion of \"reciprocal communication of a cooperative nature, extending beyond mere sexual activity.\" One can imagine societies where mutual benefits accrue that aren't reciprocally communicated, however.\n\n \u00c9mile Durkheim (1895), who established the discipline more than a century ago, saw cooperation as a key element of societies. He thought of cooperation arising when people share similar sentiments and points of view, and certainly beliefs and moral principles are essential elements of what I will talk about regarding identification in societies of humans. See also Turner & Machalek (2018).\n\n For a few of many fascinating discussions: Axelrod (2006), Haidt (2012), Tomasello et al. (2005), Wilson (2012).\n\n Dunbar et al. (2014). The premise of the social brain hypothesis has recently been put in doubt by data showing that brain size is better predicted by ecology than sociality (DeCasien et al. 2017).\n\n For the accuracy of applying the word friendship to animals, see Seyfarth & Cheney (2012).\n\n Dunbar (1996), 108. Dunbar's number is usually described in terms of positive relationships, but arguably one's knowledge of enemies should be factored in as well (De Ruiter et al. 2011).\n\n This is true for other species, too. As biologist Schaller (1972, 37) points out, companionship among lions has no influence on the composition of a pride.\n\n Dunbar (1993), 692. It is worth repeating the sentence in full: \"How is it that, despite these apparent cognitive constraints on group size, modern human societies are nonetheless able to form super-large groups (e.g., nation states)?\" Dunbar presented the human ability to categorize society members by their social roles as a solution to this question, but knowing what people do doesn't explain the memberships of societies and the distinct boundaries between them.\n\n Turnbull (1972). Some doubt his interpretations (e.g., Knight 1994).\n\n At least as of the date of the following study: European Values Study Group and World Values Survey Association (2005).\n\n Simmel (1950).\n\n A chimpanzee will sometimes be generous to an individual that's likely to return the favor in other ways (Silk et al. 2013).\n\n Jaeggi et al. (2010). Tomasello (2011; see also 2014) finds hunter-gatherers to be more cooperative than apes in every domain: \"Cooperation is simply a defining feature of human societies in a way that it is not for the societies of other great apes\" (p 36).\n\n Ratnieks & Wenseleers (2005).\n\n e.g., Bekoff & Pierce (2009); de Waal (2006).\n\n Social life can offer benefits for one group over another even when the individuals in it don't profit directly and aren't related (this is group selection) or when there are advantages for individual and group both (multilevel selection) (e.g., Gintis 2000; Nowak 2006; Wilson & Wilson 2008; Wilson 2012). I won't dwell on these alternatives because the controversies are well covered elsewhere. Group selection would seem to require the stability of societies. Still, my judgment is that, as far as most species go, societies offer sufficient benefits to individual members without the need to invoke group or kin selection.\n\n Allee (1931); Clutton-Brock (2009); Herbert-Read et al. (2016).\n\n Females sometimes carry another's infant or gang up on a female they don't like (Nakamichi & Koyama 1997).\n\n Daniel Blumstein and Christina Buesching, pers. comm.; Kruuk (1989), 109. A male marmot will also drive off competing males, but whether this is advantageous to anyone other than him is hard to say. The badgers live in distinct closed groups, but it is unclear to me whether marmots do(e.g., Armitage 2014).\n\n Henrich et al. (2004); Hogg (1993).\n\n Quoted from Columbus's log, in Zinn (2005), 1\u20132.\n\n Erwin & Geraci (2009). For more on canopy biodiversity, see Moffett (1994).\n\n Wilson (2012).\n\n Caro (1994).\n\n In this case different species may join the group (e.g., Sridhar et al. 2009). Flocks that are distinct societies will be addressed in Chapter 6.\n\n e.g., Guttal & Couzin (2010); Krause & Ruxton (2002); Gill (2006); Portugal et al. (2014).\n\n Anne Magurran, pers. comm.; Magurran & Higham (1988).\n\n Hamilton (1971).\n\n This kind of behavior had also been described in an insect before (Ghent 1960).\n\n Costa (2006), 35.\n\n Rene van Dijk, pers. comm.; van Dijk et al. (2013 & 2014).\n\n For descriptions of how societies deal with issues of fairness and \"free riders,\" see for example Boyd & Richerson (2005).\n\nChapter 2\n\n My thanks to Stephen Abrams, Ivan Chase, and Carsten Schradin for general advice on fish. Bshary et al. (2002); Schradin & Lamprecht (2000 & 2002).\n\n Barlow (2000), 87.\n\n I have space to suggest just a few key books, when any are available, for the species that follow. For meerkats, I thank Andrew Bateman, Christine Drea, G\u00f6ran Spong, and Andrew Young; for horses, Joel Berger, Wayne Linklater, Dan Rubenstein, Allen Rutberg (see _The Domestic Horse_ by Mills & McDonnell 2005); for gray wolves, Dan Stahler, David Mech, and Kira Cassidy (see _Wolves: Behavior, Ecology, and Conservation_ by Mech & Boitani 2003); for painted dogs, Scott Creel, Micaela Gunther, Markus Gusset, and Peter Apps (see _The African Wild Dog_ by Creel & Creel 2002); for lions, Jon Grinnell and Craig Packer (see _The Serengeti Lion_ by Schaller 1972); for hyenas, Christina Drea, Kay Holekamp, and Kevin Theis (see _The Spotted Hyena_ by Kruuk 1972); for the bottlenose dolphin of the eastern US seaboard, Randall Wells (who has published many technical articles about them); for ringtailed lemurs, Lisa Gould, Anne Mertl-Millhollen, Anne Yoder, and the sadly deceased Alison Jolly (see _Ringtailed Lemur Biology_ , Jolly et al. 2006); for baboons (by which in this book I mean the savanna species\u2014the yellow, chacma, and olive baboons), Susan Alberts, Anthony Collins, and Peter Henzi (see _Baboon Metaphysics_ by Cheney & Seyfarth 2007, and _A Primate's Memoir_ by Sapolsky 2007); for mountain gorillas, Stacy Rosenbaum; for chimpanzees, Michael Wilson and Richard Wrangham (see _The Chimpanzees of Gombe_ by Goodall 1986 and _The Mind of the Chimpanzee_ by Lonsdorf et al. 2010); and for bonobos, Isabel Behncke-Izquierdo, Takeski Furuichi, Martin Surbec, Nahoko Tokuyama, and Frans de Waal (see _Behavioural Diversity in Chimpanzees and Bonobos_ by Boesch et al. 2002, and _The Bonobos_ by Furuichi and Thompson 2007).\n\n I thank Jennifer Verdolin, Linda Rayor, and Con Slobodchikoff for prairie dog advice. Rayor (1988); Slobodchikoff et al. (2009); Verdolin et al. (2014).\n\n I thank Elizabeth Archie, Patrick Chiyo, Vicki Fishlock, Diana Reiss, and Shermin de Silva for help with elephants. About everything one needs to know about the savanna species is summarized in Moss et al. (2011).\n\n De Silva & Wittemyer (2012); Fishlock & Lee (2013).\n\n Benson-Amram et al. (2016).\n\n Macdonald et al. (2004); Russell et al. (2003).\n\n Silk (1999).\n\n Laland & Galef (2009); Wells (2003).\n\n Mitani et al. (2010); Williams et al. (2004).\n\n e.g., Cheney & Seyfarth (2007), 45.\n\n Only the Florida dolphins studied by Randall Wells are covered in this book: the bottlenose dolphins elsewhere may differ in their behavior and sometimes belong to other species.\n\n Linklater et al. (1999).\n\n Palagi & Cordoni (2009).\n\n e.g., Gesquiere et al. (2011); Sapolsky (2007).\n\n Van Meter (2009).\n\n That's not to say fantasies of success don't keep people going, but the daydreams of James Thurber's Walter Mitty rarely play out in reality and obsessions with being king are pathological unless one is in line for the throne. People are prone to believe in their potential to achieve goals wildly in excess of actual chances, yet seem no less happy when they don't succeed (Gilbert 2007; Sharot et al. 2011).\n\n That bonobos sometimes work together to catch large prey is shown by Surbeck & Hohmann (2008).\n\n Hare & Kwetuenda (2010).\n\n Brewer (2007), 735.\n\nChapter 3\n\n Unlike Aureli et al. (2008), I see little confusion as to the species for which the term \"fission-fusion societies\" is justified.\n\n The difficulty of attacking large parties applies to predators as well as enemies, though leopards are indifferent to the attempts of chimps to stop them and may be an exception (Boesch & Boesch-Achermann 2000; Chapman et al. 1994).\n\n Marais (1939).\n\n Strandburg-Peshkin, pers. comm.; Strandburg-Peshkin et al. (2015).\n\n Bates et al. (2008); Langbauer et al. (1991); Lee & Moss (1999).\n\n East & Hofer (1991); Harrington & Mech (1979); McComb et al. (1994).\n\n Fedurek et al. (2013); Wrangham (1977).\n\n Wilson et al. (2001 & 2004).\n\n The loud bonobo calls have complex functions (Schamberg et al. 2017).\n\n Slobodchikoff et al. (2012).\n\n e.g., Thomas (1959), 58.\n\n Bramble & Lieberman (2004).\n\n Evans (2007).\n\n Stahler et al. (2002).\n\nChapter 4\n\n Leticia Avil\u00e9s, pers. comm.; Avil\u00e9s & Guevara (2017).\n\n King & Janik (2013).\n\n Boesch et al. (2008).\n\n Zayan & Vauclair (1998).\n\n Seyfarth & Cheney (2017), 83.\n\n Pokorny & de Waal (2009).\n\n de Waal & Pokorny (2008).\n\n Miller & Denniston (1979).\n\n Struhsaker (2010).\n\n Schaller (1972), 37, 46.\n\n This fact has been overlooked, e.g., Tibbetts & Dale (2007) review individual recognition but curiously ignored its role as a requirement and aid for living in societies.\n\n Breed (2014).\n\n Lai et al. (2005).\n\n Jouventin et al. (1999).\n\n de Waal & Tyack (2003) and Riveros et al. (2012) call these \"individualized societies.\"\n\n Mentioned by Furuichi (2011). Randall Wells tells me female bottlenose dolphins, who largely stick to a contained part of their community's territory, sometimes can be almost as isolated from other group members as the female chimpanzees described here.\n\n Rodseth et al. (1991).\n\n Quoted in Jenkins (2011).\n\n Berger & Cunningham (1987).\n\n e.g., Beecher et al. (1986).\n\n The excretion differs from one mongoose to the next and is used by the animals to distinguish individuals, but the intriguing possibility exists that the scent also has a group-specific component (Rasa 1973; Christensen et al. 2016).\n\n Estes (2014), 143.\n\n Joel Berger, Jon Grinnell, and Kyle Joly, pers. comm.; Lott (2002).\n\n Whether societies generally attain this maximum population or achieve a lower size limit owing to factors other than memory will be determined by the rules of society reproduction for the species in question, a subject for Chapter 19.\n\n \"Troops\" is a sensible word, since they seem to be of a kind with\u2014are homologous to\u2014the troops of other monkeys (Bergman 2010). The animals may also recognize some of the members of the troop that most recently split off from their own (such divisions are described in Chapter 19). See also le Roux & Bergman (2012).\n\n Machalek (1992).\n\nChapter 5\n\n For more about leafcutters see Moffett (1995 & 2010). For a detailed treatment of ants generally, see H\u00f6lldobler & Wilson (1990). I thank University of California Press for permission to adapt some passages from Moffett (2010) for use in this book.\n\n de Waal (2014). For example, typical of articles comparing chimpanzees and humans, Layton & O'Hara (2010) spends far more time discussing differences than substantial similarities.\n\n Monkeys and human infants younger than 18 months fail this test of self-awareness. For these and other issues, see Zentall (2015).\n\n Tebbich & Bshary (2004).\n\n de Waal (1982).\n\n Beck (1982).\n\n McIntyre & Smith (2000), 26.\n\n e.g., Sayers & Lovejoy (2008); Thompson (1975).\n\n B\u0103descu et al. (2016).\n\n I give termites and honeybees short shrift in favor of my beloved ants, but for those who want to know more about them I suggest Bignell et al. (2011) and Seeley (2010).\n\n The pivotal role of scaling applies to the size of organisms as well as to societies. See Bonner (2006) and other works by that author.\n\n The description of ant market economies is adapted, with permission, from Moffett (2010). See Cassill (2003); Sorensen et al. (1985); and, for honeybees, Seeley (1995).\n\n Wilson (1980).\n\n For some other insect societies depending on agriculture and domesticated foods, see Aanen et al. (2002); and Dill et al. (2002) _._\n\n Bot et al. (2001); Currie & Stuart (2001).\n\n Moffett (1989a).\n\n Branstetter et al. (2017); Schultz et al. (2005); Schultz & Brady (2008).\n\n Mueller (2002).\n\nChapter 6\n\n Barron & Klein (2016).\n\n A few other ants have supercolonies, including one discovered by Magdalena Sorger and myself in Ethiopia with colonies kilometers wide (Sorger et al. 2017). For more details on Argentine ants and a critical review of the literature on which this chapter is based, see Moffett (2010 & 2012).\n\n Violence within a supercolony occurs in one situation. Each spring, for reasons unclear, workers mass-execute the queens, sparing enough to maintain colony growth. This exception proves the rule: social integrity is reflected in how well the ants manage this situation, with a colony operating smoothly as its queens are butchered\u2014even the queens don't protest (Markin 1970).\n\n Injaian & Tibbetts (2014).\n\n There are however rare cases of ant foundress queens recognizing each other individually, by scent (d'Ettorre & Heinze 2005).\n\n The ants can also distinguish workers by the tasks they are doing at the time (Gordon 1999).\n\n Dangsheng Liang, pers. comm.; Liang & Silverman (2000).\n\n After first using the term anonymous society (Moffett 2012), I found Eibl-Eibesfeldt (1998) had employed this phrase to describe any society with a big population. In my usage, even a small society can be anonymous if it is demarcated through the use of labels that _potentially_ allow some members not to know others.\n\n Brandt et al. (2009). Fighting between supercolonies continues unabated even after the ants are fed an identical diet in a uniform lab setting for a year (Suarez et al. 2002).\n\n Haidt (2012) _._\n\n Czechowski & Godzi\u0144ska (2015).\n\n In some cases the colony odors arise primarily from the queens (Hefetz 2007).\n\n Slaves are less aggressive toward outsiders than are free-living ants; one interpretation of this is that the diversity of markers in their colonies may lead to sloppiness in identification (Torres & Tsutsui 2016).\n\n Elgar & Allan (2006).\n\n Thanks to Stan Braude and Paul Sherman for advice on this species. Braude (2000); Bennett & Faulkes (2000); Judd & Sherman (1996); Sherman et al. (1991).\n\n Braude & Lacey (1992), 24.\n\n Burgener et al. (2008).\n\n I thank Russell Paul Balda, John Marzluff, Christine Dahlin, and Alan Kamil for insights into this species. See Marzluff & Balda (1992); Paz-y-Mi\u00f1o et al. (2004).\n\n I thank Mauricio Cantor and Shane Gero for advice on sperm whales. Cantor & Whitehead (2015); Cantor et al. (2015); Christal et al. (1998); Gero et al. (2015, 2016a, 2016b).\n\n Unlike the sperm whales, Florida's bottlenose dolphins don't seem to use vocalizations to identify their societies (which reach a population of a couple hundred and seem to rely on individual recognition). Still it's within the realm of possibility that differences in culture\u2014learned behavior such as fishing techniques\u2014can similarly act to separate the communities in some populations. For many years one Australian dolphin community, belonging to a different species than the Sarasota dolphins, followed trawlers to scrounge off of fish. These boat-dependent dolphins lived near another community that hunted down fish the normal way, far from boats. The two groups became one after the trawling stopped (Ansmann et al. 2012; Chilvers & Corkeron 2001).\n\n The ants are densest at the leading edge of an invasion, but this may reflect the rich untapped food there rather than indicate a weakening of a supercolony away from its borders. Elsewhere in the world some Argentine ant populations have declined, albeit prognoses of eventual collapse of supercolonies (Queller & Strassmann 1998) appear at best premature (Lester & Gruber 2016).\n\nChapter 7\n\n The gelada is an exception to the tentativeness or discomfort between societies, moving with general indifference among other \"units\" (Chapter 17).\n\n e.g., Cohen (2012): McElreath et al. (2003); Riolo et al. (2001).\n\n Womack (2005). Synonyms for \"marker\" include words like \"label\" and \"tag.\"\n\n de Waal & Tyack (2003); Fiske & Neuberg (1990); Machalek (1992).\n\n For details on different levels of human social connections, see Buys & Larson (1979); Dunbar (1993); Granovetter (1983); Moffett (2013); Roberts (2010).\n\n This is the cultural version of the idea of an extended phenotype proposed by Dawkins (1982).\n\n Wobst (1977).\n\n Alessia Ranciaro, pers. comm.; Tishkoff et al. (2007).\n\n Simoons (1994).\n\n Wurgaft (2006).\n\n Baumard (2010); Ensminger & Henrich (2014).\n\n Poggi (2002).\n\n Iverson & Goldin-Meadow (1998).\n\n Darwin (1872).\n\n Marsh et al. (2003). People in prolonged social contact can converge in facial appearance as well, perhaps through repeated similar use of the same facial muscles (Zajonc et al. 1987).\n\n Marsh et al. (2007).\n\n Sperber (1974).\n\n Eagleman (2011).\n\n Bates et al. (2007).\n\n Allport (1954), 21.\n\n Watanabe et al. (1995).\n\n Nettle (1999).\n\n Pagel (2009), 406.\n\n Larson (1996).\n\n Tajfel et al. (1970).\n\n Dixon (2010), 79.\n\n Sometimes the language spoken by a Pygmy group does not correspond to the farmers to which they are currently connected, suggesting the Pygmies sometimes migrate (Bahuchet 2012 & 2014). Equally mysterious are Bushmen who parted with their mother tongue to speak versions of the language of Khoikhoi cattle herders, once called Hottentots (Barnard 2007).\n\n Giles et al. (1977); van den Berghe (1981).\n\n Fitch (2000); Cohen (2012).\n\n Flege (1984); Labov (1989).\n\n JK Chambers (2008).\n\n Cited by Edwards (2009), 5.\n\n Dixon (1976).\n\n Barth (1969); McConvell (2001).\n\n Heinz (1975), 38. In fact, hunter-gatherer societies are considered \"loose\" because of this wide latitude in allowable behavior (Lomax & Berkowitz 1972).\n\n Guibernau (2013). Of course, any form of membership puts some expectations on behavior, as the conclusion of this book makes clear.\n\n Kurzban & Leary (2001); Marques et al. (1988).\n\n Vicki Fishlock and Richard Wrangham, pers. comm.\n\n Despite their normal tight conformity, even ants that don't take slaves can be manipulated experimentally to receive foreign individuals into their society\u2014different ant species included (Carlin & H\u00f6lldobler 1983).\n\n My viewpoint on superorganisms\u2014which ties societies to the key feature underlying all organisms, the united identity of their components\u2014is adapted from Moffett (2012).\n\n Berger & Luckmann (1966), 149.\n\n For the research using tokens, see Addessi et al. (2007).\n\n Darwin (1871), 145.\n\n Tsutsui (2004).\n\n Gordon (1989). Chimpanzees, and likely many other animals, can do the same thing (Herbinger et al. 2009), though, again, in the ant instance, it's by recognizing their group rather than identifying an individual as a foreigner the way a chimp does.\n\n Spicer (1971), 795\u2013796.\n\n e.g., see discussions in Henshilwood & d'Errico (2011).\n\n Geertz (1973).\n\n Womack (2005), 51.\n\n Leafcutter ants buck this trend with their unusually large brains (Riveros et al. 2012).\n\n Geary (2005); Liu et al. (2014); e.g., relative to body size, Bushmen have exceptionally large braincases (Beals et al. 1984).\n\n Pointed out by Gamble (1998), 431.\n\n Among the theories of social groups framed by psychologists, the inductive and deductive groups of Postmes et al. (2005) may be most similar to my distinction between individual recognition and anonymous societies. The distinction between common bond and common identity groups is also intriguing (Prentice et al. 1994).\n\n Berreby (2005).\n\nChapter 8\n\n As with many terms of anthropology, there is a mind-numbing range of definitions for \"band\" and alternative choices for the word, such as hordes, overnight camps, or local groups.\n\n There have been many other names applied to these mobile hunter-gatherer societies, most of them confusing, but \"band societies\" has a good pedigree (e.g., Leacock & Lee 1982) and emphasizes the primacy of fission-fusion condition over egalitarianism, hunting, gathering, or proficiency with tools or fire. Elsewhere I called them \"multiband societies\" (Moffett 2013), a term I simplify here.\n\n Binford (1980), 4.\n\n e.g., Headland et al. (1989); Henn et al. (2011).\n\n Roe (1974); Weddle (1985).\n\n Behar et al. (2008).\n\n Ganter (2006).\n\n Meggitt (1962), 47.\n\n Curr (1886), 83\u201384.\n\n Thanks to Thomas Barfield for advice. While Asia's \"horse nomads\" had leaders, they acted in a more egalitarian, hunter-gatherer-like manner while in their scattered camps (Barfield 2002).\n\n Hill et al. (2011).\n\n Wilson (2012).\n\n Pruetz (2007).\n\n I thank Fiona Stewart and Jill Pruetz for advice on savanna chimpanzees. Hernandez-Aguilar et al. (2007); Pruetz et al. (2015).\n\n For a description of the importance of fire and food sharing, see Wrangham (2009).\n\n See, e.g., Ingold (1999) and the \"unbounded social landscape\" of Gamble (1998).\n\n Wilson (1975), 10.\n\n Birdsell (1970).\n\n Wiessner (1977, xix) points out that \"even San [Bushmen] of a different language group... are foreign people and to be regarded with suspicion.\"\n\n Arnold (1996); Birdsell (1968); Marlowe (2010). Thanks to Brian Hayden for advice on the size of the Western Desert band societies.\n\n e.g., Tonkinson (2011). Note that because of the hardships, the people of the region gave up their original mode of hunting and gathering long ago.\n\n Meggitt (1962), 34.\n\n Tonkinson (1987), 206. We consider some of these issues in Chapters 17 and 18.\n\n These societies were distinguished by language, the importance placed on spiritual tales, or dreamings, and other matters (Brian Hayden and Brian Codding, pers. comm.). Their alliances were fragile, it would seem, because some took to fighting anyway (Meggitt 1962).\n\n Renan (1990).\n\n Johnson (1997).\n\n Dixon (1976), 231.\n\n e.g., Hewlett et al. (1986); Mulvaney (1976); Verdu et al. (2010).\n\n Murphy & Murphy (1960).\n\n e.g., Heinz (1994); Mulvaney & White (1987).\n\n Stanner (1979), 230.\n\n Stanner (1965) introduced the term \"estate\" to describe the area for which each band had primary rights. This sense of a local home did vary. The Hadza transferred more fluidly from band to band and the bands moved more readily across expanses of the Hadza territory; but even for them, individuals still stayed within that part of the overall region they knew best (Blurton-Jones 2016).\n\n As described for the !K\u00f5 Bushmen by Heinz (1972). See also Chapter 17.\n\n For ants, e.g., Tschinkel (2006). We see the same separation today with the entrenched troops of people during battles (Hamilton 2003).\n\n e.g., Smedley & Smedley (2005).\n\n Malaspinas et al. (2016).\n\n Bowles & Gintis (2011), 99; see also Bowles (2006).\n\n Guenther (1976).\n\n Lee & DeVore (1976). The word \"San\" continues to have derogatory connotations in the Kalahari. I prefer Bushmen, which had few negative associations when initially coined by Dutch explorers, although another alternative is the less familiar Bantu word \"Basarwa.\"\n\n Schapera (1930), 77.\n\n Coren Apicella, pers. comm.; Hill et al. (2014).\n\n Silberbauer (1965), 62.\n\n Schladt (1998) estimates that a century ago, the number of Khoisan (Bushman and the related Khoikhoi cattle herder) languages was 200.\n\n Such traits convey an emblemic style (Wiessner 1983). Wiessner (1984) found less of a connection to particular Bushmen peoples for the styles of their beaded headbands, but the beads don't have an ancient pedigree; rather they were acquired through trading with Europeans.\n\n Wiessner (1983), 267.\n\n Sampson (1988).\n\n Gelo (2012).\n\n Broome (2010), 17.\n\n Spencer & Gillen (1899), 205.\n\n Cipriani (1966).\n\n F\u00fcrniss (2014).\n\n Clastres & Auster (1998), 36.\n\nChapter 9\n\n Tonkinson (2002), 48; Hayden (1979).\n\n So while outsiders experienced hunter-gatherers asking for possessions as begging, hunter-gatherers saw such gifts as generous invitations to partake in relationships of sharing that assured everyone was taken care of (Earle & Ericson 2014; Peterson 1993).\n\n Endicott (1988).\n\n See arguments in Wiessner (2002).\n\n Sahlins (1968) first described hunter-gatherers as \"affluent,\" an idea that has been disputed (Kaplan 2000), the difference in opinion arising in part because work and leisure are impossible to distinguish for people who socialize as they carry out many of their tasks.\n\n Morgan & Bettinger (2012).\n\n Elkin (1977).\n\n Bleek (1928), 37.\n\n Chapman (1863), 79.\n\n Keil (2012).\n\n They also favor learning from dominant individuals (Kendal et al. 2015).\n\n Wiessner (2002).\n\n Blurton-Jones (2016); Hayden (1995).\n\n Baumeister (1986).\n\n Pelto (1968): Witkin & Berry (1975).\n\n In the Ache, men from different bands got together to fight, though even then not as a team, and the combatants often ended up fighting others in their own band (Hill & Hurtado 1996).\n\n Ellemers (2012).\n\n Finkel et al. (2010).\n\n Lee (2013), 124.\n\n Lee & Daly (1999), 4.\n\n e.g., Marshall (1976). One practical matter was that a band had too few children of a similar age who could play together in a competitive way (Draper 1976).\n\n Boehm (1999).\n\n de Waal (1982). In a similar fashion, lower-ranking olive baboons have jointly driven a repressive alpha female entirely out of their troop (Anthony Collins, pers. comm.).\n\n Ratnieks et al. (2006).\n\n Dominance of one sex over the other exists in many species, with females coming out on top in spotted hyenas, ringtail lemurs, and the bonobo, while males take control in chimpanzees and baboons.\n\n Tuzin (2001), 127.\n\n Schmitt et al. (2008).\n\n Thomas-Symonds (2010).\n\n Bousquet et al. (2011).\n\n H\u00f6lldobler & Wilson (2009); Seeley (2010); Visscher (2007). The dominant painted dogs have somewhat more influence than the other pack members (Walker et al. 2017).\n\n Rheingold (2002); Shirky (2008).\n\nChapter 10\n\n Ian McNiven and Heather Builth, pers. comm.; Broome (2010); Builth (2014); Head (1989); McNiven et al. (2015).\n\n Cipriani (1966).\n\n Brink (2008).\n\n In the elephant instance the gatherings are made up of many core groups, or societies, though in hunter-gatherers multiple band societies could similarly get together to forge alliances and trade (Hayden 2014).\n\n Guenther (1996).\n\n The inevitable could be put off. The Cheyenne put together police squads to oversee the fairness of their joint buffalo hunts, a force that disbanded when the hunts ended (MacLeod 1937).\n\n Rushdie (2002), 233.\n\n Denham et al. (2007).\n\n Mitchell (1839), 290\u2013291.\n\n Clastres (1972). This article uses an alternative name for the Ache, the Guayaki.\n\n Lee (1979), 361.\n\n Hawkes (2000).\n\n Morgan & Bettinger (2012).\n\n Roscoe (2006).\n\n My descriptions of coastal Northwest Indians owes considerably to correspondence with Kenneth Ames and Brian Ferguson. Ames (1995); Ames & Maschner (1999); Sassaman (2004).\n\n Some tribes also managed the environment, for example by stocking salmon for a time in artificial pools or raising butter clams in rock terraces still visible at low tides (Williams 2006).\n\n Patrick Saltonstall and Amy Steffian, pers. comm.; Steffian & Saltonstall (2001). American naturalist Edward Nelson, who lived in the 1870s among Yupik-speaking peoples of southwestern Alaska, wrote that the labrets, made of stone and thus painful to keep in place, \"were removed and carried in a small bag until we approached a village at night, when they were taken out and replaced, that the wearer might present a proper appearance before the people\" (Nelson 1899, 50)\u2014equivalent to a person bringing out a national flag during an international event.\n\n Townsend (1983).\n\n Johnson (1982).\n\n Silberbauer (1965).\n\n Van Vugt & Ahuja (2011).\n\n Bourjade et al. (2009).\n\n Peterson et al. (2002).\n\n Fishlock et al. (2016).\n\n Watts et al. (2000).\n\n Baumeister et al. (1989).\n\n e.g., Hold (1980).\n\n Dawson (1881); Fison & Howitt (1880), 277.\n\n Hann (1991), xv.\n\n William Marquardt, pers. comm.; Gamble (2012); Librado (1981).\n\n Hayden (2014).\n\n Van Vugt et al. (2008).\n\n Hogg (2001); Van Knippenberg (2011).\n\n For example, Passarge (1907) wrote that Bushmen told him they once had hereditary chiefs who went unnoticed by colonists because they displayed little of the pomp and circumstance of European leaders. Another anthropologist wrote, \"Both Naron and Auen [Bushmen societies] had chiefs when the old men were young. They seem to have directed the movement of their people from place to place, to have ordered the burning of the veld, and in particular to have led in war. Fights were frequent both between the opposing Bushman tribes, Naron and Auen, and against other natives who were gradually encroaching from all sides\" (Bleek 1928, 36\u201337).\n\n Andersson (1856), 281.\n\n Rather than chiefs, a better word for the leaders of the =Au\/\/ei might be \"Big Men\" (Chapter 22), though at least in some cases their position was inherited (Mathias Guenther, pers. comm.; Guenther 1997 & 2014).\n\n Ames (1991).\n\n Testart (1982).\n\n Durkheim (1893) distinguished the \"mechanical solidarity\" of people who perform similar work in technologically simple societies from the \"organic solidarity\" of societies with division of labor.\n\n This is a by-product of what's described as self-domestication: apes such as humans and bonobos have evolved to be tolerant of, and ineffectual without, others of their kind (Hare et al. 2012). Self-domestication is associated with a reduction of impulsive violence (Wrangham 2019). Baumeister et al. (2016) argue that by specialization, people have made themselves increasingly irreplaceable. Yet because many individuals are likely to perform all but the most specialized tasks across a large population, only a tiny minority of people is truly irreplaceable today.\n\n Originally proposed by Brewer (1991).\n\n Even today societies differ in the most comfortable, or optimal, level of distinctiveness their people strive for. The focus on _differences_ is sharpest in Western cultures where individualism and capitalism reign, though even among them a marketer can tell you people fall in categories and are less distinct than they think (JR Chambers 2008).\n\n Hayden (2011).\n\n Fried (1967), 118. Potlatches existed pre-contact and may have grown more elaborate after Europeans put a stop to the chronic wars of the Pacific Northwest, suggesting the feasts became an alternative to fighting as demonstrations of a chief's importance.\n\n Tyler (2006).\n\n An early statement of this point of view was presented by Hayden et al. (1981).\n\n To my knowledge this point was first made in Testart (1982).\n\n For examples from South America, see Bocquet-Appel & Bar-Yosef (2008); Goldberg et al. (2016).\n\n Berndt & Berndt (1988), 108.\n\n Cipriani (1966), 36.\n\n Mummert et al. (2011).\n\n O'Connell (1995).\n\n Roosevelt (1999).\n\n There is an irony in the contempt industrialized nations display toward hunter-gatherer peoples, given that our gut reaction to outsiders evolved over millennia spent as hunter-gatherers. So-called primitive cultures have been associated with animals and children, as if a reliance on hunting and gathering were evidence of the arrested mental facilities of a bygone era (Jahoda 1999; Saminaden et al. 2010).\n\nChapter 11\n\n Marean (2010).\n\n Behar et al. (2008).\n\n Mercader et al. (2007).\n\n Villa (1983).\n\n Curry (2008).\n\n Harlan (1967) collected enough wild wheat with stone tools to show that a prehistoric family in Turkey could have gathered a year's supply of this grain, so would have had the option to put down roots.\n\n Price & Bar-Yosef (2010); Trinkaus et al. (2014).\n\n Jerardino & Marean (2010).\n\n d'Errico et al. (2012).\n\n Henshilwood et al. (2011).\n\n This view has been effectively countered by McBrearty & Brooks (2000).\n\n Kuhn & Stiner (2007), 40\u201341.\n\n Wadley (2001).\n\n The most convincing evidence of markers differentiating societies comes late\u2014the multiplicity of ivory, antler, wood, tooth, and shell jewelry across Europe between 37,000 and 28,000 years ago (Vanhaeren & d'Errico 2006).\n\n Brooks et al. (2018).\n\n Rendell & Whitehead (2001); Thornton et al. (2010).\n\n Coolen et al. (2005).\n\n van de Waal et al. (2013).\n\n Bonnie et al. (2007); Whiten (2011). The grooming behavior can be passed on from mother to offspring (Wrangham et al. 2016).\n\n McGrew et al. (2001).\n\n One female has failed to \"correctly\" clasp the hands of other chimps since joining her community two decades ago, yet her companions groom her all the same (Michio Nakamura, pers. comm.).\n\n Brown & Farabaugh (1997); Nowicki (1983).\n\n Paukner et al. (2009).\n\n It remains to be firmly proven whether chimpanzees react to the group-specific characteristics of the pant-hoots, recognize the callers as individuals from slight differences in their pant-hoots, or both (Marshall et al. 1999; Mitani & Gros-Louis 1998).\n\n Crockford et al. (2004). The pinyon jays, with their durable flock societies (Chapter 6), may learn their _rack_ or _kaw_ vocalizations this way.\n\n Boughman & Wilkinson (1998); Wilkinson & Boughman (1998). Meerkats have contact calls that differ from one clan to another but in this species the animals themselves don't seem to grasp the difference (Townsend et al. 2010).\n\n Herbinger et al. (2009).\n\n Taglialatela et al. (2009). One essential experiment, comparing the chimp's reactions to the pant-hoots of group members versus foreigners, has not yet been done.\n\n Fitch (2000). A colony password is also hypothesized for a bird species (Feekes 1982).\n\n Zanna Clay, pers. comm.; Hohmann & Fruth (1995). In fact the spider monkey learns a whinny that is similarly specific to its community (Santorelli et al. 2013).\n\n The protolanguage would consist of \"nothing more than an inventory of calls expressing unanalyzed meanings\" (Kirby 2000, 14).\n\n Steele & Gamble (1999).\n\n Aiello & Dunbar (1993).\n\n Grove (2010).\n\n I prefer this expression to \"release from proximity\" (see Chapter 4 and Gamble 1998) given, for example, that the low density of savanna chimps demonstrates that distance per se is not the issue.\n\n Or at least ordinarily: cheaters like the colony-infiltrating spider are rare in the Argentine ant, suggesting identification with a supercolony is a tough nut to crack\u2014maybe their members are so precisely alike that even the slightest deviation from the \"norm\" sets off an alarm.\n\n Fiske (2010); Boyd & Richerson (2005).\n\n Johnson et al. (2011).\n\n Near-hairlessness is open to other explanations, such as making swimming easier, reducing parasites, or keeping the body cool (Rantala 2007).\n\n Lewis (2006), 89.\n\n Turner (2012), 488. See also Thierry (2005).\n\n Gelo (2012).\n\n Kan (1989), 60.\n\n The tattoos stop the kidnapping of the women (White 2011); for other examples of marking the skin, see Jablonski (2006).\n\n Pabst et al. (2009) argues the tattoos could have had medicinal value; even so they could have been associated with the man's tribe.\n\n Alan Rogers, pers. comm.; Rogers et al. (2004).\n\n Berman (1999).\n\n Jolly (2005).\n\n Chance & Larsen (1976).\n\n Boyd & Richerson (2005).\n\n Foley & Lahr (2011).\n\n Tennie et al. (2009). Chimpanzees also create rudimentary symbolic cultures, in the sense that a behavior can have different meanings in different communities; for example, noisily tearing leaves with the teeth is an invitation for sex in one community and for play in another (Boesch 2012).\n\n Tindale & Sheffey (2002). In the last decade, to give an example, our reliance on GPS has made us less adept at a talent honed by hunter-gatherers, spatial navigation (Huth 2013).\n\n Henrich (2004b); Shennan (2001). There is some controversy over how to interpret the simplicity of Tasmanian culture, including their inability to make fire (Taylor 2008).\n\n Finlayson (2009); Mellars & French (2011).\n\n Hiscock (2007).\n\n Aim\u00e9 et al. (2013).\n\n Powell et al. (2009). Some deny connections between social complexity and population density and interaction rates, and certainly other factors may come into play (Vaesen et al. 2016).\n\n Wobst (1977).\n\n Moffett (2013), 251.\n\nChapter 12\n\n Wiessner (2014).\n\n Hasson et al. (2012). Such coupling also occurs between monkey brains (Mantini et al. 2012).\n\n Harari (2015).\n\n For an excellent review of many general issues, see Banaji & Gelman (2013).\n\n Eibl-Eibesfeldt (1998), 38.\n\n Callahan & Ledgerwood (2013).\n\n Testi (2005); see also Testi (2010).\n\n Bar-Tal & Staub (1997); Butz (2009); Geisler (2005). In fact, the mere sight of a national flag can cause people to feel more nationalistic (Hassin et al. 2007), though this response differs between societies (Becker et al. 2017).\n\n Helwig & Prencipe (1999); Weinstein (1957); Barrett (2007).\n\n Billig (1995), Ferguson & Hassin (2007), Kemmelmeier & Winter (2008) _._\n\n Barnes (2001).\n\n Thanks to S\u00f6ren Krach and Helen Gallagher for confirming this expectation of human responses to playing a game on a computer. As robots become more lifelike, we treat them increasingly as human (Chaminade et al. 2012; Takahashi et al. 2014; Wang & Quadflieg 2015).\n\n Parr (2011).\n\n Henrich et al. (2010b).\n\n e.g., Ratner et al. (2013).\n\n Schaal et al. (2000).\n\n Cashdan (1998); Liberman et al. (2016).\n\n Or at least the race of his or her parents (Kelly et al. 2005).\n\n Kinzler et al. (2007); Nazzi et al. (2000); Raki\u0107 et al. (2011).\n\n Kelly et al. (2009); Pascalis & Kelly (2009). This effect is reversible in older children with radical changes in environment, but is difficult and requires considerable time (Anzures et al. 2012; Sangrigoli et al. 2005). We can capitalize on an appropriately aged baby's skills to beef up how well it recognizes those of other races or ethnicities\u2014exposure to as few as three faces of the group can do the trick (Sangrigoli & de Schonen 2004).\n\n Oddly enough given the effort that chicks invest in imprinting, it's unknown whether the mother hen can recognize her own brood (Bolhuis 1991).\n\n Of course, the chick only learns its mother rather than a social group, and the ants don't use their imprinting as a starting point to learn the members of their colony individually. Still, though the complexity of how people distinguish groups is greater, the genetic underpinnings of how we do it might not be much different (e.g., Sturgis & Gordon 2012).\n\n Pascalis et al. (2005); Scott & Monesson (2009); Sugita (2008).\n\n Rowell (1975).\n\n Atran (1990).\n\n Hill & Hurtado (1996) _._\n\n Keil (1989).\n\n Gil-White (2001).\n\n This reaches extremes in situations of identity fusion (Chapter 15; Swann et al. (2012).\n\n Martin & Parker (1995).\n\n Different cultures and ethnicities have different ways of categorizing the offspring of mixed marriages (e.g., Henrich & Henrich 2007).\n\n Hammer et al. (2000).\n\n Madon et al. (2001) looks at how stereotypes have changed over the last few decades.\n\n MacLin & Malpass (2001) _._\n\n Appelbaum (2015).\n\n Levin & Banaji (2006).\n\n MacLin & MacLin (2011).\n\n Ito & Urland (2003), Todorov (2017).\n\n Asch (1946), 48.\n\n Castano et al. (2002)\n\n Jewish Telegraphic Agency (1943).\n\n Greene (2013).\n\n Wiessner (1983), 269.\n\n Silberbauer (1981), 2.\n\n German sociologist Georg Simmel contributed to the confusion by defining \"stranger\" in a nontraditional way, as a group member who doesn't fit in\u2014that is, someone who acts strangely (e.g., McLemore 1970). Most dictionaries give priority to using \"xenophobia\" in describing negative reactions to a _foreigner_ , whether or not that individual has been encountered previously, and that's how I prefer to use the word.\n\n Azevedo et al. (2013). The more distinct the races, the less empathy (Struch & Schwartz 1989). Responses to pain are also stronger for animals that we see as more like us (Plous 2003).\n\n Campbell & de Waal (2011).\n\nChapter 13\n\n Macrae & Bodenhausen (2000), 94.\n\n Lippmann (1922), 89.\n\n Devine (1989).\n\n Bonilla-Silva (2014) provides one fascinating discussion.\n\n Banaji & Greenwald (2013), 149. The test has its critics (e.g., Oswald et al. 2015).\n\n Baron & Dunham (2015).\n\n Hirschfeld (2012), 25.\n\n Aboud (2003); Dunham et al. (2013).\n\n Harris (2009).\n\n Bigler & Liben (2006); Dunham et al. (2008)\n\n Hirschfeld (1998).\n\n Karen Wynn, pers. comm.; Katz & Kofkin (1997).\n\n Edwards (2009).\n\n Kinzler et al. (2007), 12580\n\n Amodio (2011), 104; see also, e.g., Phelps et al. (2000).\n\n Thus far this work has been done on people of differing political points of view, rather than national identities per se (Nosek et al. 2009).\n\n Beety (2012); Rutledge (2000).\n\n Cosmides et al. (2003); Kurzban et al. (2001); Pietraszewski et al. (2014). These authors propose that this cognitive machinery evolved to detect alliances within societies, but among recent hunter-gatherers and in early humans such alliances were likely to be fluid and not at all connected to any identifying traits (markers, including racial differences).\n\n Wegner (1994).\n\n Monteith & Voils (2001).\n\n MacLin & MacLin (2011).\n\n Haslam & Loughnan (2014), 418.\n\n Greenwald et al. (2015). To give just one of many circumstances where this occurs, minorities often receive medical care inferior to whites under the same doctor (Chapman et al. 2013).\n\n Treating ethnic friends as special is called subtyping (Wright et al. 1997).\n\n Mild disgust can be a means of differentiation rather than discrimination (Brewer 1999; Douglas 1966; Kelly 2011).\n\n See Chapter 17. Bandura (1999); Jackson & Gaertner (2010); Vaes et al. (2012); Viki et al. (2013).\n\n Steele et al. (2002).\n\n Fiske & Taylor (2013), Phelan & Rudman (2010).\n\n Gilderhus (2010).\n\n Kelley (2012); e.g., many Norwegian traditions are similar inventions (Eriksen 1993).\n\n Leibold (2006).\n\n Beccaria (1764).\n\n Haslam et al. (2011b).\n\n Renan (1990), 11. See also Hosking & Sch\u00f6pflin (1997); Orgad (2015).\n\n For more on memory, see Bartlett & Burt (1933); Harris et al. (2008); Zerubavel (2003).\n\n Gilderhus (2010); L\u00e9vi-Strauss (1972).\n\n Berndt & Berndt (1988).\n\n Billig (1995); Toft (2003).\n\n Maguire et al. (2003); Yates (1966).\n\n Lewis (1976).\n\n Joyce (1922), 317. We will see in section IX that, given a history of conquest and immigration, what counts as \"the same\" people has become a complicated question.\n\n Bar-Tal (2000).\n\n Hence the mandate of ISIS to lay claim to a state (Wood 2015). See Section IX for how this plays out for races and ethnicities.\n\n McDougall (1920).\n\n Bigelow (1969).\n\nChapter 14\n\n Wilson (1978), 70. See also Read (2011). As Claude L\u00e9vi-Strauss (1952, 21) wrote, \"Humankind ceases at the border of the tribe.\"\n\n Gombrich (2005), 278.\n\n Giner-Sorolla (2012), 60.\n\n Freud (1930).\n\n Smith (2011).\n\n Aristotle describes slavery as justified for captives taken in battle (Walford & Gillies 1853, 12).\n\n Orwell (1946), 112. Anthropomorphism and dehumanization are related (Waytz et al. 2010).\n\n David Livingstone Smith, pers. comm.; Haidt & Algoe (2004); Lovejoy (1936); Smith (2011).\n\n Costello & Hodson (2012) studied how six-to-ten-year-old white Canadians perceived black children. Those who saw a greater divide between people and animals tended to be more prejudiced.\n\n This reinforces my point that it was the society, not the camp or band, upon which these people erected their primary identities. In general when the names of outgroups are simplified descriptions, say of their \"humanness,\" relations with outsiders tend to be more acrimonious than if the names have a more nuanced meaning (Mullen et al. 2007).\n\n Haslam & Loughnan (2014).\n\n Ekman (1992). Some claim different categories of basic emotions, e.g., Jack et al. (2014) propose that surprise can't be distinguished from fear, or disgust from anger. Yet as psychologist Paul Bloom points out to me, disgust and anger are both aversive, negative emotions, but they are elicited by different stimuli, evoke different reactions and brain responses, and have different evolutionary histories and developmental trajectories.\n\n Haidt (2012) _._\n\n Bosacki & Moore (2004).\n\n Chimpanzees read each other's expressions (Buttelmann et al. 2009; Parr 2001; Parr & Waller 2006).\n\n Haslam (2006) proposes that we dehumanize everyone we meet to varying degrees. He sees the belief that outsiders lack basic human traits as _animalistic dehumanization_ , or dehumanization to the animal level. This creates species-like boundaries between groups not seen in the _mechanistic dehumanization_ of people, such as when doctors or lawyers are viewed as calculating and dehumanized to the level of inanimate objects or, more accurately, machines. Mart\u00ednez et al. (2012) show that mechanistic dehumanization can emerge at a society level.\n\n Wohl et al. (2012).\n\n Haidt (2003); Opotow (1990).\n\n Jack et al. (2009); Marsh et al. (2003).\n\n Both races more often misidentify an object as a weapon when it is in a black person's hands than when a white person is holding it. Ackerman et al. (2006); Correll et al. (2007); Eberhardt et al. (2004); Payne (2001).\n\n Hugenberg & Bodenhausen (2003).\n\n That was true at least if the liar was quiet, when onlookers apparently missed subtle cues of affect that differ across cultures; when speaking, awkward pauses in uttering sentences may give liars away (Bond et al. 1990). See also Al-Simadi (2000).\n\n Ekman (1972).\n\n Kaw (1993).\n\n The \"stereotype content\" account of dehumanization described here (Fiske et al. 2007) was developed separately from the infrahumanization model, with its focus on secondary emotions.\n\n Vaes & Paladino (2010).\n\n Clastres (1972).\n\n Koonz (2003).\n\n Goff et al. (2008); Smith & Panaitiu (2015).\n\n Haslam et al. (2011a).\n\n Haidt et al. (1997).\n\n Amodio (2008); Kelly (2011); Harris & Fiske (2006).\n\n Fear of contamination, expressed as disgust and a belief that group members share \"some fundamental bodily essence in common\" (Fiske 2004), may be an ancient kind of \"behavioral immune system\" (Schaller & Park 2011, 30; O'Brien 2003). People show more fear of immigrants after being shown pictures of diseased persons (Faulkner et al. 2004).\n\n Freeland (1979). This idea doesn't work with all parasites, since diseases that spread by feces rather than direct contact might transfer readily between territories; also, once a disease breaches a territory, the concentrated population tied to that space can promote its spread.\n\n McNeill (1976) _._ Syphilis may have been brought back to Europe from America, but to a much less widely destructive effect than smallpox had in the Americas.\n\n Heinz (1975), 21.\n\n Tajfel & Turner (1979).\n\n Bain et al. (2009).\n\n Koval et al. (2012).\n\n Reese et al. (2010); Taylor et al. (1977).\n\n As far as I can tell there is little research on this question, although one study shows that infants maintain more eye contact with same-race individuals (Wheeler et al. 2011) and there is a classic study demonstrating that white people maintain less eye contact with black job applicants (Word et al. 1974).\n\n Mahajan et al. (2011); but see also Mahajan et al. (2014).\n\n Another option is that the link we make between outsiders and the sense of disgust arose in humans (D Kelly 2013).\n\n Henrich (2004a); Henrich & Boyd (1998); Lamont & Molnar (2002); Wobst (1977).\n\n Gil-White (2001).\n\n Reviewed by Kleingeld (2012).\n\n Leyens et al. (2003), 712.\n\n Castano & Giner-Sorolla (2006).\n\n Wohl et al. (2011).\n\nChapter 15\n\n Orwell (1971), 362.\n\n Goldstein (1979).\n\n Bloom & Veres (1999); Campbell (1958).\n\n Assessments of the warmth and competence of social groups were described in the last chapter. Callahan & Ledgerwood (2016).\n\n Those alternative responses depend on the relative strength of the groups and how much they compete (Alexander et al. 2005).\n\n McNeill (1995); Seger et al. (2009); Tarr et al. (2016); Valdesolo et al. (2010).\n\n Barrett (2007); Baumeister & Leary (1995); Guibernau (2013).\n\n Atran et al. (1997); Gil-White (2001).\n\n Brewer & Caporael (2006); Caporael & Baron (1997).\n\n The idea that a society is more than the sum of its members, now widely considered to be correct, was originally called the \"nationalistic fallacy\" by Allport (1927).\n\n Sani et al. (2007).\n\n Castano & Dechesne (2005).\n\n Best (1924), 397.\n\n Wilson (2002).\n\n de Dreu et al. (2011); Ma et al. (2014).\n\n People with a strong group identity express group emotions most strongly (Smith et al. 2007).\n\n Adamatzky (2005).\n\n Hayden (1987). Gatherings of bands from different societies for trading and alliances would have proceeded with greater caution (Chapter 18).\n\n Marco Iacoboni pers. comm. and Iacoboni (2008) _._ When we consciously mimic others, however, perceived status may trump race (Elizabeth Losin, pers. comm. and Losin et al. 2012).\n\n Rizzolatti & Craighero (2004).\n\n Field et al. (1982). Dance could have gotten its start through such imitation (Laland et al. 2016).\n\n Parr & Hopkins (2000).\n\n Animal rallies grade into more immediate responses to a stimulus, such as the screams that race through a a chimpanzee community, mobilizing the animals to scare off an enemy or predator. Preston & de Waal (2002); Spoor & Kelly (2004).\n\n Wildschut et al. (2003).\n\n For explanations other than \"rallying\" for the ants' behavior, see Moffett (2010).\n\n Watson-Jones et al. (2014).\n\n Chimpanzees and bonobos ape each other when something useful comes out of it, for example, using a stick to catch a termite meal, but seldom imitate an act completely unconnected with a practical purpose. Some behaviors come close, though\u2014as when the bonobos at the San Diego Zoo took to the custom of clapping hands at intervals when grooming each other (de Waal 2001).\n\n Thanks to Harvey Whitehouse for advice on identity fusion. Whitehouse et al. (2014a); Whitehouse & McCauley (2005).\n\n Documented for the Libyan civilians turned revolutionaries who rose up against Gaddafi (Whitehouse et al. 2014b).\n\n The sting feels \"like walking over flaming charcoal with a 3-inch nail embedded in your heel\" (Schmidt 2016, 225).\n\n Bosmia et al. (2015).\n\n Fritz & Mathewson (1957); Reicher (2001); Willer et al. (2009).\n\n Hood (2002), 186.\n\n Barron (1981).\n\n Hogg (2007).\n\n Caspar et al. (2016); Milgram (1974).\n\n Mackie et al. (2008).\n\n Kameda & Hastie (2015).\n\n Fiske et al. (2007).\n\n Staub (1989).\n\n People who wish to believe something, prejudices included, ignore contrary evidence so long as they can cling to _anything_ that corroborates their point of view (Gilovich 1991).\n\n Especially troublesome were radio programs that described the violence as everyday behavior (Elizabeth Paluck, pers. comm.; Paluck 2009).\n\n Janis (1982).\n\n This is called Asch conformity after the psychologist Solomon Asch (e.g., Bond 2005).\n\n Redmond (1994), 3.\n\n Hofstede & McCrae (2004).\n\n Wray et al. (2011).\n\n Masters & Sullivan (1989); Warnecke et al. (1992).\n\n Silberbauer (1996).\n\nChapter 16\n\n Marlowe (2000).\n\n Non-kin are found in cores across all populations, but join cores most often where there has been poaching (Wittemyer et al. 2009).\n\n Sometimes one or even more of these outsiders breed under the noses of the senior, \"alpha\" pair (Dan Stahler, pers. comm.; Lehman et al. 1992; Vonholdt et al. 2008).\n\n The matter is controversial for Gunnison's prairie dogs. Hoogland et al. (2012) found adult females of a coterie tend to be maternal close kin in Colorado, but Verdolin et al. (2014) encountered few relatives among adults in Arizona, a possible regional difference.\n\n Horses can leave a band if something goes very wrong\u2014perhaps an overbearing stallion drives the females away (Cameron et al. 2009).\n\n Bohn et al. (2009); McCracken & Bradbury (1981); Gerald Wilkinson (pers. comm.).\n\n Many of a male chimpanzee's allies are likely to be childhood friends having another mother, though this needs study (Ian Gilby, pers. comm.; Langergraber et al. 2007 & 2009).\n\n Massen & Koski (2014).\n\n Sai (2005).\n\n Heth et al. (1998). Note that hamsters are social, but have no societies.\n\n That's true at least when the baby is a male (Parr & de Waal 1999).\n\n Alvergne et al. (2009); Bressan & Grassi (2004).\n\n Cheney & Seyfarth (2007).\n\n Chapais (2008); Cosmides &Tooby (2013); Silk (2002).\n\n Thanks to Elizabeth Archie for insights on baboon matrilines. Because what counts as a matrilineal \"group\" (\"network\" is a better word) depends on the perspective of each female, the one and only category every baboon shares is the troop itself\u2014its society.\n\n The possibility that the males recognize their physical similarity to their offspring has also been suggested (Buchan et al. 2003).\n\n This is especially true for women, but for men, too, when they unite against outsiders (Ackerman et al. 2007).\n\n Weston (1991); Voorpostel (2013).\n\n Apicella et al. (2012); Hill et al. (2011).\n\n Schelling (1978). People often develop an affinity with genetically similar others, kin or not, perhaps because subtle resemblances in attitude grease the path to friendship (Bailey 1988; Christakis & Fowler 2014).\n\n Silberbauer (1965), 69.\n\n Lieberman et al. (2007). Hence children raised together on a kibbutz don't marry each other even though no prohibitions exist against it (Shepher 1971).\n\n Hill et al. (2011).\n\n Hirschfeld (1989).\n\n Tincoff & Jusczyk (1999). These words might have originated to match the first babbling sounds uttered by most infants (Matthey de l'Etang et al. 2011).\n\n As it turns out David Haig was paraphrasing himself (Haig 2000). See also Haig (2011).\n\n Everett et al. (2005); Frank et al. (2008).\n\n Frank et al. (2008). Chagnon (1981) describes people recognizing kin categories for which they lack words.\n\n Woodburn (1982).\n\n Gould (1969).\n\n Cameron (2016). While just a small proportion of Comanche were captives at any one time, for example, a need for fresh warriors led to many of this tribe having foreign blood (Murphy 1991).\n\n So says Ferguson (2011, 262), referring to the work of Chaix et al. (2004).\n\n Barnard (2011). Not that being described as kin is always positive. Some Africans used kin metaphors not to suggest closeness but to convey dominance over slaves (Kopytoff 1982).\n\n Tanaka (1980), 116.\n\n As suggested by studies like that on monkeys by Chapais et al. (1997).\n\n Often those who point to the entitativity of families (e.g., Lickel et al. 2000) have allowed each person to decide what, to them, is a family. I find this problematic. Perceiving the particular family members you happen to be close to as a tight group seems trivial, and no different than imagining your close buddies as a tight group.\n\n This sense of obligation, rather than genetic relatedness as such, is how I would explain the results of Hackman et al. (2015) that show high levels of sacrifice toward mates and kin.\n\n West et al. (2002).\n\n This can make it hard to disown just part of a family (Jones et al. 2000; Uehara 1990).\n\n Many societies simplify the issue of inheritance by the people keeping track of their descent from a particular ancestor (Cronk & Gerkey 2007).\n\n Johnson (2000). Meanwhile, longer life spans have created more intricate networks of kin than had lived together before (Milicic 2013).\n\n e.g., Eibl-Eibesfeldt (1998).\n\n Barnard (2011).\n\n Johnson (1987); Salmon (1998). Unlike van der Dennen (1999), I believe such metaphors tap our beliefs in essences (Chapter 12) rather than kinship as such.\n\n Breed (2014). Hannonen & Sundstr\u00f6m (2003) describe an instance of nepotism (favoring kin) for ants, but their evidence is weak.\n\n Eibl-Eibesfeldt (1998); Johnson (1986). In my view that ape couldn't have confounded its embryonic society with its kin and allies\u2014it would have kept separate track of them from the start.\n\n Barnard (2010).\n\nChapter 17\n\n Voltaire (1901), 11. Thanks to Michael Wilson for the notion of a \"Pankind.\"\n\n Toshisada Nishida (1968), a remarkable Japanese researcher working in Uganda, was first to detect the communities.\n\n Wrangham & Peterson (1996).\n\n Mitani et al. (2010); Wilson & Wrangham (2003); Williams et al. (2004).\n\n Aureli et al. (2006).\n\n Douglas Smith, Kira Cassidy (pers. comm.); Mech & Boitani (2003); Smith et al. (2015).\n\n Quoted in McKie (2010).\n\n Wrangham et al. (2006).\n\n Wendorf (1968).\n\n Morgan & Buckley (1852), 42\u201344 _._\n\n Europeans perverted this spiritual component of trophy taking by paying for scalps (Chacon & Dye 2007).\n\n Boehm (2013).\n\n e.g., Allen & Jones (2014); Gat (2015); Keeley (1997); LeBlanc & Register (2004); Otterbein (2004); DL Smith (2009).\n\n Moffett (2011).\n\n Gat (1999); Wrangham & Glowacki (2012).\n\n This cycle is often driven by spur-of-the-moment gut responses, although some societies, Bedouin tribes for example, codify it (Cole 1975).\n\n Genetic analysis indicates Aborigines have stuck to the general regions they first occupied upon settling Australia despite environmental changes since that time (Tobler et al. 2017), although that obviously doesn't mean the individual societies didn't move around within an area. Still, many accounts suggest that land tenure by band societies was ancient and respected (LeBlanc 2014).\n\n Burch (2005), 59.\n\n de Sade (1990), 332.\n\n Guibernau (2007); van der Dennen (1999).\n\n Bender (2006), 171.\n\n Sumner (1906), 12.\n\n Johnson (1997).\n\n Bar-Tal (2000), 123.\n\n This bias is reflected even in the behavior of small groups of children (Dunham et al. 2011).\n\n For general descriptions of our ineptitude with risk, see Gigerenzer (2010); Slovic (2000).\n\n Fabio Sani, pers. comm.; Hogg & Abrams (1988).\n\n e.g., \"War is conditioned by human symbol systems,\" Huxley (1959), 59.\n\n e.g., Wittemyer et al. (2007).\n\n At least in captivity (Tan & Hare 2013).\n\n Furuichi (2011).\n\n Wrangham (2014 & 2019).\n\n Hrdy (2009), 3.\n\n Hare et al. (2012); Hohmann & Fruth (2011).\n\n The closest thing to friendship between units can be detected after one unit splits in two. These so-called \"teams\" remain near each other and grunt pleasantly in each other's direction\u2014but even these feeble signs of a shared bond fade away over several months (Bergman 2010).\n\n Pusey & Packer (1987).\n\n Boesch (1996); Wrangham (1999). Killings are most common in parts of population-dense Kibale, Uganda, where this strategy of keeping to large parties doesn't exist (Watts et al. 2006). A possible explanation for the violence of chimpanzees is that most populations under study today are restricted to patches of forests where resources and space are limited, but a recent analysis discounts this hypothesis (Wilson et al. 2014).\n\n Wrangham (2019) describes this reduction of \"reactive aggression\" and the traits that accompany it.\n\n Pimlott et al. (1969); Theberge & Theberge (1998).\n\n Mahajan et al. (2011 & 2014).\n\n Brewer (2007); Cashdan (2001); Hewstone et al. (2002).\n\nChapter 18\n\n Sperm whales are the exception that proves the rule that societies have a difficult time working together. However in this case the collaborating societies (units) are part of a greater social entity\u2014clans of whales that share the same hunting traditions (Chapter 6).\n\n The collaboration between the eel-fishing people extended to lavish intergroup gatherings, and the eels themselves were widely traded. Other names for the Gunditjmara are the Gournditch-mara or the somewhat more inclusive term Manmeet (Howitt 1904; Lourandos 1977).\n\n Timothy Shannon, pers. comm.; Shannon (2008).\n\n Dennis (1993); Kupchan (2010).\n\n e.g., Brooks (2002).\n\n Rogers (2003).\n\n e.g., Murphy et al. (2011).\n\n Gudykunst (2004).\n\n Barth (1969); Bowles (2012).\n\n Yellen & Harpending (1972).\n\n Marwick (2003); Feblot-Augustins & Perl\u00e8s (1992); Stiner & Kuhn (2006).\n\n Dove (2011).\n\n Laidre (2012).\n\n Moffett (1989b).\n\n Breed et al. (2012).\n\n Whallon (2006).\n\n Bushmen have been said to rarely steal since they recognize each other's few possessions and can identify the thief from their spoor, though I suspect this applies only to thievery _within_ a society (or \"ethnolinguistic group\": Marshall 1961; Tanaka 1980).\n\n Cashdan et al. (1983).\n\n Dyson-Hudson & Smith (1978).\n\n Bruneteau (1996); Flood (1980); Helms (1885).\n\n Bushmen cultivated special partners they could depend on for exchanging goods when traveling through their society's territory (Wiessner 1982).\n\n Binford (2001); Gamble (1998); Hamilton et al. (2007).\n\n Cane (2013).\n\n Jones (1996).\n\n Pounder (1983).\n\n Mulvaney (1976); Roth & Etheridge (1897). Words traveled, too: In advance of Europeans exploring the interior of Australia, Aborigines speaking many languages had heard about domestic animals and had already adopted words such as \"yarraman\" for horse and \"jumbuk\" for sheep (Reynolds 1981).\n\n Fair (2001); Lourandos (1997); Walker et al. (2011).\n\n Kendon (1988); Silver & Miller (1997).\n\n Newell et al. (1990).\n\n An interaction sphere could continue step by step over long distances (Caldwell 1964).\n\n Perhaps these differences ameliorated competition during years of resource shortages (Milton 1991).\n\n Blainey (1976), 207.\n\n Haaland (1969).\n\n Franklin (1779), 53.\n\n Gelo (2012).\n\n Orton et al. (2013).\n\n Bahuchet (2014), 12.\n\n Boyd & Richerson (2005); Richerson & Boyd (1998); Henrich & Boyd (1998).\n\n Leechman (1956), 83. See van der Dennen (2014).\n\n Vasquez (2012).\n\n Turner (1981); Wildschut et al. (2003).\n\n Homer-Dixon (1994); LeVine & Campbell (1972).\n\n Pinker (2011); Fry (2013).\n\nChapter 19\n\n Durkheim (1982 [1895]), 90.\n\n Best known are examples among the primates, e.g., Malik et al. (1985); Prud'Homme (1991); Van Horn et al. (2007).\n\n Serengeti authority Craig Packer filled me in on the subject: \"Lions definitely restrict their cooperative behavior to individuals they know and recognize. When prides become too large, they don't all seem to know each other that well anymore and thus split apart.\"\n\n Confusingly, the breakups of societies have been referred to as \"fissions.\" Because fission connotes the functionally very different workaday fission events in fission-fusion societies, in which groups routinely separate but then freely come back together, another term is in order, which is why I use division, albeit Sueur et al. (2011) offers another option, irreversible fission.\n\n Joseph Feldblum, pers. comm.; Feldblum et al. (2018).\n\n Williams et al. (2008); Wrangham & Peterson (1996).\n\n See for example the references in Van Horn et al. (2007).\n\n See Sueur et al. (2011).\n\n Takeshi Furuichi, pers. comm.; Furuichi (1987); Kano (1992).\n\n e.g., Henzi et al. (2000); Ron (1996); Van Horn et al. (2007).\n\n In the honeybee, the younger workers swarm off with the original queen to start a new nest, leaving the old ones behind to await the birth of her successor, who takes over the original hive; there's no dispute about who goes where. Sometimes the bees ally themselves to more than one new queen and the hive breaks into several parts. I appreciate the advice of Rapha\u00ebl Boulay, Adam Cronin, Christian Peeters, and Mark Winston. Cronin et al. (2013); Winston & Otis (1978).\n\n Jacob Negrey, pers. comm.; Mitani & Amsler (2003).\n\n Stan Braude, pers. comm.; O'Riain et al. (1996).\n\n Sugiyama (1999). Bonobo males are also known to leave their community, but are thought to join neighboring groups, a behavior believed impossible for aggressive chimps (Furuichi 2011).\n\n Brewer & Caporael (2006).\n\n Dunbar (2011) implies this form of societal founding in humans by claiming that early societies tended to have 150 members that represented the living descendants of one couple five generations in the past, but there is no evidence that this ant-like (or termite-like) founding was common.\n\n Peasley (2010).\n\n Confusingly \"budding\" is also used (e.g. for Argentine ants, see Chapter 5), when a society's own members move into unoccupied areas, rather than to form a distinct and separate society.\n\n McCreery (2000); Sharpe (2005).\n\n Generally called \"coalescent societies,\" see Chapter 22 (Kowalewski 2006; Price 1996).\n\n For humans, e.g., Cohen (1978).\n\n Fletcher (1995); Johnson (1982); Lee (1979); Woodburn (1982). This is also true for tribal villages and settled hunter-gatherers (Abruzzi 1980; Carneiro 1987).\n\n Marlowe (2005).\n\n A hint of such rebelliousness is seen these days in corporate splits, when employees forced into new relationships by their superiors continue to value their former identities and endeavor not to lose them (Terry et al. 2001).\n\n e.g., Hayden (1987).\n\nChapter 20\n\n The Aborigines retained this view at least until the 1950s (Meggitt 1962, 33).\n\n Barth (1969).\n\n Alcorta & Sosis (2005), 328.\n\n Diamond (2005). Rather than starving, they may have simply moved elsewhere, some experts believe (Kintisch 2016; McAnany & Yoffee 2010).\n\n Karen Kramer, pers. comm.; Kramer & Greaves (2016). For another example, consider Chapter 6 on the Pathans in Barth (1969).\n\n There turns out to be no particular location for \"standard American English,\" which is better interpreted as an absence of extreme speech patterns than a specific accent (Gordon 2001).\n\n Proposed for language changes by Deutscher (2010).\n\n Menand (2006), 76.\n\n Thaler & Sunstein (2009). Leaders, in other words, tend to be \"prototypical\" (Hais et al. 1997).\n\n Cipriani (1966), 76.\n\n Bird & Bird (2000).\n\n Pagel (2000); Pagel & Mace (2004).\n\n e.g., Newell (1990).\n\n Langergraber et al. (2014).\n\n Boyd & Richerson (2005). Borderland residents also had to put their identities on conspicuous display to accentuate their dissimilarities with outsiders (Bettinger et al. 2015; Conkey 1982, 116; Giles et al. 1977, Chapter 1). While contact with outsiders can cause people to flaunt their identities, and sometimes form alliances with other societies for their protection, it isn't true that \"tribes make states and states make tribes,\" proposed by Whitehead (1992), who thought that distinct tribes came into being only after colonialism forced locals to build their group identities for their own self-protection.\n\n This boundary regulation brings to mind the situation described for chimpanzees. Those apes show this sort of adjustment to neighbors with their pant-hoots, with the calls appearing most dissimilar for communities that are immediate neighbors and consequently that have to take the most care to avoid any befuddlement about who's who. Even then, there's no evidence the pant-hoots show regional variation within the territory depending on which neighbor the apes confront\u2014that's not likely because male chimps in particular tend to move around their territory rather than stay within part of it most of the time, as human bands do (Crockford et al. 2004).\n\n cf. Read (2011).\n\n Poole (1999), 16.\n\n Packer (2008).\n\n This is suggested by the fact that trust is higher in smaller groups, which may tolerate greater deviance among people who are known well (Jolanda Jetten, pers. comm.; La Macchia et al. 2016), whereas lesser-known members of a society may be trusted less (Hornsey et al. 2007).\n\n Psychologists call this pluralistic ignorance (Miller & McFarland 1987). An example was the assumption by American whites in the 1960s that other whites supported segregation, which paradoxically led to bigoted practices few thought right (O'Gorman 1975).\n\n Forsyth (2009).\n\n The Comanche subgroups could be described as being well on their way toward acting as separate societies (Daniel Gelo, pers. comm.; Gelo 2012, 87).\n\n With no prior experience even a Chihuahua will pick out a 100 kilogram mastiff as a member of its own kind from a photograph alone (Autier-D\u00e9rian et al. 2013).\n\n Dollard (1937), 366.\n\n Note, however, that if our ancestors distinguished their societies by vocalizations prior to the existence of a vocabulary, a scenario put forward in Chapter 11, there may never have been a time when all humans spoke one language.\n\n e.g., Birdsell (1973).\n\n Dixon (1972).\n\n Cooley (1902), 270.\n\n Presuming that the Ypety Ache people started to eat human flesh before that split.\n\n Birdsell (1957).\n\n Kim Hill, pers. comm.; Hill & Hurtado (1996).\n\n Quoted in Lind (2006), 53.\n\n Sani (2009).\n\n Shown experimentally, e.g., by Bernstein et al. (2007).\n\n e.g., Hornsey & Hogg (2000).\n\n Erikson (1985).\n\n Pagel (2009); Marks & Staski (1988).\n\n Abruzzi (1982); Boyd & Richerson (2005).\n\n Darwin (1859), 490.\n\nChapter 21\n\n Atkinson et al. (2008); Dixon (1997).\n\n Billig (1995); Butz (2009).\n\n The idea originated with Tajfel & Turner (1979); e.g., see Van Vugt & Hart (2004).\n\n Connerton (2010); van der Dennen (1987).\n\n Goodall (2010), 128\u2013129.\n\n Russell (1993), 111.\n\n Goodall (2010), 210.\n\n See also Roscoe (2007).\n\n Prud'Homme (1991).\n\n Gross (2000).\n\n Gonsalkorale & Williams (2007); Spoor & Williams (2007).\n\n As is true of oppressed races in modern nations (e.g., Crocker et al. 1994; Jetten et al. 2001).\n\n Boyd & Richerson (2005); studied for small groups by Hart & van Vugt (2006).\n\n For primates, see Dittus (1988); Widdig et al. (2006). For hunter-gatherers, see Walker (2014); Walker & Hill (2014).\n\n e.g., Chagnon (1979).\n\n That's true even when the group in question is far more trivial than a society: kids joining a new clique of playmates, for example, overwhelmingly find new friends within that clique even when the members of the clique were chosen at random by the researchers (Sherif et al. 1961). As Muzafer Sherif (1966, 75), pioneer in studying such trivial competing groups, put it, \"Freedom to choose friends on the basis of personal preferences turns out to be freedom to choose among persons selected according to the rules of membership of the organization\"\u2014with the membership of interest in this book being that of the society itself.\n\n Taylor (2005).\n\n Binford (2001); RL Kelly (2013a); Lee & Devore (1968).\n\n This would lead to a band society breaking down most often beyond that size: Birdsell (1968) gave the typical number for division as a thousand.\n\n Wobst (1974); Denham (2013).\n\n How these dolphin communities form in the first place is a mystery (Randall Wells, pers. comm.; Sellas et al. 2005).\n\n Eventually the amoebas run out of steam for dividing at all, and then languish in the same culture dish indefinitely (Bell 1988; Danielli & Muggleton 1959).\n\n Birdsell (1958).\n\n Hartley (1953), 1.\n\n For estimates of numbers of languages in the past, see Pagel (2000).\n\nChapter 22\n\n Kennett & Winterhalder (2006); Zeder et al. (2006).\n\n There is a slight increase in population size for some species, but hardly what might be expected from the resource opportunities in cities (Colin Chapman, Jim Moore, and Sindhu Radhakrishna, pers. comm.; Kumar et al. 2013; Seth & Seth 1983).\n\n Bandy & Fox (2010).\n\n Wilshusen & Potter (2010). That for example most Yanomami would experience divisions during their lifetimes can be deduced from the graphs in Hunley et al. (2008) and Ward (1972).\n\n Olsen (1987).\n\n Flannery & Marcus (2012).\n\n The Enga tribes are formally called phratries. The clans of a tribe intermarry and tend to be on good terms unless a clan outgrows its gardens. At that point fights can turn nasty: the Enga have never been magnanimous about diplomacy (Meggitt 1977; Wiessner & Tumu 1998).\n\n Scott (2009).\n\n I thank Luke Glowacki for advice on the Nyangatom. The generations to which the males belong are determined in a curious manner (Glowacki & von Rueden 2015).\n\n Chagnon (2013).\n\n \"Tribe\" is a word with a confusing history. I employ it here because the word has been used by some others to describe a group of villages sharing a language and culture, and because no other word is in general use for such a society (Stephen Sanderson, pers. comm.; Sanderson 1999). Hutterites behave much like a village society in North America, where they belong to three sects that still see themselves \"of a kind,\" though each finds the other sects misguided (Simon Evans, pers. comm.).\n\n Smouse et al. (1981); Hames (1983).\n\n Many tribes carry out swidden, or slash-and-burn agriculture. A village clears a plot of land to garden, then travels to clear another forest patch when harvests decline.\n\n Harner (1972). The J\u00edvaro also tried to cajole other tribes to participate in their assault on the Spaniards, but these others contributed little (Redmond 1994; Stirling 1938).\n\n With their many spats, villagers seldom stuck together long enough to invent different customs. The pair of villages resulting from a split would be pretty much indistinguishable in mode of life, as was true when people drifted away from a hunter-gatherer band\u2014the split was more like people changing neighborhoods than a shift in their social roots (albeit slight differences in word usage can arise between villages after they split: Aikhenvald, pers. comm.; Aikhenvald et al. 2008).\n\n Kopenawa & Albert (2013).\n\n e.g., Southwick et al. (1974).\n\n e.g., Jaffe & Isbell (2010).\n\n Exceptions include fusions of army ant colonies after one looses a queen (Kronauer et al. 2010) and acacia ant colonies after fights (Rudolph & McEntee 2016). In termites, colony mergers have been demonstrated between colonies of some \"primitive\" (basal) species in which workers transform into reproductives following the death of the original queen and king (e.g., Howard et al. 2013). Claims of mergers for more \"advanced\" termites (Termitidae) are difficult to evaluate and as far as is known occur rarely if ever between mature colonies in nature (Barbara Thorne, pers. comm.).\n\n Moss et al. (2011).\n\n Ethridge & Hudson (2008).\n\n Gunnar Haaland, pers. comm.; Haaland (1969).\n\n Brewer (1999 and 2000).\n\n The same is true for other such alliances, including those between some chiefdoms in North America and the league of states formed in China during the sixth century (Schwartz 1985).\n\n Robert Carneiro originally took this point of view but later backed off it to allow for the fusion of some groups into chiefdoms (Carneiro 1998). I believe such \"mergers\" should be interpreted in terms of formerly sovereign groups (e.g., independent villages), often of the _same_ society, uniting under a political umbrella to accomplish a task; however, to fully incorporate those villages into a single entity would have required power plays by a chief that amounted to a kind of subjugation.\n\n Bowles (2012).\n\n Bintliff (1999).\n\n Barth (1969).\n\n There have been arguments about how much this would have been so; for example, some claim captives taken by the Iroquois were fully assimilated over time, while others view this as impossible (Donald 1997). I suspect full acceptance would be a more accurate description than full assimilation, in that differences would still be evident.\n\n Chagnon (1977), 155.\n\n Jones (2007) claims that this infant theft is an antecedent to slavery in humans, which I doubt\u2014for sure the little monkey isn't forced into child labor.\n\n Boesch et al. (2008).\n\n Anderson (1999).\n\n Biocca (1996), xxiv.\n\n Brooks (2002).\n\n Escapees were typically caught before they got far\u2014often by tribe next door (Donald 1997).\n\n Patterson (1982).\n\n Cameron (2008).\n\n Clark & Edmonds (1979).\n\n Mitchell (1984), 46.\n\n Some commoners sold themselves into slavery during hard times, especially when the slaves of elites led better lives than the poorest free people (Garnsey 1996).\n\n Perdue (1979), 17.\n\n Marean (2016).\n\n e.g., Ferguson (1984).\n\n See Chapter 15 and Abelson et al. (1998).\n\n Adam Jones, pers. comm.; Jones (2012).\n\n Confino (2014).\n\n Haber et al. (2017).\n\n Stoneking (2003).\n\n Grabo & van Vugt (2016); Turchin et al. (2013).\n\n Carneiro (1998 & 2000). See Chapter 22, note 25.\n\n Oldmeadow & Fiske (2007). As will come up later, this perceived legitimacy of statuses applies to the relationships between ethnic groups and races as well.\n\n e.g., Anderson (1994).\n\nChapter 23\n\n Liverani (2006). Mesopotamian sites of the Ubaid period (5500\u20134000 BC) show a more rudimentary state organization. For a general discussion of the emergence of early states, see Scarre (2013) and Scott (2017).\n\n Spencer (2010).\n\n e.g., Alcock et al. (2001); Parker (2003).\n\n Tainter (1988).\n\n Bettencourt & West (2010); Ortman et al. (2014).\n\n Richerson & Boyd (1998); Turchin (2015).\n\n Wright (2004), 50\u201351.\n\n Birdsell (1968) called this \"density of communication.\"\n\n Freedman (1975).\n\n Hingley (2005). Some peripheral areas may have been little inclined to Romanization.\n\n Spirituality provided a \"moral master plan\" that made leaders unnecessary (Hiatt 2015, 62).\n\n Atran & Henrich (2010); Henrich et al. (2010a).\n\n DeFries (2014).\n\n Tilly (1975), 42.\n\n I thank Eric Cline for suggesting Minoa as an example of a relatively pacifist ancient society.\n\n RL Kelly (2013b), 156.\n\n Mann (1986).\n\n Such large chiefdoms could approach the level of organization of a state; in fact some experts claim certain ones were states (e.g., Hommon 2013).\n\n Carneiro (1970 & 2012) expertly dispatches other theories about the rise of civilizations. I admit to having simplified and adjusted his views as I see fit; for example I agree that issues of social status could also have played into state formation (Chacon et al. 2015; Fukuyama 2011).\n\n Brookfield & Brown (1963).\n\n Lowen (1919), 175.\n\n de la Vega (1966, written 1609), 108.\n\n Faulseit (2016).\n\n Diamond (2005).\n\n Currie et al. (2010); Tainter (1988).\n\n e.g., Joyce et al. (2001).\n\n Marcus (1989).\n\n Chase-Dunn et al. (2010); Gavrilets et al. (2014); Walter et al. (2006).\n\n Johnson & Earle (2000).\n\n Beaune (1991); Gat & Yakobson (2013); Hale (2004); Reynolds (1997); Weber (1976).\n\n Kennedy (1987).\n\n Frankopan (2015).\n\n Yoffee (1995).\n\n More support for her hypothesis comes to light every year (Roosevelt 2013).\n\n Some of these wars have undertaken to seize control of the whole society rather than fragment it. Holsti (1991); Wallensteen (2012); Wimmer & Min (2006).\n\n Kaufman (2001).\n\n Bookman (1994).\n\n Southerners did often come from different parts of Great Britain than northerners, giving this distinction a crude basis in reality (Fischer 1989; Watson 2008).\n\n Allen Buchanan, Paul Escott, and Libra Hilde, pers. comm.; Escott (2010); McCurry (2010); Weitz (2008).\n\n Carter & Goemans (2011).\n\n This lack of commitment can hold true for many kinds of groups (Karau & Williams 1993).\n\n Kaiser (1994); Sekulic et al. (1994).\n\n Joyce Marcus, pers. comm.; Feinman & Marcus (1998). Chiefdoms and early states were more ephemeral, by some measures with the former lasting at most 75 to 100 years (Hally 1996).\n\n Cowgill (1988), 253\u2013254.\n\n Claessen & Skaln\u00edk (1978).\n\nChapter 24\n\n Alcock et al. (2001).\n\n Isaac (2004), 8. For many insights into the transformations of ethnic groups in societies, I can recommend the classic Van den Berghe (1981).\n\n Malpass (2009), 27\u201328. Thanks to Michael Malpass for advice on the Inca.\n\n Noy (2000).\n\n My arguments are similar to Cowgill (1988), except I prefer the word \"domination,\" where he uses \"subjugation,\" given that both dominated and incorporated groups would have been subjugated initially.\n\n Yonezawa (2005).\n\n Brindley (2015).\n\n Francis Allard, pers. comm.; Allard (2006); Brindley (2015).\n\n Hudson (1999).\n\n Their great wall similarly kept out (and set apart) the Chinese from the \"primitive\" steppe nomad foreign societies at the periphery (Fiskesj\u00f6 1999).\n\n Cavafy (1976).\n\n Spickard (2005), 2. Assimilation, and the related word \"acculturation,\" have been employed in varied nuanced ways by anthropologists and sociologists, but I will use only the former term here.\n\n Smith (1986). This \"group dominance perspective\" is well supported (Sidanius et al. 1997).\n\n The major exceptions to the subjugated undergoing the preponderance of change were among the pastoral nomads, who were vastly outnumbered by the culturally more intricate societies they took over. Genghis Khan and successors drew freely from the civilizations they conquered. They permitted a wide latitude in behavior among their own people and the peoples they controlled while for the most part clinging to their nomadic traditions (nicely described by Chua 2007).\n\n e.g., Chapter 8 of Santos-Granero (2009).\n\n Hornsey & Hogg (2000); Hewstone & Brown (1986).\n\n Aly (2014). Use of yellow badges to make Jews stand out was discussed in Chapter 12.\n\n Mummendey & Wenzel (1999).\n\n The capital had by far the most impact (Mattingly 2014).\n\n In pronouncedly multiethnic societies there can be a struggle over what factors into the superordinate identity (Packer 2008; Schaller & Neuberg 2012).\n\n e.g., Vecoli (1978).\n\n Joniak-L\u00fcthi (2015).\n\n States that are more manifestly polyethnic, like the United States, aren't considered nations on this basis. I have preferred the colloquial usage of \"nation\" in this book (Connor 1978).\n\n Sidanius et al. (1997).\n\n Seneca (1970), written first century AD.\n\n Klinkner & Smith (1999), 7.\n\n Devos & Ma (2008).\n\n Huynh et al. (2011), 133.\n\n Gordon (1964), 5.\n\n Deschamps (1982).\n\n Yogeeswaran & Dasgupta (2010).\n\n Jost & Banaji (1994); Kamans et al. (2009).\n\n Sidanius & Petrocik (2001).\n\n Cheryan & Monin (2005); Wu (2002).\n\n Ho et al. (2011).\n\n Devos & Banaji (2005).\n\n Marshall (1950), 8.\n\n Deschamps & Brown (1983).\n\n Ehardt & Bernstein (1986); Samuels et al. (1987).\n\n Lee & Fiske (2006); Portes & Rumbaut (2014).\n\n Bodnar (1985).\n\n Jost & Banaji (1994); Lerner & Miller (1978).\n\n Fiske et al. (2007), 82. See also Major & Schmader (2001); Oldmeadow & Fiske (2007).\n\n Jost et al. (2003), 13.\n\n Paranjpe (1998).\n\n Hewlett (1991), 29.\n\n Mo\u00efse (2014).\n\n An infant captured by the Comanche could be treated as Comanche right away (Rivaya-Mart\u00ednez, pers. comm.; Rivaya-Mart\u00ednez 2012).\n\n Cheung et al. (2011).\n\n Cameron (2008); Raijman et al. (2008).\n\n That was true even before 212 AD (Garnsey 1996).\n\n Engerman (2007); Fogel & Engerman (1974).\n\n Lim et al. (2007).\n\n At the same time interactions between groups can also stimulate people to find new ways to distinguish themselves (Hogg 2006; Salamone & Swanson 1979).\n\n For the Romans, see Insoll (2007), 11. The Greeks themselves were of several ethnicities (Jonathan Hall, pers. comm.; Hall 1997).\n\n Smith (2010).\n\n Noy (2000).\n\n Greenshields (1980).\n\n Portes & Rumbaut (2014).\n\n At first the Indians often needed official permission to travel, with the US government going so far as to crack down on church attendance outside the reserve (Richmond Clow, pers. comm.).\n\n Schelling (1978).\n\n Christ et al. (2014); Pettigrew (2009).\n\n Paxton & Mughan (2006).\n\n Thompson (1983).\n\n Hawley (1944), Berry (2001).\n\n Park (1928), 893.\n\nChapter 25\n\n First-generation American immigrants don't necessarily score high in measures of patriotism, but that typically changes for their offspring (Citrin et al. 2001).\n\n Quoted in Beard (2009), 11.\n\n Arguably this might be more than analogy, in that many norms govern issues that we see today as matters of health, such as what to eat and how to prepare it, and outsiders failing to comply with the local ways could in fact spread sickness (Fabrega 1997; Schaller & Neuberg 2012).\n\n Dixon (1997).\n\n Gaertner & Dovidio (2000).\n\n This brings us back to the connection between specialization and social cohesion that was developed by Durkheim (1893), brought up in Chapter 10, including endnote 39.\n\n This again is positive distinctiveness (Chapter 21). Evidence for such roles is scarce because the information was seldom recorded for early states, for example Roman graves indicated the deceased's ethnicity but not job or vice versa (David Noy, pers. comm.).\n\n Esses et al. (2001). While competing with the majority is a road to failure, conflict between minorities is also costly (Banton 1983; Olzak 1992). Majority peoples have often gained by promoting rivalries that kept minorities at loggerheads rather than dissatisfied with the people in power. Of course this works between societies, too. The Romans were masters of divide and rule, splitting troublesome Macedonia into four provinces and provoking them to fight.\n\n Noy (2000). The status of black Americans has improved when they are needed as soldiers in wartime (Smith & Klinkner 1999).\n\n Boyd (2002).\n\n Abruzzi (1982).\n\n Turnbull (1965); Zvelebil & Lillie (2000).\n\n See Cameron (2016) for this and other examples.\n\n Appave (2009).\n\n Sorabji (2005).\n\n Suetonius (1979, written AD 121), 21.\n\n McNeill (1986).\n\n Dinnerstein & Reimers (2009), 2.\n\n Light & Gold (2000).\n\n Bauder (2008); Potts (1990).\n\n Meanwhile, colonialism, and the founding of nations that ensued it, led many people to lose their original tribal identities in favor of broad ethnic categories in their native regions as well (e.g., the Ewe, Shona, Igbo, and other ethnic groups in Africa: Iliffe 2007).\n\n e.g., Gossett (1963, Chapter 1).\n\n PC Smith (2009), 4, 5.\n\n Brindley (2010).\n\n Dio (2008, written second century AD), 281.\n\n Sarna (1978).\n\n Curti (1946).\n\n Crev\u00e8coeur (1782), 93.\n\n Matthew Frye Jacobson, pers. comm.; Alba (1985); Painter (2010).\n\n Alba & Nee (2003); Saperstein & Penner (2012).\n\n Leyens et al. (2007).\n\n Freeman et al. (2011).\n\n Smith (1997).\n\n Smith (1986).\n\n Bloemraad et al. (2008).\n\n Ellis (1997).\n\n Levinson (1988); Orgad (2011); Poole (1999).\n\n Harles (1993).\n\n Gans (2007); Huddy & Khatib (2007). To top that off, given the convenience of travel, communications, and trade nowadays, immigrants are seldom severed from their ancestral homeland and its traditions, though those connections are likely to fade in their children (Levitt & Waters 2002).\n\n Bloemraad (2000); Kymlicka (1995).\n\n Ironically, Lucan was born in what is modern-day Spain. Quoted in Noy (2000, 34), who discusses racism throughout the history of the Roman Empire.\n\n Michener (2012); Volpp (2001).\n\n van der Dennen (1991).\n\n Jacobson (1999).\n\n Alesina & La Ferrara (2005), 31\u201332.\n\n May (2001, 235) makes this point for the tribes of Papua New Guinea: \"The welfare of village people today depends in part on their ability to capture a share of the goods and services flowing from the state; chiefly or bigman leadership will only be effective to the extent that it ensures access to these benefits, and that implies its articulation with the state.\"\n\n Harlow & Dundes (2004); Sidanius et al. (1997).\n\n Bar-Tal & Staub (1997); Wolsko et al. (2006) _._\n\n Marilynn Brewer pointed out to me some overlap between my views with Shah et al. (2004).\n\n e.g., Van der Toorn et al. (2014).\n\n Barrett (2007); Feshbach (1991); Lewis et al. (2014); Piaget & Weil (1951).\n\n Patriotism and nationalism are loosely associated with liberal and conservative views, respectively, but differ especially in their extreme expressions. Ultra liberals, for example, can violently oppose any free speech going against their ideology, while fiscal conservatives can support free trade and positive group relations. Since nationalists can harbor patriotic feelings, my discussion of patriots applies to those high on patriotism but low on nationalism.\n\n Bar Tal & Staub (1997).\n\n Feinstein (2016); Staub (1997).\n\n See Schatz et al. (1999), who refer to nationalism as \"blind patriotism.\"\n\n Blank & Schmidt (2003); Devos & Banaji (2005); Leyens et al. (2003).\n\n Andrew Billings, pers. comm.; Billings et al. (2015); Roth\u00ec et al. (2005).\n\n De Figueiredo & Elkins (2003); Viki & Calitri (2008).\n\n e.g., Raijman et al. (2008).\n\n Greenwald et al. (2015).\n\n Smith et al. (2011), 371.\n\n Jandt et al. (2014); Modlmeier et al. (2012).\n\n Feshbach (1994).\n\n Hedges (2002); Junger (2016).\n\n Turchin (2015).\n\n In contrast patriots try to pull together varied groups by appealing to their common fate (Li & Brewer 2004).\n\n e.g., Banks (2016); Echebarria-Echabe & Fernandez-Guede (2006).\n\n Competitiveness between groups only worsens the matter (Esses et al. 2001; King et al. 2010).\n\n Bergh et al. (2016); Zick et al. (2008).\n\n Described by Sidanius et al. (1999).\n\nChapter 26\n\n Hayden & Villeneuve (2012), 130.\n\n Gaertner et al. (2006).\n\n The descendants of the mutineers, who might be looked upon as forming a kind of coalescent society, are now part of an overseas territory of the United Kingdom.\n\n For ants and humans, this separation could potentially continue for generations because of their employment of markers of identity (Chapters 5\u20137). Opinions about the length of Viking isolation vary (Graeme Davis, pers. comm.; Davis 2009).\n\n Weisler (1995). Isolation, and complete ignorance of outsiders, has been claimed of one tribe on mainland New Guinea (Tuzin 2001).\n\n Royce (1982), 12.\n\n Cialdini & Goldstein (2004).\n\n Nichole Simmons, pers. comm.\n\n Jones et al. (1984).\n\n Differences between the groups had begun to emerge before they were encouraged to compete (Sherif et al. 1961). There are questions about how much of the behavior of the children was manipulated by the researchers (Perry 2018).\n\n Carneiro (2004); Turchin & Gavrilets (2009).\n\n e.g., China (Knight 2008).\n\n Aikhenvald (2008), 47.\n\n Seto (2008).\n\n Jackson (1983).\n\n McCormick (2017); Reese & Lauenstein (2014).\n\n Goodwin (2016).\n\n Chollet (2011), 746, 751. See also Linder (2010); Rutherford et al. (2014).\n\n Leuchtenburg (2015), 634.\n\n Gellner (1983), 6. Gellner went on to say, \"Having a nation is not an inherent attribute of humanity, but... has now come to appear as such\" (ibid.). See also Miller (1995).\n\n Many new immigrants face considerable stress in adjusting to their situation (Berry & Annis 1974).\n\n Lyons-Padilla & Gelfand (2015).\n\nConclusion\n\n Reynolds (1981).\n\n Druckman (2001).\n\n Gelfand et al. (2011).\n\n Blanton & Christie (2003); Jetten et al. (2002); Maghaddam (1998). There is little difference in people's overall sense of happiness (well-being) across different countries (Burns 2018).\n\n Deschamps (1982); Lorenzi-Cioldi (2006).\n\n Cosmides et al. (2003).\n\n Brewer (2009).\n\n Easterly (2001).\n\n Christ et al. (2014).\n\n Alesina & Ferrara (2005); Hong & Page (2004). Accepting those who seem strange will be the greatest challenge for socially dominating people (Asbrock et al. 2012).\n\nDummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content Dummy content\n\n# Index\n\nAborigines, Australian, , 103\u2013106, , , , , , , , , 209\u2013210, , 233\u2013236, , , ,\n\nsettled societies of, 122\u2013123,\n\nAche peoples, 107\u2013108, 110\u2013111, , , , ,\n\nacculturation,\n\nAfrican wild dogs. _See_ dogs, painted\n\nagricultural practices,\n\nof ants, , , , ,\n\napproached by band societies, , 136\u2013137\n\nas often little different from practices of settled hunter-gatherers,\n\navoided by most hunter-gatherers,\n\nband societies impacted by, 98\u201399, 137\u2013138\n\nbrain size since advent of,\n\ndomestication and,\n\nnegative impacts with development of, 137\u2013138\n\npower structures with development of, 123\u2013124\n\nstate societies relation to rise of, , , , ,\n\nsupporting huge populations, ,\n\nalliances, between societies 227\u2013238,\n\nof band societies, , 229\u2013232,\n\nbenefits of, 228\u2013230, ,\n\nas a factor in accepting ethnic groups, ,\n\nof humans, 227\u2013238,\n\nof Iroquois Indians, , , , , ,\n\nto get a job done rare in animals, , 228\u2013229,\n\nof Pygmies with farmers, , 321\u2013322\n\nsociety mergers relation to, 282\u2013283,\n\nwith trade and commerce, 231\u2013232\n\nof the Tukanoans,\n\nAmerican Creed,\n\nAmerican Indians, , , , , , , , , , , , , , 235\u2013236, , , , , , , 322\u2013323, , , , , , ,\n\n_See also_ Calusa, Chumash, Iroquois, Pacific Northwest tribes\n\nanimal societies, terms for,\n\nAndaman Islanders, , , , , ,\n\nanonymous societies\n\narchaeological evidence of early, 141\u2013143,\n\nof ants and other social insects, , 69\u201371, 75\u201378\n\nof band societies, , , 267\u2013268\n\nrole of cultural richness in, 157\u2013158\n\nevolution of, , 141\u2013142, 146\u2013147, , 157\u2013157, , 355\u2013357\n\nof humans, 79\u201393\n\nindividual recognition societies compared with, 69\u201370, , , 92\u201393, , 249\u2013250\n\nof nonhuman vertebrates, 71\u201375\n\nsocial relationships in human, 79\u201380\n\nsocietal markers in, 69\u201375, , 79\u201393, 356\u2013357\n\nsociety size and density, , ,\n\nants\n\nabsence of societal factions,\n\nagricultural practices of, , , ,\n\nanonymous societies of, 69\u201371, 75\u201378,\n\nbreeding and society lifecycle, 75\u201377, , , ,\n\nconflict among, 67\u201368, , 223\u2013224\n\ncollective decision-making of,\n\ndivision of labor for, , 62\u201364, ,\n\nflexibility and adaptability of, 70\u201371\n\nhousing and infrastructure of, , 60\u201361\n\nhumans contrasted\/compared with, , , 57\u201358, 60\u201365, , , , , ,\n\nindividual recognition absent in, 68\u201369,\n\nresource provision for, 61\u201362,\n\nin Sater\u00e9-Maw\u00e9 rituals,\n\nas sisterhoods,\n\nslavemaking, 70\u201371, ,\n\nsocietal markers for, 66\u201367, 69\u201371, , 344\u2013345,\n\nsocieties as superorganisms,\n\nsociety size for, 57\u201358, 60\u201365, , ,\n\nspiders insertion into colony of,\n\n_See also_ Argentine ants; army ants; harvester ants; leafcutting ants; trap-jaw ants\n\napartheid,\n\nArawak tribe, 23\u201324, , 229\u2013230\n\narchaeological evidence for societies, 141\u2013147, 153\u2013156\n\nArgentine ants, , 64\u201371, 75\u201378, , , , , ,\n\nsocietal markers for, 66\u201367, 69\u201370, , , , 344\u2013345\n\nspecies other than with supercolonies,\n\narmy ants, , , ,\n\nAsh conformity,\n\nassimilation, , 314\u2013318, 322\u2013323, , , , , ,\n\nas never carried to completion, , , ,\n\nbaboons, savanna, , , 39\u201340, , , , , , 206\u2013207, , , , 320\u2013321, , ,\n\nband (a society of horses). _See_ horses\n\nband (an encampment of hunter-gatherers), 97\u201398, 100\u2013103, , , 112\u2013114, , , , , , , , , , ,\n\nbeing part of a larger society,103\u2013104\n\nchimp or bonobo parties compared to,\n\nevolutionary origins of,\n\nsize and dynamics of, , , 120\u2013121, , ,\n\nsocial organization of, 112\u2013121\n\nvillages of some tribes comparable to, 279\u2013280,\n\nband societies, hunter-gatherer\n\nadaptability of, ,\n\nas an ancient condition for human ancestors, ,\n\nas anonymous societies,\n\nas affluent,\n\nagricultural practices impacts on, 98\u201399, 137\u2013138\n\nalliances among, , 105\u2013107, 229\u2013232, , , , ,\n\nalternative names for,\n\narchaeological evidence of, 143\u2013146\n\nboundaries and delineations of, 103\u2013106\n\ncommunication across,\n\nconflict and violence between, 130\u2013131, 220\u2013222\n\ncosmologies of,\n\ndecision-making in, 119\u2013120\n\ndefinition of, 97\u201398,\n\ndivision of labor in, , 112\u2013117,\n\negalitarianism in, 117\u2013119,\n\nethnicity and race for, 106\u2013108,\n\nEuropean contact with and impact on, 99\u2013100, ,\n\nfamilies\/kinship in, , 209\u2013210\n\nfreedom in, , 115\u2013116\n\ngathering together of the bands of, 124\u2013125\n\nG\u00f6bekli Tepe, 143\u2013144\n\n\"imagined communities\" of,\n\nindividualism in, 115\u2013116,\n\njack-of-all-trades strategy of, 114\u2013117,\n\nlack of materialism in, ,\n\nleisure time for, , , ,\n\nminimal goods and infrastructure in,\n\nnovelties and fads adoption by, 254\u2013255\n\noral narrative function for, 180\u2013181\n\npeace within and between, 105\u2013106, 229\u2013237\n\npower and leadership in, 117\u2013119, , , , 130\u2013131, , 321\u2013322,\n\nprotection and security for, , 120\u2013121\n\nraids between, 221\u2013222,\n\nresolving conflicts in, 117\u2013119, , ,\n\nrituals in,\n\nsettled societies compared with,\n\nsettled times for, , 136\u2013137\n\nsharing in, , , , , , ,\n\nsize and structure of, 97\u201398, 100\u2013101, , , 266\u2013267\n\nslavery in,\n\nsocial differentiation into groups in, 114\u2013116,\n\nsocietal markers for, 107\u2013111, ,\n\nsociety benefits for, 120\u2013121\n\nsociety divisions with, 248\u2013250, 256\u2013257, , 265\u2013266,\n\nsociety membership\/citizenship for, 116\u2013117\n\nstrangers existing within,\n\nstate societies (nations) compared to, , , 104\u2013105, , 317\u2013318\n\nterritories of, 104\u2013105, , , , , 255\u2013256, , , ,\n\ntrade and commerce for, , 233\u2013236,\n\ntribes\/village societies compared to, , 279\u2013280\n\nview of the earth by, 99\u2013100\n\nwild food cultivation practices of, 125\u2013126\n\n_See also_ Aborigines; Ache; Andaman Islanders; American Indians; Bushmen; Hadza; Inuit; Pygmies\n\nbarbarians, , , , ,\n\nbats, greater spear-nosed, ,\n\nbachelor males, when not part of a society, , , , , ,\n\nbias. _See_ prejudice and bias\n\nBig Man, , ,\n\n_See also_ leadership\n\nbirds, 25\u201327, , , 72\u201374\n\nblack sheep effect, , , , ,\n\nblind patriotism. _See_ nationalism\n\nbody adornment and marking, , , , 153\u2013155, ,\n\nbonobos (societies called communities), , , , , , , , , , , ,\n\ncooperation and conflict within communities of, , , 35\u201336,\n\ncooperation and conflict between communities of, , , ,\n\nfriendships between communities,\n\nhuman relation to\/comparison with, , 58\u201359, , , , , , , ,\n\nkinship patterns of, , , , 206\u2013207\n\npeacefulness of, , , , , , , , ,\n\nsocietal divisions for, 242\u2013246,\n\nsocietal factions with,\n\nstrangers, response to,\n\nterritoriality of,\n\ntolerance between societies,\n\nboundaries, social, , , , , 103\u2013106, , , , 258\u2013259, , , 315\u2013317, , , 348\u2013349, ,\n\n_See also_ territories\n\nboundary regulation, ,\n\nbrain size and social intelligence, , 19\u201321, 30\u201331, , , , 90\u201393, , , ,\n\n_See also_ cognitive abilities; social brain hypothesis\n\nbudding, , ,\n\nBushmen, , , 98\u201399, , , , , , , , , , , 128\u2013131, 145\u2013146, , , , , , 189\u2013190, 201\u2013202, , , , , , , , , , , ,\n\nchoice of name, versus San, ,\n\nCalusa, , ,\n\ncannibalism, 110\u2013111, ,\n\ncarrying capacity,\n\nchiefdoms\n\nagricultural practices and accelerated rise of,\n\nconflict and violence between, , ,\n\ndefinition of,\n\nethnicity and race in,\n\npower structures and leadership in, 289\u2013291,\n\nmergers between, 394\u2013395\n\nsettled hunter-gatherers as,\n\nstate societies relation to, 288\u2013292, 297\u2013298,\n\nas a turning point in societies acting as a unit, 288\u2013290\n\nchiefs. _See_ leadership\n\nchildren. _See_ infants and children; teenagers\n\nchimpanzees (chimps; societies called communities), , , , , , , , , , , , , , , , ,\n\nconflict and violence between communities of,, 216\u2013220, , , 262\u2013264, , ,\n\nconflict and violence within communities of, , 262\u2013264,\n\ncooperation among, , ,\n\ndechimpization, 263\u2013264,\n\ndominance relationships in, , , ,\n\nfamilial recognition for,\n\nfamily structure for, ,\n\nfertility of,\n\nfission-fusion in, 38\u201341, , ,\n\nforeigner treatment by, ,\n\nhumans relation to and comparison with, , , 58\u201360, , , , , 219\u2013220, , , ,\n\nhunter-gatherer bands compared to parties of,\n\nindividual recognition of, , , ,\n\nkinship patterns of, , , , 206\u2013207\n\nlimits of social networks, 19\u201321\n\npant-hoot communication significance for, , , , 148\u2013150, ,\n\npossibility of social markers in,\n\nsensory limitations impacts for,\n\nsocial learning, , 148\u2013149, , ,\n\nsocietal divisions for, 243\u2013246, , 262\u2013264,\n\nsocietal markers lacking in, 148\u2013149\n\nsociety transfers for, , ,\n\nsociety mergers for, 281\u2013282\n\nsociety size and stability of, 20\u201321, ,\n\nterritories of, , , ,\n\nChinese dynasties, 311\u2013313, , , , 334\u2013335,\n\nChumash, , . ,\n\ncichlids, shell-dwelling, 29\u201330,\n\ncircumscription, 297\u2013298,\n\ncitizenship. _See_ society membership\/citizenship\n\nCivil War, US, , , , , ,\n\ncivilizations, definition of 5, ,\n\nclass struggle,\n\nclothing, 127\u2013128, 154\u2013155\n\ncoalescent societies, , , ,\n\ncognitive abilities\/intelligence\n\nfission-fusion and,\n\nhaving a home base and, 101\u2013102\n\nof prehistoric humans,\n\nsocietal markers relation to, 90\u201392\n\nof species without societies,\n\n_See also_ brain size\n\ncollapse, of societies, ,\n\ncollective memory\/intelligence, , 156\u2013157, ,\n\ncollective personalities ,\n\ncollectivism, ,\n\ncolonialism, , ,\n\nComanches, , , , 322\u2013323, , , , ,\n\ncommon bond and common identity groups,\n\ncomparisons, making, 60\u201361\n\ncompetence, in psychology, , , , , , ,\n\ncompetition, 32\u201333, , , 226\u2013227, 269\u2013270\n\ncomplex hunter-gatherers. _See_ hunter-gatherers, settled societies\n\nconflict and violence\n\nband societies toward outsiders, 130\u2013131, 220\u2013222\n\nbetween ant colonies, 67\u201368, , 223\u2013224\n\nwithin ant colonies, ,\n\nof chimps, , , 216\u2013220, , , 262\u2013264, , , ,\n\nbetween societies of other vertebrates, , 224\u2013226\n\ncivil, historical predominance of, 301\u2013302,\n\ncompetition role in, , 226\u2013227\n\nwith ingroup\/outgroup dynamics, , 200\u2013202, 220\u2013221, 223\u2013224, ,\n\ninternational, ethnic treatment impacted by, 338\u2013339\n\nleadership for and during, with outsiders, 130\u2013131, ,\n\nleadership for and during, within societies, , ,\n\nwith nationalism, , 217\u2013218, 222\u2013223\n\npeace through,\n\npsychology of, 222\u2013223, , 337\u2013338,\n\nreactive aggression,\n\nresource provision and, ,\n\nRobbers Cave experiment on, , ,\n\nin Rwanda, ,\n\nsocietal markers relation to, , ,\n\nwith societal breakups, 243\u2013245, , , , 262\u2013266, 269\u2013270, 299\u2013302\n\nspatial considerations in societies with, , ,\n\nstate societies built from, , 297\u2013298\n\nterritories role in, , , 218\u2013219, 221\u2013222\n\nwithin and between chiefdoms, 288\u2013290, ,\n\nwithin and between tribal\/village societies, 279\u2013281\n\nof wolves, , ,\n\nconformity, 88\u201389, 178\u2013179, , , , ,\n\ncooperation\/collaboration between societies,\n\npossible to maintain societal boundaries with, ,\n\nto get a job done rare in animals, , 228\u2013229,\n\nnationalists and,\n\namong bonobos, ,\n\ncooperation\/collaboration within societies, , ,\n\nof ants and other social insects, , 61\u201362, , ,\n\namong chimps, ,\n\nin band societies, , , , , , ,\n\ncommunication for, 149\u2013150\n\nviewed as the defining feature of societies, , , , ,\n\ndifficulties as societies grow, , ,\n\nin families and raising young, 25\u201326, 31\u201332\n\nin foreigner exclusion,\n\nin humans, e.g., , 200\u2013201,\n\nleaders and,\n\nof meerkats, ,\n\nminimum necessary of, ,\n\nin the origin of societies,\n\nover a distance, ,\n\nof pinyon jays, ,\n\nfor protection and security, 26\u201327\n\nselectivity of in societies, , , ,\n\nas society benefit, , , , , 34\u201336, 228\u2013229\n\nsociety membership relation to, 27\u201328,\n\nof sperm whales, , ,\n\nof painted dogs, , ,\n\nwith respect to the size of a society, ,\n\n_See also_ alliances; friendships; kin selection\n\ncooperation\/collaboration without societies, 26\u201327,\n\ncooperative breeding, ,\n\nconquering societies, , 287\u2013290,\n\naggression by, 297\u2013298\n\ncontrol and assimilation of the conquered, , 311\u2013314,\n\ncyclic rise and fragmentation of, 299\u2013301\n\ndisease spread by, , , ,\n\nemergence of ethnicities and races in,\n\ncollective identity\/personality, 199\u2013200, , , , , ,\n\nself-identity relation to, , 195\u2013197, 357\u2013359\n\nin social insects,\n\nsocietal factions and,\n\nas a societal marker, 201\u2013202\n\ncopying others. _See_ mimicry\n\ncosmopolitanism,\n\ncrime, , , 337\u2013338, 343\u2013344, 352\u2013353\n\ncross-cutting ties,\n\ncults, gangs, and criminal organizations, , , 352\u2013353,\n\ncultural hybrids, across regions 301\n\ncultural ratcheting, 155\u2013158, ,\n\nculture, definition of, ,\n\nDarwin, Charles, , ,\n\ndefinition, utility of a,\n\ndehumanization, 184\u2013189,\n\nanimalistic,\n\nrole in atrocities and the guilt that follows,\n\ncategorization of others in,\n\nequivalence among chimps for, , 263\u2013264,\n\nemotions and, 186\u2013187,\n\nin bring outsiders into a society, 286\u2013287,\n\narising in society divisions, ,\n\nperceived degree of, and difference in, humanness , , , ,\n\n_See also_ disgust; Great Chain of Being; prejudice and bias; stereotypes\n\nDeVore, Irven, 108\u2013109\n\ndifferences. _See_ similarities\n\ndiscontinuity effect,\n\ndisease\n\nrole in conquests, , , ,\n\ninteractions with foreigners and, , , ,\n\ndisgust, , 188\u2013189, , , , , , 384\u2013385\n\ndiversity, , , , ,\n\nanonymous societies relation to, 157\u2013158,\n\nnationalist view of, ,\n\nperceived as part of one society, , ,\n\nof species,\n\nas society benefits, , , ,\n\nin state societies, , , , , 304\u2013305, , , , ,\n\n_See also_ ethnicity and race\n\ndivide and rule,\n\ndivision of labor\n\nfor animals, 62\u201364, ,\n\nbetween ethnicities and races, 329\u2013332\n\nwithin hunter-gatherer societies, , 112\u2013117, 132\u2013134\n\nin simplifying interactions between individuals, 132\u2013133,\n\njack-of-all-trades strategy in, 114\u2013117,\n\nbetween societies, 235\u2013237,\n\nrelationship to society population, 62\u201363, 133\u2013134, ,\n\nrole in social solidarity, , ,\n\n_See also_ optimal distinctiveness; social identity\n\ndivisions. _See_ societal factions and divisions\n\ndogs, domesticated,\n\ndogs, painted (African wild dogs, societies called packs) , , , , , , , ,\n\nDolezal, Rachel,\n\ndolphins, bottlenose, , 33\u201334, , , 120\u2013121, , , 268\u2013269, ,\n\ndomestication, as a requirement of large-scale agriculture,\n\ndominance\n\nwithin animal societies, , , , , , , , , , , ,\n\nbetween animal societies, , 47\u201348, 224\u2013225\n\nbetween human societies, 236\u2013237, ,\n\nas a factor in divisions, ,\n\namong ethnicities and races, , 320\u2013323\n\nrelation to leadership, , 128\u2013129, ,\n\nover slaves, ,\n\nin simplifying social life,\n\n_See also_ dominant people; conquering societies; power structures and leadership; reverse dominance hierarchy\n\ndominant people\/group\n\ncharacteristics of, , , ,\n\nfounder advantage,\n\ngreater freedom of,\n\nas primary members and beneficiaries of the society, 319\u2013320, , , 361\u2013362\n\nshifting membership,\n\nrelation to other ethnic groups, 315\u2013320, 323\u2013326, , , , , , ,\n\ntypical numerical dominance of,\n\n_See also_ majority\n\nDurkheim, \u00c9mile, , , , ,\n\neccentricity, 88\u201389, ,\n\necho principle, 156\u2013157\n\negalitarianism, 117\u2013119, , , 129\u2013131, , , , , ,\n\n_See also_ inequality\n\nelderly, role of, 120\u2013121\n\nelephants, African savanna , ,\n\nsocieties are called cores,\n\nbond groups, ,\n\ncompared with other elephant species, , 30\u201331\n\ndiscord within cores of,\n\ndivision of cores, ,\n\nfamilies in, ,\n\nfission-fusion, ,\n\nherd as an aggregation of cores,\n\nhuman societal markers recognized by,\n\nintermixing of cores into herds, , ,\n\nkinship within cores of, 204\u2013205\n\nleadership,\n\nmales not belonging to societies, ,\n\npeacefulness and antagonism between cores of, 224\u2013225\n\nsociety mergers for,\n\ntransfer between cores for, 43\u201344\n\nemblemic style, . _See_ markers, societal\n\nemotions\n\nin animals, ,\n\ncultural differences in, as a marker,\n\nperception of differences in expression between groups, 186\u2013189, 191\u2013192, ,\n\nresponse to symbols and other markers, , , , ,\n\nsecondary emotions, 186\u2013187,\n\n_See also_ group emotion\n\nempathy, 171\u2013172, 196\u2013197,\n\nempires,\n\nemployment, 330\u2013332\n\n_See also_ division of labor\n\nenemies, within societies , ,\n\nentitativity, 194\u2013196, , , , , ,\n\nessences, 166\u2013167, , , , ,\n\nethnic neighborhoods\/boroughs\/ghettos, 325\u2013326,\n\nethnicity and race\n\nabsence within societies of nonhuman animals, , ,\n\namalgamation into larger categories, 332\u2013335,\n\nassimilation of, 314\u2013318\n\nassimilation impacts on identity of, , , 332\u2013335\n\nin chiefdoms,\n\nChinese dynasties recognition\/treatment of, , , 311\u2013313, , , , 334\u2013335\n\ncivilizing,\n\ncompetition between,\n\nconflation of, 332\u2013335, 342\u2013343, ,\n\ndefinition of, as ethnic group,\n\ndifficulty perceiving as individuals, 176\u2013177, 193\u2013195, 199\u2013202, , 319\u2013320\n\ndifficulties in disregarding, 177\u2013178, ,\n\ndifficulties in reading emotions and truthfulness, 187\u2013188, ,\n\ndivision of labor with, 330\u2013332\n\ndominance and status relationships between, 189\u2013191, 318\u2013323, 359\u2013360\n\ndominant\/conquering ethnicity's treatment of, , , 310\u2013326, 332\u2013335\n\ndomination versus incorporation,\n\nempathy within, 171\u2013172, 196\u2013197,\n\nas having some of the same social primacy as societies,\n\nhunter-gatherer sense of, , 106\u2013108, ,\n\nidentity markers relation to, , 167\u2013170, 174\u2013177, , , , 315\u2013318, ,\n\nImplicit Association Test findings on, , , ,\n\nIncan Empire treatment of, 310\u2013312,\n\ninfants and young people's recognition of, 165\u2013166, , , ,\n\nintegration into dominant society, 323\u2013326\n\ninternational conflicts impact on treatment of, 338\u2013339\n\nlate occurrence in history, , 106\u2013108, , , ,\n\nmelting pot ideal with,\n\nnationalism\/patriotism relation to, 339\u2013343\n\nperceived as biological species, , , , , ,\n\nperceived as clear categories, 168\u2013170,\n\nperceived as dangerous, , ,\n\nperceived as expendable, ,\n\nas a product of the expansion of state societies, 6\u20137, , ,\n\npsychology associated with, 167\u2013202, , 222\u2013223, , , , , 337\u2013338,\n\nrequirement in maintain differences,\n\nresponses to crimes committed by minority, 337\u2013338, 343\u2013344\n\nretaining connections to ancestral homelands,\n\nRoman Empire treatment of, , , , 324\u2013325,\n\nsensitivity to differences between, , , ,\n\nas parts of one society, , ,\n\nsociety membership\/citizenship for, 318\u2013320, ,\n\nas \"societies within a society,\" ,\n\nstatus relationships among and within, 320\u2013323\n\nstereotypes on, , 178\u2013179, , 193\u2013194, , , 342\u2013343\n\nterritoriality and spatial issues with, , 323\u2013326, 303\u2013305, 323\u2013326, , ,\n\nethnicity and race ( _continued_ )\n\nuniversality within nations of multiple, ,\n\nvaried perception of equality\/allegiances, 178\u2013179, , 289\u2013290, 319\u2013323, 328\u2013329, , , 335\u2013343\n\n_See also_ immigration; minorities; races; racism; stereotypes\n\nethnocentrism. _See_ racism and ethnocentrism\n\nethnolinguistic groups,\n\nEuropean Union (EU), 350\u2013351\n\nfactions. _See_ societal factions and divisions\n\nfads. _See_ novelties and fads\n\nfamilies and kinship, , 203\u2013213, ,\n\nin band societies, , 209\u2013211\n\nbenefits for knowing, , 204\u2013205, ,\n\nbirds social structure as, 25\u201326, , ,\n\nof chimps and bonobos, , , , 206\u2013207\n\ncollaboration\/cooperation in, 25\u201326, 31\u201332\n\nfictive kin, 209\u2013212,\n\ninfants limited recognition and understanding of, 205\u2013209\n\nfor insect societies,\n\nkin selection, , 204\u2013205,\n\nlanguage and learning about, 207\u2013209, ,\n\navoidance of mating with kin, ,\n\nrecognizing, 206\u2013207,\n\nrelationship bonds outside of, ,\n\n\"relationship IQ,\" 207\u2013209,\n\nsocial support relationships often more important than,\n\nin societal factions and divisions, 265\u2013266\n\nsocietal markers relation to,\n\nsocieties as extended, , , 211\u2013212\n\nsocieties containing multiple lines of kin,\n\nin societies of other mammals, 74\u201375, 204\u2013205,\n\nsocieties relation to, , 203\u2013213\n\nsociety relationships originated from relations to,\n\ntreating non-relatives as family members,\n\ntreatment of death in hunter-gatherers, 209\u2013210\n\nunderstanding among society members of, 205\u2013212\n\nfarming. _See_ agricultural practices\n\nfemale-based societies,\n\nfeudal societies,\n\nfictive kin, 209\u2013212,\n\nfission, contrasted to division, ,\n\nfission-fusion societies, 38\u201342, 100\u2013102\n\nin band societies, 100\u2013102\n\nas having clear boundaries,\n\ncompetition in, . 42\n\ncomplexity compared to settled societies,\n\nflexibility in humans, ,\n\ngathering together dynamics in, 38\u201340, 124\u2013125\n\nindividual recollection in, , ,\n\nintelligence in,\n\nprotection and security for, , ,\n\nas the original condition in human ancestors, ,\n\nspecies with, , , , ,\n\nsocial flexibility in, ,\n\nterritories of, 38\u201339, ,\n\nof tribal villages, 277\u2013278,\n\n_See also_ band societies\n\nflags, national, 162\u2013164,\n\nflexibility and adaptability, 70\u201371, , 123\u2013124, , , , 175\u2013176, 251\u2013253,\n\nfood supply and storage, 125\u2013127, 276\u2013277\n\nforaging, 31\u201332, , , , 61\u201362, , , , , , , 125\u2013126, , , ,\n\nforeigners and foreignness\n\nassimilation of, , , 314\u2013318,\n\nband societies approach to, ,\n\nboundaries defining, ,\n\ncleanliness relation to sense of, 188\u2013189\n\ncollaboration in exclusion of,\n\ndehumanization of, 184\u2013186, 188\u2013191, , 286\u2013287, ,\n\ndifference from strangers, ,\n\ndisgust applied to, , 188\u2013189, , ,\n\nfear of, , , , ,\n\nidentity impacted by ignorance of, , , 346\u2013347, ,\n\ninternationalization impact on, 348\u2013349,\n\nmisunderstandings of, 187\u2013188\n\nnationalistic response to, , 340\u2013342\n\nprehistoric humans approach to, , , ,\n\npsychology on creation of and response to, 170\u2013172, 185\u2013191, , ,\n\nranking and perceived inferiority of, , , 186\u2013191,\n\nrecognition or detection of, , , , , , , ,\n\nrelationships with, , , , , , , , , , , 228\u2013238, , ,\n\nsnap judgments of, 176\u2013179\n\nsocietal factions and the invention of of, 255\u2013257, 259\u2013261, 263\u2013266, 270\u2013271\n\nsocietal markers and, 1\u20133, , 86\u201387, , , , , , , , , ,\n\nin society, role of, 1\u20132, , , 88\u201389\n\nsocietal marker creation in response to, , 157\u2013158\n\nsubconscious alarms about, , 171\u2013172\n\ntrade and commerce with, 227\u2013237\n\ntreating citizens as, , ,\n\nperceived unity and uniformity of, 194\u2013196\n\nunavoidability of the existence of, , 346\u2013348\n\n_See also_ ethnicity and race; ingroup\/outgroup dynamics; society transfers\n\nforgetting. _See_ memory\n\nfragmentation, of state societies, 298\u2013305, ,\n\nfriendships\n\nbetween societies, 23\u201324, ,\n\nwithin societies, , , ,\n\nwithout societies,\n\nsimilar to kin relationships, ,\n\nfreedom, 38\u201339, , , , , , , , , 359\u2013360\n\nof movement, , , , , 323\u2013325\n\nrelationship to happiness, ,\n\nfrom slavery or subjugation, , , , ,\n\nfree riders,\n\nFur tribe, , 282\u2013283\n\nFutuna chiefdoms, , ,\n\ngangs, cults, and criminal organizations, , , 352\u2013353,\n\ngathering together, of hunter-gatherer bands, 124\u2013125\n\ngeladas (societies called units), , , 225\u2013226,\n\ngenocide, , , , 200\u2013201,\n\ngestures, 82\u201383,\n\nG\u00f6bekli Tepe, 143\u2013144\n\nGoodall, Jane, , , , , , ,\n\ngorillas, mountain (societies called troops), , ,\n\nGreat Chain of Being, 185\u2013186, , , , ,\n\nGreeks, ancient, , , ,\n\ngroup emotion, , 199\u2013200, ,\n\ngroup coordination signal,\n\ngroup selection, ,\n\ngroup self,\n\ngroupthink,\n\nguilt, admission of,\n\nhair styling and growth, , 153\u2013154, ,\n\nHadza, , , ,\n\nHan\n\ndynasty, ,\n\nethnic group, , , , , , ,\n\nharvester ants, ,\n\nhealth. _See_ disease\n\nHenderson Islanders, 346\u2013347,\n\nherds, wild animal, , ,\n\nselfish herds,\n\nherders, nomadic. _See_ pastoralists\n\nhierarchies. _See_ status and hierarchies\n\nhome base, importance to human evolution, 101\u2013102\n\nhoneybees, , , , , , 264\u2013265,\n\nhorses (societies called bands), , , , , , 47\u201348, , , , , , ,\n\nimportance for humans of domesticated, , , ,\n\nhuman body, as a billboard for our identities, 153\u2013154,\n\nhuman prehistory\n\nabsence of societies with ethnicities and races in, , 309\u2013310\n\ncultural ratcheting evidence in, 155\u2013158\n\necho principle in, 156\u2013157\n\nfission-fusion and settlements in,\n\nforeigners treatment in, , , ,\n\nintelligence in,\n\nprejudices role in, 191\u2013192\n\nsocietal marker evidence for, , 147\u2013148, 151\u2013155\n\nsocieties of, , 147\u2013148, 151\u2013155, 275\u2013281,\n\nhunter-gatherer settled societies\n\naggression by, , ,\n\narchaeological evidence of, 143\u2013144\n\nband societies compared with, 123\u2013124\n\ncomplexity of social life in, , 133\u2013134\n\ndynamics in rise of, 136\u2013138\n\nfood dependability and storage as a requirement for, 126\u2013127, ,\n\ngoods and infrastructure for, , ,\n\ninequality in,\n\nleadership, power, and status in, 128\u2013130, , ,\n\nnomadic times for, , 136\u2013137\n\nownership concept and impacts with, 134\u2013135\n\npotential existence of in early _Homo sapiens_ , ,\n\nresolving conflicts in, , ,\n\nslavery in, , , , ,\n\nsocietal markers in, 278\u2013279,\n\nspecialization in jobs\/talents arising with, 132\u2013134\n\n_See also_ Calusa; Chumash; eel fishermen; Pacific Northwest tribes; tribes\n\nHutterites, 277\u2013278,\n\nhyenas, spotted (societies called clans), , , 40\u201344, , , , , , , ,\n\nidentity. _See_ social identity\n\nidentity fusion,\n\nimagined communities, , , , ,\n\nimmigration\n\nas seen as damaging,\n\ncitizenship qualifications and, 335\u2013339\n\ningroup\/outgroup dynamics and, 327\u2013329, , 338\u2013339\n\nobstacles and hardship with, 328\u2013329,\n\norigins of,\n\nqualifications for citizenship of, , 335\u2013339,\n\nracial\/ethnic identity issues with, 332\u2013335\n\nroles and contributions in society with, 329\u2013332\n\nimperial overstretch, ,\n\nImplicit Association Test, , , ,\n\nimprinting, , , ,\n\nIncan Empire, 310\u2013312,\n\nincest, , ,\n\nindividual recognition, 46\u201349,\n\nin anonymous societies, 68\u201369, , , 176\u2013178, 193\u2013194\n\nin ants, 68\u201369,\n\nin fission-fusion societies, ,\n\nof foreigners, 46\u201347,\n\nincomplete within human societies, ,\n\nmemory constraints and, 50\u201351, , ,\n\nof offspring,\n\nof primates, , , , 89\u201390, 92\u201393, , ,\n\n_See also_ individual recognition societies\n\nindividual recognition societies\n\nanonymous society compared with, 69\u201370, , , 92\u201393, , 249\u2013250\n\ncognitive burden in, 91\u201392\n\ndefined, ,\n\n\"friend of a friend\" framework in, 50\u201351,\n\nmembership and size limitations in, 45\u201352,\n\nrequirement for outsiders in,\n\n_See also_ individual recognition\n\nindividualism, , , , , , , , , ,\n\nin band societies, 115\u2013116, ,\n\noptimal distinctiveness relation to, , , , 267\u2013268, 329\u2013330,\n\nin societal factions and divisions, role of, 267\u2013268\n\nindividualized societies,\n\nindividuating others, , , , ,\n\ninductive\/deductive groups,\n\ninequality, racial and ethnic, , , , 320\u2013323, 328\u2013329, , 333\u2013343,\n\n_See also_ egalitarianism; status and hierarchies\n\ninfants and children,\n\nculture absorption ability of, 175\u2013176\n\nethnicity and race recognition by, 165\u2013166, ,\n\nidentity marker recognition for, , 164\u2013166, 174\u2013176\n\nkinship recognition and understanding for, 205\u2013209\n\nsociety benefits in care of, 120\u2013121\n\nsociety membership\/citizenship for, , ,\n\ninfrahumanization,\n\ninfrastructure, , 60\u201361, , , , 299\u2013300\n\ningroup\/outgroup dynamics, as considered in this book, ,\n\n_See also_ ethnicity and race; societies\n\ningroup-overexclusion effect, ,\n\ninnate human tendencies, , , , , , ,\n\ninheritance of status or goods, 134\u2013135, , , , ,\n\ninnovation, , 155\u2013158, , , ,\n\ninsect societies. _See_ social insects\n\nintegration, 323\u2013326,\n\nintelligence. _See_ cognitive abilities\/intelligence\n\ninteraction sphere, ,\n\ninternational relations, , 237\u2013238,\n\ninternationalization, , 348\u2013349,\n\nIroquois, , , , , ,\n\nInuit (Eskimo), , , , , ,\n\nJefferson, Thomas, , 335\u2013336\n\nJews, , , , ,\n\nJ\u00edvaro, , ,\n\nKanchu Misogi ritual, 197\u2013198\n\nkinship. _See_ families and kinship\n\nkin selection, , 204\u2013205,\n\nlabels\/labeling. _See_ markers, societal\n\nlactose intolerance, 81\u201382\n\nlanguage, , , ,\n\ndiversity of, ,\n\nas identity and societal markers, 85\u201388, , , , , , , ,\n\nimmigration impacts on,\n\nkinship understanding and, 207\u2013209,\n\nsign,\n\nsociety lifecycle and, , , , , ,\n\nsuperiority reflected in, 85\u201386, , ,\n\nunderstanding multiple, , , , , ,\n\nleadership\n\nin animals, 118\u2013119, 128\u2013129, ,\n\nin band societies, 117\u2013119, , , , 130\u2013131, , , 321\u2013322, ,\n\ncharacteristics of, , 129\u2013130, ,\n\nin chiefdoms, 288\u2013290,\n\nvalue during conflict between societies, 130\u2013131,\n\nvalue during conflicts within societies, ,\n\nin conquering societies, 288\u2013290, , 311\u2013312\n\ndysfunction of, , ,\n\ngroups acting without,\n\nin hunter-gatherer settled societies, 128\u2013130,\n\nidentity markers of, ,\n\nin maintaining social identity, 294\u2013295,\n\nlegitimacy of, 289\u2013290\n\nfor nationalists, 340\u2013341\n\nrequired for keeping large populations together, , ,\n\nin society defection and takeovers,\n\nin state societies, 291\u2013292, ,\n\nas symbols, ,\n\nin tribal or village societies, 278\u2013279, ,\n\nleafcutting ants, 57\u201358, , 62\u201364,\n\nlemurs, ringtailed, , , , , ,\n\nlions, , , , , , , , , , , , , , , ,\n\n\"loners\" (species without societies), , , 246\u2013247, 347\u2013348\n\n\"loose\" societies,\n\nmacaques, 92\u201393, , , ,\n\nMacedonia,\n\nmagic number, 266\u2013268\n\nmajority (ethnic group), 318\u2013319, , ,\n\n_See also_ dominant people\n\nMaori,\n\nMarean, Curtis,\n\nmarginal man,\n\nmarkers, ethnic. _See_ markers, societal\n\nmarkers, societal\n\nexpected equivalence to those marking race and ethnicity, ,\n\nfor ants, 69\u201371, , 89\u201390, , , 344\u2013345, ,\n\nfor band societies, 107\u2013111,\n\nfor chimps and bonobos, 148\u2013150\n\nfor nonhuman vertebrates, 71\u201374, 79\u201380, 148\u2013150,\n\nfor prehistoric humans, 141\u2013155,\n\nfor settled hunter-gatherers, 127\u2013128,\n\nrole in anonymous societies, 69\u201375, 76\u201377, 79\u201380, , 356\u2013357\n\nrole in detecting alliances,\n\nautomatic reaction to,\n\nbody adornment and marking as, , , , 153\u2013155,\n\nchanges over time, 251\u2013258, , ,\n\ncontact with ethnicities\/foreign societies impact on, , 157\u2013158, ,\n\nclothing as, , , 127\u2013128, 154\u2013155,\n\ncollective personality as, 201\u2013202\n\ncomplexity and broadcasting of, 152\u2013155, 157\u2013158\n\nconflict and violence in relation to,\n\ncognitive load\/cost of social surveillance and, 91\u201392,\n\ncreation in response to foreigners, 157\u2013158\n\nculture and norms relation to, , , 154\u2013155\n\ncultural ratcheting role in, 155\u2013156, ,\n\ndangers of misidentifying, , , , , ,\n\ndiet as,\n\ndiscernment of, psychology on, 164\u2013166, 169\u2013172\n\ndiversity of in humans, 80\u201383\n\nin divisions and society lifecycles, 241\u2013242, , 249\u2013250, , 261\u2013262, 264\u2013265\n\neffects of echo principle on, 156\u2013157\n\nelephants recognition of human,\n\nemotional expression as,\n\nemotional response to,\n\nethnicity and race relation to, 107\u2013108, 167\u2013170, ,\n\nacceptance of foreign, , , 234\u2013237, 348\u2013349\n\nflags as, 162\u2013164,\n\ngestures and nonverbal accents as, 82\u201383,\n\ngroup-coordination signals as, 149\u2013150\n\ngrowing up and learning to show, , , , , , , 322\u2013323\n\nhair growth and style as, 153\u2013154,\n\nimportance decided by the members, , 87\u201388,\n\nin identifying people as bona fide humans,\n\ninfants\/offspring recognition of, , , 164\u2013166, 174\u2013176,\n\nintelligence required for, 90\u201393\n\ningroup-overexclusion effect and, ,\n\nlanguages as, 85\u201388, ,\n\nbecoming mandatory in human evolution, 154\u2013155\n\nmimicry of, ,\n\nnarcissism of small differences,\n\nnationalism\/patriotism and, 80\u201382, 162\u2013164, 193\u2013194, , , ,\n\nneed not be difficult to produce, 90\u201393\n\nnovelty and fads in generating, 253\u2013255\n\nof Pacific Northwest tribes, 127\u2013128\n\npasswords and initial evolution of, 149\u2013152\n\nphysical differences relation to, , 127\u2013128, , 153\u2013155, 166\u2013170, , ,\n\nnonhuman primates treatment of, 89\u201390\n\nsocial preservation of over time, ,\n\nreliance on in humans, 351\u2013352, 356\u2013357\n\nrituals as extreme, ,\n\nslaves given,\n\nstories as, 179\u2013180\n\nstability and population interactions for, 156\u2013157, 293\u2013294\n\nsubconscious recognition of, 82\u201383, 84\u201385, 170\u2013172\n\nsymbols, as a kind of marker, 90\u201391, , ,\n\nstate societies accommodation of, , 304\u2013305,\n\ntattoos as, , , 153\u2013154, ,\n\n_See also_ group coordination signal; markers, for groups other than societies; symbols\n\nmarkers, for groups other than societies, , , , , 132\u2013133, ,\n\nmarketplace. _See_ trade and commerce\n\nmarmots,\n\nmaroon societies,\n\nmatrilines, , , 320\u2013321,\n\nMayan civilizations,\n\nmeerkats (societies called clans), , , , , , , , , ,\n\nmemory\n\nconnected to space, ,\n\nand forgetting, , , 180\u2013181, , , ,\n\nand maintaining group identity, , , ,\n\nindividual recognition and constraints of, 48\u201351, ,\n\nin recognizing group members versus outsiders,\n\nof shared stories, 179\u2013182\n\nof survival skills,\n\n_See also_ collective memory\n\nmergers. _See_ society mergers\n\nMesopotamia, , ,\n\nmigrant workers, international,\n\nmilitary, , 199\u2013200, , , ,\n\nmimicry and copying, , , , , ,\n\nmind-to-mind coupling, ,\n\nminimal groups. , , ,\n\nminorities. _See_ ethnicity and race; immigration\n\nmirror recognition, 58\u201359,\n\nmirroring behavior, , ,\n\nMitkidis, Panos, 198\u2013199\n\nmob rule, 200\u2013201\n\n\"modern\" humans,\n\nmole rats, 71\u201372, , 128\u2013129\n\nmonkeys (societies called troops), , , , , , , , , , 89\u201390, 92\u201393, , , , , , , , , , ,\n\n_See also_ baboons, geladas, macaques, matrilines, spider monkeys, vervets\n\nMount Eccles Aborigines, 122\u2013123, , , , , ,\n\nmovement and separation. _See_ fission-fusion; spatial factors\n\nmultilevel selection,\n\nnames given to human societies, 85\u201386, , ,\n\nnarratives, societal. _See_ stories\n\nnationalism, ideology, , 162\u2013164, 184\u2013185, 193\u2013195, 199\u2013200, 217\u2013218, 222\u2013223, , , 357\u2013359,\n\n_See also_ , nationalists\n\nnationalist, psychologists view of, , 339\u2013343,\n\nnationalistic fallacy,\n\n_See also_ nationalism, patriotism\n\nNations, , 104\u2013105, , ,\n\n_See also_ state societies\n\nNazis,\n\nJews treatment by, , , ,\n\nnationalism of, 193\u2013195\n\nNuremberg rallies, 193\u2013194, ,\n\nNeanderthals,\n\nin the origin of human \"species thinking\" about societies,\n\nreasons for extinction,\n\nNeolithic revolution,\n\nnetworks, contrasted with memberships, , , , , , , , , , 320\u2013321, ,\n\n9\/11, , , ,\n\nnomadism. _See_ band societies; fission-fusion societies; pastoralists\n\nno-man's land,\n\nnonverbal accents,\n\nnorms of behavior, , ,\n\ndivergence from, , 254\u2013255, , , 341\u2013342\n\nnationalist thought on divergence from, 341\u2013342\n\nsocietal markers relation to, , , 154\u2013155\n\nnovelties and fads, 253\u2013255\n\nNyangatom, 278\u2013279\n\noffspring. _See_ infants and children\n\nopen-group species,\n\noptimal distinctiveness, , , , 267\u2013268, 329\u2013330,\n\noutgroups. _See_ foreigners; ingroup\/outgroup dynamics\n\noverlap of generations, ,\n\nownership, , 132\u2013135, 181\u2013182,\n\nPacific Northwest tribes\n\nidentity markers and diversity of, 127\u2013128\n\nleadership and hierarchies of,\n\nnomadic-settled flux of, , 136\u2013137\n\nownership and status relation with, 132\u2013135\n\nslavery among, , , , ,\n\ntrade and commerce of, 235\u2013236\n\nparties (chimps or bonobos), definition of,\n\nparties (hunter gatherer), definition of,\n\npasswords, 150\u2013152\n\npastoralists, 81\u201382, , , , 278\u2013279, ,\n\nPatriotism and patriots, , , , , 339\u2013343,\n\npeace\/peacefulness\n\nwithin ant colonies, 67\u201368, , , 344\u2013345,\n\nbetween animal societies, , 224\u2013225, 281\u2013282\n\nbetween human societies, 221\u2013222, 227\u2013238\n\nwithin human societies, , , , ,\n\nwithin hunter-gatherer societies, 105\u2013106,\n\nconflict and violence leading to,\n\nsocietal divisions with, , ,\n\nuniversal identity and, 344\u2013345\n\npeer\/social pressure, 178\u2013179,\n\npermeability of societies, 45\u201346\n\n_See also_ society transfers\/defections\n\npersonalities, , , ,\n\nin ants,\n\nin group formation, , , ,\n\nof leaders,\n\n_See also_ entitativity, collective personalities\n\nphysical appearance\/differences,\n\nfamilial recognition of, 205\u2013206,\n\nidentity markers relation to, , 127\u2013128, , 153\u2013155, 166\u2013170, , ,\n\nprejudice on, 192\u2013194\n\npinyon jay, 73\u201374,\n\nplant trap,\n\npolicing, ,\n\npositive distinctiveness, , ,\n\npotlatches, , , ,\n\nprosocial behavior,\n\nprairie dogs, , , , , , , , , , 249\u2013250, ,\n\ncoteries as societies,\n\ncolony or town as a collection of societies,\n\nprehistoric humans. _See_ human prehistory\n\nprejudice and bias,\n\ndon't require an adversarial relationship,\n\nas not consciously recognized, , , ,\n\nin international diplomacy,\n\nmaking individual exceptions to,\n\nwillingness to believe about maligned groups, ,\n\nin membership acceptance, 337\u2013338\n\namong nationalists, 340\u2013341\n\nin human prehistory and evolution, role of, , 191\u2013192\n\nin other primates, ,\n\nrelationship to phobias, 190\u2013191\n\nlinks between for different groups,\n\nrational basis for,\n\nabout our own selves,\n\nstereotypes relation to, , , 191\u2013192\n\nability to sublimate, ,\n\nas unavoidable and difficult to change, , , , ,\n\n_See also_ Implicit Association Test; stereotypes\n\nprotection and security, 26\u201327, , , , , 120\u2013121\n\nas society benefit, 29\u201330, , 32\u201334, , , 120\u2013121, 339\u2013340\n\nas contrasted to providing resources, 12\u201313, , , , , 339\u2013340\n\n_See also_ providing resources\n\nproviding resources\n\nfor ants, 61\u201362\n\ngenerosity with between societies, 232\u2013233\n\nsocietal factions and,\n\nas society benefit, 31\u201332, , , , ,\n\nsociety expansion relation to, 287\u2013288\n\nas contrasted to protection, 12\u201313, , , , , 339\u2013340\n\n_See also_ protection and security\n\npsychology, ,\n\nof conflict and violence, 222\u2013223, , 337\u2013338,\n\nof ethnicity and race, 167\u2013202, , 222\u2013223, , , , , 337\u2013338,\n\nof foreignness creation and response, 170\u2013172, 185\u2013191, , ,\n\nof identity marker discernment, 164\u2013166, 169\u2013172\n\nof marginalized and minority peoples, 321\u2013322\n\nof mind-to-mind coupling,\n\nof nationalism and nationalists, 162\u2013164, 184\u2013185, 193\u2013195, ,\n\non stereotype creation, , 173\u2013179\n\n_See also_ competence; disgust; emotions; empathy; essences; optimal distinctiveness; positive distinctiveness; prejudice and bias; self-esteem; stereotypes; warmth\n\npluralistic ignorance, ,\n\nPygmies, African, , , , , , 236\u2013237, 321\u2013322,\n\nraces, 106\u2013108, , , 332\u2013335\n\n_See also_ ethnicity and race\n\nracism and ethnocentrism, ,176\u2013179, 190\u2013192, 195\u2013196, , 222\u2013223, , , , , 320\u2013323, , , , 357\u2013361\n\nby children, 174\u2013175, ,\n\ninternational relations impact on,\n\nin nationalists and patriots, 339\u2013343\n\nstereotypes role in, 177\u2013178\n\nsubconscious, , , , , ,\n\nrallies, 193\u2013194, ,\n\nin animals, ,\n\nrelationships, 79\u201380\n\nin ants, 68\u201369,\n\ndifferent degrees of social relationship, , 195\u2013196\n\nDunbar's number on, 19\u201321,\n\nin individual recognition societies, 49\u201350,\n\nbetween society members that interact the least, 49\u201350\n\nmind-to-mind coupling in,\n\nsocietal divisions recasting, 262\u2013268\n\nas society benefit, , 34\u201336\n\nbetween societies, , 228\u2013238\n\n_See also_ families\/kinship; friendships\n\nrelease from familiarity, ,\n\nreligion and spirituality, , 180\u2013181, , ,\n\nreverse dominance hierarchy, 117\u2013118,\n\nrights,\n\nrituals, , 197\u2013199\n\nRobbers Cave experiment, , ,\n\nrole models,\n\nRoman Empire, , , , ,\n\nbreakdown of, ,\n\nethnicity and race in, , 311\u2013312, , 324\u2013325, ,\n\nmarkers of identity in, ,\n\nslavery in, ,\n\nRomani (gypsies), ,\n\nRomanization, , ,\n\nRwanda, , ,\n\nsafety. _See_ protection and security\n\nSater\u00e9-Maw\u00e9 ritual,\n\nSan. _See_ Bushmen\n\nSeattle, Chief, , ,\n\nsecurity. _See_ protection and security\n\nself-domestication,\n\nself-esteem, , 189\u2013190, 267\u2013268,\n\nseparation. _See_ fission-fusion; spatial factors\n\nsettled societies. _See_ hunter-gatherer settled societies; tribes; state societies\n\nShoshone, Western, , , , ,\n\nsimilarities\n\nin making comparisons, 60\u201361\n\nbetween individuals, ethnicities, and societies, , , , 329\u2013330\n\nsimple hunter-gatherers. _See_ band societies\n\nslavemaking ants, , 70\u201371, , , , ,\n\nslavery, , 284\u2013287, , ,\n\namong American Indians, , 284\u2013285, , , , , , 322\u2013323\n\nassimilation of, ,\n\ndehumanization with, 286\u2013287, 322\u2013323\n\neconomic value of, , , ,\n\nemancipation and social mobility, 322\u2013323\n\nas a means of increasing society populations, , ,\n\npower differences and, 286\u2013287\n\nprimate practices of, ,\n\npsychological gains from, 286\u2013287,\n\nin slave societies,\n\nsocietal markers and, , ,\n\nin society expansion, 284\u2013287, ,\n\nsociety membership out of, 322\u2013323,\n\nUS Civil War motivations and,\n\n_See also_ slavemaking ants\n\nsmart mob,\n\nsocial brain hypothesis, 19\u201320, , ,\n\nsocial differentiation\/groupiness, , , , , , , , , 133\u2013134, , , ,\n\nsocial identity\n\nin ants, ,\n\ncollective\/national identity relation to self, , 195\u2013197, 357\u2013359\n\ncompromises in, by ethnicities, 314\u2013316\n\ncosts of misidentification, , , , , ,\n\nof families,\n\nas the defining feature of a society, 17\u201319, , , , , , ,\n\nflexibility of, 252\u2013253\n\nabsence of foreigners impacts on, 346\u2013347\n\nrelating to groups other than societies\/ethnicities, , , , , , , , 133\u2013134,\n\nimmigration impacts on, 332\u2013335\n\nin individual recognition versus anonymous societies, 69\u201370, , ,\n\noptimal distinctiveness in approach to, , , , 267\u2013268, 329\u2013330,\n\nperceived as fixed,\n\nsocietal factions based on clashes of, , 264\u2013265, 303\u2013304\n\nshifts in, , 99\u2013100, , 259\u2013260,\n\nin society birth and expansion, 75\u201376, 261\u2013262,\n\nsociety membership significance for, , ,, , 50\u201351,\n\nin state societies, 292\u2013296, 304\u2013305\n\nbroader role in survival,\n\ntheft of, , ,\n\nuniversal, 344\u2013345, 348\u2013351\n\n_See also_ collective identity\/personality; division of labor; ethnicity and race; leadership; positive distinctiveness; markers, societal; society membership; status and hierarchies\n\nsocial insects , ,\n\nanonymous societies for, ,\n\ncollaboration\/cooperation of, ,\n\negalitarianism for,\n\nindividual recognition by, 68\u201369\n\nkinship in societies of,\n\nsociety lifecycle in, 75\u201376, , , , ,\n\nsociety mergers for, , ,\n\n_See also_ ants; honeybees; termites; wasps\n\nsocial learning, , , 147\u2013149,\n\nsocial substitution, ,\n\nsocietal factions and divisions\n\nfor army ants,\n\nfor band societies, , , 248\u2013249, 256\u2013257, 258\u2013259, 265\u2013268, 302\u2013303\n\nfor bonobos and chimps, 242\u2013246, , 262\u2013264, ,\n\nfor other nonhuman vertebrates, , , , , , ,\n\nfor honeybees, , ,\n\nbalance of being alike and different in, 267\u2013268\n\nbeneficial outcome of, , 244\u2013246, 248\u2013249, 261\u2013262, 267\u2013268,\n\ncauses and factors influencing, 241\u2013245, 248\u2013250, 255\u2013260, 263\u2013264, 267\u2013268, , 302\u2013304\n\ncollective identity and,\n\ncompetition role in, , , , , 269\u2013270\n\nconflict and violence with, 243\u2013245, , , , 262\u2013266, 269\u2013270, 299\u2013302\n\nfamily and kinship role in, 265\u2013266\n\nforeignness role in, 255\u2013257, 259\u2013260, 263\u2013266,\n\nidentity clashes driving, , , 300\u2013304\n\nrole of ignorance in, 259\u2013260\n\npeaceful, , , ,\n\nrelationships recast in, 262\u2013266\n\nsocietal markers in, 249\u2013250, 259\u2013260, 264\u2013265, , , ,\n\nsociety size relation to, , , , , , 266\u2013268, ,\n\nspatial and territoriality factors in, , , , , , , 255\u2013259,\n\nin state societies, 302\u2013304,\n\nin tribal or village societies, 280\u2013281\n\n_See also_ society, death of; fragmentation\n\nsocieties\n\nas an ancient human universal, , , ,\n\naversion to talk about, 4\u20135\n\ncells within organisms as, ,\n\nseen as cooperative groups, , , , ,\n\ncults and gangs compared to, , , 352\u2013353,\n\ndefining, , , 17\u201319, 27\u201328\n\ndifficulty proving the existence of, 67\u201368,\n\nas ephemeral, 241\u2013242, 268\u2013271, 304\u2013305,\n\nas extended families, , , 211\u2013212\n\nfuture for, 361\u2013362\n\ninevitability and necessity of for humans, , 344\u2013353,\n\nexisting in isolation, 345\u2013348\n\nmembership in, as distinguished from being social,\n\nmovement and separation within and between, 37\u201344\n\nin prehistory, , 147\u2013148, 151\u2013155, 275\u2013281,\n\nrarity of, among animal species,\n\nendurance in the face of stress, 21\u201322, 249\u2013250\n\nunavoidability of multiple, , ,\n\n_See also specific societies and topics_\n\nsociety, death of, , 268\u2013271\n\nsociety benefits,\n\nfor band societies, 120\u2013121\n\ncooperation as, , , , , 34\u201336, 228\u2013229\n\ndiversity as, , , ,\n\nkinship role in,\n\nprotection and security as, 12\u201313, 29\u201330, , 32\u201334, , , 120\u2013121, , 339\u2013340\n\nrelationships as, 34\u201336\n\nresource provision as, 31\u201332, , , , , , , 339\u2013340\n\nsocial learning as, , , 147\u2013148\n\nfrom population size, , , 62\u201365, 221\u2013222, , , ,\n\nsociety expansion\n\nconflict and violence in creation of, , , 287\u2013288, 297\u2013298\n\nconquering societies role in, 287\u2013290,\n\nfood supply and spatial factors in, 276\u2013277,\n\nforeigners adoption in, 283\u2013284,\n\nidentity role in, 261\u2013262, 293\u2013295,\n\nrole of incorporating outsiders in, , 283\u2013290\n\nminimal role of mergers in, 281\u2013283, 394\u2013395\n\nresource provision relation to, 287\u2013288\n\nslavery in, 284\u2013287\n\nsocietal factions role in, ,\n\ninto state societies, 291\u2013298\n\nsociety lifecycle\n\nbudding of a few individuals as part of, , ,\n\nfor ants, 75\u201376, , , , ,\n\nfor bonobos and chimps, 242\u2013246, , 262\u2013264\n\nfor band societies, , , 248\u2013249, 256\u2013257, 258\u2013259, , , 302\u2013303\n\nfor chiefdoms,\n\nfor tribes, 280\u2013281\n\nneed fulfillment role in, , 360\u2013361\n\ngeneral ignorance about,\n\nlanguages and, , , ,\n\nrole of lone individuals and mated pairs in, 246\u2013247\n\nrole of markers in, 241\u2013242, , 249\u2013250, , 261\u2013262,\n\nother than through society division, 246\u2013248\n\nfor state societies, 298\u2013305, ,\n\n_See also_ societal factions and divisions; society, death of\n\nsociety membership\/citizenship, , ,\n\nin anonymous societies, , , , ,\n\nfor band societies, 108\u2013111, 116\u2013117\n\nfor children\/offspring, ,\n\ncooperation relation to, 27\u201328,\n\ncriteria for, , , 84\u201385, 335\u2013339, ,\n\ndual,\n\nimmigration and qualifications of, , 335\u2013339,\n\nin individual recognition societies, 45\u201352,\n\nfor marginalized and ethnic people, , , , , 318\u2013320, , , 336\u2013338\n\nfrom slavery, 322\u2013323\n\nstability and predictability of, , , , , ,\n\n_See also,_ identity; society transfers\n\nsociety mergers, non-aggressive, , , 281\u2013283, , , , , 394\u2013395\n\n_See also,_ conquering society\n\nsociety transfers\/defections\n\naggressive takeovers,\n\nin animals, , 38\u201339, 43\u201346, , , , , , , 281\u2013282, 315\u2013316, ,\n\nin humans, , , 282\u2013284, , ,\n\n_See also_ immigration; slavery; subjugation\n\nsociety size\n\nanonymous societies response to, ,\n\nfor ants, 57\u201358, 60\u201365, ,\n\nfor band societies, , , ,\n\nfor chimps, 20\u201321, ,\n\nand control of the society, , , 296\u2013297, , , 338\u2013339,\n\ncultural ratcheting relation to,\n\ndivision and dysfunction relation to, , , , 266\u2013268,\n\nDunbar's number relationship to, 19\u201321\n\nidentity marker complexity relation to, 157\u2013158\n\nindividual recognition relation to, 51\u201353,\n\nmate choice impacted by, 266\u2013267\n\nand resource availability, 276\u2013277,\n\nsociety benefit from, , , 62\u201365, 221\u2013222, , 267\u2013268, ,\n\nspecialization impacted by, , 132\u2013134\n\nsurvival skills impacted by, 156\u2013157, ,\n\n_See also_ society expansion\n\nsociety stability\n\nfor Argentine ants, , ,\n\nand breakups, 241\u2013271, 299\u2013305\n\nfor chimps, 20\u201321, ,\n\nof US, 360\u2013361\n\nsolidarity, social, , ,\n\nspatial factors\n\nanxieties created with, , 39\u201340\n\nchiefdom management of,\n\nin conflict management, , , ,\n\nfor ethnic and racial integration, 323\u2013326\n\nin societal factions and breakups, 255\u2013256, 299\u2013303,\n\nfreedoms with separation as a, 38\u201339,\n\ngathering together dynamics and, 38\u201340, 124\u2013125,\n\nfor isolated society members, , 371\u2013372\n\nmovements within and between societies, 37\u201344,\n\nin social complexity, , 63\u201365, , , , , 276\u2013277, ,\n\nstate societies management of, , 293\u2013294, 324\u2013326,\n\n_See also_ fission-fusion societies; territories\n\nspecies\n\nbiological diversity,\n\nethnic groups and societies perceived as, , , , , ,\n\n_See_ also different species by name\n\nspecialization. _See_ division of labor; optimal distinctiveness; social differentiation; social identity\n\nspider monkeys, , ,\n\nspiders, , , ,\n\nspirituality and religion, , 180\u2013181, ,\n\nstate societies (nations)\n\nagricultural practices in relation to rise of, , ,\n\nband societies compared to, , , , 317\u2013318\n\nbureaucracies as key to,\n\nchiefdoms relation to, 288\u2013292, 297\u2013298, ,\n\nconflict and violence in birth and maintenance of, , , 297\u2013298\n\nconquering societies evolution into, , 297\u2013298\n\ndefining, 291\u2013292\n\ndiversity in, , , , 304\u2013305, , ,\n\nearliest, ,\n\nefficiency of, ,\n\nvaried perception of equality\/allegiances within, 320\u2013323, 328\u2013329, , , 335\u2013343\n\nglobal success and regional patterns in, ,\n\nas \"imagined communities,\" , , ,\n\ninfrastructure in, 292\u2013293\n\ninnovations in, ,\n\nlifecycle, factions, and fragmentation of, 298\u2013305, ,\n\nlongevity of,\n\nmarkers accommodation in, , 304\u2013305,\n\nmarkers and identity in, 292\u2013296, 304\u2013305\n\npower, status, and leadership in, 291\u2013292, 294\u2013296\n\nreligion in,\n\nself-identity relation to identity of, , 195\u2013197, 357\u2013359\n\nsocieties expansion into, , 291\u2013298\n\nspatial management in, , 293\u2013294, 324\u2013326,\n\ntaxes, tributes, and laws in,\n\n_See also_ ethnicity and race; nations\n\nstatus and hierarchies\n\nevidence of ancient differences in,\n\nwithin animal societies, , , , , , , , 320\u2013321\n\nbetween human societies, 184\u2013191, , ,\n\nwithin human societies, , 134\u2013135, ,\n\nbetween ethnicities and races, 185\u2013188, , , , , , , , 320\u2013323, , , , ,\n\namong hunter-gatherers, 117\u2013120 134\u2013136,\n\nof immigrants, 328\u2013329\n\nchanges in, 320\u2013322, ,\n\nmarkers for, ,\n\nof slaves, , , 322\u2013323\n\n_See also_ dominance; dominant people; egalitarianism; leadership; reverse dominance hierarchy; matrilines; status quo\n\nstatus quo\n\nseen as deserved between individuals, , , 289\u2013290\n\nseen as deserved between ethnic and racial groups, , 340\u2013341\n\nstereotypes,\n\nas a product of categorization, , ,\n\nchanging over the long haul, ,\n\nchildren learning, 174\u2013175\n\ncollective personality impacting, 201\u2013202\n\nconflict and violence relation to,\n\nin dividing people into neat categories, ,\n\non ethnicity and race, 173\u2013175, 178\u2013179, , 193\u2013194, , , , 342\u2013343\n\nin treating others as individuals, 176\u2013177, 193\u2013194, , ,\n\nprejudice and bias relation to, , , , 191\u2013192\n\nas self-fulfilling prophecies, 178\u2013178, , ,\n\npsychology of creating, , 173\u2013179\n\nracism and, , 177\u2013178, ,\n\nsnap judgments of as normal, 176\u2013179\n\nsocial pressure impact on, 178\u2013179\n\n_See also_ prejudice and bias\n\nstories\/narratives\n\nfor hunter-gatherers, , 181\u2013182\n\nmaking markers more memorable\/meaningful, , , 182\u2013183, , ,\n\nin uniting people,\n\nto put one's people in the best light,\n\nas after-the-fact rationalizations,\n\nas integral to society functioning, ,\n\nin establishing societal history and boundaries, , ,\n\nstrangers, , , , , , , , , , , , , , , , , ,\n\nchimp and bonobo response to, , , ,\n\ndifference from foreigners, , ,\n\nsubjugation, , , , , , , , , 310\u2013312, , , , ,\n\nsubtyping,\n\nsupercolonies. _See_ Argentine ant.\n\nsuperiority, over outgroups 184\u2013190, , ,\n\nsuperordinate identity, , ,\n\nsuperorganism, ,\n\nswarm intelligence, ,\n\nSwitzerland, ,\n\nsymbols, 90\u201391, , , , , , , , , , , , , , , ,\n\n_See also_ markers, societal\n\nTasmanian effect, 156\u2013157, , , ,\n\ntattoos, , , 153\u2013154, ,\n\nteenagers, as trend-starters, 254\u2013255\n\ntermites, , , , , ,\n\nterritories\n\nabsence in some species, ,\n\nof band societies, 104\u2013105, , , , , 255\u2013256, , , ,\n\nas a characteristic of human societies, , , ,\n\nas a means of defining societies,\n\nof ants, , 67\u201370,\n\nof bonobos, ,\n\nof chimps, , , , , , , , , ,\n\namong other nonhuman vertebrates, , , , , , , ,\n\nof chiefdoms, ,\n\nconflict and violence relation to, , , 218\u2013219, 221\u2013222\n\ndiseases and, ,\n\nwith ethnicity and race integration, , 323\u2013326, , 338\u2013339\n\nfission-fusion societies approach to, 38\u201339, , , 255\u2013256\n\npreference for one's own,\n\nsize and control of, , , 296\u2013297, , , 338\u2013339,\n\nsocietal factions relation to, , 255\u2013257, , 302\u2013303\n\nsocietal markers, trade, and, , ,\n\nsocieties approach to, , 38\u201339, , , , , 181\u2013182,\n\nstories and memory role with, 181\u2013182\n\nof tribes, , , , ,\n\nterrorism, 222\u2013223, ,\n\ntheft, , , , ,\n\ntolerance\n\nexamples of animal, 39\u201340, , , ,\n\nfor eccentricity and variation in identity, , 88\u201389, , , , , ,\n\nspatial, 124\u2013125, 293\u2013294\n\ntrade and commerce relation to, 232\u2013233\n\nTower of Babel, ,\n\ntrade and commerce\n\nand marketplaces, in ants, 61\u201362\n\nfor band societies, 233\u2013234\n\nbeginnings of, between societies, 230\u2013232\n\nbetween society similarities\/differences role in, 234\u2013237,\n\ninter-societal tolerance rising with, 232\u2013233\n\ntransfer chain, ,\n\ntrap-jaw ants, 63\u201364,\n\ntribes (tribal societies), , 2\u20133, , , , , , ,\n\ndefined as a type of society, , , ; includes settled hunter-gatherers, ,\n\ndivisions of, 280\u2013281\n\ndynamics of constituent villages, 277\u2013280\n\nidentity in horticultural, 280\u2013281\n\nas a loosely used word, 2\u20133\n\nsocial organization of horticultural, 277\u2013281\n\ngrowth into larger societies difficult for,\n\ntransformation into chiefdoms, 288\u2013290\n\nvillages of, comparable to hunter-gatherer bands, 279\u2013280,\n\n_See also_ American Indians; Pacific Northwest tribes; pastoralists; hunter-gatherer settled societies; _specific tribes_\n\nTukanoans, 349\u2013350\n\nUganda, , , , 245\u2013246,\n\nUnited States (US), 17\u201318, , ,\n\nCivil War, , , , ,\n\ndiversity policies in, ,\n\nimmigration response from, 335\u2013336\n\nminority treatment in, 319\u2013320, 335\u2013336, , , 342\u2013343\n\nsociety stability of, 360\u2013361\n\nuniversal society\/identity, 344\u2013345, 348\u2013351\n\nUruk,\n\nUSSR,\n\nVenezuela, , , 303\u2013304\n\nvervet monkeys, ,\n\nVikings, , 345\u2013346, ,\n\nvillage societies, . _See_ tribes\n\nviolence. _See_ conflict and violence\n\nvocal matching. _See_ social learning\n\nVoltaire, 217\u2013218,\n\nWalbiri, , , , ,\n\nwar. _See_ conflict and violence\n\nwarmth, in psychology, , , , , ,\n\nwasps, social, ,\n\nwaste disposal\/management, , ,\n\nWestern Desert, Australian, 103\u2013104,\n\nwhales, sperm, 74\u201375, , ,\n\nwolves, gray (societies called packs), , , , , , , , , , , , , 204\u2013205, , , , 245\u2013247\n\nxenophobia, ,\n\nYanomami, , 279\u2013281, 284\u2013285\n\nYugoslavia, , \n\n# Contents\n\n 1. Cover\n 2. Title Page\n 3. Copyright\n 4. Dedication\n 5. Epigraph\n 6. Introduction\n 7. SECTION I: AFFILIATION AND RECOGNITION\n 1. Chapter 1: What a Society Isn't (and What It Is)\n 2. Chapter 2: What Vertebrates Get out of Being in a Society\n 3. Chapter 3: On the Move\n 4. Chapter 4: Individual Recognition\n 8. SECTION II: ANONYMOUS SOCIETIES\n 1. Chapter 5: Ants and Humans, Apples and Oranges\n 2. Chapter 6: The Ultimate Nationalists\n 3. Chapter 7: Anonymous Humans\n 9. SECTION III: HUNTER-GATHERERS UNTIL RECENT TIMES\n 1. Chapter 8: Band Societies\n 2. Chapter 9: The Nomadic Life\n 3. Chapter 10: Settling Down\n 10. SECTION IV: THE DEEP HISTORY OF HUMAN ANONYMOUS SOCIETIES\n 1. Chapter 11: Pant-Hoots and Passwords\n 11. SECTION V: FUNCTIONING (OR NOT) IN SOCIETIES\n 1. Chapter 12: Sensing Others\n 2. Chapter 13: Stereotypes and Stories\n 3. Chapter 14: The Great Chain\n 4. Chapter 15: Grand Unions\n 5. Chapter 16: Putting Kin in Their Place\n 12. SECTION VI: PEACE AND CONFLICT\n 1. Chapter 17: Is Conflict Necessary?\n 2. Chapter 18: Playing Well with Others\n 13. SECTION VII: THE LIFE AND DEATH OF SOCIETIES\n 1. Chapter 19: The Lifecycle of Societies\n 2. Chapter 20: The Dynamic \"Us\"\n 3. Chapter 21: Inventing Foreigners and the Death of Societies\n 14. SECTION VIII: TRIBES TO NATIONS\n 1. Chapter 22: Turning a Village into a Conquering Society\n 2. Chapter 23: Building and Breaking a Nation\n 15. SECTION IX: FROM CAPTIVE TO NEIGHBOR... TO GLOBAL CITIZEN?\n 1. Chapter 24: The Rise of Ethnicities\n 2. Chapter 25: Divided We Stand\n 3. Chapter 26: The Inevitability of Societies\n 16. Conclusion: Identities Shift and Societies Shatter\n 17. Acknowledgments\n 18. About the Author\n 19. Also by Mark W. Moffett:\n 20. Praise for \"The Human Swarm\"\n 21. References\n 22. Notes\n 23. Index\n\n# Navigation\n\n 1. Begin Reading\n 2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzsmuy b/data_all_eng_slimpj/shuffled/split2/finalzzsmuy new file mode 100644 index 0000000000000000000000000000000000000000..8130f43fec8a9e4470337133f2e1e90fbde327b6 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzsmuy @@ -0,0 +1,5 @@ +{"text":"\nPerspective of dietetic and antioxidant\n\nmedicinal plants\n\nDilip De Sarker \nManas Ranjan Saha \nSubrata Saha\n\nNotion Press\n\n5 Muthu Kalathy Street, Triplicane,\n\nChennai - 600 005\n\nFirst Published by Notion Press 2015\n\nCopyright \u00a9 Dilip De Sarker, 2015\n\nAll Rights Reserved.\n\nISBN: 978-93-84878-95-5\n\nThis book has been published in good faith that the work of the author is original. All efforts have been taken to make the material error-free. However, the author and the publisher disclaim the responsibility.\n\nNo Part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording, Xerox or by any information storage and retrieval system, without permission in writing from the authors.\nContents\n\n_Title_\n\n_Copyright_\n\n1. Foreword\n\n2. Acknowledgements\n\n3. Introduction\n\n4. Abbreviations\n\n5. A Note on Biodiversity and Endemism\n\n6. Description of Plants A-Z\n\n7. Bibliography\n\n8. References Cited\n\n9. About the Authors\n\n10. Figures\n\n11. Index to English names\n\n12. Index to Vernacular names\nUniversity of Kalyani\n\nFACULTY OF SCIENCE\n\nForeword\n\nThe cultural heritage of any country or society is imbibed in the food habit of the people of that country or society. India with its diverse culture is no exception. However, with the advent of global network of fast\u00adfood companies the great heritage of diverse food habit is in stake. Our grandmothers used to prepare tasteful food ('Sak' and 'Sabji') using local resources. Such traditional preparations are our heritage and continued to so. These food in our regular dishes contributed vitamin, minerals, proteins, carbohydrates, antioxidants and immunomodulators that we know today. On the other side these herbs are available in the wild state and uncultivated.\n\nNotwithstanding the downfall of rich recipe heritage, the poorest of the poor still collect, prepare and consume such food in the face of habitat destruction, loss of plant species and environment degradation.\n\nDr. De Sarker, Associate Professor in Botany, Raiganj University has been working in the field of Botany for last 35 years, whom I know from his graduation days. This book is unique of its kind, bears the endevour of more than 20 years by his research group. This book entitled \"Perspective of Dietetic and antioxidant medicinal plants\", published by The Notion Press, an International publisher, portrays the medicinal uses of more than 300 important plants, their antioxidant potentiality and distribution status in the Upper Gangetic West Bengal. Folk use of each plant has been beautifully written. Such publication is a genuine contribution to our rich heritage. The book also describes the current status of plant exploration and extinction pressure. I am sure that the book will find an important place in dietetics, horticulture, medical botany and pharmacology.\n\n(S.C. Santra)\nAcknowledgements\n\nThe Senior author likes to express his indebtedness to the following persons, institutions, organizations, business houses, as the instances may, for their invaluable help.\n\nStaff of the AASM garden for medicinal plants, Raiganj, West Bengal, and Sri Tanmoy Choudhury and Sri Manas Ranjan Saha for the basic work.\n\nLate Smt. Renuka De Sarker, my mother, for sharing her expertise on traditional knowledge.\n\nUnited States Department of Agriculture data base for information related to nutrition.\n\nState Council for Science and Technology, Govt. of West Bengal, Bikash Bhaban, Kolkata.\n\nAll the scientists whose tireless work and references were used in various aspects of the book.\n\nMicrosoft Corporation, WA, USA for using the WINDOWS softwares.\n\nGoogle Corporation, Mountain View, CA, USA. For internet Browser and G-mail.\n\nDr. Sreejata De Sarker , my wife for her inspiration throughout the work.\n\nMs. Gracy Preety Gomes , the Project Manager & Mr. Naveen Valsakumar, The Co-founder, and other staff members of Notion Press, Chennai for bringing out this publication.\n\nProf. S.C. Santra, Kalyani University, West Bengal for writing the foreword.\n\nProf. S.C. Roy, Department of Botany, North Bengal University ,Kabiraj Uttam Chakraborty, Sri Debjoy Bhattacharjee, Sri Alok Kumar Sarkar, Late Sri Souren Nag for furnishing the traditional knowledge.\n\nLibrary & Department of Botany, Raiganj University, West Bengal providing the working place.\nIntroduction\n\nThe knowledge of curing various human ailments using available plants is perhaps as old as to the dawn of human civilization. It cannot be denied that human races survived through the ages overcoming various diseases and ailments using useful plants and as such the knowledge gathered throughout the thousands of years, is our present legacy. It is our duty and obligation too to gather such knowledge if now still left undiscovered and disseminate them for the present and future mankind to have a better peaceful life.\n\nRural India relies on local traditional medical practitioner for various ailments. Such treatments are ill organized. Some of the local medical practitioners called as \"Kabiraj\" \/\"Baidya\"- who uses his experiences and inherited knowledge of medicinal plants from several sources \u2013 mostly from family knowledge. Tribal healers often use medical knowledge derived from earlier generations \u2013 which is quite different from \u0100yurvedic medical knowledge. Thus there is distinct knowledge - dichotomy between \u0100yurvedic knowledge and knowledge of medicinal plants and their uses by tribals of various catagories. \u0100yurveda means \u2013 the Knowledge (Veda) of the life span (Ayus). It teaches how one may utilize the span of life apportion by nature. In other terms \u0100yurveda, an ancient science of life is deeply rooted in Indian culture. Most commonly \u0100yurveda has been applied related to medical matters and thus it is justified to say it as medicine provided to the various ailments. Meulenbeld(1990) observes, \"The classical treatises on \u0100yurveda clearly states that this science can be applied to all living organisms.\n\nThe Charak Samhita begins with the verse- athato dirghajiviyiyamadhyam vyakhyasyamah\/\/(Charak Samhita, Su,1.1)\n\n\"So then we shall explain the lesson about Longevity.\" It transpires from this verse and etymologically \u0100yurveda signifies the knowledge of longevity. The Indian indigenous system of medical treatment has a long history of about 3000 yrs. Where \"Vedic\" peoples had expertise for utilizing local plants. The ancient system of treatment could be distinctly divided into medicine (Charak Samhita-1000 BC \u2013 100 AD) and Surgery- (Sushrata Samhita\\- 800-700 BC). However, this system of medical practice confronted with the arrival of colonial medicine. Today's medical treatments are almost the contribution of western medicine knowledge. Ancient medicine was not solely based on empiricism and this is evident from the fact that some medicinal plants which were used in ancient times still have their place in modern therapy (Das and Mondal, 2012) . Herbal medicine is still the mainstay of about 70- 80% of the Indian population, and the major part of the traditional therapy involves the use of plant extract and their active constituents.(Akerele,1993).A large number of Wild Plants and their purified extracts have shown beneficial therapeutic potentials due to the presence of Antioxidant compounds, phenolics and flavonoids etc. There is extensive evidence to implicate free radicals in the development of degenerative diseases. Almost all organisms possess antioxidant defenses that protect against oxidative damage and numerous damage removal and repair enzymes to remove or repair damaged molecules.\n\nHowever, the natural antioxidant mechanisms can be inefficient, hence dietary intake of antioxidant compounds become important. Although, synthetic antioxidants, such as butylated hydroxyanisole (BHA), butylated hydroxytoluene (BHT) and tertiary butyl hydroquinone (TBHQ) have been widely applied in food processing, they have been reassessed for their possible toxic and carcinogenic components formed during degradation. In addition, it has been suggested that there is an inverse relationship between dietary intake of antioxidant rich foods and incidence of number of human diseases. Therefore, search into the isolation of natural antioxidant sources is important. The survey on naturally growing plants (eatable) and folk use of plants of minor food of this district has been done so far and a preliminary analysis revels that there are about 40 plants are in regular use as food adjunct by the local ethnic peoples. Some of the plants are regularly used, some are moderately used and some are occasionally used. These plants constitute a major food source for some poor habitants. Most of the plants are regularly sold in local market and therefore a source of income to the downtrodden. Better understanding of these plants will throw light on conservation, utilization and nutrition aspects.\n\nFood security exists when all people, at all times, have physical and economic access to sufficient, safe and nutritious food that meets their dietary needs and food preferences for an active and healthy life (World Food Summit, 1996). India's high economic growth rate in the past decade has not been fully reflected in the health status of its people, with 22 per cent of its population undernourished. According to the National Family Health Survey 2005-06, 40.4 per cent of children under the age of three are underweight, 33 per cent of women in the age group of 15-49 have a body mass index below normal and 78.9 per cent of children in the age group of 6-35 months are anemic. Under the above context an investigation has been made to find out applicability of wild indigenous plants \/ plant parts as dietary supplement to have the food security for the rural outreach people. The inspiration of writing this book starts here. Several plant species have been recorded for dietary supplement showing good nutritional value in the Northern part of Bengal, India. (De Sarker et al. 2013). These wild plants are collected traditionally from native wild source and are directly brought in the local market for sale. No systematic investigation relating to the money generated from the sales and maintenance of the livelihood of the poor peoples involved in such business. Mostly these wild vegetables are eaten as boiled or fried vegetables alone or in association with other vegetables commonly potato or brinjal. The dietetic value along with anti-oxidant value are not well documented.\n\nDe Sarker (2013) presented a paper entitled : Evaluating edible wild plants to improve food security: A study of their economic, medicinal and dietetic support at the Global summit on Medicinal and Aromatic plants, Miri, Malaysia, Dec,10-13,2013.\n\nSome of the plants are well known medicinal plants, therefore , role of these plants as medicinal source in regular diet is also unknown. How far these plants are protective against several crucial ailments by the way of regular intake in the diet is also unknown. Therefore, in depth research are awaiting for the role of these plants in terms of dietary support as well as medicinal support in daily food intake.\n\nOne of the most promising recent alternatives to classical antibiotic treatment is the use of immunomodulators for enhancing host defence responses. Several types of immunomodulators have been identified, including substances isolated and purified from natural sources such as plants including microorganisms.\n\nIn West Bengal in the year 2014-15 alone there are about 2000+ reported case in W.B., in U.P. there are about 1200+ reported case in Bihar there are about 1500+ cases , in Odisha also there are about 1300+ cases. In W.B. alone 265+ unfortunate death cases have been registered for this year(2014-15) ending in the month of August,2014. Since , there is no specific standard medication is available , enhancement of human immune system with the help of herbals may be a way of remedies. However, vaccination against JE is available but it is still inadequate and over and above the JE virus is continually mutating creating new strain of infection.\n\nUnder the circumstances, herbals which are known for treatment of syndromes of encephalitis including JE may be screened for immunomodulatory(IM) roles and finally active compounds may be isolated for further research and human use.\n\nNotwithstanding many short comings, we have tried to answer some of the above questions in this volume. Hopefully reader would use the data for research , home remedy, nutrition as a ready reference and daily use as well.\n\nWhile describing the plants minimum words have been used to minimize the space avoiding repetition. In doing so, nutrition status have been indicated using star marks ranging from one star to the maximum of five. The all sources for the nutrition have been from peoples' experience. Best available reference\/s have been indicated in case of antioxidant potentiality as superscript for a particular plant.\n\n2nd, March, 2015, Raiganj | Dilip De Sarker\n\n---|---\n\u2022 Meulenbeld, G.J.(1990). Conformities and divergences of Basic Ayurvedic concepts in veterinary texts, JEAS, 1(1990)1-6.\n\n\u2022 Das P K and Mondal A K (2012), \"A report to the rare and endangered medicinal plants Resources in the Drydeciduous Forest Areas of Paschim Medinipur District, West Bengal, India\", International Journal of Drug Discovery and Herbal Research (IJDDHR), Vol. 2, No. 2, pp. 418-429\n\n\u2022 Akerele, Olayiwola : 1993 Nature's medicinal bounty : don't throw it away : http:\/\/apps.who.int\/iris\/handle\/10665\/51722#sthash.iMiu7r09.dpuf\n\n\u2022 De Sarker, D.(2013) Evaluating edible wild plants to improve food security: A study of their economic, medicinal and dietetic support. Global summit on Medicinal and Aromatic plants, Miri, Malaysia, Dec,10-13, 2013\nAbbreviations\n\nAbbreviations used,\n\nB. -Bengali; N. -Nepali; E.- English, for indicating nutrition, one star means -low, two star -moderate,three star-good,four star-very good, five star-excellant.\n\nCNC. - Chopra, Nair and Chopra ; USDA- United States Department of Agriculture\nA Note on Biodiversity and Endemism\n\nVariation is a natural phenomenon and occurs continuously in each and every biological species. So said Charles Darwin in 1859. This variation comes about in an process where genes recombine \u2013 and mutation originates. These small variations, in course of time, segregate so as to give rise to new species.\n\nOften the small variants at a sub-species level are to be found scattered within a particular geographical region. The distinguished Russian agrobotanist, N.I Vavilov(1887-1943), in 1935-1940 studied these variations and their relation to the origin of crop species and listed 11 centres in the world where crop species diversified. He named \"Hindustan\" the second major centre and the Bengal-Sikkim-Assam region constitutes a small but very important part of it.\n\nBiodiversity of a species is defined as a collection of all variant forms of biological units below the species level. The region extending from Sikkim to sub-Himalayan West Bengal and the old Gangetic alluvium range of Malda and West Dinajpur comprise the epicentre of numerous endemic plant species. These variants are the real source of germplasm, considered by scientists to be the most important wealth of any country. Undoubtedly, the germplasm source area accounts for countless important genes.\n\nBoth Sikkim and sub-Himalayan West Bengal constitute a unique ecological niche. Bound by the lofty peaks of the Himalayas in the north and the dry areas in the west, this region receives an annual rainfall of around 250-300 cm. Altitudinal variations (500-1,800 ft.), temperature fluctuations, soil conditions and low intensity light account for this zone being a cradle of many plant taxa. Not too far away, the Assam-Burma centre has been hailed by celebrated botanists like Takhtajan as the cradle of flowering plants.\n\nHowever, large scale deforestation is causing irreparable damage to the biodiversity of this area. Among the biotically diverse, but partially-explored sanctuaries are the Senchal Wildlife Sanctuary, Mahananda Game Sanctuary, Singalilla National Park, Neora Valley National Park, Jaldapara National Park and Kulik Bird Sanctuary at Raiganj.\n\nThough floristically Sikkim and Darjeeling are identical, the former is somewhat richer in biodiversity, with over 4000 recorded flowering plants and endemics, 600 species of butterfly, 1,200 species of moth and insects, 144 species of mammals, 550 species of birds, 48 species of fishes and 39 species of reptiles. There are about 175 species of wild plants that are edible. Besides, 43 endemic grasses and 41 endemic orchid species are also to be found here.\n\nHowever, the list cannot be complete as some parts of this region are inaccessible and, therefore, unexplored, the recently-discovered Neora Valley region forest being one of them. It was only sometime back that the presence of Royal Bengal Tigers in this area was reported.\n\nThe other areas in this region awaiting botanical documentation include the rocky gorge upstream of the Gulma river, some parts of Mahananda valley, the dense subtropical forests of Latpanchar, the upper range of the Nandikhola river, the Ghora Mara forests, Zemu glacier and Lachen valley in Sikkim.\n\nApart from the hilly tracts of sub-Himalayan West Bengal and Sikkim, other North Bengal districts are also rich in biodiversity and a number of economically important crops and vegetables have originated here. Atleast 30 varieties of rice originated from erstwhile West Dinajpur district, including the Tulaipangi, Aalai, Changa, Jhingasal, Parijat, Katari Bhog, Bansphool, Masuri varieties. It is another matter that most of these, barring a few exceptions, are not cultivated anymore, especially after the introduction of high-yielding varieties. These varieties constitute the rice biodiversity of this region and many an important gene are found in them. Though rice production has increased two to three times after the high-yielding varieties were introduced, these run a high endemic risk which may wipe out the entire crop in a fortnight.Prof. S.C. Roy, Department of Botany has a personal collection of 130 rice varieties from North Bengal Region. He is trying DNA profiling of the germplasm collection along with wild species.\n\nAt Raiganj, a unique variety of bringal, Bekaur, is found, each of which weighs atleast a kilogram, not to mention anything of its taste. Any discussion on Malda, however, cannot be complete without the mango. Malda is also famous for silk production. A unique variety of silk cocoon is grown here called \"Nisari\", which is temperature tolerant and disease resistant. Many progressive breeds have been developed using the genome of \"Nistari\".\n\nMalda district may be considered a centre diversification for mango germplasm and has yielded countless varieties \u2013 Fajlee, Surma Fajlee, Lakshman Bhog, Gopal Bhog and Kheer Sapati, to mention a few. Our scientists should identify and isolate genes of these fragrant and tasty varieties and patent them immediately. For we definitely would not want to see a multinational biotech company patenting, say, the Surma Fajlee gene \u2013 not an impossibility! Malda is also known for house of various pulses. \"Sona Moog\" a variety of Phaseolus mungo is found cultivated here for more than a century.\n\nJalpaiguri and Cooch Behar are also equally important as far as the origin of vegetable crops are concerned. Palwal (or Patol) may have originated in the dense jungles of Assam-Cooch Behar and wild Kakrol are still to be found in the Terai region of Jalpaiguri. We also come across several varieties of cucumbers in the local markets. Jalapiguri is also famous for a Basmati like rice variety called \"Kalo Nunia\" having a black coulour husk, distinct from all other rice varieties.\n\nBreeders know the importance of wild genes in improving crop species and their value cannot be measured in money. The recognition of sovereign right on genetic diversity will work in favour of biodiversically rich yet technologically poor country like India. Though acres of our natural forest have been destroyed, we should work towards preserving our heritage of biodiversity or whatever is left of it. For only that can strengthen us.\n\nAbelmoschus (Malvaceae)\n\nA. moschatus Medik.\n\nB.\\- Lata Kasturi; E.- Bamia Moschata.\n\nParts used: Ripe fruit\/ Whole plant.\n\nMedicinal uses:\n\nRipe fruit extract most commonly used in male-sexual disorder. A major constituent of some ayurvedic preparations like Dashamularistha and Chyavanprash. Plant used as a tonic, blood purifier, blood enhancer; for rheumatism, colitis, dropsy, dyspepsia, irregular menstruation; seeds used in flatulence, as sex stimulant, increases semen secretion and also used for swelling of gums and eye disorders.\n\nFolk diatetic :\n\nYoung shoot eaten as vegetable by local tribes.\n\nNutrition : ****\n\nAntioxidant Potentiality: High activity1.\n\nBiodiversity status: Less frequent.\n\nAbroma (Malvaceae)\n\nA. augusta (L.) L. f.\n\nB.- Ulatkambal; E.- Devil's Cotton.\n\nParts used: leaf & Root bark.\n\nMedicinal uses:\n\nRoot bark is useful in physical weakness, gonorrhoea, leucorrhoea, increases semen secretion, induces menstruation and is an effective stimulant. The root has also been applied to treat herpes. Leaf and stem extract used in Gonorrhea\n\nFolk dietetic:\n\nFresh Juice stimulant\n\nNutrition : *****\n\nAntioxidant Potentiality: High activity2,3.\n\nBiodiversity status: Frequent.\n\nAbrus (Leguminosae)\n\nA. precatorius L.\n\nB.- Kunch; E.- Indian Licoric. Parts used: Seed.\n\nMedicinal uses:\n\nSeeds used as purgative, emetic and abortifacient.\n\nFolk dietetic:\n\nConsidered as poisonous plant.\n\nAntioxidant Potentiality: High activity4,5.\n\nBiodiversity status: Less-frequent.\n\nAbutilon (Malvaceae)\n\nA. indicum (L.) Sweet\n\nB.- Patari; E.- Indian Abutilon.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe root diuretic and used for leprosy. The whole plant anthelminthic and laxative. The leaves are used as aphrodisiac, pulmonary and sedative and it can also be used to treat ulcers, headache and gonorrhoea.\n\nAntioxidant Potentiality:\n\nModerate activity6,7.\n\nBiodiversity status: Frequent.\n\nAcacia (Leguminosae)\n\nA. auriculiformis Benth.\n\nB.- Akashmani; E.- Earleaf Acacia.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeeds are used for treating intestinal worms of cow.\n\nAntioxidant Potentiality: High activity8.\n\nBiodiversity status: Frequent.\n\nA. concinna (Willd.) DC.\n\nB.- Shikakai; E.- Shikakai.\n\nParts used: Burk, fruit and leaf.\n\nMedicinal uses:\n\nUsed as a hair tonic\/vitalizer. the bark of Shikakai is high on saponins , a good cleaning agent, traditionally used as a detergent. Shikakai is used in Bengal for poisoning fish and are documented to be potent marine toxins. Shikakai is naturally low in pH, and does not strip hair of its natural oils. leaf infusion in anti-dandruff preparations.\n\nFolk dietetic:\n\nLeaves taste acidic and are used in chutneys.\n\nNutrition : ****\n\nAntioxidant Potentiality: Moderate activity9.\n\nBiodiversity status: Frequent.\n\nA. nilotica L.\n\nB.- Babla; E.- Gum Arabic Tree.\n\nParts used: Barks & gum.\n\nMedicinal uses:\n\nThe gum is one of the major constituents of an ayurvedic preparation, Kushmandadi Chu-rna, a sex stimulant. The gum is used to consolidate watery semen. Bark is used to prepare another Ayurvedic preparation i.e. Mrita-sanjibanisura. Treats cough, gum swelling, diarrhoea, dysentery, indigestion, acidity, gonorrhoea, leucorrhoea, diabe-tes and leprosy.\n\nFolk dietetic:\n\nTender leaves uses as cattle feed to induce lactation.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity10,11.\n\nBiodiversity status: Frequent.\n\nAcalypha (Euphorbiaceae)\n\nA. indica Mull. Arg.\n\nB.- Muktojhuri; E.- Indian Acalypha.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are used in cold and cough and ear pain.\n\nAntioxidant Potentiality: Moderate activity12,13.\n\nBiodiversity status: Frequent.\n\nAchras (Sapotaceae)\n\nA. sapota L.\n\nB.- Sabeda; E.- Sapota.\n\nParts used: Bark, seed & Fruit.\n\nMedicinal uses:\n\nPrerventive against biliousness and fever. Hepatic disorders, appetizer, hypertension, diseases arising out of malnutrition,astringent, fever.\n\nFolk dietetic : Bark as tonic\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity14,15.\n\nBiodiversity status: Frequent.\n\nAchyranthes (Amaranthaceae)\n\nA. aspera L.\n\nB.- Apang; E.- Chaff-flower.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nMajor constituent of an ayurvedic preparation, Apamargabati, for leucorrhoea. Also used in piles, palpitation and vein rheumatism. The root extract is used to cure asthma, blisters, flatulence, liver pain, toothache, nerve weakness and even in night blindness. The plant is useful in treating leucorrhoea combined with infections and bleeding and also to dissolve abdominal fats. Methanolic extract shows anti-carcinogenic activity .\n\nAntioxidant Potentiality: Moderate activity16,17.\n\nBiodiversity status: Frequent.\n\nAcmella (Asteraceae)\n\nA. oleracea (L.) R.K.Jansen\n\n(syn: Spilanthus achmella (L.) Murray\n\nB.- Rasun Sak; E.- Toothache Plant.\n\nParts used: Flowers.\n\nMedicinal uses:\n\nUsed in toothache (local anasthetic) and eaten as vegetable to treat body pain. (During post delivery) Also used as mosquito repellent.\n\nFolk dietetic:\n\nUsed as fried vegetables. Nutrition: ***\n\nAntioxidant Potentiality: High activity18,19.\n\nBiodiversity status: Frequent.\n\nAcorus (Acoraceae)\n\nA. calamus L.\n\nB.- Bach; E.- Sweet Sledge.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nThe root extract clears the vocal cord as well as is used as sedative, analgesic and hypertensive. Rhizome used in asthma, flatulence, goiter, enhances memory; rhizome paste used as emollient on rectum and ear in tonsillitis.\n\nFolk dietetic:\n\ncleaned rhizome eaten raw, clears vocal sound. Folk artists use commonly to rejuvenate vocal sound frequency.\n\nNutrition : ***\n\nAntioxidant Potentiality: Moderate activity20,21.\n\nBiodiversity status: Frequent.\n\nAegle (Rutaceae)\n\nA. marmelos (L.) Corr. Serr.\n\nB.- Bel; E.- Bengal Quince.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nFruits are used in the treatment of diarrhoea, dysentery, swelling and gastric irritability and acts as an appetizer. The root used in dys-pepsia, stomachalgia, cardiospasm, seminal weakness and uropathy. Stem bark used in treating fever and also used in the treatment of constipation. Flowers prevent vomiting.\n\nFolk dietetic: Ripe fruit pulp eaten.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity22,23.\n\nBiodiversity status: Frequent.\n\nAerva (Amaranthaceae)\n\nA. lanata (L.) Juss. Ex Schult.\n\nB.- Chaya; E.- Mountain knot grass.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction used in gonorrhoea.\n\nAntioxidant Potentiality: Moderate activity24,25.\n\nBiodiversity status: Frequent.\n\nA. sanguinolenta (L.) Blume.\n\nB.- Bishalyakarani. E. -Climbing Wool-plant\n\nParts used: Leaves & stems.\n\nMedicinal uses:\n\nAlcoholic extract of leaves stop bleeding from cuts and wounds. It is also useful in treating various skin disorders. used as wound healing and antiinflammatory for injuries from falls, rheumatic arthritis and pain in muscles , the whole plant was used as diuretic and demulcent, tender shoot of the plant used as decoction form for galactogue to nursing mother and decoction of whole plant was taken twice a day to expel intestinal worms. anti-cancerous\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nAeschynomene (Leguminosae)\n\nA. indica L.\n\nB.- Shola; E.- Indian joint-vetch. Parts used: Roots\n\nMedicinal uses:\n\nUsed to treat kidney stones and urinary troubles.\n\nAntioxidant Potentiality: Potent phenolic & flavonoid activity26.\n\nBiodiversity status: Frequent.\n\nAgeratum (Asteraceae)\n\nA. conyzoides L.\n\nB.- Paravanga\/Chikasunga; E.- Billygoat Weed.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nDecoction of the plant prevents bleeding from cut and wounds.\n\nAntioxidant Potentiality: High activity27,28.\n\nBiodiversity status: Frequent.\n\nAlbizia (Leguminosae)\n\nA. lebbeck (L.) Benth.\n\nB.- Siris; E.- Lebbeck Tree.\n\nParts used: Root and Bark.\n\nMedicinal uses:\n\nBark is useful in treating diarrhea; Root is useful in eczema, chronic asthma, leprosy.\n\nFolk dietetic:\n\nUsed as cattle feed.\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity29,30.\n\nBiodiversity status: Frequent.\n\nA. odoratissima Benth.\n\nB.- Karoi; E.- Fragrant Albizia.\n\nParts used: Bark.\n\nMedicinal uses:\n\nUseful in loosening of teeth and also in sweating.\n\nAntioxidant Potentiality: Moderate activity31.\n\nBiodiversity status: Frequent.\n\nAllium (Alliaceae)\n\nA. cepa L.\n\nB.\u2013 Piaj; E.\u2013 Onion.\n\nParts used: Bulb, leaf and inflorescence\n\nMedicinal uses: stimulating, diuretic, aphrodisiac, anti-bacterial, heart stimulant, reduces blood sugar\n\nFolk dietetic:\n\nEaten raw as salad & generally with carbohydrate food. As curry additive, fried, boiled, backed, in soup, in pickles etc.\n\nNutrition:\n\nGood, small one are better, ****\n\nAntioxidant Potentiality: High activity.31a\n\nBiodiversity status: Cultivated throughout India.\n\nA. sativum L.\n\nB.\u2013 Rosun; E.\u2013 Garlic.\n\nParts used: Bulb,\n\nMedicinal uses\n\nstimulating, diuretic, aph-rodisiac, anti-bacterial, heart stimulant, reduces blood sugar, controls hypercholesterolemia and many more good effects.\n\nFolk dietetic:\n\nBulb salad & generally with carbohydrate food. As curry additive, fried, boiled, backed, in soup, in pickles etc.\n\nNutrition:\n\nGood, single clove better, ****. Contains anticancer nutraceuticals.\n\nAntioxidant Potentiality: High activity.31a\n\nBiodiversity status: Cultivated\n\nA. wallichi kunth.\n\nN.- Gopa sak ; E.\u2013 Himalayan Onion.\n\nParts used: arial parts\n\nMedicinal uses : stimulating, expectorant\n\nFolk dietetic:\n\nThe leaf & young inflorescence used as vegetable. Sometimes used substitute of onion by villagers.\n\nNutrition: ****\n\nAntioxidant Potentiality: not reported\n\nBiodiversity status: Wild & Cultivated at 1000ft to 7000ft.\n\nAlocasia (Araceae)\n\nA. macrorrhizos (L.) G.Don\n\nB.- Mankachu; E.- Giant Taro.\n\nParts used: Rhizome and petiole.\n\nMedicinal uses:-\n\nUsed in body pain and gout and ash of the petiole used in blister and carbuncle.\n\nFolk dietetic:\n\nLeaf base and corm eaten as vegetables. Nutrition: ***\n\nAntioxidant Potentiality: Low activity32,33.\n\nBiodiversity status: Frequent.\n\nAloe (Xanthorrhoeaceae)\n\nA. vera (L.) Burm. f.\n\nB.- Ghritakumari; E.- Aloe.\n\nParts used: fleshy leaves\n\nMedicinal uses:\n\nUseful as laxative; for wound healing, eczema, skin burns, skin care and ulcers. Plant juice used in increasing strength and stamina, post natal weakness and in leucorrhoea. Used externally, it is very effective on sunburn, as well as a variety of skin diseases (eczema, pruritus, psoriasis, acne) \u2013 it is extremely constructive and protective. Leaf gel as after shave. Used to remove undesired spot on mouth, age lines and post operative scars.\n\nAntioxidant Potentiality: High activity34-36.\n\nBiodiversity status: Rare in locality. Sometimes cultivated\n\nAlpinia (Zingiberaceae)\n\nA. zerumbet (Pers.) B. L. Burtt and R. M. Smith\n\nB.- Bon-Elach; E.- Shell Ginger.\n\nParts used: Rhizomes.\n\nMedicinal uses:\n\nRhizomes are stimulant, carminative and also used in rheumatism.\n\nAntioxidant Potentiality: High activity37.\n\nBiodiversity status: Rare; sometimes cultivated.\n\nAlstonia (Apocynaceae)\n\nA. scholaris R. Br.\n\nB.- Chatim; E.- Devil Tree of India.\n\nParts used: Bark.\n\nMedicinal uses:\n\nMajor constituent of an ayurvedic preparation, Kalmegh, a health tonic. The bark is useful in chronic diarrhoea, dysentery, leprosy and in body weakness, asthma, cold and cough, used as blood purifier and blood enhancer. An infusion of the bark is given for skin diseases such as eczema, ulcers, acne, ringworm and is very useful in case of malaria. It brings down fever steadily to normal in a short time without caus- ing perspiration. Latex of plant is useful in dental problems.\n\nAntioxidant Potentiality: Moderate activity38,39.\n\nBiodiversity status: Frequent.\n\nAlternanthera (Amaranthaceae)\n\nA. philoxeroides (Mart.) Griseb.\n\nB.- Salincha; E.- Alligator weed.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed as pot herb. Folk dietetic:\n\nFried, eaten as sak.\n\nNutrition: ***\n\nAntioxidant Potentiality: Moderate activity40.\n\nBiodiversity status: Frequent.\n\nA. sessilis (L.) DC.\n\nB.- Sanchi; E.- Sessile Joyweed.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe plant is used in dysentery, flatulence, constipation and also in diabetes. The roots are applied externally to cure inflamed wounds.\n\nFolk dietetic:\n\nMixed with potato, fried and eaten. Nutrition: ***\n\nAntioxidant Potentiality: High activity41,42.\n\nBiodiversity status: Frequent.\n\nAmaranthus (Amaranthaceae)\n\nA. spinosus L.\n\nB.- Kanta-khuria\/Kanta-note; E.- Spiny Amaranth.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe infusion of the plant is used as diuretic, laxative and as an emollient on boils. A decoction is useful for improvement of digestion. Leaves are applied as poultice to relieve bruises, abscesses, burns, wounds and inflammations. Root is used to treat gonorrhoea, eczema, intermittent fevers, leucorrhoea and as a remedy for colic and chronic diarrhoea.\n\nFolk dietetic:\n\nWidely eaten as vegetables, fried mixed with pulses or fish. (Several varieties available). Boiled, mixed with salts given to cattle to induce lactation.\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity43,44.\n\nBiodiversity status: Frequent.\n\nA. viridis L.\n\nB.- Ban note; E.- Green Amaranth.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nA decoction of the entire plant is used to stop dysentery, inflammation and even purifies blood. The root juice is used to treat piles, irregular menstruation, cold and cough, constipation, lucorrhoea and burning sensation during urination\n\nFolk dietetic: Used as cattle feed. Nutrition: ****\n\nAntioxidant Potentiality: High activity45,46.\n\nBiodiversity status: Frequent.\n\nAmorphophallus (Araceae)\n\nA. margaritifer (Roxb.) Kunth. (Syn: Arum margaritiferum)\n\nB.- Hochi\/Gajer mul. Parts used: Corms. E.- Margarifer arum\n\nMedicinal uses:\n\nCorms are eaten in body pain and corm paste is applied externally against arthritis. Antioxidant Potentiality: Not Reported.\n\nBiodiversity status: Rare; Endangered.\n\nA. paeoniifolius (Dennst.) Nicolson. (Syn: A. campanulatus)\n\nB.\\- Ol; E.- Elephant Foot Yam.\n\nParts used: Petiole, bulbil & rhizome.\n\nMedicinal uses:\n\nMainly used as vegetable. Fresh petiole and bulbils are used to treat gout. The fresh rhizome pieces are taken to cure piles, dropsy and herpes.\n\nFolk dietetic:\n\nLeaf and petiole are boiled, excess water decanted off, fried and eaten. Corm sliced and boiled and eaten as curry. Lime juice is used to minimize the throat irritation.\n\nNutrition: ****\n\nAntioxidant Potentiality: Moderate activity47,48.\n\nBiodiversity status: Frequent.\n\nAmpelocissus (Vitaceae)\n\nA. latifolia (Roxb.) Planch.\n\nB.- Goaliar lata; E.- Forest Grapes.\n\nParts used: Whole plant.\n\nMedicinal uses:-\n\nWhole plant is crushed and paste is applied as emollient against gout, cuts and wounds.\n\nAntioxidant Potentiality: High activity49.\n\nBiodiversity status: Frequent.\n\nAnanas (Bromeliaceae)\n\nA. comosus (L.) Merr.\n\nB.- Anaras; E.- Pineapple.\n\nParts used: Leaves & fruit.\n\nMedicinal uses:\n\nTender leaves and leaf bases are used for the treatment of worms, cold and cough flatulence and also used to dissolve calculi. Unripe fruit is used as abortifacient.\n\nFolk dietetic: White leaf bases mixed with fresh turmeric given to adults and children against intestinal worms. Eaten as fruit. Nutrition: ****\n\nAntioxidant Potentiality: High activity50.\n\nBiodiversity status: Frequent.\n\nAndrographis (Acanthaceae)\n\nA. paniculata (Burm.f.) Wall. ex Nees.\n\nB.- Kalmegh; E.- King of Bitters.\n\nParts used: Leaves & stems.\n\nMedicinal uses:\n\nMajor constituents of some ayurvedic preparations like Kalmegh and Salsa, both are health tonic. Used as remedy to cure fever, cough, asthma, tuberculosis, intermittent fever, malaria, carbuncles, diabetes, hepatitis, jaundice, chronic dysentery, diarrhoea, uric acid, skin diseases and even ulcers; as a blood purifier as well as enhancer; for boils and herpes. The plant extract exhibits antifungal activities like dandruff control.\n\nFolk dietetic:\n\nSometime eaten as sag.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity51.\n\nBiodiversity status: Frequent.\n\nAnnona (Annonaceae)\n\nA. squamosa L.\n\nB.- Ata; E.- Custard Apple.\n\nParts used: Leaf, bark, root & fruit.\n\nMedicinal uses:\n\nA decoction of the leaves is used as a remedy for cold and for clear urination and killing lice. Bark decoction is used to control diarrhoea, while the root is used in the treatment of dysentery. Fruit is used as appetizer.\n\nFolk dietetic:\n\nFruit eaten.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity52.\n\nBiodiversity status: Frequent.\n\nAphanamixis (Meliaceae)\n\nA. polystachya (Wall.) R.N. Parker\n\nB.- Prithviraj; E.- Pithraj tree.\n\nParts used: Bark & seed.\n\nMedicinal uses:\n\nBark is used in an ayurvedic preparation, Rohitakarishta, a health tonic. Bark is used in the treatment of spleen, liver diseases, tumour and abdominal complaints, stomach worms, intermittent fever. It is also used as blood purifier and enhancer. Seed-oil is used in rheumatism.\n\nAntioxidant Potentiality: High activity53.\n\nBiodiversity status: Frequent.\n\nAreca (Palmae)\n\nA. catechu L.\n\nB.- Supari; E.- Areca Nut\/Betel Nut.\n\nParts used: Fruit & root.\n\nMedicinal uses:\n\nThe nut is used against tapeworm problem and as a sexual stimulant. Betel nut increases salivation, for dental hygiene. Root is used as blood purifier, blood enhancer, leprosy, in insomnia, stangury, blood dysentery and even at ulcerative colitis.\n\nFolk dietetic:\n\nChewing betel nut is a culture of many Asian countries. However, betel nut, pan masala, causes oral cancer. Chewing betel nut banned in Taiwan.\n\nAntioxidant Potentiality: High activity54,55.\n\nBiodiversity status: Frequent.\n\nArgemone (Papaveraceae)\n\nA. mexicana L.\n\nB.- Siyal Kanta; E.- Mexican Poppy.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nSeed oil is useful in dropsy, herpes, stangury; root paste applied in wasp and bee stings.\n\nFolk dietetic:\n\nYoung shoots are collected and boiled, excess water is drained off, fried and eaten.\n\nNutrition: ***\n\nAntioxidant Potentiality: Moderate activity56,57.\n\nBiodiversity status: Frequent.\n\nArgyreia (Convolvulaceae)\n\nA. nervosa (Burm. f.) Bojer. (Syn: A. speciosa)\n\nB.- Briddha Darak; E.- Elephant Creeper.\n\nParts used: Seeds & root.\n\nMedicinal uses:\n\nRoots used in flatulence, stangury, poisonous insect stings, as stomachic; seeds effective against mental disorders, filarial and enhance memory.\n\nAntioxidant Potentiality: Moderate activity58.\n\nBiodiversity status: Less Frequent.\n\nAristolochia (Aristolochiaceae)\n\nA. indica L.\n\nB.- Iswarmul; E.- Indian Birthwort.\n\nParts used: Leaves & root.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation Saibachurna, a body weakness tonic. It is an analgesic, nervine tonic and useful in hypertension and acts as memory enhancer. It induces menstruation. Root acts as tonic, stimulant, emetic and is used in severe fever and leucoderma. Root-decoction is also used in impotency.\n\nAntioxidant Potentiality: Low activity59,60.\n\nBiodiversity status: Frequent.\n\nArtemisia (Asteraceae)\n\nA. annua L.\n\nB.- Janglee Karpur E.-Sweet Wormwood.\n\nParts used:- Whole plant.\n\nMedicinal uses:\n\nPlant juice is used in diabetes. Antioxidant Potentiality:\n\nHigh activity61.\n\nBiodiversity status: Frequent.\n\nA. vulgaris L.\n\nB.- Nagdona; E.- Mugwort.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe decoction of plant is used to treat intestinal worms; leaf decoction in stangury, irregular menstruation, headache, applied externally for ear pus, herpes, sores and scabies; root decoction prevents pregnancy.\n\nAntioxidant Potentiality: High activity61.\n\nBiodiversity status: Frequent.\n\nArtocarpus (Moraceae)\n\nA. heterophyllus Lam.\n\nB.\\- Kathal; E.- Jack Fruit.\n\nParts used: Ripe fruits, seed & leaves.\n\nMedicinal uses:\n\nVarious parts of the plant are used in treating small pox, swelling and infection, flatulence, dyspepsia, anorexia, tonic, clears constipation, skin problems.\n\nFolk dietetic:\n\nRipe fruit eaten raw; seed fried and eaten.\n\nNutrition: *****\n\nAntioxidant Potentiality: Low activity62.\n\nBiodiversity status: Frequent.\n\nA. lacucha Buch.-Ham.\n\nB.- Dehuya; E.- Lakoocha.\n\nParts used: Fruit & bark.\n\nMedicinal uses:\n\nUnripe fruits used to clear constipation; reduces abdominal fats; ripe fruits in anorexia, herpes, as laxative. Bark decoction used in jaundice.\n\nAntioxidant Potentiality: Moderate activity63.\n\nBiodiversity status: Frequent.\n\nAsparagus (Asparagaceae)\n\nA. racemosus Willd.\n\nB.- Satamul; E.- Asparagus.\n\nParts used: Root.\n\nMedicinal uses:\n\nMajor constituents of some ayurvedic preparations like Dashamularistha, Aswagand- haristha, Chyavanprash. The roots are diuretic, also used as sex stimulant, sperm enhancer, tonic and for constipation. It is also used in stangury, biliary colic, nervous disorders, rheumatism, stomach ulcers, chronic fevers, herpes, acidity, impotence, infertility, leucorrhoea and menopause.\n\nFolk dietetic:\n\nRoot made into wholesome vegetables. Nutrition: ****\n\nAntioxidant Potentiality: Moderate activity64,65.\n\nBiodiversity status: Rare in Locality.\n\nAverrhoa (Oxalidaceae)\n\nA. carambola L.\n\nB.- Kamranga; E.- Carambola.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nFruits useful in asthma, bronchitis, cold and cough, anorexia, abdominal fat, chronic fever, piles and as a laxative.\n\nFolk dietetic:\n\nFruit sliced and made into chutney or ripen\n\nfruit eaten as raw. Nutrition: ****\n\nAntioxidant Potentiality: High activity66.\n\nBiodiversity status: Frequent.\n\nAzadirachta (Meliaceae)\n\nA. indica A. Juss.\n\nB.\\- Neem; E.- Indian Lilac.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituents of some ayurvedic preparations like Panchatiktaghritaguggul, Neembadi Tel, Saribadi Salsa, Kalmegh, Punarnabashtak Pachan, Nabakarshik Pachan. The decoction of leaves as well as bark is used in toothache, liver pain, vomiting, jaundice, cough and cold, stomach worms, remittent fever, diabetes. It is also used as an appetizer and is very useful in gangrene. Bark extract is used effectively in ulcers, herpes, leprosy, eczema, rheumatism and also acts as a blood purifier, blood enhancer. Neem leaves were found to inhibit tumor promotion. also anti-diabetic.\n\nFolk dietetic: Leaf fried, eaten with rice.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity67,68.\n\nBiodiversity status: Frequent.\n\nBacopa (Plantaginaceae)\n\nB. monnieri (L.) Wettst.\n\nB.- Brahmi; E.- Water hyssop.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituents of some ayurvedic preparations like Brahmigrita, Brahmirasayan. Acts as a memory enhancer, cognitive enhancer for mental disorder, improves vitality, as carminative, tonic, eyesight enhancer, in stangury, cold and cough of infants, rheumatism, epilepsy and watery semen.\n\nFolk dietetic:\n\nFried with Ghee, eaten as side dish. Decoction of young branch mixed with honey, eaten for improvement of brain function.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity69,70.\n\nBiodiversity status: Frequent.\n\nBaliospermum (Euphorbiaceae)\n\nB. solanifolium (Burm.) Suresh. (Syn: B. montanum)\n\nB.\u2013 Danti; E.- Wild sultan seed. wild croton\n\nParts used:- Root and seeds.\n\nMedicinal uses:-\n\nRoots are used in dropsy and jaundice. The seed oil is used externally for rheumatism.\n\nAntioxidant Potentiality: High activity71.\n\nBiodiversity status: Frequent.\n\nBambusa (Poaceae)\n\nB. tulda L.\n\nB.- Bans; E.- Bamboo.\n\nParts used: Tender leaves.\n\nMedicinal uses:\n\nTender leaves are used in treatment of gout, piles and its poultice is used for boils.\n\nFolk dietetic:\n\nYoung Shoots when grown within a earthen pot becomes white and soft. This soft shoots made into cooked vegetables or boiled in water & washed and eaten with chili. Sometimes pickle is prepared.\n\nNutrition: ****\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nBarleria (Acanthaceae)\n\nB. lupulina Lindl.\n\nB.- Kanta Bishalyakarani; E.- Hophead.\n\nParts used:- Leaves & stems.\n\nMedicinal uses:-\n\nThe crushed leaves stop bleeding from wounds.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nB. prionitis L.\n\nB.- Jhinti; E.- Porcupine flower.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nIt cures fever, respiratory diseases, toothache, joint pains and a variety of other ailments. A mouthwash made from root is used to relieve toothache and treat bleeding gums.\n\nAntioxidant Potentiality: Moderate activity72.\n\nBiodiversity status: Frequent.\n\nBarringtonia (Lecythidaceae)\n\nB. acutangula (L.) Gaertn.\n\nB.- Hizal; E.- Freshwater Mangrove.\n\nParts used: Bark & seed.\n\nMedicinal uses:\n\nBark used in blood purification, dyspepsia and also used for watery semen treatment. Seed powder used as emetic, for nasal blockage, head- ache; seed used in conjunctivitis.\n\nAntioxidant Potentiality: High activity73.\n\nBiodiversity status: Frequent.\n\nBasella (Basellaceae)\n\nB. alba L. (Syn: B. rubra)\n\nB.- Pui; E.- Malabar Spinach.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf juice used in constipation particularly in children and pregnant women. Leaf burnt into ashes, used as toothpaste to cure pyorrhoea. Boiled soup of leafy branches eaten in piles, night blindness.\n\nFolk dietetic:\n\nEaten as curry mixing with potato and brinjal.\n\nNutrition: *****\n\nAntioxidant Potentiality: Moderate activity74.\n\nBiodiversity status: Frequent.\n\nBauhinia (Leguminosae)\n\nB. purpurea L.\n\nB.- Rakta Kanhcan; E.- Purple camel's foot.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituents of an ayurvedic preparation, Patrangasab. Herbs is very useful to treat breast tumor; it also treats leucorrhoea, uterine tumour, dyspepsia, burning sensation of body, nervous weakness for memory stimulation and in anorexia.\n\nAntioxidant Potentiality: High activity75,76.\n\nBiodiversity status: Frequent.\n\nBixa (Bixaceae)\n\nB. orellana L.\n\nB.- Sindur; E.- Lipstick Tree.\n\nParts used: Seed and Fruit.\n\nMedicinal uses:\n\nSeeds are used as a good medicine for gonorrhea, anti\u00adpyretic. Leaves used in jaundice. Cancer76a, kidney diseases, malaria, colic, digestive aid, debility, appetite stimulant, gastric ulcer,cuts and wounds.\n\nFolk dietetic: Fruit eaten\n\nNutrition : ****\n\nAntioxidant Potentiality: Moderate activity77,78.\n\nBiodiversity status: Frequent.\n\nBlumea (Asteraceae)\n\nB. lacera (Burm.f.) DC.\n\nB.-Kukurmutha\/Kukursunga; \nE.- Malay Blumea.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction is used for better hair growth.\n\nAntioxidant Potentiality: Low activity79.\n\nBiodiversity status: Frequent.\n\nBoerhavia (Nyctaginaceae)\n\nB. diffusa L.\n\nB.- Punarnaba; E.- Tar Vine.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed in the preparation of some ayurvedic preparations like Salsa, Panchatikta- ghritaguggul and Punarnabashtak Pachan. The whole plant acts as appetizer, blood enhancer, tonic and diuretic when eaten as vegetable . Used to treat chronic cough, asthma, dropsy, chronic rheumatism, to dissolve calculi, ankle sprain and stangury. The plant also effective against liver and kidney ulcers.\n\nFolk dietetic:\n\nYoung branches are fried and eaten as sak or mixed in fish-curry.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity80,81.\n\nBiodiversity status: Frequent.\n\nBombax (Malvaceae)\n\nB. ceiba L.\n\nB.- Simul; E.- Silk Cotton Tree.\n\nParts used: Flowers, roots & gum.\n\nMedicinal uses:\n\nThe roots are major constituent of an ayurvedic preparation, Simul Mul Churna, a sex stimutant and flower is used to prepare another medicine, Patrangasab, a breast tumour curing preparation. The gum (Mochras) is also used as sex stimulant and sperm enhancer. Root has also sex stimulant and tonic properties. Flowers (3-5) are used to prevent pregnancy of cows.\n\nFolk dietetic:\n\nFlowers used as cattle feed. Nutrition: ****\n\nAntioxidant Potentiality: High activity82.\n\nBiodiversity status: Frequent.\n\nBorassus (Arecaceae)\n\nB. flabellifer L.\n\nB.- Tal; E.\\- Sugar Palm.\n\nParts used:- Leaves, fruits & flower.\n\nMedicinal uses:-\n\nTender leaves are useful in insomnia; fruit juice for talkative persons, in leucorrhoea along with bleeding; fruit endosperm is useful in hiccups; male flower is used in acidic dyspepsia. Folk dietetic:\n\nFruit used in various forms. The phloem sap is sugary and taken as drink or made molasses.\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity83.\n\nBiodiversity status: Frequent.\n\nButea (Leguminosae)\n\nB. monosperma (Lam.) Taub.\n\nB.- Palash; E.\\- Bastard Teak.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaves and barks used as an astringent, antidiarrhoeal, anti dysenteric and febrifuge. It is used for intestinal parasites and geriatric tonic. Seeds are used as sexual stimulant and increases semen secretion.\n\nAntioxidant Potentiality: Low activity84.\n\nBiodiversity status: Frequent.\n\nCaesalpinia (Leguminosae)\n\nC. crista L.\n\nB.- Nata; E.\\- Yellow Nicker.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeed oil is useful for hair growth, nasal allergy, tapeworms and burning sensation of body.\n\nAntioxidant Potentiality: High activity85.\n\nBiodiversity status: Frequent.\n\nCajanus (Leguminosae)\n\nC. cajan (L.) Millsp.\n\nB.- Arahar; E.- Pigeon Pea.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction used in cold and cough, burning sensation of hands and feet, piles, lack of appetite and jaundice.\n\nFolk dietetic:\n\nYoung shoot is used sag. And also used as cattle feed.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity86.\n\nBiodiversity status: Frequent.\n\nCalamus (Aracaceae)\n\nC. rotang L.\n\nB.- Bet; E.- Rattan Palm.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoot decoction used in chronic fever and antidote to snake venom.\n\nAntioxidant Potentiality: Low activity87.\n\nBiodiversity status: Frequent.\n\nCalotropis (Apocynaceae)\n\nC. gigantea (L.) Dryand.\n\nB.- Akanda; E.- Crown Flower.\n\nParts used: Leaves & root bark.\n\nMedicinal uses:\n\nAll the parts of the plant are poisonous.\n\nHowever, root bark is useful in treating asthma; leaf paste in centipede stings. Leaves (warm) are well known for retaining heat and also treat gout.\n\nAntioxidant Potentiality: Moderate activity88.\n\nBiodiversity status: Frequent.\n\nCamellia (Theaceae)\n\nC. sinensis (L) Kuntze.\n\nParts used : Young leaf\n\nB.\u2013 Cha Gachh E.- Tea plant\n\nMedicinal uses:\n\nBlack tea ; stimulates CNS, refreshing, diuretic, restorative, suppressive of hunger. Chinese traditional system of medicine recommends for cardiac health, angina, asthma (wikipaedia). Green tea: contains antioxidant neutraceuticals like catechins and polyphenols. Anti-carcinogenic, effective in lowering LDL \u2013cholesterol level, inhibit blood clot, rheumatoid arthritis, cardiovascular diseases, infections and impaired immune function. The major and most chemo-preventive pharmacological effects owe to (-)-epigallocatechin-3-gallate. In black tea EGCG stands low.\n\nFolk dietetic: as energy drink. Often mixed with rice and taken by tea garden workers. Relieves high altitude sickness. Butter in hot tea taken by Tibetans. Lepchas and Bhotias drink tea with flour of Barley or wheat. Lemon tea now a popular drink.\n\nAntioxidant Potentiality of green tea: High88a\n\nNutrition : *****\n\nBiodiversity: cultivated , origin in china\n\nCanna L. (Cannaceae)\n\nC. edulis Ker-Gawler.\n\nN. Phul Tarul E. Hill Canna\n\nFolk dietetic: Fleshy rhizomes washed and boiled, eaten as staple food by villagers in Hill areas. 'Chhyang'- a traditional fermented beverage prpared from bolied rhizome.\n\nNutrition : ****\n\nCannabis (Cannabaceae)\n\nC. sativa L.\n\nB.- Bhang; E.- Marijuana.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are known for its stimulative and narcotic properties. Also used in gout, abcesses, general debility, insom\u00adnia. Precaution should be taken as the plant is narcotic, especially the seeds.\n\nFolk dietetic:\n\nLeaf dried and used as narcotic. Dried leaves mixed with milk, sugar made into a preparation called 'bhang'.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity89.\n\nBiodiversity status: Frequent.\n\nCapparis (Capparaceae)\n\nC. zeylanica L.\n\nB.- Hingsralata; E.- Ceylon caper.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoot is used in leucorrhoea. Antioxidant Potentiality: Moderate activity90.\n\nBiodiversity status: Frequent.\n\nCapsicum (Solanaceae)\n\nC. annum L.\n\nB.- Lanka; E.- Chilly.\n\nParts used: Leaves & fruits.\n\nMedicinal uses:\n\nUsed in lowering blood pressure (2-3 green chilies); as home remedy, when doctor not available Leaves are used in wasp stings. used for problems with digestion,intestinal gas, stomach pain, diarrhea, and cramps, for heart and blood vessels including poor circulation, excessive blood clotting, high cholesterol, and preventing heart disease, relief of toothache, seasickness, alcoholism, malaria, and fever.\n\nFolk dietetic: Used as vegetables used to help people who have difficulty swallowing.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity91.\n\nBiodiversity status: Cultivated throughout. Several varieties available with relative hotness.\n\nCardiospermum (Sapindaceae)\n\nC. halicacabum Linn.\n\nB.- Latafatki; E.\\- Balloon Vine\/Heart Pea.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation i.e. Panchatiktagritaguggul, it is a blood purifier and enhancer tonic. It is used in the treatment of nervous disorder, stiffness of the limbs. The leaves are rubefacient, they are applied as a poultice in the treatment of rheumatism and lumbago. The leaf juice has been used for the treatment of earache, herpes, skin ulcers, gangrene, leprosy and osteoarthritis. The root is laxative and used in kidney stones, clears constipation, irregular menstruation and is a stomachic and sudorific.\n\nFolk dietetic:\n\nyoung shoots as vegetable also cattle feed.\n\nAntioxidant Potentiality: Low activity92.\n\nBiodiversity status: Frequent.\n\nCarica (Caricaceae)\n\nC. papaya L.\n\nB.- Pepe; E.- Papaya.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nUnripe fruit latex is used to treat bleeding piles, tapeworms, dysentery, and herpes. Whole unripe fruit is used in jaundice, gastric ulcers. Ripened fruits are used for flatulence.\n\nFolk dietetic:\n\nGreen fruit used as vegetables. Ripe fruit eaten raw.\n\nNutrition: *****\n\nAntioxidant Potentiality: Moderate activity93-94.\n\nBiodiversity status: Frequent.\n\nCarissa (Apocynaceae)\n\nC. carandas L.\n\nB.- Karamcha; E.- Christ's Thorn.\n\nParts used: Roots & fruits.\n\nMedicinal uses:\n\nThe ripe fruit is useful in treatment of anorexia, Jaundice, edema, tumor, snake bite, stimulant, diarrhea, burning sensation, biliousness, skin diseases, cold and cough and even effective in gastric ulcers.\n\nFolk dietetic:\n\nRipe fruit eaten and also made chutny. Nutrition: ****\n\nAntioxidant Potentiality: High activity95.\n\nBiodiversity status: Frequent.\n\nCassia (Leguminosae)\n\nC. alata L. (Syn: Senna alata)\n\nB.- Dadmari; E.- Candlebrush\/Wild Senna.\n\nParts used: Leaves & Root.\n\nMedicinal uses:\n\nDecoction of leaves and flowers are used in bronchitis, asthma, ringworm. The roots are used eczema, patches etc.\n\nAntioxidant Potentiality: High activity96.\n\nBiodiversity status: Frequent.\n\nC. fistula L.\n\nB.\\- Bandarlathi\/Sondal; E.- Golden Shower Tree.\n\nParts used: Fruits, seed & leaves.\n\nMedicinal uses:\n\nThe pulp of the seed pod is used as a mild laxative; tender leaves are used as an astringent, gout, in constipation; fruits are tonic and increases vitality.\n\nAntioxidant Potentiality: High activity97,98.\n\nBiodiversity status: Frequent.\n\nC. occidentalis L.\n\n(Syn: Senna occidentalis)\n\nB.- Kalkasunda; E.- Coffee Senna.\n\nParts used: Leaves, flower & root bark.\n\nMedicinal uses:\n\nLeaves are useful in lack of appetite, clears the vocal cord; leaf decoction used in hysteria, diarrhoea, whooping cough; flower in acidity and gastric problems; root bark in high fever.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity99.\n\nBiodiversity status: Frequent.\n\nC. sophera L.\n\n(Syn: Senna sophera)\n\nB.- Kalkasunda; E.- Sophera Senna.\n\nParts used: Leaves, flower & root bark.\n\nMedicinal uses:\n\nSame as C. occidentalis.\n\nFolk dietetic: Used as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Moderate activity100.\n\nBiodiversity status: Frequent.\n\nC. tora L.\n\n(Syn: Senna tora)\n\nB.- Chakunda; E.- Chinese Senna\/Sicklepod.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeed powder is used in poisonous insect stings, guinea worms, asthma with eczema, leucoderma, headache; seed emollient is applied in herpes.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity101.\n\nBiodiversity status: Frequent.\n\nCatharanthus (Apocyanaceae)\n\nC. roseus (L.) G. Don.\n\nB.- Nayantara; E.- Madagascar Periwinkle.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nIt prevents blood cancer, high blood pressure; useful in toothache, diabetes, stomachache and menstruation problems and acts as sedative, hypotensive, tonic. It is also used for wasp stings.\n\nAntioxidant Potentiality: High activity102.\n\nBiodiversity status: Frequent.\n\nCayratia (Vitaceae)\n\nC. trifolia (L.) Domin\n\nB.- Choto goaliarlota; E.- Eared Cyphostemma.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe leaf juice is used in irregular menstruation.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity103.\n\nBiodiversity status: Frequent.\n\nCeiba (Malvaceae)\n\nC. pentandra (L.) Gaertn.\n\nB.- Swet Simul; E.- Kapok Tree.\n\nParts used: Gum & root.\n\nMedicinal uses:\n\nGum is used as tonic, laxative. The root juice is used to cure diabetes.\n\nAntioxidant Potentiality: High activity104.\n\nBiodiversity status: Frequent.\n\nCelosia (Amaranthaceae)\n\nC. argentea L.\n\nB.- Morog Phul; E.- Plumed Cockscomb.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeeds used in diarrhoea, blood diseases, mouth sores and diseases of the eye.\n\nAntioxidant Potentiality: Low activity105.\n\nBiodiversity status: Less Frequent.\n\nCentella (Apiaceae)\n\nC. asiatica (L.) Urban\n\nB.- Thankuni; E.- Indian Pennywort.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituents of an ayurvedic preparation, Bhubaneswarbati. It also cures chronic dysentery, blood stool. Used for memory boosting, neurological disorders, irregular menstruation. It also used in rheumatism, fever, ulcers, leprosy, skin and mouth ulcer.\n\nFolk dietetic:\n\nEaten raw for treatment of amoebiasis. Green leaves are crushed to make Chutny with coconut-flakes, salt and pepper.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity106,107.\n\nBiodiversity status: Frequent.\n\nCereus (Cactaceae)\n\nC. triangularis (L.) Haw.\n\nB.\u2013 Night queen. E.- Queen of the night.\n\nMedicinal uses:\n\nThe fruit is used in syrups, ice cream, fruit salads and even beverage enzymes. Unopened flower-buds are cooked and eaten as vegetables.\n\nAntioxidant Potentiality: Low activity108.\n\nBiodiversity status: Frequent.\n\nChenopodium (Amaranthaceae)\n\nC. album L.\n\nB.- Batua; E.- White Goosefoot.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nIt is useful for bleeding piles, rheumatism, dysentery, biliousness and appetizer. Leaf juice used for constipation and treats stomach worms.\n\nFolk dietetic: Eaten as fried vegetables (aerial parts). Nutrition: ***\n\nAntioxidant Potentiality: High activity109.\n\nBiodiversity status: Frequent.\n\nC. ambrosioides L.\n\n(Syn: Dysphania ambrosioides)\n\nB.- Chandan Batua; E.- Epazote.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed to prevent flatulence, asthma, hysteria and shows carminative properties. It is also eaten as a vegetable.\n\nFolk dietetic:\n\nEaten as fried vegetables (aerial parts). Nutrition: ***\n\nAntioxidant Potentiality: Low activity110.\n\nBiodiversity status: Frequent.\n\nChrozophora (Euphorbiaceae)\n\nC. rottleri (Geis.) A. Juss. ex Spreng.\n\nB.\u2013 Sahodebi; E.- Rottler's Chrozophora.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nJuice of the fruit is given in case of cough and colds.\n\nAntioxidant Potentiality: Low activity111.\n\nBiodiversity status: Frequent.\n\nChrysopogon (Poaceae)\n\nC. aciculatus (Retz.) Trin.\n\nB.\\- Chor Kanta; E.- Love Grass.\n\nParts used: Root.\n\nMedicinal uses:\n\nFresh root used in stomachache and gastric disorder.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nCinnamomum (Lauraceae)\n\nC. tamala (Buch.-Ham.) Nees & Eberm.\n\nB.\\- Tejpata; E.- Indian Bay-Leaf.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nLeaves and bark are useful in flatulence, stangury, anorexia, body odour, for cold and cough, herpes, for hypercholesterolemia, stimulant pain in gums and prevents vomiting.\n\nFolk dietetic: Leaf used in curry. Leaf rolled and fired and smoked like cigarette instantly relieves sore throat and pain in gum. Nutrition: ***\n\nAntioxidant Potentiality: High activity112.\n\nBiodiversity status: Frequent and cultivated.\n\nC. verum J. Presl (Syn: C. zeylanicum)\n\nB.- Daruchini; E.- Ceylon Cinnamon.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nUseful hyppercholesterolemia, diabetes, to reduce nervous tension, to increase memory, colds, to stop vomiting, headache, rheumatism in flatulence, anorexia, bronchitis, cold and cough, eczema, herpes, cardiac disorder and fever.\n\nFolk dietetic:\n\nBark used in curry. Nutrition: ***\n\nAntioxidant Potentiality: High activity113.\n\nBiodiversity status: Cultivated.\n\nCissus (Vitaceae)\n\nC. quadrangularis L.\n\nB.- Harjora; E.- Veldt Grape.\n\nParts used: Stem\n\nMedicinal uses:\n\nUsed to treat fracture of bones, constipation, biliousness, asthma, irregular menstruation, intestinal worms and rheumatism and also act as a laxative.\n\nAntioxidant Potentiality: High activity114.\n\nBiodiversity status: Frequent.\n\nCitrus (Rutaceae)\n\nC. aurantifolia (Christm.)Swingle. (Syn. C.medica var.acida).\n\nB.\u2013 nebu \/lebu; E.\u2013 The lemon.\n\nParts used: Fruit, leaf\n\nMedicinal uses: Fruit appetizer, refreshing, anti-vomiting, in \nJaundice, in exhaustion \n(specially in sportsl), in piles, in malnutrition. Lemon drops & rose water for skin nourishment; antiradical, antioxidant, anti- \ninflammatory and anti-cancerous114a\n\nFolk dietetic:\n\nLeaves contain volatile geranial, limonene, and neral ; twisted leaf smell for vomiting & unconsciousness. Fruit in digestion; alleviates anxiety and nervousness; relieves stress related disorders as in insomnia or nervous originated digestive disorders.\n\nAntioxidant Potentiality: High activity114b.\n\nNutrition : *****( USDA)\n\nBiodiversity status: cultivated and wild in North eastern sates & Indian W. peninsula.\n\nC. reticulata Blanco.\n\nParts used : Fruit,Fruit peel\n\nB.\u2013 Kamala lebu E.-The orange\n\nMedicinal uses:\n\nsweet, tonic, laxative, aphrodisiac, astringent, enhances CNS.\n\nFolk dietetic: fruit juice drop given to ailing baby for curing Jaundice. For quick recovery from illness. Antioxidant Potentiality:\n\nHigh activity114c\n\nNutrition: *****(USDA)\n\nBiodiversity : Cultivated\n\nC. sinensis (L.) Osbeck.(syn. c.aurentium L. var.aurentium)\n\nParts used : Fruit\n\nB.-Musambi E.- Sweet orange\n\nMedicinal uses:\n\nlike orange\n\nFolk dietetic:\n\nallays thirst,Fresh rind rubbed on face to remove acne.(CNC)\n\nAntioxidation potential : Good114d\n\nNutrition : ****(USDA).\n\nBiudiversity: Cultivated and wild\n\nC. maxima (Burm.) Merril.\n\nParts used : fruit\n\nB.-Batabi lebu\/ Chokotara E.- The pomelo\n\nMedicinal uses:\n\nnutritive, cardiotonic, refreshing. Leaves in epilepsy,anti-diabetic,\n\nFolk dietetic: have nutraceuticals\n\nAntioxidation potential : good114e\n\nNutrition : ****\n\nBiodiversity: Cultivated and wild\n\nClerodendrum (Lamiaceae)\n\nC. indicum (L.) Kuntze\n\nB.- Bhamot\/Bramhajosthi; E.- Tubeflower.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is used in rheumatism, as a blood purifier, sudorific, chronic stomachic, in dropsy, imtermittent fever, jaundice, bronchial pain, cough and cold, asthma; root bark in asthma, mumps; roots and leaves useful in tapeworm.Though not proven, but people use the dried stem as a garland to cure a special type of boil in children.\n\nAntioxidant Potentiality: Low activity115.\n\nBiodiversity status: Frequent.\n\nC. infortunatum L. (Syn. C. viscosum)\n\nB.- Ghentu\/Titvat; E.- Hill Glory Bower.\n\nParts used: Apical bud & Leaf.\n\nMedicinal uses:\n\nThe bud and leaf decoction destroys tape-worm, hair lice; useful in skin problems, cough and cold, liver disorder and also acts as blood purifier and reduces malarial fever.\n\nAntioxidant Potentiality: High activity116.\n\nBiodiversity status: Frequent.\n\nClitoria (Leguminosae)\n\nC. ternatea L.\n\nB.- Aparajita; E.- Butterfly Pea.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaf paste use to treat bleeding piles, alleviates swelling and pain. Root used in toothache.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity117.\n\nBiodiversity status: Frequent.\n\nCoccinia (Cucurbitaceae)\n\nC. grandis (L.) Voigt\n\nB.- Telakucha; E.- The Ivy Gourd.\n\nParts used: Leaves and Fruit.\n\nMedicinal uses:\n\nThe juice of the roots and leaves is used for diabetes, cold and cough. The plant is used internally in the treatment of gonorrhoea and also as a blood purifier. Leaf decoction used in diabetes. Fruits are bitter; sometimes used in diabetes.\n\nFolk dietetic:\n\nLeaf and young branched are boiled, fried in cooking oil with paanch phoran, eaten as sak. and eaten. Fruits of one variety called \"Kudri\" (C. grandis var. kudri) is eaten as vegetable like pointed gourd and also used in treatment of diabetes. Leaves are chewed every morning to keep blood sugar under control in diabetes.\n\nNutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity118,119.\n\nBiodiversity status: Frequent.\n\nCocculus (Menispermaceae)\n\nC. hirsutus (L.) W. Theob.\n\nB.- Jaljomani; E.- Broom Creeper.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nIt is used as mild laxative, digestive, appetizer, and in anorexia, asthma, cough and cold, carmination, in mouth ulcers, pruritus, dyspepsia, colic, flatulence, bronchitis, gout, intermittent fever, hypertension, herpes and general weakness. The leaf juice relieves from stangury. The juice of leaves is salutary in burns and carbuncles. It is a sex stimulant and it delays the ejaculation.\n\nAntioxidant Potentiality: High activity120.\n\nBiodiversity status: Frequent.\n\nCocos (Arecaceae)\n\nC. nucifera L.\n\nB.- Narikel; E.- Coconut.\n\nParts used: Fruit & root.\n\nMedicinal uses:\n\nBurnt roots are used in gum swelling and toothache; fruit water is used in dysentery, con- stipation, stangury, dyspepsia.Folk dietetic:\n\nGreen coconut water is used as anti-dehydrate in summer months. Solid endosperm used in various food preparation.Nutrition: *****\n\nAntioxidant Potentiality: High activity121,122.\n\nBiodiversity status: Frequent.\n\nCoix (Poaceae)\n\nC. lacryma-jobi L.\n\nB.-Gargare\/Gabeduk; E.- Job's Tear.\n\nParts used: Seed and root.\n\nMedicinal uses:\n\nSeeds are used in urinary retention, and as a general tonic. Root is used for menstrual disorders.\n\nFolk dietetic :\n\nEaten as substitute of bread millet during food crisis. As per Classical Sanskrit lit. flour of the seeds made into bread or boiled as burley potently reduces fat in body.\n\nTo reduce blood cholesterol bread prepared from grounded flour of grains to be taken for 1-2 months. Galactogauge.\n\nAntioxidant Potentiality: High activity123.\n\nBiodiversity status: Less Frequent.\n\nColocasia (Araceae)\n\nC. esculenta (L.) Schott.\n\nB.- Kachu; E.- Green Taro.\n\nParts used: Corm & Petiole.\n\nMedicinal uses:\n\nJuice of corm is used against scorpion sting as well as the cut end of a petiole is used to stop bleeding from wounds.\n\nFolk dietetic:\n\nYoung leaf and petiole boiled, fried mixed with onion and eaten. Corm sliced, fried eaten or cooked as vegetables. (Several local varients are available)\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity124.\n\nBiodiversity status: Frequent.\n\nCommelina (Commelinaceae)\n\nC. benghalensis L.\n\nB.-Kanchire; E.- Whiskered Commelina.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nIt is useful in skin diseases, ringworm, eczema, blisters, pus in ear, rheumatic colic and acts as a stomachic.\n\nAntioxidant Potentiality: Moderate activity125.\n\nBiodiversity status: Frequent.\n\nCorchorus (Tiliaceae)\n\nC. olitorius L.\n\nParts used : Young Leaf\n\nB.\u2013 Mitha pat E.- Jute plant\n\nMedicinal uses:\n\nleaf responsible for the antidiabetes and antihypertension, diuretic, tonic.\n\nFolk dietetic:\n\nas leafy vegetable, leaf fried in cooking oil with 'panch foron' and eaten as 'sak', popular among villagers. Leaf soup appetizer, improves liver dysfunction.\n\nAntioxidant Potentiality : High125a\n\nNutrition : ****\n\nBiodiversity : Cultivated,origin S.E.Asia\n\nCordyline (Agavaceae)\n\nC. fruticosa (L.) A.Chev.\n\nB.- Agniswar; E.- Ti Plant.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nRhizome eaten with betel as a remedy in case of diarrhoea.\n\nAntioxidant Potentiality: High activity126.\n\nBiodiversity status: Frequent.\n\nCoriandrum (Apiaceae)\n\nC. sativum L.\n\nB.- Dhaniya; E.- Coriander.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituent of an ayurvedic preparation, Vaskaralabana. The plant is used in the treatment of ulcers, cough, insomnia, vomiting, diarrhoea, dysentery and biliousness. The seed is used in menstrual disorders, for stomachache, conjunctivitis, headache, carminative, jaundice and fever. The leaves are used as appetizer.\n\nFolk dietetic:\n\nCrushed leaf mixed with salt and batter, fried in oil and eaten. Chutny and leaf salad is prepared.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity127.\n\nBiodiversity status: Frequent.\n\nCostus (Zingiberaceae)\n\nC. speciosus (Konig) Smith.\n\n(Syn: Cheilocostus speciosus (J. Koenig) C.D.Specht)\n\nB.-Basikaran\/Kuttus; E.- Crepe Ginger.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nRhizome is used for impotency, gonorrhoea and used in birth control.\n\nFolk dietetic14a: To hypnotize somebody, debility. Leaf juice is taken twice daily for 14 days.\n\nAntioxidant Potentiality: High activity128.\n\nBiodiversity status: Frequent.\n\nCrateva (Capparaceae)\n\nC. religiosa G.Forst.\n\nB.\\- Barun; E.- Sacred Garlic Pear.\n\nParts used: Bark & wood.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Barunadya Louha, a health tonic and Barunastak Pachan. The bark is used for rheumatism, jaundice, burning sensation of body and goiter. It is also used as antiinflammatory, diuretic, dropsy, demulcent and tonic. The bark-wood is also useful against kidney and vesicle calculi.\n\nAntioxidant Potentiality: Moderate activity129.\n\nBiodiversity status: Frequent.\n\nCroton (Euphorbiaceae)\n\nC. bonplandianum Baill.\n\nB.- Churchuri; E.- BanTulsi.\n\nParts used: Latex.\n\nMedicinal uses:\n\nThe watery latex of the plant stops bleeding.\n\nAntioxidant Potentiality: High activity130.\n\nBiodiversity status: Frequent.\n\nCurculigo (Hypoxidaceae)\n\nC. orchioides Gaertn.\n\nB.- Talmuli; E.- Common Curculigo.\n\nParts used: Rhizome & root.\n\nMedicinal uses:\n\nMajor constituent of an ayurvedic preparation, Dashamularistha. The rhizomes of the plants are used for increasing vitality. Root is aphrodisiac, appetizer, and useful in treatment of piles, biliousness, fatigue, blood related disorders. Root paste is used in arthritis and gout.\n\nAntioxidant Potentiality: Low activity131.\n\nBiodiversity status: Rare.\n\nC. recurvata W. T. Ait. (Syn. Molineria capitulata)\n\nB.- Banshmora; E.- Recurved-Leaved Curculigo.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoot is used for piles and rheumatism.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Rare.\n\nCurcuma (Zingiberaceae)\n\nC. amada Roxb.\n\nB.- Amada; E.- Mango Ginger.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nRhizome used in flatulence, anorexia, dyspepsia and gout. Folk dietetic:\n\nUsed in cooking. Nutrition: ***\n\nAntioxidant Potentiality: High activity132.\n\nBiodiversity status: rare.\n\nC. caesia Roxb.\n\nB.- Kala Halud; E.- Black Turmeric.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nRhizome used as carminative, stimulant as well as in sprains and bruises.\n\nAntioxidant Potentiality: Low activity133,134.\n\nBiodiversity status: Less Frequent.\n\nC. longa L.\n\nB.- Halud; E.- Turmeric.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nIt has anticancerous property. Used as an antiseptic; treats stammers, filaria, allergy, asthma, conjunctivitis, cuts and wounds, sprain, boils and even clears skin. It is anthelminthic in nature, antibacterial, antifungal and antivirul.\n\nFolk dietetic:\n\nEaten raw against pox and measles.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity135.\n\nBiodiversity status: Frequent.\n\nC. zedoaria (Christm.) Roscoe\n\nB.- Shothi; E.- White Turmeric.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nRhizome is given to young children as a healthy alternative to other farinaceous products, it also prevent loose motion of children.\n\nFolk dietetic:\n\nGiven as baby food. Nutrition: *****\n\nAntioxidant Potentiality: High activity136.\n\nBiodiversity status: Frequent.\n\nCuscuta (Convolvulaceae)\n\nC. reflexa Roxb.\n\nB.- Swarnalata; E.- Giant Dodder.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe seed has been found to have positive effects on sperm health and motility and has antioxidant benefits. The whole plant is used in several eye disorders.\n\nAntioxidant Potentiality: High activity137.\n\nBiodiversity status: Frequent.\n\nCymbopogon (Poaceae)\n\nC. citratus (DC.) Stapf.\n\nB.- Lemon Ghas; E.- Lemon Grass.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nAromatic plant, used as an disinfectant and insect repellant.\n\nAntioxidant Potentiality: Low activity138.\n\nBiodiversity status: Not frequent.\n\nCynodon (Poaceae)\n\nC. dactylon (L.) Pers.\n\nB.- Durba; E.- Bermuda Grass.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe leaf extract is applied to fresh cuts and wounds. It is also used to treat inflamed boils, abscess, skin diseases, bleeding piles and herpes. The plant paste applied on the upper eyelid to relieve redness of the eye due to sunburn by Munda tribes.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ****\n\nAntioxidant Potentiality: Low activity139.\n\nBiodiversity status: Frequent.\n\nCyperus (Cyperaceae)\n\nC. kyllingia Endl.\n\n(Syn. Rhynchospora colorata (L.) H.Pfeiff.)\n\nB.- Nirbish; E.- White Kyllinga.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe whole plant is used against snakebites along with Eleucine indica.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ****\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nC. rotundus L.\n\nB.- Mutha Ghas; E.- Nut Grass.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nIt is used to treat fever, dyspepsia, dysentery, inflammations, epilepsy, wounds, \npyorrhoea and other maladies. It acts as an antidote against bites of wasps. The burnt tubers are used to treat wounds, bruises, carbuncles etc.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ****\n\nAntioxidant Potentiality: Low activity140.\n\nBiodiversity status: Frequent.\n\nDalbergia (Leguminosae)\n\nD. sissoo DC.\n\nB.- Sisu; E.- Sissoo.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nTender leaves are useful in vomiting, excessive menstruation; bark is used for reducing abdominal fats and enema.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity141.\n\nBiodiversity status: Frequent.\n\nDatura (Solanaceae)\n\nD. metel L.\n\nB.- Kalo Dhutra; E.- Devil's Trumpet.\n\nParts used: Leaves & root.\n\nMedicinal uses:\n\nLeaf paste applied for promoting hair growth; root used in rheumatism, joint pain and gout. Seed is poisonous.\n\nAntioxidant Potentiality: Low activity142.\n\nBiodiversity status: Frequent.\n\nD. stramonium L.\n\nB.- Sada Dhutra; E.- Jimson Weed.\n\nParts used: Seed\n\nMedicinal uses:\n\nAll parts of the plant contain dangerous levels of poison, and may be a fatal if ingested by humans or animals. Seed paste used for poisoning of fish.\n\nAntioxidant Potentiality: Low activity143.\n\nBiodiversity status: Frequent.\n\nDaucus (Umbelliferae)\n\nD. carota L. var. carota D.C.\n\nB. Gajor, E. Carrot\n\nParts used : Tap root, young branches, seeds\n\nMedicinal uses:\n\nOrange coloured varieties are source of carotene, a precursor of vitamin A. also contains minerals,protein and carbohydrates. Cures night blindness, Seeds aphrodisiac and stimulant, carminative, seed paste used in early baby birth also stimulate birth pain. Root Anti-helmintic, relives thirst\n\nFolk dietetic:\n\nMixed with sugar\/honey a country liquor prepared called 'Kanji', a good appetizer. Root taken as raw as well as cooked in curries, made into pickles and sweetmeat. Root juice taken to reduce body weight and fat\n\nAntioxidant Potentiality: High activity143(a).\n\nBiodiversity status: Cultivated all over India, central asia thought to be its centre of origin. Wide variety are found, from black and red to yellow.\n\nDelonix (Leguminosae)\n\nD. regia (Hook.) Raf.\n\nB.- Krishnachura; E.\\- Flame Tree.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nBark, flower and leaf paste are used in piles. Bark is used in boils; bark and leaves prevent vomiting, intestinal worms; root bark induce menstruation.\n\nAntioxidant Potentiality: Moderate activity144.\n\nBiodiversity status: Frequent.\n\nDendrocalmus ( Poaceae)\n\nD. hamiltoni Fn.\n\nN. Choya bans. E. Tufted Bamboo\/Hamilton Bamboo\n\nFolk dietetic:\n\nYoung shoots up to 2 ft. cut into pieces and boiled. A few burning wood coal added to it. Cooked as vegetables. Made into pickles.\n\nNutrition : ***\n\nBiodiversity: wild & cultivated\n\nDendrophthoe ( Loranthaceae)\n\nD. falcata (L. f.) Ettingsh.\n\nB.- Dharua; E.- Honey suckle mistletoe.\n\nParts used: Leaf & stem bark.\n\nMedicinal uses:\n\nStem bark is used for wounds and menstrual disorders. Leaf paste used in various inflammation.\n\nFolk dietetic:\n\nLeaf used as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity145.\n\nBiodiversity status: Frequent.\n\nDesmodium (Leguminosae)\n\nD. gangeticum (L.) DC.\n\nB.- Salpani; E.- Ticktree\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituent of an Ayurvedic preparation, Salsa, a health tonic. Plant is used for treatment of fever, dropsy, liver disorders, stangury, diabetes, rheumatism and also as a blood purifier and stomachic.\n\nAntioxidant Potentiality: High activity146.\n\nBiodiversity status: Frequent.\n\nDillenia (Dilleniaceae)\n\nD. indica L.\n\nB.- Chalta; E.- Elephant Apple.\n\nParts used: Fruits & root bark.\n\nMedicinal uses:\n\nUnripe fruit juice is used in hysteria; ripe fruits used as laxative, in watery semen, ano\u00adrexia, for improving lactation, general debility; root bark in food poisoning.\n\nFolk dietetic:\n\nParts of fruit used as cattle feed to induce lactation. Fleshy calyx cut into pieces added in 'dal' or made into chutny.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity147.\n\nBiodiversity status: Frequent.\n\nDioscorea (Dioscoreaceae)\n\nD. alata L.\n\nB.- Kham Alu; E.\\- Purple yam.\n\nParts used: Tubers.\n\nMedicinal uses:\n\nThe tubers are useful in haemorrhoids, leprosy, intestinal worms, seminal weakness, post menopausal syndrome and general weakness. Folk dietetic:\n\nAerial bulbul and underground tubers eaten as a substitute of potato. Since, it contains diagenin too much intake causes sterility & lower sperm count.\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity148.\n\nBiodiversity status: Frequent.\n\nD. bulbifera L.\n\nB.- Lata alu E.- Air Yam\/ Air Potato.\n\nParts used: Tubers.\n\nMedicinal uses:\n\nUseful in treatment of herpes, leisons, fistula, diabetes, urinary caliculi.\n\nFolk dietetic:\n\nUnderground tubers eaten as a substitute of potato. Various dish is prepared while adding the pieces of tubers. Since it contains diagenin, too much intake causes sterility.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity149.\n\nBiodiversity status: Frequent.\n\nD. esculenta (Loureiro) Burkill\n\nB.- Kanta alu; E.- Lesser Yam.\n\nParts used: Tubers.\n\nMedicinal uses:\n\nTubers is consumed by the tribal people.\n\nAntioxidant Potentiality: Low activity150.\n\nBiodiversity status: Frequent.\n\nD. oppositifolia L.\n\nB.- Kham alu; E.-Yam\n\nParts used: Tubers.\n\nMedicinal uses:\n\nTubers is used as vegetable and used as substitute of potato.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nD. pentaphylla L.\n\nB.- Kanta alu; E.- Five-Leaf Yam.\n\nParts used: Tubers.\n\nMedicinal uses:\n\nTuber is consumed by the tribal people.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nDiospyros (Ebenaceae)\n\nD. discolor Willd.\n\nB.- Chayna Gab; E.- Velvet Apple.\n\nParts used: Leaves, bark & fruit.\n\nMedicinal uses:\n\nThe bark and leaves are used for itching and other skin ailments as well as eyewash. The decoction of bark is used for cough, fever, dysentery and diarrhoea.\n\nFolk dietetic:\n\nFruit eaten fresh or can be dried and mixed with other fruits in salads, as cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: High \nactivity151.\n\nBiodiversity status: Frequent.\n\nD. malabarica (Desr.) Kostel.\n\nB.- Gab; E.- Indian Persimmon.\n\nParts used: Bark & fruit.\n\nMedicinal uses:\n\nThe bark is used against boils and tumours, and the juice of the fresh bark is useful in bilious. The unripe fruit is astringent and when ripe, beneficial in blood diseases, and leprosy.\n\nFolk dietetic:\n\nFruit is eaten raw as salad, also as cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity152.\n\nBiodiversity status: Frequent\n\nDiplasium (Athyriaceae)\n\nD.esculentum (Retz.)Sw.\n\nB.\u2013Dhekipata\/Dhekiasak; E.\u2013 Vegetable Fern.\n\nParts used: Younf Frond\/leaf and rhizome\n\nMedicinal uses:\n\ncough, asthma, phthisis, fever, dyspepsia, stomachache, diarrhea and as antidysenteric,\n\nFolk dietetic:\n\nYoung immature fronds are wiped with a cloth to remove red petiolar scales and cut into pieces fried in cooking oil and eaten as sak. Also mixed with fishes and made into fish- curry \u2013most popular in Assam and North-east, also in sub Himalayan India.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity.149a\n\nBiodiversity status: Frequent.\n\nNote: One of the unique fern edible almost all southern slope states of India, uncultivated and collected from wild source with immense market value\n\nEchinochloa (Poaceae)\n\nE.crusgalli (L.) P.Beauv.(syn. Panicum crus-galli L.\n\nB.\u2013 Burashama; E.- Barnyard Millet, Barnyardgrass .\n\nParts used: Seeds, ,Young Leaves.\n\nMedicinal uses\n\nfolk remedy for carbuncles, haemorrhages, sores,spleen, cancer wounds.\n\nFolk dietetic :\n\nSeeds are boiled and eaten as a substitute of rice in time of scarcity of food. Some parts of Northern India grains used to make pudding. The young shoots are eaten as a vegetable during food scarcity. Used as fodder.\n\nNutrition: almost like rice, *****\n\nAntioxidant Potentiality:\n\nModerate activity showing antihypercholesterol activity in animal model.152a\n\nBiodiversity status: Frequent, especially in wet places, throughout.\n\nEclipta (Asteraceae)\n\nE. prostrata (L.) L.\n\nB.- Kesut; E.- False Daisy.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation, Kalokesurja Taila, a hair tonic. The decoction of the whole plant helps to calm the mind from excessive activity, headache and soothes noisy sleep. It is mainly used in hair oils.\n\nAntioxidant Potentiality: Moderate activity153.\n\nBiodiversity status: Frequent.\n\nEichhornia (Pontederiaceae)\n\nE. crassipes (Mart.) Solms\n\nB.- Kachuripana; E.- Water Hyacinth.\n\nParts used: Flowers.\n\nMedicinal uses:\n\nFlowers are used in skin diseases of horses.\n\nFolk dietetic:\n\nUsed as cattle feed either alone or cut into pieces, boiled and mixed with khesari dal (Cajanus cajan) for lactation and nutrition. Nutrition: **\n\nAntioxidant Potentiality: Low activity154.\n\nBiodiversity status: Frequent.\n\nElaeocarpus (Elaeocarpaceae)\n\nE. serratus L.\n\nB.- Jalpai; E.- Indian Olive.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves used in rheumatism and antidote to poison.\n\nFolk dietetic:\n\nFruits eaten; made into chutny and pickle. Nutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity155.\n\nBiodiversity status: Frequent.\n\nElephantopus (Asteraceae )\n\nE. scaber L.\n\nB.- Hastipada; E.- Prickly-leaved elephant's foot.\n\nParts used: roots and leaves.\n\nMedicinal uses:\n\nRoots and leaves are used in diarrhoea, dysentery recently reported as anti-inflamatory.\n\nAntioxidant Potentiality: Low activity156.\n\nBiodiversity status: Frequent.\n\nEleusine (Poaceae)\n\nE. indica (L.) Gaertn.\n\nB.- Jabra ghas; E.- Indian Goose Grass.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe roots are used in case of dysentery. Leaves are used against snake bite 10 gms. of leaf mixed with 10gms.leaves of Kylinga nemoralis and made paste and given orally to patient as antidote in snake bite (Late G.R. Bakshi of Raiganj used this practice).\n\nFolk dietetic: Used as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity157.\n\nBiodiversity status: Frequent.\n\nEnhydra (Asteraceae)\n\nE. fluctuans Lour.\n\nB.- Hyalencha; E.- Marsh Herb.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe leaves are used as vegetable. The leaves and stem are used as blood strengthening tonic and also used in flatulence and constipation.\n\nFolk dietetic:\n\nEaten as cooked\/fried vegetables.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity158.\n\nBiodiversity status: Frequent.\n\nErvatamia (Apocynaceae)\n\nE. coronaria (Jacq.) Stapf.\n\n(Syn. Tabernaemontana divaricata)\n\nB.- Kathmalati\/Tagar; E.- East Indian Rosebay.\n\nParts used: Bark.\n\nMedicinal uses:\n\nStem bark used in skin diseases.\n\nAntioxidant Potentiality: Low activity159.\n\nBiodiversity status: Frequent.\n\nErythrina (Leguminosae)\n\nE. variegata L.\n\nB.- Madar; E.\\- Coral Tree.\n\nParts used: Root & bark.\n\nMedicinal uses:\n\nUsed in physical weakness, musculoskeletal disorder, fever and dissolves kidney stones.\n\nFolk dietetic:\n\nLeaves as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity160.\n\nBiodiversity status: Frequent.\n\nEucalyptus (Myrtaceae)\n\nEucalyptus sp.\n\nB.- Eucalyptus; E.- Eucalyptus.\n\nParts used: Bark.\n\nMedicinal uses:\n\nEucalyptus oil is known to be a good medicine for relieving nasal congestion in cold.\n\nAntioxidant Potentiality: High activity161.\n\nBiodiversity status: Frequent.\n\nEupatorium (Asteraceae)\n\nE. odoratum L.\n\n(Syn: Chromolaena odorata)\n\nB.\u2013 German Lata; E.- Siam Weed.\n\nParts used: Leaf.\n\nMedicinal uses:\n\nLeaves used to stop bleeding from wounds.\n\nAntioxidant Potentiality: Moderate activity162.\n\nBiodiversity status: Frequent.\n\nE. triplinerve Vahl.\n\n(Syn: Ayapana triplinervis)\n\nB.\\- Ayapan; E.- Yapana\/White Snakeroot.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe decoction of the whole plant helps to calm the mind from excessive activity, headache and soothes noisy sleep.\n\nFolk dietetic:\n\nSometimes consumed as vegetable. Nutrition: **\n\nAntioxidant Potentiality:\n\nModerate activity163.\n\nBiodiversity status: Frequent.\n\nEuphorbia (Euphorbiaceae)\n\nE. hirta L.\n\nB.- Dudhe Jhar\/Dudhiya; E.- Asthma Weed.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is used to treat abdominal pain, bronchitis, asthma and intestinal amoebic dysentery.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Moderate activity164.\n\nBiodiversity status: Frequent.\n\nE. neriifolia L.\n\nB.- Patsaji; E.- Manasa Sij.\n\nParts used: Stem.\n\nMedicinal uses:\n\nUsed in leucorrhoea.\n\nAntioxidant Potentiality:\n\nModerate activity165.\n\nBiodiversity status: Frequent.\n\nE. tirucalli L.\n\nB.- Jata Lanka; E.- Pencil Tree, petrolium plant.\n\nParts used: Stem.\n\nMedicinal uses:\n\nJuice of the stem is useful in gonorrhoea, whooping cough, asthma, dropsy and leprosy Latex is toxic to fish, birds and rats. Reported to be anti-cancerous.\n\nAntioxidant Potentiality: High activity166.\n\nBiodiversity status: Frequent.\n\nEuryale (Nymphaeaceae)\n\nE. ferox Salisb.\n\nB.- Makhna; E.- Fox nut.\n\nParts used: Seed.\n\nMedicinal uses:\n\nTreatment in conditions like spermatorrhea, prevents early ejaculation, neuralgia, incontinence, Local Kabirajes prescribe in preventing early discharge of semen and restores sexual vigor in older men. Makhna helps in conditions like arthritis,dysfunction of respiration. This herb has antioxidant properties and helps in digestion specially in persons unable to ingest any food. rejuvenates respiratory system and prevents frequent urination, increases libido and helps in female infertility\n\nFolk dietetic:\n\nSeeds are fried and made into popcorn (locally known Makhna khoi) and given to patients with adverse stomach illness.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity166a.\n\nBiodiversity status: Less frequent and cultivated.\n\nEvolvulus (Convolvulaceae)\n\nE. nummularius (L.) Mill.\n\nB.- Musakarni; E.- Roundleaf Bindweed.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nPlant ash is used on cuts and bruises.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity167.\n\nBiodiversity status: Frequent.\n\nFicus (Moraceae)\n\nF. benghalensis L.\n\nB.- Bot; E.-Banyan tree.\n\nParts used: Bark, aerial root, tender leaves and latex.\n\nMedicinal uses:\n\nRoot used for gonorrhoea; bark in piles of females, watery semen, burning sensation of body, leucorrhoea; bark paste is used as emollient in sprains; tender leaves used in cracked feet, boils; gum useful in toothache.\n\nFolk dietetic:\n\nYoung buds consumed as vegetable.\n\nNutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity168.\n\nBiodiversity status: Frequent.\n\nF. bejamina L.\n\nB.- Pakur; E.- Weeping Fig.\n\nParts used: Leaves, bark & root.\n\nMedicinal uses:\n\nRoot and bark are used for gonorrhoea, nerve weakness, mouth ulcers; leaves used in treating pus in ear.\n\nFolk dietetic:\n\nLeaves as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Moderate activity169.\n\nBiodiversity status: Frequent.\n\nF. racemosa L.\n\nB.- Dumur; E.- Cluster Fig Tree.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nThe fruit is used as a vegetable. Unripe fruits used in dropsy, flatulence, anaemia, liver and kidney disorders, chronic rheumatism anorexia, and also act as blood purifier, tonic. Fruit reported as anticarcinogenic. After chemotherapy in patients used as dietetic food for early recovery.\n\nFolk dietetic:\n\nConsumed as vegetable. Made in pickle or marmalade or dried use; Given to cattle for inducing lactation.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity170.\n\nBiodiversity status: Frequent.\n\nF. hispida L.f.\n\nB.- Khoksha; E.- Hairy Fig.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nUsed only in absence of Ficus glomerata for the same reasons. Folk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity171.\n\nBiodiversity status: Frequent.\n\nF. religiosa L.\n\nB.- Aswattha; E.- Buddha Tree.\n\nParts used: Bark.\n\nMedicinal uses:\n\nUseful in fever, herpes, sores, boils, wound, granulation and pigmentation, splint in fracture and dislocations.\n\nFolk dietetic:\n\nYoung buds collected, boiled and the excess water is decanted off and fried and eaten.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity172.\n\nBiodiversity status: Frequent.\n\nFlacourtia (Flacourtiaceae)\n\nF. indica (Burm. f.) Merr.\n\nB.- Bainchi\/Paniaia; E.- Governor's Plum.\n\nParts used: Fruits, leaves & roots.\n\nMedicinal uses:\n\nThe fruits and leaves are used against diarrhoea. Fruits are used in jaundice, burning sensation of body, prevent vomiting; root oil in tooth-ache, itches and acne.\n\nAntioxidant Potentiality: High activity173.\n\nBiodiversity status: Frequent.\n\nGarcinia (Clusiaceae)\n\nG. mangostana L.\n\nB.- Chayna Gab; E.- Mangosteen.\n\nParts used: Fruit rind, bark & leaves.\n\nMedicinal uses:\n\nThe fruit rind is ground and used in the treatment of diarrhoea, dysentery and for skin diseases. A tea made from the leaves and bark is used to lower fever and for urinary disorders.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity174.\n\nBiodiversity status: Frequent.\n\nGeodorum (Orchidaceae)\n\nG. densiflorum (Lam.) Schlt.\n\nB.\u2013 Vui orchid; E.- Nodding Swamp Orchid.\n\nParts used: Bulbs.\n\nMedicinal uses:\n\nBulb paste mixed with sugar used for diabetes.\n\nAntioxidant Potentiality: High amount phyto-chemicals present175.\n\nBiodiversity status: Endangered.\n\nGlinus (Molluginaceae)\n\nG. oppositifolius (L.) Aug. DC.\n\nB.- Gima Sak; E.- Bitter Cumin.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed in hepatic disorder, urinary disorders of female, indigestion and act as appetiser. The decoction used in clearing uterus.\n\nFolk dietetic:\n\nFried eaten or mixed with potato\/brinjal to prepare curry. Nutrition: ****\n\nAntioxidant Potentiality: High activity176.\n\nBiodiversity status: Frequent.\n\nGloriosa (Colchicaceae)\n\nG. superba L.\n\nB.- Ulatchandal; E.- Glory Lily.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoot used in ulcers, leprosy, piles, inflammations, abdominal pain, itching and used as abortifacient.\n\nAntioxidant Potentiality: High activity177.\n\nBiodiversity status: Frequent.\n\nGlycosmis (Rutaceae)\n\nG. pentaphylla (Retz.) DC.\n\nB.- Atiswar; E.- Orangeberry.\n\nParts used: Root & stem.\n\nMedicinal uses:\n\nRoot paste is used in bone fracture, ankle pain, gout and rupturing boils. Stem used as tooth-brush.\n\nFolk dietetic:\n\nTender leaves consumed by goat. Nutrition: ***\n\nAntioxidant Potentiality: High activity178.\n\nBiodiversity status: Frequent.\n\nGmelina (Lamiaceae)\n\nG. arborea Roxb.\n\nB.- Gamari; E.- Goomar teak.\n\nParts used: Fruit, bark & root.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Dashamularistha and Salsa. The root and bark are stomachic, astringent, galactagogue, laxative and anthelmintic; Bark is useful in leprosy, and in skin ulcers. The fruit is used as sex stimulant and for watery semen.\n\nAntioxidant Potentiality: High activity179.\n\nBiodiversity status: Frequent.\n\nGossypium (Malvaceae)\n\nG. herbaceum L.\n\nB.- Karpas; E.- Levant Cotton.Parts used: Root.\n\nMedicinal uses:\n\nRoot decoction is used in dysentery, leucorrhoea; root bark in psychic disorders and as a stomachic.\n\nAntioxidant Potentiality: High activity180.\n\nBiodiversity status: Frequent.\n\nGymnema (Apocynaceae)\n\nG. sylvestre (Retz.) R.Br.ex Schult\n\nB.- Gurmar; E.- Gymnema.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe decoction\/dust of the leaves is used as an ingredient in most ayurvedic preparations for treatment of diabetes. Oral intake of fresh leaves (3-4) helps in decreasing of diabetes.\n\nAntioxidant Potentiality:\n\nModerate activity181.\n\nBiodiversity status: Rare.\n\nHelicteres (Malvaceae)\n\nH. isora L.\n\nB.- Aatmora; E.- Indian Screw Tree.\n\nParts used: fruit, root & bark.\n\nMedicinal uses:\n\nFruits used in treating intestinal complaints, colic pains and flatulence. Roots used in diabetes and in convulsions. Root and bark used as antigalactogogue to expectorant.\n\nAntioxidant Potentiality: High activity182.\n\nBiodiversity status: Rare in locality.\n\nHeliotropium (Boraginaceae)\n\nH. indicum L.\n\nB.- Hatisur; E.\\- Indian Heliotrope.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf juice used in conjunctivitis, dysentery and cough. Fresh leaf decoction is applied to wounds, boils and pruritus.\n\nAntioxidant Potentiality: Moderate activity183.\n\nBiodiversity status: Frequent.\n\nHelminthostachys (Ophioglossaceae)\n\nH. zeylanica (L. ) Hook.\n\nB.- Ekbir; E.- Helminthostachys.\n\nParts used: Rhizome & Leaves.\n\nMedicinal uses:\n\nRhizome used in sciatica, dysentery aperients. Leaf juice relieves blister on tongue.\n\nAntioxidant Potentiality:\n\nModerate activity184.\n\nBiodiversity status: Rare in locality.\n\nHemidesmus (Apocynaceae)\n\nH. indicus (L.) R. Br. ex Schult.\n\nB.- Anantamul; E.- Indian Sarsaparilla.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nMajor constituents of an ayurvedic preparations like Saribadi Salsa, a health tonic. The plant is used as blood enhancer and blood purifier and also reduces abdominal problems like stomachache, dyspepsia, anorexia and skin diseases. Decoction of the root is used to promote hair growth. Root paste is applied to treat rheumatic joints and boils and kidney stones.\n\nFolk dietetic:\n\nAs vegetables by tribes.\n\nNutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity185.\n\nBiodiversity status: Frequent.\n\nHemigraphis (Acanthaceae)\n\nH. hirta T. Ander.\n\nB.\u2013 Dudhiya; E.- Hairy Hemigraphis.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe decoction is used in dysentery and mouth ulcer.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nHibiscus (Malvaceae)\n\nH. mutabilis L.\n\nB.- Sthal Padma; E.- Cotton Rosemallow.\n\nParts used: Flower, bark & stem.\n\nMedicinal uses:\n\nUsed in ayurvedic preparation like Chyavanprash. Bark and stem useful in gonorrhoea, urinary problems; as a tonic and flatulence. Flower is used in stangury.\n\nFolk dietetic:\n\nLeaf and flowers used as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity186.\n\nBiodiversity status: Frequent.\n\nH. rosa-sinensis L.\n\nB.- Jaba; E.- China Rose.\n\nParts used: Flower.\n\nMedicinal uses:\n\nFlower is one component of Ashokaristha and Patrangasab, both are female tonic. Flower is used to treat menstruation, and tumour in uterine.\n\nFolk dietetic:\n\nflowers consumed by goat.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity187.\n\nBiodiversity status: Frequent.\n\nHolarrhena (Apocyanaceae)\n\nH. pubescens (L.) Wall.\n\nB.\\- Kurchi; E.- Indrajao.\n\nParts used: Bark & seed.\n\nMedicinal uses:\n\nMajor constituent of an ayurvedic preparation, Kutjaristha, a tonic for chronic dysentery and piles. Bark is used for chronic diarrhoea, chronic dysentery and is also useful in treatment of bleeding piles, skin diseases, and urinary troubles. Seed (named as 'Indrajab') is used in diabetes and guniea worm.\n\nAntioxidant Potentiality: High activity188.\n\nBiodiversity status: Frequent.\n\nHouttuynia (Saururaceae)\n\nH. cordata Thunb.\n\nB.- Anstanag; E.\\- Chameleon Plant.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are used to treat cold, measles, dysentery, indigestion and skin ailments.\n\nAntioxidant Potentiality: Low activity189.\n\nBiodiversity status: Less Frequent.\n\nHygrophila (Acanthaceae)\n\nH. auriculata (Schumach.) Heine. (Syn: H. spinosa)\n\nB.- Kulekhara; E.- Marsh Barbel.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nCooked leaves given for increasing haemoglobin content in blood. The leaf decoction is used in dropsy and insomnia. Seeds are used to dissolve stones in kidney and gall bladder. The plant is used in jaundice, wounds and herpes.\n\nFolk dietetic:\n\nAerial parts eaten as vegetables. Nutrition: *****\n\nAntioxidant Potentiality: Low activity190.\n\nBiodiversity status: Frequent.\n\nHygroryza (Poaceae)\n\nH.aristata (Retz.) Nees ex Wight & Arn.\n\nB.- Uri dhan\/ N\ufffdbar , E.- Bengal wild rice\n\nParts used: seed grains.\n\nMedicinal uses\n\nTo treat weakness of semen 10 grams boiled grains for 3months. Boiled grain applied for skin's natural texture and moisture and in sore throat (Bhattacharyya, 2008). Useful in biliousness, astringent, cooling, anti-inflammatory ; antioxidant190a\n\nFolk dietetic :\n\nTo reduce body fat 25 gms of boiled grains to be taken twice a day for 2 months. This 'aranyadhanya' (in Sanskrit literature) collected from wild source eaten substitute to rice\/wheat. Grains fried made into 'khoi' eaten with milk.\n\nNutrition: whole grain as good as rice, ***** .\n\nAntioxidant Potentiality:\n\nactivity high190a\n\nBiodiversity status: aquatic, common in ponds and ditches, throughout\n\nImperata (Poaceae)\n\nI. cylindrica (L.) Raeusch. (Syn: I. arundinacea)\n\nB.- Ulu ghas; E.- Cogon Grass. Parts used: Root.\n\nMedicinal uses:\n\nRoots are used in stangury, dropsy, flatulence and excessive thirst.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nIndigofera (Leguminosae)\n\nI. astragalina DC.\n\nB.- Neel; E.- Silky Indigo.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is used in flatulence, dropsy, stangury, piles; root bark is used in piercing pain.\n\nFolk dietetic:\n\nAs cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity191.\n\nBiodiversity status: Frequent.\n\nIpomoea (Convolvulaceae)\n\nI. aquatica Forssk.\n\nB.- Kalmi; E.- Water Spinach.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaf decoction reduces increases lactation. It is useful in treatment of boils, gonorrhoea and also relieves painful stings.\n\nFolk dietetic:\n\nEaten as fried\/boiled or cooked with other vegetables or with fish\/meat.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity192.\n\nBiodiversity status: Frequent.\n\nI. paniculata R.Br.\n\n(Syn: Jacquemontia paniculata )\n\nB.- Bhui Kumra; E.- Milky Yam. Parts used: Rhizome.\n\nMedicinal uses:\n\nThe rhizome is used in case of urination, problem, digestive problem and it induces menstruation as well as it boosts sexual desire.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity statuts: Frequent.\n\nI. quamoclit L.\n\nB.- Tarulata; E.- Star Glory.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nIt is used in case of dysentery, blood coughing, piles and body weakness.\n\nAntioxidant Potentiality:\n\nModerate activity193.\n\nBiodiversity status: Frequent.\n\nIxora (Rubiaceae)\n\nI. coccinea L.\n\nB.- Lal Rangan ; E.- Jungle Geranium.\n\nParts used: Roots & flowers.\n\nMedicinal uses:\n\nRoots and flowers together used in case of stomachache, diarrhoea and dysentery.\n\nFolk dietetic:\n\nFlowers consumed by goat Nutrition: **\n\nAntioxidant Potentiality:\n\nModerate activity194.\n\nBiodiversity status: Frequent.\n\nJasminum (Oleaceae)\n\nJ. multiflorum (Burm. f.) Andre.\n\nB.- Jui; E.- Star Jasmine.\n\nParts used: Leaves & root.\n\nMedicinal uses:\n\nRoot bark decoction is used to prevent tapeworm; leaves used in herpes.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nJ. sambac (L.) Ait.\n\nB.- Beli; E.- Arabian Jasmine.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction used in anorexia; leaves used in mouth ulcers, acne.\n\nAntioxidant Potentiality: Low activity195.\n\nBiodiversity status: Frequent.\n\nJatropha (Euphorbiaceae)\n\nJ. curcas L.\n\nB.- Sada Varenda; E.- Barbados Nut.\n\nParts used: Latex.\n\nMedicinal uses:\n\nLatex is used to treat toothache. Seed used for the production of biodiesel.\n\nFolk dietetic: Cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity196.\n\nBiodiversity status: Frequent.\n\nJ. gossypiifolia L.\n\nB.- Lal Varenda; E.- Bellyache Bush.\n\nParts used: Leaf, Latex.\n\nMedicinal uses:\n\nHighly toxic to people and animals. Medicinal use not reported here.\n\nFolk dietetic: Cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity197.\n\nBiodiversity status: Frequent.\n\nJusticia (Acanthaceae)\n\nJ. adhatoda L.\n\nB.- Basak; E.- Malabar Nut.\n\nParts used: Leaves & stems.\n\nMedicinal uses:\n\nA decoction of the leaves is used for cold and cough, bronchitis, tuberculosis and other lung and bronchiole disorders and even chicken pox. The leaves are applied to wounds for their antibacterial and anti-inflammatory properties.\n\nAntioxidant Potentiality: Low activity198.\n\nBiodiversity status: Frequent.\n\nJ. gendarussa Burm.\n\nB.- Jagatmadan; E.- Gandarusa.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nIt is beneficial for respiratory disorders like cough, cold, bronchitis and chronic rheumatism.\n\nAntioxidant Potentiality: Low activity199.\n\nBiodiversity status: Frequent.\n\nJ. spicigera Schltdl.\n\nB.- Lal Basak; E.- Mohintli.\n\nParts used: Leaves & stems.\n\nMedicinal uses:\n\nActs as stimulant & antidysenteric. Used as an anti-inflammatory for the treatment of uterine cancer, gastrointestinal disorders like nausea, stomachache, cramps and diarrhea.\n\nAntioxidant Potentiality: Moderate activity200.\n\nBiodiversity status: Less Frequent.\n\nKaempferia (Zingiberaceae)\n\nK. angustifolia Roscoe.\n\nB.- Mudunirbish; in malaysia known as Kunchi pepet.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nExcellent fragrance and is usually used as medicine to treat cold, stomach-ache, and dysentery, while its rhizome is used for coughs and as mastication. Rhizome used as vegetable. Nutrition: ***\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Less Frequent.\n\nK. galanga L.\n\nB.- Ekangi\/Bhui Chapa; E.- Aromatic Ginger.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nUsed as expectorant, carminative, diuretic. Used in catching fish.\n\nAntioxidant Potentiality: Low activity201.\n\nBiodiversity status: Less Frequent.\n\nKalanchoe (Crassulaceae)\n\nK. pinnata (Lam.) Pers. (Syn. Bryophyllum pinnatum)\n\nB.- Patharkuchi; E.- Air Plant.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe leaves are used to dissolve kidney stones, calculus and the leaf juice is also used to treat cold and cough, flatulence, dropsy, acidity, abdominal pain of children, epilepsy and acts as anti-diuretic.\n\nFolk dietetic: As cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: Moderate activity202.\n\nBiodiversity status: Frequent.\n\nLagerstroemia (Lythraceae)\n\nL. speciosa (L.) Pers.\n\nB.- Jarul; E.- Queen Crape Myrtle.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation like Dashamularistha. Bark used as a tonic, blood purifier, blood enhancer, cold and cough. Bark and leaves are purgative; decoction of dried leaves is used in diabetes.\n\nAntioxidant Potentiality: High activity203.\n\nBiodiversity status: Frequent.\n\nLannea (Anacardiaceae)\n\nL. coromandelica (Houtt.) Merr.\n\nB.- Jiga; E.\\- Indian Ash Tree.\n\nParts used: Bark & gum.\n\nMedicinal uses:\n\nBark is useful in diarrhoea and dysentery, cardiac diseases. The gum is used as sperm enhancer.\n\nAntioxidant Potentiality:\n\nModerate activity204.\n\nBiodiversity status: Frequent.\n\nLantana (Verbenaceae)\n\nL. camara L.\n\nB.- Chotra; E.- Lantana.\n\nParts used: Leaf.\n\nMedicinal uses:\n\nThe plant decoction is used in rheumatism and for the treatment of measles.\n\nAntioxidant Potentiality: High activity205.\n\nBiodiversity status: Frequent.\n\nLasia (Araceae)\n\nL. spinosa (L.) Thw.\n\nB.- Jokha\/Kanta Kachu; E.- Lasia.\n\nParts used: Root & petiole.\n\nMedicinal uses:\n\nRoot and Petioles are used in throat infection of cattle.\n\nAntioxidant Potentiality:\n\nModerate activity206.\n\nBiodiversity status: Frequent.\n\nLawsonia (Lythraceae)\n\nL. inermis L.\n\nB.- Mehendi; E.- Henna.\n\nParts used: Leaf & flower.\n\nMedicinal uses:\n\nThe leaf paste are useful in shoulder pain, burns, leucorrhoea, dandruff, mouth ulcers, steam and is also anti inflammatory. It is also used as hair tonic. Flowers are useful in insomnia.\n\nFolk dietetic:\n\nTender leaves for cattle feed. Nutrition:***\n\nAntioxidant Potentiality: High activity207.\n\nBiodiversity status: Frequent.\n\nLeea (Vitaceae)\n\nL. macrophylla Roxb. ex Hornem.\n\nB.- Hastikarnapalash; E.- Leea.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoot decoction used to control ringworm and root paste externally applied for pain. Exhibits anti-cancerous effect Leaf juice is recognized as anti-inflammatory agent and used in boils, arthritis, gout and rheumatism . The dried powder of root with clarified butter is prescribed in morning as age sustainer.\n\nAntioxidant Potentiality:\n\nLow activity (presence of phenolics)208.\n\nFolk dietetic: Leaves of the plant are used as vegetable by tribal people.\n\nNutrition : Root ****\n\nBiodiversity status: Rare.\n\nLeucas (Lamiaceae)\n\nL. aspera (Willd.) Link.\n\nB.- Dandakalash\/Dulfi\/Swetdron; E.- Common Leucas.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaves are used to treat ulcers in tongue and also in biliousness, anorexia, and it acts as a stomachic.\n\nAntioxidant Potentiality: Low activity209.\n\nBiodiversity status: Frequent.\n\nLimonia (Rutaceae)\n\nL. acidissima Groff.\n\nB.- Kat Bel; E.- Wood Apple.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nBark and fruits are used as tonic, laxative, blood enhancer, culminates acidity, lack of appetite, dysentery; Juice of unripe fruits are useful in treating acne.\n\nFolk dietetic:\n\nAs fruit. Nutrition: ****\n\nAntioxidant Potentiality: Moderate activity210.\n\nBiodiversity status: Frequent.\n\nLitsea (Lauraceae)\n\nL. glutinosa (Lour.) C.B. Rob.\n\nB.\u2013 Darodmoyda\/Pipulti; E.- Indian laurel.\n\nParts used: Leaves and bark.\n\nMedicinal uses:\n\nLeaf decoction used in diarrhea, dysentery and also for excessive semen flow in young boys; decoction of bark applied to sores, scabies and in pains.\n\nFolk dietetic:\n\nTender leaves as goat feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity211.\n\nBiodiversity status: Frequent.\n\nMadhuca (Sapotaceae)\n\nM. longifolia (Koen. ex L.) Mac Bride\n\nB.- Mahua; E.- Mahua.\n\nParts used: Seed, fruit, flower & bark.\n\nMedicinal uses:\n\nSeed oil used in headache, rheumatism, cuts and wounds; flowers and fruits are used in cold and cough, anorexia, insomnia; bark is used in diabetes. Used as a preservative in Ayurvedic preparations as it produces alcohol.\n\nFolk dietetic:\n\nFlowers used in preparation of indigenous liquor by tribes. Tender leaves as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity212.\n\nBiodiversity status: Frequent.\n\nMallotus (Euphorbiaceae)\n\nM. philippensis (Lam.) Muell. Arg.\n\nB.- Kamalaguri; E.- Rottlera.\n\nParts Used: Fruits.\n\nMedicinal uses:-\n\nThe dried fruits are dusted and applied externally to rupture boils and in puss.\n\nAntioxidant Potentiality: Moderate activity213.\n\nBiodiversity status: Frequent.\n\nMangifera (Anacardiaceae)\n\nM. indica L.\n\nB.- Aam; E.- Mango.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nSeeds and tender leaves are the major constituents of an ayurvedic preparation, Asho- karistha. Leaf extracts are used as antiseptic in the treatment of burns, dandruff control and early graying of hair. The bark is useful in dysentery and diarrhoea Ripe fruit for hemorrhage from internal organs, laxative, diuretic, astringent. Bark in uterine bleeding (CNC).\n\nFolk dietetic:\n\nFruits of varied taste available in several varieties. 'amsatya' a dried mango pulp produce often made - which can be stored for several years. Green mango slices without seed, sun dried and slightly fermented called 'amchur' can also be kept for years. 'amsatya' best eaten with milk. 'amchur' eaten as chatney or mixed in curry. Fresh green eaten as chatney or raw. The dried kernel pieces used as poultry feed, pig feed or cattle feed. pulp rich in soluble sugars (70-90%), mostly fructose (60-67% ).Crude fibre 2%-16%. Green mango grilled and made into delicious chatney.\n\nLeaves as cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity214.\n\nBiodiversity status: Frequent Cultivated.\n\nManihot (Euphorbiaceae)\n\nM. esculenta Crantz.\n\nB.- Tapioca; E.- Cassava.\n\nMedicinal uses:\n\nTubers are eaten as vegetable.\n\nNutrition: ***\n\nAntioxidant Potentiality: Moderate activity215.\n\nBiodiversity status: Frequent.\n\nMarsilea (Marsileaceae)\n\nM. quadrifolia L.\n\nB.- Susni; E.- Europian Waterclover.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed in asthma, inosomnia, blood pressure, epilepsy, insect stings as well as reduces inflammation of urinary tracts.\n\nFolk dietetic:\n\nFried leaf and petiole is mixed with potato and eaten.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity216.\n\nBiodiversity status: Frequent.\n\nMartynia (Martyniaceae)\n\nM. annua L.\n\nB.- Baghnakhi; E.- Devill's Claw.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf juice used as gargle for sore throat.\n\nAntioxidant Potentiality:\n\nModerate activity217.\n\nBiodiversity status: Frequent.\n\nMelia (Meliaceae)\n\nM. azedarach L.\n\nB.- Ghora Neem; E.- Indian Lilac.\n\nParts used:- Bark & fruit.\n\nMedicinal uses:-\n\nMajor constituent of an ayurvedic preparation, Patrangasab. Bark is effective in breast tumour, biliousness, piles, leucorrhoea and constipation. Seed-oil is used in rheumatism. Wood-extract is used in asthma.\n\nAntioxidant Potentiality: High activity218.\n\nBiodiversity status: Frequent.\n\nMelothria (Cucurbitaceae)\n\nM. heterophylla (Lour.) Cogn. (Syn: Solena amplexicaulis)\n\nB.- Rakhal Sosa; E.- Melothria. Parts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation like Mritasanjibanisura. It is a blood purifier and enhancer. It is also used in anorexia and insomnia.\n\nFolk dietetic:\n\nFruit used as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity219.\n\nBiodiversity status: Frequent.\n\nMentha (Lamiaceae)\n\nM. arvensis L.\n\nB.- Pudina; E.- Wild Mint\/Corn Mint.\n\nParts used: Leaves, flower & oil.\n\nMedicinal uses:\n\nActs as carminative, pain killer and mouth refresher and used in jaundice. Leaf extract mixed with salt given in dehydration.\n\nFolk dietetic:\n\nIngredient of chutny. Leaf made into paste mixing with salt and pepper and eaten raw with rice by indigenous people.\n\nNutrition: *****\n\nAntioxidant Potentiality: Low activity220.\n\nBiodiversity status: Less frequent and cultivated.\n\nMesua (Calophyllaceae)\n\nM. ferrea L.\n\nB.- Nageswar\/Nagkesar; E.- Indian Rose Chestnut.\n\nParts used: Flower, stamens & seed.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparation like Bhaskarlaban and Salsa. Flowers is used as carminative, clears constipation, relieves vomiting. Also used as blood purifier and blood enhancer. The stamens and flowers are used in the form of powder to treat diarrhoea, cough, leucorrhoea; seed oil is used in rheumatism, sores, scabies.\n\nAntioxidant Potentiality: Low activity221.\n\nBiodiversity status: Frequent.\n\nMichelia (Magnoliaceae)\n\nM. champaca L. (Syn: Magnolia champaca)\n\nB.- Chapa; E.- Champac\n\nParts used: Leaves, bark & flowers.\n\nMedicinal uses:\n\nThe leaves are used in stomachache; bark in headache; flower in vomiting, flatulence, boils.\n\nFolk dietetic:\n\nTender leaves as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity222.\n\nBiodiversity status: Frequent.\n\nMikania (Asteraceae)\n\nM. scandens (L.) Willd.\n\nB.- Assam Lata; E.- Climbing hempvine .\n\nParts used: Leaf.\n\nMedicinal uses:\n\nLeaves used to stop bleeding from wounds. Used in stomach ulcers, inflammation and gastritis problems.\n\nFolk dietetic:\n\nUsed as vegetables by tribes. Extensively consumed by rhino and other wild herbivores. Nutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity222(a).\n\nBiodiversity status: Frequent.\n\nMimosa (Leguminosae)\n\nM. pudica L.\n\nB.- Lajjabati; E.- Touch-Me-Not.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaves are used in toothache and gum bleeding, ear pus, piles, dysentery; root is used in leucorrhea; Leaf decoction is used in breast cancer.\n\nFolk dietetic:\n\nTender leaves as cattle & goat feed. Nutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity223.\n\nBiodiversity status: Frequent.\n\nMimusops (Sapotaceae)\n\nM. elengi L.\n\nB.- Bakul; E.- Bullet Wood Tree.\n\nParts used: Bark, flower.\n\nMedicinal uses:\n\nThe bark, flowers, fruits and seeds are aused as astringent, cooling and anthelmintic. Bark decoction is used in pyorrhea, bad odour of mouth, bleeding of gums and loose teeth. Extract of flowers are used in leucorrhoea and menorrhagia. The bark is useful in stangury.\n\nFolk dietetic:\n\nFruits are eaten by poor village children. Favoured by birds\n\nLeaves as cattle feed. Nutrition: **\n\nAntioxidant Potentiality: High activity224.\n\nBiodiversity status: Frequent.\n\nMirabilis (Nyctaginaceae)\n\nM. jalapa L.\n\nB.- Sandhyamalati; E.- Four 'O' Clock plant.\n\nParts used: Leaves & root.\n\nMedicinal uses:\n\nLeaves are used in cough; leaf paste used as emollient on boils; root used in piles and general debility.\n\nAntioxidant Potentiality: Low activity225.\n\nBiodiversity status: Frequent.\n\nMollugo (Molluginaceae)\n\nM. pentaphyla L.\n\nB.\u2013 Khetpapra; E.- Five Leaved Carpetweed.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUsed as pot herb; as laxative and stomachic.\n\nNutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity226.\n\nBiodiversity status: Frequent.\n\nMomordica (Cucurbitaceae)\n\nM. charantia L.\n\nB.- Karola; E.- Bitter Gourd.\n\nParts used: Leaves & fruits.\n\nMedicinal uses:\n\nLeaf decoction is used to treat small pox, malarial fever, bilousness; leaves and fruits in anorexia, diabetes, allergy and guinea worms. Decoction of fruit used in hyperglycemia.\n\nFolk dietetic:\n\nAs wholesome vegetable. Nutrition: *****\n\nAntioxidant Potentiality: High activity227.\n\nBiodiversity status: Frequent and cultivated.\n\nM. cochinchinensis (Lour.) Spreng.\n\nB.- Baro Kakrol; E.- Spiny Bittergourd.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nLeaf decoction used in anaemia; root paste stops hairfall; seeds in normal cold and cough. And also used as like M. charantia.\n\nFolk dietetic: Like M. dioica.\n\nNutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity227(a).\n\nBiodiversity status: Rare.\n\nM. dioica Roxb. ex Willd.\n\nB.- Kakrol; E.\u2013 Spiny Gourd.\n\nParts used: Fruit and Root.\n\nMedicinal uses:\n\nToasted root is used to stop bleeding from piles and in urinary disorders. Fruit used in diabetes.\n\nFolk dietetic:\n\nAs wholesome vegetable. Nutrition: *****\n\nAntioxidant Potentiality: High activity228.\n\nBiodiversity status: Frequent.\n\nMoringa (Moringaceae)\n\nM. oleifera Lam.\n\nB.- Sajna; E.- Moringa.\n\nParts used: Leaves & root bark.\n\nMedicinal uses:\n\nUseful to treat high blood pressure, tumour, hiccups, piles, leprosy, ringworms, cold and fever, swelling of gums and eye problems. Leaf juice is used in diabetes.\n\nFolk dietetic:\n\nLeaves fried and eaten as side dish; flower mixed with batter fried eaten. Leaf decoction given to stop hiccup.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity229.\n\nBiodiversity status: Frequent.\n\nMucuna (Leguminosae)\n\nM. pruriens (L.) DC.\n\nB.- Alkusi; E.- Velvet Bean\/Cowhage.\n\nParts used: Seed & roots.\n\nMedicinal uses:\n\nThe seeds are the main constituent of an ayurvedic preparation, Banarboti, a highly sex stimulating preparation. It is also used in another Ayurvedic preparations like Salsa and Mritasanjibanisura. It is also used for menorrhagia and spermatorrhoea. It is one of the best tonics to improve vitality. The decoction of pod is a powerful nervine tonic, anthelmintic, aphrodisiac. The root powder is used as laxative.\n\nFolk dietetic:\n\nLeaves as cattle feed. Nutrition: *****\n\nAntioxidant Potentiality: Moderate activity230.\n\nBiodiversity status: Frequent.\n\nMurraya (Rutaceae)\n\nM. koenigii (L.) Sprengel.\n\nB.- Curry Pata; E.- Curry Plant.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nLeaves are digestive, stimulant and also used for diarrhoea, dysentery and to check vomiting. Bark paste is antiseptic, applied to skin eruptions.\n\nFolk dietetic:\n\nUsed as spices in various dishes and mixed with batter fried and eaten.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity231.\n\nBiodiversity status: Frequent.\n\nM. paniculata (L.) Jack.\n\nB.- Kamini; E.- Orange Jessamine.\n\nParts used: Leaves & root\n\nMedicinal uses:\n\nRoot bark is used in acute dysentery; boiled mashed leaves used in dropsy; leaf decoctions along with root paste relieve pains on application.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nMusa (Musaceae)\n\nM. paradisiaca L.\n\nB.- Kala; E.- Banana.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nAn unripe fruit helps in treating diarrhoea.\n\nFolk dietetic:\n\nAs fruit. Leaves as fodder. Young core of inflorescence bases (locally known as vadal\/ thod) cut into pieces, fried and eaten as vegetable meal served on the leaf believed to be hygienic. The flowers in spadix is also eaten.\n\nNutrition: ****\n\nAntioxidant Potentiality: Low activity232.\n\nBiodiversity status: Frequent.\n\nM. sapientum var. paradisiaca L.\n\nB.\u2013 Kach Kala ; E.\u2013 The Plantain.\n\nParts used: Fruit, Inflorescence, stem core\n\nMedicinal uses:\n\nAn unripe fruit helps in treating diarrhea and amoebiasis. Nutritious , Inflorescence used in diabetes.\n\nFolk dietetic: fruit boiled in rice given to patients having fever. The young flower petals cut and made into delicious curry. The white core of the tree cut into pieces sold in market for vegetable. Grilled unripe fruit given to post natal care.\n\nNutrition: **** (USDA)\n\nAntioxidant Potentiality: high232a\n\nBiodiversity status: Frequent.\n\nNelumbo (Nelumbonaceae)\n\nN. nucifera Gaertn.\n\nB.- Padma; E.- Lotus.\n\nParts used: Whole plant, seed\n\nMedicinal uses:\n\nThe leaves are useful in rectal and uterus prolepsis. The flower decoction used in angina pectoris. The nectar is quite useful in cataract.\n\nFolk dietetic: Young seeds are eaten like nut. Often the fruit ovary cooked eaten. Nectar is given to child for development of immunity.\n\nNutrition: ****\n\nAntioxidant Potentiality: Moderate activity233.\n\nBiodiversity status: cultivated \/less Frequent.\n\nNeolamarckia (Rubiaceae)\n\nN. cadamba (Roxb.) Bosser.\n\nB.- Kadam; E.- Kadam.\n\nParts used: Leaves, bark & flower.\n\nMedicinal uses:\n\nThe leaf decoction is used to treat hydrocyl, mouth ulcers, round and thread worms. The bark paste is used for tumour and flower used to prevent bad odour in mouth cavity.\n\nFolk dietetic:Flower as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity234.\n\nBiodiversity status: Frequent.\n\nNerium (Apocynaceae)\n\nN. oleander L.\n\nB.- Karabi; E.- Oleander.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are used to treat diabetes along with other plant leaf-juice.\n\nAntioxidant Potentiality: High activity235.\n\nBiodiversity status: Frequent.\n\nNyctanthes (Oleaceae)\n\nN. arbor-tristis L.\n\nB.- Shiuli\/Sephali; E.\\- Night Jasmine.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nLeaves and bark are useful in remittent fever, anorexia, cynus, cold and cough. Leaves are also useful in diabetes, guinea worms, burning sensation of body, biliousness, sciatica and acts as stomachic.\n\nFolk dietetic:\n\nYoung leaves mixed with mustard oil fried and eaten.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity236.\n\nBiodiversity status: Frequent.\n\nNymphaea (Nymphaceae)\n\nN. rubra Roxb.ex.Salisb\n\nB.- Lal Shaluk; E.- Red Water Lily.\n\nParts used: Flower & root (tuber).\n\nMedicinal uses:\n\nA major constituent of some aurvedic preparations like Dashamularistha and Chyavanprash. The roots (tubers) are useful in irregular and excessive menstruation; tubers and flower are useful in post pregnancy weaknesses; flower stalk in anorexia, digestion. Seeds are used to clear constipation. It is also used as blood purifier.\n\nFolk dietetic:\n\nThe long petiole peeled cut into pieces, steamed and cooked and eaten; seeds are used during famine and draught; sand fried seeds made into popcorns; also used as cakes with molasses.\n\nNutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity237.\n\nBiodiversity status: Frequent.\n\nN. nauchali Burm.f.\n\nB.- Neel Shaluk; E.- Blue Water Lily.\n\nParts used: Flower & root (tuber).\n\nMedicinal uses:\n\nThe roots (tubers) are useful in irregular and excessive menstruation; tubers and flower are useful in post pregnancy weaknesses; flower stalk in anorexia, digestion. Seeds are used to clear constipation. Used as blood purifier.\n\nFolk dietetic:\n\nUsed like N. rubra\n\nNutrition: *****\n\nAntioxidant Potentiality: Low activity238.\n\nBiodiversity status: Frequent.\n\nOcimum (Lamiaceae)\n\nO basilicum L.\n\nB.- Babui Tulsi; E.- Sweet Basil.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe leaves are used to treat cold and cough, dysentery, diarrhoea. Flowers are used as carminative, diuretic.\n\nAntioxidant Potentiality:\n\nModerate activity239.\n\nBiodiversity status: Frequent.\n\nO. gratissimum L.\n\nB.\\- Ram Tulsi; E.- African Basil.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nSame as O. basilicum. Antioxidant Potentiality: High activity240. Biodiversity status: Frequent.\n\nO. sanctum L.\n\nB.- Radha Tulsi; E.- Sacred Basil. Parts used: Leaves & seeds.\n\nMedicinal uses:\n\nUsed as an expectorant for cold and cough, bronchitis and also used in scorpion stings. Antioxidant Potentiality: Low activity241. Biodiversity status: Frequent.\n\nO. tenuiflorum L.\n\nB.- Krishna Tulsi; E.- Holy Basil.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nSame as O. basilicum. Antioxidant Potentiality: High activity242. Biodiversity status: Frequent.\n\nOldenlandia (Rubiaceae)\n\nO. corymbosa L.\n\nB.- Khet Papra; E.- Old World Diamond Flower.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nPlant used in liver disorder, skin ulcer, spleen disorder, acne, as blood purifier, in burning sensation of body and thirstiness.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity243.\n\nBiodiversity status: Frequent.\n\nO. diffusa (Will.) Roxb.\n\nB.- Khet Papra; E.- Slender Oldenlandia.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is useful in elephantiasis, fever, inflammations, asthma and ulcers. Plant also used in skin ulcer, spleen disorder and acne.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity244.\n\nBiodiversity status: Frequent.\n\nOphioglossum (Ophioglossaceae)\n\nO. vulgatum L.\n\nB.- Aktir; E.- Adder's tongue.\n\nParts used: Leaves & rhizome.\n\nMedicinal uses:\n\nLeaves and rhizomes are used as a poultice for wounds. A tea made from the leaves is used as a remedy for internal bleeding and vomiting. Folk dietetic: Testy, fried and eaten especially in Darjeeling hills. Nutrition: ****\n\nAntioxidant Potentiality: High activity (Present study).\n\nBiodiversity status: Endangered.\n\nOroxylum (Bignoniaceae)\n\nO. indicum (L) Vent.\n\nB.- Sona; E.- Indian Calosanthes.\n\nParts used: Bark, root & seed.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Dashamularistha and Salsa, a health tonic. The root and bark are stomachic, astringent, galactagogue, laxative and anthelmintic; They are useful in rheumatism like gout and sciatica, leprosy, blood diseases and even in skin ulcers, cold and cough. The seed is used as sex stimulant and for condensation of watery semen.\n\nAntioxidant Potentiality: High activity244.\n\nBiodiversity status: Less frequent.\n\nOryza (Poaceae)\n\nO. sativa L.\n\nB.- Dhan \/Dhanya; E.- the Rice plant\n\nParts used: Rice grains\/ in different forms\n\nMedicinal uses\n\nNutritious, digestive. De-husked grain with bran layer important for antioxidation in intestine space. Rice poultice used in burn-wounds. Oryzanol present in bran layer is anti-lipidomic. Tocoferol good for skin. In Weakness due to malnutrition. Brown rice contains oryzanol in deeper layers of the grain.\n\nFolk dietetic: Flattened rice along with curd prescribed for blood dysentery. Rice washed water (Sun dried rice) demulcent, coolant, in intestine inflammation. Rice mixed with tea taken by tribes of North Bengal. Fermented rice often eaten in villages of rural India. Straw as fodder.\n\nNutrition: ***** .(USDA) 80% carbohydrate. 1.2% fibre. & others.\n\nAntioxidant Potentiality:\n\nmoderate activity244a.\n\nBiodiversity status: More than hundred varieties Cultivated \/O.sativa var. fatua is wild.\n\nOxalis (Oxalidaceae)\n\nO. corniculata L.\n\nB.- Amrul; E.- Yellow Sorrel.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nUsed to treat cold and cough, sciatica and also enhances appetite and improves digestion. Folk dietetic:\n\nBoiled and mixed with rice and eaten by indigenous people. Chutney prepared from leaves, eaten for treatment of acidity and gastritis.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity246.\n\nBiodiversity status: Frequent.\n\nPaederia (Rubiaceae)\n\nP. foetida L.\n\nB.\\- Gandha Vadal; E.- Stinkvine.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are useful against loose motion, dysentery and improves digestive capacity. It is also used in rheumatism. A soup prepared with green leaves along with green banana and fig is ideal diet for people suffering from stomach troubles and intestinal disorders.\n\nFolk dietetic:\n\nAs cattle feed to stop loose motion of cattle;\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity247.\n\nBiodiversity status: Frequent\n\nPandanus (Pandanaceae)\n\nP. amaryllifolius Roxb.\n\nB.- Payespata; E.- Pandan.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nAn aromatic plant. The leaves are used in cooking for good smell.\n\nFolk dietetic:\n\nUsed in rice preparation to enhance the smell.\n\nNutrition: **\n\nAntioxidant Potentiality: High activity248.\n\nBiodiversity status: Rare in locality.\n\nPaspalum ( Poaceae)\n\nP. scrobiculatum L\n\nB.\u2013 Kodo Dhan E.\u2013 Kodo Millet\n\nParts used: Fruit grains\n\nMedicinal uses:\n\nIn scorpion sting, occasionally develops narcotic properties(Chopra, Nayar, Chopra, 1956)\n\nFolk dietetic: Chhaang, a raw beer is prepared from the grains. The drink is popular in Bhutan, Sikkim, Nepal and North Eastern hills. Nutritive and refreshing, very popular for the Trekkers. Commonly believed to cure common cold, fevers, allergic rhinitis, and alcoholism and others. The grain grounded made into flour used in pudding, eaten substitute of rice. Contains 30% fibre.\n\nNutrition: like rice grain , *****\n\nAntioxidant Potentiality: High activity due to polyphenols247a.\n\nBiodiversity status: Cultivated in sub-Himalayan region , wild\n\nPergularia (Apocynaceae)\n\nP. daemia (Forssk.) Chiov.\n\nB.- Ajashringi\/Chagalbati.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Salsa and Dashamularistha, both are health tonic. It is used for the treatment of liver disorders as well as in bronchitis, bodyache, abdominal tumours, food poisoning, constipation, anorexia, leprosy and also acts as a blood purifier and blood enhancer.\n\nAntioxidant Potentiality: High activity249.\n\nBiodiversity status: Frequent.\n\nPersicaria (Polygonaceae)\n\nP. chinensis (L.) H. Gross.\n\nB.- Moicharan; E.- Chinese Knotwood.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nUsed for eye diseases and as a poultice for stomachache.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Less Frequent.\n\nP. glabra (Willd.) M.Gomez (Syn. Polygonum glabrum)\n\nB.- Bish Kantal; E.- Denseflower Knotweed. Parts used: Roots.\n\nMedicinal uses:\n\nThe root is used in case of rheumatism along with other ingredients.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nPhlogacanthus (Acanthaceae)\n\nP. pubinervius T. Ander.\n\nB.- Singha Puccha; E.- Red Nongmangkha.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nUsed to treat bronchitis, tuberculosis and other lung disorders and even chicken pox. Decoction of the leaves is used for cough and other symptoms of cold.\n\nAntioxidant Potentiality: High activity250.\n\nBiodiversity status: Rare.\n\nPhoenix (Arecaceae)\n\nP. sylvestris (L.) Roxb.\n\nB.- Khejur; E.- Date Palm.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nEnhances memory and sleep, nutritive, effective in cold and cough and chronic bronchitis.\n\nFolk dietetic:\n\nAs fruit. Phloem sap collected to prepare indigenous fermented liquor or Khejur Gur. Nutrition: *****\n\nAntioxidant Potentiality: Low activity251.\n\nBiodiversity status: Frequent.\n\nPhyla (Verbenaceae)\n\nP. nodiflora (L.) Greene\n\nB.- Bhuiokhra; E.- Frog fruit.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nFresh juice is applied to bleeding gums. Infusion of leaves and tender stalk is given to children in indigestion and to women. (Post delivery)\n\nAntioxidant Potentiality: High activity252.\n\nBiodiversity status: Frequent.\n\nPhyllanthus (Phyllanthaceae)\n\nP. acidus (L.) Skeels\n\nB.- Labli; E.- Otaheite Gooseberry.\n\nParts used: Leaves & fruit.\n\nMedicinal uses:\n\nThe fruits are used as liver tonic to enrich the blood. The syrup is used as a stomachic; and the seeds are cathartic. A decoction of the leaves is given as sudorific and in cases of gonorrhoea.\n\nAntioxidant Potentiality: High activity253.\n\nBiodiversity status: Frequent.\n\nP. emblica L.\n\nB.- Amloki\/Amla; E.- Indian Gooseberry.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nOne of the ingredient of Trifolachurna, a ayurvedic preparation. Fruit is useful in acidity and conjunctivitis; fruit decoction helps in culminating fever occurring from poisonous stings, alopecia, skin diseases, to maintain healthy hair, hair loss and graying of hair, sore throat, pain, sex stimulant, useful in leucorrhoea, biliary colic, insomnia and also in diabetes.\n\nFolk dietetic:\n\nAs fruit; made into chutny, pickle; the soft part of the fruit mixed with salt, eaten as appetizer.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity254.\n\nBiodiversity status: Frequent.\n\nP. fraternus Webster (Syn. P. amarus)\n\nB.- Bhui Amla; E.- Bhumiamala.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is used in gastric ulcer, stangury, cough, burning sensation of body, hiccups, jaundice, piles and applied as an emollient on sores, scabies.\n\nAntioxidant Potentiality:\n\nModerate activity255.\n\nBiodiversity status: Frequent.\n\nP. reticulatus Poir.\n\nB.- Panishitki\/panijuli; E.- Black-Honey Shrub.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction is used as remedy for bleeding of gum. Leaves are diuretic to infants.\n\nAntioxidant Potentiality: Moderate activity256.\n\nBiodiversity status: Frequent.\n\nPhysalis (Solanaceae)\n\nP. peruviana L.\n\nB.- Tepari; E.- Cape Gooseberry.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is used in stangury, dropsy, dysentery, dyspepsia, anorexia; fruits are diuretic.\n\nAntioxidant Potentiality: High activity257.\n\nBiodiversity status: Frequent.\n\nP. minima L.\n\nB.- Tepari; E.- Ground Cherry.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe juice of the leaves used for earache or pus.\n\nFolk dietetic: as cattle feed\n\nNutrition: ****\n\nAntioxidant Potentiality:\n\nModerate activity258.\n\nBiodiversity status: Frequent.\n\nPiper (Piperaceae)\n\nP. betel L.\n\nB.- Pan; E.- Betel leaf.\n\nParts used: Leaves. Medicinal uses:\n\nLeaf decoction is used in pyrrohoea, herpes, boils and control of hair lice.\n\nFolk dietetic:\n\nLeaf is chewed with areca nut. Leaf & betel nut reported to be inducer to cancer.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity259.\n\nBiodiversity status: Frequent.\n\nP. chaba Hunter\n\nB.- Chai; E.- \"Dee Plee\" pepper. Parts used: Fruits.\n\nMedicinal uses:\n\nFruits are used in cold and cough and in haemorrhoidal infection as well as carminative.\n\nAntioxidant Potentiality: Low activity260.\n\nBiodiversity status: Frequent.\n\nP. longum L.\n\nB.- Pipul; E.- Long Pepper.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe plant is useful in bronchitis, cold and cough, asthma, tonsilitis, goitre, pharyngitis, dysentery, constipation, piles, and also acts as a laxative, blood purifier, antidote and appetizer.\n\nFolk dietetic:\n\nAs spices. Nutrition: ***\n\nAntioxidant Potentiality: Low activity260.\n\nBiodiversity status: Frequent.\n\nP. nigrum L.\n\nB.- Golmorich; E.- Black Pepper.\n\nParts used: Seed.\n\nMedicinal uses:\n\nIt is also used in cold and cough, dysentery, dropsy, controls stomach worms, in insomnia and even poisonous stings. Also used as a tonic, laxative, blood enhancer, appetizer, culminates fever, gonorrhoea and bleeding piles.\n\nFolk dietetic:\n\nAs spices.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity261.\n\nBiodiversity status: Frequent.\n\nPistia (Araceae)\n\nP. stratiotes L.\n\nB.- Topa Pana; E.- Water Lettuce.\n\nMedicinal uses:\n\nThe root is burned and the ash is mixed with sesame oil and inhaled for two months to prevent goiter.\n\nFolk dietetic:\n\nAs cattle feed to induce lactation; One of the ingredient of poultry feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity262.\n\nBiodiversity status: Frequent.\n\nPlumbago (Plumbaginaceae)\n\nP. indica L.\n\nB.- Lalchita; E.- Scarlet Leadwort.\n\nParts used: Root.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Panchatiktaghritaguggul and also in Louhasab, a tonic. Root is useful in ulcers, dyspepsia, cough, gonorrhoea, diabetes, tapeworms, piles, liver and spleen disorders and also used as a chronic stomachic and arthritic, blood enhancer and purifier. It is also effectively used against tumours.\n\nAntioxidant Potentiality: High activity263.\n\nBiodiversity status: Less Frequent.\n\nP. zeylanica L.\n\nB.- Sadachita; E.- White Leadwort.\n\nParts used: vRoot.\n\nMedicinal uses:\n\nThe root is used as an appetizer, as chronic stomachic and arthritic, blood enhancer, for gastric ulcers, tapeworms and is antibacterial. Tincture of root bark used in ulcers.\n\nAntioxidant Potentiality: High activity264.\n\nBiodiversity status: Less Frequent.\n\nPlumeria (Apocynaceae)\n\nP. rubra L. var. acutifolia Bailey\n\nB.- Gorur Chapa; E.- Frangipani\/Plumeria.\n\nParts used: Leaves & latex.\n\nMedicinal uses:\n\nLeaves are used in the treatment of inflammations as well as the latex is used in rheumatism.\n\nAntioxidant Potentiality: High activity265.\n\nBiodiversity status: Frequent.\n\nPolyalthia (Annonaceae)\n\nP. longifolia Benth. & Hook. f.\n\nB.- Debdaru; E.- Indian Fir Tree.\n\nParts used: Bark.\n\nMedicinal uses:\n\nBark is used in dropsy, sleepiness, hiccups and quite effective in dysentery.\n\nAntioxidant Potentiality: High activity266.\n\nBiodiversity status: Frequent.\n\nPolygonum (Polygonaceae)\n\nP. plebejum R. Brown.\n\nB.- Chemti Sak; E.- Small Knotweed.\n\nParts used: Root.\n\nMedicinal uses:\n\nRoots are used in pneumonia.\n\nFolk dietetic:\n\nFried, eaten. Nutrition: ***\n\nAntioxidant Potentiality: Low activity267.\n\nBiodiversity status: Frequent.\n\nPongamia (Leguminosae)\n\nP. pinnata (L.) Pierre\n\nB.- Karanja; E.- Indian Beech.\n\nParts used: Bark & seed.\n\nMedicinal uses:\n\nSeed oil is useful for hair growth, nasal allergy, stomachache, fever, herpes and burning sensation of body. Stem helpful in reducing toothache when used as a brush.\n\nAntioxidant Potentiality:\n\nModerate activity268.\n\nBiodiversity status: Frequent.\n\nPortulaca (Portulacaceae)\n\nP. oleracea L.\n\nB.- Nonta Sak; E.- Common Purslane.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nIt is used to treat infections or bleeding of the genito-urinary tract as well as dysentery. The fresh herb is applied topically to relieve sores and insect bites on the skin. It increases digestion, improves cough and cold; the leaf juice is used in headache and also used in burning sensation of hand and leg.\n\nFolk dietetic:\n\nEaten as curry or fried eaten as sag. Nutrition: ****\n\nAntioxidant Potentiality: High activity269.\n\nBiodiversity status: Frequent.\n\nPouzolzia (Urticaceae)\n\nP. zeylanica (L) Benn.\n\nB.- Luchipata; E.- Graceful Pouzolzsbush.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are used in ulcers, syphilis and gonorrhoea. Leaf juice is used as galactagogue. Poultice of the herb is applied to sores, boils.\n\nAntioxidant Potentiality: Moderate activity270.\n\nBiodiversity status: Frequent.\n\nPremna (Lamiaceae)\n\nP. corymbosa (Burm.f.) Rottl. & Willd.\n\nB.- Gonal; E. Premna.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are used as antidiabetic.\n\nAntioxidant Potentiality: High activity271.\n\nBiodiversity status: Frequent.\n\nPsidium (Myrtaceae)\n\nP. guajava L.\n\nB.- Peyara; E.- Guava.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nTender leaves are used in pyrrohoea, diarrhoea, anorexia and as a stomachic; bark in dysentery.\n\nFolk dietetic: As fruit.\n\nNutrition: *****\n\nAntioxidant Potentiality:\n\nModerate activity272.\n\nBiodiversity status: Frequent.\n\nPterocarpus (Leguminosae)\n\nP. santalinus L.f.\n\nB.- Rakta Chandan; E.- Red Sandalwood.\n\nParts used: Bark.\n\nMedicinal uses:\n\nThe bark powder is used in headache, bleeding from gum, irregular menstruation, mumps, ringworm, carbuncle and abscess.\n\nAntioxidant Potentiality: High activity273.\n\nBiodiversity status: Rare in locality.\n\nPterospermum (Malvaceae)\n\nP. acerifolium (L.) Willd.\n\nB.- Muchkunda Chapa\/Kanak Chapa; E.-Bayur Tree\/Karnikar Tree.\n\nParts used: Leaves & bark.\n\nMedicinal uses:\n\nThe leaves stop bleeding. Bark used in itches, acne; flower used in headache. The bark and flowers together help reduce body fat.\n\nAntioxidant Potentiality: High activity274.\n\nBiodiversity status: Less Frequent.\n\nPunica (Lythraceae)\n\nP. granatum L.\n\nB.- Dalim; E.- Pomme Granate.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nFruit peel used in diarrhoea, blood stool, dysentery; fruit in dyspepsia, lack of appetite, and insomnia. Tender leaves useful in leucorrhoea. Root bark in tapeworms, enlargement of liver; flower decoction during bleeding of nose.\n\nFolk dietetic:\n\nAs fruit. Nutrition: *****\n\nAntioxidant Potentiality: Low activity275.\n\nBiodiversity status: Frequent.\n\nPutranjiva (Putranjivaceae)\n\nP. roxburghii Wall.\n\nB.\\- Putranjiva; E.- Putranjiv.\n\nParts used: Bark & seed.\n\nMedicinal uses:\n\nBark used in excessive thirst, seed powder is used for dysentery and rheumatism.\n\nAntioxidant Potentiality: High activity276.\n\nBiodiversity status: Less Frequent.\n\nQuisqualis (Combretaceae)\n\nQ. indica L.\n\n(Syn: Combretum indicum)\n\nB.- Madhabilata; E.- Rangoon Creeper.\n\nParts used: Roots.\n\nMedicinal uses:\n\nThe decoction of the roots are used to treat rheumatism.\n\nAntioxidant Potentiality: Moderate activity277.\n\nBiodiversity status: Frequent.\n\nRauvolfia (Apocyanaceae)\n\nR. serpentina (L.) Benth. ex Kurz.\n\nB.- Sarpagandha; E.- Snake Root.\n\nParts used: Roots.\n\nMedicinal uses:\n\nMajor constituent of some ayurvedic preparations like Sarpagandha Churna and Madhugandha. The extract of the root is used to treat high blood pressure, hypertension, insomnia, weakness and nervous disorders. It is used to improve heart function and also used as an antidote against bites of reptiles (snakes).\n\nAntioxidant Potentiality:\n\nModerate activity278.\n\nBiodiversity status: Rare in locality.\n\nR. tetraphylla L.\n\nB.- Sarpagandha; E.- Wild Snake Root.\n\nParts used: Roots.\n\nMedicinal uses:\n\nIt is used as adulterant to R. serpentina.\n\nAntioxidant Potentiality:\n\nModerate activity279.\n\nBiodiversity status: Frequent.\n\nRicinus (Euphorbiaceae)\n\nR. communis L.\n\nB.- Rayri; E.- Castor Oil plant.\n\nParts used: Root, seed & leaves.\n\nMedicinal uses:\n\nRoot is one of the main constituents of some ayurvedic preparations like Punarnabastak Pachan and Saribadi Salsa. The root extract and seed oil is used for curing chronic arthritis, gout, dropsy, acute diarrhoea, and dysentery. The leaves are heated and applied to a woman's breasts to improve lactation.\n\nAntioxidant Potentiality:\n\nModerate activity280.\n\nBiodiversity status: Frequent.\n\nRosa (Rosaceae)\n\nR. indica L.\n\nB.- Golap; E.- Rose.\n\nParts used: Flower.\n\nMedicinal uses:\n\nFlower paste is used against acne. Rose water is useful as skin cleanser and eye soother.\n\nAntioxidant Potentiality: High activity281.\n\nBiodiversity status: Frequent.\n\nRumex (Polygonaceae)\n\nR. maritimus L.\n\nB.\\- Jungli palak; E.- Golden Dock.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe leaves are applied externally to burns. It is diuretic, enhances appetite. Leaf decoction stops vomiting, burning sensation. Leaf juice used in antidote to insect bites.\n\nFolk dietetic:\n\nCrushed and used as chutney.\n\nNutrition: ***\n\nAntioxidant Potentiality:\n\nModerate activity282.\n\nBiodiversity status: Frequent.\n\nSaccharum (Poaceae)\n\nS. spontaneum L.\n\nB.- Kashphul; E.- Wild Sugar\u00adcane.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nRoots are astringent, emollient, refrigerant, diuretic, aphrodisiac and useful in treatment of gynecological troubles, burning sensations and piles.\n\nAntioxidant Potentiality: High activity283.\n\nBiodiversity status: Frequent.\n\nS. officinarum L.\n\nB.- Akh; E.-Sugarcane.\n\nParts used: Stem.\n\nMedicinal uses:\n\nStem juice is very helpful in case of jaundice and also in kidney stone. Swelling of prostate gland,laxative, diuretic,aphrodisiac. To treat underweight baby. It is a folk remedy for arthritis, bedsores, boils, cancer, colds, cough, diarrhea, dysentery, eyes, fever, hiccups, inflammation, laryngitis, opacity, penis, skin, sores, sore throat, spleen, tumors, and wounds (Duke and Wain, 1981).\n\nFolk dietetic:\n\nStem. Fresh cane stems are often chewed, especially by poorer people.\n\nFresh juice have cooling effect and contains antioxidant nutraceuticals. sap collected to prepare akh Gur.\n\nNutrition: *****\n\nAntioxidant Potentiality: High activity284.\n\nBiodiversity status: Frequent Cultivated.\n\nSagittaria (Alismataceae)\n\nS. sagittifolia L.\n\nB.\\- Koukha\/Patajhanjhi; E.- Duck Potato.\n\nParts used: Leaf.\n\nMedicinal uses:\n\nLeaf is used to treat a skin disorders; act as antimicrobial agent.\n\nFolk dietetic:\n\nStarchy tubers edible; leaves and petioles used as vegetable.\n\nNutrition: ****\n\nAntioxidant Potentiality:\n\nNot Reported.\n\nBiodiversity status: Frequent.\n\nSapindus (Sapindaceae)\n\nS. trifoliatus L.\n\nB.- Ritha; E.- Soapnut.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nFruit powder used in stomachache, asthma, enhances menstruation. Fruit is useful for hair growth and has antilice action;\n\nAntioxidant Potentiality: Low activity285.\n\nBiodiversity status: Frequent.\n\nSaraca (Leguminosae)\n\nS. indica L.\n\nB.- Ashok; E.- Ashoka Tree.\n\nParts used: Bark & fruit.\n\nMedicinal uses:\n\nBark is the main component of an ayurvedic preparation like Ashokarishtha, a health tonic for women. It is effective against leucorrhoea, dysmenorrhoea and haemorrhoids and gout . The dried fruits are used in cases of spermatorrhoea, and urinary disorders.\n\nAntioxidant Potentiality: Low activity286.\n\nBiodiversity status: Frequent.\n\nScoparia (Scrophulariaceae)\n\nS. dulcis L.\n\nB.- Chinimichri; E.- Sweet Broom Weed.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nUseful in leucorrhoea, mouth ulcer and is anti-diabetic.\n\nAntioxidant Potentiality: High activity287.\n\nBiodiversity status: Frequent.\n\nSemecarpus (Anacardiaceae)\n\nS. anacardium L.f.\n\nB.- Vela; E.- Marking Nut.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeeds are used as blood purifier, treats herpes, gangrene, and leprosy. The seeds are effectively used against tumours and applied for abortion. Oil is used in rheumatism.\n\nAntioxidant Potentiality:\n\nModerate activity288.\n\nBiodiversity status: Frequent.\n\nSesbania (Leguminosae)\n\nS. grandiflora Pers.\n\nB.- Bakphul; E.- Hummingbird Tree.\n\nParts used: Leaves & flower.\n\nMedicinal uses:\n\nFlower juice is useful in treating cold and cough; leaf decoction is used in night blindness and epilepsy.\n\nFolk dietetic:\n\nFlowers fried and eaten. Nutrition: **\n\nAntioxidant Potentiality:\n\nModerate activity289.\n\nBiodiversity status: Frequent.\n\nS. sesban Merr.\n\nB.- Jayanti; E.- Egyptian River\u00adheme.\n\nParts used: Leaves & seed.\n\nMedicinal uses:\n\nLeaves are used in cold and cough and act as appetizer; seed paste is useful in insect bites.\n\nAntioxidant Potentiality: High activity290.\n\nBiodiversity status: Frequent.\n\nSetaria (Poaceae)\n\nS. italica Beauv. (syn. Panicum italicum L.)\n\nB.\u2013 Kang\/ Kaon, E.\u2013 Foxtail millet\n\nParts used: seed grains.\n\nMedicinal uses:\n\nTo improve sexual desire in both sexes and to reduce body fat; 20 grams boiled grains for 60 days(Bhattachartta, 2008). Useful in biliousness, cooling, anti-inflammatory ; antioxidant291a\n\nFolk dietetic : Alternative to major cereals. To reduce body fat 20 gms of boiled grains to be taken twice a day for 60 days. Grains fried in sand made into 'khoi' eaten with milk or pudding is made from grain.\n\nNutrition: whole grain contains good nutraceuticals better than rice, ***** .\n\nAntioxidant Potentiality: high291a\n\nBiodiversity status: cultivated or wild, North Bengal plains and drier parts of India\n\nS. glauca (L.) Beauv. (syn. Panicum glauca L.)\n\nB.- Pinginatchi E.- Yellow foxtail\n\nParts used : seeds, leaf\n\nMedicinal uses:\n\nNot known\n\nFolk dietetic : Low grade beer produced from the seeds. Plant used as fodder.\n\nNutrition : ***\n\nAntioxidant potentiality: not known\n\nBiodiversity: Common widely distributed.\n\nS. verticillata (L.) Beauv.(syn. Panicum vertcillatum L.)\n\nB.- Silnaja; E.- Hooked Bristle grass.\n\nParts used : seeds, leaf\n\nMedicinal uses:\n\nNot known\n\nFolk dietetic : Low grade beer produced from the seeds. During great Bengal famine the seeds were eaten. Plant used as fodder.\n\nNutrition : ***\n\nAntioxidant potentiality: not known\n\nBiodiversity: Common, weedy.\n\nSida (Malvaceae)\n\nS. acuta Burm.f.\n\nB.- Berala; E.- Common wireweed.\n\nParts Used: Roots.\n\nMedicinal uses:\n\nRoots are used in biliousness, urinary disorder, stomachache and fever as well as act as aphrodisiac.\n\nAntioxidant Potentiality: High activity291.\n\nBiodiversity status: Frequent.\n\nS. cordifolia Wight & Arnott\n\nB.- Swet Berala; E.\\- Indian Ephedra.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nUseful in dog bites, blisters and women's breast irritation.\n\nFolk dietetic:\n\nLeaves as goat feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: High activity292.\n\nBiodiversity status: Frequent.\n\nS. rhombifolia L.\n\nB.- Peet Berala; E.- Arrowleaf Sida.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Salsa and Chyavanprasha, both are health tonic. It is used as blood purifier and enhancer as well as sperm enhancer. Useful in blisters and hydrocyl. Roots are used in haemoptysis and stangury.\n\nFolk dietetic:\n\nLeaves as goat feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity293.\n\nBiodiversity status: Frequent.\n\nSmilax (Smilacaceae)\n\nS. ovalifolia Roxb. ex D.Don\n\nB.- Kumarilata\/Baghnokha; E.- Kumarika.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nThe roots are used for venereal diseases, rheumatic swellings, urinary problems and dysentery.\n\nAntioxidant Potentiality: Low activity294.\n\nBiodiversity status: Frequent.\n\nSolanum (Solanaceae)\n\nS. indicum L. Sans.\n\nB.\\- Brihati; E.- Indian Nightshade.\n\nParts used: Leaves, fruits & seeds.\n\nMedicinal uses:\n\nFruits are used in ankle sprain, gout, cold and cough, dyspepsia; fruit juice in uterus itching; tender leaves for anorexia, insect bites; seeds for asthma.\n\nAntioxidant Potentiality: High activity295.\n\nBiodiversity status: Frequent.\n\nS. nigrum L.\n\nB.- Kakmachi; E.- Black Night\u00adshade.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nMature leaf decoction is useful to treat dropsy, general debility, thread worms, stangury, prickly heat, dysentery, anaemia and in diuretic.\n\nAntioxidant Potentiality:\n\nModerate activity296.\n\nBiodiversity status: Frequent.\n\nS. xanthocarpum Sch. and Weldl.\n\nB.- Kantikari; E.- Yellow - Berried Nightshade.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nIt is a major component of few aurvedic preparations like, Saribadi Salsa, Chyavanprasha, Punarnabastak Pachan etc. It is useful in treating worms, enlargement of liver, muscular pain and stone in the urinary bladder. Dried fruit treats dental infections. Roots and seeds are used as an expectorant in asthma, cough and pain in chest. Seeds are helpful for treating irregular menstruation and dysmenorrheal. Fruits also facilitate seminal ejaculation, itching and fever.\n\nFolk dietetic:\n\nUsed as cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity297.\n\nBiodiversity status: Frequent.\n\nSpinacia (Amaranthaceae)\n\nS. oleracea L.\n\nB.- Palong; E.- Spinach.\n\nParts used: Leaves & root.\n\nMedicinal uses:\n\nUsed as a wholesome vegetable. Leaf decoction is useful in asthma and roots in insect bites.\n\nFolk dietetic:\n\nTesty and eaten as mixed curry and also as fried sag.\n\nNutrition: *****\n\nAntioxidant Potentiality: Moderate activity298.\n\nBiodiversity status: Frequent.\n\nSpondias (Anacardiaceae)\n\nS. pinnata (L.f.) Kurz.\n\nB.- Amra; E.- Wild Mango.\n\nParts used: Bark, leaves, root & fruit.\n\nMedicinal uses:\n\nRoots are useful in regulating menstruation. Bark is used in dysentery, diarrhoea, cold and cough, vomiting and rheumatism. Leaves are aromatic and used in dysentery. The ripe fruits are antiscorbutic.\n\nFolk dietetic:\n\nAs fruit; Fruit is used to prepare chutny and pickle.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity299.\n\nBiodiversity status: Frequent.\n\nStephania (Menispermaceae)\n\nS. hernandiifolia (Wild.) Walp.\n\nB.- Aknadi\/Talalati\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of Salsa, an ayurvedic preparation. Useful in boils, herpes, stangury, sprains, leucorrhoea and acts as a stomachic. The raw tender leaves are well known for preventing pregnancy.\n\nAntioxidant Potentiality: Moderate activity300.\n\nBiodiversity status: Frequent.\n\nSterculia (Malvaceae)\n\nS. foetida L.\n\nB.- Kathbadam; E.- Java-Olive.\n\nParts used: Seed.\n\nMedicinal uses:\n\nSeeds are used as carminative and laxative.\n\nAntioxidant Potentiality: High activity301.\n\nBiodiversity status: Rare.\n\nStevia (Asteraceae)\n\nS. rebaudiana (Bertoni) Bertoni\n\nB.- Stevia; E.- Sweetleaf.\n\nParts used: Leaves. Medicinal uses:\n\nThe leaves are used for diabetic patients as an alternative source of sugar.\n\nAntioxidant Potentiality: High activity302.\n\nBiodiversity status: Rare and cultivated.\n\nStreblus (Moraceae)\n\nS. asper Lour.\n\nB.- Shewra; E.- Siamese Rough Bush.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaves are boiled in water and the decoction is used to prevent loose motion.\n\nAntioxidant Potentiality: High activity303.\n\nBiodiversity status: Frequent.\n\nStreptolirion (Commelinaceae)\n\nS. volubile Edgew.\n\nMedicinal Uses:\n\nJuice prepared from the root used externally on wounds of vital organs\n\nFolk Dietetic: Young inflorescence boiled and eaten as vegetable curry at villages.\n\nNutrition : ***\n\nStrychnos (Loganiaceae)\n\nS. nux-vomica L.\n\nB.- Kuchila; E.- Poison Nut.\n\nParts used: Seeds.\n\nMedicinal uses:\n\nIt is mainly used to induce vomiting as well as in abdominal pain, constipation, intestinal irritation, insomnia.\n\nAntioxidant Potentiality: High activity304.\n\nBiodiversity status: Rare in locality.\n\nSyzygium (Myrtaceae)\n\nS. cumini (L.) Skeels\n\nB.- Jam; E.- Jambolana.\n\nParts used: Leaves & seeds.\n\nMedicinal uses:\n\nLeaf decoction cures blood dysentery, vomiting, cuts, wounds and scabies. Seed powder is useful in diabetes and diarrhoea.\n\nFolk dietetic:\n\nAs fruit; leaves as cattle feed. Nutrition: *****\n\nAntioxidant Potentiality: Moderate activity305.\n\nBiodiversity status: Frequent.\n\nS. jambos (L.) Alston\n\nB.\\- Golap Jam; E.- Rose Apple.\n\nParts used: Bark & fruit.\n\nMedicinal uses:\n\nThe bark is used as astringent and fruits are used in liver disorders.\n\nFolk dietetic:\n\nAs fruit. Nutrition: ****\n\nAntioxidant Potentiality: High activity306.\n\nBiodiversity status: Frequent.\n\nTagetes (Asteraceae)\n\nT. erecta L.\n\nB.- Halud Ganda; E.- Marigold.\n\nParts used: Leaves & flowers.\n\nMedicinal uses:\n\nIt is used to treat stomachache, parasites, diar\u00adrhoea, vomiting, indigestion, hiccups, dropsy, cold and cough and toothache. Leaf paste used to stop bleeding from cuts and wounds.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: High activity307.\n\nBiodiversity status: Frequent.\n\nT. patula L.\n\nB.- Rakta Ganda; E.\\- French Marigold.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nLeaf decoction used to stop bleeding from cuts and wounds, piles, nose, during coughing and also in uterine infection.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality:\n\nHigh activity308.\n\nBiodiversity status: Frequent.\n\nTamarindus (Leguminosae)\n\nT. indica L.\n\nB.- Tentul; E.- Tamarind.\n\nParts used: Leaves & fruits.\n\nMedicinal uses:\n\nTender leaves are useful for cold and cough, chronic dysentery, sprain, mouth ulcers and rheumatic pain. Fruit is useful in gastric problems.\n\nFolk dietetic:\n\nAs fruit; made into pickle, chutny. Nutrition: ****\n\nAntioxidant Potentiality: Moderate activity309.\n\nBiodiversity status: Frequent.\n\nTamilnadia (Rubiaceae)\n\nT. uliginosa (Retz.) Tirveng. & Sastre\n\nB.- Pandal; E.- Divine Jasmine.\n\nMedicinal uses:\n\nFruits are eaten as vegetables.\n\nAntioxidant Potentiality: Moderate activity310.\n\nBiodiversity status: Frequent.\n\nTectona (Lamiaceae)\n\nT. grandis L.f.\n\nB.- Segun; E.- Teak Tree.\n\nParts used: Fruit, seed & bark.\n\nMedicinal uses:\n\nFruits are useful in stangury; bark powder in acidity, headache and seed oil used in herpes. Antioxidant Potentiality: High activity311. Biodiversity status: Frequent.\n\nTerminalia (Combretaceae)\n\nT. arjuna (Roxb.ex DC.) Wt. & Arn.\n\nB.- Arjun; E.\\- Arjuna.\n\nParts used: Bark & tender leaves.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Arjunaristha, Trifalachurna and Dashamularistha. The powder of the bark is an effective sex stimulant and useful in blood pressure, high cholesterol and in asthma. It is useful in removing calculi or stones formed in the urinary system. It helps to promote blood formation and relieve from fever.\n\nAntioxidant Potentiality: Moderate activity312.\n\nBiodiversity status: Frequent.\n\nT. bellirica (Gaertn.) Roxb.\n\nB.- Bahera; E.- Beleric\/Bastard Myrobalan.\n\nParts used: Fruits.\n\nMedicinal uses:\n\nA major constituent of some a.yurvedic preparations like Avayaristha and Trifalachurna, a tonic. The fruit powder is useful in chronic dysentery, piles and whooping cough. It expels stones in the digestive, urinary and respiratory tracts. It is a strong rejuvenator of the body, especially for the hoarseness of voice, vision and hair. It shows a strong action in preventing heart and liver fat congestion.\n\nAntioxidant Potentiality: Moderate activity313.\n\nBiodiversity status: Frequent.\n\nT. chebula Retz.\n\nB.- Haritaki; E.- Black Myrobalan\/Chebulic Myrobalan.\n\nParts used: Fruits.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like aritakikhanda,Bhuvaneswarboti, Trifalachurna and Chatursa\u00admaboti. It is used to treat piles, wounds, ulcer, leprosy, inflammation, cough, asthma, skin disorders and also in chronic diarrhoea. The seed decoction is used as gargle in sore throat and ulceration of gums. It is a good appetizer, improves digestion, acts as a liver stimulant and a stomachic. It is useful in urethral discharge like spermatorrhoea, vaginal discharge like leucorrhoea. Infusion of fruits cures conjunctivitis.\n\nFolk dietetic:\n\nAs fruit. Seed crushed and eaten. Nutrition: ***\n\nAntioxidant Potentiality: High activity314.\n\nBiodiversity status: Frequent.\n\nThunbergia (Acanthaceae)\n\nT. coccinea Wall.\n\nB.- Thunbergia; E.- Scarlet Thunbergia.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nJuice of plant is applied to cuts and wounds. Leaves and roots are used for bone fracture.\n\nAntioxidant Potentiality: Not Reported.\n\nBiodiversity status: Frequent.\n\nTinospora (Menispermaceae)\n\nT. sinensis (Lour.) Merr. (Syn: T. cordifolia)\n\nB.- Gulancha; E.- Giloy.\n\nParts used: Stem & leaf.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation, Amritaristha. The latex decoction produces a type of sugar called 'gulancha michchri' and another preparations like Kalamegh, Panchatiktoghritaguggul and Salsa. It is used to treat severe illness, skin ulcers, arthritis, gout, liver and eye diseases, urinary problems, anaemia, tumour, herpes, diarrhoea, dysentery, diabetes, gonorrhoea and even reduces blood cholesterol level.\n\nFolk dietetic:\n\nAs cattle feed to induce lactation.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity315.\n\nBiodiversity status: Frequent.\n\nTragia (Euphorbiaceae)\n\nT. involucrata L.\n\nB.- Bichuti; E.- Nose Burns.\n\nParts used: Roots.\n\nMedicinal uses:\n\nThe roots are crushed along with stem of harjora and ginger, then applied externally on broken leg and ankle pain.\n\nAntioxidant Potentiality:\n\nModerate activity316.\n\nBiodiversity status: Frequent.\n\nTrapa (Trapaceae)\n\nT. bispinosa Roxb.\n\nB. Paniphal, E. Water chestnut\n\nParts used : Fruit, whole plant\n\nMedicinal uses:\n\naphrodisiac, astringent, ap\u00ad\u00adpe\u00adtizer, anti-pyretic, constipating, diuretic, coolant, nutritive, anti-diarrheal, and tonic. Having immunomodulatory and anti-diabetic properties316a.\n\nFolk dietetic: fruits are edible raw or cooked;dried ones are baked and eaten. They are also grated into flour and made into cakes. The nutritive value of the kernels is shown by analysis to be equal to that of rice.316a\n\nNutrition : *****\n\nAntioxidant Potentiality: High-activity316a.\n\nBiodiversity status: Cultivated all over India, Introduced from Europe.\n\nTrichosanthes (Cucurbitaceae)\n\nT. tricuspidata Lour. (Syn: T. bracteata)\n\nB.- Makal; E.- Redball Snakegourd.\n\nParts used: Roots.\n\nMedicinal uses:\n\nThe Roots are crushed and paste is applied as emollient in case of headache.\n\nAntioxidant Potentiality:\n\nModerate activity317.\n\nBiodiversity status: Frequent.\n\nT. cucumerina L.\n\nB.\u2013 Bon chichinga; E.-Common Fringed- Flower Vine.\n\nParts used: Leaves and fruit.\n\nMedicinal uses:\n\nLeaf decoction used in biliousness, skin diseases such as eczema. Fruits are used as anthelmintic, emetic and purgative.\n\nFolk dietetic:\n\nAs cattle feed and fruit as vegetable by tribal.\n\nNutrition: ****\n\nAntioxidant Potentiality: High activity318.\n\nBiodiversity status: Frequent.\n\nT. dioica Roxb.\n\nB.- Patal; E.- Pointed Gourd.\n\nParts used: Fruit, root & leaves.\n\nMedicinal uses:\n\nRoasted fruit clears marks of chicken pox and relieves from infected and pus formed nails. Leaves are used as a laxative. Roots are highly laxative and used to control bile.\n\nFolk dietetic:\n\nLeaf and young branches boiled, fried and eaten as side dish. Fruit is used as very popular wholesome vegetables Seeds anti-diabetic.\n\nNutrition: ****\n\nAntioxidant Potentiality: Moderate activity319.\n\nBiodiversity status: Frequent and cultivated Assam bengal thought to be centre for origin (De Sarker, 2012).\n\nTridax (Asteraceae)\n\nT. procumbens (L.) L.\n\nB.- Kesriya; E. - Tridax Daisy.\n\nMedicinal uses:\n\nWhole plant is crushed and the paste is applied to dry the pusses. It is also used in cuts and wounds.\n\nAntioxidant Potentiality: High activity320. Biodiversity status: Frequent.\n\nTylophora (Apocynaceae)\n\nT. indica (Burm.f.) Merr.\n\nB.- Antamul; E.- Indian Ipecac.\n\nParts used: Root.\n\nMedicinal uses:\n\nDried root used in dysentery, bronchitis and asthma. The plant is used as substitute of Ipecac.\n\nAntioxidant Potentiality: High activity321.\n\nBiodiversity status: Frequent.\n\nTyphonium (Araceae)\n\nT. trilobatum (L.) Schott.\n\nB.\u2013 Kharkon kachu\/Chet Kachu; E.\\- Bengal Arum.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nUsed as vegetable to treat physical weakness, pain, anorexia and enhances lactation.\n\nFolk dietetic:\n\nLeaves and petiole cut into pieces, cooked as vegetable.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity322.\n\nBiodiversity status: Frequent.\n\nTriticum (Poaceae)\n\nT. estivum L.\n\nParts used: seed, leaf\n\nB.\u2013 Gam \/ Godhum E.\u2013 The bread wheat plant\n\nMedicinal uses:\n\nAccording to Hartwell (1967\u20131971), the seeds are used in folk remedies for cancers, corns, tumors, warts, and whitlow. demulcent, diuretic, emollient, excipient, intoxicant, laxative, useful as a poultice, restorative, sedative. Wheat grass (extract concentrate) has been shown to posses anti-cancer activity, anti-ulcer activity, antioxidant activity, anti-arthritic activity, and blood building activity in Thalassemia Major(Singh,N., et.al.,2012)\n\nFolk dietetic : used as a shampoo and vulnerary, common wheat is a folk remedy for cancer, diarrhea, dysentery, ecchymosis, epistaxis, fertility, fever, gravel, hematuria, hemoptysis, hemorrhage, incontinence, leprosy, leucorrhea, menorrhagia, neurasthenia, nightsweat, perspiration, scald, tumor, warts, whitlow, and wounds (Duke and Wain, 1981, ref. 283a).\n\nAntioxidant Potentiality wheat \/ leaf: High283a\n\nNutrition : ***** (USDA)\n\nBiodiversity: widely cultivated.\n\nUraria (Leguminosae)\n\nU. picta (Jacq.) DC.\n\nB.- Prisniparni; E.\u2013 Lagopoides.\n\nParts used: Roots & leaf.\n\nMedicinal uses:\n\nThe root has aphrodisiac properties and the decoction is used for cough and fevers. Leaves are crushed and juice is used as antiseptic externally.\n\nAntioxidant Potentiality: High activity323.\n\nBiodiversity status: Rare.\n\nUrena (Malvaceae)\n\nU. lobata L.\n\nB.- Hegra; E.- Caesar weed.\n\nParts used: Root.\n\nMedicinal uses:\n\nThe root paste is used in case of dog bite.\n\nAntioxidant Potentiality: High activity324.\n\nBiodiversity status: Frequent.\n\nUrtica (Urticaceae)\n\nU. dioica\n\nB.\u2013 Bichuti pata E.\u2013 Stinging Nettle\n\nParts used : Young leaf, root\n\nMedicinal uses\n\nHaving Immunomodulatory property,324a Antioxidant, antimicrobial, antiulcer and analgesic activities\n\nFolk dietetic: The young shoot and the inflorescences are made into curry alone or mixed with potato and eaten particularly in hilly areas. Nettle soup is a common use of the plant, particularly in Northern and Eastern Europe. Native Americans and used as a cooked plant in spring when other food plants were scarce.324b\n\nNutrition: ****\n\nAntioxidant potentiality:High activity324c\n\nBiodiversity status : Common, at plains and hills\n\nVanda (Orchidaceae)\n\nV. tessellata (Roxb.) Hook ex Don.\n\nB.- Rasna; E.- Checkered Vanda.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of some ayurvedic preparations like Dashamularistha and Salsa, health tonic.\n\nAntioxidant Potentiality: Moderate activity325.\n\nBiodiversity status: Frequent.\n\nVangueria (Rubiaceae)\n\nV. spinosa Roxb. (Syn. Meyna spinosa)\n\nB.- Mayna Kanta; E.- Muyna.\n\nParts used: Fruit.\n\nMedicinal uses:\n\nFruits used in biliousness and hepatic congestion.\n\nAntioxidant Potentiality: Potent phytochemical present326.\n\nBiodiversity status: Frequent.\n\nVernonia (Asteraceae)\n\nV. anthelminctica (L.) Willd. (Syn: Baccharoides anthelmintica)\n\nB.- Somraji; E.- Purple fleabane.\n\nParts used: Seed.\n\nMedicinal uses:\n\nThe seeds are used to treat leucoderma and other skin diseases. The seeds are also used as blood enhancer, blood purifier and treats piles, guinea worms, toothache, diabetes and dysentery.\n\nAntioxidant Potentiality: High activity327.\n\nBiodiversity status: Frequent.\n\nV. cinerea Less.\n\nB.- Sahadebi; E.- Common Vernonia.\n\nParts used: Whole Plant.\n\nMedicinal uses:\n\nThe leaves are eaten as a potherb. A decoction of the plant is used in diarrhoea, stomach-ache and for cough and colic. The juice of the plant is useful in urinary problem of children.\n\nAntioxidant Potentiality: High activity328.\n\nBiodiversity status: Frequent.\n\nVitex (Lamiaceae)\n\nV. negundo L.\n\nB.- Nishinda; E.- Vitex Plant.\n\nParts used: Leaves.\n\nMedicinal uses:\n\nThe leaf decoction treats boils, dandruff, gout, flatulence, leprosy, itches, gout, asthma, tonsilitis, pharyngitis and also reduces abdominal fats and guinea worms.\n\nFolk dietetic:\n\nTender leaves fried eaten along with rice; as cattle feed.\n\nNutrition: ***\n\nAntioxidant Potentiality: Low activity329.\n\nBiodiversity status: Frequent.\n\nVitis (Vitaceae)\n\nV. vinifera L.\n\nB.- Angur; E. Grape\n\nParts used : Fruits, Dried fruits, Seeds, Leaves\n\nMedicinal uses:\n\nLeaves astringent, paste of young branches used to cure skin infection; Fruits laxative, fruit juice energy enhancer, recovers weakness. Dry fruit: Universally used as dry food for wellness of body both mental and physical, anti-cough, diuretic, appetizer. Principal Ingredient for Dhrakharista.\n\nFolk dietetic: Mature Fruit good source of minerals and vitamins Young branches eaten as vegetable,\n\nFruits used in preparation of liquor.\n\nNutrition: Fruits : dry or fresh source of one of the best nutrition. Fruit &Seeds contains ant-cancer nutraceuticals. *****\n\nAntioxidant Potentiality: High activity329a.\n\nBiodiversity status: Cultivated , thought to have primary centre of origin at North-Western Himalayas.\n\nWedelia (Asteraceae)\n\nW. calendulacea (L.) Less.\n\n(Syn. Sphagneticola calendulacea)\n\nB.- Bhringaraj; E.- Trailing Eclipta.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nA major constituent of an ayurvedic preparation, Mahavringaraj Tel. The decoction of the whole plant helps to calm the mind from excessive stress, headache and soothes noisy sleep. The leaves are used in dyeing grey hair and in promoting the growth of hair. It is also used as nervine tonic and in cough.\n\nAntioxidant Potentiality: High activity330.\n\nBiodiversity status: Frequent.\n\nWithania (Solanaceae)\n\nW. somnifera (L.) Dunal\n\nB.- Aswagandha; E.- Indian Ginseng.\n\nParts used: Root and whole plant.\n\nMedicinal uses:\n\nA main constituent of some ayurvedic preparations like Aswagandharistha, a sperm enhancer tonic and Salsa. The roots are used to treat nervous disorders, intestinal infections, cold and cough, asthma, boils, chronic bronchitis, leucoderma and also acts as a sexual stimulant, and helps in erectile dysfunction.\n\nAntioxidant Potentiality: Moderate activity331.\n\nBiodiversity status: Rare in locality.\n\nXanthium (Asteraceae)\n\nX. strumarium L.\n\nB.- Chulatka; E.- Clotbur.\n\nParts used: Whole plant.\n\nMedicinal uses:\n\nExtracts of the whole plant is used as laxative, digestive and also in rheumatism, pruritis, leucoderma and other skin diseases.\n\nFolk dietetic:\n\nAs cattle feed. Nutrition: ***\n\nAntioxidant Potentiality: Low activity332.\n\nBiodiversity status: Frequent.\n\nZingiber (Zingiberaceae)\n\nZ. officinale Roxb.\n\nB.- Ada; E.- Ginger.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nUsed as appetizer; used in cold and fever, chicken pox, nephritis, hiccups, cuts and wounds.\n\nFolk dietetic:\n\nRaw chewed as mouth freshener and irritation in throat. Mixed with tea as zinger tea. dried flakes to prevent habit of smoking and chewing betel. Contains anti cancerous nutraceuticals.\n\nAs spices; Nutrition: ****\n\nAntioxidant Potentiality: Moderate activity333.\n\nBiodiversity status: Cultivated\n\nZ. zerumbet (L.) Roscoe ex Sm.\n\nB.- Jabakusum; E.- Shampoo Ginger.\n\nParts used: Rhizome.\n\nMedicinal uses:\n\nThe decoction of rhizome is used to cool the mind as well as it relieves from headche.\n\nAntioxidant Potentiality: Low activity334.\n\nBiodiversity status: Rare in locality.\n\nZiziphus (Rhamnaceae)\n\nZ. jujuba Mill.\n\n(Syn. 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J Func Foods. 5: 62 -70.\n 7. 336. De Sarkar, D. 2012. The Origin of Trichosanthes dioica Roxb. In Biodiversity and Conservation \u2013 fundamentals and applications. Eds. H. Saha et. al. 2012.pp. 115-117, Kolkata.\n\nAbout the Authors\n\n**Professor Dilip De Sarker** (b.1955) joined as Lecturer in Botany in 1981 at Raiganj University College (Now declared as a University). Within a short time, he earned reputation both in teaching and research. He was a bright student throughout his career and stood 1st in first class in his graduation. Being a student of famous scientist, Professor A.K.Sharma, Dr. De Sarker started research in plant cytogenetics and later on shifted his attention to medicinal plants. He has guided several Ph.D. students and has 75 research articles and two books to his credit. He has delivered many lectures in India and abroad. He was a British Council Research Fellow in 1995 and did collaborative research at Jodrel Laboratory, Royal Botanic Gardens, Kew, U.K. (he can be reached at: dilipdesarker.rnj@gmail.com)\n\n**Manas Ranjan Saha** (b.1986) is currently a research associate at Bio-informatics Centre, North Bengal University. Mr. Saha has surveyed more than hundred plants with reference to antioxidant activity, anti-carcinogenic property, and anti-diabetic property. The results of the findings have been included in this publication. He is a devoted research worker who also works in bio-informatics and is pursuing his Ph.D. at North Bengal University.\n\n**Subrata Saha** (b.1969) is an assistant Professor at Raiganj University, West Bengal. He did his post-graduation at North Bengal University. A social worker by nature, he has contributed to various social programmes including health issues of the poor. He has several international and national research publications.\n\nIndex to English Names\n\nA\n\nAdder's tongue\n\nAfrican Basil\n\nAir Plant\n\nAir Yam\/ Air Potato\n\nAlligator weed\n\nAloe\n\nArabian Jasmine\n\nAreca Nut\n\nArjuna\n\nAromatic Ginger\n\nArrowleaf Sida\n\nAsoka tree\n\nAsparagus\n\nAsthma Weed\n\nB\n\nBalloon Vine\n\nBamboo\n\nBamia Moschata\n\nBanana\n\nBanTulsi\n\nBanyan tree\n\nBarbados Nut\n\nBarnyard grass\n\nBarnyard Millet\n\nBastard Myrobalan\n\nBastard Teak\n\nBayur Tree\n\nBeleric\n\nBellyache Bush\n\nBengal Arum\n\nBengal Quince\n\nBengal wild rice\n\nBermuda Grass\n\nBetel leaf\n\nBetel Nut\n\nBhumiamala\n\nBillygoat Weed\n\nBitter Cumin\n\nBitter Gourd\n\nBlack Myrobalan\n\nBlack Nightshade\n\nBlack Pepper.\n\nBlack Turmeric\n\nBlack-Honey Shrub\n\nBlue Water Lily\n\nBread wheat\n\nBroom Creeper\n\nBuddha Tree\n\nBullet Wood Tree\n\nButterfly Pea.\n\nC\n\nCaesar weed\n\nCandlebrush\n\nCape Gooseberry\n\nCarambola\n\nCarrot\n\nCassava\n\nCastor oil plant\n\nCeylon caper\n\nCeylon Cinnamon\n\nChaff-flower\n\nChameleon Plant\n\nChampac\n\nCheckered vanda\n\nChilly\n\nChina Rose\n\nChinese date\n\nChinese Knotwood\n\nChinese Senna\n\nChrist's Thorn\n\nClimbing hempvine\n\nClimbing Wool-plant\n\nClotbur\n\nCluster Fig Tree\n\nCoconut\n\nCoffee Senna\n\nCogon Grass\n\nCommon Curculigo\n\nCommon Leucas\n\nCommon Purslane\n\nCommon vernonia\n\nCommon wireweed\n\nCoral Tree\n\nCoriander\n\nCorn Mint\n\nCotton Rosemallow\n\nCowhage\n\nCrepe Ginger\n\nCrown Flower\n\nCurry Plant\n\nCustard Apple\n\nD\n\nDate Palm.\n\nDee Plee pepper\n\nDenseflower Knotweed\n\nDevil Tree of India\n\nDevil's Trumpet\n\nDevill's Claw\n\nDevils's cotton\n\nDivine Jasmine\n\nDuck potato\n\nE\n\nE.-Common Fringed- Flower Vine\n\nEared Cyphostemma\n\nEarleaf Acacia\n\nEast Indian Rosebay\n\nEgyptian Riverheme\n\nElephant Apple\n\nElephant Creeper\n\nElephant Foot Yam\n\nEpazote\n\nEucalyptus\n\nEuropian Waterclover\n\nF\n\nFalse Daisy\n\nFive Leaved Carpetweed\n\nFive-Leaf Yam\n\nFlame Tree\n\nForest Grapes\n\nFour 'O' Clock plant\n\nFox nut\n\nFoxtail millet\n\nFragrant Albizia\n\nFrangipani\n\nFrench Marigold\n\nFreshwater Mangrove\n\nFrog fruit\n\nG\n\nGandarusa\n\nGarlic\n\nGiant Dodder\n\nGiant Taro\n\nGiloy 103\n\nGinger\n\nGlory lily\n\nGolden dock\n\nGolden Shower Tree\n\nGoomar teak.\n\nGovernor's Plum\n\nGraceful Pouzolzsbush\n\nGrape\n\nGreen Amaranth\n\nGreen Taro\n\nGround Cherry\n\nGuava\n\nGum Arabic Tree\n\nGymnema\n\nH\n\nHairy Fig.\n\nHairy Hemigraphis\n\nHamilton Bamboo\n\nHeart Pea\n\nHelminthostachys\n\nHenna\n\nHill Canna\n\nHill Glory Bower\n\nHimalayan Onion.\n\nHoly Basil\n\nHoney suckle mistletoe\n\nHooked Bristle grass.\n\nHophead\n\nHummingbird Tree\n\nI\n\nIndian Abutilon\n\nIndian Acalypha\n\nIndian Ash Tree\n\nIndian Bay-Leaf\n\nIndian Beech\n\nIndian Birthwort\n\nIndian Calosanthes\n\nIndian Ephedra\n\nIndian Fir Tree\n\nIndian Ginseng\n\nIndian Goose Grass\n\nIndian Gooseberry\n\nIndian Heliotrope\n\nIndian Ipecac\n\nIndian joint vetch\n\nIndian laurel\n\nIndian Licoric\n\nIndian Lilac\n\nIndian Lilac\n\nIndian Nightshade\n\nIndian Olive\n\nIndian Pennywort\n\nIndian Persimmon\n\nIndian Rose Chestnut\n\nIndian Sarsaperilla\n\nIndian Screw tree\n\nIndrajao\n\nJ\n\nJack Fruit\n\nJambolana\n\nJava Olive\n\nJimson weed\n\nJob's Tear\n\nJungle Geranium\n\nJute plant\n\nK\n\nKadam\n\nKaempferia\n\nKapok Tree\n\nKarnikar Tree\n\nKing of Bitters\n\nKodo Millet\n\nKumarika\n\nL\n\nLagopoides\n\nLakoocha.\n\nLantana\n\nLasia\n\nLebbeck Tree\n\nLeea\n\nlemon\n\nLemon Grass\n\nLesser Yam\n\nLevant Cotton\n\nLipstick Tree\n\nLong Pepper\n\nLotus\n\nLove Grass\n\nM\n\nMadagascar Periwinkle\n\nMahu\u0101\n\nMalabar Nut\n\nMalabar Spinach\n\nMalay Blumea\n\nManasa Sij.\n\nMango\n\nMango Ginger\n\nMangosteen\n\nMargarifer arum\n\nMarigold\n\nMarijuana\n\nMarking nut\n\nMarsh Barbel\n\nMarsh Herb\n\nMelothria\n\nMexican Poppy\n\nMilky Yam\n\nMohintli\n\nMoringa\n\nMountain knot grass\n\nMugwort\n\nMuyna\n\nN\n\nNight Jasmine\n\nNodding swamp orchid\n\nNose Burns\n\nNut Grass\n\nO\n\nOld world diamond flower\n\nOleander\n\nOnion\n\norange\n\nOrange Jessamine\n\nOrangeberry\n\nOtaheite Gooseberry\n\nP\n\nPandan\n\nPapaya\n\nPencil Tree\n\nPergularia\n\nPigeon Pea.\n\nPineapple\n\nPithraj tree\n\nPlantain\n\nPlumed Cockscomb\n\nPlumeria\n\nPointed gourd\n\nPoison nut.\n\nPomelo\n\nPomme Granate\n\nPorcupine flower\n\nPremna\n\nPrickly-leaved elephant's foot\n\nPurple camel's foot\n\nPurple fleabane\n\nPurple yam\n\nPutranjiv.\n\nQ\n\nQueen Grape Myrtle\n\nQueen of the night\n\nR\n\nRangoon creeper\n\nRattan Palm\n\nRecurved-Leaved Curculigo\n\nRed Nongmangkha\n\nRed Sandalwood\n\nRed Water Lily.\n\nRedball Snakegourd\n\nRice plant\n\nRose\n\nRose Apple\n\nRottler's Chrozophora\n\nRottlera\n\nRoundleaf Bindweed\n\nS\n\nSacred Basil\n\nSacred Garlic Pear\n\nSapota\n\nScarlet Leadwort\n\nScarlet thunbergia\n\nSessile Joyweed\n\nShampoo ginger\n\nShell Ginger\n\nShikakai\n\nSiam Weed\n\nSiamese Rough Bush.\n\nSicklepod\n\nSilk Cotton Tree\n\nSilky Indigo\n\nSissoo\n\nSlender Oldenlandia\n\nSmall Knotweed\n\nSnake root\n\nSoapnut\n\nSophera Senna\n\nSpinach\n\nSpiny Amaranth\n\nSpiny Bittergourd\n\nSpiny Gourd\n\nStar Glory\n\nStar Jasmine\n\nStephania\n\nStinging nettle\n\nStinkvine\n\nStreptolirion\n\nSugar Palm\n\nSugarcane\n\nSweet Basil\n\nSweet broom weed\n\nSweet orange\n\nSweet Sledge\n\nSweet Wormwood\n\nSweetleaf\n\nT\n\nTamarind\n\nTar Vine\n\nTea plant\n\nTeak tree\n\nThe Ivy Gourd\n\nTi Plant\n\nTicktree\n\nToothache Plant\n\nTouch-Me-Not\n\nTrailing eclipta\n\nTridax Daisy\n\nTubeflower\n\nTufted Bamboo\n\nTurmeric\n\nV\n\nVegetable Fern\n\nVeldt Grape\n\nVelvet Apple\n\nVelvet Bean\n\nVitex plant\n\nW\n\nWater chestnut\n\nWater hyacinth\n\nWater hyssop\n\nWater Lettuce\n\nWater Spinach\n\nWeeping Fig\n\nWhiskered Commelina\n\nWhite Goosefoot\n\nWhite Kyllinga\n\nWhite Leadwort\n\nWhite Snakeroot\n\nWhite Turmeric\n\nwild croton\n\nWild Mango\n\nWild Mint\n\nWild Senna\n\nWild Snake root\n\nWild sugar cane\n\nWild sultan seed\n\nWood Apple\n\nY\n\nYam\n\nYapana\n\nYellow - Berried Nightshade.\n\nYellow foxtail\n\nYellow Nicker\n\nYellow Sorrel\nIndex to Vernacular Names\n\nA\n\n\u0100am\n\nAatmor\u0101\n\n\u0100d\u0100\n\nAgniswar\n\nAjashringi\n\nAkanda\n\n\u0100kashmani\n\n\u0100kh\n\n\u0100kn\u0100di\n\nAktir\n\n\u0100lkusi\n\n\u0100m\u0100d\u0100\n\n\u0100ml\u0100\n\n\u0100mloki\n\n\u0100mr\u0100\n\n\u0100mrul\n\nAnantamul\n\n\u0100n\u0100ras\n\n\u0100ngur\n\nAnstan\u0101g\n\nAntamul\n\n\u0100pang\n\nApar\u0101jit\u0101\n\nArhar\n\nArjun\n\nAshok\n\nAssam Lata\n\nAswagandh\u0101\n\nAswattha\n\n\u0100t\u0100\n\nAtiswar\n\nAyapan\n\nB\n\nBabl\u0101\n\nB\u0101bui Tulsi\n\nBach\n\nB\u0101ghnakhi\n\nB\u0101ghnokh\u0101\n\nBaher\u0101\n\nBainchi\n\nBakphul\n\nBakul\n\nBan note\n\nB\u0101ndarl\u0101thi\n\nB\u0101ns\n\nBanshmora\n\nBaro K\u0101krol\n\nBarun\n\nB\u0101sak\n\nBasikaran Kuttus\n\nBatabi lebu\n\nBatu\u0101\n\nBel\n\nBeli\n\nBerala\n\nBet\n\nBh\u0101mot\n\nBhtng\n\nBhringar\u0101j\n\nBhui \u0100ml\u0100\n\nBhui Ch\u0101pa\n\nBhui Kumra\n\nBhuiokhra\n\nBichuti\n\nBichuti p\u0101t\u0101\n\nBish K\u0101nt\u0101l\n\nBishalyakarani\n\nBon chiching\u0101\n\nBon-Elach\n\nBot\n\nBrahmi\n\nBramhajosthi\n\nBriddha Darak\n\nBrihati\n\nBur\u0101sham\u0101\n\nC\n\nCha G\u0101chh\n\nCh\u0101galbati\n\nChai\n\nCh\u0101kunda\n\nCh\u0101lt\u0101\n\nChandan Batu\u0101\n\nCh\u0101p\u0101\n\nCh\u0101tim\n\nChaya\n\nChayna Gab\n\nChayn\u0101 G\u0101b\n\nChemti Sag\n\nChet Kachu\n\nChikasunga\n\nChinimichri\n\nChokotara\n\nChor K\u0101nt\u0101\n\nChoto goaliarlota\n\nChotr\u0101\n\nChoya bans\n\nChul\u0101tk\u0101\n\nChurchuri\n\nCurry P\u0101t\u0101\n\nD\n\nD\u0101dm\u0101ri\n\nD\u0101lim\n\nDandakalash\n\nDanti\n\nD\u0101rodmoyd\u0101\n\nD\u0101ruchini\n\nDebd\u0101ru\n\nDehuy\u0101\n\nDh\u0101n\n\nDhaniya\n\nDh\u0101nya\n\nDharu\u0101\n\nDhekias\u0101k\n\nDhekip\u0101t\u0101\n\nDudhe Jhar\n\nDudhiya\n\nDudhiya\n\nDulfi\n\nDumur\n\nDurb\u0101\n\nE\n\nEk\u0101ngi\n\nEkbir\n\nEucalyptus\n\nG\n\nGab\n\nGabeduk\n\nGajer mul\n\nG\u0101jor\n\nGam\n\nG\u0101m\u0101ri\n\nGandha V\u0101d\u0101l\n\nGargare\n\nGerman Lat\u0101\n\nGhentu\n\nGhor\u0101 Neem\n\nGhritakumari\n\nGima S\u0101k\n\nGoaliar lata\n\nGodhum\n\nGolap\n\nGol\u0101p J\u0101m\n\nGolmorich\n\nGon\u0101l\n\nGop\u0101 s\u0101k\n\nGorur Ch\u0101p\u0101\n\nGulancha\n\nGurm\u0101r\n\nH\n\nHalud\n\nHalud G\u0101nd\u0101\n\nHaritaki\n\nH\u0101rjor\u0101\n\nHastikarnapal\u0101sh\n\nHastipada\n\nHatisur\n\nHegr\u0101\n\nHingsralata\n\nHizal\n\nHochi\n\nHy\u0101lench\u0101\n\nI\n\nIswarmul\n\nJ\n\nJab\u0101\n\nJab\u0101kusum\n\nJ\u0101br\u0101 ghas\n\nJagatmadan\n\nJaljomani\n\nJalp\u0101\n\nJ\u0101m\n\nJanglee Karpur\n\nJ\u0101rul\n\nJ\u0101t\u0101 Lanka\n\nJayanti\n\nJhinti\n\nJig\u0101\n\nJokh\u0101\n\nJui\n\nJungli palak\n\nK\n\nK\u0101ch Kal\u0101\n\nKachu\n\nKachuripan\u0101\n\nKadam\n\nK\u0101km\u0101chi\n\nK\u0101krol\n\nKal\u0101\n\nK\u0101l\u0101 Halud\n\nK\u0101lkasund\u0101\n\nK\u0101lmegh\n\nKalmi Sag\n\nKalo Dhutra\n\nKamala lebu\n\nKamal\u0101guri\n\nK\u0101mini\n\nK\u0101mr\u0101nga\n\nKanak Ch\u0101p\u0101\n\nKanchire\n\nK\u0101ng\n\nKanta \u0100lu\n\nK\u0101nt\u0101 Bishalyakarani\n\nK\u0101nt\u0101 Kachu\n\nKanta-khuria\n\nKanta-note\n\nKantikari\n\nK\u0101on\n\nKarabi\n\nKaramcha\n\nKaranja\n\nKaroi\n\nKarol\u0101\n\nK\u0101rp\u0101s\n\nK\u0101shphul\n\nK\u0101t Bel\n\nK\u0101th\u0101l\n\nK\u0101thb\u0101d\u0101m\n\nKathmalati\n\nKesriya\n\nKesut\n\nKham \u0100lu\n\nKham \u0100lu\n\nKharkon kachu\n\nKhejur\n\nKhet P\u0101apr\u0101\n\nKhetp\u0101pr\u0101\n\nKhoksha\n\nKodo Dh\u0101n\n\nKoukha\n\nKrishna Tulsi\n\nKrishnachur\u0101\n\nKuchil\u0101\n\nKukurmuth\u0101\n\nKukursung\u0101\n\nKul\n\nKulekh\u0101r\u0101\n\nKumarilat\u0101\n\nKunch\n\nKurchi\n\nL\n\nLabli\n\nLajjab\u0101ti\n\nL\u0101l B\u0101sak\n\nL\u0101l Rangan\n\nL\u0101l Sh\u0101luk\n\nL\u0101l Varend\u0101\n\nL\u0101lchita\n\nLanka\n\nLata \u0100lu\n\nLat\u0101 Kasturi\n\nLat\u0101fatki\n\nLebu\n\nLemon Ghas\n\nLuchipata\n\nM\n\nMadar\n\nMadhabilat\u0101\n\nMahu\u0101\n\nM\u0101k\u0101l\n\nM\u0101khn\u0101\n\nM\u0101nkachu\n\nMayn\u0101 K\u0101nt\u0101\n\nMehendi\n\nMith\u0101 p\u0101t\n\nMoicharan\n\nMorog Phul\n\nMuchkunda Ch\u0101p\u0101\n\nMudunirbish\n\nMuktojhuri\n\nMus\u0101karni\n\nMusambi\n\nMuth\u0101 Ghas\n\nN\n\nNagdona\n\nNageswar\n\nN\u0101gkesar\n\nNarikel\n\nN\u0101t\u0101\n\nNayant\u0101r\u0101\n\nNebar\n\nNeel\n\nNeel Sh\u0101luk\n\nNeem\n\nNight queen\n\nNirbish\n\nNishind\u0101\n\nNont\u0101 S\u0101k\n\nO\n\nO l9\n\nP\n\nPadma\n\nP\u0101kur\n\nPal\u0101sh\n\nPalong\n\nP\u0101n\n\nPand\u0101l\n\nPani\u0101i\u0101\n\nPanijuli\n\nP\u0101niphal\n\nP\u0101nishitki\n\nParavanga\n\nP\u0101t\u0101jh\u0101njhi\n\nPatal\n\nPatari\n\nPatharkuchi\n\nP\u0101ts\u0101ji\n\nP\u0101yesp\u0101t\u0101\n\nPeet Ber\u0101l\u0101\n\nPepe\n\nPeyar\u0101\n\nPhul Tarul\n\nPiaj\n\nPinginatchi\n\nPipul\n\nPipulti\n\nPrisniparni\n\nPrithviraj\n\nPudin\u0101\n\nPui\n\nPunarnab\u0101\n\nPutranjiva\n\nR\n\nR\u0101dh\u0101 Tulsi\n\nR\u0101kh\u0101l Sos\u0101\n\nRakta Chandan\n\nRakta G\u0101nd\u0101\n\nR\u0101kt\u0101 Kanhcan\n\nR\u0101m Tulsi\n\nR\u0101sn\u0101\n\nRasun Sag\n\nRayri\n\nRith\u0101\n\nRosun\n\nS\n\nSabed\u0101\n\nS\u0101d\u0101 Dhutra\n\nS\u0101d\u0101 Varend\u0101\n\nS\u0101d\u0101chit\u0101\n\nSahadebi\n\nSahodebi\n\nSajna\n\nSalincha\n\nS\u0101lp\u0101ni\n\nSanchi\n\nSandhy\u0101m\u0101lati\n\nSarpagandh\u0101\n\nSatamul\n\nSegun\n\nSeph\u0101li\n\nShewra\n\nShikakai\n\nShiuli\n\nShola\n\nShothi\n\nSilnaj\u0101\n\nSimul\n\nSindur\n\nSingha Puccha\n\nSiris\n\nSisu\n\nSiyal K\u0101nt\u0101\n\nSomr\u0101ji\n\nSona\n\nSondal\n\nStevia\n\nStreptolirion\n\nSthal Padma\n\nSup\u0101ri\n\nSusni\n\nSwarnalat\u0101\n\nSwet Ber\u0101l\u0101\n\nSwet Simul\n\nSwetdron\n\nT\n\nTagar\n\nT\u0101l\n\nT\u0101l\u0101lati\n\nT\u0101lmuli\n\nTapioca\n\nTarulat\u0101\n\nTejpat\u0101\n\nTel\u0101kuch\u0101\n\nTentul\n\nTepari\n\nTh\u0101nkuni\n\nThunbergia\n\nTitvat\n\nTop\u0101 Pan\u0101\n\nU\n\nUlatchandal\n\nUlatkambal\n\nUlu gh\u0101s\n\nUridh\u0101n\n\nV\n\nVel\u0101\n\nVui orchid\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nMoney\n\nWorld Bank\n\n\u00a9 2010 by iMinds Pty Ltd.\n\nAll Rights Reserved.\n\n## Money: World Bank\n\nThe World Bank is an international financial institution of the UN whose purpose is the alleviation of poverty through providing leveraged loans and technical assistance to the world's poorest countries. The World Bank was created at the Bretton Woods Conference in 1944 with the primary role of financing the post-war reconstruction of European and Asian countries devastated in WWII. Today the World Bank consists of the International Bank of Reconstruction and Development (IBRD) and the International Development Association (IDA) which form part of the five institutions which make up the World Bank Group. The World Bank Group plays a leading coordination role in international economic affairs due to it being a multilateral institution with extensive experience and partnerships with countless UN agencies, aid agencies, NGOs and national, regional and local governments.\n\nThe IBRD, the original institution of the World Bank, is a triple A rated financial institution which provides loans, guarantees and analytical and advisory services to middle-income and creditworthy low-income countries. These countries represent 70% of the world's poor. The Bank uses its triple A credit rating to source funds from the world's financial markets and provides these at a small mark-up to qualifying countries for approved projects. While these medium-term, usually 10 year, loans are substantially lower than those in the private sector they are often attached with strict conditions and monitoring programs. Critics argue that these conditions impede on the borrowing country's freedom. The IBRD maintains that, unlike other banks, it is driven by development impact rather than profit maximization. Despite this it has been able to maintain a positive net-income since 1948.\n\nBy 1960 it became evident that the world's poorest countries could not afford to borrow capital for development or maintain the credit ratings required by the IBRD. This led to the formation of the IDA in 1960 as an institution of the World Bank that provides credit and grants to low income countries. As of 2005, this includes nations with a per capita income of less than US$1,025. There are 82 low income countries with a total of 2.6 billion people, accounting for half of the population of the developing world. These credits take the form of long-term, 20, 35 or 40 year, zero interest loans designed to assist the countries in meeting the IDS's development goals. The IDA is financed through government contributions from high income member countries every three years as well as from the net income of the IBRD. US$170 billion has been lent to 108 countries since 1960 increasing annually to over US$9 billion in the 2008-09 financial year. The US, UK, Japan, Germany, France, Italy and Canada have made the greatest contributions in the latest round.\n\nThe World Bank's current portfolio includes 1,617 operations with US$106 billion in IBRD loans outstanding and US$113 billion in outstanding IDA credits. Due to the World Financial Crisis its financing has increased significantly in the past three years. Loans, grants, equity investments, and guarantees increased 54% between the 2007-08 and 2008-09 financial years and a total of US$71.4 billion was lent between June 2008 and June 2010. This includes US$53.1 billion committed by the **IBRD and US** $18.3 billion by the IDA. The provision of capital is however not the only function of the World Bank. It also provides technical and advisory support to developing countries with a philosophy that money alone cannot achieve development goals. This is illustrated by the Bank's significant increase in expenditure and focus on knowledge and advisory activities with their proportion of total expenditure growing from 24 to 29% since 2007.\n\nThe World Bank is a cooperative that is owned and governed by its 185 member countries whose voting power, unlike the UN's, is determined by each countries share in the Bank's capital, giving wealthy countries greater power in the organization. The United States is the largest shareholder in the Bank by a significant margin followed by Japan, Germany, France and the United Kingdom. Due to their dominant share and a long standing informal agreement the President of the World Bank is always a US national. While the bank maintains that it seeks to act cooperatively and develop agreement between all country members when possible, its large size and financial power has led to ongoing and often widely held critiques. These include concerns about its undemocratic governance reflecting global imbalances in power, its narrow neoclassical economic ideology, its financing of a number of projects that have resulted in displaced populations and environmental damage and the integrity of the Bank's ethics. Despite these critiques the invaluable role the World Bank plays in financing development and poverty projects and providing advisory and knowledge services is widely regarded.\n\nIn pursuing its development goals the World Bank follows a number of core strategies which guide its work. The Millennium Development Goals, which identify and qualify eight specific targets to be met by 2015, are the Bank's fundamental objectives. These goals include the eradication of extreme poverty and hunger, achieving universal primary education, promoting gender equality and empowering women, reducing child mortality, improving maternal health, combating HIV\/Aids, malaria and other diseases, ensuring environmental sustainability and developing a global partnership for development. The World Bank directs its finance towards projects that will help build the foundations towards addressing these goals. Central to this strategy is the philosophy that countries must become the leaders and owners of their own development and poverty reduction policies. This philosophy was adopted in 1999 to address the historical shortcomings of many development projects which were narrowly focused and run from the top down with little local consultation or participation in the affected country. Examples of recent World Bank financed projects range from a US$200 million education project to ensure education for all in Vietnam to the far more controversial proposal to lend US$3.75 billion to the South African utility company Eskom to build a 4,800 MW coal plant, the world's fourth largest.\n\nThe success of the World Bank in facilitating development and eradicating poverty is a contentious question that is difficult to assess. Since its foundation 30 countries have graduated from IDA qualification indicating rising per capita incomes. The World Bank itself however acknowledges that as of 2008 40% of its projects in the world's most fragile states have produced unsatisfactory progress towards development outcomes. Conversely the Bank claims that 90% of IBRD projects in middle income countries have proved successful. Progress towards the Millennium Development Goals is uneven with China and India achieving strong poverty reduction while little progress has been seen in Sub-Saharan Africa. Despite its debatable progress the World Bank remains the world's premier development institution entrusted with the largest pool of world aid. It is deeply entrenched in every facet of world development and plays a vital role in the reduction of world poverty. \n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n## Contents\n\nCover\n\nAbout the Book\n\nAbout the Author\n\nAlso by Michel Houellebecq\n\nTitle Page\n\nEpigraph\n\nPart One: Thai Tropic\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\nPart Two: Competitive Advantage\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\nPart Three: Pattaya Beach\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nCopyright\n\n## About the Book\n\n* * *\n\nMichel is a civil-servant at the Ministry of Culture. When his father is murdered, Michel takes a leave of absence to go on a package tour to Thailand. Infuriated by the shallow hypocrisy and mediocrity of his fellow travellers, only the awkward Valerie attracts his attention. Too bashful to pursue her, Michel prefers the uncomplicated pleasures of Thai massage parlours and sex with local women.\n\nBack in Paris, he calls Valerie and they plunge into a passionate affair, which strays into S&M, partner-swapping and sex in public. Michel quits his job, and tries to help Valerie and her boss, Jean-Yves, in their ailing travel business, by offering travel packages based on sex tourism in the third world. When their project comes to fruition and the three return to Thailand, Michel discovers that sex is neither the most consuming nor the most dangerous of human passions...\n\n## About the Author\n\n* * *\n\nPoet and novelist Michel Houellebecq is the author of two previous novels, _Whatever_ ( _Extension du domaine de la lutte_ ) and the international bestseller _Atomised_ ( _Les Particules \u00e9l\u00e9mentaires_ ), winner of the Prix Novembre and the 2002 International IMPAC Dublin Literary Award. He lives in Ireland.\n\n##### ALSO BY MICHEL HOUELLEBECQ\n\n_Whatever_\n\n_Atomised_\n\n## Michel Houellebecq\n\n## PLATFORM\n\n### TRANSLATED FROM THE FRENCH BY \nFrank Wynne\n\n_Plus sa vie est inf\u00e2me, plus l'homme y tient; elle est alors une protestation, une vengeance de tous les instants_.\n\n###### Honor\u00e9 de Balzac\n\n_(The more contemptible his life, the more a man clings to it; it thus becomes a protest, a retribution for every moment.)_\n\n## Part One\n\n## _Thai Tropic_\n\n### 1\n\nFATHER DIED LAST year. I don't subscribe to the theory by which we only become _truly adult_ when our parents die; we never become truly adult.\n\nAs I stood before the old man's coffin, unpleasant thoughts came to me. He had made the most of life, the old bastard; he was a clever cunt. 'You had kids, you fucker...' I said spiritedly, 'you shoved your fat cock in my mother's cunt.' Well, I was a bit tense, I have to admit; it's not every day you have a death in the family. I'd refused to see the corpse. I'm forty, I've already had plenty of opportunity to see corpses; nowadays, I prefer to avoid them. It was this that had always dissuaded me from getting a pet.\n\nI'm not married, either. I've had the opportunity several times, but I never took it. That said, I really love women. It's always been a bit of a regret, for me, being single. It's particularly awkward on holiday. People are suspicious of single men on holiday, after they get to a certain age: they assume that they're selfish, and probably a bit pervy; I can't say they're wrong.\n\nAfter the funeral, I went back to the house where my father lived out his last years. The body had been discovered a week earlier. A little dust had already settled around the furniture and in the corners of the rooms; I noticed a cobweb on the window frame. So time, entropy, all that stuff, was slowly taking the place over. The freezer was empty. The kitchen cupboards mostly contained single-serving Weight Watchers meals-in-a-bag, tins of flavoured protein and energy bars. I wandered through the rooms nibbling a magnesium-enriched biscuit. In the boiler room, I rode the exercise bike for a while. My father was over seventy and in much better physical shape than I was. He did an hour of rigorous exercise every day, lengths of the pool twice a week. At weekends, he played tennis and went cycling with people his age; I'd met some of them at the funeral. 'He coached the lot of us!...' a gynaecologist exclaimed. 'He was ten years older than us, and on a two kilometre hill, he'd be a whole minute ahead.' Father, father, I said to myself, how great was your vanity! To the left of my field of vision I could make out a weightlifting bench, barbells. I quickly visualised a moron in shorts \u2013 his face wrinkled, but otherwise very like mine \u2013 building up his pectorals with hopeless vigour. Father, I said to myself, Father, you have built your house upon sand. I was still pedalling but I was starting to feel breathless, my thighs ached a little, though I was only on level one. Thinking back to the ceremony, I was aware that I had made an excellent general impression. I'm always clean-shaven, my shoulders are narrow and when I developed a bald spot at about the age of thirty, I decided to cut my hair very short. I usually wear a grey suit and sober ties, and I don't look particularly cheerful. With my short hair, my lightweight glasses and my sullen expression, my head bowed a little to listen to a Christian funeral-hymn medley, I felt perfectly at ease with the situation \u2013 much more at ease than I would have done at a wedding, for example. Funerals, clearly, were my thing. I stopped pedalling, coughed gently. Night was falling quickly over the surrounding meadows. Near the concrete structure which housed the boiler, you could make out a brownish stain which had been poorly cleaned. It was there that my father had been discovered, his skull shattered, wearing shorts and an 'I love New York' sweatshirt. He had been dead for three days, according to the coroner. There was the possibility, very remote, that what happened was an accident, he could have slipped in a puddle of oil or something. That said, the floor of the room was completely dry; and the skull had been broken in several places, some of the brain had even spilled on to the floor; in all probability, what we were dealing with was murder. Captain Chaumont of the Cherbourg police was supposed to come over to see me that evening.\n\nBack in the living room, I turned on the television, a 32-inch Sony widescreen with surround sound and an integrated DVD player. There was an episode of _Xena: Warrior Princess_ on TF1, one of my favourite series: two very muscular women wearing metallic bras and miniskirts made of animal hide were challenging each other with their sabres. 'Your reign has gone on too long, Tagrath\u00e2!' cried the brunette, 'I am Xena, warrior of the Western Plains!' There was a knock at the door; I turned the sound down.\n\nOutside, it was dark. The wind gently shook the branches dripping with rain. A girl of about twenty-five, she looked north-African, was standing in the doorway. 'I'm A\u00efcha,' she said, 'I cleaned for Monsieur Renault twice a week. I've just come to get my things.'\n\n'Well...' I said, '... well.' I made a vague gesture, something intended to be welcoming. She came in, glanced quickly at the television screen: the two warriors were now wrestling right next to a volcano; I suppose the spectacle had its stimulating side, for certain lesbians. 'I don't want to disturb you,' said A\u00efcha, 'I'll only be five minutes.'\n\n'You're not disturbing me,' I said, 'in fact, nothing disturbs me.' She nodded her head as though she understood, her eyes lingered on my face; she was probably gauging my physical resemblance to my father, possibly inferring a degree of _moral_ resemblance. After studying me for a few moments, she turned and climbed the stairs that lead to the bedrooms. 'Take your time,' I said, my voice barely audible. 'Take all the time you need...' She didn't answer, didn't pause in her ascent; she had probably not even heard me. I sat down on the sofa again, exhausted by the confrontation. I should have offered to take her coat; that's what you usually do, offer to take someone's coat. I realised that the room was terribly cold \u2013 a damp, penetrating cold, the cold of a cellar. I didn't know how to light the boiler, I had no wish to try, now my father was dead. I had intended to leave straight away. I turned over to FR3 just in time to catch the last part of _Questions pour un champion_. At the moment when Nad\u00e8ge from Val-Fourr\u00e9 told Julien Lepers that she was going to risk her title for the third time, A\u00efcha appeared on the stairs, a small travel bag on her shoulder. I turned off the television and walked quickly towards her. 'I've always admired Julien Lepers.' I told her, 'Even if he doesn't know the actual town or village the contestant is from, he always manages to say something about the d\u00e9partement or the region; he always knows a bit about the climate and the local beauty spots. Above all, he understands life: the contestants are human beings to him, he understands their problems and their joys. Nothing of what constitutes human reality for the contestants is entirely strange or intimidating to him. Whoever the contestant is, he manages to get them to talk about their work, their family, their hobbies \u2013 everything, in fact, that in their eyes goes to make up a life. The contestants are often members of a brass band or a choral society, they're involved in organising the local f\u00eate, or they devote themselves to some charitable cause. Their children are often there in the studio. You generally get the impression from the programme that these people are happy, and you feel better, happier yourself. Don't you think?'\n\nShe looked at me unsmilingly; her hair in a chignon, she wore little makeup, her clothes were pretty drab \u2013 a serious girl. She hesitated for a moment before saying in a low voice which was a little hoarse with shyness: 'I was very fond of your father.' I couldn't think of anything to say; it struck me as bizarre, but just about possible. The old man must have had stories to tell: he'd travelled in Colombia, Kenya or I don't know where; he'd had the opportunity to watch rhinoceros through binoculars. Every time we met, he limited himself to making ironic comments about the fact that I was a civil servant, about the job security that went with it. 'Got yourself a cushy little number, there...' he would say, making no attempt to hide his scorn; families are always a bit difficult. 'I'm studying nursing,' A\u00efcha went on, 'but since I stopped living with my parents I have to work as a cleaner.' I racked my brains to think of an appropriate response: should I enquire as to how expensive rents were in Cherbourg? I finally opted for a 'I see...', into which I tried to introduce a certain worldly wisdom. This seemed to satisfy her and she walked to the door. I pressed my face to the glass to watch her Volkswagen Polo do a U-turn in the muddy track. FR3 was showing some rustic made-for-TV movie set in the nineteenth century, starring Tch\u00e9ky Karyo as a farm labourer. Between piano lessons, the daughter of the landowner \u2013 he was played by Jean-Pierre Marielle \u2013 accorded the handsome peasant certain liberties. Their clinches took place in a stable; I dozed off just as Tch\u00e9ky Karyo was energetically ripping off her organza knickers. The last thing I remember was a close-up of a small group of pigs.\n\nI was woken by pain and by the cold; I had probably fallen asleep in an awkward position, my cervical vertebrae felt paralysed. I was coughing heavily as I stood up, my breath filling the glacial air of the room with vapour. Bizarrely, the television was showing _Tr\u00e8s P\u00eache_ , a fishing programme on TF1; I had obviously woken up, or at least reached a sufficient level of consciousness to work the remote control; I had no memory of doing so. Tonight's programme was devoted to silurids \u2013 huge fish with no scales which had become more common in French rivers as a result of global warming; they were particularly fond of the areas around nuclear power plants. The report was intended to shed light on the truth behind a number of myths: it was true that adult silurids could grow to as much as three or four metres; in the Dr\u00f4me, specimens larger than five metres had been reported; there was nothing particularly improbable about this. However there was no question of the animals ever behaving carnivorously, or attacking bathers. The public suspicion of silurids seemed, to some extent, to have rubbed off on the men who fished for them; the small group of silurid anglers was not well liked by the larger family of anglers. They felt they suffered as a result and wanted to take advantage of this programme to improve their negative image. It was true they could hardly suggest gastronomy as their motive: the flesh of the silurid was completely inedible. But it was an excellent catch, intelligent and at the same time requiring sportsmanship; it was not unlike pike fishing, and deserved a wider following. I paced around the room a little, unable to get warm; I couldn't bear the idea of sleeping in my father's bed. In the end I went upstairs and brought down pillows and blankets, settled myself as best I could on the sofa. I switched off just after the credits of 'The Silurid Demystified'. The night was opaque, the silence also.\n\n### 2\n\nALL THINGS COME to an end, including the night. I was dragged from my saurian lethargy by the clear, resonant voice of Captain Chaumont. He apologised, he hadn't had time to come by the previous evening. I offered him coffee. While the water was boiling, he set up his laptop on the kitchen table and hooked up a printer. This way he could have me re-read and sign my statement before he left; I made a murmur of approval. The police force was so completely snowed under with administrative work that it did not have enough time to dedicate to its real task: investigation. That at least was what I had concluded from various television documentaries. He agreed, warmly this time. This interview was getting off to a good start, in an atmosphere of mutual trust. 'Windows' started up with a cheerful little sound.\n\nThe death of my father occurred in the evening or the night of November 14th. I was working that day; I was working on the 15th, too. Obviously, I could have taken my car and killed my father, having driven there and back overnight. What was I doing on the evening or the night of November 14th? Nothing, as far as I knew, nothing significant. At least, I had no memory of anything, though it was less than a week before. I had neither a regular sexual partner nor any real close friends: in which case, how can you be expected to remember? The days go by, and that's it. I gave Chaumont an apologetic look; I would have liked to help him out, or at least point him in the right direction. 'I'll have a look in my diary...' I said. I wasn't expecting anything of this; but curiously, there was a mobile number written in the space for the 14th beneath a name \u2013 'Coralie'. Who was Coralie? The diary was completely useless.\n\n'My brain is a mess...' I said with a disappointed smile. 'But, I don't know, maybe I was at a private view.'\n\n'A private view?' He waited patiently, his fingers hovering some inches above the keyboard.\n\n'Yes, I work for the Ministry of Culture. I plan the financing for exhibitions, or sometimes shows.'\n\n'Shows?'\n\n'Shows... contemporary dance...' I felt completely desperate, overcome with shame.\n\n'Generally speaking, then, you work on cultural events?'\n\n'Yes, that's it... You could put it like that.' He looked at me with a compassion tinged with seriousness. He had an awareness of the existence of a cultural sector, a vague but definite awareness. He must have had to meet people from all walks of life in his profession; no area of society could be completely alien to him. Police work is a human science.\n\nThe rest of the interview proceeded more or less normally; I had watched a few made-for-TV movies, so I was prepared for this kind of conversation. Did I know any enemies my father might have? No, but no friends either, to be honest. In any case, my father wasn't _important_ enough to have enemies. Who stood to gain by his death? Well, me. When did I last visit him? August, probably. There's never much to do in the office in August, and my colleagues have to go on holiday because they have children. I stay in Paris, I play solitaire on the computer and around the 15th I take a long weekend off; that was the extent of my visits to my father. On that subject, did I have a good relationship with my father? Yes and no. Mostly no, but I came to see him once or twice a year; that in itself wasn't too bad.\n\nHe nodded. I could feel my statement was coming to an end; I would have liked to say more. I felt overcome by a feeling of irrational, abnormal pity for Chaumont. He was already loading paper into his printer. 'My father was very sporty!' I said brusquely. He looked up at me enquiringly. 'I don't know...' I said, spreading my hands in despair, 'I just wanted to say that he was very athletic.' He shrugged disappointedly and pressed 'Print'.\n\nAfter I'd signed my statement, I walked Captain Chaumont to the door. I was aware that I had been a disappointing witness, I told him. 'All witnesses are disappointing...' he said. I considered this aphorism for a while. Before us stretched the endless monotony of the fields. Chaumont climbed into his Peugeot 305; he would keep me informed of any developments in the investigation. In the public sector, the death of a parent or grandparent entitles one to three days' leave. As a result, I could very easily have taken my time going home, bought some local camembert; but I immediately took the motorway for Paris.\n\nI spent the last day of my compassionate leave in various travel agencies. I liked holiday brochures, their abstraction, their way of condensing the places of the world into a limited sequence of possible pleasures and fares; I was particularly fond of the star-ratings system, which indicated the intensity of the pleasure one was entitled to hope for. I wasn't happy, but I valued happiness and continued to aspire to it. According to the Marshall model, the buyer is a rational individual seeking to maximise his satisfaction while taking price into consideration; Veblen's model, on the other hand, analyses the effect of peer pressure on the buying process (depending on whether the buyer wishes to be identified with a defined group or to set himself apart from it). Copeland demonstrates that the buying process varies, depending on the category of product\/service (impulse purchase, considered purchase, specialised purchase); but the Baudrillard and Becker model posits that a purchase necessarily implies a series of signals. Overall, I felt myself closer to the Marshall model.\n\nBack at the office I told Marie-Jeanne that I needed a holiday. Marie-Jeanne is my colleague; together we work on exhibition proposals, together we work for the benefit of the contemporary arts. She is a woman of thirty-five, with lank blond hair, her eyes are a very light blue; I know nothing about her personal life. Within the office hierarchy, she has a position slightly senior to mine; but this is something which she ignores \u2013 she likes to emphasise teamwork within the office. Every time we receive a visit from a really important person \u2013 a delegate from the Department of Plastic Arts or someone from the Ministry \u2013 she insists on this notion of teamwork. 'And this is the most important man in the office!...' she exclaims, walking into my office; 'He's the one who juggles the figures and the financial statements... I would be completely lost without him.' And then she laughs; the important visitors laugh in turn, or at least smile good-naturedly. I smile too, insofar as I can. I try to imagine myself as a juggler; but in reality it's quite enough to master simple arithmetic. Although strictly speaking Marie-Jeanne does nothing, her work is, in fact, the most complicated job: she has to keep abreast of movements, networks, trends; having assumed a level of cultural responsibility, she constantly runs the risk of being thought reactionary, even obscurantist; it is an accusation from which she must defend herself and the institution. She is also in regular contact with artists, gallery owners and the editors of obscure reviews, obscure, at least, to me; these telephone calls keep her happy, because her passion for contemporary art is real. As far as I'm concerned, I'm not actively hostile to it: I am not an advocate of _craft_ , nor of a return to figurative painting; I maintain the disinterested attitude appropriate to an accounts manager. Questions of aesthetics and politics are not my thing; it's not up to me to invent or adopt new attitudes, new affinities with the world \u2013 I gave up all that at the same time I developed a stoop and my face started to tend towards melancholy. I've attended many exhibitions, private views, many performances that remain unforgettable. My conclusion, henceforth, is that art cannot change lives. At least not mine.\n\nI had informed Marie-Jeanne of my bereavement; she greeted me sympathetically, she even put her hand on my shoulder. My request to take some time off seemed completely natural to her. 'You need to take stock, Michel,' she reckoned, 'you need to turn inward.' I tried to visualise the movement she was suggesting and I concluded that she was probably right. 'C\u00e9cilia will put the provisional budget to bed,' she went on; 'I'll talk to her about it.' What precisely was she alluding to, and who was this C\u00e9cilia? Glancing around me, I noticed the design for a poster and I remembered. C\u00e9cilia was a fat, redhead who was always gorging herself on Cadburys and who'd been in the department for two months: a temp, work experience maybe, someone pretty insignificant at any rate. And it was true that before my father's death I had been working on a provisional budget for the exhibition, 'Hands Up, You Rascals!', due to open in Bourg-la-Reine in January. It consisted of photographs of police brutality taken with a telephoto lens in Yvelines; but we weren't talking documentary here, more a process of the theatricalisation of space, full of nods to various cop shows featuring the Los Angeles Police Department. The artist had favoured a 'fun' approach rather than the social critique you'd expect. An interesting project, all in all, not too expensive nor too complicated; even a moron like C\u00e9cilia was capable of finalising the provisional budget.\n\nUsually, when I left the office, I'd take in a peepshow. It set me back fifty francs, maybe seventy if I was slow to ejaculate. Watching pussy in motion cleared my head. The contradictory trends of contemporary video art, balancing the conservation of national heritage with support for living creativity... all of that quickly evaporated before the facile magic of a moving pussy. I gently emptied my testicles. At the same moment, C\u00e9cilia was stuffing herself with chocolate cake in a p\u00e2tisserie near the Ministry; our motives were much the same.\n\nVery occasionally, I would take a private room at five hundred francs; that was if my dick wasn't feeling too good, when it seemed to me to resemble a useless, demanding little appendage that smelled like cheese. Then I needed a girl to take it in her hands, to go into raptures, however faked, over its vigour, the richness of its semen. Be that as it may, I was always home before seven-thirty. I'd start with _Questions pour un champion_ which I had set my video to record; then I would continue with the national news. The mad cow disease crisis was of little interest to me, mostly I survived on Mousline instant mash with cheese. Then the evening would continue. I wasn't unhappy, I had 128 channels. At about two in the morning, I'd finish with Turkish musicals.\n\nA number of days went by like this, relatively peacefully, before I received another phone call from Chaumont. Things had progressed significantly, they had found the alleged killer; actually, it was more than a allegation, for in fact the man had confessed. They were going to stage a re-enactment in a couple of days. Did I want to be present? Oh yes, I said, yes.\n\nMarie-Jeanne congratulated me on this courageous decision. She talked about the grieving process, the mysteries of the father-son relationship. She used socially acceptable terms from a limited catalogue, what was more important, even surprising, was that I realized that she was fond of me, and it felt good. Women really do have a handle on affection, I thought as I boarded the Cherbourg train; even at work, they have a tendancy to establish emotional ties, finding it difficult to orient themselves, let alone thrive, in a universe completely stripped of such emotional ties, they find it difficult to thrive in such an atmosphere. This was a weakness of theirs, as the 'psychology' column _of Marie-Claire_ continually reminded them: it would be better if they could clearly separate the professional from the emotional, but they simply could not do it, and the 'true stories' column of _Marie-Claire_ confirmed with equal regularity. Somewhere near Rouen, I reviewed the essential facts of the case. Chaumont's breakthrough was the discovery that A\u00efcha had been having 'intimate relations' with my father. How often, and how intimate? He didn't know, and it had no significance to his continuing inquiry. One of A\u00efcha's brothers had quickly confessed that he had come 'to demand an explanation' of the old man, things had got out of hand, and he had left him for dead on the concrete floor of the boiler room.\n\nIn principle, the re-enactment was to be presided over by the examining magistrate, a brusque, austere little man, dressed in flannel trousers and a dark polo-neck, his face permanently clenched in a rictus of irritation; but Chaumont quickly established himself as the real master of ceremonies. Briskly and cheerfully he greeted the participants, gave each a little word of welcome, and led them to their places: he seemed remarkably happy. This was his first murder case and he'd solved it in less than a week; in this whole banal, sordid story, he was the only true hero. Clearly overcome, a black band covering her face, A\u00efcha sat on a chair trying to look small. She barely looked up when I arrived, pointedly looking away from where her brother was standing. Her brother, flanked by two policemen, stared at the floor with an obstinate air. He looked just like a common little thug; I didn't feel the slightest sympathy for him. Looking up, his eyes met mine; no doubt he knew who I was, He knew my role, he had undoubtedly been told: according to his brutal view of the world, I had a right to vengeance, I deserved an accounting for the blood of my father. Aware of the rapport establishing itself between us, I stared at him, not turning away; I allowed hatred to overwhelm me slowly, my breathing became easier, it was a powerful, pleasurable sensation. If I had had a gun, I would have shot him without a second thought. Killing that little shit not only seemed to me a morally neutral act, but something positive, beneficial. A policeman made some marks on the floor with a piece of chalk, and the re-enactment began. According to the accused, it was very simple: during the conversation, he had become angry and pushed my father roughly; the latter had fallen backwards, his skull had shattered on the floor; he panicked, he fled.\n\nOf course he was lying, and Chaumont had no trouble establishing this. An examination of the victim's skull clearly indicated a furious attack; there were multiple contusions, probably the result of a series of kicks. Furthermore, my father's face had been scraped along the ground, almost sufficient to force the eye from its socket. 'I don't remember...' said the accused man; 'I lost it.' Watching his nervous arms, his thin, horrible face, it wasn't difficult to believe him: he hadn't planned this, he was probably excited by the impact of the skull on the ground and the sight of first blood. His defence was lucid and credible, he would probably come across well in front of a jury: a two-or three-year suspended sentence, no more. Chaumont, pleased with the way the afternoon had gone, began to bring things to a conclusion. I got up from my chair and walked over to one of the picture windows. It was getting dark: a flock of sheep were bringing their day to a close. They too were stupid, possibly even more stupid than A\u00efcha's brother; but violence had not been programmed into their genes. On the last night of their lives they would bleat in terror, their hoofs would scrabble desperately; there would be a gunshot, their lives would seep away and their flesh would be transformed into meat. We parted with a round of handshakes; Chaumont thanked me for coming.\n\nI saw A\u00efcha the following day; on the advice of the estate agent, I had decided to have the house thoroughly cleaned before it was viewed. I gave her the keys, then she dropped me off at Cherbourg station. Winter was taking hold of the farmlands, clouds of mist hung over the hedges. We were uncomfortable being together. She had been familiar with my father's genitals, which tended to create a certain misplaced intimacy. It was all rather surprising: she seemed like a serious girl, and my father was hardly a ladies' man. He must have had certain traits, certain characteristics that I had failed to notice; in fact I was finding it difficult to remember his face. Men live alongside one another like cattle; it is a miracle if once in a while they manage to share a bottle of booze.\n\nA\u00efcha's Volkswagen stopped in front of the station; I was aware that it would be best to say a few words before we parted. 'Well...' I said. After a few seconds, she spoke to me in a subdued voice: 'I'm going to leave the area. I've got a friend who can get me a job as a waitress in Paris; I can continue my studies there. In any case, my family think I'm a whore.' I made a murmur of comprehension. 'There are a lot more people in Paris...' I finally ventured with difficulty; I'd racked my brains, but that was all I could think of to say about Paris. The acute poverty of my response did not seem to discourage her. 'There's no point expecting anything from my family,' she went on with suppressed fury. 'They're not only poor, they're bloody stupid. Two years ago, my father went on the pilgrimage to Mecca; since then, you can't get a word out of him. My brothers are worse: they encourage each other's stupidity. They get blind drunk on pastis and all the while they strut around like the guardians of the one true faith, and they treat me like a slut because I prefer to go out and work rather than marry some stupid bastard like them.'\n\n'It's true, Muslims on the whole aren't up to much...' I said with embarrassment. I picked up my travel bag, opened the door. 'I think you'll do alright...' I muttered without conviction. At that moment I had a vision of migratory flows crisscrossing Europe like blood vessels; Muslims appeared as clots that were only slowly reabsorbed. A\u00efcha eyed me sceptically. Cold air rushed into the car. Intellectually, I could manage to feel a certain attraction to Muslim vaginas. I managed a little forced smile. She smiled in turn, a little more sincerely than I had. I shook her hand for a long time. I could feel the warmth of her fingers. I carried on shaking her hand until I could feel the gentle pulse of her blood at the hollow of her wrist. A few feet from the car, I turned around to give a little wave. We had made a connection in spite of everything; in the end, in spite of everything, something had happened.\n\nSettling into my seat on board the intercity, it occurred to me that I should have given her money. Actually, it was better that I hadn't, it would probably have been misinterpreted. Strangely, it was at that moment that I realised for the first time that I was going to be a rich man; well, relatively rich. The money in my father's accounts had already been transferred. For the rest, I had left the sale of the car to a local garage and of the house to an estate agent; everything had been arranged as simply as possible. The value of these assets would be determined by the market. Of course there was some room for negotiation: ten per cent one way or the other, no more. The taxes that were due were no mystery either: a quick look through the carefully thought-out little brochures available from the Tax Office would be enough.\n\nMy father had probably thought of disinheriting me several times; in the end, he must have given up on the idea, considering it too complicated, too much paperwork for an uncertain result (it is not easy to disinherit your children, the law offers you very limited possibilities: not only do the little shits ruin your life, afterwards they get to profit from everything you've managed to save, despite your worst efforts). He probably thought that there was no point \u2013 after all, what the fuck did it matter to him what happened after he was dead? That's how he looked at it, in my opinion. In any case, the old bastard was dead, and I was about to sell the house in which he had spent his last years; I was also going to sell the Toyota Land Cruiser which he had used for hauling cases of Evian from the Casino G\u00e9ant in Cherbourg. I live near the Jardin des Plantes, what would I want with a Toyota Land Cruiser? I could have used it to ferry ricotta ravioli from the market at Mouffetard, that's about it.\n\nIn cases of direct inheritance, death duties are not very high \u2013 even if the emotional ties aren't very strong. After tax, I could probably expect about three million francs. To me, that represented about fifteen times my annual salary. It also represented what an unskilled worker in western Europe could expect for a lifetime of work; it wasn't so bad. You could make a start with that, you could try.\n\nIn a few weeks I would surely get a letter from my bank. The train approached Bayeux. I could already imagine the course of the conversation. The clerk at my branch would have noticed a substantial credit balance on my account, which he would very much like to discuss with me \u2013 who does not need a _financial adviser_ at one time or another in his life? A little wary, I would want to steer him towards safe options; he would greet this reaction \u2013 such a common one \u2013 with a slight smile. Most novice investors, as financial advisers well know, favour security over earnings; they often laugh about it among themselves. I should not misunderstand him: when it came to managing their capital, even some elderly and otherwsie worldly people behave like complete novices. For his part, he would try to steer me in the direction of a slightly different approach \u2013 while, of course, giving me time to consider my options. Why not, in effect, put two-thirds of my holdings into investments where there would be no surprises but a low return? And why not place the remaining third in investments that were a little more adventurous, but which had the potential for significant growth? After a few days' consideration, I knew, I would defer to his judgement. He would feel reassured by my support, would put together the papers with a flash of enthusiasm, and our handshake at the moment we parted company would be warm.\n\nI was living in a country distinguished by placid socialism, where ownership of material possessions was guaranteed by strict legislation, where the banking system was surrounded by powerful state guarantees. Unless I were to venture beyond what was lawful, I ran no risk of embezzlement or fraudulent bankruptcy. All in all, I needn't worry any more. In fact, I never really had: after serious but hardly distinguished studies, I had quickly found a career in the public sector. This was in the mid-eighties, at the beginning of the modernisation of socialism, at the time when the illustrious Jack Lang was distributing wealth and glory to the cultural institutions of the State; my starting salary was very reasonable. And then I had grown older, standing untroubled on the sidelines through successive policy changes. I was courteous, well-mannered, well-liked by colleagues and superiors; my temperament, however, was less than warm and I had failed to make any real friends. Night was falling quickly over Lisieux. Why, in my work, had I never shown a passion comparable to Marie-Jeanne's? Why had I never shown any real passion in my life in general?\n\nSeveral more weeks went by without bringing me an answer; then, on the morning of December 23rd, I took a taxi to Roissy airport.\n\n### 3\n\nAND NOW, THERE I was on my own like an idiot, a few feet from the Nouvelles Fronti\u00e8res desk. It was a Saturday morning during the Christmas holidays; Roissy was heaving, as usual. The minute they have a couple of days of freedom, the inhabitants of western Europe dash off to the other side of the world, they go halfway round the world in a plane, they behave \u2013 literally \u2013 like escaped convicts. I don't blame them, I was preparing to do just the same.\n\nMy dreams are run-of-the-mill. Like all of the inhabitants of western Europe, I want to _travel_. There are problems with that, of course: the language barrier, poorly organised public transport, the risk of being robbed or conned. To put it more bluntly, what I really want, basically, is to be a _tourist_. We dream what dreams we can afford; and my dream is to go on an endless series of 'Romantic Getaways', 'Colourful Expeditions' and 'Pleasures \u00e0 la Carte' \u2013 to use the titles of the three Nouvelles Fronti\u00e8res brochures.\n\nI immediately decided to go on a package tour, but I hesitated quite a bit between 'Rum and Salsa' (ref: CUB CO 033, 16 days\/14 nights, 11,250FF based on two sharing, single supplement 1,350FF) and 'Thai Tropic' (ref THA CA 006, 15 days\/13 nights, 9,950FF based on two sharing, single supplement 1,175FF). Actually, I was more attracted by Thailand; but the advantage of Cuba is that it's one of the last Communist countries, though probably not for much longer \u2013 it has a sort of 'endangered r\u00e9gime' appeal, a sort of political exoticism, to put it in a nutshell. In the end, I chose Thailand. I have to admit that the copy in the brochure was very well done, sure to tempt the average punter:\n\n_A package tour with a dash of adventure, which will take you from the bamboo forests of the River Kwai to the island of Ko Samui, winding up, after crossing the spectacular isthmus of Kra, at Ko Phi Phi, off the coast of Phuket. A cool trip to the tropics_.\n\nAt 8.30 a.m. on the dot, Jacques Maillot slams the door of his house on the Boulevard Blanqui in the 13th arrondissement, straddles his moped and begins a journey across the capital from east to west. Direction: the head office of Nouvelles Fronti\u00e8res on the Boulevard de Grenelle. Every other day, he stops at four or five of the company's agencies: 'I bring them the latest brochures, I pick up the post and generally take the temperature,' explains the boss, full of beans, always sporting an extraordinary multicoloured tie. It's a crack of the whip for the agents: 'On the days after my visit, there's a tremendous boost in sales at those agencies...' he explains with a smile. Visibly under his spell, the journalist from _Capital_ goes on to marvel: who could have predicted in 1967 that a small business set up by a handful of student protestors would take off like this? Certainly not the thousands of demonstrators who, in May 1968, marched past the Nouvelles Fronti\u00e8res office on the Place Denfert-Rochereau in Paris. 'We were in just the right place, right in front of the cameras...' remembers Jacques Maillot, a former boy scout and left-wing Catholic by way of the National Students Union. It was the first piece of publicity for the company, which took its name from John F. Kennedy's speech about America's 'new frontiers'.\n\nA passionate liberal, Jacques Maillot successfully fought the Air France monopoly, making air transport more accessible to all. His company's odyssey, which in thirty years had made it the number one travel agency in France, has fascinated the business press. Like FNAC, like Club Med, Nouvelles Fronti\u00e8res \u2013 born at the dawn of the leisure society \u2013 might stand as a symbol of the new face of modern capitalism. In the year 2000, for the first time, the tourist industry became \u2013 in terms of turnover \u2013 the biggest economic activity in the world. Though it required only a moderate level of physical fitness, 'Thai Tropic' was listed under 'adventure tours': a range of accommodation options (simple, standard, deluxe); group numbers limited to twenty to ensure a better group dynamic. I saw two really cute black girls with rucksacks arriving, I dared to hope that they'd opted for the same tour: then I looked away and went to collect my travel documents. The flight was scheduled to last a little more than eleven hours.\n\nTaking a plane today, regardless of the airline, regardless of the destination, amounts to being treated like shit for the duration of the flight. Crammed into a ridiculously tiny space from which it's impossible to move without disturbing an entire row of fellow passengers, you are greeted from the outset with a series of embargos announced by stewardesses sporting fake smiles. Once on board, their first move is to get hold of your personal belongings so they can put them in overhead lockers \u2013 to which you will not have access under any circumstances until the plane lands. Then, for the duration of the flight, they do their utmost to find ways to bully you, all the while making it impossible for you to move about, or more generally to move at all, with the exception of a certain number of permitted activities: enjoying fizzy drinks, watching American videos, buying duty-free products. The unremitting sense of danger, fuelled by mental images of plane crashes, the enforced immobility in a cramped space, provokes a feeling of stress so powerful that a number of passengers have reportedly died of heart attacks while on long-haul flights. The crew do their level best to maximise this stress by preventing you from combating it by habitual means. Deprived of cigarettes, reading matter and, as happens more and more frequently, sometimes even deprived of alcohol. Thank God the bitches don't do _body searches_ yet; as an experienced passenger, I had been able to stock up on some necessities for survival: a few 21-mg Nicorette patches, sleeping pills, a flask of Southern Comfort. I fell into a thick sleep as we were flying over the former East Germany.\n\nI was awoken by a weight on my shoulder, and warm breath. I sat my neighbour upright in his seat without undue manhandling; he groaned softly, but didn't open his eyes. He was a big guy, about thirty, with light brown hair in a bowl cut; he didn't look too unpleasant, nor too clever. In fact, he was rather endearing, wrapped up in the soft blue blanket supplied by the airline, his big manual labourer's hands resting on his knees. I picked up the paperback which had fallen at his feet: a shitty Anglo-Saxon bestseller by one Frederick Forsyth. I had read something by this halfwit, full of heavy-handed eulogies to Margaret Thatcher and ludicrous depictions of the USSR as the _evil empire_. I'd wondered how he managed after the fall of the Berlin Wall. I leafed through his new opus: apparently, this time, the roles of the bad guys were played by Serb nationalists; here was a man who kept up to date with current affairs. As for his beloved hero, the tedious Jason Monk, he had gone back into service with the CIA, which had formed an alliance of convenience with the Chechen mafia. Well! I thought, replacing the book on my neighbour's knees, what a charming sense of morality bestselling British authors have. The page was marked with a piece of paper folded in three, which I recognised as the Nouvelles Fronti\u00e8res itinerary: I had, apparently, just met my first tour companion. A fine fellow, I was sure, certainly a lot less egocentric and neurotic than I was. I glanced at the video screen, which was showing the flight path: we had probably passed Chechnya, whether or not we had flown over it; the exterior temperature was -53\u00b0C, altitude 10,143 metres, local time 00.27. Another screen replaced the first: we were flying directly over Afghanistan. Through the window, you could see nothing but pitch black of course. In any case the Taliban were probably all in bed stewing in their own filth. 'Goodnight, Talibans, good night... sweet dreams...' I whispered before swallowing a second sleeping pill.\n\n### 4\n\nTHE PLANE LANDED at Don Muang airport at about 5 a.m. I woke with some difficulty. The man on my left had already stood up and was waiting impatiently in the queue to disembark. I quickly lost sight of him in the corridor leading to the arrivals hall. My legs were like cotton wool, my mouth felt furry; my ears were filled with a violent drone.\n\nNo sooner had I stepped through the automatic doors than the heat enveloped me like a mouth. It must have been at least 35\u00b0C. The heat in Bangkok has something particular about it, in that it is somehow _greasy_ , probably on account of the pollution; after any long period outdoors, you're always surprised to find that you're not covered with a fine film of industrial residue. It took me about thirty seconds to adjust my breathing. I was trying not to fall too far behind the guide, a Thai woman whom I hadn't taken much notice of, except that she seemed reserved and well-educated \u2013 but a lot of Thai women give that impression. My backpack was cutting into my shoulders; it was a Lowe Pro Himalaya Trekking, the most expensive one I could find at Vieux Campeur; it was guaranteed for life. It was an impressive object, steel grey with snap clasps, special Velcro fastenings \u2013 the company had a patent pending \u2013 and zips that would work at temperatures of -65\u00b0C. Its contents were sadly pretty limited: some shorts and tee-shirts, swimming trunks, special shoes which allowed you to walk on coral (125FF at Vieux Campeur), a wash bag containing medicines considered essential by the _Guide du Routard_ , a JVC HRD-9600 MS video camera with batteries and spare tapes, and two American bestsellers that I'd bought pretty much at random at the airport.\n\nThe Nouvelles Fronti\u00e8res coach was parked about a hundred metres further on. Inside the powerful vehicle \u2013 a 64-seat Mercedes M-800 \u2013 the air-conditioning was turned up full; it felt like stepping into a freezer. I settled myself in the middle of the coach, on the left by a window. I could vaguely make out a dozen other passengers, amongst them my neighbour from the plane. No one came to sit beside me. I had clearly missed my first opportunity to integrate into the group; I was also well on my way to catching a nasty cold.\n\nIt wasn't light yet, but on the six-lane motorway which led to downtown Bangkok, the traffic was already heavy. We drove past buildings alternately of glass and steel with, occasionally, a massive concrete structure reminiscent of Soviet architecture: the head offices of banks, chain hotels, electronics companies \u2013 for the most part Japanese. Past the junction at Chatuchak, the motorway rose above a series of ring roads circling the heart of the city. Between the floodlit buildings, we began to be able to distinguish groups of small, slate-roofed houses in the middle of wasteland. Neon-lit stalls offered soup and rice; you could see the tinplate pots steaming. The coach slowed slightly to take the New Phetchaburi Road exit. There was a moment when we saw an interchange of the most phantasmagoric shape, its asphalt spirals seemingly suspended in the heavens, lit by banks of airport floodlights; then, after following a long curve, the coach joined the motorway again.\n\nThe Bangkok Palace Hotel is part of a chain along the lines of Mercure, sharing similar values as to catering and quality of service; this much I discovered from a brochure I picked up in the lobby while waiting for the situation to unfold. It was just after six in the morning \u2013 midnight in Paris I thought, for no reason \u2013 but activities were already well under way, the breakfast room had just opened. I sat down on a bench; I was dazed, my ears were still buzzing violently and my stomach was beginning to hurt. From the way they were waiting, I was able to identify some of the group members. There were two girls of about twenty-five, pretty much bimbos \u2013 not bad-looking, all things considered \u2013 who cast a contemptuous eye over everyone. On the other hand, a couple of retirees \u2013 he could have been called _spirited_ , she looked a bit more miserable \u2013 were looking around in wonderment at the interior d\u00e9cor of the hotel, a lot of gilding, mirrors and chandeliers. In the first hours in the life of a group, one generally observes only phatic sociability, characterised by the use of standard phrases and by limited emotional connection. According to Edmunds and White1, the establishment of micro-groups can only be detected after the first excursion, sometimes after the first communal meal.\n\nI started, on the point of passing out, lit a cigarette to rally my forces. The sleeping pills really were too strong, they were making me ill, but the ones I used to take couldn't get me to sleep any more; there was no obvious solution. The OAPs were slowly circling round each other. I got the feeling that the man was a bit full of himself; as he was waiting for someone specific with whom to exchange a smile, he turned an incipient smile on the world. They had to have been a couple of small shopkeepers in a previous life, that was the only explanation. Gradually, the members of the group made their way to the guide as their names were called, took their keys and went up to their rooms \u2013 in a word, they dispersed. It was possible, the guide announced in a resonant voice, for us to take breakfast now if we wished; otherwise we could relax in our rooms; it was entirely up to us. Whatever we decided, we were to meet back in the lobby for the trip along the _khlongs_ at 2 p.m.\n\nThe window in my room looked directly out onto the motorway. It was six-thirty. The traffic was very heavy, but the double glazing let in only a faint rumble. The street lights were off, the sun hadn't yet begun to reflect on the steel and glass; at this time of the day, the city was grey. I ordered a double espresso from room service, which I knocked back with a couple of Efferalgan, a Doliprane and a double dose of Oscillococcinium; then I lay down and tried to close my eyes.\n\nShapes moved slowly in a confined space; they made a low buzzing sound \u2013 like machines on a building site, or giant insects. In the background, a man armed with a small scimitar carefully checked the sharpness of the blade; he was wearing a turban and baggy white trousers. Suddenly, the air became red and muggy, almost liquid; from the drops of condensation forming before my eyes I became conscious that a pane of glass separated me from the scene. The man was on the ground now, immobilised by some invisible force. The machines from the building site had surrounded him; there were a couple of JCBs and a small bulldozer with caterpillar tracks. The JCBs lifted their hydraulic arms and brought their buckets down together on the man, immediately slicing his body into seven or eight pieces; his head, however, still seemed animated by a demonic life-force, an evil smile continued to crease his bearded face. The bulldozer in its turn advanced on the man, his head exploded like an egg; a spurt of brain and ground bone was splashed against the glass, a few inches from my face.\n\n1 _Sightseeing Tours: A Sociological Approach_ , Annals of Tourism Research, vol. 23, pp. 213\u201327 (1998)\n\n### 5\n\n###### _Essentially, tourism, as a search for meaning, with the ludic sociability it favours, the images it generates, is a graduated encoded and untraumatising apprehension system of the external, of otherness_.\n\n###### Rachid Amirou\n\nI WOKE UP at about noon, the air-conditioning was making a low buzzing sound; my headache was a little better. Lying across the king-size bed, I was aware of the mechanics of the tour, the issues at stake. The group, as yet amorphous, would transform itself into a vibrant community; as of this afternoon I would have to start positioning myself, for now I had to choose a pair of shorts for the trip along the _khlongs_. I opted for a longish pair in blue denim, not too tight, which I complemented with a _Radiohead_ tee-shirt; then I stuffed some odds and ends into a knapsack. In the bathroom mirror, I contemplated myself disgustedly; my anxious bureaucratic face clashed horribly with what I was wearing; I looked exactly like what I was: a forty-something civil servant on holiday, trying to pretend he's young; it was pretty demoralising. I walked over to the window, opened the curtains wide. From the twenty-seventh floor, the view was extraordinary. The imposing mass of the Marriott Hotel rose up on the left like a chalk cliff, striated by horizontal black lines: rows of windows half-hidden behind balconies. The sun, at its zenith, harshly emphasised planes and ridges. Directly ahead, reflections multiplied themselves into infinity on a complex structure of cones and pyramids of bluish glass. On the horizon, the colossal concrete cubes of the Grand Plaza President were stacked on top of one another like the levels of a step pyramid. On the right, above the green, shimmering space of Lumphini Park, you could make out, like an ochre citadel, the angular towers of the Dusit Thani. The sky was a pure blue. Slowly I drank a Singha Gold while meditating on the notion of irreparability.\n\nDownstairs, the guide was doing a sort of roll-call, so she could hand out breakfast vouchers. That's how I discovered the two bimbos were called Babette and L\u00e9a. Babette had curly blond hair \u2013 well, not naturally curly, it had probably been _waved_ ; she had beautiful breasts, the slut, clearly visible under her see-through top \u2013 an ethnic print from Trois Suisses, most likely. Her trousers, in the same fabric, were just as see-through; you could easily make out the white lace of her panties. L\u00e9a, very dark, was skinnier; she made up for this with the pretty curve of her bum, nicely accentuated by her black cycling shorts, and with a thrusting bust, the tips of which were squeezed into a bright yellow bustier. A tiny diamond adorned her slender navel. I stared attentively at the two sluts so that I could forget them forever.\n\nThe distribution of the vouchers continued. The guide, S\u00f4n, called each of the group members by their first names; it made me sick. We were _adults_ , for fuck's sake. I felt a ray of hope when she referred to the OAPs as 'Monsieur et Madame Lobligeois'; but immediately she added with a delighted smile 'Josette and Ren\u00e9'. It seemed unbelievable, but true nevertheless. 'My name is Ren\u00e9,' confirmed the old man, addressing himself to no one in particular. 'Tough...' I muttered. His wife shot him a look as if to say: 'Shut up, Ren\u00e9, you're annoying everyone.' I suddenly realised that he reminded me of the character Monsieur Plus in the Bahlsen biscuit ads. It might have been him, too. I directed this question to his wife: had they, in the past, ever worked as extras? Absolutely not, she informed me, they had run a _charcuterie_. Yeah, that would probably fit too. So, this cheery jolly little fellow was a former pork-butcher (in Clamart, his wife explained); a modest establishment devoted to feeding the proletariat had been the previous theatre for his antics and quips.\n\nThen there were two other couples, less distinctive, who seemed to be connected in some obscure way. Had they already been on holiday together? Had they met each other over breakfast? At this point in the tour, anything was possible. The first couple was also the more unappealing. The man looked a bit like a young Antoine Waechter, if you can imagine such a thing, but his hair was darker and he had a neatly trimmed beard; actually, he didn't so much look like Antoine Waechter as like Robin Hood, though he looked Swiss, or to be more precise, he had something of the Jura about him. All in all, he didn't look much like anything, but he seemed a real jerk. Not to mention his wife, wearing dungarees, serious, a good milker. It was inconceivable that these people had not yet reproduced, I thought; they'd probably left the child with their parents in Lon-le-Saulnier. The second couple, a little older, seemed rather less serene. Skinny and nervous, with a moustache, the man introduced himself to me as a naturopath, and, faced with my ignorance, went on to explain that he practised healing using plants or other natural means wherever possible. His wife, thin and curt, worked in social services, reintegrating, I don't know, first offenders or something in Alsace; they looked like they hadn't fucked for thirty years. The man seemed inclined to tell me about the benefits of natural medicines; but still dazed from this first encounter, I went and sat on a bench nearby. From where I sat, I could barely make out the last three members of the group, who were half hidden by the pork-butcher couple. There was some fifty-year-old thug called Robert, with a particularly harsh expression; a woman, of the same age, with curly black hair framing a face that was nasty, world-weary and flabby, whose name was Josiane; and another woman, a bit younger, almost unnoticeable, of about twenty-seven who followed Josiane with a sort of canine docility and whose name was Val\u00e9rie. Anyway, I'll get back to them; I'll have far too much time to get back to them, I thought glumly as I walked towards the coach. I noticed that S\u00f4n was still staring at her list of passengers. Her face was tense, words formed on her lips involuntarily; it was clear she was anxious, almost distraught. Counting, it appeared there were thirteen people in the group; and Thais are frequently superstitious, even more so than the Chinese: the numbering of storeys in a building or houses in a street often goes straight from twelve to fourteen, simply to avoid mentioning the number thirteen. I took a seat on the left-hand side about halfway down the coach. People establish points of reference pretty quickly on this kind of group outing: in order to feel relaxed, they need to find a place and stick to it, maybe leave some personal odds and ends around, actively inhabiting the space in some way.\n\nTo my great surprise, I saw Val\u00e9rie take a seat beside me, even though the coach was about three-quarters empty. Two rows behind, Babette and L\u00e9a exchanged a couple of scornful words. They'd better calm down, those sluts. I discreetly fixed my attention on the young woman: she had long black hair, a nondescript face, a face that could be described as _unexceptional_ : not pretty, not ugly, strictly speaking. After brief but intense consideration, I managed awkwardly: 'Not too hot?' 'No, no, here in the coach is fine,' she replied quickly, without smiling, relieved simply that I had started a conversation. Though what I'd said was remarkably stupid: actually, it was freezing in the bus. 'Have you been to Thailand before?' she went on by way of conversation. 'Yes, once.' She froze in a waiting posture, ready to listen to an interesting anecdote. Was I about to recount my previous trip to her? Maybe not right away. 'It was good...' I said eventually, adopting a friendly tone to compensate for the banality of what I was saying. She nodded in satisfaction. It was then that I realised that this young woman was in no way submissive to Josiane: she was just submissive _in general_ , and maybe just ready to look for a new master; maybe she'd already had enough of Josiane \u2013 who, sitting two rows in front of us, was furiously leafing through the _Guide du Routard_ , throwing dirty looks in our direction. Romance, romance.\n\nJust past Payab Ferry Pier, the boat turned right into the Khlong Samsen and we entered a completely different world. Life had changed very little here since the nineteenth century. Rows of teak houses on stilts lined the canal; washing dried under awnings. Some of the women came to their windows to watch us pass, others stopped in the middle of their washing. Children splashed and bathed between the stilts; they waved at us excitedly. There was vegetation everywhere: our pirogue cut a path through masses of water-lilies and lotuses; teeming, intense life sprang up all around. Every free patch of earth, air or water seemed to be immediately filled with butterflies, lizards, carp. We were, S\u00f4n told us, in the middle of the dry season; even so the air was completely, unrelentingly humid.\n\nVal\u00e9rie was sitting beside me; she seemed to be enveloped by a great sense of peace. She exchanged little waves with the old men who sat smoking their pipes on the balconies, the children bathing, the women at their washing. The ecologists from the Jura seemed at peace too; even the naturopaths seemed reasonably calm. Around us, only faint sounds and smiles. Val\u00e9rie turned to me. I almost felt like taking her hand; for no particular reason, I didn't. The boat stopped moving entirely: we were rapt in the momentary eternity of a blissful afternoon; even Babette and L\u00e9a had shut up. They were a bit spaced out, to use the expression L\u00e9a later employed on the jetty.\n\nWhile we were visiting the Temple of Dawn, I made a mental note to buy some more Viagra when I found a chemist that was open. On the way back, I found out that Val\u00e9rie was Breton and that her parents had owned a farm in Tr\u00e9gorrois; I didn't really know what to say, myself. She seemed intelligent. I liked her soft voice, her meek Catholic fervour, the movement of her lips when she spoke; her mouth was obviously pretty hot, just ready to swallow the spunk of a true friend. 'It's been lovely, this afternoon...' I said finally in desperation. I had become too remote from people, I had lived alone too long, I didn't know how to go about it any more. 'Oh, yes, lovely...' she replied all the same; she wasn't demanding, she really was a nice girl. Even so, as soon as the coach arrived at the hotel, I ran straight to the bar.\n\nThree cocktails later, I was beginning to regret my behaviour. I went out and walked round the lobby. It was 7 p.m.; no one from the group was around. For about four hundred baht, those who wished could have dinner and a show of 'traditional Thai dance'; those interested were to assemble at 8 p.m. Val\u00e9rie would definitely be there. For my part, I had already had a vague experience of traditional Thai dance, on a trip with Kuoni three years previously: 'Classic Thailand, from the \"Rose of the North\" to the \"City of Angels\".' Not bad, really, but a bit expensive and terrifyingly cultural; everyone involved had at least a masters degree. The thirty-two positions of the Buddha in Ratanakosin statuary, Thai-Burmese style, Thai-Khmer, Thai-Thai, they didn't miss a thing. I had come back exhausted and I'd constantly felt ridiculous without a _Guide Bleu_. Right now, I was beginning to feel a serious need to fuck. I was wandering round the lobby, with a sense of mounting indecision, when I spotted a sign saying 'Health Club', indicating the floor below.\n\nThe entrance was lit by neon and a long rope of coloured lights. On the white background of an electric sign, three bikini-clad sirens, their breasts a little larger than life, proffered champagne flutes to prospective customers; there was a heavily stylised Eiffel Tower in the far distance \u2013 not quite the same concept as the fitness centres of Mercure hotels. I went in and ordered a bourbon at the bar. Behind a glass screen, a dozen girls turned towards me; some smiled alluringly, others didn't. I was the only customer. Despite the fact that the place was small, the girls wore numbered tags. I quickly chose number 7: firstly because she was cute, also because she wasn't engrossed in the programme on the television or deep in conversation with her neighbour. Indeed, when her name was called, she stood up with evident satisfaction. I offered her a coke at the bar, then we went to one of the rooms. Her name was O\u00f4n, at least that was what I heard, and she was from the north somewhere, a little village near Chiang Mai. She was nineteen.\n\nAfter we had taken a bath together, I lay down on the foam-covered mattress; I realised at once that I wasn't going to regret my choice. O\u00f4n moved very nicely, very lithely; she'd used just enough soap. At one point, she at length caressed my buttocks with her breasts; it was a personal initiative, not all the girls did that. Her well-soaped pussy grazed my calf like a small hard brush. I was somewhat surprised to find I got hard almost immediately; when she turned me over and started to stroke my penis with her feet, I thought for a minute that I wouldn't be able to hold back. But with a supreme effort, tensing the abductor muscles in my thighs, I managed.\n\nWhen she climbed on top of me on the bed, I thought I would be able to hold out for a long time yet; but I was quickly disillusioned. She might have been very young, but she knew what to do with her pussy. She started very gently with little contractions on the glans, then she slipped down an inch or so, squeezing a little harder. 'Oh no, O\u00f4n, no!...' I cried. She burst out laughing, pleased with her power, then continued to slide down gently, contracting the walls of her vagina with long, slow compressions; all the while looking me in the eyes in obvious amusement. I came well before she got to the base of my penis.\n\nAfterwards we chatted a bit, entwined on the bed; she didn't seem to be in any hurry to get back out on stage. She didn't have many clients, she told me; the hotel was aimed at groups of terminal cases, ordinary people, who were pretty much blas\u00e9. There were a lot of French people, but they didn't really seem to like body massage. Those who patronised the place were nice enough, but they were mostly Germans and Australians. A few Japanese too, but she didn't like them \u2013 they were weird, they always wanted to hit you or tie you up, or else they just sat there masturbating, staring at your shoes; it was pointless.\n\nAnd what did she think about me? Not bad, but she would have liked it if I'd been able to hold out a little longer. 'Much need...' she said in English, gently shaking my sated penis between her fingers. Otherwise, she thought I seemed like a nice man. 'You look quiet...' she said. There she was somewhat mistaken, but I suppose it was true that she'd done a good job of calming me. I gave her three thousand baht, which, as far as I remembered, was a good price. From her reaction I could tell that, yes, it was a good price. 'Kr\u00f4p khun kh\u00e2t!' she said with a big smile, bringing her hands together in front of her forehead. Then she took my hand and accompanied me to the exit; at the door we kissed each other on the cheeks several times.\n\nAs I climbed the stairs I ran into Josiane, who was apparently hesitating about whether to go downstairs. She had changed into an evening dress, a black shift dress with gold piping, but it didn't make her the least bit more appealing. Her plump, shrewd face was turned towards me, unblinkingly. I noticed that she'd washed her hair. She wasn't ugly, you might even say she was pretty \u2013 I had fancied Lebanese women like her \u2013 but her basic expression was unmistakably nasty. I could easily imagine her trotting out tired political positions; she hadn't a flicker of compassion that I could make out. I had nothing to say to her, either. I lowered my head. A little embarrassed, maybe, she spoke: 'Anything interesting downstairs?' I found her so infuriating that I nearly said: 'A bar full of hookers', but in the end I lied, it was easier. 'No, no, I don't know, some kind of beauty salon...'\n\n'You didn't go to the dinner and show...' the bitch remarked. 'Neither did you,' I snapped back. This time her response was slower in coming, she became snotty. 'Oh no, I don't really like that sort of thing...' she went on, curving her arm like an actress playing Racine. 'It's all a bit touristy...' What did she mean by that? Everything is touristy. Once again, I stopped myself from putting my fist through her fucking face. Standing in the middle of the stairway, she was in my way; I had to show patience. A passionate letter-writer on occasion, St Jerome also knew how to display the virtues of Christian patience when circumstances called for it; that is why he is considered to be a great saint and a Doctor of the Church.\n\nThis 'traditional Thai dance' show was, according to her, just about Josette and Ren\u00e9's level, people she thought of, in her heart of hearts, as white trash; I realised, rather uncomfortably, that she was looking for an ally. True, the tour would soon head deep inland, we would be divided into two tables at meals; it was time to take sides. 'Well...' I said, after a long silence. At that moment, like a miracle, Robert appeared above us. He was trying to get downstairs. I smoothly stepped aside, climbing a couple of steps. Just before rushing off to the restaurant, I turned back: Josiane, still motionless, was staring at Robert, who was walking briskly towards the massage parlour.\n\nBabette and L\u00e9a were standing next to the trays of vegetables. I nodded in minimal acknowledgement before serving myself some water spinach. Obviously they too had decided that the 'traditional Thai dance' was _tacky_. As I went back to my table, I noticed the tarts were sitting a couple of feet away. L\u00e9a was wearing a _Rage Against the Machine_ tee-shirt and a pair of tight denim shorts, Babette something unstructured in which different coloured stripes of silk alternated with transparent fabric. They were chattering enthusiastically, talking about different hotels in New York. Marrying one of those girls, I thought, that would be _radically_ hideous. Did I still have time to change tables? No, it would have been a bit obvious. I took a chair opposite so that at least I could sit with my back on them, I bolted my meal and went back up to my room.\n\nA cockroach appeared just as I was about to get into the bath. It was just the right time for a cockroach to make an appearance in my life; couldn't have been better. It scuttled quickly across the porcelain, the little bugger; I looked around for a slipper, but actually I knew my chances of squashing him were small. What was the point in trying? And what good was O\u00f4n, in spite of her marvellously elastic vagina? We were already doomed. Cockroaches copulate gracelessly, with no apparent pleasure; but they also do it repeatedly and their genetic mutations are rapid and efficient. There is absolutely nothing we can do about cockroaches.\n\nBefore getting undressed, I once more paid homage to O\u00f4n and to all Thai prostitutes. They didn't have an easy job, those girls; they probably didn't come across a good guy all that often, someone with an okay physique who was honestly looking for nothing more than mutual orgasm. Not to mention the Japanese \u2013 I shivered at the thought, and grabbed my _Guide du Routard_. Babette and L\u00e9a could never have been Thai prostitutes, I thought, they weren't worthy of it. Val\u00e9rie, maybe; that girl had something, she managed to be both maternal and a bit of a slut, potentially at least, I mean; for the moment she was just a nice friendly, serious girl. Intelligent, too. I definitely liked Val\u00e9rie. I masturbated gently so I could read in peace, producing just a couple of drips.\n\nIf it was intended in principle to prepare you for a trip to Thailand, in practice the _Guide du Routard_ had strong reservations about, and as early as the preface, felt duty-bound to denounce, sexual tourism, that 'repulsive slavery'. All in all, these backpacking _routards_ were bellyaching bastards whose goal was to spoil every little pleasure on offer to tourists, whom they despised. In fact, they seemed to like themselves more than anything else, if one was to go by the sarcastic little phrases scattered throughout the book, in the style of: 'Ah, my friends, if you had been there back in the hippy days!...' The most excruciating thing was probably their stern, dogmatic, peremptory tone, quivering with repressed indignation: 'We're far from prudish, but Pattaya we don't like. Enough is enough.' A bit further on, they laid into 'pot-bellied Westerners' who strolled around with little Thai girls; it made them 'literally puke'. Humanitarian Protestant cunts, that's what they were, they and the 'cool bunch of mates who had helped to make this book possible', their nasty little faces smugly plastered all over the back cover. I flung the book hard across the room, missing the Sony television by a whisker, and wearily picked up _The Firm_ , by John Grisham. It was an American bestseller, one of the best; meaning one of those that had sold the most copies. The hero was a young lawyer with a bright future, a talented, good-looking boy who worked eighty hours a week; not only was this shit so obviously a proto-screenplay it was obscene, but you had the feeling the author had already given some thought to the casting, the part had obviously been written for Tom Cruise. The hero's wife wasn't bad either, even if she didn't work eighty hours a week; but in this case, Nicole Kidman wouldn't fit, it wasn't a part for someone with curly hair; more like someone with a blow-dry. Thank God the lovebirds didn't have any children, which meant we were spared a number of gruelling scenes. It was a suspense thriller, well, there was a little suspense: as early as Chapter Two, it was obvious that the guys running the firm were bastards, and there was no way the hero was going to die at the end; nor his wife for that matter. But, in the meantime, to prove he wasn't joking, the author was going to sacrifice a couple of sympathetic minor characters; all that was left was to find out which ones. That might make it worth a read. Maybe it would be the hero's father: his business was going through a bad patch, he was having trouble adjusting to the new matrix management; I had a feeling that this would be his last Thanksgiving.\n\n### 6\n\nVAL\u00c9RIE HAD SPENT the early years of her life in Tr\u00e9m\u00e9ven, a hamlet a few kilometres north of Guingamp. In the '70s and early '80s, the government and local councils had nurtured an ambition to create a massive production centre for pork products in Brittany, capable of rivalling those of Britain or Denmark. Encouraged to adopt intensive farming methods, the young farmers \u2013 including Val\u00e9rie's father \u2013 became heavily indebted to the Cr\u00e9dit Agricole. In 1984, pork prices began to collapse; Val\u00e9rie was eleven years old. She was a well-behaved girl, a bit lonely, a good student; she was about to enter her _second year_ at the secondary school in Guingamp. Her older brother, also a good student, had just passed his _bac_ ; he had enrolled in preparatory classes in agronomy at the lyc\u00e9e in Rennes.\n\nVal\u00e9rie remembered Christmas 1984; her father had spent the day with the accountant from the National Farmers' Union. He was silent for much of Christmas dinner. During dessert, after two glasses of champagne, he spoke to his son. 'I can hardly recommend that you take over the farm,' he said. 'For twenty years now I've been getting up at dawn and finishing the day at eight or nine o'clock; your mother and I, we've barely had a holiday. I'd be as well off selling the place now, with all the machinery and the farm buildings, and investing the money in tourist property: I could spend the rest of my days working on my tan.'\n\nIn the years that followed, pork prices continued to plummet. There were farmers' protests, marked by a desperate violence; tons of slurry were dumped on the Esplanade des Invalides, a number of pigs were gutted in front of the Palais Bourbon. At the end of 1986, the government announced emergency relief followed by a recovery plan for pig-breeders. In April 1987, Val\u00e9rie's father sold his farm \u2013 for a little more than four million francs. With the money from the sale, he bought a large apartment in Saint-Quay-Portrieux, where he planned to live, and three studio flats in Torremolinos. He had a million francs left over which he invested in unit trusts and was even able \u2013 it was his childhood dream \u2013 to buy a small yacht. Sadly, and with some disgust, he signed the farm bill of sale. The new owner was a young guy, about twenty-three, single, from Lannion, just out of agricultural college; he still believed in the plans to revive the industry. Val\u00e9rie's father was forty-eight, his wife, forty-seven; they had dedicated the best years of their lives to a hopeless task. They lived in a country where, compared to speculative investment, investment in production brought little return; he understood that now. In their first year, the rents from the studio flats brought in more money than all his years of work. He took up crosswords, took the yacht out into the bay, sometimes fishing. His wife found it easier to adapt to their new life and was a great support to him; she started to want to read again, to go to the cinema, to go out.\n\nAt the time of the sale, Val\u00e9rie was fourteen, she was just starting to wear makeup; in the bathroom mirror she watched her breasts as they gradually swelled. The night before they moved out, she spent a long time walking around the farm buildings. There were still a dozen pigs in the main sty, which came up to her grunting softly. They were being picked up that night by a wholesaler and would be slaughtered in a few days time.\n\nThe summer that followed was a strange period. Compared to Tr\u00e9m\u00e9ven, Saint-Quay-Portrieux was almost a small town. When she walked out of her door, she couldn't He on the grass, letting her thoughts float with the clouds, flow with the river. Among the holidaymakers there were boys, who turned to look at her as she passed; she never really managed to relax. Towards the end of August, she met B\u00e9r\u00e9nice, a girl from the secondary school at Saint-Brieuc. B\u00e9r\u00e9nice was a year older than she, she already wore makeup and designer skirts; she had a pretty, angular face and very long hair which was an extraordinary strawberry blond. They got into the habit of going to the beach at Saint-Marguerite together; they would get changed in Val\u00e9rie's room before they set off. One afternoon, as she was taking off her bra, Val\u00e9rie noticed B\u00e9r\u00e9nice staring at her breasts. She knew that she had superb breasts, round and high, so swollen and firm that they looked artificial. B\u00e9r\u00e9nice stretched out her hand, traced the curve and the nipple. Val\u00e9rie opened her mouth and closed her eyes as B\u00e9r\u00e9nice's lips approached her own; she abandoned herself completely to the kiss. She was already wet when B\u00e9r\u00e9nice slipped a hand into her panties. Impatiently she took them off, fell back on the bed and parted her thighs. B\u00e9r\u00e9nice knelt in front of her, placed her mouth over her pussy. Her stomach quivered with warm spasms, she felt her mind floating in the endless space of the sky; she had never imagined pleasure like this could exist.\n\nEvery day until they went back to school, they did it again. Once in the afternoon, before they went to the beach; then they would lie side by side in the sunshine. Little by little, Val\u00e9rie would feel desire mounting in her skin, she would take off her top so that B\u00e9r\u00e9nice could see her breasts. They would practically run back to the bedroom and make love a second time.\n\nFrom their first week back at school, B\u00e9r\u00e9nice began to distance herself from Val\u00e9rie, avoided walking back from school with her; shortly afterwards she started going out with a boy. Val\u00e9rie accepted the separation without any real sorrow \u2013 that's the way things go. She had taken to masturbating every morning when she woke up. Each time, in a few short minutes, she would reach orgasm; it was something marvellous, something simple happening within her and which began her day with joy. About boys she had more reservations; having bought a couple of issues of _Hot Video_ at the station kiosk, she knew what to expect from their anatomy, their organs, various sexual practices; but she felt a slight repugnance for their body hair, their muscles; their skin looked thick and not at all soft. The brownish, wrinkled skin of their balls, the brutally anatomical look of the glans when the foreskin was retracted, red, shiny... none of these things was especially attractive. In the end, however, she slept with a boy in his final year, a tall blond guy, after spending the night in a club in Paimpol; she did not find it particularly pleasurable. She tried again several times with others while she was in her last couple of years at school. It was easy to seduce boys: all you had to do was wear a short skirt, cross your legs, wear a low-cut or a see-through blouse that showed off your breasts. None of these experiences proved especially conclusive. Intellectually, she could understand the triumphant yet gentle feeling some girls experienced when they felt a cock pushing deep into their pussies; but she herself felt nothing of the sort. It had to be said that condoms didn't help; the sound the latex made, flaccid and repetitive, constantly brought her down to earth, prevented her from drifting into the nebulous infinity of sensual pleasure. By the time she sat her _bac_ , she had more or less given up.\n\nTen years later, she still hadn't really started again, she thought sadly as she woke in the bedroom of the Bangkok Palace. It was not quite daylight. She turned on the overhead light and contemplated her body in the mirror. Her breasts were as firm as ever, they hadn't changed since she was seventeen. Her arse was amazingly round too, without a trace of fat; unquestionably she had a very beautiful body. Nonetheless, she slipped on a baggy sweatshirt and a shapeless pair of shorts before going downstairs to breakfast. Before she closed the door, she glanced at herself one last time in the mirror: her face was very average, a little rounded, nice but nothing more than that; the same was true of her limp, black hair which fell untidily on her shoulders; and her brown eyes weren't much of an asset either. No doubt she could have made more of herself, a bit of makeup, a different hairstyle, a trip to the beauty salon. Most women her age spent at least a couple of hours a week there; she didn't think it would make much difference in her case. What she was lacking, essentially, was the desire to seduce.\n\nWe left the hotel at seven; the traffic was already heavy. Val\u00e9rie gave me a little nod and took a seat in the same row on the other side of the aisle. No one in the bus was talking. Slowly, the grey megalopolis woke up; mopeds carrying couples, sometimes with a baby in the mother's arms, weaved between the crowded buses. A light haze still hung in some of the alleys by the river. Soon the sun would burst through the morning clouds, it would start to get hot. At Nonthaburi, the urban fabric began to fray and we could see the first rice fields. Buffalo standing motionless in the mud followed the bus with their eyes exactly as cows would do. The ecologists from the Jura seemed a bit restless; they'd probably wanted to take a couple of pictures of the buffalo.\n\nThe first stop was Kanchanaburi, which all the guide books agree is a lively, animated city. To the Michelin, it's a 'marvellous starting point from which to explore the surrounding region'; the _Guide du Routard_ , on the other hand, considers it a 'good base camp'. The tour programme indicated a journey of several miles along the 'railway of death' which snaked alongside the River Kwai. I'd never really got to the bottom of this River Kwai story, so I tried to pay attention to what the guide was saying. Luckily Ren\u00e9, Michelin Guide in hand, was following the story, always ready to correct this point or that. In short, after they entered the war in 1941, the Japanese decided to build a railway connecting Singapore and Burma, with the long-term objective of invading India. This railway had to cross Malaysia and Thailand. Come to think of it, what were the Thais doing during the Second World War? Well, now you come to mention it, not a lot. They were 'neutral', S\u00f4n informed me diplomatically. In reality, Ren\u00e9 explained, they'd signed a military pact with the Japanese without actually declaring war on the Allies. That was the way of wisdom. Demonstrating, once again, the celebrated 'subtlety of mind' which had made it possible for them to spend two centuries caught in a vice-like grip between the colonial powers of France and England without actually surrendering to either, and to remain the only country in South-East Asia never to have been colonised.\n\nBe that as it may, by 1942 work had begun on the section along the River Kwai, marshalling sixty thousand English, Australian, New Zealand and American prisoners of war, as well as 'countless' Asian forced labourers. In October 1943, the railway was completed, but sixteen thousand of POW's had died \u2013 from a variety of causes including lack of food, the hostile climate and the innate viciousness of the Japanese. Shortly afterwards, an allied bombing raid destroyed the bridge over the River Kwai, a crucial element of the infrastructure \u2013 thereby rendering the railway completely useless. In short, a lot of people copped it for very little. Things have changed little since then \u2013 it is still impossible to get a decent rail connection between Singapore and Delhi.\n\nIt was in a state of mild distress that I began the visit to the JEATH Museum, built to commemorate the appalling suffering of the allied POWs. Certainly, I thought, what had happened was thoroughly regrettable; but, let's face it, worse things happened during the Second World War. I couldn't help thinking that if the prisoners had been Polish or Russian there would have been a lot less fuss.\n\nA little later, we were required to endure a visit to the cemetery for the allied prisoners of war \u2013 those who had, in a manner of speaking, made the ultimate sacrifice. There were white crosses in neat rows, all identical; the place radiated a profound monotony. It reminded me of Omaha Beach, which hadn't really moved me either, had actually reminded me, in fact, of a contemporary art installation. 'In this place,' I said to myself, with a feeling of sadness which I felt was somewhat inadequate, 'In this place, a bunch of morons died for the sake of democracy.' That said, the cemetery at the River Kwai was much smaller, you could even imagine counting the graves; actually, I gave up pretty quickly. 'There can't be sixteen thousand graves...' I concluded aloud. 'You're quite right,' Ren\u00e9 informed me, still armed with his Michelin Guide. 'The number of dead is estimated at sixteen thousand; but the cemetery contains only five hundred and eighty-two graves. They are considered to be (he read, running his finger under the words) the _five hundred and eighty-two martyrs to democracy_.'\n\nWhen I got my third gold star at the age of ten, I went to a p\u00e2tisserie to stuff my face with _cr\u00eapes aux Grand Marnier_. It was a little private party; I had no friends with whom I could share my joy. I was staying with my father in Chamonix, as I did every year at that time. He was an alpine guide and a committed mountaineer. His friends were like him, men who were brave and manly; I never felt comfortable around them. I've never really felt comfortable around men. I was eleven the first time a girl ever showed me her pussy; I was immediately filled with wonder, I adored this small, strange, cleft organ. She didn't have much pubic hair, she was about the same age as me; her name was Martine. For a long time, she stood with her thighs apart, holding her knickers to one side so I could look; but when I tried to move my hand towards it, she got scared, she ran off. It all seemed very recent to me; I didn't feel that I had changed much. My enthusiasm for pussy had not waned, in fact I saw in it one of my few remaining recognisable, fully human qualities; as for the rest, I didn't really know anymore.\n\nA short while after we had boarded the coach again, S\u00f4n spoke. We were now heading towards our accommodation for the night, which, she was keen to emphasise, was of exceptional quality. No TV, no video. No electricity, candles. No bathroom, the river. No mattresses, mats. Absolutely back to nature. Back to nature, I mentally noted, seemed to consist principally of privations; the ecologists from the Jura \u2013 who, I had discovered on the train \u2013 against my will \u2013 were called \u00c9ric and Sylvie \u2013 were drooling with excitement. 'French cuisine tonight,' concluded S\u00f4n for no apparent reason. 'We now eat Thai. Small restaurant too, beside river.'\n\nThe place was charming. Trees shaded the tables. Near the entrance was a sunlit pool full of turtles and frogs. I watched the frogs for a long time; once again, I was struck by the extraordinary abundance of life in the tropics. White fish swam between two pools. On the surface were water-lilies and water-fleas. Insects continuously settled on the water-lilies. Turtles observed all this with a placidity characteristic of their species.\n\nS\u00f4n came to let me know that the meal had begun. I walked towards the dining room by the river. They had laid two tables for six; all the places were taken. I glanced around me, a little panicked, but Ren\u00e9 quickly came to my rescue. 'No problem! Come and join our table!' he called generously, 'We can add another place on the end.' So I sat at what was apparently the _established couples_ table: the ecologists from the Jura, the naturopaths \u2013 who, I now discovered, answered to the names of Albert and Suzanne \u2013 and the two senior citizens and former pork-butchers. This arrangement, I quickly came to believe, was not based on any real affinity but on the urgent situation which presented itself when they were shown to the tables; the couples had instinctively banded together; all in all, lunch was nothing more than an _observation round_.\n\nThe conversation first moved to the subject of massage, a subject which seemed dear to the naturopaths. The previous evening, Albert and Suzanne, forsaking traditional dance, had enjoyed an excellent back massage. Ren\u00e9 smiled a lewd smile; Albert's expression quickly let him know that his attitude was completely inappropriate. Traditional Thai massage, he thundered, had nothing whatever to do with who knows what kind of practices; it was the expression of a centuries-old, perhaps millennia-old, civilisation and, as it happened, was completely consistent with Chinese teachings on the points of acupuncture. They practised it themselves at their surgery in Montb\u00e9liard, without, naturally, attaining the dexterity of Thai practitioners; the night before, they had had, he concluded, an excellent lesson. \u00c9ric and Sylvie listened, fascinated. Ren\u00e9 coughed slightly in embarrassment; it was true that the Montb\u00e9liard couple did not, in fact, exude even the slightest impression of lewdness. Who could possibly have proposed the idea that France was the country of _debauchery_ and _libertinage_. France was a sinister country, utterly sinister and bureaucratic.\n\n'I had a back massage too, but the girl finished on my balls...' I interrupted without much conviction. Since I was chewing cashew nuts at the time, no one heard, with the exception of Sylvie who shot me a horrified look. I took a mouthful of beer and looked her straight in the eyes, not in the least embarrassed: was this girl even capable of _correctly_ handling a cock? That remained to be seen. In the meantime, I waited for my coffee.\n\n'It's true they're cute, the little girls...' commented Josette, taking a slice of papaya and adding to the general unease. The coffee was slow in coming. What do you do at the end of a meal if you're not allowed to smoke? I sat quietly as the boredom increased. We concluded the conversation, not without difficulty, with some remarks about the weather.\n\nI saw my father once again, confined to his bed, struck down by sudden depression \u2013 a terrifying thing in such an active man; his mountaineering friends stood around awkwardly, powerless in the face of the disease. The reason he played so much sport, he told me once, was to stupefy himself, to stop himself thinking. He had succeeded: I was convinced that he had managed to go through his whole life without ever really questioning the human condition.\n\n### 7\n\nON THE BUS, S\u00f4n continued her commentary. The border region which we were about to enter was partly populated by Burmese refugees of Karen origin, but this should present no problems. Karen tribe good, deemed S\u00f4n, brave, children good study in school, no problem. Nothing like some of the northern tribes, which we would not have the opportunity to meet on our tour; according to her, we weren't missing much. Particularly in the case of the Akha tribe, which she seemed to have something against. In spite of the government's best efforts, the Akhas seemed incapable of giving up growing opium poppies, their traditional activity. Akhas bad, S\u00f4n stressed forcefully: apart grow poppy and pick fruit, know how to do nothing; children not good study in school. Many money spend for them, no result. They are completely useless, she concluded, demonstrating her consummate ability to summarise.\n\nSo, as we arrived at the hotel, I watched these famous Karens curiously, as they busied themselves by the river's edge. Seen close up, I mean without machine-guns, they didn't seem particularly nasty; the most obvious thing about them was that they clearly adored their elephants. Bathing in the river, scrubbing the backs of their elephants seemed to be their greatest pleasure. It's true that these weren't Karen rebels but ordinary Karens \u2013 those who had fled the combat zone because they were sick of the whole thing and who were more or less indifferent to the cause of Karen independence.\n\nA brochure in my hotel room gave me some information about the history of the resort, which was the product of a wonderful human adventure: that of Bertrand Le Moal, backpacker _avant la lettre_ who, having fallen in love with this place, had 'laid down his pack' here at the end of the 60s. With furious energy, and the help of his Karen friends, little by little he had built this 'ecological paradise', which an international clientele could now enjoy.\n\nIt's true the place was superb. Small, beautifully sculpted chalets made of teak connected by a pathway decked with flowers, overhung the river, which you could feel pulse under your feet. The hotel was situated at the bottom of a steep valley, the sides of which were shrouded in dense jungle. When I stepped out on to the terrace there was a profound silence. It took me a moment or two to understand why: all at once every bird had stopped singing. It was the hour when the jungle readies itself for night. What sort of large predators would there be in a jungle like that? Not many, probably \u2013 two or three leopards \u2013 but there was probably no shortage of snakes and spiders. The light was fading fast. On the far bank, a lone monkey leaped from tree to tree; he gave a short call. You could feel he was fretful, anxious to rejoin his group.\n\nI went back into the room, lit the candles. The furniture was minimal: a teak table, two rustic wooden bedsteads, sleeping bags and mats. I spent a quarter of an hour methodically rubbing myself with Cinq sur Cinq insect repellent. Rivers are all very well, but you know what they're like: they attract mosquitoes. There was a bar of citronella too, which you could melt; it seemed to me a worthwhile precaution.\n\nWhen I came down to dinner, it was completely dark; garlands of multicoloured lights were strung between the houses. So there _was_ electricity in the village, I noted, they simply hadn't thought it necessary to install it in the rooms. I stopped for a moment and leaned on the guardrail to look down at the river; the moon was up and shimmered on the water. Opposite, you could vaguely make out the dark mass of the jungle; from time to time, the raucous cry of a nocturnal bird could be heard.\n\nHuman groups of more than three people have a tendency, apparently, to split into two hostile sub-groups. Dinner was served on a pontoon in the middle of the river; this time, the tables had been laid for eight. The ecologists and the naturopaths were already installed at one table; the former pork-butchers were currently all alone at the second. What could have brought about the rift? Maybe the massage discussion at lunch, which, let's face it, hadn't gone too well. In addition, that morning, Suzanne, soberly dressed in a white linen tunic and trousers \u2013 nicely cut to emphasise her angular features \u2013 had burst out laughing when she saw Josette's flower-print dress. Whatever the reason, the divisions had begun. In a rather cowardly move, I slowed my pace so as to let Lionel, my neighbour from the plane, who also had the neighbouring chalet, overtake me. He made his choice quickly, barely aware of doing so; I didn't even get the impression it was a choice based on elective affinity, more a sort of class solidarity (since he worked at Gaz de France and was therefore a civil servant, while the others had been small shopkeepers) a solidarity based on level of education. Ren\u00e9 welcomed us with evident relief. In any case, our decision was not critical at this stage of the game: had we joined the others we would have forcefully confirmed the isolation of the former pork-butchers; whereas this way, we were really only balancing out the table numbers.\n\nBabette and L\u00e9a arrived shortly after and without a second thought sat at the neighbouring table.\n\nQuite some time later \u2013 our first courses had already been served \u2013 Val\u00e9rie appeared on the edge of the pontoon; she looked around her uncertainly. At the other table, there were still two empty places beside Babette and L\u00e9a. She hesitated a little longer, made a little start and came and sat on my left.\n\nJosiane had taken even longer than usual getting ready; she must have had trouble putting on her makeup by candlelight. Her black velvet dress wasn't bad, a bit low cut, but not excessive. She also hesitated for a moment, then came and sat opposite Val\u00e9rie.\n\nRobert arrived last, a little unsteady. He'd probably been boozing before the meal; I'd seen him with a bottle of Mekong earlier. He dropped heavily on to the bench next to Val\u00e9rie. A short but fearful cry went up from somewhere close by in the jungle; perhaps some small mammal had just breathed its last.\n\nS\u00f4n moved between the tables to check that everything was okay, that we had all settled in nicely. She was having dinner elsewhere, with the driver \u2013 a less than democratic arrangement which had already earned Josiane's disapproval at lunchtime. But, basically, I think it suited her just fine, even if she had nothing against us: despite her best efforts, she seemed to find long discussions in French a bit tiring.\n\nAt the next table the conversation purred happily, discussing the beauty of the location, the joy of being at one with nature, far from civilisation, the essential values, etc. 'Yeah, it's top,' confirmed L\u00e9a. 'And y'know, we're really bang in the middle of jungle... I can't believe it.'\n\nOur table was having a little more difficulty finding common ground. Opposite me, Lionel was eating placidly, making no effort whatsoever. I glanced ner-vously from side to side. At one point I saw a big bearded guy coming out of the kitchens and shouting angrily at the waiters; it must be none other than the famous Bertrand Le Moal. To my mind, his greatest achievement so far was to have taught the Karens the recipe for _gratin dauphinois_. It was delicious, and the roast pork was perfectly done, crisp but tender. 'All we're missing is a drop of wine...' Ren\u00e9 said sadly. Josiane pursed her lips scornfully. One didn't need to ask what she thought about French tourists who couldn't leave the country without their drop of wine. A little awkwardly, Val\u00e9rie came to Ren\u00e9's defence. With Thai food, she said, you never felt the need; but right now, a little wine would be rather appropriate. In any case, she herself only drank water.\n\n'If you go abroad,' Josiane barked, 'It is in order to eat the _local_ food and to observe _local_ customs!... If not, you might as well stay at home.'\n\n'I agree!' shouted Robert. She paused, cut off in midflow, and looked at him hatefully.\n\n'Sometimes I find it a bit too spicy...' confessed Josette timidly. 'It doesn't seem to bother you...' she said, addressing me, probably to ease the tension.\n\n'No, no, I love it. The spicier it is, the better I like it. Even in Paris I eat Chinese all the time,' I hastily responded. And so the conversation was able to move on to the Chinese restaurants that had so multiplied in Paris just recently. Val\u00e9rie liked to have lunch in them, they were very reasonable, much better than eating fast food, and probably much healthier too. Josiane had nothing to say on the subject, she had a staff cafeteria; as for Robert, he probably thought the subject was beneath him. In short, everything proceeded more or less peacefully until dessert.\n\nIt all came to a head over the sticky rice. It was a light golden colour, flavoured with cinnamon \u2013 I think the recipe was original. Taking the bull by the horns, Josiane decided to tackle the question of sex tourism head on. For her, it was absolutely disgusting, there was no other word for it. It was a scandal that the Thai government tolerated such things. The international community had to do something. Robert listened to her with a half-smile which I didn't think boded well. It was scandalous, but it was hardly surprising; it was obvious that most of these places (brothels, that was the only word for them) were owned by generals; that told you what kind of protection they had.\n\n'I'm a general...' interrupted Robert. She was speechless, her lower jaw dropped miserably. 'No, no, I'm only joking...' he said with a slight grimace. 'I've never even been in the army.'\n\nShe did not find this funny in the least. She took a moment to pull herself together, then launched back into the fray with renewed energy.\n\n'It's absolutely shameful that fat yobs can just come over here and take advantage of these girls' poverty with impunity. Of course you know they all come from the north and the northeast, the poorest regions of the whole country.'\n\n'Not all of them...' he objected. 'Some of them are from Bangkok.'\n\n'It's sexual slavery!' screamed Josiane, who hadn't heard. 'There's no other way to describe it!...'\n\nI yawned a little. She shot me a black look, but went on, calling on the others to give their verdict: 'Don't you think it's disgraceful that any fat old yob can come over here and have it off with these kids for next to nothing?'\n\n'It's hardly next to nothing...' I protested modestly. 'I paid three thousand baht, which is about what you'd pay in France.' Val\u00e9rie turned and looked at me, surprised. 'You paid a bit over the odds...' observed Robert. 'Still, if the girl was worth it...'\n\nJosiane's whole body was trembling, she was starting to unsettle me a little. 'Well!' she shrieked in a very shrill voice, 'It makes me sick, that any fat pig can pay to shove his cock into a kid!'\n\n'Nobody's forcing you to come with me, madam...' Robert replied calmly.\n\nShe got up, trembling, her plate of rice in her hand. All conversation at the next table had stopped. I really thought she was going to chuck the plate in his face, and in the end I think it was only fear that stopped her. Robert looked at her with the most serious expression, the muscles under his polo-neck tense. He didn't look like the sort of person to let himself be pushed around; I could well imagine him punching her. She viciously slammed down her plate, which broke into three pieces, turned on her heel and vanished into the darkness, walking quickly towards the chalets.\n\n'Tsk...' he said softly.\n\nVal\u00e9rie was stuck between him and me; he stood up gracefully, walked around the table and sat where Josiane had been sitting, in case Val\u00e9rie, too, wished to leave the table. She, however, did nothing; at that moment, the waiter brought the coffees. After she had taken two sips, Val\u00e9rie turned to me again. 'So is it true you've paid for girls?...' she asked gently. Her tone was intrigued, but without any real reproach.\n\n'They're not as poor as all that, these girls,' added Robert; 'they can afford mopeds and clothes, some of them even have their tits done. It's not cheap getting your tits done. It's true they help their parents out, too...' he concluded thoughtfully.\n\nAt the next table, after a few whispered comments, everyone quickly left \u2013 doubtless out of solidarity. We remained the sole masters of the place, in a sense. The moon now bathed the whole pontoon, which gleamed a little. 'Are they that good, those little masseuses?...' asked Ren\u00e9 dreamily.\n\n'Ah, monsieur!' exclaimed Robert, deliberately grandiloquent, but, it seemed to me, basically sincere, 'they are marvellous, positively marvellous! And you haven't been to Pattaya yet. It's a resort on the east coast...' he went on, '... completely dedicated to lust and debauchery. The Americans were the first to go there, during the Vietnam war; after that, a lot of English and Germans; now, you get a lot of Russians and Poles. There, they have something for everyone, they cater for all tastes: homosexuals, heterosexuals, transvestites... It's Sodom and Gomorrah combined. Actually, it's better, because they've got lesbians, too.'\n\n'Aaah, aaah...' the former pork-butcher seemed thoughtful. His wife yawned placidly, excused herself and turned to her husband; she clearly wanted to go to bed.\n\n'In Thailand,' Robert concluded, 'everyone can have what they desire, and everyone can have something good. People will talk to you about Brazilian girls, or about Cubans. I'm well-travelled, monsieur, I have travelled for pleasure and I have no hesitation in telling you: in my opinion, Thai girls are the best lovers in the world.'\n\nSitting opposite, Val\u00e9rie listened to him earnestly. She disappeared shortly after, followed by Josette and Ren\u00e9. Lionel, who hadn't said a word all evening, also got to his feet; I did likewise. I didn't really feel like pursuing a conversation with Robert. So I left him alone in the dark, a picture of apparent sobriety, ordering a second cognac. He seemed to have a sophisticated and subtle intelligence; unless, of course, he was a relativist, which always gives one the impression of complexity and subtlety. In front of my chalet, I said good night to Lionel. The atmosphere was heavy with the buzzing of insects; I was more or less sure that I wouldn't get a wink of sleep.\n\nI pushed the door and lit the candle again, more or less resigned to continue reading _The Firm_. Mosquitoes flew close, some of them charred their wings in the flame, their bodies sank into the melted wax; not one of them settled on me. Despite the fact that I was filled to the dermis with nutritious, delicious blood, they automatically turned tail, unable to break through the olfactory barrier of carbonic dimethylperoxide. Roche-Nicolas laboratories, the creators of Cinq sur Cinq, were to be congratulated. I blew out the candle, relit it, watching the ever-more teeming ballet of these sordid little flying machines. On the other side of the partition, I could hear Lionel snoring gently through the night. I got up, put another block of citronella on to melt, then went for a piss. A round hole had been made in the floor of the bathroom; it flowed straight into the river. You could hear the lapping of the water and the sound of fins; I tried not to think about what might be down there. Just as I was going back to bed, Lionel let out a long series of farts. 'Too right, my boy!' I commended him enthusiastically. 'As Martin Luther said, there's nothing like farting in your sleeping bag!' My voice resounded strangely in the dark, above the murmuring of the river and the persistent drone of the insects. Simply being able to hear the real world was a torment. 'The kingdom of heaven is like unto a cotton bud!' I shouted again into the night. 'Let he who has ears to hear, hear!' In his bed, Lionel turned over and moaned gently without waking. I didn't have much in the way of choice: I'd have to take a sleeping pill.\n\n### 8\n\nCARRIED BY THE current, tufts of grass floated downriver. The birdsong started up again, rising from the light mist which swathed the jungle. Far off to the south, at the mouth of the valley, the strange contours of the Burmese mountains were silhouetted in the distance. I had seen these curved, bluish forms before, but cut through with sudden indentations. Perhaps in the landscapes of the Italian primitives, on a visit to a museum when I was at school. The group was not awake yet; the temperature was still pleasant at this hour. I had slept very badly.\n\nAfter the disaster of the previous evening, a certain benevolence floated around the breakfast tables. Josette and Ren\u00e9 seemed to be in good form; on the other hand the ecologists from the Jura were in a terrible state, I noticed, as they shambled in. The proletariat of a previous generation, who had no hang-ups about enjoying modern comforts when they were available, proved to be much more resilient in truly uncomfortable circumstances than their offspring, who championed 'ecological' principles. \u00c9ric and Sylvie clearly hadn't got a wink all night; in addition, Sylvie was completely covered in red blisters.\n\n'Yes, the mosquitoes really got me,' she confirmed bitterly.\n\n'I've got some soothing lotion if you want. It's very good; I can go and get it.'\n\n'That would be nice, thanks; but let's have coffee first.'\n\nThe coffee was revolting, weak, almost undrinkable; from that point of view at least, we were working to American standards. The young couple looked completely bloody stupid, it almost pained me to see their 'ecological paradise' crumbling before their eyes; but I had a feeling that everything was going to cause me pain today. I looked to the south again. 'I'm told Burma is very beautiful,' I said in a low voice, mostly to myself. Sylvie solemnly agreed: it was indeed, very beautiful, she'd also heard as much; that said, she _forbade_ herself from going to Burma. It was impossible to think that one's money would go to supporting a dictatorship like that. Yes, yes, I thought, money. 'Human rights are extremely important,' she exclaimed almost despairingly. When people talk about 'human rights', I usually get the impression that they're being ironic; but that wasn't true in this case, at least I don't think so.\n\n'Personally, I stopped going to Spain _after_ the death of Franco,' interrupted Robert, taking a seat at our table. I hadn't seen him arrive. He seemed to be in excellent form, his formidable ability to infuriate well-rested. He informed us that he'd gone to bed dead drunk and consequently had slept like a log. He had almost chucked himself in the river a couple of times on his way back to the chalet; but in the end it hadn't happened. ' _Insh'allah_.' he concluded in a booming voice.\n\nAfter this parody of a breakfast, Sylvie walked back with me to my room. On the way, we met Josiane. She was serious, withdrawn and did not even look at us; she seemed to be far from the road to forgiveness. I discovered that she taught literature in civvy street, as Ren\u00e9 amusingly put it; I wasn't a bit surprised. She was exactly the kind of bitch who'd made me give up studying literature many years before.\n\nI gave Sylvie the tube of soothing lotion. 'I'll bring it straight back,' she said. 'You can keep it, I don't think we'll come across any more mosquitoes; as far as I know they hate the seaside.' She thanked me, walked to the door, hesitated, turned round: 'Surely you don't approve of the sexual exploitation of children!...' she exclaimed anguishedly. I was expecting something of the kind. I shook my head and answered wearily: 'There's not that much child prostitution in Thailand. No more than in Europe, in my opinion.' She nodded, not really convinced, and walked out. In fact, I had access to rather more detailed information, courtesy of a strange publication called _The White Book_ , which I'd bought for my previous trip. It was apparently published \u2013 no author's or publisher's name was given \u2013 by an association called Inquisition 2000. Under the pretence of denouncing sexual tourism, it gave all the addresses, country by country \u2013 each revealing chapter was preceded by a short and vehement paragraph calling for respect for the Divine plan and the reintroduction of the death penalty for sex offenders. On the question of paedophilia, _The White Book_ was unequivocal: it formally advised against Thailand, which no longer had anything to recommend it. It was much better to go to the Philippines or, better still, to Cambodia \u2013 the journey might be dangerous, but it was worth the effort.\n\nThe Khmer Kingdom was at its apogee in the twelfth century, the era in which Angkor W\u00e2t was built. After that, it pretty much fell apart; since then Thailand's principal enemy had been the Burmese. In 1351, King Ramathibodi I founded the village of Ayutthaya. In 1402, his son Ramathibodi II invaded the declining Angkor empire. Thirty-two successive sovereigns of Ayutthaya marked their reigns by building Buddhist temples and palaces. In the sixteenth and seventeenth centuries, according to the accounts of French and Portuguese travellers, it was the most magnificent city in all Asia. The wars with the Burmese continued and Ayutthaya fell in 1767, after a siege lasting fifteen months. The Burmese looted the city, melted down the gold statues and left nothing but ruins in their wake.\n\nNow, it was very peaceful; a light breeze stirred up dust between the temples. Not much remained of King Ramathibodi, apart from a couple of lines in the Michelin Guide. The image of the Buddha, on the other hand, was very much in evidence and had retained all of its significance. The Burmese had shipped in Thai craftsmen so that they could construct identical temples several hundred kilometres away. The will to power exists, and it manifests itself in the form of history; it is, in itself, radically unproductive. The smile of the Buddha continued to float above the ruins. It was three o'clock in the afternoon. According to the Michelin Guide, you needed to set aside three days for a complete visit, one day for a quick tour. We had three hours; it was time to get out the camcorders. I imagined Chateaubriand with a Panasonic camcorder at the Coliseum, smoking cigarettes \u2013 B&H probably, rather than Gauloises Lights. Faced with a religion this radical, I expect his views would have been slightly different; he would have had a lot less respect for Napoleon. I was sure that he would have been capable of writing an excellent _G\u00e9nie du bouddhisme_.\n\nJosette and Ren\u00e9 were a bit bored during the visit; I got the impression that pretty quickly they were just going round in circles. Babette and L\u00e9a were the same. The ecologists from the Jura, on the other hand, seemed to be in their element, as did the naturopaths; they deployed an impressive array of photographic equipment. Val\u00e9rie was lost in thought, walking down the alleys, across the flagstones, through the grass. That's culture for you, I thought: it's a bit of a pain in the arse, but that's good; everyone is returned to his own nothingness. That said, how did the sculptors of the Ayutthaya period _do it_? How did they manage to give their statues of the Buddha such a luminous expression of understanding?\n\nAfter the fall of Ayutthaya, the Thai kingdom entered a period of great stability. Bangkok became the capital and the R\u00e2ma dynasty began. For two centuries (actually, up to the present day) the kingdom knew no serious foreign wars, nor any civil or religious wars for that matter; it also succeeded in avoiding any form of colonisation. There had been no famines, either, nor great epidemics. In such circumstances, when lands are fertile and bring forth abundant harvests, when sickness seems to relax its grip, when a peaceable religion extends its laws over hearts and minds, human beings grow and multiply; in general, they live happily. Now, things were different. Thailand had become part of the free world, meaning the market economy; for five years it had been suffering a terrible economic crisis which had reduced the currency to less than half its previous value and brought the most successful businesses to the brink of ruin. This was the first real tragedy to strike the country for more than two centuries.\n\nOne after another, in a silence that was pretty striking, we went back to the coach. We left at sunset. We were due to take the night train from Bangkok, destination Surat Thani.\n\n### 9\n\nSURAT THANI \u2013 POPULATION 42,000 \u2013 is distinguished, according to the guidebooks, by the fact that it is of no interest whatever. It is, and this is the only thing you can say about it, an obligatory stop on the way to the Koh Samui ferry. Nonetheless, people live here, and the Michelin Guide informed us that for a long time the city has been an important centre of metallurgical industries \u2013 and that, more recently, it has played a significant role in machine assembly.\n\nAnd where would we be without machine construction? Iron ore is mined in obscure regions of the country and transported here by freighter. Machine tools are then produced, mostly under the supervision of Japanese companies. Their assembly takes place in cities like Surat Thani: resulting in coaches, train carriages, ferries; all produced under licence from NEC, General Motors or Fujimori. The products serve in part to transport western tourists, such as Babette and L\u00e9a.\n\nI was entitled to speak to them, I was a member of the same tour; I could hardly presume to be a potential lover, which limited possible conversation from the off; I had, nevertheless, purchased the same outbound ticket; I was therefore at liberty, to some extent, to make contact. Babette and L\u00e9a, it turned out, worked for the same PR agency; for the most part, they organised events. Events? Yes. For institutions or private companies keen to develop their corporate sponsorship programmes. There was certainly money to be made there, I thought. Yes and no. Nowadays, companies were more 'human rights' focused, so there had been a slowdown in investment. But it was still pretty okay. I enquired about their salaries: pretty good. They could have been better, but still pretty good. About twenty-five times the salary of a metalworker in Surat Thani. Economics is a mystery.\n\nAfter we arrived at the hotel, the group broke up, at least I suppose it did; I didn't feel much like eating with the others; I was a bit fed up with the others. I drew the curtains and lay down. Curiously, I fell asleep immediately and dreamed of an Arab girl dancing in a metro carriage. She didn't look anything like A\u00efcha, at least I don't think so. She was standing against the central pole, like the girls in go-go bars. Her breasts were covered by a miniscule strip of cotton which she was slowly lifting. With a smile, she freed her breasts completely; they were swollen, round, copper-coloured, magnificent. She licked her fingers and stroked her nipples. Then she put her hand on my trousers, eased down my flies and took out my penis, and began to jerk me off. People crowded past us, got off at their stations. She got on all fours on the floor, lifted up her mini-skirt; she wasn't wearing anything underneath. Her vulva was welcoming, surrounded by black hair, like a gift; I started to penetrate her. The carriage was half full, but no one paid any attention to us. Such things could never happen under any normal circumstances. It was the dream of a starving man, the ludicrous dream of man already grown old.\n\nI woke up at about five o'clock, noticed the sheets were completely covered in semen. A nocturnal emission... very touching. I noticed too, to my great surprise, that I still had a hard on; I put it down to the weather. A cockroach lay on its back in the middle of the bedside table; you could easily make out the detail of its legs. This one didn't have to worry any more, as my father would have said. My father, for his part, had died in late 2000; good thing too. Consequently, his existence was entirely contained within the twentieth century, of which he was a hideously representative element. I myself had survived in middling condition. I was in my forties, well, in my early forties, after all, I was only forty; I was about half way there. My father's death gave me a certain freedom; I hadn't had my last word yet.\n\nSituated on the east coast of Ko Samui, the hotel perfectly evoked the sort of tropical paradise you see in travel agents' brochures. The hills surrounding it were covered by thick jungle. The low-rise buildings, bordered by greenery, sloped down to an immense oval swimming pool with a jacuzzi at each end. You could swim up to the bar, which was on an island in the middle of the pool. A few yards further on was a beach of white sand and the sea. I looked around warily at my surroundings; from here, I recognised Lionel in the distance splashing in the waves like a handicapped dolphin. Then I turned back and headed for the bar along a narrow bridge overlooking the pool. With studied casualness, I familiarised myself with the cocktail menu; happy hour had just begun.\n\nI had just ordered a Singapore Sling when Babette made her appearance. 'Well, well...' I said. She was wearing a generously cut two-piece bathing suit, figure-hugging shorts and a wide wraparound top in a symphony of light and dark blue. The fabric seemed to be exceptionally sheer; it was a swimsuit which clearly only came into its own when wet. 'Are you not going to swim?' she asked. 'Euh...' I said. L\u00e9a appeared in turn, more classically sexy in a bright red vinyl one-piece, with black zips open to reveal her skin (one of them ran across her left breast, giving a glimpse of nipple), and cut very high on the thighs. She nodded to me before joining Babette at the water's edge; when she turned round, I was in a position to observe that she had perfect buttocks. The girls had been suspicious of me at the beginning; but since I had spoken to them on the ferry they had come to the conclusion that I was a harmless human being and moderately amusing. They were right: that was about it.\n\nThey dived in together. I turned round to ogle a bit. The guy at the next table was the spitting image of Robert Hue. When wet, Babette's swimsuit really was spectacular: you could easily make out her nipples and the crack of her bum; you could even see the slight swelling of her pubic hair, even though she had opted to cut it quite short. Meanwhile, people were working, making useful commodities; or sometimes useless commodities. They were productive. What had I produced in the forty years of my existence? To tell the truth, not very much. I had managed information, facilitated access to it and disseminated it; sometimes, too, I had carried out bank transfers (on a modest scale; I was generally happy to pay the smaller invoices). In a word, I had worked in the service sector. It would be easy to get by without people like me. Still, my ineffectuality was less flamboyant than that of Babette and L\u00e9a; a moderate parasite I had never been a high-flyer in _my job_ , and had never felt the need to pretend to be.\n\nAfter dark, I went back to the hotel lobby, where I ran into Lionel; he was sunburned from head to toe and delighted with his day. He had done a lot of swimming; he'd never dared dream of somewhere like this. 'I had to save pretty hard to pay for the trip,' he said, 'but I don't regret it.' He sat on the edge of a sofa; he was thinking about his daily life. He worked for Gaz de France in the southeastern suburbs of Paris; he lived in Juvisy. He often had to call on people who were very poor, old people whose systems weren't up to standard. If they didn't have money the to pay for the necessary modifications, he was forced to cut off their gas. 'There are people who live in conditions...' he said, 'you can't imagine.'\n\n'You get to see strange things sometimes...' he went on, shaking his head. As for himself, things were okay. The area he lived in wasn't great, actually it was downright dangerous. 'There are places that are best avoided,' he said. But in general, things weren't too bad. 'We're on holiday,' he concluded before heading off to the dining room. I picked up a couple of brochures and went off to my room to read them. I still didn't feel like eating with the others. It is in our relations with other people that we gain a sense of ourselves; it's that, pretty much, that makes relations with other people unbearable.\n\nI'd found out from L\u00e9a that Ko Samui wasn't just a tropical paradise, it was also pretty phat. Every night at the full moon there was a massive rave on the tiny neighbouring island of Ko Lanta; people came from Australia or from Germany to attend. 'A bit like Goa...' I said. 'Much better than Goa,' she interrupted. Goa was completely _past it_ ; if you were looking for a decent rave now, you had to go to Ko Samui or to Lombok.\n\nI didn't ask as much. All I wanted right now was a decent body massage, followed by a blowjob and a good fuck. Nothing too complicated on the face of it; but looking through the brochures I realised with a feeling of profound melancholy that it didn't at all seem to be the speciality of the place. There was a lot of stuff like acupuncture, massage with essential oils, vegetarian food or tai-chi; but body massage or go-go bars, _nada_. On top of everything, the place had a painfully American, even Californian, feel about it, focused on 'healthy living' and 'meditation activities'. I glanced through a letter to _What's On: Samui_ from a reader, Guy Hopkins; he was a self-confessed 'health addict' and had been coming to the island regularly for twenty years, ' _The aura that backpackers spread on the island is unlikely to be erased quickly by upmarket tourists_ ,' he concluded; it was depressing. I couldn't even set off in search of adventure as the hotel was miles from anywhere; in fact everything was miles from anywhere, since there was nothing here. The map of the island indicated no identifiable centre: several chalet resorts like ours, set on tranquil beaches. It was then that I remembered with horror that the island had had a very good write-up in the _Guide du Routard_. Here was a place where they had managed to avoid a certain moral slide: I was caught like a rat in a trap. Even so, I felt a vague satisfaction, however theoretical, at the notion that I felt up to fucking. Half-heartedly, I picked up _The Firm_ again, skipped forward two hundred pages, skipped back fifty; by chance I happened on a sex scene. The plot had developed a fair bit: Tom Cruise was now in the Cayman Islands, in the process of setting up some kind of money-laundering scheme, or in the process of unmasking it, it wasn't too clear. Whatever the deal was, he was getting to know a stunning mixed-race girl, and the girl wasn't exactly backward in coming forward. 'She unsnapped something and removed her skirt, leaving nothing but a string around her waist and a string running between her legs'. I unzipped my trousers. This was followed by a weird passage that was difficult to grasp psychologically: 'Something said run. Throw the beer bottle into the ocean. Throw the skirt on to the sand. And run like hell. Run to the condo. Lock the door. Lock the windows. Run. Run. Run.' Thankfully, Eilene didn't see things quite that way: 'In slow motion, she reached behind her neck. She unhooked her bikini top, and it fell off, very slowly. Her breasts, much larger now, lay on his left forearm. She handed the top to him. \"Hold this for me.\" It was soft and white and weighed less than a millionth of an ounce.' I was jerking off in earnest now, trying to visualise mixed-race girls wearing tiny swimsuits in the dark. I ejaculated between two pages with a groan of satisfaction. They were going to stick together; didn't matter, it wasn't the kind of book you read twice.\n\nIn the morning, the beach was deserted. I went for a swim just after breakfast; the air was warm. The sun would soon begin its ascent across the sky, increasing the risk of skin cancer in individuals of Caucasian descent. I intended to stay long enough for the maids to make up my room, then I would head back, he beneath the sheets and put the air-conditioning on full; with the greatest serenity I contemplated this free day.\n\nTom Cruise, on the other hand, was still plagued with worries about his affair with the mixed-race girl; he even considered telling his wife (who, and this was the problem, was not content simply to be loved; she wanted to be the sexiest, the most desirable woman in the world). The idiot behaved as though the future of his marriage was at stake. 'If she was cool and showed a trace of compassion, he would tell her he was sorry, so very sorry, and that it would never happen again. If she fell all to pieces, he would beg, literally beg for forgiveness and swear on the Bible that it was a mistake and would never happen again.' Obviously, it came to much the same thing; but in the end the hero's unremitting remorse, though it was of no interest whatever, began to interfere with the story \u2013 which was pretty serious: we had a bunch of extremely nasty Mafiosi, the FBI, maybe even the Russians. It was enough to make you angry, and in the end it made you sick.\n\nI had a go with another American bestseller, _Total Control_ , by David G Baldacci; but that was even worse. This time, the hero wasn't a lawyer but a young computer genius who worked a hundred and ten hours a week. His wife, on the other hand, was a lawyer and worked ninety hours a week: they had a kid. This time the bad guys were a 'European' company which had resorted to fraudulent practices in order to corner a market. Said market should have been the territory of the American company for which our hero was working. During a conversation with the bad guys from the European company, the bad guys \u2013 without the least compunction \u2013 smoked several cigarettes; the atmosphere literally stank of them, but the hero managed to survive. I made a small hole in the sand to bury the two books; the problem now was that I had to find something to read. Life without anything to read is dangerous: you have to content yourself with life and that can lead you to take risks. At the age of fourteen, one afternoon when the fog was particularly dense, I had got lost while skiing; I had had to make my way across avalanche corridors. What I remember most were the low, leaden clouds, the utter silence on the mountain. I knew the drifts of snow could shear away at any moment if I made a sudden movement, or even for no apparent reason, some slight rise in temperature, a breath of wind. If they did I would be carried with them, dragged hundreds of metres on to the rocky ridges below; I would die, probably on impact. Despite this, I wasn't in the least afraid. I was annoyed that things had turned out this way, annoyed for myself and for everyone else. I would have preferred a more conventional death, more official in a way, with an illness, a funeral, tears. Most of all, I regretted never having known my wife's body. During the winter months, my father rented out the first floor of his house; this year the tenants were a couple of architects. Their daughter, Sylvie, was also fourteen; she seemed to be attracted to me, at least she did her best to have me around. She was slender, graceful, her hair was black and curly. Was her pubic hair black and curly too? These were the thoughts that flitted through my mind as I plodded across the mountainside. I've often wondered about that, since: faced with danger, even death, I don't feel anything in particular, no rush of adrenalin. I had searched for the sensations which attract 'extreme sports' fanatics in vain. I am not remotely brave, I run away from danger if at all possible; but if push comes to shove, I greet it with the placidity of a cow. There's probably no point in searching for meaning in this, it's just a technical matter, a question of hormone levels; other human beings apparently similar to me, seem to feel nothing in the presence of a woman's body, something which plunged me, at the time and still, plunges me into a state of agitation I can't control. In most circumstances in my life, I have had about as much freedom as a vacuum cleaner.\n\nThe sun was beginning to get hot. I noticed that Babette and L\u00e9a had arrived on the beach; they had settled themselves about ten metres away from me. Today, they were topless and dressed simply, identically, in white thongs. Apparently they'd met some boys, but I didn't think they were going to sleep with them: the guys weren't bad, reasonably muscular, but not that great either; all in all, pretty average.\n\nI got up and gathered my things. Babette had put her copy of _Elle_ next to her towel. I glanced towards the sea; they were swimming and laughing with the boys. I stooped quickly and stuffed the magazine into my bag; then I walked on along the beach.\n\nThe sea was calm; the view stretched out to the east. Cambodia was probably on the other side, or maybe Vietnam. There was a yacht, midway to the horizon; perhaps there are millionaires who spend their time sailing back and forth across the oceans of the world; a life at once monotonous and romantic.\n\nVal\u00e9rie approached, walking along the water's edge, amusing herself by taking a sidestep now and then to avoid a stronger wave. I quickly propped myself up on my elbows, becoming painfully conscious that she had a magnificent body and was very attractive in her rather sensible two-piece swimsuit; her breasts filled out the bikini top perfectly. I gave a little wave, thinking that she hadn't seen me, but in fact she was already looking in my direction; it's not easy to catch women out.\n\n'You're reading _Elle_?' she asked, a little surprised, quietly ironic.\n\n'Euh...' I said.\n\n'May I?' she sat down beside me. Easily, with the familiarity of a regular reader, she skimmed through the magazine: a quick look at the fashion pages, another at the front pages. ' _Elle_ reads', ' _Elle_ goes out'...\n\n'Did you go to another massage parlour last night?' she asked, with a sidelong glance.\n\n'Um... no, I couldn't find one.'\n\nShe nodded briefly and went back to reading the cover story: 'Are you programmed to love him forever?'\n\n'Is it any good?' I asked after a silence\n\n'I haven't got a lover,' she replied seriously. This girl completely unsettled me.\n\n'I don't really understand this magazine,' she continued without a pause; 'All it talks about is fashion and new trends: what you should see, what you should read, the causes you should campaign for, new topics of conversation... The readers couldn't possibly wear the same clothes as the models, and why on earth would they be interested in new trends? They're mostly older women.'\n\n'You think so?'\n\n'I'm sure. My mother reads it.'\n\n'Maybe the writers simply write about the things they're interested in, not what interests their readers.'\n\n'Economically, that shouldn't be viable; normally things are done to satisfy the customer's tastes.'\n\n'Maybe it does satisfy the customers tastes.'\n\nShe pondered, 'Maybe...' she replied hesitantly.\n\n'You think when you're sixty you won't be interested in new trends any more,' I insisted\n\n'I certainly hope not...' she said sincerely.\n\nI lit a cigarette. 'If I'm going to stay, I'll have to put on sunscreen,' I said in a melancholy voice.\n\n'We're going for a swim! You can put on sunscreen after.' In a flash she was on her feet and pulling me towards the shore.\n\nShe was a good swimmer. Personally, I can't say that I know how to swim, I can float on my back for a bit but I get tired quickly. 'You get tired quickly,' she said. 'It's because you smoke too much. You should do some sport. I'm going to sort you out...' She twisted my bicep. Oh no, I thought, no. In the end, she calmed down and went back to sunning herself after she'd vigorously dried her hair. She was pretty like that, with her long black hair all tousled. She didn't take off her top, it was a pity; I would have really liked her to take off her top. I would have liked to see her breasts, here, now.\n\nShe surprised me looking at her breasts and smiled quickly. 'Michel...' she said after a moment's silence. I jumped at the use of my first name. 'Why do you feel so old?' she asked, looking me straight in the eyes.\n\nIt was a good question; I choked a little.\n\n'You don't have to answer straight away...' she said gently, 'I've got a book for you,' she went on, taking it from her bag. I was surprised to recognise the yellow cover of the 'Masque' series, and a title by Agatha Christie, _The Hollow_.\n\n'Agatha Christie?' I said, bewildered.\n\n'Read it anyway. I think you'll find it interesting.'\n\nI nodded like an idiot. 'Are you not coming to lunch?' she asked after a moment, 'It's one o'clock already.'\n\n'No... No, I don't think so.'\n\n'You don't much like being in a group?'\n\nThere was no point in answering; I smiled. We picked up our things, we left together. On the way, we met Lionel, who was wandering around like a lost soul; he gave us a friendly wave, but already it seemed as if he wasn't having so much fun. It isn't for nothing that single men are so rare at holiday camps. You can see them, nervously, on the periphery of the recreational activities. Most often, they turn and leave; sometimes they launch into them, and participate. I left Val\u00e9rie by the restaurant tables.\n\nIn every Sherlock Holmes story you immediately recognise the characteristics of the character; but, as well as that, the author never fails to introduce some new peculiarity (the cocaine, the violin, the existence of his older brother, Mycroft, the taste for Italian opera... certain services rendered long ago to the crowned heads of Europe... the first case Sherlock Holmes ever solved when he was still an adolescent). Each new detail that is revealed casts new areas of shadow, and in the end developed a character who was truly fascinating: Conan Doyle succeeded in creating a perfect mixture of the pleasure of discovery and the pleasure of recognition. I always felt that Agatha Christie, on the other hand, put too much emphasis on the pleasure of recognition. In her initial descriptions of Poirot, she had a tendency to limit herself to a couple of stock phrases, restricted her character's most obvious traits (his mania for symmetry, his patent-leather boots, the care he lavishes on his mustachios); in the more mediocre other books, you even get the impression that the phrases had been copied directly from one novel to another.\n\nThat said, _The Hollow_ was interesting for other reasons. Not simply for the ambitious character of Henrietta, the sculptor, in whom Agatha Christie tried to portray not only the agony of creation (the scene where she destroys a statue just after labouring to finish it because she senses that it is lacking something), but that suffering which is particular to being an artist; that inability to be _truly_ happy or unhappy, to _truly_ feel hatred, despair, ecstasy or love; the sort of aesthetic filter which separates, without the possibility of remission, the artist from the world. The author had put much of herself into her character, and her sincerity was obvious. Unfortunately, the artist, separated in a way from the world, sensing things only in a vague, ambiguous, and consequently less intense manner, became as a result a less interesting character.\n\nFundamentally conservative, and hostile to any idea of the social redistribution of wealth, Agatha Christie adopted very clear-cut ideological positions throughout her career as a writer. In practise, this radical theoretical engagement nonetheless made it possible for her to be frequently cruel in her descriptions of the English aristocracy, whose privileges she so staunchly defended. Lady Angkatell is a burlesque character, only barely credible and often almost terrifying. The author is clearly fascinated with her creation, who has clearly forgotten even those rules which apply to ordinary human beings; she must have enjoyed writing sentences like: 'But then one doesn't exactly _introduce_ people \u2013 not when somebody had just been killed' \u2013 but her sympathies did not lie with Lady Angkatell. On the other hand, she paints a warm portrait of Midge, forced to work as a salesgirl during the week, and who spends her weekends among people who haven't the faintest idea of what work really is. Spirited, lively, Midge loves Edward hopelessly. Edward, for his part, thinks himself a failure: he hasn't succeeded at anything in his life, _not even at becoming a writer_ ; he writes short stories of disenchanted irony for obscure journals read only by bibliophiles. Three times he proposes marriage to Henrietta, without success. Henrietta is John's mistress, she admires his strength, his radiant personality; but John is married. His murder shatters the delicate balance of unfulfilled desire between the characters: Edward finally realises that Henrietta will never want him, that he can never measure up to John; but nor can he bring himself closer to Midge, and his life seems to be completely ruined. It is at this point that _The Hollow_ becomes a strange, poignant book; these are deep waters, with powerful undercurrents. In the scene in which Midge saves Edward from committing suicide, and in which he proposes to her, Agatha Christie achieves something beautiful, a sort of Dickensian sense of wonder.\n\nHer arms closed round him firmly. He smiled at her, murmuring:\n\n'You're so warm, Midge \u2013 you're so warm.'\n\nYes, she thought, that was what despair was. A cold thing, a thing of infinite coldness and loneliness. She'd never understood until now that despair was a cold thing. She had always thought of it as something hot and passionate, something violent, a hot-blooded desperation. But that was not so. This was despair \u2013 this utter outer darkness of coldness and loneliness. And the sin of despair, that priests talked of, was a cold sin, the sin of cutting oneself off from all warm and living human contacts.\n\nI finished reading at about nine o'clock; I got up and walked to the window. The sea was calm, myriads of luminous specks danced on the surface; a delicate halo surrounded the circular face of the moon. I knew there was a full-moon rave party tonight at Ko Lanta; Babette and L\u00e9a would probably go, with a good many other guests. Giving up on life is the easiest thing to do, putting one's own life to one side. As preparations for the evening continued, as taxis pulled up at the hotel, as everyone began to bustle in the corridors, I felt nothing more than a sad sense of relief.\n\n### 10\n\nA NARROW STRIP of mountainous land separating the gulf of Thailand from the Andaman sea, the isthmus of Kra, is divided to the north by the border between Thailand and Burma. At Ranong, in the far south of Burma, it measures barely twenty-two kilometres across; after that it progressively widens to become the Malay peninsula.\n\nOf the hundreds of islands which speckle the Andaman sea, only a few are inhabited, and not one of the islands on the Burmese side is open to tourists. On the Thai side, on the other hand, the islands of Phang Nga bay bring in 43 per cent of the country's annual tourist revenue. The largest of these is Phuket, where resorts were developed in the middle of the 80s, mostly with Chinese and French capital (South-East Asia quickly became one of the key areas of expansion for the Aurore group). It is probably in the chapter on Phuket that the _Guide du Routard_ reaches the pinnacle of its loathing, its vulgar \u00e9litism and aggressive masochism. 'For some,' they announce first off, 'Phuket is an island on the way up; for us, it is already on the way down.'\n\n'It was inevitable that we'd get here in the end,' they go on, 'to this \"pearl of the Indian Ocean\"... Only a few years ago we were still singing the praises of Phuket: the sun, the unspoiled beaches, the relaxed rhythms of life. At the risk of putting a spanner in the works, we'll come clean: we don't like Phuket any more! _Patong_ , the most famous of the beaches, has been covered in concrete. Everywhere the clientele has become predominantly male, hostess bars are springing up everywhere and the only smiles are the ones you can buy. As for the backpacker chalets, they've had a JCB face lift to make way for hotels destined for lonely pot-bellied Europeans.'\n\nWe were due to spend two nights at Patong Beach; I settled myself confidently on the coach, perfectly prepared to adopt my role as a lonely pot-bellied European. The end of the trip was the highlight of the tour: three days at our leisure in Ko Phi Phi, a destination usually thought of as paradise itself. 'What to say about Ko Phi Phi?' lamented the travel guide, 'It's as if you asked us about a lost love... We want to say something wonderful about it, but there's a lump in our throat.' For the manipulative masochist, it is not enough that he is unhappy; others must be unhappy too; I chucked my _Guide du Routard_ into the bin at the service station. Western masochism, I thought. A mile or so later, I realised that I now didn't have anything to read; I was going to have to tackle the last part of the tour without a scrap of printed matter to hide behind. I glanced around me, my heartbeat had accelerated, the outside world suddenly seemed a whole lot closer. On the other side of the aisle, Val\u00e9rie had reclined her seat; she seemed to be daydreaming or sleeping, her face was turned toward the window. I tried to follow her example. Outside the landscape unfolded, made up of diverse vegetation. In desperation, I borrowed Ren\u00e9's Michelin Guide; I thus learned that rubber plantations and latex played a key role in the economy of the region: Thailand is the third largest rubber producer in the world. That muddle of vegetation, then, served to make condoms and tyres; human ingenuity was truly remarkable. Mankind can be criticised from a variety of standpoints, but that's one thing you can't take away from him: we're unquestionably dealing with an ingenious mammal.\n\nSince the evening at the River Kwai, the seating at table had become definitive. Val\u00e9rie had joined what she called the 'yob camp'. Josiane had thrown her lot in with the naturopaths, with whom she shared certain values \u2013 such as techniques for promoting calm. At breakfast, I was able to observe from a distance a veritable calm competition between Albert and Josiane, under the watchful eyes of the ecologists \u2013 who, living in their godforsaken hole in Franche-Comt\u00e9, obviously had access to fewer techniques. Babette and L\u00e9a, though they were from the \u00cele-de-France, didn't have much to say for themselves other than an occasional: 'That's cool...' calm was still a medium-term goal for them. All in all, they had a well balanced table, equipped with a _natural leader_ of each sex, capable of fostering team spirit. On our side, things had a bit more trouble gelling. Josette and Ren\u00e9 regularly provided a commentary on the menu; they had become very familiar with the local food, Josette even intended to take home some recipes. From time to time they carped about the people at the other table, whom they considered to be pretentious, and poseurs; that wasn't going to get us very far, and I was usually impatient for the dessert to arrive.\n\nI gave Ren\u00e9 his Michelin Guide back; Phuket was still a four-hour drive away. At the restaurant bar, I bought a bottle of Mekong. I spent the next four hours fighting back the feeling of shame that was stopping me from taking it out of my bag and quietly getting rat-arsed; shame won out in the end. The entrance to the Beach Resortel was decorated with a banner which read: WELCOME TO THE FIREMEN OF CHAZAY. 'Now that's funny,' said Josette, 'Chazay \u2013 that's where your sister lives...' Ren\u00e9 couldn't remember. 'It is, it is...' she insisted. Before I got my room-key, I just had time to hear her say: 'So, that crossing the isthmus of Kra thing was just a day wasted'; and the worst thing was, she was right. I threw myself on to the king-size bed and took a long swig of alcohol; and then another.\n\nI woke up with an appalling headache and spent quite a while throwing up into the toilet bowl. It was five in the morning: too late for the hostess bars, too early for breakfast. In the drawer of the bedside table there was a Bible and a copy of the teachings of the Buddha, both in English. 'Because of their ignorance,' I read, 'people are always thinking wrong thoughts and always losing the right viewpoint and, clinging to their egos, they take wrong actions. As a result, they become attached to a delusive existence.' I wasn't really sure that I understood, but the last sentence perfectly described my current state; I was sufficiently relieved that I was able to wait until breakfast time. At the next table there was a group of gigantic black Americans that could easily have been mistaken for a basketball team. Further along there was a table of Hong Kong Chinese \u2013 recognisable by their filthy manners, which are difficult for Westerners to stomach, and which threw the Thai waiters into a state of panic, barely eased by the fact that they were used to it. Unlike the Thais, who behave in all circumstances with a finicky, even pernickety propriety, the Chinese eat rapaciously, laughing loudly, their mouths open, spraying bits of food everywhere, spitting on the ground and blowing their noses between their fingers \u2013 they behave quite literally like pigs. To make matters worse, that's an awful lot of pigs.\n\nAfter a few minutes' walking the streets of Patong Beach I realised that everything the civilised world had produced in the way of tourists was gathered here on the two-kilometre stretch of the seafront. Before I had walked thirty metres, I'd encountered Japanese, Italians, Germans, Americans, not to mention a couple of Scandinavians and some rich South Americans. 'We're all the same, we all head for the sun,' as the girl in the travel agency had told me. I behaved like a typical, average tourist: I rented a sun-lounger with a fitted mattress, a parasol; I consumed a number of bottles of Sprite; I went for a dip, in moderation. The waves were gentle. I went back to the hotel at about five o'clock, averagely satisfied with my free day but intent nonetheless on carrying on. _I was attached to a delusive existence_. I still had the hostess bars to come, but before heading to the relevant district, I idled outside the restaurants. In front of Royal Savoy Seafood, I noticed a couple of Americans gazing at a lobster, with exaggerated concentration. 'Two mammals in search of a crustacean', I thought. A waiter came to join them, all smiles, probably praising the freshness of the produce. 'That makes three,' I continued mechnically. The crowd flowed incessantly, single men, families, couples; it all conveyed an impression of innocence.\n\nSometimes, when they've had a bit to drink, the German senior citizens get together in groups and intone slow, infinitely sad songs, much to the amusement of the Thai waiters, who gather round them making appreciative little cries.\n\nFalling in step behind three chaps in their fifties, vigorously trading shouts of 'Ach' and 'Ja', I found myself, all of a sudden, in the street of hostess bars. Young girls in short skirts billed and cooed, competing with each other to try to convince me to go the Blue Nights, the Naughty Girl, the Classroom, the Marilyn, the Venus... In the end I opted for the Naughty Girl. The place was still pretty empty: about ten or so Westerners, each sitting alone at their tables \u2013 young, twenty-five to thirty-year-olds, mostly English and American. On the dance floor, a dozen girls swayed gently to some sort of retro disco beat. Some of them wore white bikinis, others had taken their tops off and were wearing only G-strings. They were all about twenty, they all had golden brown skin, supple, exciting bodies. An elderly German was sitting in front of a Carlsberg at the table on my left: big belly, white beard, glasses, he looked a lot like a retired university professor. He stared at the bodies moving before his eyes, completely hypnotised; he was so still that for a moment I thought he was dead.\n\nSeveral smoke machines started up, the music changed, replaced by something slow and Polynesian. The girls left the stage, to be replaced by a dozen others wearing garlands of flowers around their hips and busts. Slowly, they turned round, the garlands occasionally revealing a breast or the top of the buttocks. The old German still stared at the stage; at one point he took off his glasses to wipe them, his eyes were moist. He was in paradise.\n\nStrictly speaking, the girls didn't solicit; but you could invite one of them to have a drink with you, talk a little and in due course pay the establishment a bar fee of five hundred baht to take the girl to your hotel, after negotiating a price. For a whole night, I think the price was about four or five thousand baht \u2013 about a month's salary for an unskilled Thai worker; but Phuket is an expensive resort. The elderly German signalled discreetly to one of the girls who was waiting, still wearing a white G-string, to go back on stage. She came over immediately, settled herself casually between his thighs. Her curved, youthful breasts were at the same level as the old man's face; he was roaring with pleasure. I heard her call him 'Papa'. I paid for my Tequila sour and left, a little embarrassed; I had the feeling I'd witnessed one of the old man's last pleasures. It was too moving, too intimate.\n\nJust next to the bar, I found an open-air restaurant where I sat and had a plate of crabmeat and rice. At almost every table sat a couple, always a western man and a Thai woman. Most of the guys looked Californian, the way you imagine Californians to look, at any rate they were all wearing flip-flops. Actually, they could have been Australian \u2013 it's easy to get the two mixed up; whatever they were, they looked healthy, sporty, well-fed. They were the future. It was at that point, seeing all these young, immaculate Anglo-Saxons with their brilliant futures, that I realised just how important sex tourism would be to the future of the world. At the next table, two Thai women of about thirty, shapely, generously proportioned, were chatting excitedly. Two shaven-headed English men, who looked like post-modern convicts, sat opposite; they barely sipped their beers and said nothing. A little further along, a couple of German dykes in dungarees, rather chubby, with short red hair, had treated themselves to the company of a delightful adolescent girl with long black hair and an innocent face, wearing a colourful sarong. There were also a couple of lone Arabs of indeterminate nationality, their heads wrapped in the sort of tea-towel you see Yasser Arafat wearing when he's on television. In short, all the rich or moderately wealthy world was here, all answering 'present!' to the gentle and constant roll-call of Asian pussy. The strangest thing was that you had the impression, the minute you set eyes on each couple, of knowing whether things would work out or not. More often than not, the girls were bored, wore sulky or resigned expressions, glancing around at the other tables. But some of them, their eyes turned to their companions in an attitude of loving expectancy, hung on their partners' words, responded eagerly; in such cases you could imagine things would go further, that a friendship might develop, or perhaps a more lasting relationship: I knew that marriages were not rare, especially with Germans.\n\nMyself, I didn't much feel like striking up a conversation with some girl in a bar; in general these conversations, overly focused on the character and price of sexual services to come, were a disappointment. I preferred massage parlours, where you begin with sex, sometimes an intimacy develops, sometimes not. In certain cases you think about extending your stay at the hotel and that's when you find out that the girl isn't always keen: sometimes she's divorced, she has children who need to be looked after; it's sad, but it's good. As I finished my rice, I sketched out the plot of a pornographic adventure film called _The Massage Room_. Sirien, a young girl from northern Thailand, falls hopelessly in love with Bob, an American student who winds up in the massage parlour by accident, dragged there by his mates after a boozy evening. Bob doesn't touch her, he's happy just to look at her with his lovely, pale-blue eyes and tell her about his country \u2013 North Carolina, or somewhere like that. They continue to see each other regularly, whenever Sirien isn't working, but, sadly, Bob must return to finish his final year at Yale. Ellipsis. Sirien waits expectantly while continuing to satisfy the needs of her numerous clients. Though pure at heart, she avidly wanks and sucks paunchy, moustached Frenchmen (supporting role for Gerard Jugnot), fat, bald Germans (supporting role for some German actor). Finally, Bob returns and tries to free her from her hell; but the Chinese mafia don't see things in quite the same light. Bob persuades the American ambassador and the president of some humanitarian organisation opposed to the exploitation of young girls to intervene (supporting role for Jane Fonda). What with the Chinese mafia (mention the Triads) and the collusion of Thai generals (political angle, appeal to democratic values), there would be a lot of fight scenes and chase sequences through the streets of Bangkok. At the end of the day, Bob carries her off. In the penultimate scene, Sirien gives an honest account of the extent of her sexual experience. All the cocks she has sucked as a humble massage parlour employee, she has sucked in the anticipation, in the hope of sucking Bob's cock, into which all the others were subsumed \u2013 well, I'd have to work on the dialogue. Cross-fade between the two rivers (the Chao Phraya, the Delaware). Closing credits. For the European market, I already had a trailer in mind, sort of: 'If you liked _The Music Room_ , you'll love _The Massage Room_ '. It was all a bit vague, but first I would need backers. After I paid, I got up and walked a hundred and fifty metres, dodging a variety of propositions, and found myself in front of the Pussy Paradise. I pushed the door and went in. Three metres in front of me I spotted Robert and Lionel, sitting with a couple of Irish coffees. At the back, behind a glass screen, about fifty girls sat on terraced benches, each wearing a numbered tag. A waiter quickly approached me. Turning his head, Lionel saw me and looked shamefaced. Robert also turned and with a slow wave motioned to me to join them. Lionel was biting his lip, he didn't know what to do with himself. The waiter took my order. 'I'm right wing...' Robert said, for no apparent reason; 'but watch your step...' He wagged his index finger as though warning me. Since the start of the trip, I'd noticed, he had assumed I was a leftie, and had been waiting for a favourable opportunity to have a conversation with me; I had no intention of playing that little game. I lit a cigarette; he looked me up and down gravely. 'Happiness is a delicate thing,' he announced in a sen-tentious voice, 'It is difficult to find within ourselves, and impossible to find elsewhere.' After a few seconds, he added confidently, 'Chamfort'. Lionel looked at him admiringly; he seemed to be completely under his spell. I thought his quotation was debatable: if you reversed the words 'difficult' and 'impossible' we'd probably have been a little closer to the truth; but I had no desire to pursue the conversation, it seemed to me imperative for us to get back to a normal tourist situation. On top of everything, I was starting to feel a surge of desire for number 47, a slim little Thai girl, a bit skinny maybe, but with full lips and a gentle appearance; she was wearing a red miniskirt and black stockings. Aware that my attention had wandered, Robert turned to Lionel. 'I believe in truth,' he said in a low voice, 'I believe in truth and in the importance of proof.' Listening distractedly, I was surprised to discover that he had a degree in maths and that in his youth he had written a number of promising papers on Lie groups. I reacted excitedly to this news: there were, in other words, certain areas of human intelligence in which he had been the first clearly to see the truth, to discover absolute, demonstrable certainties. 'Yes...' he agreed almost apologetically, 'Of course, it was all proved again in more general terms.' After that he had been a teacher, mostly teaching candidates for the Grandes \u00c9coles; he had derived little pleasure from spending his mature years coaching a bunch of young arseholes obsessed with getting into the \u00c9cole Polytechnique, or the \u00c9cole Centrale \u2013 and even then, only the most talented of them. 'In any case,' he added, 'I didn't have the makings of a creative mathematician. It is a gift given to very few.' Towards the end of the seventies, he sat on a government committee on the reform of maths teaching \u2013 a load of bullshit, by his own admission. Now, at fifty-three, having taken retirement three years earlier, he devoted himself to sex tourism. He had been married three times. 'I'm racist...' he said cheerfully. 'I've become racist... One of the first effects of travel,' he added, 'is to reinforce or create racial prejudice; because how do you imagine other people before you meet them? You imagine they are just like you, it goes without saying; it's only little by little that you realise that the reality is somewhat different. When he can, a Westerner _works_ ; he often finds his work frustrating or boring, but he pretends to find it interesting: this much is obvious. At the age of fifty, weary of teaching, of maths, of everything, I decided to see the world. I had just been divorced for the third time; as far as sex was concerned, I wasn't expecting much. My first trip was to Thailand; immediately after that I left for Madagascar. Since then, I haven't fucked a white woman, I've never even felt the desire to do so. Believe me,' he added placing a firm hand on Lionel's forearm, 'you won't find a white woman with a soft, submissive, supple, muscular pussy any more; that's all gone now.' Number 47 noticed that I was staring at her; she smiled at me and crossed her legs high up, revealing a pair of red suspenders. Robert continued to expound his theory. 'At the time when the white man thought himself superior, racism wasn't dangerous. For colonials, missionaries and lay teachers in the nineteenth century, the Negro was a big animal, none too clever, a sort of slightly more evolved monkey. At worst, they considered him a useful beast of burden, capable of performing complex tasks; at best a frustrated soul, coarse, but, through education, capable of elevating himself to God \u2013 or at least western reason. In both cases, they saw in him a 'lesser brother', and one does not feel hatred for an inferior \u2013 at most a sort of cordial contempt. This benevolent, almost humanist racism has completely vanished. The moment the white man began to consider blacks as _equals_ , it was obvious that sooner or later they would come to consider them to be _superior_. The notion of equality has no basis in human society,' he went on, lifting his index finger again. For a moment, I thought he was going to cite sources \u2013 La Rochefoucauld or I don't know whom \u2013 but in the end, he didn't. Lionel furrowed his brow. 'Once white men believed themselves to be inferior,' Robert went on, anxious that he be clearly understood, 'the stage was set for a different type of racism, based on masochism: historically, it is in circumstances like these that violence, inter-racial wars and massacres break out. For example, all anti-Semites agree that the Jews have _a certain_ superiority: if you read anti-Semitic literature, you're stuck by the fact that the Jew is considered to be more intelligent, more cunning, that he is credited with having singular financial talents \u2013 and, moreover, greater communal solidarity. Result: six million dead.'\n\nI glanced at number 47 again: anticipation is exciting, something you'd like to prolong; but there's always the risk that the girl will go off with another customer. I signalled discreetly to the waiter. 'I am not a Jew!' exclaimed Robert, thinking I was about to object. I could, in fact, have made several objections: we were in Thailand, after all, and the yellow races have never been considered by the White man to be 'lesser brothers', but to be civilised peoples, members of different, complex, possibly dangerous civilisations; I could also have pointed out that we were here to fuck and that these discussions were wasting time; in fact, that was my primary objection. The waiter came over to our table; with a swift gesture, Robert motioned to him to bring another round of drinks. 'I need a girl,' I said in English, my voice shrill, 'girl forty-seven'. He leaned towards me, his face anxious, quizzical; a Chinese group had just sat down at the next table, they were making an appalling racket. 'The girl number four seven!' I shouted, enunciating each syllable. This time he understood, smiled broadly and went to the microphone where he uttered a few words. The girl got up, stepped down and walked towards a side door smoothing her hair. 'Racism,' Robert went on, giving me a quick glance, 'seems to be characterised firstly by an accumulation of hostility, a more aggressive sense of competition between males of different races; but the corollary is an increased desire for the females of the other race. What is really at stake in racial struggles,' Robert said simply, 'is neither economic nor cultural, it is brutal and biological: it is competition for the cunts of young women.' I sensed that it wouldn't be long before he moved on to Darwinism; at that moment, the waiter came back to our table accompanied by number 47. Robert looked up at her, considered for a moment. 'Good choice...' he concluded soberly, 'she has something of the slut about her.' The girl smiled shyly. I slipped a hand under her skirt and stroked her arse as though to protect her. She snuggled against me.\n\n'It's true that round my way, it's not the whites that make the law any more...' Lionel said, for no apparent reason.\n\n'Exactly,' agreed Robert forcefully. 'You're scared, and you're right to be scared. I predict an increase in racial violence in Europe in years to come; it will all end in civil war,' he said, frothing at the mouth a little; 'It will all be settled with Kalashnikovs.' He gulped back his cocktail; Lionel began to look at him a little nervously. 'I don't give a fuck about any of it anymore!' Robert added, slamming his glass down on the table. 'I'm a Westerner, but I can live wherever I want, and for the time being, I'm still the one with the money. I've been in Senegal, Kenya, Tanzania, the Ivory Coast. It's true the girls are less expert than Thai girls, they're less gentle, but they're nicely curved and they have a sweet-smelling snatch.' He was obviously lost in his memories for a moment as he suddenly fell silent. 'What is your name?' I took the opportunity to ask number 47. 'I am Sin,' she said. The Chinese at the next table had made their choices, they headed upstairs, chuckling and laughing; relative silence was restored. 'They get on all fours, the little nigger girls, show you their pussies and their arses,' Robert continued thoughtfully; 'and inside, their pussies are completely pink...' he murmured. I also got to my feet. Lionel shot me a grateful look; he was visibly happy that I was the first to leave with a girl, it made things less embarrassing for him. I nodded to Robert to take my leave. His dour face, fixed in a bitter rictus, scanned the room \u2013 and beyond, the human race \u2013 without a hint of affability. He had made his point, at least he had had the opportunity; I sensed that I was going to forget him pretty quickly. All of a sudden he seemed to me to be finished, a broken man; I had the impression that he didn't even want to make love to these girls any more. Life can be seen as a process of gradually coming to a standstill, a process evident in the French bulldog \u2013 so frisky in its youth, so listless in middle age. In Robert, the process was already well advanced: he possibly still got erections, but even that wasn't certain. It's easy to play the smart aleck, to give the impression that you've understood something about life; the fact remains that life comes to an end. My fate was similar to his, we had shared the same defeat; but still I felt no active sense of solidarity. In the absence of love, nothing can be sanctified. On the inside of the eyelids patches of light merge; there are visions, there are dreams. None of this now concerns man, who waits for night; night comes. I paid the waiter two thousand baht and he escorted me to the double doors leading upstairs. Sin held my hand; she would, for an hour or two, try to make me happy.\n\nObviously, it's rare to come across a girl in a massage parlour who wants to make love. As soon as we were in the room, Sin went down on her knees in front of me, took down my trousers and my underpants and took my penis between her lips. I immediately started to get hard. She brought her lips closer, slowly pushed back the foreskin with short thrusts of her tongue. I closed my eyes, I felt a dizzying rush, I thought I was going to come in her mouth. She stopped suddenly, undressed, smiling as she did so, folded her clothes and placed them on a chair. 'Massage later...' she said, lying on the bed; then she parted her thighs. I was already inside her, and I was thrusting forcefully in and out when I realised I'd forgotten to put on a condom. According to reports by _M\u00e9decins du monde_ , one third of all prostitutes in Thailand are HIV positive. Even so, I can't say that I felt a shudder of fear; I felt slightly annoyed, no more. Clearly those ad campaigns warning us about AIDS had been a complete failure. I went a bit limp, even so. 'Something wrong?' she was worried, she propped herself up on her elbows. 'Maybe... a condom,' I said, embarrassed. 'No problem, no condom... I'm OK!' she told me cheerfully. She took my balls in the palm of one hand, slipping the other palm on to my prick. I lay down on my back, surrendering myself to the caress. The movement of her palm quickened, I felt the blood rush back to my penis. Anyway, they probably had medical check-ups or something. As soon as I was hard, she climbed on top and went straight down on me. I laced my hands behind her back; I felt invulnerable. She started to move her pelvis slowly, her pleasure mounted, I parted my thighs to penetrate her more deeply. The pleasure was intense, almost intoxicating, I breathed very slowly to hold myself back, I felt reconciled. She lay down on top of me, rubbing her pubis hard against mine; I moved my hands to stroke the nape of her neck. At the moment of orgasm, she became still, gave a long moan and then collapsed on my chest. I was still inside her, I could feel her pussy contracting. She had a second orgasm, a very powerful contraction from deep inside her. Involuntarily, I hugged her to me and ejaculated with a roar. She stayed motionless, her head on my chest, for about ten minutes; then she got up and suggested I take a shower. She dried me very delicately, patting me with the towel as you would a baby. I sat down on the sofa and offered her a cigarette. 'We have time...' she said 'we have a little time...'. I learned that she was thirty-two. She didn't enjoy her work, but her husband had abandoned her, leaving her with two children. 'Bad man,' she said, 'Thai men, bad men.' I asked her if she had any friends among the other girls. Not really, she told me; most of the girls were young and brainless, they spent everything they earned on clothes and perfume. She was not like that, she was serious, she put her money in the bank. In a couple of years she would be able to give this up and go back to live in her village; her parents were old now, they needed help.\n\nAs I was leaving, I gave her a two thousand baht tip; it was ridiculous, it was far too much. She took the money incredulously, and bowed to me several times, her hands together over her chest. 'You good man,' she said. She slipped on her mini skirt and her stockings; she had two hours left before they closed. She accompanied me to the door, bowed again, her hands together. 'Take care,' she said again; 'be happy.' I walked back out into the street, a little pensive. The following morning we were due to leave at eight o'clock for the last leg of the trip. I wondered how Val\u00e9rie had spent her free day.\n\n### 11\n\n'I BOUGHT SOME presents for my family,' she said. 'I found some beautiful shells.' The boat sped through the turquoise waters, between chalk crags covered with thick jungle; it was exactly how I imagined the scenery of _Treasure Island_. 'When all's said and done, nature is, well...' I said. Val\u00e9rie turned an attentive face towards me; she had tied her hair up in a chignon, but a couple of stray curls fluttered in the wind on either side of her face. 'In the end, nature sometimes...' I went on, discouraged. There should be lessons in _conversation_ , the way there are ballroom dancing lessons; I'd probably spent too much time doing accounting, I had lost the knack. 'You realise that it's December 31st...' she observed, unruffled. I looked around on all sides at the endless azure, the turquoise ocean; no, I really hadn't realised. Human beings must have had a lot of courage to colonise cold regions.\n\nS\u00f4n stood up to address the group: 'We now approaching Ko Phi Phi. I tell you before, here cannot go. You put swimsuit on, go now? Walk, not deep, walk. Walk in water. Not take luggage, luggage later.' The pilot rounded a headland and cut the engine, the boat continued to drift into a small cove which carved a curve into the middle of cliffs shrouded in jungle. The clear green water broke on a beach of white sand so perfect it seemed unreal. In the middle of the jungle, before the first slopes, you could make out wooden huts built on stilts, their roofs thatched with palm leaves. The group fell silent for a moment. 'Earthly paradise...' said Sylvie softly, choked with genuine emotion. It was hardly an exaggeration. That said, she was no Eve. I was no Adam either.\n\nOne by one the group members got up, stepped over the edge of the boat. I helped Josette down to her waiting husband. She had hitched her skirt up to her waist and was having trouble getting over the side, but she was thrilled, she was virtually wetting herself with excitement. I turned round; the Thai boatman waited, leaning on his oar, for all the passengers to disembark. Val\u00e9rie sat with her hands crossed in her lap; she shot me a sidelong glance and smiled in embarrassment. 'I forgot to put on my swimsuit...' she said at last. I lifted my hands slowly in a gesture of helplessness. 'I can go...' I said stupidly. She bit her lips in irritation, got up, took off her trousers in a single movement. She was wearing lace panties, very sheer, not at all in the spirit of the trip. Her pubic hair peeked out at the sides, it was quite thick, very black. I didn't turn away, that would have been stupid, but nor was my gaze insistent. I got out of the boat on the left-hand side, offered her my arm to help her down; she jumped down from the boat. We were in up to our waists in the water.\n\nBefore going to the beach, Val\u00e9rie looked again at the shell necklaces she was taking back for her nieces. Immediately after graduation, her brother had got a job as a research engineer with Elf. After a few months of on-site training, he had left for Venezuela \u2013 his first assignment. A year later, he married a local girl. Val\u00e9rie had the impression he hadn't had much previous sexual experience; at least he had never brought girls home. That's often the way with boys who study engineering; they haven't got time to go out, to have girlfriends. They spend their free time on trivial hobbies, complex role-playing games or chess on the internet. They get their degrees, find themselves their first jobs and discover everything at once: money, professional responsibility, sex; if they are posted to a tropical country, it's rare for them to resist. Bertrand had married a very dark mixed-race girl with a superb body; several times when they were on holiday at her parents, on the beach at Saint-Quay-Portrieux, Val\u00e9rie had felt a violent surge of desire for her sister-in-law. She found it difficult to imagine her brother making love. Still, they had two children now and seemed to be a happy couple. It wasn't difficult to buy presents for Juana, she adored jewellery, and pale stones stood out beautifully against her dark skin. On the other hand, she hadn't found anything for Bertrand. When men have no vices, she thought, it's very difficult to guess what might make them happy.\n\nI was leafing through a copy of _Phuket Weekly_ I'd found in the hotel lobby when I saw Val\u00e9rie walking along the beach. Further along, a group of Germans were swimming in the nude. She hesitated for a moment, then walked towards me. The sun was dazzling; it was about midday. One way or another, I would have to learn to play the game. Babette and L\u00e9a walked past wearing shoulder-bags but otherwise they too were completely naked. I registered this information without reacting. By contrast, Val\u00e9rie's eyes followed them for a while with shameless curiosity. They settled themselves not far from the Germans. 'I think I'll go for a swim...' I said. 'I'll go in later,' she replied. I entered the water effortlessly. It was warm, translucent, deliciously calm; tiny silver fish swam close to the surface. The slope was very gentle, I could still touch bottom a hundred metres from shore. I slipped my cock out of my trunks, closed my eyes and visualised Val\u00e9rie's vagina as I had seen it that morning, half exposed through her lace panties. I was hard, that in itself was something; it could be considered motive enough in itself. Besides, you have to live, you have to relate to other people; I was generally too uptight, and had been so for far too long. Perhaps I should have taken up some hobby in the evenings \u2013 badminton, choral singing or something. Even so, the only women I was still able to remember were the ones I'd fucked. That's not nothing either; we build up memories so that we will feel less alone at the moment of death. I shouldn't think like that. 'Think positive', I murmured in English to myself, panicked, 'think different'. I made my way back to the beach, stopping every ten strokes, breathing deeply to try and calm myself. The first thing I noticed as I stepped on to the sand was that Val\u00e9rie had taken off her bikini top. At that moment, she was lying on her stomach, but she would turn over, it was as inexorable as the movement of the planets. Where was I exactly? I sat down on my towel, hunched over slightly. 'Think different', I reminded myself. I had seen breasts before, I had stroked them, licked them; nonetheless, I found myself in a state of shock. I was sure that she had magnificent breasts; but it was worse than I had imagined. I couldn't tear my eyes from the nipples, the areolas; it was impossible for her not to notice me staring \u2013 even so, she said nothing for what seemed to me several long seconds. What exactly does go on in women's heads? They adapt to the rules of the game so easily. Sometimes, when they look at themselves naked in a full-length mirror, you can see a sort of realism in their eyes, a dispassionate assessment of their personal powers of seduction which no man could ever achieve. I was the first to lower my eyes.\n\nAfter that, an indeterminate period of time elapsed; the sun was still directly overhead, the light extremely bright. I was staring at the white, powdery sand. 'Michel...' she said softly. I looked up quickly as though I'd been struck. Her exceptionally brown eyes stared deeply into mine. 'What have Thai girls got that Western women don't?' she asked plainly. Yet again I was unable to hold her gaze. Her chest rose and fell to the rhythm of her breathing; I thought I saw her nipples harden. Right there, right then, I wanted to reply: 'Nothing'. Then I had an idea; not a very good idea.\n\n'There's an article in English about it in here, sort of an advertorial'; I handed her the copy of _Phuket Weekly_. 'Find your longlife companion... Well-educated Thai ladies, that one?' 'Yes, a bit further on there's an interview.' Cham Sawanasee, smiling, black suit and dark tie, answered the 'Ten questions you could ask' on the working of the Heart to Heart agency which he managed.\n\n'There seems to be,' noted Mr Sawanasee, 'a near-perfect match between the Western men, who are unappreciated and get no respect in their own countries, and the Thai women, who would be happy to find someone who simply does his job and hopes to come home to a pleasant family life after work. Most Western women do not want such a boring husband.\n\n'One easy way to see this,' he went on, 'is to look at any publication containing \"personal\" ads. The Western woman wants someone who looks a certain way, and who has certain \"social skills\", such as dancing and clever conversation, someone who is interesting and exciting and seductive. Now go to my catalogue, and look at what the girls say they want. It's all pretty simple, really. Over and over they state that they are happy to settle down FOREVER with a man who is willing to hold down a steady job and be a loving and understanding HUSBAND and FATHER. That will get you exactly nowhere with an American girl!\n\n'As Western women do not appreciate men,' he concluded, not without a certain cheek, 'as they do not value traditional family life, marriage is not the right thing for them. I'm helping modern Western women to avoid what they despise.'\n\n'What he's saying makes sense...' Val\u00e9rie remarked sadly, 'There's a market there, all right...' She put down the magazine, still thoughtful. At that moment Robert passed in front of us; he was walking along the beach, hands clasped behind his back, looking serious. Val\u00e9rie abruptly turned to look the other way.\n\n'I don't like that guy...' she whispered angrily.\n\n'He's not stupid...' I made a rather noncommittal gesture.\n\n'He's not stupid, but I don't like him. He goes out of his way to shock people, to make himself unpleasant; I don't like that. At least you try to fit in.'\n\n'Really?' I shot her a surprised look.\n\n'Yes. It's obvious you don't find it easy, you're not cut out for this kind of holiday; but at least you make an effort. Deep down, I think you're rather a nice boy.'\n\nAt that moment I could have, I should have, taken her in my arms, stroked her breasts, kissed her lips; stupidly, I didn't. The afternoon dragged on, the sun moved over the palms; we said nothing of any significance.\n\nFor dinner on New Year's Eve, Val\u00e9rie wore a long dress of sinuous, slightly transparent green material, the top of which was a bustier which showed off her breasts. After dessert, there was a band out on the terrace. A weird old singer did slow-rock cover versions of Bob Dylan songs in a nasal whine. Babette and L\u00e9a had apparently joined the German group; I heard shouts coming from their end. Josette and Ren\u00e9 danced together in a tender embrace like the nice little proles they were. The night was hot; emerald moths clustered around the multi-coloured paper lanterns which hung from the balustrade. I felt suffocated; I drank whisky after whisky.\n\n'What that guy was saying, the interview in the magazine...'\n\n'Yes?...' Val\u00e9rie looked up at me; we were sitting side by side on a rattan bench. Under her bustier, her breasts were more rounded, as though they were being offered in their own little shells. She had put on makeup; her long hair was free and floated about her shoulders.\n\n'It's mostly true of American women, I think. For Europeans, it's less clear cut.'\n\nShe pulled a face, clearly unconvinced, and said nothing. Obviously, I should have just asked her to dance. I drank another whisky, leaned back on the bench and took a deep breath.\n\nWhen I woke, the room was almost empty. The singer was still humming in Thai, half-heartedly accompanied by the drummer; no one was listening any more. The Germans had disappeared, but Babette and L\u00e9a were deep in conversation with two Italians who had appeared from who knows where. Val\u00e9rie had left. It was three in the morning local time; 2001 had just begun. In Paris, it would officially begin in three hours' time; it was exactly midnight in Teheran, five in the morning in Tokyo. Humanity in all its different forms was entering the third millennium; for my part, I had pretty much blown my entrance.\n\n### 12\n\nI WENT BACK to my cabin, mortally ashamed; laughter was coming from the garden. I came across a small grey toad, sitting stock still in the middle of the sandy path. It did not hop away, it had no defensive reflex. Sooner or later someone would accidentally step on it; its spinal column would snap, its pulped flesh would seep into the sand. The walker would feel something soft beneath his foot, utter a blunt curse, wipe off his shoes, rubbing them on the ground. I pushed the toad forward with my foot: unhurried, he made his way to the edge of the path. I pushed him again: he regained the relative safety of the lawn; I had probably prolonged his survival by a few hours. I felt I was barely better off than he was: I hadn't grown up sheltered by the cocoon of a family, nor by anything that might have concerned itself with my fate, supported me in times of misery, enthused about my adventures and my successes. Nor had I established a unit of that sort: I was single, childless; no one would have thought to come and seek support on my shoulder. Like an animal, I had lived and I would die alone. For several minutes I wallowed in gratuitous self-pity.\n\nFrom another point of view, I was a compact, resilient object, of a larger size than the average animal; my life expectancy was comparable to that of an elephant, or of a crow; I was much more difficult to destroy than a small batrachian.\n\nFor the two days that followed, I remained holed up in my cabin. From time to time I went out, hugging the walls, as far as the mini-market to buy pistachios and some bottles of Mekong. I couldn't face running into Val\u00e9rie again at the breakfast buffet or on the beach. There are some things that one can do, others that seem too difficult. Gradually, everything becomes too difficult: that's what life comes down to.\n\nOn the afternoon of January 2nd, I found a Nouvelles Fronti\u00e8res customer satisfaction questionnaire slipped under my door. I filled it out scrupulously, generally ticking the boxes marked 'Good'. It was true, in some sense, that everything had been good. My holiday had 'gone smoothly'. The tour had been 'cool' but with a hint of adventure; it lived up to the description in the brochure. In the 'personal comments' section, I wrote the following quatrain:\n\n_Shortly after waking, I feel myself transported_\n\n_To a different universe, its contours ruled and picked_\n\n_I know about this life, its details are all sorted_\n\n_It's very like a questionnaire, with boxes to be ticked_\n\nOn the morning of January 3rd, I packed my suitcase. When she saw me on the boat, Val\u00e9rie suppressed an exclamation; I looked away. S\u00f4n said her goodbyes at Phuket airport; we were early, the plane would not leave for three hours. After the check-in formalities, I wandered around the shopping arcade. Even though the departures hall was completely roofed-over, the shops were built in the form of huts, with teak uprights and roofs thatched with palm leaves. The choice of products ranged from international standards (scarves by Herm\u00e8s, perfumes by Yves Saint-Laurent, bags by Vuitton) to local products (shells, ornaments, Thai silk ties); every item had a barcode. All in all, airport shops still form part of the national culture; but a part which is safe, attenuated, one which fully complies with international standards of commerce. For the traveller at the end of his journey it is a halfway house, less interesting and less frightening than the rest of the country. I had an inkling that, more and more, the whole world would come to resemble an airport.\n\nPassing the Coral Emporium, I suddenly had the urge to buy a present for Marie-Jeanne; after all, she was all I had in the world. A necklace, a brooch? I was rummaging in a tray when I noticed Val\u00e9rie a couple of metres away from me.\n\n'I'm trying to choose a necklace...' I said hesitantly.\n\n'For a brunette or a blonde?' There was a trace of bitterness in her voice.\n\n'Blonde, blue eyes.'\n\n'In that case, you'd be better off with a pale coral.'\n\nI handed my boarding card to the girl at the counter. As I was paying, I said to Val\u00e9rie in a rather pitiful tone of voice: 'It's for a colleague at work...' She gave me a strange look as though she were in two minds whether to slap me or burst out laughing; but she walked a little way with me to the shop entrance. Most of the group were sitting on benches in the hall; apparently they had done their shopping. I stopped, took a long breath, turned to Val\u00e9rie.\n\n'We could see each other in Paris...' I said finally.\n\n'You think so?' she said scathingly.\n\nI didn't reply, I simply looked at her again. At one point I intended to say, 'It would be a pity...' but I'm not sure whether I actually uttered the words.\n\nVal\u00e9rie looked around, saw Babette and L\u00e9a on the nearest bench and turned away in irritation. Then she took a notepad out of her bag, tore off a page and quickly wrote something on it. As she gave me the piece of paper, she started to say something, gave up, and turned and rejoined the group. I glanced at the piece of paper before pocketing it: it was a mobile phone number.\n\n## Part Two\n\n## _Competitive Advantage_\n\n### 1\n\nTHE PLANE LANDED at Roissy at eleven o'clock; I was one of the first to collect my luggage. By half-past twelve I was home. It was Saturday; I could go out and do some shopping, buy some ornaments for the house, etc. An icy wind swept down the Rue Mouffetard and nothing really seemed worth the effort. Animal rights militants were selling yellow stickers. After Christmas, there's always a slight fall-off in domestic food consumption. I bought a roast chicken, two bottles of Graves and the latest copy of _Hot Video_. It was hardly an ambitious selection for my weekend, but it was all I deserved. I devoured half the chicken, the skin was charred and greasy, slightly revolting. Shortly after three o'clock I phoned Val\u00e9rie. She answered on the second ring. Yes, she was free this evening; for dinner, yes. I could collect her at eight; she lived on the Avenue Reille, near the Parc Montsouris.\n\nShe answered the door wearing a pair of white tracksuit bottoms and a short tee-shirt. 'I'm not ready...' she said, pulling her hair back. The movement raised her breasts; she wasn't wearing a bra. I put my hands on her waist, leaned my face closer to hers. She parted her lips, immediately slid her tongue into my mouth. A wave of violent excitement shuddered through me; I almost fainted, immediately got a hard on. Without moving her pubis from mine, she pushed the front door, which closed with a dull thud.\n\nThe room, lit by a single lamp, seemed huge. Val\u00e9rie took me by the waist and, feeling her way, led me to her bedroom. By the bed, she kissed me again. I lifted her tee-shirt to stroke her breasts; she whispered something I didn't catch. I knelt in front of her, slipping down her tracksuit bottoms and her panties, then pressed my face to her sex. The slit was damp, the labia parted, she smelled good. She let out a moan and fell back on the bed. I undressed quickly and entered her. My penis was on fire, spasms of intense pleasure coursed though it. 'Val\u00e9rie,' I said, 'I'm not going to be able to hold out for long, I'm too excited.' She pulled me to her and whispered in my ear: 'Come...' At that moment, I felt the walls of her pussy close on my penis. I felt as though I was disappearing into space, only my penis was alive, a wave of extraordinarily intense pleasure coursing through it. I ejaculated lengthily several times; right at the end, I realised I was screaming. I could have died for such a moment.\n\nBlue and yellow fish were swimming around me. I was standing in the water, balancing a few metres beneath the sunlit surface. Val\u00e9rie was a little way off; she too was standing, a coral reef in front of her, she had her back to me. We were both naked. I knew that this weightlessness was due to a change in the density of the ocean, but I was surprised to discover that I could breathe. In a few short strokes I was beside her. The reef was stippled with star-shaped organisms of phosphorescent silver. I placed a hand on her breasts, the other on her lower abdomen. She arched herself, her buttocks brushed against my penis.\n\nI awoke precisely in that position; it was still dark. Gently, I parted Val\u00e9rie's thighs so I could penetrate her. At the same time, I wet my fingers so I could rub her clitoris. I realised she was awake when she began to moan. She pushed herself on to her knees on the bed. I started to push into her harder and harder \u2013 I could tell she was about to come, her breaths came faster and faster. At the moment of orgasm she jerked and let out a heart-rending cry; then she was still, as though exhausted. I withdrew and lay beside her. She relaxed and wrapped herself around me; we were bathed in sweat. 'It's nice to be woken by pleasure...' she said, putting a hand on my chest.\n\nWhen I woke again, it was daylight; I was alone in the bed. I got up and crossed the room. The other room was as vast as I had imagined, with a high ceiling. Above the sofa, bookshelves ran along a mezzanine. Val\u00e9rie had gone out; on the kitchen table she had left some bread, cheese, butter and jam. I poured myself a coffee and went back to lie down. She returned ten minutes later with croissants and _pains au chocolat_ and carried a tray into the bedroom. 'It's really cold out...' she said, getting undressed. I thought about Thailand.\n\n'Val\u00e9rie,' I said hesitantly, 'what do you see in me? I'm not particularly handsome, I'm not funny; I find it difficult to understand why anyone would find me attractive.' She looked at me and said nothing; she was almost naked, she had kept only her panties on. 'It's a serious question,' I insisted. 'Here I am, some washed-up guy, not very sociable, more or less resigned to his boring life. And you come to me, you're friendly, you're affectionate and you give me so much pleasure. I don't understand. It seems to me you're looking for something in me that isn't there. You're bound to be disappointed.' She smiled, I got the impression she was about to say something; then she put her hand on my balls, brought her face towards me. Immediately I was hard again. She wound a lock of hair around the base of my penis, then started to jerk me off with her fingertips. 'I don't know...' she said, without stopping what she was doing. 'It's nice that you're unsure of yourself. I wanted you so badly when we were on the trip. It was awful, I thought about it every day.' She pressed harder against my balls, enveloping them in the palm of her hand. With her other hand she took some raspberry jam and spread it on my penis; then she began conscientiously to lick it off with wide sweeps of her tongue. The pleasure was becoming more and more intense, I parted my legs in a desperate effort to hold myself back. As though it was a game, she started to jerk me off more quickly, pressing my cock to her mouth. When her tongue touched the tip of the glans, I ejaculated violently into her half-open mouth. She swallowed with a little moan, then wrapped her lips around the head of my penis to get the last drops. I was flooded with unbelievable serenity, like a wave coursing through each of my veins. She took her mouth away and lay down beside me, coiling herself around me.\n\n'I almost knocked on your bedroom door that night, New Year's Eve, but in the end I didn't have the nerve. By then, I was convinced that nothing would happen between us; the worst thing was that I couldn't even bring myself to hate you for it. On package tours people talk to each other a lot, but it's a forced camaraderie; they know perfectly well they'll never see each other again. It's very rare for them to have a sexual relationship.'\n\n'You think so?'\n\n'I know so; there have been studies on the subject. It's even true of 18\u201330 holiday clubs. It's a big problem for them, because that's their whole selling point. Numbers have been falling consistently for ten years now, even though prices are dropping. The only possible explanation is that it's become more or less impossible to have a sexual relationship on holiday. The only destinations making any money are the ones with a large homosexual clientele like Corfu or Ibiza.'\n\n'You're very up on all this...' I said, surprised.\n\n'Of course, I work in the tourist industry.' She smiled. 'That's another thing about package tours, people don't talk about their professional lives much. It's a sort of recreational parenthesis, completely focused on what the organisers call the \"pleasure of discovery\". Tacitly, everyone agrees not to talk about serious subjects like work and sex.'\n\n'Where do you work?'\n\n'Nouvelles Fronti\u00e8res.'\n\n'So you were there in a professional capacity? To do a report or something like that?'\n\n'No, I really was on holiday. I got a big discount, obviously, but I took it as holiday time. I've been working there for five years and this was the first time I've been away with them.'\n\nAs she made a tomato and mozzarella salad, Val\u00e9rie talked to me about her work. In March 1990, three months before her _bac_ , she started to wonder what she was going to do with her education \u2013 and, more generally, with her life. After much effort, her brother had managed to get a place on a geology course at Nancy; he had just got his degree. His career as a geological engineer would probably take him into the mining sector or the oil rigs; either way, he'd be a long way from France. He was keen on travelling. She too was keen on travelling, well, more or less; eventually she decided to take a BTS diploma in tourism. She didn't really think the intellectual commitment necessary for university was in her nature.\n\nIt was a mistake and one that she quickly realised. The level of her BTS class seemed extremely low to her, she passed her continuous assessment without the slightest effort and could reasonably have expected to get her diploma without even thinking about it. At the same time, she enrolled in a course which would give her the equivalent of a DEUG university diploma in literature and human sciences. Once she had passed her BTS, she began a masters' in sociology. Here too she was quickly disappointed. It was an interesting field, there must have been discoveries to be made; but the methodologies suggested to them and the theories advanced seemed to her to be ridiculously simplistic: the whole thing smacked of ideology, imprecision and amateurism. She quit her course halfway through the academic year without a qualification and found a job as a travel agent at a branch of Kuoni in Rennes. After a couple of weeks, just as she was about to rent a studio flat, she realised: the trap was sprung; from now on she was in the world of work.\n\nShe stayed a year at the Rennes branch of Kuoni, where she proved to be a very good saleswoman. 'It wasn't difficult,' she said, 'All you had to do was get the customers to talk a bit, take an interest in them. It's pretty rare, in fact, people who take an interest in others.' Then the management had offered her a position as assistant tour planner at their head office in Paris. It involved working on concepts for the tours, preparing the itineraries, the excursions, negotiating rates with hoteliers and local contractors. She had proved to be pretty good at this too. Six months later she replied to a Nouvelles Fronti\u00e8res ad offering a similar sort of position. It was at that point that her career really took off. They had put her in a team with Jean-Yves Frochot, a young business graduate who basically knew nothing at all about tourism. He took to her immediately, trusted her and although in theory he was her boss, he gave her a lot of room for initiative.\n\n'The good thing about Jean-Yves is that he was ambitious on my behalf? Every time I've needed to negotiate a pay rise or a promotion, he's negotiated it for me. Now, he's Head of Products worldwide \u2013 he's responsible for supervising the entire range of Nouvelles Fronti\u00e8res tours and I'm still his assistant.'\n\n'You must be pretty well paid.'\n\n'Forty thousand francs a month. Well, it's calculated in euros now. A bit more than six thousand euros.'\n\nI looked at Val\u00e9rie, surprised, 'I wasn't expecting that...' I said.\n\n'That's because you've never seen me in a suit.'\n\n'You have a suit?'\n\n'There's not much point, I do almost all my work by phone. But, if I need to, yes, I can wear a suit. I even have a pair of suspenders. We can try them out some time, if you like.'\n\nIt was then, somewhat incredulously, that I realised that I was going to see Val\u00e9rie again, and that we would probably be happy together. It was so unexpected, this joy, that I wanted to cry; I had to change the subject.\n\n'What's he like, Jean-Yves?'\n\n'Normal. Married, two kids. He works a lot, he takes work home at weekends. I suppose he's a typical young executive, pretty intelligent, pretty ambitious; but he's nice, not at all fucked up. I get along well with him.'\n\n'I don't know why, but I'm glad you're rich. It's not important, really, but it makes me happy.'\n\n'It's true I'm successful, I have a good salary; but I pay 40 per cent tax and my rent is ten thousand francs a month. I'm not so sure I've done all that well: if my results fall off, they wouldn't think twice about firing me; it's happened before. If I had shares, then yes, I really would be rich. In the beginning, Nouvelles Fronti\u00e8res was just a discount flight agency. If they've become the biggest tour operator in France, it's thanks to the concepts and the value-for-money of the tours; to a large extent to our work, Jean-Yves's and mine. In ten years, the value of the company has increased twenty-fold; since Jacques Maillot still holds a 30 per cent share, I can honestly say that he's grown rich because of me.'\n\n'Have you ever met him?'\n\n'Several times; I don't like him. On the face of it, he's a stupid trendy Catholic populist, with his multi-coloured ties and his mopeds; but deep down he's a ruthless, hypocritical bastard. Jean-Yves had a call from a head-hunter before Christmas; he's probably met up with him by now to find out more. I promised I'd call him when I got back.'\n\n'Well call him then, it's important.'\n\n'Yes...' she seemed a bit doubtful, the mention of Jacques Maillot had depressed her. 'My life is important too. Actually, I feel like making love again.'\n\n'I don't know if I'll be able to get it up straight away.'\n\n'Then go down on me. It'll do me good.'\n\nShe got up, took off her panties and settled herself on the sofa. I knelt in front of her, parted her lips and started to lick her clitoris gently. 'Harder...' she murmured. I slipped a finger into her arse, pressed my face to her and kissed the nub, massaging it with my lips. 'Oh, yes...' she said. I increased the force of my kisses. Suddenly, without my expecting it, she came, her whole body shuddering violently.\n\n'Come here to me...' I sat on the sofa. She snuggled against me, laying her head on my thighs. 'When I asked you what Thai women have that we don't, you didn't really answer; you just showed me that interview with the director of the marriage bureau.'\n\n'What he said was true: a lot of men are afraid of modern women, because all they want is a nice little wife to look after the house and take care of the kids. That sort of thing hasn't disappeared really, it's just that in the West it's become impossible to express such a desire; that's why they marry Asian girls.'\n\n'Okay...' She thought for a moment. 'But you're not like that; I can tell that it doesn't bother you at all that I have a high-powered job, a large salary; I don't get the impression that that scares you at all. But still you went off to the massage parlours and you didn't even try to pick me up. That's what I don't understand. What have the girls over there got? Do they really make love better than we do?'\n\nHer voice had changed slightly on these last words; I was rather touched and it took me a minute before I could answer. 'Val\u00e9rie,' I said at last, 'I have never met anyone who makes love as well as you; what I've felt since last night is almost unbelievable.' I said nothing for a moment before adding: 'You can't possibly understand, but you're an exception. It's very rare now to find a woman who feels pleasure and who wants to give pleasure. On the whole, seducing a woman you don't know, fucking her, has become a source of irritations and problems. When you think of all the tedious conversations you have to put up with to get a girl into bed, only to find out that she's a second-rate lover who bores you to fuck with her problems, goes on at you about her exes \u2013 incidentally giving you the impression that you're not exactly up to scratch \u2013 and with whom you absolutely must spend the rest of the night at the very least, it's easy to see why men might prefer to save themselves the trouble by paying a small fee. As soon as they're a bit older or a bit more experienced, men prefer to steer clear of love; they find it easier just to go and find a whore. Actually, not a Western whore, they're not worth the effort, they're real human debris, and in any case, most of the year the men haven't got time, they've got too much work. So, most of them do nothing; and some of them, from time to time, treat themselves to a little sex tourism. And that's the best possible scenario: at least there's still a little human contact in going to visit a whore. There're also all those guys who find it easier just to jerk off on the internet or watching porn films. As soon as your cock has shot its little load, you're perfectly happy.'\n\n'I see...' she said after a long silence. 'I see what you're saying. And you don't think that men or women are capable of changing?'\n\n'I don't think we can go back to the way things were, no. What will probably happen is that women will become much more like men. For the moment, they're still very hung up on romance; whereas at heart, men don't give a shit about romance, they just want to fuck. Seduction only appeals to a few guys who haven't got particularly exciting jobs and nothing else of interest in their lives. As women attach more importance to their professional lives and personal projects, they'll find it easier to pay for sex too; and they'll turn to sex tourism. It's possible for women to adapt to male values; they sometimes find it difficult, but they can do it; history has proved it.'\n\n'So, all in all, things are in a bad way.'\n\n'A very bad way...' I agreed solemnly.\n\n'So, we were lucky.'\n\n'I was lucky to meet you, yes.'\n\n'Me too...' she said, looking me in the eyes. 'I was lucky too. The men I know are a disaster, not one of them believes in love; so they give you this big spiel about friendship, affection, a whole load of stuff that doesn't commit them to anything. I've got to the point where I can't stand the word 'friendship' any more, it makes me physically sick. Or there's the other lot, the ones who get married, who get hitched as early as possible and think about nothing but their careers afterwards. You obviously weren't one of those; but I also immediately sensed that you would never talk to me about friendship, that you would never be that vulgar. From the very beginning I hoped we would sleep together, that something important would happen; but it was possible that nothing would happen, in fact it was more than likely.' She stopped and sighed in irritation. 'Okay...' she said wearily 'I'd probably better go and call Jean-Yves.'\n\nI went into the bedroom to get dressed while she was on the phone. 'Yeah, the holiday was great...' I heard her say. A little later she yelled: 'How much?...' When I came back into the room she was holding the receiver looking thoughtful; she had not yet dressed.\n\n'Jean-Yves met the guy from the recruitment agency,' she said. 'They've offered him a hundred and twenty thousand francs a month. They're prepared to take me as well; according to him, they're prepared to go as high as eighty thousand. He has a meeting tomorrow to discuss the job.'\n\n'Where would you be working?'\n\n'It's with the leisure division of the Aurore group.'\n\n'Is it a big company?'\n\n'Too right it is; it's the biggest hotel chain in the world.'\n\n### 2\n\n###### _Being able to understand a customer's behaviour in order to categorise him more effectively, offering him the right product at the right time, but above all persuading him that the product he is offered is adapted to his needs: that is what all companies dream of_.\n\n###### Jean-Louis Barma, _What Companies Dream Of_\n\nJEAN-YVES WOKE AT five in the morning, looked over at his wife who was still sleeping. They had spent a terrible weekend with his parents \u2013 his wife couldn't stand the countryside. Nicolas, his ten-year-old son, loathed the Loiret too, as he couldn't bring his computer there; and he didn't like his grandparents, he thought they smelled. It was true that his father was slipping, increasingly, it seemed he was unable to look after himself, scarcely interested in anything apart from his rabbits. The only tolerable aspect of these weekends was his daughter, Ang\u00e9lique: at three, she was still capable of going into raptures over cows or chickens; but she was teething at the moment and had spent the greater part of the nights crying and whimpering. Once they got back, after three hours stuck in traffic jams, Audrey had decided to go out with some friends. He had heated up something from the freezer while he watched some mediocre American film about an autistic serial killer \u2013 it was apparently based on a true story. The man had been the first mentally ill person to be executed in the state of Nebraska for more than sixty years. His son hadn't wanted any dinner, he had immediately launched himself into a game of _Total Annihilation_ \u2013 or maybe it was _Mortal Kombat II_ , he got them mixed up. From time to time, he went into his daughter's room to try and quieten her howls. She fell asleep around one o'clock; Audrey still wasn't home.\n\nShe had come home in the end, he thought, making himself a coffee at the espresso machine; this time, at least. The law firm she worked for numbered _Lib\u00e9ration_ and _Le Monde_ among its clients; one way or another she had started hanging out with a group of journalists, television presenters and politicians. They went out quite a lot, sometimes to strange places \u2013 once when he was leafing through one of her books he came across a card for a fetish bar. Jean-Yves suspected that she slept with some guy once in awhile; in any case she and Jean-Yves didn't sleep together any more. Curiously, for his part, he didn't have affairs. Although he was aware he was handsome in a blond, blue-eyed way more common in Americans, he never really felt like taking advantage of the opportunities which might have presented themselves \u2013 in any case they were pretty rare: he worked twelve to fourteen hours a day and at his level of seniority you didn't really meet many women. Of course there was Val\u00e9rie; he had never thought of her other than as a colleague before. It was odd to think of her in this new light; but he knew it was an unimportant daydream: they had been working together for five years now and in situations like that things happened straight away, or they didn't happen at all. He admired Val\u00e9rie a lot, her astonishing organisational abilities, her infallible memory; without her, he realised, he would never have got to where he was \u2013 or at least not as quickly as he had. And today, he might well be about to take a decisive step. He brushed his teeth, shaved carefully, before picking out a rather sober suit. Then he pushed open the door to his daughter's room; she was asleep, blonde like he was, in a pair of pyjamas decorated with chicks.\n\nHe walked to the R\u00e9publique Fitness Club, which opened at seven. He and Audrey lived on the Rue du Faubourg-du-Temple, a rather trendy area which he hated. His meeting at the head office of the Aurore group was not until ten o'clock. Audrey could take care of getting the children dressed and driving them to school for once. He knew that when he got home tonight he would have a half-hour of nagging coming to him; as he walked along the wet pavement among the empty boxes and the vegetable peelings, he realised that he couldn't care less. He realised, also, for the first time with absolute clarity, that his marriage had been a mistake. This kind of realisation, he knew, usually precedes divorce by about two or three years \u2013 it's never an easy decision to take.\n\nThe big black guy at reception gave him a not very convincing 'How's things, boss?'. He handed him his membership card, nodded and took a towel. When he had met Audrey, he was only twenty-three. Two years later they got married, partly \u2013 but only partly \u2013 because she was pregnant. She was pretty, stylish, she dressed well \u2013 and she could be very sexy when she wanted to. Besides, she had ideas. The emergence of American-style judicial proceedings in France did not seem to her to be a regression, on the contrary, she thought it was progress, towards better protection for citizens and civil liberties. She was capable of expounding fairly lengthily on the subject, she was just back from doing work experience in the United States. In a nutshell, she had conned him. It was strange, he thought, how he had always felt the need to be impressed by women intellectually.\n\nHe started off with half-an-hour working through different levels on the Stairmaster, then twenty lengths of the pool. In the sauna, which was deserted at this time of day, he started to relax \u2013 and took the opportunity to run through in his mind what he knew about the Aurore group. Novotel-SIEH had been founded in 1966 by G\u00e9rard P\u00e9lisson and Paul Dubrule \u2013 one a graduate of the \u00c9cole Centrale, the other completely self-taught \u2013 using capital borrowed entirely from family and friends. In August 1967, the first Novotel opened its doors in Lille. It already included many of the characteristics which were to emerge as the hallmarks of the group: the rooms highly standardised, locations on the outskirts of cities \u2013 to be more precise, off the motorway, at the last exit before the city itself, above average standards of comfort for the time \u2013 Novotel was one of the first chains routinely to offer en-suite bathrooms. It was an immediate success with business travellers: in 1972, the chain already numbered thirty-five hotels. This was followed in '73 by the creation of Ibis, the takeover of Mercure in '75 and of Sofitel in '81. At the same time, the group prudently diversified into catering, acquiring the Courtepaille chain and the Jacques Borel International group, already well established in the group-catering and self-service restaurant sectors. In 1983, the company changed its name to become the Aurore group. Then, in '85, they created the Formules 1 \u2013 the first hotels with absolutely no personnel and one of the greatest successes in the history of the hotel business. Already well established in Africa and the Middle East, the company got a foothold in Asia and set up its own training centre \u2013 the Aurore academy. In 1990, the acquisition of Motel 6, comprising 650 locations throughout the United States, made the group the largest in the world; it was followed in '91 by a successful takeover bid for the Wagons Lits group. These acquisitions were costly and in '93, Aurore faced a crisis: the shareholders considered the company's debts to be too high, and the buyout of the M\u00e9ridien chain fell through. Thanks to the transfer of a number of assets and a recovery plan for Europcar, Len\u00f4tre and the Societ\u00e9 des Casinos Lucien Barri\u00e8re, the situation was turned around by the 1995 financial year. In January '97, Paul Dubrule and G\u00e9rard P\u00e9lisson resigned the presidency of the group which they ceded to Jean-Luc Espitalier, a graduate of the _\u00c9cole Normale d'Administration_ , whose career was described by the financial magazines as 'atypical'. However they remained members of the supervisory board. The transition went well and, by the end of 2000, the group had reinforced its position as world leader, consolidating its lead over Marriott and Hyatt, numbers two and three respectively. Of the ten largest hotel chains in the world, nine were American and one French \u2013 the Aurore group.\n\nAt nine-thirty, Jean-Yves parked his car in the car park of the group head office at \u00c9vry. He walked for a while in the frosty air, to unwind while waiting for the appointed time. At ten o'clock precisely, he was shown into the office of \u00c9ric Leguen, executive vice-president for hotels and member of the board of directors. He was forty-five, a graduate of the \u00c9cole C\u00e9ntrale, with a degree from Stanford.\n\nTall and sturdy, with blond hair and blue eyes, he looked a little like Jean-Yves \u2013 though ten years older and with something more confident in his attitude.\n\n'M Espitalier, our president, will meet with you in fifteen minutes,' he began. 'In the meantime, I'll explain why you're here. Two months ago, we bought the Eldorador chain from Jet Tours. It's a little chain of about a dozen beach hotel\/holiday clubs spread over the Maghreb, black Africa and the West Indies.'\n\n'It's showing a loss, I believe.'\n\n'No more than the sector as a whole.' He smiled briefly. 'Well, yes, actually, a bit more than the sector as a whole. To be quite frank, the purchase price was reasonable, but it wasn't peanuts; there were a number of other groups in the running: there are still a lot of people in the industry who believe that the sector will pick up again. It's true that, at the moment, Club Med is the only one managing to hold its own; strictly confidentially, we had actually thought of making a takeover bid for Club Med. But the prey was a little too large, the shareholders would never have gone with it. In any case, it wouldn't have been a very friendly thing to do to Philippe Bourguignon, who is a former employee...' He gave a rather phony smile, as though he was trying to suggest that this was perhaps \u2013 but not definitely \u2013 a joke. 'Anyway,' he went on, 'what we are proposing is that you take over the management of all of the Eldorador resorts. Your objective, obviously, being to bring them back quickly to breakeven and then to make them profitable.'\n\n'That's not an easy task.'\n\n'We're very aware of that; we feel that the level of remuneration offered is sufficiently attractive. Not to mention the career prospects within the group, which are huge: we have offices in 142 countries, we employ more than a hundred and thirty thousand people. On top of that, most of our senior executives quickly become shareholders in the group: it's a system we firmly believe in. I've written up some details for you with some sample calculations.'\n\n'I would also need more detailed information on the circumstances of the hotels in the group.'\n\n'Of course; I'll give you a detailed dossier a little later. This is not simply a tactical acquisition; we believe in the potential of the organisation: geographically the resorts are well sited, the general condition excellent \u2013 there's very little in the way of improvements to be made. At least, that's my opinion, but I don't have any experience of the leisure sector. We'll be working together, obviously; but you will make the decisions on these matters. If you want to get rid of a hotel or acquire another, the final decision in the matter is yours. That's how we work at Aurore.'\n\nHe thought for a moment before going on: 'Of course, it's no accident that you're here. The industry has carefully followed your career at Nouvelles Fronti\u00e8res; you might even say you have something of a following. You haven't systematically sought to offer the lowest prices, nor the highest level of service; in each case you've matched a price that is acceptable to customers very closely with a certain level of service; that's exactly the philosophy we follow within every chain in the group. And something equally important, you've had a hand in creating a brand with a very strong image; that's something we haven't always been able to do at Aurore.'\n\nThe telephone on Leguen's desk rang. The conversation was very brief. He got up and led Jean-Yves along the beige-tiled corridor. Jean-Luc Espitalier's office was vast, it must have been at least twenty metres wide; the left-hand side was taken up by a large conference table with some fifteen chairs. As they approached, Espitalier stood and welcomed them with a smile. He was a small man, quite young \u2013 certainly no older than forty-five \u2013 his hair receding at the front, he looked oddly unobtrusive, almost retiring, as though he was trying to soften the importance of his role with irony. You probably shouldn't count on it, thought Jean-Yves; ENA graduates are often like that, they develop a veneer of humour which turns out to be deceptive. They settled themselves in armchairs around a low table in front of his desk. Espitalier looked at him for a long time with his curious, shy smile before beginning to speak.\n\n'I have a lot of respect for Jacques Maillot,' he said eventually; 'He's built up a first-rate company, very original and with a real ethos. It doesn't happen often. That said \u2013 and I don't want to play the prophet of doom here \u2013 I think French tour operators need to prepare themselves for a rough ride. Very soon \u2013 it's inevitable at this stage, and in my opinion it's only a matter of months \u2013 British and German tour operators are going to make inroads into the market. They have two to three times the level of financial backing, and their tours are 20 to 30 per cent cheaper, for a comparable or a better standard of service. Competition will be tough, very tough. To be blunt, there will be casualties. I'm not saying Nouvelles Fronti\u00e8res will be one of them; it's a group with a strong identity and level-headed shareholders, it can weather this. Nevertheless, the years ahead are going to be tough for everyone.\n\n'At Aurore, we don't have that problem at all,' he went on with a little sigh, 'we are the uncontested world leader in the business hotel sector, which fluctuates very little; but we are still poorly established in the leisure hotel sector, which is more volatile, more sensitive to economic and political fluctuations.'\n\n'As a matter of fact,' interrupted Jean-Yves, 'I was rather surprised by your acquisition. I thought your main development priority was still business hotels, particularly in Asia.'\n\n'That is still our main priority,' replied Espitalier calmly. 'In China alone, for example, there is extraordinary potential in the business hotel sector. We have the experience, we have the know-how: imagine concepts like Ibis and Formule 1 rolled out across the country? That said... how should I put this?' He thought for a moment, looked at the ceiling, at the conference table to his right before looking back at Jean-Yves. 'Aurore is a discreet group,' he said at length; 'Paul Dubrule used to say that the sole secret of success in the market was to be timely. Timely means not too early: it's very rare for true innovators to reap the full profits of their innovation \u2013 that's the story of Apple and Microsoft. But obviously, it also means not too late. That's where our discretion has served us well. If you do your development work in the shadows, without making waves, by the time your competitors wake up and decide to move on to your patch, it's too late: you have your territory sewn up, you have acquired a crucial competitive advantage. Our reputation has not kept pace with our actual significance; for the most part, this has been done deliberately.\n\n'That time is gone,' he went on with another sigh; 'Everyone now knows that we are number one in the world. At that point it becomes useless \u2013 even dangerous \u2013 to count on our discretion. It's essential for a group of Aurore's size to have a public image. The business hotel sector is a dependable market, which generates guaranteed regular, substantial, revenues. But it's not, how shall I put it, it's not really _fun_. People rarely talk about their business trips, there's no pleasure in telling people about them. To build a positive image with the general public there are two possibilities open to us: tour operating or 18\u201330-style holiday clubs. Becoming a tour operator is further from our core business, but there are a number of very healthy businesses likely to change hands in the near future \u2013 we very nearly went down that road. And when Eldorador presented itself, we decided to seize the opportunity.'\n\n'I'm just trying to understand your objectives,' said Jean-Yves. 'Are you more focused on profit or public image?'\n\n'That's a complex issue...' Espitalier hesitated, shifting slightly in his chair. 'Aurore's problem is that it has a very weak shareholder base. That, in fact, is what started the rumours of a takeover bid in 1994 \u2013 I can tell you now,' he went on with a confident gesture, 'that they were completely unfounded. That would be even more true now: we have no debt whatever, and no international company, even outside the hotel business, is large enough to mount a bid. What remains true is that, unlike Nouvelles Fronti\u00e8res for example, we do not have a coherent shareholder base. At heart, Paul Dubrule and G\u00e9rard P\u00e9lisson were less capitalists than they were entrepreneurs \u2013 and great entrepreneurs in my opinion, among the greatest the century has produced. But they did not seek to keep a controlling share in their business; it is this which puts us in a delicate position today. You know as well as I do that it is occasionally necessary to sanction prestige spending, something which will improve the strategic position of the group without making a positive impact on revenues in the short term. We also know that it is sometimes necessary to temporarily shore up a loss-making sector because the market hasn't matured or because it is going through a short-term crisis. This is something that the new generation of shareholders finds difficult to accept: the focus on rapid returns on investment has been deeply unconstructive and damaging.'\n\nHe raised his hand discreetly, seeing that Jean-Yves was about to interrupt. 'Mind you,' he went on, 'our shareholders are not imbeciles. They are perfectly aware that in the current climate it would not be possible to bring a chain like Eldorador back to breakeven in the first year \u2013 probably not even in two years. But come the third year they'll want to look very hard at the figures \u2013 and they won't be long in coming to their conclusions. At that point, even if you have a magnificent plan, even if the potential is vast, I won't be able to do anything.'\n\nThere was a long silence. Leguen sat motionless, he had lowered his head. Espitalier stroked his chin sceptically. 'I see...' Jean-Yves said at last. After a couple of seconds he added calmly: 'I'll give you my answer in three days.'\n\n### 3\n\nI SAW A lot of Val\u00e9rie over the two months that followed. In fact, with the exception of a weekend she spent at her parents, I think I probably saw her every day. Jean-Yves had decided to accept the Aurore group's offer; she had decided to follow him. The first thing she said to me, I remember, was: 'I'm about to move into the 60 per cent tax bracket.' She was right: her salary was going from forty thousand francs a month to seventy-five thousand; after tax, the increase was less spectacular. She knew that she would have to put a lot of work in from the moment she took up her job at the group early in March. For the time being, at Nouvelles Fronti\u00e8res, everything was fine: they had both tendered their resignations, they were gradually handing over the reins to their successors. I advised Val\u00e9rie to save, to open a savings account or something; in fact, we didn't think about it much. Spring was late, but that was of no importance. Later, thinking about this happy time with Val\u00e9rie, a time of which, paradoxically, I have so few memories, I would say that man is clearly not intended to be happy. To truly arrive at the practical possibility of happiness, man would have to transform himself \u2013 transform himself _physically_. What can one compare with God? In the first place, obviously, a woman's pussy; but also perhaps the steam in a hammam. Something, at any rate, in which spirit becomes possible, because the body is sated with contentment, with pleasure, and all anxiety is abolished. I now know for certain that the spirit is not born, that it needs to be brought forth, and it will be a difficult birth, something of which we now have only a vague and harmful idea. When I brought Val\u00e9rie to orgasm, when I felt her body quiver under mine, I sometimes had the impression \u2013 fleeting but irresistible \u2013 of attaining a new level of consciousness, where every evil had been abolished. In those moments of suspension, almost of motionlessness, when the pleasure in her body mounted, I felt like a god on whom depended tranquillity and storms. It was the first joy \u2013 indisputable, perfect.\n\nThe second joy which Val\u00e9rie brought me was the extraordinary gentleness, the natural goodness of her nature. Sometimes, when she had been working long hours \u2013 and over the months they would become longer and longer \u2013 I felt that she was tense, emotionally drained. Never once did she turn on me, never once did she get angry, never once did she lapse into the unpredictable hysterics which sometimes make the company of women so oppressive, so pathetic. 'I'm not ambitious, Michel...' she would tell me sometimes. 'I feel happy with you, I think you're the love of my life, and I don't ask for anything more than that. But that's not possible: I have to ask for more. I'm trapped in a system from which I get so little, which I know is futile; but I don't know how to get out. Just once, we should take time to think; but I don't know when we'll be able to take time to think.'\n\nFor my part, I was doing less and less work, at least I was doing my work only in the strictest possible sense. I was home in plenty of time to watch _Questions pour un champion_ and shop for dinner; I spent every night at Val\u00e9rie's place. Curiously, Marie-Jeanne didn't seem to hold my flagging professional attention against me. True, she enjoyed her work and was more than happy to take on her share of overtime. What she wanted more than anything, I think, was for me to be nice to her \u2013 and I was nice through all those weeks, I was gentle and peaceful. She had liked the coral necklace I had brought her from Thailand, she wore it every day now. As she worked on the files for the exhibitions she would sometimes look at me in a way that was strange and difficult to decipher. One morning in February \u2013 I remember it very well, it was my birthday \u2013 she said to me straight out: 'You've changed, Michel... I don't know, you seem happy.'\n\nShe was right; I was happy, I remember that. Of course there are lots of things, a whole series of inevitable troubles, decline and death, of course. But remembering those months, I can bear witness: I know that happiness exists.\n\nJean-Yves, on the other hand, was not happy, that was obvious. I remember the three of us having dinner together in an Italian, or rather a Venetian, restaurant, something pretty trendy anyway. He knew that we would go home later and fuck, and we could fuck with love. I didn't really know what to say to him \u2013 everything there was to be said was too obvious, too blunt. His wife obviously didn't love him, she had probably never loved anyone; and she would never love anyone, that too was patently clear. He hadn't had much luck, that was all. Human relationships aren't nearly as complicated as people make out: they're often insoluble but only rarely _complicated_. Now, of course, he would have to get a divorce; it wouldn't be easy but it had to be done. What else could I possibly say? The subject was dealt with long before we finished the _antipasti_.\n\nAfterwards they talked about their careers within the Aurore group: they already had some ideas, a number of possible objectives for the Eldorador takeover. They were intelligent, competent, much-admired in their industry; but they could not afford to make a mistake. To fail in this new position would not be the end of their careers: Jean-Yves was thirty-five, Val\u00e9rie, twenty-eight; they would be given a second chance. But the industry would not forget that first blunder, they would have to start again at a significantly lower level. In the society we lived in, the most important consideration in any position was represented by the _salary_ , and more generally the financial benefits; the prestige and distinction of the post tended nowadays to occupy a much less significant position. There existed, however, a highly developed system of fiscal redistribution which allowed the useless, the incompetent and the dangerous \u2013 a group of which, in some sense, I was a part \u2013 to survive. In short, we were living in a mixed economy which was slowly evolving towards a more pronounced liberalism, slowly overcoming the prejudice against usury \u2013 and in more general terms, against money \u2013 which persists in traditionally Catholic countries. They could expect no real benefits from such a change. A number of young Hautes \u00c9tudes Commerciales business graduates, much younger than Jean-Yves \u2013 some of them still students \u2013 had thrown themselves headlong into market speculation without ever considering looking for paid employment. They had computers connected to the internet, sophisticated market-tracking software. Quite frequently, they formed groups or clubs in order to be able to make more substantial investments. They lived with their computers, worked in shifts twenty-four hours a day, never took holidays. The goal of each and every one was extraordinarily simple: to become billionaires before they turned thirty.\n\nJean-Yves and Val\u00e9rie were part of an intermediary generation for whom it still seemed difficult to imagine a career outside business, or possibly the public sector; a little older than they were, I was in more or less the same position. The three of us were caught up in a social system like insects in a block of amber; there wasn't the slightest possibility that we could turn back.\n\nOn the morning of March 1, Val\u00e9rie and Jean-Yves officially took up their positions at the Aurore group. A meeting had already been scheduled for March 4 with the principal executives who would be working on the Eldorador project. Senior management had requested a long-term study of the future of holiday clubs from Profiles, a well-known consultancy working in the field of behavioural sociology.\n\nDespite himself, as he walked into the 23rd floor conference room for the first time, Jean-Yves was quite impressed. There were about twenty people there, every one of whom had several years' experience with Aurore behind them; and it was now up to him to lead the group. Val\u00e9rie sat immediately to his left. He had spent the weekend studying the files; he knew the names, precise responsibilities and professional history of every person sitting at the table; yet he could not help feeling a little anxious. A grey day settled over the graceful suburbs of Essonne. When Paul Dubrule and G\u00e9rard P\u00e9lisson had decided to set up their head office in \u00c9vry, they had been influenced by cheap land and the proximity of the motorway to Orly Airport and the south; at the time, it was a quiet suburb. Now, the local communities had the highest crime rate in all France. Every week, there were attacks on buses, police cars, fire engines; there was not even an exact figure for assaults or robberies; some people estimated that to get the true figure, you had to multiply the number of reported crimes by five. The company premises were watched over twenty-four hours a day by a team of armed guards. An internal memo advised that public transport was best avoided after a certain time. For employees who had to work late and who did not have their own cars, Aurore had negotiated a discount with a local taxi firm.\n\nWhen Lindsay Lagarrigue, the behavioural sociologist, arrived, Jean-Yves felt he was on familiar territory. The guy was about thirty, with a receding hairline, his hair tied back in a ponytail; he wore an Adidas tracksuit, a Prada tee-shirt and a pair of battered Nikes: in short, he looked like a behavioural sociologist. He began by handing out copies of a very slim file, mostly made up of diagrams with arrows and circles; his briefcase contained nothing else. The front page was a photocopy of an article from the _Nouvel Observateur_ , more precisely, it was an editorial from the travel section, entitled 'Another Way to Travel'.\n\n'In the year 2000,' Lagarrigue began, reading the article aloud, 'mass tourism has had its day. We dream of travel as of individual fulfilment, but we have ethical concerns.' This opening paragraph seemed to him symptomatic of the changes that were occurring. He talked about this for a few minutes, then asked those present to concentrate on the following sentences: 'In the year 2000, we worry about whether tourism is respectful of others. Being affluent, we want our travels to be more than simply selfish pleasures, we want them to bear witness to a certain sense of solidarity.'\n\n'How much did we pay this guy for the study?' Jean-Yves asked Val\u00e9rie discreetly.\n\n'A hundred and fifty thousand francs.'\n\n'I don't believe it... Is this asshole just going to read out a photocopy of an article from the _Nouvel Obs_?'\n\nLinsday Lagarrigue went on, loosely paraphrasing the article, then he read a third passage, in an absurdly emphatic voice: 'In the year 2000,' he declared, 'we want to be nomads. We travel by train or by ship, over rivers and oceans; in an age of speed, we are rediscovering the pleasures of slothfulness. We lose ourselves in the silent infinities of the desert, and then, without a break, plunge into the tumult of great cities. But always with the same passion...' Ethical, individual fulfilment, solidarity, passion: these, according to him, were the key words. In this new mood, it was hardly surprising that the holiday club, based on selfish isolation, on the standardisation of needs and desires, was beset by chronic problems. The days of the _sun worshippers_ were over: what travellers today were looking for was authenticity, discovery, a sense of sharing. More generally, the Fordist production-line model of leisure travel \u2013 typified by the famous '4 S': Sea, Sand, Sun... and Sex \u2013 was doomed. As the work of Michky and Braun had shown so spectacularly, the industry as a whole would have to begin to consider its activities from a post-Fordist perspective.\n\nThe behavioural sociologist clearly knew his job; he could have gone on for hours. 'Excuse me...' Jean-Yves interrupted him in a tone of barely suppressed irritation.\n\n'Yes...' the behavioural sociologist gave him a winning smile.\n\n'I think that every person at this table, without exception, is aware that the holiday-club model is undergoing some problems at the moment. What we want from you isn't so much an endless description of the nature of those problems, but rather an attempt, however slight, to indicate the beginnings of a solution.'\n\nLindsay Lagarrigue was open-mouthed; he had not anticipated an objection of this kind. 'I think...' he mumbled eventually, 'I think that in order to solve the problem it is important to define it and to have some sense of what has caused it.' Another empty phrase, thought Jean-Yves furiously; not only empty, but, as it happened, untrue. The causes were clearly part of general shifts in society which were beyond their powers to change. They had to adapt to this new business climate, that was all. How could they adapt to it? This moron clearly hadn't the faintest idea.\n\n'What you're telling us, broadly speaking,' went on Jean-Yves, 'is that the holiday-club model is obsolete.'\n\n'No, no, not at all...' The behavioural sociologist was beginning to lose his footing. 'I think... I simply think that it requires thought.' 'And what the hell are we paying you for, asshole?' retorted Jean-Yves under his breath before addressing all those present:\n\n'All right, we'll try to give it some thought. I'd like to thank Monsieur Lagarrigue for his contribution; I don't think we'll be needing you again today. I suggest we break for ten minutes for coffee.'\n\nPiqued, the behavioural sociologist packed away his diagrams. When the meeting resumed, Jean-Yves picked up his notes and began:\n\n'Between 1993 and 1997, as you know, Club Med went through the worst crisis in its history. Competitors and imitators had multiplied, they had ripped off the Club formula wholesale while undercutting the Club considerably: numbers were in freefall. How did they manage to turn the situation around? Chiefly, by dropping their prices. But they didn't drop them to the same level as their rivals: they knew that they had the advantage of being the original, they had a reputation, an image; they knew their customers would accept a certain price differential \u2013 which they set, according to destination, and after meticulous research, at between 20 and 30 per cent \u2013 for the real Club Med experience, the 'original' if you like. This is the first idea I propose that we explore in the coming weeks: is there room in the holiday-club sector for something different from the Club Med formula? And, if so, can we begin to visualise what that something might be, what its target market might look like? The question is far from simple.\n\n'I've come,' he went on, 'as you probably know, I've come here from Nouvelles Fronti\u00e8res. And, although it's not what the group is best known for, we also had a stab at the holiday-club sector: the Paladiens. We began to experience problems with the clubs at about the same time as Club Med; we resolved those problems very quickly. How? Because we were the largest tour operator in France. At the end of their discovery tour of the country, our customers, for the most part, wanted to spend some time at the beach. Our tours often have the reputation, justifiably, of being tough, of requiring a high level of physical fitness. Having won their stripes as 'travellers' the hard way, our customers were generally delighted to be back in the shoes of ordinary tourists for a while. In fact, the formula was so successful that we decided to include a beach supplement as standard in most of the tours \u2013 which allowed us to bump up the length of the tours; as you all know, a day at the beach is much less expensive than a travel day. Given this, it was easy for us to favour our own hotels. This is the second thing I propose we consider: it's possible that the future of holiday clubs depends on closer links with tour operators. You'll have to use your imaginations here, too, and don't limit yourselves to the players currently operating in the French market. I'm asking you to explore a new field; we may have a lot to gain from alliances with the major Northern European tour-operators.'\n\nAt the end of the meeting, a woman of about thirty, blonde, pretty face, approached Jean-Yves. Her name was Marylise Le Fran\u00e7ois, she was the marketing manager. 'I wanted to tell you how much I appreciated your intervention...' she said. 'It had to be done. I think you've managed to remotivate people. Now everyone knows that there is somebody at the controls, we'll really be able to get back down to work.'\n\n### 4\n\nIT QUICKLY BECAME apparent that it would not be quite so simple. Most of the British and particularly the German tour operators already had their own chains of holiday clubs and they weren't interested in allying themselves to another group. All attempts in that direction proved futile. On the other hand, the Club Med seemed to have hit on the definitive formula for a holiday club; since their inception, none of their rivals had proved able to offer anything really new.\n\nTwo weeks later, Val\u00e9rie finally had an idea. It was almost 10 p.m.; she had collapsed into an armchair in the middle of Jean-Yves's office and was sipping a hot chocolate before heading home. They were both exhausted, they had spent the whole day working on the financial report on the clubs.\n\n'You know,' she sighed, 'I think we might be making a mistake in trying to separate the tours from the relaxation.'\n\n'What do you mean?'\n\n'Remember at Nouvelles Fronti\u00e8res: even before we added the beach supplements to the end of the holiday, whenever there was a day at the beach in the middle of a tour, the customers always enjoyed it. And the thing people complained about most often was having to change hotels all the time. What we really want, in fact, is to alternate the excursions with time at the beach \u2013 a day touring, a day relaxing and so on. Coming back to the hotel every night, or the following night if the excursion is long, but not having to pack or check out of your room.'\n\n'Resorts already offer additional excursions; and I'm not sure they have much take-up for them.'\n\n'Yes, but there's a supplement, and the French hate paying supplements. On top of that, you have to make the reservation after you get there: people hesitate, they dither, they can't decide, and in the end they do nothing. Actually, they like the excursions as long as you do all the work for them; and above all they love things that are _all-inclusive_.'\n\nJean-Yves thought for a moment. 'You know, what you're suggesting is not a bad idea. On top of that, we should be able to get it under way pretty quickly: this summer, I think. We could offer the new formula as a complement to the ordinary packages. We could call it Eldorador Discovery, something like that.'\n\nJean-Yves consulted Leguen before implementing the idea; he quickly realised that the other man had no desire to express an opinion one way or the other. 'It's your responsibility,' Leguen said solemnly. Listening to Val\u00e9rie tell me how she spent her days, I realised I didn't know very much about the world of the senior executive. Her double act with Jean-Yves was in itself remarkable. 'Under normal circumstances,' she told me, 'his assistant would be some girl who dreamed of getting his job. That complicates office politics: it means that sometimes its better to fail, as long as you can pass the blame on to somebody else.' In this case, they were in a healthier position; no one in the group wanted their jobs; most of the executives thought the takeover of Eldorador had been a mistake.\n\nFor the rest of the month, she spent a lot of time working with Marylise Le Fran\u00e7ois. The catalogues for the summer holidays had to be ready by the end of April at the very latest, in fact even that was cutting it fine. She rapidly realised that the Jet Tours marketing for the resorts had been absolutely appalling. 'An Eldorador holiday is like that magical moment in Africa, when the heat begins to fade and the whole village gathers around the talking-tree to listen to the elders...' she read aloud to Jean-Yves. 'Honestly, can you believe this stuff? With photos of the holiday reps leaping about in their ridiculous yellow uniforms. It's complete rubbish.'\n\n'What do you think of the slogan, \"Eldorador, live life more intensely\"?'\n\n'I don't know; I don't know what to think anymore.'\n\n'It's too late for the standard packages, the catalogues have already gone out. One thing's for sure though, we'll have to start from scratch with the \"Discovery\" catalogue.'\n\n'What I think we need to do,' interrupted Marylise, 'is to play up the contrast between physical effort and luxury. Mint tea in the middle of the desert, but on priceless carpets...'\n\n'Yeah, the magical moments...' said Jean-Yves jadedly. He got up from his chair with effort. 'Don't forget to put \"magical moments\" in there somewhere; oddly enough it works every time. Okay, I'll leave you to it; I have to get back to my fixed costs...'\n\nVal\u00e9rie was well aware that there was no question but that Jean-Yves had the most thankless aspects of the work. She herself knew very little about hotel management \u2013 it simply brought back vague memories of studying for her BTS. 'Edward Yang owns a three-star hotel and restaurant and believes that it is his responsibility to satisfy his customers to the best of his ability; he is constantly seeking to innovate and to respond to customer needs. From experience, he knows that breakfast is very important, the most important meal of the day, and plays a decisive role in establishing the image of the hotel.' This had been part of a written test in her first year. Edward Yang arranges for a statistical analysis of his customers, focusing particularly on the number of guests per room (single people, couples, families). You had to break down the analysis, calculate chi squared; the section concluded with the following question: 'To sum up, do family circumstances correlate with the consumption of fresh fruit at breakfast?'\n\nRummaging through her files, she managed to find a BTS question which corresponded pretty closely to their present situation.\n\n'You have just been appointed marketing manager of the international arm of the South America group. The company has recently purchased a hotel-restaurant in the West Indies, a four-star establishment with 110 rooms on the sea-front in Guadeloupe. Opened in 1988 and renovated in 1996, it is currently experiencing serious problems. The occupancy rate is only 45%, far below the anticipated breakeven point.'\n\nHer answer was marked 18\/20, which seemed to be a good sign. At the time, she remembered, it all seemed like a fairytale to her, and not a very plausible one. She couldn't imagine herself as marketing director of _South America_ , or of anything else for that matter. It was a game, an intellectual game that was neither very interesting nor very difficult. Now, it was no longer a game; or perhaps it was, but their careers were the stakes.\n\nShe would come back from work so exhausted that she hadn't the energy to make love, barely enough energy to suck me off; she would be half asleep with my penis still in her mouth. When I penetrated her, it was usually in the morning when we woke. Her orgasms were more muted, more restrained, as though muffled by a curtain of fatigue; I think I loved her more and more.\n\nAt the end of April the catalogues were printed and distributed to five thousand travel agents \u2013 almost the entirety of the French network. Now, they needed to deal with the infrastructure for the tours so that everything would be ready for July 1. Word of mouth was very important in launching a new product of this kind: a tour cancelled or poorly organised could mean a lot of lost customers. They had decided not to invest in a major advertising campaign. Curiously, although Jean-Yves had specialised in marketing, he hadn't much faith in advertising. 'It can be useful for refining your image,' he said, 'but we're not at that point just yet. For the moment, the most important thing is for us to get good distribution and ensure the product has a reputation for reliability.' On the other hand, they invested hugely in information for the travel agents; it was crucial that they offer the product quickly and spontaneously. Val\u00e9rie took most of the responsibility for this; it was familiar territory for her. She remembered the sales-pitch mnemonic SURE \u2013 Strategic Planning, Understanding, Response management, Execution excellence; she remembered too the reality, which was infinitely more simple. But most of the salesgirls were very young; most of them had barely passed their BTS diploma; it was easier to speak to them in their own language. Talking to some of the girls, she realised that Jean-Louis Barma's typology was still being taught in colleges. ( _The technician consumer_ : product centred, sensitive to quantitative aspects, attaches great importance to technical aspects of the product. _The devout consumer_ : trusts the salesperson blindly because he does not understand the product. _The complicit consumer_ : happy to focus on points he has in common with the salesperson if the latter knows how to establish a good interpersonal relationship. _The manipulative consumer_ : a manipulator whose strategy is to deal directly with the supplier and so get the best deal. _The developing consumer_ : attentive to the salesperson, whom he respects, to the product offered, aware of his needs, he communicates easily.) Val\u00e9rie was five or six years older than these girls; she had long risen above their current level and she had achieved a degree of professional success which most of them hardly dared dream of. They looked at her with a sort of childlike admiration.\n\nI had the key to her apartment now; in general, while I was waiting for her in the evenings, I read August Comte's _Course in Positive Philosophy_. I liked this tedious, dense book; often I would reread a page three or four times. It took me almost three weeks to finish Lesson Fifty: 'Preliminary Considerations on Social Statics, or the General Theory of the Spontaneous Order of Human Society'. I certainly needed some sort of theory to help me take stock of my social status.\n\n'You work far too much, Val\u00e9rie...' I told her one evening in May as she was lying, huddled up with exhaustion, on the living-room sofa. 'You have to get something out of it. You should put some money aside, otherwise one way or the other you'll just end up spending it.' She agreed that I was right. The following morning, she took two hours off and we went to Porte d'Orl\u00e9ans branch of the Cr\u00e9dit Agricole to open a joint account. She gave me power of attorney and the following day I went back to talk to a financial adviser. I decided to put aside twenty thousand francs a month from her salary, half of it in a life assurance policy, the other half into a savings account. I was at her place pretty much all the time now; it made no sense to hang on to my apartment.\n\nIt was she who made the suggestion at the beginning of June. We had made love most of the morning, taking long breaks curled up together between the sheets. Then she would wank me or suck me off and I would penetrate her again; neither of us had come. Each time she touched me I quickly got a hard on, her pussy was constantly wet. She was feeling good, I could see calm flooding her face. At about nine o'clock, she suggested we have dinner in an Italian restaurant near the Parc Montsouris. It wasn't quite dark yet; it was a warm evening. I had to go to my place afterwards, if I intended to go to work in a shirt and tie as I usually did. The waiter brought us two house cocktails.\n\n'You know, Michel...' she said as soon as he had gone, 'You could just move into my place. I don't really think we need to go on playing at being independent. Or, if you prefer, we can get a flat together.'\n\nIn point of fact, yes, I did prefer; let's say it gave me a greater sense of this as a new beginning. A first beginning in my case, truth be told; and in her case too, I suppose. It becomes habit, being alone, being independent; it's not always a good habit. If I wanted to live something that resembled the conjugal experience, now, evidently, was the time. Of course I knew the drawbacks of the set-up; I know that desire becomes dulled more quickly when couples live together. But it becomes dulled anyhow, that's one of the laws of life; only then does it become possible for the union to move on to a different level \u2013 so many people have believed. But that evening my desire for Val\u00e9rie was far from dulled. Before leaving her, I kissed her on the mouth; she opened her lips wide, abandoning herself completely to the kiss. I slipped my hand into her tracksuit bottoms, into her panties, put my hand on her buttocks. She leaned her head back, looked left and right, the street was completely quiet. She knelt down on the pavement, opened my flies, took my penis into her mouth. I leaned against the park railings. Just before I came, she moved her mouth away and continued to masturbate me with two fingers, slipping her other hand into my trousers to stroke my balls. She closed her eyes; I ejaculated over her face. At that moment I thought she was going to burst into tears; but she didn't, she simply licked at the semen trickling down her cheeks.\n\nThe very next morning, I started going through the small ads; somewhere in the southern _arrondissements_ would be best for Val\u00e9rie's work. A week later, I had found it: it was a large two-bedroom on the thirtieth floor of the Opale tower near the Porte de Choisy. I had never had a beautiful view of Paris before; I had never really looked for one, to be honest. When we were about to move in, I realised that I didn't feel the least attachment to anything in my apartment. I could have felt a certain joy, something like intoxication, at this freedom; on the contrary, I felt slightly scared. I had managed, it seemed, to live for forty years without forming the most tenuous of attachments to a single object. All told, I had two suits which I wore alternately. Books, sure, I had books; but I could easily have bought them again, not one of them was in any way precious or rare. Several women had crossed my path; I didn't have a photograph or a letter from any of them. Nor did I have any photos of myself: I had no memory of what I might have been like when I was fifteen, or twenty or thirty. I didn't really have any personal papers: my identity could be contained in a couple of files which would easily fit into a standard-size cardboard folder. It is wrong to pretend that human beings are unique, that they carry within them an irreplaceable individuality; as far as I was concerned, at any rate, I could not distinguish any trace of such an individuality. As often as not it is futile to wear yourself out trying to distinguish individual destinies and personalities. When all's said and done, the idea of the uniqueness of the individual is nothing more than pompous absurdity. We remember our own lives, Schopenhauer wrote somewhere, a little better than a novel we once read. That's about right: a little, no more.\n\n### 5\n\nVAL\u00c9RIE WAS AGAIN overwhelmed with work in the last two weeks of June; the problem with working with a number of countries is that with the time differences you could almost be working twenty-four hours a day. The weather became increasingly warmer, heralding a magnificent summer; until now, we had had little opportunity to take advantage of it. After work, I liked to go and wander round Tang Fr\u00e8res, I even made an attempt to take up Eastern cooking. But it was too complicated for me, there was a completely new balance to understand between ingredients, a special way of chopping vegetables, it was almost a different mind-set. In the end I settled for Italian, something which was much more my level. I would never have believed that some day I would take pleasure in cooking. Love sanctifies.\n\nIn his fiftieth sociology lesson, Auguste Comte tackles that 'strange metaphysical aberration' which conceives the family as the template for society. 'Founded chiefly upon attachment and gratitude, the domestic union satisfies, by its mere existence, all our sympathetic instincts quite apart from all idea of active and continuous cooperation towards any end unless it be that of its own institution. When, unhappily, the coordination of employments remains the only principle of connection, the domestic union degenerates into mere association, and in most cases will soon dissolve altogether.' At the office I continued to do the bare minimum; all the same, I had two or three important exhibitions to organise; I got through them without any difficulty. Office work isn't very difficult \u2013 you simply have to be reasonably meticulous and be decisive. I had rapidly realised that you did not necessarily have to make _the right decision_ , it was sufficient, in most cases, to make _any old decision_ , as long as you made it quickly \u2013 if you work in the public sector, at least. I binned some projects and green lighted others: I did this based on insufficient information. In ten years, not once had I asked for additional information and, in general, I didn't feel the slightest remorse. Deep down, I had pretty little respect for the contemporary art scene. Most of the artists I knew behaved exactly like entrepreneurs: they carefully reconnoitred emerging markets, then tried to get in fast. Just like entrepreneurs, they had been at the same few colleges, they were cast from the same mould. There were some differences, however: in the art market, innovation was at a greater premium than in most other professional sectors; moreover, artists often worked in packs or networks, in contrast to entrepreneurs who were solitary beings surrounded by enemies \u2013 shareholders ready to drop them at a moment's notice, executives always ready to betray them. But in the artists' proposals I dealt with, it was rare for me to come across a sense of genuine inner desire. At the end of June, however, there was the Bertrand Bredane exhibition, which I had passionately supported from the outset \u2013 to the great surprise of Marie-Jeanne, who had become accustomed to my meek indifference and was herself repelled by works of that nature. He was not exactly a young artist, he was already forty-three and, physically, he was knackered \u2013 he looked a little like the alcoholic poet in _Le Gendarme de Saint-Tropez_. He was chiefly famous for leaving rotting meat in young girls' panties, or breeding flies in his own excrement and then releasing them into the galleries. He had never been really successful, he didn't have the right connections, and he stubbornly persisted in a rather dated 'trash' aesthetic. I sensed in him a certain authenticity, but maybe it was simply the authenticity of failure. He seemed a little unbalanced. His most recent project was even worse \u2013 or better, depending on one's point of view \u2013 than his earlier work. He had made a video following the fate of the bodies people donate to medical science after their death \u2013 being used for dissection practice in medical schools, for example. A number of genuine medical students were to mingle with the audience and from time to time, flash a severed hand or an eyeball that had been gouged out \u2013 to play, in fact, the kind of practical jokes of which medical students are apparently so fond. I made the mistake of taking Val\u00e9rie to the opening even though she'd had an exhausting day. To my surprise, it was pretty well attended and the crowd included a number of major celebrities: could it be that Bertrand Bredane's moment had arrived? After about half an hour, Val\u00e9rie had had enough and asked me if we could leave. A medical student rushed up to her holding a severed dick in his hand, the testicles still fringed with hair. She turned her head away, sickened, and led me to the exit. We sought refuge in the Caf\u00e9 Beaubourg.\n\nHalf an hour later, Bertrand Bredane made his entrance, accompanied by two or three girls and some other people, among whom I recognised the director of sponsorship at a major venture capital firm. They took the table next to us; I couldn't not go and say hello to them. Bredane was visibly pleased to see me, it was true that that evening I'd given him a particularly warm handshake. The conversation dragged on, Val\u00e9rie came and sat with us. I don't know who suggested we go for a drink at Bar-Bar; probably Bredane himself. I made the mistake of accepting. Most of the partner-swapping clubs which had tried to introduce an S&M night had failed. Bar-Bar, on the other hand, had specialised in sadomasochistic practices since it opened, and, though it didn't have a particularly strict dress code \u2013 except on certain nights \u2013 had been packed from the start. As far as I was aware, the S&M scene was a pretty particular milieu, made up of people who were no longer really interested in ordinary sexual practices, and consequently disliked going to regular orgy clubs.\n\nNear the entrance, a chubby-faced woman of fifty-something, gagged and handcuffed, swung in a cage. Looking more closely, I discovered she was shackled, her heels attached to the bars of the cage with metal chains; she was wearing nothing but a leatherette corset on to which spilled her large sagging breasts. She was, as was the custom of the place, a slave whose master was going to auction her off for the evening. She didn't seem to find it terribly amusing. I noticed that she turned this way and that, trying to hide her arse, which was completely riddled with cellulite; but it was impossible \u2013 the cage was open on all four sides. Maybe she did this for a living; I knew it was possible to make between one to two thousand francs a night by renting yourself out as a slave. My impression was that she was a junior white-collar worker, maybe a switchboard operator for the Social Security, who was doing this to make ends meet.\n\nThere was only one table free, near the entrance to the first torture chamber. Immediately we sat down, a bald, pot-bellied middle-manager in a three-piece suit came by on a leash, led by a black, bare-arsed dominatrix. She stopped at our table and ordered him to strip to the waist. He obeyed. She took a pair of metal clamps from her bag; for a man, his breasts were pretty fat and flabby. She closed the clamps on his nipples, which were red and distended. He winced in pain. She tugged on his leash; he got back on all fours and followed her as best he could; the pasty folds of his belly wobbled in the dim light. I ordered a whisky, Val\u00e9rie an orange juice. She stared stubbornly at the table, not watching what was going on around her, nor taking part in the conversation. In contrast, Marjorie and G\u00e9raldine, the two girls I knew from the Plastic Arts Delegation, seemed to be very excited. 'It's tame tonight, very tame...' muttered Bredane, disappointed. He went on to explain to us that, some nights, customers had needles pushed through their balls or the heads of their cocks; once he'd even see a guy whose dominatrix had torn out a fingernail with a pair of pliers. Val\u00e9rie flinched in revulsion.\n\n'I find the whole thing completely disgusting...' she said, unable to contain herself any longer.\n\n'Why _disgusting_?' G\u00e9raldine protested. 'As long as the participants are freely consenting, I don't see the problem. It's a contract, that's all.'\n\n'I don't believe you can _freely consent_ to humiliation and suffering. And even if you can, I don't think it's reason enough.'\n\nVal\u00e9rie was really angry. For a moment I thought about moving the conversation on to the Arab\u2013Israeli war, then I realised that I didn't give a shit what these girls thought; if they never phoned me again, it would simply reduce my workload. 'Yeah, I find these people a little disgusting...' I upped the ante, 'And I find you disgusting too...' I said more quietly.\n\nG\u00e9raldine didn't hear, or she pretended not to hear. 'If I'm a consenting adult,' she went on, 'and my fantasy is to suffer, to explore the masochistic part of my sexuality, I don't see any reason why anyone should try and stop me. We are living in a democracy...' She was getting angry too, I could sense that it wouldn't be long before she mentioned human rights. At the word 'democracy', Bredane shot her a slightly contemptuous look; he turned to Val\u00e9rie. 'You're quite right...' he said gravely, 'it's completely disgusting. When I see a man agreeing to have his nails torn out with a pair of pliers, then have someone shit on him, and eat his torturer's shit, I find that disgusting. But, it's precisely what is disgusting in the human animal that interests me.'\n\nAfter a few seconds, Val\u00e9rie asked in an agonised voice: 'Why?...'\n\n'I don't know,' Bredane answered simply. 'I don't believe we have a dark side, because I don't believe in any form of damnation, nor in benediction for that matter. But I have a feeling that as we get closer to suffering and cruelty, to domination and servility, we hit on the essential, the intimate nature of sexuality. Don't you think so?...' He was talking to me now. No, actually, I didn't think so. Cruelty is a primordial part of the human, it is found in the most primitive peoples: in the earliest tribal wars, the victors were careful to spare the lives of some of their prisoners to let them die later, suffering hideous tortures. This tendency persisted, it is constant throughout history, it remains true today: as soon as a foreign or civil war begins to erase ordinary moral constraints, you find human beings \u2013 regardless of race, people, culture \u2013 eager to launch themselves into the joys of barbarism and massacre. This is attested, unchanging, indisputable, but it has nothing whatever to do with the quest for sexual pleasure \u2013 equally primordial, equally strong. So, all in all, I didn't agree; but I was aware, as always, that the discussion was pointless.\n\n'Let's take a look round...' said Bredane after he'd finished his beer. I followed him, along with the others, into the first torture chamber. It was a vaulted cellar, the brickwork exposed. The atmospheric music consisted of a series of very deep chords on an organ, overlaid with the shrieks of the damned. I noticed that the bass speakers were huge; there were red spotlights all over the place, masks and torture implements hung from iron racks; the conversion must have cost them a fortune. In an alcove, a bald, almost fleshless guy was chained by all four limbs, his feet trapped in a wooden contraption which kept him about a foot off the ground, his arms were raised by a pair of handcuffs attached to the ceiling. A booted, gloved dominatrix, dressed completely in black latex, circled him armed with a whip of fine lashes encrusted with precious stones. First, and for a long time, she thrashed his buttocks with heavy strokes; the guy was facing us, completely naked; he screamed in pain. A small crowd gathered around the couple. 'She must be at level two...' Bredane whispered to me. 'Level one is where you stop when you see first blood.' The guy's cock and balls hung down, stretched and almost contorted. The dominatrix circled round him, rummaged in a bag on her belt and took out a number of hooks which she stuck into his scrotum; a little blood beaded on the surface. Then, more gently, she began to whip his genitals. It was a very close thing: if one of the lashes caught on a hook, the skin of the scrotum could rip. Val\u00e9rie turned her head and pressed herself against me. 'Let's go...' she said, her voice pleading; 'let's go, I'll explain later.' We went back to the bar; the others were so fascinated by the spectacle that they didn't notice us leave. 'The girl who was whipping that guy...' she told me quietly, 'I recognised her. I've only ever seen her once before, but I'm sure it's her... It's Audrey, Jean-Yves's wife.'\n\nWe left immediately. In the taxi Val\u00e9rie was silent, devastated. She remained silent in the lift and until we reached the apartment. It was only when the door closed behind us that she turned to me:\n\n'Michel... you don't think I'm too conventional?'\n\n'No, I hate that stuff too.'\n\n'I can understand that torturers exist: I find it disgusting, but I know there are people who take pleasure in torturing others; what I don't understand is that victims exist. It's beyond me that a human being could come to prefer pain to pleasure. I don't know \u2013 they need to be re-educated, to be loved, to be taught what pleasure is.'\n\nI shrugged my shoulders as if to suggest that the subject was beyond me \u2013 something which now happened in almost every aspect of my life. The things people do, the things they are prepared to endure... there was nothing to be made of all this, no overall conclusion, no meaning.\n\nI undressed in silence. Val\u00e9rie sat on the bed beside me. I sensed that she was still tense, preoccupied by the subject.\n\n'What scares me about it all,' she said, 'is that there's no physical contact. Everyone wears gloves, uses equipment. Skin never touches skin, there's never a kiss, a touch or a caress. For me, it's the very antithesis of sexuality.'\n\nShe was right, but I suppose that S&M enthusiasts would have seen their practices as the apotheosis of sexuality, its ultimate form. Each person remains trapped in his skin, completely given over to his feelings of individuality; it was one way of looking at things. What was certain, in any case, was that that kind of place was increasingly fashionable. I could easily imagine girls like Marjorie and G\u00e9raldine going to them, for example; although I had trouble imagining them being able to abandon themselves sufficiently for penetration, or indeed any kind of sexual scenario.\n\n'It's more straightforward than you might think...' I said at length. 'There's the sexuality of those who love each other, and the sexuality of those who don't love each other. When there's no longer any possibility of identifying with the other, the only thing left is suffering \u2013 and cruelty.'\n\nVal\u00e9rie pressed against me. 'We live in a strange world...' she said. In a sense, she was still innocent, protected from human reality by her insane working hours, which left her barely enough time to do the shopping, sleep, start again. She added: 'I don't like the world we live in.'\n\n### 6\n\n###### _It became apparent from our research that consumers have three major expectations: the desire to be safe, the desire for affection and the desire for beauty_.\n\n###### Bernard Guilbaud\n\nON JUNE 30, THE reservation figures from the travel agencies arrived. They were excellent. Eldorador Discovery was a success, it had immediately achieved better results than Eldorador Standard \u2013 which continued to slide. Val\u00e9rie decided to take a week's holiday: we went to her parents at Saint-Quay-Portrieux. I felt rather old to be playing the role of the fianc\u00e9 brought to meet the parents; after all, I was thirteen years older than she was, and this was the first time I had ever been in such a situation. The train stopped at Saint-Brieuc, her father was waiting for us at the station. He kissed his daughter warmly and hugged her to him for a long time; you could see that he missed her. 'You've lost a bit of weight...' he told her. Then he turned to me and offered me his hand, without really looking at me. I think he was intimidated too: he knew I worked for the Ministry of Culture, while he was just a farmer. Her mother was much more talkative; she grilled me at length about my life, my work, my hobbies. In the event, it wasn't so difficult. Val\u00e9rie was at my side; from time to time she answered for me and we would exchange looks. I couldn't imagine how I might behave in a situation like this if one day I had children; I couldn't really imagine much about the future.\n\nThe evening meal was a real feast: lobster, saddle of lamb, several cheeses, a strawberry tart and coffee. For my part, I was tempted to see this as evidence of acceptance, although obviously I knew that the menu had been planned in advance. Val\u00e9rie took the brunt of the conversation, mostly talking about her new job \u2013 about which I knew just about everything. I let my gaze wander over the curtain material, the ornaments, the family photos in their frames; it was touching and a little frightening.\n\nVal\u00e9rie insisted on sleeping in the room she had had as a teenager. 'You'd be better off in the guest room,' her mother insisted; 'two of you will be pushed for space.' It was true the bed was a little narrow, but I was very moved, as I pushed Val\u00e9rie's panties down and stroked her pussy, to think that this was where she slept when she was only thirteen or fourteen. Wasted years, I thought. I knelt at the foot of the bed, took off her pants completely and turned her towards me. Her vagina closed over the tip of my penis. I pretended to penetrate her, going in a couple of inches and pulling back in quick, short thrusts, squeezing her breasts between my hands. She came with a muffled cry, then burst out laughing. 'My parents...' she whispered, 'they're not asleep yet.' I penetrated her again, harder this time so that I could come. She watched me, her eyes shining, and placed a hand over my mouth just as I came inside her with a hoarse moan.\n\nLater, I studied the furniture in the room curiously. On a shelf, just above the _Biblioth\u00e8que Rose_ series, there were several little exercise books, carefully bound. 'Oh, those,' she said; 'I used to do them when I was about ten, twelve. Have a look if you like. They're Famous Five stories.'\n\n'How do you mean?'\n\n'Unpublished Famous Five stories, I used to write them myself, using the same characters.'\n\nI took them down: there was _Five in Outer Space, Five on a Canadian Adventure_. I suddenly had an image of a little girl full of imagination, a rather lonely girl, whom I would never know.\n\nIn the days that followed, we didn't do much beyond going to the beach. The weather was beautiful, but the water was too cold to swim in for long. Val\u00e9rie lay in the sun for hours at a time; she was recovering gradually: the last three months had been the hardest of her working life. One evening, three days after our arrival, I talked to her about it. It was at the Oceanic Bar; we'd just ordered cocktails.\n\n'You won't have so much work now, I suppose, now that you've launched the package?'\n\n'In the short term, no,' she smiled cynically. 'But we'll have to come up with something else pretty quickly.'\n\n'Why? Why not just stop at that?'\n\n'Because that's how the game goes. If Jean-Yves were here he'd tell you that that's the capitalist principle: if you don't move forward, you're dead. Unless you have a decisive competitive advantage which you can bank on for a couple of years; but we're not there yet. The principle of Eldorador Discovery is good \u2013 it's clever, canny if you like, but it's not really innovative, it's just a good mix of other concepts. The competition will see that it works and before you know it, they'll be doing the same thing. It's not that difficult to do; the hard part was setting it up in so little time. But I'm sure that Nouvelles Fronti\u00e8res, for example, would be able to offer a similar package by next summer. If we want to keep our advantage, we have to innovate again.'\n\n'And it never ends?'\n\n'I don't think so, Michel. I'm well paid, I work in an industry I understand; I accept the rules of the game.'\n\nI must have looked serious; she put her hand on my neck. 'Let's go and eat...' she said. 'My parents will be waiting for us.'\n\nWe went back to Paris on Sunday evening. Val\u00e9rie and Jean-Yves had a meeting with \u00c9ric Leguen on Monday morning. He made a point of personally expressing the group's satisfaction with the first results of their recovery plan. As a bonus, the board of directors had unanimously decided to allocate shares to each of them, exceptional for executives who had been with the company less than a year.\n\nThat evening, the three of us had dinner in a Moroccan restaurant on the Rue des \u00c9coles. Jean-Yves was unshaven, his head was nodding and he looked a little puffy. 'I think he's started drinking,' Val\u00e9rie said to me in the taxi. 'He had a dreadful holiday with his wife and kids on the \u00cele de R\u00e9. They were supposed to be there for two weeks, but he left after a week. He told me he couldn't bear his wife's friends any more.'\n\nIt was true he didn't look at all well: he didn't touch his tagine, he constantly poured himself more wine. 'Here we go!' he said sarcastically, 'Here we go! We're getting into serious money now!' He shook his head, drained his wineglass, 'Sorry...' he said pitifully, 'sorry, I shouldn't talk like that.' He placed his hands, trembling slightly, on the table, waited; slowly he stopped trembling. Then he looked Val\u00e9rie straight in the eye.\n\n'You know what happened to Marylise?'\n\n'Marylise Le Fran\u00e7ois? No, I haven't seen her. Is she sick?'\n\n'Not sick, no. She spent three days in hospital on tranquillisers, but she's not sick. Actually, she was attacked, raped on the train to Paris, on her way home from work last Wednesday.'\n\nMarylise returned to work the following Monday. It was obvious she had been badly shaken; her movements were slow, almost mechanical. She told her story easily, too easily, it didn't seem natural: her voice was neutral, her face expressionless, rigid, it was as if she was unthinkingly repeating her police statement. Leaving work at 10.15 p.m., she had decided to take the 10.21 p.m. train, thinking it would be quicker than waiting for a taxi. The carriage was three-quarters empty. Four guys came up to her and immediately started insulting her. As far as she could tell, they were West Indian. She tried to talk to them, make a joke; for her trouble she got a couple of slaps which knocked her half-unconscious. At that point, they jumped her, two of them holding her down on the floor. Violently, brutally, they penetrated her every orifice. Every time she tried to make a sound, she was punched or slapped. It had gone on for a long time, the train had stopped at several stations; passengers got off, warily changed carriages. As the guys took turns raping her, they continued to taunt and insult her, calling her a slut and a douche-bag. By the end there was no one in the compartment. They ended up in a circle around her, spitting and pissing on her, then they shoved her with their feet, until she was half-hidden under one of the seats, then they calmly got off the train at the Gare de Lyon. Two minutes later, the first passengers to board called the police, who arrived almost immediately. The superintendent wasn't really surprised; according to him she'd been relatively lucky. Quite often when they had used the girl, these guys would end up shoving a piece of wood studded with nails into her vagina or her anus. The line was classed as dangerous.\n\nAn internal memo reminded employees of the usual safety measures, repeated that taxis were at their disposal should they need to work late and that fares would be entirely covered by the company. The number of security guards patrolling the grounds and the car park was increased.\n\nThat evening, as her car was being repaired, Jean-Yves drove Val\u00e9rie home. As he was stepping out of his office, he looked out over the chaotic landscape of houses, shopping centres, tower-blocks and motorway interchanges. Far away, on the horizon, a layer of pollution lent the sunset strange tints of mauve and green. 'It's strange,' he said to her, 'here we are inside the company like well-fed beasts of burden. And outside are the predators, the savage world. I was in S\u00e3o Paulo once, that's where evolution has really been pushed to its limits. It's not even a city any more, it's a sort of urban territory which extends as far as the eye can see, with its favelas, its huge office blocks, its luxury housing surrounded by guards armed to the teeth. It has a population of more than twenty million, many of whom are born, live and die without ever stepping outside the limits of its terrain. The streets are dangerous there: even in a car someone might pull a gun on you at a traffic light, or you might wind up being tailed by a gang; the really well-equipped gangs have machine-guns and rocket launchers. Businessmen and rich people use helicopters to get around almost all the time; there are helipads pretty much everywhere, on the roofs of banks and apartment blocks. At ground level the street is left to the poor \u2013 and the gangs.'\n\nAs he turned on to the motorway heading south, he added in a low voice: 'I've been having doubts lately. More and more now I have doubts about the sort of world we're creating.'\n\nA couple of days later, they returned to the subject. After he had parked on the avenue de Choisy, Jean-Yves lit a cigarette; he was silent for a moment, then he turned to Val\u00e9rie: 'I feel really terrible about Marylise... the doctors said she could go back to work, and it's true that in a sense, she's back to normal, she's not having panic attacks. But she never takes the initiative, it's as if she's paralysed. Every time there's a decision to be made, she comes and asks me; and if I'm not there, she's capable of waiting for hours without lifting a finger. For a marketing manager, it's not good enough; it can't go on like this.'\n\n'You're not going to fire her?'\n\nJean-Yves stubbed out his cigarette, stared out of the car for a long time; he gripped the steering wheel. He seemed to be more and more tense, unsettled; Val\u00e9rie noticed that even his suit was sometimes stained nowadays.\n\n'I don't know,' he whispered at last, with difficulty. 'I've never had to do anything like this. I couldn't fire her, that would be really shitty; but I'll have to find her another job where she has fewer decisions to make, fewer dealings with people. To make matters worse, ever since it happened, she's become more and more racist in her reactions. It's understandable, it's not hard to understand, but in the tourist industry it's just not acceptable. In our advertising, our catalogues, all our marketing material, we portray the locals as warm, welcoming, friendly people. That's the way it is: it really is a professional obligation.'\n\nThe following day, Jean-Yves broached the subject with Leguen, who had fewer qualms, and, a week later, Marylise was transferred to the accounts department to replace an employee who had just retired. Another marketing manager needed to be found for Eldorador. Jean-Yves and Val\u00e9rie handled the job interviews together. After they had seen about ten candidates, they had lunch in the company cafeteria to discuss the appointment.\n\n'I'd be quite tempted to go with Noureddine,' said Val\u00e9rie. 'He's incredibly talented, and he's already worked on quite a variety of projects.'\n\n'Yes, he is the best of the bunch; but I wonder if he might be a bit overqualified for the job. I can't really see him doing marketing for a travel company, I see him in something more prestigious, more _arty_. He'll get bored here, he won't stay. Our target market really is middle-of-the-road. And his parents being _beurs_ , that could cause problems. To appeal to people, we have to use a lot of clich\u00e9s about Arab countries: the hospitality, the mint tea, the festivals, the Bedouins... I've found that kind of thing doesn't really go down too well with Arabs here; in fact a lot of them don't really like Arab countries.\n\n'That's racial discrimination...' Val\u00e9rie said sardonically.\n\n'Don't be stupid!' Jean-Yves was getting angry; since he had come back from holiday, he was clearly overstressed, he was beginning to lose his sense of humour. 'Everyone does it. A person's origins are part of their personality, you have to take them into consideration, it's obvious. For example, I'd happily take a Moroccan or Tunisian immigrant \u2013 even one much more recent than Noureddine \u2013 to handle the negotiations with local suppliers. They have a foot in both camps which is a real advantage \u2013 the people they deal with are always wrong-footed. On top of that, they come across as someone who's made it in France, so the guys respect them immediately, they don't think they're going to be ripped off. The best negotiators I've ever had have always been people like that. But here, for this job, I'd be more tempted to take Birgit.'\n\n'The Danish girl?'\n\n'Yes. Purely as a designer; she's also very talented. She's really anti-racist \u2013 I think she lives with a Jamaican guy \u2013 she's a bit stupidly enthusiastic about anything exotic, on principle. She has no intention of having children just yet. All in all, I think she's the right fit.'\n\nThere was perhaps another reason, too, Val\u00e9rie realised some days later when she surprised Birgit putting her hand on Jean-Yves's shoulder. 'Yeah, you're right...' he admitted as they had coffee by the vending machine, 'my rap-sheet is getting worse, now I'm getting into sexual harassment... Look, it's only happened once or twice and it won't go any further than that \u2013 in any case, she's got a boyfriend.' Val\u00e9rie looked at him quickly. He needed a haircut. 'I wasn't having a go at you...' she said. Intellectually, he hadn't slipped at all: he was still capable of flawless assessments of situations and people, had an excellent eye for a financial set-up; but he seemed increasingly like a man who was unhappy, adrift.\n\nThey began to assess the customer-satisfaction surveys; a large number had been filled in, thanks to a prize draw in which the first fifty won a week's holiday. At first glance, the reasons for their dissatisfaction with Eldorador Standard were difficult to establish. The customers were satisfied with the accommodation and the location, satisfied with the food, satisfied with the activities and the sports offered; but that said, fewer and fewer of them were returning customers.\n\nBy chance, Val\u00e9rie happened on an article in _Tourisme Hebdo_ analysing consumers' new values. The author claimed to use the Holbrook and Hirschman model, which focuses on the emotion the consumer feels when faced with a product or service; but the conclusions were nothing new. The 'new consumers' were described as being less predictable, more eclectic, more sophisticated, more concerned with humanitarian issues. They no longer consumed to 'seem', but to 'be': _more serenity_. They had balanced diets, were careful about their health; they were slightly fearful of others and of the future. They demanded the right to be unfaithful out of curiosity, out of eclecticism; they favoured things which were solid, durable, authentic. They had ethical leanings: _more solidarity_ , etc. She had read all these things a hundred times: behavioural psychologists and sociologists transplanted the same words from one article to another, one magazine to another. In any case, they had already taken all these factors into account. The Eldorador villages were built of traditional materials, following the architectural traditions of the host country. The self-service menus were balanced, with ample room given over to selections of fresh vegetables, fruit, the Cretan diet. Among the activities on offer were yoga, relaxation therapy and Tai Chi. Aurore had signed the ethical tourism charter, gave regularly to the WWF. None of these things seemed sufficient to halt the decline.\n\n'I think people are just lying,' said Jean-Yves, having reread the summary of the customer surveys a second time. 'They say they're satisfied, they tick the box marked \"Good\" every time, but in reality they've been bored stiff for the whole holiday and they feel too guilty to admit it. I'm going to end up selling off all the resorts we can't convert to the Discovery formula and really go for it on the activity holidays: add four-by-fours, hot air balloon trips, traditional feasts in the desert, trips in dug-out canoes, scuba diving, white-water rafting, the works...'\n\n'We're not the only ones in the market.'\n\n'No...' he agreed, disheartened.\n\n'We should try spending a week at one of the clubs, incognito, not for any particular reason, just to see what the atmosphere is like.'\n\n'Yeah...' Jean-Yves sat up in his armchair, took a sheaf of listings. He flicked quickly through the pages. 'Djerba and Monastir are a disaster, but I think we're going to drop Tunisia altogether anyway. It's already too built-up, the competition are prepared to drop their prices to ludicrous levels; given our positioning, we could never follow them.'\n\n'Have you got any offers to buy?'\n\n'Oddly enough, yes. Neckermann are interested. They want to get into the Eastern European market: Czechoslovakia, Hungary, Poland... very bottom of the market, but the Costa Brava is already saturated. They're interested in our Agadir resort as well; it's a reasonable offer. I'm quite tempted to sell to them; even with southern Morocco, Agadir isn't taking off \u2013 I think people will always prefer Marrakech.'\n\n'But Marrakech is awful.'\n\n'I know... The strange thing is that Sharm-el-Sheikh isn't doing all that well. It's got a lot going for it: the most beautiful coral reefs in the world, trips to the Sinai desert...'\n\n'Yeah, but it's in Egypt.'\n\n'And...?'\n\n'I don't think people have forgotten the terrorist attack in Luxor in 1997. After all, there were fifty-eight dead. The only way you're going to sell Sharm-el-Sheikh is to take out the word \"Egypt\".'\n\n'What do you want to put in its place?'\n\n'I don't know, \"The Red Sea\" maybe?'\n\n'OK, \"The Red Sea\" it is.' He made a note and went back to leafing through the figures. 'Africa is doing well... that's strange. Cuba comes rather low. But Cuban music, that whole Latin vibe is hip isn't it? The Dominican Republic is always full for example.' He read the description of the Cuban resort: 'The Guardalavaca hotel is almost new, it's competitively priced. Not too sporty, not too family-oriented. \"Live the magic of Cuban nights to the wild rhythms of salsa...\" Numbers are down 15 per cent. Maybe we could go and have a look at the place: either there or Egypt.'\n\n'Wherever you want, Jean-Yves...' she answered wearily. 'In any case, it will do you good to get away without your wife.'\n\nAugust had settled over Paris; the days were hot, almost stifling, but the good weather didn't last: after a day or two a storm would come, the air would suddenly become cool. Then the sun would come out, the mercury in the thermometer, and the pollution index, would begin to rise again. To tell the truth, it was of limited interest to me. I had given up on peep-shows since I had met Val\u00e9rie; I had also given up, many years ago, on the urban adventure. Paris had never been a moveable feast for me, and I could think of no reason why it should become one. Still, ten or twelve years ago, when I was starting out in the Ministry of Culture, I used to go out to clubs and bars that were 'unmissable'; all I remember was a slight but persistent feeling of unease. I had nothing to say, I felt completely incapable of starting a conversation with anyone at all, I didn't know how to dance either. It was in such circumstances that I started to become an alcoholic. Alcohol didn't let me down, never once in my life: it has been a constant support to me. After about ten gin-and-tonics, I occasionally \u2013 pretty rarely, all in all it must have happened four or five times \u2013 managed to find the requisite energy to ask a girl to share my bed. The results, in general, were pretty disappointing: I couldn't get it up, and I usually fell asleep after a couple of minutes. Later, I discovered the existence of Viagra; elevated blood-alcohol levels limited its effectiveness a lot, but if you boosted the dosage, you could still get somewhere. The game, in any case, wasn't really worth the candle. In fact, before Val\u00e9rie, I had never met a single girl who could come close to a Thai prostitute; or maybe when I was very young, I managed to feel something when I was with girls of sixteen or seventeen. But in the world I moved in it was a complete disaster. The girls weren't remotely interested in sex, only in seduction \u2013 and even then it was a kind of \u00e9litist, trashy, bizarre seduction that was not the least bit erotic. In bed, they were simply incapable of the least thing. Either that, or they needed fantasies, a whole lot of fastidious, kitschy scenarios, the mere mention of which was enough to make me sick. They liked to talk about sex, that much was true, in fact it was their only real topic of conversation; but they didn't have the slightest sensual innocence. Actually, the men weren't much better. In any case, the French have a penchant for talking about sex at every possible opportunity without ever doing anything; but it was seriously starting to depress me.\n\nAnything can happen in life, especially nothing. But this time at least something had happened in my life: I had found a lover and she made me happy. Our August was very quiet. Espitalier, Leguen and most of the other senior executives at Aurore had gone away on holiday. Val\u00e9rie and Jean-Yves had decided not to make any important decisions before the Cuba trip at the beginning of September; it was a break, a period of calm. Jean-Yves was a bit better. 'He finally decided to go see a whore,' I learned from Val\u00e9rie; 'He should have done it long ago. He's drinking less now, he's calmer.'\n\n'All the same, from what I remember, hookers aren't up to much.'\n\n'Yeah, but this is different, these girls work via the internet. They're pretty young, some of them are still students. They don't take many clients, they pick and choose, they don't do it just for the money. At least, he told me it's pretty good. If you want we can try it sometime. A bisexual girl for the two of us: I know men are turned on by all that and, actually, I like girls too.'\n\nWe didn't do it that summer; but the simple fact that she had suggested it was tremendously exciting. I was lucky. She knew the different things that kept male desire alive \u2013 well, not completely, that was impossible, but let's say enough to make love from time to time, while waiting for everything to come to an end. In fact, being aware of such things is nothing, it's so easy, so pathetic and easy; but she enjoyed doing these things, she took pleasure in them, she enjoyed seeing the desire rising in my eyes. Often, in a restaurant, when she came back from the toilet, she would place the panties she had just taken off on the table. Then she liked to slip a hand between my legs to make the most of my erection. Sometimes she would open my flies and jerk me off right there, hidden by the tablecloth. In the mornings, too, when she woke me with fellatio and handed me a cup of coffee before taking me into her mouth again, I would feel a dizzying rush of gratitude and gentleness. She knew how to stop just before I came, she could have kept me on the brink for hours. I lived inside a game, a game which was tender and exciting, the only game left to adults; I moved through a universe of gentle desires and limitless moments of pleasure.\n\n### 7\n\nAT THE END of August, the estate agent in Cherbourg phoned to tell me he had found a buyer for my father's house. The guy wanted me to drop the price a little, but he was prepared to pay cash. I accepted immediately. Very shortly, I would, therefore, receive a little more than two million francs. At the time, I was working on a proposal for a touring exhibition in which frogs were to be released onto playing cards spread out in a mosaic-tiled enclosure \u2013 some of the tiles had been engraved with the names of great men of history, such as D\u00fcrer, Einstein or Michelangelo. The lion's share of the budget was allocated to buying the decks of cards: they needed to be changed fairly frequently; the frogs had to be changed too, from time to time. The artist wanted, at least for the inaugural exhibition in Paris, to use Tarot cards; in the provinces, he was prepared to make do with ordinary playing cards. I decided to go to Cuba for a week with Val\u00e9rie and Jean-Yves in early September. I had intended to pay my way, but she told me she would sort things out with the group.\n\n'I won't get in the way of your work...' I promised.\n\n'We're not really going to work, you know, we'll just behave like ordinary tourists. We're not going to do anything much, but that in itself is very important: we're going to try and work out what's going wrong, why there's no atmosphere at the resort, why people don't come back thrilled from their holidays. You won't be in the way at all; on the contrary, you could be very useful.'\n\nWe took the mid-afternoon flight to Santiago de Cuba on Friday September 5. Jean-Yves hadn't been able to stop himself bringing along his laptop, but he seemed relaxed in his pale-blue polo neck, ready for a holiday. Shortly after take-off, Val\u00e9rie put her hand on my thigh; she relaxed, her eyes closed. 'I'm not worried, we'll work out what's wrong...' she'd said to me as we were leaving.\n\nThe transfer from the airport took two and a half hours. 'Negative number one...' noted Val\u00e9rie; 'we must check and see if there's a flight into Holguin.' In front of us in the coach, two little old ladies of about sixty, with blue-grey perms, twittered constantly, pointing out items of interest as we passed: men cutting sugar cane, a vulture wheeling over the fields, two cows returning to their byre... They had the air of ladies determined to be interested in everything, they seemed dry and difficult; I got the impression they wouldn't be easy customers. Sure enough, when the rooms were being allocated, twitterer A doggedly insisted on having a room next door to twitterer B. This sort of demand had clearly not been anticipated, the receptionist couldn't understand at all, the resort manager had to be sent for. He was about thirty, with a head like a ram and a stubborn air, his narrow brow furrowed with worry lines, in fact he looked a lot like the actor Nagui. 'No problem, okay...' he said when the issue had been explained to him; 'No problem, okay, my dear lady. This evening is not possible, but tomorrow we have some people leaving and we will change your room.'\n\nA porter took us to our ocean-view chalet, turned on the air conditioning and left with a dollar tip. 'There we go...' said Val\u00e9rie sitting down on the bed. 'The meals are served buffet style. It's an all-in package including snacks and cocktails. The disco opens at eleven. There's a supplement for massages and for lighting the tennis-courts at night.' The aim of tourist companies is to make people happy, for a specified price, for a specified period. The task can be an easy one, or it can prove impossible \u2013 depending on the nature of the people, the services offered and other factors. Val\u00e9rie took off her trousers and her blouse. I lay down on the other twin bed. A source of permanent, accessible pleasure, our genitals exist. The god who created our misfortune, who made us short-lived, vain and cruel, has also provided this form of meagre compensation. If we couldn't have sex from time to time, what would life be? A futile struggle against joints that stiffen, caries that form. All of which, moreover, is as uninteresting as humanly possible \u2013 the collagen which makes muscles stiffen, the appearance of microbic cavities in the gums. Val\u00e9rie parted her thighs above my mouth. She was wearing a pair of sheer tanga briefs in purple lace. I pushed the fabric aside and wet my fingers in order to stroke her labia. For her part, she undid my trousers and took my penis in the palm of her hand. She began to massage my balls gently, unhurriedly. I grabbed a pillow so my mouth would be at the same level as her pussy. At that moment, I saw a maid sweeping the sand from the terrace. The curtains and the window were wide open. As her eyes met mine, the girl burst out laughing. Val\u00e9rie sat up and motioned to her to come in. She stayed where she was, hesitant, leaning on her broom. Val\u00e9rie got up, walked towards her and held out her hands. As soon as the girl was inside, she started to open the buttons of her blouse: she was wearing nothing underneath but a pair of white cotton panties; she must have been about twenty, her body was very brown, almost black, she had a firm little bust and finely curved buttocks. Val\u00e9rie drew the curtains; I got up in turn. The girl's name was Margarita. Val\u00e9rie took her hand and placed it on my penis. She burst out laughing again, but started to jerk me off. Val\u00e9rie quickly took off her bra and panties, lay down on the bed and started to stroke herself. Again, Margarita hesitated for a moment, then she took off her panties and knelt between Val\u00e9rie's thighs. First she looked at her pussy, stroking it with her hand, then she brought her mouth closer and began to lick it. Val\u00e9rie put her hand on Margarita's head to guide her as she continued to jerk me off with her other hand. I felt that I was going to come; I backed off and went to look for a condom in my wash bag. I was so excited that I had trouble finding one. As I put it on, my vision seemed almost blurred. The little black girl's arse rose and fell as she bobbed over Val\u00e9rie's pubis. I penetrated her in one thrust, her pussy was open like a fruit. She moaned quietly, pushed her buttocks towards me. I started to thrust in and out of her any old how, my head was spinning, shudders of pleasure coursed through my body. It was getting dark, you could hardly see anything in the room now. From far, far away, as though from another world, I heard Val\u00e9rie's rising cries. I pressed my hands hard against Margarita's arse, thrusting into her harder and harder. At the moment Val\u00e9rie screamed, I came in turn. For a second or two I had the impression of weightlessness, of floating in space. Then the feeling of gravity returned, I suddenly felt exhausted. I collapsed on the bed into their arms.\n\nLater, I vaguely saw Margarita getting dressed. Val\u00e9rie rummaged in her bag to give her something. They kissed on the doorstep; outside, it was dark. 'I gave her forty dollars...' said Val\u00e9rie lying down again beside me; 'That's the price Western men pay. To her, it's a month's salary.' She turned on the bedside lamp. Silhouettes passed by, formed shadow puppets against the curtains; we could hear the murmur of conversation. I placed a hand on her shoulder.\n\n'It was great...' I said in a tone of incredulous wonder. 'It was really great.'\n\n'Yes, she's very sensual, that girl. She was really good when she went down on me too.'\n\n'It's strange, what sex costs...' I went on. 'I get the impression that it doesn't really depend on a country's standard of living. Obviously, depending on the country, what's on offer is completely different; but the basic price is always pretty much the same: the amount Westerners are prepared to pay.'\n\n'Do you think that's what they call _supply-side economics_?'\n\n'I've no idea...' I shook my head. 'I've never really understood anything about economics; it's like I have a mental block.'\n\nI was very hungry, but the restaurant didn't open until eight o'clock; I drank three pi\u00f1a coladas at the bar while watching the pre-dinner entertainment. The effects of orgasm dissipated only slowly, I was a bit tipsy and from a distance all the reps looked like Nagui. Actually, they didn't, some of them were younger, but they all had something odd about them, a shaven head, a goatee or dreadlocks. They gave terrifying cries and from time to time grabbed members of the audience to force them onstage. Thankfully, I was too far away to be in any serious danger.\n\nThe bar manager was pretty tiresome; he was, for want of a better word, useless: every time I needed something, he simply waved contemptuously in the direction of the waiters. He looked a bit like an elderly bullfighter, with his scars and his small, contained pot belly. His yellow swimsuit hugged his penis very precisely; he was well hung, and he was determined to let it be known. As I was heading back to my table, having obtained, with extreme difficulty, my fourth cocktail, I saw the man approach one of the neighbouring tables, occupied by a compact group of fifty-something _qu\u00e9b\u00e9coises_. I had already noticed them when they arrived: they were thickset and tough, all teeth and blubber, talking incredibly loudly; it wasn't difficult to understand how they had managed to bury their husbands so quickly. I had a feeling that it wouldn't be wise to let them go in front of you in the queue at the self-service, or to grab a bowl of cereal that one of them had her eyes on. As the ageing hunk approached the table, they shot him amorous glances, almost becoming women again in the process. He strutted extravagantly in front of them, accentuating his coarseness at regular intervals by gestures through his swimsuit, as though to confirm the physical existence of his meat and two veg. The fifty-something _qu\u00e9b\u00e9coises_ seemed thrilled by his suggestive company; their aged, worn-out bodies still craved sunshine. He played his part well, whispered softly into the ears of these old creatures, referring to them, Cuban fashion, as ' _mi corazon_ ' or ' _mi amor_ '. Nothing more would come of this, that was clear: he was content to arouse some last quivers in their ageing pussies; but perhaps that was sufficient for them to go home with the impression that they had had a wonderful holiday. For them to recommend the holiday club to their girlfriends. I sketched out the plot of a socially aware pornographic film entitled _Senior Citizens on the Rampage_. It portrayed two gangs operating in a holiday club, one a group of elderly Italian men, the other of pensionettes from Quebec. Armed with nunchakus and ice picks, both gangs submit naked, bronzed teenagers to the most vile indecencies. Eventually, of course, they come face to face in the middle of a Club Med yacht; one after another the crew members, quickly rendered help-less, are raped before being thrown overboard by the bloodthirsty pensionettes. The film ends with a mammoth orgy of pensioners, while the boat, having slipped its moorings, sails straight for the South Pole.\n\nEventually, Val\u00e9rie joined me: she was wearing make-up and a short, white, see-through dress; I still wanted her. We found Jean-Yves at the buffet. He seemed relaxed, almost languid, and desultorily informed us of his first impressions. His room was acceptable, the entertainment seemed a little intrusive; he had just been up by the sound system and it was almost unbearable. The food wasn't up to much, he added, staring bitterly at his piece of boiled chicken. All the same, everyone seemed to be helping themselves generously, coming back to the buffet again and again; the OAPs in particular were astonishingly rapacious \u2013 you'd almost have thought they had spent the afternoon exhausting themselves at water sports and beach volleyball. 'They eat, they eat...' Jean-Yves observed wearily; 'What else do you expect them to do?'\n\nAfter dinner, there was a show where audience participation was once again called for. A woman of about fifty launched into a karaoke version of 'Bang-Bang' by Sheila. It was pretty brave of her; there was a smattering of applause. For the most part, however, the show was run by the reps. Jean-Yves looked as though he was ready to fall asleep; Val\u00e9rie calmly sipped on her cocktail. I looked at the next table: the people gave the impression that they were a little bored but they applauded politely at the end of each song. Customer dissatisfaction with holiday clubs didn't seem to me too difficult to understand; it appeared to be staring us in the face. The clientele was made up of OAPs or people 'of a certain age' and the reps seemed to be trying to doing their utmost to take them to heights of pleasure they could no longer attain, at least not that way. Val\u00e9rie and Jean-Yves, even I myself, in some sense, still had professional responsibilities in the real world; we were sober, respectable employees, each exhausted by routine worries, and suchlike. Most of the people sitting at these tables were in the same position: they were managers, teachers, doctors, engineers, accountants; or retired people who had once been employed in those professions. I couldn't understand how the reps could possibly expect us to launch ourselves enthusiastically into get-to-know-you evenings or song contests. I couldn't work out how at our age, in our position, we were supposed to have kept alive our sense of fun. At best, the entertainment had been designed to amuse the under-fourteens.\n\nI tried to let Val\u00e9rie know my thoughts, but the holiday rep had started speaking again; he was holding the microphone too close and it made a terrible row. Now they were performing an improvisation inspired by Lagaf, or maybe by Laurent Baffie; whichever it was, they were sauntering around carrying palm fronds while a girl dressed as a penguin followed them, laughing at everything they said. The show ended with the club anthem and some silly dances; a few people in the front row moved about half-heartedly. Standing beside me, Jean-Yves stifled a yawn. 'Shall we go check out the disco?' he suggested.\n\nThere were about fifty people, but the reps were pretty much the only ones dancing. The DJ played a mix of techno and salsa. Eventually, a number of middle-aged couples tried a salsa. The organiser with the palm fronds wandered between the couples on the dance floor, clapping his hands and shouting: ' _Caliente! Caliente!_ '; I got the impression they found him more embarrassing than anything else. I took a seat at the bar and ordered a pi\u00f1a colada. Two cocktails later, Val\u00e9rie nudged me with her elbow, pointing to Jean-Yves. 'I think maybe we can leave him to it,' she whispered into my ear. He was talking to a very pretty girl of about thirty, probably Italian. They were very close, shoulder to shoulder; their faces inches from one other.\n\nThe night was hot, muggy. Val\u00e9rie took me by the arm. The rhythm of the disco died away; we could hear the drone of walkie-talkies, guards patrolled the inside of the compound. Past the pool, we turned left towards the ocean. The beach was deserted. Waves gently licked the sand a few feet from us; we could no longer hear a sound. Arriving at the chalet, I undressed and lay down on the bed to wait for Val\u00e9rie. She brushed her teeth, undressed in turn and came to join me. I pressed myself against her naked body. I placed one hand on her breasts, the other in the hollow of the belly. It was sweet.\n\n### 8\n\nWHEN I WOKE up I was alone and I had a slight headache. I staggered out of bed, lit a cigarette; after a couple of drags, I felt a bit better. I slipped on a pair of trousers, went out on to the terrace, which was covered in sand \u2013 it must have been windy during the night. Day had only just broken; the sky seemed cloudy. I walked a few metres towards the sea, and spotted Val\u00e9rie. She was diving straight into the waves, swimming a few strokes, getting up and diving again.\n\nI stopped, pulling on my cigarette; the wind was a little chilly, I hesitated to join her. She turned, saw me and shouted: 'Come on!' waving to me. At that moment, the sun burst from between two clouds, lighting her from the front. Light gleamed on her breasts and her hips, made the foam in her hair and her pubic hair sparkle. I stood rooted to the spot for a second or two, conscious that this was an image that I would never forget, that it would become one of those images which apparently flash before you in the few seconds which precede death.\n\nThe cigarette butt burned my fingers, I threw it on to the sand, undressed and walked towards the sea. The water was cool, very salty; it was a rejuvenating experience. A band of sunshine glimmered on the surface of the water, running straight to the horizon; I held my breath and dived into the sunlight.\n\nLater, we huddled together in a towel, watching day break over the ocean. Little by little, the clouds dispersed and the patches of light grew. Sometimes, in the morning, everything seems simple. Val\u00e9rie threw down the towel, offering her body to the sun. 'I don't feel like getting dressed...' she said. 'A bit...' I ventured. A bird glided low, scanning the surface of the water. 'I really like swimming, I really like making love...' she told me again. 'But I don't like dancing, I don't know how to enjoy myself, and I've always hated parties. Do you think that's normal?'\n\nI hesitated for a long time before replying. 'I don't know...' I said at last. 'All I know is that I'm the same.'\n\nThere weren't many people at the breakfast tables, but Jean-Yves was already there, sitting with a coffee in front of him, cigarette in hand. He hadn't shaved, and it looked as if he hadn't had much sleep; he gave us a little wave. We sat down opposite him.\n\n'So, everything go well with the Italian girl?' asked Val\u00e9rie, making a start on her scrambled eggs.\n\n'Not really, no. She started telling me all about her job in marketing, her problems with her boyfriend, how that was why she'd come on holiday. She got on my nerves, I went to bed.'\n\n'You should give the chambermaids a go...'\n\nHe smiled vaguely, stubbed out his cigarette in the ashtray.\n\n'So, what are we up to today?' I asked. 'I mean... well, this is supposed to be a discovery holiday.'\n\n'Oh, yes...' Jean-Yves wearily pulled a face. 'Well, kind of. I mean, we didn't have time to get much set up. This is the first time I've worked with a socialist country; it seems it's a bit difficult getting things arranged at the last minute in socialist countries. Anyway, this afternoon, there's something involving dolphins...' He stopped himself, tried to be a little more precise. 'Well, if I've got it right, it's a dolphin show, and afterwards you can go swimming with them. I suppose you climb on their backs or something like that.'\n\n'Oh yeah, I know,' said Val\u00e9rie. 'It's crap. Everyone thinks that dolphins are these sweet, friendly mammals and stuff. Actually, it's not true, they live in highly structured hierarchical groups with a dominant male and they're really aggressive: they often fight to the death. The only time I ever tried swimming with dolphins, I was bitten by a female.'\n\n'Okay, okay...' Jean-Yves spread his hands in a gesture of appeasement. 'Whatever the deal is, this afternoon there's dolphins for those who are interested. Tomorrow and the day after we're on a two-day trip to Baracoa; that should be pretty good, at least, I hope so. And then...' he thought for a moment; 'And then that's it. Actually, no, on the last day, before we head off to the airport, there's a lobster lunch and a visit to the cemetery in Santiago.'\n\nA few seconds' silence followed this pronouncement. 'Yeah...' Jean-Yves continued, 'I think we fucked up choosing this as our destination.'\n\n'In fact...' he went on after a moment's thought, 'I get the impression things aren't going too well at this resort. Well, I mean, not just from my point of view. Last night, at the disco, I didn't get the impression there were many couples getting together, even among the young people.' He was silent again for a few seconds ' _Ecco_...' he concluded, with a gesture of resignation.\n\n'The sociologist was right...' said Val\u00e9rie, thoughtfully.\n\n'What sociologist?'\n\n'Lagarrigue. The behavioural sociologist. He was right when he said we're a far cry from the days of the sun worshippers.'\n\nJean-Yves finished his coffee, shook his head bitterly. 'Really...' he said disgustedly, 'I really never thought that one day I'd feel nostalgic about the days of the sun worshippers.'\n\nTo get to the beach, we had to suffer an ambush of people hawking shitty handicrafts; but it was okay, there weren't too many, and they weren't too persistent \u2013 you could get rid of them with smiles and apologetic waves of the hand. During the day, Cubans had access to the hotel beach. They haven't got much to offer or to sell, Val\u00e9rie explained to me; but they try, they do their best. Apparently, no-one in this country could get by on just their wages. Nothing really worked: there was no petrol for the engines or spare parts for machines. Hence the sense of a rustic utopia which you noticed crossing the countryside: farmers working with oxen, getting about in horses and carts... But this was no utopia, nor some environmentalist re-creation: it was the reality of a country which could no longer sustain itself in the industrial age. Cuba still manages to export some agricultural produce like coffee, cocoa and sugar cane; but industrial output has fallen almost to zero. It's difficult to find even the most basic consumer products: soap, paper, biros. The only well-stocked shops sell imported products and you have to pay in dollars. So, everyone in Cuba gets by thanks to some secondary, tourist-related work. The privileged work directly for the tourist industry; the others try to get their hands on dollars, one way or another, in other services or through smuggling.\n\nI lay down on the sand to think. The bronzed men and women weaving between the tourists thought of us purely as wallets on legs, there was no point in deluding oneself; but it was just the same in every third-world country. What was particular about Cuba was this glaring problem with industrial production. I myself was completely incompetent in matters of industrial production. I was perfectly adapted to the information age, that is to say good for nothing. Like me, Val\u00e9rie and Jean-Yves knew only how to manage information and capital; they used their knowledge intelligently, competitively, while I used mine in more mundane, bureaucratic ways. But if, for example, a foreign power were to impose a blockade, not one of the three of us, nor anyone I knew, would have been capable of getting industrial production up and running again. We had not the least idea about casting metal, manufacturing parts, thermoforming plastics. Not to mention more complex objects like fibre optics or microprocessors. We lived in a world made up of objects whose manufacture, possible uses and functions were completely alien to us. I glanced around me, panic-stricken by this realisation: there was a towel, a pair of sunglasses, sun screen, a paperback by Milan Kundera. Paper, cotton, glass; complex machines, sophisticated manufacturing processes. Val\u00e9rie's swimsuit, for example, I was incapable of grasping the manufacturing process which had gone into making it: it was made of 80 per cent latex, 20 per cent polyurethane. I slipped two fingers under her bikini; under the artificial fibre construction I could feel the living flesh. I slipped my fingers in a little further, felt the nipple harden. This was something I could do, that I knew how to do. Little by little the heat became sweltering. Once in the water, Val\u00e9rie took off her bikini. She wrapped her legs around my waist and lay, floating on her back. Her pussy was already open, I smoothly penetrated her, thrusting inside her to the rhythm of the waves. There was no alternative. I stopped just before I came. We came back to dry ourselves in the sun.\n\nA couple passed us, a big black guy and a girl with very white skin, a nervous face and close-cropped hair, who looked at him as she talked, laughing too loudly. She was obviously American, maybe a journalist with the _New York Times_ or something like that. In fact, looking more closely, there were quite a lot of mixed couples on the beach. Further off, two big blond, slightly overweight guys with nasal accents laughed and joked with two superb girls with coppery skin.\n\n'They're not allowed to bring them back to the hotel...' said Val\u00e9rie, following my gaze. 'There are rooms you can rent in a village nearby.'\n\n'I thought Americans weren't allowed to come to Cuba.'\n\n'They're not, in theory, but they travel via Canada or Mexico. In fact, they're furious that they've lost Cuba. You can see why...' she said pensively. 'If ever there was a country in need of sex tourism, it's theirs. But for the moment, American companies are subject to the blockade, they're simply not allowed to invest. In any case, the country will end up becoming capitalist again, it's just a matter of years; but until then, the field is open for Europeans. That's why Aurore doesn't want to give up on it, even though the holiday club is having problems: now's the time to get an edge on the competition. Cuba represents a unique opportunity in the Caribbean\u2013West Indies zone.\n\n'Yep...' she went on cheerfully after a moment's silence. 'That's how we talk in my line of work... in the world of the global economy.'\n\n### 9\n\nTHE MINIBUS TO Baracoa left at eight in the morning; there were about fifteen people on board. They had already had an opportunity to get to know each other and were full of enthusiasm for the dolphins. The retirees (the majority), the two speech therapists who took their holidays together and the student couple, naturally, expressed their enthusiasm in slightly different lexical registers; but all would have felt able to agree on the following: a unique experience.\n\nAfterwards, the conversation turned to the features of the resort. I shot a glance at Jean-Yves sitting alone in the middle of the minibus. He had placed a notepad and a pen on the seat next to him. Leaning forward a little, his eyes half-closed, he was concentrating on getting down everything that was said. It was at this stage, obviously, that he hoped to glean a generous harvest of useful observations and impressions.\n\nOn the subject of the resort, too, there seemed to be a consensus of opinion among the members. The reps were unanimously considered 'nice', but the activities themselves were not very interesting. The rooms were good, except those close to the sound system, which were too noisy. As for the food, it really wasn't up to much.\n\nNone of those present took part in the early morning aerobics, or the salsa or Spanish lessons. In the end, what they liked best was the beach; all the more so as it was quiet. 'Activities and sound levels considered irritating', noted Jean-Yves on his pad.\n\nThe chalets received general approval, especially as they were far from the disco. 'Next time, we'll insist on a chalet!' a heavy-set retired man said emphatically; he was in the prime of life and evidently used to giving orders, in fact he had spent his entire career marketing the wines of Bordeaux. The two students were of the same opinion. 'Disco unnecessary', noted Jean-Yves, thinking despondently of all the useless investment.\n\nAfter the Cayo Saetia junction, the road got steadily worse. There were potholes and cracks which sometimes covered half the road surface. The driver was forced to zigzag continuously; we rattled around in our seats, pitched from left to right. The passengers reacted with shouts and laughter. 'It's okay, they're good-natured...' Val\u00e9rie said to me quietly. 'That's the great thing about Discovery Tours: you can subject them to horrible conditions; to them it's all part of the adventure. In this case, it's our fault: for this kind of trip you need a four-wheel drive.'\n\nJust before Moa, the driver swerved to the right to avoid an enormous rut. The vehicle skidded slowly and came to a halt in a pot-hole. The driver restarted the engine and revved hard: the wheels spun in the brownish mud, the minibus did not move. Desperately he tried several times, to no effect. 'Well...' said the wine merchant, folding his arms in a jovial manner, 'we'll have to get out and push.'\n\nWe got out of the vehicle. Before us stretched a vast plain encrusted with cracked brown mud, which looked unsanitary. Pools of stagnant water, which appeared almost black, were surrounded by tall grasses, withered and bleached. In the background, a huge factory of dark brick dominated the landscape, its twin chimney-stacks spewing out thick smoke. Rusted pipes ran from the factory and appeared to zigzag aimlessly through the middle of the plain. On the hard shoulder, a metal sign depicting Che Guevara exhorting the workers to the revolutionary development of the forces of production was itself beginning to rust. The air was pervaded by an appalling stench which seemed to rise from the mud itself rather than the pools of water.\n\nThe rut was not too deep and thanks to our concerted efforts we easily got the minibus back on the road. Everyone boarded the bus again, congratulated themselves. A little later we had lunch in a seafood restaurant. Jean-Yves consulted his notebook with a worried air; he hadn't touched his meal.\n\n'With the discovery holidays,' he concluded after considerable reflection, 'I think we're off to a good start; but with the standard resort, I really don't see what we can do.'\n\nVal\u00e9rie observed him calmly, sipping her iced coffee; she looked as though she didn't give a fuck.\n\n'Obviously,' he continued, 'we could just fire the team of reps; it would reduce our total wage bill.'\n\n'That would be a good start, yes.'\n\n'You don't think it's a bit radical as an idea?' he asked anxiously.\n\n'Don't worry about that. Being a rep at a holiday club village is no education for young people. It makes them stupid and lazy, and anyway it leads nowhere. The only thing they're fit for afterwards is to be a holiday club manager \u2013 or a TV presenter.'\n\n'Okay, then, I reduce the overall wage bill; but then again, they're not all that well paid. I'd be surprised if it saved us enough to be competitive with the German clubs. Anyway, I'll run up a spreadsheet simulation this evening, but I'm not convinced.'\n\nShe nodded in indifferent assent, something like: 'Go ahead and simulate, it can't do any harm.' She was really surprising me at this point, I thought she was cool. It's true we were fucking quite a lot, and there's no doubt that fucking is calming: it puts things in perspective. For his part, Jean-Yves looked ready to rush to his spreadsheet; I even wondered whether he was going to ask the driver to get his laptop out of the boot. 'Don't worry, we'll find a solution...' Val\u00e9rie said to him, shaking him affectionately by the shoulder. That seemed to calm him for a while; he quietly went back and took his seat on the minibus.\n\nOn the last leg of the journey, the passengers talked mostly about Baracoa, our final destination; they already seemed to know pretty much everything about the city. On 28 October 1492, Christopher Columbus dropped anchor in the bay, impressed by its flawlessly circular form. 'This is the most beautiful land human eyes have ever seen,' he had noted in his logbook. At the time, the region was solely inhabited by the Tainos Indians. In 1511, Diego Velazquez founded the city of Baracoa; it was the first Spanish city in the Americas. For more than four centuries, being accessible only by boat, it remained isolated from the rest of the island. In 1963, the construction of the Farola viaduct made it possible to establish a road link with Guantanamo.\n\nWe arrived at about three o'clock; the city stretched along the bay which did indeed form an almost perfect circle. The satisfaction of the group was universal and was expressed in appreciative exclamations. In the end, what all lovers of journeys of discovery seek is confirmation of what they've already read in their guidebooks. All in all, they were a dream audience: Baracoa, with its modest one star in the _Michelin Guide_ , was unlikely to disappoint them.\n\nThe El Castillo Hotel, situated in a former Spanish fortress, dominated the city. Viewed from above, it seemed magnificent; but to be honest, no more so than other cities. In truth it was actually quite nondescript, with its seedy tower blocks of blackened grey, so squalid that they looked uninhabited. I decided to stay by the pool, as did Val\u00e9rie. There were about thirty rooms, all occupied by tourists from Northern Europe, who all seemed to have come for much the same reasons. I first noticed two rather plump English women in their forties; one of them wore glasses. They were accompanied by two easy-going mixed-race guys, who were twenty-five, tops. They seemed comfortable with the situation, talked and joked with the fatties, held their hands, slipped their arms around their waists. For my part, I would have been completely incapable of doing this kind of work; I wondered if they had some kind of trick, something they could think about when they needed to get an erection. At some point, the English women went up to their rooms while the two guys stayed and chatted by the pool; if I was truly interested in human nature, I would have struck up a conversation, tried to find out a bit more. Still, maybe you just had to jerk off properly, an erection could probably be a purely mechanical reflex; biographies of male prostitutes would undoubtedly have enlightened me on this point, but the only thing I had at my disposal was _Discourse_ on the Positive Spirit. As I was leafing through the subsection entitled 'Popular politics, ever social, must above all become moral', I noticed a young German girl coming out of her room, accompanied by a big black guy. She looked exactly the way we imagine German girls, long blonde hair, blue eyes, a firm, pleasing body, big breasts. As a physical type, it's very attractive; the problem is it doesn't last: by the age of thirty there's work to be done, liposuction, silicone. Anyway, for the time being, things were fine, in fact she looked positively sexy \u2013 her suitor had been very lucky. I wondered whether she paid as much as the English women did, if there was a going rate for men as there was for women; here again research needed to be done, enquiries made. It was too exhausting for me, I decided to go up to my room. I ordered a cocktail which I sipped slowly on the balcony. Val\u00e9rie was sunning herself, taking a dip in the pool from time to time, I noticed she'd struck up a conversation with the German girl.\n\nShe came up to see me at about six; I'd fallen asleep with my book. She took off her swimsuit, showered and came to me, a towel wrapped around her waist; her hair was slightly damp.\n\n'You're going to think I'm obsessed with this, but I asked the German girl what black guys have that white guys don't. It's true, though: white women clearly prefer to sleep with Africans and white men with Asians. It's pretty obvious after a while. I need to know why, it's very important for my work.'\n\n'There are white men who like black women...' I observed.\n\n'It's not as common; sexual tourism is much rarer in Africa than it is in Asia. Of course, tourism in general is rarer, to be honest.'\n\n'What was her answer?'\n\n'Standard stuff: black guys are laid-back, virile, they have a sense of fun; they know how to enjoy themselves, they're not hung up, you never have any trouble with them.'\n\nThe German girl's reply was banal, true, but it provided the basis for a workable theory: all things considered, white men were repressed Negroes searching for some lost sexual innocence. Obviously it in no way explained the mysterious attraction which Asian women seemed to wield; nor the sexual prestige which, by all accounts, white men enjoyed in black Africa. I sketched out the basis of a more complex, more questionable theory: generally speaking, white people want to be tanned and to dance like Negroes; Negroes want to lighten their skin and straighten their hair. All humanity instinctively tends towards miscegenation, a generalised undifferentiated state, and it does so first and foremost through the elementary means of sexuality. The only person, however, to have pushed the process to its logical conclusion is Michael Jackson: he's neither black nor white any more, neither young nor old and, in a sense, neither man nor woman. Nobody could really imagine his private life; having grasped the categories of everyday humanity, he had done his utmost to go beyond them. This was why he could be considered a star, possibly the greatest \u2013 and, in fact, the first \u2013 in the history of the world. All the others - Rudolph Valentino, Greta Garbo, Marlene Dietrich, Marilyn Monroe, James Dean, Humphrey Bogart \u2013 could at best be considered talented artists; they had done no more than imitate the human condition, had aesthetically transposed it. Michael Jackson was the first to have tried to go a little further.\n\nIt was an appealing theory, and Val\u00e9rie listened attentively as I explained it; I, on the other hand, was not entirely convinced. Did this mean that the first cyborg, the first individual to accept having elements of artificial, extra-human intelligence implanted into his brain, would immediately become a star? Probably, yes: but that actually had very little bearing on the subject. Michael Jackson might well be a star, but he was certainly not a sex symbol; if you wanted to encourage the sort of mass tourism that would warrant heavy investment, you had to turn to more basic forces of attraction.\n\nA little later, Jean-Yves and the others returned from their tour of the city. The local history museum was chiefly devoted to the customs of the Tainos, the first inhabitants of the region. It appeared that they had led a peaceable existence, dedicated to agriculture and fishing; conflicts between neighbouring tribes were practically nonexistent; the Spanish had had no difficulty in exterminating these creatures, who were ill-prepared for combat. Today, nothing of them remains apart from some minimal genetic traces in the physiognomy of a handful of individuals; their culture has completely disappeared, it might just as well have never existed. In a number of drawings made by the missionaries, who had attempted \u2013 more often than not in vain \u2013 to sensitise them to the message of the Gospel, they can be seen ploughing, or busying themselves cooking at the fire; bare-breasted women suckle their children. All of this gave the impression, if not of Eden, then at least of a slow pace of history; the arrival of the Spanish had speeded things up significantly. After the classic conflicts between the colonial powers who led the field at the time, Cuba gained its independence in 1898, only to fall immediately under American control. Early in 1959, after a civil war lasting many years, the revolutionary forces led by Fidel Castro overthrew the regular army, forcing Batista to flee. Considering that the whole world was forcibly divided into two camps at the time, Cuba had been quickly compelled to make overtures to the Soviet block and establish a Marxist-style regime. Deprived of logistical support after the collapse of the Soviet Union, now that regime was drawing to a close. Val\u00e9rie slipped on a short skirt slit up one side and a little black lace top; we had time for a cocktail before dinner.\n\nEveryone was gathered around the swimming pool, watching as the sun set over the bay. Near the shore, the wreck of a freighter slowly rusted. Other, smaller, boats floated, almost motionless, on the waters; it all exuded a powerful sense of decline. Not a sound drifted up from the streets of the city down below; a few streetlights flickered hesitantly into life. At Jean-Yves's table sat a man of about sixty, his face gaunt and exhausted, his expression gloomy; and another much younger man \u2013 no more than thirty \u2013 whom I recognised as the hotel manager. I had seen him several times during that afternoon moving nervously between the tables to make sure that everyone was happy; his face seemed to be ravaged by constant, needless worry. Seeing us approach, he got up quickly, brought two chairs over, called a waiter, ensured that the latter arrived without delay; then hurried to the kitchens. The old man at his side shot a cynical look at the swimming pool, the couples sitting at tables and, apparently, at the world at large. 'The poor people of Cuba...' he said after a long silence. 'They've nothing left to sell except their bodies.' Jean-Yves explained that this man lived nearby; he was the hotel manager's father. More than forty years before, he had fought in the revolution, he had been a member of one of the first companies to rally to the Castro uprising. After the war, he had worked in the nickel works at Moa, at first as a worker, then as a foreman, eventually \u2013 after he had gone back to university \u2013 as an engineer. His status as a revolutionary hero had made it possible for his son to obtain an important position in the tourist industry.\n\n'We have failed...' he said in a dull voice; 'and we deserved to fail. We had great leaders \u2013 exceptional, idealistic men who put the good of the country before their own personal gain. I remember _comandante_ Che Guevara, the day he came to open the cocoa-processing plant in our village; I can still remember his noble, honest face. No one could ever say that the _comandante_ had lined his pockets, that he tried to get favours for himself or his family. Nor could it be said of Camilo Cienfuegos, or any of the revolutionary leaders, not even of Fidel. It's true Fidel likes power, he wants to keep an eye on everything; but he is disinterested, he has no magnificent properties, no Swiss bank accounts. So, Che was there, he inaugurated the factory, he made a speech in which he urged the people of Cuba to win the peace through production, after the war for independence; it was just before he went to the Congo. We could easily win such a battle. The land here is fertile, the earth is rich and well irrigated, everything grows in abundance: coffee, cocoa, sugar cane, tropical fruit of every kind. The subsoil is rich in nickel ore. We had an ultra-modern factory, built with help from the Russians. In less than six months, production had fallen to half its normal level: all the factory workers stole chocolate, raw or in bars, gave it to their families, sold it to strangers. It was the same in all the factories, all over the country. If the workers couldn't find anything to steal, they worked badly, they were lazy, they were always sick, they were absent for the slightest reason. I spent years trying to talk to them, to persuade them to try a little harder for the sake of their country: I met with nothing but disappointment and failure.'\n\nHe fell silent; the last of the day floated above the Yunque, a mountain peak mysteriously truncated in the form of a table, which towered over the hills and which long ago had made a considerable impression on Christopher Columbus. What could possibly incite human beings to undertake tedious, tiresome tasks? This seemed to me the only political question worth posing. The old factory worker's evidence was damning: in his opinion, only the need for money; in any case, the revolution had obviously failed to create the _new man_ , driven by more altruistic motives. And so, like all societies, Cuba was nothing more than a system painstakingly rigged so as to allow some people to avoid tedious and tiresome tasks. Except that the system had failed, no one was fooled any longer, no one was sustained any more by the hope of one day rejoicing in communal labour. The result was that nothing functioned, no one worked or produced the slightest thing any longer, and Cuban society had become incapable of ensuring the survival of its own members.\n\nThe other members of the tour got up and headed towards the tables. I racked my brain desperately for something optimistic to say to the old man, some vague message of hope; but no, there was nothing. As he so bitterly foresaw, Cuba would soon become a capitalist country again, and nothing would remain of the revolutionary hopes he had nurtured \u2013 only a sense of failure, futility and shame. No one would respect or follow his example, his life would in fact become an object of revulsion to future generations. He would have fought, and afterwards worked his whole life, completely in vain.\n\nDuring the meal, I drank quite a bit and, by the end, I found I was completely smashed; Val\u00e9rie looked at me a little anxiously. The salsa dancers were getting ready for their show; they were wearing pleated skirts and multicoloured sheathes. We took our seats on the terrace. I knew more or less what I wanted to say to Jean-Yves; had I chosen an opportune moment? I felt that he was a little distraught, but relaxed. I ordered one last cocktail, lit a cigar before turning to him.\n\n'You really want to find a new formula that would save your holiday clubs?'\n\n'Of course I do, that's why I'm here.'\n\n'Offer a club where the people get to fuck. That's what they're missing more than anything. If they haven't had their little holiday romance, they go home unsatisfied. They wouldn't dare admit it, they might not even realise it, but the next time they go on holiday, they go with a different company.'\n\n'They can fuck all they like, everything has been set up to encourage them to; that's the basic principle of holiday clubs; why they don't actually fuck, I haven't the faintest idea.'\n\nI swept the objection aside with a wave of my hand. 'I don't know either, but that's not the problem; there's no point trying to find out the causes of this phenomenon, always supposing the phrase actually means something. Something must be happening to make Westerners stop sleeping with each other; maybe it's something to do with narcissism, or individualism, the cult of success, it doesn't matter. The fact is that from about the age of twenty-five or thirty, people find it very difficult to meet new sexual partners; although they still feel the need to do so, it's a need which fades very slowly. So they end up spending thirty years of their lives, almost the entirety of their adult lives, suffering permanent withdrawal.'\n\nHalfway along the path to inebriation, just before mindlessness ensues, one sometimes experiences moments of heightened lucidity. The decline of western sexuality was undoubtedly a major sociological phenomenon which it would be futile to attempt to explain by such and such a specific psychological factor; glancing at Jean-Yves, I realised however that he perfectly illustrated my thesis, so much so that it was almost embarrassing. Not only did he not fuck any more and didn't have the time to go looking, but he no longer really wanted to, and, worse still, he felt this decay written on his flesh \u2013 he was beginning to smell of the stench of death. 'But...' he objected after a long moment of hesitation, 'I've heard wife-swapping clubs are quite successful.'\n\n'No, actually, they're doing less and less well. There are a lot of clubs opening up, but they close almost immediately because they haven't got the customers. As a matter of fact, there are only two clubs making a go of it in Paris, Chris et Manu and 2 + 2, and even they are only full on Saturday night: for a city of ten million people, it's not much, it's a lot less than at the beginning of the 1990s. Wife-swapping clubs are a nice formula, but they're seen as more and more pass\u00e9, because people don't want to swap anything any more, it doesn't suit modern sensibilities. In my opinion, wife-swapping has as much chance of surviving today as hitch-hiking did in the 1970s. The only thing that is doing any business at the moment is S&M...' At that point, Val\u00e9rie shot me a panicked look, she even gave me a kick in the shins. I looked at her, surprised. It took me a few seconds to work it out: no, of course I wasn't going to mention Audrey; I gave her a little reassuring nod. Jean-Yves hadn't noticed the interruption.\n\n'Therefore,' I went on, 'you have several hundred million Westerners who have everything they could want but no longer manage to obtain sexual satisfaction: they spend their lives looking, but they don't find it and they are completely miserable. On the other hand, you have several billion people who have nothing, who are starving, who die young, who live in conditions unfit for human habitation and who have nothing left to sell except their bodies and their unspoiled sexuality. It's simple, really simple to understand: it's an ideal trading opportunity. The money you could make is almost unimaginable: vastly more than from computing or biotechnology, more than the media industry; there isn't a single economic sector that is comparable.'\n\nJean-Yves didn't say anything; at that moment, the band began the first number. The dancers were pretty and smiling, their pleated skirts whirled, amply revealing their tanned thighs; they illustrated my point perfectly. For a moment, I thought he wouldn't say anything, that he would simply digest the idea. However, after about five minutes, he said:\n\n'It doesn't really work for Muslim countries, your idea...'\n\n'No problem, you just leave them with their Eldorador Discovery. You could even steer them towards something much tougher, with trekking and environmental activities, a survivor kind of thing maybe, which you could call Eldorador Adventure: it would sell really well in France and in Anglo-Saxon countries. On the other hand, the sex-oriented clubs could do well in Germany and the Mediterranean countries.'\n\nThis time, he smiled broadly. 'You should have been in business...' he said half-seriously. 'You're an ideas man...'\n\n'Ideas, yeah...' My head was spinning a little, I could no longer make out the dancers, I finished my cocktail in one gulp. 'I might have ideas, but I wouldn't be able to throw myself into balance sheet or budget forecasts. So, yeah, I'm an ideas man...'\n\nI don't remember much about the rest of the evening, I must have fallen asleep. When I woke up, I was lying on my bed; Val\u00e9rie lay naked beside me, breathing gently. I woke her as I moved to reach for a pack of cigarettes.\n\n'You were pretty drunk back there...'\n\n'Yes, but I was serious about what I was saying to Jean-Yves.'\n\n'I think he took it seriously...' She stroked my belly with her fingertips. 'And actually, I think you're right. Sexual liberation in the West is over.'\n\n'You know why?'\n\n'No...' she hesitated, then went on: 'No, actually, not really.'\n\nI lit a cigarette, propped myself up on the pillows and said: 'Suck me.' She looked at me, surprised, but placed her hand on my balls and brought her mouth towards me. 'There!' I exclaimed triumphantly. She stopped what she was doing and looked at me in surprise. 'You see, I say \"Suck me\" and you suck me. When actually, you didn't feel the desire to do so.'\n\n'No, I hadn't thought of it; but I enjoy doing it.'\n\n'That's precisely what's so extraordinary about you, you enjoy giving pleasure. Offering your body as an object of pleasure, giving pleasure unselfishly: that's what Westerners don't know how to do any more. They've completely lost the sense of giving. Try as they might, they no longer feel sex as something _natural_. Not only are they ashamed of their own bodies, which aren't up to porn standards, but for the same reasons they no longer feel truly attracted to the body of the other. It's impossible to make love without a certain abandon, without accepting, at least temporarily, the state of being in a state of dependency, of weakness. Sentimental adulation and sexual obsession have the same roots, both proceed from some degree of selflessness; it's not a domain in which you can find fulfilment without losing yourself. We have become cold, rational, acutely conscious of our individual existence and our rights; more than anything, we want to avoid alienation and dependence; on top of that we're obsessed with health and hygiene: these are hardly ideal conditions in which to make love. The way things stand, the commercialisation of sexuality in the East has become inevitable. Obviously, there's S&M too. It's a purely cerebral world with clear-cut rules and a prior contract. Masochists are just interested in their own sensations, they try to see how far they can plunge into pain, a bit like people who do extreme sports. Sadists are something else, they will take things as far as they possibly can regardless \u2013 it's a very ancient human propensity: if they can mutilate or kill, they will do so.'\n\n'I really don't want to think about it again,' she said shivering; 'It really disgusts me.'\n\n'That's because you've remained sexual, animal. You're normal, in fact, you're not much like Westerners. Organised S&M with its rules could only exist among cultured, cerebral people for whom sex has lost all attraction. For everyone else, there's only one possible solution: pornography featuring professionals; and if you want to have real sex, third-world countries.'\n\n'Okay...' She smiled. 'Is it okay if I go back to sucking you off?'\n\nI leaned back on the pillows and let it happen. I was vaguely conscious at that moment of being at the beginning of something: from an economic point of view, I knew I was right; I estimated that potential clients might run to 80 per cent of Western adults. But I knew that people sometimes find it difficult, strangely, to accept simple ideas.\n\n### 10\n\nWE HAD BREAKFAST on the terrace, by the swimming pool. As I was finishing my coffee, I saw Jean-Yves emerge from his room accompanied by a girl I recognised as one of the dancers from the previous evening. She was black and slender, with long, graceful legs, she couldn't have been more than twenty. For a fleeting moment, he looked embarrassed, then came over to our table and introduced Angelina.\n\n'I've thought about your idea,' he announced straight off. 'What I'm worried about is how feminists will react.'\n\n'Some of the clients will be women,' said Val\u00e9rie.\n\n'You think so?'\n\n'Oh, yes, I'm sure of it...' she said a little bitterly. 'Look around you.'\n\nHe glanced at the tables around the pool: there were indeed a number of single women accompanied by Cuban men; almost as many as there were single men in the same situation. He asked Angelina something and translated her reply:\n\n'She's been a _jinetera_ for three years; most of her clients are Italian or Spanish. She thinks it's because she's black: Germans and Anglo-Saxons are happy with Latino girls, to them that's exotic enough. She has a lot of friends who are _jineteros_ : their customers are mostly English and American women, and some Germans too.\n\nHe took a sip of coffee, thought for a moment:\n\n'What are we going to call these clubs? We need to think of something evocative, something very different from Eldorador Adventure, but all the same, not too explicit.'\n\n'I thought maybe Eldorador Aphrodite,' said Val\u00e9rie.\n\n'\"Aphrodite\"...' he repeated the word thoughtfully. 'It's not bad; it doesn't sound as vulgar as \"Venus\". Erotic, sophisticated, a little exotic: yes, I like it.'\n\nAn hour later, we headed back towards Guardalavaca. A couple of metres from the minibus, Jean-Yves said his goodbyes to the _jinetera_ ; he seemed a little sad. When he got back on to the bus, I noticed the student couple giving him black looks; the wine merchant, on the other hand, clearly looked as though he didn't give a damn.\n\nThe return trip was pretty gloomy. Of course there was still the diving, the karaoke evenings and the archery; the muscles tire, then relax; sleep comes quickly. I remember nothing of the last days of the trip, nor of the last excursion, except that the lobster was rubbery and the cemetery disappointing: this despite the fact that it housed the tomb of Jos\u00e9 Marti, father of the nation, poet, politician, polemicist, thinker. He was depicted in a bas-relief sporting a moustache. His coffin, bedecked with flowers, lay at the foot of a circular pit on the walls of which were engraved his most notable _pens\u00e9es_ \u2013 on national independence, resistance to tyranny, justice. Nonetheless, you didn't get the sense that his spirit still animated the place; the poor man seemed quite simply dead. That said, he was not an unpleasant stiff; you felt you would have liked to meet him, if only to be ironic about his rather narrow and earnest humanist; but it hardly seemed likely, he seemed to be well and truly stuck in the past. Could he rise up once more and galvanise his homeland to greater heights of the human spirit? One didn't really imagine so. All in all, it was a disappointment letdown, as indeed all republican cemeteries are. It was irritating, all the same, to realise that Catholics are the only people who have succeeded in creating a functional funeral system. It's true that the means they use to make death magnificent and affecting consists quite simply in denying it. Difficult to fail with arguments like that. But here, in the absence of the risen Christ, you needed nymphs, shepherds, tits and arse, basically. As it was, you couldn't imagine Jos\u00e9 Marti romping about in the great meadows of the hereafter; he looked more like he had been buried in the ashes of everlasting _ennui_.\n\nThe day after we got back, we found ourselves in Jean-Yves's office. We hadn't slept much on the plane; my memory of that day is of an atmosphere of blissful enchantment, rather strange, in the deserted building. Three thousand people worked there during the week, but on that Saturday there were just the three of us, apart from the security guards. Close by, on the forecourt of the \u00c9vry shopping centre, a pair of rival gangs faced each other with Stanley knives, baseball bats and containers of sulphuric acid; that evening the number of dead would stand at seven, among them two onlookers and a member of the riot squad. The incident would be the subject of considerable debate on national radio and television; but at that moment we knew nothing about it. In a state of excitement which seemed slightly unreal, we set down our manifesto, our platform for dividing up the world. The suggestions that I was about to make might possibly result in millions of francs worth of investment or hundreds of jobs; for me it was very new and very unsettling. I felt a bit crazy all afternoon, but Jean-Yves listened to me attentively. He was convinced, he told Val\u00e9rie later, that if I was given free rein I was likely to have a brainwave. In short, I brought a note of creativity while he remained the decision-maker; that was his way of looking at things.\n\nThe Arab countries were the quickest to deal with. In view of their absurd religion, all possible sexual activity seemed to be ruled out. Tourists who opted for these countries would have to content themselves with the dubious delights of adventure. In any case, Jean-Yves had decided to sell off Agadir, Monastir and Djerba, which were making too much of a loss. That left two destinations which could reasonably be classified under the category 'adventure'. The tourists in Marrakech would do a bit of camel trekking. Those at Sharm-el-Sheikh could observe the goldfish or take an excursion into the Sinai to the site of the Burning Bush where Moses had 'flipped his lid', to use the colourful expression of an Egyptian I had met three years earlier on a felucca trip to the Valley of the Kings. 'Admittedly,' he'd said emphatically, 'it's a very impressive rock formation... but to go from that to affirming the existence of the one God!...' This intelligent and often funny man seemed to have a fondness for me \u2013 probably because I was the only Frenchman in the group \u2013 as for some obscure cultural or sentimental reasons he nurtured a lifelong, and, by then, it has to be said, a highly notional, passion for France. In speaking to me, he had literally saved my holiday. He was about fifty, always impeccably dressed, very dark skinned, with a little moustache. A biochemist by training, he had emigrated to England as soon as he had completed his studies and had been brilliantly successful working in genetic engineering there. He was revisiting his native land, for which, he said, he still had great affection; on the other hand he could not find words harsh enough to revile Islam. Above all, he wanted to convince me, Egyptians were not Arabs. 'When I think that this country invented everything!...' he exclaimed gesturing broadly towards the Nile valley. 'Architecture, astronomy, mathematics, agriculture, medicine' (he was exaggerating a little, but he was an Oriental and needed to persuade me quickly). 'Since the appearance of Islam, nothing. An intellectual vacuum, an absolute void. We've become a country of flea-ridden beggars. Beggars covered in fleas, that's what we are. Scum, scum!...' (with a wave, he shooed away some boys who had come to beg for small change). 'You must remember, _cher monsieur_ ,' (he spoke five foreign languages fluently: French, German, English, Spanish and Russian), 'that Islam was born deep in the desert amid scorpions, camels and wild beasts of every order. Do you know what I call Muslims? The losers of the Sahara. That's what they deserve to be called. Do you think Islam could have been born in such a magnificent place?' (with genuine feeling, he motioned again to the Nile valley). 'No, _monsieur_. Islam could only have been born in a stupid desert, among filthy Bedouins who had nothing better to do \u2013 pardon me \u2013 than bugger their camels. The closer a religion comes to monotheism \u2013 consider this carefully, _cher monsieur_ \u2013 the more cruel and inhuman it becomes; and of all religions, Islam imposes the most radical monotheism. From its beginnings, it has been characterised by an uninterrupted series of wars of invasion and massacres; never, for as long as it exists, will peace reign in the world. Neither, in Muslim countries, will intellect and talent find a home; if there were Arab mathematicians, poets and scientists, it is simply because they lost the faith. Simply reading the Koran, one cannot help but be struck by the regrettable mood of tautology which typifies the work: \"There is no other God but God alone\", etc. You won't get very far with nonsense like that, you have to admit. Far from being an attempt at abstraction, as it is sometimes portrayed, the move towards monotheism is nothing more than a shift towards mindlessness. Note that Catholicism, a subtle religion, and one which I respect, which well knew what suited human nature, quickly moved away from the monotheism imposed by its initial doctrine. Through the dogma of the Trinity and the cult of the Virgin and the Saints, the recognition of the role played by the powers of darkness, little by little it reconstituted an authentic polytheism; it was only by doing so that it succeeded in covering the earth with numberless artistic splendours. One God! What an absurdity! What an inhuman, murderous absurdity!... A god of stone, _cher monsieur_ , a jealous, bloody god who should never have crossed over from Sinai. How much more profound, when you think about it, was our Egyptian religion, how much wiser and more humane... and our women! How beautiful our women were! Remember Cleopatra, who bewitched great Caesar. See what remains of them today...' (randomly he indicated two veiled women walking with difficulty carrying bundles of merchandise). 'Lumps. Big shapeless lumps of fat who hide themselves beneath rags. As soon as they're married, they think of nothing but eating. They eat and eat and eat!...' (his face became bloated as he pulled a face like de Fun\u00e8s). 'No, believe me, _cher monsieur_ , the desert has produced nothing but lunatics and morons. In your noble Western culture, for which, by the way, I have great admiration and respect, can you name anyone who was drawn to the desert? Only pederasts, adventurers and crooks, like that ludicrous colonel Lawrence, a decadent homosexual and a pathetic poseur. Like your despicable Henry de Monfreid, an unscrupulous trafficker, always ready to compromise his principles. Nothing great or noble, nothing generous or wholesome; nothing which has contributed to the progress of humanity or raised it above itself.'\n\n'Okay, Egypt gets adventure...' Jean-Yves concluded simply. He apologised for interrupting my story, but we had to move on to Kenya. A difficult case. 'I'd be quite tempted to put it in with \"Adventure\"...' he suggested, having consulted his files.\n\n'Pity...' sighed Val\u00e9rie. 'Kenyan woman are very pretty.'\n\n'How do you know that?'\n\n'Well, not just Kenyan women, African women in general.'\n\n'Yeah, but there are women everywhere. In Kenya, you've got rhinoceros, zebras, gnus, elephants, buffalo. What I suggest is that we put Senegal and the Ivory Coast into \"Aphrodite\", and leave Kenya in \"Adventure\". In any case, it's a former English colony, which is terrible for its erotic image, but okay for adventure.'\n\n'They smell good, the women of the Ivory Coast...' I observed dreamily.\n\n'What do you mean by that?'\n\n'They smell of sex.'\n\n'Yes...' he chewed unconsciously on his pen. 'That could be good for an ad. 'Something like \"The Ivory Coast, the realm of the scents\" \u2013 with a girl in a grass skirt sweating, her hair tousled. I'll make a note of it.'\n\n'\"And the nude slaves imbued with fragrance...\" Baudelaire, it's public domain.'\n\n'We'd never get away with it.'\n\n'I know.'\n\nThe rest of the African countries posed fewer problems. 'In fact, in general you never have any problems with Africans. They'll fuck for free, even the fat ones. You just have to put condoms in the clubs, that's all; from that point of view they can be a bit stubborn.' He underlined PROVIDE CONDOMS twice in his notebook.\n\nTenerife took us even less time. The club's takings were average, but, according to Jean-Yves, it was crucial to the Anglo-Saxon market. You could easily throw together an adventure circuit with a climb to the summit of Mount Tiede and a trip on a hydroplane to Lanzarote. The hotel set-up was reasonable, it could be made viable.\n\nWe came to the two clubs which would be the chain's chief assets: Boca Chica in the Dominican Republic and Guardalavaca in Cuba. 'We could provide king-size beds...' suggested Val\u00e9rie. 'Done,' said Jean-Yves immediately. 'Private jacuzzis in the suites...' I suggested. 'No,' he cut me off, 'We're strictly mid-market.' One thing led effortlessly to another, with no hesitations, no doubts; we would have to liaise with the resort managers to standardise the local prostitution rates.\n\nWe paused briefly to go for lunch. At that very moment, two teenagers from the Courtili\u00e8res housing estate were smashing in a sixty-year-old woman's head with a baseball bat. I ordered _maquereau au vin blanc_ to start.\n\n'Have you got anything planned for Thailand?' I asked.\n\n'We've got a hotel in construction in Krabi. It's the new, hot destination after Phuket. We could easily speed up the building work, it could be ready by January 1st. It would be good to have a high-profile opening.'\n\nWe devoted the afternoon to developing the various innovative aspects of the Aphrodite clubs. The central point, obviously, was authorised access for local prostitutes, male and female. Clearly, there was no question of offering to accommodate children; the best thing would be to restrict admission to the clubs to the over-sixteens. An ingenious idea, suggested by Val\u00e9rie, was to list the single-room tariff as the basic catalogue price and to offer a discount of 10 per cent for double occupancy; to reverse, in short, the standard system. I think I was the one who suggested that we put forward a gay-friendly policy, and to circulate rumours that homosexuals accounted for 20 per cent of visitors to the clubs: that kind of information was enough to get them to come; and if you wanted a place to have an atmosphere _of sex_ , they had it down to a fine art. The issue of the overall slogan for the advertising campaign kept us busy for some time. Jean-Yves hit on a solution that was basic and effective: 'Going on holiday: time to go wild'; but in the end, I got a unanimous vote for 'Eldorador Aphrodite: Because pleasure is a right'. Since the NATO intervention in Kosovo, the notion of rights had become very persuasive, Jean-Yves explained to me in a half-joking tone; but he was quite serious: he had just read an article on the subject in _Strat\u00e9gies_. Every recent campaign based on the idea of rights had been a success: the right to innovation, the right to excellence... The right to pleasure, he concluded sadly, was a new one. In fact, we were beginning to feel a little tired. He dropped us off at 2 + 2 before heading home. It was Saturday night, the place was quite full. We met a really nice black couple; she was a nurse, he was a jazz drummer \u2013 he was doing well, he recorded regularly. He admitted that he spent a lot of his time working on his technique, all his time in fact. 'There's no secret to it...' I said a bit foolishly, but, strangely, he agreed; without intending to, I had hit upon a profound truth. 'The secret is there is no secret,' he said to me with conviction. We finished our drinks and headed up to the rooms. He suggested a double penetration to Val\u00e9rie. She agreed, as long as I was the one to sodomise her \u2013 you had to take it very gently with her, I was used to it. J\u00e9r\u00f4me agreed and lay down on the bed. Nicole stroked his cock to keep him hard, then slipped on a condom. I pushed Val\u00e9rie's skirt up to her waist. She wasn't wearing anything underneath. In a single movement, she impaled herself on J\u00e9r\u00f4me's prick, then lay down on top of him. I spread her cheeks, lubricated her a little, and then started to fuck her up the arse with short, careful strokes. At the point when the head of my cock was completely inside her, I felt her rectal muscles contract. I stiffened immediately, breathed deeply, I had almost come. After a few seconds, I pushed in deeper. When I was halfway in, she started to move back and forth, rubbing her pubis against J\u00e9r\u00f4me's. There was nothing more for me to do; she started a long, modulated groan, her arse opened and I pushed into her up to the hilt. It was like sliding down an inclined plane \u2013 she came surprisingly quickly. Then she became still, panting, happy. It was not that it was particularly more intense, she explained to me later; but when everything went well, there was a point when the two sensations fused, it became something gentle and irresistible, like being warm all over.\n\nNicola had been watching us, fingering herself all the time; she was starting to get really excited and immediately took Val\u00e9rie's place. I didn't have time to change my condom. 'With me, you can just go for it,' she whispered in my ear; 'I really liked to be fucked hard up the arse.' Which is what I did, closing my eyes to lessen the excitement, trying to concentrate on pure sensation. Everything went smoothly, I was agreeably surprised by my own stamina. She, too, came very quickly with loud, hoarse cries.\n\nThen Val\u00e9rie and Nicole knelt down to suck us off while we talked. J\u00e9r\u00f4me was still touring, he told me, but he didn't like it so much any more. As he got older, he felt the need to stay home more, to look after his family \u2013 they had two children \u2013 and to work on his drumming by himself. Then he talked to me about new time-signatures, 4\/3 and 7\/9; to be honest I didn't really understand very much. Right in the middle of a sentence he gave a cry of surprise, his eyes rolled back: he came all at once, ejaculating violently into Val\u00e9rie's mouth. 'Ha, she got me there...' he said, half-laughing, 'she got me good.' I felt I was not going to hold out much longer either: Nicole had a most particular tongue, large and soft, eager; she licked slowly, the ascent was insidious, but almost irresistible. I motioned to Val\u00e9rie to come nearer and explained to Nicole what I wanted: she was to close her lips round my glans, rest her tongue and remain motionless while Val\u00e9rie jerked me off and licked my balls. She agreed, closed her eyes, waiting for the ejaculation. Val\u00e9rie started immediately, her fingers quick and vigorous: already she seemed to be back on top form. I spread my arms and legs as far as I could, closed my eyes. The feeling mounted with sudden jolts, like bolts of lightning, then exploded just before I ejaculated into Nicole's mouth. For a brief moment I felt almost concussed, points of lights flashed beneath my eyelids; a little later I realised that I had been on the brink of passing out. I opened my eyes with difficulty. Nicole still had the tip of my cock in her mouth, she sucked up the last drops of semen. Val\u00e9rie had slipped her arm around my neck, she was looking at me tenderly, mysteriously; she told me I had screamed very loudly.\n\nA little later, they drove us home. In the car, Nicole had another surge of desire. She slipped her breasts out of her basque, lifted her skirt and lay down on the back seat, laying her head on my thighs. I masturbated her thoughtfully, confidently, expertly controlling her sensations, I felt her hard nipples and her wet pussy. The scent of her sex filled the car. J\u00e9r\u00f4me drove carefully, stopped at the red lights; through the windows, I could make out the lights of the Place de la Concorde, the obelisk, then the Pont Alexandre III, Les Invalides. I felt good, at peace, but still a little energetic. She came as we neared the Place d'ltalie. We went our separate ways after exchanging phone numbers.\n\nJean-Yves, meanwhile, feeling a little depressed after he had left us, had parked on the Avenue de la R\u00e9publique. The excitement of the day had subsided; he knew that Audrey would not be home, but he was actually rather glad of that. He would run into her briefly tomorrow morning, before she went out rollerblading; since coming back from holiday, they slept in separate rooms.\n\nWhy go home? He pushed back in his seat, thought about trying to turn on the radio but didn't. Gangs of young people, boys and girls, went past on the street; they looked like they were having fun, at least they were yelling. Some of them were carrying cans of beer. He could have got out, mingled with them, maybe started a fight; there were many things he could have done. In the end, he would go home. In some sense he loved his daughter, at least he supposed he did; he felt for her something organic and potentially blood-stained for her which corresponded to the definition of the word. He felt nothing of the kind for his son. In fact, the boy might not even be his; his reasons for marrying Audrey had been rather minimal. For her, at any rate, he felt nothing more than contempt and disgust; too much disgust, he would have preferred to feel indifferent; at the moment he still keenly felt that she should be made to _pay_. I'm more likely to be the one to pay, he thought suddenly, bitterly. She would get custody of the children and he would be landed with huge alimony payments. Unless he tried to get custody of the children, unless he fought her on that; but no, he decided, it wasn't worth it. It was too bad for Ang\u00e9lique. He would be better off on his own, he could try to start a new life, which meant, more or less, find some other girl. Saddled with two kids, it would be tougher for Audrey, the bitch. He consoled himself with the thought that it would be hard for him to do worse, and that, at the end of the day, she would be the one to suffer as a result of the divorce. She was already no longer as beautiful as when he had met her; she had style, she dressed fashionably, but knowing her body as he did, he knew she was already over the hill. On top of that, her career as a lawyer was far from being as brilliant as she made out; and he had a feeling that having custody of the children would not help matters. People drag their progeny around with them like a millstone, like some terrible weight which hinders their every move \u2013 and which, as often as not, effectively winds up killing them. He would have his revenge later: at the point, he thought, when it had become a matter of complete indifference to him. For some minutes more, parked near the bottom of the now deserted avenue, he practised feeling indifferent.\n\nHis worries came crashing down on him all at once as soon as he had walked through the door of the apartment. Johanna, the babysitter, was sprawled on the sofa watching MTV. He hated this listless, absurdly trendy pre-adolescent; every time he saw her he wanted to smack her round the face, to wipe the expression off her nasty, sulky, careless face. She was the daughter of one of Audrey's friends.\n\n'Everything OK?' he shouted. She nodded casually. 'Could you turn it down?' She looked around for the remote control. Exasperated, he turned the television off; she shot him a hurt look.\n\n'What about the children, everything go alright?' He was still shouting though there was no longer a sound in the apartment.\n\n'Yeah, I think they're asleep.' She curled up, a little scared.\n\nHe went up to the first floor and pushed open the door to his son's bedroom. Nicolas looked round at him abstractedly, and then went back to his game of _Tomb Raider_. Ang\u00e9lique, on the other hand, was sleeping like a log. He went downstairs, a little calmer.\n\n'Did you bathe them?'\n\n'Yeah... no, I forgot.'\n\nHe wandered into the kitchen and poured himself a glass of water. His hands were shaking. On the worktop, he saw a hammer. A couple of slaps wouldn't have been enough for Johanna; smashing her skull in with hammer blows would be much better. He toyed with this idea for a while; thoughts crisscrossed his brain rapidly, barely controlled. In the hallway, he noticed in terror that he was holding the hammer. He placed it on a low table, looked in his wallet for the taxi fare for the babysitter. She took it, mumbling thanks. He slammed the door behind her in a gesture of uncontrolled violence; the sound reverberated through the entire apartment. Something was clearly not right in his life. In the living room the drinks cabinet was empty; Audrey wasn't even capable of looking after that. Thinking of her, a wave of hatred coursed through him and he was surprised at its intensity. In the kitchen he found an open bottle of rum; that would probably do. In his bedroom he dialled in turn the numbers of three girls he had met on the internet: each time, he got an answering machine. They had probably gone out, fucking on their own account. It's true they were sexy, cool, fashionable, but they were costing him two thousand francs a night; it became humiliating after a while. How had he come to this? He should go out, make friends, spend less time on his work. He thought about the Aphrodite clubs again, realising for the first time that it might be difficult to get the idea past his superiors; there was a fairly negative attitude to sex tourism in France at the moment. Obviously, he could try getting a toned down version past Leguen, but Espitalier wouldn't be fooled; he sensed a treacherous shrewdness in the man. Anyway, what choice did they have? Their mid-market positioning made no sense up against Club Med \u2013 he would have no problem in proving that. Rummaging through his desk drawers he found the Aurore mission statement, drafted ten years earlier by the founder, and displayed in every hotel in the group:\n\nThe spirit of Aurore is the art of marrying know-how, tradition and innovation with rigour, imagination and humanism, to attain a certain form of excellence. The men and women of Aurore are the repositories of a unique cultural heritage: the art of welcoming. They know the rituals and the customs which transform life into the art of living, and the simplest of services into a privileged moment. It is a profession, it is an art: it is their gift. Creating the best in order to share it, getting in touch with the essential through hospitality, devising spaces of pleasure: these are what make Aurore a taste of France throughout the world.\n\nHe suddenly realised that this nauseating spiel could just as easily apply to a chain of well-run brothels; maybe there was a card here he could play with the German tour operators. Defying all reason, Germans still thought of France as the country of romance, of the art of love. If a major German tour operator agreed to include the Aphrodite clubs in their catalogue, it would mark a turning-point; no one in the industry had yet succeeded in achieving such a thing. He was already in contact with Neckermann over the sale of the North African clubs. But there was also TUI, who had turned down their initial approaches because they were already well established in the bottom end of the market; they might be interested in a more targeted product.\n\n### 11\n\nFIRST THING MONDAY morning, he set about making some initial approaches. From the start, he was lucky: Gottfried Rembke, president of the board of TUI, was coming to spend a few days in France at the beginning of the month; Rembke would pencil them in for lunch. In the meantime, if they could put their proposal in writing he would be delighted to give it his careful consideration. Jean-Yves went into Val\u00e9rie's office to tell her the news; she froze. The annual turnover of TUI was six billion francs, three times that of Neckermann, six times that of Nouvelles Fronti\u00e8res; they were the largest tour operator in the world.\n\nThey devoted the rest of the week to writing up a sales pitch that was as detailed as possible. Financially, the project didn't require substantial investment: there were some small changes in furnishings, the hotels would definitely have to be redecorated to give them a more 'erotic' feel \u2013 they had quickly settled on the term 'friendly tourism', which would be used in all of the business documentation. The most important point was that they could expect a significant reduction in their fixed costs: no more sporting activities, no more children's clubs. No more salaries to be paid to registered paediatric nurses or windsurfing instructors; nor to specialists in ikebana, ceramics or painting on silk. After running a first financial simulation, Jean-Yves realised to his surprise that, allowing for depreciation, the annual costs of the clubs would drop by 25 per cent. He redid the calculations three times and each time got the same result. It was all the more striking because the catalogue rates he intended proposing were 25 per cent above the category norm \u2013 essentially pegging the rates with those of the mid-range Club Med. Profits leapt by 50 per cent. 'Your boyfriend's a genius...' he told Val\u00e9rie, who had just come into his office.\n\nThe atmosphere in the office at this time was a little odd. The clashes which had taken place on the streets of \u00c9vry the previous weekend were not uncommon; but the death-toll \u2013 seven \u2013 was particularly high. Many of the employees, especially those who had worked there longest, lived in the vicinity of the offices. At first they had lived in the apartment blocks that had been built at much the same time as the offices; later, as often as not, they had borrowed in order to build. 'I feel sorry for them,' Val\u00e9rie told me; 'I really do. They all dreamed of setting themselves up out of town, somewhere peaceful; but they can't just leave now, they'd end up losing a chunk of their pensions. I was talking to the switchboard operator: she has three years before she retires. Her dream is to buy a house in the Dordogne; she's from there originally. But a lot of English people have moved there and the prices there now are outrageous, even for some miserable dump. And on the other hand, the price of her house here has collapsed, everyone knows that it's a dangerous suburb nowadays, she'd have to sell it for a third of its value.\n\n'Another thing that surprised me is the second-floor secretarial pool. I went up there at half-past five to get a memo typed up: they were all on the internet. They told me that they all do their shopping that way now, it's safer; they go home, lock themselves in and wait for the delivery man.'\n\nIn the weeks that followed, this obsessive fear did not fade, if anything it increased slightly. In the papers now it was teachers being stabbed, nursery school teachers being raped, fire engines attacked with Molotov cocktails, handicapped people thrown through the windows of trains because they had 'looked the wrong way' at some gang leader. _Le Figaro_ was having a field day: reading it every day, you got the impression of an unstoppable escalation to civil war. True, this was the run-up to an election and law and order was the only issue likely to bother Lionel Jospin. In any case, it seemed very unlikely that the French would vote for Jacques Chirac again: he seemed to be such an idiot it was affecting the country's image. Whenever you saw this lanky half-wit, hands clasped behind his back, visiting some country fair, or taking part in a heads of state summit, you felt sort of ashamed, you felt embarrassed for him. The Left, obviously incapable of curbing the rising tide of violence, behaved well: they kept a low profile, agreed that the figures were bad, very bad even, called on others not to make political capital of it, reminded people that when they'd been in power the Right hadn't done any better. There was just one little slip, a ridiculous editorial by Jacques Attali. According to him, the violence of young people on housing estates was a 'cry for help'. The shop windows of the Champs-Elys\u00e9es, he wrote, constituted so many 'obscene displays flaunted at their misery'. Neither should it be forgotten that the suburbs were a 'mosaic of peoples and ethnicities, who had come with their traditions and their beliefs to forge new cultures and to reinvent the art of living together'. Val\u00e9rie stared at me in surprise: this was the first time I had burst out laughing while reading _L'Express_.\n\n'If he wants to get elected,' Jean-Yves said, handing her the article, 'Jospin would be well advised to shut him up until the second round.'\n\n'You're clearly getting a taste for strategy...'\n\nDespite everything, I too was beginning to feel anxiety gnawing at me. Val\u00e9rie was working late again, it was rare for her to get home before nine o'clock. It might be wise to buy a gun. I had a contact, the brother of an artist whose exhibition I had organised two years before. He wasn't really part of the scene, he'd just been involved in a couple of scams. He was more of an inventor, a sort of jack-of-all-trades. He had recently told his brother that he'd discovered a way of forcing the new identity cards which were supposed to be impossible to fake.\n\n'Out of the question,' Val\u00e9rie said immediately. 'I'm not in any danger: I never leave the office during the day and at night I always take a cab home, regardless of what time I leave.'\n\n'There's still the traffic lights.'\n\n'There's only one set of traffic lights between the office and the motorway. After that, I take the exit at Porte d'Italie and I'm almost home. Our area isn't dangerous.'\n\nIt was true: in Chinatown, strictly speaking, there were very few assaults or rapes. I didn't understand how they managed it, did they have their own neighbourhood watch? In any case, they had noticed us as soon as we moved in; there were at least twenty people who regularly greeted us. It was rare for Europeans to move in here, we were in a very small minority in the building. Sometimes, posters written in Chinese characters seemed to extend invitations to meetings or parties; but what meetings? what parties? It's possible to live among the Chinese for years without understanding anything about the way they live.\n\nNevertheless, I phoned my contact who promised to ask around. He called me back two days later. I could have a serious piece, in very good nick, for ten thousand francs \u2013 the price included a healthy quantity of ammunition. All I would have to do was clean it regularly to make sure it didn't jam if ever I needed to use it. I talked to Val\u00e9rie again, who refused again. 'I couldn't,' she said, 'I wouldn't have the courage to pull the trigger.' 'Even if your life was in danger?' She shook her head, 'No...' she repeated, 'It's not possible.' I didn't insist. 'When I was little,' she told me later, 'I couldn't even kill a chicken.' To be honest, neither could I; but a man, now that seemed significantly easier.\n\nCuriously, I was not afraid for my own sake. It's true I had very little contact with the barbarian hordes, except perhaps occasionally at lunchtime when I went for a walk around the Forum des Halles, where the subtle infiltration of the security forces (the riot squad, uniformed police officers, security guards employed by local shopkeepers) eliminated all danger, in theory. So I wandered casually through the reassuring topography of uniforms; I felt as though I was in Thoiry safari park. In the absence of the forces of law and order, I knew, I would be easy prey, though of little interest; very conventional, my middle-management uniform had very little to tempt them. For my part, I felt no attraction for this youthful product of the _dangerous classes_ ; I didn't understand them, and made no attempt to do so. I didn't sympathise with their passions nor with their values. For myself, I wouldn't have lifted a finger to own a Rolex, a pair of Nikes or a BMW Z3; in fact, I had never succeeded in identifying the slightest difference between designer goods and non-designer goods. In the eyes of the world, I was clearly wrong. I was aware of this: I was in a minority, and consequently in the wrong. There _had_ to be a difference between Yves Saint-Laurent shirts and other shirts, between Gucci moccasins and Andr\u00e9 moccasins. I was alone in not perceiving this difference; it was an infirmity which I could not cite as grounds for condemning the world. Does one ask a blind man to set himself up as an expert on post-impressionist painting? Through my blindness, however involuntary, I set myself apart from a living human reality powerful enough to incite both devotion and crime. These youths, through their half-savage instincts, undoubtedly discerned the presence of beauty; their desire was laudable, and perfectly in keeping with social norms; it was merely a question of rectifying the inappropriate way in which it was expressed.\n\nThinking about it carefully, however, I had to admit that Val\u00e9rie and Marie-Jeanne, the only two long-term female presences in my life, manifested a complete indifference to Kenzo blouses and Prada handbags; in fact, as far as I could make out, they bought any old brand at random. Jean-Yves, the highest paid individual I knew, exhibited a preference for Lacoste polo-necks, but he did it somewhat mechanically, out of habit, without even checking to see whether the reputation of his favourite brand had not been surpassed by some new challenger. Some of the women at the Ministry of Culture whom I knew by sight (though I regularly forgot their names, their job titles, even their faces, between each encounter) bought designer clothes; but they were invariably by some young, obscure designer who had only one outlet in Paris, and I knew perfectly well that they would not hesitate to abandon them if by chance they ever found a wider public.\n\nThe power of Nike, Adidas, Armani, Vuitton was, nonetheless, indisputable; I could find proof of this whenever I needed simply by glancing through the business section of _Le Figaro_. But who, exactly \u2013 aside from youths on housing estates \u2013 assured the success of these brands? Clearly there had to be whole sectors of society who were still alien to me; unless, more prosaically, they were bought by rich people in the third world. I had travelled little, lived little and it was becoming increasing clear that I understood little about the modern world.\n\nOn September 27, there was a meeting of the eleven Eldorador holiday club managers, who had come to \u00c9vry for the occasion. It was a routine meeting which took place every year on the same date, to assess the figures for the summer and consider improvements which might be made. However, this time, it had particular significance. Firstly, three of the resorts were about to change hands \u2013 the contract with Neckermann had just been signed. Secondly, the managers of four of the remaining villages \u2013 those which fell into the 'Aphrodite' category \u2013 had to prepare themselves to fire half of their staff.\n\nVal\u00e9rie was not present for the meeting; she had a meeting with an Italtrav representative to present the scheme to him. The Italian market was much more fragmented than those of Northern Europe. Italtrav might well be the largest tour operator in Italy, but its turnover was less than a tenth of TUI's; an agreement with them would, nonetheless, bring in valuable customers.\n\nShe came back from her appointment at about 7 p.m., Jean-Yves was alone in his office, the meeting had just ended.\n\n'How did they take it?'\n\n'Badly. I know how they feel, too; they must think they're next for the chop.'\n\n'Are you intending to replace the resort managers?'\n\n'It's a new project; we'd be better off starting out with new teams.'\n\nHis voice was very calm. Val\u00e9rie looked at him in surprise: lately he had become more assured \u2013 and tougher.\n\n'I'm convinced that we're going to be a success, now. When we broke for lunch, I was talking to the manager at Boca Chica, in the Dominican Republic. I wanted to be clear in my mind about something: I wanted to know how he managed to have 90 per cent occupancy regardless of season. He dithered, he seemed embarrassed, talked about team work. In the end, I asked him straight out if he was allowing girls to go up to the guest rooms; I had a hard time getting him to admit it, he was afraid I was going to put him on a disciplinary. I had to tell him that it didn't bother me at all, that in fact I thought it was an interesting initiative. At that point he confessed. He thought it was stupid that guests were renting rooms a mile away, often with no running water, and with the risk of being ripped off, when they had every comfort right there. I congratulated him and I promised him he'd keep his job as resort manager, even if he's the only one who does.'\n\nIt was getting dark; he turned on a lamp on his desk, was silent for a moment.\n\n'For the others,' he went on, 'I don't feel the slightest remorse. They're all pretty much the same. They're all former reps, they joined at the right time, they got to have it off with anyone they wanted without doing a fucking stroke of work and they thought that becoming manager of the resort meant they could bum around in the sun until they retired. Their days are over \u2013 tough. Now, I need real professionals.'\n\nVal\u00e9rie crossed her legs and looked at him in silence.\n\n'By the way, the meeting with Italtrav?'\n\n'Good. No problems. He knew at once what I meant by 'friendly tourism', he even tried to make a pass at me... That's the good thing about Italians, at least they're predictable... Anyway, he promised he'd include the clubs in his catalogue, but he said we shouldn't get our hopes up: Italtrav has a strong presence because it's a conglomerate of a lot of specialised tour companies; in its own right it hasn't got a very strong image. In fact, it operates as a distributor: we can get on their list, but it will be up to us to make a name for ourselves in the market.'\n\n'What about Spain? How far have we got?'\n\n'We've got a good relationship with Marsans. They're much the same, except they're more ambitious: for a while now they've been trying to get a foothold in France. I was a bit worried that we'd be competing with their products, but apparently not, they think what we're doing is complementary.'\n\nShe thought for a moment and then continued:\n\n'What are we going to do about France?'\n\n'I'm still not sure... Maybe I'm being stupid, but I'm really worried about stirring up a moralistic press campaign. Obviously, we could do some focus groups, test the market...'\n\n'You've never believed in that stuff.'\n\n'No, that's true...' he hesitated for a moment. 'Actually, I'm tempted to do a minimal launch in France, just through the Auroretour network. Put ads in a couple of carefully targeted magazines like _FHM_ or _L'\u00c9cho des Savanes_. But really, for the first stage, I want to focus on Northern Europe.'\n\nThe meeting with Gottfried Rembke took place the following Friday. The night before, Val\u00e9rie made herself a cleansing mask and went to bed early. When I woke up at eight o'clock, she was already ready. I was impressed by the results. She was wearing a black suit with a short, tight skirt which hugged her arse magnificently; under the jacket, she was wearing a lilac blouse in lace, close fitting and, in places, transparent, and a scarlet push-up bra which showed off her breasts. When she sat opposite the bed, I discovered she was wearing black stockings, faded towards the top, held in place by suspenders. Her lips were emphasised in a dark, almost purplish, red and she had tied her hair up in a chignon.\n\n'Does this do the trick?' she asked mockingly\n\n'That does it _in spades_. Well, well, women...' I added, 'when you show yourselves to your best advantage...'\n\n'This is my corporate seductress outfit. I put it on for you, in a way, too; I knew you'd like it.'\n\n'Re-eroticising the workplace...' I muttered. She handed me a cup of coffee.\n\nUntil she left, I did nothing but watch her come and go, stand and sit. It wasn't much, I suppose, actually it was quite simple, but it did the trick, no doubt about it. She crossed her legs, a dark band appeared high up on her thighs, accentuating the contrasting sheerness of the nylon. She crossed them a little more, a band of lace was revealed a little higher up, then the fastener of the suspenders, the bare, white flesh, the curve of the buttocks. She uncrossed them, everything disappeared again. She leaned over the table: I could feel the palpitation of her breasts through the fabric. I could have spent hours watching her. It was a simple joy, innocent and eternally blessed; a pure promise of pleasure.\n\nThey were supposed to meet at 1 p.m. at Le Divellec, a restaurant on the Rue de l'Universit\u00e9; Jean-Yves and Val\u00e9rie arrived five minutes early.\n\n'How are we going to raise the subject?' Val\u00e9rie asked anxiously as she stepped out of the taxi.\n\n'I dunno... just tell him we want to open up a chain of brothels for Huns... Jean-Yves gave a weary grin. 'Don't worry about it, don't worry about it, he'll ask all the questions.'\n\nGottfried Rembke arrived at 1 p.m. precisely. The moment he walked into the restaurant, handed his coat to the waiter, they knew it was him. The solid, stocky body, the gleaming scalp, the open expression, the vigorous handshake: everything about him radiated ease and enthusiasm; he was precisely what one imagined a head honcho, more especially a German head honcho, looked like. You could imagine him eagerly throwing himself into each new day, leaping out of bed, doing half an hour on an exercise bike before driving to the office in his spanking new Mercedes, listening to the financial news. 'This guy seems perfect...' muttered Jean-Yves as he got to his feet, all smiles, to greet him.\n\nFor the next ten minutes, in fact, Herr Rembke spoke of nothing but food. It turned out that he knew France very well, the culture, the cuisine; he even owned a house in Provence. 'Perfect, the guy's perfect...' thought Jean-Yves as he studied his _consomm\u00e9 de langoustines au cura\u00e7ao_. 'Rock and roll, Gotty,' he added to himself, dipping his spoon into the soup. Val\u00e9rie was wonderful: she listened attentively, her eyes sparkling as though charmed by him. She wanted to know where, exactly, in Provence, whether he had time to visit often, etc. She had chosen the _salmis d'\u00e9trilles aux fruits rouges_.\n\n'So,' she went on without changing her tone, 'you'd be interested in the proposal.'\n\n'The way I see it,' he said thoughtfully, 'we know that \"friendly tourism\"' \u2013 he stumbled a little on the expression \u2013 'is one of the primary motivating factors of our compatriots when they holiday abroad \u2013 and, moreover, one can understand why, after all, what more delightful way to travel? However, and this is somewhat curious, up until now, no major group has actively taken an interest in the sector \u2013 apart from a number of attempts, all hopelessly inadequate, marketed to a homosexual clientele. Essentially, surprising as it may seem, we are dealing with a virgin market.'\n\n'It's much discussed. I think that attitudes still have a long way to go...' interrupted Jean-Yves, realising as he did so that what he was saying was ridiculous. 'On both sides of the Rhine...' he concluded miserably. Rembke gave him a frosty look, as though he thought Jean-Yves was taking the piss; Jean-Yves hunched over his food again, determined not to say another word until the meal was over. In any case, Val\u00e9rie was getting along brilliantly. 'Let's not project French problems on to the Germans...' she said, ingenuously crossing her legs. Rembke fixed his attention on her once more.\n\n'Our compatriots,' he went on, 'forced to fend for themselves, often find themselves at the mercy of intermediaries of dubious honesty. More generally, the sector remains marked by rank amateurism \u2013 which represents a considerable loss of earnings for the industry as a whole.' Val\u00e9rie agreed eagerly. The waiter arrived with a _saint-pierre r\u00f4ti aux figues nouvelles_.\n\n'Equally,' he went on having glanced at his dish, 'your proposal interests us because it represents a compete reversal of the traditional view of the holiday club. A formula which was conceived in the 1970s does not correspond to the expectations of contemporary consumers. Relationships between individuals in the West have become more difficult \u2013 a fact which, needless to say, we all deplore...' he continued, glancing again at Val\u00e9rie, who uncrossed her legs with a smile.\n\nWhen I got back from the office at a quarter-past six, she was already home. I felt a twinge of surprise: I think this was the first time since we lived together. She was sitting on the sofa, still wearing her suit, her legs slightly apart. Staring into space, she seemed to be thinking of happy, gentle things. Though I did not know it at the time, I was witnessing the professional equivalent of an orgasm.\n\n'Did it go well?' I asked.\n\n'Better than well. I came straight home after lunch, I didn't even bother dropping into the office; I really couldn't see what else we could do this week. Not only is he interested in the project, but he intends to make it one of his key products as of next winter. He's prepared to finance printing the catalogue and an advertising campaign targeted specifically at the German market. He believes that, on his own, he can guarantee to fill all the existing clubs; he even asked whether we had any other projects in the works. The only thing he wants in return, is exclusivity in his own market \u2013 Germany, Austria, Switzerland and the Benelux countries; he knows that we've been in touch with Neckermann too.'\n\n'I've booked a weekend,' she added, 'in a thalassotherapy centre in Dinard. I think I need it. We could drop in on my parents as well.'\n\nThe train pulled out of the Gare Montparnasse an hour later. Quite quickly, as the kilometres passed, the accumulated tension faded and she was back to normal, that is rather sexual and playful. The last buildings of the outer suburbs disappeared behind us; the TGV approached maximum speed just as we came to the Plain of Hurepoix. A sliver of daylight, an almost imperceptible reddish tinge, hung in the air to the west over the dark mass of grain silos. We were in a first-class carriage arranged in small compartments; on the tables which separated our seats, small yellow lamps already glowed. Across the corridor, a woman of about forty, very upper-middle class but pretty stylish with her blonde hair tied up in a chignon, was leafing through _Madame Figaro_. I had bought the same paper and was trying without much success to interest myself in the financial supplement. For some years I had nurtured the theory that it was possible to decode the world, to understand its evolution, by setting aside everything dealing with current affairs, politics, the society pages and the arts; that it was possible to form an accurate image of the thrust of history purely by reading the financial news and the stock prices. I therefore forced myself to read the _Figaro_ financial section daily, supplemented by even more forbidding publications like _Les \u00c9chos_ or _La Tribune Desfoss\u00e9s_. Up to this point, my theory had remained impossible to judge. It was possible, in other words, that historic news was concealed within these editorials. With their measured tones, their columns of figures; but the reverse might just as easily be true. The only definite conclusion I had categorically come to: economics was unspeakably boring. Looking up from a short article which attempted to analyse the fall of the Nikkei, I noticed that Val\u00e9rie had begun crossing and uncrossing her legs; a half-smile flitted across her face. 'Descent into hell for Milan stock exchange,' I managed to read before putting down the paper. I suddenly got an erection when I discovered she had found a way to take off her panties. She came and sat beside me, pressed herself against me. Taking off her suit jacket, she draped it across my lap. I glanced quickly to my right: our neighbour still appeared to be engrossed in her magazine, specifically in an article on the garden in winter. She too was wearing a suit with a tight skirt and black tights; she looked like a posh tart, as they say. Sliding her hand under her jacket, Val\u00e9rie placed it on my penis; I was wearing only a pair of thin cotton trousers, the sensation was terribly precise. It was, by now, completely dark. I sat back in my seat, slipped a hand under her blouse. Pushing her bra aside, I encircled her right breast with the palm of my hand and began to stimulate her nipple with my thumb and forefinger. Just as we reached Le Mans, she undid my flies. Her movements were now absolutely brazen, I was convinced that our neighbour was missing nothing of our little game. As far as I'm concerned, it is impossible to resist masturbation by a truly expert hand. Just before Rennes I ejaculated, unable to suppress a muffled cry. 'I'll have to get this suit cleaned,' Val\u00e9rie said calmly. Our neighbour glanced across, making no attempt to conceal her amusement.\n\nEven so, at the station at Saint-Malo I was a little embarrassed when I noticed that she was boarding the same shuttle bus for the thalassotherapy centre; but not so Val\u00e9rie: she even struck up a conversation with her about the various treatments. For myself, I've never really worked out the respective merits of mud baths, high-pressure showers and seaweed wraps; the following day, I was happy just to mess around in the pool. I was floating on my back, vaguely aware of the underwater currents supposedly massaging my a back, when Val\u00e9rie joined me. 'Our neighbour from the train...' she said, all excited, 'she came on to me in the jacuzzi.' I registered the information without reacting. 'Right now she's alone in the hammam,' she added. I followed her at once, wrapping myself in a bathrobe. Near the entrance to the hammam, I took off my swimming trunks; my erection was visible beneath the towelling robe. I followed Val\u00e9rie in, letting her make her way through steam so dense you couldn't see a couple of metres ahead of you. The air was saturated with a strong, almost intoxicating scent of eucalyptus. I stopped and stood still in the hot, whitish emptiness, then I heard a moan coming from the far end of the room. I untied the belt of my robe and walked towards the sound. Beads of perspiration formed on the surface of my skin. Kneeling in front of the woman, hands placed on her buttocks, Val\u00e9rie was slowly licking her pussy. She really was a very beautiful woman, with perfectly rounded silicone-enhanced breasts, a harmonious face, a wide, sensual mouth. Unsurprised, she turned to look at me and closed her hand around my penis. I came a little closer, went behind her and stroked her breasts, rubbing my penis against her buttocks. She opened her thighs and bent forward, leaning on the wall for support. Val\u00e9rie rummaged in the pocket of her robe and handed me a condom; with her other hand, she continued to masturbate the woman's clitoris. I penetrated the woman in one swift thrust, she was already wide open; she bent forward a little further. I was thrusting in and out of her when I felt Val\u00e9rie's hand slip between my thighs, then close over my balls. Then she leaned forward and began licking the woman's pussy, with each thrust, I could feel my cock rubbing against her tongue. I desperately tensed my pelvic muscles at the point when the women came in a series of long, contented moans, then slowly I pulled out. My whole body was sweating, I was panting involuntarily, I felt a little faint and had to sit down on a bench. The clouds of steam continued to undulate through the air. I heard the sound of a kiss and I looked up: they were entwined, breast to breast.\n\nWe made love a little later, in the late afternoon, again that evening and once more the following morning. Such frenzy was a little unusual; we were both conscious of the fact that we were about to enter a difficult time, when Val\u00e9rie would once more be stupefied with work, problems and calculations. The sky was an immaculate blue, the weather almost warm; it was probably one of the last fine weekends before the autumn. After making love on Sunday morning, we took a long stroll on the beach. I looked in surprise at the neoclassical, slightly kitsch hotel buildings. When we arrived at the far end of the beach, we sat down on the rocks.\n\n'I suppose it was important, that meeting with the German,' I said; 'I suppose it's the beginning of a new challenge.'\n\n'This will be the last time, Michel. If this is a success, we'll be set up for a long time.'\n\nI shot her a doubtful and slightly sad look. I didn't believe in that line of reasoning: it reminded me of history books in which politicians declared that this would be the war to end all wars, the sort that was supposed to lead to a permanent peace.\n\n'It was you who told me,' I said gently, 'that capitalism, by its very nature, is a permanent state of war, a constant struggle which can never end.'\n\n'That's true,' she agreed without hesitation, 'But it's not always the same people doing the fighting.'\n\nA gull took off, gained altitude and headed out to the ocean. We were almost alone at this end of the beach. Dinard was clearly a very quiet resort, at least at this time of year. A labrador came up and sniffed us, then turned tail; I couldn't see its owners.\n\n'I promise you,' she insisted, 'if this works as well as we hope, we can roll out the concept in lots of countries. In Latin America alone there's Brazil, Venezuela, Costa Rica. Apart from that we can open clubs in Cameroon, Mozambique, Madagascar, the Seychelles. In Asia too there are obvious possibilities: China, Vietnam, Cambodia. In two or three years, we can become an uncontested market leader; and no one will dare invest in the same market: this time we'll get it, our competitive advantage.'\n\nI didn't reply, I couldn't think of anything to say; after all, it had originally been my idea. The tide was coming in; waves crashed on to the beach and died at our feet.\n\n'On top of that,' she went on, 'this time we're going to insist on a decent share package. If it's a success, they can't possibly refuse. And when you're a shareholder, you don't have to fight any more: other people do the fighting for you.'\n\nShe stopped, looked at me, hesitant. It made sense, what she was saying, it had a certain logic. The wind was getting up a bit; I was starting to feel hungry. The restaurant at the hotel was excellent: they had impeccably fresh shellfish, and delicious, delicate fish dishes. We headed back, walking across the wet sand.\n\n'I've got money...' I said suddenly. 'Don't forget that I've got money.' She stopped and looked at me in surprise; I hadn't expected to say these words.\n\n'I know it's not the done thing to be a kept woman,' I went on, a little embarrassed, 'but there's nothing forcing us to do things the way everybody else does.'\n\nShe stared calmly into my eyes. 'When you've got the money from the house, at best it'll come out to three million francs, maximum...' she said.\n\n'Yes. That's right, something like that.'\n\n'It's not enough, at least not quite. We need just a little more.' She began walking again and said nothing for a long moment. 'Trust me...' she said, as we stepped under the glass roof of the restaurant.\n\nAfter the meal, just before heading to the station, we paid a visit to Val\u00e9rie's parents. She was about to be submerged with work again, she explained; she probably wouldn't be able to visit again before Christmas. Her father looked at her with a resigned smile. She was a good daughter, I thought, an attentive and caring daughter; she was also a sensual lover, affectionate and audacious; and, if need be, she would no doubt be a wise and loving mother. ' _Her feet are of fine gold, her legs like the columns of the temple of Jerusalem_.' I continued to wonder what exactly I had done to deserve a woman like Val\u00e9rie. Nothing, probably. I observe the world as it unfurls, I thought; proceeding empirically, in good faith, I observe it; I can do no more than observe.\n\n### 12\n\nAT THE END of October, Jean-Yves's father died. Audrey refused to accompany him to the funeral; actually, he had been expecting as much, he had asked her only for the sake of propriety. It would be a modest funeral: he was an only child, there would be some family, very few friends. His father would have a brief obituary in the ESAT alumni newsletter and that would be it, the end of the line. He had hardly seen anyone recently. Jean-Yves had never really understood what had moved him to retire to this undistinguished area, rural in the most depressing sense of the word and to which, moreover, he had no ties. Probably a last vestige of the masochism which had dogged him more or less his whole life. Having been a rather brilliant student, he had become bogged down in a lacklustre career as a manufacturing engineer. Though he had always dreamed of having a daughter, he had consciously limited himself to only one child \u2013 in order, he maintained, to give the boy a better education; the argument didn't stand up, he earned a very good salary. He gave the impression of being accustomed to his wife rather than truly loving her; perhaps he was proud of his son's professional successes \u2013 but, truth be told, the fact was he never spoke of them. He had no hobby, no leisure activity to speak of, apart from breeding rabbits and doing the crossword in _La R\u00e9publique du Centre-Ouest_. We are probably wrong to suspect that each individual has some secret passion, some mystery, some weakness; if Jean-Yves's father had had to express his innermost convictions, the profound meaning he ascribed to life, he could probably have cited nothing more than a slight disappointment. Indeed, his favourite expression, what Jean-Yves remembered him saying most often, what best encapsulated his experience of the human condition, was limited to the words 'You get old'.\n\nJean-Yves's mother seemed reasonably affected by her bereavement \u2013 he had, after all, been her lifelong companion \u2013 without really seeming to be shattered for all that. 'He'd gone downhill a lot...' she remarked. The cause of death was so vague that one might well have been talking about a general fatigue, or even despondency. 'He had no interest in anything anymore...' his mother said again. That, more or less, was her funerary oration.\n\nAudrey's absence was, of course, noticed, but during the ceremony his mother refrained from mentioning it. The evening meal was a frugal affair \u2013 in any case, she had never been a good cook. He knew she would broach the subject at some point. Bearing in mind the circumstances, it was quite difficult to avoid the issue as he usually did, by turning on the television, for example. His mother finished putting away the dishes, then sat opposite him, her elbows on the table.\n\n'How are things, with your wife?'\n\n'Not great...' He expanded on this for a few minutes, getting ever more bogged down in his own boredom; in conclusion, he indicated that he was thinking of divorce. His mother, he knew, hated Audrey, whom she accused of keeping her from her grandchildren; actually, it was quite true, but her grandchildren weren't too keen to see her either. If things had been different, it's true, they could have become used to it; Ang\u00e9lique at least, in her case it wasn't too late. But it would have meant different circumstances, a different life, all things that were difficult to imagine. Jean-Yves looked up at his mother's face, her greying chignon, her harsh features: it was difficult to feel a rush of tenderness, of affection for this woman; as far back as he could remember, she had never really been one for hugs; it was equally difficult to imagine her in the role of a sensual lover, a slut. He suddenly realised that his father must have been bored shitless his whole life. He felt terribly shocked by this, his hands tensed on the edge of the table: this time it was irreparable, it was definitive. In despair, he tried to recall a moment when he had seen his father beaming, happy, genuinely glad to be alive. There was one time, possibly, when he had been five and his father had been trying to show him how Meccano worked. Yes, his father had loved engineering, truly loved it \u2013 he remembered his father's disappointment the day he had announced he was going to study marketing; perhaps it was enough, after all, to fill a life.\n\nThe next day, he made a quick tour of the garden, which, to tell the truth, seemed quite anonymous to him; it evoked no memories of his childhood. The rabbits shifted nervously in their hutches, they hadn't been fed yet: his mother was going to sell them immediately, she didn't like looking after them. In reality, they were the real losers in this whole business, the only real victims of this death. Jean-Yves took a sack of feed granules, poured a couple of handfuls into the hutches; this gesture, at least, he could make in memory of his father.\n\nHe left early, just before the Michel Drucker programme, but that did not stop him getting caught up, just before Fontainebleau, in endless traffic jams. He tried a number of different stations, and ended up turning off the radio. From time to time the queue moved forward a few metres; he could hear nothing but the purring of engines, the splat of solitary raindrops against the windscreen. His mood was attuned to this melancholy emptiness. The only positive element of the weekend, he thought, was that he would never have to see Johanna the babysitter again. The new one, Eucharistie, had been recommended by a neighbour; she was a girl from Dahomey, serious, worked hard at school. At fifteen she was already in two years from graduation. Later, she hoped to be a doctor, possibly a paediatrician; in any case she was very good with the children. She had succeeded in tearing Nicolas away from his video games and getting him to bed before ten o'clock \u2013 something that they had never been able to do. She was wonderful with Ang\u00e9lique, fed her, bathed her, played with her; the little girl obviously adored her.\n\nHe arrived at half-past ten, exhausted from the journey; Audrey was, as far as he could remember, in Milan for the weekend; she would fly back the following morning and go straight to work. Divorce was seriously going to cramp her lifestyle, he thought with malicious satisfaction; it was easy to understand why she should want to put off broaching the subject. On the other hand, she did not go so far as to feign any affection, any rush of tenderness; that was something that could be counted in her favour.\n\nEucharistie was sitting on the sofa, she was reading a paperback of _Life: A User's Manual_ , by Georges Perec; everything had gone okay. She accepted a glass of orange juice; he poured himself a cognac. Usually, when he came home, she would tell him about the day, what she and the children had done together; this would last for a few minutes before she went. This time, too, she did so, as he poured himself a second cognac; he realised he hadn't been listening to a word. 'My father died...' he said, realising the fact again as he said the words. Eucharistie stopped abruptly and looked at him hesitantly; she did not know how to react, but he had clearly succeeded in capturing her attention. 'My parents were never happy together...' he continued, and this second observation was even worse: it seemed to deny his existence, to deprive him of a certain right to life. He was the fruit of an unhappy, mismatched union, something which would have been better if it had never been. He looked around him anxiously: in a few months at most he would leave this apartment, he would never again see these curtains, this furniture; everything already seemed to be fading, losing its solidity. He could just as easily be in the showroom of a department store after it closed; or in a photo from a catalogue, in something, at any rate, which had no real existence. He stood up unsteadily, walked over to Eucharistie and hugged the young girl's body violently in his arms. He slipped a hand under her pullover: her flesh was living, real. All of a sudden he came to himself and stopped, ashamed. She too had stopped struggling. He looked straight into her eyes, then kissed her on the mouth. She responded to his kiss, pushed her tongue against his. He slipped his hand higher under her pullover to her breasts.\n\nThey made love in the bedroom without a word; she had undressed quickly and then crouched on the bed, so he could take her. Even after they had come, they remained silent for some minutes and avoided mentioning the subject afterwards. She told him about her day again, what she and the children had been up to; then she told him that she could not stay the night.\n\nIn the weeks that followed, they did it again many times, every time she came over, in fact. He had more or less been waiting for her to broach the subject of the legality of their relations: after all, she was only fifteen, he was thirty-five; he could, at a pinch, have been her father. But she did not seem in the least inclined to see things in that light: in what light, then? In the end he realised, in a rush of emotion and of gratitude: in the simple light of pleasure. His marriage manifestly cut him off, he was out of touch; he had quite simply forgotten that certain women, in certain circumstances, make love _for pleasure_. He was not Eucharistie's first: she had already been with a boy the year before, a guy in his final year with whom she'd lost touch afterwards; but there were things she was unaware of, fellatio for example. The first time, he held back, was hesitant about coming in her mouth; but he quickly realised that she enjoyed it, or rather that it amused her to feel his semen spurt out. Usually, he had no trouble bringing her to orgasm; for his part it was immensely pleasurable feeling her firm, supple body in his arms. She was intelligent, curious; she was interested in his work and asked him lots of questions: she was almost everything Audrey was not. The universe of business was, to her, a curious, exotic world whose customs she wanted to learn; she would not have asked all these questions of her father, who, in any case, would have been unable to answer \u2013 he worked in a public hospital. In short, he thought, with a strange feeling of relativism, theirs was a relationship of equals. Even so, he was lucky that his first child had not been a daughter; in certain circumstances he found it difficult to imagine how \u2013 and more especially why \u2013 incest might be avoided.\n\nThree weeks after their first time, Eucharistie announced that she had met a boy. Under the circumstances it was best for them to end it; at any rate, it complicated matters. He seemed so desolate at the news that, the next time she came round, she offered to continue giving him blowjobs. In all honesty, he couldn't really see how that was less serious; but in any case he had more or less forgotten how he had felt when he was fifteen. When he got home he would talk for a long time about one thing and another; it was always she who decided on the moment. She would strip to the waist, allow him to caress her breasts; then he would lean back against the wall and she would kneel in front of him. From his moans she could tell precisely when he was going to come. She would then take her mouth away; with small, precise movements she would direct his ejaculation, sometimes towards her breasts, sometimes towards her mouth. In those moments she had a playful, almost childlike, expression; thinking back on it, he realised gloomily that her love life was just beginning, that she would make many lovers happy; their paths had crossed, that was all, and that in itself was a happy accident.\n\nThe second Saturday, at the moment when Eucharistie, eyes half-closed, mouth wide open, was beginning to jerk him off vigorously, he suddenly noticed his son popping his head round the door. He started, turned his head away; when he looked up again the child had disappeared. Eucharistie hadn't noticed anything; she slid her hand between his thighs, delicately squeezed his balls. At that moment, he had a strange sensation of immobility. Suddenly, it occurred to him, like a revelation, an impasse. There was too much overlap between the generations, fatherhood no longer had any meaning. He drew Eucharistie's mouth towards his penis; without quite understanding completely, he sensed that this would be the last time, he needed her mouth. As soon as her lips closed over him, he spurted several times, shoving his cock deep into her throat as shudders coursed through his body. Then she looked up at him; he kept his hands on the girl's head. She kept his penis in her mouth for two or three minutes, her eyes closed, running her tongue slowly over the head. Shortly before she left, he told her that they wouldn't do it again. He didn't really know why; if his son said anything it would surely do him a lot of harm when the divorce settlement was decided, but there was something else that he wasn't able to identify. He told me all of this a week later, in an irritating tone of self-reproach, begging me not to say anything to Val\u00e9rie. I found him a little annoying, to tell the truth; I really couldn't see what the problem was. However, purely out of friendship I pretended to take an interest, to weigh up the pros and cons, but I couldn't take the situation seriously, I felt a little as if I was on Mireille Dumas' TV show.\n\nFrom a professional point of view, on the other hand, everything was going well, he informed me with satisfaction. There had almost been a problem with the Thailand club a couple of weeks earlier: there had to be at least one hostess bar and one massage parlour to cater for customer expectations. This would be a little difficult to justify in the budget for the hotel. He telephoned Gottfried Rembke. The boss of TUI rapidly found a solution; he had an associate on the ground, a Chinese building contractor based in Phuket, who could sort out the building of a leisure complex just beside the hotel. The German tour operator seemed to be in a great mood, apparently things were looking good. At the beginning of November, Jean-Yves received a copy of the catalogue destined for the German market; he immediately noticed that they hadn't pulled any punches. In every photo the local girls were topless, wearing miniscule G-strings or see-through skirts; photographed on the beach or right in the hotel rooms, they smiled teasingly, ran their tongues over their lips: it was almost impossible to misunderstand. In France, he remarked to Val\u00e9rie, you would never get away with something like this. It was curious to note, he mused, that as Europe became ever closer, and the idea of a federation of states was ever more current, there was no noticeable standardisation of moral legislation. Although prostitution was accepted in Holland and Germany and was governed by statute, many people in France were calling for it to be criminalised, even for punters to be prosecuted as they were in Sweden. Val\u00e9rie looked at him, surprised: he had been odd lately, he launched increasingly frequently into aimless, unproductive ruminations. She herself coped with a punishing workload, methodically and with a sort of cold determination; she regularly took decisions without consulting him. It was something she was not really used to, and at times I sensed she felt lost, uncertain; the board of directors would not get involved, affording them complete freedom. 'They're waiting, that's all, they're waiting to see whether we fall flat on our faces,' she confided one day, with suppressed rage. She was right, it was obvious, I couldn't disagree with her; that was the way the game worked.\n\nFor my part, I had no objection to sex being subject to market forces. There were many ways of acquiring money, honest and dishonest, cerebral or, by contrast, brutally physical. It was possible to make money using one's intellect, talent, strength or courage, even one's beauty; it was also possible to acquire money through a banal stoke of luck. Most often, money was acquired through inheritance, as in my case; the problem of how it had been earned fell to the previous generation. Many very different people had acquired money on this earth: former top athletes, gangsters, artists, models, actors; a great number of entrepreneurs and talented financiers; a number of engineers, too, more rarely a few inventors. Money was sometimes acquired mechanically, by simple accumulation; or, on the other hand, by some audacious coup crowned with success. There was no great logic to it, but the possibilities were endless. By contrast, the criteria for sexual selection were unduly simple: they consisted merely of youth and physical beauty. These features had a price, certainly, but not an infinite price. The situation, of course, had been very different in earlier centuries, at a time when sex was essentially linked to reproduction. To maintain the genetic value of the species, humanity was compelled seriously to take into account criteria like health, strength, youth and physical prowess \u2013 of which beauty was merely a handy indicator. Nowadays, the order of things had changed: beauty had retained all of its value, but that value was now something marketable, narcissistic. If sex was really to come into the category of tradable commodities, the best solution was probably to involve money, that universal mediator which already made it possible to assure an exact equivalence between intelligence, talent and technical competence; which had already made it possible to assure a perfect standardisation of opinions, tastes and lifestyles. Unlike the aristocracy, the rich made no claim to being different in kind from the rest of the population; they simply claimed to be richer. Essentially abstract, money was a concept in which neither race, physical appearance, age, intelligence nor distinction played any part, nothing in fact, but money. My European ancestors had worked hard for several centuries; they had sought to dominate, then to transform the world, and, to a certain extent they had succeeded. They had done so out of economic self-interest, out of a taste for work, but also because they believed in the superiority of their civilisation: they had invented dreams, progress, utopia, the future. Their sense of a mission to civilise had disappeared in the course of the twentieth century. Europeans, at least some of them, continued to work, and sometimes to work hard, but they did so for money, or from a neurotic attachment to their work; the innocent sense of their natural right to dominate the world and direct the path of history had disappeared. As a consequence of their accumulated efforts, Europe remained a wealthy continent; those qualities of intelligence and determination manifested by my ancestors I had manifestly lost. As a wealthy European, I could obtain food and the services of women more cheaply in other countries; as a decadent European, conscious of my approaching death, and given over entirely to selfishness, I could see no reason to deprive myself of such things. I was aware, however, that such a situation was barely tenable, that people like me were incapable of ensuring the survival of a society, perhaps more simply we were unworthy of life. Mutations would occur, were already occurring, but I found it difficult to feel truly concerned; my only genuine motivation was to get the hell out of this shithole as quickly as possible. November was cold, bleak; I hadn't been reading Auguste Comte that much recently. My great diversion when Val\u00e9rie was out consisted of watching the movement of the clouds through the picture window. Immense flocks of starlings formed over Gentilly in the late afternoon, describing inclined planes and spirals in the sky; I was quite tempted to ascribe meaning to them, to interpret them as the heralds of an apocalypse.\n\n### 13\n\nONE EVENING, I met Lionel as I was leaving work; I hadn't seen him since the 'Thai Tropic' trip almost a year earlier. Curiously, however, I recognised him at once. I was a little surprised that he had made such a strong impression on me; I couldn't remember having said a word to him at the time.\n\nThings were going well, he told me. A large cotton disc covered his right eye. He'd had an accident at work, something had exploded; but it was okay, they'd managed to treat him in time, he would recover 50 per cent of the sight in his eye. I invited him for a drink in a caf\u00e9 near the Palais-Royal. I wondered whether I would recognise Robert or Josiane or the other members of the group as easily \u2013 yes, probably. It was a slightly distressing thought; my memory was constantly filling up with information that was almost completely useless. As a human being, I was particularly proficient in the recognition and storage of images of other humans. _Nothing is more useful to man than man himself_. The reason why I had invited Lionel was not particularly clear to me; the conversation would obviously drag. To keep it going, I asked whether he'd had the opportunity to go back to Thailand. No, and it wasn't for lack of wanting, but unfortunately the trip was rather expensive. Had he seen any of the other members of the group again? No, none of them. Then I told him I had seen Val\u00e9rie, whom he might perhaps remember, and that we were now living together. He seemed happy at this news; we had clearly made a good impression on him. He didn't get the chance to travel much, he told me; and that holiday in Thailand in particular was one of his fondest memories. I started to feel moved by his simplicity, his na\u00efve longing for happiness. It was at that point that I did something which, thinking back on it even today, I'm tempted to describe as _good_. On the whole, I am not good, it is not one of my characteristics. Humanitarians disgust me, the fate of others is generally a matter of indifference to me, nor have I any memory of ever having felt any sense of solidarity with other human beings. The fact remains that, that evening, I explained to Lionel that Val\u00e9rie worked in the tourist industry, that her company was about to open a new club in Krabi and that I could easily get him a week-long stay at 50 per cent discount. Obviously, this was pure invention; but I had decided to pay the difference. Maybe, to a degree, I was trying to _show off_ ; but it seems to me that I also felt a genuine desire for him to be able, even if only for a week, to once again feel pleasure at the expert hands of young Thai prostitutes.\n\nWhen I told her about the meeting, Val\u00e9rie looked at me somewhat perplexed; she herself had no memory of Lionel. That really was the problem with the boy, he wasn't a bad guy, but he had no personality: he was too reserved, too humble, it was difficult to remember anything at all about him. 'Okay...' she said, 'I mean, if it makes you happy; in fact he doesn't even have to pay the 50 per cent, I was going to talk to you about this, I'm going to get invitations for the week of the opening. It will be on January 1st.' I called Lionel the following day to tell him that his trip would be free; this was too much, he couldn't believe me, I even had a bit of trouble getting him to accept.\n\nThe same day, I received a visit from a young artist who had come to show me her work. Her name was Sandra Heksjtovoian, something like that, in any case some name that I was never going to remember; if I'd been her agent, I would have advised her to call herself Sandra Hallyday. She was a very young girl, wearing trousers and a tee-shirt, fairly unremarkable, with a roundish face and short, curly hair; she had graduated from the Beaux Arts in Caen. She worked entirely on her body, she explained to me; I looked at her anxiously as she opened her portfolio. I was hoping she wasn't going to show me photos of plastic surgery on her toes or anything like that \u2013 I'd had it up to here with things like that. But no, she simply handed me some postcards which she had had made, with the imprint of her pussy dipped in different coloured paints. I chose a turquoise and a mauve; I was a little sorry I hadn't brought photos of my prick to return the favour. It was all very pleasant, but, well, as far as I could remember, Yves Klein had already done something similar more than forty years ago; I was going to have trouble championing her cause. Of course, of course, she agreed, you had to take it as an _exercice de style_. She then took a more complex piece out of its cardboard packaging: it consisted of two wheels of unequal sizes linked by a thin strip of rubber; a handle made it possible to operate the contraption. The strip of rubber was covered with small plastic protuberances which were more or less pyramid-shaped. I turned the handle and ran my finger along the moving ribbon; it produced a sort of friction which was not unpleasant.\n\n'They're casts of my clitoris,' the girl explained; I immediately removed my finger.\n\n'I took photos using an endoscope, while it was erect, and put it all on a computer. Using 3-D software I reproduced the volume, I modelled the piece using ray-tracing, then I sent the co-ordinates to the factory.' I got the feeling she was allowing herself to be dominated a bit too much by technical considerations. I turned the handle again, more or less unconsciously. 'It cries out to be touched, doesn't it?' she went on with satisfaction. 'I had thought of connecting it to a resistor so it could power a bulb. What do you think?' To be honest, I wasn't in favour of the idea, it seemed to detract from the simplicity of the object. She was quite sweet, this girl, for a contemporary artist; I almost felt like asking her to come to an orgy some night, I was sure she'd get along well with Val\u00e9rie. I realised just in time that, in my position, such a thing risked being construed as sexual harassment. I considered the contraption despondently. 'You know,' I said, 'I'm really more involved in the financial aspect of the projects. For anything to do with the aesthetics, you'd be better off making an appointment to see Mlle Durry.' On a business card, I wrote down Marie-Jeanne's phone number and extension; after all, she must know a thing or two about this whole clitoris business. The girl looked a little disconcerted, but even so, handed me a small bag filled with plastic pyramids. 'I'll give you these casts,' she said, 'The factory made a lot of them.' I thanked her and walked her back to the service entrance. Before saying goodbye, I asked her if the casts were life size. Of course, she told me, it was all part of her artistic methodology.\n\nThat same evening, I examined Val\u00e9rie's clitoris carefully. I had never really paid it any serious attention; whenever I had stroked or licked it, it was as part of a more overall plan, I had memorised the position, the angles, the rhythmic movement to adopt. But now I examined the tiny organ at length as it pulsed before my eyes. 'What are you doing?' she asked, surprised, after five minutes spent with her legs apart. 'It's an artistic methodology...' I said, giving a little lick to soothe her impatience. The girl's cast lacked the taste and the smell obviously; but otherwise there was a resemblance, it was undeniable. My examination complete, I parted Val\u00e9rie's pussy with both hands and licked her clitoris with short, precise thrusts of my tongue. Was it the waiting that had stimulated her desire? More precise, more attentive movements on my part? The fact remains that she came almost immediately. Actually, I thought that Sandra was a pretty talented artist; her work encouraged one to see the world in a new light.\n\n### 14\n\nAS EARLY AS the beginning of December, it was clear that the Aphrodite clubs were going to be a huge success, and probably a success on a _historic_ scale. November is traditionally the most difficult month for the tourist industry. In October, there are still a number of late-season departures; in December, the Christmas period takes over; but rare, extremely rare, are those who consider taking a holiday in November, apart from some particularly hard-nosed and cynical senior citizens. Yet, the first results which came back from the clubs were excellent: the formula had been an immediate success, you might even go so far as to talk about a deluge. I had dinner with Jean-Yves and Val\u00e9rie the night the initial figures came in; he stared at me, almost bizarrely, the results had so exceeded his expectations: taken as a whole, the occupancy for the month was 95 per cent, regardless of destination. 'Ah yes, sex...' I said, embarrassed. 'People need sex, that's all, it's just that they don't dare admit it.' All of this made us inclined to be contemplative, almost silent; the waiter brought the _antipasti_. 'The Krabi opening is going to be unbelievable...' Jean-Yves went on. 'Rembke phoned me, everything's been booked out for three weeks. What's even better is that there's been nothing in the press, not a line. A discreet success, as massive as it is confidential; exactly what we were aiming for.'\n\nHe had finally decided to rent a studio flat and leave his wife; he would not get the keys until January 1st, but he was a lot better, I sensed he was already more relaxed. He was relatively young, handsome and extremely rich: all of these things do not necessarily make life easier, I realised, a little alarmed; but they help, at least, in awakening desire in others. I still could not understand his ambition, the furious energy he invested in making a success of his career. It wasn't for the money I don't think: he paid high taxes and didn't have expensive tastes. Neither was it out of commitment to the company, nor from a more general altruism: it was difficult to imagine the development of global tourism as a noble cause. His ambition existed in its own right, it couldn't be pinned down to one specific source: it was probably more like the desire to build something, rather than to a taste for power or a competitive nature \u2013 I had never heard him talk about the careers of his former friends at the HEC business school, and I don't think he gave them a second thought. All in all, it was a respectable motive, not unlike the one that explains the advance of human civilisation. The social reward bestowed on him was a large salary; under other regimes it might have taken the form of an aristocratic title, or of privileges like those accorded to the members of the _nomenklatura_ ; I didn't get the impression that it would have made much difference. In reality, Jean-Yves worked because he had a taste for work; it was something both mysterious and clear.\n\nOn December 15, two weeks before the opening, he received an anxious phone call from TUI. A German tourist had just been kidnapped with a Thai girl; the kidnapping had taken place in Hat Yai, in the extreme south of the country. The local police had received a confused message, written in an approximate English, which expressed no demands \u2013 but indicated that the two young people would be executed for behaviour in contravention of Islamic law. For some months there had indeed been an increase in the activities of Islamic movements, supported by Libya, in the border area with Malaysia; but this was the first time that they had attacked people.\n\nOn December 18, the naked, mutilated bodies of the young people were thrown from a van, right in the middle of the main square of the town. The young girl had been stoned to death, she had been beaten with extraordinary violence; everywhere her skin was ripped open, her body was little more than a swelling, barely recognisable. The German's throat had been cut and he had been castrated, his penis and testicles had been stuffed into his mouth. This time, the entire German press picked up the story, there were even some brief articles in France. The papers had decided not to publish photographs of the victims, but they quickly became available on the usual internet sites. Jean-Yves telephoned TUI every day: up until now, the situation was not alarming; there had been few cancellations, people stuck to their holiday plans. The prime minister of Thailand made repeated reassurances: it was undoubtedly an isolated incident, all known terrorist groups condemned the kidnapping and the executions.\n\nAs soon as we arrived in Bangkok, however, I felt a certain tension, especially around the Sukhumvit area where most of the Middle-Eastern tourists stayed. They came mainly from Turkey or Egypt, but sometimes also from more hard-line Muslim countries such as Saudi Arabia or Pakistan. When they walked through the crowds, I could feel the hostile stares directed towards them. At the entrance to most of the hostess bars, I saw signs: 'NO MUSLIMS HERE'; the owner of a bar in Patpong had even clarified his line of reasoning, writing in a decorative hand the following message: ' _We respect your Muslim faith: we don't want you to drink whisky and enjoy Thai girls_.' The poor things were hardly to blame, in fact it was obvious that in case of a terrorist attack, they would be the first to be targeted. On my first visit to Thailand, I had been surprised by the presence of people from Arab countries, in fact, they came for exactly the same reasons as Westerners, with one slight difference: they threw themselves into debauchery with much more enthusiasm. Often, in the hotel bars you'd find them around a bottle of whisky at ten in the morning; and they were first to arrive as soon as the massage parlours opened. In clear breach of Islamic law and probably feeling guilty about it, they were, for the most part, courteous and charming.\n\nBangkok was as polluted, noisy, stifling as always; but I was just happy to be back. Jean-Yves had two or three meetings with bankers, or at some ministry, anyway, I only vaguely followed what was going on. After two days, he informed us that his meetings had been very conclusive: the local authorities were as obliging as possible, they were prepared to do anything to attract the smallest amount of Western investment. For a number of years, Thailand had been unable to alleviate its economic crisis, the stock exchange and the currency were at historic lows, government debt had reached 70 per cent of the gross domestic product. 'They're so deep in shit that they're not even corrupt any more...' Jean-Yves told us. 'I had to grease a few palms, but not many, nothing at all compared to what was going on five years ago.'\n\nOn the morning of December 31, we took the plane to Krabi. As we got out of the minibus, I ran into Lionel, who had arrived the previous evening. He was delighted, he told me, absolutely delighted; I had a bit of trouble stemming his torrent of gratitude. But, as I arrived at my chalet, I too was struck by the beauty of the landscape. The beach was immense, immaculate, the sand as fine as powder. Over a distance of thirty metres, the ocean veered from azure to turquoise, from turquoise to emerald. Vast chalk crags covered with lush green forests rose out of the water as far as the horizon, losing themselves in the light and the distance, giving the bay a depth that seemed unreal, cosmic.\n\n'Isn't this the place where they filmed _The Beach_?' Val\u00e9rie asked me.\n\n'No, I think that was at Ko Phi Phi; but I haven't seen the film.'\n\nAccording to her, I hadn't missed much; apart from the landscapes it had nothing to recommend it. I vaguely remembered the book, which tells the story of a bunch of backpackers in search of an unspoiled island; the only clue they have is a map drawn for them by an old traveller in a shitty hotel on Khao San Road, just before he commits suicide. First, they go to Ko Samui \u2013 much too touristy; from there they go to a neighbouring island, but there are still too many people for their liking. In the end, by bribing a sailor, they finally arrive on their island, situated in a nature reserve and therefore, in theory, inaccessible. It's at this point that things start to go wrong. The early chapters of the book perfectly illustrate the curse of the tourist, caught up in a frenetic search for places which are 'not touristy', which his very presence undermines, forever forced to move on, following a plan whose very fulfilment, little by little, renders it futile. This hopeless situation, comparable to a man trying to escape his own shadow, was common knowledge in the tourist industry, Val\u00e9rie informed me: in sociological terms it was known as the double bind paradox.\n\nThe holidaymakers who had chosen the Krabi Eldorador Aphrodite, at any rate, did not look ready to succumb to the double bind paradox: although the beach was huge, they had all chosen more or less the same area. As far as I had been able to make out, they seemed to conform to the expected breakdown of clientele. Val\u00e9rie had the precise figures: 80 per cent Germans, mostly senior executives or professionals, 10 per cent Italians, 5 per cent Spaniards and 5 per cent French. The surprise was that there were a lot of couples. They looked pretty much like the sort of swinging couples that you might have run into on the Cap d'Agde: most of the women had silicone-enhanced breasts, a lot of them wore a gold chain around their waists or ankles. I also noticed that almost everyone swam in the nude. All of this made me fairly confident; you never have any trouble from people like that. In contrast to a 'backpackers' paradise', a resort dedicated to wife-swapping, which only comes into its own when visitor numbers are high, is not paradoxical by definition. In a world where the greatest of luxuries is acquiring the wherewithal to avoid other people, the good-natured sociability of middle-class German wife-swappers constitutes a form of particularly subtle subversion, I said to Val\u00e9rie, just as she was taking off her bra and panties. Immediately after undressing, I was a little embarrassed to discover that I had a hard on, and I lay down on my stomach beside her. She parted her thighs, serenely baring her sex to the sun. A few metres to our right was a group of German women who seemed to be discussing an article from _Der Spiegel_. One of them had shaved her public hair, you could easily make out her slender, delicate slit. 'I really go for that type of pussy...' Val\u00e9rie said in a low voice. 'It makes you feel like slipping a finger inside.' I really went for them too; but to our left was a Spanish couple where the woman, by contrast, had a really thick, black, curly pubic bush; I could really go for that too. As she lay down, I could make out the thick, plump lips of her pussy. She was a young woman, no more than twenty-five, but her breasts were heavy, with large, prominent areolas. 'Come on, turn over on to your back...' Val\u00e9rie whispered into my ear. I did as I was told, kept my eyes closed, as though somehow the fact that I could see nothing diminished the enormity of what we were doing. I felt my cock stand up, the glans emerging from its sheath of protective skin; concentrating purely on the sensation, the warmth of the sun on the mucous membranes was immensely pleasurable. I did not open my eyes when I felt a thread of suntan lotion trickle on to my torso, then on to my stomach. Val\u00e9rie's fingers moved in short, light touches. The fragrance of coconut filled the air. At the point when she began to rub oil into my penis, I opened my eyes suddenly: she was kneeling by my side, facing the Spanish woman who had propped herself up on her elbows to watch. I threw my head back, staring at the blue of the sky. Val\u00e9rie placed the palm of one hand on my balls, slipped her index finger into my anus; with her other hand she continued to jerk me off steadily. Turning my head to the left, I saw that the Spaniard was busying herself with her own guy's penis; I turned back to stare at the azure. At the point when I heard footsteps approaching across the sand, I closed my eyes again. First there was the sound of a kiss, then I heard whispering. I no longer knew how many hands or fingers stroked and wrapped around my prick; the sound of the backwash was very gentle.\n\nAfter the beach, we made a tour of the leisure centre; it was getting dark, the multicoloured signs of the go-go bars lit up one by one. Around a dozen bars arranged around in a circular piazza surrounded a huge massage parlour. In front of the entrance, we met Jean-Yves, who was just leaving, escorted to the door by a girl wearing a long dress. She had large breasts, pale skin and looked a little Chinese. 'Is it nice inside?' Val\u00e9rie asked him.\n\n'It's amazing: a bit kitsch, but very lavish. There are fountains, tropical plants, waterfalls; they've even put up statues of Greek goddesses.'\n\nWe settled ourselves on a deep sofa upholstered with gold thread before choosing two girls. The massage was very pleasant, the hot water and the liquid soap dissolved all traces of suntan oil from our skin, The girls moved gracefully: to soap us they used their breasts, their buttocks, their inner thighs: straight away Val\u00e9rie started to moan. Once again I marvelled at the richness of a woman's erogenous zones.\n\nAfter drying ourselves we lay down on a large, circular bed, two thirds of its circumference encircled by mirrors. One of the girls licked Val\u00e9rie, easily bringing her to orgasm. I knelt over her face; the other girl caressed my balls, jerking me off in her mouth. At the point when she felt I was about to come, Val\u00e9rie motioned to the girls to come closer: while the first girl licked my balls, the other kissed Val\u00e9rie on the mouth; I ejaculated over their half-joined lips.\n\nThe guests for the New Year's Eve party were mostly Thais connected in one way or another with the tourist industry. None of the directors of Aurore had come; the head of TUI had also been unable to get away, but he had sent a subordinate who clearly had no power whatever but seemed thrilled at the opportunity. The buffet was exquisite, a mixture of Thai and Chinese cuisine. There were crispy little _nems_ with basil and lemongrass, deep-fried parcels of water spinach, prawn curry with coconut milk, fried rice with cashew nuts and almonds, an unbelievably delicious Peking duck which melted in the mouth. French wines had been imported for the occasion. I chatted for a while with Lionel, who seemed to be basking in contentment. He was accompanied by a ravishing girl from Chiang Mai whose name was Kim. He had met her in a topless bar on the first night and they had been together ever since. I could easily see what this big, slightly clumsy boy saw in the fragile creature, so delicate she seemed almost unreal; I couldn't see how he could ever have found such a girl in his own country. They were a godsend, these little Thai whores, I thought; a gift from heaven, nothing less. Kim spoke a little French. She had been to France once, Lionel marvelled; her sister had married a Frenchman.\n\n'Really?' I enquired 'What does he do for a living?'\n\n'He's a doctor,' his face clouded a little. 'Obviously, with me it wouldn't be the same kind of life.'\n\n'You've got job security...' I said optimistically. 'Everyone in Thailand dreams of being a civil servant.'\n\nHe looked at me, a little doubtful. It was true, though, the public sector fascinated the Thais. It's true that in Thailand civil servants are corrupt; not only do they have job security, they're rich too. You can have everything you want. 'Well, I hope you have a nice evening...' I said, making my way towards the bar. 'Thank you...' he said, blushing, I don't know what possessed me to play the man of the world at that moment; decidedly, I was getting old. I did have some doubts about the girl: Thai girls from the north are usually very beautiful, but sometimes they're a bit too conscious of the fact. They spend their time staring at themselves in the mirror, keenly aware that their beauty alone constitutes a crucial economic advantage; and as a result they become useless, capricious creatures. On the other hand, unlike some cool western chick, Kim was not in a position to realise that Lionel himself was a bore. The principal criteria for physical beauty are youth, absence of handicap and a general conformity to the norms of the species; they are quite clearly universal. The ancillary criteria \u2013 vaguer and more relative \u2013 were more difficult to appreciate for a young girl from a different culture. For Lionel, the exotic was a wise choice, possibly even the only choice. Anyway, I thought, I've done my best to help him.\n\nA glass of Saint-Est\u00e8phe in hand, I sat on a bench to look at the stars. The year 2002 would mark France's entry into the single currency \u2013 amongst other things: there would also be the World Cup, the presidential elections, various high-profile media events. The rocky crags of the bay were lit up by the moon; I knew there would be a firework display at midnight. A few minutes later, Val\u00e9rie came and sat beside me. I took her in my arms, put my head on her shoulder; I could barely make out the features of her face, but I recognised the scent, the texture of her skin. At the moment when the first rocket exploded, I noticed that her green, almost transparent dress was the same one she had worn a year before at the New Year's Party on Ko Phi Phi; when she pressed her lips against mine, I felt something strange, as though the very order of things had been upturned. Strangely, and without in the least deserving it, I had been given a second chance. It is very rare, in life, to have a second chance; it goes against all the rules. I hugged her fiercely to me, overwhelmed by a sudden desire to weep.\n\n### 15\n\n###### _If love, then, cannot triumph, how can the spirit reign? All practical supremacy belongs to action_.\n\n###### Auguste Comte\n\nTHE BOAT SKIMMED over the immensity of turquoise, and I didn't have to worry about what I was doing. We had left early in the direction of Ko Maya, sailing past the outcrops of coral and the immense chalk crags. Some of them had eroded to form circular islets whose central lagoons could be reached via narrow channels carved into the rock. Inside these islets the water was still, emerald green. The pilot cut the engine. Val\u00e9rie looked at me, we remained silent, unmoving; moments passed in utter silence.\n\nThe pilot dropped us on the island of Ko Maya, in a bay protected by high rocky walls. The beach stretched out at the foot of the cliffs, it was about a hundred metres long, narrow and curved. The sun was high in the sky. It was already eleven o'clock. The pilot started up the engine and headed back in the direction of Krabi; he was to come back and pick us up in the late afternoon. As soon as he rounded the entrance to the bay, the roar died away.\n\nWith the exception of the sexual act, there are few moments in life in which the body exults in the simple pleasure of being alive, fills with joy at the simple fact of its presence in the world; January 1st was, for me, completely filled with such moments. I have no memory of anything other than that bliss. We probably swam, we must have warmed ourselves in the sun and made love. I don't think we spoke, or explored the island. I remember Val\u00e9rie's scent, the taste of salt drying on her pubis; I remember falling asleep inside her and being woken by her contractions.\n\nThe boat came back to collect us at five o'clock. On the terrace of the hotel overlooking the bay, I had a Campari and Val\u00e9rie a Mai Tai. The chalk crags were almost black in the orange light. The last of the bathers were returning, towels in hand. A few metres from the shore, entwined in the warm water, a couple were making love. The rays of the setting sun struck the gilded roof of a pagoda halfway up. In the peaceful air, a bell tolled several times. It's a Buddhist custom, when one has accomplished a good deed or a meritorious action, to commemorate the act by ringing a temple bell; how joyful is a religion which causes the air to resound with human testimony to good deeds.\n\n'Michel...' said Val\u00e9rie after a long silence, looking straight into my eyes. 'I want to stay.'\n\n'What do you mean?'\n\n'To stay here permanently. I was thinking about it as we were coming back this afternoon: it's possible. All I need is to be appointed resort manager. I've got the qualifications for it, and the necessary skills.'\n\nI looked at her, saying nothing; she put her hand on mine.\n\n'Only, you'd have to agree to give up your work. Would you?'\n\n'Yes.' I must have taken less than a second to answer, without a hint of hesitation; never have I been faced with a decision that was so easy to make.\n\nWe spotted Jean-Yves coming out of the massage parlour. Valerie waved to him; he came and sat at our table: she explained her plan.\n\n'Well...' he said hesitantly, 'I suppose we could manage it. Obviously Aurore are going to be a bit surprised, because what you're asking for is a demotion. Your salary will be cut in half at least; there's no other way of doing it given the other employees.'\n\n'I know,' she said; 'I don't give a damn.'\n\nHe looked at her again, shaking his head in surprise. 'It's your choice... If that's what you want... After all,' he said, as if he were only just realising it, 'I'm the one who runs the Eldorador resorts; I've got the right to appoint whoever I like as resort manager.'\n\n'So, you'd agree to it?'\n\n'Yes... yes, I can't stop you.'\n\nIt's a curious sensation, feeling your life teetering on the brink of a radical change; all you have to do is stay there, do nothing, to feel the sensation. Throughout the meal I remained silent, pensive, so much so that eventually Val\u00e9rie became worried.\n\n'Are you sure this is what you want?' she asked. 'Are you sure you won't miss France?'\n\n'No, I won't miss anything.'\n\n'There's nothing to do here, there's no cultural life.'\n\nI was already aware of this; inasmuch as I'd had occasion to give the matter any thought, culture seemed to me to be a necessary compensation for the misery of our lives. It was possible to imagine a different sort of culture, one bound up with celebration and lyricism, something which sprang from a state of happiness; I was doubtful \u2013 it appeared to me to be a highly theoretical proposition, and one which could no longer have any real significance in my life.\n\n'There's TV5...' I said indifferently. She smiled; it was well known that TV5 was in fact one of the worst television channels in the world. 'Are you sure you won't get bored?' she insisted.\n\nIn my life, I had known suffering, oppression, anxiety; I had never known boredom. I could see no objection to the endless, imbecile repetition of sameness. Of course, I harboured no illusions about being capable of getting to that point: I knew that misery is robust, it is resourceful and tenacious; but it was not a prospect that caused me the least concern. As a child, I could spend hours counting sprigs of clover in a meadow: in all the years of searching I had never found a four-leafed clover; I never felt any disappointment or any bitterness; to tell the truth, I could just as well have been counting blades of grass \u2013 all of those sprigs of clover, with their three leaves, seemed endlessly identical, endlessly splendid to me. One day, when I was twelve, I had climbed to the top of an electricity pylon high in the mountains. As I was going up, I didn't once look down at my feet. When I reached the platform at the top, the descent seemed complicated and dangerous. The mountain ranges stretched as far as the eye could see, crowned with eternal snows. It would have been much simpler to stay there, or to jump. I was stopped, _in extremis_ , by the thought of being crushed; but otherwise, I think I could have rejoiced endlessly in my flight.\n\nThe following day I met Andreas, a German who had been living in the area for ten years. He was a translator, he explained, which made it possible for him to work alone; he went back to Germany once a year for the Frankfurt Book Fair; if he had queries, he made them via the internet. He'd had the opportunity to translate a number of American bestsellers \u2013 among them _The Firm_ \u2013 which in themselves guaranteed him a healthy income; the cost of living here was low. Until now, there had been almost no tourism. He found it surprising to see so many compatriots descending on the place; he greeted the news unenthusiastically, but with no real displeasure either. His ties with Germany had in fact become very tenuous, despite the fact that his work obliged him to use the language constantly. He had married a Thai girl whom he'd met in a massage parlour, and they now had two children.\n\n'Is it easy, here, to have... um... children?' I asked. I felt as though I was asking something incongruous, as if I'd asked whether it was difficult to acquire a dog. To be honest, I had always felt a certain repugnance for young children; as far as I was concerned they were ugly little monsters who shat uncontrollably and screamed insufferably. But I was aware that it was something most couples _do_ ; I did not know whether it made them happy, at any rate they didn't dare complain about it. 'Actually,' I said glancing around the resort, 'with as much space as this, it might be feasible: they could wander between the chalets, they could play with bits of wood or whatever.'\n\nAccording to Andreas, yes, it was particularly easy to have children here; there was a school in Krabi, it was even within walking distance. And Thai children were very different from European children, a lot less quick-tempered and less prone to tantrums. For their parents, they felt a respect bordering on veneration, it came to them quite naturally, it was part of their culture. Whenever he visited D\u00fcsseldorf, he was quite literally frightened by the behaviour of his nephews.\n\nTo tell the truth, I was only half convinced by this idea of cultural immersion. For reassurance I reminded myself that Val\u00e9rie was only twenty-eight; in general, women don't get broody until about thirty-five. But, in the end, yes, if necessary, I would have her child; I knew the idea would come to her, it was unavoidable. After all, a child is like a little animal, admittedly with certain malicious tendencies; let's say, a bit like a small monkey. It might even have its advantages, I thought; eventually I would be able to teach it to play _Mille Bornes_. I nurtured a genuine passion for the game of _Mille Bornes_ , a passion which remained, in general, unsatisfied; who could I invite to play with me? Certainly not my work colleagues; nor the artists who came to show me their portfolios. Andreas, maybe? I gauged him quickly: no, he didn't look the type. That said, he seemed serious, intelligent; it was a friendship worth cultivating. 'Are you thinking of moving here... permanently?'\n\n'Yes, permanently.'\n\n'It's better to look at it like that,' he said, nodding his head. 'It's very difficult to leave Thailand; I know that if I had to do it now, it's something I'd find very hard to deal with.'\n\n### 16\n\nTHE DAYS PASSED with terrifying speed; we were supposed to go back on January 5. The night before, we met up with Jean-Yves in the main restaurant. Lionel had declined the invitation; he was going to watch Kim dance. 'I really like watching her dance almost naked in front of men...' he told us, 'knowing that later on, I'll be the one to have her.' Jean-Yves looked at him as he walked away. 'He's making progress, the gas man...' he noted, sarcastically. 'He's discovering perversion.'\n\n'Don't make fun of him...' Val\u00e9rie protested. 'I think I finally understand what you see in him,' she said turning to me; 'He's an endearing boy. Anyway, I'm sure he's having a fabulous holiday.'\n\nIt was getting dark; lights winked on in the villages around the bay. A last ray of sun lit up the golden roof of the pagoda. Since Val\u00e9rie had informed him of her decision, Jean-Yves hadn't broached the subject again. He waited until the end of the meal to do so; he ordered a bottle of wine.\n\n'I'm going to miss you...' he said. 'It won't be the same. We've been working together for more than five years. We've worked well together, we've never had a serious row. Without you, I would never have made it.'\n\nVal\u00e9rie didn't say anything. There was nothing to say; broadly speaking, it was true.\n\n'Now,' he said thoughtfully, 'we'll be able to extend the formula. One of the most obvious countries is Brazil. I've also been thinking about Kenya again: the ideal thing to do would be to open another club, further inland, for the safaris, and leave the beach club as an 'Aphrodite' resort. One of the other immediate possibilities is Vietnam.'\n\n'You're not afraid of the competition?' I asked.\n\n'There's no risk there. The American chains wouldn't dare get involved in something like this. What I was a bit afraid of was the reaction of the French press; but for the moment, there's been nothing. It has to be said that most of our customers are foreign, from Germany and Italy \u2013 they're more relaxed about this sort of thing.'\n\n'You're going to be the biggest pimp in the world...'\n\n'Not a pimp,' he protested. 'We don't take a penny from what the girls earn; we just let them work, that's all.'\n\n'Anyhow, there's no connection,' interrupted Val\u00e9rie; 'they're not really part of the hotel staff.'\n\n'Well, yes...' Jean-Yves said hesitantly. 'Here they're not connected; but I've heard that in the Dominican Republic the waitresses are only too happy to go upstairs.'\n\n'They're doing it of their own free will.'\n\n'Oh, yes, that's the least you can say.'\n\n'Well...' Val\u00e9rie extended a conciliating gesture to the world, 'don't let the hypocrites grind you down. You're there, you provide the framework, using the Aurore know-how, and that's all.'\n\nThe waiter brought lemongrass soup. At the neighbouring tables were German and Italian men accompanied by Thai girls, some German couples \u2013 accompanied or otherwise. Everyone quietly living together, with no apparent problem, in a general atmosphere of pleasure; this resort manager job promised to be pretty easy.\n\n'So, you're really going to stay here...' Jean-Yves said again; clearly he was having trouble believing it. 'It's surprising; I mean, in a way, I understand, but... what's surprising is that you're giving up the chance of making more money.'\n\n'More money to do what?' said Val\u00e9rie emphatically. 'Buy Prada handbags? Spend a weekend in Budapest? Eat white truffles in season? I've earned a lot of money, I can't even remember where it's gone: yes, I've probably spent it on stupid things like that. Do you know where your money goes?'\n\n'Well...' He thought. 'Actually, up to now, I think it's mostly Audrey who's spent it.'\n\n'Audrey's a stupid bitch,' she retorted, mercilessly. 'Thank God you're getting divorced. It's the most intelligent decision you've ever made.'\n\n'It's true, deep down she is very stupid...' he replied, unembarrassed. He smiled, hesitated a moment: 'You really are a strange girl, Val\u00e9rie.'\n\n'It's not me who's strange, it's the world around me. Do you really want to buy yourself a Ferrari cabriolet? A holiday home in Deauville, which will only get burgled anyway? To work ninety hours a week until you're sixty? To pay half of everything you earn in tax to finance military operations in Kosovo, or recovery plans for the inner cities? We're happy here; we have everything we need in life. The only thing the Western world has to offer is designer products. If you believe in designer products, then you can stay in the West; otherwise, in Thailand you can get excellent fakes.'\n\n'It's your position that's strange; you've worked for years at the centre of Western civilisation, without ever believing in its values.'\n\n'I'm a predator,' she replied calmly, 'a sweet little predator \u2013 my needs are not very great; but if I've worked all my life, it's only been for the cash; now, I'm going to start living. What I don't understand is other people: what's stopping you, for example, from coming to live here? You could easily marry a Thai girl: they're pretty, gentle, good in bed; some of them even speak a bit of French.'\n\n'Well, urn...' He hesitated again. 'Up until now, I've enjoyed having a different girl every night.'\n\n'You'll grow out of it. In any case, there's nothing stopping you visiting massage parlours after you're married; that's what they're there for.'\n\n'I know. I think... Fundamentally, I think I've always had trouble making the important decisions in my life.'\n\nA little embarrassed by this admission, he turned to me:\n\n'What about you, Michel, what are you going to do here?'\n\nThe response closest to the truth was probably something like 'Nothing'; but it's always difficult to explain that kind of thing to an active person. 'The cooking...' replied Val\u00e9rie on my behalf. I turned to her, surprised. 'Yes, yes,' she insisted, 'I've noticed that from time to time you have vague creative aspirations in that area. It's just as well, I don't like cooking; I'm sure that here you'll be able to make a start.'\n\nI tasted a spoonful of my curried chicken with green peppers; as it happened, I could imagine doing something similar with mangoes. Jean-Yves nodded thoughtfully. I looked at Val\u00e9rie: she was a good predator, more intelligent and more tenacious than I was; and she had chosen me to share her lair. It is possible to suppose that societies are dependent, if not on a common goal, then at least on a consensus \u2013 sometimes described in western democracies as a _weak consensus_ , by certain leader-writers whose political positions are very entrenched. As someone of pretty weak temperament myself, I had done nothing to change that consensus; the idea of a common goal seemed less clear. According to Immanuel Kant, human dignity consists in not accepting to be subject to laws except inasmuch as one can simultaneously consider oneself a legislator; never had such a bizarre fantasy crossed my mind. Not only did I not vote, but I had never considered elections as anything more than excellent television shows \u2013 in which, to tell the truth, my favourite actors were the political scientists: J\u00e9r\u00f4me Jaffr\u00e9 in particular delighted me. Being a political leader seemed to me a difficult, technical, wearing task; I was quite happy to delegate whatever powers I had. In my youth, I had encountered militants, who considered it necessary to force society to evolve in this or that direction; I had never felt any sympathy or any respect for them. Gradually, I had even learned to distrust them: the way they got involved in popular causes, the way they treated society as though it was something they played an active role in, seemed suspicious. What did I, for my part, have to reproach the West for? Not much \u2013 but I wasn't especially attached to it (and I was finding it more and more difficult to understand how one could feel attached to an idea, a country, anything in fact other than an individual). Life was expensive in the West, it was cold there; the prostitution was of poor quality. It was difficult to smoke in public places, almost impossible to buy medicines and drugs; you worked hard, there were cars, and noise, and the security in public places was very badly implemented. All in all, it had numerous drawbacks. I suddenly realised to my embarrassment that I considered the society I lived in more or less as a natural environment \u2013 like a savannah, or a jungle \u2013 whose laws I had to adapt to. The notion that I was in any way in solidarity with this environment had never occurred to me; it was like an atrophy in me, an emptiness. It was far from certain that society could continue to survive for long with individuals like me; but I could survive with a woman, become attached to her, try to make her happy. Just as I turned to give Val\u00e9rie another grateful look, I heard a sort of click to my right. Then I noticed an engine noise coming from the sea, which cut out immediately. At the front of the terrace, a tall blonde woman stood up, screaming. Then came the first burst of gunfire, a brief crackle. She turned towards us, bringing her hands up to her face: a bullet had hit her in the eye, the socket was now no more than a bloody hole; then she collapsed without a sound. Then I saw our assailants, three men wearing turbans, moving swiftly in our direction, machine-guns in hand. A second round of gunfire broke out, a little longer; the noise of crockery and broken glass mingled with screams of pain. For several seconds, we must have been completely paralysed; few people thought to take shelter under the tables. At my side, Jean-Yves gave a brief yelp, he had just been hit in the arm. Then I saw Val\u00e9rie slide gently from her chair and collapse on the ground. I rushed to her and put my arms around her. From that point on, I saw nothing. The bursts of machine-gun fire followed one after another in a silence disturbed only by the sound of exploding glasses; it seemed to me to go on for ever. The smell of gunpowder was very intense. Then everything was silent again. I noticed that my left arm was covered in blood; Val\u00e9rie must have been hit in the chest or the throat. The streetlamp beside us had been blown out and I could barely see a thing. Lying about a metre from me, Jean-Yves tried to get up and groaned. Just then, from the direction of the leisure complex, came an enormous explosion which ripped through the entire area and echoed around the bay for a long time. At first I thought my eardrums had burst, but some seconds later, in the midst of my daze, I became aware of a concert of dreadful screams, the genuine screams of the damned.\n\nThe emergency services arrived ten minutes later; they had come from Krabi. They went first to the leisure complex. The bomb had exploded in the middle of Crazy Lips, the largest of the bars, at peak time; it had been hidden in a sports bag left near the dance floor. It was a very powerful homemade dynamite device triggered by an alarm clock; the bag had been stuffed with bolts and nails. Under the force of the blast, the thin brick walls separating the bar from the other establishments had been blown out; a number of the metal girders which held up the whole building had buckled from the force of the blast, the roof was threatening to collapse. Faced with the extent of the catastrophe, the first thing the rescue workers did was to call for back-up. In front of the entrance to the bar a dancer crawled along the ground, still wearing her white bikini, her arms severed at the elbows. Nearby, a German tourist sitting in the midst of the rubble held his intestines as they spilled from his belly; his wife lay near him, her chest gaping, her breasts half torn off. Inside the bar a blackish smoke hung in the air; the ground was slippery, covered with blood seeping from human bodies and mutilated organs. A number of the dying, their arms or legs severed, tried to crawl towards the exit, leaving behind them a bloody trail. Bolts and nails had gouged out eyes, ripped off hands, torn faces to shreds. Some of the bodies had literally exploded from within, their limbs and viscera strewing the ground for several metres.\n\nWhen the rescue workers reached the terrace, I was still holding Val\u00e9rie in my arms; her body was warm. Two metres in front of me, a woman lay on the ground, her bloody face peppered with shards of glass. Others remained in their seats, mouths wide open, frozen in death. I screamed at the rescue workers: two nurses came over immediately, gently took Val\u00e9rie and placed her on a stretcher. I tried to stand up, but fell backwards; my head hit the ground. It was then that I heard, very distinctly, someone say in French: 'She's dead.'\n\n## Part Three\n\n## _Pattaya Beach_\n\n### 1\n\nIT WAS THE first time for a long while that I had woken up alone. The hospital in Krabi was a small, bright building; the doctor came to see me in mid-morning. He was French, a member of M\u00e9decins du Monde; they had arrived on the scene the day after the attack. He was a man of about thirty, a little stooped, with a worried expression. He told me that I had been asleep for three days. 'Actually, you weren't really asleep,' he went on; 'sometimes you appeared to be awake. We spoke to you several times, but this is the first time we've managed to make contact.' Make contact, I thought. He told me, too, that the death toll of the attack had been horrifying: at the moment, the dead numbered one hundred and seventeen; it was the most murderous attack ever to take place in Asia. A number of the injured were still in a extremely critical condition, considered too weak to be moved; Lionel was among them. Both of his legs had been severed, a piece of metal had lodged in the pit of his stomach; his chances of survival were remote. Others who had been seriously injured had been transported to Bumrungrad Hospital in Bangkok. Jean-Yves had only been slightly hurt: a bullet had fractured his humerus; it had been possible to treat him on the spot. Me, I was absolutely fine, not even a scratch. 'As for your friend...' the doctor said, 'her body has already been repatriated to France. I spoke to her parents on the phone: she will be buried in Brittany.'\n\nHe fell silent; he was probably waiting for me to say something. He watched me out of the corner of his eye; he seemed increasingly worried.\n\nTowards noon, a nurse arrived with a tray; she took it away an hour later. She told me I really should start to eat again, that it was vital.\n\nJean-Yves came to see me sometime in the afternoon. He too looked at me strangely, a little sidelong. He talked mostly about Lionel; he was dying now, it was only a matter of hours. He had asked for Kim a lot. Miraculously, she was unhurt, but seemed to have got over it rather quickly: as he was taking a stroll in Krabi the previous evening, Jean-Yves had seen her on the arm of an Englishman. He had said nothing about this to Lionel, who didn't seem to harbour any illusions anyway; at least he had been fortunate enough to have met her. 'It's strange...' Jean-Yves said to me, 'he seems happy.'\n\nAs he was leaving my room, I realised that I hadn't said a word; I really didn't know what to say to him. I knew perfectly well that something was wrong, but it was a vague feeling, difficult to put into words. It seemed to me that the best thing to do was keep quiet until the people around me realised their mistake; it was just a bad patch I had to get through.\n\nBefore he left, Jean-Yves looked up at me, then shook his head discouraged. It appears, at least this is what they told me immediately afterwards, that I talked a lot, all the time in fact, whenever I was left alone in the room; as soon as someone came in, I fell silent.\n\nA few days later they transported us to Bumrungrad Hospital in an air ambulance. I didn't really understand the reasons for the transfer; in fact I think it was mostly so the police would have an opportunity to question us. Lionel had died the night before; crossing the corridor I saw his body wrapped in a shroud.\n\nThe Thai police were accompanied by an embassy attach\u00e9 who acted as an interpreter; unfortunately, I had little to tell them. What seemed to most obsess them most was whether the attackers had been of Arab or Asian origin. I could well understand their preoccupation \u2013 it was important to know whether an international terrorist network had established a foothold in Thailand or whether they were dealing with Malay separatists \u2013 but all that I could do was repeat that everything had happened so quickly, that I had seen only shadows; as far as I knew, the men could have been of Malay appearance.\n\nThen I had a visit from some Americans, who I think were from the CIA. They spoke brutally, in an unpleasant tone; I felt as though I was a suspect myself. They hadn't thought it necessary to bring an interpreter, so I couldn't understand most of their questions. At the end, they showed me a series of photographs, purportedly of international terrorists; I did not recognise any of these men.\n\nFrom time to time, Jean-Yves came to see me in my room, sat at the foot of the bed. I was aware of his presence, I felt a little more tense. One morning, three days after we arrived, he handed me a small sheaf of papers: they were photocopies of newspaper articles. 'The board of Aurore faxed them to me yesterday,' he said; 'They made no comment.'\n\nThe first article, taken from the _Nouvel Observateur_ , was headlined 'A VERY SPECIAL CLUB'; it was two pages long, very detailed, and illustrated with a photograph taken from the German advertising campaign. The journalist accused the Aurore group in no uncertain terms of promoting sex tourism in third-world countries, and added that, in the circumstances, the reaction of the Muslims was understandable. Jean-Claude Guillebaud dedicated his editorial to the same subject. Interviewed by telephone, Jean-Luc Espitalier had declared: 'The Aurore group, a signatory of the world charter for ethical tourism, in no way sanctions such activities; those responsible will be disciplined.' The dossier continued with a vehement but poorly documented article by Isabelle Alonso, from the _Journal de dimanche_ , entitled: 'THE RETURN OF SLAVERY'. Fran\u00e7ois Giroud picked up the theme in his weekly diary: 'Faced,' he wrote, 'with the hundreds of thousands of women who have been sullied, humiliated, reduced to slavery throughout the world \u2013 it is regrettable to have to say this \u2013 what do the deaths of a few of the well-heeled matter.' The terrorist attack in Krabi had naturally given the story considerable impact. _Lib\u00e9ration_ ran a front-page story in which it published photos of the repatriated survivors, taken when they landed at Roissy, with the headline: 'NOT SO INNOCENT VICTIMS'. In his editorial, J\u00e9r\u00f4me Dupuy singled out the Thai government for its lenient attitude to prostitution and drugs, as well as for its frequent breaches of democracy. As for _Paris-Match_ , under the headline 'CARNAGE AT KRABI', came a full account of the 'night of horror'. They had managed to procure photos, which, it has to be said, were of pretty poor quality \u2013 black and white photocopies sent by fax \u2013 they could have been photos of just about anything, you could barely make out the bodies. In the same issue, they published the confessions of a sex tourist \u2013 who actually had nothing to do with the events: he was a freelance who operated mostly in the Philippines. Jacques Chirac had immediately made a statement in which, though he expressed his revulsion for the attack, he condemned the 'unacceptable behaviour of some of our fellow citizens abroad'. Speaking in the wake of the events, Lionel Jospin reiterated that a law existed to crack down on sex tourism, even when the victims were consenting adults. The articles which followed, in _Le Figaro_ and _Le Monde_ , wondered what means should be used to fight this plague, and the position the international community should adopt.\n\nIn the days that followed, Jean-Yves tried to get in touch with Gottfried Rembke by telephone; eventually, he succeeded. The head of TUI was sorry, truly sorry, but there was nothing he could do. In any case, as a tourist destination, Thailand was out of the question for several decades. Above and beyond that, the articles in the French press had had certain repercussions in Germany; it's true that opinion there was more divided, but the majority of the public nonetheless condemned sex tourism. Under the circumstances, he preferred to withdraw from the project.\n\n### 2\n\nI NO MORE understood the reasons for my return to Paris than I had the reasons for my transfer to Bangkok. I was little liked by the hospital staff, they probably found me too inert; even in hospital, even on your deathbed, you are forced to play the part. Medical personnel like patients to put up a certain amount of resistance, to show a wilfulness which they can do their utmost to break, for the good of the patient, naturally. I manifested nothing of the sort. You could roll me on to my side ready for an injection and come back three hours later: I would still be in exactly the same position. The night before my departure, I banged roughly into one of the doors in the hospital corridor as I was trying to find the toilets. In the morning, my face was covered in blood, there was a gash above my eyebrow; I had to be cleaned, have a dressing put on. It hadn't occurred to me to call a nurse; in fact, I hadn't felt a thing.\n\nThe flight was a neutral period of time; I'd lost even the habit of smoking. By the baggage carousel, I shook Jean-Yves's hand; then I took a taxi to the Avenue de Choisy.\n\nI immediately noticed that something wasn't right, that it would never be right. I didn't unpack. I walked around the apartment, a plastic bag in one hand, picking up all the photos of Val\u00e9rie I could find. Most of them had been taken at her parents' in Brittany, on the beach or in the garden. There were also a few erotic photos that I had taken in the flat; I liked to watch her masturbate, I found her movements beautiful.\n\nI sat on the sofa and dialled a number which I had been given to use in case of emergencies, twenty-four hours a day. It was a sort of crisis unit which had been set up especially to care for the survivors of the attack. It was based in a wing of the Sainte-Anne Hospital.\n\nMost of the people who had asked to go there really were in a sad state: despite massive doses of tranquillisers, they had nightmares every night, every time there were screams, worried shouts, tears. When I met them in the corridors I was struck by their distressed, panic-stricken faces; they seemed to be literally eaten up by fear. And that fear, I thought, would end only with their lives.\n\nFor my part, more than anything, I felt terribly weary. In general I only got up to drink a cup of Nescaf\u00e9 or nibble a few biscuits; meals were not compulsory, nor were the therapeutic activities. Even so, I underwent a series of tests, and three days after my arrival I had an interview with a psychiatrist; the tests had revealed 'extremely weakened reactivity'. I was not in pain, but I did, in fact, feel weakened; I felt weaker than it was possible to feel. He asked me what I intended to do. I replied, 'Wait'. I showed myself to be reasonably optimistic; I told him that all this sadness would come to an end, that I would find happiness again, but that I had to wait a while. He didn't seem really convinced. He was a man of about fifty, with a plump, cheerful face, absolutely clean-shaven.\n\nAfter a week, they transferred me to a new psychiatric hospital, this time for a lengthy stay. I had to stay there for a little over three months. To my great surprise, I met the same psychiatrist there. It was hardly surprising, he told me; this was where his surgery was based. Helping crisis victims was only a temporary assignment, something of a speciality in his case, in fact \u2013 he had already been on a committee set up after the bombing of the Saint-Michel RER station.\n\nHe didn't really talk like a typical psychiatrist, at least I found him bearable. I remember he talked to me about 'freeing oneself from attachments': it sounded like some Buddhist bullshit. Freeing what? I was nothing more than an attachment. Inclined to the transitory by nature, I had become attached to a transitory thing, as was my nature \u2013 none of this demanded any particular comment. Had I been inclined towards the eternal by nature, I went on, in order to fuel the conversation, I would have become attached to things eternal. Apparently his technique worked well with survivors haunted by fears of mutilation and death. 'These sufferings do not belong to you, they are not truly yours; they are merely passing phantoms in your mind,' he told people; and in the end, they believed him.\n\nI don't know at what point I began to become aware of the situation \u2013 but in any case, it was only episodic. There were still long periods \u2013 in fact there still are \u2013 when Val\u00e9rie is categorically not dead. In the beginning I could consciously prolong these without the slightest effort. I remember the first time I found it difficult, when I truly felt the weight of reality; it was just after a visit from Jean-Yves. It was a fraught moment; there were memories which I found difficult to deny. I didn't ask him to come back.\n\nMarie-Jeanne's visit, on the other hand, did me good. She didn't say much, she talked a bit about the atmosphere at work; I told her straight away that I wouldn't be coming back, because I was going to move to Krabi. She acquiesced without comment. 'Don't worry,' I told her, 'everything will be fine.' She looked at me with mute compassion; strangely, I actually think that she believed me.\n\nThe visit from Val\u00e9rie's parents was probably the most painful; the psychiatrist must have told them that I was going through a period of denial. As a result, Val\u00e9rie's mother cried almost the whole time; her father didn't seem very comfortable either. They had also come to iron out some practical details, to bring me a suitcase containing my personal belongings. They imagined I wouldn't want to keep the apartment in the 13th arrondissement. 'Of course not', I said, 'of course we'll deal with that later.' At that point Val\u00e9rie's mother began to cry again.\n\nLife goes by effortlessly in an institution: there, for the most part, human needs are satisfied. I had rediscovered _Questions pour un champion_ , it was the only show I watched, I no longer took any interest in the news. A lot of the other residents spent the entire day in front of the television. I wasn't all that keen, really: everything was moving too quickly. I believed that, if I could remain calm, avoid thinking as much as possible, matters would sort themselves out in the end.\n\nOne morning in April, I found out that matters had, in effect, sorted themselves out and that I would soon be able to leave. This seemed to me to complicate things rather: I would have to find a hotel room, create a neutral environment. At least I had money, that was something at least. 'You have to look on the bright side,' I said to one of the nurses. She seemed surprised, perhaps because this was the first time I had ever spoken to her.\n\nThere is no specific treatment for denial, the psychiatrist explained to me at our last interview; it is not really a disorder of mood, but a problem of perception. He had kept me in hospital all this time chiefly because he was worried about the possible risk of a suicide attempt \u2013 they are quite common in cases of sudden, brutal realisations; but now I was out of danger. I see, I said. I see.\n\n### 3\n\nA WEEK AFTER being discharged from hospital, I took a flight back to Bangkok. I had no particular plans. If we had an ideal nature, we could satisfy ourselves with the movements of the sun. The seasons were too distinct in Paris, they were a source of agitation, of insecurity. In Bangkok, the sun rose at six o'clock, it set at six o'clock; in the intervening time, it followed an unchanging course. There was, apparently, a monsoon season, but I had never witnessed it. The bustle of the city existed, but I couldn't clearly grasp the rationale behind it, it was more a sort of natural state. Undoubtedly all of these people had a destiny, a life, inasmuch as their incomes permitted; but for all I knew, they could just as easily have been a pack of lemmings.\n\nI took a room at the Amari Boulevard; most of the guests in the hotel were Japanese businessmen. This was where we had stayed, Val\u00e9rie, Jean-Yves and I, on our last visit; it wasn't really a good idea. Two days later, I moved to the Grace Hotel; it was only about ten metres down the road, but the atmosphere was noticeably different. It was probably the last place in Bangkok where you could still meet Arab sex tourists. They hugged the walls, staying holed up in the hotel \u2013 which had a discotheque and its own massage parlour. You spotted them in the surrounding alleys where there were stalls selling kebabs and long-distance call centres; but, further afield, nothing. I realised that without intending to, I had moved closer to the Bumrungrad Hospital.\n\nIt is certainly possible to remain alive animated simply by a desire for vengeance; many people have lived that way. Islam had wrecked my life, and Islam was certainly something which I could hate; in the days that followed, I devoted myself to trying to feel hatred for Muslims. I was quite good at it, and I started to follow the international news again. Every time I heard that a Palestinian terrorist, or a Palestinian child or a pregnant Palestinian woman had been gunned down in the Gaza Strip, I felt a quiver of enthusiasm at the thought that it meant one less Muslim. Yes, it was possible to live like this.\n\nOne evening, in the hotel coffee-shop, a Jordanian banker struck up a conversation with me. A man of amiable disposition, he insisted on buying me a beer; perhaps his enforced seclusion in the hotel was beginning to get to him. 'I understand how people feel, you know; you can't hold it against them...' he told me. 'It has to be said, we were asking for it. This isn't a Muslim country, there's no reason to spend hundreds of millions building mosques. To say nothing of the bomb attack, of course...' Seeing that I was listening to him attentively, he ordered another beer and become bolder. The problem with Muslims, he told me, was that the paradise promised by the prophet already existed here on earth: there were places on earth where young, available, lascivious girls danced for the pleasure of men, where one could become drunk on nectar and listen to celestial music; there were about twenty of them within five hundred metres of our hotel. These places were easily accessible. To gain admission, there was absolutely no need to fulfil the seven duties of a Muslim, nor to engage in holy war; all you had to do was pay a couple of dollars. It wasn't even necessary to travel to realise such things; all you needed was satellite TV. For him, there was no doubt, the Muslim way was doomed: capitalism would triumph. Already, young Arabs dreamed of nothing but consumer products and sex. They might try to pretend otherwise, but secretly, they wanted to be part of the American system: the violence of some of them was no more than a sign of impotent jealousy; thankfully, more and more of them were turning their backs on Islam. He himself had been unlucky: he was an old man now, and he had been forced to spend his whole life compromising with a religion he despised. I was in much the same boat: there would come a day when the world was delivered from Islam; but for me, it would come too late. I no longer really had a life; I had had a life, for a few months \u2013 that in itself was something, not everyone could say as much. The absence of the will to live is, alas, not sufficient to make one want to die.\n\nI saw him again the next day, just before he left for Amman; it would be a year before he could come back. On the whole, I was glad that he was leaving; I sensed that otherwise he would have wanted to talk to me again. The prospect gave me a bit of a headache: I found it very difficult now to tolerate intellectual debate; I no longer had any desire to understand the world, nor even to know it. Our brief conversation, however, had made a profound impression on me; in fact, he had convinced me from the outset, Islam was doomed. As soon as you thought about it, it seemed obvious. This simple thought was sufficient to dispel my hatred. Once again I ceased to have any interest in the news.\n\n### 4\n\nBANGKOK WAS STILL too much like a normal city, there were too many businessmen, too many tourists on package holidays. Two weeks later, I caught a bus for Pattaya. It had been bound to end this way I thought, as I boarded the vehicle; it was then that I realised that I was wrong, nothing in this story had been determined. I could easily have spent the rest of my life with Val\u00e9rie in Thailand, in Brittany, or indeed anywhere at all. Growing old is no joke; but growing old alone is worse than anything.\n\nAs soon as I had put down my luggage on the dusty floor of the bus station, I knew I had arrived at the end of my journey. A scrawny old junkie with long grey hair, a large lizard perched on his shoulder, was begging outside the turnstiles. I gave him a hundred baht before drinking a beer at the Heidelberg Hof directly opposite. A few pot-bellied, moustachioed German pederasts minced around in their flowery shirts. Near them, three Russian teenage girls, who had attained a pinnacle of sluttishness, squirmed as they listened to their ghetto-blaster. They writhed and rolled about on their chairs, the sleazy little cocksuckers. In a few minutes' walk through the streets of the town, I encountered an impressive variety of human specimens: rappers in baseball caps, Dutch dropouts, cyberpunks with red hair, Austrian dykes with piercings. There is nothing lower than Pattaya, it is a sort of cesspit, the ultimate sewer where the sundry waste of western neurosis winds up. Whether you're homosexual, heterosexual, or both, Pattaya is the last-chance saloon, the one beyond which you might as well give up on desire. The hotels are distinguished, naturally, by different levels of comfort and price, but also by the nationality of their clientele. There are two large communities, the Germans and the Americans (among whom probably some Australians and possibly even some New Zealanders conceal themselves). You also get quite a lot of Russians, recognisable because they dress like rednecks and behave like gangsters. There is even an establishment intended for the French, called Ma Maison. The hotel has only a dozen rooms, but the restaurant is very popular. I spent a week there before I realised that I was not particularly attached to _andouillettes_ or _cuisses de grenouille_ ; that I could live without following the French championship via satellite, and without leafing daily through the arts pages of _Le Monde_. In any case, I needed to find long-term accommodation. A standard tourist visa in Thailand only lasts for one month; but to get an extension, all you have to do is cross the border. A lot of the travel agencies in Pattaya offer a day return to the Cambodian border. After a three-hour trek in a minibus, you queue for an hour or two at customs, have lunch in a self-service restaurant on Cambodian soil (lunch is included in the price, as are tips for customs officials), then you start on your return journey. Most residents have been doing this every month for years; it's much easier than trying to get a long-term visa.\n\nYou don't come to Pattaya to start your life over, but to end it in tolerable conditions. Or, if you want to put it less brutally, to take a rest, a long rest \u2013 one which may prove permanent. These were the terms used by a homosexual of about fifty I met in an Irish pub on Soi 14; he had spent the greater part of his career as a designer working for the popular press and had managed to put some money aside. Ten years earlier, he had noticed that things were going badly for him: he still went out to clubs, the same clubs as always, but more and more often he came home empty-handed. Of course, he could always pay; but if it had to come to that, he would rather pay Asians. He apologised for this remark, hoped I would not infer any racist connotation. No, no, of course, I understood: it's less humiliating to pay for someone who looks nothing like any of those you have seduced in the past, who brings back no memories. If sex has to be paid for, it is best that, in a certain sense, it is undifferentiated. As everyone knows, one of the first things you feel in the presence of another race is that inability to differentiate, that feeling that physically, everyone looks more or less alike. The effect wears off after a few months, and it's a pity, because it bears out a reality: human beings do, in fact, look very much alike. Of course, we can distinguish between males and females; we can also, if we choose, distinguish between different age categories; but any more advanced distinction comes close to pedantry, probably a result of boredom. A creature that is bored elaborates distinctions and hierarchies. According to Hutchinson and Rawlins, the development of systems of hierarchical dominance within animal societies does not correspond to any practical necessity, nor to any selective advantage; it simply constitutes a means of combating the crushing boredom of life in the heart of nature.\n\nSo, the former designer was quietly living out the last years of his queer life treating himself to pretty, slender, muscular, dark-skinned boys. Once a year, he went back to France to visit his family and a few friends. His sex life was less frenetic than I might imagine, he told me: he went out once or twice a week, no more. He had been settled here in Pattaya for six years now; the profusion of varied, exciting and inexpensive sexual opportunities provoked a paradoxical calming of desire. Every time he went out, he was certain of being able to fuck and suck magnificent young boys who, for their part, would jerk him off sensitively and expertly in return. Confident of this fact, he spent more time getting ready to go out and he enjoyed these encounters in moderation. I realised then that he imagined I was in the throes of the erotic frenzy of my first weeks here, that he saw in me a heterosexual counterpart to his own case. I refrained from correcting him. He proved to be friendly, insisted on buying the beers, gave me a number of addresses for long-term accommodation. He had enjoyed talking to a Frenchman. Most of the homosexual residents were English; he was on good terms with them, but from time to time, he wanted to speak his own language. He had no real contact with the little French community which gathered at Ma Maison \u2013 mostly a crowd of straight, ex-colonial, ex-army thugs. If I was going to live in Pattaya, maybe we could go out together some night, no strings, obviously; he gave me his mobile number. I wrote it down, though I knew that I would never call him. He was pleasant, friendly, interesting if you like; but I simply wasn't interested in human relationships any more.\n\nI rented a room on Naklua Road, a little outside the bustle of the city. It had air-conditioning, a fridge, a shower, a bed and some bits of furniture; the rent was three thousand baht a month \u2013 a little more than five hundred francs. I informed my bank of this news, wrote a letter of resignation to the Ministry of Culture.\n\nThere was nothing much left for me to do in this life. I bought a number of reams of A4 paper with the intention of putting the elements of my life in order. It's something people should do more often before they die. It's curious to think of all these human beings who live out their whole lives without feeling the need to make the slightest comment, the slightest objection, the slightest remark. Not that these comments, these objections, these remarks are addressed to anyone in particular, nor intended to have any sort of meaning; but, even so, it seems to me to be better, in the end, that they be made.\n\n### 5\n\nSIX MONTHS LATER, I am still here in my room on Naklua Road, and I think that I have more or less finished my work. I miss Val\u00e9rie. If by chance it was my intention, when I began writing these pages, to lessen the feeling of loss, or to make it more bearable, I would by now be certain of my failure: Val\u00e9rie's absence has never been more painful to me.\n\nAt the beginning of my third month here, I decided in the end to go back to the massage parlours and the hostess bars again. In principle, the idea didn't really fill me with enthusiasm; I was afraid it would be a total fiasco. Nonetheless, I managed to get a hard-on, and even to ejaculate; but I never once experienced any pleasure. It wasn't the girls' fault, they were just as expert, just as gentle; but it was as though I was anaesthetised. I continued to go to a massage parlour once a week, to some extent on principle; then I decided to stop. It was, after all, a form of human contact \u2013 that was the drawback. Even if I didn't in the least believe that my ability to feel pleasure would return, it was possible that the girl would come, especially as the numbness in my penis meant that I could keep going for hours if I didn't make a little effort to interrupt the proceedings. I might get to the point where I wanted her to come, it could become an issue; and I didn't wish to have anything more to do with issues. My life was an empty space, and it was better that it remain that way. If I allowed passion to penetrate my body, pain would follow quickly in its wake.\n\nMy book is almost at an end. More and more often now, I stay in bed for most of the day. Sometimes I turn on the air-conditioning in the morning and turn it off at night and between the two absolutely nothing happens. I've become accustomed to the purring of the machine, which I found irritating at first; but I've also become accustomed to the heat; I don't really have a preference.\n\nFor a long time now, I've stopped buying French newspapers; I suppose that by this time the presidential elections have taken place. The Ministry of Culture, somehow or other, must be getting on with its work. Perhaps Marie-Jeanne still thinks about me from time to time, when she's working on the budget for an exhibition; I haven't tried to get in touch. I don't know what's become of Jean-Yves either; after he was fired from Aurore, I suppose he must have started his career again much further down, and probably in something other than tourism.\n\nWhen your love life is over, life in general takes on a sort of conventional, forced quality. One retains a human form, one's habitual behaviour, a sort of structure; but one's heart, as they say, isn't in it.\n\nMopeds drive down Naklua Road, sending up clouds of dust. It is noon already. Coming from outlying districts, the prostitutes arrive for work in the downtown bars. I don't think I'll go out today. Or maybe I will, late in the afternoon, to gulp down a soup at one of the stalls set up at the crossroads.\n\nWhen one gives up on life, the last remaining human contacts are those you have with shopkeepers. As far as I'm concerned, these are limited to a few words spoken in English. I don't speak Thai, which creates a barrier around me that is suffocating and sad. It is obvious that I will never really understand Asia, and actually it's of not great importance. It's possible to live in the world without understanding it: all you need is to be able to get food, caresses and love. In Pattaya, food and caresses are cheap by Western, and even by Asian, standards. As for love, it's difficult for me to say. I am now convinced that, for me, Val\u00e9rie was simply a radiant exception. She was one of those creatures who are capable of devoting their lives to someone else's happiness, of making that alone their goal. This phenomenon is a mystery. Happiness, simplicity and joy lies within them; but I still do not know how or why it occurs. And if I haven't understood love, what use is it to me to have understood the rest?\n\nTo the end, I will remain a child of Europe, of worry and of shame; I have no message of hope to deliver. For the West, I do not feel hatred; at most I feel a great contempt. I know only that every single one of us reeks of selfishness, masochism and death. We have created a system in which it has simply become impossible to live; and what's more, we continue to export it.\n\nIt's getting dark, the multicoloured fairy lights wink on at the entrances to the beer bars. The German OAPs settle in, placing their thick hands on the thighs of their young companions. More than any other people, they are acquainted with worry and shame, they feel the need for tender flesh, for soft, endlessly refreshing skin. More than any other people, they are acquainted with the desire for their own annihilation. It is rare to come across the vulgar, smug pragmatism of Anglo-Saxon sex tourists among them, that manner of endlessly comparing goods and prices. It is equally rare for them to exercise, to look after their bodies. In general, they eat too much, drink too much beer, get fat; most of them will die pretty soon. They are often friendly, they like to joke, to buy a round, to tell stories; but their company is soothing and sad.\n\nI understand death now; I don't think it will do me much harm. I have known hatred, contempt, decay and other things; I have even known brief moments of love. Nothing of me will survive, and I do not deserve for anything of me to survive; I will have been a mediocre individual in every possible sense.\n\nI imagine, I don't know why, that I will die in the middle of the night, and I still feel a little anxious at the thought of the suffering which will accompany the severing of all ties with the body. I find it difficult to envisage the cessation of life as completely painless and unconscious; naturally, I know that I'm wrong. Nonetheless, I have trouble convincing myself of that fact.\n\nThe locals will find me a few days later, quite quickly in fact; in this climate, corpses quickly start to stink. They won't know what to do with me, and will probably contact the French embassy. I'm far from being destitute, the case will be easy to deal with. There will certainly be quite a lot of money left in my account; I don't know who will inherit it \u2013 the state probably, or some distant relatives.\n\nUnlike other Asian peoples, the Thais don't believe in ghosts, and have little interest in the fate of corpses; most of them are buried in communal graves. Since I will have left no specific instructions, that is what will become of me. A death certificate will be drawn up, a box will be ticked in a registry office, far from here, in France. A few street hawkers, accustomed to seeing me in the area, will shake their heads. My apartment will be rented out to another resident. I'll be forgotten. I'll quickly be forgotten.\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\nEpub ISBN: 9781448105472 \nVersion 1.0\n\n1 3 5 7 9 10 8 6 4 2\n\nVINTAGE \n20 Vauxhall Bridge Road, \nLondon SW1V 2SA\n\nVintage is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com\n\nCopyright \u00a9 Flammarion 2001 \nTranslated from the French, _Plateforme_ \nTranslation copyright \u00a9 Frank Wynne\n\nMichel Houellebecq has asserted his right under the Copyright, Designs and Patents Act, 1988 to be identified as the author of this work\n\nThis book is sold subject to the condition that it shall not by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser\n\nFirst published in Great Britain in 2002 by William Heinemann\n\npenguin.co.uk\/vintage\n\nA CIP catalogue record for this book is available from the British Library\n\nISBN 9780099437888\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n\u00cdndice\n\nPortada\n\nIntroducci\u00f3n\n\nEl gueto jud\u00edo de Venecia. El exilio crea un hogar\n\nEl extranjero\n\nNotas\n\nCr\u00e9ditos\n\nNotas\n\n### INTRODUCCI\u00d3N\n\nHe aqu\u00ed dos ensayos sobre lo que significa ser extranjero. El primero se sit\u00faa en Venecia, en los albores del siglo XVI, cuando la ciudad se convirti\u00f3 en sede de un imperio comercial mundial y muchos de los extranjeros que se necesitaban para gobernar ese imperio eran mal vistos en la ciudad. Es lo que ocurri\u00f3 con los alemanes, los griegos, los turcos y sobre todo con los jud\u00edos, que fueron los peor considerados. \u00bfQu\u00e9 significaba forjarse la vida en un medio hostil? Me hice por primera vez esta pregunta cuando visit\u00e9 el gueto jud\u00edo de Venecia en los a\u00f1os sesenta. Las islas silenciosas y vac\u00edas que hab\u00edan formado el gueto, sus casas desmoronadas y sus sinagogas con el frente desfigurado, se hallaban a\u00fan bajo la influencia del espectro de las expulsiones y los asesinatos en masa de la Segunda Guerra Mundial. Pero mucho antes, en el esplendor del Renacimiento, los jud\u00edos exiliados de Espa\u00f1a hab\u00edan conseguido hacerse all\u00ed un hogar. Las formas en que lo lograron muestran algo, me parece, de c\u00f3mo otros exiliados y otros migrantes, forzados a vivir en el aislamiento, son capaces de crear una comunidad por s\u00ed mismos.\n\nEl segundo ensayo versa sobre extranjeros y extranjeridad de una \u00e9poca cronol\u00f3gicamente m\u00e1s cercana a nuestros d\u00edas. El ensayo gira en torno a la vida de Aleksandr Herzen, el gran reformista ruso del siglo XIX que pas\u00f3 gran parte de su vida de exiliado en Gran Breta\u00f1a, o bien en el Continente, cambiando constantemente de ciudad. Isaiah Berlin me cont\u00f3 la historia de Herzen una noche, o mejor dicho un atardecer que se hizo noche mientras el fil\u00f3sofo se sumerg\u00eda en las circunstancias, las estrategias de supervivencia y los sentimientos de Herzen. Berlin tambi\u00e9n era un ruso en el exilio, pero su destino no pudo haber sido m\u00e1s dispar, pues Berlin se construy\u00f3 un nicho propio en el coraz\u00f3n del _establishment_ brit\u00e1nico, mientras que Herzen fue siempre un extra\u00f1o, dondequiera que viviera.\n\nM\u00e1s que sucumbir a la nostalgia y la autocompasi\u00f3n, Herzen trat\u00f3 de dar sentido al desplazamiento y lo cierto es que abraz\u00f3 esta idea como un modo de vida. Esta adopci\u00f3n hizo de \u00e9l un hombre moderno. El desplazamiento y la dislocaci\u00f3n se convirtieron en emblemas del arte moderno, as\u00ed como en hechos conductores en econom\u00eda y en pol\u00edtica. Mientras Berlin expon\u00eda la historia de Herzen, me preguntaba en qu\u00e9 sentido el relato de su vida se relacionaba con la creaci\u00f3n de la dislocaci\u00f3n en las artes; el segundo ensayo explora esta conexi\u00f3n.\n\nAunque con afinidad tem\u00e1tica, estos dos ensayos siguen caminos distintos. El primero es directamente un relato hist\u00f3rico, mientras que el segundo tiene m\u00e1s de experimento sobre c\u00f3mo se conecta la vida de un hombre singular \u2013una vida dedicada a la pol\u00edtica\u2013 con las pr\u00e1cticas art\u00edsticas. El estudio sobre Venecia que aqu\u00ed se presenta es una versi\u00f3n extendida de la historia del gueto que apareci\u00f3 en mi libro _Carne y piedra_. El estudio sobre Herzen tambi\u00e9n es una versi\u00f3n ampliada de un ensayo que escrib\u00ed para un _Festschrift_ en honor de mi amigo Joseph Rykwert. Agradezco a Lucasta Miller su solicitud de que compusiera este librito, as\u00ed como sus sugerencias relativas a la edici\u00f3n.\n\n_Londres, 2011_\n\n## El gueto jud\u00edo de Venecia\n\n## El exilio crea un hogar \nLa historia ha hecho del pueblo jud\u00edo un pueblo experimentado en materia de exilio. En Europa Occidental, los jud\u00edos sobrevivieron durante m\u00e1s de tres milenios en peque\u00f1as c\u00e9lulas, mezclados entre gentes extra\u00f1as y opresoras; fueron un pueblo desplazado en m\u00faltiples ocasiones, pero un pueblo sostenido por su fe dondequiera que viviese, un pueblo que cre\u00f3 comunidades all\u00ed donde le toc\u00f3 instalarse, comunidades peque\u00f1as y segregadas en las que la propia segregaci\u00f3n resultaba inseparable de su sensaci\u00f3n de identidad. Tendemos a considerar la segregaci\u00f3n como una imposici\u00f3n del poder que convierte a los segregados en v\u00edctimas pasivas, pero la formaci\u00f3n del gueto jud\u00edo en la Venecia renacentista sugiere una historia m\u00e1s compleja. Es la historia de exiliados que fueron segregados ciertamente contra su voluntad, pero que sobre la base de tal separaci\u00f3n crearon nuevas formas de comunidad y supieron, como actores sociales, sacar provecho de la segregaci\u00f3n misma.\n\nLos jud\u00edos de la Venecia renacentista, as\u00ed como los jud\u00edos de la Roma renacentista que siguieron sus pasos, obtuvieron cierto grado de autodeterminaci\u00f3n en guetos aislados. Pero esta segregaci\u00f3n acentu\u00f3 su otredad. En efecto, dejaron de mezclarse en el espacio urbano y su vida se fue haciendo cada vez m\u00e1s enigm\u00e1tica para los poderes dominantes al otro lado de los muros del gueto. Las fantas\u00edas sobre los jud\u00edos hac\u00edan las veces de conocimiento cotidiano acerca de su vida y terminar\u00edan por ahogar el gueto. Para los propios jud\u00edos, el gueto interpon\u00eda obst\u00e1culos a su contacto con el mundo exterior, pues su propia judeidad parec\u00eda correr peligro cuando se aventuraban fuera de los muros del gueto. La exposici\u00f3n a los otros entra\u00f1aba para ellos la amenaza de una p\u00e9rdida de identidad.\n\nEn cierto sentido, \u00e9sta es la historia de la mayor\u00eda de los grupos de personas desplazadas y forzadas al aislamiento, pero la Venecia del Renacimiento convirti\u00f3 esta realidad en una historia especial y al mismo tiempo en algo m\u00e1s amplio, pues la experiencia de los jud\u00edos en el gueto de Venecia dej\u00f3 impresa una modalidad permanente de v\u00ednculo entre cultura y derechos pol\u00edticos. Gracias a su comercio, Venecia era sin duda la ciudad m\u00e1s internacional del Renacimiento, la puerta de comunicaci\u00f3n entre Europa y el Este, as\u00ed como entre Europa y \u00c1frica; en gran medida, una ciudad de extranjeros. Pero, a diferencia de la Roma antigua, no era un poder territorial; el gran n\u00famero de extranjeros que en Venecia iban y ven\u00edan no formaban parte de un imperio com\u00fan ni de una naci\u00f3n-Estado. Adem\u00e1s, a los extranjeros residentes en la ciudad \u2013alemanes, griegos, turcos, d\u00e1lmatas, lo mismo que a los jud\u00edos\u2013 les estaba vedada la ciudadan\u00eda oficial de la ciudad; eran inmigrantes permanentes. De este marco hist\u00f3rico de individuos sin ciudadan\u00eda deriv\u00f3 un conflictivo conjunto de c\u00f3digos y de derechos.\n\nPor un lado, los derechos humanos se conceb\u00edan sin localizaci\u00f3n concreta, eran derechos de contrato que se aplicaban a todas las partes con independencia de su lugar de origen, de su lugar de residencia en la ciudad o de qui\u00e9nes fueran. En este sentido, el derecho de contrato veneciano se diferenciaba del londinense de la misma \u00e9poca. En Londres, la validez de un contrato estaba restringida a la gente que pertenec\u00eda a la misma Commonwealth, lo que significaba comunidad geogr\u00e1fica, pol\u00edtica y, despu\u00e9s de la Reforma, religiosa. En Venecia, en cambio, los derechos econ\u00f3micos se reg\u00edan por un principio distinto, pues se pensaba que el simple acto de establecer un contrato generaba derechos, mientras que en la Inglaterra isabelina los derechos de contrato eran derechos que el Estado otorgaba a las partes contratantes.\n\nDe alguna manera, los venecianos asociaron lugar y derecho en la ejecuci\u00f3n de los contratos. En Venecia, en el \u00e1rea que rodeaba el puente de Rialto, se desarroll\u00f3 un conjunto de pr\u00e1cticas culturales, muy parecidas a las que m\u00e1s tarde, y sobre la base del ejemplo veneciano, se desarrollar\u00edan en la City de Londres, de modo que los contratos verbales pudieran ser realmente vinculantes. Para los venecianos, la sacralidad del contrato derivaba tanto de rituales de negociaci\u00f3n como del deseo de las partes de resultar dignas de confianza en futuras negociaciones; adem\u00e1s, la confianza verbal iba unida al uso de un capital no sometido a impuestos ni registrado, que las partes contratantes deseaban mantener oculto a los ojos del Estado, lo que consegu\u00edan dejando la menor constancia posible en papel. Estos v\u00ednculos verbales establec\u00edan la sacralidad del contrato al margen de la ley escrita vigente en el Palacio Ducal, situado en la Plaza San Marcos. El derecho veneciano era famoso por sus elaborados registros, por su intento de dejar todo asentado en papel, consecuencia de la enorme burocratizaci\u00f3n de su Estado.\n\nLa sacralidad del contrato verbal \u2013es lo que argumentar\u00e9\u2013 estableci\u00f3 una conexi\u00f3n entre los derechos econ\u00f3micos y los derechos de libertad de palabra. Los contratos verbales pod\u00edan ser revisados en cualquier momento por las partes contratantes si, por ejemplo, los barcos se perd\u00edan en el mar o se produc\u00eda un cambio en el valor de las mercanc\u00edas en el puente de Rialto. La libre expresi\u00f3n como principio hund\u00eda sus ra\u00edces en el deseo de hacer del contrato verbal un instrumento maleable. La econom\u00eda dio a la libertad de palabra una dimensi\u00f3n distinta de la que Milton le hab\u00eda atribuido en la _Areopag\u00edtica_ , donde se trataba de la libertad de expresi\u00f3n desde los puntos de vista de la moral sexual y la herej\u00eda religiosa.\n\nLa creaci\u00f3n en Venecia de espacios-gueto, para los jud\u00edos al igual que para otros extranjeros, asoci\u00f3 lugar y derecho en un nuevo sentido, originando lo que podr\u00eda denominarse derechos locales. La naturaleza de estos derechos locales se relacionaba con la protecci\u00f3n de la violencia. La ciudad proteger\u00eda a un jud\u00edo o a un turco de las turbas cristianas en Cuaresma o en otros momentos de intensa pasi\u00f3n religiosa \u00fanicamente si los que no eran venecianos estaban en su lugar, esto es, encerrados en el espacio reservado para el extra\u00f1o. Una vez que una persona se aventuraba a internarse en una zona de la ciudad a la que no pertenec\u00eda, perd\u00eda todo derecho a ser protegida de ataques, fueran del tipo que fuesen. De esta suerte, lugar y cuerpo estaban concretamente unidos.\n\nLas protecciones que proporcionaba el gueto constitu\u00edan para los jud\u00edos una experiencia nueva. Para ellos, ser \u00abjud\u00edo\u00bb se convert\u00eda entonces en una experiencia espacial. Evidentemente, la limitaci\u00f3n de la garant\u00eda de integridad del cuerpo humano al espacio segregado y reservado a otros cuerpos igualmente marginados, fortalec\u00eda los lazos comunales. En t\u00e9rminos m\u00e1s generales, desde la Baja Edad Media la concepci\u00f3n de la sociedad como cuerpo colectivo hab\u00eda sido un lugar com\u00fan. La existencia de espacios segregados, como los guetos venecianos, daba un nuevo significado a esa imagen medieval. Las fuerzas que desarrollaban el capitalismo moderno arrancaban a la gente de sus lugares geogr\u00e1ficos y sociales tradicionales en la sociedad; \u00fanicamente quienes eran oficialmente marginados estaban obligados a ocupar un lugar fijo. La creencia en la comunidad org\u00e1nica, creencia que se manten\u00eda en oposici\u00f3n a las fuerzas del capitalismo moderno, vino, en tiempos m\u00e1s cercanos, a encarnarse en los marginados; la segregaci\u00f3n adquir\u00eda as\u00ed un valor humano positivo, como si se hubiera mantenido al segregado libre de contagio. El espacio del gueto se idealiz\u00f3 como una comunidad \u00abreal\u00bb, como un espacio org\u00e1nico. Los jud\u00edos de la Venecia renacentista fueron los primeros en pensar que su segregaci\u00f3n, ir\u00f3nicamente, conten\u00eda este aspecto positivo.\n\nEl aparato gubernamental de Venecia desempe\u00f1aba un papel decisivo en lo concerniente a la formaci\u00f3n tanto de los derechos sin localizaci\u00f3n como de los derechos locales, puesto que la ciudadEstado supervisaba los contratos y patrullaba la ciudad; la ciudad-Estado actuaba con todo rigor para garantizar los contratos verbales, aun cuando \u00e9stos cometieran fraude fiscal contra la ciudad. Venecia contaba con la fuerza de polic\u00eda m\u00e1s sofisticada de Europa y castigaba severamente la violencia informal, incluso con fines l\u00edcitos, como en el caso de asesinato de una esposa ad\u00faltera. Pero en ambos casos la estructura del Estado s\u00f3lo se expresaba como si actuara, por un lado, simplemente como un mecanismo, como un instrumento de crecimiento econ\u00f3mico y, por otro lado, como regulador de la protecci\u00f3n corporal. La Commonwealth de Venecia, por tomar el t\u00e9rmino ingl\u00e9s, se defin\u00eda en el Renacimiento tard\u00edo en t\u00e9rminos de ceremonial c\u00edvico y del funcionamiento burocr\u00e1tico del propio gobierno. El lenguaje de los derechos no locales y de los derechos locales perge\u00f1ados por el gobierno era un lenguaje que se refer\u00eda a individuos en el seno de la ciudad m\u00e1s que a una representaci\u00f3n de lo que la ciudad-Estado de Venecia era en s\u00ed misma. En este sentido, la ciudadEstado encontraba cierta protecci\u00f3n; su propia legitimidad no se ve\u00eda cuestionada cuando fallaban la maquinaria de la econom\u00eda o la regulaci\u00f3n de la violencia. En consecuencia, en esta primera \u00abciudad global\u00bb del mundo moderno el sistema de gobierno y la ciudad crec\u00edan por separado.\n\nEl lenguaje que dio forma al gueto de la Venecia renacentista, as\u00ed como la construcci\u00f3n de derechos derivada de la formaci\u00f3n del gueto, aclaran nociones comunes de lugar y derecho. El dualismo que aparece por primera vez en Venecia es el de derechos econ\u00f3micos de contrato y derechos corporales ante la violencia. La protecci\u00f3n econ\u00f3mica no ten\u00eda localizaci\u00f3n; los derechos del cuerpo depend\u00edan del cuerpo segregado en el espacio. En el mundo moderno, a menudo el Estado utiliza este dualismo para dar distinto valor a los derechos humanos. Para los grupos oprimidos, el Estado est\u00e1 mejor dispuesto a garantizar los derechos locales que los derechos universales. Proteger\u00e1 a los cuerpos vulnerables en la medida en que \u00e9stos permanezcan en su lugar de pertenencia, pero brinda menos protecci\u00f3n a la extensi\u00f3n del poder econ\u00f3mico a estos cuerpos en el conjunto de la sociedad. En efecto, el Estado se servir\u00e1 de los valores de la comunidad para limitar el acceso econ\u00f3mico al conjunto de la sociedad. Los grupos oprimidos pueden coadyuvar a esta limitaci\u00f3n de su propia libertad centr\u00e1ndose en la coherencia de la comunidad, en el icono de un cuerpo org\u00e1nico de los oprimidos.\n\n### _La formaci\u00f3n del gueto veneciano_\n\nHoy es f\u00e1cil imaginar que los jud\u00edos hab\u00edan vivido siempre en Europa en las condiciones de aislamiento propias del espacio del gueto. En realidad, del Concilio Lateranense de 1179 en adelante, la Europa cristiana trat\u00f3 de impedir que los jud\u00edos vivieran entre cristianos. Roma es el prototipo de la ejecuci\u00f3n pr\u00e1ctica de dicho Concilio. Desde el comienzo de la Edad Media, al igual que otras ciudades de Europa, como Frankfurt, Roma tuvo lo que hoy se denomina un gueto. Es cierto que se pod\u00eda cerrar unas cuantas calles del barrio jud\u00edo de Roma, pero el desorden del tejido urbano medieval era tal que resultaba pr\u00e1cticamente imposible aislar por completo a los jud\u00edos. Adem\u00e1s, en la mayor\u00eda de las otras ciudades europeas los jud\u00edos no viv\u00edan en comunidades estrechamente compactas, sino m\u00e1s bien en peque\u00f1as c\u00e9lulas dispersas, lo cual se deb\u00eda en parte a una simple forma de protecci\u00f3n, pues s\u00f3lo pasando inadvertidos o mediante el anonimato pod\u00edan escapar a la persecuci\u00f3n.\n\nEn Venecia, la naturaleza f\u00edsica de la ciudad, construida sobre el agua, hizo finalmente posible poner en pr\u00e1ctica la regla prescrita por el Concilio de Letr\u00e1n. Las calles de la ciudad son canales que separan grupos de edificios en un vasto archipi\u00e9lago de islas. En la construcci\u00f3n del gueto jud\u00edo, los padres de la ciudad se limitaron a utilizar la ecolog\u00eda isle\u00f1a de la ciudad para crear un espacio de segregaci\u00f3n. Esos muros de agua sugirieron luego al papa Pablo IV la utilizaci\u00f3n en Roma de muros de piedra dentro de la ciudad para hacer efectiva la segregaci\u00f3n; m\u00e1s tarde, el papa Sixto V ampli\u00f3 y regulariz\u00f3 los primeros muros del gueto de esta ciudad. A partir de esta diferencia social amurallada, un nuevo principio del dise\u00f1o urbano europeo, el espacio del gueto, cristaliz\u00f3 como forma urbana moderna. En el mundo moderno, los autom\u00f3viles y las autopistas hacen las veces de muros de segregaci\u00f3n.\n\nAnte todo, \u00bfpor qu\u00e9 constitu\u00edan los jud\u00edos y otros extranjeros un \u00abproblema\u00bb para Venecia? \u00bfQu\u00e9 los hab\u00eda llevado all\u00ed? El comercio de especias era un buen ejemplo del tipo de comercio que hab\u00eda forjado la riqueza de Venecia, al precio de atraer a la ciudad a jud\u00edos y otros extranjeros. La primera especia que tuvo Venecia bajo su control fue la sal, que suministraba el medio m\u00e1s elemental de preservaci\u00f3n de los alimentos. La sal veneciana era secada en las marismas de la costa y luego vendida tierra adentro. Sin embargo, lo que enriqueci\u00f3 a la Venecia del Renacimiento no fue el comercio de la sal, que de acuerdo con el antiguo modelo medieval de la ciudad se vend\u00eda en sus alrededores inmediatos; la riqueza le lleg\u00f3 m\u00e1s bien de un comercio como el del azafr\u00e1n, para el que el mercado local inmediato era peque\u00f1o, pero el mercado de toda Europa era inmenso.\n\nHacia el a\u00f1o 1000, Venecia se hab\u00eda establecido como potencia hegem\u00f3nica en la totalidad del mar Adri\u00e1tico, que era a su vez una de las rutas a Jerusal\u00e9n; de esta manera, Venecia se convirti\u00f3 en una ciudad decisiva en las cruzadas europeas por la recuperaci\u00f3n de Tierra Santa. En la Tercera Cruzada, la ciudad hab\u00eda adquirido derechos de comercio con el Este, derechos que emple\u00f3 para importar especias: pimienta, que en parte proven\u00eda de la India y en parte de la costa oriental de \u00c1frica a trav\u00e9s del puerto de Alejandr\u00eda; azafr\u00e1n y nuez moscada, de Persia, y canela, de Ceil\u00e1n. Los cruzados hab\u00edan regresado del Este con el sabor de estas especias en su memoria y la llegada de ellas cambi\u00f3 la dieta europea. El comercio de las especias lleg\u00f3 a constituir una parte tan importante de la econom\u00eda veneciana, que se crearon administraciones especiales, como la Oficina del Azafr\u00e1n, para regular el comercio.\n\nLos jud\u00edos ricos hab\u00edan ido llegando al norte de Italia desde Alemania durante dos siglos antes de la instauraci\u00f3n del primer gueto, y muchos de ellos se involucraron en el proceso por el que los beneficios obtenidos en el comercio de bienes extranjeros, como las especias, se volcaron en inversiones en diamantes, oro y plata. En el V\u00e9neto, la banca jud\u00eda se desarroll\u00f3 a partir de la conversi\u00f3n de las ganancias en este tipo de activos, que se pod\u00edan volver a vender con rapidez.\n\nTambi\u00e9n la persecuci\u00f3n religiosa impuls\u00f3 a los jud\u00edos hacia Venecia. Alrededor del a\u00f1o 1300, los pogromos se hab\u00edan vuelto muy severos en Alemania y llevaron a los jud\u00edos a las ciudades del norte de Italia de Padua, Verona y Venecia. Como consecuencia de las guerras de la Liga de Cambrai, en 1509 los prestamistas jud\u00edos abandonaron la tierra firme; alrededor de quinientos huyeron de Padua y Mestre \u2013el territorio continental de Venecia\u2013 a Venecia propiamente dicha. Sin embargo, en 1515 la presencia de estos jud\u00edos fue percibida como una indignidad moral. En Venecia, a partir de 1510, los ataques contra ellos se fueron haciendo cada vez m\u00e1s enconados. Estos ataques eran dirigidos, entre otros, por fray Lovato de Padua, cuya fuerza oratoria condujo en 1511 a los venecianos a la destrucci\u00f3n de las casas de los jud\u00edos que viv\u00edan cerca de Campo San Paolo; dos a\u00f1os antes hab\u00eda propugnado la captura de todo el dinero de los prestamistas \u00aby no dejarles nada con que vivir\u00bb.\n\nSin embargo, no se expuls\u00f3 a los jud\u00edos. La econom\u00eda dio muestras de ser una fuerza tan poderosa como la religi\u00f3n. En palabras de un ciudadano destacado: \u00abLos jud\u00edos son a\u00fan m\u00e1s necesarios para una ciudad que los panaderos, y sobre todo para una ciudad como \u00e9sta.\u00bb2 No s\u00f3lo proporcionaban servicios bancarios a los comerciantes, sino tambi\u00e9n pr\u00e9stamos a los pobres y al Estado. Adem\u00e1s, era imposible expulsar a los jud\u00edos sin costes directos para el gobierno veneciano, pues \u00e9ste habr\u00eda perdido personas que pagaban cuantiosos impuestos. Del conflicto entre religi\u00f3n y econom\u00eda surgi\u00f3 la estrategia de compromiso de segregaci\u00f3n espacial en la ciudad, que, en palabras del historiador Brian Pullan, era una estrategia \u00abde segregaci\u00f3n de la comunidad jud\u00eda, aunque no de expulsi\u00f3n\u00bb.\n\nComo consecuencia de los desastres de comienzos del siglo XVI, la ciudad comenz\u00f3 a explorar la posibilidad de utilizar el Gueto Nuevo como lugar de segregaci\u00f3n. El significado original en italiano de la palabra _ghetto_ era \u00abfundici\u00f3n\u00bb (derivada de _gettare, \u00ab_ arrojar\u00bb). El Gueto Nuevo y el Gueto Viejo eran los antiguos distritos de fundici\u00f3n de Venecia, lejos del centro ceremonial de la ciudad; hacia 1500, sus funciones manufactureras hab\u00edan dado paso al Arsenal. El Gueto Nuevo era una superficie romboidal de tierra \u00edntegramente rodeada de agua; los edificios formaban una pared en todo el borde con un espacio abierto en el centro. El Gueto Nuevo se caracterizaba por ser una isla en la ciudad, pues s\u00f3lo dos puentes lo un\u00edan al tejido urbano; si se cerraban los puentes, quedaba absolutamente incomunicado.\n\nLa propuesta de utilizar el Gueto Nuevo fue hecha por Zacaria Dolfin en 1515. \u00c9ste era su plan para segregar a los jud\u00edos:\n\nEnviar a todos a vivir en el Gueto Nuevo, que es como un castillo, poner puentes levadizos y cerrarlo con un muro; s\u00f3lo tendr\u00edan una puerta, que los mantendr\u00eda encerrados y all\u00ed permanecer\u00edan; durante la noche, ir\u00edan dos buques del Consejo de los Diez y all\u00ed se quedar\u00edan, a sus expensas, para su mayor seguridad.\n\nY as\u00ed fue, en efecto, como vivieron los jud\u00edos en el gueto despu\u00e9s de 1516. Cuando se abr\u00edan los puentes, por la ma\u00f1ana, algunos jud\u00edos iban a la ciudad, sobre todo al \u00e1rea que rodeaba Rialto, donde circulaban entre la gente normal, mientras que los cristianos iban al gueto a pedir dinero prestado o a vender comida y hacer negocios. Cuando oscurec\u00eda, los jud\u00edos deb\u00edan estar en el gueto, los cristianos ten\u00edan que abandonarlo y se levantaban los puentes. Adem\u00e1s, por la noche se cerraban las ventanas de los edificios del gueto que daban al exterior y, como les hab\u00edan retirado todos los balcones, la pared del gueto se asemejaba a la muralla de un castillo, con su foso alrededor.\n\n\u00c9sta fue la primera fase de la segregaci\u00f3n de los jud\u00edos. La segunda consisti\u00f3 en la expansi\u00f3n del barrio jud\u00edo al Gueto Viejo, el antiguo distrito de fundici\u00f3n. Eso tuvo lugar en 1541. En aquella \u00e9poca los venecianos pasaban por una dif\u00edcil situaci\u00f3n financiera: sus tarifas aduaneras hab\u00edan sobrepasado las de otras ciudades y el comercio deca\u00eda. Por eso decidieron distender un poco las barreras fronterizas, lo que tuvo como consecuencia que los jud\u00edos orientales, en particular los de Ruman\u00eda y Siria, fueran a la ciudad a hacer negocios. Algo m\u00e1s que buhoneros y algo menos que comerciantes burgueses, hac\u00edan negocio con todo lo que pudiera caer en sus manos, y de esa manera, hasta 1541, se desplazaron constantemente de un lugar a otro. Sanuto, relevante figura ciudadana, describi\u00f3 crudamente la actitud de los venecianos respecto de los comerciantes jud\u00edos: \u00abNuestros compatriotas nunca quisieron que los jud\u00edos tuvieran tiendas y comerciaran en esta ciudad, sino que compraran y vendieran y se volvieran a marchar.\u00bb Esta vez los jud\u00edos no se marcharon, sino que quisieron quedarse y hasta estaban dispuestos a pagar un precio por ello. Se transform\u00f3 el Gueto Antiguo en un espacio jud\u00edo para alojarlos, se sellaron sus paredes externas y se retiraron los balcones. A diferencia del primer gueto, el segundo ten\u00eda una peque\u00f1a plaza y muchas callejuelas.\n\nEste espacio de segregaci\u00f3n no era el primero que conoc\u00eda la ciudad. El gueto no hizo m\u00e1s que llevar al extremo la respuesta que Venecia ven\u00eda dando a sus extranjeros desde hac\u00eda dos siglos. Era una respuesta que se hab\u00eda aplicado incluso a correligionarios cristianos, los alemanes encerrados en su propia temprana versi\u00f3n de un gueto, el Fondaco dei Tedeschi, la \u00abf\u00e1brica de los alemanes\u00bb, en cuyo caso la palabra _f\u00e1brica_ se empleaba en el sentido de un edificio en el que la gente viv\u00eda y trabajaba. Esta combinaci\u00f3n era t\u00edpica de las casas urbanas medievales. El Fondaco dei Tedeschi era una de esas casas por excelencia, con la caracter\u00edstica a\u00f1adida de que todos sus habitantes eran alemanes.\n\nEn su forma primitiva, el Fondaco dei Tedeschi era una casa de segregaci\u00f3n s\u00f3lo parcial. En principio, se daba por supuesto que nadie saldr\u00eda de ella una vez ca\u00edda la noche, pero en realidad la oscuridad nocturna result\u00f3 ser para los alemanes el momento de mayor actividad del d\u00eda, pues a cubierto de ella contrabandeaban bienes en ambos sentidos para evitar el pago de derechos aduaneros. En consecuencia, en 1479 el gobierno tom\u00f3 medidas para asegurar que este lugar de segregaci\u00f3n se convirtiera en un edificio de aislamiento. En efecto, decret\u00f3 que, al oscurecer, se cerraran las ventanas y se echara el pestillo a las puertas del Fondaco, desde fuera. Fue \u00e9se el a\u00f1o en que la casa de los alemanes cristianos se convirti\u00f3 en el prototipo de los espacios urbanos m\u00e1s extensos que m\u00e1s tarde mantendr\u00edan encerrados a los jud\u00edos.\n\nEl gueto de Venecia requiere otro marco hist\u00f3rico, que es lo que marca su oposici\u00f3n al del gueto romano que el papa Pablo IV comenz\u00f3 a construir en 1555. El gueto de Pablo estaba destinado, ante todo, a mantener encerrados a todos los jud\u00edos en el mismo lugar, de manera que los sacerdotes cristianos pudieran convertirlos sistem\u00e1ticamente, casa por casa, sin dejar a ninguno la posibilidad de eludir la palabra de Cristo. Esto traduc\u00eda en t\u00e9rminos espaciales la idea de la oveja descarriada; todos juntos, ser\u00edan recuperables. Pero a este respecto el gueto romano fue un tremendo fracaso, pues en todo un a\u00f1o s\u00f3lo unos veinte jud\u00edos, de una poblaci\u00f3n de cuatro mil, sucumbieron a la conversi\u00f3n por la v\u00eda de la concentraci\u00f3n espacial. De modo m\u00e1s duradero, los muros del gueto romano partieron en dos una zona previamente controlada por familias de comerciantes romanos que traficaban con la comunidad de residentes jud\u00edos; al apoderarse del espacio del gueto romano para la conversi\u00f3n, el Papa debilit\u00f3 tambi\u00e9n el dominio territorial que estas familias cristianas ten\u00edan sobre la ciudad.\n\nLos guetos venecianos, por el contrario, no fueron concebidos como espacios de conversi\u00f3n, ni estaban estrat\u00e9gicamente situados en el centro de la ciudad. En efecto, eran geogr\u00e1ficamente marginales, pues ocupaban una vieja instalaci\u00f3n industrial abandonada, y el encierro de la comunidad jud\u00eda en ese p\u00e1ramo ten\u00eda la intenci\u00f3n de marcar la irremediable diferencia inherente a su judeidad. Pero \u00bfqu\u00e9 significaba ser \u00abjud\u00edo\u00bb? \u00bfQu\u00e9 significado tendr\u00eda para los jud\u00edos, en lo relativo a su identidad como tales, el hecho de vivir en un espacio de segregaci\u00f3n? Para entender tal cosa es \u00fatil comparar a los jud\u00edos con otro grupo marginal al que los venecianos no pod\u00edan controlar mediante la segregaci\u00f3n espacial, el de las cortesanas que formaban el estrato superior de la prostituci\u00f3n en la ciudad.\n\n### _La cortesana y el jud\u00edo_\n\nEs posible que la prostituci\u00f3n sea la profesi\u00f3n m\u00e1s antigua del mundo, pero en el curso de la historia ha adoptado muchas formas. En el Renacimiento, la cortesana adopt\u00f3 la apariencia de una especie de prostituta de alto nivel; era una mujer joven que no s\u00f3lo vend\u00eda su cuerpo, sino que proporcionaba placeres sociales a sus clientes, como la asistencia a conciertos, cenas y representaciones teatrales privadas. La palabra _cortesana_ comenz\u00f3 a usarse a finales del siglo XV como femenino de _cortesano_ ; tal como se la usaba en italiano, estas mujeres eran las _cortigiane_ que proporcionaban placer a los _cortigiani_ , es decir, hombres de la nobleza, soldados, administradores y la poblaci\u00f3n par\u00e1sita de las cortes renacentistas. La corte era un escenario pol\u00edtico y sus cenas, recepciones y reuniones diplom\u00e1ticas, momentos de aburrid\u00edsima seriedad. La cortesana aliviaba a los hombres de ese mundo oficial de modo muy parecido a aquel en que la antigua prostituta griega se convert\u00eda en la compa\u00f1era de placer de los hombres en encuentros sociales festivos.\n\nLas j\u00f3venes que de alguna manera entraban en la prostituci\u00f3n lo hac\u00edan alrededor de los catorce a\u00f1os. Aretino afirma haberle o\u00eddo decir a una jovencita: \u00abEn un mes aprend\u00ed todo lo que hay que saber de la prostituci\u00f3n: c\u00f3mo despertar la pasi\u00f3n, atraer a los hombres, seducirlos y dejar plantado a un amante; c\u00f3mo llorar cuando deseaba re\u00edr y re\u00edr cuando por dentro lloraba, y c\u00f3mo vender una y otra vez mi virginidad.\u00bb M\u00e1s tiempo llevaba convertirse en cortesana. Para ello era necesario establecer una red de clientes de clase alta, enterarse de los chismes de la ciudad y de la corte con el fin de divertirlos y adquirir una casa y ropa que les agradara.\n\nA diferencia del sistema japon\u00e9s de la geisha, donde las artes sociales estaban codificadas en rituales estrictos que se aprend\u00edan y se transmit\u00edan de generaci\u00f3n en generaci\u00f3n, de la misma manera en que se daba formaci\u00f3n a un abogado, la prostituta del Renacimiento que esperaba convertirse en cortesana ten\u00eda que hacerse a s\u00ed misma. En cierto sentido, su problema se asemejaba al del cortesano que ten\u00eda necesidad de manuales de comportamiento como _El libro del cortesano_ , de Castiglione, que le explicaba c\u00f3mo comportarse en esta nueva instituci\u00f3n, la corte, que era un centro de poder distinto del antiguo castillo feudal, m\u00e1s cosmopolita y menos obligado ante la tradici\u00f3n que ante los cambios pol\u00edticos y financieros internacionales. _El libro del cortesano_ , de Castiglione, indicaba a un hombre c\u00f3mo actuar en un mundo de extra\u00f1os. Hab\u00eda muchos libros vulgares que intentaban dar an\u00e1loga instrucci\u00f3n a la cortesana, pero la verdadera educaci\u00f3n de \u00e9sta consist\u00eda en aprender a imitar a las mujeres de las clases altas, a vestir, hablar y escribir como ellas.\n\nAl aprender a \u00abhacerse aceptar\u00bb, las cortesanas creaban un problema peculiar, problema que, aunque en menor grado, tambi\u00e9n ocasionaban las prostitutas de un nivel social m\u00e1s bajo, pues si ten\u00edan \u00e9xito y aprend\u00edan el arte de disfrazarse, pod\u00edan ir a cualquier sitio. Lo m\u00e1s grave no era que se infiltraran en los c\u00edrculos de mujeres virtuosas, sino que las sustituyeran no s\u00f3lo present\u00e1ndose y hablando como virtuosas damas cristianas, sino sirviendo tambi\u00e9n como plenas compa\u00f1eras er\u00f3ticas de sus hombres. Por esta raz\u00f3n se consideraba a la cortesana una amenaza, la amenaza de una mujer con la misma apariencia que cualquier otra, pero sexualmente libre. En una proclama promulgada en 1543, el gobierno veneciano declaraba que las prostitutas se mostraban \u00aben las calles, las iglesias y en todo lugar tan enjoyadas y bien vestidas, que con harta frecuencia son tomadas por damas de la nobleza y por ciudadanas, porque su atav\u00edo no se diferencia del de las mujeres antes mencionadas, y son confundidas con ellas no s\u00f3lo por los extranjeros, sino tambi\u00e9n por los habitantes de Venecia, incapaces de distinguir lo verdadero de lo falso...\u00bb.\n\nEn la \u00e9poca de Shakespeare ya hac\u00eda siglos que Venecia contaba con gran n\u00famero de prostitutas que viv\u00edan del comercio con los marineros que visitaban la ciudad. El simple volumen de dinero que cambiaba de manos en la industria sexual v\u00e9neta durante el Renacimiento fue convirti\u00e9ndola poco a poco en \u00abuna fuente leg\u00edtima de beneficios de honorables empresarios de buena familia\u00bb. No faltaban casos de mujeres de indudable buena cuna que se hac\u00edan prostitutas, pero eran relativamente raros. Lo m\u00e1s normal era que las prostitutas de \u00e9xito respondieran al perfil de jovencita que aprend\u00eda a asimilarse y a ganarse as\u00ed el acceso a clientes de clase m\u00e1s alta, como oficiales de marina, mercaderes o miembros de las delegaciones diplom\u00e1ticas que visitaban la ciudad. Para la realizaci\u00f3n de estos contactos, las prostitutas contaban con nobles venecianos que les prestaban secretamente su asistencia en beneficio propio.\n\nEn Roma, la prostituci\u00f3n experiment\u00f3 una r\u00e1pida irrupci\u00f3n tras el regreso del papado y un declive igualmente r\u00e1pido. En 1566, el papa P\u00edo V, en pleno auge de la Contrarreforma, trat\u00f3 de expulsar de Roma toda prostituci\u00f3n, y, tras el fracaso de su empe\u00f1o, intent\u00f3 destruir la aceptaci\u00f3n social de las cortesanas, empresa en la que tuvo mejor fortuna. Venecia, por su lado, en la medida en que segu\u00eda siendo una ciudad-puerto de gran actividad, tuvo que tolerar a las prostitutas como parte de su econom\u00eda de servicios, exactamente de la misma manera en que tuvo que tolerar a los jud\u00edos, que prestaban dinero y cambiaban billetes extranjeros. Para una prostituta joven, la posibilidad de convertirse en cortesana con una clientela tan estable y rica era una perspectiva tentadora.\n\nAnte esta realidad econ\u00f3mica, la ciudad intent\u00f3 tratar a las prostitutas de la misma manera que trataba a los jud\u00edos: \u00abNo s\u00f3lo se procur\u00f3 muchas veces confinar [a las prostitutas] en barrios especiales, sino que en una ocasi\u00f3n hubo incluso un intento abortado de obligarlas a llevar una prenda amarilla para que resultaran claramente reconocibles.\u00bb Los jud\u00edos de Venecia fueron los primeros a los que se les oblig\u00f3 a llevar una insignia amarilla en 1397; a las prostitutas y a los proxenetas se les orden\u00f3 en 1416 que usaran bufandas amarillas. Aunque la ley no se lo imped\u00eda, rara vez las jud\u00edas sal\u00edan del gueto con los adornos o las joyas que usaban otras mujeres respetables. Los jud\u00edos tem\u00edan provocar rechazo si una persona con la se\u00f1al amarilla exhib\u00eda cualquier muestra de riqueza, de modo que las jud\u00edas, en las pocas ocasiones en que se aventuraban a ir a otros lugares de la ciudad, se destacaban tanto por la sencillez de su vestimenta como por la prenda amarilla. Las autoridades trataron de marcar de la misma manera a las prostitutas. El decreto de 1543, ya mencionado, defin\u00eda en estas palabras los aspectos de la apariencia de una mujer virtuosa, que una prostituta no pod\u00eda adoptar: \u00abEn consecuencia, se proclama que ninguna prostituta puede vestir ni llevar en ninguna parte de su persona oro, plata o seda, ni tampoco lucir collares, perlas, joyas o simples anillos en las orejas o en las manos.\u00bb Encontramos en esto la curiosa iron\u00eda, sobre la cual volveremos m\u00e1s adelante, de que la mujer virtuosa pod\u00eda llevar toda clase de vestimenta seductora, mientras que la puta deb\u00eda parecer sobria, al igual que los jud\u00edos.\n\nLos c\u00f3digos de vestimenta destinados a advertir a la gente respetable que se hallaba en presencia de elementos marginales puso fin a una lucha que se hab\u00eda librado durante casi sesenta a\u00f1os para confinar a las prostitutas en un espacio acotado de la ciudad cerca de Rialto, siguiendo, una vez m\u00e1s, el modelo de segregar la diferencia que se hab\u00eda llevado al extremo con los jud\u00edos. En un principio, los venecianos hab\u00edan imaginado algo as\u00ed como prost\u00edbulos administrados por el Estado, prop\u00f3sito con el cual compraron dos casas. Pero para las prostitutas era m\u00e1s lucrativo el trabajo por su cuenta con proxenetas, que consegu\u00edan clientes de toda la ciudad y proporcionaban habitaciones o creaban prost\u00edbulos clandestinos donde escapar a la vigilancia estatal; estos lugares ilegales para la pr\u00e1ctica del sexo il\u00edcito pod\u00edan evadir los impuestos que impon\u00eda el Estado, que se calculaban cuidadosamente sobre cada transacci\u00f3n sexual. El proyecto del prost\u00edbulo regido por el Estado qued\u00f3 en nada, pero el deseo de confinar a las prostitutas \u2013que, despu\u00e9s de todo, ofend\u00edan a la moral cristiana\u2013 no cej\u00f3, mientras que la \u00abindustria del sexo\u00bb continuaba traspasando sus fronteras legales. En consecuencia, el obligar a prostitutas y proxenetas a llevar vestimentas \u00abjud\u00edas\u00bb era la segunda estrategia oficial de control, pero volvi\u00f3 a fracasar, pues la vestimenta cumpl\u00eda m\u00e1s una funci\u00f3n publicitaria que de advertencia.\n\nEn la \u00e9poca de Shakespeare, la ciudad llegaba a la convicci\u00f3n de que era imposible controlar la prostituci\u00f3n por medio de la segregaci\u00f3n. Se aprob\u00f3 una ley que prohib\u00eda a las prostitutas establecerse a lo largo del Gran Canal, ubicaci\u00f3n de la ciudad a la que ten\u00edan acceso gracias a sus ping\u00fces ganancias; el \u00fanico resultado de esto fue que emplearon su dinero para infiltrarse en otras zonas respetables de la ciudad. En los primeros a\u00f1os del siglo XVII, todos los intentos de regular el aspecto exterior de las prostitutas hab\u00eda terminado en fiasco. Los c\u00f3digos \u00abjud\u00edos\u00bb fueron reemplazados por edictos que prohib\u00edan a las prostitutas vestir seda blanca, una tela destinada exclusivamente a j\u00f3venes damas solteras y a ciertas monjas, o que se adornaran las manos con anillos de mujeres casadas. Todo fue in\u00fatil; el Estado no pudo controlar el sexo en la ciudad.\n\n\u00bfPor qu\u00e9 la instauraci\u00f3n de guetos funcion\u00f3 bien para ciertos tipos de diferencia en la ciudad, pero no para otros? \u00bfPor qu\u00e9, en el caso de Venecia, ser\u00eda posible controlar mediante el aislamiento a los marginales religiosos y \u00e9tnicos, pero no a los marginales sexuales?\n\nLas cortesanas no ten\u00edan ning\u00fan inter\u00e9s, econ\u00f3mico ni personal, en que se las aislara, y por eso se resistieron por todos los medios a que se dispusiera de su libertad de movimiento. Los jud\u00edos, en cambio, afrontaban una realidad m\u00e1s compleja. Protestaron contra la severidad de esta forma de aislamiento, pero tambi\u00e9n colaboraron con las autoridades en la negociaci\u00f3n de las condiciones en las que habr\u00edan de vivir en el gueto. No entraron pasivamente en el gueto.\n\nA cambio del aislamiento inherente a la segregaci\u00f3n, los jud\u00edos obtuvieron la garant\u00eda de integridad f\u00edsica dentro de los muros del gueto mientras permanecieran en \u00e9l. Esta garant\u00eda ven\u00eda dada por el propio espacio f\u00edsico. En efecto, el espacio aislado protegi\u00f3 a los jud\u00edos, por ejemplo, en 1534, cuando fueron blanco de una ola de ataques durante la Cuaresma; se alzaron los puentes, se cerraron las ventanas, y la turba de fan\u00e1ticos no pudo llegar hasta ellos. En sus tratos con todas las comunidades extranjeras, la voluntad de la ciudadEstado era perseguir los cr\u00edmenes violentos contra los extranjeros en tanto \u00e9stos se hallaran en sus respectivos barrios, pero en la pr\u00e1ctica se negaba a perseguir ataques violentos a extranjeros que se produjeran fuera de esos espacios cerrados.\n\nEn segundo lugar, mientras que el Estado no ten\u00eda nada que ofrecer a la cortesana en compensaci\u00f3n por el uso del distintivo amarillo, ten\u00eda en cambio algo positivo que ofrecer al jud\u00edo por el cumplimiento de esa misma obligaci\u00f3n. En compensaci\u00f3n por el ingreso de los jud\u00edos en el gueto, el Estado les permit\u00eda construir sinagogas. La sinagoga, que en la Edad Media consist\u00eda en una congregaci\u00f3n lit\u00fargica en una casa de familia o en un edificio cualquiera, estaba en el gueto protegida por un Estado cristiano. El edificio de la sinagoga se convirti\u00f3 en la instituci\u00f3n p\u00fablica definitoria de la vida comunitaria y poco despu\u00e9s el gueto lleg\u00f3 a ser el emplazamiento id\u00f3neo para sinagogas que representaban a diferentes grupos confesionales, esto es, sefard\u00edes y asquenazis, y hubo incluso una para los diecinueve jud\u00edos chinos que viv\u00edan en Venecia a mediados del siglo XVI.\n\nEsta concentraci\u00f3n de diferentes tipos de sinagogas en el gueto urbano tuvo profundas consecuencias en el concepto mismo de identidad jud\u00eda. La variedad de ramas del juda\u00edsmo renacentista respond\u00eda a la variedad de los materiales sociales que las constitu\u00edan. Los asquenazis no hablaban la misma lengua que los sefard\u00edes, ni compart\u00edan una cultura com\u00fan, aparte de que entre ellos hab\u00eda tambi\u00e9n importantes diferencias doctrinarias. Los jud\u00edos levantinos inclu\u00edan en su seno a individuos pertenecientes a diversas sectas cism\u00e1ticas cuya lengua, costumbres y actividad econ\u00f3mica guardaban escasa semejanza con las de los jud\u00edos de Europa Occidental; en cuanto a los jud\u00edos chinos, es casi imposible imaginar alguna ligaz\u00f3n entre su condici\u00f3n y la de los otros jud\u00edos. Pero, forzados a vivir todos en el mismo espacio denso y limitado, estaban unidos de hecho, y esta circunstancia reforz\u00f3 la \u00fanica caracter\u00edstica que ten\u00edan en com\u00fan, la de \u00abser jud\u00edos\u00bb.\n\nLa elaboraci\u00f3n de esta identidad se produjo de maneras muy concretas. Efectivamente, los diferentes tipos de jud\u00edos cooperaban para proteger sus intereses desarrollando formas de representaci\u00f3n colectiva tales que les permit\u00edan dirigirse al mundo exterior como \u00abjud\u00edos\u00bb. En el gueto veneciano, lo mismo que poco despu\u00e9s en el gueto romano, los jud\u00edos constituyeron organizaciones fraternales que se reun\u00edan en las sinagogas, pero que s\u00f3lo se ocupaban de cuestiones puramente seculares relativas al gueto. Dado que estaban destinados a vivir en el aislamiento, gran parte de este trabajo secular ten\u00eda por finalidad la autoayuda ante las dificultades de las condiciones materiales del propio gueto.\n\nEl aumento de la poblaci\u00f3n del gueto fue poco a poco convirtiendo este espacio cerrado en un lugar sucio e insalubre, donde la peste encontraba un excelente caldo de cultivo. Los jud\u00edos trataron de protegerse recurriendo a sus propios m\u00e9dicos, pues la medicina era la \u00fanica profesi\u00f3n liberal que se les permit\u00eda practicar en la ciudad. Pero el conocimiento m\u00e9dico, incluso en su forma primitiva, ten\u00eda un gran enemigo en la ley de la vivienda a la que los jud\u00edos estaban sometidos y que no les permit\u00eda poseer en el gueto edificios en propiedad, sino \u00fanicamente construir sobre los existentes y dividir interiormente las casas en apartamentos cada vez m\u00e1s peque\u00f1os. Las fraternidades jud\u00edas controlaban el espacio del gueto, pero era un espacio en permanente degradaci\u00f3n.\n\nEn Venecia, la econom\u00eda asentada sobre la base de las especias produjo un profundo efecto en la vida religiosa del gueto, debido al consumo de caf\u00e9. Tradicionalmente, en la Edad Media las oraciones y los estudios religiosos de los jud\u00edos ten\u00edan lugar por la ma\u00f1ana. La aparici\u00f3n del caf\u00e9, del que muy pronto se pudo disponer en la ciudad, fue bien recibida por los jud\u00edos como una manera de sacar provecho de su segregaci\u00f3n espacial. Lo emplearon como est\u00edmulo para mantenerse despiertos y desplazaron los momentos de plegaria y de estudio a las horas de la noche en que deb\u00edan permanecer encarcelados.\n\nPor esos medios, debido a su encierro en el espacio del gueto, los jud\u00edos, al igual que los alemanes, los turcos y los griegos de Venecia, experimentaron un sentimiento de solidaridad rec\u00edproca; en el aislamiento, estos extra\u00f1os \u00abconstruyeron\u00bb un sentido de car\u00e1cter colectivo. Por el contrario, para las prostitutas habr\u00eda carecido por completo de sentido forjar una solidaridad en esa forma, pues en su caso no hab\u00eda una naci\u00f3n, una memoria ni una antropolog\u00eda a partir de la cual pudieran ellas constituirse como pueblo.\n\nEl espacio de identidad jud\u00eda se relacionaba con el cuerpo de otras maneras, mediante su modo de hacer dinero, es decir, mediante la usura. Tal como se la practicaba en Venecia desde el siglo XII, la usura consist\u00eda en prestar dinero a tasas de entre el quince y el veinte por ciento, que, en conjunto, eran menores que las que se pagaba en los pa\u00edses m\u00e1s septentrionales. Con independencia de d\u00f3nde se las exigiera ni de qui\u00e9n lo hiciera, se las juzgaba como dinero jud\u00edo y, por tanto, una forma vil de riqueza. En la _\u00c9tica a Nic\u00f3maco_ , Arist\u00f3teles hab\u00eda condenado la usura como \u00abuna ganancia de dinero a partir del propio dinero\u00bb, como si el dinero pudiera reproducirse como un animal, y esta conexi\u00f3n \u00abantinatural\u00bb con el dinero era ofensiva. (Un pr\u00e9stamo veneciano \u00ablimpio\u00bb en esta etapa primitiva del capitalismo significaba un pr\u00e9stamo con baja tasa de inter\u00e9s variable y sin seguridad para el prestamista de recuperar su dinero.) Como pr\u00e1ctica antinatural, la usura era comparada con el sexo il\u00edcito. Un contempor\u00e1neo de Shakespeare declaraba en _The Seven Deadly Sins of London_ que \u00abel usurero vive de la lascivia del dinero, [que] es la madama de sus propias bolsas\u00bb; otro cr\u00edtico cristiano de los jud\u00edos escrib\u00eda que el usurero \u00absomete su dinero al acto antinatural de la procreaci\u00f3n\u00bb. En otros pasajes, el dinero obtenido mediante usura se comparaba con los excrementos, y al jud\u00edo con un copr\u00f3fago.\n\nDesde los tiempos prehel\u00e9nicos, el juda\u00edsmo se hab\u00eda interesado en los rituales relativos a la purificaci\u00f3n del cuerpo. Los ba\u00f1os del _Mikvah_ para las mujeres eran precisamente uno de esos rituales, como lo era tambi\u00e9n la limpieza de la casa antes de dar inicio a importantes festividades religiosas. En la Edad Media, los jud\u00edos empezaron a sobrevivir econ\u00f3micamente mediante pr\u00e1cticas que les hac\u00edan parecer sucios a ojos de los dem\u00e1s; entonces, la cuesti\u00f3n de la limpieza corporal, codificada en religi\u00f3n desde dentro y estigmatizada por el prejuicio social desde fuera, pod\u00eda abordarse de un nuevo modo en el gueto. La separaci\u00f3n espacial pod\u00eda convertirse en una virtud si se le quitaba a la comunidad la m\u00e1cula de la contaminaci\u00f3n. La lucha contra la peste, producida por el propio espacio del gueto, adoptaba ahora un car\u00e1cter religioso. En efecto, daba una nueva dimensi\u00f3n, una dimensi\u00f3n visual, a la palabra hebrea _Kadosh_. Como observa Kenneth Stow, _Kadosh_ \u00absignifica literalmente separar o separado. \u00c9se es su sentido original, b\u00edblico. Su relaci\u00f3n con lo divino se halla en el Lev\u00edtico: \"Sed santos _(kadoshim)_ , pues yo, el Se\u00f1or vuestro Dios, soy santo _(kadosh)\"\u00bb_.14 Espacialmente segregados, empezaron a pensar en considerar el gueto como el escenario de una nueva tarea, la tarea de purificaci\u00f3n, la purificaci\u00f3n de una comunidad vali\u00e9ndose de su separaci\u00f3n. Pero \u00bfc\u00f3mo pod\u00eda coexistir esta tarea con su vida econ\u00f3mica fuera del gueto y con los m\u00e1rgenes de intercambio social con cristianos que coexist\u00edan con su propia marginalidad? \u00bfC\u00f3mo pod\u00eda una vida en el gueto ser a la vez una vida urbana? Estas preguntas dieron forma al tema de los derechos.\n\n### _Dos vidas de gueto_\n\nLa intriga de _El mercader de Venecia_ , de Shakespeare, gira en torno a una circunstancia que nos resulta extra\u00f1a cuando pensamos en ella. Shylock, rico prestamista jud\u00edo de Venecia, ha prestado tres mil ducados a Bassanio por tres meses, y Antonio, amigo de Bassanio, se ha comprometido a devolver el dinero prestado. Si no lo hace, Shylock, que odia al arist\u00f3crata cristiano Antonio y todo lo que \u00e9ste representa, quiere una libra de su carne como compensaci\u00f3n. La fortuna se vuelve contra Antonio y los barcos que transportan toda su riqueza naufragan en una tormenta. Por tanto, Shylock reclama su libra de carne.\n\nLo extra\u00f1o es que ni Antonio ni las autoridades cristianas que en ese momento entran en acci\u00f3n en la obra se sientan obligados a mantener la palabra dada a un jud\u00edo. El sistema econ\u00f3mico, como hemos se\u00f1alado, hac\u00eda amplio uso de los contratos verbales; tambi\u00e9n se llevaban registros escritos, pero lo decisivo era el car\u00e1cter vinculante de la palabra dada. En el sistema bancario de Rialto, tal como lo describe el historiador Frederick Lane, las transferencias se realizaban en estos t\u00e9rminos: \u00abel banquero [...] sentado tras un banco bajo el p\u00f3rtico de una iglesia en Rialto, con su gran diario desplegado ante \u00e9l. El pagador ordena verbalmente al banquero que realice una transferencia a la cuenta de la persona que recibe el pago\u00bb. Rialto, que era el centro de la actividad econ\u00f3mica, se llenaba todos los d\u00edas de chismes y de noticias; el capitalismo se practicaba sin estad\u00edsticas ni garant\u00edas gubernamentales de ning\u00fan tipo; la confianza personal en el otro era decisiva a la hora de cerrar tratos comerciales.\n\nFuera del teatro, el p\u00fablico ingl\u00e9s de Shakespeare, lo mismo que Sanuto en Venecia un siglo antes, trataba a los jud\u00edos como animales semihumanos en quienes no se pod\u00eda confiar. En la obra, Shakespeare presenta estos prejuicios haciendo del prestamista jud\u00edo un verdadero can\u00edbal, puesto que Shylock propone extraer una libra de carne de la cadera de Antonio si el noble no puede pagar. La pr\u00e1ctica antinatural de la usura es as\u00ed revelada en su consecuencia corporal m\u00e1s inhumana.\n\nDe tal manera, es de suponer que el p\u00fablico esperara que el Duque (el Dogo) de Venecia, a modo de poderoso _deus ex machina_ , entrara en escena y mandara al can\u00edbal a la c\u00e1rcel, o por lo menos que declarara inmoral el contrato y, por tanto, nulo. Sin embargo, cuando uno de los personajes secundarios de _El mercader de Venecia_ dice que est\u00e1 seguro de que el Duque resolver\u00e1 la cuesti\u00f3n exactamente de esa manera, Antonio responde:\n\nEl Duque no puede impedir a la ley que siga su curso,\n\ny para explicar por qu\u00e9, remite a las razones del auge de la ciudad\n\na causa de las garant\u00edas comerciales que los extranjeros encuentran cerca de nosotros en Venecia; suspender la ley ser\u00eda atentar contra la justicia del Estado, puesto que el comercio y la riqueza de la ciudad dependen de todas las naciones.\n\nAl poner estas palabras en boca del Duque, Shakespeare, m\u00e1s que presentar a su p\u00fablico las relaciones de \u00e9ste con la peque\u00f1a poblaci\u00f3n de jud\u00edos existente entre la poblaci\u00f3n londinense, expone un argumento que explica a su p\u00fablico algo relativo a su propia vida. La riqueza de la ciudad necesita que la gente acepte las impurezas morales. El Duque de Shakespeare va incluso m\u00e1s all\u00e1: en honor de las \u00abgarant\u00edas comerciales\u00bb, la gente debe honrar los contratos, piense lo que piense de las cualidades morales de las partes contratantes, o incluso del tipo de moral que \u00e9stas representan. Es menester tolerar hasta el canibalismo, si es que ambas partes han convenido en \u00e9l y esto no afecta a terceros. De esta manera, el contrato se mostraba como una fuerza que generaba sus propios derechos, reclamaciones y t\u00edtulos; ninguna ley superior ni ning\u00fan gobierno exterior pueden intervenir en su funcionamiento.\n\nEl contrato can\u00edbal entre Shylock y Antonio es un contrato verbal y son precisamente sus cualidades verbales las que Shakespeare hace enunciar al Duque en su enfrentamiento con Shylock. M\u00e1s que ordenarle que desista, el Duque razona con Shylock; si es preciso violar el contrato, eso s\u00f3lo puede hacerse a trav\u00e9s del di\u00e1logo. Shylock puede a su vez burlarse de estos alegatos ducales e insultar, porque sabe que \u00abest\u00e1 en su derecho\u00bb de decir lo que se le ocurra. Esto es justamente lo que quiere decir Porcia, la mujer que terminar\u00e1 por cortar este nudo gordiano, cuando declara que \u00abno hay fuerza en Venecia que pueda alterar un decreto establecido\u00bb. El \u00abdecreto\u00bb es el acuerdo al que Antonio y Shylock han llegado a trav\u00e9s de la negociaci\u00f3n verbal, y la libertad de Shylock para hablar con el Duque en tono insultante, combativo, deriva de la misma fuente, es decir, de la legitimidad y la protecci\u00f3n que proporcionan las palabras. La acci\u00f3n de la pieza dramatiza una conexi\u00f3n que comienza a tomar forma en el Renacimiento, esto es, la conexi\u00f3n entre la libertad de palabra y el car\u00e1cter sagrado del contrato. Esta libertad econ\u00f3mica se desprende de la libertad de expresi\u00f3n.\n\nPuesto que Shakespeare es un dramaturgo, la libertad de palabra que desvela en este contrato can\u00edbal sirve a la finalidad art\u00edstica de profundizar la imagen estereot\u00edpica del jud\u00edo avaricioso. Desde este punto de vista, _El mercader de Venecia_ marca un fuerte contraste con la pieza de Marlowe titulada _El jud\u00edo de Malta_. Marlowe convierte a Barrab\u00e1s, el jud\u00edo malt\u00e9s, en una figura divertida, simplemente despreciable por su avaricia. Shylock, en cambio, es un ser humano m\u00e1s complejo, su codicia se amalgama con el odio a los cristianos que le niegan la elemental consideraci\u00f3n debida a otro ser humano. El parlamento m\u00e1s famoso de Shylock es un intento de restaurar la dignidad corporal del cuerpo de los jud\u00edos:\n\n\u00bfEs que un jud\u00edo no tiene ojos? \u00bfEs que un jud\u00edo no tiene manos, \u00f3rganos, proporciones [...]? Si nos envenen\u00e1is, \u00bfno nos morimos? Y si nos ultraj\u00e1is, \u00bfno nos vengaremos?\n\nEl parlamento apela a la universalidad del cuerpo, a la igualdad de derechos que asiste a todos los cuerpos, comprendido el derecho de vengarse. Y a su vez, la apelaci\u00f3n de Shylock es lo que presta el marco al desenlace en el cuarto acto, cuando Porcia termina por disolver el contrato de Shylock.\n\nEn _El mercader de Venecia_ Shakespeare nos sorprende con el desenlace. Construye un gran drama de derecho versus moral y de pronto, en el cuarto acto, disipa todas sus tensiones. Porcia, disfrazada de abogado, dice a Shylock que su reclamaci\u00f3n es justa, pero que debe ajustarse estrictamente a sus t\u00e9rminos y coger una libra de carne, pero ni una gota de sangre, pues no estaba especificado en el contrato; adem\u00e1s, s\u00f3lo puede extraer una libra de carne, ni una onza m\u00e1s ni una menos. Puesto que Shylock no puede ser un can\u00edbal tan cient\u00edfico, el juego toca a su fin. Muchos cr\u00edticos han visto en esto un pobre final, pues el conflicto principal es eludido mediante una especie de triqui\u00f1uela de abogado. Sin embargo, es completamente coherente con el sentido general de la obra, pues los poderes operantes en la Venecia de Shakespeare son poderes que a\u00edslan a las personas de la textura de la circunstancia, de las cargas locales.\n\nEn ning\u00fan otro pasaje de la obra resultan m\u00e1s evidentes estos poderes de aislamiento de la circunstancia y del lugar que en la figura de la hija de Shylock, Jessica. En el momento en que se enamora de un cristiano, Jessica abandona a su padre, su casa y su fe. No da casi muestras de aflicci\u00f3n por dejar el mundo de su padre, ni tampoco por robarle, que es lo que hace en realidad cuando se apodera de las joyas de Frankfurt para pagarse los placeres de su luna de miel. Contado de esta manera, parece una criatura despreciable y, sin embargo, en la obra resulta absolutamente encantadora. Esto es as\u00ed porque Shakespeare no imagina la \u00abjudeidad\u00bb como una maldici\u00f3n racial, ni como una identidad racial o ni siquiera cultural. El peso de \u00abser jud\u00edo\u00bb es en realidad ligero en _El mercador de Venecia_ ; aqu\u00ed, el hecho de \u00abser jud\u00edo\u00bb se asemeja m\u00e1s bien al de usar cierto tipo de vestimenta, que uno puede quitarse si, por ejemplo, se enamora.\n\nLa misma ligereza conlleva la triqui\u00f1uela contractual que Porcia aplica a Shylock. A mi juicio, se trata precisamente de la visi\u00f3n \u00e9tica que Shakespeare ten\u00eda de los derechos de contrataci\u00f3n: al liberar a hombres y mujeres de la fuerza de la circunstancia, el trabajo de las palabras les arrebata tambi\u00e9n la densidad de la experiencia. Tanto la acci\u00f3n \u2013la habilidad de Porcia en el manejo de la ley\u2013 como el personaje \u2013la judeidad de Jessica\u2013 se vuelven ingr\u00e1vidos bajo el trabajo de las palabras; se prescinde del odio mortal de Shylock y se olvida tanto su vileza como la nobleza de su gran parlamento sobre el cuerpo del jud\u00edo; su hija le ha robado y lo ha abandonado. Es dif\u00edcil imaginar todo esto como una soluci\u00f3n cristiana de los acontecimientos. Tuvo raz\u00f3n Shakespeare en decir que _El mercader de Venecia_ era una comedia, pues el bien y el mal terminan por no tener consecuencia alguna. Es ya un lugar com\u00fan decir que _El mercader de Venecia_ es una pieza acerca del intercambio; pero en esta comedia, el sentido impl\u00edcito del intercambio es la indiferencia.\n\nComo lectores modernos conocemos la realidad social de la indiferencia inherente al _ethos_ de la sacralidad de los contratos. Lo que perturba al lector moderno con la mente puesta en los derechos humanos son las conexiones que en este drama se dan entre la sacralidad del contrato y la libertad de expresi\u00f3n. \u00c9ste era el emblema de Venecia para Shakespeare y sus contempor\u00e1neos, el de un lugar de riqueza adquirida, a sus ojos, por medios nuevos y modernos, una ciudad internacional tan independiente del localismo como de Europa y del poder del gobierno cristiano. Esta visi\u00f3n de una ciudad libre de ataduras culturales no es, por supuesto, la Venecia de los guetos jud\u00edos ni de los Fondaci de alemanes, turcos o albaneses. _El mercader de Venecia_ se puede leer, me parece, como un tipo de premonici\u00f3n de la modernidad, del significado de la libertad, experiencialmente despojado de contenido concreto, propio del mundo moderno; el gueto de Venecia ofrece otra premonici\u00f3n.\n\nLeon (Judah Aryeh) Modena, que vivi\u00f3 de 1571 a 1648, pas\u00f3 la mayor parte de su vida adulta en el gueto de Venecia. Fue escriba, poeta, rabino, m\u00fasico y l\u00edder pol\u00edtico, estudioso de lat\u00edn, griego, franc\u00e9s, ingl\u00e9s y, sorprendentemente para nosotros, jugador compulsivo; su autobiograf\u00eda, titulada _The Life of Judah_ , es un juego de palabras, pues se supon\u00eda que el juego era el pecado de Judas.\n\nNacido fuera de la ciudad, Modena lleg\u00f3 a Venecia en 1590, cuando ten\u00eda diecinueve a\u00f1os; tres a\u00f1os despu\u00e9s, ya casado, decidi\u00f3 convertirse en rabino y cuando alcanz\u00f3 su objetivo hab\u00eda cumplido ya treinta y ocho. Su vida en los a\u00f1os intermedios fue agitada; escribi\u00f3 mucho mientras viajaba de un lugar a otro, pero se sent\u00eda insatisfecho. Fue un cabal ejemplo de jud\u00edo errante y s\u00f3lo cuando entr\u00f3 en el mundo cerrado del gueto de Venecia, rodeado de jud\u00edos de todo tipo que llevaban una vida p\u00fablica activa, empez\u00f3 a sentirse verdaderamente en casa.\n\nCuando, en 1609, se orden\u00f3 finalmente en Venecia, su vida adopt\u00f3 un car\u00e1cter intensamente local. En la Venecia renacentista, el rabino iba a la sinagoga tres veces por d\u00eda \u00abpara dirigir el servicio, recitar las plegarias por los enfermos y los muertos, predicar todos los Sabbath por la ma\u00f1ana antes de extraer la Torah del arca y leerla, y los lunes y los jueves para comentar dos o tres pasajes de la ley despu\u00e9s de su lectura y luego devolverla al arca\u00bb.19 En virtud de sus dones intelectuales y su incesante producci\u00f3n escrita, sus sermones alcanzaron fama internacional y comenz\u00f3 a atraer al gueto a muchos cristianos que iban para o\u00edrlo hablar.\n\nLeon Modena fue el rabino que m\u00e1s atrajo a los cristianos, pero no el \u00fanico. Puesto que las sinagogas adoptaron la forma de instituci\u00f3n p\u00fablica \u2013esto es, de edificios apartados de las retorcidas callejuelas de la vida jud\u00eda, frente a la esquina sudeste del Gueto Nuevo y la calle principal del Gueto Viejo, f\u00e1ciles de encontrar y de visitar\u2013, se convirtieron en lugares que, dentro del gueto, tend\u00edan a lo largo del d\u00eda ocasionales puentes sobre el abismo cultural entre jud\u00edos y cristianos. A condici\u00f3n de que el orador tuviera la suficiente elocuencia, los cristianos, aunque no con frecuencia, ir\u00edan al gueto a o\u00edrlo hablar en la sinagoga. A semejanza de los turistas modernos que van a Harlem, estos visitantes del gueto practicaban en general una especie de voyeurismo. Los cristianos que, como Paulo Scarpi, asist\u00edan para o\u00edr con seriedad a Modena, pagaron un precio por ello; a Scarpi se le deneg\u00f3 un obispado por \u00abhaberse entendido con jud\u00edos\u00bb.\n\nLos talentos personales de Modena fueron un ejemplo que sirvi\u00f3 de prueba de la medida en que un hombre ilustrado pod\u00eda romper el aislamiento del gueto. Su reputaci\u00f3n aument\u00f3 durante la d\u00e9cada de 1620 y en 1628 asumi\u00f3 el control de la academia musical jud\u00eda (l'Accademia degli Impediti), con la que ofreci\u00f3 recitales de m\u00fasica coral jud\u00eda y salmos en la sinagoga sefard\u00ed. \u00abLa nobleza cristiana de Venecia acudi\u00f3 en gran n\u00famero a este acontecimiento espectacular\u00bb, escribe su bi\u00f3grafo reciente, \u00aby las autoridades tuvieron que intervenir para controlar a la multitud.\u00bb\n\nModena, como se\u00f1ala la historiadora moderna Natalie Davis, \u00abse diferencia de Shylock pr\u00e1cticamente en todo, pues es un jud\u00edo que arriesga su dinero con prodigalidad [...] que disfruta de la admiraci\u00f3n de los cristianos...\u00bb. Y sin embargo, cuando su vida se acercaba al final, las mismas condiciones del gueto que le hab\u00edan permitido prosperar \u2013su establecimiento formal en el interior de una ciudad y su magn\u00e9tica atracci\u00f3n para los jud\u00edos que vagaban por doquier\u2013 se convirtieron en una trampa para \u00e9l.\n\nCuando una grave peste azot\u00f3 Venecia entre 1629 y 1631, la segregaci\u00f3n del gueto que imped\u00eda a los jud\u00edos marcharse en busca de lugares menos insalubres oblig\u00f3 a quienes se hallaban bajo el cuidado pastoral de Modena a padecer con particular intensidad los estragos de la enfermedad. Tres a\u00f1os despu\u00e9s, en 1634, el nieto de Modena, de catorce a\u00f1os a la saz\u00f3n, pens\u00f3 que la gran fama de su abuelo le servir\u00eda para obtener su libertad personal como tip\u00f3grafo, pero aquella fama demostr\u00f3 tener l\u00edmites personales y el muchacho fue enviado a la c\u00e1rcel. El propio Modena se top\u00f3 con los l\u00edmites de la consideraci\u00f3n con la que los cristianos le trataban, pues por la publicaci\u00f3n de su magna obra sobre los ritos religiosos jud\u00edos fue llevado ante la Inquisici\u00f3n veneciana en 1637 y \u00fanicamente gracias a sus relaciones personales con el Gran Inquisidor pudo salvar de las llamas su vida y el libro, que, no obstante, sigui\u00f3 siendo vituperado por dignatarios menores de la Iglesia. Para la Inquisici\u00f3n, el libro de Modena sobre el ritual jud\u00edo constitu\u00eda una amenaza porque sacaba al dominio abierto y p\u00fablico de la antropolog\u00eda algo que hab\u00eda estado confinado en las sombras de la fantas\u00eda. Ante todo, en estos a\u00f1os finales de su vida, Modena hubo de descubrir la terrible conexi\u00f3n entre la fantas\u00eda psicosocial y la segregaci\u00f3n espacial.\n\nEn marzo de 1636, un grupo de jud\u00edos recibi\u00f3 y ocult\u00f3 en el gueto bienes robados en otra parte de Venecia. Modena describe c\u00f3mo la polic\u00eda organiz\u00f3 la b\u00fasqueda de seda, ropa de seda y oro ocultos. \u00abEn Purim, el recinto del gueto fue clausurado para realizar con grandes prisas una requisa casa por casa.\u00bb Dada la organizaci\u00f3n del espacio, para hacer tal cosa bastaba con levantar unos cuantos puentes y cerrar unas cuantas puertas. Modena protest\u00f3 contra esto con el argumento de que \u00abcuando un individuo comet\u00eda un delito, ellos [los cristianos] lanzaban su c\u00f3lera contra toda la comunidad...\u00bb, para agregar esta reveladora conclusi\u00f3n: \u00abllam\u00e1ndonos banda de ladrones y diciendo que en el gueto se oculta todo tipo de delitos\u00bb.\n\nEn la \u00e9poca en que Shakespeare escribi\u00f3 _El mercader de Venecia_ , los jud\u00edos de la ciudad se hab\u00edan convertido en verdaderos enigmas para sus contempor\u00e1neos cristianos, que ya no los ve\u00edan de manera rutinaria entre ellos, salvo en las escasas ocasiones en que el propio Modena los atra\u00eda al gueto. Entonces tom\u00f3 cuerpo el rumor no comprobado acerca de la vida que llevaban los jud\u00edos. En parte, esto se deb\u00eda simplemente al hecho de que los jud\u00edos que llegaban en n\u00famero creciente a la ciudad desaparec\u00edan de inmediato en el gueto. Pero al tener encerrados a los diferentes, lo \u00fanico que el resto de la ciudad pod\u00eda saber de ellos era producto de la fantas\u00eda. En 1636, la fantas\u00eda de que todos los jud\u00edos estaban implicados en una banda de ladrones se convirti\u00f3 en el curso de dos o tres d\u00edas en convicci\u00f3n inquebrantable del p\u00fablico veneciano. Dada la inexistencia de trato cotidiano con la mayor\u00eda de los jud\u00edos, no hab\u00eda experiencia capaz de poner l\u00edmite a la fantas\u00eda. A partir del robo se atribuyeron a los encerrados tras los muros del gueto tambi\u00e9n otros delitos, como el secuestro de ni\u00f1os en el gueto y la pr\u00e1ctica de la misa negra. El gueto, a ojos de los cristianos, se hab\u00eda convertido en un espacio de ocultamiento. Se pensaba que, detr\u00e1s de sus puentes levantados, sus ventanas cerradas y su vida al margen del sol y del agua, justamente por tratarse de una vida dif\u00edcilmente visible, el delito y la idolatr\u00eda se ensa\u00f1aban.\n\nComo en los d\u00edas siguientes el rumor creci\u00f3, los jud\u00edos fueron v\u00edctimas del mayor pogromo conocido en Europa desde que Adriano, el emperador romano, lanzara la guerra contra los jud\u00edos de Jerusal\u00e9n, quince siglos antes. Los venecianos hab\u00edan ocultado la diferencia y en un pogromo como el de 1636 \u2013pues no fue otra cosa\u2013 los jud\u00edos experimentaron las consecuencias extremas de la pol\u00edtica de aislamiento. En un espacio-gueto percibido desde fuera, la fantas\u00eda convierte la diferencia en una otredad incomprensible.\n\nAs\u00ed fue como los derechos locales de los que hab\u00edan gozado los jud\u00edos tocaron a su fin. El aparato del Estado no pudo proteger sus cuerpos, porque la pasi\u00f3n popular, movilizada por la fantas\u00eda, demostr\u00f3 ser m\u00e1s fuerte que los poderes de polic\u00eda del Estado. Los historiadores discuten hasta qu\u00e9 punto la polic\u00eda form\u00f3 parte de la turba, pero lo cierto es que el hacinamiento de tantos jud\u00edos, al dar tanta consistencia espacial a la comunidad, permiti\u00f3, una vez violentadas las puertas, atacarlos como a animales acorralados para la matanza.\n\nEn el gueto de Venecia, Modena, el jud\u00edo errante, el cosmopolita por excelencia, hab\u00eda por fin abrazado el juda\u00edsmo al armonizar fe y espacio. Tras el pogromo de 1636 comenz\u00f3 a arrepentirse de la vida con la que tan c\u00f3modo se hab\u00eda sentido. Su yerno Jacob, a quien le un\u00eda un v\u00ednculo personal muy estrecho, hab\u00eda sido desterrado a Ferrara como parte de un castigo generalizado a los jud\u00edos durante la persecuci\u00f3n de 1636. En 1643, enfermo, perseguido por su propia mujer, solo, Leon Modena pidi\u00f3 a las autoridades que permitieran el regreso de Jacob. \u00c9stas, todav\u00eda bajo la influencia de los temores que hab\u00edan provocado la gran persecuci\u00f3n, se negaron. Entonces Modena sucumbi\u00f3 a un severo brote de su vicio de toda la vida, \u00ablos juegos de azar. Persist\u00ed en esto [durante dos meses] con una p\u00e9rdida de dinero, y de honor, de tal magnitud como nunca hab\u00eda experimentado, m\u00e1s el a\u00f1adido de ri\u00f1as familiares\u00bb. Toda su vida hab\u00eda jugado no por placer sino por desesperaci\u00f3n, cuando sus propios esfuerzos parec\u00edan incapaces de garantizar su supervivencia. He aqu\u00ed al viejo rabino, al estudioso de renombre mundial, sentado a la mesa de juego, tirando los dados y barajando cartas semana tras semana, como si los simples rituales de la apuesta pudieran aliviarle el dolor.\n\nCerca del final, la memoria estalla en esta terrible confesi\u00f3n de desamparo:\n\n\u00bfQui\u00e9n me dar\u00e1 conocimiento de palabras de lamentaci\u00f3n, de queja y de aflicci\u00f3n tales como para poder hablar y escribir acerca de cu\u00e1nto peor fue mi suerte que la de cualquier otra persona? Habr\u00e9 de sufrir y de soportar lo que comenz\u00f3 a desolarme el d\u00eda en que nac\u00ed y continu\u00f3 sin respiro durante setenta y seis a\u00f1os enteros.\n\nLos espacios-gueto que Venecia cre\u00f3 para sus jud\u00edos, alemanes, persas y griegos, as\u00ed como el que trat\u00f3 de crear para sus cortesanas, resultan ser una gran iron\u00eda de la forma urbana. Si nos atenemos estrictamente a su poblaci\u00f3n, podemos decir que Venecia era la ciudad con mayor diversidad de la Europa renacentista; ninguna otra ten\u00eda una poblaci\u00f3n tan cosmopolita. Las condiciones econ\u00f3micas que crearon la riqueza de la Venecia renacentista dotaron a la ciudad de gran abundancia de seres humanos, pero esta ciudad de lujo material se neg\u00f3 a utilizar esa abundancia como objeto de experiencia cotidiana. Una conclusi\u00f3n ser\u00eda que nos hallamos ante una relaci\u00f3n de causa y efecto. Desde este punto de vista, la sociedad no pod\u00eda soportar la diversidad que la econom\u00eda hab\u00eda llevado a la ciudad, argumento que se oye hoy con frecuencia en ciudades mestizas como Nueva York, Berl\u00edn o Londres. Lo que gravita en contra de este argumento es que el gueto no es algo que a los jud\u00edos les sucede como v\u00edctimas pasivas. A trav\u00e9s de sus confraternidades, que hac\u00edan las veces de mediadoras con el mundo exterior, la gradual identificaci\u00f3n de fe y espacio hizo que los jud\u00edos se convirtieran en hombres y mujeres del gueto, un espacio a la vez de represi\u00f3n y de identificaci\u00f3n.\n\nPodr\u00eda decirse que para los jud\u00edos de Venecia no hab\u00eda opci\u00f3n, pero \u00e9sa no es la cuesti\u00f3n. Los jud\u00edos que rechazaron esa forma de vida abandonaron la ciudad. La dicotom\u00eda que marc\u00f3 la historia moderna de los jud\u00edos comenz\u00f3 con la posibilidad de vivir territorialmente, bajo las condiciones que les impon\u00edan la Venecia y Roma renacentistas. Los jud\u00edos que se rebelaron contra la guetizaci\u00f3n tuvieron que padecer las consecuencias de una vida desarraigada; los que aceptaron esa forma de vida tuvieron que sufrir las de su arraigo. Lo que Modena lleg\u00f3 a ver al final de sus d\u00edas fue que aquel espacio arraigado era en s\u00ed mismo un motivo de debilitamiento; el derecho local qued\u00f3 superado, abolido por la falta de l\u00f3gica espacial de la otredad.\n\n### _Conclusi\u00f3n_\n\nNo he intentado con este breve esbozo escribir una historia ingenua. Me pareci\u00f3 que la historia de los jud\u00edos de la Venecia del Renacimiento, as\u00ed como la incompatibilidad del concepto de derechos con su presencia en la ciudad, guardaba relaci\u00f3n con el pensamiento actual sobre los derechos humanos.\n\nEn estos \u00faltimos a\u00f1os, los an\u00e1lisis de los derechos humanos se orientaron a la sociedad civil, esto es, a las pr\u00e1cticas religiosas, econ\u00f3micas y comunitarias. Estos an\u00e1lisis han tratado de ofrecer una concepci\u00f3n de los derechos humanos como pr\u00e1cticas sociales en s\u00ed mismos, como pr\u00e1cticas que trascienden las formulaciones legales y la hegemon\u00eda del Estado. Pocos pensadores modernos consentir\u00edan en seguir a Hannah Arendt hasta su \u00faltima y amarga consecuencia, la del divorcio de la pol\u00edtica respecto de la esfera de la sociedad civil. Sin embargo, hay una tendencia a aceptar la distinci\u00f3n de Arendt, pero desde el otro lado, es decir, con la idea de que el marco de las pr\u00e1cticas sociales resulta ser un correctivo de la formulaci\u00f3n meramente legal de los derechos.\n\nLa historia de los jud\u00edos de Venecia, tal como la he relatado, podr\u00eda constituir un motivo de reflexi\u00f3n para quienes est\u00e1n dispuestos a aceptar esta visi\u00f3n correctiva, pues se trata de una historia en la que los poderes discursivos de los seres humanos, as\u00ed como la organizaci\u00f3n espacial de las vidas humanas, corrompen la mera experiencia del derecho. Las formulaciones culturales de derechos en el discurso y en el espacio pueden resultar ilusorias o autodestructivas. El Estado es, a mi juicio, un correctivo necesario de las normas de derecho en el seno de la sociedad civil.\n\nPero he tenido tambi\u00e9n otra raz\u00f3n para centrar este ensayo en la relaci\u00f3n entre lugar y derecho. La ideolog\u00eda del lugar \u2013en la forma de celebraciones de los ideales locales o en la forma de ideales de una cultura com\u00fan en un grupo particular\u2013 se erige hoy contra los reclamos de universalizaci\u00f3n de la Ilustraci\u00f3n. Los pensadores de la Ilustraci\u00f3n, como Kant, creyeron ilusoriamente que lo social pod\u00eda subsumirse en lo pol\u00edtico; esta ilusi\u00f3n se ve hoy contradicha por celebraciones de cohesi\u00f3n social en s\u00ed misma y por s\u00ed misma, particularmente entre los grupos oprimidos. Que los derechos efectivos puedan ser resultado del simple acto de cohesi\u00f3n social me parece un error tan grave como ha demostrado serlo la noci\u00f3n kantiana de ciudadan\u00eda universal.\n\nLos derechos sin localizaci\u00f3n de los contratos, tal como aparec\u00edan en el capitalismo veneciano, eran defectuosos por la mutabilidad misma de la palabra hablada. La lucha por conseguir la especificidad en la ley escrita me parece la \u00fanica manera de contrarrestar esta verbal \u00ablevedad del ser\u00bb, para adaptar una frase de Kundera. Los jud\u00edos de Venecia pose\u00edan tanto los derechos verbales como los espaciales, pese a lo cual no fueron incorporados al funcionamiento codificado del Estado. M\u00e1s que derechos de legitimaci\u00f3n, los suyos eran derechos de tolerancia. A mi juicio, \u00fanicamente el Estado puede realizar esta operaci\u00f3n de legitimaci\u00f3n, y \u00fanicamente puede hacerlo rescatando las palabras del dominio de la comprensi\u00f3n hablada. \n\n## El extranjero \n### _El espejo de Manet_\n\n\u00c9douard Manet fue un pintor de la ciudad, pero no un pintor realista tal como este t\u00e9rmino se entiende com\u00fanmente. No trataba de lograr con la pintura el efecto de sorprender la vida al desnudo, como hicieron los fot\u00f3grafos de su \u00e9poca. El registro de Manet tampoco comparte gran cosa con el esp\u00edritu de los indignados retratos urbanos de protesta que nos presenta Zola de putas, ni\u00f1os abandonados o familias comiendo ratas asadas. El arte de Manet es capaz de lograr una impresionante declaraci\u00f3n pol\u00edtica directa, como lo demuestra el cuadro realizado en 1868 titulado _La ejecuci\u00f3n del Emperador Maximiliano_ , pero la visi\u00f3n que el artista tiene de la ciudad se vale de otros medios para obtener sus efectos.\n\nEn su registro de la vida que ve\u00eda en Par\u00eds, Manet empleaba detalles visuales que perturban la mirada, que la arrastran de objeto en objeto dentro del cuadro y que a menudo sugieren que la verdadera historia a la que la pintura se refiere tiene lugar en otro sitio, fuera de la tela. En su pintura de la ciudad, Manet es un artista del desplazamiento; y es precisamente a trav\u00e9s de su comprensi\u00f3n del desplazamiento como nos habla en t\u00e9rminos sociales, tanto hoy como en su d\u00eda; su arte desaf\u00eda ciertos supuestos con los que describimos a gente desplazada, ya sea econ\u00f3mica, ya sea pol\u00edticamente, es decir, los inmigrantes, los exiliados, los expatriados.\n\nEstas palabras aluden a los diferentes motivos por los que es posible que una persona viva en el extranjero, pero hoy en d\u00eda el resultado de tales desplazamientos parece un destino com\u00fan. Ser extranjero es vivir a disgusto fuera del propio pa\u00eds; nos referimos al inmigrante que siente el impacto de la cultura y se aferra a s\u00ed mismo, al exiliado que hiberna con indiferencia en una ciudad que apenas lo roza, al expatriado que pronto sue\u00f1a con el retorno... Tales im\u00e1genes ti\u00f1en de sentimientos la necesidad de ra\u00edces y de valores que el coraz\u00f3n experimenta. M\u00e1s a\u00fan, niegan a los que se han convertido en extranjeros la voluntad y la capacidad para hacer algo humano a partir de la experiencia del desplazamiento, aun cuando se hubieran visto inicialmente forzados a emigrar. Pese a ser un pintor que se siente absolutamente c\u00f3modo en su ciudad y a su inter\u00e9s por los olores y las sombras de su vida cotidiana, Manet imagina lo que la experiencia misma del desplazamiento tiene de positivo. Bajo su pincel se abre la dualidad de \u00ablo propio\u00bb y \u00ablo extra\u00f1o\u00bb, pues la representaci\u00f3n imaginaria de los lugares familiares se vuelve ella misma cada vez m\u00e1s rara y extra\u00f1a.\n\nLa agudeza de Manet para captar el desplazamiento se expresa con toda amplitud en su \u00faltima gran obra, _El bar del Folies-Berg\u00e8re_ , realizada en el invierno de 1881-1882. Es interesante la historia de esta pintura. En 1879, Manet se ofreci\u00f3 al Concejo Municipal de Par\u00eds como pintor de murales para la nueva sede del ayuntamiento. Estos murales del Par\u00eds moderno mostrar\u00edan el efecto de las nuevas construcciones \u2013puentes de acero, alcantarillado de cemento, edificios de hierro forjado\u2013 en la vida de la ciudad. La propuesta de Manet fue rechazada, pero lo significativo es que la gran obra que realiz\u00f3 con posterioridad a esta negativa no presenta ninguna de las escenas previstas para sus murales de Par\u00eds, sino que gira m\u00e1s bien en torno a algo aparentemente m\u00e1s sentimental, m\u00e1s _kitsch_ incluso, un cuadro del Folies-Berg\u00e8re. El prop\u00f3sito de Manet era infundir a esta escena banal la fuerza de todos los cambios que \u00e9l sent\u00eda que se estaban operando por entonces en Par\u00eds, cambios que hab\u00edan dado lugar al desarrollo de una sensibilidad moderna.\n\nPara nosotros es importante entender qu\u00e9 era y qu\u00e9 no era el Folies-Berg\u00e8re en la \u00e9poca de Manet. Era un lugar de permisividad sexual; entre la multitud se paseaban por igual prostitutas y prostitutos y se bailaba el canc\u00e1n, que en su versi\u00f3n del siglo XIX no ten\u00eda nada que ver con su descendiente m\u00e1s moderno y adecentado (el canc\u00e1n, que se introdujo en Par\u00eds en la d\u00e9cada de 1830, lo bailaban en general mujeres sin ropa interior bajo sus faldas cortas y sueltas, de modo que cada vez que levantaban una pierna dejaban al descubierto sus montes de Venus). Pero el Folies-Berg\u00e8re no era en s\u00ed mismo un prost\u00edbulo, aunque estaba estrat\u00e9gicamente situado cerca de varias casas de este tipo, por lo cual las mujeres respetables pod\u00edan acudir al local a divertirse, lo que hac\u00edan en n\u00famero tambi\u00e9n respetable. Por tanto, se trataba de un lugar rayano en la indecencia, pero de un lugar p\u00fablico, lleno de gente ruidosa que beb\u00eda y flirteaba en un ambiente perfumado por cigarros, caf\u00e9 y Beaujolais barato. Los parisinos iban al Folies cuando deseaban relajarse. Era c\u00f3modo y acogedor, un hogar alejado, muy alejado, de los rigores del hogar familiar.\n\nLa escena que Manet ha aislado nos muestra una mujer de pie detr\u00e1s de una barra. Pensativa, triste, seria, es una figura aislada en medio del ruido (la figura del cuadro se basa en Suzon, una camarera del Folies-Berg\u00e8re a quien Manet conoc\u00eda). El espectador es atra\u00eddo al interior de la escena por el uso que hace Manet de los espejos, que crean una experiencia especial de desplazamiento.\n\nEl cuadro representa a la camarera mirando directamente de frente al espectador. El espejo delante del cual se encuentra ella de pie tambi\u00e9n se opone directamente al espectador. Manet refuerza esta alineaci\u00f3n plenamente frontal colocando los brazos extendidos y las manos sobre la barra del bar, ladeadas como un bailar\u00edn de ballet que girara las piernas hacia fuera, con los pies en l\u00ednea recta, manteniendo el cuerpo completamente de frente. A la derecha de esta figura vemos su espalda reflejada en el espejo, en donde la masa plana de su vestido negro es exactamente de la misma medida que el cuerpo, de modo que su figura reflejada carece del efecto de reducci\u00f3n propio de la perspectiva, pues el reflejo se muestra en el mismo plano dimensional que el cuerpo. Digo que vemos su reflejo en un espejo, aunque esto es imposible desde el punto de vista \u00f3ptico; no podemos verla directamente de frente y al mismo tiempo ver su reflejo a la derecha. Hoy el espectador acepta esta imposibilidad, que parece visualmente l\u00f3gica, aunque \u00f3pticamente imposible. Sin embargo, Charles de Feir, en su _Guide du Salon de Paris 1882_ , hablaba en nombre de muchos contempor\u00e1neos de Manet al considerar este extra\u00f1o espejo como una se\u00f1al de la defectuosa t\u00e9cnica del pintor.\n\nEn muchas de las \u00faltimas pinturas de Manet, la sensaci\u00f3n de desplazamiento \u00f3ptico del espectador se ve reforzada por alg\u00fan gesto arbitrario y aparentemente menor, que contribuye as\u00ed a separar la escena del hecho representacional. En _El bar del Folies-Berg\u00e8re_ esto se produce gracias a la manera en que Manet pinta dos l\u00e1mparas de gas reflejadas en el espejo; son dos discos de un blanco puro colocados sobre el plano del cuadro, l\u00e1mparas que no proyectan sombra, que no est\u00e1n rodeadas de penumbras como suele suceder con las luces reflejadas, que no est\u00e1n pintadas del todo. Una vez m\u00e1s, los contempor\u00e1neos de Manet vieron en estas extra\u00f1as luces una se\u00f1al de imperfecci\u00f3n del pintor. En _L'Illustration_ , Jules Compte dijo de ellas que \u00abprobablemente Monsieur Manet ha elegido un momento en que las l\u00e1mparas no funcionaban adecuadamente, pues nunca hemos visto una luz menos resplandeciente...\u00bb.\n\nEn la actualidad podemos apreciar que estos discos blancos tienen la misma finalidad que el reflejo desplazado del vestido negro de la camarera, esto es, organizar la pintura de tal manera que nos concentremos \u00fanicamente en la experiencia relevante de profundidad y de efecto de retroceso. En el rinc\u00f3n superior derecho del cuadro, reflejado en el espejo, vemos el hombre al que mira la camarera y que a su vez la mira intensamente a los ojos. Sin embargo, as\u00ed como la espalda de la camarera no podr\u00eda verse reflejada a su derecha, este decidido caballero de sombrero de copa que le formula una pregunta con la mirada y que provoca en ella esa expresi\u00f3n tan triste, no puede tener presencia \u00f3ptica, pues en tal caso impedir\u00eda por completo nuestra visi\u00f3n directa y sin obst\u00e1culos de Suzon, que mira directamente al frente. La pintura est\u00e1 organizada de tal manera que el espectador, usted o yo, est\u00e1 delante de ella. Pero, por supuesto, ni usted ni yo nos parecemos a la persona reflejada en el espejo. A causa de la posici\u00f3n completamente frontal del personaje en relaci\u00f3n con el espectador, es imposible mirar la figura sin que se produzca esta perplejidad reflexiva. El dramatismo creado por Manet en esta pintura es \u00e9ste: miro al espejo y veo a alguien que no soy yo.\n\nEste aspecto del cuadro llam\u00f3 la atenci\u00f3n de los contempor\u00e1neos de Manet. Alguno trat\u00f3 de superar la perplejidad con una broma (el _Journal Amusant_ del 27 de mayo de 1882 presentaba un grabado del cuadro en el cual se hab\u00eda introducido el caballero reflejado en el espejo, de pie ante la camarera y obstruyendo nuestra mirada), pero la mayor\u00eda de los cr\u00edticos reaccion\u00f3 con irritaci\u00f3n a las perturbadoras preguntas que el cuadro de Manet planteaba al espectador: \u00ab\u00bfEs ver\u00eddico este cuadro? No. \u00bfEs bello? No. \u00bfEs atractivo? No. Entonces, \u00bfqu\u00e9 es?\u00bb El malestar de los cr\u00edticos pod\u00eda tener mucho que ver con el relato que narra la propia pintura, el de las proposiciones de un hombre a una joven camarera que le responde con una mirada de infinita tristeza.\n\nEsta historia, por supuesto, se ajusta como anillo al dedo a un serm\u00f3n victoriano. Es cierto que Edgar Degas pint\u00f3 m\u00e1s directamente el mensaje moral de la joven solitaria en un ambiente p\u00fablico saturado de vicio, por ejemplo, en _La absenta_ , de 1876, pero en la pintura de Manet, la perturbaci\u00f3n \u00f3ptica exime a la mujer de servir a una finalidad tan claramente moralizadora. Una cuesti\u00f3n que surge en relaci\u00f3n con este cuadro es la de si la visi\u00f3n que del mismo puedan tener hombres y mujeres con otras vestimentas y de otras \u00e9pocas y lugares ser\u00e1 inseparable de la historia que en \u00e9l se expone. Con la misma t\u00e9cnica pict\u00f3rica se ha dado particular relieve a los objetos colocados sobre la barra. Las botellas est\u00e1n pintadas con todo detalle y contrastan con los discos abstractos de ese espejo que nos muestra una imagen de uno mismo distinta de aquella en la que habr\u00edamos preferido reconocernos. Aunque el espejo cruza la pintura de un extremo al otro, Manet s\u00f3lo permite ver el reflejo de dos de los densos conjuntos de objetos de esa abigarrada colecci\u00f3n, a pesar de que deber\u00eda verse el de _todos_. Estos fantasmas \u00f3pticos de botellas, flores y frutas parecen los objetos m\u00e1s s\u00f3lidos del cuadro.\n\nEs as\u00ed como opera el desplazamiento en _El bar del Folies-Berg\u00e8re_. El desplazamiento crea valor, valor reflexivo, lo que quiere decir valor que se da al espectador como inherente a lo que ve y al propio mundo f\u00edsico, cuya naturaleza y formas nos vemos forzados a evaluar contemplando su transmutaci\u00f3n en un espejo distorsionante. Por el contrario, hay en esos objetos una solidez, aunque meramente ilusoria, que no ha sido sometida a este desplazamiento. De haber sido Manet un fil\u00f3sofo \u2013a lo que se hubiera negado enf\u00e1ticamente\u2013 habr\u00eda podido se\u00f1alar que lo fundamental de su pintura era que la solidez de las cosas no desplazadas, al igual que la de los yos que no han tenido nunca la experiencia del desplazamiento, puede ser en realidad la mayor de las ilusiones. Esta pintura entra\u00f1a, por cierto, la promesa modernista de que la perturbaci\u00f3n infundir\u00e1 valor a la experiencia. Pero \u00bfc\u00f3mo podr\u00eda esta promesa de desplazamiento trasladarse de la tela a la calle?\n\n### _Un cambio en el exilio_\n\nSi fu\u00e9ramos capaces de recorrer las calles del Par\u00eds de la juventud de Manet \u2013las calles comprendidas entre la Rue de Rivoli y el Boulevard SaintMichel de norte a sur, y entre lo que son hoy los puentes de Saint-Michel y Carrousel de este a oeste\u2013 ver\u00edamos que el m\u00e9todo de pintura de Manet plasmaba en la tela una escena de la vida real.\n\nEsta secci\u00f3n de Par\u00eds conten\u00eda una multitud de extranjeros mezclados entre los estudiantes de las facultades de bellas artes, medicina y derecho de la Universidad de Par\u00eds. El contingente mayor y m\u00e1s antiguo estaba formado por centroeuropeos, polacos y bohemios que hab\u00edan sido desplazados de forma permanente de sus respectivos pa\u00edses en la d\u00e9cada de 1830. En la d\u00e9cada siguiente llegaron a esta zona de la ciudad los emigrados pol\u00edticos italianos, a los que en 1846 se uni\u00f3 un contingente de griegos. La mayor\u00eda estaba en Par\u00eds por razones pol\u00edticas; muchos de ellos eran intelectuales, aunque entre los griegos hab\u00eda un gran n\u00famero de marineros que hab\u00edan sido capturados una generaci\u00f3n antes, durante la guerra de Independencia.\n\nPodr\u00edamos concebir ese mundo como un mundo premoderno de extranjeros. Los parisinos idealizaron la resistencia que en otros lugares opon\u00edan los burgueses locales a la explotaci\u00f3n de la aristocracia y la realeza. Aunque los franceses no son en general particularmente accesibles para los forasteros, recibieron con simpat\u00eda a polacos y griegos; las revueltas en esos dos pa\u00edses se percib\u00edan m\u00e1s como revoluciones de la clase media que como revueltas de los pobres. Durante la d\u00e9cada de 1830, las universidades de Francia estuvieron abiertas a los extranjeros y se produjo la primera codificaci\u00f3n moderna del derecho de asilo pol\u00edtico (seg\u00fan la cual un individuo puede solicitar este estatus a trav\u00e9s de una tramitaci\u00f3n estatal establecida, sin tener que suplicarlo como favor a un gobernante). En estas condiciones, los emigrantes de las d\u00e9cadas de 1830 y 1840 trataron de movilizar a los parisinos a favor de sus diversas causas, con la esperanza de obtener dinero y presionar a la opini\u00f3n p\u00fablica para que impulsara a la acci\u00f3n al gobierno franc\u00e9s. Hoy se conoce el lado elegante de esos esfuerzos, como la m\u00fasica que Chopin escrib\u00eda a modo de _pi\u00e8ces d'occasion_ para conciertos de beneficencia, pero hab\u00eda tambi\u00e9n una adhesi\u00f3n m\u00e1s popular del p\u00fablico, como el proselitismo que los marineros griegos hac\u00edan entre los estibadores y los transportistas de los _quais_ del Sena en busca de ayuda, y con tal fortuna que en los muelles se usaba la ropa de trabajo griega como se\u00f1al de solidaridad. Adem\u00e1s, la polic\u00eda de Par\u00eds, en general, daba su aprobaci\u00f3n, pensando que los intereses extranjeros desviar\u00edan a los trabajadores franceses de sus motivos locales de descontento, desviaci\u00f3n que, con respecto al proletariado parisino, se hab\u00eda conseguido efectivamente durante las guerras napole\u00f3nicas.\n\nEra, pues, curiosa la situaci\u00f3n de esta naci\u00f3n xen\u00f3foba que encontraba atractivos a los extranjeros perseguidos, pero que era tambi\u00e9n un escenario de consecuencias hist\u00f3ricas, pues fue en Par\u00eds donde resultaron por primera vez patentes los cambios que producir\u00edan la imagen m\u00e1s moderna del extranjero como figura necesariamente sufriente. Esos cambios, parad\u00f3jicamente, se debieron al desarrollo del nacionalismo moderno; fue el nacionalismo lo que hizo que quienes abandonaban sus respectivas naciones parecieran pacientes sometidos a una amputaci\u00f3n quir\u00fargica.\n\nEs verdad, por supuesto, que a partir de los antiguos griegos se pens\u00f3 que la pertenencia a una naci\u00f3n era necesaria para dar forma a un ser humano completo; a los extranjeros de las ciudadesEstado griegas \u2013los metecos\u2013, los ciudadanos griegos los consideraban j\u00f3venes inconsistentes, porque no pod\u00edan ejercer el privilegio adulto del voto. Pero el significado de \u00abnaci\u00f3n\u00bb ha experimentado enormes variaciones en el curso de la historia occidental. Unas veces, la nacionalidad fue inseparable de una determinada pr\u00e1ctica religiosa, otras, se consustanci\u00f3 con dinast\u00edas aristocr\u00e1ticas, y aun otras incluy\u00f3 la red de socios comerciales de una ciudad matriz.\n\nEl nacionalismo que comienza a descubrir su voz en la Revoluci\u00f3n de 1848 distingue una nueva versi\u00f3n de la identidad colectiva en nuestra civilizaci\u00f3n, pues la nacionalidad se convierte entonces en un fen\u00f3meno antropol\u00f3gico del que la actividad pol\u00edtica es, en el mejor de los casos, simple sierva. La naci\u00f3n se convierte en un _ethos_ , en gobierno del _nomos_ , en t\u00e9rminos griegos, o, en otras palabras, en el puro gobierno de la costumbre; e interferir la sacralidad de la costumbre con la toma de decisiones pol\u00edticas o la negociaci\u00f3n diplom\u00e1tica es poco menos que un crimen. Debido a este gran cambio en el sentido de la nacionalidad, los exiliados que viv\u00edan en Par\u00eds en 1848 se vieron obligados a pensar de nuevo lo que significaba estar mucho tiempo desplazados de \u00abcasa\u00bb. Su vida cotidiana en el extranjero perdi\u00f3 poco a poco contacto con los rituales y las costumbres de su pa\u00eds y el _nomos_ pas\u00f3 a ser recuerdo en lugar de actividad. Se ver\u00edan forzados a buscar sentido a su vida en el hecho mismo del desplazamiento, en su condici\u00f3n de extranjeros, o sea, a contemplar los recuerdos de la naci\u00f3n como algo parecido al espejo de Manet.\n\nLa Revoluci\u00f3n de 1848 dur\u00f3 cuatro meses, de febrero a junio. Comenz\u00f3 en Par\u00eds, pero en marzo sus repercusiones se hab\u00edan de sentir en toda Europa Central, donde surg\u00edan movimientos que proclamaban la superioridad de las rep\u00fablicas nacionales sobre los parcelamientos del territorio realizados por las dinast\u00edas y los diplom\u00e1ticos en el Congreso de Viena de 1815. Los acontecimientos tuvieron algo de la misma \u00edndole explosiva que m\u00e1s tarde, en los \u00faltimos meses de 1989, caracterizar\u00eda la liberaci\u00f3n de la hegemon\u00eda rusa que se extendi\u00f3 por esas mismas naciones.\n\nLos doctrinarios de \u00abla naci\u00f3n\u00bb que empezaron a ejercer influencia p\u00fablica en 1848 empleaban otra clase de lenguaje que el de quienes, antes que ellos y en sus respectivos pa\u00edses, hab\u00edan defendido los reg\u00edmenes constitucionales, la democracia u otros ideales pol\u00edticos, haci\u00e9ndose eco de los ideales de la Revoluci\u00f3n Norteamericana y de la Revoluci\u00f3n Francesa. El lenguaje de los eslav\u00f3filos o de los Hijos del \u00c1tica era, como he dicho, un triunfo de la antropolog\u00eda sobre la pol\u00edtica. En 1848, los revolucionarios nacionalistas rechazaban la idea de naci\u00f3n como c\u00f3digo pol\u00edtico, pues lo que para ellos conformaba una naci\u00f3n eran la tradici\u00f3n, las formas de comportamiento y las actitudes morales de un _volk_ ; los elementos constitutivos de la vida nacional eran, por tanto, lo que un pueblo come, la manera en que se mueve cuando baila, las expresiones dialectales de su lenguaje, las formas de sus plegarias. El derecho es incapaz de legislar los placeres de ciertas comidas y las constituciones no pueden ordenar la fervorosa creencia en determinados santos; en resumen, el poder no puede crear la cultura.\n\nLa doctrina del nacionalismo que cristaliz\u00f3 en 1848 a\u00f1ade un imperativo geogr\u00e1fico al concepto de cultura propiamente dicho, pues el h\u00e1bito, la fe, el placer, el ritual, todo, depende de su representaci\u00f3n en un territorio determinado. Adem\u00e1s, el lugar que alimenta los rituales es un lugar formado por personas que se parecen a nosotros, personas con quienes podemos compartir sin explicar. De esta manera, \u00abterritorio\u00bb se convierte en sin\u00f3nimo de \u00abidentidad\u00bb.\n\nEs importante comprender que los anales de la Europa de mediados del siglo XIX estaban llenos de nacionalistas revolucionarios que predicaban a un p\u00fablico a veces receptivo, a veces indiferente, pero siempre un p\u00fablico que o\u00eda algo _nuevo_ , eso es, la alabanza de rituales y creencias normales y la celebraci\u00f3n de la vida cotidiana como virtud colectiva. Un c\u00f3digo m\u00e1s antiguo de honor nacional, por ejemplo, habr\u00eda considerado degradante esta celebraci\u00f3n de la vida cotidiana. En virtud de este c\u00f3digo m\u00e1s antiguo se vest\u00eda a un soldado de infanter\u00eda de franela azul y roja con adornos dorados, charreteras y botones ceremoniales estampados. No importaba que fuera una vestimenta in\u00fatil, o, peor a\u00fan, que lo fuera para el uso militar; no importaba que el soldado se muriera de hambre en los barracones; esta ropa ceremonial le daba un lugar en algo m\u00e1s grande e importante que \u00e9l, glorificaba su condici\u00f3n de franc\u00e9s. An\u00e1logamente, en tiempos de paz, monarcas como Luis XIV trataban de legitimar sus pol\u00edticas mediante sofisticados ceremoniales; esos \u00abavances\u00bb, \u00abgiros\u00bb y \u00abaudiencias\u00bb infund\u00edan un contraste dram\u00e1tico a la gloria del Estado, lo mismo que sus magn\u00edficos y altos edificios, tan elevados, y por eso tan \u00abantinaturales\u00bb, en relaci\u00f3n con el dominio de la vida cotidiana. El honor nacional hab\u00eda que buscarlo en el artificio.\n\nPor el contrario, la ideolog\u00eda de la naci\u00f3n que predicaban Kossuth, Manzoni, Garibaldi, Mickiewicz o Louis Blanc, seg\u00fan la cual la gente encontraba su honra personal en la vida ordinaria, comerciando, divirti\u00e9ndose, rezando o cosechando, significaba que el honor deb\u00eda buscarse en la autenticidad antes que en el artificio.\n\nEl esp\u00edritu de este nuevo nacionalismo se hace visible casi al mismo tiempo que se imprimen los textos revolucionarios, en febrero y marzo de 1848. En los carteles que Chodluz y otros realizaron en la primavera de 1848 convocando a la unidad nacional se muestra al Pueblo respondiendo a la llamada al alzamiento en ropa de trabajo o vestimenta campesina. El sentido de estas im\u00e1genes es m\u00e1s complejo que la simple identificaci\u00f3n del Pueblo con los pobres, pues en los carteles revolucionarios de 1790 y 1791 los pobres se representaban a menudo en uniformes militares o con los colores de sus clubs pol\u00edticos. Dos generaciones m\u00e1s tarde, en respuesta a un gran acontecimiento hist\u00f3rico, el Pueblo no viste para la ocasi\u00f3n. Tampoco en los carteles de 1848 las masas exhiben expresiones particularmente dram\u00e1ticas de odio o de exaltaci\u00f3n patri\u00f3tica; todo se hace con el prop\u00f3sito de dar a entender que el pueblo no es consciente de s\u00ed mismo, que simplemente es lo que es. Las cl\u00e1sicas figuras aleg\u00f3ricas que blasonaban los carteles de las revoluciones de 1830, como _La Libertad guiando al pueblo_ , de Delacroix, son ya realmente cosas del pasado. Para los nacionalistas revolucionarios de 1848, la falta de conciencia de s\u00ed mismo del _volk_ , la ausencia de un espejo, era una fuente de virtud que se opon\u00eda a los vicios de la autoconciencia y la autoalienaci\u00f3n del burgu\u00e9s cosmopolita, cuya actitud mental descansaba en un juego de espejos que reflejaban interminables vacilaciones y reconsideraciones.\n\nEsta imagen antropol\u00f3gica de un _volk_ es un acontecimiento que marca una \u00e9poca en la iconograf\u00eda y la ret\u00f3rica sociales modernas. El nacionalismo del siglo XIX estableci\u00f3 lo que hoy podr\u00edamos llamar regla b\u00e1sica de la modernidad para tener una identidad. La identidad m\u00e1s fuerte es la que se tiene sin conciencia de tenerla, la que corresponde simplemente a lo que se _es_. En otras palabras, uno es m\u00e1s uno mismo cuanto menos conciencia tiene de serlo.\n\nEs importante entender que esta f\u00f3rmula es en efecto una regla para el ejercicio del poder, aun cuando hable en nombre de una unidad cultural, de un alma del pueblo, con independencia de cualquier r\u00e9gimen pol\u00edtico. Los grandes imperialistas del siglo XIX, hombres como Livingstone, Stanley y Rhodes, se adhirieron a esta visi\u00f3n antropol\u00f3gica y apoyaron tambi\u00e9n el extendido punto de vista acerca del car\u00e1cter sagrado de la cultura cotidiana y creyeron en la primac\u00eda del _nomos_ , s\u00f3lo que, para ellos, la conclusi\u00f3n derivada de todo eso era el principio de no contaminar a los \u00abnativos\u00bb por un contacto excesivo con los amos extranjeros, porque eso habr\u00eda debilitado la integridad de la cultura nativa. Rhodes hablaba en serio. Lo que ocurre es simplemente que esta regla b\u00e1sica del _nomos_ moderno, la que sostiene que uno es m\u00e1s uno mismo cuanto menos conciencia tiene de serlo, puede servir tanto al levantamiento revolucionario, como en 1848, como a la orquestaci\u00f3n de formas en que una naci\u00f3n que domina a otras trata, sin embargo, de impedir la \u00abcontaminaci\u00f3n\u00bb cultural. Del mismo modo, tambi\u00e9n un Estado moderno puede sacar provecho de la virtud antropol\u00f3gica. Sus instituciones pueden legitimarse como reflejos del impulso popular m\u00e1s que como construcciones, que podr\u00edan ser problem\u00e1ticas y requerir an\u00e1lisis constantes. Instituciones como la polic\u00eda civil o el comit\u00e9 revolucionario de vecinos pueden ser declarados \u00f3rganos permanentes de espontaneidad, las consecuencias de lo que \u00abtodo el mundo\u00bb desea ver brotar de la vida del pueblo.\n\nLa celebraci\u00f3n que Rousseau hizo del \u00abbuen salvaje\u00bb un siglo antes de 1848 era un amargo juego de palabras. Parece ser que Rousseau qued\u00f3 muy impresionado por la figura disecada de un indio americano que un taxidermista exhibi\u00f3 en Par\u00eds en 1741 con toda su vestimenta ceremonial. Rousseau imagin\u00f3 que este \u00absalvaje\u00bb era un hombre con una capacidad de reflexi\u00f3n m\u00e1s aguda y profunda que los empelucados, charlatanes e irreflexivos parisinos que visitaban la tienda del taxidermista. El buen salvaje, idealizaci\u00f3n dieciochesca de la persona aut\u00e9ntica, piensa. Una vez m\u00e1s, la distinci\u00f3n entre los revolucionarios del siglo XVIII y los nacionalistas del XIX ven\u00eda marcada por una diferencia en la conciencia geogr\u00e1fica. Las doctrinas pol\u00edticas de 1789 trascend\u00edan lo local. Efectivamente, para creer en la libertad, la igualdad y la fraternidad que proclamaba la Revoluci\u00f3n Francesa no hac\u00eda falta vivir en Par\u00eds ni ser franc\u00e9s. En sus _Reflexiones de un ciudadano universal del mundo_ , escritas en 1784, Kant sosten\u00eda que un ser humano desarrolla en casa la mayor parte de lo que siente una persona y extrae est\u00edmulo de una diversidad de personas distintas. Este \u00abciudadano universal\u00bb busca el est\u00edmulo en escenarios extra\u00f1os a \u00e9l y de todos ellos recoge lo que hay de com\u00fan, de universal.\n\nPor cierto que, en materia de ideolog\u00eda, ning\u00fan cambio se produce como un simple paso de una forma de creencia a otra. En los escritos de Manzoni sobre el campesinado italiano, sus campesinos aparecen a veces como los aut\u00e9nticos italianos porque, apartados de las ciudades que fueron sede del poder austroh\u00fangaro, conservaron los modos de vida de la Italia anterior, libre. Al ser conscientes guardianes de lo que es en realidad una cultura superior, se asemejan en esto al buen salvaje de Rousseau. Entonces el campesinado \u2013a veces Manzoni escribe como m\u00e1s tarde lo har\u00e1 Tolst\u00f3i\u2013 es moralmente superior porque los campesinos no tienen conciencia de s\u00ed mismos en el tiempo y la historia y est\u00e1n libres del corrosivo veneno del exceso de reflexi\u00f3n, de pensar m\u00e1s all\u00e1 de los confines de la vida tal como se da. El campesino no mira al espejo de la historia; el campesino es, sin m\u00e1s. El Pueblo es silencioso.\n\nEn la ret\u00f3rica del nacionalismo que tom\u00f3 forma en el siglo XIX, la espontaneidad y la carencia de autoconciencia cosmopolita de la gente iban unidas a su vez a una concepci\u00f3n del tiempo nacional. Tambi\u00e9n la naci\u00f3n _es_ , sin m\u00e1s. Para la ret\u00f3rica del nacionalismo, los rituales, las creencias y la moral de un pueblo representan m\u00e1s bien formas de ser que de hacer, de acuerdo con la distinci\u00f3n de Heidegger; los rituales, las creencias y la moral que crean el ideal nacional se celebran como probados por el tiempo y como factores permanentes de cohesi\u00f3n, pues pertenecen a la tierra misma, a la unidad de los seres humanos con \u00absu\u00bb suelo. Esta noci\u00f3n de ser nacional tambi\u00e9n entra\u00f1a cierto tipo de silencio. En los llamamientos de Luis Kossuth a la revuelta magiar, la interacci\u00f3n multisecular de los magiares con los turcos, eslavos y alemanes cuya historia los hab\u00eda llevado a vivir entre ellos quedaba excluida de la explicaci\u00f3n de aquello en que consiste ser magiar, pese a que, de hecho, estos encuentros hist\u00f3ricos ti\u00f1eron la pr\u00e1ctica de la religi\u00f3n, crearon una cocina compleja y alteraron la estructura de la propia lengua h\u00fangara. En lugar de esta historia, Kossuth predic\u00f3 una versi\u00f3n de la cultura magiar como si de generaci\u00f3n en generaci\u00f3n hubiera permanecido al mismo tiempo inmutable e independiente. El corolario del tiempo nacional, que es tiempo del ser, era el concepto de pureza nacional.\n\nComo Isaiah Berlin ha mostrado en su estudio _Vico and Herder_ , los dos precursores dieciochescos del nacionalismo del siglo XIX, el encuadre de la naci\u00f3n en t\u00e9rminos antropol\u00f3gicos, empez\u00f3 por la m\u00e1s liberal de las razones, la afirmaci\u00f3n de la dignidad de las diferencias humanas. Para Herder, en palabras de Berlin: \u00ablas diferencias [del pueblo] son lo m\u00e1s importante, pues son ellas las que hacen de \u00e9l lo que es, lo que le da existencia por s\u00ed mismo\u00bb. Es f\u00e1cil olvidar hasta qu\u00e9 punto es osada y reciente la simple afirmaci\u00f3n de que los seres humanos son productos de culturas particulares. Maquiavelo aconsejaba al o\u00eddo de su Pr\u00edncipe poniendo ejemplos de emperadores y reyes antiguos, gobernantes que, aunque muertos hac\u00eda miles de a\u00f1os, pod\u00edan a\u00fan servir como modelos, porque la naturaleza humana no cambia, o eso era al menos lo que pensaban Maquiavelo y sus contempor\u00e1neos.\n\nEn el siglo XVIII, la afirmaci\u00f3n de que los seres humanos encarnan una cultura espec\u00edfica era algo m\u00e1s que un recurso para tomar en serio las variaciones antropol\u00f3gicas, era un ataque a lo que hoy llamamos \u00abeurocentrismo\u00bb. Voltaire cre\u00eda que \u00abes una terrible arrogancia afirmar que, para ser feliz, todo el mundo deber\u00eda hacerse europeo\u00bb. En diferentes lugares, diferentes pueblos encuentran diferentes maneras de buscar la felicidad, el m\u00e1s dif\u00edcil de los logros. Sin embargo, entre la afirmaci\u00f3n de la diferencia en el siglo XVIII y la misma afirmaci\u00f3n en el XIX hay una brecha que puede exponerse as\u00ed: para Voltaire, el conocimiento de que hay otros que no s\u00f3lo no mueren por ingerir comidas que a nosotros nos dan miedo, sino que incluso las saborean con placer, deber\u00eda hacernos reflexionar acerca de nuestras convicciones, deber\u00eda despertar en nosotros el deseo de probar lo prohibido. La percepci\u00f3n de valores diferentes deber\u00eda volver m\u00e1s cosmopolita al sujeto que los percibe. Herder, sin embargo, se adelant\u00f3 a su tiempo al comprender que la percepci\u00f3n de la diferencia, debido a la ausencia de humanidad com\u00fan a la que recurrir conjuntamente, puede volver m\u00e1s etnoc\u00e9ntrica a la gente.\n\nEn el lugar y fuera del lugar; la virtud de ser uno mismo en el lugar propio y el defecto de verse a s\u00ed mismo en otro lugar. Es precisamente aqu\u00ed donde empiezan los problemas de ser extranjero. En la temprana primavera de 1848, a \u00abDaniel Stern\u00bb (seud\u00f3nimo con el que firmaba Marie d'Agoult, en un tiempo compa\u00f1era de Franz Liszt, cuyas cr\u00f3nicas de 1848 son un vivo registro del levantamiento) le parec\u00eda que \u00abla colonia extranjera se vaciar\u00e1 en unos d\u00edas, cuando nuestros amigos regresen a los lugares que los reclaman\u00bb. Dado el triunfalismo con que se anunciaban los nacionalismos en la prensa, esa expectativa parece l\u00f3gica. El interrogante pol\u00edtico que este nacionalismo planteaba a los que se hab\u00edan vuelto extranjeros \u2013emigrados, expatriados o exiliados\u2013 era el siguiente: \u00bfpor qu\u00e9 no est\u00e1is en casa con los vuestros? \u00bfC\u00f3mo pod\u00e9is ser rusos en otra parte? Sin embargo, a finales de abril de 1848, Daniel Stern hab\u00eda observado que, curiosamente, pocos emigrados hab\u00edan regresado a su casa. \u00abA\u00fan se los ve discutiendo en el Palais Royal, recibiendo emisarios del exterior, lanzando bravatas; rebosan de esperanza, pero ninguno ha hecho las maletas.\u00bb\n\nTal vez el m\u00e1s grande de los exiliados del siglo XIX haya sido un hombre que, pese a su breve aparici\u00f3n en este escenario, dej\u00f3 expresada en prosa indeleble su percepci\u00f3n de la maldita relaci\u00f3n entre el nacionalismo y la condici\u00f3n de extranjero. Aleksandr Herzen era hijo ileg\u00edtimo de un noble ruso ya entrado en a\u00f1os y una joven alemana (de ah\u00ed su nombre, m\u00e1s o menos equivalente a _herzlich_ , \u00abde coraz\u00f3n\u00bb). Inspirado por la sublevaci\u00f3n de 1825, fue un joven activo en la pol\u00edtica radical rusa, tal como esto se entend\u00eda entonces, es decir que fue un propulsor de la monarqu\u00eda constitucional y las reformas liberales. Por esta causa sufri\u00f3 el exilio interno y finalmente la expulsi\u00f3n del imperio ruso. Al igual que otros miembros de su generaci\u00f3n, pens\u00f3 en un primer momento que era un exiliado temporal y esperaba regresar a su tierra natal cuando las circunstancias pol\u00edticas lo hicieran posible. Pero cuando, finalmente, esta posibilidad se dio, se ech\u00f3 atr\u00e1s. No fueron la asimilaci\u00f3n social, el amor a la cultura europea ni v\u00ednculos personales \u2013como los de su amigo Turgu\u00e9niev con Pauline Viardot\u2013 lo que le retuvo. Manten\u00eda su apasionado inter\u00e9s por los asuntos de su pa\u00eds, pero ya no se sent\u00eda capaz de vivir en \u00e9l. Deambul\u00f3 por las capitales de Europa Occidental y pas\u00f3 sus \u00faltimos a\u00f1os en Londres, donde public\u00f3 un famoso bolet\u00edn informativo sobre la realidad rusa con el t\u00edtulo de _The Bell_.\n\nHay cierto tipo de pensamiento social, falsamente humanitario, que propone una relaci\u00f3n inversa entre conciencia y circunstancia. Seg\u00fan esta manera de pensar, los padecimientos convierten a los pobres en v\u00edctimas intelectuales de sus necesidades y su pensamiento es mero c\u00e1lculo de supervivencia, mientras que las sutilezas de la conciencia, las complejidades de la interpretaci\u00f3n, son lujos de rico. Para este tipo de pensamiento, el hijo bastardo de un arist\u00f3crata no puede servir como gu\u00eda de los dilemas que afrontaban las sucesivas oleadas de emigrantes que abandonar\u00edan Europa en el siglo XIX, y mucho menos a\u00fan de los interrogantes que afrontan hoy los jornaleros mexicanos, los comerciantes coreanos, los jud\u00edos sovi\u00e9ticos y otros extranjeros. Herzen el amigo de John Stuart Mill; Herzen el t\u00edmido con esa timidez alimentada por la asistencia a tantos acontecimientos formales; Herzen el curioso de lugares a los que, no obstante, sabe que no pertenece...; este polifac\u00e9tico Herzen hace su entrada en esta historia en abril de 1848. Fue en ese momento de dilaci\u00f3n cuando Herzen se uni\u00f3 a la colonia de emigrados de Par\u00eds; lo hizo para alejarse de Roma, que se hallaba entonces en sus primeros momentos de despertar nacionalista.\n\nNo hay que pensar que Herzen o los otros emigrados parisinos que no respondieron de inmediato al llamamiento de su propia naci\u00f3n fueran cobardes; la vida de muchos emigrados estaba marcada por una larga serie de encarcelamientos y torturas, en particular a manos de la polic\u00eda austr\u00edaca. En parte, la explicaci\u00f3n de su inmovilismo debe buscarse en un mal conocido, el de quedar al margen de los acontecimientos. Su red de contactos mutuos en el extranjero estaba anticuada, as\u00ed como lo estaban sus planes pol\u00edticos, pues las constituciones y las agencias gubernamentales no ten\u00edan cabida en la nueva ret\u00f3rica del Pueblo. Pero, como observ\u00f3 Daniel Stern, algo m\u00e1s que eso hab\u00eda sucedido a los extranjeros en el exilio. \u00abEs como si se hubieran mirado al espejo y hubiesen visto otro rostro que el que esperaban ver\u00bb, dijo. Esto desconcert\u00f3 a los emigrantes y a la propia Stern; algo hab\u00eda en ellos que se resist\u00eda a regresar, algo los reten\u00eda.\n\nLa imagen de Daniel Stern se asemeja asombrosamente al espejo de Manet en la gran diferencia entre lo reflejado y lo que esper\u00e1bamos ver. Esta conexi\u00f3n es exactamente lo que Herzen recoger\u00eda en el curso de su vida en su af\u00e1n por entender de qu\u00e9 manera el nacionalismo hab\u00eda forzado a la gente a mirar algo semejante al espejo de Manet para encontrar una imagen llevadera de s\u00ed misma. En ese espejo de desplazamiento, espejo deformante, el ritual, la creencia, el h\u00e1bito y los signos del lenguaje se mostrar\u00edan completamente distintos que en el pa\u00eds de origen. Es posible, por cierto, que el extranjero tenga con su cultura una relaci\u00f3n m\u00e1s inteligente, m\u00e1s comprensiva, que la persona que nunca ha salido de ella y no conoce nada m\u00e1s que lo que tiene delante, que no se ha visto obligada a sopesar las diferencias entre una cultura y otra. Pero no es esto lo que apremia a quien se convierte en extranjero; lo m\u00e1s acuciante es la necesidad de manejar creativamente la propia condici\u00f3n de desplazado, tratar los materiales de identidad de la misma manera en que un artista trata los objetos inertes que constituyen el tema de su pintura. Uno tiene que hacerse a s\u00ed mismo.\n\nEsto era posiblemente lo que Herzen sent\u00eda cuando le\u00eda las informaciones de prensa sobre una ola de violencia en Eslovaquia contra los gitanos, que hab\u00edan entrado en gran n\u00famero en el pa\u00eds, creyendo en 1848, como en 1990, que una naci\u00f3n que se levanta contra sus amos tambi\u00e9n puede prometerles libertad; cuando beb\u00eda vino en el Caf\u00e9 Lamblin, que un siglo y medio despu\u00e9s sigue sirviendo el mismo brebaje adulterado que tanto le disgustaba; cuando sus compa\u00f1eros confabulaban, telegrafiaban, discut\u00edan y no se marchaban, o cuando encabez\u00f3, en un mes de marzo, un contingente de extranjeros hasta la Asamblea Nacional en apoyo a los \u00abderechos del hombre\u00bb. Tendr\u00edan que encontrar una nueva manera de ser rusos.\n\n### _La segunda herida_\n\nLa circunstancia en torno a la cual gira todo el relato en la leyenda de Edipo Rey no parece contener demasiado inter\u00e9s art\u00edstico en s\u00ed misma, sino ser \u00fanicamente un eslab\u00f3n en la maquinaria de la intriga. Hab\u00eda en los tobillos del rey una marca, consecuencia de una herida de la infancia. El nombre Edipo significa en griego \u00abque tiene los tobillos perforados\u00bb. El rey hab\u00eda vagado sin rumbo y hab\u00eda perdido contacto con sus or\u00edgenes; cuando los personajes de la leyenda llegan al momento en que deben conocer la verdadera identidad del rey, son capaces de recuperar esta verdad observando su cuerpo. El proceso de identificaci\u00f3n comienza cuando un mensajero declara: \u00abLas articulaciones de tus pies te lo pueden atestiguar.\u00bb\n\nSi la prueba que busca el rey Edipo no fuera la del incesto, prestar\u00edamos m\u00e1s atenci\u00f3n a esta cicatriz. Pese a las grandes migraciones de su vida, su cuerpo contiene una evidencia permanente de qui\u00e9n es \u00abrealmente\u00bb. Los viajes del rey no han dejado en su cuerpo marcas comparables. Esto quiere decir que su experiencia migratoria cuenta poco en relaci\u00f3n con su origen.\n\nEsta cicatriz de Edipo parece ser en la cultura occidental una fuente de las marcas indelebles que el siglo XIX leer\u00eda luego en el cuerpo colectivo de la naci\u00f3n. El origen se convierte en destino. Al mirar hacia atr\u00e1s a los inicios de nuestra civilizaci\u00f3n, parecer\u00eda que el exilio, el desposeimiento, la migraci\u00f3n hubiesen sido mucho menos importantes que las marcas de los or\u00edgenes y de la pertenencia. Se podr\u00eda interpretar la negativa de S\u00f3crates a exiliarse como una demostraci\u00f3n de la creencia de que incluso la muerte como ciudadano era m\u00e1s honorable que el exilio; o entender que el sentido de la observaci\u00f3n de Tuc\u00eddides de que los extranjeros carecen de lenguaje no es literalmente el de que no sepan hablar bien, sino el de que su lenguaje cuenta poco en la _polis_ , que s\u00f3lo es la charla intrascendente de quienes no pueden votar.\n\nSin embargo, las marcas en los tobillos de Edipo no eran las \u00fanicas marcas que hab\u00eda en su cuerpo. A las heridas que otros le infirieron en un principio, responder\u00e1 arranc\u00e1ndose los ojos. Si prescindimos de toda la carga sexual de esta leyenda y la analizamos simplemente como un relato, la segunda herida equilibra la primera, pues \u00e9sta marca sus or\u00edgenes, mientras que aqu\u00e9lla marca su historia posterior. Herido dos veces, se ha convertido en un hombre cuya vida puede leerse literalmente en su cuerpo y en esas condiciones se lanza al mundo como un errante. Edipo piensa que si abandona Tebas tal vez pueda regresar a sus or\u00edgenes, a la monta\u00f1a, a \u00abese Citeron que es llamado m\u00edo, el que mi padre y mi madre, en vida, dispusieron que fuera leg\u00edtimamente sepultura para m\u00ed\u00bb, aunque este retorno no se producir\u00e1. En efecto, cuando comienza _Edipo en Colono_ , ha llegado en cambio al _deme_ (\u00abaldea\u00bb) de Colono, a mil seiscientos metros al noroeste de Atenas, donde el or\u00e1culo de Delfos le ha dicho que morir\u00e1; la profec\u00eda se cumplir\u00e1 de un modo muy diferente del que Edipo hab\u00eda imaginado al inicio de la obra. Las dos heridas del cuerpo de Edipo son, pues, una inocultable cicatriz del origen y las cicatrices del errante que no parecen curarse.\n\nEsta segunda herida sin curar significa en la civilizaci\u00f3n occidental tanto como la cicatriz de origen, que marca el valor que se atribuye a la pertenencia a un lugar particular. Los propios griegos hab\u00edan visto en el viaje inacabado de Edipo un eco de las leyendas hom\u00e9ricas, sobre todo la de Odiseo. En la pr\u00e1ctica griega \u2013m\u00e1s tarde codificada en el derecho romano\u2013, hab\u00eda ciertas circunstancias en las que el exilio extranjero era en realidad honroso, m\u00e1s honroso que el camino elegido por S\u00f3crates, pues el _exsilium_ daba a una persona convicta de una acusaci\u00f3n capital la posibilidad de escoger el exilio en lugar de la muerte, elecci\u00f3n que ahorraba a sus amigos y parientes la verg\u00fcenza y el dolor de presenciar su ejecuci\u00f3n. Pero en _Edipo en Colono_ , al describir a Edipo como una figura que ha sido ennoblecida por su desarraigo, S\u00f3focles introduce una dimensi\u00f3n moral al simple acto de migrar. La obra convierte a Edipo en el _meteco_ , el extranjero, una figura de grandeza tr\u00e1gica antes que un forastero de rango inferior al ciudadano.\n\nEl hecho de hacerse extranjero implica un desplazamiento respecto de las propias ra\u00edces. Este desarraigo tiene un valor moral positivo, esencial a la tradici\u00f3n judeocristiana. El pueblo del Antiguo Testamento se conceb\u00eda como gente errante, desarraigada. El propio Yahv\u00e9 del Antiguo Testamento era un Dios errante, su Arca de la Alianza era transportable y, para decirlo en los t\u00e9rminos teol\u00f3gicos de Harvey Cox: \u00abCuando, finalmente, el Arca fue capturada por los filisteos, los hebreos comenzaron a darse cuenta de que ni siquiera en ella ten\u00eda Yahv\u00e9 una localizaci\u00f3n [...] Yahv\u00e9 viajaba con su pueblo a donde fuese.\u00bb Era m\u00e1s un dios del tiempo que del lugar, un dios que promet\u00eda a sus seguidores un significado divino para sus malhadados viajes.\n\nEl permanente desplazamiento de un lugar a otro y la exposici\u00f3n eran sentimientos de consecuencias tan poderosas para la fe entre los primeros cristianos como lo hab\u00edan sido para los jud\u00edos del Antiguo Testamento. El autor de la _Ep\u00edstola a Diogneto_ , en el momento de mayor gloria del imperio romano, declaraba lo siguiente:\n\nLos cristianos, en efecto, no se distinguen de los dem\u00e1s hombres ni por su tierra ni por su habla ni por sus costumbres. Porque ni habitan ciudades exclusivas suyas [...] ni llevan un g\u00e9nero de vida aparte de los dem\u00e1s [...] habitan sus propias patrias, pero como forasteros [...] toda tierra extra\u00f1a es para ellos patria, y toda patria, tierra extra\u00f1a.\n\nEsta imagen del errabundo lleg\u00f3 a ser una de las maneras en que San Agust\u00edn defin\u00eda las dos ciudades en _La ciudad de Dios_ :\n\nHoy se recuerda de Ca\u00edn que fund\u00f3 una ciudad; pero Abel, que s\u00f3lo fue un peregrino, no fund\u00f3 ninguna, porque la ciudad de los santos es soberana y celestial, aunque produzca en la tierra los ciudadanos, en los cuales yerra como en un peregrinaje por el tiempo hasta que llegue el Reino de la eternidad.\n\nEsta preferencia del \u00abperegrinaje por el tiempo\u00bb al establecimiento en un lugar extrae su autoridad de la negativa de Jes\u00fas a permitir a Sus disc\u00edpulos construir monumentos en Su honor, y de Su promesa de destruir el Templo de Jerusal\u00e9n. As\u00ed pues, la cultura judeocristiana gira, en sus mismas fuentes, en torno a experiencias de desplazamiento. La nuestra es una cultura religiosa de la segunda cicatriz.\n\nLa raz\u00f3n de la atribuci\u00f3n de este valor al desarraigo proviene de una profunda desconfianza respecto de la antropolog\u00eda de la vida cotidiana; _nomos_ no equivale a \u00abverdad\u00bb. Las cosas ordinarias, en s\u00ed mismas, son ilusorias, tan ilusorias para los \u00f3rficos y Plat\u00f3n como para San Agust\u00edn. Esta devaluaci\u00f3n de la conducta cotidiana aparece tambi\u00e9n en ese memorable momento de _Edipo en Colono_ en el que Edipo habla al joven Teseo:\n\n\u00a1Oh querid\u00edsimo hijo de Egeo! La vejez y la muerte a su tiempo s\u00f3lo a los dioses no alcanzan. El tiempo, que todo lo puede, arrasa todas las dem\u00e1s cosas. Se consume el vigor de la tierra, se consume el del cuerpo, perece la confianza, se origina la desconfianza y no permanece el mismo esp\u00edritu ni entre los hombres amigos ni entre una ciudad y otra.\n\nPara unos, pronto, para otros, m\u00e1s tarde, los placeres se vuelven amargos y, posteriormente, dulces. Asimismo, si a Tebas por ahora le van bien sus relaciones contigo, el tiempo incalculable en su curso engendra d\u00edas y noches sin cuento durante los cuales se pueden romper por la lanza, con un peque\u00f1o motivo, los amistosos acuerdos de hoy.\n\nEsta segunda cicatriz, la marca del extranjero, es, por tanto, un estigma moral precisamente porque no se cura. Tanto en el pensamiento cl\u00e1sico como en el judeocristiano, quienes hab\u00edan roto las cadenas de las circunstancias, quienes viv\u00edan una vida desarraigada, pod\u00edan devenir seres humanos consecuentes. Vagando por el mundo se transformaban a s\u00ed mismos. Se liberaban de la participaci\u00f3n ciega y de esa manera les era posible indagar a fondo, elegir por s\u00ed mismos o, como el ciego rey griego y el m\u00e1rtir cristiano, sentirse finalmente en presencia de un Poder Superior. Las dos cicatrices en el cuerpo del rey Edipo representan un conflicto fundamental en nuestra civilizaci\u00f3n entre, por un lado, los llamamientos de la verdad del lugar y los comienzos y, por otro lado, las verdades que se descubren al convertirse en extranjero.\n\n\u00c9ste es el contexto en el que deber\u00edan encuadrarse las pasiones del nacionalismo moderno, con su \u00e9nfasis en el hecho de compartir entre personas semejantes la dignidad de la vida cotidiana, el valor de la identidad. Estas pasiones abogan por la comunidad a expensas de la autotransformaci\u00f3n. Desde las leyendas hom\u00e9ricas, a trav\u00e9s de la tragedia, y desde los profetas del Antiguo Testamento, a trav\u00e9s de los primeros prelados cristianos, ha habido una pasi\u00f3n antit\u00e9tica por las experiencias autotransformadoras a expensas de la comunidad, una pasi\u00f3n por el desplazamiento.\n\n### _El espejo de Herzen_\n\nEl 27 de junio de 1848, la Revoluci\u00f3n tocaba a su fin en Par\u00eds. Las tropas barrieron la ciudad disparando indiscriminadamente a la multitud, desplegando ca\u00f1ones que dispararon al azar contra barrios de clase obrera. Hab\u00edan llegado las fuerzas del orden. Herzen, al igual que los otros extranjeros que hab\u00edan permanecido en Par\u00eds por decisi\u00f3n personal, fue obligado a abandonar la ciudad; fue a Ginebra, luego otra vez a Italia, despu\u00e9s nuevamente a Francia, para llegar finalmente a Londres en agosto de 1852. Era entonces un hombre de edad mediana achacoso, cuya mujer hab\u00eda quedado er\u00f3ticamente comprometida en otro lugar, un hombre que se hab\u00eda pronunciado p\u00fablicamente contra los eslav\u00f3filos que dominaban el discurso radical en su pa\u00eds y que hablaba un ingl\u00e9s titubeante y en el estilo de sus lecturas de Sir Walter Scott. \u00ab[P]oco a poco comenc\u00e9 a darme cuenta de que no ten\u00eda absolutamente ning\u00fan lugar adonde ir ni ning\u00fan motivo para ir a ninguna parte.\u00bb No es exagerar su sufrimiento decir que en ese momento Herzen se convirti\u00f3 en algo as\u00ed como una figura tr\u00e1gica, un hombre que sent\u00eda el dolor de la segunda marca de la falta de hogar, que no curar\u00eda.\n\nLo instructivo de los escritos de Herzen es su manera de dar sentido a la conducta cotidiana en tales condiciones, de dar sentido a la condici\u00f3n de extranjero. \u00abGradualmente, una revoluci\u00f3n se fue produciendo en m\u00ed.\u00bb En parte, empez\u00f3 a convertir en virtud su aislamiento en el exilio: \u00abEra consciente de un poder en m\u00ed mismo [...] Me hice m\u00e1s independiente de los dem\u00e1s.\u00bb Y as\u00ed comenz\u00f3 a reconstruir su modo de ver su mundo circundante: \u00abSe hab\u00eda acabado el baile de disfraces, todos se hab\u00edan quitado las capas con capuchas, las guirnaldas hab\u00edan ca\u00eddo de las cabezas y las m\u00e1scaras hab\u00edan dejado los rostros al descubierto.\u00bb Para explicar las consecuencias de esta nueva visi\u00f3n de los otros en esta crisis personal del exilio, Herzen recurri\u00f3 a la misma iconograf\u00eda de visi\u00f3n distorsionada que hab\u00eda evocado Daniel Stern. \u00abVe\u00eda rasgos distintos de los que hab\u00eda supuesto.\u00bb\n\nAntes que ver en el exilio un motivo para la trascendencia espiritual del mundo, como habr\u00eda hecho un cristiano, Herzen se mantuvo con los pies en la tierra y trat\u00f3 de entender qu\u00e9 deb\u00eda hacer un extranjero con su propia nacionalidad. La naci\u00f3n, para una persona que se hab\u00eda convertido en extranjera, entra\u00f1aba dos peligros, el del olvido y el del recuerdo. El primero era una condici\u00f3n en la que el extranjero se sent\u00eda humillado por el deseo de asimilarse; el otro, una condici\u00f3n en la que se sent\u00eda destrozado por la nostalgia.\n\nEn la d\u00e9cada de 1850, Herzen lleg\u00f3 a tener experiencia personal de estos peligros, ejemplificados por dos hombres que resurgieron de su pasado en la d\u00e9cada de 1830 y comienzos de la de 1840. Iv\u00e1n Golovin fue, al igual que Herzen, un refugiado pol\u00edtico de aquellos a\u00f1os, pero al principio le hab\u00eda parecido un individuo despreciable, un bandido de poca monta, expulsado de la bolsa de valores de Par\u00eds tras unos pocos a\u00f1os de trabajo, un explotador de sus correligionarios exiliados, un hombre que viv\u00eda a salto de mata. Herzen ve\u00eda sus defectos personales magnificados por su conducta en el exilio: \u00ab\u00bfPara qu\u00e9 se fue de Rusia? \u00bfQu\u00e9 hace en Europa? [...] Desarraigado de su suelo natal, no pod\u00eda encontrar un centro de gravedad.\u00bb La importancia del car\u00e1cter de Golovin fue magnificado en las reflexiones de Herzen en Londres. \u00abEl car\u00e1cter de Golovin\u00bb, escribi\u00f3 entonces, \u00ablleva el sello de toda una clase de personas\u00bb, aquellas cuyo verdadero deseo de asimilarse las ha llevado a la p\u00e9rdida de s\u00ed mismas, aquellas\n\nque llevan vidas n\u00f3madas, con o sin papeles de identidad, en peque\u00f1os espacios o en grandes ciudades, que comen siempre bien, que todo el mundo conoce y acerca de las cuales se sabe todo, salvo dos cosas, de qu\u00e9 viven y para qu\u00e9 viven. Golovin era un oficial ruso, un petimetre franc\u00e9s, un estafador ingl\u00e9s o un _Junker_ alem\u00e1n, como nuestro compatriota Nozdrev Jlestakov [personaje de G\u00f3gol].\n\nEn el extranjero, estas personas ven que sus nuevos compatriotas no pueden entender c\u00f3mo es el lugar de donde ellas proceden, o bien, lo que es comprensible, no les interesa, puesto que es todo tan lejano en el espacio y en el tiempo; en una palabra, tan extra\u00f1o. As\u00ed, hombres como Golovin, por temor a alejar o aburrir a los dem\u00e1s, act\u00faan como si nada de eso hubiese existido nunca.\n\nHerzen era demasiado civilizado para considerar que esos extranjeros que trataban de asimilarse fueran por necesidad moralmente corruptos. M\u00e1s bien los ve\u00eda como gente que se hab\u00eda embarcado en una especie de amnesia voluntaria, y \u00e9l tem\u00eda que, a partir de esa voluntad de olvido, derivaran otros actos de negaci\u00f3n. En la pintura que el extranjero hace de su vida hay grandes zonas repintadas de blanco.\n\nTal vez, la idea que Herzen se hac\u00eda pensando en Golovin pudiera reformularse de la siguiente manera: el deseo de asimilaci\u00f3n puede vivirse como una fuerza que da lugar a un sentimiento de verg\u00fcenza de uno mismo y que, por tanto, debilita el yo. Por supuesto, la capacidad de asimilaci\u00f3n requiere recursos econ\u00f3micos y ventajas educativas y ocupacionales de los que un aspirante a \u00abnuevo norteamericano\u00bb, pongamos por caso, probablemente no dispone. Pero una persona consumida por el deseo de asimilarse tambi\u00e9n puede ejercer la autocensura y dejar de lado todo su abanico de experiencias y observaciones vividas; la autocensura da por supuesto que en el propio pasado hay algo vergonzoso, inaceptable, que es menester ocultar a los dem\u00e1s. Para el extranjero, este ciclo de autocensura y verg\u00fcenza puede empezar simplemente con su necesidad de inhibir el gesto de tocar a los dem\u00e1s cuando les habla o de eliminar su propio aliento con olor a comidas extranjeras. La verg\u00fcenza por el hecho de tener un aliento distinto debido a las comidas de su antiguo pa\u00eds se ve reforzada por el simple temor de respirar a la cara de personas que no se alimentan de esas comidas. Es probable que sentirse avergonzado de uno mismo conduzca a la p\u00e9rdida de criterio propio, cuando no de la probidad moral, que Herzen observaba en Golovin. Es la raz\u00f3n por la que, a nuestro juicio, el famoso crisol de razas _(melting pot)_ del mito norteamericano encuentra mejor representaci\u00f3n en la imagen de fusi\u00f3n _(meltdown)_ de las facultades \u00e9ticas del yo.\n\nGolovin es una figura importante en el intento de Herzen por comprender qu\u00e9 significa ser ruso fuera de Rusia, es decir, el intento de comprender c\u00f3mo realizar un desplazamiento humano de la nacionalidad propia. En una carta famosa que envi\u00f3 Golovin el 1 de febrero de 1866 desde Par\u00eds al editor del _Moscow News_ , declaraba: \u00abYo era hombre antes de ser ruso.\u00bb Herzen reproduce esta carta al final de su retrato de Golovin y en la primera edici\u00f3n de _Pasado y pensamientos_. Es una iron\u00eda con resonancias, sin duda. En la era de la Ilustraci\u00f3n, una declaraci\u00f3n semejante podr\u00eda haber provenido de Kant; en este caso, su autor era un especulador de bolsa y un artista de la extorsi\u00f3n sin otra preocupaci\u00f3n que caer bien all\u00ed donde se encontrase. Pero las revelaciones del exilio no pueden terminar de esta manera. Para el extranjero, el conocimiento de que viene de otro lugar, m\u00e1s que fuente de verg\u00fcenza, deber\u00eda servir de advertencia.\n\nPara Herzen, el individualismo econ\u00f3mico era el gran peligro de la era de expansi\u00f3n capitalista que \u00e9l ve\u00eda en sus comienzos. Nacionalismo y capitalismo pod\u00edan marchar de consuno, como una y otra vez sostiene Herzen, con su firme convicci\u00f3n socialista, en _The Bell_. Por el contrario, las esperanzas de Herzen en un movimiento socialista se apoyaban en los inmigrantes. Su mero desplazamiento les daba la experiencia, o al menos la posibilidad, de mirar m\u00e1s all\u00e1 de s\u00ed mismos y mantener una relaci\u00f3n de cooperaci\u00f3n con quienes han sufrido un desplazamiento similar.\n\nComo lector de Herzen, es aqu\u00ed donde lo encuentro m\u00e1s convincente. A Herzen le habr\u00eda parecido perfectamente comprensible que los grupos \u00e9tnicos de los Estados Unidos modernos fueran el coraz\u00f3n del liberalismo de tipo norteamericano, versi\u00f3n d\u00e9bil de la socialdemocracia europea. Habr\u00eda explicado esta relaci\u00f3n entre inmigraci\u00f3n y liberalidad, pienso, argumentando que las cicatrices del desplazamiento hab\u00edan creado una disposici\u00f3n liberal en quienes eran conscientes de su condici\u00f3n de extranjeros, a diferencia de los Golovin que s\u00f3lo deseaban olvidar. La creencia de Herzen en que la pr\u00e1ctica del socialismo es m\u00e1s f\u00e1cil para los extranjeros es una idealizaci\u00f3n del desplazamiento, sin duda, pero es un ideal fundado en una duda profunda y profundamente esc\u00e9ptica acerca de que los males del individualismo posesivo pudieran curarse mediante relaciones comunales de tipo nacionalista, homog\u00e9neas, autorreferenciales. S\u00f3lo el conocimiento de la diferencia y la experiencia del desplazamiento pueden interponer una barrera de experiencia a los apetitos del individualismo posesivo.\n\nCuando se lee a Herzen como a un escritor que habla de nuestro tiempo a\u00fan en su per\u00edodo embrionario, es imposible no pensar en la diferencia entre liberalismo y pluralismo. La moderna regla b\u00e1sica de la identidad amenaza constantemente con limitar la libertad personal a la pr\u00e1ctica cultural, lo que quiere decir que las necesidades personales s\u00f3lo se legitiman en la medida en que pueden identificarse con lo que _hacen_ la comunidad mexicana, los ancianos rusos o las j\u00f3venes negras. Con la aplicaci\u00f3n de esta regla, el liberalismo puede verse degradado a mero pluralismo, que se convierte a su vez en una simple cuesti\u00f3n de definici\u00f3n de las fronteras entre comunidades que comparten territorios colindantes; dentro de cada uno de ellos la gente vive como si nunca se hubiera marchado de su pa\u00eds, como si nada hubiera sucedido. Parad\u00f3jicamente, lo necesario para superar este autoencierro pluralista en la etnicidad es la viva conciencia de uno mismo como extranjero. Herzen relata esta an\u00e9cdota, que sit\u00faa en Inglaterra: \u00ab\"Cuando se lo escucha\", me dijo una vez un hombre de gran val\u00eda, \"uno oye hablar a un espectador marginal.\" Pero \u00bfsabe usted?, yo no llegu\u00e9 a Europa como marginal, sino que me he vuelto marginal.\u00bb Y, por esta misma raz\u00f3n, en su reciente libro _Am\u00e9rica inmigrante_ , Alejandro Portes y Ruben Rumbaut declaran lisa y llanamente: \u00abLa asimilaci\u00f3n, entendida como transformaci\u00f3n r\u00e1pida de inmigrantes en norteamericanos \"como los dem\u00e1s\", nunca ha tenido lugar.\u00bb La afirmaci\u00f3n de estos autores va m\u00e1s all\u00e1 de una afirmaci\u00f3n sociol\u00f3gica; es la afirmaci\u00f3n de una conciencia necesaria, ilustrada.\n\nLa nostalgia, el peligro opuesto a la amnesia, parece una condici\u00f3n m\u00e1s simple. En realidad, eso era lo que le parec\u00eda a Herzen en Ginebra en 1850, cuando acababa de dejar Par\u00eds con los otros refugiados de Europa Central. Por primera vez, muchos de ellos comprendieron que se hallaban en exilio permanente, lo cual disparaba los peligros de la nostalgia:\n\nTodos los _\u00e9migr\u00e9s_ , separados del medio al que han pertenecido, cierran los ojos para evitar ver verdades amargas, y se adaptan cada vez m\u00e1s a un cerrado c\u00edrculo fantasmal formado por recuerdos inertes y esperanzas para siempre irrealizables...\n\ny una vez m\u00e1s,\n\nAl abandonar su tierra natal con oculto resentimiento, con el pensamiento constante en el regreso a la ma\u00f1ana siguiente, los hombres no dan un paso adelante, sino que retroceden continuamente al pasado.\n\nDe todo lo cual concluye Herzen que el exiliado pod\u00eda terminar encadenado a su propia capacidad de recordar, esto es, a esas \u00abpreguntas, pensamientos y recuerdos que construyen una tradici\u00f3n opresiva y esclavizante\u00bb.\n\nQuince a\u00f1os despu\u00e9s, en Londres, Herzen vuelve a ocuparse en sus memorias del tema de la nostalgia del emigrado, esta vez alterado por su propia transformaci\u00f3n en el exilio. Herzen habla de sus encuentros con el padre Vlad\u00edmir Pecher\u00edn en un breve retrato digno de Ch\u00e9jov. Pecher\u00edn es una persona acerca de la cual Herzen, como toda persona de su generaci\u00f3n, ten\u00eda noticias. A mediados de la d\u00e9cada de 1830, el joven Pecher\u00edn hab\u00eda obtenido la c\u00e1tedra de griego en la Universidad de Mosc\u00fa y en los a\u00f1os siguientes se sinti\u00f3 asfixiado en su pa\u00eds; en palabras del propio Herzen, \u00abestaba rodeado de silencio y de soledad, todo era muda sumisi\u00f3n sin esperanza ninguna, sin dignidad humana, y al mismo tiempo tremendamente aburrido, est\u00fapido y mezquino\u00bb. Pecher\u00edn, el joven profesor de lenguas cl\u00e1sicas, decidi\u00f3 emigrar, lo cual no sorprendi\u00f3 a ninguno de sus contempor\u00e1neos, que compart\u00edan con \u00e9l la sensaci\u00f3n de ahogo en la Madre Rusia. Pecher\u00edn zarp\u00f3 rumbo a Inglaterra, desembarc\u00f3... y entr\u00f3 repentinamente en un monasterio jesuita. Esto s\u00ed que sorprendi\u00f3 a otros j\u00f3venes de su entorno, que no entend\u00edan c\u00f3mo hab\u00eda podido rebelarse contra un sistema autoritario s\u00f3lo para someterse a otro.\n\nCuando Herzen desembarc\u00f3 en Inglaterra busc\u00f3 a Pecher\u00edn para conocerlo y para preguntarle si pod\u00eda reeditar en _The Bell_ algunos de sus poemas de juventud. Se encontraron en el monasterio jesuita de Saint Mary's, en Clapham. Los dos rusos empiezan hablando en franc\u00e9s y luego, pese a los temores de Pecher\u00edn de no recordar su lengua materna, pasan al ruso; Pecher\u00edn est\u00e1 \u00e1vido de noticias, niega valor a sus poemas rusos, pero ans\u00eda conocer la opini\u00f3n de este hombre m\u00e1s joven. Despu\u00e9s de su encuentro comienzan a mantener correspondencia. El jesuita converso escribe en franc\u00e9s sobre materialismo, ciencia y fe, y lo hace con vehemencia ante este extra\u00f1o, suponiendo que ninguna frontera se interpone en su descubrimiento mutuo, como hubiera hecho cualquier franc\u00e9s, fuera o no devoto.\n\nHerzen nos cuenta todo esto como introducci\u00f3n al relato de un suceso que leer\u00eda en los peri\u00f3dicos dos a\u00f1os despu\u00e9s, en 1855. Un monje jesuita, que la prensa presentaba como el \u00abreverendo padre Wladimir Petcherine, ruso de nacimiento\u00bb, estaba procesado por quemar una Biblia protestante en un mercado de una ciudad irlandesa. He aqu\u00ed el resumen de Herzen de lo que sucedi\u00f3 en el juicio:\n\nEl orgulloso juez brit\u00e1nico, considerando lo absurdo de la acci\u00f3n y el hecho de que el acusado fuese ruso y de que Inglaterra y Rusia estuvieran en guerra [la guerra de Crimea], se limit\u00f3 a una paternal exhortaci\u00f3n a un comportamiento decente en las calles en el futuro.\n\nM\u00e1s fascinante a\u00fan que la historia que Herzen relata es el hecho de que en 1865, cuando escribi\u00f3 esta parte de sus memorias, la hab\u00eda interpretado mal. En realidad, Pecher\u00edn expuso que hab\u00eda quemado literatura pornogr\u00e1fica, no una Biblia, y fue absuelto. La sensaci\u00f3n que se ten\u00eda en el momento era la de un jesuita que emprend\u00eda una \u00abacci\u00f3n directa\u00bb ante el descubrimiento de la obscenidad; al futuro primer ministro Gladstone, muy interesado en las condiciones de prostituci\u00f3n en la Inglaterra moderna, le intrig\u00f3 esta \u00abacci\u00f3n directa\u00bb contra la pornograf\u00eda. Hay una raz\u00f3n para que, al recordar esta historia del proceso a Pecher\u00edn, Herzen la altere (no digo que con intenci\u00f3n de enga\u00f1ar). Para Herzen se trata de una historia de la manera en que los desplazados de su tierra natal permanecen presos del pasado. Para \u00e9l todo era perfectamente coherente. En efecto, llega un mensajero ruso en busca de testimonios de la vida pasada de Pecher\u00edn, la vida de un joven en Mosc\u00fa, cuando el zar Nicol\u00e1s, inducido por el clero, hab\u00eda organizado requisas policiales de escritos her\u00e9ticos en las universidades. Para Herzen, el quid de la cuesti\u00f3n es que Pecher\u00edn sufri\u00f3 algo as\u00ed como un ataque de atavismo. La joven v\u00edctima de la ortodoxia se ha convertido en polic\u00eda de la herej\u00eda.\n\nPecher\u00edn es un ejemplo particularmente representativo de un desastre que Herzen observaba cada vez con mayor temor durante sus a\u00f1os de exilio; es lo que Freud llamar\u00eda m\u00e1s tarde \u00abel retorno de lo reprimido\u00bb. El retorno de lo reprimido es much\u00edsimo m\u00e1s peligroso para el extranjero que el anhelo expl\u00edcito del pasado. Este retorno de lo reprimido tiene lugar en aquellas personas que no hacen nada por transformar la parte de s\u00ed mismas que vive en el recuerdo. El extranjero debe afrontar recuerdos de su pa\u00eds; tiene que desplazar, que deformar la memoria, a fin de no verse repentinamente presa del pasado y volver a sentir, reactivados, los agravios recibidos mucho tiempo antes, que ahora, sin embargo, desempe\u00f1an otro papel en ese antiguo drama. Pero \u00bfc\u00f3mo tiene que darse la transformaci\u00f3n para que el drama vuelva a escribirse?\n\nEl consejo de Herzen al extranjero acerca de c\u00f3mo comportarse en los pa\u00edses en donde vive, que poco a poco va tomando cuerpo en las p\u00e1ginas de sus memorias, es m\u00e1s o menos el siguiente: \u00abParticipar, pero no identificarse.\u00bb Esta advertencia sugiere c\u00f3mo es posible para un extranjero imponerse en el juego de segregaci\u00f3n del pluralismo. El impulso a participar es la reivindicaci\u00f3n de los derechos que se tiene como animal pol\u00edtico, como _z\u00f3on politik\u00f3n_ , dondequiera que se viva. En lugar de la antigua m\u00e1xima \u00abnada humano me es ajeno\u00bb, la de la identidad moderna podr\u00eda ser \u00abnada ajeno a m\u00ed es real\u00bb. El presidente japon\u00e9s Nakasone dijo en cierta ocasi\u00f3n: \u00abS\u00f3lo quienes se entienden pueden tomar decisiones juntos.\u00bb La reivindicaci\u00f3n que un extranjero hace de su derecho a participar m\u00e1s all\u00e1 de lo que corresponde a su identidad nacional, es una manera de forzar a la sociedad dominante a reconocer que hay un dominio p\u00fablico m\u00e1s all\u00e1 de las fronteras de la antropolog\u00eda. Es tambi\u00e9n la \u00fanica manera de sobrevivir si se est\u00e1 personalmente preso en una balcanizada y desigual ciudad de diferencias.\n\nHerzen encontr\u00f3 una manera de crear una imagen tal de \u00abhogar\u00bb que hiciera soportable el anhelo de \u00e9l. Dice que, en Londres, se ha vuelto de pronto italiano:\n\nY ahora estoy en Londres, donde la suerte me ha tra\u00eddo, y aqu\u00ed me quedo porque no s\u00e9 qu\u00e9 hacer conmigo. Una raza extra\u00f1a revolotea confusamente a mi alrededor, envuelta en la pesada respiraci\u00f3n del oc\u00e9ano, un mundo que se disuelve en el caos [...] y esa otra tierra, lavada por el mar de intenso azul bajo el intenso azul del dosel del cielo [...] es la \u00fanica regi\u00f3n luminosa que ha quedado hasta que el lejano lado de la tumba [...] \u00a1Oh, Roma! \u00a1Cu\u00e1nto amo volver a tus enga\u00f1os! \u00a1Con qu\u00e9 anhelo recorro d\u00eda tras d\u00eda el tiempo en que estuve de ti intoxicado\n\n\u00abHogar\u00bb no es aqu\u00ed un lugar f\u00edsico, sino una necesidad desplazable; sea cual fuere el lugar donde uno se encuentre, el hogar se hallar\u00e1 siempre en otro sitio. Como la vida de Herzen se desarrolla en Inglaterra, una tierra sin sol y de gente extremadamente pr\u00e1ctica, aunque afable, el hogar que necesita cambiar\u00e1 de pa\u00eds. Un lugar de nieve por uno de sol, la aldea \u00edntima en las afueras de Mosc\u00fa por los l\u00e1nguidos caf\u00e9s de Roma. Herzen siempre tendr\u00e1 un hogar, en la medida en que pueda cambiar su aspecto. Este conocimiento ir\u00f3nico, y ligeramente amargo, de su necesidad de \u00abhogar\u00bb le lleg\u00f3 a un Herzen ya mayor; \u00e9l mismo reconoci\u00f3 que nunca se sentir\u00eda completo. Finalmente, lleg\u00f3 a avenirse con la insuficiencia, que es permanente; la cicatriz no se cura. Y este mismo poder para desplazar el \u00abhogar\u00bb fue lo que esperaba para los otros que no hicieron las maletas cuando, en marzo de 1848, se abrieron las fronteras, los que no regresaron al amado mundo de su infancia, su lengua, su suelo.\n\nTal vez la modulaci\u00f3n que he introducido en la voz de Herzen, voz de un hombre antes curioso que censor, haya sido injusta; como escritor, Herzen comprendi\u00f3 que lo mejor que se puede hacer con los aspectos morales de los relatos de vidas individuales es dejarlos impl\u00edcitos. Sin embargo, si he cometido con \u00e9l esta injusticia es s\u00f3lo a causa de sus p\u00e1ginas que describen las desastrosas argucias de banqueros emigrados, la rabia de poetas serbios que leen fieles traducciones inglesas de su obra, la lucha de muchos emigrados pol\u00edticos por impedir que los ideales socialistas se disuelvan en el \u00e1cido de la pseudorreligiosidad eslava, todos ellos retratos de extranjeros que luchan por construir una vida fuera de su pa\u00eds que, sin embargo, no los arranca del pasado y que parecen emblem\u00e1ticos de los peligros de otras afirmaciones de identidad racial, sexual o religiosa.\n\nEn la sociedad moderna, la antropolog\u00eda se ha convertido en una amenaza para la libertad. En sentido antropol\u00f3gico, el hombre o la mujer se abstraen de las impurezas y las dificultades anejas a la experiencia de la _diferencia_. Su _nomos_ es la solidaridad racial, la etnicidad, la pr\u00e1ctica sexual, la edad, toda una sociedad de identidades autorreferenciales. Pero el extranjero, consciente de su condici\u00f3n de extranjero, no puede abstraerse tan f\u00e1cilmente de ellas. Esta persona tiene que rescatar, si es posible, algo de su propio viaje. Las palabras de Daniel Stern sobre su observaci\u00f3n de la reticencia de los extranjeros a abandonar Par\u00eds podr\u00eda entenderse como este imperativo: \u00abMirar al espejo y ver a otra persona.\u00bb\n\nLo mismo que Manet, Herzen trat\u00f3 de comprender el desplazamiento no como algo que ha fallado, sino como un proceso con su propia forma y sus propias posibilidades. Sobre todo Herzen vio que su desplazamiento de Rusia hab\u00eda dado lugar a un nuevo tipo de libertad en su vida, una libertad interior independiente del lugar, libertad que \u00e9l sent\u00eda con mucha fuerza, pero como algo tan nuevo, tan moderno, que no acertaba a definir. Lo cierto es que esa misma incapacidad para decir clara y precisamente qui\u00e9n era vino a a\u00f1adirse a su sensaci\u00f3n de libertad. En esto fue tal vez el primero, emblem\u00e1tico y, por las cualidades de su introspecci\u00f3n y del cuestionamiento de su condici\u00f3n, el m\u00e1s grande de los extranjeros.\n\nEn cierto modo, los individuos desplazados siempre tienen la tentaci\u00f3n de idealizar sus ra\u00edces como s\u00f3lidas y seguras, de hacer fotograf\u00edas del pasado mientras el presente se despliega como una pel\u00edcula formada por escenas que cambian abruptamente. No fue casual que la pasi\u00f3n del nacionalismo que se extendi\u00f3 por Europa en 1848 adoptara esa forma antropol\u00f3gica. Ese a\u00f1o marc\u00f3 un punto de inflexi\u00f3n en el que ingentes masas de seres humanos comenzaron a sentir los intranquilizadores efectos de la industrializaci\u00f3n y la rapidez de la migraci\u00f3n urbana. El blanco declarado de los levantamientos nacionales eran las dinast\u00edas del _Ancien R\u00e9gime_ : los Habsburgo (en especial su rama menor de la Casa de Saboya), los Hohenlohe y la penetraci\u00f3n de los Hohenzollern y los Hohenstaufen en la aristocracia rusa. Pero lo que perturbaba a quienes en 1848 apuntaban al pasado era en realidad un presente cuyos horrores pod\u00edan percibir, pero no nombrar con claridad. Los italianos que se sublevaban contra los Habsburgo eran habitantes del norte que viv\u00edan en ciudades como Mil\u00e1n, en las que se estaban dando los primeros pasos importantes de la producci\u00f3n industrial; los polacos, los bohemios y los b\u00e1varos que se rebelaban contra sus respectivos monarcas viv\u00edan en lugares donde, en los a\u00f1os treinta y cuarenta, las peque\u00f1as granjas se cerraban o se incorporaban a grandes fundos, tierras en las que un inmenso n\u00famero de j\u00f3venes abandonaban el campo. Desde el inicio del desarrollo mercantil posterior a 1815, las ciudades a las que llegaban los inmigrantes eran cada vez menos lugares de poblaci\u00f3n \u00abnativa\u00bb establecida, de posici\u00f3n, h\u00e1bitos o domicilios arraigados; la imaginer\u00eda del \u00abnativo\u00bb contra el \u00abforastero\u00bb era utilizada por personas que migraban constantemente dentro de la naci\u00f3n, \u00absin descanso hasta la muerte\u00bb, como las describi\u00f3 Tocqueville.\n\n\u00c9stas eran las condiciones en las que el ideal del _ser_ nacional atra\u00eda a los desplazados. La migraci\u00f3n urbana y su econom\u00eda constitu\u00edan una de las fuerzas creadoras del nacionalismo, imagen de un lugar fijo necesario para los que experimentaban el desplazamiento. \u00abUn mundo que se disuelve en el caos\u00bb: contra \u00e9l, la tierra se erige como medida de permanencia; su ser se opone a las dificultades del devenir personal.\n\nEl \u00abcaos\u00bb de la reorganizaci\u00f3n econ\u00f3mica y la migraci\u00f3n de trabajadores que empez\u00f3 a mediados del siglo XIX dif\u00edcilmente remiten en un mundo globalizado. Los motivos de la idealizaci\u00f3n cultural ser\u00e1n tan poderosos para nosotros como para la gente cuya vida transcurri\u00f3 durante la gran era del capitalismo industrial, o tal vez m\u00e1s. En la \u00e9poca en la que Kant celebraba el \u00abciudadano universal\u00bb resultaba inconcebible la migraci\u00f3n en masa y se pensaba que el capital, invertido en la propiedad rural, se manten\u00eda estable sin dificultad. La \u00e9poca del \u00abciudadano universal\u00bb del siglo XVIII, que produjo ideas constitucionales de an\u00e1loga aplicabilidad dondequiera que fuese \u2013tanto en la adusta y provinciana Am\u00e9rica como en la Francia de las mil reverencias combinadas con sonrientes iron\u00edas\u2013, fue una era que enalteci\u00f3 el equilibrio. La base de su imaginaci\u00f3n social era la estabilidad. En un mundo material desequilibrado, surge en cambio la necesidad de una existencia con una localizaci\u00f3n espec\u00edfica.\n\nEl extranjero es la figura que tiene que capear los peligros que acechan en esta necesidad. Puesto que el extranjero no puede convertirse en ciudadano universal, no puede quitarse de encima el manto del nacionalismo y, en consecuencia, la \u00fanica manera que tiene de tratar ese pesado equipaje cultural es someterlo a cierto tipo de desplazamiento que aligere su carga. Y en este esfuerzo por desplazar la imaginer\u00eda de la cultura y las tradiciones populares, el extranjero se ve envuelto en un trabajo af\u00edn al del artista moderno cuyas energ\u00edas, en el \u00faltimo siglo, no se han dedicado tanto a representar objetos como a desplazarlos.\n\n### _La fractura de \u00abEl gran vidrio\u00bb_\n\nComenc\u00e9 este ensayo con una obra de arte y quisiera terminarlo con otra. En 1926, _El gran vidrio_ de Duchamp se rompi\u00f3 despu\u00e9s de su primera exposici\u00f3n p\u00fablica, en el Brooklyn Museum de Nueva York. Las explicaciones de lo que ocurri\u00f3 var\u00edan; unos dicen que un operario dej\u00f3 caer accidentalmente esta construcci\u00f3n de vidrio, mientras que otros sostienen que un conserje pens\u00f3 que se trataba de un desecho y lo tir\u00f3 al cubo de la basura. En cualquier caso, el resultado fue sorprendente, pues los paneles de vidrios rotos parec\u00edan oportunos a\u00f1adidos al alambre, el polvo, el papel aluminio y la pintura que constitu\u00edan la estructura. En realidad, la ruptura de las placas de vidrio parec\u00eda dar renovada importancia a estos elementos pintados, pegados o espolvoreados sobre ellos.\n\nLa gran pasi\u00f3n de Duchamp era el ajedrez; en 1928 form\u00f3 parte del equipo ol\u00edmpico de ajedrez de Francia y es el inventor de numerosos finales de partida. Esta misma pasi\u00f3n estrat\u00e9gica se manifiesta en sus empe\u00f1os visuales. En la multitud de capas superpuestas de indicios, referencias y falsas aperturas que se imprimen en esta construcci\u00f3n en el curso de diez a\u00f1os, de 1913 a 1923, _El gran vidrio_ ha pretendido ser la imagen de mayor complejidad intelectual de nuestro tiempo. Si bien se trata de significados oscuros, una dificultad creada por la propia decisi\u00f3n que su autor adopt\u00f3 en 1923 de dejar incompleta la obra, Duchamp, lo mismo que en una partida de ajedrez, estaba siempre comprometido en la actividad intencional de moverse por una raz\u00f3n. Lo oscuro y lo incompleto no son lo mismo que el movimiento irreflexivo que significa el jaque mate para el jugador de ajedrez profesional. La ruptura del medio en el que Duchamp produce un movimiento resulta, pues, extraordinariamente notable por su efecto de unir los indicios.\n\nEl t\u00edtulo original completo de la obra es _La mari\u00e9e mise \u00e0 nu par ses c\u00e9libataires, m\u00eame (La novia desnudada por sus solteros, incluso)_ , lo que puede sugerir que los complicados movimientos en los que Duchamp se halla involucrado son tanto er\u00f3ticos como \u00f3pticos. En la \u00e9poca en que Duchamp dej\u00f3 de trabajar en ella, _El gran vidrio_ conten\u00eda un conjunto de moldes m\u00e1licos, molinillos de chocolate, anillos de suspensi\u00f3n, corbatas, toboganes, tijeras, bisagras y otros signos que relacionaban el mundo de la novia, en el panel de vidrio superior, con el mundo de los solteros en el panel inferior (tal como los calific\u00f3 el propio Duchamp).\n\nEsa conexi\u00f3n entre ambos paneles de vidrio ser\u00eda en cierto sentido tan familiar a cualquier ni\u00f1o franc\u00e9s hoy como lo era en la juventud de Duchamp. En las ferias rurales de Francia suele haber una caseta en la que se coloca una mu\u00f1eca novia rodeada de pretendientes, tallados en bolos; la persona que es capaz de derribar todos los bolos con un ligero ovillo de cordel se lleva la mu\u00f1eca novia. En 1916, Duchamp hab\u00eda creado una obra que se titulaba _\u00c0 bruit secret (Con ruido oculto)_ , en la que exactamente ese ovillo de cordel aparece encerrado entre dos planchas de bronce unidas por cuatro bulones. Oculto en este ovillo se halla un objeto peque\u00f1o que cascabelea cuando el ovillo se sacude, a semejanza, otra vez, del que se emplea en una feria rural para derribar a los solteros. Esto, como he dicho, ha sido siempre un juego popular en las ferias rurales, y Duchamp parece haber elegido bien el t\u00edtulo, un juego de conquista sexual que se juega con la misma concentraci\u00f3n, la misma seriedad e incluso el mismo distanciamiento con que se juegan otros juegos.\n\nTal vez la conexi\u00f3n moralizante que viene a la mente al contemplar _El bar del Folies-Berg\u00e8re_ no sea tan importante como la relaci\u00f3n entre el uso del espejo en la pintura de Manet y el empleo del vidrio en la construcci\u00f3n de Duchamp. Sabemos bien, por el propio Duchamp, qu\u00e9 es lo que intentaba con el empleo del vidrio. Duchamp buscaba desafiar \u00abla evidencia sensorial habitual que permite tener una percepci\u00f3n ordinaria de un objeto\u00bb. Con la utilizaci\u00f3n del vidrio pudo otorgar a los objetos una naturaleza subversiva respecto de esta comprensi\u00f3n retiniana. Efectivamente, los objetos ya no existen como experiencia f\u00edsica, puesto que el ojo puede ver a trav\u00e9s de ellos, a trav\u00e9s del vidrio claro; el vidrio ha sido despojado de su sustancialidad. En la realidad establecida por el vidrio, uno entender\u00eda en cambio, \u00abdesde el punto de vista de la masa, un plano (que genera la forma del objeto por medio del paralelismo) compuesto por elementos de luz\u00bb. En otras palabras, el uso del vidrio era la manera en que Duchamp ayudaba a la mente a \u00abver\u00bb \u2013en el sentido en que la expresi\u00f3n \u00abya veo lo que quieres decir\u00bb se refiere a un acto mental\u2013 con independencia de lo que percibe el ojo.\n\nEl vidrio era, pues, el medio a trav\u00e9s del cual Duchamp trataba de obtener poder sobre el mundo f\u00edsico, la ruptura de la dependencia de la mente respecto de la informaci\u00f3n que proporciona la retina. Pero la ruptura del vidrio en 1926 vino repentinamente a desvelar que por esa v\u00eda era imposible ganar poder sobre el mundo f\u00edsico. Lo que quedaba al descubierto no era exactamente el desvanecimiento de una ilusi\u00f3n, sino m\u00e1s bien la cosa f\u00edsica que reivindicaba Duchamp, las grietas y las tablillas de vidrio que reivindicaban la realidad de la experiencia sensorial m\u00e1s all\u00e1 de las estratagemas del artista. De la misma manera, el espejo de Manet recupera la realidad de las botellas y las frutas que se hallan sobre la barra y que, si se las ve en s\u00ed mismas, sin la operaci\u00f3n de desplazamiento, son objetos fantasma sin realidad retiniana.\n\nEs posible que el volver a ver a los dem\u00e1s y a uno mismo como seres humanos concretos, particulares, no como tipos culturales, dependa tambi\u00e9n de esos giros inesperados; no de la destrucci\u00f3n de los marcos de referencia, sino de algo as\u00ed como su fractura. Este giro inesperado, este desplazamiento, es lo que da a un extranjero la posibilidad de convertirse en ruso en cualquier lugar, y a una obra de arte moderno, en moderna. \n\n### NOTAS\n\n### EL GUETO JUD\u00cdO DE VENECIA\n\n. Cita tomada de Brian Pullan, _Rich and Poor in Renaisssance Venice_ , Oxford, Blackwell, 1971, p. 484.\n\n. Ib\u00eddem, p. 495.\n\n. Ib\u00eddem, p. 486.\n\n. Cita tomada de Benjamin Ravid, \u00abThe Establishment of the Ghetti of Venice\u00bb, en Gaetano Cozzi (ed.), _Gli Ebrei e Venezia_ , Mil\u00e1n, Edizioni di Comunit\u00e0, 1987, p. 215.\n\n. Ib\u00eddem.\n\n. Pietro Aretino, _Ragionamenti_ , cita y traducci\u00f3n en Georgina Masson, _Courtesans of the Italian Renaisssance_ , Nueva York, St. Martin's Press, 1975, p. 24.\n\n. Cita tomada de Masson, _Courtesans_ , p. 152.\n\n. Guido Ruggiero, _The Boundaries of Eros: Sex Crime and Sexuality in Renaissance Venice_ , Nueva York, Oxford University Press, 1985, p. 9.\n\n. Benjamin Ravid, _op. cit_.\n\n. Masson, _Courtesans_ , p. 152.\n\n. V\u00e9ase Elliott Horowitz, \u00abCoffee, Coffeehouses, and the Nocturnal Rituals of Early Modern Jewry\u00bb, _AJS Review_ , 14, 1988, pp. 17-46.\n\n. Arist\u00f3teles, _\u00c9tica a Nic\u00f3maco,_ 1258b.\n\n. Cita tomada de L. C. Knights, _Drama and Society in the Age of Jonson_ , Nueva York, Norton, 1968, p. 165.\n\n. Kenneth R. Stow, \u00abSanctity and the Construction of Space: The Roman Ghetto as Sacred Space\u00bb, en Menachem Mor (ed.), _Jewish Assimilation, Acculturation and Accommodation_ , Lanham, Creighton, 1989, p. 54.\n\n. Frederick Lane, _Venice: A Maritime Republic_ , Baltimore, Johns Hopkins University Press, 1973, p. 147.\n\n. William Shakespeare, _El mercader de Venecia_ , III, 3, trad. de Astrana Mar\u00edn, Madrid, Aguilar, 1974, vol. I.\n\n. Ib\u00eddem, IV, 1.\n\n. Ib\u00eddem, III, 1.\n\n. Howard Adelman, \u00abLeon Modena: The Autobiography and the Man\u00bb, en Mark R. Cohen (ed.), _The Autobiography of a Seventeenth-Century Venetian Rabbi: Leon Modena's \u00abLife of Judah\u00bb_ , Princeton, Princeton University Press, 1988, p. 28; este art\u00edculo incluye un excelente suplemento biogr\u00e1fico a la autobiograf\u00eda del propio Modena, pp. 19-38.\n\n. Ib\u00eddem, p. 31.\n\n. Natalie Z. Davis, \u00abFame and Secrecy: Leon Modena's Life as an Early Modern Autobiography\u00bb, en Cohen (ed.), _The Autobiography of a Seventeenth-Century Venetian Rabbi_ , p. 68.\n\n. Leon Modena, \u00abThe Life of Judah\u00bb, _op. cit_., p. 144.\n\n. Ib\u00eddem, p. 14.\n\n. Ib\u00eddem, p. 159.\n\n. Ib\u00eddem, p. 162.\n\n### EL EXTRANJERO\n\n. Charles de Feir, _Guide du Salon de Paris 1882_ , Par\u00eds, 1882, p. 23; para la lista completa de cr\u00edticas contempor\u00e1neas de esta pintura, v\u00e9ase T. L. Clark, _The Painting of Modern Life: Paris in the Art of Manet and his Followers_ , Princeton, Princeton University Press, 1986, pp. 310-311.\n\n. Jules Compte, cita y traducci\u00f3n en T. J. Clark, _The Painting of Modern Life_ , p. 240.\n\n. Henri Houssaye, \u00abLe Salon de 1882\u00bb, _L'Art fran\u00e7ais depuis dix ans_ , Par\u00eds, 1883, p. 242; he utilizado la traducci\u00f3n de Clark, aunque la expresi\u00f3n del original franc\u00e9s es mucho m\u00e1s enf\u00e1tica; v\u00e9ase Clark, _The Painting of Modern Life_ , p. 243.\n\n. Isaiah Berlin, _Vico and Herder: Two Studies in the History of Ideas_ , Londres, Hogarth Press, 1976, p. xxiii [trad. esp.: _Vico y Herder: dos estudios en la historia de las ideas,_ Madrid, C\u00e1tedra, 2000].\n\n. Ib\u00eddem, pp. 197-198.\n\n. Daniel Stern, _\u0152uvres_ , Par\u00eds, 2.\u00aa ed., 1873, vol. 6, p. 353.\n\n. Ib\u00eddem, p. 224.\n\n. Ib\u00eddem, p. 466.\n\n. S\u00f3focles, _Edipo Rey_ , 1030, en S\u00f3focles, _Tragedias_ , trad. de Assela Alamillo, Madrid, Gredos, 1981, p. 350.\n\n. Ib\u00eddem, 1450, p. 365; _Oedipus Tyraneus_ , 1453.\n\n. En \u00e9sta y en las tres oraciones siguientes me he tomado la libertad de citarme a m\u00ed mismo; todas ellas abren mi libro _The Conscience of the Eye_ , Nueva York, Knopf, 1991 [trad. esp.: _La conciencia del ojo_ , Barcelona, Versal, 1991].\n\n. Harvey Cox, _The Secular City_ , ed. rev., Nueva York, Macmillan, 1966, p. 49 [trad. esp.: _La ciudad secular_ , Barcelona, Pen\u00ednsula, 1973].\n\n. Traducci\u00f3n y cita tomada de Jaroslav Pelikan, _Jesus through the Centuries_ , New Haven, Yale University Press, 1985, pp. 49-50. Traducci\u00f3n castellana en Iglesia viva, n.o 237, p. 125, en http:\/\/www.iglesiaviva.org\/237\/23750-PAGINA.pdf.\n\n. San Agust\u00edn, _La ciudad de Dios_ , libro XV, cap. 1 (final), en http:\/\/www.buscadoresdedios.es\/wp-content\/ uploads\/2008\/01\/la-ciudad-de-dios.pdf.\n\n. S\u00f3focles, _Edipo en Colono_ , 605-610, en S\u00f3focles, _Tragedias_ , ed. cit., p. 535.\n\n. Herzen, _My Past and Thoughts_ , Nueva York, Knopf, 1968, III, p. 1024 [trad. esp.: _Pasado y pensamientos_ , Madrid, Tecnos, 1994].\n\n. Ib\u00eddem.\n\n. Ib\u00eddem, p. 1025.\n\n. Ib\u00eddem, p. 1399.\n\n. Ib\u00eddem.\n\n. Es una intuici\u00f3n que podr\u00eda apoyarse en un extenso estudio sobre inmigrantes mexicanos y sobre norteamericanos de origen mexicano seg\u00fan el cual \u00ab... a mayor nivel de aculturaci\u00f3n (o \"americanizaci\u00f3n\"), mayor predominio del [...] abuso o dependencia del alcohol o las drogas, fobia y personalidad antisocial\u00bb: Alejandro Portes y Ruben Rumbaut, _Immigrant America: A Portrait_ , Berkeley (CA), University of California Press, 1990, p. 169 [trad. esp.: _Am\u00e9rica inmigrante,_ Rub\u00ed (Barcelona), Anthropos, 2010].\n\n. En Herzen, _My Past and Thoughts_ , III, p. 1418.\n\n. Ib\u00eddem, III, p. 1065.\n\n. Portes y Rumbaut, _Immigrant America_ , p. 141.\n\n. Herzen, _My Past and Thoughts_ , II, p. 686.\n\n. Ib\u00eddem, III, p. 1386.\n\n. Ib\u00eddem, III, p. 1397.\n\n. Cf. Sigmund Freud, _New Introductory Lectures on Psychoanalysis_.\n\n. Observaci\u00f3n _off-the-record_ en el Council of Foreign Relations (pero \u00bfpor qu\u00e9 tiene que ser as\u00ed?).\n\n. Herzen, _My Past and Thoughts_ , II, p. 655.\n\n. Cita en Gloria Moure, _Marcel Duchamp_ , Nueva York, Rizzoli, 1988, p. 21.\n\n. Ib\u00eddem. \n_T\u00edtulo de la edici\u00f3n original_ : \nThe Foreigner\n\nEdici\u00f3n en formato digital: diciembre de 2013\n\n\u00a9 de la traducci\u00f3n, Marco Aurelio Galmarini, 2014\n\n\u00a9 Richard Sennett, 2011\n\n\u00a9 EDITORIAL ANAGRAMA, S.A., 2014 \nPedr\u00f3 de la Creu, 58 \n08034 Barcelona\n\nISBN: 978-84-339-3455-0\n\nConversi\u00f3n a formato digital: Newcomlab, S.L.\n\nanagrama@anagrama-ed.es\n\nwww.anagrama-ed.es\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n**BEHIND \nENEMY \nLINES** \nWorld War II\n\nby Carol Matas\n\n_To the 168 Allied airmen shot down and sent to \nBuchenwald, and to all the other airmen to whom \nwe owe our freedom. \nAnd also to my family who, along with myself, \nmight not even exist had Hitler prevailed_.\n\n# Table of Contents\n\nCover\n\nTitle Page\n\nDedication\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\n\nChapter Sixteen\n\nChapter Seventeen\n\nChapter Eighteen\n\nChapter Nineteen\n\nChapter Twenty\n\nChapter Twenty-One\n\nChapter Twenty-Two\n\nChapter Twenty-Three\n\nChapter Twenty-Four\n\nChapter Twenty-Five\n\nChapter Twenty-Six\n\nChapter Twenty-Seven\n\nChapter Twenty-Eight\n\nEpilogue\n\nHistorical Note\n\nImages and Documents\n\nCredits\n\nAcknowledgments\n\nAbout the Author\n\nOther Books in the I AM CANADA Series\n\nCopyright\n\n# Chapter One\n\n_June 3, 1944_\n\nFrom my rear turret I got a glimpse of our attacker, a twin-engine Ju88, coming in for the kill.\n\n\"Corkscrew port! Go!\" I shouted over the intercom to the skipper.\n\nThere was flak everywhere \u2014 little black cloud bursts all around.\n\nSkipper relayed back, \"Down port!\"\n\nWe dove. I'd say 1,000 feet easy.\n\nI started to fire and kept firing as we corkscrewed.\n\n\"Changing,\" Skipper said. He pulled out of the dive.\n\nThen \"Up port,\" came over my intercom as he climbed as steeply as he could manage with the lumbering Lanc.\n\nI was almost out of ammo when suddenly the Ju broke away. I swung around, straining to see.\n\n\"Rolling.\" Lew was converting the climb to starboard. \"Up starboard.\"\n\nBefore Lew could call \"Changing\" I felt the hit. A starboard engine flared. I couldn't see a thing. I was ready with my Browning but now had to wait for Stan to feather the engine. Sure enough he did and the fire went out. That's when I saw the two fighters, but too late. We took another hit. There was an explosion so loud my ears started to ring.\n\nFor a moment I held my breath, not knowing how bad it was, just that the entire plane had shaken as if we were one of those souvenir snow globes and someone had picked us up and dropped us. But then that unseen hand picked us up again and started to shake us and shake us.\n\nThe skipper said, \"Starboard wing's all lighted up, fellows. Bail out, bail out.\" His voice was calm, as if he were telling us to have a cup of tea.\n\nI decided to get out the rear door \u2014 turning my turret would take too much time \u2014 plus this way the others could follow me out. But the door wouldn't come loose, it was jammed. _\"For fanden, fandens ogsaa!\"_ I muttered in Danish.\n\nThat's when Max appeared with an axe and broke the lock \u2014 he must have been there and come back with the axe, but I'd been trying too hard to open the door to even notice him.\n\nI snapped my parachute pack onto my harness and then Max pushed me out, the bright white lights bursting through the darkness all around me.\n\nThe wind jerked me away from the plane with such force that for a moment I couldn't even think what I had to do next. Then I heard my training sergeant's voice. \"Dreamboy. Hey, Dreamboy. Don't forget to pull your cord when you jump. Dreamboy. Hear me?\" Yup. I heard him. I reached for the rip cord. Panic washed over me. It wasn't there. It wasn't there! I was going to smash into the earth. Any second now. Did I even have time to deploy it? And then I heard another voice, my sister's. \"Sam. Can't you tell right from left? Honestly! Lucky thing they aren't training you as a navigator!\"\n\nI reached for the rip cord on the other side. And there it was! I pulled.\n\nThe parachute deployed. It tugged me back up before I started to float slowly down. I tried to get my bearings. I was high enough up still \u2014 that was good.\n\nThe night was bright. A cloudless sky. The moon almost full.\n\nMy chute began to sway in a circular motion. I couldn't control it. Then I looked down and saw what looked like water. I knew that if I landed there I'd probably drown, pulled down by the weight of the harness and the canopy. The day they'd trained us in parachute control I'd been playing cards in the back of the room with Max, because who wanted to think about the fact that maybe we'd need to bail out. We preferred to ignore what Max liked to call \"negative thoughts.\" Facts, more like it, but too late now.\n\nAs I got closer to the water I realized it wasn't water but a large field of wheat. The noise of the aircraft faded. So did the bursts of anti-aircraft fire. The quiet of the night surrounded me. Time seemed to slow. It almost seemed peaceful. And then with a pretty hard bump, I was on the ground. I rolled and then managed to right myself. I seemed to be in one piece.\n\nI hit the release button on my chute and wriggled out of it. One thing I had paid attention to \u2014 the orders about what to do if you survive a crash. I knew I only had minutes.\n\nI had an escape kit with maps and a compass and a passport photo tucked into one of the large front pockets of my battledress jacket, but I had no time to use any of that now. First thing was to hide the parachute so the enemy wouldn't know there was an airman alive somewhere around here. But there was nowhere to hide it. I pulled off my leather gauntlets and heated gloves and finally the silk gloves, so I could reach into my pants pocket. I grabbed my knife and began to cut up the chute so I could hide it better. It seemed to take forever and the silk was harder to slice than I would have thought, but it didn't help that my hands were shaking. I gave up when I had it in a few pieces. I dug a shallow pit with my hands and then stomped on the chute to flatten it down. I knew I had to get out of there fast. The Germans on the ground would have seen _H Hall_ go down in flames and they'd be out after us even before our kite's position was radioed in by the Ju88 pilot.\n\nI was starting to sweat. And that's when I realized I needed to get out of some of the gear I'd put on before the flight, just to keep from freezing to death in the minus-30 temps we'd get in the turret. I started by getting out of my lined leather flight suit, then stripped off layer after layer, which seemed to take forever, until I was down to my battledress jacket, my trousers, plus my thick cable-knit sweater that I tied around my waist. I had to dig a shallow pocket in the earth all over again and stuff the rest of my gear in as best I could.\n\nIt was still quiet. Too quiet. Where were my crew? Had any made it? There was a huge ball of flame not far off, which had to be our downed kite \u2014 it was bright enough that I could easily see where I was. Hunkering low, I decided to run away from that light. The farther away from the evidence the better. I dashed through the wheat field and suddenly got this overwhelming feeling of strangeness. The air smelled sweet. The stars were shining. And the fact that a war was raging all around seemed almost impossible.\n\nWhen I reached the edge of the field I saw a copse of trees just ahead. I lunged into it. And it was only a few seconds later that the quiet was pierced by the sound of cars travelling along a road somewhere close by. I had no doubt who they were looking for. I decided the best thing to do was to keep moving. That's when I heard the groan. The night was so quiet I couldn't have missed it. I inched cautiously over toward the sound \u2014 after all, it could as easily be a German as one of us. But when I peered around a tree, there was Bill, our navigator. He was trying to get up but couldn't.\n\n\"Need a hand?\" I whispered.\n\nHe broke into a grin when he saw me. That was Bill. It would take more than getting shot down and \u2014 maybe a broken leg? \u2014 to dampen his spirits.\n\n\"I propose we get out of here,\" he said.\n\nBill had been in law school before signing up; sometimes he still talked like a lawyer.\n\n\"Let me look at that,\" I said.\n\nI'd spent a lot of time going on call with Pops over the years, watching him treat people. And picked up a thing or two. I examined Bill's leg as gently as I could.\n\n\"Sprained ankle,\" I said.\n\n\"Lucky for me that it's the football player who found me, then.\"\n\n\"Lucky for you,\" I agreed, as I put my arm under his and shouldered his weight. Still, he couldn't help but put his foot down as we walked \u2014 it must have been excruciating, but he never uttered a peep of protest or cried out or anything.\n\nI could hear Coach's words in my head as I half-carried Bill. \"Someday you'll thank me for this, Fred.\" Coach could never bother to say _Frederiksen_. I always wondered why he didn't just call me Sam. Surely that was short and sweet. But no, it was always, \"Pick it up, Fred!\" Or, \"Are you pretending to push a pram, Fred?\" as we ran round and round the track at the start of football season and then did more push-ups than seemed reasonable.\n\nBill and I staggered on until just before dawn and by then I _was_ thanking Coach. We saw some farmhouses but didn't stop at any of them, since we wanted to get as far away as possible from the crash site. Finally we reached another copse of trees and I decided we needed to stop. I was exhausted. I helped Bill settle on the ground and I sat beside him, each of us resting against a tree.\n\n\"We need to get you to a doctor,\" I said.\n\nBut he was out cold.\n\n# Chapter Two\n\n_June 4, 1944_\n\nI woke with a start. I must have finally dozed off for an hour or so because now it was obviously morning.\n\nStanding over me was a boy of maybe thirteen or fourteen. He had his hands on his hips and was glaring at me. \" _R\u00e9veille-toi_ , _idiot!_ \" he hissed. I'd actually been very good at French in high school. I knew right away that he was saying, \"Wake up, idiot!\"\n\n\"Have some respect for your elders,\" I snapped back in English. All right, maybe eighteen wasn't that much older than he was, but after all I'd been through the night before, I was feeling pretty rough. I guess I should have been worried or maybe even scared that I'd been found, but for some reason I wasn't. I'm not sure the boy understood what I said, but I'm pretty sure he understood the tone.\n\n\"Shhhh,\" he cautioned. \"There are Germans everywhere.\" Or something close to that at any rate. Tenses were always my downfall, so he could have said \"are\" or \"were\" \u2014 I couldn't be sure.\n\nI replied in French. \"Where are we?\"\n\nAnd he obviously didn't mind my accent because he answered right away. Something about Perry?\n\nHe pointed at Bill and said, \"He needs a doctor.\"\n\nI looked over. Bill was out cold and his colour was terrible. I scrambled over to him and saw to my horror that he'd been bleeding. His jacket was thick with blood. Gingerly I peeled back a corner of the jacket and saw that he'd taken a hit from shrapnel on his right side just over his ribs. He must have already lost a lot of blood.\n\n\"Wait here,\" said the boy and then ran off.\n\nWell, I couldn't move Bill now, but could I trust this kid?\n\nI looked around. The spot we'd found wasn't as protected as I'd thought. We were sitting in a copse of about half a dozen trees and were quite near a road \u2014 a small one, but a road nevertheless. As I peered out from behind the tree I could see a village not far away and a small farmhouse just to the south. In fact we were surrounded by people. People who could spot us and turn us in. I'd heard stories from downed airmen of amazing heroics from ordinary Frenchmen and women who had risked their lives to help these fellows get back to England. On the other hand, we had all heard about the collaborators. Established escape routes out of France were compromised time and again by someone who wanted a dozen eggs in exchange for the life of an Allied airman. I scrambled away from the trees toward a high wall. From there perhaps I could see the boy before he saw me and decide whether he was going to turn us in or help us.\n\nQuickly I surveyed my surroundings. The wall I was leaning against was an old village wall. I pulled myself up and peeked over. Just beyond the wall was a road and a main street. Shopkeepers were starting to open up. I ducked back down as fast as I could. I realized that I was far too exposed where I was, so I made a run back into the trees.\n\nI needed to scout the rest of the land around us. As a gunner, my training was mostly honing my skills as a lookout \u2014 in fact, I realized suddenly, last night had been the first time I'd actually shot my guns in combat. My job for the most part was to spot the enemy and tell the skipper and hope he could lose the fighters before I had to shoot. We all knew that a slow Lancaster bomber was no match for a fast German fighter \u2014 and my Browning was no match for their guns.\n\nI could see a rather large farmhouse and barn to the west. I wondered if that was where the boy had come from. But what had he been doing out so early? Had he been looking for survivors of the crash? I thought we'd put a good distance between ourselves and the downed plane, but maybe not far enough.\n\nHe's either gone to get help, I thought, or to get some advantage with the Germans by turning us in.\n\nI watched the large farmhouse for a few minutes but saw no activity. The small farmhouse was equally quiet. I took stock. I was very thirsty. I was hungry. Well, that was normal at least. And Bill was out cold. I needed to make a decision. I thought back to the boy and the look on his face when he spoke to me. He seemed genuinely concerned for our well-being \u2014 and perhaps too young to put on such a convincing act? I decided to risk it.\n\nI sat down against the tree. I'd better be ready for a long day, I thought. And a run, carrying Bill possibly, if need be. I suddenly remembered my first-aid kit and pulled it out of my trouser pocket. There was a dressing I could use on Bill, but I was lucky I had more knowledge about medicine than most because of Pops being a doctor. I realized that if I pulled the material from Bill's jacket away from the wound, it would start the bleeding all over again. A doctor would be able to clean the wound out properly and then bandage it, so I decided to leave it alone.\n\nI took out my escape pack and had a look through it, placing the items on the ground in front of me. There was a map of France; some French money, which might come in quite useful so I slipped it into my back pocket; a small compass and a plastic water bottle which I also put aside for later; tablets for purifying water; a tiny razor and shaving soap; malted-milk tablets \u2014 two of which I ate right then, although I could hardly swallow them I was so thirsty. And of course some pictures of me in civvies so I could use them in a fake passport. I used the razor to cut the insignia off my battledress jacket but would need a regular shirt as soon as I could beg, borrow or steal one.\n\nOut of the corner of my eye I noticed something. I looked up and saw activity from the large farmhouse to the west. A young woman came out and walked over to the barn. Just as that was happening I heard a roar from the road and looked over there. I couldn't see them, but there must have been trucks and cars parked there because I didn't hear the sound of just one car. And the only people who had gas for that many cars and trucks were Germans. Had I stumbled onto some kind of headquarters?\n\nI shoved everything into my pack and put it back in my jacket. And then I saw a German soldier walk out of the front door of the farmhouse. My heart caught in my throat, my breath came in short gasps. It was one thing to see the enemy from a kite. It was quite another to see them up close and real. For a minute I almost thought I'd throw up.\n\nIt seemed that for now we were trapped where we were. If we moved we'd be spotted for sure. I hoped the boy wasn't from that farmhouse.\n\nThat hope didn't last long. The boy walked out just after the soldier did, and then sauntered away toward the road leading to the village. Had he already given us away?\n\nBill moaned and stirred. I had to have a plan. I took out the map and tried to figure out where in France we could be.\n\nAs I studied the map I suddenly remembered where we'd been just before we'd crashed. We'd been flying a typical zigzag route home, hard on us because our instinct was to head straight for the Channel and away from the enemy. Just before the shooting started we were about to turn east toward the coastline and the Channel. I stared at the map. It was full of small villages. I scanned for anything that looked like Perry and then stopped when I noticed a small village called La Perri\u00e8re, just north of Le Mans. If I could get Bill patched up, maybe we could make our way south until we got to Spain, the way they explained it to us in our briefings, and then back to England. I knew there was an invasion coming soon. I wanted to be part of it. And so did Bill.\n\nBut who was I kidding? Bill was in no shape to go anywhere and if we couldn't get him to a doctor so the wound could get cleaned, it could become infected and Bill could die.\n\nThe sound of trucks and cars moving along the road made it clear to me that we were well and truly trapped, with the road on one side of us and the open fields on the other. There was nothing to do but wait. And hope.\n\n# Chapter Three\n\n_June 4, 1944_\n\nIt suddenly occurred to me that Mom would be getting a telegram saying I was missing in action. And if I couldn't get word to anyone soon, after a certain amount of time had passed there would be another telegram saying I was presumed dead. I thought about our house on Scotia Street, and how the telegram would be delivered. It might be a hot day, so maybe Mom would be on the front porch shelling peas, or in the kitchen baking, or maybe not even at home because she was working for the Red Cross every day packing care packages for POWs. And Pops probably wouldn't be there. He'd be at his office or out on a call \u2014 he was extra busy now that so many doctors were in the fight. Mom always said thank goodness for his ulcer! And what would poor Jenny say when she got home from school?\n\nOne big adventure. That's what I'd thought when Air Marshal William Bishop came to our school and talked to us about joining the RCAF as flyers. By the time he was finished I couldn't wait! I guess I never really thought I'd end up like this. Shouldn't I have known? Shouldn't I have had a premonition or something? Instead, just yesterday, everything had seemed normal. Max had been teasing me, as usual. \"Talking to yourself, Sam? Praying?\"\n\nI'd looked up from my camp bed, embarrassed. I'd been muttering aloud. _\"What do we do today, Sam?\" \"Change the world, Pops.\"_\n\n\"Nah,\" I answered quickly. \"Just hummin' a tune.\"\n\nI wasn't going to tell Max that it _was_ a kind of habit, the way we started the day at our house, my dad asking me the same question every morning at breakfast, and me giving the same answer \u2014 the one he'd taught me to give \u2014 change the world, make it a better place.\n\nI started to sing \"Boogie Woogie Bugle Boy\" at the top of my lungs just to prove to Max that I'd been singing, not talking to myself.\n\n\"Hey, cut that out!\" Phil yelled.\n\nSo I toned it down as I threw on my gear and grabbed my kit.\n\nCome to think of it though, I _was_ nervous. And so was most of the crew, because our flight time had been changed at the last minute. I knew I wasn't the only one to worry when that happened. Did it mean good luck or bad? Were we now doomed to be shot down or was the change the thing that was going to save us? And since this was only our fifth sortie into enemy territory, maybe there was no such thing yet as \"usual.\" We knew the invasion was coming soon, but not yet. It was June 4, 1944, and we were part of the vanguard preparing for that invasion, hitting rail yards and roads and making sure the enemy couldn't get to the Front when the invasion came. Our target was the railway yards in Trappes. Our way home was a pretty complicated route and not straight back by any means, which made me even more nervous.\n\nI stood on the tarmac and looked up at _H Hall_ and felt my stomach turn over. I suddenly thought about Jenny, and how she clung to me the last time I left home and how she cried. And how she gave me the little Pooh Bear that I always kept in my pocket for luck. Winnie for Winnipeg.\n\nI reached into my pocket to give the bear a squeeze, the way I always did before an op. It wasn't there! And no time to go back for it. My heart sank. Actually I guess I should have known right then and there that trouble was coming for me, for us.\n\nWhen it was time to go, Lew turned to us and, as he'd done just before every flight, said, \"Tallyho, lads!\"\n\nWe replied in unison, \"Tallyho, Skipper!\"\n\nI winked at Max. We loved the way the British flyers talked, so calm and collected, and the way they acted and reacted too. I had all the confidence in the world in Lew and was happy to put my life in his hands.\n\nEverything last night had seemed as routine as it could be, though, considering we were flying over enemy territory, being shot at and then dropping 16,000 pounds worth of bombs! I mean, after our drop we had no hits by flak, although as usual it was pretty darn scary flying through a blizzard of ack-ack. Still, no fighters on our tail, although there were plenty of them around us. At around 0200 hours it was quiet, and I was beginning to think that we were going to be lucky with our fifth op.\n\nLucky?\n\nAs dad would say, _\"Ikke rigtig\"_ \u2014 not really.\n\n\"Penny for your thoughts.\"\n\nI almost jumped out of my skin. Bill was awake.\n\n\"I was just thinking about how we ended up here \u2014 I had a kind of bad feeling...\" I replied.\n\n\"Me too,\" he said, his voice quite weak. \"But then I do every trip.\" He groaned. \"Where are we?\"\n\nCount on the navigator wanting to know exactly where we were, no matter how weak he was. \"A small village called La Perri\u00e8re,\" I answered.\n\nI hated to worry him, but figured he'd better be prepared. \"A kid found us \u2014 he said he's gone for help, but I saw a German soldier come out of the same farmhouse he did. We might be in for it.\"\n\nBill looked down at his ribs and then over to me.\n\n\"Am I hit bad?\"\n\nI nodded. \"You need a doctor.\"\n\n\"I guess I could turn myself in \u2014 worst case, a prisoner-of-war camp and some medical help.\" He thought for a moment. I didn't say anything. Only he could make this decision.\n\n\"What did you make of the kid?\"\n\n\"Seemed okay.\"\n\n\"I think I'll take my chances,\" Bill said.\n\n# Chapter Four\n\n_June 4, 1944_\n\nIt was only minutes later that the boy turned up again.\n\nI searched for the words. How did I ask him if he'd turned us in? I said, \"Your sister is friends with the Boche?\"\n\nHe actually spat. \"Francine. She is,\" he replied. And then he used some words I didn't recognize and I'm pretty sure they weren't complimentary.\n\nOf course this could have been an act to get us to trust him. I looked at him more closely. He was tall and thin with dark curly hair and big brown eyes. He had an air about him \u2014 like he noticed things, like he was sharp.\n\n\"You need to stay here until night,\" he told us, noting that Bill was awake. \"Then I'll take you to that farmhouse\" \u2014 he pointed to the small one south of us \u2014 \"and get him a doctor.\"\n\nHe noticed my map. \"Ah, excellent!\" he said. And then he tapped where we were \u2014 just as I thought \u2014 the village of La Perri\u00e8re. He moved his finger to a spot on the map just to the north of the village. \"A farmhouse just south of this crossroads,\" he said. \"The Resistance will meet you there.\"\n\n\"No,\" I shot back. \"I'm staying with my friend.\"\n\nHe shook his head. \"One man to hide is enough in that place.\"\n\nHe put a small basket down beside us and uncovered it. There was a carafe of wine, a couple of hunks of bread and a large piece of cheese. \"Wait until dark,\" he said.\n\nHe looked around, got up and then sauntered away toward the town.\n\n\"Think it's poisoned?\" I said to Bill.\n\n\"It would be if this were a Bogart movie,\" he replied. I was glad he still had his sense of humour. \"Anyway, I'm not hungry \u2014 you go crazy.\"\n\n\"What about the kid?\" I asked. \"Do we trust him?\"\n\n\"He could have handed us in by now,\" Bill said. \"Unless he wants us to lead him to other downed airmen. But with me injured there's not much chance of that, so all in all I'd say, yeah, probably.\"\n\nI wished the kid had brought us some water. I was so thirsty that after I forced Bill to take a few sips of wine I drank the whole carafe in a few gulps. Then I devoured most of the food after making sure Bill got some small bits down him.\n\nIt was hot, and with that and the wine, I guess I must have dozed off a few times, and so did Bill before night finally fell. In between I planned my route so that once it was night I'd know where I was going after I got Bill to the farmhouse.\n\nFinally it was dark enough to move. I helped Bill up and put his arm round my shoulder. This time he was even weaker than last night, and heavier. We staggered through the field until I found a rough path that led to the farmhouse. We got there and knocked. The door flew open and an elderly man and his wife welcomed us in. She spoke mostly in French, but I was able to make out the gist of it.\n\n\"Please, please,\" she said. \"You are our guests now. Up the stairs a room is ready.\" She was very short and thin and her white hair so wispy it stood almost straight up. Her husband was small too. The woman led the way.\n\nIt wasn't easy getting Bill up those stairs and he couldn't help but let out a few small groans as he put too much weight on his ankle. I eased him onto the bed. Right away a young man I assumed was a doctor slipped into the room. \"Let's have a look,\" he said.\n\nHe shooed us all out and I went downstairs to share a glass of beer with the older man. His wife brought me some bread and sausage. It was pretty greasy but I was way too hungry to care.\n\nThey watched me eat with pleasure and then the doctor called down for help. I wanted to go but the wife wouldn't let me. \"A woman's job,\" she said. Her husband and I sat silently together until she returned.\n\n\"The doctor has cleaned the wound out,\" she said. \"We'll take care of him. You go now.\"\n\nI wasn't leaving without saying goodbye so I went up the stairs two at a time and entered the room. Bill was shirtless, his ribs bandaged, his ankle bandaged, looking pretty pale.\n\n\"It appears I have landed a cushy assignment here,\" he said to me.\n\n\"Looks like it,\" I said.\n\n\"I'll be drinking wine and telling war stories in no time,\" he added. \"My only wish would be a glass of cold water.\"\n\nI grinned. \"Looks like you'll have to make do with beer.\" I paused. \"Take it easy.\"\n\n\"You too.\"\n\nI left the room, thanked my hosts, and slipped out into the dark.\n\nI'd never felt so alone as I did at that moment, standing there in the night without Bill. But I had no choice. So far the kid had been as good as his word. Now I had to find the Resistance.\n\nI made my way north, giving a wide berth to the nearby farmhouse and the German's car parked outside. I kept off the roads and travelled across the fields. It was slow going and different than my panicked run of the night before, but I'd made it through the crash without injuries so I sure didn't want to break a leg now. I heard the planes overhead about an hour into my walk, the crack of anti-aircraft fire and the sounds of cars and trucks on the road nearby. I wondered how the fight was going and when the Allied invasion would start. I took note that it was now June 4 and reminded myself to keep track of the date.\n\nIt was another beautiful night, warm and clear, without any chill in the air at all. I must have been walking for a few hours when I spotted a farmhouse just south of a crossroads, as the boy had described. Of course, I couldn't be positive it was _the_ farmhouse, but it was in about the right place. There were no lights on. The house was surrounded by a ring of trees and I stood behind one of them so I could observe any movement.\n\n\" _Arr\u00eatez-vous!_ \" said a voice behind me, low and quiet. I spun around. The moon and the stars gave enough light for me to see that a man was pointing a gun at my chest. I threw up my hands, just like in the movies, and said, \"Don't shoot!\"\n\n# Chapter Five\n\n_June 4, 1944_\n\n\"Come with me,\" the fellow said, changing from French to perfect English \u2014 English with a very upper-class British accent.\n\nI really had little choice. He motioned me toward the barn beside the house. I hoped against hope that he was part of the Resistance, or another downed flyer, and that I was about to find help. Or be able to offer help.\n\nWe moved quietly to the barn and as we got there the door opened and another man let us in. I could see that there was a small light on, a hand-held lamp near the centre of the barn, and that's where we headed. The man with the gun asked me my name and my rank. He was as tall as me, over six feet but thinner, and had a hat pulled well over his eyes. He spoke softly so that his voice wouldn't carry. I had been instructed that I could freely give out the information he'd requested, but nothing else unless I was certain I was with friends.\n\n\"Sergeant Sam Frederiksen,\" I said. I also spoke quietly.\n\nAnd then the other fellow spoke. He looked to be older than me but not old \u2014 maybe twenty-five or so? He had red hair and blue eyes and freckles and a round pleasant face. But his expression was grim when he looked at me. He was stocky and quite a bit shorter than me, but something about him made it clear he was not to be trifled with. He had a distinct American accent, and asked me such a crazy question that for a minute I thought he must be joking.\n\n\"Who's the best player on the Yankees?\"\n\n\"What?\" I said.\n\n\"You heard me.\"\n\n\"I don't know,\" I answered honestly.\n\n\"A plant,\" he said to the British fellow with the gun. \"Shoot him.\"\n\nI heard the gun being cocked. \"Wait a minute, wait a minute,\" I gasped, heart pounding. \"Just because I don't follow baseball doesn't mean I'm a plant! I'm Canadian! Ask me anything about hockey \u2014 anything!\"\n\nThe Yank looked thoughtful for a moment. \"Who's the best player on the Boston Bruins?\"\n\n\"Bill Cowley,\" I answered without a moment's hesitation.\n\nThe Yank scowled at me.\n\nMy heart sank. Was this it? Would I be killed by my own people?\n\n\"And the Montreal Canadiens won the Stanley Cup over the Chicago Black Hawks, four to zero,\" I said.\n\n\"Anyone knows that,\" said the Yank.\n\n\"Maurice Richard, nicknamed The Rocket, scored five goals in the first round, and in game two of the finals, a hat trick,\" I said. And for good luck I added something that I doubted any double agent would know. \"And the Winnipeg RCAF Bombers lost the Grey Cup to the Hamilton Flying Wildcats, twenty-three to fourteen, last fall.\"\n\nThe Yank nodded. \"Okay,\" he said, \"but as for who's the top player on the Bruins \u2014 it's obviously Herb Cain, not Cowley.\"\n\n\"That,\" I replied with some dignity, \"is a matter of opinion.\"\n\nThe Yank winked at the other fellow and put out his hand to me. \"Ben Webber,\" he said.\n\n\"Flight Lieutenant John Thompson,\" said the Brit as he put his gun away. \"The others won't be telling you their names. Best that way. And no titles between us, just first names.\" And as he said that, out of the gloom appeared six or seven more men.\n\n\"He's all right,\" John said to them in French.\n\nThe meeting didn't last long. And it took place in French spoken far too quickly for me to catch all of it. When the men left I was introduced to the fellow whose barn we were in, Raymonde.\n\nJohn pointed to a bale of hay, indicating that was where I'd be sleeping. I lay down, but although I was tired I was far too revved up to sleep.\n\nAnd there was one more thing keeping me awake. I was hungry. Really hungry. Pops used to tease me that one day he was going to submit me to medical researchers. He said no normal kid could eat the volume of food that I did and stay so fit. He thought if we could bottle my metabolism, we'd all make a fortune. In fact one of the best things about my gunner training was that I was able to take it in Manitoba, so I was often able to take my leave at home, and even to bring my pals along with me. Mom loved it when I brought friends home and she was able to cook up great big Danish dinners of _frikadeller_ and parsley sauce and potatoes, and for dessert, cream cakes, which she'd manage by saving up all the cream from the milk and never letting Pops drink it in his coffee. Sometimes she would even make a _wienerbroed_.\n\nThinking about food made me think about Max. Danes and Jews had one thing in common for sure \u2014 food! Max used to go on and on about Montreal smoked meat sandwiches and bagels and kosher dills and something called brisket, which he swore would outmatch _frikadeller_ any day. \"Fried meatballs?\" he'd scoff. \"Try a hunk of meat cooked slow all day in sauce. Or if you want meatballs, try our sweet and sour.\"\n\nThinking about Max, I hoped against hope he hadn't been caught. Or killed. But for him caught might be worse. He'd told me some pretty gruesome stories about the Germans and how there were rumours that they were taking Jewish people to some kind of camps and killing them, but I told him over and over that no officer would ever condone such a thing. They're soldiers like we are, after all. And we soldiers have a code of honour that is never broken.\n\nI must have finally drifted off or that blasted rooster couldn't have scared the heck out of me as dawn broke. Being a city boy, I had no idea just how loud the darn things were when they crowed in the morning, but I found out. I leaped up, staring around me wildly, until I realized where I was and what that awful noise was. Then I sank back into the hay.\n\nWe were brought breakfast soon after by a girl who looked to be around my age. She had her hair under a scarf and her cheeks were pale in the morning light, but she looked so pretty, and she smiled at me, and well, that picked me up.\n\nI gobbled up the eggs and fresh bread and jam and boy oh boy did it taste good. It was gone in no time, though, and there were no seconds on offer. And my stomach was still rumbling.\n\nJohn brought out a radio and placed it in front of Ben. \"This was dropped by our chaps about a week ago along with some guns and explosives,\" he explained to me. \"We're lucky Ben's been able to contact home and get some instructions. We're just waiting for our marching orders.\"\n\n\"Have you heard about any of my crew?\" I asked.\n\n\"The plane went up in flames when it hit the ground,\" said John, \"and by the time we got people there everyone was gone \u2014 except for two bodies \u2014 chaps that didn't make it out.\"\n\nI nodded. I knew who one of them would be \u2014 Lew, who would have stayed with the kite to give the rest of us a fighting chance.\n\nI suddenly realized I didn't even know if Lew had a sweetheart or brothers and sisters. Our talk had always been about the ops, the plane, the crew. I hoped that whoever else had escaped wasn't already in some French prison.\n\nWhile I'd been lost in thought, Ben had been using the radio. He got up, hid it back in the hay and then said, \"We have our work cut out for us. Starting tonight.\"\n\n# Chapter Six\n\n_June 5, 1944_\n\nIt was then that the young boy who had first found me made an appearance. John spoke to him quickly, and he nodded and left. \"A very good courier,\" John commented.\n\nHe and Ben got down to some serious planning for the night to come. I listened and tried to learn something.\n\nAfter a very small lunch of some bread and cheese, and finally some water, which I was thankful for, I was put to work \u2014 making explosives!\n\nI worked separating the material that the boy had brought with him into 5-inch pieces that looked like sausages as much as anything, but made me a lot more nervous than a sausage would have. I didn't know when one would blow up and take my hand off... or my arm. Ben noticed how carefully I was handling the stuff and assured me I didn't have to worry. To prove his point he picked up a handful of it and threw it against the barn door. I almost had a heart attack! But nothing happened. He explained that it was a chemical reaction that would set it off. Tonight when we used it we would pinch the aluminum cover over it and that would begin the reaction.\n\nAs soon as the sun went down we headed out. It was warm and I figured that was why I was sweating, but in reality I was feeling pretty shaky. Yes, I'd had nerves before a flight, but this was different. I had no idea at all what to expect and no training to fall back on.\n\nWe had to pick our way forward because of the blackout, moving through the countryside without any light. As we continued we gained more members. I carried the explosives in a bag over my right arm. It seemed there was to be no apprenticeship. I was still in the fight, more Army now than Air Force, that was all.\n\nWe travelled for about an hour in the pale light of the moon, until we began to see more farmhouses and even some small villages. I could hear the sound of guns and then the sound of the planes, too, and anti-aircraft fire. I could even see some explosions off in the distance. More kites downed, no doubt. And then I saw the trains ahead and could hear one just leaving the yard. John was obviously in command, along with Raymonde, who was directing his men to certain areas. John pointed right at me and Ben and motioned for us to move ahead and place the explosives. I suppose the other men were covering us.\n\nWe got to the tracks. Trains loomed around us in the dark like monsters from my dreams when I was little. We could hear their engines chugging in the distance. I emptied the bag of explosives, still scared that the stuff would blow up right under us.\n\nJohn and Ben, now joined by Raymonde, quickly placed the small devices along the tracks. It seemed to take them forever, and I kept thinking, just drop it and run! Finally they motioned for me to follow them and we headed away from the tracks. I could hear a train getting closer and closer and I wondered if some poor Frenchmen were going to die tonight because of my work.\n\nWe were running down a small gravel road when the first explosion went off. I could hear brakes grinding and then the sounds of metal crashing into metal and then some screams of pain and the angry sound of men shouting. I glanced back anxiously as we sprinted away, expecting to be surrounded any moment by Germans, but the sounds faded and no others took their place. Slowly the French fighters melted away into the countryside, and finally we arrived safely back at the same farmhouse we had started from. We hustled into the barn.\n\nBen contacted headquarters and reported in. Then I was ordered to get some rest. That was a welcome order and easy to follow. I fell into a deep sleep.\n\nI dreamed my sister Jenny was crying because her best friend Liz had said something mean to her. I tried to console Jenny, telling her that she should tell Liz straight not to talk that way, and that they should make up, but Jenny cried harder and harder and then suddenly I woke up and realized that there really was someone crying nearby. The barn was still in darkness. I got up and followed the sound, which I finally realized was coming from the hayloft. Slowly I climbed the small ladder and then immediately stumbled over someone's legs.\n\n\"Ouch!\" she protested.\n\n\"Sorry, sorry,\" I said. By then I had fallen over and was sitting beside her and could see it was the girl my age, from the farmhouse, the same one who had brought us breakfast yesterday morning.\n\n\"I heard you crying,\" I said. Or tried to say in French.\n\n\"My fianc\u00e9,\" she replied, sobs catching in her throat. \"He's been arrested. Michel just ran over from the village to tell us. My fianc\u00e9 and nine others.\" She paused. And with a definite edge to her voice she added, \"Did you think no one would suffer for your little outing last night?\"\n\n\"I'm... I'm sorry,\" I said to her.\n\n\"Sorry!\" She gave a kind if snort, as if my apology was no use to her at all.\n\n\"Now wait a minute,\" I replied. \"I ended up here because the Boche shot down my plane and there were six other men in it and two of them are dead for sure and maybe more. Do you want us to leave the Boche alone? You want us to leave them here so that they can pick up your fianc\u00e9 or you or your family and do whatever they like to you at any time? Don't you want to be free?\" I'm sure my French wasn't very good, but she seemed to understand me just fine.\n\n\"But he's innocent!\" she cried.\n\n\"You're all in this war,\" I said. \"There's the enemy and there's the rest and no one is just a bystander.\" I paused. \"Well, I guess there are some who like the Germans, like the way they run things. Is your fianc\u00e9 like that?\"\n\n\"No!\" she answered.\n\n\"Well then, he'll see it as being at war, won't he? Like being a soldier.\"\n\n\"Perhaps,\" she said.\n\nIt was getting light in the barn. The rooster crowed then and I could hear the others getting up.\n\n\"I'd best go fetch your breakfast,\" she said. She brushed away her tears and hurried off.\n\nAs soon as she had climbed down the ladder I followed her.\n\nBen saw me and gave me a wink, but I had something other than romance on my mind.\n\n\"We need to think about getting out of here,\" I said to Ben and John.\n\n\"Oh?\" said John.\n\n\"The girl.\" I tilted my head toward where she'd gone.\n\n\"Marie-Claire,\" John said.\n\n\"I guess. Her fianc\u00e9 has been taken hostage by the Germans. She's mad as all get-out. I'm pretty sure she blames us, and \u2014\"\n\n\"And if she gives _us_ up,\" said Ben, finishing my thought, \"in return maybe they'll let _him_ go.\"\n\n\"That _is_ one of the main reasons they take hostages,\" John confirmed. \"It often works. I agree with young Sam here. It's time we moved along.\"\n\nJohn told me to leave my battledress jacket behind and gave me a shirt he had scrounged from somewhere. He passed me some twine to tie my escape pack around my waist. After the girl had brought us our breakfast \u2014 which I made sure to wolf down \u2014 and she was back in her house, we simply walked off, Ben with the radio packed in a burlap sack.\n\nBut where could we go now?\n\n# Chapter Seven\n\n_June 6, 1944_\n\nWe were only down the road a few hundred yards when Raymonde rode up to us on his bicycle and told us to follow him. After a ten-minute walk he turned down a rutted country lane \u2014 a good thing because the roads were starting to get busy. We could see a horse and wagon coming toward us and soon I had no doubt we would be seeing Germans too. All over the roads were the long metallic strips we called window that we dropped from our planes to confuse the enemy's radar. It was everywhere! You couldn't walk a foot without stepping on it. I wondered how the people even farmed without it getting tangled up in everything.\n\nAfter half an hour we found ourselves at another farmhouse, this one with a large greenhouse. Raymonde motioned us to go inside. It was full of flowers \u2014 orchids and roses and even some cactus plants. We sat on the floor and waited. A tiny green lizard scampered down one of the wooden slats and along an irrigation pipe.\n\nSoon Raymonde came in with an older man, maybe my _farfar_ 's age, with a shock of white hair and bright blue eyes. He said, \"Welcome. It will be my honour to be your host.\" I felt like I was at some fancy dinner party or something and could just picture him in a tuxedo looking like Cary Grant would if he were a grandfather.\n\nRaymonde said, \"I'll be sending the boy over with instructions for tonight's raid.\"\n\n\"Will he be safe?\" I asked. \"Your daughter knows all the men by sight, and most by name.\"\n\nWith a pained look, he answered, \"He has temporarily left home to live with some relatives \u2014 my daughter has not been informed of this or where anyone else has been moved. You did the correct thing to leave \u2014 she is very distraught and not thinking straight.\"\n\nWell, I figured there were worse places to be stuck. For a little while I just imagined I was on vacation in France, lounging around in a beautiful, colourful little oasis, a peaceful paradise far away from any danger. Of course I had to block out the sound of Ben's radio as he conferred with London about where we were and what our next op would be.\n\nBen looked up from his radio with a look on his face like a kid whose hockey team had just won the Stanley Cup. \"It's started. The invasion. Normandy!\"\n\nI almost let out a whoop of joy but contented myself with shaking hands with everyone there. The rest of the day flew by as we discussed strategy and what they might be doing and how the battle might go.\n\nCary Grant's wife brought us a small supper of vegetable stew and fresh bread. As we ate, John told me how lucky we were to be fed so well, because the Germans had gutted the country, taking all the food and sending it to the Front for their soldiers or back to their homeland. Starvation or near starvation was everywhere. I hadn't understood that every bite we took was food right out of our hosts' own mouths.\n\nThen, before I knew it, the boy \u2014 I knew his name was Michel because the girl had let it slip \u2014 arrived and told us he'd be taking us to the rendezvous, because he knew the area so well. As soon as it got dark enough we were going to collect a drop of supplies from the RAF.\n\nBen told Michel the coordinates of the drop and then we followed him, walking for almost an hour to a clearing in the woods. By the time we arrived at the rendezvous, the same men from the night before had constructed a makeshift landing strip using small fires. I hadn't been able to see it at all from the road, and I hoped the Germans couldn't see it either.\n\nWe heard the drone of a plane. I knew right away it was a Lysander, just from the sound, and sure enough it soon roared in for a perfect landing. The men rushed over to the plane, heaved a number of containers out and began lugging them into the woods. As I heard the engine rev up for takeoff I helped unpack one of the boxes. There were lots of guns. I was actually nervous about having to use one. It would be completely different from firing a Browning while in our kite. This time it would be up close, at a man I could actually see. I wondered if I would freeze up. Then there was another worry \u2014 though I'd had some basic training, I hoped I wouldn't shoot off my own foot or anything.\n\nI thought we'd be going right back to the greenhouse but we didn't. We headed instead toward Raymonde's house, which I figured out because I had made sure to always keep a keen eye on where we were going, and I had a pretty good sense already of the countryside around. I caught up with John and asked him what was happening.\n\n\"Our worst fears \u2014 at least Raymonde's. His daughter, Marie-Claire, went to the local mayor to plead for her fianc\u00e9's life. The mayor turned her right over to the Germans and it wasn't long before she gave up her father and mother. Raymonde thinks the blasted Hun have sent her back home as _bait_ in order to catch him. He wants help getting her away from the farm.\"\n\n\"But if we take her with us she'll just escape and hand us in,\" I protested.\n\n\"He thinks she must have learned the hard way that you can't trust the Hun. Anyway she's his daughter \u2014 he's not thinking clearly.\" He paused. \"I figure we'd better help him. Others in the Resistance won't be as kind to her.\"\n\n\"They'd kill her?\" I asked.\n\n\"In a minute,\" he answered. \"About a week ago they caught a collaborator, beat him in front of the whole town, then executed him.\" He shrugged. \"I wish I could have stopped them. Rather not see that again.\"\n\n\"But Raymonde got away,\" I said. \"How?\"\n\n\"He has a contact in the mayor's office,\" John answered. \"The fellow managed to get word to Raymonde, but his wife was picked up in town before he could reach her. One of his sons is working in Germany \u2014 slave labour \u2014 and the other is off fighting with the Resistance somewhere. Right now Marie-Claire is the only family he has left.\"\n\nI thought about the pale young girl only my age and how she probably was so desperate to save her fianc\u00e9 that she'd put everyone else she loved at risk. Enough to break a person, I thought. How could you live with a choice like that?\n\n\"Hurry! Hurry!\" I heard from Raymonde just ahead of us. I smelled the fire before I saw it. We reached the trees that edged his property and saw that the entire place was up in flames. Two Germans stood beside a big black car. In front of the farmhouse door stood Marie-Claire. She was screaming for help. But the Germans had guns pointed at her. They were going to force her to run into a bullet or burn alive!\n\nRaymonde was about to rush ahead when John grabbed him and held him back. He told one of the other men to hold Raymonde there. Then quickly he gave everyone a job. Three of the men were to draw the enemy's fire from the trees just to the west. They moved off to take their positions. John and Ben would take the Germans down. I was to run to the house and rescue Marie-Claire. For a split second I wondered if she deserved to be rescued and why we were all risking our lives for a traitor.\n\n\"Want to end up like those blockheads?\" Ben said to me, pointing at the Germans.\n\nShame washed over me. I knew right away what he meant. And I knew the answer. A young girl was about to die and we could save her. That was all there was to consider.\n\n\"No,\" I answered.\n\n\"Me neither,\" he said. \"Let's get going.\"\n\n# Chapter Eight\n\n_June 6, 1944_\n\nQuick as I could I made my way along the trees and got as close to the house as possible. I wasn't quite set when a low whistle from John signalled the others to begin. Gunfire erupted from the trees. The Germans turned west to return fire. At that moment I saw John and Ben step out from behind the trees in order to get the best shots they could, even though it made them easy targets for the Germans' gunfire. I dashed out from the bushes and ran straight for the house. I felt completely exposed \u2014 which I was. Marie-Claire had stopped screaming and was watching, her back against the door, unsure what was happening.\n\n\"Your father's waiting,\" I screamed to her. _\"Viens ici! Viens ici!\"_ I held out my hand to her as I ran forward. Shots rang out but I had no idea whether any of them were aimed at me or at her. I kept running. _\"Maintenant!!\"_ I screamed.\n\nFinally she seemed to realize that this was her only chance to escape and she staggered toward me. She seemed too weak to run at speed \u2014 she might already be suffering from smoke inhalation, I thought. I'd sometimes gone with Pops on house calls to fires, and knew how easy it was to be overcome by the smoke and even to die from it before the flames ever got close enough to burn a person. She fell into my arms so I picked her up and carried her back to the trees. As soon as we had some cover I put her down. I could feel something wet as I did. Blood. Had she been shot? Had I? With all the adrenaline coursing through my body, I might not even feel it.\n\n\"Can you walk?\" I asked her in French.\n\nShe was coughing but she managed a yes. I put her arm around my neck and half-carried, half-dragged her to the spot where we'd gathered before. Raymonde grabbed her from me and hugged her tight.\n\n\"I think she might be wounded,\" I cautioned.\n\nThe gunfire had stopped. Everyone else had made it back.\n\nBen nodded to me. \"That's it for them,\" he said with satisfaction, clearly meaning that the Germans were dead.\n\n\"Let's get her back to the greenhouse,\" John said.\n\nRaymonde and another of the men half-carried Marie-Claire as we hurried as fast as possible back to some sort of safety.\n\nBen walked beside me as we went. \"We might need to move out of this area altogether now,\" he said. \"The Krauts are nothing if not efficient. They'll flood this place with troops right away, looking for the fighters who did this. The greenhouse is quite a ways from the farmhouse, but probably not far enough. I bet there'll be a search of every single house in this area.\"\n\n\"And all the men they'll use for that won't be at the Front fighting our guys,\" I said.\n\nI couldn't see his face in the dark, but I heard a smile in his voice.\n\n\"We might not be able to be at the Front or in the skies, but we're still doing what we can.\" He slapped me on the shoulder. \"Good work tonight.\"\n\n\"I'm just glad I didn't have to use a gun,\" I confessed.\n\n\"I'm sure your buddies are too,\" he chuckled.\n\nWhen we arrived at the greenhouse, Marie-Claire was gently settled on the floor. Her father tried to comfort her but had no idea what to do \u2014 and it seemed the others didn't either.\n\n\"May I have a look?\" I asked.\n\nI bent over her. She had been shot, all right. In the arm. From what I could tell it looked like the bullet had gone right through, so all that was needed was to stop the bleeding. I asked if we could get a clean cloth and some alcohol to clean the wound. Soon the man who owned the house appeared, along with his wife. She helped Marie-Claire up and declared that the girl would be spending the night in the house, not out here on a cold floor, and that she would tend to the girl's wound. John was about to object, worried for the older couple's safety, but shut his mouth when he saw the look on the woman's face. I suspected he knew when he was about to lose an argument.\n\nOnce Marie-Claire was taken away, we sat with Raymonde and the others and discussed what the next move should be. After some back and forth, the group decided to move farther south, away from this immediate area. Ben checked in with London and was told where we would be most useful.\n\n\"Let's be off, lads,\" John said.\n\nIt was around midnight. I began to think of sleep as something as precious as food, but something I'd be even less likely to get.\n\nWe followed Raymonde down a small country road. We could hear traffic on the main road \u2014 possibly the enemy was out looking for us already \u2014 as well as planes in the distance. More of ours, I hoped.\n\nWe walked all night. Just before dawn we arrived in a small village and there Raymonde split us up. Ben and John and I were taken to a house; the other men, he said, would go to a farm just on the outskirts. We were to meet that night in the woods due east.\n\nJohn knocked at the door and a boy of about twelve opened it. He nodded and waved us in. His grandmother was already up, brewing the chicory they used instead of coffee. She motioned for us to sit at the table, as if this were an everyday occurrence. Maybe it was! We sat down and she fed us hot chicory and warm bread and butter and jam. I was starving and ate every last bite, even scooping up the crumbs with my fingers. Then she motioned for us to go into a back room, which was a bedroom with two twin beds. John and Ben took one look and each claimed a bed, just taking the time to remove their boots. That was seniority, I thought to myself, but I was so tired all I wanted to do was lie down. I collapsed onto the floor and was asleep in minutes.\n\n# Chapter Nine\n\n_June 7, 1944_\n\nScreams coming from the front of the cottage woke me. I looked around, groggy and not quite alert, when Ben hissed, \"Grab your gun!\" Then he strode over to the door, gun raised, John right behind him. Ben opened the door a crack and peeked out. He seemed to pause for a moment as if he couldn't believe what he was seeing, then shook his head and whispered, \"The old girl is crazy!\"\n\nJohn pushed him aside to look. I could hear the boy yelling to his grandmother to be quiet. After a few moments she stopped and the young boy came into the back room. He was very apologetic.\n\n\"She hates the Boche,\" he said. \"She can't help herself.\"\n\n\"She was yelling at the Boche?\" I asked in disbelief.\n\n\"Yes,\" the boy said. \"The good thing is, she does it all the time. Whenever they come into the village, she stands at the window and screams at them, and shakes her fists, so they don't pay her any mind. Maybe it's even safer here \u2014 they'd never imagine she would draw so much attention to herself if she were hiding the Resistance.\"\n\nJohn laughed. \"What a cover!\"\n\nWe were invited into the front room to eat, the boy assuring us that the Boche had left. Soon a very pretty young woman appeared. She informed us that we were now on orders to disrupt the Boche's communications, so we'd be disabling telephone wires. She left to spread the word to the other Resistance groups, then came back to be our guide. Ben told me that it was often young women who were the couriers between various Resistance groups, because the Krauts found it harder to pin anything on them.\n\nOur guide walked us through the fields and onto a narrow country lane. We travelled for a half-hour in a light drizzle, so by the time we reached the small clearing we were damp and cold. Still, all my discomfort was forgotten when I saw that we were joining a much larger group of about ten young men. Our guide murmured to me, \"The Jewish Resistance.\" I was surprised that they had their own group, but when I thought about it, it made sense. Especially if what Max had told me had any truth to it \u2014 the Germans were out to hurt them and the best defence is always a good offence.\n\nJohn had the details of the strike ordered by London, so he briefed the leader of the other group. Raymonde and his men then made an appearance too. I could see that this was to be a very large operation. When the men had finished conferring, John told me and Ben that we'd be working with the Jewish group to cut the wires, while Raymonde and his men kept watch. I ended up walking along with a fellow about my age who spoke excellent English. I asked him how his group came about.\n\n\"My friends were all joining the Resistance,\" he said, \"which was a good thing, so...\"\n\n\"But you and your friends could only have been \u2014 what?\" I interrupted, looking at him. \"Fourteen or fifteen?\"\n\n\"Hah!\" he replied. \"Many joined at thirteen! After all, as soon as the Germans marched into Paris they started passing anti-Jewish laws, first taking away the rights of Jews who were foreign nationals, then \u2014\"\n\nAgain, I interrupted. \"What do you mean taking away rights?\"\n\nHe looked at me as if I were some sort of idiot who knew nothing. \"Rights. Citizenship! Jews became non-people and then they were sent to the camps. After the Germans finished with the foreign Jews, they started with the naturalized Jews. And not with the lesser known, you know, so people might not take notice. It's as if they wanted to shout it out: 'We can do whatever we want with you, and no one \u2014 especially your French brothers \u2014 will stop us.' How right they were.\"\n\nHe paused for a moment, so lost in thought I wasn't sure he was going to continue, but eventually he did.\n\n\"The Jewish judges and lawyers and businessmen were taken first. And then the government started passing laws so that Jews were forbidden to go to a theatre or a restaurant or a swimming pool or a park. We could shop between three and four in the afternoon and we all had to wear a yellow star on our clothes.\"\n\n\"But why did anyone put up with it? I can tell you, if anyone had tried to do that at home \u2014 in Canada \u2014 we would have made their life a misery!\"\n\n\"Really?\" he said. \"Then I wonder why your country won't take in any Jewish children with visas. We tried to get a large group out of a camp. They had visas for Canada, the Germans said they could go, and Canada refused to take them.\"\n\n\"I don't believe you!\" I said. \"We're here fighting _beside_ you.\" I wanted to take a swipe at him. Everything he was saying seemed so crazy.\n\n\"I don't deny you are here and fighting and I'm pretty damn happy about that,\" he said. \"But you asked why I joined a Jewish group instead of a regular group, and I'm trying to tell you. There are special reasons for us. We're always on the lookout, for instance, for Jewish children that can be saved or hidden.\"\n\n\"And have you been able to do that?\" I asked.\n\n\"Yes, but the less you know about it the better,\" he said. \"If you're ever caught, I don't need you spilling any secrets.\"\n\n\"I would never do that!\" I protested.\n\n\"Anyone will if tortured long enough.\"\n\n\"Abe!\" called their leader. \"Up here.\"\n\nHe tipped his hat to me then and wished me luck.\n\nThe rest of the operation that night, I just did as I was told and tried not to get in anyone's way, but all I could think about was what Abe had told me. If it were true, I'd been living in some kind of dream world and things were so much worse than I could ever have imagined.\n\nWe had a couple of close calls with patrols. Abe and his fighters took out some Germans who got too close to us while I was atop a telephone pole! I was helpless up there and if they hadn't been such good shots I probably would have bought it right then. On the other hand, I got so good at scrambling up the telephone poles that by night's end I was nicknamed \" _Grand Singe_.\"\n\n# Chapter Ten\n\n_June 29, 1944_\n\nIt was a little after sunrise when the pretty girl, Louisa, came to the house quite breathless, and said, \"We need to move you.\"\n\nMy heart sank. Grandmama and her grandson Alain had become like a second family to us. We ate with them, told stories to the young boy, who doted on every word and wished he were old enough to go with us every night as we cut telephone wires, blew up bridges and set explosives on rail lines. I felt safe here in a strange way, and didn't want to go anywhere \u2014 except straight back to England \u2014 and I doubted that was in the cards. We'd been at their house for weeks now and somehow I had got used to it.\n\n\"What's happened?\" John asked.\n\n\"The Boche are doing a massive sweep in this area,\" Louisa said. \"House by house. Ripping out floorboards, going into attics. They seem determined to find every last downed flyer and every last Resistance fighter.\"\n\n\"Desperation,\" John said, with satisfaction. \"The Allies are getting closer and closer. The Hun can sense they're losing.\"\n\n\"Desperation or not,\" Louisa said, \"you need to move.\"\n\nShe explained that they were going to try to send us on the escape line through Spain. I felt uneasy. At least we knew we could trust Grandmama and Louisa and Alain. But once we were sent along the escape line, who knew who we'd run into. I hoped at least I'd be able to stay with John and Ben. I didn't like the idea of being on my own again, perhaps wandering from stranger's house to stranger's house.\n\n\"Now!\" she urged.\n\nI think we had all assumed that we'd be waiting for nightfall, but she meant that we needed to go immediately! Ben ran and packed up his radio. I had nothing to pack except my small escape kit, which I quickly tucked into my jacket pocket \u2014 a jacket Louisa had given me, in fact, in exchange for my cable-knit sweater.\n\n\"You don't want anything on you that shows you're an RCAF airman,\" John instructed me.\n\nOn the other hand, I didn't want to leave my ID tags in the house to be discovered.\n\nLouisa said, \"Leave them. The boy will bury everything.\"\n\nWe went outside to find four bicycles waiting for us. \"You'll follow me,\" she said.\n\nAnd off we went in full daylight. But we didn't go down any main roads, instead sticking to back roads \u2014 which made for a very bumpy ride. My teeth were rattling and my head felt like it might shake off from my neck at any moment. But that was nothing compared to the feeling of being out in the open for anyone to see. I hoped we just looked like three ordinary farm workers. Luckily, we ran into no Germans, although we did pass some farmers who glanced at us and then quickly went back to work.\n\nThe day was becoming hotter and hotter and it must have been late morning by the time we pedalled up to a farmhouse near a small village. Suddenly it all seemed familiar to me and I realized it was the same village where I had met the young boy I now knew as Michel, who had sent me on my way that first day. And before I knew it we were driving up to his barn.\n\n\"Wait a minute,\" I called to Louisa. \"I know this place. His sister is far too friendly with the Boche.\"\n\n\"You'll only be here one night,\" she said. \"And this is the one place they won't search.\"\n\nI supposed that might be true. Why should they when one of their own could just walk into the barn and discover us!\n\nThe barn door opened and Michel stood there grinning. When he saw me his grin widened.\n\n\"Idiot!\" he exclaimed.\n\nBen laughed. \"Your reputation precedes you!\"\n\n\"Very funny!\" I said. \"Can't say I like the idea of hiding in the place of a known collaborator.\"\n\nThe boy understood the word \"collaborator\" and spat, the same thing he'd done when first talking about his sister. \"Francine is afraid now,\" he said in French. \"She begs me for my protection when the Boche leave. I pretend I will help her. But there is no help for her once we have our country back.\"\n\nI could see in his eyes that there would be no mercy for his sister, not even from her own family.\n\n\"Tomorrow after the Boche leave, a car will come and pick you up,\" he said. \"You'll be driven into Paris.\"\n\nAt that Louisa wished us luck and got back on her bike. Michel closed the door but reappeared a short time later with lunch. Ben contacted London and, in a very quick transmission that he hoped wouldn't be traced, told them that we were on our way to the Comet Line, the code word for the Spanish route. Then he buried the radio in the hay at the back of the barn.\n\nJohn had managed to pick up a deck of cards at some point over the last couple weeks and we spent the rest of the day playing poker, using hay as money. It passed the time and kept our minds off the uncertainty to come.\n\nWe all heard the German's car drive up to the farm just after dinner. To be on the safe side we retreated to the back of the barn, out of sight, just in case he did a cursory search. He didn't though. That didn't mean I slept well. I doubt any of us did. I lay there with my eyes open, my gun at the ready.\n\n# Chapter Eleven\n\n_June 30, 1944_\n\nI guess I drifted off again because I awoke with a start to the sound of a car's engine revving up. I gripped my gun so hard it's a miracle I didn't shoot someone by accident. But the car drove off and soon after, Michel arrived with breakfast. We'd just finished eating when another car arrived. It was a big black Opel like the ones the Nazis drove. We all piled into the back and then off we went.\n\nNow I was even more nervous. Here we were in plain sight of the Germans. None of us had fake papers. No ID at all in fact. The driver was a middle-aged man who was well dressed but seemed to have little interest in talking to us \u2014 perhaps because he was very nervous too. I hoped that was the reason and that we hadn't just gotten into a car that was taking us right into the enemy's hands.\n\nWe finally drove off the small gravel road and onto the main highway. Immediately we encountered the Germans \u2014 mostly trucks with soldiers standing up in the open cargo areas. Rows and rows of trucks as far as the eye could see. Anti-aircraft guns were positioned all along the road, just waiting for our fellows. It seemed they must be expecting an attack. And then we saw the Tiger tanks! I'd heard of them of course, these formidable weapons, but to see them this close was unbelievable! We could almost touch them! And by their black uniforms I could see the troops belonged to the Panzer divisions.\n\nI couldn't understand why our driver was taking this route.\n\n\"I say, old boy,\" John said to the driver. \"Do you really think this is wise?\"\n\nThe fellow didn't reply so John repeated his question in French.\n\n\"No other way into the city,\" the driver replied. \"Do they look interested in us?\" he asked.\n\n\"Not especially,\" John answered.\n\n\"They aren't in the least,\" the fellow answered. \"They're heading west to fight. They couldn't care less about a car with a few men in it. On the other hand, the Boche in the area we just left, the ones searching for Resistance fighters, they would be _very_ interested. We have more to fear from a bombing raid courtesy of _your_ fellows right now.\"\n\nI supposed that was true. And not long after, I did hear the drone of planes. Must be the Yanks, I thought. They did the daylight raids. I saw that Ben was thinking the same thing as I was when a slow grin lit up his face. He didn't care if we were caught in a raid \u2014 he just wanted these Germans stopped.\n\nBut before any bombs started to fall and the anti-aircraft fire began, we finally made it into Versailles. Not far from there we entered a suburb and came to an old stone house set in a few acres of land. A middle-aged woman met us at the door and ushered us in. It turned out to be the home of the driver of the car. I wondered how he had managed to get fuel and keep his car, and even how he'd kept the house out of the hands of the Germans.\n\nI found out quickly.\n\n\"You have an urgent message from the hospital,\" his wife said.\n\nHe nodded. Then he turned around and left once again.\n\nSo he was most likely an important doctor, a surgeon even. Someone the Germans had more use for alive than dead. And in his spare time he helped out the Resistance. Or, I thought, he could be a collaborator and soon we'd be picked up by a few Gestapo agents. That thought made my blood run even colder than it had in the car.\n\nBut for the moment we were certainly treated well. His wife invited us to share a late dinner with her and then showed us to our rooms as if we were house guests. We were even each offered a bath! The water was lukewarm but the luxury of a bath was something I had not even dreamed of.\n\nA bit later the doctor returned and offered us some red wine. John was very circumspect with his conversation, never really giving anything away, just talking in generalities about the war.\n\nAnd for the first time since being shot down I slept in a real bed. At first it felt so wonderful \u2014 the clean sheets bringing back memories of home \u2014 that I couldn't sleep. When I finally did, I dreamed of eating Mother's chocolate cake and of her hanging sheets out to dry in the backyard while the weeping birch swayed in the wind and the warm air surrounded me like a baby in a warm blanket.\n\n# Chapter Twelve\n\n_July 1, 1944_\n\nAt breakfast, the doctor told us that it was too dangerous to keep us all together in one spot and we were to be separated. He would be taking me to a different safe house immediately. I didn't want to go off on my own. But I had no choice. We were at their mercy for good or ill and I had to hope it was for good.\n\nJohn shook my hand and said, \"Good luck, old chap.\"\n\nBen also shook my hand and, with a wink, said, \"Take it easy, kid.\"\n\nI saluted my two friends and left with the doctor.\n\nHe drove me right into the heart of Paris. I sat in the front seat with him this time. And perhaps I looked as miserable as I felt because he broke his stony silence and talked to me.\n\n\"You really are safer apart from each other,\" he said.\n\nI nodded, but was still unconvinced.\n\n\"The Boche are getting more desperate, you see,\" he explained. \"They know it's almost over. The dream of a thousand-year Reich will never happen. But they hope at least for good terms. They hope to salvage something.\"\n\n\"Or maybe,\" I said, \"Hitler _doesn't_ think it's over. And they are fighting so hard because he says they have to. Whenever I've seen him in the newsreels he looks crazy.\"\n\n\"Crazy?\" The doctor thought for a moment. \"All murderers are crazy,\" he agreed, \"otherwise they couldn't murder. But clinically insane? I'm not so sure. And do not forget, Hitler is a murderer who has the support of millions. And there are so many who murder on his orders. Mass insanity? Perhaps. But I fear it is something much worse.\"\n\nNow that he was talking he was really talking!\n\n\"What?\" I asked, very curious.\n\n\"I suspect that Hitler has learned to prey on people's primal instinct \u2014 fear \u2014 and that even if we defeat _him_ there will always be others who will do the same.\"\n\nI really didn't know what to say. I don't think he actually expected an answer. It was almost as if he were thinking aloud.\n\n\"Yes,\" he said, \"make people afraid enough and they will follow you and do almost anything.\"\n\n\"What were Germans so afraid of?\" I asked.\n\nThe doctor shook his head. \"That's the irony. They were afraid of the riots and the turmoil, the messiness of democracy. Hitler took a yearning for simplicity and order and turned it into a force that could not be stopped. It was easy, no? Orderly, no? No more mess. Democracy, my young friend, will always be at the mercy of those who say things run much easier and smoother without it! Just let us do our jobs, they say, without all this turmoil.\"\n\n\"But,\" I replied, \"after this, if we win, who would say that?\"\n\nHe glanced over at me then, and almost smiled. \"Ah,\" he said, \"to be young! Now, enough of my ramblings. Look about you. We are driving into Paris. When you get home you can tell your family about seeing the most beautiful city in the world.\"\n\nWe stopped outside a large apartment building on the Champs \u00c9lys\u00e9es. The street was beautiful, broad and tree lined. The building we stopped at looked pretty grand. As I stepped out of the car, for one brief moment I felt like a tourist \u2014 until a German convoy drove past us and the doctor motioned me to hurry along. As he took me up four flights of stairs he told me that the plans had changed. We were not to be smuggled out through Spain, but taken to an airfield outside the city and picked up by a British plane. I wondered why we had to come into the city at all, but remembered that the countryside where we had been had become too dangerous. I was just relieved that there was a plan for us at all.\n\nThe doctor knocked three times on a door when we reached the fourth floor. I almost laughed. It really was like a Bogart movie.\n\nThere was an anxious couple of minutes before the door opened. When it did I couldn't believe my own eyes. Max! Max was standing there, leaning heavily on a cane. I almost toppled him when I grabbed him in a big bear hug. He returned the hug. Then we just stood there for a moment beaming at each other.\n\n\"Get in, get in,\" urged the doctor, looking nervously down the hallway.\n\nI hurried inside and the doctor shut the door. \"I see you two know each other,\" he said.\n\n\"We were in the same crew!\" I exclaimed.\n\n\"Well, then, you won't be bored waiting for the next step,\" he said. \"Someone will bring you food. Don't go outside for any reason \u2014 none. Do as you're instructed by whoever contacts you next. Someone else will be by eventually to take you to the rendezvous site.\" And before I could properly thank him he had left.\n\nI quickly took in my surroundings. We were in a small flat with a window at the back of the building. There appeared to be at least one bedroom \u2014 and our own bathroom, which was good, as sharing one down the hall would only increase our chances of being caught.\n\nI grinned at Max. \"So? What happened to your leg?\"\n\n\"Want the whole story?\"\n\n\"I bet we'll have enough time for the whole story and thirty more like it,\" I said. \"Looks like I'm stuck here with you for days!\"\n\n\"Looks like that.\" Max grinned back. He settled down on the couch and I took a chair opposite. Slowly he raised his leg and put it up on the couch. He looked thin. The crew used to call us Mutt and Jeff. Where I'm tall, big and muscled, he's short, thin and wiry. Where I'm blond and blue-eyed \u2014 a Viking, according to him \u2014 he's got dark curly hair and brown eyes. So when we'd go to London on leave together you'd think I'd be the one to get the girls! But it was always the other way round. They'd crowd around Max as if he were a honey pot and they were the bees.\n\n\"Did anyone else make it?\" I asked before he started.\n\n\"I know for sure that Stan and James made it,\" he said, \"and now you and me. Well, that's four out of seven. I saw the two of them run for it, but they didn't see me and I didn't want to call to them and maybe alert the Germans. I broke my ankle when I hit the ground. I saw some other chutes too \u2014 so maybe everyone got out. But I also heard gunfire and wonder if the skipper got away in time or...\"\n\nI shook my head. \"He didn't make it,\" I told him. \"I was told they'd recovered two bodies. But Bill made it out and we stayed together until we found him a doctor and some help.\"\n\nThere was a pause for a moment as Max took that news in, both the good about Bill and the bad about Lew. It sounded like Phil must have been the other one who didn't make it. Still, five out of a crew of seven surviving a crash \u2014 I guess that was more than we could have hoped for.\n\nThen I asked, \"How on earth did you get away after breaking your ankle?\"\n\n\"I knew I couldn't stay there so I just had to move. Wasn't pretty.\" He grimaced. \"I basically crawled on all fours all that night and even then I only reached some woods by morning. The next night I crawled again. I found a stream, though, so that saved me. I decided to stay near the stream. On the third day an old granny saw me \u2014 she was out picking berries. She told me to stay put. Well, I didn't know if she would come back with the enemy or with the Resistance so I crawled again and tried to hide. Some men came that night but I was afraid to show myself. Anyway, the next morning I could see a farmhouse not too far away and that afternoon one of the farmer's sons came down to the stream for water. But I soon realized that he was looking for something \u2014 someone \u2014 and I figured I'd best take a chance. I can tell you that by then my ankle was throbbing and aching something fierce. So this boy helped me into the barn and called a doc and, you know, that doctor came every single day after he set my ankle and checked on me. And not long after, a few fellows from the Resistance showed up and I starting helping out, making bombs. Have you seen those little ones, like sausages?\"\n\nI nodded. \"I made some too.\"\n\n\"So that was my job for weeks and then suddenly they said they had to move me and I got here just before you did! I was pretty nervous, I can tell you,\" he said. \"Didn't know if this fellow was to be trusted. Talk about a cold fish.\"\n\n\"I had the same reaction when I first met him,\" I agreed, \"but he grows on you.\"\n\n\"Tell me how you got here,\" he said.\n\nI filled him in as briefly as I could about what had happened to me since the plane had been downed.\n\nWhen I was finished he said, \"So just showing off then, eh?\"\n\nI smiled. \"Danes _never_ show off,\" I said.\n\n\"Danish Canadians obviously do,\" he countered.\n\nI paused before asking the next question. \"Did anyone find out about...\"\n\n\"Me being Jewish?\" he said, finishing my sentence. \"I decided that might be a good thing to keep to myself.\" He paused. \"But you know how you feel like a fish in a fish pond just waiting to be scooped up?\"\n\nI nodded. \"That's the way I've been feeling most of the time.\"\n\n\"I feel more like a walking target,\" he said, \"with a big bull's eye on my back.\"\n\n# Chapter Thirteen\n\n_July 2, 1944_\n\n\"Montreal smoked meat!\" Max said.\n\n\"Danish meatballs!\" I countered.\n\nWe had been occupying ourselves with the game, \"Who makes better food, Danes or Jews?\" for over an hour. We really shouldn't have been talking about food \u2014 it was just making us hungrier. There was no food in the apartment and no one had arrived with any.\n\n\"I thought maybe that doctor was going to take me straight to the Germans,\" Max said, finally changing the subject.\n\n\"I wondered too,\" I admitted, \"until he started talking. No, I'm pretty sure he's okay.\"\n\nAnd it was just then that we heard the three knocks on the door. My heart started pounding. Was this it? Were we going to the rendezvous already? And why was it always in broad daylight?\n\nI went to the door and cautiously opened it a crack.\n\n\"The doctor sent us,\" said a lady, maybe in her thirties. She had blond hair and wore a little black hat on her head. She slipped into the apartment. \"Time to go.\"\n\nThere was no point in questioning her. How would we know whether she was on the up and up or not? And surely the only people who knew about us were the doctor and his contacts.... Of course I also knew that those contacts could go rotten at any point. Max and I exchanged a look.\n\n\"Both of us?\" I asked.\n\n\"Oh yes, both,\" she answered. \"Come now, we must hurry.\"\n\nMaybe it was because she had knocked just when Max had been wondering whether or not we could trust the doctor, but I suddenly had a funny feeling in my gut. I looked at Max. He just shrugged. I'm not sure whether the shrug meant \"What choice do we have?\" or \"I think she's okay.\" We only had two choices \u2014 go along or run for it. But how long could we last in Paris, with no papers? My French was passable and had certainly improved over the last few weeks. Max's was excellent since he was from Montreal, but his accent would certainly give him away.\n\nWe grabbed our jackets and followed her down the stairs. When we got to the street she told us to stick near her. She walked just ahead of us for a number of blocks and then turned into a small apartment building. She hurried up the stairs to the second floor and opened one of the doors. \"I'll bring you some food later,\" she said. \"Don't go out unless you need to use the loo, end of the corridor.\"\n\nAnd she was gone.\n\n\"Whew,\" I said to Max. \"I was suddenly sure she was going to turn us in!\"\n\nWe looked around. It was a very small flat with an old couch, a rickety old chair and seemingly nothing else.\n\n\"Not exactly the Ritz,\" Max said.\n\nI went to the small sink in the kitchen for some water, but there was none. \"And no running water,\" I sighed. \"We've certainly come down in the world, Max.\"\n\nHe sat down slowly on the couch and put up his leg.\n\n\"You kept up pretty well,\" I said to him.\n\n\"Not much choice,\" he replied, but I could see that now he was paying the price. His face looked a bit green.\n\nIt was hours later that the woman returned with a basket of bread, cheese and wine. She rushed off again before we could ask her anything.\n\n\"Have they never heard of water?\" I said to Max grumpily as I drank some of the wine, which did very little for my thirst, as the day had just gotten hotter and hotter and the small room almost unbearable.\n\nIt was mid-afternoon when a knock at the door sounded and a different woman altogether came into the room. She was quite a bit older, stout and had a grim nod for both of us. \"Come on,\" she said. \"We're on our way out of the city.\"\n\nNow this was the news we'd been waiting for! My spirits rose and I forgot all about being hot and tired and thirsty. I figured that maybe by tonight I could be back at base being debriefed.\n\nI could see Max thought it was pretty swell too.\n\nWe followed her down the stairs into a waiting car. Our guide got into the front passenger side. A young man was driving but he had his cap pulled over his eyes and I couldn't get a clear view of him. The car screeched away from the curb and I thought that perhaps the driver could be a little more discreet, but he didn't seem to worry about that. In fact, he careened from street to street as if drunk, and that feeling I'd had in my gut earlier came back full force. This couldn't be right.\n\nI had just had that thought when three large black cars pulled up around us. Our driver slammed on the brakes. A feeling of dread washed over me.\n\n\"We're for it,\" I murmured to Max.\n\n# Chapter Fourteen\n\n_July 2, 1944_\n\nThe back doors were thrown open and I was roughly pulled out on my side as Max was on his. Two huge men patted me down and searched me while two others held machine guns on me. Of course I had no gun, no weapon of any kind \u2014 why were they so scared of me? What could I do to them?\n\nThey pushed Max and me back into the car. It jerked ahead again, except now the woman had a gun pointed at us.\n\n\"Traitors!\" Max said in disgust.\n\nShe didn't answer.\n\n\"Where are we going?\" I asked.\n\nShe didn't answer that either.\n\nI felt so disappointed and so scared that for a moment tears burned at my eyes. And then I got mad. We had been _this_ close to getting away...\n\nDid the doctor know he had a breach in his cell \u2014 or worse, was he part of that breach? Somehow I still didn't think it was him. But now I wished I'd listened to my gut. However, at least I'd had a chance to do some damage to the Germans before getting caught.\n\nMax was silent. I'm sure he was wondering how long he could keep his secret. And would they even believe us when we told them we should be treated as prisoners of war? We had no identity tags. They could say we were spies and just have us shot.\n\nIt was a long drive. I don't think I really saw a thing as we moved through the city. Finally we entered a courtyard and the car stopped. Armed guards dragged us out of the car. I could see we were at a prison, a very large one. The windows were covered in heavy bars and more armed guards marched along the walkways above us.\n\n\"This is Fresnes Prison, I bet,\" Max said quietly to me.\n\nThe words were barely out of his mouth when the guard standing beside him hit him on the back of the head with his rifle butt. Max lurched forward. I managed to catch him before he fell. Then both of us were pushed from behind. One of the guards yanked open a huge steel door and then we were inside. A small entry gave way to another steel door. When that one slammed shut behind us I was quite sure there would be no escape from this place.\n\nWe were marched up a flight of stairs into a small room with only one chair and a small desk. An empty chair. The guards stayed with us, guns trained on us at all times. I glanced over at Max. His head was bleeding.\n\nA Gestapo agent sauntered into the room and sat down behind the desk. He looked at me. Then he spoke in English. Quite perfect English.\n\n\"Name?\"\n\n\"Sam Frederiksen.\"\n\n\"Rank?\"\n\n\"Sergeant.\"\n\n\"Serial number? Service number?\"\n\nI gave him my service number.\n\n\"Place of birth?\"\n\n\"According to the Geneva Convention governing captured prisoners,\" I said, \"I am only required to give you my name, rank and service number, all of which I have done.\"\n\n\"Squadron number?\"\n\n\"According to the Geneva Convention,\" I repeated, \"I am only required to give you my name, rank and service number.\"\n\nOne of the guards jabbed me so hard in the lower back, just over my kidneys, that I fell forward. Max made a grab for me but I waved him away.\n\nThe guard came in front of me then and hit me so hard across the face that I reeled away in the other direction toward the door. \"You'll be dead soon anyway,\" he growled, \"so it doesn't matter.\"\n\nThen they turned their attention to Max. He got the same questions and gave them the same answers.\n\n\"You've been working with the Resistance,\" the Gestapo agent said quietly. He had a long face and a hawk nose and a perfect English accent. I suspected he was not German, but had no idea what his nationality was \u2014 perhaps French? And perhaps someone who had attended Oxford or Cambridge, because his accent was very like John's.\n\n\"I was shot down over France and have tried to evade capture as per my orders,\" I countered. I suddenly realized that if I admitted we had anything to do with the Resistance I'd be shot! I'd lose my status as a POW. Max took my lead and repeated what I had said.\n\n\"Bridges blown up, rail yards, roads, you know nothing of this?\"\n\n\"Nothing,\" I replied. And then I added, \"And from now on I will only give you my name, rank and service number.\" I said that because I remembered in our briefings we were told that if we were captured, not to get tangled up in trying to be clever during an interrogation. The interrogators were experts at getting information, even information you didn't realize you might be giving away. Best to stick to the basics.\n\nThe man behind the desk then said, \"We want to know how you got to Paris. Just tell us that and we won't bother asking you about anything else.\"\n\nSo they didn't know about the doctor! I vowed right then that I would die before I gave him away. And I gave them my name, my rank and my service number.\n\n# Chapter Fifteen\n\n_July 2, 1944_\n\nA guard knocked Max's cane out from under him and forced him to stand straight on his bad leg. Whenever Max faltered the guard hit Max's leg with the butt of his gun. But Max just picked himself up, stood up straight and stared at the man behind the desk.\n\nThe Gestapo agent asked me and Max the same questions over and over and we each gave the same answer over and over and in between we were hit or punched. I started to lose track of time. My main reaction, outside of the pain, was disbelief. We were prisoners of war, after all. We were flyers. How on earth could they justify treating us in this brutal manner? I'm not even sure I really felt the pain, in fact, I was just so shocked.\n\nFinally the Gestapo agent nodded to the guards and we were led away, Max without his cane and compass, both of us without our watches and our jackets. We were marched through corridor after long corridor and steel door after steel door until we were put into a cell with two other men. Both were French \u2014 one very young, perhaps my age, the other quite old, maybe in his sixties. The older one looked rough. His face was a mess of dried blood and he was hunched over in pain. The younger one, seeing my reaction to the state of the older fellow, said, \"He's just returned from his daily interrogation. He gets taken over to Gestapo headquarters every day. My turn tomorrow, I think.\"\n\nI introduced myself and Max but our two roommates didn't seem inclined to tell us much. I guessed why immediately. They might think we were spies. So Max and I huddled together and tried to figure out what had happened.\n\n\"Obviously a serious breach in the Resistance,\" he said.\n\n\"But probably not the good doctor,\" I replied.\n\n\"Probably not,\" Max agreed. \"I suppose we'll never know for sure.\" He paused. \"I think it was the last lot. The first woman seemed okay to me, and she must be a more direct contact of the doctor. I bet it was the last two who turned us in \u2014 and if they keep doing that, no one will ever be able to tell the doctor his contacts are compromised, and he'll keep sending people right into the Nazis' hands.\"\n\n\"But I bet he has other routes,\" I said. \"Maybe Ben and John went another way altogether. In fact, maybe that's why the doctor split us up \u2014 he suspected there was a breach somewhere. I mean, when we don't turn up for the flight it's bound to get back to him, right? Then he'll know something is up.\"\n\nMax sighed. \"Doesn't look like there's any way out of here. I just hope we'll be on our way to a POW camp today or tomorrow.\"\n\nOnly minutes after Max said that, another team of guards opened the cell door, took us out and marched us through still more corridors. We entered a large room. At the front of it were many tables, and seated behind the tables were men with mounds of papers surrounding them. I was pushed into a small cubicle at the back of the room and locked up. As I sat there I suddenly felt the pain of the interrogation. My face started to burn. My lower back, where I'd been punched, started to ache. I was thirsty and hungry and felt sick all at the same time. I wondered if I was going to throw up and if so, whether anyone would let me out of the tiny cubicle. I had a pretty strong hunch that they wouldn't so I took lots of deep breaths \u2014 my mom's recipe for avoiding vomiting. It worked, and by the time I was let out of the cell and marched up to the desk I was feeling a bit less woozy.\n\n\"Name.\"\n\nI gave my name. And my rank and my service number and that seemed to be all the fellow wanted. I saw Max being shoved toward a desk as I was pushed by a guard and marched out of the room. I was taken through more hallways. After a very long walk I was thrown into a tiny cell, empty of anyone else, and the door was slammed shut behind me. The walls were a dirty white. There was a small table and chair and a cot, all bolted into the floor, and a toilet in the corner, but there was no washstand or other running water. At the end of the cell was a window. It was barred, and on the inside of the bars was frosted glass. Quickly I checked the door to hear if there was anyone outside the cell. The door had a peephole, but only so they could see in, not so I could see out. It sounded all clear.\n\nThe glass was framed with putty. The only moveable objects in the room were a spoon and a bowl. I pried away at the putty using the spoon until the frosted piece finally came out. Heaven! Fresh air flowed in. And on top of that I could hear voices! Men and women calling to each other!\n\nI peered out the window. I was on the ground floor and could ony see a courtyard, empty at present, and on the other side the high walls and windows of the prison.\n\nI tried to catch what the voices were saying. Much of it was in French but I heard some English right off the bat. \"A cobra is a deadly snake.\" Then laughter. The voice was American. Was that a code name for a new offensive? I hoped so. The rumour was that Canadians were taking heavy casualties in and around Caen. I heard men shout their names and then others shout back if they knew them.\n\nI kept waiting for Max to be thrown in with me but that didn't happen. Finally I heard movement outside. Quickly I shut the window and sat on my bunk. The door didn't open, but a tray was pushed in through a narrow opening at the bottom. The only thing on the tray was a small bowl of fake coffee. No bread, no water, no anything!\n\nI slurped it up and after a while there was a rap on the door and the clattering of a cart. I put the bowl back on the tray and shoved it through the slit. I suppose that was correct because I heard nothing after that.\n\nI went back to the window then and opened it up. I heard men shout their names and others shout back if they recognized them. That's when I heard, \"Max from Canada.\"\n\nI yelled out, \"Sam from Canada!\"\n\n\"You okay?\" Max yelled.\n\n\"Sure!\" I yelled back. \"You?\"\n\n\"Never better!\" he yelled.\n\nAnd then others took up the messages and the banter.\n\nIt was getting dark and it had been a long day. I was aching all over. And thirsty. The fact that I wasn't hungry at all said everything about the shape I was in.\n\nI thought maybe it was time to lie down. I looked warily at the bed, wondering if there were lice and such in the mattress. After all, there was no running water, there was no way to stay clean. Probably the bed was the dirtiest place in the cell. I decided not to risk it. When I was in Grade Five, head lice had gone around my school. What a nightmare. I thought I would scratch my head off. I certainly didn't want to get infested \u2014 and these were the only clothes I had too. So I decided that the cold floor would probably be a better choice than the mattress. But the floor was so cold that after about a half hour I was chilled through. So I sat on the chair with my legs up on the table and drifted off that way.\n\nI don't know how long I'd been asleep when I woke with a start and cried out in pain. I had the most awful cramp in my calf. I leaped up and for a moment I didn't know where I was. I felt a fear so deep and so intense I cried out. After a few moments I came to myself. And then I heard the knocking through the wall in Morse code.\n\n\"All right?\"\n\n\"All right,\" I answered back.\n\nBut I wasn't. Would I be dragged out for interrogation at any minute? How much pain could I endure? What would they do to me? It was said that every man had a breaking point. What would mine be?\n\nFinally morning dawned and with relief I realized that I had made it through my first night with no one coming for me and with no torture.\n\nA tray was passed through to me. It was the same meal as the night before \u2014 fake coffee and nothing else. And then the day began to drag along. I could only live in dread of being tortured for so long. I needed to take my mind off it, so I decided to do something with my time. After all, what if they actually did beat me to death or simply shot me? No one would ever know what had happened to me.\n\nI noticed that there was an old rusty nail in the corner of the cell. I decided to use it to carve a message in the wall beside the bed.\n\n_Sam Frederiksen \nWinnipeg \nJuly 3, 1944_\n\nI had just finished my masterpiece when lunch arrived. It was a small bowl of something \u2014 some sort of soup, which tasted like water, and a piece of black bread. I ate it in practically one gulp. I wondered if they would just starve us all to death. It occurred to me that being this hungry was actually painful. It hurt. Of course everything hurt \u2014 and sleeping that way hadn't helped. I wondered if I would need to take a chance on the mattress after all.\n\nI paced up and down the little cell... up and down... up and down. Surely they would move me soon. I decided to throw the question out to the rest.\n\nI shouted out the window. \"Are they taking us to POW camps? Anyone know?\"\n\n\"We see fellows taken away,\" someone shouted, \"but no idea where to. We hear shots. Could be taken to firing squads.\"\n\nI wished I hadn't asked.\n\nI paced again. I definitely needed to take my mind off all this. I thought about my father's anatomy book I'd been studying. I decided to see if I could name all the body parts \u2014 or how many I could name. I thought I'd start with skeletal and then go to muscular.\n\nAnd after that I could try to name all the players on every single NHL team.\n\nBut before I forgot, I needed to mark the date. I needed to keep track of the time somehow. Underneath my name I placed a scratch with the nail and put a line through it and then another, since this was my second day.\n\nI peered out the window. The day was grey and there was a light rain falling. It reminded me of my very first operation with my crew in _H Hall_. The weather had been exactly like this that day, and I remember tilting my face into the rain while I waited on the tarmac. I remember wondering if that would be the last time I would feel the rain on my cheeks. I had pictured dying in a flaming wreck, but I never pictured this!\n\nWe had flown to France on that first operation and encountered no problems until we dropped our load, but fighters were all around us. And that's when a Halifax, trying to avoid a fighter on its tail, dipped and pitched and ended up underneath us and then rose right under us and into us. Boy oh boy, when I saw it coming up toward us I figured we were done, but it just kind of bumped us right under my turret and then dove again.\n\nSkipper wasn't sure what damage had been done to _H Hall_ but he knew we couldn't land at Skipton when we got back. We couldn't take the chance that we were going to crash, and runways had to be kept clear, so we landed at Woodbridge and taxied in right past a Halifax that was in flames. We learned right afterward that the whole crew of the Halifax had made it out, and sure enough, it was the same plane that had nipped us. The thing I remember most about the whole thing was how calm Lew was the whole time and how it made the rest of us calm too. Lew. If I wasn't careful I was going to start to cry. Well, why not? No one could see me or hear me. So I did. I had a good bawl and actually felt a bit better after.\n\nAs I breathed in the smell of the fresh rain I thought, \"One day I'll stand in the rain a free man.\" It wasn't so much a prayer, because my father didn't believe in any of that and brought us up the same way. But it was definitely a wish. And just in case my father was wrong, I said a short prayer. \"If you do exist, God, I'd love a little help.\"\n\nI figured it wouldn't hurt to ask.\n\n# Chapter Sixteen\n\n_August 15, 1944_\n\nI had just finished marking off the date \u2014 44 marks, which made it August 15 \u2014 when my cell door opened and I was ordered out the door. I felt a wash of mixed emotions. I was thrilled just to be out of that cell. I'd gone over all my anatomy charts in my head, the NHL lists, every song I could remember, done multiplication in my head, division, calculus even. I'd pretended to eat every single meal my mom ever cooked for me \u2014 but that became too painful so I stopped that game. I listed all the girls I'd liked over the last three years and made imaginary columns of good traits and bad and then had given each a score. The best part of my days, though, was listening and participating in the chatter. Mostly it was rumours that every day got more and more hopeful. The Allies were getting closer and closer to Paris and then soon we'd be free. Sometimes it was arguments over baseball teams or hockey players, and those were great too. But we had only so much energy for yelling back and forth, and often there was just silence.\n\nI'd look out the window at the concrete and try to imagine what I would do the day I got back to England. And the day I got back to Winnipeg. I had to stop those daydreams though, because they all seemed to be about food. Being alone was the worst thing I think I'd ever experienced.\n\nSo when they took me out of the cell, somewhere in my mind I knew I might be going to a firing squad, but I almost didn't care. When they pushed me into a large cell with four others, I almost cried like a baby \u2014 especially when right away I saw that one of them was Max! I grabbed him in a huge hug until he almost had to push me off. \"Where have you been hiding?\" he asked.\n\n\"All alone in a cell,\" I replied. \"What about you?\"\n\n\"Right here. Allow me to introduce,\" he said, nodding toward the other three men in the cell. \"Lester Wiebe, also RCAF.\"\n\nWe shook hands. \"Just call me Les,\" the fellow said. I was pretty sure I recognized him from the Mess back at Skipton.\n\n\"Trent Fox, RCAF,\" Max continued.\n\nI did know Trent. He was from Calgary. \"How are you?\" I asked.\n\nHe shrugged. \"Seeing you must mean that things are changing \u2014 but for better or for worse, we don't know.\"\n\n\"Louis Meyer,\" Max said, and then muttered, \"Jewish and a Frenchman too.\"\n\nI nodded and shook his hand. He was a thin tall man in his twenties. His face was puffed up black and blue.\n\n\"I see they're treating you well.\"\n\n\"Well as can be expected,\" he answered in French. \"At least I'm still alive.\"\n\n\"And finally...\" Max said, and from behind Trent, James, our wireless operator, stepped out!\n\nI couldn't believe my eyes and ended up giving him a big hug too.\n\n\"Don't know why you're so happy to see me,\" he complained. \"I'm stuck in this rotten place with you!\"\n\nJames was a character. He had been studying English at the University of Toronto before he'd signed up and he always had a quote to throw into the conversation, or some interesting story about Oscar Wilde. I remember when I first met him he said to me, \"I suppose you're familiar with the most famous playwright of our day, George Bernard Shaw, who wrote _Arms and the Man_?\"\n\n\"Yes,\" I said, \"I studied him at school.\"\n\n\"Did you know,\" he replied, \"that he is working on a sequel called _Legs and a Woman_?\"\n\nI smiled as I thought of that and grinned at him.\n\n\"James, I'm certainly not happy about you being stuck here,\" I said. \"Not at all. But I'm very happy you're alive! What happened to you after we were shot down?\"\n\n\"This and that,\" he said.\n\nAnd suddenly I realized that although the fellow called Louis was probably just who he said he was, none of us could openly admit to being with the Resistance, just in case there was a plant listening. Or in case someone was tortured and then gave us away.\n\n\"There's a lot of that going around,\" I answered him.\n\nI turned to Max. \"So you've been here the whole time,\" I said, just confirming what I already knew from our shouts back and forth to each other.\n\nMax nodded grimly. \"We've had the privilege of being in a cell next to one where they've been torturing both men and women. We hear them scream morning, noon and night. Every day they've taken each of us out for 'questioning.'\"\n\nThat's when I looked at them more closely. They all looked bad. Max had bruises on his arms and I could see now that all the men were bruised and probably black and blue under their clothes.\n\nI shook my head. What could you say? I wondered why they had been treated like that and I had been left alone.\n\n\"Have you heard the news?\" Max asked.\n\nI knew what he meant right away. I'd heard the news too, called out over the yard from cell to cell.\n\n\"The Allies are about to surround Paris,\" I said.\n\n\"And soon we will be liberated,\" said Louis with feeling.\n\n\"Where does all this information come from?\" I asked.\n\n\"Some from the French guards,\" Louis said. \"But we have friends who know the tunnels underneath this prison. They get in just long enough to tell us the latest and then they get out.\"\n\n\"And what's the scuttlebutt on what's going to happen to us?\" I asked, thinking that it certainly didn't sound like Louis was a spy.\n\n\"That's all we've been discussing,\" James said. \"Trouble is, the Germans won't want to just leave us behind so we can be freed to fight against them. Seems there's quite a lot of airmen stuck here as well as Resistance fighters,\" he explained. \"This prison can hold thousands \u2014 Louis here says perhaps there are as many as thirty-five hundred prisoners in here right now.\n\n\"So, that leaves two possibilities. Either they shoot us and that takes care of that, or they take us out of here and ship us east into Germany. But can they spare enough men to do that?\"\n\n\"See, we've already fulfilled part of our job,\" said Max. \"After all, every man they use to guard us is one less that goes to the Front.\"\n\n\"In which case, if I were them I'd shoot us,\" I said.\n\n\"Ah, but there's one problem with that,\" said James. \"I think they know they're going to lose. And if they shoot such a lot of prisoners with so many witnesses, they will certainly be found guilty of a terrible crime. Or if they try to shoot so many in the Resistance, maybe the city will rise up against them.\"\n\n\"Hah!\" exclaimed Louis. \"As if those cretins care about that!\"\n\n\"Maybe some do and some don't,\" said James. \"At any rate, I suspect we'll find out soon enough. There may not be much time left before Paris is taken from them.\"\n\nShortly after that we were given our so-called lunch, and a bit later the door to the cell was thrown open and we were told it was time to go. We straggled out, hoping for the best.\n\nWe were led back through more corridors and one steel door after another, through the inner courtyard that had been my \"view\" and finally into the outer courtyard where we'd first arrived. There were already at least a hundred men standing there and a large group of women was herded in behind us. We were watched over not by the soldiers who had been guarding us, but by a new group in grey uniforms with what looked like side-by-side lightning bolts stitched onto their collars \u2014 SS! They carried machine guns over their shoulders, grenades on their belts, pistols in their holsters. Most of them were under thirty, fit and mean-looking. They shoved the women in behind us with no mercy and then made us stand there as the courtyard filled up with more and more prisoners, until there were at least a few hundred and more coming in every minute.\n\nMax was beside me the whole time. We made sure to stick close together, along with James and the others.\n\n\"At least they aren't going to shoot us,\" said Max. \"Way too many here for that.\"\n\n\"So?\" I said. \"Are we headed to Germany?\"\n\n\"I'll never make it out of Germany alive,\" Max whispered to me. \"I'll have to try to escape.\"\n\n\"I'm with you,\" I said.\n\n# Chapter Seventeen\n\n_August 15, 1944_\n\nWe were put on a bus and driven through the city with a young Gestapo officer as our \"tour guide.\" If it hadn't been so clearly sickening, I think I would have laughed. He talked away, showing us the sites, with no idea how disgusting it was for us to listen to him gab on as if he and his gang owned the city and could talk about it as if they belonged there and weren't occupiers and thugs! And this just after telling us \u2014 with a smile \u2014 that if anyone tried to escape they would be shot on the spot. I reminded him that we were RCAF flyers and should be treated as such, and he just said, \"You are sounding very foolish. You were in a prison with Resistance fighters, therefore you are Resistance fighters and we are treating you exactly the way we treat them.\"\n\nMax and I exchanged glances then, and my heart sank. Did that mean we were not on our way to a POW camp? That's what we had been hoping for, after all. That's the way we _should_ be treated, no matter how we were caught or what their suspicions. We were still flyers! I looked at the Gestapo officer and his steely demeanour and was sure that he had no mercy in him at all.\n\nWe were lucky, I suppose, to be on a bus. The rest of the transport consisted mostly of large trucks with no windows, so those men had no idea where they were headed. I followed where we were going quite closely and figured we were moving into the eastern part of Paris, and then we were driven into a rail yard, a very large rail yard. We pulled up along a siding and were yelled at to _\"Raus! Raus!\"_ which by then I knew meant \"Get out.\" In front of us were cattle cars \u2014 not passenger trains but cattle cars! On the outside of each one was written _40 Hommes \u2013 8 Chevaux_. They were small old boxcars made of wood. The sinking sensation I'd felt when the SS guard said we were to be treated the same as the Resistance got worse when the Germans screamed at us to get in. For a moment I wondered whether or not I should step up and demand we be treated with the respect we deserved, but those guards looked positively crazed, screaming and yelling at everyone, and by then there must have been the entire prison emptying out onto the siding.\n\nThat's when I heard it. The sound of thunder. I looked at the sky. It was around midday by then and I couldn't see a cloud anywhere. In fact it was getting hotter by the minute. That sound could be only one thing then. Artillery fire! That's how close the Allies were! No wonder the Germans were so frantic to get us away. I suppose we were lucky they hadn't just shot us.\n\nI noticed the Germans starting to swing themselves up into the boxcars on either side of us. Max and I were two of the last to be shoved into the cattle car and I swear there must have been almost a hundred men in there already. It was so packed that once in we couldn't move from our spot near the door and almost fell out again. And then the huge door was pulled shut and bolted and immediately it felt like there was no air at all. So many unwashed bodies in such close quarters.\n\nIt wasn't long afterward that the train started with a jolt and everyone fell against everyone else. I said I was sorry to a poor fellow whose toe I stepped on and then realized it was James when he said, \"'I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams.'\"\n\nDespite the terror of what was happening, I had to smile. I remembered that quote from _Hamlet_ when we'd studied it at school. And for one split second I was back at St. John's High, sitting at my desk behind Sadie Kobrinsky, her dark hair curling down her back, and me wondering if I had any chance at all of her saying yes if I asked her to go skating with me over the weekend. And how she had said yes, and less than two years later when she heard I was going overseas, she came to the house and kissed me right on the lips and wished me good luck. Right then I decided that if I made it home I was going to look up Sadie and see if she was free for a skate on the river.\n\nThe train jolted again, this time coming to a stop. And then it started again. I was taller than most and could see over everyone's heads. There was a bit of air coming in from windows beside the doors on either side. I suppose you could call them windows. They were really rough openings covered by steel bars and barbed wire. Still, at least some air got through \u2014 not enough to kill the smell, but enough to keep us alive.\n\nIt was barely possible to sit, let alone lie down, but at first there was lively chatter as friends who hadn't seen each other in weeks or months got caught up on what had happened to them, how they had been betrayed, or what had happened in Fresnes. But as the day wore on the chatter wound down. The ride started to take on the feel of a waking nightmare. For a bathroom there was one bucket at the end of the car and soon the smell was overwhelming \u2014 and when anyone had to get there it was almost impossible to move through the packed bodies. I was so tired that at times I almost fell asleep standing right there, but was always jostled awake by another stop or start.\n\n\"It must be all that bridge and rail work we did,\" Max said proudly. \"We've blown up so much they can hardly make any headway.\"\n\nI was pretty sure he was right. We were going so slowly, in fact, I wondered if there was someone walking ahead of the train looking for missing tracks or blown-up sections. That speculation turned out to be justified when the train screeched to a halt in a tunnel and didn't start up again. Smoke from the engine started to fill up the car. I began to cough. So did everyone around me. As the air became more difficult to breathe I started to panic and wondered if we would all suffocate right there and then. The air turned black. Tears streamed down my face. It occurred to me that we should get low, the way you do in a fire, but there was no room to sit. There seemed nothing to do but bang on the doors and yell for them to be opened, which those of us close enough to the doors did.\n\nFinally, well over an hour later, the doors were thrown open and we were screamed at to get out. We staggered onto the dirt track, still in the dark tunnel, gasping for air.\n\n_\"Raus! Raus!\"_\n\n\"I'll _raus_ them,\" I muttered.\n\nThe guards grabbed their kit from their boxcars and gave it to us to carry. I had to shoulder a bag that must have weighed 50 pounds, as did Max. James was pushed to an outside line and in English we were told that if anyone tried to escape, those on the outside line would be shot first. And then we started to walk.\n\nIn a few minutes we were out of the tunnel and could see that we were in the countryside with a river on one side and above us, hills. People were riding bikes along our route. I suspected they were Resistance fighters \u2014 especially since many were women and I thought they might be looking for relatives in the group. That's when Max whispered to me, \"I heard that the head of the Paris Resistance was in Fresnes when we were. Maybe he's on this transport. Maybe the Resistance is going to try to free us.\"\n\n\"If that means we don't have to go back into those trains again I'd risk anything,\" I said. \"Even a firefight.\"\n\n\"Quiet!\" screamed a guard.\n\nThey loved to scream. I wanted to tell them they sounded like little children with all their screaming \u2014 Allied servicemen would never stoop so low \u2014 but I decided to keep my mouth shut.\n\nIt was a long walk and I was getting weak from the lack of food and the heat and the coughing that still racked my body. When we finally reached a town and another train station and saw another train waiting for us, it turned out to be more cattle cars. I wasn't sure I could get inside one of them again, but what choice did I have? If I ran for it, even if I made it, someone else would die for me.\n\nBut that's when I saw something I hadn't expected. On the platform was a small group of women wearing Red Cross armbands. They were giving out both food and water. I was handed a piece of bread with some jam on it, and water to drink. It was great \u2014 even if the bread tasted more like sawdust than actual bread. And the water really helped calm down my cough.\n\nI looked up to the hills. Quite a lot of locals were standing there watching us, but we were shoved back into the cattle cars, so it seemed that there was to be no rescue. Had the Resistance not been able to muster enough men that fast? I suspected that was the case. After all, we were guarded by elite troops, and lots of them. What would be the point of a bloodbath?\n\nThe day dragged on as the train stopped and started. As night drew on James told me that he'd had an idea. He suggested that if everyone sat with their knees up we might all be able to sleep. Since I was one of the tallest I was best able to convey the message, so I yelled at the top of my voice and we got a very good response. Soon everyone was seated and we did manage to finally get some sleep.\n\n# Chapter Eighteen\n\n_August 16, 1944_\n\nBy morning nothing had changed. We received no food until late in the day. When the train stopped and the doors were finally opened we saw the Red Cross on the platform again. This time we weren't allowed out. Food and water was passed in to us. The water came in a large metal can and we had to pass it from person to person until everyone had a drink. But when it came back empty a Frenchman near me grabbed it and then crouched down with it. Intrigued, I bent over to see what he was going to do. He took the metal top and used it to lever up the nails that were holding down the wooden planks on the floor of the car. That's when the guards started calling for the water, which was okay because the fellow was done. There was a plank that was now off, but could be put back as needed, so if the guards checked from below it would look like the plank was in place. We passed the can back out and then the guards shut the doors.\n\nA group of Resistance fighters had gathered together and had begun to discuss the best way to use the escape hatch.\n\n\"We need to be careful about when to go,\" one said. \"If we jump when the train is stopped they'll catch us. If the train is going too fast, we die. It needs to be just as the train is picking up steam, leaving the station.\"\n\n\"Can we send one of ours?\" I asked the fellow who had just spoken.\n\nHe consulted with his friends and then said, \"First five of us go, then you can choose five, then us five and so on.\"\n\nI turned to Max. \"You need to be the first of us.\"\n\nHe nodded.\n\nMy heart was pounding as we waited to get to another station and the first escape.\n\nIt was full dark when the train screeched to a halt. We were stopped for a few minutes, which felt like hours to me, the Frenchman poised over the opening on the floor. Finally the train lurched forward. The man waited about a half minute and then dropped. He was gone. We replaced the plank.\n\nThe train picked up too much speed right after the drop for anyone else to attempt an escape. So we waited. But happily we didn't come to a screeching halt followed by rapid gunfire! I hoped that the fellow had made a successful escape.\n\nIt was hours later before our next stop. Every minute seemed like an age to me, especially because there were many others in the line ahead of Max and then, hopefully, me. Most of the men in the car didn't even realize what was happening, which was just as well, I thought, since the lineup would be immense and fights might even break out about who was to go next. Although we had all followed the same pattern as the night before, sitting with knees up, those of us around the escape hatch, by staying low and just shuffling positions with each other, managed to keep what was happening secret.\n\nThe train started to slow and came to another stop. The second Frenchman got into place and we took the plank off for him. We were stopped for about ten minutes and then started again. He waited until we had picked up some speed and then he dropped through.\n\nThree more went with no problem as the night wore on, and finally it was Max's turn. The train was stopped, Max was poised and ready to go when flashlights combed the outside of our car, the light beaming through the windows near the two doors on either side. This went on for a few minutes.\n\n\"Do you think they found the last guy?\" I asked James, who was in line to go after me.\n\n\"No chance,\" he replied. \"They'd be tearing apart the cars if they had. But maybe they noticed something \u2014 a shadow \u2014 and they didn't know whether it was a man or an animal or a tree...\"\n\n\"Should we keep going?\" I asked.\n\n\"I'm going,\" Max said. \"It's my only chance. Otherwise I'm a dead man.\"\n\nI certainly wasn't going to argue with him. He had to make that life-or-death decision himself.\n\nMax shook my hand.\n\n\"I'll be right behind you,\" I assured him. \"But don't wait around for me. Just get out of here and find our guys. And then do some damage to those Nazis.\"\n\n\"I'll do my best,\" he said.\n\nThe flashlights were turned off and shortly after, the train lurched forward.\n\n\"Ready?\" I said.\n\n\"Ready!\"\n\nAnd then he was gone.\n\nWe put the plank back in place and I waited for my turn. I started thinking about what to do when I dropped. Should I stay on the track until the train was far away, or roll and run the minute it passed over me? From what I could see out our windows, every time the train stopped the SS dropped from their cars and surrounded each prisoner car so that no one could escape through a door or one of the windows. Then as the train started they grabbed the rails and leaped onto their cars. If they were looking back they might see me get up, so staying put until the train was out of sight would be the ticket.\n\nAnd then I vowed to avoid all farmhouses, all help and to somehow make it to the Allied lines.\n\nFinally the train came to a stop. And that's when someone near one of the windows called, \"They're looking underneath the cars with flashlights!\"\n\nI looked at our plank. We'd be fine. It was in place and no daylight would show when they beamed their flashlights on it.\n\nBut then James said, \"Sam. Sam! Look.\"\n\nHe was pointing to the edge of the plank. It was sticking slightly up.\n\n\"We need to take it out and put it back in again,\" I hissed.\n\n\"No,\" he said. \"If they're underneath they'll see it right away. Press down on it, just try to push it back.\"\n\nI pressed down on it and so did James and some others, most of us getting up, not caring who saw us at that point and practically jumping on it to make it fit back in properly.\n\nSuddenly we heard, _\"Ach! Ja!\"_ from below us.\n\n_\"For fanden,\"_ I swore.\n\n\"Damn and blast!\" James echoed.\n\nAnd then there was nothing left to do but wait.\n\nMinutes later the door to our car was pushed open and about ten SS with machine guns pointed at us pushed us out of their way until they found the loose plank. They had brought boards and nails and within minutes they had put down a new plank. Then screaming words like _Schweinehund_ and shoving their way through the men, they exited the car and rammed the doors shut behind them. We sat back down. I could hardly breathe, but whether that was from the fear of discovery or the horrible disappointment, I wasn't sure.\n\n\"Never mind,\" said James. \"Another time.\"\n\nI nodded, but couldn't even manage an answer, I was so overcome.\n\n# Chapter Nineteen\n\n_August 17, 1944_\n\nMorning came and went. We didn't stop at any more stations and were given no food or water. Finally \u2014 it must have been about noon \u2014 the train stopped and the doors opened. We weren't at a train station though. We were ordered out.\n\nWe straggled out, almost too stiff to manage the jump to the ground. Once we were all gathered in front of the boxcar, we could see that we were in the middle of a forest and that we were the only prisoners who had been \"allowed\" out. SS troops formed a line in front of us, then raised their submachine guns and pointed them at us.\n\nThis is it, I thought. Punishment for last night.\n\nAs if he could read my thoughts, James said, \"It's our duty to escape. We have been ordered to try to escape at every opportunity despite the repercussions. Chin up. We die proud.\"\n\nI stood straight then and thought, He's right. None of this is my fault. It's those Germans right there who started all this and none of us would be here if not for them. Damn it, but I'm not going to die shaking in my boots.\n\n_\"Ausziehen! Ausziehen!\"_ they started to scream at us.\n\nOne of our fellows called out, \"They want us to take our clothes off.\"\n\nFor a moment no one complied. Then an SS guard shoved a man to the ground, kicked him and started pulling his jacket off.\n\nSo, reluctantly, we started to take our clothes off. Once we were down to our skivvies the guards kept yelling \u2014 they obviously wanted us naked. Now I was sure we were about to get shot, and instead of feeling embarrassed I got mad. Really, really mad. It wasn't enough for them to kill us, they had to humiliate us first.\n\nI started to hum \"You Are My Sunshine.\" My mom had sung that song while making dinner the last time I was home for leave before I shipped out. It made me think of home and family and love and hot food and cozy fires, and if these were my last moments, that's what I wanted to think about, not the hate-filled monsters in front of me. Somehow I felt my anger seep away and I stood there, naked like the rest, but ready.\n\nAnd then something shocking happened. The Germans ordered us back into the train. Still naked.\n\nWe scrambled back into the boxcar. Once we were inside they shut the doors and the train started up again. As before, we were stuffed together, but this time with no clothes. Now I _was_ embarrassed! Still, there was nothing for it. Everyone started talking at once and soon the reason for the punishment made its way through the car until all the prisoners knew about the escapes. From what I could gather, most of the men were happy that at least some had got away, and didn't begrudge the attempt. It was our duty, after all, to make an escape if at all possible.\n\nJames and I were discussing all this when the unmistakable sound of a rifle shot ripped through the car just as the train came to another stop. The doors were opened and the guards once again pushed their way on board. It didn't take them long to find the person who had been shot. The word went through the car quickly. He'd been leaning up against the window, using the bar to hold himself up so he wouldn't bump into others \u2014 maybe he was more conscious of that now we were all naked \u2014 and the guards had seen his hand up against the opening and shot him! As he was dragged out past me I was sure I knew him. And then it came to me. It was the young boy who had first called me an idiot. Michel! He was covered in dirt, his hair awry, and he was thin, as if he hadn't eaten in weeks. No wonder I hadn't recognized him earlier. He must have been swept up in the raids somehow and put in Fresnes prison as well. He was just a kid! How could they throw him in prison? And now what?\n\nI squeezed myself in between the men to get to the opening of the cattle car and see what was happening to him. The guards yelled at him, accusing him of trying to escape. Because he had his hand on the window opening? Perhaps this was just another way to punish us or to demonstrate that no more escape attempts would be tolerated?\n\nA guard ordered him to walk ahead. Maybe they were actually going to get him help for his injury. Everything was quiet. The forest even seemed still. No birds sang and no wind blew through the trees. It was as if the world held its breath.\n\nI watched as the young boy, so thin I could have counted his ribs, staggered forward. And then the silence was shattered by machine-gun fire. Michel's arms flung out wildly and then he crumpled to the ground, shot in the back.\n\n\"You!\" a voice said in English. A guard was pointing at me.\n\nI pointed to myself.\n\n\"Yes, you. And beside you, that guy. Get down.\"\n\nI leaped down onto the ground. James, who had been standing next to me, jumped down as well. I thought of taking a run at the guards, who stood with machine guns still raised \u2014 the murderers of a boy. I thought it would be worth it if I could take out one or two. But I knew that I'd be cut down before I took a step toward them.\n\nI remembered that Max had told me the Germans were murdering Jews. I hadn't believed it. But what I'd just seen was evil in its purest form. And it made me feel something I'd never before felt. Despair. I despaired for the entire human race.\n\nWe stood there stark naked. One of the guards handed me a shovel and pointed to a spot beside the railroad track and motioned for me to dig. He did the same to James. We were to dig a grave.\n\nI was surprised they were even bothering to bury Michel. I thought about his poor mother and how she would never even know what had happened to him. I wondered what had happened to his sister Francine, or what would happen to her after the war, and thought again about the mother whose daughter was a collaborator and whose missing son was a hero. I felt tears burning at my eyes, but didn't want to show any weakness to the Hun.\n\nI was standing on wood chips and stones because we were near the tracks. Somewhere in my head I knew I was feeling pain and yet I wasn't really feeling it. I just dug and dug and then when there was a shallow grave the guard motioned for us to throw Michel in. Gently James and I picked up his bullet-riddled body and placed it in the ground. I wanted to say a prayer, but I didn't know any except the one we said in school, the Lord's Prayer, so I said that in my head as we began to cover the body with dirt. It was so quiet you could hear a pin drop. Not a sound came from the open car. Not a person spoke. The guards puffed away on cigarettes and sat on tree stumps as we worked.\n\nAnd when we were done James murmured, \"The rest is silence.\"\n\nI put the shovel down on the ground, looked at James, and we walked back to the train car. We lifted ourselves in and shortly afterward the doors were yanked shut. The train started up.\n\nSlowly the men started to talk again, more than likely about the murder they had just witnessed. I began to feel a stinging on my feet and looked down to see they were bleeding, as were James's. There was nothing to do about it though \u2014 we didn't even have clothes to use to wrap the wounds.\n\n\"The usual rules don't apply to these men,\" James said. \"I doubt they've even been taught about the Geneva Convention, unless it was to be mocked as weak, something to ignore. If we cannot somehow become prisoners of the Luftwaffe, I fear for us all.\"\n\nI could only nod in agreement. The train chugged along and we did stop finally for some bread and water and then somehow got into sitting positions for the night. That's when I noticed that the train was picking up speed. Somehow I just knew we had crossed the border and were out of France. And any chance for the Resistance to free us was gone.\n\nI felt numb. Hungry and dirty and surrounded by the rank smell of unwashed bodies and the open-bucket latrine, I was lost in despair.\n\n\"They win if we give up,\" James said, sensing my mood.\n\n\"I know,\" I said. \"But what does it say about the human race?\"\n\n\"It says that we are capable of that kind of brutality \u2014 worse than animals, really, who don't kill out of malice. But we are also capable of great good and even heroism. 'What a piece of work is man!'\"\n\n\"I suppose it would be easier,\" I said, \"if I believed in God. I could believe that it was all His will and maybe I could accept it.\"\n\n\"I believe in God,\" said James, \"and you know, it doesn't make this any easier. I just wonder how He could allow such things. But perhaps He doesn't get involved \u2014 just watches from on high and hopes for the best.\"\n\nHe thought for a moment. \"I feel closest to Him when studying Shakespeare. When I see a spirit that can describe the human condition the way Shakespeare does, then I see God.\"\n\n\"Can you recite any of the speeches from his plays?\" I asked.\n\n\"I know most of _Hamlet_ by heart,\" he said.\n\n\"Really? Can you say it for me?\"\n\n\"I can,\" he said, and he started to recite the play. Soon the men beside me stopped talking and started to listen, then the men beside them and the men beside them, until the entire car was transfixed.\n\nWe all listened more intently when he got to Hamlet's famous speech, \"To be, or not to be: that is the question: Whether 'tis nobler in the mind to suffer the slings and arrows of outrageous fortune, Or to take arms against a sea of troubles, and by opposing end them?\"\n\nI'd always wondered about that speech, so I asked, \"Does that mean to kill yourself?\"\n\n\"Well, that's where the debate begins,\" James said. \"I personally don't believe this is only, or even, about suicide, but about how we approach life. What happens when you take arms against a sea? Isn't that hopeless? But even if it is, perhaps what's important is the effort? We need to try.\"\n\n\"But doesn't he go on to say that he fears death more than he fears life?\" someone piped up.\n\n\"Well, yes,\" James agreed and then went on with his recitation, but with many interruptions as we stopped to discuss the words and the meaning.\n\nIt was such a welcome break in the horror \u2014 I might have gone crazy if he hadn't taken my mind off things in that way.\n\n# Chapter Twenty\n\n_August 20, 1944_\n\nIt was the sixth day when the train stopped at a station and someone peered out the window and called out, \"Weimar.\"\n\nSuddenly the door was opened and our clothes were tossed into the car. It took quite a while to find whose clothes were whose, but we finally sorted out one filthy pair of pants from another and most of us got our own clothes and got dressed. I began to hope that we were on our way to a POW camp and that they didn't want us to arrive naked \u2014 a clear violation of the Geneva Convention. We peered out the windows and saw a barbed-wire fence with men behind it dressed in blue and white striped pyjamas. They were surrounded by guards and looked so emaciated that I was amazed that they could even stand. I remember being told of millions of French men who had been dragged into Germany as slave labour and I wondered if that was what we were seeing and if these men would be able to hold on a little while longer until the war was over.\n\nThe train started again and chugged slowly up a hill. About a half-hour later we stopped and the doors were opened. We were herded out of the train, thousands of us, it seemed. We stood together just outside the main gate to \u2014 where?\n\nIn front of us was a high barbed-wire fence. Guards were screaming orders and dogs on leashes were snarling at us. Inside the gates I saw more prisoners in striped pyjamas, so emaciated they were little more than skeletons. What was even more disturbing was that they didn't seem to notice or care that a new group had arrived. They hardly glanced at us, and when they did their eyes looked dull and devoid of any curiosity at all.\n\nThere were guard towers spaced about a hundred yards apart, and behind us I saw woods filled with SS and their dogs, leading men in pyjamas \u2014 perhaps work parties? How would we ever escape from here? It looked impossible.\n\nThe steel gates were opened then and we were shouted at and screamed at to move inside. We had to walk a gauntlet down the tracks, between two lines of SS, their guard dogs snarling and growling at us. We immediately found ourselves in a large open area and the first thing I noticed was that there were few SS once we were inside. That could only mean one thing \u2014 the security was so tight that they didn't need to waste manpower on us once we were there.\n\nWe were marched past a huge building with a chimney that billowed smoke and gave out a sickly sweet stench. I wondered if the kitchen was situated there, and hoped that at least it meant we'd eat now, although the smell was really vile. There was a grapevine of news that started up right away, and before long I heard that we were in a place called Buchenwald. Then as we were walking, a guard came up beside me and said, \"The only way out of here is through that chimney.\" When he saw the look on my face he laughed. I thought I might be sick, even though I had no food in my stomach to throw up. Was that smell _human flesh?_ What kind of nightmare was this?\n\nWe were taken to a large grey building and split into groups of about fifty. I was in the second group. We were herded into a big room and told to strip. I had just gotten my clothes back! We were pushed into a line and I heard groans and moans coming from the front \u2014 I wondered if this was the beginning of an interrogation or even torture. As I got to the front, though, I saw a barber who was shaving heads. From the cries coming from big strong men, I figured this was not going to be any fun at all.\n\nWhen it was my turn the \"barber\" didn't so much shave my head as tear the hair out from the roots. I too cried out despite my determination not to.\n\nFrom there we went straight into a shower room. Although the water was freezing cold and hurt my head where I was raw from the shave, still, it was the first time I'd had any water on me since the doctor's house and it felt wonderful, even with no soap. I made sure to scrub my feet because I was worried that with the open wounds I might get an infection that could turn deadly very quickly.\n\nThen I was shoved into another line, where I heard more cries and groans. When I reached the front I was confronted by a guard who was seated with a tub between his knees. He took a brush, dipped it in the tub and covered me with something orange, which burned as if ants were biting me. From the smell I figured it was some kind of disinfectant. I suddenly remembered jumping into a pile of leaves when I was about ten years old, not realizing it was filled with ants, and the agony that had caused. This was about a hundred times worse.\n\nThen I was handed a shirt and pants and clogs. I noticed that I was one of the lucky ones, because only a few men were given shoes. I didn't turn them down though, because of the cuts on my feet. No one was given clothes that fit and since I was taller than most I looked around and found a fellow who was swimming in his. We switched both tops and bottoms, and I felt that at least I could manage.\n\nA tin bowl was then shoved into my hands and I was told to get in line. I did. I noticed that our group was now much smaller. We were then taken to a desk where I gave my name, rank and service number.\n\n\"I demand to be recognized under the rules of the Geneva Convention.\"\n\nThe man behind the desk didn't even bother looking up.\n\nHe said, \"You are a _Terrorflieger_ and will be held in a _kleines Lager_. You will have no access to the Red Cross and no recognition as a prisoner of war.\" Then he waved me away.\n\nDespite the terrors of the train ride, I had held on to the hope that we were on our way to a POW camp. Now we appeared to be in one of the dreaded concentration camps we'd heard so many horrible rumours about and perhaps no one even knew we were here! If the Red Cross didn't know about us, there would be no parcels. If the Luftwaffe didn't know, we'd never be treated as POWs. We could die in this camp and there would be no help for us. I thought about the German POWs at home who were being treated honourably, with three hot meals a day and books to read and clean clothes and water and soap and clean bathrooms! And this was how we were to be treated!\n\nThe worst was the feeling of total and complete helplessness.\n\nI followed the others out of the building and could see that it might be the downed air crews who were being herded together. We were taken past more and more long, low grey buildings and through two more gates until the guards stopped us in an open area surrounded by barbed wire and left us there. Lester, Trent and James joined me and they had picked up some news.\n\nTrent spoke German well. He pointed to some men who were in the middle of an argument and told me that one of them was called a _Kapo_ , a prisoner who acted as a guard for the Nazis, and that the other two men were high-ranking Allied officers. \"I hear that one,\" he said, pointing, \"is a squadron leader from New Zealand, and that one,\" he pointed to the other fellow, \"is a colonel from the States.\"\n\nWhat a relief! I couldn't believe our luck to have high-ranking officers here who were used to command and who could take charge, and who could hopefully lead us through and then out of this mess we were in.\n\nIt seemed that our new leaders weren't winning _this_ argument, though. Word soon spread that we had missed the evening meal, that there were no barracks for us, and we were to sleep outside. We were apparently being held in a quarantine camp before being placed in a hut. The area we were in was called \"Little Camp\" in English \u2014 I suppose that's what _kleines Lager_ meant. There were five big tents, but we weren't allowed into those, so I guessed they must be full. It looked like we had one option and that was to drop right where we were onto the rocky ground. There were already what looked like thousands gathered on the ground all around us, with more arriving every minute. Soon the Resistance fighters who'd been on the train with us were marched in. They huddled together beside us.\n\nBlankets were given out but there was only one for about every three men. I lay down but there was no way to get comfortable. The ground was made up of hard, lumpy rock and stones. It was a warm night at least. I hoped we wouldn't still be here in a month when the fall weather would make it impossible to be outside.\n\nI tried not to think too far ahead. Just get through tonight, I told myself. One thing at a time. Pretend you're camping at Winnipeg Beach. In the morning you'll get up and go for a swim in the lake and in the afternoon you'll eat hot dogs and french fries. And somehow I drifted off.\n\n# Chapter Twenty-One\n\n_August 21, 1944_\n\nWhen I woke up it was still dark and I expected to see a tent over my head and to have Jenny kicking at me through her sleeping bag, the way she always did. Was it possible to awake from a dream into a nightmare? Because instead of Jenny I looked up to see James shaking me to get up, saying we had to go to something called an _Appell_ or roll call. I followed the others to a large square where we, along with the rest of the camp, had to line up and be counted. It was just becoming light. The _Kapo_ s counted us once, but apparently the numbers didn't fit so they counted us again. The sun came up and began to beat down on our heads. At first I was thankful for the heat, because the damp from sleeping on the rocks had seeped into my bones. But by the time they had counted us three more times I began to feel dizzy. And there were thousands of emaciated prisoners to be counted.\n\nAfter about the third count, a prisoner was pushed onto a block of wood, which was set up in front of the rest of us. His shirt was taken off, and he was whipped! Whipped! I couldn't believe it. Each time I thought I'd seen the worst, I'd witness something else that I just couldn't credit. And then we were counted again and once more again. If anyone fell out of line they were kicked hard.\n\nFinally we were allowed to leave and were herded back to our enclosure. Once there the squadron leader called us to order.\n\n\"Attention!\"\n\nI stood to attention, as did the rest of the men who'd been milling about hoping for some food.\n\nThe squadron leader then went on to make an impassioned speech, telling us that we were in a fine fix and that the \"goons\" \u2014 a term that seemed to fit the Germans perfectly, I thought \u2014 had completely violated the Geneva Convention and were determined to treat us like common thieves and criminals. He told us that we would conduct ourselves as befitted our training and as representatives of our countries. He ordered us to march to all roll calls as a unit. Then he directed us to get organized by reporting to a commanding officer for each country and to give our names and details to that officer.\n\nQuickly I gathered with the other Canadians and was very happy to recognize some faces I hadn't seen yet. As I gave my home address in Winnipeg and my other particulars, my spirits started to improve. At least we would be under military command now and would work together for the best outcome for all of us. It seemed we were twenty-six Canadians in a group of one hundred and sixty-eight Allied airmen.\n\nIt was shortly afterward that \"lunch\" appeared. Not quite the hot dogs and french fries I'd been dreaming of the night before. We put our bowls out and the _Kapos_ slopped some liquid into them. I looked at it and gagged. Worms were swimming all over the top. James actually laughed out loud.\n\n\"Well, well,\" he said, \"at least they've given us some protein. Must be a mistake. Don't let them see or they'll take it away!\"\n\nI had to smile, despite myself.\n\n\"Drink up,\" he said. \"Beggars can't be choosers.\"\n\nI remembered eating a worm when I was a kid, as a dare. This couldn't be worse, could it?\n\nI drank it down. I gagged once or twice but managed not to throw it back up again. Then I was given a piece of bread \u2014 more sawdust than bread, it turned out. But I choked that down too, although later on my stomach didn't thank me.\n\n\"Have you heard?\" James said.\n\n\"What?\"\n\n\"The goons\" \u2014 that term seemed to be catching on quickly \u2014 \"have ordered us all to work in the factories here. Apparently they're making parts for the German planes and guns and such.\"\n\n\"I imagine we say yes or we die,\" I said. \"I think I might rather die. I couldn't in good conscience help the Nazis in their war effort.\"\n\n\"Easy to say,\" said Les, \"until you're standing in front of a firing squad.\"\n\n\"I want to see what's really happening in that place with the chimneys. I heard someone call it the crematorium,\" I said. \"Anyone else?\"\n\n\"Why?\" Trent asked.\n\n\"A guard told me that was the smell of human flesh. He said the only way out of here was up those chimneys,\" I answered. \"But I can't believe it. I want to know what kind of a place this is. My dad always said, 'Knowledge is power, ignorance is never bliss.'\"\n\n\"I'm up for a little trip,\" James said.\n\n\"Me too,\" agreed Trent.\n\nLes said he was going to help finish up the lists that needed to be made.\n\nThe three of us started out. It was quite easy to get back to the building we wanted to find out about.\n\nThere was a small brick building attached to the larger building with the chimney. We looked through a window. That's when I saw at least fifty naked bodies piled one on top of each other.\n\nI turned away, retching.\n\n\"We're in hell,\" I muttered aloud. \"This must be hell.\"\n\nA man who looked exactly like a walking skeleton hobbled up and stood in front of us. He had a green triangle attached to his pyjama top over a yellow triangle, so together it made a six-pointed star. He spoke German, but Trent translated for us.\n\n\"New here?\" he asked.\n\n\"We're Canadian airmen,\" I answered. \"We shouldn't be here!\"\n\n\"None of us should be here,\" he replied.\n\n\"What is that? What happened to them?\" I asked, pointing to the dead bodies.\n\nHe shrugged. \"Murder, starvation, beaten to death, worked to death in the factories where we're forced to help the German war machine. Maybe one of the experiments in the medical block \u2014 injected with experimental vaccine, or killed for their tattoos. The commandant's wife likes to use decorated skins to make lampshades and book covers.\"\n\nWas he joking with me? Some sort of horrible sick joke? Maybe he'd been driven mad.\n\nI wondered if the star on his shirt meant something. I'd noticed all prisoners at the roll call wore different triangles. \"What does that mean?\" I asked him, pointing to his star.\n\n\"Jewish,\" he said, \"and the other one \u2014 green \u2014 criminal.\"\n\nWere we talking to a murderer? I guess he read my mind because he almost smiled. He said, \"All Jews are criminals or political or something else \u2014 and that way, with one triangle on top of the other... well, it's ironic, but we end up with a Jewish star.\" He paused. \"I come every day to see if I recognize any bodies. It's the only way to find out who's been killed or who's died. You'll do the same pretty soon.\"\n\nHe pointed over to another building. \"That's where they do the experiments. They shrink heads. But I wouldn't advise going over there to look. You can come here, no problem, but that's a place I'd stay away from.\"\n\nI wanted to ask him something but didn't know how. Finally I said, \"A good friend of mine is Jewish and he told me that Jews were being rounded up. He told me there was torture, even murder.\"\n\n\"But you didn't believe it?\"\n\n\"I don't know what to believe, now, after seeing that,\" I said, jerking my head toward the bodies.\n\n\"We didn't believe it either,\" the man said sadly. \"My name is Leo. I'm from Frankfurt. All the Jews were rounded up there and shipped east. In Poland they were either killed on the spot or sent to extermination camps.\"\n\n\"Extermination camps?\" I said.\n\n\"Camps built just to kill Jews. I've heard they use gas because they can kill faster that way than by shooting them.\" He paused. \"It's my birthday today. I'm twenty-two years old.\"\n\n\"But... you're close to _my_ age?\" That didn't seem possible to me.\n\n\"I look ninety, I'm sure,\" he said. \"We're slave labour here \u2014 Jews, Poles, Russians, Czechs \u2014 and when we're done and can't work and fall down on the way to the rock quarry, or slump over in the factory, then they shoot us or hang us because we're no more use to them. I think I might last another week.\"\n\nAll the time Trent had been translating, and when he said that his voice dropped and it looked like he could hardly get the words out.\n\n\"Is there anything we can do for you?\" James asked.\n\n\"Yes,\" he replied, \"if you would.\"\n\n\"What?\" James said.\n\n\"Remember me.\"\n\n\"What is your full name?\" asked James.\n\n\"Leo Cohen.\" And then he shook each of our hands and added, \"Maybe there will be no Jews left in Europe when the war is over. Before we are liberated we will all be killed. So it would be nice to be remembered.\"\n\nAnd with that he drifted away and we walked back to our part of the camp.\n\n\"Maybe ignorance _is_ bliss,\" James said.\n\nTrent looked shaken. \"I'd rather not have seen that.\"\n\n\"Or heard what he had to say,\" added James.\n\n\"But there are millions of Jews in Europe, aren't there?\" I asked. \"How could they kill them all? It's not possible! And why? Why?\"\n\nNo one answered me. And I couldn't understand it. I couldn't understand anything at all, it seemed.\n\n# Chapter Twenty-Two\n\n_August 24, 1944_\n\nIt was our fourth day in Buchenwald and certain things had been established. We were under strict orders by our commanders to always behave in a way that would bring credit to our uniforms \u2014 even if we weren't wearing them. We were to stay together in this area, only leaving to fetch food rations or visit the latrines (that word didn't describe the horror and filth of what we were forced to use). We were to avoid the SS, could meet with other prisoners, but only if we didn't talk politics, and no one was to attempt an escape for the moment. In the meantime, every day at noon the two commanders would march off in military form and try to get a meeting with the Camp Commandant, to insist we be recognized as POWs. But so far, no luck. Every day they came back with nothing to report.\n\nStill, the grapevine at the camp was better than a telephone system and soon it was well known that our group of flyers had arrived. We began to get visits from other prisoners. We quickly learned what all the different triangles meant that were stitched to men's tops, such as red for political prisoners, pink for homosexuals, purple for Jehovah's Witnesses. Alongside the triangle was the letter signifying their country, _F_ for France and so on. It seemed that there were prisoners from all over the world. We also learned what and who the _Kapos_ were. Just regular prisoners, but given special privileges \u2014 more food, parcels from home, a separate room \u2014 as rewards for taking control over the other men. It was these _Kapos_ who advised our leaders to keep a low profile and not make too big a fuss. They promised that they would try to find us indoor accommodations as soon as something opened up.\n\nOur commanders were absolutely firm on one point. We would not work in the factories, no matter what. We were not slave labour, but prisoners of war, period.\n\nMy spirits were pretty darn low. I had barely slept, was always hungry and yet my gut was a mess. The frequent trips to the _Abort_ \u2014 the latrine \u2014 were the stuff of horror. I was itchy and filthy, and finding it harder and harder to believe in the good world out there with people in it like Mom and Pops and Jenny.\n\nI stared up at the clear blue sky and imagined that it was the same blue sky over Winnipeg and that I was out walking by the Red River, looking into its muddy waters, dreaming about nothing in particular. That's what I missed so much. The days when there was nothing much to worry about, days when I thought I was bored. What I'd give to be bored like that again!\n\nI heard the sirens before I could see anything. I hoped this wasn't a false alarm and that I'd actually get to see some kites in the air \u2014 some of our boys! More sirens went off and the din was amazing. And then I saw them.\n\nIt had to be the Yanks! They were fearless. They always flew the daylight missions. Probably Forts. I judged the bombers to be about 30 miles away. I also counted hundreds! And knowing my directions well, plus remembering the route we took to reach Buchenwald, I figured they were heading for Weimar and the rail yards. It was beautiful, and yet we could hear no noise at all from them. I saw orange and yellow bursts of flame and black puffs of smoke. Then ack-ack fire and two planes went down right away.\n\nI found myself glued to the spot, thrilled that we were not alone, that almost above us were flyers ready to destroy the Germans! And that's when I saw about sixty or so planes break away from the formation and head straight for us! I looked around, ready to run, and quickly realized that there was absolutely nowhere _to_ run! But I didn't care! I yelled at the top of my lungs, \"Come on, fellows! Do your best! Wipe this place to the ground!\"\n\n\"Hold on there,\" James said, coming up beside me. \"I'd rather not be bombed to smithereens by a bunch of cowboys.\" At that he pulled me down to the ground. There was no cover for us, not even the flimsy barracks, so we just lay there and waited. It wasn't long before we saw the smoke flare from the lead plane that told the others where to drop their load. We heard the sound of the bombs, a whistle, a screech, then a rumble like a train and then the earth shook and then shook again and shook again.\n\nI risked a look up to see another large group of planes break off and head our way. And suddenly I worried that I'd left my name in those markings on the wall of Fresnes prison. What if someone saw that and had contacted my parents? What if they now hoped I was alive and then never knew I'd died here. After all, no one knew we were here, outside of the goons in charge. Like Michel on the train, I'd be missing in action forever.\n\nAnd yet, despite all that, I didn't care! After what I'd seen in the last few days I was happy they were going after the camp. And it was a good target for sure. Those factories were using slave labour to pump out guns and armaments and airplane parts. And those poor prisoners wouldn't have to work in the plants if there were no plants to work in.\n\nThat's when the second wave hit, first the white puff of smoke and then the thud, whistle, screech and rumble of a train. The ground shook more and more until black smoke started to rise up and the air seemed to heat up as if we were in some kind of hot-air bath. I could feel stuff hitting me, nothing big enough to really hurt me, but there was debris flying everywhere. I wondered if this would be what it felt like to be in a hurricane or maybe in a tornado with things whipped all around you and the very earth beneath you shaking. And of course at any second something very big could land on you or crush you and that would be that. But slowly the sound of the bombs faded and another sound took its place \u2014 the whooshing sound of a wind that had probably risen from the heat of the blast and from fire, which now seemed to be everywhere. From the way things had gone up in flames, I could tell that the Yanks had dropped incendiaries.\n\nI sat up and looked around. I couldn't see the sky anymore, the air was so black, but I couldn't hear any more planes or the telltale _whoosh_ of the bombs coming down either. It was most likely over. Quickly everyone checked, and outside of one fellow who'd caught a piece of shrapnel in the shoulder, we seemed all right.\n\n\"See,\" I said to James, \"Now that was a fine piece of flying!\"\n\nBefore we had a chance to celebrate that we'd made it through alive, guards rushed into our compound and shouted at us. \"Follow. _Raus! Raus!\"_\n\nMy heart sank. Were they going to make examples of us? It would make sense \u2014 if you thought the way the Nazis thought. We were flyers. Flyers had just destroyed a good part of the camp. We should die.\n\n# Chapter Twenty-Three\n\n_August 24, 1944_\n\nJames, Trent, me and some other flyers I didn't know too well fell in and followed one of the guards \u2014 in formation, though, as we'd been ordered by our commanders. Others in our group were also being rushed out.\n\nThere was an inferno in the section of the camp where the factories had been. We were told that we were to help put out the fires, and that we'd been chosen because we were used to acting as a unit and could work together quickly. So someone _had_ noticed that we were acting like POWs, not regular prisoners. And at least they weren't about to shoot us.\n\nThere was no water to be had, so one of our men suggested that we pull down the building next to the one on fire, to create a firewall. The guards seemed pleased with that idea and, using our hands, we pulled at the hot wood and literally broke apart the building next to the fire. I was one of the lucky ones because I had my clogs \u2014 most of the others had to work in their bare feet.\n\nI wasn't sure we should even be helping at all, but decided that if our leaders were all right with it, so was I. And the guards seemed so frantic that I had no doubt we would have been shot if we'd said no. We must have worked for hours until the fire seemed ready to burn itself out. Word spread that most of the factories had been levelled, the SS barracks had been destroyed and the commandant's wife was dead. That was all good. Not as good was that when the factories were levelled, we knew that they'd been filled with workers and there must have been a huge loss of life.\n\n\"Come!\" said a voice in front of me.\n\nI looked up to see a guard pointing to the small group of us \u2014 five, including Trent and James \u2014 that had been working right near the edge of the fire, pulling down boards. My heart started beating hard again, wondering if _this_ was it \u2014 were we going to be killed now? We followed him into a small hut and the guard motioned to a rough wood table \u2014 on it was a loaf of white bread and some jars of jam. He pointed to the benches and we sat down. Then he cut thick slices off the bread, passed them to us and invited us to spread them with preserves. I took as big a scoop as I could. It looked like apple and when I bit into it I realized it was. It was the best thing I'd tasted in months. And then he poured hot coffee for us into mugs \u2014 real coffee, not the acorn water we were given every morning for breakfast.\n\nSilently I thanked the Yanks again. A beautiful bombing run and to top it off, a real meal. What a day!\n\nWe slept outside again as usual that night, but for the first time since arriving I was able to get some rest despite the rocks sticking into me and the cold seeping into my bones. Except, when I fell asleep I dreamed. And I dreamed about Max. I dreamed that he was Leo Cohen and that Leo Cohen was Max and that they both ended up on that pile of corpses. But neither of them was really dead. And Max kept calling, \"I'm not dead! I'm not dead! Sam! Get me out of here!\" I woke up in a sweat despite the cold night air.\n\nWhat if Max had been caught? What if they had discovered he was a Jew? What if he had been thrown in with the Jewish prisoners instead of being recognized as an Allied airman? Somehow I had to find out if he was in the camp. But how? There were tens of thousands of inmates. And we were now under strict orders not to wander around the camp, so how could I do reconnaissance? And to make my search even more difficult, because our few meagre possessions had started to disappear at night when we slept, the leaders decided that our area would be out of bounds for anyone outside of Air Force, so we mounted a guard each night. I didn't want to slip past our own men.\n\nI sat up shivering the rest of the night and watched the sun peek over the horizon the next morning. Right after our morning cup of acorn coffee, the _Kapos_ herded us into another compound. We marched in formation again and that reminded me that yes, I was RCAF!\n\nA guard began to question us about what our non-military jobs were, because the camp needed men to help them rebuild. They wanted builders and electricians and anyone who could work. The squadron leader stepped forward and set the example, followed by the colonel. He gave his name, rank and service number and refused to offer any more information about himself and what he could do. When it was my turn, the guard stood over me, glaring. I stared back and calmly gave my name, rank and service number. I was not going to work for them no matter what!\n\nAn SS officer arrived then and started to yell and scream and threaten our leaders. They stood firm and stared ahead, perfectly calm. Eventually the Nazi spat in disgust and stalked off. And we marched back to our rocky, miserable home.\n\n# Chapter Twenty-Four\n\n_August 29, 1944_\n\nTen days. Ten days out in the open when James came up to me with an actual smile on his face.\n\n\"We're getting into a hut,\" he said.\n\nAnd that actually made me smile too.\n\nBy some miracle it hadn't rained the whole time we'd been outside, but today the clouds were gathering. I'd been staring at the sky all morning, transfixed, sure that our luck was about to run out and that in no time we'd all be soaked through. I was getting weak from lack of food and there were the sores on the bottom of my feet that just wouldn't heal. Others in our group had bad coughs, and some gruesome infections. I was lucky that so far I'd been spared the worst. My Viking constitution, James joked.\n\nWhen we got inside I looked around. Row after row of bunks or shelves lined both sides of the hut. There was a large wash basin at the front, big enough for about six to wash at a time, three tables with benches, and in the middle of the barracks, or _Lager_ as they called it, a wood stove. There was a notice board at the front door, stating that there were 757 prisoners, and we added 168. We soon learned that there were 400 boys aged eight to fourteen, and the rest were Poles.\n\nAt least, I thought, we were out of the elements. It wasn't long after we got inside that the rain began to pelt down.\n\nWithin an hour of being inside, a young boy struck up a conversation with me. Or tried \u2014 he spoke no English. Although I kept telling myself that nothing should surprise me, I wondered why on earth children were being held as prisoners.\n\nHe spoke German rapidly and pointed to his brown triangle. I called Trent over and he translated for us. It turned out that these boys were Romani \u2014 what we called Gypsies. They were treated the same as the Jews \u2014 not considered pure of race \u2014 and had been rounded up just as the Jews had. They had been working as slave labour but were afraid that any day they would be shipped off to be murdered. Thrown away like garbage. The boy had large brown eyes and would grow up to be a dashing young man. But now he wouldn't get that chance unless a miracle happened.\n\nAs we spoke I started to scratch. And scratch. The boy laughed. He pointed to my pants and motioned for me to take them off. I did. He showed me the little critters already hiding inside the seams. I was really revolted. Bedbugs! I guess there had been one good thing about being outside. He taught me how to go through the seams and crunch the little beggars between my fingernails.\n\n\"What's your name?\" I asked, Trent translating for me.\n\n\"Luca,\" he answered.\n\n\"Where are your parents?\" I asked.\n\n\"Don't know where my mother and sister are,\" he said. \"Many were taken by train somewhere if they couldn't work. My papa worked for a few months, but he died.\"\n\nLuca stayed with me and Trent for the rest of the day and the day after that, and showed us the ropes. He seemed to enjoy being the one in the know, the one who could tell us what was what. Those first days consisted mostly of trying to control the bedbugs and the lice. We threw out all the straw and bedding and washed everything down over and over and over again. Luca showed us his pants, the pockets turned out so that bugs couldn't live there, and we all quickly caught on until everyone was wearing their pants with the pockets inside out.\n\nI was trying to teach Luca some English on our third morning in the barracks when the SS stormed in and started screaming at the Romani boys.\n\n_\"Raus! Raus!\"_\n\nI stood in front of Luca, thinking that maybe I could push him under a bunk and then hide him with the other prisoners later \u2014 maybe the Russians would take him \u2014 but an SS guard shoved me aside, picked Luca up by the collar and threw him into the crowd of other boys. I watched them go and felt such despair and disgust, I wondered if it was even worth carrying on.\n\n# Chapter Twenty-Five\n\n_September 30, 1944_\n\nJames came up to me as I was playing cards with Trent and some others, and said, \"I have some news that might cheer you up. Some Danish policemen have just arrived. You should go see them.\"\n\n\"Cheer me up? I can't be happy that anyone has been captured and sent _here_ of all places,\" I replied.\n\n\"I understand,\" he said. \"And you know I didn't mean I was happy about it either. Still, go see them.\"\n\nAs I made my way over to the new group clustered together on the rocks, I couldn't help but realize just how long we'd already been stuck in this hellhole. Weeks, not days as we had hoped. And the brutality and horror of what we saw every day seemed to worsen as time went on, not lessen. Just that morning we had seen a _Kapo_ beaten to death right in front of our hut.\n\nI tried to put the image of that out of my mind as I approached some of the newcomers and spoke to a few of them. It _was_ good to hear the language my grandfather had taught me when I was young. Also they didn't arrive like us \u2014 with nothing. They were wearing their own clothes, carrying suitcases, just like normal people! They even had Red Cross packages under their arms. There must have been a thousand of them. I was able to speak to a few fellows in Danish, but soon found it was unnecessary, as many of them spoke excellent English.\n\nA fellow called Lars told me how they had been captured.\n\n\"There was an air raid,\" he said, \"so of course we all responded. When we were at our posts the SS popped up out of nowhere, and we were arrested.\"\n\n\"The French _gendarmes_ weren't ever arrested,\" I said. \"So why were you?\"\n\nLars chuckled. \"I suppose we weren't as good at carrying out the Germans' orders,\" he answered. \"We would forget to arrest people. We were very slow carrying out their orders. When they held their rallies no one would show up, including the police. Finally they said we were part of the Resistance. Some were, I'm sure, but most of us just tried to do our best for our people.\"\n\n\"So no one came to the rallies the Nazis held?\" I asked.\n\n\"Maybe a few Nazi sympathizers. That's it. I suppose the Nazis were a little angry about the strikes, too,\" he added.\n\n\"Strikes?\"\n\n\"Yes, and how we scuttled our navy so they couldn't use it and how we managed to get so many of our Jewish neighbours out of the country and safely to Sweden.\" He paused then added, \"Anyway, we just did what anyone would have done under the same circumstances.\"\n\nBut was that true? From my own experience and my own betrayal, it seemed that some of the French had not behaved that way at all.\n\nI wandered back to my group. Trent asked me if I had heard anything about Max, as everyone knew I was always on the lookout for him. I said I hadn't and Trent assured me that was a good thing. I hoped so. I was beginning to believe that Max, at least, had escaped.\n\nJust after our so-called dinner we were told by our leaders that we were going to the movies! We marched into a real theatre that had about five hundred seats. Once I had sat down between Trent and James, for one brief moment it felt like everything was normal and we were on leave together at the movies. In a way, that made the nightmare quality of being in the camp even worse though.\n\nWe were then forced to sit through a newsreel about how well the enemy was doing, with lots of bloody shots of our troops being slaughtered. It was sickening. There was a movie after the newsreel, all in German of course, but Trent kept up a running commentary, making up silly things that the actors never would have said and keeping me and James in stitches. When the two leads kissed passionately, Trent had the actress say, \"My, you have bad breath.\" And in return, the actor said, \"That's nothing compared to your B.O.\" And Trent hardly stopped for breath, managing to write an entire movie script in the moment.\n\nThat night I slept \u2014 after meeting the Danes and going to the movies, I felt more human than I had in ages.\n\n# Chapter Twenty-Six\n\n_October 19, 1944_\n\nIt was too crowded to be in the hut so we stayed outside on the rocks during the day. We'd sleep inside at night, squashed together on the shelves. We had so little space that if one person had to turn over, everyone had to turn over. It would have been funny if it hadn't been so uncomfortable. But it's all what you are used to \u2014 compared to the hard rocks outside, it was far better.\n\nThe days grew monotonous, the routine only interrupted by bursts of brutality we were forced to witness. Sometimes we'd play cards, created from pieces of scrap paper someone had gathered and marked up well enough to do the trick. Sometimes we'd relive old hockey games, arguing about our teams and who would win the next cup.\n\nAlex, a Russian who had made friends with us shortly after we'd arrived, came up to me while I was sitting on the rock pile, my face turned into the sun. He had been one of the prisoners who had helped us get more blankets, extra food and shoes when we flyers first arrived.\n\nHe sat down wearily and said, matter of factly, \"Five hundred a day \u2014 that's how many of us Russians they are killing. Over the other end of the camp \u2014 just outside at the west? Used to be a horse stable there. Now they use it as a killing factory. The crematorium is going full steam ahead again \u2014 all fixed up after the Allies' raid.\" He shook his head sadly. \"October twenty-fourth. That's your time.\" He patted my shoulder. \"Sorry. But better you know.\"\n\nI knew he meant that was the day we were to be executed. I also knew that today was the nineteenth. So, less than a week left?\n\nI thought back to a week ago. One morning we had been woken by \"Ten hut!\" and saw it was the American colonel.\n\nOf course we were all squashed together on the bunks, so getting up and scrambling into formation was quite a feat, but we finally managed. Our leaders instructed us to speak to some officers that had appeared out of nowhere \u2014 officers we realized were from the Luftwaffe. I was afraid I was going to be asked about the Resistance and I vowed that I wouldn't give up any information even at this late date, when hopefully no one in France would be in danger. But when it was my turn I was just asked my name, rank and service number and where I'd been captured. I looked at the Luftwaffe officers and a small hope started to grow \u2014 perhaps we would make it out alive. Now I wondered if that hope was to be dashed completely.\n\nI wanted to find Lars and ask him if he had heard anything, but first I had to go to the infirmary and visit Les. The open wound in his arm had become so infected that you could see the bone.\n\nI sat on his cot and asked him how he was doing.\n\n\"Every day an SS goon walks through checking the patients,\" he said to me. \"If you look too weak and he points at you, you'll get a needle and be dead in a couple hours. The doctor has been moving me from bed to bed so the goon doesn't realize I've been here so long.\"\n\nI didn't know what to say so I waited for him to go on.\n\n\"Yesterday I saw them wrap a guy in a wet rubber blanket and in a few hours his pneumonia got worse and he suffocated to death.\"\n\nI thought to myself that only a few months ago, if someone had told me that one man would do that to another, I would have laughed at him. Now I knew better. Now I didn't doubt for a moment that what Les was telling me was the truth.\n\n\"But there's hope,\" I said. \"The Luftwaffe has been here. Someone at least knows about us now. Don't give up.\"\n\n\"Canadians don't give up,\" he said, but his words said one thing, his eyes another.\n\nI certainly wasn't going to tell him what Alex had just told me. I hurried over to the Danes and searched out Lars.\n\n\"Ah, my friend!\" he said when he spotted me heading toward him. \"Just the person I wanted to see. Look, I have some clean cloths here and some antiseptic! Let me have a look at your feet. Plus we all just got more packages from the Red Cross and I want you to take some.\"\n\nI'd been hobbling around for days. The sores that I got when James and I had to dig Michel's grave had never healed, and no wonder \u2014 we were living in pure filth.\n\nLars had managed to scrounge some clean water and he bathed my wounds, cleaned them up and wrapped them in clean cloths. It reminded me of my mother and how she would take care of me when I was little and how much sweetness and goodness there was in the world \u2014 even here in this hell. Then he gave me a clean pair of pants and a clean shirt.\n\n\"You want to be a doctor, correct?\" he asked me.\n\nI nodded.\n\n\"Have you heard this one? A waiter walks over to Dr. Jensen's table. He says, 'I have saut\u00e9ed liver, frog's legs and boiled tongue.'\n\n\"The doctor answers, 'I'm here to eat, not to listen to your problems!'\"\n\nIt was so silly I had to laugh. How could Lars keep his sense of humour in the midst of all this?\n\nI asked him something else though. \"How is it that you're receiving Red Cross packages and food and clothes, and no one else has such luck?\"\n\n\"Luck?\" he said. \"I don't think it is luck, my friend. From the moment we were arrested, the Danish Red Cross has kept track of us, insisted on visits, insisted on knowing where we were. They are a real thorn in the Germans' side!\"\n\n\"The hardest thing for most of us,\" I said, \"is that no one knows where we are, that our parents have no idea what's happened to us and might never know. Every day our commanders try to get a message to the Luftwaffe, and it seemed as if maybe we'd made progress because some of them actually came to see us.\" I paused, barely able to get the words out. \"But now I hear we're to be executed.\"\n\n\"I've heard that too,\" Lars said quietly. \"But where there's life, there's hope. And without hope we have nothing. So don't give up.\"\n\nWhen I didn't answer he said, \"Besides, Danes don't give up. We're very stubborn!\"\n\nI felt so downhearted that for a few moments I didn't speak. I guess I'd been hoping he'd tell me that he'd heard we were going to be saved and sent to a POW camp! When I did speak it was difficult to express to him what I'd been dwelling on lately.\n\n\"There's something I've been worried about,\" I said.\n\n\"Go on.\"\n\nI wasn't sure exactly how to say it or even what I meant, so I struggled to put this feeling into words. \"I was brought up to always see the best in people. And I worry... well, I worry that maybe if I'd been a German, I would have supported Hitler because he promised to make everyone's life better, and maybe I would have trusted him. I mean, could I have ended up being a Nazi? Or supporting them like so many Germans did?\"\n\nLars thought about what I had said before he replied.\n\n\"I think that there is one very simple way to discover whether someone deserves your trust or not \u2014 like a leader or someone in power, or even a friend. 'Do unto others as you would have others do unto you.' Would you want to be treated the way Hitler treated his political rivals? Rounded up? Sent away? Beaten. Murdered. Would you want to be treated the way he treated the Jewish people? And I don't think that you would have followed him. Because seeing the best in people doesn't mean you go blind!\"\n\n\"No, I guess it doesn't.\"\n\nAnd to make me feel better he gave me some cheese. Real cheese! Along with chocolate and biscuits. And despite the terror, eating some real food did actually make me feel better.\n\n# Chapter Twenty-Seven\n\n_October 20, 1944_\n\nSomehow I had managed to drop into a deep sleep, so it was a shock to be abruptly woken up by SS troopers storming into our hut and ordering us to follow them. What was going to happen? Was the Russian right? Were we going straight to the execution hut outside the gates?\n\nThe first thing I noticed was that Les and about a dozen others weren't there, maybe too sick to go, so there were only about a hundred and fifty of us moving out. At that point I didn't know whether Les and the others were the lucky ones or we were. We formed up as a unit and, as we did every day going to the roll call, marched in military formation to a building used as a storehouse. I was handed the clothes I wore when I'd arrived here \u2014 they had actually been cleaned and disinfected, so they weren't too bad. And anyway, why give them to me if they were just going to shoot me? On the other hand, anything was possible with the SS, so I still wasn't sure that we were safe.\n\nWe then marched to roll call. There we were separated from the others and counted over and over again. It was raining and I was soaked through but I barely felt it. All I could feel was my heart beating. It was so loud I thought the entire camp could hear it. Was it possible that our leaders had pulled off a miracle and had us declared POWs? Or was it another Nazi ploy and the goons were hoping that was what we'd think and then we'd go quietly to our deaths? Oh yes. I think I knew them well by now, how they thought. They wouldn't want to bother fighting us, or troubling with a group who might still put up a show if we discovered that we were to be slaughtered. No, they would lie to us. They would tell us we were going to a POW camp. And when we were pacified they would mow us down the way they did Michel.\n\nRumours swirled from one man to the next \u2014 some said we were going to a POW camp near the Austrian border. Others said we were being sent to another concentration camp. Still others said we were going straight to a firing squad.\n\nAn hour went by. Then another hour, then another hour, then another hour and on and on until we'd been standing in the rain for six hours. I remembered that one day we'd stood for five hours in the sun and some of the men had fainted and we'd had to hold them up and pretend they were fine so they weren't taken away. No one fainted today but that was because we refused. We refused to give in!\n\nSometime in the afternoon SS troops appeared. They had machine guns and dogs and grenades. And then the huge gate opened. Opened. This was it.\n\nWe marched out the main gate and were herded to a railway siding. We were yelled at to stop once we reached a train and the train's boxcars.\n\nAnd that's when something happened that was totally unexpected. The SS guards just marched off, and in their place, men in the blue uniforms of the Luftwaffe took their place!\n\nFor the first time I felt that maybe, just maybe, we were not going to be shot.\n\nThe doors to the cattle cars were pulled open. About thirty of us were told to get in. We did. And so did two guards. By this time I was shaking so hard from excitement, fear and cold that I thought my teeth would rattle out. The cattle car was freezing and we huddled together for warmth. Finally the train jerked and began to move. Could this be true? Were we really leaving this nightmare behind?\n\nThe conditions on the cattle car were the exact ones as before, but at least there was plenty of room. James and Trent were on board with me and after a while going over and over what might be happening to us, we got bored and looked for something to occupy our minds and keep ourselves from going crazy wondering about the possibilities.\n\nI was talking hockey to James when the rest of the men got involved and we decided to make hockey teams. We were too weak to actually get up and move about, so we talked ourselves through an entire game. There might have been an unrealistic number of, \"He shoots, he scores!\" shouted out, but our guards left us alone and at that point we were happy for small miracles.\n\n# Chapter Twenty-Eight\n\n_October 22, 1944_\n\nIt was around noon when the train screeched to a halt. We'd been travelling for two and a half days. I looked out the opening and saw the word _Sagan_ but couldn't remember having seen it on any of my maps. I wasn't sure if we were still in Germany or if we were now in Poland. The guards were packing up their gear and we were told to get off the train.\n\nWe jumped out, formed up and then began a march down a small path beside the train. I looked around. There was a pine forest on one side, the train on the other. The sun was shining, but it was cool. I had just a shirt and pants on, and since I had no fat left at all, or muscle either for that matter, it was only minutes before I started to shiver. Also I still wasn't quite convinced we were going to a camp. At that I almost laughed out loud. What other Allied airmen would be so happy to get to a POW camp?\n\nIt must have been about a half-mile when we turned a corner and there it was, the camp! It was a series of long low buildings behind barbed-wire fencing. There were watchtowers all around with guards whose rifles were pointed right at us. There was an outer platform with machine guns affixed to it and searchlights as well. At one time I might have found the sight intimidating, frightening even. Now there was almost something comforting in it because it was the way I expected a POW camp to look.\n\nThe guards counted us at least three times before finally opening the gate and letting us in. We heard right away that we were, in fact, in Stalag Luft III. We were told by an officer that we would be admitted and then sent into a building where we would get rid of our clothes and be taken to the showers.\n\nA shower! Heaven! The water wasn't hot but it was warm, and there was a sliver of soap. I scrubbed and scrubbed until I felt like I must be shining. I couldn't help but notice all the open sores I had all over my body, things I'd tried to ignore while in Buchenwald.\n\nI was then issued military kit, but it was a mix of Army and Air Force. I didn't care. It was clean. I was given my own boots back! Imagine \u2014 they had kept track of them from our arrival at Buchenwald. Now _that_ was organization. And a coat. A warm winter coat! And a new towel and a new toothbrush!\n\nWe were split up then and assigned to quarters, depending on what was available. James and I were sent with eight others to a central barracks. As we walked I saw men playing football! That they had the energy for it said everything to me. And sure enough, the other inmates we saw looked almost healthy and well fed, at least compared to the prisoners of Buchenwald. They were smoking! They had full heads of hair! Some even had moustaches! Still, if you compared them to the way we all were before being captured, then you could see how they too had suffered at the hands of the goons.\n\nWe still had more paperwork to do \u2014 the goons took our pictures and wrote down our details. Then a fellow called Larry took us for a tour \u2014 there was even a music room. We were now separated from the Americans, which I thought was a shame as I'd grown quite close to them all during our two months in Buchenwald.\n\nWe were split up again. James and I said we'd meet up later and Larry took me to my living quarters. When I walked in I felt like I'd landed in heaven. I saw books, playing cards \u2014 real ones \u2014 a chess board, pipes, wash basins, water and soap. Someone had just made tea and offered me a cup. Real tea? As I drank, I was peppered with questions about where we'd come from. But when I answered them I was met with looks that said what I was telling them was hardly to be believed. And yet, it's not that they thought I was a liar, I don't think \u2014 maybe they just didn't want to believe it. I assured them that I hadn't wanted to believe it either, but that they should check with the others, especially our squadron leader, and they would see that it was true.\n\nThere was one thing and one thing alone that I wanted to do, and that was to write a letter home. I asked one of the fellows and he gave me a form and a pencil. The form was only 6 inches wide and 10 inches long. It had twenty-four ruled lines. There was so much to say. But in reality I knew the censors would read it and that I couldn't write anything about France because I wouldn't want to give anything away. I couldn't write anything about Buchenwald because they would just censor it. I couldn't even write anything about my ops. I sat there and thought about what I wished I could say, and some of it surprised even me.\n\nThinking back over the last few months, I realized that everything I thought I had known was wrong. I wondered how these Nazis \u2014 thugs, really \u2014 were able to not only take over in Germany, but almost take over the world. Was it because we couldn't imagine the evil they represented? I thought that might be it. I simply hadn't believed that the actions and deeds of the Germans were possible. Even after witnessing it myself I found it hard to believe! The world could not imagine such evil. And because we denied it, we weren't ready... and before we knew it \u2014 well, it was almost too late.\n\nSomeone gave me a second cup of tea. I began to write:\n\n_October 22, 1944_\n\n_Dear Mother, Pops and Jenny:_\n\n_I am in a POW camp. It's been a hard time. There is one thing I know for certain \u2014 this was all worth it. All I want now, though, is to come home. I dream of_ frikadeller _and_ wienerbroed _and hot dogs and french fries and swimming at the beach and tobogganing in the winter and going to school_.\n\n_I just want to come home_.\n\n_With love_ ,\n\n_Sam_\n\n# Epilogue\n\nSam's parents had remained hopeful that he was still alive because they'd received a letter from Max, who had managed to evade the Germans and had eventually reached the Allied lines. That had kept their hopes alive until Sam himself was able to write them. He endured many more hardships, including a forced march from Stalag Luft III in January 1945, in which some of the POWs died. Sam had other close calls before he made it safely back to England \u2014 just two days before Victory in Europe (VE) Day, May 8, 1945. On that day, while all around him people celebrated, Sam could think only of those who had perished.\n\nWhen Sam finally returned home to his family \u2014 who were delirious with joy \u2014 they believed his story, of course, but so many others didn't that he soon stopped telling people he had been in a concentration camp.\n\nIt was hard for Sam that people didn't believe him. But he had other things on his mind. He wanted to go to university and study medicine. And he wanted to go out with Sadie Kobrinsky, so he asked her and she said yes right away. The two were never parted again. When their first son was born they named him Leo after Leo Cohen, the young man Sam had met in Buchenwald, so that Leo would be remembered, just as Sam had promised.\n\nAlthough Max returned to Montreal after the war, he and Sam remained fast friends for the rest of their lives.\n\n# Historical Note\n\n**Canada's Contribution to the Air War**\n\nCanada entered the Second World War on September 10, 1939, one week after Britain and a full two years before the United States. During the First World War, many flyers had been recruited and trained in Canada for Britain. Similarly, when the Second World War broke out, Canada became a training ground for all Commonwealth flyers, under the British Commonwealth Air Training Plan (BCATP).\n\nCanada played a major role in the training of aircrew for the Royal Air Force (RAF), the Royal Australian Air Force, and the Royal New Zealand Air Force, as well as for its own Royal Canadian Air Force (RCAF). It had the open spaces that enabled it to train pilots and other aircrew far away from the range of German aircraft, something Britain lacked.\n\nThe Royal Canadian Air Force ran the training program, in conjunction with the Canadian Flying Clubs Association, aviation companies and the Department of Transport. Though there were limited aircraft, trainers and airfields, training began on April 29, 1940. Aircrew candidates were transferred to Canada for training. Eventually, over one hundred and thirty-one thousand pilots, navigators, bomb aimers, wireless (radio) operators, air gunners and flight engineers graduated from the BCATP. Almost seventy-three thousand of these were Canadians.\n\nBCATP graduates went where they were most needed for the war effort, with the RAF having the key say in this. Most squadrons had a mix of Commonwealth air forces represented, both aircrew and especially ground crew. No. 6 RCAF Bomber Group, established in January 1943, was still part of RAF Bomber Command.\n\nStatistics from the Bomber Command Museum of Canada website indicate just how dangerous it was to be part of an aircrew fighting for the Allies. Only German U-boat forces and Allied Merchant Navy sailors had a lower chance of surviving the war than RAF Bomber Command aircrew:\n\n * For every 100 men who joined, 45 were killed, 6 were wounded and 8 became prisoners of war.\n * 120,000 airmen served; over 55,000 died.\n * Over 10,000 Canadians were killed.\n * A pilot's average lifespan was 6 weeks. Pilots who started flying at the start of the war had only a 10 percent chance of surviving.\n * Between March of 1943 and February 1944 there was only a 16 percent survival rate for the crews of Halifax bombers.\n\nSir Arthur Harris of the RAF, the person who sent the men to battle, said this: \"There are no words with which I can do justice to the aircrew who fought under my command. There is no parallel in warfare to such courage and determination in the face of danger over so prolonged a period.\"\n\nHe described them as having \"a clear and highly conscious courage, by which the risk was taken with calm forethought.... It was, furthermore, the courage of the small hours, of men virtually alone, for at his battle station the airman is virtually alone. It was the courage of men with long-drawn apprehensions of daily 'going over the top.' Such devotion must never be forgotten!\"\n\n**The French Resistance**\n\nThe French Resistance is a term that encompasses all the anti-German movements in France during the Second World War. There were as many as nine major Resistance networks by the time Sam and his crew were shot down in the summer of 1944.\n\nResistance to the German forces began with individuals rejecting their own government's collaboration with the Germans who occupied France after the invasion on May 10, 1940. The prime minister of France, Marshal P\u00e9tain, negotiated France's surrender to Germany and authorized the French delegation to sign an armistice on June 22, 1940. Part of the agreement allowed an \"independent\" French government headquartered in Vichy, which came to be known as Vichy France (versus Occupied France in the north, which included Paris). The government of Vichy, with P\u00e9tain as Chief of State, collaborated with the Germans.\n\nAt first Resistance fighters did what they could with the few resources they had \u2014 they cut telephone lines, derailed trains or sent them to the wrong locations, killed or kidnapped German officers or soldiers when they could, and published underground newspapers. But soon many of these individuals joined together and formed all sorts of different groups, from the Communists to Jewish groups to the Maquis. The latter were fighters who hid in the forests and around the countryside, especially in the mountainous regions of France. Many had originally been conscripted to work in German factories and were to be sent away to Germany or further east to Poland, but instead joined a Resistance group.\n\nBy 1943 some of these Resistance groups had joined together under the leadership of Charles de Gaulle, a French officer who had fled to London and had begun working with the British government. The Resistance managed to send vital intelligence to the Allies about German troop movements. It went on the attack whenever possible, and it was the main reason so many Allied airmen survived after their planes were shot down.\n\nThe Resistance set up a number of escape routes to help return Allied flyers to Britain so that they could carry on the fight. An estimated three thousand American flyers and twenty-five hundred British flyers were sent to freedom along these escape routes. According to Don Lawson's _The French Resistance_ , for every man who escaped, a Resistance operator lost his or her life.\n\nSecrecy was of the utmost importance in the Resistance. Members of the small groups \u2014 cells \u2014 often used false names so that any member who was captured could not give away the identities of the others under torture.\n\nGerman reprisals for the Resistance's guerilla tactics were harsh. Successful raids against German targets sometimes meant that local civilians would be punished or killed by the German forces, to instill fear in the population and undercut local support for the Resistance.\n\nAfter the Allied landing of June 6, 1944, in Normandy, the Resistance played a large role in the war effort, moving from sabotage, guerilla actions and maintaining escape routes to actual fighting alongside the Allied forces and supporting their advance against the Germans. But there were also French men and women who worked diligently for P\u00e9tain's Vichy government, especially after 1942 when the entire country was occupied by German forces. These \"Vichyites\" would infiltrate the Resistance, sabotage their efforts, and betray them to the German authorities.\n\nThe men and women of the Resistance endured hardship, possessed great courage, and made a great impact with often-limited resources. American General Dwight D. Eisenhower said that the Resistance had been \"of inestimable value in the campaign.\"\n\n**The Holocaust**\n\nThe United States Holocaust Memorial Museum defines the Holocaust as the \"systematic, bureaucratic, state-sponsored persecution and murder of approximately six million Jews by the Nazi regime and its collaborators.\"\n\nAdolf Hitler was voted into office by the German people. Once in the Reichstag (parliament), he cleverly manoeuvred and orchestrated events so that he could gain more power. He had his men set fire to the parliament buildings and then blamed the fire on the party he considered his main enemy, the Communists. He made people so afraid of the Communists that he easily took power, manoeuvring to have himself named Chancellor, with the promise to \"save\" the people from the Communists.\n\nOften leaders will use fear of others to gain power for themselves \u2014 blaming others and depicting their opponents as evil and dangerous. Hitler did this not only with Communists, but also with Jews. Jews became the scapegoats for all that was wrong in Germany. A bad economy? It must be the Jews' fault. Germany seen as weak by other countries? The Jews' fault. Jews, of course, had no more effect on the economy than any other Germans. But because of long-standing anti-Semitism (prejudice against or hatred of Jews), Hitler was able to perpetuate this lie.\n\nHitler took this foundation of anti-Semitism and expanded on it. He passed laws that restricted Jews from teaching and from professions such as medicine and law. He passed further laws declaring that Jews could not own businesses; that they could not shop in most places; that they could not walk in a park, swim in a pool, attend school. The courts failed to strike down these laws. Judges upheld them and lawyers argued that they were legal. When no one stood up for the Jews, Hitler began to round them up and to send them to concentration camps. He told the world that he wanted to be rid of Jews and offered to give them away to other countries. But no one wanted them. Canada took in only five thousand during the whole of the Second World War.\n\nFinally Hitler decided to be done with Jews once and for all. He devised the final solution to what he called \"the Jewish problem.\" The final solution was simple. He would exterminate them all.\n\nDeath camps, as they came to be known, were places where those who had been rounded up by the Nazi regime were sent to be murdered. At first the locations were work camps, and then often work and death camps, where people were separated on arrival \u2014 some sent to do work, some sent to death in the gas chambers, some worked to death.\n\nBuchenwald concentration camp was established in 1937. It was built in Germany near the city of Weimar. When it first opened, political prisoners were sent there, but in 1938 ten thousand Jews were imprisoned, hundreds dying almost immediately due to the harsh treatment they received. As well as Jews and political prisoners, others were also sent there \u2014 Jehovah's Witnesses, Roma and Sinti (groups that some people used to call \"Gypsies\"), and later, prisoners of war, resistance fighters, foreign forced labour and those deemed criminals by the state.\n\nBy 1945 one hundred and twelve thousand prisoners were incarcerated. Many were forced to work for the German war machine and Buchenwald became important for its factories, which churned out munitions and other material for the war effort, using forced labour.\n\n**168 Allied Airmen**\n\nEach of the 168 Allied airmen who ended up in Buchenwald had a different story, in that each was shot down on a different day, at a different time, and had different experiences while hiding in France, trying to evade the enemy. But each was somehow betrayed, or captured by the Germans, and sent to Fresnes Prison in Paris. Even in prison each had a different experience: some were isolated, some were tortured, some were with friends. Some were there for days, others for weeks, even months. From the moment they were put on the train to Buchenwald, their stories intertwined. According to the Geneva Convention, captured Allied airmen should never have been sent to a concentration camp but directly to a prisoner-of-war camp. However, they suffered the same fate as the Resistance fighters.\n\nOn October 20, the airmen were removed from Buchenwald and on October 22 they were interned in POW camp Stalag Luft III. On January 28, 1945, Stalag Luft III was evacuated and ten thousand Allied airmen were forced to march away from their liberators, the Russian and American troops that were drawing near. The POWs, or as they called themselves, _Kriegies_ (short for _Kriegsgefangener_ , or war prisoner in German) had no ability to hold off the evacuation, since they were at the mercy of their German guards. Still, the _Kriegies_ walked as slowly as they could and delayed as much as possible, following orders from their group captain, in the hope of being overtaken by the Allies.\n\nAt one point RAF fighters mistook the column for retreating German soldiers and strafed them. Some had the presence of mind to dive into a ditch when they saw the planes coming, but others stayed on the road to wave at their liberators. Because of that, some of the _Kriegies_ were killed by their own side.\n\nOn April 23 some of the airmen found themselves at a large country estate where they were allowed to wait until Allied troops could reach them. They were liberated on May 2 by General Montgomery's 11th Armoured Division. Their war was over!\n\nOn May 6 they boarded a transport to England and landed in an airfield north of London. Some of the men were then assigned transport to Bournemouth, and a billet in a hotel where they could clean up and get into new clothes. Then they went out to celebrate Victory in Europe (VE) Day on May 8.\n\nThis book is inspired by the true story of the flyers who were shot down, evaded the Germans, were captured and imprisoned in Paris, sent to Buchenwald, and then at the last moment \u2014 just before they were to be executed \u2014 sent to a POW camp. The flyers were from all the Allied countries.\n\nCanada's National Film Board created a documentary about these men: _The Lucky Ones: Allied Airmen of Buchenwald_. A more recent documentary film by Mike Dorsey titled _Lost Airmen of Buchenwald_ uses archival and often secret footage of the plight of these airmen. It held its premiere in 2011, sixty-five years after the captured Allied airmen marched out of Stalag Luft III.\n\n# Images and Documents\n\n_**Image 1.** RCAF training prepared students for deadly combat. This photo, taken upon graduation from a school in Mossbank, Saskatchewan, indicates with an \"X\" those young men who were killed overseas during the war_.\n\n_**Image 2.** Smoke billows below this Lancaster on a bombing run over its target_.\n\n_**Image 3.** An RAF first-aid kit such as this was given to each flyer. It contained items like burn cream, wound dressings and morphine capsules for pain_.\n\n_**Image 4.** Members of the French Jewish Resistance group Arm\u00e9e Juive stand in their military uniforms_.\n\n_**Image 5.** The small and light Type A MK III suitcase radio was used by operatives of the British Strategic Operations Executive (SOE) to help resistance efforts in Nazi-occupied countries. Its transmission range was approximately 800 km_.\n\n_**Image 6.** SS and police officials during a roll call of Polish concentration camp prisoners_.\n\n_**Image 7.** Concentration camp prisoners stand during roll call. Each wears a striped hat and uniform bearing coloured, triangular badges and identification numbers_.\n\n_**Image 8.** Former prisoners of the \"little camp\" in Buchenwald stare out from the wooden bunks in which they slept three to a bed. Elie Wiesel, Nobel Peace Prize winner and writer of dozens of books about the Holocaust, is in the second row of bunks from the bottom, seventh from the left, next to the vertical beam_.\n\n_**Image 9.** American troops discovered human remains in Buchenwald's crematoria_.\n\n_**Image 10.** Northeastern Europe in 1943\u20131944_.\n\n# Credits\n\nGrateful acknowledgment is made for permission to reprint the following:\n\nCover cameo (detail): _Pte. John Lewis, 1_ _st_ _Royal Canadian Regiment_ , Paul E. Tomelin \/ Canada, Ministry of National Defence, Library and Archives Canada, PA-146992.\n\nCover scene: _Lancaster MK II flying on one engine;_ Charles. E. Brown, \u00a9 Royal Air Force Museum, 5981-6.\n\nCover details: (front cover) Aged journal \u00a9 Shutterstock\/Bruce Amos; aged paper \u00a9 Shutterstock\/Filipchuck Oleg Vasilovich; Tape \u00a9 Phase4Photography; belly band \u00a9 ranplett\/istockphoto; (back cover) label \u00a9 Shutterstock\/Thomas Bethge.\n\n**Image 1** : _RCAF Air Gunners' School, 1941_ , George Metcalf Archival Collection, Canadian War Museum, 19870036-002.\n\n**Image 2** : _A Lancaster over its target_ , Collections Canada, PL-144407.\n\n**Image 3** : RAF Air Crew Personnel, First-Aid Kit, courtesy of Richard Archer, Mons Military Antiques.\n\n**Image 4** : _Group portrait of members of the French Jewish Resistance group Arm\u00e9e Juive dressed in their military uniforms_ , United States Holocaust Memorial Museum, courtesy of Arnold Einhorn, #49171.\n\n**Image 5** : Type A MK III suitcase radio, courtesy of Richard Brisson, Collector, Ottawa, Ontario.\n\n**Image 6** : _SS and police officials speak among themselves during a roll call of Polish prisoners_ , United States Holocaust Memorial Museum, courtesy of Robert A. Schmuhl, #13129.\n\n**Image 7** : _Prisoners standing during a roll call_ , United States Holocaust Memorial Museum, courtesy of Robert A. Schmuhl, #10105.\n\n**Image 8** : _Former prisoners of the \"little camp\" in Buchenwald stare out from the wooden bunks in which they slept three to a \"bed,\"_ United States Holocaust Memorial Museum, courtesy of National Archives and Records Administration, College Park, #74607.\n\n**Image 9** : _Human remains found by American troops in the crematoria ovens of Buchenwald_ , United States Holocaust Memorial Museum, courtesy of Harold Royall, #82856.\n\n**Image 10** : Map by Paul Heersink\/Paperglyphs.\n\n# Acknowledgments\n\nMy heartfelt thanks to the veterans who helped me with this manuscript, first and foremost John Harvie, Lancaster navigator, one of the 168 airmen shot down over France and later incarcerated in Buchenwald. His book _Missing in Action_ was an inspiration, as was his tireless help through many phone calls and emails. I'm sorry to say that he has since passed away. The book _A Fighter Pilot in Buchenwald_ by Joe Moser, as told by Gerald R. Baron, is also a wonderful memoir. I listened to a podcast by Harold Bastable, who grew up in Winnipeg and gave talks to students about his wartime experiences. I was also able to read his personal memoir thanks to the generosity of his family. Ed Carter-Edwards answered a question no one else could. Finally, the book _168 Jump into Hell_ by Arthur Kinnis and Stanley Booker was written because after the war many did not believe that Allied airmen were actually imprisoned in Buchenwald. So the airmen banded together once again and set out to put the record straight.\n\nI would be remiss to leave out the experts who read the manuscript and helped me correct mistakes and pointed me in the right direction \u2014 many thanks to Carl Christie and all his Air Force contacts, and to Professor Robert Young, Trish McNorgan and fact-checker extraordinaire Barb Hehner. I also thank the following airmen who so kindly emailed me with answers to some of my questions: Andrew Christie, Stu Beaton, Robert Vincent, Ernest Cable, Jim Buckland, Jim Bell, William Carr, Fred Aldworth, Jim Shilliday, Cal Shermerhorn, Ernie Drouin, E.V. \"Dusty\" Titheridge, James Popplow. If there are any mistakes they are mine and mine alone.\n\nAnd a big thank you to my editor, Sandy Bogart Johnston, for her calm and cool and her tireless attention; my husband Per Brask; and my friend Perry Nodelman for reading the manuscript and for all his suggestions. I also thank the Manitoba Arts Council for the grant, which helped tremendously, giving me the time to research this large project.\n\nLastly, a word to my readers. This is a book of fiction, although it is based on the story of 168 airmen who were shot down in France in World War II. My character Sam is a fictional character inserted into this group, as are his friends, Max, James, Trent, etc. On the night that Sam is shot down, a Lancaster bomber was actually lost in an op over Trappes, France, but Sam Frederiksen's crew is not based on that one.\n\n# About the Author\n\nCarol Matas's parents were born in Canada, but her grandparents and great-grandparents immigrated here from various eastern European countries, thus escaping some of the horrifying events of World War II. Carol's father-in-law, Olaf Brask, was a fighter in the Danish Resistance during that war. Her book _Jesper_ is inspired by his story.\n\nCarol is perhaps best known for her Holocaust novels, such as _Daniel's Story_ (shortlisted for the Governor General's Award and winner of the Silver Birch Award), _After the War_ and _The Garden_ (both winners of the Jewish Book Award), _Lisa_ (a Geoffrey Bilson Award winner), _Jesper, Greater Than Angels_ and _In My Enemy's House_. She has written two Dear Canada books, _Footsteps in the Snow_ , about Winnipeg's Red River Settlement, and _Turned Away_ , which received the Margaret McWilliams Award from the Manitoba Historical Society. That story highlights Canada's refusal to allow any Jews from Europe into the country during World War II, as well as the actions of Canadian soldiers who fought at Hong Kong and were either killed or made prisoners of war by the Japanese.\n\nCarol interviewed several of the 168 Allied airmen whose story was the springboard for this novel. When one of the few remaining Lancaster bombers that are still flying landed in Winnipeg in spring 2011, she climbed right inside it, to see what it might have been like for Sam Frederiksen in his position as a gunner.\n\nCarol lives with her family in Winnipeg, Manitoba.\n\n# Other Books in the\n\n# Series\n\n_Prisoner of Dieppe_ \n _World War II_ \nHugh Brewster\n\n_Blood and Iron_ \n _Building the Railway_ \nPaul Yee\n\n_Shot at Dawn_ \n _World War I_ \nJohn Wilson\n\n_Deadly Voyage_ \n _RMS_ Titanic \nHugh Brewster\n\nFor more information please see the I AM CANADA \nwebsite: www.scholastic.ca\/iamcanada\nWhile the events described and some of the characters in this book \nmay be based on actual historical events and real people, \nSam Frederiksen is a fictional character created by the author, \nand his journal is a work of fiction.\n\nCopyright \u00a9 2012 by Carol Matas. All rights reserved.\n\nA Dear Canada Book. Published by Scholastic Canada Ltd. \nSCHOLASTIC and I AM CANADA and logos are trademarks \nand\/or registered trademarks of Scholastic Inc.\n\nAll rights reserved under International and Pan\u2013American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read this e-book on-screen. No part of this publication may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher, Scholastic Canada Ltd., 604 King Street West, Toronto, Ontario M5V 1E1, Canada.\n\nISBN 978-1-4431-1925-2\n\nFirst eBook edition: January 2012\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvvxv b/data_all_eng_slimpj/shuffled/split2/finalzzvvxv new file mode 100644 index 0000000000000000000000000000000000000000..e071540fe3ac32e17d6755cc8500f314830341a5 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzvvxv @@ -0,0 +1,5 @@ +{"text":"At 35 weeks pregnant, I'm all about not leaving the house. Or the couch. But my family insists on things like toilet paper, dinner, diapers\u2026 I mean. Sheesh.\nPeople, this is a game-changer. Local friends, I use Byerly's \u2013 I shop online and pay either $5 to drive through and pick them up or $10 for home delivery. You could use Coborn's Delivers (also based in the Twin Cities) or Amazon Fresh, as well as others I'm sure. My mom has ordered entire Thanksgiving dinners from Byerly's before and had them delivered! Brilliant.\nKind of. I've used StitchFix several times in the past, including a maternity fix, and it's been really fun! Other favorite places to order clothing from include Old Navy, Kohl's, Target and Zulily (they have much more than clothing too \u2013 from toys to household decor to shoes, there's always a new deal on zulily!) Stock up on gifts, treat yo'self to a new top, or (like me lately) fill in the gaps of your wardrobe without having to hit the mall.\nOf course, Amazon is always a good bet for quick delivery. Target also has a subscription delivery service for items such as diapers, and actually ePantry has diapers + wipes too! With a preschooler, a toddler and a newborn in tow this winter (two of them in diapers), I'm sure we'll be upping our diaper delivery orders.\nePantry is my one and only in this department. We order most of our cleaning supplies from them for monthly delivery (with the ability to completely customize or skip a month) I also order Sam's vitamins, toothbrushes, facial tissues, toilet paper diapers + more from them! I never run out of these staple items and I don't have to leave the house! AND, stay tuned for an incredible deal from ePantry that I get to share with you next week. It's a good one, friends.\nThere are tons more delivery sites out there \u2013 these are just the ones that serve our family well. What are some of your favorites? I'd love to hear about how they make life (and the holidays) less stress for you!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"k swiss infant shoes white silver tennis child boys kids wholesale trade p.\nfake peony flowers pink and white artificial silk wholesale flower for sale.\ncrystal ceiling chandelier high quality led lights diy fan.\nchalkboard home decor beautiful fall idea autumn the creative corner craft link party.\nturtle decorations for home cartoon wall art mural decal sticker decor kids babies room nursery decoration homecoming ideas.\nallure window treatments photo of treasure valley id united states.\ngrey front door 6 panel painted smooth fiberglass with best colours n.\n48 french doors x exterior in white primer primed left hand lowes interior.\nnautical sconces indoor inspirational pictures sconce lighting.\npetmate 2 door kennel deluxe top load up carriers totes dog supplies pet two small wonderful medium sup.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Popular social activist and co-convener of 'our mumu don do' crusade, Prince Deji Adeyanju, has been arrested by men of the Nigeria Police Force.\nIrked by the allegation of police involvement in the Tuesday's debacle, Adeyanju took to his social media handles and accused the security agency of complicity.\n\"\u202aThis is why we protest at 11am today. #PoliceNotPoliticians\".\nWheres our freedom of speech!\nI swear the system is annoying. Why arrest him?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Cosmah received some help from the stewards in becoming a stakes winner, as she inherited her lone black-type win by disqualification. Nonetheless, she was a filly with genuine ability, and she was an even better producer. The Kentucky Broodmare of the Year for 1974, she produced four stakes winners, including a champion and a champion sire, and left a lasting impact through her daughters as well.\nRated at 110 pounds on the Experimental Free Handicap for American juveniles of 1955, 6 pounds below co-champion juvenile fillies Doubledogdare and Nasrina.\nA blocky bay mare standing 15.3 hands, Cosmah was powerfully built but was blessed with a kind disposition. She had a strong, straight hind leg.\nCosmiah (1959, by Olympia) produced the minor stakes winner Muriesk (by Nashua). She is the second dam of three stakes winners, none of particular note.\nTosmah (1961, by Tim Tam) was the American co-champion 2-year-old filly of 1963, the American champion 3-year-old filly of 1964 and the American co-champion handicap female of 1964. She produced only four foals including 1973 New Jersey Futurity winner La Guidecca (by Royal I. J.).\nMaribeau (1962, by Ribot) won the 1965 Fountain of Youth Stakes. He sired 13 stakes winners from 194 named foals.\nFathers Image (1963, by Swaps) placed in a number of important races but scored his only stakes win in the 1966 City of Miami Handicap. He sired eight stakes winners from 142 named foals in the United States and Japan.\nRoyal Match (1964, by Turn-to) produced stakes winners Matchless Native (by Raise a Native) and Rexson (by Bold Bidder) and is the second dam of grade II winner John's Gold. She is the third dam of English Group II winner Catrail and English Group III winner Life at the Top.\nPerfecta (1965, by Swaps) is the second dam of 1983 Poule d'Essai des Poulains (French Two Thousand Guineas, FR-I) winner and French champion miler L'Emigrant and French Group II winner Salpinx. She is the third dam of English Group II winner Zalazl, Grade II winner Sauvage, German Group III winner Eyeq and Grade III winner Sir Alfred.\nQueen Sucree (1966, by Ribot) produced 1974 Kentucky Derby (USA-I) winner Cannonade (by Bold Bidder), multiple Grade III winner Circle Home (by Bold Bidder) and European stakes winners Del Sarto (by Bold Bidder) and Wassl Touch (by Northern Dancer). She is the second dam of multiple Grade I winner Stephan's Odyssey, 1986 Acorn Stakes (USA-I) winner Lotka, Australian Group II winner Rasheek and English Group III winner Burooj. She is also the third dam of 2010 Serbian champion 3-year-old filly Vatrena Maca, Grade II winner Blazonry and Grade III winners Lotta Dancing and Sonic West.\nLove of Learning (1967, by Hail to Reason) is the second dam of 1991 Santa Ana Handicap (USA-IT) winner Annual Reunion.\nHalo (1969, by Hail to Reason) won the 1974 United Nations Handicap (USA-IT) and was the American champion sire of 1983 and 1989. His 63 stakes winners include 1989 American Horse of the Year Sunday Silence, who led the Japanese general sire list 13 times. Halo also sired Southern Halo, only stakes-placed on the track but a 10-time champion sire in Argentina.\nLa Dame du Lac (1973, by Round Table) produced Irish Group III winners Lake Como and Single Combat (both by Nijinsky II) and Irish stakes winners Nazoo (by Nijinsky II), Miznah (by Sadler's Wells) and Heeremandi (by Royal Academy). She is the second dam of two-time American champion turf female Flawlessly, 2001 Prix Saint-Alary (FR-I) winner Nadia, multiple English Group II winner Zindabad and Grade III winner Perfect. She is also the third dam of 2014\/15 Chilean champion imported sprinter Genial Boy, Japanese Group II winner Kongo Rikishio, Brazilian Group II winner Gol Tricolor, multiple Grade III winner Laura's Pistolette and Irish Group III winner An Tadh.\nHonor Tricks (1976, by Bold Bidder) is the second dam of English Group III winner Brancaster and the third dam of French Group III winner Meleagros.\nFoaled in Kentucky, Cosmah was bred by Henry Knight. She was owned by Eugene Mori and was trained by F. Zitto. Following the birth of Maribeau (Cosmah's fifth foal), Mori sold the mare to John R. Gaines.\nCosmah is inbred 5x5 to 1906 Derby Stakes winner Spearmint and to 1913 English champion 2-year-old male and 1919 English\/Irish champion sire The Tetrarch. She is a half sister to Natalma (by Native Dancer), whose four stakes winners include the epochal sire Northern Dancer, a champion in Canada (where he was the 1964 Horse of the Year) and the United States. Natalma is also the second dam of 1972 Canadian Horse of the Year La Prevoyante and the third dam of another great stallion in 1989 Ladbroke Sprint Cup (ENG-I) winner Danehill, who won nine sire championships in Australia, three in England\/Ireland and two in France. In addition to her other accomplishments, Natalma is also the second dam of French Group III winner Coup de Folie and the third dam of 1989 French champion 2-year-old male and important sire Machiavellian, 1993 French champion 2-year-old filly Coup de Genie and1992 Prix Jacques Le Marois (FR-I) winner Exit to Nowhere.\nReturning to Cosmah, she is also a half sister to stakes winner Folklore (by Native Dancer) and to Bubbling Beauty (by Hasty Road), whose son Arctic Tern (by Sea-Bird) won the 1977 Prix Ganay (FR-I) and was the French champion sire of 1986. Cosmah was produced from 1950 Vineland Handicap winner Almahmoud, a full sister to 1954 Test Stakes winner Dispute and 1946 Will Rogers Stakes winner Burra Sahib. Almahmoud is also a full sister to Avatara, second dam of 1974 Jersey Derby (USA-I) winner Better Arbitor.\nAlmahmoud and her siblings were produced from Arbitrator, an unraced daughter of 1934 Belmont Stakes winner Peace Chance and 1924 Futurity Stakes winner Mother Goose (by Chicle). Mother Goose, generally considered the American co-champion 2-year-old filly of 1924 and the namesake for the modern Grade I race, also produced the St. Germans filly Gosling, a minor stakes winner as a juvenile; Gooseflesh (by Mad Hatter), second dam of 1953 Comely Handicap winner Home-Made; and Wes (by Mad Hatter), second dam of 1961 Dixie Handicap winner Hunter's Rock (by Hunter's Moon IV).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Delicious real. Fibre from 5 whole grains, oats, wheat, rice, corn and barley. With 9 vitamins and minerals. Source of calcium & vitamin D. No artificial colours of flavours. Suitable for vegetarians. Kosher.To produce 100g of this product we have used 77.2g of Whole Grain.\nWhat's the suggested portion size? Kids 25-30g Adults 30-45g Enjoy a serving of Nestl\u00e9 Cereals with the following items: 125ml Milk A glass of water 1 Fresh fruit 1 of 5-a-day Keep your Cheerios fresh! Roll down the bag and close carton after use.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaafmz b/data_all_eng_slimpj/shuffled/split2/finalzzzaafmz new file mode 100644 index 0000000000000000000000000000000000000000..b0eb925a6bd05f1371ff4afc3ed048db0425fee1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaafmz @@ -0,0 +1,5 @@ +{"text":"You don't have to stress yourself. There's an easy process!\nBuilding a house and renovating an old home is never an easy feat. There are stressful events at every stage of the process. Your pulse races with the thought of construction mishaps and costly mistakes. Decision making is also harder, knowing that you're going to live with your choice for a long period of time.\nDon't let the tension get you. Read this essential advice that will walk you through this important life event.\nFor a project to run smoothly, one must invest their time and effort in it.\nFirst, envision your future home. Know what you want and what needs to be done. Get inspiration from home improvement articles, nearby houses for sale, and virtual showrooms of residential properties sold online. All of these will help you manage your expectations. It will give you an idea of the designs and styles which will fit your budget.\nAn architect and engineer working on a client's new home.\nNext, decide if you need the help of a professional. There are architects, building designers, interior decorators, structural engineers, landscape architects, and surveyors\u2014each one of them provides services that can contribute to the success of your project. Get a recommendation from your peers or search for them online. Examine their portfolios to see if they can provide the output that you want. If you're comfortable with the skills of the person you wish to hire, call them and schedule a consultation.\nSuperdraft tip: The first person to hire is always the designer. You and your architect or draftsman should draw the plan before your hire other pros on the job. The designer will normally advise if you are in need of another person for a specific job. You can also ask him or her to put up a team of people for your project. Of course, you have to review them all before you proceed to work. Work with them closely during the design process.\nThe last set of people you need are the builders. Aim to get quotes from at least three best builders in your town. Don't always go with the cheapest value. Examine and see who has the most competitive offer from the three. Be smart about your decisions\u2014that's the only way you can save time and money in the process.\nPrepare a detailed outline of works complete with the agreed time frame, construction costs, list of products to use, and the finish of the surfaces. Both parties must sign and have the copy of the document. If possible, prepare a formal and legal contract with the contractor. Do this to protect you from builders who failed to do their jobs.\nThere are times when you have to let the contractors take over.\nMajor construction work requires homeowners to move out of their home until the project is completed. You and your family should stay somewhere safe, where everyone can continue their daily lives. However, you, the head of the project, must pay a visit with the contractors every day. Check on your project and the people you hired.\nIf you choose to stay, you should make way for the ongoing construction. Seal rooms so the dust won't spread from one area to another. Talk about dust and waste management with the head of the builders. Ask them to put protective sheets around the area where you will stay. Aside from a room to sleep, you should assign an area where you can temporarily station a microwave and a sink. This is where everyone can have their lunch without worrying about the dust and dirt getting on their food.\nThis is the easiest yet the most important part of your job during construction.\nMake it a habit to inspect the construction site regularly. As the homeowner and head of this project, you need to see how much of the work is being done on a daily basis.\nSuperdraft tip: Visit the site before or after the workday. Inspection, while the contractors are working, is not recommended. The chaotic surrounding is not conducive to someone who needs his\/her 100% attention.\nIf it's not possible to meet them every day, conduct a weekly catch-up with the key members of the construction team. Ask about their team's progress as well as the problems that came up while you were gone. Find solutions to a pending construction issue. Take note of everything especially the changes with the design, building materials, and the workload of your people. This will help you manage your finances properly.\nEffective communication alleviates almost 90% of potential construction issues. Both parties must have a clear line of communication with each other to avoid miscommunication. If you encountered something that's unclear, ask questions. A text, call, email, or a friendly chat is enough, as long as you're dealing well with the contractors. Again, keep a record of all dealings that happened.\nSuperdraft tip: Your architect can also manage a construction project for you. This way, all the instructions and requests of the builder will be channelled to you by a professional. You can ask him for advice, in the event that you're unsure of what you're going to do.\nCouples decide together, but it helps if only one person communicates with the construction team.\nExtra Superdraft tip: Couples building their family home must agree on who will deal with the contractors. The builders should know who to approach when they are in need of something. Having a point person will help avoid misunderstandings.\nYou plan for your project because you don't want to make sudden and costly design changes during construction. Based on our experience, even a minor change in the design can lead to delays and budget overruns. It is frustrating when you blow your finances.\nSo, plan all your moves wisely in the beginning. Changes and revisions will come along the way. Go back to your vision every time you are required to make a decision. Weigh the pros and cons of your actions. Most importantly, make sure that you can still sustain the project after.\nChanging building materials in pursuit of more updated interiors is okay.\nOne way to cut the stress of building a house is through escaping. Make sure that you enjoy a relaxing downtime every day. Do yoga, run in the park, exercise, or anything that can get your mind off the construction site. Get all the lost energy back by doing something different. Taking a break will also help you make better decisions.\nThis is the last and the most important piece of advice that you need to hear.\nYou should not stop, despite all the delays, turmoils, and budget overflows. Find a way to get back on track. Focus on finishing your project and you will get there. Once you see your home standing on the ground, all the hardships will feel worthwhile.\nHire top-notch architects from Adelaide and nearby areas.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"You're going to love the Watch Ya' Mouth Throwdown Edition! Are you looking for a fun game to play during holiday parties? Or, maybe you just want to find something that will keep the kids busy while you bake, wrap, clean and cook to get ready for the holidays? Either way, keep reading. I will receive a product for this post.\nMy kids were introduced to this game last year when a friend received it. I think we may have to add this to the Christmas list for ourselves this year!\nIt does look like loads of fun!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This statistic shows the results of a survey conducted by Cint on the distribution of videogames played in Finland in 2017 and 2018, by game type. In 2017, 8.01 percent of respondents stated that they play first person shooter games.\n* Number of respondents interviewed corresponds to year 2018s.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The Whaley K1, a versatile portable integrated system, comes with DLP technology. The K1 brings a 120-inch screen, hi-fi and smart OS all together. It's not only perfect for making great parties and fun trips, but also provides an easy solution for conferences and events because of its integration, mobility and long-endurance design. The K1 is an easy-to-use device with three interaction approaches such as the basic selections by keys, the content searching by voice and the complementary functions by postures.\nThe versatile portable wireless projector with a distinctive tilt angle feature is a built-in DLP technology that combines 120-inch screens, high-fidelity sound and intelligent operating systems into a variety of meetings and events for users, friends gather or travel to provide an easy-to-use solution. In the function, it has all the features of wireless projection equipment, can cast high-definition picture quality, although the technology is not a revolutionary breakthrough, but through the sound, gestures and gently hit three interactive way to control, to provide users with a relaxed control experience.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Do you feel like you're in 'Prison' waiting for Dexter Season 5? Need a 'Break'?\nFeel like you're in 'Prison' waiting for Dexter's return? Need a 'Break'?\nDexter Season 5 ended in December. We were sad. We were impatient. But we were going to be strong. We have to put on a brave face. There is no choice but to endure the many, many months ahead before our favorite show returns to tantalize us with 12 weeks of pure televison bliss.\nWe whined. We cursed. Some of his cried ourselves to sleep at night. Some of his took up a hobby. Others, just, waited, stoicly. It's now April. What? only April? There's still 5 more months of a Dexter-less existence, before Season 6 finally arrives and gives meaning back to our pitiful lives.\nSure, there's other shows airing right now. While most are crap by comparison, some are tolerable, and a few are even, well, good.\nSo, what is a diehard Dexter fan to do? Especially now that Netflix is no longer streaming Dexter?\nSpeaking for myself, I am very picky about my choice of TV watching. I can watch one episode of something new, but it takes a certain magic formula to really grab my interest.\nThe premise. Is it something that hasn't already been done to death?\nThe main character(s). Is he or she multi-faceted in their personality? Can they convey the meaning behind their lines with simple facial expression?\nThe supporting cast. Can they each hold their own enough that I can watch a full episode about just them?\nThe writing. Is the show, exciting, witty, smart, and not predictable enough that I'm guessing the end?\nThe storylines. Are they unique, with twists and turns, with a whole of \"whoa I SO didn't see that one coming!\" ?\nDexter, of course, meets all of the above criteria.\nI do watch other shows. I love Law & Order, especially SVU, but also Criminal Intent. I love me some House. I was absolutely hooked on Criminal Minds before they started killing off their cast members. I couldn't get enough of The Sopranos. And Lie to Me, if you haven't seen it, had two really strong first seasons that were a joy to watch. However, Dexter has been my main man since he started dismembering the guilty in 2006.\nI almost feel like I'm betraying him, recommending another show. Especially since this show is no longer on the air. But because I feel for you, the Dexter fan starving for quality TV during this hiatus, I feel it absolutely necessary to share this little recommendation with you.\nI admit, I never thought twice about watching this show, when it was on the air and in the height of it's glory. It wasn't my style, wasn't my preferred genre, and just didn't at ALL interest me. Yawn. But in my own moment of Dexter-deprivation, I noticed this show was streaming all 4 seasons on Netflix. What could it hurt, right? Seemed innocent enough.\nOh how wrong I was. Not only did I immediately know, within the first 5 minutes \"oh snap, this is going to be GOOD!', but I shamefully admit to having watched almost 2 full seasons in the time-span of a week. Yes, it's that good! It's so good, you may find yourself staying up all night and calling in sick that next morning to watch \"just ONE more episode\".\nIf you're like me, just the title alone sounds like nothing more than some dirty convicts breaking of the pen, right? This show is not what it seems.\nFrom Netflix: When his brother, Lincoln (Dominic Purcell), is wrongly convicted of murdering the brother of the U.S. Vice President, engineer Michael Scofield (Wentworth Miller) resolves to bust his innocent sibling out of the notorious Fox River Penitentiary in this intense Fox action-drama series. Chronicling the prison break and its aftermath, the conspiracy-rich storyline takes the Scofields and their accomplices throughout the United States and beyond.\nMichael, a squeaky clean engineer, commits an armed robbery so he can intentionally get arrested and put into the same prison as his brother Linc, who is awaiting his fast approaching execution for a murder he didn't commit. This isn't just any framed murder. The cover up goes all the way to the White House.\nThe details of this show are highly intricate, starting with the fact that Michael is wearing the blue prints for the prison escape all over his upper body, in the form of highly intricate tattoos. Another interesting aspect of the show (and Michael's) appeal is that; while he's nothing short of devious and genius, masterminding the escape, he's also a man with a big moral compass. He breaks many laws and rules, but he will not kill another man.\nThen there are the characters. And OH are they characters. Whether it's the mysterious old man, rumored to be a legendary bank robber with $5 million hidden away somewhere; or the disgustingly vile T-Bag who would just as soon slice you than look at you; the endearing Fernando who did it all for love; or even the delightfully deranged Haywire, who's eyes can't look straight ahead if they tried \u2026. Don't be fooled. There's much more than meets the eye when it comes to this oddball collection of caged criminals.\nMy favorite character is Teddy Bagwell, aka T Bag. In researching this show, I've found I'm not the only one. Many fans have found that he is the one you love to hate, and then hate to love.\nThe entire cast of characters are so well developed, that each and every episode another layer peels off, revealing that they each have their own fascinating story to tell.\nAnd don't even think about trying to predict an outcome. Just when we think we're clever and know what's coming next, the plot twists and turns so fast that we humbly accept the fact that this show is much smarter than we are. My advice? Just enjoy the ride!\nWhere Can I Watch Prison Break Streaming Online?\nHave I sold you yet? I promise, I won't reveal any spoilers, but I will say this much: The scene where they finally attempt their escape, is the most EXCITING, heart-pounding, blood-pressure challenging scene I've ever seen, on ANY show. Ever.\nSo you may be asking yourself, where can I watch Prison Break, online? As of this writing, Prison Break is streaming all 4 seasons on Netflix. I cannot guarantee how long that will be the case! Make sure you don't miss out on this television masterpiece \u2026 Watch Prison Break.\n\u2026\u2026\u2026.EW is reporting the exclusive shocking and unfortunate news that Michael C. Hall have filed for divorce. Both reps add that the split was amicable.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabhmp b/data_all_eng_slimpj/shuffled/split2/finalzzzabhmp new file mode 100644 index 0000000000000000000000000000000000000000..2d3af65aa670ffbfffffda1ec1c29f9fbc16e5bf --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabhmp @@ -0,0 +1,5 @@ +{"text":"Rain Worthington herself kindly sent this CD to me for review. There seems to be very little biographical information online about this very interesting composer. Her current credentials available on her website are impressive but there is little to be found on her training and influences. (Time to talk to the webmaster I suppose) This appears to be the first disc dedicated solely to her music and it is an great one.\nThere are seven tracks representing seven orchestral pieces dating from 2001 to 2012. Like the album title, all are given poetic titles which make no reference to musical structure. No doubt there are stories behind each of these but there is no information in the liner notes as to the inspirations or intended meanings of this music.\nThis reviewer is fond of detailed liner notes but their absence is hardly a hindrance to the appreciation of these little tone poems for various sized orchestras. There is a remarkable consistency of style here suggesting that this composer has a clearly established voice. Clearly Worthington is a master of orchestration as she wields complex orchestral pallets where dissonance as well as orchestration contribute to the color in a basically tonal framework.\nThe music here is about motive and gesture rather than melody per se. Perhaps that is a \"minimalist\" method but the impression to this listener suggests influences as diverse as Claude Debussy, Jacob Druckman, Daniel Asia, Toru Takemitsu, and Anton Bruckner. This is beautiful though not \"pretty\" music with complex emotional content that leans in somber, wistful, sometimes haunting, and even nostalgic directions much as an impressionist painting.\nThe lovely cover photograph, a time exposure of a night scene with a blurring of stars over a lake, works as a metaphor for the album's content.\nThree orchestras (none of them American) under three different conductors are employed in this very satisfying release. Multiple listenings will definitely contribute new layers of understanding for the listener. This listener found that initial impressions of seemingly simple, pretty romantic\/impressionistic leanings give way to things deeper and perhaps even disturbing.\nOne of the conductors, Robert Ian Winstin (1959-2010) was an American composer and conductor (whose compositional efforts deserve some serious review) is at the helm of the Czech Philharmonic Orchestra on two tracks. Petr Vronsky leads the Moravian Philharmonic Orchestra on four tracks and Ovidiu Marinescu leads the Russian Philharmonic on one track. It is amazing how well these orchestras handle the challenges here. These are some excellent musicians.\nKudos to producer Bob Lord for his bold choice in releasing this disc. Hopefully we will soon be seeing more of this composer's work in the near future. My thanks to Ms. Worthington for sending this my way.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. This book features an ugly elementary and complete treatment of determinants early in the book. 9780134022697 0134022696 linear algebra and its applications plus new mymathlab with pearson etext access card package 5e.\nSchneider and linear algebra gems assets for undergraduate mathematics with d.\nLinear algebra and its applications. Linear algebra and its applications books a la carte edition plus mymathlab with pearson etext access code card 5th edition. Articles in press latest issue special issues all issues sign in to set up alerts. 1 16 of 132 results for linear algebra and its applications linear algebra and its applications 5th edition jan 3 2015. 14 7 1 4 7 1 4 7 0 3 5 035 0 3 5 25 9 0 3 5 2 0 0 0 2 gg g hh h kkg kgh let b denote the number k 2g h. Linear algebra and its applications is a biweekly peer reviewed mathematics journal published by elsevier and covering matrix theory and finite dimensional linear algebra.\n9780134022697 0134022696 linear algebra and its applications plus new mymathlab with pearson etext entry card package deal deal 5e. The book doesnt assume a lot of previous knowledge. Linear algebra and its applications 5th edition 579 pages. Now is the time to redefine your true self using sladers free linear algebra and its applications answers. With typical linear algebra texts the course is relatively simple for school youngsters by means of the early ranges as supplies is launched in a well known concrete setting.\n9780134022697 0134022696 linear algebra and its applications plus new mymathlab with pearson etext access card package 5e. Study of complex systems edit most physical phenomena are modeled by partial differential equations. Cheggs linear algebra experts can. Taylor calculus and its applications with lj. With traditional linear algebra texts the course is relatively easy for students during the early stages as material is presented in a familiar concrete setting.\nLinear algebra and its applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic arithmetic combinatorial geometric or numerical aspects. I had to read linear algebra and its applications by david lay for the linear algebra 1 class in my first semester in university. Replace the curve by its tangent line t the surface by a plane and the problem becomes linear. Powerpoint slides download only for linear algebra and its applications 4th edition download download compressed powerpoint files applicationzip 98mb instructors maple manual download only for linear algebra and its applications 4th edition. Now is the time to make today the first day of the rest of.\nSchneider and linear algebra gems assets for undergraduate mathematics with d. Thus it might be considered as linear algebra done wrong. Gilbert strang linear algebra pdf. The power of this subject comes when you have ten variables or 1000 variables instead of two. So this is a gentle introduction to linear algebra.\nLinear algebra and its applications 5th edition linear algebra and its applications 5th linear algebra and its applications 4th edition linear algebra and its applications strang linear algebra and its applications 4th. Gilbert strangs linear algebra and its applications gives both an intuitive feel to linear algebra and the geometry that accompanies most linear algebra ideas in two or three dimensions which can then be generalized to n dimensions. Unlike static pdf linear algebra and its applications 5th edition solution manuals or printed answer keys our experts show you how to solve each problem step by step. It also publishes articles that give significant applications of matrix. Shed the societal and cultural narratives holding you back and let free step by step linear algebra and its applications textbook solutions reorient your old paradigms.\nTaylor calculus and its applications with lj. Renowned professor and author gilbert strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. This is a basic subject on matrix theory and linear algebra. Linear algebra and its applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic arithmetic combinatorial geometric or numerical aspects. Review of linear algebra and its applications 4th edition i am an independent math tutor and have one student i work with who is currently taking linear algebra and is using this text.\n6 chapter 1 linear equations in linear algebra 25. Linear algebra and its applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic arithmetic combinatorial geometric or numerical aspects. Working with curved lines and curved surfaces the rst step is always to linearize. Then the third equation represented by the augmented matrix above. Strang provides a multitude of examples of each idea and a great set of exercises for the reader to try.\nLay is also co author of several mathematics texts including introduction to functional analysis with angus e. Of linear algebra for its own sake. This is why there are numerous applications some fairly unusual. While the mathematics is there the effort is not all concentrated on proofs. With traditional linear algebra texts the course is relatively easy for students during the early stages as material is presented in a familiar concrete setting.\nMathematica manual download only for linear algebra and its applications linear algebra and its applications 5th edition download download compressed pdf files applicationzip 363mb download download compressed notebook files applicationzip 13mb. Emphasis is given to topics that will be useful in other disciplines including systems of equations vector spaces determinants eigenvalues similarity and positive definite matrices. Algebra and its applications 4th edition linear algebra with applications 5th edition student study guide for linear algebra and its applications linear algebra and its applications schaums outline of linear algebra 5th edition. Linear algebra and its applications. Lay is also co author of several mathematics texts including introduction to functional analysis with angus e.\n612 solved problems 25 videos schaums outlines introduction to linear algebra 5th edition elementary. Linear algebra and its applications 5th edition uploaded by. Linear algebra is thus a fundamental part of functional analysis and its applications which include in particular quantum mechanics wave functions. I have done this because of the usefulness of determinants. Lay and steven r.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"GLOBAL AUTOMOTIVE SCANNER MARKET: TRENDS, OPPORTUNITIES AND FORECASTS (2016-2021) ##.\nThe automotive market in Germany is one of the most mature markets in the world.\nThe company' s business operations comprise one business segment: automobile tires, automobile tubes, and automobile flaps.\nThey were first used in automobiles by the Michelin brothers who went on to become one of the prominent tire vendors in Europe.\nRange of goods: - - - - tire goods for automotive and agricultural equipment; tire for airplanes; tire for bicycles and motorcycles; the company is also famous for its maps and guides \" ViaMichelin\".\nIt is one of the world' s largest automotive manufacturing companies that manufactures, develops, and sells passenger cars, LTs, industrial tires, and other related products.\nIt also sells other leading tire brands and automotive products and services.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This was one of those stories that demanded to be written. Set in the Eerie Side of the Tracks universe, Christmas Magic takes place between Witchfire Burning and Light Magic. It was fun to write because I really love Quinn and the other characters in the story. It is now available for download from Amazon.\nOh my. I'm not sure I dare even try to do a post today. How can I follow Dave's post yesterday? I don't know whether to smack his sad puppy nose with a rolled up newspaper or just give in to the laughter. Maybe both. VBEG. Add to that a laptop that has decided it really, really likes give off-the-wall error messages (usually at the most inopportune times) and I find myself wishing I followed my own advice and had a backup post ready to go. I don't, so I will try to push through before the laptop goes wonky yet again.\nOver the last week or so, I've had occasion to talk with several authors who are looking at dipping their toe in the indie market. Each of them had one common concern \u2014 promotion. They recognized that their publishers weren't giving the sort of promotion they had hoped for but feared it was better than what they could do for themselves. Even when I pointed out that they are already doing most of their own promo now, it took a while for the lightbulb to go on. They weren't tying their FB posts, their tweets, their blogs, etc., as promotion. One of them actually talked about it as their author platform because that was the terminology the publisher had used. When he finally realized it was promotion, it was almost a V-8 forehead smacking moment. The look on his face was priceless. I have a feeling if we hadn't been sitting in a public place, he might have banged his head against the desk.\nThe first rule you have to remember when you go indie is that it is a business, your business. That means you have to make informed decisions about what you are considering doing and you have to look at what the potential return on investment happens to be. Whether you are talking about editorial services, cover creation or promotion, you have to ask yourself one very important question: will you be able to not only make back the money you spend but then make a profit? The next question you have to ask is equally important: how long will it take for you to make back that money?\nSo, when I saw this post over at The Passive Voice yesterday, I quickly emailed the link to the authors I had been talking with. It is the tale of an author who learned the lesson of \"if it sounds too good to be true, it probably is,\" the hard way.\nLet me start by saying that I almost passed out when I saw that the author in question spent $10,000 to launch a book. Of that, $3,500 was for cover design, interior design layout and proofreading. That, alone, was enough for me to bang my head against the table. I can't speak about the interior design because I haven't looked beyond the first couple of pages of the sample. However, the cover design is nothing innovative and nothing to stand out from all the other books like it. Then I noted \u2014 and did pound my head against the desk \u2014 that she paid for proofreading but not editing. What? If she has a publisher, and it sort of sounds like she does, then they should have done the editing AND proofing. If she doesn't have one, then why pay for proofing and not editing?\nThe rest of the money, by far the bulk of it, was given over for promotion. Promotion that did not come through. You can read the post at TPV. Then follow the link to the original post. There were so many warning signs that the author ignored or let pass. I get wanting to avoid conflict. I get that she wanted to believe the promo folks would actually come through at the end. But there comes a point wher eyou have to remind yourself that this is YOUR business and you have to act in a businesslike manner. That means demanding accountings before the proverbial excrement hits the fan. That means doing the hard thing, even when you would prefer to keep your head down and avoid making the hard decisions.\nI'm not saying that all those who promise to promote your book are con artists. I'm sure there are some out there who give a good value for your dollar. But, before you agree to go with one of them \u2014 and certainly before you give them any money \u2014 read every word of the contract. If you don't understand it, have an attorney look at it. Ask for references and google the names of not only the company but any employee you might be working with. If you start working with someone and that person quits answering your emails or calls, escalate the matter to their supervisor asap. Don't wait until it is almost time for your book to come out. Most of all, make sure there is an out clause in the contract, something that will allow you to vacate the contract for cause \u2014 and without accruing a penalty. There is one other bit of advice I'd give. Make sure they are willing to give regular accountings of not only what they have done but the time spent doing it, monies spent, etc. You are their client. That means they work for you and should answer to you and not the other way around.\nFinally, remember that for all the money you spend before a book is published, you have to make that money back before you can say you have made any sort of a profit. So do the math and try to figure out how many books you would have to sell in order to become profitable. This brings up one last \"don't do this\" warning. Don't sign away a percentage of your roylaties. That leaves you with the onus of having to keep the books in such a way that you are giving regular accountings to someone else. It means you might be paying someone years after they last did anything for you. It also means you leave yourself open to a call for an accounting and that will put the burden on you to prove you only sold so many books. You pay for a service at the time the service is rendered, not in perpetuity.\nSince we're talking promotion, I guess I ought to do a bit myself.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"It's likely you'd heard of Vampire: The Masquerade\u2014Bloodlines before last week's sequel announcement, even if you've never played it. It's a cult hit with a vocal following, and often brought up in best-of lists\u2014one of those beloved PC games cited as somewhat broken, but full of amazing ideas. If you missed out at any point between 2004 and now, GOG has it for $5\/\u00a33.79 right now in its Spring sale.\nI'll admit it, reader: Bloodlines is on my pile of shame, and I've owned the GOG version for about three years, installing it on various laptops and never quite setting aside the time to give it a try. That'll change in the next couple of (blessedly) quieter months. Chris's preview of Vampire: The Masquerade\u2014Bloodlines 2 really got me in the mood for it, as did Jody's piece on the experience of being a vampire in the first game. Join me in experiencing it for the first time, if you wish.\nThe bonus with the GOG edition is that it'll install with the basic version of Wesp5's Unofficial Patch (currently 10.2, though you can download version 10.3 here).\nThe sequel, meanwhile, will be out in 2020.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzablww b/data_all_eng_slimpj/shuffled/split2/finalzzzablww new file mode 100644 index 0000000000000000000000000000000000000000..e77cc8a7a2b5159907c5addbf66a2746ee9e4c95 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzablww @@ -0,0 +1,5 @@ +{"text":"Our award winning Timber Software helps businesses meet today's challenges and is scalable for any growth strategy. A strong product roadmap ensures that our software is in line with the market requirements and supports 'best practice' processes for your business in the timber industry.\nEase of use - Quick and easy order entry, which we know is vital when processing what can be complex timber orders, is at the heart of the system.\nTraceability - Critical in the timber industry, our timber management software can follow the supply chain from the initial source, through any number of internal or third party processes.\nTimber Works Orders - You can build timber works orders on the fly, from pre-existing kits or bills of materials, or create configurations for special orders.\nPurchasing and Imports - Whether you import timber and stock it at the quay for onward transportation to your branches or a mill, or you purchase manufactured product through distribution, we provides comprehensive features to support the procurement cycle.\nWith our timber merchants software you can monitor performance in real time to get the best and the most accurate insights. This can enable you to spend more time reviewing your business requirements and processes, meaning more time to concentrate on how you can improve your business.\nKey performance indicators provide an overview of sales, margin data, stock levels and supplier positions, and because information is provided in real time, you can take appropriate actions accordingly.\nDrill down facilities, which provides information about margins, orders, refund\/credit information and customer purchase history.\nCustomer information on request to enable your staff to provide better service and fully respond to their needs and requirements.\nLearn how our timber management software can help manage your business.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"My therapist tells me things I don't like to hear.\nIn this last session we were talking about how sometimes people in helping professions become the subject of people's ire for no discernible reason.\nFor people like me, well, it really bothers me. I'm happy for you to dislike me if I've ticked you off or made an unpopular decision. That makes sense.\nBut many times pastors end up being the subject of people's disdain simply because, well, humans need enemies. And pastors are pretty easy pickings, most days.\nThey (usually) care, and it's always better to dislike someone who cares if they're liked or not. What good is a grudge if no one feels it but you?\nAnd sometimes people just don't like you for being you. And that, folks, is the the hardest to take. Because there's not a darn thing you can do about it. And so you just have to let your skin get tough\u2026and go to therapy.\nI mean, go ahead and choose your own expletive. He used one I can't write on a public blog that my mom will (probably) read.\nWe all have a Huckleberry, by the way. Or even a few of them.\nOur Huckleberries are usually that not for something they did, but usually for this indiscernible reason that we just can't seem place.\nWe just don't like them. We just don't.\nWhen pastors get this kind of flack, there are all sorts of reasons.\nIt may be because they're not the previous pastor. Or not like the previous pastor enough to pass muster.\nMost of my Huckleberries are my Huckleberries not for anything they did, but mostly because of me.\nThey are that because of my own baggage that I put on them and force them to carry, even though they didn't ask for it. I have to have somewhere to put it, and they're usually an ideal spot in my mind: they don't have to consent to carry it.\nThe Biblical model for this whole human practice, by the way, is the Scapegoat. It's a totally human, and apparently ancient, thing that we do.\nWhile having scapegoats, having Huckleberries, seem to be an important part of what it means to be a human with issues (and we all have issues), scapegoats (or, as I prefer it, Huckleberries) prevent you from ever confronting your own crap.\nAnd instead, the Huckleberry becomes the embodiment of our issues. Our issues with legs on. Our issues that can talk and smile and do good\u2026which makes us dislike them all the more.\nSee, we all know this intellectually. We know this. We know it's a problem; we know it's a manufactured malady that we create to deal with life.\nAnd so will our issues.\nPart of what the helping professions do, I think, is take it on the chin for folks who just need a Huckleberry. It's just true. And I say that with no amount of romanticism or martyrdom or any of that useless mess.\nThe world doesn't need any more martyrs. What I'm trying to talk about is truth.\nAnd the truth is that as long as we use religion as the harbor for our misplaced issues, it can never do what it's intended to do: free us.\nInstead it just becomes the storehouse for the issues we hoard away. A living museum of our personal problems transferred from one person to another.\nSo here's an idea: let's all start unloading our scapegoats and taking back our own issues. Leave your pastor, your musician, your teacher, your social worker, your doctor, your parents, your whomever out of your issues.\nLet's all start working through them, piece by piece, and clear out the rummage sale of religious baggage out there a bit so that the church can be a place of healing. The church has enough issues of its own, they don't need yours!\nBut the trick is, of course, that you can't store them anywhere else, either. You have to start sorting them out, bit by bit.\nI mean, it's worth a try.\nBecause as long as you have a Huckleberry, you're stuck working through your stuff from afar.\nBecause, in all honesty: you're your own Huckleberry.\nThis entry was posted in Church Life, Uncategorized and tagged Community, Complaints by Timothy Brown. Bookmark the permalink.\nI think I can readily agree with your premise but I am confused about the Tombstone parallel. I saw that movie years ago and all this time I thought being someone's huckleberry was some kind of endearment. It's not??\nI struggle with this because pastors, doctors, caregivers, etc. sometimes are doing a poor job. Or maybe they are doing the best they can, but it is not what your church or your loved one needs. And when it comes time to discuss the helper's performance, people are afraid to voice specific criticisms, so it all gets swept under the rug of \"I just don't like them.\" Somehow, someone has to be able to look beyond the generic sentiment and see if there is a real concern that needs to be addressed.\nCertainly poor performance can cause legitimate criticism.\nAnd just as frequently I believe that people can be poor patients, patrons, and parishioners. Relationships that are ill defined cause unrealistic expectations to be created.\nNeeded this today, Tim. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Running a business is time consuming. You likely are so busy managing staff, looking for growth opportunities and satisfying your clients that you have little time to search for technology solutions that can benefit your business. Even worse, you may be so frustrated with your current IT system that you make an impulsive technology purchase that causes more harm than good. To help you avoid this pitfall, we've listed five IT investment mistakes every business owner needs to be aware of.\nIt's the job of every marketer to make you believe the newest technology on the market will resolve all your problems. And while the latest cloud or virtualization offering is likely to make things better for many individuals and organizations, it isn't going to work for everyone.\nDon't let the flash and hype of a new product deceive you. Take the time to think about the results you're trying to achieve with technology. Make a list of them, and when you're done match those criteria with the product that fits. Any good IT provider will be happy to serve as your consultant to ensure you make the best choice.\nAs technology evolves, it is inevitably becoming simpler to use. Consumers want user friendly products and solutions that are easy to implement, and nowadays that's what they're getting \u2013 at least most of the time. Because of this belief that all products are going to be plug-and-play, many business owners hold the misguided assumption that any new technology they implement is automatically going to synchronize with their other IT. It is simply not true.\nThough many technologies are compatible with one another, your business is taking a big risk \u2013 that could result in massive downtime and wasted money \u2013 if you implement a new tool that doesn't integrate well into your current system and workflow. Be smart, do some research or consult with an IT professional before making a purchase.\nNow that you've found the perfect fit technology and you're sure it will integrate into your current IT setup, you go ahead and purchase it. You let out a sigh of relief as you kick back and let your sparkly new IT solution power your company to new levels of success and profits in a SMB \"happily ever after\" fantasy. Sound too good to be true? That's probably because it usually is.\nDon't forget that not all of your employees are going to instinctively know how to use the new technology. Consult with your IT provider to see if they offer support and training. If not, you may want to look elsewhere or find an alternative solution before you buy.\nMore and more IT solutions are packaged with pay-as-you-go monthly pricing. While this is a great way to help you avoid large upfront capital investment, if you implement too many different technologies too fast \u2013 and without thinking about the recurring costs \u2013 you could quickly run out of money before having properly created a complete technology platform.\nThink about what you're comfortable spending on IT before you open your wallet. Do some research, and either draft a budget on your own or acquire the assistance of a consultant to help you along.\nIt's wise to consult with the employees who will be using the new technology you implement, on a daily basis. It's even wiser to do it before you purchase it.\nThe truth is that not all of your employees may be on board with the new product. They may actually even know some downsides to it you weren't aware of. Regardless, it's smart to consult with them beforehand, or you may find yourself in a constant fight getting them to adopt it.\nNeed to consult with an IT professional to create a complete technology solution for your business? We're happy to serve you in any way possible.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"It's just that this is not where the story halts; it's not the whole story. You can say that the Greeks are not independent while the Americans are, but that is true in name only. What does that famous American independence consist of?\nHow free and independent are Americans really today? In 1956, Dwight Eisenhower famously talked about the military-industrial complex that the nation should beware of, but 62 years later is seems safe to say his warning was not heeded.\nDonald Trump ran on a non-interventionist platform, but the military-industrial complex appears to have him gift-wrapped up and ready for delivery a year and a half into his presidency. Under Trump, the US have dropped more bombs than even under Obama, no small feat.\nIt is of course well understood that if you justify such action properly through the media, the people will buy about anything when it comes to waging war abroad, but it's still not what people voted for. They are not independent of the war machine.\nAnd you can take this one step further. You can ask how independent a nation can really be if its citizens never learn to have independent thought, if they are deliberately never taught how to think for themselves. How can you be independent if and when other people define your thought and views?\nThe baby is taught history, which is the study of the ancient, leftover propaganda from whichever civilization happened to win the wars in a given place at a given time. The baby is taught geography, so that later on when its country begins bombing another country, the baby's country won't be embarrassed if its citizens cannot find that country on a globe. The baby is taught obedience, and the importance of performing meaningless tasks in a timely manner.\nThis prepares the baby for the half century of pointless gear-turning it will be expected to undertake after graduation. The baby is taught that it lives in a free country, with a legitimate electoral system which facilitates meaningful elections of actual representatives in a real government. It is never taught that those elections, representatives and government are all owned and operated by the very rich, who use them to ensure policies which make them even richer while keeping everyone else as poor as possible so that they won't have to share political power.\nAnd so you get to, have to, wonder if this is what the Founding Fathers would have wanted. Were they intending for America to be a nation filled with obedient sheeple? Since they themselves were revolting against a power that wanted for them to be just that, it doesn't seem likely.\nBeyond the issue of slavery, which would take another 90 years to come on the agenda, and another 90 after that to lead to the abolishment of racial separation (and likely another 90 for the next step), the Fathers appear to have sought actual independence. There are precious few signs that they would have looked kindly upon the military-industrial complex.\nOr the existing political parties for that matter, replete with 'life-long' politicians who label themselves 'public servants' but whose careers depend entirely on lobbyists and the corporate powers they serve, which donate the campaign cash needed to get the 'public servants' elected and re-elected.\nPower corrupts absolutely. And it crushes independence. Since all US politicians need corporate money to build and sustain their careers, isn't it ironic that they, too, would celebrate Independence Day? Independent from what?\nSure, you can celebrate that you no longer depend on British rule, and their tea, but if and when you simply swap one dependency for another, what does the word independence even mean any more?\nAt least that's something the Greeks have an answer to. They depend on Europe's largest nations and their banks. They had a chance to break that dependency, and voted to do exactly that, only to see their Independence Day crushed by the first prime minister in decades who was not part of an age-old corrupt system.\nTheir quiet despair should be shared by Americans too, who are free and independent men and women in name only, and who depend on Washington and Wall Street's deeply entrenched power brokers perhaps more then ever in their history.\nThe Greeks have no options left; they voted for the one independent party there was three years ago, and were betrayed with a kiss. America voted Trump into the White House and though he seemed independent, he also seems to have been eaten alive by the Deep State and the war machine.\nIndependence appears to mean, first of all, not relying on entrenched power blocks for your lives and livelihoods, the ability to make your own decisions, to reap the fruit of your own labor. Has America been further away from that at any time since 1776 than it is today?\nI hardly ever go out in the morning, the first 7-8 hours of every single day are taken up by reading and writing. But today I did, to feel the mood in the city. Not sure I got it, though. Everything's quiet. It may not help that I'm staying smack in the middle of the Acropolis tourist area (still haven't figured out why 90% of them are American).\nNot sure if many Greeks even really understand what is going on, and who can blame them, they have every reason to be scared more than anything else.\nAnd I'm still trying to wrap my head around the trouble just about everyone seems to have with the simplest and most basic exercise in direct democracy that's taking place right now. The referendum here today has been called manipulative, opaque, some say there's not enough time, others claim Tsipras is merely trying to save face in the face of defeat, courts have been called in to rule on its legality.\nBut this place here is where democracy started. And votes were held just like the one today, all the time. To be rid of despot rule, to let the people decide. And sure, in the beginning it wasn't all the people, just the alleged wise men, but it was a start. So why do we now find this simplicity so hard to stomach?\nPut another way: is this because for Americans the 4th of July is these days more about stuffing your faces surrounded by your equally overweight families than it is about honoring the Founding Fathers? It's quite possible, that our troubles with processing and absorbing direct democracy are somehow linked to that.\nThat Americans and Europeans have precious little understanding and appreciation left of what happened that led to the US celebrating, commemorating, its Independence Day in the first place. And therefore can't see how and why it couldn't have been accomplished without the example set right here in Greece many many years earlier.\nMaybe that's why a thousand pundits feel free to question the very principle of democracy. Or to at least try and hang all sorts of conditions and reservations on it. But it's not that hard really: a government asks its people what they think about a certain issue.\nThat's a democratically elected government's prerogative. It couldn't really get any more basic than that. And of all the freedoms we have, maybe the one that makes us question the very principle those very freedoms are derived from, is not the best choice. Maybe there are better and more productive freedoms to occupy ourselves with.\nAnd it can't be that the unfolding Greek drama hasn't given us enough material to hold against the light of democratic principles. The Troika machinations, culminating in the oppression of data vital to the negotiations, from those same negotiations, is just one example. A damning one, though, but still.\nFor democracy to function, it must first of all be allowed to function. That requires revealing all relevant information. It also requires all parties who are not party to a vote to keep their mouths shut. If you look at it from that point of view, Brussels and Berlin seem to have little understanding and respect for what democracy is. For them it seems to be something to be manipulated with impunity.\nAnd that does matter: democracy, to function, needs to be respected. Mere lip service doesn't cut it.\nWhatever the result of the vote is today, Greece is in for more hard times. A No vote would lead the little, little people in Brussels to engage in more strong arm tactics. And I see no reason to doubt that voting Yes is tantamount to sticking one's head in a noose.\nWho would want to live at the mercy of an institution populated by little people who actively try to keep vital numbers behind in a discussion held against the backdrop of hunger, suicide and despair in a country whose interests it is supposed to serve? But that's just me. And I don't have a vote.\nIf you look through Greek history, the country could claim an entire calendar full of Independence Days. The US has just the one, and it owes it to the ancient Greeks. Maybe that's something to ponder when waking up from those glucose-induced stupors this morning.\nThat like it or not, this is where the democracy was born that allowed for America to become a nation of free people. The same democracy celebrated from sea to shining sea every Fourth of July. And also the same democracy that is under threat, in Greece, in Europe as a whole, and very much in the US too.\nIt looks to me that we've all become quite far removed from what Independence Day is about, in Brussels, Berlin and Washington. And we should feel lucky if Athens today can give us back some of what has been lost in the translation and erosion of history.\nDemocracy is a fragile child. It needs to be fed and nurtured and caressed around the clock. Or it will wither away before our very eyes. The Greeks taught us all a valuable lesson before. Here's hoping they can again.\nAnd at the same time add yet another Independence Day to their long and rich calendar.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We hope that you will be delighted with your order but we understand that there are occasions that you may want to return items or that a parcel arrives damaged or faulty.\nIn this case, you must contact us within 14 days of receipt of your order on 01872 223220 or by email at Customer Care. We do however pride ourselves on our commitment to the highest levels of customer service and will consider cases on a case by case basis, subject to the situation, after this period.\nPlease note that we are not responsible for return postage, with the exception of faulty items or goods damaged in transit.\nOnce you have contacted us, you must take reasonable care of the products that you wish to return and not use them.\nPlease ensure all items are wrapped securely and we recommend that you use insured registered delivery and keep the Proof of Posting receipt as we cannot be responsible for return parcels that fail to reach us.\nIn order to qualify for a refund, the returned products must be unused, in perfect condition, undamaged and in the original packaging. All fragile items must be securely packaged.\nPlease note that large unwanted items will need to be collected by one of our couriers and the cost for this will be deducted from your refund - simply call us on 01872 223220 and a member of our team will be able to advise you.\nIf you are a UK\/EU consumer, the above comply with the Consumer Protection (Distance Selling) Regulations 2000.\nIf I buy something as a Christmas present, what is your refund policy?\nTo help you with your Christmas shopping, any products bought from the 1st December can be returned up to 14th January. This means you can shop early without worrying and should they need it, this will give the gift recipient extra time to return unwanted presents. The extended Christmas returns period applies to all products that can normally be returned for exchange or refund, in line with our usual returns policy. Any purchases made from the 1st January onwards will be eligible for the normal returns period.\nstay in the loop - sign up for our regular email newsletter and never miss out!\nthe art of \"just enough\"","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacejz b/data_all_eng_slimpj/shuffled/split2/finalzzzacejz new file mode 100644 index 0000000000000000000000000000000000000000..0dab16deb75d9768c61378022d63d9311bb1acd5 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacejz @@ -0,0 +1,5 @@ +{"text":"With the charm of a quaint European town and the hustle and bustle of a busy North American city, Old Montr\u00e9al is truly the crossroads of the past and the future. Narrow cobblestone roads lead to the doorsteps of Montr\u00e9al's hippest art galleries, cutting edge design companies, pioneering advertising agencies, and coolest fashion designers.\nWhen it came time for us to decide where to relocate, the choice was clear. Just steps away from historic Place d'Youville, in a building that dates back to the 1920s, our studios boast the best of what leading edge technology and old world charm have to offer.\nPlease contact us if you'd like more information and don't hesitate to drop by and see us if you're ever in the neighbourhood. The Clear Perceptions espresso bar is always open!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Please Note - This is an ex-demo or returned item that may have been assembled or slightly used. It has been checked and tested to insure its in full working order with all of the parts and accessories, but it may have slight marks, tatty packaging or signs of minor use.\nGRADED ITEM - THIS ITEM MAY HAVE BEEN PREVIOUSLY SHIPPED TO A CUSTOMER AND RETURNED TO US. IT IS CHECKED, TESTED AND FULLY WORKING, BUT MAY HAVE SIGNS OF PREVIOUS ASSEMBLY OR USE.\nThis 12 volt tough and rugged 4x4 is one of the hottest around and now your kids can drive one, too!\nRealistic 'key' start - Turn the realistic key to switch the car on.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"You know how way back in the day -- the Golden Age of gaming, as we oftentimes call it -- you could buy a PC game and then download a software development kit for it and mod the heck out of it? Well, those days are long gone.\nIn place of modding tools in today's industry we have SDKs from game engine developers, such as Unity Technologies, Epic Games and soon id Software. Capcom's senior vice president, Christian Svensson explains why, on Capcom's front, there isn't official mod support and it all boils down to resources.\nI love the modding scenes that have popped up around our games and at some point in time I'd love to see us do more actively to enable those efforts.\nThat said, there's a lot of hand wringing within the legal and IP depts when it comes to modding though. The concept of a derivative work starts to muddy the water of who owns the product or the mods. Secondly, modding opens up the community to infringe upon others' IP (say, a skin for Ryu made to look like Goku from DragonballZ). If we're enabling that ability, we also need to be able to police it and remove those infringing pieces from the community as we could have the liability in those cases, which takes time and resources. I think there are ways to work around that via a EULA and stuff, however there's a large hurdle to get folks past.\nFirstly, what is Sven talking about EULA, legality and protecting IP? It's a freaking modding community. Where the heck has Capcom been for the past 20 years of modding where pop-cultural entities are creatively modded into a game? Heck, all the top mods for Max Payne are recreations of The Matrix or Equilibrium (they're quite awesome, too). Modders tossed 3D Realms' Duke Nukem into Doom and modders also tossed the Doom Guy into Duke Nukem 3D. Has Capcom just been out of touch to not have seen Spider-Man and Superman in Grand Theft Auto IV or Link from The Legend of Zelda in Elder Scrolls V: Skyrim?\nMod tools or not, modders have been \"infringing\" on intellectual property for years. Like all of a sudden it starts if Capcom decides to officially support it the way DICE used to, the way id Software used to, and the way Epic Games used to? It's just tools, they don't need to get into the modding community any further than just \"opening\" up the game more for the community.\nHow about I just say what his PR spiel prevented him from saying: \"Our legal department can't seem to find a loophole for us to make money on mod creations without infringing on the copyrights of other material, so we're not going to waste resources on something that won't make us more money from those endeavors.\"\nLastly, there's the emotional connection some of the producers have to their games. Rationally or not, they don't want people changing the experience or the characters they've labored for 2+ years over. I don't agree but I know I've had exactly this conversation with a few producers in Japan on that front.\nHaha, really? There's so much wrong with that sentiment that it's not worth discussing.\nAnyways, it's funny because the modding community made Resident Evil 4 actually playable for the PC given that the textures, shader support, controls and graphics were completely screwed up and it was no where near as polished as the GameCube or PS2 version. If it wasn't for the modding community I don't think anyone could ever bother to play Resident Evil 4 on PC, which is sad.\nBut for those of you hoping for official mod support for Street Fighter X Tekken or Resident Evil 6 on PC, it doesn't sound like it's going to happen. Oh well, I guess you modders will just have to stick with doing what you do best all on your own.\nOh, and closing out the interview, one user noted that Capcom should no longer call it DLC since it's clearly not downloadable (although it is disc-locked content) and that maybe they should call it time-release content. Svensson briefly responded with \"Noted. Thank you for the suggestion.\"","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"R. M. Molfino, Muscolo, G. G., Puig, D., Recchiuto, C. T., Solanas, A., and Williams, A. M., \"An embodied-simplexity approach to design humanoid robots bioinspired by taekwondo athletes\", in A. Natraj et al. (Eds.): TAROS 2013, LNAI 8069, Springer-Verlag Berlin Heidelberg, 2014, pp. 1\u20132.\nP. Fedele, Federighi, P., Molfino, R. M., Muscolo, G. G., Recchiuto, C. T., and Rufa, A., \"High Energy Efficiency Biped Robots controlled by the Human Brain for people with ALS disease\", in 17th IEEE Mediterranean Electrotechnical Conference, 2014.\nV. Cipolla, Frediani, A., Molfino, R. M., Muscolo, G. G., Oliviero, F., Puig, D., Recchiuto, C. T., Rizzo, E., Agusti, S., and Stewart, P., \"A personal robotic flying machine with vertical takeoff controlled by the human body movements\", in A. Natraj et al. (Eds.): TAROS 2013, LNAI 8069, Springer-Verlag Berlin Heidelberg, 2014, pp. 1\u20132.\nA. Casals, Fedele, P., Marek, T., Molfino, R. M., Muscolo, G. G., and Recchiuto, C. T., \"A robotic suit controlled by the human brain for people suffering from quadriplegia\", in A. Natraj et al. (Eds.): TAROS 2013, LNAI 8069, Springer-Verlag Berlin Heidelberg., 2014, pp. 1\u20132.\nR. M. Molfino, Zoppi, M., Dinale, A., and Muscolo, G. G., \"A Robotic Vehicle for Freight Delivery in Urban Areas\", in The 16th International Conference on Harbor, Maritime & Multimodal Logistics Modelling and Simulation, 2014, pp. 154-159.\nP. Matteucci and Molfino, R. M., \"DEVELOPMENT OF A ROBOTIC CUTTING DEVICE FOR CONTAMINATED STRUCTURE DECOMMISSIONING\", in Mobile Robotics, 2012, pp. 811-818.\nR. M. Molfino and Zoppi, M., \"A robotic system for underwater eco-sustainable wire-cutting\", Automation in Construction, vol. 24, pp. 213-223, 2012.\nR. M. Molfino and Zoppi, M., \"Robotic grippers for handling fabrics: design cycle and example applications\", in Proceedings of the Internation IFToMM Mech. and Machine Science Conference CCAMMS07, 2007, pp. 16\u201326.\nR. M. Molfino, Costo, S., Cepolina, F., and Zoppi, M., \"AirEOD\", Int. J. Industrial Robots, vol. 33, pp. 228-235, 2006.\nR. M. Molfino and Zoppi, M., \"Heavy duty uuvs for off-shore plants decomissioning\", in Proc. of the 9th International Conference on Climbing and Walking Robots, 2006, pp. 1\u20137.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Re: Budget for Master Bedroom?\nWe really needed a new couch when our cat peed on our old one. We already re-stuffed the cushions once when she did it before.\nThis couch was expensive brand (i only know expensive bc I googled the brand and their couches start at $5000) that we got cheap at Goodwill. But in that case we had a gross looking old one, so I *had* one when I Ran into that one. But we needed a new couch immediately this time and I just couldn't figure out how to get one used for any decent price without the free that it has hidden odors.\nWe got a cheap one from IKEA. We definitely wanted cheap in case the cat struck again OR just bc we have animal(s) and children. It was the fake leather so EASY to clean. And sure enough the cat (who only has back claws) got scared adn jumped on and off the sofa and put tiny claws marks on it. It's not really noticable but I did say - and that is why we bought an inexpensive new from Ikea. So one thing I would factor in, what is your household like. Are you the type of home where you sigh and say and that is why we can't have nice things. Or do you plan to keep it for a long, long time.\nI love ikea. We just got new curtains from there for $25 total for both panels. They may not be the best quality but IKEA things seem to last longer for us than Walmart things. Tho dh has a computer desk from them that he had before I met him. So its over 20 years old and still looks fantastic. Its glass and metal and like new looking.\nAlso, we are looking to rehome our cat if anyone is interested.\nHey, we're also looking to rehome our cat!\nWe have the Hemnes bed, night tables and tall dresser. I love them!\nWe don't have night table lights, we have a cat who would knock them over and break them.\nDo you have a furniture outlet or a freight type store? Around here, we have something called American Freight that sells decent quality furniture for discount prices.\nI'm not sure, but that's a great idea. I'll check. Thank you!!","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacfqd b/data_all_eng_slimpj/shuffled/split2/finalzzzacfqd new file mode 100644 index 0000000000000000000000000000000000000000..88deaa1e2f8bbf7bbecf1b7d94496766c5cd8328 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacfqd @@ -0,0 +1,5 @@ +{"text":"We will discuss Dementia and Lyme disease on March 27, 2018 at 8PM EST. I will start the discussion with the legendary singer and songwriter Kris Kristofferson story.\nComments are no longer live. This is an archive of a Lyme Hangout held on 3\/27\/2018. You can still view the video below. For the next Lyme Hangout where you can participate, Save Your Seat by clicking here.\nFor several years, legendary singer and songwriter Kris Kristofferson was believed to be suffering from Alzheimer's disease or dementia. However, the musician's memory problems were later identified as undiagnosed, untreated Lyme disease.\nAccording to a story in the June 6, 2016, issue of Rolling Stone, doctors had believed the 79-year-old singer's \"increasingly debilitating memory loss was due to either Alzheimer's, or due to dementia brought on by blows to the head from the boxing, football and rugby of his teens and early twenties.\" He reportedly could not remember what he was doing from one moment to the next, the magazine reports.\nFor 10 years, Kristofferson was treated unsuccessfully for Alzheimer's disease. After he was diagnosed and treated for Lyme disease, his life changed. Kristofferson's wife, Lisa, explained that once her husband stopped taking medication for Alzheimer's and depression; and underwent three weeks of treatment for Lyme disease; his memory and mood changes were shocking. \"All of a sudden he was back,\" she says. \"There are still bad days, but some days he's perfectly normal and it's easy to forget that he is even battling anything,\" she told Rolling Stone.\nFrom Kris Kristofferson: An Outlaw at 80. by Neil Strauss in Rolling Stone at http:\/\/www.rollingstone.com\/music\/features\/kris-kristofferson-an-outlaw-at-80-20160606. Last accessed 6\/12\/16.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"At Janssen, we don't want cost to get in the way of treatment you need. We can help you explore options to lower your out-of-pocket cost for DARZALEX\u00ae. No matter what type of insurance you have\u2014or even if you don't have insurance\u2014Janssen CarePath can help explain your medication insurance coverage and benefits, and help find programs that may help you pay for DARZALEX\u00ae.\nIf you have commercial or private health insurance and need help paying for DARZALEX\u00ae, the Janssen CarePath Savings Program may be able to help. If you don't have commercial or private health insurance, you may find help from the programs and resources found on JanssenPrescriptionAssistance.com.\nIf you are eligible, you may be able to lower your out-of-pocket medication costs for DARZALEX\u00ae. Depending on your health insurance plan, savings may apply toward co-pay, co-insurance, or deductible. With the Janssen CarePath Savings Program for DARZALEX\u00ae, eligible patients pay $5 for each infusion for your medication costs, with a $20,000 maximum program benefit each calendar year. Not valid for patients using Medicare, Medicaid, or other government-funded programs to pay for their medications. Terms expire at the end of each calendar year and may change. There is no income requirement. For full eligibility requirements, please visit Darzalex.JanssenCarePathSavings.com.\nCreate an online account and enroll at MyJanssenCarePath.com or call a Janssen CarePath Care Coordinator at 844-55DARZA (844-553-2792), Monday-Friday, 8:00 AM - 8:00 PM ET.\nJanssen CarePath can provide information about other resources that may be able to help with your out-of-pocket medication costs for DARZALEX\u00ae. Call Janssen CarePath at 844-55DARZA (844-553-2792) or visit JanssenPrescriptionAssistance.com for more information about affordability programs and independent foundations\u2020 that may have funding available.\nIf you are enrolled in Medicare and using Medicare Insurance to pay for your medication, our Medicare Resource Guide provides additional information on sources for cost support that may be available to you.\n\u2020 Independent co-pay assistance foundations have their own rules for eligibility. We cannot guarantee a foundation will help you. We only can refer you to a foundation that supports your disease state. This information is provided as a resource for you. We do not endorse any particular foundation.\nThe Johnson & Johnson Patient Assistance Foundation, Inc. (JJPAF) is an independent, nonprofit organization that is committed to helping eligible patients without insurance coverage receive prescription products donated by Johnson & Johnson operating companies. To see if you might qualify for assistance, please contact a JJPAF program specialist at 1-800-652-6227 (Monday-Friday, 9:00 AM - 6:00 PM ET) or visit the foundation website at www.JJPAF.org.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The Library of Congress > Chronicling America > The Ironton journal.\n245 14 $a The Ironton journal.\n362 1 $a Ceased in 1878?\n580 $a Triweekly ed.: Tri-weekly journal (Ironton, Ohio : 1867).\n651 0 $a Lawrence County (Ohio) $v Newspapers.\n651 4 $a Ironton (Ohio) $v Newspapers.\n752 $a United States $b Ohio $c Lawrence $d Ironton.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We know that those involved in estate management want a responsive, pro-active, no-nonsense and transparent service. Whilst cost might not ultimately be the deciding factor when you select your legal services partner, our regional placement puts us at the absolute forefront with the ability to provide a competitive and flexible fee structure for all aspects of the service.\nOur team will develop a unique knowledge of your business, objectives and how you like things to be done. This will be your work, done your way. We will explain the reasoning behind our advice in a clear \"no-jargon\" manner so that you will have a fully informed view of any risks associated with a transaction, matter or approach.\nBelow is a non-exhaustive list of the types of matters we can assist you with, whether it be transactional, litigious or contentious, or in relation to a tenant debt, we have an expert to assist.\nThis brochure outlines the services we offer to the real estate sector, with case studies from Bowmer & Kirkland, Land Registry, Marston's plc and Peveril Homes.\nWe can assist and provide advice at all stages of the landlord and tenant relationship which gives us a unique birds-eye view of any property portfolio that we are instructed on. This end-to-end approach means that our team will be aware of any issues that arise at the outset, enabling them to deal swiftly with any issues during the term, and with the necessary knowledge to achieve the required results when a lease comes to an end.\nIn developing this offering, we contacted Estate Managers and Directors for various well-known and national companies across a multitude of industries including retail, property management and the licensed trade. We asked them what was most important to them in choosing a legal services partner for real estate management and property litigation work. The results which came back gave us the confidence that our service was, in fact exactly what clients are looking for.\nWe know that even when you have a large portfolio of properties, just one problem tenant can be extremely stressful, causing issues throughout the estate and disrupting other tenants.\nThe most common problem faced by any landlord is getting their property back. This comes with a minefield of constantly changing legislation and legal loopholes.\nWe recognise that what works for one client, may not necessarily work for all clients and therefore we have tailored our approach to offer a range of pricing structures to suit your business. The fee packages below are an illustration of what we can offer, however, we are happy to consider alternatives.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"the simplest demo... need some help with multiple rounds, earning money, a shop, a system of spawning characters, and the rest i can handle..\nplease pm any bugs and glitches found..\nSometime I plan to remake this game, preferably after Oncoming Storm has been finished. Watch this space!\nYou are a SWAT Agent and in a normal night were you are organizing your things for tomorow job you hear a sound of pistols and go investigate.\nYou follow the trail of the bad guys and find the mafia boss of Giligan City!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadrxz b/data_all_eng_slimpj/shuffled/split2/finalzzzadrxz new file mode 100644 index 0000000000000000000000000000000000000000..5056f8c504de5339bd7eb7eb4d21ba3cf4eebbf9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadrxz @@ -0,0 +1,5 @@ +{"text":"As \"Help!\" turns 50, these psychedelic cinema songfests will have you shouting just that.\nHelp!, the second film starring the Beatles, hit theaters on July 29, 1965. 50 years later, it's still an effervescent onslaught of off-the-wall comedy, innovative moviemaking, and one wondrous classic song after another\u2014 all resulting in ninety-six minutes of pure joy.\nDirector Richard Lester, who oversaw the Beatles' big-screen debut, A Hard Day's Night, also returned for Help! For their first collaboration, Lester shot in black-and-white, using an almost documentary style to chronicle one London day in the life of the humanity's most explosive pop phenomenon. It is flawless.\nHelp!, while also perfect, is an entirely different undertaking. The movie launches the group into a splashy, madcap, full-color, globetrotting series of adventures, utilizing wild gags and flights of surrealism to spoof James Bond, Western interest in Eastern spirituality, avant-garde film techniques, and, of course, Beatlemania itself.\nIn terms of influencing other films, the difference between A Hard Days Night and Help! is, interestingly, akin to something Bruce Springsteen once said about one key difference between Beatles and the Rolling Stones.\nAs a result, Help! immediately changed the look, sound, and feel of mass media, directly impacting everything from commercials to kiddie shows to underground art pieces. NBC's The Monkees was essentially Help!: The TV Series. And as mid-'60s popular music conquered human consciousness as nothing had since the dawn of cinema, rock movies became all the rage.\nIn honor of Help!'s 50th anniversary, here's a look at ten movie musicals that share either direct ties to the Beatles or aesthetic connections to the band's fantastically fanciful second movie. Some are brilliant, some are brutal, and some are just blah. Each, in its own way, will make you cry Help!\nBritish Invasion popsters Herman's Hermits send themselves up in Hold On!, a flimsy but pleasant cobbling together of Beatles movie tropes and Hermit charm on a cheap \"teensploitation\" budget.\nHold On! mostly focuses on the Hermits keeping it together while Herman Hysteria rages in Los Angeles. In the meantime, a NASA scientist tails the group for some cockamamie rocket project. The big climax involves the band flying via hypersonic jet from Hollywood to Cape Kennedy and back again for a Rose Bowl concert.\nOddly, the squeaky-clean Hermits got to make a second movie two years later, after acid rock and confrontational uncensored cinema took hold. Mrs. Brown, You've Got a Lovely Daughter (1968) is about Herman and the boys' misadventures at a dog-racing track. So perhaps they weren't entirely immune from the chemical influences of the time, either.\nFledgling Aussie songbird Olivia-Newton John stars in Toomorrow as (wouldn't you just know it!?) Olivia. It's a cosmic romp produced by record biz mogul Don Kirshner, who also served as musical overlord to bubblegum phenoms the Monkees and the Archies.\nOlivia fronts a pop group at the London College of the Arts called (again, no surprise) Toomorrow. Their sound is so far out that it gets picked up across vast galaxies by an alien race who believe the music of Toomorrow holds the key to their future. Naturally, they kidnap Olivia and her mates.\nToomorrow artlessly fuses together sunshine-pop lip-synch performances, zany shenanigan music montages, and jolts of slightly too serious sci-fi to a point that it all almost becomes kind of fascinating. Almost.\nAlas, Kirshner's gambit to hit pink, chewy AM-radio pop gold yet again crapped out. He had far better luck a few years hence when he switched to FM airwaves and signed hit-making prog-stompers Kansas.\nThe Spencer Davis Group, featuring dynamic young keyboardist Steve Winwood on vocals, agrees to spend the night in a haunted mansion inherited by their stuffed shirt manager Algernon Rowthorpe (played by popular UK media personality Nicholas Parsons).\nTo steady their nerves and appease the spectral presence of the title, they play music. Later on, Spencer Davis, Steve Winwood, and the gang hold a concert to refurbish the beat-up spook manor, and they're joined by the Paramount Jazz Band, the Three Bells, St. Louis Union, and the Lorne Gibson Trio.\nYep, that's what happens in The Ghost Goes Gear. That, and nothing else. Gear, no?\nAmidst the first wave of the British Invasion, the Dave Clark Five provided the closest (pre-Stones) thing the Beatles had to a real rival.\nThus, when the DC5 got to make their own movie, they went the black-and-white route of A Hard Day's Night, but from there the resemblance between the two films veers off in entirely unexpected directions.\nTo begin with, the Dave Clark Five don't play themselves, or even a rock band, in Having a Wild Weekend. Instead, they're professional stuntmen. Dave Clark plays Steve, a cynical crash-and-tumble pro who takes off from a meat commercial gig with Dinah (Barbara Ferris), the ad's Swinging London fashion model star, in tow.\nEn route to their spontaneous countryside getaway, Steve and Dinah come across beatniks, potheads, soldiers conducting war games with live ammunition, and other living signs on the times. Satirical barbs are launched at the middle class, the advertising industry (cool people really hated billboards back then), and, as usual, pop stardom; in addition, Wild Weekend boasts unexpectedly moving dramatic passages and bold flourishes of original filmmaking.\nDirector John Boorman went on to a brilliantly eccentric career, making Point Blank (1967), Deliverance (1972), Zardoz (1974), Exorcist II: The Heretic (1977), Excalibur (1981), and Hope and Glory (1987).\nA \"gonk\" is a furry doll sporting googly eyes and flat feet that became a staple of the 1960s British mod scene. Gonks Go Beat taps this toy fad and filters a standard Romeo and Juliet scenario through some (charmingly) inane science fiction that tweaks establishment mores and stands with the (also charmingly) lamest of touches.\nIn the future, Earth is split into Beatland, where everybody dresses cool and listens to rock-and-roll; and Ballad Island, where they're all pent-up squares who still play crooner records. Romance is forbidden between denizens of the opposing districts, under penalty of banishment to Planet Gonk, which is overrun with fuzzy toys who really dig Dixieland jazz.\nNaturally, a Beatland Boy falls for a Ballad Island gal (and she for him), prompting each one's warring homeland to wage a battle through music.\nOn-screen performers include Lulu (\"To Sir With Love\"), the Nashville Teens (\"Tobacco Road\"), and, in a casting move way more curious than any of those plot details, the heavy blues Graham Bond Organisation, featuring Ginger Baker and Jack Bruce who, just a few years later, would rock the planet off its axis as two\/thirds of Cream.\nIn between their first breaking big as the Australian front of the '60s British Invasion and their late-'70s stint as Saturday Night Fever's disco stud-gods supreme, the Brothers Gibb took quite the bountiful psychedelic side trip.\nOne result was magnificent: the Bee Gees' masterful 1969 two-disc concept album, Odessa. The other was Cucumber Castle.\nAiring on TV over two nights at Christmas, Cucumber Castle stars only two Brothers Gibb, Barry and Maurice, as Robin temporarily quit the group before filming.\nThe story takes place in a medieval fairy tale milieu, with Barry and Maurice playing the songs to a dying king (UK comic Frankie Howerd, who went on to be Mean Mr. Mustard in the Sgt. Pepper movie\u2014which is coming up).\nBarry declares himself the King of Cucumbers. Maurice announces he's the King of Jelly. Their slapstick rivalry involves playing tennis in full suits of armor. Lulu co-stars as a maid. Blind Faith jams in Hyde Park while the regal siblings are dressed as chickens, and the camera provides peeks at Mick Jagger, Donovan, and Marianne Faithfull in the crowd.\nWhatever everyone involved was smoking, it likely wasn't cucumbers and\/or jelly.\nMade for a turned-on \"head\" movie circuit that had revived golden oldies like Reefer Madness and was launching new classics like the Rocky Horror Picture Show, the jarring, feature-length All This and World War II consists of nothing but actual 1940s military footage intercut with golden age Hollywood moments, all of which plays out over a continuous roll out of Beatles songs covered by other famous musicians.\nThat soundtrack is where the movie goes from grotesque to groovy. Among its highlights are Elton John's hit version of \"Lucy in the Sky With Diamonds,\" along with \"When I'm Sixty-Four\" by Keith Moon; \"She's Leaving Home\" by Bryan Ferry; \"Come Together\" by Tina Turner; and \"Get Back\" by Rod Stewart. It also contains multiple tracks by the Bee Gees, the Four Seasons, and Leo Sayer.\nSo declared the posters for Beatlemania, a long-running 1970s Broadway hit in which Fab Four imitators worked their way through the group's hits amidst a multimedia stage show that used sophisticating lighting and projected images to acclaimed effect.\nThe posters for this inevitably cheeseball movie version proclaimed the same thing, but, up close on a movie camera, the four Beatlemaniacs look like dudes in costumes and the '60s images that came off powerful as ethereal projections look clumsy and dull when they fill the entire screen.\nNonetheless, Beatlemania is a nifty relic and well worth a gander for completists and curiosity seekers.\nOne of Hollywood's all-time most notorious mega-bombs casts the Bee Gees and Peter Frampton as the titular figures who were immortalized on the Beatles' most famous album.\nHere, Sgt. Pepper and his merry band are buried under mounds of movie and record industry hubris, topped only by the tsunamis of late-'70s bad habits that blew through of every aspect (and up every nostril) of this mammoth disaster's crash-and-burn inferno.\nSgt. Pepper's plot regards the Lonely Hearts Club band leaving idyllic Pepperland, where the town center is dominated by a giant, drippy cheeseburger for Los Angeles, where they fall prey to evil corporate overlords (the irony was indisputably lost on the film's makers).\nAlong the way, A-list guests perform Beatles songs in elaborate production numbers, Pepperland falls from grace, Billy Preston as the good sergeant himself restores order, and, for the grand finale, a hodgepodge of almost entirely unrecognizable rock stars and oddball celebrities \"recreate\" the original album's famous cover.\nThe Monkees was essentially television's weekly half-hour version of the Beatles' Help! It was an immediate hit, generating a rapid-fire succession of radio smashes and even bona fide \"Monkeemania,\" while the TV series itself proved to be a fertile visual incubator of cutting edge ideas that later fully flowered during the heyday of music videos.\nOf course, rock purists griped that the Monkees were a fake band that relied on outside writers (albeit the very best in business, including Neil Diamond, Harry Nilsson, and Tommy Boyce and Bobby Hart) to churn out pop product for mainstream nincompoops. Mostly it was kids that dug the Monkees, but still.\nThe four performers hired to play the band were each uniquely talented and even artistically serious individuals who immediately resented the devil's deal they felt they'd made. They were the biggest group in the world for a while, with millions screaming for them, but their high-minded rock-and-roll friends shunned them as traitors and sell-outs.\nAs Monkeemania ebbed in 1968, the group's members\u2014Micky Dolenz, Michael Nesmith, Davy Jones, and Peter Tork\u2014teamed with screenwriter Jack Nicholson (yes, that Jack Nicholson) and director Bob Rafelson (who'd go on to Five Easy Pieces) to give the Monkees a proper Viking funeral by way of the freakiest, heaviest, goofiest, angriest, most amazingly off-the-goddamned-wall rock musical to ever scorch movie screens.\nHead begins and ends with the Monkees committing suicide. In between, a relentless succession of scenarios set up jokes that come with no punchlines\u2014but they just keep coming.\nFrank Zappa walks a talking cow. Mousekateer Annette Funicello shows up as \"Minnie.\" Sonny Liston k.o.'s Davy Jones. A tough cop go-go dances. Peter Tork unlocks the meaning of life from a nelting ice cream cone. Stanley Kubrick favorite Timothy Carey teaches the Monkees, \"Boys, don't never, but never, make fun o' no cripples!\" Real footage of Vietnam atrocities gets intercut with frantic images of fans tearing apart fake Monkees. The entire group gets trapped as particles of dandruff in Victor Mature's hair.\nBoth the movie and the record bombed, of course, which could be set to have been part of the design and which, when you see it, was certainly inevitable.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"There are no people in stigerland's collective.\nSubscribe to a podcast of everything huffduffed by stigerland.\nMarvel er endelig tilbake p\u00e5 kidolerettet og endelig har vi f\u00e5tt en kvinnelig superhelt i hovedrollen!\nMed: Jostein Hakestad, Stian Askeland, Are Ness Fl\u00f8gstad og Ida Doris Joynt.\nSonic the Hedgehog is Sega's flagship character. He is in a ton of games, including some that might not be so obvious. Let's find all of the games Sonic is secretly hiding in on the Sega Genesis!\nLearn about the history of a unique NES accessory, the Aladdin Deck Enhancer! In the early 90's, the 16-bit war between Sega and Nintendo was heating up. But one company was thinking about the past: Camerica. In January of 1993, they announced the Aladdin Deck Enhancer, a device for your 8-bit NES that played proprietary compact cartridges. Why? Watch the video to find out!\nHere's a quick analysis of the Garmin acquisition of Tacx and their trainers. From the winners to losers, and how it'll ripple across the industry. I dive into all the nuances.\nThe Super Mario Land series was Nintendo's attempt at bringing the Super Mario franchise to the Game Boy. But it was also the first time Mario creator Shigeru Miyamoto would not be involved. Learn all about the history of this series and the impact it has left on the Mario franchise!\nCES 2019 Sports Tech Wrap-Up and Thoughts!\nThis is like a livestream, except, not live. But, all in one shot! Here's all my thoughts on CES 2019, sports tech, and all sorts of randomness with it. From the products that were announced, to what's missing here. Plus, which tech products I think will do well, and which ones will failboat.\nI cover wearables, sports tech platforms, action cams, drones, and all sorts of totally random things that have really nothing to do with sports tech\u2026oh, and don't miss the end!\nVi runder av 2018 med en \u00e5rets spill-spesial av podcasten v\u00e5r. I tillegg g\u00e5r vi gjennom de filmene og TV-seriene vi likte best i 2018. Takk til alle som har fulgt oss i \u00e5r! Vi gleder oss veldig til et spennende 2019. God jul og godt nytt \u00e5r!\nA documentary detailing the complete history of Nintendo's Punch-Out!! video game series. In 1983, Nintendo had a problem: They had too many arcade monitors. Rather than toss them, they made a new game that could use 2 monitors. That game was Punch-Out!!, one of Nintendo's most beloved franchises.\nPC players definitely missed out on some of the biggest AAA games this year but the platform is still the best around for 3rd party and creative independent games. We discuss our personal favorites along with what we think the future of the PC holds when compared to consoles.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Dead One Percenters: 17 Down.\nThe American-Turkish Council (ATC) is a business association dedicated to enhancing the promotion of US-Turkish commercial, defense, technology, and cultural relations. Its diverse membership includes Fortune 500s, multinationals, U.S. and Turkish companies, non-profit organizations, and individuals with an interest in U.S.-Turkish relations. Guided by member interests, the ATC strives to enhance growing ties between the U.S. and Turkey by increasing investment and trade through high level government relations, effective commercial engagement, and bilateral cultural education.\nATC's advisory board also includes representatives of a number of defense, pharmaceutical, consulting, and technology firms, including BAE Systems and Motorola.\nThis is the group that has been pinpointed as the instigators of the coup. The report on Stew Webb's YouTube channel points out that the Russian Federation is responsible for stopping the coup, and that they left with all computers, documents and cell phones. The coup is the product of Mossad and CIA operatives. Israel had formerly been allied with Turkey, but when Erdogan began to cooperate with Putin last week, a hastily put-together coup was arranged to prevent Putin from continuing to get evidence of systemic crimes by government officials.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Visit the form below to determine if your household is within one of the compost collection service areas and reserve a green cart for your home.\n**If you already have a green cart at your home, please do not sign up here. Instead call 311 (720-913-1311) or email DenverRecycles.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The NASBP Golf Tournament is scheduled on Tuesday, April 16 at 1:00 pm. The tournament will be held at the Omni Barton Creek Coore Crenshaw Course which is approximately 11 miles from the JW Marriott Austin, shuttles to and from the course will be provided. There will only be one tournament which will be a scramble format.\nIf you plan to play with a specific foursome, include the names of your foursome on the Meeting Registration Form. Foursome requests are due by Friday, March 29, 2019.\nRegister early, since the course has limited space and golfers will be accommodated on a first come, first served basis.\nThe Tournament Fee is $315 per player. Tournament fees include transportation to courses, tournament coordination, range balls, a boxed lunch, golf carts, and prizes. Rental Clubs are available for rental, to be purchased at the pro-shop on day of play.\nTournament Cancellation Policy: Golf cancellations must be received in writing by 5:00pm on Friday March 15, 2019 to NASBP at bjones@nasbp.org. Any Golf Tournament cancellations received after March 15 will only be refunded if another player fills the space. Refunds will be issued after the meeting and may be subject to a 5% processing fee.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafufq b/data_all_eng_slimpj/shuffled/split2/finalzzzafufq new file mode 100644 index 0000000000000000000000000000000000000000..c3dd493677d37cfc96679df7c5fb415efda647d1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafufq @@ -0,0 +1,5 @@ +{"text":"Declutters the floor (Park your car) - No ladders!\nImagine an organized garage - THE Only SAFETY-CERTIFIED LIFT!\nOur installer is returning tomorrow to complete the other two sets. The first set looks awesome!\nWOW!! Great Product. Excellent customer service!! Great product. Would recommend this to anyone.\n\u2022 Remove the dangers of ladders. No time-consuming and backbreaking lifting!\nIn addition to all those benefits our unbeatable pricing, and by blending durability, highest quality parts and an impressive design, our storage option brings maximum value - without the unsafety of other organization systems.\nHow was the Auxx-Lift Series developed? Auxx-lift has been developed over the last 4 years and has undergone many revisions and product testings to UL-standards to make it a safe and trouble-free product. It is a revolutionary home improvement and storage product. We adopted special features such as a remote-control.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"I belong to a Middle Class, Joint Family with Moderate Values. I have completed my B.Tech. Previously working in pvt. sector and presently preparing for public sector bank exams.\nMy relative is self-employed with a Bachelor's degree currently living in Chandigarh. She belongs to a Middle class, Nuclear family with Moderate values.\nMasters in English literature . I am working with Indian air force. I belong to a mediocre family. My hobbies and interest include listening songs and cooking. I am a very simple person who believes in leaving happily and spreading happiness around.\nMy name is RITU SIWACH and I have completed my B.A.,LL.B.(Hons.) and LL.M. Now I am preparing for competitive exams. I belong to a Upper Middle Class, Nuclear Family with Moderate Values.\nMy daughter has a Bachelor's degree and She is employed in private Sector as a Legal Associate in Noida.\nMy sister is a journalist with a Master's degree working with All India Radio as a Senior Journalist. She is currently living in Delhi. We come from an upper middle class, nuclear family background with moderate values having our roots in Meerut.She has high moral and social values. She has managed her work and family very well. Even accompanied Prime Minister of India in 2 of his official foriegn visits.\nMy name is Gautam Siwach and I am currently self Self Employed. I have completed my B.Tech.. I grew up in a Middle class, Joint family with Moderate.\nMy name is Shubham Kumar Singh and I am working as IES in the Government Sector with Indian Railways. I have completed my B.Tech.. I grew up in a Upper Middle Class, Nuclear Family with Liberal Values.\nA passionate Medical Device Engineer with an undying love for travel and food. I am a pretty chilled out, down to earth guy who enjoys meeting and having decent conversations with people from all walks of life.\nMy name is Anil and I am work as an Officer in the Government Sector. I have completed my LL.M.. I have been raised up in a Upper Middle Class, Joint Family with Moderate Values.\nMy son is employed in government Sector as a doctor with a Bachelor's degree. He currently lives in Meerut. We belong to an upper middle class, nuclear family with moderate values.\nMyself, Deepak Siwach and I have completed my schooling from APS (Bareilly and Meerut). Presently , I'm working in a Public Sector Bank . I have done graduation (B.Com). I grew up in a nuclear family with Moderate values.\nMy name is Aman Chaudhary and I am currently Self Employed. I belong to a Upper Middle Class, Nuclear Family with Liberal Values. I have completed my B.Com..","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Cryptolia Limited has been deemed an exaggerated Scam. And we have the all the proof you need exposing why nobody should trust them. If you were thinking about joining Cryptolia, think again. This investment program is not what you think it is.\nIf you take a few minutes to see what Cryptolia has to offer, you can see they're offering a means for \"easy\" wealth. In essence, they're tricking clients a simple solution for making money through cryptocurrencies without doing any work.\nFor rookie investors, this idea sounds appealing since most people don't know much about the cryptocurrency market. However our investigation throughout their website shows many scamming factors commonly found within other ICO scams.\nSo before making the mistake in handing these cooks your money, read our review first. Learn the truth they don't want you knowing about their Cryptolia Scam.\nWhat are dealing with exactly? Cryptolia Limited represents itself as a form of crypto-based investment front. Meaning you as an investor, you \"invest\" into one of their plans. In return, you expect them to invest your money into various cryptos for a returned profit.\nSounds easy, however it isn't that simple\u2026.nor are they being honest.\nThe problem is their advertised results are extremely outrageous. Ranging 125% per day, or 10500% in 3 months (Depending on which plan you choose).\nBut lets be real & clarify for a moment these numbers are insane & unrealistic. Its virtually unethical to promise investors any fixed rate of income. This is far beyond what any legitimate company could ever offer. And improbable to deliver on a consistent basis as they 'promise'.\nSuch \"easy\" wealth claims presented by this fraud (and many scams like it) are known as 'door busters'. Used to catch your attention and fool you into believing you can become rich very quickly without any effort.\nIn these situations, ask yourself this simple question: \"if a company such as Cryptolia can really generate 'guaranteed' & consistent 125% daily payouts, you honesty believe there wouldn't be a hidden catch or motive\". Certainly not!\nSpeculations are one thing, but its vital we dig deeper into this corrupt Cryptolia scam. Let's review the specifics of their illegitimate methods for manipulating beginners.\nBecause Cryptolia Limited represents itself as an 'investment' firm, requiring members to invest \"x\" amounts of money for promised \"y\" returns, by law they must operate under the conditions of proper License and Regulation.\nWhile trying to validate their operations, our team simply could not verify any certificates confirming any licenses in which Soxasox should be mandated to hold.\nOn their platform, there's a section stating their 'licensed' UK Company. Unfortunately that doesn't solidify any LEGAL validations either.\nWe've confirmed through the FCA & the FTC Cryptolia Limited isn't licensed. Thereby they are trading and investing for its members illegally.\nI like calling these type of people \"faceless scammers\". Because in all honesty we have no idea who's really behind this Cryptolia investment app.\nIf you step back for a moment, you'll realize the entirety of Cryptolia.Net contains no useful directories. They're described as team of \"professionals\" with collaborative experience. But nothing which discloses any developers, founders or CEOs.\nAlmost every scam you can imagine either creates fake aliases or keep quiet altogether in efforts to maintain anonymous as these criminals profit from your losses. Cryptolia is just another vague scam hiding in the shadows to disguise their shady activities.\nTransparency is important when you're about to \"invest\" your money. Traders should fee comfortable & safe in their decisions. Yet the level of mysteriousness surrounding Cryptolia simply beckons more suspicions rather than approval.\nWhile we've found many ICOs a bit more subtle, advertising average returns of 2%-5% daily, Cryptolia takes it a step beyond with their own unproven claims.\nAs stated, they guarantee up to 10500% on your investment. So if you made a $1000 investment with their top plan, they're saying youll receive $105,000 in 3 months. Sounds impressive!\u2026..but highly unlikely.\nDon't forget the value of bitcoin, ethereum & other Altcoins are constantly moving up and down. Yet these scammers 'guarantee' consistent returns of high amounts. However these promises are unrealistic because of never-ending market fluctuations.\nIf you think about it, if their results were real, there would tons of wealthy investors all over the internet accredited to Cryptolia. But to this day we're trying to find any happy members pleased with their services. So far the number of people who joined are NOT satisfied.\nThere's more beyond phony claims and unregistered companies happening throughout the Cryptolia Limited Scam. Like most fraudulent investments, their main goal is to accumulate as much money from registrations possible, then run away with at your expense.\nCryptolia.Net also contains an affiliated program where traders can recruit other people to join. In return, you will receive a percentage from their deposits. Their system is set up with different entry levels and downlines. Basically sharing similar characteristics of a Pyramid\/Ponzi Scheme.\nIn reality, these crooks do NOT really invest in the crypto market. Instead they profit from your deposits, & promote their services in the form of affiliations to gather more victims. Allowing them to profit from your losses.\nWhat concerned us the most about Cryptolia was constant negative member feedback. The problem most people have faced vary from case to case. While some may have had a great experience, many more are facing serious issues.\nThese complaints revolve around the same group issues too. From denied payouts, unauthorized account activities, ignore withdrawals requests, and of course failure to produce their advertised results.\nRemember those crazy results they promised? Sadly members aren't even reaching close to the bottom scale of their profit range. But simply for the fact people who try to get their money back are faced with excuses from Cryptolia. This cannot stand.\nThese reasons alone was more than enough cause which motivated us to warn others about this damaging software. Hopefully today's Cryptolia Review will help prevent further losses from innocent investors.\nFinal Review Verdict: Here's what we have so far. Unlicensed Company, Unrealistic Results, and Negative comments from active members. Safe to assume this program cannot be trusted.\nWe thank our readers for taking time in reading our transparent & honest Cryptolia Review. Hope it helps others from falling for this shady scam. Please comment below by sharing any feedback, input and dealings you might have with this faulty ICO system.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"What a weekend we had! We started out on Friday with a big surprise---my sister's boyfriend Matt got her a puppy for her 21st birthday (which isn't until Feb 28)! \"Maggie May\" is a 13 week old miniature beagle and she is just as precious as she can be, or at least her pictures are! I only have some pics that MMM sent me from her phone so when we meet her in person this weekend I promise to take some with my camera so you can see the little darling!! Also on Friday, Forrest and I became members of the 9'o clock Dance Club. Another young couple from Gadsden, Sarah Kathryn and Mark Haller, also became members and we had a great time hanging out with them. This club is different from the others that we are in, in that it includes dinner. I thought it was SO much fun! Cocktails start at 7, and then everyone sits down to a yummy steak dinner at 8. It was nice to actually get to sit down and talk with people during dinner. It probably helped that we had a really fun table of people. We were seated with the Hallers, the Boyds, the Martins, and our neighbor Melissa, whose hubby was away hunting for one last weekend. We were laughing and cutting up all through dinner and just had a blast! At 9, us and the Hallers participated in the \"lead out\" of new members, which was fun and super low key. After that, it was time for dancing. The Gadsden State Show Band performed so it was more jazzy than \"dancy\" but that was nice because we were able to sit and talk with people while the band played. After the party we went to the Martin's house for a drink and to unwind before heading home and calling it a night. We were glad to get to know some nice new friends!!\nOn Saturday morning Forrest woke up early and headed to Birmingham for some lectures put on by UAB for their alumni weekend. He and his friend Richard had a great time and they even got to meet Dr. Dorfman, a celebrity dentist who was on \"The Dr's\" and \"Extreme Makeover\". Forrest really enjoyed his lecture! Forrest told me that last week Dr. Dorfman saw Eva Longoria, Usher, and Anthony Hopkins at his office. How cool!!\nWhile Forrest was learning more about dentistry, I was also in Birmingham NOT learning with my mom. We met my friend Randi for lunch at O'Carr's, my most favorite lunch place in Birmingham. Can you tell why??\nMom and I always get the fruit and cheese plate and a sandwich to split. Delicious!! After lunch we all headed over to Vestavia to see \"The King's Speech\". Mom and I developed a strong \"thing\" for Colin Firth years ago and there was no way we were going to let his Oscar worthy performance pass us by! It was a great movie and really made me interested in the Royal Family's business during WWII. When I think about leaders during WWII I think of Churchill. The King never really crossed my mind (probably because AL public school's don't bother to teach anything about him.) It all seemed very scandalous and soap opera like so I am definitely going to have to check into the history behind the movie!! When it was over I was very sad to say good bye to Randi! I get to see her lots during the fall, since she lives in Tuscaloosa and we are always down there for games, but it is harder during the spring. We have started meeting for lunch once a month to make sure we are caught up on each other's lives! Love you Randi!!!\nAfter the movie Mom and I went to my grandmother's house and Forrest and Davis (who had ridden to Bham with Forrest to mountain bike while Forrest was at the conference) were there too. While we picked up my grandmother and headed to Mass, Forrest and Davis got ready to meet Jonathan and Denise at J. Clydes for dinner. When we were at Supper Club a week or so ago, J and D mentioned that they wanted to go see Ira Glass in Birmingham at UAB. As luck would have it, Forrest and Davis already had tickets for the show, and even better than that, Forrest gets discounts tickets since he is a UAB student. The date was set after that! They all had a great time at the show and said Ira Glass was really humorous. Now we are itching to see David Sedaris later this spring! While they were at the show, I went back to Gadsden with my mom and then went to meet Farrah for dinner at El Tapatio. Other than the fact that we really wanted a girls' night, we needed to get together to discuss our table for the Girlfriend Gala later this spring. We had so much fun brainstorming! I can't wait to see our ideas come to life! We stayed at the restaurant for 2 1\/2 hours, just laughing and having a ball!\nOn Sunday, I didn't get out of bed until 1pm!! This was not because I wasn't awake, it was just because I had nothing better to do! I loved every minute of it! Forrest got up and worked out and even took Sonny on a run before I woke up, but once I was awake I wanted him to come hang with me! We ate lunch in bed and just lounged around until we couldn't lounge any more! When we decided we were being too lazy, I dropped Forrest off at a bike race that was going on near the park and I went to the office to work on a photo book that I got free from Shutterfly. We got back home and decided it was too pretty a day to stay indoors, so we took the dogs on a walk, something we haven't done in so long (well, I guess except our walk in the snow). The weather was great and it felt so nice to do something active outside for a change. Now we are all spoiled and have already started to discuss trips to the lake! Come on warm weather, head our way!!\nLater that night, we went to my mom and dad's for dinner to celebrate my cousin Sara's 16th birthday! We always call her \"Super Bowl\" Sara because she was born right in the middle of the ballgame! It was great to spend time with the \"other\" McCords, especially since we got to have cake and ice cream! :) She also told us that because she took and passed driver's ed, she doesn't even have to take the driving test! Now this is a new rule!! I remember driving out to take the test on the little course---SCARY! Now they have taken the fun out of it!! :) Apparently all you have to worry about on the day you get your license is what you are going to wear for the picture. I am not sure what I think about this new rule at all!!!\nAnyway, hope you all have a great week!\nThanks!! I will definitely have to check that out!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Sensitive skin is more prone to redness and irritation and needs more tender loving care than other skin types. Our Tamanu-Jasmine Soothing Moisturizer is a balancing blend of calming antioxidant herbs and oils ideally suited to nourish imbalanced skin, prevent aging effects and sensitive skin reactions, and maintain normal balanced moisture retention.\nNon-allergenic grapeseed oil provides the base for the OM PUR blend of healing, calming, regenerative and antioxidant herbs. Tamanu oil from Tahiti is featured for it's extraordinary ability to heal damaged skin, regenerate tissue and calm inflammation. Andiroba oil from the Amazon Rain Forest is included for its exceptional anti-inflammatory, antimicrobial and insect repelling properties. Schisandra extract has profound anti-inflammatory, antiirritant and antioxidant properties. The high level of antioxidant oils included provide natural sun protection and contain optimal nutrients to nourish sensitive and normal skin.\nThe natural aroma of jasmine essential oil blended with purifying and calming lavender and sandalwood oils provide the finishing touch to our soothing facial moisturizer.\nOM PUR's oil moisturizers are 100% pure and 100% active containing no 'filler' ingredients. Every drop is alive and highly energetic. Immediate and long term beneficial results are visibly noticed. A cream or lotion by contrast, has a large percentage of filler type ingredients, thereby limiting it's effectiveness compared with the 100% live ingredients in an active oil. It is not possible for a cream to be 100% pure or 100% active as it is with oil moisturizers.\nOther points that define the superior value of oil moisturizers are: (1) Oils do not require preservatives, cream and lotions do. (2) Oils can be cold-processed to maintain their full integrity and activity. Creams and Lotions must be prepared with high temperatures to create a bond between the water and oil ingredients. High temperatures during manufacturing diminishes or weakens any active ingredients included in the formulas.\nWarming up to using an oil as your moisturizer is easy when you understand the differences and experience the results. However, for cream users who find it difficult to make the switch, we have created a light cream as a transitional moisturizer that is made with only a small amount of wax. It has been our experience that when an individual who has always used a cream moisturizer switches to an OM PUR oil moisturizer for at least 2 weeks, the lighter feeling after the oil has penetrated, and the obvious results are so significantly different, they never want to go back to using a cream again.\nUse morning and night for best results. Apply after cleansing or exfoliating, or after misting or after applying the Pomegranate-Vitamin C Serum. For a lighter application, blend 2 pumps of oil with 2-4 pumps of Lavender-Neroli Toning Mist in your palm and apply together. The oil may also be blended with the serum.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagfzf b/data_all_eng_slimpj/shuffled/split2/finalzzzagfzf new file mode 100644 index 0000000000000000000000000000000000000000..86d64b98b92db09e5ea366238913d185571b5e4e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagfzf @@ -0,0 +1,5 @@ +{"text":"Last December, after Flash Professional CC was retired in favour of Animate CC, some tech publications proclaimed \"Flash Is Dead, Long Live Adobe\", or \"The Final Nail in the Flash Coffin\". The truth is more complex and nuanced. In fact, Animate CC is just the new name for one of the products in the Creative Cloud suite of products.\nThat said, it is a well-known fact that Flash has fallen out of favour with the emergence of new technologies and, to be honest, due to bad decisions on Adobe's part.\nFlash technology was born in 1995 in San Diego, California, when FutureWave enhanced its SmartSketch vector-based drawing software by adding frame-by-frame animation features and re-released it as FutureSplash Animator. The new software, available for Mac and PC, was a hit. In 1996, it was acquired by Macromedia, which, at the time, was well known for Director, its interactive CD-ROM development tool.\nFutureSplash was rebranded as Macromedia Flash. Flash was a two-part system: a graphics and animation editor, and a free player known as Macromedia Flash Player, which quickly became available as a free browser plug-in. Subsequent versions added programming capabilities with HyperTalk-based language (Apple HyperCard's scripting language).\nThanks to its browser plug-in status, Flash abetted countless crimes against Webanity over the last twenty years: sites resembling CD-ROM ports with esoteric, user-unfriendly ergonomics that broke all accessibility rules; \"splash\" pages that had us scrambling for the merciful \"Skip Intro\" button; unsolicited audio components; and invasive ads, just to name a few. But Flash also helped with uploading of videos on Web pages, besting QuickTime, Real, and Windows Media solutions, as well as developing an entire system of on-line games (Armor Games, for example, in 2005).\nMacromedia was acquired by Adobe in 2005 and Macromedia Flash was rebranded as Adobe Flash in 2007 with the release of the CS3 software suite. Flash inherited Adobe After Effects' features, offering new 3D capabilities. Adobe also released Adobe AIR (Adobe Integrated Runtime), a runtime engine for distributed autonomous applications that are developed with Adobe Flash Pro and can run outside a web browser.\nThe first sign of trouble for Flash on the Web appeared a dozen years ago when power users started deactivating the plug-in and using extensions to activate animations only on demand in order to spare themselves the worst of the ads and to save bandwidth (the \"ClickToFlash\" feature has since become standard in some browsers).\n\" tag, which broke the monopoly of Flash, as well as Steve Jobs' 2010 decision to not include the reader in new mobile devices (iPad and iPhone).\nUnder assault on all sides, including and especially on the security front, Flash is losing ground to Web standardisation solutions (HTML5 Canvas). In early 2015, after extensive tests, YouTube switched its default video player to HTML5, sending shockwaves through the industry. Now, navigators such as Safari on Mac OS no longer offer the Flash plug-in for their standard installation.\nIf anything is dying, it's the Web extension, which has morphed into a vector for malware, though not for the animation application. Adobe decided to rebrand its application in order to distance it from an obsolete technology. Though Flash animations on the Web are a thing of the past, the animation solution is still alive and kicking under the name Animate, with no connection to any particular platform. In any case, Flash had become one format among many for sharing animations, Adobe having made considerable efforts with exporting HTML5 and WebGL. And Animate still makes it possible to create autonomous applications with AIR, which is still an important market, especially for mobile applications.\nIn the end, the field that has the most to lose from the death of the Flash reader is probably gaming, with the standard solutions, and especially their browser implementation, still lagging behind what could be done with Flash.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The PowerLine\u00ae EC Series of In-Line centrifugal fans incorporate the latest state-of-the-art, energy saving EC motor technology and are most efficient where conditions vary during the course of the day. They feature fully integrated, infinitely variable speed control which eliminates the need for external VSDs, current overloads and motor phase protection.\nOptional matching sensors monitor the ambient conditions and provide real time feedback to the fan. The fan's on-board microprocessor adjusts the speed and therefore modulates the ventilation rate to match the specific requirements of the area. The PowerLine\u00ae EC fans are a simple \"plug and play\" system which means installers do not need to have specialised control programming knowledge.\nThey feature easy to fit 35mm TDF profile flange connections, a robust yet lightweight galvanised steel construction and are available in 7 sizes ranging from 315 to 630mm diameter.\nCommercial and industrial supply or exhaust applications such as shopping centres, office buildings, exhibition centres, hotels, health centres, schools and universities.\nAll models supplied standard with 0-10V control input. Diameter sizes from 450mm and above can be pre-configured to suit specific sensors and specific applications.\nRobust, yet lightweight galvanised steel construction.\nCentrifugal fans suit applications where medium to high air pressure is required.\nEasy to fit 35mm TDF profile flange connections.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"From Donvale to the foothills of the Dandenong's, Hoskins Wallace P\/L strives to offer a premium personalised Real Estate service. Whether buying, selling or leasing our dedicated staff of experienced people are always there to assist you.\nAs a strong independent Real Estate company, with two prominently located offices in Donvale and Croydon, Hoskins Wallace P\/L has earnt an enviable reputation of integrity in the industry for almost 50 years.\nEstablished in 1960, when Springvale Road was just a dirt track amongst the orchards, the company grew as the new homes sprouted. In 1984, John Hoskins joined the company and now as Director, leads a quality team of highly respected and professional salespeople and support staff.\nHe chose Maroondah Highway, Croydon as a natural expansion of the business to service his ever-increasing client base.\nA Premium Real Estate Experience through Personal Service, the latest technology and the highest integrity!\nDonvale Office: 67 Mitcham Rd, Donvale, Vic 3111.\nCroydon Office: 142 Maroondah Hwy, Croydon, Vic 3136.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"El\u00eassar obtained Lord Waycrest's Signet.\nRanch\u00ec obtained Lord Waycrest's Signet.\nAvarispala earned the achievement Level 120 for 10 points.\nFrostxtq obtained Port Pillager's Belt.\nFrostty obtained Petticoat of the Self-Stylized Azerite Baron.\nFrostty purchased item Relinquished Azerite Chestpiece.\nDrunkentank obtained Leggings of the Aberrant Tidesage.\nFrostty obtained Ignition Mage's Fuse.\nFrostxtq obtained Overseer's Lost Seal.\nMatux obtained Hyo'gi Basket Binders.\nLinlanta obtained Vessel of Skittering Shadows.\nEl\u00eassar obtained Reins of the Drake of the South Wind.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Total weight gain: Around 5--another check up tomorrow.\nSleep: So much better. Still limited to only my sides, which stinks, but I get a good quality night of sleep!\nBest moment this week: Hitting 70+ degrees finally! So nice!\nMiss Anything? Being able to reach down comfortably to tie shoes, paint nails, shave, etc.\nFood cravings: Sweets--Baklava....Lemon Bars....Ice cream...I'll take it all!\nAs her digestive system preps for the outside world, she's manufacturing meconium -- the tarry black substance you'll find in her first dirty diaper (ew)!\nIf it's a girl, she's already got a lifetime's supply of eggs in her womb -- about six million of them!\nSymptoms: Tired. More out of breath. Breathing takes a little more labor and my stomach can't hold as much food!","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagmrm b/data_all_eng_slimpj/shuffled/split2/finalzzzagmrm new file mode 100644 index 0000000000000000000000000000000000000000..d83181ac613a860a4c617b8701786f375348acda --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagmrm @@ -0,0 +1,5 @@ +{"text":"How many of you have tired yourself to find real information about HGH? I hope almost each one of you and that is probably one of the reasons that you are on this page\u2026 Even though we see that a lot of people these days have become conscious about their health and wish to know about HGH, it won't be wrong to say that the reasons behind their inquiry is different.\nWhile some of you may be looking for HGH related information to lose that extra fat or get shiny, healthy hair, others may be interested in having a strong immune system and metabolism. Anyways, whatever may be the reason behind your inquisition about HGH, you all would be happy to know that the hormone is really beneficial for overall health and life.\nFirst let us understand the main concept of HGH and then move forward to check out the Best Human Growth Hormone Supplements available. This knowledge will help you to get complete information about the best HGH releasers that you can choose from.\nHuman growth hormone serves as the FOUNTAIN OF YOUTH and plays a very important role in every aspect of our lives. The hormone is released or secreted by the brain's pituitary gland and is responsible for the growth of one's body. This production of the hormone gets reduced when we get older and this results in facing many health problems. Some of the problems related with less production of HGH include wrinkles, reduced energy levels, decreased sexual desire and more.\nSo, when you start facing such problems, it is best to head over to some natural way out. This comes in the garb of HGH Pills.\nIt is indeed a question that crops up in our minds every now and then. But have you ever thought of it? Certainly not! Or maybe yes! Well, let us understand what these supplements are. HGH supplements or what are also called as the HGH releasers are the dietary supplements that are made of natural, safe and effective ingredients. These ingredients then in turn trigger or stimulate the production of growth hormone in our pituitary gland.\nBut a word of caution: ALL Human Growth Hormone Supplements SUPPLEMENTS ARE NOT THE SAME. Yes! You may think that the ingredients in these releasers are somewhat same and so the results will be similar. You are wrong. You need to understand the fact that the releasers that are filled with artificial or synthetic hormone are not the real products.\nWhich HGH supplements to opt for?\nThus, you need to understand which type of HGH Releases to go with if you really want to enjoy their benefits. You need to ensure that you get quality, authentic, genuine and all safe supplements only. But how to find out that a particular supplement is genuine or not? A nice thought indeed!\nYou can search for the customer reviews, feedback and testimonials of the supplements and then make out which ones are the best. You need to go through the detailed and unbiased reviews and check for yourself the releasers that are liked by the customers. This way, you will be able to make a nice choice and take the best decision possible.\nWhere to Buy HGH safely?\nMost important question comes in mind is where can I buy HGH supplements? So you have scrutinized each and every product in the market and found out which one to go with, you can take the online way because effective HGH for Sale options are mostly available over the Internet. You can straightway go to the manufacturer website to buy HGH supplements.\nWith this, you will be certain of getting the real product and with all benefits offered by the manufacturer. You will be amazed to know that the top HGH releasers are available in different packs to make your purchase easier.\nSo, what are the best HGH releasers that we recommend? Here are the top 3 releasers that we have shortlisted.\nGenF20 Plus has become the Number 1 HGH supplement in the market now. Available with 67 days money back guarantee and different monthly packages, you can easily Buy GenF20 Plus from its official website. The benefits offered by the releaser are many like improved memory, mood, sleep, libido, better metabolism, increased energy and more. The user reviews of this product have been positive. If you feel giving it a try, buy from the official site only.\nOfficial Website of GenF20 Plus is the best place to buy GenF20Plus. If you are planning to buy it, must read GenF20 Plus Review to know all about this HGH Supplement.\nThe HGH supplement is very useful and effective, specially for bodybuilding. With positive customer experiences, you can give HyperGH 14X a try. Made with natural and safe ingredients, it has become a much liked product in the HGH market. Buy HyperGH 14X from its company's main site and enjoy the varied benefits that the product has in store. As the product is free from any ill effects and negative feedback, you can try it for your own benefit.\nYou can Buy it from OFFICIAL WEBSITE of HyperGH 14X. You must read HyperGH 14X Review before purchasing it.\nThe oral supplement has gained much popularity all around. Providing a large number of advantages to the users, Sytropin has become a much preferred HGH releaser. With its Money back guarantee scheme, people find it easy to try this product and see if it is of any real worth. No side effects have been reported so far with Sytropin.\nYou can buy Sytropin from the official site and have a great time using the safe HGH releaser. You can even go through an unbiased Sytropin review before making the choice.\nSo, with all these HGH supplements in the market, you can certainly say hello to good health and quality life. My final suggestion would be, choose the HGH supplement as per your need, if you are facing aging related issues, mantel illness, laziness etc, GenF20 Plus would be the Best pick, however if you are looking for HGH for Bodybuilding, there is no other option than HyperGH 14x. On other hand Sytropin is bit cheaper than the above 2 so if you are low on budget, Sytropin can still do miracle for you.\nHGH-X2 by Crazybulk Review-Is it the Best Legal HGH Supplement?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"When you notice an email discussing a review, could you jot down the email number, date, the reviewer, the reviewee. Please don't editorialize, just note it for reference.\nAs noted previously, the Climategate letters and documents show Jones and the Team using the peer review process to prevent publication of adverse papers, while giving softball reviews to friends and associates in situations fraught with conflict of interest. Today I'll report on the spectacle of Jones reviewing a submission by Mann et al.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"B Corps are a global movement of more than 2,700 companies in 60 countries--like Patagonia, Ben & Jerry's, Kickstarter, Danone North America, and Eileen Fisher--that are using the power of business as a force for good. B Corps have been certified to have met rigorous standards of social and environmental performance, accountability, and transparency. This book is the authoritative guide to the what, why, and how of B Corp certification.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"With a deadline for a column looming, Samantha had writer's block.\n\"See you tomorrow.\" she said to her roommate Jack. Grabbing her loaded framepack, she pulled her blonde ponytail through the back of her ball cap.\n\"Have fun and be careful.\" Jack tossed her cell phone to her.\n\"Don't worry!\" She slipped the phone into her jacket pocket and left the apartment.\nShe threw her pack in the back of her black Toyota and started the hour-long trip into the mountains. As she drove higher in elevation, the road narrowed. After a series of hairpin turns it ended altogether.\nSamantha pulled over next to a grove of Aspens. She closed her eyes and inhaled the fresh pine fragrance.\nThrowing the bulging pack over her shoulders, she started up the trail. Shadows crept across the path.\nHalfway to her destination, something streaked through the understory. Knowing that mountain lions stalked this area, she continued on edge.\nAfter a steep incline, she found herself overlooking a placid mountain lake fed by the glaciers farther up the mountain pass.\nShe unstrapped the tent and she assembled it in the fading light. Thunderclouds gathered in the distance.\nAs she scouted around for kindling, an animal broke out into the open. Samantha breathed a sigh of relief. It was a mutt.\n\"Here boy!\" It came closer, but shied away from her outstretched hand.\nShe gave up with the dog and continued to search for dry wood. After amassing a pile, she lit the leaves and twigs. At first the fire sputtered, then it crackled, warming her cold fingers.\nAfter filtering water from the icy lake, she boiled it over the fire. Then she poured it into her dehydrated dinner and made green tea. As lightning flashed in the distance, she swung around and scanned her surroundings now engulfed in darkness. Samantha heard rustling in the underbrush, but there was no sign of the dog.\nLightning flickered across the sky followed by a low rumble. A few rain drops fell. She gathered up her pack and entered the tent as the sky opened up. Samantha peeked out and saw her fire extinguished and smoldering.\nShe snuggled down into her sleeping bag and was almost asleep when a twig snapped. Samantha pulled on her jacket and hiking boots and turned on her flashlight. She ducked out of the tent and stood up in the swirling drizzle.\nWhile walking away from the tent, she heard something behind her. She spun on her heels and the beacon from her flashlight showed pine branches swaying in the breeze.\nThe wind picked up and began to howl. She headed towards the lake.\nThen she heard the crunching of rocks on the path. She never had time to turn around.\nSamantha screamed as she was thrown to the ground.\n\"Hey pretty girl. Let's have some fun!\" the man whispered in her ear.\nSamantha kicked, clawed, and finally twisted away from him, but slipped in the mud and fell. He pounced on top of her. She could smell the alcohol on his breath and the stench of an unbathed body.\nHe struggled to flip her over.\nShe felt around for a weapon. Her fingers clawed the ground, breaking nails and ripping her flesh. She found a rock, spun around and smacked him in the head.\nShe struggled to her feet. While navigating the rugged terrain in pitch darkness, she stumbled back towards the trail.\nHe grabbed her ponytail and pitched her to the ground again.\nSamantha hit her head against a boulder. She felt searing pain and the warmth of blood as it trickled down her forehead. Trembling, she turned towards her attacker. A fair-skinned man with blood-shot eyes laughed with a mouth full of rotting teeth.\n\"No!\" she moaned. Tears slid down Samantha's cheeks.\nHe reached out and touched her face with a grimy finger.\n\"Be a good little girl.\" The man straddled her, when out of nowhere the mutt jumped through the air and knocked him over.\nWithout looking back, Samantha forced herself to get up and run. She heard the vagrant scream and the dog growl behind her. Woozy and with wobbly legs, she plowed ahead and found the trail.\nShe ran as far as she could and then slowed. While shivering, she pulled her cell phone out of her zipper pocket and dialed. No signal.\nHer head throbbed. She reached up and felt the gaping wound. My wallet and keys are still in the backpack.\nAfter what felt like hours, she made out the familiar lines of her truck. She kept a spare key under the carriage. While feeling around for the metal case, she heard the sound of footsteps on the path.\nShe found it and pulled out the key.\nHer heart banged away while fumbling with the lock. She finally slid it in, opened the truck door, and slammed it. Samantha hit the door locks just as the vagrant tried the door.\nHis face was inches from hers through the glass.\nShe turned on the engine, jammed the truck into drive, and careened away.\nSamantha began the first hairpin turn. With blurry eyes, she struggled to keep the car on the road.\nAs she approached the straight-away, the vagrant ran out in front of her. She screamed and hit him head on. He rolled off the hood. In her rear view mirror, she saw him get back on his feet and limp off into the forest.\nSamantha shook while driving back to her apartment. She dragged herself out of the truck and climbed the stairs with the rhythm of her heart clanging in her ears. She knocked on the door.\n\"I got my story.\" She collapsed into his arms and passed out.\nIn the post Striking Gold \u2013 A Photo Essay, I mentioned that I'd experiencing deja-vu. I wrote The Shattering before knowing a place like it existed. Talk about shivers!\nReally good, though I have to question Samantha's sanity at going camping in an isolated area all by herself.\nI know women like that here in Boulder. They grew up with it. In fact one of my friends grew up in Aspen where they had each kid winter camp by themselves for a night! She treks all over the world now! I like to have my husband with me and my mighty dog (15 lb.) Roxy! Hahaha!\nWhoa. Probably should have read this right before bed. I am officially creeped out. Great storytelling, Susie.\nThanks Darla! I am sorry to say that I am so glad! Hahaha! Or should I say Muahahaha!\nVery very creepy stuff Susie. I WAS contemplating a camping trip, but I'm thinking a hotel and a nice hike might be a better idea now.\nThere you go! Although I would doubt that you would have any problems out there unless\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\nThat's it! Staycation it is\u2026.\nHahaha! Sorry I messed up your tranquil moment! You know me and my twisted tales. Yep. She bumped him pretty good, but like so many predators, he survived\u2026.\nThanks so much for reading Barbara!\nThat's crazy! Camping out in the wilderness by one's self. *shakes head* This piece is a nice argument for safety in camping. And like Tameri, I worry about her keys and address. How does the \"ever after\" go?\nThanks for reading it Kristi!\nIt feels odd to ask if we like to camp after you just finished telling that story. I'm not sure I want to go camping ever again.\nHahaha! My point exactly. Do they have campgrounds in Shanghai?\nI had to ask about camping afterward for kicks and mostly giggles. I would love to camp up there, but will definitely get the willies!\nOohhh! You were so brave. Sounds like some great book and blog fodder!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Melt the syrup, butter and brown sugar over a lowish heat in a saucepan.\nAdd the vanilla, stir, then take off the heat and let it stand for 10 minutes.\nTake the pastry-lined flan dish out of the freezer, and arrange the nuts on it.\nWhisk the eggs into the slightly cooled sugary syrup until it looks like a caramel mixture, then pour it over the nuts.\nBake in the oven for 40 minutes, or until the filling has set and the pastry is golden.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagxij b/data_all_eng_slimpj/shuffled/split2/finalzzzagxij new file mode 100644 index 0000000000000000000000000000000000000000..a4ea0afc0a999c6f6b86cae86417a53e0fc72ead --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagxij @@ -0,0 +1,5 @@ +{"text":"This home is perfect! The 4 bedrooms \/ 2 bathroom split floor plan offers many possibilities! There are 2 bedrooms and a full bath in the back of the house. There is even a separate entrance to the back of the house is preferred. Upon entering you will be in the living room with a cozy fireplace. The dining room has the space for a large dining room table. The kitchen is has was previously remodeled with cabinets, countertops, microwave, flat top electric stove, and dishwasher. Condenser units to both AC systems have been replaced. With all of the upgrades done the original charm still remains within the home. The welcoming porch and original trim around the doors and windows have withstood the test of time. There is a guest room attached to the home that has the washer and dryer hook up. It could be a home office!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"B&Q Online is more than just another UK DIY store - it's a completely unique DIY shopping experience!\nClick here to go directly to the B&Q web site.\nFocus DIY Store, UK: The Focus DIY website also provides a wide range of UK DIY, but a wider range of DIY products is usually available at the B&Q online store.\nHomebase DIY, UK: Homebase DIY store is operated by GUS, the same company as operates Argos, not to be confused with Home Depot, the American DIY store.\nKays Catalogue Store - Top quality products, also including Belts, Cufflinks, Socks, Underwear and More!\nBuy affordable fashions online at Savile Row - shirts, suits, ties and other luxury UK clothing at highly competitive prices.\nHarry Potter fans will also enjoy browsing the Harry Potter and The Deathly Hallows UK store.\nSelect the following B&Q link to browse or buy DIY at the B&Q UK Website.\nGreat deals on toys, home-wares and more, from Woolworths UK.\nB&Q Discount Coupons: Catalogue Shopping, Buy T-Mobile at CarphoneWarehouse, AA Motorcyle Insurance Plan, Jos. A. Bank Online Clothes Store, PC World UK, Chapters-Indigo, Canada, John Lewis, AA Home Insurance Quotes B&Q DIY .","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We are so pleased to tell you that Broadreach Nature + are joining us at Paws in the park 2016.\nBroadreach Nature + are kindly holding a stand, sponsoring the smiliest face competition and adding to the goody bag. They do food and health supplements for any life stage of dogs and cats. Grain free and hypo-allergenic foods available. The company founder was motivated by satisfying the needs of her own much loved dogs. Such a great reason to start a company and make sure the product is right!\nWe look forward to seeing everyone at Braybrook Park, 11am 20th Aug. The support for this show is immense. Its going to be an amazing day. Thank you to all our supporters!\nThis entry was posted in News and tagged Broadreach, dogs, nature, Paws, suppliments by Donna. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"WHY RENT WHEN YOU CAN OWN AND LIVE IN THIS LUXURY ESTATE THIS SEASON!!!\nCITIVIEW ESTATES is a 4 Bedroom Detached Duplex with Penthouse fully completed internally and externally at Arepo, 6 minutes from Alausa in Ikeja with a unique Infrastructures of underground electricity cables, streetlights on every street, interlocked\/tarred road on every street, Walkways, Network of durable covered drainages, recreational areas, Green areas, CCTV, Perimeter fence, Gate house etc.\nIt is an already built Duplex with an interior and exterior fully completed and painted for you to pay and move in and it is is a home of hundreds of residents.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"In the bowling shot, the moment you release the ball is crucial. This is because the smallest of movements of your hands or wrists can have a major effect on the trajectory of the ball. It often means the difference between a gutter and a strike! This article will teach you what you need to know about releasing a bowling ball.\nIt's a common for bowlers to think about the release as a stand-alone motion, but this is a mistake. Instead, you should understand that a bowling release is closely connected to the rest of your stroke. From approach to delivery, the bowling swing needs to be one smooth, unified motion.\nWhen making your approach (which means the steps you take up to the foul line) you want to swing your arm like a pendulum straight back and forth, not to either side.\nOnce you make your last step and start to slide with your foot, the ball should be at its lowest point near your leg. Make sure to keep your toes pointed straight ahead towards the target spot.\nAt this point, it's time to actually release the ball.\nThe first step is to remove your thumb from its hole. Just about a split-second later, you release your middle and ring fingers.\nYou must be sure to keep your wrist and arm straight when you release the ball, they should not move to either side. Also be sure to keep your shoulders aimed straight ahead.\nThe swing isn't done when our fingers come out, though, as the follow-through is also very important. Once you have sent the ball down the lane, stay in balance and let your arm continue to swing. It should finish aimed directly at your target, with your shoulder remaining square.\nIt's crucial to understand that you need to release the bowling ball at the correct moment during your armswing. Many common problems are because of this, such as dropping the ball too early which results in a lack of control and power.\nAlthough the techniques of a smooth stroke and proper timing are the same no matter what bowling style you use, a hook bowler will have a slightly different motion when releasing the ball.\nAfter removing the thumb from the thumb-hole, a hook bowler should rotate his or her forearm slightly, instead of keeping it completely straight.\nWhen removing the middle and ring fingers from the ball, you should lift them up and to the side of the bowling ball. In a hook bowling follow through, you should look like you are giving someone a handshake.\nThis motion should be counter-clockwise for right-handers and clockwise for lefties. This lift action puts the right type of spin on the ball so that it travels down the lane straight at first before curving inside to hit the pocket.\nAs with any aspect of bowling, practice is key to success. If you work on making your bowling release as smooth as can be, your scores can really start to improve.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaipme b/data_all_eng_slimpj/shuffled/split2/finalzzzaipme new file mode 100644 index 0000000000000000000000000000000000000000..23118fd438cf6b02a37c20f345cf0537c0f6dad7 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaipme @@ -0,0 +1,5 @@ +{"text":"The Care Provider Alliance has reported that providers in all counties are being asked to produce documentation on their contingency planning for Brexit. They have developed a template which may be of assistance to providers.\nHMRC's Social Care Compliance Scheme Employer Guide September 2018\u2026A Little Clarity\u2026Finally!!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Take a look at some of the members rides, Fishboy and some others' cars have what you may be looking for.\nHey jonboy, some black slipstreams would look nice on your car! They look just like the spoon rims, lightweight and great quality!\nOnce I get a pic I'll post mine.\nPlease do so. I think the Absolute would look nice on the new si.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"To apply for CCRF, the borrower must either be a Corporation, Limited Liability Company or Limited Liability Partnership (not the same as a Limited Partnership) formed and operating in the USA.\nThe borrower must have been in business for a minimum of 12 months and have minimum of 6 month credit card processing history.\nOwners must have a social security number. If the owner is not a US citizen, we must see the copy of the owner's permanent resident alien card (a.k.a., Green card) or Visa (in the passport) & I-94 page on his\/her passport.\nWe cannot make personal loans under any circumstances.\nEating\/drinking establishments, i.e. restaurants, bars, taverns, etc, are the best candidates for CCRF because; 1) 60 - 70% of their revenue is generated via VS\/MC, 2) Traditional lending sources do not like the restaurant industry, 3) their gross margin is usually high enough to absorb the daily payment.\nRetail, automotive repair, full service spas, dentists, cosmetic surgeons are also a good source of new deals.\nWe consider internet merchants, furniture stores & gas stations a high risk.\nWe currently cannot work with adult industry businesses.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Thomas and Sa\u00efd go to Assisi, in Umbria. Since the twelfth century, this city is marked by the spirit of one man and one woman who instilled an experience of peace on a universal scale: Francis and Clare from Assisi.\nFor about thirty years international and inter-religious meetings for peace are held here on a regular basis, gathering religious leaders from all over the world.\nIn October 1986, Pope John Paul II held the \"World Day of Prayer for Peace of Assisi\", the first time such an event had ever occurred. Major religious leaders, such as the Dalai Lama and the Archbishop of Canterbury joined with the Pope to call for peace, unity and interreligious understanding.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Because this is Fire Emblem everything has a strength and a weakness. Weapons and magic do not escape this rule.\nWith weapons it's rather simple, all weapons fall into three main categories; Sword, Lance, and Axe. Knives fall under Sword, and Staffs do not appear on this triangle... circle... thing.\nSword is strong against axe and weak against lance.\nLance is strong against sword and weak against axe.\nAxe is strong against lance and weak against sword.\nWith magic it is the same way, where all magic is put into three categories with the exception of healing magic which is not present.\nAnima magic is strong against light magic and weak against dark magic.\nLight magic is strong against dark magic and weak against anima magic.\nDark magic is strong against anima magic and weak against light magic.\nOn a special note, Anima magic means all the elemental magic: Fire, Lightning, Wind, and Ice.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajndz b/data_all_eng_slimpj/shuffled/split2/finalzzzajndz new file mode 100644 index 0000000000000000000000000000000000000000..6485fcaef31d3c18d874f0d12a26a3174043c9a3 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajndz @@ -0,0 +1,5 @@ +{"text":"Halong Bay Cruise Dragon Legend is the best way to get some insight of Vietnamese culture. This is the time for you to enjoy and learn awesome things about Vietnam. Dragon Legend Cruise designed in a way that satisfies the expectation of tourists while portraying Vietnamese culture! With amazing Hung King dynasty decorative features and handmade cultural products. It gives guest the feel of cultural experience and distinctive stay in this South East Asian country.\nSet sailing \"off the beaten track\" of Halong Bay with Dragon Legend cruise now. You will get free round transfer & kayak!\n2. Any person who has symptoms of hallucinations, sleepwalking, fear of being enclosed, epilepsy and so forth.\nHave a unique experience in the Natural Wonder of the World. This is the best for discover the natural beauty and local life on a limited time schedule. You will get to visit an enchanting cave, go kayaking through a magnificent limestone islands.\nSpend 1 night on Dragon Legend that takes you deep into natural wonder to experience a memorable experience and relaxing on the majestic bay.\nTravel out Hanoi city to Halong Bay and get 3,5 hours on car to reach Hon Gai harbor. The cruise offers luxury shuttle bus and pick up in Hanoi Old Quarter (optional: arrange a private car).\nReach Hon Gai harbor at noon time. Check-in, enjoy the welcome drink & cruise briefing.\nHave lunch in the outdoor dining room (if the weather is permitted) while the cruise sails to the magnificent Bai Tu Long Bay, admire the fascinating formations on the way.\nReturn the boat and have yourself fresh or indoor Mineral Seawater Pool, Fitness Room or enjoying the spa and massage services while cruising to Cap La area. Have dinner at restaurant.\nFree time for relaxing, communicating with the staffs, evening activities (watching movie listening to music, squid fishing\u2026). Overnight on board.\nIf you travel by a private car will not stop at Yen Duc village and no enjoy water puppet show.\nMore chance to enjoy the sunrise on bay with majestic scene. Start your day with a cup of tea or coffee. Have breakfast in the fresh air.\nThen you start cruising towards the harbor. Enjoy the scenic on the way. Checkout cabin and pack your luggage.\nDriver welcomes at the harbor and transfers back Hanoi city. If you travel on shuttle bus you will be driven to Yen Duc village in Dong Trieu province for a short break in this typical Vietnamese agricultural village which is rich in culture and traditions.\nEnjoy water puppet show performed by local artists. You arrive in Hanoi at 17.00 pm. Say goodbye after enjoying a wonderful time with us. Wishing you to have a wonderful Halong bay 1 night tour. We hope have chance to serve you again in the near future. Happy Travel!\nDragon Legend offers the luxurious journey to Bai Tu Long Bay and get the untouched area in the bay. It brings you to the natural beauties and experience local life at floating village. Join different activities of kayaking, visiting cave. Especially you will have chance to enjoy BBQ lunch on the beach.\nLet to take the deep natural beauties and have a memorial experience on the majestic bay of Vietnam and the world.\nWake up early and start a new day with Tai Chi on the sundeck. Have a morning view of Bay's landscape with a cup of tea or coffee.\nHave breakfast in the fresh air while cruising to Vung Vieng Fishing Village and visit Vung Vieng fishing village by a rustic row-boat.\nAfter that you continue to viisit Thien Canh Son cave. Having a BBQ lunch on the beach.\nAfternoon you return the boat and start cruising to Hai Quan area, kayak to explore the mystery Karst Mountain and get more the beauties of the bay. Having dinner on board.\nFree time for communicating with staffs, evening activities (squid fishing, beauty spa\u2026) and service on board. Overnight on board.\nEnjoy water puppet show performed by local artists. You arrive in Hanoi at 17.00 pm. Say goodbye after enjoying a wonderful time with us and wishing you to have a lovely holiday in in our enchanting country.\nYou do not like to share your space with other? So Halong Bay 3 days tour private is the best choice.\nWe are a young professional Asian American couple and enjoyed this trip very much. We chose the 2 day\/1 night package with the \"luxury\" van transport. This is the first Halong Bay cruise that we have taken so are unable to compare to others. The Ship itself is very large so even though there were 35+ passengers, it never felt crowded. There was always enough seating in the dining area as well as on the sun deck.\nThe Luxury van ride so nice since. It's only $36 a person, had wifi with complimentary water. Our cabin was amazing and large. It definitely felt very Luxurious. It included a large tub and a separate shower. The bathroom had large windows which allowed you to take in the scenery while relaxing in the tub. The bedroom area was very large as well and also included a sitting area. The bedroom\/sitting room area also had large windows (that opened) for you to relax and take in the scenery.\nThe Staff was very friendly and always made an effort to communicate and made sure that we were happy with everything. The food was pretty good and abundant for what you can get on a cruise. Overall - this might be a little pricier than the other cruises out there. But there really is nothing that I could complain about. We went during a time when it was raining however, the scenery is beautiful even during less ideal weather.\nFantastic Cruise in Halong Bay. Well worth it. The ship was clean and new and the staff was wonderful. Very friendly and helpful. Tom was our tour guide for the cruise who was great. The staff truly cares to know all individuals on board. I would recommend Dragon Legend Cruise to all my old and new friends.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Adobe Cares: Now experience the same excellence in Adobe customer support that you have come to expect of our tools and technology for over two decades!\nWith a reputation for excellence and a portfolio of many of the most respected and recognizable software brands, Adobe is committed to continually drive innovation while helping you to be more productive and offer great 'end-to-end' user experiences for our tools and technology.\nAnd this philosophy of commitment to excellence and innovation not only touches our tools and technology, but infiltrates deeper in the way we do business\u2026and not least in the way we engage with our valued customers while offering product support. And not only when someone has purchased our products, but also during the 30-day trial period when evaluating our products.\nWhenever you might require assistance and guidance from Adobe while using our tools and technology, we are eager and well-prepared to be at your service; and that is Adobe's Customer Care! We believe in standing behind our products with a \"best-in-class\" support platform that we have invested in upfront, so that we can leverage our multiple delivery channels to ensure high touch, high quality service.\nWe are fully aware that support experiences in the past have not always been up to the expectations of our valued customers, and also up to Adobe's own high standards. This could be attributed to various external factors, and we have analyzed the reasons in detail to take comprehensive corrective action. To overcome this 'perceivable gap' in the support organization, we've now moved away from the earlier vendor-based model to a more holistic \"Adobe\"-centric model by bringing our product support completely \"in-house\" for the Adobe Print and Publishing Business Unit, which Technical Communications is a part of. And we have invested heavily putting together a world-class support infrastructure that can serve your needs in the best possible way.\nWe've recently gone live with exceptionally talented support staff, strong processes, seasoned leadership and state-of-the-art technology infrastructure\u2026to overcome the support challenges, and enable our customers to leverage Adobe support in getting issues resolved faster and smarter. The dedicated team of highly responsive support professionals have been handpicked by the Adobe leadership, and trained extensively to give you the best possible support experience. We are cognizant of the fact that building a world-class support organization from scratch is indeed a challenge and there will be room for continuous improvement. But we are committed and determined to offer you the best, and we look up to you for your candid feedback on how we can get better.\nWe truly appreciate your time and consideration, and reaffirm our commitment to offering you the best end-to-end product experience to boost your productivity and contribute to your success!\nShare the post \"Adobe Cares: Experience Excellence in Adobe Customer Support\"","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"For better output effect, you can adjust video brightness, contrast, saturation, volume and hue. Check \"Deinterlacing\" can help you get the advanced video.\nClip the video length from the source DVD with the \"Trim\" function. And you can crop the video frame to change the video play screen.\n4Videosoft Mac DVD to iPad 3 Converter 5.0 Video Converters software developed by 4Videosoft Studio. The license of this video converters software is shareware$, the price is 29.95, you can free download and get a free trial before you buy a registration or license. Do not use illegal warez version, crack, serial numbers, registration codes, pirate key for this video converters software 4Videosoft Mac DVD to iPad 3 Converter. If you want to get a full and unlimited version of 4Videosoft Mac DVD to iPad 3 Converter, you should buy from original publisher 4Videosoft Studio.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"I have just had a rather disturbing thought about the weather, or its function in human communication.\nWhat if the reason why people blither on meaninglessly about the weather is not, as I had generally assumed, a sort of mutual grooming by which they indicate that they are high enough in each others' regard that they're willing to coexist, and even communicate meaningfully when there is something worth saying?\nBy anarag, Fri, 2016-03-04 18:10.\n\"But it is important to say we simply don't have any evidence in this paper to suggest that any carbon coming from these seeps is entering the atmosphere.\"\nThis quote comes from Professor Adam Skarke of Mississippi State University. Context: this is an article about recent research into releases of methane from clathrates in the seabed off the American coastline. As we know, a warming ocean is highly likely to lead to a greater trend of release of methane than has hitherto been the case, because the equilibrium level of clathrate formation and melting changes with temperature. The released methane mostly is oxidised in the sea, adding a 2:1 molecular ratio of water and dissolved carbon dioxide to it.\nBy anarag, Thu, 2014-08-28 12:59.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"All materials for nominations and submissions must be uploaded online in one complete package. Any incomplete documentation will not be accepted.\nProjects that have previously been recognized by ACSA will not be considered.\nIn support of the American Institute of Architecture Students' (AIAS) resolution on unpaid interns, ACSA does not allow partners of firms who do not pay their interns, in accordance with all applicable laws, to submit work for ACSA publications or to receive ACSA awards. (Intern refers to those individuals participating in IDP or equivalent required training and includes working students.) Nomination or submission to the ACSA Awards Program constitutes your understanding of this agreement.\nCurrent ACSA Board of Directors members may not submit nominating or supporting letters.\nAward winners are expected to register and attend the Awards Session(s) at the 106th ACSA Annual Meeting in Denver, CO.\nBy submitting your project, you certify that you have granted ACSA permission to use all graphics included.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalagk b/data_all_eng_slimpj/shuffled/split2/finalzzzalagk new file mode 100644 index 0000000000000000000000000000000000000000..221de2c28832026989d7d8f06d6de68299705346 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzalagk @@ -0,0 +1,5 @@ +{"text":"Studio 360 was an exciting executive search conducted by Filcro Media Staffing for a talented executive producer with the capacity and desire to create this compelling and intriguing program pertaining to the arts and culture.\nFilcro Media Staffing was honored to have been chosen as the executive search firm of record that identified, recruited and attracted the talented executive producer responsible for creating this highly successful media franchise and then managing Studio 360 for an additional (9) nine years.\nFocusing on issues and ideas (rather than events), Studio 360 is a weekly, one-hour radio program that explores the impact of arts and culture on the quality of individual and national life. It features new voices and well-known personalities, people who are deeply engaged in the arts \u2013 not just as critics and experts, but as lovers and doers who can take listeners behind the scenes and into the creative process. The arts are the program's focal point and the lens through which it explores the world at large. Studio 360 offers numerous access points to the arts with the goal of providing a compelling listening experience even for those who feel least engaged in cultural affairs.\nKurt Andersen, columnist for the New Yorker, co-founder of Spy magazine, and author of the critically acclaimed novel, Turn of the Century, is the program's host. Described by New York Times journalist Christopher Lehmann-Haupt as a person who is \"endlessly energetic [and] seems to know just about everything,\" Andersen brings wit, charisma, and a flair for incisive cultural commentary to the program.\nEach week, Andersen guides listeners through a broad array of subjects \u2013 probing, musing, and asking questions with genuine curiosity and a healthy dose of irreverence. A typical program combines documentary, discussion, commentary, and humor interwoven in artful and surprising ways.\nPoet laureate Robert Pinsky, gangsta rapper Snoop Dog, and \"poetry activist\" Bob Holman discussing the rise of poetry in today's America; and Marcia Tucker, singer, comedian, and Founding Director of the New Museum, on the changing depictions of the Madonna throughout history.\nBusiness of the Arts: explorations of the business, financial, and policy side of the arts.\nDesign for the Real World: from the ring of a telephone to the shape of a light bulb, this feature explores the design of items listeners encounter in their daily lives.\nChoreographer Jacques D'Amboise teaching Appalachian dance along the Appalachian Trail, helping children to rediscover their heritage; and a city mayor sharing his personal reflections as an attendee at a NEA-sponsored urban design forum.\nThe woman who programs music on Air Force One explaining the factors that influence her choices; and an established artist who tells us about the work of a younger artist that they have \"discovered\" and what they see or hear in the work.\nStudio 360 captures a diverse audience by providing access to arts and culture in a contemporary context that is highly informative and fun. While the program is aimed specifically at 35-49 year-old news\/information listeners (the largest growth area for public radio), PRI is confident that Studio 360's thoughtful, searching conversation about the arts will transcend demographic distinctions and appeal to a broad audience.\nStudio 360 Wins a Peabody - Congratulations!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"MySQL TimeStamps can have 2- 14 digits (where 2 = YY and 10 would equal YYMMDDHHSS).\nUse indexes in the database tables will increase speed of queries.\nUse lowercase names for tables, in the singular form, no abbreviations (unless common standards). The primary key should end in _id.\nForeign key should be the name of the referenced table + the name of the referenced field.\nAs of this writing, there exists a single usable CSV format that Microsoft Excel can both read and write safely across platforms. Tab delimited UTF-16LE with leading Byte Order Mark.\nUsing Excel to output into this format to begin with, you have to use the Save As dialog and choose \"UTF-16 Unicode Text (.txt)\". Frustratingly, it includes a .txt extension by default which can be confusing for non-technical users. However, if you provide clear instructions to Windows users to save their filename in quotes with a CSV extension such as \"example.csv\" it is workable.\nIf you operate an application with a CSV exporter please use this as your default export format. It will save us all many headaches in the long run.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"ELECTRAFIN\u00ae IS AN ELECTRIC PROPULSION SYSTEM WHICH MOUNTS ONTO NEARLY ALL SUP'S AND MANY KAYAKS.\nCurrent Drives pitch their product, the ElectraFin, on the premiere episode of CNBC's Adventure Capitalists.\nESUP\u2122 BY CURRENT DRIVES IS A ZERO-EMISSION, ALL INCLUSIVE MOTORIZED STAND-UP PADDLE BOARD DESIGNED TO TACKLE STRONG CURRENTS, WIND, FATIGUE AND DOLDRUMS.\nThe 34\" inch wide platform and 6\" inch thickness is perfect for first time paddlers, fishing, yoga SUP-ers, white water, and touring.\nThe folding design of the propeller eliminates drag from the fin. Higher top speeds can also be achieved with the new propeller thanks to the enhanced hydrodynamic blade design and the V2 Propeller can be used on all existing ElectraFin kits.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Watch Yeh Rishta Kya Kehlata Hai 11th April 2019 Video Episode 2903 latest Video by Star Plus In HD. Indian Tv Serial Yeh Rishta Kya Kehlata Hai episode 2903 by HotStar Complete Episode. Star Plus drama Yeh Rishta Kya Kehlata Hai 11th Apr 2019 Online Today New Episode.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Note that the column for grouping can only start from column 1, since column 0 is the placeholder for the expand \/ contract nodes. The GroupColumn property has the additional benefit that it returns -1 when grouping is not active. Otherwise it returns the current grouped column.\nTo undo the effect of grouping, the UnGroup method can be used, or as an alternative, the GroupColumn property can be set to -1.\nThe next step, is to write the event handler to set the grouping when a column header is right clicked.\nHere, grouping is done by setting the GroupColumn property to the column that is clicked. Instead of doing UnGroup when grouping is active and must change to another column, the new group column number is incremented if the grouping is currently active and the current grouped column is located before the clicked column. After this check, the GroupColumn property is set when it differs from the current grouped column.\nThe concept is simple : if a row contains a node, ie. it is a group header, then its background color is set to dark gray and the font color to white.\nTo show vertical borders only, the OnGetCellBorder event is used while the normal gridline display is turned off by setting GridLineWidth to 0.\nA border is drawn at the left side of the cell if it isn't a group header and the border color is set to clSilver.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanaxj b/data_all_eng_slimpj/shuffled/split2/finalzzzanaxj new file mode 100644 index 0000000000000000000000000000000000000000..11e908d3249a59dc4058a3bfc03a2e5b1facc7a3 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanaxj @@ -0,0 +1,5 @@ +{"text":"Reasons to Buy: Cheapest option with a secondary reflector.\nNotable Drawbacks: Manual zoom only. No advanced features.\nThe secondary reflector on this is far less powerful than that on the Metz flashes, but is still interesting. Even when you add in the cost of the required 5050DXR\/PX module, this is the cheapest you'll find that feature. The interface is designed for manual control, even including a mechanical calculator for flash power. This is somewhat unique for the flashes in this guide \u2014 generally they are newer models designed primarily for automatic control, with manual flash as an afterthought (if available at all).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"\"Pave Paradise\" Installed at Stamp Gallery, UMD-College Park. 5\u2032 x 6\u2032 x1\u2032 Floor, 5\u2032 x 3\u2032 x 1\u2032 Wall. Plywood, MDF, conversion varnish, artificial tree.\nHome \u00bb Sculpture \u00bb \"Pave Paradise\" Installed at Stamp Gallery, UMD-College Park. 5\u2032 x 6\u2032 x1\u2032 Floor, 5\u2032 x 3\u2032 x 1\u2032 Wall. Plywood, MDF, conversion varnish, artificial tree.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"You probably reached this site in your quest to find a clinical psychologist in Sandton, Johannesburg.\nConsulting the right psychotherapist for you, has a significant bearing on what you accomplish from psychotherapy.\nReasons for seeking psychotherapy are wide-ranging but broadly are motivated by a desire to address some aspect of life.\nIf you are contemplating psychotherapy you are welcome to browse through my website to see if you think you would be comfortable working with me.\nPlease contact me should you have any queries.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This year's Willy Wonka family costume theme has been my favorite family themed costume we have ever done. I loved making each of the costumes! Today, I am sharing the main man himself, Willy Wonka! Only there is a slight twist. It is a Willy Wonka costume for girls! If you missed the Wonka Bar costume and the Golden Ticket costume, check those out too!\nThere is nothing that screams Willy Wonka more than a classic Wonka Bar! That is why I was excited to be a Wonka Bar in our Willy Wonka family Halloween costume theme. Yesterday, I shared the Wonka Golden Ticket costume. Today is all about the no sew Wonka Bar costume!\nWilly Wonka is a family favorite at our house. We love the book, Charlie and the Chocolate Factory and the movies. That's why we knew we needed our family Halloween costume to be a Willy Wonka family costume. The Golden Ticket is such an important part of the story that we had to have that as one the costumes. So to kick off the no sew Halloween costumes for 2018, here is the Golden Ticket costume!\nDIY No Sew Ninja Turtle Costume for Girls!\nIt is officially the time of year that I look forward to each and every year. When I first started making my family costumes, I didn't mean to create the no sew theme that I did. But I am sure glad that I didn't start out by sewing every costume because that would have never lasted! If you are new around these parts, this is the 5th year that I have created a new sew costume for my daughter. We end up having a family theme with all of us, but I usually just share the tutorial for her costume. I will be sharing our entire family look later, but today I am excited to show you my very own little Ninja Turtle! Here is my DIY No Sew Ninja Turtle Costume for Girls!\nThis costume was not only easy to make, but it was so fun too! The best part was definitely putting it on my daughter and seeing her instantly turn into a ninja. Its like she literally thought she was a Ninja Turtle!\nChoosing our family Halloween costume each year is quite the debate. The last two years haven't been too bad, but this year it was much more difficult because 3 1\/2 year olds seem to change their minds every 5 minutes! Eventually, we narrowed it down to being Minions, Wreck it Ralph characters, or Powerpuff Girls characters. It took a long time to decide, but we finally decided on Wreck it Ralph characters! I kind of love that we are going with this theme for a few reasons\u2026 1. I love this movie and I don't think it got the love that it deserves. 2. This movie is a few years old and there aren't going to be a million other people dressed like us. 3. This Vanellope Von Schweetz costume is out. of. CONTROL! After we decided what we were going to be, I asked my daughter about the Vanellope Von Schweetz costume and that is when I realized that she wasn't thinking about the hoodie that Vanellope wears 99% of the time in the movie, she was thinking of the Vanellope Von Schweetz princess dress that she wears for about 30 seconds in the entire movie! So, I mapped it all out and made it happen. Introducing the Vanellope Von Schweetz Princess Costume!\nI am excited to be a part of the Really Awesome Handmade Costume Series with C.R.A.F.T. this year! Did you know that she has another website called Really Awesome Costumes that is full of the coolest costumes you've ever seen? She does! If you need to find a great DIY costume for yourself, your kids, your husband, etc. that is the place to look! The costume I am about to share will be featured on there tomorrow!\nWell, here is the moment we have all been waiting for. Little girl's costume for the year\u2026.\nElmo! Or as she would say it\u2026 \"Melmo!\" A few victories with this costume include\u2026 The fact that she recognizes it as Elmo. It costed less to make than the store bought Elmo costumes. It is 1000000000% cuter than the store bought Elmo costumes. It was will keep her warm during the freezing Idaho Halloween. It was totally no sew!\nFirst, the fleece into strips about 2-3 inches wide. Cut those strips into fringes. I folded the strips over a few times so I didn't have to cut so many fringes.\nLay out the shirt and pants and glue the fringe onto it. Overlap the fringe just a little bit as you work your way up. Make sure to cover every part of the clothing with fringe. I put a cereal box in the shirt so that the glue wouldn't go through the shirt onto the other side. Let the glue dry for 24 hours.\nFind round objects in your house that you can trace to make the perfect sizes for the eyes and nose. Trace them onto the felt and cut them out. The nose is a little more oval, but you can cut the circle and just cut a little bit off the sides to make it more of an oval shape.\nTake all of the shapes and glue them onto the hackle pad. Glue the hackle pad onto the headband and let it dry.\nOnce you finish that, you are done! Your little munchkin can now dress up as that lovable red, furry monster!\nWe dressed as Sesame Street characters as a family with our little Elmo!\nFor Cookie Monster, we just purchased a blue shirt and blue beanie. I cut out felt eyes and a glued them to the hat. For my Big Bird costume, we actually borrowed it from an incredibly talented neighbor! The costume would be insanely simple to recreate though. The dress was made of yellow fleece and was literally just a straight seam up the side and yellow boas glued to the collar and along the bottom. The socks were dyed orange because she couldn't find any orange knee socks when she made the costume. To make the headband, she stuffed a circular piece of fleece for the head. She took a cereal box, cut it apart, and rolled it into the shape of a beak and then glued yellow fabric over it. She glued it onto the head. The eyeballs were tennis balls with black felt glued to them. She glued it all onto a headband and added some feathers to the top of it for detail. It was so sweet of her to let me borrow it!\nIf you happened to miss out on her handmade Hedwig the owl costume from last year, check it out too! I used a lot of the same concepts on both costumes! Super easy!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Composition : Holarrhena antidysenterica (bark) 10%, Eclipta alba (leaves) 20%, Tephrosia purpurea (leaves) 20%, Tinospora cordifolia (stems) 10% Azadirachta indica (bark) 10%, Phyllanthus niruri (whole plant) 20%, Plumbago zeylanica (roots) 10%.\nThe Best Herbal Care for Liver Disorders Drug Induced Hepatitis, Viral Hepatitis, Liver Cirrhosis, Fatty Liver, All Liver Disorders. Infants: 1-2 Tea Spoon Twice a Day after meal or Children: 12 Tea Spoon Twice a Day after meal or.\nAdminister LivOrdain-FS Powder to Poultry birds @ 0.5 to 1.0 kg per ton of feed. As Tonic Broilers : LivOrdain-FS Powder @ 0.5 to 1.0 kg per ton of feed. LivOrdain-FS Powder mixed feed should be given from day one till date of marketing. For Layers: LivOrdain-FS Powder @ 0.5 to 1.0 kg mixed in one ton of feed. Breeders : LivOrdain-FS Powder @ 1.0 to 2.0 kg mixed in one ton of feed.\nBenefits : Nerves and Brian Health to improve your cognitive functions, and protect you from neurological disorders.\nBest Before : 36 Month from the date of mfg.\nThe Quality product made nt of fresh squid abd cuttle fish offals and fresh peeling waste. The product has great valuein shreep feed inndustries which is used as attractant.\nWe feel enough pleasure and pride to introduce ourselves as the leading growers, dealers and exporters of liver extract powder in India.\nWe are leading supplier of huge range of Liver Extract Powder.\nAndhra Agrovet Therapeutics Pvt. Ltd.\nComposition : Holarrhena antidysenterica (bark) 10%, Eclipta alba (leaves) 20%, Tephrosia purpurea (leaves) 20%, Tinospora cordifolia (stems) 10% Azadirachta indica (bark) 10%, Phyllanthus niruri (whole plant) 20%, Plumbago zeylanica (roots) 10%. Dosage : About 4 gms. powder should be given to the patient twice daily, half an hour before meals with water.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzandwx b/data_all_eng_slimpj/shuffled/split2/finalzzzandwx new file mode 100644 index 0000000000000000000000000000000000000000..538a04d41a93f35f70bd9186699c7fb0ff33ce44 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzandwx @@ -0,0 +1,5 @@ +{"text":"\u0418\u043c\u0430\u0442\u0435 \u0432\u044a\u043f\u0440\u043e\u0441 \u043a\u044a\u043c OYO Flagship 13847 HCMR Complex?\nLocated in the East Delhi area, OYO Flagship 13847 HCMR Complex is the ideal point of departure for your excursions in New Delhi and NCR. From here, guests can make the most of all that the lively city has to offer. With its convenient location, the property offers easy access to the city's must-see destinations.\nAll guestrooms feature a variety of comforts. Many even provide mirror, towels, internet access \u2013 wireless, fan, internet access \u2013 wireless (complimentary) to please the most discerning guest. The property's host of recreational offerings ensures you have plenty to do during your stay. OYO Flagship 13847 HCMR Complex is an ideal place of stay for travelers seeking charm, comfort, and convenience in New Delhi and NCR.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Maybe it's the murderous heat that bumps off our better angels, but something about summer brings out the outlaw in the Chronicle crew. Thus, we've decided July is Crime Month, and for four weeks, our writers and editors will be toasting the unlawful with reviews of recent crime fiction and nonfiction; interviews with authors in the field; plugs for crime-related shows to stream, screenings, podcasts, and events; appreciations of classic crime sagas and their creators; and more.\nWhere in the World Is Bunny Lake?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Local bus operator Metrobus is launching new buses that will cut emissions and improve passenger experience.\nThey are running on Gatwick Direct route 200 which serves Horsham, North Heath, Roffey, Bewbush, Gossops Green, Ifield, Langley Green, Manor Royal and Gatwick Airport.\nThe new fleet comes from a \u00a3500,000 investment to deliver additional comfort and features to passengers whilst providing environmentally conscious buses on the route. Designed and built in Britain, the new buses are powered by Euro 6 engines aimed at ultra-low emissions.\nWith improved seating for added comfort, free Wi-Fi on every bus and next stop information with regular updates, the enhancements for Gatwick Direct route will provide users with an even better service than before.\nNick Hill, Metrobus' Head of Commercial Development said: \"We are delighted to roll out these shiny new buses on Gatwick Direct route 200, which feature extensive route branding and are among the first to feature our new brighter livery.\"","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Photo boxers: (Left to right) Sam Barzilaye, Laura Roumanos, and Dave Shelley are co-foundres of Photoville's pop-up photo exhibition that will be housed in about 30 shipping containers in Brooklyn Bridge Park.\nFreight and frames: Photoville, a photo exhibit housed in 30 shipping containers in DUMBO and funded with a significant portion of donations from Kickstarter, showcases the changing nature of art festivals.\nTo think outside of the box, sometimes you have to get inside of it.\nA photography collective in DUMBO is throwing a show in one of Brooklyn's marquee outdoor spaces housed entirely in shipping containers \u2014 an unconventional move taken in part due to the rising costs of art spaces on other prime spots of Brooklyn turf.\nThe festival features photos from more than 30 photographers, from Brooklyn-based Russell Frederick's preservation-minded shots of Bedford-Stuyvesant, to the images captured by war photographers such as Lynsey Addario, Tyler Hicks and Joao Silva in Afghanistan and Iraq, to a collection of slice-of-life images taken behind prison walls.\nBut even with such significant names from the photography world on the roster of participants, some of the most important contributors to the show have been the public itself, who gave more than $30,000 to the project after its organizers set up a page on the website Kickstarter when some pledged donations from sponsors fell through.\nIn a way, the crowd-sourced nature of the project's financing \u2014 roughly 20 percent of which came from the 252 people who donated on Kickstarter \u2014 reflects the egalitarian spirit of the festival itself.\nThe show will include an exhibit made up entirely of community-sourced photographs taken by people riding the East River ferry, as well as \"The Fence,\" a collection of 240 photographs solicited from photographers around New York that will be blown-up and printed on weather-proof material to form a wall of pictures that will span from Pier 1 to Pier 3 through August.\nArtists agreed, saying it was important to them to showcase their work outside the controlled world of the art gallery.\nPhotoville at the Pier 3 Uplands of Brooklyn Bridge Park [Brooklyn Bridge Park Greenway at Joralemon Street in DUMBO. Contact United Photo Industries at (718) 215\u20139075. www.photovillenyc.org] June 22\u201325, and June 28\u2013July 1, various times. Free.\nGuys where the exhibit is going to be is not in DUMBO. That section of the park is considered Brooklyn Heights.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This shirt is part of our Small Batch program and only a couple dozen were made.\nAbout The Black & Navy Gingham: After a wave of colorful ginghams, we are getting back to fundamentals with Small Batch #13. Navy, white, and black are colors that anyone can wear with anything. Our ginghams are lightweight and made from the best cotton available on God's green earth which means it will hold its color, shape and size better than anything in its price range.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapuwg b/data_all_eng_slimpj/shuffled/split2/finalzzzapuwg new file mode 100644 index 0000000000000000000000000000000000000000..ac555e114b973ff411bee80ff37f52d4c232e1e4 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapuwg @@ -0,0 +1,5 @@ +{"text":"In the 17th century the fort Fortaleza de Santiago was built along the beach, on 3 June an English fleet defeated a Spanish galley fleet and in addition captured a large Portuguese carrack and severely damaged the fort. InCarrefour sold its Portuguese retail ventures existing under the Carrefour ensign to Sonae, Continente used to have one branch in Dubai, which was later purchased by Carrefour.\nIn the lower corner of the browser, you will see how much time is left back from downloading. This same year, the company Sonae IM was started.\nDownload video format Download audio format By Downloading this video I agree that I will not upload this video anywhere else without proper permission from the creator. Its GDP amounts to Subsequently, the new settlers observed a black cloud suspended to the southwest.\nDigital technologies are changing the way that consumers pay for goods. An authoritarian military junta came to power in and ruled untilBrazils current constitution, formulated indefines it as a democratic federal republic.\nThe popular appellation eclipsed and eventually supplanted the official Portuguese name, early sailors sometimes also called it the Land of Pingo doce receitas online dating.\nLisbon \u2014 Lisbon is the capital and the largest city of Portugal, with a population ofwithin its administrative limits in an area of Its position as the capital has formed through constitutional convention, making its position as de facto capital a part of the Constitution of Portugal.\nDuring its first two decades of existence, Sonae stayed as a small to medium-size business, focused on the derivatives market.\nThe Carthaginians, Romes adversary in the Punic Wars, were expelled from their coastal colonies and it suffered a severe setback in BC, when a rebellion began in the north 5. It's important to know that you are responsible for downloading the legal responsibility!\nClose to the Moorish castle is the church Nossa Senhora do Castelo and it stands on the spot where king Sancho I sevyn streeter and kid ink dating forensic a Romanesque chapel in the early 13th century, leading to the creation in of the parish of Nossa Senhora do Castelo de Sesimbra.\nThe region is noted for its Madeira wine, gastronomy, historical and cultural value, its flora and fauna, landscapes which are classified as a UNESCO World Heritage Site and embroidery artisans.\nThe download time depends on the Internet connection but is usually a fast process. Supermarket \u2014 A supermarket, a large form of the traditional grocery store, is a self-service shop offering a wide variety of food and household products, organized into aisles.\nHumberto Delgado Airport serves over 20 million passengers annually, as ofand the motorway network, the city is the 7th-most-visited city in Southern Europe, after Istanbul, Rome, Dating website australia oasis, Madrid, Athens and Milan, with 1, tourists in Sonae \u2014 Sonae is a Portuguese retail company with two big partnerships in the shopping centres areas and the Software and Information Systems, Media and Telecommunications.\nInthe company changed its name to Ahold. Most of the headquarters of multinationals in the country are located in the Lisbon area and it is also the political centre of the country, as its seat of Government and residence of the Head of State.\nShopping generally refers to the act of buying products, sometimes this is done to obtain final goods, including necessities such as food and clothing, sometimes it takes place as a recreational activity.\nBesides professional fishing and sport fishing, the most significant revenues in Sesimbra come from tourism, the town is famous for its beaches, fish restaurants and nightlife. Portugal was born as result of the Christian Reconquista, and inAfonso Henriques was proclaimed King of Portugal, in the 15th and 16th centuries, Portugal established the first global empire, becoming one of the worlds major economic, political and military powers.\nShortly after, independence was granted to almost all its overseas territories, Portugal has left a profound cultural and architectural influence across the globe and a legacy of over million Portuguese speakers today. The first Portuguese settlers began colonizing the islands around orgrain production began to fall and the ensuing crisis forced Henry the Navigator to order other commercial crops to be planted so that the islands could be profitable.\nThe Indo-European Celts invaded in the 1st millennium BC, mixing with the Pre-Indo-European population and this indigenous settlement maintained commercial relations with the Phoenicians, which would account for the recent findings of Phoenician pottery and other material objects 4.\nIts annual New Year celebrations feature the largest fireworks show in the world, Madeira is the second richest region of Portugal by GDP per capita, only surpassed by Lisbon.\nAhold \u2014 Koninklijke Ahold N. Brazil is a power in Latin America and a middle power in international affairs.\nThis active town twinning began in and there are events, such as a produce market from each of the other countries. These practices were by nature very labor-intensive and therefore also quite expensive, the shopping process was slow, as the number of customers who could be attended to at one time was limited by the number of staff employed in the store.\nThe hypermarket Continente chain is in all over continental Portugal as well on Madeira. As brazilwood produces a red dye, it was highly valued by the European cloth industry and was the earliest commercially exploited product from Brazil.\nThe following year, an expedition, under the captaincy of Zarco, Vaz Teixeira. There is some debate as to the effectiveness of this tactic, to maintain a profit, supermarkets make up for the lower margins by a higher overall volume of sales, and with the sale of higher-margin items bought by the intended higher volume of shoppers.\nTheir investigation revealed it to be the island they called Madeira. The former chief marketing officer was sentenced to 46 months in prison, the former CFO was sentenced to six months of home detention and three years probation. Lisbon lies in the western Iberian Peninsula on the Atlantic Ocean, the westernmost areas of its metro area is the westernmost point of Continental Europe.\nLater this legend was the basis of the naming of the city of Machico, knowledge of some Atlantic islands, such as Madeira, existed before their formal discovery and settlement, as the islands were shown on maps as early as During the s, the company began a period of rapid growth and it was also during the 80s that Sonae began its fast growth and, inSonae Investimentos SGPS, S.\nYou can do this easily with the buttons below.\nIt includes the islands of Madeira, Porto Santo, and the Desertas, the region has political and administrative autonomy through the Administrative Political Statue of the Autonomous Region of Madeira provided for in the Portuguese Constitution.\nIn antiquity, exchange involved direct selling, merchants or peddlers, the Phoenicians, noted for their seafaring skills, plied their ships across the Mediterranean, becoming a major trading power by 9th century BCE.\nDuring the reign of King Edward III of England, lovers Robert Machim and they were driven off their course by a violent storm and their ship went aground along the coast of an island, that may have been Madeira. Customers usually shop by placing their selected merchandise into shopping carts or baskets, a larger full-service supermarket combined with a department store is sometimes known as a hypermarket.\nMadeira \u2014 Madeira is a Portuguese archipelago situated in the north Atlantic Ocean, southwest of Portugal.\nPortugal is a country with a high-income advanced economy and a high living standard. InSonae gave back to the community Macys dep store A French itinerant medicine vendor on stage selling his wares Goods are displayed in a rustic manner in Marrakech's souk market Fresh produce markets have existed since ancient times.\nThe company is the second biggest retail company in Latin America by revenue, the company operates its e-commerce through Cnova Brasil, a subsidiary of its subsidiary Viavarejo. Nowadays, shareholders of S. The grocery chain expanded through the first half of the 20th century, Albert Heijn became the largest grocery chain in the Netherlands during this time, and expanded into liquor stores and health and beauty care stores in the s.\nIt was restored between andthe walls of the church are inlaid with beautiful azulejos.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Friends often say: \"You are really lucky to have a job where you can see all the latest in appliances.\nThe \"yes\" is the part of me that loves to see all the new innovations. The \"no\" part is the desire to constantly redo my kitchen appliances.\nThis year's crop of new kitchen standouts is better than ever because of technology, especially in refrigerators. Two of the coolest models are from GE and Samsung.\nGE's Caf\u00e9 Series refrigerator with Keurig K-Cup brewing system has a two-fold advantage: Its hot water dispenser makes brewing quicker without having to fill the container and it leaves more room on the counter to get rid of the cluttered look. The brewer is also dishwasher safe, another bonus. And this is something all the techies will love (I do) \u2013 GE connected technology allows you to preset the brew time through your cell phone. Suggested selling price: $3,300.\nSamsung, the company that makes everything from high-definition televisions and cell phones to appliances, has developed the Family Hub, the one appliance that is on top of my must-have list. The computer on the door has a built-in camera to show you what's missing in the fridge, allows direct grocery ordering for those of us who hate to go to the grocery store and an integrated touchscreen that shows calendars, notes and photos. My biggest love: It has four doors and the bottom two are what Samsung calls \"FlexZone\" because if you, like me, don't use much frozen food, it allows more refrigerator space when one of the bottom sections is converted from freezer space. I have seen it advertised for as high as $5,599.\nThere is one caveat: If you are replacing your old refrigerator rather than doing a complete remodel with a designer, you need to measure carefully. Many of the new refrigerators are taller than those in the past. For my kitchen, I would have to remove overhead cabinets.\nFor those of us watching our weight and wanting to eat healthier (I plead guilty on this one), several companies offer steam ovens. For example, Gaggenau recently launched the new 400 series Combi-steam oven in 24- and 30-inch widths. It has an automatic cleaning system and a sous-vide cleaning mode. The French term may sound intimidating, but it isn't. It is French for vacuum and means you can seal ingredients in a plastic bag and cook in a water bath, a healthy way to maintain the food's vitamins, minerals and color. You can save up to 50 recipes with the TFT touch screen. The oven also can steam, bake, simmer, braise and broil. A fixed inlet and outlet water connection and new cleaning cartridge make cleanup easy. Starting price: $6,999.\nMy dream appliances have already cost me an estimated $12,598.\nPerhaps you haven't noticed, but wall ovens have received a makeover. The concept that a variety of manufacturers have embraced is French door ovens. Viking introduced a French door oven this year that not only looks good, it is easy to open with one hand in small spaces such as a galley kitchen. It has a large oven cavity and three racks including one full extension rack that I love because of access to larger items such as a turkey. Viking boasts that it has the largest convection fan in the industry and even heat distribution because of the broiling element is ribbon shaped under glass. It sells for about $4,999.\nIf I would have to replace my electric range, I would still need a cooktop. My answer would be induction. It cooks faster, doesn't heat up the kitchen and is good for parents of small children who love to turn on the knobs because the stovetop is not hot until the pans are placed on the burner. Chefs like it because unlike a standard electric cooktop that has temperature fluctuations this one maintains it temperature and has little heat loss.\nWhen Zach Bell was executive chef at Caf\u00e9 Boulud on Palm Beach island, he told me it had a single burner unit in the area where salads and cold appetizers are prepared. Chefs like it because unlike a standard electric cooktop that has temperature fluctuations this one maintains it temperature and has little heat loss.\n\"I use it for the flexibility and speed it gives me in the cold appetizer station,\" Bell says.\nAlthough at first consumers were told they would have to buy special pans, you can use any pan as long as it attracts a magnet. Magnetism is how induction works. The magnet heats the pan so it becomes the cooking surface, not the burner. The cooktops typically start at around $1,000 and up according to the brand.\nOne of the best innovations is the freestanding range with induction cooktop. This wasn't possible in the past because the induction cooktop had more depth. But be prepared to pay. One from Samsung was advertised at $3,699.\nTotal cost of my dream kitchen with induction cooktop: Estimated $18,597.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Ubereats Promo Code & Ubereats Coupon UberEATS is an on-demand meal delivery service ubereats promo code 2017. Originally powered by the Uber ride sharing app it is now available as a stand ubereats is an online meal ordering and delivery service. Discounts average $18 off with a UberEATS promo code or coupon it partners with restaurants in dozens of cities around the world. 50 UberEATS coupons now on RetailMeNot under this post you will get all the latest and working ubereats promo code and coupons. UberEats Discount Code: JEREMYA15 -- Use this special promo code to score a discounted or FREE meal when using UberEats for the first time! 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Click and save big with latest 2011 Ubereats trust coupons. com Promo Codes & Coupon Codes com for savings.\ncom coupons on June 2017 atlanta ubereats via living social.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"A perceived anti-abortion tilt of the U.S. Supreme Court has inspired state lawmakers to move to outlaw abortion entirely if Roe V. Wade ever falls. But the rush to regulate has exposed divisions among lawmakers who consider themselves staunch abortion opponents.\nOklahoma may soon see more money to help fight the AIDS epidemic.\nTo get a medical marijuana license in Oklahoma, patients need a recommendation from their doctor \u2014 paperwork attesting that, in their medical opinion, a patient would benefit from the drug. But what happens when there's not a doctor in the area willing to sign off?\nOklahoma has the second highest uninsured rate in the country and some of the poorest health outcomes.\nThe bill is endorsed by both the state's optometric association and Walmart, who just a few months ago were on opposing sides of State Question 793, which voters ultimately defeated. Both sides spend millions of dollars on the campaign.\nStateImpact Oklahoma is a collaboration of KGOU, KOSU, KWGS and KCCU.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"For a complete outfit, pair with other items from the Surf collection.\nEvolutive Pants : What are they !?\nSubscribe to Coton's Vanille VIP CLUB to get our Exclusive promos and sales alers!!\nEnter your email to receive exclusive promotions to Coton Vanille fans!!!","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarsxu b/data_all_eng_slimpj/shuffled/split2/finalzzzarsxu new file mode 100644 index 0000000000000000000000000000000000000000..b76587b51e5c34a705dfb3afbdac42ad11d5c24b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarsxu @@ -0,0 +1,5 @@ +{"text":"And mostly based on recent PCW results and the time Remaining till new year.\nStop whining. Only constructive Arguments.\nAdded Possibility Number 5 and 6 As somebody requested in Steam.\nI can't understand why the fewer teams in the division, the more divisions need time to finish the season?\nI think 4 div split is the most beneficial in case if we can ensure the shortest season. Why do you need the group stage and playoff in case if only 4 teams playing in a single group? Isn't it seems more reasonable to play Bo5 Double Elimination then? It can take only 3 weeks for a season then without the playoff.\nThe argument for Playoffs b4 was Playoffs are the real interesting thing where everyone knows now its important based on the title already.\nJust a Bo5 Double elimination well the more play the better right and the losing team would only play 2 Matches in that and be out pretty bad.\nAnd the thing about the 4 Division Setup is if only 1 Team disbands in that your ending up with just 3 Teams constantly playing the same enemy s (could be boring) if you suddenly end up with 2 teams disbanding just play 2 matches and done in a division pretty bad.\nI think for us very bad if we get in div where is pell mell or elogain, them impossible to win, we played some pcw with pubstars, they for us hard too... idk. I like possibility 6, i dont understand why changed the seeding.\nLet's revisit this in another 2 weeks and see which teams fold.\n1, 2 and 3 look okay to me. the others are bad because of at least one division having 4 teams... if 1 team folds, the entire division suffers because of it.\nI think seeding 1 is probably the better of them all, having 6 teams in division 1 will be good for people to improve and for viewers to see variety in the high skill matches.\nIt also means that Elogain aren't in division 2 because they have had some bad games recently, they were finalists in div 1 last season, there is no reason for them to be in div 2 imo.\nThank you that you guys appropriate the Challenge.\nI think for us very bad if we get in div where is pell mell or elogain, them impossible to win, we played some pcw with pubstars, they for us hard too... idk. I like possibility 6, i don't understand why changed the seeding.\n3) Comp mod is still in maintenance so we cant play atm anyway time to take the chance and resort.\n4) A division size with 7 Teams takes at least 6 Weeks to play each team once when you assume each team plays 1 each week and time is running short Christmas is coming soon i would like to finish inb4.\nThis is honestly my biggest fear about these seeding's with 4 Teams.\nAnd Placeholder might be slightly ahead of EloGain.\nWith the 3 Teams Div. Same problem as i mentioned about a 4 Teams Div. setup might be very fast true but its first of all not enough variety in enemy's and if a team disbands the entire div suffers more then on a bigger group.\nThank you guys for your feedback please keep it coming and don't stop the discussion.\nI'm all for #2 or #3 (only difference between #2 and #3 is placement of 8bitgamers which could be determined by a seeding match).\nI strongly agree having 4-team divs is too risky, which cuts out #4 and #5.\nMy grievance with #1 being that pubstars probably won't be challenged enough (famous last words Kappa). Give us some challenge pls, kthxbye.\nidk what to do with them they say they want to play div2 but could they not compete in div 1 ?\nAlso im unsure if Saunabois should stay in div3 with Ypsi added to the roaster.\nIm happy for feedback will add a Possiblity 7 8 and 9 soon.\nOne person on that new team is active on three different rosters.\nFor OOP the majority of players in that team have played div 1 before so I don't see why they should not play div 1.\nSaunabois lost 3-1 in a scrim against For the Lolz yesterday but that was without ypsi.\nit is a fucjing mess. if anyone should go to div1 it is team awesome who have mulyiple players in inbite gathers.\nIs Ypsi really allowed to play for a div 3 team, or is he just coaching?\nYpsi IS a d3 team.\nOk as i said i Created a few new Seeding's cause we got some late joiners with OOP in there.\nPlease just Judge every team based on their 6 Strongest players equally like idc if they have in the comments \"only comm\" or \"inactive\" or \"only pistol and welder\" or whatever.\n2nd of all i need feedback which team should play in what division regarding Saunabois and the Team OPP.\nwrong team you mentioned there.\nPlease keep talking about what seeding it should be.\nPossibility 11 so far, if there's more team joining it could look differently, but as it is now.. 11.\n\"im so sad i tried hard in a game and now people think im good and do t let me stomp lower divs weee\"\nThe only player who ever managed to pull that was ryssk and we all know how he managed that. Taaawtastic!\nyes you know everything the old and wise bauer at it once again.\ni like the pun u did with the name its clever.\nyour avatar won't load for me, can you try re-uploading?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"One of the world's most distinguished brain scientists is to give a series of public lectures at the University of St Andrews.\nRodolfo Llinas, Professor of Neuroscience and Chairman of the Department of Physiology and Biophysics at New York University, will give three separate lectures in the School of Biology on Tuesday 28 and Wednesday 29 November and Friday 1 December 2000.\nThe lectures, being held as part of the centenary year celebrations of the Bute Medical Buildings, will all be held at 5.15pm in Lecture Theatre B of the Bute Medical Buildings, St Andrews.\nNOTE TO EDITORS \u2013 Should you wish to send a reporter\/photographer to any of the events, please contact Claire Grainger, contact details below.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This website is using cookies. To achieve a Gold competency level, Solvusoft goes through extensive independent analysis that looks for, amongst other qualities, a high level of software expertise, a successful customer service track record, and top-tier customer value. As a Gold Certified Independent Software Vendor ISV , Solvusoft is able to provide the highest level of customer satisfaction through delivering top-level software and service solutions, which have been subject to a rigourous and continually-audited approval process by Microsoft. He is a lifelong computer geek and loves everything related to computers, software, and new technology. News Blog Facebook Twitter Newsletter. Solvusoft's close relationship with Microsoft as a Gold Certified Partner enables us to provide best-in-class software solutions that are optimized for performance on Windows operating systems. Gigabyte HD drivers are tiny programs that enable your Graphics Card hardware to communicate with your operating system software.\nMaintaining updated Gigabyte HD software prevents crashes and maximizes hardware and system performance. Subscription auto-renews at the end of the term Learn more.\nFurthermore, installing the wrong Gigabyte drivers can make these problems even worse. As a Gold Certified Independent Software Vendor GigabytSolvusoft is able to provide the highest level of customer satisfaction through delivering top-level software and service solutions, which have been subject to a rigourous and continually-audited approval process by Microsoft. Gigabyte hd 5570 achieve a Gold competency level, Solvusoft goes through extensive independent analysis that looks for, amongst other qualities, a high level of software expertise, a successful customer service track gigabyte hd 5570, and top-tier customer value.\nUsing outdated or corrupt Gigabyte HD drivers can cause system errors, crashes, and cause your computer or hardware to fail.\nSolvusoft's close relationship with Microsoft as a Gold Certified Partner enables us to provide best-in-class software solutions that are optimized for performance on Windows operating systems. Jay Geater is the President and CEO of Solvusoft Corporation, a global software company focused on providing innovative gigabyte hd 5570 software.\nYou are downloading trial software. Microsoft Gold Certified Company Recognized gigabyte hd 5570 best-in-class capabilities as an ISV Independent Software Vendor Solvusoft is recognized by Microsoft as a leading Independent Software Vendor, achieving the highest level of completence and excellence in software development.\nGigabyte HD Drivers Download. Gigabyte HD drivers are tiny programs that enable your Graphics Card hardware to communicate with your operating system software. News Blog Facebook Twitter Newsletter.\nThis tool will download and update the correct Gigabyte HD driver versions automatically, protecting you against installing the wrong HD drivers. He is a lifelong gigabyte hd 5570 geek and loves everything related to computers, software, and new technology. Gigabyte HD Drivers: Windows XP, Vista, 7, 8, ud Downloads: If you are inexperienced gigabyte hd 5570 updating Gigabyte device drivers manually, we highly recommend downloading the Gigabyte HD Driver Utility.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"each time an address is submitted with the form on any given page-to have that saved instead of overwriting the previous entry.\nYou can use add_option to add each time. But in this case you need new option name every time. And interesting thing is, if you use new option name every time, you can use update_option too.\nUpdate_option updates the value if option_name exists, if not it will add the data.\nThanks, how do I implement that?\nNo, I don't think you did answer my question. You didn't even work with me to really see what the question was> Please refund my money!!!\nYou need to change py_menu_item every time. The value will be saved based on py_menu_item so it will always overwrite if the name is same.\nWhere are you implementing this? For a logged in user? If yes, then you can use user_id to save new data.\nNo, not a logged in user. It is just a form on user-facing-page that anyone can use (my preference to do it that way).\nAnytime any random person (to be secured with Captcha) comes to the site and enters an address, the address is to be saved so that it can later be called in search.\nthe above link answered the question, much quicker than WPMU and did it correctly. Please refund my money!!!\nHe suggested sanitize_key( $py_username ) instead of $current->user_login. I suggested $current->user_login because I thought the page is for logged in user only. The logic is to add a unique keyword with py_menu_item so that it will not be overwritten!\nAlso according to stackoverflow, are you sure $py_username is always unique? If not, then it might overwrite again.\nIMO, the best solution will be if you create an extra table in the database. So, there can be columns like ID, address, username, address_result etc.\nAnd about refund, if you please contact our billing department here: https:\/\/premium.wpmudev.org\/contact\/ they should assist you further.\nSome more notes, as stackoverflow is not a private forum and I think many people (may be hundreds or even thousands) are waiting there to answer a question in first place.\nBut in our forum we process all the threads based on a queue. So first come, first serve. We are human here and when we stuck in a thread, it needs time, and it takes some time to come back in older threads.\nWe are really happy to see you in the community!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Michaels Wedding Cake toppers \u2013Wedding cake options are not quite unlimited, and the wedding cake has come to symbolize the wedding more than any supplementary visual element. There are a range of options, but don't let them limit or crush you. These options are just opportunities to make the tone and feel of the reception you have planned and your cake is a ration of the romance and happiness of the memories you are creating for your wedding day.\nWedding cakes today range from the tiered cakes of archaic and the cascade cakes of the eighties to individual cakes accurately contracted on a tiered cake dish to see taking into account a wedding cake. At truly extravagant weddings, miniature versions of the cake are served to guests. whatever your budget, a magical cake can be found to ration with your guests.\nAlthough important, there's more to rule than just taste. Many different flavors, shapes and textures are affable in cakes today and they are a postscript of you and your prudence of personal style.\nThere are few limitations upon space for cakes and fillings. Although there are many icing options, the two most popular are butter cream and fondant. Butter cream is serene and creamy and can be easily colored. Fondant is made of flavored corn syrup and gelatin, but it is draped more than the cake then again of monster smoothly applied next a knife. The result is a shiny, porcelain-like surface that is easily decorated. Butter cream frosting does not preserve in the works with ease at outdoor summer weddings or in non let breathe conditioned locales.\nWhen choosing your wedding cake, as always, friends and intimates recommendations are a fine pretension to locate the right baker for your wedding cake. Cakes may be ordered from a bakery or caterer, but some hotels and restaurants may then be accomplished to pay for a wedding cake. However, you will probably be enlarged off ordering the cake from a baker who specializes in wedding cakes. The best people create works of art, which taste wonderful and are made from open ingredients, not a mix.\nBefore visiting the cake maker, you will need to know the number of people attending the reception, as this determines the size of your cake and whether new ones will be necessary. Bakers often base their charges on a per slice basis. If the cake you desire is out of your price range, rule having a smaller explanation made for you to display and cut and sustain sheet cakes next a similar decoration. Communicate to the baker the type of reception and decorations and give him colors to be used if possible.\nAs for decorations, cake designs tend to follow bridal fashion. Remember, in imitation of wedding cakes, the more labor-intensive your cake design is, the more it will cost. The cake summit can be conventional or imaginative. It may enhance blithe flowers, bells, adore birds, a bridal couple or rings. plan to discuss the summit you will be using subsequent to your baker.\nThe groom's cake can be any size, shape, spread or theme. It can be sliced and served at the reception or boxed for guests to say you will home. At the reception, spotlight the wedding cake on a separate table covered like a tablecloth and garlanded taking into consideration flowers and greens. Your cake knife and toasting glasses should praise your overall setting.\nThe tradition of sharp the cake behind the bride and groom feeding each additional cake and toasting each supplementary after the cake cutting, makes a beautiful wedding tradition captured in photographs. A reputable baker or caterer will be glad to con you photographs of similar to work. He or she will as a consequence arrange for a tasting of two or three flavors.\nCan it be made bearing in mind well-ventilated fruit in season (this will help save the cost down)?\nWhat nice of cake and flavors are comprehensible (is the cake made taking into account lighthearted ingredients)?\nAre there any other charges or deposits (some bakers encounter deposits upon plates, columns, and accessories. locate out exactly what these items will cost and get a written proposal)?\nRelated Posts of \"Michaels Wedding Cake toppers\"","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarvsb b/data_all_eng_slimpj/shuffled/split2/finalzzzarvsb new file mode 100644 index 0000000000000000000000000000000000000000..37a6c22c47ed369ee56e6d9cdb30b4d18ed07ec8 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarvsb @@ -0,0 +1,5 @@ +{"text":"A single stem of glowing flora to infuse your cards with wild warmth. Black-Eyed Susans come in a variety of forms, so you won't want to limit yourself to straight-up yellow! Although straight-up yellow delivers rays of friendliness to a card. :) Enjoy this botanical study of Rudbekias and have fun layering it various ways to create a whole field of flowers \u2014 it's easy with digital stamps to multiply!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This is kind of a fun little technique to try, but it can give you truly valuable insight into what might be happening in your business.\nThe idea is to look at your website with fresh eyes to help you find some relatively easy-to-fix problems that could be costing you clients.\nWhy 5 seconds? Because stats say a business has anywhere from two-tenths of a second to 15 seconds to make an impression on their website visitors that will lead them down a path to buying.\nThat means that even on the generous end, there's no time to diddle around, slowly dripping out the information your prospects need to decide if you might be a good fit for them. Five seconds is just a happy medium to test your site.\nIf you don't think you can be objective, ask your spouse \/ partner or a friend to help. Better yet, pair up with a designer friend you trust and review each others' sites!\n1. Start with your Home page.\n2. Set a timer for 5 seconds \u2013 don't try to count because you need to concentrate on your site.\n3. Open your website on your computer (the actual site your prospects see \u2013 not your admin page where you make edits).\n4. Scan the page during those 5 seconds, scrolling down if you have time.\n5. At the 5 second mark, close your browser.\n\u00b7 Is it obvious what you do?\n\u00b7 Is it obvious what geographic area you serve?\n\u00b7 Is the kind of design work you really want to do visually represented?\n\u00b7 Is it obvious how to get in touch with you?\n\u00b7 Is there anything on this page that is not consistent with your aesthetic, your brand values, and \/ or your brand voice?\n\u00b7 Is it obvious how you serve your clients uniquely?\n7. Repeat for every page of your site.\nIf you answered any of these questions in the negative, you've got a money leak!\nSimply put, every page of your site needs to offer every bit of that info as obviously as possible.\nThere's no guarantee a visitor with enter your site through your Home page. With back links coming from blog posts, Pinterest pins, and so on, plus SEO and metadata on your site page titles (just trust me on that one, OK?) you really never know onto which of your site pages a visitor may stumble first. And you certainly can't count on them looking at every page on your site \u2013 especially if you don't give them relevant info that makes them want to stay!\nLook, I know a lot of designers feel that if someone is serious about hiring them, they'll take a good long look at the website. And that's true \u2013 but only after they've already done a quick (maybe just 5 seconds) scan of your site on their way to narrowing down which designers they want to look at more closely. It's just how people buy., so why not play the game to win (win your ideal clients' business, that is!)?\nLet me know in the comments below how your site fared with this little test.\nAnd of course, feel free to reach out to me at deb@debmitchellwriting.com if you want help fixing your website's money leaks!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Watercolor is a medium that embodies the spontaneous nature of the world around us. The design shown here was created using watercolor paints and a frenetic approach. This pattern is created using an installation of one \"A\" panel and then a \"B\" panel. Then you continue with an \"A\" and so on.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"A Technavio study, released by Research and Markets, predicts the global private and public cloud market in financial services industry will experience a compound annual growth rate of 13.12 percent from 2016 to 2020.\nThe 107-page market research report incorporates analysis and inputs from industry professionals and covers the market landscape and its growth forecast over the next four years, Research and Markets said in August.\nResearch and Markets says the growing demand for unlimited virtual storage and big data will drive the market and data security will pose a challenge.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Have you been thinking of starting or changing your exercise routine?\nOnline personal training is a great way to work with a trainer at a fraction of the 1-on-1 investment.\nThe holidays are a busy time of the year - stay on track, stay accountable.\nSkip the gym and workout at home (or) utilize the equipment you have access to at your local gym.\nAfter writing up workout routines for over 2 decades, I choose to simplify -but if you prefer complex and complicated - I can show you how to make it more difficulty (notice i didn't say more effective).\nSimple doesn't mean easy. You still need to put in the work and stay consistent.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasekb b/data_all_eng_slimpj/shuffled/split2/finalzzzasekb new file mode 100644 index 0000000000000000000000000000000000000000..907bc1cc26117b8a38d625dc4e10eb47b5e6bba2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasekb @@ -0,0 +1,5 @@ +{"text":"As every food lover in town surely knows by now, Boulder has in the last few years become a proving ground for inventive pizzerias cooking pies the old fashioned Neapolitan way, in a scorching hot wood-fired oven, with thin crusts and fresh toppings. Passionate about the alchemical combinations of crust, cheese and toppings that make authentic Italian-style pizzas such a wonderment (not to mention the longstanding challenge of crafting quality crusts at altitude), these establishments have put all their considerable culinary talents and financial resources to making world class pies. Boulder is the happier for it.\nWith such a luxury of riches to sample, we here at EDB decided to turn our inaugural staff kick-off party into an Artisanal Pizza Contest. Which of the great pizzerias in town, new or old, makes the best pie? From the start we recognized that it's a loaded question: pizza elicits such strong personal reactions, often drawn from our deepest, dearest childhood memories. And with several top notch contenders vying for the title, the stakes were high. Nevertheless, despite the risk of having our first meeting devolve into foodfight and fisticuffs, we dedicated ourselves stomach and soul to answering this delicate, delicious question.\nOur ground rules were simple, and limited contenders to a manageable seven: we only sampled pies that were cooked in an Italian style (wood-fired) oven, made in the classic Italian, thin crust, single-serving style. We ordered a simple margherita pizza from each (tomatoes, basil and mozzarella), then gathered at our secret EDB lair in downtown Boulder and got down to business. Each pie was rated on looks, crust, cheese, sauce and overall quality. Aware that all these pizzas shine brightest when eaten straight out of the oven, we coordinated pick up times just before meeting, so all were sampled under the same conditions.\nThe establishments we tested featured two longtime Boulder mainstays (Proto's in North Boulder and Laudisio's in 29th Street); three celebrated entrants from the artisanal pizza boom of late 2010\/early 2011 (Basta, da Lupo, and Locale); a young, yet pedigreed, operation that arrived late last year (Tossa), and finally, a bit of a wild card, the pizza counter at resurrected Boulder natural foods mecca Alfalfa's, who do use a wood-fired oven and thus passed our entry requirements.\nOnce we cracked the pie boxes, all the getting-to-know-you chatter subsided and the crew took to tasting and rating in earnest. Each of the judges brought their own unique experience to the challenge: Christy, hailing from the northwest (farthest from bella Italia) has recent (we trust) fond memories of her nuptials there last year; her bar was high. Sara and Marsha, professional photographers, were especially attuned to the plate presentation, but were unforgiving in their taste assessments. (Screw up their pie and they will make you look silly, girlfriend.) Holly, recently returned from menu-consulting in the Middle East, was happy to have pizza of any kind, although her dietitian instincts and sophisticated palate kicked in immediately. Shannon let out her inner French philosopher-gourmand and deconstructed each pie with inspired acumen. Allan, who has been biking and eating his way through Italy for a decade, had a wealth of native examples for comparison. He rated the wares across the board surprisingly highly: a testament to the overall quality of product in the Republic. Finally, LV Torio and yours truly had to suppress fond memories of Sally's and Modern Apizza in our hometown of New Haven, not to mention our Campania-blooded nonni's fried dough \"pizzas\" \u2014 both very different beasts from the Neapolitan-inspired classics we were eating.\nOverall: 1. Locale, 2. Proto's, 3. Basta.\nThe results were decisive: Locale won the overall rating by a wide margin. It is perhaps not a surprise that Locale swept top honors, with their imported cheese, wheat, toppings, prosciutto slicer, and (who knows?) ground water. They have a clear vision for each pizza, their cooks are well trained, and they bring deep care to their constructions, which comes through in the taste. Proto's, proudly out-pointing most of the new kids on the block, can stand tall knowing they are still riding atop the wave they helped usher in a decade ago. And Basta, only slightly less heralded than winner Locale, is well-deserving of their medal showing. Our judges were unanimous that their sauce is especially brilliant, making a trip to the Peloton mandatory.\nWe should mention that Laudisio's and da Lupo suffered from split opinions. Each had their strong advocates, but equally opinionated detractors. You should taste their wares yourself in the event you are among the sizable contingents who love them. We found the pizzas from Alfalfa's and Tossa to be something other than the traditional Italian style, whether it was ingredient or proportion, and it resulted in their downgrading across our panel.\nIf you have a comment on the proceedings, we'd love to hear from you. If you are from one of the establishments we reviewed, and would like to laud or dispute our findings, please leave a comment here and let us know where we got it right or screwed up.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Less than 2,000 tickets remaining for the St. Jude Dream Home Giveaway!\n\"Fewer than 2000 tickets remain for the Mississippi Gulf Coast St. Jude Dream Home.\nDon't miss out on your chance to win a car and a new home! Tickets are going to go fast; don't miss out!\"\nSt. Jude Dream Home Giveaway happens on April 7th, 2019!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Hurry. . . . Haaarrrrrd!\nYesterday was one magnificent day indeed. Church was, as always, alive and great with good numbers and good community. The real sense of community came out in the afternoon when we had our first ever bonspiel. We all assembled at Beachgrove to curl. It was quite an event. We had over 30 curlers and another 20 or so spectators.\nAt one point I was confused with figure skating and did a couple of triple axles \u2026except I did not land them. I was not alone in slipping and falling. We had a couple of people go down on the ice. One woman is a little on the banged up side. She is in a lot of pain and I pray that she is feeling better soon.\nI have three really good bruises for my folly yesterday and I used muscles that I did not know I had. I must confess that I am stiff this morning. I won't be surprised next Sunday is half the congregation is bandaged or limping to church.\nThe mishaps aside this was a great day For St. Mark's. We had all age groups participate and there was a real sense of joy and laughter in the group. We were being church at the curling rink. Four sheets of ice filled with fun, fellowship and freestyle skating\u2026.a great day for the parish. One thing is for certain I am no Brad Gushue and Catherinanne is no Colleen Jones. Yet we both really enjoyed ourselves and Catherinanne really caught on well.\nGot photos \u2026 want them posted\u2026email them to me!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Jericho Ilagan fondly recalls how he and his brother Jensen would play basketball back in their youth. They would go out in a small court just in front of their home, and everything else wouldn't matter as they enjoyed the game.\n\"Lumaki kami ni Jensen naglalaro ng basketball. Yung bahay namin may half-court sa harap,\" shared Jericho, who is also the Chief Operations Officer of the ASEAN Basketball League.\nFrom those moments, dreams were born. And one of the things that Jericho and Jensen dreamed about was having their own basketball court \u2013one that was not just set up on a sandlot or some other makeshift space; but an actual one.\nNow, the two have realized that, as they have opened the Phenom Sports Center in San Isidro, Makati.\n\"Nakakatuwa na ngayon, meron ng Phenom, where everybody can play,\" beamed Jericho after the ceremonial ribbon-cutting alongside Jensen in the sports center's grand opening last October 22.\n\"Very fulfilling for someone who's a basketball-loving person, for a basketball enthusiast,\" said Jensen, on the other hand.\nPhenom isn't just the typical basketball court. Jensen proudly shared that the \"Phenom Sports Center will be the first complete athletic training facility,\" as it has equipment that can help athletes improve.\nA few members of the media were able to experience what Phenom can offer, from measuring the biometrics a la Draft Combine, to hitting the weights room, to undergoing drills made specifically for basketball.\nThere's also a presence of a sports psychologist in the facility, whom athletes can easily approach for mental skills for enhanced performance, as sports psychology has proven to be a vital part of an athletes' development.\n\"Actually, marami naman ding mga sports center in Metro Manila. Some have a basketball court, some have a gym, but this is the first facility where we can customize a program for you,\" said Jensen.\nBut even if their dream has now come into fruition, Jensen made it known that they won't be stopping there.\n\"Now our goal is to somewhat bring back to the community, whereas with this kind of facility, we'll be able to train, develop, and produce athletes that would perform well not just in the collegiate level but also in the professional,\" he said.\nPhenom Sports Center is located at JM Business Center, Zanzibar Street, Barangay San Isidro, Makati.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Do these shows leave a realistic amount of time before we're cruising around the galaxy? Interstellar travel makes less sense in the 21st century than it would in the 51st. Social dynamics: Do people behave the way we'd expect, given the timeframe and the technology? Historically, major advances in science have brought massive social change in their wake. No wonder the show stayed in space and rarely delved into the utopia back on Earth, which always felt more like an cool thought experiment than a realistic look at where social progress might take us. Technology: Netflix's Lost in Space, the most recent entry on this list, may also have the most realistic depiction of technology. Technology: Since its first broadcast in November 1963, Doctor Who has played fast and loose with science and technology. The show has offered multiple futures for our planet, often showing technologies \u2013 such as self-aware plastic, instant cloning and \"Fountain of youth\" de-aging devices \u2013 far beyond where we're at now. There are no aliens, and, apart from the assumed progress in space travel, no magical technologies along the lines of Star Trek's replicators. Technology: Red Dwarf was a space sitcom with a simple premise: A mining ship from the 22nd century suffers a radiation leak.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasoyr b/data_all_eng_slimpj/shuffled/split2/finalzzzasoyr new file mode 100644 index 0000000000000000000000000000000000000000..45533dbd27716d4861742abee9738ed2e7cebfa7 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasoyr @@ -0,0 +1,5 @@ +{"text":"Greenhaus respects your privacy, and we are committed to protecting it through our compliance with this policy. This policy describes the types of information we may collect from you or that you may provide when you visit the website www.greenhaus.com (\"Website\") and our practices for collecting, using, maintaining, protecting and disclosing that information.\nAny of the information we collect from you may be used in one of the following ways: to personalize your experience on our Website; to improve our Website; to provide you with information you request; to improve customer service; and to send you periodic emails. The email address you provide may also be used to send you information and updates about Greenhaus.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"If you do not see something that might interest you listed below.\nAll specials must be mentions by guest at time of reservation to receive the special rates.\nJust the two of you. ESCAPE!\n3 night weekend (or midweek) starting at $276.00 for two people in a room with one queen bed.\nOther lodging upgrades available or just make it two nights.\nA gift certificate for either Eddie B's White Spruce or Black Forest Pub n Grille.\n\u2022 3:30 PM: Art Exhibition - ONE HOUR SHOWDOWN between Brian Defferding and another artist (Tim Sievert, hopefully)? Winner will be voted on by the group's applause and will receive a prize.\nClick on below image for more information.\nTwo nights lodging, 1\/2 day fishing with a guide, while the other half visits Artists Galleries.\nCertificates for a local restaurant.\nBasket with a Bottle of local wine and other local products.\nA Masseuse to pamper you.\n2 hour Guided Pontoon boat tour on the Eagle River Chain of lakes.\nPackage Available From 06 May to 17 June 2016. Check again for the fall.\nDeposit of one half the cost at time of booking.\nTour a farm during Sugar Maple harvest time.\nSee if a cottage is available.\nCall 715 479 4011 for more details.\nClick on below image for more information about the festival.\nStarting at $256.00 for a room with a two beds (one Queen-size and a full-size bed) for 2 adults and two children.\nA day at the Northwoods Children's Museum.\nThis Package available all year.\nSunday, March 28th -- It's our Easter Bunny Sunday!\nTurkey & Traditions Days, This two-day event is always held the Friday and Saturday following Thanksgiving.\nCheck out Northwoods Children's Museum for other activities.\nEnjoy a weekend with some of your friends from United AirLines.\nBring some food to share with your fellow employees.\nRun up to the casino for a few hours of fun.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Putting those Lessons Learned from Google to Use!\nLast week, Twitter made changes to its Terms of Service including a provision that allows the company to \"leave the door open for advertising.\" Per Twitter co-founder Biz Stone, \"We'd like to keep our options.\"\nDespite the fact that advertising on Twitter appeared inevitable, Connectual Managing Partner Aaron Goldman says it makes sense that Twitter would change the ToS now rather than wait until the ads go live. \"That's a lesson learned from Facebook who changed their terms in tandem with implementing Beacon and we know the uproar that caused,\" he says.\nI didn't share a POV with MediaPost on what type of advertising Twitter might introduce although I've explored the topic of Twitter monetization from other -- non-advertising -- sources on my digital marketing blog.\n1. In-Text. Similar to Vibrant Media, keywords in tweets could be purchased by advertisers and hyperlinked to marketer websites or trigger video-plays on roll-over. Pricing here would be CPC for click-outs and\/or CPM or CPE (engagement) for roll-overs.\n2. Contextual In-Stream. Sponsored tweets could be embedded within the Twitter stream -- and therefore pushed out to Twitter apps like TweetDeck-- attached to \"regular\" tweets and targeted to keywords within the tweets. Eg, a Tweet about pizza could trigger a Dominos sponsored tweet and both tweets would appear in a stream together. The sponsored tweet would be clearly labeled as an ad and include the keyword that triggered it so there's no confusion. And these would have to be capped at a set number of ads delivered per user per day to limit the annoyance factor. Pricing could be hybrid CPM\/CPC\/CPA.\n3. Contextual On-Site. Similar to Google AdSense -- perhaps even using AdSense -- text ads would appear on the right rail of pages on Twitter.com with targeting based on keywords within tweets. Importantly, because a single Twitter stream may include tweets on a variety of topics, each ad would appear adjacent to the tweet that triggered it with the keyword highlighted. Some sort of design treatment would separate church and state but very clearly show the tweet aligned with the corresponding ad. These ads would be priced on a CPC using a Google-like quality score to factor in CTR.\n4. Sponsored Hash-Tags. Twitter could give its users the ability to use special advertiser hash-tags that would link to a branded Twitter page. Right now when you click on a hash-tag, it takes you to a search results page for that keyword\/phrase. Sponsored hash-tags would take you to a page branded for the advertiser with a column showing all tweets referencing the brand but include other elements like images, video, maps, contact info, etc. Almost like a Facebook fan page. Advertisers would then tag all their tweets with their sponsored hash-tag and encourage their followers to RT or use the hash-tag as appropriate when discussing their brand. To encourage participation, advertisers could run contests and pick winners from among everyone who used the sponsored hash-tag in a tweet that day or that week. Advertisers would be charged either a flat fee inclusive of a branded search-results page or a CPC based on clicks on sponsored hash-tags.\nAs you may have noticed, each of my suggestions falls in line with the marketing lessons learned from Google that I've laid out in my recent Search Insider columns -- \"Everything I Need to Know About Marketing I Learned from Google\" and \"More On: Everything I Need to Know About Marketing I Learned from Google.\"\n1. Relevance rules. Ads should always be targeted to the content of the tweets. And, over time, they should be tailored to individual users based on their behavior \u2013 what ads they click on, what topics do they tweet about most often, what registration data have they provided, etc.\n2. Don't interrupt. Twitter ads should never disrupt the experience. They should be there for people to interact with on their own terms.\n3. Tap the wisdom of the crowds. Using quality-scores to determine ad effectiveness and, in turn, placement and pricing will ensure the people have a voice in what ads they see. And enticing them with contests or other rewards for interacting with ads and brands will only stimulate the almighty WOM that advertisers are hoping for.\n4. Act like content. Ads should feel like tweets, be embedded in hash-tags or otherwise appear as familiar Twitter features.\n5. Test everything. Twitter should not just try one ad format. It should experiment with multiple models and see how the ecosystem -- advertisers, agencies, users, etc. -- responds.\n6. Let the data decide. The response of Twitter users -- via feedback thru tweets and feedback thru clicks -- should determine which formats to use and which ads to show within a given format.\n7. Mindset matters. Fortunately for Twitter, many people its platform to talk about brands they like and don't like and solicit input from others. Accordingly, the mindset on Twitter is often commercial \u2013 similar to Google and unlike Facebook, where the mindset is mostly social communication and entertainment. This should bode well for advertising effectiveness.\nThere's no doubt Twitter has a difficult task ahead when it comes to introducing ads into an environment that has been mostly ad-free. I say \"mostly\" because there's no shortage of brands who have set up self-promotion Twitter accounts.\nThat said, Twitter is showing signs that it understands this challenge and is willing to take things one step at a time. That first step, of course, was changing its Terms of Service. As I told MediaPost, that was a lesson learned from Facebook. What comes next is where the rubber meets the road. To that end, let's hope Twitter takes into account the lessons learned from Google.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Kota Kinabalu, Malaysia: The Borneo pygmy elephant may not be native to Borneo after all. Instead, the population could be the last survivors of the Javan elephant race \u2013 accidentally saved from extinction by the Sultan of Sulu centuries ago, a new publication suggests.\nThe origins of the pygmy elephants, found in a range extending from the north-east of the island into the Heart of Borneo, have long been shrouded in mystery. Their looks and behaviour differ from other Asian elephants and scientists have questioned why they never dispersed to other parts of the island.\nBut a new paper published today supports a long-held local belief that the elephants were brought to Borneo centuries ago by the Sultan of Sulu, now in the Philippines, and later abandoned in the jungle. The Sulu elephants, in turn, are thought to have originated in Java.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Every company can find, in the various celebrations held throughout the year, a great opportunity to increase the number of customers and therefore of sales, as well, they can also take advantage of these special dates to strengthen their bond with their clientele.\nOne of the most anticipated celebrations worldwide, both for clients and for companies, is Christmas. This is because this date has a special meaning and is closely linked to the end of the year. To create a Christmas atmosphere, you can use various decorations in all areas of the company, and this simple fact changes the attitude of customers and at the same time of the employees.\nAnother thing you can do is share a Christmas greeting either to write it down in a Christmas card, or post it on the website of the company or on any of the social networks. Up next we offer you a list of Christmas messages that will be of much use for you. Read them chose the one you like the most.\nShare some of these Christmas greetings that they will please everyone who receives them.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaufcm b/data_all_eng_slimpj/shuffled/split2/finalzzzaufcm new file mode 100644 index 0000000000000000000000000000000000000000..760ff859b9c95a69ec5255de1ce2bef43876ee4c --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaufcm @@ -0,0 +1,5 @@ +{"text":"Thank you for linking to the Homecoming Blog! My name is Fran Roach and I am the Special Events Overall. If you have any more ideas or suggestions for this year's Homecoming celebration, I would love to hear them at eventshomecoming@psu.edu.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"OSAM: Autorun Manager is a powerful tool for controlling autorun items.\nIs CD\/DVD image utility that can make and burn ISO files.\nManage CD recording, make ISO-files, bootable ISO, and more.\nFree ISO Creator to make ISO from any types of files and folders.\nDivX The Launcher is a soft wich help you to make an autorun.\nOkoker ISO Maker is a professional make and burn ISO tool.\nRT Windows ISO Maker is a handy tool to make making bootable windows iso files.\nIt allows to make backup copies of DVD movies as ISO files or to DVD folders.\nFree ISO Grabber is an ISO ripper program to rip ISO image from CD\/DVD discs.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Wall Street Tax Act would tax sale of stocks, bonds and derivatives, and exempt short-term debt and IPOs.\nEleven U.S. representatives and four U.S. senators are co-sponsors of new legislation that would tax financial transactions in order to curb high frequency trading and begin to address economic equality in the U.S. The Wall Street Tax Act, according to primary sponsors Rep. Peter DeFazio (D-Ore) and Sen. Brian Schatz (D-Hawaii), could generate $777 billion in new revenue over ten years.\nBill's provisions. The bill calls for a tax on the sale of stocks, bonds, and derivatives at 0.1 percent, or 10 basis points, so a stock trade of $1,000 would incur a tax of one dollar. The tax would apply to the fair market value of equities and bonds, and the payment flows under derivatives contracts. The proposal includes an exemption for initial public offerings and short-term debt, which is defined as debt having a maturity of fewer than 100 days.\nThe bill is similar to ones that DeFazio introduced in the last two Congresses, with the amount of the tax lowered from 0.3 percent to 0.1 percent in the current version. DeFazio said that high-frequency traders front-run the market and drive up prices for individuals, pension funds and other value investors. In his view, the new legislation will curb unnecessary speculation and generate much-needed revenue to help the federal government fund national priorities.\nReducing risk. DeFazio and Schatz said in a news release that by increasing transaction costs slightly, the bill will help redirect investment that has poured into transactions without economic value into more productive areas of the economy. In their view, the bill also will also reduce the risk of financial crashes and limit the risks that high-speed arbitrage pose to the financial system.\nThey noted that the wealthiest 10 percent of Americans own about 85 percent of stock market wealth. The bipartisan Tax Policy Center estimates that the proposed tax would only apply to the highest earners in the country, with almost half of those affected belonging to the wealthiest one percent.\nThe Wall Street Tax Act is supported by a number of organizations, including Americans for Financial Reform and the Take on Wall Street campaign. A representative of Take on Wall Street applauded the proposed legislation, saying that if working families have to pay sales tax on nearly every purchase, Wall Street traders should have to pay sales taxes when buying millions of dollars in stocks.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Banner Stands - Hightech Signs, Inc.\nPop-up banner stands are a convenient way to display banners. Hightech Signs offers many styles and sizes for both indoor and outdoor use. Banners stands are portable and easy to assemble, perfect for trade shows and conventions. Our banner stands work great alongside a full wall display.\nWe also carry feather stands, sometimes called flying banners or beach flags, these work with custom printed banners and are popular for both indoor and outdoor use. These stands have the ability to bend and rotate with the wind, which avoids the need for a weighted base. The eye-catching movement makes feather stands a popular outdoor marketing option.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"For whom do I fight?\nMust I fight my King?\nAnd thus it all began ... \"It was a curious field of battle for the opening of the Revolution: 20 miles of winding hilly road connecting a port city and several country towns\" (quote from U.S. National Park Service brochure titled \"Minute Man National Historical Park, Massachusetts.\").\nBattle of Lexington. 19 April 1775 British Red Coats fired on local (\"rebel\") militia in Lexington, Massachusetts in the first overtly hostile act. Later, at the Battle of Concord, the \"rebels\" sent the Red Coats into full retreat, and fired at them from behind every tree and boulder! Herein lies the tale as seen, told, and experienced from South Carolina. Many hesitating Georgians and Carolinians made up their minds after the Battle of Lexington and blood was spilled. As long as there was a chance for peace, it was a large step to begin a rebellion, but the choice was made for them by the British. When the news arrived that fighting had actually begun in Massachusetts, a number of men broke into the powder magazine in Savannah and took six hundred barrels of powder.\nNorth Carolina, April 1775. In April, 1775, the people of North Carolina drove the royal governor out of the colony, and in May the citizens of Charlotte, Mecklenburg County, North Carolina, declared the British government suspended. Part of the British plan for 1776 included the invasion of North Carolina under General Clinton and Sir Peter Parker. There were many Scottish highland royalists (aye, and more is the shame for those of us with Scottish blood!) in the colony, who hastened to raise a force to cooperate with the British. Why on Earth they did this, with all the Scottish hatred of the British, I simply do not understand. About 1600 of these loyalists under Donald MacDonald marched down the Cape Fear River, and on 27 February 1776 at Moore's Creek Bridge, fiercely attacked 1000 patriot riflemen under Colonels Caswell and Lillington. Despite the odds, patriots won a brilliant and overwhelming victory, capturing several hundred of the loyalists. Just deserts. Sir Peter Parker decided the locals were not particularly friendly, could give a golly gee whiz about his \"Sir,\" and he kept right on going to South Carolina! We meet him again in the defense of Charleston June 1776.\nSouth Carolina: The Provincial Congress committed the country to war, and adjourned to re-assemble on the 20th of June 1775. But the battle of Lexington intervened, removing any hope of a final reconciliation with the mother country. The Provincial Congress instead met on the first of June 1775.\nFrom Simms' 1844 Book, The Life of Francis Marion, is quoted a contemporary letter from Isaac Marion, a brother of the Swamp Fox, who dwelt at Little River, the Northern boundary of South Carolina. It was written to accompany the express, which brought the news of the battle of Lexington. The following is the letter of Isaac Marion, addressed to the Committee of Safety of Little River.\nBoundary, May 9, 1775, Little River.\nYour obliged humble servant, &c.\nThe members of this body passed the following act of association, with considerable opposition (emphasis by FOC): \"The actual commencement of hostilities against this Continent by the British troops, in the bloody scene of the 19th of April last, near Boston -- the increase of arbitrary imposition from a wicked and despotic ministry -- and the dread of insurrections in the Colonies -- are causes sufficient to drive an oppressed people to the use of arms. We, therefore, the subscribers, inhabitants of South Carolina, holding ourselves bound by that most sacred of all obligations, the duty of good citizens to an injured country, and thoroughly convinced, that, under our present distressed circumstances, we shall be justified before God and man, in resisting force by force -- do unite ourselves, under every tie of religion and honor, and associate as a band in her defence, against every foe -- hereby solemnly engaging, that, whenever our Continental and Provincial Councils shall deem it necessary, we will go forth, and be ready to sacrifice our lives and fortunes to secure her freedom and safety. This obligation to continue in force, until a reconciliation shall take place between Great Britain and America, upon Constitutional principles -- an event which we most ardently desire. And, we will hold all those persons inimical to the liberty of the Colonies, who shall refuse to subscribe to this association.\"\nSecond Provincial Congress. In July, 1775, a second Provincial Congress was called.\nFall of the Royalist Government in Georgia, August 1775. Believing that they at last they had the support of the majority of the people, the leaders in Georgia proceeded to take control from the royalists. In August 1775, the royalist officers of militia were dismissed, the port of Savannah was closed to British ships, and in December all the courts were taken over by a Council of Safety. This Council was a body of fifteen leading republicans, created by a public meeting held at Savannah in June 1775. To complete the destruction of the royalist government, in January 1776, (British) Governor Wright was arrested by (Patriot) Major Joseph Habersham. He was confined to his house under guard, but in a few weeks escaped to one of the king's vessels in the harbor.\nCharleston: First British Invasion of the South, 1776. South Carolina, ever the firebrand state, acting under the lead of Henry Laurens, seized the royal arsenal and munitions of war long before the news of Lexington and Concord had reached that colony. Some months later, the royal governor withdrew from the colony and John Rutledge was vested by the Colonial Assembly with the power of Governor. Under Rutledge 5000 Carolina militia were enlisted for the defense of that colony. Charleston, its leading seaport, was then one of the most prosperous American cities. In June 1776, British commanders Clinton and Parker prepared to reduce it to submission. In the meantime, however, Colonel William Moultrie, with a kind of careless skill, had constructed fortifications of palmetto logs at the southern end of Sullivan's Island, at the entrance to the harbor of Charleston. Palmetto, for those of you no longer are familiar with our fair state, is a funny looking tree that thrives near the coast with a fuzzy trunk that I always thought of as looking like some kind of palm tree. Here was placed a defending force of about 1200 men, while several hundred men and a battery (cannon) were sent to defend the northern end of the island against a possible landing party.\nDefense of Charleston, 28 June 1776. The British attacked on the morning of 28 June 1776, and bombarded Fort Sullivan (the palmetto log construct described above) from a fleet of war vessels, aided by a landing force of several thousand men under Sir Henry Clinton. The landing party was promptly repulsed by the guns of the battery at the north end of the island. The British fleet, however, began a heavy ten hour bombardment. The Americans returned fire causing great damage, while the shots of the British bounced off or sank almost harmlessly into the soft spongy, but strong enough!, palmetto logs. As at Bunker Hill, the patriots had a meager powder supply, but used it to good advantage, firing with the cool precision of trained soldiers, and did terrible damage to the British ships. The flagship of Admiral Parker was made a wreck, he himself was wounded, and only one of the other nine vessels was immediately able to put to sea after the battle. No more disastrous defeat of a British armament is reported in history (take that Admiral Reuyter!). During the hottest part of the bombardment, the flagstaff on Fort Sullivan was cut down and the flag fell outside the walls of the fort. Sergeant William Jasper (after whom Jasper County was later named) leaped down, secured the flag, attached it to a sponge staff, and replaced it on the ramparts. He was later offered a lieutenant's commission in recognition of his inspiring act, but he modestly refused it. Lord William Campbell, the expelled royal governor of South Carolina, while on the flagship of Sir Peter Parker, was mortally wounded. The South Carolina Historical Society has an excellent description of the Battle of Sullivan's Island (off this web site, use your back arrow to return). And another South Carolina Historical Society web site devoted to the Battle of Sullivan's Island (off this web site, use your back arrow to return).\nIn addition to the damage to their ships, the British lost 200 men in the conflict, while the American loss was but 12 killed and 25 wounded. The British ship Aciwon ran aground on the site of present day Fort Sumter and was burned by the the victorious Americans, not, however, before its guns were first redirected at the other British ships. Just as the engagement at Bunker Hill and the evacuation of Boston freed New England from British armies, so the successful defense of Fort Moultrie (so named after the battle, this is the same palmetto log construct) for a period of three years saved South Carolina and the other southern colonies from invasion. This first decisive patriot victory preceded the Declaration of Independence but a few days, and, becoming known at about the same time, began things on a strong footing. It unfortunately was not to remain so.\nSimms' (1844) Description of The Battle of Fort Sullivan in the Defense of Charleston: With the formation of new regiments, under the resolves of the Council of Safety, Francis Marion, newly promoted to a Major, was ordered with his Regiment, under command of Col. Wm. Moultrie, to take post at Fort Sullivan, on the island of that name, which stands at the entrance of Charleston harbor, and within point blank shot of the channel. Fort Sullivan, better known as Fort Moultrie, was yet to be built, it was only an outline upon the sand. Palmetto rafts lay around it, waiting to be moulded into form. The structure was a simple wall, behind which young beginners might train guns to do mischief to a veteran enemy in front, the vaunted British Navy. Its form was square, with a bastion at each angle, sufficiently large, when finished, to cover a thousand men. It was built of logs, laid one upon another in parallel rows, at a distance of sixteen feet, bound together at frequent intervals with timber, dovetailed and bolted into the logs. The spaces between were filled up with sand. The merlons were walled entirely by palmetto logs, notched into one another at the angles, well bolted together and strengthened with pieces of massy timber. Such was the plan of the work; but, with all the diligence of the officers, and all the industry of the men, it remained unfinished at the perilous moment when a powerful British fleet appeared before its walls. The defence was confided to Col. Moultrie. The force under his command was four hundred and thirty-five men, rank and file, comprising four hundred and thirteen of the Second Regiment of Infantry, and twenty-two of the Fourth Regiment of Artillery. The whole number of cannon mounted on the fortress was thirty-one, of these, nine were French twenty-sixes; six English eighteens; nine twelve and seven nine pounders. The unfinished condition of the fort, really presented little more than a front towards the sea, literally a facade.\nOn the 20th of June, 1776, a day ever memorable in the annals of South Carolina, the nine British ships of war commanded by Sir Peter Parker, drew up abreast of the fort, dropped anchor, with springs upon their cables, and commenced a terrible bombardment. The fire of the enemy was promptly answered, and with such efficiency of aim as to be long remembered by the survivors. Having but five thousand pounds of powder, with which to maintain a conflict that raged for eleven hours, with unabated violence, it became necessary, not only that the discharge from the fort should be timed, but that every shot should find its target. In order to do this the guns were trained by the field-officers in person; hence, perhaps, the terrible fatality of their fire. The Bristol, a 50 gun ship, commanded by Commodore Sir Peter Parker, lost 44 men killed and thirty wounded. Sir Peter himself lost an arm. The Experiment, another 50 gun ship, had 57 killed and 30 wounded.\nTo these two vessels in particular, the attention of the fort was directed. The words, passed along the line by officers and men, were -- \"Look to the Commodore -- look to the fifty gun ships.\" The smaller vessels suffered comparatively little. Their loss of men was small. The injury to the vessels themselves was greater, and one of them, the Acteon, ran aground, and was subsequently burnt, but not before it's guns were directed at the enemy. The Carolinians lost but twelve men killed and twice that number wounded. One of the former was the brave Macdonald. When borne from the embrasure where he received his mortal wound, he cried out to those around him -- \"Do not give up -- you are fighting for liberty and country.\" The want of powder was severely felt. But for this, judging from the effects of the fire from the fort, the British Commodore himself would have been killed, or his fleet would have been destroyed. So slow, at one time, were the discharges -- so great the interval of time between them, -- that the British were of opinion that the place was abandoned. But a new supply of powder was obtained by Marion, who, with a small party, leaving the fort, proceeded to the armed schooner Defence, lying in Stop Gap Creek, and seized upon her powder, by which the fire was kept up until a supply of five hundred weight was received from the city.\nThis new powder supply caused a renewal of the conflict in all its fury. The garrison fought with a coolness which would have done honor to veterans. The day was very warm, and the men partially stripped in response. Moultrie states that when the action begun, some of the men took off their coats and threw them upon the top of the merlons. He saw a shot take one of them and throw it into a small tree behind the platform. It was noticed by our men, and they cried out, \"look at the coat!\" A little incident that speaks volumes for their coolness. Moultrie himself and several of his officers smoked their pipes during the action, only removing them when it became necessary to issue orders. In the hottest fire of the battle the flag of the fort was shot away, and fell without the fort. Jasper, one of Marion's men, instantly sprang after it upon the beach, between the ramparts and the enemy, and binding it to a sponge staff, restored it to its place, and succeeded in regaining his own safety. The coolness, the cavalier indifference, displayed by the Carolinians throughout the combat, is not its least remarkable feature. There is something chivalric in such deportment, which speaks for larger courage than belongs to ordinary valor. Mere bull-dog resolution and endurance is here lifted, by a generous ardor of soul, into something higher than a mere passive virtue. The elasticity of spirit which it shows might be trained to any performance within the compass of human endowment.\nDeclaration of Independence Signed. And I bet you thought this started it all! The Provincial Congress met again in January, 1776 to chose delegates to the Continental Congress to sign the Declaration of Independence on 4 July 1776. The signers of the Declaration of Independence almost all suffered enormously before this war was over.\nSummary. After the Battle of Lexington the hesitation of the majority of Georgians and Carolinians about entering the revolutionary movement ended. The patriots organized, held meetings and Provincial Congresses, and in the summer of 1775 overthrew the royalist government at Savannah, by arresting the governor and expelling the royalist officers from the militia. They then sent delegates to the Continental Congress, at Philadelphia, and the delegates there signed the Declaration of Independence. North and South Carolina likewise successfully threw out the British.\nCampaigns in South Georgia 1778-1783. In the early days of the war, the military activity of Georgia was directed against Florida. Florida had been ceded to Great Britain in 1763. St. Augustine was strongly fortified and again threatened the safety of Georgia and South Carolina. The southern border of Georgia was constantly raided by bands of British, Indians and royalists. Three separate expeditions were organized against St. Augustine, and all ended in failure.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawkpd b/data_all_eng_slimpj/shuffled/split2/finalzzzawkpd new file mode 100644 index 0000000000000000000000000000000000000000..622862c83b0dbd23d79669c6aa0ec6ea381fb67d --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawkpd @@ -0,0 +1,5 @@ +{"text":"I don't have a big budget but I can offer future projects for sure.\nHello client! I am an expert in this major with profound knowledge and skills in writing a marketing plan and I would like to help on this paper. Hire me for quality work and prompt delivery.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Getting an LLM in Chicago places you in one of the world's great cities in which to study law. As the home to hundreds of law-related organizations, from governmental agencies to major law firms to advocacy groups, taking an LLM in Chicago provides many opportunities for students to experience the law in action.\nAn LLM in Chicago also offers students a wide range of entertainment and cultural activities, from the Art Institute of Chicago and the legendary downtown blues clubs to the Chicago sports scene, which includes the Cubs, White Sox, Bulls, Bears and Blackhawks. An LLM in Chicago offers something for every taste.\nChicago has two universities offering LLM programs for those wishing to take an LLM in Chicago.\nOpened in 1902, the University of Chicago Law School is one of the premier law schools in the United States. For those taking an LLM in Chicago, the University of Chicago Law School offers two graduate degrees at the masters level: Master of Laws (LLM) and Master of Comparative Law (M.Comp.L.) While most students elect to receive the LLM, the M.Comp.L. will be awarded at the student's option if the same degree requirements are met.\nThe Northwestern University School of Law is a private law school that is also located in Chicago. The law school was independently founded in 1859 and is well-regarded, particularly in international law. The nine-month general LLM program offers graduates of foreign law schools interested in an LLM in Chicago an opportunity to expand their knowledge of American law and legal processes, continue their studies in international law, and engage in comparative legal research.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Photoshop elements, specifically the organizer app needs a way to Batch Rename by just changing the Metadata like is done in the Photo Downloader>Advanced Rename when importing files from an external device.\nAs it is now, other than a simple Common Base Rename, to do any other kind of renaming of files in bulk, one needs to use Process Multiple Files in the editor which requires resaving the item, which in the case of jpegs is destructive and may result in loss of photo\/document quality.\nIn PSE, you have three situations in which you can batch rename.\n- in the downloader 'advanced' option. As shown in your screeshot, it's the more powerful tool.\n- in the editor, with 'process multiple files'. Less options, but nearly always, you only rename copies in another folder, so no risk of quality loss.\n- in the 'export as new files' of the Organiser, with only the very limited renaming option you are mentioning.\nOf course, all Elements users would love to have the best option of the downloader available for the other two situations. That has already been requested, I don't remember when... But is that going to happen?\n... I would use the downloader to reimport the files. I would first select my files (perhaps in an album?) and export them temporarily to a card, then reimport them in a new folder in a new empty catalog. I'll give it a try later.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"CAMBRIDGE -- The Actors' Shakespeare Project is on a roll. This season's production of ''King Lear\" with Alvin Epstein is heading to New York, while the home fires are ablaze with a production of the rarely performed ''All's Well That Ends Well\" that could turn the most hardened Shakespeare-phobe into a phile.\nPhat chance, you say? Rest assured: This is Shakespeare the way it's supposed to be performed. The troupe plays to the crowd hilariously, speaks Elizabethan verse beautifully, and posits a smart interpretation of the play.\nDirector Benjamin Evett, who also runs the company, adds salacious clowning and old English songs (which sound like an early '60s Joan Baez playlist) to a love story that veers between Elizabethan notions of romantic love and contemporary fears about obsessive love, a la ''Fatal Attraction.\"\nOf all the delights in this production, it's mandatory to start with John Kuntz. He's in classic form with his libidinous take on the clown, Lavatch -- groping Paula Plum's breasts at any opportunity, coming out into the audience to flirt lasciviously.\nAnd the clowning is what Kuntz does in his spare time. He also plays one of the romantic leads, Bertram, the object of Helena's affections. He doesn't return her love but instead wants to cut a rakish swath through the female population of France.\nIt's here that Kuntz's acting and Evett's direction are particularly impressive. Kuntz makes Bertram a fairly sympathetic character rather than a young cad. After Helena (Jennie Israel) cures the King of France from his ailments, he tells her she can have anyone she wants as her mate, and she opts for the high-born Count of Rossillion, Bertram.\nHis ''Who, me?\" reaction is conveyed smartly by Kuntz. That he eventually accepts her is often seen as a traditional happy ending, though Evett, Kuntz, and Israel provide a more ambiguous spin.\nIsrael's portrayal of Helena adds to the sense of mystery. Her quivering voice and heaven-bent eyes bespeak a certain formality that seems somewhat out of place with the more relaxed style of acting of her colleagues. On the other hand, the Shakespearean poetry comes across best when she is speaking, and perhaps some of her formality is meant to convey her obsessiveness with Bertram.\nAnother aspect of the company is that it's always on the move. Here Evett makes excellent use of the tight Cambridge Family YMCA Theatre space. White banners with blue emblems offer all the backdrop that's needed as the actors move in and out fluidly, sometimes changing onstage. It isn't a short production, but you probably won't be tempted to look at your watch.\nThe music adds to the charm -- the actors are very nice singers -- though I would have opted for solo piano rather than piano and violin. But the production lives up to its title. After this year's ''King Lear\" and ''All's Well That Ends Well,\" Actors' Shakespeare Project has to be listed as a local treasure.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Please login or register to add your review.\nDeily is a play on the Latin word for God. The mission of Deily is to facilitate the sharing our faith and understanding, amongst all religious sects in order to learn more about ourselves and one other, making the world a better place.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawrax b/data_all_eng_slimpj/shuffled/split2/finalzzzawrax new file mode 100644 index 0000000000000000000000000000000000000000..a86ef96dc35bc3b5b5d975560cc92efd3ed79fb2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawrax @@ -0,0 +1,5 @@ +{"text":"The DCISD school board met in special session on Monday night at 5 pm to consider new hires, resignations\/retirements, a reassignment, and maybe the most important item, to consider a new Athletic Director.\nThere were no questions concerning the new hires until the name of a possible new head baseball coach came up. New board member Cody Williams questioned the hiring of any new coach until an athletic director was in place to recommend an individual. Robbie Underhill was quick to agree with Williams. However, Jess Roberts, who was a former athletic director before his move to Denver City, noted it was getting late in the year and felt like, based on the recommendations of superintendent Gary Davis, there should be no issue in the hiring of any new coach.\n\"Timing is not the best as far as AD issues or the hiring of new coaches,\" Roberts said.\nThis led to several minutes of discussion among the board members including Williams questioning the qualifications of the new baseball coach. Roberts assured Williams he had family in Ft. Stockton (the current location of the coach in question) and he comes highly recommended. Davis agreed.\nJunior high principal Billy Moore said he had interviewed the new coach for a teaching position on his campus and he felt like he was more than qualified for the position.\nBoard President Brad Woosley then suggested they should skip the approval of the new hires and the athletic director until they could address concerns during an executive meeting.\nBefore going into executive session the board unanimously approved the following resignations\/retirements: Rodney Curtis (high school science and baseball coach), Chad Dennis (swimming pool director and baseball coach), Jarryd Taylor (high school history and football coach), Marian Miller (Kelley Elementary), Regina Cummings (high school math aide), Ernestine Dominquez (Mustang Cafeteria), Laura Taylor (junior high teacher and coach), Sharon Scott (Elementary crossing guard), Steve Taylor (Athletic Director), and Lisa Landry (High School Theater Arts and One Act Play director). Carla Cruz was reassigned to a junior high ELA teacher.\nAt 5:20 the board went into Executive Session which ended at 7:47. When they returned to open session the new hires were unanimously approved after a motion from David Mitchell and second by Robbie Underhill. These included Mary (Katie) Huse (2nd grade Math\/Science), DeeAnna Carrasco (4th grade ELA), B.J. Green (DCISD Diagnostician), Sheri Parsons (Life Skill Aide in junior high), Sonya Sanchez (Kindergarten) Angelita Sainz (High School Custodian), Tina King (Junior High Counselor), Twyla Jones (4th grade Math), Steven Taylor (High School Social Studies, assistant football and baseball coach), Tracy Taylor (4th Grade ELA), and Chris Gonzales (6th grade Science, head baseball coach).\nThe board then unanimously approved a motion to open the Athletic Director's position to all applicants. According to the job posting on the school website it will remain open until filled.\nThe board will next meet in regular session on June 19 when action could be taken on the Athletic Director position. One local application is expected to be submitted for the position.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Have you wondered about m-Learning and the relationships of mobile phones to e-Learning? This class will explore pedagogical connectivity in urban or natural spaces around you for an introduction. We will use Google Earth and Maps to track environmental territories and conceptually overlay ways of spatial occupation documented through video, drawing, and text. In this context we explore ideas of the \"digital fl\u00e2neur,\" media presence, and discuss implications of mobility, social media connectivity, and Twitter-flux. This is an experimental framework open to app exploration and the extension of learning and assignment production in the environment. For those interested, work may be produced from smartphones; but mobile phone use is not a requirement. The class may be completed from laptop and\/or desktop computers.\nQuestions can be attached to the most recent post.\nSite Admin \u2022 Urban Drift \u2022 m-Learning with apps \u2022 Dennis Dollens Copyright \u00a9 2016 All Rights Reserved. Powered by WordPress MU.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Rural hotel nestled 50 metres from Izki Natural Park, surrounded by woods. It is located just 20 minutes from Vitoria-Gasteiz by car. It boasts a small spa, Wi-Fi in communal areas and a lounge with a chimney and pool table. In addition, it offers independent apartments equipped with kitchen facilities. The hotel also contains a restaurant which serves traditional cuisine.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"FUZHOU, China, Nov. 8, 2018 \/PRNewswire\/ -- Pingtan Marine Enterprise Ltd. (NASDAQ:PME), (\"Pingtan\" or the \"Company\"), a global fishing company based in the People's Republic of China (PRC), today announced its unaudited financial results for the third quarter and nine months ended September 30, 2018.\nOctober 2018: The Company announced that it will soon complete the modification and rebuilding project for its 27 new fishing vessels, and the first batch of 4 vessels have completed all inspections and the modification and rebuilding project. The new fishing vessels are expected to be put in operation in batches starting November 2018.\nNovember 2018: The first batch of 4 vessels that have completed the modification and rebuilding project just departed the port from Fuzhou on November 7 EST and will sail to the international waters of the Indian Ocean for operation.\nMr. Xinrong Zhuo, Chairman and CEO of the Company, commented, \"We were pleased to deliver strong operating and financial results during the third quarter of 2018. In early January this year, we announced that Pingtan has deployed two large scaled new squid jigging vessels into the international waters of the Atlantic Ocean; after months of operation, the two squid jigging vessels further enriched our product mix by providing Argentina squid and Peru squid products and increased harvest volume. Despite the average unit sale price decrease due to different sales product mix and sufficient market supplies this season, Pingtan was able to achieve an increase in sales volume by 279% and an increase in revenue by 168.8% for this quarter as compared to the same period in the previous year, which is an improvement in our operation performance. In late October, we were pleased to announce that we will soon complete the modification and rebuilding project for our 27 new fishing vessels and that these vessels are expected to be placed to designated water for operation in batches. The first batch of 4 fishing vessels that completed the modification and rebuilding project, all large scaled light luring seine fishing vessels, just departed the port from Fuzhou last night New York time for voyage and will sail to the international waters of the Indian Ocean for operation. The remaining 23 vessels shall be in service successively. As we entered the fourth quarter, management continues to observe market changes so that we may quickly adjust our sales strategy in preparation for the coming peak season during the Spring Festival in 2019. We expect to demonstrate improved performance in all aspects of operation in the future.\"\nAs previously disclosed in our Forms 10-K and 10-Q filed since 2015, in early December 2014, the Indonesian government introduced a six-month moratorium on issuing new fishing licenses and renewals so that the country's Ministry of Maritime Affairs and Fisheries (\"MMAF\") could combat illegal fishing and rectify ocean fishing order. In February 2015, Pingtan ceased all fishing operations in Indonesia. During the moratorium, the Company was informed that fishing licenses of four vessels operated through PT. Avona, one of the local companies through which Pingtan conducts business in Indonesia, and the fishery business license of PT. Dwikarya, the other local company through which the Company conducts business in Indonesia, were revoked. As a result and because license renewal was prohibited due to the general moratorium, all local fishing licenses of the Company's vessels in Indonesia are presently inactive.\nIn November 2015, the Indonesian government announced that the moratorium had concluded. As the MMAF has not implemented new fishing policies and resume the license renewal process, the Company does not know when exactly licensing and renewal will start. Since the Company historically derived a majority of its revenue from this area, this ban has caused a significant drop in the Company's production and financial results will continue to be adversely affected.\nIn September 2017, the Company was informed that the fishing licenses of 13 vessels deployed to the Indo-Pacific waters were suspended and the vessels were docked in the port by the Ministry of Agriculture and Fisheries of the Democratic Republic of Timor-Leste (\"MAF\"). The MAF alleged and investigated whether false statements were made during the licensing process and the vessels were simultaneously registered in Indonesia. The Company disputed these allegations and the government of Timor-Leste eventually agreed to release these vessels as no evidence was presented to support such allegations. The 13 vessels have returned to China for regular maintenance.\nAs of September 30, 2018, among the Company's 140 vessels, 12 are located in the Bay of Bengal in India; 11 are located in international waters; 13 have returned to China from the Democratic Republic of Timor-Leste due to the reason described above; and 27 vessels are in the modification and rebuilding project. The remaining 77 vessels were licensed by the Ministry of Agriculture and Rural Affairs of the People's Republic of China (\"MOA\") to operate in the Arafura Sea in Indonesia. The vessels in Indonesian waters, however, are not in operation because the licenses are currently inactive due to either the moratorium discussed above, the revocation of the fishery business license of the local entity through which the vessels operate, or, with respect to four vessels, the revocation of the local fishing licenses.\nRevenue increased by 168.8% to $14.7 million from $5.5 million as a result of increase in sales activities from recovery of production capacity.\nGross profit increased by 10,886.4% to $9.4 million from a gross loss of $87,004.\nNet income increased by 110.4% to $13.7 million from net income of $6.5 million.\nNet income attributable to owners of the Company was $12.6 million, or $0.16 per basic and diluted share, an increase of 111.7% from net income attributable to owners of the Company of $6.0 million, or $0.08 per basic and diluted share.\nRevenue for the three months ended September 30, 2018 was $14.7 million, increasing 168.8% from $5.5 million of the same period in 2017. The increase was mainly attributable to an increase in sales activities from recovery of production capacity. Sales volume in the three months ended September 30, 2018 increased by 279.3% to 5,827,555 kg from 1,536,369 kg in the three months ended September 30, 2017. Due to sufficient market supplies of fish products, however, the average unit sale price decreased by 29.2% in the three months ended September 30, 2018 as compared to the three months ended September 30, 2017.\nFor the nine months ended September 30, 2018, the Company's revenue decreased to $39.2 million from $46.8 million for the nine months ended September 30, 2017. The decrease in revenue was primarily attributable to the different sales mix and decreased average unit sale price, as compared to the same period in 2017. Sales volume increased by 26.5% to 15,893,704 kg from 12,566,227 kg in the nine months ended September 30, 2017. Average unit sale price decreased by 34.0% as compared to the nine months ended September 30, 2017. The decrease was due to the same reasons described above.\nThe Company's gross margin was 63.9% for the three months ended September 30, 2018, compared to (1.6) % in the prior-year period. The significant increase was primarily attributable to an increased number of deliveries into warehouse that averaged down selling cost and lowered the unit production cost of fish.\nFor the nine months ended September 30, 2018, gross margin increased to 56.0% from 28.7% for the nine months ended September 30, 2017. The increase was primarily due to the same reasons described above.\nGross profit for the three months ended September 30, 2018 was $9,384,636, representing an increase of $9,471,640, or 10,886.4%, as compared to gross loss of $87,004 for the three months ended September 30, 2017.\nGross profit for the nine months ended September 30, 2018 was $21,940,724, representing an increase of $8,481,267, or 63.0%, as compared to gross profit of $13,459,457 for the nine months ended September 30, 2017. The increase was due to the same reasons described above.\nSelling expenses were $0.3 million for the three months ended September 30, 2018, compared to $0.2 million in the prior-year period. The increase was primarily due to the increase in storage fees and shipping and handling fees as a result of an increased number of deliveries from Customs to warehouse, and larger warehouses were rented as more fish were delivered for inventory compared to the prior year period.\nFor the nine months ended September 30, 2018, selling expenses were $1.2 million, compared to $0.8 million in the same period of 2017. The increase was primarily due to the same reasons described above.\nFor the nine months ended September 30, 2018, general and administrative expenses were $8.3 million compared to $6.9 million in the same period of 2017, an increase of 19.9%. The increase was primarily due to the increase of $495,493 in salary for the crews on non-operating vessels booked as general and administrative expenses, an increase of $ 840,740 in depreciation of these vessels, an increase of $127,000 in insurance, and an increase of $126,000 in bank service charge.\nNet income for the three months ended September 30, 2018 was $13.7 million compared to net income of $6.5 million in the same period of 2017, an increase of 110.4%.\nFor the nine months ended September 30, 2018, net income was $18.1 million, compared to $27.2 million in prior year period, a decrease of 33.6%.\nFor the three months ended September 30, 2018, net income attributable to owners of the Company was $12.6 million, or $0.16 per basic and diluted share, compared to net income attributable to owners of the Company of $6.0 million, or $0.08 per basic and diluted share, in the same period of 2017.\nFor the nine months ended September 30, 2018, net income attributable to owners of the Company was $16.4 million, or $0.21 per basic and diluted share, compared to net income attributable to owners of the Company of $24.8 million, or $0.31 per basic and diluted share, in the same period of 2017.\nPingtan also announced that it will discuss financial results in a conference call on Friday, November 9, 2018, at 8:30 AM EST.\nTo listen to the live webcast, please go to http:\/\/www.ptmarine.com and click on the conference call link at the top of the page or go to: https:\/\/78449.themediaframe.com\/dataconf\/productusers\/pme\/mediaframe\/26971\/indexl.html. This webcast will be archived and accessible through the Company's website for approximately 30 days following the call.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Caleb Price shovels snow off the ice at the Arlington Square Ice Rink on Tuesday afternoon.\nThe National Weather Services said the powerful Pacific storm that came into the area Tuesday was expected to last through early Thursday.\nUp to three feet of much-needed snow is expected to fall in the mountains. Three years of devastating drought has left the Sierra Nevada snowpack \u2014 which counts for most of the state's water supply \u2014 at just 24 percent of normal for this time of year. In Carson City, the National Weather Service called for an 80 percent chance of rain today.\nSnow levels are running around 7,000 feet around Lake Tahoe this morning, according to the National Weather Service.\nAs of 9:30 a.m., almost one inch of preciptation was measured at the Carson City Airport in the past 24 hours.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayyyl b/data_all_eng_slimpj/shuffled/split2/finalzzzayyyl new file mode 100644 index 0000000000000000000000000000000000000000..b019c93091f7f04e099931d051cd126f2353b9bf --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzayyyl @@ -0,0 +1,5 @@ +{"text":"[S77] Social Security Death Index, Ancestry.com, (Name: - Provo, UT, USA: The Generations Network, Inc., 2006. - Social Security Administration. Social Security Death Index, Master File. Social Security Administration.: Social Security Administration. Social Security Death Index, Master File. Social Security A;), Number: 307-10-8718; Issue State: Indiana; Issue Date: Before 1951.\n[S447] Loyd Chester Hays, Death Certificate, Record Type: Death Certificate, Name Of Person: Loyd C. Hays, File No: 1980-526, (Name: 9 October 1980;).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This page displays sold house prices for Churchward Drive in Telford.\nChurchward Drive in Telford TF3 has of a mix of terraced houses, semi-detached houses and detached houses. Properties on Churchward Drive typically have values around \u00a3200,000 - \u00a3250,000, but larger detached houses can command upwards of \u00a3300,000.\nMap showing Churchward Drive in Telford.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"One of the key factors in this year's presidential election will be the state of the economy. Everyone knows it was bad when Obama took office. The question is whether he put into place the policies needed to help it recover. If the economy continues to lag through November, Obama's chances for reelection look dim. If, however, voters believe that an economic recovery is underway they may decide to give him four more years. That is why economic statistics such as GDP growth and the unemployment rate are so important. Unemployment is a lagging indicator. We would normally expect to see it fall well after economic growth had picked up. The significant decline in the unemployment rate over the past six months is therefore surprising. Does it signal that the economy is better off than we think?\nToday's jobs numbers were not good. On the surface 120,000 new jobs were created and the unemployment rate dropped to 8.2 percent. But a deeper look at the numbers shows significant problems. Although 120,000 new jobs is very good for an economy running at full employment its far below what we would hope for in a recovery, especially after such a steep recession.\nBut the government actually releases two surveys, one of households and one of establishments. The household report shows that the number of employed workers fell by 31,000. That points to a deeper mystery. According to the household survey, the civilian noninstitutional population (those 16 years or older and not in an institution or on active duty) grew by 169,000 in March. Despite the loss of jobs and the growing number of workers, the number of unemployed persons fell by 133,000. If none of these people got jobs, where did they go? They apparently left the workforce. The number of people over 16 not in the work force rose by 333,000 last month. As a result the labor participation rate fell slightly to 63.8. Yet the number of these people who currently want a job actually fell by 79,000. So while the establishment data point to a mediocre rise in jobs, the household data indicate that we lost jobs and that people may be leaving the workforce for an indefinite period. Some of these may never return. Others may be in school or taking time off. But it is not a good trend.\nThere is one very important caveat. The numbers above are adjusted for seasonal variations. If we use unadjusted numbers the number of employed workers actually rose by 728,000 the unemployed fell by 526,000 and those not in the labor force fell by 34,000. So the true picture depends a lot on the accuracy of the government's seasonal adjustments.\nalso, P. 2 is there supposed to be an \"it is' after employment?\nI don't understand the difference between adjusted and unadjusted figures.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"I got back late last night from the funeral of my wife's grandfather. I am exhausted and overwhelmed with work that needs to be done at work and home, but am working to get back into the swing of things with my blogging. I wanted to share with everyone a few things I learned while away.\n1. I love the south and never want to go farther north than Virginia (unless it is for a bear hunt). The funeral was in New Jersey. The press of people, the sense of urgency, and what I perceived to be rudeness was defeating. I know many wonderful people from the north, but I'm not sure how they deal with it!\n2. The North has quite a different view of the Civil War than the South. Not trying to be political, but just a funny observation. The Civil war monuments I took note of in New Jersey were dedicated to those who fought in the \"rebellion\". The monuments down here are dedicated to those who fought \"defending their land\". To the north it is the \"War of the Southern Rebellion\" to the south it it the \"War of the Northern Aggression.\" Makes sense that the two sides would see things differently, but I had never actually noticed it on memorials.\nCrappy cellphone picture, but you can see they call it the \"War of the Rebellion\"\nMy brother-in-law was complaining about how bad his shotgun kicked with 12ga 3 1\/2\" magnums when loaded with buckshot for deer. I tried to explain to him that 3\" magnums didn't kick as bad and were almost as effective. 3 1\/2\" magnum loads, when loaded with 00 buck, only have 3 more pellets and have the same muzzle velocity. \"No\", he said, \"You need them to get the extra range\".\nMagnum shotgun shells do not give you more range! They give you more pellets, making them more effective within their limited range. Some magnum shells actually have less muzzle velocity than their 2 3\/4\" counterparts.\nAnother common misconception is that animals cannot see a red light beam. Don't even try arguing this one! Hunters will think you are crazy! Red lights are more effective when night hunting because animals eyes are less sensitive to it and the red filter casts a subdued light instead of a harsh one. It is a myth that animals only see in black and white, they see a limited range of color much like a color blind person. There is debate among the scientific community about what colors they actually see, but it is generally accepted that most animals cannot see the color red well. Does this mean that everything that is red disappears? No, of course not. It just appears as a different color, most likely green or blue. So while an animal may not see a red light well, they can see it.\n4. Networking is the best way to find property to hunt on. I was invited to hunt on some private land in western Virginia. I didn't ask to hunt on it, it was offered to me after a couple hours of chatting about hunting. This made the difficult weekend well worth it for me.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The iiNet copyright case was decided this week. It concerns the copyright liability of an Internet service provider (ISP) for the conduct of its customers.\nI liked iinet response that the appeal was disappointing and that the legal action to date had not prevented one illegal download.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayzqh b/data_all_eng_slimpj/shuffled/split2/finalzzzayzqh new file mode 100644 index 0000000000000000000000000000000000000000..d538788a80cd8374ea4775c1082a44ffc5b24f9b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzayzqh @@ -0,0 +1,5 @@ +{"text":"pencil signed with butterfly, lower right margin; signed in plate, \"Whistler\", lower right.\ndated \"1859\" within plate, lower right. In pencil on verso: \"a 34644 W43\" and \"#4\"???\niii\/iv, before the vertical scratches were removed from the hat and face of the pipe smoker at the back.\nA rare impression of this etching, in the third state of four, annotated with the pencil \"butterfly\" signature and \"imp\", indicating that Whistler printed this impression himself.\nIt is most likely that it was drawn in an inn or public house, or in a local 'ordinary' (a cheap restaurant) in the Rotherhithe or Wapping area, where Whistler was working on the 'Thames set' etchings. Rotherhithe shows such a pub, the 'Angel' in Cherry Gardens, Rotherhithe.\n\"When his friends came,\" Mr Armstrong writes us, \"they dined at an ordinary there used to be (sic). People who had business at the wharves in the neighbourhood dined there, and Jimmie's descriptions of the company were always humorous.\" Mr Ionides drove down once for a dinner-party Whistler gave at his inn: \"The landlord and several bargee guests were invited. Du Maurier was there also, and after dinner we had songs and sentiments. Jimmie proposed the landlord's health - he felt flattered, but we were in fits of laughter.\" ' \"\n\"Longshoreman\" was printed in comparatively large numbers, considering that it was not published. Over forty impressions have been recorded. Impressions of the first and second states were printed in black ink on ivory 'antique' (pre-1800) laid paper. Most impressions of the third state were printed in black ink, on a variety of papers including buff; grey wove; and cream 'modern' (post-1800) laid; off-white laid; cream laid paper taken from a book. Quite a large proportion are on Asian papers, mostly cream Japanese and ivory Japan. Some of these impressions may have been printed and sold in the 1870s.\nThe final state was usually printed in black ink, again on a variety of papers. Some of these were probably printed in the 1890s or after Whistler's death, possibly for the New York print dealer Frederick Keppel (1845-1912).\nCited from: Margaret F. MacDonald, Grischka Petri, Meg Hausberg, and Joanna Meacock, James McNeill Whistler: The Etchings, a catalogue raisonn\u00e9, University of Glasgow, 2012, on-line website at http:\/\/etchings.arts.gla.ac.uk.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This may, or may not have actually happened. But Og is alive and well in modern Minnesota.\nI present as my evidence the fact that people \"don't expect a winter this bad\" when it hits.\nI have been around this neighborhood for over 90% of my life. Yet each year I am forced to deal with millions of others who appear to have slicked their memory banks from the previous year.\nSeriously? You have two missing buses out there in sub-zero cold and you're not aware of it? Nobody called in and said, \"Hey, Mikey, I'm broken down on the east side. Send help.\" Two of them off the radar and \u2026 well, I guess the MTC doesn't plan on winter being an issue.\nThe same lack of preparedness goes for a lot of hipster types I see downtown. No hat, no heavy coat, no gloves. They dash from their office to the bus when it pulls up and board without the encumbering Carhartt coat. The'd best hope they aren't on the bus that fails on my route.\nHere's a newsflash: the stuff keeps literally forever. A couple of the bags at the bottom of the pile are probably 10 years old.\nAlmost enough salt to make it through a Minnesota winter. This is my personal pile.\nThis wouldn't be a huge deal if it wasn't every year that the same mopes didn't shovel\/salt their sidewalk. This year, in some kind of bonus round to accompany the extreme cold, several downtown buildings (worth hundreds of millions of dollars) have failed to shovel their sidewalks\/pedestrian plazas, and completely neglected the application of salt. Consequently, you are taking your life in your hands walking around the capitol city of Minnesota.\nI do have a solution: appoint me the Czar of winter in Minnesota. In order for you to get your tax refund, or occupancy certificate, I must be satisfied that you have a good hat for your head, a decent shovel\/snowblower (those broom things that just push the snow into the street will be outlawed), and 40 pounds of salt for every 50 square feet of sidewalk\/steps. That should cover you for the full year.\nIn addition, as Czar, if you wish to drive between September and June, you must have new wipers, a battery that tests properly, tread on your tires, cold weather washer fluid in the tank, and 1\/4 of a tank of gas at all times. You are also prohibited from operating your vehicle until the snow is removed from all windows, the hood, and the roof of your car.\nFailure to comply will result in your car being crushed while you watch. With your groceries inside.\nSee, all this prep stuff I do each year is easy as can be. I can be very helpful if you'll just let me.\nNow, if you'll excuse me, I have to go put on my long underwear and stocking cap: I've got a formal banquet to attend tonight.\nThis entry was posted in Uncategorized and tagged battery, bus, commotion in the pews, freezing, frostbite, ice, joseph courtemanche, julie, metropolitan transit commission, occupancy certificate, og, polar vortex, salt, tax refund, tread on tires, washer fluid, windshield wiper, winter czar. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"There might be a more detailed post coming but for now, just popping by to say I finished up all of the work early and hit publish on Blood & Bone Book Two: The Convergence Theory.\nMages & Werewolves, oh my!\n69000 \/ 80000 words. 86% done!\nI'm still pretty blown away by this word count. I know in the grand scope of writing it's actually not very impressive but for myself, it's pretty damn monumental. Prior to this story, the longest thing I'd ever written and come close to finishing hit 21,000 before going on hiatus.\nYou'll never find a married witch. Women are just smarter about these things. Occasionally you may run across a warlock trying to make a go of it, but this inevitably ends in disaster. Heavy magic users tend to share a common, and disastrous, personality type: narcissistic, self-absorbed and forgetful. And if you think they make terrible spouses, then the truth is, they make even worse mates. This is something every wolfcub knows, in his gut.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Transaction Data includes details about payments to and from you and other details of products and services you have purchased from us. We do not store card details.\nIn light of the services which we provide to our customers, it will often be necessary for us to use information about your health. Even in the broadest sense of the word, 'health' information is a special category of personal data which requires us to have an additional lawful basis for using this information. Where we use information about your health for the purpose of providing our services to you (or in preparation for us to provide services to you), the additional lawful basis on which we shall rely will be that such use is necessary in order for us to provide you with health care or treatment.\nAs a medical clinical we use your contact data, personal identity data, health data to enable us to provide healthcare or treatment.\nWe use transaction details (not stored card details) to allow us to recover debts due to us and marketing and communications to allow us to provide and deliver our services. We use your insurance details for the purpose of billing for treatment and consultations.\nWe do not send your data to any other party and it is only used within Southface Skin Clinic.\nRequest erasure of your personal data. This enables you to ask us to delete or remove personal data where there is no good reason for us continuing to process it.. Note, however, that we may not always be able to comply with your request of erasure for specific legal reasons which will be notified to you, if applicable, at the time of your request.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"LOS ANGELES -- UCLA jumped on the board early and added two more goals en route to a 3-0 win against Kentucky to advance to the round of 16 in the NCAA tournament.\nCaprice Dydasco scored at the 4:15 mark off an assist from Sam Mewis to give the second-seeded Bruins an early lead. The 1-0 score held through halftime and until Courtney Proctor doubled the lead in the 71st minute, assisted by Mewis and Chelsea Stewart. Dydasco added the final goal 10 minutes later, assisted by Proctor.\nDydasco's goals came on just two shots. Overall, UCLA took 20 shots and 12 were on goal. Kentucky had five, and forced Bruin keeper Katelyn Rowland to make one save.\nUCLA will meet Pac-12 rival Stanford in the round of 16 on Sunday. The Bruins won the regular-season matchup 2-1 in double overtime.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaznls b/data_all_eng_slimpj/shuffled/split2/finalzzzaznls new file mode 100644 index 0000000000000000000000000000000000000000..7b4f5c39b4d2122744c8a2420bb9e13816035385 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaznls @@ -0,0 +1,5 @@ +{"text":"Where's the evidence? There is NONE!\nthere was \"intelligence that regime forces took precautions consistent with chemical weapons use\" in the immediate prelude to the Ghouta attacks. He did not explain where that information had come from.\nThe White House has claimed to have obtained intercepted phone calls that provide further evidence against the Assad regime, and administration officials also told CBS News that intelligence agencies detected activity at known Syrian chemical weapons sites the week before the Ghouta attack.\nThere simply ARE \"plausible alternative scenarios,\" for example, the so-called rebels did it.\nWhat scale? What was the scale of the attack and why couldn't they have done it? Answer: no answer.\nMost Americans have concluded, with a high degree of confidence, that our government has lied to get us into wars before and is doing it again. I conclude with very high degrees of slightly varying confidence that the British ruling class is willing to fight to the last American.\n\"At this juncture, any belated decision by the regime to grant access to the UN team would be considered too late to be credible, including because the evidence available has been significantly corrupted as a result of the regime's persistent shelling and other intentional actions over the last five days,\" the official said.\nThis lovely little theme has been used by Kerry among others to explain any lack of evidence. Very Rumsfeldian epistemology here.\nSo, what we have is: a limited but growing body of highly sensitive intelligence which is believed with some degree of varying confidence.\nThe BS is deep this time, and while Americans might get fooled once, or twice, or 50 times, this might be the \"red line\" we've been waiting for.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"History is an interesting topic to discuss and to learn more about. Therefore there are many people that like to learn more about at least one aspect of history that holds their interest. But that data is not always easy to gain access to. So if you're looking for some old records, where can you go to? One of the company that has put a lot of them online is Forces War Records. But what kind of information will you be able find in these records exactly? Will it help you on your search? And what will the people that work there be like? To find that out, the best course of action it to read the reviews of people that have searched through the website of Forces War Records before. They can tell you the things that you need to be aware of before deciding to access Forces War Records.\nForces War Records is a website that offers historic records. They have various UK offices that contain a total of 70 employees. The company was founded by a group of people that wanted to find some more information about their ancestry and their ancestors themselves, but were not able to find what they were looking for in the already exiting public records and genealogy sites. They offer an online base of all kinds of historic records. If you are looking for information about war casualties and other war records \u2013 for instance because one of your ancestors might have fought in that war- you might find what you are looking for in their fighter and bomber command losses, feel air arm data, medal rolls, individual battle data, prisoner of war records or other war casualties lists. For WWI of World War II you can take a look at the special records about these major wars. But they also offer records for other wars and battles like their African War database, Crimean War database, Fleet Air arm data, RN lossess, RFC lists, RAF lists, Royal Marines database, Indian Mutiny database and even Napoleonic War databases that include Waterloo.\nOn their website, you can access all of the records that they have put online. And in their online shop you can find medals, memorial scrolls, heraldry scrolls, but also CD's, DVD's etc. You can even hire a researcher.\nSo what is your opinion on Forces War Records ? How accessible do you find their website? Is it easy enough to find the records that you are looking for? Are the records and databases complete enough? Have you found some interesting facts and\/or figures that you might like to share? What do you think of their shop? And what are the people that work at Forces War Records like? Please leave your own review and help others to find out what kind of website Forces War Records really is.\nHow complete and thorough are the records of Forces War Records? Read reviews about their casualty lists, war databases, army records, history etc.\nI love my country, as I love my mother. FWR helped. me a lot to know more about our great warriors,our forces,wins etc as best as possible.\nThey have done wonderful work to compile relevant information for common nationals.\nYou should know great history of those warriors who died for us and nation unconditional.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"This week starts with a funny quote and then a serious quote by Abraham Lincoln. The FINAL installment of Money:Master the Game by Tony Robbins is discussed. Annuities and the financial product that will create an income stream for life is explained and the debut of a new financial product for those under the age of 50 with want this can participate in. Ray Dalio's advice on asset allocation is exposed as well! The interesting case is about impacted tooth numbers 17, 18 and 19 with a pretty cool radiograph which can be seen in the interesting cases section.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The motoscope tiny is a very compact, analogue 49 mm speedometer.\nThe speed is digitally measured and precisely displayed via stepping motor and indicator.\nThe sophisticated technology provides for a classic look without compromise. The dial completely does without display sections or LCD and has a convincing classical look. The ideal solution for purists and minimalists.\nThe speedometer is compatible with the 3 wires speed sensors as the M12 sensor and the universal sensor M12.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"A stunning mix of natural materials, the Kingston Table Lamp is a great way to add some bohemian inspiration to a modern setting. A black rattan disc houses a globe shaped vintage bulb that is included with this shipment and adds even more decorative appeal to the look. Hang this pendant over the dining room table for the perfect mood lighting. A black fabric covered cord adds to the modern look that is finished off with a black metal canopy.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazyqi b/data_all_eng_slimpj/shuffled/split2/finalzzzazyqi new file mode 100644 index 0000000000000000000000000000000000000000..d61691dab7376d481da90dbcbf65af32d8020696 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazyqi @@ -0,0 +1,5 @@ +{"text":"man cave decor ideas charming chandeliers man cave idea sports man cave decor ideas.\ncool room decor for guys guys room dorm room ideas guys guys room ideas guy bedroom ideas full size of cool room painting ideas for teenage guys.\nmodern rustic bathroom cool bathroom vanity mix of rustic and modern just need to find one with one sink that is as long modern rustic bathroom design ideas.\n5 foot bathtub freestanding tub 5 foot with jets ft 1 best 5 ft bathtub.\nfrench country decorating ideas modern french country decorating ideas french country living room ideas pictures of modern french with regard to french country decor design french country french count.\ncountry kitchen decor country kitchen decorating ideas on a budget.\noutdoor wooden bar cool idea for an outdoor wooden bar with an overhang outdoor wooden swivel bar stools.\nhome game room manly guys home game room ideas with pool table and bar home bar game room ideas.\nfire pit designs outdoor fire pit grill designs implementation of outdoor fire outdoor cooking fire pit ideas garden fire pit ideas uk.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The name itself, makes clear what TEO stands for \u2013 high quality products with a timeless yet unique design suitable for everyday life. In close collaboration with the Taiwanese manufacturer Etano Home, TEO creates long lasting products carefully made by valuable materials like mouth blown glass or hand spun metal. TEO offers lighting,furniture and home accessories, the collections vivid color and material range adds personality to your home.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"If you are looking for Maintenance Free, you must see the Somerset Villa in Easton Park. Built by Nardo Homes and READY FOR IMMEDIATE OCCUPANCY. They offer stunning elevations w\/ full porch or stone entry option. Every Villa comes standard w\/ full landscaping & sprinkler system, just the start of all the standard features. One step inside & you'll see what sets these homes apart. All homes will have 9' ceilings, can lighting, quartz countrs, & ALL appliances. As you tour the home you will see the attention to detail including 4 inch wide base trim, casing around all windows & built-in bench in mudrm. Garages will be fully finished w\/ insulation, drywall, paint & a pull down ladder for attic storage. Energy Saving Features include High Efficient Furnace, Blow-in insulation in the exterior walls & ceilings & LoE Double hung windows. Optional 2bdrm, 3bdrm & double master suite layouts ensure you will find a plan that works.\nI was searching for a Property and found this listing (MLS\u00ae #440311). Please send me more information regarding 2116 Cherrywood Lane, Chesterton, IN, 46304. Thank you!\nI'd like to request a showing of 2116 Cherrywood Lane, Chesterton, IN, 46304 (MLS\u00ae #440311). Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Gordon Arms, 107-109 Gordon Road, Fareham, Hants in 1973. Now closed and converted to a dwelling. Reported to have closed in 2013 and police searched for the body of a murder victim in the disused pub in 2014. The Ford Cortina Automatic is almost new and registered as ORV 336M so registered between August 1973 and July 1974. The car in the drive of the adjoining house is a Ford Anglia. There is a gap between the pub and the next property and a sign sending you down the alleyway to \"Saloon and Off Sales.\"\nFor commercial use of this photograph please visit www.alamy.com and search for RFFCBK but for a one off print, please see below.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"As part of the Sally & Seymour Olshin Adult Education Program, the Am Hasefer program organizes periodic lectures by notes teachers, Rabbis, authors and other relevant presenters. The lectures are well publicized and are open to the whole Great Neck Community.\nLook out for the schedule as announced by the Shul office.\nThe Kaddish, widely known as a prayer for the dead, does not speak of death. So why, then, is the Kaddish recited after the death of a close relative? And how did the Kaddish, a prayer of unknown origin, become one of the most famous and familiar prayers in Jewish liturgy? Author Leon H. Charney will shed light on the prayer's fascinating history.\nLeon H. Charney has enjoyed a distinguished career as an advisory to world leaders. He aided Israeli Prime Minister Golda Meier in bringing Soviet Jewish dissidents to Israel, and helped President Jimmy Carter broker the Camp David Accords. He is the author of three books and hosts The Leon Charney Report, a weekly, national television show.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbconr b/data_all_eng_slimpj/shuffled/split2/finalzzzbconr new file mode 100644 index 0000000000000000000000000000000000000000..758c227117bcb00400e51c155937339c23c08579 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbconr @@ -0,0 +1,5 @@ +{"text":"Today the Oldenburg Association or Verband has over 220 approved sires and 7000 mares in addition to the 96 sires and 1300 mares that are part of the \"Oldenburg International\" breeding program for show jumping. These figures make Oldenburg one of the largest studbooks in Germany. Oldenburg is the largest studbook in terms of breeding area.\nEach autumn, the Oldenburg Verband holds the \"Stallion Days\" in Vechta (the capital of the Vechta district in Lower Saxony, Germany) during which the young stallions undergo their licensing evaluation. After the results of the licensing are announced, many are auctioned off to new homes at stallion stations, or as gelding prospects bound for performance homes. The \"Old Stallion Parade\" occurs on the last day, showcasing all the fully approved, performance tested stallions. However, this event is not just a pageant, as the offspring of mature stallions are subject to intense scrutiny. The best stallions of their age class, based on their offspring, receive a \"premium\" or award for their achievements in breeding.\nThe Oldenburg Verband places special emphasis on mare lines, many of which trace back to the Alt-Oldenburg ancestors. Selected from the mare inspections throughout the year, the best young mares are invited to the Elite Broodmare Show in Rastede. There they compete not only for the States Premium \u2013 originally a bribe to keep breeders from exporting high-quality broodmares \u2013 but for the title of Champion Mare.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Put the needle on a pedestal! Embrace a little discomfort! Sing, \"Hail to the Chief\" to a next-generation dermal injectable named Radiesse that works wonders in softening facial lines.\nEvery time I opened a magazine, there were glossy advertisements jumping out at me for a myriad number of cosmetic products and procedures, claiming amazing results. The veritable smorgasbord of treatments available only served to confuse me, and, of course, enabled me to dither around and delay making an appointment to find out which one to choose.\nSifting through all of the options, I was forced to do my research (which is a good thing) to become familiar with brand names and understand precisely what they can do. Just because we hear a lot about one product, it might not be the one that benefits your area of concern. A product for your forehead might not work on your lips, so it is important to listen to your medical professional.\nIn my case, I was desperately feeling the need for Gorilla Glue to keep the corners of my mouth in an upward curve. Being a person who enjoys a good laugh, the years of mirth, fits of giggles, chortles, snickers, and to say nothing of smirks accompanying witty repartee, have taken their toll. Expressionless faces hold no charm for me, yet the growing chasms of smile lines running rampant down my cheeks, formed explicit parentheses around my lips, specifically gained from a lifetime experience of effortless beaming and grinning.\nFrankly, if it hadn't been for another family member I'd still have slack, droopy areas in my face. Once I'd seen the positive results of FDA-approved Radiesse, I couldn't wait to get a consultation and an appointment.\nDermal fillers restore a natural, more youthful look without major cosmetic surgery, and most of us just want to tweak nature and enhance our looks by softening those aging smile lines, nasty wrinkles around the nose and mouth and the deepening Nasolabial folds (they run from your nose to your lips).\nRadiesse is made of unique calcium-based microspheres suspended in a natural gel that is injected into the skin, just below the surface. Forming tiny \"scaffolds\" to support and stimulate your own collagen, it temporarily fills out facial folds and rebuilds the foundation of the skin.\nAh, the word \"temporarily\". How long will it last, you ask? It is a longer-term enhancement, but eventually the microspheres will break down and naturally absorb into your body. More than likely, it won't happen for a year or more, and that's enough to make you smile and enjoy the reprieve.\nFortified with a Tylenol and a good dose of apprehension, I found myself sitting in the medical office of Drs. Mark and Jeannine Koffard at Riviera Laser Studios (www.rivieralaser.com) in Dana Point. Even though I knew it was no big deal, my heart changed from a steady pitty-pat to a nervous boom-diddy-boom in a very short space of time. Nitrous Oxide (laughing gas) is offered for the worst part \u2013 the numbing injections - which are no worse than you get at the dentist. Sitting there with flabby, rubbery lips and trying to enunciate clearly was a challenge, although in no time at all, Dr. Mark was working hard at filling up those crevasses with shots I could not feel. He injected small amounts using a very fine needle \u2013 his weapon of mass construction - as I tried to remember to breathe. The lines at the corner of my lips needed a lot of work and we knew that I would be bruised and slightly tender. The numbness wore off gently and a light application of an icepack helped the swelling go down.\nYes, these are the side effects and, let's face it; my purple-hued chin wasn't a pretty sight for a week, yet the excellent results became more apparent as each day passed. Interested friends wanted digital images on a daily basis to monitor my progress, but I wasn't going that route \u2013 that's blackmail material.\nRadiesse stimulates the body to produce new collagen, so not only was I graced with instant gratification of facial volume; I knew that the results would only get better. The smile lines that made me frown were plumped up and much less noticeable, not, may I emphasize, erased. Achieving a natural look is what Dr.Mark and I had in mind.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The Great Barrier Reef exhibit is the grand finale of your Aquarium visit. The second largest tank at 700,000 litres, and by far the most colourful, the reef is holds over 70 species of fish. The tank is home to some fabulous characters, including Samson the grouper, Cooper the Humphead Wrasse, and three Zebra Sharks to name a few.\nCoral reefs like the Great Barrier Reef are incredibly important to the marine ecosystem as they are considered hotspots of biodiversity. The majority of the warmer waters of our oceans contain little life, due to the low abundance of plankton. Coral reefs are diverse habitats that provide homes and support food chains for many species. About one-third of all marine species live on coral reefs \u2013 the Great Barrier Reef alone houses over 1,000 species of fish.\nCoral reefs are under threat from a combination of pressures \u2013 increasing sea temperatures, acidification of the world's oceans, overfishing and pressure from tourism. Although large swathes of coral reef have been lost in the last 20 years, other areas are proving surprisingly resilient and there are numerous scientific and conservation projects that focus on finding ways to help the reef survive and thrive. The National Marine Aquarium is working with partners globally to contribute to reef conservation through raising awareness and finding ways that culturing coral in captivity can assist in understanding these fascinating and valuable animals.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Click Start > Administrative Tools > Management Consoles > Citrix AppCenter.\nIn the left pane, expand XenApp, expand the server and then click Policies.\nIn the Policies pane, click the User tab and then click New.\nIn Name, type a name for the policy and then click Next.\nUnder Categories, click All Settings.\nUnder Settings, in Auto connect client drives, click Add.\nClick Enabled, click OK and then click Next.\nUnder Categories, click All Filters.\nUnder Filters, in Access Control, click Add.\nIn the New Filter dialog box, click Add.\nIn Connection Type, select With Access Gateway.\nIn AG Farm, type the virtual server name.\nIn Access Condition, type or select the session policy name that is configured on NetScaler Gateway, click OK two times, click Next and then click Create to complete the wizard.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The Life & Songs Of Emmylou Harris black t-shirt (unisex) features an alternate screen printed logo from the show on the front and all of the artist names intertwined in the roots of a tree on the back. The Life and Songs of Emmylou Harris CD and DVD captures a once-in-a-lifetime concert celebrating one of the most influential women in music whose contribution as a singer and songwriter spans 40 years. Click here for full tracklist and samples.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdbnm b/data_all_eng_slimpj/shuffled/split2/finalzzzbdbnm new file mode 100644 index 0000000000000000000000000000000000000000..b958aa654528617fd47e4b571f8b7a0305f925ec --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdbnm @@ -0,0 +1,5 @@ +{"text":"On this date in 1922, Justice William Rufus Day retired as an Associate Justice of the United States Supreme Court. Justice Day was born on April 17, 1849 in Ravenna, Ohio. A graduate of the Univesity of Michigan School of Law, Day spent the next couple of decades of his career as a litigator in Canton, Ohio. He became friends with William McKinley who would later be elected President of the United States in 1896. When McKinley went to Washington, Day followed and served as Assistant Secretary of State and later as Secretary of State. In 1899, Day resigned and returned to Ohio where he was appointed to the U.S. Court of Appeals for the Sixth Circuit. On January 29, 1903, President Theodore Roosevelt nominated him for the U.S. Supreme Court. On the Court, Day was known to generally approve federal government regulation of transportation and selling, but not manufacturing, under the Commerce Clause. After almost twenty productive years on the Court, Day retired and accepted President Hardings appointment to the American-German Claims Commission. He only served on that Commission for about 6 months when ill health forced him to resign in May, 1923. Justice Day passed away on July 9, 1923 from lung congestion at age 73. Solicitor General John W. Davis once noted that Day's \"greatest value was in conference. His tact, his personal charm, his knowledge of the law, and his honest-mindedness were of extreme importance in discussions...\"","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Magna sweets vividly demonstrates how even the very last bottle can become a high-impact promotional messenger thanks to the BottleBag. Punched with a hole, the individually imprinted promotional bag made of white OPP laminate film that was distinguished in the category Give-aways can simply be hung over a bottle. What's more, containing fruit gums or optionally nuts, the item is also equipped with a small in-between meal snack. Whether for a trade show or event, conference or seminar, wellness hotel or trendy club \u2013 the BottleBag can be implemented wherever bottles are found, whereby it can also be hung over the door handle or handlebars of a bike. The large-surface advertising messages are guaranteed to attract attention.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Interpret \"Vox Luminis, Lionel Meunier\"\nfounded in 2004 in Namur, Belgium, is an ensemble specialising in the performance of 16th \u2013 18th century vocal music.\nThe ensemble has been praised for its seamless blend of high quality individual voices, exquisite tuning and clarity of sound. Critics have also commented on the ensemble's enthusiasm in sharing its passion for early music with an audience. The majority of the group met at one of the most significant centers for early music in Europe: the Royal Conservatory in The Hague.\nVox Luminis has performed in festivals and venues in Belgium (Nuits de Septembre de Li\u00e8ge, festival de Stavelot, Juillet Musical de Saint-Hubert, Automne Musical de Spa, MA festival Bruges, festival van Vlanderen Gent, Laus Polifoniae Antwerpen, festival des midi-Minimes, Et\u00e9 Musical de Roisin, Soci\u00e9t\u00e9 Philharmonique de Namur\u2026); France (festival d'Ambronay, festival de Saintes, Rencontres Musicales de V\u00e9zelay, festival de Saint-Michel en Thi\u00e9rarche, Midsummer festival d'Hardelot, festival Contrepoint 62, Festival Musique et M\u00e9moire de Luxeuil, Musique et Natures en Bauges, festival Bach en Combrailles\u2026); Germany (Ratingen Bachtage festival, Muziekfest Stuttgart); The Netherlands (Oude Muziek Utrecht, Den Haag, Delft); Portugal (Centro Cultural B\u00e9lem-Lisbon) and Croatia (Varazdin Baroque Evenings).\nVox Luminis records exclusively for the Belgian Label Ricercar. The ensemble's first CD, released at the end of 2007, features four vocal works by Domenico Scarlatti, including his famous 10-part Stabat Mater. In May 2010 the ensemble released its second album, Samuel Scheidt: Sacrae Cantiones, which features several world premieres. Vox Luminis' most recent CD, released in June 2012, created a buzz with an interpretation of Heinrich Sch\u00fctz's Musicalische Exequien. This recording has received several prestigious awards such as the Gramophone 'Recording of the Year', the 'Baroque Vocal Gramophone Award' and the 'International Classical Music Award'.\nOther recordings have also received many awards throughout the world including, Gramophone's 'Editor's Choice', Diapason d'Or, Choc de Classica, Muse d'Or Baroque, Preis der Deutschen Schallplatten Kritik, Joker de Crescendo and Prix Caecilia.\nVox Luminis is very grateful to have the support of the Federation Wallonie-Bruxelles and the City of Namur.\nwas born in France. He began his musical education in the music school of his town, Clamecy where he studied solfege, recorder and trumpet. Being passionated for music from his young age, he continued his studies in IMEP, Namur at the age of 18, where he gratuated with high distinction from recorder in 2004 as a student of Tatiana Babut du Mares. He had masterclasses with Jean Tub\u00e9ry, working at the same time with Hugo Reyne. The same year he started vocal studies in the Royal Conservatoire of the Hague with Rita Dams and Peter Kooij as his teachers.\nHe is the artistic director of the ensemble Vox Luminis, group that he created in 2004 with his friend pianist.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Creative, passionate and clever are some of the numerous qualities of the participants in Egypt's 2019 Pitch Competition, which is the first phase of The AIM Startup. The winners will participate in the AIM Startup in Dubai next April. The 2019 edition focuses on digitisation to empower startups and small and medium-sized enterprises.\nEgypt Pitch Competition was hosted by Creative Hub Egypt, an futuristic platform managed by the Industrial Modernization Centre (IMC). It offers services and spaces to designers, entrepreneurs, startups, and freelancers in order to share design knowledge, develop businesses, and spread innovation.\nFrom 21 startups, a jury consisted from the AIM Startup team and other experts, had to select five winners. The jury includes Programme Manager of Electronics Industry at ITIDA Yasser Abdel Bary, Entrepreneurship Development Senior Advisor at USAID SEED Project, Mohamed Sultan, Adjunct Professor at the American University in Cairo, Ramez Daoud, and of Entrepreneurship Centre Director at the Arab Academy for Science, Wael El Desouki Bedda.\nYasser Abdel Bary informed Daily news Egypt that the jury members were impressed with the work of the startups, \"they came well-prepared and have a clear plan to grow and profit. Yet, we had to choose five startups according to the AIM Startup's criteria\" he added.\nAIM Startup was launched in 2017 as an initiative of the UAE Ministry of Economy to connect promising startups with investors and business partners from other parts of the world.\nThe winning startups of this year tackled many issues such as a device that allows travellers to use their SIM card for making or receiving calls and SMSs with zero roaming fees; a recycling machine for beverage cans and plastic bottles, and a smart internet of things (IoT) platform that tracks garment production lines.\nDNE interviewed the five winners, and asked them about their next move, the obstacles they faced ,and what inspired to create their startups.\nAmr Elgalby, the founder of VoxEra, was travelling on a business trip to Georgia, where he needed to keep calling and sending SMSs to his clients in Egypt. Once he came back to the country, he had to pay a huge sum of money to cover his roaming fees. Elgalby is a voice over IP engineer, so he thought of a solution to reduce the fees, and ended up with VoxEra, the device with its four flashing indicator lights and connectivity ports, and looks a little like a home router. Once a SIM card is inserted into its designated rear slot, the VoxEra works its magic: all calls placed while the user is travelling are charged at domestic rates.\nAmr Elraiany, the business developer of VoxEra, rendered the presentation before the jury, as Elgalby was in China following the production process of the device. Elraiany told DNE that the biggest issue they had to deal with was manufacturing the device because it was the team's first experience in manufacturing a hardware.\nThe startup is now working on the second pitch to deliver their pre-orders. The next move is to expand the market targeting the Gulf area, which has a large group of foreigners. As a first phase, VoxEra accepted a startup in Chile, so the team will be in Chile for some months to further develop the product.\nElraiany highlighted that the startup at this point is handling the service very well and covering the production line, yet getting increased investments into the firm will give it a bigger and faster chance to enter more markets.\nGarment IO is a smart IoT platform tailored to boost the productivity of the garment manufacturing industry. It is a plug and play software and hardware solution providing garment factory managers and owners a complete overview on production processes, worker performance, and production cost calculations, all the way down to the transaction details, real-time, and is accessible online.\nThe Founder of Garment IO, Ahmed Nounou, previously had a garment factory, and when he met Sabae, they discussed the issues in the garment industry, and from there they got the idea for their startup.\nGarment IO helps garment manufacturers and factories track, analyse, organise, manage, and optimise their production processes. It provides the hardware and software as a service for a periodic fee that the factory pays per installed device.\nThe startup is only targeting the garment manufacturing industry, which is sized at $420bn in 2017 (excluding raw material), employing over 70 million employees worldwide. \"We are focusing on doing one thing better than others\" Sabae added.\nThe biggest challenge Sabae and Nounou faced was the hardware, especially importing the required electronics into Egypt, besides getting some approvals from a few government agencies. Sabae said that they will soon start selling pre-orders, \"we are already functioning in terms of hardware and software, and we are planning to quickly penetrate the market, and once we establish ourselves in Egypt, we will expand globally\".\nEpicVR designs and implements virtual reality arcade games, from the games themselves, operating software, exterior design and production, and required hardware. It sells the complete product to arcades and family entertainment centres.\nEpicVR's team are engineers who are passionate gamers and interested in virtual reality (VR), and when it was available for developers, they decided to make their business model of it, a profitable business out of it.\nEpicVR was founded in August 2015. Aiming to deliver this amazing experience to the masses, they started focusing on creating out-of-home arcade games that are sold to game centres in Egypt and the MENA region.\nDNE interviewed Ahmed Atif and Ahmad Amin, co-founders of the startup, who are focusing on their first potential market in the region, Saudi Arabia, \"we showcase our product in many exhibitions, but one of the biggest challenges we faced was connecting with the owners of entertainment centres, because our model is a B2B model,\" Atif said.\nThey managed to corporate with someone who works in the arcade industry since the 1980s, as they seek to gain clients who trust their product, \"we are now present in many centres, but the number of our team is not large enough, so we aim to expand our team\" Amin added.\nAnnoyed by the amount of rubbish in Cairo's streets, Islam El-Refaey, founder and CEO of ZeroPrime Technologies, thought of a way to recycle the rubbish, as he has a background in aerospace and mechanical engineering, and is interested in applying technological solutions especially for social causes.\nEl-Refaey, who established his startup 2017, worked on designing and implementing smart sustainable city infrastructures. His first solution for the scattered city rubbish was CanBank, which is a machine that gives users different rewards of their choice in exchange for their used beverage cans or plastic bottles. The machine is available in corporates, universities, clubs, and malls to encourage them to recycle.\nWideBot is the first Arabic-focused bot builder platform that gives a tool to businesses in the MENA to build smart chatbots without coding in a few minutes, while helping them to engage with their audiences by advanced artificial intelligence (AI) to maximise conversions and increase loyalty.\nMohamed Nabil, the CEO and co-founder of WideBot, is looking to attract more clients to his platform, \"we are working to improve our platform, by adding more Arabic slang terms and making it more accurate. We struggled during the developing process of our platform, because there is not a lot of data available in Arabic\" Nabil said.\nNext month, the tech startup will open its office in Dubai. Nabil believes they can build the most powerful AI tool that supports Arabic language, \"we need some investment in our project in order to improve the team's productivity, and hire more employees in our company\" he added.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Dark is the underworld. Darker are the lands of the Shadow Elves. Far underneath the rugged broken lands, even deeper than the deepest caverns of the trolls, live these pale faced warriors. Their law is that of their Immortal Rafiel, and their mystical strength that of their Soul Crystals. Together, they will guide the people of the shadows to the lands under the sun.\nGAZ13: \"The Shadow Elves\" (1990), by Carl Sargent and Gary Thomas, is the thirteenth book in the \"GAZ\" series of Gazetteers for the Known World. It was published in May 1990.\nContinuing the \"GAZ\" Sourcebooks. Like its immediate predecessors, \"The Shadow Elves\" is broken into two books. A player's guide gives considerable background on the Shadow Elves and also details some new character classes, while the GM's Book describes their history and the geography, flora, fauna, and NPCs of the area. Like two of the immediately previous Gazetteers, GAZ10: \"The Orcs of Thar\" (1988) and GAZ12: \"The Golden Khan of Ethengar\" (1989), \"The Shadow Elves\" once more described a more antagonistic and villainous land.\nUnfortunately, \"The Shadow Elves\" was the last racial sourcebook in the \"GAZ\" series and also the last sourcebook in this classic format, as the \"GAZ\" series was on the verge of ending. That's because higher numbered supplements tended to sell less for TSR, and \"The Shadow Elves\" bore the number 13. Beyond that, the \"GAZ\" supplements had now detailed the entire square of land which had been the line's focus for the last year or two; in fact, \"The Shadow Elves\" had to dig below one of the previous realms to detail something new!\nOnly one more \"GAZ\" supplement would follow, GAZ14: \"The Atruaghin Clans\" (1991), which appeared over a year later. However, TSR wasn't yet done with the Known World. They simultaneously were beginning a new series of sourcebooks, starting with Hollow World Campaign Set (1990), which detailed new realms far below the Known World.\nExpanding Basic D&D. Like all of the later \"GAZ\" modules, \"The Shadow Elves\" provided considerable expansions to the Basic D&D rules, including skills and new classes. The skills were somewhat increased in power from the early Basic D&D skills, as they included options like \"Disguise\" and \"Tracking\", which effectively provided shadow elves with new class skills \u2014 a trend that had begun in \"The Orcs of Thar\". The character classes included a shadow elf class and a shadow elf shaman \u2014 but unlike the shaman in \"The Golden Khan of Ethengar\" this one was just a spell-caster, not a spirit magician.\nLove It or Hate It? GAZ1: \"The Grand Duchy of Karameikos\" (1987) and GAZ3: \"The Principalities of Glantri\" (1987) are fairly universally recognized as the best books in the \"GAZ\" series, but \"The Shadow Elves\" may have gotten the most positive attention of the later books.\nExpanding the Known World. The land of the shadow elves lies below the Broken Lands of \"The Orcs of Thar\". Besides detailing that land, this Gazetteer also describes the culture of the Shadow Elves, who had been briefly introduced in GAZ5: \"The Elves of Alfheim\" (1988). Fans of the Known World like to make it very clear that the shadow elves aren't drow; though they're an underground elf race that's antagonistic to surface elves, they're not necessarily evil like drow are (usually). The shadow elves were the fourth major elf culture detailed for the Known World, following the elves of Alfheim, Glantri, and Minrothad.\n\"The Shadow Elves\" also provides the first major reference to the new Hollow World setting through an extensive description of the shadow elves' interactions with \"the land of the red sun\". a one-sentence aside in DDA1: \"Arena of Thyatis\" (1990) had preceded it in the previous month, while it would be followed by the release of the Hollow World boxed set itself, just two months later in July.\nLike the nearby regions of the \"Golden Khan\", the lands of the Shadow Elves also have deep connections to the ancient past of Blackmoor. The elves even follow an immortal named Rafiel, who was once a nuclear physicist in Blcakmoor.\nAbout the Creators. \"The Shadow Elves\" was written by two authors who are better known for their work in other settings. Sargent had burst onto the D&D scene in 1989 with contributions to four different books, but he'd earn his reputation with the From the Ashes books (1992-1993) for Greyhawk. Thomas similarly began writing for D&D in 1989, but had already gained attention for his work on Traveller through Digest Group Publications.\nWhere is Gaz 6 & 14? Where is Dawn of the Emperors?\nThis title was added to our catalog on October 14, 2014.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdipj b/data_all_eng_slimpj/shuffled/split2/finalzzzbdipj new file mode 100644 index 0000000000000000000000000000000000000000..4ce1a9f672bf9d99376bd36bc96ff2e84b495892 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdipj @@ -0,0 +1,5 @@ +{"text":"\"The Rebellious Rise\" centers around controversial court cases based on the most heated issues in America's history -- from the lawyers' viewpoints. Follow us on Facebook to keep up to date.\nThis series will be produced, written by a mix of races and genders. Our team spans from 24-70 in age, made up of women, men, transgender, and LGBT in front of and behind the camera. The stories address class, gender, equal rights of all Americans in the courtroom.\nWhen I was a little boy, I never thought about the vital role history played in my life. Every decision I make now is based on my past experiences.\nHistory is the witness that testifies to the passing of time; it illuminates actuality, gives strength and energy to memory, provides guidance in daily life. All people and peoples are living histories. Communities speak languages that are inherited from the past. We live in societies with complex cultures, traditions, and religions that have not been created on the spur of the moment.\n\"The Rebellious Rise\" is an episodic television show that explores infamous courtroom trials based on the most controversial issues in America's history.\n\"The Rebellious Rise\" of American history is an episodic doc series television show based on pivotal moments in the history of America in the courtroom, starting from the 19th century until today. \"The Rebellious Rise\" of American History will examine how the US Government and the Court System dealt with some of the important moments such as slavery, human rights, gender equality and much more. Few people know all of its hidden stories. \"The Rebellious Rise\" is modeled after the format of a show like \"The People v. O.J. Simpson\" consisting of a combination of interviews with people facing various social issues, experts examining important court cases and re-enactment of some of the cases. All these are important moments in the history of the United States. Most, if not all, ended in the courtroom where the US Government was forced to act to change the situation.\n\"The Rebellious Rise\" will educate and enrich people's knowledge of what happened in the courtroom in the 19th century, what is going on now and how the Government dealt with it. The audience will have the chance to experience the tension in the courtroom at that point, through re-enactment. The show, which is about the people, The viewers will be able to see that it is about human weaknesses and how facts, sometimes do not matter, and that's good sometimes and bad other times in life.\n\"The Rebellious Rise\" depicts legal history via re-enactments, and also re-enacting anecdotes that need a good writer to help bring the stories to life.\nThis show is incredibly precious to me and your support as an integral part in bringing it to life. Anything you can contribute will be greatly appreciated, as will helping me spread the story.\nShare the link and spread the word. You will also get a list of Best courtroom Films!\nYou will be invited to an exclusive video chat where you can ask us all your wildest questions about the film or our personal lives and a copy of the film. Excluding traveling expense and lodging.\nYou will get to be an extra in our show pilot.\nThis Stella team is bursting with talent- I am so lucky to be working on this project with these lovely people. I trust implicitly, both personally and professionally.\nI am Bernard Abagali; I am the executive producer of the show 'The Rebellious Rise.' Originally from Ghana, West Africa. I just finished my second year as Video Coordinator for Wagner College's Athletic Department. I was the Producer of all live events for the athletic department, and also I had the opportunity to serve as the producer for a bi-weekly football Coaches' Show, in which we would interview the Seahawks' head coach and several players while highlighting the previous games and scouted the upcoming matches.\nI saved as video Coordinator \/ Production intern for Richmond University Medical Center Palliative Care in Staten Island. During this time I served as the liaison between the hospital administration and video staff. I managed and organized all schedules, including coordinating video shoots, as well as acted as a second shooter.\nI hold a BA in Media and Society from Hobart College. I am also graduating in may with my MBA in Media Management at Wagner College in Staten Island, NY.\nWhile an undergrad, I studied in Rome, Italy, where I helped produce commercials for restaurants throughout the city. I have also created a short documentary \"Understanding\" about Tiara Trymore Gendi, a young transgendered woman from Harare, Zimbabwe. I am also working on another short documentary about a transgendered woman who lives in NY.\nPaula Landry, MBA, serves as an expert adviser to The Rebellious Rise.\nAs a film producer, her films have debuted at Sundance, and AMFEST, winning awards from Columbia Pictures, Time Warner Showtime, and WorldFest Houston International Film Festival. As a media consultant she has worked with Christie's, Entertainment Weekly, Pearson Television, advertising agencies and media entrepreneurs. She is the author of Scheduling and Budgeting Your Film: A Panic-Free Guide (Focal Press) and co-author of This Business of FILM with Steve Greenwald.\nSarah Friedland received her BA from Sarah Lawrence College and the International School of Film and Television in Cuba and her MFA from the Integrated Media Art Program at Hunter College. Her documentary films and installations are concerned with personal stories that reveal larger histories and intricacies about place and society. Friedland's works with her long-time production partner, Esy Casey have screened widely in the US and abroad and have been supported by grants from the Jerome Foundation, the Paul Newman Foundation, The Princess Grace Foundation, the Ford Foundation, and the Center for Asian American Media.\nJohn Ryan Andersen is an entertainer. Whether that is in front or behind the camera in short films, theater or improv John has a great passion for the entertainment industry. He has a B.A. in Mass Communications with a concentration in Television and Film and minor in Hospitality Management from Saint John's Universtiy and a recent M.B.A. in Media Management from Wagner College. John is currently pursuing an actor career full time as well as running his Youtube Channel Some Reel Good Movies. He was very happy and excited to be aiding Bernard Abagali in this venture.\nStephen Greenwald has been professionally involved in the motion picture and media industries for over thirty years as an attorney, financier, corporate executive, and producer. He has lectured and taught at colleges and law schools in the US and Israel and served as President of Metropolitan College of New York. Stephen is the co-author of This Business Of Film (Random House).\nYou're now following The Rebellious Rise ! Hungry for more great series? Try our streaming subscription!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Ion channels control many essential functions in the cell, including chemical signalling, regulation of pH, transcellular transport and regulation of cell volume. Abnormal function of ion channels can cause a diverse range of diseases, socalled channelopathies. The number of diseases related to ion channel malfunction are increasing and with that the amount of potential drugable ion channel targets.\nExamples of channelopathies are Cystic fibrosis (CFTRT, TMEM16), epilepsy (Kv7.2\/KCNQ2), but ion channels are involved in diseases in many areas such as Cardiac, Genitourinary, Metabolic, Gastrointestinal, Stroke, Pain\/Inflammation, Psychiatric Disorders, Pulmonary\/Respiratory and Seizures\/Convulsions. More and more evidence shows that ion channels could be a relevant target for cancer treatments as well.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Although developed global economies remain generally sluggish, we are still of the opinion that the chances of a double dip recession or a deflationary environment are relatively low. Buoyed by strong corporate earnings and balance sheets, robust third quarter equity market returns reversed losses suffered in the second quarter and improved year to date returns. Reflective of the uneven nature of this recovery, the surge in equity prices occurred despite continued strong demand for bonds and gold which would generally be expected to ease in such an environment.\nInvestors continue to be risk adverse resulting in an unrelenting demand for bonds. This has resulted in corporations being able to raise record levels of cash, driving corporate debt default rates near pre-crisis lows and providing them with the liquidity needed to weather the current cycle as well as re-invest in their businesses. This is likely the main reason for employment statistics which improved during the quarter, albeit at a tepid rate. This lackluster improvement will hopefully persist until consumer attitudes and spending can step up. Even as households reduce their debt level, such consumer involvement will be critical as monetary and fiscal policy strategies have been primarily exhausted. This combination of slow growth and limited policy responses leaves financial markets pushing out the timeframe for both economic expansion and inflation.\nThe speed and duration of the global economic expansion will be the key variable to investment performance in the short to intermediate term. Although we remain guardedly optimistic that improvement should continue on a sporadic and gradual basis, investment policy should concede the many unknown factors which will have an impact on investment performance. For this reason, we continue to suggest a policy that allocates your investment assets based on the envisioned timing of required withdrawals from the portfolio. Short to intermediate requirements should be reflected by cash and fixed income exposure, and longer term holding reflected in a portfolio of diversified equities.\nAs the table below illustrates, equity market returns recovered strongly across the board for the previous quarter bolstering the full year statistics.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Sweet Streams Chocolate Fountains is RI's Favorite Chocolate Fountain Company!\nI was looking forward to the chocolate fountain at my wedding for months. While waiting to be announced as husband and wife, I could smell the chocolate from the other room. It was just wonderful and well worth the wait. It was very nice to finally meet you. I have already referred my friends!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The beauty of a celebrant ceremony is that it can conducted anywhere.\nThere so many hidden gems, stunning private estates, beautiful outdoor locations, unlicensed and alternative venues out there to suit every style and taste. And if you or your family have a large garden, then your perfect wedding venue may be closer to home than you think-you don't get more personal than that!\nOne fabulous thing about what I do, is that I get to work at incredible venues that host all types of celebrations. And , in my opinion, some of the best weddings have been at places that aren't \"traditional\" wedding venues.\nDo we want an indoor or outdoor ceremony and\/or reception?\nDo we want a city or country feel?\nAre we fixed on a date, or are we flexible with our wedding date?\nDo we want our ceremony and reception to be in the same place?\nIs there a plan B in case of bad weather and does the venue have shaded areas?\nHow many guests do we expect, and can this venue cater for that amount of bodies?\nWhere are we going to get our photos taken? \u202fOnsite, or close by?\nIs there accommodation available locally for the guests who have come from afar? Or does the venue have space for camping?\nIs the venue easy to find and get to, by those wishing to arrive by taxi or other forms of public transport?\nDoes the venue reflect us as a couple?\nIs the venue willing to let us decorate it in the style we want?\nAre there any restrictions? Some venues will require heel protectors due to the type of flooring; some may have noise restrictions due to the proximity of local residents; some may not allow fireworks due to neighbouring farmland.\nWhat does the venue include in its fees? Bar service, PA system, electricity hook ups for outdoor caterers, cake table etc.\nWhat type of atmosphere are we trying to create? \u202fRelaxed, upmarket, black tie, rustic, traditional, vintage?\nDoes the venue have an event planner? Or do we want to organise it all ourselves?\nWhat type of food do we want at our reception? \u202fSit down, share platters, canap\u00e9s, buffet?\nDo we have to use the venues suppliers, or can we select our own to provide services? \u202fCaterer, Celebrants, Photographers, DJs?\nIf you are getting married in Devon, then here are a selection of venues that you might want to take a look at. They are ran by very lovely, accommodating people, all with a passion for weddings and all ready to meet you and chat through your wedding ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdsap b/data_all_eng_slimpj/shuffled/split2/finalzzzbdsap new file mode 100644 index 0000000000000000000000000000000000000000..61a7fb4ef1734c37a8d6313952225d6ca8bfe27d --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdsap @@ -0,0 +1,5 @@ +{"text":"A lone watchkeeper was 'distracted' by social media on his mobile just before his boat collided with another in Shoreham, resulting in three deaths, a report has found.\nThe Marine Accident Investigation Branch has released a report into the late-night collision between a commercial fishing vessel, Vertrouwen, and a motor cruiser, on which four men were rod fishing for mackerel, on Sunday, August, 6, 2017.\nTraiam Dumitrache, 50, from London; Mircea Ilie, 40, of Elm Grove, Brighton and Irinel Popovici, 41, of Barcombe Road, Brighton, drowned after their boat, James 2, was swamped by the wash and sank.\nA fourth man was rescued by commercial fishermen after clinging to a buoy for more than five hours.\nThe report concluded that an effective lookout was not being kept on either vessel.\n\"Neither vessel was maintaining a proper lookout by sight and hearing,\" the report states.\n\"Vertrouwen's skipper was a lone watchkeeper and he allowed himself to become distracted by using his mobile phone and working on his laptop computer.\nNone of the sea anglers on James 2, who had shared a drink of whiskey, were wearing lifejackets and the group were 'ill-prepared to make a short trip offshore at night', the report found.\nThe collision occurred because Vertrouwen's lone watchkeeper did not see James 2 either visually or by radar and, by the time the sea anglers realised they were in danger of being run down, they were unable to get out of the way, the report found.\nThe Marine Accident Investigation Branch has issued a safety flyer to the fishing industry highlighting the importance of maintaining a proper lookout and avoiding wheelhouse distractions.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Three out of four students live at the School, which means life at SSM is a 24\/7 experience for adults and students.\nShattuck-St. Mary's students relaxing in their dorms and Morgan Refectory.\nLearning to live independently and managing one's time is great preparation for college. Most students have roommates, often from a different state or even a different country. Sharing the experience of being away from home with people from all over the world is a strong start to life-long friendships.\nBridging the distance between parents and students is accomplished in a number of ways at SSM. Each student is assigned an advisor \u2013 a faculty mentor who keeps a watchful eye on the student, meets with them regularly and offers academic counseling.\nParents are also able to log on to our intranet to view their child's schedule, assignment and test grades, attendance, and comments from teachers. And, we have webcasts available for a variety of on-campus events from senior speeches, to athletic competitions to performing arts events.\nTwice a year, families are invited to campus to meet with teachers, coaches and most importantly spend time with their children. Special activities and events are planned for these weekends \u2013 one typically in October and the other typically in February.\nMost of our teachers and some of our coaches live in our residence halls, some with their families and pets, and serve as on-site parents for our students. They have baking and cooking sessions in their kitchens, spontaneous tutoring sessions, and hang-out time when students need a dose of home.\nYes, the campus dogs are well-loved!\nOur school is full of young families with little giggling munchkins running the halls. It feels like home!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Notes Based on Spider-Man and His Amazing Friends cartoon; Reprints Spider-Man and His Amazing Friends #1 with a new cover.\nCover Details - \"The Triumph of the Green Goblin!\"\n20 page Spider-Man; Iceman; Firestar story \"The Triumph of the Green Goblin!\"\nSynopsis Green Goblin busts up a college costume party to steal his secret Goblin formula.\nNotes Based on Spider-Man and His Amazing Friends cartoon.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"I've shared the idea of Information Pyramids previously. It's one of my favorite strategies for getting students to summarize a main idea and to include some specific supporting details. Since sharing it, many teachers have told me of their successes with it. So I am referencing it again to encourage those of you struggling to teach main idea to try it.\nMain idea asks students to step back from the reading and to identify the gist of what something was about. What's the point? What's the big picture topic? Using the Information Pyramid, students are forced to think more inferentially and not to look for literal answers in the text.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"\u0531\u0566\u0563\u0561\u0575\u056b\u0576 \u057a\u0561\u0580\u056f \u0539\u0565\u056b\u0562\u056c \u056c\u0565\u057c\u0561\u0576 \u057e\u0580\u0561.\n\u0554\u0565\u0575\u0583\u0569\u0561\u0578\u0582\u0576\u056b \u054d\u056b\u0569\u056b \u0532\u0578\u0582\u0578\u056c \u0569\u0561\u0572\u0561\u0574\u0561\u057d\u056b \u057f\u0565\u057d\u0561\u0580\u0561\u0576\u0568 \u0570\u0575\u0578\u0582\u057d\u056b\u057d\u056b\u0581 .\n\u2191 A.G. Rebelo, C. Boucher, N. Helme, L. Mucina, M.C. Rutherford et al. 2006. Fynbos Biome, in: L. Mucina & M.C. Rutherford (eds). The Vegetation of South Africa, Lesotho and Swaziland.\n\u2191 Archived 10 August 2015 at the Wayback Machine.\n\u2191 Archived 10 February 2010 at the Wayback Machine.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdzzw b/data_all_eng_slimpj/shuffled/split2/finalzzzbdzzw new file mode 100644 index 0000000000000000000000000000000000000000..a59c455c24dfe29ff3d697cb12e93618ab99db69 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdzzw @@ -0,0 +1,5 @@ +{"text":"Hart Schaffner Marx teams up with Autism Workforce to help change lives!\nWe outfitted the on-air talent for the NHL on NBC. Take a look as we meet the guys for fittings and follow them to Madison Square Garden for the Stanley Cup playoffs.\nA behind the scenes look at our Fall 2014 photoshoot.\nWe hooked up with NBC and outfitted the entire Football Night in America team with custom Hart Schaffner Marx suits.\nWe spent a weekend in DUMBO, Brooklyn hanging out with other U.S. Made brands at the most recent Northern Grade Pop-Up Event.\nGet an inside look at how a Hart Schaffner Marx suit is made, from testing to steaming and everything in between.\nGet a behind the scenes look at this year's 2013 photo shoot.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The other day I went to a yoga class and I was pleasantly surprised to see an elderly woman unfolding her yoga mat to take the class. Wow, what an inspiration! About halfway through the class, I glanced over, whilst huffing and puffing from the 'Downward Dog' pose into the plank. There she was, not even breaking a sweat. And here I was trying to think of a way to leave the class halfway before it was through. At the end of the class I waved goodbye \u2013 I hope to see her again to find out how she got to be a yoga superstar!\nDo You Also Know of a Superstar Senior?\nWe all know a golden ager that does incredible things (like yoga at 80) and has an inspiring story to tell. It could be your mom, a doting grandpa, great-grandmother, a volunteer, or even a school teacher from our childhood past that shaped out future in a positive way. This page is dedicated to all the amazing, wise, seniors out there that inspire us on a daily basis!\nAre you a Superstar Senior? Or do you know of one? Sure you do!\nShare Your Inspiring Stories About Amazing Seniors, right here!\nIf you are a senior and have a great story to share about your accomplishments or successes, I'd love to hear from you! The same goes if you have a story to share about someone amazing, like the elderly woman I met the other day at Yoga, or maybe a volunteer, or someone that is still working well-past retirement simply because they love their work!\nSend in your Superstar Senior stories and with your permission, I may just publish it on my very next blog! Please include a picture of the person (and setting, if applicable) that you are writing about. Also, don't forget to ensure you have their permission to write about them and send in their picture \u2013 some people are shy and of course, I want to respect their privacy.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"It hasn't always been this way. The original Ner Tamid, described in Parshat Tetzaveh,1illumined the most sacred physical location known to our ancestors, the Ark of the Covenant in the mishkan (Tabernacle) and then the Holy of Holies in the Temple. Far from a Divine miracle, it was the responsibility of human hands to ensure that the light illuminating God's presence would never go dark.\nWendi Geffen is a rabbi at North Shore Congregation Israel in suburban Chicago. Dedicated to social justice and its Jewish textual roots, she regularly works to empower the synagogue and her larger community to act on the Jewish imperative to pursue tzedek. Wendi is also the creator of Beyond and Back, a Chicago-wide program that engages young adults to explore their Jewish identity through text study and dialogue. A graduate of Emory University, she was ordained at Hebrew Union College and received honors for academic achievement in Bible. Wendi can be reached at geffenajws@gmail.com.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"China PEXJaw Crusher with Good Quality and Cheap. China PEXJaw Crusher with Good Quality and Cheap Price, Find details about China Jaw Crusher, Mining Machine from PEXJaw Crusher with Good Quality and Cheap PriceShanghai DingBo Heavy Industry Machinery Co., Ltd.\ndurable stone jaw crusher, wholesale various high quality durable stone jaw crusher products from global durable stone jaw crusher suppliers and durable stone jaw. jaw crusher and rock crusher from Manufacturer. ... china good quality supplier mineral jaw.\nDurable Brick Crusher For Sale, Wholesale Various High Quality Durable Brick Crusher . durable and efficient brick jaw crusher for crushing stone .sbm hot sale mobile impact rock crusher , durable brick crusherquarry mining crusher.\n&#;&#;We can sell jaw crusher and export jaw crusher all over the world. We export pad printer and sell pad printer. We are a Leading manufacturer of pad printer in China.\nRock Crusher Spare Parts Movable Jaw Crusher Crushing Stone Line Swing And Fixed Jaw Crusher Plate Liner . Overview . Quick Details . Place of Origin: jiangsu, China ... Durable and Wear-resistant . Customized: Yes . Color: ... China Good Quality Rock Crusher Spare Parts Supplier.\nGood Quality Jaw Crusher,mini jaw crusher for sale,Jaw Crusher price Durable Quality Large Capacitytph Primary PE Stone Jaw Crusher And Secondary Stone Jaw PEXsmall jaw crusher\/Small Stone Crusher for mining.\n&#;&#;Extec, Extec Suppliers and Manufacturers . 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A wide variety of good warranty jaw crusher options are available to you, such as jaw crusher, impact crusher, and cone crusher.\nJaw Crusher,stone Crusher. ... We list many of China's best JAW CRUSHERS manufacturers and also have a number of other good companies that supply equipment & related supplies to dealers and distributors\/vendors.\ndurable good quality cone stone crusher . jaw crusher for sale stone crusher machine Durable Quality, Sdsy Series Stone Crusher For Sale. crushers,price for mobile stone crusher,scrushing machine. Chat With Sales. High Quality Durable Use Jaw Crusher Stone.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"For many of us, worrying about getting pick-pocketed while carrying our backpack can be an unpleasant experience. Even something as simple as enjoying a warm cup of coffee at a terrace can result in your bag being seized from you all in the blink of an eye. This is exactly why they strive to protect your luggage from theft. They want nothing more than for their clients to travel safely, comfortably and without having to be concerned about unwanted incidents. Whether traveling far or near, by bike, by foot or public transport, the Bobby Bizz is the perfect traveling sidekick.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbempq b/data_all_eng_slimpj/shuffled/split2/finalzzzbempq new file mode 100644 index 0000000000000000000000000000000000000000..bc0a256236faea4d722d469576e1c061e011b590 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbempq @@ -0,0 +1,5 @@ +{"text":"Gunship have released a new video and it tells the story of Ernest Cline's Ready Player One in five minutes. What makes it extra special is that the whole video is done in the style of a classic point and click adventure game, in particular a Lucasarts SCUMM\u2122 adventure. This has become one of my favourite music videos of all time. The song that accompanies is perfect and has a sense of Ready Player One intrigue about it.\nSeriously though, how good would a Ready Player One adventure game be? I could think of two ways this could be done. The first would be in classic style, so keeping the retro theme just like in the music video. Tim Schafer are you listening? The second way would be an episodic adventure in the style of TellTale Games. After all, you can do anything and be anyone in The Oasis so why not have each episode with a different character, story and scenario?\nWould you like to see a Ready Player One game? Let us know in the comments and what kind of game it should be.\nStream & Download 'Art3mis & Parzival' here - http:\/\/smarturl.it\/HITH004DL Find the hidden clues in this video to win GUNSHIP's Holy Arcade Machine Of Antioch! You must use your cunning to pass the trials that GUNSHIP themselves have laid down. Head to http:\/\/www.gunshipmusic.com\/gate to play.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Well, I'm bored, so I took some photos of carvings from many years ago. (over 20} The carousel horse is the result of following instructions from a Ken Hughes book. The bears are inspired by an Art McKellips book.\nNice work. I never could carve. I would whittle and whittle and the bear never came out of the block of wood, just shavings.\nI know what you mean Chris. I started out making clay models from pictures then carving the wood taking exact measurements where possible and careful extrapolations when necessary.\nHave tried whittling so many times. I can do a boot for a key chain, but man, its a patience thing I'm sure.\nUsed to have an old cowboy friend who would carve a boot, while you talked to him at lunch. Then he would take out a sheet of sand paper from his shirt pocket, no bigger than your thumb and sand it, hand it to ya and just smile.\nGood job you did with these sir.\nGreat carvings, they look great and adorable especially the bears.\nDan, those are super cool!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Today is our first Easter without John's grandmother, Mary Kelemen. She was an amazing lady, and though she slowed down in the last few of her 93+ years, to the end, Easter was the holiday we continued to celebrate at her home, with formal dinner, fixings, and eveything that entailed. She was raised in the Russian Orthodox Church, and their Easter is often on a different day than most Christians celebrate it. This year, for instance, today is regular Easter, while the Russian church does not celebrate Easter until next Sunday, April 8. The latest I recall celebrating Easter with Mary was May 5.\nA couple years ago, I was let in on the family secret: the recipe for hrudka, a traditional Slovak dish made and served specifically for the Easter meal. It's a sweet egg cheese \u2013 and I never liked it. Let it be stated that I may be the fussiest eater on the planet, something I never realized until I married into John's family. \"You don't like that, either?\" they'd exclaim through the years, as I just shook my head apologetically.\nBaked ham: symbolic of the great joy and abundance of the Easter season.\nKielbasa (pronounced kil-bah-see in the Kelemen house, for some unknown reason): a spicy, garlicky, smoked pork sausage that originated in Poland and the Ukraine. It symbolizes God's favor and generosity.\nRed beet horseradish: symbolic of the Passion of Christ, yet typically sweetened with a little sugar since, after all, Christ did rise again.\nSalt: used for flavor and to serve as a reminder to Christians of their duty to others.\nButter, molded into the shape of a lamb (ours always had cloves for the eyes): symbolic of the goodness of Christ which Christians should exhibit toward all other living creatures.\nYay-ech-nick: egg-based cheese made into a \"ball\" and cut into slices. It's supposed to be served with bitter herbs that indicate the moderation with which Christians should approach all things, but we never had the herbs, so that point was a bit lost on us, perhaps.\nHard-boiled eggs: Mary had the coolest shrink-wraps for her eggs with amazing Russian patterns on them. She would tell us stories about her mom and her aunts painting the same designs by hand on actual eggshells. The eggs symbolize the rebirth of Christ.\nPotato salad: not a Czech tradition, but a family one, as John's stepmom, Gayle, makes the most amazing potato salad you've ever tasted.\nMy family did not have quite as many Easter traditions, at least as they relate to the meal. Every year, we did attend all three days of the Triduum services, the days, beginning with Holy Thursday, leading up to Easter Sunday.\nHoly Thursday commemorates the Last Supper. In the Catholic tradition, Jesus' washing of his disciples' feet before the start of that meal is reenacted during Holy Thursday Mass. Twelve parishoners are selected to be seated at the front of the church as the priests literally bathe their feet. Our family had that honor on two occasions during my growing up years.\nGood Friday is the day that commemorates Christ's death on the cross \u2013 the holiest times being noon to 3 p.m., recognizing the time he is supposed to have actually spent on the cross before dying. Good Friday is the only day of the year when no Mass is said anywhere in the world \u2013 in honor of Christ being gone for those three days. Most churches have a Stations of the Cross ceremony during the afternoon, where the entire Passion of Christ is reenacted through a prayerful walk around the church, stopping at statues or paintings that depict various events throughout that time of Christ's life.\nThough Easter is supposed to be the Big Day, Holy Saturday is really the Catholic church's Super Bowl: it is the day all the new people who have been studying and preparing to become Catholic for the last six months to a year are baptized and receive communion for the first time. In college, I sponsored a gal to convert to Catholicism. Mary also sponsored a woman who converted to the Russian church, so we had that in common.\nYou get a sense of how deep the theological indoctrination runs \u2013 and not necessarily in a bad way \u2013 in that I have not attended these services for more than a dozen years, yet I can still describe them with great detail.\nMy family's Easter Sunday traditions were milder: an egg hunt in our backyard, followed by a midday meal featuring a ham. This continued even into our adulthood, as my mom really loved the searching and finding game. Who knew what happened to the Easter baskets after the hunt was over, so each year, I would head to the thrift store to get new ones for all of us. Mom's was, without a doubt, the blingiest basket I could find.\nI'm not sure I ever heard the Stanfields' Easter traditions described. If I did, it's the one detail I failed to commit to memory. I imagine it involves Mass on Sunday morning and a large family meal. I do know that my kiddo has taken after me in the sweet tooth department, so he probably loves the candy and jelly beans.\nSpeaking of Easter candy\u2026 my sister Corina also had a sweet tooth, and a few years ago, she sent her daughter, Samantha, to the store a day or two after Easter to grab a couple bags of discounted Easter candy. She gave Samantha a twenty, expecting her to return with about $15 in change. Things did not go well when Sam came back with a giant bag laden with every candy imaginable and no change. \"They were all on sale and I couldn't decide, so I got one of each,\" was her explanation.\nEverything being different this year, John, his stepmom, and I have decided to take ourselves out for Easter brunch. If you ever decide to try this, I recommend making your reservation before 9 p.m. on Good Friday, as most people do. Your choices will be (understandably) limited if you wait. It will be quiet, but nice. I'm sure we'll express gratitude for the abundance with which we've been blessed \u2013 but we'll also acknowledge all the people recently gone from our table.\nWishing you Easter blessings and a beautiful spring!\nNext Post What's at the Top of Your Lifetime Playlist?\nMy family's Easter traditions were less formal, but I did wake up singing Easter hymns today!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Its wide adoption stunned the financial industry \u2013 and eventually, the banks themselves began to realize the potential of blockchain technology. Now, various studies report that 69% of the leading banks around the world are experimenting with blockchain technology, with research claiming that if they use blockchain, banks could save approximately 8-12 billion USD every year.\nIt made it possible for 'pirates' around the world to share music, videos, digital art, games, writings and more \u2013 all without asking for permission or giving credit and compensation to the original owners. Additionally, the creative industry is also plagued with a lot of 'middlemen' such as music labels, digital publishers, stores, etc., who take a huge chunk out of artists' revenue.\nFirst of all, blockchain can give creators the ability to store their original works and mark them with metadata which includes information such as their name, the date on which the work was created \u2013 and so much more.\nPlus, with blockchain, you can create 'smart contracts' which are highly intelligent programs that execute instructions automatically on the creators' behalf. This means creators can fully control how their content is distributed \u2013 and how much they are paid.\nAnd because of this, blockchain has the potential to positively impact the publishing industry, the music industry, the fashion industry \u2013 and all other creative industries that want a more secure method of protecting their IP (intellectual property).\nBefore construction starts, the terms and conditions can be written down in a blockchain, which both parties can agree to. This can include specifications of the building, the quantity of materials to be used, the deadline for milestones, when each payment will be released, and a host of other things.\nAs a result, the real estate industry will become safer \u2013 and will progress faster, thanks to blockchain.\nBecause with blockchain, where a limited number of bitcoins are created which give each coin its value, similarly, a limited number of trading cards can be created as well \u2013 each with its own unique identity. And since these cards are stored in a blockchain network, they cannot be copied or counterfeited. And in this way, each card will carry a specific value and a guarantee of authenticity.\nThe health industry is one of the most complex and important industries in the world. And it should be because it literally deals with people's lives. A strong, integrated healthcare system can benefit both the hospitals and their patients \u2013 and blockchain can help the health industry with that.\nEach person can be assigned a unique identity \u2013 and all their medical records can be stored under that identity. Thanks to artificial intelligence (AI) and the internet of things (IoT), medical equipment can autonomously upload sensitive health data in the blockchain network, where it will be stored forever.\nAccording to the Financial Times, legal disputes regarding family inheritance has increased every single year from 2012 to 2016. In 2016 alone, the number of inheritance related cases increased 36% compared to the previous year.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The venue for this clash with the league leaders, Acton, was North Acton Playing Fields which filled us with a certain level of trepidation because generally when a ground has the words 'playing fields' in the title it is usually just that, a field with people playing various different sports and not just cricket. Well I can report that North Acton Playing Fields did not disappoint on the field front, in fact all it was missing was a flock of sheep nibbling away at the grass. Actually a flock of sheep might have improved the playing conditions given that shorter grass in the outfield would have helped double our score!\nSo when faced with a wet cabbage patch of a wicket the thing you really need to do is win the toss and bowl first. I therefore apologise to all the Teddington players involved as I lost the toss and unsurprisingly we were inserted.\nWe decided that survival was the main order of the day hoping that if we batted enough overs we could get some kind of score we could defend and so we set a new world record for dot balls faced as we battled away valiantly building our total around a really excellent innings from Stefan Bullon who crafted 88 not out against a bowling attack that gave us nothing. I cannot commend Stef's innings too highly, his determination and concentration were exemplary and set a standard that sadly no-one else could match with no other batsman reaching double figures.\nActon knew how to bat on this wicket and sadly we did not bowl well enough to cause them too much trouble. Credit where it's due \u2013 Acton bowled better than us, they fielded better than us and they definitely batted better than us. The best team won.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzberel b/data_all_eng_slimpj/shuffled/split2/finalzzzberel new file mode 100644 index 0000000000000000000000000000000000000000..74ec877606f392c9bdb26485ba94d3eb1b21336d --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzberel @@ -0,0 +1,5 @@ +{"text":"Dana VanVeller is coming home March 16 with a cookbook.\nThe Sarnia native, and her co-author Lindsay Anderson, are set to celebrate the release of their first cookbook, Feast: Recipes and Stories from a Canadian Road Trip, March 16, 7 p.m., at The Book Keeper book store.\nThe Northern Collegiate graduate left Sarnia as a teenager to travel and continue her education, before settling in Vancouver a few years ago.\n\"I've kind of always been interested in food,\" VanVeller said.\nSo, the friends came up with the idea of taking a road trip through every province and territory, in an attempt to search out Canadian food culture.\nIn the summer of 2013, the set off on a five-month, 37,000-kilometre journey across the country, blogging along the way at edibleroadtrip.com.\n\"At any given time, we'd be writing about where we'd been,\" VanVeller said.\nThey made it from coast to coast, and even spent time in Nunavut.\nDuring their time in Ontario, they drove along Lake Huron to Sarnia, where they stayed a few days before heading out on the road again.\nShe took Anderson for bridge fries, \"of course,\" and ice cream in her hometown.\n\"The whole go-to-the-bridge culture is so Sarnia, it's awesome,\" VanVeller said.\nThey also dropped by the Sarnia Farmers' market for butter tarts VanVeller looks forward to when she's home for a visit, along with the spring rolls.\n\"Our lakes are hugely influential,\" she added, about Sarnia and Ontario's food culture.\n\"I grew up eating at Purdy's\" and fishing with her grandfather, VanVeller said.\nAlong with what the lakes provide, she said her childhood meals in Sarnia included peaches, barbequed corn and Ontario's other summer produce.\nVanVeller said they didn't start out with a Canadian food theory they wanted to prove, other than a belief the country must have a lot of interesting food stories.\n\"We weren't trying to find two or three Canadian dishes that are quintessential, and can absolutely represent all of Canada,\" she added.\nInstead, they found that Canada's food cultures are extremely regional, which wasn't surprising given the distance between its regions, communities and coasts.\nWhile they ended up surprised by how diverse the country's food culture is, they didn't feel like that's a bad thing.\n\"Ultimately, our diversity is what's so interesting about us, and that is our strength,\" VanVeller said.\nDuring that time, they won a food blog award.\n\"That kind of gave us more recognition, and something to stand on to submitted a book proposal to a publisher,\" VanVeller said.\nThe book, being officially released March 7 by Penquin Random House Canada, includes 114 recipes, including 90 from people they met during the trip.\n\"It feels incredible to come home, to my hometown, to a beloved bookstore, and share what the last four years of my life has been like,\" VanVeller said.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"To reduce the chance of an accident with a large truck or RV, motorists must be familiar with a big rig's physical capabilities and how they maneuver.\nLarge trucks take longer to stop than a car traveling at the same speed. The average passenger vehicle traveling at 55 mph can stop in about 400 feet. However, a large truck traveling at the same speed can take almost 800 feet to stop. Don't pull in front of a large truck and suddenly slow down or stop. The trucker will not be able to stop quickly enough to avoid crashing into you.\nFor all turning vehicles, the rear wheels follow a shorter path than the front wheels. The longer the vehicle, the greater the difference. This is why big rig drivers must often swing wide to complete a right turn. When you follow a big rig, look at its turn signals before you start to pass. If you think the truck is turning left\u2014wait a second and check the turn signals again. The driver may actually be turning right.\nTrucks are designed to transport products and they are not as maneuverable as passenger vehicles. Large trucks have longer stopping and starting distances. They take more space for turns and they weigh more. On multilane highways and freeways, large trucks usually stay in the center portion of the lane to help the flow of traffic. This also increases the trucker's options in case he or she must change lanes to avoid a hazard. Here are some of the most common mistakes passenger vehicle drivers must avoid when driving around large trucks.\nCutting off a truck in traffic or on the highway to reach your exit or turn. Cutting into the open space in front of a truck is dangerous. Trying to beat a truck through a single-lane construction zone, for example, removes the truck driver's cushion of safety and places you in danger. Slow down and take your turn entering the construction zone. Don't speed up to pass a truck so you can exit the roadway. Take a moment to slow down and exit behind a truck\u2014it will only take you a few extra seconds.\nDon't linger alongside a truck when passing. Always pass a large truck on the left side and after you pass the truck, move ahead of it. Don't linger. If you linger beside the truck, you make it very difficult, if not impossible, for the trucker to take evasive action if an obstacle appears in the road ahead.\nFollowing too closely or tailgating. When you follow behind a truck and you cannot see the truck driver's side view mirrors, the trucker has no way of knowing you are there. Tailgating a truck, or any vehicle, is dangerous because you take away your own cushion of safety. Where will you go when the vehicle in front of you stops quickly?\nNever underestimate the size and speed of an approaching tractor-trailer. A large tractor-trailer often appears to be traveling at a slower speed because of its large size. Many passenger vehicle vs. large truck accidents take place at intersections because the passenger vehicle driver did not realize how close the truck was or how quickly it was traveling.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Comfortable, breathable socks - perfect for trekking, hiking and winter sports. The individual foot parts are reinforced, comfort on the toes and heel padding is ensured. 3\/4 height, ideal for trekking shoes. Composition: 51% wool, 22% fabric with silver ions, 10% polypropylene, 5% polyamide, 2% elastane.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"If you find that you need new gutters in Canoga Park, to call the expert contractors at Rain Gutter Specialists. We work hard to provide all of the local residential and commercial customers with the best prices on all new gutters. We can even custom create gutters to meet the needs of your home. Choose us for gutter sales and service in Canoga Park, CA.\nWhen you need to have gutters installed in Canoga Park, take a moment and call our trusted team. Our experts are experienced, licensed, and trained to provide you with the services and help you need in regards to gutters installation and replacement in Canoga Park. We can quickly and safely remove and recycle your old gutters for you.\nRain Gutter Specialists carry a wide selection of gutters from copper to aluminum and even seamless options. Our seamless option is the best option out there and you will enjoy gutters that are free from leaks. If you would like to receive a quote for copper, seamless or aluminum gutters services in Canoga Park, call our office now.\nIn addition to our rain gutters, we also carry a wide selection of accessories to include rain chains, headers, leaders, rain barrels, and more. If you would like to discuss what options are available to you, call our office now.\nThe team at Rain Gutter Specialists wants your residential or commercial property to look amazing and be free from any possible water damage caused from rain. Call for rain gutters service and sales in Canoga Park.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Well, fast forward a couple of decades or more and take a look at what's on today's menu. We're looking at a 600 Watt-second (Ws) flash with an exchangeable battery that will give you more than 500 full-power pops on each charge. If that's not enough, you can buy another battery for $150. In addition, the flash is full TTL and HSS with Nikon, Canon, and Sony; it has a nine-stop range; and you can control this unit and several others with the included R2 transmitter. And this entire wonder light will only cost you $750.\nHow can that be? Is it that perfect? Will it cook your breakfast?\nFor the answers to those questions and much more, let's take the Flashpoint XPLOR 600 monolight for a spin.\nWhen I cut open the shipping box for the Flashpoint XPLOR 600 monolight, I was delighted to see a nice nylon case that contained everything. The little suitcase has a carry handle plus a shoulder strap for easy portability\u2014nice. Checking it out I found the light, the battery, the flash tube, the charging cords, the R2 transmitter, a basic instruction booklet for the light and the transmitter, and a standard seven-inch reflector using the popular Bowens mount.\nPulling the XPLOR 600 out of the case, I proceeded to mount it on a light stand and immediately ran into one of the very few issues I had with this product. The light sits on a ratchet handle. I loosened it up and went to turn the handle so I could mount the light but it was a no-go. I thought it was broken! As a reviewer, the last thing I wanted to do was to return a product that I busted up but I forced it and it finally moved. As I used it more I got used to the very heavy ratchet but it is quite inelegant and I always felt like I was going to break it. While it doesn't affect performance, it hurts usability, so I'd suggest to Flashpoint that they take a look at another option.\nHere is the same image with the same exposure but with the Flashpoint XPLOR 600 monolight added. Because I wanted shallow depth of field, I just set my exposure and let the flash do the work. This is TTL with HSS also, no exposure compensation. Technical info: Nikon D3, Tamron 28-75mm f\/2.8 lens at 40mm, ISO 200, 1\/1250 second, f\/4.5.\nAs for the build of the XPLOR 600, it appears to be metal where warranted, like the Bowens mount and the battery mount, and plastic in the main body area to ostensibly save on both weight and cost.\nHere is a nice image of Brianne Terrence taken at dusk with just daylight. While pretty, the light is flat and uninteresting. Technical info: Nikon D3, Nikkor 80-200mm f\/2.8 lens, ISO 400, 1\/250 second, f\/4.\nBy adding a flash we can add texture and depth to an image. In this case I used a white umbrella on the Flashpoint XPLOR 600 monolight, camera right and on a 13-foot stand. This gave me the texture and depth I was looking for but also by exposing properly for the flash, I was able to darken the background since I gave it less exposure. In this instance I did adjust the TTL, using almost minus two stops compensation, due to the great deal of sky in the image.\nThe first thing I did with the XPLOR 600 was plug in the flash tube and charge up the battery. A full charge from a spent battery takes four hours. The battery is pretty substantial, as you'd expect from something that gives you that many flashes. It also has a very handy button on it that you press to see how much power you have left. All in all, this battery is a winner with quick charging, a solid connection to the light, tons of power, and a great price.\nTurning our attention to the body of the XPLOR 600 monolight, we see the big and bright LCD panel on the side next to several buttons that control the various flash functions. Hold the on\/off button down for two seconds to turn everything on and light up the screen.\nHere are some of the things you'll notice immediately. The XPLOR 600 can handle five groups and 32 channels, which should be enough for just about anyone. Of course, I had one light so I did not have the opportunity to use multiple groups. It should be noted that there are many options that allow you to use the XPLOR 600 with several other speedlights with no modification and even with Nikon and Canon with adapters. That is beyond the scope of this review though.\nThis image of bodybuilder Jon Arnold was taken using TTL with no compensation. Thanks to Iron Empire gym of Dover, New Hampshire, for the use of their facilities in all the photos of Arnold. I even had them turn the overhead lights off because they were so distracting. This image was taken with me lying on my back. Technical info: Nikon D3, Sigma 17-35mm f\/2.8-4 lens at 25mm, ISO 400, 1\/125 second, f\/5.6.\nYou'll also notice that LED technology now allows you to have modeling lights. Incandescent bulbs would kill your battery in about one minute but with this 10W-rated bulb it should be bright enough in the studio and to use for focus assist in some situations. It even has three different levels to help save battery power. Outside in bright light it is of very little use so I just shut it off.\nOther controls are what you'd expect, like mode, group, channel, and menu. The one that stands out is the \"H\" button for High Speed Sync, as it has a couple of unique features. First, if you are using the R2 remote, the transition from regular to HSS is seamless\u2014you don't need to turn it on at all. Just set whatever speed you like up to 1\/8000 second and the flash will adjust automatically. Just remember that all those flashes pulsing during the exposure will need to be rather close to your subject to get enough firepower. I used it beachside on a sunny day within 10 feet and didn't experience any issues.\nI went old-school manual on this photo of Jon Arnold. No TTL, I locked in the power to give me f\/10 and kept it that way for a series of images. Confession: I'm so used to using location lights with no modeling light to go by that I neglected to even turn it on and you can see light placement is critical! In this series I used a round white box that mounted via umbrella. I didn't crop the light out of the corner so you could see its placement. Technical info: Nikon D3, Tamron 28-75mm f\/2.8 lens at 48mm, ISO 200, 1\/60 second, f\/10.\nThe second unique feature about the \"H\" button is that it will display the flash speed at different levels when set to manual. The duration at full power is only 1\/220 second\u2014not very fast. Clicking down through the power settings reveals faster flash durations at lower power settings. To speed things up you can use HSS in some instances to just boost shutter speed but you'll have to decide if this is an issue for you.\nMy \"scientific\" testing of the XPLOR 600 resulted in popping it a few times to get a Guide Number (GN) and to check recycle time. I measured f\/22 at 10 feet with the standard reflector for a GN of 220 at ISO 100, a little higher than the 183 listed in the spec sheet. I'm guessing they are conservative in their rating. It's much tougher to measure the recycle time at full power. Using the stopwatch on my iPhone I got approximately two seconds. Figure a little over that and you'll be good.\nAs far as the R2 transmitter, it does require a few minutes spent with the manual. It has three major buttons and a wheel for adjustments, but you can figure it out pretty quickly. Some nice features are a test button, an on\/off button for AF assist, and even a hot shoe on top so you can also fire a speedlight while using the R2. It's also powered by two AA batteries, which are available everywhere.\nAlthough you'll find specific comments in the photo captions, here is a summation of the Flashpoint XPLOR 600 monolight. At $750, it meets or exceeds many other lights in this class at a fraction of the price. Accessories like additional batteries are also very aggressively priced. The XPLOR 600 does special tricks like allowing you to configure it (with low-priced options) as a two-piece light so you can hang just the head on a boom and also be able to combine two power sources with one head for a 1200 Ws light! You also have the ability to use other speedlights to be able to control several flash units from the camera. The one feature that I hope is looked at in the future is the ability to shoot TTL and once you have the exposure nailed, you have the option to lock that exposure down. It's a very useful feature that would make a terrific light even better.\nHere's why having a method to lock in TTL is a great feature. Both images were taken using TTL with no compensation. In the image on the left, our exposure is accurate since our subject fills most of the frame. When I backed off and included more background area the flash was trying to expose for that also. While I could easily compensate from camera position, having a method to lock in the exposure would save time and result in more correct exposures.\nOne more thing I should address: low-priced units like this are often inconsistent in color temperature. Flashpoint states a 200K variation. While I suspect it may be slightly more, I have no way of checking. In any case, unless you are shooting very critical work with multiple units it is not an issue. Simply adjust the Raw file in Lightroom or Photoshop and you're done.\nI can't recommend the Flashpoint XPLOR 600 monolight highly enough. I know what's out there and have tested many of the units. If you want an all-in-one monolight that is high powered and low priced, plus has a great battery along with HSS and TTL, this is it. I just hope they change the handle.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbewqm b/data_all_eng_slimpj/shuffled/split2/finalzzzbewqm new file mode 100644 index 0000000000000000000000000000000000000000..22ef67580a4cde9928b353914dfa8cf843f5e658 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbewqm @@ -0,0 +1,5 @@ +{"text":"Join us as we speak with Nadia Morris who will be discussing innovation and entrepreneurship inside a large organization. Nadia will share how AT&T has innovated beyond a telephone service company, and how various business units work together to keep the company relevant in a wide variety of industries. She share specifically about her work at the AT&T Foundry for Connected Health, a group dedicated to technology solutions that can help improve quality of life through digital health innovations that bridge the clinical care environment with patients' daily lives.\nPrior to her post in Houston, Nadia was a Lead Engineer at the Foundry in Palo Alto. She has been awarded over 45 US patents, covering areas such as network security, streaming video, radio networking, and mobile application development.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Human technology has been shaping the land and waterways for generations in the Pioneer Valley. On Sunday, September 16 from 10 am to noon, join us for a walk along the Holyoke Canals to learn about their history and impact on our environment.\nThis guided walking tour will be led by Robert Comeau. Bob explains that the Holyoke Canal System was built to generate mechanical power at low cost for the textiles and paper mills. Tunnels below the mills allowed water to pass through hydro-mechanical turbines that powered looms. We will visit two of the three canals, one spillway, the dam, and some mills on this tour. For the first 100 years the environment was negatively impacted by the canals and mills. Since the 1950's however there has been great improvements in their interactions with fish and the river water.\nThis walking tour will cover about 3 miles during the two hour program, and will be mostly flat terrain.\nRobert Comeau was a high school science teacher for 15 years and a retailer for the last 15 years. He has led many tours for the past three years on the canal system and Holyoke's neighborhoods.\nThe program is now full, but if you would like to be added to the waiting list, please email kari@kestreltrust.org.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We think every company is socially responsible to treat each employee with respect, and contribute to economic growth and welfare improvement through its healthy business development.\nUnfortunately large numbers of welfare organizations do not receive enough financial assistance from the government. Although there are various funds, support from them is not sufficient for these welfare organizations.\nAs a member of the society, we decided to establish \"ATAGO Fund\" in order to directly contribute to them.\nWe made a donation to a NPO organization in May, 2008 and June, 2009.\nThe organization have supported mentally handicapped people so that they can lead serf-reliance and community-based life.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"How many times have you replaced your kids Easter choc in the last few weeks?\nIs it worth paying lots of money for purple shampoo and conditioner?\nDo you stick cotton tips in your ears?\nIf you can't save $2 a month for Easter eggs you need to learn to budget Do you agree ?\nFor fun: what's your \"type\" of man\/woman\/partner ?\nMen make me sick sometimes.\nMy todo portable hairbrush,is it battery operated only for the light or should something else happen?\nWhy was my tax question \/ statement deleted?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Universal has doubts about aspects of the scenario of Grease Paint, which it proposes to film. They refer the question to Joy, who duly asks Hays to take it up with the Catholics and with Herron re the characterization of an Austrian.\nUNIVERSAL.\t1928. Letter from Joy to Hays Re \"Grease Paint\", 03-23-1928 :I am sending herewith a copy of the scenario called \"Grease Paint. \"Please have someone consult Catholic authority and also ask Col. Herron to check up the character of Franz the Viennese. You will note that Universal contemplates it as a picture. They sent it to me here in Chicago by air-mail, because they have some doubt about the attitude of the Catholic church towards it. It seems to me that the ending might be changed a bit so that it would appear that the training which Fernandi had received in the monastery made it possible for him to forgive the sins committed against him by Franz and Eva. This happens just before Franz dies. Please wire me your findings.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfrht b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrht new file mode 100644 index 0000000000000000000000000000000000000000..60de733a6e0a1d5dd1bd4d841e4042b1fbd0e743 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrht @@ -0,0 +1,5 @@ +{"text":"NIGERIAN ARMY INSTITUTE OF TECHNOLOGY AND ENVIRONMENTAL STUDIES NAITES HAS JUST ANNOUNCED ITS 2017\/2018 MATRICULATION CEREMONY DATE AND SCHEDULE.\nThis is to inform the general public and students at the Nigerian Army Institute of Technology and Enviromental studies NAITES that the school management has just announced ts 2017\/2018 Matriculation Ceremony date.\nThe 2017\/2018 matriculation ceremony of the Nigerian Army Institute of Technology and Environmental Studies is scheduled to hold on February 24th 2018 and March 7th 2018 at its Biu (Borno State) and Makurdi (Benue State) Campuses respectively.\nThe general public is by this notice informed of this upcoming event.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Grace Roots Blog: Forgive Others In Order To Be Forgiven By God?\nForgive Others In Order To Be Forgiven By God?\nIn Matthew 6:14-15, Jesus made a very strong statement: \"For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses.\"\nOver the years I've heard this quoted quite often in the church and used as a principle that Christians are to live by. After all, Jesus said it, so of course it applies to us... right?\nBack up just a minute. Do people really consider what Jesus is actually saying here? As I said, these are strong words. Very strong words! Jesus is telling His listeners that GOD WILL NOT FORGIVE THEM if they don't forgive others!\nSo is it true that if we don't forgive others, God won't forgive us? If that's the case, then how many believers are actually forgiven? And if they aren't forgiven, are they actually saved? What if a person dies, unforgiven? I'm just making the point that I don't think Christians really think about all of this stuff when they go around using scriptures like this.\nBut anyway, fortunately there's good news here. Very good news!\nLet's remember that these words of Jesus (and so many of His other words) were spoken for a particular reason at a particular time under a particular covenant to a particular group of people! Without getting into all of that right now (although I'm certainly willing to go there, and I've written about it plenty in the past, such as here and here), in short I'll just say that these words of Jesus were spoken before the Cross, to the Jews who were under the Old Covenant. They do not pertain to Christianity (in our co-resurrected lives with Christ), and in fact they go against New Covenant life. Old Covenant principles (such as \"if you do this, God will bless you,\" and \"if you do that, God will punish you\") were always against us and contrary to us! (see Col 2:14).\nBut now, with the post-Cross resurrected life of Christ, we have a new reality - a New Covenant reality - with teachings that say we are already forgiven of all sins, not based upon what we do or don't do, but rather upon the Blood of Jesus! We're now encouraged and exhorted to forgive others, not commanded to forgive others with the threat of not being forgiven if we don't. Why are we encouraged to forgive others? Because God has already forgiven us completely. Since we've been the recipients of such wonderful mercy and grace, let's turn around and show the same to others.\nWe can now stop worrying about whether or not we're forgiven, based upon whether or not we've forgiven others. Instead, we can fully experience and enjoy the fullness of God's grace and mercy, and our redemption and forgiveness that's not based at all upon what we do, but that's based solely upon the finished work of Jesus!\nGreat stuff, Joel. Right on the mark!\nToo often people are left with the false impression that Scripture can't be FULLY reconciled. Yet under a new covenant economy, there is no conflict.\nThanks for slicing the truth thinly, Brother!\nYep indeed, Mike. I formerly did a lot of head-scratching when I came across verses that didn't seem to make sense together, such as the words of Jesus here and the words of Paul elsewhere. But in light of two completely different covenants, and different reasons for each, it all makes a lot of sense now.\nWhat about the parable of the unmerciful servant, and when Peter asked the Lord how many times should he forgive his brother?\nWe are still capable of offending the Lord. Why would 1 Thess 5:19 warn us about quenching the Holy Spirit? Why would God say \"do not\" if we're already forgiven? Ultimately, why would we even read the bible about sins if we're already forgiven?\nThis strange doctrine is ultimately teaching that we can commit sin and not worry about a thing. This is insane.\nI want to make clear that what I'm not saying here is that we shouldn't forgive others. I'm simply saying that before the Cross, Jesus said to his hearers that God would not forgive them if they did not forgive others, and that is a very, very strong statement! But now, after the Cross, the Blood of Jesus has been shed for the remission of sin, and we stand forgiven because of that. God's forgiveness of us is not based upon our forgiveness of others, nor on any of our other actions, but is based upon His grace and the finished work of Jesus.\nAnd so now, we don't forgive others in order for God to forgive us, but rather we forgive others because we've received such great forgiveness from God, through the once-for-all sacrifice of Jesus.\nWe are now \"the righteousness of God.\" (2 Cor 5:21). The gospel reveals the righteousness of God (Rom 1:17) that has been given to us as a gift, apart from our works. Now, of course we don't always walk according to who we truly are. Again, we are the righteousness of God. That is who we are. That is our new identity, not based upon our actions but based upon the gift of God. And so when we don't \"behave\" according to this new nature of ours, it's not that we're no longer the righteousness of God, and it's not that we're no longer completely forgiven. It's just that we've not fully appropriated the truth of who God says we are and has in fact made us to be.\nThe New Testament exhortations to do good and to not to evil, are not based upon whether or not we remain righteous (again, a gift, not something we do) or forgiven, but rather are based upon the fact that we are righteous and forgiven. Because we are new creations (2 Cor 5:17), joined as one spirit with the Lord (1 Cor 6:17), partaking in the very nature of God (2 Peter 1:4), then lets live like it! If we don't, we're still righteous and forgiven, because it's not based upon what we do, but it's based upon the gift of God.\nThis isn't a license to sin. This is a license to live according to our righteous identity!\nYeah it is a very strong statement. People didn't want to hear these kind of statements. Jesus and the Apostles died (except John) for statements that people didn't want to hear. Jesus IS the new covenant and He would not have said this or told the parable ot the two servants if this didn't also pertain to those under the new covenant. It would have been redundant!\nWell, the prophets also died for making statements - statements that strongly supported the Old Covenant but didn't apply to the New Covenant.\nJesus came as one who was \"born of a woman, born under the law, to redeem those who were under the law.\" A lot of what Jesus said was spoken to those who were under the Old Covenant (the Jews). The Old Covenant had a very big purpose in the grand scheme of God's larger plan, but its purpose was not for Gentiles.\nAnd so we can look at the things that Jesus spoke to the Jews as things that He meant for the Jews, under that covenant, and not as applying to the New Covenant.\nAlong with what I said in my earlier comment, in the New Covenant we are already forgiven of all sin. Jesus' blood has provided for complete forgiveness of all sins. It's not through us having forgiven others that we've been forgiven, but rather through us trusting in the finished work of Christ.\nIf we are forgiven only if we forgive others, then the blood of Christ did not accomplish what it was supposed to accomplish! Thank God that the Blood of Jesus accomplished for us the remission of sins - something that could not be accomplished through the condition of us having to first forgive others, as Jesus told the Jews under the Old Covenant.\nwow maybe we should do away with the 4 gospels because as you say everything he taught was under the law This is ridiculous Grace & Truth came by Jesus He is the last Adam Firstborn from the Dead Hebrews tells he's the Author and The Finisher of out faith in other words he's our example of born again Child of God is supposed to live by the way what happens if I choose not to Forgive???\nI most definitely did not say that everything Jesus taught was law-based. We need to read each thing that He said in context.\nIn our modern day Christianity we take all the words that we read at face value, without taking into consideration the context. There were no Christians around when Jesus taught. He very often really was speaking to Jews who were under the Old Covenant. He really was \"born of a woman, born under the law, to redeem those who were under the law\" (Galatians 4:4). All of this needs to be considered when reading each particular happening in His earthly ministry.\nNo doubt Jesus had New Covenant talk in the things He said at times. But remember, the people were still under the Old Covenant when He walked the earth. Much of His ministry was as a Jew, under the Old Covenant.\nThe Cross, of course, changed everything. But the purpose and intent of the Law\/Old Covenant had to be magnified and understood by people in order to show just how much the New Covenant changed all of it.\nAnonymous, in addition to what I said above, I'll say that you asked a great question: \"What happens if I choose not to forgive?\"\nSo indeed, what happens? According to this teaching from Jesus, you then not forgiven by God, and therefore you cannot be justified and made righteous. So salvation is not only by faith, but it is by what you do. It is not by grace, but it is by your works.\nSo is it really not the blood of Jesus and His resurrection that saves us? Is it really a matter of something that we have to do, in addition to the finished work of Jesus?\nThere is nothing that our works can add to the blood of Jesus. If so, then we are saying that the blood of Jesus was not enough.\nIn the above two verses we see that we are exhorted to forgive others just as the Lord has forgiven us. We see that forgiveness of others is not a requirement for God's forgiveness of us. Rather, it's just the opposite. We forgive others because God has forgiven us. We forgive and show grace to others in response to what God has already provided us with.\n\"12 So speak and so act as those who are to be judged under the law of liberty. 13 For judgment is without mercy to one who has shown no mercy. Mercy triumphs over judgment. 14 What good is it, my brothers, if someone says he has faith but does not have works? Can that faith save him?\" - Jas 2:12-14 ESV (a rhetorical question to which the answer is an emphatic \"No!\".\nThe problem, Joel, is that you fail to understand the essence of being 'born again' as well as what grace really is. The problem of sin is serious. God didn't send his Son to the cross so that we could all get 'Get Out of Hell Free' cards. Grace is not a gift that is like we get on our birthday with pretty wrapping and a bow, that we receive once, open, and it's a done deal...it's the free gift of salvation that's ours forever. Grace is not a thing...it's a resource that's only found 'in Christ'...by faith. And faith isn't simply an intellectual acknowledgement that something is true. Faith MUST produce a change in behavior. That's why the gospel that John the Baptist and Jesus preached began with \"Repent...for the Kingdom of God is at hand\". Jesus died not only for our sins, but also 'for the sins of the whole world'. Well, by your line of reasoning, the WHOLE WORLD will be saved because it has nothing to do with whether we make any changes in our attitudes or behaviors. Why would God make a distinction? But Jesus taught that Salvation was a very narrow path and that not many would get in. Jesus is 'the author of eternal salvation....TO EVERYONE WHO OBEYS HIM'!!!! Simply saying I'm a Christian doesn't make me one. There's overwhelming evidence in the New Testament, even if you exclude the gospels, that there is a threat of jeopardizing our salvation by accepting doctrines that deviate from the original message that Jesus preached: \"REPENT!\"\nYou are also in error to conclude that the New Covenant began AFTER the crucifixion.\nThe old covenant was given on Mount Sinai BEFORE it was ratified with blood. Jesus began his ministry saying \"Repent for the Kingdom of God is HERE...NOW!\" And he demonstrated it by forgiving sins and healing in response to faith combined with contrition.\nThe cheap grace that you are advocating is lulling people into a false sense of security.\nIt appears that you have a LOT of misconceptions about me, and also some things in which we simply disagree upon. I'll address a few of these things.\nYou are right\u2026 this is not the same as the demons who believe. This is an active faith. We are new creations in Christ. We have new desires\u2026 to walk as the new creations that we are!\nIt's true that whoever isn't holy won't see the Lord. But it's not our works that make us holy. It's the blood of Jesus. By the blood of Jesus, we are a holy priesthood (1 Pet 2:5), a royal priesthood and a holy nation (1 Pet 2:9). It's because of this (the fact that we are already holy, apart from our behavior) that we are implored to \"be holy in all your conduct\" (1 Pet 1:15). We don't \"do\" in order to be holy. We \"do\" because we are holy.\nAnd indeed, the problem of sin is of the uttermost seriousness! It's so very serious that God sent Jesus as a human being to shed His blood and die on a cross for us! That's how utterly serious sin is! There's no getting around how serious sin is. And there was absolutely NO other way to deal with it, than with the blood of Jesus. THAT'S how serious the blood of Jesus is!\nTo obey the gospel is to put aside any and all ideas and notions of self-dependence or self-righteousness, and to depend solely upon the finished work of Jesus. Again, it's a serious, serious matter, and God provided on e way as the way to deal with it \u2013 the blood of Jesus. The writer of Hebrews told the Jews that if they reject this one sacrifice, by going back to the blood of bulls and goats, which could never take a way sins but only served as a reminder of sins (Heb 10:3), then they have it even worse than someone who rejected MOSES! (Heb 10:28). By trusting in anything other than the one perfect sacrifice of Jesus, they have trampled the Son of God underfoot and counted the BLOOD of the Covenant by which they were sanctified as a COMMON THING, and have insulted the Spirit of GRACE. That's serious stuff! Rejecting the blood of Jesus by going back to bulls and goats nullifies grace and puts Christ back on the cross over and over again! To \"sin willfully\" (to willfully miss the mark) in the context of Hebrews 10 (and all of Hebrews) means to reject the sacrifice of Jesus, through unbelief.\nAs for the timing of the New Covenant, Jesus most certainly preached the New Covenant at times (and He also preached a LOT of the Old Covenant), and He Himself was God in the flesh and therefore the kingdom was truly at hand, but the New Covenant was not in force until His death, as was testified by the writer of Hebrews in Chapter 9:16-22. \"Without the shedding of blood, there is no remission of sin,\" and so the New Covenant, in which our sins were taken away and remembered no more, could not be in effect until the death of Jesus.\nAll of what I'm saying here, and all of what I shared in the original post, is not cheap grace. It was very costly grace. God became a man and shed His blood for us. (WOW!!!) Through His poverty we became rich. It cost Him a lot, and we receive what He paid for, freely by grace through faith.\nI want you to know that I'm not attacking you. I just take issue with the fact that a definition of grace and 'unconditional love' has crept into the church which causes folks to explain away all the warnings in the New Testament about persisting in carnal living. And that to the extent where, as you have, they take straight-forward statements by Jesus himself and explain them away. Jesus was telling a parable about a guy whose huge debt was forgiven, who then found a guy who owed him a relatively insignificant amount and would not extend the same mercy. Then the parable ends and Jesus explains the meaning in plain Hebrew: \"If you don't forgive, you won't be forgiven.\" If this was the only passage in the NT that indicated that there are some 'gotchas' to this whole Good News thing, I might be more inclined to accept your explanation...but taken all together, I have to conclude that the kind of grace and what constitutes 'being saved' that you are expounding is not scriptural.\nRight before James talks about the kind of faith that doesn't lead to salvation, he says: \"For judgment is without mercy to one who has shown no mercy. Mercy triumphs over judgment.\" - Jas 2:13 ESV It doesn't say 'Judgement is mostly without mercy'...it's without ANY mercy. Jesus says \"The way is hard, and not many find it\". If the faith that leads to salvation were so sloppy that we can just explain away Jesus' warning about not forgiving others, then his statement about few finding the way that leads to life would be meaningless. To me, you're saying that as long as I believe that Jesus death on the cross guarantees my ticket into heaven, then I can hold onto unforgiveness, sleep with hookers, smoke crack...doesn't matter...I believe in Jesus. But here's where I know you'll launch into some circular logic about 'if one really believes, their life will not be able to help itself, it will change'. I think not. Otherwise all of the exhortations about putting the flesh to death, about those who practice these things will not inherit the kingdom, to him who overcomes, and on and on and on...would be meaningless. Why should I work out my salvation with fear and trembling? That sounds like a drag. And how will you explain how Matthew 25 doesn't apply to you and I? It relates directly to what James is saying to believers about their faith...if it doesn't bear fruit, it's on track to get thrown into the fire. God will patiently work with a person who continues to live a carnal, fruitless life to try to get them to repent and start being salt and light, but ultimately, if they don't start producing fruit, they will be thrown into the burn pile. Jesus makes it clear, the epistle writers make it clear. The only ones who don't make it clear are the ones afraid to speak the truth in love. Salvation doesn't occur as the result of a one time decision to repent and to believe in the advent, life, death & resurrection of Jesus. That only puts one on the right path. The medicine only works if we keep taking it. That's why there are so many exhortations about how Christians must live and continue to pursue holiness through confession, repentance and good works. It's not those things that save us. It's ONLY Jesus that saves us. But remaining 'in Christ' requires that we cooperate with the Holy Spirit; that we 'abide in Christ' by letting the Holy Spirit convict us of the sin and selfishness that still lives in our carnal nature and let God's work of salvation have it's way in our hearts. God's love IS unconditional, but his salvation is not. Otherwise, since Jesus died for the sins of the whole world, the whole world would be saved, since it's not God's will that ANY should perish.\nThe writer of Hebrews wrote an entire epistle about the narrow gate. The narrow gate leads to life, Jesus said. There is only one way that leads to life, and that is the blood of Jesus.\nAll throughout the epistle, the writer of Hebrews makes a contrast between two things. In Jesus' words, the contrast was between the narrow gate and the wide gate. In the book of Hebrews, the contrast was made between the blood of bulls and goats and the superior blood of Jesus. The contrast was made between the priesthood of the Levites and the superior priesthood of Jesus. The point all throughout the book of Hebrews is that the finished work of Jesus is superior to anything and everything. Jesus Christ IS the way. Any other way is part of the \"broad\" way. The \"wide\" gate.\nChrist Himself is the narrow gate. We must enter through the narrow gate? Yes! And the narrow gate has nothing to do with our own works, and everything to do with the finished work of Jesus.\nWayne, you're saying that we're ONLY saved by Jesus. But then you're ADDING to Jesus by saying that it's up to us to abide and be convicted of sin and all this other stuff in order to \"remain\" (your own word) in Christ.\nYou are adding to the finished work of Jesus.\nThere's no doubt that we have the ability now, as new creations, to cooperate with the Holy Spirit and walk in step with Him. And as God loves us, we can respond in love to Him! But our response (or lack thereof) doesn't have anything to do with whether or not we \"remain\" in Him. We remain in Him, not because of our works, but because of HIS! And we can respond to His love and grace, and walk it out, all because of HIM!\nIf it's up to US, we're completely and utterly doomed. Break the law even once, and we're guilty of ALL. That's you, me, the entire world!\nThank GOD for His grace that not only saved us, but KEEPS us! As we grow in GRACE (not our own works), we grow in so many wonderful things that could never be accomplished through our own attempts at trying to maintain our salvation.\nSalvation is God's life in us, for now and evermore. It's not our works that we do for Him, it's His work that He's done for us, and that He Himself continues to do in and through us.\nBack to the book of Hebrews. The writer was a Jew and he was writing to Jews. All their life, up until the cross of Jesus Christ, what they had known was law, law and more law. As far as they knew, they had to keep up this \"law\" thing. Their hope was in the blood of bulls and goats. Even though they sinned, and fell short of the glory of God, their hope was in the yearly sacrifice of bulls and goats. Once a year, their sins were transferred to a \"scapegoat\" (Leviticus 16), and they were good for another year. But yet, as the writer of Hebrews says, the problem was that while this \"atoned\" for their sins, the blood of bulls and goats could never TAKE AWAY their sins (Hebrews 10:4). And in fact, it only served as a \"REMINDER of sins every year\" (Hebrews 10:3).\nBut along came Jesus, sent by God to not only \"atone\" for sins, but to TAKE THEM AWAY. It was so much more than what was accomplished through a bull or goat. It was a PERFECT sacrifice, and it took away ALL sin, ONCE AND FOR ALL.\nTHIS is the Jesus that we proclaim!! The Jesus through Whom sin was not only atoned for, for a year, but through Whom our sin was TAKEN AWAY! This is the reality of what was accomplished through the blood of Jesus.\nOur works are so very TINY compared to the finished work of Jesus! NOTHING compares to what Jesus did! Our \"work\" is to believe in Him! (John 6:29). \"HE HIMSELF is the PROPITIATION for our sins\" (1 John 2:2). We can't do anything to match or add to what Jesus did. We only believe, and receive freely... as a gift... what was accomplished through HIS work.\nSo now, if you want to keep trying to add your own attempts at remaining in Him through something you DO... go for it! As for me, I'm resting in what He has done and in what He has freely given me.\nThis is grace. This is the very power of God in me that has nothing to do with licentiousness, but rather has everything to do with LIFE.\nI think I know where Wayne is coming from. He's worried about many in the church using grace as a \" license to sin\". I could elaborate but I'm going to try and get directly to the point using an example in scripture. In galatians as well as Acts it appears to me Peter \"used\" grace as a \"license to sin\" by basically being racist and not walking according to the gospel by not mingling with Gentiles at least up until his vision and NOTE even after playing a hypocrite so Judaizers wouldn't know he began to do this even though according to the law he wasn't supposed to (interesting this one of the DO NOTS in the law). Yet God was always with him. So I think this bears out what Paul was saying that \"if righteousness comes by the law then Christ died in vain.\" What other sins do we see in the church \"worse \" than this? Do we tell people you need to do such and such to stay saved or do we tell them to trust in Jesus' blood and let the Holy Spirit worry about their works?\nBy the way\u2026I am finding it a bit tough to read your words with the translucent backdrop\u2026not complaining, just a tip.\nHey there LS... great to hear from you, as always! Spot on with your comments. Indeed, it's not about self-help or principles or morality, but rather about the very life of Christ being \"formed\" (\"fashioned\") in us (Gal 4:19). Even \"values\" misses the point of the life of Christ in us.\nDo you think that the \"formation\" and \"fashioning\" seems to \"mystical\" to some people that have a felt need for a check list?\nThank you so much Joel! Since delving into trying to understand the difference between the covenants, I often wondered about this verse. You are the first person who has explained it the way that I suspected it might be!\nFor years, I never thought to question the words of Jesus and if they apply to me as a Christian or not; I was taught they all apply. I agree with you now... we must always look to who the audience is when we read scriptures. (And I'm not trying to be rebellious).\nUnfortunately (fortunately?), this changes many things. So... what is your take on the Beatitudes?\nBTW - I have a friend named J Slick and you were really confusing me for awhile. I wondered if it was him - until I watched a video!\nLS, I think you're onto something there. A checklist is so much easier for some people to follow, rather than having the ongoing experience of the actual living life of Christ at work in them, being formed\/shaped\/fashioned in them on an ongoing basis.\nCathy, I think you are so very right... this changes many things! And indeed, this look at things is not out of rebellion, but out of a real desire to understand Christ and His ministry while He was on the earth, compared to His different ministry to us as He lives in and through us, and as He in fact has become our very life.\nI just recently was part of a conversation on Facebook with someone who believes we are only to follow the words of Paul. (And not Peter, John, James, etc). I don't fully agree with him about all of that, but his main point was that Jesus had a pre-Cross ministry to the Jews, and that much of what He said was indeed His ministry to the Jews and not to anyone else.\nSo many of His words were spoken to them alone and for them alone, as part of God's overall ministry to them. Now, I do think that Jesus did also speak New Covenant things at times, as He looked ahead to what would ultimately be accomplished through the Cross and His resurrection from the dead, and we can glean so much out of all of that.\nI'm still in an ongoing process of working a lot of this stuff out, so I don't have a specific answer about the Beatitudes. But I will say that my podcast host and I had some things to say about the teachings of Jesus that will perhaps not sit well with many in the church today. :) Again, as you say, this is not to be rebellious, but it's an honest look at what we truly think Jesus' earthly ministry was about, as compared to the post-Cross reality of life in Him.\nThe podcast is entitled \"The Unattainable, Insurmountable, Unachievable Teachings of Jesus.\" :) (You can click on the link if you want to hear it. It will take you to our podcast page).\nThat's right, Anonymous, we have been forgiven (past tense) of all sins - past, present and future! Through one sacrifice, 2,000 years ago, all our sin was taken away.\nSo you're saying that Jesus taught the Lord's Prayer as sort of a temporary prayer? I think there's a much more sound explanation. If forgiveness is once for all, then to withhold forgiveness is to choose to not live under forgiveness. We either accept or reject forgiveness, for all.\nWhat I'm saying is that what is known as \"the Lord's Prayer\" is an Old Covenant prayer.\nThe New Covenant didn't begin until the death of Jesus. As I said earlier, Jesus was \"born of a woman, born under the law, to redeem those who were under the law.\" (Gal 4:4-5). He first had a ministry that was specifically to the Jews (those who were under the law), and much of what He said was directed to them, and was part of that Old Covenant ministry.\nIn order to see more of a foundation of what I'm saying, pay attention to the audience Jesus is speaking to each time you read the accounts of His life in Matthew, Mark, Luke and John. Put yourself in the shoes of those He is speaking to in each given situation. Most of the time, He was speaking to Jews, and they understood His words as words that were exclusively to them. There wasn't yet a thing called Christianity, nor was there a New Covenant. There was still a reason for Jesus ministering under the Old Covenant.\nThe Old Covenant led to the New Covenant, but the Old Covenant is not the New Covenant. The New Covenant is an entirely different covenant. While there were specific reasons for the Old Covenant, everything changed with the New Covenant.\nWhere did you get the idea that The Lords prayer was the old covenant prayer? In context The disciples asked Jesus how to pray because they witnessed how effective his prayer life was If its an old covenant prayer show me where in the old testament people prayed like that The jews where God's chosen people born again people are God's Sons & daughters That's why he refer's God as our Heavenly Father Guys be careful of false doctrine!\nWhere in the Old Covenant were they told to cut off their right hand if it caused them to sin, or pluck out their right eye if it caused them to sin? But yet I don't see any New Covenant Christians living that way. :) Is your righteousness greater than that of the Pharisees and scribes? Are you perfect as your Father in heaven is perfect?\nThese are questions that everyone needs to answer before calling the Sermon on the Mount (which includes what is known as the Lord's Prayer) a teaching for Christians.\nGood word again Joel!! Now in the New Covenant in the Blood of Jesus (this side of the cross)we forgive BECAUSE we have BEEN forgiven!Eph 4:32....& Sin no longer has DOMINION over us Because of Our New nature (the nature of God).\nCame across your blog while doing a search on this topic. Really appreciate your thoughts on this...Hallelujah! Very well written and explained...Thank you!\nI Ran across your thread when I was studying out some things I've been learning about the mixture of covenants we have been taught. My question for you is can you help me understand Mark 11? I understand Mark 11:25 is not new covenant thinking and was written to people before the death of Christ therefore it is not considered New covenant and I agree with that.\nThat's really a great question. I think my best answer to that at this time would be to say that all is fulfilled in Jesus.\n1John 1:9 is confusing me.\nIs this verse is for us christians?\nIt appears to me that the first chapter of 1 John was written to address the unbelievers in the church that John was writing to. Rather than explaining it all here, I'll direct you to a series I wrote about this, entitled \"One John One Nine.\" I hope this will be of help to you.\nIt's doubtful that there were many, if any, unbelievers in the church when this letter was written. The 'church' is by definition an 'assembly of believers'. It wasn't the practice then to invite people to church to get them saved. By identifying yourself as a disciple of Christ in John's time you invited persecution unlike the situation today where identifying yourself as a Christian might only elicit derision from the popular media who justly see Christianity as a bunch of self-righteous hypocrites, based on the public buffoonery that so many celebrities and television preachers have displayed. The entire New Testament was written for the benefit of believers, not unbelievers. Again, Jesus died not only for our sins (believers), but also for the sins of the whole world. So, if only a few will be saved (according to Jesus), what is the distinction, Joel? Let's just become Universalists. After all why wouldn't this schmaltzy, sentimental grace that you preach apply to everyone who sincerely clings to a faith regardless of it's basis in fact or fiction? Why would there be so many references to false doctrine in the New Testament? According to your gospel, doctrine shouldn't matter either...in fact, it would seem, nothing really matters. The demons believe, so they must be among the redeemed as well!\nThe word \"church,\" in and of itself, is the word ekklesia, and it simply means an assembly or an assembling of people together. Indeed, the purpose of \"the\" church assembling together was so that believers could encourage one another and spur one another on towards love and good deeds, etc. But at the time John wrote his epistle, it's been historically noted that there were unbelievers known as Gnostics who had also become part of some of the assemblies, and John was addressing them.\n\"Gnosticism was a teaching that blended Eastern mysticism with Greek dualism (which claimed that the spirit is completely good, but matter is completely evil)... Based on the concept that matter is evil and spirit is good, some Gnostics concluded that if God was truly good He could not have created the material universe. Therefore, some lesser god had to have created it... The dualistic views of Gnosticism were also reflected in the prevalent belief that Jesus did not have a physical body.\"\nHe is telling the Gnostics that indeed they had seen and touched Jesus - He had truly come in the flesh - and then he said he was declaring this to them so that \"you also may have fellowship with us; and truly our fellowship is with the Father and with His Son Jesus Christ.\" He then went on in the last few verses to tell the unbelievers\/Gnostics how it is that they could come into the light and be forgiven and cleansed of all unrighteousness.\nIt's not until the second chapter that John redirects his thoughts and begins talking to the believers. \"My little children...\" And he even tells them quite directly, \"I write to you, little children, because your sins are forgiven you for His name's sake.\" (1 John 2:12).\nSo John was writing to a mixed crowd\u2026 believers and unbelievers\u2026 who were assembling together, and at the time each group knew what was being addressed to whom.\nJesus tells the churches in Revelation to repent. He's speaking to believers. We are to strive for holiness 'without which no one will see the Lord'. We 'have not yet struggled against sin to the point of shedding blood...'. Etc. Wheat and tares, brother. Why would Jesus tell us to disfellowship 'brothers' who won't listen even to the church after they've been confronted with the truth about their sin? Sin is definitely not a non-issue for believers.\nThe Gnostics didn't even believe that sin was an issue, so they couldn't believe that they needed to be cleansed of sin, and thereby they were rejecting the true Jesus. They didn't believe in Christ, or His sacrifice for sin. John wrote to address this.\nI didn't say that sin is a non issue for believers. In fact, above I quoted Peter who called upon believers to be holy in their conduct! I think you are assuming certain things about me that just aren't true.\nExcellent commentary, Joel! I understand that Scripture so much better now! God bless you.\nAlthough it's true that Jesus blood is associated with sanctification (I find one instance), it is overwhelmingly more associated with the same process whereby we are exhorted to \"work our your salvation with fear and trembling\". Otherwise it would be meaningless to to tell us to \"strive for the holiness without which no one will see the Lord\". Grace isn't a 'free pass' or a thing that we receive once and then stick in our pocket and present whenever the cops show up, it's a resource. It's the means through which we are able to work out our salvation. And our salvation is a gift which receive 'by grace, through faith'. And if faith is something we do (ala James), then grace is only received in conjunction with action on our part. A 'gift', yes, but not one that is received without conditions. The medicine is placed on the counter in front of us. It is a prescription with unlimited refills, but we must come boldly to the pharmacy of grace when we run out or the cure will not take its course. I know this sounds like salvation by works, but it's not. God is the \"Author and Finisher of our faith\", but just as we must respond to the conviction of the Holy Spirit when we are first made to understand the Gospel and drawn into that place of decision to accept or reject the gift of salvation, we must continue to respond to the Holy Spirit as he seeks to complete the work. We do it in fits and starts and God's grace is always sufficient, but our salvation doesn't happen without our participation...regardless of what 8 Point Calvinists believe.\nWithout holiness no one will see the Lord does not mean the Lord won't see you. It means others may struggle seeing Him in your life or you may be deceived by sin and not see the true you in His Blood.....Abraham's messy life would've caused most purity seekers to think he didn't know the Lord..ie..they couldn't see it....yet God saw what no eye had seen, what no ear had heard, what He prepared for those who really loved God. Because He first loved them...The sign of circumcision came AFTER not before. ..Most seek purity over the Pure One and wonder why they keep stumbling over the Stone in Zion and never enter God's Rest....try Psalms 32:1-2...try Romans 4:7-8....when Jesus said consequently the sons are EXEMPT He was allegorically referring to what God has been trying to say through Joel and others...Grace is first a Noun...a Person...JESUS! GO FIGURE.\nexcellent stuff Joel....Praise be to Jesus who has rescued my soul.\nThis is one of those verses that can't be taken alone, by itself, but we need the rest of the epistle, as well as the rest of scripture.\nIn Chapter 2, John says, \"These things I write to you that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous.\"\nJohn knows that believers will sin. I believe that what he's talking about in Chapter 5 is the intentional practice of sinning. A believer will sin, perhaps many times, but it's not out of a will or desire to sin. It may be out of weakness, but it's not out of a desire to defy God. In our spirit, which is the core of who we are, we are spotless and blameless, and we have no desire to sin and we are one with God. Because of this, our aim and goal is never sin, but rather we live from a state of righteousness (righteousness that we've received as a gift).\nI agree. I truly feel that the Holy Spirit has shown me this Truth.\nWhat about the scriptures where Jesus says, \" I have not come to take away the Old Law, but to add too.\"\nAnd then, those also, presented the scriptures of Matthew and James about bearing fruits, and faith without works?\nI don't let others shake my faith and i know it is Truth as the Holy Spirit dwells and gives discernment. I just don't find scripture to show absolute for them. I just know Jesus said, It is Finished. He Conquered Sin. When we accept him in our hearts we are saved.\nHe says, he remebers out sins no more.\nWhat is the Judgement Day for? I believe its Only about Do You Believe, or not? He says, every knee shall bow and every mouth shall confess,that I am Jesus Christ Your Lord and Savior.\nIt is so simple, but Its made confusing and contradictive,by so many.The Holy Spirit convicts and brings forth a repentance and Praise and Compassion and Love. You don't change by your might and power, You change by the Holy Spirit's Power living within you. Thats the faith alive (not faith without works)bearing fruit. Anyway, We each have our own Walk, Our own Choice, Our own Measure of faith, to leave as the measure he gave us or to add to it and grow and learn and receive as well as give and share the word and receive the word. Learn about receiving JOY as we have Sorrows and Trials and Judgements and Demands from others.\nIndeed, Jesus said \"It is finished,\" and He meant it and it was true. His shed blood took away our sin (not just covered it temporarily, but actually took it away forever). With His finished work, the law was \"wiped out,\" \"taken out of the way\" and \"nailed to the cross.\" (Col 2:14).\nWhat Jesus had previously said was, \"Do not think that I have come to destroy the Law or the Prophets. I did not come to destroy but to fulfill.\"\nAnd that's exactly what He did. He fulfilled the Law and the Prophets. After that, the law was then \"wiped out,\" \"taken out of the way\" and \"nailed to the cross.\" But He had to fulfill it first.\nAs believers who have been born again, we do abide in Him, and He abides in us. He is the Vine, we are the branches. Branches don't work to produce fruit. Branches abide in the Vine, which is where the Life comes from. It's God's fruit that He produces in and through us. It's not our fruit that we strive to produce for Him.\nIn James, when he says that faith without works is dead, he's not talking about works of the law. The two examples James uses are Abraham offering Isaac, and Rahab the harlot protecting spies. The main point of is that there is a response that comes with faith that is alive, but it's not about trying to do works in order to prove anything.\nAs to Judgment Day, there are two groups of people on that day - sheep and goats. Since there is no condemnation for those who are in Christ Jesus, there will be no judgment or condemnation of them on that day either. They will be with the Lord forever. For those who don't believe, Judgment Day will be very bad.\nWOW!!!! How confusing (for me) it is to read all of this. I guess my foremost thought is, if Jesus is not talking to us why do I need to read what does not pertain to me? I am absolutely no bible scholar, but I didn't think God was the God of confusion.\nIndeed, God is not the author of confusion. The confusion comes when we don't look at the words of Jesus in the context that I wrote about in this blog post, as well as the articles that I linked to within this blog post.\nHere's the point: None of this is confusing if we look at it all in context. The law was given to Israel, and it was exclusive to them. Gentiles were aliens and strangers from it (See Eph 2:12). It's the same with many of the words of Jesus. He was ministering to the people of Israel who were still under the Old Covenant. When He said, \"For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses,\" He was speaking Old Covenant talk. One huge reason why we know this is because in the New Covenant, we've not been forgiven based upon whether or not we've forgiven others, but rather we've got tons of verses that say that we've been forgiven based upon the blood of Jesus, and our faith in His finished work.\nGreat testimony, Adams! The truth will always make us free, and it's wonderful to know that the truth of the New Covenant has set you free in regards to forgiveness.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"On September 26, Ambassador Liu Xiaoming met at the Chinese Embassy with Cheung Kin-chung, Chief Secretary for Administration of Hong Kong SAR Government and his delegation that is visiting the UK.\nAmbassador Liu noted that since its return to China, Hong Kong has made remarkable achievements, including the newly-opened Guangzhou\u2013Shenzhen\u2013Hong Kong Express Rail Link, the successful completion of Hong Kong-Zhuhai-Macau Bridge, and the full-scale construction of Guangdong-Hong Kong-Macau Big Bay Area. This year marks the 40th anniversary of China's reform and opening up, in the process of which, Hong Kong has been playing its special role. He hoped that Secretary Cheung could share the success stories of \"One Country, Two Systems\" and Hong Kong's prosperity in various fields to the British society.\nThe Ambassador pointed out that at present, China-UK relations on the whole are developing well, and Hong Kong has increasingly become a bridge between China and Britain. He hoped that Hong Kong would give full play to its own strengths in financial and legal services and seize the historical opportunities of Belt and Road Initiative and the construction of Guangdong-Hong Kong-Macau Big Bay Area to accelerate its development and help China-UK relations to bear more \"Golden Fruits\" in the \"Golden Era\".\nAmbassador Liu stressed that the recent decision of the Hong Kong SAR government to ban the Hong Kong National Party in accordance with relevant laws and legal procedures is a necessary move to safeguard national security and Hong Kong's rule of law, and we strongly support it. We firmly oppose the irresponsible remarks of some foreign governments and call on them to respect the Hong Kong SAR government's decision to outlaw the \"Hong Kong independence\" group, stop interfering in Hong Kong's affairs and China's internal affairs, and fulfill their commitments of not supporting \"Hong Kong independence\" through concrete actions.\nSecretary Cheung said that he was born in Hong Kong and has been working there for nearly 50 years. He has witnessed the progress in Hong Kong. With the full support of the Central Government, the various undertakings in Hong Kong have been advanced steadily, and the development of innovation and technology is particularly rapid. Being a fervent participant and a beneficiary of China' reform and opening up, Hong Kong will make the most of the close links with the Mainland to carry out closer cooperation. He expressed his thanks to the Central Government for its effective measures to oppose and stop the foreign intervention in Hong Kong's affairs to guarantee Hong Kong's long-term prosperity and stability. Secretary Cheung said that during his visit in the UK, he would show Hong Kong's achievements comprehensively to people from all walks of life in the UK in order to enhance understanding and reduce misunderstandings. Looking ahead, Hong Kong stands ready to participate in the Belt and Road Initiative and the construction of Guangdong-Hong Kong-Macao Big Bay Area to contribute to the development of the country and China-UK \"Golden Era\".","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"\"Can the ICC be certain that the Project will remain a 100% merchant project that will be paid for only by persons who voluntarily agree to use the line?\n\"Is the Project currently at a stage where a lender, be it a conventional or project finance lender, would be expected to commit to lend the required additional capital to complete the Project?\n\"If I may offer an analogy, I can say with great certainty that a person wishing to become a violinist in the Chicago Symphony Orchestra needs to acquire a violin, find a violin instructor, apply him or herself conscientiously to years of lessons and practice, and apply for an audition for an open seat at the orchestra. It would be misleading, however, to say that any individual who follows those steps will with great certainty gain a seat in the violin section of the Chicago Symphony Orchestra.\n\"Q. Do you believe the cost of RICL's proposed project meets the least-cost criterion that is defined in Section 8-406 of the Act?\n\"Q. Has RICL demonstrated that the RICL Project's benefits outweigh its costs so that, on net, it is expected to be beneficial to Illinois consumers?\n\"Which, if any, of the impacts that you just listed would you exclude from an economic analysis of the benefits and costs of the project?\n\"Q. Why would you exclude an increase in employment, revenues of manufacturing and service enterprises, landowners' wealth, and tax revenues from an economic analysis of the benefits and costs of the project?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Ahead of World Water Day, we interview Tim Wainwright, Chief Executive of WaterAid, an international non-profit organization founded in 1981, to find out about how the not-for-profit tackles the water crisis around the world.\nQuestion: Firstly, we would like to know briefly your career path and your current role in WaterAid.\nAnswer: I was delighted to join WaterAid as Chief Executive in May 2017. I've spent most of my career working in international development, after starting out in operational research and marketing at British Airways. I've worked in Beijing and Bangkok, as well as here in the UK, with Oxfam and VSO, and before joining WaterAid I was chief executive of a smaller charity working on disability in development. I've also been a chair of Bond, the consortium of UK international development agencies.\nI really believe water, sanitation and hygiene are the foundation of all progress in development. If we are going to reach everyone, everywhere with services that last, this means we need to work with civil society, governments and the private sector \u2013 everyone has a role to play.\nI really believe water, sanitation and hygiene are the foundation of all progress in development.\nQ: What type of activities does WaterAid carry out related to water and sanitation?\nA: WaterAid works in 28 countries across South and Southeast Asia, Sub-Saharan Africa, Latin America and Oceania to deliver clean water, decent toilets and good hygiene. We work with local partners to help provide services to households, communities, schools and healthcare centres, and work with governments at all levels and the private sector to try to create large-scale change which will help fulfil the UN Sustainable Development Goal 6, which pledges water and sanitation for everyone, everywhere by 2030.\nSince 1981, WaterAid has reached 25.8 million people with clean water and 25.1 million with decent toilets.\nQ: What are WaterAid's main achievements since its foundation in 1981? And what are the goals your organization hopes to achieve this coming year?\nA: WaterAid is working to make clean water, decent toilets and good hygiene normal for everyone, everywhere within a generation. Since 1981, WaterAid has reached 25.8 million people with clean water and 25.1 million with decent toilets. And we currently work in 34 countries, including 28 countries with programme work, and offices to support programming, advocacy and fundraising in Sweden, Australia, Japan, Canada and the United States as well as the UK.\nQ: What are the main difficulties WaterAid finds in providing access to clean water and sanitation?\nA: We work in some very remote and rural areas, so we do face some challenges. First is the challenge of accessing water in areas that are very hard to reach: some of the communities we work in are accessible only on foot, so it's hard to bring in machinery to drill deep wells. But the bigger challenges often come in convincing governments that water, sanitation and hygiene are so important that they need to be a political and financial priority. Most politicians are happy to open a new road or a new school; it isn't as exciting to open a new set of community latrines! But we know that water, sanitation and hygiene are critical to good health, education and better productivity for communities and entire countries, and working with our partners, we are making progress.\nWe believe that everyone has a role to play if we are to reach everyone everywhere by 2030 with clean water and decent toilets.\nQ: What partners does WaterAid work with to deliver clean water and decent toilets?\nA: We work with partners at all levels \u2013 from local to regional to national governments, local water utilities, civil society groups, other non-governmental organisations, UN bodies and the private sector. We believe that everyone has a role to play if we are to reach everyone everywhere by 2030 with clean water and decent toilets.\nQ: With regards to World Water Day, why do you think it necessary to highlight the marginalized groups' difficulty in accessing safe water?\nA: It's so important to highlight the needs of the poorest and most marginalised people because they are the ones who suffer most when there is no safe water or decent toilets, and also, they are the ones most likely to be left out if there isn't enough water to go around. Women and girls, older people, disabled people, people who live in remote and rural locations, and other people who face discrimination because of their caste, religion or any other reason all face extra challenges when it comes to accessing clean water and decent toilets, and may face extra health or educational challenges when these are not available.For instance, we've worked in communities where disabled people are literally hidden away because they're believed to be cursed or to bring bad luck; a really important part of our work is ensuring those myths are dispelled and that their voices and needs are heard, included and met. Our goal is to ensure that no one is left behind: a community only really benefits from clean water and good sanitation if everyone is reached.\nQ: Does WaterAid carry out educational projects related to water, hygiene and sanitation? And if so, which are the most relevant?\nA: Schools are an important part of our community work on water, decent toilets and good hygiene. Children who are thirsty at school have difficulty learning; they may not be able to concentrate if they know there is no safe, private place to go to the toilet when they need to, and children who are frequently unwell from dirty water also miss many lessons. As girls grow older and begin menstruating, having access to a decent, private toilet and clean water becomes even more important so they can care for themselves properly at school \u2013 otherwise, they may simply stay at home, missing lessons, falling behind and heightening the risk that they may drop out completely. In addition to ensuring schools have a clean water supply and decent, separate toilets for boys and girls, we also create school hygiene clubs where pupils can learn about the health benefits of good hygiene and how to keep clean. They often take these lessons home to ensure everyone in a family benefits.\nOur goal is to ensure that no one is left behind: a community only really benefits from clean water and good sanitation if everyone is reached.\nA: We need everyone to play a role if we are to reach everyone, everywhere with clean water and decent toilets. We know it's possible but we'll need political priority and financing to get there. We want people to support our work in any way they can: this might be going on a World Water Day walk, holding a bake sale or participating in a sporting event to fundraise and raise awareness, or writing a letter to their MP showing their support for international aid for water, sanitation and hygiene. It can also include making careful choices in food and clothing purchases, thinking about what the water 'footprint' of those items might be, and choosing things that are less water-intensive, and produced by companies that can demonstrate they are being responsible about their use of water in production.And don't take our own water supply for granted \u2013 the UK will soon be facing its own problems with water scarcity so be thoughtful about your own use of water: don't waste water by taking long showers or letting the tap run during toothbrushing.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgqqi b/data_all_eng_slimpj/shuffled/split2/finalzzzbgqqi new file mode 100644 index 0000000000000000000000000000000000000000..4725ae30d99d4cc2625adbbafa7bb2377223d56b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgqqi @@ -0,0 +1,5 @@ +{"text":"A percentage of all the proceeds from the CD sale will be donated to the music department at Dilworth Academy, Pittsburgh Public Schools elementary Arts & Humanities magnet school located in Highland Park.\nSave a few dollars on the CD and know that your purchase will help benefit elementary music education right in your back yard.\nThe pre-release sale is for a limited time only! Order your CD today!\n*The pre-release sale will only last for 3 weeks. Then allow 4-6 weeks for delivery.\nRSVP for pre-release sale to add comments!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Looking for a SPLIT-function in MySQL, I came across this one. I tried it and I must have have done something not right, because MySQL threw an error at the function. I am not a MySQL guru and since this is a one time Q&D conversion-action, I only took the SUBSTRING code and created a query with which one can split the contents of an old field into separate columns and directly insert the results into a new, normalized table.\nMy example is about a TEXT-column I want to get rid of and of which I want to transfer the contents to a separate table. This column contains email addresses separated by a comma. Thus, first, I had to find the maximum number of email addresses used in that column, so I found this query and added MAX() around it.\nNow that I have all used email address associated with the IDs of the original rows, I can now delete the old column and change all my LIKE-queries into LEFT JOINs. Much better, because email addresses change.\nA new project and a server in the basement, drove us to research how we could replicate MySQL databases of two other servers onto the one in the basement. Quest: we need multiple instances of MySQL on the server, with each its own setup. Our findings and how to set this up on Mac OS X Client (your regular desktop Mac) or Mac OS X Server (your co-located Xserve, Mac Pro or Mac Mini Server) are described on this page.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"PACIFIC OCEAN (Mar. 04, 2019) - Naval Aircrewman (Helicopter) 2nd Class Brenden Pieszak hoists Naval Aircrewman (Helicopter) 2nd Class Ty Nichols, both assigned to the \"Island Knights\" of Helicopter Sea Combat Squadron (HSC) 25, from the deck of the MV Arcadia while providing emergency assistance to a cruise passenger in distress during a MEDEVAC. HSC-25 provides a multi-mission rotary wing capability for units in the U.S. 7th Fleet area of operations and maintains a Guam-based 24-hour search-and-rescue and medical evacuation capability, directly supporting U.S. Coast Guard and Joint Region Marianas. HSC-25 is the Navy's only forward-deployed MH-60S expeditionary squadron.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The fabulous clothing types of the 20th Century may possibly have been relegated to the archives history, with a handful of bespoke ones good only as show in the museums of classic arts, but a very good quantity of trendy types like the poodle skirt (1 of the prime fashion styles of the 70s) and the trumpet skirts of the fifties are reappearing, albeit in modified forms, on the fashion runway. My mother was former fashion designer Anne Fogarty, and it is to my dismay that I did not stick to in her footsteps (I am a teacher). Music is a representation of fashion that expresses the abstract style idea into relatable harmony for viewers. Donated or borrowed fashions could come from style students at a nearby fashion college , a fashion boutique, or even your own closet.\nStyle Me Girl lets you shop thousands of distinct designs, select different poses to create awesome snapshots, decorate your pictures with accessories, unlock fashion genres and play mini games for coins. Protected, comfortable versatile and fashion fierce perform boots from factory floor to fabulous. Alumna Audra Noyes describes how the training she received as a student helped open doors to positions at Jeanne Lanvin and John Galliano, and eventually led to the debut of her personal luxury line for the duration of Paris Fashion Week.\nSince fakes are distinguishable by their poorer good quality, there is nevertheless a demand for luxury goods, and as only a trademark or logo can be copyrighted, many fashion brands make this a single of the most visible aspects of the garment or accessory. In the 80s jeans went from wise, sporty, or a casual ranch weekend put on to a hot fashion trend that became glamorous wardrobe need to haves.\nFashion trend is influenced by various aspects and incidents happening in our atmosphere. The designer was excited about this comfort but also about the artistic freedom it supplied her in producing complex geometric styles central to her show that would have been nearly not possible to develop any other way. Following reading 20 or far more blogs and magazines, per day, I came up with a collection of fashion tricks and tips. Most of the political events, laws, variables are affecting fashion in a way of generating fashion trend.\nAnd as fashion appears cyclic, we can be bold sufficient to don special designs from any era. Every thing should be put in spot and deadlines need to be met, just like it ought to be if functioning as a young fashion designer who has the job to full a mission. I agree on the fact that use of organic cotton by fashion business can modify the globe for better. Playing the game is like a sort of quest that need to be implemented to completion, and it may possibly take some time to comprehensive which tends to make fashion designing games much far more fun to play.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Brightcove technology partner Kiosked offers a platform for creating, distributing and tracking interactive and shoppable video.\nVisual content is at the heart of content marketing. In fact, eConsultancy recently published a study stating that almost three quarters of digital marketers agree that brands are becoming publishers. Digital media, and especially video, helps brands to develop their relationship with consumers. Also, according to Internet Retailer, 52 percent of consumers say that watching product videos makes them more confident in their online purchasing decisions. Beyond that, people who watch product videos are 85 percent more likely to purchase the product. Purchase impulses are born independent of time and place, a phenomenon we refer to as \"everywhere commerce.\" Below, we've outlined the related trends that you need to know about everywhere commerce.\nConnecting with consumers in a meaningful way is key to growing sales and strengthening customer loyalty. Content marketing is not just a much-used buzz word; it's an ideal way for brands and online merchants to make important connections with their customers. Content marketing means connecting with consumers in meaningful ways with branded or subtly branded content that tells your story with an authentic voice.\nIn fact, at the 2013 Digiday Agency Summit, Razorfish North American leader Shannon Denton spoke about the imminent collision between commerce and content as commerce becomes the front door for the brand. As Denton explains it, online retailers must break down the silos between commerce and marketing, ultimately creating an ever-present shopping experience for the consumer: \"We have to make the commerce experience the best brand experience, too. We have to reimagine what an online store looks like, and we have to think about how we work storytelling into commerce.\"\nDeliver your content to consumers where they are \u2013 don't make the consumer come to you. It is crucial for your consumers to be able to access content anytime and on any device. Creating an optimal cross-screen experience is more viable than ever with responsive design.\nThe shopping experience must be ubiquitous. Consumers actually want to connect with brands and products, but they want this connection to be meaningful, in the places they want to engage and when they want to interact.\nAt the heart of your content strategy should be visual content\u2014images and videos. And to truly make your content a valuable resource, it must produce results (in retail speak this means sales!). Interactive, shoppable, visual content is one of the most promising trends in content marketing for retailers. A significant and meaningful shift in shoppable content can be seen with video. Video is moving from passive brand awareness to active sales channel through the use of interactive video.\nThe challenge of integrating a great brand story with your e-commerce model, or monetizing your media if you are a publisher, is possible through advanced technology that brings visual content and commerce together for a seamless shopping experience. Content marketing, interactive video, and the ubiquitous shopping experience should be part of any online merchant's marketing strategy. To learn more, register here for our webinar about interactive video which is taking place on February 18, 2014.\nJaakko Iso-Jarvenpaa is a Senior Partner Manager for Kiosked, based in London - he is building a global partner network of companies who share Kiosked's mission to turn all visual content online smarter. Prior to joining Kiosked, Jaakko worked at Google training digital and media agencies on how to better utilise Google's advertising solutions for their clients. Jaakko received his M.Sc. in Business from Turku School of Economics in Finland with focus on network theories and systems thinking. He is passionate about combining analytics and data with design thinking and behavioural science. To contact Jaakko and learn more about Kiosked, email him at jaakko@kiosked.com, follow on Twitter at @jaakko_ij or visit www.kiosked.com.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgrbl b/data_all_eng_slimpj/shuffled/split2/finalzzzbgrbl new file mode 100644 index 0000000000000000000000000000000000000000..f1a003a81b2d0fc22f89978ae55cbf133682a173 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgrbl @@ -0,0 +1,5 @@ +{"text":"Growing up, we often come to the table of faith with preconceived notions of God in our heads. I pictured him resembling Gandalf from Lord of the Rings, complete with a long flowing beard and walking staff. Other people pictured him as a kindly father, graciously delighting in his children. Others might have looked upon God with a more negative image, as a grumpy old man or someone far worse.\nThe psalmist in Psalm 54 (and elsewhere) saw the wickedness in the world and could not look anywhere else but to God. Even though he was surrounded by his enemies and had fallen into despair, he knew that his only hope was in God. Though he was poor and needy, he found solace in God because it was the only place he could have turned.\nWe in Christ must also cling to our faithful Father with the same tenacity as the psalmist. And it is in this reliance on God that we have the grace and room to give to our community. In Matthew 25:14-30, we find the two diligent workers and the one not so diligent worker. While the first two were congratulated and praised for their faithfulness in a \"few things,\" the last worker was rejected. Did you catch the worker's excuse? It was his belief that the Master was a harsh man and thus did not trade it for more.\nI wonder how our view of God might affect our ability to live out the transformation that came about through the Gospel. Even the view of a Gandalf in the sky can cause a different view of our lives here and now. Instead of looking to the Lamb who was slain and clinging onto the hope that is tied to Him, we might choose to live in fear. Instead of giving generously in light of the Lamb, we hold onto our lives with a clenched fist, believing the lie that God is harsh.\nGod is love, and He cared for us so much that He walked on this Earth. Let us also walk in the steps of the Messiah this season and beyond.\nWhat was your picture of God as a child?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Live, Eat, Feel Better-for FREE; RIGHT NOW!\nThis works very well for pregnancy sickness - I don't call it morning sickness as can happen 24\/7 for several months for many women! It also works for people of all ages who don't want to take a motion sickness pill or suffer needlessly!\n1) Eat a few salted plain crackers and drink some water (or any other type of CLEAR beverage, other than coffee as it leads to more dehydration).\n2) Eat a Baked Potato.\n-Wash well, poke a fork at least 1\/3 of the way into the potato in 2 or 3 spots on each side and both ends, pat dry with paper towel, put potato on a fresh folded paper towel square on a microwave safe plate and cook on high for 3-6 minutes depending on potato size.\n-About halfway through cooking time, stop process, open door and using an oven mitt if needed turn potato over on paper towel and finish cooking. This makes for a \"crunchy thick crust\" like as if it was slow-baked in the oven.\n-Using an oven mitt, pickup the potato and gently squeeze to see if done--if it doesn't have any \"give\" it needs another 1-3 minutes depending on size but overcooking will make it rock hard too).\n-Once done, cut it open and put a little bit of salt on top (salt helps balance your electrolytes and a small amount every day helps our brain and body to function properly!).\n-You can add some margarine or butter, as it adds in some \"fats\" and make the potato easier to eat, but it is not necessary.\n-Enjoy it and eat the skin\/crust as well as it has the most nutrients anyway and is very tasty as well as adds \"roughage\" that you need.\n3) Have a glass of Sprite, 7Up, Ginger ale, etc., if you can tolerate it or plain tea with a teaspoon of sugar and a teaspoon of lemon or lime juice.\nAs I travel I meet people who are nowhere near taking in what they normally do at home for fluids despite being in extreme heat, and many have felt literally sick from the heat, lack of enough fluids and even not eating properly. Any or all of these scenarios can leads to nausea, diarrhea, constipation, and just being plain old irritable at times. No matter where you are or what you are doing, a variety of fluids is necessary at all times to keep you moving and functioning properly as the human body is mostly water anyway!\n1) Clamato Juice -- the actual brand name juice. A single serving size is about a 493ml bottle (if you can buy and store a large bottle in the fridge it is cheaper in volume) and you will feel better within about 10 minutes.\n2) Gatorade works awesomely, costs about $1.50-$2US equivalent for the single serving size bottle and I have had no trouble finding it in over 9 countries so far.\n-Half a bottle every day (or every other day depending on your situation) is good to help with maintenance of fluids.\n-I also drink PowerAde or Vitamin Water now and then as both have other benefits but they are high in other things too, so check labels if you have dietary issues.\n3) Mix 100% pure orange juice (or any kind you prefer with no sugar added) with Sprite, 7UP, Ginger ale, etc in equal amounts.\n-It refreshingly lightens the juice consistency and adds the carbonation bubbles that help keep things moving too.\n-If you really want to clean out your kidneys as well use 100% pure cranberry JUICE (not cocktail as that has sugar and other items added) and add Sprite, etc., as directed.\n4) **Don't use diet pop or you are defeating the purpose as there are chemical artificial sweeteners added in all diet foods that actually do damage to your kidneys especially.\n-If when you have \"to go\"--you have to go NOW...chances are you are eating waaayyyyy too many artificial sweeteners, which hide in many foods.\n-Switching back to real sugar as it is only 16 calories per teaspoon of white sugar is much less harmful to your body, except in cases of diabetes and other medical conditions that require very limited amounts of any foods.\n-** I have many severe intolerances that border on allergies myself, so I understand the issues all too well. Reading labels will save you unexpected reactions and resultant additional health issues. Speak to your medical professional-obviously-for all medical information. This is only info on techniques that work for some, but you need to use common sense when it comes to your health!\n-The reduced fat\/salt\/sugar\/etc. \"diet\" or \"healthy\" foods work the same way. Removing\/reducing one ingredient necessitates adjustments\/increases of other ingredients to keep it edible. Many times, it is healthier to eat the regular version.\n-Check the nutrition label on all foods so that a quick comparison between your usual brand and what's on sale will tell you which the better deal is really health-wise due to \"added ingredients\".\n-As an example, some brands of potato chips use yeast extract and soy protein in their flavorings!\n-**If you have allergies of any kind, this is helpful for all products as shampoo, all kinds of soaps, make-up, toothpaste, deodorant and lotions use common allergens such as gluten, soy and\/or lactose for thickening.\n5) Step up your plain tap water intake for a few days to flush it all out and that feeling will go away shortly. Sweeteners cause the bladder to contract continually instead of just when it is full, so it overworks the bladder wall muscles, which can to lead to urinary incontinence in both men and women.\n-**There are helpful additives in most tap water such as fluoride in tiny amounts that are required by the body to function properly.\n-If you don't drink tap water, make sure you are getting your fluoride and trace minerals from a food source or vitamins to maintain a healthy body and mind.\n-These minerals are added to tap water for decades now in many countries, as people didn't get enough essential minerals so many got very sick or died needlessly.\nThis is especially helpful for kids of all ages (as long as they are old enough to eat mashed fruit) and everyone else who gets any kind of stomach bug with diarrhea as that usually also leads to dehydration and messed up electrolytes.\nYou can buy Pedialyte or energy drinks like PowerAde, but they have many other things you don't need, are very expensive over the long run and many times hard to get\/keep down due to the aftertaste.\n1) Eat a banana! If you don't like bananas or can't even look at food without feeling more nauseated then peel a few, push a popsicle stick in one end of each one (this is more for the kids), pop them in the freezer for an hour or more and instantly--banana popsicles!\n-They are soothing, a cold treat and medicine all rolled into one--and let's face it, when you feel like crap--who doesn't want a popsicle? There is no chewing involved either, which can generate more gastric acid to keep \"moving\" things along too fast at times.\n-Usually diarrhea goes hand in hand with nausea so using this tip and adding some crackers and fluids works great to \"slow the flow\" within about 15 minutes.\n2) Prevention: Eat half a banana a day regularly to help keep things moving as well as to help balance your vitamins and minerals as they are loaded in potassium too!\n-1\/2 a banana cut up; 2\/3 cup Greek Activia yogurt; 1\/4 cup nuts (your choice, raw\/unsalted preferred); 1\/3 cup cereal (gluten\/soy\/lactose free options do exist but not many); 1 Tablespoon Hemp\/Flax\/Chia seeds and 1 Tablespoon coconut. Alternatively, use the banana (add in other fruits as well), yogurt, hemp\/flax\/chia seeds and add pre-made granola or make your own ahead of time and freeze in baggies to keep fresh.\nAgain usually these symptoms all show up together to have a big party in your body and it makes you want to take yourself out back and get a bit slap happy over your own attitude at times, but some simple tips and extremely cheap items will fix most of what ails you!\nConstipation is a term used to describe hard stools due to a lack of enough fluids and roughage to keep things \"moveable\" in the first place (or certain medications, foods and other situations also can cause it). It is often used as a general term to describe a complete lack of \"movement\" at all by many people!\n1) The fastest, cheapest, most natural way is to drink 100% pure prune juice.\n-There are several kinds so check the nutrient labels as some have 90% iron and some only 15% - mostly water in other words - so go with the high iron content because the more pure it is, the less you need to drink.\n-Chill well in the fridge for a few hours or pour it over ice, as it is easier to get\/keep down cold than warm let me tell you!\n-It is very thick, so either add very cold water to dilute the intense sweetness and thickness of the nectar and end up drinking twice as much of the same \"flavor\", or chug it down a few swallows at a time along with a cup of room temperature\/warm water, alternating between the two and finishing both.\n-Drink about a cup and a half (about 375 ml) and a full cup (250ml) of water along with it.\n-Stay near a bathroom and have lots of TP handy as this works very well to sort of \"liquefy\" things, often within 30-60 minutes I have found personally and it is totally natural!\n-Continue to drink more water and several other types of clear fluids over the next several hours as well to rebalance your body.\n- I haven't actually tried it, but I would assume you can make \"popsicles\" with prune juice just like you would with any other type of juice for kids of all ages who simply can't tolerate the consistency or intense taste but don't want to take medication or supplements which can harm your bowels in the long run.\n-Fill small containers (or popsicle trays and sticks), fill with juice, place sticks in and freeze. It will cut out having to drink it and the cold is soothing when you don't feel well as it is!\n2) Warm up and eat a can (or 2 depending on severity of constipation) of brown beans - also called pork and beans - and add 1Tablespoon of olive\/sesame\/coconut\/etc. oil;, garlic; peppers; onions and spices such as turmeric which is excellent for digestion and metabolism.\n-Add in some cooked hamburger, tomato sauce\/paste, brown sugar, celery, etc., and make a big pot of chili. Portion it out once cold into baggies or containers, freeze and then re-heat in the microwave on power 7 for 2-3 minutes while making a salad to go with it and dinner is served within 5 minutes-fresh, hot, no additives, portioned out, and almost NO dishes afterwards!!\n-All the fiber from the beans and the oil helps to move things along as well as causes gas, which also pushes everything \"down the pike\" so to speak.\n3) Drink lots of fluids--again just water is not sufficient in any way.\n-Milk based fluids can tend to irritate the whole digestive system when you are not feeling good from creating extra mucus when you have a cold already anyway, to causing cramping and diarrhea when you are already sick; so it is best to avoid too much dairy when not feeling good.\n-Normally though you want 2 to 3 cups of either 2% cow's milk or Silk Unsweetened Almond Milk (gluten\/soy\/lactose free) a day to get your required calcium, vitamin D, etc.\n-I dilute it with half water so it is not so thick and goes farther.\n4) Eat a salad as a meal twice a day for a few days in a row.\n-Ideally use romaine or some \"healthy green\" type of lettuce but iceberg works just as well.\n-Shred\/Cube a 1 ounce serving of hard cheese, a handful of nuts (your choice but raw\/unsalted), a handful of currants\/cranberries\/etc.\n-Add a serving of cooked meat (hamburger, chopped chicken, tuna, pork, etc.) along with 2 capfuls of lemon or lime juice and a small amount of dressing (use your favorites).\n-I usually add in about 2 tablespoons of hemp\/chia\/flax seed to boost the fiber content and really get things moving quicker!\n-Normally, a salad a day will help to keep things moving, is good \"roughage\" and most greens have a high water content as well.\n5) Relax and stop worrying about what you are \"missing\" at that moment while feeling sick, as you wouldn't be able to enjoy it anyway.\n-Go to bed or relax on the sofa with your feet up and just rest.\n-Watch some \"laugh out loud\" comedy and get a real belly workout! It is said that 30 minutes of \"snort coffee out your nose\" type of unexpected and real laughter exercises your abdomen as well as your lungs and cardiovascular system just as good as a 60 minute power walk.\n-Don't knock it till you've tried it!\n-Get up and dance around to your favorite bouncy type of music and sing your heart out (decide to be comfortable enough to not care what other people think as they can't solve your issues anyway so who cares what they think?) and \"shake your booty\" enough to get things moving as well!\n-Or go for a good power walk, swim, whatever type of \"exercise\" you like to do and adding in your favorite type of music to keep up a steady beat while you exercise helps too.\n6) Have a very warm bath with 1-2 cups (250-500 mg) of baking soda for about 20 minutes.\n-Try to lie down in the water so that your stomach and back are soaking and if you can't do that, then lie on one side for 10 minutes and roll over to the other side for another 10 minutes so that your entire back and stomach get a good soaking.\n-The baking soda works by reverse osmosis and helps to relax the muscles as well as pull enough water from your body to move things along, usually within an hour or so after your bath.\n---**This is especially helpful for babies and small children who don't want to eat or drink anything but are in discomfort and you want to avoid medications and strong laxatives, etc.\nI hope this helps you and those you know and love in some way, may you soon find what makes YOU happy in YOUR life and have an awesome day! NEVER GIVE UP!\nPlease LIKE\/SHARE\/FOLLOW\/POST\/+1\/TWEET a link to this Update so that everyone can live\/eat\/feel better; faster\/easier\/cheaper!","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"They may have refought the battle as a wargame but these pictures are not from a wargame\u2026they are nice, but it's a collection of diorama shots, perhaps depicting scenes from the battle, not much else.\n\"We often blend in the bases using Photoshop to create a diorama look\"\nFantastic pics \u2013 drool, drool! Surely the gold standard for all our fantasy set ups \u2013 amazing terrain, beautifully painted (and an endless supply) of great Perry figs.\nYes, I can confirm it was a wargame!\nI was lucky enough to take part and even managed to be on the losing side!\nExcellent scenary' troops and photography.\nLovely stuff in the ' Grand manner'.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"List of Delaware newspapers for news and information on local business sports, jobs, education, lifestyles, travel, real estate, and more.\nLeading newspaper published in Delaware.\nCommunity newspaper published in Bethany Beach, Delaware.\nNewspaper based in Georgetown, DE. Sussex Countian was founded in 1886.\nMonthly bilingual (Spanish and English ) newspaper covering New Castle, Kent and Sussex counties.\nNewspaper serving Laurel, Bethel, Delmar& Sharptown.\nRegional newspaper serving Inland Bays, Atlantic Ocean, Dewey Beach, Lewes, Rehoboth Beach, and Milton.\nNewspaper based in Newark, Delaware.\nWeekly newspaper serving Seaford, Blades, Bridgeville & Greenwood. Seaford Star published by the Morning Star Publications, Inc.\nStudent newspaper of the University of Delaware.\nDPA a network journalist and news media professionals, was founded in 1977. DAP is an Affiliate of the National Federation of Press Women (NFPW).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Growing up, my travel baseball team won three consecutive state championships, despite my efforts.\nMax Jensen joined Good Sports in February of 2016 as a Partnerships Assistant. Max attended Central College in Pella, IA where he received his degree in Business Management, Nonprofit Management, and Communication Studies and played catcher on the baseball team. Prior to Good Sports, Max worked on the development team at a youth shelter in Iowa, and served as an AmeriCorps volunteer. Max is a die-hard Braves, Packers, and Iowa State fan, making him the only person in the northeast to support that strange trio of teams.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhbls b/data_all_eng_slimpj/shuffled/split2/finalzzzbhbls new file mode 100644 index 0000000000000000000000000000000000000000..35e49972bfda051b1cff7167718921adc948155a --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhbls @@ -0,0 +1,5 @@ +{"text":"This 2 bedroom, 2 bath condo on the first floor of popular Midtown Village features a fenced in yard! Unit boasts high end finishes including stainless steel appliances, hardwood floors, granite countertops, and custom cabinetry. Condominiums adjoin the popular Midtown Village Retail center, and are just 6\/10s a mile from campus. Retailers and restaurants are only steps away from your door. This complex features awesome amenities including a pool, putting green, fitness center, tanning bed, clubhouse with kitchen for entertaining, and two parking garages. Safety features include controlled gate access and security cameras. HOA fees are $285 a month and cover water, internet, and cable. Pictures shown are of another unit with same floor plan and interior features. .","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We proudly present Agape Nursing Home, the first privately built, high end care facility for the elderly in Belgrade, Serbia, located in a peaceful and quiet quarter of the city, Zplenska 86G of Vozdovac. Our facility offers the best of care and service for up to 52 residents. From care, hygiene, and nutrition to medical services, occupational therapy, and active social interaction, all is provided for as to extend high-end services to our clients in order for them to feel, safe, comfortable, and well-taken care of.\nOur clients are cared for by experienced staff consisting of Social Workers, General Physicians, Physiotherapists, medical technicians, nurses, caregivers, and other dedicated staff whose concern is the health, comfort, and wellbeing of our clients.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Located on the south coast of the island, the Amari Galle Sri Lanka is a luxury resort perfect for your next holiday destination. The hotel features rooms and suites with spacious and flexible living spaces that combine elegant simplicity with the newest technology, decorated with influences of the local culture. The resort has a free-form outdoor pool, and a spa and fitness centre, ensuring you are able to rejuvenate and relax during your holiday. An array of dining options is available at the hotel, whether you want to give local Asian flavours a try, or enjoy a variety of international dishes at your convenience. A holiday at Amari Galle Sri Lanka, filled with adventure and relaxation along the scenic coastline, will certainly be one to remember.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Stepping into the nursing field and becoming a certified nursing assistant is a rewarding decision for those students, who really want to become health care or medical professionals. CNAs suggest that they assist the senior nurses and doctors in the hospitals, nursing homes and other health care facilities. Certified nursing assistants have to practice nursing under the guidance and supervision of their senior nurses. They perform various responsible tasks such as feeding and bathing the patients, providing them with medication, helping them to recover the disabilities, repositioning them whenever required, keeping their health records, recognizing their vital signs and communicating between the patients and doctors. All of these jobs require highly skilled knowledge of nursing which is acquired after taking CNA classes. The nursing programs are offered by various schools, community colleges, vocational schools, health care facilities, medical schools, etc., in all the states.\nCNA training in the state of Alabama was started after the passing of the Omnibus Finances \/ Budget Reconciliation Act (OBRA \u2013 1987). There is an inventory of certified nursing assistants in the Alabama division of Well being and they are legally allowed to work as nursing assistants in long term care facilities. As per the OBRA, the requirement for being a certified nursing aide is the completion of 75 hours of training materials and the Alabama state nursing exam; which are followed by the specified time of nursing practice.\nHanceville Nursing & Rehab Center, Inc.\nTerrace Manor Nursing and Rehabilitation Center, Inc.\nUnlike most of other states, the Alabama state nursing board has a few unique educational requirements. It does not mandate to possess a high school diploma or a GED (general educational development) test certificate. The students here can also attend the training program even when they are pursuing their high school degree or diploma. The age of the students must be 17 years or above. This allows the students to prepare for the jobs while taking education.\nThe training program covers the advanced technical as well as clinical aspects of nursing which are indeed necessary for the students to become successful as certified nursing assistants.\nThe candidates who have completed the course work of certified nursing assistant become eligible to appear for the Alabama state nursing exam. This exam comprises of a two-part test. The first part of the test asks to answer the technical questions based on classroom session; while the second part asks the students to perform clinical skills. The students have to prove their competency in both of these tests so as to pass the exam successfully. Those who fail in exam once, can retake it; which can be tried thrice within the period of 2 years. Otherwise, they have to re-enroll in the Alabama state for certified nursing assistant training. After passing in the exam, they get the certificate within a short time.\nThere are so many training programs in the state of Alabama which prepare the students for the evaluation of certified nursing assistant proficiency. The only important thing is that the students must make sure that the respective training program is approved by the Alabama Department of health and holds recognition at the time of employment. The reason behind this is that if the programs are state approved, they tend to render the standard nursing knowledge and train the students much effectively as compared to other programs.\nGenerally, all the Alabama state approved programs of certified nursing assistant provide the students with all the necessary tools such as course material, 2 sets of uniforms, lab coat, a pair of shoes, a watch, stethoscope, a background check proof, TB screen report, etc. Some of them also provide with the payment of the exam fees. No matter the course instructors are accredited or not, the courses do offer CPR training during the nursing course work is going on. These are some of the advantages of Alabama nursing aide training programs.\nOnce you get the certification and licensure of being a certified nursing aide in Alabama, your name gets registered on the Alabama state nurse aide registry. To maintain the certificate and license active on this registry, one has to actively continue the nursing practice in the period of 24 months. Failing to do so, the license and certificate is declared as 'inactive' or 'expired' by the registry and then, the respective candidates have to reapply for the same. Usually, this period of renewal is 2 years. The candidates get to know about the renewal information well before the date of deactivation or expiry. Hence, one has to keep updating the personal information regarding mailing and postal addresses on the registry.\nAttending the nursing aide classes in Alabama is very convenient and reliable to ensure the candidates about becoming successful nursing assistant professionals.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"PIRATES returned to winning ways and kept the heat on SGB Premiership leaders Somerset with a 54-36 victory against Swindon at Wimborne Road.\nThree late 5-1s allowed Poole to pull away from Alun Rossiter's side, who were hamstrung by the scoreless efforts of Polish racer Tobiasz Musielak and reserve Jack Smith.\nPirates were without Danish duo Nicolai Klindt and Frederik Jakobsen due to racing commitments in their homeland.\nRider replacement was utilised in place of the former while Ipswich's Mark Riss guested for the latter.\nBefore the action got under way, a minute's applause was held out of respect for New Zealand's double world champion Ronnie Moore, who died on Saturday at the age of 85.\nFirst blood went to Swindon with David Bellego and Troy Batchelor getting a 4-2 as all four riders hugged the inside line.\nThe third-place Riss almost blew a Poole heat advantage when he ran wide and was momentarily off balance but the German recovered.\nMusielak blazed through the tapes next and went off 15m in the re-run, the Pole just unable to catch the third-place Richie Worrall.\nChris Harris got stuck behind Riss after missing the gate in heat four and by the time he got past, Nick Morris was out of sight.\nWoryna swept round everyone in the first two turns en route to heat five success, claiming the notable scalps of Bellego and Batchelor with a lengthy measure of goggle tape trailing behind him.\nKurtz then bounced back from a disappointing first outing with a straightforward win, Morris swooping under Grajczonek for second.\nFormer Poole flier Adam Ellis picked himself out of the air fence to win a re-run heat seven after Musielak had been excluded for his part in the spill.\nPirates extended to six points their lead with a 4-2 in heat eight, a result which would have been a maximum had Bellego not forced his way past Riss after a hard-fought battle.\nSwindon team boss Alun Rossiter threw in Batchelor for a tactical ride which paid off, Worrall trailing at the back with seemingly little power at his disposal.\nAnd Musielak's nightmare meeting continued when he went through the tapes for a second time and was replaced by reserve Smith. Kurtz and Grajczonek notched the first 5-1 in the re-run.\nHarris kept his nerve in the face of fierce early pressure from Batchelor and Bellego but Ellis's tapes-to-flag effort next closed to six points the gap.\nA race-long pursuit of Batchelor by Harris ended with the Poole man sweeping round his rival in thrilling fashion on the final bend.\nWorrall did superbly to force his way into second from fourth in heat 14 and Harris showed he was not winding down by battling past Batchelor in the final showdown.\nPirates 54: 1 Brady Kurtz (0-3-3-3-3) 12, 2 Josh Grajczonek (2-3-1-3-2*) 11+1, 3 Richie Worrall (1-0-0-2-2*) 5+1, 4 Nicolai Klindt r\/r, 5 Chris Harris (2-2-3-2*-2*) 11+2, 6 Kacper Woryna (3-3-1*-2-0-3) 12+1, 7 Mark Riss, guest (1-1*-1-0) 3+1.\nSwindon 36: 1 Troy Batchelor (1-1*-3-2-1-1) 9+1, 2 David Bellego (3-2-2-1*-0) 8+1, 3 Adam Ellis (2-3-1-3) 9, 4 Tobiasz Musielak (0-fx-0) 0, 5 Nick Morris (3-2-1-0) 6, 6 Zach Wajtknecht (2-0-0-1-1) 4, 7 Jack Smith (0-0-0) 0.\nBomber produced two of the best moves of the night \u2013 both of which were on Troy Batchelor.\nThe fans' favourite recovered from an indifferent meeting at Somerset to get back in double figures, a feat achieved by no fewer than four riders.\nBrady Kurtz reeled off four wins after a scoreless opening outing and Kacper Woryna and Josh Grajczonek also went big at Wimborne Road.\nGrajczonek can feel much more positive about his form at a crucial time of the season \u2013 and Pirates will need him if they are to make an impression in the play-offs.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhoxr b/data_all_eng_slimpj/shuffled/split2/finalzzzbhoxr new file mode 100644 index 0000000000000000000000000000000000000000..1f598d7c8e80a9229e67acf1b291f8677d53efc9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhoxr @@ -0,0 +1,5 @@ +{"text":"Attention Builders, developers and owner users! Located in a prime area of Studio City surrounded by Multi Million Dollar homes. Close to shops, restaurants, entertainment & the recently updated river walk. This cape cod features a formal living room with a fireplace, dining room and an office with garage access. Master bedroom & family room have French doors that lead to the backyard. It's one of the largest lots in the neighborhood, and it has tons of room for a pool.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Will Colorado Stay Ahead Of The Pack With Sports Betting?\nIt's still 2018, but Colorado officials are already eyeing a 2020 sports betting launch.\nLike many others, the Centennial State is starting to put the gears into motion so it does not miss out on what could be a roughly $18 billion national market, if all states get involved in it.\nLet's be blunt: Colorado should stay ahead of the pack, similar to how it did with recreational marijuana.\nColorado's push into sports betting might involve putting the issue up to voters. However, Colorado Attorney General Cynthia Coffman weighed in on the matter this summer, putting forth guidance that the state could launch sports betting with just the legislature's approval.\nAdvocates for the industry would of course prefer not to put the issue on the ballot. It's not that there's fear of voters rejecting it; it just means that Colorado would have to wait until 2020.\nNumerous studies have shown that a majority of Americans want sports betting to be legal.\nIf it waits, Colorado would be leapfrogged by other states. Currently, sports betting is active in six states \u2014 Nevada, New Jersey, Delaware, Mississippi, West Virginia and New Mexico \u2014 and many more are set to tackle the issue next year.\nIt's worth noting that five states went active with sports betting within about four months of the U.S. Supreme Court ruling on PASPA in May. The floodgates have just started to open.\nColorado might go ahead with a sports betting ballot question thanks to the mobile betting component, though, according to the Denver Post. Why? Because the state is already starting to indicate that mobile betting would be seen as a significant form of gambling expansion.\nAll sports betting states need mobile, and a must-have for mobile is allowing bettors to sign up and verify their identity with an operator remotely.\nWith Colorado's casinos being located in just a few cities, some residents would potentially need to drive as far as eight hours to sign up for a mobile sports betting account.\nSome gamblers might make a fun road trip out of it, but the vast majority of potential customers would likely stay away from such a hassle.\nThe state has more than 30 casinos, according to the American Gaming Association, and all are located in three towns (Black Hawk, Central City and Cripple Creek).\nThose are the only places in the state where it is currently legal to develop and operate a casino.\nColorado does have racetracks, which, like New Jersey, could operate sportsbooks. However, the state's commercial casino industry has successfully opposed those facilities expanding their gambling options.\nUnlike some other states that are moving toward sports betting, Colorado's casino gambling market is on the upswing. According to the AGA, the state's 2017 gaming revenue of $828.1 million was the highest ever recorded by Colorado's commercial casino industry, surpassing the previous record set in 2007.\nAccording to a 2017 Oxford Economics study commissioned by the AGA, Colorado's sports betting market could see as much as $5.7 billion worth of bets per year and about $338 million in revenue from that.\nThat's based on a best-case scenario in terms of \"convenient availability\" of sports betting and a low tax rate. If there was \"limited availability\" and a high tax rate, the Colorado projection shrinks to a handle of $970 million and $63 million in annual sports betting win.\nConvenient availability is defined as brick-and-mortar plus mobile, with the offering of in-play wagering. Limited availability would be if Colorado only allowed sportbooks at its existing casinos.\nColorado has a gaming revenue tax of more than 14%, and that's likely what lawmakers would seek for sports betting. According to the study, a low tax rate would be 6.75 percent (what Nevada has), while a high tax rate would be 15 percent.\nWhy Is NBA Scoring Soaring, And Can Bettors Take Advantage Of It?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Orange Tree organise customised company events to suit your specification whether it be for staff development, team building, fun days or client entertainment. In addition we run events for the general public for fun days, parties, stag and hen weekends.\nWraysbury Lake - Orange Tree Events runs corporate team building days based on a raft building event specifically designed for corporate clients requiring a relaxed fun day out or full on team building. The morning activity can be team-building games or a wide range of other activities. Full catering and bar facilities available.\nMonkey Island Estate - Monkey Island Estate lies on a picturesque island in the River Thames in the historic village of Bray, Berkshire. The island has a rich history centuries old, and has been the haunt of monarchs, aristocrats and artists, along with writers, famous performers and Berkshire locals and now is home to one of the finest hotels in Bray. The two grade I listed heritage pavilions which form the property were originally commissioned as a fishing retreat in 1723 by the third Duke of Marlborough. These have been completely restored and renovated, with beautifully landscaped gardens and picturesque views of the River Thames. Set across seven acres, Monkey Island Estate features 27 exquisite guestrooms and three sumptuous suites, designed by the award-winning New York-based Champalimaud Design. The furnishings juxtapose timeless glamour and traditional features with a modern, countryside style, creating a unique blend of past and present. It's the perfect base from which to explore the area or just immerse yourself in history against the backdrop of the delightful village of Bray.\nBeaumont House - Beaumont House is a superb venue set in 40 acres of landscaped grounds with and excellent location which makes it an ideal base for both business and pleasure. Dating back to the 17th century, Beaumont House has expanded over the years and today offers guests a tranquil location in which to meet and relax. We have a huge area to run team-building and fun activities and it is close tothe River Thames for water-based activities.\nRunnymede Hotel - This 4 star hotel is in Egham, Surrey, on the banks of the River Thames near Windsor, just minutes from Heathrow Airport. The Runnymede~on~thames is the ideal location for meetings and events, weddings and parties plus short weekend hotel breaks. Team-building events can be run on site or at a mearby lake activity centre by ORANGE TREE EVNTS Ltd.\nEnglemere - Secluded and elegant, Englemere is the ideal venue for a meeting or conferences. This is also an excellent corporate venue with great team-building facilities run by ORANGE TREE EVENTS Ltd. With the varied and well-furnished rooms and excellent catering, we can ensure that your event is a great success.\nGreat Fosters Hotel - Gracing an estate of more than fifty acres in what was once the heart of Royal Windsor Forest in Surrey, Great Fosters is the most distinguished of hotels \u2013 a Grade One listed building blessed with the grandeur of 16th century England and luxurious modern facilities. This hotel is ideal for companies wishing to have meetings with or without accommodation and a team-building event alongside at any time of the year.\nTaplow House - A stunning Georgian retreat situated on the border between Berkshire and Buckinghamshire. Set in 6 acres of beautiful gardens and boasting Europe's largest tulip trees Taplow House Hotel is the perfect setting for a company meeting with great facilities for both indoor and outdoor team building facilities provided by ORANGE TREE EVENTS Ltd.\nConference Venues Malta - Business Meetings Malta, will organise your conference, congress, business meeting, product launch, sports tournament or concert in Malta. With our unique expertise in the organisation of business meetings, events, incentives and conferences in Malta, we are offering the finest selection of quality hotels, restaurants and conference facilities. We are a fully licensed DMC Call today for our free Conference Venue finding service.\nCrown Plaza Marlow- The stunning lakeside location of our accommodation in Marlow, set in 5 acres of beautiful countryside, makes it an ideally placed conference destination, within easy reach of the M4, M40 and M25 motorways and Heathrow Airport, as well as a spectacular wedding and events venue. The ideal location of our Marlow Hotel offers endless possibilities for walks along the River Thames, trips into London, Windsor and Henley, as well as retail therapy in Marlow's boutiques.\nAngelPie Hen, Stag and Fun Shirts including many Personalised Products with no minimum orders.\nRouteOne Design is a manufacturer of custom clothing. The website offers you the ability to design your own clothes.\nAdult Costumes - Your online Halloween party store for all unique adult costumes, child costumes and costume wigs. Our prices are guaranteed and are directly from the manufacturers. We pride ourselves on our competitive prices and superior services. Fast shipping too!\nPrinted circuit board manufacturer, Custom Circuit Boards, is a USA PCB manufacturing company located in Phoenix.\nTerranium displays : Here at Jar & Fern we lovingly hand make Terrariums, Succulent Globes and Indoor Plant Arrangements.\nWe can create beautifully curated plant arrangements for weddings, corporate events or simply to arrange around your home.\nMarquee : MarqueeHire.org has information on Marquees and Marquee Hire.\nHuman Resources Consulting Services Company - This page contains links to sites that we either partner with or they provide valuable information that we feel may be of interest to you or your organization.\nClaimsHeaven - - information and resources for legal problems in the UK.\nLaw Firm Search Marketing- Increase the visibility of your firm with law firm search marketing from lead To Conversion. With experience in attorney search engine optimization, LTC has the tools to help bring the clients to you.\nLondon Serviced Office Finder - Serviced office space across The City and central London searchable by area, Tube station and placename. It might come in handy for your business-minded visitors.\nTeam Building Events - : Revolutionizing the blunt team building events in the corporate sector, Focus adventure exemplifies the power of experiential and adventurous-based team building approach in the creation of positive team dynamics.\nTeam Building - Motivational espionage themed team building days.\nTuition bangalore ThinkVidya.com has the most comprehensive listing of tutors providing tuitions in Bangalore. Some tutors provide private tuitions at home or tutor's home, some provide group tuitions, while others provide online tuitions. It is quite easy to find the best tutor for your specific need by posting your requirement on ThinkVidya.com.\nit outsourcing There are different types of hosting, business hosting and Managed Dedicated Server Hosting is one of them. This type of hosting service is a generic, should be taken when a company needs more private care into their online presence. In the managed dedicated hosting provider monitors server performance around the clock, all server management tasks and remove any burden from the customer on the shoulder, so that they focus on their core business to achieve their goals. Many benefits are of server hosting and all areas of business. The first and most important advantage is that you get great value for your money. You pay about the same amount or less than a basic shared hosting, but you get a number of features usually found only in dedicated server hosting. Looking for a reliable and secure web server hosting services, when you hear a small personal website, corporate site or a large educational website? Supporters.dk is the ultimate destination host for your website, since some types of it konsulenter offers for your web hosting server.\nLink Building Services : India's No 1 one way link building services provider.\nGoogle Maps Optimization : Specializing in Google maps optimization, Lead to Conversion optimized your Google Places profile to ensure visibility in Google maps. Google local optimization is essential for local businesses.\nIf you would like a free entry to this link page please enter the link information below and advise by e-mail of the location and your website information and we will add your site.\nDetails: ORANGE TREE EVENTS Ltd provide event management, corporate team building, raft building and sailing events in the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Plentex Limited, formerly Plenty River Corporation Limited, listed on the ASX in 1983 and operated as a minerals exploration company in Australia and several overseas countries. In 2006 the Company acquired mining and exploration interests in North Queensland held by privately owned Georgetown Mining Limited and several ASX listed companies. In accordance with ASX Listing Rule requirements, trading in the Company's securities following these acquisitions was suspended in November 2006. The Company was unsuccessful in attempts to raise funds which would have led to its reinstatement of trading in late 2007 due to the onset of the global financial crisis and in May 2009 reluctantly sold its gold mining project and exploration interests in North Queensland to the Australian subsidiary of the German company, Deutsche Rohstoff AG, which subsequently brought the gold mining project into production.\nIn October 2009, Plentex signalled an exciting new direction with an investment in Victorian based private company, Blue Sundial Pty. Ltd., which it ultimately acquired in September 2011, with the aim to become Australia's first commercial supplier of algae as a sustainable resource for multiple industries. Through collaborations with the South Australian Research & Development Institute (SARDI), Flinders University, and the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Plentex identified that the most promising commercial opportunities available to it for algae were in high value nutraceutical markets and as a fish oil replacement in aquafeed.\nOver the next four years both opportunities were developed in tandem. Further investigation revealed significant opportunities in aquafeed even without algae input. In late 2013, the Company's biotechnology pathway was dramatically advanced when the opportunity emerged to acquire an existing range of algae-based anti-fungal products that could generate immediate cashflow. A spin-off company, Xerion Limited, was formed in May 2014 to finance and develop the Company's expanding biotechnology-related business interests.\nThe year 2013 was also marked by Plentex's establishment in conjunction with Flinders Partners Pty Ltd (the commercial arm of Flinders University) of Nest Group Limited whose business platform was to be the commercialisation of technology developed in selected partner universities. In June 2014, Nest Group Limited was acquired by Nideux Limited (subsequently renamed UnIPartners Limited) resulting in Plentex becoming UnIPartners then largest shareholder.\nIn late 2014 Plentex established Plentex Philippines Inc. to focus on the development of agri-business opportunities in the Philippines which would become Plentex's core business and also Protemax Pty Ltd as the corporate vehicle for its proposed Victorian based aquafeed\/petfood manufacturing activities.\nPlentex expects to be recapitalised and to recommence trading on the ASX in 2017 after which it plans a dual listing on the Philippines Stock Exchange.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Is mild to moderate wrinkling affecting the way you feel when you look in the mirror? As we age, wrinkles and other textural changes begin to take hold as collagen and moisture are lost \u2013 especially living in Fort Lauderdale with the constant sun exposure. It doesn't have to be this way, VIP Aesthetics has your answer.\nSublative is used to treat mild to moderate wrinkles and textural irregularities and to help restore your skins natural beauty. By directing a matrix of RF energy into the skin, Sublative encourages your skin to heal and correct itself. New collagen and healthy skin cells are triggered to grow and those wrinkles and imperfections in your skins tone and texture fade away as your body heals itself.\nDuring your sublative treatment, you may feel a warm, prickly feeling \u2013 but there should be no pain. Afterwards, you may look and feel a bit sunburned. Topical cooling ointments are available to those who need them.\nSome patients begin to see results after their first treatment, but typically a series of at least three treatments is needed to truly see how amazing your skin can be. The treatments are spaced between four and six weeks apart, but our staff will help you determine what the best treatment protocol for your individual needs will be.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhwar b/data_all_eng_slimpj/shuffled/split2/finalzzzbhwar new file mode 100644 index 0000000000000000000000000000000000000000..ff3e021ee62e6ac6a2baa8e98bbff1bba89c5a40 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhwar @@ -0,0 +1,5 @@ +{"text":"Marie Antoinette Wedding Invitations \u00bb Big Tips For The Major Day And Above, When you're planning for a wedding ceremony, there are plenty of points to juggle. It could be a challenging task Marie Antoinette Wedding Invitations even with the aid of friends, family and skilled wedding ceremony planners. Below are a few sound advice to successfully feel assured about every facet of the wedding, from the centerpieces on the hire charges.\ntonic wedding on pinterest wedding cards card from Marie Antoinette Wedding Invitations, When using your family on the spot wedding ceremony, a way for anyone to economize is simply by cooking food your own personal food. from: pinterest.com.\nsweet 15 invitations on pinterest quince invitations from Marie Antoinette Wedding Invitations, When using your loved ones on your vacation spot wedding, a method for anyone to save money is actually by food preparation your very own food. from: pinterest.com.\nSelect Marie Antoinette Wedding Invitations your wedding attire prior to decide on whatever else, including jewellery, shoes or boots, purse, your hair or cosmetics layout. Take your time trying on gowns and appear about at distinct locations before you make your choice. When you can find what you prefer Marie Antoinette Wedding Invitations on the internet, that can save you a lot of cash also!\n\u00bb Make certain that every person inside your about Marie Antoinette Wedding Invitations wedding ceremony is aware how they're arriving at the marriage web site and back home through your wedding! This is especially important if you're will be serving liquor, and Marie Antoinette Wedding Invitations important if it's a wide open nightclub. If you're worried about any person obtaining as well inebriated, supply cost-free cab rides to anyone who doesn't deliver an automobile, or travel them house inside your limo.\npink color significance archives the wedding specialists from Marie Antoinette Wedding Invitations, When taking your family on your vacation spot wedding ceremony, an easy method for all to economize is as simple as food preparation your personal food. from: theweddingspecialists.net.\npretty pink wedding ideas on pinterest pink weddings from Marie Antoinette Wedding Invitations, When getting your family in your destination wedding party, a method for everyone to save cash is as simple as preparing food your own foods. from: pinterest.com.\n1000 ideas about siegelstempel auf pinterest policy from Marie Antoinette Wedding Invitations, When getting your loved ones on the location wedding ceremony, a method for anyone to spend less is as simple as food preparation your own foods. from: de.pinterest.com.\nMarie Antoinette Wedding Invitations \u00bb The morning prior to deciding to have your wedding, you need to indulge yourself whenever possible by looking at a health spa for a massage. This will allow you to reduce the pressure and tension from the days leading approximately your wedding event so that you will feel convenient strolling along the aisle.\nMarie Antoinette Wedding Invitations \u00bb If you have a reduced plan for your wedding, don't involve blossoms inside your Marie Antoinette Wedding Invitations decorations. For centerpieces you may use branches you locate on the move within the woodland, and surrounds the storage containers you put them in with gemstones from the close by river or creek. Produce a bouquet from community plants and flowers and blossoms for a locavore touch!\nLet you to While an absolutely coordinated place placing with dishes, dishes, and cutlery can produce a extraordinary, elegant, and timeless physical appearance for receptions and rehearsal dishes, it may also be very costly. Instead, take into account deciding on dishes, bowls, and items that talk about a frequent theme, like a particular shade, all flowery images, or perhaps a mismatched shabby fashionable appearance.\nWhen acquiring estimates from suppliers for the wedding event, stay away from mentioning that you will be arranging a wedding ceremony if you don't need to. Some providers Marie Antoinette Wedding Invitations mark up the price of solutions for marriage ceremonies, so you may spend far more ultimately than you would for a in the same way size occasion with the same requires Marie Antoinette Wedding Invitations.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"I am left with the corpse, the living dead man\u2026the angry lonely man, the bitter man with the nightmares, the murder man, the man who cries in the shower.\nIn one big bang they have taken it all from me, in one clean sweep, and now I am in this place around all the others like me, and though I keep trying not to feel sorry for myself, I want to cry. There is no shortcut around this thing. It is too soon to die even for a man who has died once already.\nAs promised, my Top Five summer books. With any luck, I'll finish the list before June 21, 2007.\nAs you probably remember, Born on the Fourth of July by Ron Kovic was made into a movie with Tom Cruise, back before he was crazy. I mean, crazy in love with Katie Holmes. Or crazy in love with the idea of being crazy in love with Katie Holmes.\nAnyway, the movie was good and deserved all its awards, but the book is phenomenal, riveting, heart-rending, and more important now than ever.\nThis end was no beginning. It was starting to become very clear that there would be no change in his condition, no reconciliation with the half of his body that seemed so utterly lost forever. He was in the rain, trapped, and there was no one. It was ugly and cold and final.\nHe eventually protested the war with many of his former comrades as part of the Vietnam Veterans Against the War. He was spit upon, beaten, even arrested.\n[T]hese ignorant, arrogant men and women who never saw the things we saw, never had to grieve over the loss of their bodies or the bodies of their sons and daughters, never had to watch as so many friends and fellow veterans were destroyed by alcoholism and drugs, homelessness, imprisonment, neglect and rejection, torture, abandonment, and betrayal, in the painful aftermath of the war.\nThese leaders have never experienced the tears, the dread and rage, the feeling that there is no God, no country, nothing but the wound, the horrifying memories, the shock, the guilt, the shame, the terrible injustice that took the lives of more than 58,000 Americans and over two million Vietnamese.\nNearly 20,000 U.S. troops have been wounded in Iraq since the beginning of military action in March 2003. 9,100 of them were unable to return to duty because of their injuries.\nEvery warrior accepts the possibility of death. It's the living death of the wounded soldier that no one imagines and everyone would rather ignore.\nBut if you give Born on the Fourth of July a chance, it will put you in the perspective of another human being in a way that few books can.\nDare to care. Then do something with those feelings. Me, I'm going to find a disabled vets group to donate 10% of royalties from Voice of Crow (October 2007), which features a main character forever crippled by war. That, and try to tell the man's story as honestly as a lucky person like me possibly could.\nThanks for posting that. I never even realized there was a book. I'm going to go check it out now.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"A conviction can inhibit your freedom and damage your reputation, relationships, job prospects or immigration status. However, with the help of an experienced lawyer, you may be able to avoid or minimize such far-reaching penalties.\nMinneapolis criminal defense attorney Kristine Zajac helps clients fight for justice. She can give you a down-to-earth assessment of your options. As your advocate, she will take a personal interest in the outcome of your case, working to avoid lasting consequences.\nIf you have been accused of a crime, it is essential to get legal representation as soon as possible. Zajac Law Firm is ready to act quickly to protect your rights after an arrest and during a criminal trial.\nCriminal charges have a ripple effect. They often intersect with other areas of Minnesota law, such as immigration and family law. If you are a noncitizen, you must take extra care in resolving any criminal charges. A misstep can jeopardize your immigration status, potentially leading to deportation or permanent immigration bars.\nLikewise, if you are involved in a divorce or child custody case, a criminal conviction can endanger your interests. Astute legal guidance is critical in navigating these complex issues.\nLawyer Kristine Zajac is familiar with all the ways these areas of law overlap. She can help you minimize the impact on your immigration status or family law case.\nDo not leave yourself vulnerable when facing criminal allegations. Call 612-789-2300 or send a message to the Zajac Law Firm to arrange a free consultation to discuss your case. The firm is conveniently located in downtown Minneapolis.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The pNeuton model S ventilator provides mask CPAP plus volume or pressure ventilation for use in the pre-hospital and hospital environments. The built-in CPAP (Continuous Positive Airway Pressure) system allows you to start with mask CPAP and if the patient needs it, provide ventilation as well.\nWith simple-to-use controls \u2013 Tidal Volume, Respiratory Rate and Pressure Limit \u2013 pNeuton S can provide volume-targeted or pressure-limited ventilation. The built-in high output demand flow CPAP system insures critical care ventilation needs are optimized for spontaneous and mandatory breaths.\nFully pneumatic and MRI compatible, the pNeuton S does not use batteries or electrical power. And it can help you care for the most difficult patients, from trauma site to the emergency room, for inter-facility transport, wherever the patient may be located; including the MRI.\nThe pNeuton ventilators are critical care devices designed for use by healthcare professionals under the direction of a physician.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Through health and wellness, VivaCore helps individuals and businesses in Edmonton reach their potential.\nWe service individuals through personal training, weight loss coaching, running and triathlon coaching and more. We service businesses through wellness retreats, lunch and learns, benefits referral, facility enhancement as well as through comprehensive strategic wellness plans that motivate and empower employees to better reach the organization's mission, become an employer of choice and gain a competitive edge.\nBoost employee motivation and performance with lunch and learn (s). Choose from a variety of topics including stress management, nutrition, physical activity, psychological safety and work-home balance.\nBuild a stronger, higher performing team with a half day, one day or multi day wellness retreat. Each retreat is customized to your business needs and can cover topics and activities such as team building, leadership, engagement, strategic planning and more.\nGood strategic planning necessitates knowledge about current employee motivation, performance, engagement, health and wellness. Make an informed strategic plan to boost your competitive edge by finding out what drives your employees.\nCreate a psychologically safe workplace by having a Mental Health First Aid Train the Trainer certify Mental Health First Aiders.\nImprove your organization's bottom line and become an employer of choice. Offer a tailored health and wellness strategy that will attract, retain, motivate and inspire employees to reach your organization's mission to rise against competitors.\nWhether you want to feel better, lose weight, get sculpted or improve your cardiovascular health Victoria will help you reach your goal.\nPregnancy, child birth and labour creates a number of considerations for your fitness goals. Let a specialist help you reach your goals the healthy way and get you looking and feeling better than before your little one arrived!\nLook and feel your best with a personalized weight loss plan based on philosophies of sustainability and wellness.\nHave a Team Canada triathlete provide support, motivation and a personalized plan to help you achieve your goal of completing a Sprint or Olympic distance triathlon. Get advice on equipment, training and nutrition.\nWhether you are a beginner or intermediate runner, Victoria will help you reach your goals. Take it from someone who has placed top four in all races from 5 km to half marathons over the last three years.\nWhether you want to reach your personal or business potential let VivaCore support your goals to create your competitive edge. Health and wellness is the foundation of success.\nPersonal and business health resources and tools coming soon!\nFrom the blog: Christmas Treats in the Office!","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbigbv b/data_all_eng_slimpj/shuffled/split2/finalzzzbigbv new file mode 100644 index 0000000000000000000000000000000000000000..f3053bb6e122da52f9fafd776ea22f6129bc18b0 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbigbv @@ -0,0 +1,5 @@ +{"text":"It was once on the so-called \"Black List\" survey of unproduced scripts well-regarded by industry insiders. Well, so much for that. It is about to have a miniscule opening, one month after it released on VOD. These was the same circumstances faced by Kevin Spacey's most recent film, Billionaire Boys Club, which the trades trumpeted far and wide as the death knell of the disgraced actor's career. So, are they going to say the same thing about Black List scripts, because fair is fair isn't it? Actually, there is nothing fair about stupid stories that cherry-pick low opening grosses for films that are primarily seeking VOD business. It would be particularly easy to beat up on this unfortunate genre picture, because it was dropped by a major distributor, but there are worse films hitting theaters this week than Stefan Ruzowitzky's Patient Zero (trailer here), which opens in a few theaters somewhere this Friday.\nRose is okay. In fact, Morgan is definitely attracted to her, creating a lot of awkward guilt for the \"Infected Talker,\" since his wife Janet is one of the Infected they are transfusing, in the hope they can eventually be cured. Of course, they have heightened hearing, so she knows. In contrast, it is the jerkweed martinet in charge of bunker security who really chafes Morgan, but his nemesis will be \"The Professor,\" a leader the Infected, who does not seem to follow any of the rules Morgan and Rose thought they had figured out.\nIn fact, those rules are somewhat interesting and probably give is a sense of what the original Black List script read like. The Infected can't process music, so Morgan, a former record store owner, uses classic rock and soul as part of the interrogation process. They also lost their ability to deceive\u2014or did they? Unfortunately, just when the rules are fully established, they are thrown out and the film loses its distinctive vibe, becoming predictable and cliched.\nAs you might expect, it is a good deal of fun to watch Stanley Tucci act all sinister as \"The Professor\" (but Mary Ann remains unaccounted for). Matt Smith also fully commits as the brainy but prickly Morgan, using a lot of his old Doctor Who acting muscles. Natalie Dormer is fine as Dr. Rose, but it too bad Agyness Deyn doesn't get more screen time as Janet, the Infected, because her portrayal is really quite complex and emotionally engaging.\nYes, Patient Zero clearly owes debts to Day of the Dead, 28 Days and Weeks Later, as well as scores of other zombie and pandemic movies, but Ruzowitzky keeps it moving along nicely. His output has been a little uneven since The Counterfeiters won the Academy Award, but his films usually have intriguing elements, especially the intense psycho-thriller, Cold Hell. Sure, it craters during the third act, but isn't that always the way? In any event, it is far better than any reports of its minimal opening might suggest. Recommended as a VOD distraction for fans of apocalyptic horror, Patient Zero opens somewhere this Friday (9\/14).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Expert advice, inspiring stories, upcoming events and other great information delivered to your inbox once a month.\nOur clinic in Le Sueur is a community-based health care provider that brings the resources and expertise of Mayo Clinic to the Le Sueur community.\nFor a list of the services we provide in Le Sueur, visit our Medical Services page. If you need a medical service that is not available, search our system-wide medical services to find the most convenient location for you.\nLearn more about Mayo Clinic Health System or view and print our Fact Sheet.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Shop and choose from a beautiful variety of affordable herbs, vegetables, perennials, annuals, natives, and hanging baskets. The sale includes the always popular homegrowns section featuring plants grown and donated by local gardeners. Experts will be available throughout the day offering selection guidance. Activities include a special Mother's Day kid's craft, refreshments and music. A fun day for the whole family! Parking is available on nearby streets and curbside assistance for purchases is provided. Watch for the Plant Sale Catalog mid-March. Place your orders early for home delivery (20-mile radius).","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The diplomatic cable demonstrates an undeniable desire to capitalize on the Hariri resignation for the purpose of weakening Hezbollah and Iran and combatting their influence in the region and the rest of the world. In the cable, the Israelis wish to paint Iran and Hezbollah as responsible for any and all chaos in Lebanon as a result of the Hariri resignation as well as for the launching of a missile by Houthi militants toward the airport in Riyadh. Thus, the propaganda narrative desired by the Israelis is that both Iran and Hezbollah are forces of instability and destabilization in the region.\nIt has become increasingly clear to anyone observing the situation in the Middle East that Israel is attempting to bring down Hezbollah in Lebanon and possibly engage Iran militarily in the near future. As always, it will do so by coordinating with Saudi Arabia's influence and financial assistance as well as the military might of the United States.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"We're not just ambitious about growing as a firm; we want to be the best UK law firm to work for and we invest a lot into achieving this.\nPeople are central to our success and we're passionate about making sure that each and every individual who works here feels valued and has the opportunity to learn, grow and excel. We don't do unnecessary hierarchy or work in siloes. Instead, our culture is one of open minds, collaboration and refreshing perspectives, where people achieve great things, together.\nBelow, you'll get a good insight into life at TLT; our awards, activities and, of course, our people.\nWe're investing in our people. Learn how we'll support your development.","meta":{"redpajama_set_name":"RedPajamaC4"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbllsc b/data_all_eng_slimpj/shuffled/split2/finalzzzbllsc new file mode 100644 index 0000000000000000000000000000000000000000..99c5ddab46557c77d7a18bfcd2cb384cc79ddd65 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbllsc @@ -0,0 +1,5 @@ +{"text":"Had a fun packed great afternoon of football.\nOn the 31st October we came along to the Billingham Schools Football tournament as reporters and photographers and what a good afternoon we had.\nWe had two strong teams from Year 5 and Year6. Mrs Tattersdill PE Leader said \" We like to involve as many children as possibly in these events, that is why we bring two teams\".\nAt the beginning of the day we were ready for a day at school due to the sorrowful sky ( it was pouring down) and our sorrowful spirits! Suddenly the sky burst with light and so did our happiness as we realised we were going.\n\" I have positive energy\" said Ashley.\nAs soon as we got there we could see competition was tough, but we quickly started our warm up.Mr Sheehan called all the teams over to give them a format of the day!\nFor most children in the B team it was there first competitive football match.At the beginning our spirits were high but they shrank during the afternoon as the matches didn't go quite as we expected. We didn't give up and showed good team spirit. We went into the final match down but our spirits soon went high when we scored a goal!! We won our last match. Ended on a high.\nThe A team went through their matches winning like no tomorrow. Mr Sheehan announced we had made it to the finals!! Yeah!!\n\"I am so excited, I hope we win\" said Joel one of our defenders.\nWe had the final a very tense game and sadly we lost 1-0.\n\"Well done to both teams and all the best in next round\" said Mr Dos Santos a parent spectator.\n\" Great team spirit and fair play from all teams ,great to see\" said Mr Sheehan Competition Manager.\nAlthough we didn't win we are through to the Stockton FInals, we are so excited for another chance to compete against other schools.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"OASIS provides support for Commercial and Non-Commercial requirements, which means any contract type may be utilized.\nServices under OASIS span 28 NAICS codes and 6 exceptions or 34 codes - exceptions in all. These 34 codes\/exceptions are allocated among 7 NAICS code pools. All codes\/exceptions in a pool share a common small business (SB) size standard. Agencies can solicit task order proposals from contractors holding awards in the pool with the NAICS code that matches the agency's requirement.\nOASIS offers more meaningful proposed rate comparisons at the task order level \u2013 apples to apples, and 104 OASIS LCATs (individual and groups and priced at the master contract level) cover 127 OMB SOC occupations and over 1,000 typical industry job titles.\nThe remainder of the 840 SOC occupations can be added and priced at the task order level as ancillary labor.\nCan index pricing for up to 640 precise geographic locations.\nOASIS has no program ceiling, a five-year base and one five-year option, and provides for long term planning for complex program requirements.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"The world of e-commerce is very likely to change soon. And the XYO Network is going to be the one to do it.\nThey propose a new platform that features blockchain technology that will usher in the new era of smart contracts. The XYO Network is the first-ever blockchain, trustless location finding service. But, why is that important?\nToday, you use technology for more things than ever before. There is even a huge amount of contracts happening every day that do not require middlemen. You no longer have to have a lawyer or an escrow for a multitude of various contracts. Smart contracts are taking over, and they are doing it quite impressively.\nHowever, those smart contracts are not without flaw. One of the flaws that are very important is that not everything is blockchain based. Namely, the data input maintenance relies on a single, centralized source. The location oracles, specifically are the one posing an issue. Namely, having only one source of data you also have only one potential source of failure and only one target a malicious third party would have to attack. This means that, currently, smart contracts can be tricked and prompted to execute at wrong times. This is why the brand new smart network is necessary. Adding layers to the verification process for location finding is of utmost importance for some smart contracts.\nBack in 2012, XY-The Findables Company started finding ways to decentralize the location data. So, they created a network of devices that use Bluetooth, GPS, and Mobile phone beacons. These devices interact all the time, and with their service, they wanted to up the location finding game. All of this lead to the creation of the XY Oracle Network.\nThe XY Oracle Network is a decentralized, trustless and identityless network. But, what does this mean for you? Being decentralized, you will not have to worry about large corporations trying to profit from your queries. It runs on a completely transparent system that works by the rules.The network is actually free to use as it is an open source and it provides fair competition since everyone can use it. It is also trustless in the same way Bitcoin is. The participants are those that share the power. There is no entity there to collect the fee. There is no reason to trust or not trust the company as everything is 100% transparent and you are in the driver's seat. And lastly, the network works anonymously. Your privacy is secure and their cryptographic method called \"Proof of Origin\" is there to keep you safe.\nThis network essentially eliminates the risk of attacks and location spoofing. But, how does it actually work? As a user of the network, you can send out a transaction that is called a \"query\". Then the aggregator will return with the answer for your location query. There are four important components of that process. The \"sentinels\" that are there to collect the data, the \"bridges\" that relay it to the \"archivists\" which serve as storage until the \"diviners\" which are the aggregators need the answer.\nThis network ensures that these answers are as accurate as possible.\nThe XYO Tokens will start selling on the 20th March 2018. The sale will last for exactly two months. At the beginning of the sale, 1 Ethereum will get you 100,000 tokens, however, that will reduce to 33,333 over time. The network will burn the extra tokens that nobody buys.\nThis Network has real potential when it comes to interacting with the real world. In fact, the possibilities are incredible. Just imagine having smart contracts in your e-commerce trades that only activate the payment once the package gets to the location. You will no longer have to worry about buying things online and never receiving them. The XYO Network proposes huge changes for the world of e-commerce and the world of smart contracts in general.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Travel expenses will quickly increase the cost of HIPAA consulting support. RMHG offers remote HIPAA compliance assessments and consulting support to keep expenses down, depending on the size and complexity of your organization. Contact RMHG at if you are interested in learning if your organization qualifies for remote HIPAA support.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Natural Gluten-free Red Velvet Cheesecake Brownies came out of my desire to create a cheesecake brownie, and red velvet felt like a more congenial base than a traditional dark chocolate one. Plus I wanted to have a go at a red velvet product that didn't contain any food colouring.\n1Preheat oven to 180 degrees c (160 degrees c for fan assisted). Butter a 9\u00d713 brownie or roasting tin and line with greaseproof paper, bottom and sides.\n2In a large bowl combine the eggs, sugars, vanilla, cacao, beetroot juice, buttermilk and salt. Whisk until well blended. I do this by hand. You can use a stand mixer but mix sparingly \u2013 brownies should have texture (but not lumps). Add the vinegar, whisk, then add the flour and give it a final blend. Pour this mixture into the prepared tin and set aside.\n3To make the cheesecake, combine all of the ingredients in the bowl of a stand mixer (or mix by hand) and mix well, for about 2 minutes.\n4Pour the cheesecake filling over the top of the brownie mixture levelling it off gently with a spatula. Then take a chopstick and \"swirl\" the cheesecake gently into the brownie mix, moving up and down then across the pan. The red should start to show through and you'll have a swirl pattern on top.\n5Bake for 35 minutes. You want the brownie to set but there should be a little wobble at the centre. If you over-bake these they start to resemble cake.\n6Cool completely in the tin. Trim the edges and cut into 12 large or 15 medium squares. I store these in the fridge and bring to room temperature before eating.","meta":{"redpajama_set_name":"RedPajamaC4"}}